{"text": "In Fig.\u00a0\u22121)\u201d. In Fig. E (V vs. RHE)\u201d should be replaced with \u201cVoltage (V)\u201d. The number \u201c1.4\u201d and \u201c1.6\u201d should be replaced with 1.6 and 2.0, respectively. The corresponding data analysis and conclusions in the manuscript are not affected and thus not to be changed. The correct Fig.\u00a0In the original publication, the label text \u201cPt/C\u201d in Fig."}
{"text": "Myroxylon balsamum (L.) Harms var. pereirae (Royle) Harms  is a botanical balsam that has a long history of medicinal use, particularly as antiseptic and for wound healing. Except for a Chinese article discussing the reception of balsam in China and Japan, no scientific studies on its impact in China and Japan and the channels of transfer from the Americas to Asia exist.Connections between China and the new Spanish colonies in America are known for an exchange of silver for silks and porcelains. That also medicinal drugs and medicinal knowledge crossed the Pacific Ocean is hardly known or discussed. Commiphora gileadensis  as a medicinal category and discusses the specific medicinal properties of Myroxylon balsamum (L.) Harms var. pereirae (Royle) Harms. The section \u201cHistorical research and uses\u201d provides a brief survey on some historical analyses of balsam. Aim, design, setting: (2) Applying a comparative textual and archaeological analysis the article critically examines Chinese and Japanese sources  to show (i) what Chinese and Japanese scholars knew about balsam, (ii) where and how it was used, and (iii) to identify reasons why the \u201cdigestion\u201d of knowledge on balsam as a medicinal developed so differently in China and Japan.Description: (1) This section provides a general introduction into Myroxylon balsamum (L.) Harms var. pereirae (Royle) Harms was partly a highly valued substance imported from the Americas into China and Japan. But the history of the reception of medicinal knowledge on Peruvian balsam was significantly different in China and Japan.This chapter discusses the introduction of \u201cPeruvian balsam\u201d into, its uses as a medicinal as well as its scholarly reception in early modern China and Japan and introduces the channels of transmission from Spanish America to Asia. It is shown that Myroxylon balsamum was continuously updated, especially through mediation of Dutch physicians; Japanese scholars, doctors and pharmacists possessed a solid knowledge on this balsam, its origin and its medicinal uses. In China, on the contrary, there was no further \u201cdigestion\u201d or development of the knowledge on either Myroxylon balsamum (L.) Harms var. pereirae (Royle) Harms or Commiphora gileadensis. By the late nineteenth century, related medicinal and even geographic knowledge had mostly been lost. The interest in \u201cbalsam\u201d in late Qing scholarship was pure encyclopaedic and philosophic.In Japan, the knowledge on  Kunyu Wanguo quantu \u5764\u8f3f\u842c\u570b\u5168\u5716 , following Ricci\u2019s format, drew his own map Wanguo quantu \u842c\u570b\u5168\u5716, included in his geographical treatise Zhifang waiji \u8077\u65b9\u5916\u7d00 . Aleni also mentions Peru (Bolu \u5b5b\u9732), and speaks of \u201cextremely rich gold and silver\u201d ores (gu jinyin zuiduo \u6545\u91d1\u9280\u6700\u591a)  plant, or the \u201cBalm of Judea\u201d was frequently mentioned ) by Duan Chengshi \u6bb5\u6210\u5f0f (801 or 802\u2013863) as a product of the Byzantine Empire (Fulin guo \u62c2\u6797\u570b) . Its bark has a greenish-whitish colour. The leaves are fine and always two face one another. The flowers are upright like rape turnips [Brassica rapa var. rapa], and the buds are reddish like the colour of ripe black pepper (hujiao \u80e1\u6912). Cutting its branches, a sap [leaks] like oil. Taken against scabies and sores, there is no disease it cannot heal. Its oil is very valuable and is as expensive as gold.\u201d\u963f\u52c3\u53c3\u51fa\u62c2\u6797\u570b\u9577\u4e00\u4e08\u9918\u76ae\u9752\u767d\u8272\u8449\u7d30\u5169\u5169\u76f8\u5c0d\u82b1\u4f3c\u8513\u83c1\u6b63\u9ec4\u5b50\u4f3c\u80e1\u6912\u8d64\u8272\u65ab\u5176\u679d\u6c41\u5982\u6cb9\u4ee5\u5857\u75a5\u766c\u7121\u4e0d\u7625\u8005\u5176\u6cb9\u6975\u8cb4\u50f9\u91cd\u65bc\u91d1.This description attests to the fact that already Duan Chengshi knew about the excellent properties of balsam against skin diseases. And he, too, mentions the high price of balsam.aboshen   \u2026[Ar.] \u2018Ud-e balas\u0101n [wood of C. opobalsamum] ([subtext] this is a black-colored myrrh) \u2026[Pr.] balas\u0101n    to smear the hand. \u2026Use oil of [Ar.] Balas\u0101n  seeds  [[Pr.]  seed\u201d]) .aboshen appears in the sources until Ming times, and the Chinese knew balsam as a product originating from the eastern Mediterranean and the Arab worlds. And also the medicinal qualities of balsam, especially against skin diseases and sores, were already known in China during Tang times. It was only with the advent of the Jesuits that balsam from the New World, that is, from the new Spanish colonies in America, was introduced\u2014the balsam of Peru.As we will see below, the expression Actually, the balsam of Peru\u2014that belongs to a quite different family, Fabaceae, that is interestingly nowhere close to the family of the Old World balsam, Burseraceae\u2014originally came from the part of Central America known as Salvador or balsam coast, so the name is at least misleading. Albert Hale traces this misnomer back to European ignorance of the American continent. Next to Mexico, Peru was the land of El Dorado and everything originating from the Pacific \u201cwas apt to be called Peruvian, whether from Peru itself, from Chile, or the mysterious Potos\u00ed\u201d . I wouldhoitziloxitl\u201d . In Peru, there is a balsam tree that produces an oil; when the tree is cut with a knife, the oil leaks out, applied on dead corpses, these do not decompose (\u5b5b\u9732\u570b\u6709\u5df4\u723e\u5a46[=\u6492]\u6469\u6a39\u4e0a\u6cb9\u4ee5\u5200\u53d6\u4e4b\u5857\u5c38\u4e0d\u6557) . On the Da Qing huidian shili \u5927\u6e05\u6703\u5178\u4e8b\u4f8b (Collected Statutes of the Qing Dynasty: Precedents), in 1727, the Portuguese royal envoy Alexandre Metello de Souza e Menezes (1687\u20131766) presented the Yongzheng Emperor several gifts, among them \u201cSt. Thomas, Peruvian and Brazilian balsam oil\u201d  . Xu. XuHaigupage 40b .Fig. 4aHaiguo tuzhi \u6d77\u570b\u5716\u5fd7  says: \u201cThere is a tree that produces a grease (oil) which is extremely fragrant and is called balsam; it is used to treat all kinds of injuries, within one night, the muscles and flesh are totally recuperated as before. Applied on smallpox, they do not leave scars. Applied to dead corpses, these do not decompose even after ten thousands of years\u201d (\u6709\u6a39\u751f\u8102\u818f\u6975\u9999\u70c8\u540d\u62d4\u723e\u6492\u5f4c\u5085\u8af8\u50b7\u640d\u4e00\u665d\u4e00\u591c\u808c\u8089\u5fa9\u5408\u5982\u6545\u5085\u75d8\u4e0d\u7622\u4ee5\u5857\u5c38\u5343\u842c\u5e74\u4e0d\u673d\u58de)  the right syndrome\u201d    has memo\u96cd\u6b63\u5341\u4e8c\u5e74\u4e8c\u6708\u521d\u516d\u65e5\u5167\u5927\u81e3\u6d77\u671b\u5949\u65e8:\u64da\u984d\u9644\u7b56\u6de9\u594f\u7a31,\u5df4\u723e\u6492\u6728\u6cb9\u8ecd\u524d\u6df1\u70ba\u9069\u7528,\u723e\u5c07\u6cb9\u591a\u591a\u6599\u7406\u4e9b,\u7528\u76db\u8336\u932b\u74f6\u76db\u88dd,\u52d9\u671f\u5805\u56fa\u5305\u88f9,\u5e36\u8207\u984d\u9644\u7b56\u6de9\u61c9\u7528\u3002\u6b3d\u6b64\u3002\u65bc\u672c\u6708\u521d\u4e5d\u65e5\u53f8\u5eab\u5e38\u4fdd\u5c07\u5df4\u723e\u6492\u6728\u6cb9\u4e8c\u5341\u65a4\u76db\u5728\u9020\u8fa6\u8655\u505a\u5f97\u5927\u932b\u74f6\u4e8c\u5341\u74f6\u3001\u5c0f\u932b\u74f6\u56db\u74f6\u3001\u76db\u8336\u932b\u74f6\u5341\u516d\u74f6,\u4ee5\u4e0a\u5171\u56db\u5341\u74f6 .Faced with its usefulness as a remedy in the army, it is not surprising that the available Peruvian balsam from Spanish America circulated more widely and was more widely applied during Qing times than the balsam that originated from the Middle East.\u20131795) decreed that the Portuguese missionary Manuel de Mattos (1725\u20131764) provide \u201cbalsam fragrance\u201d \u5df4\u62c9\u85a9\u55ce\u9999  Bai Shixiu asked for 8 qian (ca. 30 g) of balsam oil to use it for balsam fragrance to prevent colds   because he died in a sitting posture  will dry by itself; this is the special way of Theravada  to reach the status of [unification with nature]. Is this not sufficient to startle? Bones and hard matter do not decay, the will congeals and the essence is bound to become a demon. If one burns what is not evil, this is the origin of the Buddhist \u015aar\u012bra relic. What is said in the explorations of thing is that hair and organ of a boy can be stuck to the \u015aar\u012bra relic, when frankincense remains long, it can bear a \u015aar\u012bra.\u201d\u201cWhen Li Yu  ingested posture , nao min\u5408\u5a46\u5f8b\u4e5f\u4e7e\u6f92\u7d0d\u9f3b\u800c\u8028\u9999\u718f\u70ba\u8089\u8eab\u8005\u4e5f\u5be1\u6b32\u606c\u6fb9\u6c14\u76e1\u800c\u901d\u5fc3\u4e0d\u4fc2\u6200\u7156\u5f9e\u9802\u8131\u5c38\u70ba\u907a\u8715\u4e45\u4e4b\u81ea\u4e7e\u5c0f\u4e58\u5c08\u81f4\u5e38\u7136\u8c48\u8db3\u602a\u4e4e\u9aa8\u525b\u8005\u4e0d\u58de\u5fd7\u51dd\u7cbe\u7d50\u6709\u6210\u584a\u71d2\u4e0d\u58de\u8005\u6b64\u820d\u5229\u5b50\u4e4b\u6240\u7531\u4f86\u4e5f\u7269\u7406\u6240\u66f0\u7ae5\u7537\u9aea\u6839\u53ef\u9ecf\u8d77\u820d\u5229\u4e73\u9999\u4e45\u7559\u80fd\u751f\u820d\u5229.\u7559\u5c4d\u6cd5 \u674e\u9810\u670d\u7389\u5176\u5c38\u6691\u4e2d\u4e0d\u6ba0\u9ec4\u91d1\u585e\u7ac5\u4ea6\u4e0d\u8150\u704c\u6f92\u66f4\u6548\u6216\u4ee5\u8336\u7042\u6f92\u7d0d\u9f3b\u4ea6\u53ef\u6691\u4e2d\u4e0d\u6ba0\u6f22\u66f8\u8a00\u6c59\u8fb1\u5442\u96c9\u5c38\u4ee5\u6c34\u9280\u6582\u4e5f\u5b5b\u9732\u570b\u6709\u4e00\u62d4\u723e\u6492\u6469\u6a39\u8102\u5857\u5c38\u5343\u5e74\u4e0d\u673d\u78ba\u8fa8\u9332\u66f0\u6709\u5f92\u4ee5\u5750\u8131\u8ca9\u5176\u5e2b\u8005Why Fang Yizhi was so interested especially in the embalming of dead corpses in ancient Peru is an interesting question. We know from ancient, especially Han and pre-Han period, tombs that members of the ruling and social elites in China also frequently embalmed their dead. Lady Dai from the Mawangdui \u99ac\u738b\u5806 tomb complex (168 BCE) is perhaps one of the most well-known examples. Well preserved mummies were also discovered from the Ming and Qing dynasties . The pracirujano), requested the authorities to pay for the medicines and the labour to embalm the cadaver of the archbishop of the Philippines (1729\u20131732), Carlos Berm\u00fadez de Castro (1678\u20131729)  took care of a cargo of animals for the Spanish King, among them one elephant, four small deer from Batavia, two deer from Coromandel and a dove from Ternate; accordingly, the elephant and three of the four deer reached their destination alive and the others embalmed  .F.FKunyu Keiy\u014d kibun, he explicitly states that he discussed the location of Peru with Dutch people. Dutch medicine applied various balsams in its treatments. So Dutch physicians offered another channel to obtain more knowledge on the uses of balsam as a medicinal.All this demonstrates that Arai Hakuseki intensively engaged with the knowledge about these foreign countries and their products. In Hon\u2018yaku Chiky\u016b zenzu ryakusetsu \u7ffb\u8b6f\u5730\u7403\u5168\u5716\u7565\u8aaa (Translation and Brief Account of the Entire World Map) in 4 juan [Katsuragawa Hosh\u016b \u6842\u5ddd\u752b\u5468 (1751\u20131809), a Japanese physician and Rangaku scholar who served the Tokugawa Shogunate \u5fb3\u5ddd\u5e55\u5e9c (1603\u20131868) as a physician and translator of Dutch, translated the world map by Willem Janszoon Blaeu (1571\u20131638) into Japanese, n 4 juan . He alson 4 juan  and numen 4 juan .Fig. 7aOranda ih\u014d san\u2019y\u014d \u548c\u862d\u91ab\u65b9\u7e82\u8981  [In 1817, Ema Sh\u014dsai \u6c5f\u99ac\u677e\u658e  translated and printed the Recipes) . He alsoRecipes) .Fig. 8EShinyaku Oranda neikai y\u014dh\u014d \u65b0\u8b6f\u548c\u862d\u5167\u5916\u8981\u65b9  that also explicitly refers to balsam of Peru (\u62d4\u723e\u6492\u8b28\u767e\u9732\u6bd4\u4e9e\u6ce5) [Ensei ih\u014d meibutsuk\u014d \u9060\u897f\u91ab\u65b9\u540d\u7269\u8003  that provides even more information on the Peruvian balsam: \u201cBo\u2019ersamo Poli, also called barusamu peruviana in Latin, and zuid indiaanse balsam in Dutch\u201d (\u62d4\u723e\u6492\u8b28\u5b5b\u9732 \u53c8\u300c\u30d0\u30eb\u30b5\u30e0.\u30da\u30eb\u30d3\u30a2\u30cb\u300d\u7f85\u300c\u30b9\u30ef\u30c6.\u30a4\u30f3\u30c7\u30a2\u30fc\u30f3\u30bb.\u30d0\u30eb\u30bb\u30e0\u300d\u862d). They describe the tree and also its medicinal uses and indications: it strengthens the nerves, prevents decomposition, can be used as a laxative, with cleaning functions, to sweat out excretions, can dissipate cold tuberculosis and is also effective against pneumoconiosis, asthma, gonorrhoea and other deficiency diseases [Yoshio Shunz\u014d \u5409\u96c4\u4fca\u85cf (1787\u20131843) translated a manual written for physicians on board of ships by the seventeenth-century\u00a0Dutch official surgeon, Jan Kouwenburg, \u6492\u8b28\u767e\u9732\u6bd4\u4e9e\u6ce5) . In the \u6492\u8b28\u767e\u9732\u6bd4\u4e9e\u6ce5)  translatdiseases . This isChaxiang shi congchao\u00a0\u8336\u9999\u5ba4\u53e2\u9214, he copies Fang Yizhi\u2019s Wuli xiaoshi and adds his remarks: \u201cI do not know where the country of Peru is located, and also do not know what kind of tree this balsam is\u201d  [Yu Yue \u4fde\u6a3e (1821\u20131907) was a very prominent Qing scholar, specialized in philology and textual studies, including history. In his scholarly notes, \u64a4\u6469\u4ea6\u4e0d\u77e5\u4f55\u6a39) . That anWhat conclusions can we draw regarding the transfer of knowledge related to the use and application of Peruvian balsam?\u20131730) and Jean-Fran\u00e7ois Gerbillon (1654\u20131707) even set up a pharmaceutical laboratory with all the necessary equipment at the request of Kangxi [Knowledge about the substance was brought to China through European missionaries. Initially, balsam was probably only used in the hospitals and pharmacies of European missionaries. But the knowledge gradually reached the court and the social and ruling elites of China. Especially the Kangxi Emperor possessed a very positive attitude towards Western medicine, and even wished to have European physicians and pharmacists at court. He definitely sponsored the dissemination of Western medicine in China. The French Jesuits Joachim Bouvet , as it was very effective in curing external knife, sword, arrow and other injuries as well as skin diseases, including smallpox . This isfugu \u5fa9\u53e4). After the Opium Wars (1839\u20131842 and 1856\u20131860), many also felt threatened by the West and by Western traditions. In this context, starting in the second half of the nineteenth century, the idea of using \u201cChinese learning as substance, and Western learning for practical use\u201d  became very widespread and popular among Chinese thinkers. This may explain, at least partly, why their interests in Western knowledge were strictly limited. If such knowledge was not directly related to Western technology and military systems that could be used to strengthen the country , it remained basically encyclopaedic and philosophic.After the Qianlong era, however, the \u201cnew\u201d knowledge gradually was lost\u2014in a complete disparity with Japan where the knowledge was continuously updated. Living in an environment where contacts to the outside world were the exception rather than the rule, many Japanese physicians were eager to learn more about medical practices from abroad. Especially through the mediation of Dutch physicians and their medicinal knowledge\u2014then among the most advanced in the early modern world\u2014Japanese scholars, doctors and pharmacists came to have a solid knowledge about the balsam of Peru, its origin and its medicinal uses. By contrast, most Chinese physicians, medical theorists and intellectuals were preoccupied with metaphysical and cosmological interpretations of their own classical traditions, seeking a \u201creturn to antiquity\u201d ("}
{"text": "Autism\u200c \u200cspectrum\u200c \u200cdisorders\u200c \u200c(ASDs)\u200c \u200care\u200c \u200cneurodevelopmental\u200c \u200cdisorders\u200c \u200cthat\u200c \u200cpresent\u200c \u200cwith\u200c \u200csocial\u200c\u00a0skills\u200c \u200cand\u200c \u200ccommunication\u200c \u200cchallenges,\u200c \u200crestricted\u200c \u200cinterest,\u200c \u200cand\u200c \u200crepetitive\u200c \u200cbehavior.\u200c \u200cThe\u200c \u200cspecific\u200c cause\u00a0\u200cof\u200c \u200cautism\u200c \u200cis\u200c \u200cnot\u200c \u200cwell\u200c \u200cunderstood\u200c \u200cyet.\u200c \u200cHowever,\u200c \u200cnumerous\u200c \u200cstudies\u200c \u200cindicated\u200c \u200cthat\u200c \u200cenvironmental\u200c \u200cand\u200c \u200cgenetic\u200c \u200cfactors,\u200c \u200cdysregulated\u200c \u200cimmune\u200c \u200cresponse,\u200c \u200cand\u200c \u200calterations\u200c \u200cto\u200c \u200cthe\u200c \u200cbalance\u200c \u200cand\u200c \u200ccontent\u200c \u200cof\u200c \u200cthe\u200c \u200cgut\u200c \u200cmicrobiota\u200c \u200care\u200c \u200cimplemented\u200c \u200cin\u200c \u200cthe\u200c \u200cdevelopment\u200c \u200cof\u200c \u200cautism.\u200c \u200cMany\u200c \u200cnon-pharmacological\u200c \u200cinterventions\u200c \u200care\u200c \u200cnominated\u200c \u200cto\u200c \u200cmanage \u200cautism,\u200c \u200cincluding\u200c \u200cfamily\u200c \u200csupport\u200c \u200cservices\u200c \u200cand\u200c \u200cpsychoeducational\u200c \u200cmethods\u200c. Moreover,\u200c \u200cdifferent\u200c \u200cpharmacological\u200c \u200ctherapy\u200c \u200cmodalities\u200c \u200care\u200c \u200crecommended\u200c \u200cfor\u200c \u200cchildren\u200c \u200cwith\u200c \u200cASD.\u200c \u200cLearning\u200c \u200cmore\u200c \u200cabout\u200c \u200cthe\u200c \u200cbrain,\u200c \u200cimmune\u200c \u200csystem, \u200cand\u200c \u200cgut\u200c \u200cconnections\u200c \u200ccould\u200c \u200cassist\u200c \u200cin early\u200c \u200cdiagnosis\u200c \u200cand\u200c \u200ctreatment\u200c \u200cof\u200c \u200cthis\u200c \u200cdevastating\u200c \u200cneurodevelopmental\u200c \u200cdisorders\u200c \u200cas\u200c \u200can\u200c \u200cearly\u200c \u200cintervention\u200c \u200cin\u200c \u200cASD\u200c \u200ccould\u200c \u200cimprove\u200c \u200ca\u200c \u200cchild's\u200c \u200coverall\u200c \u200cdevelopment.\u200c We\u200c \u200cgathered\u200c \u200cdata\u200c \u200cfrom\u200c \u200crelevant\u200c \u200cpreviously\u200c \u200cpublished\u200c \u200carticles\u200c \u200con\u200c \u200cPubMed\u200c \u200cto\u200c \u200cevaluate \u200cthe\u200c \u200crole\u200c \u200cof\u200c \u200cthe\u200c \u200cgut\u200c \u200cmicrobiota\u200c \u200cand\u200c \u200cthe\u200c \u200cimmune\u200c \u200csystem\u200c \u200con\u200c \u200cthe\u200c \u200cdevelopment\u200c \u200cof\u200c \u200cautism.\u200c Compared to other body areas, the gut has the most significant number of microorganisms, more than a trillion, with various species . Gut MicAutism spectrum disorders (ASDs) refer\u00a0to a wide range of disorders present with social skills and communication challenges, restricted interest, and repetitive behavior ,8. The rThe precise etiology of autism is not yet known ,14. HoweMany non-pharmacological interventions are nominated to manage autism, including family support services and psychoeducational methods ,25. DiffLearning more about the brain, immune system, and gut connections could assist early diagnosis and treatment of this devastating neurodevelopmental disease as an early intervention in autism could improve a child's overall development. This literature review article aims to evaluate the neurobiology of autism and the role of the gut microbiota and immune system response in developing autism in children. We searched Pubmed as the primary source to gather the related studies. We selected articles based on details and relevancy and included randomized clinical trials, animal studies, systematic reviews, and observational studies.The gut microbiota and autism\u00a0Research\u00a0indicates that there are discrepancies in the composition of the gut flora between autistic children and controls, with an excess quantity of members of Clostridium and Sutterella genus found in the gut of children with autism .\u00a0FurtherFinegold et al. examined stool specimens from 33 children with autism aged two to nine years with gastrointestinal (GI) abnormalities and 13 control children with no autism or GI disorders to assess the amount of Clostridium perfringens (C. perfringens) and C. perfringens toxin genes in their gut . They coTomova et al. investigated fecal microbiota changes and the frequently existing digestive tract disorders in children with autism . They exInvestigations showed a microbial shift in the digestive tract of children with autism, with a significant increase in the number of Desulfovibrio species, Lactobacillus species, and C. perfringens. Likewise, animals with similar microbial changes presented with a clinical picture of autism. Concluding, the gut microbiota is necessary to maintain the digestive system's physiological state; alteration of this intestinal flora has been associated with autism.The influence of TNF-\u03b1, IL-8, and IL-6 on autismAutism is associated with central nervous system inflammation, which leads to an imbalance in the plasma level of some inflammatory markers . IdentifAlzghoul et al. conducted a study in Jordanian children to measure and compare TNF-\u03b1, IL-8, and IL-6 in autistic children, their unaffected siblings, and unrelated healthy controls . After iWei et al. had immunohistochemistry studies that indicated the association of IL-6 with autism . They emSeveral studies measured and compared the plasma level of TNF-\u03b1, IL-8, and IL-6 in autistic and control subjects. Results indicated a statistically significant increase in the level of TNF-\u03b1, IL-8 (p < 0.001), and IL-6 (p < 0.05). The influence of these inflammatory cytokines on autism is proposed.Other inflammatory factors associated with autismStudies indicated that brain inflammation has a significant role in the pathophysiology of autism. Theoharis C Theoharides et al. reported that mast cells (MCs) are involved in the development of autism, which worsens with stress .\u00a0StimulaAutism is also related to dysregulation of some other inflammatory factors . In 2018Interleukin-4 (IL-4) and interleukin-10 (IL-10) are recognized mostly as anti-inflammatory cytokines . HoweverAhmad et al., in their clinical trial, assessed the effect of T cell immunoglobulin and mucin domain-3 (TIM-3) signaling in the development of autism . They reThere is an intimate and complicated relationship between the brain and the immune system; the pathophysiological changes in autism have been correlated to certain inflammatory factors\u2019 elevation. Understanding each element\u2019s role is critical for early testing and diagnosis of autism and could be studied to develop future interventions.Several studies showed a significant variation in the gut microbiota in children with autism compared to control children with a remarkable amount of Desulfovibrio species, lactobacillus species, and C. perfringens. Research\u00a0also reported that some inflammatory cytokines\u2019 plasma levels were higher in autistic children with a notable amount of tumor necrosis factor-alpha (TNF-\u03b1), interleukin 8 (IL-8), interleukin 6 (IL-6). Further, there is an elevation of some anti-inflammatory cytokines like Interleukin-4 (IL-4) and interleukin-10 (IL-10) in children with autism. These could be targeted areas to manage autism and provide better developmental outcomes. However, it is not clear yet if the gut microbiota treatment and the blunt of the inflammatory cytokine signals courses could treat autism. More research is needed to elaborate on that."}
{"text": "Scientific Reportshttps://doi.org/10.1038/s41598-020-71725-0, published online 9 September 2020Correction to: The Supplementary Information file that accompanies this Article contains errors in Table S1, where the citation information under columns \"First Author\", \"Publication Year\" and \"Journal\" in row 86 is incorrect.\u201cHavm\u221a\u220fller\u201d, \u201c2018\u201d and \u201cReport\u201d.should read:\u201cHavm\u00f8ller\u201d, \u201c2019\u201d and \u201cPlosOne\u201d."}
{"text": "Two families of dependence measures between random variables are introduced. They are based on the R\u00e9nyi divergence of order  X and Y taking values in the finite sets X and Y, but of their joint probability mass function (PMF) The solutions to many information-theoretic problems can be expressed using Shannon\u2019s information measures such as entropy, relative entropy, and mutual information. Other problems require R\u00e9nyi\u2019s information measures, which generalize Shannon\u2019s. In this paper, we analyze two R\u00e9nyi measures of dependence, r \u03b1 (see  ahead); opy (see  ahead). The measures ings see : J\u03b1The measures ormation , and botThe rest of this paper is organized as follows. In The measure  Hayashi  ; r \u03b1 (see  ahead); y|x)\u03b1]1\u03b1=minQYD\u03b1(\u2212H\u03b1(X|Y)=\u03b1\u03b1\u22121log\u2211|x)\u03b1]1\u03b1,I\u03b1c\u225cy|x)\u03b1]1\u03b1=minQYD\u03b1(Equation  and 8)..I\u03b1s(\u00b7), vided in  =m2J\u03b1=XY\u2225QXQY)=minQY\u03b1\u03b1\u2212y)1\u2212\u03b1]1\u03b1\u2265minQY\u03b1\u03b1\u2212QY(y)1\u2212\u03b1=I\u03b1cWe next prove :18)J\u03b1(J\u03b1((18)J\u03b1XY\u2225PXQY)=I\u03b1s in .\u2003\u25a1X and Y; =maxPXI\u03b1cProposition\u00a02For every conditional PMF .By , the lef\u03b1>1maxPXI\u03b1c((PXPY|X)\u2264maxPXI\u03b1s\u03b1>1maxPXI\u03b1c((PXPY|X)\u2264maxPXI\u03b1sf-divergence f-divergence.  ,12,13. EXY\u2225PXPY)=\u2211x,yP(x)ample in , and a cXY\u2225PXPY)=\u2211x,yP(x)ample in .In this section, we discuss the operational meaning of  testing  and of Kencoding .0)Under the null hypothesis, 1)Under the alternative hypothesis, Consider the hypothesis testing problem of guessing whether an observed sequence of pairs was drawn IID from some given joint PMF n sufficiently large, Associated with every deterministic test or whichlim infn\u2192or whichlim infn\u2192The measure lows: In  . For a fixed For large values of exponent ,18, whicnstraint  is given)=1].In  (second rs as in  and 31)EQ, the on, the sequences n tends to infinity .To better understand the role of Then, by  H0\u00a0For all (ii)\u00a0The mapping (iii)\u00a0The mapping . Let P The relative entropy (or Kullback\u2013Leibler divergence) is defined asThe R\u00e9nyi divergence of order \u03b1 ,22 is den one asD\u03b1(P\u2225Q)\u225c1With this extension to Proposition\u00a04.Let P and Q be PMFs. Then,(i)\u00a0For all (ii)\u00a0For all (iii)\u00a0For every (iv)\u00a0The mapping (v)\u00a0The mapping Proof.\u00a0Part (i) follows from the definition of ventions , and ParThe R\u00e9nyi divergence for negative found in  \u225c\u03b1Proposition\u00a05.Let P and Q be PMFs. Then,(i)\u00a0For all (ii)\u00a0For all (iii)\u00a0For every (iv)\u00a0The mapping (Part (i) differs from   follows from the definition of (P\u2225Q) in  and the (P\u2225Q) in . For \u03b1\u2208. We first establish the relation between Proposition\u00a06Let P and Q be PMFs, and let where the PMFs =mXY\u2225QXQY)=minQX,QYY\u2225QXQY\u02dc)=minQX,QY\u2225Q\u02dcXQ\u02dcY)=minQX,QY=Q\u02dcXQ\u02dcY;  and (66)XY\u2225QXQY)=J1\u03b1, so  holds. OP(Y), so  is finitgiven by  is well-QY\u2208P(Y),D\u03b1(PXY\u2225QXQY\u2208P(Y),D\u03b1(PXY\u2225QXtion 4),  and 76)\u03b1\u2208\u222a(The case e RHS of  is infine RHS of  is infine RHS of  is finitgiven by  is well-QY\u2208P(Y),D\u221e(PXY\u2225QXQY\u2208P(Y),D\u221e(PXY\u2225QXtion 4),  and 78)\u03b1=\u221e is anWe state the properties of Theorem\u00a01.Let X, (Lemma\u00a01)For every The following properties of the mutual information I  (Chapter(Lemma\u00a02)For all (Lemma\u00a03)For all (Lemma\u00a04)If (Lemma\u00a012)If the pairs (Lemma\u00a013)For all (Lemma\u00a014)For every Moreover:(Lemma\u00a05)(Lemma\u00a06)Let where (Lemma\u00a07)(Lemma\u00a08)For all Thus, being the minimum of concave functions in \u03b1, the mapping (Lemma\u00a09)The mapping (Lemma\u00a010)The mapping (Lemma\u00a011)If The minimization problem in the definition of (Lemma\u00a015)For every For (Lemma\u00a016)Let with the conventions of . The caswith the conventions of . (If \u03b1=1(Lemma\u00a020)For every The measure (Lemma\u00a017)For all where and is given explicitly as follows: for with the conventions of ; and forwith the conventions of . For eve(Lemma\u00a018)For all whereFor every (Lemma\u00a019)For all where the minimization is over all PMFs given by ; and Galfunction  is definWe now move on to the properties of Theorem\u00a02.Let X, (Lemma\u00a021)For every The following properties of the mutual information (Lemma\u00a022)For all (Lemma\u00a023)For all (Lemma\u00a034)If the pairs (Lemma\u00a035)For all Unlike the mutual information, (Lemma\u00a036)There exists a Markov chain Moreover:(Lemma\u00a024)For all where wer mean  For where in the RHS of (102), we use the conventions . The ine(Lemma\u00a026)(Lemma\u00a027)Let where (Lemma\u00a028)(Lemma\u00a029)The mapping (Lemma\u00a030)The mapping (Lemma\u00a031)The mapping (Lemma\u00a032)If (Lemma\u00a033)For every In this section, we prove the properties of Lemma\u00a01.For every Proof.\u00a0Let Lemma\u00a02.For all Proof.\u00a0X and Y are independent, then X and Y are independent.\u2003\u25a1The nonnegativity follows from the definition of Lemma\u00a03.For all Proof.\u00a0X and Y.\u2003\u25a1The definition of Lemma\u00a04.If Proof.\u00a0Let  becauseJ\u03b1\u2264D\u2225Q^XQ^Z)\u2264D\u03b1(PXY\u2225Qlow thatD\u03b1(PXZ\u2225Q^\u2225Q^XQ^Y)=J\u03b1,X;Y), soJ\u03b1=DX, Y, and Z form a Markov chain. If The proof of  is based,z\u2032|x,y)=\u2211yPXY(x\u2032Z\u2032|XY asAX\u2032Z\u2032|XY(|Y(z\u2032|y)=\u2211yPXY=\u2211yQ^X(x\u2032Z\u2032|XY asAX\u2032Z\u2032|XY(|Y(z\u2032|y)=Q^X(x\u2032)Q),where . Finally),where :120)D\u03b1D\u03b1AX\u2032Z\u2032|Xz\u2032|x\u2032,y)=PXZ=Q^X(x\u2032)QX\u2032Z\u2032|XY)\u2264D\u03b1=ws from Q^Y(y)\u2264maxQX,QYy)Q^Y(y)\u2264maxQX,QYuiAi,jvj\u2264\u2211i,j|ui|y)Q^Y(y)\u2264maxQX,QY2=1, and .\u2003\u25a1Lemma\u00a07.Proof.\u00a0This follows from Proposition 8 because Lemma\u00a08.For all Thus, being the minimum of concave functions in \u03b1, the mapping Proof.\u00a0For \u03b1=1,  holds be\u03b1\u2208,(1\u2212\u03b1)J\u03b1(XXY\u2225QXQY)=minQX,QYRXY\u2225PXY)=minRXYJ\u03b1(XXY\u2225QXQY)=sup=minRXY\u2208RRXY\u2225PXY)=minRXY\u2208Pf defined next takes on finite values only. A brief proof of Ky Fan\u2019s minimax theorem appears in \u03b4\u2264{\u2211x,yP(xy)]2\u03b4}12\u2264{\u2211x,yP(xion 4); \u22641,where,D1\u22122|1\u2212\u03b1|y)]2\u03b4}12=2\u2212\u03b4D1\u22122\u03b4=J\u03b1\u2264D\u03b1,D1\u22122|1\u2212\u03b1|=J\u03b1\u2264D\u03b1(PXY\u2225PLemma\u00a011.If Proof.\u00a0We show below that  holds fo). Thus,  holds alFix ). Then,J\u03b1=mx)1\u2212\u03b1]1\u03b1=minQX\u03b1\u03b1\u2212First consider the case X(x)1\u2212\u03b1\u03b1=2\u03b1\u22121\u03b1\u22121lher with  and (172her with .Now consider the case QX\u2208P(X),\u2211xPX(x)QXQX\u2208P(X),\u2211xPX(x)QX(x)QX(x)\u2264\u2211x\u2265minQY\u03b1\u03b1\u2212f]\u2265\u03bbminQY\u03b1\u03b1)f=\u03bbJ\u03b1(PXPYPX\u2032)PY|X=minQYminy)1\u2212\u03b1]1\u03b1=minQY\u03b1\u03b1\u2212Lemma\u00a015.For every For Proof.\u00a0We establish  for \u03b1\u2208[1ablishes  for \u03b1\u2208{1le where  is violaWe begin with the case where )]1\u2212\u03b1]\u03bb\u2032\u2264\u03bb\u2211x,yP(x\u2032(y)1\u2212\u03b1]\u2264\u2211x,yP2(1\u2212\u03b1)\u2264\u2211x,yP2(1\u2212\u03b1)\u2264\u2211x,yP]1\u2212\u03b1\u2264\u2211x,yP\u03bb\u2032]1\u2212\u03b1=\u2211x,yP,D\u03b1PXY\u2225,D\u03b1PXY\u2225(0.Lemma\u00a016.Let with the conventions of . The caswith the conventions of . (If \u03b1=1Proof.\u00a0If low from  and 78)ablishes  and J\u03b1;  holds be\u03b1; \u2264\u03bbD\u03b1(PXY\u2225QX\u2032Q^Y\u2032)=\u03bb\u03d5\u03b1(QX)+X be uniformly distributed over Finally, we show that the mapping \u2208,\u03d5\u03b1,\u03d5\u03b1\u222a,  follows stablish  as folloBecause  holds foWe now establish . Let RXYe RHS of . Then, fY\u03c8\u221e(RXY)\u2265maxRXY\u03c8\u03b1),where , \u03c8\u221e(RXY)X\u00d7Y). By , \u03b1\u21a6\u03c8\u03b1(RXY\u03c8\u03b1(RXY)\u2265\u03c8\u03b1(RXY*)ity, and  follows We now show that 1,\u221e). By , \u03b1\u21a6\u03c8\u03b1(RXWe next show that if ). Then,\u03c8\u03b1(RXY)\u2264J),where , and \u2264D\u03b1\u2264D(RXY\u2225QXRXY\u2225PXY)\u2264D\u03b1(PXY\u2225QRXY\u2225PXY)\u2264D(RXY\u2225QX). Then,J\u03b1=\u03c8\u2225QX*QY*)=J\u03b1,RXY\u2225PXY)\u2264D\u03b1\u225cPXe RHS of  is finite RHS of  holds be RHS of =mx)1\u2212\u03b1]1\u03b1=log\u2211xPX( (Q^X asQ^X(x)\u225cPXx)1\u2212\u03b1]1\u03b1\u2264\u03b1\u03b1\u22121log\u2211 RHS of =mf (RX asRX(x)\u225cPX( (Q^X asQ^X(x)\u225cPXx)1\u2212\u03b1]1\u03b1\u2264\u03b1\u03b1\u22121log\u2211Lemma\u00a020.For every Proof.\u00a0First consider rve thatJ\u03b1\u2264D+0.5Q^Y)\u22640.5D\u03b1(PX+0.5Q^Y)\u22640.5D\u03b1, and  in the pY), and , then  in the p), then (D(RXY\u2225RXR), then =\u03b1Lemma\u00a021.For every Proof.\u00a0Let Lemma\u00a022.For all Proof.\u00a0X and Y are independent, then X and Y are independent.\u2003\u25a1The nonnegativity follows from the definition of Lemma\u00a023.For all Proof.\u00a0X and Y.\u2003\u25a1The definition of Lemma\u00a024.For all where wer mean  =J\u03b1=minQX,QY.Hence,  follows \u03b1=minQX,QY and QY,D1\u03b1(P\u02dcXY\u2225Lemma\u00a025.For where in the RHS of (292), we use the conventions . The ineProof.\u00a0We first prove . Recall  and QY,\u03940. Let We next establish . To that\u03b1\u2208,K\u03b1+H\u03b1\u2208,K\u03b1+H\u03b1\u2208,K\u03b1+H),where  follows (y)]\u03b1\u22121\u03b1=minQX,QY(y)]\u03b1\u22121\u03b1=minQX,QYFinally, we provide an example for which (292) holds with strict inequality. Let Lemma\u00a026.Proof.\u00a0The claim follows from Proposition 8 because Lemma\u00a027.Let where Proof.\u00a0Let 2.Then,K2=J2=\u22122log\u03c31og\u03c31(\u03b2B)=\u22122log\u03b2\u03c31og\u03b2\u03c31(B)=\u22122log\u03c31+H\u2032=K\u03b1\u2032\u03b1\u2265minQX,QY and QY,M\u03b1\u22121\u03b1+HThe monotonicity extends to \u03b1\u2208,K\u03b1+H\u2265H\u03b1\u2265H\u221e\u2265H\u221e=K\u221e+Lemma\u00a031.The mapping Proof.\u00a0Because \u03b1\u2208,1\u22121\u03b1K\u03b1=minRXY1\u2212\u03b1H\u03b1=minRXY1\u2212\u03b1H\u03b1=minRXY1\u2212\u03b1\u2208,1\u22121\u03b1K\u03b1,K\u221e+H+H\u03b1\u2264\u2212logM\u03b1\u22121Q^X,Q^Y)=\u03b1\u03b1\u22121H\u221e in  and H\u221e in ; and =K\u221e+\u03b1\u2208,K\u221e+H1H\u221e=K\u221e+It remains to show the continuity at e RHS of  ahead, 1ed as in  and ]\u03b4]1\u03b4\u22642\u03b4]12\u03b4\u22642\u03b4]12\u03b4=M2\u03b4,\u2212logM2|\u03b1\u2212Lemma\u00a032.If Proof.\u00a0We first treat the cases For \u03b1=0,  holds beor \u03b1=0, =lpp(PXY)|=log|suppFor \u03b1=1,  holds beFor \u03b1=\u221e,  holds beNow let <2, then  holds be2, then =J\u03b1=H12\u2212\u03b1(X\u02dc\u2032)\u03b1]12\u2212\u03b1=2H\u03b12\u2212\u03b1=J\u03b1=1\u03b1\u22121H\u221e(XXPX(x\u2032)\u03b1=\u03b1\u03b1\u22121H\u221e(XLemma\u00a033.For every Proof.\u00a0Let X be uniformly distributed over We next show that for (P\u2225Q) in  that \u03940|=K0|=log|suppNow let oint PMFP\u02dcX1X2Y1Y2. Then,K\u03b1=K\u03b1=J1\u03b1=l becauseK\u03b1=J\u03b1\u2264log|X|,Lemma\u00a036.There exists a Markov chain Proof.\u00a00001Using Lemma 27, we see that Let the Markov chain"}
{"text": "Rhinodontodesalashanensissp. nov., Trachyphloeosomahonzasp. nov., T.jirkasp. nov., and T.martinsp. nov. are described from China, illustrated and compared with similar species. The genus Rhinodontodes and the species Rhinodontodessubsignatus Voss, 1967 and Rhinodontusmongolicus Borovec, 2003 are recorded from China for the first time. Keys to all Chinese genera of Trachyphloeini, and to the Chinese species of Rhinodontodes and Trachyphloeosoma, are provided. Trachyphloeini Gistel, 1848 is a medium-sized tribe of entimines containing small wingless, terricolous species with body size 1.3\u20136.8 mm, having limited ability to migrate. They are mostly xerothermophilous, associated with steppe habitats, xeric grasslands, stony or sandy places, ranging to sandy semideserts ; Rhinodontodessubsignatus Voss, 1967, based on a single specimen from Mongolia. The holotype was later examined by IZCAS and ZIN, and we are able to discuss characters used for the definition of Rhinodontodes. Voss described the genus as similar to Rhinodontus in its elongated epistome, but distinguishable by tarsomere 3 being wider than tarsomere 2, claws parallel in basal half, apical part of protibiae not distinctly enlarged laterally, rounded, without spines and narrower pronotum, 1.34\u20131.42 \u00d7 as wide as long, with anterior margin not distinctly narrower than posterior one. Trachyphloeini confirmed Rhinodontodes as related to Rhinodontus and Pseudocneorhinus, sharing the character states of epistome projected anteriorly and ocular lobes with short setae with Rhinodontus, and having as an autapomorphy, rostrum continuous with head, not separated by any furrow. Some of the characters previously used to distinguish Rhinodontodes are not unique, in comparison with newly known Pseudocneorhinus described in P.bifasciatus Roelofs, 1880 also have the epistome projected anteriorly .Holotype. China \u2013 Inner Mongolia Autonomous Region \u2022 1 \u2642; \u0410\u043b\u0430\u0448\u0430\u043d\u044a, \u0414\u044b\u043d-\u044e\u0430\u043d\u044c-\u0438\u043d, V.08 \u0435\u043a. \u041a\u043e\u0437\u043b\u043e\u0432\u0430 ; Pseudocneorhinus alashanicus Typ. m.; G. Suvorov det.; ZIN. Paratypes. CHINA \u2013 Inner Mongolia Autonomous Region \u2022 1 \u2642 1 \u2640; same data as for holotype; ZIN.Body length: 3.94\u20134.31 mm, holotype 3.94 mm.Body  1.4\u20131.6 \u00d7 as long as tarsomere 3. Claws fused at basal third, moderately and regularly divergent apicad.Penis ; Inner Mongolia Autonomous Region \u2022 2 \u2642\u2642 2 \u2640\u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u6c34\u78e8\u6c9f\u6b63\u6c9f ; 27 Jul. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1965104, IOZ(E)1965108, IOZ(E)1965105, IOZ(E)1965110; \u2022 3 \u2640\u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u54c8\u62c9\u4e4c\u9752\u6811\u6e7e ; 30 Jul. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1965107, IOZ(E)1965109, IOZ(E)1965113; \u2022 1 \u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u4e3b\u5cf0\u5cf0\u9876 ; 3134 m a.s.l.; 17 Aug. 2010; 38\u00b049.8'N, 105\u00b056.4'E; \u6797\u7f8e\u82f1 [M.Y. Lin leg.]; IZCAS, IOZ(E)1965112; \u2022 1 \u2642; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u6c34\u78e8\u6c9f\u6b63\u6c9f ; 2025 m a.s.l.; 16 Aug. 2010; 38\u00b055.8'N, 105\u00b053.4'E; \u6797\u7f8e\u82f1 [M.Y. Lin leg.]; IZCAS, IOZ(E)1965114 \u2022 2 \u2640\u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u5f3a\u5c97\u5cad ; 8 Aug. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1965106, IOZ(E)1965111; \u2022 1 \u2642 1 \u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u53e4\u62c9\u672c ; 6 Aug. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1965102, IOZ(E)1965101; \u2022 1 \u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u5317\u5bfa ; 13 Aug. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1965103; \u2022 1 \u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u6c34\u78e8\u6c9f ; 25 Jul. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1941122.China \u2013 RBSC; \u2022 2 \u2640\u2640; Bayan-Chong, Aimak, Ich-Bogdo-Ula, srednegorie [central mountains]; 2500 m a.s.l.; 3 Jul. 1973; G. Medvedev leg.; ZIN.Mongolia \u2022 1 \u2640; 40 km W Dalanzadgad, Gobi Gurvansaikhan NP, Yolyn am env.; 28\u201330 Jun. 2003; 1700\u20132000 m a.s.l.; Z. Jindra leg.; R.subsignatus come from Mongolia and also from China, Inner Mongolia Autonomous Region. The four males and 11 females from China differ somewhat from the three females from Mongolia, which share with the holotype slender, subparallel-sided, semi-erect elytral setae, while material from Mongolia has wider, subspatulate, semi-appressed elytral setae. Mongolian and Chinese specimens are almost identical in all other characters thus we assume the shape of elytral setae is a variable character of the species. This is the first record of R.subsignatus from China ; Rhinodontus currently contains five valid species from China and Mongolia ; Qinghai Prov. \u2022 2 \u2640\u2640; \u042e. \u0441\u043a\u043b. \u0445\u0440. \u0411\u0443\u0440\u0445\u0430\u043d-\u0411\u0443\u0434\u0434\u0430: \u0434o\u043b. o\u0437. \u0410\u043b\u044b\u043a-\u043do\u0440. 30.V.1900. \u0415\u043a\u0441\u043f. \u041ao\u0437\u043bo\u0432\u0430. ; ZIN.China \u2013 This species was described based on three females from China, Xinjiang and Gansu. This is the first additional locality since the original description.Rhinodontuscrassiscapus differs from all other species of the genus by its very short, distally thickened scape and by its long raised elytral setae being longer than one half of the interval width.Taxon classificationAnimaliaColeopteraCurculionidaeFaust, 18908F6F4056-F0C6-5E77-BDE5-EF41EBA5E089Rhinodontusignarus Faust, 1890: 455 ; Rhinodontusproximuscentralis Voss, 1967: 276 .Inner Mongolia Autonomous Region \u2022 2 \u2640\u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u54c8\u62c9\u4e4c\u9752\u6811\u6e7e ; 30 Jul. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1965140, IOZ(E)1965142; \u2022 1 \u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u5f3a\u5c97\u5cad ; 8 Aug. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1965141; \u2022 6 \u2640\u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u54c8\u62c9\u4e4c\u4e3b\u5cf0 ; 1 Aug. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1965143\u20131965148.China \u2013 ZIN; \u2022 1 \u2640; Centralnyi Aimak, Dzorgol-Khairkhan, Uver-Undzhul-Ul hill; 16 Jul. 1973; G. Medvedev leg.; ZIN.Mongolia \u2022 1 \u2640, Centralnyi Aimak, Dzorgol-Khairkhan, 30 km NE Undzhul; 16 Jul. 1973; G. Medvedev leg.; R.ignarus.Nine females from China (Inner Mongolia) have the spermatheca with a shorter ramus and more slender collum and nine spines at the protibial apex in comparison with previously known material, including the type specimens, of the species having only eight spines. Due to the lack of males of this population we currently retain it as conspecific with Taxon classificationAnimaliaColeopteraCurculionidaeBorovec, 200347E16381-005D-55D2-90D3-30D68AFE87AARhinodontusmongolicus Borovec, 2003: 36 ; Inner Mongolia Autonomous Region \u2022 1 \u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u54c8\u62c9\u4e4c\u4e3b\u5cf0; 1 Aug. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1965118; \u2022 1 \u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u5f3a\u5c97\u5cad ; 8 Aug. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1965131; \u2022 1 \u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u6c34\u78e8\u6c9f\u6b63\u6c9f ; 27 Jul. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1965133; \u2022 1\u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u54c8\u62c9\u4e4c\u9752\u6811\u6e7e ; 30 Jul. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1965137; \u2022 1 \u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u6c34\u78e8\u6c9f ; 25 Jul. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1941074.China \u2013 ZIN; \u2022 1 \u2640; Ara-Khangaiskyi Aimak, 20 km NE Tevshrulakh; 17 Jun. 1975; Emelianov leg.; ZIN; \u2022 1 \u2640; Uver Khangaiskyi Aimak, Orkhon, 15 km W Bat-Ulgyi; 22 Sept. 1981; Korolas leg.; ZIN.Mongolia \u2022 1 \u2640; Uver Khangaiskyi Aimak, Arc-Bogdo Mts., 20 km S Khovda; 12\u201313 Aug. 1967; Kerzhner leg.; Rhinodontusmongolicus is very easy distinguishable from all other species of Rhinodontus by the prominent sulci covering eyes in dorsal view and by the slender antennal scape.The species was described based on 17 females from Mongolia, Ulaan Baatar and these are the first additional specimens since the original description. This is also the first record of the species in China Fig. . RhinodoTaxon classificationAnimaliaColeopteraCurculionidaeVoss, 19675248D370-C7EE-5AEC-83FF-00CE501EE92DRhinodontusproximus Voss, 1967: 275 ; Inner Mongolia Autonomous Region \u2022 1 \u2640; \u963f\u62c9\u5584\u5de6\u65d7\u8d3a\u5170\u5c71\u5357\u5bfa\u96ea\u5cad\u5b50; 11 Aug. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1965149; \u2022 4 \u2640\u2640; \u963f\u62c9\u5584\u5de6\u65d7\u6c34\u78e8\u6c9f ; 25 Jul. 2010; \u9ec4\u946b\u78ca [X.L. Huang leg.]; IZCAS, IOZ(E)1941092\u20131941095. \u2013 Gansu Prov. [Kan-Ssu Prov.] \u2022 1 \u2640; 1884; O. Potanin leg.; ZIN.China \u2013 45\u00b055.1'N, 101\u00b010.4'E; 2050 m, a.s.l.; 8 Jun. 2013; M. Ko\u0161\u0165\u00e1l leg.; MKBC; \u2022 2 \u2640\u2640; Uver Khangaisk Aimak, Arc-Bogdo Mts., 20 km S Khovda; 12\u201313 Aug. 1967; Kerzchner leg.; ZIN; \u2022 2 \u2640\u2640; Iuzhno-Gob. Aimak, Ukh-Shankhai; 12 Jun. 1972; ZIN; \u2022 3 \u2640\u2640; Iuzhno-Gob. Aimak, 25 km SW Bulgan; 5 Aug. 1971; ZIN; \u2022 1 \u2640; Iuzhno-Gob. Aimak, Navtgar-Ul hill, 35 km NW Iamat-Ul; 9 Aug. 1971; Emelianov leg.; ZIN; \u2022 3 \u2640\u2640; Vostochno-Gob. Aimak, Nomt-Ul hill, 30 km SSE Shokhoi-Nur lake; 26 Jun. 1971; Emelianov & Kozlov leg.; ZIN; \u2022 1 \u2640; Baian-Kchongor. Aimak, 20 km ESE Uldzint; 9 Jul. 1970; Emelianov leg.; ZIN; \u2022 2 \u2640\u2640; Iuzhno-Gob. Aimak, Tachilga-Ul hill, 35 km NNE Dalan-Deadagad; 10 Aug. 1971; Kerzhner leg.; ZIN; \u2022 1 \u2640; Centralnyi Aimak, Dzorgol-Khairkhan, Uver-Undzhul-Ul hill; 16 Jul. 1973; G. Medvedev leg.; ZIN; \u2022 1 \u2640; Iuzhno-Gob. Aimak, Khuryn-Khalkha-Nur, 25 km W No\u00ebn; 20 Jun. 1973; G. Medvedev leg.; ZIN.Mongolia \u2022 1 \u2640; Bayankhongor aym., Khangayan Nuruu Mts., Tsagaan-Ovoo 25 km W; R.ignarus, but differs by possessing eight or nine spines at apex of protibia, tarsal claws connate only in the very short basal part, and also the more slender antenna.This species was described from four specimens from two localities in Mongolia, later recorded also from China. It is very similar to Taxon classificationAnimaliaColeopteraCurculionidaeBorovec, 200369834EBD-F4CA-58B8-89DE-37081AC6336CRhinodontussawadai Borovec, 2003: 40 .Rhinodontussawadai : Xinjiang Autonomous Region \u2022 1 \u2640; Polu; 13 May 1890; ZIN.China \u2013 Rhinodontussawadai can be distinguished from other species of the genus by its wider rostrum, curved scape, missing prominences above eyes, and less enlarged outside apex of protibia.This species was described based on three females from China, Xinjiang; this is the first additional specimen since the original description. Taxon classificationAnimaliaColeopteraCurculionidaeWollaston, 18692D61DFD1-3A97-57E0-AD58-904BEFFD9632Trachyphloeosoma Wollaston, 1869: 414 ; Trachyphloeini by Entiminae, surveyed the Japanese species of the genus. China is the most northwestern part of the range of the genus, and Trachyphloeosoma was first recorded from this country only in 2009 by Borovec, based on one male from Yunnan, and subsequently by T.advena Zimmerman, 1956 and was listed under this name in the Palaearctic catalogue  23 Jun. 2016; J. H\u00e1jek & J. R\u016f\u017ei\u010dka leg.; sift \u266f05, border of old orchard, wet debris under trees; NMPC; \u2022 2 \u2640\u2640; same data as for preceding; RBSC; \u2022 1 \u2640; same data as for preceding; IZCAS; \u2022 2 \u2640\u2640; Tengchong city, Laifeng Shan Forest Park; 25\u00b001.24'N, 98\u00b028.94'E; 1800 m a.s.l.; (CH05) 22 Jun. 2016; J. H\u00e1jek & J. R\u016f\u017ei\u010dka leg.; sift \u266f04, dense mixed forest above tombs near track, wet debris in terrain depressions; NMPC.Body length: 1.87\u20132.39 mm, holotype 2.13 mm.Body . The Czech name Jan has its nickname \u201cHonza\u201d. The specific name is a noun in apposition.China, Yunnan Fig. .Trachyphloeosomahonza sp. nov. shares with T.martin sp. nov. short and robust protibiae, short and wide rostrum and subspatulate setae. It is easily distinguished from T.martin sp. nov. by elytral setae on all elytral intervals, dorsal margin of antennal scrobes directed towards middle of eye and female sternite VIII lacking fenestra, while T.martin sp. nov. has elytral setae only on odd intervals, dorsal margin of scrobes directed above dorsal margin of eye and female sternite VIII with longitudinal fenestra. In comparison with non-Chinese species, T.honza sp. nov. is similar to T.advena Zimmerman, 1956, known from Japan, Korea and introduced to U.S.A. and T.ryukyuensis Morimoto, 2015, known from Japan, in the funicle being 7-segmented and body covered by appressed setae and elytra with raised setae on all intervals. It is possible to distinguish T.honza sp. nov. from both by short subspatulate setae, distinctly shorter than width of an elytral interval , elytral setae distinctly bent backwards in lateral view (perpendicularly erect in T.advena and T.ryukyuensis) and plate of sternite VIII in females without fenestra (with fenestra in T.advena and T.ryukyuensis).Taxon classificationAnimaliaColeopteraCurculionidae38BF82DD-1327-54EE-BFAE-7167A564531Bhttp://zoobank.org/766B872B-D8F0-4E20-8DD4-D27E44BAEA02China, Jiangxi, Jinggangshan Mts., Xiangzhou.Holotype. China \u2013 Jiangxi Prov. \u2022 1 \u2640; Jinggangshan Mts., Xiangzhou ; 26\u00b035.5'N, 114\u00b016.0'E; 374 m a.s.l.; 26 Apr. 2011; Fik\u00e1\u010dek & H\u00e1jek leg.; sifting, accumulation of moist leaf litter along the stream and on the steep slope above the stream in the sparse secondary forest; [MF08]; NMPC. Paratypes. China \u2013 Jiangxi Prov. \u2022 1 \u2640; the same data as holotype; NMPC; \u2022 1 \u2640; same data as holotype; IZCAS.Body length: 2.06\u20132.44 mm, holotype 2.06 mm.Body , elytra long, oval, 1.42\u20131.46 \u00d7 longer than wide  and protibiae slender, distinctly curved inwards at apical portion  and also plate of sternite VIII in females without fenestra (with fenestra in T.advena and T.ryukyuensis).Taxon classificationAnimaliaColeopteraCurculionidae8CDC37CF-558E-5FBC-A722-D174303EC669http://zoobank.org/0F81305F-227E-49C0-95E3-B4FB14BB3F1CChina, Hainan, Limushan Mts.Holotype. China \u2013 Hainan Prov. \u2022 1 \u2642; Limushan Mts., mountains above frst. admin. Centre; 19\u00b010.5\u201319\u00b010.9'N, 109\u00b044\u2013109\u00b045'E; 650\u2013900 m a.s.l.; 6 May 2011; Fik\u00e1\u010dek leg.; sifting \u2013 small accumulations of moist leaf litter along an on the trail in secondary forest partly with Cyathea and bamboo; MF19; NMPC. Paratypes. China \u2013 Hainan Prov. \u2022 1 \u2640; the same data as holotype; NMPC; \u2022 1 \u2640; same data as for preceding; IZCAS.Body length: 1.63\u20132.31 mm, holotype 1.63 mm.Body  as long as tarsomere 3, strikingly widened apicad with very long, strongly divaricate claws, as long as part of onychium projecting beyond lobes of tarsomere 3.Abdominal ventrites 1.14\u20131.19 \u00d7 longer than wide, sparsely roughly punctate; ventrite 2 slightly longer than ventrite 1 and distinctly longer than ventrites 3 and 4 combined; suture between ventrites 1 and 2 sinuate, the others straight. Metaventral process as wide as transverse diameter of metacoxa.Penis  known from St. Helena, where it is apparently introduced (but region of origin not yet known). Trachyphloeosomamartin sp. nov. is similar to them in having raised elytral setae only on odd intervals, but distinguished from them by a more slender and longer rostrum, 1.38\u20131.42 \u00d7 wider than long (1.56\u20131.73 \u00d7 in T.roelofsi and T.setosum), longer and more slender elytra, 1.44\u20131.48 \u00d7 longer than wide, (1.19\u20131.27 \u00d7 in T.roelofsi and T.setosum), and also by the different shape of the spermatheca, with collum distinctly longer than wide (isodiametric in T.roelofsi), or long and irregularly curved cornu (short and regularly curved in T.setosum).Taxon classificationAnimaliaColeopteraCurculionidaeSharp, 18966CE3DF4D-8704-5EC3-BD3F-D218E9B9E146Trachyphloeopssetosusnon Wollaston, 1869). Roelofs 1873: 166 (TrachyphloeosomaroelofsiTrachyphloeopssetosus Roelofs);  Sharp, 1896: 92  Beitou Twnsh., Taipei Co., S. Samau Mt.; 3 Jan. 2009; S. V\u00edt leg.; dead leaves; NMPC; \u2022 1 \u2640; Rd. Jhuzihhu/Shuiwei, Yangmingshan Mts., slopes E of Mt. Datun, Taipei Co.; 650 m a.s.l.; 24 Oct. 2007; S. V\u00edt leg.; putresc. base of Cryptomeria (?); NMPC.China \u2013 Trachyphloeosomasetosum Wollaston, 1869 from St. Helena with Sharp\u00b4s material of T.roelofsi from Japan and found the two series represent only one species and placed them in synonymy. However, T.roelofsi as an independent species, and distinguished it from T.setosus Wollaston from St. Helena. Trachyphloeosomaroelofsi is thus known from Japan and Taiwan, while T.setosus is assumed as a species introduced to St. Helena without knowledge of its original country.This species was described by Sharp from Nagasaki, Japan."}
{"text": "Yazdana shirkuhensis gen. & spec. nov.  is described and illustrated from the high alpine zone of this mountain. Molecular phylogenetic analyses of nuclear and plastid DNA sequence data show that Y. shirkuhensis is related to Cyathophylla and Heterochroa (tribe Caryophylleae). The newly described genus and species accentuate Shirkuh Mts. as a center of endemism, which harbors a high number of narrowly distributed species, mostly in high elevations reaching alpine habitats. As this area is currently not protected, a conservation priority is highlighted for high elevations of Shirkuh Mts.Although mountain ranges are often recognized as global biodiversity hotspots with a high level of endemism, diversity and biogeographic connections of isolated and weakly explored mountains remain poorly understood. This is also the case for Shirkuh Mts. in central Iran. Here,  C. Wu & C. Y. Wu, Saponaria L.), of which seven are found in Iran  are located in the southern part of the Irano\u2010Turanian region, with a dry and continental climate Zohary,\u00a0, and ann2.2Plant material of the new taxon was collected in early July 2012 and, in the course of a trip dedicated to re\u2010collect this species, in mid July 2019. For the molecular investigation, leaves of six individuals  were used. For detailed morphological investigation, 18 individuals (3 from 2012 and 15 from 2019) were collected as vouchers.2.3rps16 intron and the nuclear ribosomal ITS  regions were amplified and sequenced using the primers rpsF and rpsR2  and the addition of 3% DMSO (Sigma\u2010Aldrich) for ITS.Total genomic DNA was extracted using the DNeasy Plant Mini Kit  according to the manufacturer's protocol. The plastid https://www.geneious.com). The 12 newly obtained sequences (six for each marker) were added to the Caryophylleae data set of Madhani et al. (rps16 data set contained 119 accessions representing 86 species (Doc. S1). Following Madhani et al. (Silene (five species sampled) was selected as outgroup. Nuclear and plastid DNA sequence data were analyzed separately and jointly. Combinability of the markers was assessed with the incongruence length difference (ILD) test  were analyzed using maximum likelihood (ML) and Bayesian methods. The best\u2010fit substitution models for the ITS and the rps16 data, determined using the Akaike Information Criterion (AIC) as implemented in jModelTest 2.1.4  using 1000 bootstrap replicates, obtained by the rapid bootstrap algorithm  in Shirkuh Mts. at elevations above 1400\u2009m a.s.l. were recorded, and their distribution patterns in different mountain ranges of Iran . The new taxon is placed in tribe Caryophylleae with strong support as closely related to Cyathophylla and Heterochroa . Additional specimens: Iran, Yazd, Shirkuh Mt., 31.610\u00b0 N, 54.068\u00b0 E, 4000\u2009m a.s.l., on scree grounds, 5 July 2012, J. Noroozi 2827 (WU).Type: Cyathophylla and Heterochroa, but it differs from Cyathophylla by non\u2010perfoliate leaves, bicolored petals, and capsules \u00b1enclosed in the calyx, and from Heterochroa in being annual, possessing dark purple stems, having basal spathulate leaves and capsules \u00b1enclosed in the calyx. A comparison among the three genera is provided in Table\u00a0Diagnosis: This monotypic genus is similar to Note: The generic and the specific names are published here simultaneously via a single diagnosis , emarginate, 4\u20134.5\u2009mm long, only slightly clawed; stamens 10, enclosed, 3.5\u20134\u2009mm, developing non\u2010simultaneously ; styles 2, 1.5\u20132\u2009mm; ovary with a short gynophore; ovules 6\u201310; capsules oblong, slightly shorter than or subequal to the calyx, opening with 4 valves; seeds 1\u20138, reniform to reniform\u2010roundish, black, 1\u2009\u00d7\u20090.8\u2009mm.Etymology: The generic name refers to the city of Yazd in central Iran, whereas the specific name refers to Shirkuh Mts. in the vicinity of Yazd.Distribution: It is found only in Shirkuh Mts., immediately below the highest summit (4050\u2009m a.s.l.) on the northern slope.Yazdana, based on six vegetation plots (each of 5\u2009\u00d7\u20095\u2009m), are presented in Table\u00a0Senecio has been discovered  without particular attention to the population size. In 2019, the location was well explored to find more individuals of the species and to make more detailed observations on its ecology and accompanying species. The species grows only in the northern slope and in a few scree patches from 3950 up to 4050\u2009m a.s.l. The size of the population was estimated to have been between 100 to 200 individuals in this year. Its conservation status is, thus, given as Critically Endangered  according to IUCN criterion B  hilum; embryo straight\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202621b.reniform\u2010roundish or comma\u2010shaped, with lateral hilum; embryo curved or hook\u2010shaped\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.5Seeds reniform, reniform\u2010oblong, 2a.PsammosileneLeaves with short petiole, ovate; stamens 5; capsules membranous, nearly indehiscent\u2026\u2026\u2026\u2026\u2026..2b.Leaves sessile, linear, subulate, grass\u2010like; stamens (5)10; capsules papery, dehiscent\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..33a.Velezia); calyx tube long tubular, teeth straight\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.Dianthus (incl. Velezia)Calyx without membranous commissures, with 35 or more veins, rarely 5\u201015\u2010nerved Seeds >1.5 mm, with thin margin, smooth on surface\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20264b.PetrorhagiaSeeds <1.5 mm, with thickened margin, reticulate on surface\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..5a.Seeds comma\u2010shaped (or oblong), with hook\u2010shaped embryo\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202665b.reniform\u2010roundish, or reniform\u2010oblong, embryo curved\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.7Seeds reniform, 6a.GraecobolanthusPetals turning abruptly downward and becoming clearly deflexed (Greece)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20266b.Bolanthus (incl. Phrynella)Petals recurved gradually \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20267a.DiaphanopteraCalyx bladdery inflated, or turbinate, constricted at teeth, commissural region membranous hyaline, sometimes wing\u2010like\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..7b.cylindrical, or tubular, if inflated, commissural regions papery or leafy and main veins with leafy wings\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.8Calyx campanulate, 8a.AcanthophyllumBracteoles present, leafy, papery or rarely membranous; calyx papery in texture or only membranous at intervals\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.8b.Bracteoles absent\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.99a.Gypsophila (cf. Vaccaria)Calyx bladdery inflated, nerves prominent and thick, costate, or winged, midveins 5; bracteoles membranous hyaline\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..9b.cylindrical, or obconical, not much inflated, lateral nerves obscure, not prominent and thick, midveins 5 or more; bracteoles absent\u2026\u2026\u2026\u2026\u2026..10Calyx tubular, campanulate, 10a.Calyx obscurely nerved or with 15\u201025 nerves, commissures absent or present; petals inconspicuous, or clawed, mostly with appendages\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..1110b.10 nerved, with membranous commissures; petals not or only indistinctly clawed, without appendages\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202612Calyx 5\u201011a.CyathophyllaPlants annual; inflorescences congested; capsule slightly longer than the calyx; coronal scales absent\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.11b.SaponariaPlants annual, biennial or perennial; inflorescences usually lax; capsule mostly shorter than the calyx; coronal scales mostly present\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202612a.PetroanaLeaves fleshy, spathulate; flowers very small: calyx <4mm, corolla<5 mm; seed testa with swollen cells tuberculate on periclinal wall, testa cells polygonal\u2010oblong, moderately elongated \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.12b.lanceolate, or ovate; flowers small or large; seed testa variously shaped, with or without tubercles\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202613Leaves not fleshy or subfleshy, linear, 13a.Petals bicolored\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..1413b.Petals always concolored, variously colored; leaves slender, in few species triquetrous, then the plants mostly caespitose, paired at nodes\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.1514a.Petals red on the outer surface, white or pink on the inner surface; leaves triquetrous, mostly 3 or 4 at each node \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..Balkana14b.YazdanaPetals greenish at base, white to white tinged with purple at apex, leaves slender, paired at each node (Iran)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.15a.The stigmatic surface terminal; ovules fewer than 24\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..1615b.The stigmatic surface extending along the inner side of styles; ovules 24\u201036\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..2016a.AcanthophyllumStem nodes with small lateral shoots in leaf axils giving a verticillate appearance; leaves acerose, spiny, or terminating to a spine\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..16b.G. acantholimoides and G. pinifolia\u2026\u2026\u2026\u2026\u2026..17Lateral shoots in leaf axils absent; leaves not spiny except in 17a.Capsules shorter than the calyx\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.1817b.Capsules exceeding the calyx\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..1918a.Bolanthus confertifoliusPlants annual, shorter than 10 cm, covered by long glandular hairs\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..18b.GypsophilaPlants perennial, if annual then taller than 10 cm and glandular hairs absent or short\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202619a.Acanthophyllum (cf. A. cerastioides)Plants perennial; capsules \u00b1indehiscent\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202619b.Plants annual or perennial; capsules dehiscent\u2026\u2026\u2026\u20262020a.HeterochroaCalyx without membranous commissural intervals or with very narrow ones, calcium oxalate crystals absent; stamens shorter than the petals\u2026\u2026\u2026\u2026\u2026\u2026.20b.Calyx with membranous commissural intervals encompassing calcium oxalate crystals; stamens longer (or sometimes shorter) than petals\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262121a.PsammophiliellaAnnual plants with fibrous roots, puberulent below and glabrous in inflorescence \u2026\u2026\u2026\u2026\u2026\u2026. 21b.GypsophilaAnnual or perennial plants with tap root, variously hairy\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.Identification key: To allow 3.2A total of 125 plant species endemic to the Iranian Plateau are recorded from Shirkuh Mts. above 1400\u2009m a.s.l. Fig.\u00a0, with 13Acantholimon horridum Bunge, Plumbaginaceae; Echinops cervicornis Bornm., Asteraceae), three species between 2000 and 2500\u2009m a.s.l. , three species between 2500 and 3000\u2009m a.s.l. , one species between 3000 and 3500\u2009m , and four species between 3500 and 4050\u2009m a.s.l. . A. issatissensis was described as a new species  in a single trip demonstrates that the plant diversity of this area is still poorly explored. Therefore, detailed studies of flora and vegetation of the Shirkuh Mts., especially in high elevations, are highly recommended. Naturally, the area size decreases sharply from lowlands to high elevations. Whereas plant diversity increases until mid elevation (ca. 2000\u2009m a.s.l.), it decreases gradually until the nival zone , voucher (herbarium), GenBank accession numbers for ITS and rps16, respectively. Species names follow the taxonomic treatment suggested in the present study.Click here for additional data file.Doc. S2. All phylogenetic trees obtained from three datasets using Maximum likelihood and Bayesian approaches.Click here for additional data file.Supporting information.Click here for additional data file.Supporting information.Click here for additional data file.Supporting information.Click here for additional data file.Supporting information.Click here for additional data file.Supporting information.Click here for additional data file."}
{"text": "Nature Communications 10.1038/s41467-020-18011-9, published online 25 August 2020.Correction to: M. smegmatis Sdh2\u2019, where the Sdh2 apparent catalytic constant (kcat) value was incorrectly stated as (3.16\u2009\u00b1\u20090.07)\u2009\u00d7\u2009104 s\u22121, rather than the correct value of 3.16\u2009\u00b1\u20090.07\u2009s\u22121. The original version of this Article also contained an error in the Results and discussion section \u2018QD-binding site contributes to the structure-based drug discovery\u2019, where the kcat values of the QD1 and QD mutants were incorrectly stated as (2.16\u2009\u00b1\u20090.04)\u2009\u00d7\u2009104 s\u22121 and (1.48\u2009\u00b1\u20090.04)\u2009\u00d7\u2009104\u2009s\u22121, which has been corrected to 2.16\u2009\u00b1\u20090.04\u2009s\u22121 and 1.48\u2009\u00b1\u20090.04\u2009s\u22121, respectively. This has been corrected in both the PDF and HTML versions of the Article.The original version of this Article contained an error in the second last sentence of the Results and discussion section \u2018Purification and characterization of"}
{"text": "There are errors in the mean values in Row \u201cHead throws\u201d of column \u201cCONT\u201d: Original \u201c1.94\u201d, Corrected \u201c2.53\u201dRow \u201cHead throws\u201d of column \u201cB&W\u201d: Original \u201c0.72\u201d, Corrected \u201c0.89\u201dRow \u201cHead throws\u201d of column \u201cB\u201d: Original \u201c1.89\u201d, Corrected \u201c2.14\u201dRow \u201cEat beats\u201d of column \u201cCONT\u201d: Original \u201c14.8\u201d, Corrected \u201c13.2\u201dPlease see the correct There are also errors in the Supporting Information, S1 File(XLSX)Click here for additional data file."}
{"text": "Correction to: Scientific Reports 10.1038/s41598-018-21282-4, published online 16 February 2018This Article contains errors.As a result of errors during the figure assembly, in Figure\u00a02 different frames from the same original image for \u201cBurn\u2009+\u2009ITF/7\u00a0days\u201d sample were used also for the image of the \u201cControl/1\u00a0day\u201d sample and the image for \u201cBurn\u2009+\u2009ITF/5\u00a0days\u201d sample. An incorrect image was also used for \u201cControl/3\u00a0day\u201d inset. The correct Figure\u00a02 appears below as Figure\u00a0Additionally, a Supplementary Information file containing original data for Figure\u00a0Supplementary Information."}
{"text": "Scientific Reports 10.1038/s41598-018-29166-3, published online 18 July 2018Correction to: 2 should be given as \u2018\u00b5M\u2019 instead of \u2018\u00b5g L\u22121\u2019. The correct Figures\u00a0In Figures 1 and 2B, the units for CO"}
{"text": "While there are currently two approved gene-based therapies for SMA, availability, high cost, and differences in patient response indicate that alternative treatment options are needed. Optimal therapeutic strategies will likely be a combination of SMN-dependent and -independent treatments aimed at alleviating symptoms in the central nervous system and peripheral muscles. Kr\u00fcppel-like factor 15 (KLF15) is a transcription factor that regulates key metabolic and ergogenic pathways in muscle. We have recently reported significant downregulation of Klf15 in muscle of presymptomatic SMA mice. Importantly, perinatal upregulation of Klf15 via transgenic and pharmacological methods resulted in improved disease phenotypes in SMA mice, including weight and survival. In the current study, we designed an adeno-associated virus serotype 8 (AAV8) vector to overexpress a codon-optimized Klf15 cDNA under the muscle-specific Spc5-12 promoter (AAV8-Klf15). Administration of AAV8-Klf15 to severe Taiwanese Smn\u2212/\u2212;SMN2 or intermediate SmnB/\u22122 SMA mice significantly increased Klf15 expression in muscle. We also observed significant activity of the AAV8-Klf15 vector in liver and heart. AAV8-mediated Klf15 overexpression moderately improved survival in the SmnB/\u22122 model but not in the Taiwanese mice. An inability to specifically induce Klf15 expression at physiological levels in a time- and tissue-dependent manner may have contributed to this limited efficacy. Thus, our work demonstrates that an AAV8-Spc5-12 vector induces high gene expression as early as P2 in several tissues including muscle, heart, and liver, but highlights the challenges of achieving meaningful vector-mediated transgene expression of Klf15.Spinal muscular atrophy (SMA) is a neuromuscular disease caused by loss of the  As aeletions . While aeletions , humans med SMN2 , which adegraded , 6. Impo\u2122, was approved in December 2016 by the Food and Drug Administration (FDA) and in June 2017 by the European Medicines Agency\u00a0(EMA) [SMN2 exon 7 inclusion [SMN1 via an adeno-associated virus serotype 9 (AAV9) gene therapy that received FDA approval in May 2019 [The first genetic therapy for SMA, nusinersen/Spinrazacy\u00a0(EMA) . This annclusion . ZolgensMay 2019 . AdditioMay 2019 . While tMay 2019 \u201315.Klf15) in skeletal muscle of SMA mice during disease progression [Klf15 expression in presymptomatic SMA mice and found that its neonatal upregulation via pharmacological (prednisolone) or transgenic (muscle-specific Klf15 overexpression) interventions significantly improved several disease phenotypes in SMA mice [Klf15 in skeletal muscle of SMA may have resulted in compensatory mechanisms [Klf15 in skeletal muscle of neonatal SMA mice via a self-complementary adeno-associated virus serotype 2/8 and the Spc5-12 promoter. While this strategy led to substantial Klf15 expression in skeletal muscle of SMA mice and control littermates, there were no associated significant improvements in disease phenotypes. Nevertheless, AAV8-Klf15 injections resulted in pronounced expression as early as postnatal day 2 in several tissues, including muscle, liver, and heart, highlighting the potential of this specific viral construct for efficient perinatal delivery.Skeletal muscle with reduced levels of SMN displays both cell-autonomous and non-autonomous defects , 17 and gression . KLF15 igression \u201322. SpecSMA mice . Howeverchanisms . In thisSmn\u2212/\u2212;SMN2 2Hung/J, RRID: J:59313) [SmnB/\u22122 [ad libitum, 12\u2009h light: 12\u2009h dark cycle) at the Biomedical Sciences Unit, University of Oxford, according to procedures authorized by the UK Home Office . The viral constructs were diluted in sterile 0.9% saline and administered at the indicated dose at postnatal day (P) 0 by a facial vein intravenous injection [SmnB/\u22122 mice, weaned mice were given daily wet chow at the bottom of the cage to ensure proper access to food. Sample sizes were determined based on similar studies with SMA mice.Wild-type (WT) FVB/N mice were used for initial expression screening. The Taiwanese J:59313)  and the [Smn2B/\u2212  mice (genjection . LittersKlf15 sequence:The generation of the self-complementary adeno-associated virus serotype 2/8 (scAAV2/8) vectors and quality control were performed by Atlantic Gene Therapies . The synthetic Spc5-12 promoter  was usedagcgcttcaccacaggctcggccaggccagcatggttgatcatctgctgcctgtggacgagacattcagcagccctaagtgttctgtgggctacctgggcgacagactggcctctagacagccttaccacatgctgccctctccaatcagcgaggacgactccgatgtgtctagcccttgtagctgtgcctctcctgacagccaggccttctgtagctgttactctgctggacctggacctgaggctcagggctctatcctggatttcctgctgagcagagctacactcggctctggcggaggatctggcggaatcggagattcttctggccctgtgacatggggctcttggagaagggctagcgtgcccgtgaaagaggaacacttctgcttccctgagttcctgagcggcgacaccgatgacgtgtccagacctttccagcctacactggaagagatcgaagagttcctcgaagagaacatggaagccgaagtgaaagaagcccctgagaacggctcccgcgacctggaaacatgttctcagctgtctgccggctctcacagaagccatctgcaccctgaaagcgccggcagagagagatgtacacctcctccaggtggaacatctggcggcggagcacaatctgctggcgaaggacctgctcatgatggacctgtgcctgtgctgctgcaaatccagcctgtggctgtgaagcaagaggctggaacaggaccagcttctcctggacaggctcctgaatctgtgaaggtggcccagctgctggtcaacatccagggacaaacattcgccctgctgcctcaggtggtgcccagcagtaatctgaacctgcctagcaagttcgtgcggatcgctcctgtgccaatcgctgctaagcctatcggatctggctctcttggtcctggaccagctggactgctcgtgggacagaagttccctaagaaccctgccgccgagctgctgaagatgcacaagtgtacattccccggctgctccaagatgtataccaagtcctctcacctgaaggcccacctgagaaggcataccggcgagaagcctttcgcttgcacatggcctggatgtggctggcggttcagcagatctgatgagctgagcaggcaccgcagatctcacagcggagtgaagccataccagtgtcctgtgtgcgagaagaagttcgccagaagcgaccacctgtccaagcacatcaaggtgcacagattccctagaagcagcagagccgtgcgggccatcaattgactgcagaagctt.5. Cells were harvested 3 days post-transduction for molecular analyses .C2C12 myoblast cells  were maiKlf15 sequence . RNA polymerase II polypeptide J (PolJ) was used as a validated stably expressed housekeeping gene [Quadriceps muscles, liver, and heart were harvested at the indicated time points during disease progression and immediately flash frozen. RNA was extracted with the RNeasy MiniKit (Qiagen). For C2C12 cells, the media was removed and cells were washed with phosphate buffered saline (PBS) before being directly lysed as per instructions within the RNeasy MiniKit (Qiagen). Reverse transcription was performed using the High-Capacity cDNA Reverse Transcription Kit (ThermoFisher Scientific). qPCR was performed using SYBR green Mastermix (ThermoFisher Scientific) and primers for the codon-optimized ing gene  . The two primer-probe sets yielded similar results (data not shown) and only data with primer set 2 are shown. No significant difference was detected between Smn+/\u2212;SMN2 and Smn\u2212/\u2212;SMN2 mice for any of the tissues (data not shown) and mice from both genotypes were therefore pooled. No amplification could be seen in untreated samples (data not shown).In order to establish virus penetration into the tissue, TaqMan (ThermoFisher Scientific) qPCRs were performed using two primer-probe sets (Integrated DNA Technologies) specific to the AAV8 capsid . The results of the two primer-probe sets were each efficiency-adjusted and normalized to 5) and untreated cells were trypsinized and washed in fluorescence-activated cell sorting (FACS) buffer . Cells were pelleted by centrifugation at 300\u2009\u00d7\u2009g for 5\u2009min. The final cell pellet was resuspended in 200\u2009\u03bcl of FACS buffer and the cell suspension was further diluted 1:2 in FACS buffer before detection in the Cytek DxP8 flow cytometer (Cytek\u00ae Biosciences). Cell viability was tested by adding 1\u2009\u00b5l Sytox Red (Thermofisher) to the final cell suspension for 5\u2009min at RT. The gating strategy included gating around the Sytox Red (RedFL1 channel) negative population following FCS/FCSW doublet exclusion. The remaining population was assessed for a GFP-shift by recording in the BluFL1 channel. A total of 10,000 cells was recorded for each sample replicate. Data were analyzed using Flowjo 10 software (TreeStar Inc.).Differentiated C2C12 cells 3 days post-transduction .C2C12 cells transduced with the Quadriceps muscles were harvested at the indicated time points during disease progression, fixed in 4% paraformaldehyde, cryopreserved in 30% sucrose, and cryosectioned at a thickness of 12\u2009\u03bcm. The sections were immunostained with chicken anti-GFP antibody  and detected with Alexa-488-conjugated anti-chicken secondary antibody . Images were taken with an Olympus Fluoview FV1000 confocal microscope and processed with Fiji [t test or a two-way ANOVA followed by an uncorrected Fisher\u2019s LSD multiple comparison test was used. Outliers were identified via the Grubbs\u2019 test and subsequently removed. Instances of outlier removal are detailed in the relevant figure legends. For the Kaplan\u2013Meier survival analysis, a log-rank test was used.All statistical analyses were performed using GraphPad Prism version 8.1.1 software. Exact sample sizes as well as description of sample collection, number of times the experiments were replicated and statistical measures and methods can be found in the figure legends. When appropriate, a Student\u2019s unpaired two-tailed Klf15 expression in skeletal muscle, we utilized a self-complementary adeno-associated virus serotype 2/8 driven by the synthetic muscle-specific promoter Spc5-12 [Klf15, henceforth termed AAV8-Klf15). This combination of AAV and promoter has previously been\u00a0successfully used for gene delivery to muscles for treatment of the muscle disorder Duchenne muscular dystrophy\u00a0(DMD) [GFP construct was also generated (henceforth termed AAV8-GFP).To specifically induce  Spc5-12  (scAAV2/hy\u00a0(DMD) . A contrGFP construct in differentiated C2C12 myoblasts [GFP (multiplicity of infection (MOI) 1E105) for 3 days and assessed for GFP expression compared with untreated cells. Both flow cytometry and live imaging analyses confirm the abundant presence of GFP in AAV8-transduced cells  demonstrate a significant increased expression of Klf15 mRNA compared with untreated cells  to P0 WT pups via a facial vein intravenous injection [Klf15 in the Taiwanese Smn\u2212/\u2212;SMN2 SMA mice [GFP expression at both P2 and P7, albeit variable between animals, with increased levels at the later time point  to both genotypes (data not shown). Seeing as this dose was not harmful with AAV8-GFP, the adverse effects are most likely due to the supraphysiological levels of Klf15. We therefore reduced the AAV8-GFP and AAV8-Klf15 dose to 2E10 vg/pup for subsequent administrations, which still allowed for an age-dependent increased expression of GFP , post-weaned mice demonstrate a growth-dependent gain in weight  and ~10 times (Klf15) higher than in skeletal muscle for both time points. Similarly, we find a significant upregulation of GFP and Klf15 mRNA in the hearts of AAV8-GFP-  and ~50 times (Klf15) higher at P2, while it was ~70 times (GFP) and ~10 times (Klf15) higher at P7 compared with skeletal muscle at those respective time points. Furthermore, qPCR analysis of AAV8 content in respective tissues, using capsid-specific primers, indeed demonstrates increased AAV8 presence in heart and liver compared with muscle, particularly at P2  or pharmacological (prednisolone) approaches results in improved disease phenotypes [Klf15 in skeletal muscle in perinatal mice by driving its expression via an AAV8-Spc5-12 vector. We find that while neonatal administration of the AAV8-Klf15 construct leads to significant increased levels of Klf15 in muscle, this has no overt effect on survival or weight gain in the severe Taiwanese SMA mice, while we observe a small improvement in the lifespan of the intermediate SmnB/\u22122 mice.We have recently demonstrated that enotypes . Here, wKlf15 in severe SMA mice may be due to several compounding factors. While the levels of Klf15 expression achieved with AAV8-Klf15 are similar to the levels observed in transgenic SMA mice overexpressing muscle-specific Klf15 at P2 (~5\u201310 fold greater than control littermates), the amounts measured in P7 AAV8-Klf15-treated animals are significantly greater than the transgenic mice   [Klf15 induction ceased at P7, specifically in SMA mice [Klf15 increase in P7 animals may be limited by compensatory inhibitory mechanisms due to already significantly increased Klf15 levels in symptomatic SMA mice compared with controls [Klf15 expression in presymptomatic stages only, which is not easily achieved as the kinetics of AAV-mediated overexpression require several days for efficient transgene activity. To achieve optimal expression at early presymptomatic postnatal stages may therefore require prenatal delivery.We have previously shown that administration of prednisolone to SMA mice also increases ontrols) . HoweverSMA mice , suggestcontrols . It is tKlf15 specifically in skeletal muscle, our analysis of heart and liver demonstrates significantly higher activity in these tissues (Fig.\u00a0Klf15 in the liver and heart of symptomatic mice [Klf15 was used throughout this study, its toxicity may have been due to batch-specific impurities [Klf15 was dose-dependent in both compromised SMA mice and healthy littermates and not observed in parallel experiments with AAV8-GFP, it is most likely that the adverse effects were directly linked to the supraphysiological levels of Klf15 in several key metabolic tissues such as muscle, heart and liver. Thus, the dose-dependent effects of AAV8-Spc5-12-mediated Klf15 upregulation are probably due to a complex interplay between tissue- and age-dependent beneficial and adverse signaling pathways that should be considered and evaluated in future investigations.While our AAV8 construct was designed to overexpress ues Fig.\u00a0. Tropismtic mice , which mpurities . HoweverGFP construct also demonstrated some non-negligible effects on disease phenotypes of SMA mice (Fig.\u00a0Surprisingly, the AAV8-ice Fig.\u00a0. While wKlf15 administration in SMA mice might be explained by several experimental conditions that most likely reduced our ability to increase Klf15 specifically in skeletal muscle at physiological levels and with the optimal timing, without influencing the function of other tissues and systems. In the experimental paradigms tested here, the positive, albeit small, effect on survival and weight was restricted to the milder SmnB/\u22122 SMA mouse model. Future investigations will require endeavors to further optimize a muscle-specific construct by either reconfiguring the AAV/promoter combination and/or inhibiting its expression in non-skeletal muscle tissues.In summary, the limited impact of AAV8-Supplementary Figure 1Supplementary Figure 2Supplementary Figure Legends"}
{"text": "Scientific Reports 10.1038/s41598-019-51841-2, published online 29 October 2019Correction to: In this Article, there is a repeated typographical error in the legends of Figure 1, 2 and 3 where,\u00a7p\u2009\u2009<\u2009\u20090.05 vs. Fasting group; #p\u2009\u2009<\u2009\u20090.05 vs. all other conditions.\u201d\u201cshould read:#p\u2009\u2009<\u2009\u20090.05 vs. Fasting group; \u00a7p\u2009\u2009<\u2009\u20090.05 vs. all other conditions.\u201d\u201c"}
{"text": "Smart meters are of the basic elements in the so-called Smart Grid. These devices, connected to the Internet, keep bidirectional communication with other devices in the Smart Grid structure to allow remote readings and maintenance. As any other device connected to a network, smart meters become vulnerable to attacks with different purposes, like stealing data or altering readings. Nowadays, it is becoming more and more popular to buy and plug-and-play smart meters, additionally to those installed by the energy providers, to directly monitor the energy consumption at home. This option inherently entails security risks that are under the responsibility of householders. In this paper, we focus on an open solution based on Smartpi 2.0 devices with two purposes. On the one hand, we propose a network configuration and different data flows to exchange data (energy readings) in the home. These flows are designed to support collaborative among the devices in order to prevent external attacks and attempts of corrupting the data. On the other hand, we check the vulnerability by performing two kind of attacks . We conclude that, as expected, these devices are vulnerable to these attacks, but we provide mechanisms to detect both of them and to solve, by applying cooperation techniques. The interconnection of devices in electricity networks to support the exchange of data has become an essential aspect that electricity companies need to face. On\u00a0the one hand, because\u00a0it will enhance the self-knowledge of the infrastructure by a constant monitoring of data. On\u00a0the other hand, because\u00a0national and European regulations have strongly encouraged companies to update their systems to improve the efficiency of the energy consumption. This new infrastructure, usually known as Smart Grid, combines advances in both electric engineering and information and communication technology. Smart Grid leads to a more unified and simplified system for control, maintenance and management of the electricity grid, including generation, transmission, distribution, storage and trade. This new philosophy takes into account an important aspect in energy production. The\u00a0growing popularity of photovoltaic facilities and other energy systems has increased the number and variety of energy producers: customers cannot be considered as just consumers anymore, but\u00a0also producers. This would entail a more efficient delivering of energy, by\u00a0reducing costs and harmful emissions. Besides, the\u00a0advantages of energy real-time readings are twofold: for consumers and for energy companies. On\u00a0the one hand, consumers would be aware of their energy consumption, allowing them to adopt new consumption strategies. On\u00a0the other hand, energy companies would infer consumption patterns and predict needs and potential peaks of activity to stablish appropriate energy plans and the best\u00a0fees.In this context, smart meters can be considered one of the key elements in the Smart Grid since (i)\u00a0they allow measuring energy consumption in much more detail than a conventional meter (fine-grained accurate readings) and (ii) they can communicate this information back to the provider and also to other devices or applications in the so-called smart home. A\u00a0good overview of the smart meters evolution is provided in\u00a0, by\u00a0detaHowever, privacy is not the only concern. In\u00a0this promising scenario, a\u00a0new threat arises: like\u00a0any other device connected to a network, the\u00a0electric devices in the smart grid are also vulnerable to attacks  . Smart gWithin this scenario, some approaches have arisen with the aim of providing mechanisms able to support a more secure context for the Smart Grid and to prevent unauthorized access to this sensitive information. In\u00a0fact, one of the main changes in the Smart Grid is the bidirectional flow of information, so the information goes from smart meters to the power operator and vice\u00a0versa. Thus, new communications schemes are needed to support the transmission of energy readings in short time intervals, taking into account the constrained resources of the metering devices and the aforementioned security aspects. Authentication, then, plays an important role in the smart grid domain by providing a variety of security services including credentials\u2019 privacy, session-key (SK) security, and\u00a0secure mutual\u00a0authentication.Encryption is the primary security measure and, recently, the\u00a0Security as a Service (SECaaS) model has introduced different cloud-based solutions for Encryption as a Service (EaaS). One of them is ES4AM\u00a0, that ofOther approaches for KMSs are based on a Physically Unclonable Function (PUF), although\u00a0there are not so many proposals in the specialized literature. One of them is\u00a0, where PCollectors are other critical elements in the Smart Grid and are the focus of the proposal in\u00a0. The\u00a0autfaced in\u00a0: unusualfaced in\u00a0 the authAlthough there are substantial differences among policy contexts and market penetrations across countries\u00a0, currentOur contribution is two-fold. On\u00a0the one hand, we propose an infrastructure based on Smartpi 2.0 devices and a protocol to exchange data (energy readings) in the home. These devices collaborative work to prevent external attacks and attempts of corrupting the data. With\u00a0this aim, we have defined different data flows using the open-source software provided with these devices (Node-RED\u00a0). On\u00a0theThis paper is organized as follows. We have emulated a simple architecture that any user may have at home to monitor the energy consumption. 2 production. These products  are created to form a network with standardized interfaces that is easy to configure, which supports the energy connection between providers and consumers, and\u00a0that includes different sensors. Their\u00a0modular design and their combination of hardware and software offer a flexible and suitable\u00a0solution.The Smartpi 2.0 is a device that was designed by the German company nD-enerserve GmbH, which\u00a0is specialized in products for energy management and optimization of self-consumption for smart homes and industrial environments. Besides\u00a0the Smartpi 2.0, the\u00a0company has developed other products like a unit to control power generation and power consumption or a screen for displaying data about energy efficiency or COMore specifically, the\u00a0Smartpi 2.0 consists of a Raspberry Pi 3 Model B+ and an expansion module that allows the device to read amperage and, as\u00a0a consequence, to\u00a0read the power consumed. The\u00a0device has four inputs: L1, L2, L3 and N ; this way, power can be measured in three-phase systems. For\u00a0single-phase systems, only L1 and N need to be connected. One interesting advantage is that the Raspberry Pi can be powered via the three voltage inputs, so an external power supply is not required. The\u00a0voltage measurement also allows determining the direction of the energy flow, which offers a versatile measurement of both power generation and power consumption. The\u00a0device has the following range of operation: Voltage\u00a0(0\u2013390\u00a0V), Amperage (0\u2013100 A), Precision (2%) and Consumption (10 W). The\u00a0technical characteristics of the basic Smartpi device, the\u00a0Raspberry Pi 3 Model B+, are detailed in _msgid, a\u00a0random identifier for each message created; (ii) topic, a\u00a0property used for fragmenting and reassembling messages; and (iii) payload, the\u00a0content of the\u00a0message.For measurement management and communication between the devices, we have used the software that is included by default in the Smartpi 2.0 by the manufacturer (Smartpi version 0.18.5 and Raspberries version 0.20.5): Node-RED\u00a0. This isGo is activated, a\u00a0message is introduced in the flow, processed by node Hello ! and displayed in the console thanks to node display.According to their role in the information flow, nodes are classified into three types: (i) Input nodes, which introduce information in the flow that is usually gathered from a sensor or from an incoming IP packet; (ii) Output nodes, which do not forward the information to another node but to a database (to be stored) or to a console (to be debugged), for\u00a0this the message is sent as an IP packet that exits the flow; and (iii) Intermediate nodes, which are all the other nodes that receive the message (input), modify the information and send the message (ouput). Node-RED messages (Javascript objects) are stored within the Raspberries in MongoDB instances, a\u00a0NoSQL database that stores information in JSON format. Converting these Javascript objects into JSON format (and vice\u00a0versa) is extremely simple. MongoDB works with collections that group the data together. These collections work like tables in SQL databases, grouping objects with the same structure. Since JSON keys are always strings, there is no need for the keys of two different objects to be the same, as\u00a0long as their structure is the same in terms of\u00a0arrays.X, is assigned with the IP address 192.168.4.X. Node 1 with IP address 192.168.4.1 is the Smartpi 2.0, which will be the access point. Additionally, and\u00a0in order to feed the devices with information, we have used synthetic data, since\u00a0using real readings are not relevant for these experiments. In\u00a0fact, in\u00a0a real context (using Smartpi devices), the\u00a0only modification needed would be replacing the input of data by the real measurements of the sensors if they are connected to a real power\u00a0grid.In order to perform our experiments, we configured a network connecting the devices, as\u00a0We have defined two basic flows for the devices interconnected in yyyy-mm-dd hh:mm:ss and the field \u201cvalue\u201d is the energy consumption registered. This new Javascript object is stored as a JSON element in the database. Secondly, the\u00a0message is again modified to include the node identifier, such as it would follow the next Javascript object: true, if\u00a0the target node has processed the packet without problems, or\u00a0false, if\u00a0any of the fields was incorrect, which will generate an error message in the\u00a0flow.The first flow, reading flow , obtainstrue. Otherwise, an\u00a0error message would be generated and the message false would be returned to the device where the reading came\u00a0from.The second one, reception flow , gathersThese two new flows allow all the devices in the domestic network to share their energy readings. This is key for the next flow, defined to try to protect the network against external attempts of corrupting the readings by injecting false readings in the system. Therefore, the\u00a0third flow, defence\u00a0flow, was\u00a0designed to work as a defence against unauthorised alterations in the database. The\u00a0main objective of this flow is to support collaborative work among the domestic devices. The\u00a0underlying idea is that each device compares its own energy readings with the previous ones locally obtained. When anomalies are detected, the\u00a0device asks for the readings from its neighbours to compare the\u00a0data.Therefore, the\u00a0designed defence flow is composed of two parts or steps. The\u00a0first one focuses on the local analysis of the data, whereas the second one focuses on a procedure to collaboratively decide if an unusual energy reading is, indeed, a\u00a0right energy reading or a potentially altered\u00a0one.At the beginning of this defense flow , a\u00a0queryhttps://www.statsmodels.org/stable/index.html) is a useful tool for this tasks) in order to obtain and to compare consumption trends. Second, we have checked the consistency and fluctuation of the energy data by using a hybrid approach that combines (i) approximate entropy 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{"text": "Page 1, line 30: \u201cD8788\u201d should read \u201cD7070.\u201dVolume 6, no. 1, e01257-17, 2018, Page 1, line 31: \u201cD7070\u201d should read \u201cD8788.\u201dPage 2, line 3: \u201cD7070\u201d should read \u201cD8788.\u201dPage 2, line 4: \u201cD8788\u201d should read \u201cD7070.\u201d"}
{"text": "The author name \u201cGemma Fern\u00e1nez-Garc\u00eda\u201d should be \u201cGemma Fern\u00e1ndez-Garc\u00eda\u201d.Page 2, first line: \u201c257 proteins\u201d should be \u201c301 proteins\u201d.The authors wish to make the following corrections to this paper :The authThe authors would like to apologize for any inconvenience caused to the readers by these changes."}
{"text": "Scientific Reports 10.1038/s41598-019-54029-w, published online 25 November 2019Correction to: 42. The incorrect versions of Equations 1 and 2 appear below:In the original PDF version of this Article, there were typesetting errors in Equations 1 and 2 where dash signs were incorrectly typeset as minus signs. As a result, the symbols \u2018t-tau\u2019, \u2018p-tau\u2019, \u2018\u03b1-syn\u2019 and \u20181-3\u2019 were incorrectly typeset as \u2018t \u2013 tau\u2019, \u2018p \u2013 tau\u2019, \u2018\u03b1 \u2013 syn\u2019 and \u20181 \u2013 3\u2019 respectively. Secondly, in Equation 2 the notation for amyloid was incorrectly typeset as AbEquation\u00a0(1)Equation\u00a0(2)PDCDm)1 column of Table 4. In the row \u201cMaximum total\u201d,Furthermore, in the original HTML and PDF versions of this Article there were errors in the (11\u201d\u201c12now reads:\u201c12 [11]\u201dIn the row \u201cCut-off\u201d,5\u201d\u201c6now reads:\u201c6 [5]\u201dThese errors have now been corrected in the PDF and HTML versions of the Article."}
{"text": "Ed in relation to the polarization is studied. In particular, the value of permittivity for Ed (\u03b5d) in prototypical situations of ferroelectrics, including Mehta formula, is examined by ab initio calculations. By using spontaneous polarization PS corresponding to accurate experiment ones, we show \u03b5d\u2009=\u20091, which suggests that the results of \u03b5d\u2009\u226b\u20091 indicate hidden mechanisms; \u03b5d\u2009=\u20091 suggests that the effect of Ed is significant to induce intriguing important phenomena overlooked by \u03b5d \u226b 1. A bridge between \u03b5d\u2009=\u20091 and \u03b5d\u2009\u226b\u20091, i.e. the consistency of \u03b5d\u2009=\u20091 with conventional results is presented. The exact electrostatic equality of head-to-head\u2013tail-to-tail domains to free-standing ferroelectrics is deduced. Hence, most stoichiometric clean freestanding monodomain ferroelectrics and head-to-head\u2013tail-to-tail domains are shown unstable regardless of size, unless partially metallic. This verifies the previous results in a transparent manner. This conclusion is shown consistent with a recent hyperferroelectric LiBeSb and \u201cfreestanding\u201d monolayer ferroelectrics, of which origin is suggested to be adsorbates. In addition, this restriction is suggested to break in externally strained ultrathin ferroelectrics. The macroscopic formulas of Ed are found valid down to a several unit-cells, when electronic and atomic-scale surface effects are unimportant and accurate PS is used.Electrostatics of depolarization field  PS that is useful in various applications, for which high insulativity is desired. Therefore, ideal insulativity of FE is assumed in most studies. In such high insulativity FEs, the depolarization field Ed exists universally even in the absence of external electric field Eext, owing to the charge \u2212\u2009\u2207\u00b7PS originating from inhomogeneity or the existence of surface; For a homogeneous PS, Ed disappears for infinite FE without surface or FE with no surface effect and\u00a0ideal metal electrodes.Ferroelectrics (FEs) have reversible spontaneous polarization Ed are unnecessary in ab initio calculations, they are indispensable for non ab initio examinations of PS configurations, stability of domains, and critical thicknesses of FEs17. These formulas use the permittivity for Ed (\u03b5d), of which difference affects critically the results. No controversy exists for \u03b5d\u2009=\u2009\u03b5f used in combination with an initial PS that is an ideal bulk PS for no macroscopic field in FE as in Kittel models2. Here, \u03b5f is static linear permittivity \u03b5r of FE. Otherwise, the choice of \u03b5d is controversial.Although analytical formulas of P under Eext is P\u2009=\u2009PS\u2009+\u2009(\u03b5f\u2009\u2212\u20091)\u03b50Eext (\u03b50: vacuum permittivity) and the permittivity of FE underEd is still \u03b5f\u2009\u226b\u20091 ] thin films by using Ed\u2009=\u2009\u2212\u2009PS/\u03b5f\u03b50 with PS from remnant polarization measurements (7\u00a0\u03bcC/cm2) and \u03b5d\u2009=\u2009\u03b5f\u2009=\u200910. In the electrostatic study of Tian et al.7, Ed in BiFeO3 thin films was estimated by using \u03b5f\u2009=\u200960 (in Eq. (1) of Ref.7), where this P is initially a total polarization of a single domain state. Kim et al.8 estimated Ed in BaTiO3 (BTO) ultrathin films through the formula by Mehta4 with the experimental remnant P obtained from pulse train methods and \u03b5d\u2009=\u2009\u03b5f\u2009=\u200980 from \u03b5f\u2009\u2212\u2009Eext curves. In the Ginzburg\u2013Landau\u2013Devonshire (GLD) theory of BTO ultrathin films by Jo et al.9, Ed was given through the Mehta formula4 with the same P as P in the GLD equation and \u03b5d\u2009=\u2009\u03b5f\u2009=\u200980 of Ref.8, where P in GLD theory is the total polarization. Schroeder et al.10 estimated Ed in HfO2 and PZT ultrathin films through the Mehta formula4 with experimental PS and \u03b5d\u2009=\u2009\u03b5f\u2009=\u200920\u2013300. Similar analyses with \u03b5d\u2009=\u2009\u03b5f\u2009\u226b\u20091 are frequently employed17.Examples of , a primitive considerations show \u03b5d\u2009=\u20091 for PS, i.e. a total PS (\u2261\u2009P(Ef(Eext\u2009=\u20090)))13, where Ef is the total macroscopic electric filed in FE. For freestanding FEs, for example, Ef(Eext)\u2009=\u2009Eext\u2009\u2212\u2009P(Ef(Eext))/\u03b50 or Ef(Eext)\u2009=\u2009Eext/\u03b5f\u2009\u2212\u2009P(Ef(Eext\u2009=\u20090))/\u03b50 )/\u03b50 \u201c\u201d. This i\u03b5d is due to explanations based on macroscopic quantities. Because macroscopic explanations are abstract, they are unsuitable to bridge the gap between two conflicting views of \u03b5d. On the other hand, ab initio estimation of \u03b5d is considered as the clearest method for this problem but is not reported to our knowledge. Hence, we clarify \u03b5d in the formulas of Ed, by ab initio simulations in which ab initio PS is exactly P(Ef(Eext\u2009=\u20090)), which is considered as PS obtained by accurate experiments of ideal samples. Here, the standard theoretical assumptions: pure, stoichiometric, clean FEs are used.We think that the existence of this controversy on Ed is related to fundamental issues such as stability of monodomains, critical thicknesses of FE, and the emergence of ferroelectricity in superlattices. Some of these subjects require the consideration of other effects such as strain-induced FE and electronic effect at electrodes11. To avoid the complexity, we concentrate on free-standing insulating FE and its electrostatic identicals, i.e. head-to-head\u2013tail-to-tail (HH\u2013TT) domains. Thus, we estimate the value of \u03b5d in the formula of Ed in a clear manner.11, it may be argued that the formula of Ed based on electrostatics is not possible for nm-FEs. We resolve this by focusing on the formulas of Ed and using ab initio PS in the formulas. Therefore, PS in these formulas contains the effects of the interactions in slabs or superlattices, whereas the absence of these effects in conventional studies has limited the applicability. The use of nonlinear \u03b5f is often better than linear \u03b5f but can be approximated by an average linear \u03b5f12. Therefore, the conclusions below are applicable also to the nonlinear \u03b5f (\u226b\u20091).As expected from the electronic interaction at the electrodeEext\u2009=\u20090, where FEs with thickness lf are in periodic slabs  superlattice, respectively, while the latter mimics an inhomogeneous FE. Iadj stands for both vacuum and insulator, which is dielectric or FE having different PS. The polarization, thickness, and permittivity of Iadj are PI, lI, and \u03b5I, respectively, and the thickness of slab is lSC\u2009=\u2009lf\u2009+\u2009lI (lV). The angles of the polarization P of FE and Iadj to the slab direction are \u03b8 and \u03b8I, respectively.For simplicity, we discuss 1-dimensional (1D) cases with ect Fig.\u00a0 but to t\u03d5) of these models are represented by Fig.\u00a0Ed (Edab initio) was obtained from the envelope of the peak tops of atomic electrostatic potential, of which example is Fig.\u00a03 (BTO) capacitors are examined, where metal layers are standard electrode materials for FEs: SrRuO3 or Pt and ~\u200920\u00a0\u00c5  BTO in vacuum is metallic, when FE is enforced18. To achieve insulativity, we used tetragonal (P4mm) SrTiO3 (STO) of which a-axis lattice constant increased by 0.5% and decreased by 0.01% from that of the theoretical cubic phase. For these a-axis lattice constants, bulk STO was FE19. We refer to them as STO1.005 and STO.9999, respectively, of which bulk PS\u2019s were 3.56\u00a0\u03bcC/cm2 and 6.15\u00a0\u03bcC/cm2, respectively, by VASP28.Accurate estimation of Ed are obtained in a following manner. The normal component of P of FE (P\u22a5) under Ed is P\u22a5\u2009=\u2009PS cos \u03b8\u2009+\u2009(\u03b5d\u2009\u2212\u20091) \u03b50Ed in standard approaches17. The equation of continuity of electric flux is PScos \u03b8\u2009+\u2009\u03b5d\u03b50Ed\u2009=\u2009PI cos \u03b8I\u2009+\u2009\u03b50EI, where EI is the macroscopic electric field in Iadj. The validity of this continuity in the presence of peaks at the surfaces\u00a0/\u03b50(\u03b5d\u2009+\u2009lf/lI). In the present study, PS and PI in Eq.\u00a0\u03b50EI, Eq.\u00a0.For \u2260\u20090, Eq.\u00a0 is Ed\u2009=\u2009Ed in FE capacitors, because equations of continuity of electric flux similar to the above hold; A short-circuited FE capacitor is modelled as a perfect-metal/insulator(lI/2)/FE/insulator(lI/2)/perfect-metal29, where the perfect metal refers to a metal with zero screening length and the thickness of each screening layer \u03bb is lI/2. Assuming PI\u2009=\u2009\u03b5I\u03b50EI in screening layer, Eq.\u00a0 is the distance between an outermost electron density and a center of ion position   lf is thd layer \u201c\u201d; The ef\u03b5d was examined through the comparison of ab initio Ed with Ed of Eq.\u00a0(\u03b5d. For FE/vacuum (PI\u2009=\u20090), PS in Eq.\u00a0 t t\u03b5d was S in Eq.\u00a0 was the I in Eq.\u00a0 and (2) Ed\u2019s by Eq.\u00a0, which, with \u03b5d\u2009=\u20091, quantitatively agrees with bandgap Eg that decreases with lf and lV in Fig.\u00a0In Fig.\u00a0Ed\u2019s by Eq.\u00a0. Equatulation \u201c with \u03b5d\u2009Ed by Eq.\u00a0) and Ed\u2009\u2261\u2009Ef(Eext\u2009=\u20090). When the potential difference between the electrodes is V, Eq.\u00a0(Ef(V)\u2009=\u2009V/(lf\u2009+\u2009lI/\u03b5I)\u2009\u2212\u2009P(Ef(Eext))/\u03b50(1\u2009+\u2009\u03b5Ilf/lI)13. EI, the field in the screening layer of the electrode, is EI(V)\u2009=\u2009V/\u03b5I(lf\u2009+\u2009lI/\u03b5I)\u2009+\u2009P(Ef(Eext))lf/\u03b50lI(1\u2009+\u2009\u03b5Ilf/lI).Using the total electric field in FE s V, Eq.\u00a0 changes Ef(V) can be written as Ef(Eext)\u2009=\u2009Eext\u2009+\u2009Ed(Ef(Eext)), where Eext\u2009\u2261\u2009V/(lf\u2009+\u2009lI/\u03b5I) (=\u2009V/lf for lf\u2009\u226b\u2009lI/\u03b5I) and Ed(Ef(Eext))\u2009\u2261\u2009\u2212\u2009P(Ef(Eext))/\u03b50(1\u2009+\u2009\u03b5Ilf/lI) similar to Eq.\u00a0(r to Eq.\u00a0.\u03b5f is defined by \u03b5f\u2009\u2212\u20091\u2009=\u2009{P(Ef(Eext))\u2009\u2212\u2009PS}/\u03b50(Ef(Eext)\u2009\u2212\u2009Ed), where Ed\u2009=\u2009Ef(Eext\u2009=\u20090)\u2009=\u2009\u2212\u2009PS/\u03b50 for lf\u2009\u226b\u2009lI/\u03b5I, and \u03b5f (\u226b\u20091) is linear for |Eext|\u2009\u226a\u2009|PS|/\u03b50. Hence, P(Ef(Eext))\u2009=\u2009PS\u2009+\u2009(\u03b5f\u2009\u2212\u20091)(\u03b50Ef(Eext)\u2009+\u2009PS) for lf\u2009\u226b\u2009lI/\u03b5I.P(Ef(Eext)) in the expression of Ef(Eext) yields Ef(Eext)\u2009=\u2009Eext/\u03b5f\u2009\u2212\u2009PS/\u03b50 for lI/\u03b5I\u2009\u226b\u2009lf (freestanding), suggesting that the measured permittivity is \u03b5f.The substitution of this Q\u2009=\u2009\u03b5I\u03b50(EI(V)\u2009\u2212\u2009EI(V\u2009=\u20090))\u2009=\u2009V/(lf/\u03b5f\u03b50\u2009+\u2009lI/\u03b5I\u03b50)13 from the above expression of EI(V). This is equal to \u0394Q\u2009=\u2009CV, where C\u2009=\u2009(Cf\u22121\u2009+\u2009CI\u22121)\u22121  is a series capacitance per area. In particular, for lf\u2009\u226b\u2009\u03b5flI/\u03b5I, \u0394Q\u2009=\u2009\u03b5f\u03b50V/lf\u2009=\u2009Cf V. Therefore, the permittivity of FE under Ed\u2009=\u2009Ef(Eext\u2009=\u20090)\u2009=\u2009\u2212\u2009PS/\u03b50 is \u03b5f.We show an example: \u0394\u03b5d\u2009=\u20091 shows D\u2009=\u2009PS\u2009\u2212\u2009PS/(1\u2009+\u2009\u03b5Ilf/lI), while D = Q. Because lI/\u03b5I\u2009~\u20090.1\u00a0\u00c5, the difference between the real PS and the measured PS is detectable only for lf\u2009<\u200910\u00a0\u00c5. As for potential difference, Eq.\u00a0(Ed\u2009=\u2009\u2212\u2009PSlI/(\u03b5I\u03b50lf) for lf\u2009>\u200910\u00a0\u00c5, because lI/\u03b5I is short ~\u20090.1\u00a0\u00c5. Therefore, the potential difference across the capacitor is independent of the FE thickness lf, when the quality of FE is independent of lf and FE is ideally stoichiometric.Additionally, Eq.\u00a0 with \u03b5d\u2009nce, Eq.\u00a0 is well Here, we discuss only monodomain FE Fig.\u00a0c.P in standard GLD theories34 are formulated with a single total polarization, \u03b5d\u2009=\u20091 should be used in standard GLD theories.Because the polarization P in response to Eext is \u03b5f (\u226b\u20091) even for FE under Ed. By the same logic, the change of P by built-in internal field Ebi is also expressed by \u03b5f (\u226b\u20091), where Ebi is not due to P or a dipole that is not expressed by P. Ebi exists in FEs by various mechanisms such as the diffusion potentials at pn and Schottky junctions and chemical orders, e.g. LaAlO3 in the polar catastrophe model.The preceding results have shown that the permittivity that expresses the change of PS\u2009=\u20090 and Ed\u2009=\u20090 in a bulk cubic BTO. However, Ebi\u2009\u2260\u20090, when the surfaces of a cubic BTO slab are asymmetrically terminated to form a dipole, e.g., BaO/TiO2/BaO/\u2026/TiO2/BaO/TiO2. Hence, to achieve Ebi\u2009=\u20090, the present study used chemically symmetric slabs , for which Eq.\u00a0 are bulk Eg normalized by 2\u00a0eV and PS of bulk FE at TC normalized by 10\u00a0\u03bcC/cm2, respectively, and the unit of lf and lV is \u00c5. PS of bulk FE at TC approximates the critical PS of FE that is about to become paraelectric by Ed12. Equations\u00a0, Eq.\u00a0(lf\u2009<\u20091.8\u00a0\u00c5(Eg/2\u00a0eV)/(PS(TC)/10\u00a0\u03bcC/cm2). This suggests that freestanding FEs with normal bulk\u00a0properties are FEs with metallic layer or insulating paraelectrics15 as explained by the following GLD analysis. This conclusion is valid also for HH\u2013TT domains with \u03b8\u2009=\u20090.The giant permittivity and large piezoelectric coefficients of FE are regarded as an \u20090), Eq.\u00a0 yields lP as the order parameter. We approximate the polarization possibly missed in such GLD theory29 by an extra permittivity \u03b5NG\u2009\u2212\u2009136, while \u03b5NG is speculatively close to electronic permittivity29. The GLD energy F of an insulating FE is F\u2009=\u2009(T\u2009\u2212\u2009T0)P2/2C\u03b50\u2009+\u2009\u03b2P4/4\u2009+\u2009\u03b3P6/4\u2009\u2212\u2009PEd/2, where T0, C, \u03b2, \u03b3, and \u03b8 are Curie\u2013Weiss temperature, Curie constant, and GLD coefficients, respectively. The effect of strain can be incorporated in T0 and \u03b237. Curie temperature TC is T0\u2009+\u2009\u0394T, where \u0394T\u2009=\u20093\u03b22/16\u03b3C\u03b50. For 2nd order transition, \u03b3\u2009=\u20090, \u03b2\u2009>\u20090, and TC\u2009=\u2009T0. The effect of Ed\u2009=\u2009\u2212\u2009PS/\u03b50\u03b5NG is the change of T0 to T0\u2009\u2212\u2009C/\u03b5NG in F, where Eq.\u00a0/C\u03b50\u2009+\u20093\u03b2P2\u2009+\u20095\u03b3P4}\u03b50 at T\u2009<\u2009TC. Stable state satisfies \u2202F/\u2202P\u2009=\u2009(T\u2009\u2212\u2009T0)/C\u03b50\u2009+\u2009\u03b2P2\u2009+\u2009\u03b3P4\u2009=\u20090. Therefore, \u03c7GL\u2009=\u20091/(4T0/C\u03b50\u2009\u2212\u20092\u03b2P2)\u03b50\u2009<\u2009C/4T0 at T\u2009=\u20090 (We assume T0\u2009>\u20090), and TC\u2009=\u2009T0\u2009+\u2009\u0394T\u2009=\u2009T0\u2009+\u20093\u03b22/16\u03b3C\u03b50. Because \u0394T\u2009\u226a\u2009T0 in almost all FEs34, we may assume TC\u2009<\u20092T0. Under this assumption, \u03c7GL\u2009<\u2009C/2TC at T\u2009=\u20090, and C\u2009<\u2009\u03b5NGTC means \u03c7GL\u2009<\u2009\u03b5NG/2 at T\u2009=\u20090. \u03b5f\u2009=\u2009\u03c7GL\u2009+\u2009\u03b5NG\u2009<\u20093\u03b5NG/2 at T\u2009=\u20090, which is <\u00a07.5 for \u03b5NG\u2009=\u20095.Similarly, for FE undergoing 1st order FE transition, it is known that \u03b5f\u2019s of FEs undergoing 2nd and 1st transitions appear too small for experimentally observed bulk metal oxide FEs. Therefore, freestanding insulating FEs satisfying C\u2009<\u2009\u03b5NGTC are unlikely to exist, unless \u03b5NG is far larger than 5; That is, for freestanding FE materials, there exists virtually one choice between a partial loss of insulativity and a loss of FE.These \u03b8\u2009=\u20090), C\u2009<\u2009\u03b5NGTC was shown, while \u0394\u03d5\u2009~\u2009PS(Ed)lf/\u03b50. Therefore, FE materials having a very large \u03b5NG (\u226b\u20095) can retain FE and remain insulating, when ultrathin. Such FE materials are unlikely to exist. Alternatively, we may consider electrically freestanding FE or FE with clean surface that is not mechanically freestanding. In this case, T0 (~\u2009TC) of common FEs increases to T0eff by inplane\u00a0strain, while ab initio calculations shows that T0eff is much larger than those of standard GLD theories37. Therefore, heavily strained FE materials may retain FE and remain insulating, when ultrathin (Formula estimating an effective T0 from ab initio PS is Ref. [88] of Ref.37). The above indicates that \u03b5f of such FE is extremely low for Ed\u2009=\u20090 but can be large for Ed\u2009\u2260\u20090, because the coefficient of the first term GLD energy F is (T\u2009\u2212\u2009T0eff\u2009+\u2009C/\u03b5NG)/2C\u03b50 (freestanding).For freestanding insulating FE  in vacuum  that retains FE down to monolayer. This appears to contradict both the reports of metallicity at HH\u2013TT domains of BFO and the present results, esp. the single choice between insulating paraelectric and partially metallic FE.Mechanically freestanding FE is customarily referred to as freestanding; Ji et al.\u03b5d\u2009=\u2009\u03b5f\u2009=\u2009100, the potential difference \u0394\u03d5 across freestanding insulating BFO of 1\u20134 unit-cell thickness with a moderate PS\u2009~\u200920\u00a0\u03bcC/cm2 is 0.09\u20130.36\u00a0V by Eq.\u00a0 and may move ions43, those by Ji et al.42 are not that of freestanding FE. Consequently, all the measurements of Ji et al.42 do not contradict the conclusion of the present paper.Hence, we shall look at the measurements of Ji et al.Ed-limited domain and size effect, which was later attributed to adsorbates44. This agrees with recent ab initio study45. Further, photoemission spectroscopy in UHV showed that SrTiO3 surface was covered by adsorbates even in ultrahigh vacuum (UHV)46. Actually, the free surface with PS \u22a5 surface is insulator-like in air and metallic in UHV when cleaned14. Because the insulating freestanding FE42 was exposed to air and water, we suggest adsorbates as its hidden mechanism.More importantly, the interdisciplinarity of nano FE hides true mechanisms. In the present case, \u201cfreestanding\u201d is defined by surface science and electrostatics. For example, Fong et al. found monodomain FE of 3 unit-cell thickness as opposed to Ed, especially, the value of permittivity \u03b5d in the formula of Ed by ab initio simulations, where ab initio PS corresponded accurately to experimental PS. For this, the standard theoretical assumptions: pure, insulating, stoichiometric, and clean FEs were used. To validate the analyses of Ed based on electrostatics, we concentrated on the formulas of Ed for accurate ab initio total P(Ef(Eext\u2009=\u20090)) that contained various atomic effects and corresponded to experimental PS. Further, we focused on the simplest cases of Ed: freestanding 1D-FE, HH\u2013TT domains, and superlattices that mimicked inhomogeneous FE and FE/dielectric.We studied the electrostatics of \u03b5d\u2009=\u20091\u2009\u00b1\u20090.06\u20131\u2009\u00b1\u20090.2. That is, \u03b5d\u2009=\u20091 should be applied to experimental and standard-GLD PS\u2019s. A contradiction between \u03b5d\u2009=\u20091 and \u03b5d\u2009=\u2009\u03b5f was resolved by a bridge; Even under Ed, the permittivity for Eext and built-in field Ebi was \u03b5f. Therefore, if a study requires \u03b5d\u2009\u226b\u2009117, the value of PS is incorrect, the values of the parameters are inappropriate, or, most likely, hidden screening mechanisms exist48.The present ab initio simulations showed lI\u2009=\u2009\u221e), Eq.\u00a0(Ed\u2009=\u2009\u2212\u2009PS/\u03b50 (or Ed\u2009=\u2009\u2212\u2009PScos \u03b8/\u03b50), while, for HH\u2013TT insulating domains, Eq.\u00a0, Eq.\u00a0 yields Eins, Eq.\u00a0 with PI\u2009Ed and the FE free energy of insulating freestanding and HH\u2013TT FEs scale linearly with lf. This implies that the stability of 1D-freestanding and HH\u2013TT insulating FEs is independent of size15, when the energy increase by surface effect and domain walls energy is ignored. A strain effect to overcome this restriction was suggested.Consequently, both the electrostatic energy of lfPS/\u03b50\u2009<\u2009Eg/e by \u03b5d\u2009=\u20091, the insulativity required an extremely small bulk PS\u2009\u226a\u20091\u00a0\u03bcC/cm2 or paraelectricity  required a partially metallic FE. This conclusion verified the previous results41 in a material-independent manner and was confirmed also for hyper-FE LiBeSb that was reported to be insulating in FE/paraelectric38. This conclusion appeared inconsistent with \u201cfreestanding\u201d monolayer BFO42. But, the examinations of experimental procedures42 suggested adsorbates as a hidden mechanism46.Because ity Fig.\u00a0. AlternaEd 28, which was used also for BTO/Pt. In the slab calculations, graphic processing units acceleration50 was used. The supercells were produced by VESTA51.The ab initio calculations with VASP\u03b5d, accurate estimations of a total polarization PS under Ed are essential. Because we compare Eqs.   \u03b5d, accuPSfor given ion positions is accurate but only possible for insulators. For example, the present Berry phase calculations yields PS of bulk BaTiO3 that agree with experimental PS within 4%, when experimental ion positions and lattice constants of at 303\u00a0K are used37.Berry phase calculation of PS, the dipole moment of a whole slab was calculated with Berry phase; We treated these slabs as unit-cells to apply Berry phase calculations directly, unlike conventional approaches. PS was obtained by dividing the dipole moment by the volume of FE part of the slab. These PS\u2019s were referred to \u201crigorously calculated PS\u2019s of the slab\u201d and obtained for all the FE/vacuum and BTO/STO slabs. Here, STO1.005 and STO.9999 slabs are insulating, allowing accurate Berry phase calculations.Therefore, to obtain accurate PS of the unit-cell that possessed exactly the same ion positions as those in the slab was calculated with Berry phase and, then corrected with atomic polarization by Ed by the procedures in Ref.29. These PS\u2019s agree perfectly with \u201crigorously calculated PS\u2019s of the slab\u201d, which further confirmed the accuracy of the present PS\u2019s of FE/vacuum and BTO/STO. These corrected PS\u2019s29 were used for capacitors. Therefore, in the present study, PS\u2019s are accurate total PS\u2019s and self-consistent with Ed. Hence, PS\u2019s in Figs. Additionally, Equations\u00a0 and 2)  2) are a\u03c3+R\u2009=\u2009\u03c3+L and \u03c3\u2212R\u2009=\u2009\u03c3\u2212L as expected from their origin, where \u03c3+R, \u03c3\u2212R, \u03c3+L, and \u03c3\u2212L are positive and negative charge densities that yield the right and left peak, respectively. Because of the charge neutrality of FE, \u03c3+R\u2009+\u2009\u03c3+L\u2009+\u2009\u03c3\u2212R\u2009+\u2009\u03c3\u2212L\u2009=\u20090, i.e. \u03c3\u2212R\u2009+\u2009\u03c3+R\u2009=\u20090. Therefore, the continuity of the electric fluxes DFE in FE and DI in Iadj .Because surface buckling in vacuum is electrostatically dipole due to ion displacements, the arguments exactly the same as the above hold. Therefore, the electric flux of the inside lf (lfeff) was estimated from the planer averaged electron density \u03c1 profiles29. Below, z\u2009=\u20090 corresponds to the position of bottom ion. Because \u03c1 at z\u2009=\u2009\u2212\u20090.8\u00a0\u00c5 was same as the minimum \u03c1 of inner part in all the \u03c1\u2013z curves, the region of z\u2009=\u20090\u2009~\u2009\u2212\u20090.8\u00a0\u00c5 was considered as a part of FE (\u03bbsmear\u2009=\u20090.8\u00a0\u00c5), and lfeff was lfeff\u2009=\u2009lf\u2009+\u20092\u03bbsmear. In addition, \u03bbsmear\u2009=\u2009c/2  was also tested, and lV\u2009=\u2009lSC\u2009\u2212\u2009lfeff. For BTO/STO, lf was defined as the distance between the top and bottom Ti ions of BTO , for which the quantum mechanical smearing29 may be responsible. The estimations with lT-B\u2009\u2212\u20091.5ucBTO were also tested. The effective thicknesses of the screening layer, i.e. the effective passive layer lI/2\u03b5I of BTO/SrRuO3 and BTO/Pt were estimated as 0.1\u00a0\u00c5 and approximately 0.05\u00a0\u00c5, respectively29.For FE capacitors,"}
{"text": "Scientific Reports 10.1038/s41598-020-60750-8, published online 12 March 2020Correction to:The Article contains errors.In the Results section under the subheading \u201cIdentifying differentially expressed language features prior to a hospital visit\u201d \u201cDictionary-based: Prior to ED visits, patients less more likely to post about leisure (d\u2009=\u2009\u2009\u2212\u20090.225), associated words such as \u2018family\u2019, \u2018fun\u2019, \u2018play\u2019, \u2018nap\u2019, internet slang (netspeak) (d\u2009=\u2009\u2009\u2212\u20090.374) words such as \u2018:)\u2019, \u2018fb\u2019, \u2018ya\u2019, \u2018ur\u2019, and informal language (d\u2009=\u2009\u2009\u2212\u20090.345) with words such as \u2018u\u2019, \u2018lol\u2019, \u2018smh\u2019, \u2018da\u2019. Patients also use personal pronouns less (d\u2009=\u2009\u2009\u2212\u20090.345) prior to ED visits compared to random time windows.\u201dshould read:\u201cDictionary-based: Prior to ED visits, patients were less likely to post about leisure (d\u2009=\u2009\u2009\u2212\u20090.225) with words such as \u2018fun\u2019, \u2018play\u2019, \u2018nap\u2019, internet slang (netspeak) (d\u2009=\u2009\u2009\u2212\u20090.374) such as \u2018u\u2019, \u2018da\u2019, \u2018smh\u2019, and informal language (d\u2009=\u2009\u2009\u2212\u20090.345) with words such as \u2018lol\u2019, \u2018:)\u2019, \u2018b\u2019.\u201dIn Table 2 under \u201cCategories that decrease in usage before emergency visit\u201d, the row \u201c1st person singular\u201d is a duplication of row \u201cinformal speech\u201d.In Table 2 under \u201cChange in linguistic topics\u201d, the row \u201ceven, still, tho, though, yet, blah\u201d is a duplication of the row \u201ckids, child, their, children, mother, father\u201d.The correct Table 2 appears below as Table 1In Table 3 under \u201cCategories that decrease in usage before inpatient visit\u201d, the row \u201c1st person singular \u2019, \u2018b\u2019)\u201d is a duplication of the row \u201cInformal Speech \u2019, \u2018b\u2019)\u201d.In Table 3 under \u201cChange in Linguistic Topics\u201d, the row \u201ceven, still, tho, yet, blah, mad\u201d is a duplication of the row \u201ckids, child, their, children, mother, father\u201d.In Table 3 under \u201cTopics that decrease in usage before inpatient visit\u201d, the row \u201cbetter, feeling, little, hope, bit, type\u201d is a duplication of the row \u201clol, ha, ctfu, lmao, funny, haha\u201d and the row \u201cno, what, matter, how, always, end\u201d is a duplication of the row \u201c:), show, crew, awesome, fashion, guys\u201d.The correct Table 3 appears below as Table 2These corrections do not affect the conclusions of the Article."}
{"text": "We confirm Kreps' conjecture if the consumer's utility function U has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function U with asymptotic elasticity equal to 1, for \u03b6 such that E[\u03b63]>0.We examine Kreps' conjecture that optimal expected utility in the classic Black\u2013Scholes\u2013Merton (BSM) economy is the limit of optimal expected utility for a sequence of discrete\u2010time economies that \u201capproach\u201d the BSM economy in a natural sense: The  The price process for the stock is generated as follows: For an i.i.d. sequence {\u03b6j;j=1,2,\u2026}, where each \u03b6k has the distribution of \u03b6, the law for the price of the stock at time k/n is\u03be(0)\u22610 and S(0)\u22611.) At time 1, the bond pays a consumption dividend of 1, and the stock pays a consumption dividend of S(1) defined as\u00a0above.Fix a random variable \u03b6 with mean 0, variance 1, and bounded support. For \u03a9=C0. Let \u03c9 denote a generic element of \u03a9, with \u03c9(t) the value of \u03c9 at time t. Endow \u03a9 with the sup\u2010norm topology; let F denote the Borel \u03c3\u2010field, and let {Ft;t\u2208} denote the standard filtration on \u03a9. For each n, let Pn be the probability measure on \u03a9 such that the joint distribution of (\u03c9(0),\u03c9(1/n),\u2026,\u03c9(1)) matches the distribution of (\u03be(0),\u03be(1/n),\u2026,\u03be(1)), and such that \u03c9(t) for k/n<t<(k+1)/n is the linear interpolate of \u03c9(k/n) and \u03c9((k+1)/n). And let S:\u03a9\u2192R+ be defined by S=e\u03c9(t).Embed this model into the standard state space Pn\u21d2P, where P is Wiener measure on C0; that is, \u03c9 under P is a standard Brownian motion, starting at \u03c9(0)=0, and S(\u03c9) under P is geometric Brownian motion, so that P, together with the riskless bond, prescribes the simple continuous\u2010time economy of Black and Scholes  the information she possesses at time k/n, on which basis she trades, is [only] the history of the stock price up to and including time k/n, and (b) any purchase of stock after time 0 is financed by the sale of bonds, and the proceeds of any sale of stock are invested in bonds), seeking to maximize the expectation of a utility function U:\u2192R applied to the final dividend generated by the portfolio she holds at time\u00a01.We imagine an expected\u2010utility\u2010maximizing consumer who is endowed with initial wealth nth discrete\u2010time economy (where the stock and bond trade (only) at times 0,1/n,2/n,\u2026,(n\u22121)/n), does the optimal expected utility she can attain approach, as n\u2192\u221e, what she can optimally attain in the continuous\u2010time BSM\u00a0economy?The question that forms the basis for this paper is: If we place this consumer in the un(x) be the supremal expected utility she can attain in the nth discrete\u2010time economy if her initial wealth is x, and let u(x) be her supremal expected utility in the BSM economy. Kreps (liminfnun(x)\u2265u(x). And he proves limnun(x)=u(x) in the very special cases of U having either constant absolute or relative risk aversion. But he only conjectures that the second \u201chalf,\u201d or limsupnun(x)\u2264u(x) is true for general (concave and differentiable) U.Let y. Kreps  obtains asymptotic elasticity of utility from Kramkov and Schachermayer (limnun(x)=u(x) if the utility function U has asymptotic elasticity less than 1. However, we show by example that if the asymptotic elasticity of U is 1, it is possible that u(x) is finite while limnun(x)=\u221e, both for all x>0.Employing the notion of hermayer   like the continuous\u2010time limit, at least for the BSM continuous\u2010time limit. But what if \u03b6 has support of, say, size three, but there are only the two securities? Markets are incomplete for any finite n; does this incompleteness mean very different economic outcomes? Or, if the probability laws Pn that govern the discrete\u2010time security\u2010price processes converge weakly to P, is it then true that limnun(x)=u(x)?Our interest, motivated by the discussions in Kreps , is in cU has constant relative risk aversion with risk\u2010aversion parameter less than 1/2, the optimal strategy in the BSM economy is to short\u2010sell bonds, leveraging to achieve a (fixed) fraction greater than 100% of current wealth in the risky asset. Suppose that, in our special discrete\u2010time setting, where the security\u2010price process is driven by scaled copies of a single random variable \u03b6, \u03b6 has support that is unbounded below. Trying to achieve such a leverage strategy in any of the finite\u2010time economies would give a positive probability of bankruptcy, which is incompatible with these utility functions. The best an investor can do for large enough n in these circumstances is to hold 100% of her wealth in the risky asset, which results in limnun(x)<u(x).It is already known that weak convergence is insufficient. Merton  observesPn to P alone does not preclude the possibility of asymptotic arbitrage =\u221e, even when u(x) is finite valued =u(x) for all x>0 if U has asymptotic elasticity less than 1. But if U has asymptotic elasticity of 1, even if markets are complete for each n, convergence can fail, and fail in spectacular\u00a0fashion.By assuming in our setting that \u03b6 has bounded support, we avoid the first problem. And, in our setting, asymptotic arbitrage is precluded n=1\u221e are general semi\u2010martingales and the limiting market S is a continuous semi\u2010martingale. Also, the utility function U in Bayraktar et\u00a0al. ((Sn) is induced by a single  random variable \u03b6 and S is geometric Brownian motion. In this more general setting, they make assumptions sufficient to show that limnun(x)=u(x). The key assumptions in Bayraktar et\u00a0al. =\u221e and limx\u2192\u221eU\u2032(x)=0. Moreover, without loss of generality and for notational convenience later, we assume unless otherwise specified that U is normalized so that limx\u2192\u221eU(x)>0, without precluding the possibility that limx\u2192\u221eU(x)=\u221e.  can be either finite or \u2212\u221e.)The utility function We always assume the followingV denote the conjugate function to U:I:\u2192 be the inverse of U\u2032; that is, I(y)=(U\u2032)\u22121(y). Then for every y\u2208, V(y)=U(I(y))\u2212yI(y).Let y\u2192V(y) is strictly convex, continuously differentiable, and strictly decreasing.The function V(\u221e)=U(0) and V(0)=U(\u221e), where the values of U and V at 0 and \u221e are interpreted as the limits as x and y approach 0 and \u221e, respectively.V\u2032(y)=\u2212I(y), so limy\u21920V\u2032(y)=\u2212\u221e and limy\u2192\u221eV\u2032(y)=0.U(x)=infy>0[V(y)+xy],forx>0.We let asymptotic elasticity of U, defined in Kramkov and Schachermayer , is defined byU(x)=x\u03b1/\u03b1 for \u03b1\u2208, then AE(U)=\u03b1.The notion of hermayer , plays aU implies that AE(U)\u22641 in all cases; if U is bounded above and if U(\u221e)>0, then AE(U)=0. But if U(\u221e)=\u221e, AE(U) can equal 1; an example is where U(x)=x/ln(x) for sufficiently large x.The concavity of (U)<1, which derives from a comparison of the average and marginal utilities provided by U as the argument of U approaches \u221e: AE(U)<1 is equivalent to:x0. As(U)<1 is equivalently:x0.Many of our results depend on the assumption that AElimx\u2192\u221e\u2212xU\u2032\u2032(x)/U\u2032(x), exists and is strictly positive, then limx\u2192\u221exU\u2032(x)/U(x) exists and is less than 1; that is, U has asymptotic elasticity less than 1. As it is believed to be \u201ccommon\u201d for economic agents to have nonincreasing relative risk aversion, this belief implies that agents with this common property have asymptotic elasticity less than\u00a0one.As noted in Schachermayer , the con4P\u2217; that is, a probability measure on \u03a9 that is probabilistically equivalent to P and such that {S;t\u2208} is a martingale . This measure P\u2217 has Radon\u2013Nikodym derivative with respect to P given byX that she can afford, where what she can afford is given by the single budget constraint EP\u2217[X]\u2264x, where EP\u2217[\u00b7] denotes expectation with respect to P\u2217. Hence, with wealth x, the consumer's problem is tou(x) is the supremum of expected\u2010utility level that the consumer can achieve in the BSM economy, starting with wealth x.As is well known, the continuous\u2010time BSM economy admits a unique equivalent martingale measure (emm) denoted by Z:C0\u2192 byZ is the unique continuous (in \u03c9) version of the random variable dP\u2217/dP. Of course, EP\u2217[X]=EP[X\u00b7Z] for any random variable X such that (at least) one of the expectations makes sense. And, in this notation u(x)=sup{EP[U(X)]:EP[X\u00b7Z]\u2264x}.It is convenient for later purposes to define the density function 4.1.u(x)\u2265U(x) .4.2.x\u2192u(x) is continuously differentiable, strictly increasing, and concave.4.3.u(x0)<\u221e for any x0>0, then u(x)<\u221e for all x>0.From 4.1 and 4.2, if 4.4.u(x)<\u221e, and if the consumer's problem has a solution , then there exists y(x)>0 such that the solution has the form X(\u03c9)=I(y(x)\u00b7Z(\u03c9)), where y(x)=u\u2032(x) and (as noted earlier) I=(U\u2032)\u22121.If We have the following from Cox and Huang , Karatzax>0, then it has a solution for all wealth levels x\u2032>0 such that x\u2032<x.If the consumer's problem has a solution at wealth level x, u(x)<\u221e and yet the supremum that defines u(x) is not attained by any contingent claim X.  this is true for some finite x, let x\u00af be the infimum of all x for which there is no solution (but u(x)<\u221e); there is a solution at x\u00af, and so the range of x for which there is a solution is the interval  is continuously differentiable and the \u201cLagrange multiplier function\u201d x\u2192y(x)=u\u2032(x) is continuous and strictly decreasing on .4.5.v be the conjugate function to u. That is,y\u2192v(y) is convex and nonincreasing, and it is strictly decreasing and continuously differentiable where it is finite.Let The function u(x)\u2261\u221e, in which case v(y)\u2261\u221e. But suppose u(x)<\u221e for some, and therefore for all, x>0. While u(x) is necessarily concave, differentiable, and strictly increasing, it is not in general true that limx\u2192\u221eu\u2032(x)=0. That is, the marginal  utility of wealth need not approach zero as the wealth level goes to \u221e. Roughly speaking, this can happen when a consumer can purchase ever larger amounts of consumption on events of ever smaller probability, but where the ratio of the amount purchased to the probability of the event approaches infinity at a rapid enough rate. This idea was exploited by Kramkov and Schachermayer (U that satisfies AE(U)=1, by choosing, based on U, specific measures that are different from but play an analogous role to P\u2217 and P. Here, P\u2217 and P are fixed\u2014they come from BSM\u2014so we show this sort of possibility through the selection of specific utility functions U.Of course, it may be that hermayer  for any v and, in particular, for v around the value y0, where y0=limx\u2192\u221eu\u2032(x). To the left of y0 (y<y0), we have v(y)=\u221e. To the right, v(y) is finite. But what is the limiting behavior of v as y approaches y0 from the right? In theory, we could have limy\u2198y0v(y)=\u221e. Or limy\u2198y0v(y)<\u221e, and limy\u2198y0v\u2032(y)=\u221e. Or limy\u2198y0v(y)<\u221e, and limy\u2198y0v\u2032(y)<\u221e. All of these are possible. In fact, giving the full catalog, we have Proposition\u00a0Proposition 4.1U satisfies the conditions (3.1). It is possible that u(x)=\u221e . But if u(x)<\u221e for some x, hence for all x, it must be that x\u2192u(x) is strictly increasing. Moreover, we have the following\u00a0possibilities.(a)U, limx\u2192\u221eu\u2032(x)=0, in which case v(y) is finite for all y>0. (As limy\u21920v(y)=limx\u2192\u221eu(x) and limy\u2192\u221ev(y)=limx\u21920u(x), the function v can have limit \u221e or a finite limit as y approaches 0; and v can have limit \u2212\u221e or a finite limit as y\u2192\u221e.)For some utility functions (b)U, limx\u2192\u221eu\u2032(x)>0. If we denote limx\u2192\u221eu\u2032(x) by y0, then v(y)=\u221e for y<y0, while v(y)<\u221e for y>y0. As for the behavior of v as y\u2198y0, we have the following possibilities:For other utility functions (i)limy\u2198y0v(y)=\u221e;(ii)limy\u2198y0v(y)<\u221e and limy\u2198y0v\u2032(y)=\u221e;(iii)limy\u2198y0v(y)<\u221e and limy\u2198y0v\u2032(y)<\u221e.\u00a0y0>0.Moreover, all are possible for any value of Finally, AE(u)\u2264 AE(U); hence, AE(U)<1 implies limx\u2192\u221eu\u2032(x)=0. That is, asymptotic elasticity less than 1 removes the cases given by part b.Assume that Admitting this is possible (we show that it is), consider the implications for a is simple to show: Take utility functions with constant relative risk aversion, for which solutions are well known and fit case a. And the final assertion needs no proof; it derives from ,For the sequence of discrete\u2010time economies as described in Section\u00a0u(x)=\u221e, this proposition still applies, implying that limn\u2192\u221eun(x)=\u221e.If X satisfies EP[U(X)]=z and EP\u2217[X]=x (so that u(x)\u2265z), then for every \u03b5>0, there exists N such that, for all n>N, the consumer in the nth discrete\u2010time economy can synthesize a claim Xn for an initial investment of x such that EPn[U(Xn)]\u2265z\u2212\u03b5.Kreps (u(x)<\u221e and, for the given x, a solution to the consumer's problem exists  is a max). We then know, as the solution is of the form X=I(yZ) for some multiplier y>0 (see (4.4) above), that the solution X:\u03a9\u2192 is a continuous function of \u03c9. By truncating the solution X, we get approximately u(x), with what is a bounded and continuous claim. Hence, we conclude thatSuppose we know that u(x)<\u221e but no solution exists and where u(x)=\u221e are a bit more delicate, because we don't know, a priori, that we approach the upper bound  with bounded and continuous contingent claims. But we can show this is so. Suppose for some level z, there is a measurable contingent claim X such that EP[U(X)]=z and EP\u2217[X]=x. In this context, of course X\u22650.The cases where \u03b5>0. We first replace X with a bounded claim X\u2032, bounded away from \u221e above and away from 0 below, in two steps. First, for \u03b1<1 but close to 1, let X\u03b1:=\u03b1X+(1\u2212\u03b1)x. Of course, EP\u2217[X\u03b1]=x. And by a double application of monotone convergence (split EP[U(X\u03b1)] into EP[U(X\u03b1)1{X\u03b1\u2265x}]+EP[U(X\u03b1)1{X\u03b1<x}]), we have lim\u03b1\u21921EP[U(X\u03b1)]=EP[U(X)]=z. So, choose \u03b1o close enough to 1 so that EP[U(X\u03b1o)]\u2265z\u2212\u03b5/4. Of course, X\u03b1o is bounded below by (1\u2212\u03b1o)x. As for the upper bound, cap X\u03b1o at some large \u03b2. That is, let X\u03b1o,\u03b2 be X\u03b1o\u2227\u03b2. For large enough \u03b2o, this is bounded above and will satisfy EP>z\u22123\u03b5/4. It may be that EP\u2217[X\u2032]>x, but the last \u03b5/4 is used to replace X\u2032 with X\u2032\u2212(EP\u2217[X\u2032]\u2212x), giving a bounded and continuous contingent claim that costs x (or less) and provides expected utility z\u2212\u03b5, at which point Proposition 5.2 of Kreps  where E[\u03b63] is the third moment of \u03b6, and that limnbn=1/8. (The notation E[\u00b7] is used to denote expectations over \u03b6.) Of course, Pn\u21d2P  and, for this specific emm, Pn\u2217\u21d2P\u2217.We fix one particular emm for each EPn[\u00b7] denotes expectation with respect to Pn and EPn\u2217[\u00b7] denotes expectation with respect to the specific emm Pn\u2217. In words, we allow the consumer any consumption claim X she wishes to purchase, subject only to the constraint that she can afford X at the \u201cprices\u201d given by dPn\u2217/dPn.So, suppose we pose the following problem for the consumer:Zn be the function on C0 given by Zn(\u03c9)=exp[\u2212an\u03c9(1)\u2212bn]. Hence, Zn is a specific version of the random variable dPn\u2217/dPn and the constraint EPn\u2217[X]=x can be rewritten as EPn[ZnX]=x. That is, we can think simply of a consumer facing complete markets with Zn the \u201cpricing kernel\u201d for contingent claims. Using this interpretation, we denote the supremal utility the consumer can obtain in the problem (4.1) asLet nth discrete\u2010time economy; in problem (limnunZn(x)=u(x), we will know that limsupnun(x)\u2264u(x). This, together with Proposition\u00a0limnun(x)=u(x). So, this is what we set out to\u00a0do.The point of this is that the problem  relaxes  problem  she face7Z is the function Z(\u03c9)=exp(\u2212\u03c9(1)/2\u22121/8), which is a version (the unique continuous version) of dP\u2217/dP. DefineunZ(x) is the supremal expected utility that the consumer can attain if she faces complete markets and \u201cprices\u201d Z in the nth discrete\u2010time economy. That is, moving from the consumer's problem in the BSM model to the problem described by Equation =su. DefineunZ(x)=su8Theorem 8.1U satisfies conditions (3.1) and that AE(U)<1. Then, for all x>0, the value function x\u2192u(x) is finite\u2010valued andSuppose that the utility function  The proof of Theorem\u00a0Lemma 8.2limn\u2192\u221eEPn[exp(\u03b3\u03c9(1)]=EP[exp(\u03b3\u03c9(1))].For any constant \u03b3, In this section, we prove the following result.Pn, \u03c9(1)=\u2211k=1n\u03b6k/n for {\u03b6k} an i.i.d. sequence of random variables with the law of \u03b6. Hence,exp(\u03b3\u03b6/n) is 1+\u03b3\u03b6/(n)+\u03b32\u03b62/(2n)+o(1/n), where the o(1/n) term is uniform in the value of \u03b6 because \u03b6 has bounded support. Therefore,e\u03b32/2, which is (of course) EP[exp(\u03b3\u03c9(1))].\u25a1Under Step 1.(U)<1, u(x)<\u221e for all x>0.Because AENow we turn to the proof of Theorem\u00a0not guarantee that the optimal expected utility is finite. The result here strongly depends on the price processes being given by the BSM\u00a0model.This step rates a remark: For \u201cgeneral\u201d price processes as investigated, for instance, in Kramkov and Schachermayer , having (U)<1, but only for 0<y\u2264y0, for some y0>0. If we have that V(\u221e)=U(0)<0, we get the bound for all y>0. And, for purposes of this theorem, it is without loss of generality to shift U by a constant. So, if U(0)\u22650, simply replace U with U(x)\u2212U(0)\u2212c, for a suitable constant c>0. Then we have, and henceforth assume, .The second key to Step 1 is the following bound.hermayer  establisThereexisy\u2192v(y) as well as the primal value function x\u2192u(x) have finite values; and we deduce as well from part c of Proposition\u00a0(u)\u2264 AE(U)<1.Step 2.u\u221eZ(x) for x>0 byu\u221eZ(x)<\u221e for all x>0.Using the notation , define  Define for each n the conjugate function vnZ, which isHence, we may estimatev(y)=EPVyvnZ(y) is uniformly bounded in n for fixed y, which, by standard arguments concerning conjugate functions, proves that u\u221eZ(x) is finite for each x.Step 3.limnunZ(x) exists and equals u(x) for all x>0.Indeed, We show that limnvnZ(y)=v(y) for all y>0, which proves Step 3, again using standard arguments concerning conjugate\u00a0functions.Lemma\u00a0vnZ(y) and v(y):V were a bounded function , the conclusion would follow immediately from Pn\u21d2P. But V is typically not bounded, and so we must show that the contributions to the expectations from the \u201ctails\u201d can be uniformly controlled. We do this by showing the following two uniform bounds.Compare U(0) is finite, it follows that V(y)\u2265V(\u221e)=U(0) for all y, so taking M=\u2212U(0) immediately works. The (slightly) harder case is where U(0)=\u2212\u221e. In this case, recall that, by the Inada conditions, limy\u2192\u221eV\u2032(y)=\u2212limy\u2192\u221e(U\u2032)\u22121(y)=0. As V is convex, this implies that for large enough M, |V(y)|\u2264\u03b5y, provided V(y)\u2264\u2212M. But thenlimn\u2192\u221eEPn[Z]=EP[Z]=1, which shows Assertion (Begin with Assertion . If U(0)ssertion .L that give the bound EPn\u2192EP and(2)Pn({\u03c9(1)\u2265B})\u2192P({\u03c9(1)\u2265B}), for all B>0.And to show the (uniform) Inequality : For thehe bound  and for V(y)\u2264Ly\u2212\u03b1 for all y>0, and son.And lety and \u03b5 both >0, and find M such that (VM(yZ):=max{\u2212M,min{M,V(yZ)}}; that is, VM(yZ) is V(yZ) \u201ctruncated\u201d at \u00b1M. This truncated function is bounded and continuous, so Pn\u21d2P implies EPn[VM]\u2192EP[VM]. And the differences EPn[V(yZ)\u2212VM(yZ)] are uniformly bounded by 2\u03b5. Hence, vn(y)=EPn[V(yZ)]\u2192EP[V(yZ)]=v(y) for all y, which implies that unZ(x)\u2192u(x) for all x>0.Step 4.y>0 and \u03b5>0, there exists M>0 such thatFor every  The parallel to the uniform inequality (vnZ(y)=EPn[V(yZ)]; here we are uniformly controlling the right\u2010hand tail of the integral vnZn(y)=EPn[V(yZn)]. And the same proof works; indeed, in this case we even have EPn[Zn]=1 (as opposed to EPn[Z]\u21921 before).Step 5.C>1 such that, for all n,There is a constant  (The reason for this step is to prove a uniform bound for EPn[V(yZn)] analogous to the uniform bound (Having shown the two uniform bounds  and 12)12), whatuch that  and (12)equality  is obvioequality  uniformlan=1/2+d/n+o(1/n), for d=E[\u03b63]/24, and bn=1/8+o(1). Hence,K such that |\u03b6|\u2264K with probability 1, and so |\u03c9(1)|\u2264Kn, Pn\u2010a.s. The ability to find a constant C>1 that gives the inequalities in /C\u2264Zn(\u03c9), on the support of Pn. As V is a decreasing function, this implies that, for all y>0,Pn. Therefore, for any M>0,M such that the inequality in /ality in  is satisV(0)=U(\u221e) and/or V(\u221e)=U(0) are finite valued. So we first give the argument in the case where V(0)=U(\u221e)=\u221e and V(\u221e)=U(0)=\u2212\u221e, and then sketch how to handle the easier cases where one or the other is\u00a0finite.The argument for this step changes a bit when \u03b5>0 and y>0, and pick M>0 large enough, so that )=M\u2032 and V(yZ(w2))=\u2212M\u2032, where by Z(w), we temporarily mean exp(\u2212w/2\u22121/8). This implies that if \u03c9 is such that \u03c9(1)\u2208, then V(yZ(\u03c9))\u2208. Moreover, by the continuity and monotonicity of V, Z, and Zn (the latter two viewed as functions of \u03c9(1)), and the fact that for fixed w, Zn(w)\u2192Z(w), there exists n0 such that for all n>n0, \u03c9(1)\u2208 implies that V(yZn(\u03c9))\u2208.Fix  so that , 12), , so there is a Lipschitz constant \u2113 such thatn>n0 and w\u2208. And, on the event \u03c9(1)\u2208, there is n1 such that for all n>n1, |Zn(\u03c9)\u2212Z(\u03c9)|<\u03b5/\u2113.The functions D={\u03c9(1)\u2208}, and for n>max{n0,n1}, we have |V(yZ(\u03c9))\u2212V(yZn(\u03c9))|\u2264\u03b5, and EPn[|V(yZ(\u03c9))\u2212V(yZn(\u03c9))|\u00b71D]<\u03b5. By construction, the complement of D is a subset of the union of the four events on which we have uniformly controlled the integrals of V(yZ(\u03c9)) and V(yZn(\u03c9)), so for n>max{n0,n1}, EPn[|V(yZ(\u03c9))\u2212V(yZn(\u03c9))]<5\u03b5, uniformly in n, which proves Step\u00a07.Hence, on the event V(0) and/or V(\u221e) are finite, the argument needs a bit of modification. Suppose V(0)<\u221e. This is relevant when yZ and yZn are both close to zero, which is for paths \u03c9 where \u03c9(1) is large. And for those paths, Zn(\u03c9) can be quite far from Z(\u03c9). However, even if these terms are far apart, V(yZ(\u03c9)) and V(yZn(\u03c9)) will be close together, as each is close to the finite V(0). A similar argument works for cases where V(\u221e) is finite.Step 8.limn\u2192\u221eunZn=u(z) for all z>0.Combine Steps 7 and 3 to conclude that Step 3 shows that vnZ(y)\u2192v(y) for all y>0 (which is how we concluded that unZ(x)\u2192u(x)). Step 7 then implies that vnZn(y)\u2192v(y) for all y>0. This, in turn, implies limn\u2192\u221eunZn(x)\u2192u(x) by standard arguments on conjugate functions.Step 9.\u25a1Combine Steps 8 and Proposition\u00a0When Pn faces complete markets and prices given by Z: Comparing this with the BSM model, the conjugates vn and v to optimal expected utility functions un and u are the expectations of a fixed function for different probability measures. So, after controlling the tails of the integrals that define these conjugate functions, we have a more or less standard consequence\u2010of\u2010weak\u2010convergence result in Step 3. In Step 7, both the probability assessments and the prices (for one of the two problems being compared) change with n. Although the pairs of problems being compared differ only in the prices, because both the integrand and the integrating measure Pn change with n, a level of finicky care is\u00a0required.This proof clarifies why we introduced the analogous problem, where a consumer with probability assessment 9U that satisfy the conditions (3.1) and have asymptotic elasticity less than one, everything works out nicely within the context of the BSM model and the discrete\u2010time approximations to BSM that we have posited.Theorem 1 guarantees that for utility functions U for which AE(U)=1. In such cases, it may be that things work out in the sense of Theorem 1. But it is also possible that limsupn\u2192\u221eun(x)>u(x). That is, when AE(U)=1, Kreps' conjecture can fail. In this section, we provide an example to illustrate this failure in stark fashion: In this example, u(x)<\u221e while limsupn\u2192\u221eun(x)=\u221e, both for all x>0.It is natural to ask, then, what can be said if we maintain (3.1) and these specific models of the financial markets, but we look at utility functions V taking the form\u03b1k,\u03b2k>0 and the sequences {\u03b1k} and {\u03b2k} are chosen so that the sum defining V(y) is finite for all y>0.In this example =\u03b2y\u2212\u03b1.Lemma 9.1\u03b1>0 and \u03b2>0, denote by V\u03b1,\u03b2(y) the function V\u03b1,\u03b2(y):=\u03b2y\u2212\u03b1 for y>0. For the utility function U\u03b1,\u03b2 that is conjugate to V\u03b1,\u03b2, if x0=\u2212V\u03b1,\u03b2\u2032(y0)=\u03b2\u03b1y0\u2212\u03b1\u22121 for given y0, thenU\u03b1,\u03b2 is\u03b1>0, \u03b1/(1+\u03b1)\u2208, and AE=\u03b1/(1+\u03b1)<1.For We begin with some standard facts about conjugate pairs V\u03b1,\u03b2.Lemma 9.2U\u03b1,\u03b2 is given by Equation \u2010distributed, so that Y\u22121/8,1/4=\u2212Y/2\u22121/8 has the law of ln(dP\u2217/dP)=ln(Z). Hence, the random variable \u03b2[yexp(\u2212Y/2\u22121/8)]\u2212\u03b1 has the law of V\u03b1,\u03b2(yZ), and so\u25a1Let e(\u03b1+\u03b12)/8 recurs occasionally, so to save on keystrokes, let \u03d5(\u03b1):=e(\u03b1+\u03b12)/8.The factor L\u03b6(\u03bb) the Laplace transform of the law of \u03b6; that is,Y a standard  Normal variate and writeYn be the scaled sum of n independent copies of \u03b6,LYn(\u03bb) converges to LY(\u03bb). On the other hand, if E[\u03b63]>0, by considering\u2014similarly as in the proof of Lemma\u00a0exp(\u03bbY) around \u03bb=0, it follows that, for small enough \u03bb0>0,V\u03b1,\u03b2(y)=\u03b2y\u2212\u03b1 as above, its conjugate U\u03b1,\u03b2, the corresponding value functions for the BSM economy u\u03b1,\u03b2 and its conjugate v\u03b1,\u03b2, as in Lemma\u00a0Zn. In this section, we do not require value functions for discrete\u2010time economies in which the consumer faces prices Z, so to simplify notation, we write v\u03b1,\u03b2n for the conjugate\u2010to\u2010the\u2010value\u2010function for the nth discrete\u2010time economy\u2014that is,u\u03b1,\u03b2n to denote that primal value function .Denote by k, the ratio v\u03b1,\u03b2n(k)/v\u03b1,\u03b2(1/k). From Equations , letLemma 9.3k, there exists n large enough so that, for \u03b1:=2\u03bb0n1/2, M\u226522k.For each integer Consider, for integer quations  and 23)k, the rak. Without loss of generality, set \u03b2=1. For given n and \u03b1=2\u03bb0n1/2, calculate the denominator and numerator in the M\u00a0separately.Fix Y is a standard  Normal variate and E denotes the expectation with respect to Y.For the denominator, we havey and \u03b2 before specializing to y=k and \u03b2=1:\u03b6j's are i.i.d. copies of \u03b6, and E denotes expectation with respect to these random\u00a0variables.And for the numerator, which we calculate for general We therefore have thatbn\u21921/8), so as n\u2192\u221e, this term, raised to the power n1/2 is bounded above by Gn1/2 for some constant G. And the term within the second set of square brackets converges to L\u03b6(\u03bb0)/LY(\u03bb0), which, per Equation ,nk\u2265k and nk/k is increasing, and we do so. As \u03b1k=2\u03bb0nk1/2, this implies that limk\u2192\u221e\u03b1k=\u221e.For each LetEquation ,(29)v\u03b1ky>0, and the function has all the properties required to be conjugate to a utility function U that satisfies Condition (3.1).U the conjugate (utility) function to V, and for u the  utility\u2010of\u2010wealth function corresponding to this U within the BSM economy, u(x)<\u221e for all x.Clearly, the sum is well defined for all V\u03b1k,\u03b2k(y)\u2264V(y). This implies that, for each k,Concerning the discrete\u2010time economies, begin by noting that all the terms in the sum  are posiyk>0, letNow we enlist Lemma\u00a0yk=k1/2. As v\u03b1k,\u03b2knk(k)>2k and v\u03b1k,\u03b2knk(y) is decreasing in y, we know that v\u03b1k,\u03b2knk(k1/2)>2k. As \u03b1k=2\u03bb0nk1/2 and nk/k is, by construction, nondecreasing, we know that \u03b1k/k1/2 is nondecreasing. Putting these two observations together, we know thatChoose yk=k1/2, yk=k1/2, u\u03b1k,\u03b2knk is concave and has value 0 at x=0. So, Equation >k1/2x. Hence, from Inequality (unk(x). But as k increases toward \u221e, the intervals  over which this is true expand to all of  asymptotically generate infinite expected utility, although she can only generate finite expected utility in the BSM economy. The limit in Equation  is precisely what she can attain in the nkth discrete\u2010time economy, even with the synthesizability constraint\u00a0imposed.But this final step is easy. The properties of \u03b6 that are used to get to Equation  >LY(\u03bb0) for some \u03bb0>0,\u00a0fails.For such \u03b6, the above reasoning does not apply. To the contrary, in the specific case of the symmetric random walk, we havelimsupnun(x)>u(x), for some utility function U =1). Or it may be that equality holds true, in the case of the symmetric binomial . We leave this question\u00a0open.It may still be true for the symmetric binomial that 10Z=dP\u2217/dP=e\u2212\u03c9(1)/2\u22121/8, the law of Z is that of exp, where Y\u22121/8,1/4 is Normal variate with mean \u22121/8 and variance 1/4. By standard calculations, then, the law of Z has density functionBecause Consider the functionV0 shows that it is decreasing in y as long as ln(y)<3/8, and a similar computation shows that V0 is also convex on  for small enough z0>0. Hence, we may define a function V:R++\u2192R++ that coincides with V0 on the interval  and is extended to all of  and that is convex, decreasing, differentiable, and satisfies V\u2032(\u221e)=0. The conjugate function to this V, denoted U, therefore satisfies the conditions of Assumption Differentiating y for which the value functionZ=dP\u2217/dP. Write the expectation  asw=yz.We want to determine the range of values of strictly positive functionv(y)=EPVyK depending on y,4ln(y)\u2264\u22121; that is, if y\u2264e\u22121/4.By straightforward calculation, we find that, for some constant u and v, this demonstrates, for y0=e\u22121/4, the possibility that v(y)=\u221e for y\u2264y0 and is finite for y>y0; moreover, as y\u2198e\u22121/4=y0,v(y)\u2197\u221e. This is possibility b(i) in Proposition\u00a0y0>0, it suffices to pass from the function V0(y) toBy the duality between \u03d5(\u03b1):=e(\u03b12+\u03b1)/8. Begin by defining, for y>0,\u03b1k and \u03b2k are given byy>0 and, in fact, does so faster than geometrically past some k0 (that depends on y).For possibility b(iii) in Proposition\u00a0for y>0,V(y):=\u2211k=V is strictly positive, convex, and twice (and more) continuously differentiable. And from Lemma\u00a0\u03b1k and \u03b2k, Equations (v(y)=\u221e for y<1 and is finite for y\u22651. And v\u2032(y) is finite for y\u22651. This, then, is the possibility b(iii) in Proposition\u00a0y0=1.Moreover, it is evident that om Lemma\u00a0v(y)=EPVy(\u03b1k)y\u2212\u03b1kandv\u2032(y)=b(ii) and examples where the pole is y0\u22601.We leave to the reader the construction of an example of possibility"}
{"text": "Scientific Reports 10.1038/s41598-020-62327-x, published online 26 March 2020Correction to: This Article contains errors in the Results section.Under the subheading \u2018Systematic Bias\u2019,H\u2009=\u2009120\u00b0 and \u03b8F\u2009=\u200990\u00b0, \u03b8H\u2009=\u200990\u00b0 and \u03b8F\u2009=\u200960\u00b0, and \u03b8H\u2009=\u200960\u00b0 and \u03b8F\u2009=\u200930\u00b0, i.e., postures under which the difference between the finger and head angles was \u221230\u00b0 (\u03b8H\u2009\u2212\u2009\u03b8F\u2009= \u221230\u00b0).\u201d\u201cInterestingly, the systematic bias was approximately zero for postures of \u03b8should read:H\u2009=\u2009120\u00b0 and \u03b8F\u2009=\u200990\u00b0, \u03b8H\u2009=\u200990\u00b0 and \u03b8F\u2009=\u200960\u00b0, and \u03b8H\u2009=\u200960\u00b0 and \u03b8F\u2009=\u200930\u00b0, i.e., postures under which the difference between the finger and head angles was \u221230\u00b0 (\u03b8F\u2009\u2212\u2009\u03b8H\u2009= \u221230\u00b0).\u201d\u201cInterestingly, the systematic bias was approximately zero for postures of \u03b8Under the subheading \u2018Nonsystematic Bias\u2019,2\u2009=\u2009, t\u2009=\u2009, p\u2009=\u2009, df = 22, data from participant #1.\u201d\u201cHowever, no correlation was found between any of the three postures and the amplitude of the nonsystematic bias; indicating that the amplitude of the nonsystematic bias is not posture-related .\u201dshould read:\u201cIn particular, the slope is negative for the systematic bias and positive for the phase of the nonsystematic bias given changes in the finger posture and finger-head posture. Similarly, the slope is positive for the systematic bias and nonsignificant for the phase of the nonsystematic bias given changes in the head posture .\u201d"}
{"text": "Bacillus amyloliquefaciens has been used to synthesize the immunomodulatory \u03b3\u2010D\u2010glutamyl\u2010L\u2010tryptophan (\u03b3\u2010D\u2010Glu\u2010L\u2010Trp) and the kokumi\u2010active \u03b3\u2010D\u2010glutamyl peptides. The optimum yield of \u03b3\u2010D\u2010Glu\u2010L\u2010Trp was 55.76\u00a0mM in corresponding to a minimum yield of by\u2010product (\u03b3\u2010D\u2010Glu\u2010\u03b3\u2010D\u2010Glu\u2010L\u2010Trp) in the presence of 75\u00a0mM D\u2010Gln and 100\u00a0mM L\u2010Trp. The glutaminase has a low Km values for the donors (D\u2010Gln and L\u2010Gln:5.53 and 0.98\u00a0mM), but high ones for the acceptors \u2010L\u2010Val/L\u2010Phe/L\u2010Met, ranging from 32.51 to 193.05\u00a0mM). The highest Km value appearing when n\u00a0=\u00a02 \u2010L\u2010Val/L\u2010Phe/L\u2010Met) suggested the rising difficulty for synthesis when the number of donor increases in the reaction mixtures. The \u03b3\u2010[D\u2010Glu]\u2010L\u2010Val/L\u2010Phe/L\u2010Met at 5\u00a0mM can impart the blank chicken broth an enhancing monthfulness, thickness, and umaminess taste.Glutaminase of  Glutaminase of Bacillus amyloliquefaciens has been used to synthesize the immunomodulatory \u03b3\u2010D\u2010glutamyl\u2010L\u2010tryptophan (\u03b3\u2010D\u2010Glu\u2010L\u2010Trp) and the kokumi\u2010active \u03b3\u2010D\u2010glutamyl peptides. The optimum yield of \u03b3\u2010D\u2010Glu\u2010L\u2010Trp was 55.76\u00a0mM in corresponding to a minimum yield of byproduct (\u03b3\u2010D\u2010Glu\u2010\u03b3\u2010D\u2010Glu\u2010L\u2010Trp) in the presence of 75\u00a0mM D\u2010Gln and 100\u00a0mM L\u2010Trp. The glutaminase has a low Km values for the donors (D\u2010Gln and L\u2010Gln:5.53 and 0.98\u00a0mM), but high ones for the acceptors \u2010L\u2010Val/Phe/Met, ranging from 32.51 to 193.05\u00a0mM). The highest Km value appearing when n=2 \u2010L\u2010Val/Phe/Met) suggested the rising difficulty for synthesis when the number of donor increases in the reaction mixtures. The \u03b3\u2010[D\u2010Glu]\u2010L\u2010Val/Phe/Met at 5\u00a0mM can impart the blank chicken broth an enhancing monthfulness, thickness, and umaminess taste. D\u2010Gln, L\u2010Gln, L\u2010Val, L\u2010Trp, L\u2010Met, L\u2010Phe, Gly\u2010Gly, and \u03b3\u2010glutamyl\u2010p\u2010nitroanilide (\u03b3\u2010GpNA) were purchased from Sigma\u2010Aldrich . L\u2010Glutaminase of Bacillus amyloliquefaciens was acquired from Amano Enzyme China Ltd. . The HPLC grade acetonitrile and formic acid were purchased from CapitalBio Corporation .Commercial synthetic 2.2\u03b3\u2010D\u2010Glu\u2010L\u2010Trp: a reaction mixture that contained D\u2010Gln , L\u2010Trp , and glutaminase (0.05 U/ml) was placed in a water bath shaking table for incubating for 8\u00a0hr at 37 \u00baC, and then the mixture was inactivate enzyme in a boiling water bath at 90 \u00baC for 15\u00a0min. The optimum ratio of substrate was also analyzed including the various D\u2010Gln  and a fixed L\u2010Trp (100\u00a0mM). The isolation of \u03b3\u2010D\u2010Glu\u2010Dipeptides was carried in accordance with the method of Suzuki with some modification , and then purified \u03b3\u2010D\u2010Glu\u2010L\u2010Trp was dissolved in D2O and then analyzed with a Bruker 600\u00a0MHz Superconducting Fourier Transform NMR Spectrometry . HPLC  equipped with XSelect HSS T3 column  was used to determine the content of peptides and the L\u2010Trp. Acetonitrile and water with 0.1% formic acid were as the mobile phase. The other parameter contained: 1\u00a0ml/min flow rate, 220\u00a0nm detection wavelength (\u03bb), 40\u00b0C column temperature, and 20\u00a0\u03bcL injection volume. The objective products in the HPLC chromatogram were preliminarily identified on the basis of the retention time of external standard compounds.: Synthesis of \u03b3\u2010D\u2010Glutamyl peptides: The reaction mixture contained one of the substrate mixtures of D\u2010Gln:L\u2010Met\u00a0=\u00a075:100, D\u2010Gln:L\u2010Val\u00a0=\u00a075:100 or D\u2010Gln:L\u2010Phe\u00a0=\u00a075:100, 0.05 U/ml enzyme, incubated in a water bath shaking table at 37\u00b0C for 12\u00a0hr, then the mixtures were heated to the point of inactivating enzyme in a boiling water bath at 90 \u00baC for 15\u00a0min. These \u03b3\u2010D\u2010Glu\u2010peptides in the mixtures were analyzed by UPLC\u2010Q\u2010TOF MS/MS system  with an Agilent ZORBAX RRHD SB\u2010C18  110 column, referring to Yang's methods  of glutaminase for the synthesis of \u03b3\u2010D\u2010glutamyl peptides.Enzymatic kinetic parameters  and a fixed L\u2010Trp (25\u00a0mM).The apparent K\u03b3\u2010GpNA, \u03b3\u2010glutamyl acceptors and enzyme in the borate\u2010NaOH  buffer solution, then adding 0.1\u00a0M HCl to terminate the reaction after incubation at 37 \u00baC for 30\u00a0min. The content of p\u2010nitroaniline was determined by a UV spectrophotometer  at 410\u00a0nm. The release of 1\u00a0\u03bcmol of p\u2010nitroaniline per min from \u03b3\u2010GpNA was defined as the amount of one unit (U) of enzyme through the transpeptidation reaction.The substrate specificity of the enzyme with different acceptors were measured as described previously with some modification  were added into a blank chicken broth to evaluate the flavor characteristic. Taste profiles mainly included mouthfulness, thickness and umaminess. Thereinto, mouthfulness describes an overall perception associated with food texture, structure and morphological complexity, and thickness which was measured as the increased taste intensity evaluated 10\u00a0s after food tasting describes rich complexity. The panels were consisted of 10 males and 7 females (age 30\u201340) from a professional company. 10\u00a0ml of solutions was given to these panelist individually and then put into the mouth for 15\u00a0s before swallowing, finally recorded the taste experienced in 25\u00a0s. Panelists were asked to rate the intensity of a given taste on a scale from 0 (not detectable) to 5 (strongly detectable).5\u00a0mM of these synthetic 2.3p\u00a0<\u00a0.05 between samples using SPSS 16.0 statistical software.Statistical analysis was performed in triplicate and analyzed by Microsoft Excel 2000. Duncan test was used to determine significant differences at 33.1\u03b3\u2010D\u2010Glu\u2010L\u2010Trp.\u03b3\u2010D\u2010Glu\u2010L\u2010Trp (12.245\u00a0min) indicating that \u03b3\u2010D\u2010Glu\u2010L\u2010Trp could be enzymatically synthesized in the action of a commercial glutaminase of Bacillus amyloliquefaciens : \u03b4 7.61\u22127.55 , 7.41 , 7.19\u22127.13 , 7.08 , 4.71\u22124.64 , 3.81\u22123.72 , 3.74\u22123.69 , 3.67 , 3.59 , 3.31 , 3.16 , 2.35\u22122.27 , 1.95\u22121.87 .Three main intense chromatographic peaks were observed, the most intense peak corresponded to lu\u2010L\u2010Trp 2.245\u00a0min\u03b3\u2010D\u2010Glu\u2010\u03b3\u2010D\u2010Glu\u2010L\u2010Trp (12.389\u00a0min), suggesting that there was a significant coexisting product (\u03b3\u2010D\u2010Glu\u2010\u03b3\u2010D\u2010Glu\u2010L\u2010Trp) in the postenzymatic reaction mixture. The production of \u03b3\u2010D\u2010Glu\u2010L\u2010Trp and \u03b3\u2010D\u2010Glu\u2010\u03b3\u2010D\u2010Glu\u2010L\u2010Trp, as well as the consumption of L\u2010Trp with time\u2010course are showed in Figure\u00a0\u03b3\u2010D\u2010Glu\u2010L\u2010Trp would reach a maximum yield of 60.90\u00a0mM after 3\u00a0hr incubation (its yield increasing during the first 3\u00a0hours but without significant differences) and then leveled off, the result was closed to that of GGT\u2010catalyzing , finally maintained stability (p\u00a0>\u00a0.05). The observation illustrates there was a time difference between obtaining the maximum of \u03b3\u2010D\u2010Glu\u2010L\u2010Trp (3\u00a0hr of incubation) and \u03b3\u2010D\u2010Glu\u2010\u03b3\u2010D\u2010Glu\u2010L\u2010Trp (5\u00a0hr), so the concentration of by\u2010product can be minimized through controlling the incubation time. Correspondingly, the consumption of L\u2010Trp can be summarized in roughly three stages during the synthesis of \u03b3\u2010D\u2010Glu\u2010L\u2010Trp: firstly, L\u2010Trp sharply dropped to 43.12\u00a0mM after 1\u00a0hr reaction, then dropped at a lower rate from the 1st to 5th h, and finally reached a plateau (L\u2010Trp at about 28.36\u00a0mM) after the 5th h .The other two intense peaks were the L\u2010Trp (7.762\u00a0min) and \u03b3\u2010D\u2010Glu\u2010L\u2010Trp and \u03b3\u2010D\u2010Glu\u2010\u03b3\u2010D\u2010Glu\u2010L\u2010Trp (a 2\u2010fold difference of glutamic acid residue), it can be speculated\u00a0the concentration of D\u2010Gln would influence the production of by\u2010product more than that of objective product. Therefore, the effect of D\u2010Gln  with a fixed L\u2010Trp (100\u00a0mM) was analyzed for reducing the yield of \u03b3\u2010D\u2010Glu\u2010\u03b3\u2010D\u2010Glu\u2010L\u2010Trp . The yield of \u03b3\u2010D\u2010Glu\u2010L\u2010Trp was 55.85\u00a0mM, with a below detectable limit of \u03b3\u2010D\u2010Glu\u2010\u03b3\u2010D\u2010Glu\u2010L\u2010Trp.When comparing the sequence characteristic between p Figure\u00a0. With a \u03b3\u2010D\u2010Glutamyl Phenylalanine, Methionine, and Valine.\u03b3\u2010D\u2010Glu\u2010L\u2010Phe, \u03b3\u2010D\u2010Glu\u2010L\u2010Val, and \u03b3\u2010D\u2010Glu\u2010L\u2010Met were with the significant signals at m/z 295.1295, 247.1299, and 279.1016, respectively in positive ESI mode , [AA(Acceptor)\u2010COOH]+ (y\u2010CO2 type ion), [AA(Acceptor)\u2010NH3]+ (y\u2010CO2 type ion), [AA(Acceptor)+H]+ (y type ion) in each of the mass spectrum. Their yield was 30.23\u00a0\u00b1\u00a04.18, 23.12\u00a0\u00b1\u00a03.97, 17.19\u00a0\u00b1\u00a02.04\u00a0mM, respectively \u2010L\u2010AA as an acceptor to participate in the \u03b3\u2010glutamyl transfer reaction to form the by\u2010products (\u03b3\u2010D\u2010[Glu]n\u2010L\u2010AA) \u2010L\u2010Trp/Phe/Met/Val) during \u03b3\u2010glutamyl transpeptidation by glutaminase\u2010catalysis were getting bigger with the increased \u201cn,\u201d that is, the lowest Km with L\u2010Trp/L\u2010Phe/L\u2010Met/L\u2010Val , while the highest  with \u03b3\u2010[D\u2010Glu](n=1)\u2010L\u2010Trp, \u03b3\u2010[D\u2010Glu](n=2)\u2010L\u2010Phe, \u03b3\u2010[D\u2010Glu](n=2)\u2010L\u2010Met and \u03b3\u2010[D\u2010Glu](n=2)\u2010L\u2010Val. Thus, it was more and more difficult for the synthesis of \u03b3\u2010[D\u2010Glu]n\u2010L\u2010Trp/L\u2010Phe/L\u2010Met/L\u2010Val became more difficult with the increase of the number of \u03b3\u2010D\u2010glutamyl residues in the acceptor, which were consistent with \u03b3\u2010[L\u2010Glu]n\u2010L\u2010Phe/L\u2010Met/L\u2010Val synthesis. The Km values of \u03b3\u2010[D\u2010Glu]\u2010L\u2010Trp/L\u2010Phe/L\u2010Met/L\u2010Val during the hydrolysis reaction by glutaminase\u2010catalysis were in the range of 36.73\u00a0~\u00a0122.73\u00a0mM, which were lower than those for the synthesis of \u03b3\u2010[D\u2010Glu]n\u2010L\u2010Trp/L\u2010Phe/L\u2010Met/L\u2010Val. Moreover, the affinities of glutaminase for the same acceptors with the L\u2010Gln as the donor were obviously lower than that of D\u2010Gln. The affinities of glutaminase for the acceptors illustrate L\u2010Trp is the most natural acceptor, indicating the glutaminase is very suitable for the synthesis of \u03b3\u2010D\u2010Glu\u2010L\u2010Trp.The results of the Km values of glutaminase show high Km values for acceptors, but low ones for s, Table . The Km \u03b3\u2010glutamyl acceptors with the D\u2010Gln as the \u03b3\u2010glutamyl donor  an enhancing mouthfulness, thickness and umami taste with the taste\u2010enhancing score of 0.60\u00a0~\u00a01.07  and \u03b3\u2010[D\u2010Glu]n\u2010L\u2010Phe/L\u2010Met/L\u2010Val have been synthesized in the presence of D\u2010Gln catalyzed by the glutaminase of B. Amyloliquefaciens. Differ from the GGT enzyme, this glutaminase can catalyze the binding of D\u2010Gln to the synthetic \u03b3\u2010D\u2010glutamyl di\u2010/tri\u2010peptides. When comparing the D\u2010Gln and L\u2010Gln as the donors, the number and content of the by\u2010products with the D\u2010Gln as the donor were significantly less than that of L\u2010Gln in the \u03b3\u2010glutamyle peptides synthesis. The Km values of glutaminase of B. Amyloliquefaciens for D\u2010Gln and L\u2010Gln (the donors) were 5.53 and 0.98\u00a0mM, and for the acceptors were range from 32.51 (L\u2010Trp) to 193.05\u00a0mM . The optimal yield of \u03b3\u2010D\u2010Glu\u2010L\u2010Trp was 55.85\u00a0mM with a below detectable limit of \u03b3\u2010D\u2010Glu\u2010\u03b3\u2010D\u2010Glu\u2010L\u2010Trp under the reaction condition: 75\u00a0mM D\u2010Gln and 100\u00a0mM L\u2010Trp, incubating for 2\u00a0hr at 37\u00b0C (pH 10.0). The addition of the \u03b3\u2010[D\u2010Glu]n\u2010L\u2010Phe/Met/Val can impart the blank chicken broth an enhancing kokumi taste.The authors declare that they do not have any conflict of interest.This study does not involve any human or animal testing."}
{"text": "During the early Byzantine period, the therapeutic herb \u201cColchicum autumnale\u201d, or \u201cermodaktylon\u201d was introduced in the treatment of podagra (gout). Podagra presented throughout the Byzantine period a disease with high incidence, since 14 out of the total 86 Emperors seem to have suffered from it. The lead pipes of the city of Constantinople\u2019s sewer system, utensils, but also the production of the sweetening grape syrup sapa contributed to its appearance. Although Alexander of Tralles considered to be the physician who discovered the properties of the plant, Severus Iatrosophista, Theodosius the Philosopher and Jacobus Psychrestos, were the healers who introduced ermodaktylon as the pioneering treatment of podagra in the early Byzantine period. Colchicum autumnale (Greek: \u03ba\u03bf\u03bb\u03c7\u03b9\u03ba\u03cc \u03c4\u03bf\u03c5 \u03c6\u03b8\u03b9\u03bd\u03bf\u03c0\u03ce\u03c1\u03bf\u03c5), or bitter hermodactyl , or cochlicon (Greek: \u03ba\u03bf\u03c7\u03bb\u03b9\u03ba\u03cc\u03bd), or articulorum (the soul of the joints), later named as saffron, or autumn crocus, was most probably known since the early rhizotomi (Greek: \u03c1\u03b9\u03b6\u03bf-\u03c4\u03cc\u03bc\u03bf\u03b9), the pharmacobotanists practicing herbal medicine since the Prehippocratic era . Although it was Pedanius Dioscorides (ca 1st century AD) who firstly mentioned Colchicum as a therapeutic plant with strong poisonous action , it was Alexander of Tralles (Greek: \u0391\u03bb\u03ad\u03be\u03b1\u03bd\u03b4\u03c1\u03bf\u03c2 \u03bf \u03a4\u03c1\u03b1\u03bb\u03bb\u03b5\u03b9\u03b1\u03bd\u03cc\u03c2) (525\u2013605 AD) who was wrongfully credited with having introduced it intothe treatment of gout.7He himself, mentioned Severus Iatrosophista, Theophilus the Philosopher, and Jacobus Psychrestos who were the first healers to use the plant Colchicum. Although Alexander had proposed the use of cantharides as a blister to treat podagra, he had adopted previous knowledge to emphasise on a new promising herbal treatment. Alexander gave us six recipes containing hermodactyl; the final two of which he had attributed to other physicians: one to Jacobus Psychrestus, and another to Theodosius the Philosopher.8The plant\u2019s name comes from Colchis, a region in Asia Minor, were Colchicum was endemic at the era .9Podagra  disease, a complex form of arthritis currently known as gout, was apparently accumulated among Byzantine Emperors, and common citizens of Constantinople. It has been considered that gout may be a disease caused by lead poisoning: a contributing cause for this accumulation may have been exposure to high levels of lead, originating from Constantinople\u2019s water pipes, wine containers and cooking pots used for producing the sweetening grape syrup sapa (Greek: \u03ad\u03c8\u03b7\u03bc\u03b1). Hippocrates (ca 460-370 BC), 10 eons earlier, suggested that podagra (Greek: \u03c0\u03bf\u03b4\u03ac\u03b3\u03c1\u03b1) was the result of the excess-accumulation of the body humours (4 humours theory) inside the articular capsule of the joints. The disease was called \u201cpodagra\u201d, meaning in Greek severe pain grabbing the leg. As podagra seems to have been a significant, widespread and invalidating disease at the era, a disease that was prevalent in Byzantium, but also a well-known entity in ancient Greece.10Our historical note presents the first Byzantine physicians who introduced Colchicum autumnale as a treatment for podagra. Although Colchicum autumnale is no longer recommended due to its significant toxicity, its active components, colchicine in particular, are still widely used. The plant contains colchicine , and smaller amounts of secondary alkaloids, such as demecolcine. German botanist Hyeronimus Bock (Latinized: Tragus) (1498\u20131554 AD) mentioned in his treatise \u201cKreuterbuch\u201d during 1550, that Arab and Byzantine physicians employed the plant in the early Byzantine period.11Theodosius the Philosopher, an eminent scholar of the same era, had also proposed the same plant as a botanic cure against podagra.3Severus Iatrosophista (Greek: \u03a3\u03b5\u03c5\u03ae\u03c1\u03bf\u03c2 \u03bf \u0399\u03b1\u03c4\u03c1\u03bf\u03c3\u03bf\u03c6\u03b9\u03c3\u03c4\u03ae\u03c2), was a 5th century medicophilosopher, known for his masterpiece \u201cDe Clysteribus\u201d (Greek: \u03a0\u03b5\u03c1\u03af \u0395\u03bd\u03b5\u03c4\u03ae\u03c1\u03c9\u03bd); suggesting clysters for various diseases such as skin rash. He was the first Byzantine physician to describe the allergic shock from food and drugs, while he had noted women\u2019s hysteria proposing acute intervention. He had also presented a thorough report upon the surgical instruments availliable in that period. He is considered by Alexander of Tralles the first to introduce hermodactyl (Colchicum).Figure 4).12Jacobus studied podagra, \u00ab \u03c4\u03b1\u03c2 \u03b6\u03b5\u03bf\u03cd\u03c3\u03b1\u03c2 \u03c6\u03bb\u03b5\u03b3\u03bc\u03bf\u03bd\u03ac\u03c2 \u03c4\u03c9\u03bd \u03c0\u03bf\u03b4\u03ce\u03bd \u00bb , to be put upon the inflamed joint.14Today it is known that the plants of the genus Colchicum contain colchicine, a toxic natural product and secondary metabolite, which is still in use for the treatment of gout.2Jacobus was a scholar with a broad spectrum of knowledge concerning human physiology. Thus, he was also considered as an exceptional dietician, proposing a moistens temperate diet (Greek: \u03c5\u03b3\u03c1\u03b1\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03b5\u03cd\u03ba\u03c1\u03b1\u03c4\u03bf\u03bd \u03b4\u03af\u03b1\u03b9\u03c4\u03b1) for the organism\u2019s homeostasis to be balanced.15Although Jacobus was an eminent physician in the imperial court of the Byzantine empire, his work was lost, and only fragments survived inside the treatises of other important medical figures such as Damascius the philosopher (458\u2013550 AD), Alexander of Tralles, and Aetius of Amida (ca mid 5th-mid 6th century AD). His pioneering thought to use colchicine to treat podagra should grant him a place among the significant figures of the history of rheumatology.The Byzantine physician Jacobus Psychrestos , was the offspring of Hesychios from Damascus, a known physician of the era. He had exercised both medicine and philosophy in Constantinople during the 5th century AD. He was considered an authority (Greek: \u03ac\u03c1\u03b9\u03c3\u03c4\u03bf\u03c2), capable to treat a plethora of diseases. He had soon become the personal head physician (Greek: \u03b1\u03c1\u03c7\u03af\u03b1\u03c4\u03c1\u03bf\u03c2) of the Byzantine emperor Leo the 1st, known as the Thracian (Greek: \u039b\u03ad\u03c9\u03bd \u0391\u2019 \u1f41 \u0398\u03c1\u1fb7\u03be) (401\u2013474 AD) , thus, perhaps, suggesting the speed of cure that the herbal drug provided for the patients. Although by many researchers Alexander of Tralles is considered to be the physician who introduced this drug in the treatment of podagra, it was the personal physician of Emperor Leon the 1st, Jacob Psychrestus, who epitomized Severus\u2019 and Theodosius\u2019 proposals and systematized its use; quickly gaining substantial fame.It appears that a great number of Sovereigns (14 of the 86 Byzantine Emperors) of the Byzantine Empire and officials of the State and leaders of the Church suffered from what in most cases seems to have been the podagra disease. The texts of the most Byzantine writers referred to the main medicine for arthritis during that era, hermodactylus, a constituent of the herb Colchicum autumnale. Hermodactylus means, in the Hellenic language, the \u201cfinger of Hermes\u201d; the Olympian messenger god of the ancient Greeks (In the history of medicine, a plethora of forgotten physicians, such as our three Byzantine healers, still seek their rightful place in medical history. Rheumatology owes them an honourable citation."}
{"text": "The contributions of \u03b3\u03b4 T cells to immune (patho)physiology in many pre\u2010clinical mouse models have been associated with their rapid and abundant provision of two critical cytokines, interferon\u2010\u03b3 (IFN\u2010\u03b3) and interleukin\u201017A (IL\u201017). These are typically produced by distinct effector \u03b3\u03b4 T cell subsets that can be segregated on the basis of surface expression levels of receptors such as CD27, CD44 or CD45RB, among others. Unlike conventional T cells that egress the thymus as na\u00efve lymphocytes awaiting further differentiation upon activation, a large fraction of murine \u03b3\u03b4 T cells commits to either IFN\u2010\u03b3 or IL\u201017 expression during thymic development. However, extrathymic signals can both regulate pre\u2010programmed \u03b3\u03b4 T cells; and induce peripheral differentiation of na\u00efve \u03b3\u03b4 T cells into effectors. Here we review the key cellular events of \u201cdevelopmental pre\u2010programming\u201d in the mouse thymus; and the molecular basis for effector function maintenance vs plasticity in the periphery. We highlight some of our contributions towards elucidating the role of T cell receptor, co\u2010receptors (like CD27 and CD28) and cytokine signals (such as IL\u20101\u03b2 and IL\u201023) in these processes, and the various levels of gene regulation involved, from the chromatin landscape to microRNA\u2010based post\u2010transcriptional control of \u03b3\u03b4 T cell functional plasticity. This ordered rearrangement results in timed production of defined \u03b3\u03b4 T cell populations, which leave the thymus and populate different epithelial\u2010rich tissues in the adult animal  \u03b3 chains. Early work on the genetics of TCR rearrangement during fetal, neonatal and adult thymocyte development revealed an organized and sequential rearrangement of specific + \u03b3\u03b4 T cells are generated.+ \u03b3\u03b4 T cells develop exclusively in the fetal thymus and have no junctional diversity due to absence of terminal deoxynucleotidyl transferase expression. DETC are found in the adult skin, while V\u03b36+ \u03b3\u03b4 T cells localize to diverse tissues such as, eg, uterine epithelia, tongue and meninges.+ \u03b3\u03b4 T cells is restricted to the confined window of fetal development and cannot be induced in adult animals.+ and V\u03b34+ \u03b3\u03b4 T cells develop from late fetal life onwards, throughout adulthood. These show higher junctional diversity and localize to diverse sites including peripheral lymphoid tissues, where they represent the majority of \u03b3\u03b4 T cells in the adult mouse. The distinctive developmental phases, orchestration of TCR rearrangements and specialized tissue localization suggest a diversity of physiological roles for \u03b3\u03b4 T cells encompassing features of both innate and adaptive \u2013 or \u201cadaptate\u201d \u2013 immune surveillance.The first T cells produced in the fetal thymus are dendritic epidermal T cells (DETC), a specialized subset of \u03b3\u03b4 T cells expressing an invariant V\u03b35V\u03b41 TCR. Subsequently, V\u03b36Listeria monocytogenes.E coli infection.Staphylococcus aureus and Candida albicans, among other infections, or detrimental roles in inflammatory diseases and in cancer, where they can promote angiogenesis, immune suppression and tumor cell growth  and the majority of V\u03b46.1/V\u03b46.3+ \u03b3\u03b4NKT cells (liver and spleen); and post\u2010natally generated cells expressing more polyclonal TCR\u03b3\u03b4 (mostly V\u03b31+ or V\u03b34+) with junctional diversity .IFN\u2010\u03b3\u2010producing (\u03b3\u03b4IFN) T cells, including a fetal/ perinatal\u2010derived subgroup expressing invariant and semi\u2010invariant TCRs with no junctional diversity, which is made up by V\u03b35+ or V\u03b34+ TCRs, although minor populations expressing V\u03b31 and V\u03b32/3 chains have been described in the thymus and most notably in the liver.+ \u03b3\u03b417 T cells are dominated by one invariant V\u03b36V\u03b41 clone lacking additional N\u2010nucleotide insertions and few semi\u2010invariant V\u03b36V\u03b41 clones,+ \u03b3\u03b417 T cells present a more oligoclonal population encompassing multiple (semi)invariant TCRs.\u03b3\u03b417 T cells usually expressing V\u03b36In this review, we will focus on the biology of effector \u03b3\u03b4 T cells making IFN\u03b3 or IL\u201017, from their early steps of differentiation in the thymus to their functional properties in the periphery, and the underlying molecular mechanisms.2highCD4\u2212CD8\u2212CD45RBlowCD44highCD62Llow, implicating a memory\u2010like phenotype.Over the last decade or so, the field has put much effort into establishing markers to identify and analyze the two main effector \u03b3\u03b4 T cell subsets. The initial description of \u03b3\u03b417 T cells characterized them as CD3\u2212 in the steady state. We found that CD27\u2212 \u03b3\u03b4 T cells were CD44highCD62Llow and contained all IL\u201017 producers but very few \u03b3\u03b4IFN T cells. In contrast, CD27+ \u03b3\u03b4 T encompassed most CD122+ \u03b3\u03b4 T cells, had lower expression of CD44 and produced IFN\u03b3 but essentially no IL\u201017.Plasmodium berghei). Most interestingly, we found that the effector phenotypes were already established during thymic development, and since embryonic life. Our detailed analysis of \u03b3\u03b4 T cell development revealed that manipulating CD70\u2010CD27 signals on developing \u03b3\u03b4 T cells in fetal thymic organ cultures (FTOC) impacted effector phenotype acquisition. In particular, CD27 signals were required for expression of the lymphotoxin\u2010\u03b2 receptor (LT\u03b2R) and downstream genes previously associated with IFN\u03b3 production in \u03b3\u03b4 T cells.+ while \u03b3\u03b417 cells are exclusively Ly\u20106C\u2212.+ \u03b1\u03b2 T cells, the authors suggested \u03b3\u03b4 T cell na\u00efve\u2010like and memory\u2010like subsets sharing characteristics with adaptive \u03b1\u03b2 T cells.CD27 had been previously used to characterize functional subsets of \u03b1\u03b2 T cells and natural killer (NK) cells.+ \u03b3\u03b4 T cells in the adult thymus and peripheral tissues was especially prominent and resulted in establishment of CCR6 as a \u03b3\u03b417 T cell marker. Of note, in the neonatal thymus only V\u03b36+ \u03b3\u03b417 thymocytes express CCR6, whereas the vast majority of V\u03b34+ \u03b3\u03b417 thymocytes do not; it was suggested that CCR6 expression might be induced extrathymically on dermal V\u03b34+ \u03b3\u03b4 T cells.\u2212) \u03b3\u03b4 T thymocytes.\u2212 V\u03b34+ \u03b3\u03b4 T cells are mostly absent from the thymus. Therefore, it remains possible that CCR6 induction occurs intrathymically on V\u03b34+ cells that promptly leave to peripheral sites. Interestingly, the expression of the two scavenger receptors SCART1 and SCART2 was shown to be mutual exclusive on V\u03b36+ and V\u03b34+ dermal \u03b3\u03b417 T cells, respectively, possibly due to different ontogenic origins.\u2212CD44highCD45RB+) and IL\u201017 (CD24\u2212CD44highCD45RB\u2010) committed subsets, as well as their developmental trajectories from immature (CD25+CD24+) \u03b3\u03b4 thymocytes.Besides lacking CD27 expression, \u03b3\u03b417 T cells were found to express SCART1 and SCART2 and CCR6 in the adult mouse.+ \u03b3\u03b4 T cells within the adult thymus.+ \u03b3\u03b4 T cells in adult mice; and confirmed that CD24\u2212 \u03b3\u03b4 T cells in the adult thymus mostly represent long\u2010lived resident effector cells generated during fetal/ perinatal life.Different thymic trajectories for the generation of \u03b3\u03b4IFN cells have been recently described by introducing transient surface markers, namely CD117, CD200 and CD371, as useful tools to segregate CD2433.1Among the surface markers that segregate the two \u03b3\u03b4 T cell effector subsets \u2010based subsetting described above. The synthesis of T10/T22\u2010tetramers  for flow cytometry analysis enabled the first study of a minor but antigen\u2010specific \u03b3\u03b4 T cell subset within wildtype (ie non\u2010transgenic) mice.+ \u03b3\u03b417 T cells were decreased in early life but V\u03b34+ \u03b3\u03b417 T cells developed normally, resulting in normal \u03b3\u03b417 T cell abundance in the periphery of adult mice. In contrast, \u03b3\u03b4IFN T cells were diminished throughout life in the thymus and in peripheral lymphoid organs of CD3DH mice. Most importantly, CD122+ NK1.1+ \u03b3\u03b4IFN T cells (mostly V\u03b31+) were virtually absent, but their thymic development could be rescued upon injection of an agonist CD3 antibody. The reduction in peripheral \u03b3\u03b4IFN T cells in CD3DH mice associated with less susceptibility to P berghei ANKA\u2010driven experimental cerebral malaria, an inflammatory syndrome dependent on IFN\u03b3 and \u03b3\u03b4 T cells.A subsequent study from our laboratory has provided additional data supporting a role of TCR signaling in the development of specific \u03b3\u03b4 T cell effector subsets. The research was based on a new mouse model, haploinsufficient for both CD3\u03b3 and CD3\u03b4 , where we found both TCR\u03b3\u03b4 surface expression and TCR\u03b3\u03b4 signal strength to be substantially reduced.The CD3DH phenotype is surprising considering that single haplodeficient mice did not show a similar impairment, and because several studies on TCR composition reported that CD3\u03b4 was not incorporated into the murine TCR\u03b3\u03b4 at least on mature \u03b3\u03b4 T cells.+NK1.1+ \u03b3\u03b4IFN T cells appear to be especially dependent on TCR\u03b3\u03b4 signaling during development. It may be that these CD122+NK1.1+ \u03b3\u03b4 T cells represent a unique subset that has yet to be fully explored; we next summarize additional findings that support this possibility. In the thymus of adult wildtype mice, CD122+NK1.1+ \u03b3\u03b4 T cells are found among the CD44highCD24low mature \u03b3\u03b4 T cells .Bcl11b knockout mouse, two hepatic \u03b3\u03b4 T cell subsets with different developmental requirements were described: a NK1.1+CD5\u2212 subset generated early in newborn mice producing exclusively IFN\u03b3 rapidly upon infection; and a NK1.1\u2212CD5+ subset that comprised both IFN\u03b3 and IL\u201017 producers.Bcl11b deficiency resulted in a complete loss of the NK1.1\u2212CD5+ \u03b3\u03b4 T cell subset , whereas the NK1.1+CD5\u2212 proved to be Bcl11b\u2010independent and retained CD122, high PLZF expression and IFN\u03b3 production upon stimulation. These observations suggest that the NK1.1+ \u03b3\u03b4IFN T cell subset has specific developmental properties and represents a unique innate\u2010like population.It is also intriguing that CD122Another important clue on the role of TCR signaling in effector \u03b3\u03b4 T cell differentiation came from the analysis of Skint\u20101\u2010deficient mice. V\u03b35V\u03b41 TCR\u2010expressing DETC were shown to rely on encountering Skint\u20101 during development in the embryonic thymus to embark on the IFN\u03b3 effector program. Absence of Skint\u20101 expression in a naturally occurring mutant strain resulted in V\u03b35V\u03b41 T cells that atypically produced IL\u201017 instead of IFN\u03b3.A cell\u2010intrinsic program indeed appears to drive the development of \u03b3\u03b417 T cells in the fetal/ perinatal thymus. Thus, IL\u201017 expression in early precursors prior to TCR rearrangement has been documented in the fetal thymus; and \u03b3\u03b417 T cells appear to require only weak or even no TCR signaling for their development.+ \u03b3\u03b417 T cell development.As a final point on TCR signaling, several studies have reported variable expression and dependency on associated proteins by different \u03b3\u03b4 T cell subsets. The main Src family kinase Lck was surprisingly poorly expressed in mature \u03b3\u03b417 T cells in the thymus3.2The identification of pre\u2010programmed \u03b3\u03b4 T cell effector subsets and their key contributions to peripheral immune responses highlighted, once more, the importance of understanding thymic development. How \u03b3\u03b4 T cell effector subsets acquire their effector functions during thymic development remains an area of active research and controversies, particularly whether \u03b3\u03b417 and \u03b3\u03b4IFN T cells arise from common or distinct thymic progenitors.\u2212CD8\u2212) stage 2 (DN2), in which they simultaneously rearranged and transcribed Trb, Trg and Trd,In the adult mouse, bipotential thymocytes were found up to the DN (double negative/CD4prior to TCR expression. One notable example is the expression of the interleukin 7 receptor (IL\u20107R), since intrathymic injection and reconstitution of fetal thymic lobes with purified DN2 populations from adult mice demonstrated that the progeny of IL\u20107Rhigh DN2 cells presented a higher \u03b3\u03b4 T cell to \u03b1\u03b2 T ratio compared to that of IL\u20107Rlow DN2 cells.TCR\u03b3 locus,+ \u03b3\u03b417 T cells, as these were absent in a spontaneous Sox13 mutant mouse strain, while other \u03b3\u03b4 T cell subsets appeared to develop normally.In support of pre\u2010commitment, cell heterogeneity within the DN2 subset has been described and linked to different lineage biases loxP site flanked version of the gene of interested and the expression of Cre recombinase from a specific promoter whose activity is restricted to a certain cell type or tissue. The heterogeneity within the \u03b3\u03b4 T cell lineage and our incomplete understanding of its developmental trajectories impose limitations to the use of several conditional KO mice for the analysis of \u03b3\u03b4 T cells. Indeed, conditional deletion using the proximal promoter of Lck\u2010driven Cre (pLckCre) was shown to occur in DN2 cells and to efficiently target the \u03b1\u03b2 T cell lineage,On the other hand, TCR signal \u201cstrength\u201d emerged as a key determinant based on studies using transgenic TCR\u03b3\u03b4 expression, in which the manipulation of downstream signaling mediators had major effects on \u03b3\u03b4 vs \u03b1\u03b2 T cell commitment. In such settings, \u03b3\u03b4 T cells required stronger TCR signals to develop than their \u03b1\u03b2 T cell counterparts.+ and IL\u201017+ \u03b3\u03b4 T cells, c\u2010kitlow DN3 cells failed to give rise to \u03b3\u03b417 T cells.An interesting possibility is the existence of various points of divergence from the \u03b1\u03b2 T cell lineage path for discrete \u03b3\u03b4 T cell subsets. Support for such a \u201cmultiple branching\u2010off model\u201d first came from studies on E17 fetal thymi. While DN2 cells expressing high levels of c\u2010kit developed into both IFN\u03b3+ \u03b3\u03b417 T cells derive from CD44+CD25\u2212CD24+c\u2010Kit\u2212 (DN1d) but not DN2 precursors.+ \u03b3\u03b417 T cell development.+ \u03b3\u03b417 T cells, which was inferred to be TCR\u2010independent from the analysis of mutant mice.More recently, a progenitor fate analysis and single\u2010cell RNA sequencing of discrete DN1 subpopulations, as previously defined by Petrie and colleagues,high DN2 cells can develop into \u03b3\u03b417 T cells.+ cells, whereas Spidale et al employed DN1 and DN2 precursors from 10\u2010day\u2010old thymi, which were introduced into FTOC, and focused on V\u03b34+ \u03b3\u03b417 T cell development. Since \u03b3\u03b417 T cell thymocyte development is supposed to be terminated in 10\u2010day\u2010old mice,+ \u03b3\u03b417 T cell development excluding a DN2 stage, while directly assessing the differences to earlier fetal V\u03b36+ \u03b3\u03b417 T cell development.In contrast with the conclusions of Spidale et al,Tcrg locus, may develop into IL\u201017 producers by default, with no or low TCR signaling involved \u2013 unless the rearranged TCRs encounter their ligands, as it is the case for V\u03b35V\u03b41 DETC precursors, and potentially for V\u03b31V\u03b46 \u03b3\u03b4NKT cells. Ligand encounter does not seem to positively select the TCR repertoire, but instead divert the effector fate towards IFN\u03b3 production. Later on, a second wave of T cell progenitors enters the thymus; these differentiate towards \u03b3\u03b4IFN cells upon TCR signaling but are no longer prone to differentiate towards a default \u03b3\u03b417 fate, instead generating na\u00efve, uncommitted \u03b3\u03b4 T cells that can display adaptive\u2010like behavior in the periphery.In sum, there may be two separate waves of \u03b3\u03b4 T cell development that differ substantially with regard to thymic precursors and dependence on TCR signaling. The first wave in the fetus is made up by a set of progenitor cells whose origin is not yet fully elucidated, but some reports suggest they may arise in the yolk sac.Next, we will focus on the current view of the molecular mechanisms that underlie the acquisition and maintenance  of \u03b3\u03b4 T cell effector functions.4+ T helper cells.The acquisition of the capacity to secrete IFN\u03b3 and IL\u201017 during thymic development distinguishes \u03b3\u03b4 T cells from their \u03b1\u03b2 T helper cell counterparts, whose effector cell differentiation  occurs in peripheral lymphoid organs upon activation in specific inflammatory milieus.Some of the key questions addressed over the past decade have been:How peripheral \u03b3\u03b4 T cell subsets expressing IFN\u03b3 or IL\u201017 compare at the molecular level to their Th1 and Th17 counterparts, respectively;Whether the molecular determinants identified in peripheral \u03b3\u03b4 T cell subsets are imprinted during thymic development;What are the relative contributions of epigenetic, transcriptional and post\u2010transcriptional (including microRNA\u2010based) mechanisms to the regulation of IFN\u03b3 and IL\u201017 expression in effector \u03b3\u03b4 T cell subsets.In the next sections, we review the currently available data that provide some answers to these broad and still outstanding questions.4.1+ T helper cell subsets, where genes encoding transcription factors exhibit a large spectrum of epigenetic marks and allow for functional plasticity.Epigenetic mechanisms operating at the chromatin level control the maintenance of transcriptional networks ensuring autonomous maintenance of lineage phenotypes in differentiated cells, even through mitotic divisions.+ and IL\u201017+ \u03b3\u03b4 T cells, we isolated CD27+ (\u03b3\u03b427+) and CD27\u2212 (\u03b3\u03b427\u2212) \u03b3\u03b4 T cells, respectively, from peripheral organs of C57BL/6 mice and subjected them to Chip\u2010seq (chromatin immunoprecipitation followed by deep sequencing) analysis of activating H3K4me2 and repressive H3K27me3 marks.+ vs \u03b3\u03b427\u2212 T cell subsets were not segregating between Th1 and Th17 cells, suggesting that lineage\u2010specific mechanisms operate in \u03b3\u03b4 T cell differentiation.+T cells,\u2212 T cells but not in \u03b3\u03b427+ T cells nor in either CD4+ helper T subset.\u2212/\u2212\u03b3\u03b427\u2212 T cells were enriched for IL\u201017 producers when compared with wildtype controls, although the underlying mechanism is yet to be clarified.Recurring to CD27 levels to segregate IFN\u03b3Bcl11b, Id3 or Etv5) and survival  displayed, as expected, very similar histone marking in \u03b3\u03b427+ and \u03b3\u03b427\u2212 T cells.Ifn\u03b3 and its transcriptional regulators Tbx21, Eomes and Hlx, were all \u201cprimed\u201d for expression in both \u03b3\u03b4 T cell subsets  between the two subsets, whereas Ifng was only mildly (<10\u2010fold) higher in \u03b3\u03b427+ cells compared to \u03b3\u03b427\u2212 cells. Importantly, the key epigenetic and transcriptional signatures observed in peripheral \u03b3\u03b4 T cell subsets were also present in their thymic counterparts,Our detailed analysis of chromatin marks in genes involved in global \u03b3\u03b4 T cell biology revealed that those implicated in \u03b3\u03b4 T cell development  in highly inflammatory settings, ie, upon stimulation with high amounts of IL\u20101\u03b2 and IL\u201023, or in an ovarian cancer microenvironment.+ T cells, IL\u201017\u2010producing \u03b3\u03b427\u2212 cells are endowed with functional plasticity, which can be deployed (to co\u2010express IFN\u03b3) under quite specific conditions. Subsequent work from our group showed that the polyfunctional \u03b3\u03b427\u2212 (\u03b3\u03b417) T cell population can also secrete IL\u201017F, IL\u201022 and GM\u2010CSF upon IL\u20101\u03b2 and IL\u201023 stimulation,+ T cells, Th17 cells are substantially more prone to plasticity than their Th1 or Th2 counterparts. Thus, Th17 cells are able to transdifferentiate into Th1, regulatory T (Treg) and follicular helper T (Tfh) cells,Tbx21 or Gata3, also underlying their functional plasticity.Building on these molecular analyses, we challenged the functionalities of \u03b3\u03b4 T cell subsets in vitro and in vivo. We found that while \u03b3\u03b4274.2Following the genome\u2010wide analysis of \u03b3\u03b4 T cell subsets,+ T cells both in vitro and in vivo upon infection with murid herpes viruses, while also impairing IFN\u03b3 production by \u03b3\u03b427\u2212CCR6+ T cells stimulated with IL\u20101\u03b2 or IL\u201023 in vitro or during Listeria infection in vivo.\u2212/\u2212 \u03b3\u03b427+ cells compared to controls,T\u2010bet deficiency was found to significantly reduce IFN\u03b3 expression by peripheral \u03b3\u03b427Rorc) deficiency completely abolished the production of IL\u201017 by \u03b3\u03b427\u2212\u00a0T cells.+ T cells, IL\u201017 production had been shown to be co\u2010regulated by the auxiliary transcription factors ROR\u03b1 and BATF,On the other hand, ROR\u03b3t (+ and V\u03b34+), the IFN\u2010\u03b3 producers  and DETCs (V\u03b35+), with Rorc, Maf,\u00a0Sox13, and\u00a0Sox4 associated with the IL\u201017A producers; and Tcf7\u00a0(TCF\u20101),\u00a0Lef1, Tbx21\u00a0(T\u2010bet) and Eomes\u00a0with the IFN\u03b3 producers. These results were confirmed in additional studies that showed, for example, that Sox13, Sox4, and the Ets family member ETV5, are key regulators of the development of \u03b3\u03b417\u00a0T cells.Of relevance, in \u03b3\u03b4 T cells, an additional and critical layer of TF\u2010mediated regulation occurs during thymic development,Sox13, Maf and Rorc knockout mice, Sagar and colleagues have shown a sequential activation of these factors during both fetal and adult \u03b3\u03b417 cell differentiation.+ Rorc+ Il17a+ Il17f+ \u03b3\u03b4 T cells in the fetal thymus, and displayed reduced levels of Maf, Blk and Roc in \u03b3\u03b4 T cells from adult thymus, whereas Maf\u2010deficient fetal thymi lacked Rorc+ Il17a+ \u03b3\u03b4 T cells, and Rorc\u2010deleted \u03b3\u03b4 T cells did not show reduced Sox13 or Maf expression. Thus, during thymic \u03b3\u03b417 cell development, Sox13 acts upstream of c\u2010MAF which is essential for ROR\u03b3t function in orchestrating the \u03b3\u03b417 program.A recent study has added an important temporal dimension to the role of some of these transcription factors regulating thymic \u03b3\u03b417 cell differentiation. Upon performing single\u2010cell analyses of + T helper cell differentiation. This is the case for BATF and IRF4, which promote opening of chromatin at Th17 cell\u2010specific loci, allowing access to ROR\u03b3t.+ \u03b3\u03b4 T cells has not been addressed, STAT3 was shown to be dispensable for the generation of IL\u201017\u2010producing \u03b3\u03b4 T cells.A more detailed discussion on the transcriptional networks operating in effector \u03b3\u03b4 T cell subsets, especially during thymic pre\u2010programming, is provided by Anderson and colleagues elsewhere in this issue. Of note, an important regulatory function of TFs is to act as chromatin remodelers, mostly as promoters of open chromatin, thus leaving specific T cell loci accessible to other TFs/regulatory factors. Although, so far, no studies have specifically addressed this issue in \u03b3\u03b4 T cells, several TFs have been implicated in chromatin remodelling during CD4Other issues that require further elucidation are (i) which extracellular cues, including potential TCR ligands, may feed in and regulate the transcriptional programs required for effector \u03b3\u03b4 T cell differentiation; and (ii) how the transcriptional networks described separately for thymic and peripheral \u03b3\u03b4 T cells are integrated, and potentially cross\u2010talk, within the cell. Future studies, based on single\u2010cell analysis, will likely contribute to improving our understanding of these phenomena.4.3Our group has been recently addressing the role of non\u2010coding RNAs, with special attention to microRNAs, in effector \u03b3\u03b4 T cell differentiation.+ T cells show reduced proliferation and survival after in vitro stimulation, but increased frequencies of IFN\u03b3 producers, implicating miRNAs in T helper cell differentiation.MicroRNAs (miRNAs) constitute a fundamental layer of post\u2010transcriptional regulation, acting as negative regulators of expression for most mammalian genes by promoting the degradation of mRNAs or preventing their translation.Interestingly, the development of \u03b3\u03b4 T cells is not impaired by miRNA ablation; on the contrary, there is a substantial increase of \u03b3\u03b4 T cells in the double negative thymic compartment of mice conditionally lacking Dicer in early thymocytes.\u2212 cells  and in vivo, upon Listeria monocytogenes infection.More recently, our group identified miR\u2010146a as functionally relevant for \u03b3\u03b4 T cell differentiation, and an important determinant of the limited functional plasticity of \u03b3\u03b427\u2212V\u03b34\u2212/V\u03b36+ \u03b3\u03b4 T cells) compared to \u03b3\u03b427+T cells, in accordance with the expected inverse correlation between the levels of a given miRNA and its target mRNA. In fact, the high expression of Nod1 in \u03b3\u03b427+ T cells is consistent with Nod1\u2010mediated promotion of IFN\u03b3 production in CD4+ and CD8+ T cells, given that Nod1\u2010deficient \u03b1\u03b2 T cells have impaired IFN\u03b3 responses in vivo.\u2212/\u2212 \u03b3\u03b427\u2212 cells were unable to differentiate into IL\u201017+ IFN\u03b3+ double producers, in opposition to the phenotype (accumulation of IL\u201017+ IFN\u03b3+ cells) of miR\u2010146a\u2212/\u2212 \u03b3\u03b427\u2212 cells .\u2212/\u2212 with Nod1\u2212/\u2212 mice  revealed that heterozygous Nod1 reduction (in miR\u2010146\u2212/\u2212 Nod1+/\u2212 mice) prevented the accumulation of double producers among \u03b3\u03b427\u2212 T cells observed in miR\u2010146\u2212/\u2212 Nod1+/+ mice. These results collectively indicate that Nod1 is the key miR\u2010146a target implicated in the regulation of \u03b3\u03b427\u2212 cell plasticity. Of note, until now, only one other miRNA has been implicated in T helper cell plasticity, miR\u201010a, which restricts regulatory CD4+ T cells from acquiring Th17 and follicular helper T cell characteristics.A differential Argonaute\u20102 (Ago2) RIP\u2010seq strategy allowed us to identify Nod1 as a novel target of miR\u2010146a in \u03b3\u03b4 T cells,In sum, we have shown that miR\u2010146a limits \u03b3\u03b4 T cell plasticity by targeting Nod1, an intracellular pattern recognition receptor that is an important mediator of endoplasmic reticulum (ER) stress\u2010induced production of pro\u2010inflammatory cytokines,4.4Peripheral effector \u03b3\u03b4 T cell responses rely either on the activation and expansion of thymically  pre\u2010programmed cells; or on activation and de novo differentiation of effectors from na\u00efve \u03b3\u03b4 T cells exported from the adult thymus.\u2212CD27 pathway, besides critical during \u03b3\u03b4 thymocyte development,+ \u03b3\u03b4 T cell responses via expansion of \u03b3\u03b427+ cells.+ cells, thus controlling their responses to viral and parasitic infections in vivo.Plasmodium berghei infection, which contrasted with the \u03b3\u03b4IFN\u2010specific effect of CD27.Our group has shown that the CD70+ V\u03b31+ T cells, which protected mice from Listeria infection in an IFN\u03b3\u2010dependent manner.+ V\u03b36+ T cells in mucosal tissues in the steady\u2010state and upon Listeria infection, which associated with reduced bacterial clearance, and could be rescued upon administration of an agonist anti\u2010CD30 antibody.Other \u201ccostimulatory\u201d (or inhibitory) receptors reported to differentially impact on \u03b3\u03b4IFN and \u03b3\u03b417 T cells are CD137, CD30, BTLA and PD\u20101. Agonist anti\u2010CD137 (4\u20101BB) antibodies promoted the expansion of IFN\u03b3\u2212 cells in vitro, even in the absence of TCR stimulation.\u2212 T cells,Another important molecular layer of \u03b3\u03b4 T cell activation and differentiation are cytokines, including usual suspects like IL\u20102Finally, IL\u20101\u03b2 and especially IL\u201023 can also drive de novo differentiation of \u03b3\u03b417 cells from na\u00efve peripheral \u03b3\u03b4 T cells. While the acquisition of the IL\u201017\u2010producing capacity of \u201cnatural\u201d or thymic \u03b3\u03b417 T cells occurs exclusively during fetal/ perinatal development,5This review focused on several layers of regulation of mouse effector \u03b3\u03b4 T cell differentiation, while highlighting our group's main contributions. Despite the significant progress made over the past decade, various issues remain incompletely understood.The signals involved in dictating effector cell commitment continue to be an area of intensive research, now benefiting from full transcriptomic comparisons and pseudotime alignments of single thymocytes through the use of single\u2010cell RNA sequencing approaches. These may allow the dissection of novel TCR\u2010dependent vs independent thymic \u03b3\u03b4 T cell developmental pathways, as well as resolve the contradiction of having low or no TCR signalling in \u03b3\u03b417 T cell development within a \u03b3\u03b4 T cell lineage promoted (at the \u03b1\u03b2/\u03b3\u03b4 bifurcation) by strong TCR signals. Moreover, such single\u2010cell approaches will permit further validation of the proposed model suggesting that \u03b3\u03b417 and \u03b3\u03b4IFN T cells arise from distinct thymic progenitors.bona fide ligands remains a priority,Besides the TCR, for which the identification of il17a\u2010GFP Ifng\u2010YFP), which we are using to define the full mRNA and miRNA transcriptomes of \u201cpure\u201d effector \u03b3\u03b4 T cell subsets. By combining with single\u2010cell technologies it will be possible to enquire the potential heterogeneity even within IL\u201017\u2010 or IFN\u03b3\u2010expressing populations.Following the seminal research performed on the basis of surface markers or TCR V\u03b3 chain usage, we believe future studies should employ cytokine reporter mice to isolate pure populations of IL\u201017\u2010 or IFN\u03b3\u2010expressing cells, so that cellular and molecular properties can be directly associated to effector functions (within heterogeneous \u03b3\u03b4 T cell subsets). We have generated mice with reporter gene markers for both cytokines .Tcrd\u2010Cre showing limitations to delete genes in most \u03b3\u03b4 T cell populations (except DETC).Another critical aspect will be how to best assess the function of particular genes within the \u03b3\u03b4 T cell lineage. Indeed, a major limitation of our (and other group's) studies so far is the analysis of \u03b3\u03b4 T cell phenotypes in full KO animals, rather than having specific gene ablation in \u03b3\u03b4 T cells. Although some promoters, such as Upcoming research should further elucidate the extracellular signals that drive peripheral effector \u03b3\u03b4 T cell responses. A current view that requires more experimental support is that \u03b3\u03b417 cells respond to innate signals, whereas \u03b3\u03b4IFN cells may be involved in adaptive\u2010like responses, including antigen specificity . This is an outstanding topic also in human \u03b3\u03b4 T cell biology.Finally, while this review was focused on IL\u201017 and IFN\u03b3 production as hallmark effector functions of murine \u03b3\u03b4 T cells, one should highlight their versatility in mice and humans: besides being highly cytotoxic, as acknowledged for decades and currently being explored for cancer immunotherapy,The authors declared no conflicts of interest."}
{"text": "Two correlated sources emit a pair of sequences, each of which is observed by a different encoder. Each encoder produces a rate-limited description of the sequence it observes, and the two descriptions are presented to a guessing device that repeatedly produces sequence pairs until correct. The number of guesses until correct is random, and it is required that it have a moment (of some prespecified order) that tends to one as the length of the sequences tends to infinity. The description rate pairs that allow this are characterized in terms of the R\u00e9nyi entropy and the Arimoto\u2013R\u00e9nyi conditional entropy of the joint law of the sources. This solves the guessing analog of the Slepian\u2013Wolf distributed source-coding problem. The achievability is based on random binning, which is analyzed using a technique by Rosenthal. X is drawn from a finite set X equal to x?\u201d until the guess is correct. The guessing order is determined by a guessing function G, which is a bijective function from G proceeds as follows: the first guess is the element X. Ar\u0131kan , then  follows \u00b7a+b+c3p\u22644p\u00b7ap+bpLemma\u00a06.Let a, b, c, and d be nonnegative real numbers. Then, for all Proof.\u00a0If 1], then  follows Lemma\u00a07\u00a0.Let Proof.\u00a0This is a special case of  p\u22121]\u2264E[X]\u00b72p\u2212E[Xp\u22121])\u22642p\u22121E[X]We now consider two cases depending on which term on the RHS of  achieves implies  because p, then:E[Xp]\u22642pE[X]p,so  holds alIn this section, we prove a nonasymptotic and an asymptotic converse result .Theorem\u00a02.Let Proof.\u00a0We view  as three because:E\u2264\u2211x,yP\u03d5g(y\u2032)\u03c1=\u2211x,yP\u03c1]\u22641+4\u03c1\u2211x,y])with:\u03c8=)\u03d5g(y\u2032),\u03b23=\u03b231|U|\u03c1\u2264\u2211x,yP\u22642\u2212\u03c1\u03f5,wh]\u03c1\u02dc; and . In the \u2212\u03c1\u03f5.From , 82), \u22642\u2212\u03c1\u03f5.Frotisfying .We now consider  when \u03c1>1e did in . Insteade RHS of  by:93)\u03c1>1. Unlips as in \u201382),\u2211xx, y, f, and g is implicit in our notation.Bounding e RHS of , is moreosenthal  (Proof og(y\u02dc)\u03c1\u22121=Ef,g\u2211x\u2032\u22602+\u03b33)\u03c1\u22121=\u2211x\u2032\u2260x,y\u20322+\u03b33)\u03c1\u22121\u2264\u2211x\u2032\u2260x,y\u20322+\u03b23)\u03c1\u22121\u2264\u2211x\u2032\u2260x,y\u2032g[\u03b41\u03c1\u22121]=4\u03c1\u22121{+\u2211x]}with:\u03b31=\u03b31\u03d5f(x\u02dc),\u03b42=\u03b42\u2211xWith the help of  and 120120, we nTo prove this, we consider four cases depending on which term on the RHS of  achievesum, then  holds bee LHS of  achievese LHS of  follows e LHS of  because e LHS of  achievese LHS of  follows \u03b23]\u03c1,so  holds alHaving established , we now We now study the terms on the RHS of  separatee RHS of ,(127)\u2211x\u03c1>1,\u2211x,yP\u00b7\u03c1\u22121\u03c1\u2264\u2211x,yP1\u03c1x,y)]\u03c1\u02dc; \u201390) and and\u03c1>1,x,y)]\u03c1\u02dc; \u201382). In. In\u03c1>1,x,y)]\u03c1\u02dc; :(135)\u2211xFrom , 127), , 127), , 135), , 135), aFinally, , (99), . Then, \u20138) hold hold(RX,n, there exist encoders Using Theorem 3 with n tends to infinity, the RHS of (n tends to infinity, which implies that the rate pair Because e RHS of  tends to"}
{"text": "We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schr\u00f6dinger equation as a model example, we show that the solution of the initial value problem on the circle can be expressed in terms of the solution of a Riemann\u2013Hilbert problem whose formulation involves quantities which are defined in terms of the initial data alone. Our approach provides an effective solution of the problem on the circle which is conceptually analogous to the solution of the problem on the line via the inverse scattering transform. In \u00a7In \u00a7(c)k-plane will be denoted by Dj, j\u2009=\u20091, \u2026, 4, and \u03a3 will denote the contour f on a contour from the left and right will be denoted by f+ and f\u2212, respectively. We will use A will be denoted by [A]1 and [A]2, respectively.The four open quadrants of the complex 2.Before turning to the periodic problem, we recall some aspects of the analysis of the NLS equation on a finite interval presented in , which w\u03bc is a 2\u2009\u00d7\u20092-matrix-valued eigenfunction, the matrices Q and q of  is a smooth solution of \u2009\u2208\u2009\u2009\u00d7\u2009, where 0\u2009<\u2009L\u2009>\u2009\u221e and 0\u2009<\u2009T\u2009<\u2009\u221e is some fixed final time. Following \u2009\u00d7\u2009. The solution q can be obtained from m via the relationMoreover, m obeys the symmetriesP of poles is empty, then uniqueness follows by standard considerations because the jump matrix has unit determinant. The problem with a non-empty set P can be transformed into a problem for which P is empty. Indeed, if the set \u03b7 and \u03be can be regularized in the standard way; see, e.g. 1\u2009=\u2009A[\u03bc2]1/\u03b1 has at least a simple zero at kj and it then follows from (m]2 has (at most) a simple pole at kj. This proves  as k\u2009\u2192\u20090. The behaviour of m as k\u2009\u2192\u2009\u221e is a consequence of   m and thmmetries  also impuence of . Finallyuence of  follows uence of .\u2003\u25aa4.m depends on the final time T via the function \u0393\u2009=\u2009B/A. However, the solution q is independent of T. This suggests that it should be possible to eliminate the T dependence from the RH problem 3.3. In this section, we define a new T-independent RH problem\u2014henceforth called the RH problem for m. The basic idea of this deformation is to replace \u0393 by a new T-independent function a(k) and b(k) alone, this will lead to our main result.The RH problem 3.3 for (a)q of the NLS equation on the interval  whose boundary values have decay as t\u2009\u2192\u2009\u221e. In this case, it can be shown that the functions A, B, T\u2009\u2192\u2009\u221e; we denote these limits by T-independent function x-periodic initial value problem.As motivation for the definition of the RH problem for relation  leads toWe first need to give a careful definition of  root in  is fully(b)k) is given byWe define the function The function fined in  is relat2:k) is the trace of the so-called monodromy matrix and features heavily in the classical approach to the x-periodic problem. In that framework, 1 and is well studied. In what follows, we collect some well-known facts about In order to make the definition  of \u0393~ pree, e.g. , and defq0\u2009\u2261\u20090, we have a\u2009=\u20091, thus \u0394(k)\u2009=\u20092cos (kL), and hence 4\u2009\u2212\u2009\u03942\u2009=\u20094sin 2(kL) has a double zero at each of the points n\u03c0/L, 2 has a similar structure for large k for any initial datum q0\u2009\u2208\u2009L2 in the following sense: For \u03c0/(4L) centred at n\u03c0/L. Then, there is an integer N\u2009>\u20090 such that, counted with multiplicity, 4\u2009\u2212\u2009\u03942 has exactly two roots in each disk n|\u2009>\u2009N and exactly 4N\u2009+\u20092 roots in the disk {|k|\u2009<\u2009N\u03c0/L\u2009+\u2009\u03c0/(4L)}. Using this result and employing the lexicographic ordering of complex numbersn|. The symmetry \u03bb\u2009=\u20091 and \u03bb\u2009=\u2009\u22121 separately.In the case of vanishing initial datum \u03bb\u2009=\u20091), the periodic spectrum nth spectral gap whenever it is non-empty. For \u03bb\u2009=\u20091, we define nth interval n|.In the defocusing case (i.e. \u03bb\u2009=\u2009\u22121), z1, z2) denote the open straight-line segment from z1 to z2, i.e.N\u2009>\u20090 be as in the Counting Lemma so that there are 4N\u2009+\u20092 roots counted with multiplicity in the disk {|k|\u2009<\u2009N\u03c0/L\u2009+\u2009\u03c0/(4L)}. An even number, say 2M, of these 4N\u2009+\u20092 roots have odd multiplicity; let kL)\u2009+\u2009o(1) as k\u2009\u2192\u2009\u00b1\u221e, an even number, say k|\u2009<\u2009N\u03c0/L\u2009+\u2009\u03c0/(4L)} ordered lexicographically:k\u2009=\u2009constant. We define n|\u2009>\u2009N:In the focusing case 2 in  is singl\u2009\u221e in R,  implies hat (see  for a mofined by  with the(c)g(k) bygj denotes the restriction of g to Dj for j\u2009=\u20091, \u2026, 4. We introduce m is the solution of RH problem 3.3. The function g is defined in such a way that v except that \u0393 is replaced by v on \u03bb\u2009=\u2009\u22121, then j\u2009=\u20091, \u2026, 4. It is quite remarkable that all these jumps can be expressed completely in terms of a and b alone.Define the function pression  as v exc2 has roots of even multiplicity. Let Let k approaches where m, in general, has singularities at the poles of \u0393, \u03b7 and \u03be, it turns out that Whereas the function We assume the following:\u2014has no poles on the contour\u2014In D1, has at most finitely many polesand these poles are all simple.\u2014In D3, has at most finitely many polesand these poles are all simple.k0 is a zero of k tends to infinity in k0, we see that k0 to the same order as k near k0. It follows, in particular, that Regarding assumption 4.2, we note that there exist large families of initial conditions for which it can be shown explicitly that uence of ). Thus, Let \u2014\u2014m as k approaches The limits of \u2014k\u2009\u2192\u2009\u221e, \u03c0/(4L) centred at n\u03c0/L defined in  are nowhere zero.In order to show that finition  of the mThe following identities are valid:In particular, the functionis non-zero for allk, then \u03942\u2009=\u2009\u03942\u2009\u2212\u20094 at k, which is a contradiction. Thus The identities follow by a direct computation using the definitions  and 4.44.4 of \u0393~(d)q of the x-periodic NLS equation in terms of the solution of the RH problem 4.4. Since the formulation of this RH problem only involves quantities defined in terms of the initial datum, the theorem provides an effective solution of the IVP for the x-periodic NLS equation.The following theorem, which is the main result of the paper, provides an expression for the solution Suppose q is a smooth solution of (1.1) forwhich is x-periodic of period L\u2009>\u20090, i.e. q\u2009=\u2009q. Define a(k) and b(k) by (2.3) and letbe the function defined in terms of a and b by (4.3). Suppose assumption 4.2 holds.Then the RH problem 4.4 has a unique solutionfor each \u2009\u2208\u2009\u2009\u00d7\u2009\u2009\u00d7\u2009[0, \u221e). Choose T\u2009\u2208\u2009 and define m and g defined using T as final time. We will show that x, t).Fix \u2013(3.12) to show that \u03bcj as its starting point.Let us show that \u03bcj as follows:In light of the symmetries , it is einitions  and  as k\u2009\u2192\u2009\u221e provided that we can show thatg, it is enough to prove for j\u2009=\u20091, 2 thatk\u2009\u2192\u2009\u221e in the closed upper half-plane It only remains to show that symmetry  of g, itstimates  imply4.a\u2009=\u20091\u2009+\u2009O(k\u22121) and k\u2009\u2192\u2009\u221e in The estimates  also impombining  and (4.3g1. As k\u2009\u2192\u2009\u221e in O(k\u22121) and g1. As for the non-zero off-diagonal elementB/A givesx\u2009\u2264\u2009L and T\u2009>\u2009t, this yieldsj\u2009=\u20091.Let us consider her with , this yishowing  for the relation  for \u0393\u2009=\u2009ent (cf. )4\u2212\u03942\u22124\u03bb\u2208ZDn.By , 2.7), , g1. As \u2208ZDn.By , 4.26)  g1. As k\u2208ZDn.By , we havetilizing , 4.28)  g1. As kproof of  for j\u2009=\u2009g2. We haveO(k\u22121) as x\u2009\u2265\u20090 and T\u2009>\u2009t, we conclude thatWe next consider nfinity,  givesa\u2212over, by , a\u2212\u03bbb\u0393~\u00afng as in , we deduproof of .g\u2009\u2212\u2009I is exponentially small as k\u2009\u2192\u2009\u221e along any ray Finally, by , 4.35),,4.35), aequation  then folequation . This co5.N is an integer and the constant q0\u2009>\u20090 can be taken to be positive due to the phase invariance of N be an integer. Direct integration of the x-part of the Lax pair (q given by (a and b:r(k) denotes the square roota, b and \u0394 are entire functions of k even though r(k) has a branch cut. The periodic spectrum 2\u2009=\u20094sin 2(Lr) and consists of the two simple zeros \u03bb\u00b1 defined by\u03bb\u2009=\u20091, then all zeros are real; if \u03bb\u2009=\u2009\u22121, then the zeros are real for |n|\u2009\u2265\u2009L q0/\u03c0 and non-real for |n|\u2009<\u2009L q0/\u03c0. The function r so that r(k)\u2009=\u2009k\u2009+\u2009\u03c0N/L\u2009+\u2009O(k\u22121) as k\u2009\u2192\u2009\u221e. Then, using  as k approaches (a)\u03bb\u2009=\u20091, thenIf 5.4) and .(b)\u03bb\u2009=\u2009\u22121, thenIf 5.4) and .The limits of lows see :(a)If \u03bb\u2014k\u2009\u2192\u2009\u221e, \u2014k\u2009\u2192\u2009{0,\u2009\u2212\u2009\u03c0N/L, \u03bb\u00b1}, Find a 2\u2009\u00d7\u20092-matrix valued function It is easy to verify that the jump matrices in RH problem 5.1 satisfy the following consistency conditions at the origin:(b)The RH problem 5.1 for \u03bb\u2212, \u03bb+). Let us orient the contour\u03bb\u2009=\u20091 and upward if \u03bb\u2009=\u2009\u22121. It follows that k\u2009\u2192\u2009\u221e, and that f(k)\u2009\u2261\u2009f is defined byThe jump matrices ng (5.4)\u2013 show thak) by performing another transformation. Define \u03b4(k)\u2009\u2261\u2009\u03b4 by\u03b4 satisfies the jump relation \u03b4+\u03b4\u2212\u2009=\u2009f on  and lim k\u2192\u221e\u03b4(k)\u2009=\u2009\u03b4\u221e, where\u03b4(k)\u2009=\u2009O((k\u2009\u2212\u2009\u03bb\u00b1)1/4) and \u03b4(k)\u22121\u2009=\u2009O((k\u2009\u2212\u2009\u03bb\u00b1)\u22121/4) as k\u2009\u2192\u2009\u03bb\u00b1. Consequently, k\u2009\u2192\u2009\u221e, (iii) k\u2009\u2192\u2009\u03bb\u00b1, and (iv) Q\u2009\u223c\u20091 as k\u2009\u2192\u2009\u221e. Since The jump matrix in  can be mq for all t. Indeed, . This gives\u03bb\u2009=\u20091, then the substitutions s\u2009=\u2009\u2212\u03c0N/L\u2009+\u2009\u03c3 and \u03c3\u2009=\u2009q0sin\u03b8 give\u03bb\u2009=\u2009\u22121, then the substitutions s\u2009=\u2009\u2212\u03c0N/L\u2009+\u2009i\u03c3 and \u03c3\u2009=\u2009q0sin\u03b8 yield\u03b4\u221e into  of \u2009=\u2009q0\u2005e\u03c0N/L) x of  correspo\u03bb\u2212, \u03bb+). More generally, whenever the Riemann surface defined by We have shown that the single exponential solutions  can be colutions  is the g"}
{"text": "Scientific Reports 10.1038/s41598-019-39832-9, published online 01 March 2019Correction to: \u03b8\u2009=\u2009\u03c0/4\u2019 should read \u2018\u03b8\u2009=\u2009\u03c0/2\u2019.This Article contains an error in the legend of Figure 5, where \u2018"}
{"text": "Scientific Reports 10.1038/s41598-020-69659-8, published online 29 July 2020Correction to: This Article contains errors in Table 1.In the column, \u201cCompany\u201d,\u201cIRE EliT \u201dshould read:\u201cIRE EliT \u201dIn the column, \u201cColumn matrix\u201d,\u201cUnspecified\u201dshould read:2\u201d\u201cTiO68Ga elution yield (%)\u201d,In the column, \u201cInitial \u201c\u2009>\u200965\u201dshould read:\u201c\u2009>\u200980\u201dAs a result, in the Introduction,\u20133%\u00a068Ge breakthrough18. According to its brochure, metal content per elution is less than 1\u00a0ppm and\u2009\u2264\u200910\u00a0\u00b5g/GBq of\u00a068Ga for Fe, Cu, Ni, Zn, Pb and Al. The Eckert & Ziegler GalliaPharm\u00ae\u00a0and the IRE ELiT Galli Eo\u00ae\u00a0generators are both GMP grade and have type II drug master files on file with the FDA.\u201d\u201cAnother emerging generator is Galli Eo ,\u00a0however, the column contains an unspecified resin. The generator is eluted with 0.1\u00a0M HCl, and has more than 67% elution yield and less than 1\u2009\u00d7\u200910should read:2. The generator is eluted with 0.1\u00a0M HCl, and has more than 67% elution yield and less than 1\u2009\u00d7\u200910\u20133%\u00a068Ge breakthrough18. According to its brochure, metal content per elution is less than 1\u00a0ppm and\u2009\u2264\u200910\u00a0\u00b5g/GBq of\u00a068Ga for Fe, Cu, Ni, Zn, Pb and Al. The Eckert & Ziegler GalliaPharm\u00ae\u00a0and the IRE ELiT Galli Eo\u00ae\u00a0generators are both GMP grade and have type II drug master files on file with the FDA. Recently, the IRE ELiT generator has been authorized in several European countries  and its initial elution yield has been improved .\u201d\u201cAnother emerging generator is Galli Eo  which is based on TiO"}
{"text": "Scientific Reports 10.1038/s41598-019-55074-1, published online 11 December 2019Correction to: This Article contains a repeated typographical error where the unit \u201c\u00b5m\u201d is incorrectly given as \u201cm\u201d or \u201cum\u201d and \u201c\u00b0\u201d is incorrectly given as \u201cdeg\u201d."}
{"text": "Scientific Reports 10.1038/s41598-019-42604-0, published online 17 April 2019.Correction to: The Article contains errors in Table 2, where the units used for components \u2018EGF\u2019 and \u2018FGF\u2019,\u201c\u03bcg/m\u201dshould read:\u201cng/ml\u201dAdditionally, the units used for component \u2018NRG-1\u2032,\u201cng.ml\u201dshould read:\u201cng/ml\u201dIn the same table, the \u2018Human colon epithelia\u2019 entry for WCM,\u201c\u2014\u201dshould read:\u201c0.5\u00d7\u201dIn addition, the \u2018Human colon carcinoma\u2019 entry for WCM,\u201c0.5\u00d7\u201dshould read:\u201c\u2014\u201d"}
{"text": "Correction to: BMC Infect Dishttps://doi.org/10.1186/s12879-019-4668-xIn the Abstract:ng/L vs. 0.250 ng/L, p = 0.005)\u2019 should be replaced with \u2018testosterone \u2019\u2018testosterone \u2019 should be replaced with \u2018testosterone \u2019\u2018testosterone :\u2018\u03bcg/mL\u2019 should be replaced with \u2018\u03bcg/dL\u2019unit of DHEA-S in Table\u00a04 (4th row):\u03bcg/mL\u2019 should be replaced with \u2018\u03bcg/dL\u2019\u2018After publication of the original article , we were"}
{"text": "Scientific Reports 10.1038/s41598-019-53693-2, published online 11 December 2019Correction to: This Article contains a typographical error in the Results and Discussion section under subheading \u2018Experimental analysis of the optical properties\u2019 where,\u201c{L}_{w}\u2009\u2009=\u2009\u20092.4\u2009nm\u201dshould read:w\u2009=\u20092.4\u2009nm\u201d\u201cL"}
{"text": "Scientific Reports 10.1038/s41598-019-54688-9, published online 24 December 2019Correction to: This Article contains errors in Table 4.Under column ESR age (ka) in scenario 1,\u201c1330\u00b113\u201dshould read:\u201c1330\u00b1134\u201dUnder column ESR age (ka) in scenario 2\u201c128\u00b195\u201dshould read:\u201c1285\u00b195\u201d(c)\u201d where,Finally, there is an error in the heading \u201c% of total dose rate(c)\u201d\u201c% of total dose rateshould read:(b)\u201d\u201c% of total dose rate"}
{"text": "Scientific Reports 10.1038/s41598-019-56215-2, published online 23 December 2019Correction to: This Article contains errors in the Methods sections under subheading \u2018MRI protocol\u2019.3, resolution\u2009=\u2009125 \u00d7\u2009125\u2009\u00d7\u2009372 \u00b5m3, TE/TR\u2009=\u200922/425\u2009ms, gradient pulse duration/diffusion time (\u03b4/\u0394)\u2009=\u20094/12\u2009ms, b-values\u2009=\u20092000 and 4000\u2009s/mm2, 20 diffusion directions, number of averaged excitations\u2009=\u20092, and scan time\u2009=\u20097\u2009hours and 40\u2009minutes for each b-value.\u201d\u201cImages were acquired with a three-dimensional diffusion-weighted spin-echo sequence and the following parameters: field-of-view\u2009=\u200960\u2009\u00d7\u200960\u2009\u00d7\u2009160\u2009mmshould read:3 (adjusted to fit sample), resolution\u2009=\u2009125\u2009\u00d7\u2009125\u2009\u00d7\u2009372 \u00b5m3, TE/TR\u2009=\u200922/425\u2009ms, gradient pulse duration/diffusion time (\u03b4/\u0394)\u2009=\u20094/12\u2009ms, b-values\u2009=\u20092000 and 4000\u2009s/mm2, 20 diffusion directions, number of averaged excitations\u2009=\u20092, and scan time\u2009\u2248\u20097-10\u2009hours for each b-value (depending on the field-of-view).\u201d\u201cImages were acquired with a three-dimensional diffusion-weighted spin-echo sequence and the following parameters: field-of-view \u2248 6\u2009\u00d7\u20096\u2009\u00d7\u200916\u2009mm"}
{"text": "Integral proteins can reach the plasma membrane via different routes. Here, Lucken-Ardjomande H\u00e4sler et al. identify three proteins that are associated with dynamic intracellular tubules, closely aligned with the ER, and involved in the transport of specific cargos whose export is particularly sensitive to ER stress. In addition to the classical pathway of secretion, some transmembrane proteins reach the plasma membrane through alternative routes. Several proteins transit through endosomes and are exported in a Rab8-, Rab10-, and/or Rab11-dependent manner. GRAFs are membrane-binding proteins associated with tubules and vesicles. We found extensive colocalization of GRAF1b/2 with Rab8a/b and partial with Rab10. We identified MICAL1 and WDR44 as direct GRAF-binding partners. MICAL1 links GRAF1b/2 to Rab8a/b and Rab10, and WDR44 binds Rab11. Endogenous WDR44 labels a subset of tubular endosomes, which are closely aligned with the ER via binding to VAPA/B. With its BAR domain, GRAF2 can tubulate membranes, and in its absence WDR44 tubules are not observed. We show that GRAF2 and WDR44 are essential for the export of neosynthesized E-cadherin, MMP14, and CFTR \u0394F508, three proteins whose exocytosis is sensitive to ER stress. Overexpression of dominant negative mutants of GRAF1/2, WDR44, and MICAL1 also interferes with it, facilitating future studies of Rab8/10/11\u2013dependent exocytic pathways of central importance in biology. In eukaryotic cells, the ER is the birthplace of the majority of membrane proteins, secreted proteins, and lipids. Despite the canonical belief that they follow the same route from the ER through the ER-Golgi intermediate compartment (ERGIC), Golgi, and TGN to reach the plasma membrane, differences between individual cargos exist. Lipids can be transferred between membranes at contact sites . Some prRabs are regulators of intracellular transport whose GTP-GDP cycle drives membrane trafficking processes forward. Rab8 controls the export of MMP14 , Rab11 mDrosophila plasmatocytes  can participate both in endocytic and exocytic routes. On the endocytic side, OPHN1 regulates clathrin- and Endophilin-dependent endocytosis in neuronal cells , while Gatocytes . Conversatocytes  and in catocytes ; GRAF1c atocytes ; GRAF1 aatocytes ; and GRAatocytes .Little is known about the molecular mechanisms governing GRAF1/2\u2013mediated pathways as among the direct binding partners identified , which contains the PPKPP motif  was reported to inhibit Rab11-mediated recycling of the Transferrin receptor . In agreWe have shown that WDR44 and MICAL1 bind to the SH3 domain of GRAF1/2. But even though the membrane-binding regions of GRAF1/2 (GRAF1/2 BAR-PH) did not colocalize with endogenous MICAL1, endogenous WDR44 was found on the same structures . Live imMICAL1 G3W, WDR44 \u0394C, GRAF2 BAR-PH, and GRAF1 BAR-PH are thus four dominant negative proteins affecting the normal functioning of different components of Rab8/10/11\u2013 and MICAL1/GRAF/WDR44\u2013mediated trafficking.To gain more insight into the identity of MICAL1/GRAF/WDR44 tubules, we first examined colocalization of endogenous WDR44 with dextran as a marker of clathrin-independent endocytosis. There was none . EndogenIn transfected HeLa cells, WDR44 tubules and patches colocalized with VAPA and VAPB . WDR44 tBy similarly with WDR44, GRAF1b/2 tubules colocalized with VAPA and VAPB . ColocalTo identify the compartments with which WDR44 tubules communicate, we used BFA, a drug that leads to the tubulation of certain organelles and the mixing of groups of intracellular compartments . IncubatpF1KA0819, and overexpression of MICAL-L1 inhibited GRAF1b/2 tubulation , a G protein\u2013coupled receptor that was reported to reach the plasma membrane in a Rab1-independent and Rab8-dependent manner -anchored protein GFP-GPI was not inhibited by overexpression of Rab8a T22N, Rab10 T23N, MICAL1 G3W, GRAF1/2 BAR-PH, or WDR44 \u0394C  and with the fact that we did not see endogenous MICAL1 tubules under resting conditions. Alternatively, other proteins might substitute for MICAL1, although we have shown that none of the other MICAL family members interact with GRAF1b/2. Contrasting with our original expectation, but in agreement with others (2B-AR and of a GPI-anchored protein are unaffected by incubation of the cells with thapsigargin. As inhibition of ER to Golgi transport invariably leads to ER stress (2)\u2013generating enzyme PIPKI\u03b3, something that was reported to be abolished by ER stress  a route inhibited by ER stress followed by E-cadherin, MMP14, CFTR, and CFTR \u0394F508 that may or may not use COPII-coated vesicles and at some stage depends on Rab8, Rab10, GRAF2, and WDR44; and (3) a stress-induced pathway followed by CFTR \u0394F508 that bypasses the Golgi and is also dependent on Rab10.In this study, we report the identification of three novel proteins\u2014GRAF2, WDR44, and MICAL1\u2014participating in the export of a subset of neosynthesized proteins. E-cadherin, MMP14, CFTR, and CFTR \u0394F508 were previously found to reach the plasma membrane in a Rab8- and/or Rab11-dependent manner . We now h others , we founR stress , we now R stress . NeverthIn recent years, proteins of the GRAF family have attracted attention because they are mutated in cancer  and haveOverexpression of Rab8 is known to induce a decrease in F-actin and focal adhesions in a RhoA-dependent manner . We now 2\u2013enriched microdomains of the plasma membrane ; pCI GRAF2res \u0394BAR-GFP was recombined from pDONR GRAF2res \u0394BAR C-tag ; pCI GRAF1b-GFP, pCI GRAF1b-RFP, and pCI GRAF1b-myc were recombined from pDONR GRAF1b C-tag (made from a PCR of a human brain cDNA library using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGA\u200bAGG\u200bAGA\u200bTAG\u200bAAC\u200bCAT\u200bGGG\u200bGCT\u200bCCC\u200bAGC\u200bGCT\u200bCGA\u200bG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bGAG\u200bGAA\u200bCTC\u200bCAC\u200bGTA\u200bATT\u200bCTC\u200bAGG\u200bG-3\u2032); pGex 4T2 GRAF1 SH3 was recombined from pDONR GRAF1 SH3 N-tag ; pCI GRAF1 BAR-PH\u2013GFP and pCI GRAF1 BAR-PH\u2013RFP were recombined from pDONR GRAF1 BAR-PH C-tag ; pCI myc-WDR44, pCI GFP-WDR44, and pCI RFP-WDR44 were recombined from pDONR WDR44 N-tag (made from a PCR of IMAGE 4837674 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGC\u200bGTC\u200bGGA\u200bAAG\u200bCGA\u200bCAC\u200bC-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bAAG\u200bATA\u200bCAT\u200bTTT\u200bTTC\u200bTTT\u200bTAT\u200bTAA\u200bCAA\u200bACA\u200bCTT\u200bTG-3\u2032); pCI WDR44-GFP and pCI WDR44-RFP were recombined from pDONR WDR44 C-tag (made from a PCR of IMAGE 4837674 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGA\u200bAGG\u200bAGA\u200bTAG\u200bAAC\u200bCAT\u200bGGC\u200bGTC\u200bGGA\u200bAAG\u200bCGA\u200bCAC\u200bC-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bAGA\u200bTAC\u200bATT\u200bTTT\u200bTCT\u200bTTT\u200bATT\u200bAAC\u200bAAA\u200bCAC\u200bTTT\u200bG-3\u2032); pCI GFP-WDR44 \u0394FFAT was recombined from pDONR WDR44 \u0394FFAT N-tag ; pCI GFP-WDR44 \u0394N1 was recombined from pDONR WDR44 \u0394N1 N-tag ; pCI GFP-WDR44 \u0394N2 and pGex 4T2 WDR44 \u0394N2 were recombined from pDONR WDR44 \u0394N2 N-tag ; pCI GFP-WDR44 \u0394C, pCI myc-WDR44 \u0394C, pCI RFP-WDR44 \u0394C, and pGex 4T2 WDR44 \u0394C were recombined from pDONR WDR44 \u0394C N-tag ; pCI GFP-WDR44 \u0394Pro was recombined from pDONR WDR44 \u0394Pro N-tag (deletion of aa 210\u2013257 of WDR44 by mutagenesis of pDONR WDR44 N-tag using 5\u2032-GGC\u200bCAC\u200bTTC\u200bTTC\u200bCAC\u200bAGC\u200bGG-3\u2032 and 5\u2032-AGA\u200bAAA\u200bAGG\u200bAAA\u200bAGC\u200bGAA\u200bTTG\u200bGAA\u200bTTT\u200bG-3\u2032); pCI RFP-WDR44res was recombined from pDONR WDR44res N-tag (mutagenesis of pDONR WDR44 N-tag using 5\u2032-GTG\u200bGCT\u200bTTG\u200bTGG\u200bAAC\u200bGAG\u200bGTA\u200bGAC\u200bGGT\u200bCAG\u200bACA\u200bAAA\u200bTTG\u200bATC-3\u2032 and 5\u2032-GAT\u200bCAA\u200bTTT\u200bTGT\u200bCTG\u200bACC\u200bGTC\u200bTAC\u200bCTC\u200bGTT\u200bCCA\u200bCAA\u200bAGC\u200bCAC-3\u2032); pCI RFP-WDR44res \u0394Pro was recombined from pDONR WDR44res \u0394Pro N-tag (mutagenesis of pDONR WDR44 \u0394Pro N-tag using 5\u2032-GTG\u200bGCT\u200bTTG\u200bTGG\u200bAAC\u200bGAG\u200bGTA\u200bGAC\u200bGGT\u200bCAG\u200bACA\u200bAAA\u200bTTG\u200bATC-3\u2032 and 5\u2032-GAT\u200bCAA\u200bTTT\u200bTGT\u200bCTG\u200bACC\u200bGTC\u200bTAC\u200bCTC\u200bGTT\u200bCCA\u200bCAA\u200bAGC\u200bCAC-3\u2032); pCI myc-MICAL1, pCI GFP-MICAL1, pCI RFP-MICAL1, and pGex 4T2 MICAL1 were recombined from pDONR MICAL1 N-tag (made from a PCR of IMAGE 5756097 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGC\u200bTTC\u200bACC\u200bTAC\u200bCTC\u200bCAC\u200bCAA\u200bCC-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bAGC\u200bCCT\u200bGGG\u200bCCC\u200bCTG\u200bTCC-3\u2032); pCI MICAL1-GFP was recombined from pDONR MICAL1 C-tag (made from a PCR of IMAGE 5756097 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGA\u200bAGG\u200bAGA\u200bTAG\u200bAAC\u200bCAT\u200bGGC\u200bTTC\u200bACC\u200bTAC\u200bCTC\u200bCAC\u200bCAA\u200bCC-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bGCC\u200bCTG\u200bGGC\u200bCCC\u200bTGT\u200bCC-3\u2032); pCI untagged MICAL1 (uMICAL1) was recombined from pDONR MICAL1 (made from a PCR of pDONR MICAL1 N-tag using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGA\u200bAGG\u200bAGA\u200bTAG\u200bAAC\u200bCAT\u200bGGC\u200bTTC\u200bACC\u200bTAC\u200bCTC\u200bCAC\u200bCAA\u200bCC-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bAGC\u200bCCT\u200bGGG\u200bCCC\u200bCTG\u200bTCC-3\u2032); pGex 4T2 MICAL1 \u0394N was recombined from pDONR MICAL1 \u0394N N-tag ; pCI GFP-MICAL1 \u0394MO was recombined from pDONR MICAL1 \u0394MO N-tag ; pCI GFP-MICAL1 \u0394MO\u0394CH and pGex 4T2 MICAL1 \u0394MO\u0394CH were recombined from pDONR MICAL1 \u0394MO\u0394CH N-tag ; pCI myc-MICAL1tail was recombined from pDONR MICAL1tail N-tag ; pCI myc-MICAL1Pro was recombined from pDONR MICAL1Pro N-tag ; pCI GFP-MICAL1 G3W, pCI RFP-MICAL1 G3W, and pCI myc-MICAL1 G3W were recombined from pDONR MICAL1 G3W N-tag (MICAL1 G91W G93W G96W by mutagenesis of pDONR MICAL1 N-tag using 5\u2032-CAA\u200bGTG\u200bCCT\u200bGGT\u200bGGT\u200bGTG\u200bGGC\u200bTTG\u200bGCC\u200bTTG\u200bCTG\u200bGCT\u200bGCG\u200bGGT\u200bCGC\u200bTGT\u200bGG-3\u2032 and 5\u2032-CCA\u200bCAG\u200bCGA\u200bCCC\u200bGCA\u200bGCC\u200bAGC\u200bAAG\u200bGCC\u200bAAG\u200bCCC\u200bACA\u200bCCA\u200bCCA\u200bGGC\u200bACT\u200bTG-3\u2032); pCI uMICAL1 G3W was recombined from pDONR MICAL1 G3W (mutagenesis of pDONR MICAL1 using 5\u2032-CAA\u200bGTG\u200bCCT\u200bGGT\u200bGGT\u200bGTG\u200bGGC\u200bTTG\u200bGCC\u200bTTG\u200bCTG\u200bGCT\u200bGCG\u200bGGT\u200bCGC\u200bTGT\u200bGG-3\u2032 and 5\u2032-CCA\u200bCAG\u200bCGA\u200bCCC\u200bGCA\u200bGCC\u200bAGC\u200bAAG\u200bGCC\u200bAAG\u200bCCC\u200bACA\u200bCCA\u200bCCA\u200bGGC\u200bACT\u200bTG-3\u2032); pCI myc-MICAL1 PPAPP, pCI GFP-MICAL1 PPAPP, and pCI RFP-MICAL1 PPAPP were recombined from pDONR MICAL1 PPAPP N-tag (MICAL1 K832A by mutagenesis of pDONR MICAL1 N-tag using 5\u2032-GGA\u200bGCC\u200bTCC\u200bACC\u200bCGC\u200bGCC\u200bTCC\u200bCCG\u200bCAG\u200bC-3\u2032 and 5\u2032-GCT\u200bGCG\u200bGGG\u200bAGG\u200bCGC\u200bGGG\u200bTGG\u200bAGG\u200bCTC\u200bC-3\u2032); pCI RFP-Rab8a, pCI GFP-Rab8a, and pCI myc-Rab8a were recombined from pDONR Rab8a N-tag (made from a PCR of pEGFP cRab8 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGC\u200bGAA\u200bGAC\u200bCTA\u200bCGA\u200bTTA\u200bCCT\u200bG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bACA\u200bGAA\u200bGAA\u200bCAC\u200bATC\u200bGGA\u200bAAA\u200bAGC-3\u2032); pCI RFP-Rab8a T22N and pCI GFP-Rab8a T22N were recombined from pDONR Rab8a T22N N-tag (mutagenesis of pDONR Rab8a N-tag using 5\u2032-GGG\u200bGGT\u200bGGG\u200bGAA\u200bGAA\u200bCTG\u200bTGT\u200bCCT\u200bGTT\u200bCC-3\u2032 and 5\u2032-GGA\u200bACA\u200bGGA\u200bCAC\u200bAGT\u200bTCT\u200bTCC\u200bCCA\u200bCCC\u200bCC-3\u2032); pCI RFP-Rab8a Q67L and pCI GFP-Rab8a Q67L were recombined from pDONR Rab8a Q67L N-tag (made from a PCR of pEGFP cRab8 Q67L using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGC\u200bGAA\u200bGAC\u200bCTA\u200bCGA\u200bTTA\u200bCCT\u200bG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bACA\u200bGAA\u200bGAA\u200bCAC\u200bATC\u200bGGA\u200bAAA\u200bAGC-3\u2032); pCI RFP-Rab8b and pCI GFP-Rab8b were recombined from pDONR Rab8b N-tag (made from a PCR of IMAGE 4701429 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGC\u200bGAA\u200bGAC\u200bGTA\u200bCGA\u200bTTA\u200bTCT\u200bC-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bAAA\u200bGTA\u200bGCG\u200bAGC\u200bAAC\u200bGAA\u200bAG-3\u2032); pCI RFP-Rab10 and pCI GFP-Rab10 were recombined from pDONR Rab10 N-tag (made from a PCR of a 293T cDNA library using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGC\u200bGAA\u200bGAA\u200bGAC\u200bGTA\u200bCGA\u200bCCT\u200bGC-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bAGC\u200bAGC\u200bATT\u200bTGC\u200bTCT\u200bTCC\u200bAGC\u200bC-3\u2032); pCI GFP-Rab10 T23N and pCI RFP-Rab10 T23N were recombined from pDONR Rab10 T23N N-tag (mutagenesis of pDONR Rab10 N-tag using 5\u2032-GGA\u200bGTG\u200bGGG\u200bAAG\u200bAAC\u200bTGC\u200bGTC\u200bCTT\u200bTTT\u200bC-3\u2032 and 5\u2032-GGA\u200bATC\u200bCCC\u200bGAT\u200bCAG\u200bGAG\u200bCAG\u200bC-3\u2032); pCI GFP-Rab10 Q68L was recombined from pDONR Rab10 Q68L N-tag (mutagenesis of pDONR Rab10 N-tag using 5\u2032-GGG\u200bATA\u200bCAG\u200bCAG\u200bGCC\u200bTGG\u200bAGC\u200bGAT\u200bTTC\u200bAC-3\u2032 and 5\u2032-ATA\u200bTCT\u200bGTA\u200bGCT\u200bTGA\u200bTCT\u200bTCT\u200bTTC\u200bC-3\u2032); pCI GFP-Rab11a and pCI RFP-Rab11a were recombined from pDONR Rab11a N-tag (PCR of IMAGE 3510339 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGG\u200bCAC\u200bCCG\u200bCGA\u200bCGA\u200bCGA\u200bG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bAGA\u200bTGT\u200bTCT\u200bGAC\u200bAGC\u200bACT\u200bGCA\u200bCC-3\u2032); pCI GFP-Rab11a S25N, and pCI RFP-Rab11a S25N were recombined from pDONR Rab11a S25N N-tag (mutagenesis of pDONR Rab11a N-tag using 5\u2032-GAG\u200bATT\u200bCTG\u200bGTG\u200bTTG\u200bGAA\u200bAGA\u200bATA\u200bATC\u200bTCC\u200bTGT\u200bCTC\u200bG-3\u2032 and 5\u2032-CGA\u200bGAC\u200bAGG\u200bAGA\u200bTTA\u200bTTC\u200bTTT\u200bCCA\u200bACA\u200bCCA\u200bGAA\u200bTCT\u200bC-3\u2032); pCI GFP-Rab11a Q70L was recombined from pDONR Rab11a Q70L N-tag (mutagenesis of pDONR Rab11a N-tag using 5\u2032-GAT\u200bATG\u200bGGA\u200bCAC\u200bAGC\u200bAGG\u200bGCT\u200bAGA\u200bGCG\u200bATA\u200bTCG\u200bAGC-3\u2032 and 5\u2032-GCT\u200bCGA\u200bTAT\u200bCGC\u200bTCT\u200bAGC\u200bCCT\u200bGCT\u200bGTG\u200bTCC\u200bCAT\u200bATC-3\u2032); pCI GFP-Rab1 S25N was recombined from pDONR Rab1 S25N N-tag ] using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCTC\u200bCAG\u200bCAT\u200bGAA\u200bTCC\u200bCGA\u200bATA\u200bTG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bAGC\u200bAGC\u200bAAC\u200bCTC\u200bCAC\u200bCTG\u200bAC-3\u2032); pCI GFP-Rab1 and pCI RFP-Rab1 were recombined from pDONR Rab1 N-tag (mutagenesis of pDONR Rab1 S25N N-tag using 5\u2032-GCG\u200bACT\u200bCTG\u200bGGG\u200bTTG\u200bGAA\u200bAGA\u200bGTT\u200bGCC\u200bTCC\u200bTTC\u200bTTA\u200bGG-3\u2032 and 5\u2032-CCT\u200bAAG\u200bAAG\u200bGAG\u200bGCA\u200bACT\u200bCTT\u200bTCC\u200bAAC\u200bCCC\u200bAGA\u200bGTC\u200bGC-3\u2032); pCI GFP-Rab1 Q70L was recombined from pDONR Rab1 Q70L N-tag (mutagenesis of pDONR Rab1 N-tag using 5\u2032-CAA\u200bATA\u200bTGG\u200bGAC\u200bACA\u200bGCA\u200bGGC\u200bCTA\u200bGAA\u200bAGA\u200bTTT\u200bCG-3\u2032 and 5\u2032-CGA\u200bAAT\u200bCTT\u200bTCT\u200bAGG\u200bCCT\u200bGCT\u200bGTG\u200bTCC\u200bCAT\u200bATT\u200bTG-3\u2032); pCI myc-VAPA, pCI RFP-VAPA, and pCI GFP-VAPA were recombined from pDONR VAPA N-tag (made from a PCR of IMAGE 2822547 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGC\u200bGTC\u200bCGC\u200bCTC\u200bAGG\u200bGG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bACA\u200bAGA\u200bTGA\u200bATT\u200bTCC\u200bCTA\u200bGAA\u200bAGA\u200bATC\u200bC-3\u2032); pCI GFP-VAPA \u0394CC was recombined from pDONR VAPA \u0394CC N-tag (deletion of aa 151\u2013225 of VAPA by mutagenesis of pDONR VAPA N-tag using 5\u2032-CAG\u200bTGG\u200bAAC\u200bAGC\u200bTTT\u200bGCT\u200bAGG-3\u2032 and 5\u2032-CTT\u200bCCT\u200bTCA\u200bCTT\u200bCTT\u200bGTT\u200bGTA\u200bATT\u200bGC-3\u2032); pCI GFP-VAPA \u0394MSP was recombined from pDONR VAPA \u0394MSP N-tag ; pCI GFP-VAPA \u0394TM was recombined from pDONR VAPA \u0394TM N-tag ; pCI GFP-VAPA DD was recombined from pDONR VAPA DD N-tag (VAPA K94D M96D by mutagenesis of pDONR VAPA N-tag using 5\u2032-GAG\u200bTAA\u200bACA\u200bCGA\u200bCTT\u200bTGA\u200bCGT\u200bACA\u200bGAC\u200bAAT\u200bTTT\u200bTGC\u200bTCC\u200bACC\u200bA-3\u2032 and 5\u2032-TGG\u200bTGG\u200bAGC\u200bAAA\u200bAAT\u200bTGT\u200bCTG\u200bTAC\u200bGTC\u200bAAA\u200bGTC\u200bGTG\u200bTTT\u200bACT\u200bC-3\u2032); pCI GFP-VAPB and pCI myc-VAPB were recombined from pDONR VAPB N-tag (made from a PCR of IMAGE 3543354 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGC\u200bGAA\u200bGGT\u200bGGA\u200bGCA\u200bGG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bACA\u200bAGG\u200bCAA\u200bTCT\u200bTCC\u200bCAA\u200bTAA\u200bTTA\u200bC-3\u2032); pCI GFP-MICAL2 was recombined from pDONR MICAL2 N-tag (made from a PCR of IMAGE 5275364 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGG\u200bGGA\u200bAAA\u200bCGA\u200bGGA\u200bTGA\u200bGAA\u200bG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bAGC\u200bCAA\u200bGAA\u200bGTG\u200bGGT\u200bGTA\u200bGC-3\u2032); pCI GFP-MICAL3 was recombined from pDONR MICAL3 N-tag (made from a PCR of IMAGE 100000333 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGA\u200bGGA\u200bGAG\u200bGAA\u200bGCA\u200bTGA\u200bGAC\u200bC-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bAGG\u200bGCC\u200bGTG\u200bTGG\u200bCGG\u200bG-3\u2032); pCI GFP-MICAL-L1 was recombined from pDONR MICAL-L1 N-tag (made from a PCR of IMAGE 100015878 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGC\u200bTGG\u200bGCC\u200bGCG\u200bGGG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bAGC\u200bTCT\u200bTGT\u200bCTC\u200bTGG\u200bGGG\u200bAC-3\u2032); pCI GFP\u2013MICAL-L2 was recombined from pDONR MICAL-L2 N-tag (made from a PCR of IMAGE 5521653 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGC\u200bGGC\u200bCAT\u200bCAG\u200bGGC\u200bG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bACT\u200bGGG\u200bAGG\u200bGGC\u200bTGC\u200bTTT\u200bTGC-3\u2032); pCI E-cadherin\u2013GFP and pCI E-cadherin\u2013RFP were recombined from pDONR E-cadherin C-tag (made from a PCR of a human kidney cDNA library using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGA\u200bAGG\u200bAGA\u200bTAG\u200bAAC\u200bCAT\u200bGGG\u200bCCC\u200bTTG\u200bGAG\u200bCCG\u200bC-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bGTC\u200bGTC\u200bCTC\u200bGCC\u200bGCC\u200bTCC\u200bG-3\u2032); pCI Occludin-GFP was recombined from pDONR Occludin C-tag (made from a PCR of a human kidney cDNA library using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGA\u200bAGG\u200bAGA\u200bTAG\u200bAAC\u200bCAT\u200bGTC\u200bATC\u200bCAG\u200bGCC\u200bTCT\u200bTGA\u200bAAG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bTGT\u200bTTT\u200bCTG\u200bTCT\u200bATC\u200bATA\u200bGTC\u200bTCC\u200bAAC\u200bC-3\u2032); pCI MMP14-GFP was recombined from pDONR MMP14 C-tag (made from a PCR of a human kidney cDNA library using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGA\u200bAGG\u200bAGA\u200bTAG\u200bAAC\u200bCAT\u200bGTC\u200bTCC\u200bCGC\u200bCCC\u200bAAG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bGAC\u200bCTT\u200bGTC\u200bCAG\u200bCAG\u200bGG-3\u2032); HA\u2013\u03b12B-AR\u2013GFP and HA\u2013\u03b12B-AR\u2013RFP were recombined from pDONR HA\u2013\u03b12B-AR C-tag (made from a PCR on IMAGE 9020527 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGA\u200bAGG\u200bAGA\u200bTAG\u200bAAC\u200bCAT\u200bGTA\u200bCCC\u200bATA\u200bCGA\u200bTGT\u200bTCC\u200bAGA\u200bTTA\u200bCGC\u200bTGA\u200bCCA\u200bCCA\u200bGGA\u200bCCC\u200bCTA\u200bC-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCA\u200bGGC\u200bCGT\u200bCTG\u200bGGT\u200bCCA\u200bC-3\u2032); pDONR Sar1 N-tag was made from a PCR on IMAGE 712455 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCTC\u200bTTT\u200bCAT\u200bCTT\u200bTGA\u200bGTG\u200bGAT\u200bCTA\u200bC-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bAGT\u200bCAA\u200bTAT\u200bACT\u200bGGG\u200bAAA\u200bGCC\u200bAGC-3\u2032); pCI myc-Sar1 H79G, pCI GFP-Sar1 H79G, and pCI RFP-Sar1 H79G were recombined from pDONR Sar1 H79G N-tag (mutagenesis of pDONR Sar1 N-tag using 5\u2032-CTT\u200bTTG\u200bATC\u200bTTG\u200bGTG\u200bGGG\u200bGCG\u200bAGC\u200bAAG\u200bCAC\u200bGTC-3\u2032 and 5\u2032-GAC\u200bGTG\u200bCTT\u200bGCT\u200bCGC\u200bCCC\u200bCAC\u200bCAA\u200bGAT\u200bCAA\u200bAAG-3\u2032); pCI Calnexin-GFP was recombined from pDONR Calnexin C-tag ; pCI GFP-STX16 was recombined from pDONR STX16 N-tag (made from a PCR of a 293T cDNA library using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGC\u200bCAC\u200bCAG\u200bGCG\u200bTTT\u200bAAC\u200bCG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bATC\u200bGAG\u200bACT\u200bTCA\u200bCGC\u200bCAA\u200bCG-3\u2032); pCI GFP-VAMP3 and pCI RFP-VAMP3 were recombined from pDONR VAMP3 N-tag (made from a PCR on IMAGE 3544610 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCTC\u200bTAC\u200bAGG\u200bTCC\u200bAAC\u200bTGC\u200bTGC-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bATG\u200bAAG\u200bAGA\u200bCAA\u200bCCC\u200bACA\u200bCG-3\u2032); pCI TGN46-RFP was recombined from pDONR TGN46 C-tag (made from a PCR on IMAGE 4272702 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCGA\u200bAGG\u200bAGA\u200bTAG\u200bAAC\u200bCAT\u200bGCG\u200bGTT\u200bCGT\u200bAGT\u200bTGC\u200bC-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bGGA\u200bCTT\u200bCTG\u200bGTC\u200bCAA\u200bACG\u200bTTG\u200bGTA-3\u2032); pCI GFP-EHD1 was recombined from pDONR EHD1 N-tag (made from a PCR on pEGFP C3 mEHD1 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCTT\u200bCAG\u200bCTG\u200bGGT\u200bGAG\u200bCAA\u200bGG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bACT\u200bCGT\u200bGCC\u200bTCC\u200bGTT\u200bTGG\u200bAG-3\u2032); and pCI GFP-EHD3 was recombined from pDONR EHD3 N-tag (made from a PCR on SKB-LNB hEHD3 using 5\u2032-GGG\u200bGAC\u200bAAG\u200bTTT\u200bGTA\u200bCAA\u200bAAA\u200bAGC\u200bAGG\u200bCTT\u200bCTT\u200bCAG\u200bCTG\u200bGCT\u200bGGG\u200bTAC\u200bGG-3\u2032 and 5\u2032-GGG\u200bGAC\u200bCAC\u200bTTT\u200bGTA\u200bCAA\u200bGAA\u200bAGC\u200bTGG\u200bGTC\u200bCCT\u200bACT\u200bCGG\u200bCAA\u200bCTT\u200bTCC\u200bTCT\u200bTGG\u200bACG-3\u2032).In general, cDNAs were amplified from cDNA libraries, IMAGE clones, or plasmids and were integrated in the entry vector pDONR201 for cloning using the Gateway recombination system (Thermo Fisher Scientific). For expression in mammalian cells, the entry clones were recombined with modified pCI (Promega) vectors, engineered to be Gateway compatible and to express proteins with a GFP (EGFP protein), RFP (tagRFP protein), or myc tag in N-terminal or C-terminal position. The empty destination (DEST) vectors pCI N-GFP DEST and pCI N-RFP DEST were used for expression of soluble EGFP and tagRFP, respectively. For expression of GST-tagged proteins in bacteria, entry clones were recombined with a modified pGex 4T2 vector  engineered to be Gateway compatible. Mutagenesis was performed by PCR amplification of parent clones, digestion of the template with DpnI, ligation, and transformation. Apart from Rab8a and Rab1, which were canine, EHD1 which was from mouse, and Sar1 which was from rat, all the other constructs were human. pCI GRAF2-GFP and pCI GRAF2-RFP were recombined from pDONR GRAF2 C-tag  using the primers 5\u2032-GTA\u200bCAA\u200bGTC\u200bCGG\u200bACT\u200bCAG\u200bATC\u200bTCG\u200bAGC\u200bTCA\u200bGAG\u200bGTC\u200bGCC\u200bTCT\u200bGGA\u200bAAA\u200bGGC\u200bC-3\u2032 and 5\u2032-CCG\u200bGTG\u200bGAT\u200bCCC\u200bGGG\u200bCCC\u200bGCG\u200bGTA\u200bCCC\u200bTAA\u200bAGC\u200bCTT\u200bGTA\u200bTCT\u200bTGC\u200bACC\u200bTCT\u200bTCT\u200bTCT\u200bGTC\u200bTCC\u200bTCT\u200bTT-3\u2032 and inserted into pEGFP-C1 (Takara). The plasmids Str-KDEL_SBP-EGFP-Ecadherin, Str-KDEL_SBP-EGFP-GPI, and Str-Ii_VSVG-SBP-EGFP were gifts from Franck Perez . VSVG was excised with XmaI and EcoRI and replaced with human E-cadherin using primers 5\u2032-CTT\u200bGCC\u200bACA\u200bACC\u200bCGG\u200bGAG\u200bGCG\u200bCGC\u200bCAT\u200bGGG\u200bCCC\u200bTTG\u200bGAG\u200bCCG\u200bC-3\u2032 and 5\u2032-CTC\u200bGTC\u200bCAT\u200bGGA\u200bATT\u200bCAG\u200bGTC\u200bGTC\u200bCTC\u200bGCC\u200bGCC\u200bTCC-3\u2032 and the In-Fusion system (Takara Bio) to make Str-Ii_E-cadherin\u2013SBP-EGFP. Str-Ii_VSVG-SBP\u2013Fusion Red was made after excision of EGFP from Str-Ii_VSVG-SBP-EGFP using SbfI and XbaI and insertion of Fusion Red using primers 5\u2032-TCA\u200bACG\u200bTGA\u200bACC\u200bACC\u200bTGC\u200bAGG\u200bTAT\u200bGGT\u200bGAG\u200bCGA\u200bGCT\u200bGAT\u200bTAA\u200bGG-3\u2032 and 5\u2032-TGA\u200bTCA\u200bGTT\u200bATC\u200bTAG\u200bAGT\u200bTTC\u200bATT\u200bTAC\u200bCTC\u200bCAT\u200bCAC\u200bCAG\u200bCGC-3\u2032 and the In-Fusion system.pF1KA0819) was a gift from Anna Akhmanova ; HA-GLUT4-GFP was a gift from Sam Cushman ; pEGFP-wtCFTR, and pEGFP-\u0394F508-CFTR were gifts from Ineke Braakman .GFP-GPI (GPI anchoring region of hCD58 fused to GFP) was a gift from Ben Nichols ; GFP-MICAL3FL . Corresponding oligonucleotides were annealed, phosphorylated, and ligated into pLKO.1-puro (Sigma), which had been predigested with BglII and HindIII. For each gene, at least three different shRNAs were cloned and tested. The best ones were used: shControl (5\u2032-ccg\u200bgCG\u200bTAC\u200bGCG\u200bGAA\u200bTAC\u200bTTC\u200bGAt\u200btca\u200baga\u200bgaT\u200bCGA\u200bAGT\u200bATT\u200bCCG\u200bCGT\u200bACG\u200bttt\u200bttg-3\u2032 and 5\u2032-aat\u200btca\u200baaa\u200baCG\u200bTAC\u200bGCG\u200bGAA\u200bTAC\u200bTTC\u200bGAt\u200bctc\u200bttg\u200baaT\u200bCGA\u200bAGT\u200bATT\u200bCCG\u200bCGT\u200bACG-3\u2032); shGRAF2a (5\u2032-ccg\u200bgGC\u200bACA\u200bGAT\u200bCTG\u200bGAG\u200bGGA\u200bAAt\u200btca\u200baga\u200bgaT\u200bTTC\u200bCCT\u200bCCA\u200bGAT\u200bCTG\u200bTGC\u200bttt\u200bttg-3\u2032 and 5\u2032-aat\u200btca\u200baaa\u200baGC\u200bACA\u200bGAT\u200bCTG\u200bGAG\u200bGGA\u200bAAt\u200bctc\u200bttg\u200baaT\u200bTTC\u200bCCT\u200bCCA\u200bGAT\u200bCTG\u200bTGC-3\u2032); shGRAF2b (5\u2032-ccg\u200bgCG\u200bTTG\u200bAAA\u200bCAC\u200bGAG\u200bGTA\u200bTAt\u200btca\u200baga\u200bgaT\u200bATA\u200bCCT\u200bCGT\u200bGTT\u200bTCA\u200bACG\u200bttt\u200bttg-3\u2032 and 5\u2032-aat\u200btca\u200baaa\u200baCG\u200bTTG\u200bAAA\u200bCAC\u200bGAG\u200bGTA\u200bTAt\u200bctc\u200bttg\u200baaT\u200bATA\u200bCCT\u200bCGT\u200bGTT\u200bTCA\u200bACG-3\u2032); shGRAF1 (5\u2032-ccg\u200bgAG\u200bGAA\u200bGTC\u200bCAA\u200bGAG\u200bAGA\u200bAAt\u200btca\u200baga\u200bgaT\u200bTTC\u200bTCT\u200bCTT\u200bGGA\u200bCTT\u200bCCt\u200bttt\u200bttg-3\u2032 and 5\u2032-aat\u200btca\u200baaa\u200baaG\u200bGAA\u200bGTC\u200bCAA\u200bGAG\u200bAGA\u200bAAt\u200bctc\u200bttg\u200baaT\u200bTTC\u200bTCT\u200bCTT\u200bGGA\u200bCTT\u200bCCT-3\u2032); shWDR44a (5\u2032-ccg\u200bgGG\u200bAAT\u200bGAA\u200bGTA\u200bGAT\u200bGGT\u200bCAt\u200btca\u200baga\u200bgaT\u200bGAC\u200bCAT\u200bCTA\u200bCTT\u200bCAT\u200bTCC\u200bttt\u200bttg-3\u2032 and 5\u2032-aat\u200btca\u200baaa\u200baGG\u200bAAT\u200bGAA\u200bGTA\u200bGAT\u200bGGT\u200bCAt\u200bctc\u200bttg\u200baaT\u200bGAC\u200bCAT\u200bCTA\u200bCTT\u200bCAT\u200bTCC-3\u2032); shWDR44b (5\u2032-ccg\u200bgTG\u200bATG\u200bAAA\u200bCCT\u200bGTG\u200bAGA\u200bAAt\u200btca\u200baga\u200bgaT\u200bTTC\u200bTCA\u200bCAG\u200bGTT\u200bTCA\u200bTCA\u200bttt\u200bttg-3\u2032 and 5\u2032-aat\u200btca\u200baaa\u200baTG\u200bATG\u200bAAA\u200bCCT\u200bGTG\u200bAGA\u200bAAT\u200bCTC\u200bTTG\u200bAAT\u200bTTC\u200bTCA\u200bCAG\u200bGTT\u200bTCA\u200bTCA-3\u2032); shMICAL1 (5\u2032-ccg\u200bgGC\u200bATT\u200bGAT\u200bCTG\u200bGAG\u200bAAC\u200bATt\u200btca\u200baga\u200bgaA\u200bTGT\u200bTCT\u200bCCA\u200bGAT\u200bCAA\u200bTGC\u200bttt\u200bttg-3\u2032 and 5\u2032-aat\u200btca\u200baaa\u200baGC\u200bATT\u200bGAT\u200bCTG\u200bGAG\u200bAAC\u200bATt\u200bctc\u200bttg\u200baaA\u200bTGT\u200bTCT\u200bCCA\u200bGAT\u200bCAA\u200bTGC-3\u2032); shRab8a1 (5\u2032-ccg\u200bgGA\u200bACA\u200bAGT\u200bGTG\u200bATG\u200bTGA\u200bATt\u200btca\u200baga\u200bgaA\u200bTTC\u200bACA\u200bTCA\u200bCAC\u200bTTG\u200bTTC\u200bttt\u200bttg-3\u2032 and 5\u2032-aat\u200btca\u200baaa\u200baGA\u200bACA\u200bAGT\u200bGTG\u200bATG\u200bTGA\u200bATt\u200bctc\u200bttg\u200baaA\u200bTTC\u200bACA\u200bTCA\u200bCAC\u200bTTG\u200bTTC-3\u2032) cotransfected with shRab8a2 (5\u2032-ccg\u200bgGA\u200bGAA\u200bTTA\u200bAAC\u200bTGC\u200bAGA\u200bTAt\u200btca\u200baga\u200bgaT\u200bATC\u200bTGC\u200bAGT\u200bTTA\u200bATT\u200bCTC\u200bttt\u200bttg-3\u2032 and 5\u2032-aat\u200btca\u200baaa\u200baGA\u200bGAA\u200bTTA\u200bAAC\u200bTGC\u200bAGA\u200bTAt\u200bctc\u200bttg\u200baaT\u200bATC\u200bTGC\u200bAGT\u200bTTA\u200bATT\u200bCTC-3\u2032); shRab10 (5\u2032-ccg\u200bgGA\u200bATA\u200bGAC\u200bTTC\u200bAAG\u200bATC\u200bAAt\u200btca\u200baga\u200bgaT\u200bTGA\u200bTCT\u200bTGA\u200bAGT\u200bCTA\u200bTTC\u200bttt\u200bttg-3\u2032 and 5\u2032-aat\u200btca\u200baaa\u200baGA\u200bATA\u200bGAC\u200bTTC\u200bAAG\u200bATC\u200bAAt\u200bctc\u200bttg\u200baaT\u200bTGA\u200bTCT\u200bTGA\u200bAGT\u200bCTA\u200bTTC-3\u2032); shRab11a (5\u2032-ccg\u200bgAG\u200bTTT\u200bAAT\u200bCTG\u200bGAA\u200bAGC\u200bAAt\u200btca\u200baga\u200bgaT\u200bTGC\u200bTTT\u200bCCA\u200bGAT\u200bTAA\u200bACT\u200bttt\u200bttg-3\u2032 and 5\u2032-aat\u200btca\u200baaa\u200baAG\u200bTTT\u200bAAT\u200bCTG\u200bGAA\u200bAGC\u200bAAt\u200bctc\u200bttg\u200baaT\u200bTGC\u200bTTT\u200bCCA\u200bGAT\u200bTAA\u200bACT-3\u2032); shVAPA (5\u2032-ccg\u200bgCC\u200bACA\u200bGAC\u200bCTC\u200bAAA\u200bTTC\u200bAAt\u200btca\u200baga\u200bgaT\u200bTGA\u200bATT\u200bTGA\u200bGGT\u200bCTG\u200bTGG\u200bttt\u200bttg-3\u2032 and 5\u2032-aat\u200btca\u200baaa\u200baCC\u200bACA\u200bGAC\u200bCTC\u200bAAA\u200bTTC\u200bAAt\u200bctc\u200bttg\u200baaT\u200bTGA\u200bATT\u200bTGA\u200bGGT\u200bCTG\u200bTGG-3\u2032); and shVAPB (5\u2032-ccg\u200bgCG\u200bGAA\u200bGAC\u200bCTT\u200bATG\u200bGAT\u200bTCt\u200btca\u200baga\u200bgag\u200baAT\u200bCCA\u200bTAA\u200bGGT\u200bCTT\u200bCCG\u200bttt\u200bttg-3\u2032 and 5\u2032-aat\u200btca\u200baaa\u200baCG\u200bGAA\u200bGAC\u200bCTT\u200bATG\u200bGAT\u200bTCt\u200bctc\u200bttg\u200baaG\u200bAAT\u200bCCA\u200bTAA\u200bGGT\u200bCTT\u200bCCG-3\u2032).For lentivirus production, pMD2.G  and PSPAX2  were used. Both were gifts from Didier Trono .All constructs were verified by sequencing.Purified rabbit polyclonal \u03b1-GRAF1 and \u03b1-GRAF2 antibodies were described previously , as wellThe secondary reagents for Western blotting were goat \u03b1\u2212mouse and \u03b1\u2212rabbit IgG-HRP conjugates (Bio-Rad). HRP-Protein A (Thermo Fisher Scientific) was used for analysis of beads when antibodies used for the immunoprecipitation and for Western blotting were raised in the same species.The secondary antibodies used for immunofluorescence were Alexa Fluor 488, 514, 546, or 647 conjugates of goat \u03b1\u2212mouse, \u03b1\u2212rabbit, \u03b1\u2212rat, and \u03b1\u2212sheep IgG antibodies .The following reagents were used: Nocodazole, Cytochalasin D, BFA, D-biotin, methanol, and DMSO (Sigma); Alexa Fluor 546\u2013Transferrin, Alexa Fluor 647\u2013Transferrin, Alexa Fluor 546\u2013dextran 10.000 MW fixable, and Alexa Fluor 546\u2013phalloidin (Thermo Fisher Scientific); Puromycin and C4 ; VX-809 ; and thapsigargin (Santa Cruz Biotechnology).HeLa (ECACC 93021013), 293T , HT 1080 , U-87 MG (ECACC 89081402), COS-7 (ECACC 87021302), undifferentiated 3T3 L1 cells (ECACC 86052701), and BSC1 (ECACC 85011422) cells were grown in DMEM-GlutaMAX supplemented with 10% FBS. NIH 3T3 cells (ECACC 93061524) were grown in DMEM-GlutaMAX with 10% newborn calf serum. SH-SY5Y cells (ECACC 94030304) were grown in a 1:1 mixture of MEM and Ham\u2019s F-12 supplemented with GlutaMAX, 1% nonessential amino acids, and 15% FBS. hTERT-RPE1 cells  were grown in a 1:1 mixture of DMEM and Ham\u2019s F-12 with GlutaMAX, 0.25% sodium bicarbonate, and 10 \u00b5g/ml hygromycin. Unless otherwise stated, cell culture products were from Thermo Fisher Scientific. Cells were routinely checked for mycoplasma contamination. When export of MMP14 was tested, HT 1080 cells were grown on dishes coated with collagen from calf skin (Sigma). For biotinylation experiments using 293T cells, they were grown on dishes coated with Poly-L-Lysine (Sigma). HeLa, 293T, and HT 1080 cells were transfected with Genejuice (Merck Millipore) or polyethylenimine (PEI) Max Linear 40 kD (Generon). For PEI transfection of 35-mm dishes, in general, 1 \u00b5g of each DNA was diluted in 100 \u00b5l OptiMEM, then complexed with 3 \u00b5g PEI prediluted in 100 \u00b5l OptiMEM. The mix was vortexed, incubated at RT for 15\u201320 min, and added dropwise to cells. For colocalization purposes, cells were generally examined 16\u201324 h after transfection. For all transfections of CFTR \u0394F508, C4 (5 \u00b5M) and VX-809 (5 \u00b5M) were added 12 h after transfection.res-RFP, pCI RFP-WDR44res, or pCI RFP\u2013WDR44res \u0394Pro. Cells were split, transfected after 52\u201356 h, and collected 36 h later. For knockdown of protein expression in HT 1080 cells, cells were infected with lentivirus, selected 24 h later with puromycin , and split. Cells were transfected 72 h after the original infection with pCI MMP14-GFP, alone or together with pCI RFP-WDR44res or pCI RFP\u2013WDR44res \u0394Pro, and analyzed 16 h later. To make lentivirus, 293T cells were grown on Poly-L-Lysine\u2013coated plates and cotransfected with shRNA, pMD2.G, and PSPAX2. Medium was changed to DMEM supplemented with 4% FBS and 25 mM Hepes after 24 h, and lentivirus was harvested as tissue culture supernatant after a further 48\u201360 h.For knockdown of protein expression in HeLa cells, cells were transfected with an shRNA-encoding plasmid, selected 24 h later with puromycin , and then split. Experiments were performed after 88\u201396 h of knockdown. If shRNA cells were to be transfected, transfection was performed 72 h after the transfection of the shRNA. For knockdown of protein expression in 293T cells, cells were cotransfected with an shRNA-encoding plasmid and pCI N-RFP DEST, pCI GRAF2Cells grown on glass coverslips were fixed in 3.2% PFA diluted in culture medium . For immunostaining of cytoplasmic proteins, coverslips were then washed in PBS and blocked in 5% normal goat serum (NGS) and 0.1% saponin (Sigma) in PBS. All further incubations were done in 1% NGS and 0.1% saponin in PBS. Coverslips were mounted on slides in a buffered PVA glycerol mountant containing 2.5% DABCO (Sigma). For staining of F-actin, Alexa Fluor 546\u2013labeled phalloidin was added to the primary antibody dilution . A similar protocol was followed for immunostaining of surface proteins under nonpermeabilizing conditions, but saponin was omitted from the buffers.2 95:5 and 100% humidity.For live imaging, cells were grown on glass-bottom culture dishes (MatTek). Immediately before imaging, medium was replaced with phenol red\u2013free DMEM containing 5% FBS and placed in a temperature-controlled chamber on the microscope stage with air/COMost imaging data were acquired using a fully motorized inverted microscope  equipped with a CSU-X1 spinning disk confocal head  using a 60\u00d7 oil immersion lens  under control of Volocity 6.0 (PerkinElmer). 14-bit digital images were acquired with a cooled EMCCD camera . Three 50-mW solid-state lasers  coupled to individual acoustic-optical tunable filters (AOTF) were used as a light source to excite fluorescent proteins and dyes, as appropriate. Imaging of RUSH proteins was performed with a Zeiss LSM 780 using standard photomultiplier tubes, a 63\u00d7 oil immersion lens of NA 1.4, and ZEN acquisition software.ImageJ 1.48s  was used to process the images, adjusting only the brightness and contrast of the different channels. Due to the high dynamic range of the signals and the fact that we were particularly interested in the dim structures in the cell\u2019s periphery, we adjusted the scaling of the images to bring this information into the dynamic range of a computer display. No filters were applied. We followed the principles of Color Universal Design. On the merged channel, images acquired in the red channel are colored in red; images acquired in the green channel are colored in cyan; and images acquired in the far red channel are colored in magenta. Representative images of live cells and fixed cells collected in a minimum of three independent experiments are shown. Unless otherwise mentioned, images are from a single focal plane. For confocal stacks, maximum intensity projections of images acquired at 0.3-\u00b5m intervals, in single channels and as a merged image, are shown.For quantification of the proportion of HeLa cells with RFP\u2013Rab8a/10 tubules, cells were fixed 20 h after transfection. The number of cells with at least one tubule was counted and expressed as percentage of the total number of cells examined (minimum 25 cells per sample per experiment). For quantification of the total skeletal length of RFP\u2013Rab8a/10 structures and for quantification of the total volume of GRAF1b/2\u2013GFP structures per cell, cells were fixed 20 h after transfection. Confocal stacks were acquired. Each cell was identified as a region of interest. Volocity was used to identify fluorescent objects and measure their combined skeletal length or volume. Manders colocalization coefficients were quantified using Volocity, setting thresholds for each cell using a region of cytosol. For quantification of the proportion of HeLa cells with endogenous WDR44 tubules, cells were fixed 24 h (overexpression) or 96 h (shRNA) after transfection. The number of cells with at least one WDR44 tubule was counted and expressed as a percentage of the total number of cells examined (minimum 100 cells per sample per experiment). For quantification of the total WDR44 length per cell, confocal stacks were acquired. Each cell was identified as a region of interest. WDR44 objects with a nonspherical shape (shape factor <0.5) were identified using the Volocity software, and their skeletal length was measured. The total skeletal length per cell was quantified. For displaying fluorescence intensity profiles, fluorescence intensities were measured on a 1-px line using ImageJ software. Values were then normalized using the minimum and maximum values measured along that line. For detailed numbers of independent experiments, see the figure legends.For super-resolution imaging, cells were plated on high-precision coverslips . Coverslips were processed as for all other immunofluorescence experiments, but antibodies were used at a higher concentration: mouse \u03b1-myc (1:100), rabbit \u03b1-Calnexin (1:200), Alexa Fluor 488 conjugated \u03b1-mouse (1:100), and Alexa Fluor 514 \u03b1-rabbit (1:100). Coverslips were mounted in Prolong Diamond Antifade Mountant  curing for more than 48 h to achieve the best possible refractive index for STED microscopy.The microscope was a Leica TCS SP8 STED 3X equipped with a pulsed white light laser and HyD hybrid detectors enabling gating. Gating was adjusted to detect signal emitting between 1.7 and 6.8 ns after the excitation pulse. Excitation of Alexa Fluor 488 was performed at 488 nm and of Alexa Fluor 514 at 535 nm. Suitable clean-up filters were in place. The depletion laser was at 592 nm.The primary lens was a 100\u00d7/1.46 NA , chromatically corrected across the used spectrum. Sampling frequency was set to 20 nm, and the depletion laser was operated at settings that allowed a lateral resolution below 60 nm. As the photon budget was limited, line averaging (16\u00d7) and frame accumulation (8\u00d7) were used to achieve a suitable signal-to-noise ratio. ImageJ was used to process the images and to measure fluorescence intensity profiles across tubules.Escherichia coli cells (Bioline), and protein expression was induced with isopropyl-\u03b2-D-thiogalactoside . Cells were resuspended in bacterial lysis buffer  and lysed by freezing/thawing and sonication. Lysates were cleared by centrifugation  and incubated with glutathione sepharose beads for 45 min at 4\u00b0C under rotation. Beads were pelleted and washed three times in bacterial lysis buffer. Purified recombinant proteins were analyzed by electrophoresis on 4%\u201312% NuPAGE Novex Bis-Tris gels (Thermo Fisher Scientific) and Coomassie staining.pGex 4T2, pGex 4T2 GRAF1 SH3, pGex 4T2 GRAF2 SH3, pGex 4T2 WDR44 \u0394C, pGex 4T2 WDR44 \u0394N2, pGex 4T2 MICAL1, pGex 4T2 MICAL1 \u0394N, and pGex 4T2 MICAL1 \u0394MO\u0394CH were transformed in BL21(DE3) pLysS g), and protein concentration of the supernatants was determined with a Bradford assay (Bio-Rad). Equivalent protein amounts of GST, GST-GRAF1 SH3, or GST-GRAF2 SH3 bound to glutathione sepharose beads were incubated with a volume of lysate corresponding to \u223c7 mg protein or with the same volume of cell lysis buffer for 30 min at 4\u00b0C under rotation. Beads were pelleted  and washed five times in cell lysis buffer without protease inhibitors. For identification of bound proteins by Western blot, beads were directly incubated with sample buffer, heated at 95\u00b0C, and loaded on gels. For identification of bound proteins by mass spectrometry, beads were incubated with Thrombin . Eluted proteins were collected in the supernatant after centrifugation , sample buffer was added, and proteins were separated using 4%\u201312% NuPAGE Novex Bis-Tris gels. Protein bands were revealed by Coomassie staining. Bands were excised, and proteins were identified by liquid chromatography-tandem mass spectrometry . Results were analyzed with Scaffold 3 (Proteome Software).For pull-downs of cell lysates, beads were washed three times in cell lysis buffer . In parallel, 15-cm dishes of HeLa cells or rat brains were lysed in the same buffer . Lysates were cleared by centrifugation  and collected in the supernatant after centrifugation. WDR44 and MICAL1 domains bound to glutathione sepharose beads were then incubated in the presence or absence of an equal amount of GRAF2 SH3 for 30 min at 4\u00b0C and washed three times in bacterial lysis buffer. Proteins bound to the beads were separated using 4%\u201312% NuPAGE Novex Bis-Tris gels and revealed by Coomassie staining.g), and protein concentration was determined using a Bradford assay. Rat brains were homogenized in the same buffer using a Dounce homogenizer, but lysates were cleared at 100,000 g, 30 min. When lysates were to be directly analyzed, sample buffer was added to equal protein amounts (75 \u00b5g protein per sample); in general, for immunoprecipitation of endogenous proteins, 150 \u00b5g protein was incubated with purified antibodies or serum in a total volume of 100 \u00b5l and incubated 3 h at 4\u00b0C under rotation. 12.5 \u00b5l of a 50% suspension of Protein G Sepharose 4 Fast Flow beads  was added for another 30 min before beads were spun down , washed three times in lysis buffer, and finally resuspended in sample buffer. Samples were heated 3 min at 95\u00b0C, loaded on 4%\u201312% NuPAGE Novex Bis-Tris gels, and analyzed by Western blot. Proteins were transferred to polyvinylidene fluoride membranes; powder milk was used as blocking agent. Secondary antibodies were detected using Amersham ECL detection reagent , and results were developed on x-ray film. The same protocol was followed for the coimmunoprecipitation of endogenous proteins from SH-SY5Y cells but using a volume of lysate corresponding to 100 cm2 of a confluent layer of cells for each immunoprecipitation.Untreated cells or shRNA-expressing cells (96 h after transfection) were scraped in cold PBS and pelleted. They were resuspended in lysis buffer  and incubated 30 min at 4\u00b0C. Lysates were cleared . The pellets were resuspended in the same volume of lysis buffer and cleared at low speed , and a volume equivalent to 125 cm2 of a confluent layer of cells was used for each immunoprecipitation.For coimmunoprecipitation of endogenous proteins from HeLa cells, cells were resuspended in Triton X-100\u2013containing lysis buffer  and incubated 30 min at 4\u00b0C. Lysates were centrifuged . 10 \u00b5l of each supernatant was collected, and 5 \u00b5l sample buffer was added (lysates). The rest of the supernatants were incubated with 12.5 \u00b5l of a 50% suspension of Protein G Sepharose 4 Fast Flow beads for 30 min at 4\u00b0C under rotation. Beads were then spun down  and washed three times with 500 \u00b5l lysis buffer. Beads were resuspended in 10 \u00b5l sample buffer. Beads and lysates were heated 3 min at 95\u00b0C, loaded on 4%\u201312% NuPAGE Novex Bis-Tris gels, and analyzed in parallel by Western blot. Membranes from immunoprecipitations done with rabbit \u03b1-GFP antibody were first probed with mouse \u03b1-myc antibody. Corresponding beads were then probed with mouse \u03b1-GFP antibody while lysates were probed with rabbit \u03b1-GFP antibody. Membranes from immunoprecipitations done with mouse \u03b1-myc antibody were first probed with rabbit \u03b1-GFP antibody. Corresponding beads were then probed with rabbit \u03b1-myc antibody while lysates were probed with mouse \u03b1-myc antibody. For coimmunoprecipitation of uGRAF2 with rabbit \u03b1-GFP, uGRAF2 was detected in the beads with rabbit \u03b1-GRAF2 and HRP\u2013Protein A. In general, the membranes corresponding to the beads were chosen to show the level of expression of the baits, but when the background was too strong, the lysates were shown. For immunoprecipitations using \u03b1-myc\u2013coated beads, a similar procedure was followed, but 100-mm dishes were used. Cells were resuspended in 200 \u00b5l lysis buffer, 15 \u00b5l of each lysate was loaded per well, and 5 \u00b5l of \u03b1-myc\u2013coated magnetic beads  were used per sample. In all cases, results displayed are representative of a minimum of three independent experiments.On the first day, 293T cells were plated in six-well dishes and a few hours later were transfected with the plasmid coding for the prey. The next day, cells were transfected with plasmids encoding the baits. 20\u201324 h later, the medium was removed, and cells were detached by pipetting in 1 ml cold PBS. Cells were spun down, and pellets were resuspended in 100 \u00b5l lysis buffer  and incubated on ice for 30 min. Lysates were cleared  and resuspended in 1% NGS in PBS containing DyLight 650 \u03b1-HA antibody (5 \u00b5g/ml) or unconjugated \u03b1-HA (1 \u00b5g/ml) and incubated for 90 min at RT. Cells were spun down, washed three times in 1% NGS in PBS, and resuspended in Alexa Fluor 647\u2013conjugated \u03b1-mouse antibody in 1% NGS in PBS. Cells were incubated for another hour, washed three times, fixed once more in 3.2% PFA, and resuspended in PBS. Cells were analyzed by FACS using an LSRFortessa analyzer (BD Biosciences), and results were analyzed with FlowLogic 7.2.1 (Inivai Technologies Pty. Ltd.). After comparison with undifferentiated cells, differentiated cells were selected based on their light-scattering properties. GFP or GFP/RFP\u2013positive cells were selected. Background was subtracted from the GFP and Alexa Fluor 647 fluorescence intensities. The ratio of the geometric mean of the Alexa Fluor 647 fluorescence intensity to the geometric mean of the GFP fluorescence intensity was calculated and expressed as a ratio to the value obtained for the corresponding insulin-treated control. This was used as a measurement of HA-GLUT4-GFP export efficiency. For all cotransfections of nonfluorescent proteins, coexpression of myc-tagged proteins or of MICAL1 was verified in parallel following a similar protocol but using buffers containing 0.1% saponin and \u03b1-myc (2 \u00b5g/ml) or \u03b1-MICAL1 (1.9 \u00b5g/ml). 50\u20131,000 differentiated cells were analyzed in each independent experiment.To induce differentiation, 3T3 L1 cells were grown to confluence in DMEM-GlutaMAX with 10% newborn calf serum. They were then fed with DMEM-GlutaMAX with 10% FBS, 0.5 mM 3-isobutyl-1-methylxanthine (IBMX), 0.25 \u00b5M dexamethasone, 1 \u00b5g/ml bovine insulin , and 5 \u00b5M troglitazone . 2\u20133 d later, medium was replaced with the above without IBMX for another 2\u20133 d. Cells were then maintained in DMEM-GlutaMAX with 10% FBS for 2\u20133 d. HA-GLUT4-GFP was used to examine export of GLUT4 . For trag) and incubated in blocking solution (5% NGS in PBS) for 30 min at RT. All further incubations and washes were done in PBS with 1% NGS. Cells were pelleted and incubated with unconjugated \u03b1-Integrin-\u03b21 antibody (5 \u00b5g/ml) for 90 min and washed three times. They were then incubated with Alexa Fluor 546\u2013conjugated \u03b1-mouse antibody (2 \u00b5g/ml) for 1 h, washed three times, and fixed once more in 3.2% PFA before being resuspended in PBS. Cells were analyzed by FACS using an LSRFortessa analyzer, and results were analyzed with FlowLogic 7.2.1. Within each experiment, the geometric means of the Alexa Fluor 647 and the Alexa Fluor 546 intensities were normalized by the values obtained for the shControl uptake sample. For each shRNA, the ratio of the normalized intensity of internalized Integrin-\u03b21 (Alexa Fluor 647) to the one of surface Integrin-\u03b21 (Alexa Fluor 546) was calculated as a measurement of the internalized Integrin-\u03b21 ratio. Any defect in degradation of the internalized Integrin-\u03b21 antibody or in the recycling of the receptor should lead to a higher internalized Integrin-\u03b21 ratio. Within each experiment, between 1,000 and 30,000 cells were analyzed in each sample.shRNA-expressing HeLa cells (96 h after transfection) were incubated for 1 h with Alexa Fluor 647\u2013conjugated \u03b1-Integrin-\u03b21 antibody (1 \u00b5g/ml) at 37\u00b0C. Cells were then washed in PBS, acid stripped  for 45 s , and eit2B-AR\u2013GFP, HA\u2013\u03b12B-AR\u2013RFP, GFP-GPI, SBP-GFP\u2013E-cadherin, SBP-GFP-GPI, E-cadherin\u2013SBP-GFP, and VSVG-SBP-GFP in HeLa cells, of MMP14-GFP in HT 1080 cells, and of GFP\u2013Extope CFTR and GFP\u2013Extope CFTR \u0394F508 in 293T cells. HeLa and HT 1080 cells were grown in 35-mm dishes. Unless otherwise specified in the figure legend, they were delicately scraped in 1 ml culture medium 16 h after transfection. 293T cells were grown in 12-well dishes and collected 36 h after transfection by pipetting. In all cases, cells were fixed in PFA . Following fixation, all further incubations and centrifugations were done at RT. Cells were spun down , resuspended in blocking solution (5% NGS in PBS), and incubated 30 min. Cells were then pelleted and resuspended in 50 \u00b5l of buffer (1% NGS in PBS) containing the appropriate primary antibody  and incubated for 90 min under gentle rocking. Cells were washed three times in 250 \u00b5l wash buffer (0.1% BSA in PBS). When unlabeled primary antibodies were used, cells were then incubated in a dilution of Alexa Fluor 647\u2013labeled secondary antibody (2 \u00b5g/ml) in buffer (1% NGS in PBS) and incubated for 1 h under gentle rocking; cells were then washed three times in 250 \u00b5l wash buffer. After the final washes, cells were fixed once more in 3.2% PFA in PBS (20 min), washed, and resuspended in PBS. Cells were analyzed by FACS using an LSRFortessa analyser, and results were analyzed with FlowLogic 7.2.1. For single transfections, GFP-positive or RFP-positive cells were selected; for cotransfections, GFP- and RFP-positive cells were selected. Background was subtracted from the GFP, RFP, and Alexa Fluor 647 fluorescence intensities. The ratio of the geometric mean of the Alexa Fluor 647 fluorescence intensity to the geometric mean of the GFP or RFP (depending on the tag of the export candidate) fluorescence intensity was calculated and expressed as a ratio to the value obtained for the control. This was used as a measurement of protein export efficiency. Within each experiment, between 1,000 and 20,000 cells were analyzed in each sample.This assay was used to quantify the export of E-cadherin\u2013GFP, E-cadherin\u2013RFP, HA\u2013\u03b1g), and 5 \u00b5l of each supernatant was removed (lysate). 10 \u00b5l of a 50% suspension of Neutravidin agarose beads (Thermo Fisher Scientific) was added to each sample, and they were then incubated 30 min at 4\u00b0C under rotation. Beads were pelleted , washed three times in lysis buffer, and resuspended in sample buffer. Lysates and beads were heated 3 min at 95\u00b0C and analyzed by Western blot. For quantification of protein export efficiency, band intensities in the beads and in the lysates were measured in triplicate using ImageJ software; background was removed. Within each experiment, the intensity of the band in the beads was normalized by the corresponding value in the lysates and expressed as a ratio to the chosen reference.Cells were delicately washed three times in cold PBS. Fresh EZ-Link Sulfo-NHS-LC-Biotin  was added to each well, and cells were gently rocked for 30 min at 4\u00b0C. Reaction was stopped by adding 1 ml of 100 mM glycine + 0.1% BSA in PBS. Cells were scraped, pelleted, washed delicately in Tris buffer , and lysed  for 30 min on ice. Lysates were cleared  is detailed in the figure legends. In all cases, significant adjusted P values are represented as *, P < 0.05; **, P < 0.01; ***, P < 0.001; and ****, P < 0.0001.Statistical data analysis was performed using Prism version 8.0.0 for Mac, GraphPad Software. For all quantifications provided, the means and SEM are shown. Unless otherwise stated, statistical data analysis was performed with one-way ANOVA of unmatched data. Gaussian distribution was assumed. Homoscedasticity was tested using a Brown-Forsythe test. The number of independent experiments (n = 24 for GFP-transfected cells. In n = 24 for shControl-transfected cells. In t tests between shGRAF2a cells cotransfected with RFP and RFP-GRAF2res and between shWDR44a cells cotransfected with RFP and RFP-WDR44res or RFP\u2013WDR44res \u0394Pro were used. In t test between untreated and BFA-treated GFP-transfected cells and Dunnett\u2019s multiple comparisons tests between BFA-treated transfected cells using BFA-treated GFP-transfected cells as control were used.In"}
{"text": "The gender of rats used for the experiment was wrongly mentioned as \u201cfemale.\u201d This should be corrected as \u201cmale.\u201d\u03bcg/ml\u201d should be corrected to \u201c20.93\u2009\u00b1\u20096.98\u2009\u03bcg/ml.\u201dIn Figure 4 legend, the value of serum uric acid in normal rats \u201c21.93\u2009\u00b1\u20096.98\u2009In the article titled \u201cExcretory Function of Intestinal Tract Enhanced in Kidney Impaired Rats Caused by Adenine\u201d , there wThe authors would like to apologize for any inconvenience caused."}
{"text": "A theory of cultural structures predicts the objects observed by anthropologists. We here define those which use kinship relationships to define systems. A finite structure we call a partially defined quasigroup  on a dictionary  allows prediction of certain anthropological descriptions, using homomorphisms of pdqs onto finite groups. A viable history (defined using pdqs) states how an individual in a population following such history may perform culturally allowed associations, which allows a viable history to continue to survive. The vector states on sets of viable histories identify demographic observables on descent sequences. Paths of vector states on sets of viable histories may determine which histories can exist empirically. The structures described here and their consequences imply much of what may be predicted about empirical cultures. Anthropologists very often draw illustrations of structures using methods discussed here, but based on intuition, thus have little notion of what their commonly used diagrams might predict. While ,2 defineOur notion of studying viable minimal structures\u2014which are the smallest minimal cultural structures that can \u201creproduce\u201d the ascribed social relations in one generation\u2014follows from . Our hiss = 4. The minimal structure is not intended to illustrate the actual empirical relations of each empirical generation of individuals, instead it shows \u201chow the rule operates\u201d\u2014it describes the minimal representation of the kinship and marriage rules  and let * be a partially defined binary operation on D, such that when x, y \u2208 D:(1)\u00a0If there exists an a \u2208 D such that a*x and a*y are defined and a*x = b and a*y = b then x = y, we call such object  a partially defined quasigroup, or pdq.(2)\u00a0If  is a pdq and * is fully defined on D, then  is a (complete) quasigroup.(3)\u00a0The pair L =  is a natural language with dictionary D whenever  is a pdq.(4)\u00a0If L =  is a natural language, a kinship terminology is a quasigroup subset k \u2286 L.Definition\u00a02.Let X, Y and Z be non-empty finite sets and let ,  and  be quasigroups with binary relations *, \u00b0 and \u25aa respectively. Then:(1)\u00a0A function f: X \u2192 Y is a homomorphism if f for all b, c \u2208 X, f(b * c) = f(b) \u00b0 f(c).(2)\u00a0If f: X \u2192 Z and g: Y \u2192 Z are homomorphisms then f and g are isotopic.k of a quasigroup (a kinship terminology k) onto itself forms a group (the symmetric group on k). If ,  and  are complete pdqs then (isotopic) homomorphisms classify kinship terminologies by the form of the pdq onto which they are mapped. For example, the isotopic terminologies classified as Dravidian /2\u03b4y = 2 \u2212 2\u03b1\u03c7n where y = v(t). When s\u03b1 \u2260 s\u03c7 and at least one of \u03b1s, \u03c7s is > 3, \u03b1\u03c7n > 2, then exp[r(t)] is concave down; so, r(t) is also concave down.\u2003\u2610To show Proof\u00a03.H. Since sn increases as s increases, then Lemma 1 part 1 shows that the largest value of r(t) will be set by that two-history subset \u03b1, \u03c7 of H having the largest difference between their structural numbers, hence the largest \u03b1\u03c7n in Equation (3). Combining the sn of the largest s with any other combination of sn values will result in a smaller \u03b1\u03c7n hence smaller er(t), from Equation (3).\u2003\u2610There is a finite number of two-history pairs in Observation\u00a01.H acting on a finite non-empty descent sequence G. Let \u03b1, \u03b2, \u03c7 \u2208 tH \u2286 H act on generation Gt \u2208 G. See Assume a finite non-empty set of viable histories np = 2, so includes the modal demography of each history in H; that is, the modal demography for each of \u03b1, \u03b2 and \u03c7 each appear on the line np = 2, since spsn = 2. If for any \u03b1, \u03b1v(t) = 1, then n(t) = \u03b1n, p(t) = \u03b1p, so ret)( = \u00bdn(t)p(t) = \u00bd\u03b1p\u03b1n = \u00bd2 = 1 so r(t) = 0; this occurs if all histories in tH have the same structural number . When that does not occur we have a set of two or more histories in tH each with 0 < \u03b1v(t) < 1 and thus r(t) > 0; see also Lemma 1. When H is full, assume \u03b1, \u03b2 and \u03c7 have distinct structural numbers, at least two of \u03b1, \u03b2, \u03c7 have structural numbers >3, and \u03b1 is has the lowest structural number of those three histories. Then the modal demography of \u03b1, \u03b2, \u03c7 have  \u2260  \u2260 , and the computation of r(t) appears in the values in the triangle area of tH is not full then events in the triangle area might not occur. Even if paths allow \u03b1 with \u03b2, \u03b1 with \u03c7 and \u03b2 with \u03c7 (thus the boundaries of the triangle area), values of r(t) within the triangle only occur if all three histories are allowed by H, which may be prohibited by not-full H. We call such area an un-accessed region. Thus, we study change using both full and non-full sets of histories. The curved bottom-line in Definition\u00a06.\u03b1, p\u03b1) be the modal demography for history \u03b1 \u2208 H. Let G be a finite non-empty descent sequence using H, and let Gt \u2208 G be the generation at time t. Let Ht be a face of H specified at t. Let St := { s\u03b1|\u03b1 \u2208 Hi} be the set of structural numbers St \u2286 S of histories Ht available at t. Let At := {|s\u03b1 \u2208 Si, \u03b1 \u2208 Hi,  = } be the set of modal demographies At of the histories in Ht; and let v(t) be the vector state of Gt. List the histories in H in a defined order from \u03b1 to \u03c7. Then for all \u03b1 \u2208 Ht and all  \u2208 At, let:Let H be a finite non-empty set of histories and let  be a row vector;(n2.\u00a0t) :=  be a column vector;|p3.\u00a0for all \u03b1, \u03c7t|pt) as a square matrix then for all \u03b1, \u03c7\u2208 H, arranging the sum of the inner product  at t;\u2208 H, H(t): = [n4.\u00a0T as V(t) := [v\u03b1\u03c7(t)] is a probability picture  of the vector state of a descent sequence at t;the square matrix we get by arranging the products of v(t)v(t)5.\u00a0\u03b1\u03c7(t + \u03b5) \u2212 v\u03b1\u03c7(t)] := [\u0394\u03b1\u03c7(t)]. (Notice that \u22121 \u2264 \u0394\u03b1\u03c7(t) \u2264 1)for \u03b5 \u2265 0 let V(\u0394(t)) := V(t + \u03b5) \u2212 V(t) =  at all entriesLet H be a finite non-empty set of full viable histories. Let G be a non-trivial descent sequence using H, let H. Proof of Lemma\u00a02.i\u03a3jvi(t)jv(t)ipjn. From the definition of modal demography, ip = 2/in and jp = 2/jn. The values on the diagonal of \u00bdH(t) are for each history \u03b1, \u00bd\u03b1p\u03b1n = 1. We thus examine the off-diagonal products \u03b1p\u03c7n and \u03c7p\u03b1n. Then ipjn = 2in/jn and jpin = 2jn/in. Thus, 2ni/jn = 2jn/in occurs only if in = jn, in which case ipjn = jpin = 2. This occurs only if all histories i and j have the same structural number or both have s = 2 or 3, and thus \u00bdH(t) = [1] in all entries. Otherwise stated, in this case the value from Equation (3) is \u03b1\u03c7n = 2.\u2003\u2610Assume the premises. Equations (4)\u2013(7) simply rearrange terms in \u00bd\u03a3H we can compute a proposed population growth rate r(t).\u03b1n = \u03c7n occurs if structural numbers \u03b1s, \u03c7s are <4 or whenever \u03b1s = \u03c7s; so r(t) = 0. Otherwise, then Lemma 2 implies Lemma 1, which says that ret) > 0. This occurs since \u03b1p\u03c7n does not equal \u03c7p\u03b1n; thus from Lemma 1 and  where r11 = 1 + r3, r22 = 1 \u2212 r3, r21 = r1 + ir2, r12 = r1 \u2212 ir2. That isLet r:z0, z1, z2, z3, be complex numbers such that R = z01 + z1\u03a31 + z2\u03a32 + z3\u03a33 where:Following  1, \u03a32, and \u03a33 are the standard Pauli spin matrices, where for R then z0 = 1, z1 = r1, z2 = r2 and z3 = r3. Note that \u22121 \u2264 r1, r2, r3 \u2264 1. From Definition\u00a08.t \u2208 H, let Gt \u2208 G be a generation of G using a face Ht \u2208 H at t, and let v(t) be the vector state of GtLet H be a finite non-empty set of viable histories, let G be a non-trivial viable descent sequence using histories H. 2tH = {\u03b1, \u03c7} t\u2286 H be a two-history subset of tH. Let R(t) = \u00bd[ijr(t)] be a projection, let r1(t), r2(t), and r3(t) be real numbers such that r1(t)2+r2(t)2 + r3(t)2 = 1, such that 0 \u2264 r1(t) < 1, 0 \u2264 r2(t) < 1, and such that \u03b1v(t) = \u00bdr1(t) = \u00bd(1 + r3(t)). Then R(t) is the status of tG.Let unit circle C is meant a set of points  in the plane R2 which satisfy the equation x2 + y2 = 1.A Theorem\u00a01.t, and let v(t) be the vector state of Ht. Let 2Ht = {\u03b1, \u03c7} \u2286 Ht be a non-empty subset of Ht. Let t2 > t1 > t0 define a path of v\u03b1(t) from t = t0 to t = t2 such that v\u03b1(t0) = 1 changes monotonically to v\u03b1(t1) = 0 and then monotonically back to v\u03b1(t2) = 1. That is, let r3 move from r3(t0) = 1 to r3(t1) = \u2212 1 and then back to r3(t2) = 1. Let O(t) = (n(t), p(t), r(t)). Then:Assume the premises of Definition 8. Let H be a finite non-empty set of viable histories having structural numbers s < 152 (see [(1)\u00a0trR(t) = 1;(2)\u00a0\u03c7(t) = \u00bd(1 \u2212 r3(t));v(3)\u00a02Ht is given by the main diagonal of R(t);the vector state v(t) of (4)\u00a03 = 0r(t) is a maximum when r. Theorem\u00a02.1(t) = 0. Then: (i) R(t) has \u03a3\u03a3ijrij(t) = 1; and (ii) the sum \u03a3\u222br(t)dv(t) = 0 when summed over all paths  of for pairs 2HtLet r.Theorem\u00a03.2(t) = 0. Then \u03a3\u222br(t)dv(t) = 0 when summed over all paths  of all pairs 2HtLet r.Proof of Theorem\u00a01.2tH = {\u03b1, \u03c7} \u2286 tH is the vector state v(t) = (\u03b1v(t), \u03c7v(t)) where \u03c7v(t) = 1 \u2212 \u03b1v(t). R(t) is a status and since in a status \u03b1v(t) = \u00bdr11 = \u00bd(1 + r3), and since \u03b1v(t) + \u03c7v(t) = 1 in a two-history state, then \u03c7v(t) = \u00bdr22 = \u00bd(1 \u2212 r3). In addition, also then \u03b1v(t) + \u03c7v(t) = \u00bd(1 + r3) + \u00bd(1 \u2212 r3) = 1 = trR(t), which establishes Theorems 1, 2, and 3. Establishing 4: We find r(t) is a maximum when r3 = 0, given Theorem 1(1) and 1(3), and Lemma 1(2), so when r3 = 0 then v(t) = . Assume the premises of Theorem 1. In a two history system, r1(t) = 0 soi\u03a3jrij(t) = \u00bd2 = 1 which establishes 1(1). Let tH = {\u03b1, \u03c7}. At time t, tH picks a set of modal demographies At = {, } and v(t) acts as a linear operator on At; so we getLet O(t) = (n(t), p(t), e(t)) are the predicted results at t; when \u03b1s \u2260 \u03c7s then  \u2260 . R(t) is an idempotent Hermitian matrix per Definition 7, and under the premises has r1 = 0. Then r22 + r32 = 1. We have a two history system with vector state v(t) = (\u03b1v(t), \u03c7v(t)) where \u03c7v(t) = 1 \u2212 \u03b1v(t), and where \u03b1v(t) = \u00bdr11 = \u00bd(1 + r3(t)). We let t2 > t1 > t0 define a path from t = t0 to t = t2 such that \u03b1v(t0) = 1 changes monotonically to \u03b1v(t1) = 0 and then monotonically again to \u03b1v(t2) = 1, which occurs as r3(t) moves monotonically from r3(t) = 1 to r3(t) = \u22121 and then back to r3(t) = 1. At each t, given r3(t), we compute r2(t)2 = 1 \u2212 r3(t)2. Then r2(t)2 + r3(t)2 = 1 traces a unit circle C. Theorems 2 and 3 then follow from Green\u2019s theorem. \u25a1From Lemma 2, Observation\u00a02.t \u2208 G with the sub-populations Gt using the set of histories using face Ht \u2208 H, and let v(t) be the vector state of Gt. Assume history \u03b1 \u2208 Ht, \u03b1 \u2208 Ht+1, and v\u03b1(t) = v\u03b1(t + 1) = 1. Then from t to t + 1, v(t) forms a loop. That is, the minimal descent sequence of any viable pure system \u03b1 also forms a loop, indicated also since the minimal structure of \u03b1 is a group. So any pure system is a loop. Diagrams like Let G be a population, G.sn, sp) pairs by the structural number of the identified history; or systems undergoing change in their culture. In that case the  pairs are changing and we can predict that rate of change yielding both the n(t) and p(t) for the given t, and the value of the adiabatic growth rate r(t) at t. An example of this prediction of rate of change in western Europe for about 1000 years from about AD 1000 to 1950 is given in [Following the examples of ,2 we hern(t), p(t) and r(t) of the society per generation. The methods of [Thus, study of the homotopy groups resulting from Definitions 5, 7 and 8 may thus tell us a lot about the possible paths of the empirical demography of cultures. Definitions 7 and 8 assume no physical model, but we can use their math to study the changes in vector states on histories on the predicted thods of ,29 and mthods of  in sociathods of , which ithods of , are quithods of , Postulathods of ,31 are aIn this paper, in , and in Our study also helps identify what cannot be predicted by this method. For example, sociologists and anthropologists use relationship studies to describe how individuals are \u201crelated\u201d; the minimal structure defined here based on assignments made based on the \u201cprinciples\u201d used to arrange or avoid marriages, given the natural language and the history; they do not define which specific individuals are in fact assigned to each relation. In contrast, in genetic inbreeding experiments Sewall Wright  at diagrThe ability to derive population measures from the language-based statement of rules is something new to science, and should be explored. Many other things also affect population change, and are not explored here."}
{"text": "N = p2q where p and q are balanced large primes. Suppose e, \u03d5(N)) = 1 where \u03d5(N) = p(p \u2212 1)(q \u2212 1) and d < N\u03b4 be its multiplicative inverse. From ed \u2212 k\u03d5(N) = 1, by utilizing the extended strategy of Jochemsz and May, our attack works when the primes share a known amount of Least Significant Bits(LSBs). This is achievable since we obtain the small roots of our specially constructed integer polynomial which leads to the factorization of N. More specifically we show that N can be factored when the bound N = p2q.This paper presents a cryptanalytic approach on the variants of the RSA which utilizes the modulus  Next, let e be a random integer such that gcd) = 1 where \u03d5(N) = (p \u2212 1)(q \u2212 1) is the Euler totient function. Let d be its multiplicative inverse of e such that ed \u2261 1 mod \u03d5(N). Let  be publicised for encryption purpose while p, q, \u03d5(N), d are kept private. For decryption process, private parameter d is needed. The mathematical difficulty of the RSA cryptosystem relies on the hardness of solving the integer factorization problem on N = pq, solving the key equation ed \u2212 k\u03d5(N) = 1 and solving the RSA diophantine key equation that is, C \u2261 Me mod N. Up until today, the RSA cryptosystem has remained secure.Secure communication up till the 70\u2019s was executed through symmetrical ways. In other word, both of the encryption and decryption processes used the same key. Later in 1978, the first assymetric cryptosystem went public and solved the problematic issue of distributing keys. This cryptosystem used different keys to encrypt and decrypt the data. It is known as the RSA cryptosystem d could bN = prq for r \u2265 2 is utilized. This type of modulus provides advantage for both key generation and the decryption algorithms provided the Chinese Remainder Theorem is utilized  fo fo\u03c9,n\u2208Z+Theorem 1 [Letbe the lattice generated by a set of basisand has the dimension \u03c9. The reduced basisproduced by the LLL algorithm satisfiesfor all 1 \u2264 i \u2264 \u03c9.heorem 1  LetLbeLLL algorithm has been extensively applied in order to find reduced basis vectors in a lattice. For instance, [LLL algorithm to find a reduced basis of the lattice generated by the modular polynomial, [Since its invention, nstance,  introducynomial,  managed ynomial,  describeTheorem 2 [Letbe a polynomial with at \u03c9 monomials. Suppose thatwhereandThenholds over integers.heorem 2  Leth. These results will be used to approximate the bound for one of the variables in our polynomial.The following results by  show an Lemma 1Let N = p2q with q < p < 2q. ThenLemma 2Let N = p2q with q < p < 2q. ThenN = p2q consists of two primes that share a known amount of their LSBs.The following lemma is reformulated from result . It consLemma 3Let N = p2q be the modulus and suppose that p \u2212 q = 2bu for a known value of b. Let p = 2bp1 + u0and q = 2bq1 + u0where u0is a solution to p3 \u2261 N (mod 2b). Ifp2 + pq \u2212 p = 2b3s + s0 \u2212 v whereProof. Suppose that p \u2212 q = 2bu. Then q = p \u2212 2bu andp3 \u2261 N (mod 2b). Let u0 be a solution of the modular form p3 \u2261 N (mod 2b). Then, p \u2261 u0 (mod 2b) is a solution which implies that p = 2bp1 + u0 where p1 is a positive integer. Now we have,q1 = p1 \u2212 u. Using N = p2q, we getp3\u22122bup2.Hence, fu0, 2b3) = 1, we multiply (From  we deducN = p2q which works when there has a known amount of LSBs shared between the primes p and q.This section presents the attack on modulus Theorem 3Let N = p2q be modulus such that p \u2212 q = 2bu where 2b \u2248 N\u03b1. Let e be a public exponents satisfying e \u2248 N\u03b3and ed \u2212 k\u03d5(N) = 1. Suppose that d < N\u03b4. Then N can be factored in time polynomial ifProof. Suppose we have public exponent e and key equationp \u2212 q = 2bu. Then, from Lemma 3, p2 + pq \u2212 p can be rewritten in the form p2 + pq \u2212 p = 2b3s + s0 \u2212 v where u0 is a solution of the modular equation p3 \u2261 N (mod 2b). Thus, substitutes  is a root of f and can be solved by using Coppersmith\u2019s technique [e1, e2) = N\u03b3.max(d) < X1 = N\u03b4.max(p \u2212 q = 2bu with 2b \u2248 N\u03b1 and p2 + pq \u2212 p = 2b3s + s0 \u2212 v with 2b3s \u2212 v < X3 = N2/3.stitutes  from Lempq\u2212p))=1.Suppose s\u2212v)\u22121=0.We transechnique  due to iS and M be defined as:fm\u22121 satisfiesM areW satisfiesa4 is coprime with R. We want to work with a polynomial that has constant term 1, thus we define g and h as:g and h with dimensionm = 3 and t = 1.The bounds of the variables are fixed as follows:1x2i2x3i3The monog and h and their combinations share the root  modulo R. A new basis with short vectors is produced after applying the LLL algorithm to the lattice i = 1, 2, let fi be two short vectors of the reduced basis. Each fi shares the roots . Then by Theorem 3, we havefi for i = 1, 2 to fulfill the condition ofNext, defineposed by , we forc\u03c9 = |M| and |M| \u2212 |M\u2216S| = |S|, we gett = \u03c4m, then,m3, the inequality . By Assumption 1 in Section 2, the solution of the roots can be extracted using resultant technique. By using the third root 2b3s \u2212 v, we compute p2 + pq \u2212 p = 2b3s + s0 \u2212 v. This value is then used to find \u03d5(N) and since \u03d5(N) = p(p \u2212 1)(q \u2212 1) we can factor out p by taking the gcd). By knowing the value of p, we can factor the modulus N.Using 0)Using , we getjsj<W|S|.Using  = 1 where \u03d5(N) = pr\u22121(pr \u2212 1)(q \u2212 1). Note that in these former attacks, their primes do not share any amount of LSBs. Their bounds depend only on the value of r. We compare the results with various values of \u03b3 = logN(e). Our corollary is as follow.We compare our bounds with these three former attacks,  and 19 tCorollary 1Let N = p2q be the modulus where q < p < 2q. Let e be a public exponent satisfying ed \u2212 k\u03d5(N) = 1 for \u03d5(N) = p(p \u2212 1)(q \u2212 1). Suppose that d < N\u03b4. Then N can be factored in time polynomial if\u03b4 of [r = 2. We describe their bound for d as in the Note that the bounds for \u03b4 of  and 19 r\u03b4 increases inversely proportional to the value of \u03b3.Remark 1 In Corollary 1, it states that N can be factored ife = N\u03b3. We have\u03b3. A direct calculation shows that \u03b3, this translates into d \u2248 N.N = p2q where the primes share an amount of LSB(s). Based on the result of Nitaj et al. [p2 + pq \u2212 p which is the value of N \u2212 \u03d5(N). We use the result from our lemma in our theorem which then yields a set of integer polynomials. By applying the extended strategy of Jochemsz and May, one is able to determine the small roots of our integer polynomial and thus factoring the modulus N. We show that N can be factored when d < N\u03b4 for N = p2q.We describe an attack related to partial key exposure. Our attack works upon the modulus j et al. , we refoS1 Appendix(PDF)Click here for additional data file."}
{"text": "Heaney LG, Busby J, Hanratty CE, et al. Composite type-2 biomarker strategy versus a symptom\u2013risk-based algorithm to adjust corticosteroid dose in patients with severe asthma: a multicentre, single-blind, parallel group, randomised controlled trial. Lancet Respir Med 2020; 9: 57\u201368\u2014In this Article, in table 1, column \u201c1\u201d, the threshold ranges quoted should read \u201c\u226515\u201329\u201d, \u201c\u2265150\u2013299\u201d, and \u201c\u226545\u201354\u201d, and in column \u201c2\u201d, the ranges should read \u201c\u226530\u201d, \u201c\u2265300\u201d, and \u201c\u226555\u201d. These corrections have been made to the online version as of Jan 4, 2021 and will be made to the printed version."}
{"text": "Given a 1\u2010ipm \u03bc, denote by G\u03bc the associated random graph model. Let \u21331,\u2a7ep(G) denote the collection of 1\u2010ipms \u03bc on G for which each edge is included in G\u03bc with probability at least p. For G=Z2, Balister and Bollob\u00e1s asked for the value of the least p\u22c6 such that for all p\u2009>\u2009p\u22c6 and all \u03bc\u2208\u21331,\u2a7ep(G), G\u03bc almost surely contains an infinite component. In this paper, we significantly improve previous lower bounds on p\u22c6. We also determine the 1\u2010independent critical probability for the emergence of long paths on the line and ladder lattices. Finally, for finite graphs G we study fG1,\u2009(p), the infimum over all \u03bc\u2208\u21331,\u2a7ep(G) of the probability that G\u03bc is connected. We determine fG1,\u2009(p) exactly when G is a path, a complete graph and a cycle of length at most 5.A probability measure  Spanning subgraphs of G are called configurations. In a configuration H, an edge is said to be open if it belongs to H, and closed otherwise. A bond percolation model on the host graph G is a probability measure \u03bc on the spanning subgraphs of G, that is, on the space of configurations. Given such a measure, we denote the corresponding random graph model by G\u03bc, and refer to it as the \u03bc\u2010random graph or \u03bc\u2010random configuration.Let \u03bc where the states (open or closed) of edges in subsets F1,\u2009F2 of E in a \u03bc\u2010random configuration are independent provided that the edges in F1 and F2 are \u201csufficiently far apart.\u201d To make this more precise, we make use of the following definition.In this paper, we study bond percolation models Definition 1F1,\u2009F2\u2009\u2286\u2009E are k\u2010distant if F1\u2009\u2229\u2009F2\u2009=\u2009\u2205 and the shortest path of G from an edge in F1 to an edge in F2 contains at least k edges. A bond percolation model \u03bc on G is k\u2010independent if for any pair  of k\u2010distant edge sets, the intersections G\u03bc\u2229F1 and G\u03bc\u2229F2 are independent random variables.Two edge sets \u03bc is 0\u2010independent if each edge of G\u03bc is open at random independently of all the others, that is, \u03bc can be viewed as a product of Bernoulli measures on the edges of G. A well\u2010studied 0\u2010independent model is the Erd\u0151s\u2010R\u00e9nyi random graph Gn,\u2009p, where the host graph is G\u2009=\u2009Kn, the complete graph on n vertices, and where \u03bc, known as the p\u2010random measure, sets each edge to be open with probability p, independently of all the others.So for example site percolation on the square integer lattice. In this case the host graph is the square integer lattice Z2 (where two vertices are joined by an edge if they lie at distance 1 apart), and the measure \u03bc=\u03bcsite(\u03b8) is obtained by switching each vertex of Z2on at random with probability \u03b8, independently of all the others, and by setting an edge to be open if and only if both of its endpoints are switched on. Site percolation measures may be defined more generally on any host graph in the natural way.In this paper, we focus on the next strongest notion of independence, namely 1\u2010independence. Measures that are 1\u2010independent have the property that events determined by vertex\u2010disjoint edge sets are independent. For many 1\u2010independent models, the randomness can be thought to \u201creside in the vertices.\u201d An important example of a 1\u2010independent model is that of v\u2009\u2208\u2009V(G) a state Sv at random, and set an edge uv to be open if and only if f\u2009=\u20091, for some deterministic function f (which may depend on u and v). We refer to such measures as vertex\u2010based measures , it is well\u2010known that a graph G may support many 1\u2010independent measures which cannot be realized as vertex\u2010based measures or as general \u201cblock factors,\u201d see for instance\u00a0G, it is not feasible to generate or simulate the collection of 1\u2010independent measures of G.An important point to note is that while all 0\u2010independent bond percolation models are a product of Bernoulli measures on the edges of Definition 2\u03bc on a host graph G, the (lower)\u2010edge\u2010probability of \u03bc is Given a bond percolation model p\u2010random measure has edge\u2010probability p, while a site percolation measure with parameter \u03b8 has edge\u2010probability \u03b82. The collection of k\u2010independent bond percolation models \u03bc on a graph G with edge\u2010probability d(\u03bc)\u2a7ep is denoted by \u2133k,\u2a7ep(G).So for instance a Remark 3\u03bc\u2208\u2133k,\u2a7ep(G), we may readily produce a measure \u03bc\u02dc\u2208\u2133k,\u2a7ep(G) such that \u03bc\u02dc({eis open})=p for all e\u2009\u2208\u2009E(G) via random sparsification: independently delete each edge e of G\u03bc with probability p/\u03bc({eis open})\u2208. The resulting bond percolation model on G is clearly k\u2010independent and has the property that each edge is open with probability exactly p; the corresponding bond percolation measure \u03bc\u02dc thus has the required properties.Given a measure 1.2Percolation theory is the study of random subgraphs of infinite graphs. Since its inception in Oxford in the 1950s, it has blossomed into a rich theory and has been the subject of several monographs\u00a0G, and let \u03bc be a 0\u2010independent bond percolation model on G. We say that percolation occurs in a configuration H on G if H contains an infinite connected component of open edges. By Kolmogorov's zero\u2010one law, for G and \u03bc as above, percolation is a tail event whose \u03bc\u2010probability is either zero or one. This allows one to thus define the Harris critical probability pc0,\u2009(G) for 0\u2010independent percolation: In the most fundamental instance of this problem, consider an infinite, locally finite connected graph Problem 4Given an infinite, locally finite connected graph G, determine pc0,\u2009(G).p0,c(Z2) to be 1/2.One of the cornerstones of percolation theory\u2014and indeed one of the triumphs of twentieth century probability theory\u2014is the Harris\u2010Kesten theorem, which established the value of Let\u03bcbe the p\u2010random measure onZ2. Then. Z2 if the assumption of 0\u2010independence is weakened to k\u2010independence. In particular, how much can local dependencies between the edges postpone the global phenomenon of percolation?In this paper, we focus on the question of what happens to the Harris critical probability in Definition 5G be an infinite, locally finite connected graph and let k\u2208N0. The Harris critical probability for k\u2010independent percolationG is defined to be: Let Problem 6Determinep1,c(Z2).Problem\u00a0renormalisation, which entails reducing a 0\u2010independent model to a 1\u2010independent one (possibly on a different host graph), trading in some dependency for a boost in edge\u2010probabilities. Renormalisation arguments feature in many proofs in percolation theory; a powerful and particularly effective version of such arguments was developed by Balister, Bollob\u00e1s, and Walters\u00a0A standard technique in percolation is Z2  and Monte\u2010Carlo simulations to estimate the probabilities of bounded events, has been applied to give rigorous confidence intervals for critical probabilities/intensities in a wide variety of settings: various models of continuum percolation\u00a0Z3\u00a0Their method, which relies on comparisons with 1\u2010independent models on k\u2010independent bond percolation models for k\u2a7e1 are as natural an object of study as the more widely studied 0\u2010independent ones.From a more practical standpoint, many of the real\u2010world structures motivating the study of percolation theory exhibit short\u2010range interactions and local dependencies. For example a subunit within a polymer will interact and affect the state of nearby subunits, but perhaps not of distant ones. Similarly, the position or state of an atom within a crystalline network may have a significant influence on nearby atoms, while long\u2010range interactions may be weaker. Within a social network, we would again expect individuals to exert some influence in esthetic tastes or political opinions, say, on their circle of acquaintance, and also expect that influence to fade once we move outside that circle. This suggests that many very different 1\u2010independent models with edge\u2010probability p, and they tend to be harder to study than 0\u2010independent ones due to the extra dependencies between edges. In particular simulations are often of no avail to formulate conjectures or to get an intuition for 1\u2010independent models in general. Moreover, while the theoretical motivation outlined above is probabilistic in nature, the problem of determining a critical constant like p1,c(Z2) is extremal in nature\u2014one has to determine what the worst possible 1\u2010independent model is with respect to percolation\u2014and calls for tools from the separate area of extremal combinatorics.Despite the motivation outlined above, 1\u2010independent models remain poorly understood. To quote Balister and Bollob\u00e1s from their 2012 paper: \u201c1\u2010independent percolation models have become a key tool in establishing bounds on critical probabilities [\u2026]. Given this, it is perhaps surprising that some of the most basic questions about 1\u2010independent models are open.\u201d There are in fact some natural explanations for this state of affairs. As remarked on in the previous subsection, there are In this paper, we continue Balister and Bollob\u00e1s's investigation into the many open problems and questions about and on these measures. Before we present our contributions to the topic, we first recall below previous work on 1\u2010independent percolation.1.3k\u2010independent models by 0\u2010independent ones were given by Liggett, Schonmann and Stacey\u00a0p1,c(Z2)<1. Balister, Bollob\u00e1s and Walters\u00a0Z2 with edge\u2010probability at least 0.8639, the origin has a strictly positive chance of belonging to an infinite open component. This remains to this day the best upper bound on p1,c(Z2). In a different direction, Balister and Bollob\u00e1s\u00a0p1,c(G)\u2a7e12 for any infinite, locally finite connected graph G. In the special case of the square integer lattice Z2, they recalled a simple construction due to Newman which gives \u03b8site is the critical value of the \u03b8\u2010parameter for site percolation, that is, the infimum of \u03b8\u2208 such that switching vertices of Z2 on independently at random with probability \u03b8 almost surely yields an infinite connected component of on vertices. Plugging in the known rigorous bounds for 0.556\u2a7d\u03b8site\u2a7d0.679492\u00a0p1,c(Z2)\u2a7e0.5062, while using the nonrigorous estimate \u03b8site\u22480.592746 \u2a7e0.5172.Some general bounds for stochastic domination of k\u2010independent setting by Mathieu and Temmel\u00a0With regards to other lattices, Balister and Bollob\u00e1s completed a rigorous study of 1\u2010independent percolation models on infinite trees\u00a01.4p1,c(Z2) with the following theorems.In this paper, we make a three\u2010fold contribution to the study of Problem\u00a0Theorem 7For alld\u2208N\u2a7e2, we have thatd\u2009=\u20092 given in \u03b8site again denote the critical threshold for site percolation. Then the following holds.Theorem Theorem 8\u03b8site\u2a7e0.556 into Theorem p1,c(Z2)\u2a7e0.531136, which does slightly worse than Theorem \u03b8site\u22480.592746 yields a significantly stronger lower bound of p1,c(Z2)\u2a7e0.554974.Substituting the rigorous bound p1,c(Z2), and in particular to establish some 1\u2010independent analogs of the Russo\u2010Seymour\u2010Welsh (RSW) lemmas on the probability of crossing rectangles, we investigate the following problems. Let Pn denote the graph on the vertex set {1,\u20092,\u2009\u2026\u2009n} with edges {12,\u200923,\u2009\u2026\u2009,\u2009(n\u2009\u2212\u20091)n}, that is, a path on n vertices. Given a connected graph G, denote by Pn\u2009\u00d7\u2009G the Cartesian product of Pn with G. A left\u2010right crossing of Pn\u2009\u00d7\u2009G is a path from a vertex in {1}\u2009\u00d7\u2009V(G) to a vertex in {n}\u2009\u00d7\u2009V(G). We define the crossing critical probability for 1\u2010independent percolation on Pn\u2009\u00d7\u2009G to be Pn\u2009\u00d7\u2009G, there is a strictly positive probability of being able to cross Pn\u2009\u00d7\u2009G from left to right.Second, motivated by efforts to improve the upper bounds on Problem 9Givenn\u2208Nand a finite, connected graph G, determine p1,\u2009\u00d7(Pn\u2009\u00d7\u2009G).n\u2009\u2192\u2009\u221e in Problem\u00a0G be an infinite, locally finite connected graph. The long paths critical probability for 1\u2010independent percolation on G is G.Problem\u00a0Problem 10Given an infinite, locally finite, connected graph G, determine p\u2113p1,\u2009(G).G consists of a vertex or an edge p1,\u2113p(Z)=34, and(ii)p1,\u2113p(Z\u00d7P2)=23.p1,\u2009\u00d7(Pn\u2009\u00d7\u2009G) and p1,\u2113p(Z\u00d7G) for a variety of graphs G. We summarize the latter, less technical, set of results below. Let Cn and Kn denote the cycle and the complete graph on n vertices respectively.Note that part (i) of Theorem Theorem 12We have that(i)0.5359\u2026=4\u221223\u2a7dp1,\u2113p(Z\u00d7Cn)\u2a7dp1,\u2113p(Z\u00d7Pn)\u2a7d23for alln\u2a7e3;(ii)p1,\u2113p(Z\u00d7K3)\u2a7d116(13\u22125512814\u22122513+12814\u22122513)=0.63154\u2026\u00a0;(iii)p1,\u2113p(Z\u00d7C4)\u2a7d(3\u22123)/2=0.63397\u2026\u00a0;(iv)p1,\u2113p(Z\u00d7C5)\u2a7d0.63895\u2026\u00a0;(v)0.5359\u2026=4\u221223\u2a7dlimn\u2192\u221ep1,\u2113p(Z\u00d7Kn)\u2a7d59=0.5555\u2026.Pn\u2009\u00d7\u2009G to the probability of a given copy of G being connected in that model. This motivated our third contribution to the study of 1\u2010independent models in this paper, namely an investigation into the connectivity of 1\u2010independent random graphs.A key ingredient in the proof of Theorems\u00a0Definition 13G be a finite connected graph. For any p\u2009\u2208\u2009, we define the k\u2010independent connectivity function of G to be Let Problem 14Given a finite connected graph G, determine fG1,\u2009(p).G is a path, a complete graph or a cycle on at most 5 vertices.We resolve Problem\u00a0Theorem 15Givenn\u2208N\u2a7e2and p\u2009\u2208\u2009, let\u03b8=\u03b8(p):=1+4p\u221232andpn:=143\u2212tan2\u03c0n+1. We have thatTheorem 16Givenn\u2208N\u2a7e2and p\u2009\u2208\u2009, let\u03b8=\u03b8(p):=1+2p\u221212andpn:=12(1\u2212tan2(\u03c02n)). We have thatIn particular,Theorem 17For p\u2009\u2208\u2009 we have thatTheorem 18For p\u2009\u2208\u2009 we have that\u2133k,\u2a7dp(G) denote the collection of 1\u2010independent measures \u03bc on G such that supe\u2208E(G)\u03bc{eis open}\u2a7dp. Set We also consider the opposite problem to Problem\u00a0Problem 19Given a finite connected graph G, determine FG1,\u2009(p).G is a path, a complete graph or a cycle on at most 5 vertices.We resolve Problem\u00a0Theorem 20For alln\u2208Nwithn\u2a7e2,F1,Pn(p)=p\u230an2\u230b.Theorem 21For alln\u2208Nwithn\u2a7e2,F1,Kn(p)=1\u2212f1,Kn(1\u2212p).Theorem 22Theorem 23\u03bc on Kn, Pn, C4 and C5\u2014i.e., the range of values \u03bc({connected}) can take if every edge is open with probability p. In Figure G with plots of fG1,\u2009(p), FG1,\u2009(p) and f0,G(p):=\u03bcGpis connected, where Gp is the 0\u2010independent model on G obtained by setting each edge of G to be open with probability exactly p, independently at random.Together, Theorems\u00a01.5Our first set of results, Theorems\u00a0Z\u00d7G. This result is used in Sections\u00a0In Section fG1,\u2009(p) and FG1,\u2009(p) when G is a path, a complete graph or a short cycle. We apply these results in Section In Sections\u00a01.6N for the set of natural numbers {1,\u20092,\u2009\u2026}, N0 for the set N\u222a{0}, and N\u2a7ek for the set of natural numbers greater than or equal to k.We write n]:\u2009=\u2009{1,\u20092,\u2009\u2026\u2009n}. Given a set A, we write Ar) where V\u2009=\u2009V(G) and E\u2009=\u2009E(G)\u2009\u2286\u2009V(G)(2) denote the vertex set and edge set of G respectively. Given a subset A\u2009\u2286\u2009G, we denote by G[A] the subgraph of G induced by A. We also write N(A) for the set of vertices in G adjacent to at least one vertex in A.We set ][[spimspace=\"0.3em\"]] mod[[spimspace=\"0.3em\"]] 6, then we color v Blue.If v is a vertex in Tk, where k\u2009\u2261\u20091[[spimspace=\"0.3em\"]][[spimspace=\"0.3em\"]] mod[[spimspace=\"0.3em\"]] 6, then we color v Red with probability q/2 and color it Blue otherwise.If v is a vertex in Tk, where k\u2009\u2261\u20092[[spimspace=\"0.3em\"]][[spimspace=\"0.3em\"]] mod[[spimspace=\"0.3em\"]] 6, then we color v Red with probability q and put it in the Inwards state I otherwise.If v is a vertex in Tk, where k\u2009\u2261\u20093[[spimspace=\"0.3em\"]][[spimspace=\"0.3em\"]] mod[[spimspace=\"0.3em\"]] 6, then we color v Red.If v is a vertex in Tk, where k\u2009\u2261\u20094[[spimspace=\"0.3em\"]][[spimspace=\"0.3em\"]] mod[[spimspace=\"0.3em\"]] 6, then we color v Blue with probability q/2 and color it Red otherwise.If v is a vertex in Tk, where k\u2009\u2261\u20095[[spimspace=\"0.3em\"]][[spimspace=\"0.3em\"]] mod[[spimspace=\"0.3em\"]] 6, then we color v Blue with probability q and put it in the Inwards state I otherwise.If Let Tk\u2009+\u20093,\u2009Tk\u2009+\u20094,\u2009Tk\u2009+\u20095 are the same as those for Tk,\u2009Tk\u2009+\u20091,\u2009Tk\u2009+\u20092 respectively, except with red and blue interchanged. See Figure T0,\u2009T1,\u2009T2 and T3 when d\u2009=\u20092. Now suppose that e\u2009=\u2009{v1,\u2009v2} is an edge in Zd. First we say that the edge e is open if either both v1 and v2 are Blue or both v1 and v2 are Red. We also say the edge e is open if, for some k, we have that v1\u2009\u2208\u2009Tk, v2\u2009\u2208\u2009Tk\u2009+\u20091, and v2 is in state I. In all other cases we say that the edge e is closed. It is clear that this gives a 1\u2010independent measure on Zd as it is vertex\u2010based, and it is also easy to check that every edge is present with probability at least 4\u221223.Note that the rules for \u03bc, and let G:=Zd. We claim that in G\u03bc, for all k\u2009\u2261\u20090[[spimspace=\"0.3em\"]][[spimspace=\"0.3em\"]] mod[[spimspace=\"0.3em\"]] 3, there is no path of open edges from Tk to Tk\u2009+\u20093. Suppose this is not the case, and P is some path of open edges from a vertex in Tk to Tk\u2009+\u20093. We first note that P cannot include a vertex in state I, as such a vertex would be in Tk\u2009+\u20092 and would only be adjacent to a single edge. Thus every vertex of P is either Blue or Red. However, as one end vertex of P is Blue and the other end vertex is Red, and there are no open edges with different colored end vertices, we have that such a path P cannot exist. As a result, every component of G\u03bc is sandwiched between some Tk\u2009\u2212\u20093 and Tk\u2009+\u20093, where k\u2009\u2261\u20090[[spimspace=\"0.3em\"]][[spimspace=\"0.3em\"]] mod[[spimspace=\"0.3em\"]] 3, and so is of finite size. Thus we have that p1,c(Zd)\u2a7e4\u221223.Call this measure G, and a vertex set A\u2009\u2286\u2009V(G), let A\u00af be the closure of A under 2\u2010neighbor bootstrap percolation on G. That is, let A\u00af:=\u222ai\u2a7e0Ai, where A0:\u2009=\u2009A and for i\u2a7e1G has the finite 2\u2010percolation property if, for every finite set A\u2009\u2286\u2009V(G), we have that A\u00af is finite.The construction in Theorem Corollary 24If G has the finite 2\u2010percolation property, thenp1,c(G)\u2a7e4\u221223.V(G) in the following way: pick any vertex v and set T0:\u2009=\u2009{v}. For k\u2a7e1 let w\u2009\u2208\u2009Tk, then w is only adjacent to vertices in Tk\u2009\u2212\u20091,\u2009Tk and Tk\u2009+\u20091. Moreover, w is adjacent to at most one vertex in Tk\u2009\u2212\u20091\u2014this is the crucial property needed for our construction. Since G has the finite 2\u2010percolation property, each Tk is finite. Thus we can use the Tk to construct a nonpercolating 1\u2010ipm on G in the exact same fashion as done for Zd in Theorem I are still dead ends, being incident to a unique edge), which in turn shows that p1,c(G)\u2a7e4\u221223.Partition p1,c3,4,6,4\u2a7e0.52981682. As this is less than 4\u221223, we have that our construction gives the (rigorous) improvement of p1,c3,4,6,4\u2a7e4\u221223.An example of a lattice with the finite 2\u2010percolation property is the lattice , where here we are using the lattice notation of Gr\u00fcnbaum and Shephard Proof of Theorem 8\u03b5\u2009>\u20090 sufficiently small so that q:=\u03b8site(Z2)\u2212\u03b5 is strictly larger than 1/4. For each vertex v\u2208Z2, we assign to it one of three states: On, L or D, and we do this independently for every vertex. We assign v to the On state with probability q, we assign it to the L state with probability 12(1\u2212q), and else we assign it to the D state with probability 12(1\u2212q).Fix e is open if both of its vertices are in the On state. If a vertex is in state L, then the edge adjacent and to the left of it is open. Similarly, if a vertex is in state D, then the edge adjacent and down from it is open. All other edges are closed. See Figure We now describe which edges are open based on the states of the vertices. We first say that the edge Z2 as it is vertex\u2010based, and every edge is present with probability q2+12(1\u2212q). Call this measure \u03bc and let G:=Z2. We will show that every component of G\u03bc has finite size. We begin by first proving an auxiliary lemma. Let t\u2208, and let us define another 1\u2010independent measure on Z2, which we call the left\u2010down measure with parameter t. In the left\u2010down measure, each vertex of Z2 is assigned to one of three states: Off, L or D, and we do this independently for every vertex. For each vertex v\u2208Z2, we assign it to state L with probability t, we assign it to state D with probability t, and we assign it to state Off with probability 1\u2009\u2212\u20092t. As above, if a vertex is in state L, then the edge adjacent and to the left of it is open, while if a vertex is in state D, then the edge adjacent and down from it is open. All other edges are closed. We use \u03bdt to denote the left\u2010down measure with parameter t.It is easy to see that this is a 1\u2010independent measure on Lemma 25If0\u2a7dt\u2a7d38, then all components inG\u03bdtare finite almost surely.z:=1\u22121\u22122t. As 0\u2a7dt\u2a7d38 we have that 0\u2a7dz\u2a7d12. We start by taking a random subgraph of Z2 where every edge is open with probability z, independently of all other edges. We then further modify it as follows. For each vertex v\u2009=\u2009 we look at the state of the edge e1 from v to the vertex , and the state of the edge e2 from v to the vertex . If at least one of e1 or e2 is closed, we do not change anything. However, if both e1 and e2 are open, with probability 12 we close the edge e1, and otherwise we close the edge e2. We do this independently for every vertex v of Z2.Let \u03bdt, the left\u2010down measure with parameter t. Indeed, to each vertex v\u2009=\u2009 as above we may assign a state Off if both the edge e1(v) to the vertex to the left of v and the edge e2(v) to the vertex below v are closed, a state L if e1(v) is open and a state D if e2(v) is open. The probabilities of these three states are (1\u2009\u2212\u2009z)2\u2009=\u20091\u2009\u2212\u20092t, t and t respectively, and since the vertex states depend only on the pairwise disjoint edge sets {e1(v),e2(v)}v\u2208Z2, they are independent of one another just as in the \u03bdt measure.It is easy to see that this is an equivalent formulation of \u03bdt to the 0\u2010independent bond percolation measure \u03be on Z2 with edge\u2010probability z. In this coupling we have that if an edge e is open in G\u03bdt, then it is also open in G\u03be. As z\u2a7d0.5 we have that all components in G\u03be are finite by the Harris\u2010Kesten theorem, and so we also have that all of the components in G\u03bdt are finite too.Thus we have coupled 0\u2a7dt<12, then almost surely all components in G\u03bdt are finite. We make no use of this stronger result in this paper, so we omit its proof. It is also clear that when t=12, every vertex in G\u03bdt is part of an infinite path consisting solely of steps to the left or steps downwards, and so percolation occurs in G\u03bdt at this point.By considering an appropriate branching process it is possible to prove the stronger result that if \u03bc, where every vertex is in state On, L or D. Recall that our aim is to show that all components have finite size in G\u03bc. Consider removing all vertices in state L or D, and also any edges adjacent to these vertices. What is left will be a collection of components consisting only of edges between vertices in the On state, which we call the On\u2010sections. The black edges in Figure q<\u03b8site(Z2), we have that almost surely every On\u2010section is finite. Similarly, consider removing all edges in the On\u2010sections. What is left will be a collection of edges adjacent to vertices in the L or D states. We call these components the LD\u2010sections; the dashed red edges in Figure LD\u2010sections. As each vertex is in state L with probability 12(1\u2212q)\u2a7d38 and in state D with the same probability, Lemma LD\u2010section is finite.Let us return to our original 1\u2010independent measure v in state L orient the open edge to the left of it away from v, while for each vertex v in state D orient the open edge below it away from v. This gives a partial orientation of the open edges of G\u03bc, in which every vertex in state L or D has exactly one edge oriented away from it, and vertices in state On have no outgoing edge. Furthermore, if v1 is a vertex in the On state and v2 is a vertex in the L or D state, then the edge between them is oriented from v2 to v1. Since the LD\u2010sections are almost surely finite, this implies the LD sections under this orientation consist of directed trees, each of which is oriented from the leaves to a unique root, which is in the On state. In particular, every LD\u2010section attaches to at most one On\u2010section. As such, almost surely every component in G\u03bc consists of at most one On\u2010section, and a finite number of finite LD\u2010sections attached to it. Thus almost surely every component in G\u03bc is finite.For each vertex 3G be a finite connected graph. Set v(G):\u2009=\u2009|V(G)|. Recall that for any 1\u2010independent bond percolation measure \u03bc\u2208\u21331,\u2a7ep(G), we have \u03bc(G\u03bcis connected)\u2a7ef1,G(p).Let Theorem 26If p satisfiesfor some\u03b1\u2208. For any n\u2208N, the restriction of \u03bc to [n]\u2009\u00d7\u2009V(G) is a measure from \u21331,\u2a7ep(Pn\u00d7G), and clearly all such measures can be obtained in this way. Furthermore, for every n\u2208N, the restriction of \u03bc to {n}\u2009\u00d7\u2009V(G) is a measure from \u21331,\u2a7ep(G), and in particular the subgraph of (Z\u00d7G)\u03bc induced by {n}\u2009\u00d7\u2009V(G) is connected with probability at least fG1,\u2009(p).Consider an arbitrary measure \u03bc\u2010random graph (Z\u00d7G)\u03bc. For n\u2a7e1 let Yn be the event that [n]\u2009\u00d7\u2009V(G) induces a connected subgraph. For n\u2a7e2, let Xn be the event that [n\u2009\u2212\u20091]\u2009\u00d7\u2009V(G) induces a connected subgraph and at least one vertex in {n}\u2009\u00d7\u2009V(G) is connected to a vertex in {n\u2009\u2212\u20091}\u2009\u00d7\u2009V(G). For n\u2009=\u20091, set X1 to be the trivially satisfied event occurring with probability 1. For n\u2a7e1, let Vn be the event that {n}\u2009\u00d7\u2009V(G) induces a connected subgraph, and for n\u2a7e2 let Hn be the event that at least one of the edges from {n\u2009\u2212\u20091}\u2009\u00d7\u2009V(G) to {n}\u2009\u00d7\u2009V(G) is present.We consider the Xn\u2009=\u2009Yn\u2009\u2212\u20091\u2009\u2229\u2009Hn and Xn\u2009\u2229\u2009Vn\u2009\u2286\u2009Yn. From here, we obtain the following inclusions:Xn\u2009+\u20091)c\u2009\u2229\u2009Yn\u2009=\u2009(Hn\u2009+\u20091)c\u2009\u2229\u2009Yn,(Yn\u2009\u2229\u2009Yn\u2009\u2212\u20091\u2009\u2287\u2009(Vn\u2009\u2229\u2009Xn)\u2009\u2229\u2009Yn\u2009\u2212\u20091, andYn)c\u2009\u2229\u2009Xn\u2009\u2286\u2009(Vn)c\u2009\u2229\u2009Xn. and (b) we have, We begin by establishing two inductive relations for the sequences xn\u2a7d\u03b1f1,G(p) for all n. Indeed x1\u2009=\u20090, and if xn\u2a7d\u03b1f1,G(p), then by Second, using (c), Finally, we have that G, fG1,\u2009(p) is a nondecreasing function of p with fG1,\u2009(1)\u2009=\u20091. Thus the function )2 is also nondecreasing in p and attains a maximum value of 1 at p\u2009=\u20091. On the other hand, the function 4(1\u2009\u2212\u2009p)v(G) is strictly decreasing in p and is equal to 4 at p\u2009=\u20090. Thus there exists a unique solution p\u22c6\u2009=\u2009p\u22c6(G) in the interval  to the equation For any finite connected graph Theorem Corollary 27Let G be a finite connected graph. Let p\u22c6\u2009=\u2009p\u22c6(G) be as above. Then\u03b1=1/2.Apply Theorem 4In this section we prove a lemma that we shall use in Sections\u00a0Lemma 28Let G be a finite graph, and let\ud835\udcac:={QH(\u03b8):H\u2286G}be a set of polynomials with real coefficients, indexed by subgraphs of G. Given\u03b8\u2208C, let\u03bc\u03b8be the following function from subgraphs of G toC:Suppose there exists a nontrivial intervalI\u2286Rsuch that, for all\u03b8\u2208I, the function\u03bc\u03b8defines a 1\u2010ipm on G. Suppose further that there exists a setX\u2286Csuch that, for all\u03b8\u2208Xand all H\u2009\u2286\u2009G,\u03bc\u03b8(H)is a nonnegative real number. Then\u03bc\u03b8is a 1\u2010ipm on G for all\u03b8\u2208X.\u03bc\u03b8 is a measure on G for all \u03b8\u2208X. As \u03bc\u03b8(H) is a nonnegative real number for all \u03b8\u2208X and all H\u2009\u2286\u2009G, all that is left to prove is that \u03b8 with real coefficients, and is equal to zero for all \u03b8 in the interval I. By the fact that a nonzero polynomial over any field has only finitely many roots, the polynomial is identically zero and so \u03b8.We start by proving that \u03bc\u03b8 is a 1\u2010ipm on G for all \u03b8\u2208X. To do this we must show that the following holds true for all \u03b8\u2208X, for all A,\u2009B\u2009\u2286\u2009V(G) such that A and B are disjoint, and all G1 and G2 such that G1 is a subgraph of G[A] while G2 is a subgraph of G[B]: \u03b8 with real coefficients\u2014the left hand side, for example, can be written as \u03bc\u03b8 is a 1\u2010ipm on G for all \u03b8\u2208I, we have that these two polynomials agree on I, and so must be the same polynomial. Thus We now show that 5n\u2208N\u2a7e2 and p\u2009\u2208\u2009, we let \u03b8=\u03b8(p):=1+4p\u221232 and pn:=143\u2212tan2\u03c0n+1. Let gn(\u03b8):=\u2211j=0n\u03b8j(1\u2212\u03b8)n\u2212j.In this section we prove Theorem \u03bdp\u2208\u21331,\u2a7ep(Pn) as follows. Let us start with the case p\u2a7e34. For each vertex of Pn, we set it to state 0 with probability \u03b8, and set it to state 1 otherwise, and we do this independently for every vertex. Recall that for each j\u2009\u2208\u2009[n] we write Sj for the state of vertex j; in this construction, the states are independent and identically distributed random variables. We set the edge {j,\u2009j\u2009+\u20091} to be open if Sj\u2a7dSj+1, and closed otherwise. Thus, as p=\u03b8+(1\u2212\u03b8)2, we have that each edge is open with probability p. Moreover (Pn)\u03bdp will be connected if and only if there exists some j\u2009\u2208\u2009[n\u2009+\u20091] such that Sk\u2009=\u20090 for all k\u2009<\u2009j, while Sk\u2009=\u20091 for all k\u2a7ej. Therefore (Pn)\u03bdp is connected with probability gn(\u03b8). As this construction is vertex\u2010based, it is clear that it is 1\u2010independent.We begin by constructing a measure p<34 we have that \u03b8 is a complex number, and so the above construction is no longer valid. However, as discussed in Section p\u2009\u2208\u2009. For each subgraph G of Pn, set QG(\u03b8) to be the polynomial \u03bdp((Pn)\u03bdp=G) for all \u03b8\u2208. The following claim, together with Lemma \u03bdp is a 1\u2010ipm on Pn for all p\u2009\u2208\u2009.When Claim 29p\u2208 such {j,\u2009j\u2009+\u20091} is not an edge, and so Sj\u2009=\u20091 while Sj\u2009+\u20091\u2009=\u20090. Note that if j\u2a7e2, then the edge {j\u2009\u2212\u20091,\u2009j} is present in G regardless of the state of vertex j\u2009\u2212\u20091. Similarly, if j\u2a7dn\u22122, then the edge {j\u2009+\u20091,\u2009j\u2009+\u20092} is present in G regardless of the state of vertex j\u2009+\u20092. If we write G1:=G{1,\u2026,j\u22121} and G2:=G{j+2,\u2026,n}, then we have that QG1\u03b8p and QG2\u03b8p are positive real numbers for all p\u2208 by q1\u2009=\u20090 and by the recurrence relation qk=min1\u2212p1\u2212qk\u22121,1 for k\u2009\u2265\u20092, which corresponds exactly to the equality case in inequality We now prove that this construction is optimal with respect to the connectivity function. Note that the following proof involves essentially following the proof of Theorem Proof of Theorem 15f1,Pn(p)\u2a7e0 for all p, and so all that remains to show is that f1,Pn(p)\u2a7egn\u03b8p for all p\u2009\u2208\u2009.The above construction discussed shows that \u03bc\u2208\u21331,\u2a7ep(Pn). For k\u2009\u2208\u2009[n], let Xk be the event that the subgraph of (Pn)\u03bc induced by the vertex set [k] is connected, and let Hk be the event that the edge {k\u2009\u2212\u20091,\u2009k} is not present in (Pn)\u03bc. Applying random sparsification as in Remark\u00a0k, the event Hk occurs with probability exactly 1\u2009\u2212\u2009p.Let q2\u03bc:=\u03bc(X2)c=1\u2212p, and for k\u2009>\u20092 let qk\u03bc:=\u03bcXkc|Xk\u22121. We have that Let \u03bc((Pn)\u03bc=Pn)=\u220fj=2n(1\u2212qn\u03bc). Thus to show that the previous construction is optimal with respect to the connectivity function it is enough to show that equality holds for inequalities \u03bc=\u03bdp. In the measure \u03bdp, we have that every edge is present with probability exactly p, thus \u03bdp(Hk)=1\u2212p and so equality holds in Note that \u03b8(p), and so it is sufficient to show that equality holds for p\u2a7e34, as that will show they are the same polynomial . Suppose that the event (Hk\u2009\u2229\u2009Xk\u2009\u2212\u20092) occurs. As Hk has occurred we have that Sk\u2009\u2212\u20091\u2009=\u20091 while Sk\u2009=\u20090. As Sk\u2009\u2212\u20091\u2009=\u20091, we have that edge {k\u2009\u2212\u20092,\u2009k\u2009\u2212\u20091} is open, regardless of Sk\u2009\u2212\u20092. Thus, as Xk\u2009\u2212\u20092 has occurred we also have that Xk\u2009\u2212\u20091 has occurred. Therefore (Hk\u2009\u2229\u2009Xk\u2009\u2212\u20091) has also occurred, and so we are done.Both the left and right hand sides of \u03bc\u2208\u21331,\u2a7ep(Pn): \u03bc=\u03bdp. This leads us to another way to define gn\u03b8p: let g1\u03b8p:=1, g2\u03b8p:=p, and for all n\u2a7e3 we have that We remark in similar fashion to the above proof that the following holds for any Proof of Theorem 11f1,P1(p)=1 into equation\u00a0p\u22c6(P1)=34 and apply Corollary\u00a0p1,\u2113p(Z)\u2a7d34.(i). For the upper bound we plug p<34 be fixed. As the sequence (pn)n\u2208N is monotone increasing and tends to 3/4 as n\u2009\u2192\u2009\u221e, there exists N\u2208N such that p\u2009<\u2009pN. We showed in Theorem \u03bdpN\u2208\u21331,\u2a7epN(PN) such that the probability PN\u03bdpN is connected is equal to zero.For the lower bound, let \u03bd\u2208\u21331,\u2a7ep(Z). For each i\u2208Z, we let the subgraphs Z\u03bd[(i(N\u22121)+[N])] on horizontal shifts of PN by i(N\u2009\u2212\u20091) be independent identically distributed random variables with distribution given by \u03bdpN. This gives rise to a 1\u2010independent model \u03bd on Z with edge\u2010probability at least p (in fact at least pN). Furthermore, all connected components of Z\u03bd have size at most 2(N\u2009\u2212\u20091)\u2009\u2212\u20091. In particular, p1,\u2113p(Z)\u2a7ep. Since p<34 was chosen arbitrarily, this gives the required lower bound p1,\u2113p(Z)\u2a7e34.We use this measure to create a measure 6Z\u00d7P2 with edge\u2010probability close to 2/3 for which with probability 1 there are no open left\u2010right crossings. The idea of this construction is due to Walters and the second author\u00a0In this section we construct a family of 1\u2010ipms on segments of the ladder V(PN\u2009\u00d7\u2009P2) as [N]\u2009\u00d7\u2009[2]. As in the case of the line lattice, we independently assign to each vertex  a random state Sn,\u2009y)n\u2208N,(rn)n\u2208N,(sn)n\u2208N are suitably chosen sequences of real numbers, ensuring that the Sn,\u2009y)\u2009\u2208\u2009[N]\u2009\u00d7\u2009[2] as follows:n\u2009\u2208\u2009[N\u2009\u2212\u20091] and y\u2009\u2208\u2009[2], the horizontal edge {,\u2009} is open in G\u03bc if and only if S\u2a7dS,for each n\u2009\u2208\u2009[N], the vertical edge {,\u2009} is open in G\u03bc if and only if S\u2212S))=0.for each Let us begin by giving an outline of our construction. We write the vertex set n,\u20091),\u2009} to be open can be rephrased as if and only if either Sn,\u20091)n\u2208N,(rn)n\u2208N,(sn)n\u2208N and taking N sufficiently large, one can ensure that in addition \u03bc satisfies d(\u03bc)\u2265p and \u03bc(\u2203open left\u2010right crossing)=0. In particular, with this construction we prove the following result.Clearly the bond percolation measure Theorem 30Fixp\u220812,23. Then there existsN\u2208Nsuch that for alln\u2a7eN,p:=23\u2212\u03b5, with \u03b5\u22080,16. We start by defining the sequences pnn\u2208N, rnn\u2208N and snn\u2208N iteratively as follows. We set p1\u2009=\u2009r1\u2009=\u20091 and s1\u2009=\u20090. Then for n\u2208N, we let Fix Lemma 31The following hold for alln\u2208N:(i)pn,\u2009rn\u2009\u2208\u2009,(ii)sn\u2009\u2208\u2009,(iii)pn+1\u2a7dpn,(iv)rn+1\u2a7drn,(v)rn+1+sn+1\u2a7drn+sn.n. By definition of our sequences, p1=r1=1\u2a7ep=p2=r2, s1\u2009=\u20090, and 0<s2=(2p\u22121)(1\u2212p)p<1\u2212p=r1+s1\u2212r2, and thus (i)\u2010(v) all hold in the base case n\u2009=\u20091.We prove the lemma by induction on n\u2a7dN, for some N\u2a7e1. Since pN and rN\u2009+\u2009sN both lie in , the definition of pN\u2009+\u20091 and rN\u2009+\u20091 implies these also both lie in . This establishes (i) for n\u2009=\u2009N\u2009+\u20091. By construction, sN+1\u2a7e0, and by the inductive hypotheses (ii) and (v), we have n\u2009=\u2009N\u2009+\u20091.Suppose now (i)\u2010(v) hold for all pN\u2009+\u20092\u2009=\u20090, then pN+2\u2a7dpN+1 trivially holds (since pN+1\u2a7e0 by (i)). On the other hand, suppose pN+2=1\u22121\u2212prN+1+sN+1>0. Then we have rN\u2009+\u20091\u2009+\u2009sN\u2009+\u20091\u2009>\u20091\u2009\u2212\u2009p, which by our inductive hypothesis (v) implies rN+sN\u2a7erN+1+sN+1>1\u2212p. The definition of pN\u2009+\u20091 then implies n\u2009=\u2009N\u2009+\u20091. Arguing in exactly the same way (using the inductive hypothesis (iii) instead of (v)), we obtain that rN+2\u2a7drN+1. Hence (iv) holds for n\u2009=\u2009N\u2009+\u20091.If n\u2009=\u2009N\u2009+\u20091, which is the most delicate part of the induction. We begin by recording two useful facts, the second of which we shall reuse later.Finally we consider (v) for Claim 32pN\u2009+\u20092\u2009=\u20090 or rN\u2009+\u20092\u2009=\u20090, then sN\u2009+\u20092\u2009=\u20090.If pN\u2009+\u20092\u2009=\u20090, then by construction sN\u2009+\u20092\u2009=\u20090 and so we are done. If rN\u2009+\u20092\u2009=\u20090, then by construction pN\u2009+\u20091\u2009\u2264\u20091\u2009\u2212\u2009p, which by our inductive hypothesis (iii) implies pN\u2009+\u20092\u2009\u2264\u20091\u2009\u2212\u2009p and hence sN+2=max1\u22122rN+2+rN+2\u2212(1\u2212p)PN+2,0=max1\u22121\u2212ppN+2,0=0.If Claim 33pN\u2009+\u20092 and rN\u2009+\u20092 are both strictly positive, then for all i\u2009\u2208\u2009{2,\u2009\u2026\u2009,\u2009N\u2009+\u20091}, we have si\u2009>\u20090.If i\u2009\u2208\u2009{2,\u2009\u2026\u2009,\u2009N\u2009+\u20091}. By our inductive hypotheses (iii) and (iv)  and since i\u2a7e2, we have 0<pN+2\u2a7dpi\u2a7dp2=p and 0<rN+2\u2a7dri\u2a7dr2=p. Since ri\u2009+\u20091\u2009>\u20090, we in fact have pi\u2009>\u20091\u2009\u2212\u2009p. We also have that y\u2009\u2208\u2009, the function x\u2009\u21a6\u2009f is a nonincreasing function of x. Thus if ri\u2a7e1/2, we have y\u2009\u2208\u2009, the function x\u2009\u21a6\u2009f is strictly increasing in x. Therefore if ri\u2009<\u20091/2, we have f\u2009>\u20090, and thus si=max1pif,0>0.Fix sN\u2009+\u20092\u2009=\u20090, then (v) follows immediately from (iv). Thus we may assume that sN\u2009+\u20092\u2009>\u20090, whence by Claim\u00a0pN\u2009+\u20092\u2009>\u20090 and rN\u2009+\u20092\u2009>\u20090. By Claim\u00a0pi, ri and si are all strictly positive for i\u2009\u2208\u2009{2,\u20093\u2009\u2026\u2009,\u2009N\u2009+\u20092}. By definition of our sequences we thus have for all i\u2009\u2208\u2009[N\u2009+\u20091] that i\u2009\u2208\u2009{2,\u2009\u2026\u2009,\u2009N\u2009+\u20091} that: With these results in hand, we return to the proof of (v). If Claim 34sN\u2009+\u20092\u2009>\u20090, for all integers i\u2009\u2208\u2009[N\u2009+\u20091] we have Under our assumption that p1\u2009=\u20091 and p2\u2009=\u2009p, our claim holds for i\u2009=\u20091. Suppose it holds for some i\u2a7dN. Then by rearranging terms, we have pi\u2009+\u20092 given by i\u2009+\u20091 as well.Since i\u2009\u2208\u2009[N\u2009+\u20091], we can write ri\u2009+\u20091\u2009+\u2009si\u2009+\u20091 as a function of pi: It follows from Claim\u00a0pi\u2009>\u2009(1\u2009\u2212\u2009p) , the expression above is an increasing function of pi. By our inductive hypothesis (iii) that pN+1\u2a7dpN it follows that rN+2+sN+2\u2a7drN+1+sN+1 and we have verified that (v) holds for n\u2009=\u2009N\u2009+\u20091.For p=23\u2212\u03b5, for some fixed \u03b5\u22080,16.Recall that Lemma 35We have that pn\u2009=\u2009rn\u2009=\u2009sn\u2009=\u20090 for alln\u2a7eN\u03b5, where N\u03b5:\u2009=\u2009\u23082\u03b5\u22121\u2309.m\u2009\u2208\u2009[N\u03b5\u2009\u2212\u20091] such that rm\u2009=\u20090. Then sm\u2009=\u20090, so pm\u2009+\u20091\u2009=\u20090 and sm\u2009+\u20091\u2009=\u20090 by construction and rm\u2009+\u20091\u2009=\u20090 by Lemma pn\u2009=\u2009rn\u2009=\u2009sn\u2009=\u20090 for all n\u2208N\u2a7em+1, as required.Suppose first that there exists rn\u2009>\u20090 for all n\u2009\u2208\u2009[N\u03b5\u2009\u2212\u20091] and there exists some m\u2009\u2208\u2009[N\u03b5\u2009\u2212\u20092] such that pm\u2a7d1\u2212p. Then rm\u2009+\u20091\u2009=\u20090, and thus by the argument above, we have that pn\u2009=\u2009rn\u2009=\u2009sn\u2009=\u20090 for all n\u2208N\u2a7em+2, as required.Suppose instead that pn\u2009>\u20091\u2009\u2212\u2009p and rn\u2009>\u20090 both hold for all n\u2009\u2208\u2009[N\u03b5\u2009\u2212\u20092]. By Claim\u00a0sn\u2009>\u20090 for all n\u2009\u2208\u2009{2,\u2009\u2026\u2009,\u2009N\u03b5\u2009\u2212\u20093}. This allows us in turn to apply Claim\u00a0n in this interval and to deduce that p1\u2009=\u20091. As such, it follows from inequality\u00a0pn\u2a7d1\u2212(n\u22121)3\u03b54 for all n\u2009\u2208\u2009[N\u03b5\u2009\u2212\u20092]. In particular, as N\u03b5\u2009=\u2009\u23082\u03b5\u22121\u2309 and \u03b5\u22080,16, we have Finally, suppose N\u2009=\u2009N\u03b5 be the integer constant whose existence is given by Lemma G\u03bc on the graph G\u2009=\u2009PN\u2009\u00d7\u2009P2 from independent random assignments of states Sn,\u2009y) in V(G)\u2009=\u2009[N]\u2009\u00d7\u2009[2], as described at the beginning of this section.Now let Sn,\u2009y)\u2009\u2208\u2009[N]\u2009\u00d7\u2009[2], and so \u03bc is a well\u2010defined 1\u2010ipm. We recall here for the reader's convenience the state\u2010based rules governing which edges are open in G\u03bc:n\u2009\u2208\u2009[N\u2009\u2212\u20091] and y\u2009\u2208\u2009[2], the horizontal edge {,\u2009} is open if and only if S\u2a7dS,for each n\u2009\u2208\u2009[N], the vertical edge {,\u2009} is open if and only if either Sn,\u20091)\u2a7ep.We have that n,\u2009y)\u2009\u2208\u2009[N\u2009\u2212\u20091]\u2009\u00d7\u2009[2], consider the horizontal edge {,\u2009}, . If n\u2009+\u2009y is even, then by definition of rn\u2009+\u20091, n\u2009+\u2009y is odd, then by definition of pn\u2009+\u20091, n,\u20091),\u2009}, n\u2009\u2208\u2009[N], we have pn\u2009=\u20090, then rn\u22121\u2a7d1\u2212p by definition of pn, whence rn\u2a7d1\u2212p by Lemma 1\u2212rn\u2a7ep. On the other hand if pn\u2009\u2260\u20090, then by definition of sn the expression above is at least p. Thus each horizontal edge and each vertical edge is open in G\u03bc with probability at least p, and d(\u03bc)\u2a7ep as claimed.For \u2009=\u2009S\u2009=\u20092. Furthermore, by Lemma N, pN\u2009=\u2009rN\u2009=\u2009sN\u2009=\u20090, whence SN,\u20091),\u2009} of \ud835\udcab is traversed from left to right.Let i\u2009\u2208\u2009[\u2113], we have Si\u2009\u2208\u2009{1,\u20092}. Indeed, by construction Sv0=2. Suppose there exists some 1\u2a7di<\u2113 such that Svj\u2208{1,2} for all j\u2009<\u2009i. If Svi=2, then the edge vivi\u2009+\u20091 can be open in G\u03bc only if Svi+1\u2208{1,2}. What is more, Svi+1 can be equal to 1 if and only if vivi\u2009+\u20091 is a vertical edge. On the other hand, suppose Svi=1. Then vi\u2009\u2212\u20091vi was a vertical edge , and so vi\u2009+\u20091\u2009=\u2009vi\u2009+\u2009. But then vivi\u2009+\u20091 open in G\u03bc implies Svi+1=2. Thus for every vertex vi of \ud835\udcab, we have that Svi is indeed in state 1 or 2.We claim that for all v\u2113\u2009\u2209\u2009{N}\u2009\u00d7\u2009[2] (\u2009=\u20090). Thus there is no open path in G\u03bc from {1}\u2009\u00d7\u2009[2] to [N]\u2009\u00d7\u2009[2].This implies in particular that \u03bc is an element of \u21331,\u2a7ep(PN\u00d7P2) for which n\u2a7eN, we may extend \u03bc to an element \u03bc\u2032\u2208\u21331,\u2a7ep(Pn\u00d7P2) by letting every edge in Pn\u2009\u00d7\u2009P2\u2009\u2216\u2009PN\u2009\u00d7\u2009P2 be open independently at random with probability p. In this way we obtain a 1\u2010independent bond percolation measure \u03bc\u2032 on Pn\u2009\u00d7\u2009P2 with edge\u2010probability p for which there almost surely are no open left\u2010right crossings of Pn\u2009\u00d7\u2009P2, giving the required lower bound on p1,\u2009\u00d7(Pn\u2009\u00d7\u2009P2).Thus We conclude this section by proving Theorem Proof of Theorem 11P2  is f1,P2(p)=p. Thus the constant p\u22c6(P2) defined by equation\u00a0x2\u2009=\u20094(1\u2009\u2212\u2009x)2, namely p\u22c6(P2)=23. By Corollary\u00a0p1,\u2113p(Z\u00d7P2)\u2a7d23.(ii). Trivially, the 1\u2010independent connectivity function of the path on 2 vertices p\u220812,23. In the proof of Theorem N\u2208N and \u03bc\u2208\u21331,\u2a7ep(PN\u00d7P2) such that(i)\u03bc(\u2203open left\u2010right crossing)=0;(ii)\u03bc{,}and{,}are open=1.For the lower bound, fix \u03bd\u2208\u21331,\u2a7ep(Z\u00d7P2). Let G:=Z\u00d7P2. For each i\u2208Z, we let the subgraphs G\u03bd(i(N\u22121)+[N])\u00d7[2] on horizontal shifts of the ladder PN\u2009\u00d7\u2009P2 by i(N\u2009\u2212\u20091) be independent identically distributed random variables with distribution given by \u03bc. Thanks to property (ii) recorded above, the random subgraphs agree on the vertical rungs {1\u2009+\u2009i(N\u2009\u2212\u20091)}\u2009\u00d7\u2009P2 of the ladder, and this gives rise to a bona fide 1\u2010independent model \u03bd on Z\u00d7P2 with edge\u2010probability p. Furthermore, property (i) implies all connected components in G\u03bd have size at most 4(N\u2009\u2212\u20091)\u2009\u2212\u20092\u2009=\u20094N\u2009\u2212\u20096. In particular, p1,\u2113p(Z\u00d7P2)\u2a7ep. Since p<23 was chosen arbitrarily, this gives the required lower bound p1,\u2113p(Z\u00d7P2)\u2a7e23.We use this measure to create a measure 7n\u2208N\u2a7e2 and p\u2009\u2208\u2009, we let \u03b8=\u03b8(p):=1+2p\u221212 and pn:=121\u2212tan2\u03c02n. Let gn(\u03b8):=\u03b8n+(1\u2212\u03b8)n.In this section we will prove Theorem 7.1\u03bdp\u2208\u21331,\u2a7ep(Kn) that shows f1,Kn(p)\u2a7dgn(\u03b8) for p\u2a7e12. We call this measure the Red\u2010Blue construction. We think of Kn as the complete graph on vertex set [n], and we color each vertex Red with probability \u03b8 and color it Blue otherwise, and we do this independently for all vertices. The edge {i,\u2009j}\u2009\u2208\u2009[n](2) is open if and only if i and j have the same color. As p=\u03b82+(1\u2212\u03b8)2, we have that each edge is present in (Kn)\u03bdp with probability p. Note that (Kn)\u03bdp will either be either a disjoint union of two cliques, in which case it is disconnected, or the complete graph Kn, in which case it is connected. This latter case occurs if and only if every vertex receives the same color, and so the probability that (Kn)\u03bdp is connected is equal to gn(\u03b8). As this construction is vertex\u2010based, it is clear that it is 1\u2010independent.Before proving Theorem p<12 then \u03b8 is a complex number, and so the Red\u2010Blue construction is no longer valid. However, as discussed in Section p\u2009\u2208\u2009. Given j\u2009\u2208\u2009{0,\u20091,\u2009\u2026\u2009,\u2009n}, let j\u2009=\u20090 or j\u2009=\u2009n we have that gn,0(\u03b8) and gn,n(\u03b8) are each equal to gn(\u03b8), and so we just write the latter instead. Given some A\u2009\u2286\u2009[n], let HA be the disjoint union of a clique on A with a clique on [n]\u2009\u2216\u2009A. Note that when A\u2009=\u2009\u2205 or [n] we have that HA is equal to Kn]\u2009\u2216\u2009A, let \u03bcp be the following function on subgraphs G of Kn: p\u220812,1 this function matches the Red\u2010Blue construction given above, and so by defining \u03bdp((Kn)\u03bdp=G):=\u03bcp(G) for all subgraphs G\u2009\u2286\u2009Kn, we obtain a measure \u03bdp which is a 1\u2010ipm defined without making reference to states of vertices. The following claim, together with Lemma \u03bdp is a 1\u2010ipm on Kn for all p\u2009\u2208\u2009.If Claim 38p\u2208pn,12 and all j\u2009\u2208\u2009{0,\u2009\u2026\u2009,\u2009n} we have that gn,j\u03b8p is nonnegative real number.For all j\u2009=\u2009n. As p\u2a7d12, we have that \u03b8 and 1\u2212\u03b8 are complex conjugates, and so gn(\u03b8(p)) is a real number for all p in this range. By writing \u03b8=rei\u03c6, where r:=1\u2212p2 and \u03c6:=arctan1\u22122p, we can write p\u2208 implies 0\u2264\u03c6\u2264\u03c02n, which in turn gives cos(n\u03c0)\u22650. By\u00a0gn\u03b8p is a nonnegative real number for all p\u2208, which proves the claim when j\u2009=\u2009n. For general j\u2009\u2208\u2009{0,\u2009\u2026\u2009,\u2009n}, we have that gn,j\u03b8p\u2208 for all p\u2208; at this stage we are using the fact that (pn)n\u2a7e2 forms an increasing sequence, and so p\u2a7epn implies that p\u2a7eps for all s\u2a7dn.Let us begin with the case gn\u03b8pn=0, we have that the probability (Kn)\u03bdpn is connected is equal to 0. As \u03bdpn\u2208\u21331,\u2a7ep(Kn) for all p\u2a7dpn, we have that f1,Kn(p)=0 for all p\u2a7dpn. We now prove that this construction is optimal with respect to the connectivity function.Note that as this proof shows that 7.2Proof of Theorem 16f1,Kn(p)\u2a7e0 for all p, and so all that remains to show is that f1,Kn(p)\u2a7egn(\u03b8) for p\u2009\u2208\u2009. We will prove this result by induction on n. The inequality is trivially true when n\u2009=\u20092, so let us assume that n\u2009>\u20092 and that the inequality is true for all cases from 2 up to n\u2009\u2212\u20091. First, we note that gn(\u03b8)=gj(\u03b8)gn\u2212j(\u03b8)\u2212gn,j(\u03b8) for all j\u2009\u2208\u2009{0,\u20091,\u2009\u2026\u2009,\u2009n}. Thus, if we multiply both sides of this equation by nj and sum over all j\u2009\u2208\u2009{0,\u20091,\u2009\u2026\u2009,\u2009n}, we have that \u03bc\u2208\u21331,\u2a7ep(Kn) and let C be the event that (Kn)\u03bc is connected. Given A\u2009\u2286\u2009[n], let XA be the event that (Kn)\u03bc[A] and (Kn)\u03bc[Ac] are each connected, where Ac\u2009=\u2009[n]\u2009\u2216\u2009A. Moreover, let YA be the event that (Kn)\u03bc[A] and (Kn)\u03bc[Ac] are each connected, and there are no edges between A and Ac in (Kn)\u03bc. For all A\u2009\u2286\u2009[n], we have that A\u2009=\u2009\u2205 or A\u2009=\u2009[n], the above equation is trivially true due to the fact that C,\u2009X\u2205,\u2009Xn][ are all the same event. As \u03bc is 1\u2010independent we have that if A is a nonempty proper subset of [n], then, by induction on n, we have (pn)n\u2a7e2 forms an increasing sequence, and so p\u2a7epn implies that p\u2a7eps for all s\u2a7dn. We are also using the fact that g1(\u03b8)=1 for all \u03b8\u2208. We proceed by summing n], and then applying C,\u2009Y\u2205 and Yn][ are all the same event to A\u2009\u2286\u2009[n], the events YA and YAc are the same event, and so \u2211A\u2286[n]\u03bc(YA)=2\u22111\u2208A\u2286[n]\u03bc(YA). Moreover, the set {YA \u2009:\u20091\u2009\u2208\u2009A\u2009\u2286\u2009[n]} consists of pairwise disjoint events, and so \u22111\u2208A\u2286[n]\u03bc(YA)\u2a7d1. Thus The previous constructions discussed show that (2n\u22124)\u03bc(C)\u2a7e(2n\u22124)gn(\u03b8). As n\u2009>\u20092, we have that \u03bc(C)\u2a7egn(\u03b8) and so we are done.We apply 7.3fk,\u2009G(p) analogously to fG1,\u2009(p) for k\u2208N0. For k\u2009=\u20090, f0,Kn(p) is exactly the probability that an instance of the Erd\u0151s\u2010R\u00e9nyi random graph Gn,\u2009p contains a spanning tree. As far as we know, there is no nice closed form expression for this function.Clearly we can define f1,Kn(p) exactly, which is the other interesting case, as for k\u2a7e2 the connectivity problem is trivial.In this section, we have computed Proposition 39For allk,n\u2208N\u2a7e2, we have that\u03bc\u2208\u2133k,\u2a7ep(Kn). Since any subgraph of Kn with at least n2\u2212(n\u22121) edges is connected, we can apply Markov's inequality to show that For the lower bound, consider G obtained as follows. Let x:=1\u2212p2. With probability min, select a vertex i\u2009\u2208\u2009[n]\u2009=\u2009V(Kn) uniformly at random, and let G be the subgraph of Kn obtained by removing all edges incident with i. Otherwise, let G be the complete graph Kn. It is easy to check that G is a 2\u2010independent model with edge\u2010probability p and that G is connected if and only if G\u2009=\u2009Kn, an event which occurs with probability 1\u2212min=max0,1\u2212n(1\u2212p)/2.For the upper bound, consider the random graph 88.1fG1,\u2009(p), for any graph G, as a (possibly nonlinear) program.In this subsection we describe how we can represent the problem of finding G on vertex set [n], let \u210b=\u210b(G) be the set of all labeled subgraphs of G. Throughout this section we treat these subgraphs as subsets of E(G), and always imagine them to be on the full vertex set [n]. For each labeled subgraph of G we write Given a graph \u03bc:\u210b\u2192R\u2a7e0, we have \u03bc\u2208\u21331,\u2a7ep(G) if and only if the following three conditions all hold:\u03bc is a probability measure on labeled subgraphs of G,G is open in G\u03bc with probability at least p,Every edge of S,T\u2208\u210b such that S and T are supported on disjoint subsets of [n], \u03bc(S)\u00b7\u03bc(T)=\u03bc(S\u222aT).Given nonempty Recall that for a function fG1,\u2009(p), and as randomly deleting edges cannot increase the probability of being connected, we may assume that in fact every edge of G is open in G\u03bc with probability exactly p (by applying random sparsification as in Remark\u00a0\u2211H\u2208\u210b\u03bc(\u0124)=1,e\u2009\u2208\u2009E(G), we have that\u2211H\u2208\u210b1(e\u2208H)\u03bc(\u0124)=p,For all edges S,T\u2208\u210b, such that S and T are supported on disjoint subsets of [n], we have that For all nonempty As we are interested in determining A\u2009=\u2009A(G) be a matrix which has columns indexed by \u210b, and a row for each piece of information given by one of the above conditions. That is:A\u2205,\u2009H:\u2009=\u20091.We have a row for the empty set such that e\u2009\u2208\u2009G; the entry Ae,H:=1(e\u2208H);We have a row for S,T\u2208\u210b\u2216{\u2205} supported on disjoint subsets of [n]; the entry A{S,T},H:=1((S\u222aT)\u2286H)\u2212\u03bc(T)\u00b71(S\u2286H).We have a row for each pair Let q\u2009=\u2009q(G) be a vector with indexing the same as the rows of A; let q\u2205:\u2009=\u20091, qe:\u2009=\u2009p for e\u2009\u2208\u2009G, and qS,\u2009T}{:\u2009=\u20090 for each pair S,T\u2208\u210b\u2216{\u2205} supported on disjoint subsets of [n]. Then a vector w, whose entries are indexed by \u210b, which satisfies wH\u2a7e0 for all H\u2208\u210b, and also Aw\u2009=\u2009q corresponds precisely to a 1\u2010ipm \u03bc on G with \u03bc{eopen}=p for all edges e\u2009\u2208\u2009E(G).Let c be a vector indexed by \u210b defined by cH:=1(His connected). Just to make it clear, we say that H\u2208\u210b is connected if it contains a spanning tree of [n]. Then for a given value of p the vector w(p) satisfying Aw(p)\u2009=\u2009q and cTw minimal corresponds to a measure \u03bc\u2208\u21331,\u2a7ep(G) such that \u03bc(His connected)=f1,G(p).Let G is a graph on [5], and S and T are nonempty subgraphs of G supported on disjoint subsets of [5], then one of S and T must consist of precisely one edge of G. By choosing T to be this subgraph, we can always choose S and T for \u03bc(T)=p. Thus for any choice of p, we can turn the problem of finding fG1,\u2009(p) into the following linear program: S and T such that \u03bc(T) Observe that for any graph with five vertices or fewer, any partition of the graph into two parts has that one part must have at most two vertices in it. In particular, if  \u03bc(T) in  is an una\u2217: p, for example using the software Maple, and the LPSolve function it contains. However we of course wish to find solutions for all values of p\u2009\u2208\u2009.The duality theorem states that the asymmetric dual problem has the same optimal solution A\u2009=\u2009(aij), w\u2009=\u2009(wj), c\u2009=\u2009(cj), q\u2009=\u2009(qi) and x\u2009=\u2009(xi) any solutions w and x must satisfy \u2211jaijwj=qi, \u2211iaijxi\u2a7dcj, xi\u2009\u2265\u20090 and wi\u2a7e0. Thus we have \u2211iqixi=\u2211jcjwj and so the inequality must be an equality, that is j we either have wj\u2009=\u20090 or \u2211iaijxi=cj. Thus in our attempt to obtain a function for all p, it seems reasonable to look at an optimal solution for one value of p and see which wj have been set to zero; assume for these indices that we always have wj\u2009=\u20090 and attempt to directly solve the equations that result from this. This motivates the following method:p to obtain a solution w(p) and a set J\u2009\u2009:\u2009=\u2009{j\u2009\u2208\u2009[|w|]\u2009:\u2009wj(p)\u2009=\u20090}.Solve Aw)i\u2009=\u2009qi,\u2009wj\u2009=\u20090 \u2009:\u2009i\u2009\u2208\u2009[|w|],\u2009j\u2009\u2208\u2009J} to obtain functions of p for all wk, k\u2009\u2208\u2009[|w|], which we write as wk\u2032(p).Solve the set of equations {(ATx)i\u2009=\u2009ci \u2009:\u2009i\u2009\u2208\u2009[|w|]\u2009\u2216\u2009J} to obtain functions of p for all xk, k\u2009\u2208\u2009[|w|], which we write as xk\u2032(p).Solve the set of equations {(w\u2217(p):\u2009=\u2009 cTw\u2032(p) and x\u2217(p):\u2009=\u2009 qTx\u2032(p).Write P\u2009\u2286\u2009 of values of p, check that (ATx\u2032)i(p)\u2a7dci and wi\u2032(p)\u2a7e0, for all i\u2009\u2208\u2009[|w|].For a certain interval By writing P which works above, the conditions above ensure that the w\u2032(p) and x\u2032(p) obtained in this way are feasible solutions to w\u2217(p)\u2009=\u2009x\u2217(p), then by the duality theorem we have fG1,\u2009(p)\u2009=\u2009w\u2217(p). Furthermore, a measure \u03bc on the subgraphs of G which is extremal is given directly by w\u2032(p). In the following subsection we give, as examples, two results which are proved using the above method.For the given interval 8.2In this subsection we prove Theorems\u00a0Proof of Theorem 17C4 and p\u2208 an extremal construction is given by the measure \u03bc, defined by C4 and p\u2208 an extremal construction is given by the measure \u03bc, defined by For \u03bc\u2208\u21331,\u2a7ep(C4), we have by 1\u2010independence that \u03bc above , this gives a second and perhaps more insightful proof of Theorem f1,C5(p), we do not have a combinatorial proof, and our result relies solely on linear optimisation.We can in fact give a direct combinatorial proof of the lower bound in Theorem Proof of Theorem 18C5 and p\u2208 an extremal construction is given by the measure \u03bc, defined by C5 and p\u2208 an extremal construction is given by the measure \u03bc, defined by For 8.3f1,Cn(p) for n\u2a7e6.We can use Markov's inequality to derive the following simple lower bound on Proposition 40Forn\u2208N, withn\u2a7e6, and p\u2009\u2208\u2009, we havef1,Cn(p)\u2a7enp\u2212(n\u22122)2.n\u2009=\u20096.A small adjustment to this argument gives the following improvement for Proposition 41For p\u2009\u2208\u2009 we have thatf1,C6(p)\u2a7e\u2212p3+3p2\u22121.Proof of Proposition 40\u03bc\u2208\u21331,\u2a7ep(Cn). Note that G\u03bc is connected if and only if it has at most one closed edge. Thus by Markov's inequality, we have Let Proof of Proposition 41X be the number of closed edges in G\u03bc. Cyclically label the edges of C6 as e1,\u2009\u2026\u2009,\u2009e6. Then by simple counting, Let 9f1,Kn(p) and hence prove Theorem In this section, we derive our results for maximizing connectivity in 1\u2010independent modes. First of all Theorem Proof of Theorem 21G on Kn with edge\u2010probability at least 1\u2009\u2212\u2009p, observe that the complement Gc of G in Kn is a 1\u2010independent model in which every edge is open with probability at most p. Furthermore, Gc is connected whenever G fails to be connected. This immediately implies \u03bdp we constructed to obtain the upper bound on f1,Kn(p) in the proof of Theorem \u03bdp\u2010random graph is connected if and only if its complement fails to be connected. This immediately implies that we have equality in Given a 1\u2010independent model F1,Pn(p).For paths, a simple construction achieves the obvious upper bound for Proof of Theorem 20\u03bc\u2208\u21331,\u2a7dp(Pn), we have by 1\u2010independence that F1,Pn(p)\u2a7dp\u230an2\u230b. For the lower bound, we construct a 1\u2010ipm as follows. For each integer i: 1\u2a7di\u2a7dn/2, we assign a state On to the vertex 2i with probability p, and a state Off otherwise, independently at random. Then set an edge of Pn to be open if one of its endpoints is in state On, and closed otherwise. This is easily seen to yield a 1\u2010ipm \u03bc on Pn in which every edge is open with probability p, and for which For any measure F1,Pn(p)\u2a7ep\u230an2\u230b, as claimed.Thus Cn appears to be slightly more subtle. For the 4\u2010cycle, as in the previous section, we can give two proofs, one combinatorial and the other via linear optimisation.The case of cycles Proof of Theorem 17\u03bc\u2208\u21331,\u2a7dp(C4), we have by 1\u2010independence that p)2\u2009=\u20092p\u2009\u2212\u2009p2, we obtain F1,C4(p).The theorem immediately follows from an application of the linear optimisation techniques from Section p. For p\u2208 consider the measure \u03bc defined by \u03bc\u2208\u21331,\u2a7dp(C4) and that \u03bc({connected})=1\u2212(1\u2212p(2\u2212p))=2p\u2212p2, which is maximal for p in that range.For the lower bound, we give two different constructions, depending on the value of p\u2208, consider the measure \u03bc defined by \u03bc\u2208\u21331,\u2a7dp(C4) and that \u03bc({connected})=\u03bc{\u2a7e3edges open}=2p2, which is maximal for p in that range.For Proof of Theorem 18We simply apply the linear optimisation method from Section p\u2208 an extremal construction is given by the measure \u03bc, defined by p\u2208 an extremal construction is given by the measure \u03bc, defined by p\u2208 an extremal construction is given by the measure \u03bc, defined by For 10Combining Corollary\u00a0Proof of Theorem 12Z\u00d7Cn has the finite 2\u2010percolation property. Thus, as described after the proof of Theorem p1,\u2113p(Z\u00d7Cn)\u2a7e4\u221223. For the upper bound in part (i), since the long paths critical probability is nondecreasing under the addition of edges, we have G\u2009=\u2009K3,\u2009C4,\u2009C5, we plug in the value of fG1,\u2009(p) in equation\u00a0p\u22c6(G) and apply Corollary\u00a0For the lower bound in part (i), we note that p1,\u2113p(Z\u00d7Kn)n\u2208N is nonincreasing in  and hence tends to a limit as n\u2009\u2192\u2009\u221e. For the lower bound in (v), observe that for any n\u2208N the graph Z\u00d7Kn has the finite 2\u2010percolation property\u2014indeed for any finite k, the closure of a copy of Pk\u2009\u00d7\u2009Kn under 2\u2010neighbor bootstrap percolation in Z\u00d7Kn is equal to itself. We construct a 1\u2010ipm \u03bc on Z\u00d7Kn as in Corollary\u00a0T0\u2009=\u2009{0}\u2009\u00d7\u2009V(Kn) and hence Tk\u2009=\u2009({k}\u2009\u00d7\u2009V(Kn))\u2009\u222a\u2009({\u2212\u2009k}\u2009\u00d7\u2009V(Kn)). It is easily checked that \u03bc\u2010almost surely, all components  in a \u03bc\u2010random graph have length at most 5n. Since by construction d(\u03bc)=4\u221223, this proves n\u2208N. For the upper bound, we perform some simple analysis. By solving a quadratic equation, we see that p\u2208. Then by Theorem p and all n sufficiently large, we have that p\u22c6(Kn)\u2009<\u2009p for all n sufficiently large, which by Corollary\u00a0p1,\u2113pZ\u00d7Kn<p.In part (v), we begin by noting that as we are considering an increasing nested sequence of graphs, the sequence 1111.1p1,c(Z2) can be improved. This problem is, we suspect, very hard in general. However, it may prove more tractable if we restrict our attention to a smaller family of measures.The most obvious open problem about 1\u2010independent percolation is of course whether the known lower and upper bounds on Definition 42G be a graph. A G\u2010partition is a partitioned set \u2294v\u2208V(G)\u03a9v, with nonempty parts indexed by the vertices of G. A G\u2010partite graph is a graph H on a G\u2010partition V(H)=\u2294v\u2208V(G)\u03a9v whose edges are a subset of the union of the complete bipartite graphs \u2294uv\u2208E(G){\u03c9u\u03c9v:\u03c9u\u2208\u03a9u,\u03c9v\u2208\u03a9v} corresponding to the edges of G.Let G\u2010partite graph H on a G\u2010partition \u2294v\u2208V(G)\u03a9v, we have a natural way of constructing 1\u2010independent bond percolation models: given a family X=Svv\u2208V(G) of independent random variables with Sv taking values in \u03a9v, the X)\u2010random subgraph obtained by setting uv to be open if and only if SuSv\u2009\u2208\u2009E(H).Given a Definition 43G be a graph. A measure \u03bc\u2208\u21331,\u2a7ep(G) is said to be vertex\u2010based if there existG\u2010partition \u2294v\u2208V(G)\u03a9v,a G\u2010partite graph H, andan associated Sv)v\u2009\u2208\u2009V (G) with Sv taking values in \u03a9v,a collection of independent random variables \u2010random subgraph H[X] has the same distribution as the \u03bc\u2010random graph G\u03bc.such that the  denote the collection of all vertex\u2010based measures on G with edge\u2010probability at least p.Let Problem 44Determineinf{p\u2208:\u2200\u03bc\u2208\u2133vb,\u2a7ep(Z2),\u03bc({percolation})=1}.\u03a9v have bounded size.Vertex\u2010based measures arise naturally in renormalising arguments, and are thus a natural class of examples to consider. A special case of Problems\u00a0Definition 45\u03bc on a graph G is N\u2010uniformly bounded if it as in Definition\u00a0v\u2009\u2208\u2009V(G), |\u03a9v|\u2a7dN. Furthermore, a vertex\u2010based measure \u03bc on a graph G is uniformly bounded if it is N\u2010uniformly bounded for some N\u2208N.A vertex\u2010based measure \u2133N\u2212ubvb,\u2a7ep(G) and \u2133ubvb,\u2a7ep(G) denote the collection of all vertex\u2010based measures on G with edge\u2010probability at least p that are N\u2010uniformly bounded and uniformly bounded respectively.Let Problem 46(i)ForN\u2208N, determine(ii)Determinep1,c(Z2), is arguably that of giving bounds on p1,\u2113p(Z2) and closely related variants. Such problems, which correspond to new questions in extremal graph theory, are discussed in the subsections below. For these problems too we believe restrictions to the class of uniformly bounded vertex\u2010based 1\u2010ipms could be both fruitful and interesting in their own right.Finally, let us note that the second most obvious problem arising from our work, besides that of improving the bounds on 11.2Z2, it is natural to ask about bounds on pc1,\u2009(G) for some of the other commonly studied lattices in percolation theory.Beyond Problem 47Give good bounds on the value of pc1,\u2009(G) when G is one of the eleven Archimedean lattices in the plane or the d\u2010dimensional integer latticeZd.G is the triangular lattice or the honeycomb lattice , or the cubic integer lattice Z3 (which is important in applications). A challenge in all cases is finding constructions of nonpercolating 1\u2010independent measures with high edge\u2010probability\u2014indeed, our arsenal of constructions for 1\u2010independent percolation problems is so sparse that any new construction could be of independent interest.This problem is particularly interesting when Zd+1 contains a copy of Zd, whence the sequence p1,c(Zd)d\u2208N is nonincreasing in  and converges to a limit. Balister and Bollob\u00e1s asked for its value:In a different direction, we can observe that Problem 48 Determinelimd\u2192\u221ep1,c(Zd).4\u221223.Note that by Theorem 11.3G be an infinite, locally finite connected graph, and v0 a fixed vertex of G. Given a bond percolation model \u03bc on G, we let Cv0 denote the connected component of G\u03bc containing v0.Let \u03bc is 0\u2010independent, then \u03bc({percolation})=1 if and only if \u03bc({|C0|=\u221e})>0. However this need not be true for a 1\u2010independent measure. Indeed, consider the 1\u2010ipm on Z2 obtained by taking the measure constructed in the proof of Theorem \u2113\u221e ball of radius 3 around the origin and setting every other edge to be open independently at random with probability 4\u221223. Then in this model percolation occurs almost surely, but the origin is contained inside a component of order at most 28.If G for percolation to occur somewhere with probability 1 and for it to occur anywhere with strictly positive probability. Indeed, if p1,c(Z2) were strictly less than 3/4, then one could obtain examples of such a graph G by attaching a long path to the origin in Z2.Thus in principle there are different edge\u2010probability thresholds in 1\u2010independent percolation on a graph Temperley critical probability, which in 0\u2010independent percolation is the threshold pT at which E|Cv|=\u221e for any vertex v (and every 0\u2010independent measure with edge\u2010probability >pT). In general this threshold is different from the Harris critical probability. Again for 1\u2010independent percolation we have that the threshold for some vertex v\u2009\u2208\u2009V(G) to satisfy E|Cv|=\u221e and for the threshold for all vertices of G to satisfy this are different.Another critical edge\u2010probability of interest is the Problem 49Given an infinite, locally finite connected graph G, determine the following four critical probabilities:It follows from their definition that these four critical probabilities satisfy G with p1,H1(G)=12. For any p: 12<p<34, we have shown in Theorem N such that f1,PN(p)=0. Attach one end of a path of length N to an arbitrary vertex of G to form a graph G1, and let v denote the other end of the path. Then there exist 1\u2010ipm \u03bc\u2208\u21331,\u2265p(G1) such that with probability 1 the component of v in a \u03bc\u2010random graph has order at most N, which is finite. Thus we have G2 obtained from the line lattice by attaching to each vertex i\u2208Z a collection of 2i|+2| leaves. Clearly, p1,H1(G2)=p1,H1(Z)=1. Now consider a 1\u2010ipm \u03bc\u2208\u21331,\u226534(G2). By Theorem G\u2009=\u2009K1 and \u03b1=12, for any path P of length i in G2, the \u03bc\u2010probability that all edges in P are open is at least 2i\u2009+\u20091)\u2212(. Thus for any v0\u2009\u2208\u2009V(G2), the expected size of |Cv0| is In general, these four critical probabilities are all different. Indeed, Balister and Bollob\u00e1s showed in\u00a0 What isp1,F(Z2)?11.4An obvious problem is to tighten the bounds in Theorem Problem 54Determinelimn\u2192\u221ep1,\u2113pZ\u00d7Kn.p1,\u2113p(Z\u00d7Pn) is a nonincreasing function of n (since Z\u00d7Pn+1 contains Z\u00d7Pn as a subgraph). In this paper, we have given constructions showing that for all integers n\u2a7e3, p1,\u2113p(Z\u00d7Pn)n\u2208N tends to a limit in the interval  as n\u2009\u2192\u2009\u221e.In a similar vein, the sequence Problem 55DetermineZ2 .An in principle different but related problem is determining the value of the long paths critical probability in Problem 56Determinep1,\u2113pZ2.k\u2010independent versions of the long paths critical probability. Defining pk,\u2113pG mutatis mutandis, it is straightforward to adapt our arguments and constructions from Section Z by k\u2010independent ones.We can also ask for Theorem 57 For anyk\u2208N0, we havewith the convention that 00\u2009=\u20091.k fixed, it is easy to construct a 3k\u2010ipm \u03bc on Z2 with d(\u03bc)=1\u22122k and no open path of length more than (2k\u2009+\u20091)2. Indeed, build a random graph model as follows:Z2 open;begin with all edges of \u2208Z2, choose Hij\u2009\u2208\u2009[k\u2009+\u20091] uniformly at random and then for all j\u2032: j(k\u2009+\u20091)\u2009\u2212\u2009k\u2009\u2264\u2009j\u2032\u2009\u2264\u2009j(k\u2009+\u20091)\u2009+\u2009k, set the horizontal edge {(i(k\u2009+\u20091)\u2009+\u2009Hij\u2009\u2212\u20091,\u2009j\u2032),\u2009(i(k\u2009+\u20091)\u2009+\u2009Hij,\u2009j\u2032) to be closed;independently for each \u2208Z2, choose Vij\u2009\u2208\u2009[k\u2009+\u20091] uniformly at random and then for all i\u2032: i(k\u2009+\u20091)\u2009\u2212\u2009k\u2009\u2264\u2009i\u2032\u2009\u2264\u2009i(k\u2009+\u20091)\u2009+\u2009k, set the vertical edge {\u2009+\u2009Vij\u2009\u2212\u20091),\u2009\u2009+\u2009Vij) to be closed.independently for each Given k\u2010independent, has edge probability at least k2(k+1)2\u22651\u22122k and that every connected component has order at most (2k\u2009+\u20091)2.It is easy to check that this random graph model is 3Corollary 58For any fixedk\u2208N,\u25a1[[spimspace=\"1em\"]]limk\u2192\u221epk,\u2113p(Z2)=1 (and in fact a similar construction shows this remains true in Zd).In particular we have G does not imply that for every \u2113\u2208N every vertex of G has a strictly positive probability of being part of a path of length at least \u2113. Thus we may actually define a second long paths critical probability, Finally, as in Section Problem 59Determinep1,\u2113p2(Z2).p1,\u2113p2(Z2)\u2a7e4\u221223, and we know it is upper\u2010bounded by p1,H2(Z2)\u2a7d0.8639. As in Section Our construction in the proof of Theorem n\u2010uniformly bounded Z2\u2010partite graph H with partition \u2294v\u2208Z2\u03a9v. A transversal subgraph of H is a subgraph of H induced by a set of distinct representatives S for the parts of H, that is, a set of vertices of H such that |S\u2229\u03a9v|=1 for all v\u2208Z2. The G\u2010partite density of H is Given an Question 60Suppose H is an n\u2010uniformly boundedZ2\u2010partite graph in which in every transversal subgraph the connected component containing the origin is\u2026(a)of size at most C, for some constantC\u2208N.\u2026 (b)finite.\u2026 How large can dG(H) be?This question can be viewed as a problem from extremal multipartite graph theory. Plausibly some tools from that area, in particular the work of Bondy, Shen, Thomass\u00e9 and Thomassen\u00a011.5f1,Cn(p) for cycles Cn of length at most 5. It is natural to ask what happens for longer cycles.We determined in Sections\u00a0Problem 61Determinef1,Cn(p)forn\u2208N\u2a7e6.f1,C6(p) is nonlinear. Nevertheless, one can use software, such as Maple and its contained NLPSolve function, to try to estimate the answer. This suggests the following:f1,C6(p) becomes nonzero is approximately p\u2009=\u20090.59733;The threshold at which p just above this threshold, the best \u201casymmetric\u201d (see the next subsection for a definition) measure is better than the best \u201csymmetric\u201d measure; e.g. at p\u2009=\u20090.62 we have f1,C6(0.62) is approximately 0.007, but is as high as 0.11 when restricted to \u201csymmetric\u201d measures.For As mentioned in Section More generally, one can ask what happens in cycles if we have higher dependency or if we try to maximize connectivity rather than minimize.Problem 62Determinefk,Cn(p)for all p\u2009\u2208\u2009,k\u2208Nand integersn\u2a7ek+2.Problem 63Determinefk,Cn(p)for all p\u2009\u2208\u2009,k\u2208Nand integersn\u2a7ek+2.Qn and in the n\u2009\u00d7\u2009n toroidal grid Cn\u2009\u00d7\u2009Cn. Progress on either of these would likely lead to progress on other problems in 1\u2010independent percolation as well.Beyond paths, cycles and complete graphs, the 1\u2010independent connectivity problem is perhaps most natural to study in the hypercube graph Problem 64Determinef1,Qn(p)for alln\u2a7e3.Problem 65Determinef1,Cn\u00d7Cn(p)for alln\u2a7e3.fG1,\u2009(p) can be required to have \u201cnice\u201d properties. For C4 and p\u2009\u2208\u2009 another extremal construction for f1,C4(p) is given by the measure \u03bc, defined by \u03bc\u2208\u21331,\u2a7ep(G)symmetric if for any pair of labeled subgraphs S and T of G such that there exists an automorphism of G mapping S to T, then \u03bc(\u015c)=\u03bc(T^). Note that the above measure is an example of a nonsymmetric extremal construction for f1,C4(p), whereas the measure given at the end of Section In a different direction, we can ask whether the extremal measures attaining Question 66For any G and any p\u2009\u2208\u2009, does there always exist a symmetric measure\u03bc\u2208\u21331,\u2a7ep(G)which achieves fG1,\u2009(p)?fG1,\u2009(p) attained via our method in Section G is yes(see the appendix for a proof of this fact). Another natural question is when the extremal connectivity can be attained by vertex\u2010based measures.If the program for solving Question 67For which G and which p does there exist a vertex\u2010based measure\u03bc\u2208\u21331,\u2a7ep(G)which achieves fG1,\u2009(p)?"}
{"text": "M of genus 0. We consider this connection of independent interest; it is natural to expect that similar ideas can be used for other matrix valued OPs, as long as the corresponding Riemann surface M is of genus 0. The rest of the method consists of two parts, and mainly follows the lines of a previous work of Charlier, Duits, Kuijlaars and Lenells. First, we perform a Deift\u2013Zhou steepest descent analysis to obtain asymptotics for the scalar valued OPs. The main difficulty is the study of an equilibrium problem in the complex plane. Second, the asymptotics for the OPs are substituted in the double contour integral and the latter is analyzed using the saddle point method. Our main results are the limiting densities of the lozenges in the disordered flower\u2010shaped region. However, we stress that the method allows in principle to rigorously compute other meaningful probabilistic quantities in the\u00a0model.We analyze a random lozenge tiling model of a large regular hexagon, whose underlying weight structure is periodic of period 2 in both the horizontal and vertical directions. This is a determinantal point process whose correlation kernel is expressed in terms of non\u2010Hermitian matrix valued orthogonal polynomials (OPs). This model belongs to a class of models for which the existing techniques for studying asymptotics cannot be applied. The novel part of our method consists of establishing a connection between matrix valued and scalar valued OPs. This allows to simplify the double contour formula for the kernel obtained by Duits and Kuijlaars by reducing the size of a Riemann\u2013Hilbert problem. The proof relies on the fact that the matrix valued weight possesses eigenvalues that live on an underlying Riemann surface  As the size of the hexagon tends to infinity (while the size of the lozenges is kept fixed), the local statistical properties of this model are described by universal processes.A lozenge tiling of a hexagon is a collection of three different types of lozenges =\u220fp\u2208Twp. Provided that at least one tiling has a positive weight, this defines a probability measure on the set of tilings by  size nHn:={ it suffices to analyze the asymptotic behavior of the correlation kernel. However, until recently,By putting points on the paths as shown in , each tiY) to a 2p\u00d72p Riemann\u2013Hilbert (RH) problem. This RH problem is related to certain orthogonal polynomials (OPs), which are nonstandard in two aspects:p\u00d7p matrix valued,the OPs and the weight are the orthogonality conditions are non\u2010Hermitian. The size of the RH problem, the size of the weight, and the size of the OPs all depend on p, but quite interestingly not on q.The main result of Ref. p\u00d7q periodic weightings are rather unexplored up to now. To the best of our knowledge, the model considered in Ref. Y (of size 2 \u00d7 2) with a nonstandard saddle point analysis of the double contour integral. However, because p=1, the associated OPs are scalar (this fact was extensively used in the proof) and it is not clear how to generalize these techniques to the case p\u22652.Lozenge tiling models of the hexagon with 1.U) to 2 \u00d7 2 RH problem related to scalar OPs. This formula allows for a simpler analysis than the original formula from Ref.  First, we establish a connection between matrix valued and scalar valued OPs. In particular, in Theorem\u00a02.U via the Deift\u2013Zhou steepest descent method. The construction of the equilibrium measure and the associated g\u2010function is the main difficulty. The remaining part of the RH analysis is rather standard. Second, we perform an asymptotic analysis of the RH problem for 3. Third, the asymptotics for the OPs are substituted in the double contour integral and the latter is analyzed using the saddle point method. The first step is the main novel part of the paper. The remaining two steps were first developed in Ref. The aim of this paper is precisely to develop a method to handle a situation involving matrix valued OPs. We will implement this method on a particular lozenge tiling model with 2 \u00d7 2 periodic weightings, which presents one simply connected liquid region , six frozen regions, and six staircase regions ; it will be presented in more detail in Section\u00a0Our main results, which are stated in Theorem\u00a01.1M, which turns out to be of genus 0. This fact is crucial to obtain the new formula for the kernel in terms of scalar OPs. We expect that ideas similar to the ones presented here can be applied to other tiling models with periodic weightings, as long as the corresponding Riemann surface M is of genus\u00a00.The eigenvalues and eigenvectors of the 2 \u00d7 2 orthogonality weight play an important role in the first step of the analysis. They are naturally defined on a two\u2010sheeted Riemann surface p=q=1). In fact, it is expected that the smallest periods that lead to the presence of a gas phase are either p=2,q=3 or p=3,q=2. For models that present gas phases, we expect M to have genus at least 1, and then new techniques are required. This is left for future\u00a0works.Lozenge tiling models of large hexagons with periodic weightings can feature all of the three possible types of phases known in random tiling models: the solid, liquid, and gas phases. A solid region  is filled with one type of lozenges. In the liquid and gas phases, all three types of lozenges coexist. The difference between these two phases is reflected in the correlations between two points: in the liquid region, the correlation decay is polynomial with the distance between the points, while in the gas region the decay is exponential. It is known that there is no gas phase for the uniform measure :|x|+|y|\u2264n+1} with 2 \u00d7 1 or 1 \u00d7 2 rectangles , where n>0 is an integer which parameterizes the size of the covered region. Uniform domino tilings of the Aztec diamond features four solid regions and one liquid region. The associated discrete point process is determinantal, and turns out to belong to the class of Schur processes  do not belong to the Schur class, but have been studied using other techniques based on some connections with Hahn polynomials:The Aztec diamond is a well\u2010studied tiling model.n\u2192+\u221e. This same model was analyzed soon afterward in Ref. n. The analysis of Refs. 2\u00d7k periodic Aztec diamond, for an arbitrary k. The class of models for which the formula from Ref. The doubly periodic Aztec diamond exhibits all three phases. It still defines a determinantal point process, but it falls outside of the Schur process class. However, Chhita and Young found in Ref. However, lozenge tiling models of the finite hexagon cannot be represented as models with infinitely many paths (as opposed to the Aztec diamond and the infinite hexagon). In particular, they do not belong to the class of models studied in Ref. 1.3dimer coverings, which are perfect matchings of a certain bipartite graph. We refer to Ref. In addition to being in bijection with nonintersecting paths, lozenge tilings of the hexagon are also in bijection with 2In this section, we present a lozenge tiling model with 2 \u00d7 2 periodic weightings. We also introduce the necessary material to invoke the double contour formula from Ref. 2.1For the presentation of the model and the results, it is convenient to define the hexagon and the lozenges as in  and 3)..3). HoweHn is mapped by this transformation to a hexagon whose six sides are of equal length. Above the definition . In the figures, we will assign the colors red, green, and yellow for the three lozenges in (4), from left to right,\u00a0respectively.so that finition  of Hn, w2.2Hn has corners located at , , , , , and . We normalize the lozenges such that they cover each a surface of area 1, and the vertices of the lozenges have integer coordinates. We recall that each lozenge tiling of Hn gives rise, through , and the bottom left vertex has coordinates . We denote the paths bypj(0)=j+12 and ending positions pj(2n)=n+j+12. The particular 2 \u00d7 2 periodic lozenge tiling model that we consider depends on a parameter \u03b1\u2208,) is an edge of Gn, thenThe regular hexagon  through , to a sy0, n), n,n, >0 for all T, and thus we have a well\u2010defined probability measure via =0 for all T. So in this case,  and let n\u22651 be an integer. There exists a unique tiling Tmax of Hn such that W(T)\u2264\u03b1W(Tmax) for all T\u2260Tmax. Furthermore,Let For any values of ightings  are suchis case,  does notGn, and we omit the details.\u25aaThe proof of Proposition\u00a0\u03b1\u21920, the randomness disappears because the tiling Tmax becomes significantly more likely than any other tiling. Therefore, our model interpolates between the uniform measure over the tilings (for \u03b1=1) and a particular totally frozen tiling Tmax (as \u03b1\u21920), see Figures\u00a0It follows from Proposition\u00a0W(T) of a tiling T can alternatively be defined\u00a0asSeveral tiling models in the literature   \u03b1\u2208, and whose edges are of the form e=,) with x2=x1+1 and y2\u2212y1\u2208{0,1}. The weighting  for the weight associated to the edge e=,) of G\u221e. This weight can be obtained from =Tx,x+1 for all y1,y2\u2208Z. The weightings Tx,x+1 can be represented as two 2 \u00d7 2 block Toeplitz matrices  that are infinite in both directions. These two infinite matrices can be encoded in two 2 \u00d7 2 matrix symbols Ax,x+1(z), whose entries )i+1,j+1, 0\u2264i,j\u22641, are given byIt will be convenient for us to define eighting  was defined from  and onlyTx,x+1 from its symbol byGn is then obtained by taking the following product 2Nz2N2.4Y(z)=Y defined byY is characterized as the unique solution to the following RH\u00a0problem.RH problems for scalar OPs have been introduced by Fokas, Its, and Kitaev in Ref. (a)Y:C\u2216\u03b3\u2192C4\u00d74 is analytic.(b)Y(z) as z approaches \u03b3 from inside and outside exist, are continuous on \u03b3 and are denoted by Y+ and Y\u2212, respectively. Furthermore, they are related byThe limits of (c)z\u2192\u221e, we have Y(z)=(I4+O(z\u22121))zNI20202z\u2212NI2.As 2.5K denote the associated kernel. By definition of determinantal point processes, for integers k\u22651, and x1,\u2026,xk,y1,\u2026,yk with \u2260 if i\u2260j we haveProposition 2(from Ref. 20) Let \u03b1\u2208 is the unique bivariate polynomial of degree \u2264N\u22121 in both variables w and z, which satisfies the following reproducing property:P of degree \u2264N\u22121. Because it satisfies  is called a reproducing\u00a0kernel.From Ref. roperty:12\u03c0i\u222b\u03b3P=0 or \u03a8\u2032(\u03b6)=0. In the liquid region, Proposition\u00a0s, lying in the upper half plane. This saddle plays an important role in our analysis, and some of its properties are stated in Propositions\u00a0s in Theorem\u00a0Remark 1\u03b1=1, our model reduces to the uniform measure and the kernel can be expressed in terms of scalar\u2010valued OPs. However, our approach is based on the formulas . Because the limiting densities for the lozenges in this case are already well\u2010known,\u03b1\u2208 to avoid unnecessary\u00a0discussions.If formulas  and 22)\u03b1=1, our The new double contour formula for the kernel in terms of scalar OPs is stated in Theorem\u00a03.1W byWe define the scalar weight (a)U:C\u2216\u03b3C\u2192C2\u00d72 is analytic, where \u03b3C is a closed curve surrounding c and c\u22121 once in the positive direction, but not surrounding 0.(b)U(\u03b6) as \u03b6 approaches \u03b3C from inside and outside exist, are continuous on \u03b3C and are denoted by U+ and U\u2212, respectively. Furthermore, they are related byThe limits of (c)\u03b6\u2192\u221e, we have U(\u03b6)=(I2+O(\u03b6\u22121))\u03b62N00\u03b6\u22122N.As U to the above RH problem is unique (provided it exists), and can be expressed in terms of scalar\u2010valued OPs as follows:p2N and q2N\u22121 are polynomials of degree 2N and 2N\u22121, respectively, satisfying the following conditions:RU is defined byTheorem 1x\u2208{1,\u2026,2N\u22121}, y\u2208Z, and \u03b5x\u2208{0,1}, we have\u03b3C is a closed curve surrounding c and c\u22121 once in the positive direction that does not go around 0, and where HK and HK are given byFor It is knownRemark 2RY from , they are ordered as follows:The function given by , r1, r3.3, x,y\u2208{0,1,\u2026,4N}, are denoted\u00a0byThe densities for the three types of lozenges at a point \u2211j=13Pj=1. Because our model is 2 \u00d7 2 periodic, P1, P2, and P3 depend crucially on the parity of x and y, and it is convenient to consider the following matrices:x,y\u2208{0,1,\u2026,2N\u22121}. Let {}N\u22651 be a sequence satisfying lies in the hexagon belongs to the liquid region L\u03b1\u2282H.and satisfy 3.4\u2208H, there are in total eight saddles for the double contour integral (Q(\u03b6) is given by . This saddle plays a particular role in the analysis of Section\u00a0Proposition 3\u2208Ho (the interior set of H). Then, there exists at most one solution \u03b6=s to  to  in C+={\u03b6For each integral , which agiven by . FollowiDefinition 1L\u03b1\u2282H bys:L\u03b1\u2192C+ by \u21a6s.We define the liquid region \u2208L\u03b1 and s=r+ for all \u03b1\u2208. We now describe some properties of \u21a6s. Consider the following three circles:R0=\u03b1, R\u03b1=(1\u2212\u03b1)\u03b1, and R1=1\u2212\u03b1\u03b1 , \u03b8\u03b1\u2208, and \u03b80\u2208. We also defineRemark 3A, the notation A\u00af stands for the closure of A.For a given set It is clear from  and 41)41) that \u03c0]}\u2282\u03b3\u03b1,4\u03a30={\u03b6\u2208C:|Proposition 4\u21a6s satisfies s=s, and(a){\u03be=0}\u2229L\u03b1 onto \u03a31\u00af\u2229C+,it maps (b){\u03b7=\u03be2}\u2229L\u03b1 onto (\u03b31\u2216\u03a31)\u2229C+,it maps (c){\u03b7=\u03be}\u2229L\u03b1 onto \u03a30\u00af\u2229C+,it maps (d){\u03b7=\u2212\u03be}\u2229L\u03b1 onto (\u03b30\u2216\u03a30)\u2229C+,it maps (e){\u03b7=0}\u2229L\u03b1 onto \u03a3\u03b1\u00af\u2229C+,it maps (f){\u03b7=2\u03be}\u2229L\u03b1 onto (\u03b3\u03b1\u2216\u03a3\u03b1)\u2229C+.it maps The map Q(\u03b6)1/2 with a branch cut joining r\u2212 to r+ along \u03a31, such that Q(\u03b6)1/2\u223c12\u03b6 as \u03b6\u2192\u221e, and denote the associated Riemann surface by R\u03b1:1, and the sheets are ordered such that w=Q(\u03b6)1/2 on the first sheet and w=\u2212Q(\u03b6)1/2 on the second sheet. For each solution \u03b6 to (w satisfying w2=Q(\u03b6), and such thatDefinition 2\u21a6w is defined by w2=Q), such that  and w=w.The map uch that  holds wiBy definition, the saddles lie in the complex plane. We show here that they can be naturally projected on a Riemann surface. Define ion \u03b6 to , there eProposition 5\u21a6,w) is a diffeomorphism from L\u03b1 toL\u03b1l={\u2208L\u03b1\u2223\u03be<0} to the upper half\u2010plane of the first sheet of R\u03b1, and it maps L\u03b1r={\u2208L\u03b1\u2223\u03be>0} to the upper half\u2010plane of the second sheet. Moreover, its inverse \u21a6=,\u03b7) is explicitly given byThe map \u21a6, we conclude that L\u03b1 is symmetric with respect to the origin. Also, this equation has real coefficients, so s and s\u00af are both solutions whenever \u2208L\u03b1. At the boundary \u2202L\u03b1 of the liquid region, s and s\u00af coalesce in the real line, so \u2202L\u03b1 is part of the zero set of the discriminant of \u21a6. As we will see, these regions are frozen (or semifrozen).From Propositions\u00a0After clearing the denominator in , we get(\u03b6\u2212r1)2(\u03b6(\u03b6\u2212r1)2. Denote0(\u03b6\u2212r1)2\u2208L\u03b1. We obtain the following limits:\u03d5k,ij, 1\u2264i,j,k\u22642, and \u03d53,11(l), \u03d53,11(r), \u03d53, 12, \u03d53, 21, \u03d53,22(l), and \u03d53,22(r) are the angles represented in Figure\u00a0Let tisfying  with \u2208L\u03b1. We havej=1,2,3.Let tisfying  with \u2208{0,\u2026,2N\u22121} be such that =\u2208Fj,\u03b1, j\u2208{1,\u2026,6}. In Figure\u00a0From Figure\u00a0\u2208Fj,\u03b1, j=1,\u2026,6, respectively. Corollary\u00a0depending on whether 3.6M. The proof of Theorem\u00a0M is of genus 0. The proof of Theorem\u00a0In Section\u00a0U. This analysis goes via a series of transformations U\u21a6T\u21a6S\u21a6R. The first transformation U\u21a6T uses a so\u2010called g\u2010function, which is obtained in Section\u00a0In Section\u00a0R\u03b1 and play a central role in the large N analysis of the kernel. In Section\u00a0N\u03a6={\u03b6\u2208C:Re\u03a6(\u03b6)=Re\u03a6(s)}, which is of crucial importance to find the contour deformations that we need to consider for the saddle point analysis.The functions \u03a6 and \u03a8 denote the restrictions of the phase function \u039e to the first and second sheets of The proofs of Propositions\u00a0\u03b1\u2208, even though it will not be written\u00a0explicitly.As mentioned in Remark\u00a044.1M given byM are real, Proposition\u00a0M has at least six zeros on the real line. This can be proved by a direct inspection of the values of M(\u03b6) at \u03b6=\u2212\u221e,r1,0,\u03b1c,r2,\u03b1c,c,r3,c\u22121,+\u221e:\u2208Ho, whereM are 1\u2212(\u03be\u2212\u03b7)2>0. We conclude the following:\u03b7\u2260\u2212\u03be, M has at least one simple root on  and at least one simple root on ,if \u03b7\u2260\u03be2, M has at least one simple root on  and at least one simple root on ,if \u03b7\u22602\u03be, M has at least one simple root on  and at least one simple root on .if  Finally, other computations show that M\u2032(r1)=0 if \u03b7=\u2212\u03be, that M\u2032(r2)=0 if \u03b7=\u03be2 and that M\u2032(r3)=0 if \u03b7=2\u03be. So M has at least 6 real zeros (counting multiplicities) for each \u2208Ho.By , the sad\u03b6\u2212r2)2\u03b6\u2212r2(\u03b6\u2212r+)(\u03b64.2w satisfies w2=Q(\u03b6). This can be rewritten as\u03be,\u03b7\u2208R, we geta1,a2,a3,a4>0, we have\u21a6,w) is a bijection from L\u03b1 to R\u03b1+. This mapping is clearly differentiable, and therefore it is a diffeomorphism. Replacing \u21a6 in the right\u2010hand side of \u21a6. This implies the symmetry s=s. It remains to prove that \u2208L\u03b1l is mapped to a point ,w) lying in the upper half plane of the first sheet. The proof of this claim is splitted in the next two lemmas.Lemma 1Im(2\u03b6(\u03b6\u2212c)(\u03b6\u2212c\u22121)(\u03b6\u22121)(\u03b6+1)Q(\u03b6)1/2)=0 if and only if \u03b6\u2208R\u222a\u03a31\u00af.We have We start with the proof of Proposition\u00a0itten as\u2212(\u03b6\u2212\u03b1), there are eight solutions \u03b6\u2208C to f(\u03b6)=x. The claim follows if we show that all these solutions lie on R\u222a\u03a31. First, note that the function f is positive on the real line, has poles at \u22121,\u03b1c,\u03b1c\u22121,1, and zeros at r1,r2,r3. Because \u22121<r1<\u03b1c<r2<\u03b1c\u22121<r3<1, the equation f(\u03b6)=x has at least six real solutions (counting multiplicities) for each x\u2208. Writing \u03b6=c\u22121+R1eit\u2208\u03b31, t\u2208, some computations show thatt\u21a6f(c\u22121+R1eit) is even, positive, and decreases from f(c\u22121+R1) to 0 as t increases from 0 to \u03b81, which finishes the proof.\u25aaConsider the function Lemma 2\u2208R\u03b1+ such that w=Q(s)1/2  is in the first sheet). Then, \u03be=\u03be<0.Let s\u2208\u03a31\u00af, which implies that the sign of s\u2212c\u2212(s\u22121)\u21a6 that lead to simpler proofs of (b)\u2013(f). We only sketch the proof of (e). First, we rewrite  has the same sign as\u03b6\u2208R\u222a\u03a3\u03b1, which proves part (e). We omit the proofs of parts (b), (c), (d), and (f).This finishes the proof of Proposition\u00a0 rewrite  as\u03be+\u2212(\u03b6te (\u03be+\u2212\u03b6\u2212(\u03b6+1)2Nz\u22122N. To do this, we follow an idea of Duits and KuijlaarsA. Then, we use this factorization to perform a first transformation Y\u21a6X on the RH\u00a0problem.To describe the behavior of 5.1A(z) defined in (z\u2212z\u2212) such that it is analytic in C\u2216, with an asymptotic behavior at \u221e given byA are in the columns of the following matrix:\u039b(z)=diag(\u03bb1(z),\u03bb2(z)) is the matrix of eigenvalues. The matrix E(z) is analytic for z\u2208C\u2216, and satisfies\u03c31=0110.The matrix fined in  has the 5.2Remark 4detY\u22611. Note however that the Y\u21a6X transformation does not preserve the unit determinant. Indeed, because detE(z)=\u2212(1\u2212\u03b1)(z\u2212z+)(z\u2212z\u2212), we have detX(z)=(1\u2212\u03b1)2(z\u2212z+)(z\u2212z\u2212).By standard arguments,The first transformation of the RH problem diagonalizes the 2 \u00d7 2 upper right block of the jumps, and is defined byE given by X:C\u2216\u2192C4\u00d74 is analytic, where we recall that \u03b3 is a closed contour surrounding 0 once in the positive direction.(b)X are given byZ:=\u03b3\u2229. Depending on \u03b3, Z can be the empty set, a finite set, or an infinite set. If Z contains one or several intervals, then on these intervals the jumps areThe jumps for (c)z\u2192\u221e, we have X(z)=(I4+O(z\u22121))zNE(z)0202z\u2212NE(z). As z\u2192z\u2212 or as z\u2192z+, X(z)=O(1)E(z)0202E(z).As 6A obtained in A obtaineproperty  of RY imM associated to the eigenvalues and eigenvectors of A. This Riemann surface is of genus 0 and therefore there is a one\u2010to\u2010one map between it and the Riemann sphere .Now, we introduce the Riemann surface 6.1M is defined byz\u2010plane glued along . On the first sheet we require y=z+O(1) as z\u2192\u221e, and on the second sheet we require y=\u2212z+O(1) as z\u2192\u221e. To shorten the notations, a point  lying on the Riemann surface will simply be denoted by z when there is no confusion, that is, we will omit the y\u2010coordinate. If we want to emphasize that the point  is on the j\u2010th sheet, j\u2208{1,2}, then we will use the notation z(j). With this convention, the two points at infinity are denoted by \u221e(1) and \u221e(2). The function y satisfies\u03bb1(z) and \u03bb2(z) are defined on the z\u2010plane (see (M as follows:M with two simple poles at \u221e(1) and \u221e(2) and no other poles. Using ((2) and 1\u03b1(2), and because M has genus 0, \u03bb has no other zeros. From , defined byM to the Riemann sphere. The first sheet of M is mapped by  is given byz is on the first sheet if |\u03b6|>1 and on the second sheet if |\u03b6|<1. By definition, the above function z(\u03b6) vanishes at \u03b6(0(1)) and \u03b6(0(2)). Because it has simple poles at \u03b6=0 and \u03b6=\u221e, and because z(\u03b6)=z+\u2212z\u22124\u03b6+O(1) as \u03b6\u2192\u221e, (z(\u03b6) and \u03b6(z) satisfyz\u2208M, Imz=0, z\u2209, follows the straight line segments , , , , the function \u03b6(z) increases from \u2212\u221e to +\u221e. In particular, we havec byThe Riemann surface ane (see ), and toatisfiesy(1\u03b1(2))=os. From , the matfined by\u03b6=2z\u2212(z++given byz=z++z\u22122+fined by\u03b6=2z\u2212(z++6.2w(j) on the j\u2010th sheet of M and z(k) on the k\u2010th sheet, we define RM(w(j),z(k)) bye1=10 and e2=01. Note that RM:M\u2217\u00d7M\u2217\u2192C is scalar valued, with M\u2217=M\u2216{\u221e(1),\u221e(2)}. It is convenient for us to consider formal sums of points on M, which are called divisors in the literature. More precisely, a divisor D can be written in the formD\u22650 if n1,\u2026,nk\u22650. The divisor of a nonzero meromorphic function f on M is defined byz1,\u2026,zk1 are the zeros of f of multiplicities n1,\u2026,nk1, respectively, and zk1+1,\u2026,zk2 are the poles of f of order nk1+1,\u2026,nk2, respectively. Given a divisor D, we define L(\u2212D) as the vector space of meromorphic functions on M given byLN:=L(\u2212DN) is the vector space of meromorphic functions on M, with poles at \u221e(1) and \u221e(2) only, such that the pole at \u221e(1) is of order at most N\u22121, and the pole at \u221e(2) is of order at most N. Similarly we define LN\u2217=L(\u2212DN\u2217). From the Riemann\u2013Roch theorem, we haveLemma 3(a)z\u21a6RM\u2208LN for every w\u2208M\u2217,(b)w\u21a6RM\u2208LN\u2217 for every z\u2208M\u2217,(c)RM is a reproducing kernel for LN in the sense thatf\u2208LN, where \u03b3M is a closed contour surrounding once 0(1) and 0(2) on the Riemann surface M in the positive direction (in particular \u03b3M visits both sheets).We haveFor RM and RX given by   and at most N+1 at \u221e(2). Therefore, we have shown thatw\u2208M\u2217 a linear combination of the functions fj, so belong to L(\u2212(DN+\u221e(1)+\u221e(2))) as a function of z. By definitions of RM and RX, the numerator vanishes for z=w(1) and for z=w(2). Thus, the division by z\u2212w in ((1) and \u221e(2) by one, and therefore z\u21a6RM\u2208LN as claimed in part (a). Now, we turn to the proof of part (b). First, we note thatdetY\u22611, by using condition (c) of the RH problem for X, we havegj are also analytic in M\u2217. On the other hand, by using the asymptotics Y(w)=I4+O(w\u22121) as w\u2192\u221e together with the fact that detY\u22611, we can obtain asymptotics for X\u22121(w) as w\u2192\u221e using  in (z=w (on any sheet). Finally, let us take P(w)=p(w)e1T=p(w)10 in =11=e1T+e2T, it givesek, we obtain(1) and \u03b3(2) for the projections of \u03b3 on the first and second sheets of M, respectively. Using  and 0(2) on M in the positive direction, and thusz\u2208M\u2217. Let us now take P(w)=p(w)e2T=p(w)01 in (w\u2208M\u21a611+\u03b1(\u03bb(w)\u2212\u03b12\u2212w) on M. By proceeding in a similar way as for =p1(z)+p2(z)(\u03bb(z)\u2212\u03b12\u2212z)withp1,p2twopolynomialsofdegree\u2264N\u22121}. Because z\u21a6\u03bb\u2212\u03b12\u2212z has a simple pole at \u221e(2) (and no other poles), we conclude that L\u2286LN. Note also that dimL=dimLN=2N, and thus we have L=LN. This finishes the proof.\u25aaUsing the definitions of given by  and 68)RM and RXgiven by ) with pry =\u2211y =\u2211z(2),andgj(w)=y=\u2211(w)10 in , with p y. Using , the abo(w)01 in , and notand thusp(z)=12\u03c0i6.3z=z(\u03b6) and w=w(\u03c9) byz (resp. w) is on the first sheet if |\u03b6|>1 (resp. |\u03c9|>1), and on the second sheet if |\u03b6|<1 (resp. |\u03c9|<1). We define RU in terms of RM as follows:Proposition 6W and c be defined as in  is meromorphic on M, with a pole of order at most N\u22121 at \u221e(1) and a pole of order at most N at \u221e(2). Because z(0)=\u221e(2) and z(\u221e)=\u221e(1), we conclude that for each \u03c9\u2208C, the function \u03b6\u21a6RM(w(\u03c9),z(\u03b6)) is meromorphic on C\u222a{\u221e}, with a pole of order at most N\u22121 at \u221e and another pole of order at most N at 0. Therefore, for each \u03c9\u2208C, the function \u03b6\u21a6RU is a polynomial of degree at most 2N\u22121. From part (b) of Lemma\u00a0\u03b6\u2208C, the function \u03c9\u21a6RU is a polynomial of degree at most 2N\u22121. So we have proved that RU is a bivariate polynomial of degree \u22642N\u22121 in both \u03c9 and \u03b6.From part (a) of Lemma\u00a0\u03c9(0(1))=c\u22121, \u03c9(0(2))=c, (\u2202\u03c9w)(c\u22121)>0, and (\u2202\u03c9w)(c)<0. In particular, the map w\u21a6\u03c9(w) is conformal in small neighborhoods of 0(1) and 0(2). Because conformal maps preserve orientation, the curve \u03b3M which surrounds both 0(1) and 0(2) once in the positive direction, is mapped by w\u21a6\u03c9(w) onto a curve \u03b3C on the complex plane, which surrounds c and c\u22121 once in the positive direction. Furthermore, because \u03c9(\u221e(2))=0, the curve \u03b3C does not surround 0. By changing variables \u21a6 in ) is meromorphic on the Riemann sphere, with a pole of degree at most N at \u03b6=0 and a pole of degree at most N\u22121 at \u03b6=\u221e. In other words, \u03b6\u21a6\u03b6Nf(z(\u03b6))=:p(\u03b6) is a polynomial of degree at most 2N\u22121. By multiplying the above equality by \u03b6N, we thus have\u25aaNow, we turn to the proof of . It can  in , and by  in , 76), a, a\u03c9(0(1) in , we obtatituting  in the aRU in terms of the solution U to the 2 \u00d7 2 RH problem presented in Section\u00a0Proposition 7RU defined by  and E(z)\u22121 asRX\u21a6RM transformation given by (\u03b3M is a closed contour surrounding once 0(1) and 0(2) on M in the positive direction. By performing the change of variables w=w(\u03c9) and z=z(\u03b6) as in  is defined for \u03c9,\u03b6\u2208C and \u03b5x\u2208{0,1} byNow, using the results of Sections\u00a0given in  to rewri we haveK\u22121=11(z)\u22121 asE(w)=11\u03bb\u22121=11iven in  as in  and 87)x\u2208{1,\u2026,2N\u03b6) as in  and 77)x\u2208{1,\u2026,2N\u03b6) as in , and alsgiven by , we getng again  and (77)finition  of W, weiven by x\u2208{1,\u2026,2N7Pj, j=1,2,3, defined in , j=1,2,3, in terms of RU. In the rest of this section, we follow Ref. \u03b7\u2264\u03be2\u22640 of the liquid region for the proof of Theorem\u00a0This section is about the lozenge probabilities fined in . In Sect7.1q and q\u223c are given byPj, j=1,2,3, are obtained via a series of lemmas. Let us first recall that the paths pj:{0,1,\u2026,4N}\u2192Z+12, j=0,\u2026,2N\u22121 are defined in ..(98)K|>1 whenever \u03b6\u2208\u03b3\u223cC and \u03c9\u2208\u03b3C.For The next lemma establishes a double integral formula for the expectation value of the height function.X be the random variable that counts the number of paths going through the point , x\u223c,y\u223c\u2208{0,1,\u2026,4N}. Because X\u2208{0,1}, we have P=1)=E). Also, note that the identity =1)=K. Thus, by definition \u222aC, where C denotes a circle oriented positively centered at a of radius r. We see from |\u2192+\u221e as \u03b6 tends to c or c\u22121. Thus, by choosing r sufficiently small, we can make sure that \u03b3\u223cC lies in the interior region of \u03b3C, and that\u03b5>0. Therefore, uniformly for \u03b6\u2208\u03b3\u223cC and \u03c9\u2208\u03b3C, we have\u03b5y=0 follows after combining , , X. By . By , for \u03b5x,,formula  follows ,formula  with E be a smooth parameterization of a curve \u03c3, satisfying \u03b6\u2032(t)\u22600 for all t\u2208. \u03c3 is a trajectory of the quadratic differential Q(\u03b6)d\u03b62 if Q(\u03b6(t))\u03b6\u2032(t)2<0 for every t\u2208, and an orthogonal trajectory if Q(\u03b6(t))\u03b6\u2032(t)2>0 for every t\u2208. \u03c3 is critical if it contains a zero or a pole of Q. Note that these definitions are independent of the choice of the\u00a0parameterization.Let r+ and r\u2212 are simple zeros of Q, there are three critical trajectories  emanating from each of the points r\u00b1. Recall the definitions of \u03b30,\u03b3\u03b1,\u03b31,\u03a30,\u03a3\u03b1, and \u03a31 given in Section\u00a0Lemma 8\u03a30\u00af, \u03a3\u03b1\u00af, and \u03a31\u00af are three critical trajectories of Q(\u03b6)d\u03b62 joining r\u2212 with r+, and \u03b30\u2216\u03a30, \u03b3\u03b1\u2216\u03a3\u03b1, and \u03b31\u2216\u03a31 are each the union of two critical orthogonal trajectories of Q(\u03b6)d\u03b62. An illustration is shown in Figure\u00a0The arcs Because t\u21a6\u03b6=\u03b6(t)=c\u22121+R1eit, t\u2208, be a parameterization of \u03b31. Writing r\u00b1=c\u22121+R1e\u00b1i\u03b81 with \u03b81\u2208, and noting that \u03b6\u2032=iReit, we have(\u03b6\u2212r2)=2R1eit2cost2, andt\u2208, positive for t\u2208\u222a, and zero for t=\u2212\u03c0,\u2212\u03b81,\u03b81,\u03c0. We conclude that \u03a31\u00af is a critical trajectory and that \u03b31\u2216\u03a31 is the union of two orthogonal critical trajectories. The statement about \u03a3\u03b1\u00af, \u03b3\u03b1\u2216\u03a3\u03b1, \u03a30\u00af, \u03b30\u2216\u03a30 can be proved a similar way, and we provide less details. For \u03b6=\u03b6(t)=R0eit, t\u2208, after long but straightforward computations, we obtain\u03a30\u00af is a critical trajectory and that \u03b30\u2216\u03a30 is the union of two orthogonal critical trajectories. For \u03b6=\u03b6(t)=\u03b1c\u22121+R\u03b1eit, t\u2208, we obtain\u03a3\u03b1\u00af is a critical trajectory and that \u03b3\u03b1\u2216\u03a3\u03b1 is the union of two orthogonal critical trajectories. This finishes the proof.\u25aaLet )2.Using , we show, we getQ(\u03b6)(\u03b6\u2032)2ions in (Q(\u03b6)(\u03b6\u2032)2e obtainQ(\u03b6)(\u03b6\u2032)28.3g\u2032 can be expressed asQ(\u03b6)1/2. To obtain a g\u2010function with the desired properties (a), (b), and (c), it turns out that the branch cut needs to be taken along the critical trajectory \u03a31 (as in Section\u00a0Definition 3Q1/2 as(\u03b6\u2212r+)(\u03b6\u2212r\u2212) is taken on \u03a31 such thatWe define As can be seen in , g\u2032 can Q1/2.Definition 4\u03d5:C\u2216\u2192C by\u2208iR for \u03b6\u2208\u03a31. Therefore, Re\u03d5 is single\u2010valued and continuous in C\u2216{0,\u03b1c,\u03b1c\u22121,c,c\u22121}, and Re\u03d5 is also harmonic in C\u2216. Finally, by combining Definition\u00a0We first state some basic properties of \u03d5. By \u2013154), Q, Q154), N\u03d5 of Re\u03d5. This will be useful in Section\u00a0g\u2010function. Let us defineLemma 9N\u03d5 divides the complex plane in three regions. The sign of Re\u03d5 in these regions is as shown in Figure\u00a0We haveIn the rest of this subsection, we determine the zero set \u03d5\u2032=Q\u03b11/2 changes sign when it crosses each of the nine points r1, 0, \u03b1c, r2, \u03b1c\u22121, c, r3, c\u22121, c\u22121+R1. Because \u03d5\u2032(\u03b6)=2\u22121\u03b6\u22121+O(\u03b6\u22122) as \u03b6\u2192\u221e, we have \u03d5\u2032>0 on the intervals\u03d5\u2032<0 on the intervalsRe\u03d5 admits no other zeros on \u222a\u222a. On the intervals  and , Re\u03d5 admits a local minimum at r1 and r3, respectively, and on the interval , it admits a local maximum at r2. Thus, (Re\u03d5 is strictly monotone on each of the curves (\u03b30\u2216\u03a30)\u2229C+, (\u03b3\u03b1\u2216\u03a3\u03b1)\u2229C+, and (\u03b31\u2216\u03a31)\u2229C+. The expressions (Re\u03d5 is strictly increasing on (\u03b30\u2216\u03a30)\u2229C+ oriented from r+ to r1, strictly increasing on (\u03b3\u03b1\u2216\u03a3\u03b1)\u2229C+ oriented from r+ to r3, and strictly decreasing on (\u03b31\u2216\u03a31)\u2229C+ oriented from r+ to r2. In particular this proves , , (167)N\u03d5ressions , togethe. Thus, (Re\u03d5(r1)>0how thatN\u03d5\u2229R=\u22600 for \u03b6\u2208\u03a31, we must have \u03c3\u2229\u03a31=\u2205. Also, in view of (C\u2216(R\u222a\u03a31), and the max/min principle for harmonic functions would then imply that Re\u03d5 in constant on the whole bounded region delimited by \u03c3. By .\u25aa8.4Definition 5\u03a31=supp(\u03bc) is given by 1/2 denotes the limit of Q(\u03be)1/2 as \u03be\u2192\u03b6\u2208\u03a31 with \u03be in the exterior of the circle \u03b31.We define the measure \u03bc bygiven by , and is Proposition 13The measure \u03bc defined in  is a pro\u222b\u03a31d\u03bc by residue calculation. Because Q+=\u2212Q\u2212, we haveC is a closed curve surrounding \u03a31 once in the positive direction, but not surrounding any of the poles of Q. By deforming C into another contour C\u223c surrounding 0, \u03b1c, \u03b1c\u22121, c, and c\u22121, we pick up some residues:P={0,\u03b1c,\u03b1c\u22121,c,c\u22121}. By combining Definition\u00a0Q(\u03b6)1/2=12\u03b6+O(\u03b6\u22122) as \u03b6\u2192\u221e, we find1. Let \u03b6(t)=c\u22121+R1eit, \u2212\u03b81<t<\u03b81, be a parameterization of \u03a31. Consider the functionQ(\u03b6(t))(\u03b6\u2032(t))2<0 for t\u2208 by Lemma\u00a0t=\u2212\u03b81 and equals 1 for t=\u03b81 by (\u25aaWe compute given by1\u03c0iQ\u2212(\u03b6(tfunctiont\u21a6\u222br\u2212\u03b6(t) we find\u222b\u03a31d\u03bc(\u03b6)=given by1\u03c0iQ\u2212 has a branch cut along =log|\u03b6|+O(\u03b6\u22121), as \u03b6\u2192+\u221e.The \u2113\u2208C byWe define the variational constant g is consistent with (g satisfies (Proposition 14g and \u03d5 are related byg\u2010function satisfies the properties (a), (b), and (c) listed at the beginning of Section\u00a0\u03b3C=\u03b31.The functions we have10g+(\u03b6)+g\u2212The next proposition shows, among other things, that Definition\u00a0ent with , and thaatisfies .Proposi\u03b6\u2208C\u2216\u03a31, we haveC is a closed curve surrounding \u03a31 once in the positive direction, but not surrounding any of the poles of Q, and not surrounding \u03b6. By deforming C into another contour C\u223c surrounding 0, \u03b1c, \u03b1c\u22121, c, c\u22121, and \u03b6, we obtainP={0,\u03b1c,\u03b1c\u22121,c,c\u22121}. By deforming C\u223c to \u221e, noting that Q(\u03be)1/2=O(\u03be\u22121) as \u03be\u2192\u221e, the integral on the right\u2010hand side of =0. Then,  is fulfilled. For \u03b6\u2208 is also fulfilled. For \u03b6\u2208\u03b31\u2216\u03a31\u00af, by (\u03b6\u2208\u03a31 we haver\u2212 to r+. So (\u25aaWe first prove . For a fe obtain\u222b\u03a31d\u03bc(\u03be)\u03b6 side of , and we /2.Using  and \u03d5\u2032=Q0. Then,  follows 0. Then, . Becausen \u03a31, by  we haveom which  and T:C\u2216\u03b31\u2192C2\u00d72 is analytic.(b)T are given byBy using , 140), , 140), a(c)\u03b6\u2192\u221e, we have T(\u03b6)=I+O(\u03b6\u22121). As \u03b6 tends to r+ or r\u2212, T(\u03b6) remains bounded.As T will be important for the saddle point analysis of Section\u00a0Proposition 15T(\u03b6)=O(N1/6) and T\u22121(\u03b6)=O(N1/6) as N\u2192\u221e, uniformly for \u03b6\u2208C\u2216\u03b31. In addition, for every \u03b4>0 fixed, we have T(\u03b6)=O(1) and T\u22121(\u03b6)=O(1) as N\u2192\u221e uniformly forWe have The following estimates for The rest of this section is devoted to the proof of Proposition\u00a09.2\u03b6\u2208\u03a31, the jumps for T can be factorized as follows:\u03d5+(\u03b6)+\u03d5\u2212(\u03b6)=0 for \u03b6\u2208\u03a31. We define the lenses \u03b3+ and \u03b3\u2212 byT\u21a6S transformation is given by S(\u03b6)=T(\u03b6)W(\u03b6), whereS satisfies the following RH\u00a0problem.Note that for (a)S:C\u2216(\u03b31\u222a\u03b3+\u222a\u03b3\u2212)\u2192C2\u00d72 is analytic.(b)S are given byThe jumps for (c)\u03b6\u2192\u221e, we have S(\u03b6)=I+O(\u03b6\u22121). As \u03b6 tends to r+ or r\u2212, S(\u03b6) remains bounded.As 9.3S in different regions of the complex plane. By Lemma\u00a0Re\u03d5(\u03b6)>0 for \u03b6\u2208\u03b3+\u222a\u03b3\u2212, Re\u03d5(\u03b6)<0 for \u03b6\u2208\u03b31\u2216\u03a31\u00af, and Re\u03d5(\u03b6)=0 for \u03b6\u2208\u03a31. So the jumps for S on \u03b3+\u222a\u03b3\u2212\u222a(\u03b31\u2216\u03a31\u00af) are exponentially close to the identity matrix matrix as N\u2192\u221e, uniformly outside fixed neighborhoods of r\u2212 and r+. By ignoring these jumps, we are left with the following RH problem, whose solution is denoted as P(\u221e). We will show in Section\u00a0P(\u221e) is a good approximation to S away from r+ and r\u2212.In this subsection, we find good approximations to (a)P(\u221e):C\u2216\u03a31\u00af\u2192C2\u00d72 is analytic.(b)P(\u221e) are given byThe jumps for (c)\u03b6\u2192\u221e, we have P(\u221e)(\u03b6)=I+O(\u03b6\u22121). As \u03b6\u2192\u03b6\u2217\u2208{r+,r\u2212}, P(\u221e)(\u03b6)=O((\u03b6\u2212\u03b6\u2217)\u22121/4).As P(\u221e)(\u03b6) as \u03b6\u2192\u03b6\u2217\u2208{r+,r\u2212} has been added to ensure existence of a solution. This RH problem is independent of N, and its unique solution is given bya(\u03b6):=(\u03b6\u2212r+\u03b6\u2212r\u2212)1/4 is analytic in C\u2216\u03a31 and such that a(\u03b6)\u223c1 as \u03b6\u2192\u221e.The condition on the behavior of P(\u221e) is not a good approximation to S in small neighborhoods of r+,r\u2212; this can be seen from the behaviors\u03b4>0 in Proposition\u00a0Dr+ and Dr\u2212 be small open disks of radius \u03b4/2 centered at r+ and r\u2212, respectively. We now construct local approximations P(r+) and P(r\u2212)  to S in Dr+ and Dr\u2212, respectively. We require P(r\u00b1) to satisfy the same jumps as S inside Dr\u00b1, to remain bounded as \u03b6\u2192r\u00b1, and to satisfy the matching condition\u03b6\u2208\u2202Dr\u00b1. The density of \u03bc vanishes like a square root at the endpoints r+ and r\u2212, and therefore P(r\u00b1) can be built in terms of Airy functions.z\u2208Dr\u00b1.Note that 9.4S\u21a6R of the steepest descent is defined byS and P(r\u00b1) satisfy the same jumps inside Dr\u00b1, R is analytic inside (Dr+\u2216{r+})\u222a(Dr\u2212\u2216{r\u2212}). Furthermore, S and P(r\u00b1) remain bounded near r\u00b1, so the singularities of R at r\u00b1 are removable. We conclude that R is analytic inR\u2212\u22121R+ are O(N\u22121) on \u2202Dr+\u222a\u2202Dr+, and by Lemma\u00a0R\u2212\u22121R+=O(e\u2212cN) on (\u03b31\u222a\u03b3+\u222a\u03b3\u2212)\u2216(Dr+\u222aDr\u2212) for a certain c>0. It follows by standard theoryR and R\u22121 remain bounded as N\u2192\u221e.The final transformation \u222a\u2202Dr\u2212.By , the jumined by16R(\u03b6)=S(\u03b6lytic inC\u2216(\u03b31\u222a\u03b3+\u222aW(\u03b6) and W(\u03b6)\u22121 are bounded as N\u2192+\u221e, uniformly for \u03b6\u2208C. Proposition\u00a0Inverting the transformations  and 196196, we g10N\u2192+\u221e will be in the form e2N\u2212\u03a6), for a certain function \u03a6 which is described below. The analytic continuation of \u03a6 to the second sheet of R\u03b1 is denoted \u03a8\u2014it will also play a role in the saddle point analysis and is presented\u00a0below.In Section\u00a0The content of this section is a preparation for the saddle point analysis of Section\u00a010.1Definition 7\u2208H and \u03b6\u2208C\u2216, we define \u03a6 and \u03a8 byFor ave used  and (179and \u03a8 by\u03a6(\u03b6)=\u03a6 and \u03b6\u21a6e\u00b12N\u03a8 have no jumps along \u2208H, Re\u03a6 and Re\u03a8 are harmonic on C\u2216, and well defined and continuous on C\u2216{0,\u03b1c,\u03b1c\u22121,c,c\u22121}. For \u2208Ho, we note the following basic properties of \u03a6:In the formulas that will be used in Section\u00a0\u03a6\u2032 and \u03a8\u2032. For the saddle point analysis, it will be important to know: (1) the sign of |s\u2212c\u22121|\u2212R1 and (2) whether \u03a6\u2032(s)=0 or \u03a8\u2032(s)=0. We summarize the different cases in the next lemma.Lemma 10\u2208L\u03b1 and s=s. Then, we have(a)\u03a6\u2032(s)=0 and |s\u2212c\u22121|<R1 if and only if \u03be<0 and \u03b7<\u03be2,(b)\u03a6\u2032(s)=0 and |s\u2212c\u22121|>R1 if and only if \u03be<0 and \u03b7>\u03be2,(c)\u03a8\u2032(s)=0 and |s\u2212c\u22121|<R1 if and only if \u03be>0 and \u03b7>\u03be2,(d)\u03a8\u2032(s)=0 and |s\u2212c\u22121|>R1 if and only if \u03be>0 and \u03b7<\u03be2,(e)|s\u2212c\u22121|=R1 if and only if \u03be=0 or \u03b7=\u03be2.Let Because the saddle points are the solutions to , it foll\u25aaThis is an immediate consequence of Propositions\u00a010.2\u03b7\u2264\u03be2<0. We have represented N\u03a6 for different values of  in Figures\u00a0\u03a6\u2032 and \u03a8\u2032. From , , and . This determines the location of six saddles. The remaining two are s and s\u00af, and we already know from Lemma\u00a0\u03a6\u2032(s)=0=\u03a6\u2032(s\u00af). Therefore, \u03a6\u2032\u22600 on \u222a. Because \u03a6\u2032(\u03b6)\u2208R for \u03b6\u2208R\u2216{0,\u03b1c,\u03b1c\u22121,c,c\u22121}, this implies by \u2229C+ is either the empty set or a\u00a0singleton.We show with the next two lemmas that the set \u03b6\u2208C\u2216{0,\u03b1c,\u03b1c\u22121,c,c\u22121}, we define the following functions:Lemma 11(\u03a3\u03b1\u222a\u03a31\u00af)\u2229C+ from left to right, then(1)Ref1 is strictly decreasing on \u03a3\u03b1\u2229C+ and constant on \u03a31\u2229C+,(2)Ref2 is constant on \u03a3\u03b1\u2229C+ and strictly decreasing on \u03a31\u2229C+,(3)Ref3 is strictly decreasing.If \u03b6 moves along For ddtRef1(\u03b1c\u22121+R\u03b1e\u2212it) has the same sign as sint. In particular, Ref1(\u03b6) is strictly decreasing along \u03a3\u03b1\u2229C+ as \u03b6 moves from left to right. Another (and simpler) computation givesRef1 is constant on \u03a31. The proofs for f2 and f3 are similar, so we omit them.\u25aaA long and tedious computation shows that Corollary 3\u03b7\u2264\u03be2<0, the function \u03b6\u21a6Re\u03a6(\u03b6) is strictly decreasing as \u03b6 moves along (\u03a3\u03b1\u222a\u03a31\u00af)\u2229C+ from left to\u00a0right.For Re\u03d5=0 on \u03a3\u03b1\u222a\u03a31. Therefore, from the expression  for the open and bounded region delimited by \u03c3.\u03a6\u2032(s)=0, there are four curves {\u0393j}j=14 emanating from s that belongs to N\u03a6. By Corollary\u00a0N\u03a6\u2229(\u03a3\u03b1\u222a\u03a31\u00af)\u2229C+ is either the empty set or a singleton, so at least three of the \u0393j's, say \u03931,\u03932,\u03933, do not intersect (\u03a3\u03b1\u222a\u03a31\u00af)\u2229C+. The curves \u0393j, j=1,2,3 cannot lie entirely in C+; otherwise the max/min principle for harmonic functions would imply that Re\u03a6 is constant within the region int(\u0393j). Therefore, \u0393j, j=1,2,3 have to intersect R. Note that \u03a6(\u03b6)\u00af=\u03a6(\u03b6\u00af) implies that N\u03a6 is symmetric with respect to R. In particular, the curves \u0393j, j=1,2,3 join s with s\u00af. The next lemma states that \u03934 is not contained in the region int(\u03a3\u03b1\u222a\u03a31\u00af).Lemma 12N\u03a6\u2229(\u03a3\u03b1\u222a\u03a31\u00af)\u2229C+ is a\u00a0singleton.Because 4 lies entirely in int(\u03a3\u03b1\u222a\u03a31\u00af), and denote pj for the intersection point of \u0393j with R. We assume without loss of generality that p1<p2<p3<p4. There is at most one pj inside each of the intervalspj's, and each of them leads to a contradiction. Let us treat the caseRe(\u03a6(\u03b6)\u2212\u03a6(s)) changes sign as \u03b6 crosses N\u03a6\u2216{s,s\u00af}, by \u2229N\u03a6=\u2205, and \u03c32 is a closed curve surrounding either c or c\u22121, such that int(\u03c32)\u2229N\u03a6=\u2205. Because N\u03a6 intersects both  and  exactly once, \u03c31 surrounds \u03b1c and \u03c32 surrounds c\u22121. Then, the max/min principle implies that Re\u03a6 is constant on int(\u03933\u222a\u03934\u00af)\u2216int(\u03c32), which is a contradiction. The four other cases than \u2192+\u221e as \u03b6\u2192\u221e, so \u03934 intersects the real line, and then by symmetry ends at s\u00af. So each of the \u0393j's intersects R. We denote pj for the intersection point of \u0393j with R, and choose the ordering such that p1<p2<p3. We recall that Re(\u03a6(\u03b6)\u2212\u03a6(s)) is harmonic for \u03b6\u2208C\u2216 and changes sign as \u03b6 crosses N\u03a6\u2216{s,s\u00af}. Therefore, by (int(\u03931\u222a\u03932\u00af) must contain at least one of the singularities \u03b1c and \u03b1c\u22121, and int(\u03932\u222a\u03933\u00af) must contain at least one of the singularities c and c\u22121. There are still quite a few cases that can occur. The figures provide a fairly good overview (though not complete) of what can happen:1.\u03b1c,\u03b1c\u22121,c\u2208int(\u03931\u222a\u03932\u00af), c\u22121\u2208int(\u03932\u222a\u03933\u00af). In Figure\u00a02.\u03b1c,\u03b1c\u22121\u2208int(\u03931\u222a\u03932\u00af) and c,c\u22121\u2208int(\u03932\u222a\u03933\u00af). In Figures\u00a03.\u03b1c\u22121\u2208int(\u03931\u222a\u03932\u00af) and c\u2208int(\u03932\u222a\u03933\u00af). In Figures\u00a0 Furthermore, \u03934 intersects both \u03a31 and  in Figure\u00a0\u03a3\u03b1 and  in Figure\u00a0\u03a3\u03b1 and  in Figure\u00a0\u03b3\u03b6\u2217 and \u03b3\u03c9\u2217 as described in the following proposition. These contours are illustrated for two different situations in Figures\u00a0Proposition 16\u2208L\u03b1 with \u03b7<\u03be2<0. There exist contours \u03b3\u03b6\u2217 and \u03b3\u03c9\u2217 such that\u03b3\u03c9\u2217\u2282int(\u03a3\u03b1\u222a\u03a31\u00af), it surrounds \u03b1c and \u03b1c\u22121, and it goes through s and s\u00af in such a way that\u03b3\u03b6\u2217\u2282int(\u03b31), surrounds c and c\u22121, and it goes through s and s\u00af in such a way thatLet Lemma\u00a0now from  that Re\u03a6fore, by , the reg\u03b7=\u03be2, we know from Proposition\u00a0s lies on \u03b31\u2216\u03a31\u00af. For the saddle point analysis, we will need \u03b3\u03b6\u2217 lying inside \u03b31 (not necessarily strictly inside). To prove existence of such a contour \u03b3\u03b6\u2217, we need to know that Re\u03a6(\u03b6)\u2212Re\u03a6(s) is strictly negative for \u03b6\u2208\u03b31\u2216\u03a31\u00af .Lemma 13\u03b7=\u03be2<0. For \u03b6\u2208\u03b31\u2216(\u03a31\u00af\u222a{s})\u2229C+, we have Re\u03a6(\u03b6)<Re\u03a6(s).Let If \u03b6=c\u22121+R1eit. For t\u2208, we havea1,a2 are given by a1=\u03b12+(1\u2212\u03b1)1\u2212\u03b1+\u03b122(1\u2212\u03b1) and a2=2\u22123\u03b1+2\u03b12+\u03b132(1\u2212\u03b1)1\u2212\u03b1+\u03b12 and satisfy a1>a2>1. The expression , such that Re(\u03a6\u2032(\u03b6)d\u03b6)=0, and this must be s. This implies that Re\u03a6(\u03b6)\u2212Re\u03a6(s) is of constant sign on \u03b31\u2216(\u03a31\u00af\u222a{s})\u2229C+. By (Re(\u03a6\u2032(\u03b6)d\u03b6)>0 at t=\u03b81 , so the claim is proved.\u25aaLet  we haveRe(\u03a6\u2032(\u03b6)d we haveRe(\u03a6\u2032(\u03b6)d\u03b3\u03b6\u2217 and \u03b3\u03c9\u2217 as described in Proposition\u00a0Proposition 17\u2208L\u03b1 with \u03b7=\u03be2<0. There exist contours \u03b3\u03b6\u2217 and \u03b3\u03c9\u2217 such that\u03b3\u03c9\u2217\u2282int(\u03a3\u03b1\u222a\u03a31\u00af), it surrounds \u03b1c and \u03b1c\u22121, and it goes through s and s\u00af in such a way that\u03b3\u03b6\u2217\u2282int(\u03b31)\u00af, surrounds c and c\u22121, and it goes through s and s\u00af in such a way thatLet Therefore, we can find contours 11 in the lower left part of the liquid region, that is for \u2208L\u03b1\u2229{\u03b7\u2264\u03be2\u22640}. We divide the proof in three subcases: \u03b7\u2264\u03be2<0, \u03b7<\u03be2=0, and \u03b7=\u03be=0.Remark 6 lies in the other quadrants of the liquid region. Note however that this is not needed, thanks to the symmetries of Section\u00a0By adapting the analysis of this section and of Section\u00a0In this section, we prove Proposition\u00a011.1I is defined in  another formula for I in terms of the second column of U. Therefore, the choice of \u03b3C will matter. To be able to use the steepest descent of Section\u00a0\u03b3C=\u03b31. Recall that T is expressed in terms of U via  is uniformly bounded as \u03b6 and \u03c9 stay bounded away from r+ and r\u2212. We will need the analytic continuation in \u03c9 of R\u223cT from the interior of \u03b31 to the bounded region delimited by \u03a31\u222a\u03a3\u03b1\u00af , and by (Re\u03d5(\u03c9)<0 for \u03c9\u2208int((\u03b31\u2216\u03a31)\u222a\u03a3\u03b1), so R\u223cT,a remains bounded as N\u2192+\u221e, uniformly for \u03b6 and \u03c9 bounded away from r+ and r\u2212, as long as \u03c9\u2208int(\u03a31\u00af\u222a\u03a3\u03b1). Our next goal is to prove the following.Proposition 18 be coordinates inside the hexagon, such that \u03be:=xN\u22121 and \u03b7:=yN\u22121 satisfy \u2208L\u03b1 with \u03b7\u2264\u03be2<0. Take \u03b3\u03b6\u2217 and \u03b3\u03c9\u2217 as in Proposition\u00a0\u03b7<\u03be2, and as in Proposition\u00a0\u03b7=\u03be2 I is defiof U via , and def, and by  it is giRemark 7\u03b3\u03b6\u2217 intersects \u03b31\u2216\u03a31\u00af whenever \u03b7=\u03be2. We do not indicate whether we take the + or \u2212 boundary values in the integrand of  and R\u223c\u2212T denote the limits of R\u223cT as \u03c9\u2032\u2192\u03c9 from the interior and exterior of \u03b31, respectively.where Remark 8x,y\u2208{1,2,\u2026,2N\u22121}, we definem appear in the integrand of (q and q\u223c are defined in (H satisfies the conditions stated in Proposition\u00a0g(\u03c9) is bounded for \u03c9 in compact subsets and satisfies g(\u03c9)\u223clog(\u03c9) as \u03c9\u2192\u221e. Therefore, the following properties hold:(i)\u03b6\u21a6m is analytic in C\u2216{\u03c9,0,c,c\u22121},The function (ii)\u03c9\u21a6m is analytic in (C\u222a{\u221e})\u2216.The function  The statement that \u03c9\u21a6m is analytic at \u221e deserves a little computation: because x,y\u2208{1,2,\u2026,2N\u22121}, we have m=O(\u03c9\u22121\u22122N+N\u2212y+x)=O(\u03c9\u22122) as \u03c9\u2192\u221e.For grand of . We recafined in , that H \u03b7<\u03be2, Proposition\u00a0\u03b3\u03b6 lies strictly inside \u03b31, so in this case we can (and do) take \u03b3\u03b6=\u03b3\u03b6\u2217 in ). After a small computation, we find that this residue is the first term on the right\u2010hand side of \u2208L\u03b1\u2229{\u03b7\u2264\u03be2<0}, and define \u03beN:=xN/N\u22121 and \u03b7N:=yN/N\u22121. By \u2208L\u03b1\u2229{\u03b7\u2264\u03be2<0} for all large enough N  in , we getsN=s, \u03a6N(\u03b6):=\u03a6, and the contours \u03b3\u03b6\u2217 and \u03b3\u03c9\u2217 also depend on N, even though this is not indicated in the notation. Because \u03b3\u03b6\u2217 and \u03b3\u03c9\u2217 do not pass through r+ and r\u2212, Proposition\u00a0C1>0, and where D\u03b5 is the union of two small disks of radii \u03b5>0 surrounding s and s\u00af. Because sN and sN\u00af are simple zeros of \u03a6N\u2032, we have the estimatesC2>0. Therefore, the left\u2010most term in  in  by 2\u222b\u03b3ich give .\u25aa11.2\u2208L, such that \u03b7\u2032<\u03be\u20322<0. In this case, the set N\u03a6 contains four curves emanating from s: three of these curves, namely, \u03931, \u03932, and \u03933, lie in int(\u03a31\u222a\u03a3\u03b1\u00af), the other curve \u03934 intersects once \u03a31\u222a\u03a3\u03b1\u00af. Denote pj for the intersection of \u0393j with R, and recall that the ordering for \u03931, \u03932, and \u03933 is such that p1<p2<p3.Let us briefly recall first the situation for \u2192 with \u03b7<0  tends to a point s=s lying on \u03a31. In this limit, both \u03933 and \u03934 tend to the arc1 tends to \u03a31\u2216\u03a3s. Thus, the case \u03be=0 gives less freedom for the contour deformations and the saddle point analysis is more involved. To handle this case, we need information about both N\u03a6 and N\u03a8, whereN\u03a6 and N\u03a8 are represented in Figure\u00a0Lemma 14\u03be=0, we have \u03a31\u2282N\u03a6 and \u03a31\u2282N\u03a8.For As Re\u03d5(\u03b6)=0 for \u03b6\u2208\u03a31, by Definition\u00a0\u03b6\u2208\u03a31.\u25aaBecause \u03b3\u03b6\u2217 and \u03b3\u03c9\u2217 as follows \u00af, is such that (\u03a31\u2216\u03a3s)\u2282\u03b3\u03c9\u2217, surrounds \u03b1c and \u03b1c\u22121, and it satisfies\u03b3\u03b6\u2217\u2282int(\u03b31)\u00af, is such that \u03a3s\u2282\u03b3\u03b6\u2217, surrounds c and c\u22121, and it satisfiesWe choose {\u2208L\u03b1\u2229{\u03b7<\u03be2=0}, and define \u03beN:=xN/N\u22121 and \u03b7N:=yN/N\u22121. By Lemma\u00a0\u2208L\u03b1 satisfies \u03b7N<0 and \u03beN=0 for all N. The proof of Proposition\u00a0\u03b3\u03b6\u2217 and \u03b3\u03c9\u2217, and as in , \u03a6(\u03b6)=\u03a6, and the contours \u03b3\u03b6\u2217 and \u03b3\u03c9\u2217 depend on N. We also take the + boundary value in (\u03c9\u2208\u03b31. Because Re\u03a6(\u03b6)=Re\u03a6(s) for all \u03b6\u2208\u03a3s and Re\u03a6(\u03c9)=Re\u03a6(s) for all \u03c9\u2208\u03a31\u2216\u03a3s\u00af, we need additional deformation of\u00a0contours.Let tisfying  with =\u03a8\u2212(\u03b6) and the jumps for T =\u03a6\u2213(\u03b6) for \u03b6\u2208\u03a31, and because \u03a31\u2282N\u03a6\u2229N\u03c8, the signs of\u03b6\u2208\u03a31, provided \u03b5=\u03b5(\u03b6)\u2208R is small enough (\u03b5 nonnecessarily positive), see also the signs around \u03a31 in Figure\u00a0Re\u03a8(\u03b6)<Re\u03a8(s) for \u03b6\u2208\u03a3s,out. For the first term in (e2N(\u03a6+(\u03b6)\u22122\u03d5+(\u03b6)), and by Definition\u00a0\u03a8=\u03a6\u22122\u03d5. Therefore, we deform \u03a3s inward to \u03a3s,in, and this contour is chosen such that Re\u03a8(\u03b6)<Re\u03a8(s) for \u03b6\u2208\u03a3s,in, see Figure\u00a0We first treat the contour deformations in the \u03b6\u2010variable. Recall the definition  of R\u223cT,aps for T  to obtais for T (e2N\u03a6(\u03b6)TT(s for T (e2N\u03a6(\u03b6)T(\u03a31\u2216\u03a3s outward to (\u03b31\u2216\u03a3s)ext, see Figure\u00a0Re\u03a8(\u03c9)>Re\u03a8(s) for \u03c9\u2208(\u03b31\u2216\u03a3s)ext. Because \u03a6+(\u03c9)=\u03a8\u2212(\u03c9) on \u03a31, the exponential factor of the integrand is e\u22122N\u03a8(\u03c9) there. Also, for \u03c9\u2208\u03a31\u2216\u03a3s, by (e\u22124N\u03d5(\u03c9) remains bounded for \u03c9\u2208(\u03b31\u2216\u03a3s)ext.In the \u03c9\u2010variable, we simply analytically continue the integrand and deform 1\u2216\u03a3s, by  we haves,s\u00af, and that \u03c9 stays away from s,s\u00af,r+,r\u2212. By a similar analysis as the one done in (s or s\u00af is O(N\u221212) as N\u2192+\u221e. When \u03c9 is close to r\u00b1, we know by Proposition\u00a0T\u22121(\u03c9)=O(N1/6). Because \u03a6\u2032(r\u00b1)\u22600\u2260\u03a8\u2032(r\u00b1), the contribution to =r+, s\u00af=r\u2212, and \u03a6=\u2212\u03a8=\u03d5 =\u2208L\u03b1 with \u03b7\u2032<0, part of contour \u03b3\u03c9\u2217 lies in the region int(\u03a3\u03b1\u222a\u03931\u00af), see Figures\u00a0\u03b7\u2032\u2192\u03b7=0, \u03931 tends to \u03a3\u03b1, so we need additional contour deformations to handle this case. Consider the contours \u03b3\u03b6\u2217:=\u03b31 and \u03b3\u03c9\u2217=\u03b3\u03b1. By Lemma\u00a0At the center of the hexagon, we have {=}N\u22651, so that \u03beN:=xN/N\u22121=0 and \u03b7N:=yN/N\u22121=0 for all N. In the same way as done in Proposition\u00a0\u03c9\u2208\u03a3\u03b1.For simplicity, we consider the sequence position\u00a0I is given by the second line of (\u03a3\u03b1 outward so that Re\u03d5(\u03c9)>0, and for the first term, we deform \u03a3\u03b1 inward so that Re\u03d5(\u03c9)<0.For  line of , and thur+ and r\u2212. For \u03b6 and \u03c9 close to r\u00b1, by Proposition\u00a0T(\u03b6)=O(N1/6) and T\u22121(\u03c9)=O(N1/6). The contribution to (r+ and r\u2212 is thus bounded byC1,C2,C3>0 and for all large enough N. This finishes the proof of Proposition\u00a0On the deformed contours, the integrand is uniformly exponentially small, as long as \u03b6 and \u03c9 are bounded away from ution to  when \u03b6 a"}
{"text": "Given a finite set of European call option prices on a single underlying, we want to know when there is a market model that is consistent with these prices. In contrast to previous studies, we allow models where the underlying trades at a bid\u2013ask spread. The main question then is how large (in terms of a deterministic bound) this spread must be to explain the given prices. We fully solve this problem in the case of a single maturity, and give several partial results for multiple maturities. For the latter, our main mathematical tool is a recent result on approximation by\u00a0peacocks. Put differently, we want to detect arbitrage in given prices. We do not consider continuous call price surfaces, but restrict to the  case of finitely many strikes and maturities. Therefore, consider a financial asset with finitely many European call options written on it. In a frictionless setting, the consistency problem is well understood: Carr and Madan . To define the payoff of the call options, we use an arbitrary reference price process that evolves within the bid\u2013ask spread. We show =B(t)\u22121 for the time zero price of a zero\u2010coupon bond maturing at\u00a0t, and kt,i=D(t)Kt,i for the discounted strikes. The symbol Ct(K) denotes a call option with maturity\u00a0t and strike\u00a0K.Our time index set will be T=1 bid and ask are S_1=$99, respectively, S\u00af1=$101. Then, the agent might wish to exercise the option to obtain a security for $99 instead of $100, or he may forfeit the option on the grounds that spending $100 would earn him a position whose liquidation value is only $99. The exercise decision cannot be nailed down without making further assumptions. In practice, the quoted ticker price of the underlying is the last price at which an actual transaction has occurred. This price then triggers cash\u2010settled options. However, this approach is not feasible in our setup, which does not include an order\u00a0book.In the presence of a bid\u2013ask spread on the underlying, it is not obvious how to define the payoff of an option; this issue seems to have been somewhat neglected in the transaction costs literature. Indeed, suppose that an agent holds a call option with strike $100, and that at maturity SC. This process evolves within the bid\u2013ask spread. It is not a traded asset by itself, but just serves to fix the call option payoff (StC\u2212K)+ for strike\u00a0K and maturity\u00a0t. This payoff is immediately transferred to the bank account without any costs.Definition 2.1model consists of a finite probability space  with a discrete filtration (Ft)t\u2208T and three adapted stochastic processes S_, S\u00af, and SC, satisfyingA In the literature on option pricing under transaction costs, it is usually assumed that the bid and ask of the underlying are constant multiples of a mid\u2010price (often assumed to be geometric Brownian motion). This mid\u2010price is then used as trigger to decide whether an option should be exercised, followed by physical delivery Bichuch, . The assS_t and S\u00aft denote the bid, respectively, ask price of the underlying at time\u00a0t. Note that, in our terminology, the initial bid and ask are part of the given prices ), but allow any adapted process inside the bid\u2013ask spread. We now give a definition for consistency of option prices, allowing for (arbitrarily large) bid\u2013ask spreads on both the underlying and the options.Definition 2.2consistent with the absence of arbitrage, if there is a model (in the sense of Definition\u00a0E\u2208,1\u2264i\u2264Nt,t\u2208T\u2217.(S\u2217)t\u2208T such that S_t\u2264St\u2217\u2264S\u00aft for t\u2208T and such that (D(t)St\u2217)t\u2208T is a P\u2010martingale(Ft)t\u2208T. The pair  is called a consistent price system.There is a process The prices\u00a0As for the reference price process\u00a0S\u2217 is also called a shadow price. According to Kabanov and Stricker E1{s>0})s\u2208T as bid price process (and similarly for the ask price), and (B(s)E)s\u2208T as the process in the second part of Definition\u00a0The process\u00a0Stricker   is \u201ctoo expensive,\u201d and without frictions, buying C2(k)\u2212C1(k) would be an arbitrage opportunity (upon selling one unit of stock if C1(k) expires in the money). In particular, the first condition from Corollary 4.2 in\u00a0Davis and Hobson \u2212C1(k): the short call \u2212C1(k) finishes in the money with payoff \u2212(c+1). This cannot be compensated by going short in the stock, because its bid price stays at\u00a02. The payoff at time t=2 of this strategy, with shorting the stock at time\u00a0t=1, isBut with spreads we can choose As mentioned in Section\u00a0Definition 2.4\u03b5\u22650. Then the prices\u00a0Let Our focus will thus be on a stronger notion of consistency, where the discounted spread on the underlying is bounded. Hence, our goal becomes to determine how large a spread is needed to explain given option prices.SC, made for tractability, and makes sense given the actual size of market prices and spreads . With the same justification, in our main results on \u03b5\u2010consistency we will assume that all discounted strikes\u00a0kt,i are larger than\u00a0\u03b5. If \u03b5=0 and the bid and ask prices in\u00a0The bound\u00a0SC. However, it is not hard to show that choosing SC=12(S_+S\u00af) yields almost the same notion of \u03b5\u2010consistency.Proposition 2.5\u03b5\u22650 and assume that we are interested in arbitrage\u2010 free models where, in addition to the requirements of Definition\u00a0arithmetically \u03b5\u2010consistent. For \u03b5\u22650, the prices are arithmetically 2\u03b5\u2010consistent if and only if they are \u03b5\u2010consistent.Let As mentioned above, we do not insist on any specific definition of the reference price\u00a0S_t,S\u00aft,StC,St\u2217. We define new bid and ask prices S_t\u2032:=StC\u2227St\u2217 and S\u00aft\u2032:=StC\u2228St\u2217. Then\u00a0S\u00aft\u2032\u2212S_t\u2032\u2264B(t)\u03b5. Therefore, the model consisting of S_t\u2032,S\u00aft\u2032,StC,St\u2217 is \u03b5\u2010consistent. Conversely, assume that the given prices are \u03b5\u2010consistent. Then there exist processes SC and S\u2217 on a probability space  such that |StC\u2212St\u2217|\u2264B(t)\u03b5 a.s. We then simply set S_t=StC\u2212B(t)\u03b5 and S\u00aft=StC+B(t)\u03b5, and have thus constructed an arithmetically 2\u03b5\u2010consistent model.\u25a1First, assume that there exists an arithmetically 2\u03b5\u2010consistent model with corresponding stochastic processes p\u2208 and p\u226012.Note that the statement of Proposition\u00a0(D(t)StC)t\u2208T does not have to be a martingale, as\u00a0SC is not traded on the market. The option prices give us some information about the marginals of the process\u00a0SC, though. On the other hand, the process (D(t)St\u2217)t\u2208T has to be a martingale, but we have no information about its marginals, except that |St\u2217\u2212StC|\u2264\u03b5B(t). This impliesW\u221e denotes the infinity Wasserstein distance, and L the law of a random variable. The distance W\u221e is defined on\u00a0M, the set of probability measures on\u00a0R with finite mean, by and random pairs  with marginals . See\u00a0Gerhold and G\u00fcl\u00fcm \u2264\u03b5 is equivalent to the existence of a probability space with random variables X\u2032\u223cLX, Y\u2032\u223cLY such that |X\u2032\u2212Y\u2032|\u2264\u03b5 a.s. t\u2208T\u2217 in M is called a peacock, if \u03bcs\u2264c\u03bct for all s\u2264t in\u00a0T\u2217 . These notions are useful for constructing models for \u03b5\u2010consistent prices, as made explicit by the following lemma. As is evident from its proof, the sequence\u00a0(\u03bct) consists of the marginals of a (discounted) reference price, whereas\u00a0(\u03bdt) gives the marginals of a martingale within the bid\u2013ask spread. The proof uses a coupling result from our companion paper t\u2208T\u2217 and (\u03bdt)t\u2208T\u2217 in M such that:(i)R\u03bct\u2208 for all t\u2208T\u2217 and i\u2208{1,\u22ef,Nt}, and \u03bct)=1 for t\u2208T\u2217,(ii)(\u03bdt)t\u2208T\u2217 is a peacock and its mean satisfies E\u03bdT\u2208, and(iii)W\u221e\u2264\u03b5 for all t\u2208T\u2217.For For (\u03bct)t\u2208T\u2217 and (\u03bdt)t\u2208T\u2217 be as above. Recall that Strassen's ((S\u223ct)t\u2208T such that \u03bdt is the law of S\u223ct for t\u2208T\u2217.Let rassen's  Theorem M^ and\u00a0S^C such that M^t\u223c\u03bdt,D(t)S^tC\u223c\u03bct, and |M^t\u2212D(t)S^tC|\u2264\u03b5 for t\u2208T\u2217. As in the the proof of Theorem\u00a09.2 in\u00a0Gerhold and G\u00fcl\u00fcm (finite probability space with these properties. The sufficiency statement now easily follows from Lemma\u00a09.1 in\u00a0Gerhold and G\u00fcl\u00fcm ((S\u02c7t)t\u2208T and\u00a0(StC)t\u2208T\u2217 satisfyingS\u02c7 is a martingale,S\u02c7t\u223c\u03bdt and D(t)StC\u223c\u03bct for t\u2208T\u2217,|S\u02c7t\u2212D(t)StC|\u2264\u03b5 for t\u2208T\u2217. It then suffices to defineS_t\u2264St\u2217\u2264S\u00aft holds for t\u2208T and not just T\u2217.From\u00a0(iii), and the remark before Definition\u00a0nd G\u00fcl\u00fcm , it is end G\u00fcl\u00fcm . Indeed,t\u2208T\u2217, define\u00a0\u03bct as the law of D(t)StC, and \u03bdt as the law of\u00a0St\u2217. It is then very easy to see that the stated conditions are satisfied. As for the finite support condition, note that the probability space in Definition\u00a0\u25a1Conversely, assume now that the given prices are \u03b5\u2010consistent. For tth trading period, for example, becomesDefinition 2.8(i)semistatic portfolio, or semistatic trading strategy, is a triple\u03c610\u2208R, \u03c6t0:3t\u2192R are Borel measurable for t\u2208T\u2217, analogously for \u03d51, and \u03c6t,i\u2208R for t\u2208T\u2217,i\u2208{1,\u22ef,Nt}. Here, \u03c6t0 denotes the investment in the bank account, \u03c6t1 denotes the number of stocks held in the period from t\u22121 to\u00a0t, and \u03c6t,i\u2208R is the number of options with maturity t\u2208T\u2217 and strike Kt,i which the investor buys at time zero.A (ii)self\u2010financing, if1\u2264t<T and s_u,suC,s\u00afu\u2208, 1\u2264u\u2264t, whereA semistatic portfolio is called (iii)initial portfolio value of a semistatic portfolio\u00a0\u03a6 is given byFor prices\u00a0(iv)liquidation value at time\u00a0T is defined asThe To prepare for the central notions of model\u2010independent and weak arbitrage, we now define semistatic trading strategies in the bank account, the underlying asset, and the call options. Here, semistatic means that the position in the call options is fixed at time zero. The definition is model\u2010independent; as soon as a model , 1\u2264t\u2264T, that satisfyL\u03a6(sT)\u22650.Let t\u2208T\u2217,cf.\u00a0, we haveHaving defined semistatic portfolios, we can now formulate two useful notions of arbitrage.Definition 2.10\u03b5\u22650. The prices weak arbitrage opportunity with respect to spread\u2010bound\u00a0\u03b5 if there is no model\u2010independent arbitrage strategy (with respect to spread\u2010bound \u03b5), but for any model satisfying\u00a0r\u03a6 is nonpositive,Let \u03b5\u22650 and write only model\u2010independent arbitrage, meaning model\u2010independent arbitrage with respect to spread\u2010bound \u03b5, and similarly for weak arbitrage. The notion of weak  arbitrage was first used in\u00a0Davis and Hobson , but we think of C(\u03b5B(1)) as the underlying. The choices for r_0=S_0\u22122\u03b5 and r\u00af0=S\u00af0 made in Theorem\u00a0\u03b5B(1) has to satisfyD(1)E[S1C] has to lie in the closed interval , which implies that the price of an option with strike B(1)\u03b5 has to lie in the interval . Therefore, in the proof of Theorem\u00a0Ct(\u03b5B(t)) as a reference to the underlying and \u2212Ct(\u03b5B(t)) as a reference to a short position in the underlying plus an additional deposit of\u00a02\u03b5 in the bank\u00a0account.We fix 0\u2264i<j<l\u2264N, and that its payoff is nonnegative. A call spread is a portfolio of a long and a short call, where the latter has a larger strike.Theorem 3.1\u03b5\u22650 and consider prices as at the beginning of Section\u00a0T=1 and k1>\u03b5 All butterfly spreads have nonnegative time\u20100 price, that is,(ii)The call prices satisfy(iii)All call spreads have nonnegative time\u20100 price, that is,(iv)If a call spread is available for zero cost, then the involved options have zero bid, respectively, ask price, that is,Let Moreover, there is a model\u2010independent arbitrage, as soon as any of the conditions (i)\u2013(iii) is not satisfied. Finally, if (i)\u2013(iii) hold but\u00a0(iv) fails, then there is a weak arbitrage\u00a0opportunity.Before we formulate the main result for a single maturity, we recall that a butterfly contract (with maturity\u00a01) is defined byThis theorem is proved in Appendices\u00a0\u03b5=0 and r_i=r\u00afi=ri, the conditions from Theorem\u00a0Remark 3.2r_i\u2265r_j or r\u00afi\u2265r\u00afj for i<j, as shown in the following example.Note that in contrast to the frictionless case, it is not required that bid or ask prices decrease as the strike increases, in order to get models that are \u03b5\u2010consistent with the absence of arbitrage. This means that we do not have to require \u03b5=0 (no spread on the underlying), and the prices are given by S_0=S\u00af0=5,r\u00afi=i+5,r_i=1+i2,ki=i for i=1,2. We assume that the bank account is constant until maturity. These prices and a possible choice of shadow prices ei:=D(1)E[(S1C\u2212Ki)+] are shown in Figure\u00a0\u03bc=\u03b45, where \u03b4 denotes the Dirac delta. This example shows that, in our setting, prices that are admissible from a no\u2010arbitrage point of view do not necessarily make economic sense: As the payoff of C(K2) at maturity never exceeds the payoff of C(K1), the utility indifference ask\u2010price of C(K2) should not be higher than the utility indifference ask\u2010price of C(K1).Consider two call options, where For d Hobson .Remark i,j,l>0, as these conditions depend on\u00a0\u03b5 only for i=0 i\u2208{1,\u22ef,Nt} can then be extended to a call function of a measure \u03bct StC is given by \u03bct. Then we have thats\u2208 and set \u03bdt=\u03b4s (Dirac delta) for all t\u2208T\u2217. Clearly, (\u03bdt)t\u2208T\u2217 is a peacock, and we set St\u2217=B(t)s, which implies D(t)St\u2217\u223c\u03bdt. Finally, we define S_t=St\u2217\u2227StC and S\u00aft=St\u2217\u2228StC, and have thus constructed an arbitrage\u2010free model.\u25a1We define random variables (\u03bdt) from Lemma\u00a0W\u221e,M introduced before Definition\u00a0\u03b5>0, a measure \u03bc\u2208M, and m\u2208, the setR\u03bc of\u00a0\u03bc (see\u00a0W\u221e\u2010distance \u03b5 of a given sequence of measures.Theorem 4.2.(Theorem\u00a03.5 in\u00a0Gerhold and G\u00fcl\u00fcm ) Let \u03b5>0 and (\u03bcn)n\u2208N be a sequence in M such that(\u03bdn)n\u2208N such thatm\u2208I and for all N\u2208N, x1,\u22ef,xN\u2208R, we have\u03c3n=sgn(xn\u22121\u2212xn) depends on\u00a0xn\u22121 and\u00a0xn. In this case, it is possible to choose E\u03bd1=E\u03bd2=\u22ef=m.To prepare for our main result on \u03b5\u2010consistency in the multiperiod model, we now recall the main result of\u00a0Gerhold and G\u00fcl\u00fcm , which gof\u00a0\u03bc see\u00a0 as follond G\u00fcl\u00fcm  gives an\u03bct from Lemma\u00a0DSC) has to be assumed, but the existence of the peacock (\u03bdt) can be replaced by fairly explicit conditions, using Theorem\u00a0Theorem 4.3\u03b5\u22650 the prices S\u00af0\u2212S_0\u2264\u03b5 and there is a sequence of finitely supported measures (\u03bct)t\u2208T\u2217 in M such that:(i)R\u03bct\u2208 for all t\u2208T\u2217 and i\u2208{1,\u22ef,Nt}, and \u03bct)=1 for t\u2208T\u2217,(ii)N\u2208{1,\u22ef,T\u22121} and x1,\u22ef,xN\u2208R\u03c3n is as in Theorem\u00a0\u03bcn:=\u03bcT for n>T.There isFor We can now give a partial solution to the multiperiod \u03b5\u2010consistency problem. The existence of the measures\u00a0\u25a1Immediate from Lemma\u00a0SC in Definitions\u00a0Theorem 4.4p\u2208The prices satisfy Definition\u00a0(ii)The prices are consistent with the absence of arbitrage.Let As we allow an arbitrary reference price process\u00a0Definition 4.5p\u2208 and two probability measures \u03bc,\u03bd on R, we define the modified Prokhorov distance asFor For the proof of Theorem\u00a0p by\u00a0h in the right\u2010hand side.) Note that d0P=W\u221e. A well\u2010known result, which was first proved by Strassen  with random variables X\u223c\u03bc and Y\u223c\u03bd such thatGiven measures  in Theorem\u00a0Theorem 4.7.(Theorem\u00a08.3 in\u00a0Gerhold and G\u00fcl\u00fcm ) Let (\u03bcn)n\u2208N be a sequence in M, \u03b5>0, and p\u2208n\u2208N with mean m such thatThe following result shows that, unlike for\u00a0Proof of Theorem 4.4(\u03bct)t\u2208T\u2217 as in the proof of Proposition\u00a0R\u03bct\u2208 for i\u2208{1,\u22efNt} and t\u2208T\u2217. Now we pick s\u2208. Then by, Theorem\u00a0(\u03bdt)t\u2208T\u2217 with mean s such that dpP\u2264\u03b5 for all t\u2208T\u2217. We can now use Proposition\u00a0(S\u223ctC)t\u2208T\u2217 and (S\u223ct\u2217)t\u2208T\u2217 whose marginal distributions are given by (\u03bct)t\u2208T\u2217 , respectively, (\u03bdt)t\u2208T\u2217, such that (S\u223ct\u2217)t\u2208T\u2217 is a martingale and such thatp=0, but the proof trivially extends to p\u2208. We then simply put\u25a1(i) implies\u00a0(ii) by definition. To show the other implication, we define probability measures nd G\u00fcl\u00fcm  for the 5t\u2208T\u2217 the option prices can be associated to a measure \u03bct, such that E\u03bct\u2208 \u2013 (6) in\u00a0Cousot, (\u03bct)t\u2208T\u2217 is a\u00a0peacock.The main result of the preceding section Theorem\u00a0 gives se\u03b5>0), it turns out that we need conditions that involve all maturities simultaneously , x=, I= and J= are given, such that(i)xt\u2208R for all t\u2208{1,\u22ef,u},(ii)\u03c31\u2208{\u22121,1} and \u03c3t=sgn(xt\u22121\u2212xt) for all t\u2208{2,\u22ef,u},(iii)jt\u2208{0,\u22ef,Nt} and kt,jt=xt+\u03b5\u03c3t for all t\u2208{1,\u22ef,u},(iv)it\u2208{0,\u22ef,Nt} and either kt,it\u2264xt\u22121+\u03b5\u03c3t or it=0 for all t\u2208{2,\u22ef,u}. Then, we define a CVB with these parameters as the contractCVBu are given byu as the maturity of the\u00a0CVB.Fix If we consider a bid\u2013ask spread on the underlying and want to check for \u03b5\u2010consistency according to Definition\u00a0Lemma 5.2\u03b5\u22650. For all parameters u,\u03c3,x,I,J as in Definition\u00a0r\u03a6=\u2212r_uCVB, such that for all models satisfying\u00a0t\u2208{2,\u22ef,u+1} one of the following conditions holds:(i)\u03c6t0\u22650 and \u03c6t1=0, or(ii)\u03c6t0\u2265kt,j\u2212\u03b5\u03c3t and \u03c6t1=\u22121. In particular, all corresponding cash flows are\u00a0nonnegative.Fix \u03c6t0,\u03c6t1 are of course the same as in\u00a0\u03c6t0,\u03c6t1 inductively. As we are defining a model\u2010independent strategy, we could also use the deterministic dummy variables\u00a0(S_u)u\u2264t,(SuC)u\u2264t,(S\u00afu)u\u2264t, though. We just have to keep in mind that \u03c6t0,\u03c6t1 have to be constructed as functions of (S_u)u\u2264t,(SuC)u\u2264t,(S\u00afu)u\u2264t, without using the distribution of these random\u00a0vectors.The arguments of u<T, therefore we excluded the case u=T.Moreover, note that later on in Theorem\u00a0Proof of Lemma 5.2r_uCVB. We have to keep in mind that if it=0 for some t\u2208{2,\u22ef,u}, then the corresponding expression in\u00a0jt=0 for some t\u2208{1,\u22ef,u}, then the expression\u00a0\u2212Ct in\u00a0Kt,i instead of Kt,it and Kt,j instead of Kt,jt.Assume that we buy the contract2\u2212CVBu, we will call this scenario\u00a0A, or we have one short position in the underlying  and \u03c6t+10\u2265kt,j\u2212\u03b5\u03c3t; we will refer to this as scenario\u00a0B. Note that scenarios\u00a0A and\u00a0B are not disjoint, but this will not be a\u00a0problem.We will show inductively that after we have traded at time \u03c31=\u22121 and afterward with the case \u03c31=1. We start with t=1 and first assume that j1>0. If C1 expires out of the money, then we do not trade at time\u00a01 and obtain \u03c620=2\u03b5\u22650, so we are in scenario\u00a0A. Otherwise we sell one unit of the underlying, and thusD(t)=B(t)\u22121. If j1=0 then k1,j=\u03b5. We do not close the short position in this case and we get that \u03c620=4\u03b5\u2265k1,j\u2212\u03c31\u03b5, so we also get to\u00a0scenario\u00a0B.We will first deal with the case where t\u22121 we are in scenario\u00a0A, and in part\u00a0B, we will assume that at the end of period\u00a0t\u22121 we are in\u00a0scenario\u00a0B.For the induction step, we split the proof into two parts. In part\u00a0A, we will assume that after trading time\u00a0t we can end up either in situation\u00a0A or\u00a0B. First we assume that jt,it>0, and so both expressions in\u00a0t denote options (and not the underlying). Under these assumptions,\u00a0\u03c6t0 satisfiesKt,i\u2264Kt,j or if both options expire out of the money, then \u03c6t+10\u22650, and we are in situation\u00a0A. So, suppose that Kt,i>Kt,j and that StC>Kt,j. This also implies that \u03c3t=1. If this is the case, we go short one unit of the underlying, and\u00a0\u03c6t+10 can be bounded from below as follows:jt=0 and it>0. Then we have that kt,j=\u03b5. After trading time\u00a0t, we end up in scenario\u00a0B,jt>0 and it=0. As kt,j>\u03b5, we can close the long position in the underlying and end up in scenario\u00a0A at the end of time\u00a0t,jt=it=0 is easily handled, because the long and the short position simply cancel out. We are done with\u00a0part\u00a0A.Part A: We will show that after we have traded at time\u00a0t\u22121 we are in scenario\u00a0B, and thus \u03c6t0=kt\u22121,j\u2212\u03b5\u03c3t\u22121. First we will consider the case where jt,it>0. If at time\u00a0t the option with strike Kt,j expires in the money, we do not close the short position and havext\u22121\u2264xt and xt\u22121>xt, and always assume that Ct expires out of the money. If xt\u22121\u2264xt, then we also have that kt,i\u2264kt,j and that \u03c3t=\u22121. We close the short position to end up in scenario\u00a0A,xt\u22121>xt and \u03c3t=1, we do not trade at time\u00a0t to stay in scenario\u00a0B,Part B: Assume that after we have traded at time\u00a0jt=0 and\u00a0it>0. As before, we have kt,j=\u03b5, and we can close one short position to stay in scenario\u00a0B,jt>0 and it=0, then we distinguish two cases: either Ct expires out of the money, in which case we cancel out the long and short position in the underlying and haveCt expires in the money. Then we sell one unit of the underlying and hence we end up in scenario\u00a0B,kt\u22121,j\u2212\u03b5\u03c3t\u22121=xt\u22121\u2265kt,i\u2212\u03b5\u03c3t, and that kt,i=\u03b5.We proceed with the case where\u00a0jt=it=0 is again easy to handle, because the long and the short position cancel out and we are in scenario\u00a0B at the end of the (t+1)\u2010st\u00a0period.The case where u we are either in scenario\u00a0A or scenario\u00a0B, which proves the assertion if \u03c31=\u22121.Thus, after we have traded at time\u00a0\u03c31=1 is similar. We will first show that after trading at time 1 we can either be in scenario\u00a0A or scenario\u00a0B, and the statement of the proposition then follows by induction exactly as in the case \u03c31=\u22121.The proof for j1>0. Then, if the option C1 expires out of the money, we are in scenario\u00a0A; otherwise we go short in the underlying and havej1=0, then we also have that kj,1=\u03b5, and hence we are in scenario\u00a0B.\u25a1First we assume that \u2212CVBu, such that\u00a0 only depends on \u03c3u,ku,j (the investor might have some surplus in the bank account). In the following, we will use this strategy and only write \u2212CVBu, respectively, r_uCVB instead of \u2212CVBu, respectively, r_uCVB. In the case where \u03c6u0\u22650 and \u03c6u1=0, we will say that the CVB expires out of the money; otherwise we will say that it expires in the\u00a0money.According to Lemma\u00a0\u03b5\u22650.Theorem 5.3\u03b5\u22650, s,t,u\u2208T such that s<t and s<u and i\u2208{0,\u22ef,Nt}, j\u2208{0,\u22ef,Ns}, l\u2208{0,\u22ef,Nu}. Fix prices as at the beginning of Section\u00a0kt,1>\u03b5 for all t\u2208T. Then, for all CVBs with maturity s\u2208T and parameters ks,j and \u03c3s, the following conditions are necessary for \u03b5\u2010consistency:(i)(ii)\u00a0(iii)\u00a0(iv)\u00a0Let The next theorem states necessary conditions for the absence of arbitrage in markets with spread\u2010bound s<t\u2264u and that i,l>0. Similarly, the other cases can be dealt with. In all four cases (i)\u2013(iv), we will assume that until time\u00a0s we followed the trading strategy described in Lemma\u00a0(i)\u03b8Ct+(1\u2212\u03b8)Cu\u2212CVBs, making an initial profit. If the calendar vertical basket CVBs expires out of the money, then we have model\u2010independent arbitrage. Otherwise we have a short position in the underlying at time\u00a0s. To close the short position, we buy\u00a0\u03b8 units of the underlying at time\u00a0t, and we buy 1\u2212\u03b8 units of the underlying at time\u00a0u. The liquidation value of this strategy at time\u00a0u is then nonnegative,If\u00a0(ii)CVBs expires out of the money, then we leave the portfolio as it is. Otherwise we immediately enter a short position and close it at time\u00a0u. The liquidation value is then nonnegative,Next, assume that\u00a0(iii)Ct\u2212CVBs for negative cost. Again we can focus on the case where CVBs expires in the money. We sell one unit of the underlying at time\u00a0s and close the short position at time\u00a0t. The liquidation value of this strategy at time\u00a0t is nonnegative,If (iv)CVBs expires in the money is zero, we could simply sell CVBs and follow the trading strategy from Lemma\u00a0CVBs expires in the money with positive probability, then we can use the same strategy as in the proof of\u00a0(iii). At time\u00a0t, the liquidation value of the portfolio is positive with positive probability.\u25a1We will show that an \u03b5\u2010consistent model cannot exist, if\u00a0We will assume that \u03b5=0, then CVBs has the same payoff as \u2212Cs. Keeping this in mind, it is easy to verify that the conditions from Theorem\u00a0Note that, if Conjecture 5.4Given the conditions stated in Theorem\u00a0It remains open whether (i)j1\u2208{0,\u22ef,N1} and \u03c31\u2208{\u22121,1} and set x1=k1,j1\u2212\u03b5\u03c31.Pick (ii){x1,\u22ef,xt\u22121}, {\u03c31,\u22ef,\u03c3t\u22121}, {j1,\u22ef,jt\u22121} and {i2,\u22ef,it\u22121} first pick jt\u2208{0,\u22ef,Nt}.Given (iii)\u03c3t distinguishing the following cases:Choose kt,jt\u2265xt\u22121+\u03b5 set \u03c3t=\u22121;if kt,jt\u2264xt\u22121\u2212\u03b5 set \u03c3t=1;if kt,jt=xt\u22121 pick \u03c3t\u2208{\u22121,0,1};if kt,jt\u2208\u2216{xt\u22121} pick \u03c3t\u2208{\u22121,1}.if (iv)xt=kt,jt\u2212\u03c3t\u03b5 and pick it\u2208{0,\u22ef,Nt} such that either kt,it\u2264xt\u22121+\u03c3t\u03b5 or it=0.Set (v)Repeat steps\u00a0(ii)\u2013(iv).Theorem\u00a06We define the notion of \u03b5\u2010consistent prices, meaning that a set of bid and ask prices for call options and the underlying can be explained by a model with bid\u2013ask spread bounded by\u00a0\u03b5. For a single maturity, we solve the \u03b5\u2010consistency problem, recovering the trichotomy consistency/weak arbitrage/model\u2010independent arbitrage from the frictionless case\u00a0(Davis & Hobson,"}
{"text": "Te author \u201c\u6b66\u98ce\u7389\" should be \u201c\u6b66\u51e4\u7389\".https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5999742/"}
{"text": "Scientific Reports 10.1038/s41598-019-39754-6, published online 11 March 2019Correction to: s variable is not scaled by a factor of 104. As a result, in the Results section under the subheading \u2018Uplift Rate\u2019,This Article contains a repeated error, where the QQs_min\u2009\u2248\u20096\u00a0m3/yr and Qs_max\u2009\u2248\u200960\u00a0m3/yr, with a response time of ~\u20090.93\u00a0Myr for both increased and decreased uplift rate .\u201d\u201cIn this case, should read:Qs_min\u2009\u2248\u20096\u2009\u00d7\u2009104\u00a0m3/yr and Qs_max\u2009\u2248\u200960\u2009\u00d7\u2009104\u00a0m3/yr, with a response time of ~\u20090.93\u00a0Myr for both increased and decreased uplift rate .\u201d\u201cIn this case, In the Results section under the subheading \u2018Precipitation\u2019,Qs_max (19\u00a0m3/yr) is attained rapidly with a response time of ~\u20090.3\u00a0Myr. During a decrease in precipitation, Qs_min (~\u20092\u00a0m3/yr) is also achieved rapidly but with a more prolonged response time (~\u20090.9\u00a0Myr).\u201d\u201cIn our model, should read:Qs_max (19\u2009\u00d7\u2009104\u00a0m3/yr) is attained rapidly with a response time of ~\u20090.3\u00a0Myr. During a decrease in precipitation, Qs_min (~\u20092\u2009\u00d7\u2009104\u00a0m3/yr) is also achieved rapidly but with a more prolonged response time (~\u20090.9\u00a0Myr).\u201d\u201cIn our model, In Figure\u00a01 in panel (b), the y-axis label,s (m3yr-1)\u201d\u201cQshould read:s (\u00d7\u2009104\u00a0m3yr\u22121)\u201d.\u201cQIn Figure\u00a03 in panel (a-b), the y-axis label,s (m3/yr)\u201d\u201cQshould read:s (\u00d7\u2009104\u00a0m3yr\u22121)\u201d.\u201cQs_min\u2019 and \u2018Qs_max\u2019,In Supplementary Table DR2, the units for \u2018Magnitude of Change\u2019 for both \u2018Q3yr\u22121)\u201d\u201c(mshould read:4\u00a0m3yr-1)\u201d.\u201c\u201d\u201cQshould read:s (\u00d7\u2009104\u00a0m3/yr)\u201d.\u201cQIn Supplementary Scenarios 1.1-1.15, Scenarios 2.1-2.15, and Animations, in the graph \u2018Change in Qs and provenance (% Source A) over time\u2019, the unit is omitted for the y-axis, wheres\u201d\u201cQshould read:s (\u00d7\u2009104 m3/yr)\u201d.\u201cQ"}
{"text": "The combination of light intensity and NSC exhibited significant effects on photosynthetic pigment, nutritional quality, mineral content and antioxidant capacity. That a higher light intensity were readily accessible to higher chlorophyll a/b showed in lettuce of treatment of 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20093/4NSC and 450 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC. Lower total N contents, higher content of soluble protein, vitamin C, soluble sugar and free amino acid exhibited in lettuce under treatment of 250 and 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC or 3/4NSC. With increasing NSC and LED irradiance, the content of total P and K in lettuce increased and decreased, respectively. The highest and lowest total Ca content were found in treatment of 150 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC and 450 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC, respectively, and higher content of total Mg and Zn was observed under 250 \u03bcmol m\u22122 s\u22121\u2009\u00d7\u20091/4NSC and 150 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20093/4NSC, respectively. The antioxidant contents generally decreased with increasing NSC level. The higher antioxidant content and capacity occurred in lettuce of 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC treatment. The interaction of 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC might be the optimal condition for lettuce growth in plant factory.Light and nutrient are important factors for vegetable production in plant factory or greenhouse. The total 12 treatments which contained the combination of four light intensity (150, 250, 350 and 450 \u03bcmol\u2009\u00b7\u2009m Lactuca sativa L.) is one of the most important vegetable in the world, which is main crop in plant factory. The secondary metabolite in vegetable plays key roles for human body health, such as phenolic compound, vitamin A and C, and carotenoid. These compounds have a function on nutrition and health care1, which could enhance anti-oxidation ability of human body, and the suppression of inflammatory disease and cancer3. Nutritional quality of lettuce is affected by light and nutrient solution in plant factory. Light  plays a crucial role in improvement of plant nutrition quality5. The proper ratio of red and blue light is essential for plant growth and development6. Light intensity not only positively regulates lettuce biomass and morphology, but also nutrition quality and activities of anti-oxidative enzymes8. The content of soluble sugars and ascorbic acid in lettuce increases with increasing light intensity9. Different light intensity is required by different plant for nutritional quality and growth. The highest content of lutein, \u00df-carotene and chlorophyll in leave shows at 335 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121 for kale, but 200 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121 for spinach10. In ten leafy vegetables , higher leaf dry matter, content of protein, K, Ca and Mg, hydrophilic antioxidant activity, and lipophilic antioxidant activity are observed under low light intensity than high intensity11.Lettuce  exhibits the highest weight recovery efficiency in combination of low light intensity and nutrient supply condition14. The combination of high light illumination and low nitrogen contributes to increased vitamin C content and reduced nitrate content15. High CO2 nutrient supply and monochromatic LED synergistically enhance both the lettuce biomass and some amino acids content16. However, it is not yet available research which the interaction between light intensity and nutrient solution regulates vegetable growth and phytonutrients.In hydroponic systems, the arrangement of nutrient solution could modify the quality and yield of vegetable. Nitrate content in lettuce is closely related to solution nitrate concentration\u22122\u2009\u00b7\u2009s\u22121) based on red and blue LED light were investigated. This is aim to provide valuable insights into the optimal combination of light intensity and nutrient solution to improve quality and yield of lettuce in plant factory.In this study, the changes of mineral element contents, nutritional quality, antioxidant activity of lettuce in response to different interaction of nutrient solution concentration  and light intensity  seeds were sowed in sponge block with 1/4 strength nutrient solution concentration (NSC). The full-strength NSC were composed of the following elements: 210.0\u2009mg\u2009\u00b7\u2009L\u22121 N, 31.0\u2009mg\u2009\u00b7\u2009L\u22121 P, 234.0\u2009mg\u2009\u00b7\u2009L\u22121 K, 160.0\u2009mg\u2009\u00b7\u2009L\u22121 Ca, 48.0\u2009mg\u2009\u00b7\u2009L\u22121 Mg, 64.0\u2009mg\u2009\u00b7\u2009L\u22121 S, 5.6\u2009mg\u2009\u00b7\u2009L\u22121 Fe, 0.5\u2009mg\u2009\u00b7\u2009L\u22121 B, 0.5\u2009mg\u2009\u00b7\u2009L\u22121 Mn, 0.05\u2009mg\u2009\u00b7\u2009L\u22121 Zn, 0.02\u2009mg\u2009\u00b7\u2009L\u22121 Cu, 0.01\u2009mg\u2009\u00b7\u2009L\u22121 Mo. The experiment was performed in a growth chamber in South China Agricultural University and the seedlings with three expended true leaves were transplanted in hydroponic system as follow: 22\u201325/14\u201318\u2009\u00b0C (day/night), 60\u201380% relative humidity, the nutrient solution aeration rate of 15\u2009min/h, the nutrient solution was renewed every 10 days. Three light intensity  including 150, 250 and 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121, basing on LED: red (660\u2009\u00b1\u200910\u2009nm): blue (460\u2009\u00b1\u200910\u2009nm)\u2009=\u20092:1  until to turn white to extract chlorophyll a, b and carotenoid. The absorbance of extract liquor was determined by UV-1200 spectrophotometer  at 645\u2009nm (OD645), 663\u2009nm (OD663) and 440\u2009nm (OD440). The content of pigment was calculated as follow: chlorophyll a concentration (mg/L)\u2009=\u200912.7\u2009\u00d7\u2009OD663 \u22122.69\u2009\u00d7\u2009OD645; chlorophyll b concentration (mg/L)\u2009=\u200922.9\u2009\u00d7\u2009OD645 \u22124.86\u2009\u00d7\u2009OD663; total chlorophyll concentration (mg/L)\u2009=\u20098.02\u2009\u00d7\u2009OD663\u2009+\u200920.20\u2009\u00d7\u2009OD645; carotenoid concentration (mg/L)\u2009=\u20094.7\u2009\u00d7\u2009OD440 \u22120.27\u2009\u00d7\u2009total chlorophyll concentration; chlorophyll a content (mg/g)\u2009=\u2009(chlorophyll a\u2009\u00d7\u200925\u2009mL)/0.5\u2009g; chlorophyll b content (mg/g)\u2009=\u2009(chlorophyll b\u2009\u00d7\u200925\u2009mL)/0.5\u2009g; carotenoid content (mg/g)\u2009=\u2009(carotenoid\u2009\u00d7\u200925\u2009mL)/0.5\u2009g.Pigment content was measured colorimetrically according to Gratani18. A total of 0.5\u2009g fresh lettuces was ground into pulp by liquid nitrogen with 5\u2009mL distilled water. The extract solution was centrifuged at 10000\u2009rpm for 10\u2009min at 4\u2009\u00b0C, and 0.05\u2009mL supernatant was combined with 0.95\u2009mL distilled water and 5\u2009mL Coomassie brilliant blue G-250 solution . After 2\u2009min, the soluble protein content was detected at 595\u2009nm by UV-spectrophotometer .Soluble protein content in lettuce was examined by Coomassie brilliant blue G-250 dye method19. 1\u2009g fresh lettuces were heated on the boiling water bath with 10\u2009mL distilled water for 30\u2009min. After the solution cooling, the extracting solution was filtered by volumetric flask. 0.1\u2009mL sample solution was mixed with 0.4\u2009mL 5% salicylic and sulfuric acid and 9.5\u2009mL 8% NaOH. The nitrated content of mixed solution was determined by UV-VIS spectrophotometer  at 410\u2009nm.Nitrate content was determined by ultraviolet spectrophotometry method20. 0.5\u2009g fresh leaves were ground into pulp with 3\u2009mL 1% oxalic acid, 1\u2009mL 30% zinc sulfate and 1\u2009mL 15% potassium ferrocyanide. 10\u2009mL extracting solution was mixed with 1\u2009mL phosphate-acetic acid, 2\u2009mL 5% vitriol and 4\u2009mL ammonium molybdate. After 15\u2009min, the mixed solution was determined at 500\u2009nm by UV-VIS spectrophotometer .Vitamin C was performed by using the 2, 6-dichlorophenol indophenol titration method21. 0.5\u2009g fresh leaves were heated on boiling water bath with 10\u2009mL distilled water for 30\u2009min. 0.1\u2009mL supernatant was mixed with 1.9\u2009mL distilled water, 0.5\u2009mL anthrone ethyl acetate and 5\u2009mL vitriol. After shaking, the soluble sugar was detected by UV-VIS spectrophotometer  at 630\u2009nm.Soluble sugar content was performed by anthronesulfuric acid colorimetry method19. 1\u2009g fresh leaves were ground into pulp with 10\u2009mL deionized water, which were heated on a water bath at 80\u2009\u00b0C for 30\u2009min. The extract solution was centrifuged at 13000\u2009g for 10\u2009min. The 0.2\u2009mL supernatant was mixed with 0.8\u2009mL 5% (w/v) salicylic acid  and 19\u2009mL 4\u2009mol\u00b7L\u20131 NaOH. The nitrate content was determined by UV-VIS spectrophotometer  at 410\u2009nm.Free amino acid content was determined colorimetrically22. 1.0\u2009g fresh lettuces exacted by 20\u2009mL 60% alcohol (pH\u2009=\u20093.0) were heated on the boiling water bath for 2\u2009h. The exacting solution was filled in volumetric flask. The certain volume of exacting solution using the same extractant to dilute was determined by UV-VIS spectrophotometer  at 535\u2009nm.Anthocyanin content was determined by spectrometric method23. 1\u2009g fresh leaves were pulverized with liquid nitrogen. The sample powder was heated on boiling bath with 8\u2009mL 80% methanol for 60\u2009min, and then the exacting solution was centrifuged with 12000\u2009rpm for 10\u2009min. The supernatant was moved to evaporation flask at 40\u2009\u00b0C with 3\u20134\u2009rpm for 5\u20136\u2009min. 10\u2009mL distilled water was added to the exacting solution with putting into centrifuge with 8000\u2009rpm for 20\u2009min. 1\u2009mL supernatant was mixed with 7\u2009ml distilled water, 0.5\u2009mL foline-phenol and 11.5\u2009mL 26.7% sodium carbonate. The absorbance of mixed solution was measured at 760\u2009nm by spectrophotometer .Polyphenol content was determined by Folin-Cioealteu methodet al.24. The extracted method of flavonoid identified with polyphenol. 1\u2009mL extract solution was added to 11.5\u2009mL 30% alcohol and 0.7\u2009mL 5% NaNO2. After 5\u2009min, the reaction solution was mixed with 0.7\u2009mL 10% Al(NO3)3, and then 6\u2009min later, the mixture was added 5\u2009mL 5% NaOH. Finally, 10\u2009min later, the mixed solution was determined by spectrophotometer  at 510\u2009nm.Flavonoid content was determined according to Jia et al.23. 0.5\u2009g sample solution which identified with exacting polyphenol was added 2.5\u2009mL 65 \u03bcmol\u2009\u00b7\u2009L\u22121 DPPH solution. After 30\u2009min, the mixed solution was determined by spectrophotometer  at 517\u2009nm.The 2, 2-diphenyl-1-picrylhydrazyl (DPPH) radical scavenging rate was performed by basing on Tadolini 25. 0.4\u2009mL sample solution which identified with exacting polyphenol was mixed with 3.6\u2009mL mixed solution  for 10\u2009min at 37\u2009\u00b0C. The mixed solution was determined by spectrophotometer  at 593\u2009nm.The value of ferric-reducing antioxidant power (FRAP) was performed according to Benzie and Strain26, Mo-Sb Colorimetry27, and flame photometry method28, respectively, while Ca, Mg and Zn content were performed by using atomic absorption spectrophotometry method29.Fresh lettuce was heated to de-enzyme at 105\u2009\u00b0C for 1\u2009h, then dried at 75\u2009\u00b0C. The kiln-dried sample was smashed and stored to measure mineral element content. Total N, P and K was determined by using Ojeda\u2019sp\u2009\u2264\u20090.05 and p\u2009\u2264\u20090.01 level.All the assays were analyzed in triplicates.Variance analysis of one-way in single light intensity or NSC factor, and two-way in the combination of light intensity and NSC was performed by using SPSS17.0 software to determine the significance at \u22122\u2009\u00b7\u2009s\u22121 treatment. A tendency which plant and shoot weight increased at first then decreased with increasing light intensity was observed. Lettuce fresh and dry weight under the combination of 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20093/4NSC was the highest  and dry weight (DW) of plant and shoot Table\u00a0. There w\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/2NSC, whereas the lowest under 450 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/2NSC. Chlorophyll b content descended with light intensity increasing. A lower chlorophyll a/b was observed under low light intensity treatments (150 and 250 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121), but no difference under different NSC. Chlorophyll a/b significantly improved with increasing NSC under 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121, along with the highest at 3/4NSC. Hence, 150 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC was beneficial to accumulation of chlorophyll a and b, and total chlorophyll, while the 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20093/4NSC treatment contributed to the promotion of chlorophyll a/b.As shown in Table\u00a0p\u2009\u2264\u20090.01), NSC (p\u2009\u2264\u20090.01) and their interaction (p\u2009\u2264\u20090.01)  Table\u00a0. There went Fig.\u00a0. These gent Fig.\u00a0.Figure 1\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC. Two-way ANOVA analysis confirmed that nitrate content in lettuce was strongly associated with light intensity (p\u2009\u2264\u20090.01), NSC (p\u2009\u2264\u20090.01) and their interaction (p\u2009\u2264\u20090.01)  Table\u00a0.p\u2009\u2264\u20090.01), NSC level (p\u2009\u2264\u20090.01) and their combination (p\u2009\u2264\u20090.01) exhibited a significant difference on vitamin C content , NSC (p\u2009\u2264\u20090.01) and their interaction (p\u2009\u2264\u20090.01)  Table\u00a0. It incrNSC Fig.\u00a0.p\u2009\u2264\u20090.01), NSC (p\u2009\u2264\u20090.01) and their combination (p\u2009\u2264\u20090.01) significantly affected the content of free amino acid in lettuce  with 1/4NSC or 3/4NSC could increase the content of soluble protein and sugar, vitamin C, and free amino acid, and reduce nitrate content in lettuce.In general, 250 \u03bcmol\u2009\u00b7\u2009mp\u2009\u2264\u20090.01), NSC level (p\u2009\u2264\u20090.01) and their combination (p\u2009\u2264\u20090.01) exhibited a significant effect on the content of total N, P and K in lettuce   Fig.\u00a0.Figure 2p\u2009\u2264\u20090.01), NSC level (p\u2009\u2264\u20090.01) and their interactions (p\u2009\u2264\u20090.01)  Table\u00a0. At 1/4Nely Fig.\u00a0. The higs\u22121 Fig.\u00a0. The lowNSC Fig.\u00a0.p\u2009\u2264\u20090.01), NSC level (p\u2009\u2264\u20090.01) and their combination (p\u2009\u2264\u20090.01)  Table\u00a0. AnthocyNSC Fig.\u00a0. With insed Fig.\u00a0, the higNSC Fig.\u00a0.Figure 3p\u2009\u2264\u20090.01), NSC (p\u2009\u2264\u20090.01) and their crosstalk (p\u2009\u2264\u20090.01) (Table\u00a0\u22122\u2009\u00b7\u2009s\u22121) and the highest flavonoid content, FRAP and DPPH in lettuce were found under 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC. These meant that 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC was the most efficient condition for the improvement of antioxidant component content and capacity in lettuce, including anthocyanin, polyphenol, flavonoid, FRAP and DPPH.Two-way analysis revealed that flavonoid content, FRAP and DPPH in lettuce were significantly related to light intensity (1) Table\u00a0. Under 31) Table\u00a0, FRAP (F1) Table\u00a0 and DPPH1) Table\u00a0 in lettup\u2009\u2264\u20090.01), flavonoid  and DPPH  were higher than anthocyanin content . Moreover, the content of anthocyanin, polyphenol and flavonoid were greatly relevant to DPPH (p\u2009\u2264\u20090.01), the highest coefficient was found in polyphenol content (r\u2009=\u20090.952), while the minimum in anthocyanin content. Hence, the antioxidant activity mainly derived from polyphenol, flavonoid, anthocyanin in lettuce under light intensity\u2009\u00d7\u2009NSC condition.To obtain a more detailed understanding of antioxidant role in lettuce under light intensity\u2009\u00d7\u2009NSC, the correlation analysis was performed between antioxidant content and DPPH, FRAP Table\u00a0. The coe\u22122 s\u22121\u20098, while in lettuce increased under 200\u2009\u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u200915. Plant biomass and leaf number of lettuce increased  combined with higher nitrogen (15\u2009mmol\u2009\u00b7\u2009L\u22121 or 23\u2009mmol\u2009\u00b7\u2009L\u22121) could enhance the content of chlorophyll a and b in lettuce, while the combination of 220 \u03bcmo\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20097\u2009mmol\u2009\u00b7\u2009L\u22121 reduced the content of chlorophyll a and b15. In this study, 150 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC was conducive to chlorophyll accumulation, whereas the treatment of 450 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/2NSC was unfavourable for chlorophyll accumulation (Table\u00a030. This might be due to that the chlorophyll contents were more affected by light intensity (p\u2009\u2264\u20090.01) than NSC (p\u2009\u2264\u20090.05)  could lead to lower nitrate content but higher content of vitamin C, soluble sugar, soluble protein, and anthocyanin in lettuce31. 220\u2013330 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121 was the most suitable irradiance level for growth and nutritional quality of Brassica microgreens32. Similarly, lettuce under higher irradiance of 250 or 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC had lower nitrate content, and higher contents of soluble protein, vitamin C, soluble sugar and free amino acid  treatment in comparison with high or low EC33. It was possible that higher light intensity could promote nitrate accumulation through increasing photosynthetic production34. The increase of light intensity could induce accumulation of soluble sugars9. In lettuce, soluble sugar content increased under 150 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121 to 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121 and decreased under 450 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121, but was negatively regulated by NSC level  and 1/4NSC or 3/4NSC could be beneficial to improve nutrition quality in lettuce.In plant factory, higher light intensity  than high intensity (800\u20131200 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121)11. In lettuce, total N content was the highest under 450 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20093/4NSC while the lowest under 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/2NSC  but the P content decreased10. However, the mineral contents in spinach were significant different under different irradiance levels, content of Ca and Fe decreased at low light levels10. Mineral nutrient played a crucial role in photosynthesis, carbohydrate content in plant36. P, K, Ca, Mg, Zn in lettuce mainly were accumulated under higher light intensity (350 and 450 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121)\u2009\u00d7\u20091/4 or 3/4NSC , NSC level (p\u2009\u2264\u20090.01) and their combination (p\u2009\u2264\u20090.01) (Table\u00a0\u22122\u2009\u00b7\u2009s\u22121) than higher irradiance (350 and 450 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121), the highest contents were exhibited under 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC , low light intensity (200\u2013400 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121) could improve antioxidant activity for ten leafy vegetables15. Brassica microgreens possessed the highest anthocyanin content under 330\u2013440 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121 irradiance32. Anthocyanin content in lettuce decreased with increasing NSC level , and decreased at first and then increased under higher irradiance (350 and 450 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121) , flavonoid , DDPH , anthocyanin  and FRAP  Table\u00a0. These wvel Fig.\u00a0, while i\u22121) Fig.\u00a0. Plant iAP Table\u00a0. Thus, p\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20093/4NSC or 1/4NSC was conducive to growth of lettuce, while the irradiance of 250 and 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4 or 3/4NSC contributed to increased content of soluble protein and sugar, vitamin C, nitrate and free acid, and 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC exhibited a dramatically effect on improving antioxidant content and capacity. The 350 \u03bcmol\u2009\u00b7\u2009m\u22122\u2009\u00b7\u2009s\u22121\u2009\u00d7\u20091/4NSC treatment was the more suitable condition for lettuce production in plant factory.In plant factory, light and nutrient solution are the most effective factors improving yield and quality of vegetable. This study clearly demonstrated that the combination of light intensity and nutrient solution could significantly affect growth and quality of lettuce. The interaction of 350 \u03bcmol\u2009\u00b7\u2009mSupplementary Information."}
{"text": "This paper is devoted to the homogenization of a nonlinear transmission problem stated in a two\u2010phase domain. We consider a system of linear diffusion equations defined in a periodic domain consisting of two disjoint phases that are both connected sets separated by a thin interface. Depending on the field variables, at the interface, nonlinear conditions are imposed to describe interface reactions. In the variational setting of the problem, we prove the homogenization theorem and a bidomain averaged model. The periodic unfolding technique is used to obtain the residual error estimate with a first\u2010order corrector. The coupling of the species arises via nonlinear transmission conditions at the interface, which model surface reactions. Nonlinear interface reactions are relevant, for instance, in electrochemistry, see, eg, Landstorfer et al\u03b5>0. The main objective is to derive a macroscopic model for vanishing \u03b5, where both phases are connected sets. The limit bidomain model is given via two coupled parabolic equations defined in the macroscopic domain describing the diffusion of the two species in each phase and reactions at the interface. In the case of connected\u2010connected domains, we exploit the existence of a continuous extension operator from the periodic domain to the whole domain following.The characteristic length scale of the periodic cell is given by the homogenization parameter A qualitative homogenization result for reaction\u2010diffusion systems with nonlinear transmission conditions has recently been obtained in Gahn et al.O(\u03b52).Within elecktrokinetic modeling  optimal for H1\u2010estimates up to an Lipschitz boundary, whereas in Fatima et al,\u03b51/4. For this task, we apply the Poincar\u00e9 inequality in periodic domains \u03a9\u2282Rd is a d\u2010dimensional hyperrectangle, d\u2a7e2, ie, it isThe reference domain (D2)unit cellY=d consists of two open, connected subsets Y1 and Y2, which have Lipschitz continuous boundaries \u2202Y1, \u2202Y2 and are disjoint by the interface \u0393=\u2202Y1\u2229\u2202Y2. We assume the reflection symmetry, ie,k=1,\u2026,d, i=1,2. This assumption allows us to define periodic functions on Yi in n1 and n2 denote the unit normal vectors at the respective boundaries \u2202Y1 and \u2202Y2. Every normal is chosen outward from the domain, and it does not depend on scaling by \u03b5.The (D3)\u03b5>0, we introduce the decomposition of a point x\u2208Rd asx\u03b5\u2208Zd and the fractional part x\u03b5\u2208Y. According to \u03b5 an admissible parameter, if the reference domain \u03a9 from (D1) can be partitioned periodically into the local cells as follows:For (D4)\u03a91\u03b5 and \u03a92\u03b5 and their interface \u0393\u03b5 are determined viani\u03b5 at \u2202\u03a9i\u03b5 coincide with the normal vectors ni at \u2202Yi for i=1,2 and do not depend on the scaling \u03b5. The interface \u0393\u03b5 is a Lipschitz continuous manifold.As a consequence of (D1) to (D3), the periodic components We make the following geometric assumptions.\u03b5>0, time t\u2208 with the final time T>0 fixed, the space variable x\u2208\u03a91\u03b5\u203e\u22c3\u03a92\u03b5\u203e in the two\u2010component domain, we consider a nonlinear transmission problem for ui\u03b5, i=1,2, such that\u2202t stands for the time derivative, \u2207 for the spatial gradient, and \u201c\u2009 \u00b7\u2009\u2032\u2032 for the scalar product in Rd. Below, we explain in detail the terms entering the system (4). We note that |\u0393\u03b5|=O(1/\u03b5); therefore, the scaling \u03b5 in (A1)Ai(y)\u2208L\u221e, i=1,2, are symmetric, uniformly bounded and elliptic: There exist 0<\u03b1\u2a7d\u03b2 such thatThe diffusivity matrices For admissible Ai\u03b5(x)=Ai{x\u03b5} according to the notation The matrices entering gi:R2\u21a6R, i=1,2, describe interface reactions and are assumed to satisfy(G1)Kg>0 such thatthe uniform growth condition: there exists (G2)Lg\u2a7e0 such thatui,vi\u2208R, i=1,2.the Lipschitz continuity: There exists In the transmission conditions uiin\u2208L2(\u03a9), i=1,2.The linear diffusion equations\u00a0ui\u03b5\u2208Ui\u03b5, i=1,2, in the search (solution) spacevi from the test spaceH1(\u03a9i\u03b5)\u2217 in Ui\u03b5 stands for the topologically dual space to H1(\u03a9i\u03b5), and \u27e8\u00b7,\u00b7\u27e9\u03a9i\u03b5 denotes the duality between them.We introduce the variational formulation of the problem (4) as follows: find 3This section provides the existence of weak solutions in the sense of variational formulation for the microscopic problem Theorem 1(Well\u2010posedness)(i)ui\u03b5\u2208Ui\u03b5 to the nonlinear transmission problem \u03b5\u2208 for \u03b50>0 sufficiently small.The unique solution (ii)uiin>0 everywhere in \u03a9\u203e, the solution ui\u03b5 is positive at least locally in time, and ui\u03b5\u2a7e0 at any time under the assumption of the positive production rate from Roub\u00cd\u010dek(ui\u03b5)\u2212=\u2212min stands for the negative part of the function.Under assumptions on positivity of the initial data (i)uim0=uiin, m0\u2208N, i=1,2.To prove existence of the solution, we apply the Tikhonov\u2010Schauder fixed point theorem. We iterate m>m0, m\u2208N, a solution uim\u2208Ui\u03b5 can be found, which satisfies the initial data vi\u2208Vi\u03b5, using the notation gim\u22121:=gi for short. We can test vi=uim leading to2\u03b4Ktr>0, the trace theorem C=Ktr4\u03b4Kg2T\u03b5|\u0393\u03b5|=O(1) with a constant Ktr from the trace theorem Kg from \u27e8\u2202tuim,uim\u27e9=12ddt\u2016uim\u2016L2(\u03a9i\u03b5), using the uniform ellipticity Ai\u03b5 and the estimate \u03b4<\u03b1, applying Gr\u00f6nwall inequality, we obtaint\u2208, we conclude\u2016\u2202tuim\u2016L2\u2217)2=O(1) uniformly with respect to m\u2192\u221e and \u03b5\u21920, and the continuous embedding of the solution in C) holds; see Dautray and Lions.For M:Ui\u03b5\u21a6Ui\u03b5 defined when solving ui\u03b5\u2208Ui\u03b5, i=1,2, and a subsequence still denoted by m such that as m\u2192\u221eM in the weak topology is justified using the Lipschitz continuity of the nonlinear term gi in Therefore, the mapping wi\u03b5:=ui1,\u03b5\u2212ui2,\u03b5, i=1,2, of two solutions of vi=wi\u03b5:To prove uniqueness, we consider the difference Igi\u03b5 is estimated due to the Lipschitz continuity wi\u03b5\u22610, which proves ui1,\u03b5\u2261ui2,\u03b5.The integral (ii)ui\u03b5, we decompose the solution into the positive and the negative parts as: ui\u03b5=(ui\u03b5)+\u2212(ui\u03b5)\u2212 and substitute it in the Equation\u00a0vi=(ui\u03b5)\u2212. The assumption of the positive production rate Ai\u03b5 and the nonnegativity of the initial data lead to the estimate:(ui\u03b5)\u2212\u22610 and ui\u03b5\u2a7e0. If ui\u03b5(0)=uiin>0 everywhere in \u03a9\u203e, then ui\u03b5(t)>0 at least for t sufficiently small, which follows by the continuity of the solution. This completes the proof.To prove the nonnegativity of  \u25a1\u03a9i\u03b5.We note that Theorem 4\u03a9i\u03b5\u203e and Yi\u203e, i = 1,2, up to the boundaries.Following Cioranescu et al,Definition 1u(x)\u2208L2(\u03a9i\u03b5), the unfolding operator T\u03b5:L2(\u03a9i\u03b5)\u21a6L2), i=1,2, in the domain is defined byu(x)\u2208L2(\u2202\u03a9i\u03b5), the operator T\u03b5:L2(\u2202\u03a9i\u03b5)\u21a6L2), i=1,2, is performed on the boundary by\u03c6\u2208L2), the averaging operator T\u03b5\u22121:L2)\u21a6L2(\u03a9i\u03b5), i=1,2, in the domain is defined by\u03c6\u2208L2), the operator T\u03b5\u22121:L2)\u21a6L2(\u2202\u03a9i\u03b5), i=1,2, on the boundary is expressed byFor T\u03b5\u22121 is used for a left inverse operator of T\u03b5 according to Lemma H1(\u03a9i\u03b5), the restriction of the unfolding operator T\u03b5 is well\u2010defined as the mapping H1(\u03a9i\u03b5)\u21a6L2), and for functions in L2), the restriction of the averaging operator T\u03b5\u22121 is well\u2010defined as L2)\u21a6H1(\u22c3\u03bb\u2208I\u03b5Yi\u03bb), where Yi\u03bb is from H1(\u22c3\u03bb\u2208I\u03b5Yi\u03bb) and H1(\u03a9i\u03b5) do not coincide because functions from H1(\u22c3\u03bb\u2208I\u03b5Yi\u03bb) are discontinuous while they can have jumps across the interface \u0393\u03b5.Abusing the notation The operator properties are collected below in Lemma Lemma 1T\u03b5 and T\u03b5\u22121)(Properties of the operators  For arbitrary x\u21a6u(x)\u2208H1(\u03a9i\u03b5)\u2229L2(\u2202\u03a9i\u03b5) and \u21a6\u03c6\u2208L2\u2229L2(\u2202Yi)), i=1,2, and the extension by zero: \u016b(x)=u(x) for x\u2208\u03a9i\u03b5, otherwise \u016b(x)=0 for x\u2208\u03a9\u2216\u03a9i\u03b5\u203e, the following properties hold:(i)T\u03b5: (T\u03b5\u22121T\u03b5)u(x)=u(x);invertibility of (ii)T\u03b5\u22121:(iia)(T\u03b5T\u03b5\u22121\u03c6)=\u03c6(y) for x\u2208\u03a9, if \u03c6(y) is a constant or periodic function of the argument y\u2208Yi,(iib)(T\u03b5T\u03b5\u22121\u016b)=(T\u03b5\u22121\u016b)(x)=|Yi||Y|\u27e8T\u03b5u\u27e9Yi(x) for x\u2208\u03a9, where is the average \u27e8\u00b7\u27e9Yi=1|Yi|\u222bYi(\u00b7)dy;invertibility of (iii)T\u03b5(F(u))=F(T\u03b5u) for any elementary function F;composition rule: (iv)\u03b5T\u03b5(\u2207u)=\u2207y(T\u03b5u), and \u2207(T\u03b5\u22121\u03c6)(x)=T\u03b5\u22121(\u2207\u03c6+1\u03b5\u2207y\u03c6)(x) for x\u2208Yi\u03bb and \u03c6\u2208H1(\u03a9\u00d7Yi);chain rules: (v)integration rules:(vi)T\u03b5:boundedness of (T\u03b5T\u03b5\u22121\u016b)=(T\u03b5\u22121\u016b)(x) for x\u2208\u03a9 and z\u2208Y:T\u03b5\u22121\u016b=|Yi||Y|\u27e8T\u03b5u\u27e9Yi as a consequence of the definition \u03c6\u2261\u016b(x) for all \u03c6\u2208L2). The proof of the other properties can be found in other studies.The property (iib) follows in a straightforward manner from the calculation of  \u25a15In this section, we collect some auxiliary tools used later in the derivation of the residual error estimates.Lemma 2 For u(x)\u2208H1(\u03a9i\u03b5), the following Poincar\u00e9 inequality holds \u2208H1(Yi) in the unit cell with connected subsets Yi for i=1,2:\u03c6\u2208L2). Choosing \u03c6=T\u03b5u givesT\u03b5\u27e8T\u03b5u\u27e9Yi=\u27e8T\u03b5u\u27e9Yi. For all \u2208\u03a9\u00d7Yi, we have\u03b5x\u03b5+\u03b5y\u03b5=x\u03b5 for all y\u2208d. This shows, in particular, that y\u21a6T\u03b5\u27e8T\u03b5u\u27e9Yi is constant for a.e. x\u2208\u03a9.We recall the Poincar\u00e9 inequality for a function  \u25a1\u03c6\u2208L2):Ktr>0. After the substitution of \u03c6=T\u03b5u for the function u(x)\u2208H1(\u03a9i\u03b5), there follows  For u(x)\u2208H1(\u03a9i\u03b5), there exists a continuous extension \u0169\u2208H1(\u03a9) from the connected set \u03a9i\u03b5 to \u03a9 such that \u0169=u in \u03a9i\u03b5 andu=0 on \u2202\u03a9i\u03b5\u2229\u2202\u03a9, then \u0169\u2208H01(\u03a9) exists satisfying \u2202\u03a9 is argumented in H\u00f6pker.Indeed, the assertion holds in accordance with previous studies, \u25a1Below, we recall the auxiliary result from Fellner and KovtunenkoLemma 4\u03a9i\u03b5) (which have no jumps across the interface \u0393\u03b5), the asymptotic estimate\u03b5\u21920 for i=1,2.Yi byNi=(y), i=1,2, fromNki\u2208H#1(Yi) for i=1,2 and k=1,\u2026,d. In (30), the notation \u2202yNi(y)\u2208Rd\u00d7d for y\u2208Yi stands for the matrix of derivatives with entries (\u2202yNi(y))kl=\u2202Nki\u2202yl, k,l=1,\u2026,d, and I\u2208Rd\u00d7d denotes the identity matrix. The system (30) admits the weak formulation: find vector\u2010functions Ni\u2208H#1(Yi)d such that\u03c6\u2208H#1(Yi). A solution of Yi.Based on the geometric assumptions (D1) to (D4), we define the space of periodic functions in the cells Ni of the cell problem Ai admit the following asymptotic representation formulated in the lemma below; see Fellner and KovtunenkoBased on the solution Lemma 5(Asymptotic formula for periodic diffusivity matrices)(i)Ni of the cell problem Ai0\u2208Rsymd\u00d7d given by the averagingd\u2010by\u2010d matrix:d\u2010by\u2010d matrix Bi(y) is periodic and has the following divergence form in the cell Yi:bklm(i) are skew\u2010symmetric:Bi is divergence\u2010free:\u27e8Bi\u27e9Yi=0. At the interface, the condition holds:For the solution (ii)Ni\u2208W\u221e1,(Yi)d. For varying function vi\u2208Vi\u03b5 and fixed ui0\u2208L2), the following integral form corresponding to the averaged equation\u00a0ui1:=ui0+\u03b5(T\u03b5\u22121Ni)\u00b7\u2207ui0 is approximated as follows:Assume that (i)Ni of For the vector\u2010valued solution (ii)vi\u2208Vi\u03b5 and ui0\u2208L2) be given. To prove IAi0 in y:Let Ai0=T\u03b5Ai0 holds. Then, expressing Ai0 from \u03b5T\u03b5(\u2207ui0)=\u2207y(T\u03b5ui0), and the notation of the corrector ui1:=ui0+\u03b5(T\u03b5\u22121Ni)\u00b7\u2207ui0, we rearrange the following terms:IAi0 is performed equivalently byIBi is written component\u2010wisely as follows:Bi and the fact that it is divergence\u2010free, the term IBi is integrated by parts as follows:\u2202Yi\u2216\u0393 vanishes after rewriting the integral again in macrovariables because of vi=0 on \u2202\u03a9i\u03b5\u2229\u2202\u03a9 and because jumps across the cell boundary of vi and \u2207ui0 are zero , while Bi is periodic.For the constant matrix, the identity Yi in \u27e8Bi\u27e9Yi=0 as follows:I1i and I2i in the macrovariable x in all local cells using the integration rules (20) and (21) and then apply to the result the Cauchy\u2010Schwarz inequality and the Poincar\u00e9 inequality The integral over \u03a9\u00d7k,l,m will refer to both x as well as y coordinates. We are starting fromx\u2208\u03a9i\u03b5:\u03bb=\u230ax\u03b5\u230b. First, there are some constants 0<K1\u2a7dK2 such thatK3>0 such thatIB1 from K4>0:Below, the indices  \u25a1\u03b5\u21920.With these preliminaries, in the next section, we homogenize the nonlinear transmission problem 6ui0, i=1,2, in the time\u2010space domain \u00d7\u03a9 fromAi0 are defined in ui0\u2208U0 in the spacev\u2208V0:=L2). In \u03a9 implies the duality between H1(\u03a9) and its topologically dual space H1(\u03a9)\u2217.We state the averaged bidomain diffusion problem determining the functions gi. Moreover, the a priori estimate like i=1,2):Ni and of its gradient in order to prove the residual error estimate, which is a standard assumption for cell problems; see, ie, Zhikov et al.The solvability of Theorem 2 Let the cell problem Ni\u2208W\u221e1,(Yi), and the macroscopic solution be such that ui0\u2208L2)\u2229L\u221e), \u2202t(\u2207ui0)\u2208L2)d, i=1,2. Then the solution ui\u03b5 of the inhomogeneous problem ui0 of the averaged problem \u00d1i\u2208W1,\u221e(Y) is a periodic extension of Ni to Y, satisfy the residual error estimate:12 is determined in ui\u03b5\u2212ui1 \u00d7\u03a9i\u03b5, it follows the variational equation in two subdomains for i=1,2:We start with derivation of an asymptotic equation for the difference \u2212ui1 see . Multiplv\u2208V0 and vi\u2208Vi\u03b5. With a special choice of vi, it can be equal to v. For test functions vi=v\u2208V0\u2282Vi\u03b5, i=1,2, we subtract 0 is given by the formula k, k=1,2,3, in the right\u2010hand side of We choose follows:1\u222b0T\u27e8\u2202t(u\u2202tui1=\u2202t[ui0+\u03b5(T\u03b5\u22121Ni)\u00b7\u2207ui0] implying thatgiv the restriction operator from Lemma gi leads toK7=|\u0393|Lg)We use the Cauchy\u2010Schwarz inequality and the expansion of the time\u2010derivative of the corrector v by piecewise constant average \u27e8T\u03b5v\u27e9(x):=\u27e8T\u03b5v\u27e9Yj(x) for x\u2208\u03a9j\u03b5, j=1,2. For this task, we decompose Ii in 4 and rewrite the third term using |\u0393|=\u222b\u0393d\u03c3y. Based on the boundedness gi, from the Cauchy\u2010Schwarz inequality, it follows the error estimateK8=\u03b5T|\u0393\u03b5|Ktr(1+KP)+|\u0393||Y|T|\u03a9|KP. Here, we have used the Poincar\u00e9 inequality Ji implies thatgi, using the mean inequality5 in ui1\u2208H1(\u03a9), according to Griso,5 holds:Yj and 5 is estimated byIn the following, we aim at substitution of \u03b7\u03a9(x) be a smooth cutoff function with a compact support in \u03a9 and equals one outside an \u03b5\u2010neighborhood of the boundary \u2202\u03a9 such that |\u03b7\u03a9|\u2a7d1 and \u03b5|\u2207\u03b7\u03a9|\u2a7dC\u03b7. For further use, we employ the following functions wi\u2208V0\u2282Vi\u03b5 expressed equivalently in two ways as\u0169i\u03b5\u2208H01(\u03a9) is the uniform extension of ui\u03b5\u2208Ui\u03b5 according to Lemma Let \u0169i\u03b5\u2212ui1 with the help of substitution of the test function v=wi from k, k=0,1,\u2026,5. This implies the following structure of the bounds:\u03b1k=O(1) and Uk=O(1) for k=0,1,\u2026,5.We will derive the estimates for v=wi from M is not an error term; in contrary, it enters with the factor \u2212\u03b41 the left\u2010hand side of the estimate The asymptotic equation\u00a06 is estimated by integration by parts with respect to timeM(ui\u03b5\u2212ui1) is evaluated by Young inequality with the weight \u03b41>0 and using the boundedness property of Ai with the upper bound \u03b2 from U7=O(1), in particular, because 1\u2212\u03b7\u03a9\u22600 on a O(\u03b5)\u2010set using the fact that 1\u2212\u03b7\u03a9\u22600 on a set of measure O(\u03b5), thus compensating \u2207\u03b7\u03a9=O(\u03b5\u22121) here.ErrJi(\u27e8T\u03b5wi\u27e9) and the uniform positive definiteness Ai with the lower bound \u03b1>0, from \u03b18:=|\u0393|2\u2211j=121|Yj|2, andi=1,2 we rearrange the terms such that\u03b3:=\u03b1\u22124\u03b52KtrLg2\u22123\u03b2\u03b412, \u03b110:=2(KtrLg2+\u03b18), and the error Err9 implies\u03b41 small enough such that \u03b3>0. Therefore, applying Gr\u00f6nwall inequality leads toTherefore, using the inequality 7Compared with previous results in the literature on multiscale diffusion equations, in the paper, we derived the macroscopic bidomain model that is advantageous for numerical simulation; we first proved the homogenization result supported by residual error estimate of the asymptotic corrector due to the nonlinear transmission condition at the microscopic level, which appears to describe interface chemical reactions.\u03a91\u03b5 and \u03a92\u03b5. While in the connected domain \u03a91\u03b5 the uniform extension is applicable, the disconnected domain \u03a92\u03b5 allows a discontinuous Poincar\u00e9 estimate (see Kovtunenko and ZubkovaFor further generalization of the obtained result, we suggest to consider the case of connected\u2010disconnected domains This work does not have any conflicts of interest."}
{"text": "Zhou CC, Wang J, Bu H, Zhongguo Fei Ai Za Zhi, 2020, 23(2): 65-76.In the version of this article initially published, error appeared on page 65. The author address should be \u201c400037 \u91cd\u5e86\uff0c\u91cd\u5e86\u65b0\u6865\u533b\u9662\uff08\u6731\u6ce2\uff09\u201d. And another error appeared on page 66. The first sentence in part 1 should be \u201c\u5e74\u9f84\u6807\u5316\u7387\uff08age standardized rate, ASR\uff0931.5/10\u4e07\u548c14.6/10\u4e07\u201d.https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7049793"}
{"text": "We prove an extremal result for long Markov chains based on the monotone path argument, generalizing an earlier work by Courtade and Jiao. Shannon\u2019s entropy power inequality (EPI) and its variants assert the optimality of the Gaussian solution to certain extremal problems. They play important roles in characterizing the information-theoretic limits of network information theory problems that involve Gaussian sources and/or channels   2) are ewhere \u201c\u2248\u201d means that the two sides are equal up to an additive constant.Remark\u00a02.Conjecture 1 corresponds to the special case where The rest of the paper is organized as follows. n-dimensional zero-mean Gaussian random (column) vectors with covariance matrices In this section, we consider the Gaussian version of the optimization problem in  defined Given  implies\u03a31=((1\u2212\u03bb)\u03a3Z\u22121)\u22121,\u03a32= vectors with covariance matrices Let hat (cf. \u201311))(1).. U and V be two arbitrary random objects jointly distributed with Let ction in  does notFor ures see a:(20)U\u2194X\u03bb\u2208,DX=DX|X\u03bb,U,where , 23) is is\u03bb\u2208 that, \u03a3Z\u02dc\u227a\u03a3Z,DX|W,U=DXed (cf. (\u03a3Z\u02dc=\u03a3Z\u22121+ we haveDY\u2aafD2\u22121+(veraging \u201319) tha thah\u03bb=\u2212\u03bcnditions:\u2212\u03bc(1\u2212\u03bb)2D\u2212D\u00af1)=0,\u03a02\u2032(DY\u2212D\u00afced from \u201316); it; ith\u03bb=\u2212\u03bca 2 and (dh\u03bbd\u03bb\u2264\u2212n2erefore,min(DX,DYWe have generalized an extremal result by Courtade and Jiao. It is worth mentioning that recently Courtade  found a"}
{"text": "On PDF page 3, paragraph 3: \u201cFig. S2B\u201d should be \u201cFig. S1B.\u201d On PDF page 6, Fig. 3 legend: \u201crnfABCDEF\u201d should be \u201crnfABCDEG.\u201d On PDF page 8, 9th line from the bottom: \u201cRS10900\u201d should be \u201cRS10900\u201d. On PDF page 13: \u201cG.T.\u201d should be \u201cG.E.T.\u201d These changes do not change the conclusions of the paper.Volume 11, no. 1, e03221-19, 2020."}
{"text": "Scientific Reports 10.1038/s41598-020-66688-1, published online 30 June 2020Correction to: This Article contains errors in the average and standard deviation calculations of postoperative weight patients for the effect of Roux-en-Y gastric bypass reported in Table 1.The \u2018Weight (kg)\u2019 value for \u2018Postoperative\u2019 patients \u2018With cholecystectomy (n\u2009=\u20097),\u2019\u201c112.3\u2009\u00b1\u200910.89\u201dshould read:\u201c95\u2009\u00b1\u200910.55\u201dAdditionally, the \u2018Weight (kg)\u2019 value for \u2018Postoperative\u2019 patients \u2018Without cholecystectomy (n\u2009=\u200921),\u2019\u201c108.17\u2009\u00b1\u200914.50\u201dshould read:\u201c90.7\u2009\u00b1\u200912.40\u201dThe correct Table"}
{"text": "Halictidae, described by Ferdinand Morawitz from the collection of Aleksey Fedtschenko deposited in the Zoological Museum of the Moscow State University and in the Zoological Institute, Russian Academy of Sciences, St. Petersburg (Russia), are critically reviewed. Precise information with illustrations of types for 43 taxa is provided. Lectotypes are here designated for the following seven nominal taxa: Halictusaprilinus Morawitz, 1876, H.cingulatus Morawitz, 1876, H.laevinodis Morawitz, 1876, H.limbellus Morawitz, 1876, H.nasica Morawitz, 1876, H.rhynchites Morawitz, 1876 and H.vulgaris Morawitz, 1876.The type specimens of the family  Apidae genuinae\u201d (1875), Morawitz treated a total of 255 species of numerous genera, of which many species were described as new. In this second part, \u201cAndrenidae\u201d (1876), the remaining bees were dealt with, including the species of the difficult genera Andrena, Halictus and Hylaeus, totalling 183 species  and in the Zoological Institute of the Russian Academy of Sciences, St. Petersburg, Russia (ZISP). To this day, these remain some of the most important manuscripts on bees of this region.More than 140 years ago (1876), the second part of Ferdinand Morawitz\u2019s critical study on the bees collected by the Aleksey Fedtschenko 1869\u2013 species . The spe species  and the The entomological literature uses various, often obscure, terms and names for Central Asian regions and countries. The term \u201cTurkestan\u201d has a particularly special use in entomology, widely adopted by th and early 19th Centuries as Lesser Bukharia, as opposed to Greater Bukharia, where the Bukhara Khanate was situated. In Europe, these lands came to be referred to in the 18th and 19th Centuries as Turkestan, i.e. \u201cthe Land of Turks,\u201d which was the original Iranian name for the territory east of Fergana and Bukhara where nomadic Turkic tribes roamed. Subsequently, when the Turkic tribes occupied the enormous territory from the Caspian Sea to Lop Nor, the name Turkestan acquired a new meaning, so broad that it was deemed necessary to distinguish such areas as Western\u2014Bukhara, or Russian Turkestan \u2013 and Eastern or Chinese Turkestan  in Chinese sources, was referred to in the Russian and European historiography of the 18urkestan . AccordiHalictidae is represented in Nomioides Schenck, 1866, Halictus Latreille, 1804, Sphecodes Latreille, 1804 and Nomia Latreille, 1804), comprising 73 species (Nos. 327\u2013399). Only 30 species were previously known, while the remaining 43 were newly described  , who visited ZMMU from 26.03.1975 to 01.04.1975  received a red label with the inscription \u201cLectotypus Halictidae described by Morawitz, only five specimens received such labels: Sphecodesnigripennis Morawitz, 1876, S.pectoralis Morawitz, 1876, S.rufithorax Morawitz, 1876, Nomiaedentata Morawitz, 1876 and N.rufescens Morawitz, 1876  and require corrections by subsequent authors  . Square brackets are used for English translations and when information is added to specimen label information  or published data . Photographs were made using a combination of a stereomicroscope Olympus SZX10 and a digital camera (Olympus OM-D and Canon EOS70D).All of the material listed below was examined for this study. In the following list, the taxa are treated in alphabetical order of the names used in the original descriptions. Each entry includes the name of the taxon in its original combination, the complete reference to the original description of the species  and a list of type specimens present in the collections of the Illustrations were obtained by montaging from an image series that covers different focal planes into a single in-focus image with the Helicon Focus 6. The final illustrations were post-processed for contrast and brightness using Adobe\u00ae Photoshop\u00ae software.Halictus and Lasioglossum follow Sphecodes follow Nomiinae follow Nomioidinae follow The classification and current species status for Halictus Latreille, 1804Genus Taxon classificationAnimaliaHymenopteraHalictidae1.Morawitz, 187606D27E1F-7F9E-54A9-B19F-6B73690F7A82Halictusalbitarsis Morawitz, 1876: 217 , 246, \u2640.Samarkand (Uzbekistan).Uzbekistan: Samarkand, Tashkent.39\u00b039'N, 66\u00b057'E] // Halictusalbitarsis Mor., [N]372 [handwritten by F. Morawitz] // LasioglossumEvylaeusalbitarsis Mor., \u2640 = lectotype, det A.W. Ebmer 1993 // Syntypus Lectotypus <red label> [ZMMU].\u2640, designated by Halictusalbitarsis Mor., F. Morawitz det. [handwritten by F. Morawitz] [ZMMU]; 1 \u2640, <golden circle>, 11.[III.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // albitarsis Mor., Typ. [handwritten by F. Morawitz]; 3 \u2640, 11.[III.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] //Halictusalbitarsis Mor. [handwritten by F. Morawitz]; 1 \u2640, 11.[III.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // Paralectotypus Halictusalbitarsis Mor., design. Warncke, [19]82 <identical red label on each paralectotype specimens, labelled by Yu. Astafurova> [ZISP].(12 \u2640). 6 \u2640, 11.[III.1871], 24., 25.[V.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // [N]372; 1 \u2640, 23.[III.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // [N]372 // Lasioglossum (Sphecodogastra) leucopymatum , replacement name for Halictusalbitarsis Morawitz, 1876 .The secondary designation of the lectotype by Description of male. Kazakhstan, Turkmenistan, Uzbekistan, Afghanistan .Taxon classificationAnimaliaHymenopteraHalictidae2.Morawitz, 18763DD5E1FF-FF28-5BBA-885B-0EDB0E436896Halictusannulipes Morawitz, 1876: 217 (key), 221, \u2640.Karatyube Mt., 15 km S Samarkand (Uzbekistan).Uzbekistan: Samarkand, Karatyube Mt. (= 15 km S Samarkand).39\u00b030'N, 66\u00b052'E] // Halictusannulipes Mor., [N]332 [handwritten by F. Morawitz] // F. Morawitz det. 18.7.5. // Lectotypus Halictusannulipes Mor. 1876, design. Warncke, 1982 <red label, labelled by Yu. Astafurova> [ZMMU].designated by H.annulipes F. Morawitz det. 1875 [ZMMU]; 1 \u2640, 20[.V.1869] // \u0417\u0435\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430] [Zeravshan River valley] // annulipes Mor. Typ. [handwritten by F. Morawitz] // Paralectotypus Halictusannulipes Mor., 1876, design. Warncke, 1982 <identical red label on each paralectotype specimen, labelled by Yu. Astafurova> [ZISP].(2 \u2640). 1 \u2640, 17. [V.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a  // [N]332 // Lasioglossum  annulipes .Halictusmetopias Vachal , Afghanistan, Iran, Tajikistan, Uzbekistan, Kazakhstan, Pakistan .Taxon classificationAnimaliaHymenopteraHalictidae3.Morawitz, 1876B05FBBBF-88E9-58A6-9110-D1C63147BFBBHalictusaprilinus Morawitz, 1876: 216 (key), 228, \u2640.Kattakurgan (Uzbekistan).Uzbekistan: Katty-Kurgan.39\u00b053'N, 66\u00b015'E] // \u2640 // H.aprilinus n. sp., \u2640, F. Morawitz det., typus [handwritten by F. Morawitz] // Lectotypus Halictusaprilinus Mor., design. Astafurova et Proshchalykin, 2020 <red label> [ZMMU].\u2640, <golden circle>, 28. [IV.1869] .3 \u2640, 28. [IV.1869] // \u041a\u0430\u0442\u0442\u044b\u043a\u0443\u0440\u0433\u0430\u043d\u044a [Kattakurgan] // [N]343 // Paralectotypus Lasioglossum (Sphecodogastra) aprilinum .Halictusinexspectatus  .Taxon classificationAnimaliaHymenopteraHalictidae4.Morawitz, 18767D96A7D5-B70D-5CC6-824D-E5ADF1DA2B43Halictusatomarius Morawitz, 1876: 218 (key), 254, \u2640.Tashkent (Uzbekistan).Uzbekistan: Tashkent.41\u00b018'N, 69\u00b016'E] // Halictusatomarius Mor. [handwritten by F. Morawitz] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] / / Syntypus <red label> // LasioglossumEvylaeusatomarium (Mor.), \u2640, Lectotypus, det. A.W. ZISP].\u2640, designated by Halictusatomarius Mor. [handwritten by F. Morawitz] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] [ZISP].1 \u2640, 8.[VIII.1870] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // Lasioglossum (Evylaeuss.l.)politumatomarium  , 226, \u2642.Osh (Kyrgyzstan).Uzbekistan: Samarkand, Dzhyuzak [=Jizzakh], Sokh; Kyrgyzstan: Osh.40\u00b032'N, 72\u00b047'E] // Halictuscariniventris Mor., [N]341 [handwritten by F. Morawitz] // Lectotypus Halictuscariniventris Mor., design. Bl\u00fcthgen <red label, labelled by Yu. Astafurova> [ZMMU].\u2642, designated by Halictuscariniventris, design. Bl\u00fcthgen <identical red labels on each paralectotype specimen, labelled by Yu. Astafurova> [ZMMU].(4 \u2642). 2 \u2642, 4., 7.[VII.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434 [Samarkand] // [N]341; 2 \u2642, 18., 22.[VII.1870] // \u0414\u0436\u044e\u0437\u0430\u043a\u044a [Dzhyuzak] // [N]341 // Paralectotypus Halictus  pollinosacariniventris Morawitz, 1876 , Russia , Turkey, Israel, Iran, Afghanistan, Pakistan, Central Asia, Mongolia, North China .Taxon classificationAnimaliaHymenopteraHalictidae6.Morawitz, 1876ED601DFE-909A-5859-AE9F-EB0E15954C64Halictuscingulatus Morawitz, 1876: 218 (key), 245, \u2640.Samarkand (Uzbekistan).Uzbekistan: Samarkand, Dzham [near Samarkand], Aksay [near Samarkand].39\u00b039'N, 66\u00b057'E] // Halictuscingulatus Mor., [N]371 [handwritten by F. Morawitz] // Lectotypus Halictuscingulatus Mor., design. Astafurova et Proshchalykin, 2020 <red label> [ZMMU].\u2640, 18.[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a ; 1 \u2640, the same labels, but 21.[V.1869] [ZMMU]; 1 \u2640, <golden circle> // 18.[V.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // cingulatus Mor. Typ. [handwritten by F. Morawitz]; 1 \u2640, 18.[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]; 1 \u2640, 20.[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Halictuscingulatus Mor., \u2640, CoType, F. Morawitz det. [handwritten by F. Morawitz]; 1 \u2640, 27.[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]; 1 \u2640, 16.[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Halictuscingulatus Mor., \u2640, F. Morawitz det. [handwritten by F. Morawitz]; 1 \u2640, 30.[III.1869] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430]  // Halictuscingulatus Mor. [handwritten by F. Morawitz]; 2 \u2640, 12.[III.1869] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430]  // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Halictuscingulatus Mor., design. Astafurova et Proshchalykin, 2020 <identical red labels on each paralectotype specimen> [ZISP].(22 \u2640). 11 \u2640, 27., 28. [II. 1869], 18., 21. [V. 1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // [N]371; 1 \u2640, 12.[V.1969] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430]  // [N]370; 1 \u2640, 16.[V.1969] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430]  // [N]370 , Dzham Gorge, Urgut; Tajikistan: Iori.39\u00b030'N, 67\u00b052'E] // croceipes Mor. Typ [handwritten by F. Morawitz] // croceipes Mor. Bl\u00fcthgen det. 1935 // Lectotype <red label> // Lasioglossum (Evylaeus) croceipes (Mor.), \u2640, Lectotypus, det. A.W. ZMMU].\u2640, designated by croceipes Mor., [N]338 [handwritten by F. Morawitz] // croceipes Mor. Bl\u00fcthgen det. 1935; 3 \u2640, 8.[VIII.1871] // \u0422\u0430\u043a\u0430 [Taka] // [N]338; 1 \u2640, 13.[V.1869] // \u0414\u0436\u0430\u043c\u0441\u043a\u043e\u0435 \u0443\u0449. [Dzham Gorge] // [N]338 [ZMMU]; 1 \u2640, 1.[VI.1869] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430]  // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // croceipes Mor., Bl\u00fcthgen det. 1935 // Paralectotypus Halictuscroceipes Mor., design. ZISP].(6 \u2640). 1 \u2640, 20.[V.1869] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430]  // Lasioglossum  croceipes .ZMMU from the Zeravshan River valley, neither with Warncke\u2019s lectotype label.The lectotype designation by Turkey, Iran, Afghanistan, Central Asia, Kazakhstan .Taxon classificationAnimaliaHymenopteraHalictidae8.Morawitz, 1876D1065512-A6E2-5834-99CF-F239AE07EA09Halictusdesertorum Morawitz, 1876: 217 (key), 228, \u2640.Kattakurgan (Uzbekistan).Uzbekistan: Katty-Kurgan.39\u00b053'N, 66\u00b015'E] // Halictusdesertorum Mor., [N]344 [handwritten by F. Morawitz] // Lectotypus Halictusdesertorum Mor., design. Warncke <red label, labelled by Yu. Astafurova> [ZMMU].\u2640, designated by Halictus  desertorum Morawitz, 1876.Description of male. Turkey, southern Kazakhstan, Turkmenistan, Uzbekistan, Pakistan .Taxon classificationAnimaliaHymenopteraHalictidae9.Morawitz, 187623A7E390-6AFD-58F8-8C57-54164BF4725BHalictusdeterminatus Morawitz, 1876: 217 (key), 233, \u2640.30 km SSE Samarkand, Sangu-dzhuman Pass (Uzbekistan).Uzbekistan: \u201cOn the road to Sangy-Dzhuman [pass] and the Kulbasy Mountain\u201d.39\u00b027'N, 67\u00b014'E] // Halictusdeterminatus Mor., [N]351 [handwritten by F. Morawitz] // Lectotypus H.determinatus Mor., \u2640, design. Pesenko [1]981 <red label> [ZMMU].\u2640, designated by ZMMU]; 2 \u2640, \u0421\u0430\u043d\u0433\u044b-\u0414\u0436\u0443\u043c\u0430\u043d\u044a [Sangu-Dzhuman] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // determinatus F. Mor., \u2640 [handwritten by F. Morawitz] // Paralectotypus H.determinatus Mor., \u2640, design. Pesenko [1]981 <identical red labels on each paralectotype specimen, labelled by Yu. Pesenko> [ZISP].(6 \u2640). 3 \u2640, the same label as in the lectotype; 1 \u2640, 25.[V.1869] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430] [Zeravshan River valley] // [N]351 .39\u00b024'N, 67\u00b001'E] // Halictusequestris Mor., [N]366 [handwritten by F. Morawitz] // Lectotypus Halictusequestris Mor., design. Warncke [19]82 <red label> [ZMMU].\u2640, designated by Lasioglossum (Lasioglossum) equestre .Halictusferghanicus , 243, \u2642.Shakhimardan (Uzbekistan).Uzbekistan: near Shakhimardan.39\u00b058'N, 71\u00b047'E] // Halictusferghanicus Mor., [N]367 [handwritten by F. Morawitz] // Lectotypus Halictusferghanicus Mor., \u2642, design. Warncke [19]82 <red label, labelled by Yu. Pesenko > [ZMMU].\u2642, designated by ferghanicus Mor., Typ. [handwritten by F. Morawitz]; 1 \u2642, 2.[VII.1871] // \u0428\u0430\u0433\u0438\u043c\u0430\u0440\u0434\u0430\u043d\u044a [Shagimardan] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Halictusferghanicus M. [handwritten by F. Morawitz] // Lasioglossumequestre Mor., Pesenko det., 1985 // Paralectotypus, Halictusferghanicus Mor., design. Warncke, [19]82 <identical red labels on each paralecotype specimen, labelled by Yu. Pesenko> [ZISP].(2 \u2642). 1 \u2642, <golden circle> // 2.[VII.1871] // \u0428\u0430\u0433\u0438\u043c\u0430\u0440\u0434\u0430\u043d\u044a [Shagimardan] // Lasioglossum (Lasioglossum) equestre  , 230, \u2642.30 km SE Kozhatogai, Turkistan Province (Kazakhstan).Uzbekistan: steppe between Tashkent and Syrdarya River.41\u00b047'N, 68\u00b023'E] // Halictusfucosus Mor., [N]345 [handwritten by F. Morawitz] // Halictussenilis Evers. v. fucosus Mor., \u2642, P. Bl\u00fcthgen det. // Holotypus <red label> [ZMMU].\u2642, 18.[V.1871] // \u0421\u0442\u0435\u043f\u044c \u043c.[\u0435\u0436\u0434\u0443] \u0421.[\u044b\u0440] \u0434.[\u0430\u0440\u044c\u0435\u0439] \u0438 \u0422.[\u0430\u0448\u043a\u0435\u043d\u0442\u043e\u043c] 361 [handwritten by F. Morawitz] // Holotypus <red label> [ZMMU].\u2642, 26.[VI.1871] // \u0427\u0438\u0431\u0443\u0440\u0433\u0430\u043d\u044a  // 25.[V.1869] // funerarius Mor. Typ. [handwritten by F. Morawitz] // Lectotypus H.funerarius Mor., \u2640, design. Pesenko [1]981 <red label> // Zoological Institute St. Petersburg INS_HYM_0000159 [ZISP].\u2640, designated by H.funerarius Mor. design. Pesenko [1]981 <red labels> .(8 \u2640). 8 \u2640, the same label as in lectotype // Paralectotypus Halictus  funerarius Morawitz, 1876.A rare montane Central Asian species: Kazakhstan, Uzbekistan, Tajikistan, Western Kyrgyzstan, Iran, north-eastern Afghanistan and north-western China .Taxon classificationAnimaliaHymenopteraHalictidae15.Morawitz, 1876871A8FE5-8CA9-5E23-9138-F676B77806EFHalictusfuscicollis Morawitz, 1876: 217 (key), 229, \u2640.50 km NW Chardara, Kyzylkum Desert .Kazakhstan: \u201cKyzyl-Kum Steppe, near Baybek\u201d.41\u00b044'N, 67\u00b054'E] // Halictusfuscicollis Mor., [N]345 [handwritten by F. Morawitz] // HalictusVestitohalictusfuscicollis Mor., \u2640, Lectotypus, design. A.W. Ebmer 1994 // Lectotypus Halictusfuscicollis Mor., design. Warncke <red label, labelled by Yu. Pesenko> [ZMMU].\u2640, designated by fuscicollis Mor. Typ. [handwritten by F. Morawitz] // Paralectotypus <red label>, labelled by Yu. Astafurova [ZISP].1 \u2640, <golden circle> Kisilkum [handwritten by F. Morawitz] // Halictus  fuscicollis Morawitz, 1876.Halictusflavocallosus  .Taxon classificationAnimaliaHymenopteraHalictidae16.Morawitz, 1876F8F496CE-73D6-5D1A-897C-ADA4523E76EFHalictushyalinipennis Morawitz, 1876: 218 , 220 , 253\u2013254, \u2640, \u2642.Tashkent (Uzbekistan).Kazakhstan: Chardara, steppe between Tashkent and Syr-Darya; Uzbekistan: Dzhamsk Gorge, Dzhizmansk Gorge, Ulus, Dzham, Urgut, Keles, Samarkand, Soch, Shakhimardan, Uch-Kurgan; Kyrgyzstan: Alay, Osh, Gulsha, Taka.41\u00b018'N 69\u00b016'E] // hyalinipennis F. Mor., \u2640 [handwritten by F. Morawitz] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Lectotypus Halictushyalinipennis Morawitz, 1876, \u2640, design. Astafurova & Proshchalykin 2018 <red label> [ZISP].\u2640, designated by ZISP]; 1 \u2642, \u041e\u0448\u044a [Osh] // 1.[VIII.1871]; 1 \u2642, \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // 4.[VII.1869] // [N]384; 3 \u2642, \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // 7.[VII.1869]; 2 \u2640, \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // 3., 21.[III.1869]; 16 \u2640, \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // 26., 27.[III.1871], 21., 23.[V.1871] and 1.,3.,5.,10.[VI.1871]; 1 \u2640, \u0427\u0430\u0440\u0434\u0430\u0440\u0430 [Chardara] // 27.[IV.1871]; 1 \u2640, \u0423\u0447\u044c-\u041a\u0443\u0440\u0433\u0430\u043d\u044a [Uch-Kurgan] // 15.[VII.1871]; 2 \u2642, \u0428\u0430\u0433\u0438\u043c\u0430\u0440\u0434\u0430\u043d\u044a [Shagimardan] // 2.[VII.1871] 5 \u2640, \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430] [Zeravshan River valley], 3., 11., 18., 23.[III.1871] // Paralectotypus Halictushyalinipennis Morawitz, 1876, design. Astafurova & Proshchalykin 2018 <identical red labels on each paralectotype specimen> [ZMMU].. 1 \u2640, \u0428\u0430\u0433\u0438\u043c\u0430\u0440\u0434\u0430\u043d\u044a [Shagimardan] // 3.[VII.1871] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]; 1 \u2642, \u0422\u0430\u043a\u0430 [Taka] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]; 1 \u2640, 1 \u2642, \u0421\u043e\u0445\u044a [Sokh] // 29. [29.VI.1871] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]; 1 \u2642, \u0423\u0447-\u041a\u0443\u0440\u0433\u0430\u043d\u044a [Uch- Kurgan] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]; 2 \u2640, \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]  // 25.[III.1869] // laevinodis Mor., Typ. [handwritten by F. Morawitz] // Lectotype Halictuslaevinodis, design. Astafurova et Proshchalykin, 2020 <red label> [ZISP].\u2640, <golden circle> // \u0421\u0430\u043d\u0433\u044b \u0414\u0436\u0443\u043c\u0430\u043d\u044a ; 1 \u2640 25.[III.1869] // \u0421\u0430\u043d\u0433\u044b \u0414\u0436\u0443\u043c\u0430\u043d\u044a [Sangy Dzhuman] // [N]375 // Paralectotype Halictuslaevinodis, design. Astafurova et Proshchalykin, 2020 <identical red labels on each paralectotype specimen> [ZMMU].(2 \u2640). 1 \u2640, 25.[III.1869] // \u0421\u0430\u043d\u0433\u044b \u0414\u0436\u0443\u043c\u0430\u043d\u044a [Sangy Dzhuman] // Lasioglossum  laevinode .ZMMU.Description of male. Bl\u00fcthgen, 1934b: 154, Fig. Kazahkstan, Uzbekistan, Tajikistan, Kyrghyzstan, Afghanistan .Taxon classificationAnimaliaHymenopteraHalictidae18.Morawitz, 18760CB23044-6369-5D2D-A43E-C7004EE64026Halictuslimbellus Morawitz, 1876: 218 (key), 249, \u2640.Samarkand (Uzbekistan).Uzbekistan: Samarkand; Tajikistan: Peti.39\u00b039'N, 66\u00b057'E] // Halictuslimbellus Mor., [N]377 [handwritten by F. Morawitz] // Lectotypus Halictuslimbellus Mor., Astafurova et Proshchalykin, 2020 <red label> [ZMMU].\u2640, 5.[IV.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a ; 1 \u2640, <golden circle> // 5.[IV.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // Halictuslimbellus F. Mor., Typ. [handwritten by F. Morawitz]; 1 \u2640, \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // limbellus Mor., Typ. [handwritten by F. Morawitz] // Paralectotypus Halictuslimbellus Mor., Astafurova et Proshchalykin, 2020 <identical red labels on each paralectotype specimen> [ZISP].(3 \u2640). 1 \u2640, 5.[IV.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // [N]377 // Typus <red label>  // Halictuslongirostris Mor., [N]356 [handwritten by F. Morawitz] // Lectotypus Halictuslongirostris Mor., design. Warncke [19]82 <red label, labelled by Yu. Astafurova> [ZMMU].\u2642, designated by ZMMU]; 1 \u2640, 2.[VII.1871] // \u0428\u0430\u0433\u0438\u043c\u0430\u0440\u0434\u0430\u043d\u044a [Shagimardan] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Halictuslongirostris Mor. [handwritten by F. Morawitz] // Paralectotypus Halictuslongirostris Mor., design. Warncke < identical red labels on each paralectotype specimen, labelled by Yu. Astafurova> [ZISP].(2 \u2640). 1 \u2640, 25.[V.1869] // \u0421\u0430\u043d\u0433\u044b \u0414\u0436\u0443\u043c\u0430\u043d\u044a [Sangy Dzhuman] // [N]356; 1 \u2640, 12.[VII.1870] // \u0424\u0430\u043d\u044a [Fan] // [N]356 .Kekh .39\u00b057'N, 71\u00b007'E] // Halictusmaculipes Mor., [N]373 [handwritten by F. Morawitz] // Lectotypus Halictusmaculipes Mor., design. Warncke <red label, labelled by Yu. Astafurova> [ZMMU].\u2640 designated by Lasioglossum  maculipes .Male unknown.Turkey, Turkmenistan, Tajikistan, Uzbekistan, Iran, Afghanistan .Taxon classificationAnimaliaHymenopteraHalictidae21.Morawitz, 187613F40084-10DA-56C1-A4F2-92B7254C56F6Halictusmelanarius Morawitz, 1876: 241, \u2642.Shakhimardan (Uzbekistan).Near Shakhimardan.39\u00b058'N, 71\u00b047'E] // Halictusmelanarius Mor., [N]364 [handwritten by F. Morawitz] // Lasioglossumfallax (Mor.) syn: melanarium (Mor.) det. A.W. Ebmer 1979 // Holotypus <red label> [ZMMU].\u2642, 9.[VII.1871] // \u0428\u0430\u0433\u0438\u043c\u0430\u0440\u0434\u0430\u043d\u044a , Sangy-Dzhuman; Tajikistan: Pyandzhikent.39\u00b020'N, 67\u00b019'E] // 25.[V.1869] // minor Mor. [handwritten by F. Morawitz] // Lectotypus H.minor Mor., design. Pesenko, [1]981, \u2640 <red label> // Zoological Institute St. Petersburg INS_HYM_0000164 [ZISP].\u2640, designated by minor Mor. [handwritten by F. Morawitz] // Syn.: jarkandensis Strand, \u2640 // Paralectotypus H.minor Mor., design. Pesenko, [1]981, \u2640 <red label >[ZISP]; 1 \u2640, 25.[V.1869] // \u0421\u0430\u043d\u0433\u044b \u0414\u0436\u0443\u043c\u0430\u043d\u044a [Sangy Dzhuman] // Halictusminor Mor., [N]352 [handwritten by F. Morawitz]; 1 \u2640, 24.[V.1869] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430]  // [N]352 // Paralectotypus H.minor Mor., design. Pesenko, [1]981, \u2640 <identical red labels on each paralectotype specimen> [ZMMU].(3 \u2640). 1 \u2640, \u0421\u0430\u043d\u0433\u044b \u0414\u0436\u0443\u043c\u0430\u043d\u044a [Sangy Dzhuman] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Halictus  minor Morawitz, 1876.Description of male. Azerbaijan, Afghanistan, Iran, Kazakhstan, Central Asia, Altai, Pakistan, north-western and northern China, northern India , b.Taxon classificationAnimaliaHymenopteraHalictidae23.Morawitz, 18762CE15736-D68B-502E-9EAE-6318C69E3B1FHalictusmodernus Morawitz, 1876: 217 (key), 235, \u2640.Samarkand (Uzbekistan).Uzbekistan: near Samarkand.39\u00b039'N, 66\u00b057'E] // Halictusmodernus Mor., [N]354 [handwritten by F. Morawitz] // Holotypus <red label> [ZMMU].\u2640, 5.[VII.1870] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a  near draw-well Chakany; Uzbekistan: steppe between Katty-Kurgan and Ulus, Samarkand, Murzarabat, Chinaz, Sokh.39\u00b039'N, 66\u00b057'E] // Halictusnasica Mor., [N]346 [handwritten by F. Morawitz] // Lectotypus Halictusnasica Mor., design. Astafurova et Proshchalykin, 2020 <red label> [ZMMU].\u2640, 9.[VI.1869] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430] ; 1 \u2640, 28.[IV.1871] // \u041a\u0438\u0437\u0438\u043b\u044a\u043a\u0443\u043c\u044a [Kizilkum] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Halictusnasica Mor. [handwritten by F. Morawitz]; 1 \u2640, <golden circle> // 9.[VI.1869] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430]  // nasica Mor. Typ. [handwritten by F. Morawitz]; 3 \u2642, 28.[VI.1871] // \u0421\u043e\u0445\u044a [Sokh] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]; 1 \u2642, 9.[VI.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Halictusnasica Mor., design. Astafurova et Proshchalykin, 2020 <identical red labels on each paralectotype specimen> [ZISP].. 14 \u2640, the same label as in the lectotype; 11 \u2640, 2 \u2642, 9., 13.[VI.1869], 4., 7.[VII.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // [N]346; 12 \u2642, 24.[VII.1870], 29.[VIII.1870] // \u041c\u0443\u0440\u0437\u0430\u0440\u0430\u0431\u0430\u0434\u044a [Murzarabad] // [N]346; 1 \u2640, 9 \u2642, 28.[IV.1871] // \u041a\u0438\u0437\u0438\u043b\u044a\u043a\u0443\u043c\u044a [Kizilkum] //[N]346; 17 \u2642, 25.[VII.1870] // \u0427\u0438\u043d\u0430\u0437\u044a [Chinaz] // [N]346  // Halictusnigrilabris Mor., [N]378 [handwritten by F. Morawitz] // Lectotypus Halictusnigrilabris Mor. design. Warncke [1]982 <red label, labelled by Yu. Pesenko> [ZMMU].\u2642, designated by Halictusnigrilabris F. Morawitz [handwritten by F. Morawitz] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Halictusnigrilabris Mor., design. Warncke [1]982 <red label, labelled by Yu. Pesenko> // Zoological Institute St. Petersburg, INS_HYM 0000076 [ZISP].1 \u2642, \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430] [Zeravshan River valley] // Lasioglossum (Lasioglossum) nigrilabre .Halictussubprasinus .Tajikistan: Iori George, Iskander River; Uzbek enclave in Kyrgyzstan: Karakazuk; Kyrgyzstan: Alay.39\u00b060'N, 71\u00b050'E] // Halictusnigripes Mor., [N]380 [handwritten by F. Morawitz] // nigripes Mor., \u2642, lecto-holotype, Bl\u00fcthgen det. 1933 // Lectotypus Halictusnigripes Mor., design. Bl\u00fcthgen [19]34 <red label> labelled by Yu. Astafurova [ZMMU].\u2642, designated by nigripes Mor. \u2642, Lecto-Paratype, Bl\u00fcthgen det., 1933 [ZMMU]; 1 \u2642, <golden circle> // 23.[VII.1871] // \u0410\u043b\u0430\u0439 [Alay] // nigripes Mor. Typ., [N]380 [handwritten by F. Morawitz]; 1 \u2642, 22.[VII.1871] // \u0410\u043b\u0430\u0439 [Alay] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Halictusnigripes Mor. [handwritten by F. Morawitz] // Paralectotypus Halictusnigripes Mor., design. Bl\u00fcthgen <identical red labels on each paralectotype specimen, labelled by Yu. Astafurova> [ZISP].(3 \u2642). 1 \u2642, 23.[VII.1871] // \u0410\u043b\u0430\u0439 [Alay] // [N]380 // Lasioglossum  melanopus , replacement name for Halictusnigripes Morawitz, 1876 .Halictuspseudonigripes Bl\u00fcthgen, 1934.The specimens from Iskander and Iori George in the Morawitz type series are the holotype and paratypes of Halictusattritus  .Taxon classificationAnimaliaHymenopteraHalictidae27.Morawitz, 1876EBA6D4CC-95C9-543D-957F-98FFF03A1131Halictusobscuratus Morawitz, 1876: 218 (key), 245, \u2640.Sangy-dzhuman Pass, 30 km SSE Samarkand [Uzbekistan].Uzbekistan: Samarkand, Aksay; Tajikistan: Iori Gorge, Varzaminor [=Ayni], Sangy-Dzhuman Pass.39\u00b020'N, 67\u00b019'E] // Halictusobscuratus Mor., [N]370 [handwritten by F. Morawitz] // Lectotypus Halictusobscuratus Mor., design. Warncke [19]82 <red label, labelled by Yu. Astafurova> [ZMMU].\u2640, designated by obscuratus [handwritten by F. Morawitz]; 1 \u2640, [7.VI.1870] // \u0412\u0430\u0440\u0437\u0430\u043c\u0438\u043d\u043e\u0440\u044a [Varzaminor] // [N]370; 1 \u2640, 16.[V.1869] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430]  // N[370]; 1 \u2640, 2.[VI.1869] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430]  // N[370]; [ZMMU]; 1 \u2640, <golden circle> // 25.[V.1869] // \u0421\u0430\u043d\u0433\u044b \u0414\u0436\u0443\u043c\u0430\u043d\u044a [Sangy Dzhuman] // obscuratus Mor., Typ. [handwritten by F. Morawitz]; 1 \u2640, \u0421\u0430\u043d\u0433\u044b \u0414\u0436\u0443\u043c\u0430\u043d\u044a [Sangy Dzhuman] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // obscuratus Mor.[handwritten by F. Morawitz]; 1 \u2640, \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430] [Zeravshan River valley] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Halictusobscuratus F. Morawitz, \u2640 [handwritten by F. Morawitz] // Paralectotypus Halictusobscuratus Mor., design. Warncke <identical red labels on each paralectotype specimen, labelled by Yu. Astafurova> [ZISP].(7 \u2640). 1 \u2640, 3.[IV.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // [N]370; 1 \u2640, 27.[II.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // [N]370 // Lasioglossum (Sphecodogastra) obscuratumssp.obscuratum .Description of male. Europe (except North), Cyprus, Azerbaijan, Russia (North Caucasus), Turkey, Syria, Jordan, Israel, Iran, Afghanistan, Central Asia, Kazakhstan .Taxon classificationAnimaliaHymenopteraHalictidae28.Morawitz, 1876F01DBEB8-81F4-50B8-A2AE-D44EEF813413Halictuspalustris Morawitz, 1876: 217 (key), 234, \u2640.Iskanderkul Lake (Tajikistan).Tajikistan: \u201cnear Iskander-Kul Lake\u201d.39\u00b004'N, 68\u00b022'E] // Halictuspalustris Mor., [N]353 [handwritten by F. Morawitz] // Lectotypus Halictuspalustris Mor., \u2640, design. Warncke [1]982 <red label>, labelled by Yu. Pesenko [ZMMU].\u2640, designated by palustris Mor., Typ. [handwritten by F. Morawitz] // Paralectotypus H.palustris Mor., \u2640, design. Pesenko [1]981 <red label> [ZISP].1 \u2640, <golden circle> // 15.[VI.1870] // \u0418\u0441\u043a\u0430\u043d\u0434\u0435\u0440\u044a [Iskander] // Halictus  palustris Morawitz, 1876.Description of male. Kazakhstan, Uzbekistan, Tajikistan, Kyrgyzstan, China (Xinjiang) .Taxon classificationAnimaliaHymenopteraHalictidae29.Morawitz, 1876303C41DE-B945-5FF1-877F-EFB22CE12547Halictuspectoralis Morawitz, 1876: 218 (key), 251, \u2640.Gulcha (Kyrgyzstan).Kyrgyzstan: Gulsha [Gulcha].40\u00b019'N, 73\u00b026'E] // Halictuspectoralis Mor., [N]381 [handwritten by F. Morawitz] // Holotype H.pectoralis Mor., 1876 <red label, labelled by Yu. Pesenko> [ZMMU].\u2640, 10.[VIII.1871] // \u0413\u0443\u043b\u044c\u0448\u0430 369 [handwritten by F. Morawitz] // Lectotype Halictuspicipes Mor., \u2640, design. Pesenko [1]985 <red label> [ZMMU].\u2640, designated by ZMMU]; 1 \u2640, <golden circle>, the same label as in the lectotype; 1 \u2640, \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430] [Zeravshan River valley] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Halictuspicipes Mor., \u2640 [handwritten by F. Morawitz] // Paralectotypus Hal.picipes Mor., design. Pesenko [1]985 <identical red labels on each paralectotype specimen> [ZISP].(3 \u2640). 1 \u2640, the same label as in the lectotype ; Kyrgyzstan: Alay, Kichi-alay.39\u00b058'N, 71\u00b047'E] // [N]334 // Lectotypus Halictusrhynchites Mor., design. Astafurova et Proshchalykin, 2020 <red label> [ZMMU].\u2642, 7.[VII.1871] // \u0428\u0430\u0433\u0438\u043c\u0430\u0440\u0434\u0430\u043d\u044a 334 [handwritten by F. Morawitz]; 1 \u2640, 2 \u2642, 26.[VI.1871] // \u0427\u0438\u0431\u0443\u0440\u0433\u0430\u043d\u044a [Chiburgan] // [N]334; 1 \u2640, 21.[VI.1871] // \u0427\u0438\u0431\u0443\u0440\u0433\u0430\u043d\u044a [Chiburgan] // [N]334; 2 \u2640, 28.[VII.1871] // \u041a\u0438\u0447\u0438-\u0410\u043b\u0430\u0439 [Kichi-Alay] // [N]334 [ZMMU]; 2 \u2640, 1 \u2642, \u0427\u0438\u0431\u0443\u0440\u0433\u0430\u043d\u044a [Chiburgan] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // rhynchites F. Mor. [handwritten by F. Morawitz]; 1 \u2642, \u041a\u0447\u0438-\u0410\u043b\u0430\u0439 [Kchi-Alay] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // rhynchites F. Mor. [handwritten by F. Morawitz]; 1 \u2640, < golden circle> // 22.[VII.1871] // \u0410\u043b\u0430\u0439 [Alay] //rhynchites Mor., Typ. [handwritten by F. Morawitz]; 1 \u2640, 22.[VII.1871] // \u0410\u043b\u0430\u0439 [Alay] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Halictusrhynchites Mor. [handwritten by F. Morawitz] // Paralectotypus Halictusrhynchites Mor., design. Astafurova et Proshchalykin, 2020 <identical red labels on each paralectotype specimen> [ZISP].. 1 \u2642, the same label as in a lectotype // Lasioglossum (Sphecodogastra) rhynchites .ZMMU.The lectotype designation by Turkey, Afghanistan, southern Kazakhstan, Turkmenistan, Uzbekistan, Kyrgyzstan .Taxon classificationAnimaliaHymenopteraHalictidae32.Morawitz, 18765F934AAA-0C72-5C79-B6AC-56423DAD631BHalictusscutellaris Morawitz, 1876: 218 (key), 238, \u2640.Bairkum .Kazakhstan: Bayrakum [= Bairkum]; Tajikistan: Pendzhikent, Iori.42\u00b005'N, 68\u00b010'E] // Halictusscutellaris Mor., [N]359 [handwritten by F. Morawitz] // Lectotypus Halictusscutellaris Mor., \u2640, design. Pesenko [1]985 <red label> [ZMMU].\u2640, designated by ZMMU]; 1 \u2640, \u0411\u0430\u0439\u0440\u0430\u043a\u0443\u043c\u044a [Bayrakum] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // scutellaris F. Mor., \u2640 [handwritten by F. Morawitz]; 2 \u2640, 4.[V.1871] // \u0411\u0430\u0439\u0440\u0430\u043a\u0443\u043c\u044a [Bayrakum] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Hal.scutellaris Mor., design. Pesenko [1]985 <identical red labels on each paralectotype specimen> [ZISP].(5 \u2640). 1 \u2640, the same label as in the lectotype; 1 \u2640, 30.[V.1869] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430]  // N[359] , Iskander [River]; Kyrgyzstan: Osh.39\u00b039'N, 66\u00b057'E] // Halictussogdianus Mor., [N]342 [handwritten by F. Morawitz] // sogdianus Mor., \u2640, Lecto-Holotype, Bl\u00fcthgen det. 1931 // Lecto-Type <red label> // Lectotypus Halictussogdianus Mor., design. Bl\u00fcthgen 1934 <red label> [ZMMU].\u2640, designated by Halictussogdianus Mor. design. Bl\u00fcthgen <identical red labels on each paralectotype specimen, labelled by Yu. Astafurova> [ZMMU].(5 \u2640). 1 \u2640, 21.[VI.1870] // \u0418\u0441\u043a\u0430\u043d\u0434\u0435\u0440\u044a [Iskander] // [N]342; 1 \u2640, 2.[VIII. 1871] // \u041e\u0448\u044a [Osh] // [N]342; 2 \u2640, 4.[VII.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // [N]342; 1 \u2640, 18.[VII.1870] // \u0414\u0436\u044e\u0437\u0430\u043a\u044a [Dzhyuzak] // [N]342 // Paralectotype Halictus  pulvereus Morawitz, 1874 , Cyprus, Turkey, Iran, Afghanistan, Central Asia, Mongolia, north-western China .Taxon classificationAnimaliaHymenopteraHalictidae34.Morawitz, 1876FDA0BA5F-415E-5762-AF30-281E7AD30236Halictustrifasciatus Morawitz, 1876: 218 (key), 240, \u2640.Bairkum .Kazakhstan: Bayrakum.42\u00b005'N, 68\u00b010'E] // Halictustrifasciatus Mor., [N]362 [handwritten by F. Morawitz] // Lectotypus Halictustrifasciatus Mor., design. Warncke [19]82 <red label, labelled by Yu. Pesenko> [ZMMU].\u2640, designated by Lasioglossum (Lasioglossum) lebedevi Ebmer, 1972, replacement name for Halictustrifasciatus Morawitz, 1876 .Male unknown.Southern Kazakhstan  is doubtTaxon classificationAnimaliaHymenopteraHalictidae35.Morawitz, 187606CD3366-2926-552D-8046-EEA89F2366B4Halictusvaripes Morawitz, 1876: 217 , 220 , 223, \u2640, \u2642.Jizzakh (Uzbekistan).Uzbekistan: Katty-Kurgan [= Kattakurgan], Dzhyuzak [= Jizzakh], Karatyube [= Karatepa near Samarkand], Urgut, Sangy-Dzhuman; Kyrgyzstan: near Osh.40\u00b007'N, 67\u00b051'E] // [N]337 // Halictusvaripes Mor., \u2640, Lecto-Holotype, P. Bl\u00fcthgen det. // Typus <red label> // Lectotypus Halictusvaripes Mor., 1876, design. Bl\u00fcthgen, 1955 <red label> [ZMMU].\u2640, designated by Halictusvaripes Mor. \u2642, lecto-Paratype, Bl\u00fcthgen det.; 1 \u2642, the same label, but 14.[VII.1870] // Halictusvaripes Mor. \u2642, lecto-Holotype, Bl\u00fcthgen det.; 1 \u2640, 20. [VI.1869] // \u041a\u0430\u0442\u0442\u044b-\u041a\u0443\u0440\u0433\u0430\u043d\u044a [Katty-Kurgan] // Halictusvaripes Mor. \u2640, lecto-Paratype, Bl\u00fcthgen det. // Paralectotype Halictusvaripes Mor., design. ZMMU].. 1 \u2642, the same labels as in the lectotype // Halictus (Seladonia) lucidipennis Smith, 1853 , 250, \u2640.Samarkand (Uzbekistan).Uzbekistan: Tashkent, Samarkand, Katty-Kurgan [Kattakurgan].39\u00b039'N, 66\u00b057'E// Hylaeus [sic!] vulgaris Mor., [N]379 [handwritten by F. Morawitz] // Lectotypus Halictusvulgaris Mor., design. Astafurova et Proshchalykin, 2020 <red label> [ZMMU].\u2640, 3.[III.1869] ; 1 \u2640, 3.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // \u043a. \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 // Halictusvulgaris Mor. [handwritten by F. Morawitz]; 1 \u2640, 5.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]; 2 \u2640, 8.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]; 1 \u2640, 1.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus, Halictusvulgaris Mor., design. Astafurova et Proshchalykin <identical red labels on each paralectotype specimen> [ZISP].(257 \u2640). 26 \u2640, 4., 20., 23., 30.[III.1869], 3., 11., 19.[IV.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u043d\u0434\u044a [Samarkand]; 5 \u2640, 28.[IV.1869] // \u041a\u0430\u0442\u0442\u044b\u043a\u0443\u0440\u0433\u0430\u043d\u044a [Kattykurgan]; 222\u2640, 10., 11., 24., 26., 27., 28., [II.1871], 23., 24. [III.1871], 1., 2., 3., 5., 8., 10., 11.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] .41\u00b010'N, 68\u00b006'E] // Sphecodesnigripennis Mor. [handwritten by F. Morawitz] // Lectotypus, ZMMU].\u2640, designation by Sphecodesnigripennis Mor., design. Warncke <red label, labelled by Yu. Astafurova> [ZMMU].1 \u2640, 21.[V.1869] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430] [Zeravshan River valley] // Paralectotype Sphecodesgibbus (Linnaeus 1758) , Israel, Jordan, Russia (east to Yakutia), Turkey, Iran, Pakistan, Central Asia, Kazakhstan, Mongolia, NW China, India .Taxon classificationAnimaliaHymenopteraHalictidae38.Morawitz, 1876168597F0-F09A-5EC4-B44F-55A8A2EA0C4ESphecodespectoralis Morawitz, 1876: 256, \u2640.Shardara District of Turkistan Province (Kazakhstan).Kazakhstan: coasts of Kosaral Lake; Kyzylkum [desert] near Chakany Well.41\u00b010'N, 68\u00b006'E// Sphecodespectoralis Mor. [handwritten by F. Morawitz] // Lectotypus, ZMMU].\u2640, designation by Sphecodespectoralis Mor., design. Warncke <red label, labelled by Yu. Astafurova> [ZMMU].1 \u2640, 28.[VI.1871] // \u041a\u044b\u0437\u044b\u043b\u044a\u043a\u0443\u043c\u044a [Kyzylkum] // Paralectotype Sphecodespectoralis Morawitz, 1876.Sphecodescristatus sensu Meyer (non Hagens 1882)  see : 475.South Kazakhstan, Central Asia, China  , b, 2020Taxon classificationAnimaliaHymenopteraHalictidae39.Morawitz, 1876871E77F6-9DDB-5017-9259-074246B9574DSphecodesrufithorax Morawitz, 1876: 255, \u2640, \u2642.Bairkum .Kazakhstan: Bayrakum [Bairkum]; steppe between Syr-Darya River and Tashkent.42\u00b005'N, 68\u00b010'E] // Sphecodesrufithorax F. Moraw., \u2640 [handwritten by F. Morawitz] // Lectotypus Warncke, 1975 <red label> [ZMMU].\u2640, designated by Sphecodesrufithorax F. Moraw., \u2640 [handwritten by F. Morawitz] // F. Morawitz det., Typ.; 1 \u2640, 20.[V.1871] // \u0421\u0442\u0435\u043f\u044c \u043c.[\u0435\u0436\u0434\u0443] \u0421.[\u044b\u0440] \u0434.[\u0430\u0440\u044c\u0435\u0439] \u0438 \u0422.[\u0430\u0448\u043a\u0435\u043d\u0442\u043e\u043c] [Steppe between Syrdarya River and Tashkent] // Sphecodesrufithorax Mor., \u2642 [handwritten by F. Morawitz] // Paralectotypus Sphec.rufithorax Mor., design. Warncke, [19]92 <red label, labelled by Yu. Astafurova> [ZMMU]; 1 \u2640, the same labels [ZISP].. 1 \u2640, \u0411\u0430\u0439\u0440\u0430\u043a\u0443\u043c\u044a [Bairkum] // \u043a\u043e\u043b.[\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Sphecodesolivieri Lepeletier, 1825 , Turkey, Caucasus, Iran, Pakistan, Central Asia, Kazakhstan, NW China .Nomia Latreille, 1804Genus Taxon classificationAnimaliaHymenopteraHalictidae40.Morawitz, 1876D6C3CB5F-DCAE-5E9E-8BD0-AAAB4FB031EANomiaedentata Morawitz, 1876: 259, \u2640, \u2642.Jizzakh (Uzbekistan).Uzbekistan: Samarkand, Dzhyuzak.40\u00b007'N, 67\u00b051'E] //Nomiaedentata Mor. [handwritten by F. Morawitz] // Lectotypus, ZMMU].\u2640, designated by Nomiaedentata Mor., design. Warncke <red label> [ZISP]; 1 \u2642, 8.[VII.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // Paralectotypus Nomiaedentata Mor., design. Warncke <red label> [ZMMU].(2 \u2642). 1 \u2642, the same labels as in the lectotype // Paralectotypus, Pseudapisedentata .North Africa, Saudi Arabia, Turkey, Azerbaijan, Kazakhstan, Central Asia, Iraq, Iran, Afghanistan, Pakistan, India .Taxon classificationAnimaliaHymenopteraHalictidae41.Morawitz, 187665869E10-D5BC-5910-AE38-7623BFCCFE47Nomiarufescens Morawitz, 1876: 261, \u2640.Zeravshan River valley (Uzbekistan).Uzbekistan: \u201cAykul Lake\u201d in Zeravshan River valley.39\u00b056'N, 66\u00b049'E] // Nomiarufescens Mor. // Lectotypus, ZMMU].\u2640, designated by Pseudapisrufescens .Description of male. Turkey, Central Asia, Kazakhstan .Nomioides Schenck, 1866Genus Taxon classificationAnimaliaHymenopteraHalictidae42.Morawitz, 1876A04B3F23-E067-5F3E-8AE2-0316930DAE54Nomioidesparviceps Morawitz, 1876: 215, \u2642.Bairamali (Turkmenistan).Uzbekistan: Samarkand.39\u00b039'N, 66\u00b057'E], 13.VI[1869], lost  parviceps Morawitz, 1876.Nomioidesconjungens .Tajikistan: Murzarabat; Uzbekistan: Sokh, Samarkand.39\u00b039'N, 66\u00b057'E] // turanica Mor., Typ. [handwritten by F. Morawitz] // Lectotypus Nom.turanica Mor., \u2642, design. Pesenko, 1976 // Zoological Institute St. Petersburg, INS_HYM_0000131 [ZISP].\u2642, designated by Nomioidesturanica n. sp. F. Morawitz det.; 1 \u2642, <gold circle> 8.[VII.1870] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // Nomioidesturanica Mor. [handwritten by F. Morawitz] [ZMMU]; 3 \u2642, \u0421\u043e\u0445\u044a [Sokh] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Nomioidesturanica Mor. // Paralectotypus \u2642 Nom.turanica Mor., design. Pesenko, 1976 <identical red labels on each paralectotype specimen> [ZISP].(5 \u2642). 1 \u2642, 28.[VI.1871] // \u0421\u043e\u0445\u044a [Sokh] // Nomioides (Nomioides) turanicus Morawitz, 1876.North Africa, Central Asia, Iran, Pakistan ."}
{"text": "Scientific Reports 10.1038/s41598-022-16666-6, published online 28 July 2022Correction to: The original version of this Article contained errors in Figure 1. During preparation of Figure 1 some of the data has been inappropriately superimposed. As a result, in Figure 1A a section of the spectrum for the \u201c0.6\u00a0\u03bcM Sn\u201d condition overlapped with a corresponding section in the spectrum of the \u201c0.3\u00a0\u03bcM Sn\u201d condition. A section of the spectrum for the \u201c0.8\u00a0\u03bcM Ag\u201d condition overlapped with corresponding sections in the spectra for \u201c0.4\u00a0\u03bcM Ag\u201d in Figure 1B, and \u201c0.6\u00a0\u03bcM Sn\u201d and \u201c0.3\u00a0\u03bcM Sn\u201d conditions in Figure 1A. The original incorrect Figure\u00a0The original Article has been corrected."}
{"text": "Page 16, Acknowledgments, \u201c82372592\u201d should read \u201c82372559.\u201dVolume 8, no. 4, e00415-23, 2023, Page 17, Funding, \u201c82372592\u201d should read \u201c82372559.\u201d"}
{"text": "Correction to: Pharmacological Reports (2023) 10.1007/s43440-023-00480-6 Figure\u00a03A panel legend has the incorrect units for ADIPEG20 (\u201cmg/ml\u201d should be \u201cng/ml\u201d)Figure\u00a03D legend says incorrectly \u201c75ong/ml\u201d but should be \u201c750\u00a0ng/ml\u201dFigure\u00a04B legend has the incorrect units for argininosuccinate (\u201cmg/ml\u201d but \u201c\u00b5g/ml\u201d)Figure\u00a07B legend has the incorrect units for the scale bar (\u201cmM\u201d but needs to be \u201c\u00b5m\u201d)In this article following figures had incorrect units.The original article has been corrected."}
{"text": "Correction: Neural Dev 15, 3 (2020)10.1186/s13064-020-00139-5The authors would like to correct errors in the original publication of the article .1. Page 2, \u201cthe G4 embryonic stem cells derived from 129S6/SvEvTac\u2009\u00d7\u2009C57BL/6Ncr F1 embryos\u201d corrected to \u201cthe G4 embryonic stem cells derived from a 129S6/SvEvTac\u2009\u00d7\u2009C57BL/6Ncr F1 embryo\u201d.Lrig1 expression in the V-SVZ quiescent neural stem cells (qNSC\u2019s) [12]\u201d corrected to \u201cIn addition to our line of investigation , others have also previously observed Lrig1 expression in the V-SVZ quiescent neural stem cells (qNSC\u2019s) [2. Page 6, \u201cIn addition to our line of investigation , others have also previously observed (qNSC\u2019s) \u201d. An add3. Page 7, \u201cWhen heterozygotes of this C57BL/6J congenic mouse line were interbred\u201d corrected to \u201cWhen heterozygotes of this C57BL/6J congenic mouse line were intercrossed\u201d.Lrig1 expression domain\u201d corrected to \u201cconsistent with the known Lrig1 expression domains\u201d.4. Page 9, \u201cconsistent with the known 5. Page 11, Fig.\u00a04 legend, \u201cAn ependymal cell at the lateral surface\u201d corrected to \u201cAn ependymal cell at the ventricular surface\u201d.6. Page 13, \u201cas well as EdU\u2009+\u2009Ki-67\u2009+\u2009, Ascl1\u2009+\u2009, or EdU\u2009+\u2009Ki-67\u2009+\u2009Dcx\u2009+\u2009proliferating cells (data not shown)\u201d corrected to \u201cas well as EdU\u2009+\u2009Ki-67\u2009+\u2009Ascl1\u2009+\u2009or EdU\u2009+\u2009Ki-67\u2009+\u2009Dcx\u2009+\u2009proliferating cells (data not shown)\u201d.7. Page 13, \u201cvery rare singlet RFP\u2009+\u2009cells with \u03b1/\u03b2 morphologies that were EdU\u2009+\u2009Ascl1\u2009+\u2009and Ki-67\u2009+\u2009\u201d corrected to \u201cvery rare singlet RFP\u2009+\u2009cells with \u03b1/\u03b2 morphologies that were EdU\u2009+\u2009Ascl1\u2009+\u2009Ki-67\u2009+\u2009\u201d.8. Page 13, \u201c(2) we did not observe proliferating clusters of these cells during our experiments\u201d corrected to \u201c(2) we did not observe proliferating clusters of these cells during our experiments.\u201d. A missing period was added.9. Page 14, \u201c\u201d was corrected to \u201c\u201d. One of the two p-values was corrected.10. Page 16, Fig.\u00a06 legend, \u201cGfap- \u03b1/\u03b2 subtype cells\u201d corrected to \u201cGfap- \u03b1 subtype cell\u201d.11. Page 16, Fig.\u00a06 legend, \u201cGfap\u2009+\u2009tanycytes\u201d corrected to \u201cGfap\u2009+\u2009tanycyte\u201d.12. Page 16, Fig.\u00a06 legend, \u201cGfap- tanycytes\u201d corrected to \u201cGfap- tanycyte\u201d.13. Page 17, Additional file legend, \u201csuch as FGF-2, EGF, and BMP4\u201d corrected to \u201csuch as FGF2, EGF, and BMP4\u201d."}
{"text": "Whole exome sequencing (WES) can also detect some intronic variants, which may affect splicing and gene expression, but how to use these intronic variants, and the characteristics about them has not been reported. This study aims to reveal the characteristics of intronic variant in WES data, to further improve the clinical diagnostic value of WES. A total of 269 WES data was analyzed, 688,778 raw variants were called, among these 367,469 intronic variants were in intronic regions flanking exons which was upstream/downstream region of the exon (default is 200 bps). Contrary to expectation, the number of intronic variants with quality control (QC) passed was the lowest at the +2 and \u22122 positions but not at the +1 and \u22121 positions. The plausible explanation was that the former had the worst effect on trans-splicing, whereas the latter did not completely abolish splicing. And surprisingly, the number of intronic variants that passed QC was the highest at the +9 and \u22129 positions, indicating a potential splicing site boundary. The proportion of variants which could not pass QC filtering  in the intronic regions flanking exons generally accord with \u201cS\u201d-shaped curve. At +5 and \u22125 positions, the number of variants predicted damaging by software was most. This was also the position at which many pathogenic variants had been reported in recent years. Our study revealed the characteristics of intronic variant in WES data for the first time, we found the +9 and \u22129 positions might be a potentially splicing sites boundary and +5 and \u22125 positions were potentially important sites affecting splicing or gene expression, the +2 and \u22122 positions seem more important splicing site than +1 and \u22121 positions, and we found variants in intronic regions flanking exons over \u00b1\u200950\u00a0bps may be unreliable. This result can help researchers find more useful variants and demonstrate that WES data is valuable for intronic variants analysis.The online version contains supplementary material available at 10.1186/s12920-023-01542-7. The exome has been defined traditionally as the sequence encompassing all exons of protein coding genes in the genome, it covers 1\u20132% regions of the genome. The method of sequencing all the exons is known as whole exome sequencing (WES) [Exonic variants are often emphasized in WES data, however, intronic variants had been found to affect gene activity and protein production, leading to genetic disorders. Intronic variants mainly regulate biological activities by dysregulating mRNA splicing \u20137. For iIn this study, we analyzed 269 whole exome sequencing data to describe characteristics of intronic variants in data from conventional WES testing, which include: (1) the number of intronic variants in the WES data and whether the variant count of different positions in intronic regions flanking exons which are defined as the region upstream/downstream of the exon (default is 200 bps) have significant difference. (2) The proportion of false variants which cannot pass quality control (QC) and the number of pass variants which can pass QC at the different intron position, and whether these proportion and variants number between different position have significant different. (3) Whether the false proportion between intronic variants of flanking regions and exonic variants have significant different. (4) The number of deleterious variants which are predicted to be damaging by prediction software and the deleterious variants proportion of pass variants, and whether the deleterious variants number and proportion at different position of flanking regions have significant different. We hope that the result can help researchers better understand the characteristics of intronic variants in conventional WES testing data and find some meaningful intronic variants for genetic disease diagnosis or research study, finally improve the clinical diagnostic value of WES.https://github.com/OpenGene/fastp) [https://github.com/bwa-mem2/bwa-mem2) [https://github.com/samtools/samtools) [https://github.com/biod/sambamba) [https://github.com/broadinstitute/gatk/releases) [http://pcingola.github.io/SnpEff/) [https://annovar.openbioinformatics.org/) [https://github.com/WGLab/Phen2Gene) [https://cadd.gs.washington.edu/) [https://www.openbioinformatics.org/annovar/spidex_download_form.php) [http://www.liulab.science/dbscsnv.html) [The 269 WES data was from our WES genetic testing project for patients with adult genetic disorders, including cardiovascular diseases, nervous system diseases, digestive disease, endocrine disease, reproductive system disease, etc. Table . Patiente/fastp)  and selfwa-mem2) , samtoolamtools)  and sambambamba) . Bwa-memeleases) , the HapSnpEff/) , Annovarcs.org/) , Phen2Geen2Gene) , CADD (hon.edu/) , SPIDEX orm.php) , dbscSNVnv.html) , and selRaw variants were filtered by HardFilter module, and the QC filter parameters included 1) SNV: QD\u2009<\u20092.0 or FS\u2009>\u200960.0 or MQ\u2009<\u200940.0 or MQRankSum\u2009<\u2009\u2212\u00a012.5 or ReadPosRankSum\u2009<\u2009\u2212\u00a08.0 or DP\u2009<\u200920 or DV\u2009<\u20098; (2) INDEL: QD\u2009<\u20092.0 or FS\u2009>\u2009200.0 or ReadPosRankSum\u2009<\u2009\u2212\u00a020.0 or DP\u2009<\u200920 or DV\u2009<\u20098. These QC pass variants were called Pass variants. Variants which could not pass QC filtering would be counted as False variants, which might be caused by sequencing errors and PCR amplification. The proportion of false variants were called FP. To find the relationship between the FP and the sequencing depth of different positions, we calculated the average sequencing depth for each position of intronic region , and the damaging filter parameters included 1) population frequency (GnomAD_AF_POPMAX/In-House Database)\u2009<\u20090.05; (2) Not Benign/Likely_Benign variants in CLINSIG; (3) CADD score\u2009>\u200910 and SPIDEX dpsi_zscore\u2009>\u2009=\u20092 and dbscSNV\u2009>\u20090.6. These variants were called Deleterious variants  If the raw variants number of different positions in intronic regions flanking exons had significant difference. (2) If the FP of different positions in intronic regions flanking exons had significant difference (3) If the pass variants number of different positions in intronic regions flanking exons had significant difference. (4) If the deleterious variants number of different positions in intronic regions flanking exons had significant difference. (5) If the FP between variants in intronic regions flanking exons and variants in exonic region had significant different. When the p-value was 0.05 or lower, the result was trumpeted as significant, but if it was higher than 0.05, the result was non-significant.The average sequencing depth of all 269 WES was more than 100X in target region and 99% region was with more than 20X sequencing depth. If the variant was detected in multiple samples, the number of this variant was still counted as 1. Finally, 688,778 unique raw variants detected from 20,791 genes were called from 269 WES data, which contain 377,807 intronic variants and 310,971 exonic variants , and this distance was more and more small with the position away from nearby exons was more and more far expect at the +18 and \u221218 positions, +20 and \u221220 positions, +40 and \u221240 positions and so on. If the region was within 20\u00a0bp or more than 150\u00a0bp from nearby exon, the variants number at different position in these regions was always significant difference (p\u2009<\u20090.01), expect when we compared the first (\u00b1\u20091) position with the +2 and \u22122 positions, +3 and \u22123 positions, +185 and \u2212185 positions to +195 and \u2212195 positions, and so on Fig.\u00a0a.Fig. 7SWe compared the FP and the number of pass variants and deleterious variants at different positions in intronic regions flanking exons with each other, the FP and the number of pass variants was non-significant difference (p\u2009>\u20090.05) between most positions. The number of deleterious variants was significant difference (p\u2009<\u20090.05) between some positions, expect when we compared the position at \u00b1\u20091 versus \u00b1\u20092, \u00b1\u20091 versus \u00b1\u20094\u2009~\u2009\u00b1\u20095, \u00b1\u20099 versus \u00b1\u20093\u2009~\u2009\u00b1\u20098, \u00b1\u200910 versus \u00b1\u20091\u2009~\u2009\u00b1\u20092, \u00b1\u200910 versus \u00b1\u20094\u2009~\u2009\u00b1\u20097, \u00b1\u200911 versus \u00b1\u20094\u2009~\u2009\u00b1\u200910, \u00b1\u200913\u2009~\u2009\u00b1\u200914 versus \u00b1\u20094\u2009~\u2009\u00b1\u200911, \u00b1\u200918\u2009~\u2009\u00b1\u200920 versus \u00b1\u20094\u2009~\u2009\u00b1\u200911, \u00b1\u200929\u2009~\u2009\u00b1\u200932 versus \u00b1\u200923\u2009~\u2009\u00b1\u200926, \u00b1\u200935\u2009~\u2009\u00b1\u200938 versus \u00b1\u200923\u2009~\u2009\u00b1\u200926, \u00b1\u200935\u2009~\u2009\u00b1\u200938 versus \u00b1\u200930\u2009~\u2009\u00b1\u200932, \u00b1\u200940\u2009~\u2009\u00b1\u200946 versus \u00b1\u200923\u2009~\u2009\u00b1\u200926, \u00b1\u200940\u2009~\u2009\u00b1\u200946 versus \u00b1\u200929\u2009~\u2009\u00b1\u200932, \u00b1\u200940\u2009~\u2009\u00b1\u200943 versus \u00b1\u200923\u2009~\u2009\u00b1\u200926, \u00b1\u200945\u2009~\u2009\u00b1\u200946 versus \u00b1\u200940\u2009~\u2009\u00b1\u200944, 48 versus \u00b1\u200923\u2009~\u2009\u00b1\u200926, \u00b1\u200948 versus \u00b1\u200929\u2009~\u2009\u00b1\u200932, \u00b1\u200960 versus \u00b1\u200935\u2009~\u2009\u00b1\u200951, \u00b1\u200963 versus \u00b1\u200923\u2009~\u2009\u00b1\u200952, \u00b1\u200965\u2009~\u2009\u00b1\u200966 versus \u00b1\u200923\u2009~\u2009\u00b1\u200952, \u00b1\u200969 versus \u00b1\u200923\u2009~\u2009\u00b1\u200952, \u00b1\u200971\u2009~\u2009\u00b1\u200973 versus \u00b1\u200923\u2009~\u2009\u00b1\u200925, \u00b1\u200971\u2009~\u2009\u00b1\u200973 versus \u00b1\u200930\u2009~\u2009\u00b1\u200932, \u00b1\u2009123\u2009~\u2009\u00b1\u2009124 versus \u00b1\u200935\u2009~\u2009\u00b1\u200938, \u00b1\u2009128\u2009~\u2009\u00b1\u2009129 versus \u2009\u00b1\u200935\u2009~\u2009\u00b1\u200938, \u00b1\u2009123\u2009~\u2009\u00b1\u2009128 versus \u00b1\u200940\u2009~\u2009\u00b1\u200946, \u00b1\u2009123\u2009~\u2009\u00b1\u2009128 versus \u00b1\u200948, \u00b1\u2009122\u2009~\u2009\u00b1\u2009128 versus \u00b1\u200951\u2009~\u2009\u00b1\u200952, \u00b1\u2009123\u2009~\u2009\u00b1\u2009128 versus \u00b1\u200984\u2009~\u2009\u00b1\u200987, \u00b1\u2009123\u2009~\u2009\u00b1\u2009128 versus \u00b1\u200992\u2009~\u2009\u00b1\u200993, \u00b1\u2009129 versus \u00b1\u2009106\u2009~\u2009\u00b1\u2009114, \u00b1\u2009123\u2009~\u2009\u00b1\u2009128 versus \u00b1\u2009118\u2009~\u2009\u00b1\u2009120, \u00b1\u2009127 versus \u00b1\u2009122\u2009~\u2009\u00b1\u2009126, \u00b1\u2009128 versus \u00b1\u2009123\u2009~\u2009\u00b1\u2009127, \u00b1\u2009131\u2009~\u2009\u00b1\u2009143 versus \u2009\u00b1\u200923\u2009~\u2009\u00b1\u200926, \u00b1\u2009131\u2009~\u2009\u00b1\u2009143 versus \u00b1\u200929\u2009~\u2009\u00b1\u200932, \u00b1\u2009131\u2009~\u2009\u00b1\u2009133 versus \u2009\u00b1\u200935\u2009~\u2009\u00b1\u200938, \u00b1\u2009136\u2009~\u2009\u00b1\u2009\u2009~\u2009\u00b1\u2009139 versus \u00b1\u200935\u2009~\u2009\u00b1\u200938, \u00b1\u2009131\u2009~\u2009\u00b1\u2009135 versus \u00b1\u200940\u2009~\u2009\u00b1\u200946, \u00b1\u2009137\u2009~\u2009\u00b1\u2009138 versus \u00b1\u200940\u2009~\u2009\u00b1\u200946, \u00b1\u2009140\u2009~\u2009\u00b1\u2009143 versus \u00b1\u200941\u2009~\u2009\u00b1\u200946, \u00b1\u2009131 versus \u00b1\u200948\u2009~\u2009\u00b1\u200952, \u00b1\u2009134\u2009~\u2009\u00b1\u2009135 versus \u00b1\u200948\u2009~\u2009\u00b1\u200953, \u00b1\u2009140 versus \u00b1\u200948\u2009~\u2009\u00b1\u200953, \u00b1\u2009143 versus \u00b1\u200948\u2009~\u2009\u00b1\u200953, \u00b1\u2009131\u2009~\u2009\u00b1\u2009135 versus \u2009\u00b1\u200965\u2009~\u2009\u00b1\u200966, \u00b1\u2009130\u2009~\u2009\u00b1\u2009148 versus \u00b1\u200963, \u00b1\u2009137\u2009~\u2009\u00b1\u2009148 versus \u00b1\u200965\u2009~\u2009\u00b1\u200966, \u00b1\u2009168\u2009~\u2009\u00b1\u2009174 versus \u00b1\u2009106\u2009~\u2009\u00b1\u2009114, \u00b1\u2009175\u2009~\u2009\u00b1\u2009192 versus \u00b1\u2009118\u2009~\u2009\u00b1\u2009120, \u00b1\u2009178\u2009~\u2009\u00b1\u2009192 versus \u2009\u00b1\u2009123\u2009~\u2009\u00b1\u2009128, \u00b1\u2009175 versus \u00b1\u2009118\u2009~\u2009\u00b1\u2009153, \u00b1\u2009194\u2009~\u2009\u00b1\u2009200 versus \u00b1\u2009122\u2009~\u2009\u00b1\u2009128, \u00b1\u2009168\u2009~\u2009\u00b1\u2009192 versus \u00b1\u2009131\u2009~\u2009\u00b1\u2009133, \u00b1\u2009168\u2009~\u2009\u00b1\u2009174 versus \u00b1\u2009154\u2009~\u2009\u00b1\u2009166, \u00b1\u2009194\u2009~\u2009\u00b1\u2009200 versus \u00b1\u2009181\u2009~\u2009\u00b1\u2009192, \u00b1\u2009188\u2009~\u2009\u00b1\u2009192 versus \u00b1\u2009175\u2009~\u2009\u00b1\u2009187, \u00b1\u2009181\u2009~\u2009\u00b1\u2009192 versus \u00b1\u2009140\u2009~\u2009\u00b1\u2009153, \u00b1\u2009178 versus \u00b1\u2009118\u2009~\u2009\u00b1\u2009143, and so on  between these two groups, the FP in intronic regions flanking exons was much greater than exonic variants, which might result from the average sequencing depth in exonic region was much more than intronic region  cases [Introns are a hallmark of eukaryotic evolution, and a substantial intron gain has accompanied the origin of metazoan . Severald adults . Weisschvariants . Qian etvariants . Li et avariants . Fitzgermyopathy . Lin et D) cases . mRNA seD) cases . HoweverWES is more widely used than whole genome sequencing (WGS) in the clinical setting due to lower cost and more manageable data volumes. Reanalyzing WES raw data is recommended before performing WGS . To imprWe also found the farther away from the nearby exon, the less the number of pass variants. The FP in the intronic regions flanking exons generally accorded with \u201cS\u201d-shaped curve. However, the distribution of average depth at different positions in intronic regions flanking exons was inversely proportional to the FP distribution. As we thought, in the WES sequencing data the FP of variants in intronic regions flanking exons was much greater than in exonic regions. The false variants can be caused by low coverage, sequencing errors, and PCR amplification . The seqhttps://github.com/Illumina/SpliceAI) [The greatest number of deleterious variants was at the +5 and \u22125 positions, which was also the position at which many pathogenic variants had been reported in recent years \u201342, all pliceAI)  can predThrough the analysis of statistical differences, we found that the number of unfiltered intronic variants was significantly different between most positions, but the number of intronic pass variants and deleterious variants was significantly different only in some regions, and there was no obvious pattern. The results of statistical differences analysis indicated that in the process of evolution the variation rate between different intron regions has non-significant difference. Because, while the variants that do not change the amino-acid sequence, such as intronic variants, are under lower evolutionary constraints, but for flanking intronic sequences, there was a higher level of conservation in mammals , 45. ThuOur study revealed the characteristics of intronic variant in conventional WES testing data for the first time. We found that contrary to expectation, the number of intronic variants with QC passed was the lowest at the +2 and \u22122 positions but not at the +1 and \u22121 positions. The plausible explanation was that the former had the worst effect on trans-splicing, whereas the latter did not completely abolish splicing. And surprisingly, the number of intronic pass variants was the highest at the +9 and \u22129 positions, indicating a potential splicing site boundary. The FP in the intronic regions flanking exons generally accorded with \u201cS\u201d-shaped curve. At +5 and \u22125 positions, the number of variants predicted damaging by software was most which was also the position at which many pathogenic variants had been reported in recent years. And we found variants in intronic regions flanking exons over \u00b1\u200950\u00a0bps might be unreliable.This result can help researchers find more useful variants and demonstrate that WES data is valuable for intronic variants analysis. Although the 269 samples in our study is still not very sufficient, G*Power software  was usedAdditional file 1. All Annotated Pass Variants Information.Additional file 2. The Number of Raw/Pass/Deleterious Variants and The Number of Genes associated with these variants.Additional file 3. The Number Distribution of The Raw Intronic Variants.Additional file 4. The FP Distribution of The Raw Intronic Variants.Additional file 5. The Number Distribution of The Pass Intronic Variants.Additional file 6. The Number Distribution of The Deleterious Intronic Variants."}
{"text": "Nature Communications 10.1038/s41467-023-37194-5, published online 05 May 2023Correction to: R\u2032 to R\u2033 on productivity P in plot i is defined as [(R\u2033)\u2009\u2212\u2009(R\u2032)], where Pi(R\u2033) is the potential productivity outcome when R\u2009=\u2009R\u2033 and P(R\u2032) is the potential productivity outcome when R\u2009=\u2009R\u2032 (R\u2032\u2009\u2260\u2009R\u2033).\u201d and \u201cThe average causal effect of a change in biodiversity from R\u2032 to R\u2033 across all plots is [(R\u2033)\u2009\u2212\u2009P(R\u2032)], where E[\u00b7] is the expectation operator.\u201d. The correct version states \u201c[Pi(R\u2032\u2032)\u2009\u2212\u2009Pi(R\u2032)]\u201d in place of \u201c[(R\u2033)\u2009\u2212\u2009(R\u2032)]\u201d, \u201cPi (R\u2032)\u201d in place of \u201cP (R\u2032)\u201d, and \u201cE[Pi(R\u2032\u2032)\u2009\u2212\u2009Pi(R\u2032)]\u201d in place of \u201c[(R\u2033)\u2009\u2212\u2009P(R\u2032)]\u201d. This has been corrected in both the PDF and HTML versions of the Article.The original version of this Article contained errors in the Methods section \u2018Target causal effect\u2019, in which terms were omitted from the mathematical definitions of the causal effect and average causal effect. These sentences incorrectly read \u201cThe causal effect of a change in richness from"}
{"text": "The authors wish to make the following corrections to their published paper . The autText CorrectionThere was a typing error included on page 5. In Section 3.1, the first paragraph, line four, the numerical value \u201c7.53\u201d should be changed to \u201c7.5\u201d, the numerical value \u201c5.95 \u00b1 0.24 \u00b5g/g\u201d should be revised to \u201c4.33 \u00b1 0.18 \u00b5g/g\u201d, and the numerical value \u201c0.79 \u00b1 0.02 \u00b5g/g\u201d should be revised to \u201c0.58 \u00b1 0.01 \u00b5g/g\u201d.Errors in TableIn the original publication, there was a typing error in"}
{"text": "Troglonectes is a small-body loach endemic to the Guangxi and Guizhou provinces of China, showing a particular affinity for cave areas. Twenty species were recorded in this genus, including one new species. The new species, Tr. canlinensis, can be distinguished from other congenetic species by their morphological characteristics and molecular evidence. In the genus of Troglonectes, the eye, lateral line and scale present or absent, the number of branched pectoral fin rays, caudal fin rays and anal fin rays, and the depth of the upper adipose keel on the caudal peduncle are important identifying characteristics.Troglonectes is described based on specimens from a karst cave in Andong Town, Xincheng County, Liuzhou City, Guangxi, China. Troglonectes canlinensis sp. nov. can be distinguished from its congener species by the following combination of characteristics: eye degenerated into a black spot; whole body covered by scales, except for the head, throat, and abdomen; incomplete lateral line; forked caudal fin; 8\u201310 gill rakers on the first gill arch; 13\u201314 branched caudal fin rays; 8\u20139 branched dorsal fin rays; 5\u20136 anal fin rays; 9\u201310 pectoral fin rays; upper adipose keel depth mostly 1/2 of the caudal peduncle depth; and caudal fin forked.A new species of the genus  Troglonectes Zhang, Zhao, and Yang, 2016 (abbreviation is Tr. in this study in order to differ from the abbreviation of Triplophysa) are small-bodied fish that mainly occur in the Guangxi and Guizhou provinces of China, showing a particular affinity for cave areas. Troglonectes was separated from the genus Oreonectes G\u00fcnther, 1838, that Du et al. [Oreonectes into the platycephalus group, i.e., caudal fin rounded or truncated, and the furcocaudalis group, i.e., caudal fin forked. Subsequently, Zhang et al. [Troglonectes and assigned seven nominal species to Troglonectes, i.e., Tr. acridorsalis , Tr. barbatus , Tr. elongatus , Tr. macrolepis , Tr. microphthalmus , and Tr. translucens , in addition to the type species Tr. furcocaudalis . Troglonectes can be distinguished from other genera in the Nemacheilidae by possessing narrowly separated nostrils, tube-shaped anterior nostril, tip of the anterior nostril extending into the barbel, dorsal fin origin anterior to the pelvic fin origin, and caudal fin forked or truncated [Oreonectes placed in Troglonectes, some species of Paracobitis and Triplophysa were also placed in Troglonectes based on their morphology and molecular evidence. Chen et al. [P. longibarbatus Chen, Yang, Sket, and Aljancic, 1998 from Libo County, Guizhou, but Du et al. [Triplophysa, based on the morphological characteristics, elongated barbel-like anterior nostril, and sexual dimorphism present in males. Li et al. [P. maolanensis Li, Ran, and Chen, 2006 and T. jiarongensis Lin, Li, and Song, 2012 from Guizhou Province, respectively. However, Huang et al. [P. longibarbatus, P. maolanensis, and T. jiarongensis in the Troglonectes. Subsequently, Luo et al. [T. jiarongensis as a synonym of Tr. elongatus. Additionally, Huang et al. [O. daqikongensis Deng, Wen, Xiao, and Zhou, 2016 and O. shuilongensis Deng, Wen, Xiao, and Zhou, 2016 also belong to the genus Troglonectes due to the forked caudal fin, dorsal fin originating anterior to the pelvic fin origin, and presence of caudal crests. Luo et al. [T. huanjiangensis Yang, Wu, and Lan, 2011, T. lihuensis Wu, Yang, and Lan, 2012, T. lingyunensis , and O. retrodorsalis Lan, Yang, and Chen, 1995 in the Troglonectes based on molecular analysis. Zhao et al. [T. hechiensis Zhao, Liu, Du, and Luo, 2021, and stated that 17 species were contained within the Troglonectes. Luo et al. [Karstsinnectes Zhou, Luo, Wang, Zhou, and Xiao, 2023 , and placed O. acridorsalis and Heminoemacheilus parvus Zhu and Zhu, 2014 in this genus. In conclusion, 19 species of Troglonectes have been recorded in China, including Tr. barbatus, Tr. daqikongensis, Tr. donglanensis, Tr. dongganensis, Tr. duanensis, Tr. elongatus, Tr. furcocaudalis, Tr. hechiensis, Tr. huanjiangensis, Tr. jiarongensis, Tr. lihuensis, Tr. lingyunensis, Tr. longibarbatus, Tr. macrolepis, Tr. maolanensis, Tr. microphthalmus, Tr. retrodorsalis, Tr. shuilongensis, and Tr. translucens.Cave loaches of the genus i et al.  and Lin i et al.  describeTroglonectes were collected from a cave in Andong Town, Xincheng County, Liuzhou City, Guangxi Zhuang Autonomous Region (hereinafter referred to as Guangxi), China. Morphological and molecular evidence supported these loach specimens representing a new species of Troglonectes. Hence, the new species is described herein.In July 2022, 10 specimens of Troglonectes canlinensis sp. nov. were collected by F.G. Luo and euthanized rapidly by an overdose of clove oil anesthetic. The right-side pectoral fin and pelvic fin of one specimen were removed and preserved in 99% ethanol. The specimens for the morphological study were stored in 10% formalin, then transferred to 75% alcohol for long-term preservation in the Kunming Natural History Museum of Zoology, Kunming Institute of Zoology (KIZ), Chinese Academy of Science (CAS), China.All care and use of experimental animals complied with the relevant laws of the Chinese Laboratory of Animal Welfare and Ethics (GB/T 35892-2018). Specimens of b reference sequences of twenty-four Nemacheilidae and two Botiidae species from the NCBI GenBank database for phylogenetic tree reconstruction. Parabotia fasciata Dabry de Thiersant, 1872 and Leptobotia elongata , two species of Botiidae, were used as outgroups. To test the phylogenetic position of Troglonectes canlinensis sp. nov., Bayesian inference (BI) analysis was performed using MrBayes on XSEDE (v3.2.7a) and the CIPRES Science Gateway [Counts and measurements followed Du et al. ,9, Tang  Gateway . Two runTroglonectes canlinensis sp. nov. , Andong Town, Xincheng County, Guangxi Zhuang Autonomous Region, China; 24\u00b018.57\u2032 N, 108\u00b059.61\u2032 E, 179 m a.s.l.; collected by F.G. Luo, 20 July 2022.Paratypes. KIZ-GXNU202207\u201309, 9 ex., 29.9\u201354.3 mm SL, collected with holotype.Troglonectes canlinensis sp. nov., T. duanensis, T. lingyunensis, T. macrolepis, T. hechiensis, and T. retrodorsalis share their whole trunk being scaled, except for the head and area between the pectoral fins and pelvic fins; other species of Troglonectes have scaleless bodies or bodies scaled after the dorsal fin origin in Tr. furcocaudalis. However, the new species can be distinguished from T. duanensis by the incomplete lateral line (vs. absent), from T. lingyunensis and T. macrolepis by the eye being present (vs. eye reduced to black pigment), from T. hechiensis by the 8\u201310 inner-gill rakers on first gill arch (vs. 14), and from T. retrodorsalis by the tip of the anterior nostril being elongated to barbel-like and the nostril barbel length being nearly twice the nostril tube length (vs. nostril barbel length being less than 1/2 of the tube length).Diagnosis. Troglonectes canlinensis sp. nov. are given in Description. The morphometric data of the type specimens of Body elongated, slightly flattened in front, strongly compressed in back. Dorsal profile convex and ventral profile straight in live specimen, but it inversed in preserved specimens. From snout to dorsal fin origin, the body depth increases to its maximum, maximum body depth of 18.2\u201321.3% SL. Head slightly depressed and flattened, maximum head width greater than the deepest head depth. Anterior and posterior nostrils adjacent, distance less than the diameter of the posterior nostril. Eyes reduced, eye diameter 7.5\u201311.6% of the lateral head length. Mouth inferior, snout obtuse, upper and lower lips with small furrows and without papillae, median of the lower lip with a V-shaped notch. Three pairs of barbels, inner, outer, and maxillary barbels, extend vertically to the posterior margin of the anterior nostril, anterior margin of the eye, and preopercle, respectively.Distal margin of dorsal fin truncates, origin anterior to the pelvic fin origin, predorsal length of 54.2\u201358.6% SL. Tip of pectoral fin reaching halfway to the pelvic fin origin. Tip of pelvic fin far away from the anus. Anus with close-set anal fin base. Caudal fin forked, upper part slightly longer than the lower part. Upper and lower edges of the caudal peduncle with caudal adipose keels, upper adipose keel height mostly 1/2 of the caudal peduncle depth. Caudal peduncle length 90.2\u2013119.0% of its depth. Body trunk covered by tiny scales, except for the ventral surface before the pelvic fin origin. Lateral line incomplete. Cephalic lateral line system with 3 + 3 supratemporal, 6 supraorbital, 3 + 8 infraorbital, and 7\u201311 preoperculo-mandibular pores.Stomach U-shaped, intestine long, after stomach, with a bend. Swim bladder divided into two chambers. Anterior chamber covered by dumbbell-shaped bony capsule, and posterior chamber developed.Coloration. Dorsal surface and trunk of body yellowish brown, abdomen gray and translucent, stomach and intestine visible from outside. Fin membrane hyaline.Troglonectes canlinensis sp. nov. lives in a karst cave, where water accumulates to form a pool. Most specimens were collected in the rainy season. During the winter, the pool dries up and the cave opening is too narrow for human access. The water temperature was 20 \u00b0C during the survey period in July 2022.Distribution and habitat. The new species was collected from Andong Township, Xincheng County, Laibin City, Guangxi Zhuang Autonomous Region, China . canlinensis\u201d is derived from the pinyin of \u201ccan\u201d and \u201clin\u201d, which refer to resplendence and forest, respectively, with \u201ccanlin\u201d symbolizing health and tenacious vitality. Troglonectes canlinensis sp. nov. is valuable and rare and requires strong vitality to maintain a viable population. We suggest the common Chinese name \u201c\u707f (c\u00e0n) \u6797 (l\u00edn) \u6d1e (d\u00f2ng) \u9cc5 (q\u012bu)\u201d.Etymology. The specific name \u201cTroglonectes species formed a monophyletic group, sister to the genus Paranemachilus. Troglonectes canlinensis sp. nov. was sister to the clade including T. dongganensis, T. duanensis, T. macrolepis, T. microphthalmus, and T. translucens, with bootstrap values of 100. Additionally, the species of Troglonectes were divided into two sub-clades: sub-clade 1 contained species with truncated caudal fins, i.e., Tr. shuilongensis, Tr. retrodorsalis, and Tr. hechiensis; sub-clade 2 contains species with forked or emarginated caudal fins, i.e., Tr. elongatus, Tr. jiarongensis, Tr. dongganensis, Tr. longibarbatus, Tr. daqikongensis, Tr. barbatus, Tr. furcocaudalis, Tr. duanensis, Tr. donglanensis, Tr. microphthalmus, Tr. macrolepis, and Tr. canlinensis sp. nov.Genetic comparisons. The molecular phylogenies based on BI analysis showed that b revealed that the average uncorrected p-distances interspecies of Troglonectes ranged from 0.2% to 12.2% .TroglonectesIdentification Key to Species of 1. Eye present\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u03872\u2013. Eye degenerated or absent\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u03875Tr. furcocaudalis2. Body scaled after dorsal fin origin\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u2013. Whole body scaled except for head and thorax\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u03873Tr. duanensis3. Caudal fin forked\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u2013. Caudal fin truncated\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u03874Tr. hechiensis4. Caudal peduncle length 12.0\u201313.6% SL\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387Tr. retrodorsalis\u2013. Caudal peduncle length 10.8\u201312.0% SL\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u03875. Eye degenerated with black pigment\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u03876\u2013. Eye absent\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387106. Body scaleless\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u03877\u2013. Whole body scaled except for head and thorax\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u03878Tr. microphthalmus7. Upper adipose keel height larger than caudal peduncle depth\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387Tr. donglanensis\u2013. Upper adipose keel height mostly 1/2 the caudal peduncle depth\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387Tr. lingyunensis8. Posterior chamber of swim bladder degenerated\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u2013. Posterior chamber of swim bladder developed\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u03879Tr. macrolepis9. Total of 12\u201313 inner gill rakers on first gill arch\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387Tr. canlinensis sp. nov.\u2013. Total of 8\u201310 inner gill rakers on first gill arch\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387Tr. shuilongensis10. Caudal fin truncated\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u2013. Caudal fin emarginated or forked\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u03871111. Caudal fin emarginated\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u038712\u2013. Caudal fin forked\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u038714Tr. jiarongensis12. Lateral line complete\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u2013. Lateral line incomplete or absent\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u038713Tr. translucens13. Lateral line incomplete\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387Tr. lihuensis\u2013. Lateral line absent\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u038714. Lateral line absent\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u038715\u2013. Lateral line complete or incomplete\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u038716Tr. barbatus15. Standard length 2.6\u20133.5 times the lateral head length\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387Tr. huanjiangensis\u2013. Standard length 4.3\u20134.9 times the lateral head length\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u038716. Lateral line complete\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u038717\u2013. Lateral line incomplete\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u038719Tr. maolanensis17. Dorsal fin with six branched rays, anal fin with four branched rays\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u2013. Dorsal fin with eight or nine branched rays, anal fin with six branched rays\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u038718Tr. daqikongensis18. Standard length 10.1\u201314.0 times the caudal peduncle depth\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387Tr. longibarbatus\u2013. Standard length 14.5\u201318.1 times the caudal peduncle depth\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387Tr. dongganensis19. Pelvic fin origin opposite the dorsal fin origin\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387Tr. elongatus\u2013. Pelvic fin origin anterior to the dorsal fin origin\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387\u0387Mitochondrial differentiation. The pairwise comparisons of ctyTroglonectes is currently distributed in the Pearl River system in Guangxi and Guizhou Provinces, and is endemic to China. Although Zhang et al. [Troglonectes divided into sub-clade 1 contains species with truncated caudal fins and sub-clade 2 contains species with emarginated or forked caudal fins. Hence, the caudal fin shape and phylogenetic tree support that Troglonectes could be divided into two groups; the truncated caudal fin group contains Tr. hechiensis, Tr. retrodorsalis, and Tr. shuilongensis, and the emarginated or forked caudal fin group contains Tr. donglanensis, Tr. microphthalmus, Tr. macrolepis, Tr. canlinensis sp. nov., Tr. lingyunensis, Tr. barbatus, Tr. huanjiangensis, Tr. longibarbatus, Tr. maolanensis, Tr. daqikongensis, Tr. dongganensis, Tr. elongatus, Tr. translucens, Tr. jiarongensis, Tr. lihuensis, Tr. furcocaudalis, and Tr. duanensis. Thus, based on our BI analysis and external characteristics, the genus description for Troglonectes includes the following characteristics: anterior and posterior nostrils separated by a short distance shorter than the diameter of the posterior nostril, tip of anterior nostril elongated to barbel-like, and adipose keels on the upper and lower edges of the caudal peduncle present.The genus g et al.  mentioneTr. donglanensis and Tr. duanensis as synonyms of Tr. translucens, and Tr. jiarongensis and Tr. dongganensis as synonyms of Tr. elongatus based on morphological characteristics and a lack of genetic differences, respectively. Troglonectes dongganensis, Tr. elongatus, Tr. jiarongensis, and Tr. longibarbatus formed a monophyletic group in the phylogenetic tree, and the genetic distance was 0.4\u20131.0% (average 0.7%). However, they can be morphologically distinguished from each other by the lateral line , branched caudal fins (16 in Tr. jiarongensis and 13\u201314 in other species), and body depth . Hence, we treated T. dongganensis, T. elongatus, T. huanjiangensis, T. jiarongensis, and T. longibarbatus as valid species in this study. Additionally, Tr. donglanensis, Tr. duanensis, and Tr. translucens can be distinguished from each other by the 16 branched caudal fins in Tr. translucens (vs. 13\u201314 in Tr. donglanensis and Tr. duanensis) and the body being covered by scales and the lateral line being absent in Tr. duanensis . Thus, we propose Tr. donglanensis, Tr. duanensis, and Tr. translucens as valid species.Luo et al.  treated Troglonectes, 20 valid species were recorded, including the new species. Troglonectes canlinensis sp. nov. can be distinguished from Tr. hechiensis, Tr. retrodorsalis, and Tr. shuilongensis by its forked caudal fin (vs. truncated) and upper adipose keel height being mostly 1/2 of the caudal peduncle depth (vs. 1/4), and it can be further distinguished from Tr. shuilongensis by its degenerated eye with a black pigment (vs. absent), scaled body , incomplete lateral line (vs. complete), and 8\u221210 inner-gill rakers on the first gill arch (vs. 10\u221212); from Tr. hechiensis by the 8\u221210 inner-gill rakers on the first gill arch (vs. 14) and 9\u221210 branched pectoral fin rays (vs. 11); and from Tr. retrodorsalis by the 8\u221210 inner-gill rakers on the first gill arch (vs. 13\u221214) and 9\u221210 branched pectoral fin rays (vs. 11\u221212). Troglonectes canlinensis sp. nov. is different from Tr. translucens, Tr. jiarongensis, and Tr. lihuensis owing to its forked caudal fin (vs. emarginated) and scaled body ; it can be further differentiated from Tr. jiarongensis and Tr. lihuensis by its incomplete lateral line (vs. absent in Tr. lihuensis and complete in Tr. jiarongensis) and upper adipose keel height being mostly 1/2 of the caudal peduncle depth ; and from Tr. jiarongensis by the 13\u221214 branched caudal fin rays (vs. 16). The new species is different from Tr. furcocaudalis owing to its whole body being scaled, except for the head and thorax , 8\u221210 inner-gill rakers on the first gill arch (vs. 12\u221213), and 5\u22126 branched pelvic fin rays (vs. 7); from Tr. duanensis owing to the incomplete lateral line (vs. absent), 8\u221210 inner-gill rakers on the first gill arch (vs. 13), anal fin with 5\u22126 branched rays (vs. 6\u22127), and eye degenerated with black pigment (vs. present); from Tr. lingyunensis by the developed posterior chamber of the swim bladder (vs. degenerated), caudal fin with 13\u221214 branched rays (vs. 16), dorsal fin with 8\u22129 branched rays (vs. 6\u22127), and upper adipose keel height being mostly 1/2 half of the caudal peduncle depth (vs. 1/4); and from Tr. macrolepis by the 8\u221210 inner-gill rakers on the first gill arch (vs. 12\u221213), dorsal fin with 8\u22129 branched rays (vs. 9\u221211), pectoral fin with 9\u221210 branched rays (vs. 10\u221212), and upper adipose keel height being mostly 1/2 of the caudal peduncle depth . Troglonectes canlinensis sp. nov. is different from Tr. barbatus, Tr. huanjiangensis, Tr. longibarbatus, Tr. maolanensis, Tr. daqikongensis, Tr. dongganensis, Tr. elongatus, Tr. donglanensis, and Tr. microphthalmus owing to its scaled body ; it can be further distinguished from these species by the eye being degenerated with black pigment , lateral line being incomplete , dorsal fin having 8\u22129 branched rays , anal fin having 5\u22126 branched rays , and upper adipose keel height being mostly 1/2 of the caudal peduncle depth .Within the genus Troglonectes are highly adapted to survive in cave habitats and are found only in limited regions with relatively small populations. Ma et al. [Troglonectes have developed barbels, well-developed adipose keel on the upper and lower caudal peduncles, reduced or no eyes, lateral line, scales, and pigment; these characteristics are adaptions to cave environments. As their life histories are limited to caves, these fish are vulnerable to various threats, such as habitat degradation, hydrological alterations, environmental pollution, resource overexploitation, and non-native species introduction [Species of a et al.  mentioneoduction . Karst coduction . On 16 STroglonectes is described herein based on its morphological characteristics and molecular analysis. Additionally, the phylogenetic tree indicates species of Troglonectes divided into two sub-clades, viz. the truncated caudal fin sub-clade and emarginated or forked caudal fin sub-clade.One new species of http://zoobank.org/ (accessed on 10 February 2023).This published work and the nomenclatural acts it contains have been registered in ZooBank LSIDs (Life Science Identifiers) and can be resolved, and the associated information can be viewed through any standard web browser by appending the LSID to the prefix Publication LSID:urn:lsid:zoobank.org:pub:5DB60B6B-94EC-4E18-B734-9258F9E31D2A.Troglonectes canlinensis LSID:urn:lsid:zoobank.org:act:1FEADE1C-2EBF-4956-A052-F6926AB9124C."}
{"text": "T is sufficiently large, then uniformly for all positive integers \u2113\u2a7d(logT)/(log2T), we have|\u03b6(\u2113)(\u03c3+it)| when \u2113\u2208N and \u03c3\u2208[1/2,1) are\u00a0fixed.It is proved that if  Recall that Littlewood . In this case we obtain maxT/2\u2a7dt\u2a7dT|\u03b6(\u2113)(12+it)|\u2a7eexp{(1+o(1))logT/log2T}, losing a log3T factor compared to the above result on the longer interval .By Soundararajan's original resonance method , we can \u03b6(1+it)=\u03a9(log2t) (see  denotes the least common multiple of m and n.Let \u03c3=1 was studied by G\u00e1l , \u03c3\u2208,Let \u03b6(s) and Cauchy's integral formula.\u25a1It follows from classical convex estimates for In the following, we will derive a \u201cdouble version\u201d convolution formula, similar to Lemma of 5.3 of de la Bret\u00e8che and Tenenbaum . The proK^ of K asLemma 3\u2113\u2208N and \u03c3\u2208, satisfying the growth conditiont\u2208R\u2216{0}, thenLet whereandDefine the Fourier transform h(z):=F\u2113(z+it)F\u2113(z\u2212it)K(i\u03c3\u2212iz). h(z) is a meromorphic function in the vertical strip \u03c3\u2a7d\u211c(z)\u2a7d2, with two poles, namely, at z=1+it and z=1\u2212it. Let Y be large and consider straight line integrals for h(z). Set J1=\u222b\u03c3\u2212iY2\u2212iYh(z)dz,J2=\u222b2\u2212iY2+iYh(z)dz,J3=\u222b2+iY\u03c3+iYh(z)dz,J4=\u222b\u03c3+iY\u03c3\u2212iYh(z)dz.Define F\u2113(s)=\u2113!/(s\u22121)\u2113+1+E(s), where E(s) is an entire function. The residue theorem gives thatNote that By (2) and (9) and applying Cauchy's theorem term by term,Clearly,F\u2113(s)\u226a1+|\u03b6(\u2113)(s)|, estimates for \u03b6(\u2113)(s)  Lemma\u00a0 and 9),,F\u2113(s)\u226a1+\u03b5=1/6, then J1\u226a1/(YlogY)\u21920, as Y\u2192\u221e. Similarly for J3.\u25a1Take Lemma 4 Let \u03bc,\u03bd\u2208N and \u03b11,\u03b12\u2208 be fixed. Suppose \u03b11+\u03b12>1, thenThe following results are due to Hadamard, Landau, and Schnee .Lemma Remark 4o\u201d are under the condition when the corresponding variable tends to infinity. Namely, as T\u2192\u221e, t\u2192\u221e, or N\u2192\u221e.In this paper, if it is not stated, the limit notations \u201c\u223c\u201d and \u201c\u2113\u2208N and \u03c3\u2208 are fixed, one hasIn particular, when \u03c3=12, Ingham [A\u2033] has proved the following result on second moments of \u03b6(\u2113)(s).Lemma 5(Ingham) Let \u2113\u2208N be fixed. ThenFor 3We will use the construction of Bondarenko and Seip in .\u03b4=\u2113\u00b7(\u2113+1)\u22121. Given a positive number y and a positive integer b, defineLet x and an integer b later to make P\u2a7dT. Let M be the set of divisors of P and M\u03b4 be the set of divisors of P. Let M\u03b4\u00af be the complement of M\u03b4 in M. Note that both M and M\u03b4 are divisor\u2010closed which means k|n,n\u2208M\u21d2k\u2208M and k|n,n\u2208M\u03b4\u21d2k\u2208M\u03b4. Define the function r:N\u2192{0,1} to be the characteristic function of M, thenWe will choose a number As showed in ,1|M|\u2211nAlso in . By the prime number theorem, P\u2a7dT when T is sufficiently large. Take the choices of x,b and \u03b4=\u2113\u00b7(\u2113+1)\u22121 into the above inequality, then we are done.\u25a1By the definition of 4N=[T12] and let R(t):=\u2211n\u2a7dNr(n)n\u2212it. Define the moments as follows:Set \u03a6:R\u2192R denotes a smooth function, compactly supported in , with 0\u2a7d\u03a6(y)\u2a7d1 for all y, and \u03a6(y)=1 for 5/4\u2a7dy\u2a7d7/4. Partial integration gives that \u03a6^(y)\u226a\u03bd|y|\u2212\u03bd for any positive integer \u03bd.As in , \u03a6:R\u2192R dAlso in , SoundarSince \u03a6 is compactly supported in , we deduce thatN\u2a7dT12, for the off\u2010diagonal terms km\u2260n we have \u03a6^(Tlog(km/n))\u226aT\u22122, by the rapid decay of \u03a6^ \u00a0N=[T12], we obtainAgain, by O(\u00b7) term is absolute:By Lemma\u00a0M2, the big O(\u00b7) term above contributes at mostIn the integral of Combining this with , we haveFinally, the above formula together with (18) gives that\u03b5=(log2T)\u22121. By Stirling's formula, if T is sufficiently large, then for all positive integers \u2113\u2a7d(logT)(log2T)\u22121, we have \u00a0\u2113!\u00b7\u03b5\u2212\u2113\u00b7T\u22121+\u03b5\u2a7d(log2T)\u2113. Other big O(\u00b7) terms can be easily bounded. Together with Proposition\u00a0\u25a1Now let 55.1M of positive integers and a parameter T, we will construct a resonator R(t), following ideas from , an, anM of eas from . DefineBy Cauchy's inequality, one has the following trivial estimates :R(0)2\u2a7d\u03a6(t):=e\u2212t2/2. Its Fourier transform satisfies \u03a6^(\u03be)=2\u03c0\u03a6(\u03be).As in  , set \u03a6. Choose \u03ba\u2208 and set N:=[T\u03ba]. Fix \u03b5>0 such that \u03ba+4\u03b5<1.Let As in , chooseDefine2T\u03b2\u2a7d|t|\u2a7dT2 and |y|\u2a7d|t|2 gives the main term for I(T). We will frequently use the following trivial estimates |\u2a7dR(0) and \u03a6(\u00b7)\u2a7d1,A simple computation givesThe fast decay of \u03a6 and (23) give thatUsing (21) and (23), one can computeCombining the above estimates, one gets2T\u03b2\u2a7d|t|\u2a7dT2 and |y|\u2a7d|t|2 give T\u03b2\u2a7d|t\u00b1y|\u2a7dT.\u00a0 Again, by (21)I2(T),I3(T) as follows:K, we have the following estimates for all 0\u2a7dn\u2a7dl:Note that 0\u2a7dm\u2a7dl, one hasAnd trivially, for all  satisfying m+n=\u2113, soI2(T), so we get (Note that there are finitely many non\u2010negative integer pairs o we get .I1(T) to the GCD sums, we would like to use Fourier transform on the whole real line. So setK^(lognm)=0 if mn\u2a7eT2\u03b5. Clearly, al(n)/n\u03c3\u226a1. So one can getI1(T)=I1\u223c(T)+|M|\u00b7O(T\u03ba+4\u03b5). Thus we haveNext, in order to relate I1\u223c(T) by expanding the product of the resonator and the infinite series of G\u03c3(t), and then integrate term by term, as in , soIn this case, let M,Also, in , p. 128,o(1). And clearly, the big O(\u00b7) terms in (32) can be ignored. ThusSo the second term on the right\u2010hand side of (32) is Case 2: \u03c3\u2208.M be the set in (6) with |M|=N. Again, N=[T\u03ba], soIn this case, let Similarly as (31), we haveAnd by (4), we can get the following estimates:Hence,B).\u25a1Make \u03ba slightly larger in the beginning then one can get (6(m\u2297n) for the ordered pair of m and n.The idea of the proof is basically the same as in the proof of Proposition\u00a0P\u223c=p1b\u22121\u00b7\u2026\u00b7prb\u22121\u00a0, where pn denotes the nth prime. Define M to be the set of divisors of P\u223c, then |M|=br. By G\u00e1l's identity , then pr\u223clogN by the prime number theorem. Let b be the integer satisfying thatbr\u223cN, as N\u2192\u221e. Choose a set M\u2032\u2282N such that M\u2282M\u2032 and |M\u2032|=N.Let (e\u03b3logpr)2\u223c(e\u03b3loglogN)2 as N\u2192\u221e. The second product converges as N\u2192\u221e toFollowing Lewko\u2013Radziwi\u0142\u0142\u00a0\u00a0in , we use \u03b4=\u2113\u00b7(2\u2113+1)\u22121 and define the sets M\u03b4(1),M\u03b4(2) as follows:Next, let M\u03b4(1):={(m\u2297n)\u2208M\u00d7M|\u2200i>r\u03b4,\u03b1i=min{\u03b1i,\u03b2i}, \u00a0\u00a0where m and n have prime factorizations as \u00a0\u00a0\u00a0 m=p1\u03b11p2\u03b12\u22efpr\u03b1r,andn=p1\u03b21p2\u03b22\u22efpr\u03b2r}.M\u03b4(2):={(m\u2297n)\u2208M\u00d7M|\u2200i>r\u03b4,\u03b2i=min{\u03b1i,\u03b2i}, \u00a0\u00a0where m and n have prime factorizations as \u00a0\u00a0\u00a0 m=p1\u03b11p2\u03b12\u22efpr\u03b1r,andn=p1\u03b21p2\u03b22\u22efpr\u03b2r}.M\u03b4 to be the union of the above two sets and M\u03b4\u00af to be the complement of M\u03b4 in M\u00d7M:Then define Now we split the GCD sum into two parts:By symmetry, we haveM\u03b4(1) and G\u00e1l's identity, we haveBy the definition of br\u223cN, the first product converges to 6/\u03c02, and the third product is asymptotically equal to (e\u03b3logpr)2\u223c(e\u03b3loglogN)2 as N\u2192\u221e.Again, we have For the second product, it can be bounded asAs a result, we obtain thatHence by , 39), a, a\u2211\u2208M\u03b4\u00af, thenBy the construction of Thus\u03b4=\u2113\u00b7(2\u2113+1)\u22121, we are done.\u25a1By our choice of 7\u03b6(\u2113)(1+it)=\u03a9((log2t)\u2113+1), when \u2113\u2208N is\u00a0fixed.One can use the method of Bohr\u2013Landau \u2a7ecos(2\u03c0/q) for all integers n\u2208. HenceWrite q=8 to getTake One can compute thatN thatand for large A (only depending on \u2113) such that (\u2113+1)A\u2113\u00b7e\u2212A<1/12 and let \u03c3\u22121=A/logN. Combining with (41) gives thatNow fix a positive constant \u03b6(\u2113)(1+it)\u2260\u03a9((loglogt)\u2113+1). So f(1+it)=o(1). Clearly, f(2+it)=o(1). Then we get a contradiction with  and \u03c3\u2208R, define the following normalized log\u2010type GCD sums as:Problem 1\u0393\u03c3(\u2113)(N).Given \u03c3 and \u2113, optimize Let Remark 5\u03c3=1. Given \u2113, what is the optimal constant C\u2113 such that \u03931(\u2113)(N)\u2a7dC\u2113(log2N)2\u2113+2\u00a0? (See , or \u2113=[2T]?In our Theorem Problem 6Can one find some range for \u2113, such that the results in Theorem Remark 9a\u2113(n)\u2a7e1 for all n and \u2113. It is not clear about the moments of derivatives of the zeta function if \u2113 can depend on T. For instance, if we let \u2113=[(logT)(log2T)\u22121], then what can we say about the second moments as T\u2192\u221e,T, it also seems difficult to bound the contributions of \u0394++\u0394\u2212 .The main terms always satisfy since we have |\u03b6(\u2113)(\u03c3+it)| can be if \u2113 can be taken arbitrary large with respect to the length T of the interval .Problem 7\u03c30\u2208, decide which one of following four properties can be\u00a0true.Given Moreover, we have the following general problem, which asks how large or how small the extreme values of (A) Given any function V:\u2192, there always exists some function fV:\u2192N such that if \u2113=fV(T), then for sufficiently large T, we have(B) Given any function V:\u2192, there always exists some function fV:\u2192N such that if \u2113=fV(T), then for sufficiently large T, we have(C) There exists some function V:\u2192, such that for all function f:\u2192N, if \u2113=fV(T), then for sufficiently large T, we have(D) There exists some function V:\u2192, such that for all function f:\u2192N, if \u2113=fV(T), then for sufficiently large T, we haveMathematika is owned by University College London and published by the London Mathematical Society. All surplus income from the publication of Mathematika is returned to mathematicians and mathematics research via the Society's research grants, conference grants, prizes, initiatives for early career researchers and the promotion of\u00a0mathematics."}
{"text": "C is a closed, convex subset of a separable Banach space, C is Haar null if and only if C is Haar meagre. We then use this fact to improve a theorem of Matou\u0161kov\u00e1 and to solve a conjecture proposed by Esterle, Matheron and Moreau. Finally, we apply the main theorem to find a characterisation of separable Banach lattices whose positive cone is not Haar\u00a0null.Haar null sets were introduced by Christensen in 1972 to extend the notion of sets with zero Haar measure to nonlocally compact Polish groups. In 2013, Darji defined a categorical version of Haar null sets, namely Haar meagre sets. The present paper aims to show that, whenever  A Borel set E\u2282G is Haar null if one can find a Borel probability measure \u03bc on G such that \u03bc(gEh)=0 for all g,h\u2208G. The measure \u03bc is said to witness that E is Haar null and can always be assumed to have compact support, as every Borel probability measure on a Polish space is inner regular. This definition was given for the first time in Abelian Polish groups by Christensen in  and andG be E in a Polish group G is Haar positive and which can be expressed as follows: for every compact set K\u2282G, there are an open set V\u2286G and g,h\u2208G such that K\u2229V\u2260\u2300 and K\u2229V\u2282gEh. Indeed, suppose that E enjoys this property, let \u03bc be any Borel probability measure on G with compact support K and find an open set V\u2286G and g,h\u2208G such that K\u2229V\u2260\u2300 and K\u2229V\u2282gEh. Clearly, \u03bc(K\u2229V)>0, because K is the support of \u03bc, hence \u03bc(gEh)>0. Sets with this property are called compactivorous, can be defined in any Hausdorff topological group and were mentioned for the first time in . We havex\u2228y\u2208DX+ and, by property (2), (1\u2212t)x+ty\u2208DX+. This shows that DX+ is convex.\u25a1Pick X is called an AM\u2010space if, for every x,y\u2208X+, the equality ||x\u2228y||X=max{||x||X,||y||X} holds. Our last result states that, up to lattice isomorphisms, only separable AM\u2010spaces have nonnegligible positive cones.Theorem 4.2X be a separable Banach lattice. The following assertions are equivalent.(1)X+ is Haar positive.(2)x=(xn)n=1\u221e\u2208c0(X), there exists z\u2208X+ such that z\u2a7e|xn| for each n.For every sequence (3)k>0  such that for every integer n\u2a7e1 and any x1,\u22ef,xn\u2208X+, one hasThere is (4)Y and a lattice isomorphism T:X\u2192Y.There are an AM\u2010space Let Recall that a Banach lattice \u21d4 (2) Suppose that X+ is Haar positive and pick x=(xn)n=1\u221e\u2208c0(X). By Theorem\u00a0X+ is not Haar meagre and therefore compactivorous, hence there must be z\u2208X such that z+\u03a3(x)\u2282X+. This implies that, for each n, we have z\u2a7exn and z\u2a7e\u2212xn, hence z\u2a7exn\u2228(\u2212xn)=|xn|. Conversely, suppose that (2) holds and pick again x=(xn)n=1\u221e\u2208c0(X). Find z\u2208X+ such that z\u2a7e|xn| for each n. Then z+\u03a3(x)\u2282X+. This shows that X+ is compactivorous, therefore Haar\u00a0positive.(1) \u21d4 (3) Suppose that X fulfils assertion (3) and let k>0 be such thatn\u2a7e1 and any x1,\u22ef,xn\u2208X+. Fix x=(xn)n=1\u221e\u2208c0(X). Define the sequence y=(yn)n=1\u221e recursively in the following way: y1=|x1| and yn+1=yn\u2228|xn+1| for every n. Observe that y is an increasing sequence in X+. Choose \u03b5>0 and let n0 be such that ||xn||X<\u03b5/k for all n\u2a7en0. For every n and m with n>m\u2a7en0, we havey is a Cauchy sequence and it has therefore a limit z\u2208X+. Clearly, z\u2a7eyn for every n, thus z\u2a7e|xn| for every n. On the other hand, assume by contradiction that for each positive integer k there are xk,1,\u22ef,xk,n(k)\u2208BX\u2229X+ such that ||xk,1\u2228\u22ef\u2228xk,n(k)||X\u2a7ek2. Let y=(yn)n=1\u221e\u2208c0(X) be the sequence whose elements are x1,1,\u22ef,x1,n(1), followed by 2\u22121x2,1,\u22ef,2\u22121x2,n(2), followed by 3\u22121x3,1,\u22ef,3\u22121x3,n(3) and so on. If there were z\u2208X+ such that z\u2a7eyn for all n, then for each k it would hold that z\u2a7ek\u22121xk,1\u2228\u22ef\u2228k\u22121xk,n(k). Thus, for each k,(2) \u21d4 (4) Assume that (4) holds, that is, that there are an AM\u2010space Y and a lattice isomorphism T:X\u2192Y. Set k=||T\u22121||BL||T||BL and choose an integer n\u2a7e1 and x1,\u22ef,xn\u2208X+. ThenX which turns X into an AM\u2010space. If (3) holds, then it follows from the definition of DX+ and from the properties of lattice norms that ||x||X\u2a7dk for every x\u2208DX, which implies BX\u2286DX\u2286kBX. DX is symmetric, as |x|=|\u2212x| for every x\u2208X. Moreover, DX is convex. Indeed, if x,y\u2208DX and t\u2208, then |x|,|y|\u2208DX+ and |(1\u2212t)x+ty|\u2a7d(1\u2212t)|x|+t|y|. By the convexity of DX+ and by (2) in Lemma\u00a0|(1\u2212t)x+ty|\u2208DX+, hence (1\u2212t)x+ty\u2208DX. Let \u03bd be the Minkowski functional associated to the set DX. That is, \u03bd(x)=inf{\u03bb>0:x\u2208\u03bbDX} for every x\u2208X. Since BX\u2286DX\u2286kBX, \u03bd defines an equivalent norm on X. Moreover, one notices easily that \u03bd(x)=\u03bd(|x|) for every x\u2208X. To see that \u03bd is indeed a lattice norm, pick x,y\u2208X such that |x|\u2a7d|y|. For every \u03b5>0, we have|x|\u2208\u03bd(y)+\u03b5DX+. As \u03b5 is arbitrary, it follows that \u03bd(x)=\u03bd(|x|)\u2a7d\u03bd(y), as wished. Now, let x,y\u2208X+ and set m=max{\u03bd(x),\u03bd(y)}. For every \u03b5>0, we have x,y\u2208(m+\u03b5)DX+. By (1) in Lemma\u00a0x\u2228y\u2208(m+\u03b5)DX+. Since \u03b5 is arbitrary, this allows to conclude that \u03bd(x\u2228y)\u2a7dm. The opposite inequality is true in every Banach lattice, therefore \u03bd turns X into an AM\u2010space.\u25a1(3) Remark: The proof of the equivalence between (2), (3) and (4) in Theorem\u00a0Bulletin of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not\u2010for\u2010profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of\u00a0mathematics.The"}
{"text": "Correction: Fluids Barriers CNS 20, 20 (2023)10.1186/s12987-023-00421-8Following publication of this article , the autThese errors were introduced during typesetting.The correct values are given below and the original article has been corrected.The publisher would like to apologise for any inconvenience caused.p = 2\u2009\u00d7\u200910\u2212\u200911\u00a0m Pa\u2212\u20091\u00a0s \u2212\u20091 to 3\u2009\u00d7\u200910\u2212\u200910\u00a0m Pa\u2212\u20091\u00a0s \u2212\u20091, diffusion membrane coefficients for small solutes in the range CM = 5\u2009\u00d7\u2009102\u00a0m\u2212\u20091 to 6\u2009\u00d7\u2009103\u00a0m\u2212\u20091 and gap area fractions in the range 0.2\u20130.6%, based on a inter-endfoot gap width of 20\u00a0nm.We provide structural-functional relationships between vessel radius and resistance that can be directly used in flow and transport simulations. We estimate endfoot sheath filtration coefficients in the range L"}
{"text": "In the published article, there were errors in affiliations 1 and 2. Instead of1Positive Psychology Research Group, Institute of Psychology, Faculty of Education and Psychology, E\u00f6tv\u00f6s Lor\u00e1nd University, Budapest, Hungary\u201d and\u201c\u201c2Doctoral School of Psychology, ELTE E\u00f6tv\u00f6s Lor\u00e1nd University, Budapest, Hungary\u201d,it should be1Doctoral School of Psychology, ELTE E\u00f6tv\u00f6s Lor\u00e1nd University, Budapest, Hungary\u201d and\u201c2Institute of Psychology, ELTE E\u00f6tv\u00f6s Lor\u00e1nd University, Budapest, Hungary\u201d.\u201cAttila Ol\u00e1h should be affiliated to \u201cInstitute of Psychology, ELTE E\u00f6tv\u00f6s Lor\u00e1nd University, Budapest, Hungary\u201d.Andr\u00e1s Vargha should be affiliated to \u201cInstitute of Psychology, ELTE E\u00f6tv\u00f6s Lor\u00e1nd University, Budapest, Hungary\u201d and \u201cPerson- and Family-oriented Health Science Research Group, Institute of Psychology, Faculty of Humanities and Social Sciences, K\u00e1roli G\u00e1sp\u00e1r University of the Reformed Church in Hungary, Budapest, Hungary\u201d.Additionally, in the published article, there was an error in the Appendix/The Mental Health Test (MHT). Item 16 of the questionnaire was incorrectly written as \u201cI easily become impatient./SR\u201d. The corrected Item 16 appears below.(16) I become frustrated when something does not happen the way I planned it./SRThe authors apologize for these errors and state that they do not change the scientific conclusions of the article in any way. The original article has been updated."}
{"text": "Multiword expressions (MWEs) are sequences of words that pose a challenge to the computational processing of human languages due to their idiosyncrasies and the mismatch between their phrasal structure and their semantics. These idiosyncrasies are of lexical, morphosyntactic and semantic 11 nature, namely: non-compositionality, i.e., the meaning of the expression cannot be computed from the meanings of its constituents; discontinuity, i.e., alien elements may intervene; non-13 substitutability, i.e., at least one of the expression constituents is lexicalized and therefore, does not enter in alternations at the paradigmatic axis; and non-modifiability, in that they enter in syntactically 15 rigid structures, posing further constraints over modification, transformations, etc. The paper presents a model for representing MWEs at the level of semantics by taking into account all these inherent idiosyncrasies. The model assumes the form of a linguistic ontology and is applied to Greek verbal multi-word expressions (VMWEs); moreover, the semantics of the lexical entries under scrutiny is also represented via the semantics of their arguments based on corpus evidence. In this regard, modeling the semantics of VMWEs is placed in the lexicon-corpus interface. MWEs are highly idiosyncratic structures Gross,  and thusvia encoding the lexical semantic relations between a VMWE and other single- or multi-word entries; and (d) at the syntagmatic axis, by modeling the semantics of their arguments based on corpus evidence. In the latter case, the VMWE is taken as a whole, that is, as a complex predicate. Our goal is to treat both single- and multi-word entries in a comparable way that would be useful for Natural Language Processing (NLP) applications.Our work seeks to fill this gap by proposing a model for encoding the semantic properties of VMWEs into a lexical resource by considering all the idiosyncrasies they exhibit. The semantics of VMWEs are thus defined along the following axes: (a) the type of VMWE in terms of the degree of compositionality, (b) their mapping onto concepts or word senses already existing in an inventory, that is, a semantic lexical resource already available; (c) at the paradigmatic axis, Most Lexical Resources (LRs) dedicated to MWEs give an account only of their lexical, morphological, and syntactic idiosyncrasies. Within the Lexicon-Grammar framework, the pioneering work of Gross  toward tQualia Structure, SIMPLE lexica encode structured semantic types and semantic (subcategorization) frames. A few years later, the Brandeis Semantic Ontology (BSO) seeks to extend the English SIMPLE lexicon  is one of the earliest corpora annotated with semantic roles , marked also for fixedness. Possible syntactic properties  are also encoded at this level. In the next sections, we elaborate on the encoding at the level of semantics. Our model provides mechanisms for encoding diathesis alternations, register, and for signaling MWEs that have a literal  besides their idiomatic one, as defined in Savary et al. , and syntactic properties at the SIGNIFIER level has been extensively presented in Fotopoulou et al. . Accorditopoulou  and Minitopoulou . Accordiy et al. .via semantic relations, and (d) identifying their arguments and the roles they assume. We will elaborate on the model itself in the next paragraphs.The semantic representation of lexical items\u2014both single- and multi-word ones\u2014is achieved at the SIGNIFIED level, taking into account the following aspects: (a) coarse classification that reflects their degree of compositionality; (b) mapping onto word senses or concepts; (c) linking with other entries verbal idiomatic expressions (VIDs), that bear a meaning that cannot be computed based on the meaning of their constituents and the rules used to combine them; (b) light verb constructions (LVCs), i.e., expressions with a rather transparent meaning; (c) multi-verb constructions (MVCs), that is, expressions with coordinated lexicalised head verbs ; and (d) verb-particle constructions (VPCs) comprising a verb and one of the adverbs \u03bc\u03c0\u03c1o\u03c3\u03c4 \u03ac (=in front), \u03c0\u00ed\u03c3\u03c9 (=back), \u03c0 \u03ac\u03bd\u03c9 (up), \u03ba \u03ac\u03c4\u03c9 (=down), \u03bc \u03ad\u03c3\u03b1 (=in), \u03ad\u03be\u03c9  in Greek; these adverbs are not morphologically derived from adjectives and exhibit most - if not all - of the properties particles in other languages have Giouli et al.  \u03c0 \u03ad\u03d5\u03c4\u03c9 \u03bc \u03ad\u03c3\u03b1lit. fall1\u2212sg in (2) \u03c0 \u03ad\u03d5\u03c4\u03c9 \u03ad\u03be\u03c9lit. fall1\u2212sg out (=to get bankrupt)In terms of meaning, the classification in the afore-mentioned classes is a first step toward defining their semantics: VIDs and MVCs are non-compositional, LVCs are semi-compositional, in that they have a transparent meaning which is retained by the predicative noun, whereas VPCs present semantic ambiguity. Of course, other dimensions exist along which these different types of VMWEs can also be compared, namely, non-modifiability, and non-substitutability. In this regard, VIDs, VPCs and MVCs are syntactically rigid structures posing constraints with respect to modification, syntactic transformations, or other alternations at the paradigmatic axis etc., as opposed to the more flexible LVCs.At the next level, the semantic representation of VMWEs makes use of the SIGNIFIED branch of our ontology, and each VMWE  is mapped onto a concept. In our model lexicon, concepts are treated as instances under hierarchically organized (sub-)classes; these sub-classes are themselves subsumed under a set of top-level classes (or top ontology) and roughly correspond to the notion of semantic or lexical fields Lyons, , p. 268.via lexical semantic relations: synonymy, near-synonymy, antonymy; similarly, concepts are also linked together via semantic relations as appropriate: hypernymy-hyponymy or is_a relation, meronymy, etc. Apart from the standard lexical semantic relations, other relations are also included: entailment (entails), causation (causes), temporal order (happens_before), etc. Moreover, relations that link together words and/or concepts that belong to different grammatical categories have been defined in the resource. For example, relations of the type is_the_agent_of, feels_emotion, is_cogniser, etc. link together concepts instantiated by verbs denoting an activity, an emotion or a cognitive state and concepts instantiated by nouns denoting the actor, the experiences of the cognitive agent, etc. Similarly, relations that link together adjectives with adverbs have been used. In total, more than 100 relations have been employed so far; some of them are generic in that they are relative to more than one semantic field , whereas others are domain-specific. Examples of the latter category include \u2013 but are not limited to \u2013 the following: Is_made_of, Is_located_in, Works_in, Is_workplace_of, Has_Habitat, Is_the_Inhabitant_of, Causes, Is_the_result_of, Has_color, Is_the_color_of, Is_payment_to, Is_payment_for, Wears_garment, etc. Contrary to resources like SIMPLE  in the lexicon.Lexical entries (words) are then linked together emotion event, relative to the emotion love. It is mapped, therefore, onto the concept prototypically defined for the single-word entry \u201c\u03b5\u03c1\u03c9\u03c4 \u03b5 \u03cdo\u03bc\u03b1\u03b9\u201d .However, mapping words to concepts already defined in the lexicon is not an easy task. This is especially true for VMWEs. Moreover, in many cases, concepts already defined for single-word entries in the lexicon are perceived of as more general or neutral and only roughly correspond to the meaning load that VMWEs bear. For example, the VMWE \u03b4\u03b1\u03b3\u03ba\u03ce\u03bd\u03c9 \u03c4\u03b7 \u03bb\u03b1\u03bc\u03b1\u03c1\u00ed\u03bd\u03b1 in (3) denotes an (3) \u03b4\u03b1\u03b3\u03ba\u03ce\u03bd\u03c9 \u03c4\u03b7 \u03bb\u03b1\u03bc\u03b1\u03c1\u00ed\u03bd\u03b1lit. bite-1SG the panel\u2212SG.ACC has_troponym links a concept that bears a more \u201cgrounded\u201d or neutral sense with another one that signifies a shift in terms of quantity, intensity, quality, etc. For example, the verbs \u03b3 \u03bd\u03c9\u03c1\u00ed\u03b6\u03c9 (=to know) and \u03be \u03ad\u03c1\u03c9 (=to know) are both lexicalizations of the concept [to know]; on the contrary, the VMWE \u03c0\u03b1\u00ed\u03b6\u03c9 \u03c3\u03c4\u03b1 \u03b4 \u03ac\u03c7\u03c4\u03c5\u03bb\u03b1 (= \u03b3\u03bd\u03c9\u03c1\u00ed\u03b6\u03c9 \u03c0o\u03bb \u03cd \u03ba\u03b1\u03bb \u03ac) is mapped onto the concept [to know well]. The two concepts are then linked via the troponymy relation:Note, however, that the two lexical instances are not absolute synonyms in the sense that there are subtle differences in terms of the intensity of the emotion experienced. As a matter of fact, VMWEs are rarely exact synonyms of a single verbal predicate. Within this context, the major challenge faced was to account for these fuzzy cases and find out ways for capturing the semantic distance. To overcome this issue and represent differences in meaning, near synonymous entries are also linked using relations, both generic and specific for each semantic class. More precisely, the generic relation has_troponym(4) has_troponym relation does not reflect this difference. To remedy this shortcoming, a list of attributes (or semantic features) with either binary or scalar values have also been defined for better representing the underlying meaning. In most cases, these attributes are specific to semantic fields. For example, lexical units that belong to the semantic field emotion are assigned values for the following attributes: (a) emotion polarity, (b) emotion intensity and (c) aspect of the emotion event. In effect, these features better account for capturing the semantic distinction between near synonyms, as for example the single word verbal predicate \u03d5o\u03b2 \u03ac\u03bc\u03b1\u03b9 (=to be scared), and the VMWE in (5).Troponymy, however, is not a semantically homogenous relation  \u03bco\u03c5 \u03ba\u00f3\u03c0\u03b7\u03ba\u03b1\u03bd \u03c4\u03b1 \u03ae\u03c0\u03b1\u03c4\u03b1lit.me.01SG.GEN were-cut.03 the livers.PL.NOM fear emotion event that is more intense than the emotion conveyed by the single word; thus, the two predicates can hardly be encoded as being synonyms in the lexicon. Their semantic distance is captured by encoding them as related via the has_troponym relation, and the semantic distinction is highlighted by assigning the attribute high to the feature Intensity. This brings in mind the mechanism of Lexical Functions proposed by Mel'\u010duk , comprises the lexicalised elements \u03c0\u03b1\u00ed\u03c1\u03bd\u03c9.v (=to take) and \u03c7\u03b1\u03bc\u03c0 \u03ac\u03c1\u03b9.n (=notice). Since the semantic load of the expression is on the noun, the expression is classified as LVC. The underlying syntactic configuration of the expression is that of a verb head that is light, and its complement (direct object). This configuration is compatible with the argument structure of the verb \u03c0\u03b1\u00ed\u03c1\u03bd\u03c9.v . However, the whole expression as a lexical unit assumes the meaning of a cognitive event, and as such, it is conceived of as a predicate with two arguments: the first assumes the role of the CCOGNISER \u03c0 \u03ae\u03c1\u03b5 \u03c7\u03b1\u03bc\u03c0 \u03ac\u03c1\u03b9 [\u03c4\u03b7\u03bd \u03b1\u03bb\u03bb\u03b1\u03b3 \u03ae]THEME(6) [O \u0393\u03b9\u03ac\u03bd\u03bd\u03b7\u03db]lit. The-NOM.SG John\u2212NOM.SG took\u22123SG notice\u2212ACC.SG the ACC.SG change ACC.SGJohn realized the changeThe semantic representation of the VMWE is expected to be similar to the representation of its single-word verbal counterpart \u03ba\u03b1\u03c4\u03b1\u03bb\u03b1\u03b2\u03b1\u00ed\u03bd\u03c9.v  as shown in (7):COGNISER \u03ba\u03b1\u03c4 \u03ac\u03bb\u03b1\u03b2\u03b5 [\u03c4\u03b7\u03bd \u03b1\u03bb\u03bb\u03b1\u03b3 \u03ae]THEME(7) [O \u0393\u03b9\u03ac\u03bd\u03bd\u03b7\u03db]lit.The-NOM.SG John-NOM.SG noticed the-ACC.SG change-ACC.SGJohn realized the change.o\u03c1\u03b3\u00ed\u03b6\u03c9.v (=make furious) in Greek is an Object Experiencer verb that is, a verb in which the Experiencer of the denoted emotion event is realized as a noun phrase in accusative in Object position. The cause of the event is realized as an argument, that functions as the Subject of the verb. On the contrary, in the case of the idiomatic expression (VID) \u03b1\u03bd\u03b5\u03b2 \u03ac\u03b6\u03c9 \u03c4o \u03b1\u00ed\u03bc\u03b1 \u03c3\u03c4o \u03ba\u03b5\u03d5 \u03ac\u03bb\u03b9 (=make furious), the Experiencer is realized as a nominal complement (in genitive case), whereas the cause of the emotion is realized in Subject position:However, this is not always the case, and the argument structure of complex predicates is not realized in a uniform way. This is particularly true about VIDs. For example, the verb \u03b5\u03beCAUSE \u03bco\u03c5EXPERIENCER \u03b1\u03bd \u03ad\u03b2\u03b1\u03c3\u03b5 \u03c4o \u03b1\u00ed\u03bc\u03b1 \u03c3\u03c4o\u00a0\u03ba\u03b5\u03d5\u03ac\u03bb\u03b9(8) [O \u0393\u03b9\u03ac\u03bd\u03bd\u03b7\u03db]lit.The.nom John.nom me.gen raised3\u2212sg the.acc blood.acc to-the headJohn made me furious.o\u03cd\u03c7\u03b1 in (9) and \u03b2\u03b3\u03b1\u00ed\u03bd\u03c9 \u03b1\u03c0\u00f3 \u03c4\u03b1 \u03c1o\u03cd\u03c7\u03b1 \u03bco\u03c5 in (10) correspond to the transitive and unaccusative usage of the verb \u03b8\u03c5\u03bc\u03ce\u03bd\u03c9.v (=to make angry) depicted in (11) and (12) respectively.There is no doubt that SRL is of major importance to computational systems since it provides a shallow meaning representation that is prerequisite of inferences that are not possible from the pure surface form or even from the parse tree. This is especially true for VMWEs  [O \u0393\u03b9\u03ac\u03bd\u03bd\u03b7\u03db]lit. The.nom John.nom took-out3\u2212sg the.acc Maria.acc from the clothes hersJohn made Maria very angry)EXPERIENCER \u03b8 \u03cd\u03bc\u03c9\u03c3\u03b5(10) [H M\u03b1\u03c1\u00ed\u03b1]lit. The.nom Maria.nom got-angry3\u2212sgNotice that \u03b8\u03c5\u03bc\u03ce\u03bd\u03c9.v (=to make angry) is an Object-Experiencer predicate that enters the choative-inchoative alternation as shown below:CAUSE/AGENT \u03b8 \u03cd\u03bc\u03c9\u03c3\u03b5 [\u03c4\u03b7 M\u03b1\u03c1\u00ed\u03b1]EXPERIENCER(11) [O \u0393\u03b9\u03ac\u03bd\u03bd\u03b7\u03db]lit. The.nom John.nom made-angry3\u2212sg the.acc Maria.accJohn made Maria angry.EXPERIENCER \u03b2\u03b3 \u03ae\u03ba\u03b5 \u03b1\u03c0\u00f3 \u03c4\u03b1 \u03c1o\u03cd\u03c7\u03b1 \u03c4\u03b7\u03c2(12) [H M\u03b1\u03c1\u00ed\u03b1]lit. The.nom Maria.nom went-out3\u2212sg from the.acc clothes.acc hers.possMaria got very angry.In this regard, our model seeks to address these issues by assigning semantic roles to the arguments of the VMWEs. The encoding of semantic roles was based on empirical data retrieved from annotation.Annotation at the level of semantics has been applied manually on top of an existing Greek (EL) corpus that has already been annotated for VMWEs. More precisely, we used the latest version (edition 1.2) of the Greek (EL) section of the PARSEME corpus  have been encoded under the SIGNIFIED branch of our linguistic ontology and have been mapped onto concepts. However, the annotation has been performed only on a subset, that is, on c. 800 instances of VMWEs, that is, the ones that were also found in the PARSEME corpus. Of these, 379 instances were identified as VIDs, 7 as VPCs, and 425 as LVCs: in total, 811 VMWEs. Circa 10% of the VMMEs (80 VMWEs) were annotated by a second student annotator in view of calculating the inter-annotator agreement (IAA) between the two. Prior to the annotation proper, extensive discussions took place. A pilot annotation of c. 20 VMWEs identified problematic cases and discrepancies. After reaching a consensus in annotation, the second annotator worked alone annotating c. 80 VMWEs in 120 sentences. IAA was then calculated (Cohen-\u03ba) with respect to the number of arguments identified in each sentence, and the labels assigned to them, reaching an agreement of 0.80 and 0.75 respectively. In fact, VMWEs denoting As expected, SRL on VIDs was the most challenging. In fact, depending on the meaning SRL is occasionally straightforward:EXPERIENCER \u03ad\u03c7\u03b5\u03b9 \u03d5\u03ac\u03b5\u03b9 \u03c7\u03c5\u03bb\u00f3\u03c0\u03b9\u03c4\u03b1(13) [o \u03a0\u03ad\u03c1\u03b5\u03b8]lit. The\u2212NOM.SG Perez-NOM\u2212SG has\u22123SG eaten chilopita-ACC.SGPerez has been disappointed.o\u00ed\u03b3\u03c9 \u03c4o\u03c5\u03c2 \u03b1\u03c3\u03bao\u03cd\u03c2 \u03c4o\u03c5 A\u03b9\u00f3\u03bbo\u03c5 (=to open Aeolus bag) is semantically equivalent to the phrase \u201ccreate problems\u201d. However, only the AGENT is realized in the sentence:Problematic cases are related to a shift in meaning and the incorporation of one or more arguments into the VMWE, diathesis alternations, or difficulties in word sense identification and/or sense mapping. For example, the VID \u03b1\u03bdAGENT \u03ac\u03bdo\u03b9\u03be\u03b5 \u03c4o\u03c5\u03c2 \u03b1\u03c3\u03bao\u03cd\u03c2 \u03c4o\u03c5 A\u03b9\u00f3\u03bbo\u03c5 \u03c3\u03c4\u03b7 [M\u03ad\u03c3\u03b7 A\u03bd\u03b1\u03c4o\u03bb \u03ae]LOC(14) [O T\u03c1\u03b1\u03bc\u03c0]lit. The-NM.SG Trump-NM.SG opened-3.SG the-ACC.SG bag-ACC.SG of-the-GEN.SG Aeolos-GEN.SG in the Mid-EastTrump created problems in the Mid-Easto\u03bb \u03cd\u03c9 \u03c0\u03c5\u03c1 \u03ac (=unleash fire) to a single-word verb predicate proved to be difficult; as a result, disambiguation of their arguments proved to be problematic:Similarly, mapping the sense of VIDs like \u03b5\u03be\u03b1\u03c0AGENT \u03b5\u03be\u03b1\u03c0o\u03bb \u03cd\u03b5\u03b9 \u03c0\u03c5\u03c1\u03ac [\u03ba\u03b1\u03c4 \u03ac \u03c4\u03b7\u03c2 \u03ba\u03c5\u03b2\u03ad\u03c1\u03bd\u03b7\u03c3\u03b7\u03c2]THEME/GOAL [\u03b3\u03b9\u03b1 \u03c4o\u03c5\u03c2 \u03c7\u03b5\u03b9\u03c1\u03b9\u03c3\u03bco\u03cd\u03c2]CAUSE \u03c4\u03b7\u03c2(15) [H \u03b1\u03bd\u03c4\u03b9\u03c0o\u03bb\u00ed\u03c4\u03b5\u03c5\u03c3\u03b7]lit. The opposition\u2212NM.SG unleash-3.SG fire-ACC.SG against the-GEN.SG government-GEN.SG for the\u2212ACC.PL handlings-ACC.PL it's to-the issueThe opposition accuses/attacks the government for handling the issue.This process revealed pairs of VIDs usually with shared lexicalised elements that differ only in their fixed verb heads; these are conceived of as diathesis alternations and are encoded accordingly:AGENT \u03b2\u03b3 \u03ac\u03b6\u03b5\u03b9 \u03c3\u03c4o \u03c3\u03d5\u03c5\u03c1\u00ed [\u03c4o \u03b9\u03c3\u03c4o\u03c1\u03b9\u03ba\u00f3 \u03be\u03b5\u03bdo\u03b4o\u03c7\u03b5\u00edo]THEME(16) [H \u03c4\u03c1\u03ac\u03c0\u03b5\u03b6\u03b1]lit. The bank\u2212NOM.SG takes-3.SG to-the-ACC.SG hammer-ACC.SG the-ACC.SG hotel-ACC.SGThe bank auctions the historic hotelTHEME \u03b8\u03b1 \u03b2\u03b3o\u03c5\u03bd \u03c3\u03c4o \u03c3\u03d5\u03c5\u03c1\u00ed(17) [\u03c7\u03b9\u03bb\u03b9\u03ac\u03b4\u03b5\u03c2 \u03c3\u03c0\u00ed\u03c4\u03b9\u03b1]lit. thousands houses\u2212NOM.PL will go-out-3.PL to-the-ACC.SG hammer-ACC.SGthousands of houses will be sold at auctionWe have presented a model for representing the semantics of VMWEs by taking into account their inherent idiosyncrasies: lexical, syntactic and semantic. The model entails a holistic approach to VMWE representation and touches upon the lexicon-corpus interface beyond providing lexical semantic relations. In contrast to dictionary models that try to model the internal structure of the MWE, our approach models argument structure taking the whole MWE as a semantic predicate. We seek to provide a shallow semantic representation for VMWEs that is similar to the semantic representation of single-word verb predicates. The model assumes the form of a linguistic ontology and has already been used to encode Greek VMWEs. The encoding of semantic properties is based on empirical data drawn from a corpus annotated at the level of semantic role labeling. Future work is underway toward enriching the lexicon with more instances of VMWEs also taking into account MWEs that belong to other grammatical categories. Moreover, inter-linking entries with other lexical resources, as for example, WordNet synsets, would be the next step. Additionally, SRL on the PARSEME corpus is still ongoing with a view to training a tool for the automatic SLR that takes VMWEs into account. Moreover, the quality of the annotation will be further ensured by obtaining more annotations to calculate inter-annotator agreement.https://www.clarin.si/repository/xmlui/handle/11356/1555.Publicly available datasets were analyzed in this study. This data can be found here: The author confirms being the sole contributor of this work and has approved it for publication."}
{"text": "Arthrobotrys and Drehslerella based on morphological and multigene  phylogenetic analyses. A. hengjiangensis sp. nov. and A. weixiensis sp. nov. are characterized by producing adhesive networks to catch nematodes. Dr. pengdangensis sp. nov., Dr. tianchiensis sp. nov., and Dr. yunlongensis sp. nov. are characterized by producing constricting rings. Morphological descriptions, illustrations, taxonomic notes, and phylogenetic analysis are provided for all new taxa; a key for Drechslerella species is listed; and some deficiencies in the taxonomy and evolution study of nematode-trapping fungi are also discussed herein.Nematode-trapping fungi are widely studied due to their unique morphological structure, survival strategy, and potential value in the biological control of harmful nematodes. During the identification of carnivorous fungi preserved in our laboratory, five novel nematode-trapping fungi were established and placed in the genera  Arthrobotrys superba Corda) because of their unique survival strategy, excellent application potential in nematode control, and significance of maintaining the balance of nematode populations in the ecosystem , 2-septate conidia with tapering base, and branched conidiophores . In addition, the conidiophore of A. globospora bears only a single conidium, while the conidiophore of A. weixiensis bears 1\u20133 conidia [Notes: Phylogenetically, Drechslerella pengdangensis F. Zhang & X.Y. Yang sp. nov.  long, 2.5\u20135 \u00b5m (n = 50) wide at the base, gradually tapering upwards to the apex 2.5\u20134 \u00b5m (n = 50) wide, erect, septate, unbranched, and bearing a single conidium at the knob-like apex. Conidia: 30\u201345 \u00d7 17\u201327 \u00b5m (n = 50), ellipsoidal to subfusiform, rounded at the apex, tapering towards narrow with truncate at the base, 1\u20132-septate (mostly 2-septate), hyaline, with the largest cell located at the middle or apex of the conidia, where the base cell is tiny. Chlamydospore: not observed. Nematodes were captured with constricting rings; in the non-constricted state, the outer diameter is 19\u201328.5 \u00b5m (n = 50), and the inner diameter is 13\u201322.5 \u00b5m (n = 50).Additional specimen examined: CHINA, Yunnan Province, Nujiang City, Pengdang County, N 27\u00b056\u203216.88\u2033, E 98\u00b039\u20328.71\u2033, from terrestrial soil, 4 May 2018, F. Zhang. Living culture DL53.Drechslerella pengdangensis forms a sister lineage with another new species (Drechslerella tianchiensis) reported in this study, with 89% MLBS support. There are 15% (128/853 bp) differences in ITS sequence between them. Morphologically, Dr. pengdangensis can be easily distinguished from Dr. tianchiensis in the shape of the conidia and single conidiophore. Dr. pengdangensis is similar to Dr. doedycoides in their ellipsoidal to sub-fusiform conidia and simple conidiophore with knob-like apex [Dr. doedycoides produces 3-septate conidia, while Dr. pengdangensis never. Moreover, the base cell of conidia produced by Dr. pengdangensis is significantly smaller than those of Dr. Doedycoides [Notes: Phylogenetically, are 15% 18/853 bp are 15% 18/853 bp Drechslerella tianchiensis F. Zhang & X.Y. Yang sp. nov.  long, 2.5\u20135 \u00b5m (n = 50) wide at the base, gradually tapering upwards to the apex with 1.5\u20133 \u00b5m (n = 50) wide, erect, septate, hyaline, unbranched or producing 1\u20132 short branches near the apex, each branch bearing a single conidium. Microconidiophores 137.5\u2013245.5 \u00b5m (n = 50) long, 2\u20134 \u00b5m (n = 50) wide at the base, gradually tapering upwards to the apex with 1.5\u20133 \u00b5m (n = 50) wide, erect, septate, hyaline, unbranched, producing 3\u201312 short denticles near the apex, each denticles bearing a single conidium. Conidia two types: Maroconidia 30\u201341 \u00d7 14.5\u201324 \u00b5m (n = 50), ellipsoidal, rounded at the apex, tapering towards narrow with truncate base, 1\u20132-septate (mostly 2-septate), hyaline, with a largest cell located at the middle or apex of the conidia. Miroconidia 16\u201326.5 \u00d7 4.5\u201311.5 \u00b5m (n = 50), clavate or cylindrical, rounded at the apex, tapering towards narrow with truncate base, 0\u20131-septate (mostly 1-septate), hyaline. Chlamydospore not observed. Capturing nematodes with constricting rings, in the non-constricted state, the outer diameter is 20.5\u201327.5 \u00b5m (n = 50), the inner diameter is 14.5\u201322 \u00b5m (n = 50).Additional specimen examined: CHINA, Yunnan Province, Dali City, Yunlong County, Tianchi Nature Reserve, N 25\u00b051\u203222.50\u2033, E 99\u00b013\u203238.43\u2033, from burned forest soil, 28 May 2018, F. Zhang. Living culture XJ353.Drechslerella tianchiensis formed a sister lineage with Dr. pengdangensis (89% MLBS). Morphologically, Dr. tianchiensis is similar to Dr. hainanensis and the asexual morph of Orbilia pseudopolybrocha in their shape of macroconidia and microconidia. The difference between Dr. tianchiensis and Orbilia pseudopolybrocha is that the macro-conidiophore of the latter is simple and bears a single conidium, while some macro-conidiophore of Dr. tianchiensis produces 1\u20132 short branches near the apex and bears 1\u20132 conidia. The conidia of Dr. tianchiensis are significantly larger than those of O. pseudopolybrocha  \u00d7 14.5\u201324 (18.7) \u00b5m versus O. pseudopolybrocha, 26\u201330 \u00d7 16\u201322.2 \u00b5m) [Dr. tianchiensis can be easily distinguished from Dr. hainanensis by its 1\u20132-branch macro-conidiophore and wider microconidia  \u00d7 4.5\u201311.5 (6) \u00b5m versus Dr. hainanensis, 18.2\u201322.8 \u00d7 4.2\u20135.3 \u00b5m) [Notes: Phylogenetically, 22.2 \u00b5m) . Dr. tia\u20135.3 \u00b5m) .Drechslerella yunlongensis F. Zhang & X.Y. Yang sp. nov.  long, 2.5\u20135 \u00b5m (n = 50) wide at the base, gradually tapering upwards to the apex 1.5\u20133\u00b5m (n = 50) wide, erect, septate, unbranched, hyaline, bearing a single conidium at the apex. Conidia: 36\u201354 \u00d7 17\u201327 \u00b5m (n = 50), drop-shaped or fusiform, rounded at the apex, tapering towards narrow with truncate base, 1\u20134-septate (mostly 4-septate), hyaline, with the largest cell located at the apex or middle of the conidia. Chlamydospore: 5\u201314 \u00d7 5.5\u201310 \u00b5m (n = 50), cylindrical, globose or ellipsoidal, hyaline, and in chains when present. Nematodes were captured with constricting rings; in the non-constricted state, the outer diameter was 19.5\u201327 \u00b5m (n = 50), the inner diameter was 15\u201321.5 \u00b5m (n = 50).Additional specimen examined: CHINA, Yunnan Province, Dali City, Yunlong County, N 25\u00b052\u203227.91\u2033, E 99\u00b022\u203219\u2033, from terrestrial soil, 3 June 2018, F. Zhang. Living culture YL402.Drechslerella yunlongensis with the other four fusiform conidia-producing species . Dr. yunlongensis was 9.8% (55/559 bp), 8.1% (40/496 bp), 9.1% (51/559 bp), and 7.9% (47/596 bp) different from Dr. aphrobrocha, Dr. bembicodes, Dr. coelobrocha, and Dr. xiaguanensis in ITS, respectively. Morphologically, Dr. yunlongensis is also similar to these four species. However, the conidia of Dr. yunlongensis are bigger than those of Dr. bembicodes and Dr. xiaguanensis  \u00d7 17\u201327 (23.6) \u00b5m versus Dr. bembicodes, 36\u201343.2 (40) \u00d7 16.8\u201321.6 (20.5) \u00b5m versus Dr. xiaguanensis, 33\u201352 (42.5) \u00d7 9.5\u201328 (15.5) \u00b5m); moreover, Dr. bembicodes produces obovoid, 1-septate microconidia, while Dr. yunlongensis does not; the conidia of Dr. xiaguanensis are mostly 3-septate, while the conidia produced by Dr. yunlongensis are mostly 4-septate [Dr. yunlongensis and Dr. aphrobrocha is that Dr. aphrobrocha produces mostly 3-septate conidia, while Dr. yunlongensis produces mostly 4-septate conidia; the conidia of Dr. yunlongensis are smaller than that of Dr. aphrobrocha due to its smaller apical cell  \u00d7 17\u201327 (23.6) \u00b5m versus Dr. aphrobrocha, 40\u201357.5 (51) \u00d7 15.5\u201335 (24.6) \u00b5m) [Dr. yunlongensis can be distinguished from Dr. coelobrocha by its wider conidia  \u00b5m versus Dr. coelobrocha, 16.8\u201321.6 (19.8) \u00b5m), and shorter base and apical cells [Dr. yunlongensis produces cylindrical or ellipsoidal chlamydospores, while none of the four closely related species produces chlamydospores [Notes: The phylogenetic analysis clustered s, 33\u201352 .5 \u00d7 9.5\u2013s, 33\u201352 .5 \u00d7 9.5\u2013Arthrobotrys in this study because it has been updated in Zhang et al. [We do not update the species key of g et al. , and no 1. Conidia without super-cell\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202621. Conidia with a super-cell\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026.\u2026\u202652. Conidia 1\u20133-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026...32. Conidia 0\u20131-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026...\u20264O. tonghaiensis3. 5\u201310 conidia cluster arrangement on a cluster of short denticles (5\u201310) at the apex of conidiophore, conidia 28.5\u201339.0 \u00d7 6.0\u20138.5 \u00b5m, microconidia cylindrical\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..Dr. brochapaga3. Conidiophore produce 3\u20138 short denticles by repeated elongation, conidia are cylindrical, botuliform, 20\u201345 (30) \u00d7 5\u201312.5 (6) \u00b5m, and do not produce microconidia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.....Dr. dactyleoids4. Conidia digitiform are mostly curved, 1-septate, 35\u201351.5 (42.1) \u00d7 6.5\u20138 (7.5) \u00b5m, with 3\u201313 conidia capitate arrangement at the apex of conidiophore\u2026\u2026\u2026\u2026..Dr. yunnanensis4. Conidia are elongated and ellipsoidal, straight, 0-1-septate, 7.8\u201312.9 \u00d7 3.3\u20134.2 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026.5. Conidia are sub-fusiform to fusiform\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u202665. Conidia are ellipsoidal, elongate ellipsoidal, subellipsoidal, or obovate\u2026\u2026\u2026\u2026\u2026\u2026.12Dr. acrochaetum6. Conidia are 1\u20132-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.....6. Conidia are 1\u20135-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026....77. Conidia are 1\u20134-septate, mostly 3-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026..87. Conidia are 1\u20135-septate, mostly 4-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202610Dr. xiaguanensis8. Conidia are smaller in size, 33\u201352 (42.5) \u00d7 9.5\u201328 (15.5) \u00b5m, swollen at both ends of cells\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026..8. Conidia are bigger, sometimes more than 52 \u00b5m in length and usually greater than 15 \u00b5m in width; the cells at both ends are not enlarged\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026..9Dr. aphrobrocha9. Conidia are wider, 40\u201357.5 (51) \u00d7 15.5\u201335 (24.6) \u00b5m, 2\u20134-septate, and conidiophore occasionally bear two conidia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026....\u2026.\u2026\u2026.Dr. inquisitor9. Conidia are narrower, 42.5\u201362.5 (47) \u00d7 15\u201322.5 (16.9) \u00b5m, 1\u20134-septate, sub-fusiform, and conidiophore bear a single conidium\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026Dr. bembicodes10. Conidia are 3\u20134-septate, smaller in size, 36\u201343.2 (40) \u00d7 16.8\u201321.6 (20.5) \u00b5m, producing obovoid, 1-septate microconidia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026.10. Conidia are bigger, do not produce microconidia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202611Dr. yunlongensis11. Conidia are 1\u20134-septate, 36\u201354 (47) \u00d7 17\u201327 (23.6) \u00b5m, producing cylindrical, globose, or ellipsoidal chlamydospore\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026....Dr. coelobrocha11. Conidia are 2\u20135-septate, 45.6\u201355.2 (49.5) \u00d7 16.8\u201321.6 (19.8) \u00b5m, both ends cells are slender, and do not produce chlamydospore\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..12. Conidia are obovate and 1-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026.1312. Conidia are ellipsoidal, elongate ellipsoidal, and 0\u20133-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026....1413. Conidia are obovate, 29\u201343 (35) \u00d7 15\u201319 (16.8) \u00b5m, base cells are pyramidal, with 3\u20138 conidia capitate arrangement at the apex of conidiophore\u2026.....................Dr. anchoniaDr. polybrocha13. Conidia are obovate or sub-ellipsoidal, 35 \u00d7 24 \u00b5m, single conidium bear at the apex of conidiophore\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026...14. Conidiophore is branched or bears more than 1 conidium\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u20261514. Conidiophore is unbranched, bears a single conidium\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026..16Dr. tianchiensis15. Conidiophore is unbranched or produces 1\u20132 short branches near the apex, each branch bearing a single conidium, with conidia 30\u201341 (36.2) \u00d7 14.5\u201324 (18.7) \u00b5m, 1\u20132-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026Dr. effusa15. Conidiophore is unbranched, bearing a loose head consisting of 2\u201312 conidia, with conidia 32.5\u201345 (38.9) \u00d7 17.5\u201325 (21.4) \u00b5m, 1\u20132-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026.16. Conidiophore produces a swollen, knob-like apex\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u20261716. Conidiophore produces a truncated, non-swelling apex\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026....2017. Produces cylindrical, clavate, or bottle-shaped, 1-septate microconidia\u2026\u2026\u2026\u2026\u2026...1817. Does not produce microconidia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..19Dr. heterospora18. Macroconidia are bigger, 17.5\u201345 (34) \u00d7 17.5\u201325 (20.4) \u00b5m, 1\u20132-septate, mostly 1-septate, and microconidia are bigger, 23\u201340 (31.3)\u00d7 5\u20138 (6.8) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.....\u2026O. pseudopolybrocha18. Macroconidia are smaller, 26\u201330 \u00d7 16\u201322.2 \u00b5m, 0\u20132-septate, mostly 2-septate, and microconidia smaller, 14.7\u201323 \u00d7 3.3\u20136 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026...Dr. pengdangensis19. Conidia are bigger, 30\u201345 (38) \u00d7 17\u201327 (22.4) \u00b5m, 1\u20132-septate, and basal cells are tiny\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026...\u2026Dr. doedycoides19. Conidia are 25\u201352.5 (33.2) \u00d7 12.5\u201329 (17.3) \u00b5m, and 1\u20133-septate\u2026\u2026\u2026\u2026.Dr. stenobrocha20. Conidia are elongated and ellipsoidal, 1\u20133-septate, mostly 3-septate, 34\u201356.5 \u00d7 12.5\u201316.5 \u00b5m, and do not produce microconidia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202620. Conidia are ellipsoidal, 0\u20132-septate, and produce clavate or bottle-shaped microconidia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...21Dr. daliensis21. Macroconidia are thinner, 20\u201349.5 (38.5) \u00d7 8.5\u201315 (12) \u00b5m, 1\u20132-septate, mostly 2-septate, and microconidia wider, 6.5\u201322 (15.5) \u00d7 3.5\u20137 (5) \u00b5m\u2026\u2026\u2026.\u2026\u2026Dr. hainanensis21. Macroconidia are 32.5\u201343 \u00d7 17\u201325 \u00b5m, 0\u20132-septate, mostly 1 or 2-septate, and microconidia are 18.2\u201322.8 \u00d7 4.2\u20135.3 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026..\u2026\u2026\u2026\u2026Drechslerella produces constricting rings to capture nematodes with mechanical force, and the genera Arthrobotrys and Dactylellina catch nematodes with adhesive traps) [Dactylellina species into a stable cluster, possibly due to insufficient DNA data. We believe that as more DNA data are used, we will find more morphological or physiological features that match phylogenetic studies.Both the phylogenetic analysis in this study and previous studies divided NTF into two main clades based on the mechanisms by which they catch nematodes (the genus e traps) ,12,13,14Drechslerella in this study showed that some species with similar conidia morphology cluster stably into one branch, such as species in clade I producing fusiform conidia and species in clade II producing ellipsoidal conidia  is one crucial node to understanding the history of fungal evolution because of its unique morphological characteristics and survival strategy ,3,4,5. C conidia . Moreoveia shape . In addiDrechslerella species is that species are first roughly classified by those features that can be used to identify species and are easily distinguishable, such as whether the conidia produce a super-cell or not, the shape of the conidia , whether the conidiophore is branched or not, and the number of conidia on the conidiophore. Then, species are further classified by those features that can be used for species identification but require further measurement and observation, such as the detailed feature of macroconidia (number and position of the septum and the size of the macroconidia). Finally, morphologically similar species are distinguished by those characteristics that are uncertain whether they can be used for species identification but are differences between different species, such as the presence and features of microconidia, the detailed feature of the apex of the conidiophore, and the features of the chlamydospore. Even identifying Drechslerella species requires those morphological features that are not known to be valid, so how difficult would it be to identify the more complex Arthrobotrys and Dactylellina species based on these features alone? Therefore, follow-up research needs to systematically study all potential morphological characteristics to find more reliable characteristics for species identification.The compilation logic of the key of Orbilia [Orbilia sp.). This results in these different asexual species sharing the same sexual species name [https://www.ncbi.nlm.nih.gov/nuccore/?term=Orbilia+auricolor (accessed on 3 April 2023)). For this reason, we suggest that when reporting a pair of sexual and asexual species, it is necessary to discuss the difference between the sexual generation and known sexual species and, more importantly, consider the distinction between the asexual generation and known asexual generation. The naming of this pair of sexual and asexual species should be carefully evaluated separately, giving sexual and asexual generations different species names if necessary.Most of the sexual generations of Orbiliomycetes nematode-trapping fungi are members of Orbilia . HoweverOrbilia . AdditioOrbilia ,46, asexies name , which f"}
{"text": "Fermatean fuzzy sets (FFSs) have piqued the interest of researchers in a wide range of domains. The striking framework of the FFS is keen to provide the larger preference domain for the modeling of ambiguous information deploying the degrees of membership and nonmembership. Furthermore, FFSs prevail over the theories of intuitionistic fuzzy sets and Pythagorean fuzzy sets owing to their broader space, adjustable parameter, flexible structure, and influential design. The information measures, being a significant part of the literature, are crucial and beneficial tools that are widely applied in decision-making, data mining, medical diagnosis, and pattern recognition. This paper aims to expand the literature on FFSs by proposing many innovative Fermatean fuzzy sets-based information measures, namely, distance measure, similarity measure, entropy measure, and inclusion measure. We investigate the relationship between distance, similarity, entropy, and inclusion measures for FFSs. Another achievement of this research is to establish a systematic transformation of information measures  for the FFSs. To accomplish this aim, new formulae for information measures of FFSs have been presented. To demonstrate the validity of the measures, we employ them in pattern recognition, building materials, and medical diagnosis. Additionally, a comparison between traditional and novel similarity measures is described in terms of counter-intuitive cases. The findings demonstrate that the innovative information measures do not include any absurd cases. The idea of the fuzzy set (FS) was developed by Zadeh  in 1965,Atanassov  establisIn many situations, it is conceivable that the sum of the MG and NMG will be greater than 1. To overcome these challenges, Yager  introducIn the field of PyFS, there are various approaches for solving real-life multiattribute decision-making (MADM) situations. A number of researchers have also suggested real-world applications in a Pythagorean fuzzy environment. However, if orthopair FSs as \u23290.9,0.5\u232a, where 0.9 is the MG of specific criteria of a parameter and 0.5 is the NMG, it does not fulfill the IFS and PFS requirements. However, the cubic sum of the MG and NMG is equal to or less than one. In this context, Senapati and Yager  known to be MG.(see ). Let \ud835\udcb5=\ud835\udcb5={\u03f01, \u03f02, \u03f03,\u2026, \u03f0n} be a finite set. An IFS \u03dc over \u03f0 is follows:i \u2208 \ud835\udcb5 the functions \u2111\u03dc : \ud835\udcb5\u27f6, 1 and \u00f0\u03dc : \u03f0\u27f6 denotes the MG and NMG, respectively, which must satisfy the property 0 \u2264 \u2111\u03dc(\u03f0i)+\u00f0\u03dc(\u03f0i) \u2264 1.(see ). Let \ud835\udcb5=1, \u03f02, \u03f03,\u2026, \u03f0n} be a finite set. The HFS H on \u03f0 is defined as follows:h(\u03f0i) is a set of values contain in , which shows the MG of \u03f0i \u2208 H. The element of h(\u03f0i) is known as the HF element.(see ). Let \u03f0=\ud835\udcb5={\u03f01, \u03f02, \u03f03,\u2026, \u03f0n} be a finite set, a Fermatean fuzzy sets (FFSs) \u2131 over \ud835\udcb5 is defined as follows:i \u2208 \u03f0 the functions \u2111\u2131 : \u03f0\u27f6 and \u00f0\u2131 : \u03f0\u27f6 denote the MG and NMG, respectively, which must satisfy (\u00f0\u2131(\u03f0i))3+(\u2111\u2131(\u03f0i))3 \u2264 1. The degree of indeterminacy is given as follows:(see ). Let \ud835\udcb5=\ud835\udcb4 and \ud835\udcab be two FFSs, then the operations can be defined as follows:Addition: Multiplication: Scalar multiplication: Exponent: (see ). If \ud835\udcb4 a\ud835\udcb4 and \ud835\udcab be two FFSs, then the operations can be defined as follows:\ud835\udcb4c={\u2329\u00f0\ud835\udcb4(\u03f0i), \u2111\ud835\udcb4(\u03f0i)\u232a|\u03f0i \u2208 \ud835\udcb5};Complement: \ud835\udcb4=\ud835\udcab iff for all \u2329\u03f0i \u2208 \ud835\udcb5, \u2111\ud835\udcb4(\u03f0i)=\u2111\ud835\udcab(\u03f0i)and\u2006\u00f0\ud835\udcb4(\u03f0i)=\u00f0\ud835\udcab(\u03f0i)\u232a;Equality: \ud835\udcb4\u2229\ud835\udcab=\u2329min\u2009(\u2111\ud835\udcb4(\u03f0i), \u2111\ud835\udcab(\u03f0i)), max\u2009(\u00f0\ud835\udcb4(\u03f0i), \u00f0\ud835\udcab(\u03f0i))|\u03f0i \u2208 \ud835\udcb5\u232a;Intersection: \ud835\udcb4 \u222a \ud835\udcab=\u2329max\u2009(\u2111\ud835\udcb4(\u03f0i), \u2111\ud835\udcab(\u03f0i)), min\u2009(\u00f0\ud835\udcb4(\u03f0i), \u00f0\ud835\udcab(\u03f0i))|\u03f0i \u2208 \ud835\udcb5\u232a.Union: (see ). If \ud835\udcb4 a\ud835\udcb4, \ud835\udcab, and\u2009\u2133 be three FFSs, then the following characteristics are held:\ud835\udcb4 \u222a \ud835\udcab=\ud835\udcab \u222a \ud835\udcb4;\ud835\udcb4\u2229\ud835\udcab=\ud835\udcab\u2229\ud835\udcb4;\ud835\udcb4 \u222a (\ud835\udcab \u222a \u2133)=(\ud835\udcb4 \u222a \ud835\udcab) \u222a \u2133;\ud835\udcb4\u2229(\ud835\udcab\u2229\u2133)=(\ud835\udcb4\u2229\ud835\udcab)\u2229\u2133;\u03b1 \u222a (\ud835\udcb4 \u222a \ud835\udcab)=\u03b1\ud835\udcb4 \u222a \u03b1\ud835\udcab;\ud835\udcb4 \u222a \ud835\udcab)\u03b1=\ud835\udcb4\u03b1 \u222a \ud835\udcab\u03b1.((see ). If \ud835\udcb4, This section explains the axiomatic framework of FFSs information measures , as well as their related formulations. Simultaneously, their transformation relationships are thoroughly examined.This section introduces the idea of a distance measures for FFSs. A number that is assigned to a pair of points in a space which indicates how far those points are from one another. A distance measure is called a metric if it is always positive and also it is always symmetric.\ud835\udcb4, \ud835\udcab, and \u2133 be three FFSs on \ud835\udcb5. A distance measure \ud835\udc9f is a mapping \ud835\udc9f: FFS (\ud835\udcb5)\u00d7 FFS (\ud835\udcb5)\u27f6, carrying the following features:\ud835\udc9f \u2264 1;0 \u2264 \ud835\udc9f=\ud835\udc9f;\ud835\udc9f=0 iff \ud835\udcb4=\ud835\udcab;\ud835\udc9f=1 iff \ud835\udcb4 is a crisp set;\ud835\udcb4\u2286\ud835\udcab\u2286\u2133, then \ud835\udc9f \u2264 \ud835\udc9f and \ud835\udc9f \u2264 \ud835\udc9f.If Let \ud835\udcb4 and \ud835\udcab be two FFS s, then \ud835\udc9fi is a distance measure.\ud835\udc9f=(1/2|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|+|\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i)|+|\u03c0\ud835\udcb43(\u03f0i) \u2212 \u03c0\ud835\udcab3(\u03f0i)|);\ud835\udc9f2=(1/2|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(|(\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i))+(\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i))|);\ud835\udc9f3=(1/4|X|)(\u2211i\u2208\ud835\udcb5\u03f0(|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|+|\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i)|+|\u03c0\ud835\udcb43(\u03f0i) \u2212 \u03c0\ud835\udcab3(\u03f0i)|)+\u2211i\u2208\ud835\udcb5\u03f0(|(\u2111\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i)) \u2212 (\u2111\ud835\udcab3(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i))|))\ud835\udc9f4=(1/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|\u2228|(\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i))|);\ud835\udc9f5=(2/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|\u2228|(\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i))|/1+|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|\u2228|(\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i))|);\ud835\udc9f6=(2\u2211i\u2208\ud835\udcb5\u03f0|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|\u2228|(\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i))|/\u2211i\u2208\ud835\udcb5\u03f0(1+|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|\u2228|\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i)|));\ud835\udc9f7=1 \u2212 \u03b1(\u2211i\u2208\ud835\udcb5\u03f0(\u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i))/\u2211i\u2208\ud835\udcb5\u03f0(\u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i))) \u2212 \u03b2(\u2211i\u2208\ud835\udcb5\u03f0(\u00f0\ud835\udcb43(\u03f0i)\u2227\u00f0\ud835\udcab3(\u03f0i))/\u2211i\u2208\ud835\udcb5\u03f0(\u00f0\ud835\udcb43(\u03f0i)\u2228\u00f0\ud835\udcab3(\u03f0i))), \u03b1+\u03b2=1, \u03b1, \u03b2 \u2208 ;\ud835\udc9f8=1 \u2212 (\u03b1/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(\u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i))/(\u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i)) \u2212 (\u03b2/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(\u00f0\ud835\udcb43(\u03f0i)\u2227\u00f0\ud835\udcab3(\u03f0i))/(\u00f0\ud835\udcb43(\u03f0i)\u2228\u00f0\ud835\udcab3(\u03f0i)), \u03b1+\u03b2=1, \u03b1, \u03b2 \u2208 ;\ud835\udc9f9=1 \u2212 (1/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0((\u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i))+(\u00f0\ud835\udcb43(\u03f0i)\u2227\u00f0\ud835\udcab3(\u03f0i))/(\u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i))+(\u00f0\ud835\udcb43(\u03f0i)\u2228\u00f0\ud835\udcab3(\u03f0i)));\ud835\udc9f10=1 \u2212 (\u2211i\u2208\ud835\udcb5\u03f0(\u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i))+(\u00f0\ud835\udcb43(\u03f0i)\u2227\u00f0\ud835\udcab3(\u03f0i))/\u2211i\u2208\ud835\udcb5\u03f0(\u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i))+(\u00f0\ud835\udcb43(\u03f0i)\u2228\u00f0\ud835\udcab3(\u03f0i)));\ud835\udc9f11=1 \u2212 (1/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0((\u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i))+(1 \u2212 \u00f0\ud835\udcb43(\u03f0i))\u2227(1 \u2212 \u00f0\ud835\udcab3(\u03f0i)))/((\u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i))+(1 \u2212 \u00f0\ud835\udcb43(\u03f0i))\u2228(1 \u2212 \u00f0\ud835\udcab3(\u03f0i)));\ud835\udc9f12=1 \u2212 (\u2211i\u2208\ud835\udcb5\u03f0(\u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i))+((1 \u2212 \u00f0\ud835\udcb43(\u03f0i))\u2227(1 \u2212 \u00f0\ud835\udcab3(\u03f0i))))/(\u2211i\u2208\ud835\udcb5\u03f0(\u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i))+((1 \u2212 \u00f0\ud835\udcb43(\u03f0i))\u2228(1 \u2212 \u00f0\ud835\udcab3(\u03f0i))));Let This section introduces the idea of similarity measures for FFSs. Similarity functions take a pair of points and return a large similarity value for nearby points, and a small similarity value for distant points. One way to transform between a distance function and a similarity measure is to take the reciprocal.\ud835\udcb4, \ud835\udcab, and \u2133 be three FFSs on \ud835\udcb5. A similarity measure \ud835\udcae(\ud835\udcb4 and \ud835\udcab) is a mapping \ud835\udcae FFS (\ud835\udcb5) \u00d7 FFS (\ud835\udcb5) \u27f6, possessing the following properties:\ud835\udcae (\ud835\udcb4 and \ud835\udcab) \u2264 1;0 \u2264\ud835\udcae (\ud835\udcb4 and \ud835\udcab)\u2009=\u2009\ud835\udcae (\ud835\udcab and \ud835\udcb4);\ud835\udcae (\ud835\udcb4 and \ud835\udcab)\u2009=\u20091 iff \ud835\udcb4\u2009=\u2009\ud835\udcab;\ud835\udcae (\ud835\udcb4 and \ud835\udcb4c)\u2009=\u20090 iff \ud835\udcb4 is a crisp set;\ud835\udcb4\u2286\ud835\udcab\u2286\u2133, then \ud835\udcae (\ud835\udcb4 and \ud835\udcab) \u2264\ud835\udcae (\ud835\udcb4 and \u2133) and \ud835\udcae (\ud835\udcab and \u2133) \u2264\ud835\udcae (\ud835\udcb4 and \u2133).If Let \ud835\udcb4 and \ud835\udcab be two FFSs, then \ud835\udcaei is a distance measure.\ud835\udcae2=1 \u2212 (1/2|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i) \u2212 (\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i))|);\ud835\udcae3=1 \u2212 (1/4|X|)(\u2211i\u2208\ud835\udcb5\u03f0(|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|+|\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i)|+|\u03c0\ud835\udcb43(\u03f0i) \u2212 \u03c0\ud835\udcab3(\u03f0i)|)+\u2211i\u2208\ud835\udcb5\u03f0(|(\u2111\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i))+(\u2111\ud835\udcab3(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i))|))\ud835\udcae4=1 \u2212 (1/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|\u2228|(\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i))|);\ud835\udcae5=(1/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(1 \u2212 |\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|\u2228|\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i)|)/(1+|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|\u2228|\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i)|);\ud835\udcae6=(\u2211i\u2208\ud835\udcb5\u03f0(1 \u2212 |\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|\u2228|\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i)|))/(\u2211i\u2208\ud835\udcb5\u03f0(1+|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|\u2228|\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i)|));\ud835\udcae7=\u03b1(\u2211i\u2208\ud835\udcb5\u03f0(\u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i))/\u2211i\u2208\ud835\udcb5\u03f0(\u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i)))+\u03b2(\u2211i\u2208\ud835\udcb5\u03f0(\u00f0\ud835\udcb43(\u03f0i)\u2227\u00f0\ud835\udcab3(\u03f0i))/\u2211i\u2208\ud835\udcb5\u03f0(\u00f0\ud835\udcb43(\u03f0i)\u2228\u00f0\ud835\udcab3(\u03f0i))), \u03b1+\u03b2=1, \u03b1, \u03b2 \u2208 ;\ud835\udcae8=(\u03b1/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0((\u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i))/(\u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i)))+(\u03b2/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0((\u00f0\ud835\udcb43(\u03f0i)\u2227\u00f0\ud835\udcab3(\u03f0i))/(\u00f0\ud835\udcb43(\u03f0i)\u2228\u00f0\ud835\udcab3(\u03f0i))), \u03b1+\u03b2=1, \u03b1, \u03b2 \u2208 ;\ud835\udcae9=(1/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0((\u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i))+(\u00f0\ud835\udcb43(\u03f0i)\u2227\u00f0\ud835\udcab3(\u03f0i)))/((\u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i))+(\u00f0\ud835\udcb43(\u03f0i)\u2228\u00f0\ud835\udcab3(\u03f0i)));\ud835\udcae10=(\u2211i\u2208\ud835\udcb5\u03f0(\u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i))+(\u00f0\ud835\udcb43(\u03f0i)\u2227\u00f0\ud835\udcab3(\u03f0i)))/(\u2211i\u2208\ud835\udcb5\u03f0(\u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i))+(\u00f0\ud835\udcb43(\u03f0i)\u2228\u00f0\ud835\udcab3(\u03f0i)));\ud835\udcae11=(1/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0((\u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i))+((1 \u2212 \u00f0\ud835\udcb43(\u03f0i))\u2227(1 \u2212 \u00f0\ud835\udcab3(\u03f0i))))/((\u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i))+((1 \u2212 \u00f0\ud835\udcb43(\u03f0i))\u2228(1 \u2212 \u00f0\ud835\udcab3(\u03f0i))));\ud835\udcae12=(\u2211i\u2208\ud835\udcb5\u03f0(\u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i))+((1 \u2212 \u00f0\ud835\udcb43(\u03f0i))\u2227(1 \u2212 \u00f0\ud835\udcab3(\u03f0i))))/(\u2211i\u2208\ud835\udcb5\u03f0(\u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i))+((1 \u2212 \u00f0\ud835\udcb43(\u03f0i))\u2228(1 \u2212 \u00f0\ud835\udcab3(\u03f0i))));Let i=1,2,3,\u2026, 13, if \u03b1=\u03b2=1/2, then we have\ud835\udcaei=\ud835\udcaei;\ud835\udcaei=\ud835\udcaei;\ud835\udcaei=\ud835\udcaei;\ud835\udcaei=\ud835\udcaei.For i=1,2,\u2026, 6, we have\ud835\udcaei=\ud835\udcaei;\ud835\udcaei=\ud835\udcaei;For i=1,4,5,6 and for all \u03f0i \u2208 \ud835\udcb5, (\u2111\ud835\udcb43(\u03f0i)+\u2111\ud835\udcab3(\u03f0i))=1, and \u00f0\ud835\udcb43(\u03f0i)+\u00f0\ud835\udcab3(\u03f0i)=1, we have\ud835\udcaei=\ud835\udcaei,\u2111\ud835\udcb43(\u03f0i) \u2264 \u2111\ud835\udcab3(\u03f0i), and(\u00f0\ud835\udcb43(\u03f0i) \u2265 \u00f0\ud835\udcab3(\u03f0i))\ud835\udcaei=\ud835\udcaei,\u2111\ud835\udcb43(\u03f0i) \u2265 \u2111\ud835\udcab3(\u03f0i), and(\u00f0\ud835\udcb43(\u03f0i) \u2264 \u00f0\ud835\udcab3(\u03f0i))For \ud835\udcb4 and \ud835\udcab two FFSs on \ud835\udcb5. An entropy measure E(\ud835\udcb4) is a mapping E: FFS (\ud835\udcb5)\u27f6, carrying the following features:(i)E(\ud835\udcb4) \u2264 1;0 \u2264 (ii)E(\ud835\udcb4)=0 iff \ud835\udcb4 is a crisp set;(iii)E(\ud835\udcb4)=1 iff \u2111\ud835\udcb43(\u03f0i)=\u00f0\ud835\udcb4t(\u03f0i);(iv)E(\ud835\udcb4)=E(\ud835\udcb4c);(v)E(\ud835\udcb4) \u2264 E(\ud835\udcab) if \ud835\udcb4 is less fuzzy than \ud835\udcab, that isIf Let Let\ud835\udcb4be two FFSs, thenEiis an entropy.\ud835\udcb4, \ud835\udcab, and \u2133 be three FFSs on \ud835\udcb5. An inclusion measure \u2110 is a mapping \u2110: FFS (\ud835\udcb5)\u00d7 FFS (\ud835\udcb5)\u27f6, carrying the following features:\u2110 \u2264 1;0 \u2264 \u2110=1 iff \ud835\udcb4\u2286\ud835\udcab;\u2110=0 iff \ud835\udcb4=\u03a6 and \ud835\udcab=\u2205;\ud835\udcb4\u2286\ud835\udcab\u2286\u2133, then \u2110 \u2264 \u2110 and \u2110 \u2264 \u2110.If Let \ud835\udcb4 and \ud835\udcab be two FFS s, then \u2110i is an inclusion measure.\u21101=1 \u2212 (1/2|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i)|+|\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i)\u2228\u00f0\ud835\udcab3(\u03f0i)|);\u21102={1,\ud835\udcb4=\u2205(\u2211i\u2208\ud835\udcb5\u03f0(1+\u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i)\u2228\u00f0\ud835\udcab3(\u03f0i)))/(\u2211i\u2208\ud835\udcb5\u03f0(1+\u2111\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i))),\ud835\udcb4 \u2260 \u2205;\u21103={1,\ud835\udcb4=\ud835\udcab=\u2205(\u2211i\u2208\ud835\udcb5\u03f0(1+\u2111\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i)))/(\u2211i\u2208\ud835\udcb5\u03f0(1+\u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i)\u2227\u00f0\ud835\udcab3(\u03f0i))),others;\u21104={1,\ud835\udcb4=\ud835\udcab=\u2205(\u2211i\u2208\ud835\udcb5\u03f0(1+\u2111\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i)))/(\u2211i\u2208\ud835\udcb5\u03f0(1+\u00f0\ud835\udcb43(\u03f0i)\u2227\u00f0\ud835\udcab3(\u03f0i) \u2212 \u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i))),others;\u21105={1,\ud835\udcb4=\ud835\udcab=\u2205(1/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(1+\u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i)\u2228\u00f0\ud835\udcab3(\u03f0i))/(1+\u2111\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i)),others\u21106={1,\ud835\udcb4=\ud835\udcab=\u2205(1/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(1+\u2111\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i))/(1+\u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i)\u2227\u00f0\ud835\udcab3(\u03f0i)),others\u21107={1,\ud835\udcb4=\ud835\udcab=\u2205(1/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(1+\u2111\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i))/(1+\u00f0\ud835\udcb43(\u03f0i)\u2227\u00f0\ud835\udcab3(\u03f0i) \u2212 \u2111\ud835\udcb43(\u03f0i)\u2228\u2111\ud835\udcab3(\u03f0i)),othersLet In this section, we study the relations between inclusion, entropy, similarity measure, and distance measure of Fermatean fuzzy sets. First, according to the definitions of similarity measure and distance measure of Fermatean fuzzy sets, one should note that they are all used for estimating the degree of similarity between two Fermatean fuzzy sets. The main difference is as follows: for the similarity measure, a greater value means that the two Fermatean fuzzy sets are more similar than are a pair with a lower value. The situation for the distance measure is just the opposite, that is, the smaller the value is, the more similar these two Fermatean fuzzy sets are. So, we can obtain the following theorem.\ud835\udc9f be the Fermatean fuzzy distance measure for \ud835\udcb4, \ud835\udcab\u2208 FFS s, then \ud835\udcae=1 \u2212 \ud835\udc9f is the similarity measure of FFS s\ud835\udcb4 and \ud835\udcab. The proof is straightforward.Suppose \ud835\udcb4, \ud835\udcab \u2208 FFSs, and we order \ud835\udc9f1=(1/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|+|\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i)|+|\u03c0\ud835\udcb43(\u03f0i) \u2212 \u03c0\ud835\udcab3(\u03f0i)|), then we have\ud835\udcae=1 \u2212 (1/2|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|+|\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i)|+|\u03c0\ud835\udcb43(\u03f0i) \u2212 \u03c0\ud835\udcab3(\u03f0i)|)=\ud835\udcae1.For \ud835\udcaei=1 \u2212 \ud835\udc9fi.Also, \ud835\udc9f and \ud835\udcae be the distance and similarity measures of FFSs, for \ud835\udcb4\u2208 FFSs, then E(\ud835\udcb4)=1 \u2212 \ud835\udcae=1 \u2212 \ud835\udcae is the entropy of FFSs.Let E1) It is straightforward.(E2) If \ud835\udcb4 is a crisp set, then \ud835\udcb4=\u2205 or \ud835\udcb4=\u03a6, we have \ud835\udcae=0. Therefore, E(\ud835\udcb4)=0.(E3) E(\ud835\udcb4)=1\u21d4\ud835\udcae=\ud835\udcae\u21d4\u2111\ud835\udcb43(\u03f0i)=\u2110\ud835\udcb43(\u03f0i)=\u00f0\ud835\udcb43(\u03f0i) for \u03f0i \u2208 \ud835\udcb5.(E4) E(\ud835\udcb4)=\ud835\udcae=\ud835\udcae=E(\ud835\udcb4c).(E5) Since \u2111\ud835\udcb43(\u03f0i) \u2264 \u2111\ud835\udcab3(\u03f0i) \u2264 \u00f0\ud835\udcab3(\u03f0i) \u2264 \u00f0\ud835\udcb43(\u03f0i) implies \ud835\udcb4\u2286\ud835\udcab\u2286\ud835\udcabc\u2286\ud835\udcb4c. Therefore, according the definition of similarity measure of FFS s, we have \ud835\udcae \u2264 \ud835\udcae \u2264 E, that is, E(\ud835\udcb4) \u2264 E(\ud835\udcab). Similarly, if \u00f0\ud835\udcb43(\u03f0i) \u2264 \u00f0\ud835\udcab3(\u03f0i) \u2264 \u2111\ud835\udcab3(\u03f0i) \u2264 \u2111\ud835\udcb43(\u03f0i), then we have \ud835\udcae \u2264 \ud835\udcae \u2264 \ud835\udcae, that is, E(\ud835\udcb4) \u2264 E(\ud835\udcab). This completes the proof.=\ud835\udcae1=1 \u2212 (1/2|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)|+|\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i)|+|\u03c0\ud835\udcb43(\u03f0i) \u2212 \u03c0\ud835\udcab3(\u03f0i)|), then we have\u2009E(\ud835\udcb4)=\ud835\udcae1=1 \u2212 (1/|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0|\u2111\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i)|=E4(\ud835\udcb4) Also,\u2009E4(\ud835\udcb4)=\ud835\udcae2=\ud835\udcae3=\ud835\udcae4=\ud835\udcae13\u2009E5(\ud835\udcb4)=\ud835\udcae7=\ud835\udcae10, E6(\ud835\udcb4)=\ud835\udcae8=\ud835\udcae9, E7(\ud835\udcb4)=\ud835\udcae12E8(\ud835\udcb4)=\ud835\udcae11.\u2009For \ud835\udcb4 be an FFS, m(\ud835\udcb4), n(\ud835\udcb4)\u2208 FFSs, \u2200\u03f0i \u2208 \ud835\udcb5, m(\ud835\udcb4)(\u03f0i)=(\u2111m(\ud835\udcb4)(\u03f0i), \u00f0m(\ud835\udcb4)(\u03f0i)), and n(\ud835\udcb4)(\u03f0i)=(\u2111n(\ud835\udcb4)(\u03f0i), \u00f0n(\ud835\udcb4)(\u03f0i)) their membership and nonmembership functions are defined as follows:\u2009\u2009\u2009\u2009Let \ud835\udc9f be the distance measure and \ud835\udcae be the similarity measure of FFSs, for \ud835\udcb4 in FFSs, then E(\ud835\udcb4)=\ud835\udcae(m(\ud835\udcb4), n(\ud835\udcb4))=1 \u2212 \ud835\udc9f=1 \u2212 \ud835\udcae(m(\ud835\udcb4), n(\ud835\udcb4)) is the entropy of FFS \ud835\udcb4.Let (i)E1) It is straightforward.((ii)E2) If \ud835\udcb4 is a crisp set, then \u2200\u03f0i \u2208 \ud835\udcb5, we have \u2111\ud835\udcb4(\u03f0i)=1, \u00f0\ud835\udcb4(\u03f0i)=0, or \u2111\ud835\udcb4(\u03f0i)=0, \u00f0\ud835\udcb4(\u03f0i)=1. Therefore, we can achieve(m(\ud835\udcb4)=\u03a6, n(\ud835\udcb4)=\u2205, consequently, \ud835\udcae(m(\ud835\udcb4), n(\ud835\udcb4))=0.It implies that (iii)E3) ((iv)E4) Using the definitions of m(\ud835\udcb4) and n(\ud835\udcb4), we have m(\ud835\udcb4)=m(\ud835\udcb4c), n(\ud835\udcb4)=n(\ud835\udcb4c), hence \ud835\udcae(m(\ud835\udcb4), n(\ud835\udcb4))=\ud835\udcae(m(\ud835\udcb4c), n(\ud835\udcb4c)).((v)E5) Since \u2111\ud835\udcb4(\u03f0i) \u2264 \u2111\ud835\udcab(\u03f0i) \u2264 \u00f0\ud835\udcb4(\u03f0i) \u2264 \u00f0\ud835\udcab(\u03f0i) implies \ud835\udcb4\u2286\ud835\udcab\u2286\ud835\udcabc\u2286\ud835\udcb4c. Therefore, we have |\u2111\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i)| \u2265 |(\u2111\ud835\udcab3(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i))|. It means that n(A) \u2264n(\ud835\udcab) \u2264 m(\ud835\udcab) \u2264 m(\ud835\udcb4), so we have \ud835\udcae(m(\ud835\udcb4), n(\ud835\udcb4)) \u2264 \ud835\udcae(m(\ud835\udcab), n(\ud835\udcb4)) \u2264 \ud835\udcae(m(\ud835\udcab), n(\ud835\udcab)) that is, E(\ud835\udcb4) \u2264 E(\ud835\udcab).(\ud835\udcb4(\u03f0i) \u2264 \u00f0\ud835\udcab(\u03f0i) \u2264 \u2111\ud835\udcab(\u03f0i) \u2264 \u2111\ud835\udcb4(\u03f0i), then we haveSimilarly, if \u00f0This completes the proof.\ud835\udc9f and \ud835\udcae be the distance measure and similarity measures of FFSs, respectively, for \ud835\udcb4, \ud835\udcab\u2208 FFSs, then \u2110=\ud835\udcae=1 \u2212 \ud835\udc9f is the inclusion measure of FFSs \ud835\udcb4 and \ud835\udcab.Suppose I1) It is straightforward.(I2) If \ud835\udcb4\u2286\ud835\udcab, then \ud835\udcae=\ud835\udcae=1=\ud835\udcae.(I3) \u2110=0\u21d4\ud835\udcae=0\u21d4\ud835\udcb4=\u03a6, \ud835\udcab=\u2205.(I4) If \ud835\udcb4\u2286\ud835\udcab\u2286\ud835\udcaa, then \u2110=\ud835\udcae=\ud835\udcae and \u2110=\ud835\udcae=\ud835\udcae. Known by the similarity measure of FFSs, we have \u2110 \u2264 \u2110. Similarly, \u2110 \u2264 \u2110. This completes the proof.=\ud835\udcae=1 \u2212 \ud835\udc9f is the inclusion measure of FFSs \ud835\udcb4 and \ud835\udcab.Suppose \ud835\udcb4 and \ud835\udcab be two FFS s, then we define g \u2208 FFS s, \u2200\u03f0i \u2208 \ud835\udcb5,\u2009\u2009Let E be the entropy measure of FFSs, for \ud835\udcb4, \ud835\udcab\u2208 FFSs, then E) is the similarity measure of FFSs \ud835\udcb4 and \ud835\udcab.Suppose S1)\u2212 ( S2) are straightforward.(S3) Known by the definition of entropy of FFS s, E)=1\u21d4\u2111g(\u03f0i)=\u00f0g(\u03f0i)\u21d4|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcabt(\u03f0i)|=0, |\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i)|=0\u21d4\u2111\ud835\udcb4(\u03f0i)=\u2111\ud835\udcab(\u03f0i), \u00f0\ud835\udcb4(\u03f0i)=\u00f0\ud835\udcab(\u03f0i)\u21d4\ud835\udcb4=\ud835\udcab.(S4) If \ud835\udcb4 is a crisp set, then \u2111\ud835\udcb4(\u03f0i)=1, \u00f0\ud835\udcb4(\u03f0i)=0 or \u2111\ud835\udcb4(\u03f0i)=0, \u00f0\ud835\udcb4(\u03f0i)=1. Hence, \u2111g(\u03f0i)=1, \u2110\ud835\udcb4(\u03f0i)=0, \u00f0\ud835\udcb4(\u03f0i)=0, it implies g\u2009=\u2009\u03a6, so E)=0.(S5) Since \ud835\udcb4\u2286\ud835\udcab\u2286\ud835\udcaa, then \u2200\u03f0i \u2208 \ud835\udcb5, we have \u2111\ud835\udcb4(\u03f0i) \u2264 \u2111\ud835\udcab(\u03f0i) \u2264 \u2111\ud835\udcaa(\u03f0i), \u00f0\ud835\udcaa(\u03f0i) \u2264 \u00f0\ud835\udcab(\u03f0i) \u2264 \u00f0\ud835\udcb4(\u03f0i). Therefore, we have \u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcaa3(\u03f0i)| \u2265 |\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcab3(\u03f0i)| and |\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcaa3(\u03f0i)| \u2265 |\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcab3(\u03f0i)|.) \u2264 E).Also, we can knowE) \u2264 E). This completes the proof. \u25a1Similarly, we can prove that \u2110 be the inclusion measure of FFS s, for \ud835\udcb4\u2208 FFS s, then E(\ud835\udcb4)=\u2110 is the entropy of FFS s\u2009\ud835\udcb4.Suppose E1) It is straightforward.(E2) If \ud835\udcb4 is a crisp set, then \ud835\udcb4=\u03a6 or \ud835\udcb4=\u2205, we have \u2110=\u2110=0. Therefore, E(\ud835\udcb4)=0.(E3)\u2009E(\ud835\udcb4)\u2009=\u20091 \u21d4\u2110=1\u21d4\ud835\udcb4 \u222a \ud835\udcb4c\u2286\ud835\udcb4\u2229\ud835\udcb4c\u21d4\ud835\udcb4 \u222a \ud835\udcb4c=\ud835\udcb4\u2229\ud835\udcb4c\u21d4\u2111\ud835\udcb4(\u03f0i)=\u2110\ud835\udcb4(\u03f0i)=\u00f0\ud835\udcb4(\u03f0i)(E4)\u2009E(\ud835\udcb4)=\u2110=\u2110=E(\ud835\udcb4c).(E5) Since \u2111\ud835\udcb4(\u03f0i) \u2264 \u2111\ud835\udcab(\u03f0i) \u2264 \u2110\ud835\udcab(\u03f0i) \u2264 \u2110\ud835\udcb4(\u03f0i) \u2264 \u00f0\ud835\udcab(\u03f0i) \u2264 \u00f0\ud835\udcb4(\u03f0i) implies \ud835\udcb4\u2286\ud835\udcab\u2286\ud835\udcabc\u2286\ud835\udcb4c. Furthermore, \ud835\udcb4\u2229\ud835\udcb4c\u2286\ud835\udcab\u2229\ud835\udcabc\u2286\ud835\udcab \u222a \ud835\udcabc\u2286\ud835\udcb4 \u222a \ud835\udcb4c. \u2264 \u2110 \u2264 \u2110, so E(\ud835\udcb4) \u2264 E(\ud835\udcab).According to the definition of inclusion measure, we have \ud835\udcb4(\u03f0i) \u2264 \u00f0\ud835\udcab(\u03f0i) \u2264 \u2111\ud835\udcab(\u03f0i) \u2264 \u2111\ud835\udcb4(\u03f0i), then we can have \u2110 \u2264 \u2110 \u2264 \u2110, that is E(A) \u2264E(\ud835\udcab). This completes the proof.Similarly, if \u00f0\ud835\udcb4, \ud835\udcab\u2208 FFS s, and we order \u2110=\u21101=1 \u2212 (1/2|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(|\u2111\ud835\udcb43(\u03f0i) \u2212 \u2111\ud835\udcb43(\u03f0i)\u2227\u2111\ud835\udcab3(\u03f0i)|+|\u00f0\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i)|), then we have E(\ud835\udcb4)=\u21101=1 \u2212 (1/2|\ud835\udcb5|)\u2211i\u2208\ud835\udcb5\u03f0(|\u2111\ud835\udcb43(\u03f0i) \u2212 \u00f0\ud835\udcb43(\u03f0i)|)=E4(\ud835\udcb4)For \ud835\udcae(\u2111\ud835\udcb4(\u03f0i), \u00f0\ud835\udcb4(\u03f0i))is the entropy of FFS\ud835\udcb4.(i)E1) It is straightforward.((ii)E2) If \ud835\udcb4 is a crisp set, then we have \u2111\ud835\udcb4(\u03f0i)=\u03a6, and \u00f0\ud835\udcb4(\u03f0i)=\u2205 or \u2111\ud835\udcb4(\u03f0i)=\u2205, and \u00f0\ud835\udcb4(\u03f0i)=\u03a6. Therefore, \ud835\udcae(\u2111\ud835\udcb4(\u03f0i), \u00f0\ud835\udcb4(\u03f0i)=0.((iii)E3) Known by the definition of similarity measure of FFSs, we have((iv)E4) E(\ud835\udcb4)=\ud835\udcae(\u2111\ud835\udcb4(\u03f0i), \u00f0\ud835\udcb4(\u03f0i))=\ud835\udcae(\u00f0\ud835\udcb4(\u03f0i), \u2111\ud835\udcb4(\u03f0i))=E(\ud835\udcb4c).((v)E5) Since \u2111\ud835\udcb4(\u03f0i) \u2264 \u2111\ud835\udcab(\u03f0i) \u2264 \u00f0\ud835\udcab(\u03f0i) \u2264 \u00f0\ud835\udcb4(\u03f0i) implies \ud835\udcb4\u2286\ud835\udcab\u2286\ud835\udcabc\u2286\ud835\udcb4c. Namely, \u2111\ud835\udcb4\u2286\u2111\ud835\udcab\u2286\u00f0\ud835\udcab\u2286\u00f0\ud835\udcb4. According to the definition of similarity measure, we can have \ud835\udcae(\u2111\ud835\udcb4(\u03f0i), \u00f0\ud835\udcb4(\u03f0i)) \u2264 \ud835\udcae(\u2111\ud835\udcab(\u03f0i), \u00f0\ud835\udcab(\u03f0i)), that is, E(\ud835\udcb4) \u2264 E(\ud835\udcab).(\ud835\udcb4(\u03f0i) \u2264 \u00f0\ud835\udcab(\u03f0i) \u2264 \u2111\ud835\udcab(\u03f0i) \u2264 \u2111\ud835\udcb4(\u03f0i), then we can have E(\ud835\udcb4) \u2264 E(\ud835\udcab). This completes the proof.Similarly, if \u00f0In this part, we use numerical examples to show the feasibility and effectiveness of the innovative FFS distance measures to illustrate the applications of the established distance measures for specific FFSs in pattern recognition. Furthermore, we compare them to the existing distance measures.\u21331, \u21332, \u21333, and \u21334 be four known patterns that are illustrated by the following FFSs in X as follows:\u2112 is an unknown pattern. Its aimed is to determine the class to which \u2112 belongs. In order to do that, the distance between \u2112 and classes \u21331, \u21332, \u21333, and \u21334 are measured, and \u2112 is then allocated to the class \u2133g specified as follows:Let \ud835\udc9f1 \u2212 \ud835\udc9f13) for FFS, the distance between \ud835\udc9f, \ud835\udc9f, \ud835\udc9f, and \ud835\udc9f is determined and displayed in \u2112 belongs to a class \u21333 when \ud835\udc9f1 to \ud835\udc9f13 are used. It is clear that the cause for this difference is the first characteristic, i.e., (\u03f01). The FFNs of \u03f01 are as follows:\u21331, \u21332, \u21333, \u21334, and \u2112, respectively. It is predicted that the distance between  and  is larger than the distance between  and  is larger than the distance between  and  is larger than  and . As a conclusion, it appears that\ud835\udc9f to \ud835\udc9f13 as shown in For all the newly developed distance measures  for patients P: \u2009 with disease symptoms V:\u2009. The symptoms associated with the considered diagnosis are listed in Table 2Table 2\ud835\udc9f13. We may conclude that all the patients suffer from viral fever.Assume that a doctor would like to diagnose the condition of To illustrate the effectiveness of the novel distance measure for specific FFSs in pattern recognition, we present a numerical example and compare the novel findings to those reported in the literature.\u2133={\u21331, \u21332, \u21333, \u21334} who have the symptoms temperature, headache, stomach pain, cough, and chest pain, which are represented by S={\u03f01, \u03f02, \u03f03, \u03f04, \u03f05}. Let \u2112={\u21121, \u21122, \u21123, \u21124, \u21125} be a list of possible diagnoses. Possible disease are defined as follows: \u21121: viral fever, \u21122: malaria, \u21123: typhoid, \u21124: stomach problem, and \u21125: chest problem. The FF relation \u2133\u27f6S is illustrated by FFS, as can be seen in S\u2009\u27f6\u2112 is denoted by the FFS, as seen in \u2133j with regard to the diagnostic \u2112i and the final diagnosis findings are given in Al has malaria \ud835\udc9f ,,\u2133={\u21331, \u2133 \ud835\udc9f ) ) \u2112 belonasures \ud835\udcae , \ud835\udcae 27],,\u2112 belong \ud835\udcae ,,\u2112 belong,,C: \u2009, , },\u2009\u21332={, , },\u2009\u21333={, , },\u2009\u21334={, , }.\u2009Let, \u2112: \u2112={, , and}. Its purpose is to determine which class \u2112 belongs to. To do this, the inclusion degrees between \u2112 and classes \u21331, \u21332, \u21333, and \u21334 are measured, and \u2112 is then allocated to the class \u2133g specified as follows:The following is an unknown pattern \u21101 \u2212 \u21107) for FFS, the degree of inclusion between \u2110, \u2110, \u2110, and \u2110 are determined and displayed in \u2112 belongs to a class \u21334 when \u21101 to \u21103 and \u21105 to \u21106 are used, and \u2112 belongs to a class \u21333 and \u21332 when \u21104 and \u21107 are, respectively, used. It is clear that the cause for this difference is the first characteristic, i.e., (\u03f01). The FFNs of \u03f01 are , , , , and  for \u21331, \u21332, \u21333, \u21334, and \u2112, respectively. It is predicted that the inclusion degree between  and  is larger than the inclusion degree between  and  is larger than the inclusion degree between  and  is larger than  and . As a conclusion, it appears that \u2110 > \u2009\u2110 > \u2110 > \u2110 is more acceptable. In a similar way, we can find the previously mentioned relations for \u21102 to \u21107.For all the established inclusion measures .We constructed various formulae for FFSs information measures and analyzed the associated transformation relationships in detail.\ud835\udc9f1-\ud835\udc9f13) to pattern recognition and medical diagnosis to demonstrate their efficacy. The applications substantiate the results and also illustrate the feasibility and effectiveness of the distance measures between FFSs information.We used the established distance measures (1-S13); several counterintuitive examples of existing similarity measures are shown. We employed them to pattern recognition, construction materials, and medical diagnosis. For pattern recognition problems, we conclude that the proposed similarity measures dominate existing similarity measures. In some special situations, it has been shown that many conventional similarity measures are incapable of providing reasonable findings. However, in these specific cases, the proposed similarity measure is proficient of discriminating FFSs. A comparison of the proposed similarity measures with conventional similarity measures is performed for medical diagnosis problems. The applications emphasise the results and also illustrate the significance and reliability of the established similarity measures.We demonstrated the efficacy of the novel similarity measures (SAdditionally, we illustrated the applicability of the suggested FFS inclusion measures to pattern recognition with an example. The findings demonstrate the feasibility and effectiveness of new inclusion measures.The main findings of this research are emphasized and encapsulated as follows:The experimental findings demonstrated that the proposed measures are more reliable and can avoid the counter-intuitive situation in dealing with practical applications based on Fermatean fuzzy environment. , 51, 52.The FFSs are inappropriate to deal with situations where the cube sum of membership and nonmembership grades of exceeds 1A near future target is to unfold the application of the proposed information measures in scientific investigations for decision-making, pattern recognition, linguistic summarization, and data miningWe have also a plan to apply the presented approach to procurement planning, water desalination station selection, wind power plant site selection, and many more domains of real world problemsAdditionally, we will be further interested to immerse them in a variety of fuzzy environmentsFurthermore, since this work presents an applicative analysis of the FFS information measures, we should develop an appropriate software to effectively apply the presented information measures in a realistic situation"}
{"text": "Amanita sect. Vaginatae (Fr.) Qu\u00e9l. are challenging to delimitate due to the morphological similarity or morphostasis among different taxa. In this study, a multi-locus  phylogeny was employed to investigate the species diversity of the section in eastern China. Sixteen species were recognized, including four new species; namely, A. circulata, A. multicingulata, A. orientalis, and A. sinofulva. They were documented with illustrated descriptions, ecological evidence, and comparisons with similar species. A key to the species of the section from eastern China is provided.Species of  Amanita Pers. is a cosmopolitan genus with about 700 accepted species  ring-like zone at proximal end of marginal striations; volval remnants on pileus absent; margin striate (0.2\u20130.5 R), non-appendiculate; trama white (1A1), unchanging. Lamellae free, crowded, white (1A1); lamellar edges white (1A1); lamellulae truncate, plentiful. Stipe 9\u201318 cm long \u00d7 0.5\u20131.5 cm diam., slender, subcylindric, slightly tapering upwards, with apex slightly expanded, white (1A1), gray (1B1), brownish (2B2\u20134) to gray-brown (2C2\u20134); context white (1A1), hollow in center; basal bulb absent; volva saccate, membranous, both surfaces white (1A1). Annulus absent. Odor indistinct.Lamellar trama bilateral. Mediostratum 20\u201340 \u03bcm wide, composed of abundant, ellipsoid inflated cells (25\u201360 \u00d7 10\u201330 \u03bcm); filamentous hyphae abundant, 2\u20138 \u03bcm wide; vascular hyphae scarce. Lateral stratum composed of abundant, ellipsoid to fusiform inflated cells (20\u201340 \u00d7 10\u201325 \u03bcm), diverging at an angle of ca. 30\u00b0 to 60\u00b0 to mediostratum; filamentous hyphae abundant and 2\u20137 \u03bcm wide. Subhymenium 30\u201340 \u03bcm thick, with 2\u20133 layers of ellipsoid to fusiform or irregularly arranged cells, 5\u201310 \u00d7 5\u201310 \u03bcm. Basidia 45\u201360 \u00d7 15\u201320 \u03bcm, clavate, 4-spored; sterigmata 5\u20138 \u03bcm long; basal septa lacking clamps. Basidiospores [60/3/3] (10.5\u2013) 11\u201313 (\u201313.5) \u00d7 (9.5\u2013) 10\u201312.5 (\u201313) \u03bcm, Q = 1\u20131.15 (\u20131.21), Qm = 1.08 \u00b1 0.05, globose to subglobose, occasionally broadly ellipsoid, inamyloid, colorless, thin-walled, smooth; apiculus small. Lamellar edge appearing as a sterile strip, composed of subglobose to ellipsoid or sphaeropedunculate inflated cells (15\u201350 \u00d7 10\u201345 \u03bcm), single and terminal or in chains of 2\u20133, thin-walled, colorless; filamentous hyphae abundant, 2\u20136 \u03bcm wide, irregularly arranged or \u00b1 running parallel to lamellar edge. Pileipellis 50\u201390 \u03bcm thick; upper layer (15\u201340 \u03bcm thick) gelatinized, composed of radially arranged to interwoven, thin-walled, colorless, filamentous hyphae 2\u20135 \u03bcm wide; lower layer (35\u201350 \u03bcm thick) composed of radially arranged, filamentous hyphae 3\u20136 \u03bcm wide, colorless to brownish; vascular hyphae scarce. Interior of volval remnants on stipe base composed of longitudinally arranged elements: filamentous hyphae dominant and very abundant, 3\u201310 \u03bcm wide, colorless, thin-walled, branching, anastomosing; inflated cells rare, globose, subglobose, ellipsoid to fusiform, 50\u201380 \u00d7 40\u201350 \u03bcm, colorless, thin-walled, mostly terminal or sometimes in chains of 2\u20133. Outer and inner surface of volval remnants on stipe base similar to structure of interior part, but with inner surface gelatinized. Stipe trama composed of longitudinally arranged, clavate terminal cells, 80\u2013250 \u00d7 15\u201340 \u03bcm; filamentous hyphae scattered to abundant, 2\u201310 \u03bcm wide; vascular hyphae scarce. Clamps absent in all parts of basidioma.Habitat: Solitary to scattered on soil in subtropical mixed forests with Fagaceae and Pinaceae.Distribution: known from eastern and southwestern China.Pinus, Quercus and Keteleeria, altitude 1900 m, 20 July 2009, Li-Ping Tang 858 (HKAS 56815); same county, in a forest with trees of Pinus armandii Franch. and Keteleeria fortune (A. Murray bis) Carri\u00e8re, altitude 2010 m, 14 August 2010, Qing Cai 391 (HKAS 67955); same city, Changning County, forest type unknown, altitude 2000 m, 25 July 2009, Gang Wu 4 (HKAS 57535); Kunming, Panlong District, in a mixed forest with trees of Fagaceae and Pinaceae, altitude 1990 m, 21 August 2016, Xiao-Xia Ding 111 (HKAS 97054); same city, Wuhua District, in a mixed forest with trees of Fagaceae and Pinaceae, altitude 1990 m, 6 September 2012, Yan-Jia Hao 753 (HKAS 76411); Lincang, Fengqing County, in a mixed forest with trees of Fagaceae and Pinaceae, altitude 1800 m, 26 July 2009, Gang Wu 12 (HKAS 57543); Puer, Lancang Lahu Autonomous County, in a mixed forest with trees of Fagaceae and Pinaceae, altitude 1780 m, 29 September 2016, LC-LJW 280 (HKAS 97784); same county, in a forest dominated with trees of Fagaceae, altitude 1350 m, 31 August 2017, Zhu L. Yang 6049 (HKAS 101238).Additional specimens examined: CHINA. ANHUI PROVINCE: Huangshan, in a broad-leaved forest with trees of Fagaceae, altitude 620 m, 13 July 2018, Hong-Yu Chen 32 (HKAS 127629); same location, in a broad-leaved forest with trees of Fagaceae, altitude 610 m, 12 July 2018, Ting Guo 979 (HKAS 127639). YUNNAN PROVINCE: Baoshan, Tengchong, in a mixed forest with trees of Amanita circulata is somewhat related to A. flavidocerea   (9.5\u2013) 10\u201312 (\u201312.5) \u00d7 (9\u2013) 9.5\u201311 (\u201311.5) \u03bcm, Q = 1\u20131.13 (\u20131.15), Qm = 1.06 \u00b1 0.03, globose to subglobose, inamyloid, colorless, thin-walled, smooth; apiculus small. Lamellar edge appearing as a sterile strip, composed of subglobose, ellipsoid to clavate inflated cells (10\u201345 \u00d7 10\u201330 \u03bcm), single and terminal or in chains of 2\u20133, thin-walled, colorless; filamentous hyphae abundant, 2\u20138 \u03bcm wide, irregularly arranged or \u00b1 running parallel to lamellar edge. Pileipellis 50\u2013100 \u03bcm thick; upper layer (30\u201350 \u03bcm thick) gelatinized, composed of radially arranged to interwoven, thin-walled, colorless filamentous hyphae 2\u20135 \u03bcm wide; lower layer (40\u201350 \u03bcm thick) composed of radially arranged filamentous hyphae 4\u20137 \u03bcm wide, colorless; vascular hyphae scarce. Volval remnants on pileus composed of more or less vertically arranged elements: inflated cells very abundant to dominant, globose, subglobose, ellipsoid to fusiform, 10\u201360 \u00d7 10\u201350 \u03bcm, brown to brownish or colorless, thin-walled, mostly terminal or sometimes in chains of 2\u20133; filamentous hyphae rare, 3\u20137 \u03bcm wide, brown to brownish or colorless, thin-walled, branching, anastomosing. Volval remnants on stipe base composed of longitudinally arranged elements, becoming horizontally arranged towards upper parts: inflated cells very abundant to nearly dominant, globose, subglobose, ellipsoid, fusiform to clavate, 20\u201380 \u00d7 10\u201350 \u03bcm, brown to brownish or colorless, thin-walled, mostly terminal or sometimes in chains of 2\u20133; filamentous hyphae rare to fairly abundant, 2\u20138 \u03bcm wide, brown to brownish or colorless, thin-walled, branching, anastomosing. Stipe trama composed of longitudinally arranged, clavate terminal cells, 100\u2013400 \u00d7 15\u201340 \u03bcm; filamentous hyphae scattered to abundant, 2\u201310 \u03bcm wide; vascular hyphae scarce. Clamps absent in all parts of basidioma.Pinus plants.Habitat: Solitary to scattered on soil in subtropical broad-leaved forests dominated with Fagaceae, sometimes in mixed forests with fagaceous and Distribution: Known from eastern China.Additional specimens examined: CHINA. ANHUI PROVINCE: Huangshan, in a forest dominated with Fagaceae, altitude 1300 m, 13 July 2018, Ting Guo 1018 (HKAS 127631); same location, in a forest dominated with Fagaceae, altitude 670 m, 13 July 2018, Rui-Heng Yang 73 (HKAS 127632); same location, in a forest with Fagaceae and Pinaceae, altitude 940 m, 15 July 2018, Rui-Heng Yang 123 (HKAS 127633); same location, in a forest dominated with Fagaceae, altitude 760 m, 15 July 2018, Rui-Heng Yang 127 (HKAS 127634); same location, in a forest dominated with Fagaceae, altitude 1220 m, 17 July 2018, Rui-Heng Yang 184 (HKAS 127635); same location, in a forest dominated with Fagaceae, altitude 1200 m, 17 July 2018, Rui-Heng Yang 186 (HKAS 127636); same location, in a forest dominated with Fagaceae, altitude 940 m, 15 July 2018, Hong-Yu Chen 77 (HKAS 127637).A. multicingulata is closely related to A. liquii  and shorter striations on the pileal margin (0.3\u20130.4 R) distinguish A. cinctipes from A. multicingulata  (10\u2013) 10.5\u201313 \u00d7 9\u201312 (\u201313) \u03bcm, Q = 1\u20131.26 (\u20131.31), Qm = 1.13 \u00b1 0.07, subglobose to broadly ellipsoid, sometimes globose, inamyloid, colorless, thin-walled, smooth; apiculus small. Lamellar edge appearing as a sterile strip, composed of subglobose, ellipsoid to clavate inflated cells (10\u201350 \u00d7 10\u201330 \u03bcm), single and terminal or in chains of 2\u20133, thin-walled, colorless; filamentous hyphae abundant, 3\u20137 \u03bcm wide, irregularly arranged or \u00b1 running parallel to lamellar edge. Pileipellis 50\u2013100 \u03bcm thick; upper layer (30\u201350 \u03bcm thick) gelatinized, composed of radially arranged to interwoven, thin-walled, colorless to brownish filamentous hyphae 2\u20136 \u03bcm wide; lower layer (30\u201340 \u03bcm thick) composed of radially arranged filamentous hyphae 3\u20138 \u03bcm wide, brownish to brown; vascular hyphae scarce. Volval remnants on stipe base composed of longitudinally arranged elements: inflated cells abundant, globose, subglobose, ellipsoid, fusiform to clavate, 20\u201340 \u00d7 10\u201340 \u03bcm, yellow-brown to gray-brown, thin-walled, mostly terminal or sometimes in chains of 2\u20133; filamentous hyphae abundant, 2\u20137 \u03bcm wide, yellow-brown to gray-brown, thin-walled, branching, anastomosing. Stipe trama composed of longitudinally arranged, clavate terminal cells, 80\u2013300 \u00d7 15\u201340 \u03bcm; filamentous hyphae scattered to abundant, 2\u201310 \u03bcm wide; vascular hyphae scarce. Clamps absent in all parts of basidioma.Habitat: Solitary to scattered on soil in subtropical forests with Fagaceae and Pinaceae.Distribution: Known from eastern China.Additional specimens examined: CHINA. ANHUI PROVINCE: Huangshan, in a mixed forest with Fagaceae and Pinaceae, altitude 760 m, 14 July 2018, Rui-Heng Yang 117 (HKAS 127638).A. griseofolia can be distinguished from A. orientalis  (10.5\u2013) 11\u201313.5 (\u201318) \u00d7 (9\u2013) 9.5\u201312 (\u201313) \u03bcm, Q = (1.01\u2013) 1.05\u20131.23 (\u20131.34), Qm = 1.14 \u00b1 0.08, subglobose to broadly ellipsoid, occasionally globose or ellipsoid, inamyloid, colorless, thin-walled, smooth; apiculus small. Lamellar edge appearing as a sterile strip, composed of subglobose, ellipsoid to clavate inflated cells (10\u201330 \u00d7 10\u201325 \u03bcm), single and terminal or in chains of 2\u20133, thin-walled, colorless; filamentous hyphae abundant, 2\u20138 \u03bcm wide, irregularly arranged or \u00b1 running parallel to lamellar edge. Pileipellis 50\u201390 \u03bcm thick; upper layer (30\u201350 \u03bcm thick) gelatinized, composed of radially arranged to interwoven, thin-walled, colorless to brownish filamentous hyphae 2\u20135 \u03bcm wide; lower layer (35\u201350 \u03bcm thick) composed of radially arranged filamentous hyphae 3\u20137 \u03bcm wide, colorless to brownish; vascular hyphae scarce. Interior of volval remnants on stipe base composed of longitudinally arranged elements: filamentous hyphae dominant and very abundant, 3\u20137 \u03bcm wide, colorless, thin-walled, branching, anastomosing; inflated cells rare to fairly abundant, subglobose, ellipsoid to fusiform, 40\u201365 \u00d7 15\u201350 \u03bcm, colorless, thin-walled, mostly terminal or sometimes in chains of 2\u20133. Stipe trama composed of longitudinally arranged, clavate terminal cells, 80\u2013300 \u00d7 15\u201350 \u03bcm; filamentous hyphae scattered to abundant, 3\u201310 \u03bcm wide; vascular hyphae scarce. Clamps absent in all parts of basidioma.Pinus.Habitat: Solitary to scattered on soil in subtropical forests dominated with Fagaceae, sometimes mixed with Distribution: Known from eastern, central, and southwestern China. Based on the phylogenetic tree inferred from the ITS dataset, it also occurs in Tibet autonomous region and Hunan province . Castanea seguinii Dode and Pinus taiwanensis Hayata, altitude 840 m, Yan-Jia Hao 1609 (EFHAAU 207); same county, in a mixed forest with Fagaceae and Pinaceae, altitude 1000 m, Yan-Jia Hao 1715 (EFHAAU 313). YUNNAN PROVINCE: Nujiang Lisu Autonomous Prefecture, Lanping Bai and Pumi Autonomous County, in a subtropical forest dominated with Quercus, mixed with Pinus yunnanensis, altitude 2150 m, Gang Wu 743 (HKAS 75058).Additional specimens examined: CHINA. ANHUI PROVINCE: Liuan, Jinzhai County, in a forest dominated with Fagaceae, altitude 1110 m, 21 July 2017, Yan-Jia Hao 1520 (HKAS 100610); same county, in a mixed forest with Amanita orientifulva and A. suborientifulva can be confused with A. sinofulva. According to our multi-gene phylogenetic analysis  and is found in subalpine forests dominated by trees of Abies and Picea [A. suborientifulva set it apart from A. sinofulva [A. fulva is also similar to A. sinofulva, but differs in the globose to subglobose basidiospores and in the saccate volva, with inflated cells dominant in its outer part [Notes: analysis , the firnd Picea . The nonA. sect. Vaginatae. Five of them are most informative, viz. the color of the basidiomata, the striations on the pileal margin, the presence or absence of the annulus, the volval remnants on the stipe base, and the size of the basidiospores. In this study, A. cingulata is the only species with a white basidioma, while the other species from eastern China have basidiomata ranging from yellow, to gray, to brown. The striations on the pileal margin of A. zonata, A. pallidozonata, and A. circulata form a ring-like zone at the proximal end, while the remaining taxa in eastern China are without this zone [A. griseofolia, A. multicingulata, and A. orientalis. Species with an annulus, a ring-like zone at the proximal end, or with incomplete rings of volval remnants on the stipe base are clustered in non-monophyletic groups. Our data revealed that several macro- and microscopic characteristics could be useful for the delimitation of species in his zone . Most sphis zone ,21,50. TGiven that it is difficult to delimitate these species based solely on morphological studies, integrative taxonomy is indispensable in recognizing species of the section. This method, which delimits and describes taxa by integrating information from different types of data and methodologies , is proven to be useful for species recognition in plants, animals, and fungi ,62,63,64A. zonata, A. pallidozonata, and A. circulata are morphologically similar due to the pronounced ring-like zones at the proximal end of the marginal striations. However, they occupy different positions in the phylogenetic tree and are distantly related  [For example,  related . Followiipe base . Amanita5\u201310 \u03bcm) . Amanita sinofulva is phylogenetically close and morphologically similar to A. orientifulva. However, they are clustered into two independent lineages , and three species new to China . Thirty-nine of them are reported from the southern parts of China\u2014namely, southwestern, central, eastern, and southern China. In this study, 40 taxa of own taxa ,2,21, foA. angustilamellata (H\u00f6hn.) Boedijn, A. brunneoprocera Thongbai et al., A. cinctipes, and A. pallidocarnea (H\u00f6hn.) Boedijn are typical tropical elements restricted in the tropical areas of China. The other three species, viz. A. brunneosquamata Thongbai et al., A. suborientifulva, and A. subovalispora, extend their distribution from Southeast Asia to subtropical China. In addition, several taxa found in subtropical or subalpine temperate areas in southern parts of China are phylogenetically close to species from Southeast Asia. For example, A. circulata and A. zonata, reported from the subtropical regions of China, are sister to A. flavidocerea and A. flavidogrisea Thongbai et al. from Southeast Asia, respectively , Qm = 1.13 \u00b1 0.07\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263\u2019.Basidiospores more rounded, globose to subglobose\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..44.A. griseofoliaBasidioma more grayish; basidiospores slightly larger, 10\u201313.5 \u00d7 9.5\u201313 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026.\u2026\u2026.\u2026\u20264\u2019.A. multicingulataBasidioma more brownish; basidiospores slightly smaller, 10\u201312 \u00d7 9.5\u201311 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u20265.Pileal surface forming a distinctive ring-like zone at proximal end of marginal striations\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...65\u2019.vPileal surface without a distinctive ring-like zone at proximal end of marginal striations\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026...86.A. zonataPileal margin with relatively shorter striations, 0.15\u20130.3 R; basidiospores slightly smaller and rounder, globose to subglobose, 9\u201310.5 \u00d7 8.5\u201310 \u03bcm, Q = 1.00\u20131.11, Qm = 1.05 \u00b1 0.04\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026.6\u2019.Pileal margin with relatively longer striations; basidiospores slightly larger, globose, subglobose to broadly ellipsoid\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...77.A. pallidozonataBasidiospores slightly smaller, 10\u201312 \u00d7 9\u201311 \u03bcm; volval remnants on stipe base with abundant inflated cells in inner part\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20267\u2019.A. circulataBasidiospores slightly larger, 11\u201313 \u00d7 10\u201312.5 \u03bcm; volval remnants on stipe base mainly with abundant filamentous hyphae\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...8.Basidioma with distinctive yellow color\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026.98\u2019.Basidioma gray to brown, without yellow color\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..109.A. hamadaeStriations on pileal margin relatively longer, 0.2\u20130.5 R; basidiospores broadly ellipsoid to ellipsoid, 10\u201312 \u00d7 8\u20139 \u03bcm, Q = 1.22\u20131.37, Qm = 1.3 \u00b1 0.07\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u20269\u2019.A. croceaStriations on pileal margin relatively shorter, 0.2\u20130.3 R; basidiospores globose to subglobose, 9\u201311 \u00d7 9\u201311 \u03bcm, Q = 1\u20131.11, Qm = 1.05 \u00b1 0.04\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.10.Basidioma orange-brown to yellow-brown\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026..1110\u2019.Basidioma gray to gray-brown\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026...1211.A. suborientifulvaPileus without an umbo\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u202611\u2019.A. sinofulvaPileus with a distinctive umbo at center\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026...12.A. olivaceofuscaPileus gray-brown to yellow-brown, with distinct olivaceous tinge; basidiospores broadly ellipsoid to ellipsoid, relatively larger, 10.5\u201313 \u00d7 8.5\u201310 \u03bcm, Q = 1.05\u20131.45, Qm = 1.25 \u00b1 0.09\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202612\u2019.Pileus gray to brown, without olivaceous tinge; basidiospores globose to subglobose, or broadly ellipsoid to ellipsoid, relatively smaller\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20261313.Pileus gray, without umbo at the center; basidiospores broadly ellipsoid to ellipsoid\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u20261413\u2019.Pileus gray-brown to brown, umbonate at center; basidiospores globose to subglobose\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u20261514.A. subovalisporaPileal margin with shorter striations, 0.36\u20130.4 R\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.14\u2019.A. ovalisporaPileal margin with longer striations, 0.4\u20130.6 R\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026\u2026\u2026.......15.A. brunneoumbonataPileus dark brownish, with darker colored central disk; pileal margin with longer striations, 0.33\u20130.42 R\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..15\u2019.A. brunneoproceraPileus grayish brown; pileal margin with shorter striations, 0.18\u20130.21 R\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.."}
{"text": "Hakanp\u00e4\u00e4 et al. show that reticular adhesions are assembled at flat clathrin lattices, a process inhibited by fibronectin and its receptor, integrin \u03b15\u03b21. This novel adhesion assembly mechanism is coupled to cell migration and reveals a unique crosstalk between cell-matrix adhesions. Reticular adhesions (RAs) consist of integrin \u03b1v\u03b25 and harbor flat clathrin lattices (FCLs), long-lasting structures with similar molecular composition as clathrin-mediated endocytosis (CME) carriers. Why FCLs and RAs colocalize is not known. Here, we show that RAs are assembled at FCLs in a process controlled by fibronectin (FN) and its receptor, integrin \u03b15\u03b21. We observed that cells on FN-rich matrices displayed fewer FCLs and RAs. CME machinery inhibition abolished RAs and live-cell imaging showed that RA establishment requires FCL coassembly. The inhibitory activity of FN was mediated by the activation of integrin \u03b15\u03b21 at Tensin1-positive fibrillar adhesions. Conventionally, endocytosis disassembles cellular adhesions by internalizing their components. Our results present a novel paradigm in the relationship between these two processes by showing that endocytic proteins can actively function in the assembly of cell adhesions. Furthermore, we show this novel adhesion assembly mechanism is coupled to cell migration via unique crosstalk between cell-matrix adhesions.  Integrins are nonenzymatic dimeric transmembrane receptors that recognize ECM components. These mechanosensory proteins govern cell adhesion to the ECM maintaining correct tissue development and function, with elaborate connections to cellular homeostasis and disease . Ligand Cells can form a variety of integrin-based adhesions. Small integrin clusters engaged to the ECM, called nascent adhesions, form on the cell periphery and establish their connection to the actin cytoskeleton via adaptor proteins such as Talin. A balancing act of traction forces and signaling molecules determines whether nascent adhesions mature into larger and molecularly more complex focal adhesions FAs; . In migrRecently, a novel type of integrin-based cell adhesion was discovered . Called Intriguingly, RAs colocalize with large, persistent forms of clathrin structures at the cell membrane called flat clathrin lattices . These experiments were performed using U2Os cells with an endogenously GFP-tagged \u03b1 adaptin 2 Sigma-Aldrich subunit . AP2 is a widely used CME marker that faithfully mirrors clathrin dynamics . We usedU2Os cells on non-coated dishes presented typical and abundant FCLs , we found that cells plated on non-coated dishes displaying fewer FCLs were predominantly lying on top of an FN-rich region of the culture .Next, we asked if the reduction in FCL proportions observed in FN-coated samples is a cell-wide effect or specific to cellular regions in direct contact with the extracellular substrate. For that, we used patterned dishes containing FN-coated regions interspersed with uncoated regions, where single U2Os-AP2-GFP cells could adhere simultaneously to both an FN- and a non-coated region. In line with a contact-dependent effect, low FCL proportions were observed in cellular regions in contact with FN whereas FCL proportion was high in cellular regions contacting non-coated surfaces .The abundance of an alternative splice variant of the clathrin heavy chain containing exon 31 was recently shown to increase the frequency of clathrin plaques in myotubes . We thusAs discussed in the introduction, FCLs localize to RAs. To check how ECM composition affects these structures, U2Os AP2-GFP cells plated on FN, VTN, Col IV, Col I, or LN111 and stained with the RA component integrin \u03b1v\u03b25 and\u2014to be able to distinguish integrin \u03b1v\u03b25 on RAs or FAs\u2014were also stained with an FA marker . Cells plated overnight without coating formed abundant RAs . On FN-cNext, we used our substrate patterning strategy to check if the local FN effects on FCLs were also similar for RAs. Strikingly, cells plated on patterned FN revealed that RAs, akin to FCLs, were completely inhibited on cellular regions in contact with FN. Cellular regions in contact with non-coated surfaces displayed many FCLs colocalizing with RAs while regions in contact with FN presented no RAs or FCLs . InteresNext, we checked if the effects we see in U2Os cells are also true for other cell lines. To avoid problems of overexpression, we endogenously tagged AP2 with either Halo tag or GFP in various human cell lines: HeLa , MCF7 , HDF , Caco2 , and hMEC . These cell lines presented a large variation in the amount of FCLs and the morphology of RAs. Importantly, these cells could be divided into two groups in terms of endogenous FN secretion, and this division clearly correlated with the amount of FCLs and RAs . U2Os, HWe next evaluated the response of these cell lines to FN precoating. In low-FN-producing cell lines , RA coverage dropped significantly . Among tTo evaluate if changes in the amount of RAs are reflected in the amount of FAs, we measure FA coverage  from U2Os, Hela, and MCF7. We did not observe any clear difference in the coverage of FAs for these three cell lines plated on uncoated or FN-coated dishes . These rFor all experiments so far, we used media supplemented with serum, which is known to contain ECM components, including FN. Given that our cells are left to attach overnight in this media, it would be reasonable to expect that the FN present in serum would coat the dishes and completely mask our results. To test why this does not seem to happen , we compTaken together, the presence of FN controls the formation of RAs and FCLs in a very similar manner and in different cell lines , suggestNext, we set out to dissect the relationship between the formation of FCLs and RAs. It has been shown that integrin \u03b1v\u03b25 is required for the establishment of FCLs . We confWhile all FCLs colocalize to RAs, FCL-free areas of larger RAs are rather common e.g., , which mTo confirm these results, we expressed the AP180 C-terminal fragment (AP180ct), which acts as a strong dominant negative of CME . AP180ctNext, we set out to visualize the dynamics of AP2 during RA formation. For that, we generated a double U2Os knock-in cell line AP2-GFP and integrin \u03b25 (ITGB5)-mScarlet. RAs are remarkably stable structures  and theiTaken together, our results show that the relationship between FCLs and RAs is beyond a simple colocalization. In fact, our data reveal strict codependency, where FCLs are required for the stabilization and growth of integrin \u03b1v\u03b25 clusters, thereby establishing RAs, and vice versa, with integrin \u03b1v\u03b25 being required for the formation of FCLs .To understand the mechanism controlling the coassembly of FCLs and RAs, we turned our attention back to FN. While integrin \u03b1v\u03b25 binds to VTN at FAs and RAs, the major FN receptor is integrin \u03b15\u03b21 . First, In line with these results, integrin \u03b21 silencing in U2Os-AP2-GFP cells plated on FN resulted in a high FCL proportion and large, prominent RAs . DespiteA significant increase in RAs was also seen in cells silenced for integrin \u03b15, the \u03b1 subunit which pairs with integrin \u03b21 for FN binding . Taken tWhen bound to FN, integrin \u03b15\u03b21 translocates centripetally from FAs to form elongated structures called FBs along actin stress fibers. This movement generates long FN fibrils in a process called FN fibrillogenesis and is mediated by the cytoskeleton scaffolding protein Tensin1 . To deteNext, to determine which active integrin \u03b21 pool is more important for the inhibition of FCLs and RAs, we silenced FAs and FB components on U2Os-AP2-GFP cells and plated them on FN. In accordance with the higher accumulation of active integrin \u03b21 in FBs, silencing of Tensin1 led to a marked increase of RAs and FCLs accompanied by a reduction in the presence of active integrin \u03b21 on the membrane , cells are laterally confined. U2Os-AP2-GFP cells were plated on slides with arrow- and H-shaped micropatterns either precoated with FN or not and stained for integrin \u03b1v\u03b25 and p-Pax and imaged to measure RA coverage. In addition, to measure integrin \u03b21 activation, U2Os-AP2-GFP-ITGB5-mScarlet cells were plated similarly and stained for active integrin \u03b21. Supporting our hypothesis, we could detect clear FCL and RAs in FN-coated micropatterns , FCLs and RAs were abundant and equally distributed at the edge and away from the wound . Within The extracellular environment is a key regulator of cellular physiology with integrins playing a key role translating the chemical composition of the extracellular milieu into intracellular signals. Among various mechanisms controlling integrin function, integrin trafficking via endocytosis and exocytosis plays a major role . Thus faOur results support the idea that FCLs and RAs are two sides of the same structure . PreviouWe observed that RA formation events are rare, which led us to use non-physiological conditions\u2014a Cilengitide washout experiment\u2014to detect them. Therefore, the physiological trigger leading to the formation of FCLs and establishment of RAs remains to be understood. VTN, the ligand for integrin \u03b1v\u03b25, could be considered a good candidate. However, as we show in Recent evidence showed that EGFR activation led to the enlargement of FCLs in an integrin \u03b25 phosphorylation-dependent manner , pointinAnother key unknown aspect of RA formation by FCL and \u03b1v\u03b25 coassembly concerns how these structures can be molecularly differentiated from canonical endocytic events. The connection between integrin \u03b1v\u03b25 located in RAs and FCLs occurs primarily via the endocytic adaptors ARH and NUMB . ImportaRecently, a correlation was found between the presence of clathrin plaques and an alternatively spliced isoform of clathrin containing exon 31 in myotubes . We did E. coli infection, the CME machinery is co-opted to form a clathrin-based actin-rich adhesive structure for the bacteria called pedestal  correlated with integrin \u03b1v\u03b25 localizing to FAs over RAs . MoreoveGiven the fact that RAs are long-lasting cellular adhesion structures, it is tempting to hypothesize that these structures act as a \u201cparking brake\u201d for a cell. As the cell is triggered to migrate, this brake needs to be released for efficient cell movement. This process would be analogous to the loss of cell\u2013cell contacts, which happens during epithelial to mesenchymal transition , but insWe showed that FN regulates CCAC assembly in all the cell lines we tested. However, how these in vitro findings will operate in vivo is still unknown. Even though the ECM composition in tissues is complex, the FN effect on CCAC formation is local and strictly contact dependent, which opens the possibility that, in vivo, tissues may use focal changes in ECM composition to control these structures.U2Os, U2Os-AP2-GFP, U2Os-AP2-halo, U2Os-AP2-GFP-ITGB5-mScarlet, HeLa-AP2-GFP, and HeLa-AP2-halo were cultured in MEM supplemented with 10% FBS (Gibco) and penicillin\u2013streptomycin . HDF-AP2-halo and Caco2-AP2-halo were cultured in DMEM supplemented with 10% FBS (Gibco) and penicillin\u2013streptomycin . MCF7-AP2-halo was cultured in DMEM supplemented with 10% FBS (Gibco) and penicillin\u2013streptomycin , 2 mM glutamine, 5 \u00b5g/ml human insulin, and 1 \u00b5M sodium pyruvate. hMEC-AP2-GFP was cultured in MEGM complete medium (Lonza).The following primary antibodies were used: anti-human integrin \u03b21 clones 12G10 , mAb13 , total integrin \u03b21 , anti-human integrin \u03b1v\u03b25 clone 15F11 , anti-human integrin \u03b15 clone SNAKA51 , anti-human Tensin1 , anti-human p-paxillin Y118 , anti-human-Talin1 , anti-\u03b1-tubulin , and anti-GAPDH . Corresponding secondary antibodies raised against rabbit or mouse IgG were purchased from Jackson Immunoresearch.To compare the effects of major ECM proteins on AP2 lifetimes, the glass coverslip areas (14 mm diameter) of imaging dishes (Mattek) were precoated with 10 or 20 \u00b5g/ml (300 \u03bcl in PBS) of the following ECM proteins: recombinant human FN , recombinant human VTN , Col IV , Col I , and LN111 . Coatings were incubated overnight at +37\u00b0C, except Col I, which was incubated at RT overnight, and LN111 was incubated at +4\u00b0C overnight. Alternatively, as non-ECM protein controls, 1% BSA  or 0.3% PLL  were used to coat the dishes overnight at +37\u00b0C. Throughout this study, the standard FN coating was always performed similarly, 10 \u00b5g/ml (300 \u03bcl in PBS), overnight at +37\u00b0C.To study local vs. global effects of FN, FN was mixed with 50 ng/ml of Alexa647-labeled BSA and used to precoat the imaging dishes overnight at +37\u00b0C. The coated surface was subsequently scratched with a needle to allow partial reappearance of non-coated surface. After scratching, the dishes were heavily rinsed with PBS. 20,000 U2Os-AP2-GFP cells were seeded on patterned imaging dishes to ensure sufficient single-cell attachment to border areas.The clathrin inhibitor AP180 C-terminal fragment  cDNA, from rat origin, was described previously . This coTransient transfections were carried out with PEI MAX transfection reagent  using 70% confluent U2Os cells.https://wge.stemcell.sanger.ac.uk//). gRNAs were cloned into pSpCas9(BB)-2A-Puro (PX459) V2.0  using BbsI sites and confirmed by Sanger sequencing.Three gRNA sequences (Integrated DNA Technologies) were designed using the Welcome Sanger Institute Genome online editing tool (gRNAs were transfected with the donor template for homologous recombination and the most effective gRNA (5\u2032-TGC\u200bTAC\u200bAGT\u200bCCC\u200bTGG\u200bAGT\u200bGA-3\u2032), judged by the percentage of fluorescent cells by FACS, was used for single-clone selection, genotyping, and confirmation by microscopy.TGAGGG\u200bCAG\u200bGCG\u200bAGC\u200bCCC\u200bACC\u200bCCG\u200bGCC\u200bCCG\u200bGCC\u200bCCT\u200bCCT\u200bGGA\u200bCTC\u200bGCC\u200bTGC\u200bTCG\u200bCTT\u200bCCC\u200bCTT\u200bCCC\u200bAGG\u200bCCC\u200bGTG\u200bGCC\u200bAAC\u200bCCA\u200bGCA\u200bGTC\u200bCTT\u200bCCC\u200bTCA\u200bGCT\u200bGCC\u200bTAG\u200bGAG\u200bGAA\u200bGGG\u200bACC\u200bCAG\u200bCTG\u200bGGT\u200bCTG\u200bGGC\u200bCAC\u200bAAG\u200bGGA\u200bGGA\u200bGAC\u200bTGC-3\u2032, where C-terminal tagging with GFP is in green (codon-optimized) and short linker in purple. 150-bp homology arms (orange) were incorporated via PCR amplification from a synthesized (IDT), codon-optimized monomeric EGFP.The donor template sequence was: 5\u2032-GGC\u200bCAG\u200bCAT\u200bCCT\u200bGGG\u200bGGG\u200bCCT\u200bCGT\u200bCTC\u200bACC\u200bCCA\u200bGGG\u200bTCT\u200bCCC\u200bCTC\u200bACA\u200bCAG\u200bGTT\u200bTAC\u200bACG\u200bGTC\u200bGTG\u200bGAC\u200bGAG\u200bATG\u200bTTC\u200bCTG\u200bGCT\u200bGGC\u200bGAA\u200bATC\u200bCGA\u200bGAG\u200bACC\u200bAGC\u200bCAG\u200bACG\u200bAAG\u200bGTG\u200bCTG\u200bAAA\u200bCAG\u200bCTG\u200bCTG\u200bATG\u200bCTA\u200bCAG\u200bTCC\u200bCTG\u200bGAG\u200bGGA\u200bAGT\u200bGCA\u200bTCT\u200bGGG\u200bAGC\u200bTCA\u200bGGC\u200bGCT\u200bAGT\u200bGGT\u200bTCA\u200bGCG\u200bAGC\u200bGGG\u200bGTG\u200bAGC\u200bAAG\u200bGGC\u200bGAG\u200bGAG\u200bCTG\u200bTTC\u200bACC\u200bGGG\u200bGTG\u200bGTG\u200bCCC\u200bATC\u200bCTG\u200bGTC\u200bGAG\u200bCTG\u200bGAC\u200bGGC\u200bGAC\u200bGTA\u200bAAC\u200bGGC\u200bCAC\u200bAAG\u200bTTC\u200bAGC\u200bGTG\u200bTCC\u200bGGC\u200bGAG\u200bGGC\u200bGAG\u200bGGC\u200bGAT\u200bGCC\u200bACC\u200bTAC\u200bGGC\u200bAAG\u200bCTG\u200bACC\u200bCTG\u200bAAG\u200bTTC\u200bATC\u200bTGC\u200bACC\u200bACC\u200bGGC\u200bAAG\u200bCTG\u200bCCC\u200bGTG\u200bCCC\u200bTGG\u200bCCC\u200bACC\u200bCTC\u200bGTG\u200bACC\u200bACC\u200bCTG\u200bACC\u200bTAC\u200bGGC\u200bGTG\u200bCAG\u200bTGC\u200bTTC\u200bAGC\u200bCGC\u200bTAC\u200bCCC\u200bGAC\u200bCAC\u200bATG\u200bAAG\u200bCAG\u200bCAC\u200bGAC\u200bTTC\u200bTTC\u200bAAG\u200bTCC\u200bGCC\u200bATG\u200bCCC\u200bGAA\u200bGGC\u200bTAC\u200bGTC\u200bCAG\u200bGAG\u200bCGC\u200bACC\u200bATC\u200bTTC\u200bTTC\u200bAAG\u200bGAC\u200bGAC\u200bGGC\u200bAAC\u200bTAC\u200bAAG\u200bACC\u200bCGC\u200bGCC\u200bGAG\u200bGTG\u200bAAG\u200bTTC\u200bGAG\u200bGGC\u200bGAC\u200bACC\u200bCTG\u200bGTG\u200bAAC\u200bCGC\u200bATC\u200bGAG\u200bCTG\u200bAAG\u200bGGC\u200bATC\u200bGAC\u200bTTC\u200bAAG\u200bGAG\u200bGAC\u200bGGC\u200bAAC\u200bATC\u200bCTG\u200bGGG\u200bCAC\u200bAAG\u200bCTG\u200bGAG\u200bTAC\u200bAAC\u200bTAC\u200bAAC\u200bAGC\u200bCAC\u200bAAC\u200bGTC\u200bTAT\u200bATC\u200bATG\u200bGCC\u200bGAC\u200bAAG\u200bCAG\u200bAAG\u200bAAC\u200bGGC\u200bATC\u200bAAG\u200bGTG\u200bAAC\u200bTTC\u200bAAG\u200bATC\u200bCGC\u200bCAC\u200bAAC\u200bATC\u200bGAG\u200bGAC\u200bGGC\u200bAGC\u200bGTG\u200bCAG\u200bCTC\u200bGCC\u200bGAC\u200bCAC\u200bTAC\u200bCAG\u200bCAG\u200bAAC\u200bACC\u200bCCC\u200bATC\u200bGGC\u200bGAC\u200bGGC\u200bCCC\u200bGTG\u200bCTG\u200bCTG\u200bCCC\u200bGAC\u200bAAC\u200bCAC\u200bTAC\u200bCTG\u200bAGC\u200bACC\u200bCAG\u200bTCC\u200bAAG\u200bCTG\u200bAGC\u200bAAA\u200bGAC\u200bCCC\u200bAAC\u200bGAG\u200bAAG\u200bCGC\u200bGAT\u200bCAC\u200bATG\u200bGTC\u200bCTG\u200bCTG\u200bGAG\u200bTTC\u200bGTG\u200bACC\u200bGCC\u200bGCC\u200bGGG\u200bATC\u200bACT\u200bCTC\u200bGGC\u200bATG\u200bGAC\u200bGAG\u200bCTG\u200bTAC\u200bAAGPCR product was purified and 150 ng was used directly for transfection together with gRNAs. 70\u201380% confluent 24-well plates of U2Os cells were transfected with 2 \u00b5g PEI (1 \u00b5g/ml), 150 ng of plasmid, and 150 ng of the PCR product. In addition, hMEC-AP2-GFP were treated with 1 \u03bcM DNA-PKc inhibitor NU7441 for 48 h after transfection. 2 d after transfection, cells were treated with puromycin (1 \u00b5g/ml) to enrich for successfully transfected cells. After expansion, GFP-positive cells were sorted by FACS and single clones were expanded and genotyped.U2Os-AP2-halo and HeLa-AP2-halo cell lines were generated with the same protocol as the U2Os-AP2-GFP cell line.GGC\u200bCAG\u200bCAT\u200bCCT\u200bGGG\u200bGGG\u200bCCT\u200bCGT\u200bCTC\u200bACC\u200bCCA\u200bGGG\u200bTCT\u200bCCC\u200bCTC\u200bACA\u200bCAG\u200bGTT\u200bTAC\u200bACG\u200bGTC\u200bGTG\u200bGAC\u200bGAG\u200bATG\u200bTTC\u200bCTG\u200bGCT\u200bGGC\u200bGAA\u200bATC\u200bCGA\u200bGAG\u200bACC\u200bAGC\u200bCAG\u200bACG\u200bAAG\u200bGTG\u200bCTG\u200bAAA\u200bCAG\u200bCTG\u200bCTG\u200bATG\u200bCTA\u200bCAG\u200bTCC\u200bCTG\u200bGAGGGA\u200bAGT\u200bGCA\u200bTCT\u200bGGG\u200bAGC\u200bTCA\u200bGGC\u200bGCT\u200bAGT\u200bGGT\u200bTCA\u200bGCG\u200bAGC\u200bGGGGCA\u200bGAA\u200bATC\u200bGGT\u200bACT\u200bGGC\u200bTTT\u200bCCA\u200bTTC\u200bGAC\u200bCCC\u200bCAT\u200bTAT\u200bGTG\u200bGAA\u200bGTC\u200bCTG\u200bGGC\u200bGAG\u200bCGC\u200bATG\u200bCAC\u200bTAC\u200bGTC\u200bGAT\u200bGTT\u200bGGT\u200bCCG\u200bCGC\u200bGAT\u200bGGC\u200bACC\u200bCCT\u200bGTG\u200bCTG\u200bTTC\u200bCTG\u200bCAC\u200bGGT\u200bAAC\u200bCCG\u200bACC\u200bTCC\u200bTCC\u200bTAC\u200bGTG\u200bTGG\u200bCGC\u200bAAC\u200bATC\u200bATC\u200bCCG\u200bCAT\u200bGTT\u200bGCA\u200bCCG\u200bACC\u200bCAT\u200bCGC\u200bTGC\u200bATT\u200bGCT\u200bCCA\u200bGAC\u200bCTG\u200bATC\u200bGGT\u200bATG\u200bGGC\u200bAAA\u200bTCC\u200bGAC\u200bAAA\u200bCCA\u200bGAC\u200bCTG\u200bGGT\u200bTAT\u200bTTC\u200bTTC\u200bGAC\u200bGAC\u200bCAC\u200bGTC\u200bCGC\u200bTTC\u200bATG\u200bGAT\u200bGCC\u200bTTC\u200bATC\u200bGAA\u200bGCC\u200bCTG\u200bGGT\u200bCTG\u200bGAA\u200bGAG\u200bGTC\u200bGTC\u200bCTG\u200bGTC\u200bATT\u200bCAC\u200bGAC\u200bTGG\u200bGGC\u200bTCC\u200bGCT\u200bCTG\u200bGGT\u200bTTC\u200bCAC\u200bTGG\u200bGCC\u200bAAG\u200bCGC\u200bAAT\u200bCCA\u200bGAG\u200bCGC\u200bGTC\u200bAAA\u200bGGT\u200bATT\u200bGCA\u200bTTT\u200bATG\u200bGAG\u200bTTC\u200bATC\u200bCGC\u200bCCT\u200bATC\u200bCCG\u200bACC\u200bTGG\u200bGAC\u200bGAA\u200bTGG\u200bCCA\u200bGAA\u200bTTT\u200bGCC\u200bCGC\u200bGAG\u200bACC\u200bTTC\u200bCAG\u200bGCC\u200bTTC\u200bCGC\u200bACC\u200bACC\u200bGAC\u200bGTC\u200bGGC\u200bCGC\u200bAAG\u200bCTG\u200bATC\u200bATC\u200bGAT\u200bCAG\u200bAAC\u200bGTT\u200bTTT\u200bATC\u200bGAG\u200bGGT\u200bACG\u200bCTG\u200bCCG\u200bATG\u200bGGT\u200bGTC\u200bGTC\u200bCGC\u200bCCG\u200bCTG\u200bACT\u200bGAA\u200bGTC\u200bGAG\u200bATG\u200bGAC\u200bCAT\u200bTAC\u200bCGC\u200bGAG\u200bCCG\u200bTTC\u200bCTG\u200bAAT\u200bCCT\u200bGTT\u200bGAC\u200bCGC\u200bGAG\u200bCCA\u200bCTG\u200bTGG\u200bCGC\u200bTTC\u200bCCA\u200bAAC\u200bGAG\u200bCTG\u200bCCA\u200bATC\u200bGCC\u200bGGT\u200bGAG\u200bCCA\u200bGCG\u200bAAC\u200bATC\u200bGTC\u200bGCG\u200bCTG\u200bGTC\u200bGAA\u200bGAA\u200bTAC\u200bATG\u200bGAC\u200bTGG\u200bCTG\u200bCAC\u200bCAG\u200bTCC\u200bCCT\u200bGTC\u200bCCG\u200bAAG\u200bCTG\u200bCTG\u200bTTC\u200bTGG\u200bGGC\u200bACC\u200bCCA\u200bGGC\u200bGTT\u200bCTG\u200bATC\u200bCCA\u200bCCG\u200bGCC\u200bGAA\u200bGCC\u200bGCT\u200bCGC\u200bCTG\u200bGCC\u200bAAA\u200bAGC\u200bCTG\u200bCCT\u200bAAC\u200bTGC\u200bAAG\u200bGCT\u200bGTG\u200bGAC\u200bATC\u200bGGC\u200bCCG\u200bGGT\u200bCTG\u200bAAT\u200bCTG\u200bCTG\u200bCAA\u200bGAA\u200bGAC\u200bAAC\u200bCCG\u200bGAC\u200bCTG\u200bATC\u200bGGC\u200bAGC\u200bGAG\u200bATC\u200bGCG\u200bCGC\u200bTGG\u200bCTG\u200bTCG\u200bACG\u200bCTC\u200bGAG\u200bATT\u200bTCC\u200bGGCTGAGGG\u200bCAG\u200bGCG\u200bAGC\u200bCCC\u200bACC\u200bCCG\u200bGCC\u200bCCG\u200bGCC\u200bCCT\u200bCCT\u200bGGA\u200bCTC\u200bGCC\u200bTGC\u200bTCG\u200bCTT\u200bCCC\u200bCTT\u200bCCC\u200bAGG\u200bCCC\u200bGTG\u200bGCC\u200bAAC\u200bCCA\u200bGCA\u200bGTC\u200bCTT\u200bCCC\u200bTCA\u200bGCT\u200bGCC\u200bTAG\u200bGAG\u200bGAA\u200bGGG\u200bACC\u200bCAG\u200bCTG\u200bGGT\u200bCTG\u200bGGC\u200bCAC\u200bAAG\u200bGGA\u200bGGA\u200bGAC\u200bTGC-3\u2032, where C-terminal tagging with halo is underlined (codon-optimized) and flexible linker region (GSASGSSGASGSASG)\u00a0is bold. 150-bp homology arms  were incorporated via PCR amplification from a synthesized (IDT), codon-optimized monomeric halo tag.The donor template sequence was: 5\u2032-The most effective gRNA (5\u2032-TGC\u200bTAC\u200bAGT\u200bCCC\u200bTGG\u200bAGT\u200bGA-3\u2032) was ordered as single guide RNA (sgRNA) from Synthego, and Cas9 protein (purified in the lab) was used instead of plasmid. The donor templates for these cell lines to insert either EGFP or Halo tag were the same as above, respectively.5 cells were combined with 4 \u00b5g Cas9 protein, 45 pmol sgRNA, and 300 ng purified PCR product into a nucleofection cuvette. The nucleofection protocols were the following: MCF7-AP2-halo: EN130, buffer SF (Lonza #V4XC-2031); HDF-AP2-halo: EO114, buffer SF (Lonza #V4XC-2031); Caco2-AP2-halo: DG113, buffer SF (Lonza #V4XC-2031); HeLa-AP2-GFP: CN114, buffer SF (Lonza #V4XC-2031).2 \u00d7 10Cell lines were then treated with 1 \u00b5M DNA-PKc inhibitor NU7441 for 48 h after nucleofection.This cell line was produced by the same protocol as the U2Os-AP2-GFP cells with the following changes:ATATAACGGC\u200bACT\u200bGTTGACGGA\u200bAGT\u200bGCA\u200bTCT\u200bGGG\u200bAGC\u200bTCA\u200bGGC\u200bGCT\u200bAGT\u200bGGT\u200bTCA\u200bGCG\u200bAGC\u200bGGGGTG\u200bAGC\u200bAAG\u200bGGC\u200bGAG\u200bGCA\u200bGTG\u200bATC\u200bAAG\u200bGAG\u200bTTC\u200bATG\u200bCGG\u200bTTC\u200bAAG\u200bGTG\u200bCAC\u200bATG\u200bGAG\u200bGGC\u200bTCC\u200bATG\u200bAAC\u200bGGC\u200bCAC\u200bGAG\u200bTTC\u200bGAG\u200bATC\u200bGAG\u200bGGC\u200bGAG\u200bGGC\u200bGAG\u200bGGC\u200bCGC\u200bCCC\u200bTAC\u200bGAG\u200bGGC\u200bACC\u200bCAG\u200bACC\u200bGCC\u200bAAG\u200bCTG\u200bAAG\u200bGTG\u200bACC\u200bAAG\u200bGGT\u200bGGC\u200bCCC\u200bCTG\u200bCCC\u200bTTC\u200bTCC\u200bTGG\u200bGAC\u200bATC\u200bCTG\u200bTCC\u200bCCT\u200bCAG\u200bTTC\u200bATG\u200bTAC\u200bGGC\u200bTCC\u200bAGG\u200bGCC\u200bTTC\u200bATC\u200bAAG\u200bCAC\u200bCCC\u200bGCC\u200bGAC\u200bATC\u200bCCC\u200bGAC\u200bTAC\u200bTAT\u200bAAG\u200bCAG\u200bTCC\u200bTTC\u200bCCC\u200bGAG\u200bGGC\u200bTTC\u200bAAG\u200bTGG\u200bGAG\u200bCGC\u200bGTG\u200bATG\u200bAAC\u200bTTC\u200bGAG\u200bGAC\u200bGGC\u200bGGC\u200bGCC\u200bGTG\u200bACC\u200bGTG\u200bACC\u200bCAG\u200bGAC\u200bACC\u200bTCC\u200bCTG\u200bGAG\u200bGAC\u200bGGC\u200bACC\u200bCTG\u200bATC\u200bTAC\u200bAAG\u200bGTG\u200bAAG\u200bCTC\u200bCGC\u200bGGC\u200bACC\u200bAAC\u200bTTC\u200bCCT\u200bCCT\u200bGAC\u200bGGC\u200bCCC\u200bGTA\u200bATG\u200bCAG\u200bAAG\u200bAAG\u200bACA\u200bATG\u200bGGC\u200bTGG\u200bGAA\u200bGCA\u200bTCC\u200bACC\u200bGAG\u200bCGG\u200bTTG\u200bTAC\u200bCCC\u200bGAG\u200bGAC\u200bGGC\u200bGTG\u200bCTG\u200bAAG\u200bGGC\u200bGAC\u200bATT\u200bAAG\u200bATG\u200bGCC\u200bCTG\u200bCGC\u200bCTG\u200bAAG\u200bGAC\u200bGGC\u200bGGC\u200bCGC\u200bTAC\u200bCTG\u200bGCG\u200bGAC\u200bTTC\u200bAAG\u200bACC\u200bACC\u200bTAC\u200bAAG\u200bGCC\u200bAAG\u200bAAG\u200bCCC\u200bGTG\u200bCAG\u200bATG\u200bCCC\u200bGGC\u200bGCC\u200bTAC\u200bAAC\u200bGTC\u200bGAC\u200bCGC\u200bAAG\u200bTTG\u200bGAC\u200bATC\u200bACC\u200bTCC\u200bCAC\u200bAAC\u200bGAG\u200bGAC\u200bTAC\u200bACC\u200bGTG\u200bGTG\u200bGAA\u200bCAG\u200bTAC\u200bGAA\u200bCGC\u200bTCC\u200bGAG\u200bGGC\u200bCGC\u200bCAC\u200bTCC\u200bACC\u200bGGC\u200bGGC\u200bATG\u200bGAC\u200bGAG\u200bCTG\u200bTAC\u200bAAGTAATGT\u200bTTC\u200bCTT\u200bCTC\u200bCGA\u200bGGG\u200bGCT\u200bGGA\u200bGCG\u200bGGG\u200bATC\u200bTGA\u200bTGA\u200bAAA\u200bGGT\u200bCAG\u200bACT\u200bGAA\u200bACG\u200bCCT\u200bTGC\u200bACG\u200bGCT\u200bGCT\u200bCGG\u200bCTT\u200bGAT\u200bCAC\u200bAGC\u200bTCC\u200bCTA\u200bGGT\u200bAGG\u200bCAC\u200bCAC\u200bAGA\u200bGAA\u200bGAC\u200bCTT\u200bCTA\u200bGTG\u200bAGC\u200bCTG\u200bGGC\u200bCAG\u200bGAG\u200bCCC\u200bACA\u200bGTG\u200bCCT-3\u2032, where A = silent mutations in 5\u2032 homology arm, flexible linker region (GSASGSSGASGSASG) is bold, and mScarlet is italic.The gRNA sequence was 5\u2032-CAA\u200bATC\u200bCTA\u200bCAA\u200bTGG\u200bCAC\u200bTG-3\u2032 and the donor template was: 5\u2032-GGT\u200bTTG\u200bAGT\u200bGTG\u200bTGA\u200bGCT\u200bAAC\u200bATG\u200bTGT\u200bCCT\u200bCAT\u200bCCT\u200bCTT\u200bCCC\u200bCGC\u200bCGT\u200bGTT\u200bCTG\u200bTAG\u200bGCT\u200bTCA\u200bAAT\u200bCCA\u200bTTA\u200bTAC\u200bAGA\u200bAAG\u200bCCT\u200bATC\u200bTCC\u200bACG\u200bCAC\u200bACT\u200bGTG\u200bGAC\u200bTTC\u200bACC\u200bTTC\u200bAAC\u200bAAG\u200bTTC\u200bAAC\u200bAAA\u200bTCLentiviruses for shRNA production were produced using packaging plasmids pCMVR and pMD2.g and specific shRNAs in pLKO.1 vector as follows: 80% confluent HEK293T cells in DMEM supplemented with 10% FBS and 100 U penicillin\u2013streptomycin were transfected using PEI MAX transfection reagent. 5 h later, the medium was changed to DMEM supplemented with 4% FBS and 25 mM Hepes. Media containing lentiviral particles were harvested after 48 and 72 h, filtered (0.45 \u00b5m), aliquoted, and stored at \u221280\u00b0C.U2Os-AP2-GFP cells were transduced with lentiviral media expressing respective shRNAs in the presence of Polybren 8 \u00b5g/ml  for 5 h and replaced with culture medium. 48 h later, puromycin (1 \u00b5g/ml) was added for 24 h to allow selection of transduced cells. Experiments targeting AP2 subunits were performed otherwise similarly but without puromycin selection. All shRNA-silenced cell lines were replated on non-coated glass-bottomed imaging dishes 1 d prior to imaging.ITGB5 shRNAs: TRCN0000057744 (5\u2032-GCA\u200bTCC\u200bAAC\u200bCAG\u200bATG\u200bGAC\u200bTAT-3\u2032) and TRCN0000057745 (5\u2032-GCT\u200bGTG\u200bCTA\u200bTGT\u200bTTC\u200bTAC\u200bAAA-3\u2032).ITGB1 shRNAs: TRCN0000029644 (5\u2032-CCT\u200bGTT\u200bTAC\u200bAAG\u200bGAG\u200bCTG\u200bAAA-3\u2032) and TRCN0000029645 (5\u2032-GCC\u200bTTG\u200bCAT\u200bTAC\u200bTGC\u200bTGA\u200bTAT-3\u2032).AP2S1 shRNAs: TRCN0000060263 (5\u2032-GAC\u200bGCC\u200bAAA\u200bCAC\u200bACC\u200bAAC\u200bTTT-3\u2032), TRCN0000060266 (5\u2032-GTG\u200bGAG\u200bGTC\u200bTTA\u200bAAC\u200bGAA\u200bTAT-3\u2032), and TRCN0000060267 (5\u2032-CAC\u200bAAC\u200bTTC\u200bGTG\u200bGAG\u200bGTC\u200bTTA-3\u2032).AP2A1 targeting shRNAs: TRCN0000065108 (5\u2032-GCT\u200bGAA\u200bTAA\u200bGTT\u200bTGT\u200bGTG\u200bTAA-3\u2032) and TRCN0000065109 (5\u2032-GCA\u200bCAT\u200bTGA\u200bCAC\u200bCGT\u200bCAT\u200bCAA-3\u2032)Integrin \u03b15 (ITGA5) targeting shRNAS: TRCN0000029651 (5\u2032-CCA\u200bCTG\u200bTGG\u200bATC\u200bATC\u200bATC\u200bCTA-3\u2032) and TRCN0000029652 (5\u2032-CCT\u200bCAG\u200bGAA\u200bCGA\u200bGTC\u200bAGA\u200bATT-3\u2032).Tensin1 targeting (TNS1) shRNAs: TRCN0000002953 (5\u2032-GAG\u200bGAT\u200bAAG\u200bATT\u200bGTG\u200bCCC\u200bATT-3\u2032) and TRCN0000002954 (5\u2032-CCC\u200bAAA\u200bGAA\u200bGGT\u200bACG\u200bTGC\u200bATT-3\u2032).Talin1 targeting (TLN1) shRNA: TRCN0000123106 (5\u2032-GCC\u200bTCA\u200bGAT\u200bAAT\u200bCTG\u200bGTG\u200bAAA-3\u2032)The following sequences were targeted:Cell culture dishes were coated with 10 \u00b5g/ml of FN in a cell culture incubator or left uncoated. The next day, U2Os cells were plated to reach confluency in 24 h. RNA extraction, cDNA synthesis, and PCR followed those described by U2Os were silenced for respective proteins, as mentioned above, and lysed into 150 mM NaCl, 50 mM Tris, pH 8, and 1% NP-40 (Sigma-Aldrich). Protein concentrations were measured with the Pierce BCA kit (Thermo Fisher Scientific) and boiled in Laemmli loading buffer with 10% \u03b2-mercaptoethanol. Equal amounts of protein lysates were loaded onto SurePAGE Bis-Tris 4\u201320% gels (Genscript) and transferred either to 0.2 \u00b5m polyvinylidene difluoride (PVDF) or 0.45 \u00b5m nitrocellulose  membranes. Membranes were blocked in 5% BSA or skimmed milk for 1 h, incubated with primary antibodies at +4\u00b0C overnight, and HRP-conjugated secondary antibodies  for 1 h in RT. HRP was activated with Supersignal West Femto or Pico reagents (Thermo Fisher Scientific) and bands were detected with ChemiDoc XRS+ (BioRad).All live videos and images from fixed samples were acquired with the ONI nanoimager microscope equipped with 405, 488, 561, and 647 lasers, an Olympus 1.49NA 100\u00d7 super achromatic objective, and a Hamamatsu sCMOS Orca flash 4 V3 camera. Image acquisition software used was NimOS. Fixed samples were imaged in RT and live imaging was carried out at +37\u00b0C. Used fluorochromes were GFP, mScarlet, JF-646, Alexa-488, Alexa-594, and Alexa-647.35,000 U2Os-AP2-GFP cells were plated on precoated/non-coated areas of the dish resulting in \u223c70\u201380% confluence 16\u201320 h later, at the onset of imaging. Alternatively, shRNA-silenced cell lines were plated. After overnight culture, 25 \u00b5M Hepes was added and samples were subjected to live TIRF imaging in a preheated +37\u00b0C chamber.The ONI Nanoimager microscope set to TIRF angle was used to acquire AP2 lifetimes at the cell membrane from 300 frames (1 frame/s) with an exposure time of 330 ms. Each video represents endocytic events from two to three cells .Acute modulation of ligand binding activity for integrin \u03b21 was achieved using the function-blocking antibody mab13 (0.3 \u00b5g/ml). U2Os-AP2-GFP cells were plated on FN, as explained above, and 16\u201320 h later subjected to live TIRF imaging. 0-min sample has no mab13 added to control baseline FCL proportions. Immediately after mab13 addition, 5-min time lapses were continuously collected until 35 min; control videos (time point 0) had no mab13 added.To acutely induce the inhibition of integrin \u03b1v\u03b25, we used the small molecular inhibitor Cilengitide . U2Os-AP2-GFP cells plated on non-coated imaging dishes were treated with Cilengitide for 15 or 45 min, fixed, stained, and imaged with the ONI Nanoimager microscope at TIRF angle and analyzed for the resulting reduction of RA coverage.U2Os-AP2-GFP-ITGB5-mScarlet cells were plated on non-coated imaging dishes and 1 d later confluent monolayers were treated with 1 \u00b5M Cilengitide for 15\u201325 min, during which most FCLs and RAs were dissociated from the cell membrane. Samples were then washed twice and immediately subjected to live TIRF imaging to detect the de novo formation of FCLs and RAs. 1-h time-lapses were acquired with the ONI Nanoimager microscope at TIRF angle, at 30 s intervals, with an exposure time of 100 ms for AP2 and 300 ms for integrin \u03b25.For immunofluorescence experiments, cells were fixed with 4% paraformaldehyde\u2013PBS for 15 min in a +37\u00b0C incubator, washed with PBS, and blocked with 1% BSA-PBS. Primary antibodies diluted in 1% BSA\u2013PBS were incubated for 1 h, samples were washed with PBS, and secondary antibodies diluted in 1% BSA\u2013PBS were let to bind for 30 min. Samples were imaged with the ONI Nanoimager microscope using a TIRF angle and exposure times of 500 or 1,000 ms.Clathrin assembly at the cell membrane was reduced by silencing two subdomains of AP2 or by overexpressing AP180 ct, which acts as a dominant negative for AP2. U2Os AP2-GFP cells silenced for AP2A1 shRNA #1 and #2 or AP2S1 shRNA #1, #2, and #3, and control shRNA, and plated on non-coated imaging dishes were fixed and stained for integrin \u03b1v\u03b25 and FA marker p-Pax. Alternatively, AP180 ct was overexpressed in U2Os AP2-GFP cells, and cells were plated on non-coated imaging dishes, fixed the next day, and stained for integrin \u03b1v\u03b25 and p-Pax. RA formation (integrin \u03b1v\u03b25 adhesions without FA marker) and FAs  were imaged using TIRF microscopy.We performed time-series experiments to block integrin \u03b21 active conformation with the mab13 antibody or to inhibit integrin \u03b1v\u03b25 with Cilengitide. Experiments utilizing mab13 were performed with U2Os-AP2-GFP cells plated on FN-coated dishes to disfavor the preformation of FCLs and RAs. Mab13 (0.3 \u00b5g/ml) was added to replicate samples and fixed 15, 30, or 45 min later. Similarly, U2Os-AP2-GFP cells plated on non-coated dishes to favor the preformation of FCLs and RAs were treated with Cilengitide. Samples were stained for integrin \u03b1v\u03b25 and FA marker p-Pax and imaged using TIRF microscopy.Cilengitide washout experiments were developed to study acute reappearance of FCLs and RAs in cell cultures. Cells were treated for 15\u201325 min with 1 \u00b5M Cilengitide (or DMSO as control) and washed two times. Unwashed controls were collected. Washed samples were let to recover for indicated time points, fixed, stained for p-Pax, subjected to TIRF imaging, and analyzed for the reappearance of FCLs and RAs (RA coverage).U2Os-AP2-GFP cells silenced for integrin \u03b21, integrin \u03b25, integrin \u03b15, or Tensin1 and controls were plated either on FN-coated dishes  or on non-coated dishes (control shRNA and ITGB5 shRNAs) and fixed the next day, 16\u201320 h later. Samples were stained for integrin \u03b1v\u03b25 and FA marker p-Pax and imaged using TIRF microscopy.U2Os-AP2-GFP or U2Os-AP2-GFP-ITGB5-mScarlet cells were seeded onto micropatterned glass coverslips (CYTOO) either precoated with FN 10 \u00b5g/ml or left uncoated. Excess amount  of cells were plated and monitored for attachment to the patterns. The cells plated on FN had attached in 1 h and the cells plated on non-coated micropatterns had attached in 4 h. After attachment, excess cells were carefully rinsed and the samples were fixed 16\u201320 h later, stained, and subjected to immunofluorescence analysis with TIRF microscopy.U2Os-AP2-GFP-ITGB5-mScarlet cells were plated confluent onto non-coated imaging dishes for 2 d. Fully confluent monolayers were wounded with a micropipette tip, washed twice with fresh complete medium, and let to migrate. 0-min samples were collected directly after wounding. After fixing, samples were stained and subjected to TIRF imaging. Tile images (5 \u00d7 1), starting at the edge of the wound, were acquired with a 25% overlap and stitched using the pairwise stitching plug-in Image J.n\u201d refers to a movie, which contained two to four cells. To determine the proportion of FCLs, we used the output from u-track to count the number of pits (events lasting longer than 20 s and shorter than 120 s) and the number of FCLs  and p-Pax channels were transformed into binary masks using the Robust Automatic Threshold Selection function. The Robust Automatic Threshold Selection parameters used for segmentation  were defined by visual inspection for each experiment. The mask of the p-Pax channel was then subtracted from the mask of the ITGB5 (or \u03b1v\u03b25) channel using the \u201cimage calculation\u201d function. This calculation results in a mask containing only the RAs . RA coverage was then calculated by dividing the area covered by RAs in the RA mask (integrated density/255) by the area of each marked cell (or cells) in the image. An illustration of this method is shown in Focal adhesion coverage  was calcFor the wound healing experiment, a line on the migration front of each image was manually drawn. This line was then used as a reference to automatically draw a box 100 pixels in width (11.7 \u00b5m). RA coverage (as above) was calculated for this box, which was then moved inward in the culture in 50-pixel steps, where the RA coverage analysis was repeated. Values are normalized to the average RA coverage on the three innermost areas in the culture.Events showing the appearance of both AP2 and ITGB5 were identified by visual inspection of videos. For the generation of graphs, we selected only events where we could unambiguously ensure that significant FCLs and ITGB5 signals were not present in the region for at least 3 min. For the events where an RA was established , left, tAP2, ITGB5, and p-Pax channels were segmented using the Robust Automatic Threshold Selection function (similarly to what was done in the RA coverage experiments). Individual AP2 spots were identified as regions of interest (ROIs) using the \u201canalyze particles\u201d function. The fluorescence intensity for each AP2 spot  was then measured from the original image. Using the same AP2 ROIs in the ITGB5 and p-Pax binary masks, we then classified each AP2 spot for their colocalization with either marker. In this case, an AP2 ROI with a non-zero signal in either mask was considered colocalizing. We used full images for these analyses.Individual cells were marked and AP2, ITGB5, and p-Pax channels were segmented using the Robust Automatic Threshold Selection function. RAs were defined as ITGB5 signals not colocalizing with p-Pax. In the conditions used for these experiments (FN + mab13), RAs were primarily individual spots. The colocalization of RAs to AP2 was classified by measuring the intensity of each RA region at the segmented AP2 channel.AP2, ITGB5, and P-Pax channels were segmented using the Robust Automatic Threshold Selection function. Each segmented AP2 spot had the fluorescence intensity measured from the original image. A 3 \u00b5m \u00d7 3 \u00b5m region was drawn around each AP2 spot and used to measure the intensity of FN from the original image. Data are presented as the fluorescence for each AP2 spot.n-values and individual repeats used in analyses. For multiple comparisons, one-way ANOVA was performed followed by Tukey\u2019s multiple comparison. Pairwise comparisons were performed using two-tailed Student\u2019s t test with two-tailed distributions. Data distributions were assumed to be normal, but this was not formally tested. All graphs and statistical calculations were performed with GraphPad Prism 9.Figure legends state the exact Review HistoryClick here for additional data file.SourceData FS1is the source file for Fig. S1.Click here for additional data file.SourceData FS4is the source file for Fig. S4.Click here for additional data file.SourceData FS5is the source file for Fig. S5.Click here for additional data file."}
{"text": "Scientific Reports 10.1038/s41598-023-31831-1, published online 20 March 2023Correction to: The original version of this Article contained errors in the Results and discussion, under the subheading \u2018Mechanisms for Fe-DOMP\u00a0formation and isolation\u2019,FeCl3\u2009+\u20096NaOH\u2009+\u20096DOM\u2009\u2212\u2009COOH\u2009=\u20092Fe3\u2009+\u20096DOM\u2009\u2212\u2009COO\u2212\u2009+\u20092NaCl3\u2009+\u20096H2O.\u201d\u201c2now reads:FeCl3\u2009+\u20096NaOH\u2009+\u20096DOM\u2009\u2212\u2009COOH\u2009=\u20092Fe3\u2009+\u20096DOM\u2009\u2212\u2009COO\u2212\u2009+\u20096NaCl\u2009+\u20096H2O.\u201d\u201c2The original Article has been corrected."}
{"text": "Correction to: Neuropsychology Review10.1007/s11065-022-09536-5g\u2009=\u20090.21\u201d and \u201cg\u2009=\u20090.21\u201d were incorrect and should be corrected to \u201cHedges\u2019 g\u2009=\u20090.20\u201d and \u201cg\u2009=\u20090.20\u201d, respectively.In the original published paper, it was belatedly discovered that there was a data entry error in pages 1 and 15. The value of \u201cHedges\u2019 The original article has been corrected."}
{"text": "We prove that there exists a global weak solution of this initial\u2010boundary value problem and provide a representation for the solution in terms of the solution of a Riemann\u2013Hilbert problem. Using this representation, we obtain asymptotic formulas for the long\u2010time behavior of the solution. In particular, by restricting our asymptotic result to solutions whose initial data are close to the initial profile of the stationary one\u2010soliton, we obtain results on the asymptotic stability of the stationary one\u2010soliton under any small perturbation in H1,1(R+). In the focusing case, such a result was already established by Deift and Park using different methods, and our work provides an alternative approach to obtain such results. We treat both the focusing and the defocusing versions of the\u00a0equation.We consider the nonlinear Schr\u00f6dinger equation\u00a0on the half\u2010line  It arises in a variety of situations such as the modeling of slowly varying wave packets in nonlinear media 1 and [A]2 denote the first and second columns of a 2\u00d72 matrix A.\u2013A is n\u00d7m matrix, we define |A|\u2a7e0 by |A|2=\u03a3i,j|Aij|2. Then |A+B|\u2a7d|A|+|B| and |AB|\u2a7d|A||B|.If \u2013\u03b3\u2282C and 1\u2a7dp\u2a7d\u221e, we write A\u2208Lp(\u03b3) if |A| belongs to Lp(\u03b3). We write \u2225A\u2225Lp(\u03b3)\u2254\u2225|A|\u2225Lp(\u03b3).For a (piecewise smooth) contour \u2013C+={k\u2208C|Imk>0} and C\u2212={k\u2208C|Imk<0} denote the open upper and lower halves of the complex plane.\u2013R+=, that is, e\u03c3^3A=e\u03c33Ae\u2212\u03c33. Define the functions a(k) and b(k) byr(k) and \u0394(k) byq is the constant appearing in the Robin boundary condition  lies in H1,1(R) and depends continuously on u0 whenever \u0394 has no real zeros.Proposition 2.1 Given u0\u2208H1,1(R+) and q\u2208R, define a(k), r(k), and \u0394(k) by ((i)\u0394(k) is continuous for Imk\u2a7e0 and analytic for Imk>0.(ii)\u0394(k) obeys the symmetry(iii)\u0394(0)=\u2212iq.(iv)k\u2192\u221e, \u0394(k) satisfiesAs (v)\u0394(k)\u22600 for all k\u2208R, then r\u2208H1,1(R). Moreover, the map u0\u21a6r:H1,1(R+)\u2229{u0|\u0394(k)\u22600forallk\u2208R}\u2192H1,1(R) is continuous.If  \u0394(k) by \u20132.4). T. T, the following also hold:(a)|r(k)|<1 for all k\u2208R.(b)q<0, then \u0394 has no zeros in C\u00af+={k\u2208C|Imk\u2a7e0}.If (c)q>0, then \u0394 has no zeros on R and exactly one simple zero in C+. Moreover, this zero is pure imaginary.If (d)q=0, then \u0394(k)\u22600 for all k\u2208C\u00af+\u2216{0}, but \u0394(0)=0.If (e)a(k)\u22600 for all k\u2208C\u00af+.In the defocusing case The statements of our main theorems involve four spectral functions ondition . The fol\u03bb=\u22121. Recall that we introduced the notion of a global weak solution in H1,1(R+) in Definition\u00a0Proposition 2.2 Suppose \u03bb=1 or \u03bb=\u22121. Let q\u2208R and u0\u2208H1,1(R+). Then there exists a unique global weak solution in H1,1(R+) of the Robin IBVP for NLS with parameter q and initial data u0. Moreover, for each T>0, the data\u2010to\u2010solution mapping u0\u21a6u is continuous from H1,1(R+) to C).Before stating our main results we also need the following well\u2010posedness result for the Robin IBVP. The proof, which is given in Appendix\u00a0Remark 2.3q in  well\u2010posedness, we give an independent proof of Proposition q is constant is much easier to handle than the case considered in \u22822 of \u03bc. The second column of the Volterra Equation\u00a0=\u2211l=0\u221e\u03c8l converges absolutely and uniformly for x\u2a7e0 and Imk\u2a7e0 to a continuous function \u03c8. Using the uniform convergence of the series  satisfies \u2208C1(R+) for each k\u2208C\u00af+ and the second column of =1 follows from =\u2211uation\u00a0=\u03c8uation\u00a0=\u03c8olumn of  follows uation\u00a0=\u03c8mmetries  then folmmetries . The uniows from  and the |U0(x\u2032)|\u2a7e0, we findx to \u221e gives\u2225u0\u2225L1(R+)\u2a7d\u2225u0\u2225H1,1(R+), we obtainC>0 depends on u0 but is independent of x and k. Sincex\u2a7e0 and Imk\u2a7e0,\u25a1Let us prove . Equatio.Hence|\u03c8^olumn of  follows  obtain|\u03c8\u2212olumn of  then fololumn of .\u25a1a(k)=\u03bc22 and b(k)=\u03bc12, we immediately obtain the following properties of a(k) and b(k).Corollary 3.2u0\u2208H1,1(R+), then a(k) and b(k) defined in \u2229{u0|\u0394(k)\u22600forallk\u2208R} into H1,1(R).Lemma 3.3g\u21a6f(x)\u2254\u222bx\u221eg(x\u2032)dx\u2032 is continuous H0,1(R+)\u2192L2(R+).The linear map The next few lemmas are needed to show that g\u2208H0,1(R+). Then, by definition, g,xg\u2208L2(R+). The estimatef(x) is well defined and absolutely continuous. Also, f is bounded since limx\u2192\u221ef(x)=0. Consider the Sobolev space H1(R)={h\u2208L2(R)|h\u2032\u2208L2(R)}. A function h belongs to H1(R) if and only if k\u21a6\u27e8k\u27e9h^(k) lies in L2(R), where h^(k)=\u222bRh(x)e\u2212ikxdx denotes the Fourier transform of h. Let us extend g(x) and f(x) to the negative real axis by setting g(x)=0 and f(x)=0 for x<0. Then, since \u27e8x\u27e9g\u2208L2(R), the Fourier transform g^(k) of g(x) belongs to H1(R). Moreover, the Fourier transform f\u2032^=ikf^ is well defined as a tempered distribution and, for a test function \u03d5, we haveikf^(k)=g^(0)\u2212g^(k) in the sense of distributions. Since g^\u2208H1(R), g^ is absolutely continuous and soAssume that H1(R)\u2192L2(R). Since the Fourier transform is bounded H0,1(R)\u2192H1(R) and the inverse Fourier transform is bounded L2(R)\u2192L2(R), it would follow that the composition g\u21a6g^\u21a6f^\u21a6f is a bounded linear map H0,1(R)\u2192L2(R). Thus it only remains to show that the linear map (H1(R)\u2192L2(R). To prove this, note thatM\u03c6 of \u03c6 is defined by\u03c6\u21a6M\u03c6 is a bounded linear map L2(R)\u2192L2(R). HenceH1(R)\u2192L2(R).\u25a1Suppose that we can show that the mapg^\u21a6\u22121ik\u222bg^\u21a6\u22121ik\u222bE and \u03c80 be as in \u22121=1+K+\u22ef is the Neumann series. Properties of \u03c8 can be obtained by analyzing the operator K. In the following, Lxp) denotes the space of Lkq(R)\u2010valued Lxp(R+)\u2010functions with the normLemma 3.4(Compare with 2 of \u03bc\u223c satisfies the Volterra equationK\u2032 is the operator defined in , we have h1\u2208Lx\u221e) by Lemma\u00a0h2\u2208Lx\u221e) by Lemma\u00a0u0 has compact support, then b\u2032(k)a\u2032(k)=\u03d5\u2208L2(R) so that a(k)\u22121 and b(k) lie in H1(R). To extend this result to the case of general u0\u2208H1,1, we consider the continuity of the u0\u2010dependence.Let us now again consider the case of a general potential u0,u\u02c70\u2208H1,1(R+) and let \u03c8,\u03c8\u02c7,\u03d5,\u03d5\u02c7 be the associated solutions of the Equations\u00a0) and \u2225\u0394\u03d5\u2225Lx\u221e) in terms of \u2225\u0394u0\u2225H1,1. Note that \u0394\u03c8 satisfies the Volterra equationu0,u\u02c70 in bounded subsets of H1,1(R+). Moreover, using parts (d) and (f) of Lemma\u00a0u0,u\u02c70 in bounded subsets of H1,1(R+). Similarly, \u0394\u03d5 satisfies the Volterra equationu0,u\u02c70 in bounded subsets of H1,1(R+). The inequalities \u2225L2 and \u2225\u0394\u03d5\u2225L2 tend to zero whenever \u2225\u0394u0\u2225H1,1 tends to zero, provided that u0,u\u02c70 remain in a bounded subset of H1,1(R+). Since , a continuity argument implies that \u03bc\u2208H1(R) for any u0\u2208H1,1(R+) and that the relation . We conclude that a\u22121,b\u2208H1(R) and that the maps in  lies in the weighted Sobolev space H1,1(R), but the spectral functions a(k)\u22121 and b(k) are not in H1,1(R). To proceed, we therefore need to subtract suitable constants.Lemma 3.7u0\u2208H1,1(R+) and define b1\u2208C bykb(k)\u2212b1\u2208L2(R).Let We want to show that the reflection coefficient \u03c8=\u03c81\u03c82T, we haveb(k)=\u03c81, we deduce from (kb(k)\u2212b1\u2208L2(R). Here we have used that by the Fubini\u2013Tonelli theorem, if f\u2208Lk2) or f\u2208Lx2), then \u2225\u2225f\u2225Lx2\u2225Lk2=\u2225f\u2225Lk2\u2225Lx2.\u25a1If we expand Equation\u00a0 in compouce from  and  and let \u03c8,\u03c8\u02c7 be the associated solutions of  hold also for the spectral function a(k). Namely, if for u0\u2208H1,1(R+) we define a1:=\u2212\u03bb2i\u222b0\u221e|u0(x)|2dx, then the mappingsk(a(k)\u22121)\u2212a1\u2208L2(R); given this fact, the continuity follows by arguments similar to those used in the proof of Lemma k(a(k)\u22121)\u2212a1\u2208L2(R), note that Equation\u00a0(1\u27e8x\u27e9 is in Lx\u221e(R+)\u2229Lx2(R+), we can estimate the three terms on the right\u2010hand side of  as follows:\u2225k(a(k)\u22121)\u2212a1\u2225Lk2<\u221e.Conclusions similar to those established in Lemma Equation\u00a0 impliesEquation\u00a0, we haveuation\u00a0\u2212uation\u00a0 and to establish the continuity of the map u0\u21a6r.Lemma 3.11u0\u2208H1,1(R+) and q\u2208R, define the reflection coefficient r(k) by the formula (\u0394(k)\u22600 for all k\u2208R, then r\u2208H1,1(R). Moreover, the map u0\u21a6r:H1,1(R+)\u2229{u0|\u0394(k)\u22600forallk\u2208R}\u2192H1,1(R) is\u00a0continuous.Given  formula . If \u0394(k)We are now ready to prove that We have\u21a6fg:H1\u00d7H1\u2192H1 is continuous. Since 2k+iq2k\u2212iq\u22081+H1(R), we see thata\u22121\u2208H1(R) by Lemma\u00a0kb(k)\u2212b1\u2208L2(R) by Lemma\u00a0fa\u2208H1(R) and fb\u2208L2(R) such thata and b, we can writeb1\u2208C, fa\u2208H1(R), fb\u2208L2(R), a\u22081+H1(R), b\u2208H1(R) depend continuously on u0\u2208H1,1(R+). Thus, using Lemma\u00a0\u25a1By Lemma\u00a0o Lemma\u00a0u0\u21a6F:H1,ow thatu0\u21a6(2k\u2212i3.3\u0394(k) is continuous for Imk\u2a7e0 and analytic for Imk>0. The symmetry (k=0 and using the unit determinant relation (\u0394(0)=\u2212iq. The asymptotic formula =\u0394(k)/(2k\u2212iq) be the function defined in |<1 for all k\u2208R if \u03bb=1. This proves (a). To prove (b)\u2013(d), we employ (b(\u2212k\u00af)\u00af from the right\u2010hand side of (Let fined in . A straifined in  shows thned in (1\u2212\u03bb|r(k)e employ  to elimi side of  and use  side of  to see t\u03bb=1. Then Equation\u00a0(a(k)\u22600 for all k\u2208R, and thus (\u0394a(k)\u22600 for all k\u2208R. Let Z\u0394a and P\u0394a denote the number of zeros and poles of \u0394a in C+ counted with multiplicity, and let Za denote the number of zeros of a(k) in C+ counted with multiplicity. Recalling the asymptotic formulas 2\u03c0i|k=\u2212\u221e+\u221e denotes the winding number of f(k) around the origin as k traverses the real axis from \u2212\u221e to +\u221e. On the other hand, by \u22600 for all k\u2208R, the assertions (b)\u2013(d) about the zeros of \u0394\u00a0follow.Suppose now that Equation\u00a0 implies and thus  shows thformulas  and a of the Robin IBVP, then the RH problem of Theorem\u00a0m, and u can be recovered from m via (Proposition 4.1\u03bb=1 or \u03bb=\u22121. Let u0\u2208S(R+) be such that q\u2254\u2212u0x(0)/u0(0)\u2208R. If \u03bb=\u22121, suppose that Assumption u of the Robin IBVP for NLS with initial data u0. Then the RH problem of Theorem m and, for each \u2208 be a smooth cut\u2010off function such that \u03b7 is even, \u03b7\u22611 in a neighborhood of 0, \u03b7(x)=0 for |x|\u2a7e1, and \u222bR\u03b7(x)dx=1. Let u0e\u2208H1,1(R) be an extension of u0 to R such that u0e(x)=u0(x) for x\u2a7e0 and u0e(x)=0 for x\u2a7d\u22121. Standard mollifier arguments show that v\u03b5(x)\u2254\u222bRu0e(y)\u03b7\u03b5(x\u2212y)dy, where \u03b7\u03b5(x)\u2254\u03b7(x/\u03b5)/\u03b5, are smooth compactly supported functions such that v\u03b5(x)\u2192u0e in H1,1(R) as \u03b5\u21930. If u0(0)\u22600, then v\u03b5(0)\u22600 for all sufficiently small \u03b5, and we can obtain the desired sequence by setting\u03b5=1/(N+n) and N is large enough. If u0(0)=0, then by modifying u0e so that u0e(\u2212x)=\u2212u0(x) for all sufficiently small x>0, we may assume that v\u03b5(0)=0 for all small enough \u03b5, and then straightforward estimates show that\u03b5=1/(N+n) provides a sequence with the desired properties.\u25a1Let Lemma 6.2u0\u21a6a and u0\u21a6b are continuous H1,1(R+)\u2192L\u221e(C\u00af+).The maps \u0394\u03c8=K\u0394u0\u03c8\u02c7+Ku0\u0394\u03c8, where, by . Setting x=0 in this estimate, the lemma follows.\u25a1Using the same notation as in the proof of Lemma\u00a0here, by , 3.9), , \u0394\u03c8=K\u0394u0here, by ,| and suppose that the associated spectral functions \u0394\u02c7,a\u02c7 satisfy Assumption u0 sufficiently close to u\u02c70 in H1,1(R+). Moreover, if {\u03bei}1M are the simple zeros in C+ of \u0394(k), then the maps u0\u21a6M\u2208Z and u0\u21a6\u2208CM are continuous at u\u02c70\u2208H1,1(R+).Let q<0. The function \u0394a(k)\u22081+H1(R) defined in  by Lemma\u00a0\u0394a, and hence also \u0394, is nonzero on R for any u0 sufficiently close to u\u02c70. Moreover, \u0394a has the same number of zeros and poles in C+ as \u0394, and \u0394a\u21921 as k\u2192\u221e. Since \u0394 has no poles in C+, the argument principle applied to a large semicircle enclosing the upper half\u2010plane yieldsZ\u0394 is the number of zeros of \u0394 in C+ counted with multiplicity. Using again that \u0394a(k)\u2208H1(R) depends continuously on u0\u2208H1,1(R+), we infer that the map u0\u21a6Z\u0394\u2208Z is continuous at u\u02c70\u2208H1,1(R+).Suppose first that fined in  depends q<0, the function \u0394a has the same zeros as \u0394 in the upper half\u2010plane, and Cauchy's integral formula shows that if \u03bej\u2208C+ is a simple zero of \u0394a, then\u03b3j is a small circle in C+ around \u03bej which contains no other zeros of \u0394a. By Lemma\u00a0u0\u21a6\u0394a is continuous H1,1(R+)\u2192L\u221e(C\u00af+). Since \u0394a is analytic in C+, it follows that u0\u21a6\u0394\u02d9a is continuous H1,1(R+)\u2192L\u221e(K) where K is any compact subset of C+. Since the zeros {\u03be\u02c7j}1M of \u0394\u02c7(k) are simple by assumption, we conclude that the zeros {\u03bej}1M of \u0394(k) are simple and depend continuously on u0 for u0 in a H1,1(R+)\u2010neighborhood of u\u02c70.Since \u0394a(k) show that the above conclusions hold also if q>0 (note that the identity (\u0394a replaced by \u0394b).\u25a1Analogous arguments applied to the functionidentity  holds al\u0394 is nonempty, we replace the poles in the RH problem of Theorem\u00a0Dj\u2282C+, j=1,\u22ef,M, be small disjoint open disks centered at the zeros \u03bej, j=1,\u22ef,M, of \u0394. Let Dj\u2217 be the image of Dj under complex conjugation. Define the contour \u0393 by\u2202Dj is oriented clockwise and \u2202Dj\u2217 is oriented counterclockwise. Let{cj}1M are the residue constants defined in (m satisfies the RH problem of Theorem\u00a0m\u223c satisfies the following RH problem.RH problem 6.4m\u223c) with the following properties.(a)m\u223c:C\u2216\u0393\u2192C2\u00d72 is analytic.(b)m as k approaches \u0393 from the left (+) and right (\u2212) exist are continuous on \u0393, and satisfyv\u223c is defined byThe boundary values of (c)m\u223c\u2192I as k\u2192\u221e.If the set of zeros of fined in . Then m Definition 6.5U\u2282H1,1(R+) denote the set of all potentials u0\u2208H1,1(R+) such that the corresponding spectral functions satisfy the assumptions of Theorem \u03bb=\u22121, then U consists of all u0\u2208H1,1(R+) for which the associated spectral functions satisfy Assumption if \u03bb=1 and q<0, then U=H1,1(R+);if \u03bb=1 and q>0, then U consists of all u0\u2208H1,1(R+) for which the RH problem of Theorem\u00a0m for each \u2208[0,\u221e)\u00d7[0,\u221e).if We let We next consider the set of all initial data for which the assumptions of Theorem\u00a0h\u2208L2(\u0393), we define the Cauchy transform Ch byC+h and C\u2212h for the left and right boundary values of Cf on \u0393. Then C+ and C\u2212 are bounded operators on L2(\u0393) and given w\u2208L2(\u0393)\u2229L\u221e(\u0393), we define Cw:L2(\u0393)+L\u221e(\u0393)\u2192L2(\u0393) by Cw(f)=C\u2212(fw).Lemma 6.6u0\u2208U and q\u2208R\u2216{0}. Then RH problem m\u223c for each \u2208[0,\u221e)\u00d7[0,\u221e) and this solution admits the representationw\u223c=v\u223c\u2212I andLet For r\u2208H1,1(R). By Morrey's inequality, every function in H1,1(R+)\u2282H1(R+) is H\u00f6lder continuous with exponent 1/2. Hence, if \u03bb=\u22121, the unique existence of m\u223c follows by the same vanishing lemma arguments already used to prove Proposition\u00a0\u03bb=1 and q<0, because in this case M=0. If \u03bb=1 and q>0, then m, and hence also m\u223c, exists by assumption. Standard theory for RH problems then yields the representation formula . Moreover, the map\u03bc\u223c is defined by  is continuous. Recalling the definition \u21a6v\u2212I:[0,\u221e)2\u00d7U\u2192(L2\u2229L\u221e)(R) is continuous. On the other hand, since \u0394a\u2208L\u221e(C+) depends continuously on u0\u2208H1,1(R+) by Lemma\u00a0\u03bej depends continuously on u0\u2208U by Lemma\u00a0\u0394\u02d9(\u03bej)\u2208C depends continuously on u0\u2208U. The continuous dependence of a(\u03bej),b(\u2212\u03be\u00afj)\u2208C on u0\u2208H1,1(R+) follows from Lemma\u00a0u0\u21a6cje2i\u03b8 is continuous [0,\u221e)2\u00d7U\u2192C for each j. We conclude that \u21a6w\u223c=v\u223c\u2212I:[0,\u221e)2\u00d7U\u2192(L2\u2229L\u221e)(\u0393) is\u00a0continuous.By Proposition\u00a0finition  of v, stB(L2(\u0393)) denote the space of bounded linear operators on L2(\u0393). The set of invertible operators is open in B(L2(\u0393)) and the linear map w\u223c\u21a6Cw\u223cI lies in B(L2(\u0393)). Also, since \u2225Cw\u223c\u2225B(L2(\u0393))\u2a7dC\u2225w\u223c\u2225L\u221e(\u0393), the map w\u223c\u21a6I\u2212Cw\u223c is continuous L\u221e(\u0393)\u2192B(L2(\u0393)). Since \u21a6\u03bc\u223c\u2212I with \u03bc\u223c given by (I\u2212Cw\u223c\u21a6(I\u2212Cw\u223c)\u22121, this proves that the map in  is trivially open. Suppose therefore that \u03bb=1 and q>0. Then \u0394\u22600 on R, and \u0394 has exactly one zero \u03be1 in C+ by Proposition\u00a0a(\u03be1) is nonzero by Proposition\u00a0H1,1(R+), implying thatu0\u2010dependence explicitly for clarity. Fix u\u02c70\u2208U. By assumption, the RH problem for m\u223c corresponding to u\u02c70 has a solution for each \u2208[0,\u221e)\u00d7[0,\u221e). By standard theory for RH problem, this is equivalent to the map I\u2212Cw\u223c:L2(\u0393)\u2192L2(\u0393) being invertible for each \u2208[0,\u221e)\u00d7[0,\u221e). On the other hand, since w\u223c\u2192I\u2212Cw\u223c:L\u221e(R)\u2192B(L2(\u0393)) is continuous, the continuity of the map (F defined byV denote the open set of invertible operators in B(L2(\u0393)). Then F\u22121(V) is open. Given any compact set K\u2282[0,\u221e)2, K\u00d7{u\u02c70} is a compact subset of F\u22121(V), and hence there is a neighborhood U1\u2282H1,1(R+) of u\u02c70 such that \u2282F\u22121(V) for all \u2208K, that is, such that the solution m\u223c exists whenever u0\u2208U1 and \u2208K. On the other hand, by a Deift\u2013Zhou steepest descent analysis of the RH problem for m\u223c, one obtains that there is a neighborhood U2\u2282H1,1(R+) of u\u02c70 such that m\u223c exists for all u0\u2208U2 whenever x2+t2 is large enough. Indeed, for large x2+t2, the solution is well approximated by the global parametrix, which in this case is the solution of the RH problem corresponding to the pure stationary one\u2010soliton and always exists  is an open neighborhood of u\u02c70 such that m\u223c exists for all u0\u2208U1\u2229U2 and all \u2208[0,\u221e)2, that is, U1\u2229U2\u2282U. This shows that U is open.\u25a1If ng thatexample, ). By choProof of Theorem 1q\u2208R\u2216{0} and u0\u2208U. Let {u0(n)}n=1\u221e\u2282S(R+) be a sequence such that u0(n)\u2192u0 in H1,1(R+) as n\u2192\u221e and such that (u0(n))\u2032(0)+qu0(n)(0)=0 for each n\u2a7e1. Such a sequence exists by Lemma\u00a0U is open in H1,1(R+). Hence, passing to a subsequence if necessary, we may assume that {u0(n)}n=1\u221e\u2282U, which means that the assumptions of Proposition\u00a0u0(n). Applying Proposition\u00a0u(n) of the Robin IBVP for NLS with parameter q and initial data u0(n) for each n.Let m\u223c and \u03bc\u223c corresponding to u0, as well as the functions m\u223c(n) and \u03bc\u223c(n) corresponding to u0(n), are all well defined and satisfy (u(n) satisfies, for each n\u2a7e1,By Lemma\u00a0 satisfy  and  in H1,1(R+) of the Robin IBVP for NLS with parameter q and initial data u0. The Schwartz class solution u(n) is clearly also a solution in H1,1(R+) and by the continuity of the data\u2010to\u2010solution mapping established in Proposition\u00a0u(n)\u2192u as n\u2192\u221e for each . Hence, letting n\u2192\u221e in the (12)\u2010entry of R, and\u2208[0,\u221e)\u00d7[0,\u221e). By Lemma\u00a0\u03bc\u223c(n)\u2192\u03bc\u223c in I+L2(\u0393) as n\u2192\u221e. By Lemma\u00a0w\u223c(n)\u2192w\u223c in H1,1(R+) as n\u2192\u221e. Hence, Xn\u21920 and Yn\u21920 as n\u2192\u221e, so letting n\u2192\u221e in the (12)\u2010entry of (6.11)2,t)and2ilimk\u2192\u221ehis into , we obtahis into  for u. T77.1\u03bb=1 and q>0. Let u0\u2208H1,1(R+) and suppose that the RH problem of Theorem\u00a0m for each \u2208[0,\u221e)\u00d7[0,\u221e). By Proposition\u00a0\u0394(k) has one simple zero \u03be1\u2208iR>0, which means that m has simple poles at \u03be1 and \u03be\u00af1. We will use a Darboux transformation to remove these poles and consider a solution mreg of an associated regular RH problem without poles. Definer,rreg\u2208H1,1(R) and |r(k)|=|rreg(k)|<1 for k\u2208R. Let mreg be the unique solution of the RH problem of Theorem\u00a0r(k) replaced by rreg(k) and with no poles in C\u2216R. Such a solution exists by a standard vanishing lemma argument as described in the proof of Lemma\u00a0detmreg=1. Define B1 bym and mreg imply that B1 is an entire function of k which is O(1) as k\u2192\u221e; hence, B1\u2261B1 is independent of k. Evaluating  is given in terms of m by 12.Lemma 7.1(a)t\u2192\u221e, the function ureg defined in , where urad(1) is given by (As fined in  obeys thfined in  with r rgiven by .(b)\u03be1\u2208C+, thent\u2192\u221e uniformly for x\u2208[0,\u221e), whereIf Let e B1 byB1=\u2212kI+mnditions  satisfiee B1 byB1=\u2212kI+m of m by . Accordi of m by  and (7.5mreg has the same form as the RH problem associated to the defocusing NLS equation\u00a0on the line with reflection coefficient given by rreg(k). The lemma therefore follows from well\u2010known results (see [\u25a1The RH problem for lts (see ) in the t\u2192\u221e, we havex\u2208[0,\u221e), where usol and urad(2) are given by  of the defocusing NLS equation\u00a0whenever the initial data u0 lie in the subset U of H1,1(R+) defined in Definition\u00a0us0 clearly lie in U. Moreover, by Lemma\u00a0U is open in H1,1(R+). Hence there exists a neighborhood U of us0 such that U\u2282U. It follows that if u0\u2208U, then the global weak solution u with parameter q and initial data u0 satisfies  associated to u0 has exactly one simple zero in C+. By Lemma\u00a0\u03be1 and \u03bes1 be the zeros of u0 and us0, respectively, then |\u03be1\u2212\u03bes1|\u21920 as |u0\u2212us0|H1,1(R+)\u21920. This completes the proof of the Theorem\u00a0Suppose atisfies  as t\u2192\u221e. 7.3\u03bb=\u22121 and let q=\u03c9tanh\u03d5\u2208R\u2216{0} be the value of q associated with the stationary\u00a0one\u2010soliton.Suppose q>0. In this case, the spectral functions \u0394s and as corresponding to the stationary one\u2010soliton initial data us0 fulfill Assumption \u0394s has only one simple zero in C\u00af+ at \u03bes1:=i\u03c9/2, and as has no zeros in C\u00af+  defined in Definition\u00a0U is open, and hence the desired conclusions follow immediately from Proposition\u00a0Suppose first that C\u00af+  also has a simple zero at \u03bes2, Assumption Suppose now that u0(x) be a small perturbation of us0 in H1,1(R+). Let a(k) and \u0394(k) be the spectral functions corresponding to u0. By Lemma\u00a0\u0394(k) has two simple zeros \u03be1 and \u03be2 and these zeros are such that \u03be1\u2192\u03bes1 and \u03be2\u2192\u03bes2 as u0\u2192us0. In particular, \u03be1\u2260\u03be2 and \u03be1\u2260\u03bes2 for u0 sufficiently close to us0. The symmetry \u2192L\u221e(C\u00af+), so the same arguments used to prove Lemma\u00a0u0 is close to us0, a(k) has exactly one simple zero k1 in C+ and this zero satisfies k1\u2192\u03bes2 as u0\u2192us0. We distinguish two cases depending on whether \u03be2 coincides with \u03bes2 or\u00a0not.Let symmetry  implies \u03be2=\u03bes2). Suppose that \u03be2 coincides with \u03bes2. Equation\u00a0(k=\u03bes2=\u2212iq/2 reads\u0394(k) vanishes at \u03bes2, we deduce that a(k) also vanishes at \u03bes2. This means that both \u0394 and a have simple zeros at \u03bes2. In particular, Assumption \u03be2 of \u0394 is excluded from the list of poles of m. Indeed, consider the derivation of the residue conditions (\u03be1 is in the present situation a simple zero of \u0394(k) and a(\u03be1)\u22600. This means that m satisfies the residue conditions }, where c^1 is defined in  defined by  defined by Equation\u00a0(\u03c3^d and the reflection coefficient r(k) given by =\u2212fined in . We concfined in  holds asfined by  and uradEquation\u00a0 with thegiven by . This co\u03be2\u2260\u03bes2). Suppose that \u03be2 does not coincide with \u03bes2. In this case, we have a(\u03be2)\u22600. Indeed, evaluating  corresponding to us0, we see that bs is nonzero everywhere in the upper half\u2010plane. Furthermore, by Lemma\u00a0u0\u21a6b is continuous H1,1(R+)\u2192L\u221e(C\u00af+). Hence, b(\u03be2)\u22600 for all u0 close to us0. Since \u03be2\u2260\u2212iq/2 by assumption, we have 2\u03be2\u00b1iq\u22600, and hence (a(\u03be2)\u22600. We conclude that whenever u0 is sufficiently close to us0, Assumption m of Theorem\u00a0\u03be1,\u03be2 in C+. This means that the RH problem involves the modified discrete scattering data \u03c3^d={,}, and that we have the long\u2010time asymptotics formula  defined by  defined by (Case 2 (aluating  at \u03be2, wpression  for the  obtain0=(2\u03be2\u2212ibtain7.60=(2\u03be2\u2212 formula  as t\u2192\u221e wfined by  and uradfined by . This coProceedings of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not\u2010for\u2010profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of\u00a0mathematics.The"}
{"text": "Germline-produced Hedgehog signals require PTR-6 and PTR-16 receptors for mating-induced shrinking and death. Our results reveal an unconventional role of the piRNA pathway in transcriptional regulation of Hedgehog signaling and a new role of Hedgehog signaling in the regulation of longevity and somatic maintenance: Hedgehog signaling is controlled by the tunable piRNA pathway to encode the previously unknown germline-to-soma pro-aging signal. Mating-induced piRNA downregulation in the germline and subsequent Hedgehog signaling to the soma enable the animal to tune somatic resource allocation to germline needs, optimizing reproductive timing and survival.The reproductive system regulates somatic aging through competing anti- and pro-aging signals. Germline removal extends somatic lifespan through conserved pathways including insulin and mammalian target-of-rapamycin signaling, while germline hyperactivity shortens lifespan through unknown mechanisms. Here we show that mating-induced germline hyperactivity downregulates piRNAs, in turn desilencing their targets, including the Hedgehog-like ligand-encoding genes  Caenorhabditiselegans. The authors find that germline piRNAs influence longevity and somatic maintenance by transcriptionally regulating germline-to-soma Hedgehog signaling.This study identifies a new germline-to-soma aging signal tuned by mating in  Conversely, the status of worm, fly, mouse, and human germlines influence their somatic tissues: germline removal and ovary transplantation extend lifespan11, while germline hyperactivity decreases lifespan and leads to dramatic changes in somatic physiology in animals across great evolutionary distances14. While these studies in animals ranging from invertebrates to humans support the existence of a pro-aging signal from the germlines, the nature and identity of that signal remains elusive.Longevity is plastic and is influenced by external factors, for example, diet, and internal factors such as reproductive demands. Communication between the germline and soma allows a coordinated response to physiological and environmental challenges. Animals couple nutrient availability to reproduction18. However, concomitant removal of the somatic gonad eliminates lifespan extension and other somatic benefits, suggesting the existence of two opposing signals: a germline pro-aging signal and a somatic gonad prolongevity signal19. The somatic gonad prolongevity signal pathway has been characterized using germlineless worms28. Dafachronic acids, insulin, mTOR, and steroid signaling are required in the soma for germline loss-mediated lifespan regulation, suggesting that they mediate the somatic gonad prolongevity effect30. By contrast, the identity of the initial pro-aging signal originating from the germline remains elusive. Identifying this pro-aging signal is critical for understanding how animals tune their aging rates in response to germline activity and reproductive needs.Removal of the germline extends lifespan, increases fat accumulation, and enhances resistance to various stresses19, less is known about the role of the hyperactive germline. Mating accelerates germline proliferation and ultimately leads to somatic collapse, shrinking and early death13, suggesting that the germline pro-aging signal is substantially amplified by mating. Therefore, mated worms are an ideal system in which to identify the mysterious germline pro-aging signal.While most studies of the influence of germline on lifespan have compared germlineless animals with intact, unmated animalsWe set out to identify the underlying mechanism of hyperactive germline-induced shrinking and early death. Our transcriptional analyses of isolated germlines revealed that a specific subset of piRNAs is downregulated in response to mating, de-repressing Hedgehog-related genes. Hedgehog signaling communicates the status of the germline to somatic cells, resulting in mating-induced shrinking and early death. Thus, somatic responses to germline hyperactivity are tuned by piRNA regulation of an important developmental signaling pathway.12. Day\u20091 adult self-spermless hermaphrodites (fog-2) mated for 24\u2009h with Day 1 adult males live 40% shorter than their unmated counterparts, and shrink by up to 30%12  In addition to shrinking, mated animals also display an acceleration in age-related autofluorescence31 and intestinal barrier dysfunction , we used an equal amount of RNA, rather than an equal number of worms or germline, from each germline sample to make our sequencing libraries, and sequencing reads were normalized in our analysis. In unmated hermaphrodites, over 8,000 mRNAs were previously reported to have reduced expression in the distal germline compared with whole animals35  and small RNA sequencing (RNA-seq) Fig. . Hermaphs35 Fig. , left. Ts35 Fig. , right. sri-40   Fig. , indicat36. Endogenous siRNAs are required for maintenance of proteostasis and lifespan extension in germlineless worms37, and play critical roles in shaping the germline transcriptome35. PIWI-associated RNAs (piRNAs) are expressed predominantly in the germline and are required to maintain germline integrity and fertility38. Differential expression analysis of small RNAs from dissected mated and unmated germlines revealed 440 significantly upregulated and 296 downregulated small RNAs in the mated germline  or when shrinking is obvious (Day\u20097)  to preserve genome integritynes Fig. , suggest 42 Fig. . P granu\u20097) Fig. , suggest\u20097) Fig. , these s\u20097) Fig. . Togethe12. We mated mutant hermaphrodites with young males for 2\u2009days starting on Day\u20091 of adulthood and measured their body size on Day\u20096/7 of adulthood , matura-1 refs. \u201352, seco-3 refs. , 53), anl-2 ref. ), as weling Fig. , none ofing Fig. , indicatCaenorhabditis elegans germ granules)56, and disruption of P granules compromises piRNA-mediated silencing43. P granule distribution and assembly/disassembly dynamics depend on maternal-effect germline (MEG) proteins57. We found that meg double mutants in which P granule functionality is severely impaired57 were also resistant to mating-induced shrinking  revealed that the lack of a functional piRNA pathway in prde-1 worms eliminated most germline transcriptional differences induced by mating  of 148 significantly downregulated piRNAs identified in the mated germline (Supplementary Data 64) to the 34 genes and found that 13 are predicted to encode secreted proteins. We then ranked the final 13 genes according to their expression fold change in the germline and in the whole worm, and their consistency in expression patterns across the samples  genes, A.7 Fig. . KnockinA.7 Fig. , but notA.7 Fig.  or the iA.7 Fig. , was sufwrt-1 provided over 50% protection against mating-induced early death ; Fig. wrt-10 yielded a much milder effect . Therefore, while inhibiting germline piRNA-mediated Hedgehog signaling is beneficial to the mothers, as it prevents mated worms from shrinking and largely rescues mating-induced death, such benefits do not come without a cost to progeny. Signaling from the mated germline, while deleterious to the mother\u2019s soma and ultimately leading to her death, may be an unavoidable cost of activation of the germline upon mating that is necessary for sufficient nutrient provisioning to progeny.Finally, we wondered whether there is any generational penalty for eliminating mating-induced shrinking and death, as these processes are deleterious to the mother. While ype Fig. , knockdo14; however, the underlying mechanisms are poorly understood. Here we found that mating-induced piRNA downregulation in the germline releases suppression of the Hedgehog signaling pathway, which in turn leads to body size and lifespan decrease in mated worms . Our results suggest that there is likely a trade-off between somatic integrity and progeny production in mated animals, as demonstrated by the reduced mated brood size of animals with reduced Hedgehog signaling ine Fig. . These r78 and the reduction in energy devoted to somatic integrity maintenance, reflecting the direct cost of germline hyperactivity and substantially increased progeny production. Signaling from the mated germline, while deleterious to the mother\u2019s soma and ultimately leading to her death, may be an unavoidable cost of activation of the germline upon mating that is necessary for sufficient nutrient provisioning to progeny. Our study reveals a mechanism that can efficiently convey germline status to the soma in adulthood, allowing the animals to better organize the balance between reproduction and somatic maintenance, optimizing reproductive fitness.Mating-induced germline-to-soma piRNA-mediated Hedgehog signaling elegantly coordinates germline function and somatic aging. In an unmated female with low or no germline proliferation, the priority of the germline is to maintain its integrity until mating occurs, while avoiding unnecessary proliferation or differentiation of the germ cells. piRNAs contribute to this process by suppressing errant TE expression and developmental programs, including those regulated by Hedgehog signaling. By contrast, upon mating, the germline switches to progeny production mode, and the previous suppression of Hedgehog signaling by piRNAs is released to allow rapid germline stem cell proliferation and further differentiation Fig. . As Hedg13. Seminal fluid kills the mother by turning off her IIS/FOXO protective pathway13, which is unnecessary for successful reproduction. By contrast, germline activation upon mating is the most likely to be both necessary and shared with other animals, as germline activity correlates with shorter lifespan in worms, flies, mice, and possibly humans14. Because germline proliferation may be induced by mating in other organisms, and piRNAs are ubiquitous in animal germlines, it will be interesting to see whether mating-induced piRNA regulation of signaling from germline to somatic tissues is conserved.Of the three mating-related killing mechanisms\u2014male pheromone, seminal fluid, and sperm activation of germline proliferation\u2014the last has the highest chance of being evolutionarily conserved. Male pheromone-induced killing kills hermaphrodites only at nonphysiologically high concentrations, and kills males specifically but only in androdioecious speciesC.elegans strains used in the study were as follows:79. Worms were transferred to iced M9 buffer for dissection. Heads or tails were removed with 26G needles, allowing the distal portion of the germline to pop out of the worm. Glass capillaries pulled with an opening just large enough to fit the end of the germline were used to detach them rapidly at the ventral-to-distal bend. Dissected distal germlines were transferred immediately into 1.5\u2009ml Eppendorf tubes filled with 500\u2009\u03bcl Trizol. About 200 germlines were collected for each biological replicate.Germline dissection was modified from a previous publicationTubes containing dissected germlines (immersed in Trizol) were put in an Eppendorf MixMate Vortex Mixer at 800\u2009r.p.m. and 65\u2009\u00b0C for 1\u2009h before the isolation process. Total RNA was extracted from Trizol using the mirVana miRNA isolation kit (ThermoFisher). mRNA libraries for directional RNA-seq were prepared using the SMARTer Apollo System and were sequenced (150-nt single-end) on the Illumina NovaSeq platform. RNA samples for small RNA-seq were treated with 5\u2032\u2009polyphosphatase (Lucigen) and prepared using the SMARTer Apollo system with modifications for small RNA library preparation. Small RNA-containing libraries were Blue Pippin size selected (15\u201330\u2009nt insert size) before 75-nt single-end sequencing.C.\u2009elegans genome  using Bowtie. Count matrices were generated using featureCounts. Data were normalized with a variance-stabilizing transformation (DESeq2). DESeq2 was used for differential expression analysis. Genes at Padj\u2009<\u20090.05 were considered significantly differentially expressed. PCA was carried out using the R method (prcomp). Heatmaps were generated in R using normalized read counts (variance-stabilizing transformation). Tissue enrichment and gene ontology (GO) term enrichment analysis were performed using wormbase enrichment analysis (https://wormbase.org/tools/enrichment/tea/tea.cgi) for significantly differentially-expressed genes. Predicted targets of piRNAs were retrieved from piRTarBase (http://cosbi6.ee.ncku.edu.tw/piRTarBase/) using the default \u2018relaxed\u2019 piRNA targeting setting, see Supplementary Data PRJNA892981.FASTQC was used to assess read quality scores. Universal adapter sequences were trimmed from small RNA library sequences using Cutadapt. Reads were mapped to the glp-1(e2141) was retrieved from a previous publication34. Microarray data of mated and unmated fog-2(q71) was retrieved from a previous publication61. Hermaphrodites were mated on Day\u20091 of adulthood for 24\u2009h in a 2:1 male:hermaphrodite ratio. About 200 hermaphrodites were collected on Day\u20092/3 of adulthood for each biological replicate. RNA was extracted by the heat vortexing method. Two-color Agilent microarrays were used for expression analysis. Significantly differentially-expressed gene sets were identified using significance analysis of microarrays (SAM)80. One-class SAM was performed to identify genes that are significantly differentially expressed after mating. The lists were then compared with significantly upregulated genes in dissected germline (RNA-seq).Microarray data of mated and unmated ekl-1 and drh-3 homozygous mutants, we confirmed successful mating empirically by observing the decrease in darkness of mated sterile worms under normal light, which indicates mating-induced fat loss. About 25 synchronized Day\u20091 hermaphrodites were transferred onto each plate. The hermaphrodites were transferred daily onto new seeded plates in the first week of the lifespan assay. Afterward, they were transferred once every 2\u2009days. When RNAi was used in lifespan assay, RNAi treatment always started from eggs for all the experiments in this study. Kaplan\u2013Meier analysis with log-rank (Mantel\u2013Cox) method was performed to compare the lifespans of different groups. \u2018Bagged\u2019 worms were censored on the day of the event.NGM plates (60\u2009mm) plates were used to set up group mating. Each 60\u2009mm NGM plate was seeded with OP50 to make a bacterial lawn of around 3\u2009cm in diameter 2\u2009days before mating. All lifespan assays were performed at room temperature (20\u201321\u2009\u00b0C). About 50 hermaphrodites and 100 young (Day\u20091\u22122 of adulthood) males were transferred onto the plate. Then, 24\u201348\u2009h later, the hermaphrodites were transferred onto newly seeded 60\u2009mm NGM plates in the absence of males for lifespan assays. No 5-fluro-2\u2032-deoxyuridinevwas added to the plates. We confirmed successful mating for all worms by checking their progeny. With successful mating, about half of the progeny are male. For the only two sterile strains used in this study, ekl-1 and drh-3 homozygous mutants, we confirmed the successful mating empirically by observing the decrease in darkness of mated sterile worms under normal light, which indicates mating-induced fat loss. Images of live hermaphrodites on 60\u2009mm plates were taken on Day 6/7 of adulthood with a Nikon SMZ1500 microscope. When RNAi was used, RNAi treatment always started from eggs for all the experiments. ImageJ was used to analyze the body size of the worms. The middle line of each worm was delineated using the segmented line tool and the total length was documented as the body length of the worm. A t-test was performed to compare the body size differences between groups of worms in the same day.Mating set up was the same as described in Individual hermaphrodites after mating were transferred onto 3\u2009cm NGM plates seeded with 25\u2009ml OP50 and moved to fresh plates daily until reproduction ceased. The old plates were left at 20\u2009\u00b0C for 2\u2009days to allow the offspring to grow into adults, which were counted manually for daily production and total brood size. Between 10 and 25 plates of individual worms of each genotype per treatment were counted to account for individual variation.t-test was performed to compare the P granule density differences between mated and unmated germlines.mCherry-tagged fluorescent PGL-1 was visualized in living nematodes (YY968) by mounting young adult animals on 2% agarose pads with M9 buffer with 20\u2009mM levamisole. Fluorescent images were captured using a Nikon Ti microscope with a \u00d7100 objective. Images were processed and quantified in ImageJ. The quantification of germline granule fluorescence was performed using ImageJ. For every image, a region of interest (ROI) with a clear focus of P granules was selected manually. The area of the whole ROI was kept the same for all images. The number of puncta within the ROI was measured blindly for each germline and image. The densities of germline P granules were calculated as: the number of puncta within the ROI per the area of the whole ROI. Densities of germline granules were determined for 50\u201360 ROIs, and the mean and s.d. were calculated using GraphPad Prism. A 65 to overexpress endogenous piRNAs. The scaffold information was obtained from the website https://www.wormbuilder.org/piRNAi/. Cluster was generated using simple search option with the default setting (targeted gene: wrt-1 or wrt-10). Synthetic piRNAs were replaced by endogenous piRNAs as indicated by the text and figure legends. The sequences (about 1.5\u2009kb) were synthesized by Twist Bioscience. The injection mix consisted of 20\u2009ng\u2009\u03bcl\u20131 synthetic dsDNA piRNA overexpression cluster with adapters that were not further purified (Twist Bioscience), 2\u2009ng\u2009\u03bcl\u20131 coinjection marker (Pmyo2::mCherry::unc-54 3\u2032\u2009UTR), and 1\u2009kb DNA ladder to a final total DNA concentration of 100\u2009ng\u2009\u03bcl\u20131. F1 progeny with red pharynx were selected from injected animals. Synchronized F2 progeny from positive transgenic F1 worms were used in mating and body size measurement assays.We modified recently developed piRNAiThe sequences of piRNA overexpression clusters used in this study are as follows:wrt-1 (uppercase: piRNAs)piRNA overexpression cluster targeting cgcgcttgacgcgctagtcaactaacataaaaaaggtgaaacattgcgaggatacatagaaaaaacaatacttcgaattcatttttcaattacaaatcctgaaatgtttcactgtgttcctataagaaaacattgaaacaaaatattaagtTAGAACTTCATCTTTAGAACActaattttgattttgattttgaaatcgaatttgcaaatccaattaaaaatcattttctgataattagacagttccttatcgttaattttattatatctatcgagttagaaattgcaacgaagataatgtcttccaaatactgaaaatttgaaaatatgtttttatggcaggtgctgacggattgccagaactcaaaatatgaaatttttatagttttgttgaaacagtaagaaaatcttgtaattactgtaaactgtttgctttttttaaagtcaacctacttcaaatctacttcaaaaattataatgtttcaaattacataactgtgtgaagttgggcgcccagttgtactgtagagcttcaatgttgataagatttattaacacagtgaaacaggtaatagttgtttgttgcaaaatcggaaatctctacatttcatatggtttttaattacaggtttgttttataaaataattgtgtgatggatattattttcagacctcatactaatctgcaaaccttcaaacaatatgtgaagtctactctgtttcactcaaccattcatttcaatttggaaaaaaatcaaagaaatgttgaaaaattttcctgtttcaacattatgacaaaaatgttatgattttaataaaaacaatTGTCATAAACGTAGAATCATCttctgtttttcttagaagtgttttccggaaacgcgtaattggttttatcacaaatcgaaaacaaacaaaaatttttttaattatttctttgctagttttgtagttgaaaattcactataatcatgaataagtgagctgcccaagtaaacaaagaaaatttggcagcggccgacaactaccgggttgcccgatttatcagtggaggatctacaaggctaactgcgttatctaatgtgatgtacacggttttcatttaaaaacaaattgaaacagaaatgactacattttcaaattgtctatttttgctgtgtttattttgccaccaacaatgagataatgtgttagccttgtcaatctagtaaactcacttaatgcaattcctccagccacatatgtaaacgttgtatacatgcagaaaacggttttttggttttaatgggaacttttgacaaattgttcgaaaatcttaagctgtcccatttcagttgggtgatcgatttwrt-10 (uppercase: piRNAs)piRNA overexpression cluster targeting cgcgcttgacgcgctagtcaactaacataaaaaaggtgaaacattgcgaggatacatagaaaaaacaatacttcgaattcatttttcaattacaaatcctgaaatgtttcactgtgttcctataagaaaacattgaaacaaaatattaagtTAAATGAAAAGCTGGCTATGGctaattttgattttgattttgaaatcgaatttgcaaatccaattaaaaatcattttctgataattagacagttccttatcgttaattttattatatctatcgagttagaaattgcaacgaagataatgtcttccaaatactgaaaatttgaaaatatgtttttatggcaggtgctgacggattgccagaactcaaaatatgaaatttttatagttttgttgaaacagtaagaaaatcttgtaattactgtaaactgtttgctttttttaaagtcaacctacttcaaatctacttcaaaaattataatgtttcaaattacataactgtgtgaagttgggcgcccagttgtactgtagagcttcaatgttgataagatttattaacacagtgaaacaggtaatagttgtttgttgcaaaatcggaaatctctacatttcatatggtttttaattacaggtttgttttataaaataattgtgtgatggatattattttcagacctcatactaatctgcaaaccttcaaacaatatgtgaagtctactctgtttcactcaaccattcatttcaatttggaaaaaaatcaaagaaatgttgaaaaattttcctgtttcaacattatgacaaaaatgttatgattttaataaaaacaatTGGAATAGCGTAAACAAAAGAttctgtttttcttagaagtgttttccggaaacgcgtaattggttttatcacaaatcgaaaacaaacaaaaatttttttaattatttctttgctagttttgtagttgaaaattcactataatcatgaataagtgagctgcccaagtaaacaaagaaaatttggcagcggccgacaactaccgggttgcccgatttatcagtggaggatctacaaggctaactgcgttatctaatgtgatgtacacggttttcatttaaaaacaaattgaaacagaaatgactacattttcaaattgtctatttttgctgtgtttattttgccaccaacaatgagataatgtgttagccttgtcaatctagtaaactcacttaatgcaattcctccagccacatatgtaaacgttgtatacatgcagaaaacggttttttggttttaatgggaacttttgacaaattgttcgaaaatcttaagctgtcccatttcagttgggtgatcgatttpiRNA overexpression cluster control (uppercase: piRNAs)cgcgcttgacgcgctagtcaactaacataaaaaaggtgaaacattgcgaggatacatagaaaaaacaatacttcgaattcatttttcaattacaaatcctgaaatgtttcactgtgttcctataagaaaacattgaaacaaaatattaagtTAGAAACTGATCTCTGAAAGTctaattttgattttgattttgaaatcgaatttgcaaatccaattaaaaatcattttctgataattagacagttccttatcgttaattttattatatctatcgagttagaaattgcaacgaagataatgtcttccaaatactgaaaatttgaaaatatgttACTTTCCATAACGTCGACAAAattgccagaactcaaaatatgaaatttttatagttttgttgaaacagtaagaaaatcttgtaattactgtaaactgtttgctttttttaaagtcaacctacttcaaatctacttcaaaaattataatgtttcaaattacataactgtgtAATTCGGGAGTCCTAATTCTAactgtagagcttcaatgttgataagatttattaacacagtgaaacaggtaatagttgtttgttgcaaaatcggaaatctctacatttcatatggtttttaattacaggtttgttttataaaataattgtgtgatggatattattttcagacctcatactaatctgcaaaccttcaaacaatatgtgaagtctactctgtttcactcaaccattcatttcaatttggaaaaaaatcaaagaaatgttgaaaaattttcctgtttcaacattatgacaaaaatgttatgattttaataaaaacaatGAAAATTTTGCTGAACACCTTttctgtttttcttagaagtgttttccggaaacgcgtaattggttttatcacaaatcgaaaacaaacaaaaatttttttaattatttctttgctagttttgtagttgaaaattcactataatcatgaataagtgagctgcccaagtaaacaaagaaaatttggcagcggccgacaactaccgggttgcccgatttatcagtggaggaGATCGAGGCTTAATGAACGGAatctaatgtgatgtacacggttttcatttaaaaacaaattgaaacagaaatgactacattttcaaattgtctatttttgctgtgtttattttgccaccaacaatTAAAAGTGGCTCCGAGCTAGGtcaatctagtaaactcacttaatgcaattcctccagccacatatgtaaacgttgtatacatgcagaaaacggttttttggttttaatgggaacttttgacaaattgttcgaaaatcttaagctgtcccatttcagttgggtgatcgatttTwist Primer Set 2 Fw: 5\u2032-CAATCCGCCCTCACTACAACCG-3\u2032Twist Primer Set 2 Rev: 5\u2032-TCCCTCATCGACGCCAGAGTAG-3\u2032\u20131. Single colonies of each type were inoculated and allowed to grow overnight at 30\u2009\u00b0C to reach stationary phase (OD\u2009>\u20091.5). From the culture, 20\u2009\u03bcl of undiluted, 5\u00d7, 25\u00d7, 125\u00d7 and 625\u00d7 dilutions were spotted on the assay plates. Plates were imaged after incubating for 4\u2009days at 30\u2009\u00b0C. Quantification of yeast growth was performed according to the published protocol81.Yeast two-hybrid assay was performed using Takara Matchmaker Gold Yeast Two-Hybrid System . WRT-1 and WRT-10 cDNA were cloned in frame into the bait construct pGBKT7, whereas PTR-6 and PTR-16 cDNA were cloned in frame into the prey construct pGADT7. pGBKT7-53, which encodes the Gal4 DNA-BD fused with murine p53, and pGADT7-7, which encodes the Gal4 AD fused with SV40 large T-antigen, were used as positive interaction controls in the assay. The bait constructs and the prey constructs were cotransformed into Y2HGold yeast strain in various combinations to test potential interaction. A dominant mutant version of the AUR1 gene that encodes the enzyme inositol phosphoryl ceramide synthase is expressed in Y2HGold Yeast strain in response to positive interaction between the bait and prey, conferring strong resistance to the otherwise highly toxic drug Aureobasidin A (AbA). The final concentration of AbA used in the assay was 200\u2009ng\u2009mlC. elegans program, v.1.0 (http://worm-tissue.princeton.edu/search/multi). The tissue-specific enrichment scores were used to generate the heatmap in Fig. https://worm.princeton.edu/). The prediction scores of principal tissues were used to generate the heatmap in Fig. Tissue enrichment prediction analyses for ptr genes were performed using two methods: (1) the Tissue Expression Predictions for 82 with a few modifications. Briefly, synchronized Day\u20091 worms were set up for mating. About 50 worms were removed from the mating plate or the control plate 48\u2009h later, and suspended overnight in liquid cultures of OP50 bacteria mixed with blue dye . Animals were then washed three times with M9 buffer before imaging using Nikon Ti with RGB illumination under \u00d710 or \u00d720 magnification.The intestinal barrier function assay (Smurf assay) was performed according to the published protocol\u20131). mRNA libraries for directional RNA-seq were prepared using the SMARTer Apollo System and were sequenced (150-nt single-end) on the Illumina NovaSeq platform. Reads from germline mRNA-seq were mapped to the C. elegans RepBase database (https://www.girinst.org/server/RepBase/protected/repeatmaskerlibraries/RepBaseRepeatMaskerEdition-20181026.tar.gz) using Bowtie. Raw counts were then normalized on the total number of mapping reads and multiplied by 1,000,000, obtaining expression values indicated as reads per million mapped reads (RPM). Transposon type was determined according to RepBase\u2019s annotation.In addition to grouped germline samples collected for mRNA-seq and small RNA-seq, we collected single distal germline (four for each condition) into 10\u2009\u03bcl 0.2% Triton X100 with RNase inhibitor  to over 100. When quantifying the P granule intensities of mated and unmated worms, images were scored blindly. No randomization was necessary for this study because mated and unmated worms of specific genotype/treatment were always compared with each other.t-test was used for all comparisons to determine statistical significance in this study. Detailed information of each assay can be found in the corresponding figure legends.Lifespan data were plotted as Kaplan\u2013Meier survival curves and statistical analyses were performed using the log-rank (Mantel\u2013Cox) test and two-way ANOVA. For body size measurements and fluorescence intensity quantification, two-tailed Source Data for sequencing data and statistics displayed in the main figures are provided as Supplementary Files.Further information on research design is available in the Reporting SummarySupplementary Table 1.fog-2 whole worms with intact germlines.List of germline-specific upregulated genes that are also upregulated in mated Supplementary Table 2.List of predicted targets of 148 downregulated piRNAs in the mated germline.Supplementary Table 3.List of piRNA-targeted germline-specific mating-induced genes (genes in Supplementary Data 1 that belong to the predicted targets of piRNAs in Supplementary Data 2)."}
{"text": "Orations which has never been the subject of systematic investigation. This paper provides a starting point by focusing exclusively on medical imagery, one of the most pervasive and instrumental types of imagery in Maximus\u2019 work that has gone entirely unnoticed in the literature to date. This paper shows that Maximus uses medicine , the physician , the body (its physiology and workings) and notions of health and disease with considerable diversity and creativity, in ways that make his examples stand out in relation to earlier (Platonic) or contemporary applications of the medical parallel. It argues that the use of the medical imagery in the pedagogical context in which Maximus\u2019 Orations were performed facilitated not just clarity but also concept formation and the shaping of a moral outlook as well as the familiarisation with the proper literary references and verbal and conceptual topoi for admission into the group of the educated elite. Another main thesis is that medical imagery valorises Maximus\u2019 philosophical status and his claims to Imperial-period acculturation, thus functioning as a trademark for the rhetorical philosophy he wished to promote.Imagery is an overarching feature of Maximus of Tyre\u2019s  Orations  are an important yet little-studied source for the cultural ambience of late second-century Rome.Orations offer a fascinating window into the dynamics of philosophical rhetoric as a pedagogical practice in this age. They also cast light on Maximus\u2019 role as a philosophising orator and especially on his moral-didactic approach, given that the professed aim of his Orations is to \u2018rouse young men\u2019s souls and guide their ambitions\u2019,The forty-one lectures that make up Maximus of Tyre\u2019s Halbphilosoph)stricto sensu medical writers or medical experts. Thus, it tackles the wider role of ancient medicine beyond the more usual territories explored by the majority of modern scholars. Medicine in the Roman Imperial period (roughly the first three centuries AD) was not just a technical field, but also a vibrant area of study and intellectual engagement, which attracted the attention not just of medical authors and practising physicians but also of other educated elite men, such as sophists and philosophers. This study stresses precisely this less-explored area in the dynamics of ancient medicine as an intellectual field. What is extremely interesting in Maximus\u2019 case is that medicine seems to have interested him a lot as an educational tool, a vehicle for his self-promotion and an innovative means of advertising his public role as a professional orator. This is an aspect of ancient medicine that deserves further exploration, attesting, as it does, to the social role of medicine in Graeco-Roman antiquity.A powerful tool that Maximus deploys to that end is imagery in the broadest of senses: namely, not just the more familiar figures of speech, such as metaphors or similes, but also looser mechanisms of comparison, such as allegories, prosopopoeiae (personifications) and even fables.Orations, almost in a formulaic fashion, that his use of imagery specifically serves the purpose of clarification and illumination,Before embarking on the main part of the analysis, it seems relevant to mention that Maximus stresses many times throughout the dei, \u2018need\u2019] in almost all cases), and hence an essential factor for his performative and communicative success when addressing young upper-class listeners who sought to be acculturated as much as entertained by his rhetorical skill.eikon) is always connected either with the effectiveness of Maximus\u2019 educative discourse or with the specific cognitive abilities that it is expected to foster in them, notably that of imagination  and comparison .paideia . In that sense, imagery may be seen as an authority-conferring move on Maximus\u2019 part, especially considering that his notion of the function of the \u03b5\u1f30\u03ba\u03ce\u03bd as argued above is a modification of the way it is used in his favourite philosopher Plato,Taken together, these examples point to the overarching features of Maximus\u2019 image-making in general. First, they show that the employment of imagery is a necessary precondition for driving home any given point Maximus is making in his argument (note the use of the verb \u03b4\u03b5\u1fd6 [nosemata tes psyches), and hence the symbolic representation of philosophy as psychic therapy  targeted at the extirpation of false beliefs and the control of violent emotions through the mediation of philosophical arguments and training.logos) is the only steady aspect in human life:Set over life, however, is Reason (\u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2), which constantly adapts itself to the circumstances of the moment, like a skilled doctor whose duty is to regulate the indigence and satiety of a body that is not stable, but surges back and forth, in the turmoil of evacuation and repletion . This is precisely what the rational teaching of philosophers can do for human life, adapting its tone to suit the emotions of the moment, so as both to offer consolation in sad times and to enhance the celebrations in times of joy .The point Maximus wishes to get across here is that philosophy shares with medicine two positive elements: a) its adaptability to individual circumstances, picking up on the preceding mention of \u03ba\u03b1\u03b9\u03c1\u03cc\u03c2 nosos) as per the conventional analogy but instead stresses the lack of equilibrium in an unstable body (\u03c3\u03ce\u03bc\u03b1\u03c4\u03b9 \u03bf\u1f50\u03c7 \u1f11\u03c3\u03c4\u1ff6\u03c4\u03b9)symmetria), or internal economy, which he considers the ultimate goal of philosophy.oikonomia (\u03bf\u1f30\u03ba\u03bf\u03bd\u03bf\u03bc\u03af\u03b1) rather than of therapeia (\u03b8\u03b5\u03c1\u03b1\u03c0\u03b5\u03af\u03b1), so as to advertise the more reassuring aspects of philosophy, that is a provision for maintaining internal balance at all times and not simply restoring it when things go wrong.A ubiquitous theme in the ethical writings of the Hellenistic and Roman periods is the theorisation of moral passions as diseases of the soul , when it comes to the philosophical component of the analogy at the end of the quoted section, he gives a more balanced presentation of the conditions liable to affect his pupils in real life by referring to both sadness and happiness. This is no concession, however, because the typical analogy would have dwelt to a greater extent (if not exclusively) on the former (see the examples in note 35). So it is clear that Maximus strives to mitigate the negativity of the stereotypical therapeutic imagery. This is attuned to the kind of philosophy that is on offer here: this is no \u2018hardcore\u2019, that is doctrinal or scholastic, philosophy of the sort pursued by Plutarch, Numenius (mid 2nd century AD) or Alcinous (2nd century AD), but a \u2018soft\u2019,enkyklios paideia) and wider cultural background.Oration 27.3, which provides a definition of physical health as the state in which the blendings of the body\u2019s constituents are harmonised to produce the proportions conducive to health (\u03c4\u1f78 \u03b4\u1f72 \u03c3\u1ff6\u03bc\u03b1 \u1f01\u03c1\u03bc\u03bf\u03bd\u03af\u03b1\u03b9\u03c2 \u03ba\u03b1\u1f76 \u03ba\u03c1\u03ac\u03c3\u03b5\u03c3\u03b9\u03bd \u03b5\u1f30\u03c2 \u1f51\u03b3\u03b9\u03b5\u03af\u03b1\u03c2 \u03bc\u03ad\u03c4\u03c1\u03bf\u03bd).eukrasia), a state of good mixture predicated on balancing the four elementary qualities of hot, cold, dry and wet.poiotetai), which are indeed never mentioned as such in the Orations (of which more below), even though the author seems aware of them.Oration 27.3, too, the use of the medical analogy is straightforward and positive through an emphasis on the bodily and psychic symmetry that Maximus\u2019 philosophy can achieve for his audience.The popularising character of Maximus\u2019 philosophyTimaeus 81e\u201382a)Should we call it \u2018learning\u2019 (\u03bc\u03ac\u03b8\u03b7\u03c3\u03b9\u03bd), or should we adopt Plato\u2019s terminology and call it \u2018recollection\u2019 (\u1f00\u03bd\u03ac\u03bc\u03bd\u03b7\u03c3\u03b9\u03bd)? Or should we use both names, \u2018learning\u2019 and \u2018recollection\u2019, of the one phenomenon? Whatever the answer, the phenomenon itself resembles what can happen to the eye (\u03c4\u1f78 \u03b4\u03ad \u1f10\u03c3\u03c4\u03b9\u03bd \u03c4\u03bf\u03b9\u03bf\u1fe6\u03c4\u03bf\u03bd \u03bf\u1f37\u03bf\u03bd \u03c4\u1f78 \u03c0\u03b5\u03c1\u1f76 \u03c4\u1f78\u03bd \u1f40\u03c6\u03b8\u03b1\u03bb\u03bc\u1f78\u03bd \u03c0\u03ac\u03b8\u03bf\u03c2). The eye never ceases to possess the faculty of sight, but from time to time some mischance allows a mist to cover and embrace the organ and so block off its contact with the outside world (\u1f24\u03b4\u03b7 \u03b4\u03ad \u03c0\u03bf\u03c5 \u1f51\u03c0\u1f78 \u03c3\u03c5\u03bc\u03c6\u03bf\u03c1\u1fb6\u03c2 \u1f10\u03c0\u03b9\u03c7\u03c5\u03b8\u03b5\u1fd6\u03c3\u03b1 \u1f00\u03c7\u03bb\u1f7a\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03bc\u03c6\u03b9\u03ad\u03c3\u03b1\u03c3\u03b1 \u03c4\u1f78 \u1f44\u03c1\u03b3\u03b1\u03bd\u03bf\u03bd \u03b4\u03b9\u03b5\u03c4\u03b5\u03af\u03c7\u03b9\u03c3\u03b5\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03c4\u1f74\u03bd \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f70 \u1f41\u03c1\u03ce\u03bc\u03b5\u03bd\u03b1 \u1f41\u03bc\u03b9\u03bb\u03af\u03b1\u03bd). When medical science comes to the rescue, its task is not to implant sight in the eye, but rather to remove the blockage so as to uncover it and restore its outward passage . You must understand that the soul too has a kind of sight, the natural function of which is to discern and understand reality. The misfortune of physical embodiment covers it over with a thick mist (\u1f51\u03c0\u03bf\u03ba\u03b5\u03c7\u03cd\u03c3\u03b8\u03b1\u03b9 \u03b1\u1f50\u03c4\u1fc7 \u03c0\u03bf\u03bb\u03bb\u1f74\u03bd \u1f00\u03c7\u03bb\u03cd\u03bd), which confounds its powers of vision, removes its precise discernment, and quenches its native brightness. Reason, coming to the soul like a doctor, does not bring and implant understanding (\u03bb\u03cc\u03b3\u03bf\u03bd \u1f65\u03c3\u03c0\u03b5\u03c1 \u1f30\u03b1\u03c4\u03c1\u1f78\u03bd \u03bf\u1f50 \u03c0\u03c1\u03bf\u03c3\u03c4\u03b9\u03b8\u03ad\u03bd\u03b1\u03b9 \u03b1\u1f50\u03c4\u1fc7 \u03c6\u03ad\u03c1\u03bf\u03bd\u03c4\u03b1 \u1f10\u03c0\u03b9\u03c3\u03c4\u03ae\u03bc\u03b7\u03bd), like something the soul did not already possess; instead, it reawakens the understanding it does possess, but which is dim and constrained and torpid.That Maximus had taken his inspiration from Plato\u2019s Republic 518d for the imagery relating to the eye could scarcely have been missed by his audience, who were well versed in the Platonic tradition. Plato\u2019s point is merely that there is an art (philosophy) which effectively reawakens the soul so as to enable it to \u2018see\u2019 , but it does not produce sight in the eye as if the capacity for vision was not inherent in it already. Maximus\u2019 reworking of the eye imagery is both clearer and more developed: a) it overtly introduces medicine (absent from Plato) as a parallel system to philosophy, b) it elaborates on the blocking of vision through \u2018mist\u2019 , eliciting an imaginative involvement on the part of the audience, as this would have been an uncomfortable eye condition that they would have experienced either in themselves or vicariouslyRepublic 518d, as seen above, with another medically related imagery, that of the art of midwifery derived from Theaetetus 149a\u2013151d (cf. Phaedrus 276b\u2013277a) which is instrumental in showing that \u2018reason plays midwife to the pregnant soul\u2019 (Oration 10.4),Another issue meriting attention in this connection is that, in talking about elements in the body, Maximus seems to be drawing specifically on Plato ; nor are potters or leatherworkers, or the practitioners of still more lowly pursuits than these, answerable to any other judge of their activities than their own craft. But Socrates, whom not even Apollo\u2013he who knows the numbers of the sands and can divine the measures of the sea\u2013could convict of ignorance, has not ceased to this very day to be the object of accusation and investigation.The superior wisdom and expertise of the physician as well as that of the steersman and of other professionals  are enough to shield them from the criticism of laymen and ensure them general approbation. So why is it not counterintuitive when Socrates, whose proficiency was not some manual skill, but something more elevated involving the symmetry of life (\u1f00\u03bb\u03bb\u1f70 \u03c4\u1f78\u03bd \u03b1\u1f51\u03c4\u03bf\u1fe6 \u03b2\u03af\u03bf\u03bd \u03c3\u03c5\u03bc\u03bc\u03ad\u03c4\u03c1\u03c9\u03c2), is not appreciated? This is the question that Maximus problematises for his audience and which he accentuates through the additional emphasis on Socrates\u2019 virtues, notably his simple lifestyle (\u03b5\u1f50\u03c4\u03b5\u03bb\u03b5\u03af\u1fb3), perseverance (\u03ba\u03b1\u03c1\u03c4\u03b5\u03c1\u03af\u1fb3) and self-control (\u03c3\u03c9\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u1fc3), encouraging his listeners to be inspired by Socrates\u2019 moral grandeur.Indeed, Maximus is very keen on employing medical imagery when he wants to reinforce Socrates\u2019 (circa\u00a0470\u2013399 BC) philosophical authority and thereby capitalise on the principles from Socratic/Platonic philosophy that he expects his auditors to assimilate. For example, in Oration 8.3), and the use of medical analogies for didactic purposes becomes even more intriguing. The following case study shows that, in order to highlight Socrates\u2019 powers of persuasion, Maximus includes examples from other authorities who have, in fact, been eclipsed by Socrates\u2019 persuasive talents:What doctor ever persuaded his feverish patients that going without food and drink was a good thing?  Who ever persuaded a hedonist that his aims were worthless? Who ever persuaded the money-grubber that his object is not a good? To be sure, Socrates would have had no trouble at all in persuading the Athenians that the pursuit of Virtue is not the same thing as corrupting the young, and that knowledge of the divine is not the same thing as irregularity in religious observance.Unlike the image of the hedonist and the money-grubber, the one with the doctor involves an audience to whom the latter directs his persuasive competence, making it a better match to Socrates\u2019 relationship with his accusers. The image of the unpersuasive doctor (like that of the money-grubber) originates in the Gorgias. However, in Maximus\u2019 hands, it is transformed on two levels. First, Maximus is inspired by the famous section 456b ff., where Gorgias explains how he, as an orator, was able to persuade the patients of his brother, the physician Herodicus, to accept drugs, surgical operations or cauterisation in instances where the latter was unable to do so. Maximus supplements the element of persuasion that dominates this section from the Gorgias with references to food and drink that occur in another famous passage from this dialogue,Oration 25.5. This suggests that this was a set image, easily retrieved in an oral delivery; while focusing specifically on \u2018feverish\u2019 patients could spark some recognition in his audience (just like the mist obscuring the eye cited above) and thereby help Maximus cement the image\u2019s impact on them, making its message even more perceptible.Examples such as this could be multiplied , Maximus mentions that \u2018it is the benevolent doctor who causes the greatest pain, and the most scrupulous general, and the most reliable helmsman\u2019.No: Plato\u2019s foundation and his republic are established in purely theoretical terms; he aims for the greatest possible perfection rather than for what might be most practicable\u2013just like those sculptors who bring together beautiful elements from all over, using their art to combine details from many different bodies into one single representation, so as to produce a single, sound, well-constructed, and harmoniously beautiful artefact. You wouldn\u2019t be able to find a real body that exactly resembled such a statue, because art aims at what is most beautiful, while the things we encounter and use in everyday life fall short of what art can produce. I suppose that if human beings had the power to sculpt bodies of flesh and blood, then our craftsmen would be able to mix together, in the right proportions, the quantities of earth and fire (\u03c4\u1f70\u03c2 \u03b4\u03c5\u03bd\u03ac\u03bc\u03b5\u03b9\u03c2 \u03be\u03c5\u03bc\u03bc\u03ad\u03c4\u03c1\u03c9\u03c2 \u03b3\u1fc6\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03c5\u03c1\u03cc\u03c2) and everything else that when harmonized and co-ordinated with them constitute our bodily nature, and so presumably produce a body that had no need of drugs and quack remedies and the regimens of doctors (\u1f61\u03c2 \u03c4\u1f78 \u03b5\u1f30\u03ba\u1f78\u03c2 \u1f02\u03bd \u03c3\u1ff6\u03bc\u03b1 \u1f00\u03b4\u03b5\u1f72\u03c2 \u03c6\u03b1\u03c1\u03bc\u03ac\u03ba\u03bf\u03c5 \u03ba\u03b1\u1f76 \u03bc\u03b1\u03b3\u03b3\u03b1\u03bd\u03b5\u03c5\u03bc\u03ac\u03c4\u03c9\u03bd \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b1\u03b9\u03c4\u03b7\u03bc\u03ac\u03c4\u03c9\u03bd \u1f30\u03b1\u03c4\u03c1\u03b9\u03ba\u1ff6\u03bd). Suppose then that someone heard one of those craftsmen legislating for his theoretical creations and saying that they had no need even of a Hippocrates to heal them (\u03b4\u03b5\u03ae\u03c3\u03bf\u03bd\u03c4\u03b1\u03b9 \u03bf\u1f50\u03b4\u1f72 \u1f39\u03c0\u03c0\u03bf\u03ba\u03c1\u03ac\u03c4\u03bf\u03c5\u03c2 \u1f30\u03c9\u03bc\u03ad\u03bd\u03bf\u03c5 \u03c3\u03c6\u1fb6\u03c2), but that they ought to crown the man with wool and anoint him with myrrh and send him somewhere else, to win his reputation where sickness made his arts necessary (\u03b5\u1f50\u03b4\u03bf\u03ba\u03b9\u03bc\u03ae\u03c3\u03bf\u03bd\u03c4\u03b1 \u1f10\u03ba\u03b5\u1fd6 \u1f45\u03c0\u03bf\u03c5 \u03c4\u1f74\u03bd \u03c4\u03ad\u03c7\u03bd\u03b7\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03b1\u03bb\u03b5\u1fd6 \u1f21 \u03bd\u03cc\u03c3\u03bf\u03c2); and suppose that our hearer grew angry with the craftsman for dishonouring the art of Asclepius and the Asclepiadae (\u1f61\u03c2 \u1f00\u03c4\u03b9\u03bc\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1 \u03c4\u1f74\u03bd \u1f08\u03c3\u03ba\u03bb\u03b7\u03c0\u03b9\u03bf\u1fe6 \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u1f08\u03c3\u03ba\u03bb\u03b7\u03c0\u03b9\u03ac\u03b4\u03c9\u03bd \u03c4\u03ad\u03c7\u03bd\u03b7\u03bd). Wouldn\u2019t he be making a laughing-stock of himself by bringing an accusation against someone who was not rejecting medicine because he scorned it, but because he neither needed it for practical purposes nor welcomed it as a source of pleasure ?The functional adjustment of philosophyRepublic, both in terms of quotation and allusion,\u2026just like fever patients who gorge themselves on food and drink against doctor\u2019s orders . Comparing one evil (disease) (\u03bd\u03cc\u03c3\u1ff3) with another (exertion) (\u03c0\u03cc\u03bd\u03bf\u03c5\u03c2), they prefer to be sick and enjoy themselves (\u03b1\u1f31\u03c1\u03bf\u1fe6\u03bd\u03c4\u03b1\u03b9 \u1f21\u03b4\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u03bd\u03bf\u03c3\u03b5\u1fd6\u03bd), sooner than to exert themselves and be cured (\u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd \u1f22 \u03c0\u03bf\u03bd\u03bf\u1fe6\u03bd\u03c4\u03b5\u03c2 \u1f51\u03b3\u03b9\u03b1\u03c3\u03b8\u1fc6\u03bd\u03b1\u03b9). Many a resourceful doctor has before now tempered the bitterness of his cure with a small admixture of something sweeter; but neither Asclepius nor the Asclepiadae are indiscriminate purveyors of pleasure\u2013that is the work of caterers .Even though this section is heavily informed by Plato\u2019s technai) and \u2018knacks\u2019 : crafts, based on accurate knowledge of a subject, benefit the soul or body, like medicine which cares for the body in a genuine sense. Knacks, by contrast, based on mere imitation of crafts, seek to flatter the audience regardless of their well-being. The knack that imitates medicine is cookery, targeted at pleasing the body, thereby engendering its destruction. Maximus seems well acquainted with these and other related insights.Men were exposed to the flattery of a bogus form of medicine, when they abandoned the healing techniques of Asclepius and the Asclepiadae and reduced science to something indistinguishable from gourmet cookery, a substandard flatterer to substandard physiques . The informer imitates the orator, setting argument against argument, fortifying injustice against Justice and the base against the noble. The sophist imitates the philosopher. He is the most scrupulous imitator of them all.The juxtaposition of medicine and cookery in Plato is a motif which Maximus frequently deploys to develop with more clarity the archetypal distinction between crafts , then the supervision of reason will have to be added to this emotion in order to make of it a virtue rather than a sickness. Just as in the case of our bodily constitution, health is a certain non-rational condition of the forces of wetness, dryness, cold, and heat, neatly blended by human artifice and artfully harmonized by Nature\u2013and if you remove anything of the contribution made by Nature or by artifice, you will have upset this non-rational state and driven health away \u2013so it is in the case of Love: even if it enjoys the control of reason it remains an emotional state, and if you remove reason, you will have disturbed its equilibrium and converted it wholesale into sickness .Turning now in more detail to the image of the body and its diseases, it could be said that Maximus makes use of it chiefly in order to clarify the meaning of abstract and complex philosophical notions. To better explain that pleasure has its basis in the enjoyment of the experiencing subject, not in its own nature, he compares it with the nourishment of the body, which works exactly the other way around.Republic 444c\u2013d. Nevertheless, nowhere in this passage does Plato advance the definition of somatic health as a non-rational condition  of the forces  of hot, cold, dry and wet, blended by means of the medical art and harmonised by nature; Plato simply speaks of bodily balance in a general sense. What is more, even though Maximus refers here to hot, cold, dry and wet, he does not label them as \u2018qualities\u2019 but simply as \u2018forces\u2019, a concept that would have made perfect sense to his recipients. By the same token, in other cases, nouns referring to the constitution of the body remain unqualified by Maximus, notably through his use of substantive adjectives accompanied by the definite article: for example, \u03c4\u1f00\u03bd\u03b1\u03bd\u03c4\u03af\u03b1 for \u2018opposing elements\u2019.krasis) in Maximus never denotes \u2018mixture of elements\u2019 but rather points more generally to the body\u2019s constitution, attesting to an intuitive appreciation of the body\u2019s organic unity on the audience\u2019s part, just like \u03b4\u03cd\u03bd\u03b1\u03bc\u03b9\u03c2 (dynamis) above could hardly have been construed in its specialised meaning of \u2018capacity\u2019, but rather as referring more generally to \u2018power\u2019, as noted. Maximus could simply not afford to be obscure in oral performance, and hence any jargon had to be strictly avoided.The comparison between bodily health and love that Maximus proposes here is most probably influenced by the connection between bodily health and justice in the soul from daimon), responsible for healing the sick,Orations is the knowledge Maximus claims to possess about the invention and history of medicine. As far as its invention goes, Maximus gives a description of how medicine first became a science by means of the accumulated records of the sufferings of patients and the ensuing therapeutic measures that proved effective for the majority of them.Cutting out arrows, and smearing on soothing drugs (Iliad 11.515). But as time went on the human body slipped out of the control of this archaic form of medicine, fell prey to a more sophisticated style of living, and developed flaws in its constitution (\u1f51\u03c0\u03bf\u03bb\u03b9\u03c3\u03b8\u03b1\u03b9\u03bd\u03cc\u03bd\u03c4\u03c9\u03bd \u03b1\u1f50\u03c4\u1fc7 \u03c4\u1ff6\u03bd \u03c3\u03c9\u03bc\u03ac\u03c4\u03c9\u03bd \u03b5\u1f30\u03c2 \u03b4\u03af\u03b1\u03b9\u03c4\u03b1\u03bd \u03c0\u03bf\u03b9\u03ba\u03b9\u03bb\u03c9\u03c4\u03ad\u03c1\u03b1\u03bd \u03ba\u03b1\u1f76 \u03ba\u03c1\u1fb6\u03c3\u03b9\u03bd \u03c0\u03bf\u03bd\u03b7\u03c1\u03ac\u03bd), with the results that we now see: medical science has itself had to become more sophisticated (\u1f10\u03be\u03b5\u03c0\u03bf\u03b9\u03ba\u03af\u03bb\u03b8\u03b7 \u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u03ae), and to exchange its former simplicity for something more complex.\u2019 Come now, following Asclepius\u2019 example, let the poet and the philosopher together defend their pursuits to us.What is philosophy, if not a younger form of poetry, less formal in composition and more lucid in expression? If then these two differ from each other only in age and in superficial form, how ought one to understand the difference in what the two kinds of composer-poet and philosopher-say about the gods? Could we perhaps say that this present enquiry of ours is like someone comparing medicine in its original form with the modern form that treats the patients of today (\u03bf\u1f37\u03bf\u03bd \u03b5\u1f34 \u03c4\u03b9\u03c2 \u03ba\u03b1\u1f76 \u1f30\u03b1\u03c4\u03c1\u03b9\u03ba\u1f74\u03bd \u1f10\u03bd\u03b8\u03c5\u03bc\u03b7\u03b8\u03b5\u1f76\u03c2 \u03c4\u1f74\u03bd \u03c0\u03c1\u03ce\u03c4\u03b7\u03bd \u1f10\u03ba\u03b5\u03af\u03bd\u03b7\u03bd \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f74\u03bd \u03bd\u03ad\u03b1\u03bd \u03b4\u1f74 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fd6\u03c2 \u03bd\u1fe6\u03bd \u03c3\u03ce\u03bc\u03b1\u03c3\u03b9\u03bd \u1f10\u03c0\u03b9\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03b7\u03bd), and examining the weak and strong points of each? Asclepius would inform this enquirer that other arts and sciences remain unaffected by the passage of time: where the need remains constant, the response does not vary either. The science of medicine, however, is constrained to adapt to the physical constitutions of its patients, which do not constitute a stable and clearly defined factor, but are modified and changed by the diets that go with different styles of life ; different treatments and therapeutic regimes, adapted to the dietary habits of the moment, must be devised for different eras . \u2018So do not think\u2019, Asclepius would continue, \u2018that my famous sons, Machaon and Podalirius, were any less skilled in the art of healing than those who have set their hands to it subsequently, and thought up their various clever cures. At the time my sons were working, the bodies that their arts had to treat were not degenerate and oversophisticated and completely enervated; as a result they found them easy to cure, and their function was a simple one: The reference to Asclepius mentioned above requires further discussion because the figure of Asclepius is a potent one in Maximus\u2019 rhetorical apparatus. Beyond the fact that Maximus portrays Asclepius as the god of health , the optimum balance in the embodied soul.Oration 39.2 he expresses dissatisfaction with what he perceives as specialised areas of medicine arising from attempts to treat imbalance qua disease.Two topics merit consideration here. First, Maximus mentions that, owing to the degeneracy of the human body, medicine became more diversified so as to be able to respond to the treatment of different bodily malfunctions. The refinement of medicine is a negative thing, according to Maximus  because it militates against its authentic form as a single, unified science.Secondly, the decline of medicine is a recognisable trope in Imperial-period literature, reflecting the cultural anxieties of this era, particularly the nostalgia for the great Greek past and the aspirations of Second Sophistic authors to invoke and emulate their superior Classical predecessors. Yet, Maximus (again) does not seem to have taken the theme from his contemporary intellectual trends, which, in fact, tackled the issue rather differently: Galen, for instance, also talked about the decadence of medicine in his day but presented it in terms of the failure of medical practice and the moral bankruptcy of medics.Do you want God to care for creation as a whole? Then do not importune him; he will not listen to you if what you ask militates against the preservation of the whole. What would happen if the limbs of the body gained voices, and if when one of them grew tired of being operated on by the doctor in the interests of the body as a whole, they prayed to Medicine not to be destroyed ? Would not Asclepius reply to them like this: \u2018Miserable creatures, it is not for the whole body to be ruined to serve your interests; it is for you to perish in order that it should be saved\u2019 . Precisely the same is true of creation as a whole. The Athenians suffer from the Plague, the Spartans suffer earthquakes, Thessaly is flooded by tidal waves, and Aetna erupts. You may call such breakings-up \u2018destruction\u2019, but the true doctor knows their cause (\u1f41 \u03b4\u1f72 \u1f30\u03b1\u03c4\u03c1\u1f78\u03c2 \u03bf\u1f36\u03b4\u03b5\u03bd \u03c4\u1f74\u03bd \u03b1\u1f30\u03c4\u03af\u03b1\u03bd); he disregards the prayers of the parts and preserves the whole , for his concern is for creation at large (\u03c6\u03c1\u03bf\u03bd\u03c4\u03af\u03b6\u03b5\u03b9 \u03b3\u1f70\u03c1 \u03c4\u03bf\u1fe6 \u1f45\u03bb\u03bf\u03c5). Although God\u2019s Providence does in fact extend to particulars as well. But prayer is out of place there too, being like a patient asking his doctor for food or medicine on his own initiative (\u1f45\u03bc\u03bf\u03b9\u03bf\u03bd \u1f61\u03c2 \u03b5\u1f30 \u03ba\u03b1\u1f76 \u1f30\u03b1\u03c4\u03c1\u1f78\u03bd \u1f94\u03c4\u03b5\u03b9 \u1f41 \u03ba\u03ac\u03bc\u03bd\u03c9\u03bd \u03c6\u03ac\u03c1\u03bc\u03b1\u03ba\u03bf\u03bd \u1f22 \u03c3\u03b9\u03c4\u03af\u03bf\u03bd): if it is efficacious, the doctor will give it unasked ; if it is dangerous, he will withhold it even when asked . To sum up: nothing that falls under the heading of Providence is to be requested or prayed for.The same technique features in another instance, where this time, Maximus presents God as caring for the creation as a whole and not for particular aspects, like a sensitive doctor, whose aim is similarly to preserve the entire organism and not its individual parts. The personified Asclepius, this time, castigates the limbs for acting selfishly and disregarding the organism\u2019s status as an integral whole:qua doctor.Oration 15.4\u20135 presents close similarities with Oration 5.4 cited above in that it likens the function of political society to the function of the bodily parts and in that it frames this comparison as a fictional dialogue between the body members and a Phrygian storyteller, who plays a similar role to that of Asclepius in the passage cited above.The idea of individual body parts that should operate in the best interests of the whole organism appears in other authors of this period.Orations, the use of the medical imagery facilitates not just clarity but also concept formation and the shaping of a moral outlook as well as the familiarisation with the \u2018right\u2019 literary references and  clich\u00e9s for admission into the club of the cultivated and the philosophically woke. We have shown that Maximus works with the typical analogy of philosophy as therapy of the soul but adjusts it to the demands of his philosophical teaching, particularly the need to be accessible as well as appealing so as not to discourage the members of the social elite who have just embarked upon their engagement with philosophy at an advanced level. We have also argued that Maximus accentuates various aspects of the medical encounter between doctor and patients , the workings of the human body and the decline of the art of medicine to enable his young listeners to make sense of his argument and be persuaded by his admonitions. Moreover, we have remarked that Maximus uses the physician as an image for God and that he often connects medicine with Asclepius to stress its ideals of balance and stability but also to bestow authority on his own practice. As has been noted, Maximus\u2019 medical imagery makes no use of medical technicalities and does not show any sharp contemporary social detail. What does emerge, however, is an important rhetorical point\u00a0about what medicine as a given source domain makes available to the teacher/orator and how the skilled teacher/orator makes use of it. Medicine does not provide a fixed set of one-to-one readymade correspondences but rather a field of possibilities that is highly flexible in its application. According to need, different details from the domain can be focused on , and the same detail can be applied to a range of different targets. In that sense, a lot of Maximus\u2019 skill as an orator/teacher lies in the perceptiveness with which he sees how the resources of the field can be turned to advantage differently in each new moment of need.All in all, this contribution has attempted to explore the multifarious ways in which Maximus of Tyre adjusts the medical imagery in his rhetorical speeches designed to familiarise his young pupils with the characteristic aspects of philosophy, especially moral philosophy. We have seen that in the pedagogical context of Maximus\u2019 That Maximus\u2019 work is full of medical references and associations reflects contemporary trends, given that health talk was part of mainstream elite culture in Maximus\u2019 day, as can be seen from relevant discussions in Plutarch, Gellius (circa 125\u2013after 180 AD), Aurelius\u2019 correspondence with Fronto (circa 100\u2013late 160s AD), and of course Aelius Aristides (117\u2013181 AD). However, it also highlights Maximus\u2019 own knowledge of Greek culture, and more especially, his distinctive ways of translating and appropriating that tradition. Medical imagery, therefore, valorises Maximus\u2019 philosophical status and his claims to Second Sophistic cultural capital, functioning as a trademark for the rhetorical philosophy he wished to parade, which was, to his mind, still respectable, proper philosophy in this period, just as he was himself a major player in the contemporary philosophical tradition .Orations and is thus seen through the lens of medical anthropology which places health and disease at the forefront of human society, never at its periphery as a distant or irrelevant science. As we have seen, the medical metaphor in Maximus draws significantly on the role of medicine as experienced in the world around him and his audience, regardless of its Platonic precedents and invocations which, as we have noted, are meaningfully adjusted to Imperial-period reality. While passively reflecting what the speaker wished to say, medical images in Maximus actively affected the way his listeners thought and behaved, acting at the same time as both mirror and agent.Above all, the foregoing analysis has brought out the fact that medical imagery in Maximus does not simply reflect reality so as to make the speaker\u2019s thought and material easier for his audience to digest, but rather it actively shapes and orients the audience\u2019s moral decision-making and associated behaviour. The most important ethical behaviour Maximus sought to encourage in them through the analogy with the medical discipline is the cultivation of moderation, self-control and conscious deliberation. In that sense, the employment of medical imagery in Maximus can be correlated with the function of modern Concept Formation theory, which postulates that metaphor, including imagery, is not a decorative literary device in human culture but a conceptual tool to think and act with."}
{"text": "We thank Augello and Wu  for theiThe directions specified in our first contrast codes appear to have caused parameter estimation issues, though both sets of contrasts below produce the same model-fitted values, and other model estimates appear to remain unchanged.OriginalCorrectedContrast #1\u2014M+C+ to M+C\u2212Contrast #1\u2014M+C\u2212 to M+C+Contrast #2\u2014M+C+ to M\u2212C\u2212Contrast #2\u2014M+C+ to M\u2212C\u2212Contrast #3\u2014M\u2212C+ to M\u2212C\u2212Contrast #3\u2014M\u2212C\u2212 to M\u2212C+Models were re-specified using a contrast code set that was constructed using backward difference coding .Level 1 vs. Level 2Level 2 vs. Level 3Level 3 vs. Level 4M/C GroupsM+C\u2212 to M+C+M+C+ to M\u2212C\u2212M\u2212C\u2212 to M\u2212C+M+C\u2212\u2212(k \u2212 1)/k\u2212(k \u2212 2)/k\u2212(k \u2212 3)/kM+C+1/k\u2212(k \u2212 2)/k\u2212(k \u2212 3)/kM\u2212C\u22121/k2/k\u2212(k \u2212 3)/kM\u2212C+1/k2/k3/kModels examining associations between cannabis and methamphetamine characteristics\u2019 associations with neurocognitive performance and impairment were not impacted by the contrast coding error, as only the two groups with a lifetime for cannabis or methamphetamine were compared in each set.Models examining interactions between the substance use group and HIV status/characteristics were not impacted by the contrast codes operationalization error, as contrasts were comparison coded vs. M\u2212C\u2212 in these models to aid in the interpretation of model interactions. However, the main effects model without interactions presented in Supplementary Table S3 of the original manuscript needed to be re-estimated, and these changes are reflected in the Correction submitted by the authors. Changes to model estimates did not impact inferences discussed in the manuscript and, like in other models, the same fitted values were produced as previously described."}
{"text": "Prunus avium L.) is an economic fruit tree of the Prunus\u00a0genus in the\u00a0Roaceae\u00a0family  has been found in Shandong, Gansu and Beijing (Wang et al. 1 population of \u2018Rainier \u00d7 21\u201321\u2019 was used. The maternal parent \u2018Rainier\u2019 is a variety developed jointly by the Washington State Agricultural Experiment Station and the United States Department of Agriculture. Its parents are \u2018Bing\u2019 and \u2018Van\u2019. The paternal parent \u201821\u201321\u2019 is a sweet cherry superiors cultivated by the Cherry Research Group of Beijing Academy of Agriculture and Forestry Sciences, which is self-crossed from \u2018Stella\u2019. No symptoms of CCL were found on \u2018Rainier\u2019 and \u201821\u201321\u2019. This family had 257 individuals including 77 with CCL. CCL seedlings showed irregular leaf edge, uneven color, deep crack or rough surface, most with dwarfing symptoms (Fig.\u00a0In order to develop molecular marker of CCL, the FPrc123 amplified two bands in sweet cherry genome, named Prc123-A and Prc123-B respectively. There were three combinations of Prc123-A and Prc123-B, namely AA, AB and BB (Fig.\u00a0In addition, 25 cultivars of sweet cherry were surveyed for genotype of Prc123, and 24 cultivars had clear amplification results. Among them, 11 cultivars were Prc123-AA, including \u2018Mingzhu\u2019, \u2018\u041a\u043e\u0441\u043c\u0438\u0447\u0435\u0441\u043a\u0430\u044f\u2019, \u2018\u0414\u0438\u043b\u0435\u043c\u043c\u0430\u2019, \u2018Van\u2019, \u2018Tieton\u2019, \u2018Caihong\u2019, \u2018Py\u03c3\u0418\u041d\u041e\u0412\u0430\u042f\u0420\u0430\u041d\u041d\u042f\u042f\u2019, \u2018Zaodan\u2019, \u2018Summit\u2019, \u2018Wanhongzhu\u2019 and \u2018Sunburst\u2019, which hadn\u2019t been reported CCL in field. 13 cultivars were Prc123-AB type, including \u2018Juhong\u2019, \u2018Rainier\u2019, \u2018Lapins\u2019, \u2018Caixia\u2019, \u2018Hongdeng\u2019, \u2018Hedelfinger\u2019, \u2018Valerij Cskalov\u2019, \u2018Burlat\u2019, \u2018Hongyan\u2019, \u2018Zaolu\u2019, \u2018Hongmi\u2019, \u2018\u041a\u0440\u0443\u043f\u043d\u043e\u043f\u043b\u043e\u0434\u043d\u0430\u044f\u2019 and \u201821\u201321\u2019, in which 5 cultivars was detected CCL in field, including \u2018Hedelfinger\u2019, \u2018Hongdeng\u2019, \u2018Valerij Cskalov\u2019, \u2018Hongyan\u2019 and \u2018Hongmi\u2019. The proportion of CCL actually occurring was consistent with the proportion predicted by Prc123-AB.To sum up, this study obtained a molecular marker of CCL, Prc123 and identified genotypes of Prc123 in 24 sweet cherry main cultivars, which can be used in preventing the CCL from spreading and molecular assisted breeding for non-CCL cultivars."}
{"text": "Nature Plants 10.1038/s41477-022-01305-9. Published online 23 December 2022.Correction to: 2 sequestered by sinking seaweed in the ambient nutrient scenario were incorrect due to typographical errors. As a result, in the sentence beginning \u201cIn the optimistic case\u201d, \u201c0.85%\u201d is now \u201c0.110%\u201d, \u201c310,000 km2\u201d is now \u201c400,000 km2\u201d, and \u201cPoland\u201d now reads as \u201cZimbabwe\u201d. In the sentence following, \u201cis slightly higher: 0.036% and 0.099%\u201d now reads as \u201cis 0.035% and 0.100%\u201d and \u201c$40\u201d is now \u201c$30\u201d. Furthermore, in the penultimate sentence of this paragraph, \u201cocean areas of 0.09\u20130.10% and 0.28\u20130.37%\u201d now reads as \u201cocean areas of 0.085\u20130.100% and 0.285\u20130.410%\u201d and \u201croughly 320,000\u2013360,000\u2009km2 and 1,010,000\u20131,330,000\u2009km2\u201d is now \u201croughly 310,000\u2013360,000\u2009km2 and 1,030,000\u20131,480,000\u2009km2\u201d. Calculated values in this subsection have been revised to reflect these corrections: in the sentence beginning \u201cAverage costs at the median of Monte Carlo\u201d, \u201c$1,110\u2013$2,100\u201d is now \u201c$1,120\u2013$2,090\u201d; and in the paragraph beginning \u201cDespite being a small percentage\u201d, \u201c17%\u201d is now \u201c18%\u201d and \u201c61%\u201d is now \u201c64%\u201d. Corresponding values in the Methods subsection \u201cComparison of gigaton-scale sequestration area to previous estimates\u201d and in Supplementary Figures 10 and 11 have been updated accordingly. The errors have been corrected in the HTML and PDF versions of the article.In the version of this article initially published, in the Results subsection \u201cCosts and benefits of large-scale seaweed farming\u201d, the percentages of ocean area farmed to reach 1\u2009Gt and 3 Gt of CO"}
{"text": "Scientific Reportshttps://doi.org/10.1038/s41598-022-26919-z, published online 14 January 2023Correction to: The original version of this Article contained errors in the names of authors Filip Tulis, Michal \u0160ev\u010d\u00edk, Radoslava J\u00e1no\u0161\u00edkov\u00e1, Ivan Bal\u00e1\u017e, Michal Ambros, Lucia Zvar\u00edkov\u00e1 and Gy\u00f6z\u00f6 Horv\u00e1th, which were incorrectly given as Tulis Filip, \u0160ev\u010d\u00edk Michal, J\u00e1no\u0161\u00edkov\u00e1 Radoslava, Bal\u00e1\u017e Ivan, Ambros Michal, Zvar\u00edkov\u00e1 Lucia and Horv\u00e1th Gy\u00f6z\u00f6.The original Article has been corrected."}
{"text": "Scientific Reports 10.1038/s41598-022-21122-6, published online 24 October 2022Correction to: The original version of this Article contained a repeated error, where the pressure unit was incorrectly given as \u2018mbar\u2019 instead of \u2018mmHg\u2019.In the Results and discussion section, under the subheading \u2018Intra-catheter pressure\u2019,2O. When the abdominal pressure was adjusted to 50 cmH2O, the average pressure for Brand A was \u2212\u2009383\u2009\u00b1\u200950 mbar, \u2212\u2009323\u2009\u00b1\u200947 mbar for Brand B and \u2212\u2009330\u2009\u00b1\u200993 mbar for Brand C.\u201d\u201cThe average pressure variation for Brand A was \u2212\u2009364\u2009\u00b1\u200942 mbar, \u2212\u2009248\u2009\u00b1\u200981 mbar for Brand B and \u2212\u2009272\u2009\u00b1\u200959 mbar for Brand C at 20 cmHnow reads:2O. When the abdominal pressure was adjusted to 50 cmH2O, the average pressure for Brand A was \u2212\u2009383\u2009\u00b1\u200950 mmHg, \u2212\u2009323\u2009\u00b1\u200947 mmHg for Brand B and \u2212\u2009330\u2009\u00b1\u200993 mmHg for Brand C.\u201d\u201cThe average pressure variation for Brand A was \u2212\u2009364\u2009\u00b1\u200942 mmHg, \u2212\u2009248\u2009\u00b1\u200981 mmHg for Brand B and \u2212\u2009272\u2009\u00b1\u200959 mmHg for Brand C at 20 cmHAdditionally,2O was equal to \u2212\u2009296\u2009\u00b1\u200956\u00a0mbar  for the tests where mucosal suction was perceived by the operator. Conversely, the intra-catheter pressure variation that could be measured at the first flow-stop for Brand B at 20 cmH2O when mucosal suction was not detected by the operator was equal to\u2009\u2212\u2009180\u2009\u00b1\u200964\u00a0mbar . A similar scenario was seen for Brand C at 50 cmH2O, where the measured intra-catheter pressure variation was equal to\u2009\u2212\u2009373\u2009\u00b1\u200962\u00a0mbar  when mucosal suction was perceived by the operator, and to\u2009\u2212\u2009212\u2009\u00b1\u200945\u00a0mbar  when mucosal suction was not perceived by the operator.\u201d\u201cThe measured intra-catheter pressure variation for Brand B at 20 cmHnow reads:2O was equal to \u2212\u2009296\u2009\u00b1\u200956 mmHg  for the tests where mucosal suction was perceived by the operator. Conversely, the intra-catheter pressure variation that could be measured at the first flow-stop for Brand B at 20 cmH2O when mucosal suction was not detected by the operator was equal to \u2212\u2009180\u2009\u00b1\u200964 mmHg . A similar scenario was seen for Brand C at 50 cmH2O, where the measured intra-catheter pressure variation was equal to \u2212\u2009373\u2009\u00b1\u200962 mmHg  when mucosal suction was perceived by the operator, and to \u2212\u2009212\u2009\u00b1\u200945 mmHg  when mucosal suction was not perceived by the operator.\u201d\u201cThe measured intra-catheter pressure variation for Brand B at 20 cmHFurthermore,\u201cWhat remains to be understood is whether a pressure variation of, for example \u2212\u2009250 mbar, is sufficient to cause discomfort to the IC users, or even cause microtraumas to the bladder mucosa, and if the speed at which the peak is generated has any relevance.\u201dnow reads:\u201cWhat remains to be understood is whether a pressure variation of, for example \u2212\u2009250 mmHg, is sufficient to cause discomfort to the IC users, or even cause microtraumas to the bladder mucosa, and if the speed at which the peak is generated has any relevance.\u201dUnder the subheading \u2018In-vivo animal studies\u2019 of the same section,\u201cDuring bladder emptying, the pressure difference at first flowstop was equal to \u2212 96 mbar .\u201dnow reads:\u201cDuring bladder emptying, the pressure difference at first flowstop was equal to \u2212 96 mmHg .\u201dFinally, the error was also present in Table 1 and in Figures\u00a010, 11, 12, 13 and 15.The correct and incorrect values of Table 1 appear below.Table 1Incorrect:Correct:The original Figures\u00a0The original Article has been corrected."}
{"text": "Communications Biology 10.1038/s42003-023-05183-5, published online 03 August 2023Correction to: In the original version of the article (in the Aerosolization section on page 3 and in the sentence beginning \u2018The emission fluxes\u2026\u2019 the power sign of +4 is displayed incorrectly as \u20134 at 5 places.4 m\u20132.s\u20131 which replaces the previous incorrect version: 0.8\u20137.9\u2009\u00d7\u200910\u22124\u2009m\u22122.s\u22121.The correct version of the number is 0.8\u20137.9\u2009\u00d7\u2009104\u2009m\u22122.s\u22121\u2009\u00b1\u20091.3\u2009\u00d7\u2009104) which replaces the previous incorrect version: (2.4\u2009\u00d7\u200910\u22124\u2009m\u22122.s\u22121\u2009\u00b1\u20091.3\u2009\u00d7\u200910\u22124).In the same line, the correct version of equation is (2.4\u2009\u00d7\u2009104\u2009m\u22122.s\u22121\u2009\u00b1\u20092.4\u2009\u00d7\u2009104) which replaces the previous incorrect version: (4.8\u2009\u00d7\u200910\u22124\u2009m\u22122.s\u22121\u2009\u00b1\u20092.4\u2009\u00d7\u200910\u22124).Further down the same line, the correct version of equation is (4.8\u2009\u00d7\u200910This has been corrected in the PDF version."}
{"text": "Correction to: Environmental Microbiome (2023) 18:18 10.1186/s40793-023-00469-xFollowing publication of the original article , it was In Fig.\u00a0B, the same text elements had been lost in the left part of the figure, in which green indicated \u201cno stress\u201d and orange \u201cstress\u201d. Moreover, the headers above the columns were missing. From left to right: \u201cFunction KEGG C\u201d, \u201c# normalized reads\u2014all taxa\u201d, \u201cmost abundant in\u2014Ac BGP AP\u2013Ac BGP AP\u201d, \u201coverrepresented\u2014Ac BGP AP\u201d and \u201cFunction KEGG B\u201d. In C, the pathway labels were missing. From left to right and top to bottom: \u201cTrehalose, biosynthesis\u201d, \u201cShikimate\u201d, \u201cReductive PPP\u201d, \u201cLeucine degradation\u201d, \u201cHeme biosynthesis\u201d, \u201cABC-transporter\u201d, \u201cC10\u2013C20 & C5 Isoprenoid biosynthesis\u201d and \u201cMultidrug efflux\u201d.In Fig.\u00a0B, the headers above the columns were missing. From left to right: \u201c# normalized reads\u2014all taxa\u201d, \u201cmost abundant in\u2014Ac BGP AP\u2013Ac BGP AP\u201d and \u201coverrepresented\u2014Ac BGP AP\u201d. In, C the headers \u201cDesir\u00e9e\u201d and \u201cStirling\u201d and pathway labels were missing. Below the turquois box, the labels were: \u201cABC-Transporter\u201d, \u201cbranched-chain amino acid\u201d, \u201cUrea\u201d, \u201cGlucose oligomer / Maltoologosaccharide\u201d, \u201cChitobiose\u201d, \u201cD-Xylose\u201d, \u201cOligopeptide\u201d, \u201cRaffinose/Stachyose/ Melibiose\u201d, \u201cRibose/D-Xylose\u201d, \u201cMultiple sugar\u201d, \u201cPurin degrad. Xanthine- urea\u201d, \u201cTwo component\u201d, \u201cC4-Dicarboxylate\u201d, \u201cTetrathionate respiration\u201d, \u201cGeosmin\u201d. Below the ocher box: \u201cABC-Transporter\u201d, \u201cLipoprotein\u201d, \u201cPhospholipid\u201d, \u201cHeme\u201d, \u201cNa+\u201d, \u201cPTS\u201d, \u201cRaetz pathway\u201d. Within the boxes of each pathway, some abbreviations for genes had been lost.In Fig.\u00a0B, the y-axis label \u201crelative reads\u201d and the headers \u201cplasmids\u201d and \u201cphages\u201d were missing. In C, the y-axis label \u201cShannon index\u201d and the header \u201cplasmids ARG\u201d were missing. In D, the x-axis label \u201cBacteria\u2014FoldChange\u201d and the y-axis label \u201cPlasmid\u2014FoldChange\u201d were likewise missing. Moreover, \u201cup in no stress\u201d had been written in the green boxes and \u201cup in stress\u201d in the orange ones. In E, the x-axis label \u201cBacteria\u2014FoldChange\u201d and the y-axis label \u201cPlasmid\u2014FoldChange\u201d were missing. Finally, \u201cDesir\u00e9e\u201d had been written in the turquoise boxes and \u201cStirling\u201d in the ocher boxes.In Fig.\u00a0The error has since been corrected in the original article.The authors apologize for any inconvenience caused."}
{"text": "Coinfection of hepatitis B virus (HBV) and COVID-19 is a common public health problem throughout some nations in the world. In this study, a mathematical model for hepatitis B virus (HBV) and COVID-19 coinfection is constructed to investigate the effect of protection and treatment mechanisms on its spread in the community. Necessary conditions of the proposed model nonnegativity and boundedness of solutions are analyzed. We calculated the model reproduction numbers and carried out the local stabilities of disease-free equilibrium points whenever the associated reproduction number is less than unity. Using the well-known Castillo-Chavez criteria, the disease-free equilibrium points are shown to be globally asymptotically stable whenever the associated reproduction number is less than unity. Sensitivity analysis proved that the most influential parameters are transmission rates. Moreover, we carried out numerical simulation and shown results: some parameters have high spreading effect on the disease transmission, single infections have great impact on the coinfection transmission, and using protections and treatments simultaneously is the most effective strategy to minimize and also to eradicate the HBV and COVID-19 coinfection spreading in the community. It is concluded that to control the transmission of both diseases in a population, efforts must be geared towards preventing incident infection with either or both diseases. Illnesses caused by tiny microorganisms like viruses, bacteria, fungi, and parasites are known as infectious diseases; for instance, COVID-19 and hepatitis B diseases are infectious diseases caused by viruses . HepatitAn infectious disease known as COVID-19 is a highly contagious respiratory infection caused by SARS-CoV-2 virus, and for the first time, its outbreak was investigated in China at the end of December 2019 \u201319. On MLiteratures of some scholars mentioned in references , 28\u201332 iFor a better understanding of the spreading of communicable diseases, the concept of mathematical modelling has a fundamental impact . DiffereSome epidemiological and medical studies proved that hepatitis B virus and COVID-19 coinfection is a common public health issue. The main aim of this study is to discover the most effective control strategy from intervention strategies applied in the proposed HBV and COVID-19 coinfection model. Literatures  investedN(t) as a total human population in the study under consideration and divided it into eight distinct groups of individuals with their infection status as individuals who are susceptible to either of HBV or COVID-19 given by \u2009S(t), who are protected form COVID-19 given by \u2009CP(t), protected from HBV given by \u2009HP(t), infected with COVID-19 given by \u2009CI(t), infected with HBV given by HI(t), coinfected with HBV and COVID-19 given by C(t), recovered from COVID-19 given by CR(t), and treated from HBV infection given by HT(t) so that N(t) = S(t) + CP\u2009(t) + HP(t) + HI(t) + CI(t) + CR(t) + C(t) + HT(t).In this study, we need to construct a deterministic model on the coinfection of HBV and COVID-19. Consider \u03c11 < \u221e is the rate at which HBV infectivity increases and \u03c31 is the HBV spreading rate.Individuals who are susceptible will acquire HBV at the force of infection\u03c9 < \u221e is the rate at which COVID-19 infectivity increases and \u03c32 is the COVID-19 spreading rate.Individuals who are susceptible will acquire COVID-19 at the force of infection\u03b31, \u03b32 and (1 \u2212 \u03b31 \u2212 \u03b32) are portions of the human recruitment rate \u0393 that enters in the compartment S, CP and HP, respectively. Population is homogeneously mixing, population is not constant, HBV-treated individuals do not transmit HBV, HBV is not vertically transmitted, and HBV and COVID-19 do not transmit simultaneous dually.To construct the coinfection of HBV and COVID-19 model, let us assume the following: The parameters Using Based on Adding all differential equations given in  gives, \u2009CP(t), HP(t), HI(t), CI(t), \u2009C(t), CR(t), and HT(t) of the dynamical system  > 0, CP(0) > 0, HP(0) > 0, HI(0) > 0, \u2009CI(0) > 0, \u2009C(0) > 0, \u2009CR(0) > 0, and HT(0) > 0; then for each t > 0, we need to show that S\u2009(t) > 0, \u2009CP(t) > 0, HP(t) > 0, \u2009HI(t) > 0, CI(t) > 0, \u2009C(t) > 0, \u2009CR(t) > 0, \u2009and HT(t) > 0.Let \u2009\u03c4=sup{t > 0 : S\u2009(t) > 0, CP(t) > 0, HP(t) > 0,\u2009HI(t) > 0, CI(t) > 0, C(t) > 0,CR(t) > 0 and\u2009HT(t) > 0}.Define:\u2009S(t), CP(t), HP(t), HI(t), CI(t), C(t), CR(t), and\u2009HI(t) are continuous so that we assured that \u2009\u03c4 > 0. If \u03c4 = \u221e, then the nonnegativity holds. But, if 0 < \u03c4 < \u221e, \u2009S(\u03c4) = 0 or CP(\u03c4) = 0 or HP(\u03c4) = 0 or HI(\u03c4) = 0 or CI(\u03c4) = 0 or C(\u03c4) = or \u2009CR(\u03c4) = 0 or\u2009HI(0) = 0.The functions \u2009S(\u03c4) = M1S(0) + M1\u222b0\u03c4exp\u03bbH + \u03bbC + \u03bc))dt\u222b(((1 \u2212 \u03b31 \u2212 \u03b32)\u0393 + \u03b41CP(t) + \u03b42HP(t) + \u03b7CR(t))dt > 0, where M1 = exp\u03bc\u03c4 + \u222b0\u03c4(\u03bbH(w) + \u03bbC(w))\u2212 > 0, CP(t) > 0, HP(t) > 0, CR(t) > 0, and by the definition of \u2009\u03c4, the solution \u2009S(\u03c4) > 0; hence, S(\u03c4) \u2260 0.From the first equation of the system  we haveM1 = exp\u03b41\u03c4 + \u03bc\u03c4 + \u222b0\u03c4(\u03b4\u03bbH(w))\u2212 > 0, and from the definition of \u2009\u03c4\u2009, we proved that CP(\u03c4) > 0; hence, CP(\u03c4) \u2260 0.Similarly from the second equation of system  we have HP(\u03c4) > 0; hence, \u2009HP(\u03c4) \u2260 0; HI(\u03c4) > 0; hence, HI(\u03c4) \u2260 0; CI(\u03c4) > 0; hence, CI(\u03c4) \u2260 0; C(\u03c4) > 0; hence, C(\u03c4) \u2260 0; CR(\u03c4) > 0; hence, CR(\u03c4) \u2260 0; and HI(\u03c4) > 0; hence, HI(\u03c4) \u2260 0.In the same manner, we have all the following results: \u03c4 = \u221e, and using definition of the constant \u03c4, all the model ) \u2264 \u222bdt and the result \u2212(1/\u03bc)ln(\u0394 \u2212 \u03bcN) \u2264 t + k, where k is an arbitrary constant. After a number of steps of computations, we have determined that the final result 0 \u2264 N\u2009(t) \u2264 \u0394/\u03bc means the dynamical system  = S(t) + HP(t) + HI(t) + HT(t), with infection rate \u2009\u03bbH = (\u03c31/N1)HI and initial data \u2009S(0) > 0, \u2009HP(0) \u2265 0, \u2009HI(0) \u2265 0,\u2009and \u2009HT(0) \u2265 0. In \u03a91 is both positive invariant and global attractor of each nonnegative solution of the HBV infection system )/(\u03b12 + \u03bc)), \u0394\u03c02/(\u03b12 + \u03bc), 0, 0).The disease-free equilibrium (DFE) point of the HBV infection submodel  is calcuUsing the van den Driessche and Warmouth well-known method illustrated in  we can cFV\u22121 represented byFinally, using the method in  and afte\u211bH is defined as the average number of secondary infections caused by a single infected person during his life infectious period in a susceptible group, the submodel has a local stable DFE, \u2009EH0 = (\u0393/\u03bc((\u03b42 + \u03bc(1 \u2212 \u03b32))/(\u03b42 + \u03bc)), \u0393\u03b32/(\u03b42 + \u03bc), 0, 0) whenever \u211bH < 1 and unstable whenever \u211bH > 1.Because the calculated reproduction number of HBV submodel given by Making the submodel  equationm1 = \u03b42 + \u03bc and m2 = \u03bc + \u03bc1 + \u03b3 and substitute HI\u2217 in the incidence rate of HBV and calculated asLet us put From equation  we computhat is,\u211bH > 1.only if \u03bbH\u2217 > 0 obtained above, we can conclude that the submodel  represents the DFE of the sum-model  = I, \u2009Y0 has a global asymptotic stabilityFor  \u2208 \u03a91, where H = DUJ is an M-matrix, i.e., the off diagonal elements of H are nonnegative and \u03a91 is the region in which the system makes epidemiological sense. The DFE of the submodel  has a global asymptotic stability provided that\u2009\u211bH < 1submodel  given byLet us assume the following:EH0 = (\u0393/\u03bc((\u03b42 + \u03bc(1 \u2212 \u03b32))/(\u03b42 + \u03bc)), \u0393\u03b32/(\u03b42 + \u03bc), 0, 0) of the HBV dynamical system  = (((1 \u2212 \u03b32)\u0393(\u03b42 + \u03bc) + \u03b42\u03b32\u0393)/\u03bc(\u03b42 + \u03bc), \u03b32\u0393/(\u03b42 + \u03bc)) has a global stability and satisfies criteria (i) of Theorem 4 andUsing We computed the result given byS \u2264 N1, we can prove that S/N1 \u2264 1 and EH0 = (\u0393/\u03bc((\u03b42 + \u03bc(1 \u2212 \u03b32))/(\u03b42 + \u03bc)), \u0393\u03b32/(\u03b42 + \u03bc), 0, 0) has a global asymptotic stability whenever \u2009\u211bH < 1.Here, because \u211bH < 1 in this case the total number of population is going up.Epidemiologically, it means that the HBV single infection will die out in the community provided that \u2009HP = HI = C = HT = 0 for the dynamical system (S(0) > 0, CP(0) \u2265 0, CI(0) \u2265 0, CR(0) \u2265 0 and total number of individuals given by \u2009N2(t) = S(t) + CP(t) + CI(t) + CR(t).By making \u2009l system  the COVI\u03a92 is both positive invariant and global attractor of each nonnegative solution of the subdynamical system  = (\u0393/\u03bc((\u03b41 + \u03bc(1 \u2212 \u03b31))/(\u03b41 + \u03bc)), \u0393\u03b31/(\u03b41 + \u03bc), 0, 0).The COVID-19 subdynamical system  disease-l system  is given\u211bC = (\u03c32(\u03bc(1 \u2212 \u03b31) + \u03b41))/((\u03bc + \u03bc2 + \u03ba)(\u03bc + \u03b41)).The COVID-19 subdynamical system  effectivEC0 =  = (\u0393/\u03bc((\u03b41 + \u03bc(1 \u2212 \u03b31))/(\u03b41 + \u03bc)), \u0393\u03b31/(\u03b41 + \u03bc), 0, 0) has a local asymptotic stablty whenever \u211bC < 1 and unstable whenever \u211bC > 1.Based on the next generation matrix, the DFE point of the COVID-19 subdynamical system given by \u2009n1 = \u03b41 + \u03bc, \u2009n2 = \u03ba + \u03bc + \u03bc2, and n3 = \u03bc + \u03b7.The COVID-19 subdynamical system  endemic CI\u2217 stated in (N2\u2217\u03bbC\u2217 = \u03c32CI\u2217 and gives asWe can substitute tated in  in 24)  CI\u2217 statBy arranging equation  we deter\u211bC > 1.whenever Using  we deriv\u211bC > 1.only whenever \u211bC > 1.Hence, subdynamical system  has a un\u211bC > 1.The COVID-19 subdynamical system  has a poEC0 = (((1 \u2212 \u03b31)\u0393(\u03b41 + \u03bc) + \u03b41\u03b31\u0393)/(\u03bc(\u03b41 + \u03bc)), \u03b31\u0393/(\u03b41 + \u03bc), 0, 0) has a global asymptotic stability whenever \u2009\u211bC < 1, and the two sufficient criteria stated in The DFE point of the COVID-19 subdynamical system  given byU \u2208 \u211d2 to be the noninfected components, V \u2208 \u211d2 to be the infected components including the COVID-19 recovery group. Now we derived the matrices given byNow using the criteria in After a number of some steps of computations, we derived the following:S \u2264 S0 and CP < CP0\u2009, we can show that (S0/(S0 + CP0)S0 + CP0) \u2265 (S/N2) and EC0 = (((1 \u2212 \u03b31)\u0393(\u03b41 + \u03bc) + \u03b41\u03b31\u0393)/(\u03bc(\u03b41 + \u03bc)), \u03b31\u0393/(\u03b41 + \u03bc), 0, 0) of the COVID-19 subdynamical system  + \u03b42))/((\u03b3 + \u03bc + \u03bc1)(\u03bc + \u03b42)), (\u03c32(\u03bc(1 \u2212 \u03b31) + \u03b41))/((\u03bc + \u03bc2 + \u03ba)(\u03bc + \u03b41))}, where \u211bHC0 = max{\u211bH, \u211bC}, \u211bH to be the HBV submodel \u2009 exists if \u2009\u211bH > 1 and \u211bC > 1, i.e., \u211bHC0 > 1. We have discussed the complete model endemic equilibrium in the numerical analysis section.The dynamical system  is highlS = z1, CP = z2, HP = z3, \u2009HI = z4, \u2009CI = z5, \u2009C = z6, \u2009CR = z7,\u2009and \u2009HI = z8 such that N = z1 + z2 + z3 + z4 + z5 + z6 + z7 + z8.Assume \u2009Z = T, the dynamical system (dZ/dt = F(Z) with F = T, as\u03bbH = \u03c31/N[z4 + \u03c11z6]\u2009, 1 \u2264 \u03c11 < \u221e, \u2009\u03bbC = \u03c32/N[z5 + \u03c9z6], and \u20091 \u2264 \u03c9 < \u221e. Then, the Jacobian matrix of the complete dynamical system , is derived asF1 = \u2212(\u03c31/N0)z10, \u2009F2 = \u2212\u03c32z10, F3 = \u2212(\u03c31/N0)\u03c11z10 \u2212 \u03c32\u03c9z10, \u2009F4 = \u2212(\u03c31/N0)z20, F5 = \u2212(\u03c31/N0)\u03c11z20, F6 = \u2212\u03c32z30, F7 = \u2212\u03c32\u03c9z30, \u2009F8 = (\u03c31/N0)z10 + (\u03c31/N0)z20 \u2212 (\u03bc + \u03bc1 + \u03b3), F9 = (\u03c31/N0)\u03c11z10 + (\u03c31/N0)\u03c11z20, F10 = \u03c32z10 + \u03c32z30 \u2212 (\u03ba + \u03bc + \u03bc2), and F11 = \u03c32\u03c9z10 + \u03c32\u03c9z30.Moreover, the vector representation \u2009l system  at \u2009EHC0\u211bC > \u211bH and \u211bHC0 = 1, so that \u211bC = 1. Moreover, assume \u03c32 = \u03c3\u2217\u2009and taken as a bifurcation parameter. Calculating the expression for \u03c32 using \u211bC = 1, i.e., \u211bC = (\u03c32(\u03bc(1 \u2212 \u03b31) + \u03b41))/((\u03bc + \u03bc2 + \u03ba)(\u03bc + \u03b41)) = 1, we computed the value \u03c3\u2217 = \u03c32 = ((\u03bc + \u03bc2 + \u03ba)(\u03bc + \u03b41))/((\u03bc(1 \u2212 \u03b31) + \u03b41)).Let us consider the case at J(EHC0) of the dynamical system  of the system . Applying the Castillo-Chavez and Song criteria stated in [\u211bC = 1. In the right and left eigenvectors of J\u03b2\u2217, at the case whenever \u211bC = 1, the right eigenvector of the Jacobian of the dynamical system (\u03c32 = \u03c3\u2217 (represented by\u2009J\u03b2\u2217) corresponding to a simple zero eigenvalue is represented by u = TTherefore, every eigenvalue is negative if e system  at DFE, tated in  can be ul system  at \u03c32 = \u03c32 = \u03c32\u2217 qualifying the product \u2009y.w = 1, given as w = , are w1 = w2 = w3 = w4 = w6 = w7 = w8 = 0 and w5 = w5 > 0.Left eigenvectors corresponding to the simple zero eigenvalue at \u2009a and b given byUsing many steps of calculations, we have derived the bifurcation coefficients Then,\u211bHC0 = \u211bC = 1. Hence, only the disease-free equilibrium point given by\u211bHC0 < 1.Therefore, applying the Castillo-Chavez and Song criteria stated in  we have iven by4EHC0=S0,C\u211bHC0 = max{\u211bH, \u211bC} < 1.Note: in the subsections represented To verify the mathematical analysis results shown in the previous sections and subsections, we have carried out various sensitivity and numerical analyses. For the sensitivity and numerical analysis computations, we used parameter values adopting from different scholar studies and given the collection in y normalized forward sensitivity index which depends on a differentiable parameter \u03be is defined by SI(\u03d1) = (\u2202y/\u2202\u03d1)\u2217(\u03d1/y) [Definition: the variable \u03d1)\u2217(\u03d1/y) , 41, 55.\u211bH and \u211bC since \u211bHC0 = max{\u211bH, \u211bC}.The sensitivity analysis is used to examine the most influential parameters in the spreading of the coinfection of HBV and COVID-19. From results of sensitivity analysis among others, the one which has a larger sensitivity index in magnitude is known as the most sensitive parameter. For this study, the sensitivity indices can be computed using the model effective reproduction numbers given by Applying \u211bH = 1.82 which implies that HBV infection spreads throughout the population. Also, \u03c31 has major effect on the HBV effective reproduction number denoted by \u211bH.Using \u211bC = 3.23\u2009 which implies that the COVID-19 single infection is persistent throughout the population. Also, the sensitivity analysis given in \u03c32 is the most sensitive model parameter which has great impact on the COVID-19 transmission. Comparing sensitivity indices given in Tables \u03c31 and COVID-19 spreading rate \u2009\u03c32 are the most influential model parameters in the disease transmission, and stakeholders shall concentrate to control the values of these parameters by considering the suitable intervention strategies.In a similar manner, applying values of the model parameters stated in \u03c31 and \u03c32 are highly sensitive with respect to the HBV infection and COVID-19 infection submodel effective reproduction numbers, respectively. Also, one can conclude that portions of protections \u03b31 and \u2009\u03b32 and COVID-19 treatment rate \u03ba are more sensitive parameters and important to control the disease transmission in the community.Simulation represented in In this subsection, we carried out numerical analysis of the dynamical system . For sim\u211bHC0max{\u211bH, \u211bC} = max{0.14, 0.26\u2009} = 0.26 < 1, and the simulation result is illustrated in In this subsection, we performed the complete coinfection model numerical simulation by considering the value of the model effective reproduction number as \u211bHC0 = 3.23. The simulation result illustrated in \u211bHC0 = 3.23 > 1.In this subsection, we performed numerical simulation of the full dynamical system  using mo\u03c31 on the number of HBV and COVID-19 coepidemic individuals C. From \u03c31 from 0.00001 to 0.8 leads to a highly increase of HBV and COVID-19 coepidemic number of individuals C.Simulation illustrated in \u03c32 on the number of HBV and COVID-19 coepidemic people C. From \u03c32 increases, then the number of HBV and COVID-19 coinfectious individuals C is going up. Thus, increasing COVID-19 spreading rate \u03c32 from 0.00001 to 0.8 makes the HBV and COVID-19 coinfection C highly increases.Numerical simulation given in HI throughout the community. From the result, we can conclude that when the value of treatment rate \u03b3 is going up, then the number of HBV infectious individuals HI is going down. For the stakeholders, we recommend that they take their maximum effort to increase the value of HBV treatment rate to minimize the HBV transmission rate.Numerical simulation shown in C. From the result, we can conclude that whenever we increase the value of treatment rate \u03b8, the number of coinfectious population is going down. Thus, whenever we increase the value of \u03b8 from 0.2 to 0.8, then the number of HBV and COVID-19 coinfectious individuals decreases through time.In this subsection, numerical simulation represented in \u03c31 highest direct impact on the HBV single infection model effective reproduction number \u211bH. From the numerical result, we observed that increasing the HBV spreading rate \u03c31 has a direct impact on its effective reproduction number \u2009\u211bH. Thus, introducing protective and controlling strategies against HBV spreading is fundamental to minimized \u03c31 value less than 0.801.\u03b32 has the highest indirect impact on the HBV submodel effective reproduction number \u211bH. The simulation result from \u03b32 increases then the HBV spreading rate decreases. Thus, applying the portion \u03b32 of the human recruitment rate \u0393\u2009 to be more than 0.597 makes the value of \u211bH less than one.Simulation illustrated in \u03b3 has influential indirect impact on \u211bH. We observed the result whenever we increase the treatment rate; then, the HBV transmission decreases in the community. Thus, applying the treatment rate \u03b3 to more than 0.898 made the value of the HBV infection effective reproduction number \u211bH less than one.\u03c32 on the COVID-19 subdynamical system  on the COVID-19 effective reproduction number given by \u211bC. The result proved that whenever the COVID-19 treatment rate (\u03ba) increases, then the value of \u211bC is going down. As a result, giving the value of \u03ba more than 0.758, then the value of \u211bC is below one, and we recommend for the stakeholders to maximize the value of \u03ba.Numerical simulation represented in S), the HBV protection group (HP), and the COVID-19 protection group (CP), and this made the model highly nonlinear and challenging for the qualitative analysis of the coinfection model. The model has been mathematically analyzed both for the submodels associating the cases that each disease type is isolated and in the case when there is HBV and COVID-19 coinfection. The proposed model includes the intervention strategies, protective as well as treatment, and numerical simulation of the deterministic model is presented. In the analysis, it has been indicated that the effect of protection as well as treating the infected ones with the available treatment mechanisms affects significantly the infection control strategy and its outcome. From the simulation results, it can be concluded that applying both protective and treatment control mechanisms simultaneously at the population level yields the most effective outcomes both economically and epidemiologically. Therefore, we strongly recommended to the stakeholders regarding economic as well as health issues to give more attention and the overall effort to implement both the protective and treatment control strategies simultaneously to minimize the HBV and COVID-19 single infections as well as the HBV and COVID-19 coinfection disease transmission in the community.In this paper, we have constructed and investigated a continuous time dynamical model for the transmission of HBV and COVID-19 coinfection with protection and treatment strategies. The model incorporates three noninfectious groups, the susceptible group (Any interested scholar can modify this study by considering the limitations of this study such as formulate a model which incorporate either of stochastic method, fractional order method, optimal control theory, age structure, or environmental effects, collect real data, and validate the formulated model."}
{"text": "Nature Communications 10.1038/s41467-021-25301-3, published online 16 August 2021Correction to: 2 atm A\u22121\u2019 were incorrectly written as \u2018cm2 atm mA\u22121\u2019. In figure 6a, the y-axis \u2018Specific activity at 0.9 V (mA cm\u22122)\u2019 was incorrectly written as \u2018Specific activity at 0.9 V (A cm\u22122)\u2019. The legend for Fig. 4a inadvertently reported the wrong temperature. \u2018333 K\u2019 was incorrectly written as \u2018353 K\u2019. The original article has been corrected.In this article the units on the y-axis of Figures 4a and 6a contained an error. In figure 4a, the y-axis units \u2018cm\u22121\u00b7rad\u22122\u2019 was in incorrectly written as \u2018kJ\u00b7mol\u22121\u00b7rad\u22122\u2019. The HTML has been updated to include a corrected version of the\u00a0The units in the Supplementary Table\u00a0Updated Supplementary Information"}
{"text": "Bufo bufo) has been the subject of many folk tales and superstitions in Western Europe, and as a result, it is characterised by numerous common names (zoonyms). However, the zoonyms of the toad and its associated traditions have remained unexplored in the Balkans, one of Europe\u2019s linguistic hotspots. In the present study, it was attempted to fill this knowledge gap by focusing on Greece, where more than 7.700 individuals were interviewed both in the field and through online platforms, in order to document toad zoonyms from all varieties and dialects of Greek, as well as local non-Greek languages such as Arvanitika, South Slavic dialects, and Vlach. It was found that the academically unattested zoonyms of the toad provide an unmatched and previously unexplored linguistic and ethnographic tool, as they reflect the linguistic, demographic, and historical processes that shaped modern Greece. This is particularly pertinent in the 21st century, when a majority of the country\u2019s dialects and languages are in danger of imminent extinction\u2013and some have already gone silent. Overall, the present study shows the significance of recording zoonyms of indigenous and threatened languages as excellent linguistic and ethnographic tools that safeguard our planet\u2019s ethnolinguistic diversity and enhance our understanding on how pre-industrial communities interacted with their local fauna. Furthermore, in contrast to all other European countries, which only possess one or only a few zoonyms for the toad, the Greek world boasts an unmatched 37 zoonyms, which attest to its role as a linguistic hotspot.The common toad ( Data collected online are based on more than 4.700 informants who volunteered to participate in this study through social media groups (Facebook). Together with the field data, the total observations for this research exceed 7.700 informants, rendering this work as one of the most comprehensive zoonymic studies in Greece. Written and verbal consent was obtained from all participants  in classical and Roman antiquity, apart from a literature search, online catalogues of ancient inscriptions and names  can be found throughout the modern Hellenic world , 78Discuot found . The Sla century , 77, 78.\u03b6\u03ac\u03bc\u03c0\u03b1 in the country reflects the areas which received intense Slavic settlement (e.g. large parts of Macedonia and Thrace), the areas which retain a significant Slavophone population to this day     .Macedonia: Smixi and Perivoli.Discussion. Term of unknown etymology. In Messolonghi, the female flathead grey mullet  is also called \u03bc\u03c0\u03ac\u03c6\u03b1.Thessaly: in the Vlach villages of Malakasi .Discussion. A zoonym of unknown etymology, used only to describe unusually large toads, while the term \u03bc\u03c0\u03c1\u03bf\u03ac\u03c4\u03b9\u03ba\u03bf\u03c5 is used exclusively for frogs and small-sized toads. Potentially from Greek \u03bc\u03c0\u03ac\u03ba\u03b1\u03ba\u03b1\u03c2/\u03bc\u03c0\u03ac\u03bc\u03c0\u03b1\u03ba\u03b1\u03c2 , or kaplumba\u011fa [covered frog or toad: kapl\u0131 (cover) +\u200e ba\u011fa] [Schildkr\u00f6te [shielded toad: Schild (= shield) + Kr\u00f6te (= toad)], while in Romanian it is called broasc\u0103-\u021bestoas\u0103 [frog with shell: broasc\u0103 (= frog) + \u021bestoas\u0103 (= shell)] [+\u200e ba\u011fa] . In Germ shell)] . Taking bru\u00e1sca refers to the tortoise or the toad. This is also due to the fact that differences in naming are often accompanied by particular beliefs. For example, in Zagori, according to a local belief among Greek-speakers (probably influenced by Vlachs), the toad originally had a shell, like other tortoises, but because it fell into Communion, God cursed it, causing the toad to lose its shell and roam naked for eternity.These conceptual differences have caused particular confusion among speakers of Vlach, with different villages ardently arguing over whether bru\u00e1sca in its Hellenised form \u03bc\u03c0\u03c1\u03ac\u03c3\u03ba\u03b1 refers exclusively to the toad, it has been shown here that in many Vlach populations it denotes the tortoise (broscus (= frog), which in turn may derive from the ancient Greek \u03b2\u03c1\u03bf\u03c4\u03ac\u03c7\u03bf\u03c2 , due to its superficial similarity to the amphibian . The same process has taken place for the toadfish (family Batrachoididae), which has been named after the amphibian in English, Catalan (gripau), and Spanish (pez sapo).It is also interesting to note that in the city of Volos and in many other places, the toad zoonym Bufotes viridis), which lacks any zoonym that is specifically dedicated to this species in any region of the country. Across its distribution in Greece, the zoonyms of the frog are assigned to the green toad. The following indicative examples are provided: \u03b1\u03c6\u03bf\u03c1\u03b4\u03b1\u03ba\u03cc\u03c2/\u03bc\u03c0\u03b1\u03c1\u03b4\u03b1\u03ba\u03cc\u03c2 (Crete), \u03b2\u03cc\u03b8\u03c1\u03b1\u03ba\u03bf\u03c2/\u03b2\u03cc\u03b4\u03c1\u03b1\u03ba\u03bf\u03c2/\u03b2\u03bf\u03c1\u03c4\u03b1\u03ba\u03bf\u03cd\u03b4\u03b9\u03bd (Cyprus), \u03b2\u03b1\u03b8\u03c1\u03b1\u03ba\u03bb\u03ac\u03c2 (Mani), \u03c6\u03b1\u03c1\u03b4\u03b1\u03ba\u03bb\u03cc\u03c2 (Rhodes), \u03c6\u03bf\u03c5\u03c1\u03b4\u03b1\u03ba\u03bb\u03ac\u03c2/\u03c6\u03bf\u03c1\u03b4\u03b1\u03ba\u03bb\u03ac\u03c2/\u03c6\u03bf\u03c1\u03c4\u03b1\u03ba\u03bb\u03cc\u03c2 (Leukada), \u03c6\u03b1\u03c1\u03b4\u03b1\u03ba\u03ac\u03c2 (Tsakonia), \u03bc\u03c0\u03ac\u03ba\u03b1\u03ba\u03b1\u03c2 , \u03bc\u03c0\u03ac\u03bc\u03c0\u03b1\u03ba\u03b1\u03c2 (Kolindros in Pieria).The second member of the family Bufonidae in Greece is the green toad  , mainly in Constantinople, but also in areas of linguistic contact [\u03c6\u03bf\u03c5\u03c1\u03bd\u03cc\u03c2, \u03c6\u03bf\u03c5\u03c1\u03bd\u03cc\u03bd, \u03c6\u03bf\u03c5\u03c1\u03bd\u03af\u03b1, \u03c6\u03bf\u03c5\u03c1\u03bd\u03b9\u03ac) and late antiquity , to the Migration Period  and the contraction of the Roman Empire, the late Middle Ages , the huge expansion of transhumant pastoralism in the 17th century (\u03bc\u03c0\u03c1\u03ac\u03c3\u03ba\u03b1), and developments of more recent times .Indeed, the findings presented here provide additional evidence for processes that shaped the evolution of the Greek language itself. Although the country\u2019s standard linguistic literature  consider contact , 102\u2013104st century Western perspective. In a European context, this is particularly pertinent to Greece, as it is the continent\u2019s only country that lacks a dialectal atlas. To this end, the authors of this study have begun recording all of the country\u2019s reptile and amphibian zoonyms, where both field research and online interviews are crucial . By showcasing the utility of the toad\u2019s zoonyms in anthropological, ethnozoological, historical, and linguistic studies, it is hoped that researchers in these fields will further explore the potential of zoonyms as substrate terms in their group(s) of study.Overall, this research stresses the fundamental importance of exploring and preserving linguistic diversity on a global scale, as it represents an exceptional tool for understanding humanity\u2019s shared history, to investigate the similarities and differences on how people interacted with their surroundings, and to preserve different ways of meaning that do not conform to the 21S1 Text(DOCX)Click here for additional data file.S1 Table(CSV)Click here for additional data file."}
{"text": "The corrected In the published article, there was an error in 120 the HC value was \u201c39.1 \u00b1 15.4\u201d but should have been \u201c37.4 \u00b1 16.2.\u201d For PPI200 the HC value was \u201c50.1 \u00b1 16.5\u201d but should have been \u201c49.8 \u00b1 17.0.\u201d For PPI300 the HC value was \u201c41.8 \u00b1 18.7\u201d but should have been \u201c40.5 \u00b1 19.8.\u201d The corrected In the published article, there were also errors in The authors apologize for these errors and state that this does not change the scientific conclusions of the article in any way. The original article has been updated."}
{"text": "In the recent article by P\u00e9rez\u2010S\u00e1nchez et al.\u00a0, the authttps://orcid.org/0000\u20100002\u20101686\u20108811Anett Schibalski: https://orcid.org/0000\u20100002\u20108577\u20107980Boris Schr\u00f6der: https://orcid.org/0000\u20100002\u20102544\u2010640XSebastian Klimek: https://orcid.org/0000\u20100002\u20103420\u20100380Jens Dauber:"}
{"text": "Deinagkistrodon acutus, also known as Sharp-nosed Pit Viper, one hundred-pacer viper or five-pacer viper, is a venomous snake with significant economic, medicinal and scientific importance. Widely distributed in southeastern China and South-East Asia, D. acutus has been primarily studied for its venom. Here, we employed next-generation sequencing to assemble and annotate a highly continuous genome of D. acutus. The genome size is 1.46\u00a0Gb; its scaffold N50 length is 6.21\u00a0Mb, the repeat content is 42.81%, and 24,402 functional genes were annotated. This study helps to further understand and utilize D. acutus and its venom at the genetic level.The study of the currently known >3,000 species of snakes can provide valuable insights into the evolution of their genomes.  \u5bf9\u76ee\u524d\u5df2\u77e5 3,000 \u4f59\u79cd\u86c7\u7c7b\u7684\u7814\u7a76\u53ef\u4e3a\u5b83\u4eec\u7684\u57fa\u56e0\u7ec4\u8fdb\u5316\u63d0\u4f9b\u6709\u4ef7\u503c\u7684\u89c1\u89e3\u3002\u5c16\u543b\u876e\uff0c\u4e5f\u88ab\u79f0\u4e3a\u5c16\u9f3b\u876e\u3001\u767e\u6b65\u86c7\u6216\u4e94\u6b65\u86c7\uff0c\u662f\u4e00\u79cd\u5177\u6709\u91cd\u8981\u7ecf\u6d4e\u3001\u533b\u5b66\u548c\u79d1\u5b66\u4ef7\u503c\u7684\u6bd2\u86c7\u3002\u5176\u5e7f\u6cdb\u5206\u5e03\u4e8e\u4e2d\u56fd\u4e1c\u5357\u90e8\u548c\u4e1c\u5357\u4e9a\uff0c\u4e3b\u8981\u7528\u4e8e\u86c7\u6bd2\u7814\u7a76\u3002\u672c\u6587\u91c7\u7528\u4e8c\u4ee3\u6d4b\u5e8f\u6280\u672f\uff0c\u7ec4\u88c5\u548c\u6ce8\u91ca\u4e86\u4e00\u4e2a\u9ad8\u5ea6\u8fde\u7eed\u7684\u5c16\u543b\u876e\u57fa\u56e0\u7ec4\u3002\u57fa\u56e0\u7ec4\u5927\u5c0f\u4e3a 1.46 Gb; \u5176 scaffold N50 \u957f\u5ea6\u4e3a 6.21 Mb\uff0c\u91cd\u590d\u5e8f\u5217\u542b\u91cf\u4e3a 42.81%\uff0c\u5171\u6ce8\u91ca\u51fa 24,402 \u4e2a\u529f\u80fd\u57fa\u56e0\u3002\u672c\u7814\u7a76\u6709\u52a9\u4e8e\u5728\u9057\u4f20\u6c34\u5e73\u4e0a\u8fdb\u4e00\u6b65\u8ba4\u8bc6\u548c\u5229\u7528\u5c16\u543b\u876e\u53ca\u5176\u6bd2\u6db2\u3002 Deinagkistrodon acutus is a species of venomous pit viper, a member of the suborder Ophiopodes and the Viperidae family. It is commonly known as the Sharp-nosed Pit Viper, as well as hundred-pacer viper, five-pacer viper, Chinese moccasin, and Long-nosed Agkistrodon  weighing 781\u00a0g was obtained from Huangshan City, in Anhui (China), for genome assembly and annotation. The liver, stomach, kidney and muscle tissues were collected for RNA extraction. Additionally, two other muscle tissues were taken for DNA extraction before Whole Genome Sequencing (WGS) and single-tube long fragment read (stLFR) sequencing. We extracted the D. acutus DNA, constructed the library and performed paired-end sequencing according to the protocol described by Liu et\u00a0al.\u00a0.A specimen of .\u00a0Figure\u00a0\u00a0[10]. SaD. acutus genome. Kmerfreq from GCE  was used for k-mer frequency statistics. The output showed that 32,372,553,516 k-mer fragments (k =\u00a019) were obtained. Next, these results were input into GCE with the heterozygous mode (k-mer depth peak of 21) to evaluate genome size, heterozygosity and other parameters.We used the 25\u00d7 WGS sequencing data to estimate the size of our assembled SCR_016756). To make the assembled sequences more complete, we used GapCloser  and the WGS sequencing data to fill gaps. Also, to remove redundant sequences from the genome, we used redundans (v0.14a) . The final genome was obtained using the method described in Figure\u00a0de novo prediction and homology-based approaches to identify the repetitive regions in the genome assembly. The homology-based prediction was performed using Blastall (v2.2.26) . Specifically, we mapped the protein sequences from the UniProt database (release-2020_05) of Pseudonaja textilis, Crotalus tigris, Thamnophis elegans and Notechis scutatus to the D. acutus genome assembly. Annotation and assessment were performed according to the protocol described by Liu et\u00a0al.\u00a0.The stLFR data were used to generate the genome assembly using Supernova   to search for single-copy orthologs among the protein sequences of Rana temporaria (GCA_905171775.1), Gopherus evgoodei (GCA_007399415.1), Podarcis muralis (GCA_004329235.1), Thamnophis elegans (GCA_009769535.1) and Pseudonaja textilis (GCA_900518735.1).To construct a phylogenetic tree, we used OrthoFinder  and 185 (5.5%) genes were BUSCO fragments and deletions, respectively.We used the 164.75\u00a0Gb main result file generated by stLFR sequencing to assemble a 1.46\u00a0Gb D. acutus genome, the total length of repetitive sequences is 642\u00a0Mb, accounting for 42.81% of the genome  (443\u00a0Mb), followed by long terminal repeats (LTRs) (180\u00a0Mb), DNAs (26.43\u00a0Mb) and then short interspersed nuclear elements (SINEs) (0.94\u00a0Mb). The LINEs and LTRs contents were 29.53% and 11.99%, respectively (Table\u00a0In our me Table\u00a0. Based oLINEs) 44\u00a0Mb, follD. acutus  \u662f\u5c5e\u4e8e\u86c7\u4e9a\u76ee\u3001\u8770\u79d1\u7684\u4e00\u79cd\u6709\u6bd2\u86c7\uff0c\u5e38\u88ab\u79f0\u4e3a\u767e\u6b65\u86c7\u3001\u4e94\u6b65\u86c7\u3001\u957f\u9f3b\u876e\u7b49\uff08\u5982\u56fe, 2\u3002\u5176\u6bd2\u6db2\u4e3b\u8981\u5177\u6709\u8840\u6db2\u6bd2\u6027\uff0c\u53ef\u5bfc\u81f4\u51dd\u8840\u529f\u80fd\u5f02\u5e38\u5e76\u4fc3\u8fdb\u7ec4\u7ec7\u635f\u4f24\u3001\u6c34\u80bf\u3001\u6025\u6027\u80be\u8870\u7aed\u7b49\u53cd\u5e94\u53d1\u751f\uff0c\u4e3b\u8981\u4f5c\u7528\u4e8e\u80ba\u90e8 \u3002\u5c16\u543b\u876e\u5728\u4e2d\u56fd\u4e1c\u5357\u90e8\u3001\u8001\u631d\u548c\u8d8a\u5357\u5317\u90e8\u5e7f\u6cdb\u5206\u5e03\uff0c\u56e0\u5176\u8f83\u5927\u7684\u8eab\u4f53\u4ee5\u53ca\u6bd2\u6db2\u800c\u5177\u6709\u91cd\u8981\u7684\u5546\u7528\u53ca\u836f\u7528\u4ef7\u503c , 5\u3002\u76ee\u524d\u5c16\u543b\u876e\u7684\u7814\u7a76\u4e3b\u8981\u96c6\u4e2d\u5728\u5176\u6bd2\u6db2\u7684\u6bd2\u6027\u6210\u5206\u3001\u88ab\u54ac\u4f24\u60a3\u8005\u7684\u75c7\u72b6\u5206\u6790\u7b49\u65b9\u9762\uff0c\u4ee5\u53ca\u5bf9\u86c7\u6bd2\u7684\u5229\u7528\u8fdb\u884c\u4e86\u7814\u7a76\uff0c\u5982\u4f53\u5916\u6291\u83cc\u3001\u6bd2\u6db2\u4e2d\u7279\u5b9a\u86cb\u767d\u5177\u6709\u6297\u8840\u6813\u3001\u6297\u51dd\u8840\u6d3b\u6027\u7b49 \u20139\u3002\u9ad8\u8d28\u91cf\u7684\u57fa\u56e0\u7ec4\u6709\u52a9\u4e8e\u86c7\u6bd2\u76f8\u5173\u57fa\u56e0\u7684\u53d1\u73b0\uff0c\u8fdb\u800c\u53ef\u4ee5\u5e2e\u52a9\u7814\u7a76\u4eba\u5458\u66f4\u597d\u5730\u4e86\u89e3\u53ca\u5229\u7528\u86c7\u6bd2\u3002\u5c16\u543b\u876e YesIs the language of sufficient quality?YesPlease add additional comments on language quality to clarify if needed\rAre all data available and do they match the descriptions in the paper? YesAdditional CommentsAre the data and metadata consistent with relevant minimum information or reporting standards? See GigaDB checklists for examples <a href=\"http://gigadb.org/site/guide\" target=\"_blank\">http://gigadb.org/site/guide</a>NoAdditional CommentsI could not find the archive SRR24201538 referring to transcriptomic data in INSDC Sequence Read Archive (SRA).Is the data acquisition clear, complete and methodologically sound?YesAdditional CommentsIs there sufficient detail in the methods and data-processing steps to allow reproduction?NoAdditional CommentsA very essential methodology for a genome sequencing article is the assembly process. The authors mention: \"We used stlFR data for genome assembly, to make the assembled sequences more complete\" and provide a protocols.io https://dx.doi.org/10.17504/protocols.io.5jyl8j6e9g2w/v2 as the primary methodology reference. However, this protocol does not describe the assembly process step . It just reports the initial steps  and the final bioinformatic step (gene annotation). Please clarify in the main text how the stlFR reads were processed until reaching a draft genome and provide the detailed protocol as a supplementary file or link.Is there sufficient data validation and statistical analyses of data quality? Not my area of expertiseAdditional CommentsIs the validation suitable for this type of data?YesAdditional CommentsIs there sufficient information for others to reuse this dataset or integrate it with other data?YesAdditional CommentsAny Additional Overall Comments to the AuthorRecommendationMinor Revision Reviewer name and names of any other individual's who aided in reviewer Jia-Tang LiDo you understand and agree to our policy of having open and named reviews, and having your review included with the published papers. YesIs the language of sufficient quality?YesPlease add additional comments on language quality to clarify if needed\rAre all data available and do they match the descriptions in the paper? YesAdditional CommentsAre the data and metadata consistent with relevant minimum information or reporting standards? See GigaDB checklists for examples <a href=\"http://gigadb.org/site/guide\" target=\"_blank\">http://gigadb.org/site/guide</a>YesAdditional CommentsIs the data acquisition clear, complete and methodologically sound?YesAdditional CommentsIs there sufficient detail in the methods and data-processing steps to allow reproduction?YesAdditional CommentsIs there sufficient data validation and statistical analyses of data quality? YesAdditional CommentsIs the validation suitable for this type of data?YesAdditional CommentsIs there sufficient information for others to reuse this dataset or integrate it with other data?YesAdditional CommentsAny Additional Overall Comments to the AuthorIn this manuscript, the authors have sequenced and assembled the genome of an important venomous snake. This provides significant data to support future studies on snake evolution. While the content of the manuscript is insightful, there are certain areas where the language could be improved. For example, the title of Figure 2 appears to be incorrectly expressed. I suggest that the author perform a thorough check of the entire paper and make necessary corrections. Furthermore, in Figure 5, the authors need to explain the meaning of the numbers and dots on the phylogenetic tree. The formatting of references, including Latin names, also needs to be checked and corrected, where necessary, to ensure that they are properly italicized. Besides, it is worth noting that the first genome of the sharp-nosed pit viper has already been published . Therefore, I recommend that the authors include a comparison with older versions of the genome in this manuscript.RecommendationMinor Revision Editor\u2019s AssessmentThe sharp-nosed pit viper, Deinagkistrodon acutus is a highly venomous snake distributed across South East and Eastern Asia. To help better understand the evolution of D. acutus, and also provides a molecular basis for the understanding venom production a 1.46Gb in size reference genome was sequenced and described here. This data can be combined with already published and new venomous snake genome data to construct the evolutionary history of venomous snakes and other reptiles. After submission,feedback and improvement to the language, it is now useful to share this data to the community."}
{"text": "Following the rubric of corpus-assisted Discourse-Historical Approach (DHA) to Critical Discourse Studies (CDS), we trace (a) the main meaningful patterns, and (b) discursive and argumentation strategies (topoi) in three balanced corpora of mainstream news portals aligned with centre-right and centre-left political views. Among our main findings, the mobilisation of migrant populations is construed as an extremely polarised issue both in national and EU contexts and claims in favour of its prevention are justified on topoi of  Since the increase of the movement of refugee and migrant populations from war and conflict zones in the Middle East to the Mediterranean in summer 2015, the European public sphere has witnessed high polarization produced leading to an institutionalization (Zappettini in prep) of politics against migrant populations. To this end, we study balanced corpora of articles from mainstream online news portals in Greece, Malta and Italy, which have been identified as the EU member-states that constitute the primary entry points of migrant populations, and have, as a result, experienced highly politicised domestic debates over the \u2018refugee crisis\u2019 , outlined by the Discourse-Historical Approach (henceforth DHA), which can be seen as underlying the main reasoning patterns that justify hatred against migrant populations in the perceived peak of the \u2018refugee crisis\u2019 (06/2015\u201312/2017). Our analysis follows in Sects.\u00a0In what follows, we firstly provide a theoretical discussion of the process of institutionalization and the role of mass media in framing politics Sect.\u00a0. In Sects\u2019 06/201\u201312/2017.We ground our study within the larger theoretical framework of institutionalization in the \u201cnetworked public sphere\u201d legitimised. In this sense, discursive chains can become self-reinforcing and path-dependent processes, as specific narratives are given epistemic value over others along the chain. For example, Krzy\u017canowski . Even though a direction of causality between a newspaper\u2019s political alignment, news coverage and the political attitudes of its readership cannot be established in a linear fashion, we can see public discourses as manifested in the media as capable of prelegitimising and institutionally reinforcing specific (hatred) arguments of the political debate over migration.Importantly, as previously mentioned, not only can the discursive framing operated by the mass media over a topic such as immigration contribute to prime audiences\u2019 perceptions of it, but it can also legitimise political decision-making processes triggering a further setting of political agendas and policy deliberations Kingdon . From anOur analysis (as detailed below) therefore treats journalist practices as \u201can argumentative discourse genre\u201d  them  and Efimer\u00edda ton Syntakt\u00f3n , the Maltese Malta Independent  and Malta Today , and the Italian Corriere della Sera  and La Repubblica .Our overall dataset comprises online news portals\u2019 articles in three national contexts, namely Greece, Malta and Italy during the peak period of the \u2018refugee crisis\u2019 (2015\u20132017). We selected two mainstream online portals, i.e. portals with high traffic but different ideological orientations, in each national context. More specifically, our data come from the Greek portals refugee* and migrant*/immigrant* in all three languages that were published online from 06/2015 to 12/2017, using the European Commission\u2019s platform NewsBrief.We downloaded all the news articles comprising the terms To ensure representativeness between each national context and news portal, we created three balanced corpora of approximately 100,000 words for each country that is 113,474 words for Greece, 92,437 words for Malta, and 98,715 words for Italy, by randomly incorporating in it one out of every ten or twelve articles for every online news portal, respectively. Thus, we are confident that our balanced corpora constitute a representative sample of the mainstream ideological orientations  in the Greek, Maltese and Italian mediascape, throughout the topical two-years period of the emergence of the \u2018refugee crisis\u2019 in the Mediterranean. Table Sketch Engine softwareSketch Engine . In this way, we were able to pinpoint the main meaningful lines around which online news portals constructions on the \u2018refugee crisis\u2019 revolve in the Mediterranean mediascape. Following this mapping of the main meaningful constructions revolving around the phenomenon of migration towards the Mediterranean in the examined timeframe , the concordance analysis paved the way to a more qualitative this time investigation of the portrayals of the targeted groups in (con)text. Specifically, while analysing concordances generated by Sketch Engine we were able to initially show how refugee/ migrant populations and their actions are discursively constructed in clause complexes that constitute the concordance lists.In order to analyse our data, we drew on the methodological outlook of corpus-assisted CDS, which enables the researcher to identify the main attitudes that emerge in large corpora within a specific timeframe, all the while avoiding possible biases of data cherry-picking through an automatic software analysis  in our frequency and keyword analysis, we zoom in on the examination of the main patterns of the discursive mobilisation of the main social actors  and the actions they undertake or are undergone in this event, through concordances analysis, during which we qualitatively analyse exemplary cases-extracts that represent the main patterns identified \u201d in terms of \u201c evaluative attributions of negative or positive trails [\u2026], explicit predicates or predicative nouns/adjectives/pronouns\u201d (ibid.). This cross-examination gave us a first look at the discursive construction and evaluation of migrant populations in exemplary extracts, which are part of the first twenty concordances appearing in the relevant automatic analysis.During this step, we specifically drew on a study of \u201cdiscursive strategies\u201d outlined by the DHA  couplings can emerge on the basis of this discursive construction and evaluation  of migrant populations in online news, instantiating certain argumentative place in which an argument is developed\u201d  emerging in our corpora, we showcase the \u2018logical lines\u2019 on the basis of which claim-argument(s) pairs against migrants/migration can be legitimately developed. It is essential to highlight that we do not claim that the relevant topoi constitute hate speech per se; rather, they can legitimately be the argumentative lieu on which hatred and exclusion of migrants can be justified all the while being proliferated through media lens  across times (2015\u20132017) and places , institutionalising relevant policies that further establish hatred and exclusionary perspectives in society. In this respect, we claim that the incitement to hatred can be oblique  and discriminatory  attitudes in relation to migration. The same conflict between humanitarian vis-\u00e0-vis discriminatory attitudes becomes more evident when one looks at the keyword tables  or police , which can form the conceptual basis for both discriminatory and humanitarian viewpoints. Moreover, in these corpora, too, the migratory phenomenon is tightly related to neighbour countries  as evident by the relevant words that appear in both the frequency  to other EU member-states, corroborating the view that European South was a transit-place for migrant populations seeking to move to Northern EU member-states. Surprisingly, if one takes into account its centre-left ideological orientation, Repubblica frequently uses the discriminatory term illegal immigrant, favouring in this sense an anti-migrant portrayal that has been witnessed in recent studies on the 2015\u20132017 \u2018refugee crisis\u2019 in the Italian context  as well as crime, illegal immigrant (Table Repubblica), appear to suggest that the Italian corpus leans more towards discriminatory discourseThe frequency and keywords analysis of Italian news portals\u2019 corpora provide us with more insight. While lemmas that are similar to the previous corpora text. To that end, we analysed the first 20 out of 100 concordances of the relevant types in our corpora. Figures\u00a0and\u2013in traced concordances. In this respect, it seems that populations who should be protected by international law (i.e. refugees) are devaluated when assigned equal status to populations who migrate because of economic and/or other reasons. Moreover, the fact that the relevant populations are realized always in plural, gives an impression of a dense group with no individually evident characteristics, shifting the focus from their place of origin and the reasons for their mobilization to their massive presence on Greek soil; a representation that can facilitate the construction of a state of emergency and the necessity for countermeasures . This need to confine migration is further backed up on constructions that picture a massive number of refugees coming to or staying on Greek soil . Overall, the examined concordances provide us with preliminary evidence of the ways migrant and refugee populations are treated by the centre-right news portal in terms of a dense, dangerous and massive social body present in Greece.As realised in the concordances coming from EfSyn is in line with its centre-left orientation, and thus takes a more sympathetic position regarding migration, similar problematic constructions appear here too . According to these constructions, migrants are portrayed in terms of dehumanising metaphors ,massive, huge. Moreover, in some cases, they are also portrayed as prone to violence .Although Turning to the Maltese news portals Figs.\u00a0a\u2013d, we nIndependent is capitalizing on concordances such as: There are over 300,000 Eritrean refugees\u2013and thousands more seeking asylum\u2013across the world [doc#2], the influx of refugees to Europe was triggered in part by donors [\u2026], the burden of refugees flooding the continent [doc#51], which promote a dehumanised portrayal of the relevant populations (in line with such metaphors as: influx of refugees), foregrounding their massive presence (see the relevant numbers) and explicitly construing them as a burden in the host soil. Similar patterns appear when looking at the concordances of the word migrant* . Apart from the numbers employed to connote a massive presence of migrant populations, nominal types such as illegal migrants can also be seen as provoking a discriminatory stance against the respective populations.Specifically, while reporting on refugee populations, the centre-right Malta Today, when it comes to reporting on migrant populations, too. Concordances such as with rescues at sea occurring at a rate of over 1,000 migrants a day (doc#758), accelerate the sense of emergency because of the burden, i.e., the large numbers of migrant populations that Mediterranean member-states must rescue. This seems to be decreased only in cases when countermeasures are implemented . When it comes to representing refugee populations in this context, once more, reporting on their massive presence creates a sense of emergency , while even more problematic and unique among the concordances under study, is a construal of their mobilisation in terms of a crisis that EU officials pinpoint (see Junker focused on the refugee crisis currently faced by the European Union [doc#789]) and the Union has to deal with (see Europe\u2019s refugee crisis is \u201chere to stay\u201d [doc#777]).The same construction is favoured by the centre-left Finally, in accordance with the Maltese corpus, the concordances analysed from Italian news portals Figs.\u00a0a\u2013d make Corriere problematises refugees mobilisation by employing terms such as refugee emergency or refugee problem . On top of that, dehumanising metaphors are used also in this case (see because of the flux of refugees [doc#105]), \u201cfram[ing] migration as a threat and thus assign[ing] a special sense of urgency to the situation, indirectly calling for mobilization on the part of the in-group to defend itself\u201d , the arrival of 10,000 of migrants every week in November (doc#105), give rise to an interpretation whereby there is a massive mobilisation of migrants to the host territory, while, in other cases the emergent issue of migration is additionally pinpointed . All in all, migration is construed as an emergency and a problem, a burden for the host country/ies; a construction that implies a need to prevent it.The centre-right Repubblica seems to mostly empathise with refugee populations, as evident in constructions like was available to host refugees at home (doc#1312), [\u2026] criticizes the EU-Turkey agreement for pushing back refugees who depart from Greece, [\u2026] more than 1,000 places for hosting refugees (doc#1322). Along with these, however, concordances that aggregate refugees and present them undertaking negatively evaluative actions (see there are at least a hundred of refugees and asylum seekers who during the night [\u2026] are drunk [doc#1362]) are present, in line with previous constructions in different news portals and national settings. This portrayal is further supported by the news portal\u2019s reports on migrants. Despite its centre-left orientation, Repubblica also appears to follow discriminatory lines similar to the ones witnessed in previous analyses when migrants were portrayed as a numerous mass  that burdens the host societies (The burden of hosting migrants [doc#1212]) and accelerates a sense of emergency (the migrants\u2019 emergency [doc#1252]).On the contrary, the centre-left influx, flows) and/or negative evaluative language . Along with a use of numerals  that pinpoint their massive presence in the host soils, these can be seen to aggravate the sense of emergency that the so-called crisis gives rise to.In sum, when one looks at concordances that include the main target populations, one can easily witness constructions of threat and burden assigned to them through the use of dehumanising metaphorical vehicles \u03a3\u03ba\u03b7\u03bd\u03ad\u03c2, \u03c3\u03bb\u03af\u03c0\u03b9\u03bd \u03bc\u03c0\u03b1\u03b3\u03ba \u03ba\u03b1\u03b9 \u03b5\u03af\u03b4\u03b7 \u03ba\u03b1\u03b8\u03b1\u03c1\u03b9\u03cc\u03c4\u03b7\u03c4\u03b1\u03c2, \u03c3\u03c5\u03bd\u03bf\u03bb\u03b9\u03ba\u03ae\u03c2 \u03b1\u03be\u03af\u03b1\u03c2 300 \u03c7\u03b9\u03bb\u03b9\u03ac\u03b4\u03c9\u03bd \u03b5\u03c5\u03c1\u03ce, \u03b5\u03c4\u03bf\u03b9\u03bc\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u03bd\u03b1 \u03c3\u03c4\u03b5\u03af\u03bb\u03b5\u03b9 \u03c3\u03c4\u03b7\u03bd \u0395\u03bb\u03bb\u03ac\u03b4\u03b1 \u03bf \u03b4\u03b9\u03b5\u03b8\u03bd\u03ae\u03c2 \u0395\u03c1\u03c5\u03b8\u03c1\u03cc\u03c2 \u03a3\u03c4\u03b1\u03c5\u03c1\u03cc\u03c2, \u03c0\u03c1\u03bf\u03ba\u03b5\u03b9\u03bc\u03ad\u03bd\u03bf\u03c5 \u03bd\u03b1 \u03b2\u03b5\u03bb\u03c4\u03b9\u03c9\u03b8\u03b5\u03af \u03ba\u03b1\u03c4' \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf\u03bd \u03b7 \u03ba\u03b1\u03c4\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03c0\u03bf\u03c5 \u03ad\u03c7\u03b5\u03b9 \u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03b7\u03b8\u03b5\u03af \u03c3\u03c4\u03b1 \u03bd\u03b7\u03c3\u03b9\u03ac \u03c4\u03bf\u03c5 \u0391\u03b9\u03b3\u03b1\u03af\u03bf\u03c5 \u03b5\u03be\u03b1\u03b9\u03c4\u03af\u03b1\u03c2 \u03c4\u03b7\u03c2 \u03bc\u03b1\u03b6\u03b9\u03ba\u03ae\u03c2 \u03b5\u03b9\u03c3\u03c1\u03bf\u03ae\u03c2 \u03c0\u03c1\u03bf\u03c3\u03c6\u03cd\u03b3\u03c9\u03bd \u03ba\u03b1\u03b9 \u03bc\u03b5\u03c4\u03b1\u03bd\u03b1\u03c3\u03c4\u03ce\u03bd \u03ba\u03b1\u03b9 \u03c4\u03b7\u03c2 \u03c0\u03bb\u03ae\u03c1\u03bf\u03c5\u03c2 \u03ad\u03bb\u03bb\u03b5\u03b9\u03c8\u03b7\u03c2 \u03b4\u03bf\u03bc\u03ce\u03bd \u03c5\u03c0\u03bf\u03b4\u03bf\u03c7\u03ae\u03c2 \u03ba\u03b1\u03b9 \u03c6\u03b9\u03bb\u03bf\u03be\u03b5\u03bd\u03af\u03b1\u03c2.[\u2026] for the explosive dimensions and problems that the uncontrolled entry and stay of hundreds of illegal immigrants and refugees has created on the island in recent weeks. [...] and in view of the beginning of the tourist season, the Government must take immediate measures, because the uncontrolled entry of illegal immigrants and refugees to the Aegean islands has created an explosive atmosphere. .8(3)\u0397 \u03bd\u03ad\u03b1 \u03ba\u03b9\u03bd\u03b7\u03c4\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b7 \u03c0\u03c1\u03bf\u03c3\u03c6\u03cd\u03b3\u03c9\u03bd \u03c0\u03bf\u03c5 \u03c6\u03b9\u03bb\u03bf\u03be\u03b5\u03bd\u03bf\u03cd\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf \u03ac\u03c4\u03c5\u03c0\u03bf \u03ba\u03ad\u03bd\u03c4\u03c1\u03bf \u03c5\u03c0\u03bf\u03b4\u03bf\u03c7\u03ae\u03c2 \u03c4\u03bf\u03c5 \u0394\u03ae\u03bc\u03bf\u03c5 \u039b\u03ad\u03c3\u03b2\u03bf\u03c5 \u03c3\u03c4\u03b7\u03bd \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ae \u039a\u03b1\u03c1\u03ac \u03a4\u03b5\u03c0\u03ad \u03ba\u03b1\u03b9 \u03bf \u03c9\u03c1\u03b9\u03b1\u03af\u03bf\u03c2 \u03b1\u03c0\u03bf\u03ba\u03bb\u03b5\u03b9\u03c3\u03bc\u03cc\u03c2 \u03ba\u03b5\u03bd\u03c4\u03c1\u03b9\u03ba\u03bf\u03cd \u03b4\u03c1\u03cc\u03bc\u03bf\u03c5 \u03c4\u03b7\u03c2 \u039c\u03c5\u03c4\u03b9\u03bb\u03ae\u03bd\u03b7\u03c2 \u03ae\u03c1\u03b8\u03b1\u03bd \u03bd\u03b1 \u03c5\u03c0\u03b5\u03bd\u03b8\u03c5\u03bc\u03af\u03c3\u03bf\u03c5\u03bd \u03b3\u03b9\u03b1 \u03ac\u03bb\u03bb\u03b7 \u03bc\u03b9\u03b1 \u03c6\u03bf\u03c1\u03ac \u03ba\u03b1\u03b9 \u03bd\u03b1 \u03ba\u03ac\u03bd\u03bf\u03c5\u03bd \u03bf\u03c1\u03b1\u03c4\u03cc \u03c4\u03bf \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03b1\u03b4\u03c5\u03bd\u03b1\u03bc\u03af\u03b1\u03c2 \u03b4\u03b9\u03b1\u03c7\u03b5\u03af\u03c1\u03b9\u03c3\u03b7\u03c2 \u03c4\u03b7\u03c2 \u03c4\u03b5\u03c1\u03ac\u03c3\u03c4\u03b9\u03b1\u03c2 \u03b5\u03b9\u03c3\u03c1\u03bf\u03ae\u03c2 \u03bc\u03b5\u03c4\u03b1\u03bd\u03b1\u03c3\u03c4\u03ce\u03bd \u03c3\u03c4\u03bf \u03bd\u03b7\u03c3\u03af. [\u2026] \u039f \u03bc\u03b5\u03b3\u03ac\u03bb\u03bf\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03bc\u03b5\u03c4\u03b1\u03bd\u03b1\u03c3\u03c4\u03ce\u03bd (\u03c0\u03ac\u03bd\u03c9 \u03b1\u03c0\u03cc 4.500 \u03ac\u03c4\u03bf\u03bc\u03b1), \u03bf\u03b9 \u03ba\u03b1\u03b8\u03c5\u03c3\u03c4\u03b5\u03c1\u03ae\u03c3\u03b5\u03b9\u03c2 \u03c3\u03c4\u03b7 \u03b3\u03c1\u03b1\u03c6\u03b5\u03b9\u03bf\u03ba\u03c1\u03b1\u03c4\u03b9\u03ba\u03ae \u03b4\u03b9\u03b1\u03b4\u03b9\u03ba\u03b1\u03c3\u03af\u03b1, \u03b7 \u03b4\u03b9\u03b1\u03ba\u03bf\u03c0\u03ae \u03c5\u03b4\u03c1\u03bf\u03b4\u03cc\u03c4\u03b7\u03c3\u03b7\u03c2 \u03c3\u03c4\u03b7\u03bd \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ae, \u03c4\u03b1 \u03c0\u03c1\u03bf\u03b2\u03bb\u03ae\u03bc\u03b1\u03c4\u03b1 \u03c3\u03af\u03c4\u03b9\u03c3\u03b7\u03c2 \u03ba\u03b1\u03b9 \u03bf \u03bc\u03b9\u03ba\u03c1\u03cc\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03c0\u03c1\u03bf\u03c9\u03b8\u03ae\u03c3\u03b5\u03c9\u03bd \u03c0\u03c1\u03bf\u03c2 \u03c4\u03bf\u03bd \u03a0\u03b5\u03b9\u03c1\u03b1\u03b9\u03ac  \u03ad\u03c7\u03bf\u03c5\u03bd \u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03ae\u03c3\u03b5\u03b9 \u03ad\u03bd\u03b1 \u03b5\u03ba\u03c1\u03b7\u03ba\u03c4\u03b9\u03ba\u03cc \u03ba\u03bb\u03af\u03bc\u03b1.The international Red Cross is preparing to send tents, slip-in bags, and cleaning supplies to Greece, with a total value of 300 thousand euros, in order to improve at least the situation that has been created in the Aegean islands due to the massive influx of refugees and immigrants and the complete lack of reception and hospitality structures. .9The new mobilization of refugees, who are hosted in the informal reception centre of the Municipality of Lesvos in the Kara Tepe area, and the hourly blockade of the central road of Mytilini came to remind once again and make visible the problem of the inability to manage the huge influx of immigrants to the island. [\u2026] The large number of migrants , the delays in the bureaucratic process, the interruption of water supply in the area, the feeding problems and the small number of advances to Piraeus  have created an explosive atmosphere. .10and) and discursively constructed (nomination strategy) through nominal types that connote their massive mobilization (uncontrollable entry and stay) and openly discriminatory adjectives . They are moreover qualified, in the relevant predicates, as responsible for explosive dimensions and problems or the explosive atmosphere that their presence has created. These portrayals can give rise to a reasoning that supports different claims in favour of the prevention of the migratory mobilization, along two interweaved DHA topoi. On the one hand, a topos of threat or danger, which is realized in terms of the conditional: \u201cif there are specific dangers and threats, one should do something against them\u201d  and The large number of migrants  (extract 3), and predicates such as to improve at least the situation that has been created in the Aegean islands due to the massive influx of refugees and immigrants (extract 2) and The large number of migrants , [\u2026] have created an explosive atmosphere (extract 3) can be seen to facilitate xenophobic perspectives and claims of hatred against migrant populations; again, along the lines of the two aforementioned topoi of threat/danger and numbers.(4)Independent).11Around 100,000 migrants have entered Europe so far this year, with some 2,000 dead or missing during their perilous quest to reach the continent. Dozens of boats are launched from lawless Libya each week, with Italy and Greece bearing the brunt of the surge. [\u2026] Many more migrants from Africa and the Middle East are expected to arrive over the next three months \u2013 the summer high season for migrant departures. Today).12The number of migrants and asylum seekers who have arrived in Europe by sea so far in 2015 is now approaching a quarter of a million, according to an analysis by the International Organisation for Migration (IOM). With rescues at sea occurring at a rate of over 1,000 migrants a day this summer off Italy and Greece, the number of arrivals has already surpassed the total arrivals in 2014, the IOM said in a press release.  or The number of migrants and asylum seekers who have arrived in Europe by sea so far in 2015 is now approaching a quarter of a million (extract 5) including nominal types that construct a massive mobilization , all the while backing up standpoints in favour of a prevention of migration on a topos of numbers, as explained above.(6)Corriere).13L'emergenza profughi continua e il Comune di Bergamo, cos\u00ec come altri amministrati dal centrosinistra (e non solo), sta lavorando per trovare soluzioni riassumibili nel concetto di \"accoglienza diffusa\": gruppi poco numerosi spalmati su pi\u00f9 alloggi temporanei. Nel frattempo per\u00f2 i continui arrivi obbligano a cercare anche spazi di grande dimensione [\u2026] The refugee emergency continues and the Municipality of Bergamo, as well as other ones administered by the centre-left (and not only), is working to find solutions that can be summarized in the concept of \"widespread reception\": small groups spread over several temporary accommodations. In the meantime, however, the continuous arrivals also force us to look for large spaces [\u2026] Per il momento le strutture di accoglienza reggono le richieste, anche sotto il profilo sanitario, ma dalla frontiera si allontanano molti che cercano altre possibili strade verso il Nord Europa. Il bollettino della disperazione, oggi, conta altri 50 migranti in arrivo in Liguria da Lampedusa. Saranno ripartiti, come i 50 giunti ieri, 20 a Genova, 10 a Savona, 10 alla Spezia e 10 ad Imperia, e ospitati nelle strutture di accoglienza sul territorio.In almost the same vein, in the extracts from the Maltese news portals under study we find predicates like (8)Questi disperati per poter varcare il confine avrebbero pagato circa 150 euro ciascuno. I conducenti dei mezzi vistisi scoperti perch\u00e8 inseguiti dalle forze dell'ordine lasciavano i mezzi in autostrada, con a bordo i clandestini, in pendenza e contromano. Ormai in fondo a via Sammartini, accanto alla stazione Centrale di Milano, dove c'\u00e8 l'hub comunale per accogliere i migranti, si \u00e8 creato un campo profughi all'aperto. Sono almeno un centinaio i profughi e richiedenti asilo che di notte\u2014ma ormai anche di giorno\u2014bivaccano e dormono sul marciapiedi e sul prato che divide la strada dal naviglio della Martesama.For the moment, the reception structures are holding up against demands, even from the health point of view, but many are leaving the border looking for other possible routes to Northern Europe. The bulletin of despair today counts another 50 migrants arriving in Liguria from Lampedusa. They will be distributed again, like the 50 arrived yesterday, 20 in Genoa, 10 in Savona, 10 in La Spezia and 10 in Imperia, and hosted in the reception facilities in the area. .14Those desperate to cross the border would have paid around 150 euros each. The drivers of the vehicles having been found out by the police left the vehicles on the highway, with the illegal immigrants on board, on a slope and against the traffic. An open-air refugee camp has now been created at the end of via Sammartini, next to Milan Central Station, where there is the municipal hub to welcome migrants. There are at least a hundred refugees and asylum seekers who camp at night - but now also during the day - and sleep on the sidewalks and on the lawn that divides the road from the Martesama canal. .15The refugee emergency, the continuous arrivals (extract 6), The bulletin of despair (extract 7), illegal immigrants on board, at least a hundred refugees and asylum seekers (extract 8), which accelerate a sense of burden that the host societies need to undertake because of the construed emergency. Once more, an argumentation in favour of counteractions for the prevention of the migratory phenomenon is underpinned. This can be outlined in terms of the DHA topos of burdening or weighing down, which \u201cis to be regarded as a specific causal topos (a topos of consequence)\u201d and could \u201cbe reduced to the following conditional: if [\u2026] a \u2018country\u2019 is burdened by specific problems, one should act in order to diminish these burdens\u201d  along with others that pinpoint the European dimension of the issue . In doing so, the construction of a highly polarized national/ European discourse is favoured by the presence of lemmas that promote a humanitarian perspective  vis-\u00e0-vis others that underpin a more discriminatory and preventive one  in open/close spatial representations of Europe.To summarize our findings, following the rationale of studies belonging to CL, the first layer of our micro-level analysis  has shown that the news portals under study construed the mobilization of refugee migrant populations as a focal issue both for national and (extra)European affairs; this transpires from the presence of nominal types that refer to national contexts related to migration  seems to continuously permeate online news portals\u2019 discourses both from the centre-right and the centre-left across different social settings , we can reasonably argue that a macro-level anti-migrant discriminatory/xenophobic attitude is argumentatively justified and therefore established in the Mediterranean mediascape. According to the respective lines of reasoning, migrant populations are construed as \u2018Others\u2019 that should be discriminated and/or hated by host populations because they constitute a menacing burden to host societies. These reasoning lines constitute the logical basis that can potentially justify EU/European policies of restriction and exclusion based on the denigrating, discriminatory attitudes they cultivate against migrant populations; this is the argumentative basis of (soft) hate speech in the examined timeframe and contexts.Focusing, then, on concordances of topoi) identified in Sect.\u00a0Specifically, although none of the examined discursive instantiations falls under the definitions of prosecutable hate speech, since they do not explicitly incite to violence or hatred against them  sociodiagnostic critique and (c) prospective/retrospective critique.What we hope to have shown in this paper is that the systematic analysis of mass media discourses through the methodological lens outlined in this paper can be a first necessary step towards the establishment of a rigorous and critical scholarly stance against social inequalities that certain discursive patterns may aggravate in highly polarised contexts. It is worth mentioning that our effort does not exist in limbo. In their interdisciplinary tradition, CDS have facilitated ways of scholarly critique. For example, Reisigl and Wodak , 32\u201335 htext or discourse immanent critique \u201c[\u2026] is based on a hermeneutic exegesis with the help of specific linguistic and discourse-analytical tools  [it] aims at discovering inconsistencies, (self-)contradictions, paradoxes and dilemmas in the text-internal or discourse internal [\u2026] structures\u201d  discourse itself or from contextual, social, historical and political knowledge\u201d (Reisigl and Wodak networked publics from an institutional perspective and pointed out self-reinforcing and path-dependent discursive chains that as a whole can plausibly illustrate a way in which media can discursively motivate discriminatory/hateful policy-making and establish new exclusionary \u2018realities\u2019 in societies (see Krzy\u017canowski validity claims that Habermas\u2019s (The Theory of Communicative Action could motivate future studies along the lines described here (see Forchtner Moreover, this first angle of critique can further bermas\u2019s , 99 deveprospective/retrospective critique \u201cseeks to become practical and to change and transform things\u201d (Reisigl and Wodak Finally,"}
{"text": "The changes are as follows:In the original publication , the aut\u00ae\u201d. There are four occurrences.In the abstract, change \u201cLAE\u201d into \u201cLAE\u00ae\u201d. There are two occurrences.In the last paragraph of the introduction, change \u201cLAE\u201d into \u201cLAE\u00ae\u201d. There is one occurrence.In Section 2.1, change \u201cLAE\u201d into \u201cLAE\u00ae\u201d. There are three occurrences.In Section 2.5, change \u201cLAE\u201d into \u201cLAE\u00ae\u201d. There are three occurrences.In Section 3.2 (including Table 1), change \u201cLAE\u201d into \u201cLAE\u00ae\u201d. There are nine occurrences.In the Section 3.3.1 , change \u201cLAE\u201d into \u201cLAE\u00ae\u201d. There are five occurrences.In Section 3.3.2 , change \u201cLAE\u201d into \u201cLAE\u00ae\u201d. There are fifteen occurrences.In Section 3.3.3 (including Table 2), change \u201cLAE\u201d into \u201cLAE\u00ae\u201d. There are eight occurrences.In Section 3.3.4 (including Table 3), change \u201cLAE\u201d into \u201cLAE\u00ae\u201d. There are eight occurrences.In Section 3.3.5 (including Table 4), change \u201cLAE\u201d into \u201cLAE\u00ae\u201d. There are ten occurrences.In Section 3.3.6 (including Table 5), change \u201cLAE\u201d into \u201cLAE\u00ae\u201d. There are twelve occurrences.In Section 3.3.7 (including Table 6), change \u201cLAE\u201d into \u201cLAE\u00ae\u201d. There are five occurrences.In the conclusion, change \u201cLAE\u201d into \u201cLAEThe authors apologize for any inconvenience caused and state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated."}
{"text": "Despite extensive biochemical analysis, aspects of the mechanism of activation remain controversial, and competing theoretical models have been proposed for the binding of Ca2+ and CaM to MLCK. The models are analytically solvable for an equilibrium steady state and give rise to distinct predictions that hold regardless of the numerical values assigned to parameters. These predictions form the basis of a recently proposed, multi-part experimental strategy for model discrimination. Here we implement this strategy by measuring CaM-MLCK binding using an in\u00a0vitro FRET system. Interpretation of binding data in light of the mathematical models suggests a partially ordered mechanism for binding CaM to MLCK. Complementary data collected using orthogonal approaches that assess CaM-MLCK binding further support this conclusion.Activation of myosin light chain kinase (MLCK) by calcium ions  activation of MLCK is key to many cellular functions\u2022An experimental strategy can distinguish between three models of CaM-MLCK binding\u2022Experimental data support a partially ordered CaM-MLCK binding mechanism in silico biologyBiochemistry; Biochemical mechanism;  CaM-MLCK binding takes place as follows: Ca2+ enters the cytosol and binds to the four Ca2+ binding sites of CaM, leading to a dramatic change in CaM protein conformation.,Calmodulin (CaM) is a calcium-binding protein found ubiquitously in the cytosol of eukaryotic cells. CaM contains two globular domains joined by a flexible linker. The N-terminus and C-terminus domains each have a pair of EF-hand motifs and are each capable of binding two calcium ions , Model 2 to a partially ordered mechanism (MLCK can bind to CaM after Ca2+ is bound at the C-terminus), and Model 3 to a fully ordered mechanism (MLCK can bind to CaM only after Ca2+ is bound at both the C-terminus and N-terminus).Multiple formal models have been proposed for the activation of MLCK by Caet\u00a0al. showed that this class of models is analytically solvable for a CaM/MLCK system at both thermodynamic equilibrium and steady state and that each model predicts distinct steady-state behavior in certain Ca2+ concentration regimes.In recent work, Dexter 2+-dependent activation of MCLK both in\u00a0vivo and in\u00a0vitro.,,,,,,2+] for wild-type CaM (CaMWT) and three different CaM mutants: CaM with its N-terminus EF-hands mutated , CaM with its C-terminus EF-hands mutated  and CaM with all of its EF-hands mutated . Each of the site-directed Asp to Ala mutations we performed has been demonstrated to prevent Ca2+ binding, while having only a slight impact on the structure of each CaM EF hand.Our biochemical approach to testing the predictions of the models centers on the use of an established fluorescence resonance energy transfer (FRET) reporter that we term FR. FR has been employed to characterize CaM- and Ca2+] is determined as Model 1, but not Models 2 and 3, predicts binding under these conditions. In the second step, the binding of FR to a CaM with impaired N-terminus Ca2+  is measured at high [Ca2+]. Models 1 and 2, but not Model 3, predict binding between FR and this mutant CaM at high [Ca2+]. Finally, we use a CaM mutant with impaired C-terminus Ca2+ binding  to evaluate additional predictions of Model 2, including the absence of binding of FR to this mutant in high [Ca2+]. The predictions about binding are based on algebraic calculations with all parameters treated symbolically; as such, they do not depend on fitting the models to experimental data.We use a multi-step strategy to discriminate between these models. We first compare Model 1 to Models 2 and 3, and we then compare Model 3 to Models 1 and 2. During the first step, CaM-FR binding at zero  for each of the models. Plots of F are shown in WT, CaM21A,57A CaM94A,130A and CaM21A,57A,94A,130A , and [FR] was 22.9\u00a0nM (1.3\u00a0ng/\u03bcL). After collecting Ex 430/Em 480 and Ex 430/Em 535 data from different CaM-FR pairs . For the second test, we compared the FRET ratio of CaM21A,57A -FR pair in high [Ca2+]. We found that the FRET ratio in the maximum concentration used (39\u00a0\u03bcM [Ca2+]) is significantly higher than baseline (p\u00a0= 2.83\u221710\u22126 by a one-tailed Mann-Whitney U test), as is the area under the complete binding curve (p\u00a0= 2.83\u221710\u22126 by a one-tailed Mann-Whitney U test). These observations are sufficient to falsify Model 3, which predicts zero binding of MLCK to any CaM mutant with impaired N-terminus Ca2+ binding. As such, only the predictions of Model 2 are consistent with the full set of experimental results. As an additional test of Model 2, we examined the binding of C-terminus mutant CaM94A,130A in high [Ca2+]; consistent with the predictions of the model, we found no significant difference over baseline for the FRET ratio in 39\u00a0\u03bcM [Ca2+] or for the area under the complete binding curve .For the first model discrimination test, we compared the FRET ratio of the CaMWT and CaM21A,57A (p\u00a0= 1.16\u221710\u22126 by a one-tailed Mann-Whitney U test). This difference is also consistent with the predictions of Model 2 , so that the model predicts equal maximum binding if K2\u00a0= K4. In the reference parameter set, K2\u00a0= 1,000 and K4\u00a0= 16.7, corresponding to predicted binding of 99.9% for CaMWT and 92.0% for CaM21A,57A in 39\u00a0\u03bcM [Ca2+]. As such, the experimentally observed difference is also predicted by our mathematical analysis, assuming that the literature parameter values are correct within an order of magnitude.A striking feature of the experimental binding curves is the significant difference in maximum binding between CaM Model 2 A. AssumiThe validity of our experimental approach for model discrimination depends on the ability of the CaM-FR interaction to accurately mirror CaM-MLCK binding with sufficient sensitivity, as well as other considerations related to the experimental perturbations. To address these potential limitations, we performed a series of control experiments and tested the robustness of our experimental design and data.2+-bound CaM, we collected 460\u00a0nm-700\u00a0mm emission spectra with Ex 430 of the FR in 0 and 39\u00a0\u03bcM [Ca2+]. When FR was assayed alone, FR showed no changes in FRET ratio as a function of [Ca2+] and produced fluorescence peaks at Em 480 and Em 535 , the fraction of CaMWT bound to FR increased proportionally with free [Ca2+]. At 39\u00a0\u03bcM [Ca2+], the majority of the CaMWT input was bound to the FR  assay as an orthogonal method to measure binding between FR and CaM. Binding detection in the BLI assay is independent of FRET measurement, which enabled us to decouple binding and FRET interference. The BLI assay showed robust binding between FR and CaMWT at high [Ca2+], an intermediate degree of binding between FR and CaM21A,57A at high [Ca2+], and no detectable binding between FR and CaM94A,130A at high [Ca2+] . When we repeated our FRET-based CaM-FR binding assay using these ratios (22.9\u00a0nM FR and 11.45\u00a0nM or 13.75\u00a0nM CaM), we observed a significant increase in F480/F535 (U test), with a calculated 8.7% difference in FRET ratios between the two conditions. We therefore conclude that, under our experimental conditions, the assay is sensitive enough to detect at least a 9% increase in the fraction of CaM-bound FR.We next assessed the intrinsic sensitivity of the FRET assay. To test if our assay can detect small changes in binding, which is necessary for many of the comparisons involved in our model discrimination analysis, we measured FRET at 39\u00a0\u03bcM . As is clear from the structure of 2+] increases with the ratio of total CaM to total MLCK. To confirm our falsification of Model 1, we therefore repeated the binding experiment in zero [Ca2+] with a much higher [CaM] (35.65\u00a0\u03bcM), for which 73.5% binding is predicted with the reference parameter values and 21.7% binding is predicted with K9\u00a0= 0.0078\u00a0\u03bcM\u22121 . As in the main experiment, the FRET ratio did not increase above baseline in zero [Ca2+] (U test).The predictions of zero MLCK binding in a particular [Cao [Ca2+] , providi21A,57A depends on two parameters, K4 and K6 , before and after the addition of the buffer in which our CaM proteins were stored. The largest volume of protein added to any of our FRET-based binding experiments was 1.3\u00a0\u03bcL per assay, so we tested this volume. We did not observe a significant change in indicator dye fluorescence intensity after CaM storage buffer addition, which suggests that any changes to free [Ca2+] that are experimentally introduced are smaller than can be detected reliably with Fluo-4, a dye that has a very high Ca2+ affinity (17\u00a0nM) (U test).Finally, we confirmed that the addition of the purified proteins to the prepared Ca (17\u00a0nM)  . Model 2 makes six correct predictions and no incorrect predictions  and the fully ordered binding mechanism (Model 3). Our data instead support Model 2, which assumes partially ordered binding between CaM and MLCK . As preddictions . Model 1et\u00a0al. analyzed a class of previously developed theoretical models of CaM-MLCK binding and proposed a multi-part strategy for distinguishing between them.In recent work, Dexter 2+], which used both FR and MLCK-FLAG when possible, rule out both Model 1 and Model 3. Our data are consistent with Model 2, which makes six correct predictions. Taken together, these results strongly support a partially ordered mechanism in which MLCK can bind to CaM after Ca2+ is bound at the C-terminus (Model 2).It is important to remember that model discrimination strategies of this kind work by process of elimination. Models can be ruled out when their predictions contradict experimental data, but the failure of some models does not guarantee the correctness of others. We present here evidence sufficient to falsify all but one of the models of CaM-MLCK binding in the literature . Data coin\u00a0vivo and in\u00a0vitro contexts.,,,,,,To investigate MCLK-CaM binding, we centered our experimental design on an FR system that uses the CaM-binding domain of smooth muscle MLCK. Prior studies have demonstrated that this reporter accurately reflects CaM-MLCK binding in a variety of 2+] tested. Determining the proportionality constant, however, is not necessary for our model discrimination strategy, which relies primarily on FRET ratios measured at 0\u00a0\u03bcM [Ca2+] and 39\u00a0\u03bcM [Ca2+]. We also complement this FRET-based binding assay data with on-bead binding assays showing that in 39\u00a0\u03bcM [Ca2+], the majority of CaMWT binds to either FR or full-length MLCK protein (MLCK-FLAG) expressed using human smooth muscle MLCK gene MYLK1, while a negligible amount of CaM binds to FR or MLCK-FLAG in 0\u00a0\u03bcM [Ca2+]. We therefore conclude that FRET ratio data represent CaMWT-MLCK binding in 0\u00a0\u03bcM [Ca2+] and 39\u00a0\u03bcM [Ca2+].In HEK 293T lysates, changing myosin phosphorylation has been shown to correspond to changes in the FRET ratio produced by FR. Changes to the FRET ratio can therefore be used as a proxy for MLCK activation.MYLK1, MYLK2, MYLK3, and MYLK4.MYLK1 has alternative initiation sites that enable the expression of at least four protein products including nonmuscle (long isoform) MLCK and smooth muscle (short isoform) MLCK. MYLK2 encodes an MLCK isoform expressed solely in skeletal muscle, MYLK3 encodes a cardiac-specific MLCK (MLCK3), and the gene product(s) of MYLK4 remain largely uncharacterized.,2+.It should be noted that mammalian myosin light chain kinases are a group of serine/threonine kinases encoded by at least four genes: in\u00a0vivo.2+-CaM-MLCK interaction network may therefore prove relevant for future drug discovery efforts.In addition to providing a blueprint for future model discrimination efforts that integrate mathematical and biochemical approaches, our work may also prove useful in a translational or pharmaceutical context, as MLCK and its activation by CaM have been linked to the pathogenesis of human disease. For example, increased organization of the sarcomere, the contractile unit of the striated muscle, is observed during the onset of cardiomyocyte hypertrophy. CaM-activated MLCK has been shown to mediate sarcomere organization induced by a hypertrophic agonist in cultured cardiomyocytes and 2+ binding; binding data should not be interpreted as evidence of MCLK activation. In addition, MLCK is only one of the many binding partners of CaM; future work is needed to determine the binding mechanisms of other proteins regulated by CaM-Ca2+.Although our data consistently support partially ordered binding between MLCK and CaM, there are meaningful limitations to our study. Our experimental strategy does not investigate MLCK activation in response to CaM-Casancak@uw.edu).Information and requests for resources and reagents should be directed to and will be fulfilled by the lead contact, Dr. Yasemin Sancak  \u2022DMEM; Thermo Fisher Scientific, cat. no. 11-965-118\u2022Glutamax; Fisher Scientific, cat. no. 35-050-061\u2022PBS; Thermo Fisher Scientific, cat. no. 20012050\u2022Penicillin/streptomycin solution; VWR cat. no. 45000-652\u2022Fetal bovine serum; Life Technologies, cat. no. 26140087\u2022Genlantis MycoScope PCR Detection Kit; VWR cat. no. 10497-508\u2022SDS; Sigma-Aldrich Cat. no. L4509-1\u00a0KG\u2022BME/2-mercaptoethanol; Sigma-Aldrich Cat. no. M3148-25\u00a0ML\u2022Glycerol; Sigma-Aldrich Cat. no. G5516-1L\u2022Tris\u2013HCl: Sigma-Aldrich cat. no. T5941\u2022Bromophenol Blue; VWR Cat. no. 97061-690\u202210X Tris/Glycine Buffer; Boston BioProducts Cat. no. BP-150-4L\u2022Tris-Glycine 12% Gel, 10-well or 15-well; Bio-Rad Cat. No. 4561043 or 4561046\u2022Methanol; Sigma-Aldrich cat. no. 32213-2.5L\u2022Acetic Acid; Sigma-Aldrich cat. no. A6283-500\u00a0MLGels were stained with SYPRO Ruby Protein Gel Stain (#S12000) following manufacturer\u2019s instructions and imaged with iBrightCL 1000 on fluorescent protein gel setting.\u2022Transfection reagent; X-treme(GENE) 9, Sigma-Aldrich, cat. no. 6365779001Day 4: Cells were harvested, and FR or MLCK-FLAG was either purified  or used for on-bead binding assays.Day 1: Either 5 million (purification) or 2 million (on-bead binding assays) HEK 293T cells were plated on 15\u00a0cm or 10\u00a0cm plates, respectively. Day 2: 5\u00a0\u03bcg FR plasmid (purification), 1.5\u00a0\u03bcg FR plasmid (on-bead binding assays with FR), or 3\u00a0\u03bcg MLCK-FLAG plasmid (on-bead binding assays with MLCK-FLAG) was transfected using transfection reagent. \u202250\u00a0mM HEPES-KOH \u2022150\u00a0mM NaCl, Sigma-Aldrich, # 746398-5\u00a0KG\u20225\u00a0mM EDTA: Sigma-Aldrich, # 607-429-00-8\u20221% Triton X-100: Sigma, #X100-1L\u2022FLAG Peptide Elution Buffer: 50\u00a0mM HEPES, 500\u00a0mM NaCl, pH 7.4\u2022Protease Inhibitor Tablets; Complete, Mini, EDTA-free Protease Inhibitor Cocktail, Sigma-Aldrich cat. no. 5892953001\u2022Anti-FLAG affinity gel; Sigma-Aldrich, cat. no. A2220-5\u00a0ML\u20223X FLAG peptide; Sigma-Aldrich cat. no. F4799-4\u00a0MG\u2022Chromatography spin column; Bio-Rad cat. no. 7326204Cells were lysed using lysis buffer supplemented with proteases inhibitors. Lysates were triturated in tubes and then centrifuged at 17,000\u00a0g for 10\u00a0min. Cell supernatant was divided into three tubes with 200\u00a0\u03bcL of anti-FLAG affinity gel slurry (50:50 bead/lysis buffer). These tubes were rocked at 4\u00a0\u00b0C for 1\u00a0h, and beads were washed three times with lysis buffer. A 22.5-gauge syringe was used to aspirate all remaining liquid from the tubes. FLAG-tagged protein was eluted with 90\u00a0\u03bcL of elution buffer, and 10\u00a0\u03bcL of 3XFLAG peptide prepared at 5\u00a0mg/mL for 30\u00a0min at 30\u00a0\u00b0C. The gel/elution buffer slurry from all three tubes was then pipetted into one spin column and spun at max speed for 5\u00a0min.\u2022Low-salt LB broth: Millipore Sigma, #L3397-1\u00a0KG\u2022Isopropyl beta-D-1-thiogalactopyranoside (IPTG): Sigma-Aldrich, #I5502-5G\u2022HiTrap Q HP anion exchange chromatography column: Cytiva, # 17115301\u2022Centrifugal protein concentrator, 10K cutoff: Amicon, # UFC8010\u202220\u00a0mM Tris-HCl, Sigma-Aldrich cat. no. T5941\u2022100\u00a0mM NaCl, Sigma-Aldrich, cat. no. 746398-5\u00a0KG\u202220\u00a0mM imidazole, Sigma-Aldrich, cat. no. I5513\u20220.4\u00a0mM EGTA, Sigma-Aldrich cat. no. 324626-25\u00a0GM\u20220.5\u00a0mM TCEP, Sigma-Aldrich cat. no. C4706-2G\u202220\u00a0mM Tris-HCl, Sigma-Aldrich cat. no. T5941\u2022100\u00a0mM NaCl, Sigma-Aldrich, cat. no. 746398-5\u00a0KG\u2022200\u00a0mM imidazole, Sigma-Aldrich, cat. no. I5513\u20220.4\u00a0mM EGTA, Sigma-Aldrich cat. no. 324626-25\u00a0GM\u20220.5\u00a0mM TCEP, Sigma-Aldrich cat. no. C4706-2G\u2022100\u00a0mM NaCl, Sigma-Aldrich, cat. no. 746398-5\u00a0KG\u202220\u00a0mM Tris-HCl, Sigma-Aldrich cat. no. T5941\u20221M NaCl, Sigma-Aldrich, cat. no. 746398-5\u00a0KG\u202220\u00a0mM Tris-HCl, Sigma-Aldrich cat. no. T5941WT: 3.98\u00a0\u03bcg/\u03bcL; CaM21A,57A: 4.96\u00a0\u03bcg/\u03bcL; CaM94A,130A: 1.36\u00a0\u03bcg/\u03bcL; CaM21A,57A,94A,130A: 1.52\u00a0\u03bcg/\u03bcL. Purified protein was aliquoted, flash frozen in a liquid nitrogen dewar, and then stored at \u221280\u00a0\u00b0C until use.BL21 (DE3) bacteria transformed with a CaM expression vector were grown at 37\u00a0\u00b0C, 200 RPM until OD600 reached 0.8\u20131.0. Cultures were cooled to 18\u00b0C and supplemented with IPTG to a final concentration of 0.3\u00a0mM before incubation at 18\u00b0C, 200 RPM for 18\u00a0h. Bacteria were pelleted, frozen, and stored at \u221280\u00b0C until protein purification. Cells were lysed using sonication for 5\u00a0min at 40% power before being centrifuged at 17,400 RPM for 40\u00a0min to separate the soluble and insoluble portions of the lysate. The clarified lysate was loaded onto a Ni-NTA gravity column and then washed with 10 column volumes of lysis buffer before elution with 5 column volumes of elution buffer. The eluted protein was loaded manually into a 1\u00a0mL anion exchange chromatography column. HPLC was performed using buffers A and B. The HPLC fractions that contained the highest concentrations of protein were collected and concentrated to their final concentrations using a centrifugal filter with a 10\u00a0kDa molecular weight cutoff. Stock protein concentrations were as follows: CaM\u2022Calcium calibration buffer kit; Invitrogen cat. no. C3008MP\u2022Microplate reader; BioTek Synergy H1\u2022Black 96-well plates; Greiner Bio-One cat. no. 6550762+ buffer (14 conditions between 0\u00a0\u03bcM and 39\u00a0\u03bcM). 150\u00a0\u03bcL of the resulting mixture was pipetted into 1 well of a 96-well plate . CaM was used at a final concentration of 673\u00a0nM (13\u00a0ng/\u03bcL); FR was used at a final concentration of 22.9\u00a0nM (1.3\u00a0ng/\u03bcL). All protein preps (FR and CaM) were at sufficiently high concentrations that each protein addition added negligible\u00a0volume; stock protein concentrations were as follows: CaMWT: 3.98\u00a0\u03bcg/\u03bcL; CaM21A,57A: 4.96\u00a0\u03bcg/\u03bcL; CaM94A,130A: 1.36\u00a0\u03bcg/\u03bcL; CaM21A,57A,94A,130A: 1.52\u00a0\u03bcg/\u03bcL; FR: 0.52\u00a0\u03bcg/\u03bcL. Follow-up experiments with Ca2+ indicator dyes confirmed that the final [Ca2+] were unaffected by the addition of CaM or FR to the Ca2+ buffer  and 200\u00a0ng FR was thoroughly mixed with 150\u00a0\u03bcL of Cafer see \u201c\u201d below. \u2022Microcon concentrator column, Millipore cat. No. 424072+] using 35.65\u00a0\u03bcM [CaMWT]. CaMWT was concentrated to a stock concentration of 66.6\u00a0\u03bcg/\u03bcL using a concentrator column so that a comparable volume of protein stock (relative to previous assays) could be used. The \u201cWT].To test the robustness of the falsification of Model 1 to uncertainty in parameter values, we repeated the binding assay in zero [Ca2+] buffer along with either 11.45 or 13.74\u00a0nM [CaMWT], which resulted in a CaMWT:FR molar ratio of either 1:2 or 1.2:2. 150\u00a0\u03bcL of the resulting mixture was then read in a plate reader using the program employed by the \u201cTo determine a lower sensitivity limit for the CaM-FR binding assay, 22.9\u00a0nM FR was added to 150\u00a0\u03bcL of high  as per \u201cCaM and FR were added to either 0 or 39\u00a0\u03bcM [Ca2+]  and two controls . For assays utilizing MLCK-FLAG, assay data were collected under 4 different conditions: MLCK-FLAG bound to anti-FLAG M2 affinity gel beads and CaM under 2 different [Ca2+] (0\u00a0\u03bcM and 39\u00a0\u03bcM) and two controls .For assays utilizing FR, assay data were collected under 8 different conditions: FR bound to anti-FLAG M2 affinity gel beads and CaM under 6 different [Ca2+], while the tubes for the 6 different Ca2+ conditions were each washed with the appropriate Ca2+ buffer. 50\u00a0\u03bcL of the appropriate Ca2+ buffer was then added to each experimental tube, while 50\u00a0\u03bcL of 39\u00a0\u03bcM Ca2+ was added to each control tube. Finally, for FR assays, 0.5\u00a0\u03bcg of CaM was added to all tubes except the reporter-only condition; final CaM concentration was approximately 505\u00a0nM (10\u00a0ng/\u03bcL). For MLCK-FLAG assays, 0.25\u00a0\u03bcg of CaM was added to all tubes except the MLCK-FLAG-only condition; final CaM concentration was approximately 24\u00a0nM (5\u00a0ng/\u03bcL) (NOTE: CaM concentration for MLCK-FLAG assays was decreased to compensate for decreased MLCK-FLAG expression relative to FR expression). CaM stock was at a sufficiently high concentration to ensure the addition of CaM did not meaningfully alter Ca2+ buffer concentration  for 10\u00a0min using a 4\u00b0C centrifuge. 160\u00a0\u03bcL bead:glycerol M2 FLAG slurry was prepared for FLAG-tagged protein binding by washing three times with 1\u00a0mL 1% Triton buffer before being divided evenly between 8 tubes. The cleared lysate from the transiently transfected cells was divided evenly between 7 of the tubes, while cleared lysate from control WT HEK 293T was added to the eighth tube (control lysate). The beads were incubated with the lysate on a rocker at 4\u00b0C for 45\u00a0min. After incubation, the two control tubes were washed twice with 200\u00a0\u03bcL 39\u00a0\u03bcM [Caion see \u201c below\u201d  added to a final concentration of 1\u00a0\u03bcM. For one set of buffers, the buffer in which the CaM proteins was stored was added (\u201cdummy buffer\u201d). The amount of \u201cdummy buffer\u201d added was equal in volume to the maximum volume of CaM added per reaction . 150\u00a0\u03bcL of each of these prepared buffers was added to wells of a black 96-well plate and data were collected using the GFP filter of a plate reader.This assay was performed to confirm that the addition of CaM protein to the Ca\u2022Calcium calibration buffer kit, Invitrogen cat. no. C3008MP\u2022Chloroacetamide, Sigma cat. no. C0267\u2022TCEP, Sigma-Aldrich cat. no. C4706-2G\u2022Rappsilbers Stage tipping Paper (23), C-18 material: CDS Empore C18 Extraction Disks, Fisher cat. no. 13-110-016\u2022Formic acid for HPLC LiChropur, Sigma cat. no. 5438040100\u2022Trifluoroacetic acid (TFA), HPLC Grade, 99.5+%, Alfa Aesar, cat. no. AA446305Y\u2022Acetonitrile (ACN), Sigma-Aldrich cat. no. 271004-100\u00a0ML\u2022Mass spectrometer: LC-MS: Orbitrap Fusion Lumos Tribrid (Thermo Fisher Scientific) and EASY-nLC 1200 System (Thermo Fisher Scientific)\u2022Triethylammonium bicarbonate buffer 1.0\u00a0M, pH 8.5\u00a0\u00b1 0.1, Sigma cat. no. T7408\u2022Ethanol for HPLC, Sigma cat. no. 459828\u2022Promega trypsin, cat no. V5111\u2022Methanol for HPLC, Sigma cat. No. 494291\u20225% ACN\u20220.1% TFA\u2022MQ water\u20220.1% TFA\u202280% ACN\u2022MQ water\u20220.1% TFA, diluted in MQ water2+ calibration buffer to a final volume of 20\u00a0\u03bcL. A blank control consisting of CaM storage buffer and Ca2+ calibration buffer was also prepared. All samples were reduced using 1\u00a0mM final concentration of TCEP and alkylated using 2\u00a0mM final concentration of chloroacetamide (CAM) at 37\u00b0C for 20\u00a0min on a shaker. They were then quenched using 1\u00a0mM final concentration of TCEP at 37\u00b0C for 20\u00a0min on a shaker. Next, they were mixed with 5X gel loading buffer, boiled for 5\u00a0min, and loaded into a 10% gel  and one was comprised of the gel between \u223c10 and 17\u00a0kDa, the area under the major band. The rest of the in-gel digestion was performed as previously described in.. (2007).StageTip extraction of peptides was performed as described previously in Rappsilber et alLC-MS analyses were performed as described previously with the following minor modifications.Data.raw files were analyzed by MaxQuant/AndromedaThe intensity of peptides annotated as CaM, keratin, or bacterial protein was summed up and their relative intensities were calculated by dividing them to the total peptide intensity in the samples.\u2022Octet Red 96 \u2022BSA, Sigma-Aldrich cat. no. A3294-100G\u2022Calcium chloride, Sigma-Aldrich cat. no. 746495-1\u00a0KG\u2022Tris-HCl, Sigma-Aldrich cat. no. T5941\u2022NaCl Sigma-Aldrich, # 746398-5\u00a0KG2+ was analyzed using the Octet Red 96  following the manufacturer\u2019s procedures in duplicates. The reaction was carried out in black 96 well plates maintained at 30\u00a0\u00b0C. The reaction volume was 200\u00a0\u03bcL in each well. The Octet buffer contained 20\u00a0mM Tris-HCl, 200\u00a0mM NaCl, and 0.1% BSA, pH 8.0. The Association buffer contained 20\u00a0mM Tris-HCl, 200\u00a0mM NaCl, 1.5\u00a0mM Ca2+ and 0.1% BSA, pH 8.0. The Dissociation buffer contained 20\u00a0mM Tris-HCl, 200\u00a0mM NaCl, and 0.1% BSA, pH 8.0. The concentration of ligand\u00a0\u2013 His-CaMWT, His-CaM21A,57A or His-CaM94A,130A\u00a0\u2013 in the Octet buffer was 2\u00a0\u03bcM. The concentration of His-GST as the quench in Octet Buffer was 0.68\u00a0\u03bcM. The concentration of FR or MLCK-FLAG as the analyte in the Association buffer was 1.5\u00a0\u03bcM. Ni-NTA optical probes were loaded with His-CaMWT, His-CaM21A,57A or His-CaM94A,130A as ligands for 110\u00a0s and quenched with His-GST for 80\u00a0s prior to binding analysis. While not loaded with ligand, the control probes were quenched with His-GST. In each experiment, \u201cControl\u201d refers to the analyte (either FR or MLCK-FLAG) binding to the ligand-free probe. This control is performed to demonstrate that the association seen in the ligand bound probes is not due to non-specific binding but is, in fact, specific. The binding of the analyte (either FR or MLCK-FLAG) to the optical probes was measured simultaneously using instrumental defaults for 236\u00a0s. The dissociation was measured for 287\u00a0s. There was no binding of FR to the unloaded probes; however, slight binding of MLCK-FLAG to the unloaded probes was observed. The data were analyzed by the Octet data analysis software. The association and dissociation curves for FR binding were globally fit with a 1:1 ligand model, and the curves for MLCK-FLAG binding were locally fit for 80s. The data were plotted using Prism 7.The binding of His-CaM to FR or FL-MLCK-FLAG protein in the presence of CaBolding: mutations.atgggcagcagccatcatcatcatcatcacagcagcggcctggtgccgcgcggcagccatagcgaaaacctctacttccaatcgatggctgaccagctgactgaggagcagattgcagagttcaaggaggccttctccctctttgacaaggatggagatggcactatcaccaccaaggagttggggacagtgatgagatccctgggacagaaccccactgaagcagagctgcaggatatgatcaatgaggtggatgcagatgggaacgggaccattgacttcccggagttcctgaccatgatggccagaaagatgaaggacacagacagtgaggaggagatccgagaggcgttccgtgtctttgacaaggatgggaatggctacatcagcgccgcagagctgcgtcacgtaatgacgaacctgggggagaagctgaccgatgaggaggtggatgagatgatcagggaggctgacatcgatggagatggccaggtcaattatgaagagtttgtacagatgatgactgcaaagtgaccaaggatggagatggcactatcaccaccaaggagttggggacagtgatgagatccctgggacagaaccccactgaagcagagctgcaggatatgatcaatgaggtggcagcagatgggaacgggaccattgacttcccggagttcctgaccatgatggccagaaagatgaaggacacagacagtgaggaggagatccgagaggcgttccgtgtctttgacaaggatgggaatggctacatcagcgccgcagagctgcgtcacgtaatgacgaacctgggggagaagctgaccgatgaggaggtggatgagatgatcagggaggctgacatcgatggagatggccaggtcaattatgaagagtttgtacagatgatgactgcaaagtgaatgggcagcagccatcatcatcatcatcacagcagcggcctggtgccgcgcggcagccatagcgaaaacctctacttccaatcgatggctgaccagctgactgaggagcagattgcagagttcaaggaggccttctccctctttgccaaggatgggaatggctacatcagcgccgcagagctgcgtcacgtaatgacgaacctgggggagaagctgaccgatgaggaggtggatgagatgatcagggaggctgccatcgatggagatggccaggtcaattatgaagagtttgtacagatgatgactgcaaagtga.atgggcagcagccatcatcatcatcatcacagcagcggcctggtgccgcgcggcagccatagcgaaaacctctacttccaatcgatggctgaccagctgactgaggagcagattgcagagttcaaggaggccttctccctctttgacaaggatggagatggcactatcaccaccaaggagttggggacagtgatgagatccctgggacagaaccccactgaagcagagctgcaggatatgatcaatgaggtggatgcagatgggaacgggaccattgacttcccggagttcctgaccatgatggccagaaagatgaaggacacagacagtgaggaggagatccgagaggcgttccgtgtctttgccaaggatggagatggcactatcaccaccaaggagttggggacagtgatgagatccctgggacagaaccccactgaagcagagctgcaggatatgatcaatgaggtggcagcagatgggaacgggaccattgacttcccggagttcctgaccatgatggccagaaagatgaaggacacagacagtgaggaggagatccgagaggcgttccgtgtctttgccaaggatgggaatggctacatcagcgccgcagagctgcgtcacgtaatgacgaacctgggggagaagctgaccgatgaggaggtggatgagatgatcagggaggctgccatcgatggagatggccaggtcaattatgaagagtttgtacagatgatgactgcaaagtga.atgggcagcagccatcatcatcatcatcacagcagcggcctggtgccgcgcggcagccatagcgaaaacctctacttccaatcgatggctgaccagctgactgaggagcagattgcagagttcaaggaggccttctccctctttg\u2022Bolding: EYFP\u2022Bold italics: Calmodulin binding domain\u2022Underlining: ECFP\u2022Bold underlined italics: Flexible linker\u2022Italics: FLAG tagggtaccgccgctcgtcagaaatggcagaaaaccggacatgcggtgcgtgcgattggccgtctggctgctaccggtAtggtgagcaagggcgaggagctgttcaccggggtggtgcccatcctggtcgagctggacggcgacgtaaacggccacaagttcagcgtgtccggcgagggcgagggcgatgccacctacggcaagctgaccctgaagttcatctgcaccaccggcaagctgcccgtgccctggcccaccctcgtgaccaccttcggctacggcctgatgtgcttcgcccgctaccccgaccacatgcgccagcacgacttcttcaagtccgccatgcccgaaggctacgtccaggagcgcaccatcttcttcaaggacgacggcaactacaagacccgcgccgaggtgaagttcgagggcgacaccctggtgaaccgcatcgagctgaagggcatcgacttcaaggaggacggcaacatcctggggcacaagctggagtacaactacaacagccacaacgtctatatcatggccgacaagcagaagaacggcatcaaggtgaacttcaagatccgccacaacatcgaggacggcagcgtgcagctcgccgaccactaccagcagaacacccccatcggcaacggccccgtgctgctgcccgacaaccactacctgagctaccagtccgccctgagcaaagaccccaacgagaagcgcgatcacatggtcctgctggagttcgtgaccgccgccgggatcactctcggcatggacgagctgtacaagagcaagggcgaggagctgttcaccggggtggtgcccatcctggtcgagctggacggcgacgtaaacggccacaggttcagcgtgtccggcgagggcgagggcgatgccacctacggcaagctgaccctgaagttcatctgcaccaccggcaagctgcccgtgccctggcccaccctcgtgaccaccctgacctggggcgtgcagtgcttcagccgctaccccgaccacatgaagcagcacgacttcttcaagtccgccatgcccgaaggctacgtccaggagcgtaccatcttcttcaaggacgacggcaactacaagacccgcgccgaggtgaagttcgagggcgacaccctggtgaaccgcatcgagctgaagggcatcgacttcaaggaggacggcaacatcctggggcacaagctggagtacaactacatcagccacaacgtctatatcaccgccgacaagcagaagaacggcatcaaggcccacttcaagatccgccacaacatcgaggacggcagcgtgcagctcgccgaccactaccagcagaacacccccatcggcgacggccccgtgctgctgcccgacaaccactacctgagcacccagtccgccctgagcaaagaccccaacgagaagcgcgatcacatggtcctgctggagttcgtgaccgccgccgggatcactctcggcatggacgagctgtacaagcccgggggtggatctggtggatctggtggatctatggattacaaggatgacgatgacaag.\u2022Bold italics: Calmodulin binding domain\u2022Bold underlined italics: Flexible linker\u2022Italics: FLAG tagagaaggaaatggcagaaaacgggcaatgctgtgagagccattggaagactgtcctctatggcaatgatctcagggctcagtggcaggaaatcctcaacagggtcaccaaccagcccgctcaatgcagaaaaactagaatctgaagaagatgtgtcccaagctttccttgaggctgttgctgaggaaaagcctcatgtaaaaccctatttctctaagaccattcgcgatttagaagttgtggagggaagtgctgctagatttgactgcaagattgaaggatacccagaccccgaggttgtctggttcaaagatgaccagtcaatcagggagtcccgccacttccagatagactacgatgaggacgggaactgctctttaattattagtgatgtttgcggggatgacgatgccaagtacacctgcaaggctgtcaacagtcttggagaagccacctgcacagcagagctcattgtggaaacgatggaggaaggtgaaggggaaggggaagaggaagaagaggtcgacatggactacaaggacgacgacgacaag.atgggggatgtgaagctggttgcctcgtcacacatttccaaaacctccctcagtgtggatccctcaagagttgactccatgcccctgacagaggcccctgctttcattttgccccctcggaacctctgcatcaaagaaggagccaccgccaagttcgaagggcgggtccggggttacccagagccccaggtgacatggcacagaaacgggcaacccatcaccagcgggggccgcttcctgctggattgcggcatccgggggactttcagccttgtgattcatgctgtccatgaggaggacaggggaaagtatacctgtgaagccaccaatggcagtggtgctcgccaggtgacagtggagttgacagtagaaggaagttttgcgaagcagcttggtcagcctgttgtttccaaaaccttaggggatagattttcagcttcagcagtggagacccgtcctagcatctggggggagtgcccaccaaagtttgctaccaagctgggccgagttgtggtcaaagaaggacagatgggacgattctcctgcaagatcactggccggccccaaccgcaggtcacctggctcaagggaaatgttccactgcagccgagtgcccgtgtgtctgtgtctgagaagaacggcatgcaggttctggaaatccatggagtcaaccaagatgacgtgggagtgtacacgtgcctggtggtgaacgggtcggggaaggcctcgatgtcagctgaactttccatccaaggtttggacagtgccaataggtcatttgtgagagaaacaaaagccaccaattcagatgtcaggaaagaggtgaccaatgtaatctcaaaggagtcgaagctggacagtctggaggctgcagccaaaagcaagaactgctccagcccccagagaggtggctccccaccctgggctgcaaacagccagcctcagcccccaagggagtccaagctggagtcatgcaaggactcgcccagaacggccccgcagactccggtccttcagaagacttccagctccatcaccctgcaggccgcaagagttcagccggaaccaagagcaccaggcctgggggtcctatcaccttctggagaagagaggaagaggccagctcctccccgtccagccaccttccccaccaggcagcctggcctggggagccaagatgttgtgagcaaggctgctaacaggagaatccccatggagggccagagggattcagcattccccaaatttgagagcaagccccaaagccaggaggtcaaggaaaatcaaactgtcaagttcagatgtgaagtttccgggattccaaagcctgaagtggcctggttcctggaaggcacccccgtgaggagacaggaaggcagcattgaggtttatgaagatgctggctcccattacctctgcctgctgaaagcccggaccagggacagtgggacatacagctgcactgcttccaacgcccaaggccaggtgtcctgtagctggaccctccaagtggaaaggcttgccgtgatggaggtggccccctccttctccagtgtcctgaaggactgcgccgttattgagggccaggattttgtgctgcagtgctccgtacgggggaccccagtgccccggatcacttggctgctgaatgggcagcccatccagtacgctcgctccacctgcgaggccggcgtggctgagctccacatccaggatgccctgccggaggaccatggcacctacacctgcctagctgagaatgccttggggcaggtgtcctgcagcgcctgggtcaccgtccatgaaaagaagagtagcaggaagagtgagtaccttctgcctgtggctcccagcaagcccactgcacccatcttcctgcagggcctctctgatctcaaagtcatggatggaagccaggtcactatgactgtccaagtgtcagggaatccaccccctgaagtcatctggctgcacaatgggaatgagatccaagagtcagaggacttccactttgaacagagaggaactcagcacagcctttgtatccaggaagtgttcccggaggacacgggcacgtacacctgcgaggcctggaacagcgctggagaggtccgcacccaggccgtgctcacggtacaagagcctcacgatggcacccagccctggttcatcagtaagcctcgctcagtgacagcctccctgggccagagtgtcctcatctcctgcgccatagctggtgacccctttcctaccgtgcactggctcagagatggcaaagccctctgcaaagacactggccacttcgaggtgcttcagaatgaggacgtgttcaccctggttctaaagaaggtgcagccctggcatgccggccagtatgagatcctgctcaagaaccgggttggcgaatgcagttgccaggtgtcactgatgctacagaacagctctgccagagcccttccacgggggagggagcctgccagctgcgaggacctctgtggtggaggagttggtgctgatggtggtggtagtgaccgctatgggtccctgaggcctggctggccagcaagagggcagggttggctagaggaggaagacggcgaggacgtgcgaggggtgctgaagaggcgcgtggagacgaggcagcacactgaggaggcgatccgccagcaggaggtggagcagctggacttccgagacctcctggggaagaaggtgagtacaaagaccctatcggaagacgacctgaaggagatcccagccgagcagatggatttccgtgccaacctgcagcggcaagtgaagccaaagactgtgtctgaggaagagaggaaggtgcacagcccccagcaggtcgattttcgctctgtcctggccaagaaggggacttccaagacccccgtgcctgagaaggtgccaccgccaaaacctgccaccccggattttcgctcagtgctgggtggcaagaagaaattaccagcagagaatggcagcagcagtgccgagaccctgaatgccaaggcagtggagagttccaagcccctgagcaatgcacagccttcagggcccttgaaacccgtgggcaacgccaagcctgctgagaccctgaagccaatgggcaacgccaagcctgccgagaccctgaagcccatgggcaatgccaagcctgatgagaacctgaaatccgctagcaaagaagaactcaagaaagacgttaagaatgatgtgaactgcaagagaggccatgcagggaccacagataatgaaaagagatcagagagccaggggacagccccagccttcaagcagaagctgcaagatgttcatgtggcagagggcaagaagctgctgctccagtgccaggtgtcttctgaccccccagccaccatcatctggacgctgaacggaaagaccctcaagaccaccaagttcatcatcctctcccaggaaggctcactctgctccgtctccatcgagaaggcactgcctgaggacagaggcttatacaagtgtgtagccaagaatgacgctggccaggcggagtgctcctgccaagtcaccgtggatgatgctccagccagtgagaacaccaaggccccagagatgaaatcccggaggcccaagagctctcttcctcccgtgctaggaactgagagtgatgcgactgtgaaaaagaaacctgcccccaagacacctccgaaggcagcaatgccccctcagatcatccagttccctgaggaccagaaggtacgcgcaggagagtcagtggagctgtttggcaaagtgacaggcactcagcccatcacctgtacctggatgaagttccgaaagcagatccaggaaagcgagcacatgaaggtggagaacagcgagaatggcagcaagctcaccatcctggccgcgcgccaggagcactgcggctgctacacactgctggtggagaacaagctgggcagcaggcaggcccaggtcaacctcactgtcgtggataagccagaccccccagctggcacaccttgtgcctctgacattcggagctcctcactgaccctgtcctggtatggctcctcatatgatgggggcagtgctgtacagtcctacagcatcgagatctgggactcagccaacaagacgtggaaggaactagccacatgccgcagcacctctttcaacgtccaggacctgctgcctgaccacgaatataagttccgtgtacgtgcaatcaacgtgtatggaaccagtgagccaagccaggagtctgaactcacaacggtaggagagaaacctgaagagccgaaggatgaagtggaggtgtcagatgatgatgagaaggagcccgaggttgattaccggacagtgacaatcaatactgaacaaaaagtatctgacttctacgacattgaggagagattaggatctgggaaatttggacaggtctttcgacttgtagaaaagaaaactcgaaaagtctgggcagggaagttcttcaaggcatattcagcaaaagagaaagagaatatccggcaggagattagcatcatgaactgcctccaccaccctaagctggtccagtgtgtggatgcctttgaagaaaaggccaacatcgtcatggtcctggagatcgtgtcaggaggggagctgtttgagcgcatcattgacgaggactttgagctgacggagcgtgagtgcatcaagtacatgcggcagatctcggagggagtggagtacatccacaagcagggcatcgtgcacctggacctcaagccggagaacatcatgtgtgtcaacaagacgggcaccaggatcaagctcatcgactttggtctggccaggaggctggagaatgcggggtctctgaaggtcctctttggcaccccagaatttgtggctcctgaagtgatcaactatgagcccatcggctacgccacagacatgtggagcatcggggtcatctgctacatcctagtcagtggcctttcccccttcatgggagacaacgataacgaaaccttggccaacgttacctcagccacctgggacttcgacgacgaggcattcgatgagatctccgacgatgccaaggatttcatcagcaatctgctgaagaaagatatgaaaaaccgcctggactgcacgcagtgccttcagcatccatggctaatgaaagataccaagaacatggaggccaagaaactctccaaggaccggatgaagaagtacatggcaWT, each data point is presented as the mean\u00a0\u00b1 SD of 15 independent experiments. For CaM21A,57A, each data point is presented as mean\u00a0\u00b1\u00a0SD of 16 independent experiments. For CaM94A,130A, each data point is presented as mean\u00a0\u00b1 SD of 8 independent experiments, and for CaM21A,57A,94A,130A each data point is presented as mean\u00a0\u00b1 SD of 3 independent experiments. For reporter-only data, each data point is presented as a mean\u00a0\u00b1 SD of 13 independent experiments. Curve fits were computed using the MATLAB (version R2020a) package doseResponse. Algebraic calculations involving the mathematical models were done in Wolfram Mathematica (version 12.1).For CaMFraction of calmodulin bound to FR was quantified using ImageJ; GraphPad Prism 7 was used to plot mean and SD.Following data collection, the data were analyzed using the Octet data analysis software  and plotted using GraphPad Prism 7.U test was performed to determine the significance  of the ratio increase.Following the determination of the mean and SD using GraphPad Prism, a one-tailed Mann-Whitney U test was performed to determine the significance  of the ratio increase.Following the determination of the mean and SD using GraphPad Prism, a one-tailed Mann-Whitney U test was performed to determine the significance  of the fluorescence change.Following the determination of the mean and SD using GraphPad Prism, a two-tailed Mann-Whitney Statistical calculations were performed using MATLAB. AUC was computed using the trapz built-in MATLAB function."}
{"text": "Although the absence of the gut microbiota had marginal effects, feeding mice with a vitamin B1-deficient diet compromised the number of intestinal IL-4 + ILC2s. The decrease in the number of IL-4+ ILC2s caused by the vitamin B1 deficiency was accompanied by a reduction in IL-25\u2013producing tuft cells. Our findings reveal that dietary vitamin B1 plays a critical role in maintaining interaction between tuft cells and IL-4+ ILC2s, a previously uncharacterized immune cell population that may contribute to maintaining intestinal homeostasis.Group 2 innate lymphoid cells (ILC2s) expressing IL-5 and IL-13 are localized at various mucosal tissues and play critical roles in the induction of type 2 inflammation, response to helminth infection, and tissue repair. Here, we reveal a unique ILC2 subset in the mouse intestine that constitutively expresses IL-4 together with GATA3, ST2, KLRG1, IL-17RB, and IL-5. In this subset, IL-4 expression is regulated by mechanisms similar to but distinct from those observed in T cells and is partly affected by IL-25 signaling. Although the absence of the microbiota had marginal effects, feeding mice with a vitamin B1-deficient diet compromised the number of intestinal IL-4  H2) cells (Group 2 innate lymphoid cells (ILC2s) are predominantly localized at mucosal tissues, including skin, lungs, and the gastrointestinal tract, and play critical roles as critical sentinels against infection and tissue damage. In particular, ILC2s serve as the major effector cells against helminth infection . On the 2) cells . Epithel2) cells . ILC2-de2) cells . Importa2) cells .Recent studies have shown that ILC2s can be subdivided into at least two subtypes: natural ILC2s (nILC2s) and inflammatory ILC2 iILC2s; . nILC2s + tuft cells constitute a very small fraction of epithelial cells , and DCLK1Vitamin B1  is a water-soluble vitamin that serves as a co-factor in the metabolism of carbohydrate and amino acids to produce energy and maintain physiological homeostasis. Because mammals do not possess a biosynthetic pathway for VB1, they must obtain this vitamin from commensal microbes or through dietary supplementation . Without+ ILC2s) expressed IL-17RB and were at least in part dependent on IL-25 signaling for their development. Moreover, IL-4+ ILC2s and IL-25\u2013expressing tuft cells were significantly affected by the deficiency of dietary VB1. Therefore, dietary VB1 is required for the maintenance of intestinal tuft cells and the constitutive induction of IL-4+ ILC2s. Our study revealed a critical role of diet\u2013epithelial\u2013immune interactions in inducing a previously undescribed immune cell subset.In this study, we found that a substantial proportion of ILC2s in the intestine, but not other organs, in mice constitutively expressed IL-4. Much like iILC2s, IL-4-expressing ILC2s , expression of IL-4 and -5 was mainly observed in Thy1.2+Lin\u2212 innate cells  in the small intestinal (SI) and colonic LP and compared them with those in inguinal lymph node (iLN) and mesenteric lymph node (mLN) from specific pathogen-free  C57BL/6 (B6) mice using RT-qPCR. As expected, Tic cells . We thuste cells . IL-4\u2013expression , indicat+ ILC2s (GATA3+ Thy1.2+Lin\u2212 cells) were most abundant in the colon and SI, followed by lungs, and were rare in the skin and lymphoid organs, such as spleen  and pulmonary ILC2s (Thy1+ CD3\u2212 ST2+ cells) through RNA-sequencing (RNAseq). Colonic and pulmonary ILC2s exhibited distinct transcriptional profiles , Arginase 1 (Arg1), Il10ra, and Icosl . In both Il33\u2212/\u2212 and Tslpr\u2212/\u2212 mice (BALB/c background), colonic IL-4+ ILC2s were present at similar levels to control BALB/c mice (+ ILC2s.As colonic IL-4nchanged . NotablyB/c mice . These r+ inflammatory ILC2s (iILC2s) in the lungs  region of Il4, which serves as a BATF and RBP-J binding site and plays the role of a TFH cell-specific enhancer  mice. The frequencies of colonic and SI IL-4+ ILC2s were not significantly different between GF and SPF mice (+ ILC2 was even increased rather than decreased in GF mice (+ CD3\u2212 KLRG1+ cells) isolated from GF and SPF B6 mice revealed similar gene expression profiles  were fed a defined diet devoid of either cellulose, soy oil, corn starch, minerals, or vitamins for 4 wk. Feeding with a diet deficient in vitamins led to a significant reduction in colonic IL-4+ ILC2s, whereas deprivation of other dietary components showed marginal effects  had little impact on colonic IL-4+ ILC2 levels , and transient receptor potential melastatin 5 (Trpm5), were significantly decreased in epithelial cells obtained from mice on a VB1-deficient diet . Further, Il25 was not significantly different between organoids cultured with and without VB1 (ThTr1) and thiamin transporter-2 (ThTr2) in DCLK1+ and DCLK1\u2212 epithelial cells isolated from SPF B6 mice and found that there was no significant difference in their expression between DCLK1+ tuft cells and other epithelial cells (+ ILC2s) on tuft cell development. To this end, organoids were cultured in the presence of IL-4. The development of Pou2f3+ cells was substantially enhanced, and the expression of the tuft cell signature genes and Il25 was significantly increased in organoids following IL-4 treatment regardless of the presence of VB1  mice were kindly provided by Dr. Steven F. Ziegler (HS2\u2212/\u2212 and CNS2\u2212/\u2212 mice were generated and characterized in the Masato Kubo lab previously (Batf\u2212/\u2212 mice (Foxp3hCD2) mice  was dissolved in PBS and introduced into mice by intraperitoneal (i.p.) injection at a dose of 200 mg/kg/d or through drinking water at a concentration of 350 \u03bcg/ml. For treatment with 6-AN, mice were i.p. injected with 6-AN  or an equivalent volume of PBS on days 0 and 4 and euthanized on day 6. For treatment with lactic acid, mice were treated with drinking water supplemented with 0, 5, or 50 mM lactic acid  for 5 wk before sacrifice. To investigate the effect of IL-25 or IL-33 on the development of colonic and lung ILC2s, mice were i.p. injected with recombinant IL-25 (rIL-25) or IL-33  in PBS daily for 3 d at a dose of 200 ng/mice/d  mice . Batf\u2212/\u2212g/mice/d  and euthg for 30 min at 25\u00b0C. The fraction containing lymphocytes was collected from the interface of the two layers and washed with RPMI 1640 containing 10% FBS. For cytokine detection, the cells were stimulated with 50 ng/ml PMA and 750 ng/ml ionomycin (both from Sigma-Aldrich) in the presence of GolgiStop (BD Biosciences) at 37\u00b0C for 3.5 h. After labeling with Ghost Dye 780, the cells were permeabilized and stained with antibodies against GATA3 , Thy1.2 , leukocytes lineage markers , IL-4 , IL-5 , KLRG1 , ST2 , IL-17RB , IL-13 , and ROR\u03b3t  using the Foxp3/Transcription Factor Staining Buffer Kit (eBioscience) as per the manufacturer\u2019s instructions. ILC2 cells were defined as the Thy1.2+Lineage\u2212GATA3+ population within the live-cell gate. For epithelial cell staining, the cells were labeled with Ghost Dye 780 (Tonbo Biosciences) and then stained with anti-CD45 , EPCAM  and DCLK1 . Tuft cells were defined as the EpCAM+CD45\u2212DCLK1+ population within the live-cell gate. All data were collected on a BD LSRFortessa or FACSAria IIIu (BD Biosciences) instrument and analyzed using the Flowjo software (TreeStar).To analyze intestinal lymphocytes and epithelial cells, intestines were opened longitudinally and washed with PBS to remove luminal contents. All samples were incubated in 20 ml Hanks\u2019 balanced salt solution (HBSS) containing 5 mM EDTA for 20 min at 37\u00b0C in a shaking water bath to remove epithelial cells. After vigorous vortexing, colonic epithelial cells released into suspension were centrifuged, immediately frozen in liquid nitrogen, and stored at \u221280\u00b0C until further analysis. An aliquot of epithelial cells was washed with 10 ml of HBSS containing 5 mM EDTA, resuspended in 5 ml of 20% Percoll , and underlaid with 2.5 ml of 40% Percoll in a 15 ml Falcon tube to isolate colonic tuft cells. After the epithelial cells were removed, the muscle layer and adipose tissue were removed manually using forceps. The remaining LP layer was cut into small pieces and incubated in 10 ml of RPMI 1640 containing 4% fetal bovine serum, 0.5 mg/ml collagenase D (Roche), 0.5 mg/ml dispase (Gibco), and 40 \u03bcg/ml DNase I (Roche) for 45 min at 37\u00b0C in a shaking water bath. The digested tissues were washed with 10 ml of HBSS containing 5 mM EDTA, resuspended in 5 ml of 40% Percoll , and underlaid with 2.5 ml of 80% Percoll in a 15 ml Falcon tube. Percoll gradient separation was performed using centrifugation at 900 \u00d7 For RNA-sequencing analysis, an RNA library was prepared using a NEBNext Ultra RNA Library Prep Kit for Illumina (New England Biolabs) according to the manufacturer\u2019s instructions. After assessing the library quality, sequencing was conducted on a HiSeq 1500 system (Illumina) using single-ended 50-bp reads. The sequenced reads were mapped to the mouse reference genome  and normalized to fragments per kilobase per million reads (FPKM) values using the Tophat and Cufflinks software pipeline. The heatmaps in Gapdh, 5\u02b9-CTC\u200bATG\u200bACC\u200bACA\u200bGTC\u200bCAT\u200bGC-3\u02b9 and 5\u02b9-CAC\u200bATT\u200bGGG\u200bGGT\u200bAGG\u200bAAC\u200bAC-3\u02b9; Actb, 5\u02b9-AGC\u200bCAG\u200bACC\u200bGTC\u200bTCC\u200bTTG\u200bTA-3\u02b9 and 5\u02b9-TAG\u200bAGA\u200bGGG\u200bCCC\u200bACC\u200bACA\u200bC-3\u02b9; Il1a, 5\u02b9-GGT\u200bTAA\u200bATG\u200bACC\u200bTGC\u200bAAC\u200bAGG\u200bA-3\u02b9 and 5\u02b9-GGC\u200bTGG\u200bTCT\u200bTCT\u200bCCT\u200bTGA\u200bGC-3\u02b9; Il1b, 5\u02b9-GTG\u200bGAC\u200bCTT\u200bCCA\u200bGGA\u200bTGA\u200bGG-3\u02b9 and 5\u02b9-CGG\u200bAGC\u200bCTG\u200bTAG\u200bTGC\u200bAGT\u200bTG-3\u02b9; Il2, 5\u02b9-CAA\u200bGCT\u200bCTA\u200bCAG\u200bCGG\u200bAAG\u200bCA-3\u02b9 and 5\u02b9-GAG\u200bCAT\u200bCCT\u200bGGG\u200bGAG\u200bTTT\u200bCA-3\u02b9; Il3, 5\u02b9-CCA\u200bGGG\u200bGTC\u200bTTC\u200bATT\u200bCGA\u200bGA-3\u02b9 and 5\u02b9-CGG\u200bTTC\u200bCAC\u200bGGT\u200bTAG\u200bGAG\u200bAG-3\u02b9; Il4, 5\u02b9-TCA\u200bTCG\u200bGCA\u200bTTT\u200bTGA\u200bACG\u200bAG-3\u02b9 and 5\u02b9-CCT\u200bTGG\u200bAAG\u200bCCC\u200bTAC\u200bAGA\u200bCG-3\u02b9; Il5, 5\u02b9-TGA\u200bGAC\u200bGAT\u200bGAG\u200bGCT\u200bTCC\u200bTG-3\u02b9 and 5\u02b9-CAG\u200bTAC\u200bCCC\u200bCAC\u200bGGA\u200bCAG\u200bTT-3\u02b9; Il6, 5\u02b9-TTC\u200bTCT\u200bGGG\u200bAAA\u200bTCG\u200bTGG\u200bAAA-3\u02b9 and 5\u02b9-TGC\u200bAAG\u200bTGC\u200bATC\u200bATC\u200bGTT\u200bGT-3\u02b9; Il7, 5\u02b9-TTG\u200bCCC\u200bGAA\u200bTAA\u200bTGA\u200bACC\u200bAA-3\u02b9 and 5\u02b9-GCG\u200bAGC\u200bAGC\u200bACG\u200bATT\u200bTAG\u200bAA-3\u02b9; Il9, 5\u02b9-ACA\u200bGCT\u200bGAC\u200bCAA\u200bTGC\u200bCAC\u200bAC-3\u02b9 and 5\u02b9-GGT\u200bCTG\u200bGTT\u200bGCA\u200bTGG\u200bCTT\u200bTT-3\u02b9; Il10, 5\u02b9-AGA\u200bGAA\u200bGCA\u200bTGG\u200bCCC\u200bAGA\u200bAA-3\u02b9 and 5\u02b9-CTC\u200bTTC\u200bACC\u200bTGC\u200bTCC\u200bACT\u200bGC-3\u02b9; Il11, 5\u02b9-GGC\u200bTAC\u200bTCC\u200bGCC\u200bGTT\u200bTAC\u200bAG-3\u02b9 and 5\u02b9-CCT\u200bCCT\u200bAGG\u200bATG\u200bGCA\u200bTGA\u200bGC-3\u02b9; Il12a, 5\u02b9- GAA\u200bGAC\u200bATC\u200bACA\u200bCGG\u200bGAC\u200bCA-3\u02b9 and 5\u02b9-CAG\u200bCTC\u200bCCT\u200bCTT\u200bGTT\u200bGTG\u200bGA-3\u02b9; Il12b, 5\u02b9-TGC\u200bTGC\u200bTCC\u200bACA\u200bAGA\u200bAGG\u200bAA-3\u02b9 and 5\u02b9-CGT\u200bGAA\u200bCCG\u200bTCC\u200bGGA\u200bGTA\u200bAT-3\u02b9; Il13, 5\u02b9-TGC\u200bCAT\u200bCTA\u200bCAG\u200bGAC\u200bCCA\u200bGA-3\u02b9 and 5\u02b9-GGC\u200bGAA\u200bACA\u200bGTT\u200bGCT\u200bTTG\u200bTG-3\u02b9; Txlna, 5\u02b9-AGC\u200bTAG\u200bTGG\u200bACG\u200bCCA\u200bAGC\u200bTC-3\u02b9 and 5\u02b9-CTT\u200bCAT\u200bCAG\u200bCTC\u200bGCA\u200bCAT\u200bCC-3\u02b9; Il15, 5\u02b9-TGC\u200bTCT\u200bACC\u200bTTG\u200bCAA\u200bACA\u200bGCA-3\u02b9 and 5\u02b9-CCT\u200bCCA\u200bGCT\u200bCCT\u200bCAC\u200bATT\u200bCC-3\u02b9; Il16, 5\u02b9-TCC\u200bAAT\u200bGAC\u200bCAA\u200bGAA\u200bATC\u200bTGC-3\u02b9 and 5\u02b9-GTG\u200bCTC\u200bAGT\u200bGAC\u200bCGA\u200bGTT\u200bGG-3\u02b9; Il17a, 5\u02b9-GTT\u200bCCA\u200bCGT\u200bCAC\u200bCCT\u200bGGA\u200bCT-3\u02b9 and 5\u02b9-ATG\u200bTGG\u200bTGG\u200bTCC\u200bAGC\u200bTTT\u200bCC-3\u02b9; Il17b, 5\u02b9-TGA\u200bCTT\u200bGGT\u200bGGG\u200bATG\u200bGAC\u200bTG-3\u02b9 and 5\u02b9-CCT\u200bCCC\u200bTTG\u200bCCC\u200bTTT\u200bTCT\u200bTT-3\u02b9; Il17c, 5\u02b9-AGG\u200bAGG\u200bTGC\u200bTGG\u200bAAG\u200bCTG\u200bAC-3\u02b9 and 5\u02b9-CTG\u200bTCT\u200bCAC\u200bGGC\u200bCTG\u200bTCT\u200bTG-3\u02b9; Il17d, 5\u02b9-GCG\u200bGCG\u200bCCC\u200bTTA\u200bTTT\u200bACT\u200bTC-3\u02b9 and 5\u02b9-TGC\u200bAGC\u200bGTG\u200bTGG\u200bTGG\u200bAA-3\u02b9; Il17f, 5\u02b9-CAA\u200bCCA\u200bAAA\u200bCCA\u200bGGG\u200bCAT\u200bTT-3\u02b9 and 5\u02b9-CAG\u200bCGA\u200bTCT\u200bCTG\u200bAGG\u200bGGA\u200bAC-3\u02b9; Il18, 5\u02b9-TGG\u200bCTG\u200bCCA\u200bTGT\u200bCAG\u200bAAG\u200bAC-3\u02b9 and 5\u02b9-CAG\u200bTGA\u200bAGT\u200bCGG\u200bCCA\u200bAAG\u200bTT-3\u02b9; Il19, 5\u02b9-AGG\u200bAAG\u200bCCA\u200bCCA\u200bATG\u200bCAA\u200bCT-3\u02b9 and 5\u02b9-GTC\u200bAGG\u200bCTG\u200bCAG\u200bGAG\u200bTTT\u200bCC-3\u02b9; Il20, 5\u02b9-AGC\u200bCTC\u200bGCC\u200bAAC\u200bTCC\u200bTTT\u200bCT-3\u02b9 and 5\u02b9-TCT\u200bTCC\u200bCCA\u200bCAA\u200bTGA\u200bCAT\u200bGC-3\u02b9; Il21, 5\u02b9-GCC\u200bAGA\u200bTCG\u200bCCT\u200bCCT\u200bGAT\u200bTA-3\u02b9 and 5\u02b9-CAA\u200bAAG\u200bCTG\u200bCAT\u200bGCT\u200bCAC\u200bAG-3\u02b9; Il22, 5\u02b9-GGT\u200bGAC\u200bGAC\u200bCAG\u200bAAC\u200bATC\u200bCA-3\u02b9 and 5\u02b9-CCA\u200bATC\u200bGCC\u200bTTG\u200bATC\u200bTCT\u200bCC-3\u02b9; Il23a, 5\u02b9-TGG\u200bTTG\u200bTGA\u200bCCC\u200bACA\u200bAGG\u200bAC-3\u02b9 and 5\u02b9-CAG\u200bGCT\u200bCCC\u200bCTT\u200bTGA\u200bAGA\u200bTG-3\u02b9; Il24, 5\u02b9-ACA\u200bGAT\u200bTCT\u200bCCC\u200bCTG\u200bCCT\u200bGA-3\u02b9 and 5\u02b9- CAG\u200bAAG\u200bGCC\u200bTCC\u200bCAC\u200bAGT\u200bTC-3\u02b9; Il25, 5\u02b9-TCC\u200bAGT\u200bCAG\u200bCCT\u200bCTC\u200bTCA\u200bGA-3\u02b9 and 5\u02b9-CAA\u200bGAA\u200bTGC\u200bAAC\u200bAGC\u200bCTG\u200bGT-3\u02b9; Il27, 5\u02b9-TCT\u200bCGA\u200bTTG\u200bCCA\u200bGGA\u200bGTG\u200bAA-3\u02b9 and 5\u02b9-GAA\u200bGGG\u200bCCG\u200bAAG\u200bTGT\u200bGGT\u200bAG-3\u02b9; Ifnl2, 5\u02b9-TCC\u200bCAG\u200bTGG\u200bAAG\u200bCAA\u200bAGG\u200bAT-3\u02b9 and 5\u02b9-GGA\u200bAGA\u200bGGT\u200bGGG\u200bAAC\u200bTGC\u200bAC-3\u02b9; Ifnl3, 5\u02b9-TCC\u200bCAG\u200bTGG\u200bAAG\u200bCAA\u200bAGG\u200bAT-3\u02b9 and 5\u02b9-GGA\u200bGAT\u200bGAG\u200bGTG\u200bGGA\u200bACT\u200bGC-3\u02b9; Il31, 5\u02b9-GTG\u200bCCC\u200bCAA\u200bTAT\u200bCGA\u200bAGG\u200bAA-3\u02b9 and 5\u02b9-GCT\u200bGAA\u200bACA\u200bCGG\u200bCAG\u200bCTG\u200bTA-3\u02b9; Il33, 5\u02b9-AGA\u200bCTC\u200bCGT\u200bTCT\u200bGGC\u200bCTC\u200bAC-3\u02b9 and 5\u02b9-CCC\u200bGTG\u200bGAT\u200bAGG\u200bCAG\u200bAGA\u200bAG-3\u02b9; Il34, 5\u02b9-GGG\u200bCAA\u200bGCT\u200bGCA\u200bGTA\u200bCAA\u200bGA-3\u02b9 and 5\u02b9-CGA\u200bAGC\u200bTCT\u200bCGC\u200bTCA\u200bCTC\u200bAC-3\u02b9; Ebi3, 5\u02b9-AGA\u200bGCC\u200bACA\u200bGAG\u200bCAT\u200bGTC\u200bCA-3\u02b9 and 5\u02b9-CAC\u200bGGG\u200bATA\u200bCCG\u200bAGA\u200bAGC\u200bAT-3\u02b9; Il36rn, 5\u02b9-CTG\u200bACT\u200bGCC\u200bGAA\u200bGCT\u200bTCC\u200bTT-3\u02b9 and 5\u02b9-CCC\u200bACA\u200bAAG\u200bCAT\u200bCCA\u200bTCA\u200bGA-3\u02b9; Il36a, 5\u02b9-TGT\u200bGTG\u200bGAT\u200bCCT\u200bGCA\u200bGAA\u200bCA-3\u02b9 and 5\u02b9-ATA\u200bTTG\u200bGCA\u200bTGG\u200bGAG\u200bCAA\u200bGG-3\u02b9; Il36b, 5\u02b9-GTT\u200bGAG\u200bATG\u200bGAG\u200bGGC\u200bAAA\u200bCC-3\u02b9 and 5\u02b9-GGA\u200bGCC\u200bCTC\u200bTAT\u200bGCC\u200bATG\u200bAT-3\u02b9; Ifna4, 5\u02b9-TCC\u200bATC\u200bAGC\u200bAGC\u200bTCA\u200bATG\u200bAC-3\u02b9 and 5\u02b9-TAT\u200bGTC\u200bCTC\u200bACA\u200bGCC\u200bAGC\u200bAG-3\u02b9; Ifnb1, 5\u02b9-CCC\u200bTAT\u200bGGA\u200bGAT\u200bGAC\u200bGGA\u200bGA-3\u02b9 and 5\u02b9-ACC\u200bCAG\u200bTGC\u200bTGG\u200bAGA\u200bAAT\u200bTG-3\u02b9; Ifng, 5\u02b9-GCG\u200bTCA\u200bTTG\u200bAAT\u200bCAC\u200bACC\u200bTG-3\u02b9 and 5\u02b9-CTG\u200bGAC\u200bCTG\u200bTGG\u200bGTT\u200bGTT\u200bGA-3\u02b9; Dclk1, 5\u02b9-CAA\u200bGCC\u200bAGC\u200bCAT\u200bGTC\u200bGTT\u200bC-3\u02b9 and 5\u02b9-TTC\u200bCTT\u200bTGA\u200bAGT\u200bAGC\u200bGGT\u200bCAC-3\u02b9; Pou2f3, 5\u02b9-AGA\u200bGAA\u200bTCA\u200bACT\u200bGCC\u200bCCG\u200bTG-3\u02b9 and 5\u02b9-GGA\u200bAGG\u200bCAC\u200bGAC\u200bTCT\u200bCTT\u200bCC-3\u02b9; Trpm5, 5\u02b9-TAT\u200bGGC\u200bTTG\u200bTGG\u200bCCT\u200bATG\u200bGT-3\u02b9 and 5\u02b9-ACC\u200bAGC\u200bAGG\u200bAGA\u200bATG\u200bACC\u200bAG-3\u02b9; Tslp, 5\u02b9-CGT\u200bGAA\u200bTCT\u200bTGG\u200bCTG\u200bTAA\u200bACT-3\u02b9 and 5\u02b9-GTC\u200bCGT\u200bGGC\u200bTCT\u200bCTT\u200bATT\u200bCT-3\u02b9; ThTr1, 5\u02b9-GTT\u200bCCT\u200bCAC\u200bGCC\u200bCTA\u200bCCT\u200bTC-3\u02b9 and 5\u02b9-GCA\u200bTGA\u200bACC\u200bACG\u200bTCA\u200bCAA\u200bTC-3\u02b9; ThTr2, 5\u02b9-TCA\u200bTGC\u200bAAA\u200bCAG\u200bCTG\u200bAGT\u200bTCT-3\u02b9 and 5\u02b9-ACT\u200bCCG\u200bACA\u200bGTA\u200bGCT\u200bGCT\u200bCA-3\u02b9.Total RNA was isolated from lymphocytes, epithelial cells, and organoids using the TRIzol reagent (Invitrogen) following the manufacturer\u2019s instructions. For qPCR analysis, cDNA was synthesized using ReverTra Ace Master Mix (TOYOBO), and qPCR was performed using the Thunderbird SYBR qPCR Mix (TOYOBO) on a LightCycler 480 (Roche). The following primer pairs were used: + cells per crypt\u2013villus axis. Four images were analyzed for each replicate.For tuft cell staining, intestinal tissues were flushed with PBS and fixed in 4% paraformaldehyde overnight. Tissues were washed with PBS and incubated in 30% (wt/vol) sucrose overnight at 4\u00b0C. Colon samples were then coiled into \u2018\u2018Swiss rolls\u2019\u2019 and embedded in Optimal Cutting Temperature Compound (Tissue-Tek) and sectioned at 14 \u03bcm on a Microm HM550 cryostat (Thermo Fisher Scientific). The tissues were incubated in 2% goat serum for 1 h, followed by an incubation with the primary antibodies  and DAPI (Thermo Fisher Scientific) overnight. The tissues were then incubated with goat anti-rabbit IgG F(ab\u2019)2-AF488 secondary antibodies for 1 h and then mounted with ProLong gold antifade reagent (Thermo Fisher Scientific) on slides. Images were acquired with KEYENCE (BZ-X810) using a 10\u00d7 NA 0.45 lens. Tuft cell proportions were calculated using ImageJ software to manually quantify DCLK1Intestinal organoids were generated from crypts isolated from the colon of SPF C57BL/6 mice as previously described . BrieflyFor mucus visualization, a transverse sample was taken at the same position within colonic tissue, fixed with methanol-Carnoy\u2019s solution and embedded in paraffin. Colonic sections were incubated with blocking buffer  at room temperature for 60 min and stained with rabbit anti-MUC2 monoclonal antibody  for 120 min, followed by Alexa Fluor 546-labeled goat anti-rabbit IgG  in blocking buffer for 60 min. All sections were counterstained with 4,6-diamidino-2-phenylindole , mounted with Fluoromount/Plus (Diagnostics BioSystems) and visualized under a TCS SP5 (Leica) confocal microscope. The thickness of the colonic mucus layer and the number of goblet cells were measured using ImageJ software.C57BL/6 SPF adult mice were fed with a VB1-deficient or control diets for 3 wk. 2,4,6-Trinitrobenzene sulphonic acid  was intracolonically administered to anaesthetized mice using a thin round-tip needle. The needle tip was inserted 4 cm proximal to the anal verge, and mice were held in a vertical position for 30 s after the injection. The mice were observed daily and were sacrificed on day 2 after TNBS administration. To evaluate the severity of colitis, colons were fixed with 4% paraformaldehyde, embedded in paraffin, sectioned, and stained with hematoxylin and eosin. Images were acquired with a KEYENCE (BZ-X810) using a 10\u00d7 NA 0.45 lens. The degree of inflammation in the distal part of the colon was graded from 0 to 4 as follows: 0, normal; 1, ulcer with cell infiltration limited to the mucosa; 2, ulcer with limited cell infiltration in the submucosa; 3, focal ulcer involving all layers of the colon; 4, multiple lesions involving all layers of the colon, or necrotizing ulcer larger than 1 mm in length.t test (parametric), and one-way ANOVA followed by Tukey\u2019s post-hoc test .All statistical analyses were performed using GraphPad Prism software  or JMP software v.12  with two-tailed unpaired Student\u2019s + ILC2s in the small intestine of Il17rb\u2212/\u2212 mice (The supplementary information shows IL-4\u2212/\u2212 mice , the exp\u2212/\u2212 mice , and inf\u2212/\u2212 mice ."}
{"text": "Bolbelasmusunicornis for Belarus is given, representing the northernmost known occurrence of the species. The second recent record for Croatia is quoted from an internet source. Updated distribution maps are provided for the Czech Republic and Slovakia, and for the entire range, as well as a distribution map of the Western Palaearctic representatives of the B.unicornis species group. The species is currently known from 386 localities in 20 countries.The author provides corrections of minor errors and omissions from his initial study, as well as data from omitted and new literature, and new records based on the material studied and new observations. For some of the previously published records, details obtained subsequently by the author are added. The first record of  Bolbelasmusunicornis   of k, 1789)  containek, 1789) , 1909, PB.unicornis from southeastern Belarus, which represents the first reliable record for this country and the northernmost known point of occurrence for the species. Records from new localities in Hungary  and Slovakia  are also provided. The updated distribution maps are shown in Figs B.unicornis for each country. These data show that almost half of all known localities where the species has been recorded after 1999 are located in Hungary.New records are presented based on material and observations obtained by the author just after his initial study  was publunicornis species group with the exception of B.tauricus Petrovitz, 1973, the validity of which was questioned by B.nireus Reitter, 1895, the holotype is unclear whether it is a form of B.nireus or a different species).Fig. CEST = Central European Summer Time, FSLG = flying slowly low above the ground, FMF = faunistic map field used in grid mapping of fauna and flora in Central Europe , PR = P\u0159\u00edrodn\u00ed rezervace/Pr\u00edrodn\u00e1 rezerv\u00e1cia (Nature Reserve). Unless otherwise stated, the material has been identified or revised by the author.Within Errata, only lines of text (including headings), not spaces, are numbered; figure legends are not included in the numbering. The faunistic records are divided into paragraphs according to the largest superior administrative units or traditional regions. The countries, administrative units/traditional regions, and faunistic records are ordered according to their geographical position from east to west and from north to south. A question mark at the beginning of a faunistic record indicates dubious data. The following acronyms are used in the text: ope Fig. , PP = P\u0159The following systems are used to transliterate cited literature and geographical or personal names in the Cyrillic scripts: BGN/PCGN 1979 system for Belarusian, BGN/PCGN 1947 System for Russian, BGN/PCGN 2005 Agreement for Serbian, and BGN/PCGN 2019 system for Ukrainian.unicornis species group was compiled using data contained in For the distribution map of the Czech Republic and Slovakia, the records are divided into three time periods: pre-1960, 1960\u20131999 and post-1999 Fig. ; the disB.unicornis for each country is based on the data provided by Table ABW Adam Byk, Warsaw, PolandBKL Bence Krajcsovszky, L\u00e1batlan, HungaryBLZ Boris Lau\u0161, Zagreb, CroatiaBPK Bal\u00e1zs Pint\u00e9r, Kerepes, HungaryDJP Daniel Ju\u0159ena, Prost\u011bjov, Czech RepublicDKP David Kr\u00e1l, Praha, Czech RepublicFPT Filip Pavel, T\u00fdni\u0161t\u011b nad Orlic\u00ed, Czech RepublicFTT Filip Trnka, Tr\u0161ice, Czech RepublicGDK Gejza Dunay, Kr\u00e1\u013eovce, SlovakiaKHE Kriszti\u00e1n Harmos, Eger, HungaryLFS Luk\u00e1\u0161 Fiala, S\u00e1zava, Czech RepublicMBK Marek Bidas, Kielce, PolandMSC Miroslav Sn\u00ed\u017eek, Homole near \u010cesk\u00e9 Bud\u011bjovice, Czech RepublicMSP Milan Sl\u00e1ma, Praha, Czech RepublicOBL Olivier Boilly, Lille, FranceOSD Oleksandr Oleksiiovych Sukhenko (\u041e\u043b\u0435\u043a\u0441\u0430\u043d\u0434\u0440 \u041e\u043b\u0435\u043a\u0441\u0456\u0439\u043e\u0432\u0438\u0447 \u0421\u0443\u0445\u0435\u043d\u043a\u043e), Dnipro, UkraineRFS Rudolf Fiala, S\u00e1zava, Czech RepublicRGZ Radim Gabri\u0161, Zlat\u00e9 Hory, Czech RepublicRVO \u2020 Radovan Veigler, Olomouc, Czech RepublicSET Sebastian Tylkowski, Krak\u00f3w, PolandVGG \u2020 Vadim Gennad\u2019yevich Grach\u00ebv (\u0412\u0430\u0434\u0438\u043c \u0413\u0435\u043d\u043d\u0430\u0434\u044c\u0435\u0432\u0438\u0447 \u0413\u0440\u0430\u0447\u0451\u0432), Moscow, RussiaVNP Vladim\u00edr Nov\u00e1k, Praha, Czech RepublicALMD Aquazoo L\u00f6bbecke Museum, D\u00fcsseldorf, GermanyMNHT Civic Museum of Natural History, Trieste, ItalyNHRSEntomological Collections of the Swedish Museum of Natural History, Stockholm, SwedenNMOKNaturkundemuseum im Ottoneum, Kassel, GermanyNMPCNational Museum, Prague, Czech RepublicSGGW Department of Forest Protection, Institute of Forestry Sciences, Warsaw University of Life Sciences, Warsaw, PolandZMMUZoological Museum of the Moscow Lomonosov State University, Moscow, RussiaPage 1, line 2 of Abstract: \u201c377 localities\u201d should read \u201c378 localities\u201d.Page 1, line 3 of Abstract: \u201c152 localities\u201d should read \u201c153 localities\u201d.Page 3, line 5: among the literature cited giving the body length of B.unicornis adults to be 12\u201315 mm, three references are missing: Page 3, line 21: missing citation of Page 3, line 28: missing citation of Page 3, line 30: missing citation of Page 3, line 40: in relation to hypogeous fungi as the presumed food of B.unicornis, Page 4, line 11: a new reference Bolboceratinae as a subfamily of Geotrupidae.Page 8, line 10: a comma is missing after the name of Luciano Ragozzino.Page 8, line 25: the acronym for Milan Sl\u00e1ma should read MSP instead of MPP.Page 13, line 33: \u201cOd.armiger\u201d should read \u201cOdonteusarmiger \u201d.Page 14, legend to Fig. 1: five missing commas after \u201cfemale\u201d and \u201cmale\u201d.Page 15, line 9: \u201cPR \u010cejkovick\u00e9 \u0160pidl\u00e1ky reserve\u201d should read \u201cPP \u010cejkovick\u00e9 \u0160pidl\u00e1ky reserve\u201d.Page 15, line 16: \u201cOdonteusarmiger \u201d should read \u201cOd.armiger\u201d.Page 17, line 17: \u201cErn\u0151 Csiki obs.\u201d should read \u201cErn\u0151 Csiki leg.\u201d.Page 18, line 28: \u201cErn\u0151 Csiki obs.\u201d should read \u201cErn\u0151 Csiki leg.\u201d.Page 19, line 31: \u201c\u201d.Page 20, line 24: the acronym for Milan Sl\u00e1ma should read MSP instead of MPP.Page 22, line 7: \u201cOchodaeuschrysomeloides \u201d should read \u201cOch.chrysomeloides\u201d.Page 30, line 40: \u201c21.10\u20131.40 CEST\u201d should read \u201c21.10\u201321.40 CEST\u201d.Page 35, line 1: missing question mark before \u201cUpper Bavaria\u201d.Page 36, line 9: the cited literature listing B.unicornis from the Canton of Ticino is missing Page 40, line 34: missing \u201cleg.\u201d after A. Liana\u2019s name.Page 43, line 26: \u201cWien Umg.,\u201d should read \u201cWien Umg.\u201d.Page 44, lines 11 and 16: \u201cErn\u0151 Csiki obs.\u201d should read \u201cErn\u0151 Csiki leg.\u201d.Page 47, line 22: \u00ab\u201cPest\u201c\u00bb should read \u00ab\u201cPest\u201d\u00bb.Page 49, line 15: the acronym for Gergely Petr\u00e1nyi should read GPB instead of GBP.Page 51, lines 34\u201335: \u201cVad\u00e1sz Csaba\u201d should read \u201cCsaba Vad\u00e1sz\u201d and use the abbreviation CVK.Page 53, line 5: \u201c68 Hungarian localities\u201d should read \u201c69 Hungarian localities\u201d.Page 54, line 22: \u201c(\u0420\u0435\u043f\u0443\u0431\u043b\u0438\u043a\u0430 \u0421\u0440\u043f\u0441\u043a\u0430)\u201d should be in bold.Page 58, line 4: \u201cErn\u0151 Csiki obs.\u201d should read \u201cErn\u0151 Csiki leg.\u201d.Page 59, line 5: the record from Hammersdorf (= Sibiu-Gu\u0219teri\u021ba) is missing \u201c1 \u2642\u201d.Page 61: there should be no space between lines 4 and 5.Page 64, line 4: \u201c1 \u2640 in Hartmann [leg.]\u201d should read \u201c1 \u2640, Hartmann [leg.]\u201d.Page 67, line 36: \u201cans\u201d should read \u201cand\u201d.Page 71, lines 20\u201321: \u201cBolbelasmuskeithiBolbelasmuskeithi Miessen & Trichas, 2011\u201d.Page 72, legend to Fig. 18: \u201cB.unicornis\u201d should be in italics.Page 73, line 3: \u201c...locality is Mulhouse\u201d should read \u201c...localities are Colmar and Mulhouse\u201d.Page 77, line 30: the reference \u201cPage 83, line 38: Knautiaarvensis is missing among the plants characteristic of Central European localities with B.unicornis.Page 90, line 20: for Page 100, line 4: \u201c83\u201383.\u201d should read \u201cp. 83.\u201d.Page 102, line 25: for Page 102, line 30: for Page 102, lines 34\u201336: for Page 104, line 28: for Page 109, line 36: for Page 110, line 15: for Page 113, lines 36\u201338: the reference Page 119, lines 2\u20133: \u201c7pp. 710\u20131390\u201d should read \u201c710\u20131390\u201d.Faunistic dataPublished data\u201cElsass\u201d [= Alsace], no other data .Material examinedALMD; since this is a mountainous area that does not meet the ecological requirements of the species, it is likely that this is a confusion of locality.? \u201cSavoie\u201d , 1 \u2640 with no other data, coll. Grand Est, Haut-Rhin, Mulhouse, 1 \u2642 with no other data, OBL det. + coll.Published dataBavaria (Bayern), no other data , Provincia di Alessandria, Lerma\u2014all records from this locality published by Trentino-Alto Adige/S\u00fcdtirol, \u201cTrient\u201d [= Trento], end of October 1875, number of specimens not specified, plant materials alluviated by the flooded Adige River, Stefano de Bertolini leg. .\u201cItal\u201d [= Italy], 1 \u2642 [ex coll.] Dohrn, coll. Published dataTyrol (Tirol), no other data , Linz-Ebelsberg, bank of the badly flooded Traun River, [10.vii.1954], 28 spec., F. Linninger leg. , no other data (her data .Styria (Steiermark), no other data , \u201cWien, Umg.\u201d [= Vienna env.], 1 \u2642 , undated, A. Winkler [probably leg.], coll. ZMMU.Published dataPre\u0161ov Region (Pre\u0161ovsk\u00fd kraj), Snina District (okres Snina), Snina, July 1965, 1 \u2642 flew through an open window into a room (hostel of the Snina Forestry Plant on the outskirts of the town) after sunset, together with Odonteusarmiger , MSP leg., coll. DKP deposited in NMPC , Bratislava II District (okres Bratislava II), Bratislava-Podunajsk\u00e9 Biskupice, Kop\u00e1\u010d Island, PP Pansk\u00fd diel , FMF No. 7868, ca 132 m a.s.l., 18.vi.2022, 3 spec.  excavated from their burrows on the edge of a forest path, and 26 spec.  flying < 0.5 m above the ground at 21.30\u201321.50 CEST (sunset: 20.53 CEST), no wind, 20\u00b0C, DJP, FTT & RGZ obs.Tren\u010d\u00edn Region (Tren\u010diansky kraj), Tren\u010d\u00edn District (okres Tren\u010d\u00edn), Tren\u010d\u00edn , undated, [Rudolf] \u010cepel\u00e1k [leg.], 1 \u2642 in coll. ALMD, 1 \u2642 (ex coll. Egon Leke\u0161) in coll. ZMMU; Tren\u010d\u00edn District (okres Tren\u010d\u00edn), \u201cTrencsen, Ungarn\u201d , FMF No. 7174, 1 \u2642 and 1 \u2640 with no other data, coll. ALMD.Nitra Region (Nitriansky kraj), Nov\u00e9 Z\u00e1mky District (okres Nov\u00e9 Z\u00e1mky), Mu\u017ela-\u010cenkov env., outside edge of the flood barrier of the Danube River, 47\u00b046'24.52\"N, 18\u00b033'21.52\"E, FMF No. 8277, 108 m a.s.l., 12.vi.2022, 1 \u2642 FSLG at 21:45 CEST (= 62 min after sunset), together with Od.armiger , 20.vi.2022, 1 \u2640 FSLG at 21:25 CEST (= 38 min after sunset), together with Ochodaeuschrysomeloides , an anonymous observer from the Czech Republic obs.; Kamenica nad Hronom, [\u010cierna hora hill], FMF No. 8178, 6.vi.2010, 1 \u2642, LFS & RFS leg., OBL det. + coll.Bansk\u00e1 Bystrica Region (Banskobystrick\u00fd kraj), Rimavsk\u00e1 Sobota District (okres Rimavsk\u00e1 Sobota), Cerov\u00e1 vrchovina Mts, Hajn\u00e1\u010dka env., 48\u00b013'41.56\"N, 19\u00b058'10.57\"E, FMF No. 7785, ca 350 m a.s.l., 14.\u201315.vii.1984, 2 \u2640\u2640 FSLG after sunset, GDK leg. + det., storage of the specimens unknown; Rimavsk\u00e1 Sobota District (okres Rimavsk\u00e1 Sobota), Cerov\u00e1 vrchovina Mts, Hajn\u00e1\u010dka , 10.vi.1989, 1 \u2642 FSLG after sunset, 27.vi.1989, 1 \u2640 FSLG after sunset, RVO leg., coll. Ulrich Schaffrath deposited in NMOK , Cerov\u00e1 vrchovina Mts, Gemersk\u00fd Jablonec, FMP No. 7785\u20137885, 7.vii.2013, 1 \u2642, FPT leg., OBL det. + coll.; Rimavsk\u00e1 Sobota District (okres Rimavsk\u00e1 Sobota), Cerov\u00e1 vrchovina Mts, Gemersk\u00e9 Decht\u00e1re env., 48\u00b014'38.31\"N, 20\u00b00'56.32\"E, FMF No. 7786, ca 240 m a.s.l., 27.vi.1987, 1 \u2642 and 1 \u2640 FSLG after sunset, GDK leg. + det., storage of the specimens unknown; Rimavsk\u00e1 Sobota District (okres Rimavsk\u00e1 Sobota), Cerov\u00e1 vrchovina Mts, Jestice env., Drienkov\u00e9, 48\u00b012'38.25\"N, 20\u00b02'54.45\"E, FMF No. 7786, ca 230 m a.s.l., 6.vii.2020, 2 \u2642\u2642 and 1\u2640 FSLG after sunset, GDK obs.; Rimavsk\u00e1 Sobota District (okres Rimavsk\u00e1 Sobota), Cerov\u00e1 vrchovina Mts, Jestice env., Drienkov\u00e9, 48\u00b012'40.07\"N, 20\u00b02'53.01\"E, FMF No. 7786, 235 m a.s.l., 6.vii.2020, 1 \u2642 flying in sunlight ca 0.5 m above the ground at about 19.00 CEST (sunset: 20:42 CEST), an anonymous observer from Slovakia obs.Material examined and new observationsCentral Transdanubia (K\u00f6z\u00e9p-Dun\u00e1nt\u00fal), Veszpr\u00e9m County (Veszpr\u00e9m v\u00e1rmegye), Csopak env., ca 46\u00b059'10.83\"N, 17\u00b054'39.43\"E, ca 265 m a.s.l., 14.v.2022, 1 \u2642, Botond Balogh obs. + photo; Kom\u00e1rom-Esztergom County, L\u00e1batlan, ca 200 m a.s.l., 31.v.2022, 1 \u2640, at light, BKL obs. + photo.Central Hungary (K\u00f6z\u00e9p-Magyarorsz\u00e1g), Pest County (Pest v\u00e1rmegye), Ver\u0151ce, 47\u00b050'36.895\"N, 19\u00b02'34.01\"E, 140 m a.s.l., 10.vi.2022, 1 \u2640, at light at ca 21.00 CEST, BPK obs. + photo; Budapest, 1 \u2642 and 1 \u2640 (ex coll. Carl Bartels [1823\u20131901]) with no other data, coll. NMOK; Pest County, Csom\u00e1d, \u00d6reg-hegy, , 15.vi.2002, 1 \u2642, [at light (mercury-vapor lamp)], collector unknown, OBL det. + coll.; Pest County, Domonyv\u00f6lgy, B\u00e1r\u00e1nyj\u00e1r\u00e1s, , 21.v.2004, 1 \u2642, [at light (mercury-vapor lamp)], collector unknown, OBL det. + coll.Northern Hungary (\u00c9szak-Magyarorsz\u00e1g), N\u00f3gr\u00e1d County, Kaz\u00e1r-P\u00f3lyos, ca 48\u00b02'46.94\"N, 19\u00b052'32.37\"E, ca 270 m a.s.l., June 2021, 1 \u2640 crawling in the grass during the day, Vikt\u00f3ria Szecsk\u00f3 obs. + photo, KHE det., DJP rev.Data from the internetBaranja , locality not specified, June 2022, 1 \u2642, BLZ obs. + photo ( + photo .Published dataVojvodina (\u0412\u043e\u0458\u0432\u043e\u0434\u0438\u043d\u0430), Srem District (\u0421\u0440\u0435\u043c\u0441\u043a\u0438 \u043e\u043a\u0440\u0443\u0433), village of Vrdnik (\u0412\u0440\u0434\u043d\u0438\u043a) env., 45\u00b007'31\"N, 19\u00b048'01\"E, 235 m a.s.l., 8.\u20139.vi.2022, 5 spec. FSLG at 21.00\u201322.00 CEST, ABW, MBK and SET leg., coll. SGGW (ll. SGGW .Published dataTransylvania (Transilvania), \u201cSch\u00e4ssburg\u201d [= Sighi\u0219oara or Segesv\u00e1r], no other data, Karl Petri leg. , Karma District (\u041a\u0430\u0440\u043c\u044f\u043d\u0441\u043a\u0456 \u0440\u0430\u0451\u043d), Karma (\u041a\u0430\u0440\u043c\u0430) env., Karots\u2019ki (\u041a\u0430\u0440\u043e\u0446\u044c\u043a\u0456), 20.vii.1987, 1 spec. crawling on a sandy steppe during daylight hours, MSC leg. + det., storage of the specimen unknown.B.unicornis for Belarus. This record, along with records from northeastern Ukraine , Novhorod-Siverskyi Raion (\u041d\u043e\u0432\u0433\u043e\u0440\u043e\u0434-\u0421\u0456\u0432\u0435\u0440\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Rozloty (\u0420\u043e\u0437\u043b\u044c\u043e\u0442\u0438) env., 51\u00b041'10.03\"N, 33\u00b08'30.37\"E, [140 m a.s.l.], 31.vii.\u20131.viii.2021, 3 \u2640\u2640, at light, M. V. Leshchenko leg.  with no other data, coll. Chernivtsi Oblast (\u0427\u0435\u0440\u043d\u0456\u0432\u0435\u0446\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Bukovina (\u0411\u0443\u043a\u043e\u0432\u0438\u043d\u0430), Chernivtsi Raion (\u0427\u0435\u0440\u043d\u0456\u0432\u0435\u0446\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), \u201cBukowina, Czernowitz\u201d , 1 \u2642 , undated, coll. MNHT.Vinnytsia Oblast (\u0412\u0456\u043d\u043d\u0438\u0446\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Vinnytsia Raion (\u0412\u0456\u043d\u043d\u0438\u0446\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), \u201c\u0421\u043a\u0432\u0438\u0440[\u0441\u043a\u0438\u0439] \u0443[\u0435\u0437\u0434] \u041a\u0438\u0435\u0432[\u0441\u043a\u043e\u0439] \u0433[\u0443\u0431\u0435\u0440\u043d\u0438\u0438]\u201d , \u201c\u0418\u043b\u044c\u0438\u043d\u0446\u044b\u201d [= Illintsi (\u0406\u043b\u043b\u0456\u043d\u0446\u0456)], [ca 215 m a.s.l.], 14.vi.[year not specified], 2 \u2642\u2642 (ex coll. M. K. Tikhonravov), A[ndrey] I[vanovich] Shelyuzhko [leg.], coll. ZMMU.Kyiv Oblast (\u041a\u0438\u0457\u0432\u0441\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Kiyv (\u041a\u0438\u0457\u0432), \u201c\u041f\u043e\u043b\u0438\u0442\u0435\u0445\u043d\u0438\u043a\u201d [= probably area of the National Technical University of Ukraine], July [19]26, 1 \u2640 (ex coll. M. K. Tikhonravov), collector unknown, coll. ZMMU.Cherkasy Oblast (\u0427\u0435\u0440\u043a\u0430\u0441\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Cherkasy Raion (\u0427\u0435\u0440\u043a\u0430\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Kaniv (\u041a\u0430\u043d\u0456\u0432) env., Kaniv Nature Reserve (\u041a\u0430\u043d\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u043f\u0440\u0438\u0440\u043e\u0434\u043d\u0438\u0439 \u0437\u0430\u043f\u043e\u0432\u0456\u0434\u043d\u0438\u043a), , 20.vi.1984, 1 \u2642 , VGG leg., coll. ZMMU.Dnipropetrovsk Oblast (\u0414\u043d\u0456\u043f\u0440\u043e\u043f\u0435\u0442\u0440\u043e\u0432\u0441\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Dnipro Raion (\u0414\u043d\u0456\u043f\u0440\u043e\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Dnipro (\u0414\u043d\u0456\u043f\u0440\u043e) [Dnipropetrovsk until 19 May 2016], Tunelna Balka tract (\u0422\u0443\u043d\u0435\u043b\u044c\u043d\u0430 \u0431\u0430\u043b\u043a\u0430) [the name of an area with oak forest in the southern part of the city], June 2009, 1 \u2640, OSD leg., OBL det. + coll.; June 2011, 1 \u2642, OSD leg., OBL det. + coll.; 48\u00b025'02.7\"N, 35\u00b002'23.6\"E, 100 m a.s.l., 8.vi.2014, 2 \u2642\u2642 and 2 \u2640\u2640, OSD leg., OBL det. + coll. (part of already published record\u2014see"}
{"text": "It should read as \u201cShaalan\u201d instead of \u201cShalaan\u201d.In the article by Hamdy et\u00a0al. , the 6th"}
{"text": "Thyreophagus (Acari: Acaridae) are distributed worldwide; they inhabit concealed habitats and include several beneficial and economically important species. However, species identification is difficult because many species are poorly described or delimited and their phoretic stages are unknown or uncorrelated. Furthermore, Thyreophagus is interesting because it includes entirely asexual (parthenogenetic) species. However, among the 34 described species of Thyreophagus, the asexual status is confirmed through laboratory rearing for only two species. Here, we provide detailed descriptions of five new species from North America (four) and Europe (one) based on adults and phoretic heteromorphic deutonymphs. Four of these species were asexual, while one was sexual. For most of these mites, the asexual status was confirmed and phoretic deutonymphs were obtained through rearing in the lab. We show that asexual mites retain seemingly functional copulatory and sperm storage systems, indicating that these lineages have relatively short evolutionary lifespans. One North American species, Thyreophagus ojibwe, was found in association with the native American chestnut Castanea dentata, suggesting a possibility that this mite can be used to control chestnut blight in North America. We also provide a diagnostic key to females, males, and heteromorphic deutonymphs of the Thyreophagus species in the world.Mites of the genus  Thyreophagus  is distributed worldwide, except in Antarctica [Thyreophagus corticalis  is beneficial because it is a natural vector of a hypovirus pathogenic to the ascomycete fungus Cryphonectria parasitica that causes chestnut blight, a dangerous disease of chestnuts in temperate regions in the northern hemisphere [C. parasitica and reduces its parasitic growth, leading to the spontaneous recovery of infected chestnut trees, Castanea sativa, from the disease in Europe [C. parasitica was inadvertently introduced to North America, it caused the functional extinction of the American chestnut Castanea dentata [The genus Rondani, 874 is ditarctica ,3,4,5. TRondani, 874 is ditarctica ,6,7,8,9.tarctica ,11,12. Tmisphere . The hypn Europe ,15. When dentata . Efforts dentata .Thyreophagus live in concealed habitats, they have been generally overlooked and understudied by researchers in comparison to most other acarid mite genera. The lack of correlation of adult and deutonymphal stages for many species; the presence of old, insufficiently described, unrecognizable taxa; the lack of types; and unclear species boundaries are cited as being the major impediments to the systematics of Thyreophagus [Th. Australis , Th. Corticalis, Th. Entomophagus, and Th. Calusorum Klimov, Demard, Stinson, Duarte, W\u00e4ckers, and Vangansbeke, 2022 [Thyreophagus is the presence of both sexual and asexual species, although only a few species from the latter group are known to date: Th. Calusorum, Th. Plocepasseri , and an unnamed species from a wasp nest in Japan [Since most species of eophagus . As an eke, 2022 ,18,19,20Castanea dentata, thus opening avenues for further research investigating whether the mite also can vector a hypoviruses that can infect and control the fungal pathogen of the tree. We demonstrate that females of asexual species retain functional structures responsible for insemination and sperm storage, with no signs of morphological reductions observed in the spermatheca. This suggests that these taxa are recent and probably evolutionary short-lived asexuals, thus meriting further genomic research into the potential causes of their asexual mode of reproduction. We also provide a diagnostic key to females, males, and heteromorphic deutonymphs for all known Thyreophagus species in the world.Here, we provide detailed descriptions of five new species from North America (four) and Europe (one) based on adult and phoretic heteromorphic deutonymphs. Four of these species were asexual, while one was sexual. The asexual status for most of these mites was confirmed through lab rearing. One North American species was found in association with the native American chestnut Fallen tree branches were collected from forest litter and processed in the lab. Feeding stages of live mites were sampled in subcortical spaces under a Zeiss Stemi DV4 dissecting microscope. Voucher specimens were mounted on slides, preserved in ethanol, or cultured  using bran as a food source, with the addition of baker\u2019s yeast . The culture was harvested and subcultured several times to obtain a large number of specimens and heteromorphic deutonymphs. The specimens were preserved in ethanol, cleared in Nesbitt\u2019s fluid for 1\u20132 days, mounted in Hoyer\u2019s medium, and dried at 60 \u00b0C for 7 days . To contImages were taken using a Nikon Eclipse E800 microscope equipped with a DIC optic and a Tucsen Discovery CH30 digital camera, and a Zeiss Axio Imager.A2 equipped with a DIC optic and Axiocam 305 color camera. Images were taken from multiple focal planes and assembled in Helicon Focus 7.6.4 Pro  with subsequent manual editing to add missing fine detail from the individual focal planes (retouching). Parts of the layered images were combined in Adobe Photoshop 22.2.0. Line drawings were created in Photoshop using microphotographs as the background.The idiosomal chaetotaxy followed Griffiths et al. ; the terThyreophagus ais Klimov, Kolesnikov, Demard, Vangansbeke sp. n..urn:lsid:zoobank.org:pub:7C7FEAA7-EB57-4A12-A489-196D2BEEA5D9.Type material. Holotype: One female\u2014lab culture on bran, harvested 26 February 2021, culture started from specimens originated from the USA, Florida, Fort Pierce, branches on ground in a small wooded area, subcortical, 27\u00b025\u203234.5\u2033 N 80\u00b024\u203222.7\u2033 W, 20 November 2020, Emilie Demard coll., BMOC 20-0101-014#S1.1. Paratypes: 3f , 6 HDNs \u2014same data; 2f\u2014original wild sample (see above).Depository. Holotype, paratypes\u2014University of Michigan, Museum of Zoology, Ann Arbor, Michigan, USA.Etymology. The new species is named after the Ais people who lived in the eastern coastal area of central Florida  but went extinct after the arrival of European colonizers [lonizers .Habitat. Thyreophagus ais lives under the bark of small fallen branches of deciduous trees in wooded areas. These branches are typically in the initial stages of decomposition, with approximately 80% of their natural sapwood retaining a white color, and 20% displaying brown discoloration, indicating the ongoing decomposition process. Additionally, the bark usually exhibits boreholes from wood-boring beetles.DescriptionFemale (n = 4), 2.4 -times longer than wide. Idiosomal cuticle smooth. Subcapitular setae (h) long, widened basally; palp tibial setae (a), lateral dorsal palp tibial setae (sup), dorsal palp tarsal seta (cm) filiform; supracoxal seta elcp absent; terminal palp tarsal solenidion \u03c9 short; external part of eupathidium ul\u2019\u2019 dome-shaped; terminal eupathidium ul\u2019 not observed. Prodorsal sclerite 105  long, 96  wide, 1.1 -times longer than wide, both smoothly punctated and very finely longitudinally striated, except in anterior 1/5 (smoothly punctate) and posterior 1/4 (ornamented with distinct pattern of broken striae arranged in a triangle). Prodorsal sclerite with setae vi situated at anterior part of shield (bases touching but separated from each other), rounded anterolateral incisions, and elongate midlateral incisions (corresponding to insertion points of setae ve).Female . Idiosomscx) smooth, sword-shaped, widened, and flattened, tapering at tip. Idiosomal setae  smooth, filiform, and short; opisthosomal gland openings slightly anteriad of setal bases e2. One pair of fundamental cupules (ia) present (other cupules not observed).Grandjean\u2019s organ (GO) expanded into 13 membranous finger-like extensions. Supracoxal seta  and one pair of genital setae (g). Shape of coxal sclerites as shown in Ventral idiosoma with four pairs of coxal setae ; s flattened, button-shaped; setae wa absent. Tarsal II setation similar to that of tarsus I, except proral setae  represented by small triangular rudiments  of tarsus I wide, spiniform, with broadly rounded apex, widest at middle. Solenidia \u03c6 of tibiae I\u2013III elongate, tapering, well-extended beyond apices of respective tarsi with ambulacra; solenidion \u03c6 IV shorter, shorter than tarsus IV (with ambulacra). On genu I, solenidion \u03c3\u2019 elongate, tapering, slightly not reaching bases of \u03c6 I, about 1.8-times longer than \u03c3\u2019\u2019; \u03c3\u2019\u2019 I slightly wider than \u03c3\u2019, about half the length of tibia I. On genu II, solenidion \u03c3 very short and somewhat conical. Solenidion \u03c3 of genu III absent.Legs short. Trochanters I\u2013III each with long, filiform seta, udiments J. TarsusMale. Absent.Heteromorphic deutonymph (n = 4), widest in sejugal region; idiosomal length 210\u2013240, n = 4 width 130\u2013150, n = 4. Gnathosoma short, subcapitulum and palp fused, bearing apical palpal solenidia (\u03c9) and filiform apicodorsal setae (sup); setae h absent from subcapitular remnant, their positions marked by refractile spots.utonymph . Body elvi) (bases separated) and a single sclerite. A pair of lateral ocelli present on propodosoma, widely separated from each other ; lenses and pigmented spots present; maximal diameter of lenses 12\u201314, n = 4. External vertical setae (ve) absent; external scapular setae se situated just below eye lenses; internal scapular setae (si) distinctly posterior and medial to external scapulars (se). Supracoxal setae of legs I (scx) filiform, with extended base, satiated below se. Sejugal furrow well-developed. Propodosomal sclerite 65\u201380, n = 4, hysterosomal shield 139\u2013160, n = 4, ratio hysterosomal shield/propodosomal sclerite = 2.0\u20132.13. Hysterosomal shield with 11 pairs of simple, filiform setae , setae h3 distinctly longer than others. Opisthonotal gland openings (gla) ventral, situated on hysterosomal shield, slightly posterior to setae c3. Of four fundamental pairs of cupules, three pairs observed: ia posteriomediad of setae c2, im ventral, lateral to trochanter IV, ih ventral, lateral to posterior portion of attachment organ.Dorsum. Propodosoma and hysterosoma each covered by smoothly punctate shields; distinct linear pattern present on anterior and lateral sides of propodosomal sclerite and hysterosomal shield. Anterior propodosoma with internal vertical setae (c3) filiform, situated on ventral surface between legs II\u2013III, adjacent to region separating sternal and ventral shields. Coxal setae 1a, 3a reduced, represented by very small structures each within an alveolus. Coxal setae (4b) filiform, situated at tips anterior to coxal apodemes IV; 4a in form of small, rounded conoids. Progenital region in posterior portion of coxal fields IV; genital opening elongate; two pairs of genital papillae within genital atrium; genital papillae two-segmented, with rounded apices; genital setae (g) filiform, situated laterad of genital opening. Attachment organ posterior to coxal fields IV. Anterior suckers (ad3) round, median suckers (ad1+2) distinctly larger, with paired vestigial alveoli; pair of small refractile spots anterolateral to median suckers (ps3); lateral conoidal setae of attachment organ (ps2) situated distinctly posterior to line joining centers of median suckers, slightly anterior conoids (ps1) and slightly posterior to median suckers (ad1+2); anterior and posterior lateral and posterior median cuticular conoids well-developed; anus situated between anterior suckers (ad3).Venter. Coxal fields sclerotized, smoothly punctate. Anterior apodemes of coxal fields I fused forming sternum; sternum not reaching posterior border of sternal shield by distance exceeding its length. Posterior border of sternal shield not sclerotized. Anterior apodemes of coxal fields II curved medially. Posterior apodemes of coxal fields II weakly developed, thin. Sternal and ventral shield adjacent. Anterior apodemes of coxal fields III free, connected by thin transverse sclerotization. Posterior medial apodeme present in area of coxal fields IV, well-separated from anterior apodemes IV and genital opening. Posterior apodemes IV absent. Subhumeral setae , setae d elongate, their bases situated at level anterior to bases of setae ra and la; one spoon-shaped seta e; setae s alveolar, setae wa, aa and ba I absent; tarsus II similar to tarsus I except seta ba present and filiform, close to \u03c91. Tarsus III with eight setae  smooth; all setae, except d III more or less foliate; seta d equal to or longer than leg III. Tarsus IV similar to tarsus III, except setae q and p short, spiniform, seta r longer, filiform, seta w filiform and with distinct prong, seta d distinctly longer than legs IV. Solenidia \u03c91 on tarsi I\u2013II cylindrical, with slightly clavate apices; \u03c93 on tarsus I slightly shorter than \u03c91, with rounded apex, positioned slightly anterior to \u03c91; \u03c91 and \u03c93 separated by bulbous famulus (\u03b5); solenidion \u03c92 of tarsus I slightly widened apically, situated somewhat more basal and posterior to \u03c91 + \u03b5+\u03c93 group; solenidia \u03c6 of tibiae I\u2013III elongate, tapering; \u03c6 I longer than tarsus I; \u03c6 II shorter than tarsus II; \u03c6 III reaching tip of tarsus III without ambulacrum; \u03c6 IV short; \u03c3 of genu I elongate, tapering slightly, nearly reaching tip of tibia I; \u03c3 of genu II shorter, cylindrical, not reaching midlength of tibia II; \u03c3 of genu III absent.Legs. Legs elongate, all segments free. Trochanters I\u2013III each with long, filiform seta, filiform B; setae DiagnosisThyreophagus ais is close to Th. calusorum, Th. mauritianus , Th. gallegoi Portus and Gomez, 1979 and Th. vermicularis Fain and Lukoschus, 1982 in having a very short solenidion \u03c3 II. Th. ais is similar to Th. calusorum and Th. mauritianus by the shape of solenidion \u03c3 II, which has convex sides , but it differs from Th. calusorum by the following: sclerites of oviducts are Y-shaped ; the canal of spermatheca at the entrance to spermatheca is not widened . Th. ais differs from Th. mauritianus as follows: the cuplike portions of the sclerites of oviducts are subequal to their stems (Th. mauritianus [Th. mauritianus); solenidion \u03c91 II is three-times longer than its width (five-times longer in Th. mauritianus); and solenidion \u03c6 IV is nearly reaching the bases of setae d IV (reaching the middle of tarsus IV in Th. mauritianus). Th. ais differs from Th. gallegoi by the absence of linear sclerites near the typical sclerites of oviducts  and from Th. vermicularis by the Y-shaped paired sclerites of oviducts, which are at least three-times longer than their width .Female.  widened I vs. wid Fain, 192, Th. ga Fain, 192, Th. ga Fain, 192, Th. gaTh. ais is very similar to Th. calusorum, but differs by short, spiniform setae q and p IV , setae d III are longer than leg III , d IV distinctly longer than leg IV . Setae hT I and kT IV are always spiniform in Th. ais, but it can be variable, either spiniform or filiform in Th. calusorum .Heteromorphic deutonymph. Thyreophagus hobe Klimov, Kolesnikov, Demard, Vangansbeke, sp. n..urn:lsid:zoobank.org:pub:7C7FEAA7-EB57-4A12-A489-196D2BEEA5D9.Type material. Holotype: female\u2014USA: Florida, Fort Pierce, branches on ground in a small, wooded area, subcortical, stick3, 27\u00b025\u203234.5\u2033 N 80\u00b024\u203222.7\u2033 W, 12 October 2020, Emilie Demard, BMOC 20-0101-008. Culture maintained until 26 February 2021 but then was accidentally lost (no specimens from these cultures were preserved).Depository. Holotype\u2014University of Michigan, Museum of Zoology, Ann Arbor, Michigan, USA.Etymology. The new species is named after the Hobe Indians who lived in the eastern coastal area of central Florida (between St. Lucie and Jupiter inlets), but went extinct after the arrival of European colonizers [6, which is reminiscent of the long dorsal setae of Thyreophagus hobe.lonizers . The menHabitat. Thyreophagus hobe lives under the bark of small fallen branches found in wooded areas. These branches are typically in the initial stages of decomposition, with approximately 80% of their natural sapwood retaining a white color, and 20% displaying brown discoloration, indicating the ongoing decomposition process.DescriptionFemale (h) long, widened basally; palp tibial setae (a), lateral dorsal palp tibial setae (sup), dorsal palp tarsal seta (cm) filiform; supracoxal seta elcp present, slightly widened basally; terminal palp tarsal solenidion \u03c9 short; external part of terminal eupathidium ul\u2019\u2019 dome-shaped; terminal eupathidium ul\u2019 not observed. Prodorsal sclerite 63 long, 46 wide, 1.4-times longer than wide, with setae vi , rounded anterolateral incisions, and elongate midlateral incisions (corresponding to insertion points of setae ve); shield punctate, 3/4 of posterior central region with linear pattern. Grandjean\u2019s organ (GO) expanded anteriorly into membranous finger-shaped extensions (exact number cannot be observed on the single specimen). Supracoxal seta (scx) smooth, sword-shaped, widened and flattened, tapering at tip. Idiosomal setae  smooth, filiform, setae h2 and h3 very long ; opisthosomal gland openings slightly anteriad of setal bases e2. Four pairs of fundamental cupules  present.Female . Idiosom1a, 3a, 4a, and 4b) and one pair of genital setae (g), seta 4b unpaired in holotype . Shape of coxal sclerites as shown in Ventral surface of idiosoma with four pairs of coxal setae , setae wa absent. Tarsus II setation similar to that of tarsus I. Tarsus III with eight setae; f, d, r filiform; e and s spiniform; p, q, u, and v similar to these of tarsus I\u2013II; seta w flattened, button-shaped. Tarsus IV similar to tarsus III, except w filiform. Solenidion \u03c91 on tarsus I cylindrical, with clavate apex, slightly curved; solenidion \u03c91 on tarsus II simple, cylindrical, with clavate apex, not bent, longer than \u03c91 on tarsus I. Solenidion \u03c92 on tarsus I shorter than \u03c91, cylindrical, with rounded apex, slightly expanded at tip, situated slightly anterior and external to \u03c91. Solenidion \u03c93 on tarsus I cylindrical, with rounded apex, shorter than \u03c91 I. Famulus (\u03b5) of tarsus I spiniform, with pointed apex. Solenidia \u03c6 of tibiae I\u2013III elongate, tapering, well extending beyond apices of respective tarsi with ambulacra; solenidion \u03c6 IV shorter than tarsus IV (with ambulacra). Genual solenidia \u03c3\u2019 and \u03c3\u2019\u2019 I elongate, tapering, subequal, slightly not reaching bases of \u03c6 I. Genual solenidion \u03c3 II 7\u20139-times longer than its width, with rounded tip. Genual solenidion \u03c3 III absent.Legs short, all segments free. Trochanters I\u2013III each with long, filiform seta, Male. Absent.Heteromorphic deutonymph. Unknown.DiagnosisThyreophagus hobe differs from all known species of Thyreophagus by the absence of femoral setae wF IV and unguinal setae u, v I\u2013IV. Like Th. australis Clark, 2009, Th. hobe has very long setae h2 and h3 , but in Th. hobe, setae h1 five-times shorter than h2 and h3 .Female. Thyreophagus ojibwe Klimov, Kolesnikov, Vangansbeke, sp. n..urn:lsid:zoobank.org:pub:7C7FEAA7-EB57-4A12-A489-196D2BEEA5D9.Type material. Holotype female (f1), paratype female (f2)\u2014USA: Michigan, Michigan State University Chestnut Plantation, Castanea dentata , branch on ground (tree1.stick1), 42\u00b009\u203206.6\u2033 N 84\u00b025\u203233.2\u2033 W, 23 November 2020, P. Klimov, BMOC 20-0101-010#slide1; 1 paratype female\u2014same data, slide 2; 1 paratype female\u2014same data, C. dentata blight tissues, BMOC 20-0101-012#slide3.Non-type material. One male, two females\u2014CANADA: Ontario C.E.F., Ottawa, under bark of dead willow twigs, Salix sp. , O. Peck, 1974 CNC788949  ).Depository. Holotype, paratypes\u2014University of Michigan, Museum of Zoology, Ann Arbor, Michigan, USA.Etymology. The new species is named after the Ojibwe, North American indigenous people from what is currently southern Canada and Midwestern United States, including the state of Michigan.Habitat. Thyreophagus ojibwe lives under the bark of small fallen branches typically found around American chestnut trees. These branches are generally in the initial stages of decomposition. Additionally, mites were found on the blight-infected tissues, suggesting that these mites can likely feed on this harmful fungus and potentially transmit hypoviruses.DescriptionFemale (n = 3), 2.3 -times longer than wide. Idiosomal cuticle smooth. Subcapitular setae (h) long, widened basally; palp tibial setae (a), lateral dorsal palp tibial setae (sup), dorsal palp tarsal seta (cm) filiform; supracoxal seta elcp present, slightly widened basally; terminal palp tarsal solenidion \u03c9 short; external part of terminal eupathidium ul\u2019\u2019 dome-shaped; terminal eupathidium ul\u2019 not observed. Prodorsal sclerite 85  long, 75  wide, 1.1 -times longer than wide, with setae vi , rounded anterolateral incisions, and elongate midlateral incisions (corresponding to insertion points of setae ve); shield smoothly punctate, 3/4 of posterior central region with linear pattern. Grandjean\u2019s organ (GO) with 8 membranous, finger-shaped extensions. Supracoxal seta (scx) smooth, sword-shaped, widened and flattened, tapering at tip. Idiosomal setae  smooth, filiform; setae h2 and h3 long, 1.75-times longer than length of prodorsal sclerite; opisthosomal gland openings slightly anteriad of setal bases of e2. Four pairs of fundamental cupules  present.Female . Idiosom1a, 3a, 4a, and 4b) and one pair of genital setae (g). Shape of coxal sclerites as shown in Ventral surface of idiosoma with four pairs of coxal setae . Solenidion \u03c92 on tarsus I shorter than \u03c91, cylindrical, with rounded apex, slightly widened at tip, situated slightly anterior and external to \u03c91. Solenidion \u03c93 on tarsus I cylindrical, with rounded apex, shorter than \u03c91, longer than \u03c92. Famulus (\u03b5) of tarsus I wide, spiniform, with broadly rounded apex, widest at middle. Solenidia \u03c6 of tibiae I\u2013III elongate, tapering, well extending beyond apices of respective tarsi with ambulacra; solenidion \u03c6 IV shorter, almost reaching tip of tarsus IV (without ambulacra). Genual solenidia \u03c3\u2019 and \u03c3\u2019\u2019 I elongate, tapering, subequal, slightly not reaching bases of \u03c6 I. Genual solenidion \u03c3 II 10-times longer than its width) and somewhat conical. Genual solenidion \u03c3 III absent.Legs short, all segments free. Trochanters I\u2013III each with long, filiform seta, Male (n = 1)  . Idiosomscx) as in female. Idiosomal setae  smooth, filiform, setae h2 and h3 long ; opisthosomal gland openings slightly anteriad of setal bases e2. Four pairs of fundamental cupules  present. Opisthonotal shield solid, whole, smoothly punctated; ventral part of shield extends to anal suckers.Grandjean\u2019s organ (GO) and supracoxal seta  and one pair of genital setae (g). Shape of coxal sclerites shown in ps1-3 very short.Ventral surface of idiosoma with four pairs of coxal setae ; the diameter of the genital papillae is approximately four-times shorter than the length of genital setae ; the stem of the Y-shaped sclerites of oviducts is about 3\u20134-times shorter than the length of Y-shaped sclerites of oviducts ; ventral apical spines p and q of tarsi III\u2013IV are shorter than spines u and v III\u2013IV .Female. Th. ojibwe differs from Th. corticalis by the longer setae h2 and h3  and by spines p and q on tarsi III\u2013IV, which are shorter than spines u and v .Male. Thyreophagus potawatomorum Klimov, Kolesnikov, Vangansbeke, sp. n..urn:lsid:zoobank.org:pub:7C7FEAA7-EB57-4A12-A489-196D2BEEA5D9.Type material. Holotype female (f2) and one paratype female (f1)\u2014USA: Michigan, Ann Arbor, Hansen Nature Area, stick 5, 42\u00b016\u203203.8\u2033 N 83\u00b046\u203253.2\u2033 W, 14 October 2020, P. Klimov, BMOC 20-0101-004#slide 1; four paratype females\u2014same data , slide 2; six paratype females\u2014same data, slide 3; two paratype HDNs\u2014same data, slide 5; one paratype HDN\u2014same data, slide 6; one paratype HDN\u2014same data, slide 8.Non-type material. Two DNs, one pharate DN\u2014same data, slide 7.Depository. Holotype, paratypes\u2014University of Michigan, Museum of Zoology, Ann Arbor, Michigan, USA.Etymology. This species is named after Potawatomi, native American people of the Great Plains, upper Mississippi River, and western Great Lakes region (including the state of Michigan).Habitat. Thyreophagus potawatomorum lives under the bark of small fallen branches of deciduous trees in wooded areas. These branches are typically in the initial stages of decomposition, with approximately 80% of their natural sapwood retaining a white color, and 20% displaying brown discoloration, indicating the ongoing decomposition process. Additionally, the bark usually exhibits boreholes from wood-boring beetles.DescriptionFemale (h) long, widened basally; palp tibial setae (a), lateral dorsal palp tibial setae (sup), dorsal palp tarsal seta (cm) filiform; supracoxal seta elcp present, widened basally; terminal palp tarsal solenidion \u03c9 short, bacilliform; external part of terminal eupathidium ul\u2019\u2019 dome-shaped; terminal eupathidium ul\u2019 not observed. Prodorsal sclerite 88  long, 80  wide, 1.1 -times longer than wide, with setae vi , rounded anterolateral incisions, and elongate midlateral incisions (insertion points of setae ve). Prodorsal sclerite punctate, with longitudinal linear pattern extending anteriorly from posterior end of sclerite and covering area exceeding 75% of sclerite; anterior lateral and medial areas have only punctate patterns; lines form a triangle at posterior end of sclerite. Grandjean\u2019s organ (GO) with seven membranous finger-like processes; central process distinctly wider than remaining processes. Supracoxal seta (scx) smooth, sword-shaped, widened and flattened, tapering at tip, slightly curved. Idiosomal setae  smooth, filiform, short and slender; opisthosomal gland openings slightly anteriad of setal bases e2. Four pairs of fundamental cupules  present.Female . Idiosom1a, 3a, 4a, and 4b) and one pair of genital setae (g). Shape of coxal sclerites as shown in Ventral idiosoma with four pairs of coxal setae  represented by small triangular rudiments. Tarsus III with 10 setae; f, d, r filiform; e, w, s, u, v, p, q spiniform. Tarsus IV similar to tarsus III, except w filiform. Solenidion \u03c91 on tarsus I cylindrical, with clavate apex, bent and pointed outward, to posterior side of tarsus; solenidion \u03c91 on tarsus II simple, cylindrical, with clavate apex, not bent, shorter than \u03c91 on tarsus I. Solenidion \u03c92 on tarsus I shorter than \u03c91, cylindrical, with rounded apex, slightly widened at tip, situated slightly anterior and external to \u03c91. Solenidion \u03c93 on tarsus I cylindrical, with rounded apex, shorter than \u03c91, longer than \u03c92. Famulus (\u03b5) of tarsus I wide, spiniform, with broadly rounded apex. Solenidia \u03c6 of tibiae I\u2013III elongate, tapering, well extending beyond apices of respective tarsi with ambulacra; solenidion \u03c6 IV shorter than tarsus IV (with ambulacra). On genu I, solenidion \u03c3\u2019 elongate, with rounded tip, slightly not reaching bases of \u03c6 I, about 1.5-times longer than \u03c3\u2019\u2019; \u03c3\u2019\u2019 I distinctly wider than \u03c3\u2019, about half the length of tibia I. On genu II, solenidion \u03c3 short (more than three-times longer than its width), with rounded tip. Solenidia \u03c3 III and IV absent.Legs short, all segments free. Trochanters I\u2013III each with filiform seta, Male. Absent.Heteromorphic deutonymph (n = 1) ((sup); setae h absent from subcapitular remnant, their positions marked by somewhat refractile spots. (n = 1) . Body elvi), sitiated on apex of propodosoma, long, bases separated. A pair of lateral ocelli present on propodosoma; widely separated from each other ; lenses and pigmented spots present, maximal diameter of lenses 16. External vertical setae (ve) absent; external scapular setae se situated just below eye lenses; internal scapular setae (si) distinctly posterior and medial to external scapulars (se). Supracoxal setae of legs I (scx) filiform, situated below se. Sejugal furrow well-developed. Propodosomal sclerite 73, hysterosomal shield 160, ratio hysterosoma shield/propodosomal sclerite, length = 2.19. Hysterosoma with 11 pairs of simple, filiform setae on hysterosomal shield , setae h3 distinctly longer than others. Opisthonotal gland openings (gla) ventral; situated slightly posterior to setae 3c off hysterosomal shield. Of four fundamental pairs of cupules, only three pairs observed: ia posteriomedial of setae c2, im posterior of d2 level and ih ventral, lateral to posterior sides of attachment organ.Dorsum. Propodosomal and hysterosomal smoothly punctate; distinct linear pattern present on anterior and lateral sides of propodosomal sclerite and hysterosomal shield. Apex of propodosomal sclerite shaped as obtuse triangle. Internal vertical setae (c3) long, filiform, positioned on ventral surface between legs II\u2013III, adjacent to region separating sternal and ventral shields. Coxal setae 1a, 3a reduced, represented by minute structures each situated in an alveolus. Setae 4b, g filiform; 4a in form of small, rounded conoids, 4b longer than g. Genital region in posterior portion of coxal fields IV; genital opening elongate, there are two pairs of genital papillae inside progenital atrium; papillae two-segmented, with rounded apices. Coxal setae (4b) situated at tips anterior to coxal apodemes IV; genital setae (g) situated laterad of genital opening. Attachment organ posterior to coxal fields IV. Anterior suckers (ad3) round, median suckers (ad1+2) distinctly larger, with paired vestigial alveoli; pair of small refractile spots anterolateral to median suckers (ps3); lateral conoidal setae of attachment organ (2ps) situated slightly posterior to a line joining centers of median suckers, distinctly anterior conoidal setae (ps1) and slightly posterior to median suckers (ad1+2); anterior and posterior lateral and posterior median cuticular conoids well-developed; anus positioned between anterior suckers (ad3).Venter. Coxal fields sclerotized, smoothly punctate. Anterior apodemes of coxal fields I fused forming sternum. Sternum not reaching posterior border of sternal shield by distance exceeding its length. Posterior border of sternal shield weakly sclerotized. Anterior apodemes of coxal fields II curved medially. Posterior apodemes of coxal fields II weakly developed, thin. Sternal and ventral shields adjacent. Anterior apodemes of coxal fields III free. Posterior medial apodeme present between coxal fields IV, well-separated from anterior apodemes IV and genital opening. Posterior apodemes IV absent. Dorsal hysterosomal shield separated from ventral surface by a distinct suture on each side. Subhumeral setae , three slightly foliate seate , and one spoon-shaped seta e. Seta d I elongate, longer than tarsus (its base situated at level of setae ra and la); seta s I alveolar; setae wa, aa and ba I absent. Tarsus II similar to tarsus I except seta ba present and filiform, base of seta d posterior to level of setae ra and la. Tarsus III with eight setae  smooth, all setae, except d III foliate. Tarsus IV similar to tarsus III, except seta r longer, filiform, and setae w filiform and has a distinct prong. Solenidia \u03c91 on tarsi I\u2013II cylindrical, with slightly clavate apices, \u03c91 II longer than \u03c91 I. Solenidion \u03c93 on tarsus I slightly shorter than \u03c91, with rounded apex, positioned slightly anterior to \u03c91; famulus (\u03b5) bulbous, situated between \u03c91 and \u03c93. Solenidion \u03c92 of tarsus I thin, with rounded apex, positioned somewhat more basal and posterior to \u03c91+\u03b5+ \u03c93 group. Solenidia \u03c6 of tibiae I\u2013III elongate, tapering; \u03c6 I and III, longer than tarsus I and III, respectively; \u03c6 II shorter than tarsus II; \u03c6 IV short. Solenidion \u03c3 of genu I elongate, slightly tapering, nearly reaching tip of tibia I; \u03c3 of genu II shorter, cylindrical, not reaching midlength of tibia II; \u03c3 of genu III absent.Legs. Legs elongate, all segments free. Trochanters I\u2013III each with long, filiform seta, DiagnosisThyreophagus potawatomorum is similar to Th. spinitarsis  by the linear striations of the prodorsal sclerite extending over at least 75% of the sclerite length and by tarsi III having seven spiniform setae , but differs by the widened, vase-shaped atrium (dome-shaped in Th. spinitarsis). Th. potawatomorum is very close to Th. berxi sp. n., but differs by the following character states: setae vi are not extending beyond the anterior margin of the prodorsum (extending in Th. berxi); bases of vi touch each other (distinctly separated in Th. berxi), lines of the prodorsal sclerite are longer than the diameter of the bases of vi (shorter in Th. berxi); on the prodorsal sclerite, the anterior medial striated area differs from the posterior medial striated area ; opisthosomal setae h2 and h3 are shorter than the anus (distinctly longer in Th. berxi); a rounded capsule is present between the canal of spermatheca and the atrium (absent in Th. berxi).Females. Th. potawatomorum is close to Th. corticalis and Th. berxi by the following character states: body elongate (more than 1.7-times longer than wide), setae hT and gT I\u2013II are present, different in shape, bases of setae d are at the level with bases of setae of ra and la on tarsus I. Th. potawatomorum differs from Th. corticalis and Th. berxi by the following character states: posterior median apodeme present, well-developed ; the diameter of ocelli is about 16 ; the distance between the ocelli is 36 ; setae wa I are absent ; on tibia IV, seta kT IV without distinct prong .Heteromorphic deutonymph. Thyreophagus berxi Klimov, Kolesnikov, W\u00e4ckers, Merckx, Duarte, Vangansbeke, sp. n..urn:lsid:zoobank.org:pub:7C7FEAA7-EB57-4A12-A489-196D2BEEA5D9.Type material. Holotype female, five paratype females\u2014BELGIUM: East Flanders, Gentbrugge, Moscou, Fagus sylvatica  twig, subcortical, 51\u00b001\u203250.7\u2033 N 3\u00b044\u203254.6\u2033 E, 16 November 2019, Dominiek Vangansbeke, PBK 22-0905-033#slide1; one paratype female, same data, slide 2. One paratype female\u2014BELGIUM: Flemish Brabant, Arenberg Castle, Leuven, Fagus sylvatica (label on slide = \u201c19-1-20 Tr. Sp. LA Arenberg beech 1\u201d), 50\u00b051\u203232.6\u2033 N 4\u00b039\u203254.4\u2033 E, 19 January 2020, Jonas Merckx, PBK 22-0905-034#slide1. Six paratype females, one paratype HDN\u2014GERMANY: Saxony, Dresden, Betula  , 51\u00b005\u201938.7\u2033 N 13\u00b050\u201935.6\u2033 E, 22 November 2020, Felix W\u00e4ckers (received via Dominiek Vangansbeke), PBK 22-0905-035#slide1. Six paratype females, three paratype HDNs\u2014FRANCE: Grand Est, Verzenay, Marcus Duarte, 8 April 2023, DV_2023-037.Depository. Holotype, paratypes \u2014University of Michigan, Museum of Zoology, Ann Arbor, Michigan, USA. Paratypes (France)\u2014Royal Belgian Institute of Natural Sciences, Brussels, Belgium.Etymology. The new species is named after the Belgian entomologist Peter Berx.Habitat. Thyreophagus berxi lives under the bark of small fallen branches of deciduous trees, such as European beech and birch trees.DescriptionFemale (n = 2) , 230  wide, 2.9 -times longer than wide. Idiosomal cuticle smooth. Subcapitular setae (h) long, widened basally; palp tibial setae (a), lateral dorsal palp tibial setae (sup), dorsal palp tarsal seta (cm) filiform; supracoxal seta elcp present, widened basally; terminal palp tarsal solenidion \u03c9 short, bacilliform; external part of terminal eupathidium ul\u2019\u2019 dome-shaped; terminal eupathidium ul\u2019 small, dome-shaped. Prodorsal sclerite 98 (100) long, 94 (96) wide, 1.1 -times longer than wide, with setae vi , rounded anterolateral incisions, and elongate midlateral incisions (insertion points of setae ve). Prodorsal sclerite has three types of patterns: smoothly punctated , longitudinal lines , smoothly punctated smaller lines . Grandjean\u2019s organ (GO) with 9\u201310 membranous finger-like extensions. Supracoxal seta (scx) smooth, sword-shaped, widened and flattened, tapering at tip, slightly curved. Idiosomal setae  smooth, filiform, medium length and slender; opisthosomal gland openings slightly anteriad of setal bases e2. Only one pair of fundamental cupules (ih) observed.Female . Idiosom1a, 3a, 4a, and 4b) and one pair of genital setae (g). Shape of coxal sclerites as in Ventral surface of idiosoma with four pairs of coxal setae  represented by small triangular rudiments; setae wa absent. Tarsus III with 10 setae; f, d, r filiform; e, w, s, u, v, p, q spiniform. Tarsus IV similar to tarsus III, except w filiform. Solenidion \u03c91 on tarsus I cylindrical, with clavate apex, bent and pointed outward, to posterior side of tarsus; solenidion \u03c91 on tarsus II simple, cylindrical, with clavate apex, not bent, shorter than \u03c91 on tarsus I. Solenidion \u03c92 on tarsus I shorter than \u03c91, cylindrical, with rounded apex, slightly widened at tip, situated slightly anterior and external to \u03c91. Solenidion \u03c93 on tarsus I cylindrical, with rounded apex, as long as \u03c91, longer than \u03c92. Famulus (\u03b5) of tarsus I wide, spiniform, with broadly rounded apex. Solenidia \u03c6 of tibiae I\u2013III elongate, tapering, well extending beyond apices of respective tarsi with ambulacra; solenidion \u03c6 IV shorter, shorter than tarsus IV (with ambulacra). On genu I, solenidion \u03c3\u2019 elongate, with rounded tip, reaching bases of \u03c6 I; \u03c3\u2019\u2019 I distinctly wider than \u03c3\u2019, slightly not reaching bases of \u03c6 I. On genu II, solenidion \u03c3 short (more than three-times longer than its width), with rounded tip. Solenidion \u03c3 of genu III and IV absent.Legs short, all segments free. Trochanters I\u2013III each with filiform seta, Male. Absent.Heteromorphic deutonymph (n = 1) (sup); setae h absent from subcapitular remnant, their positions marked by somewhat refractile spots. (n = 1) . Body elvi) apical, long, bases separated. A pair of lateral ocelli present on propodosoma; ocelli widely separated from each other (distance 50); lenses and pigmented spots present, maximum diameter of lenses 23. External vertical setae (ve) absent; external scapular setae se situated just below eye lenses; internal scapular setae (si) distinctly posterior and medial to external scapulars (se). Supracoxal setae of legs I (scx) filiform, situated below setae se. Sejugal furrow well-developed. Propodosomal sclerite 80, hysterosomal shield 180, ratio hysterosomal shield/propodosomal sclerite length = 2.25. Hysterosomal shield with 11 pairs of simple, filiform setae , setae h3 distinctly longer than others. Opisthonotal gland openings (gla) ventral; situated ventrally on hysterosomal shield, slightly posterior to setae c3. Of four fundamental pairs of cupules, only three pairs observed: ia posteriomedial of setae c2, im posterior to level of d2 and ih ventral, lateral to posterior sides of attachment organ.Dorsum. Propodosomal sclerite and hysterosomal shield smoothly punctate; distinct linear pattern present on anterior and lateral sides of propodosomal sclerite. A small area of linear pattern present on hysterosomal shield. Anterior end p of propodosoma shaped as obtuse triangle. Internal vertical setae (c3) long, filiform, situated ventrally between legs II\u2013III, adjacent to region separating sternal and ventral shields. Coxal setae 1a, 3a reduced, represented by minute structures each situated in an alveolus. Setae 4b, g filiform; 4a small, rounded conoids, 4b longer than g. Genital region in posterior portion of coxal fields IV; genital opening elongate; there are two pairs of genital papillae; genital papillae two-segmented, with rounded apices. Coxal setae (4b) situated at anterior tips of coxal apodemes IV; genital setae (g) laterad of genital opening. Attachment organ posterior to coxal fields IV. Anterior suckers (ad3) round, median suckers (ad1+2) distinctly larger, with paired vestigial alveoli (not situated on a common sclerite); pair of small refractile spots anterolateral to median suckers (ps3); lateral conoidal setae of attachment organ (2ps) situated slightly posterior to line joining centers of median suckers, distinctly anterior conoidal setae (ps1) and slightly posterior to median suckers (ad1+2); anterior and posterior lateral and posterior median cuticular conoids well-developed; anus situated between anterior suckers (ad3).Venter. Coxal fields sclerotized, smoothly punctate. Anterior apodemes of coxal fields I fused forming sternum. Sternum not reaching posterior border of sternal shield by distance exceeding its length. Posterior border of sternal shield weakly sclerotized. Anterior apodemes of coxal fields II curved medially. Posterior apodemes of coxal fields II weakly developed, thin, sternal and ventral shield adjacent. Anterior apodemes of coxal fields III free. Posterior medial apodeme in area of coxal fields IV weakly expressed. Posterior apodemes IV absent. Subhumeral setae ; gT II shorter, filiform; setae kT III somewhat spiniform, kT IV and has a distinct prong. Tarsal setation 7-8-8-8. All pretarsi consisting of hooked empodial claws arising from tarsal apices, and short, paired condylophores within tarsal apices. Tarsus I with three filiform setae , three slightly foliate , and one spoon-shaped seta e; seta d elongated, longer than tarsus (its base situated at level of bases ra and la); seta s alveolar; setae wa, aa, and ba I absent; tarsus II similar to tarsus I except seta ba present and filiform, base of seta d posterior of bases ra and la. Tarsus III with eight setae  smooth; all setae, except d III foliate. Tarsus IV similar to tarsus III, except seta w filiform, with a distinct prong. Solenidia \u03c91 on tarsi I\u2013II cylindrical, with slightly clavate apices, \u03c91 II longer than \u03c91 I. Solenidion \u03c93 on tarsus I longer and thinner than \u03c91, with rounded apex, positioned slightly anterior to \u03c91; \u03c91 and \u03c93 separated by bulbous famulus (\u03b5). Solenidion \u03c92 of tarsus I thin, slightly widened apically, situated somewhat more basal and posterior to \u03c91+\u03b5+ \u03c93 group. Solenidia \u03c6 of tibiae I\u2013III elongate, tapering; \u03c6 I and III longer than tarsus I and III, respectively; \u03c6 II shorter than tarsus II; \u03c6 IV short. Solenidia \u03c3 of genu I elongate, slightly tapering, nearly reaching tip of tibia I; \u03c3 of genu II shorter, cylindrical, not reaching midlength of tibia II; \u03c3 of genu III absent.Legs. Legs elongate, all segments free. Trochanters I\u2013III each with long, filiform seta, DiagnosisThyreophagus berxi is close to Th. spinitarsis  and Th. potawatomorum sp. n. by tarsi III with seven spiniform setae  and the prodorsal sclerite with linear striation extending over at least 75% of its length. Th. berxi differs from Th. spinitarsis by the vase-shaped atrium of the spermatheca, with the width in the central part two-times shorter than the width at the entrance to spermatheca . See above for the differences from Th. potawatomorum.Female. Th. berxi is close to Th. corticalis and Th. potawatomorum by the following characters: body elongate (more than 1.7-times longer than wide), setae hT and gT I\u2013II present, different in shape, bases of seta d are at the level of bases of ra and la on tarsus I. Th. berxi differs from Th. corticalis by the following character states: the posterior medial apodeme is weakly developed ; the diameter of ocelli is about 23 ; setae wa I are absent ; seta kT IV with a distinct prong . See above for the differences from Th. potawatomorum.Heteromorphic deutonymph. FemalesTh. africanus , Th. javensis , Th. sminthurus , Th. johnstoni , Th. leclercqi , Th. rwandanus .Adults of the following species are unknown: species inquirendae): Th. aleurophagus , Th. angustus , Th. berlesianus , Th. entomophagus nominalis , Th. lignieri , Th. magnus , Th. polezhaevi , Th. ponticus .Not included .1 Very large species, body length > 1500 \u03bcm. Egypt\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026- Smaller species, body length < 700 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262u and v on tarsi III\u2013IV vestigial; seta wF IV absent. USA (Florida) \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. hobe sp. n.2 Setae u and v on tarsi III\u2013IV well-developed, spiniform; seta wF IV present\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263.- Setae s, u and v) well-developed, proral setae p and q vestigial or absent\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20264.3 Tarsus III with three ventral apical spiniform setae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20267.- Tarsus III with five ventral apical spiniform setae well-developed .4 Prodorsal sclerite wider than long, almost entirely punctate, with a few short longitudinal striations in posteromedian region; atrium of spermatheca in form of an inverted bell, with base 18\u201320 \u03bcm wide; seta w III small spine or absent\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265- Prodorsal shield distinctly longer than wide, almost entirely covered with fine longitudinal striations; atrium of spermatheca smaller, not in form of a bell; seta Th. italicus .5 Prodorsal sclerite with linear striation in posterior half of shield. Italy\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026- Prodorsal sclerite with linear striation extending over at least 75% of its length\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20266w III a very short spine; seta se not longer than prodorsal shield. Morocco\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. cooremani .6 Idiosomal length 270\u2013360 \u03bcm, width 87\u2013150 \u03bcm; atrium of spermatheca dome-shaped, wider (6 \u03bcm) than long (5 \u03bcm) and not narrowed toward its center; seta w III vestigial; seta se longer than prodorsal sclerite. Europe\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..Th. odyneri .- Idiosomal length 525\u2013675 \u03bcm, width 210\u2013280 \u03bcm; atrium of spermatheca vase-shaped, 12 \u03bcm long, maximum width 12 \u03bcm, narrowed toward the center where it 5 \u03bcm wide; narrowed toward middle and widened in its proximal part; seta w III filiform\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202687 Seta w III spiniform or vestigial\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202613- Seta 8 Anterior margin of prodorsal sclerite without paired indentations\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20269- Anterior margin of prodorsal sclerite with paired indentations\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202610Th. passerinus .9 Atrium of spermatheca weakly developed; sclerotized portion of spermatheca in form of a broad arc, much wider than long. Cuba \u2026Th. plocepasseri .- Atrium of spermatheca well-developed, vase-shaped. Kenya\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026wa I absent. Ukraine\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. annae .10 Seta wa I present, spiniform or vestigial\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202611- Seta 11 Solenidion \u03c3\u2019 I longer than \u03c3\u2019\u2019 I; prodorsal sclerite at most 1.3-times as long as wide...................................................................................................................12Th. evansi .- Solenidion \u03c3\u2019 I shorter than \u03c3\u2019\u2019 I; prodorsal sclerite 1.5-times longer than wide. Ireland\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026h2 and h3 with bases inflated, conical; base of spermatheca forming thin sclerotized arc that divided anteriorly into four short, fine sclerotized lines; genu I with solenidion \u03c3\u2019 short, 8 \u03bcm long, \u03c3\u201d 6 \u03bcm long (ratio 1.4:1); Great Britain \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. macfarlanei .12 Long terminal setae h2 and h3 with very thin bases; atrium U-shaped with thick sides, 6 \u03bcm long, 5 \u03bcm wide; genu I with solenidion \u03c3\u2019 thin, 18\u201320 \u03bcm, \u03c3\u201d slightly thickened, 12 \u03bcm long (ratio 1.58: 1); Morocco \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. athiasae .- Long terminal setae w III well-developed, spiniform\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20261413 Seta w III vestigial\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202625- Seta Th. ais). Base of spermatheca in form of a broad arc, much wider than long, or small and rounded \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20261514 Solenidion \u03c3 II very short, 2\u20133-times longer than its width; atrium absent or minute, much shorter than sclerites of oviducts .16 Solenidion \u03c3 II short and filiform or nearly conical, but with sides straight and not convex; linear sclerites near the typical sclerites of oviducts present; paired sclerites of oviducts Y-shaped, not elongated, at least three-times longer than its width. Widespread \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...Th. vermicularis .- Solenidion \u03c3 II short and filiform; linear sclerites near the typical sclerites of oviducts absent; paired sclerites of oviducts elongated, more than five-times longer than their width, V-shaped. Great Britain \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026vi nearly touching, situated in common unsclerotized area. USA (Florida)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. Calusorum .17 Paired sclerites of oviducts V-shaped; canal of spermatheca at entrance to spermatheca widened; bases of setae vi, separated, not in common area \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202618- Paired sclerites of oviducts Y-shaped; canal of spermatheca at entrance to spermatheca, uniform in width, not widened; bases of setae 1 II five-times longer than its width; solenidion \u03c6 IV reaching middle of tarsus IV. Mauritius \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. mauritianus .18 Cuplike portions of sclerites of oviducts distinctly shorter than their stems; solenidia \u03c3\u2019 and \u03c3\u2019\u2019 subequal; solenidion \u03c91 II three-times longer than its width; solenidion \u03c6 IV longer, nearly reaching bases of setae d IV. USA (Florida) \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. ais sp. n.- Cuplike portions of sclerites of oviducts subequal; solenidia \u03c3\u2019 distinctly longer than \u03c3\u2019\u2019; solenidion \u03c9wa I present\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262019 Seta wa I absent\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202621- Seta h1 less than half of length of h2. Colombia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. incanus .20 Anterior margin of prodorsal shield without paired indentations; prodorsal shield smoothly punctated; posterior hysterosomal seta h1 half of length of h2. Europe\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. spinitarsis .- Anterior margin of prodorsal shield with paired indentations; prodorsal shield with linear striation extending over at least 75% of its length; posterior hysterosomal seta 21 With one pair of large, sclerotized, funnel-like, internal structures near posterior end of body (not to be confused with sclerites oviducts) \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202622- Without paired, funnel-like structures in posterior body\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202623Th. tridens .22 Solenidion \u03c6 of tibia IV very short (4 \u03bcm); USA  \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. cracentiseta Barbosa, .- Solenidion \u03c6 of tibia IV longer (14 \u03bcm). Brazil\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026h1, h2, and h3 very long , similar in length. New Zealand\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. australis .23 Setae h2 and h3 less than 2.5-times longer than length of prodorsal shield, seta h1 less than half of length of h2\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202624- Setae vi short, not extending beyond anterior margin of prodorsum; bases of vi touching; medial area of prodorsal shield with non-uniform pattern of striations in its anterior and posterior parts; setae h2 and h3 shorter than anus; rounded sclerotized capsule at junction of canal of spermatheca and atrium present. USA (Michigan)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. potawatomorum sp. n.24 Setae vi long, extending beyond anterior margin of prodorsum; bases of setae vi distinctly separated; entire medial area of prodorsal shield uniformly striated; setae h2 and h3 longer than anus; rounded capsule at junction of canal of spermatheca and atrium absent. France, Belgium, Germany\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. berxi sp. n.- Setae h2 and h3 1.1\u20131.25-times longer than length of prodorsal shield; diameter of genital papillae approximately six-times shorter than length of genital setae. Palaearctic \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. corticalis .25 Setae h2 and h3, long, their lengths 1.75-times longer than length of prodorsal shield; diameter of genital papillae approximately four-times shorter than length of genital setae. USA (Michigan), Canada (Ontario) \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. ojibwe sp. n.- Setae Males (modified after )Th. africanus , Th. javensis , Th. sminthurus , Th. johnstoni , Th. leclercqi , and Th. rwandanus .Adults of the following species are unknown: species inquirendae): Th. aleurophagus , Th. angustus , Th. berlesianus , Th. entomophagus nominalis , Th. lignieri , Th. magnus , Th. polezhaevi , and Th. ponticus .Not included : Th. aleurophagus, Th. angustus, Th. berlesianus, Th. italicus, Th. lignieri, Th. magnus, and Th. ponticus.Not included .2 Posterior venter with sclerotized projection very poorly developed or absent. Colombia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.- Posterior venter with sclerotized projection well-developed\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263Th. cynododactylon .3 Body elongate, six-times longer than wide; large species, length > 700 \u03bcm. Egypt\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.- Body ovoid, 1.5\u20132-times longer than wide; small species, length < 500 \u03bcm. Widespread\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...4s, u, v, p, q), three filiform setae , and two suckers . Ireland\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.Th. evansi .4 Tarsus IV with five spine-like setae , three filiform setae , and two suckers . Widespread\u2026\u2026\u2026\u2026\u2026\u2026Th. entomophagus .- Tarsus IV with three spine-like setae .6 Entire width of prodorsal sclerite covered by longitudinal striation. Europe\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026- Longitudinal striation on prodorsal sclerite restricted to median region, lateral areas simply punctulate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20267h2 and h3 1.1\u20131.25-times longer than length of prodorsal shield; spines p, q, and u, v III\u2013IV subequal. Palaearctic\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..Th. corticalis .7 Setae h2 and h3 long, their lengths are 1.8-times longer than the length of the prodorsal shield; spines p and q shorter than spines u and v on tarsi III\u2013IV. USA (Michigan), Canada (Ontario). \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. ojibwe sp. n.- Setae e2. Ukraine\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..Th. annae .8 Posterior hysterosoma with a large sclerotized area extending posteriad from level of setae h1\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20269- Posterior hysterosoma unsclerotized or at most with short terminal sclerotization posterior to setae h1\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026109 Posterior idiosoma with short sclerotized area posterior to setae - Posterior idiosoma unsclerotized\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202611Th. gallegoi .10 Genu I with solenidia \u03c3\u2019 and \u03c3\u201d approximately equal in length. Widespread\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. passerinus .- Genu I with solenidion \u03c3\u2019 only half of length of \u03c3\u201d. Cuba\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026d2 and e2 much longer than distance between their alveoli. New Zealand\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. australis .11 Dorsal hysterosomal setae relatively long, setae d2 and e2 shorter than distance between their alveoli. Brazil\u2026\u2026\u2026.Th. cracentiseta .- Dorsal hysterosomal setae much shorter, setae Heteromorphic deutonymphsTh. aleurophagus, Th. angustus, Th. annae, Th. athiasae, Th. cooremani, Th. cracentiseta, Th. cynododactylon, Th. entomophagus nominalis, Th. evansi, Th. gallegoi, Th. hobe, Th. incanus, Th. italicus, Th. macfarlanei, Th. magnus, Th. mauritianus, Th. ojibwe, Th. odyneri, Th. passerinus, Th. plocepasseri, Th. polezhaevi, Th. ponticus, Th. spinitarsis, Th. tridens, and Th. vermicularis.Unknown for the following species: species inquirendae): Th. berlesianus, Th. lignieri.Not included  were corrected.Published measurements of Th. africanus .1 Dorsal surface completely striated. Afrotropical\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026- Dorsal surface smoothly punctate, without striations\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202622 Body ovoid, 1.3\u20131.5-times longer than wide\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263- Body elongate, more than 1.7-times longer than wide\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265hT I more than half the length of gT I. Europe\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.Th. leclercqi .3 Seta hT I less than half the length of gT I\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20264- Seta c3 and cp; seta kT III filiform; setae wa I\u2013II absent. Widespread\u2026Th. entomophagus .4 Opisthonotal gland openings approximately equidistant from setae c3 than to dorsolateral seta cp; seta kT III with distinct prong; setae wa I-II present. New Zealand\u2026Th. australis .- Opisthonotal gland openings much closer to ventral seta 5 Hysterosomal sclerite about 1.7-times longer than prodorsal sclerite\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20266- Hysterosomal sclerite about two-times longer than prodorsal sclerite\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20267hT I absent; posterior medial apodeme in area of coxal fields IV present; seta kT III with distinct prong. Great Britain, USA (Washington) \u2026Th. sminthurus .6 Seta hT I present; posterior medial apodeme in area of coxal fields IV absent; seta kT III without prong, filiform. USA (Maryland)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. johnstoni .- Seta hT II and gT II subequal \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202687 Setae hT II twice the length of gT II\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20269- Setae hT II and gT II filiform, both subequal to tibia II. Java\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. javensis .8 Setae hT II and gT II spiniform, shorter than half the length of tibia. Afrotropical\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. rwandanus .- Setae d situated at the same level with bases of setae ra and la; diameter of ocellus 16\u201323 \u03bcm \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026109 On tarsus I, bases of setae d distal to bases of setae ra and la; diameter of ocellus 10\u201314 \u03bcm \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.12- On tarsus I, bases of setae Th. potawatomorum sp. n.10 Width of ocellus 16 \u03bcm, distance between ocelli 36 \u03bcm; posterior medial apodeme in area of coxal fields IV present. USA (Michigan)\u2026\u2026\u2026\u2026\u2026\u2026..- Width of ocellus about 18\u201323 \u03bcm, distance between ocelli about 40\u201350 \u03bcm; posterior medial apodeme in area of coxal fields IV weakly developed or absent\u2026\u2026\u2026\u2026\u2026\u2026\u2026.11kT IV with distinct prong; width of ocellus 23 \u03bcm, distance between ocelli 50 \u03bcm; posterior medial apodeme in area of coxal fields IV weakly developed. Europe\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. berxi sp. n.11 On tibia IV, seta kT IV without prong; width of ocellus about 18\u201319 \u03bcm, distance between ocelli about 40\u201342 \u03bcm; posterior medial apodeme in area of coxal fields IV absent. Widespread\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. corticalis .- On tibia IV, seta p and q on tarsus IV foliate; seta d III shorter than leg III; setae d IV shorter or slightly longer than leg IV; diameter of ocellus 10\u201313. USA (Florida)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Th. calusorum 12 Tarsal setae p and q on tarsus IV short, spiniform; setae d III longer than leg III, d IV distinctly longer than leg IV; diameter of ocellus 12\u201314. USA (Florida)\u2026\u2026\u2026.Th. ais sp. n.- Tarsal setae"}
{"text": "In this paper, the authors introduce the anisotropic Hardy space of Musielak-Orlicz type, HA\u03c6(\u211dn), via the grand maximal function. The authors then obtain some real-variable characterizations of HA\u03c6(\u211dn) in terms of the radial, the nontangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy space HAp(\u211dn) with p \u2208 , and, as an application, the authors prove that, for a given admissible triplet , if T is a sublinear operator and maps all -atoms with q < \u221e -atoms with q = \u221e) into uniformly bounded elements of some quasi-Banach spaces \u212c, then T uniquely extends to a bounded sublinear operator from HA\u03c6(\u211dn) to \u212c. These results are new even for anisotropic Orlicz-Hardy spaces on \u211dn. Let  Moreover, there were several efforts to extend classical Hardy spaces, some of which are weighted anisotropic Hardy spaces  denote the class of Muckenhoupt weights \u2192)., Bownik  showed tt al. in , page 19t al. in , via usiand Zhou  proved ta et al.  establispaces in , weightepaces in , and, espaces in .HA\u03c6(\u211dn), via grand maximal functions and characterize these spaces via anisotropic atomic decompositions. These Hardy spaces include classical Hardy spaces Hp(\u211dn) of Fefferman and Stein  for all x, y \u2208 \u211dn, where H \u2208 2/i(\u03c6), \u221e). Then there exists a positive constant C such that, for all K \u2208 \u2124, L \u2208 2/i(\u03c6), \u221e). If k \u2264 K and x \u2212 y \u2208 Bk, thenF0K\u2217, is as in ]2/i(\u03c6), \u221e), we choose p < i(\u03c6) large enough and q > q(\u03c6) small enough such that Np \u2212 q2 > 0. Therefore, from this, f implies . This fiThe following Lemmas \u03c8 \u2208 \ud835\udcae(\u211dn) with \u222bn\u211d\u03c8(x)dx \u2260 0. Then, for any given N, L \u2208 2/i(\u03c6), \u221e), we know that there exists a positive integer m such that, for all f \u2208 \ud835\udcae\u2032(\u211dn), x \u2208 \u211dn, and integers K \u2208 \u2124+,f \u2208 \ud835\udcae\u2032(\u211dn) and K \u2208 \u2124+,K \u2192 \u221e, by the monotone convergence theorem and the continuity of \u03c6  \u21d2 \u21d225) \u21d2 . Let \u03c6 b \u2260 0. By  of Lemma, \u00b7) see , we have\u2133\u03c80f \u2208 L\u03c6(\u211dn). By L \u2208  such that (f \u2208 L\u03c6(\u211dn) for all K \u2208 \u2124+. By Lemmas m \u2208 \u2115 such thatC1 being independent of K \u2208 \u2124+. For any given K \u2208 \u2124+, letC2\u2254[2C1]p1/ with p \u2208 ). We claim thatp of \u03c6 and C2p\u2212C1 = 1/2, we havex \u2208 \u03a9K and p \u2208 ), we choose q \u2208  small enough such that 1/q > q(\u03c6), where q(\u03c6) is as in  such that, for all integers K \u2208 \u2124+ and x \u2208 \u03a9K,\u03c6 is of uniformly upper type 1 and positive lower type p with p < i(\u03c6), it follows that q and lower type p/q. Consequently, using  but is independent of K \u2208 \u2124+. This inequality is crucial, since it gives a bound of the nontangential maximal function by the radial maximal function in L\u03c6(\u211dn).Suppose uch that  holds truch that\u222b\u211dn\u03c6f(f(x) converges pointwise and monotonically to \u2133\u03c8f(x) for all x \u2208 \u211dn as K \u2192 \u221e, it follows that \u2133\u03c8f \u2208 L\u03c6(\u211dn) by  , and the monotone convergence theorem, we conclude that ||\u2133\u03c8f||L\u03c6(\u211dn) \u2264 C4||\u2133\u03c80f||L\u03c6(\u211dn), where now the positive constant C4 corresponds to L = 0 and is independent of f \u2208 \ud835\udcae\u2032(\u211dn). Combining this, (Since \u03c6(\u211dn) by , the con, \u00b7) see , and thend using , the conng this, , and Lemn established in . We denote by L\u03c6q(\u211dn) the usually anisotropic weighted Lebesgue space with the anisotropic Muckenhoupt weight \u03c6. Then we have the following technical lemma (see  and f \u2208 L\u03c6q(\u211dn), then the series \u2211ibi converges in L\u03c6q(\u211dn), and there exists a positive constant C6, independent of f and \u03bb, such that ||\u2211i|bi|||L\u03c6q(\u211dn) \u2264 C6||f||L\u03c6q(\u211dn).If The following conclusion is essentially  and q \u2208 (q(\u03c6), \u221e).(i)m \u2265 s \u2265 \u230aq(\u03c6)ln\u2061b/(i(\u03c6)ln\u2061\u03bb\u2212)\u230b and f \u2208 Hm,A\u03c6(\u211dn), then gm* \u2208 L\u03c6q(\u211dn), and there exists a positive constant C8, independent of f and \u03bb, such thatIf (ii)m \u2208 \u2115 and f \u2208 L\u03c6q(\u211dn), then g \u2208 L\u221e(\u211dn), and there exists a positive constant C9, independent of f and \u03bb, such that ||g||L\u221e(\u211dn) \u2264 C9\u03bb.If Let f \u2208 Hm,A\u03c6(\u211dn), by ibi converges in Hm,A\u03c6(\u211dn) and therefore in \ud835\udcae\u2032(\u211dn) by ti(x) is as in m \u2265 \u230aq(\u03c6)ln\u2061b/(i(\u03c6)ln\u2061\u03bb\u2212)\u230b implies that (\u03bb\u2212)m+1 > bq(\u03c6). Moreover, for any fixed x \u2208 xi + (Bt+\u2113i+2\u03c3+1\u2216Bt+\u2113i+2\u03c3) with t \u2208 \u2124+, we find thatL\u03c6qq(\u03c6)(\u2113q(\u03c6))-boundedness of the vector-valued maximal function \u2133A ), we find thatfm* \u2264 \u03bb on \u03a9\u2201, for any x \u2208 \u03a9\u2201, usingNoticing that tes with , we obtaf \u2208 L\u03c6q(\u211dn), then g and {bi}i are functions. By ibi converges in L\u03c6q(\u211dn) and hence in \ud835\udcae\u2032(\u211dn) due to the fact that L\u03c6q(\u211dn) \u2282 \ud835\udcae\u2032(\u211dn) is continuous embedding (see \u230b \u2264 s \u2264 m and height \u03bb associated with fm* as in g and bi in g\u03bb and bi\u03bb, respectively. By  as \u03bb \u2192 \u221e. Moreover, by gm*)\u03bb \u2208 L\u03c6q(\u211dn), which, together with g\u03bb \u2208 L\u03c6q(\u211dn). This finishes the proof of Let vely. By  of LemmaHA\u03c6(\u211dn) and anisotropic atomic Hardy spaces of Musielak-Orlicz type HA\u03c6,q,s(\u211dn)  see  below.\u212c\u2254{B = x + Bk : x \u2208 \u211dn, k \u2208 \u2124} be the collection of all dilated balls.Let B \u2208 \u212c and q \u2208 , let L\u03c6q(B) be the set of all measurable functions f, supported in B, such thatFor any L\u03c6q(B), ||\u00b7||L\u03c6q(B)) is a Banach space. Next we introduce anisotropic atomic Hardy spaces of Musielak-Orlicz type.It is easy to show that ((i)\u03c6, q, s) is said to be admissible, if q \u2208 (q(\u03c6), \u221e] and s \u2208 \u2124+ such that s \u2265 m(\u03c6) with m(\u03c6) as in (An anisotropic triplet (\u03c6) as in .(ii)\u03c6, q, s), a measurable function a is called an anisotropic\u2009\u2009-atom ifFor an admissible anisotropic triplet ((a)a \u2208 L\u03c6q(B) for some B \u2208 \u212c;(b)a||L\u03c6q(B) \u2264 ||\u03c7B||L\u03c6(\u211dn)\u22121;||(c)n\u211da(x)x\u03b1dx = 0 for any |\u03b1 | \u2264s.\u222b(iii)\u03c6, q, s), the anisotropic atomic Hardy space of Musielak-Orlicz type, HA\u03c6,q,s(\u211dn), is defined to be the set of all distributions f \u2208 \ud835\udcae\u2032(\u211dn) which can be represented as a sum of multiples of anisotropic -atoms, that is, f = \u2211jaj in \ud835\udcae\u2032(\u211dn), where aj for j is a multiple of an anisotropic -atom supported in the dilated ball xj + B\u2113j, with the propertyFor an admissible anisotropic triplet , where {aj}j are -atoms supported in dilated balls {xj + B\u2113j}j, andf as above withHA\u03c6,q,s(\u211dn) coincide with equivalent (quasi)norms.(i) In f = \u2211j\u03bbjaj in \ud835\udcae\u2032(\u211dn) for some -atoms, {aj}j, and {\u03bbj}j \u2282 \u2102 such that Indeed, if \ud835\udcae\u2032(\u211dn) with f = \u2211j\u03bbjaj and Conversely, if \u03c6 is as in (A\u221e(\u211dn) Muckenhoupt weight w and \u03a6(t)\u2254tp for all t \u2208 \u230b, and let f \u2208 Hm,A\u03c6(\u211dn). For each k \u2208 \u2124, as in f has a Calder\u00f3n-Zygmund decomposition of degree s and height \u03bb = 2k associated with fm* as follows:k \u2208 \u2124, {xik}i\u2254{xi}i is a sequence in \u03a9k and {\u2113ik}i\u2254{\u2113i}i is a sequence of integers such that  with respect to the norms given by \u03b6ik onto \ud835\udcabs(\u211dn) with respect to the norm associated with \u03b6jk+1 given by (\ud835\udcabs(\u211dn) such that, for all Q \u2208 \ud835\udcabs(\u211dn),To obtain the conclusion uch that  through uch that  hold forgiven by , and {Pgiven by . Moreovegiven by , namely,Lemmas \u2113jk+1 \u2264 \u2113ik + \u03c3 and If i, L is as in \u2229(\u03a9k+1)\u2201.There exists a positive constant k \u2208 \u2124, \u2211i\u2211jPi,jk+1\u03b6jk+1 = 0, where the series converges pointwise and also in \ud835\udcae\u2032(\u211dn).For every The proof of the following lemma is similar to that of \u230b and let q \u2208 (q(\u03c6), \u221e). Then, for any f \u2208 L\u03c6q(\u211dn)\u2229Hm,A\u03c6(\u211dn), there exists a sequence {aik}k\u2208\u2124,i of multiples of -atoms such that f = \u2211k\u2208\u2124\u2211iaik converges almost everywhere and also in \ud835\udcae\u2032(\u211dn), andC, independent of f, such that, for all k \u2208 \u2124 and i,\u03bb \u2208 ,Let f \u2208 Hm,A\u03c6(\u211dn)\u2229L\u03c6q(\u211dn). For each k \u2208 \u2124, f has a Calder\u00f3n-Zygmund decomposition of degree s \u2265 \u230aq(\u03c6)ln\u2061b/[i(\u03c6)ln\u2061(\u03bb\u2212)]\u230b and height 2k associated with fm*, f = gk + \u2211ibik as above. The conclusions  and \ud835\udcae\u2032(\u211dn) as k \u2192 \u221e. It follows, from gk||L\u221e(\u211dn) \u2192 0 as k \u2192 \u2212\u221e, which further implies that gk \u2192 0 almost everywhere as k \u2192 \u2212\u221e, and, moreover, by the fact that L\u221e(\u211dn) is continuously embedding into \ud835\udcae\u2032(\u211dn) (see \u230b and q \u2208 (q(\u03c6), \u221e], then HA\u03c6,q,s(\u211dn) = Hm,A\u03c6(\u211dn) = HA\u03c6(\u211dn) with equivalent (quasi)norms.Let be as in . If m \u2265 m \u2265 s \u2265 \u230aq(\u03c6)ln\u2061b/[i(\u03c6)ln\u2061\u03bb]\u230b, and all the inclusions are continuous. Thus, to finish the proof of f \u2208 Hm,A\u03c6(\u211dn) with m \u2265 s \u2265 \u230aq(\u03c6)ln\u2061b/[i(\u03c6)ln\u2061\u03bb]\u230b, ||f||HA\u03c6,\u221e,s(\u211dn)\u2272||f||Hm,A\u03c6(\u211dn), which implies that Hm,A\u03c6(\u211dn) \u2282 HA\u03c6,\u221e,s(\u211dn).Observe that, by , Definitf \u2208 Hm,A\u03c6(\u211dn)\u2229L\u03c6q(\u211dn); by To this end, let f \u2208 Hm,A\u03c6(\u211dn). By fk}k\u2208\u2115 of functions in Hm,A\u03c6(\u211dn)\u2229L\u03c6q(\u211dn) such that ||fk||Hm,A\u03c6(\u211dn) \u2264 2k\u2212||f||Hm,A\u03c6(\u211dn) and f = \u2211k\u2208\u2115fk in Hm,A\u03c6(\u211dn). By k \u2208 \u2115, fk has an atomic decomposition fk = \u2211i\u2208\u2115dik in \ud835\udcae\u2032(\u211dn), where {dik}i\u2208\u2115 are multiples of -atoms with supp\u2061dik \u2282 xik + B\u2113ik. Sincef = \u2211k\u2208\u2115\u2211i\u2208\u2115dik \u2208 HA\u03c6,\u221e,s(\u211dn) andLet HA\u03c6(\u211dn) the anisotropic Hardy space Hm,A\u03c6(\u211dn) of Musielak-Orlicz type with m \u2265 m(\u03c6).For simplicity, from now on, we denote simply by HA\u03c6(\u211dn), and, as an application, a bounded criterion on HA\u03c6(\u211dn) of quasi-Banach space-valued sublinear operators is also obtained.The goal of this section is to obtain the finite atomic decomposition characterization of q < \u221e (or continuous atoms when q = \u221e), its norm in HA\u03c6(\u211dn) can be achieved via all its finite atomic decompositions. This extends the conclusion p for all x \u2208 \u211dn and t \u2208  with p \u2208  and \u03c6 as in  as in  an admissible triplet.q \u2208 (q(\u03c6), \u221e), then ||\u00b7||HA,fin\u03c6,q,s(\u211dn) and ||\u00b7||HA\u03c6(\u211dn) are equivalent quasinorms on HA,fin\u03c6,q,s(\u211dn).If HA,fin\u03c6,\u221e,s(\u211dn) and ||\u00b7||HA\u03c6(\u211dn) are equivalent quasinorms on HA,fin\u03c6,\u221e,s(\u211dn)\u2229\u2102(\u211dn).||\u00b7||Let \u03c6) as in , and  \u2282 HA\u03c6(\u211dn) and, for all f \u2208 HA,fin\u03c6,q,s(\u211dn),f \u2208 HA,fin\u03c6,q,s(\u211dn) when q \u2208 (q(\u03c6), \u221e) and for all f \u2208 HA,fin\u03c6,q,s(\u211dn)\u2229\u2102(\u211dn) when q = \u221e, ||f||HA,fin\u03c6,q,s(\u211dn)\u2272||f||HA\u03c6(\u211dn).Obviously, by Now we prove this by three steps.Step\u2009\u20091 ). Assume that q \u2208 (q(\u03c6), \u221e]. Without loss of generality, we may assume that f \u2208 HA,fin\u03c6,q,s(\u211dn) and ||f||HA\u03c6(\u211dn) = 1. Notice that f has compact support. Suppose that supp\u2061f \u2282 B\u2254Bk0 for some k0 \u2208 \u2124, where Bk0 is as in k \u2208 \u2124, letq \u2208 (q(\u03c6), \u221e) and q = \u221e, by aik}k\u2208\u2124,i of multiples of -atoms such that f = \u2211k\u2208\u2124\u2211iaik holds almost everywhere and in \ud835\udcae\u2032(\u211dn). Moreover, by HA\u03c6,\u221e,s(\u211dn) \u2282 HA\u03c6,q,s(\u211dn) and m(\u03c6), such that x \u2208 (B*)\u2201\u2254(Bk0+4\u03c3)\u2201. Hence, for all x \u2208 (B*)\u2201, we haveOn the other hand, by Step\u2009\u20092 of the proof of  and q1 \u2208 ), by H\u00f6lder's inequality and \u03c6 \u2208 \ud835\udd38q/q1(A), we havef \u2282 B and f has vanishing moments up to order s, we know that f is a multiple of a -atom and therefore f* \u2208 L1(\u211dn). Then, by  This, together with the vanishing moments of aik, implies that \u2113 has vanishing moments up to order s and, hence, so does h by h = f \u2212 \u2113. Using \u03c7B||L\u03c6(\u211dn)\u22121 ~ ||\u03c7B*||L\u03c6(\u211dn)\u22121, we obtainC0, independent of f, such that h/C0 is a -atom, and, by \u03c6, q, s)-atom for any admissible triplet . Then, by , 116), , Step\u2009\u20092Then, by , and ). Let q \u2208 (q(\u03c6), \u221e). We first show \u2211k>k\u2032\u2211i\u03bbikaik \u2208 L\u03c6q(\u211dn). For any x \u2208 \u211dn, since \u211dn = \u222ak\u2208\u2124(\u03a9k\u2216\u03a9k+1), there exists j \u2208 \u2124 such that x \u2208 (\u03a9j\u2216\u03a9j+1). Since supp\u2061aik \u2282 B\u2113ik+\u03c3 \u2282 \u03a9k \u2282 \u03a9j+1 for k > j, applying x \u2208 (\u03a9j\u2216\u03a9j+1),f \u2208 L\u03c6q(B) \u2282 L\u03c6q(B*), we further have f* \u2208 L\u03c6q(B*). Since \u03c6 satisfies the uniformly locally dominated convergence condition, it follows that \u2211k>k\u2032\u2211i\u03bbikaik converges to \u2113 in L\u03c6q(B*).K, let\u2113K\u2254\u2211i,k)\u2208FK(\u03bbikaik. Since \u2211k>k\u2032\u2211i\u03bbikaik converges in L\u03c6q(B*), for any \u03f5 \u2208 , if K is large enough, we have that (\u2113 \u2212 \u2113K)/\u03f5 is a -atom. Thus, f = h + \u2113K + (\u2113 \u2212 \u2113K) is a finite linear combination of atoms. By ; then aik is also continuous by examining its definition (see (f*(x) \u2264 Cn,m(\u03c6)||f||L\u221e(\u211dn) for any x \u2208 \u211dn, where the positive constant Cn,m(\u03c6) only depends on n and m(\u03c6), it follows that the level set \u03a9k is empty for all k satisfying that 2k \u2265 Cn,m(\u03c6)||f||L\u221e(\u211dn). We denote by k\u2032\u2032 the largest integer for which the above inequality does not hold. Then the index k in the sum defining \u2113 will run only over k\u2032 < k \u2264 k\u2032\u2032.To prove (ii), assume that ion (see ). Since \u03f5 \u2208 . Since f is uniformly continuous, it follows that there exists a \u03b4 \u2208  such that if \u03c1(x \u2212 y) < \u03b4, then |f(x) \u2212 f(y)|<\u03f5. Write \u2113 = \u21131\u03f5 + \u21132\u03f5 with \u21131\u03f5\u2254\u2211i,k)\u2208F1\u2208F2 : b\u2113ik+\u03c3 < \u03b4, k\u2032 < k \u2264 k\u2032\u2032}.Let k \u2208 . Since \u03f5 is arbitrary, we can hence split \u2113 into a continuous part and a part that is uniformly arbitrarily small. This fact implies that \u2113 is continuous. Thus, h = f \u2212 \u2113 is a multiple of a continuous -atom by Step\u2009\u20092.On the other hand, for any fixed integer k\u2032\u2032], by  of Lemmaf. Let us use again the splitting \u2113\u2254\u21131\u03f5 + \u21132\u03f5. By -atoms and\u2113, \u21131\u03f5 are continuous and have vanishing moments up to order s and, hence, so does \u21132\u03f5 = \u2113 \u2212 \u21131\u03f5. Moreover, supp\u2061\u21132\u03f5 \u2282 B* and ||\u21132\u03f5||L\u221e(\u211dn) \u2264 C1(k\u2032\u2032 \u2212 k\u2032)\u03f5. Thus, we can choose \u03f5 small enough such that \u21132\u03f5 becomes a sufficient small multiple of a continuous -atom; that is,f = h + \u21131\u03f5 + \u21132\u03f5 is a finite linear combination of continuous atoms. Then, by -atom, we haveNow we can give a finite atomic decomposition of 2\u03f5. By , the partoms and||\u21131\u03f5||HAHA\u03c6(\u211dn) of quasi-Banach-valued sublinear operators.As an application of the finite atomic decompositions obtained in  quasi-Banach space\u2009\u2009\u212c is a vector space endowed with a quasinorm ||\u00b7||\u212c which is nonnegative, nondegenerate , and homogeneous and obeys the quasitriangle inequality; that is, there exists a constant K \u2208 ))\u2113p, Lw\u03b3-quasi-Banach space \u212c\u03b3 with \u03b3 \u2208 \u2212T(g)||\u212c\u03b3 \u2264 ||T(f\u2212g)||\u212c\u03b3.For any given T is linear, then T is \u212c\u03b3-sublinear. Moreover, if \u212c\u03b3\u2254Lwq(\u211dn) with q \u2208 , w is a Muckenhoupt A\u221e weight, and T is sublinear in the classical sense, then T is also \u212c\u03b3-sublinear.We remark that, if \u03c6, q, s) be an admissible triplet. Assume that \u03c6 is an anisotropic growth function satisfying the uniformly locally dominated convergence condition and being of uniformly upper type \u03b3 \u2208 q \u2208 (q(\u03c6), \u221e), and T : HA,fin\u03c6,q,s(\u211dn) \u2192 \u212c\u03b3 is a \u212c\u03b3-sublinear operator such that(ii)T is a \u212c\u03b3-sublinear operator defined on continuous -atoms such that\u2009T has a unique bounded \u212c\u03b3-sublinear operator extension HA\u03c6(\u211dn) to \u212c\u03b3.then Let , by \u03bbj}j=1k and -atoms {aj}j=1k supported in balls {xj + B\u2113j}j=1k such that f = \u2211j=1k\u03bbjaj pointwise. By \u03c6 is of uniformly upper type \u03b3, it follows that there exists a positive constant C\u03b3 such that, for all x \u2208 \u211dn, s \u2208 [1, \u221e), and t \u2208 ,j0 \u2208 {1,\u2026, k} such that C\u03b3|\u03bbj0|\u03b3 \u2265 \u2211j=1k|\u03bbj|\u03b3, thenHA,fin\u03c6,q,s(\u211dn) is dense in HA\u03c6(\u211dn), a density argument then gives the desired conclusion.Suppose that the assumption (i) holds true. For any  ,\u03c6\u2264Cf \u2208 HA,fin\u03c6,\u221e,s(\u211dn)\u2229\u2102(\u211dn), ||T(f)||\u212c\u03b3\u2272||f||HA\u03c6(\u211dn). To extend T to the whole HA\u03c6(\u211dn), we only need to prove that HA,fin\u03c6,\u221e,s(\u211dn)\u2229\u2102(\u211dn) is dense in HA\u03c6(\u211dn). Since H\u03c6,finp,\u221e,s is dense in HA\u03c6(\u211dn), it suffices to prove HA,fin\u03c6,\u221e,s(\u211dn)\u2229\u2102(\u211dn) is dense in H\u03c6,finp,\u221e,s in the quasinorm ||\u00b7||HA\u03c6(\u211dn). Actually, we only need to show that HA,fin\u03c6,\u221e,s(\u211dn)\u2229\u2102\u221e(\u211dn) is dense in HA,fin\u03c6,\u221e,s(\u211dn) due to Suppose now that the assumption (ii) holds true. Similar to the proof of (i), by f \u2208 HA,fin\u03c6,\u221e,s(\u211dn). Since f is a finite linear combination of functions with bounded supports, it follows that there exists l \u2208 \u2124 such that supp\u2061f \u2282 Bl. Take \u03c8 \u2208 \ud835\udcae(\u211dn) such that supp\u2061\u03c8 \u2282 B0 and \u222bn\u211d\u03c8(x)dx = 1. By (\u03c8k\u2217f) \u2282 Bl+\u03c3 for any \u2212k < l, and f\u2217\u03c8k has vanishing moments up to order s, where \u03c8k(x)\u2254bk\u03c8(Akx) for all x \u2208 \u211dn. Hence, f\u2217\u03c8k \u2208 HA,fin\u03c6,\u221e,s(\u211dn)\u2229\u2102\u221e(\u211dn).To see this, let  = 1. By , it is ef \u2212 f\u2217\u03c8k) \u2282 Bl+\u03c3 for any \u2212k < l, and f \u2212 f\u2217\u03c8k has vanishing moments up to order s. Take any q \u2208 (q(\u03c6), \u221e). By [\u03c6 satisfies the uniformly locally dominated convergence condition, we know thatf \u2212 f\u2217\u03c8k = ckak for some -atom ak, where ck is a constant depending on k and ck \u2192 0 as k \u2192 \u221e. Thus, we obtain ||f\u2212f\u2217\u03c8k||HA\u03c6(\u211dn) \u2192 0 as k \u2192 \u221e. This finishes the proof of Likewise, supp\u2061("}
{"text": "Laryngeal hypersensitivity may be an important component of the common disorders of laryngeal motor dysfunction including chronic refractory cough, pdoxical vocal fold movement , muscle tension dysphonia, and globus pharyngeus. Patients with these conditions frequently report sensory disturbances, and an emerging concept of the \u2018irritable larynx\u2019 suggests common features of a sensory neuropathic dysfunction as a part of these disorders. The aim of this study was to develop a Laryngeal Hypersensitivity Questionnaire for patients with laryngeal dysfunction syndromes in order to measure the laryngeal sensory disturbance occurring in these conditions.The 97 participants included 82 patients referred to speech pathology for behavioural management of laryngeal dysfunction and 15 healthy controls. The participants completed a 21 item self administered questionnaire regarding symptoms of abnormal laryngeal sensation. Factor analysis was conducted to examine correlations between items. Discriminant analysis and responsiveness to change were evaluated.The final questionnaire comprised 14 items across three domains: obstruction, pain/thermal, and irritation. The questionnaire demonstrated significant discriminant validity with a mean difference between the patients with laryngeal disorders and healthy controls of 5.5. The clinical groups with laryngeal hypersensitivity had similar abnormal scores. Furthermore the Newcastle Laryngeal Hypersensitivity Questionnaire (LHQ) showed improvement following behavioural speech pathology intervention with a mean reduction in LHQ score of 2.3.The Newcastle Laryngeal Hypersensitivity Questionnaire is a simple, non-invasive tool to measure laryngeal pesthesia in patients with laryngeal conditions such as chronic cough, pdoxical vocal fold movement , muscle tension dysphonia, and globus pharyngeus. It can successfully differentiate patients from healthy controls and measure change following intervention. It is a promising tool for use in clinical research and practice. Laryngeal Dysfunction Syndromes include chronic refractory cough, pdoxical vocal fold movement , muscle tension dysphonia and globus pharyngeus. These conditions present to clinicians as discrete syndromes based around a dominant manifestation of a disordered laryngeal adductor reflex, e.g. cough, vocal fold closure, vocalisation, or swallowing ,2. They Abnormal sensory experience is characterised in three ways: (1) hypersensory\u2013sensation triggered by stimuli that is sub-threshold for triggering that sensation, (2) pesthesia\u2013altered sensory experience, and (3) allodynia\u2013sensation triggered by stimuli that do not normally trigger those sensations. This characterisation can be applied to the concept of abnormal laryngeal sensation ,8.Patients with laryngeal dysfunction syndromes such as chronic refractory cough, pdoxical vocal fold movement, globus pharyngeus and muscle tension dysphonia frequently report irritation and discomfort in the laryngeal region. The focus of treatment in these conditions involves motor rather than sensory areas of dysfunction. There are standardised tests to measure sensory laryngeal dysfunction such as cough reflex sensitivity testing, Fibreoptic Endoscopic Evaluation with Sensory Testing (FEEST) and hypertonic saline challenge. While these tests provide objective reliable data they are expensive to administer and are rarely available outside of specialist treatment areas. Furthermore, these tests do not quantify the patient experience of the discomfort. Several questionnaires exist to measure quality of life in patients with voice disorders  and chroIn this study, we have developed and tested a questionnaire to document the laryngeal sensory abnormalities reported in these syndromes. The aim of this study was to develop a Laryngeal Hypersensitivity Questionnaire for patients with laryngeal dysfunction syndromes and to measure the laryngeal sensation occurring in these conditions. This paper describes the development and validation of the Newcastle Laryngeal Hypersensitivity Questionnaire which is a self-rated measure of laryngeal sensation.A total of 97 participants were studied. For the item validation and discriminant analysis, we studied 53 participants comprising healthy controls (n\u2009=\u200915) and 4 clinical groups: chronic refractory cough (n\u2009=\u200911), pdoxical vocal fold movement , globus pharyngeus (n\u2009=\u20096), and muscle tension dysphonia(n\u2009=\u20093). The case groups were recruited from consecutive referrals to the speech pathology department for assessment and treatment of their laryngeal condition. Exclusion criteria for all groups included recent (past month) upper respiratory tract infection, current smoking, untreated asthma, rhinitis or gastroesophageal reflux, significant psychological factors or neurological impairment preventing participation.For the study of questionnaire responsiveness, the questionnaire was administered to an additional group of 44 participants with laryngeal dysfunction syndromes. These participants included chronic refractory cough (n\u2009=\u200938), PVFM (n\u2009=\u20094) and globus pharyngeus (n\u2009=\u20092).1. Chronic Refractory Cough. The participants with chronic refractory cough had been referred by respiratory physicians for behavioural management of cough . The cou2. pdoxical Vocal Fold Movement. The participants with pdoxical vocal fold movement were diagnosed by either respiratory physicians or otolaryngologists. These patients had been referred for behavioural management of their respiratory symptoms which included inspiratory dyspnoea, noisy breathing and throat tightness. Asthma and other pulmonary diseases had been discounted as a reason for the respiratory problems in this group. They had positive symptoms of PVFM and a fall in FIF50 of greater than 20%.3. Globus Pharyngeus. The third group with globuspharyngeus were referred for clinical assessment and management of swallowing. These participants presented with globus sensation such as a sensation of an irritation, lump or tightness in the throat in the absence of oropharyngeal dysphagia.4. Muscle Tension Dysphonia. The fourth group included patients with muscle tension dysphonia diagnosed by otolaryngologists referred for dysphonia. These patients had a deviation in perceptual voice quality along with excessive tension in the intrinsic and/or extrinsic laryngeal muscles  in the a5. Healthy Controls. The healthy controls were recruited from the Hunter Medical Research Institute Healthy Control Register (n\u2009=\u200910), from a previous study where they served as healthy controls (n\u2009=\u20092) and by word of mouth (n\u2009=\u20093). All healthy controls had no history of voice disorder, chronic cough or extrathoracic airway hyperrresponsiveness. Voice was judged as within normal limits by a qualified speech pathologist. Healthy controls were also excluded if there was presence of asthma, presence of post-nasal drip syndrome, presence of gastro-oesophageal reflux, symptoms of breathing, cough or voice difficulty and/or swallowing difficulty.Written consent was obtained for all participants. The study was approved by the Hunter New England Research Ethics Committee.Item generation: Potential questionnaire items were generated by reviewing literature regarding sensations in chronic pain and neuropathy and from previous patient reports of laryngeal discomfort . These r1 equates to all of the time and 7 equates to none of the time (Appendix I). A lower score denotes greater impairment with sensory symptoms. This scaling is similar to the Leicester Cough Questionnaire.Item scaling: The items were rated on a 7 point Likert scale where Participants completed the 21 item questionnaire prior to their initial assessment in speech pathology. Reproducibility was assessed in a subgroup.In order to document the changes following therapy a revised 14 item version of the questionnaire was administered to a further group of 44 participants with laryngeal dysfunction syndromes.Descriptive statistics were obtained for each item. Items with a low mean rating i.e. that indicates the symptom occurs more frequently, were compared to other symptoms to determine whether there was a correlation. A correlation matrix was conducted and rotation component matrix was performed. Factor analysis and item reduction were also completed. Discriminant validity was assessed by one way anova which was conducted on the final questionnaire to determine whether there was a significant difference (1) between the clinical groups, and (2) between the clinical groups and healthy controls. Questionnaire responsiveness was assessed by comparing pre-post treatment data using a Wilcoxon test. Significance was accepted at p\u2009<\u20090.05.Participant characteristics are reported in Table\u00a0abnormal sensation in the throat, phlegm and mucous in the throat, tickle and irritation in the throat and a tickle in the throat .The items Discriminant validity was assessed by comparing mean scores between each patient group, and healthy controls, for the final 14-item questionnaire Table\u00a0. These rConcurrent validity was examined by comparing results to the Leicester Cough Questionnaire and the John Hunter Hospital Symptom and Frequency Severity Questionnaire. The Pearson Correlation Coefficient showed a correlation between the Laryngeal Hypersensitivity Questionnaire the Leicester Cough Questionnaire of .673 (p\u2009<\u2009.001) and the John Hunter Hospital Symptom Frequency and Severity questionnaire of \u2013.791 (p\u2009<\u2009.001).Participant characteristics for the responsiveness analysis are reported in Table\u00a0The Newcastle Laryngeal Hypersensitivity Questionnaire (LHQ) is a valid measure of laryngeal discomfort for patients with laryngeal hypersensitivity syndromes. The final version contains 14 items with a 7 point Likert frequency response scale. It is designed for self-administration and takes less than 5\u00a0minutes for completion. The questionnaire was useful in discriminating between patients with laryngeal hypersensitivity syndromes and healthy controls, and was able to detect a change in laryngeal hypersensitivity after speech pathology treatment.Healthy controls had a mean total score of 19.2 (SD\u2009=\u20090.7). A cut off for normal function could be considered to be 17.1 (i.e. mean minus 3 standard deviations). Questionnaire scoring is in a similar direction to the Leicester Cough Questionnaire. Subscale scores are averaged from the number of completed items for each subscale and range from 1 (worst) to 7 (best). The total score is the sum of the three subscale scores which range from 3 (worst) to 21 (best). We calculated the minimal important difference as 1.7 using both a standard statistical approach (at 0.5sd), and as the change in LHQ that corresponded to the MID for the Leicester Cough Questionnaire.The LHQ has a number of potential purposes. It could be used to quantify the patient experience of laryngeal discomfort and to measure change over time. It has the potential to discriminate between patient groups and healthy controls. It could also be used in be used in trials of speech pathology intervention for these clinical groups.most of the time although the quality of the sensation is not the same as in chronic pain. It would appear that the laryngeal sensation is similar between clinical groups and may suggest some underlying sensory neuropathy. These findings are consistent with results of quantitative sensory testing in patients with laryngeal hypersensitivity syndrome [This data also demonstrated that most patients with laryngeal dysfunction have abnormal laryngeal sensation at least syndrome .an abnormal sensation in my throat\u2019, \u2018pain in my throat\u2019, and \u2018my throat feels tight\u2019. It was beyond the scope of this study to explain causality of symptoms. Although patients had received prior treatment for associated medical conditions such as gastroesophageal reflux disease, asthma, or rhinitis, it is possible that these conditions were contributing to the symptoms. The purpose of the questionnaire was to measure the patient experience of laryngeal symptoms rather than determine the causality of symptoms.Neural hypersensitivity is best characterised in chronic pain syndromes. Symptoms indicating hypersensitivity include a spontaneously occurring sensation, termed pesthesia, increased perception of pain for a given stimulus level, termed hyperalgesia, and a situation where a normally non-painful stimulus evokes pain, termed allodynia. There is growing recognition that laryngeal symptoms in chronic cough can be interpreted in a similar fashion ,8. ThereWhen coupled with a history that these symptoms develop with nontussive triggers , or as an exaggerated response to stimuli (hypertussia), the LHQ can aid in the recognition of a laryngeal hypersensitivity syndrome with central sensitisation. This has important treatment implications as recent treatment developments successfully apply the treatment approaches used in chronic pain to chronic refractory cough, such as behavioural therapy  or gabapThese concepts can be also applied to other laryngeal hypersensitivity syndromes, such as PVFM, muscle tension dysphonia, and globus. We have recently reported evidence for laryngeal pesthesia in these conditions  as well One potential limitation of this study is the small sample size for some of the disease groups. This can be overcome by future research that assesses the LHQ in muscle tension dysphonia and globus. The sample size used reflected the referral patterns and community prevalence of these conditions. The patient mix, while heterogeneous, was reflective of overlapping symptomatology and consistent with the multiple underlying medical factors which affect this population. It is possible that this group may not be representative of any particular disorder. Full thematic analysis was not conducted in the item generation phase of the study. Further study of the MID would be useful.In conclusion, the study reports the development and validation of a tool to measure laryngeal hypersensitivity that can readily be applied in clinical practice and research. The LHQ will facilitate the recognition and assessment of laryngeal hypersensitivity in several laryngeal disorders and should be useful in as an outcome measure in clinical trials.1. There is an abnormal sensation in my throat.(circle one)All of the time. 1Most of the time. 2A good bit of the time. 3Some of the time. 4A little of the time. 5Hardly any of the time. 6None of the time. 72. I feel phlegm and mucous in my throat3. I have pain in my throat4. I have a sensation of something stuck in my throat5. My throat is blocked.6. My mouth and/or throat feels dry.7. My throat feels tight.8. There is an irritation in my throat.9. I have a sensation of something pushing on my chest.10. I have a sensation of something pressing on my throat11. There is a feeling of constriction as though needing to inhale a large amount of air.12. Food catches when I eat or drink.13. There is a tickle in my throat.14. There is an itch in my throat.15. I have a tingling sensation in my throat16. I have pins and needles in my throat17. I have a hot or burning sensation in my throat.18. I have numbness in my throat19. I have a sensation of an electric shock in my throat.20. There is a shooting sensation in my throat21. There is a freezing or painfully cold sensation in my throat.John Hunter Hospital Laryngeal pesthesia Questionnaire.1. There is an abnormal sensation in my throat. (O)(circle one)All of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1Most of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2A good bit of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3Some of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4A little of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5Hardly any of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6None of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 72. I feel phlegm and mucous in my throat (TT)(circle one)All of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1Most of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2A good bit of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3Some of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4A little of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5Hardly any of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6None of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 73. I have pain in my throat (P/Th)(circle one)All of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1Most of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2A good bit of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3Some of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4A little of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5Hardly any of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6None of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 74. I have a sensation of something stuck in my throat (O)(circle one)All of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1Most of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2A good bit of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3Some of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4A little of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5Hardly any of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6None of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 75. My throat is blocked. (O)(circle one)All of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1Most of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2A good bit of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3Some of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4A little of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5Hardly any of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6None of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 76. My throat feels tight. (O)(circle one)All of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1Most of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2A good bit of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3Some of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4A little of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5Hardly any of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6None of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 77. There is an irritation in my throat. (O)(circle one)All of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1Most of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2A good bit of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3Some of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4A little of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5Hardly any of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6None of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 78. I have a sensation of something pushing on my chest. (P/Th)(circle one)All of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1Most of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2A good bit of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3Some of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4A little of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5Hardly any of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6None of the timee\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 79. I have a sensation of something pressing on my throat (O)(circle one)All of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1Most of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2A good bit of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3Some of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4A little of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 5Hardly any of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6None of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 710. There is a feeling of constriction as though needing to inhale a large amount of air. (O)(circle one)All of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1Most of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2A good bit of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3Some of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4A little of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5Hardly any of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6None of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 711. Food catches when I eat or drink. (O)(circle one)All of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1Most of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2A good bit of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3Some of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4A little of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5Hardly any of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6None of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 712. There is a tickle in my throat. (TT)(circle one)All of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1Most of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2A good bit of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3Some of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4A little of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5Hardly any of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6None of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 713. There is an itch in my throat. (TT)(circle one)All of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1Most of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2A good bit of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3Some of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4A little of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5Hardly any of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6None of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 714. I have a hot or burning sensation in my throat (P/Th)(circle one)All of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1Most of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2A good bit of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3Some of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4A little of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5Hardly any of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6None of the time\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 7LHQ: Laryngeal hypersensitivity questionnaire; PVFM: Paradoxical vocal fold movement.The authors declare that they have no competing interests.AV and PG conceived the study and wrote the manuscript. AV did the statistical analysis. SB was responsible for collection and collation of the data. All authors read and approved the final manuscript."}
{"text": "Furthermore, we give some inclusion relations concerning the space \u2113p\u03bb(B) and we construct the basis for the space \u2113p\u03bb(B), where 1 \u2a7d p < \u221e. Furthermore, we determine the alpha-, beta- and gamma-duals of the space \u2113p\u03bb(B) for 1 \u2a7d p \u2a7d \u221e. Finally, we investigate some geometric properties concerning Banach-Saks type p and give Gurarii's modulus of convexity for the normed space \u2113p\u03bb(B).We introduce the sequence space  From the summability theory perspective, the role played by the algebraical, geometrical, and topological properties of the new Banach spaces which are defined by the matrix domain of triangle matrices in sequence spaces is well-known.w, we denote the space of all real or complex valued sequences. Any vector subspace of w is called a sequence space.By \u03bc with a linear topology is called a K-space provided that each of the maps pi : \u03bc \u2192 \u2102 defined by pi(x) = xi is continuous for all i \u2208 \u2115, where \u2102 denotes the complex field and \u2115 = {0,1, 2,\u2026}. A K-space is called an FK-space provided \u03bc is a complete linear metric space. An FK-space whose topology is normable is called a BK-space  be an infinite matrix of real or complex numbers ank, where n, k \u2208 \u2115. Then, we say that A defines a matrix transformation from \u03bc into \u03b3 and we denote it by writing A : \u03bc \u2192 \u03b3, if for every sequence x = (xk) \u2208 \u03bc the sequence Ax = {(Ax)n}, the A-transform of x is in \u03b3, whereLet \u03bc : \u03b3) denotes the class of all matrices A such that A : \u03bc \u2192 \u03b3. Thus, A \u2208 (\u03bc : \u03b3) if and only if the series on the right hand side of n}n\u2208\u2115 \u2208 \u03b3 for all x \u2208 \u03bc. The matrix domain \u03bcA of an infinite matrix A in a sequence space \u03bc is defined byA = (ank) is said to be a triangle if ann \u2260 0 for all n \u2208 \u2115 and ank = 0 for k > n. The study of matrix domains of triangles has a special importance due to the various properties which they have. For example, if A is a triangle and \u03bc is a BK-space, then \u03bcA is also a BK-space with the norm given by ||x||\u03bcA = ||Ax||\u03bc for all x \u2208 \u03bcA.The notation Nq and cNq in =\u2211k=0ed as in .x \u2208 \u2113p\u03bb(B) is given by (x = \u2211k\u03bck(x)bk)(. Since the transformation T defined from \u2113p\u03bb(B) to \u2113p by k \u2208 \u2115. That is to say that the representation (x \u2208 \u2113p\u03bb(B) is unique. Let us show that the uniqueness of representation for given by . Supposeentation  of x \u2208 \u2113\u2113p\u03bb(B) and \u2113\u221e\u03bb(B). We start with the definition of the alpha-, beta- and gamma-duals of a sequence space.In this section, we give some theorems determining the alpha-, beta- and gamma-duals of the spaces x and y are sequences and X and Y are subsets of \u03c9, then we write x \u00b7 y = (xkyk)k=0\u221e, x\u22121\u2217Y = {a \u2208 \u03c9 : a \u00b7 x \u2208 Y}, andX and Y. One can easily observe for a sequence space Z with Y \u2282 Z and Z \u2282 X that the inclusions M \u2282 M and M \u2282 M hold. The alpha-, beta- and gamma-duals of a sequence space, which are, respectively, denoted by X\u03b1, X\u03b2, and X\u03b3 are defined byX\u03b1 \u2282 X\u03b2 \u2282 X\u03b3. Also, it can be easily seen that the inclusions X\u03b1 \u2282 Y\u03b1, X\u03b2 \u2282 Y\u03b2, and X\u03b3 \u2282 Y\u03b3 hold, whenever Y \u2282 X. Now, we may begin with quoting the following lemmas \u03b1 = t\u221e\u03bb and [\u2113p\u03bb(B)]\u03b1 = tq\u03bb for 1 < p \u2a7d \u221e, where the matrix T = (tnk\u03bb) is defined via the sequence a = (an) \u2208 w byn, k \u2208 \u2115. Define the sets a = (an) \u2208 w and 1 < p < \u221e. Then, by using (n \u2208 \u2115 thatax = (anxn) \u2208 \u21131 whenever x = (xk) \u2208 \u2113p\u03bb(B) if and only if Ty \u2208 \u21131 whenever y = (yk) \u2208 \u2113p. This means that a = (ak)\u2208[\u2113p\u03bb(B)]\u03b1 if and only if T \u2208 (\u2113p : \u21131). Therefore, we get by T instead of A that a = (ak)\u2208[\u2113p\u03bb(B)]\u03b1 if and only if\u2113p\u03bb(B)]\u03b1 = tq\u03bb, for 1 < p \u2a7d \u221e. Similarly, we get from (a = (ak)\u2208[\u21131\u03bb(B)]\u03b1 if and only if T \u2208 (\u21131 : \u21131) which is equivalent to  is defined for all n, k \u2208 \u2115 byax = (anxn) \u2208 cs whenever x = (xk) \u2208 \u2113p\u03bb(B) if and only if Dy \u2208 c whenever y = (yk) \u2208 \u2113p. This means that a = (ak)\u2208[\u2113p\u03bb(B)]\u03b2 if and only if D \u2208 (\u2113p : c), where 1 \u2a7d p \u2a7d \u221e. Therefore, we derive from (\u2113p\u03bb(B)]\u03b2 = eq\u03bb\u2229d1\u03bb\u2229d4\u03bb for 1 < p < \u221e. Since beta-dual of the space \u2113p\u03bb(B) for the cases p = 1 and p = \u221e can be similarly computed, we omit the details. This completes the proof. \u2211k=0nakxkive from  and 52)(63)\u2211k=0np \u2a7d \u221e. Then, Let 1 < This may be obtained in the similar way used in the proof of \u2113p\u03bb(B) : \u2113\u221e), (\u2113p\u03bb(B) : c0), (\u2113p\u03bb(B) : c), (\u2113p\u03bb(B) : \u21131), (\u21131\u03bb(B) : \u2113p), (\u2113\u221e\u03bb(B) : \u2113p), where 1 \u2a7d p \u2a7d \u221e. Also, by means of a given basic lemma, we derive the characterizations of certain other classes. Since the characterization of matrix mapping on the space \u2113p\u03bb(B) can be proved in a similar way, we omit the proof for the cases p = 1 and p = \u221e and consider only the case 1 < p < \u221e in the proofs of theorems given in this section.In this section, we characterize the matrix classes (A = (ank), we write for brevity thatFor an infinite matrix mas (see ) which aA = (ank) be an infinite matrix. Then, the following statements hold. (i)A \u2208 (\u21131 : c0) if and only if sup\u2061n,k\u2208\u2115 | ank | <\u221e and(ii)p < \u221e. Then, A \u2208 (\u2113p : c0) if and only if (k\u2208\u2115\u2211n | ank|q < \u221e. Let 1 <  only if  holds an(iii)A \u2208 (\u2113\u221e : c0) if and only if \u2208(\u21131 : \u2113p) if and only if sup\u2061n\u2208\u2115\u2211k | ank|q < \u221e. Let 1 \u2a7d p < \u221e. Then, A = (ank)\u2208(\u2113\u221e : \u2113p) if and only if sup\u2061K\u2208\u2131\u2211n|\u2211k\u2208Kank|p < \u221e.Let 1 < A = (ank) be an infinite matrix. Then, the following statements hold.(i)p < \u221e. Then, A \u2208 (\u2113p\u03bb(B) : \u2113\u221e) if and only ifLet 1 < (ii)A \u2208 (\u21131\u03bb(B) : \u2113\u221e) if and only if ( only if  and 69)A \u2208 (\u21131(iii)A \u2208 (\u2113\u221e\u03bb(B) : \u2113\u221e) if and only if ( only if  and 69)A \u2208 (\u2113\u221eLet x \u2208 \u2113p\u03bb(B), where 1 < p < \u221e. Then, we have by ank)k\u2208\u2115 \u2208 [\u2113p\u03bb(B)]\u03b2 for all n \u2208 \u2115 and this implies the existence of Ax. Also, it is clear that the associated sequence y = (yk) is in the space \u2113p \u2282 c0.(i) Assume that the conditions \u201371) hol hol71) hmth partial sum of the series \u2211kankxk as follows:m \u2192 \u221e that\u2113p : \u2113\u221e) by n in (Ax \u2208 \u2113\u221e and hence A \u2208 (\u2113p\u03bb(B) : \u2113\u221e).Let us now consider the following equality derived by using relation  from mthby using \u201370), we, wemth pfollows:\u2211k=0mankx\u2192 \u221e that\u2211kankxk=\u2211A = (ank)\u2208(\u2113p\u03bb(B) : \u2113\u221e), where 1 < p < \u221e. Then, since (ank)k\u2208\u2115 \u2208 [\u2113p\u03bb(B)]\u03b2 for all n \u2208 \u2115 by the hypothesis, the necessity of (ank)k\u2208\u2115 \u2208 [\u2113p\u03bb(B)]\u03b2, (x \u2208 \u2113p\u03bb(B) and y \u2208 \u2113p which are connected by relation (fn on \u2113p\u03bb(B) by\u2113p\u03bb(B) and \u2113p are norm isomorphic, it should follow with  are pointwise bounded. Thus, we deduce by Banach-Steinhaus theorem that these functionals are uniformly bounded, which yields that there exists a constant K > 0 such that ||fn|| \u2a7d K for all n \u2208 \u2115. This shows the necessity of the condition ]\u03b2,  holds forelation . Let us low with  that(79ondition  which coA = (ank) be an infinite matrix. Then, the following statements hold. (i)A \u2208 (\u21131\u03bb(B) : c) if and only if ( only if  and 69)A \u2208 (\u21131(ii)p < \u221e. Then, A \u2208 (\u2113p\u03bb(B) : c) if and only if (Let 1 <  only if \u201371) hol holp < \u221e only if  also hol(iii)A \u2208 (\u2113\u221e\u03bb(B) : c) if and only if , a, aA \u2208 , where 1 < p < \u221e. Then, Ax exists and by using (k \u2208 \u2115 that n \u2192 \u221e which leads us with (k \u2208 \u2115. This shows that (\u03b1k) \u2208 \u2113q. Since x \u2208 \u2113p\u03bb(B), we have y \u2208 \u2113p. Therefore, we derive by applying H\u00f6lder's inequality that (\u03b1kyk) \u2208 \u21131 for each y \u2208 \u2113p.We consider only part (ii). Assume that nditions \u201371) and andA satnditions , and x \u2208by using , we have us with  to the f\u03f5 > 0, choose a fixed k0 \u2208 \u2115 such thatm0 \u2208 \u2115 such thatm\u2a7em0. Thus, by using n \u2192 \u2211k\u03b1kyk as n \u2192 \u221e which means that Ax \u2208 c; that is, A = (ank)\u2208(\u2113p\u03bb(B) : c).Now, for any given  follows  that theby using , we get A \u2208 (\u2113p\u03bb(B) : c), where 1 < p < \u221e. Then, since c \u2282 \u2113\u221e, A \u2208 (\u2113p\u03bb(B) : \u2113\u221e). Thus, the necessity of (x \u2208 \u2113p\u03bb(B). Since Ax \u2208 c by our assumption, we derive by  is  is A \u2208 n = xn+1 for all n \u2208 \u2115. A Banach limit L is defined on \u2113\u221e such that L(x)\u2a7e0 for x = (xk), where xk\u2a7e0 for all k \u2208 \u2115, L(Px) = L(x), L(e) = 1, where e = . A sequence x = (xk) \u2208 \u2113\u221e is said to be almost convergent to the generalized limit \u03b1 if all Banach limits of x is \u03b1 \u03b2 for all n \u2208 \u2115 and this implies that Ax exists. Besides, it follows by combining (Ax \u2208 \u21131 and so we have A \u2208 (\u2113p\u03bb(B) : \u21131).Suppose that nditions \u201371),  thaive from  holds whA \u2208 (\u2113p\u03bb(B) : \u21131), where 1 < p < \u221e. Since \u21131 \u2282 \u2113\u221e, A \u2208 (\u2113p\u03bb(B) : \u2113\u221e). Thus, Ax \u2208 \u21131 by the hypothesis, we deduce by \u2208(\u21131\u03bb(B) : \u2113p) if and only if (Let 1 \u2a7d  only if  and 69)p < \u221e. Thx \u2208 \u21131\u03bb(B). Then, y \u2208 \u21131. We have by ank)k\u2208\u2115 \u2208 [\u21131\u03bb(B)]\u03b2 for each n \u2208 \u2115 and this implies that Ax exists. Furthermore, by  : \u2113p).Suppose that the conditions , 69), a, a69), amore, by , one cane; since  and (69)e; since  as m \u2192 \u221erelation  holds. Trelation  and 95)x \u2208 \u21131\u03bbA \u2208 (\u21131\u03bb(B) : \u2113p), where 1 \u2a7d p < \u221e. Since \u2113p \u2282 \u2113\u221e, then A \u2208 (\u21131\u03bb(B) : \u2113\u221e). Thus, Ax \u2208 \u2113p by our assumption, we deduce by \u2208(\u2113\u221e\u03bb(B) : \u2113p) if and only if (Let 1 <  only if  and 69)p < \u221e. Th\u03bb and \u03bc be any two sequence spaces, let A be an infinite matrix and B a triangle matrix. Then, A \u2208 (\u03bb : \u03bcB) if and only if BA \u2208 (\u03bb : \u03bc). Let It is trivial that A = (ank) be an infinite matrix and u = (un) and v = (vn) be sequences of non-zero numbers, and define the matrix C = (cnk) by cnk = un\u2211j=0nvjajk for all n, k \u2208 \u2115. Then, the necessary and sufficient conditions in order A belongs to any of the classes (\u2113p\u03bb(B) : \u2113\u221e), (\u2113p\u03bb(B) : \u2113p), and (\u2113p\u03bb(B) : c) are obtained from respective ones in Theorems A by those of the matrix C. The spaces \u2113\u221e, \u2113p, and c are defined in  characterizes the uniformly convex spaces. In [Gurarii's modulus of convexity is defined by\u03b5 \u2a7d 2. It is easily shown that \u03b4X(\u03b5) \u2a7d \u03b2X(\u03b5) \u2a7d 2\u03b4X(\u03b5) for any 0 \u2a7d \u03b5 \u2a7d 2. Further, if 0 < \u03b2X(\u03b5) < 1, then X is uniformly convex, and if \u03b2X(\u03b5) < 1, then X is strictly convex.In the present section, we investigate some geometric properties of the space ity see , 25) def def\u2113p\u03bbaces. In , GurariiX is said to have the Banach-Saks property if every bounded sequence (xn) in X admits a sequence (zn) such that the sequence {tk(z)} is convergent in the norm in X [A Banach space orm in X , where(X is said to have the weak Banach-Saks property whenever given any weakly null sequence (xn) in X and there exists a subsequence (zn) of (xn) such that the sequence {tk(z)} is strongly convergent to zero.A Banach space In , Garc\u00eda-X with R(X) < 2 has a weak fixed point property. A Banach space \u2113p\u03bb(B) has Banach-Saks type p. The space \u03b5n) be a sequence of positive numbers for which \u2211n=1\u221e\u03b5n \u2a7d 1/2. Let (xn) be a weakly null sequence in B(\u2113p\u03bb(B)). Set u0 = x0 and u1 = xn1 = x1. Then, there exists t1 \u2208 \u2115 such thatxn) is a weakly null sequence\u201d implies that xn \u2192 0 with respect to the coordinatewise, there exists n2 \u2208 \u2115 such thatn\u2a7en2. Set u2 = xn2. Then, there exists t2 > t1 such thatxn \u2192 0 with respect to the coordinatewise, there exists n3 > n2 such thatn\u2a7en3. If we continue this process, we can find two increasing sequences (ti) and (ni) of natural numbers such thatn\u2a7enj+1 anduj = xnj. Hence,x||\u2113p\u03bb(B) < 1. Thus, ||x||\u2113p\u03bb(B)p < 1, and we haven + 1)p1/ for all n \u2208 \u2115 and 1 \u2a7d p < \u221e, we have\u2113p\u03bb(B) has Banach-Saks type p. Let (R(\u2113p\u03bb(B)) = R(\u2113p) = 2p1/, since \u2113p\u03bb(B) is linearly isomorphic to \u2113p. Note that  Thus, by Remarks p < \u221e. Then, the sequence space \u2113p\u03bb(B) has the weak fixed point property. Let 1 < \u2113p\u03bb(B) is\u03b5 \u2a7d 2. Gurarii's modulus of convexity for the normed space x \u2208 \u2113p\u03bb(B). Then, we have\u03b5 \u2a7d 2 and consider the following sequences:z = (zn) and t = (tn), we obtain the following equalities:\u03b1 \u2a7d 1Let p < \u221e, we haveTherefore, for 1 \u2a7d \u03b5 > 2, \u03b2\u2113p\u03bb(B)(\u03b5) = 1. Thus, \u2113p\u03bb(B) is strictly convex. For \u03b5 \u2a7d 2, \u03b2\u2113p\u03bb(B)(\u03b5) \u2a7d 1. Thus, \u2113p\u03bb(B) is uniformly convex. For 0 < The following statements hold. \u03b1 = 1/2, \u03b2\u2113p\u03bb(B)(\u03b5) = \u03b4\u2113p\u03bb(B)(\u03b5). For Er of order r, the method Ar, and the generalized difference matrix B in the sequence spaces \u2113p and \u2113\u221e investigated by Altay et al. [B, our corresponding results are much more general than the results given by Kiri\u015f\u00e7i and Ba\u015far [\u2113p\u03bb(B). It is obvious that the matrix Er, Ar, or B. So, the present results are new. As a natural continuation of this paper, one can study the domain of the matrix \u2113\u221e(p), c(p), c0(p), and \u2113(p).The domain of Euler means y et al. , Ayd\u0131n ay et al. , and Kiry et al. , respectnd Ba\u015far . Additio"}
{"text": "I, x(t) \u2208 S, \u2200t \u2208 I, x(0) = x0 \u2208 S, (\u2217), where S is a closed subset in a Banach space \ud835\udd4f, I = , (T > 0), F : I \u00d7 S \u2192 \ud835\udd4f, is an upper semicontinuous set-valued mapping with convex values satisfying F \u2282 c(t)(||x|| + ||x||p)\ud835\udca6, \u2200 \u2208 I \u00d7 S, where p \u2208 \u211d, with p \u2260 1, and c \u2208 C. The existence of solutions for nonconvex sweeping processes with perturbations with nonlinear growth is also proved in separable Hilbert spaces.In the Banach space setting, the existence of viable solutions for differential inclusions with nonlinear growth; that is,  G is included in the subdifferential of convex lower semicontinuous (l.s.c) function g : \u211dn \u2192 \u211d. This result has been extended in many ways by many authors  function g : \u211dn \u2192 \u211d, S is closed subset in \u211dn, and F :  \u00d7 \u211dn is a continuous set-valued mapping. The infinite dimensional case of (S at C(t) = S + v(t), where S is a fixed nonconvex closed set and v is a mapping with finite variation. Another important study of the inclusion (C(t), has been realized by the author in  \u00d7 \u210d \u2192 \u210d is an upper semi-continuous set-valued mapping with closed convex values satisfying F \u2282 c(t)(||x|| + ||x||p)\ud835\udca6, where p \u2208 \u211d, with p \u2260 1, and c \u2208 C and \ud835\udca6 is a convex compact set. The paper is organized as follows. After recalling the needed concepts in C in separable Hilbert spaces.To the best of our knowledge no existing works studied the existence of solutions for differential inclusions with nonlinear growth. The main purpose of the paper is to prove the existence of solutions for (\u2217) in Banach spaces and for  in separThis section is devoted to recall some notations and concepts needed in the paper.\ud835\udd4f be a Banach space, let S \u2282 \ud835\udd4f be a nonempty closed subset of \ud835\udd4f, and let dS(x) = inf\u2061{||x \u2212 s|| : s \u2208 S} is the usual distance function associated with S.Let r-prox-regular sets in Hilbert spaces as the class of all closed sets S satisfying the following definition. Many equivalent definitions of this class have been used for different applications; see, for example,  and p \u2265 0 (with p \u2260 1) is a constant. Thent \u2208  we gett \u2208 , and hence the proof is finished.Multiplying  by v\u2212p) yield. Let \u03b3(t) = \u222bath(s)ds, for all t \u2208  and let b1 \u2208  satisfying the inequalityt \u2208 J1 : = . Thent \u2208 .Let tisfying  with p \u2208F be a set-valued mapping satisfying the following nonlinear growth:I = , (T > 0), D is a closed nonempty set in \ud835\udd4f, c \u2208 C, and p \u2208 \u211d with p \u2260 1. Clearly, when p = 0, this assumption coincides with the well known linear growth; that is,x0 \u2208 D. To ensure the viability of the solution on the set D, we need the following classical tangential condition:K is the Bouligand tangent cone to D at x.Let  growth:F\u2282c and I1 = , and G : I \u00d7 \ud835\udd4f\u21c9\ud835\udd4f satisfying the following:G is u.s.c. with closed convex values;G \u2282 c(t)\ud835\udca6 on J \u00d7 D, for some \ud835\udca6 convex compact set in \ud835\udd4f and c \u2208 C;G\u2229K \u2260 \u2205 on I \u00d7 D.Then for every x0 \u2208 D, there exists an absolutely continuous mapping x : I \u2192 D such thatLet D be a closed subset in \ud835\udd4f and let F : D\u21c9\ud835\udd4f be an upper semicontinuous set-valued mapping with closed convex values and let r1, r2 > 0 be such that r1 < r2, and let \u03c8 :  be a continuous function such that \u03c8(s) = 1 for s \u2264 r1 and \u03c8(s) = 0 for s \u2265 r2. Let G be a set-valued mapping defined on D as follows:F satisfies the nonlinear growth on I \u00d7 D; that is, F \u2282 c(t)(||x|| + ||x||p)\ud835\udca6 on I \u00d7 D, for some c \u2208 C, p \u2208 \u211d with p \u2260 1, and \ud835\udca6 is a convex compact set in \ud835\udd4f, then G is upper semicontinuous on I \u00d7 D with closed convex values.Let G has closed convex values. Let \ud835\udca60 : = (r1 + r1p)\ud835\udca6 \u222a {0}. For any t \u2208 I and any x \u2208 D with ||x|| < r2, we have by the convexity of \ud835\udca6 the following:t \u2208 I and any x \u2208 D with x \u2209 r2\ud835\udd39, we have G is closed. To do that, we fix , yn) \u2208 gph\u2009G with G we havexs(n))n of (xn)n such that \u03c8(||xs(n)||) \u2192 0. In this case we have for n large enough ||xs(n)|| \u2264 r2 and so zs(n) is bounded which ensures that ys(n) \u2192 0 and so \u03c8 and the convergence of xn to \u03b1 > 0 and n0 \u2208 \u2115 such that \u03c8(||xn||) > \u03b1 > 0 for all n \u2265 n0. Then by continuity of \u03c8 we have F, we get G and hence the proof is achieved.Clearly, Now, we are ready to prove our main existence result under the nonlinear growth condition in Banach spaces.\ud835\udd4f be a Banach space, D \u2282 \ud835\udd4f a nonempty closed set, and F : \ud835\udd4f\u21c9\ud835\udd4f satisfying the following: F is u.s.c. on I \u00d7 D with F being closed convex for all  \u2208 I \u00d7 D;F \u2282 c(t)(||x|| + ||x||p)\ud835\udca6 on I \u00d7 D, for some c \u2208 C, and p \u2208 \u211d with p \u2260 1, and for some convex compact set \ud835\udca6 in \ud835\udd4f;F\u2229K \u2260 \u2205 on I \u00d7 D.Then for every x0 \u2208 D, there exists an absolutely continuous mapping x : I1 \u2192 D such thatI1 =  when p \u2208  and I1 =  be a continuous function such that \u03c8(s) = 1 for s \u2264 M and \u03c8(s) = 0 for s \u2265 r. Define now the set-valued mapping G on I \u00d7 D as follows:G inherits the convexity and the upper semicontinuity of the set-valued mapping F with G \u2282 c(t)\ud835\udca60, where \ud835\udca60 : = r\ud835\udca6 \u222a {0}. We have to check that G satisfies the tangential condition on D. Let t \u2208 I and let x \u2208 D\u2229M\ud835\udd39. Then G = F and so the tangential condition is satisfied from (c). Assume now that t \u2208 I and x \u2208 D with ||x|| > M. Then by (c) there exists some z \u2208 F such that z \u2208 K. Let y = \u03c8(||x||)z. Clearly, y \u2208 G and y \u2208 \u03c8(||x||)K \u2282 K, since K is a cone. So, G\u2229K \u2260 \u2205. Therefore, the tangential condition is satisfied for G on all I \u00d7 D. Consequently, all the assumptions (a), (b), and (c) in \ud835\udca60 instead of \ud835\udca6 are satisfied and hence for every x0 \u2208 D there exists a solution x on I of  = x0, and x(t) \u2208 D on I. Let us prove that x is the desired solution for (p \u2260 1. Using x(t)|| \u2264 M a.e. on I1 and so by the definition of G and \u03c8 we get G) = \u03c8(||x(t)||)F) = F) that is, x is a solution of , and let C : I\u21c9\u210d be a set-valued mapping satisfying the following Lipschitz condition for any y \u2208 \u210d and any t, t\u2032 \u2208 I:Our purpose, in this section, is to use the techniques developed previously to extend some existing results, in separable Hilbert spaces, of nonconvex sweeping processes with perturbations from the case of perturbation with linear growth to the case of perturbation with nonlinear growth. For this end let m 4.1 in .\u210d be a separable Hilbert space and let r \u2208  is r-prox-regular for every t \u2208 I and that the assumption  \u2282 \ud835\udca61 \u2282 \u03b1\ud835\udd39 for all  \u2208 I \u00d7 \u210d, for some compact set \ud835\udca61 in \u210d. Then, for any x0 \u2208 C(0), the sweeping process (SPP) with the perturbation F has at least one Lipschitz continuous solution; that is, there exists an Lipschitz continuous mapping x : I \u2192 \u210d such thatI.Let sumption  holds. LUsing the techniques from the previous section and r \u2208  is r-prox-regular for every t \u2208 I and that the assumption , a convex compact set \ud835\udca6, and k > 0 such thatL, k, p, and T are satisfied:u0 \u2208 C(0), there exists a Lipschitz continuous mapping u : I \u2192 \u210d satisfying the following sweeping process with a perturbation:Let sumption  holds. L\ud835\udca6. By our assumptions on the constants L, T we have after simple computations\u03b2 > 0 such that\u03b1 = \u03b2p\u22121)1/\u2192 to be a continuous function such that \u03c8(s) = 1 for s \u2264 \u03b1/2 and \u03c8(s) = 0 for s \u2265 \u03b1 and define the set-valued mapping G on \u210d as follows:G inherits the convexity of the values from F. Also, for any x \u2208 \u03b1\ud835\udd39, we havex \u2209 \u03b1\ud835\udd39, we have G \u2282 \ud835\udca60, for any  \u2208 I \u00d7 \u210d with G follows from the u.s.c. of F and C and G we get a Lipschitz continuous mapping x : I \u2192 \u210d such thatx is a solution of (SPP) with F. Clearly, we have\u03b1 and the assumptions on the constants L, k to deduce from (x(t)|| \u2264 \u03b1/2 which yields that \u03c8(||x(t)||) = 1 and so G) = F). This means that x is a solution of . Then\u03b1>4(LT+||ution of  and henc"}
{"text": "Under the resonant condition, we study the existence of 1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two 1-periodic orbits; the coexistence of 1-homoclinic loop and 1-periodic orbit. Moreover, we give the corresponding existence fields and bifurcation surfaces. At last, we study the stability of the homoclinic loop for the two cases of non-resonant and resonant, and we obtain the corresponding criterions.By using the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local current coordinates system of the system in the small neighborhood of the homoclinic orbit, we discuss the bifurcation problems of nondegenerated homoclinic loops. Under the nonresonant condition, existence, uniqueness, and incoexistence of 1-homoclinic loop and 1-periodic orbit, the inexistence of  With the rapid development of nonlinear science, in the studies of many fields of research and application of medicine, life sciences and many other disciplines, there are a lot of variety high-dimensional nonlinear dynamical systems with complex dynamic behaviors. Homoclinic and heteroclinic orbits and the corresponding bifurcation phenomenons are the most important sources of complex dynamic behaviors, which occupy a very important position in the research of high-dimensional nonlinear systems. We know that in the study of high-dimensional dynamical systems of infectious diseases and population ecology we tend to ignore the stability switches and chaos when considered much more the nonlinear incidence rate, population momentum, strong nonlinear incidence rate, and so forth. The existence of transversal homoclinic orbits implies that chaos phenomenon occur; therefore, it is of very important significance to study the cross-sectional of homoclinic orbits and the preservation of homoclinic orbits for the system in small perturbation.In addition, in the study of infectious diseases and population ecology systems, we sometimes require the existence of periodic orbits. And, homoclinic and heteroclinic orbits bifurcate to periodic orbits in a small perturbation means that we can get the required periodic solution only by adding a small perturbation when using the similar system which exists homoclinic or heteroclinic orbits to represent the natural system. This also explains the importance of homoclinic and heteroclinic orbits bifurcating periodic orbits in real-world applications.Therefore, by using the research methods and theoretical results of qualitative and bifurcation problems of high-dimensional systems, especially the results of homoclinic and heteroclinic orbits and their bifurcations for the systems, to study the high-dimensional infectious disease dynamics and population ecology systems to reveal the complex dynamical behavior of the nonlinear dynamical systems and the corresponding reality systems is essential.About the study of bifurcation problems of homoclinic and heteroclinic loops for two-dimensional systems, a large number of papers were obtained and achieved many good results (for some results see \u20136); but red with , paper .Using the expressions of z = h(t) into (Substitute (t) into , and uset), and using \u03a6*(t)Z(t) = I, we have Multiplying both sides by \u03a6* \u21a6 N(T) which has the following formMi = \u222b\u221e\u2212\u221e+\u03d5i*(t)gu(r(t), 0)dt, i = 1,3, 4, is said to the Melnikov vectors [Thus, from the flow of , we defi vectors , 14, 15.F0 : S0 \u21a6 S1, q0 \u21a6 q1 defined by the orbit of ) > \u03c10 > \u03bb1(\u03bc).Now, we consider the map orbit of . For con\u03c4 be the flying time from q0 to q1, and s = e\u03bb1(\u03bc)\u03c4\u2212 be called Silnikov time. By  is called Silnikov coordinate.(F1(q1) = q2, by , a, a12), aG =  = F(q0)\u2009\u2212\u2009q0 = * \u2212 * as follows:So, from the above and , we haves \u2265 0, there is a one to one correspondence between the 1-homoclinic loop and 1-periodic orbit of  of the following equation:Thus, for orbit of  and the Equation  is calle(H4)\u03bb1 < \u03c11. (Nonresonant condition) \u03bc small enough, we may assume \u03c11(\u03bc) > \u03bb1(\u03bc).Obviously, for \u03bc| is small enough, the system (Suppose that hypotheses (H1)~(H4) are valid, then if |e system  exists ngij) are all zero except g31 = w14\u03b4. Therefore, Q, \u03bc) = , (s(0) = 0, u1(0) = 0, v0(0) = 0.Consider the solution of . Let G~Q. From , we have= ,  exists as = 0, then the solution -dimensional surface L \u2282 Rl in the small neighbourhood of \u03bc = 0, such that when \u03bc \u2208 L and |\u03bc | \u226a1, the system , v0 = v0. Substituting into G1 = 0, we getFrom , for s \u2265M1 \u2260 0, then, according to the implicit function theorem, in the neighbourhood of \u03bc = 0, the equation M1\u03bc + h.o.t. = 0 defines a unique (l \u2212 1)-dimensional surface L \u2282 Rl, such that if \u03bc \u2208 L and |\u03bc | \u226a1, \u03c41\u2212. Suppose F0(q2) = q3, F1(q3) = q4 = q0, and let \u03c42 be the time from q2 to q3, s2 = e\u03bb1(\u03bc)\u03c42\u2212.We rewrite the time from G2 =  as follows:Similar to the previous discussion, we can get its associated successor function Q2 = , gij) are all zero except g41 = w14\u03b4, g62 = w14\u03b4.Denote Q2, \u03bc) = , the bifurcation equations1(0) = 0, u1(0) = 0, v0(0) = 0, s2(0) = 0, and u3(0) = 0, v2(0) = 0.Hence, s1 = s2 = 0, then the homoclinic loop of the system ~(H4) are fulfilled, e system  does not\u03bb1 = \u03c11. For convenience, we assume the resonant condition has the following form.(H5)\u03bb1 = \u03c11 = \u03bb, \u03bb1(\u03bc) = \u03bb, \u03c11(\u03bc) = \u03bb + \u03b1(\u03bc)\u03bb, where \u03b1(\u03bc) \u2208 R1, |\u03b1(\u03bc)|\u226a1, and \u03b1(0) = 0. (Resonant condition) We say that the homoclinic loop is Resonance if s \u2265 0, there is a one to one correspondence between the 1-homoclinic loop and 1-periodic orbit of  of the bifurcation equation , v0 = v0. Substituting it into G1 = 0, we getAt first, we discuss the bifurcations of 1-homoclinic loop and 1-periodic orbit. Now, the bifurcation equation has the following form:G1=\u03b4(w12G1=\u03b4(w12N(s) = s\u03b1(\u03bc)1+, V(s) = w12(s \u2212 \u03b4\u22121M1\u03bc) + h.o.t., we have the following conclusion.Denote s0, |\u03bc | \u226a1, the necessary condition of N(s) and V(s) are tangent at s0 is that \u03b1(\u03bc)M1\u03bc > 0. Meanwhile, if \u0394 = 1 and 0 < s0 \u226a 1, N(s), and V(s) are tangent at s0 if and only if \u03b1(\u03bc)(1 \u2212 w12) > 0, andSuppose (H1)~(H3) and (H5) are fulfilled, then, for 0 < N(s) and V(s) are tangent at s = s0 if and only if N(s0) = V(s0), s0 = ((1 + \u03b1(\u03bc))/\u03b1(\u03bc))\u03b4\u22121M1\u03bc + h.o.t.. Substituting it into  is iss0 = The proof is complete.1 is the surface defined by .Suppose \u03a3fined by , \u03a3\u00af2(s)fined by , \u03a32=\u03a3\u00af2 \u03bc \u2208 \u03a32,  is turne\u03b1(\u03bc) > 0, 0 < w12 < 1. If M1 \u2260 0, then, in the neighborhood of \u03bc = 0, there exists two (l \u2212 1)-dimensional surfaces \u03a31 and \u03a32, such that, for sufficiently small |\u03bc|, we have the following.\u03bc \u2208 \u03a31. The system  has a un\u03bc satisfies M1\u03bc > \u03b21(\u03bc). The system  has no 1\u03bc satisfies \u03b22(\u03bc) < M1\u03bc < \u03b21(\u03bc). The system  has exac\u03bc \u2208 \u03a32. The system  has exac\u03bc satisfies M1\u03bc < \u03b22(\u03bc). The system  has exacSuppose (H1)~(H3) and (H5) are fulfilled, s = |\u03bc|2, we can treat s as a small parameter in -dimensional surface s1 = 0 and s2 = (w12)\u03b1(\u03bc)1/ + h.o.t. > 0.Through taking a proper scale transformation, such as meter in . Accordir \u03bc = 0,  can defi\u03a3\u00af2(0),  has exacM1\u03bc < \u03b22(\u03bc), then (s1 > 0.If \u22121 \u226a \u03bc), then  has exacM1\u03bc = \u03b21(\u03bc), that is \u03bc \u2208 \u03a31, then , then (If \u03bc), then  has no n\u03b22(\u03bc) < M1\u03bc < \u03b21(\u03bc), then , then  has exacThe proof is complete.\u03b21(\u03bc) and \u03b22(\u03bc), and the corresponding \u03a31 and \u03a32, to obtain the following theorem.Similarly, we can define the corresponding \u03b1(\u03bc) < 0, w12 > 1. If M1 \u2260 0, then, in the neighborhood of \u03bc = 0, there exists two (l \u2212 1)-dimensional surfaces \u03a31 and \u03a32 such that for sufficiently small |\u03bc|, the following conclusions hold.\u03bc \u2208 \u03a31. The system  has a un\u03bc satisfies M1\u03bc < \u03b21(\u03bc). The system  has no 1\u03bc satisfies \u03b22(\u03bc) > M1\u03bc > \u03b21(\u03bc). The system  has exac\u03bc \u2208 \u03a32. The system  has exac\u03bc satisfies M1\u03bc > \u03b22(\u03bc). The system  has exacSuppose (H1)~(H3) and (H5) are fulfilled, 1 is called 2-fold 1-periodic orbit bifurcation surface, \u03a32 is called 1-homoclinic loop bifurcation surface \u03a3k-homoclinic loop and k-periodic orbit, where k > 1. We may assume that k = 2.Now, we consider the nonexistence of q0 and q1 as \u03c41, s1 = e\u03bb\u03c41\u2212. Suppose F0(q2) = q3, F1(q3) = q4 = q0, let \u03c42 be the time from q2 to q3, s2 = e\u03bb\u03c42\u2212. Similar to the previous discussion, we can get its associated successor function G2 =  as follows:We rewrite the time from Q2 = . The equation  = 0 always has a unique solution u1 = u1,\u2009\u2009v0 = v0,\u2009\u2009u3 = u3,\u2009\u2009v2 = v2, substituting it into  = 0, we gets1 = s2 = 0, then (Denote  we get\u03b4[(w12)\u2212 we get\u03b4[(w12)\u2212M1 \u2260 0, the above formula defines a (l \u2212 1)-dimensional surface L, now the 2-homoclinic loop is the 1-homoclinic loop.If s1 > 0, s2 = 0, then -dimensional surface L1, such that when |\u03bc | \u226a1 and \u03bc \u2208 L1, the system  and V(s2) be the left and right of the above formula, respectively, then N(s2) and V(s2) are tangent at some point if and only if \u03b1(\u03bc) > 0, that is, \u0393 is twisted. At present, s2 satisfies 0 < s2 < \u03b4\u22121M1\u03bc \u226a 1. Substituting  \u03b4\u22121M1\u03bc = \u03b2*(\u03bc). We notice that every 1-periodic orbit also corresponds to a solution s1 = s2 > 0 of ~H3) and (H5) are fulfilled, if \u0394 = 1,  and (H5)Now, we consider the stability of homoclinic loop \u0393.F(q0) = F* = q2 = *, if \u03bc = 0, and \u03c11 \u2260 \u03bb1, we can get the following by (According to owing by (40)n2s \u2248 \u03b4\u22121x0, u1 \u2248 sB1(0)/\u03bb1\u2212u0, substituting into , x2 \u2248 n2,1, u2 = w33n2,3, v2 = O. And by Re\u03c3(\u2212B1(0)) < 0, Re\u03c3B2(0) > 0, s \u226a 1, we can get u2 = O(sB1(0)/\u03bb1\u2212u0) \u226b u0, v2 = O(sB2(0)/\u03bb1v0) \u226a v0. Meanwhile, we make Poincar\u00e9 map F restrict at the half transversal section S0+ = { \u2208 S0, 0 \u2264 x < \u03b41 < \u03b4}, F makes the transversal line Lx = {0 \u2264 x < \u03b41, y = \u03b4, u = 0, v = 0} maps to the segment Lx\u2032 = {0 \u2264 x < \u03b3\u03b41, y = \u03b4, u = 0, v = 0} approximately, where \u03b3 is called the shrinkage (expansion) rate. So, when \u03c11/\u03bb1 > 1(<1), we can get \u03b3 = w12\u22121(\u03b4\u22121x0)\u03c11/\u03bb1\u22121 < 1(>1) for \u03b4 \u226a 1, x0 \u226a 1. Hence, we have the following.According to , we know\u03c11/\u03bb1 > 1, homoclinic loop \u0393 is weak stability, and \u0393 has a (m + 1)-dimensional stable manifold and a n-dimensional instable manifold, If \u03c11/\u03bb1 < 1, homoclinic loop \u0393 is weak instability, and \u0393 has a m-dimensional stable manifold and a (n + 1)-dimensional instable manifold.If \u03bc = 0 and \u03c11 = \u03bb1, similar to above, we can get \u03b3 = w12\u22121 not difficulty. Thus, we have the following.If w12\u22121 < 1, homoclinic loop \u0393 is weak stable, and \u0393 has a (m + 1)-dimensional stable manifold and a n-dimensional instable manifold. If w12\u22121 > 1, homoclinic loop \u0393 is weak instable, and \u0393 has a m-dimensional stable manifold and a (n + 1)-dimensional instable manifold.If Besides, by the above discussion, we can get the following.The homoclinic loop or the periodic orbit of the perturbed system have the same stability with the homoclinic loop of the unperturbed system.Cr system  \u2208 R2, r \u2265 5. We consider the stability of the homoclinic orbit \u0393 under the resonance case. Denote f(z) = (f1(z), f2(z))*, \u03c3 = exp\u2061{\u222b\u221e\u2212\u221e+(\u2202f1/\u2202x + \u2202f2/\u2202y)(r(t))dt}, Now, suppose the  system  is 2-dim\u03c3 is convergent, and Cr transformation coordinates, such that the system , H2 is Cr\u22123. So, in U, we have \u0393\u2229W0s = { : x = 0}, \u0393\u2229W0u = { : y = 0}. Thus, in U, if  \u2208 \u0393, (\u2202f1/\u2202x) + (\u2202f2/\u2202y) = xyH = 0.In fact, from \u20133, thereThe proof is complete.C2 transformation (refer [f(\u00b7) and r(t) of the divergence integration \u03c3 can be thought of the original forms of , so, soC2 tforms of .w21 = \u22121, w12 = 1/\u03c3. w21 = \u22121. And by the Liouville formula, we have w12w21\u03c3 = 1, and by w21 = \u22121, we get w12 = 1/\u03c3.According to The proof is complete.Combined with the \u03c3 < 1, the homoclinic orbit \u0393 is stable; If \u03c3 > 1, the homoclinic orbit \u0393 is instable.If"}
{"text": "A controlled model for a financial system through washout-filter-aided dynamical feedback control laws is developed, the problem of anticontrol of Hopf bifurcation from the steady state is studied, and the existence, stability, and direction of bifurcated periodic solutions are discussed in detail. The obtained results show that the delay on price index has great influences on the financial system, which can be applied to suppress or avoid the chaos phenomenon appearing in the financial system. For the last two decades, there have been growing interests in studying the complex dynamics of financial systems in both micro- and macroeconomics , 2. It iOn the other hand, delays are ubiquitous in life, so it is in the social and economic activities. There are at least two ways that time delays emerge in the dynamics of economic variables. One is the time lag between the time economic decisions are made and the time the decisions bear fruit . The othThe aim of this paper is to investigate the dynamics of a financial system by considering the effect of washout filters with time delay. By analyzing the characteristic equation of linearization of the system, we theoretically prove that the Hopf bifurcations occur in the model with delay. Furthermore, by using the theory of functional differential equation and Hassard's method , we alsoThe rest of the paper is organized as follows. In x, the investment demand y, and the price index z. The model is represented by three-dimensional ODEs:a > 0 is the saving amount, b > 0 is the cost per investment, and c > 0 is the elasticity of demand of commercial market. This model is well studied in \u03c4=\u03c4ij\u22121 and h\u2032(\u03b6i) have the same sign. Thus, from Lemmas Without loss of generality, suppose that roots of . Then, \u03c9 root of . From  into  and taki we have[d\u03bbd\u03c4]\u22121=i = 1,2,  are posiac \u2212 (c/b) > 0 and c + a \u2212 (1/b) > 0; then, one has the following: r\u2a7e0 and \u0394 = p2 \u2212 3q \u2a7d 0, then the equilibrium P0 is stable for all \u03c4\u2a7e0;if r < 0 (or r\u2a7e0 and \u0394 = p2 \u2212 3q > 0) and h\u2032(\u03b6i) \u2260 0, then, when \u03c4 \u2208 . For \u03c8 \u2208 \u21021*), define\u03b7(\u03b8) = \u03b7; let A = A(0); then, A and A* are adjoint operators. By the discussion in i\u03c9i\u03c4i are eigenvalues of A. Thus, they are also eigenvalues of A*. We first need to compute the eigenvector of A and A* corresponding to i\u03c9i\u03c4i and \u2212i\u03c9i\u03c4i, respectively.Obviously, e system , where uq(\u03b8) = Tei\u03c9i\u03c4i\u03b8 is the eigenvector of A corresponding to i\u03c9i\u03c4i. Then, Aq(\u03b8) = i\u03c9i\u03c4iq(\u03b8). It follows from the definition of A, L\u03bc\u03d5, and \u03b7 that\u03b1 = 2x*/(i\u03c9i + b), \u03b2 = i\u03c9i + a \u2212 y* \u2212 \u03b1x*, \u03b3 = 1 + (i\u03c9i + c)\u03b2, and q(0) = T.Suppose that q*(s) = Dei\u03c9i\u03c4is be the eigenvector of A* corresponding to \u2212i\u03c9i\u03c4i. By the definition of A*, we can compute \u03b1* = \u2212x*/(i\u03c9i \u2212 b), \u03b2* = i\u03c9i \u2212 a + y* + 2x*\u03b1*, and \u03b3* = \u2212(\u03b2*/(i\u03c9i \u2212 d)).Similarly, let q*(s), q(\u03b8)\u232a = 1, we need to determine the value of D. From (q*(s), q(\u03b8)\u232a = 1, In order to assure that \u2329 D. From , we haveC0 at \u03bc = 0. DefineC0, we havez and C0 in the direction of q and W is real if ut is real. We consider only real solutions. For the solution ut \u2208 C0, since \u03bc = 0, we haveq(\u03b8) = Tei\u03c9i\u03c4i\u03b8; then,In the following, we first compute the coordinates to describe the center manifold . Define\u2009\u2009\u2009\u2009z(t)= we haveW=W, we haveut1(0), ut2(0), ut3(0), and ut4(0) into the above equation and comparing the coefficients with  and W11(\u03b8). From , we haveA, we can getq(\u03b8) = q(0)ei\u03c9i\u03c4i\u03b8, and we haveE1 = (E1(1), E1(2), E1(3), E1(4))T \u2208 \u211d4 is a constant vector. In the same way, we can also obtainE2 = (E2(1), E2(2), E2(3), E2(4))T \u2208 \u211d4 is also a constant vector.In order to assure the value of \u03b8). From  and 43)g21, we ). From (W\u02d9=u\u02d9t\u2212z\u02d9). From (W\u02d9=u\u02d9t\u2212z\u02d9 we have(A\u22122i\u03c9k\u03c4kts with (H20(\u03b8)=\u2212gE1 and E2. From the definition of A and 52), we htituting  and 62)(62)H20 and W11(\u03b8). Furthermore, we can determine each gij. Therefore, each gij is determined by the parameters and delay in , then the Hopf bifurcation is supercritical  and the bifurcation exists for \u03c4 > \u03c4i (<\u03c4i); \u03b22 determines the stability of the bifurcation periodic solutions; the bifurcating periodic solutions are stable (unstable) if \u03b22 < 0 (>0); and T2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if T2 > 0 (<0).Similarly, substituting  and 63)63) into delay in . Thus, wIn this section, we present some numerical results to verify the analytical predictions obtained in the previous section. These numerical simulation results constitute excellent validations of our theoretical analysis; it is shown that the chaotic orbit can be controlled to a periodic orbit by using washout-filter-aided controller with time delay.a = 4, b = 0.1, and c = 1.a = 4, b = 0.4, and c = 1; then, system . From the algorithm of r = (1.5d \u2212 1.5k)2 \u2212 2.25k2; thus, if k > d/2, then r < 0. From d = 1 and k = 1. From the algorithm of \u03c40\u22505.5932. Thus, from P0 is asymptotically stable when \u03c4 < \u03c40, and, as \u03c4 crosses \u03c40, there are periodic orbits bifurcating from P0  .a = 4, b = 0.1, and c = 4; then, from P0 is unstable and P\u00b1 are stable (k = 2, and d = 1, from the algorithm of \u03c4 is \u03c4h = 2.3542. When \u03c4 pass through \u03c4h = 2.3542, a family of periodic orbits will bifurcate from equilibria P\u00b1, respectively (In this subsection, we choose e stable . If we cectively .a = 4, b = 0.1, and c = 1. If we choose k = d = 1, a family of periodic orbits bifurcate from the equilibria of system (\u03c4. This can be verified by From \u03c4 as bifurcating parameter, we discussed the conditions at which periodic orbits bifurcate from the equilibria P0 and P\u00b1, respectively. The stability and direction of bifurcated periodic solutions have been also investigated in detail. And the obtained results can be applied to control the chaos of this financial system.In this paper, we have investigated a financial system with time-delayed washout-filters-aided controller. Taking the time delay k is also applied to influence the dynamical behaviors of this financial system; it will be investigated in the near future.From a financial sense, the obtained results show that the delay on price index has great influence on the financial system, which can be applied to suppress or avoid the chaos phenomenon appearing in the financial system, so as to make the economic system run well. On the other hand, the control gain"}
{"text": "We study the properties of almost periodic solutions for a general discrete system of plankton allelopathy with feedback controls and establish a theorem on the uniformly asymptotic stability of almost periodic solutions. Meanwhile, {ri(n)}, {aij(n)}, {bi(n)}, {ci(n)}, {di(n)}, and {ei(n)} are bounded nonnegative sequences such thati, j = 1,2. Here, we use the notations fu = supn\u2208\u2124+f(n) and\u2009\u2009fl = infn\u2208\u2124+f(n) for any bounded sequence {f(n)}. For the simplicity and convenience of exposition, throughout this paper we let \u2124, \u2124+, and \u211d+ denote the sets of all integers, and nonnegative integers, nonnegative real numbers, respectively.Allelopathy is first used by Hans (see ) in 1937x1(n), x2(n), u1(n), u2(n)} are positive; that is,From the point of view of biology, we only consider {For convenience, we give the following definition of persistence of system .xi*, ui*, xi*, and ui* which are independent of the solutions of system (x1(n), x2(n), u1(n), u2(n)} of system  > 0 such that each discrete interval of length l(\u03b5) contains a \u03c4 = \u03c4(\u03b5) \u2208 E{\u03b5, x} such thatn \u2208 \u2124. \u03c4 is called the \u03b5-translation number of x(n).A sequence f : \u2124 \u00d7 \ud835\udd3b \u2192 \u211dk, where \ud835\udd3b is an open set in \u211dk. f is said to be almost periodic in n uniformly for x \u2208 \ud835\udd3b or uniformly almost periodic for short if, for any \u03b5 > 0 and any compact set S in \ud835\udd3b, there exists a positive integer l such that any interval of length l contains a \u03c4 for whichn \u2208 \u2124 and all x \u2208 S. \u03c4 is called the \u03b5-translation number of f.Let Next, we will introduce the following Lemmas.x(n)} is an almost periodic sequence if and only if for any sequence {hk}\u2282{hk\u2032} such that x(n + hk) converges uniformly on n \u2208 \u2124 as k \u2192 \u221e. Furthermore, the limit sequence is also an almost periodic sequence.{x(n)} satisfies x(n) > 0 andn \u2208 \u2124+, where a(n) and b(n) are nonnegative sequences bounded above and below by positive constants. ThenAssume that {x(n)} satisfiesn\u2192+\u221ex(n) \u2264 x*, bux* > al, x(n0) > 0, where a(n) and b(n) are nonnegative sequences bounded above and below by positive constants and n0 \u2208 \u2124+. ThenAssume that {In this section, we will establish sufficient conditions for the persistence of system .x1(n), x2(n), u1(n), u2(n)} of system (Assume that  and 3)  3) hold;x1(n), x2(n), u1(n), u2(n)} be any positive solution of system xi(p)/\u220fq=0n\u221211 \u2212 di(q)), i = 1,2. Since 0 < d1l < 1, we can find a g \u2208 \u2124+ such that 1 \u2212 d1l = eg\u2212; by Stolz's theorem, we have\u03b5 \u2192 0 and substituting (Let {f system . By assuApplying limsup\u2061n\u2192uch thatxi(n)\u2264xi\u2217i, j = 1,2, i \u2260 j, are satisfied, where ui* and xi* are the same as those in Assume that  and 3);;3); furt\u03b5 > 0 and n > n2, according to n2 \u2208 \u2124+ such that\u03b5 \u2192 0 in the above inequality leads to\u03b5 > 0, there exists a large enough integer n0 such thatvi = \u2211p=0n\u22121(ei(p)xi(p)/\u220fq=0n\u221211 \u2212 di(q)), i = 1,2. As 0 < diu < 1, similar to the analysis in the proof of For leads toliminf\u2061n\u2192Now, we are in a position to state If the inequalities in , 3), an, an3), aBased on f : \u2124 \u00d7 SB \u2192 \u211dk, SB = {x \u2208 \u211dk : ||x|| < B}, and f is almost periodic in n uniformly for x \u2208 SB and is continuous in x. The product system of system =f defined for n \u2208 \u2115, ||x|| < B, ||y|| < B satisfying thata(||x \u2212 y||) \u2264 V \u2264 b(||x \u2212 y||), where a, b \u2208 K, with K = {a \u2208 C : a(0) = 0}, and a is increasing;V \u2212 V| \u2264 L(||x1 \u2212 x2|| + ||y1 \u2212 y2||), where L > 0 is a constant;|V(10)\u2264\u2212aV, where 0 < a < 1 is a constant and \u0394V(10) = V, f) \u2212 V.\u0394Moreover, if there exists a solution \u03c6(n) of system (\u03c6(n)||\u2264B* < B for n \u2208 \u2124+, then there exists a unique uniformly asymptotically stable almost periodic solution p(n) of system  is periodic of period \u03c9, then there exists a unique uniformly asymptotically stable periodic solution of  = {x1(n), x2(n), u1(n), u2(n)} of system (xi* \u2264 xi(n) \u2264 xi*, ui* \u2264 ui(n) \u2264 ui*.According to \u03a9 \u2260 \u00d8.Assume that , 3), an, an3), ai, j = 1,2, i \u2260 j. From X(n) = {x1(n), x2(n), u1(n), u2(n)} of system  \u2192 xi*(n) uniformly in n on any finite subset B of \u2124 as \u03b1 \u2192 \u221e, wherem is a finite number.It is now possible to show by an inductive argument that the system  leads to3xi(n)=xi, ui(k + \u03c4\u03b1)} are uniformly bounded for large enough \u03b1.In fact, for any finite subset a1 \u2208 B, we can choose a subsequence \u03c4\u03b1(1) of {\u03c4\u03b1} such that {xi(a1 + \u03c4\u03b1(1)), ui(a1 + \u03c4\u03b1(1))} uniformly converges on \u2124+ for \u03b1 large enough.Now, for a2 \u2208 B, we can choose a subsequence \u03c4\u03b1(2) of {\u03c4\u03b1} such that {xi(a2 + \u03c4\u03b1(2)), ui(a2 + \u03c4\u03b1(2))} uniformly converges on \u2124+ for \u03b1 large enough.Similarly, for am \u2208 B, we can choose a subsequence {\u03c4\u03b1m\u22121)(} of {\u03c4\u03b1m)(} such that {xi(am + \u03c4\u03b1m), ui(am + \u03c4\u03b1m)} uniformly converges on \u2124+ for \u03b1 large enough.Repeating this procedure, for \u03c4\u03b1m)}, {aij(n)}, {bi(n)}, {ci(n)}, {di(n)}, and {ei(n)} are almost periodic sequence, for the above sequence \u03c4\u03b1, \u03c4\u03b1 \u2192 \u221e as \u03b1 \u2192 \u221e, there exists a subsequence which we still denote it by {\u03c4\u03b1} , such that\u03b1 \u2192 \u221e uniformly on \u2124+.Since {\u03b4 \u2208 \u2124, we can assume that \u03c4\u03b1 + \u03b4 \u2265 n0 for \u03b4 large enough. Letting\u2009\u2009n \u2265 0 and n \u2208 \u2124, by an inductive argument of  = {x1*(n), x2*(n), u1*(n), u2*(n)} is a solution of system (\u2124+. It is clear that 0 \u2264 xi* \u2264 xi(n) \u2264 xi*; 0 \u2264 ui* \u2264 ui(n) \u2264 ui*, n \u2208 \u2124+.By the arbitrariness of \u03a9 \u2260 \u03a6. So Before stating \u03b2 < 1, whereX(n) = {x1(n), x2(n), u1(n), u2(n)} of system  = exp\u2061pi(n); from system  = {p1(n), p2(n), u1(n), u2(n)} satisfyingX, U) \u2208 \u211d2+2, we define a normZ = {p1(n), p2(n), u1(n), u2(n)} and W = {q1(n), q2(n), v1(n), v2(n)} be any two solutions of system =p system (pi(n+1)=pf system  defined a \u2208 C, a(x) = C12x2, b \u2208 C, and b(x) = C22x2; thus condition (i) in ThenL = 4max\u2061{Ai, Bi}, i = 1,2. And In addition,V along the solutions of (epi(n) \u2212 eqi(n) = \u03be(n)Mi(n), where \u03be(n) lies between epi(n) and eqi(n).Finally, calculating the \u0394tions of  leads toj = 1,2 that\u03b2 = min\u2061{rij, rij*}, i, j = 1,2, i \u2260 j. That is, there exists a positive constant 0 < \u03b2 < 1 such that\u03b2 < 1, condition (iii) of X(n) = {x1(n), x2(n), u1(n), u2(n)} of system  = {x1(n), x2(n), u1(n), u2(n)} of system , o, o54), of system  which is(55)\u0394V2(n)=\u2211i=1"}
{"text": "An error occurred in the Figure 1 legend. The correct legend should read: \"*Ultrasound soft markers included mild ventriculomegaly (N \u200a=\u200a 3), increased nuchal translucency (N \u200a=\u200a 3), short nasal bones (N \u200a=\u200a 1), hyperechogenic bowel (N \u200a=\u200a 7), pyelectasis (N \u200a=\u200a 5).\""}
{"text": "Differential inequalities, comparison results, and sufficient conditions on initial time difference stability, boundedness, and Lagrange stability for fractional differential systems have been evaluated. The problem of stability of solutions is one of the major problems in the theory of differential equations. Lyapunov function and the Lyapunov direct method allow us to obtain sufficient conditions for the stability of a system without explicitly solving the differential equations , 2. The Only a few decades ago, it was realized that fractional calculus provides an attractive tool for modelling the real world problems. The differentiation and integration of arbitrary orders have found applications in diverse fields of science and engineering like viscoelasticity, electrochemistry, diffusion processes, control theory, heat conduction, electricity, mechanics, chaos, and fractals \u20135. RecenIn practical situations, it is possible to have not only a change in initial position but also in initial time because of all kinds of disturbed factors. When we do consider such a deviation in initial time, it causes measuring the difference between any two different solutions starting with different initial times. From this point of view, several studies have been made on this problem to explore the stability and boundedness, criteria for differential systems relative to initial time difference (ITD) by using variation of parameters and differential inequalities technique \u201313. In tf :  \u2192 \u211d be a function.Fractional calculus generalizes the derivative and the integral of a function to a noninteger order , 4, 14. q > 0 is defined asThe fractional integral (or the Riemann-Liouville (RL) type integral) of order q \u2208  = sqX(s) \u2212 \u2211k=0n\u22121sk[Dq\u2212k\u22121f(t)]t=0, where Dq represents 0Dtq.\u2112[cDqf(t)] = sqX(s) \u2212 \u2211k=0n\u22121sq\u2212k\u22121fk)((0), where cDq represents 0cDtq.cDqIqf(t) = f(t) and DqIqf(t) = f(t).IqDqf(t) = f(t) \u2212 \u2211k=1n\u22121[Dq\u2212kf(t)]t=a((t \u2212 a)q\u2212k/\u0393(k \u2212 q + 1)) and IqcDqf(t) = f(t)\u2012\u2211k=0n\u22121((t \u2212 a)k/k!)fk)((a).cDqf(t) = Dq(f(t) \u2212 \u2211k=0n\u22121((t \u2212 a)k/k!)fk)((a)).DqC = C(t \u2212 a)q\u2212/\u0393(1 \u2212 q) and cDqC = 0, where C is arbitrary constant.q < 1, f \u2208 C and J = . Then IVP's are equivalent to the following Volterra fractional integral equations:After giving definition and properties of fractional integral and derivatives, we consider fractional order IVP with (RL) and Caputo derivative, respectively;vely. In  authors t0 \u2208 \u211d+, f \u2208 C and 0 < q < 1. Suppose that the function f is smooth enough to guarantee existence, uniqueness, and continuous dependence of solutions of IVP (x(t) = x the solution of (\u03c40 \u2208 \u211d+ and y(t) = y be the solution of the system (x(t) = x is the solution on which we shall study stability and boundedness criteria with respect to it. Set \u03b7 = \u03c40 \u2212 t0 > 0 and denote S(\u03c1) = {x \u2208 \u211dn : ||x|| < \u03c1}.Consider the IVP for the system of nonlinear fractional differential equationDqcx(t)=fDqcx(t)=fDqcx(t)=fBefore giving our comparison theorem, stability, boundedness criteria, and Lagrange stability for FDE we need to introduce the following definitions.x(t) = x of ((S1)\u03f5 > 0 and t0 \u2208 \u211d+; there exist \u03b4 = \u03b4 > 0 and stable with ITD, if given (S2)S1) holds with \u03b4 and t0.uniformly stable with ITD, if (The solution  x0) of  is said (B1)\u03b1 > 0 and t0 \u2208 \u211d+; there exist \u03b2 = \u03b2 > 0 such thatequibounded with ITD, if given (B2)B1) holds with \u03b2 independent of t0;uniformly bounded with ITD, if ((A1)\u03f5 > 0 and each \u03b1 > 0 there exists T = T such thatattractive in the large with ITD, if for every (L1)Lagrange stable if (B1) and (A1) hold.The system  is said a is said to belong to the class \ud835\udca6 such thatA function x = x(t) = x is the solution of  is the solution of  = x be a solution of (x(t). Consider the solution Y(t) = Y of  = Y(t) \u2212 x(t) is a solution of the IVP (Let ution of . Study t y0) of . Set they0) of (Dqcz=f~(tx(t) of (The IVP  has a zex(t) of  is reducx(t) of .x(t). Consider the solution of (y(t) = y. Set the IVP asz = y(t + \u03b7) \u2212 x(t) is a solution of (x(t) could not be reduced to the study of stability of the zero solution of an appropriate fractional differential system.Study the stability with initial time difference of ution of  with dife IVP asDqcz=f~, if and only if the Caputo derivative of cDqm(t) exists and satisfiesm \u2208 Cq. Suppose that for any t1 \u2208  = 0 and m(t) < 0 for t0 \u2264 t < t1; then it follows thatLet m(t0) < 0 we have Dqm(t0) < 0. Therefore, we obtainFrom the relation between Riemann-Liouville and Caputo fractional derivatives we writeive from  and 18)(17)Dqcm, f \u2208 C andv(t0) < w(t0) impliesLet v(t0) < w(t0) and v(t), w(t) are continuous, there exists a t1 \u2208  = w(t1) and v(t) < w(t) for t0 \u2264 t < t1. Set m(t) = v(t) \u2212 w(t). Then m(t1) = 0 and m(t) < 0 for t1 \u2208  such that m(t0) \u2264 u0. Then one hasAssume that r(t), it is enough to prove that m(t) < u, t0 \u2264 t \u2264 T, where u is any solution of the IVPm(t) < u. And from lim\u2061\u03f5\u21920\u2061u = r(t) uniformly on each compact set t0 \u2264 t \u2264 T0 < T, we get the desired estimate m \u2208 Cq, g \u2208 C and(ii)r(t) = r of the IVPt \u2265 \u03c40;the maximal solution (iii)g is nondecreasing in t for each w and \u03c40 > t0.Then (a) m(t) \u2264 r(t + \u03b7), t \u2265 t0 and (b) m(t \u2212 \u03b7) \u2264 r(t), t \u2265 \u03c40.Assume thatw is any solution of\u03f5\u21920\u2061w = r on every compact set .(a) It is well known that if w0 = w, we have w0 = w = w = w0 + \u03f5 > w0 \u2265 m(t0). Thus we get m(t0) < w0. On the other hand, using (iii)m(t) < w0,\u2009\u2009t \u2265 t0 and hence it follows that m(t) \u2264 r(t + \u03b7),\u2009\u2009t \u2265 t0.Setting m0(t) = m(t \u2212 \u03b7) so that m0(\u03c40) = m(t0) \u2264 w0 < w0 + \u03f5 andm0(t) < w, t \u2265 \u03c40. The conclusion follows taking the limit as \u03f5 \u2192 0. The proof of theorem is complete.(b) We set In the following theorem, we obtain a comparison result in terms of Lyapunov-like functions with ITD.(i)V \u2208 C, V is locally Lipschitzian in x \u2208 \u211dn, g \u2208 C and(ii)r(t) = r of the fractional scalar differential equationt \u2265 \u03c40;the maximal solution (iii)g is nondecreasing in t for each u.Then V \u2264 u0 impliesAssume thatm(t) = V \u2212 x) so thatz = y \u2212 x so thatV is locally Lipschitzian in x and L > 0 is the Lipschitz constant and \u03f5(hq)/hq \u2192 0 as h \u2192 0, we haveDefine A comparison principle is obtained by employing the notion of Lyapunov function together with the theory of differential inequalities. In this part, one can see Lyapunov-like function as a transformation which reduces the study of stability, boundedness, and Lagrange stability properties relative to ITD of a given complicated system to a relatively simpler scalar equation.(i)V \u2208 C, V is locally Lipschitzian in x \u2208 \u211dn, g \u2208 C and(ii)r(t) = r of  of  exists f(iii)a, b \u2208 \ud835\udca6 such that b(||x||) \u2264 V \u2264 a(||x||) for  \u2208 \u211d+ \u00d7 S(\u03c1);there exists (iv)g is nondecreasing in t for each u and g = 0;Then the stability properties of the null solution of .Assume thatution of  imply th\u03f5 < \u03c1 be given. Then by definition of equistability given b(\u03f5) > 0, \u03c40 \u2208 \u211d+, there exist a \u03b41 = \u03b41 such thatu is any solution of the  > 0 as 0 < a(\u03b42) < \u03b41. Obviously lim\u2061\u03c40,y0)\u2192 \u2212 x|| = 0. Then given \u03f5 > 0 and \u03c40 \u2208 \u211d+, \u03b43 = \u03b43 > 0 such thatu0 = a(||y0 \u2212 x0||) and choose \u03b4 = min\u2061. Then we claim thatt1 > \u03c40 and a solution y with ||y0 \u2212 x0|| < \u03b4 and y0 \u2212 x0|| < \u03b4, by (iii) we havex of (Assume that the null solution of  is equisn of the . Choose uch that||y0\u2212x0||uch that||y(t1+\u03b7)uch thatu(t)<b(\u03f5)uch that||y(t1+\u03b7), (ii), In this section, Lagrange stability, which includes boundedness criteria, of fractional dynamic systems are discussed by employing comparison method.Let the assumption of \u03b1 \u2265 0 be given, and let ||y0 \u2212 x0|| \u2264 \u03b1. In view of (iii) V \u2264 a(\u03b1) = \u03b11. Assume that  such thatb(u) \u2192 \u221e as u \u2192 \u221e; we can choose a \u03b2 = \u03b2 verifying the relation\u03b2 > 0 and \u03c40 \u2208 \u211d+, \u03b41 = \u03b41 > 0 such thatu0 = V. Then we claim thatt* > \u03c40 and a solution y of (We need to prove (B1) and (S3) for . Let \u03b1 \u2265ume that  is Lagraume that  is boundhen from , given \u03b2 y0) of  such thaelations , 51), \u2265\u03b21.olds for .\u03f5 > 0, \u03b1 \u2265 0, \u03c40 \u2208 \u211d+ be given and ||y0 \u2212 x0|| \u2264 \u03b1. In view of (iii), V \u2264 a(\u03b1) = \u03b11. Since  and \u03c40 \u2208 \u211d+ there exist a T = T such thattk} \u2208 \u211d+, tk \u2192 \u221e as k \u2192 \u221e, tk > \u03c40 + T and a solution y of  of  such thas (iii), , 56), a, a\u03f5 > 0,y0) of (||y(tk+\u03b7)Firstly, differential inequalities and a new comparison principle for fractional differential equations relative to initial time difference have been developed and then stability, boundedness criteria, and Lagrange stability relative to initial time difference have been proved by employing comparison method."}
{"text": "Under sub-section \"Mouse Primary Cell Culture\", line 2,\u201c0.025% w/v\u201d should read as \u201c0.016% w/v\u201dUnder sub-section \"Mouse Primary Cell Culture\"The Krebs Buffer recipe indicated in parentheses should read asUnder sub-section \"Mouse Primary Cell Culture\", lines 4 and 5 \u201c0.008% w/v\u201d should read as \u201c0.08% w/v\u201d, and \u201c0.026% w/v\u201d should read as \u201c0.26% w/v\u201dCTUnder sub-section \"RNA Isolation and RT-qPCR\", 2-\u0394\u0394CT should read as 2-\u0394\u0394"}
{"text": "We also characterize ideals, nilradicals, and nilpotent elements of such algebras.We describe weak-BCC-algebras  in which the condition ( It holds in BCK-algebras and in some generalizations of BCK-algebras, but not in BCC-algebras. BCC-algebras satisfying this identity are BCK-algebras (cf. [One of very important identities is the identity (ras (cf.  or 7]).).x\u2217y)\u2217z ras (cf. .x\u2217y)\u2217z = (x\u2217z)\u2217y is satisfied only in the case when elements x, y belong to the same branch. We describe some endomorphisms of such algebras, ideals, nilradicals, and nilpotent elements.In this paper, we will study weak-BCC-algebras in which the condition  of type  satisfying the following axioms: x\u2217y)\u2217(z\u2217y))\u2217(x\u2217z) = 0, )BZ-algebrea \u2009x = 00\u2217is called a BCC-algebra. A BCC-algebra with the condition (vi)\u2009x\u2217(x\u2217y))\u2217y = 0\u2009x\u2217y)\u2217z = (x\u2217z)\u2217y. (One can prove (see  or 7]) ) 7]) thaG; \u2217, 0) of type  satisfying the axioms (i), (ii), (iii), (iv), and (vi) is called a BCI-algebra. A BCI-algebra satisfies also (vii). A weak-BCC-algebra is a BCI-algebra if and only if it satisfies (vii).An algebra  of type  with a relation \u2a7d defined by x\u2217y)\u2217(z\u2217y) \u2a7d x\u2217z,((ii\u2032)x \u2a7d x,(iii\u2032)x\u22170 = x,(iv\u2032)x \u2a7d y\u2009\u2009and\u2009\u2009y \u2a7d x\u2009\u2009imply\u2009\u2009x = y.An algebra , it follows that in weak-BCC-algebras, implicationsproper. Proper weak-BCC-algebras have at least four elements . But thThey are proper, because in both cases (3\u22172)\u22171 \u2260 (3\u22171)\u22172.Since two nonisomorphic weak-BCC-algebras may have the same partial order, they cannot be investigated as algebras with the operation induced by partial order. For example, weak-BCC-algebras defined by  and 5)  5) have n\u2a7e4, there exist at least two proper weak-BCC-algebras of order n which are not isomorphic.The methods of construction of weak-BCC-algebras proposed in  show tha\u2a7d) elements of G is denoted by I(G). Elements belonging to I(G) are called initial.The set of all minimal  = 0\u2217x. The main properties of this map in the case of weak-BCC-algebras are collected in the following theorem proved in , we mean the set of all elements of G such that 0\u2217xk \u2208 A; that is,Let G defined in A = {0, a} and any natural k, we have  =  = B(0),  = G. But for B = {a, e}, we get  = {d},  = B(c). The set  is empty.In the weak-BCC-algebra G defined byk-nilradical of A = {0,5} is equal to G; each k-nilradical of B = {1,4} is empty.The solid weak-BCC-algebra A its k-nilradical is also nonempty? The answer is given in the following proposition.The first question is when for a given nonempty set k-nilradical  of a nonempty subset A of a weak-BCC-algebra G is nonempty if and only if A contains at least one element a \u2208 I(G).A xk = 0\u2217ak for every x \u2208 B(a) and any positive k. So, x \u2208  if and only if 0\u2217ak \u2208 A. The last means that 0\u2217ak \u2208 A\u2229I(G) because I(G) is a subalgebra of G.From the proof of I(G); k] = G for every k..Indeed, 0\u2217I(G) has n elements, then  = G for any subset A of G containing 0, and  = \u2205 if 0 \u2209 A.If xk = 0\u2217ak for every x \u2208 B(a) and any k. Since 0\u2217ak \u2208 I(G) and I(G) is a group-like subalgebra of G, 0\u2217xk = ak\u2212 in the group (I(G); \u00b7, \u22121, 0) (I(G) has n elements, then obviously 0\u2217xn = an\u2212 = 0 \u2208 A. Hence, x \u2208 . This completes the proof.Similarly, as in previous proofs, we have 0\u2217, \u22121, 0) . If I(G)x \u2208 B(a). Then x \u2208  if and only if B(a)\u2282.Let xk = 0\u2217ak, we have x \u2208 \u21d4a \u2208 .Since 0\u2217A; k] = \u22c3{B(a) : 0\u2217ak \u2208 A}. is a subalgebra of G such that A\u2286.Let x, y \u2208 . Then 0\u2217xk, 0\u2217yk \u2208 A and 0\u2217(x\u2217y)k = (0\u2217xk)\u2217(0\u2217yk) \u2208 A, by x\u2217y \u2208 . Clearly A\u2286.Let k-nilradical of an ideal is also an ideal.In a solid weak-BCC-algebra, a A be a BCC-ideal of G. If y \u2208  and (x\u2217y)\u2217z \u2208 , then 0\u2217yk \u2208 A and A\u220b0\u2217((x\u2217y)\u2217z)k = ((0\u2217xk)\u2217(0\u2217yk))\u2217(0\u2217zk), by A\u220b(0\u2217xk)\u2217(0\u2217zk) = 0\u2217(x\u2217z)k. Thus, x\u2217z \u2208 .Let Note that the last two propositions are not true for weak-BCC-algebras which are not solid.G induced by the symmetric group S3 is not solid because S3 is not an abelian group -fold p-ideal is also an -fold p-ideal.In a solid weak-BCC-algebra, a k-nilradical of an -fold p-ideal A of G is an ideal of G. If y, (x\u2217zm)\u2217(y\u2217zn)\u2208, then 0\u2217yk, 0\u2217((x\u2217zm)\u2217(y\u2217zn)k) \u2208 A. Hence, applying xk \u2208 A. So, x \u2208 .By k-nilradical  of an ideal A does not save all properties of an ideal A. For example, if an ideal A is a horizontal ideal, that is, x \u2208 A\u2229B(0)\u21d4x = 0, then a k-nilradical  may not be a horizontal ideal. Such situation takes place in a weak-BCC-algebra defined by  means that 0\u2217x3 \u2208 A and x \u2208 B(0) which is also true for x \u2260 0.Note that in general, a fined by . In thisk-nilradicals. Below, we present the list of the main types of ideals considered in BCI-algebras and weak-BCC-algebras.Nevertheless, properties of many main types of ideals are saved by their A of a weak-BCC-algebra G is called (i)antigrouped, if(ii)associative, if(iii)quasiassociative if(iv)closed, if(v)commutative, if(vi)subcommutative, if(vii)implicative if(viii)subimplicative if(ix)weakly implicative if(x)obstinate, if(xi)regular, if(xii)strong, iffor all x, y, z \u2208 G.An ideal A of a weak-BCC-algebra G has the property \ud835\udcab if it is one of the above types, that is, if it satisfies one of implications mentioned in the above definition.We say that an ideal A of a solid weak-BCC-algebra G has the property \ud835\udcab, then its k-nilradical  also has this property.If an ideal A is antigrouped. Let \u03c62(x)\u2208. Then 0\u2217(\u03c62(x))k \u2208 A. Since, by \u03c62 is an endomorphism of each weak-BCC-algebra, we have\u03c62(0\u2217xk) \u2208 A, which according to the definition implies 0\u2217xk \u2208 A. Hence, x \u2208 .(1)\u2009\u2009A is associative. If (x\u2217y)\u2217z, \u2009y\u2217z \u2208 , then 0\u2217((x\u2217y)\u2217z)k \u2208 A and 0\u2217(y\u2217z)k \u2208 A which, in view of xk)\u2217(0\u2217yk))\u2217(0\u2217zk) \u2208 A and (0\u2217yk)\u2217(0\u2217zk) \u2208 A. Since an ideal A is associative, this implies 0\u2217xk \u2208 A; that is, x \u2208 .(2)\u2009\u2009A is quasiassociative. Similarly as in the previous case x\u2217(y\u2217z), \u2009y \u2208  means that 0\u2217(x\u2217(y\u2217z))k \u2208 A and 0\u2217yk \u2208 A. Hence, (0\u2217xk)\u2217((0\u2217yk)\u2217(0\u2217zk)) \u2208 A. This implies 0\u2217(x\u2217z)k = (0\u2217xk)\u2217(0\u2217zk) \u2208 A. Consequently, x\u2217z \u2208 .(3)\u2009\u2009A is closed. Let x \u2208 . Then, 0\u2217xk \u2208 A. Thus,x \u2208 .(4)\u2009\u2009A is commutative. Let x\u2217y \u2208 . Then, 0\u2217(x\u2217y)k \u2208 A. From this, we obtain (0\u2217xk)\u2217(0\u2217yk) \u2208 A, which gives 0\u2217(x\u2217(y\u2217(y\u2217x)))k = (0\u2217xk)\u2217((0\u2217yk)\u2217((0\u2217yk)\u2217(0\u2217xk))) \u2208 A. Hence, x\u2217(y\u2217(y\u2217x))\u2208.(5)\u2009\u2009For other types of ideals, the proof is very similar."}
{"text": "The approach used in this paperis the variational method for locally Lipschitz functions. More precisely, Weierstrass theorem and Mountain Passtheorem are used to prove the existence of at least two nontrivial solutions.A class of nonlinear Neumann problems driven by  Of the existing works in the literature, the majority deal with problems in which the potential function is smooth  \u2208 C1(\u211d)). We mention the works of Mihailescu p(x)-Laplhailescu  and Cammhailescu . Problemhailescu , 7, who n et al.  studied f Clarke . The autV(x) \u2261 1, then problem (If  problem  as folloproblem (\u2212div\u2061(|\u2207uu|p(x)\u22122\u2207u) is said to be p(x)-Laplacian, which becomes p-Laplacian when p(x) \u2261 p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian; for example, it is inhomogeneous and in general, it has not the first eigenvalue. The study of various mathematical problems with variable exponent growth conditions has received considerable attention in recent years. These problems are interesting in applications to modeling electrorheological fluids  and andu|p(xids (see ).Lp(x)(\u03a9) and the generalized Lebesgue-Sobolev space Wp(x)1,(\u03a9). In the third section, we give the assumptions on the nonsmooth potentials F,\u2009\u2009G and prove the multiplicity results for problem (\u03a9) and Wp(x)1,(\u03a9). For the details, see \u03c6(\u03b3(t)),\u2009\u2009\ud835\udcaf = {\u03b3 \u2208 C:\u03b3(0) = x0, \u03b3(1) = x1}, then c is a critical value of \u03c6 and c \u2265 inf\u2061U\u2202\u03c6.Let In this section, we will discuss the existence of weak solution of .F and G are given as follows.\u2009H(F) : F : \u03a9 \u00d7 \u211d \u2192 \u211d is a function such that F = 0 almost everywhere on \u03a9 and satisfies the following facts:\u2009t \u2208 \u211d, x \u21a6 F is measurable;(1) for all \u2009x \u2208 \u03a9, t \u21a6 F is locally Lipschitz.(2) for almost all \u2009H(G) : G : \u2202\u03a9 \u00d7 \u211d \u2192 \u211d is a function such that G = 0 almost everywhere on \u2202\u03a9 and satisfies the following facts:\u2009t \u2208 \u211d, x \u21a6 G is measurable;(1) for all \u2009x \u2208 \u2202\u03a9, t \u21a6 G is locally Lipschitz.(2) for almost all Our hypotheses on nonsmooth potential \u03c6 : X \u2192 \u211d for problem , H(G), and the following conditions hold:f1) \u2264 \u03b1+ < p\u2212 such thatthere exist \u2009x \u2208 \u03a9, all t \u2208 \u211d and w \u2208 \u2202F;for almost all g1) with 1 < \u03b2(x) \u2264 \u03b2+ < p\u2212 < p\u2217(x) such thatthere exist \u2009x \u2208 \u2202\u03a9, all t \u2208 \u211d and w \u2208 \u2202G.for almost all Suppose that \u03c6 is locally Lipschitz in X.Then, J \u2208 C1, we have t \u2208 .By Br = {x \u2208 X : ||u\u2212u0||X \u2264 r}.Let Br is w-compact. Then, we obtain that there exists a positive constant M, such that r.Note that u1, u2 \u2208 Br, we haveTherefore, for any f1) and Lebourg mean value theorem, we haveOn the other hand, by (\u03b1\u2032(x))+(1/\u03b1(x)) = 1.Hence,Obviously, it is verified thatWp(x)1,(\u03a9)\u21aaL\u03b1(x)(\u03a9) is a compact imbedding.So,c4, such thatWp(x)1,(\u03a9)\u21aaL\u03b2(x)(\u2202\u03a9) is a compact imbedding.As above, there is a positive constant \u03c6 is locally Lipschitz.Therefore, f1) and (g1) in \u03b1(x) \u2264 \u03b1+ < p\u2212 < p*(x) such thatthere exist \u2009x \u2208 \u03a9, all t \u2208 \u211d and w \u2208 \u2202F;for almost all \u03b2 \u2208 C(\u2202\u03a9) with 1 < \u03b2(x) \u2264 \u03b2+ < p\u2212 < p\u2217(x), such thatthere exist \u2009x \u2208 \u2202\u03a9, all t \u2208 \u211d and w \u2208 \u2202G, then the result of for almost all If assumptions (H(F), H(G), (f1), (g1), and the following conditions (f2)-(f3), (g2) hold:f2);where 1 < f3) and 0 < |t | \u2264\u03b4;g2) and 0 < |t | \u2264\u03b4,\u2009whereLet Then, the problem  has at lThe proof is divided into four steps as follows.Step\u2009\u20091. We will show that \u03c6 is coercive in this step.x \u2208 \u03a9, by H(F) (2), t \u21a6 F is differentiable almost everywhere on \u211d and we haveFirstly, for almost all f1), we havex \u2208 \u03a9 and t \u2208 \u211d.Moreover, from (\u03b1(x) \u2264 \u03b1+ < p\u2212 < p*(x) and 1 < \u03b2(x) \u2264 \u03b2+ < p\u2212 < p\u2217(x); then by Wp(x)1,(\u03a9)\u21aaL\u03b1(x)(\u03a9) and Wp(x)1,(\u03a9)\u21aaL\u03b2(x)(\u2202\u03a9) (compact imbedding).Note that 1 < Furthermore, there exist u|L\u03b1(x)(\u03a9) > 1, |u|L\u03b2(x)(\u2202\u03a9) > 1 and ||u|| > 1, we haveSo, for any |Hence,Step\u2009\u20092. We will show that the \u03c6 is weakly lower semicontinuous.un\u21c0u weakly in Wp(x)1,(\u03a9); by Let By Fatou's lemma, we haveThus,u0 \u2208 Wp(x)1,(\u03a9) such thatHence, by Weierstrass theorem, we deduce that there exists a global minimizer Step\u2009\u20093. In this step, we prove that \u03c6(u0) < 0.f2), we haveBy (x0 \u2208 \u03a9 and Br2(x0)\u2286\u03a9 with 2r < 1. Let \u03b7 \u2208 C0\u221e(Br2(x0)) such that \u03b7 = 1, x \u2208 Br(x0); 0 \u2264 \u03b7(x) \u2264 1 and |\u2207\u03b7 | \u22642/r. Denote s = 2\u03b4; thenSuppose that Step\u2009\u20094. We will show that there exists another nontrivial weak solution of problem ( problem .p+ < \u03c1\u2212 \u2264 \u03c1+ < p\u2217(x).Let u \u2208 X with ||u|| < 1, by c7 and For f3), we haveBy (f3), we obtain\u03bb \u2208 , 0 < |t | \u2264\u03b4.From Lebourg's mean value theorem and |\u2264\u03b4} and \u03a92 = {x \u2208 \u03a9 : |u(x)|\u2265\u03b4}.Divide u \u2208 X such that ||u|| = d, we haveFor any \u03c6 is coercive; hence, it satisfies the nonsmooth C-condition. So by the nonsmooth Mountain Pass theorem (consequence of u1 \u2208 Wp(x)1,(\u03a9) such thatu1 is another nontrivial critical point of \u03c6.Note that Using the similar and simpler arguments, we can prove the following theorems.H(F), H(G), (f1), (f2), (g1), and the following conditions (f3\u2032), (g2\u2032) hold:f3\u2032) = 0,\u2009for all x \u2208 \u03a9, 0 < |t | \u2264\u03b4;g2\u2032) = 0, for \u2009all\u2009\u2009x \u2208 \u2202\u03a9, 0 < |t | \u2264\u03b4,\u2009whereLet Then, the problem  has at l\u03c6(u0) < 0 under the assumptions of The steps are similar to those of f2), we haveStep 3\u2032. By (x0 \u2208 \u03a9 and Br2(x0)\u2286\u03a9 with 2r < 1.Suppose that \u03b7 \u2208 C0\u221e(Br2(x0)) such that \u03b7 = 1, x \u2208 Br(x0); 0 \u2264 \u03b7(x) \u2264 1 and |\u2207\u03b7 | \u2264(2/r). Denote s = 2\u03b4; thenLet H(F), H(G), (f2),\u2009\u2009Let  H(G), f,\u2009\u2009(f\u00af1),\u03c6 has a critical point, which is just the solution of problem  \u2212 \u03b8(x)) > 0, inf\u2061x\u2208\u2202\u03a9(\u03b2(x) \u2212 \u03b8(x)) > 0 and \u03b8\u2212 > 1. In the following, we will show that F satisfies hypotheses H(F) and (f1) \u2212 (f3), and G satisfies hypotheses H(G) and (g1)-(g2).Let t \u21a6 |t \u2212 \u03b4|\u03b8(x) and t \u21a6 |t \u2212 \u03b4|\u03b1(x) are convex functions; thus, F is also convex. Since t \u21a6 |t \u2212 \u03b4|\u03b8(x), t \u2192 |t \u2212 \u03b4|\u03b1(x) are locally Lipschitz, so t \u21a6 F is locally Lipschitz. Thus, t \u21a6 F is regular. ThenNote that w \u2208 \u2202F, we haveHence, for any Thus, we haveH(F) and (f1) \u2212 (f3) hold. In a similar fashion, we have that conditions H(G) and (g1)-(g2) hold.Therefore, conditions"}
{"text": "FWC-spaces, in short) which have no linear, convex, and topological structure. Using the maximal element theorem, we develop new existence theorems of solutions to variational relation problem, generalized equilibrium problem, equilibrium problem with lower and upper bounds, and minimax problem in FWC-spaces. The results represented in this paper unify and extend some known results in the literature.A maximal element theorem is proved in finite weakly convex spaces ( In 1983, by using fixed point theorems for set-valued mappings, Yannelis and Prabhakar  improvedH-spaces and obtained some minimax inequalities, variational inequalities, and quasivariational inequalities by using this maximal element theorem. Subsequently, Wu  and  = , where \u03c4 is a family of subsets of X. We can verify that  is not a topological space. For simplicity, we write X instead of . Let Y =  =  for each z = \u2211i=0n\u03bbiei \u2208 \u0394n. It is easy to see that  forms an FWC-space. Now we define a set-valued mapping T : X \u2192 2Y byN = {u0, u1,\u2026, un}\u2208\u2329D\u232a, we have T(\u03c6N(\u0394n)) = . Therefore, the composition T|\u03c6N\u2218\u03c6N : \u0394n \u2192 2T(\u03c6N(\u0394n)) is an upper semicontinuous set-valued mapping with nonempty compact contractible values. By Lemma 1 of  and  = , where \u03c4 is a family of subsets of X. It is easy to check that  is not a topological space. For simplicity, we will write X instead of . Define a set-valued mappings H : Y \u2192 2D such thatY. Therefore, H\u22121 is compactly open-valued, and hence, (ii) of P : X \u2192 2Y such thatT : X \u2192 2Y be defined byY. Let 1) of N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, we define a set-valued mapping \u03c6N : \u0394n \u2192 2X by \u03c6N(z) = {0,4} for all z \u2208 \u0394n. Then  forms an FWC-space. For each y \u2208 , we havey \u2208 Y, X\u2216P\u22121(y) is an FWC-subspace of  relative to H(y). Therefore, by x \u2208 X, T(x)\u2286P(x) and T, we can obtain T\u22121 : Y \u2192 2X as follows:T\u22121(y)\u2286P\u22121(y) for each y \u2208 Y. Hence, for each x \u2208 X, T(x)\u2286P(x). By using the same method as in Let T = P in If X, D; \u03c6N) be an FWC-space, Y a Hausdorff topological space, and K a nonempty compact subset of Y. Let H : Y \u2192 2D, Q : D \u2192 2X, and (i)u \u2208 D, H\u22121(u) is compactly open;for each (ii)N = {u0, u1,\u2026, un}\u2208\u2329D\u232a and each {ui0, ui1,\u2026, uik}\u2286N,for each (iii)one of the following conditions holds:1) is an FWC-subspace of  relative to LN, Y, andfor each Let (Then there exists X = D and Q(x) = {x} in Taking X; \u03c6N) be an FWC-space, Y a Hausdorff topological space, and K a nonempty compact subset of Y. Let H : Y \u2192 2X and (i)x \u2208 X, H\u22121(x) is compactly open;for each (ii)N = {x0, x1,\u2026, xn}\u2208\u2329X\u232a and each {xi0, xi1,\u2026, xik}\u2286N,for each (iii)one of the following conditions holds:1), Y, andfor each Let  be an FWC-space, Y a Hausdorff topological space, K a nonempty compact subset of Y, and FWC-spaces.R be a relation linking y \u2208 Y and x \u2208 X. Find x \u2208 X.Let Z be a nonempty set, Q : X \u2192 2Z a set-valued mapping, and R a relation linking y \u2208 Y and z \u2208 Z. Find x \u2208 X and each z \u2208 Q(x).Let Let  be an FWC-space, Y a Hausdorff topological space, and K a nonempty compact subset of Y. Let R be a relation linking elements x \u2208 X, y \u2208 Y such that(i)x \u2208 X, the set {y \u2208 Y : R\u2009\u2009does\u2009\u2009not\u2009\u2009hold} is compactly open;for each (ii)N = {x0, x1,\u2026, xn}\u2208\u2329X\u232a, each {xi0, xi1,\u2026, xik}\u2286N and each y \u2208 T(\u03c6N(\u0394k)), there exists for each (iii)one of the following conditions holds:1), Y, andfor each Let  is compactly open. By (ii), for each N = {x0, x1,\u2026, xn}\u2208\u2329X\u232a and each {xi0, xi1,\u2026, xik}\u2286N, we have(a)N0 \u2208 \u2329X\u232a such that N = {x0, x1,\u2026, xn}\u2208\u2329X\u232a, Y;there exists (b)N \u2208 \u2329X\u232a, there exists a subset LN of X containing N such that LN is an FWC-subspace of , Y, andfor each Therefore, by x \u2208 X.Define a set-valued mapping X = Y and T(x) = {x} for every x \u2208 X in By taking X; \u03c6N) be an FWC-space and K a nonempty compact subset of X, where X is a Hausdorff topological space. Let I is the identity mapping on X. Let R be a relation linking elements x \u2208 X, y \u2208 X such that(i)x \u2208 X, the set {y \u2208 X : R\u2009\u2009does\u2009\u2009not\u2009\u2009hold} is compactly open;for each (ii)N = {x0, x1,\u2026, xn}\u2208\u2329X\u232a, each {xi0, xi1,\u2026, xik}\u2286N and each y \u2208 \u03c6N(\u0394k), there exists for each (iii)one of the following conditions holds:1)\u2009\u2009does\u2009\u2009not\u2009\u2009hold} and \u03c6N : \u0394n \u2192 2X is a compact set-valued mapping for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a;there exists 2) andfor each Let  be an FWC-space, where X is a Hausdorff compact topological space. Let I is the identity mapping on X. Let R be a relation linking elements x \u2208 X, y \u2208 X such that the following conditions hold:x \u2208 X, the set {y \u2208 X : R\u2009\u2009does\u2009\u2009not\u2009\u2009hold} is compactly open;for each N = {x0, x1,\u2026, xn}\u2208\u2329X\u232a, each {xi0, xi1,\u2026, xik}\u2286N and each y \u2208 \u03c6N(\u0394k), there exists for each Let (x \u2208 X.Then there exists K = X. Then (iii1) of Let x \u2208 X, the set {y \u2208 X : R\u2009\u2009holds} is closed; (2) (ii) of Theorem 2.1 of Pu and Yang [x0, x1,\u2026, xn}\u2208\u2329X\u232a, there exists a continuous mapping \u03c6N : \u0394n \u2192 X such that, for each \u03bb = {\u03bb0, \u03bb1,\u2026, \u03bbn} \u2208 \u0394n, there exists i \u2208 J(\u03bb) such that R(\u03c6N(\u03bb), xi) holds, where J(\u03bb) = {i \u2208 {0,1,\u2026, n} : \u03bbi > 0}. By (ii) of Theorem 2.1 of Pu and Yang [X, \u03c6N) in Theorem 2.1 of Pu and Yang [FWC-space; (3) in X needs not to have the fixed point property, but X in Theorem 2.1 of Pu and Yang [I on X in Theorem 2.1 of Pu and Yang [N = {x0, x1,\u2026, xn}\u2208\u2329X\u232a and for every continuous mapping \u03c8 : \u03c6N(\u0394n) \u2192 \u0394n, the composition \u03c8\u2218\u03c6N : \u0394n \u2192 \u0394n is continuous, where \u03c6N coincides with the one in (ii) of Theorem 2.1 of Pu and Yang [z0 \u2208 \u0394n such that z0 = \u03c8\u2218\u03c6N(z0), which implies that It is interesting to compare and Yang  can be sand Yang , we knowand Yang  forms anand Yang  needs toand Yang , we mustand Yang . Then byX; \u03c6N) be an FWC-space, Y a Hausdorff topological space, K a nonempty compact subset of Y, and Z a nonempty set. Let Q : X \u2192 2Z be set-valued mappings and R a relation linking elements y \u2208 Y, z \u2208 Z. Assume that(i)x \u2208 X, the set {y \u2208 Y : P(y)\u22c2Q(x) \u2260 \u2205} is compactly open, where P : Y \u2192 2Z is defined by P(y) = {z \u2208 Z : R\u2009\u2009does\u2009\u2009not\u2009\u2009hold} for each y \u2208 Y;for each (ii)N = {x0, x1,\u2026, xn}\u2208\u2329X\u232a, each {xi0, xi1,\u2026, xik}\u2286N and each y \u2208 T(\u03c6N(\u0394k)), there exists R holds for each for each (iii)one of the following conditions holds:1), Y, andfor each Let (x \u2208 X and each z \u2208 Q(x).Then there exists Y and X be defined by P(y)\u22c2Q(x) = \u2205. Then by (i), for each x \u2208 X, the set N = {x0, x1,\u2026, xn}\u2208\u2329X\u232a, each {xi0, xi1,\u2026, xik}\u2286N and each y \u2208 T(\u03c6N(\u0394k)), there exists (a)N0 \u2208 \u2329X\u232a such that N = {x0, x1,\u2026, xn}\u2208\u2329X\u232a, Y;there exists (b)N \u2208 \u2329X\u232a, there exists a subset LN of X containing N such that LN is an FWC-subspace of  andfor each where Y. Therefore, by x \u2208 X; that is, x \u2208 X and each z \u2208 Q(x).Let the relation FWC-spaces and convex spaces, respectively. It follows from the previous analysis that FWC-spaces include convex spaces as special cases; (b) The class of better admissible mappings in \u212c, respectively. By \u212c is contained in Z in T in T in 1) of N = {x0, x1,\u2026, xn}\u2208\u2329X\u232a, each x \u2208 co\u2061{x0, x1,\u2026, xn} and each y \u2208 T(x), there exists Rholds for all z \u2208 Q(xj). (ii) of N = {x0, x1,\u2026, xn}\u2208\u2329X\u232a, we can define a continuous mapping \u03c6N : \u0394n \u2192 co\u2061(N)\u2286X byX; \u03c6N) forms an FWC-space. On the basis of this fact, we can see that (ii) of Theorem 3.1 of Balaj and Lin [(1) and Lin  can be s and Lin  implies X = Z and Q(x) = {x} for each x \u2208 X in (2) P : Y \u2192 2Z be defined by P(y) = {z \u2208 Z : R\u2009\u2009does\u2009\u2009not\u2009\u2009hold} for each y \u2208 Y. (i)\u2032z \u2208 Q(X), the set {y \u2208 Y : R\u2009\u2009holds} is compactly closed; for each (ii)\u2032Z is a topological space, the set-valued mapping P is lower semicontinuous, and Q has open values.Let z\u2208Q(x){y \u2208 Y : R\u2009\u2009holds} is compactly closed for each x \u2208 X. Thus, {y \u2208 Y : P(y)\u22c2Q(x) \u2260 \u2205} = Y\u2216\u22c3z\u2208Q(x){R\u2009\u2009does\u2009\u2009not\u2009\u2009hold} is compactly open. If (ii)\u2032 holds, then by the definition of a lower semicontinuous set-valued mapping, for each x \u2208 X, the set {y \u2208 Y : P(y)\u22c2Q(x) \u2260 \u2205} = Y\u2216{y \u2208 Y : P(y)\u22c2Q(x) = \u2205} is open and thus, compactly open.Suppose that (i)\u2032 is satisfied. Then \u22c2In recent years, many authors  be an FWC-space, Y a Hausdorff topological space, K a nonempty compact subset of Y, and Z a nonempty set. Let J : Y \u00d7 X \u2192 2Z, L : Y \u00d7 D \u2192 2Z, F, W : Y \u2192 2Z, Q : D \u2192 2X, and (i)x \u2208 X and each y \u2208 T(x), J\u2286F(y);for each (ii)u \u2208 D, the set {y \u2208 Y : L\u2286W(y)} is compactly closed;for each (iii)y \u2208 Y, the set {x \u2208 X : J\u2288F(y)} is an FWC-subspace of  relative to the set {u \u2208 D : L\u2288W(y)};for each (iv)one of the following conditions holds:1)\u2288W(y)} for some N0 \u2208 \u2329D\u232a and for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let (u \u2208 D.Then there exists P : X \u2192 2Y and H : Y \u2192 2D byT(x)\u2286P(x) for each x \u2208 X. By (ii), for each u \u2208 D, H\u22121(u) is compactly open. Now, we show that (iii) of N = {u0, u1,\u2026, un}\u2208\u2329D\u232a and {ui0, ui1,\u2026, uik}\u2286N such thaty* \u2208 P(\u03c6N(\u0394k)) such that y* \u2208 H\u22121(uij) for each j \u2208 {0,1,\u2026, k}; that is, uij \u2208 {u \u2208 D : L\u2288\u2009\u2009W(y*)}. By (iii), we havey* \u2208 P(\u03c6N(\u0394k)), it follows that there exists 1) of 1) and the definition of H, we know that there exists N0 \u2208 \u2329D\u232a such that Y\u2216K\u2286\u22c3d\u2208N0H\u22121(d) and for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, Y. Therefore, (iv1) of 2) of 2) and the definition of H again, we know that for each N \u2208 \u2329D\u232a, there exists a subset LN of D containing N such that Q(LN) is an FWC-subspace of  relative to LN andY. Therefore, (iv2) of u \u2208 D. This completes the proof.Define ies thatJ\u2288FFWC-spaces and FC-spaces, respectively. By the previous analysis, we know that FWC-spaces include FC-spaces as special cases; (b) The class of better admissible mappings in \u212c, respectively. By \u212c is contained in 2) of By using the same argument as in X, D; \u03c6N) be an FWC-space, Y a Hausdorff topological space, K a nonempty compact subset of Y, and Z a nonempty set. Let J : Y \u00d7 X \u2192 2Z, L : Y \u00d7 D \u2192 2Z, F, W : Y \u2192 2Z, Q : D \u2192 2X, and (i)x \u2208 X and each y \u2208 T(x), J\u2288F(y);for each (ii)u \u2208 D, the set {y \u2208 Y : L\u2288W(y)} is compactly closed;for each (iii)y \u2208 Y, the set {x \u2208 X : J\u2286F(y)} is an FWC-subspace of  relative to the set {u \u2208 D : L\u2286W(y)};for each (iv)one of the following conditions holds:1)\u2286W(y)} for some N0 \u2208 \u2329D\u232a and for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let  The class of better admissible mappings in \u212c, respectively. It follows from \u212c is contained in 2) of X, D; \u03c6N) be an FWC-space, Y a Hausdorff topological space, K a nonempty compact subset of Y, and Z a nonempty set. Let J : Y \u00d7 X \u2192 2Z, L : Y \u00d7 D \u2192 2Z, F, W : Y \u2192 2Z, Q : D \u2192 2X, and (i)x \u2208 X and each y \u2208 T(x), J\u22c2F(y) \u2260 \u2205;for each (ii)u \u2208 D, the set {y \u2208 Y : L\u22c2W(y) \u2260 \u2205} is compactly closed;for each (iii)y \u2208 Y, the set {x \u2208 X : J\u22c2F(y) = \u2205} is an FWC-subspace of  relative to the set {u \u2208 D : L\u22c2W(y) = \u2205};for each (iv)one of the following conditions holds:1)\u22c2W(y) = \u2205} for some N0 \u2208 \u2329D\u232a and for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let  be an FWC-space, Y a Hausdorff topological space, K a nonempty compact subset of Y, and Z a nonempty set. Let J : Y \u00d7 X \u2192 2Z, L : Y \u00d7 D \u2192 2Z, F, W : Y \u2192 2Z, Q : D \u2192 2X, and (i)x \u2208 X and each y \u2208 T(x), J\u22c2F(y) = \u2205;for each (ii)u \u2208 D, the set {y \u2208 Y : L\u22c2W(y) = \u2205} is compactly closed;for each (iii)y \u2208 Y, the set {x \u2208 X : J\u22c2F(y) \u2260 \u2205} is an FWC-subspace of  relative to the set {u \u2208 D : L\u22c2W(y) \u2260 \u2205};for each (iv)one of the following conditions holds:1)\u22c2W(y) \u2260 \u2205} for some N0 \u2208 \u2329D\u232a and for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let  be an FWC-space, Y a Hausdorff topological space, K a nonempty compact subset of Y, and Z a nonempty set. Let F, W : Y \u2192 2Z, Q : D \u2192 2X, and E be a nonempty set and G : Y \u2192 2E a set-valued mapping. Let \u03b6 : E \u00d7 Y \u00d7 X \u2192 Z and \u03be : E \u00d7 Y \u00d7 D \u2192 Z be two single-valued mappings. Assume that(i)x \u2208 X and each y \u2208 T(x), \u03b6(G(y), y, x)\u2288F(y);for each (ii)u \u2208 D, the set {y \u2208 Y : \u03be(G(y), y, u)\u2288W(y)} is compactly closed;for each (iii)y \u2208 Y, the set {x \u2208 X : \u03b6(G(y), y, x)\u2286F(y)} is an FWC-subspace of  relative to the set {u \u2208 D : \u03be(G(y), y, u)\u2286W(y)};for each (iv)one of the following conditions holds:1)(ivY\u2216K\u2286\u22c3u\u2208N0{y \u2208 Y : \u03be(G(y), y, u)\u2286W(y)} for some N0 \u2208 \u2329D\u232a and for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let  the class \u212c in Theorem 4.4 of Fang and Huang [2) of FWC-spaces. We emphasis that X, D, Z, and E in (1)nd Huang  as a spend Huang . In factnd Huang , we can nd Huang  imply (iZ and E in (ii)\u2032W is open in Y \u00d7 Z;the graph of (ii)\u2032\u2032G is upper semicontinuous on each compact subset of Y with nonempty compact values and for each u \u2208 D, \u03be is continuous on each compact subset of E \u00d7 Y.(2) Let H\u22121 : D \u2192 2Y is a set-valued mapping with compactly open values. Thus, these solution sets are compactly closed subsets of the compact set The solution sets of generalized equilibrium problems considered in Theorems C be a nonempty closed subset of a locally convex semireflexive topological vector space X, and let F be a real-valued function on C \u00d7 C. In 1999, Isac et al. [y \u2208 C, where c1, c2 are two real numbers with c1 \u2264 c2. Later, Li [G-convex spaces.Let c et al.  first raater, Li  introducater, Li  obtainedFWC-spaces.In this section, we apply X, D; \u03c6N) be an FWC-space and K a nonempty compact subset of a Hausdorff topological space Y. Let Q : D \u2192 2X and \u03bc and \u03bd be real-valued functions on Y \u00d7 X and Y \u00d7 D, respectively. Let g and h be real-valued functions on Y such that g(y) \u2264 h(y) for each y \u2208 Y. Assume that(i)x \u2208 X and each y \u2208 T(x), g(y) \u2264 \u03bc \u2264 h(y);for each (ii)u \u2208 D, the set {y \u2208 Y : g(y) \u2264 \u03bd \u2264 h(y)} is compactly closed;for each (iii)y \u2208 Y, the set {x \u2208 X : \u03bc > h(y)\u2009\u2009or\u2009\u2009\u03bc < g(y)} is an FWC-subspace of  relative to the set {u \u2208 D : \u03bd > h(y)\u2009\u2009or\u2009\u2009\u03bd < g(y)};for each (iv)one of the following conditions holds:1) > h(y)\u2009\u2009or\u2009\u2009\u03bd < g(y)} for some N0 \u2208 \u2329D\u232a and for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let  The condition that there are four functions in g(y) = c1 and h(y) = c2 for all y \u2208 Y, where c1 and c2 are real numbers such that c1 \u2264 c2. In this case, Let X, D; \u03c6N) be an FWC-space and K a nonempty compact subset of a Hausdorff topological space Y. Let Q : D \u2192 2X and \u03bc and \u03bd be real-valued functions on Y \u00d7 X and Y \u00d7 D, respectively. Let c1 and c2 be two real numbers such that c1 \u2264 c2. Assume that(i)x \u2208 X and each y \u2208 T(x), c1 \u2264 \u03bc \u2264 c2;for each (ii)u \u2208 D, the set {y \u2208 Y : c1 \u2264 \u03bd \u2264 c2} is compactly closed;for each (iii)y \u2208 Y, the set {x \u2208 X : \u03bc > c2\u2009\u2009or\u2009\u2009\u03bc < c1} is an FWC-subspace of  relative to the set {u \u2208 D : \u03bd > c2\u2009\u2009or\u2009\u2009\u03bd < c1};for each (iv)one of the following conditions holds:1) > c2\u2009\u2009or\u2009\u2009\u03bd < c1} for some N0 \u2208 \u2329D\u232a and for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let  be an FWC-space and K a nonempty compact subset of a Hausdorff topological space Y. Let Q : D \u2192 2X and \u03bc and \u03bd be real-valued functions on Y \u00d7 X and Y \u00d7 D, respectively. Let c be a real number. Assume that(i)x \u2208 X and each y \u2208 T(x), \u03bc = c;for each (ii)u \u2208 D, the set {y \u2208 Y : \u03bd = c} is compactly closed;for each (iii)y \u2208 Y, the set {x \u2208 X : \u03bc > c\u2009\u2009or\u2009\u2009\u03bc < c} is an FWC-subspace of  relative to the set {u \u2208 D : \u03bd > c\u2009\u2009or\u2009\u2009\u03bd < c};for each (iv)one of the following conditions holds:1) > c\u2009\u2009or\u2009\u2009\u03bd < c} for some N0 \u2208 \u2329D\u232a and for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let  in Corollary 3.1 of Li [It is interesting to compare .1 of Li  as speci.1 of Li ; (3)    be an FWC-space and K a nonempty compact subset of a Hausdorff topological space Y. Let Q : D \u2192 2X and \u03bc1 and \u03bd1 be real-valued functions on Y \u00d7 X and Y \u00d7 D, respectively. Let c be a real number. Assume that(i)x \u2208 X and each y \u2208 T(x), \u03bc1 \u2264 c;for each (ii)u \u2208 D, the set {y \u2208 Y : \u03bd1 \u2264 c} is compactly closed;for each (iii)y \u2208 Y, the set {x \u2208 X : \u03bc1 > c} is an FWC-subspace of  relative to the set {u \u2208 D : \u03bd1 > c};for each (iv)one of the following conditions holds:1) > c} for some N0 \u2208 \u2329D\u232a and for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let \u2032u \u2208 D, \u03bd1 is lower semicontmuous on each nonempty compact subset of Y.For every (iii)\u2032N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, every {ui0, ui1,\u2026, uik}\u2286N, and every y \u2208 Y, we have \u03bc1 \u2265 min\u2061j\u2264k0\u2264\u03bd1 for all x \u2208 \u03c6N(\u0394k).For every (ii)-(iii) of y \u2208 Y, N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, and {ui0, ui1,\u2026, uik}\u2286N\u2229{u \u2208 D : \u03bd1 > c} such that \u03c6N(\u0394k)\u2288{x \u2208 X : \u03bc1 > c}. Hence, there exists x \u2208 \u03c6N(\u0394k) such that \u03bc1 \u2264 c. Since {ui0, ui1,\u2026, uik}\u2286N\u2229{u \u2208 D : \u03bd1 > c}, we have \u03bd1 > c for each j \u2208 {0,1,\u2026, k}. By (iii)\u2032, we obtain the following contradiction:It is clear that (ii)\u2032 implies (ii) of P : X \u2192 2Y and H : Y \u2192 2D byu \u2208 D.c \u2208 \u211d be given. Let us define two real-valued functions \u03bc1 : Y \u00d7 X \u2192 \u211d and \u03bd1 : Y \u00d7 D \u2192 \u211d by\u03bc1 and \u03bd1 satisfy all conditions of u \u2208 D; that is, u \u2208 D, which implies that Conversely, let X, D; \u03c6N), Y be as in \u03bc1 and \u03bd1 be real-valued functions on Y \u00d7 X and Y \u00d7 D, respectively. Let c be a real number. Assume thatx \u2208 X and each y \u2208 T(x), \u03bc1 \u2264 c;for each u \u2208 D, the set {y \u2208 Y : \u03bd1 \u2264 c} is compactly closed;for each y \u2208 Y, the set {x \u2208 X : \u03bc1 > c} is an FWC-subspace of  relative to the set {u \u2208 D : \u03bd1 > c}.for each Let (u \u2208 D.Then there exists Q : D \u2192 2X by Q(u) = X for each u \u2208 D. For each N \u2208 \u2329D\u232a, let LN = D. Let 2) of Define a set-valued mapping FWC-spaces which contain G-convex spaces adopted in Theorem 3.3 of Tan [G-co(A) is compact in Theorem 3.3 of Tan [FWC-spaces without any linear, convex, and topological structure. The comparison details between 3 of Tan ; (2) The3 of Tan ; (3) the3 of Tan  is dropp3 of Tan . In fact3 of Tan  and Rema3 of Tan . Corolla3 of Tan  from topX, D; \u03c6N) be an FWC-space and K a nonempty compact subset of a Hausdorff topological space Y. Let Q : D \u2192 2X and \u03bc1 and \u03bd1 be real-valued functions on Y \u00d7 X and Y \u00d7 D, respectively. Let c be a real number. Assume that(i)u \u2208 D, the set {y \u2208 Y : \u03bd1 \u2264 c} is compactly closed;for each (ii)y \u2208 Y, the set {x \u2208 X : \u03bc1 > c} is an FWC-subspace of  relative to the set {u \u2208 D : \u03bd1 > c};for each (iii)one of the following conditions holds:1) > c} for some N0 \u2208 \u2329D\u232a and for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let , \u03bc1 \u2264 c. Hence, by u \u2208 D. This completes the proof.If (a) is false, then it follows that for each X, D; \u03c6N) be an FWC-space and K a nonempty compact subset of a Hausdorff topological space Y. Let Q : D \u2192 2X and \u03bc1 and \u03bd1 be real-valued functions on Y \u00d7 X and Y \u00d7 D, respectively. Assume that sup\u2061x\u2208X,y\u2208T(x)\u03bc1<+\u221e and the following conditions hold:(i)u \u2208 D, the set {y \u2208 Y : \u03bd1 \u2264 sup\u2061x\u2208X,y\u2208T(x)\u03bc1} is compactly closed;for each (ii)y \u2208 Y, the set {x \u2208 X : \u03bc1 > sup\u2061x\u2208X,y\u2208T(x)\u03bc1} is an FWC-subspace of  relative to the set {u \u2208 D : \u03bd1 > sup\u2061x\u2208X,y\u2208T(x)\u03bc1};for each (iii)either1) > sup\u2061x\u2208X,y\u2208T(x)\u03bc1} for some N0 \u2208 \u2329D\u232a and for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, Y or2) is an FWC-subspace of  relative to LN, Y, andfor each Let \u03bc1. By the definition of c, (a) of u \u2208 D. In particular, we have Let \u03bc1\u2032 = \u2212\u03bc1 and \u03bd1\u2032 = \u2212\u03bd1 and adjusting the corresponding conditions of u \u2208 D. In particular, we have FWC-spaces.By setting T in If X, D; \u03c6N), K, Y, and Q be as in \u03bc1 and \u03bd1 be real-valued functions on Y \u00d7 X and Y \u00d7 D, respectively. Assume that sup\u2061x\u2208X\u03bc1(T(x), x)<+\u221e and the following conditions hold:(i)u \u2208 D, the set {y \u2208 Y : \u03bd1 \u2264 sup\u2061x\u2208X\u03bc1(T(x), x)} is compactly closed;for each (ii)y \u2208 Y, the set {x \u2208 X : \u03bc1 > sup\u2061x\u2208X\u03bc1(T(x), x)} is an FWC-subspace of  relative to the set {u \u2208 D : \u03bd1 > sup\u2061x\u2208X\u03bc1(T(x), x)};for each (iii)either1) > sup\u2061x\u2208X\u03bc1(T(x), x)} for some N0 \u2208 \u2329D\u232a and for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a, Y or2) is an FWC-subspace of  relative to LN, Y, andfor each Let  = {x} for all x \u2208 X, we can obtain the following result from By taking X, D; \u03c6N) be an FWC-space, where X is a Hausdorff topological space. Let K be a nonempty compact subset of X. Let Q : D \u2192 2X be a set-valued mapping and let IX is the identity mapping on X. Let \u03bc1 and \u03bd1 be real-valued functions on X \u00d7 X and X \u00d7 D, respectively. Assume that sup\u2061x\u2208X\u03bc1<+\u221e and the following conditions hold:(i)u \u2208 D, the set {y \u2208 X : \u03bd1 \u2264 sup\u2061x\u2208X\u03bc1} is compactly closed;for each (ii)y \u2208 X, the set {x \u2208 X : \u03bc1 > sup\u2061x\u2208X\u03bc1} is an FWC-subspace of  relative to the set {u \u2208 D : \u03bd1 > sup\u2061x\u2208X\u03bc1};for each (iii)either1) > sup\u2061x\u2208X\u03bc1} for some N0 \u2208 \u2329D\u232a and \u03c6N : \u0394n \u2192 2X is a compact set-valued mapping for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a or2) relative to LN andfor each Let  \u2264 sup\u2061x\u2208X\u03bc1.Then there exists FWC-spaces which include L-convex spaces adopted in Corollary 5 of Jin and Cheng [H(A) in Corollary 5 of Jin and Cheng [X implies (i) of nd Cheng  as specind Cheng  is compand Cheng . In factX, D; \u03c6N) be an FWC-space, where X is a Hausdorff topological space. Let K be a nonempty compact subset of X. Let Q : D \u2192 2X be a set-valued mapping and let IX is the identity mapping on X. Let \u03bc1 and \u03bd1 be real-valued functions on X \u00d7 X and X \u00d7 D, respectively. Assume that inf\u2061x\u2208X\u03bc1>\u2212\u221e and the following conditions hold:(i)u \u2208 D, the set {y \u2208 X : \u03bd1 \u2265 inf\u2061x\u2208X\u03bc1} is compactly closed;for each (ii)y \u2208 X, the set {x \u2208 X : \u03bc1 < inf\u2061x\u2208X\u03bc1} is an FWC-subspace of  relative to the set {u \u2208 D : \u03bd1 < inf\u2061x\u2208X\u03bc1};for each (iii)either1) < inf\u2061x\u2208X\u03bc1} for some N0 \u2208 \u2329D\u232a and \u03c6N : \u0394n \u2192 2X is a compact set-valued mapping for each N = {u0, u1,\u2026, un}\u2208\u2329D\u232a or2) relative to LN andfor each Let  \u2265 inf\u2061x\u2208X\u03bc1.Then there exists \u03bc1\u2032 = \u2212\u03bc1 and \u03bd1\u2032 = \u2212\u03bd1, we can see that the conclusion of By setting Y; \u03c6N11) and  be two FWC-spaces, where X and Y are two Hausdorff topological spaces. Let Y \u00d7 X; \u03c6N11 \u00d7 \u03c6N22) is an FWC-space defined as in \u03bc1 be a real-valued function on Y \u00d7 X. Assume thatz, w) \u2208 Y \u00d7 X, each  \u2208 T, and each \u03b1 \u2208 \u211d, \u03bc1 \u2264 \u03b1 or \u03bc1 \u2265 \u03b1;for each  \u2208 Y \u00d7 X and each \u03b1 \u2208 \u211d, the sets {x \u2208 X : \u03bc1 \u2264 \u03b1} and {y \u2208 Y : \u03bc1 \u2265 \u03b1} are compactly closed;for each  > \u03b1} is an FWC-subspace of ;for each y \u2208 Y and each \u03b1 \u2208 \u211d, the set {w \u2208 X : \u03bc1 < \u03b1} is an FWC-subspace of .for each Let  = sup\u2061y\u2208Yinf\u2061x\u2208X\u03bc1.Then inf\u2061\u03b1 \u2208 \u211d such thaty, x) \u2208 Y \u00d7 X, there exists \u03c5 : (Y \u00d7 X)\u00d7(Y \u00d7 X) \u2192 \u211d by\u03c5, for each  \u2208 Y \u00d7 X, each  \u2208 T, we have \u03c5, ) \u2264 0. For each  \u2208 Y \u00d7 X, we havez, w) \u2208 Y \u00d7 X, the set { \u2208 Y \u00d7 X : \u03c5, ) \u2264 0} is compactly closed.It is clear that the following inequality4 of Tan , we defiy, x) \u2208 Y \u00d7 X, the set { \u2208 Y \u00d7 X : \u03c5, ) > 0} is an FWC-space of . In fact, for each  \u2208 Y \u00d7 X, we have the following:N = N1 \u00d7 N2 = {, ,\u2026, }\u2208\u2329Y \u00d7 X\u232a and each {, ,\u2026, }\u2286{ \u2208 Y \u00d7 X : \u03c5, ) > 0}\u2229N, we have\u03c6N(\u0394k) = \u03c6N11(\u0394k) \u00d7 \u03c6N22(\u0394k), it follows from (\u03c6N(\u0394k)\u2286{ \u2208 Y \u00d7 X : \u03c5, ) > 0}, which implies that for each  \u2208 Y \u00d7 X, the set { \u2208 Y \u00d7 X : \u03c5, ) > 0} is an FWC-subspace of . Thus, by X = D and \u03bc1 = \u03bd1, there exists z, w) \u2208 Y \u00d7 X. Hence, for each  \u2208 Y \u00d7 X, either x\u2208Xsup\u2061y\u2208Y\u03bc1 = sup\u2061y\u2208Yinf\u2061x\u2208X\u03bc1. This completes the proof.Now, we show that for each ( we have\u03c6N11(\u0394k)\u2286tradicts . TherefoFWC-spaces which contain G-convex spaces adopted in Theorem 4.4 of Tan [G-co\u2061(A) and each G-co\u2061(B) in Theorem 4.4 of Tan [4 of Tan ; (b) the4 of Tan  are comp4 of Tan ; (d) (ii4 of Tan ."}
{"text": "We introduce the notion of abelian fuzzy subsets on a groupoid, and we observe a variety of consequences which follow. New notions include, among others, diagonal symmetric relations, several types of quasi orders, convex sets, and fuzzy centers, some of whose properties are also investigated. Fuzzy commutative algebra, presented a fuzzy ideal theory of commutative rings, and applied the results to the solution of fuzzy intersection equations. The book included all the important work that has been done on L-subspaces of a vector space and on L-subfields of a field. The notion of a fuzzy subset of a set was introduced by Zadeh . His semX) and obtained a semigroup structure. Fayoumi  is said to be  abelian fuzzy if \u03bc(x\u2217y) = \u03bc(y\u2217x) for all x, y \u2208 X. Let  be a left-zero-semigroup; that is, x\u2217y = x for all x, y \u2208 X. Let \u03bc : X \u2192  be an abelian fuzzy subset of X. Then, \u03bc(x) = \u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(y) for all x, y \u2208 X. It follows that \u03bc is a constant map. Let  \u2208 ZBin(X), then every abelian fuzzy subset on  is a constant function. If . Then, \u03bc(x\u2217y) = \u03bc(y\u2217x) for all x, y \u2208 X. Let  \u2208 ZBin(X). By x \u2260 y in X, then  is either a left-zero-semigroup or a right-zero-semigroup. It follows that either x\u2217y = x, y\u2217x = y or x\u2217y = y, y\u2217x = x. In either cases, we obtain \u03bc(x) = \u03bc(y) for all x, y \u2208 X, proving that \u03bc is a constant function. Assume that X, \u2217) \u2208 Bin(X), we denote the set of all abelian fuzzy subgroupoids on  by A. Given a groupoid  \u2208 Bin(X). Then,  is commutative if and only if A = X. Let  is not commutative; that is, there exist x, y \u2208 X such that x\u2217y\u2009\u2009\u2260y\u2217x. If we let \u03b1 : = x\u2217y and let \u03bc : = \u03c7\u03b1 be the characteristic function of \u03b1, then \u03bc(x\u2217y) = 1, \u03bc(y\u2217x) = 0, proving that \u03bc is not an abelian fuzzy subset of . The converse is straightforward. Assume that  \u2208 Bin(X), we define a fuzzy subset \u03bc\u03b1 : X \u2192  by \u03bc\u03b1(x): = \u03b1 for all x \u2208 X, where \u03b1 \u2208 . Denote by C: = {\u03bc\u03b1 | \u03b1 \u2208 }. Then, C\u2286A for all groupoids  whatsoever. Thus, the extreme of non-commutativity is the situation C = A. Given . If \u03bc is one-one, then  is commutative. Let \u03bc \u2208 A, then \u03bc(x\u2009\u2009\u2217\u2009\u2009y) = \u03bc(y\u2217\u2009x) for all x, y \u2208 X. Since \u03bc is one-one, we have x\u2217y = y\u2217x for all x, y \u2208 X. If X, \u2217) \u2208 Bin(X), we define a set a : = { \u2208 X \u00d7 X | x\u2217y = y\u2217x}. Note that \u2208a implies \u2208a as well. If we let \u03bd : = \u03c7X,\u2217)aa, then \u03bd is an abelian fuzzy subgroupoid on X \u00d7 X. Given a\u03bc : = { \u2208 X \u00d7 X | \u03bc(x\u2217y) = \u03bc(y\u2217x)}, then X, \u2217)a\u2286a\u03bc, a = a\u03bc, if \u03bc is constant, then a\u03bc = X \u00d7 X. if Let It is straightforward. X, \u2217) \u2208 Bin(X), then there exists a fuzzy subset \u03bc of X such that a = a\u03bc. If  \u2208 Bin(X) such that a \u2260 a\u03bc for any fuzzy subset \u03bc of X. Then, there exists an element \u2208a\u03bc\u2216a. It follows that \u03bc(x\u2217y) = \u03bc(y\u2217x), but x\u2217y \u2260 y\u2217x for some x, y \u2208 X. If we let \u03b1\u2236 = x\u2217y and let \u03bc\u2236 = \u03c7\u03b1 be the characteristic function of \u03b1, then \u03bc(x\u2217y) = 1, but \u03bc(y\u2217x) = 0, which proves that \u2209a\u03bc. Assume that there exists  \u2208 Bin(X) and let \u00b7 be the usual product on the set of real numbers. If \u03bc : \u2192 is a homomorphism, then \u03bc \u2208 A and a\u03bc = X \u00d7 X. Let \u2192 is a homomorphism, then \u03bc(x\u2217y) = \u03bc(x) \u00b7 \u03bc(y) = \u03bc(y) \u00b7 \u03bc(x) = \u03bc(y\u2217x) for all x, y \u2208 X; that is, \u03bc \u2208 A.If x, y) \u2208 X \u00d7 X, then \u03bc(x\u2217y) = \u03bc(x) \u00b7 \u03bc(y) = \u03bc(y\u2217x), proving that \u2208a\u03bc. If  \u2208 Bin(X). Define a binary operation \u201c\u22c6\u201d on . Let (Bin(X). The relation \u2264 is a quasi order on X, \u2217)a\u03bc\u2286a\u03bc for any fuzzy subset \u03bc : X \u2192 , we have \u2264 for all  \u2208 Bin(X).Since \u2264 and \u2264, then a\u03bc\u2286a\u03bc and a\u03bc\u2286a\u03bc, and hence, a\u03bc\u2286a\u03bc, for any fuzzy subset \u03bc : X \u2192 . It follows that \u2264. If (X), since a\u03bc = a\u03bc for any fuzzy subset \u03bc : X \u2192  does not imply  = . Note that the relation \u2264 described in X, \u2217) \u2208 Bin(X), we defineGiven (X)\u232a\u2236 = \u2009\u2009{ |  \u2208 Bin(X)}. If we define a relation \u2264q on \u2329Bin(X)\u232a byX), \u2264q\u232a) is a partially ordered set. For partially ordered sets, we refer to [Let \u2329Bin] =  in \u2329Bin(X)\u232a, then A = A. If  = . Then, a\u03bc = a\u03bc for any fuzzy subset \u03bc : X \u2192 . It follows thatA = A. Assume that , then a\u03bc\u2229a\u03bd\u2286a\u03bb. Let x, y)\u2208a\u03bc\u2229a\u03bd, then \u03bc(x\u2217y) = \u03bc(y\u2217x) and \u03bd(x\u2217y) = \u03bd(y\u2217x) for all x, y \u2208 X. It follows thatx, y)\u2208a\u03bb.If  is a left-zero-semigroup. Define two fuzzy subsets \u03bc, \u03bd on X by \u03bc(x)\u2236 = x, \u03bd(x)\u2236 = 1 \u2212 x for all x \u2208 X. Then, \u03bc and \u03bd are one-one mappings. Let \u2208 be a real number such that 0 < \u2208<1/2, and let u\u2236 = (1/2)\u2212\u2208, v\u2236 = (1/2)+\u2208. Then, \u03bc(v) \u2212 \u03bc(u) = \u03bd(u) \u2212 \u03bd(v) \u2260 0. Let \u03bb\u2236 = (1/2)(\u03bc + \u03bd). Then,u, v)\u2208a\u03bb.Let X, \u2217) is a left-zero-semigroup and \u03bc is one-one, we have a\u03bc = { | \u03bc(x\u2217y) = \u03bc(y\u2217x)} = { | \u03bc(x) = \u03bc(y)} = \u25b3(X) = a\u03bd. This shows that \u2209\u25b3(X) = a\u03bc\u2229a\u03bd. Since . If \u03bb\u2236 = t\u03bc + (1 \u2212 t)\u03bd, where t \u2208 , and if a\u03bc = a\u03bd, then a\u03bc = a\u03bd\u2286a\u03bb. Let \u03bc, \u03bd be fuzzy subsets of . If \u03bbi\u2236 = ti\u03bc + (1 \u2212 ti)\u03bd, , where t1, t2 \u2208  with t1 \u2260 t2, then a\u03bb1\u2229a\u03bb2\u2286a\u03bc\u2229a\u03bd. Let x, y)\u2208a\u03bb1\u2229a\u03bb2, theni = 1,2. It follows that\u03bc(x\u2217y) \u2212 \u03bc(y\u2217x) = \u03bd(x\u2217y) \u2212 \u03bd(y\u2217x) = 0, so that \u2208a\u03bc\u2229a\u03bd. This proves the theorem. If , and let \u03bbi\u2236 = ti\u03bc + (1 \u2212 ti)\u03bd, , where t1, t2 \u2208  with t1 \u2260 t2. If a\u03bc = a\u03bd, then a\u03bb1\u2229a\u03bb2 = a\u03bc = a\u03bd. Let It follows immediately from \u03bc, \u03bd of X and groupoids  and  such thatWe give a pause to find some examples of fuzzy subsets X : = {a, b, c} with the tables:\u03bc, \u03bd on X by \u03bc(a) = 1, \u03bc(b) = 1/2, \u03bc(c) = 1/3 and \u03bd(a) = 1, \u03bd(b) = \u03bd(c) = 1/2. Then, it is easy to see that a = {, , } = a\u03bc, a\u03bd = a\u03bd = a\u03bd = X \u00d7 X. Hence, a\u03bc\u2286a\u03bc, a\u03bd = a\u03bd, and a\u03bc\u2286a\u03bd, a\u03bd\u2286a\u03bd. Let X, \u2217) \u2208 Bin(X), and let \u03bc be fuzzy subsets of X. Define a set Z\u03bc by\u03bc-center of . In Z\u03bc = \u2205, Z\u03bd = X. Let  \u2208 Bin(X). Then, x \u2208 Z\u03bc if and only if ({x} \u00d7 X)\u222a(X \u00d7 {x})\u2286a\u03bc. Let  \u2208 Bin(X), and let \u03bc : X \u2192  be a fuzzy subset of X. Define a set OZ\u03bc by\u03bc-center, we obtain the following. Let \u2229Z\u03bc \u2260 \u2205, then \u03bc is a constant function. If x \u2208 OZ\u03bc\u2229Z\u03bc, then \u03bc(x\u2217y) = \u03bc(y\u2217x) for all y \u2208 X. Since x \u2208 OZ\u03bc, we have \u03bc(x) = \u03bc(y) for all y \u2208 X, proving that \u03bc is a constant function. If \u03bc is a constant function, then OZ\u03bc = X = Z\u03bc. If \u03bc is a constant function, then \u03bc(x\u2217y) = \u03bc(y\u2217x) for all x, y \u2208 X. It follows that Z\u03bc = X.x \u2209 OZ\u03bc. Then, there exists an y \u2208 X such that \u03bc(x\u2217y) = \u03bc(y\u2217x), but \u03bc(x) \u2260 \u03bc(y). Since \u03bc is a constant function, we obtain \u03bc(x) = \u03bc(y) \u2260 \u03bc(x), a contradiction. Hence, OZ\u03bc = X.Assume that"}
{"text": "We prove a Liouville property of holomorphic maps from a complete K\u00e4hler manifold with nonnegative holomorphic bisectional curvature to a complete simply connected K\u00e4hler manifold with a certain assumption on the sectional curvature. Liouville property is an interesting topic in analysis since Liouville found that a bounded holomorphic function on the complex plane must be a constant function. It has been studied by many geometers. For example, Yau  studied In this paper, we prove the following Liouville property of holomorphic maps. This is, in some sense, a generalization of Yau's Liouville property for holomorphic maps. M be a complete K\u00e4hler manifold with nonnegative holomorphic bisectional curvature and N a complete simply connected K\u00e4hler manifold with sectional curvature \u2264\u2212c/(1+r)2, where c is some positive constant and r is the distance function of N to a fixed point o. Then, any holomorphic map from M to N must be constant. Let N which is guaranteed by the classical Hessian comparison. Then, by using Ni-Tam's [Our strategy to prove this result is by first showing that there are plenty of bounded pluri-subharmonic functions on the target Ni-Tam's  LiouvillIn order to show the existence of enough bounded plurisubharmonic functions on complete simply connected K\u00e4hler manifolds with negative quadratically decayed sectional curvature, we need the following technical lemma. c, there is a bounded function f \u2208 C2) such that f(0) = 0 and f(r) > 0 for r > 0;f\u2032(r) > 0 for r \u2265 0;f\u2032\u2032(r) + c/(1 + r)coth\u2061(cr/(1 + r))f\u2032(r) \u2265 0 for r \u2265 0. For any positive constant r \u2192 \u221e and ccoth\u2061c > 1. There are two positive numbers R and \u03b4, such thatr \u2265 R. Leth \u2208 C). Moreover,r \u2265 R. Letf \u2208 C2) satisfying (1) and (2). Moreover, when r \u2265 R,f is bounded.Note thatFinally,Mn, g) be a complete simply connected K\u00e4hler manifold with sectional curvature \u2264\u2212c/((1+r)2), where c is some positive constant and r is the distance function of M with respect to a fixed point o \u2208 M. Then, for any point p \u2208 M, there is a bounded continuous plurisubharmonic function \u03d5 such that \u03d5 > 0 on M\u2216{p} and \u03d5(p) = 0. Let 2, where c1 is some positive constant. By Hessian comparison, we havee1, e2,\u2026, en be a parallel unitary frame along geodesic rays emanating from p. Thenrp)\u03b1 = 0 for any \u03b1 > 1.Let f be a bounded function in C2) such that f(0) = 0 and f(r) > 0 for r > 0; f\u2032(r) > 0 for r \u2265 0; r \u2265 0.  Let \u03d5(x) = f(rp(x)). Then, by (\u03d5 is a bounded continuous pluri-subharmonic function on M with \u03d5 > 0 on M\u2216{p} and \u03d5(p) = 0. By Then, by , we haveWe are now ready to prove f : M \u2192 N be a nonconstant holomorphic map. Then, there are two points p, q in M, such that f(p) \u2260 f(q). By \u03d5 be a bounded continuous pluri-subharmonic function on N such that \u03d5(f(p)) = 0, \u03d5(f(q)) > 0. Then, \u03d5(f) is a bounded continuous pluri-subharmonic function on M that is nonconstant. This contradicts Theorem\u2009\u20093.2 in [We proceed by contradiction. Let \u2009\u20093.2 in . \u2102n constructed by Seshadri [r2log\u2061\u2061r) outside a compact subset. This means that the curvature decayed rate cannot be raised to be greater than 2 in The unitary invariant metric on Seshadri  has nega"}
{"text": "The bifurcations ofheteroclinic loop with one nonhyperbolic equilibrium and onehyperbolic saddle are considered, where the nonhyperbolicequilibrium is supposed to undergo a transcritical bifurcation;moreover, the heteroclinic loop has an orbit flip and an inclinationflip. When the nonhyperbolic equilibrium does not undergo atranscritical bifurcation, we establish the coexistence andnoncoexistence of the periodic orbits and homoclinic orbits. Whilethe nonhyperbolic equilibrium undergoes the transcriticalbifurcation, we obtain the noncoexistence of the periodic orbits andhomoclinic orbits and the existence of two or three heteroclinicorbits. In recent years, a great deal of mathematical efforts has been devoted to the bifurcation problems of homoclinic and heteroclinic orbits with high codimension, for example, the bifurcations of homoclinic or heteroclinic loop with orbit flip, the bifurcations of homoclinic or heteroclinic loop with inclination flip, and so forth; see \u20135 and thCr (r \u2265 5) systemz \u2208 \u211d4, the vector field\u2009\u2009g\u2009\u2009depends on the parameters , \u03bb \u2208 \u211d, \u03bc \u2208 \u211dl, l \u2265 2, 0 \u2264 \u03bb, |\u03bc | \u226a1, g = f(z), g = 0, and g = 0. Moreover, the parameter\u2009\u2009\u03bb\u2009\u2009governs bifurcation of the nonhyperbolic equilibrium, while\u2009\u2009\u03bc\u2009\u2009controls bifurcations of the heteroclinic orbits.Consider the following p1, p2, where \u0393 = \u03931\u22c3\u03932, \u0393i = {z = ri(t) : t \u2208 \u211d}, ri(+\u221e) = ri+1(\u2212\u221e) = pi+1, i = 1,2, r3(t) = r1(t), and p3 = p1. Furthermore, the linearization\u2009\u2009Df(p1)\u2009\u2009has real eigenvalues\u2009\u20090, \u03bb11, \u2212\u03c111, and \u2212 \u03c112\u2009\u2009satisfying \u2212 \u03c112 < \u2212\u03c111 < 0 < \u03bb11; Df(p2) has simple real eigenvalues \u03bb21, \u03bb22, \u2212\u03c121, and \u2212\u03c122 fulfilling \u2212\u03c122 < \u2212\u03c121 < 0 < \u03bb21 < \u03bb22.Assuming system  has a heH1) is the unstable  manifold of pi, and Wiuu  is the strong unstable  manifold of pi, i = 1,2. Moreover,H2) manifold W2s  of\u2009\u2009p2\u2009\u2009are fulfilling the strong inclination property. And the fourth equation implies that the stable manifold W1s is of inclination flip as t \u2192 \u2212\u221e.The following conditions hold in the whole paper:m \u2265 1 and n \u2265 1, if we assume dim\u2061(W1u) = dim\u2061(W2uu) = m and dim\u2061(W1ss) = dim\u2061(W2ss) = n, then all the results achieved in this paper are still valid.It is worthy of noting that, for any integers \u03bb \u2208 \u211d be a parameter to control the transcritical bifurcation of system  be the vector field defined on the center manifold; then by [H3) = 0, (\u2202\u03b8/\u2202x) = 0, (\u22022\u03b8/\u2202x2) > 0, (\u22022\u03b8/\u2202x\u2202\u03bb) < 0, (\u22022\u03b8/\u2202x\u2202\u03bc) = 0, where xp1 is the x component of p1.Let f system , let the then by , we may H3) is true, then system ; there are two hyperbolic saddles p10 and p11 bifurcated from p1. Denote by p10 = p1 = * and p11 = p1 + *, where \u03bbp = \u03b80\u03bb + O(\u03bb2) + O(\u03bb\u03bc) and \u03b80 = \u2212(\u22022\u03b8/\u2202x\u2202\u03bb)/(\u22022\u03b8/\u2202x2). Moreover, dim\u2061(Wp10s) = 3, dim\u2061(Wp10u) = 1, and dim\u2061(Wp11u) = dim\u2061(Wp11s) = 2.If x and \u03bb \u2192 \u2212\u03b8x\u03bb\u22121\u03bb, system , \u03bbp = \u03bb + O(\u03bb2) + O(\u03bb\u03bc), \u03bb11(0) = \u03bb11, \u03c1ij(0) = \u03c1ij, j = 1,2,\u2009\u2009i = 1,2, \u03bb2j(0) = \u03bb2j,\u2009\u2009j = 1,2.Let the neighborhood H1), we may select \u2212Ti and Ti such that\u03b4 > 0 is small enough such that {:|x | , |y | , |u | , |v | <2\u03b4} \u2282 Ui and |\u03b4u| = o(\u03b4), |\u03b4v | = o(\u03b4).From the normal form , 7), an, an7), ai = 1,2, where (Df(ri(t)))* is the transposed matrix of Df(ri(t)).Consider the linear variational systemZi(t) = (zi1(t), zi2(t), zi3(t), zi4(t)) is a fundamental solution matrix of\u2009\u2009\u2013(H3) are satisfied, thenIf conditions (1 satisfying(1) there exists a fundamental solution matrix of (9)2 has a fundamental solution matrix fulfillingwi21 < 0, w112wi33w214w242 \u2260 0, |(wi33)\u22121wij3 | \u226a1, j = 1,2, 4, i = 1,2.(2)\u2009\u2009(9)zi1(t), zi2(t), zi3(t), zi4(t)) be a new local active coordinate system along \u0393i. Given \u03a6i(t) = (\u03d5i1(t), \u03d5i2(t), \u03d5i3(t), \u03d5i4(t)) = (Zi\u22121(t))*, then \u03a6i(t) is the fundamental solution matrix of  + Zi(t)Ni(t)\u225chi(t), where Ni(t) = *, i = 1,2. Defining the cross sectionsi at t = \u2212Ti\u2009\u2009and t = Ti, respectively, i = 1,2.Let qi0 \u2208 Si0 and qi1 \u2208 Si1, thenZi(\u2212Ti) and Zi(Ti), we get their new coordinates of qi0* and qi1*; that is,Now that if Next, we divide our establishment of the Poincar\u00e9\u2009\u2009map in the new coordinate system in three steps.Fi1 : Si0 \u21a6 Si1. Put z = hi(t) into  = Zi\u22121(t), thenFi1 : Si0 \u21a6 Si1, \u21a6.First, consider the map her with  and ((21Fi0 : Si\u221211 \u21a6 Si0 (where\u2009\u2009S01 = S21). Let \u03c4i,\u2009\u2009i = 1,2 be the flying time from qi\u221211* to qi0*; set s1 = e\u03c111(\u03b1)\u03c41\u2212 and s2 = e\u03c121(\u03b1)\u03c42\u2212. By virtue of the approximate solution of system ,,Fi0 : F20 : S11 \u21a6 S21 issi, ui0, vi\u221211), i = 1,2 are called Shilnikov coordinates, andp1 undergoes a transcritical bifurcation based on the structure of orbits in U1, we may see that the equation x01 \u2248 x10/h(s1) holds only for x01 \u2265 \u03bbp. While for x01 \u2208 [\u2212\u03b2, \u03bbp)\u2009\u2009(0 < \u03b2 \u226a 1), the map F10 is well defined only if s1 = 0  asThe final step is to compose the maps Gi = Fi(qi\u221211) \u2212 qi1, i = 1,2. Combing  means that system ,,Gi = F Combing , 27), a, aGi =  Combing , we deritions of = 0 has a unique solution . And putting it into  = 0, then we obtain the following bifurcation equations:Based on the expressions of the successor functions and the implicit function theorem, we know that the equation \u22121\u03b4ln\u2061s1) = 0, it then follows thatFirstly, we consider the case ppen. By  and 25)\u03bb = 0, whppen. By  turns toFrom the above bifurcation equations, we obtain the following results immediately.H1)\u2013(H3) be true and Mi\u03bc1 \u2260 0, i = 1,2. Then, for \u03bb = 0 and 0 < |\u03bc| \u226a 1, one hasLet the conditions  = 2, there exists a codimension 2 surface\u03bc \u2208 L12, where the surface L12 has a normal plane span{M\u03bc11, M\u03bc21} at \u03bc = 0.(i) for rank\u2061(l \u2212 1)-dimensional surfacep1  near \u0393 if and only if \u03bc \u2208 L12 .(ii) there exists an l \u2212 -dimensios1 = s2 = 0 into (The result (i) will be proved by putting = 0 into .s1 = 0 and s2 > 0 in -dimensional surface L12 given by  > 0 as\u2009\u2009\u03bc \u2208 L12 and 0 < |\u03bc | \u226a1. This implies system  it has a normal vector M\u03bc11  at \u03bc = 0.There is no difficulty to see that H1)\u2013(H3) hold and Mi\u03bc1 \u2260 0, i = 1,2. Then for \u03bb = 0, \u03bc \u2208 L12, and 0 < |\u03bc | \u226a1, the periodic orbit and homoclinic loop with p1 of system (Assume the conditions H\u2013(H3) ho\u03bc \u2208 L12 and 0 < |\u03bc | \u226a1, then system \u22121s1 + \u03b4\u22121M\u03bc11\u03bc + h.o.t.>0, and \u03bc \u2208 L12, then (V1(0) = N1(0) and2, then  is reducw112w212 < 0, then V1\u2032(s1)N1\u2032(s1) < 0; it is obvious that V1(s1) = N1(s1) has no sufficiently small positive solutions.If \u03c122 > \u03bb21, then |V1\u2032(s1)|\u226a1 and |N1\u2032(s1)|\u226b1 hold for 0 < s1 \u226a 1, which shows that V1(s1) = N1(s1) has no sufficiently small positive solution.While \u03c122 \u2264 \u03bb21 and w112w212 > 0. As \u03bc \u2208 L12, we have M\u03bc11\u03bc > 0, and then, for wi12 > 0, i = 1,2 we see that\u03c122 < \u03bb21 yields that lim\u2061s1\u21920+s1\u03c122/\u03bb21\u22121 = +\u221e, lim\u2061s1\u21920+N1\u2032(s1) = +\u221e, and lim\u2061s1\u21920s1\u03c122/\u03bb21\u22121/N1\u2032(s1) = 0, which shows V1(s1) = N1(s1) has no sufficiently small positive solutions. Obviously, the conclusion is correct as\u2009\u2009\u03c122 = \u03bb21.Next, we only consider the case \u03c122 < \u03bb21, wi12 < 0, i = 1,2, there does not exist a small positive solution for V1(s1) = N1(s1).Similarly, for The proof is then completed.H1)\u2013(H3) hold and Mi\u03bc1 \u2260 0, i = 1,2. Let \u03bb = 0, \u03bc \u2208 L21, and 0 < |\u03bc | \u226a1; then p2 of system  = N2(0) as \u03bc \u2208 L21. Moreover,(i) Let \u2009s1\u2009\u2009in , we deri\u03c122 \u2265 \u03bb21 and N2\u2032(s2) \u226a 1 = V2\u2032(s2), this means V2(s2) = N2(s2) has no sufficiently small positive solutions.For w112w212 < 0, since we are interested in sufficiently small positive solutions of (V2(s2) = N2(s2) satisfying w112(s2\u03bb21/\u03c122 \u2212 \u03b4\u22121M\u03bc11\u03bc) > 0, which implies that s2\u03bb21/\u03c122 \u2212 \u03b4\u22121M\u03bc11\u03bc < 0  for w112 < 0 . It is easy to see that V2(s2) = N2(s2) has no sufficiently small positive solutions as w112w212 < 0.Now we turn to the case tions of , it suff\u03c122 < \u03bb21, we have V2\u2032(0) = 1 > 0 = N2\u2032(0), which implies that there exists an V2(s2) > N2(s2) for (ii) For w112 > 0, sow212 > 0. As a result, N2(s2) = V2(s2) has at least one solution Choosing s2 must fulfill 0 < s2 < (\u03b4\u22121M\u03bc11\u03bc)\u03c122/\u03bb21 as w112 < 0; with similar arguments in proof of (ii), we can prove that there exists a 0 < s2* \u226a 1 such that V2(s2*) = N2(s2*) for 0 < s2* < (\u03b4\u22121M\u03bc11\u03bc)\u03c122/\u03bb21 \u226a 1. It is easy to compute that N2\u2032\u2032(s2) > 0 for w212 < 0, 0 < s2 < (\u03b4\u22121M\u03bc11\u03bc)\u03c122/\u03bb21, and \u03bc \u2208 L21. Combining with the fact V2(0) = N2(0), N2\u2032(s2) > 0, and V2\u2032(s2) = 1, we immediately know that s2* is unique.(iii) This completes the proof.\u03bb > 0, when p1 undergoes a transcritical bifurcation. From \u03bb > 0, after the creation of the equilibria p10 and p11, there always exists a straight segment orbit heteroclinic to p11 and p10, its length is \u03bbp, and we denote this heteroclinic orbit by \u0393*. Moreover, x01 = \u03bbp is a critical position.Now, we turn to discussing the bifurcations of the heteroclinic loop for x01 \u2265 \u03bbp. In this case, (s = s1\u03bbp/\u03c111 (s = 0 means s1 = 0 and vice versa); by virtue of Taylor's development for \u03b4\u03bbp/(\u03b4 \u2212 (\u03b4 \u2212 \u03bbp)s1\u03bbp/\u03c111), we haveFirstly, we take into account the case is case,  becomes\u03bb = 0, we may easily obtain the following results.With similar arguments to H1)\u2013(H3) hold, 0 < \u03bb \u226a 1; thenSuppose the conditions  = 2, there exists an\u2009\u2009(l \u2212 2)-dimensional surface\u03bc \u2208 L12\u03bb and 0 < |\u03bc | \u226a1;(i) if rank\u2061(l \u2212 1)-dimensional surfacep11  if and only if \u03bc \u2208 L\u03bb12 and 0 < |\u03bc | \u226a1.(ii) there exists an l \u2212 -dimensioH1)\u2013(H3) hold, Mi\u03bc1 \u2260 0, i = 1,2, 0 < \u03bb, |\u03bc | \u226a1, and w112w212 < 0. Then, except the homoclinic loop connecting p11 , system .Suppose hypotheses H\u2013(H3) hop10 and heteroclinic loop joining p10, p2 cannot be bifurcated from \u0393, which is exactly determined by the generic condition (H1).It is easy to see that homoclinic loop connecting \u03b2 \u2264 x01 < \u03bbp. Due to Finally, we consider the case \u2212H1)\u2013(H3) are true, rank\u2061 > 0 and rank\u2061 > 0. Then, (i)p11, p2, p10 as \u03bc \u2208 \u03a31;there exists a surface5\u03a31={(ii)\u03bb, \u03bc) spacep11 and p10 for  \u2208 \u0394.there exists a region in the :\u2212\u03b2\u2264w22\u22121\u03b4 such that , then system (p11\u2009\u2009and\u2009\u2009p2 and the other is heteroclinic to p2 and\u2009\u2009p10.(i) If 2 = 0 in , then \u2208 \u0394, system (p11 and p10.(ii) If 2 > 0 in , one attg s2 in , we achip10 will go into p10 in different ways according to different fields of x01; see All the heteroclinic orbits joining"}
{"text": "Two types of distributed time delays are incorporated into the model to describe the time needed for infection of target cell and virus replication. Using the method of Lyapunov functional, we have established that the global stability of the model is determined by two threshold numbers, the basic reproduction number R0and the immune response reproduction number R0\u2217. We have proven that, if R0 \u2264 1, then the uninfected steady state is globally asymptotically stable (GAS), if R0* \u2264 1 < R0, then the infected steady state without CTL immune response is GAS, and, if R0* > 1, then the infected steady state with CTL immune response is GAS.We study the global stability of a human immunodeficiency virus (HIV) infection model with Cytotoxic T Lymphocytes (CTL) immune response. The model describesthe interaction of the HIV with two classes of target cells, CD4 Mathematical modeling and model analysis of HIV dynamics are important to discover the dynamical behaviors of the viral infection process and estimating key parameter values which leads to development of efficient antiviral drug therapies. Several mathematical models have been proposed to describe the HIV dynamics with CD4+ T cells , and\u2009\u2009hi\u2009\u2009is limit superior to this delay. The factor\u2009\u2009emi\u03c4\u2212\u2009\u2009accounts for the loss of target cells during this delay period, where\u2009\u2009mi\u2009\u2009is constant. On the other hand, to consider the delay between viral RNA transcription and viral release and maturation, we let\u2009\u2009\u03c4\u2009\u2009be the random variable; that is, the time between these two events with a probability distribution\u2009\u2009g(\u03c4)\u2009\u2009over the interval\u2009\u2009, and\u2009\u2009h3\u2009\u2009is limit superior to this delay \u2009\u2009into\u2009\u2009R+6. By the fundamental theory of functional differential equations \u2009\u2009for some constant\u2009\u2009\u03c1\u2009\u2009and let\u2009\u2009ti* \u2208 \u2009\u2009be such that\u2009\u2009xi(ti*) = 0.\u2009\u2009From (xi(t) > 0\u2009\u2009for some\u2009\u2009t \u2208 , where\u2009\u2009\u03f5 > 0\u2009\u2009is sufficiently small. This leads to a contradiction and hence\u2009\u2009xi(t) > 0, for all\u2009\u2009t \u2265 0.\u2009\u2009Further, from (yi(t) \u2265 0,\u2009\u2009i = 1,2, and\u2009\u2009v(t) \u2265 0\u2009\u2009for all\u2009\u2009t \u2208 . By a recursive argument, we obtain\u2009\u2009yi(t) \u2265 0,\u2009\u2009i = 1,2, and\u2009\u2009v(t) \u2265 0\u2009\u2009for all\u2009\u2009t \u2265 0.\u2009\u2009Now from (z(t) \u2265 0, for all\u2009\u2009t \u2265 0.First, we prove that\u2009\u20090.\u2009\u2009From  we have\u2009er, from  and 7)  xi(t) >Now from  we havei = 1,2. This implies\u2009\u2009lim\u2061sup\u2061t\u2192\u221exi(t) \u2264 \u03bbi/di,\u2009\u2009i = 1,2.Next we show the boundedness of the solutions of system \u20138). Fro. Fro8). X(t) = \u2211i=12[Fixi(t) + yi(t + \u03c4)] + (p/c)z(t + \u03c4). Then\u03c3 = min\u2061{d1, d2, a, b}. Hence\u2009\u2009lim\u2061sup\u2061t\u2192\u221eX(t) \u2264 L, where\u2009\u2009L = \u2211i=12Li = \u2211i=12(Fi\u03bbi/\u03c3). Since\u2009\u2009xi(t) > 0,\u2009\u2009yi(t) \u2265 0\u2009\u2009and\u2009\u2009z(t) \u2265 0 then\u2009\u2009lim\u2061sup\u2061t\u2192\u221e\u2211i=12yi(t) \u2264 L,\u2009\u2009lim\u2061sup\u2061t\u2192\u221eyi(t) \u2264 L,\u2009\u2009i = 1,2\u2009\u2009and\u2009\u2009lim\u2061sup\u2061t\u2192\u221ez(t) \u2264 cL/p. On the other hand,t\u2192\u221ev(t) \u2264 kGL/r. Therefore,\u2009\u2009x1(t), y1(t), x2(t), y2(t), v(t), and\u2009\u2009z(t)\u2009\u2009are ultimately bounded.Let\u2009\u2009R0\u2009\u2009and immune response reproduction number\u2009\u2009R0*\u2009\u2009of system  as.(i) If\u2009\u2009R0 > 1, then there exist\u2009\u2009E0\u2009\u2009and an infected steady state without CTL immune response(ii) If\u2009\u2009R0* > 1, then there exist\u2009\u2009E0,\u2009\u2009E1, and an infected steady state with CTL immune response\u2009\u2009E2 = .(iii) If\u2009\u2009z = 0\u2009\u2009orz = 0, then from (xi\u2009\u2009and\u2009\u2009yi\u2009\u2009asv = 0\u2009\u2009or\u2009\u2009\u2211i=12(kGFi\u03b2i\u03bbi/a(di + \u03b2iv)) \u2212 r = 0.The steady states of \u20138) sati sati8) syi\u2212rv=0,\u2211i=12cyiz0.From , where\u2009\u2009xi0 = \u03bbi/di,\u2009\u2009i = 1,2. If\u2009\u2009v \u2260 0, then we have\u03b4i = \u03b2i/di,\u2009i = 1,2.\u2009\u2009The solution of .If\u2009\u2009ng it in  leads to we have\u2211i=12kGFin write (R011+\u03b41v+n system \u20138) has  has v = hen from  we haveit into (v\u2217=kGbrc.rc.From  and 20)v = 0, then from \u2009\u2009for any\u2009\u2009u \u2208 {xi, yi, v, z, i = 1,2}. We also define a function\u2009\u2009H : \u2192[0, \u221e)\u2009\u2009as\u2009\u2009H(u) = u \u2212 1 \u2212 ln\u2061u. It is clear that\u2009\u2009H(u) \u2265 0\u2009\u2009for any\u2009\u2009u > 0\u2009\u2009and\u2009\u2009H\u2009\u2009has the global minimum\u2009\u2009H(1) = 0.In this section, we establish the global stability of the three steady states of system \u20138) empl empl8) eR0 \u2264 1, then\u2009\u2009E0\u2009\u2009is GAS.If\u2009\u2009W0\u2009\u2009as follows:\u03b3i = kGFi/a,\u2009\u2009i = 1,2.Define a Lyapunov functional\u2009\u2009W0\u2009\u2009along the trajectories of  = 0. Since\u2009\u2009yi \u2265 0\u2009\u2009for\u2009\u2009i = 1,2, then\u2009\u2009y1 = y2 = 0. Hence\u2009\u2009dW0/dt = 0\u2009\u2009if and only if\u2009\u2009xi = xi0,\u2009\u2009yi = 0,\u2009\u2009i = 1,2,\u2009\u2009v = 0, and\u2009\u2009z = 0. From LaSalle's Invariance Principle,\u2009\u2009E0\u2009\u2009is GAS.The time derivative of\u2009\u2009ories of \u20138) sati satiW0\u2009ries of  we  we (39)Wz.From , we, we56)We obtaindW2dt=\u2211i=e obtaindW2dt=\u2211i=.3.1\u2009\u2009in , the sol6We obtai obtain5dW2dt=\u2211i=hen from  we have\u2009that is,yi\u2217xi(t\u2212\u03c4er, from  we have+ T cells and macrophages, taking into account the CTL immune response. Two types of distributed time delays have been incorporated into the model to describe the time needed for infection of target cell and virus replication. The global stability of the three steady states of the model has been established by constructing suitable Lyapunov functionals and using LaSalle's Invariant Principle. We have proven thatR0 \u2264 1, then the uninfected steady state\u2009\u2009E0\u2009\u2009is GAS;if\u2009\u2009R0 > 1 \u2265 R0*, then the infected steady state without CTL immune response\u2009\u2009E1\u2009\u2009is GAS;if\u2009\u2009R0 > R0* > 1, then the infected steady state with CTL immune response\u2009\u2009E2\u2009\u2009is GAS.if\u2009\u2009In this paper, we have proposed an HIV infection model describing the interaction of the HIV with two classes of target cells, CD4"}
{"text": "The purpose of this paper is to introduce the concept of partial rectangular metric spaces as a generalization of rectangular metric and partial metric spaces. Some properties of partial rectangular metric spaces and some fixed point results for quasitype contraction in partial rectangular metric spaces are proved. Some examples are given to illustrate the observed results. In 1906, the famous French mathematician Fr\u00e9chet  introducIn this paper, we generalize the concept of rectangular metric space and extend the concept of partial metric space by introducing the partial rectangular metric space. A fixed point theorem for quasitype contraction is also proved in the partial rectangular metric space which generalizes several known results in metric, partial metric, and rectangular metric spaces. Results are illustrated by some examples.First, we recall some definitions from partial metric and rectangular metric spaces see , 3)..3]).X is a mapping p : X \u00d7 X \u2192 \u211d such that, for all x, y, z \u2208 X,(P1)p \u2265 0;(P2)x = y if and only if p = p = p;(P3)p \u2264 p;(P4)p = p;(P5)p \u2264 p + p \u2212 p.A partial metric space is a pair  such that X is a nonempty set and p is a partial metric on X.A partial metric on a nonempty set X be a nonempty set and let d : X \u00d7 X \u2192 \u211d be a mapping such that(R1)d, for all x, y \u2208 X and d = 0, if and only if x = y;0 \u2264 (R2)d = d, for all x, y \u2208 X;(R3)d \u2264 d + d + d, for all x, y \u2208 X and for all distinct points w, z \u2208 X \u2212 {x, y} (rectangular property).Then, d is called a rectangular metric on X, and  is called a rectangular metric space. A sequence {xn} in X is called convergent and converges to x \u2208 X, if, for every \u025b > 0, there exists n0 \u2208 \u2115 such that d < \u025b for all n > n0. Sequence {xn} is called a Cauchy sequence if, for every \u025b > 0, there exists n0 \u2208 \u2115 such that d < \u025b, for all n, m > n0. A rectangular metric space  is called complete, if every Cauchy sequence in X converges in X.Let In this section, we define partial rectangular metric spaces and prove some properties of partial rectangular metric spaces.X be a nonempty set and let \u03c1 : X \u00d7 X \u2192 \u211d be a mapping such that\u03c11) \u2265 0, for all x, y \u2208 X;\u03c12) = \u03c1 = \u03c1, for all x, y \u2208 X;\u03c13) \u2264 \u03c1, for all x, y \u2208 X;\u03c14) = \u03c1, for all x, y \u2208 X;\u03c15) \u2264 \u03c1 + \u03c1 + \u03c1 \u2212 \u03c1 \u2212 \u03c1, for all x, y \u2208 X and for all distinct points w, z \u2208 X\u2216{x, y}.Then, \u03c1 is called a partial rectangular metric on X and the pair  is called a partial rectangular metric space.Let X, \u03c1), if x, y \u2208 X and \u03c1 = 0, then x = y but the converse may not be true.In a partial rectangular metric space  is a partial rectangular metric space, but it is not a rectangular metric space, because \u03c1 \u2260 0, for all x > 0. Also,  is not a partial metric space because it lacks the property (P5). Indeed,Let Next proposition shows that every partial rectangular metric space induces a rectangular metric space.X, \u03c1), the pair  is a rectangular metric space, whereFor each partial rectangular metric space  and (R2). For (R3), let x, y, z, w \u2208 X, where w and z are distinct and w, z \u2208 X\u2216{x, y}; then we haveX, \u03c1r) is a rectangular metric space.By definition of X, \u03c1r) is called the induced rectangular metric space, and \u03c1r is the induced rectangular metric. In further discussion until specified,  will represent induced rectangular metric space.Here,  be a rectangular metric space and there exists x0 \u2208 X such that d \u2264 d, for all distinct x, y \u2208 X. Then, the pair  is a partial rectangular metric space, where\u03c1 is the rectangular metric d.Let (\u03c11) and (\u03c14) follow by the definition of \u03c1. For (\u03c12), note that \u03c1 = d, for all x \u2208 X; therefore, if x, y \u2208 X and \u03c1 = \u03c1 = \u03c1, we have (1/2)[d + d + d] = d = d, which implies that d = 0; that is, x = y.The properties  follows immediately. For (\u03c15), let x, y, w, z \u2208 X, \u2009x \u2260 y, \u2009w, \u2009z \u2208 X\u2216{x, y}. Then, by definition of \u03c1, we haveX, \u03c1) is a partial rectangular metric space. Now, by the definition, it is easy to verify that d is the rectangular metric induced by \u03c1.By the choice of X = {1/n : n \u2208 \u2115} \u222a {0} and define d : X \u00d7 X \u2192 \u211d byX, d) is a rectangular metric space. Note that d \u2264 d for all distinct x, y \u2208 X and so \u03c1 : X\u2009\u00d7\u2009X \u2192 \u211d defined byX, \u2009 is a partial rectangular metric space, and d is the rectangular metric induced by \u03c1.Let The proof of following proposition is straightforward and, by using it, one can obtain some more examples of partial rectangular metric space.X, d) and constant \u03b1 \u2265 0, the pair  is a partial rectangular metric space, whereFor any rectangular metric space  be a partial rectangular metric space, {xn} a sequence in X, and x \u2208 X. Then,(i)xn} is said to be convergent and converges to x \u2208 X, if lim\u2061n\u2192\u221e\u03c1 = \u03c1;the sequence {(ii)xn} is said to be Cauchy in , if lim\u2061n,m\u2192\u221e\u03c1 exists and is finite;the sequence {(iii)X, \u03c1) is said to be a complete partial rectangular metric space, if, for every Cauchy sequence {xn} in X, there exists x \u2208 X such that is a partial rectangular metric space. Consider the sequence {xn} in X, where xn = 1/n. Then, lim\u2061n\u2192\u221e\u03c1 = lim\u2061n\u2192\u221e[1/n + \u03b1] = \u03b1 = \u03c1 and lim\u2061n\u2192\u221e\u03c1 = lim\u2061n\u2192\u221e[1/n + \u03b1] = \u03b1 = \u03c1. Therefore, {xn} has two limits, namely, 0 and 2.Let X, \u03c1) be a partial rectangular metric space and let {xn} be a sequence in X. Then, the sequence {xn} converges in  and converges to x \u2208 X; that is, lim\u2061n\u2192\u221e\u03c1r = 0, if and only if lim\u2061n\u2192\u221e\u03c1 = lim\u2061n\u2192\u221e\u03c1 = \u03c1.Let  be a partial rectangular metric space and let {xn} be a sequence in X. Then, the sequence {xn} is a Cauchy sequence in  if and only if it is a Cauchy sequence in .Let ; that is,\u03c1 \u2212 \u03c1|\u2264\u03c1r, for all n, m \u2208 \u2115; therefore, the sequence {\u03c1} is a Cauchy sequence in \u211d+ and so we have lim\u2061n\u2192\u221e\u03c1 \u2208 \u211d+. Therefore, it follows from  \u2208 \u211d+.Let {that is,lim\u2061n,m\u2192\u221exn} is a Cauchy sequence in , that is, lim\u2061n,m\u2192\u221e\u03c1 \u2208 \u211d+, then again lim\u2061n\u2192\u221e\u03c1 \u2208 \u211d+ and by definition lim\u2061n,m\u2192\u221e\u03c1r = 0.Conversely, if {The proof of the following lemma follows from Lemmas A partial rectangular metric space is complete, if and only if its induced rectangular metric space is complete.In this section, some fixed point theorems in partial rectangular metric spaces are proved.X, \u03c1) be a partial rectangular metric space and let T : X \u2192 X be a mapping. For A \u2282 X, we denote the diameter of A by \u03b4[A] andA = {x} \u2282 X, then \u03b4[A] need not be zero, but \u03b4[A] = \u03c1.Let  be a complete partial rectangular metric space and let T : X \u2192 X be a mapping. Then, T is called a quasicontraction on X with constant \u03bb, if it satisfies the following property:x, y \u2208 X, where \u03bb \u2208 . Furthermore there exists a positive integer k such that k \u2208 {1,2,\u2026, n} and \u03b4[O] = \u03c1.Let  and so, by , we have \u03c1 < \u03b4[O] and so, by the definition of orbit of T and the diameter \u03b4 of a set, we must have \u03b4[O] = \u03c1, where k \u2208 {1,2,\u2026, n}.Suppose d so, by , we haveThe next lemma shows that the orbit of a quasicontraction is necessarily bounded.x \u2208 X.Suppose that all the conditions of x \u2208 X be arbitrary. Note that {\u03b4[O]} is a nondecreasing sequence of real numbers, son = 1. If n = 2, then from (\u03c13) and , we must have \u03c1 \u2264 max\u2061{\u03c1, \u03c1}. Also, since \u03c1 \u2264 \u03c1, for all y, z \u2208 X, therefore ](\u03c13) and , we havehow that\u03b4[O]n \u2265 3, then, by \u03b4[O] = \u03c1, where k \u2208 {1,2,\u2026, n}. If k = 1 or k = 2, we have finished. Suppose k \u2265 3. If Tx = x or T2x = x or T2x = Tx, then we have Tkx \u2208 {x, Tx} and  \u2264 max\u2061{\u03c1, \u03c1}, therefore, from (\u03c15), ] = \u03c1 \u2264 \u03bb\u03b4[O] \u2264 \u03bb\u03b4[O] = \u03bb\u03c1; therefore, it follows from the above inequality thatSuppose  Tx} and  holds. Sx, then  holds. Tom (\u03c15), , 21), a, an \u2265 3,In the next theorem, existence and uniqueness of fixed point of quasicontraction in complete partial rectangular metric spaces are proved.X, \u03c1) be a complete partial rectangular metric space and let T : X \u2192 X be a quasicontraction on X with constant \u03bb. Then, T has a unique fixed point u \u2208 X and \u03c1 = 0.Let  > 0. Then, from  = 0; that is, u = v. Thus, if fixed point of T exists, then it is unique. Further, if u \u2208 X is a fixed point of T and \u03c1 > 0, then, from  = 0. So, if u is a fixed point of T, then \u03c1 = 0.Let us first show that, if fixed point of en, from , we haveen, from , we haveT. For arbitrary x \u2208 X, we will show that the iterative sequence {Tnx} is a Cauchy sequence. Let m, n \u2208 \u2115 with m > n. Then, using k \u2208 {1,2,\u2026, m \u2212 n + 1} such that \u03b4[O] = \u03c1; therefore, using \u03bb \u2208 [0,1), it follows from , there exists u \u2208 X such thatu is the fixed point of T. Suppose \u03c1 > 0. Without loss of generality, we can assume that Tnx \u2260 Tn+1x, for all n \u2208 \u2115, also, there exists n0 \u2208 \u2115 such that Tnx \u2209 {u, Tu}, for all n > n0. Therefore, it follows from (\u03c15) thatn, from  \u2264 \u03c1, then, from  = 0. If \u03c1 \u2264 \u03c1, then, for sufficiently large n, we have\u03c1 = 0. Therefore, we must have Tu = u. Thus, u is the unique fixed point of T.Now, we prove existence of fixed point of her with \u03c1 that\u03c1\u2264\u03c1n, from  that\u03c1\u2264\u03c1n, from  is a complete partial rectangular metric space. Since, for all x \u2208 X, x > 0, \u2009\u03c1 = x > 0, therefore  is not a rectangular metric space. Also,  is not a partial metric space because it lacks the property (P5). Indeed,T : X \u2192 X byT is a quasicontraction with constant \u03bb \u2208 [11/14, 1). All the conditions of T has a unique fixed point u = 0. Note that the rectangular metric induced by \u03c1 is given byT is not a quasicontraction with respect to \u03c1r. Indeed, for x = 4, y = 5, we have \u03c1r = 5 and \u03c1r = 5, \u03c1r = 5, \u03c1r = 5, \u03c1r = 0, and \u2009\u03c1r = 5. Therefore, there exists no \u03bb \u2208 [0,1) such thatx, y \u2208 X. Thus, T is not a quasicontraction in the induced rectangular metric space.Let T is a Banach contraction in the partial rectangular metric space ; that is, \u03c1 \u2264 \u03bb\u03c1, for all x, y \u2208 X with \u03bb \u2208 [9/10, 1), while it is not even a quasicontraction in the induced rectangular metric space . Also  is not a partial metric space; therefore, the results of Branciari [Note that, in the above example, the mapping ranciari  and Mattranciari  are not The following corollaries generalize the Banach, Kannan, Reich, Chatterjea, and Hardy-Rogers fixed point results  in partX, \u03c1) be a complete partial rectangular metric space and let T : X \u2192 X be a mapping. Suppose that the following condition is satisfied:\u03bb \u2208 [0,1). Then, T has a unique fixed point u \u2208 X and \u03c1 = 0.Let  be a complete partial rectangular metric space and let T : X \u2192 X be a mapping. Suppose that the following condition is satisfied:x, y \u2208 X, where \u03b1 \u2208 [0, 1/2). Then, T has a unique fixed point u \u2208 X and \u03c1 = 0.Let  be a complete partial rectangular metric space and let T : X \u2192 X be a mapping. Suppose that the following condition is satisfied:x, y \u2208 X, where \u03b1, \u03b2, and \u03b3 are nonnegative constants such that \u03b1 + \u03b2 + \u03b3 < 1. Then, T has a unique fixed point u \u2208 X and \u03c1 = 0.Let  be a complete partial rectangular metric space and let T : X \u2192 X be a mapping. Suppose that the following condition is satisfied:x, y \u2208 X, where \u03b1 \u2208 [0, 1/2). Then, T has a unique fixed point u \u2208 X and \u03c1 = 0.Let  be a complete partial rectangular metric space and let T : X \u2192 X be a mapping. Suppose that the following condition is satisfied:x, y \u2208 X, where \u03b1, \u03b2, \u03b3, \u03bc, and \u03bb are nonnegative constants such that \u03b1 + \u03b2 + \u03b3 + \u03bc + \u03bb < 1. Then, T has a unique fixed point u \u2208 X and \u03c1 = 0.Let  be a complete partial rectangular metric space and let T : X \u2192 X be a mapping. Suppose that for some positive integer n, the following condition is satisfied:x, y \u2208 X, where \u03bb \u2208 [0,1). Then, T has a unique fixed point u \u2208 X and \u03c1 = 0.Let  = 0. Now, TnTu = TTnu = Tu; therefore, Tu is another fixed point of Tn and, by uniqueness, we have Tu = u. Thus, u is a fixed point of T. Since every fixed point of T is also a fixed point of Tn, the fixed point of \u2009T is unique.We note that ondition  of Theor"}
{"text": "We build a Taylor's expansion for composite functions. Some applications are introduced, where the proposed technique allows the authors to obtain an asymptotic expansion of high order in many small parameters of solutions. So, first in However, for given in , it is v\u03a9 be an open subset of \u211dn\u2009\u2009and 0 \u2208 \u03a9. Let\u2009\u2009g =  \u2208 CN+1\u2009\u2009and\u2009\u2009f \u2208 CN+1.Seek the representation formula for fog, such that for ||x|| being small enough,d\u03b1, |\u03b1| \u2264 N are calculated from the values of the given functions f; g1,\u2026, gp\u2009\u2009and of their derivatives at a suitable point.Let Next in \u03b1 =  \u2208 \u2124+n and x =  \u2208 \u211dn, we putWe use the following notations. For a multi-index The following lemma is useful to solve the Problems m, N \u2208 \u2115 and a\u03b1 \u2208 \u211d, \u03b1 \u2208 \u2124+n, 1 \u2264 |\u03b1| \u2264 N. ThenTNm), \u03b1 \u2208 \u2124+n, 1 \u2264 |\u03b1| \u2264 N are defined by the recurrent formulasLet The proof of f\u2009\u2009around the point g =  \u2208 \u211dp, we obtain thatNow, using Taylor's expansion of the function gi as follows:Similarly, we use Maclaurin's expansion of Substituting  into 1212, we gea = \u03c3i)( = (\u03c3\u03b2i), \u03c3\u03b2i)( = (1/\u03b2!)D\u03b2gi(0), 1 \u2264 |\u03b2| \u2264 N, it implies thatApplying Hence,Consequently,Clearly, the f = f \u2208 CN+1, for each\u2009\u2009fixed  \u2208  \u00d7 \u211d+; gi \u2208 CN+1, i = 1,2, 3, for each\u2009\u2009fixed  \u2208  \u00d7 \u211d+ : gi \u2208 CN+1, i = 4,5, 6, for each\u2009\u2009fixed t \u2208 \u211d+, we now investigate the  As an application of the method used in x, t) \u2208  \u00d7 \u211d+, using Taylor's expansion of the function f around the point g =  \u2208 \u211d3 \u00d7 \u211d+3 up to order N, we obtain thatH =  \u2208 \u211d6, ||H|| is small enough, Di\u03b1if = \u2202f/\u2202gi, i = 1,2,\u2026, 6.For each\u2009\u2009fixed  \u2208  \u00d7 \u211d+: we have the following.For each fixed (gi(\u03b5) = gi, i = 1,2, 3.(i) The representation formula for g1(\u03b5) as followsWe rewrite g2(\u03b5), g3(\u03b5); we writeIt is similar to t \u2208 \u211d+: we have the following.For each\u2009\u2009fixed gi(\u03b5) = gi, i = 4,5, 6.(ii) The representation formula for WriteSimilarly,Then\u2009\u2009, 24), , , 28), a, a28), aWe also need the following lemma.\u03b1 =  \u2208 \u2124+n, |\u03b1 | \u2265 1, thenFor all We haveRj = O(||\u03b5||N+1), j = 4,5, 6, soNote that Combining  and 36)36) yieldSubstituting  into 2020, we obu\u03b1 \u2208 W = {v \u2208 L\u221e) : vt \u2208 L\u221e)}, \u03b1 \u2208 \u2124+n, |\u03b1| \u2264 N, hence u\u03b1, u\u03b1x, u\u03b1t \u2208 L\u221e) \u2282 L\u221e(QT). Then from u\u03b1 \u2208 W =n = 1.(i) This result improves the one in , for n =F = \u03bc, where \u03bc \u2208 CN+1, n = p, this result is also obtained in [(ii) In case of ained in ."}
{"text": "A high-order finite difference scheme is proposed for solving time fractional heat equations. The time fractional derivative is described in the Riemann-Liouville sense. In the proposed scheme a new second-order discretization, which is based on Crank-Nicholson method, is applied for the time fractional part and fourth-order accuracy compact approximation is applied for the second-order space derivative. The spectral stability and the Fourier stability analysis of the difference scheme are shown. Finally a detailed numerical analysis, including tables, figures, and error comparison, is given to demonstrate the theoretical results and high accuracy of the proposed scheme. In the last decades, more and more attention has been placed on the development and research of fractional differential equations, because they can describe many phenomena, physical and chemical processes more accurately than classical integer order differential equations \u20134. And tO(\u03c4\u03bc + h4), where 1 \u2264 \u03bc < 2. Here, we propose a method for the time fractional differential heat equations with the accuracy of order O(\u03c42 + h4).There are many different discretizations in time variable equipped with the compact difference scheme in spatial variable. The approximations given in \u20139 are ofM\u03b1u/\u2202t\u03b1 denotes \u03b1-order modifying Riemann-Liouville fractional derivative , thenSuppose xi, and then we obtain the following truncation error for any i, where 1 \u2264 i \u2264 M \u2212 1:We use the Taylor expansion of each term about \u03ba to both side of (O(\u03c42 + h4);\u03c6j = T, \u03c6j0 = r(xj), \u03c6jk = f, 1 \u2264 k \u2264 N, 1 \u2264 j \u2264 M, and Uj = T.If we apply the operator  side of ,(8)\u03ba\u2202M\u03b1AN+1)\u00d7(N+1)( and BN+1)\u00d7(N+1)\u00d7(N+1)( and \u03b21 = ON+1)\u00d71\u03b1j+\u03b1j (1 \u2264 j \u2264 M) of the scheme  denote the spectral radius of a matrix A, that is, the maximum of the absolute value of the eigenvalues of the matrix A.Let \u03c1(\u03b1j) < 1, (1 \u2264 j \u2264 M), by inductionWe will prove that \u03b11 is a zero matrix \u03c1(\u03b11) = 0 < 1.since \u03b12 = \u2212B\u22121A, and \u03b12 is a lower triangular matrix of the following form:Moreover, \u03c1(\u03b12) = \u03c1(\u2212B\u22121A) = |(1/2h2 \u2212 w0/12)/(1/h2 + 10w0/12)| < |(1/2h2 \u2212 w0/12)/(1/2h2 + 9w0/12 + 1/2h2 + w0/12)|<|(1/2h2 \u2212 w0/12)/(1/2h2 + w0/12)|.Therefore h2 > 0, we can write \u03c1(\u03b12) < 1.Since\u03c1(\u03b1j) < 1. After some calculations we find thatAi,i = w0/12 \u2212 1/2h2, Bi,i = 10w0/12 + 1/h2 and \u03b1ji,i = \u03bbj, nonzero eigenvalue of \u03b1j for 2 \u2264 i \u2264 N + 1. h2 \u2212 w0/12 \u2265 0, then we have two subcases. (a)\u03bbj < 1,If 0 < (b)\u03bbj < 0,If \u22121 < If 1/2h2 \u2212 w0/12 < 0, then we have two subcases.(a)\u03bbj < 1,If 0 < (b)\u03bbj < 0,If \u22121 < Now, assume \u03c1(\u03b1j) < 1 then it follows that \u03c1(\u03b1j+1) < 1. So \u03c1(\u03b1j) < 1 for any j, where 1 \u2264 j \u2264 M.So, we have proved that whenever A \u2208 \u211dN\u00d7N, Am \u2192 0 as m \u2192 \u221e if and only if \u03c1(A) \u2264 1. We note that if A is normal, then ||A|| = \u03c1(A) but when the matrix A is not normal the spectral radius gives no indication of the magnitude of the roundoff error for finite M. In this case a condition of the form \u03c1(A) \u2264 1 guarantees eventual decay of the errors, but does not control the intermediate growth of the errors. Then, it is easy to understand that \u03c1(A) \u2264 1 is a necessary condition for stability but not always sufficient.It is well known that for any Ujk be the approximate solution and define \u03c1jk = ujk \u2212 Ujk, k = 0,1,\u2026, N \u2212 1, j = 1,\u2026, M \u2212 1. Then, we write \u03c1jk = dkeijh\u03b2 and obtain the following roundoff error equation for 112\u03c1k(x) can be expanded in a Fourier series as follows:dk(l) = 1/L\u222b0L\u03c1k(x)ei2\u03c0lx/L\u2212 and we introduce the following norm:0L|\u03c1k(x)|2dx = \u2211l=\u2212\u221e\u221e|dk(l)|2, we obtainWe now define the grid functions:\u03c1jk = dkeijh\u03b2, where \u03b2 = 2\u03c0l/L and L = 1. Substituting the above expression into  = (t4 + 1)x5(1 \u2212 x). The solution by the proposed scheme is given in Exact solution of this problem is The errors in It can be concluded from In this work, the compact difference scheme was successfully applied to solve the time fractional heat equations. The second order approximation for the Riemann-Liouville fractional derivative is equipped with the higher order compact difference schemes. The Fourier analysis and the spectral stability method are used to show that the proposed scheme is unconditionally stable. Numerical results are in good agreement with the theoretical results."}
{"text": "Lip\u03b1 and Lip  classes are the particular cases ofLip(\u03be(t), p) class. The main result of this paper generalizes some well-known results in this direction.Approximation theory is a very important field which has various applications in pure and applied mathematics. The present study deals with a new theorem on the approximation of functions of Lipschitz class by using Euler's mean of conjugate series of Fourier series. In this paper, the degree of approximation by using Euler's means of conjugate of functions belonging toLip( The Fourier series associated with\u2009\u2009f\u2009\u2009at the point\u2009\u2009x\u2009\u2009is given bysn. The conjugate series of .Let\u2009\u2009given byf(x)~12a0f \u2208Lip\u03b1\u2009\u2009ifA function\u2009\u2009f \u2208Lip,\u2009\u2009p > 1\u2009\u2009consider that if(Definition\u2009\u20095.38\u2009\u2009of Chandra ).\u03be(t),\u2009\u2009f \u2208Lip(\u03be(t), p),M\u2009\u2009is a positive number independent of\u2009\u2009x\u2009\u2009and\u2009\u2009t.Given a positive increasing function\u2009\u2009\u03be(t) = t\u03b1, thenLip(\u03be(t), p)\u2009\u2009coincides withLip. If\u2009\u2009p \u2192 \u221e\u2009\u2009inLip, then it coincides withLip\u03b1.In case\u2009\u2009L\u221e-norm of a function\u2009\u2009f : R \u2192 R\u2009\u2009is defined byLp-norm is defined byf : R \u2192 R\u2009\u2009by a trigonometric polynomial\u2009\u2009tn\u2009\u2009of order\u2009\u2009n\u2009\u2009under sup norm ||\u2002||\u221e is defined by (, then the degree of approximation of\u2009\u2009f\u2009\u2009by the N\u00f6rlund means of the Fourier series for f is given byTn\u2009\u2009are the\u2009\u2009\u2009\u2009means of Fourier series of\u2009\u2009f.If\u2009\u2009\u03b4\u2009\u2009and consider a function\u2009\u2009f \u2208Lip\u03b1,\u2009\u20090 < \u03b1 \u2264 1. Working in same direction we prove the following theorem.H\u00f6lland et al.  have shof : R \u2192 R\u2009\u2009is a\u2009\u20092\u03c0\u2009\u2009periodic, Lebesgue integrable and belonging toLip(\u03be(t), p)\u2009\u2009for > 1\u2009\u2009and ifx, then degree of approximation off \u2208Lip{\u03be(t), p}, by Euler  meanIf\u2009\u2009\u2009\u2009and if{\u222b01/n(t|(\u03be(1n)),{\u222b1/n\u03c0 < 1\u2009\u2009when\u2009\u20090 < x < 1. Therefore,We havekth\u2009\u2009partial sum of the conjugate series of the Fourier series  means, we getThe\u2009\u2009r series  is givenf \u2208Lip(\u03be(t), p)\u21d2\u03c8 \u2208Lip(\u03be(t), p).Clearly,\u03c8(t) \u2208Lip(\u03be(t), p), condition (t \u2265 (2t/\u03c0), lemma, and second mean value theorem for integrals, we have\u03c8(t) \u2208Lip(\u03be(t), p), and condition  = t\u03b1, then the degree of approximation of a functionf \u2208Lip,\u2009\u20091/p < \u03b1 < 1, by Euler's means\u2009\u2009\u2009\u2009of the conjugate series of the Fourier series (If\u2009\u2009r series  is givenp \u2192 \u221e\u2009\u2009in \u03b1 < 1,If"}
{"text": "We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established. F-expansion method [G\u2032/G)-expansion method [Nonlinear differential equations of integer order (NLDEs) can be used to describe many nonlinear phenomena such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, and chemical physics. In the research of the theory of NLDEs, searching for more explicit exact solutions to NLDEs is one of the most fundamental and significant studies in recent years. With the help of computerized symbolic computation, much work has focused on the various extensions and applications of the known algebraic methods to construct the solutions to NLDEs. There have been a variety of powerful methods. For example, these methods include the generalized Riccati subequation method , 2, the n method , the Expn method \u20139, and tn method , 11.Fractional differential equations are generalizations of classical differential equations of integer order and have recently proved to be valuable tools to the modeling of many physical phenomena and have been the focus of many studies due to their frequent appearance in various applications in physics, biology, engineering, signal processing, systems identification, control theory, finance, and fractional dynamics. In order to obtain exact solutions for fractional differential equations, many powerful and efficient methods have been proposed so far . But\u03be = \u03be, a given fractional partial differential equation expressed in independent variables t, x1, x2,\u2026, xn can be turned into another ordinary differential equation of integer order, whose solutions are supposed to have one of the following forms:m1, m2, n1, and n2 are unknown positive integers that will be determined by using the balance method. Determine the highest order nonlinear term and the linear term of highest order in the ordinary differential equation and express them in terms of Dt\u03b1f follows \u201324, 26):In the following, we will apply the Exp-function method to find exact solutions for the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation.q = U(\u03be), where \u03be = (k1x1\u03b1/\u0393(1 + \u03b1)) + (k2x2\u03b1/\u0393(1 + \u03b1)) + (l1y1\u03b1/\u0393(1 + \u03b1)) + (l2y2\u03b1/\u0393(1 + \u03b1)) + (ct\u03b1/\u0393(1 + \u03b1)) + \u03be0, k1, k2, l1, l2, c, \u03be0 are all constants with k1, k2, l1, l2, c \u2260 0. Then by the use of 4\u22022\u03b1q<\u03b1\u22641.In , the aut4\u22022\u03b1q<\u03b1\u22641e use of , (6) canm2, n2, we need to balance the linear term of the highest order with the highest order nonlinear term in  and UU\u2032\u2032, we have 15n2 + m2 = 14n2 + 2m2, which implies m2 = n2. Similarly, balancing the lowest order Exp-function in U(4) and UU\u2032\u2032, we have 15n1 + m1 = 14n1 + 2m1, which implies m1 = n1. For simplicity, we will proceed in two selected cases.First we suppose that the solution of  can be e term in . By simpm1 = n1 = 1, m2 = n2 = 1. Thenep\u03be together, equating each coefficient to zero, yield a set of algebraic equations. Solving these equations with the aid of the mathematical software Maple yields the following family of values of ai, bi, i = \u22121,0, 1. Let  1. ThenU(\u03be)=a\u22121etituting , eliminaFamily\u2009\u20091. Considerb1, b0 are arbitrary constants.\u03be = (k1x1\u03b1/\u0393(1 + \u03b1)) + (k2x2\u03b1/\u0393(1 + \u03b1)) + (l1y1\u03b1/\u0393(1 + \u03b1)) + (l2y2\u03b1/\u0393(1 + \u03b1)) + (ct\u03b1/\u0393(1 + \u03b1)) + \u03be0. If we especially take b0 = 2b1, then we obtain the following hyperbolic function solitary wave solution:Substituting the result above into , we can m1 = n1 = m, m2 = n2 = m, and for simplicity, we takeepn\u03be together, equating each coefficient to zero, yield a set of algebraic equations. Solving these equations yields the following result. Let  we takeU(\u03be)=a\u2212metituting , eliminaFamily\u2009\u20092. Considerbm, b0 are arbitrary constants.\u03be = (k1x1\u03b1/\u0393(1 + \u03b1)) + (k2x2\u03b1/\u0393(1 + \u03b1)) + (l1y1\u03b1/\u0393(1 + \u03b1)) + (l2y2\u03b1/\u0393(1 + \u03b1)) + (ct\u03b1/\u0393(1 + \u03b1)) + \u03be0. If we especially take b0 = 2bm, then we obtain the following solitary wave solution:Substituting the result above into , we can m1 = n1 and m2 = n2. Similar to the above, we only consider the following two cases. Now we suppose that the solution of  can be e1U(\u03be)=\u2211p=\u2212m1 = n1 = 1, m2 = n2 = 1. Theneip\u03be together, equating each coefficient to zero, yield a set of algebraic equations. Solving these equations yields the following. Let  1. ThenU(\u03be)=a\u22121etituting , eliminaFamily\u2009\u20093. Considerb1, b0 are arbitrary constants.\u03be = (k1x1\u03b1/\u0393(1 + \u03b1)) + (k2x2\u03b1/\u0393(1 + \u03b1)) + (l1y1\u03b1/\u0393(1 + \u03b1)) + (l2y2\u03b1/\u0393(1 + \u03b1)) + (ct\u03b1/\u0393(1 + \u03b1)) + \u03be0. If we especially take b0 = 2b1, then we obtain the following trigonometric function solution:Substituting the result above into , we can eipn\u03be together, equating each coefficient to zero, yield a set of algebraic equations. Solving these equations, we obtain another family of values of ai, bi, i = \u2212m, 0, m. LetU(\u03be)=a\u2212metituting , eliminaFamily\u2009\u20094. Consider\u03be = (k1x1\u03b1/\u0393(1 + \u03b1)) + (k2x2\u03b1/\u0393(1 + \u03b1)) + (l1y1\u03b1/\u0393(1 + \u03b1)) + (l2y2\u03b1/\u0393(1 + \u03b1)) + (ct\u03b1/\u0393(1 + \u03b1)) + \u03be0. If we especially take b0 = 2bm, then we obtain the following solitary wave solution:ove into :(25)U7 = U(\u03be), where \u03be = kx + (ct\u03b1/\u0393(1 + \u03b1)) + \u03be0, k, c, \u03be0 are all constants with k, c \u2260 0. Then by use of  equation with time-fractional derivative , 27: caU\u2032\u2032\u2032 and U2U\u2032, we have m2 = n2, while we obtain m1 = n1 by balancing the lowest order Exp-function in U\u2032\u2032\u2032 and U2U\u2032. Similar to the solving process for the space-time fractional Fokas equation, we will also proceed the computation under two selected cases. First we suppose that the solution of  can be em1 = n1 = 1, m2 = n2 = 1. Thenep\u03be together, equating each coefficient to zero, yield a set of algebraic equations. Solving these equations yields two families of values for ai, bi, i = \u22121,0, 1.Let  1. ThenU(\u03be)=a\u22121etituting , eliminaFamily\u2009\u20091. Considerb1, b\u22121 are arbitrary constants. Family\u2009\u20092. Considerb1, b0, b\u22121 are arbitrary constants.\u03be = kx + (ct\u03b1/\u0393(1 + \u03b1)) + \u03be0. If we especially take c = \u2212ak3, b1 = b\u22121 or c = \u2212ak3, b1 = \u2212b\u22121 in ==a\u2212metituting , eliminaFamily\u2009\u20093. Considerbm, bm\u2212 are arbitrary constants. Family\u2009\u20094. Considerbm, b0, bm\u2212 are arbitrary constants.\u03be = kx + (ct\u03b1/\u0393(1 + \u03b1)) + \u03be0. If we especially take c = \u2212am2k3, bm = bm\u2212 or c = \u2212am2k3, bm = \u2212bm\u2212 in \u2003==a\u22121etituting , eliminaFamily\u2009\u20095. Considerb1, b\u22121 are arbitrary constants. Family\u2009\u20096. Considerb1, b0, b\u22121 are arbitrary constants.Substituting the results above into , we can c = \u2212ak3, b1 = b\u22121 or c = \u2212ak3, b1 = \u2212b\u22121 in =a\u2212metituting , eliminaFamily\u2009\u20097. Considerbm, bm\u2212 are arbitrary constants. Family\u2009\u20098. Considerbm, b0, bm\u2212 are arbitrary constants.c = \u2212am2k3, bm = bm\u2212 or c = \u2212am2k3, bm = \u2212bm\u2212 in \u22121,U14(\u03be)\u2003= re rem = 2,tions by , (54) artions by , 34), , ,  ar arm = 2, We have extended the Exp-function method to solve fractional partial differential equations successfully. As applications, some generalized and new exact solutions for the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation have been successfully found. As one can see, this method is based on the homogeneous balancing principle. So it can also be applied to other fractional partial differential equations where the homogeneous balancing principle is satisfied."}
{"text": "AbstractPseudaspidapion botanicum sp. n. from China is described and figured. Its host plant is Grewia biloba G. Don var. parviflora (Bunge) Hand.-Mazz . The genus Harpapion Voss, 1966 is recorded as new for China and Vietnam and two comb. n. are proposed: Harpapion vietnamense  (from Aspidapion) and Harpapion coelebs  (from Pseudaspidapion). A key to the known species of the genus Pseudaspidapion from China is presented. Pseudaspidapion relating it to the genus Aspidapion Schilsky, 1901. Aspidapiini PageBreakAlonso-Zarazaga, 1990 of the subfamily Apioninae Schoenherr, 1823 . This genus is quite similar to Aspidapion, but it can be distinguished from the latter by relatively equal width of elytral striae and interstriae, the presence of fenestrae in the tegminal plate, and the absence of protibial mucrones in males, among other features , the Museo Nacional de Ciencias Naturales, Madrid (MNCN), the Zoological Institute of Russian Academy of Sciences, Moscow (ZIN), the Museum of Natural History, University of Wroc\u0142aw (MNHW) and the Beijing Botanical Garden, Beijing (BBG).Type specimens were obtained from ZIN on loan or belong to IZCAS and their data are summarised in Descriptions were made and photographs were taken with a CCD Qimagine MicroPublisher 5.0 RTV mounted on a Zeiss SteREO Discovery V.12. Extended focus images were generated with Auto-Montage Pro 5.03.0061 and edited with Adobe Photoshop CS 5.0 if required. Microscopic slides were studied under a Leica DM 2500 microscope and photos were taken with a Nikon CoolPix 5400. The map was made with the software ArcGIS 9.3. Drawings were made from the original photographs by using the software Adobe Illustrator CS5.0, or directly by using a drawing tube linked to the microscope.Nomenclature of the rostral parts follows The dissecting method used follows After description, the genitalia and other parts of each specimen were placed in DMHF on a plastic card for long term conservation .PageBreakLabels are described as they are (in Chinese), with pinyin romanization or comments in square brackets; labels are separated by semicolons and lines by slashes.Alonso-Zarazaga & Wang sp. n.urn:lsid:zoobank.org:act:88656E61-918F-4F1B-A3BD-6A3FA714A237http://species-id.net/wiki/Pseudaspidapion_botanicumPseudaspidapion yunnanicum, but it can be distinguished from the latter by the characters in the  This new species resembles Measurements (in mm): Standard length: 1.82. Rostrum: length: 0.71, maximum width: 0.17. Pronotum: median length: 0.49, maximum width: 0.59. Elytra: median length: 1.38, maximum width: 1.08. . Pseudaspidapion as described in With the general characters of genus Integument. Generally piceous black  semicircular, 0.69\u00d7 as long as wide, apical edge of arc with a flat and glabrous side, punctate medially with interspace microreticulate . Ninth sternite  Y-shaped and slightly winged near arm base, manubrium ca. 2.00\u00d7 as long as arms  (n=10): Standard length: 1.38\u20132.05 (mean= 1.735). Rostrum: length: 0.52\u20130.82 (mean= 0.722), maximum width: 0.13\u20130.19 (mean= 0.164). Pronotum: median length: 0.36\u20130.53 (mean= 0.479), maximum width: 0.44\u20130.68 (mean= 0.594). Elytra: median length: 1.06\u20131.54 (mean= 1.406), maximum width: 0.78\u20131.18 (mean= 1.038). Rostrum 1.44\u20131.58\u00d7 as long as pronotum, 4.00\u20134.63\u00d7 as long as wide. Ventrite 5 sometimes slightly subtriangular. Otherwise as in holotype.Female paratypes , Haidian district, Beijing (P. R. of China). Paratypes (132\u2642137\u2640): 4\u26427\u2640: : \u5317\u4eac\u4e09\u5821 [B\u011bij\u012bng S\u0101np\u00f9]; : 1964.VII.20 / leg. \u9a6c\u6587\u73cd [M\u0103 W\u00e9nzh\u0113n]; : IOZ(E)1638668-1638678; 3\u2640: PageBreak: \u5317\u4eac\u4e09\u5821 [B\u011bij\u012bng S\u0101np\u00f9] / 600m; : 1964.VIII.21 / leg. \u674e\u94c1\u751f [L\u00ed Ti\u00e9sh\u0113n]; : IOZ(E)1638679-1638681; 1\u26421\u2640: : \u5317\u4eac\u4e09\u5821  / 1979.VIII.8; : leg. \u5ed6\u7d20\u67cf [Li\u00e0o S\u00f9b\u0103i]; : IOZ(E)1638682-1638683; 5\u26425\u2640: : \u5317\u4eac\u4e0a\u65b9\u5c71 [B\u011bij\u012bng Sh\u00e0ngf\u0101ngsh\u0101n]; : 1980.VIII.2 / leg. \u5ed6\u7d20\u67cf [Li\u00e0o S\u00f9b\u0103i];  IOZ(E)1638684-1638693; 2\u2642: : \u5317\u4eac\u4e0a\u65b9\u5c71 [B\u011bij\u012bng Sh\u00e0ngf\u0101ngsh\u0101n]; : 1979.VII.25 / leg. \u9648\u5143\u6e05 [Ch\u00e9n Yu\u00e1nq\u012bng]; : IOZ(E)1638694-1638695; 2\u26422\u2640: : \u5317\u4eac\u5c45\u5eb8\u5173 [B\u011bij\u012bng J\u016by\u014dnggu\u0101n] / 500m; : 1964.VIII.20 / leg. \u674e\u94c1\u751f [L\u00ed Ti\u00e9sh\u0113n]; : IOZ(E)1638696-1638699; 1\u2642: : \u5317\u4eac\u516b\u8fbe\u5cad [B\u011bij\u012bng B\u0101d\u00e1l\u012bng] / 700m; : 1963.VII.25 / leg. \u674e\u94c1\u751f [L\u00ed Ti\u00e9sh\u0113n]; : IOZ(E)1638700; 8\u26428\u2640: : \u5317\u4eac\u9999\u5c71 [B\u011bij\u012bng Xi\u0101ngsh\u0101n]; : 1963.V.30 / leg. \u674e\u94c1\u751f [L\u00ed Ti\u00e9sh\u0113n]; : IOZ(E)1638701-1638716; 1\u2640: : \u5317\u4eac\u9999\u5c71 [B\u011bij\u012bng Xi\u0101ngsh\u0101n]; : 1964.V.5 / leg. \u9a6c\u6587\u73cd [M\u0103 W\u00e9nzh\u0113n]; : IOZ(E)1638717; 1\u2640: : \u5317\u4eac\u9999\u5c71 [B\u011bij\u012bng Xi\u0101ngsh\u0101n] / 1957.VIII.16; : IOZ(E)1638718; 1\u2642: : \u5317\u4eac\u5367\u4f5b\u5bfa [B\u011bij\u012bng W\u00f2f\u00f3s\u00ec] / 50m; : 1962.VI.25 / leg. \u738b\u6625\u5149 [W\u00e1ng Ch\u016bngu\u0101ng]; : IOZ(E)1638719; 1\u2640: : \u5317\u4eac\u5367\u4f5b\u5bfa [B\u011bij\u012bng W\u00f2f\u00f3s\u00ec]; : 1963.IX.3 / leg. \u59dc\u80dc\u5de7 [Ji\u0101ng Sh\u00e8ngqi\u00e1o]; : IOZ(E)1638720; 1\u2640: : \u5317\u4eac\u5367\u4f5b\u5bfa [B\u011bij\u012bng W\u00f2f\u00f3s\u00ec] / 50m; : 1962.VIII.31 / leg. \u8c22\u6c5d\u5fe0 [Xieruzhong]; : IOZ(E)1638721; 1\u2642: : \u5317\u4eac\u6f6d\u67d8\u5bfa [B\u011bij\u012bng T\u00e1nzh\u00e8s\u00ec] / 1975.VII.24; : leg. \u738b\u4e66\u6c38 [Wang Shuyong]; : IOZ(E)1638722; 1\u2640: : 1964.VII.20 / leg. \u9a6c\u6587\u73cd PageBreakPageBreak[M\u0103 W\u00e9nzh\u0113n]; : IOZ(E)1638723; 2\u26423\u2640: : \u5317\u4eac\u5706\u660e\u56ed [B\u011bij\u012bng Yu\u00e1nm\u00edngyu\u00e1n] / 1980. VII.7 / leg. \u59dc\u80dc\u5de7 [Ji\u0101ng Sh\u00e8ngqi\u00e1o]; : IOZ(E)1638724-1638728; 2\u264210\u2640: : \u5317\u4eac\u95e8\u5934\u6c9f\u519b\u5e84\u897f\u6768\u5768\u6751 [B\u011bij\u012bng M\u00e9nt\u00f3ug\u014du J\u016bnzhu\u0101ng X\u012by\u00e1ngt\u00fan] / 2008.VII.27 / leg. \u738b\u5fd7\u826f [W\u00e1ng Zh\u00ecli\u00e1ng]; : \u5bc4\u4e3b: \u5c0f\u82b1\u6241\u62c5\u6728 [J\u00eczh\u016b: Xi\u0103ohu\u0101bi\u0103nd\u00e0nm\u00f9] / Grewia biloba G. Don var. parviflora (Bunge) Hand.-Mazz; : IOZ(E)1638729-1638740; 5\u26428\u2640: : \u5317\u4eac\u6000\u67d4\u4e5d\u6e21\u6cb3\u9547\u6000\u4e5d\u6cb3 [B\u011bij\u012bng Hu\u00e1ir\u00f3u Ji\u016bd\u00f9h\u00e9zh\u00e8n Hu\u00e1iji\u016bh\u00e9] / 2008.VI.15 / leg. \u738b\u5fd7\u826f [W\u00e1ng Zh\u00ecli\u00e1ng]; : IOZ(E)1638741-1638753; 1\u2640: : \u5317\u4eac\u6000\u67d4\u4e09\u6e21\u6cb3 [B\u011bij\u012bng Hu\u00e1ir\u00f3u S\u0101nd\u00f9h\u00e9] / 2008.V.24 / leg. \u738b\u5fd7\u826f [W\u00e1ng Zh\u00ecli\u00e1ng];  IOZ(E)1638754; 46\u264224\u2640: : \u5317\u4eac\u9999\u5c71\u690d\u7269\u56ed\u6a31\u6843\u6c9f [B\u011bij\u012bng Xi\u0101ngsh\u0101n Zh\u00edw\u00f9yu\u00e1n Y\u012bngt\u00e1og\u014du] / 2008.VIII.19 / leg. \u738b\u5fd7\u826f [W\u00e1ng Zh\u00ecli\u00e1ng]; : IOZ(E)1638755-1638770, IOZ(E)1638784-1638790, IOZ(E)1638804-1638850; 28\u264241\u2640: : \u5317\u4eac\u690d\u7269\u56ed\u6a31\u6843\u6c9f [B\u011bij\u012bng zh\u00edw\u00f9yu\u00e1n Y\u012bngt\u00e1og\u014du] / \u5bc4\u4e3b\uff1a\u5c0f\u82b1\u6241\u62c5\u6728 [J\u00eczh\u016b: Xi\u0103ohu\u0101bi\u0103nd\u00e0nm\u00f9] / 2008.VI.1 / Leg. \u738b\u5fd7\u826f [W\u00e1ng Zh\u00ecli\u00e1ng]; : IOZ(E)1638851-1638900, IOZ(E)1638911, IOZ(E)1638928-1638945; 7\u26427\u2640: : \u5317\u4eac\u6d77\u6dc0\u767e\u671b\u5c71 [B\u011bij\u012bng H\u0103idi\u00e0n B\u0103iw\u00e0ngsh\u0101n] / 2009.VI.21 / leg. \u6768\u5e72\u71d5 [Y\u00e1ng G\u00e0ny\u00e0n]; : IOZ(E)1639912-1638918, IOZ(E)16398920-1638926; 2\u26421\u2640: : \u9655\u897f\u534e\u9634\u53bf\u5b5f\u586c [Sha\u0103nxi Hu\u00e1y\u012bn M\u00e8ngyu\u00e1n] / 450m / 1972.VIII.9; : IOZ(E)1639562-1639563, IOZ(E)1639540, deposited in IZCAS.Holotype: \u2642: : \u5317\u4eac\u690d\u7269\u56ed\u6a31\u6843\u6c9f [B\u011bij\u012bng zh\u00edw\u00f9yu\u00e1n Y\u012bngt\u00e1og\u014du] / 2008.VI.1 / Leg. \u738b\u5fd7\u826f [W\u00e1ng Zh\u00ecli\u00e1ng]; : IOZ(E)1638667, deposited in IZCAS. This is the Beijing Botanical Garden at Yingtaogou : \u5317\u4eac\u9999\u5c71\u690d\u7269\u56ed\u6a31\u6843\u6c9f [B\u011bij\u012bng Xi\u0101ngsh\u0101n Zh\u00edw\u00f9yu\u00e1n Y\u012bngt\u00e1og\u014du] / 2008.VIII.19 / leg. \u738b\u5fd7\u826f [W\u00e1ng Zh\u00ecli\u00e1ng]; : IOZ(E)1638781, IOZ(E)1638801, IOZ(E)1638771-1638779, IOZ(E)1638791-1638799, to be deposited in MNCN.1\u26421\u2640: : \u5317\u4eac\u9999\u5c71\u690d\u7269\u56ed\u6a31\u6843\u6c9f [B\u011bij\u012bng Xi\u0101ngsh\u0101n Zh\u00edw\u00f9yu\u00e1n Y\u012bngt\u00e1og\u014du] / 2008.VIII.19 / leg. \u738b\u5fd7\u826f [W\u00e1ng Zh\u00ecli\u00e1ng]; : IOZ(E)1638780, IOZ(E)1638800, to be deposited in BBG.1\u26421\u2640: : \u5317\u4eac\u9999\u5c71\u690d\u7269\u56ed\u6a31\u6843\u6c9f [B\u011bij\u012bng Xi\u0101ngsh\u0101n Zh\u00edw\u00f9yu\u00e1n Y\u012bngt\u00e1og\u014du] / 2008.VIII.19 / leg. \u738b\u5fd7\u826f [W\u00e1ng Zh\u00ecli\u00e1ng]; : IOZ(E)1638783, IOZ(E)1638803, to be deposited in ZINM.1\u26421\u2640: : \u5317\u4eac\u9999\u5c71\u690d\u7269\u56ed\u6a31\u6843\u6c9f [B\u011bij\u012bng Xi\u0101ngsh\u0101n Zh\u00edw\u00f9yu\u00e1n Y\u012bngt\u00e1og\u014du] / 2008.VIII.19 / leg. \u738b\u5fd7\u826f [W\u00e1ng Zh\u00ecli\u00e1ng]; : IOZ(E)1638782, IOZ(E)1638802, to be deposited in MNHW.The male holotype has not been dissected to avoid any damage, since there are many specimens collected with it. The new species is named after the first locality where it was found during a collecting visit: the Beijing Botanical Garden. It is a Latin adjective. The species is known for the moment only from the municipality of Beijing and the province of Shaanxi.Pseudaspidapion botanicum sp. n. was collected from Grewia biloba G. Don var. parviflora (Bunge) Hand.-Mazz   , which i to July , this bePseudaspidapion and Aspidapion in China in the recent Palaearctic catalogue  of Aspidapion vietnamense, which has been previously dissected and whose pygidium and tegmen were not conserved, shows several differences from Aspidapion, namely, the metatibial mucros are evidently elongate and knicked at their apices (subdentate at the outer margin), the rostrum is clearly dilated at the antennal insertion and distinctly constricted apicad, the setae on the front margin of pronotum are parallel to it, and the apex of the penis is distinctly curved in lateral view, dorsally dentate near the apex. All these characters suggest it is a Harpapion Voss, 1966. After a study of the type species of this genus, Harpapion considerandum , we consider that, even in the absence of these diagnostic parts, it can be unmistakeably considered a member of this genus and consequently is here transferred: Harpapion vietnamense (Korotyaev), comb. n. The same can be said about Pseudaspidapion coelebs , which show also the characters of the latter genus, and is here formally transferred as well: Harpapion coelebs , comb. n. This genus will be the subject of a forthcoming paper.During this research, we were able to study types of several of the species placed in the genera atalogue  , from Altay prefecture in northern Xinjiang PageBreak(China). Apart from the species Aspidapion panfilovi, whose placement may need a revision as mentioned above, these specimens represent the easternmost distribution record of the otherwise Western Palaearctic genus Aspidapion. Therefore, the distribution pattern of Pseudaspidapion is mainly Palaeotropical but has expanded to Philippines . The two genera can still be distinctly separated from each other. Also, we suppose that the blank distributional area of Pseudaspidapion between Shaanxi and South China represents a lack of collection records rather than an actual distributional range limit, because the vegetation between these two points includes most of the range of Grewia biloba  and many other congeneric species of the host genus.Previously, the genus c region . Thus thlippines  and to NPseudaspidapion rufopiceum , a related species, which is included below from the data in its original description.PageBreakSome of these species are described only from one sex, which makes it difficult to key all of the Chinese species. However, we have tried to provide a key to the known Chinese species (see below). Consequently, more field work is needed to gather specimens of both sexes, and complete our knowledge. We have not studied specimens of Aspidapion panfilovi Korotyaev, 1985 , which seems to be related to these species. Body length excludes rostrum.This key includes also"}
{"text": "This paper deals with the study of a generalized function of Mittag-Leffler type. Various properties including usual differentiation and integration, Euler(Beta) transforms, Laplace transforms, Whittaker transforms, generalized hypergeometric series form with their several special cases are obtained and relationship with Wright hypergeometric function and Laguerre polynomials is also established.33C45, 47G20, 26A33. In 1903, the Swedish mathematician Gosta Mittag-Leffler  was studied by Wiman n is the Pochhammer symbol (Rainville (where (ainville ) \u03b1)\u2009>\u20090,Re(\u03b2)\u2009>\u20090,Re(\u03b3)\u2009>\u20090 and In 2007, Shukla and Prajapati , Re(\u03b2), Re(\u03b3), Re(\u03b4))\u2009>\u20090where In this paper a new definition of generalized Mittag-Leffler function is investigated and defined as where Further the generalization of definition (1.7) is investigated and defined as follows where, and The definition (1.9) is a generalization of all above functions defined by (1.1)-(1.7).\u03bc\u2009=\u2009\u03bd, \u03c1\u2009=\u2009\u03c3, it reduces to Eq.  will be the result.Setting s to Eq.  defined s to Eq. , moreoveThroughout this investigation, we need the following well-known facts to study the various properties and relation formulas of the function f(z) is defined as Beta(Euler) transforms (Sneddon ) of the f(z) is defined as Laplace transforms (Sneddon ) of the f(z) is defined as Mellin- transforms of the function and the inverse Mellin-transform is given by Whittaker transform (Whittaker and Watson ) 1.16W\u03bb,\u03bc(t) is the Whittaker confluent hypergeometric function.where The generalized hypergeometric function (Rainville ) is defiWright generalized hypergeometric function (Srivastava and Manocha ) is defiFox\u2019s H-function (Saigo and Kilbas ) is giveGeneralized Laguerre polynomials (Rainville ). These \u03b3 and is defined by Incomplete Gamma function (Rainville ). This iAs a consequence of definitions (1.1)-(1.9) the following results hold:\u03b1)\u2009>\u20090, Re(\u03b2)\u2009>\u20090, Re(\u03b3)\u2009>\u20090, Re(\u03b4)\u2009>\u20090, Re(\u03bc)\u2009>\u20090, Re(\u03bd)\u2009>\u20090, Re(\u03c1)\u2009>\u20090, Re(\u03c3)\u2009>\u20090 and p, q\u2009>\u20090 andq\u2009\u2264\u2009Re(\u03b1)\u2009+\u2009p, thenIf In particular, which is (2.1.1).The proof of (2.1.2) can easily be followed from the definition (1.9). Now which proves (2.1.3). \u25a1\u03bc\u2009=\u2009\u03bd, \u03c1\u2009=\u2009\u03c3and p\u2009=\u20091 in (2.1.1) immediately leads to (2.1.4).Substituting \u03bc\u2009=\u2009\u03bd, \u03c1\u2009=\u2009\u03c3and p\u2009=\u20091 in (2.1.2) immediately leads to (2.1.5).Substituting \u03bc\u2009=\u2009\u03bd,\u03c1\u2009=\u2009\u03c3and p\u2009=\u20091 in (2.1.3) immediately leads to (2.1.6).Putting \u03b1)\u2009>\u20090, Re(\u03b2)\u2009>\u20090, Re(\u03b3)\u2009>\u20090, Re(\u03b4)\u2009>\u20090, Re(\u03bc)\u2009>\u20090, Re(\u03bd)\u2009>\u20090, Re(\u03c1)\u2009>\u20090, Re(\u03c3)\u2009>\u20090, Re(w)\u2009>\u20090; andq\u2009\u2264\u2009Re(\u03b1)\u2009+\u2009p then for If In particular, From (1.9), which is the proof of (2.2.1).Again using (1.9) and term by term differentiation under the sign summation(which is possible in accordance with the uniform convergence of the series (1.9) in any compact set which is the proof of (2.2.2). \u25a1\u03bc\u2009=\u2009\u03c1,\u03bd\u2009=\u2009\u03c3, in (2.2.1), we get (2.2.3).Setting \u03bc\u2009=\u2009\u03c1, \u03bd\u2009=\u2009\u03c3, in (2.2.2), we get (2.2.4).Setting \u03b1\u2009+\u2009p), thenIf which proves (2.3.1). \u25a1\u03bc\u2009=\u2009\u03bd, \u03c1\u2009=\u2009\u03c3, \u03b4\u2009=\u2009p\u2009=\u20091, (2.3.1) reduces to the known result of Shukla and Prajapati Shukla and Prajapati (For rajapati  (2.3.1).\u03bc\u2009=\u2009\u03bd, \u03c1\u2009=\u2009\u03c3and p\u2009=\u20091 in (2.3.1), we get (2.3.2).Setting Special Properties: Setting putting \u03bc\u2009=\u2009\u03bd, \u03c1\u2009=\u2009\u03c3and p\u2009=\u2009q\u2009=\u2009\u03b4\u2009=\u20091 in (2.3.1), we have\u03b2\u2009=\u2009\u03b3\u2009=\u2009\u03b4\u2009=\u2009q\u2009=\u20091 in (2.3.2), we have For If In particular, which proves (2.4.1).s to Now change the variable from which proves (2.4.2).Now which proves (2.4.3).q\u2009=\u2009\u03b4\u2009=\u20091 and \u03b3\u2009=\u2009q\u2009=\u2009\u03b4\u2009=\u20091 in (2.4.1) and (2.4.3) yields (2.4.4) and (2.4.5) respectively. \u25a1Putting Using (1.9) with \u0394 is a l-tupple \u0394 is a q-tupple \u0394 is a k-tupple where q+l+1Fk+p+m: (i) If q\u2009+\u2009l\u2009+\u20091\u2009\u2264\u2009k\u2009+\u2009p\u2009+\u2009m, the function q+l+1Fk+p+m converges for all finite z. (ii) If q\u2009+\u2009l\u2009+\u20091\u2009=\u2009k\u2009+\u2009p\u2009+\u2009m\u2009+\u20091, the function q+l+1Fk+p+m converges for |z|\u2009<\u20091 and diverges for |z|\u2009>\u20091(iii) If q\u2009+\u2009l\u2009+\u20091\u2009>\u2009k\u2009+\u2009p\u2009+\u2009m\u2009+\u20091, the function q+1+1Fk+p+m+1 is divergent for |z|\u2009\u2260\u20090(iv) If q\u2009+\u2009l\u2009+\u20091\u2009=\u2009k\u2009+\u2009p\u2009+\u2009m\u2009+\u20091, the function q+l+1Fk+p+m+1 is absolutely convergent on the circle for |z|\u2009=\u20091, if Convergence criterion of generalized Mittag-leffler function In this section we discuss some useful integral transforms like Euler transform, laplace transform and Whittaker transform of Mellin-Barnes integral representation of q\u2009<\u2009Re(\u03b1)\u2009+\u2009p. Then the function Let (1.9) and (1.10) be satified and z)|\u2009<\u20091; the contour of integration beginning at \u2212i\u221e and ending at +i\u221e, and indented to separate the poles of the integrand at where | arg as the sum of the residues at the poles which completes the proof. \u25a1\u03bc\u2009=\u2009\u03c1, \u03bd\u2009=\u2009\u03c3and p\u2009=\u20091, we get the Melin Barne\u2019s integral of the function Setting (Mellin transform) ofFrom Theorem 4.1, we have where is in the form of inverse Mellin-Transform (1.15). So applying the Mellin-transform (1.14) yields directly the required result. \u25a1(Euler(Beta)transform) offrom which the result follows. \u25a1Special properties: (i) For q\u2009=\u20091, (4.3.2) reduces to Tariq OSalim . 4.(ii) For \u03b4\u2009=\u2009q\u2009=\u20091 in (4.3.2), we have a\u2009=\u2009\u03b2, \u03b1\u2009=\u2009\u03c3, then (4.3.2) reduces to If \u03b1\u2009=\u2009\u03b2\u2009=\u2009\u03b3\u2009=\u2009\u03b4\u2009=\u2009q\u2009=\u20091 in (4.3.2), we have Putting (Laplace transform)from which the result follows. \u25a1q\u2009=\u20091, (4.4.2) reduces to Tariq O Salim .(Whittaker transform)\u03d5t\u2009=\u2009v in L.H.S. of Theorem 4.5, we have Substituting from which the result follows. \u25a1Special properties :(i) Putting q\u2009=\u2009\u03b4\u2009=\u20091 in (4.5.2), we have (ii) For q\u2009=\u2009\u03b3\u2009=\u2009\u03b4\u2009=\u20091 in (4.5.2), we have (iii) Now putting q\u2009=\u2009\u03b2\u2009=\u2009\u03b1\u2009=\u2009\u03b3\u2009=\u2009\u03b4\u2009=\u20091 in (4.5.2), we have If the condition (1.10) be satisfied, then (1.9) can be written as Using (4.1.1), we have from \u03b1\u2009=\u2009k, \u03b2\u2009=\u2009\u03bc\u2009+\u20091, \u03b3\u2009=\u2009\u2212\u2009m, q\u2009\u2208\u2009N with q|m and replacing z by zk in (1.6), we get Putting et alwhere zk.Note that Further for which is the required relationship."}
{"text": "Some new Hermite-Hadamard type inequalities for differentiable convex functions were presented by Xi and Qi. In this paper, we present new generalizations on the Xi-Qi inequalities. Below we recall some Hermite-Hadamard type inequalities.Let f\u2032(x)| is convex on , thenIn 1998, Dragomir and Agarwal  showed tf\u2032(x)|p/(p\u22121) is convex on  with p > 1, thenand (ii) if |f\u2032(x)|q is convex on  with q \u2265 1, thenIn 2000, Pearce and Pe\u010dari\u0107  showed tf\u2032(x)|p/(p\u22121) is convex on  with p > 1, thenIn 2004, Kirmaci  showed tf\u2032 \u2208 L and |f\u2032(x)|q is convex on  with q \u2265 1, thenIn 2010, Sarikaya et al.  showed t\u03bb, \u03bc \u2208  and if f\u2032 \u2208 L and |f\u2032(x)|q is convex on  with q \u2265 1, thenIn 2012, Xi and Qi  showed tMoreover, for other results involving the Hermite-Hadamard type inequalities, we also refer to \u201323.In this paper, we generalize the Xi-Qi inequalities.\u03bb,\u2009\u2009\u03bc \u2208 \u211d and let f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L and 0 < \u03f5 < b \u2212 a. ThenLet Integrating by part and changing variable, we haveThus,s > 0 and 0 \u2264 \u03be \u2264 1. ThenLet \u03bb,\u2009\u2009\u03bc \u2208  and let f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L and 0 < \u03f5 < b \u2212 a. If |f\u2032(x)|q is convex on  with q \u2265 1, thenLet f\u2032(x)|q is convex on  with q \u2265 1. By Suppose that |Case\u2009\u2009(q = 1). By the convexity of |f\u2032(x)| and Case\u2009\u2009(q > 1). By H\u00f6lder's inequality, we havef\u2032(x)|q and By the convexity of |\u03f5 = (b \u2212 a)/2 in It is easy to notice that if we put \u03bb,\u2009\u2009\u03bc \u2208  and let f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L. If |f\u2032(x)|q is convex on  with q \u2265 1, thenLet \u03f5 = (b \u2212 a)/3 in One can easily check that if we put \u03bb,\u2009\u2009\u03bc \u2208  and let f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L. If |f\u2032(x)|q is convex on  with q \u2265 1, thenLet \u03f5 = 2(b \u2212 a)/3 in One can easily check that if we put \u03bb,\u2009\u2009\u03bc \u2208  and let f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L. If |f\u2032(x)|q is convex on  with q \u2265 1, thenLet \u03bb = \u03bc = 0 in It is easy to notice that if we put f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L and 0 < \u03f5 < b \u2212 a. If |f\u2032(x)|q is convex on  with q \u2265 1, thenLet \u03bb = \u03bc = 1/2 in It is easy to notice that if we put f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L\u2009\u2009 and 0 < \u03f5 < b \u2212 a. If |f\u2032(x)|q is convex on  with q \u2265 1, thenLet \u03bb = \u03bc = 1 in It is easy to notice that if we put f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L and 0 < \u03f5 < b \u2212 a. If |f\u2032(x)|q is convex on  with q \u2265 1, thenLet \u03bb,\u2009\u2009\u03bc \u2208  and let f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L and 0 < \u03f5 < b \u2212 a. If |f\u2032(x)|q is convex on  with q \u2265 1, thenLet f\u2032(x)|q is convex on  with q \u2265 1. If q = 1, then, by Suppose that |q > 1. By Next, we suppose that f\u2032(x)|q and By the convexity of |This proof is completed.\u03f5 = (b \u2212 a)/2 in It is easy to notice that if we put \u03bb,\u2009\u2009\u03bc \u2208  and let f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L. If |f\u2032(x)|q is convex on  with q \u2265 1, thenLet \u03f5 = (b \u2212 a)/3 in One can easily check that if we put \u03bb,\u2009\u2009\u03bc \u2208  and let f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L. If |f\u2032(x)|q is convex on  with q \u2265 1, thenLet \u03f5 = 2(b \u2212 a)/3 in One can easily check that if we put \u03bb,\u2009\u2009\u03bc \u2208  and let f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L\u2009\u2009. If |f\u2032(x)|q is convex on  with q \u2265 1, thenLet \u03bb = \u03bc = 0 in It is easy to notice that if we put f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L and 0 < \u03f5 < b \u2212 a. If |f\u2032(x)|q is convex on  with q \u2265 1, thenLet \u03bb = \u03bc = 1/2 in It is easy to notice that if we put f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L and 0 < \u03f5 < b \u2212 a. If |f\u2032(x)|q is convex on  with q \u2265 1, thenLet \u03bb = \u03bc = 1 in It is easy to notice that if we put f : I\u2286\u211d \u2192 \u211d be differentiable on I\u00b0 and {a, b}\u2286I with a < b. Assume that f\u2032 \u2208 L and 0 < \u03f5 < b \u2212 a. If |f\u2032(x)|q is convex on  with q \u2265 1, thenLet s, q}\u2286[1, \u221e) and {a, b, wa, wb}\u2286 with a < b. Let 0 < \u03f5 < b \u2212 a.In this section, we suppose that {a, b} with weight {wa, wb} is defined byThe weighted arithmetic mean of data {a, b} with weight {wa, wb} is defined byThe weighted geometric mean of data {a, b} is defined byThe generalized logarithmic mean of data {a, b} is defined byThe identric mean of data {f(x) = xs on , we get the following:Applying f(x) = xs on , we get the following:Applying f(x) = lnx on , we get the following:Applying f(x) = lnx on , we get the following:Applying f(x) = xs on , we get the following:Applying f(x) = xs on , we get the following:Applying f(x) = lnx on , we get the following:Applying f(x) = lnx on , we get the following:Applying"}
{"text": "\u2112g(z) = (1 \u2212 z)\u22121\u222bz1\u200df(\u03b6)d\u03b6 on a class of \u201cmixed norm\u201d spaces of analytic functions on the unit disk, X = H\u03b1,\u03bdp,q, defined by the requirement g \u2208 X\u21d4r \u21a6 (1 \u2212 r)\u03b1Mp \u2208 Lq), where 1 \u2264 p \u2264 \u221e, 0 < q \u2264 \u221e, \u03b1 > 0, and \u03bd is a nonnegative integer. This class contains Besov spaces, weighted Bergman spaces, Dirichlet type spaces, Hardy-Sobolev spaces, and so forth. The expression \u2112g need not be defined for g analytic in the unit disk, even for g \u2208 X. A sufficient, but not necessary, condition is that p, q, \u03b1, and \u03bd for which 1\u00b0\u2112 is well defined on X, 2\u00b0\u2112 acts from X to X, 3\u00b0 the implication We consider the action of the operator  H(\ud835\udd3b) denote the class of all functions holomorphic in the unit disk \ud835\udd3b of the complex plane. In ).\u2112 acts as a bounded operator from Ap,\u03b2 into Ap,\u03b2 if and only if p > \u03b2 + 2.Theorem C (see ). \u2112 actsIf  for all n, and , then \u2112g \u2208 \ud835\udc9f01 if and only ifTheorem D (see ). If g^\u2112g is defined on the space in the sense of \u03c1 \u2208 .Here \u201cacts\u201d means, among other things, that g \u2208 H1 and therefore for g \u2208 Hp \u222a \ud835\udc9f01, p \u2265 1, because Hp \u2282 H1 and \ud835\udc9f01 \u2282 H1. However, .Let 1/2 \u2264 p = \u221e.See the proof of X satisfies . Besides H1, we have, for instance, the following.On the other hand, if a space atisfies  for all The space \ud835\udc9f01 is contained in H1and satisfies , but \u2112 does not map \ud835\udc9f01 into H1.Theorem F (see ). The spOne of the aims of the present paper is to extend Theorems A, B, C, H\u03b1p,q , where \u03b1 > 0 when q < \u221e, and \u03b1 \u2265 0 when q = \u221e, the class of those g \u2208 H(\ud835\udd3b) for which\u03bd be a nonnegative integer, we define the spaceH\u03b1,\u03bdp,q is given byj=0\u22121 should be interpreted as equal to zero.We denote by It is well known and easy to prove that these spaces are complete.p = q, \u03bd = 0, a more (but not most) natural norm is given byThe norm  is not tgiven by  because H\u03b1,\u03bdp,\u221e = :H\u03b1,\u03bdp is specific in that the set, \ud835\udcab, of all analytic polynomials is not dense in it. The closure of \ud835\udcab in H\u03b1,\u03bdp coincides with the \u201clittle oh\u201d spaceThe space From now on, unless specified otherwise, we suppose thatIt is sometimes more convenient to work with the Besov type spaces.\u03b2 \u2208 \u211d and choose any integer \u03bd \u2265 0 such that \u03bd \u2212 \u03b2 > 0. The space \ud835\udd05\u03b2p,q is defined asq = \u221e, we can assume that \u03bd = \u03b2. It is well known that this definition is independent of \u03bd; this follows immediately from Lemma A below. If \u03b2 > 0, then \ud835\udd05\u03b2p,q is \u201ctrue\u201d Besov spaces; if \u03b2 < 0, then \ud835\udd05\u03b2p,q = H\u03b2\u2212p,q; if \u03b2 = 0, then it is called Hardy-Bloch spaces , and, in particular, \u2112 maps the ordinary Lipschitz space into itself.This is, maybe, a new result.\u2112 does not act as a bounded operator from A\u03b21 into A\u03b21, for any \u03b2 > \u22121.In particular, \u03b2 = \u03b1p \u2212 1; that is, \u03b1 = (\u03b2 + 1)/p.This is seen from \u2112 maps the Dirichlet space \ud835\udc9f\u03b2p into itself if and only if p > 1 + \u03b2/2. In particular, \u2112 maps \ud835\udc9f\u03b21 if and only if \u22121 < \u03b2 < 0.(a) \u2112 maps \ud835\udc9f\u03b22 into itself if and only if \u03b2 < 2.Another case: (b) These facts are, maybe, new.\u2112 maps H\u03b1p,q into itself if and only if \u03b1 < 1 and p > 1/(1 \u2212 \u03b1).(a) \u03bd = 0. This is related to a result of . In the case q \u2265 1, a direct proof can be found in . Fix n and let\u03c3kP are -means of P. It follows that\u03c3kP||1 \u2264 ||P||1. Now we use Lagrange's inequality in the formP asLet f \u2208 H\u03b2,\u03bdq1,, where \u03b2 = \u03bd. In this case, by Lemma A, the function f belongs to H\u03b2,\u03bdq1, if and only iff is in H\u03b2,\u03bdq1,, because \u025bq > 1.As noted above it is enough to prove that Thus, we have proved the implication (a)\u21d2(b) of It is clear that (b) implies (a). We have already noted that (c) implies (b). The following assertion shows that (d) implies (c).q \u2264 1,\u2009\u20091 \u2264 p \u2264 2,\u2009\u2009\u03bap,\u03b1,\u03bd = \u03bd + 1 \u2212 \u03b1 \u2212 1/p = 0, and g \u2208 H\u03b1,\u03bdp,q, thenIf H\u03b1,\u03bdq1, \u2282 H\u03b1,\u03bd1,1, it is enough to consider the case q = 1. LetPn\u2217Vn = Vn, which follows from . We usesee thatLemma\u2009\u2009A\u2009p \u2264 \u221e,\u2009\u20090 < q \u2264 \u221e, and \u03bap,\u03b1,\u03bd < 0;1 \u2264 p \u2264 2,1 < q \u2264 \u221e,\u03bap,\u03b1,\u03bd = 0;1 \u2264 p \u2264 \u221e,0 < q \u2264 \u221e,\u03bap,\u03b1,\u03bd = 0. 2 < It remains to be proven that (a) implies (d); that is, that (a) does not hold in the following cases:q < \u221e.Case (1) is part of q > 1).The following assertion proves the desired result in Cases (2) and .If 1 \u2264 Cq is independent of \u025b. This inequality remains true if f\u025b,1 is replaced by f\u025b,\u03c1 with \u03c1 < 1. The function f\u025b,\u03c1(\u03c1 < 1) belongs to f\u025b,\u03c1 : \u025b > 0, \u03c1 < 1} is bounded in H\u03b1,\u03bdp,q. On the the other hand,(a) Letq \u2264 1 and p > 2 is more delicate and depends on Khinchin's inequality and a deep result of Khinchin and Kolmogorov  = sign\u2061(sin(2n\u03c0t)), n \u2265 0, 0 \u2264 t \u2264 1. If {cn} is a sequence in \u2102 such that \u2211n=0\u221e | cn|2 = \u221e, then the series \u2211n=0\u221ecnRn(t) diverges for almost all t \u2208  and moreover the sequence of its partial sums is unbounded a.e. Theorem H. \u2009\u2009We also need the following theorem of Khinchin  such that |E | = 1, which follows from Theorem H. (We can assume that E does not contain points where Rn(t) = 0 because the set of such points is denumerable). On the other hand, by g\u03c4 \u2208 H\u03b1,\u03bdp,q for at least one \u03c4 \u2208 E.Consider first the case 2 < sn|| \u2264 C||g\u03c4|| < \u221e , where C is independent of n. On the other hand, as noted before, the sequencep < \u221e.To complete the proof we consider the polynomials  \u2264 1 \u2212 3p/2, while this implies p \u2264 2, a contradiction which shows that g \u2209 H\u03b1p,\u221e, for p > 2.If For technical reasons, we introduce the spaceIn the proof of  For any sequence {ck}k=mn there is a polynomial h(z) = \u2211k=mnbkzk such that |bk | \u2265|ck| andwhere C is an absolute constant.Theorem J.\u03bap,\u03b1,\u03bd > 0. If (1) 1 < q \u2264 \u221e, 2 < p \u2264 \u221e, and \u03bd \u2212 \u03b1 \u2264 \u22121/2, or (2) 0 < q \u2264 1, 2 < p \u2264 \u221e, and \u03bd \u2212 \u03b1 < \u22121/2, then H\u03bd,\u03b1p,q\u2288\u2113\u221211.Let We haveCase (1),\u2009\u2009(p < \u221e). Let q = \u221e and letck = (k + 1)\u03be. Then\u03be + 1/2 = \u03b1 \u2212 \u03bd; that is, \u03be = \u03b1 \u2212 \u03bd \u2212 1/2 \u2265 0.We have, by \u03b5n \u2208 {\u22121,1} such that the functionH\u03b1,\u03bdp,\u221e. On the other hand,This implies that there is a sequence q < \u221e, we consider the functionIf 1 < Case (1),\u2009\u2009(p = \u221e). Choose {ck} as above and consider the function h(z) = \u2211k=0\u221ebkzk, where |bk | \u2265ck and ||\u0394nh||\u221e \u2264 C(\u2211k\u2208Inck2)1/2 . Finally we use the inequality\u03be < \u03b1 \u2212 \u03bd \u2212 1/2, and repeat the above reasoning to complete the proof.In Case (2) we choose \u03bd \u2212 \u03b1 = \u22121/2, 0 < q \u2264 1, and 2 < p \u2264 \u221e, then H\u03b1,\u03bdp,q \u2282 \u2113\u221211.If We havep \u2264 \u221e, \u03bap,\u03b1,\u03bd > 0, 0 < q \u2264 \u221e, and \u03bd \u2212 \u03b1 > \u22121/2, then H\u03b1,\u03bdp,q \u2282 \u2113\u221211.If 2 < g \u2208 H\u03b1,\u03bdp,q. Then g \u2208 H\u03b1,\u03bdp,\u221e, and hence g \u2208 H\u03b1,\u03bd\u221e2,. It follows that ||\u0394ng||2 \u2264 c2n(\u03b1\u2212\u03bd). On the other hand,\u03b1 \u2212 \u03bd < 0. This proves the proposition.Let The following proposition completes the proof of p \u2264 2, \u03bap,\u03b1,\u03bd > 0, and 0 < q \u2264 \u221e, then H\u03b1,\u03bdp,q \u2282 \u2113\u221211.If 1 \u2264 g \u2208 H\u03b1,\u03bdp,q. Choose \u03b2 so that \u03bap,\u03b1,\u03bd = \u03ba\u03b2,\u03bd2,; that is, \u03b1 = \u03b2 + 1/2 \u2212 1/p. Then, by H\u03b1,\u03bdp,q \u2282 H\u03b2,\u03bdq2,. This implies that H\u03b1,\u03bdp,q \u2282 H\u03b2,\u03bd\u221e2, which means that ||\u0394ng||2 \u2264 C2n(\u03b2\u2212\u03bd), and soLet kp,\u03b1,\u03bd = \u03bd \u2212 \u03b1 + 1 \u2212 1/p > 0; that is, \u2112 acts as an operator from H\u03b1,\u03bdp,q into H\u03b1,\u03bdp,q(\u03b1 > 0). We want to analyze the proof of p \u2264 \u221e and \u03bap,\u03b1,\u03bd > 0, so that \u2112 maps H\u03b1,\u03bdp,q into H\u03b1,\u03bdp,q by In this section we suppose that q = \u221e and \u03bd \u2212 \u03b1 \u2264 \u22121/2. Then (i) there exists a function g \u2208 H\u03b1,\u03bdp,q such that \u03be = \u03b1 \u2212 \u03bd \u2212 1/2 \u2265 0. (ii) The exponent \u03be is best possible. (iii) If \u03be > 0, then there is an \u03b7 > 0 and a function g \u2208 H\u03b1,\u03bdp,q such that Let \u03be is best possible we use g \u2208 H\u03b1,\u03bdp,\u221e if and only if ||\u0394ng||p \u2264 C2n(\u03b1\u2212\u03bd), which implies that ||\u0394ng||2 \u2264 C2\u03b1\u2212\u03bd)n follows from the proof of (iii) This follows from (i).q < \u221e and \u03bd \u2212 \u03b1 \u2264 \u22121/2. Then (i) there exists a function g \u2208 H\u03b1,\u03bdp,q such that g \u2208 H\u03b1,\u03bdp,q such that \u03be > 0 and 0 < \u03b7 < \u03be, then there exists a function g such that Let 1 < H\u03b1,\u03bdp,q \u2282 h\u03b1,\u03bdp, which implies that ||\u0394ng||2 = o(2n(\u03b1\u2212\u03bd)). If o(1), which is impossible.Assertion (i) is part of the proof of In a similar way one proves the following.q \u2264 1 and \u03be = \u03b1 \u2212 \u03bd \u2212 1/2 > 0. Then (i) if 0 < \u03b7 < \u03be, then there is a function g \u2208 H\u03b1,\u03bdp,q such that \u03b7 cannot be replaced by \u03be.Let Combining the above propositions we get the following.\u2112 map H\u03b1,\u03bdp,q into itself, and p > 2. Then the following statements are equivalent: g \u2208 H\u03b1,\u03bdp,q such that \u03b7 > 0;there is a function \u03bd \u2212 \u03b1 < \u22121/2.  Moreover, if \u03bd \u2212 \u03b1 < \u22121/2 and \u03b7 \u2208  is arbitrary, then there is g \u2208 H\u03b1,\u03bdp,q such that q = \u221e, we can take \u03b7 = \u03b1 \u2212 \u03bd \u2212 1/2.Let"}
{"text": "Notions of frontier and semifrontier in intuitionistic fuzzy topology have been studied and several of their properties, characterizations, and examples established. Many counter-examples have been presented to point divergences between the IF topology and its classical form. The paper also presents an open problem and one of its weaker forms. IFS is a sufficiently generalized notion to include both fuzzy sets and vague sets. Fuzzy sets are IFSs but the converse is not necessarily true  and \u03b1 + \u03b2 \u2264 1. An intuitionistic fuzzy point (IFP for short) x\u03b1,\u03b2)( \u2208 A if \u03b1 \u2264 \u03bcA(x) and \u03b2 \u2265 \u03b3A(x). Clearly an intuitionistic fuzzy point can be represented by an ordered pair of fuzzy points as follows:X is denoted as IFP(X).Let B = \u2329y, \u03bcB(y), \u03b3B(y)\u232a is an IFS in Y, then the preimage of B under f, denoted by f\u22121(B), is the IFS in X defined byA = \u2329x, \u03bcA(x), \u03b3A(x)\u232a is an IFS in X, then the image of A under f, denoted by f(A), is the IFS in Y defined byIf The concept of fuzzy topological space, first introduced by Chang in , was genX is a family of IFSs in X satisfying the following axioms:G1\u22c2G2 \u2208 \u03c4 for any G1, G2 \u2208 \u03c4,Gi \u2208 \u03c4 for any arbitrary family {Gi : i \u2208 J}\u2286\u03c4.\u22c3An intuitionistic fuzzy topology (IFT for short) on a nonempty set X, \u03c4) is called an intuitionistic fuzzy topological space  and members of \u03c4 are called intuitionistic fuzzy open  sets. The complement A is called an intuitionistic fuzzy closed (IFC) set in X. Collection of all IFO  sets in IFTS X is denoted as IFO(X) ).In this case, the pair , then A1\u22c3A2 \u2208 IFC(X),If \ud835\udc9c \u2282 IFC(X), then \u22c2\ud835\udc9c \u2208 IFC(X).Let X, \u03c4) be an IFTS and A = \u2329\u03bcA, \u03b3A\u232a an IFS in X. Then the fuzzy interior and fuzzy closure of A are denoted and defined asLet  be an IFTS and A, B be IFSs in X. Then the following properties hold:A\u2286A, (A\u2286Cl\u2009A),Int\u2009A = Int\u2009A, (Cl\u2009Cl\u2009A = Cl\u2009A),Int\u2009Int\u2009A\u2286B\u21d2Int\u2009A\u2286Int\u2009B, (A\u2286B\u21d2Cl\u2009A\u2286Cl\u2009B),A\u22c2B) = Int\u2009A\u22c2Int\u2009B(Cl\u2009(A\u22c3B) = Cl\u2009A\u22c3Cl\u2009B),Int\u2009(A\u22c3B)\u2287Int\u2009A\u22c3Int\u2009B, (Cl\u2009(A\u22c2B)\u2286Cl\u2009A\u22c2Cl\u2009B).Int\u2009(Let (X be an IFTS and let A \u2208 IFS(X). Then x\u03bb,\u03bc)( \u2208 IFP(X) is called an intuitionistic fuzzy frontier point  of A if A is called an IF frontier of A and denoted by Fr\u2009A. It is clear that Let A \u2208 IFS(X), A\u22c3Fr\u2009A \u2282 Cl\u2009A. However, the inclusion cannot be replaced by an equality.For each A in an IFTS X, the following hold:A is IFC then Fr\u2009A\u2286A,If A is IFO then If For an IFS (1)\u2009\u2009A\u2286A, if A is IFC set in X.(2) A is IFO implies (3) (4) Converse of (2) and (3) of X, \u03c4) be the IFTS defined by \u00c7oker (Example 3.3 d)\u2286Cl\u2009f(f\u22121(B))\u2286Cl\u2009B = B or f([f\u22121(B)]d)\u2286B gives [f\u22121(B)]d\u2286f\u22121(f[f\u22121(B)]d)\u2286f\u22121(B) or [f\u22121(B)]d\u2286f\u22121(B) implies f\u22121(B) is IF closed in X. Thus, f is IF continuous.(2)\u21d2(1) Suppose f : X \u2192 Y be an IF continuous mapping. Then Fr\u2009f\u22121(B)\u2286f\u22121(Fr\u2009B), for any IFS B in Y.Let f is IF continuous. Let B be an IFS in Y. Thenf\u22121(B)\u2286f\u22121(Fr\u2009B).Suppose that A\u2286B and B \u2208 IFCS(X). Then Fr\u2009A\u2286B.Let A\u2286B implies Cl\u2009A\u2286Cl\u2009B, we have Since X, \u03c4) and  be two IFTSs and f : X \u2192 Y, a function. Then f is said to be fuzzy open if the image of each IFS in \u03c4 is in \u03d5.Let (f : X \u2192 Y be an IFO mapping and B an IFS in Y. Then f\u22121(Fr\u2009B)\u2286Fr\u2009f\u22121(B).Let f is IFO and B is an IFS in Y. PutA is IF open and therefore f(A) is IF open in Y. This gives f\u22121(Fr\u2009B)\u2286Fr\u2009f\u22121(B).Suppose n Y. PutA=Fr\u2009f\u22121 is called an intuitionistic fuzzy semiopen set (IFSOS) ifA is called an intuitionistic fuzzy semiclosed set if the complement of A is an IFSOS.An IFS A in an IFTS  are denoted and defined asThe semiclosure and semi-interior of an IFS A in IFTS X, the following hold:sCl\u2009A = A\u22c3Int\u2009Cl\u2009A,sInt\u2009A = A\u22c2Cl\u2009Int\u2009A.For an IFS A be an IFS in X. From Int\u2009(Cl\u2009(A\u22c3Int\u2009Cl\u2009A)) = Int\u2009(Cl\u2009A\u22c3Cl\u2009Int\u2009Cl\u2009A) = Int\u2009Cl\u2009A\u2286A\u22c3Int\u2009Cl\u2009A, it follows that A\u22c3Int\u2009Cl\u2009A is an IFSC set. Hence, sCl\u2009A\u2286A\u22c3Int\u2009Cl\u2009A. Since sCl\u2009A is IFSC, we have Int\u2009Cl\u2009A\u2286Int\u2009Cl\u2009sCl\u2009A\u2286sCl\u2009A. Thus A\u22c3Int\u2009Cl\u2009A\u2286sCl\u2009A.(1)\u2009\u2009Let (2) This can be proved in a similar manner as (1).A be an IFS in IFTS X. Then the intuitionistic fuzzy semifrontier of A is defined as sFr\u2009A is an IFSC set.Let In the following theorems, we note that almost all the properties related to intuitionistic fuzzy semi-interior, intuitionistic fuzzy semi-closure and intuitionistic fuzzy semifrontier are analogous to their counterparts in Intuitionistic Fuzzy Topology, and hence proofs of most of them are not given.A and B in an IFTS X, one hassCl\u2009\u2009sCl\u2009A = sCl\u2009A,sInt\u2009\u2009sInt\u2009A = sInt\u2009A,sInt\u2009(A\u22c3B)\u2287sInt\u2009A\u22c3sInt\u2009B,sInt\u2009(A\u22c2B) = sInt\u2009A\u22c2sInt\u2009B,sCl\u2009(A\u22c3B) = sCl\u2009A\u22c3sCl\u2009B,sCl\u2009(A\u22c2B)\u2286sCl\u2009A\u22c2sCl\u2009B.For IFSs sInt\u2009A and sInt\u2009B are both IFSO sets and A\u2286A\u22c3B, B\u2286A\u22c3B implies sInt\u2009A\u2286sInt\u2009(A\u22c3B) and sInt\u2009B\u2286sInt\u2009(A \u22c3 B). In Combination, sInt\u2009A\u22c3sInt\u2009B\u2286sInt\u2009(A\u22c3B) or(5) A\u22c2B\u2286A and A\u22c2B\u2286B imply sInt\u2009(A\u22c2B)\u2286sInt\u2009A\u22c2sInt\u2009B. Conversely, sInt\u2009A\u2286A and sInt\u2009B\u2286B imply sInt\u2009A\u22c2sInt\u2009B\u2286A\u22c2B and sInt\u2009A\u22c2sInt\u2009B is IFSO. But sInt\u2009(A\u22c2B) is the largest IFSO set contained in A\u22c2B; hence sInt\u2009A\u22c2sInt\u2009B\u2286sInt\u2009(A\u22c2B). This gives the equality.(6) (7) This follows easily from (2).A\u22c2B\u2286A, A\u22c2B\u2286B(8) Since In the following theorem, (1)\u2013(5) are analogues of A in IFTS X, the following hold:A is IFSC, then sFr\u2009A\u2286A,if A is IFSO, then if A\u2286B and B \u2208 IFSC(X) ). Then sFr\u2009A\u2286B  ) denotes the class of intuitionistic fuzzy semi-closed (resp. intuitionistic fuzzy semiopen) sets in X,let sFr\u2009A\u2286Fr\u2009A,sCl\u2009sFr\u2009A\u2286Fr\u2009A.For an IFS sCl\u2009A\u2286Cl\u2009A and (6) Since (7) The converse of (2), (3), (6), and (7) of In Example 3.3 of , we chooThe following is an analogue of A be an IFS in IFTS X. Then one hassFr\u2009A = sCl\u2009A \u2212 sInt\u2009A,sFr\u2009sInt\u2009A\u2286sFr\u2009A,sFr\u2009sCl\u2009A\u2286sFr\u2009A,sInt\u2009A\u2286A \u2212 sFr\u2009A. Let To show that (2), (3), and (4) of In general topology, the following hold:In Example 3.3 of , we chooA and B be IFSs in an IFTS X. Then sFr\u2009(A\u22c3B)\u2286sFr\u2009A\u22c3sFr\u2009B.Let The converse of In Example 3.3 of , we chooHowever, we have the following theorem which is an analogue of A and B in IFTS X, one hasFor IFSs A and B in IFTS X, one hasFor IFSs The analogue of A in IFTS X, one hassFr\u2009sFr\u2009A\u2286sFr\u2009A,sFr\u2009sFr\u2009sFr\u2009A\u2286sFr\u2009sFr\u2009A.For an IFS As in the case of In Example 3.3 of , we chooA in IFTS X is called an intuitionistic fuzzy semi-Q-neighborhood of an IFP e if there exists an IFSO set B in X, such that eqB\u2286A.An IFS e = x\u03b1,\u03b2)\u2287A. Equivalently, x\u03b1,\u03b2), whenever e \u2208 A. The union of all the semi-accumulation points of A is called the intuitionistic fuzzy semiderived set of A, denoted as Asd\u2061. It is evident that Asd\u2061\u2286sCl\u2009A.An IFP A be an IFS in X, then sCl\u2009A = A\u22c3Asd.Let \u03a9 = {e | e\u2009\u2009is\u2009\u2009a\u2009\u2009semi-adherence\u2009\u2009point\u2009\u2009of\u2009\u2009A}. Then from sCl\u2009A = \u22c3\u03a9. On the other hand, e \u2208 \u03a9 is either e \u2208 A or e \u2209 A; for the latter case, by e \u2208 Asd\u2061; hence sCl\u2009A = \u22c3\u03a9 \u2282 A\u22c3Asd\u2061. The reverse inclusion is obvious.Let A in an IFTS X, A is IFSC if Asd\u2286A.For any IFS f : X \u2192 Y be a function from an IFTS X to another IFTS Y. Then f is said to be an intuitionistic fuzzy semicontinuous function if f\u22121(A) is IFSO in X for each IFO set A in Y.Let f : X \u2192 Y be a function. Then the following are equivalent:f : X \u2192 Y is intuitionistic fuzzy semi-continuous,f(Asd)\u2286sCl\u2009f(A), for any IFS A in X.Let f is intuitionistic fuzzy semi-continuous. Let A be an IFS in X. Since sCl\u2009f(A) is IFC in Y, f\u22121(sCl\u2009f(A)) is IFSC in X. A\u2286f\u22121(sCl\u2009f(A)) gives sCl\u2009A\u2286sCl\u2009f\u22121(sCl\u2009f(A)) = f\u22121(sCl\u2009f(A)). Therefore, f(Asd\u2061)\u2286f(sCl\u2009A)\u2286ff\u22121(sCl\u2009f(A))\u2286sCl\u2009f(A). Consequently, f(Asd\u2061)\u2286sCl\u2009f(A).(1)\u21d2(2) Suppose that f(Asd\u2061)\u2286sCl\u2009f(A). Letting B be any IFC set in Y, we show that f\u22121(B) is IFSC in X. By our hypothesis, f([f\u22121(B)]sd\u2061)\u2286sCl\u2009f(f\u22121(B))\u2286sCl\u2009B = B or f([f\u22121(B)]sd\u2061)\u2286B gives [f\u22121(B)]sd\u2061\u2286f\u22121(f[f\u22121(B)]sd\u2061)\u2286f\u22121(B) or [f\u22121(B)]sd\u2061\u2286f\u22121(B) implies f\u22121(B) is IFSC in X. Thus, f is intuitionistic fuzzy semi-continuous.(2)\u21d2(1) Suppose f : X \u2192 Y be a intuitionistic fuzzy semi-continuous function. Then one hasB in Y.Let f is intuitionistic fuzzy semi-continuous. Let B be an IFS in Y.Suppose that sFr\u2009f\u22121(B)\u2286f\u22121(sFr\u2009B).Then"}
{"text": "Relaxation oscillations of two-dimensional planar singular perturbed systems with a layer equation exhibiting canard cycles are studied. The canard cycles under consideration contain two turning points and two jump points. We suppose that there exist three parameters permitting generic breaking at both the turning points and the connecting fast orbit. The conditions of one  relaxation oscillation near the canard cycles are given by studying a map from the space of phase parameters to the space of breaking parameters. Let H(n) denote the maximum number of limit cycles of a general planar polynomial vector field of degree n. As mentioned in [H(n), but there are many results on the lower bounds of H(n); for example, H(3) \u2265 13, H(4) \u2265 21, and H(5) \u2265 28  and g are two smooth functions with respect to variables  \u2208 \u211d2 \u00d7 \u211dk and \u025b is a small real number. As \u025b > 0 and small, we make the time scaling t = \u03c4/\u025b and get the following equivalent standard form of slow-fast system which has the same phase portraits as the one of system ,,\u025b = 0 inn system  and 4)\u025b = 0 iny with the slow variable x acting as a parameter.System  and 4)  4) are, f = 0 defines the critical manifold S of the equilibrium of the layer equation  =  of system , and rescaling of time, system  of system (h1 and h2 are smooth and a(\u03bb0) = 0 and a(\u03bb) is smooth at \u03bb0 in case of a generic turning point.At contact points that are isolated points on the critical manifold where normal hyperbolicity is lost, the blow-up method pioneered by Dumortier and Roussarie  is a powf system  have bee, system  can, loc\u025b approaches 0 and the limit cycles that are close to slow-fast cycle are called relaxation oscillations. In 2007, Dumortier et al. [f of degree 7 bifurcate from the canard cycle consisting of two jump points by analyzing the zeros of slow divergence integral of canard cycle  is near .Consider the following smooth slow-fast Li\u00e9nard system Fa(x), g fulfill the following conditions.H1), as a = 0, the function F0(x) has five singular points: two Morse maximum points at p1, p4, two Morse minimum points at p2, p3, and one Morse maximum at p0 with \u2212x0 < p1 < p2 < p0 < p3 < p4 < x0. The parameter a is just the difference a = Fa(p4) \u2212 Fa(p1). Let \u03b10 = F0(0), \u03b11 = F0(p2), \u03b12 = F0(p3), \u03b1 = max\u2061{\u03b11, \u03b12}, and \u03b2 = F0(p4) = F0(p1). Then the values F0(\u00b1x0) are assumed to be below the minimum value min\u2061{\u03b11, \u03b12}.On the interval (\u2212H2) = 0, g = 0, g = 0, g \u2260 0, and g \u2260 0, but \u2202g/\u2202x \u2260 0, i = 2,3, \u2202g/\u2202b \u2260 0 and \u2202g/\u2202c \u2260 0, g > 0 for p1 < x < p2 or p0 < x < p3, g < 0 for p2 < x < p0 or p3 < x < p4 (see Suppose that  p4 see .H3), there exists a canard cycle containing four horizontal segments: one between the two Morse maxima x = p1, x = p4 denoted by \u03c3h, one below the left Morse maximum value and at the height y = u denoted by \u03c3l, one below the right Morse maximum value and at the height y = v denoted by \u03c3r(v), and one at the height y = w and between p2 and p3 denoted by \u03c3m, where u \u2208 , v \u2208 , w \u2208 , and the corresponding canard cycle is denoted by \u0393uvw.When In the following, we assume that the smooth functions x1, x2 is defined as follows:An essential tool to study the limit cycles bifurcated from the canard cycle is the slow divergence integral see , 13\u201315);\u201315;13\u201315uvw of system (x1(u), x2(u), respectively, denote x-coordinates of intersection points between \u03c3l and slow curve y = F0(x), where \u2212x0 < x2(u) < x1(u) < p2; let x1(w), x2(w), respectively, denote x-coordinates of intersection points between \u03c3m and y = F0(x), where p2 < x1(w) < x2(w) < p3; let x1(v), x2(v), respectively, denote x-coordinates of intersection points between \u03c3r and curve y = F0(x), where p3 < x1(v) < x2(v) < x0. By applying the slow divergence integral formula introduced in , J(v), K(w), L(v), M(u), and N(w), which are the slow divergence integrals of six slow curves contained in \u0393uvw. In detail, two of these curves are located on the left of p2 and their slow divergence integrals are functions of u, two of them are on the right of p3 and their slow divergence integrals are functions of v, and two of them are between p2 and p3 and their slow divergence integrals are functions of w , g fulfill the conditions H1, H2, H3. Let D = J(v) \u2212 I(u) + M(u) \u2212 N(w) + K(w) \u2212 L(v) denote the total slow divergence integral of \u0393uvw of system , v0 \u2208 , and w0 \u2208  such that D \u2260 0, then for \u025b > 0 and small enough  is a regular point of \u03a6\u025b, whose explicit expression will be given in uvw.If there exist u0 \u2208 , v0 \u2208 , and w0 \u2208  such that I(u0) \u2212 J(v0) > 0, M(u0) \u2212 N(w0) = 0, and I(u0) \u2212 J(v0) = K(w0) \u2212 L(v0), then for \u025b > 0 and small enough there exits (u0(\u025b), v0(\u025b), w0(\u025b)) that is a generic fold singularity of \u03a6\u025b. A relaxation oscillation bifurcates from \u0393uvw and this semistable limit cycle is generically unfolded by the parameter  for \u025b > 0 and small enough, producing a pair of hyperbolic limit cycles of system , v0 \u2208 , and w0 \u2208  such that J(v0) \u2212 I(u0) = 0, M(u0) \u2212 N(w0) = 0, L(v0) \u2212 K(w0) = 0, and J\u2032(v0)M\u2032(u0)K\u2032(w0) \u2212 I\u2032(u0)L\u2032(v0)N\u2032(w0) \u2260 0, then for \u025b > 0 and small enough there exits \u025b. A codimension 2 relaxation oscillation bifurcates from \u0393uvw and this degenerated limit cycle is generically unfolded by the parameter  for \u025b > 0 and small enough, producing system , with z \u2208 \u211dp for some p \u2208 \u2115, is called \u025b-regularly smooth in z, if f is continuous and all partial derivatives of f with respect to z exist and are continuous in .A function \u025b > 0 small enough, a limit cycle of system , J(v), K(w), L(v), M(u), and N(w) for \u025b = 0.For f system  cuts \u2211lCi\u2009\u2009, the transitions have the following expressions:l to C1: from \u2211l to C2: from \u2211m to C2: from \u2211m to C3: from \u2211r to C1: from \u2211r to C3: from \u2211where functions fl, gl, fr, gr, hm1, hm2 are \u025b-regularly smooth in \u025b-regularly smooth in From \u201315, due o(\u03b5) and are \u03b5-regularly smooth in By using the same analysis as , 15, we a, a, \u025b. So one can solve , a, \u025b-flat perturbations of the previous ones. We will continue to call them We can solve this system in an solve  to obtai term of  except tThe proof of \u025b small enough and we view \u03a6\u025b as a map from  to \u025b as follows:u, v, and w.We take u, v, w, \u025b) = det\u2061/\u2202). To find the singular points of map \u03a6\u025b, by direct computation we get the following formula about the determinant \u0394:Let \u0394 = 0 is equivalent to the equationFor \u025b > 0 and small enough by noting \u025b-regularity of function \u025b is nondegenerated at the point , so from u0v0w0.It follows from  that theSo the conclusion of the first part of In this subsection, we give the proof of the second part of First, we present the following lemma.\u025b > 0 and small enough, if there exists P0(\u025b) = (u0(\u025b), v0(\u025b), w0(\u025b)) that satisfies \u0394 = 0, then it holds thatgrad\u2009\u0394 =  is the gradient of function \u0394 and k is nonzero function at (P0(\u025b), \u025b).As E = 0 determines a surface S1 in the neighborhood of point P0 for \u025b > 0 and small. From the fact that \u0394 = 0 is equivalent to E = 0, we get that the equation \u0394 = 0 also determines the surface S1. For grad\u2009\u0394|P0(\u025b), grad\u2009E|P0(\u025b) are both the normal vectors of surface S1 at the point P0(\u025b), then we get , \u03c8, \u03d5 as follows:\u025b is given in u0, v0, w0) by P0, from assumptions in the second part of E = 0 and (\u2202E/\u2202u \u00b7 \u2202E/\u2202v)|P0,0) = 0. On surface u0(\u025b), v0(\u025b), w0(\u025b)) denoted by P0(\u025b) such that E(P0(\u025b), \u025b) = 0, P0(0) = P0; that is, \u0394(P0(\u025b), \u025b) = 0.Denote (\u03c6/\u2202v)|P0,0) from the first equation of (\u03d5/\u2202w)|P0,0), From  = 0 we compute the second derivative of u asThen, as \u0394(D/\u2202u)|P0 = M\u2032(u0) \u2212 I\u2032(u0) < 0, (\u2202D/\u2202v)|P0 = J\u2032(v0) \u2212 L\u2032(v0) > 0, and (\u2202D/\u2202w)|P0 = K\u2032(w0) \u2212 N\u2032(w0). So (\u2202E/\u2202u)|P0(\u025b),\u025b)( = (\u2202D/\u2202u)|P0 + O(\u025b) < 0, (\u2202E/\u2202v)|P0(\u025b),\u025b)( = (\u2202D/\u2202v)|P0 + O(\u025b) > 0, and (\u2202E/\u2202w)|P0(\u025b),\u025b)( = (\u2202D/\u2202w)|P0 + O(\u025b).From , we get P0(\u025b),\u025b)( = k((\u2202D/\u2202u)|P0 + O(\u025b), (\u2202D/\u2202v)|P0 + O(\u025b), (\u2202D/\u2202w)|P0 + O(\u025b)).From From , we get I(u0) \u2212 J(v0) > 0, M(u0) \u2212 N(w0) = 0, we get that \u025b \u2192 0+ and \u025b \u2192 0+.Noticing that \u025b > 0 and small enough we get thatThen for \u03d5w\u03c6v \u2260 0, d\u0394/du)|P0(\u025b),\u025b))( \u2260 0. So we get that (u0(\u025b), v0(\u025b), w0(\u025b)) is a fold point of map \u03a6\u025b. So from uvw by unfolding the parameters a, b, c.Because \u025b) (see ), we getThe conclusion of the second part of In this subsection, we give the proof of the third part of \u025b > 0 and small enough, if there exists u, v, w, \u025b) = 0, d\u0394/du = 0, and As u, v, w, \u025b) = 0 we getFrom the proof of the second part of d\u0394/du = 0 is equivalent toFrom d\u0394/du = 0 is equivalent to \u025b > 0 and small. That means that both equations d\u0394/du = 0 and So rewrite , and we n we get .The proof of Proof of the Third Part of  From Lemmas E = 0, S1 : E = 0 and l\u025b which passes through point . Therefore, for \u025b > 0 and small enough, we choose point l\u025b. Then It is easy to check that \u025b given in (u at Consider the map \u03a6given in . By applFrom From  and 26)26), we g\u025b > 0 and small enough we get thatThen for \u025b > 0 and small enough. From a, b, c), for \u025b > 0 and small enough, producing system  = (\u2212(1/6)x6 + (5/4)x4 \u2212 2x2 + ax). Here we use symbols pi, i = 1,2, 3,4 defined in p1 = \u22122, p4 = 2; the values of F0(x) at these two points are both 4/3 and the values of F0(x) at the points p2 = \u22121, p3 = 1 are both \u221211/12 and F0(0) = 0.For this system, V \u2208 , the equation F0(x) = V will have three roots on the right side of the y-axis. For the roots, the formula in the above equation contains complex number, so we choose the smallest one of the above three roots and denote it by the variable v, where v \u2208 . By noting that relation between u and v is one to one relation, we express the other two roots in the form v \u2208 . Using the same way, we take y = U, U \u2208 ; then the equation F0(x) = U will have three roots on the left side of the y-axis; we choose the biggest one of the above three roots and denote it by the variable u; we express the other two roots in the form x2(u) < x1(u) < u, u \u2208 . Also F0(x) = y is symmetrical about 0, so the straight line y = W, W \u2208 , between \u22121 and 1 will intersect the curve with two points; then we, respectively, denote their x-coordinate by \u2212w, w, where w \u2208 . The corresponding canard cycle \u0393UVW can be seen in For a given P(x) = (1/8)x8 \u2212 (3/2)x6 + 6x4 \u2212 8x2, Q1(x) = \u2212(1/9)x9 + (1/8)x8 + (9/7)x7 \u2212 (24/5)x5 \u2212 (3/2)x6 + 6x4 + (16/3)x3 \u2212 8x2, Q2(x) = \u2212(1/11)x11 + (3/10)x10 + (2/3)x9 \u2212 (13/4)x8 + (3/7)x7 + (21/2)x6 \u2212 (56/5)x5 \u2212 6x4 + 16x3 \u2212 8x2; then from the slow divergence integral formula (Let P(x) = 1/x8 \u2212 (3/M(u) \u2212 N(w) = 0; that is,Firstly, consider c1 = c2 = 0, from the fact that the function P(x) is monotone on the interval  we get c1 and c2 of obtained equation to equal zeros, we get w1(u) = 0, w2(u) = 0. Therefore, we getWhen  it into ; by settI(u) \u2212 J(v) = 0; that is,Secondly, consider P(x) is strictly monotone on the interval  and v = \u2212u as c1 = c2 = 0.By noticing that the even function c1 and c2 of obtained equation to equal zeros, we getLet it into ; then byL(v) \u2212 K(w); that is,Thirdly, consider function Putting the expressions  and 37)37) into Qi(x), i = 1,2, we get that \u03b61(u) \u00b7 \u03b62(u) \u2260 0, u \u2208 .From the definition of function Denote \u03b8(u) \u2260 0, u \u2208 , then \u03b61(u)/\u03b62(u) = \u03b71(u)/\u03b72(u).For H(u) = L(v(u)) \u2212 K(w(u)), where w(u) and v(u) are, respectively, implicitly determined by M(u) \u2212 N(w) = 0 and I(u) \u2212 J(v) = 0.Denote By direct computations, we getOn the other hand from expressions , we get H(u) = 0, then we getAs \u03b71(u)/\u03b72(u) and \u03b71\u2032(u)/\u03b72\u2032(u) on the interval  is plotted in The graph of functions u0 \u2208 , there exists c2 = \u2212(\u03b72(u0)/\u03b71(u0))c1 + O(c12), 0 < |c1 | \u226a1 such that equation H(u) = 0 holds and dH(u0)/du = c1\u03b8(u0)(\u03b71\u2032(u0) + (\u03b71(u0)/\u03b72(u0))\u03b72\u2032(u0)) + O(c12) \u2260 0 for c1 > 0 and small enough.From the graph in c2 = \u2212(\u03b72(u0)/\u03b71(u0))c1 + O(c12), 0 < |c1 | \u226a1, it holds that I\u2032(u0)L\u2032(v0)N\u2032(w0) \u2212 J\u2032(v0)K\u2032(w0)M\u2032(u0) \u2260 0, where w0 = w(u0), v0 = v(u0). So there exists  which satisfies the conditions in the third part of UVW, where U = F0(u0), V = F0(v(u0)), W = F0(w(u0)), u0 \u2208 .In other words, as"}
{"text": "We investigate the existence of the periodic solutions of a nonlinear integro-differential system with piecewise alternately advanced and retarded argument of generalized type, in short DEPCAG; that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use Green's function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results. Among the functional differential equations, Myshkis  proposediated in , in Wieniated in , the firiated in \u201321. Appliated in , 22\u201325. iated in , 26\u201334.\u2124, \u2115, \u211d, and \u2102 be the set of all integers, natural, real, and complex numbers, respectively. Denote by |\u00b7| a norm in \u211dn, n \u2208 \u2115. Fix two real sequences ti, \u03b3i, i \u2208 \u2124, such that ti < ti+1,\u2009 and\u2009ti \u2264 \u03b3i \u2264 ti+1 for all i \u2208 \u2124,\u2009and \u2009ti \u2192 \u00b1\u221e as i \u2192 \u00b1\u221e.Let \u03b3 : \u211d \u2192 \u211d be a step function given by \u03b3(t) = \u03b3i for t \u2208 Ii =  corresponds to \u03b3i = ti = i \u2208 \u2124, and \u03b3(t) = 2[(t + 1)/2] corresponds to ti = 2i \u2212 1, \u03b3i = 2i, i \u2208 \u2124. The particular case of DEPCAG, when \u03b3i = ti, i \u2208 \u2124, an only delayed situation, is considered by first time in Akhmet  the advanced intervals and Ii\u2212 =  \u00d7 D \u00d7 D1.Butris  investigA : \u211d \u2192 \u211dn\u00d7n, f : \u211d \u00d7 \u211dn \u00d7 \u211dn \u2192 \u211dn, and g : \u211d \u00d7 \u211dn \u00d7 \u211dn \u2192 \u211dn are continuous in their respective arguments. In the analysis we use the idea of Green's function for periodic solutions and convert the nonlinear integro-differential systems with DEPCAG =A(tz\u2032(t)=A(t\u03b3(t) is discontinuous. Thus, in general, the right-hand side of the DEPCAG system , i \u2208 \u2124. In other words, by a solution z(t) of the DEPCAG system (z\u2032(t) exists at each point t \u2208 \u211d, with the possible exception of the points ti \u2208 \u211d, i \u2208 \u2124, where a one-sided derivative exists, and the nonlinear integro-differential systems with DEPCAG (z(t) on each interval , i \u2208 \u2124 as well.In our paper we assume that the solutions of the nonlinear integro-differential systems with DEPCAG  are contG system  has discG system  we mean h DEPCAG  are sati\u03c9-periodicity of Green's function. The rest of the paper is organized as follows. In In this section we state and define Green's function for periodic solutions of the nonlinear integro-differential system with piecewise alternately advanced and retarded argument .I be the n \u00d7 n identity matrix. Denote by \u03a6, \u03a6 = I, t, s \u2208 \u211d, the fundamental matrix of solutions of the homogeneous system  \u2208 \u2124 be the unique integer such that t \u2208 Ii = , Green's function for (t) is a fundamental solution of (Suppose that the condition (tion for  is givenution of  and(19)N\u03c9) implies the existence of the matrix D.We note that the condition , we first give the following lemma.To prove double N\u03c9) holds. Let the matrix D be defined by  to study after the existence of periodic solutions.For N\u03c9) holds. Then Green's function G is double \u03c9-periodic; that is, G = G.Suppose that the condition z\u2032 a function such that |y1 \u2212 y2| \u2264 \u03b4 impliesLet where Lipschitz ConditionsLC) such that t, s \u2208 \u211d and for any y1, y2 \u2208 \u211dn we haveThere exists a continuous function Lf) such thatFor Moreover, Lg) such thatFor \u2009Moreover, Moreover, Invariance ConditionsMC) functions and positive constants For every Mf) and positive constants \u03c11, \u2009C1, \u2009\u03b72 for whichFor every where Mg) and positive constants \u03c12, \u2009C2 for whichFor every where \u222b\u03c4\u03c4+\u03c9[\u03ba1(s) + \u03ba2(s)]ds \u2264 C2, \u03c9 \u2208 \u211d+, \u03c4 \u2208 \u211d.where Periodic ConditionsP)(\u03c9 > 0 such thatThere exists (1)A(t), f, and g are periodic functions in t with a period \u03c9 for all t \u2265 \u03c4;(2)C = C for all t \u2265 \u03c4;(3)p \u2208 \u2124+, for which the sequences {ti}i\u2208\u2124, {\u03b3i}i\u2208\u2124 satisfy the  condition; that is,there exists \u03c9, p) condition is a discrete relation, which moves the interval Ii into Ii+p. Then we have the following consequences.(i)\u03c4 \u2208 \u211d, the interval  can be decomposed as follows:For any (ii)t \u2208  \u00d7 \ud835\udc9e\u211b \u00d7 \ud835\udc9e\u211b and by the periodicity in t, the function g is uniformly continuous on \u211d \u00d7 \ud835\udc9e\u211b \u00d7 \ud835\udc9e\u211b. Thus, for any \u03f5\u2032 = (\u03f5/cG\u03c9) > 0, there exists \u03b4 = \u03b4(\u03f5) > 0 such that z1, z2 \u2208 \ud835\udd4a, ||z1 \u2212 z2|| \u2264 \u03b4 implies |g, z1(\u03b3(t))) \u2212 g, z2(\u03b3(t)))| \u2264 \u03f5\u2032 for t \u2208 . Then ||\u212cz1 \u2212 \u212cz2|| \u2264 \u03f5. In fact, by the continuity of g,\u212c is proved.The function Step\u2009\u20092. We show that the image of \u212c is contained in a compact set.x, y \u2208 \ud835\udc9e\u211b and s \u2208 ; for the continuity of the function g, there exists M > 0 such that |g| \u2264 M. Let \u03c6n \u2208 \ud835\udd4a where n is a positive integer; then we have\u212c\u03c6n(t))\u2032 shows thatA(t) is bounded on  and \u212c\u03c6n(t), g, \u03c6n(\u03b3(t)) are bounded on  \u00d7 \ud835\udd4a \u00d7 \ud835\udd4a. Thus, the above expression yields ||(\u212c\u03c6n)\u2032|| \u2264 L, for some positive constant L. Hence the sequence (\u212c\u03c6n) is uniformly bounded and equicontinuous. Ascoli-Arzela's theorem implies that a subsequence (\u212c\u03c6nk) of (\u212c\u03c6n) converges uniformly to a continuous \u03c9-periodic function. Thus \u212c is continuous and \u212c(\ud835\udd4a) is a compact set.Let \ud835\udc9c we obtain the following.In a similar way, for N\u03c9) and (C) hold, \ud835\udc9c is defined by , (P), (Lf), (LC), (Mg) hold. Let R be a positive constant satisfying the inequality\u03c9-periodic solution in \ud835\udd4a.Suppose the hypotheses , (P), (Lg), (Mf), (MC), (C) hold. Let R be a positive constant satisfying the inequality\u03c9-periodic solution in \ud835\udd4a.Suppose the hypothesis , (P), (Lf), (LC), (Lg) hold. If\u03c9-periodic solution.Suppose the hypotheses , (P), (Mf), (MC), (Mg), (C) hold. Let R be a positive constant satisfying the inequality\u03c9-periodic solution in \ud835\udd4a.Suppose the hypotheses |/(|x| + |y|)) = 0, then the system (N\u03c9), (P3), (Mg), A(t) and g are periodic functions in t with a period \u03c9 for all t \u2265 \u03c4 and |g| \u2264 c1(|x| + |y|) + \u03c12, with 2c1\u03c9 < C2 and \u03c12 constants for |x| + |y| \u2264 R, R > 0.Considering the DEPCAG system 56)x\u2032(tkii (see ) proved x\u2032(tkii (N\u03c9), (P1), (P3), (Lg), (Mf), and cG(C1 + L2)R + cG(\u03b2 + \u03c11)\u03c9 \u2264 R with Considering the nonlinear system of differential equations with a general piecewise alternately advanced and retarded argument,x\u2032(t)=A(tP1) is satisfied by \u03c9 = \u03c91, (P2) by \u03c9 = \u03c92, and (P3) by \u03c9 = \u03c93, if \u03c9i/\u03c9j is a rational number for all i, j = 1,2, 3; then (P1), (P2), and (P3) are simultaneously satisfied by \u03c9 = l.c.m.{\u03c91, \u03c92, \u03c93}, where l.c.m.{\u03c91, \u03c92, \u03c93} denotes the least common multiple between \u03c91, \u03c92 and \u03c93. In the general case it is possible that there exist five possible periods: \u03c91 for A, \u03c92 for f, \u03c93 for g, \u03c94 for C, and the sequences {ti}i\u2208\u2124, {\u03b3i}i\u2208\u2124 satisfy the  condition. If \u03c9i/\u03c9j is a rational number for all i, j = 1,2, 3,4, 5, so, in this situation our results insure the existence of \u03c9-periodic solution with \u03c9 = l.c.m.{\u03c91, \u03c92, \u03c93, \u03c94, \u03c95}. Therefore the above results insure the existence of \u03c9-periodic solutions of the DEPCAG system (Suppose that (G system . These sP\u03c9), f, and g are periodic functions in t with a period \u03c91, \u03c92, and \u03c93, respectively, for all t \u2265 \u03c4;C = C, for all t \u2265 \u03c4;p \u2208 \u2124+, for which the sequences {ti}i\u2208\u2124, {\u03b3i}i\u2208\u2124 satisfy the  condition.There exists To determine criteria for the existence and uniqueness of subharmonic solutions of the DEPCAG system , from noAs immediate corollaries of Theorems N\u03c9), (P\u03c9), (Lf), (LC), (Mg) and (\ud835\udd4a.Suppose the hypotheses (Mg) and  hold. ThMg) and  has at lN\u03c9), (P\u03c9), (Lg), (Mf), (MC), (C) and (\ud835\udd4a.Suppose the hypotheses ( (C) and  hold. Th (C) and  has at lN\u03c9), (P\u03c9), (Lf), (LC), (Lg) and (Suppose the hypotheses (Lg) and  hold. ThLg) and  has a unN\u03c9), (P\u03c9), (Mf), (MC), (Mg), (C) and (\ud835\udd4a.Suppose the hypotheses ( (C) and  hold. Th (C) and  has at lWe will introduce appropriate examples in this section. These examples will show the feasibility of our theory.Mathematical modelling of real-life problems usually results in functional equations, like ordinary or partial differential equations, integral and integro-differential equations, and stochastic equations. Many mathematical formulations of physical phenomena contain integro-differential equations; these equations arise in many fields like fluid dynamics, biological models, and chemical kinetics. So, we first consider nonlinear integro-differential equations with a general piecewise constant argument mentioned in the introduction and obtain some new sufficient conditions for the existence of the periodic solutions of these systems.\u03bb : \u211d2 \u2192  satisfies (MC): |C| \u2264 c2(R)\u03bb|y| + h for |y| \u2264 R, where g : \u211d \u00d7 \u211dn \u00d7 \u211dn \u2192 \u211dn is continuous and C satisfies (C).The conditions of \u025b > 0, there exists \u03b4 > 0 such that |y1 \u2212 y2| \u2264 \u03b4 impliesR such thata* = supt\u2208\u211d\u2009\u2009|a(t)| and a\u2217 = inft\u2208\u211d|a(t)|.Indeed, for any \u03c0-periodic solution.Then, by Thus many examples can be constructed where our results can be applied.n\u00d7n and \u03bc : \u211d \u2192 \u211dn be two functions satisfyingti}i\u2208\u2124 and {\u03b3i}i\u2208\u2124 satisfy the  condition, A, B, \u039b, \u03ba, and \u03bc are \u03c9-periodic continuous functions, and z\u2032(t) = A(t)z(t) satisfies (N\u03c9);B is \u03c9-periodic matrix function: |B(t)| \u2264 b, g : \u211dn \u00d7 \u211dn \u2192 \u211dn is a continuous function, and |g| \u2264 c1(R)|x| + c2(R)|y| + \u03d11 for |x|, |y| \u2264 R, where (c1(R) + c1(R))\u03c9 \u2264 C2;\u03ba(x) \u2212 \u03ba(y)| \u2264 c3(R)|x \u2212 y|, where |Let \u039b : \u211d \u2192 \u211dR such that\u03c9-periodic solution of the DEPCAG system (The hypotheses of G system .Lg) and (MC). On the other hand, the periodic situation of the DEPCAG system (Note that similar results can be obtained under (G system  and 57)Lg) and Let us consider another example for second-order differential equations with a general piecewise constant argument. In this case, we can show the existence and uniqueness of periodic solutions of the following nonlinear DEPCAG system.\u03ba1, \u03ba2 \u2208 \u211d, \u03b3(t) = \u03b3i if ti \u2264 \u03b3i < ti+1, i \u2208 \u2124, and the sequences {ti}i\u2208\u2124 and {\u03b3i}i\u2208\u2124 satisfy the  condition.Consider the following nonlinear DEPCAG system:z\u2032(t) = Az(t) does not admit any nontrivial \u03c9-periodic solution; that is, the condition (N\u03c9) is satisfied. Let \u03c6(t) = (\u03c61(t), \u03c62(t)), \u03c8(t) = (\u03c81(t), \u03c82(t)) and define \ud835\udd4a = {z \u2208 \u2119\u03c9, ||z|| \u2264 R}, where R \u2208 \u211d+ satisfies the condition\u03c6, \u03c8 \u2208 \ud835\udd4a we havef we have\u03c0/\u03c9-periodic solution in \ud835\udd4a.We write the DEPCAG system  in the s"}
{"text": "As an application, we prove the existence of some new singular directions for a meromorphic function  For the module distribution of a meromorphic function, there are three main theorems, that is, the Picard theorem, the Borel theorem, and the Nevanlinna second fundamental theorem. The fundamental concept in the angular distribution is singular direction. Singular direction is a concept of localizing value distribution in \u2102 onto a sector S containing a single ray J : argz = \u03b8 emanating from the origin say. A Julia direction and a Borel direction are refinements of the Picard theorem and the Borel theorem, respectively. Corresponding to the Nevanlinna second fundamental theorem, a new singular direction, called T direction, was recently introduced in Zheng  values of a, for any positive number \u025b < \u03c0, we havex] implies the maximum integer number which does not exceed x and l is a positive integer.Theorem C was conjectured by Zheng . In 11]11], the f(z) be a meromorphic function and satisfy (f(z) concerning multiple values.Let  satisfy . Then thl + 2)/l] \u2192 2 as l \u2192 \u221e. Since Zheng [Note that the T direction of meromorphic function concerning multiple values is a refinement of the ordinary T direction since [ be a meromorphic function. A direction argz = \u03b8 is called a pseudo-T direction of f(z) if, for any number \u025b\u2009\u2009(0 < \u025b < \u03c0/2), any system aj\u2009\u2009 of distinct values, and any system kj\u2009\u2009 such that kj is a positive integer or +\u221e and thatj\u2009\u2009(1 \u2264 j \u2264 q) such thatLet f(z) be a meromorphic function and satisfy (f(z).Let  satisfy . Then thq = 3, kj = \u221e\u2009\u2009, so Theorem C is a special case of (i) In Theorem C, kj = 1\u2009\u2009, then q = 5; if kj = 2\u2009\u2009, then q = 4; if kj = l \u2265 3\u2009\u2009, then q = 3. So Theorem D is a special case of (ii) If In order to prove r) is a nonnegative increasing function in  and satisfiesE \u2282  such that \u222bE(1/rlog\u2061\u2061r)dr < 1/3, one hasSuppose that \u03a8(m\u2009\u2009(m \u2265 4) be a fixed positive integer, \u03b80 = 0, \u03b81 = 2\u03c0/m,\u2026, \u03b8m\u22121 = (m \u2212 1)2\u03c0/m, \u03b8m = \u03b80. We put \u0394(\u03b8i) = {z : |argz \u2212 \u03b8i| < 2\u03c0/m},\u2009\u0394o(\u03b8i) = {z : |argz \u2212 \u03b8i| < \u03c0/m},\u2009\u2009i = 0,1,\u2026, m \u2212 1; \u0394(\u03b8m) = \u0394(\u03b80), \u0394o(\u03b8m) = \u0394o(\u03b80). Then among these m angular domains {\u0394(\u03b8i)}, there is at least an angular domain \u0394(\u03b8i) such that for any system aj\u2009\u2009 of distinct values and any system kj\u2009\u2009 such that kj is a positive integer or +\u221e and thatj\u2009\u2009(1 \u2264 j \u2264 q) such that\u03b8i)\u2009\u2009(1 \u2264 i \u2264 m), there is a system aij\u2009\u2009 of distinct values and a system kij\u2009\u2009 such that kij is a positive integer or +\u221e and thatj\u2009\u2009(1 \u2264 j \u2264 q) we haveo(\u03b8i+1), \u0394(\u03b8i+1), we haveT = \u2211i=0m\u22121T, f) and adding two sides of the above expression from i = 0 to m \u2212 1, we can obtaini, there exists a ri, the inequality T, f) > em3 would bold for r > ri, while the inequality (Eo(\u03b8i+1)\u0394 is the set of r which consists of a series of intervals and satisfiesr0 = max\u2061\u2061{ri, i = 1,2,\u2026, m}; we have for any i, T, f) > em3; thenE = \u222ai=0m\u22121Eo(\u03b8i+1)\u0394. Therefore, there exists a sequence rn\u2032 \u2208  \u2212 E, such thatm, there is at least an angular domain \u0394(\u03b8i) such that for any system aj\u2009\u2009 of distinct values and any system kj\u2009\u2009 such that kj is a positive integer or +\u221e and thatj\u2009\u2009(1 \u2264 j \u2264 q) such that\u03b8m}, still denote it {\u03b8m}, we assume that \u03b8m \u2192 \u03b80. Put L : argz = \u03b80; then L is a pseudo-T direction that is stated in Firstly, we prove the following statement. Let equality  does notows from , 68), a, am\u2009\u2009(m uch thatlim\u2061n\u2192\u221eT(\u025b\u2009\u2009(0 < \u025b < \u03c0/2), when m is sufficiently large, we have \u0394(\u03b8m) \u2282 \u03a9. By . By , we have"}
{"text": "Apart from establishing coefficient bounds for these classes, we establish inclusion relationships involving ( \ud835\udc9c(n) denote the class of functions of the formLet f(z) \u2208 \ud835\udc9c(n) is star-like of complex order b, denoted as f(z) \u2208 S*(b) if and only if it satisfiesA function f(z) \u2208 \ud835\udc9c(n) is convex of complex order b, denoted as f(z) \u2208 C(b) if and only if it satisfiesS*(b), and C(b) are introduced and studied by Nasr and Aouf [A function and Aouf .fj given byf1\u2217f2)(z), is given byFor the two functions f(z) of the form  ase(z) = z,Given the form  and \u03b4 \u2265 Nn,\u03b4 of a function f is introduced and studied by Ruscheweyh [The concept of Neighborhood scheweyh  and extescheweyh .\u03b11, \u03b12,\u2026, \u03b1q and \u03b21, \u03b22,\u2026, \u03b2s , we define the generalized hypergeometric function qFs as\u2115 denotes the set of all positive integers and (x)k is the Pochhammer symbol defined in terms of gamma functions asgq,s defined by\ud835\udc9f\u03bb,\u03bcmf(z) : \ud835\udc9c \u2192 \ud835\udc9c is defined by\u03bc \u2264 \u03bb \u2264 1 and m \u2208 \u21150. By the above definition, it is easy to note that\u03b1i\u2032s, \u03b2j\u2032s, q, s, m, \u03bb, and \u03bc, we can deduce several operators such as S\u0103l\u0103gean derivative operator [For complex numbers ently in , an operoperator , Ruschewoperator , fractiooperator , Carlsons; z) asFsqk=\u0393 consisting of functions of the form f(z) are as given in (For 0 \u2264 given in . \u03b1 \u2264 1 we let B be the subclass of \ud835\udc9c(n), consisting of functions of the form f(z) are as given in  denote the S\u0103l\u0103gean derivative of order m given byBy specializing the parameters involved in the above definitions, we could arrive at several known as well as new classes. For example, by taking q = 2, s = 1, \u03b11 = \u03b7 \u2212 1, (\u03b7 > \u22121), \u03b12 = 1, \u03b21 = 1, one getsDmf(z) is the operator introduced and studied by Abubaker and Darus [Similarly, on taking nd Darus  given bym = 0 in the definition of the classes A and B, we could arrive at Sn and Rn which were introduced and studied by Murugusundaramoorthy et al. [Further, by taking y et al. .A and B and several inclusion relationships involving n-\u03b4 neighborhoods of analytic univalent functions with negative and missing coefficients belonging to these classes.In this paper, we establish the coefficient inequalities for the classes f \u2208 \ud835\udc9c(n) as given in  satisfies (Let the functions of form  belong t view of  and 16)A. Then, ssertion . Conversatisfies , then, iSimilarly, we prove the following.f \u2208 \ud835\udc9c(n) be as defined in  as given in (f(z) \u2208 A1 if and only ifLet the function given in . Then, ff \u2208 \ud835\udc9c(n) be as defined in  as given in  \u2208 \ud835\udc9c(n) be as defined in .Iff \u2208 A. Then, in view of .IfA1 \u2282 Nn,\u03b4(e).IfB1 \u2282 Nn,\u03b4(e).IfA2 \u2282 Nn,\u03b4(e). IfB2 \u2282 Nn,\u03b4(e). IfA\u03c3 and B\u03c3. Here, the classes A\u03c3 consist of functions f \u2208 \ud835\udc9c(n) for which there exists a function g(z) \u2208 A such thatB\u03c3, consisting of functions f(z) \u2208 \ud835\udc9c(n) for which there exists another function g(z) \u2208 B such thatIn this section, we determine the neighborhood properties of g \u2208 A andNn,\u03b4(g) \u2282 A\u03c3.If f \u2208 Nn,\u03b4(g), thenSuppose g \u2208 A, we havef \u2208 A\u03c3 for \u03c3 given by  \u2282 B\u03c3. If g \u2208 A1 andNn,\u03b4(g) \u2282 A1\u03c3. If g \u2208 B1 andNn,\u03b4(g) \u2282 B1\u03c3. If g \u2208 A2 andNn,\u03b4(g) \u2282 A2\u03c3. If g \u2208 B2 andNn,\u03b4(g) \u2282 B2\u03c3.If"}
{"text": "H space of Chen's expanded mixed element method. We study the new expanded mixed element method for convection-dominated Sobolev equation, prove the existence and uniqueness for finite element solution, and introduce a new expanded mixed projection. We derive the optimal a priori error estimates in L2-norm for the scalar unknown u and a priori error estimates in (L2)2-norm for its gradient \u03bb and its flux \u03c3. Moreover, we obtain the optimal a priori error estimates in H1-norm for the scalar unknown u. Finally, we obtained some numerical results to illustrate efficiency of the new method.We propose and analyze a new expanded mixed element method, whose gradient belongs to the simple square integrable space instead of the classical  \u03a9 is a bounded convex polygonal domain in R2 with Lipschitz continuous boundary \u2202\u03a9 and J =  and f are given functions, coefficients a = a, b = b are smooth and bounded functions, coefficient c(x) = (c1(x), c2(x)) is a bounded vector, anda0, a1, b0, b1, and d1.We consider the following Sobolev equation with convection term:Sobolev equations are a class of important evolution partial differential equations and have a lot of applications in many physical problems, such as the porous theories concerned with percolation into rocks with cracks, the heat conduction problems in different mediums, and the transport problems of humidity in soil. In , the finIn 1994, Chen , 9 develH space. In this paper, we will study the new expanded mixed element method for convection-dominated Sobolev equation. We will give the proof for the existence and uniqueness of the solution for semidiscrete scheme and a new expanded mixed projection and the proof of its uniqueness. We will prove the optimal a priori error estimates in L2-norm for the scalar unknown u and a priori error estimates in (L2)2-norm for its gradient \u03bb and its flux \u03c3. In particular, we obtained the optimal a priori error estimates in H1-norm for the scalar unknown u. Finally, we obtained some numerical results to confirm our theoretical analysis.In 2011, we developed and analyzed a new expanded mixed finite element method  for elliC will denote a generic positive constant which is free of the space-time parameters h and \u0394t. At the same time, we denote the natural inner product in L2(\u03a9) or (L2(\u03a9))2 by  with the corresponding norm ||\u00b7||. The other notations and definitions of Sobolev spaces as in  \u21a6 H01 \u00d7 (L2(\u03a9))2 \u00d7 (L2(\u03a9))2 such thatuh, \u03bbh, \u03c3h}: \u21a6 Vh \u00d7 Wh \u00d7 Wh such thatVh, Wh) is chosen as the finite element pair P1 \u2212 P02 as follows:Vh, Wh) satisfies the so-called discrete Ladyzhenskaya-Babuska-Brezzi condition.In this paper, the new expanded mixed weak formulation of  is to fiuch that(a)\u2009)2 instead of the classical H space. Obviously, the regularity requirements on the solution \u03bb = \u2207u reduced.Compared to Chen's expanded mixed weak formulation , the graThere exists a unique discrete solution to semidiscrete scheme .\u03c8i(x)}i=1n1 and {\u03c6j(x)}j=1n2 be bases of Vh and Wh, respectively. Letvh = \u03c8m, wh = \u03c6l, and zh = \u03c6s, the problems \u2009Au~\u2032(Substitute (b) into I is a unit matrix.Substituting (b) and  into 1111(a), weThus, by the theory of differential equations , 13) ha ha13) hat); then  and 11)u~(t); thvalently  has a unIn order to analyze the convergence of the method, we first introduce the new expanded mixed elliptic projection associated with our equations.uh, \u03bbh, \u03c3h): \u2192 Vh \u00d7 Wh \u00d7 Wh be given by the following mixed relations:Let , wh=\u03c3h=\u03c3~h in (b), and h=\u03bb~h in (c) to obs to get||a1/2\u03bb~h=\u2207u~h in  and usin\u2207u~h in (||u~h||\u2264Co obtain2a0\u222b0t||\u03bbstitute (\u222b0t||\u03bb~h|\u2207u~h in (||u~h||\u2264C0.From (\u03bb~h=0.Coombining (c), we g0.From )2 \u2192 Wh such thatThere exists a linear operator \u03a0h of For the linear operator \u03a0Ph : H01(\u03a9) \u2192 Vh such thatThere exists a linear operator From , 21, we h and Ph, we rewrite \u03b7, \u03b4, and \u03c1 as\u03b7m, \u03b4m, and \u03c1m are known, it is enough to estimate \u03b7e, \u03b4e, and \u03c1e. Using Lemmas Using the definition of \u03a0g Lemmas (a), wh = \u03c1e in (b), and = \u03b4e in (c) to obt to obtainwh = \u03c1e, we obtainIntegrate  with resntiating (b) and tvh = (\u03b7e)t in (zh = (\u03b4e)t in (Choose \u03b7e)t in (a) and z\u03b4e)t in (c) to obCombining  and 39)39), we hSubstitute  into 4040 to obtzh = \u03c1e in (Choose = \u03c1e in (c) and u= \u03c1e in  to obtaiwh = \u2207\u03b7e in (Choose  \u2207\u03b7e in (b) and u \u2207\u03b7e in  and CaucCombining , 42), , , and LemC independent of h such thatThere is a constant \u03b7||, ||\u03b7t||, and ||\u03b7||1, we consider the following auxiliary elliptic problem:To estimate terms ||problem:\u2212\u2207\u00b7(a\u2207\u03c7)=problem:5\u2212\u2207\u00b7(a\u2207\u03c7)From , we obta\u03b7||L2, we can obtainUsing similar method to ||Combining  and 47)47), we oUsing \u201349), we, we49), For a priori error estimates, we decompose the errors asUsing -6) and  and 6) aWe will prove the error estimates for semidiscrete scheme.u||H2 + ||ut||H2 + ||\u03bb||H1(\u03a9))2( + ||\u03bbt||H1(\u03a9))2(.Assume that vh = \u03c2 in (wh = \u03be in (zh = \u03b8 in (Choose h = \u03c2 in (a), wh h = \u03be in (b), and h = \u03b8 in (c) to obt to obtainwh = \u2207\u03c2 in  to hantiating (b) and t= \u03c2t in (a) and z= \u03b8t in (c) to obe obtain\u2009\u2009\u2212 \u03b8t in ((a)\u2009||\u03c2t| \u03b8t in ((a)\u2009||\u03c2t|e obtain||\u03c2||2+\u222b0h = \u03be in (c) and ue obtain||\u03c2||2+\u222b0 we have||\u03c2t||2+| \u2207\u03c2 in \u2212e obtain||\u03c2||2+\u222b0= \u2207\u03c2 in  and In this section, we get the error estimates of fully discrete schemes. For the backward Euler procedure, let 0 = uhn, \u03bbhn, \u03c3hn) \u2208 Vh \u00d7 Wh \u00d7 Wh,  such thatEquation  has the We will prove the theorem for the fully discrete error estimates.C independent of h and \u0394t such thatAssume that t = tn, we get the error equationsUsing , 63), a, a63), avh = \u03c2n in , (b0 \u2212 C\u0394t) > 0. Then we use Cronwall's lemma to obtainwh = \u2207\u03c2n in (wh = \u03ben in (vh = \u2202t\u03c2n in (zh = \u2202t\u03b8n in (Choose = \u03c2n in (a), wh = \u03ben in (b), and = \u03b8n in (c) to obe obtain12\u0394t(||\u03c2note that||R1n||2\u2264o obtain||\u03c2J||2+| \u2207\u03c2n in (b), we h||.From (b), we g|.From t\u03c2n in (a), and t\u03b8n in (c) to obAdding the three equations, we obtainUsing  and 74)74), we gzh = \u03ben in (Taking = \u03ben in (c), we gSubstitute  into 8080 to getCombining -24), , \u03a9 = \u00d7, J =  = 1 + 2x12 + x22, b = 1 + x12 + 2x22, and c(x) = T, and f is chosen so that the exact solution for the scalar unknown function isIn order to illustrate the efficiency of the new expanded mixed element method, we consider the following initial-boundary value problem of 2D Sobolev equation with the convection term:\u03a9 into the triangulations of mesh size h uniformly, consider the piecewise linear space Vh with index k = 1 for the scalar unknown function u and the piecewise constant space Wh with index k = 0 for the gradient \u03bb and the flux \u03c3, use the backward Euler procedure with uniform time step length \u0394t = 1/M, and obtain some convergence results for ||u\u2212uh||L\u221e(L2(\u03a9)), ||u\u2212uh||L\u221e(H1(\u03a9)), ||\u03bb\u2212\u03bbh||L\u221e((L2(\u03a9))2), and ||\u03c3\u2212\u03c3h||L\u221e((L2(\u03a9))2) with u, \u03bb, and\u03c3 in Figures uh, \u03bbh, and \u03c3h in Figures t = 1, and We divide the domain u in L2-norm, H1-norm, and the error estimates for \u03bb and \u03c3 in (L2)2-norm, which confirm the theoretical results in this paper, in It is easy to see that we obtained the optimal error estimates for L2 for the scalar unknown u and the a priori error estimates in (L2)2-norm for its gradient \u03bb and its flux \u03c3 are proved. Especially, the optimal a priori error estimates in H1-norm for the scalar unknown u are derived. Finally, some numerical results are provided to confirm our theoretical analysis.In this paper, a new expanded mixed finite element method is proposed and studied for Sobolev equation with convection-term. The proof for the existence and uniqueness of the solution for semidiscrete scheme, the new expanded mixed projection, and the proof of its uniqueness are given. The optimal a priori error estimates in u, \u03bb, \u03c3}: \u21a6 H01 \u00d7 (L2(\u03a9))2 \u00d7 (L2(\u03a9))2 such thatIn the near future, the new expanded mixed method will be applied to other evolution equations such as evolution integrodifferential equations, hyperbolic wave equations, and nonlinear evolution equations. And the new characteristic expanded mixed finite element method for Sobolev equation will be studied. The new expanded characteristic-mixed weak formulation is to find {In another article, we will give the error estimates for the new characteristic expanded mixed finite element method."}
{"text": "We construct some results on the regularity of solutions and the approximate controllability for neutral functional differential equations with unbounded principal operators in Hilbert spaces. In order to establish the controllability of the neutral equations, we first consider the existence and regularity of solutions of the neutral control system by using fractional power of operators and the local Lipschitz continuity of nonlinear term. Our purpose is to obtain the existence of solutions and the approximate controllability for neutral functional differential control systems without using many of the strong restrictions considered in the previous literature. Finally we give a simple example to which our main result can be applied. H and V be real Hilbert spaces such that V is a dense subspace in H. Let U be a Banach space of control variables. In this paper, we are concerned with the global existence of solution and the approximate controllability for the following abstract neutral functional differential system in a Hilbert space H:A is an operator associated with a sesquilinear form on V \u00d7 V satisfying G\u00e5rding's inequality, f is a nonlinear mapping of  \u00d7 V into H satisfying the local Lipschitz continuity, B : L2 \u2192 L2 and C : L2 \u2192 L2 are appropriate bounded linear mapping.Let This kind of equations arises in population dynamics, in heat conduction in material with memory and in control systems with hereditary feedback control governed by an integrodifferential law.Recently, the existence of solutions for mild solutions for neutral differential equations with state-dependence delay has been studied in the literature in , 2. As fL2-regularity remain valid under the above formulation of the neutral differential equation \u2229W1,2\u21aaC for some T > 0 by using fractional power of operators and Sadvoskii's fixed point theorem. Thereafter, by showing some variations of constant formula of solutions, we will obtain the global existence of solutions of  \u21a6 x is Lipschitz continuous, which is applicable for control problems and the optimal control problem of systems governed by nonlinear properties.In this paper, we propose a different approach from the earlier works  is the set of all m-times continuously differential functions on \u03a9.\u2009\u2009C0m(\u03a9) will denote the subspace of Cm(\u03a9) consisting of these functions which have compact support in \u03a9.\u2009\u2009Wm,p(\u03a9) is the set of all functions f = f(x) whose derivative D\u03b1f up to degree m in distribution sense belong to Lp(\u03a9). As usual, the norm is then given byD0f = f. In particular, Wp0,(\u03a9) = Lp(\u03a9) with the norm ||\u00b7||p,\u03a9.\u2009\u2009W0m,p(\u03a9) is the closure of C0\u221e(\u03a9) in Wm,p(\u03a9).\u2009p = 2 we denote Wm,2(\u03a9) = Hm(\u03a9) and W0p2,(\u03a9) = H0m(\u03a9).For \u2009p\u2032 = p/(p \u2212 1),\u2009\u20091 < p < \u221e. Wp\u22121,(\u03a9) stands for the dual space W0p\u20321,(\u03a9)* of W0p\u20321,(\u03a9) whose norm is denoted by ||\u00b7||p,\u221e\u22121,.Let Let X is a Banach space and 1 < p < \u221e,\u2009Lp is the collection of all strongly measurable functions from  to X, the pth powers of norms are integrable,\u2009Cm will denote the set of all m-times continuously differentiable functions from  to X.\u2009X and Y are two Banach spaces, B is the collection of all bounded linear operators from X to Y, and B is simply written as B(X).If \u2009X0 and X1, \u03b8,p and \u03b8 denote the real and complex interpolation spaces between X0 and X1, respectively.For an interpolation couple of Banach spaces If A be a closed linear operator in a Banach space. Then\u2009D(A) denotes the domain of (A) and R(A) the range of A;\u2009\u03c1(A) denotes the resolvent set of A, \u03c3(A) the spectrum of A, and \u03c3p(A) the point spectrum of A;\u2009x \u2208 D(A) : Ax = 0} of A is denoted by Ker\u2061(A).the kernel or null space {Let H is identified with its dual space we may write V \u2282 H \u2282 V* densely and the corresponding injections are continuous. The norm on V, H, and V* will be denoted by ||\u00b7||, |\u00b7| and ||\u00b7||\u2217, respectively. The duality pairing between the element v1 of V* and the element v2 of V is denoted by , which is the ordinary inner product in H if v1, v2 \u2208 H.If l \u2208 V* we denote  by the value l(v) of l at v \u2208 V. The norm of l as element of V* is given byV has a stronger topology than H and, for brevity, we may considerFor a be a bounded sesquilinear form defined in V \u00d7 V and satisfying G\u00e5rding's inequality:A be the operator associated with this sesquilinear form:A is a bounded linear operator from V to V* by the Lax-Milgram theorem. The realization of A in H which is the restriction of A toA. From the following inequalitiesD(A), it follows that there exists a constant C0 > 0 such thatLet V,V*)1/2,2 denotes the real interpolation space between V and V*, o, o12), o1.3.3 of ).A generates an analytic semigroup S(t) in both H and V*. The following lemma is from Lemma 3.6.2 of  \u2192 \u211d with b(0) = 0 such thatLet f is a nonlinear mapping of  \u00d7 V into H satisfying the following.(i)L1 : \u211d+ \u2192 \u211d such thatx||\u2264r and ||y||\u2264r.There exists a function (ii)t \u2208  and x \u2208 V.The inequalityLet us rewrite (Fx)(t) = f) for each x \u2208 L2. Then there is a constant, denoted again by L1(r), such thatx \u2208 L2 and x1,x2 \u2208 Br(T) = {x \u2208 L2:||x||L2 \u2264 r}.From now on, we establish the following results on the solvability of .x0 \u2208 H, k \u2208 L2 for T > 0. Then, there exists a solution x of (\u03a3) \u2282 \u03a3, \u2192 L2 asJ has a fixed point in the space L2. By Assumptions B and F, it is easily seen that J is continuous from C in itself. LetL2. From . From  it folloy0|.By , 25), a, a(42)irtue of  in Lemma.Since  and Assuch thatT13\u03b2/2, let xt*(x) = (K1x)(t) for every x \u2208 L2. Then xt* \u2208 L2 and we have lim\u2061n\u2192\u221ext*(xn) = xt*(x\u221e) since w\u2009\u2212\u2009lim\u2061n\u2192\u221exn = x\u221e. Hence,n\u2192\u221e||K1xn||L2 = ||K1x\u221e||L2. Since L2 is a Hilbert space, the relation , a, aK1 + ow thatlim\u2061n\u2192\u221e\u2061milar to  and 52)K1 + K2ondition  the contondition  exists uK2 is contractive. Thus, \ud835\udcb21(T1).So by virtue of condition , K2 is x be a solution of (x0 \u2208 H. Then we have that from (C3 such thatT].From now on we establish a variation of constant formula  of solutution of  and x0 hat from -52) it  it x be  account  there exequality . Since tequality  and 44)x be a soequality (63)|xM0b(T1) < 1, then the uniqueness of the solution of (\ud835\udcb21(T) is obtained.If ution of  in \ud835\udcb21M0L < 1ch thatM0b(T1)+ondition  exists ux0,k) \u2208 D(A) \u00d7 L2. Then by the argument of the proof of x of  \u2208 H \u00d7 H \u00d7 L2. Then the solution x of (x \u2208 \ud835\udcb21(T) \u2261 L2\u2229W1,2 and the mappingLet Assumptions B and F be satisfied and  \u2208 H \u00d7 L2, then x belongs to \ud835\udcb21(T). Let  \u2208 H \u00d7 H \u00d7 L2 and let xi \u2208 \ud835\udcb21(T) be the solution of  in place of  for i = 1,2. Let xi\u2009\u2009 \u2208 \u03a3. Then as seen in From ution of  with , let xk be the solution of  let xk be the solution of  to L2. Moreover, if one defines the operator \u2131 by\u2131 is also a compact mapping from L2 to L2.Let one assume that the embedding ution of . Then thx0,k) \u2208 H \u00d7 L2, then in view of xk \u2208 L2, we have f \u2208 L2. Consequently, by (xk \u2208 \ud835\udcb21(T) \u2282 C. With aid of (a) of xk||L2 \u2264 ||xk||\ud835\udcb21(T), we havek is bounded in L2, then so is xk in \ud835\udcb21(T) \u2261 L2\u2229W1,2. Since V is compactly embedded in H by assumption, the embeddingk \u21a6 xk is compact from L2. Moreover, we have that \u2131 is a compact mapping ofL2 to L2.If  to L2, which is called a controller. For x \u2208 L2 we setN :  if for every zT \u2208 H and \u03f5 > 0 there exists a control function u \u2208 L2 such that the solution x of  \u2212 zT | <\u03f5; that is, R(T) in H.The system  is said ,f,u) of  satisfieL2 to H byp \u2208 L2.We define the linear operator We need the following hypothesis.\u025b > 0 and p \u2208 L2, there exists a u \u2208 L2 such thatq1 is a constant independent of p.(i) For any f is a nonlinear mapping of \u2009\u00d7\u2009H into H satisfying the following.(ii) L1 : \u211d+ \u2192 \u211d such thatx | \u2264r and |y | \u2264r.There exists a function AS(t \u2212 s)Bx = S(t \u2212 s)ABx for each x \u2208 V. Therefore, the system  such thatFx)(t) = f) for t \u2265 0. Throughout this section, invoking (r1 such thatBy virtue of condition (i) of Assumption S we note that e system  is approinvoking , we can u1 and u2 be in L2. Then under the Assumption S, one has that, for 0 \u2264 t \u2264 T,M2 = eM(MNT+L1(r1)).Let x(t) = x and x2(t) = x. Then for 0 \u2264 t \u2264 T, we haveLet we havex1(t)\u2212x2T].Under Assumption S, the system  is appro\u025b > 0 and zT \u2208 D(A), there exists u \u2208 L2 such thatzT \u2208 D(A) there exists p \u2208 L2 such thatp(s) = {(zT + (Bx)(T)) \u2212 sA(zT + (Bx)(T))} \u2212 S(s)(x0 + y0)/T. Let u1 \u2208 L2 be arbitrary fixed. Since by Assumption S there exists u2 \u2208 L2 such thatw2 \u2208 L2 by Assumption S such thatt \u2264 T. Therefore, in view of u3 = u2 \u2212 w2. We determine w3 such thatt \u2264 T. Hence, we haveu* \u2208 L2 such thatwn \u2208 L2 by Assumption B, such thatun+1 = un \u2212 wn, we have\u025b > 0 there exists integer N such thatT] as N tends to infinity.We will show that ws that|zT+(Bx)ch that|S^(G(\u00b7xe system  is approA are \u03bbn = \u2212n2 and \u03d5n(y) = (2/\u03c0)1/2sinny, respectively. Moreover, (a)\u03d5n : n \u2208 N} is an orthogonal basis of H,{(b)S(t)x = \u2211n=1\u221een2t\u03d5n,\u2009\u2009\u2200x \u2208 H, t > 0,(c)\u03b1 < 1; then the fractional power A\u03b1 : D(A\u03b1) \u2282 H \u2192 H of A is given bylet 0 < LetA\u22121/2x = \u2211n=1\u221e(1/n)\u03d5n and ||A\u22121/2|| = 1.In particular, g is a real valued function belonging to C2) which satisfies the following conditions:(i)g(0) = 0, g(r) \u2265 0 for r > 0,(ii)g\u2032(r) \u2264 c(r + 1) and |g\u2032\u2032(r)|\u2264c for r \u2265 0 and c > 0. If we presentf is a mapping from the whole V to H by Sobolev's imbedding theorem  = \u03bc2r + \u03b72r2/2 (\u03bc and \u03b7 are constants). In addition, we need to impose the following conditions ons see , 25)..g\u2032(r) \u2264 (iii)b is measurable andThe function (iv)2/\u2202z2)b is measurable, b = b, andThe function (\u2202(v)C : L2 \u2192 L2 is a bounded linear operator.We define B : L2 \u2192 L2 byB is bounded linear andB1 is bounded linear with B \u2208 D(A1/2) and ||A1/2Bx|| = ||B1x||. Therefore from x of (T].Consider the following neutral differential control system:\u2202\u2202t[x(t,\u2202\u2202t[x(t,"}
{"text": "F : \u03a9 \u2192 \u211dn is a continuous quasi-bounded functional which satisfies a local Lipschitzcondition with respect to the second argument and \u03a9 is a subset in \u211d \u00d7 Crn, Crn : = C,yt\u2208Crn, and yt(\u03b8) = y(t + \u03b8), \u03b8 \u2208 . A monotone iterative method is proposed to provethe existence of a solution defined for t \u2192 \u221e with the graph coordinates lying betweengraph coordinates of two (lower and upper) auxiliary vector functions. This result is applied to scalar advanced linear differential equations. Criteria of existence of positive solutions are given and their asymptotic behavior is discussed.We study asymptotic behavior of solutions of general advanced differential systems  C, where a, b \u2208 \u211d, a < b, be the Banach space of the continuous mappings from the interval  to \u211dn equipped with the supremum normn. In the case of a = 0 and b = r > 0, we will denote this space as Crn; that is,Let \u03c3 \u2208 \u211d, A \u2265 0, and y \u2208 C, then, for each t \u2208 , we define yt \u2208 Crn by yt(\u03b8) = y(t + \u03b8), \u03b8 \u2208 .If F : \u03a9 \u2192 \u211dn is a continuous quasi-bounded functional which satisfies a local Lipschitz condition with respect to the second argument and \u03a9 is a subset in \u211d \u00d7 Crn. We recall that the functional F is quasi-bounded if F is bounded on every set of the form  \u00d7 CrLn \u2282 \u03a9, where t1 < t2, CrLn : = C, and L is a closed bounded subset of \u211dn  with A > 0 if y \u2208 C,  \u2208 \u03a9 for t \u2208 \u2286\ud835\udd43 \u2192 \ud835\udd43 is a compact monotone increasing operator on a real Banach space \ud835\udd43 with normal order cone \ud835\udd42. If u0 is a subsolution of  we denote the set of all component-wise nonnegative (positive) vectors v in \u211dn; that is, v =  with vi \u2265 0 (vi > 0) for i = 1,\u2026, n. For u, v \u2208 \u211dn, we say that u \u2264 v if v \u2212 u \u2208 \u211d\u22650n, u \u226a v if v \u2212 u \u2208 \u211d>0n, u < v if u \u2264 v, and u \u2260 v.By \u211dt* \u2208 \u211d we put \u03a9 : =  \u00d7 Crn. In the following, let t0 \u2208 \u211d and t0 > t*.For a fixed k \u2208 \u211d>0n, we consider two systems of the integrofunctional inequalitiest0, \u221e), t0 \u2208 \u211dn for \u03bb1, \u03bb2 \u2208 C, \u211dn), \u03bb1(t) < \u03bb2(t), wherei = 1,\u2026, n, t \u2208  and any function \u03bb \u2208 C with ||\u03bb||C \u2264 M, where the norm is defined by (a = t0 and b = \u03d1 + r. For any (ii)k \u2208 \u211d>0n and functions \u03bbj \u2208 C, \u211dn), j = 1,2 satisfying \u03bb1(t) \u2264 \u03bb2(t) on  into \u211dn equipped with the maximum norm and the normal conet0, \u03d1] into \u211d\u22650n. By the cone \ud835\udd42\u03d1, a partial ordering \u2264 in \ud835\udd43\u03d1 is given. For \u03bb, \u03bc \u2208 \ud835\udd43\u03d1 we say that \u03bb \u2264 \u03bc if and only if \u03bc \u2212 \u03bb \u2208 \ud835\udd42\u03d1. We introduce the operator T\u03d1 : \ud835\udd43\u03d1 \u2192 \ud835\udd43\u03d1 defined byWe prove that  has a co\u03bb, \u03bc \u2208 \ud835\udd43\u03d1 with \u03bb \u2264 \u03bc. Condition (iii) implies that T\u03d1\u03bb \u2264 T\u03d1\u03bc if \u03bb \u2264 \u03bc. LetT\u03d10\u03bc\u03d1,i = \u03bc\u03d1,i and T\u03d11\u03bc\u03d1,i = T\u03bc\u03d1,i, i = 1,2) to the fixed points of operator T\u03d1, we need to prove that T\u03d1 is continuous and compact. The first property is obvious (due to continuity of F and I). Let us prove compactness. To this end, let \u2112 be a bounded subset of \ud835\udd43\u03d1. We have to show that T\u03d1\u2112 is a relatively compact subset of \ud835\udd43\u03d1. Due to the theorem of Arzel\u00e0 and Ascoli, it suffices to show that T\u03d1\u2112 is bounded and equicontinuous. That T\u03d1\u2112 is bounded follows from condition , we have\u03d1, j = 1,2. Thus the functions j = 1,2, t \u2208 [t0, \u221e), satisfyt0, \u221e) andj = 1,2, t \u2208 [t0, \u221e). Now choosing, for example, y = I (see substitution (I is strictly increasing with respect to the second argument.Now we are in a position to apply titution ), the prF  are not explicitly mentioned. Nevertheless these assumptions can be viewed as almost necessary for validity of inequalities F  \u2192 \u211d is locally Lipschitz continuous, |d(t)| < \u025b < c, and (c + \u025b) < 1/(re).Let us consider linear advanced equation :(39)y\u02d9(x(t) = exp\u2061(\u03bbt) or z(t) = exp\u2061(\u03bbt) we get corresponding transcendental equations:c + \u025b) < 1/(re) is here substantial).To apply \u03bbx1, \u03bbx2, \u03bbx1 < \u03bbx2, and (t) defined by formula  \u2192 \u211d, j = 1,2, \u03bb1*(t) \u2264 \u03bb2*(t) for t \u2208 [t0, \u221e), satisfying the inequalitiest0, \u221e). Then there exists a solution y = y(t) of  such thatLet there be continuous functions ualities5\u03bb1\u2217(t)\u2264, where Ld is a suitable constant. The boundedness of operator T  obvious:Firstly we verify conditions  and 23)23) of Thondition ) is on [\u03be lying betweenNow, condition  holds. LT .y = y(t) on [t0, \u221e) satisfying inequalities\u03bbx1, \u03bbz1 are defined in Equation  has a pot0 is large enough for the asymptotic relations and inequalities to be valid. Now we employ \u03bb1*(t), \u03bb2*(t) satisfy inequalities (y(t) inequalities . W. Wt0 isequality  is fulfiualities  hold.\u03b3 : [t0, \u221e) \u2192 \u211d+ is bounded, locally Lipschitz continuous and r is a positive constant. This equation can be viewed in a sense as a generalization of (c = 0 and d(t) = \u03b3(t)). In the following criterion of existence of positive solutions of (Let us consider linear advanced equation :(61)y\u02d9(ation of  (with c tions of  it is imtions of  and 46)(61)y\u02d9(t)y = y(t) of , the existence of a continuous function \u03bb : [t0, \u221e) \u2192 \u211d is necessary and sufficient, satisfying inequalityt0, \u221e). The mentioned positive solution y = y(t) satisfies inequalitiesFor existence of a positive solution  y(t) of  on [t0,Sufficiency. The proof uses t0, \u221e). We set \u03bb1*(t) \u2261 0 and \u03bb2*(t) \u2261 \u03bb(t). Then inequalities (oof uses \u03bb1\u2217(t)\u2264\u03b3exp+r0\u2009ds),\u03bb(t)\u2265\u03b3(t)equality . Then inequality  is a conequality . Since tequality  must be equality  is obvioNecessity. If a positive solution y = y(t) of  exists, we set y(t) of  on [t0,i = 1,2Then both inequalities  hold sim\u03bb(t) in a suitable way we can get sufficient conditions of existence of positive solutions of . The following corollary illustrates such possibility.Taking tions of  on [t0,y = y(t) of  inequalityt0, \u221e) sufficient.For existence of a positive solution  y(t) of  on [t0,\u03bb(t) = e\u03b3(t).The corollary is a consequence of  Note that criterion  of existResults obtained in the paper are sharp in a meaning. Indeed, in accordance with [equality , guarant\u03b3(t) = 1/er. It means that inequality . Nevertheless, as it was demonstrated by advanced equation (In connection with the statement of quations  and 42)\u03bbx1, \u03bbualities  and 46)\u03bbx1, \u03bbequation . This isy = y2(t) on [t0, \u221e) satisfying inequalities\u03bbx2, \u03bbz2 are defined in Equation  has a po"}
{"text": "We will establish some sufficient conditions for nonoscillatory solutions with the property limt\u2192\u221e\u2061xi(t) = 0,  i = 1, 2,\u2026, n.We will discuss nonoscillatory solutions to the  Using the idea and method of \ud835\udd4b, which is called a close interval in \ud835\udd4b. Open intervals and half-open intervals and so forth are defined accordingly.A time scale z(t) is defined for x1(t) asA function x =  is a solution to the system (t1 \u2208 \ud835\udd4b such that functions xi(t), i = 1,2,\u2026, n, are continuously differentiable and x satisfies \ud835\udd4b. Let W be the set of all solutions x =  to the system (i=1n|xi(t)| : t \u2208  > 0 on  such that [ln\u2061x(t)]\u0394 = x\u0394(t)/X(\u03bet). Because \u03b4(t) > \u03c3(t) \u2265 \u03bet \u2265 t for all t \u2208 \ud835\udd4b\u03ba, x(\u03b4(t)) \u2265 x(\u03c3(t)) \u2265 X(\u03bet) \u2265 x(t) > 0 for all t \u2208 \ud835\udd4b\u03ba. Therefore,\u03c3(t) to \u03b4(t) on \ud835\udd4b, we get, for t \u2208 \ud835\udd4b\u03ba,t1 \u2208 \ud835\udd4b and a constant c such that \u222b\u03c3(t)\u03b4(t)q(s)\u0394s \u2265 c > 1/e holds for t \u2208 \ud835\udd4b, we get thatl \u2212 1)th equations of \ud835\udd4b, thatlth,\u2026, nth equation of , t \u2208 [t1, \u221e)\ud835\udd4b, thatn \u2212 l + 1 inequalities above and \ud835\udd4b,x1(h(t)) \u2265 z(h(t)) hold. So we have that, for t \u2265 s and s, t \u2208 [t1, \u221e)\ud835\udd4b,s = h\u22121(t)\u2208[t1, \u221e)\ud835\udd4b, using the monotonicity of z(h(t)), then we have, for some sufficiently large t2 \u2208 [t1, \u221e)\ud835\udd4b, thatt \u2208 [t2, \u221e)\ud835\udd4b,N4+ \u222a N6+ \u222a \u22ef\u222aNn\u22121+ = \u2205.get thatz(t)\u2265\u222bstp)\ud835\udd4b, thatz(t)\u2265\u222bstpote that  and the tradicts  and henc(III) x \u2208 Nn+ on [t1, \u221e)\ud835\udd4b. In this case, we have that, for t \u2208 [t1, \u221e)\ud835\udd4b,s = h\u22121(t), we can obtain thatNn+ = \u2205.ion with , and so  (IV) x \u2208 N1\u2212 on [t1, \u221e)\ud835\udd4b. In this case, we have thatt\u2192\u221ex1(t) = 0. From (t\u2192\u221e | z(t)| = L1 < +\u221e. Then it follows, by t\u2192\u221exi(t) = 0, i = 2,3,\u2026, n. The proof is complete.ave thatx1(t)>0,\u2009Similarly to n is even, \u03b9 = \u22121 in the system (x \u2208 W to (t\u2192\u221exi(t) = 0, i = 1,2,\u2026, n.Assume that e system \u201380) hol holn is q = 2. Let a(t) = 8, g(t) = 16t, h(t) = 512t, p1(t) = (1/4)t, p2(t) = (7/8)t, p3(t) = (31/32)t, p4(t) = (127/128)t, p5(t) = 511/t9, f(y) = y, K = 1, \u03b1(t) = 2t, \u03b9 = \u22121, and n = 5; that is to say, the system has the following form:We consider a system on the time scale x \u2208 W to (t\u2192\u221exi(t) = 0, i = 1,2,\u2026, 5. In fact, functions x1(t) = 1/t, x2(t) = \u22121/t3, x3(t) = 1/t5, x4(t) = \u22121/t7, and x5(t) = 1/t9 are particular components of such a kind of solutions.It follows, from x \u2208 W to , lim\u2061t\u2192\u221eq = 2. Let a(t) = 2, g(t) = 4t, h(t) = 128t, p1(t) = (1/8)t, p2(t) = (1/4)t, p3(t) = (7/32)t, p4(t) = 31/t7, f(y) = 127y, K = 127, \u03b1(t) = 16t, \u03b9 = 1, and n = 4; that is to say, the system has the following form:We also consider a system on the time scale x \u2208 W to (t\u2192\u221exi(t) = 0, i = 1,2,\u2026, 4. In fact, functions x1(t) = 1/t, x2(t) = \u22122/t3, x3(t) = 7/t5, and x4(t) = \u221231/t7 are particular components of such a kind of solutions.It follows, from x \u2208 W to , lim\u2061t\u2192\u221e"}
{"text": "A, the smallest nonnegative integer k such that rank\u2009(Ak) =rank\u2009(Ak+1) is called the Drazin index of A. In this paper, we give some results on the Drazin indices of sum and product of square matrices.For a square matrix  A \u2208 \u2102n\u00d7n, the smallest nonnegative integer k such that rank\u2061(Ak) = rank\u2061(Ak+1) is called the Drazin index of A, denoted by ind\u2061(A). The Drazin inverse of A is the unique matrix AD satisfying Ak+1AD = Ak, \u2009ADAAD = AD, and AAD = ADA, where k = ind(A) (see [A) \u2a7d 1, AD is called the group inverse of A, denoted by A#. If ind\u2061(A) = 0, then AD = A# = A\u22121. The theory of generalized inverses is an active research field in computational mathematics, and the Drazin index plays an important role in the study of the Drazin (group) inverse. Some results on the Drazin indices of matrices (operators) can be found in [For rank\u2061Ak+ is calle(A) (see ). When ifound in \u20139.A \u2208 \u2102n\u00d7n, it is known [P, \u0394 and nilpotent matrix N such that A) is the smallest nonnegative integer k such that Nk = 0; that is, ind\u2061(A) = ind\u2061(N). Hence, ind\u2061(A) is the smallest nonnegative integer k such that the group inverse of Ak exists.For any is known  that theIn this paper, we give some results on the Drazin indices of sum and product of square matrices.In order to prove our main results, we give some lemmas as follows.A \u2208 \u2102n\u00d7n and nonnegative integer p, the limit\u2009\u2009lim\u2061\u03b5\u21920+\u03b5p(A + \u03b5I)\u22121 exists if and only if p\u2a7eind(A).For any A is square. Then,Let A is square. Then,Let A is square. Then, M# exists if and only if A# exists and rank\u2061(M) = rank\u2061(A).Let A is square. Then, M# exists if and only if A#, C# exist and rank\u2061(M) = rank\u2061(A) + rank\u2061(C).Let A \u2208 \u2102m\u00d7n, \u2009B \u2208 \u2102n\u00d7m, |ind(AB) \u2212 ind(BA)|\u2a7d1.For any A, let A\u03c0 = I \u2212 AAD. We first give an upper bound for the Drazin index of the sum of two square matrices.For a square matrix P, Q be square complex matrices such that PDQ = 0 and PQP\u03c0 = 0. Then,Let U, \u0394 and nilpotent matrix N such that P) = ind(N). Suppose that Q1 is a square matrix with the same order as \u0394. By PDQ = 0 we get Q1 = 0, Q2 = 0. Then, PQP\u03c0 = 0 we get NQ4 = 0.There exist nonsingular matrices P + Q) is the smallest nonnegative integer k such that the group inverse of (P + Q)k exists. Since P + Q) = ind(N + Q4). Since NQ4 = 0, we have \u03b5(N + Q4 + \u03b5I) = (N + \u03b5I)(Q4 + \u03b5I). Let m = ind(Q4) + ind(N) \u2212 1. By P + Q) = ind(N + Q4) \u2a7d m = ind(Q4) + ind(N) \u2212 1. By Q4) \u2a7d ind(Q). Since ind(P) = ind(N), we have ind(P + Q) \u2a7d ind(P) + ind(Q) \u2212 1.Note that ind(Now we give an example to show that the upper bound in P) = ind(Q) = 2 and PDQ = 0 and PQP\u03c0 = 0. The Drazin index of P + Q) = ind(P) + ind(Q) \u2212 1.Let We can obtain the following result from P, Q be square complex matrices such that PQ = 0. Then,Let CA = 0, CB = 0 or BC = 0, DC = 0 (see [It is known that = 0 (see ). A bettA is square. Then, ind(M) \u2a7d ind(A) + ind(D) + 1 if one of the following holds:BC = 0, DC = 0;BC = 0, BD = 0;CA = 0, CB = 0;AB = 0, CB = 0.Let We only prove part (1). Parts (2)\u2013(4) can be obtained in the same way.M = P + Q. By BC = 0, DC = 0, we get PQ = 0. By Let A, B \u2208 \u2102n\u00d7n, if s\u2a7emax\u2061{ind(AB), ind(BA)}, then (AB)s and (BA)s are similar.For AB)0 and (BA)0 are similar. So we only consider the case s\u2a7e1. There exist nonsingular matrices P, Q such that Ir is the identity matrix of order r = rank\u2061(A). Suppose that B1 \u2208 \u2102r\u00d7r. Then,s\u2a7emax\u2061{ind(AB), ind(BA)}, the group inverses of (AB)s and (BA)s both exist. Lemmas X, Y such that B1sX = B1s\u22121B2 and YB1s = B3B1s\u22121. Then,AB)s and (BA)s are similar.Clearly  = s, then (AB)l and (BA)l are similar for any l\u2a7es + 1.For AB)l and (BA)l are similar for any l\u2a7es + 1.By The following corollary is a special case of A, B \u2208 \u2102n\u00d7n, if (AB)# and (BA)# both exist, then AB and BA are similar.For"}
{"text": "The functions \u03a6 and w are connected through the distributional identity tn4(h\u03bc\u2032\u03a6)(t) = 1/w(t), where h\u03bc\u2032 denotes the generalized Hankel transform of order \u03bc. In this paper, we use the projection operators associated with an appropriate direct sum decomposition of the Zemanian space \u210b\u03bc in order to derive explicit representations of the derivatives S\u03bcm\u03a6 and their Hankel transforms, the former ones being valid when m \u2208 \u2124+ is restricted to a suitable interval for which S\u03bcm\u03a6 is continuous. Here, S\u03bcm denotes the mth iterate of the Bessel differential operator S\u03bc if m \u2208 \u2115, while S\u03bc0 is the identity operator. These formulas, which can be regarded as inverses of generalizations of the equation (h\u03bc\u2032\u03a6)(t) = 1/tn4w(t), will allow us to get some polynomial bounds for such derivatives. Corresponding results are obtained for the members of the interpolation space Yn.For  The d-dimensional Fourier transform of a radial function is also radial and reduces to a 1-dimensional Hankel transform of order d/2 \u2212 1 0, \u221e. Then:(i)x, t \u2208 I, one hasfor all (ii)x \u2208 I and 0 < t \u2264 a, one hasfor Suppose that the function  Qn cf.  also satx, t \u2208 I. Equation . Hencet \u2208 I, whilet \u2264 a for some a \u2208 I.Assume now 0 < n m \u2265 n,  yields \u2264 1\u2009\u2009(x \u2208 I) and supp\u03c1 \u2282 , so that Throughout this section we will assume that the function First we prove a regularity result, along with a Hankel inversion formula and a polynomial estimate, in a general setting.f \u2208 \u210b\u03bc\u2032 be such that the distribution h\u03bc\u2032f is regular on \u210b\u03bc,n. Then, for all m \u2208 \u2124+,r \u2208 \u2124+ and a function G integrable on I for whichC is independent of t, then S\u03bcmf \u2208 \ud835\udc9e, withLet m \u2208 \u2124+ and let \u03c8 \u2208 \u210b\u03bc. We havef and since Qn\u03c8 \u2208 \u210b\u03bc,n, we may writeFix  We have\u2329h\u03bc\u2032(S\u03bcmfay write\u2329(h\u03bc\u2032f)(\u03beation of .h\u03bc\u2032f)S\u03bc,xm \u2208 \ud835\udc9e as a function of x \u2208 I ensure that\ud835\udc9e as a function of x \u2208 I \u2329h\u03bcEquations , 93), a, a93), aAt this point, let us formalize the definition of a basis distribution.\u210b\u03bc\u2032 a basis distribution if tn4(h\u03bc\u2032\u03a6)(t) = 1/w(t) for some weight w  = O(t\u03b3\u2212) as t \u2192 \u221e.We call \u03a6 \u2208 ht w cf.  such tha\u03c0\u03bc,2n\u22121. Although The existence of basis distributions is guaranteed by the next F \u2208 L\u03bc,l1 and there exists \u03b3 \u2208 \u211d such that F(x) = O(x\u03b3\u2212) as x \u2192 \u221e. Let r \u2208 \u2124+. On \u210b\u03bc,r we define the linear functional Fr by(i)Fr \u2208 \u210b\u03bc,r\u2032.(ii) Any extension Fre \u2208 \u210b\u03bc\u2032 of Fr to \u210b\u03bc satisfies(iii) If Fr2e \u2208 \u210b\u03bc\u2032 is an extension of Fr2 \u2208 \u210b\u03bc,2r\u2032 to \u210b\u03bc, then(iv) If Fr1 and Fr2 are two extensions of Fr to \u210b\u03bc, then h\u03bc\u2032(Fr1) \u2212 h\u03bc\u2032(Fr2) \u2208 \u03c0\u03bc,r\u22121.Assume h\u03bc\u2032S\u03bcr(h\u03bc\u2032Fre)(t) = (\u2212t2)rFre(t) = F(t). Therefore, part (iv) is a consequence of Note that (ii) gives a, C > 0 be such that F(x) \u2264 Cx\u03b3\u2212\u2009\u2009(x > a). To prove (i), take \u03c8 \u2208 \u210b\u03bc,r and writeF \u2208 L\u03bc,l1, for the first integral we obtaink \u2208 \u2124+ is chosen so that k > \u22122r \u2212 \u03b3 + \u03bc + 3/2. A combination of .Let nd write\u2329Fr,\u03c8\u232a=\u222b0e obtain|\u222b0aF(t)(r = 0 in (i) we obtain F \u2208 \u210b\u03bc\u2032. Next, let Fre \u2208 \u210b\u03bc\u2032 be an extension of Fr to \u210b\u03bc, and let \u03c6 \u2208 \u210b\u03bc. Since (\u2212t2)r\u03c6(t) \u2208 \u210b\u03bc,r, we may write\u03c6 \u2208 \u210b\u03bc gives (ii).Now we establish (ii). First of all, we note that specializing Fr2 \u2208 \u210b\u03bc,2r\u2032 by . Define \u03bc,2r\u2032 by , and letWhen applied to a basis distribution \u03a6, w with the properties that 1/w \u2208 L\u03bc,l1 and 1/w(t) = O(t\u03b3\u2212) as t \u2192 \u221e for some \u03b3 \u2208 \u211d. Let \u03a6 \u2208 \u210b\u03bc\u2032 satisfy tn4(h\u03bc\u2032\u03a6)(t) = 1/w(t), so that h\u03bc\u2032\u03a6 \u2208 \u210b\u03bc,2n\u2032 andm \u2208 \u2124+ one hasS\u03bcm\u03a6 \u2208 \ud835\udc9e,r \u2208 \u2124+.Pick a weight function \u03c6\u2208\u210b\u03bc,2n). Then, ff.In order to derive this result from C > 0,\u2009\u2009a > 1 be such that 1/w(t) \u2264 Ct\u03b3\u2212\u2009\u2009(t > a), with \u03b3 \u2208 \u211d. Use r \u2208 \u2124+ satisfyingLet t \u2264 a, whilet > a. SetI as long as 1/w \u2208 L\u03bc,l1 and , , t \u2264 a, a, while|S\u03bc,xm[Q2w as in S\u03bcm\u03a6 \u2208 \ud835\udc9e whenever m \u2208 \u2124+ satisfies 2m < 4n + \u03b3 \u2212 \u03bc \u2212 3/2, a condition in fact weaker than  = O(t\u03b3\u2212) as t \u2192 \u221e. Let f \u2208 Yn, so that (\u2212t2)n(h\u03bc\u2032f)(t) \u2208 L\u03bc,l1. Then h\u03bc\u2032f \u2208 \u210b\u03bc,n\u2032, and for all m \u2208 \u2124+ one hasS\u03bcmf \u2208 \ud835\udc9e, withr \u2208 \u2124+.Assume 1/h\u03bc\u2032(S\u03bcnf) \u2208 L\u03bc,w2, we haveC > 0, a > 1 such that 1/w(t) \u2264 Ct\u03b3\u2212\u2009\u2009(t > a), with \u03b3 \u2208 \u211d. For any \u03c8 \u2208 \u210b\u03bc,n, the Cauchy-Schwarz inequality givesk \u2208 \u2124+ is chosen so that 2k > \u22124n \u2212 \u03b3 + \u03bc + 3/2. This proves that h\u03bc\u2032f \u2208 \u210b\u03bc,n\u2032. Now  = |F(t)|H(t)\u2009\u2009(t \u2208 I), from (t)This ends the proof.w as in f \u2208 Yn has the property that S\u03bcmf \u2208 \ud835\udc9e whenever m \u2208 \u2124+ satisfies 4m < 4n + \u03b3 \u2212 \u03bc \u2212 3/2. Although this condition is actually weaker than (S\u03bc-derivatives.In [ker than , Theorem"}
{"text": "The functions f1(x), f2(x), g(x), h1(x), h2(x), l1(x), andl2(x) satisfy some suitable conditions. We will prove that the problem has at least two nontrivial solutions by using Mountain Pass Theorem and Ekeland's variational principle.We consider the multiplicity of nontrivial solutions of the following quasilinear elliptic system \u2212div(| Problems of this type are motivated by mathematical physics, since certain stationary waves in nonlinear Klein-Gordon or Schr\u00f6dinger equations can be reduced to this form . Probleorm  = f2(x) = 0. By applying Nehari manifold method, the author proved the problem (\u03bbp/(p\u2212q) + \u03bcp/(p\u2212q) < \u03bb1 and has at least two positive solutions when 0 < \u03bbp/(p\u2212q) + \u03bcp/(p\u2212q) < \u03bb2.In , Hsu stu\u2212div\u2061(|\u2207u\u2212div\u2061(|\u2207uThis paper is motivated by the results of \u201312. ReplH1)(hi \u2208 L\u221e(\u211dN), hi(x)|x|b\u03b3 \u2208 L\u03b8(\u211dN), \u03b8 = p*/(p* \u2212 \u03b3), i = 1,2.H2)(g(x)|x|b(\u03b1+\u03b2) \u2208 L\u03b4(\u211dN)\u2229L\u221e(\u211dN), \u03b4 = p*/(p* \u2212 \u03b1 \u2212 \u03b2).H3)(l1(x)|x|b, l2(x)|x|b \u2208 L\u03c3(\u03a9), \u03c3 = p*/(p* \u2212 1), l1(x) > 0, l2(x) > 0.Throughout this paper, we make the following hypotheses:Lbp(\u211dN) the completion of the space C0\u221e(\u211dN) with respect to the norm ||u||Lbp = (\u222bN\u211d|x|bp\u2212|u|pdx)p1/, and let Wap1,(\u211dN) the completion of C0\u221e(\u211dN) with respect to the norm ||u||Wap1, = (\u222bN\u211d | x|ap\u2212 | \u2207u|pdx)p1/.In this paper, we let \u03b8 = p*/(p* \u2212 \u03b3), \u03c3 = p*/(p* \u2212 1).We defineS > 0 such that\u221e < a < (N \u2212 p)/p, a \u2264 b < a + 1, d = a + 1 \u2212 b, and p* = pN/(N \u2212 pd).The following inequality is called Caffarelli-Kohn-Nirenberg inequality . There iX = Wap1,(\u211dN) \u00d7 Wap1,(\u211dN) denote the usual Sobolev space with the normX is a reflexive and separable Banach space endowed with the norm ||||. Let u, v) \u2208 X is a weak solution of problem  \u2208 X, there holdsWe say that  and for all  \u2208 X, there holdsJ\u03bb,\u03bc correspond to the weak solutions of problem  admits a sequence of critical points in X. Our main result is the following.It is clear that problem  has a vare holds\u2329J\u03bb,\u03bc\u2032\u2013(H3) hold. There exist c0, c1 > 0 such that if \u03bb, \u03bc > 0 satisfy 0 < \u03bbh\u03b81 + \u03bch\u03b82 < c0 and 0 \u2264 l\u03c31p/(p\u22121) + l\u03c32p/(p\u22121) \u2264 c1, then the problem (Assume that H\u2013(H3) ho In this section, we first introduce a compact embedding theory which is an extension of the classical Rellich-Kondrachov compactness theorem; see Xuan .\u03a9 \u2208 \u211dN is an open bounded domain with C1 boundary and 0 \u2208 \u03a9, N \u2265 3, \u2212\u221e < a < pN/(N \u2212 p). Then the embedding Wap1,(\u03a9)\u21aaL\u03b1\u03b3(\u03a9) is continuous if 1 < \u03b3 \u2264 Np/(N \u2212 p) and 0 \u2264 \u03b1 \u2264 (1 + a)\u03b3 + N(1 \u2212 \u03b3/p) and is compact if 1 \u2264 \u03b3 < Np/(N \u2212 p) and 0 \u2264 \u03b1 < (1 + a)\u03b3 + N(1 \u2212 \u03b3/p).Assume that  The following lemma is called the Mountain Pass Theorem, see   with J = 0. Suppose that J satisfies (PS) condition andA1) \u2265 \u03b10 when ||||X = \u03c1; there are A2) \u2208 X, ||||X > \u03c1 such that J < 0. there is .DefineH1)\u2013(H3) hold. There exist c0, c1 > 0 such that if the parameters \u03bb, \u03bc satisfy 0 < \u03bbh\u03b81 + \u03bch\u03b82 < c0, and 0 \u2264 l\u03c31p/(p\u22121) + l\u03c32p/(p\u22121) \u2264 c1, then J\u03bb,\u03bc satisfies the assumptions (A1)\u2013(A2) in Assume that  \u2208 X. By (H2) and the H\u00f6lder inequality, we haveg\u03b4 = (\u222bN\u211d(g(x) | x|b(\u03b1+\u03b2))\u03b4dx)\u03b41/, \u03b4 = p*/(p* \u2212 \u03b1 \u2212 \u03b2).Let J\u03bb,h1(x) | x|b\u03b3 \u2208 L\u03b8(\u211dN)\u2229L\u221e(\u211dN), we obtain from the H\u00f6lder inequality and (16)\u222b\u211dNg in g(z1) < 1/2p for some z1 = |||| > 0.Letg(z)\u2192+\u221e where z \u2192 0+ or z \u2192 +\u221e. Then g(z) has a minimum at z1 > 0. In order to find z1, we haveg\u2032(z1) = 0 andg(z1) < 1/2p implies thatc3 = (2g\u03b4/(\u03b1 + \u03b2))S\u03b1+\u03b2c2\u03b1+\u03b2\u2212p + (1/\u03b3)S\u03b3c2\u03b3\u2212p.Note that c0, c1, \u03b10 > 0 such that J\u03bb,\u03bc \u2265 \u03b10 with 0 < \u03bbh\u03b81 + \u03bch\u03b82 < c0 and 0 \u2264 l\u03c31p/(p\u22121) + l\u03c32p/(p\u22121) \u2264 c1. Then J\u03bb,\u03bc satisfies the assumption (A1) in Thus, it follows from  and 27)27) that A2) in \u03c61, \u03c62) \u2208 C0\u221e(\u211dN)\u2009\u2009\u00d7\u2009\u2009C0\u221e(\u211dN) such thatJ\u03bb,\u03bc\u2192\u2212\u221e (t \u2192 +\u221e), since \u03b1 + \u03b2 > p > \u03b3. Therefore, there exists t large enough, such that J\u03bb,\u03bc < 0. Then we take e =  \u2208 X and J\u03bb,\u03bc(e) < 0. The condition (A2) of We now verify (H1)-(H2) hold.Let , then there exist subsequences, still denoted {un} and {vn}, and u \u2208 Wap1,(\u211dN), v \u2208 Wap1,(\u211dN) such that as n \u2192 \u221eIf {Uk = {x \u2208 \u211dN | |x | <k}, k = 1,2,\u2026.Let un} is bounded in Wap1,(\u211dN), {un} is bounded in Wap1,(Uk) for for all k \u2265 1. By un} has a subsequence {un(1)} which converges L\u03b3(U1). Likewise, the subsequence {un(1)} is bounded in Wap1,(U2). So that it has a subsequence {un(2)} which converges L\u03b3(U2). Since {un(2)} is a subsequence of {un(1)}, 1, whereumm).Since {h1(x) | x|b\u03b3 \u2208 L\u03b8(\u211dN) gives that lim\u2061k\u2192\u221e(\u222bUkc(|h1 | |x|b\u03b3)\u03b8dx)\u03b81/ = 0.The fact Then,  implies is gives .un} is bounded in Wap1,(\u211dN), we can assume (up to a subsequence) that un\u21c0u weakly in Wap1,(\u211dN) and ||un||Wap1, \u2264 C0 for some constant C0 > 0 and all n \u2265 1.In the following, we show that\u03b5 > 0, there exists k\u03b5 > 0 so large thatWap1,(Uk\u03b5)\u21aaL\u03b3(Uk\u03b5) is compact  as n \u2192 \u221e.By , we knowpact see , we haveBy Brezis-Lieb's Lemma in , we obtaSimilarly, we can proveBy Brezis-Lieb's Lemma, it follows from  that(45H1)\u2013(H3) hold. Then J\u03bb,\u03bc defined by (PS) condition on X.Assume that } be a (PS)c sequence of J\u03bb,\u03bc in X; that is, J\u03bb,\u03bc \u2192 c, J\u03bb,\u03bc\u2032 \u2192 0, in X*. We first claim that {} is bounded in X. In fact, for large n, we obtainp > \u03b3 > 1, we conclude that {} is bounded. Then, the {un} and {vn} are bounded in Wap1,(\u211dN), respectively. Thus, there exist {} \u2208 X and a subsequence of {}, still denoted by {}, such thatJ\u03bb,\u03bc\u2032 \u2192 0 in X* implies Pn \u2192 0 as n \u2192 \u221e.Let {, we see thatTn \u2192 0, as n \u2192 \u221e.Since that ows from , 49) to toun\u21c0u, we have\u222b\u211dN|h2||vN given byun \u2212 u, vn \u2212 v)|| \u2192 0, as n \u2192 \u221e.SoTn\u2212Qn=\u222b\u211dNJ\u03bb,\u03bc satisfies (PS)c condition on X.Thus, J\u03bb,\u03bc satisfies all assumptions in u1, v1) \u2208 X such that  is a solution of problem  \u2265 \u03b10 > 0.By Lemmas u2, v2) of problem  \u2208 C0\u221e(\u211dN) \u00d7 C0\u221e(\u211dN) such that \u222bN\u211dl1(x)\u03c61dx + \u222bN\u211dl2(x)\u03c62dx > 0, and thent > 0 and thus for any open ball B\u03c4 \u2282 X such that\u03c1 > 0, such that\u03b5n \u2193 0 such thatun, vn) \u2208 B\u03c1. We now consider the functional F < F, u \u2260 un, v \u2260 vn, and thus  is a strict local minimum of F. Moreover,t \u2192 0+; then,\u03c61, \u03c62) in , we getJ\u03bb,\u03bc\u2032|| \u2264 \u03b5n.We now seek a solution <c\u03c1+\u03b5n,J\u03bb,\u03bc} \u2282 B\u03c1 such that J\u03bb,\u03bc \u2192 c\u03c1 < 0, and J\u03bb,\u03bc\u2032 \u2192 0 in X* as n \u2192 \u221e. By un, vn)} has a convergent subsequence in X, still denoted by {}, such that \u2192 in X. Thus  is a solution of  < 0. Then the proof of Therefore, there is a sequence {(ution of  with J\u03bb"}
{"text": "R be a ring and U be a Lie ideal of R. Suppose that \u03c3, \u03c4 are endomorphisms of R. A family D = {dn}n \u2208 Nof additive mappings dn:R \u2192 R is said to be a - higher derivation of U into R if d0 = IR, the identity map on R and a, b \u2208 U and for each n \u2208 N. A family F = {fn}n \u2208 Nof additive mappings fn:R \u2192 R is said to be a generalized - higher derivation -higher derivation) of U into R if there exists a - higher derivation D = {dn}n \u2208 Nof U into R such that, f0 = IR and a, b \u2208 U and for each n \u2208 N. It can be easily observed that every generalized -higher derivation of U into R is a generalized Jordan -higher derivation of U into R but not conversely. In the present paper we shall obtain the conditions under which every generalized Jordan - higher derivation of U into R is a generalized -higher derivation of U into R.Let  For any x,y \u2208 R denote the commutator xy \u2212 yx by . Recall that a ring R is prime if aRb = {0} implies that a = 0 or b = 0. An additive subgroup U of R is said to be a Lie ideal of R if  \u2286 U. A Lie ideal U of R is said to be a square closed Lie ideal of R if u2\u2208U for all u\u2208U. An additive mapping d:R\u2192R is said to be a derivation (resp. Jordan derivation) of R if d(xy) = d(x)y + xd(y)(resp. d(x2) = d(x)x + xd(x)) holds for all x,y \u2208 R. Now let D = {dn}n \u2208 N be a family of additive mappings dn:R. Following Hasse and Schimdt (\u2192 RD is said to be a higher derivation (resp. Jordan higher derivation) on R if d0 = IR(the identity map on R) and a, b \u2208 R and for each n \u2208 N. Several interesting results on higher derivation can be seen in Haetinger -derivation -derivation) of R if d(xy) = \u03c3(x)d(y) + d(x)\u03c4(y) (resp. d(x2) = \u03c3(x)d(x) + d(x)\u03c4(x)) holds for all x,y \u2208 R. For a fixed a \u2208 R, the map da:R\u2192Rgiven by da(x) = a\u03c4 (x) \u2212 \u03c3(x)a for all x \u2208 R is a - derivation which is said to be a -inner derivation determined by a.Throughout - derivation the authors together with Haetinger - higher derivation as follows: A family D = {dn}n \u2208 N of additive mappings dn:R \u2192 R is said to be a - higher derivation of R if d0 = IR and a, b \u2208 R and for each n \u2208 N - higher derivation of U into R if the corresponding conditions are satisfied for all a, b \u2208 U).Inspired by the notion of  = F(x)y + xd(y) holds for all x, y \u2208 R. Motivated by the definition of generalized derivation the notion of generalized higher derivation was introduced by Cortes and Haetinger -inner derivation if F(x) = \u03c3(x)b + a\u03c4(x) holds for some fixed a, b \u2208 R and for all x \u2208 R. If F is a generalized -inner derivation, a simple computation yields that F(xy) = \u03c3(x)db\u2212(y) + F(x)\u03c4(y), where db\u2212 is a -inner derivation. With this point of view an additive mapping F:R\u2192R is said to be a generalized -derivation on R if there exists a -derivation d:R\u2192R such that F(xy) = \u03c3(x)d(y) + F(x)\u03c4(y) holds for all x,y \u2208 R. For such an example let S be any ring and F:R\u2192R such that \u03c3,\u03c4:R\u2192R such that F is a generalized -derivation on R with associated - derivation d:R\u2192R such that An additive mapping \u03c3,\u03c4)-higher derivation in a more general setting.In view of the above definition we introduce the analogue of -higher derivation -higher derivation) of R if there exists a -higher derivation D = {dn}n \u2208 N of R such that f0 = IR and a, b \u2208 R and for each n \u2208 N.Let U be a Lie ideal of R. Then F is said to be a generalized -higher derivation -higher derivation) of U into R if the above corresponding conditions are satisfied for all a, b \u2208 U.Let Example 1.1. Let R be an algebra over the field of rationals and \u03c3,\u03c4 be the endomorphisms of R. Define n \u2208 N, where f is a generalized - derivation on R with associated -derivation \u03b4such that f\u03c3 = \u03c3f and \u03b4\u03c4 = \u03c4\u03b4-derivation ensures the existence of such f). Consider the sequence {Fn}n \u2208 N, this defines a generalized - higher derivation with associated -higher derivation f to be a generalized Jordan \u2212derivation on R which is not a generalized \u2212derivation on R then one can easily construct an example of a generalized Jordan \u2212 higher derivation on R which is not a generalized \u2212 higher derivation on R.If we choose the underlying R is a Jordan derivation but the converse need not be true in general. A well-known result due to Herstein -higher derivation of R. The main objective of the present paper is to find the conditions on R under which every generalized Jordan -higher derivation of R is a generalized -higher derivation of R. In fact our results generalize, extend and compliment several results obtained earlier in this direction.It is easy to see that every derivation on Herstein  states tHerstein  generaliHerstein  further R is a prime ring of characteristic different from 2 and U a square closed Lie ideal such that U\u2ac5\u0338Z(R). Then every Jordan higher derivation of U into R is a higher derivation of U into R. The following theorem shows that the above result still holds for arbitrary square closed Lie ideal of R that is, U may be central.Recently, Haetinger  proved tTheorem 2.1. Let R be a prime ring such that char(R) \u2260 2 and U be a square closed Lie ideal of R. Suppose that \u03c3,\u03c4 are endomorphisms of R such that \u03c3\u03c4 = \u03c4\u03c3 and \u03c4 is one-one & onto. Then every generalized Jordan -higher derivation of U into R is a generalized -higher derivation of U into R.In order to develop the proof of the theorem, we begin with the following known lemma:Lemma 2.1. . Let G1,G2,\u22ef,Gnbe additive groups, S:G1\u00d7G2\u00d7\u22ef\u00d7Gn\u2192R and T:G1\u00d7G2\u00d7\u22ef\u00d7Gn\u2192 R be the mappings which are additive in each argument. If SxT = 0 for every x \u2208 U, ai\u2208Gi, i = 1,2,\u22ef,n then S = 0 for every ai\u2208Gi, i = 1,2,\u22ef,nor T = 0 for every bi\u2208Gi, i = 1,2,\u22ef,n.Lemma 2.2. Let R be a ring and \u03c3, \u03c4 be endomorphisms of R such that \u03c3\u03c4 = \u03c4\u03c3 and F = {fn}n \u2208 Nbe a generalized Jordan -higher derivation of U into R with associated -higher derivation D = {dn}n \u2208 N of U into R. Then for all u, v, w \u2208 U and each fixed n \u2208 N we have for all u, v, w \u2208 U.(i) R is a 2-torsion free ring then,If (ii) (iii) Proof. (i) For u, v \u2208 U, n \u2208 N we have, u we obtain u, v \u2208 U. Again;  for all u, v \u2208 U.Comparing the above expressions and reordering the indices we obtain the required result. for all uv + vu = (u + v)2\u2212u2\u2212v2\u2208U, using (i) and replacing v by uv + vu we find that, (ii) Since On the other hand, R is 2-torsion free we get the required result.Comparing the equations (2.1) and (2.2) and reordering the indices and using the fact that (iii) Linearizing the above result, we have Again, R is 2-torsion free we get the required result.Comparing (2.3) & (2.4) and using the fact that n \u2208 N and for each u,v\u2208U we denote by \u03a6n the element of R such that \u03a6n = 0, then F = {fn}n \u2208 N is a generalized -higher derivation of U into R. Trivially, by using Lemma 2.2(i) we get \u03a6n = \u2212\u03a6n, for all u,v\u2208U.For every fixed \u03a6nis additive in both the arguments.It can also be seen that the function Lemma 2.3. Let R be a 2-torsion free ring and \u03c3,\u03c4be endomorphisms of R such that \u03c3\u03c4 = \u03c4\u03c3. Let F = {fn}n \u2208 Nbe a generalized Jordan -higher derivation of U into R with associated -higher derivation D = {dn}n \u2208 Nof U into R. If \u03a6m = 0, for each m<n and for all u,v\u2208U, then \u03a6n\u03c4n = 0, for all u,v\u2208U(i) \u03a6n\u03c4n(w)\u03c4n = 0, for all u,v,w\u2208U.(ii) Proof.\u2003u,v\u2208U, uv + vu\u2208U and uv\u2212vu\u2208U, we find that 2uv\u2208U. Suppose \u03b2 = 4(uv(uv) + (uv)vu)\u2208U. Using Lemma 2.2(iii) we have, (i) Since for any \u03a6m = 0 for all m<n we have, Using the fact that On the other hand, fn(\u03b2)we get \u03a6n \u00d7  = 0 for all u, v \u2208 U.Comparing (2.5) with (2.6) for \u03c7 = 4(uvwvu + vuwuv) for u, v, w \u2208 U. Then by Lemma 2.2(ii) we obtain (ii) Let Again consider, iii), we have Applying Lemma 2.2(R is 2 torsion free we find that Equating (2.7) & (2.8) and using the fact that Initially calculating the first term we have \u03a6m = 0 for all m < n. Using the hypothesis that Similarly the second term reduces to \u03c3\u03c4 = \u03c4\u03c3 we obtain Now, subtracting the equation (2.11) from (2.10) and using the hypothesis that Similarly, the difference of the last two terms of equation (2.9) yields \u03a6n\u03c4n(w)\u03c4n = 0 for all u, v \u2208 U.Thus, (2.9) becomes We are now well equipped to prove our main theorem.Proof of Theorem 2.1. Suppose that U is commutative. If U is commutative then U is also central -higher derivation reduces to generalized Jordan -derivation and hence using Lemma 2.2(iii) of . We\u2019ll U is commutative, in view of Lemma 2.2(i) of  = d(vu) = \u03c3(v)d(u) + d(v)\u03c4(u), the above equation can be rewritten as Since, Comparing the equations (2.12) and (2.13) we obtain w is central and since \u03c4 is one-one and onto hence \u03c4(w) is central but the center of a prime ring is free from zero divisors, the above equation implies that \u03a61 = 0 for all u, v \u2208 U. Let \u03a6m = 0 for all u, v \u2208 U and each m < n then from Lemma 2.2(iii), we have As u, v, w \u2208 U.for all i) and commutativity of U, we get Again by using Lemma 2.2(U is commutative we find that dj(\u03c4n\u2212j(uv)) = dj(\u03c4n\u2212j(vu)) and as dj is a -higher derivation the above equation can be written as Since Comparing equations (2.15) and (2.14) we find that \u03a6m = 0 for all u, v \u2208 U, m < n, the above equation reduces to Since w is central and as \u03c4 is one-one and onto, \u03c4n(w) is also central. But the center of a prime ring is free from the zero divisors, equation (2.16) implies that \u03a6n = 0 for all u, v \u2208 U and each n \u2208 N.Since U is non-commutative hence U\u2ac5\u0338Z(R). Using Lemma 2.3(ii) we have \u03a6n\u03c4n(w)\u03c4n = 0 for all u,v,w\u2208U. Since \u03c4 is one-one and onto, the relation yields that \u03c4n\u2212)w = 0 for all u,v,w\u2208U. Hence by Lemma 2.1, \u03c4n\u2212) = 0 for all u,v\u2208U or  = 0 for all u,v\u2208U. But since U is non-commutative, we find that \u03a6n = 0, for all u,v\u2208U and each n \u2208 N.This completes the proof of our theorem.Now consider the possibility that As a consequence of the above result we find the following corollaries. Corollary 2.1 settle the conjecture given in  derivation on U then F is a generalized  derivation on U.Corollary 2.2. Let R be a prime ring such that char(R) \u2260 2 and U a square closed Lie ideal of R. Then every Jordan higher derivation of U into R is a higher derivation of U into R.In the above theorem if the underlying ring is arbitrary, then we can prove the following:Theorem 2.2. Let R be a 2- torsion free ring and U be a square closed Lie ideal of R. Suppose that \u03c3,\u03c4 are endomorphisms of R such that \u03c3\u03c4 = \u03c4\u03c3 and \u03c4 is one-one & onto. If U has a commutator which is not a right zero divisor, then every generalized Jordan -higher derivation of U into R is a generalized -higher derivation of U into R.Proof. Let x,y\u2208U be the fixed elements such that c=0\u21d2c = 0 for every c\u2208R. We\u2019ll prove the result by induction on n.n = 0, \u03a60 = 0. Hence proceeding by induction we can assume that \u03a6m = 0 for all m<n. Using Lemma 2.3(i) we have We know that for \u03c4n\u2212) = 0 for all u, v \u2208 U. Hence in particular, \u03c4n\u2212) = 0. This implies that, The above equation implies that u by u + x in (2.17) we get Replacing v by y in (2.19) and using (2.18) we get \u03a6n = 0, for every u \u2208 U,i.e., \u03c4n\u2212) = 0. This yields that Again replacing v by v + y in (2.19), to get Replace u by x in (2.21) and using (2.18) we obtain \u03a6n = 0 for all v \u2208U. This yields that Replacing \u03a6n = 0 i.e., \u03c4n\u2212) = 0. Hence, we conclude that \u03a6n = 0 for all u,v\u2208U.This completes the proof of our theorem. \u25a1Combining (2.20), (2.21) and (2.22) we find that Some special cases of the above theorem are already known and are of great interest.Corollary 2.3.  derivation of U into R then F is a generalized  derivation of U into R.Corollary 2.4. ((Cortes and Haetinger R be a 2 torsion free ring such that R has a commutator which is not a right divisor and U a square closed Lie ideal of R. Then every generalized Jordan higher derivation of U into R is a generalized higher derivation of U into R."}
{"text": "A class related to this function will be introduced and the properties will be discussed.By using a linear operator, we obtain some new results for a normalized analytic function  A simpler definition states that a meromorphic function the formf(z)=g(z)An equivalent definition of a meromorphic function is a complex analytic map to the Riemann sphere. For example, the Gamma function is meromorphic in the whole complex plane; see , 2.Latf(z) will be discussed.In the present paper, we will derive some properties of univalent functions defined by means of the Hadamard product of Hurwitz Zeta function; a class related to this function will be introduced and the properties of the liner operator f(z) normalized byU*. For 0 \u2264 \u03b2, we denote by S*(\u03b2) and k(\u03b2) the subclasses of \u03a3 consisting of all meromorphic functions which are, respectively, starlike of order \u03b2 and convex of order \u03b2 in U*.Let \u03a3 denote the class of meromorphic functions fj(z)\u2009\u2009 defined byf1(z) and f2(z) byFor functions \u03b2 \u2260 0, \u22121, \u22122,\u2026, and \u03b1 \u2208 \u2102/{0}, where (\u03bb)n = \u03bb(\u03bb + 1)n+1 is the Pochhammer symbol. We note thatLet us define the function a \u2208 \u2102/\u21240\u2212, \u21240\u2212 = {0, \u22121, \u22122,\u2026}; t \u2208 \u2102 when z \u2208 U = U* \u2282 {0}; \u211c(t) > 1 when z \u2208 \u2202U).We recall here a general Hurwitz-Lerch-Zeta function, which is defined in , 4 by thz, t, a) include, for example, the Riemann zeta function \u03b6(t) = \u03a6, the Hurwitz zeta function \u03b6 = \u03a6, the Lerch zeta function lt(\u03b6) = \u03a6,  > 1), and the polylogarithm Lti(z) = z\u03a6. Recent results on \u03a6 can be found in the expositions [Gt,a(z) and using the Hadamard product for f(z) \u2208 \u03a3, we define a new linear operator Lt,a on \u03a3 by the following series:Important special cases of the function \u03a6 = 1 + qnzn + qn+1zn+1 + \u22ef be analytic functions in U = U* \u222a {0} with q(z) \u2260 0 for z \u2208 U. IfLet a > 0, and\u20612,then\u211c{1q(z)}>We begin with the following theorem.\u03b1 + 1 > 0, Latf(z)/Latf(z) \u2260 0 for z \u2208 U* and suppose thatLet \u20612.Then\u211c{Lat(\u03b1+1q(z) byq(z) = 1 + qnzn + qn+1zn+1 + \u22ef analytic function in U* with q(z) \u2260 0 for z \u2208 U*. It follows from (a = 1/(1 + \u03b1), we get the required result.Define the function q(z) byq(z)=Lat(ows from  that(19identity  in (19),\u03b1 = \u03b2 = 1 in Letting Gt,a(z)/z)\u2032 \u2260 0 for z \u2208 U* and suppose thatLet \u20612.Then\u211c{z/z)\u2032 \u2260 0 and t = 0 for z \u2208 U* and suppose thatf(z) is starlike in U*.Let \u03b4(\u03b1 + 1) > 0, zLatf(z) \u2260 0 for z \u2208 U* and suppose thatLet \u20612.Then\u211c(zLat(\u03b1+q(z) byq(z) = 1 + qnzn + qn+1zn+1 + \u22ef analytic function in U* with q(z) \u2260 0 for z \u2208 U*. It follows from (a = 1/(1 + \u03b1), we get the required result.Define the function q(z) byq(z)=(zLaidentity  in 30),,q(z) by\u03b1 = \u03b2 = 1 in Letting \u03b4 > 0, Gt,a(z) \u2260 0 for z \u2208 U* and suppose thatLet The bound in  is the b\u03b4 = 1, M = 1 + n/8log\u20612, and t = 0 in Letting f\u2032(z) \u2260 0 for z \u2208 U* and suppose thatLet \u03be > 0, zf(z))\u2032/Latf(z) \u2260 0 for z \u2208 U* and suppose thatLet \u20612.Then\u211c byq(z) = 1 + qnzn + qn+1zn+1 + \u22ef analytic function in U* with q(z) \u2260 0 for z \u2208 U*. Also by a simple computation and by making use of the familiar identity  byq(z)=)\u2032/Gt,a(z) \u2260 0 for z \u2208 U* and suppose thatLet \u20612.Then\u211c\u03be = 1, M = 1 + n/2log\u20612, and t = 0 in Letting zf\u2032(z)/f(z) \u2260 0 for z \u2208 U* and suppose thatLet"}
{"text": "The authors prove that the exact difference \u0394f =(z+1) - f(z)has infinitely many fixed points, if a \u2208 \u2102 and \u221e are Borel exceptional values  of f. These results extend the related results obtained by Chen and Shon.Let  We use the notations \u03c3(f) to denote the order of f(z), \u03bb, and \u03bb(1/f), respectively, to denote the exponent of convergence of zeros of f(z) \u2212 a and poles of f(z). Especially, if a = 0, we denote \u03bb = \u03bb(f). A point z \u2208 \u2102 is called as a fixed point of f(z) if f(z) = z. There is a considerable number of results on the fixed points for meromorphic functions in the plane; we refer the reader to Chuang and Yang )and Yang . It folland Yang ; we use f be a transcendental meromorphic function in the complex plane \u2102. The exact differences \u0394f are defined by \u0394f = f(z + 1) \u2212 f(z).Let f, Chen and Shon have proved the following.Recently, there are a number of papers (including \u201316) focuf be a transcendental entire function of order of growth \u03c3(f) = 1 and have infinitely many zeros with the exponent of convergence of zeros \u03bb(f) < 1. Then \u0394f has infinitely many zeros and infinitely many fixed points.Let f is less than 1, Chen and Shon have proved the following.When the order of f be a transcendental meromorphic function of order of growth \u03c3(f) \u2264 1. Suppose that f satisfies \u03bb(1/f) < \u03bb(f) < 1 or has infinitely many zeros (with \u03bb(f) = 0) and finitely many poles. Then \u0394f has infinitely many fixed points and satisfies the exponent of convergence of fixed points \u03c4(\u0394f) = \u03c3(f).Let f be a transcendental meromorphic function of order of growth \u03c3(f) < 1, is there a similar result as that in Theorem B if \u03bb(1/f) \u2265 \u03bb(f) or f has infinitely many zeros (with \u03bb(f) = 0) and infinitely many poles?A natural question is, letting In this paper, we will prove the following theorem to answer the question.f be a transcendental meromorphic function of order of growth \u03c3(f) < 1 and a \u2208 \u2102. Suppose that f satisfies \u03bb(1/f) < \u03c3(f) and \u03bb < \u03c3(f). Then \u0394f has infinitely many fixed points and satisfies the exponent of convergence of fixed points \u03c4(\u0394f) = \u03c3(f).Let From f be a transcendental meromorphic function of order of growth \u03c3(f) < 1. Suppose that f satisfies \u03bb(f) \u2264 \u03bb(1/f) < \u03c3(f). Then \u0394f has infinitely many fixed points and satisfies the exponent of convergence of fixed points \u03c4(\u0394f) = \u03c3(f).Let f satisfies \u03bb(1/f) < \u03c3(f) and \u03bb < \u03c3(f). That is to say \u221e and a are Borel exceptional values of f. If we suppose that \u221e and a are Nevanlinna deficiency values of f, is there a similar result as that in Theorem B? In the following, we give In f(z) be a meromorphic function in the complex plane \u2102 and a \u2208 \u2102\u221e = \u2102 \u222a {\u221e}. Nevanlinna's deficiency of f with respect to a is defined byLet a = \u221e, then one should replace N) in the above formula by N. If \u03b4 > 0, then a is called a Nevanlinna deficiency value of f.If f be a transcendental meromorphic function of order of growth \u03c3(f) < 1 and a \u2208 \u2102. Suppose that f satisfies \u03b4 = 1 and a is a Nevanlinna deficiency value of f. Then \u0394f has infinitely many fixed points.Let f be a transcendental entire function of order of growth \u03c3(f) < 1 and a \u2208 \u2102. Suppose that \u03b4 > 0. Then \u0394f has infinitely many fixed points.Let f(z) be a meromorphic function. If the function f(z) has finite order, thenk.Let f(z) be a meromorphic function with the exponent of convergence of poles \u03bb(1/f) = \u03bb < +\u221e and let c be a nonzero complex number. Then for each \u025b > 0, we haveLet f be a transcendental meromorphic function of order of growth \u03c3(f) < 1 and let c be a nonzero complex number. ThenLet \u03c3(f)\u2254\u03c3 < 1, then \u03bb(1/f) = \u03bb \u2264 \u03c3 < 1. Therefore, for any 0 < \u025b < 1 \u2212 \u03c3, it follows from Since the order f be a function transcendental and meromorphic in the plane which satisfiesf is transcendental.Let f be a transcendental meromorphic function of order of growth \u03c3(f) = \u03c3 < 1. Then \u0394f is transcendental.Let \u03c3(f)\u2254\u03c3 < 1, then, for any positive \u025b(0 < \u025b < 1 \u2212 \u03c3), there exists R > 0 such that for any r > R we haveSince the order Therefore,f(z) be a meromorphic function of finite order, then \u03c3(\u0394f) \u2264 \u03c3(f).Let f be a transcendental meromorphic function of order of growth \u03c3(f) < 1. Then for any \u025b > 0 and any positive integer k, there exists a set E \u2282  that depends on f and has finite logarithmic measure, such that for all z satisfying |z | = r \u2209 E \u222a  we haveLet It is easy to derive the following lemma from f be a transcendental meromorphic function of order of growth \u03c3(f) < 1. Then for any positive integer k there exists a set E \u2282  that depends on f and has finite logarithmic measure, such thatLet SincethenApplying the first fundamental theorem, we getCombining -15) we  we 15) wSincethen, we can getTherefore,Thus from f \u2212 z is a transcendental meromorphic function of order of growth \u03c3(\u0394f \u2212 z) \u2264 \u03c3(f) < 1. It follows from E \u2282  that has finite logarithmic measure, such that for any r \u2209 E we haveBy Lemmas From  and 21)21)-22),,22), we g \u2261 f \u2212 a by (Denoting f \u2212 a by  we deriv\u03c4(\u0394f) \u2264 \u03c3(f). If \u03c4(\u0394f) < \u03c3(f), by \u03bb(1/f) < \u03c3(f) and \u03bb < \u03c3(f), there exists a number \u03b7 < \u03c3(f), such that for any sufficient\u2009\u2009r we haveBy \u03c4(\u0394f) = \u03c3(f).Combining  and 25)25), we c\u03b4 = 1, then N = o). By (Since  f)). By , we can \u03b4 > 0, then there is a positive number \u03b8 < 1 such thatSince f has only a finite number of fixed points, then from (If \u0394hen from  and 27)f has onlf being transcendental. Therefore, \u0394f has infinitely many fixed points.This contradicts"}
{"text": "A mathematical model for the relationship between the populations of giant pandas and two kinds of bamboo is established. We use the impulsive perturbations to take into account the effect of a sudden collapse of bamboo as a food source. We show that this system is uniformly bounded. Using the Floquet theory and comparison techniques of impulsive equations, we find conditions for the local and global stabilities of the giant panda-free periodic solution. Moreover, we obtain sufficient conditions for the system to be permanent. The results provide a theoretical basis for giant panda habitat protection. The giant panda is a highly specialized Ursid, approximately its 99% of their diet is bamboo . Many ofAiluropoda melanoleuca) and bamboo by adding a correction term which takes into account the effect of a sudden collapse of bamboo as a food source  \u00d7 \u211d+3 \u222a ((n + l \u2212 1)T, nT] \u00d7 \u211d+3 and lim\u2061t,y)\u2192 = V exists, where t0 = (n + l \u2212 1)T+ and nT+.V is locally Lipschitzian in x.Let \u211dV \u2208 V0, for \u2208((n \u2212 1)T, (n + l \u2212 1)T] \u00d7 \u211d+3, and the upper right derivative of V with respect to the impulsive differential system (Let  x)\u2208 is continuous on ((n \u2212 1)T, (n + l \u2212 1)T]\u222a((n + l \u2212 1)T, nT] and X(t0+) = lim\u2061t\u2192t0\u2061X(t) exists, where t0 = (n + l \u2212 1)T+ and nT+. The smoothness properties of f guarantee the global existence and uniqueness of solution of system (X(t) of  with a > 0, we say that a solution Y(t) of (X(t) if Y(t) is defined on  \u00d7 \u211d+ \u222a ((n + l \u2212 1)T, nT] \u00d7 \u211d+ and the limit lim\u2061t,y)\u2192 = g exists, where t0 = (n + l \u2212 1)T+ and nT+, and is finite for x \u2208 \u211d+ and n \u2208 \u2115, and \u03c81, \u03c82 : \u211d+ \u2192 \u211d+ are nondecreasing for all n \u2208 \u2115. Let r(t) be the maximal solution for the impulsive Cauchy problem:(H):\u2009\u2009\u221e). Then V \u2264 u0 implies that V) \u2264 r(t), t \u2265 0, where x(t) is any solution of T, (n + l \u2212 1)T] \u00d7 \u211d+ \u222a ((n + l \u2212 1)T, nT] \u00d7 \u211d+ etc. see Remark 2.3 and Theorem 2.3 of  \u2192 \u211d3 a saturated solution of system . By t < T0,x] represents the largest integer not exceeding x. That is, (x1(t), x2(t), x3(t)) remains strictly positive on .Let us consider (x1(t), x2(t), x3(t)) of  \u2208 \u211d+3 are bounded.All solutions (x1(t), x2(t), x3(t)) be a solution of  and lett \u2260 nT, t \u2260 (n + l \u2212 1)T, and t > 0, we obtain thatD, it follows thatLet (By t \u2192 \u221e isV(t) is bounded in its domain. Consequently, (x1(t), x2(t), x3(t)) are bounded by a constant \u201cD/a30\u201d for sufficiently lager t.which yieldsV(t)\u2264V is eradicated, it is easy to see that the equations in (When the giant panda tions in  decouple\u03b1) + a10T > 0, then the system (x1*(t) with this notation, and the following properties are satisfied:x1(t) of (x10.Suppose that ln\u2061(1 \u2212 e system  has a pe1(t) of  startingSimilarly, we have the following \u03b2) + a20T > 0, then the system (x2*(t) with this notation, and the following properties are satisfied:x2(t) of (x20.Suppose that ln\u2061(1 \u2212 e system  has a pe2(t) of  startingx1*(t), x2*(t), 0).It follows from Lemmas x1*(t), x2*(t), 0) by means of the Floquent theory. \u03b1) + a10T > 0 and ln\u2061(1 \u2212 \u03b2) + a20T > 0 andx1*(t), x2*(t), 0) is locally stable.Suppose that ln\u2061(1 \u2212 x1*(t), x2*(t), 0) may be determined by considering the behavior of small-amplitude perturbations of the solution. DefineThe local stability of the periodic solution ( Definex1(t)=u(t) satisfieswhere \u03a6 = 0t(\u2212a30 + a31x1*(s) + a32x2*(s))ds).For the upper triangular matrix, there is no need to calculate the exact forms of \u2217 and \u2217\u2217 as it is not required in the analysis that follows. And \u0394 = exp\u2061 + a10T > 0, ln\u2061(1 \u2212 \u03b2) + a20T > 0, and condition , x2*(t), 0) is stable.are\u03bb3 > 1 and (x1*(t), x2*(t), 0) is unstable.If the reverse of  is satisx1*(t), x2*(t), 0) is globally asymptotically stable.If the conditions of \u025b1 > 0, \u025b2 > 0 small enough that if condition (x1*(t) is the periodic solution of the systemu(t) \u2192 x1*(t), t \u2192 \u221e, andx2*(t) is the periodic solution of the systemv(t) \u2192 x2*(t), t \u2192 \u221e, andn + l \u2212 1)T, (n + l)T) yields\u03c1k \u2192 0 as k \u2192 \u221e. This implies that x3(t) \u2192 0 as t \u2192 \u221e.Choose ondition  holds,refore,dx3dt\u2264x3x1(t) \u2192 x1*(t) and x2(t) \u2192 x2*(t) as t \u2192 \u221e if lim\u2061t\u2192\u221e\u2061x3(t) = 0. For \u025b3 > 0 there exists a T1 > 0 such that 0 < x3(t) < \u025b3, t > T1. Without loss of generality, we may assume that 0 < x3(t) < \u025b3 for all t \u2265 0. Then we havet \u2192 \u221e, and\u025b1 > 0, we havet large enough. Let \u025b3 \u2192 0, and we get x1*(t) \u2212 \u025b1 \u2264 x1(t) \u2264 x1*(t) + \u025b1 for large t, which implies x1(t) \u2192 x1*(t) as t \u2192 \u221e.Next, we prove that x2(t) \u2192 x2*(t) as t \u2192 \u221e. This completes the proof.Similarly, we can get that T* = ((a31/a11)ln\u2061(1 \u2212 \u03b1) + (a32/a22)ln\u2061(1 \u2212 \u03b2))/(a30 \u2212 (a31/a11)a10 \u2212 (a32/a22)a20), and we find that when T < T*, the giant panda-free periodic solution is globally asymptotically stable. That is to say, in this case, the giant panda will be extinct. In biology, when the period of bamboo flowing is smaller than the threshold T*, the bamboo cannot be revived to support giant panda again, so giant panda will die by starvation.Condition  can be rWe make mention of the definition of permanence before starting the permanence of system .m and M such that every positive solution (x1(t), x2(t), x3(t)) of system  \u2264 M, m \u2264 x2(t) \u2264 M, and m \u2264 x3(t) \u2264 M for sufficiently large t.System  is said \u03b1)+(a10 \u2212 a13M)T > 0, ln\u2061(1 \u2212 \u03b2)+(a20 \u2212 a23)T > 0, andM is upper bound of the solution of system (Suppose that ln\u20611 \u2212  \u2212 \u03b1+(a1x1(t), x2(t), x3(t)) be a solution of (M > 0 such that x1(t) \u2264 M, x2(t) \u2264 M, x3(t) \u2264 M for each solution X = (x1(t), x2(t), x3(t)) of (t. The first equation of (u*(t) is the periodic solution of system:u(t) \u2192 u*(t), t \u2192 \u221e, and\u03b2) + (a20 \u2212 a23M)T > 0 holds, we can get x2(t) > v*(t) \u2212 \u025b \u2261 m2, where v*(t) is the periodic solution of system:v(t) \u2192 v*(t), t \u2192 \u221e, andLet (m3 > 0 such that x3(t) \u2265 m3 for sufficiently large t. This can be done in the following two steps.Therefore, it is necessary only to find an Step 1. Choose m0 > 0, \u025b4 > 0, \u025b5 > 0 small enough that if condition  \u2265 m0 for some t1 > 0. Assuming the contrary, x3(t) < 0 for all t, from system (x1(t) \u2265 u(t), x2(t) > v(t), and x3(t) \u2265 w(t). By n + l \u2212 1)T, (n + l)T) yieldsThis step will show that .Thusdwdt\u2265 \u2192 \u221e as t \u2192 \u221e, which contradicts the boundedness of x3(t).Since Step \u20092. If x3(t) \u2265 m0 for all t \u2265 t1, then the proof is complete. If not, let t* = inf\u2061t\u2265t1\u2061{x3(t) < m0}, and then x3(t) \u2265 m0 for t \u2208 , (n \u2208 \u2115), such that x3(t) \u2265 m0, t \u2208 . Let T\u2032 = sup\u2061|tn+1 \u2212 tn|, n \u2208 \u2115. If T\u2032 = \u221e, there must exist a subsequence tni such that tni+1 \u2212 tni \u2192 \u221e as ni \u2192 \u221e. From Step 1, this can lead to a contradiction with the boundedness of x3(t); therefore, T\u2032 < \u221e. Thenm = min\u2061{m1, m2, m3}, and thenNext, we consider the following two subsystems:Imitating the proof of Subsystem  is permaSubsystem  is permaIn this paper, our purpose is that of considering the survival of the giant panda. From Theorems T*. When T < T*, the giant panda-free periodic solution is globally asymptotically stable. That is to say, the giant panda will be extinct if the period of bamboo flowering is smaller than the threshold T*, because the bamboo cannot be revived to support giant panda again. Comparing In this paper, we consider an impulsive differential system of the population ecology on the three populations of the giant panda and two kinds of bamboo. The local and global stability of the giant panda-free periodic solution are obtained and we find the threshold value"}
{"text": "K be an arbitrary field and X a square matrix over K. Then X is sum of two square nilpotent matrices over K if and only if, for every algebraic extension L of K and arbitrary nonzero \u03b1 \u2208 L, there exist idempotent matrices P1 and P2 over L such that X = \u03b1P1 \u2212 \u03b1P2.Let  A over a field K is a sum of two nilpotent matrices over K if and only if A is similar to a particular form. In an early paper, Pazzis (see  = {z \u2208 Z+ | 1 \u2264 z \u2264 s} for some s \u2208 Z+. Mm,n(K) denotes the space consisting of all m \u00d7 n matrices over K; Mn(K) = Mn,n(K). r(A) is the rank of A \u2208 Mm,n(K). E denotes a vector space over K and dim\u2061(E) is the dimension of E. X \u2208 Mn(K) is called s2N in Mn(K) if there exist square nilpotent N1 and N2 \u2208 Mn(K) such that X = N1 + N2, while X is called an  composite in Mn(K) if there exist idempotent P1 and P2 \u2208 Mn(K) such that X = \u03b1P1 + \u03b2P2, where \u03b1, \u03b2 \u2208 K\u2216{0}  composite in Mn(L) for every algebraic extension L of K and arbitrary nonzero \u03b1 \u2208 L (when car(K) = 2, we still use \u00b1P for the meaning of  composites).If there is no statement, the meanings of notations mentioned in this paragraph hold all over the paper. X \u2208 Mn(K), on the one hand, we will prove that X is s2N in Mn(K) implies X is \u00b1P; that is, the form provided by Botha satisfies the condition as in , Ni \u2208 Mri(K), \u2211i=1sri = n, and both the characteristic polynomial and minimal polynomial of Ni are xri. Furthermore, Ni is similar to C(xri) as follows:C(xri) = E2,1 + E3,2 + \u22ef+Eri,ri\u22121 \u2208 Mri(K).For arbitrary field ri is even, C(xri) = \u2211j=1ri/2Ej,2j\u221212 + \u2211j=1ri/2\u22121Ej+1,2j2; when ri is odd, C(xri) = \u2211j=1ri\u22121)/2/2 is s2N, and Ni is s2N follows. Hence N is s2N.When s2N \u2192 \u00b1P. Suppose X \u2208 Mn(K) is s2N in Mn(K); that is, there exist square nilpotent matrices N1 and N2 \u2208 Mn(K) such that X = N1 + N2. It will take two steps to prove X is \u00b1P.X is nonsingular, then X is \u00b1P. If X = N1 + N2 with N12 = N22 = 0, inspect the eigenspaces of N1 and N2. Note that N1 and N2 are square nilpotent matrices, their ranks satisfy the following inequality matrices. r(N1) = r(N2) = n/2.Since X is nonsingular implies 0 is not its eigenvalue. Secondly, if the inequality is strict, then intersection of eigenspaces of N1 and N2 contains nonzero vectors; that is, there exists nonzero x \u2208 Mn,1(K) such that N1x = N2x = 0, which implies that 0 is one of eigenvalues of X. This is a contradiction. Hence, r(N1) + r(N2) = n; that is, n is even and N1 and N2 are similar but not equal.At first, N1 is square nilpotent with r(N1) = n/2, we can choose n/2 linear independent vectors from the set of its column vectors which can make up a base of eigenspace of N1 and denote \u03b2 by the n \u00d7 (n/2) matrix consisting of these n/2 columns. Correspondingly, we have n \u00d7 (n/2) matrix \u03b3 with all columns from the set of columns of N2. Because 0 is the only vector in the intersection of eigenspaces of N1 and N2, n \u00d7 n matrix Because N12\u03b3 = 0 implies that nonzero column vectors of N1\u03b3 are eigenvectors of N1 and r(N1\u03b3) = n/2. Hence; N1\u03b3 and \u03b2 are equal under certain column transformation; that is, there is an invertible matrix T1 such that N1\u03b3 = \u03b2T1. Correspondingly, there is an invertible matrix T2 such that N2\u03b2 = \u03b3T2.y1 and y2 are (n/2) \u00d7 n matrices. Naturally, the following equation is true:N1 and N2 as follows:N1\u03b3 = \u03b2T1 and N2\u03b2 = \u03b3T2, the above three equations imply that N1 is similar to N2 is similar to Let X is similar to L of K and arbitrary nonzero \u03b1 \u2208 L, X is also similar to the following matrix:X is \u00b1P.Hence, X is singular and similar to Y \u2295 N, where Y is nonsingular and N is nilpotent. Then X is \u00b1P. If Y is s2N. Without loss of generality, we assume X = Y \u2295 N in the following proof since s2N holds under similarity transformations.At first, we need to prove that n1 is the same for Y and the order of n4 is the same for N. Then N12 = 0 implies the following equations are true:Let X \u2212 N1)2 = N22 = 0, we get the following equations after replacing n1 with Y \u2212 n1 and n4 with N \u2212 n4 in the previous equations:Since  is \u00b1P. If X is similar to Y \u2295 N, where Y is nonsingular and N is nilpotent, then X is \u00b1P if and only if Y is \u00b1P by Corollaries X is nonsingular. Furthermore, if X is nonsingular and similar to Y1 \u2295 Y2, where all eigenvalues of Y1 are not in K and all eigenvalues of Y2 are in K. Then X is \u00b1P if and only if Y1 is \u00b1P and Y2 is \u00b1P. It will take two steps to prove X is s2N.K) \u2260 2 and all eigenvalues of X are not in K; then for arbitrary nonzero \u03b1 \u2208 K, X is an  composite; that is, there exist idempotent matrices P1 and P2 \u2208 Mn(K) such that X = \u03b1P1 \u2212 \u03b1P2. Suppose car(Q1(0) be the eigenspace of P1 with respect to 0, Q1(1) the eigenspace of P1 with respect to 1, let Q2(0) be the eigenspace of P2 with respect to 0, and Q2(1) the eigenspace of P2 with respect to 1. Both \u03b1 and \u2212\u03b1 are not eigenvalues of X implies that Q1(0)\u2229Q2(0) = Q1(1)\u2229Q2(1) = Q1(0)\u2229Q2(1) = Q1(1)\u2229Q2(0) = {0}; then dim\u2061(Q1(0)) = dim\u2061(Q1(1)) = dim\u2061(Q2(0)) = dim\u2061(Q2(1)) = n/2 ) \u2265 n/2 implies Q1(0)\u2229Q2(0) \u2260 {0} or Q1(0)\u2229Q2(1) \u2260 {0}, etc.); that is, n is even.Let S and T are n \u00d7 (n/2) matrices with r(S) = r(T) = n/2 satisfying P1S = 0 and P2T = T; then n \u00d7 n nonsingular matrix. Let P1 and P2 as follows:P1 and P2 are idempotent implies that VP1T and UP2S are idempotent and r(P1) = r(P2) = n/2 implies that VP1T = In/2 and UP2S = 0n/2. Hence, X is similar to the following matrix:X is s2N in Mn(K).Suppose K) = 2, X is  composite for arbitrary nonzero \u03b1 \u2208 K, we can similarly prove that X is s2N in Mn(K) replacing \u2212\u03b1 with \u03b1 in the previous proof.When car(K) \u2260 2 and all eigenvalues of X are in K; then by jk = jk for every k \u2208 Z+ and arbitrary nonzero \u03b1 \u2208 K. Suppose car(x + \u03b1i)]ri = (x2 \u2212 \u03b1i2)ri with 2\u2211i=1sri = n and \u03b1i \u2208 K\u2216{0} is one of eigenvalues of X for every i \u2208 [s]. Without loss of generality, we just need to prove Xi is s2N.Moreover, Xi is similar to C((x2 \u2212 \u03b1i2)ri) as follows:x2 \u2212 \u03b1i2)ri = xri2 \u2212 ari\u221222xri\u221222 \u2212 \u22ef\u2212a2x2 \u2212 a0. We have C((x2 \u2212 \u03b1i2)ri) = E2,1 + \u22ef+Eri,2ri\u221212 + a0Eri1,2 + a2Eri3,2 + \u22ef+ari\u221222Eri\u22121,2ri2 = + = N1 + N2. Obviously, both N1 and N2 are square nilpotent matrices; that is, Xi is s2N. Hence, X is s2N in Mn(K).Since K) = 2, all blocks in the Jordan reduction of X with respect to \u03b1 \u2208 K\u2216{0} have an even size by \u03b1 are (x + \u03b1)si = ((x + \u03b1)2)si/2 = (x2 + \u03b12)si/2, where si is even. Similarly, we can also prove that X is s2N in Mn(K).When car("}
{"text": "The monotonicity of the solutions of a class of nonlinear fractional differential equations is studied first, and the existing results were extended. Then we discuss monotonicity, concavity, and convexity of fractional derivative of some functions and derive corresponding criteria. Several examples are provided to illustrate the applications of our results. Fractional calculus is a generalization of the traditional integer order calculus. Recently, fractional differential equations have received increasing attention since behavior of many physical systems can be properly described as fractional differential systems. Most of the present works focused on the existence, uniqueness, and stability of solutions for fractional differential equations, controllability and observability for fractional differential systems, numerical methods for fractional dynamical systems, and so on see the monographs \u20134 and thIt is well known that the monotonicity, the concavity, and the convexity of a function play an important role in studying the sensitivity analysis for variational inequalities, variational inclusions, and complementarity. Since fractional derivative of a function is usually not an elementary function, its properties are more complicated than those of integer order derivative of the function. The focal point of this paper is to investigate the monotonicity, the concavity, and the convexity of fractional derivative of some functions.Now we recall some definitions and lemmas which will be used later. For more detail, see \u20134. a, b] of \u211d, the fractional order integral of a function f \u2208 L1 of order \u03b1 \u2208 \u211d+ is defined byGiven an interval  can be written asRiemann-Liouville's derivative of order f is defined on the interval  and fn)((t) \u2208 L1. The Caputo's fractional derivative of order \u03b1 with lower limit a for f is defined asn \u2212 1 < \u03b1 \u2264 n.Suppose that a function \u03b1 \u2264 1, it holds thatParticularly, when 0 < \u03b1. Namely,Re(\u03b1) denotes the real parts of \u03b1.There exists a link between Riemann-Liouville and Caputo's fractional derivative of order \u03b1 < 1, it holds thatParticularly, for 0 < f :  \u2192 \u211d with  \u2282 \u211d is said to be convex if whenever t1 \u2208 , t2 \u2208 , and \u03b8 \u2208 , the inequalityA function RLDt0\u03b1f(t) and CDt0\u03b1f(t). Summarizing this paper forms the content of The rest of this paper is organized as follows. \u03b1 < 1. The paper  and u(t0) \u2265 0, then the solutions u(t) of  of  are nonng) \u2265 0 and (d/dt)g) \u2265 0 on , then the solution u(t) of  of  is nondeg) \u2264 0 and (d/dt)g) \u2264 0 on , then the solution u(t) of  of  is not iAssume that 0 < tions of  exist.Ifg) \u2265 0, it holds that (1/\u0393(\u03b1))\u222bt0t(t \u2212 s)\u03b1\u22121g)ds \u2265 0. Noting u(t0) \u2265 0, we have u(t) \u2265 0.The conclusion of (1) is obvious. In fact,  is equivd/dt)g) \u2265 0 on  and g) \u2265 0, thus t0, t1]. Hence u(t) is nondecreasing on , and (2) holds.Now we prove the validity of (2) and (3). First, by the definition of the Caputo's derivative, it holds from  that(10Similar to the proof of (2), we can prove that (3) holds. This completes the proof.g = \u03bbu, then (CDt0\u03b1u(t) = \u03bbu. The solution u(t) of CDt0\u03b1u(t) = \u03bbu isu(t0) \u2265 0, thenLemma\u2009\u20092.4 in  is a par\u03bbu, then  is CDt\u03b1 < 1. Consider the fractional differential equationt > 0, we have t + sint \u2265 0 and (t + sint)\u2032 = 1 \u2212 cos\u2061t \u2265 0. By u(t) is nondecreasing in t for t \u2265 0. Assume that 0 < \u03b1 < 1. Consider the fractional differential equationg) = \u22122, then g) < 0 and t \u2265 t0. By f(t) is not increasing. In fact, by computation we get t0, \u221e), thus f(t) is decreasing. Assume that 0 < The following fractional comparison principle is an improvement of Lemma\u2009\u20096.1 in  and Theo\u03b1 < 1 and CDt0\u03b1f(t) \u2265 CDt0\u03b1g(t) on interval . Suppose further that f(t0) \u2265 g(t0), then f(t) \u2265 g(t) on .Suppose that 0 < CDt0\u03b1f(t) \u2212 CDt0\u03b1g(t) = m(t), t \u2208 . ThenIt0\u03b1 on both sides of (m(t) \u2265 0, thus It0\u03b1(m(t)) \u2265 0. Then we havef(t) \u2265 g(t) on , and the proof is completed.Set 1]. ThenCDt0\u03b1. Theorem\u2009\u20092.6 in  such that f(t0) \u2264 0, t0, t1], then RLDt0\u03b1f(t) is nondecreasing on ; f(t0) \u2265 0, t0, t1], then RLDt0\u03b1f(t) is not increasing on ; f(t0) > 0, t0, t1] and \u03b2 \u2208  such that RLDt0\u03b1f(t) is not increasing on  and is not decreasing on ; f(t0) < 0, \u03b7 \u2208  such that RLDt0\u03b1f(t) is nondecreasing on  and RLDt0\u03b1f(t) is not increasing on .Assume that 0 < d/dt)(RLDt0\u03b1f(t)) \u2265 0 on . Thus RLDt0\u03b1f(t) is nondecreasing on . By assumptions in (2), it follows that RLDt0\u03b1f(t) is not increasing in t on . Consequently, the conclusions of (1) and (2) are true.Using formula , we havef(t0) > 0 and t \u2192 t0. By the fact that \u03b41 > 0 such that (d/dt)(RLDt0\u03b1f(t)) \u2264 0 on . On the other hand, when \u03b2 \u2208  such that (d/dt)(RLDt0\u03b1f(t)) \u2264 0 on  and (d/dt)(RLDt0\u03b1f(t)) \u2265 0 on . Therefore, the conclusion of (3) is valid.Let us prove (3). Noting formula ,(24)ddtThe proof of (4) is similar to that of (3). This completes the proof. CDt0\u03b1f(t). Now we are to investigate the monotonicity of the function \u03b1 < 1. If there exists an interval  such that t0, t1] and CDt0\u03b1f(t) is nondecreasing on . If t0, t1] and CDt0\u03b1f(t) is not increasing on . Assume that 0 < t0, t1] and d/dt)(CDt0\u03b1f(t)) \u2265 0 in t on . Hence, CDt0\u03b1f(t) is nondecreasing on interval . If d/dt)(CDt0\u03b1f(t)) \u2264 0. Hence, CDt0\u03b1f(t) is not increasing on . The proof is completed. Set The following examples illustrate applications of Theorems \u03b1 < 1. Consider RLDt0\u03b1f(t), where f(t) = et, for all t0 \u2208 \u211d. Since \u03b2 > t0 such that RLDt0\u03b1(et) is decreasing on  and is increasing on . Since (sint)\u2032\u2032 \u2264 0 for t \u2208  and (sint)\u2032|t=\u03c0/2 = 0, by CD\u03c00.5\u03b1sint is decreasing on . By similar argument, CD\u03c01.5\u03b1sint is increasing on t \u2208 . Since CDt0\u03b1sint = (1/\u0393(1 \u2212 \u03b1))\u222bt0t(t \u2212 \u03c4)\u03b1\u2212cos\u2061\u03c4d, thus (1/\u0393(1\u2009\u2212\u2009\u03b1))\u222b\u03c00.5t(t\u2009\u2212\u2009\u03c4)\u03b1\u2212cos\u2061\u03c4d\u03c4 is decreasing on  and (1/\u0393(1 \u2212 \u03b1))\u222b\u03c01.5t(t \u2212 \u03c4)\u03b1\u2212cos\u2061\u03c4d\u03c4 is increasing on t \u2208 .Assume that 0 < \u03b1 < 1. Consider CDt0\u03b1f(t); here t0 = 1 and f(t) = t \u2212 t2. Obviously, t \u2208 , CDt0\u03b1f(t) is not increasing on .Assume that 0 < RLDt0\u03b1f(t) and CDt0\u03b1f(t). By formula  \u2265 0 on , f(t0) > 0, then RLDt0\u03b1f(t) is concave on . If f\u2032\u2032\u2032(t) \u2264 0 on , f(t0) \u2264 0, then RLDt0\u03b1f(t) is convex on . Assume that 0 < CDt0\u03b1f(t). The next theorem is on the convexity and the concavity of \u03b1 < 1. If there exists an interval  such that f\u2032\u2032\u2032(t) \u2265 0 on , CDt0\u03b1f(t) is concave on . If f\u2032\u2032\u2032(t) \u2264 0 on , CDt0\u03b1f(t) is convex on . Assume that 0 < f\u2032\u2032\u2032(t) \u2265 0 on , d2/dt2)(CDt0\u03b1f(t)) \u2265 0 on . Hence, CDt0\u03b1f(t) is concave in t on . If f\u2032\u2032\u2032(t) \u2264 0, on , d2/dt2)(CDt0\u03b1f(t)) \u2264 0 on . Therefore CDt0\u03b1f(t) is convex in t on . By formula , we have\u03b1 < 1. Consider the fractional differential equation CDt0\u03b1f(t); here t0 < 0 and f(t) = t \u2212 t2. Obviously, f\u2032\u2032\u2032(t) = 0. For all t0 < 0, it holds that f\u2032\u2032\u2032(t) = 0 on . Then by CDt0\u03b1(t \u2212 t2) is convex on . Assume that 0 < RLDt0\u03b1f(t), where f(t) = et, t0 \u2208 \u211d. Obviously, Theorems RLDt0\u03b1et. Now we employ the method which is used in the proof of t \u2208 . Thus (d2/dt2)(RLDt0\u03b1et) > 0 on  and (d2/dt2)(RLDt0\u03b1et) \u2265 0 on  and is concave on [\u03b2, +\u221e). Consider the concavity and convexity of the function proof of d2dt2 and CDt0\u03b1f(t). Based on the relation between the Riemann-Liouville fractional derivative and the Caputo's derivative, we obtain the criteria on the monotonicity, the concavity, and the convexity of the functions RLDt0\u03b1f(t) and CDt0\u03b1f(t). In the meantime, five examples are given to illustrate the applications of our criteria.The main part of this paper is to study the monotonicity, the concavity, and the convexity of the functions"}
{"text": "In this paper, we use the properties of error estimation and the analytic method to study the reciprocal sums of higher power of higher-order sequences. Then we establish several new and interesting identities relating to the infinite and finite sums.Let { Fn and Ln denote the Fibonacci numbers and Lucas numbers, have been considered in several different ways; see [The so-called Fibonacci zeta function and Lucas zeta function, defined by\u03b6F(s)=\u2211n=ays; see . Ohtsukaays; see  studied Further, Wu and Zhang , 5 generk=n\u221e(1/vk))\u22121. Specifically, suppose that ||x|| = \u230ax + (1/2)\u230b (the nearest integer function) and {vn}n\u22650 is an integer sequence satisfying the recurrence formulap \u2265 q and n > k. Then we can conclude that there exists a positive integer n0 such thatn > n0.Recently, some authors considered the nearest integer of the sums of reciprocal Fibonacci numbers and other well-known sequences and obtained several meaningful results; see \u201316. In pIn , Wu and n > m, the mth-order linear recursive sequence {un} is defined as follows:ui \u2208 \u2115 for 0 \u2264 i < m and at least one of them not being zero. For any positive real number \u03b2 > 2 and any positive integer a1 \u2265 a2 \u2265 \u22ef\u2265am \u2265 1, there exists a positive integer n1 such that\u03b2 \u2192 +\u221e, there exists a positive integer n2 such thatFor any positive integer k=n\u221e(1/uks))\u22121 for all integers s \u2265 2, because it is quite unclear a priori what the shape of the result might be. In [It seems difficult to deal with  of reciprocal sums of some recursive sequences. The main purpose of this paper is using the properties of error estimation and the analytic method to study the higher power of the reciprocal sums of {un} and obtain several new and interesting identities. The results are as follows.To find and prove this result is a substantial achievement since such a complex formula would not be clear beforehand that a result would even be possible. However, there is no research considering the higher power f(x) has exactly one positive real zero \u03b1 with a1 < \u03b1 < a1 + 1;polynomial (II)m \u2212 1 zeros of f(x) lie within the unit circle in the complex plane.other Let See Lemma 1 of .m \u2265 2 and {un}n\u22650 be an integer sequence satisfying the recurrence formula .Let  formula . Then foun is given byc > 0, d > 1, a1 < \u03b1 < a1 + 1, and \u03b1 is the positive real zero of f(x). Now we prove s = 2. Suppose that for all integers 2 \u2264 s \u2264 k we haves = k + 1 we haveFrom Lemma 2 of , the clogiven byun=c\u03b1n+O, (1/\u03b1ndn)}.In this section, we shall complete the proof of Taking the reciprocal of this expression yieldsh = (a1s\u230a\u03b2n\u230b\u2212sn/\u03b1s\u230a\u03b2n\u230b\u2212sn), then for any real number \u03b2 > 2 and positive integer s we haveIf h = (1/\u03b1ndn), for any positive integer a1 \u2265 2, 1 < (\u03b1/a1) < \u03b1d holds. Then for any positive integer s withIf \u03b2 > 2 and positive integer 1 \u2264 s < \u230alog\u2061\u03b1/a1)k=n\u03b2n\u230b\u230a((\u2212a1)sk/uks))\u22121 \u2212 (\u22121)sn(uns/a1sn + un\u22121s/a1sn\u2212s)|| = 0, (n \u2265 n8). ||(\u2211(b)k=n\u03b2n\u230b\u230a(a1spk+sq/upk+qs))\u22121 \u2212 (upn+qs/a1spn+sq \u2212 upn\u2212p+qs/a1spn+sq\u2212sp)|| = 0, (n \u2265 n9). ||(\u2211(c)k=n\u03b2n\u230b\u230a((\u2212a1)spk+sq/upk+qs))\u22121 \u2212 (\u22121)spn+sq(upn+qs/a1spn+sq + upn\u2212p+qs/a1spn+sq\u2212sp)|| = 0, (n \u2265 n10). ||k=n\u221e((\u2212a1)sk/uks))\u22121 \u2212 (\u22121)sn(uns/a1sn + un\u22121s/a1sn\u2212s)|| = 0, (n \u2265 n11). ||(\u2211(e)k=n\u221e(a1spk+sq/upk+qs))\u22121 \u2212 (upn+qs/a1spn+sq \u2212 upn\u2212p+qs/a1spn+sq\u2212sp)|| = 0, (n \u2265 n12). ||(\u2211(f)k=n\u221e((\u2212a1)spk+sq/upk+qs))\u22121 \u2212 (\u22121)spn+sq(upn+qs/a1spn+sq + upn\u2212p+qs/a1spn+sq\u2212sp)|| = 0, (n \u2265 n13). ||(\u2211Let {fined by  with theh = max\u2061\u2009{(a1s\u230a\u03b2n\u230b\u2212sn/\u03b1s\u230a\u03b2n\u230b\u2212sn), (1/\u03b1ndn)}.We shall prove only (c) in Taking the reciprocal of this expression yieldsCase\u2009\u20091. If h = (a1s\u230a\u03b2n\u230b\u2212sn)/(\u03b1s\u230a\u03b2n\u230b\u2212sn), then for any real number \u03b2 > 2 and positive integer s we haveCase\u2009\u20092. If h = (1/\u03b1ndn), for any positive integer a1 \u2265 2, 1 < (\u03b1/a1) < \u03b1d holds. Then for any positive integer s with\u03b2 > 2 and positive integer 1 \u2264 s < \u230alog\u2061\u03b1/a1)(\u03b1pdp\u230b, there exists n \u2265 n10 sufficiently large so that the modulus of the last error term of identity (In both cases, it follows that for any real number identity  becomes"}
{"text": "Triticum aestivum L.) during domestication, and this enables them to grow productively in a wide range of environments. Several major genes controlling flowering time have been identified in wheat with mutant alleles having sequence changes such as insertions, deletions or point mutations. We investigated genetic variants in commercial varieties of wheat that regulate flowering by altering photoperiod response  or vernalization requirement  and for which no candidate mutation was found within the gene sequence. Genetic and genomic approaches showed that in both cases alleles conferring altered flowering time had an increased copy number of the gene and altered gene expression. Alleles with an increased copy number of Ppd-B1 confer an early flowering day neutral phenotype and have arisen independently at least twice. Plants with an increased copy number of Vrn-A1 have an increased requirement for vernalization so that longer periods of cold are required to potentiate flowering. The results suggest that copy number variation (CNV) plays a significant role in wheat adaptation.The timing of flowering during the year is an important adaptive character affecting reproductive success in plants and is critical to crop yield. Flowering time has been extensively manipulated in crops such as wheat ( Triticum aestivum) is an allohexaploid combining A, B and D genomes from three diploid ancestors Cultivated bread wheat (Photoperiod-1 (Ppd-1) loci on chromosomes 2A, 2B and 2D. The Ppd-1 genes are members of the pseudo-response regulator (PRR) family orthologous to the Ppd-H1 gene of barley Wheat's ancestors and many modern varieties are described as photoperiod sensitive. That is, they flower rapidly in long days but are late flowering in short days. Day neutral (photoperiod insensitive) varieties flower rapidly in short or long days. This allows production in environments where appropriate temperature and rainfall coincide with short day conditions or where early flowering avoids high summer temperatures, as in Southern Europe a suffix  while wild type alleles have a b suffix Ppd-A1a alleles and the single Ppd-D1a allele are deletions that remove a shared region of the promoter likely to be important for regulation Four day neutral alleles have been characterized in wheat previously. They are given an Ppd-1b (wild type) alleles have a marked diurnal fluctuation in expression, with very low transcript levels at dawn, a peak 3\u20136 h after dawn and a subsequent drop to very low levels in the dark. Ppd-1a alleles that have been studied show elevated expression throughout the 24 h period but particularly at dawn and during the dark period. This is associated with expression of TaFT1 (the wheat orthologue of the flowering inducer FLOWERING LOCUS T) in short days Ppd-1 and the induction of TaFT1 without the normal long day cue.Ppd-B1a alleles identified by genetic mapping in five diverse sources: \u2018Chinese Spring\u2019 , Vrn-2 or TaFT1  genes Vrn-1 encodes a MADS-box transcription factor whose expression increases quantitatively during vernalization. This potentiates flowering by inhibiting expression of the flowering repressor Vrn-2Vrn-A1, Vrn-B1 or Vrn-D1 genes  are the predominant cause of spring types in wheat Vernalization is another important flowering time control. The ancestors of wheat and many modern varieties are winter types that require vernalization, but during domestication there has been selection for spring types that do not. Spring types result from mutation of the Vrn-A1 was the cause; specifically a single base difference in exon 4 Vrn-A1 as the cause but in contrast to In addition to the winter/spring difference, winter types vary in the duration of cold necessary to complete vernalization. We refer to this as variation in vernalization requirement. This aspect of vernalization is much less studied, but recently it was proposed that Ppd-B1a and Vrn-A1 alleles we investigated, the changes in flowering time were associated with increased gene copy number and not with specific sequence polymorphisms. Copy number variation (CNV) is a type of mutation that has been widely studied in human genetics Arabidopsis thaliana accessions For the Ppd-B1. The photoperiod sensitive hexaploid spring wheat variety \u2018Paragon\u2019 was the recurrent parent for introgression lines containing Ppd-1a alleles from \u2018Chinese Spring\u2019, \u2018Sonora64\u2019 or \u2018R\u00e9cital\u2019. Two independent lines of each allele were developed and from each of these ten plants at BC3F4  or BC4F4  were grown in short days  in a photoperiod glasshouse. Days to ear emergence on the main stem were recorded for each plant , while \u2018Chinese Spring\u2019 had four copies. Furthermore, the other genotypes with a day neutral Ppd-B1a allele also had elevated copy numbers; two in \u2018R\u00e9cital\u2019 and three in \u2018Sonora64\u2019, \u2018Timstein\u2019 and \u2018C591\u2019 (Ppd-B1a) replaced the \u2018Mercia\u2019 2B had the same increased copy number as \u2018Chinese Spring\u2019 itself. Conversely, a line in which the 2B chromosome from the photoperiod sensitive variety \u2018Marquis\u2019 (Ppd-1Bb) replaced the \u2018Chinese Spring\u2019 chromosome had a low copy number, showing that all the additional gene copies were on chromosome 2B. These data suggested that increased copy number could be the basis of the Ppd-B1a alleles.d \u2018C591\u2019 . A \u2018MercA BAC library Ppd-B1 gene flanked by retrotransposons. 97J10 and 170M01 did not assemble readily because of sequence differences at the ends of approximately 35 kb regions containing either an intact or truncated gene. Digestion with NotI allowed BAC clone inserts to be sized. 170M01, which did not contain the truncated copy, would contain three repeat units of approximately 35 kb. PFGE of 97J10 using AarI, which does not cut within the truncated copy, placed the truncated copy at the left end of the repeating structure (copy_1), showing that the Ppd-B1a locus of \u2018Chinese Spring\u2019 comprised one truncated copy and three intact copies of the Ppd-B1 gene, all in the same orientation, within a 185 kb region  from the other varieties in Tsp509I CAPS assay of amplicons from \u2018Chinese Spring\u2019 and individual BAC clones showed that all copies had the A form. Although exon and intron sequences were identical apart from the truncation in copy_1, different haplotypes were found upstream as shown by six nucleotides below each gene copy  which was correlated with copy number, showing the copies to be genetically linked. This suggests that the \u2018R\u00e9cital\u2019 allele is derived from a third independent amplification event or from the \u2018Sonora64\u2019/\u2018Timstein\u2019 type by further mutation.\u2018R\u00e9cital\u2019 had a haploid copy number of two  and the Ppd-B1b alleles , \u2018Cheyenne\u2019 and \u2018Renan\u2019 which have wild type  alleles , S2. Thi alleles . This suPpd-B1 specific quantitative RT-PCR assay. First, \u2018Mercia\u2019 (Ppd-B1b) was compared with a \u2018Mercia(Chinese Spring 2B)\u2019 (Ppd-B1a) single chromosome substitution line. Plants were grown in short days  for 21 days and RNA samples were taken from three biological replicates, each of four plants, at 3 h intervals over a 24 h period. Expression of the \u2018Mercia\u2019 allele was similar to that of Ppd-1b alleles in previous studies Ppd-B1a allele had significantly higher levels at dawn, a higher expression peak, and elevated expression throughout the dark period \u2019, \u2018Paragon(Sonora64 Ppd-B1a)\u2019 and \u2018Paragon\u2019 introgression lines were grown in short days and sampled at the end of the dark period and at three time points in the lit period as described above  or 0 to 6 weeks  after which they were transferred to a heated lit glasshouse providing long days (18 h light). Day length was sufficient to satisfy photoperiod response, and genotyping showed that no Days to ear emergence on the main stem were recorded for five plants of each genotype per time point. Unvernalized \u2018Claire\u2019 plants flowered about 40 days before \u2018Malacca\u2019 but the acceleration of flowering per week of vernalization was greater in \u2018Malacca\u2019 so that flowering times were similar after 4 or more weeks of cold. Unvernalized plants of \u2018Hereward\u2019 and \u2018Malacca\u2019 flowered at a similar time but the 1 to 4 week treatments in \u2018Hereward\u2019 had less effect on flowering, after which the response was similar to \u2018Malacca\u2019 .Vrn-A1 on chromosome 5A To investigate the genetic basis of long vernalization requirement in \u2018Hereward\u2019 we selected a 4 week vernalization treatment  to study a population of 77 doubled haploid (DH) lines from a \u2018Malacca\u2019\u00d7\u2018Hereward\u2019 cross. Three plants of each DH line and four plants of each parent were vernalized and then grown in a heated glasshouse with supplementary lighting providing long days (18 h light). Days to ear emergence on the main stem were recorded and averaged for each genotype. QTL mapping showed one significant peak  and \u2018Hereward\u2019 (JF965397) by direct sequencing of overlapping PCR amplicons generated from gene specific primers based on the \u2018Triple Dirk C\u2019 winter allele  only fitted a 1\u22361 ratio while for \u2018Hereward\u2019 (51\u223687) they only fitted a 1\u22362 ratio, suggesting two and three copies of Vrn-A1, respectively. Full length cDNA clones from RNA of vernalized \u2018Malacca\u2019 and \u2018Hereward\u2019 plants also had the C and T forms showing that two intact variants of the gene were present and expressed. The C/T difference in exon 4 is likely to be that previously reported as distinguishing \u2018Jagger\u2019 from \u20182174\u2019 Approximately 12 kb of AY747601 . No diffAY747601 . CloningVrn-A1  confirmed a haploid copy number of two for \u2018Malacca\u2019 and three for \u2018Hereward\u2019 . The latter has a spring allele of 2 population to be used for an additional test of the genetic behaviour of the \u2018Hereward\u2019 Vrn-A1 copies  increased most rapidly in \u2018Claire\u2019 and was slowest in \u2018Hereward\u2019, while \u2018Malacca\u2019 was intermediate \u2019 lines were developed as follows. \u2018R\u00e9cital\u2019 carries Ppd-B1a and Ppd-D1a alleles 1F2 plants were assayed for Ppd-D1a as described in Ppd-D1a carriers discarded. The remainder were grown in short days . Early flowering plants would have Ppd-B1a and these were backcrossed to \u2018Paragon\u2019. BC3F2 families in short days showed 3\u22361 segregation for flowering time and early flowering plants had higher Ppd-B1 copy number, showing that Ppd-B1a was the major determinant of flowering time. BC3F3 families that were all early flowering in short days  were selected. BC3F4 plants were compared to \u2018Paragon\u2019 and previously described \u2018Paragon\u2019 BC4F4 introgression lines homozygous for the \u2018GS-100\u2019 Ppd-A1a, \u2018Chinese Spring\u2019 Ppd-B1a, \u2018Sonora64\u2019 Ppd-B1a or \u2018Sonora64\u2019 Ppd-D1a alleles \u2018Mercia\u2019 and \u2018Mercia(Chinese Spring 2B)\u2019 lines were from the John Innes Centre Germplasm Resources Unit. \u2018Paragon, annealing (55\u00b0C for 20 sec) and polymerisation (72\u00b0C for 1 min per kb of amplicon).50 ng genomic DNA in 20 \u00b5l reactions comprising 1\u00d7 PCR Buffer and 0.4 units Ppd-1 genes from a \u2018Chinese Spring\u2019 BAC library Bsu36I or BsiW1 followed by re-ligation and selection with chloramphenicol allowed recovery of the rightmost and leftmost Ppd-B1 copies, respectively, in resulting subclones. The haplotypes of individual copies was determined by standard PCR (above) and Sanger sequencing (see sequencing of Vrn-A1).Filter hybridization was used to select clones containing catttctcccgcttgtaagg and caacgttacagcttcggtca, respectively) was conducted using the Roche Expand Long template kit . This produced an approximately 9 kb fragment in \u2018Sonora64\u2019 and \u2018Timstein\u2019, which was analyzed by Sanger sequencing (below) using internal primers in a stepwise manner.Long range PCR using primers facing outwards from the ends of the gene sequence  and 97J10R2 (ccggaacctgaggatcatc) gave a 425 bp product; (b) the junction between intact Ppd-B1 copies in the \u2018Chinese Spring\u2019 allele. Primers PpdB1_F25 (aaaacattatgcatatagcttgtgtc) and PpdB1_R70 (cagacatggactcggaacac) gave a 994 bp product; (c) the junction between intact Ppd-B1 copies in the \u2018Sonora64\u2019/\u2018Timstein\u2019 allele. Primers PpdB1_F31 (ccaggcgagtgatttacaca) and PpdB1_R36 (gggcacgttaacacaccttt) gave a 223 bp product. Junction sequences and primer positions for assays are given in PCR with \u201cHot Start\u201d polymerase and buffer (Qiagen) and an initial denaturation step of 95\u00b0C for 5 minutes was used to detect; (a) the truncated vrn-A1 sequence (AY747601) was used to design a series overlapping PCR amplicons spanning the gene. Amplicons were obtained from genomic DNA using the standard PCR protocol and were directly sequenced using ABI Big Dye Mix v3.1 (Applied Biosystems Inc) under the manufacturer's conditions, with products resolved on an ABI 3730 capillary electrophoresis instrument. Segments of Vrn-A1 from \u2018IL369\u2019 were amplified using primers IL369_F1 (tgaattcatgatcggagcag) and IL369_R1 (tgctgaacttctctgcaagtg) or IL396_F2 (tgctgaacttctctgcaagtg) and IL369_R1 (\u2018Triple Dirk D\u2019 IL369_R1  and direPpd-B1 were gcgtaagttactatctctcatggtgtatc and tttgttttagtacccagtaccataccag (0.2 \u00b5M) and the probe sequence was FAM-ctgctgcttcagttcctagtttcacttgtgtcc-TAMRA (0.1 \u00b5M). Forward and reverse primers for Vrn-A1 were gcagcccacttttggtctcta and tctgccctctcgcctgtt (0.2 \u00b5M) and the probe sequence was FAM-tgtgttcgctttggttgtgcagca-TAMRA (0.1 \u00b5M). Forward and reverse primers for the TaCO2 control were tgctaaccgtgtggcatcac and ggtacatagtgctgctgcatctg (0.1 \u00b5M) and the probe sequence was VIC-catgagcgtgtgcgtgtctgcg-TAMRA (0.1 \u00b5M). Ppd-B1 plus control or Vrn-A1 plus control were analyzed together in multiplexed reactions.20 \u00b5l reactions comprised water (3 \u00b5l), AbGeneQPCR mix (10 \u00b5l), Probe plus primers (2 \u00b5l) and DNA (5 \u00b5l). Samples were denatured at 95\u00b0C for 15 min followed by 40 cycles of . Forward and reverse primers for www.kbioscience.co.uk) in reactions containing water (2 \u00b5l), KASPar mix (4 \u00b5l), primers (0.1 \u00b5l), 50 mM MgCl2 (0.064 \u00b5l) and DNA (2 \u00b5l). An activation time  was followed by 20 cycles of  followed by 24 cycles of . Fluorescence was read as an end point reading at 25\u00b0C. Primer combinations were;Polymorphisms were scored using KBioscience KASP reagents . Generic primer ccagttgctgcaactccttgagatt (0.4 \u00b5M).Exon4_C/T SNP specific primers: gaaggtgaccaagttcatgctgagtttgatcttgctgcgccG, gaaggtcggagtcaacggattctgagtttgatcttgctgcgccA (0.16 \u00b5M). Generic primer cttccccacagctcgtggagaa (0.4 \u00b5M).Exon7_C/T SNP specific primers: Ppd-B1a)\u2019, \u2018Paragon(Sonora64 Ppd-B1a)\u2019 and \u2018Paragon\u2019 introgression lines in the same way.RNA of \u2018Mercia\u2019 and \u2018Mercia (Chinese Spring 2B)\u2019 was extracted from seedlings grown for 20 days after germination in a controlled environment room with short days . Three replicate samples from each genotype were harvested into liquid nitrogen at each three-hourly time point over 24 h. RNA was sampled from 20 day old seedlings of \u2018Paragon\u2019 and \u2018Paragon(Chinese Spring http://www.invitrogen.com). DNA was removed by DNAseI digestion. cDNA was synthesized with Superscript II (Invitrogen) using the manufacturer's protocols with 5 \u00b5g of total RNA as template and a mixture of OligodT (12\u201318) (250 ng) and random hexamers (150 ng) as primers. 1/40 by volume of the final cDNA aliquot was used for real-time PCR as described below. In the case of 18S rRNA analysis, cDNA samples were diluted 1\u2236100 and 1/40 of this dilution was used as template.RNA was extracted using Trizol reagent . The forward primer was specific for Ppd-B1, and genome-specificity was confirmed using \u2018Chinese Spring\u2019 nullisomic-tetrasomic lines. The primers generate a 549 bp amplicon from cDNA. Similarly, primers taactgctcctcacaagtgc and ccggaacctgaggatcatc were used to detect expression of the truncated copy in \u2018Chinese Spring\u2019 . Primer pairs were:Samples of leaf tissue from plants of each genotype/vernalisation time combination  were harvested in the vernalisation chamber at the end of each treatment 3\u20134 h after dawn and frozen in liquid nitrogen. RNA and cDNA were prepared as above, and VrnA1-all: cttgaacggtatgagcgctat and gcatgaaggaagaagatgaag, product size 396 bp.VrnA1-exon 4 C form: gcatctcatgggagaggatC and gaatagtacgcctgtatgggctggat, product size 436 bp.VrnA1-exon 4 T form: gcatctcatgggagaggatT and gaatagtacgcctgtatgggctggat, product size 436 bp. This assay produced a low level signal in \u2018Claire\u2019, which does not contain the T form  The \u2018Chinese Spring\u2019 Ppd-B1a allele. (B) Hypothetical structure of the \u2018Sonora64\u2019/\u2018Timstein\u2019 Ppd-B1a allele. (C) Hypothetical structure of the \u2018R\u00e9cital\u2019 Ppd-B1a allele. Ppd-B1 copies are shown as large dark green rectangles (exons plus introns), the solid black line shows upstream and downstream regions and small coloured rectangles show transposable elements as in (TIF)Click here for additional data file.Figure S2PCR assays detecting junction sequences in Ppd-B1a  alleles and sequences of the junction regions. (A) to (C) Reverse colour images of PCR products from genotypes with known Ppd-B1 alleles. Genotypes with a known Ppd-B1a allele are boxed. \u2018Chinese Spring(N2BT2D)\u2019 is nullisomic for chromosome 2B (no Ppd-B1 gene is present) and tetrasomic for chromosome 2D. \u2018Chinese Spring(N2AT2B)\u2019 is nullisomic for chromosome 2A and tetrasomic for chromosome 2B. (A) Assay for the gene/transposon junction  in the truncated Ppd-B1 copy of \u2018Chinese Spring\u2019. Primer positions for this assay are underlined in the sequence below. Ppd-B1 gene sequence is to the left of the arrow  and the TREP 3161 WIS element is to the right of the arrow. A 425 bp band is produced when the junction is present. \u2026TAACTGCTCCTCACAAGTGCCGGAAGGGAAAGACGCCGACCGTGAGAACGCCATGCCATATCTTGAGCTGAGCCTAAAGAGGTCGAGATCGACCACGGAGGGTGCGGATGCGATCCAGGAGGAACAGAGGAACGTCGTGAGACGATCAGACCTCTCGGCATTCACGAGgtgcaaagcataatatcagtgtcctttgtgaatccttaaatcatccatatgttgcatactaaccgttttcattctttgcaagGTACAATACGTGCTCGTTCTCCAATCAAGGCGGGGCAGGGTTCGTCGGGAGCTGTTCGCCCA\u2191cgtgactgccaagcgttcataacgtcttggttctatgggatgggtgcttcacctagcggtccttctaggacatatgctttcttggcagctatgaggatgatcctcaggttccgg. The same primers were used for RT-PCR where a 343 bp product was produced from cDNA. The product was cloned and sequenced, confirming that intron 6 was correctly spliced. The transcript has a stop codon (double underline) close to the break point which gives a predicted protein lacking a CCT domain. (B) Assay for the junction between intact Ppd-B1 gene copies in the \u2018Chinese Spring\u2019 allele. Primer positions for this assay are underlined in the sequence below. The left primer spans the junction between Ppd-B1 gene sequence and the TREP 3457 Danae element. The junction  marks the start of the TREP 3161 WIS element sequence which contains the right primer. A 994 bp band is generated when the junction is present. \u2026aaacattatgcatatagcttgtgtcggtgtacaaaagtaggagctctgcttttgacccctttacttgtgcacgagcagtcagagccacccgccacggccacgcataacagggcagggaagggaagccggagagaagccgaaggcacaagacaaacaaagcaacgacaaggaccaaggctacaaagtgcagatgggcgaagcaggtttcccctgcaagacccttgccgggggcagcctccgcagccccggcaagcccctttgccggggcaactcgcccacaccaatggagagagccacccttgaacccacggcctccaatgtcaaccaccacgttggaccagggctcgagaggcacctccatggtggcatgcagatctttgtgaagacatagaatgctcaagaacatgtgatgattggagggtaacaatcctcaccaagatcctcaccgaataaacccacaagacccccccggcaagatccttgccgaggacggcaagcgccacggcaagacccttgccgggccacccaacgagaccctcgccagaggcaaccacgagcccactgccaggcccgcaccaaccaactccccgccgccgttcgcatgcagctgccaacccaaccagctgagcaggcacctgcgtggcaacatgcagcttctaggcccactcagcacacacctgtgtggcggcatgcagatcttcgtggaggctccaccaccgcaccacctcagctgcttgcctgcctacatggcaccgcatgcatcgctggccagggcgcgtgtcgaagcaaggaggagcggcgacggatgggacgggcgtcgctcccgtccccgataaagtgagggacacctaagccatgcat\u2191gttcatcatgtgagacggactagtcatcatcggtgaacatctccatgttgatcgtatcttccatacgactcatgttcgaactttcggtcccttgtgttccgagtccatgtctgtacatgctaggctcgtcaagttaacataagtgttttgcatgtgtaaatctgtcttacacccattgtatgtggacgttggaatctatcacacccgatcatcacgtggtgcttcgaaacaatgaacttcacacaagcattatacctggaaccgtagcataattttcctcaacattttgagctacgtgtatggagtttgcattgttatcttcaacatttctcccgcttgtaaggcctgca\u2026 (C) Assay for the junction between intact Ppd-B1 gene copies in the \u2018Sonora64\u2019/\u2018Timstein\u2019/\u2018C591\u2019 allele. Primer positions for this assay are underlined in the sequence below. The junction is shown by a vertical arrow. The TREP 3457 Danae element is to the left of the arrow, the TREP 3173 Derami element is to the right. A 223 bp band is generated when the junction is present. \u2026ccaggcgagtgatttacacatcacccctcagaaggaatgagaaaacttgcgcaagtacatccaaaggttcagccgggtgcagtacaacatccccgatgttcatcccgccgtcgtgatcagtgtgttccattagaatgtgcgcaaccgcaagatgcgtgaagagctagcgatgaacaa\u2191aagttaaattttcatgaagtgaaataaaaggtgtgttaacgtgcccatgccatctattttcaaaaataaataaataatctttaagaaaagttaaatgcataaaaaagtccaatagtttacatatacattggctcaatatagtggtgaggcatatttatttatattactcatggtcttttcatggttgatgcacttgccggtggattccatctcttcttgggcggatctgtcccatgagtttgttggcgcctttaccggaggtcaccaagctcatgg\u2026 Descriptions of repeat sequences and TREP numbers in (A) to (C) follow (TIF)Click here for additional data file.Figure S3QTL Cartographer plot of days to ear emergence in \u2018Malacca\u2019\u00d7\u2018Hereward\u2019 DH lines. Plants were vernalized for 4 weeks  and grown in lit glasshouse (18 h light). For each DH line the score was the mean of three plants. One region exceeded the significance threshold (black line at LOD 2.5) and this was on a 20 cM linkage group containing marker barc151 previously shown to be linked to Vrn-A1Vrn-A1 was scored qualitatively based on copy number scores from the TaqMan\u00ae assay (Vrn-A1 position.(TIF)Click here for additional data file.Figure S4Example trace files of directly sequenced PCR amplicons from the region of Vrn-A1 exon 4 containing the C/T variants. \u2018Hereward\u2019 and \u2018Malacca\u2019 had a C/T double peak (arrowed) while \u2018Claire\u2019 and \u2018Chinese Spring\u2019 had only the C form.(TIF)Click here for additional data file.Figure S5C and T variants in exon 4 of Vrn-A1. (A) Predicted amino acid sequence of part of the K box region of VRN-1 and related MADS-box genes. Protein sequences are from related MADS-box genes Vrn-A1 from \u2018IL369\u2019. Intr1/C/F, Intr1/AB/R, Intr1/A/F2 and Intr1/A/R3 are previously published primers for PCR assays (TIF)Click here for additional data file.File S1Quantification of Ppd-B1 expression.(DOC)Click here for additional data file."}
{"text": "The well-posedness of this problem is established in Banach spaces C0\u03b2,\u03b3(E\u03b1\u2212\u03b2) of all E\u03b1\u2212\u03b2-valued continuous functions \u03c6(t) on  satisfying a H\u00f6lder condition with a weight (t + \u03c4)\u03b3. New Schauder type exact estimates in H\u00f6lder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.The nonlocal boundary value problem for the parabolic differential equation  It is known that \u2282E\u2192E be a linear unbounded operator densely defined in E. The operator A is said to be positive in the Banach space E if its spectrum \u03c3A lies in the interior of the sector of angle \u03d5, 0 < 2\u03d5 < 2\u03c0, symmetric with respect to the real axis, and if, on the edges of this sector, S1(\u03d5) = {\u03c1ei\u03d5 : 0 \u2264 \u03c1 \u2264 \u221e} and S2(\u03d5) = {\u03c1ei\u03d5\u2212 : 0 \u2264 \u03c1 \u2264 \u221e}, and outside of the sector the resolvent (\u03bb\u2212A)\u22121 is subject to the bound\u03d5 is called the spectral angle of the positive operator A and is denoted by \u03c6. We call A strongly positive in the Banach space E if its spectral angle \u03c6 < \u03c0/2.Let A in the Banach space E, let us introduce the fractional spaces E\u03b2 = E\u03b2\u2009(0 < \u03b2 < 1) consisting of those \u03bd \u2208 E for which the normFor positive operator E with the strongly positive operator A was established. The importance of coercive inequalities (well-posedness) is well known . The derivative at the endpoints of the segment is understood as the appropriate unilateral derivatives.v(t) belongs to D = D(A(t)) for all t \u2208 , and the function A(t)v(t) is continuous on .The element v(t) satisfies the equation and the initial condition (ondition .A function  problem  if the fC(E) = C. Here, C(E) stands for the Banach space of all continuous functions \u03c6(t) defined on  with values in E equipped with the normf(t) \u2208 C(E) and v0 \u2208 D.A solution of problem  defined C(E) if the following conditions are satisfied.(1)f(t) \u2208 C(E) and any v0 \u2208 D. This means that an additive and homogeneous operator v(t) = v, v0) is defined which acts from C(E) \u00d7 D to C(E) and gives the solution of problem (C(E).Problem  is uniqu problem  in C(E).(2)v, v0), regarded as an operator from C(E) \u00d7 D to C(E), is continuous. Here, C(E) \u00d7 D is understood as the normed space of the pairs (f(t), v0), f(t) \u2208 C(E), and v0 \u2208 D, equipped with the normBy Banach's theorem, in C(E) and these properties, one has coercive inequalityMC\u2009\u2009(1 \u2264 MC < +\u221e) does not depend on\u2009\u2009v0\u2009\u2009and f(t).We say that the problem  is well C(E) for (A(t) = A, then the coercivity inequality implies the analyticity of the semigroup exp{\u2212sA}\u2009\u2009(s \u2265 0), that is, the following estimatesM \u2208 , the operator \u2212A(t) generates an analytic semigroup exp{\u2212sA(t)}\u2009\u2009(s \u2265 0) with exponentially decreasing norm, when s \u2192 +\u221e, that is, the following estimatesM \u2208 , and v0 \u2208 D, it is easy to show that the formulaIf the function  problem .sA(t)}\u2009\u2009(s \u2265 0) and the fundamental solution v of  of  and theot, s) of  which wis < s + \u03c4 < T,\u2009 0 \u2264 t \u2264 T, and 0 \u2264 \u03b1 \u2264 1, one has the inequalityM does not depend on \u03b1, t, s, and \u03c4.For any 0 < s, \u2009 \u03c4, \u2009t \u2264 T and 0 \u2264 \u03b5 \u2264 1, the following estimates hold:M\u22650 and \u03b4 > 0 do not depend on \u03b5, t, s, and \u03c4.For any 0 \u2264 s<t \u2264 T and u \u2208 D, the following identities hold:For any 0 \u2264 s<t \u2264 t + r \u2264 T, the following estimates hold:M\u22650 does not depend on t and s.For any 0 \u2264 v(t) is said to be a solution of problem (F(E) if it is a solution of this problem in C(E) and the functions\u2009\u2009v\u2032(t) and A(t)v(t) belong to F(E).A function  problem  in F(E) C(E), we say that the problem (F(E) if the following two conditions are satisfied.(1)f \u2208 F(E) and v0 \u2208 D, there exists the unique solution v(t) = v, v0) in F(E) of problem , v0) is defined which acts from F(E) \u00d7 D to F(E) and gives the solution of (F(E).For any  problem . This meution of  in F(E).(2)v, v0), regarded as an operator from F(E) \u00d7 D to F(E), is continuous. Here, F(E) \u00d7 D is understood as the normed space of the pairs (f(t), v0),f(t) \u2208 F(E), and v0 \u2208 D, equipped with the normAs in the case of the space  problem  is well F(E) equal to C0\u03b2,\u03b3(E),  space, obtained by completion of the set of all smooth E-valued functions \u03c6(t) on  with respect to the normC0\u03b2,\u03b3(E\u03b1\u2212\u03b2), 0 \u2264 \u03b3 \u2264 \u03b2 \u2264 \u03b1, 0 < \u03b1 < 1 from  for any 0 \u2264 t \u2264 t + \u03bb. Then, for any 0 \u2264 s < t \u2264 t + \u03bb and u \u2208 D, the following identity holdsAssume that I \u2212 v in E and the following estimate holds:Under the assumption of v(t) is called a solution of the problem (v(t) is continuously differentiable on the segment .v(t) belongs to D for all t \u2208 , and the function A(t)v(t) is continuous on .The element v(t) satisfies the equation and the nonlocal boundary condition (ondition .A function  problem  if the fC(E) if the following conditions are satisfied.(1)f(t) \u2208 C(E) and any \u03c6 \u2208 D. This means that an additive and homogeneous operator v(t) = v, \u03c6) is does not depend on defined which acts from C(E) \u00d7 D to C(E) and gives the solution of problem (C(E).Problem  is uniqu problem  in C(E).(2)v, \u03c6), regarded as an operator from C(E) \u00d7 D to C(E), is continuous. Here, C(E) \u00d7 D is understood as the normed space of the pairs (f(t), \u03c6),\u2009 f(t) \u2208 C(E), and \u03c6 \u2208 D, equipped with the normBy Banach's theorem in C(E) and these properties, one has coercive inequalityMC\u2009\u2009(1 \u2264 MC < +\u221e) does not depend on \u03c6\u2009\u2009and f(t).We say that the problem  is well C(E) for (A(t) = A, then the coercivity inequality implies the analyticity of the semigroup exp{\u2212sA}\u2009\u2009(s \u2265 0). Thus, the analyticity of the semigroup exp{\u2212sA}\u2009\u2009(s \u2265 0) is necessary for the well-posedness of problem (C(E). Unfortunately, the analyticity of the semigroup exp{\u2212sA}\u2009\u2009(s \u2265 0) is not sufficient for the well-posedness of problem (C(E).Inequality  is calleC(E) for .\u2009\u2009If A(t problem  in C(E). problem  in C(E).v(t) is said to be a solution of problem (F(E) if it is a solution of this problem in C(E) and the functions\u2009\u2009v\u2032(t) and A(t)v(t) belong to F(E).A function  problem  in F(E) C(E), we say that the problem (F(E) if the following two conditions are satisfied.(1)f \u2208 F(E) and \u03c6 \u2208 D, there exists the unique solution v(t) = v, \u03c6) in F(E) of problem , \u03c6) is defined which acts from F(E) \u00d7 D to F(E) and gives the solution of (F(E).For any  problem . This meution of  in F(E).(2)v, \u03c6), regarded as an operator from F(E) \u00d7 D to F(E), is continuous. Here, F(E) \u00d7 D is understood as the normed space of the pairs (f(t), \u03c6),f(t) \u2208 F(E), and \u03c6 \u2208 D, equipped with the normIf v(t) is a solution in C0\u03b2,\u03b3(E) of problem (C(E) of this problem. Hence, by , 0 \u2264 \u03b3 \u2264 \u03b2 \u2264 \u03b1, 0 < \u03b1 < 1.As in the case of the space  problem  is well  problem , then itence, by , we get ence, by :34)v(t).Using  and formv(t).Usidness of  in the sf(t) \u2208 C0\u03b2,\u03b3(E\u03b1\u2212\u03b2)\u2009\u2009. Suppose that the assumptions (\u03b1 \u2264 \u03b5 < 1. If A(0)\u03c6 + f(\u03bb) \u2212 f(0) \u2208 E\u03b1\u2212\u03b3, then for the solution v(t) in C0\u03b2,\u03b3(E\u03b1\u2212\u03b2) of the nonlocal boundary value problem (M(\u03bb) does not depend on \u03b1, \u03b2, \u03b3, \u03c6, and f(t).Suppose umptions  and 10)f(t) \u2208 C problem  the coerKm for any m = 1,2, 3,4 in E\u03b1\u2212\u03b3, separately. We start with K1. Applying the inequality , a, a38)||stimates , we obtaerefore,||K11||E\u03b1erefore,||K12||E\u03b1quality , astimates , we obtaove thatz1\u2212\u03b1+\u03b3\u222b0\u03bbquality (||K2||E\u03b1\u2212timates (z1\u2212(\u03b1\u2212\u03b3)|ove thatz1\u2212\u03b1+\u03b3\u222b0\u03bbequality , we getequality , we getstimates , (19), astimates , we obtatimates (z1\u2212(\u03b1\u2212\u03b3)|quality , \u20090 \u2264 \u03b3 \u2264 \u03b2 \u2264 \u03b1, \u20090 < \u03b1 < 1.Furthermore, the method of proof of f(t) \u2208 C0\u03b2,\u03b3(E\u03b1\u2212\u03b2)\u2009\u2009. Suppose that the assumptions (\u03b1 \u2264 \u03b5 < 1. If A(0)\u03c6 + f(\u03bb) \u2212 f(0) \u2208 E\u03b1\u2212\u03b2\u03b2,\u03b3, then for the solution v(t) in C0\u03b2,\u03b3(E\u03b1\u2212\u03b2) of the nonlocal boundary value problem (M(\u03bb) does not depend on \u03b1, \u03b2, \u03b3, \u03c6, and f(t).Suppose umptions  and 10)f(t) \u2208 C problem  the coerC0\u03b2,\u03b3(E\u03b1\u2212\u03b2)\u2009\u2009, in which coercive solvability has been established, depend on the parameters \u03b1, \u03b2, and \u03b3. However, the constants in the coercive inequalities depend only on \u03b1. Hence, we can choose the parameters \u03b2 and \u03b3 freely, which increases the number of functional spaces in which problem \u2009\u2009 which is established in the paper  is the space of the all continuous functions \u03c6(x) defined on  with the usual normAt,x acting in C\u03b2 with the domain C\u03b2+2 and satisfying nonlocal boundary conditions v = v, vx = vx for any 0 \u2264 t \u2264 T.We introduce the Banach spaces E = C with a strongly positive operator At,x defined by formula  does not depend on \u03b3, \u03b2,\u2009\u2009f,\u2009\u2009\u03c6(x).For the solution of nonlocal boundary value problem , the folE\u03b1.The proof of E\u03b1 = C\u03b1+\u03bc2 for all 0 < 2\u03b1 + \u03bc < 1, \u20090 \u2264 \u03bc \u2264 1 and 0 \u2264 x \u2264 1, \u20090 \u2264 t \u2264 T [ \u2264 t \u2264 T .t \u2264 T, x \u2208 \u211dn}, we consider the nonlocal boundary value problem for the 2mth order multidimensional parabolic equationar = ar\u2009\u2009and \u03c6(x)(x \u2208 \u211dn), \u2009ar, and f\u2009\u2009, x \u2208 \u211dn) are given sufficiently smooth functions. Here, \u03b4 > 0 is a sufficiently large number.Second, on the range {0 \u2264 Rn. Here, r \u2208 Rn is a vector with nonnegative integer components,\u2009\u2009|r| = r1 + \u22ef+rn. If \u03c6(y)\u2009\u2009 \u2208 Rn) is an infinitely differentiable function that decays at infinity together with all its derivatives, then, by means of the Fourier transformation one establishes the equalityB(\u03be) is called the symbol of the operator B and is given by\u2009\u2009n, satisfies the inequalities\u03be \u2260 0 and for any 0 \u2264 t \u2264 T. Assume that the all compatibility conditions hold which guarantees that the problem . This allows us to reduce the nonlocal boundary value problem (E = C\u03bc(Rn) of all continuous bounded functions defined on \u211dn satisfying a H\u00f6lder condition with the indicator \u03bc \u2208  with a strongly positive operator At,x = Bt,x + \u03b4I defined by  does not depend on\u2009\u2009\u03b3,\u2009\u2009\u03b2,\u2009\u2009f, \u03c6(x).For the solution of the nonlocal boundary value problem , the folAt,x in C\u03bc(Rn), and on the following theorem on the structure of fractional spaces E\u03b1(C\u03bc(Rn), At,x).The proof of E\u03b1(C\u03bc(Rn), At,x) = Cm\u03b1+\u03bc2(Rn) for all 0 < 2m\u03b1 + \u03bc < 1, \u20090 \u2264 \u03bc \u2264 1\u2009\u2009and x \u2208 \u211dn, \u20090 \u2264 t \u2264 T [ \u2264 t \u2264 T .In the present study, the well-posedness of the nonlocal boundary value parabolic problem with dependent coefficients in H\u00f6lder spaces with a weight is established. In practice, new Schauder type exact estimates in H\u00f6lder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are obtained. Moreover, applying the result of the monograph , the hig"}
{"text": "Several interesting consequences of our theorems are also given.We introduce the concept of triangular  They also gave some examples and applications of the obtained results to ordinary differential equations. In this paper, we will introduce the concept of triangular \u03b1c-admissible mappings (pair of mappings) with respect to \u03b7c nonself-mappings and establish the existence of PPF dependent fixed point theorems for contraction mappings involving triangular \u03b1c-admissible mappings (pair of mappings) with respect to \u03b7c nonself-mappings in Razumikhin class.The applications of fixed point theory are very important and useful in diverse disciplines of mathematics. In fact, fixed point theory can be applied for solving equilibrium problems, variational inequalities, and optimization problems. In particular, a very powerful tool is the Banach fixed point theorem, which was generalized and extended in various directions: modifying Banach's contractive condition, changing the space, or extending single-valued mapping to multivalued mapping  is a Banach space, I denotes a closed interval  in \u211d, and E0 =  denotes the sets of all continuous E-valued functions on I equipped with the supremum norm ||\u00b7||E0 defined byThroughout this paper, we assume that  for some c \u2208 I.A mapping S : E0 \u2192 E0 and T : E0 \u2192 E. A point \u03d5 \u2208 E0 is said to be a PPF dependent coincidence point or a coincidence point with PPF dependence of S and T if T\u03d5 = (S\u03d5)(c) for some c \u2208 I.Let T : E0 \u2192 E is called a Banach type contraction if there exists k \u2208 . Define T : E0 \u2192 E by T\u03d5 = 2\u03d5(1) for all \u03d5 \u2208 E0 and \u03b1, \u03b7 : E \u00d7 E \u2192 [0, +\u221e) by\u03b7 = x2 + y2 + |x||y| + 1/2. Then T is a triangular \u03b1c-admissible mapping with respect to \u03b7c. Indeed, if \u03b1(\u03d5(1), \u03be(1)) \u2265 \u03b7(\u03d5(1), \u03be(1)), then \u03d5(1) \u2265 \u03be(1) and so 2\u03d5(1) \u2265 2\u03be(1). That is, T\u03d5 \u2265 T\u03be which implies \u03b1 \u2265 \u03b7. Also, if\u03d5(c) \u2265 \u03c0(c) and \u03c0(c) \u2265 \u03be(c) and so \u03d5(c) \u2265 \u03be(c). That is, \u03b1(\u03d5(c), \u03be(c)) \u2265 \u03b7(\u03d5(c), \u03be(c)).Let The following lemma is necessary later on.T be a triangular \u03b1c-admissible mapping with respect to \u03b7c. Define the sequence {\u03d5n} by the following way:n \u2208 \u2115, where \u03d50 \u2208 \u211bc is such that \u03b1(\u03d50(c), T\u03d50) \u2265 \u03b7(\u03d50(c), T\u03d50). ThenLet T is a triangular \u03b1c-admissible mapping with respect to \u03b7c,\u03b1(\u03d5m(c), \u03d5m+2(c)) \u2265 \u03b7(\u03d5m(c), \u03d5m+2(c)). By continuing this process, we getSince then by  we get \u03b1\u2131 denote the class of all functions \u03b2 : [0, +\u221e)\u2192[0,1) satisfying the following condition:One of our main theorems is a result of Geraghty type  obtainedT : E0 \u2192 E, \u03b1, \u03b7 : E \u00d7 E \u2192 [0, +\u221e) be three mappings satisfying the following assertions:(i)c \u2208 I such that \u211bc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular \u03b1c-admissible mapping with respect to \u03b7c;(iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 E0;there exists (iv)\u03d5n} is a sequence in E0 such that \u03d5n \u2192 \u03d5 as n \u2192 +\u221e and \u03b1(\u03d5n(c), \u03d5n+1(c)) \u2265 \u03b7(\u03d5n(c), \u03d5n+1(c)) for all n \u2208 \u2115 \u222a 0, then \u03b1(\u03d5n(c), \u03d5(c)) \u2265 \u03b7(\u03d5n(c), \u03d5(c)) for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b1(\u03d50(c), T\u03d50) \u2265 \u03b7(\u03d50(c), T\u03d50).there exists Then, T has a PPF dependent fixed point \u03d5* \u2208 \u211bc.Let \u03d50 \u2208 \u211bc such that \u03b1(\u03d50(c), T\u03d50) \u2265 \u03b7(\u03d50(c), T\u03d50). Since T\u03d50 \u2208 E, there exists x1 \u2208 E such that T\u03d50 = x1. Choose \u03d51 \u2208 \u211bc such that\u03d5n} in \u211bc\u2286E such that,\u211bc is algebraically closed with respect to difference, it follows thatn \u2208 \u2115. This implies that the sequence {||\u03d5n\u2212\u03d5n+1||E0} is decreasing in \u211d+. Then, there exists r \u2265 0 such that lim\u2061n\u2192+\u221e||\u03d5n\u2212\u03d5n+1||E0 = r. Assume r > 0. Now, by taking limit as n \u2192 +\u221e in (n\u2192+\u221e\u03b2(||\u03d5n\u22121\u2212\u03d5n||E0). That is,\u03b2 \u2208 \u2131, lim\u2061n\u2192+\u221e||\u03d5n\u22121\u2212\u03d5n||E0 = 0 which is a contradiction. Hence, r = 0. That is,Let , we get||\u03d5n\u2212\u03d5n+1\u03d5n} is Cauchy in \u211bc. Assume the contrary; then there exist \u025b > 0 and two sequences {mk} and {nk} with k \u2264 mk < nk such thatk \u2192 +\u221e, we getNow, we prove that the sequence {On the other hand, by (iii) and , we havek \u2192 +\u221e in the above inequality and applying (k\u2192+\u221e\u03b2(||\u03d5mk\u2212\u03d5nk||E0) = 1 and since \u03b2 \u2208 \u2131, we deduce\u03d5n} is a Cauchy sequence in \u211bc\u2286E0. By the completeness of E0 we get that {\u03d5n} converges to a point \u03d5* \u2208 E0; that is, \u03d5n \u2192 \u03d5* as n \u2192 +\u221e. Since \u211bc is topologically closed, we deduce \u03d5* \u2208 \u211bc. From (iv) we have \u03b1(\u03d5n(c), \u03d5*(c)) \u2265 \u03b7(\u03d5n(c), \u03d5*(c)) for all n \u2208 \u2115 \u222a 0. Then, from (iii) we getn \u2208 \u2115. Taking limit as n \u2192 +\u221e in the above inequality, we get\u03d5* is a PPF dependent fixed point of T in \u211bc.Therefore, we getapplying  and 31)(34)||\u03d5mk\u03b7(\u03d5(c), \u03be(c)) = 1 for all \u03d5, \u03be \u2208 E0, then we deduce the following corollary.If in T : E0 \u2192 E and \u03b1 : E \u00d7 E \u2192 [0, +\u221e) be two mappings satisfying the following assertions:(i)c \u2208 I such that \u211bc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular \u03b1c-admissible mapping;(iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 E0;there exists (iv)\u03d5n} is a sequence in E0 such that \u03d5n \u2192 \u03d5 as n \u2192 +\u221e and \u03b1(\u03d5n(c), \u03d5n+1(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0, then \u03b1(\u03d5n(c), \u03d5(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b1(\u03d50(c), T\u03d50) \u2265 1.there exists Then, T has a PPF dependent fixed point \u03d5* \u2208 \u211bc.Let \u03b1(\u03d5(c), \u03be(c)) = 1 for all \u03d5, \u03be \u2208 E0, then we deduce the following corollary.If in T : E0 \u2192 E and \u03b7 : E \u00d7 E \u2192 [0, +\u221e) be two mappings satisfying the following assertions:(i)c \u2208 I such that \u211bc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular \u03b7c-subadmissible mapping;(iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 E0;there exists (iv)\u03d5n} is a sequence in E0 such that \u03d5n \u2192 \u03d5 as n \u2192 +\u221e and \u03b7(\u03d5n(c), \u03d5n+1(c)) \u2264 1 for all n \u2208 \u2115 \u222a 0, then \u03b7(\u03d5n(c), \u03d5) \u2264 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b7(\u03d50(c), T\u03d50) \u2264 1.there exists Then, T has a PPF dependent fixed point \u03d5* \u2208 \u211bc.Let c \u2208 I and S : E0 \u2192 E0, T : E0 \u2192 E, \u03b1, \u03b7 : E \u00d7 E \u2192 [0, +\u221e). We say that  is a triangular \u03b1c-admissible pair with respect to \u03b7c if, for \u03d5, \u03be, \u03c0 \u2208 E0,\u03b7(\u03d5(c), \u03be(c)) = 1, then, we say that  is a triangular \u03b1c-admissible pair. Also, if we take \u03b1(\u03d5(c), \u03be(c)) = 1, then we say that  is a triangular \u03b7c-subadmissible pair.Let The following theorem gives a result of existence of PPF dependent coincidence points.S : E0 \u2192 E0, T : E0 \u2192 E, and \u03b1, \u03b7 : E \u00d7 E \u2192 [0, +\u221e) be four mappings satisfying the following assertions:(i)c \u2208 I such that S(\u211bc) \u2282 \u211bc is algebraically closed with respect to difference;there exists (ii)S, T) is a triangular \u03b1c-admissible pair with respect to \u03b7c;((iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 \u211bc;there exists (iv)S\u03d5n} is a sequence in \u211bc such that S\u03d5n \u2192 S\u03d5 as n \u2192 +\u221e and \u03b1((S\u03d5n)(c), (S\u03d5n+1)(c)) \u2265 \u03b7((S\u03d5n)(c), (S\u03d5n+1)(c)) for all n \u2208 \u2115 \u222a 0, then \u03b1((S\u03d5n)(c), (S\u03d5)(c)) \u2265 \u03b7((S\u03d5n)(c), (S\u03d5)(c)) for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b1((S\u03d50)(c), T\u03d50) \u2265 \u03b7((S\u03d50)(c), T\u03d50);there exists (vi)S(\u211bc) is complete in \u211bc;(vii)T(\u211bc)\u2282{(S\u03d5)(c) : \u03d5 \u2208 \u211bc}.Then, there exists \u03d5* \u2208 \u211bc such that S\u03d5* \u2208 \u211bc is a PPF dependent fixed point of T and hence \u03d5* is a PPF dependent coincidence point of S and T.Let \u03d50 \u2208 \u211bc such that \u03b1((S\u03d50)(c), T\u03d50) \u2265 \u03b7((S\u03d50)(c), T\u03d50). By condition (vii), there exists \u03d51 \u2208 \u211bc such that\u03d5n} in \u211bc such thatS(\u211bc) is algebraically closed with respect to difference, it follows thatn \u2208 \u2115. This implies that the sequence {||S\u03d5n\u2212S\u03d5n+1||E0} is decreasing in \u211d+. Then, there exists r \u2265 0 such that lim\u2061n\u2192+\u221e||S\u03d5n\u2212S\u03d5n+1||E0 = r. Assume r > 0. Now by taking limit as n \u2192 +\u221e in (n\u2192+\u221e\u03b2(||S\u03d5n\u22121\u2212S\u03d5n||E0). That is,\u03b2 \u2208 \u2131, lim\u2061n\u2192+\u221e||S\u03d5n\u22121\u2212S\u03d5n||E0 = 0 which is a contradiction. Hence, r = 0. That is,Let , we get||S\u03d5n\u2212S\u03d5nS\u03d5n} is Cauchy in S(\u211bc). Assume the contrary; then there exist \u025b > 0 and two sequences {mk} and {nk} with k \u2264 mk < nk such thatk \u2192 +\u221e, we getNow, we prove that the sequence {On the other hand, by (iii) and , we havek \u2192 +\u221e in the above inequality and applying (k\u2192+\u221e\u03b2(||S\u03d5mk\u2212S\u03d5nk||E0) = 1 and since \u03b2 \u2208 \u2131, we deduceS\u03d5n} is a Cauchy sequence in S(\u211bc) \u2282 \u211bc. By the completeness of S(\u211bc), there exists \u03d5* \u2208 \u211bc such that S\u03d5n \u2192 S\u03d5* as n \u2192 +\u221e. From (iv), we have \u03b1((S\u03d5n)(c), (S\u03d5*)(c)) \u2265 \u03b7((S\u03d5n)(c), (S\u03d5*)(c)) for all n \u2208 \u2115 \u222a 0. Then from (iii) we getn \u2208 \u2115. Taking limit as n \u2192 +\u221e in the above inequality, we getS\u03d5* is a PPF dependent fixed point of T in S(\u211bc) and hence \u03d5* is a PPF dependent coincidence point of S and T.Therefore, we getapplying  and 57)(60)||S\u03d5m\u03b7(\u03d5(c), \u03be(c)) = 1 for all \u03d5, \u03be \u2208 E0, then we deduce the following corollary.If in S : E0 \u2192 E0, T : E0 \u2192 E, and \u03b1 : E \u00d7 E \u2192 [0, +\u221e) be three mappings satisfying the following assertions:(i)c \u2208 I such that S(\u211bc) \u2282 \u211bc is algebraically closed with respect to difference;there exists (ii)S, T) is a triangular \u03b1c-admissible pair;((iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 \u211bc;there exists (iv)S\u03d5n} is a sequence in \u211bc such that S\u03d5n \u2192 S\u03d5 as n \u2192 +\u221e and \u03b1((S\u03d5n)(c), (S\u03d5n+1)(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0, then \u03b1((S\u03d5n)(c), (S\u03d5)(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b1((S\u03d50)(c), T\u03d50) \u2265 1;there exists (vi)S(\u211bc) is complete in \u211bc;(vii)T(\u211bc)\u2282{(S\u03d5)(c) : \u03d5 \u2208 \u211bc}.Then, S and T have a PPF dependent coincidence point \u03d5* \u2208 \u211bc.Let \u03b1(\u03d5(c), \u03be(c)) = 1 for all \u03d5, \u03be \u2208 E0, then we deduce the following corollary.If in S : E0 \u2192 E0, T : E0 \u2192 E, and \u03b7 : E \u00d7 E \u2192 [0, +\u221e) be three mappings satisfying the following assertions:(i)c \u2208 I such that S(\u211bc) \u2282 \u211bc is algebraically closed with respect to difference;there exists (ii)S, T) is a triangular \u03b1c-subadmissible pair;((iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 \u211bc;there exists (iv)S\u03d5n} is a sequence in E0 such that S\u03d5n \u2192 S\u03d5 as n \u2192 +\u221e and \u03b7((S\u03d5n)(c), (S\u03d5n+1)(c)) \u2264 1 for all n \u2208 \u2115 \u222a 0, then \u03b7((S\u03d5n)(c), (S\u03d5)(c)) \u2264 1;if {(v)\u03d50 \u2208 \u211bc such that \u03b7((S\u03d50)(c), T\u03d50) \u2264 1;there exists (vi)S(\u211bc) is complete in \u211bc;(vii)T(\u211bc)\u2282{(S\u03d5)(c) : \u03d5 \u2208 \u211bc}.Then, S and T have a PPF dependent coincidence point \u03d5* \u2208 \u211bc.Let T : E0 \u2192 E and \u03b1 : E \u00d7 E \u2192 [0, +\u221e) be two mappings satisfying the following assertions:(i)c \u2208 I such that \u211bc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular \u03b1c-admissible mapping;(iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 E0;there exists (iv)\u03d5n} is a sequence in E0 such that \u03d5n \u2192 \u03d5 as n \u2192 +\u221e and \u03b1(\u03d5n(c), \u03d5n+1(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0, then \u03b1(\u03d5n(c), \u03d5(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b1(\u03d50(c), T\u03d50) \u2265 1.there exists Then, T has a PPF dependent fixed point \u03d5* \u2208 \u211bc.Let \u03b1(\u03d5(c), \u03be(c)) \u2265 1; then by (iii) we haveT has a PPF dependent fixed point \u03d5* \u2208 \u211bc.Let T : E0 \u2192 E and \u03b1 : E \u00d7 E \u2192 [0, +\u221e) be two mappings satisfying the following assertions:(i)c \u2208 I such that \u211bc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular \u03b1c-admissible mapping;(iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 E0, where \u03f5 \u2265 1;there exists (iv)\u03d5n} is a sequence in E0 such that \u03d5n \u2192 \u03d5 as n \u2192 +\u221e and \u03b1(\u03d5n(c), \u03d5n+1(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0, then \u03b1(\u03d5n(c), \u03d5(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b1(\u03d50(c), T\u03d50) \u2265 1.there exists Then, T has a PPF dependent fixed point \u03d5* \u2208 \u211bc.Let \u03b1(\u03d5(c), \u03be(c)) \u2265 1; then by (iii) we haveT\u03d5\u2212T\u03be||E \u2264 \u03b2(||\u03d5\u2212\u03be||E0)||\u03d5\u2212\u03be||E0. That is, all conditions of T has a PPF dependent fixed point \u03d5* \u2208 \u211bc.Let T : E0 \u2192 E and \u03b1 : E \u00d7 E \u2192 [0, +\u221e) be two mappings satisfying the following assertions:(i)c \u2208 I such that \u211bc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular \u03b1c-admissible mapping;(iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 E0, where 1 < \u03f5 \u2264 \u03c3;there exists (iv)\u03d5n} is a sequence in E0 such that \u03d5n \u2192 \u03d5 as n \u2192 +\u221e and \u03b1(\u03d5n(c), \u03d5n+1(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0, then \u03b1(\u03d5n(c), \u03d5(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b1(\u03d50(c), T\u03d50) \u2265 1.there exists Then, T has a PPF dependent fixed point \u03d5* \u2208 \u211bc.Let \u03b1(\u03d5(c), \u03be(c)) \u2265 1; then by (iii) we haveT\u03d5\u2212T\u03be||E \u2264 \u03b2(||\u03d5\u2212\u03be||E0)||\u03d5\u2212\u03be||E0. That is, all conditions of T has a PPF dependent fixed point \u03d5* \u2208 \u211bc.Let T : E0 \u2192 E and \u03b7 : E \u00d7 E \u2192 [0, +\u221e) be two mappings satisfying the following assertions:(i)c \u2208 I such that \u211bc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular \u03b7c-subadmissible mapping;(iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 E0;there exists (iv)\u03d5n} is a sequence in E0 such that \u03d5n \u2192 \u03d5 as n \u2192 \u221e and \u03b7(\u03d5n(c), \u03d5n+1(c)) \u2264 1 for all n \u2208 \u2115 \u222a 0, then \u03b7(\u03d5n(c), \u03d5(c)) \u2264 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b7(\u03d50(c), T\u03d50) \u2264 1.there exists Then, T has a PPF dependent fixed point \u03d5* \u2208 \u211bc.Let T : E0 \u2192 E and \u03b7 : E \u00d7 E \u2192 [0, \u221e) be two mappings satisfying the following assertions:(i)c \u2208 I such that \u211bc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular \u03b7c-subadmissible mapping;(iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 E0, where \u03f5 \u2265 1 and \u03c8 \u2208 \u03a8;there exists (iv)\u03d5n} is a sequence in E0 such that \u03d5n \u2192 \u03d5 as n \u2192 +\u221e and \u03b7(\u03d5n(c), \u03d5n+1(c)) \u2264 1 for all n \u2208 \u2115 \u222a 0, then \u03b7(\u03d5n(c), \u03d5(c)) \u2264 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b7(\u03d50(c), T\u03d50) \u2264 1.there exists Then, T has a PPF dependent fixed point \u03d5* \u2208 \u211bc.Let T : E0 \u2192 E and \u03b7 : E \u00d7 E \u2192 [0, +\u221e) be two mappings satisfying the following assertions:(i)c \u2208 I such that \u211bc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular \u03b7c-subadmissible mapping;(iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 E0, where 1 < \u03f5 \u2264 \u03c3 and \u03c8 \u2208 \u03a8;there exists (iv)\u03d5n} is a sequence in E0 such that \u03d5n \u2192 \u03d5 as n \u2192 +\u221e and \u03b7(\u03d5n(c), \u03d5n+1(c)) \u2264 1 for all n \u2208 \u2115 \u222a 0, then \u03b7(\u03d5n(c), \u03d5(c)) \u2264 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b7(\u03d50(c), T\u03d50) \u2264 1.there exists Then, T has a PPF dependent fixed point \u03d5* \u2208 \u211bc.Let S : E0 \u2192 E0, T : E0 \u2192 E, and \u03b1 : E \u00d7 E \u2192 [0, +\u221e) be three mappings satisfying the following assertions:(i)c \u2208 I such that S(\u211bc) \u2282 \u211bc is algebraically closed with respect to difference;there exists (ii)S, T) is a triangular \u03b1c-admissible pair;((iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 \u211bc;there exists (iv)S\u03d5n} is a sequence in \u211bc such that S\u03d5n \u2192 S\u03d5 as n \u2192 +\u221e and \u03b1((S\u03d5n)(c), (S\u03d5n+1)(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0, then \u03b1((S\u03d5n)(c), (S\u03d5)(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b1((S\u03d50)(c), T\u03d50) \u2265 1;there exists (vi)S(\u211bc) is complete in \u211bc;(vii)T(\u211bc)\u2282{(S\u03d5)(c) : \u03d5 \u2208 \u211bc}.Then, S and T have a PPF dependent coincidence point \u03d5* \u2208 \u211bc.Let S : E0 \u2192 E0, T : E0 \u2192 E, and \u03b1 : E \u00d7 E \u2192 [0, +\u221e) be three mappings satisfying the following assertions:(i)c \u2208 I such that S(\u211bc) \u2282 \u211bc is algebraically closed with respect to difference;there exists (ii)S, T) is a triangular \u03b1c-admissible pair;((iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 \u211bc where \u03f5 \u2265 1;there exists (iv)S\u03d5n} is a sequence in E0 such that S\u03d5n \u2192 S\u03d5 as n \u2192 +\u221e and \u03b1((S\u03d5n)(c), (S\u03d5n+1)(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0, then \u03b1((S\u03d5n)(c), (S\u03d5)(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b1((S\u03d50)(c), T\u03d50) \u2265 1;there exists (vi)S(\u211bc) is complete in \u211bc;(vii)T(\u211bc)\u2282{(S\u03d5)(c) : \u03d5 \u2208 \u211bc}.Then, S and T have a PPF dependent coincidence point \u03d5* \u2208 \u211bc.Let S : E0 \u2192 E0, T : E0 \u2192 E, and \u03b1 : E \u00d7 E \u2192 [0, +\u221e) be three mappings satisfying the following assertions:(i)c \u2208 I such that S(\u211bc) \u2282 \u211bc is topologically closed and algebraically closed with respect to difference;there exists (ii)S, T) is a triangular \u03b1c-admissible pair;((iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 \u211bc, where 1 < \u03f5 \u2264 \u03c3;there exists (iv)S\u03d5n} is a sequence in E0 such that S\u03d5n \u2192 S\u03d5 as n \u2192 +\u221e and \u03b1((S\u03d5n)(c), (S\u03d5n+1)(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0, then \u03b1((S\u03d5n)(c), (S\u03d5)(c)) \u2265 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b1((S\u03d50)(c), T\u03d50) \u2265 1;there exists (vi)S(\u211bc) is complete in \u211bc;(vii)T(\u211bc)\u2282{(S\u03d5)(c) : \u03d5 \u2208 \u211bc}.Then, S and T have a PPF dependent coincidence point \u03d5* \u2208 \u211bc.Let S : E0 \u2192 E0, T : E0 \u2192 E, and \u03b7 : E \u00d7 E \u2192 [0, +\u221e) be three mappings satisfying the following assertions:(i)c \u2208 I such that S(\u211bc) \u2282 \u211bc is algebraically closed with respect to difference;there exists (ii)S, T) is a triangular \u03b1c-subadmissible pair;((iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 \u211bc;there exists (iv)S\u03d5n} is a sequence in E0 such that S\u03d5n \u2192 S\u03d5 as n \u2192 +\u221e and \u03b7((S\u03d5n)(c), (S\u03d5n+1)(c)) \u2264 1 for all n \u2208 \u2115 \u222a 0, then \u03b7((S\u03d5n)(c), (S\u03d5)(c)) \u2264 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b7((S\u03d50)(c), T\u03d50) \u2264 1;there exists (vi)S(\u211bc) is complete in \u211bc;(vii)T(\u211bc)\u2282{(S\u03d5)(c) : \u03d5 \u2208 \u211bc}.Then, S and T have a PPF dependent coincidence point \u03d5* \u2208 \u211bc.Let S : E0 \u2192 E0, T : E0 \u2192 E, and \u03b7 : E \u00d7 E \u2192 [0, +\u221e) be three mappings satisfying the following assertions:(i)c \u2208 I such that S(\u211bc) \u2282 \u211bc is algebraically closed with respect to difference;there exists (ii)S, T) is a triangular \u03b1c-subadmissible pair;((iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 \u211bc, where \u03f5 \u2265 1;there exists (iv)S\u03d5n} is a sequence in E0 such that S\u03d5n \u2192 S\u03d5 as n \u2192 +\u221e and \u03b7((S\u03d5n)(c), (S\u03d5n+1)(c)) \u2264 1 for all n \u2208 \u2115 \u222a 0, then \u03b7((S\u03d5n)(c), (S\u03d5)(c)) \u2264 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b7((S\u03d50)(c), T\u03d50) \u2264 1;there exists (vi)S(\u211bc) is complete in \u211bc;(vii)T(\u211bc)\u2282{(S\u03d5)(c) : \u03d5 \u2208 \u211bc}.Then, S and T have a PPF dependent coincidence point \u03d5* \u2208 \u211bc.Let S : E0 \u2192 E0, T : E0 \u2192 E, and \u03b7 : E \u00d7 E \u2192 [0, +\u221e) be three mappings satisfying the following assertions:(i)c \u2208 I such that S(\u211bc) \u2282 \u211bc is topologically closed and algebraically closed with respect to difference;there exists (ii)S, T) is a triangular \u03b1c-subadmissible pair;((iii)\u03b2 \u2208 \u2131 such that\u03d5, \u03be \u2208 \u211bc, where 1 < \u03f5 \u2264 \u03c3;there exists (iv)S\u03d5n} is a sequence in E0 such that S\u03d5n \u2192 S\u03d5 as n \u2192 +\u221e and \u03b7((S\u03d5n)(c), (S\u03d5n+1)(c)) \u2264 1 for all n \u2208 \u2115 \u222a 0, then \u03b7((S\u03d5n)(c), (S\u03d5)(c)) \u2264 1 for all n \u2208 \u2115 \u222a 0;if {(v)\u03d50 \u2208 \u211bc such that \u03b7((S\u03d50)(c), T\u03d50) \u2264 1;there exists (vi)S(\u211bc) is complete in \u211bc;(vii)T(\u211bc)\u2282{(S\u03d5)(c) : \u03d5 \u2208 \u211bc}.Then, S and T have a PPF dependent coincidence point \u03d5* \u2208 \u211bc.Let"}
{"text": "A predator-prey system was studied that has a discrete delay, stage-structure, and Beddington-DeAngelis functional response, where predator species has three stages, immature, mature, and old age stages. By using of Mawhin's continuous theorem of coincidence degree theory, a sufficient condition is obtained for the existence of a positive periodic solution. The dynamic relationship between the predator and the prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. The traditional predator-prey model has been studied extensively , x2(t), and x3(t) are the densities of immature, mature, and the old age predator at time t, respectively; y(t) represents the density of prey at time t. \u03c4 = max\u2061t\u2208{\u03c41(t), \u03c42(t)}, \u03c41(t) \u2265 0 is delay due to prey densities and \u03c42(t) \u2265 0 is delay due to gestation of predator. The death rates of the immature, mature, and old age predator are proportional to the fact that existing immature, mature, and old age predator population with respective proportionality d1(t), d2(t), d3(t), r1(t) denote the rate of transformation from immature predator into mature predator; r2(t) is the rate of transformation from mature predator into old age predator; k(t) = (d(t)/c(t)) (0 < k(t) \u2264 1) is the transformation coefficient from prey into the immature predators. Let d1(t) + r1(t) = m(t) and d2(t) + r2(t) = n(t).In the present paper, motivated by the above work, we investigate a time delay predator-prey system with three-stage-structure for predator and Beddington-DeAngelis functional response of the form(H1)a(t), b(t), c(t), d(t), di(t) , \u03b1(t), \u03b2(t), \u03b3(t), e2(t) and ri(t)  are all positive \u03c9-periodic continuous function in R = .For ecological reasons, we consider the initial function of the system  only in f(t), t \u2208  | ui(t)|; then X and Y are Banach spaces with the norm ||\u00b7||.Define Nu = (N1(t), N2(t), N3(t), N4(t))T, u \u2208 X, Pu = (1/\u03c9)\u222b0\u03c9u(t)dt, u \u2208 X, and Qy = (1/\u03c9)\u222b0\u03c9y(t)dt, y \u2208 Y, whereLu = Nu,\u2009\u2009u \u2208 X.Let e system  can be wL = R4, Im\u2061L = {u(t) \u2208 Z, \u222b0\u03c9u(t)dt = 0} is closed in Y and dim\u2061\u2061Ker\u2061L = 4 = Codim\u2061\u2061Im\u2061L and P, Q are continuous projectors such that Im\u2061P = Ker\u2061L, Ker\u2061Q = Im\u2061L = Im\u2061(I \u2212 Q). Therefore, L is a Fredholm mapping of index zero. Furthermore, the inverse KP : Im\u2061L \u2192 Ket\u2009P\u2229Dom\u2061L exists and has the formQN and KP(I \u2212 Q)N are continuous by the Lebesgue theorem; it is not difficult to show that \u03a9 \u2282 X by the Arzela-Ascoli theorem. Hence N is L-compact on \u03a9 \u2282 X.Obviously, Ker\u2061\u03a9 \u2282 X. Corresponding to the operator equation, Lu = \u03bbNu, \u03bb \u2208 , we haveIn order to apply u = u(t) \u2208 X is a solution to . Integrating (\u03c9] leads toeu1(t) and integrating it over  lead toeu2(t) and integrating it over  give0\u03c9eu2(t)dt)2 \u2264 \u03c9\u222b0\u03c9eu2(t)2dt and from (eu3(t) and integrating it over  lead to\u03c9] leads tou(t) \u2208 X, there exist \u03bei, \u03b7i \u2208  such thatt \u2208 g(x) = x/(x + l) leads toA5 = e2M\u03b3M((r1LdLA4/mM) \u2212 nM(\u03b1M + \u03b2MA4)). From  is independent of \u03bb. From condition of \u03b6i \u2208 , i = 1,2, such that the algebraic equations,v1*, v2*, v3*, v4*)T.Suppose that ution to  for a ceegrating  over theleads to\u222b0\u03c9d(t)eudt=d\u22123\u03c9,\u222b0\u03c9[a(t)eation of  by eu1(ation of  by eu2( impliesmL\u222b0\u03c9eu1(h yieldse2L\u222b0\u03c9e2u impliesmL\u222b0\u03c9eu1(].Hence\u222b0\u03c9eu2(t)ation of  by eu3(].Hence\u222b0\u03c9eu2(t)ation of  over .Hence\u222b0\u03c9eu2(t)2.From (\u222b0\u03c9eu1(t)er with (\u222b0\u03c9eu3(t) implies\u222b0\u03c9eu4(t)m\u2212\u03c9=:l1,\u222b0\u03c9|u\u02d92(t].Hence\u222b0\u03c9eu2(t)ws from (\u222b0\u03c9|u\u02d91(tu2(t)dt,\u222b0\u03c9|u\u02d93\u2264lndt=d\u22123\u03c9,\u222b0\u03c9 \u2192 X, and a direct calculation shows that\u03a9 satisfies all the conditions of Lu = Nu has at least one solution in \u03c9-periodic solution in \u03a9. Then, there exists at least one \u03c9-periodic solution for the system  \u2261 1, \u03b2(t) \u2261 m(constant), and \u03b3(t) \u2261 0, the corresponding subsystems of the system (Chen et al.  considere system  are syste system ; that ise system . The cone system  remain te system .In a word, despite considering the three-stage-structure, people can more protect the beneficial insect from dying out and under the preconditions; the actualization of the periodic changes is more convenient and realistic. The results accord with the change in the natural environments, so the consideration of stage-structure is necessary and important."}
{"text": "Lx)(t) = x\u2032(t) + \u2211i=1mpi(t)x(t \u2212 \u03c4i(t)) = f(t), \u2009t \u2208 , x(tj) = \u03b2jx(tj \u2212 0), \u2009j = 1,\u2026, k, \u2009a = t0 < t1 < t2 < \u22ef<tk < tk+1 = b, \u2009x(\u03b6) = 0, \u2009\u03b6 \u2209 , with nonlocal boundary condition lx = \u222bab\u03c6(s)x\u2032(s)ds + \u03b8x(a) = c, \u03c6 \u2208 L\u221e; \u03b8, \u2009c \u2208 R. Various results on existence and uniqueness of solutions and on positivity/negativity of the Green's functions for this equation are obtained. The impulsive delay differential equation is considered ( Mathematical models with impulsive differential equations attract the topic attention of many researchers see \u20137); many; many7])Various comparison theorems for solutions of the Cauchy and periodic problems for ordinary differential equations with impulses have been obtained in , 21\u201324. Equations with nonlocal boundary conditions are applied in modelling different processes . NonlocIn this paper we consider the following impulsive equation:We develop the ideas presented in  and we hD of piecewise continuous functions x :  \u2192 R, isomorphic to the topological product L \u00d7 R, where L is the space of measurable essential bounded functions z :  \u2192 R, by the following equality:a = t0 < t1 < t2 < \u22ef<tk < tk+1 = b.Define the space x(t) is absolutely continuous in , i = 1,\u2026, k + 1, satisfying the equality x(ti) = \u03b2ix(ti \u2212 0). It is clear that x(t) has discontinuity of the first kind and is continuous from the right at points ti, i = 1,\u2026, k.It is clear that By , the genD for every essential bounded f and c \u2208 R, then its solution can be represented in the formG of this problem isC = 0\u2009\u2009for t < s. If the boundary value problem \u20135) is u is u5) iFrom the boundary condition , we obtaSubstituting  into 9)9), we obC can be represented in the following form:Cij = C in the rectangle ti \u2264 t < ti+1, si \u2264 s < si+1.The Cauchy function Let us take derivative in :(15)x\u2032 = 1, we obtainSince, according to the definition of the Cauchy function, Let us substitute  into 1313 as folC = 0, s > t, it follows that \u222batCds = \u222batCds + \u222btbCds = \u222babCds.Since Changing the order of integration in the third double integral, we getG of this problem is of the following form:C = 0, for t < s.Substituting now  into 1212, we obx(a)=c\u03b8\u22121tituting , we obtaOne hasC of impulsive equations (quations \u20134)..C c = 0, \u03c6(s) \u2261 \u03b3, \u03b8 = \u03b3 \u2212 1 we obtain the Corollary.Substituting W is as follows:If generalized periodic problem \u20134), 7), (7)4), , (7)4), C for several simple equations.Let us construct the Cauchy function Consider now the following auxiliary equation:K. It is known that K = estp(\u03b6)d\u03b6\u2212\u222b. For every s this function is absolutely continuous function with respect to t, and K = 0 for s > t.Let us denote the Cauchy function of the nonimpulsive equation  by K for every fixed s as a function of t is a solution ofC = 1, we obtain the following theorem.Using the fact that the Cauchy function C of (Ha(t), Heaviside function, is defined by the following equality:The Cauchy function t, s) of  and 2)  C Now let us consider the following auxiliary equation:C0 of the problem , an, anC0 of the problem  and andC0 Green's function of problem  Green's function of generalized periodic problem , an, anG0..G0 = \u03b8 + \u222bab\u03c6(t))t\u2032dt.Let us denote \u03b2 = Col{\u03b21,\u2026, \u03b2k}. The computation of integrals in formula  = \u03c6(s) + \u222bsb\u03c6(t))t\u2032dt, we obtain for a \u2264 s < t1Denoting I2 = \u03c6(s) + \u222bsb\u03c6(t))t\u2032dt, we obtain for t1 \u2264 s < t2Denoting Ik+1 = \u03c6(s) + \u222bsb\u03c6(t))t\u2032dt, we obtain for tk \u2264 s < bDenoting J = (\u03b3 \u2212 1) + \u03b3\u222bab)t\u2032dt.Let us denote The computation of integrals in formula  leads usJ1 = \u03b3\u222bsb)t\u2032dt + \u03b3, we obtain for a \u2264 s < t1Denoting J2 = \u03b3\u222bsb)t\u2032dt + \u03b3, we obtain for t1 \u2264 s < t2Denoting Jk+1(\u03b2) = \u03b3\u222bsb)t\u2032dt + \u03b3, we obtain for tk \u2264 s < bDenoting I\u22600; then Green's function G0 of problem (k = 3) is described by Let us assume that  problem , 5) in  in I \u2260 0; then Green's function W0 of generalized periodic problem (k = 3) is described by Let us assume that  problem , 7) in  in J|)|(1 \u2212 \u03b2k+1)/(1 \u2212 \u03b2)|((2k + 1)/2)\u0394 where \u0394 = (max\u2061i\u2264k+11\u2264\u2061|ti\u2212ti\u22121|)2,Let us denote \u03b2i, 1 \u2264 i \u2264 k,\u2009\u2009I \u2260 0 andf and real c.Let 0 <  problem  and thisBy \u03c8(t) = \u222babG0f(s)ds + C0(c/(\u03b8 + \u222bab\u03c6(s))s\u2032ds)) = \u222babG0f(s)ds + /I).Let us denote M : D \u2192 D by the equality (Mx)(t) = \u2212\u222babG0\u2211i=1mpi(s)x(s \u2212 \u03c4i(s))ds.Define the operator We can write equation which is equivalent to problem \u20135):(43:(435):M|| < 1, then there exists the unique solution of (x(t) = (I \u2212 M)\u22121\u03c8(t).If ||ution of  which cax|| = max\u2061a\u2264t\u2264b\u2061|x(t)|. Thus we have got the following condition for existence of unique solution of problem (It is clear that problem :(45)||MC0 can be written asDenoteIt is clear thatEstimation of the solution of the problem \u20135) can  can 5) cLet us denoteThen the problem \u20135) has  has 5) hIt is clear thatEstimating the integrals in formula , we get G0 = G1 + G2, where G1 is upper triangular and G2 lower triangular parts of G0 and estimate ||G0||. One hasLet us write From estimation of the integrals, we obtainBy (56)Q\u2264m(Substituting  into 5151 and 5 we obtai\u03a92 = |1 \u2212 \u03b2k+1|(2k + 1)\u0394/2|1 \u2212 \u03b2||J|,Let us denote \u2009\u2009\u03b2i, 1 \u2264 i \u2264 k,\u2009\u2009J \u2260 0 andf.Let 0 <  problem , 7) and and\u03b2i, By \u03c8(t) = \u222babW0f(s)ds.Let us denote T : D \u2192 D by the equality (Tx)(t) = \u2212\u222babW0\u2211i=1mpi(s)x(s \u2212 \u03c4i(s))ds.Define the operator We can write equation which is equivalent to problem \u20134), 7), (7)4), , (7)4), T|| < 1, then there exists the unique solution of (x(t) = (I \u2212 M)\u22121\u03c8(t).If ||ution of , 7) whi whiT|| <x|| = max\u2061a\u2264t\u2264b\u2061|x(t)|. Thus we have got the following condition for existence of unique solution of problem (It is clear that6||Tx||\u2264|| problem :(65)||TEstimation of the solution of the problem \u20134), 7), (7)4), , (7)4), Let us denoteThen the problem \u20134), 7), (7)4), , (7)4), It is clear thatEstimating the integrals in formula , we get W0 = W1 + W2, where W1 is upper triangular and W2, lower triangular parts, and estimate ||W0||:Let us write We obtainIt is clear thatWe obtain the following solution estimation:The proofs of the following two assertions follow from the construction of Green's functions.G0 of problem  is  is (75)0G0 of problem  is  is (76)0Estimation of Green's function  leads uspi(t) \u2264 0 for i = 1,\u2026, m; then problem  is positive for t, s \u2208 .Let conditions  and 75)75) be fuc = 0.Without losing generality we assume that Solution of problem \u20135) can  can 5) cM is positive.Using M|| < 1.The conditions of Now it is clear thatM that for every nonnegative f we get according to (\u03c8(t) \u2265 0 and x(t) \u2265 \u03c8(t) \u2265 0.It follows from the positivity of the operator rding to  \u03c8(t) \u2265 0It is clear thatx(t) for every nonnegative f(t), we obtain that G \u2265 G0 \u2265 0.From nonnegativity of pi(t) \u2265 0 for i = 1,\u2026, m then problem  is negative for t, s \u2208 .Let conditions  and 76)76) be fu problem  is uniquWe prove this assertion analogously to the proof of The proofs of the following two assertions follow from the construction of Green's functions.W0 is negative for t, s \u2208 .Let problem , 7) W0 W0(81)0W0 is positive for t, s \u2208 .Let problem , 7) W0 W0(82)0The proof of the following two theorems can be obtained analogously from the proof of pi(t) \u2264 0 for i = 1,\u2026, m; then problem  is positive for t, s \u2208 .Let conditions  and 82)82) be fu problem , 7) is  is pi(tpi(t) \u2265 0 for i = 1,\u2026, m; then problem  is negative for t, s \u2208 .Let conditions  and 81)81) be fu problem , 7) is  is pi(t"}
{"text": "This paper attempts to present a multivariable extension of generalized Humbert polynomials. The results obtained here include various families of multilinear and multilateral generating functions, miscellaneous properties, and also some special cases for these multivariable polynomials. Thix = ,\u2009\u2009y = , m =  and mi\u2009\u2009 is a positive integer. It follows from (\u03bb)k : = \u03bb(\u03bb + 1) \u22ef (\u03bb + k \u2212 1), (\u03bb)0 : = 1 is the Pochhammer symbol.In this paper, we consider the following multivariable extension of the generalized Humbert polynomials which are completely different from the polynomials introduced in . This cl of s complex variables z1,\u2026, zs\u2009\u2009(s \u2208 \u2115) and of complex order \u03bc, letak \u2260 0, \u2009\u2009\u03bc, \u03bd \u2208 \u2102); z = . Then, for p \u2208 \u2115; x = ; y = ; m = , mi \u2208 \u2115\u2009, one hasCorresponding to an identically nonvanishing function  one has\u2211n1,\u2026,nr=S denote the first member of the assertion  of s complex variables z1,\u2026, zs\u2009\u2009(s \u2208 \u2115) and for pi \u2208 \u2115, \u03bbi, \u03b7i \u2208 \u2102, z = , w = ,\u2009\u03bb : = ,\u2009\u2009\u03b7 : =  letak1,\u2026,kr \u2260 0; ni,  \u2208 \u21150; \u21150 : = \u2115 \u222a {0}. Then, one hasFor a nonvanishing function  one has\u2211l1=0n1\u22ef\u2211\u03a9\u03bc+\u03bdk(z),\u2009\u2009z = ,\u2009\u2009k \u2208 \u21150, s \u2208 \u2115, is expressed in terms of simpler functions of one and more variables, then we can give further applications of the above theorems. We first set\u03bc+\u03bdk\u03b21,\u2026,\u03b2r)((z) [As an application of the above theorems, when the multivariable function \u2026,\u03b2r)(z)  are genecitly by .\u03bc,\u03bd: = \u2211k=0\u221eak\u2009g\u2009\u03bc+\u03bdk\u03b21,\u2026,\u03b2r)((z)wk, ak \u2260 0, \u03bc, \u03bd \u2208 \u2102, z =  then, one hasIf \u039b one has\u2211n1,\u2026,nr=ak = 1,\u2009\u2009\u03bc = 0, \u03bd = 1, we find thatUsing the generating relation  for the s = 2r\u2009\u2009and\u03bbi, \u03b7i\u2009\u2009 \u2208 \u21150,\u2009\u2009u = , v =  in On the other hand, choosing 2r\u2009\u2009and1\u03c6\u03bb1+\u03b71k1,ak1,\u2026,kr \u2260 0; pi \u2208 \u2115; ni, \u03bbi, \u03b7i \u2208 \u21150, i = 1,2,\u2026, r, then, we getIf, we get\u2211l1=0n1\u22ef\u2211ak\u2009\u2009(k \u2208 \u21150), if the multivariable functions \u03a9\u03bc+\u03bdk(y) and \u03c6\u03bb1+\u03b71k1,\u2026,\u03bbr+\u03b7rkr(y),\u2009\u2009y = , (s \u2208 \u2115), are expressed as an appropriate product of several simpler functions, the assertions of Theorems Furthermore, for every suitable choice of the coefficients In this section, we now discuss some further properties of the family of multivariable polynomials given by . We starn1,\u2026,nr be a family of functions generated byni \u2265 mi \u2212 1,\u2009\u2009i = 1,2,\u2026, r; nj \u2265 0, \u2009j \u2260 i andni = 0,1,\u2026, mi \u2212 2, i = 1,2,\u2026, r; nj \u2265 0, j \u2260 i. Also, one finds thatni \u2265 mi \u2212 1, i = 1,2,\u2026, r; nj \u2265 0, j \u2260 i andni = 0,1,\u2026, mi \u2212 2,\u2009\u2009i = 1,2,\u2026, r; nj \u2265 0, j \u2260 i.Let \u03a8i = 1,2,\u2026, r. Then, by differentiating i = 1,2,\u2026rentiate  with resrelation .n1,\u2026,nr is a family of functions generated by 29), respSimilarly, as a consequence of n1,\u2026,nr\u03b1) areni \u2265 mi, i = 1,2,\u2026, r; nj \u2265 0, j \u2260 i andni = 1,2,\u2026, mi \u2212 1, i = 1,2,\u2026, r; nj \u2265 0, j \u2260 i.Other recurrence relations for the family of multivariable polynomials \u0398The generating relation  yields tni by ni \u2212 ki, i = 1,2,\u2026, r,\u2009 \u2009the right-hand side of the last equality ist1n1 \u22ef trnr completes the proof.From , we haven1,\u2026,nr\u03b1) given explicitly by  in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are as followsExpansions of \u0398ni \u2212 miki instead of ni in the last equality, we havet1n1 \u22ef trnr gives the desired relation.By , we getn1,\u2026,nr\u03b1) in series of Gegenbauer, Hermite, and Laguerre polynomials.In a similar manner, in , using tn\u03b1) and give their several properties.In this section, we discuss some special cases of the family of multivariable polynomials \u0398r = 2.\u2009Equation  is a polynomial of degree ni with respect to the fixed variable xi\u2009\u2009.\u2009 \u2009Thus, Cn1,\u2026,nr\u03b1) is a polynomial of total degree (n1 + \u22ef+nr) with respect to the variables x1,\u2026, xr. Equation  is a polynomial of degree (n1 + \u22ef+nr \u2212 2) with respect to the variables x1,\u2026, xr. In , one hasni, i = 1,2,\u2026, r, is odd; thenFor the polynomials xi = 0, i = 1,2,\u2026, r in .From the theorems and corollaries given in ni \u2265 1,\u2009\u2009i = 1,2,\u2026, r.By Corollaries Cn1,\u2026,nr\u03b1) satisfy the following addition formula:From Cn1,\u2026,nr\u03b1) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are as follows:As a result of Cn1,\u2026,nr\u03b1) given by  have the following hypergeometric representation:FBr)(\u03b1)equality  can be wr = 2, For This case yields that in .Un1,\u2026,nr\u03b1) satisfyni \u2265 mi \u2212 1, i = 1,2,\u2026, r; nj \u2265 0, j \u2260 i.From Corollaries Un1,\u2026,nr\u03b1) have the following addition formula:As result of r-variable analogue of Lagrange polynomials  which ials (see (64) satisfyi = 1,2,\u2026, r andFrom Corollaries Gn1,\u2026,nr\u03b1) have the following addition formula:By Gn1,\u2026,nr\u03b1) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are given byFrom Gn1,\u2026,nr\u03b1).We can discuss some generating functions for the multivariable polynomials Gn1,\u2026,nr\u03b1),\u2009 \u2009the following generating function holds true for k \u2208 \u21150:i = 1,2,\u2026, r, wheren\u03b1,\u03b2) denotes the classical Lagrange polynomials defined by [For the polynomials fined by (72) satisfyni \u2265 i \u2212 1,\u2009 i = 1,2,\u2026, r; nj \u2265 0, j \u2260 i.By Corollaries Hn1,\u2026,nr\u03b1) have the following addition formula:From mi = i, xi = 0, yi = \u2212xi, i = 1,2,\u2026, r in Hn1,\u2026,nr\u03b1).Furthermore, setting ak1,\u2026,kr \u2260 0; pi \u2208 \u2115; ni, \u03bbi, \u03b7i \u2208 \u21150, i = 1,2,\u2026, r, and \u03bb = , \u03b7 = ,w = , then we haveIf"}
{"text": "We present a new comparison principle by introducing a notion of upper quasi-monotone nondecreasing and obtain the practical stability criteria for set valued differential equations in terms of two measures on time scales by using the vector Lyapunov function together with the new comparison principle. Stability theory in the sense of Lyapunov is now well known. Its basic theory and applications can be found in the monographs of Lasalle and Lefschetz  and RoucThe practical stability is a very important problem in the field of application, which deals with the question of whether the system state evolves within certain subsets of the state-space. It is very useful in estimating the worst-case transient and steady-state responses and in verifying pointwise in time constraints imposed on the state trajectories. Thus practical stability is concerned with quantitative analysis as opposed to Lyapunov analysis which is qualitative in nature. There are some relative results for practical stability of various dynamic systems. We can refer to the monograph of Lakshmikantham et al.  and the Recently, the study of set differential equations in a semilinear metric space has gained much attention due to its applicability to multivalued differential inclusions and fuzzy differential equations and its inclusion of ordinary differential systems as a special case , and som\ud835\udd4b be a time scale with t0 \u2265 0 as minimal element and no maximal element. Firstly, we give some relative definitions, which can be found in  denote the set of rd-continuous mappings from \ud835\udd4b to \u211dn.Let f : \ud835\udd4b \u00d7 \u211dn \u2192 \u211dn is said to be right-dense (rd) continuous and is denoted by f \u2208 Crd\u2061 ift, x) with right-dense or maximal t;it is continuous at each  = lim\u2061s,y)\u2192 and lim\u2061y\u2192xf exist at each  with left-dense t.the limits The mapping K(\u211dn) denote the collection of all nonempty, compact, and convex subsets of \u211dn. Define the Hausdorff metricd = inf\u2061[d : y \u2208 A], A, B are bounded sets in \u211dn. We note that K(\u211dn) with this metric is a complete metric space.Let K(\u211dn) is equipped with the natural algebraic operations of addition and nonnegative scalar multiplication, then K(\u211dn) becomes a semilinear metric space which can be embedded as a complete cone into a corresponding Banach space.It is known that if the space A, B, C \u2208 K(\u211dn) and \u03bb \u2208 \u211d+.The Hausdorff metric  satisfieA, B \u2208 K(\u211dn). The set C \u2208 K(\u211dn) satisfying A = B + C is known as the Hukuhara difference of the sets A and B and is denoted by the symbol A \u2212 B.Let F: \ud835\udd4b \u2192 K(\u211dn), denoted by \u0394F, is defined byF is defined asThe forward derivative of F, G : \ud835\udd4b \u2192 K(\u211dn) be differentiable at t; then \u0394(F + G)(t) = \u0394F(t) + \u0394G(t);let \u03b1F)(t) = \u03b1(\u0394F(t)), \u03b1 \u2208 \u211d+.\u0394 are singletons only, then there is only one selector possible, namely, F itself. In this case, the integral reduces to the generalized integral from time scales into (\u211dn). The assumption n = 1 generalizes into the integral for time scales. When \ud835\udd4b is everywhere right-dense, then we have \u0394t = dt, which results in the conventional formulation for integration of set valued functions. Moreover, for a, b, c \u2208 \ud835\udd4b, we have acF(t)\u0394t = \u222babF(t)\u0394t + \u222bbcF(t)\u0394t;\u222babF(t)\u0394t is closed and convex but need not be necessarily compact.\u222bIf the sets in F, G: \ud835\udd4b \u2192 K(\u211dn). Then the Hausdorff distance between F and G, D: \ud835\udd4b \u2192 \u211d+, is \u0394-integrable andLet F: \ud835\udd4b \u2192 K(\u211dn) is set valued function and t \u2208 \ud835\udd4bk. Let \u0394HF(t) be an element of K(\u211dn) (provided it exists) with the property that given any \u03f5 > 0, there exists a neighborhood Ut of t \u22c2\ud835\udd4b for some \u03b4 > 0) such thatt \u2212 h, t + h) \u2208 Ut with 0 \u2264 h < \u03b4, where \u03bc(t) = \u03c3(t) \u2212 t. We call \u0394HF(t) the \u0394H-derivative of F at t. We say that F is \u0394H-differentiable at t if its \u0394H-derivative exists at t. Moreover, we say F is \u0394H-differentiable on \ud835\udd4bk if its \u0394H-derivative exists at each t \u2208 \ud835\udd4bk. The multivalued function \u0394HF(t): \ud835\udd4bk \u2192 K(\u211dn) is then called the \u0394H-derivative of F on \ud835\udd4bk.Assume Let\u2009\u2009F: \ud835\udd4b \u2192 K(\u211dn), \u0394HF(t) \u2208 K(\u211dn)\u2009\u2009(provided it exists). Then some easy and useful relationships concerning the \u0394H-derivative hold.(i)If the\u2009\u2009\u0394H-derivative of\u2009\u2009F\u2009\u2009at\u2009\u2009t\u2009\u2009exists, then it is unique. Hence, the\u2009\u2009\u0394H-derivative is well defined.(ii)Assume\u2009\u2009F:\u2009\u2009\ud835\udd4b \u2192 K(\u211dn) is a multivalued function and let\u2009\u2009t \u2208 \ud835\udd4bk. Then one has the following.(1)If\u2009\u2009F\u2009\u2009is\u2009\u2009\u0394H-differentiable at\u2009\u2009t, then\u2009\u2009F\u2009\u2009is continuous at\u2009\u2009t.(2)If\u2009\u2009F\u2009\u2009is continuous at\u2009\u2009t\u2009\u2009and\u2009\u2009t\u2009\u2009is right-scattered, then\u2009\u2009F\u2009\u2009is\u2009\u2009\u0394H-differentiable at\u2009\u2009t\u2009\u2009with(3)If\u2009\u2009\u2009t\u2009\u2009is right-dense, then\u2009\u2009F\u2009\u2009is\u2009\u2009\u0394H-differentiable at\u2009\u2009t\u2009\u2009if the limits\u2009exist and satisfy the equations (4)If\u2009\u2009F\u2009\u2009is differentiable at\u2009\u2009t, thenF: \ud835\udd4b \u2192 K(\u211dn), \u0394HF(t) \u2208 K(\u211dn) (provided it exists). We consider the two cases \ud835\udd4b = \u211d and \ud835\udd4b = \u2124, where \u2124 stands for the set consisting of all integers. (1)\ud835\udd4b = \u211d, then F: \u211d \u2192 K(\u211dn) is \u0394H-differentiable at t \u2208 \u211d if and only ifIf exists, that is, if and only if F is differentiable in the Hukuhara sense at t. In this case, we have(2)\ud835\udd4b = \u2124, then F: \u2124 \u2192 K(\u211dn) is \u0394H-differentiable at t \u2208 \u2124 withIf where \u0394 is the usual forward multivalued difference operator.Let K(\u211dn)N:X, Y \u2208 Kc(Rn)N, Xi, Yi \u2208 Kc(Rn), i = 1,2,\u2026, N.Consider the spaceF \u2208 Crd\u2061, X \u2208 K(\u211dn)N, X =  such that for each i, i = 1,2,\u2026, N, Xi \u2208 K(\u211dn).Consider the initial value problemFirst of all, we define the following classes of function:In order to discuss the stability of the solution of set valued differential systems , we statV \u2208 Crd\u2061 along with the solutions of DG: \ud835\udd4b \u00d7 \u211d+N \u2192 \u211dN\u2009\u2009(N \u2265 1) is said to be upper quasi-monotone nondecreasing in U, if U, W \u2208 \u211d+N and ||U||M \u2264 ||W||M imply ||G||M \u2264 ||G||M, where ||\u00b7||M = max\u2061i\u2264N1\u2264Ui.A function In the following, we will prove the comparison result in terms of vector Lyapunov functions relative to the set differential system on time scales.H1)(V \u2208 Crd\u2061, V is locally Lipschitzian in X; that is, for X, Y \u2208 K(\u211dn)N, one has A is an N \u00d7 N matrix of nonnegative elements, |V \u2212 V| =  \u2212 V1|, |V2 \u2212 V2|,\u2026, |VN \u2212 VN|). Here, by |V| one means the vector |, |V2|,\u2026, |VN|), where Vi are the components of V, i = 1,2,\u2026, N;H2) = U is the solution ofThen for any solution X(t) of ||M \u2264 ||U0||M.Assume that  X(t) of , one hasU(t) be the solution of  = V) as an application of the properties of Hausdorff metric; we obtain the estimationX(\u03c3(t)) = X(t) + Y(t), where Y(t) is the Hukuhara difference of X(\u03c3(t)) and X(t) for small h > 0 and is assumed to exist, hence,D\u0394+m(t) is the right-derivative of m(t). Hence, we haveV||M \u2264 ||U0||M, we can obtain thatLet ution of  for t \u2265 Now, we list some definitions about stability which will be used in the following discussion.h0, h \u2208 \u0393, t \u2208 \ud835\udd4b. Then one says that (i)h0 is finer than h if there exists a \u03c1 > 0 and a function \u03d5 \u2208 \ud835\udc9e\ud835\udca6 such that(ii)h0 is uniformly finer than h if in (i) \u03d5 is independent of t.Let \u03bb, A, B be positive constants . The system ||M < A, t \u2265 t0 for t0, t \u2208 \ud835\udd4b, where X(t) = X is any solution of ||M < B, t \u2265 t0 + T, t \u2208 \ud835\udd4b;practically quasi-stable if for any strongly practically stable if (i) and (ii) hold simultaneously;\u03f5 > 0 there exists T0 > 0 such that t0 + T0 \u2208 \ud835\udd4b and ||X0|| < \u03bb implies ||X(t)|| < \u03f5, t \u2265 t0 + T0.practically asymptotically stable if (i) holds and for any Let e system  is said \u03bb, A, B be positive constants , h, h0 \u2208 \u0393. The system ((PS1)h0, h)-practically stable if for any 0 < \u03bb < A, the condition h0 < \u03bb implies h) < A, t \u2265 t0, for some t0, t \u2208 \ud835\udd4b, where X(t) = X is any solution of h0, h)-practically quasi-stable if for any \u03bb, B, T > 0 and some t0 \u2208 \ud835\udd4b with t0 + T \u2208 \ud835\udd4b, the condition h0 < \u03bb implies h) < B, t \u2265 t0 + T, t \u2208 \ud835\udd4b;((PS3)h0, h)-strongly practically stable if (PS1) and (PS2) hold simultaneously;((PS4)h0, h)-practically asymptotically stable if (PS1) holds and for any \u03f5 > 0 there exists T0 > 0 such that t0 + T0 \u2208 \ud835\udd4b and h0 < \u03bb implies h) < \u03f5, t \u2265 t0 + T0., Q0, Q \u2208 \u03a3. Then we say that the system -practically stable if for any 0 < \u03bb < A, the condition Q0(||U0||M) < \u03bb implies Q(||U(t)||M) < A, t \u2265 t0, t \u2208 \ud835\udd4b, where U(t) = U is any solution of  \u2264 \u03d5), \u03d5 \u2208 \ud835\udca6, where h0 < \u03bb;A3)(V \u2208 Crd\u2061, such that V is locally Lipschitzian in X for each right-dense t \u2208 \ud835\udd4b and for a, b \u2208 \ud835\udca6. It holds thatthere exists A4) \u2208 S,for  is the solution of (A5)\u2009(\u03d5(\u03bb) < A, a(\u03bb) < b(A). Assume that ution of ;\u2009(A5)\u03d5-practical stability properties of , b(A)), we haveA2) and (A5), it follows that h \u2264 \u03d5) < \u03d5(\u03bb) < A. We claim that h < A.Assume that  is practX(t) of  < \u03bb and t1 > t0, t1 \u2208 \ud835\udd4b, such that h) \u2265 A, h) < A, t0 \u2264 t < t1. As h0 < \u03bb, A2), (A3) implies thath0 < \u03bb implies h < A, t \u2265 t0.Indeed, if this were not true, there would exist a solution X(t) of  with h0h0, h)-strongly practically stable. For given positive numbers \u03bb, A, B, and T, suppose that (a(\u03bb), b(A), b(B), and T; this means we only need to prove -practical quasi-stability of system (U0||M < a(\u03bb) implies ||U(t)||M < b(B), t \u2265 t0 + T with t0 + T \u2208 \ud835\udd4b.Next we prove that system  is )||M < a(\u03bb), if h0 < \u03bb, we have b) \u2264 ||V||M \u2264 ||U(t)||M < b(B) for all t \u2265 t0 + T if h0 < \u03bb; thus we have h) < B, t \u2265 t0 + T provided h0 < \u03bb. Hence system -strongly practically stable.From the foregoing argument, since ||e system  is -practically asymptotically stable. Now, let us suppose that  < \u03bb implies h) < B, t \u2265 t0 + T0 for system ||M < a(\u03bb) whenever h0 < \u03bb, we obtaint \u2265 t0 + T0, if h0 < \u03bb. Thus we have h < B, t \u2265 t0 + T0, provided h0 < \u03bb. Hence system -strongly practically stable.Finally, we show that system  is  is replaced by A6)-practical stability properties of system -practical stability properties of the system -practically stable. Then we have (a(\u03bb), b(A)), such thath0, h)-practical stability of (A2), (A6), we can show thatA6), we haveAssume that  is (Q0,ility of  does not"}
{"text": "We consider the class of those distributions that satisfy Gauss's principle (the maximum likelihood estimator of the mean is the sample mean) and have a parameter orthogonal to the mean. It is shown that this so-called \u201cmean orthogonal class\u201d is closed under convolution. A previous characterization of the compound gamma characterization of random sums is revisited and clarified. A new characterization of the compound distribution with multiparameter Hermite count distribution and gamma severity distribution is obtained. Gauss's principle by Campbell } exists and denotes the variance by \u03c32 = \u03c32. The standard regularity conditions for maximum likelihood estimation are supposed to hold. The vector X =  denotes a random sample of size n, which realizes the random variable X, and gk = gk, k = 1,\u2026, m, h = h, such that the following equivalent partial differential equations hold :(1):(1)X be\u03bc is called orthogonal to the parameter vector \u03b8, denoted by \u03bc\u22a5\u03b8, if one has E[(\u22022\u2113/\u2202\u03bc\u2202\u03b8k)] = E[(\u2202\u2113/\u2202\u03bc)(\u2202\u2113/\u2202\u03b8k)] = 0, k = 1,\u2026, m.The mean \u03bc, \u03b8). In this respect, one is interested in the subclass mean orthogonal property \u03bc\u22a5\u03b8. This so-called mean orthogonal class is characterized as follows.The original motivation for parameter orthogonality is improvement of maximum likelihood estimation by reparameterization. In the class tions in  are elemX be a random variable with cgf C satisfying the above assumptions. Then, one has Let This is shown in H\u00fcrlimann . Hudson  = \u2211n=0\u221ep(n)sn of N, it is very useful to consider the associated so-called cumulant pgf defined byTo show this, some preliminaries are required. First, we review conditions under which g(k), k = 1,2,\u2026, is the unique solution of the system of equations  \u2265 0, k = 1,2,\u2026, the distribution of N is compound Poisson with parameter \u03bbN and severity distribution h(k) = g(k)/\u03bbN, k = 1,2,\u2026. Otherwise, one speaks of the so-called pseudo-compound Poisson representation of the distribution.The sequence s  of the form  to , and set \u03bbN = \u2212ln\u2009Pr(N = 0). Then \u03bcN = \u03bcN\u22a5\u03b8N is equivalent to the following conditions: Let the form . SupposeThe condition  is a res\u03c3N2\u00b7\u2202CN(tatements  and 11)(12)\u03c3N2\u00b7\u2202atements .\u03c3N2 \u00b7 (\u2202\u03bbN/\u2202\u03bcN) = \u03bcN holds.If \u03bbN = G(1) = \u2211k=1\u221eg(k), CN\u2032(0) = \u2211k=1\u221ekg(k) = \u03bcN. Now, using (\u03c3N2 \u00b7 (\u2202g(k)/\u2202\u03bcN) = kg(k), k = 1,2,\u2026. It follows that \u03c3N2 \u00b7 (\u2202\u03bbN/\u2202\u03bcN) = \u2211k=1\u221e\u03c3N2 \u00b7 (\u2202g(k)/\u2202\u03bcN) = \u2211k=1\u221ekg(k) = \u03bcN. The representation  implies w, using  one seesw, using  is equivMY(t) = exp\u2061{CY(t)} be the moment generating function of Y. Expressed in terms of the mean scaled severity Z = Y/\u03bc one has MY(t) = MZ(\u03bct). The relationship (CN(t) yields the series expansion:\u03c32 \u00b7 (\u2202C/\u2202\u03bc) \u2212 ((\u2202C/\u2202t) \u2212 \u03bc) = 0 is satisfied. With the series representation for C(t) and the assumption \u03c32 \u00b7 (\u2202\u03bbN/\u2202\u03bc) = \u03bc, this equation is equivalent to the following condition (use that MZ(t) does not depend on \u03bc):\u03c32 \u00b7 (\u2202\u03bbN/\u2202\u03bc) = \u03bc, implies thatMZ(t) = (1 \u2212 \u03ba2t)\u03b1\u2212. Since \u03ba2\u03bc = \u03bcY/\u03b1, one sees thatLet tionship  for the natural parameterization  of the compound gamma distribution. It is interesting to obtain explicit parameters orthogonal to the means of N, Y, and X. By the assumption \u03bcN = \u03bcN\u22a5\u03b8N, and since Y is gamma distributed, one has \u03bcY\u22a5\u03b1. It remains to construct a parameter vector orthogonal to the mean of X such that\u03b8 = \u03b8 must be determined. This task can be solved in a unified way for a lot of counting distributions , \u03b8N > 0, p \u2208 , be a negative binomial random variable. Its cumulant pgf (G(s) = \u2212\u03b8N \u00b7 ln\u2061(1 \u2212 p \u00b7 s), \u03bbN = \u2212\u03b8N \u00b7 ln\u2061(1 \u2212 p). One has the following identity (see [s = 1 that p(\u2202\u03bbN/\u2202p) = \u03bcN. Together, this shows that (\u03bcN = \u03b8N \u00b7 p/(1 \u2212 p)\u22a5\u03b8N. Now, by \u03c32 \u00b7 (\u2202\u03bbN/\u2202\u03bc) = \u03bc is satisfied. Written in terms of the parameter \u03b1 = (\u03bcN\u03ba2)\u22121, \u03ba = \u03c3/\u03bc, the latter equation is equivalent to the condition (1/\u03bcN)\u00b7(\u2202\u03bbN/\u2202\u03bc) = \u03b1/\u03bc. With \u2202\u03bbN/\u2202\u03bc = (\u2202\u03bbN/\u2202p)\u00b7(\u2202p/\u2202\u03bc) = (\u03bcN/p)\u00b7(\u2202p/\u2202\u03bc) one obtains the differential equation (1/p)\u00b7(\u2202p/\u2202\u03bc) = \u03b1/\u03bc, which has the solution p = \u03b8 \u00b7 \u03bc\u03b1 for some \u03b8. In the coordinates , this constant is equal to\u03bc = \u03bcN \u00b7 \u03bcY\u22a5.Let lant pgf  reads G, B2(p),\u2026, be independent and identically distributed Bernoulli random variables with probability of success p \u2208 , i = 1,2,\u2026. Then N(p) = \u2211i=1NBi(p)\u2009(X(p) = 0\u2009\u2009if\u2009\u2009N = 0) is called an independent p-thinning of N.Let F be a family of count distributions. It is called closed under binomial subsampling if, for any random variable N with distribution in F, all its independent p-thinnings, for all p \u2208  and factorial cumulant generating function (fcgf) CNf(s) = ln\u2061P(s + 1). For any integer n \u2265 1 the nth factorial cumulant of N is defined and denoted by \u03ban)( = dnCNf(s)/dsn|s=0.Let There is only one count distribution family closed under convolution and binomial subsampling.F be a family of count distributions parameterized by its r first factorial cumulants \u03b8 = (\u03ba(1),\u2026, \u03bar) and assume that its pgf P(s) is continuous in \u03b8 over its parameter space. Then F is closed under convolution and binomial subsampling if and only if the pgf is of the formLet See Puig and Valero , proof or = 1 corresponds to the Poisson distribution, r = 2 is the Hermite distribution , k = 1,2,\u2026, r, solves the system in  > 0, g(r) \u2265 0, g(k) = 0, k = 2,\u2026, r \u2212 1 is the generalized Hermite by Gupta and Jain [g(k), k = 1,2,\u2026, r, under which (g(k) < 0 is preceded by a positive value and followed by at least two positive values. In particular, if at least g(1), g(r \u2212 1), g(r) are nonzero, then g(1) > 0, g(r \u2212 1) > 0, g(r) > 0, are necessary conditions for (g(k) \u2265 0 for k = 1,2,\u2026, r, then the multiparameter Hermite is compound Poisson with parameter \u03bbN = \u2211k=1rg(k) and severity h(k) = g(k)/\u03bbN, thus infinitely divisible by Feller [Some comments are in order. The case n . For arWestcott . In termystem in , that isstem in (n\u00b7p(n)=\u2211kd Valero . The speand Jain . The muland Jain  family oestcott [P(s)=exp\u2061estcott [P(s)=exp\u2061ck(\u03b8), k = 1,2,\u2026, r, be continuous real functions in the parameter vector \u03b8 over some parameter space, and set g(k) = ck(\u03b8) \u00b7 pk, k = 1,2,\u2026, r, for a parameter p > 0. Assume that the cumulant pgf G(s) = \u2211k=1rck(\u03b8)\u00b7(ps)k defines a feasible multiparameter Hermite random variable N of order r over the parameter space. Then \u03bcN = \u03bcN = \u2211k=1rk \u00b7 ck(\u03b8) \u00b7 pk\u22a5\u03b8.Let \u03bbN = G(1) = \u2211k=1rck(\u03b8) \u00b7 pk. Then one hasSet ows that  is satis\u03ba(1), \u03ba(2). Since \u03ba(1) = \u03bcN, \u03ba(2) = \u03c3N2 \u2212 \u03bcN, it can equivalently be parameterized by its mean \u03bcN and variance \u03c3N2. Consider a parameterization p > 0, \u03b8N > 0 such that g(k) = \u03b8N \u00b7 pk, k = 1,2. There exists a one-to-one mapping between  and . Since \u03bcN = g(1) + 2g(2),\u03c3N2 = g(1) + 4g(2), it is determined by the coordinate transformation:G(s) = \u03b8 \u00b7 (ps + p2s2) defines a feasible two-parameter Hermite distribution such that the corresponding random variable belongs to \u03bcN = \u03bcN = \u03b8N \u00b7 p \u00b7 (1 + 2p)\u22a5\u03b8N. Since \u03c3N2 > \u03bcN one notes that the Hermite distribution is necessarily overdispersed. As noted by Puig and Valero [r \u2265 2. Therefore, it should be useful to analyze data with this property .Suppose the Hermite distribution is parameterized by its first two factorial cumulants d Valero  overdispWe are ready for the following new characterization result.N be a counting random variable parameterized by its r first factorial cumulants \u03be = (\u03ba(1),\u2026, \u03bar) and assume that its cgf CN is continuous in \u03be over its parameter space and set \u03bcN = \u03ba(1), \u03bbN = \u2212ln\u2009Pr(N = 0). Suppose the cgf CY(t) of the severity Y exists, and let C(t) = CN(CY(t)) be the cgf of the random sum X = \u2211i=1NYi. Assume the cgf of the mean scaled severity CZ(t) = CY(t/\u03bc) is functionally independent of \u03bc, and set \u03b1 = (\u03bcN\u03ba2)\u22121, \u03ba = \u03c3/\u03bc. Assume N is closed under convolution and binomial subsampling, \u03c32 \u00b7 (\u2202\u03bbN/\u2202\u03bc) = \u03bc. Then N is a multiparameter Hermite distribution of order r, and Y is gamma distributed with cgf CY(t) = \u03b1 \u00b7 ln\u2061{\u03b1/(\u03b1 \u2212 \u03bcYt)}. Furthermore, there exists a parameterization  of N such that its cumulant pgf reads G(s) = \u2211k=1rck(\u03b8N)\u00b7(ps)k. One has \u03bcN = \u03bcN = \u2211k=1rk \u00b7 ck(\u03b8N) \u00b7 pk\u22a5\u03b8N, \u03bcY\u22a5\u03b1, \u03bc = \u03bcN \u00b7 \u03bcY\u22a5, and, in the coordinates , the constant \u03b8 is equal toLet \u03bcN, \u03bcY, \u03bc follows along the same arguments as in The result follows by combining Theorems"}
{"text": "Our result generalizes some useful results in the literature. We provide an example to support our result.We define the concept of  A large variety of the problems of analysis and applied mathematics relate to finding solutions of nonlinear functional equations which can be formulated in terms of finding the fixed points of nonlinear mappings. Heilpern  first inFuzzy sets\u201d . There are basically two understandings of the meaning of L, one is when L is a complete lattice equipped with a multiplication \u2217 operator satisfying certain conditions as shown in the initial paper \u03b1L, where \u03b1L \u2208 L\u2216{0L}. The point z \u2208 X is called a common L-fuzzy fixed point of S and T if z \u2208 [Sz]\u03b1L\u2229[Tz]\u03b1L.Let . We say that T is \u03b2-admissible if for all x, y \u2208 X we haveLet X be a nonempty set, T : X \u2192 2X, where 2X is a collection of subset of X, \u03b2 : X \u00d7 X \u2192 \u03b1L(x), where \u03b1L(x) \u2208 L\u2216{0L}, with \u03b2 \u2265 1, we have \u03b2 \u2265 1 for all z \u2208 [Ty]\u03b1L(y) \u2260 \u03d5, where \u03b1L(y) \u2208 L\u2216{0L};for each  (ii)x \u2208 X and y \u2208 [Tx]\u03b1L(x), where \u03b1L(x) \u2208 L\u2216{0L}, with \u03b2 \u2265 1, we have \u03b2 \u2265 1 for all z \u2208 [Sy]\u03b1L(y) \u2260 \u03d5, where \u03b1L(y) \u2208 L\u2216{0L}.for each Let  is \u03b2\u2131L-admissible, then  is also \u03b2\u2131L-admissible.It is easy to see that if  be a complete metric space, \u03b2 : X \u00d7 X \u2192 \u03b1L(x), [Tx]\u03b1L(x) are nonempty closed bounded subsets of X and for x0 \u2208 X, there exists x1 \u2208 [Sx0]\u03b1L(x0) with \u03b2 \u2265 1.For each (b) x, y \u2208 X, we haveFor all \u2009a1, a2, a3, a4, and a5 are nonnegative real numbers and \u2211i=15ai < 1 and either a1 = a2 or a3 = a4.where (c) S, T) is \u03b2\u2131L-admissible pair.((d) xn} \u2208 X, such that \u03b2 \u2265 1 and xn \u2192 x, then \u03b2 \u2265 1.If {Let (z \u2208 X such that z \u2208 [Sz]\u03b1L(z)\u2229[Tz]\u03b1L(z).Then there exists a1 + a3 + a5 = 0,a2 + a4 + a5 = 0,a1 + a3 + a5 \u2260 0 and a2 + a4 + a5 \u2260 0.We will prove the above result by considering the following three cases:Case 1. For x0 \u2208 X in condition (a), there exist \u03b1L(x0) \u2208 L\u2216{0L} and x1 \u2208 [Sx0]\u03b1L(x0) such that \u03b2 \u2265 1 and also there exists \u03b1L(x1) \u2208 L\u2216{0L} such that [Sx0]\u03b1L(x0) and [Tx1]\u03b1L(x1) are nonempty closed bounded subsets of X. From Now, inequality  implies a1 + a3 + a5 = 0 together with the fact that d) = 0, we getUsing x1 \u2208 [Tx1]\u03b1L(x1), which further implies thatIt follows that x0 \u2208 X and x1 \u2208 [Sx0]\u03b1L(x0) such that \u03b2 \u2265 1, we have \u03b2 \u2265 1 for all z \u2208 [Tx1]\u03b1L(x1). Since x1 \u2208 [Tx1]\u03b1L(x1), therefore \u03b2 \u2265 1 and henceBy condition (c), for Again, inequality  implies a1 + a3 + a5 = 0 and d) = 0, we getx1 \u2208 [Sx1]\u03b1L(x1) and henceSince Case 2. For x0 \u2208 X in condition (a), there exist \u03b1L(x0) \u2208 L\u2216{0L} and x1 \u2208 [Sx0]\u03b1L(x0) such that \u03b2 \u2265 1 and also there exists \u03b1L(x1) \u2208 L\u2216{0L} such that [Sx0]\u03b1L(x0) and [Tx1]\u03b1L(x1) are nonempty closed bounded subsets of X. By condition (c), we have \u03b2 \u2265 1 for all x2 \u2208 [Tx1]\u03b1L(x1). From a2 + a4 + a5 = 0 together with the fact that d) = 0, we getUsing x2 \u2208 [Sx2]\u03b1L(x2), which further implies thatIt follows that \u03b2 \u2265 1, and henceBy condition (c), we have Again, inequality  implies a2 + a4 + a5 = 0 and d) = 0, we getx2 \u2208 [Tx2]\u03b1L(x2) and henceSince Case 3. Let \u03bb = ((a1 + a3 + a5)/(1 \u2212 a2 \u2212 a3)) and \u03bc = ((a2 + a4 + a5)/(1 \u2212 a1 \u2212 a4)). Next, we show that if a1 = a2 or a3 = a4, then 0 < \u03bb\u03bc < 1.a3 = a4, then \u03bb, \u03bc < 1 and so 0 < \u03bb\u03bc < 1. Now if a1 = a2, thenx1 \u2208 X, there exists \u03b1L(x1) \u2208 L\u2216{0L} such that [Tx1]\u03b1L(x1) is a nonempty closed bounded subset of X. Since a1 + a3 + a5 > 0, by x2 \u2208 [Tx1]\u03b1L(x1) such thatx2 \u2208 X, there exists \u03b1L(x2) \u2208 L\u2216{0L} such that [Sx2]\u03b1L(x2) is a nonempty closed bounded subset of X. Since a2 + a4 + a5 > 0, by x3 \u2208 [Sx2]\u03b1L(x2) such thatx0 \u2208 X and x1 \u2208 [Sx0]\u03b1L(x0) such that \u03b2 \u2265 1, we have \u03b2 \u2265 1 for x2 \u2208 [Tx1]\u03b1L(x1). So we havex3 \u2208 X, there exists \u03b1L(x3) \u2208 L\u2216{0L} such that [Tx3]\u03b1L(x3) is a nonempty closed bounded subset of X. From x4 \u2208 [Tx3]\u03b1L(x3) such thatx1 \u2208 X and x2 \u2208 [Tx1]\u03b1L(x1) such that \u03b2 \u2265 1, we have \u03b2 \u2265 1 for x3 \u2208 [Sx2]\u03b1L(x2). So we havexn} in X such thatk = 0,1, 2,\u2026,m < n, we have\u03bb\u03bc < 1, so by Cauchy's root test, we get \u2211(2i + 1)(\u03bb\u03bc)i and \u22112i(\u03bb\u03bc)i are convergent series. Therefore, {xn} is a Cauchy sequence in X. Now, from the completeness of X, there exists z \u2208 X such that xn \u2192 z as n \u2192 \u221e. By condition (d), we have \u03b2 \u2265 1 for all n \u2208 \u2115. Now, we haven \u2192 \u221e, we have d) = 0. It implies that z \u2208 [Sz]\u03b1L(z). Similarly, by usingz \u2208 [Tz]\u03b1L(z). Therefore, z \u2208 [Sz]\u03b1L(z)\u2229[Tz]\u03b1L(z). This completes the proof.If ies thatd = |x \u2212 y|, whenever x, y \u2208 X; then  is a complete metric space. Let L = {\u03b4, \u03c9, \u03c4, \u03ba} with \u03b4\u2009\u227eL\u2009\u03c9\u2009\u227eL\u2009\u03ba, \u03b4\u2009\u227eL\u2009\u03c4\u2009\u227eL\u2009\u03ba, \u03c9 and \u03c4 are not comparable; then  is a complete distributive lattice. Define a pair of mappings S, T : X \u2192 \u2131L(X) as follows:\u03b2 : X \u00d7 X \u2192 \u03b1L(0)\u2229[T0]\u03b1L(0).Let X, d) be a complete metric space, \u03b2 : X \u00d7 X \u2192 \u03b1(x) are nonempty closed bounded subsets of X and for x0 \u2208 X, there exists x1 \u2208 [Sx0]\u03b1(x0) with \u03b2 \u2265 1.For each (b) x, y \u2208 X, we haveFor all \u2009a1, a2, a3, a4, and a5 are nonnegative real numbers and \u2211i=15ai < 1 and either a1 = a2 or a3 = a4.where (c) S, T) is \u03b2\u2131-admissible pair.((d) xn} \u2208 X, such that \u03b2 \u2265 1 and xn \u2192 x then \u03b2 \u2265 1.If {Let (z \u2208 X such that z \u2208 [Sz]\u03b1(z)\u2229[Tz]\u03b1(z).Then there exists L-fuzzy mapping A : X \u2192 \u2131L(X) defined by\u03b1L \u2208 L\u2216{0L}, we haveConsider an \u03b2 = 1 for all x, y \u2208 X in If we set X, d) be a complete metric space and S, T fuzzy mappings from X into \u2131(X) satisfying the following conditions:(a) x \u2208 X, there exists \u03b1(x) \u2208 , [Tx]\u03b1(x) are nonempty closed bounded subsets of X;for each (b) x, y \u2208 X, we havefor all \u2009a1, a2, a3, a4, and a5 are nonnegative real numbers and \u2211i=15ai < 1 and either a1 = a2 or a3 = a4.where Let (z \u2208 X such that z \u2208 [Sz]\u03b1(z)\u2229[Tz]\u03b1(z).Then there exists"}
{"text": "Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and chaotic behaviors are observed. Finally, using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of the periodic solutions.A modified Leslie-Gower predator-prey system with two delays is investigated. By choosing  The parameters r1 and r2 are the intrinsic growth rates of the prey and the predator. The value K is the carrying capacity of the prey, and \u03b3x takes on the role of a prey-dependent carrying capacity for the predator; the parameter \u03b3 is a measure of the quality of the prey as food for the predator. However, this model has attracted the attention of some authors  \u2192 R2 such that\u03c6 = T \u2208 C, defineU = (X(t), Y(t))T and Ut(\u03b8) = U(t + \u03b8), \u03b8 \u2208 . For \u03c8 \u2208 C*), define A(0) = A and the adjoint operator A* of A as\u03b7T is the transpose of the matrix \u03b7.In this section, we show that the system undergoes the Hopf bifurcation for different combinations of t al. in , we studn system  can be w system (8U\u02d9(t)= and \u03c8 \u2208 C*), in order to normalize the eigenvectors of operator A and adjoint operator A*, we define a bilinear inner product\u03b7(\u03b8) = \u03b7.For i\u03c90\u03c402\u2032 are eigenvalues of A, they will also be the eigenvalues of A*. The eigenvectors of A and A* are calculated corresponding to the eigenvalues +i\u03c90\u03c402\u2032 and \u2212i\u03c90\u03c402\u2032.Since \u00b1q(\u03b8) = Tei\u03c90\u03c402\u2032\u03b8 is the eigenvector of A corresponding to +i\u03c90\u03c402\u2032; q*(s) = (1/D)Tei\u03c90\u03c402\u2032s is the eigenvector of A* corresponding to \u2212i\u03c90\u03c402\u2032 andR = (R(1), R(2)) \u2208 R2 and S = (S(1), S(2)) \u2208 R2 are constant vectors, computed asFollowing the algorithms explained in Hassard et al. , we can W20(\u03b8) and W11(\u03b8); then gij is determined by the parameters and delays \u03c402\u2032 and \u03c41*. Thus, we can compute the following quantities:\u03c42 = \u03c402\u2032 and when Re{\u03bb\u2032(\u03c402\u2032)} > 0 which can be stated as follows: \u03bc2 gives the direction of the Hopf bifurcation: if \u03bc2 > 0\u2009\u2009(\u03bc2 < 0), the Hopf bifurcation is supercritical ;\u03b22 determines the stability of bifurcating periodic solution: the periodic solutions are stable (unstable) if \u03b22 < 0\u2009\u2009(\u03b22 > 0);T2 denotes the period of bifurcating period solutions: if T2 > 0\u2009\u2009(T2 < 0), periodic solutions increase (decrease).As a result, we know r1 = 0.8, r2 = 1, a = 1.3, K = 0.7, \u03b3 = 1, and m = 0.5. It is easy to show that system . By calculation, when \u03c41 = 0, the critical delay for \u03c42 is obtained as \u03c402 = 1.3507 and \u03c401 = 5.8228 when \u03c42 = 0.To demonstrate the algorithm for determining the existence of the Hopf bifurcation in x\u02d9(t)=0.8E\u2217 is asymptotically stable at \u03c41 = 0, \u03c42 = 1.1 < \u03c402 = 1.3507, while from E\u2217 loses stability and the Hopf bifurcation occurs at \u03c41 = 0, \u03c42 = 1.5 > \u03c402 = 1.3507. From E\u2217 is asymptotically stable when \u03c41 = 2.8 < \u03c401 = 5.8228, \u03c42 = 0, while from E\u2217 loses stability and the Hopf bifurcation occurs when \u03c41 = 6.5 > \u03c401 = 5.8228, \u03c42 = 0.We can see from \u03c41 = 1.28, when \u03c42 = 1.32 < \u03c402\u2032 = 1.9507, E\u2217 is also stable . Further, the method we used here is based on the global Hopf bifurcating theorem for general functional differential equations introduced by Wu [\u03c4 = \u03c42 and write system (zt = (zt1(\u03b8), zt2(\u03b8))T = (z1(t + \u03b8), z2(t + \u03b8))T \u2208 C. Following the work of Wu [\u2009\u2009X = C,\u2009\u2009Cl{(z(t), \u03c4, p) \u2208 X \u00d7 R+ \u00d7 R+, z(t) is a p-periodic solution of  in [m = 1,2, 3,\u2026, where mth crossing number of Assume that 1\u2013A4) in . Denote z-space and onto p-space are bounded, then the projection of \u03c4-space is unbounded. Further, we show that the projection of \u03c4-space is away from zero; then the projection of \u03c4-space must include [\u03c4, \u221e). Following this idea, we can prove our results on the global continuation of the local Hopf bifurcation.It is well known that if (ii) of the theorem is not true, then B + E > 0 hold, nontrivial periodic solutions of (If (tions of  are unifx(t), y(t) be a nontrivial solution of system  at t = 0 with \u03c6(0) > 0, \u03c8(0) > 0. Then it follows from , y(t)) is a nontrivial solution of (x(t) > 0, y(t) > 0, then we havex(t) < x(t \u2212 \u03c41)er1\u03c41 for t > \u03c41; then if t > \u03c41, we obtain\u03b5 > 0, there exists a T > 0 such that when t > T, we havet > T + \u03c4,y(t) < \u03b3(1 \u2212 m)\u03b4er1\u03c4 for sufficiently large t. Thus, the nontrivial periodic solutions lying in the first quadrant of system  < K, 0 < y(t) < \u03b3K(1 \u2212 m), respectively. System ; we defineV is well defined and continuous for all x(t) > 0, y(t) > 0. The function V satisfiesx\u2217, y\u2217) is the only extremum of the function V in the first quadrant. It is easy to see that the point  is a minimum, sincex\u2217, y\u2217) is the global minimum; that is,x(t) > 0, y(t) > 0.Assume that system  has a nol systemx\u02d9(t)=r1xatisfies\u2202V\u2202x=1x < x(t), we haveV satisfies Lyapunov's asymptotic stability theorem; we conclude thatCalculate the derivative of f system . Use Razt system  has periB + E > 0 hold, let \u03c90 and \u03c4j2\u2032\u2009\u2009 be defined in \u03c4 > \u03c4j2\u2032\u2009\u2009(j \u2265 1), system , j = 1,2,\u2026, are isolated centers.It is easy to know that the characteristic matrix of system  at the p\u03c4 \u2212 \u03c4j2\u2032| \u2264 \u03b4 and \u2208\u2202\u03a9\u03f5, then the necessary and sufficient conditions for det\u2061(\u03b7 + i(2\u03c0/p))) = 0 are \u03b7 = 0, \u03c4 = \u03c4j2\u2032, and p = 2\u03c0/\u03c90.Letlz*,\u03c4j2\u2032,2\u03c0/\u03c90) in \u03a3 is nonempty. Meanwhile, we havelz*,\u03c4j2\u2032,2\u03c0/\u03c90). Noting that 2\u03c0/\u03c90 < \u03c4j2\u2032 and applying p < \u03c4* for  belonging to lz*,\u03c4j2\u2032,2\u03c0/\u03c90) is bounded, which is a contradiction. This implies that the projection of lz*,\u03c4j2\u2032,2\u03c0/\u03c90) and show the global existence of the periodic solutions.In this paper, we investigate the effect of the time delays f system  and deri"}
{"text": "This notion was introduced independently by Kadison   d on an  Sourour  proved t\ud835\udc9c is a local derivation. But the converse is in general not true. Kadison [Of course, every derivation on an algebra  Kadison  construc\ud835\udc9c which is generated by its idempotents into any \ud835\udc9c-bimodule is a derivation. This motivated us to study local generalized -derivations on algebras generated by their idempotents.In the last few decades a lot of work has been done on local mappings on some algebras. The results show that, in many important cases, local mappings of some certain class of transformations on a given algebra are global (see ). In the\ud835\udc9c will be an algebra and \u2133 will be an \ud835\udc9c-bimodule. A linear mapping d : \ud835\udc9c \u2192 \u2133 is called a derivation ifg : \ud835\udc9c \u2192 \u2133 is a generalized derivation if there exists a derivation d : \ud835\udc9c \u2192 \u2133 such thatg is a generalized derivation associated with a derivation d . On the other hand, Nakajima [g : \ud835\udc9c \u2192 \u2133 be a linear mapping and m an element of \u2133. A pair  is called a generalized derivation if\ud835\udc9c has a unit 1 \u2208 \ud835\udc9c, then m = \u2212g(1) and g is a generalized derivation if it satisfiesg : \ud835\udc9c \u2192 \u2133 is a generalized derivation if and only if it satisfies the above condition. We refer the readers to [Before continuing, let us fix the notation and write some basic definitions which we will need in our further investigation. Throughout the paper, ,b\u2208\ud835\udc9c.In , Bre\u0161ar Nakajima  defined aders to , where t\u03b1, \u03b2 be endomorphisms of a unital algebra \ud835\udc9c and I\ud835\udc9c the identity map on \ud835\udc9c. Motivated by the above notions, we define -derivations as linear mappings d : \ud835\udc9c \u2192 \u2133 satisfyingg : \ud835\udc9c \u2192 \u2133 is called a generalized -derivation ifI\ud835\udc9c, I\ud835\udc9c)-derivation -derivation, resp.) is just a derivation . The next example will show that there exist -derivations which are not derivations.Let \ud835\udc9c be an algebra with a nontrivial central idempotent e. Let us define d(a) = ea, \u03b1(a) = a \u2212 ea for all a \u2208 \ud835\udc9c, and \u03b2 = I\ud835\udc9c. Then d is an -derivation which is not a derivation since d(ee) = eee \u2260 2eee = (ee)e + e(ee) = d(e)e\u2009\u2009+\u2009\u2009ed(e). Moreover, if \ud835\udc9c is a semiprime algebra and \u03b1 \u2260 I\ud835\udc9c is an endomorphism of \ud835\udc9c, then d = I\ud835\udc9c \u2212 \u03b1 is an -derivation but not a derivation.Let \u03b1, \u03b2)-derivations -derivations, resp.) can be self-explanatory, A linear mapping g : \ud835\udc9c \u2192 \u2133 is called a local generalized -derivation -derivation, resp.) if for every a \u2208 \ud835\udc9c there exists a generalized -derivation -derivation, resp.) ga : \ud835\udc9c \u2192 \u2133 depending on a such that g(a) = ga(a).The definition of local generalized )g(eaf)(1 \u2212 \u03b1(f)) = 0,(1 \u2212 g(eaf) = g(ea)\u03b1(f) + \u03b2(e)g(af) \u2212 \u03b2(e)g(a)\u03b1(f). Let a \u2208 \ud835\udc9c and all idempotents e, f \u2208 \ud835\udc9c. Thena \u2208 \ud835\udc9c and all idempotents e, f \u2208 \ud835\udc9c. Therefore,a \u2208 \ud835\udc9c and all idempotents e, f \u2208 \ud835\udc9c.Obviously, (ii) implies (i). So, assume that (i) holds for every g is a linear mapping from an algebra \ud835\udc9c into an \ud835\udc9c-bimodule \u2133 such that g(eaf) = g(ea)\u03b1(f) + \u03b2(e)g(af) \u2212 \u03b2(e)g(a)\u03b1(f) for every a \u2208 \ud835\udc9c and all idempotents e, f \u2208 \ud835\udc9c, thena \u2208 \ud835\udc9c and all idempotents e1,\u2026, em, f1,\u2026, fn \u2208 \ud835\udc9c.If n, we first prove that for every a \u2208 \ud835\udc9c and all idempotents e, f1,\u2026, fn \u2208 \ud835\udc9c,n = 1 is clear. Assume now that  = \u2110, \u03b2(\u2110) = \u2110, and let g : \ud835\udc9c \u2192 \u2133 be a linear mapping. If for every a \u2208 \ud835\udc9c and all idempotents e, f \u2208 \ud835\udc9c,g is a generalized -derivation. In particular, if g(1) = 0, then g is an -derivation.Let The idea of the proof comes from [p, q \u2208 \u2110 be arbitrary elements. Then, according to  = \u2110, it follows thata \u2208 \ud835\udc9c. Furthermore, if a, b \u2208 \ud835\udc9c and q \u2208 \u2110, thena, b \u2208 \ud835\udc9c. If g(1) = 0, then g is, obviously, an -derivation.Let rding to  and Lemmhand, by , we haveg be a local generalized -derivation from an algebra \ud835\udc9c into an \ud835\udc9c-bimodule \u2133. Suppose that a \u2208 \ud835\udc9c is an arbitrary element and suppose that e, f \u2208 \ud835\udc9c are idempotents. Then there exists a generalized -derivation geaf : \ud835\udc9c \u2192 \u2133 such that g(eaf) = geaf(eaf). It is also easy to see thata \u2208 \ud835\udc9c and all idempotents e, f \u2208 \ud835\udc9c. Thus, by g satisfies the condition  = \u2110, \u03b2(\u2110) = \u2110. Then every local generalized -derivation -derivation, resp.) from an algebra \ud835\udc9c into an \ud835\udc9c-bimodule \u2133 is a generalized -derivation -derivation, resp.).Let \u03b1, \u03b2 = I\ud835\udc9c, we have the next direct consequence of Taking \u2110 be a separating set of an \ud835\udc9c-bimodule \u2133 contained in the algebra generated by all idempotents in \ud835\udc9c. Then every local generalized derivation  from an algebra \ud835\udc9c into an \ud835\udc9c-bimodule \u2133 is a generalized derivation .Let \u03b1 = I\ud835\udc9c, then we have the next result for local generalized skew derivations, that is; linear mappings g : \ud835\udc9c \u2192 \u2133 with the propertyAt the end, if \u2110 be a separating set of an \ud835\udc9c-bimodule \u2133 contained in the algebra generated by all idempotents in \ud835\udc9c and let \u03b2 be an endomorphism of \ud835\udc9c such that \u03b2(\u2110) = \u2110. Then every local generalized skew derivation  from an algebra \ud835\udc9c into an \ud835\udc9c-bimodule \u2133 is a generalized skew derivation .Let \ud835\udc9c is a unital algebra and n \u2265 2, a positive integer, then Mn(\ud835\udc9c), that is, the algebra of all n \u00d7 n matrices over \ud835\udc9c, belongs to this class . Th. Th\ud835\udc9c is \u03b1, \u03b2 be automorphisms of Mn(\ud835\udc9c). Then every local generalized -derivation -derivation, resp.) from an algebra Mn(\ud835\udc9c) into any Mn(\ud835\udc9c)-bimodule is a generalized -derivation -derivation, resp.).Let \u212c is called a local matrix algebra if any finite subset of \u212c can be embedded in a subalgebra which is a matrix algebra Mn(\ud835\udc9c), n \u2265 2.The next result involves local matrix algebras: an algebra \u03b1, \u03b2 be automorphisms of \ud835\udc9c. If, for any a1, a2 \u2208 \ud835\udc9c, there exists a unital subalgebra \u212c of \ud835\udc9c which contains a1, a2 and is isomorphic to a matrix algebra, then every local generalized -derivation -derivation, resp.) from an algebra \ud835\udc9c into any \ud835\udc9c-bimodule is a generalized -derivation -derivation, resp.).Let X and Y be complex Hausdorff topological linear spaces and let \u212c be the algebra of all continuous linear mappings from X into Y. We say that a subset \ud835\udcae of \u212c is reflexive if T \u2208 \ud835\udcae whenever T \u2208 \u212c and x \u2208 X, where \ud835\udcaex. By a subspace lattice on X we mean a collection \u2112 of closed subspaces of X containing {0} and X such that, for each family {L\u03f5} of elements of \u2112, both \u22c2L\u03f5 and \u22c1L\u03f5 belong to \u2112, where \u22c1 denotes the closed linear span of {L\u03f5}. If \u2112 is a subspace lattice, then we denote the algebra of all operators on X, that leave invariant each element of \u2112 by alg\u2112. A totally ordered subspace lattice \ud835\udca9 is called a nest and the associated reflexive algebra alg\ud835\udca9 is called a nest algebra.Let \u03b1, \u03b2)-derivations on a reflexive subalgebra in a factor von Neumann algebra. The proof of the following corollaries uses Now we consider local generalized -derivation -derivation, resp.) from \u2133\u2229alg\u2112 into \u2133 is a generalized -derivation -derivation, resp.).Let \ud835\udca9 be a nest in a factor von Neumann algebra \u2133 on H and let \u03b1, \u03b2 be automorphisms of \u2133\u2229alg\ud835\udca9. Then every local generalized -derivation -derivation, resp.) from \u2133\u2229alg\ud835\udca9 into \u2133 is a generalized -derivation -derivation, resp.).Let \ud835\udc9c is topologically generated by its idempotents  and suppose that g : \ud835\udc9c \u2192 \u2133 is a continuous local generalized -derivation, where \u03b1, \u03b2 are automorphisms of \ud835\udc9c. Let a = \u2211i=1m\u03bbi\u220fj=1tieji)( for some idempotents eji) = g(a)\u03b1(b) + \u03b2(a)g(b) \u2212 \u03b1(a)g(1)\u03b2(b) and since g is continuous and \ud835\udc9c is generated by its idempotents, we have the following proposition.Suppose that \u03b1, \u03b2 be automorphisms of \ud835\udc9c. If \ud835\udc9c is topologically generated by its idempotents and \u2133 is a topological \ud835\udc9c-bimodule, then every continuous local generalized -derivation -derivation, resp.) from \ud835\udc9c into \u2133 is a generalized -derivation -derivation, resp.).Let \ud835\udca9 is a nest in a von Neumann algebra \u2133 and \ud835\udc9c = \u2133\u2229alg\ud835\udca9, then the linear span of all idempotents in \ud835\udc9c is w*-dense in \ud835\udc9c -derivation -derivation) from \ud835\udc9c into \u2133 is a generalized -derivation -derivation).Let \u03b1, \u03b2)-derivations from \ud835\udc9c into \u2133, denoted by GDer\u03b1,\u03b2, is reflexive.At the end, let us prove that the set of all continuous generalized  is reflexive.Let g : \ud835\udc9c \u2192 \u2133 be a continuous linear mapping such that for any x \u2208 \ud835\udc9c,e, f \u2208 \ud835\udc9c be arbitrary idempotents, a any element from \ud835\udc9c, and x = eaf. Then, according to above observations, there exists a sequence {gn}n=1\u221e \u2282 GDer\u03b1,\u03b2 such thatg is a generalized -derivation; that is, g \u2208 GDer\u03b1,\u03b2. The proof is completed.Let"}
{"text": "In this paper, we introduce a new iterative scheme that converges strongly to a common fixed point of a countable family of strictly pseudo-contractive mappings in a real Hilbert space which is also a solution of variational inequality problem related to quadratic minimization problems. Our results extend ones of Yao et al. , Gu et al.  and some authors. H is a real Hilbert space with inner product and norm denoted by \u2329\u00b7,\u00b7\u232a and \u2225\u00b7\u2225, respectively and let C is a nonempty closed and convex subset of H. A mapping f:C\u2192H is called a contraction on C if there exists a constant \u03c1 \u2208 , such thatwhere Therefore, fixed point algorithms can be applied to solve variational inequalities.hierarchical fixed point problem: Find x\u2217\u2208F(T) such thatThe following problem is called a S:C\u2192H be a mapping. It is known that the hierarchical fixed point problem (5) links with some monotone variational inequalities and convex programming problems; see Gu et al. , Moudafi  intoduceS,T:C\u2192C are two nonexpansive mappings, {\u03b1n} and {\u03b2n} are two sequences in . Then he showed that {xn} converges weakly to a fixed point of T which is a solution of problem (5). For obtaining a strong convergence result, in Mainge and Moudafi . Then they showed that {xn} converges strongly to a fixed point of T which is a solution of problem (5).where On the other hand, Cianciaruso et al.  introducf:C\u2192C is a contraction mapping, S and T:C\u2192C are two nonexpansive mappings, {\u03b1n} and {\u03b2n} are two sequences in . Under some certain restrictions on parameters, the authors proved the sequence {xn} generated by (8) converges strongly to x\u2217\u2208F(T), which is a unique solution of the following variational inequality:where xn} generated by (8) converges strongly to x\u2217\u2208F(T), which is a unique solution of the following variational inequality:By changing the restrictions on parameters, the authors obtained another result on the iterative scheme (8), i.e., the sequence {\u03c4 \u2208  is a constant.where f from C to H by using the metric projection of H onto C. They introduced the following iterative scheme:In 2010, Yao et al. . The authors proved the sequence {xn} generated by (11) converges strongly to x\u2217 \u2208 F(T), which is a unique solution of one of the variational inequalities (9) and (10).where In 2011, Gu et al. . The authors proved the sequence {xn} generated by (12) converges strongly to x\u2217 \u2208 F(T), which is a unique solution of one of the variational inequalities (9) and (10).where In this paper, motivated and inspired by the results of Gu et al. , we intrVi=kiI+(1\u2212ki)Ti and ki-strict pseudo-contraction mappings. Under some certain condition on parameters, we first prove that the sequence {xn} generated by (13) converges strongly to where xn} generated by (13) converges strongly to By changing the restrictions on parameters, we also prove that the sequence {\u03c4 \u2208  is a constant. It is easy to see that, if ki=0 for each i\u22651, then our algorithm (13) is reduced to algorithm (12) of Gu et al. Also our results extend the corresponding one of Yao et al. \u2192 for strong convergence and \u21c0 for weak convergence.(ii)\u03c9-limit set of {xn}.We will use the following notation: H be a Hilbert space, C is a closed convex subset of H and T:C\u2192C be a nonexpansive mapping with F(T)\u2260\u2205. If {xn} is a sequence in C weakly converging to x and if {(I\u2212T)xn} converges strongly to y, then (I\u2212T)x=y; in particular, if y=0 then x \u2208 F(T).Browder  Let H beC be a nonempty closed convex subset of a real Hilbert space H. If T:C\u2192C is a k-strict pseudo-contraction, then the mapping I\u2212T is demiclosed at 0. That is, if {xn} is a sequence in C weakly converging to x and {(I\u2212T)xn} converges strongly to 0, then (I\u2212T)x=0.Acedo and Xu  Let C bex \u2208 H and z \u2208 C be any points. Then we have the following: z=PC[ x] if and only if there holds the relation:That z=PC[ x] if and only if there holds the relation:That There holds the relation:Let PC is nonexpansive and monotone.Consequently, H be a Hilbert space, C be a closed convex subset of H, f:C\u2192H be a contraction with coefficient 0 < \u03c1 < 1 and T : C \u2192 C be a nonexpansive mapping. Then, for 0 < \u03b3 < \u03b3\u0304/\u03c1, for x,y\u2208C, I\u2212f) is strongly monotone with coefficient (1\u2212\u03c1) that isthe mapping (I\u2212T) is monotone, that isthe mapping  and {\u03b4n} is a sequence in where {or n\u2192\u221ean=0.Then limC be a closed convex subset of H. Let {xn} be a bounded sequence in H. Assume that \u03c9-limit set \u03c9w(xn)\u2282C,The weak z \u2208 C, limn\u2192\u221e\u2225xn\u2212z\u2225 exists.For each Acedo and Xu  Let C bexn} is weakly convergent to a point in C.Then {H be a real Hilbert space, C be a closed and convex subset of H, and T be a k-strict pseudo-contraction mapping on C, then F(T) is closed convex, so that the projection PF(T) is well defined.Zhou  Let H beH be a Hilbert space, C be a closed and convex subset of H, and T:C\u2192H be a k-strict pseudo-contraction mapping. Define a mapping V:C\u2192H by Vx=\u03bbx+(1\u2212\u03bb)Tx for all x \u2208 C. Then, as \u03bb \u2208 /\u03b1n\u03b2n=0;\u25cf\u2003(H8) limK > 0 such that\u25cf\u2003\u25cf\u2003(H9) there exists a constant where xn} and {yn} are bounded.Assume that (H1) holds. Then {Let So, by induction, one can obtain thatxn} is bounded. Of course {yn} is bounded too. \u25a1Hence {(i)xn} is asymptotically regular, that is,{(ii)the weak cluster points set Suppose that (H1) and (H3) hold. Also, assume that either (H4) and (H5) hold, or (H6) and (H7) hold. Then PC is a nonexpansive mapping, we haveSet yn one obtain thatBy definition of So, substituting (22) in (21), we obtainBy Proposition 1, we saySo, we haveSo, if (H4) and (H5) hold, we obtain the asymptotic regularity by Lemma 5, if instead, (H6) and (H7) hold, from (H1), we can writeBy Lemma 5, we obtain the asymptotics regularity.Vixn \u2208 C for each i \u2265 1 and In order to prove (2), since Now, fixing It follows that Now, from Lemma 9 and (27), we get \u03b2n\u21920, as n\u2192\u221e, so thatBy (H1) and (H3), it follows that i\u22651 and {\u03b1n} is strictly decreasing, one hasSince Hence, we obtainxn} is asymptotically regular and demiclosedness principle, we obtain the proposition. \u25a1Since {(i)n\u2192\u221e\u2225xn\u2212yn\u2225=0;lim(ii)n\u2192\u221e\u2225xn\u2212Viyn\u2225=0, \u2200i\u22651;lim(iii)n\u2192\u221e\u2225yn\u2212Viyn\u2225=0, \u2200i\u22651.limSuppose that the hypotheses of Proposition 2 hold. Then i), we can observe thatTo prove .To prove (ii), we observe thatSince andyn\u2212xn\u2225\u21920 and \u2225xn\u2212Vixn\u2225\u21920 as n\u2192\u221e, \u2200i\u22651, then \u2225yn\u2212Vixn\u2225\u21920, that is, we obtain (ii). To prove (iii), we can observe thatSince \u2225i) and (ii), we obtain (iii). \u25a1By  and {\u03b1n} is a strictly decreasing sequence, Vi=kiI+(1\u2212ki)Ti, {\u03b2n}\u2282 and {\u03b1n} and {\u03b2n} are sequences satisfying the conditions (H2) with \u03c4=0, (H3), either (H4) and (H5), or (H6) and (H7). Then the sequence {xn} converges strongly to a point where {First of all, since xn} is bounded. Moreover, since either (H4) and (H5) or (H6) and (H7) then {xn} is asymptotically regular. Similarly, by Proposition 2, the weak cluster points set of xn, that is, \u03c9w(xn), is a subset of Since (H2) implies (H1), thus {xn} such thatLet and Set By Lemma 3(1), we haveFrom (32) and (33), it follows thatxn+1 \u2212 z\u22252\u2225n\u22651. SinceLet xn\u2192z as n\u2192\u221e. \u25a1Hence, by Lemma 5, we conclude that f\u22610, then we get In the iterative scheme (30), if we set That isz is the unique solution to the following quadratic minimization problem:Therefore, the point By changing the restrictions on parameters in Theorem 1, we obtain the following results.C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C\u2192H be a \u03c1-contraction mapping, S:C\u2192C be a nonexpansive mapping and ki-strict pseudo-contraction mappings and \u03b10 = 1, and x1 \u2208 C and define the sequence {xn} byLet \u03b1n}\u2282 and {\u03b1n} is a strictly decreasing sequence, Vi=kiI+(1\u2212ki)Ti, {\u03b2n}\u2282 and {\u03b1n} and {\u03b2n} are sequences satisfying the conditions (H2) with \u03c4 \u2208 , (H3), (H8) and (H9). Then the sequence {xn} converges strongly to a point where {x\u2032 and x\u2217 be two solutions. Then, since x\u2032 is solution, for y=x\u2217 one hasFirst, we shows that (49) has the unique solution. Let andAdding (36) and (37), we obtainx\u2032=x\u2217. Also now the condition (H2) with 0<\u03c4<\u221e implies (H1) so the sequence {xn} is bounded. Moreover, since (H8) implies (H6) and (H7), then {xn} is asymptotically regular. Similarly, by Proposition 2, the weak cluster points set of xn, i.e., \u03c9w(xn), is a subset of so From (20)-(24), we observe that\u03b3n=(1\u2212\u03c1)\u03b1n and Let By Lemma 5, we obtainFrom (34), we haveIt follows thatLet By Lemma 4, we haveandBy Lemma 3(1), we obtainNow, from (38)-(42), it follows thatvn\u21920 and (I\u2212Vi)yn\u21920, as n\u2192\u221e, then every weak cluster point of {xn} is also a strong cluster point. By Proposition 2, {xn} is bounded, thus there exists a subsequence x\u2217. For all since vn\u21920,(I\u2212Vi)yn\u21920 for all i\u22651, and \u2225un\u2212un\u22121\u2225/\u03b1n\u21920, letting k\u2192\u221e in (44), we obtainSince \u03c9w(xn)={x\u2217}. Since every weak cluster point of {xn} is also a strong cluster point, we conclude that xn\u2192x\u2217 as n\u2192\u221e. This completes the proof.Since (49) has the unique solution, it follows that Ti=T, for all i\u22651, where T:C\u2192C is a k-strict pseudo-contraction mapping in Theorem 1, then we get the following result: \u25a1If we take C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C\u2192H be a \u03c1-contraction mapping, S:C\u2192H be a nonexpansive mapping and T:C\u2192C be a k-strict pseudo-contraction mapping such that F(T)\u2260\u2205. Let x1 \u2208 C and define the sequence {xn} byLet V=kI+(1\u2212k)T,{\u03b1n}\u2282 and {\u03b2n}\u2282 are sequences satisfying the conditions (H2) with \u03c4=0, (H3), either (H4) and (H5), or (H6) and (H7). Then the sequence {xn} converges strongly to a point z \u2208 F(T), which is the unique solution of the variational inequality:where ki=0, for all i\u22651 in Theorem 1, then we get the following result:Taking C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C\u2192H be a \u03c1-contraction mapping, S:C\u2192H be a nonexpansive mapping and \u03b10 = 1, x1 \u2208 C and define the sequence {xn} byLet \u03b1n}\u2282 and {\u03b1n} is a strictly decreasing sequence, {\u03b2n}\u2282 and {\u03b1n} and {\u03b2n} are sequences satisfying the conditions (H2) with \u03c4=0, (H3), either (H4) and (H5), or (H6) and (H7). Then the sequence {xn} converges strongly to a point where {k=0 in Corollary 2, then we get the following result:If we take C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C\u2192H be a \u03c1-contraction mapping, S:C\u2192H be a nonexpansive mapping and T:C\u2192C be a nonexpansive mapping such that F(T)\u2260\u2205. Let x1\u2208C and define the sequence {xn} byLet \u03b1n}\u2282,{\u03b2n}\u2282 and {\u03b1n} and {\u03b2n} are sequences satisfying the conditions (H2) with \u03c4=0, (H3), either (H4) and (H5), or (H6) and (H7). Then the sequence {xn} converges strongly to a point z \u2208 F(T), which is the unique solution of the variational inequality:where {Ti=T, for all i\u22651, where T:C\u2192C is a k-strict pseudo-contraction mapping in Theorem 2, then we obtain the following result:If we take C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C\u2192H be a \u03c1-contraction mapping, S:C\u2192C be a nonexpansive mapping and T:C\u2192C be a k-strict pseudo-contraction mapping and x1\u2208C and define the sequence {xn} byLet V=kI+(1\u2212k)T, {\u03b1n}\u2282, {\u03b2n}\u2282 and {\u03b1n} and {\u03b2n} are sequences satisfying the conditions (H2) with \u03c4 \u2208 , (H3), (H8) and (H9). Then the sequence {xn} converges strongly to a point where ki=0, for all i\u22651 in Theorem 2, then we get the following result:If we take C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C\u2192H be a \u03c1-contraction mapping, S:C\u2192C be a nonexpansive mapping and \u03b10 = 1, x1 \u2208 C and define the sequence {xn} byLet \u03b1n}\u2282 and {\u03b1n} is a strictly decreasing sequence, {\u03b2n}\u2282 and {\u03b1n} and {\u03b2n} are sequences satisfying the conditions (H2) with \u03c4 \u2208 , (H3), (H8) and (H9). Then the sequence {xn} converges strongly to a point where {k=0 in Corollary 5, then we get the following Corollary:If C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C\u2192H be a \u03c1-contraction mapping, S,T:C\u2192C be nonexpansive mappings and x1\u2208C and define the sequence {xn} byLet \u03b1n}\u2282, {\u03b2n}\u2282 and {\u03b1n} and {\u03b2n} are sequences satisfying the conditions (H2) with \u03c4 \u2208 , (H3), (H8) and (H9). Then the sequence {xn} converges strongly to a point where {\u03b1n=n\u03b8\u2212 and \u03b2n=n\u03c9\u2212 . Since |\u03b1n\u2212\u03b1n\u22121|\u2248n\u03b8\u2212 and |\u03b2n\u2212\u03b2n\u22121|\u2248n\u03c9\u2212, it is not difficult to prove that (H8) is satisfied for 0<\u03b8,\u03c9<1and (H9) is satisfied if \u03b8+\u03c9\u22641.Prototypes for the iterative parameters are, for example, Theorem 1 and Theorem 2 extend and improve the result of Gu et al.  from the"}
{"text": "Then we introduce bifuzzy soft Lie subalgebras and investigate some of their properties.We introduce the concept of ( The concept of Lie groups was first introduced by Sophus Lie in nineteenth century through his studies in geometry and integration methods for differential equations. Lie algebras were also discovered by him when he attempted to classify certain smooth subgroups of a general linear group. The importance of Lie algebras in mathematics and physics has become increasingly evident in recent years. In applied mathematics, Lie theory remains a powerful tool for studying differential equations, special functions, and perturbation theory. It is noted that Lie theory has applications not only in mathematics and physics but also in diverse fields such as continuum mechanics, cosmology, and life sciences. A Lie algebra has nowadays even been applied by electrical engineers in solving problems in mobile robot control .bifuzzy sets. The elements of the bifuzzy sets are featured by an additional degree which is called the degree of uncertainty. This kind of fuzzy sets have now gained a wide recognition as a useful tool in the modeling of some uncertain phenomena. Bifuzzy sets have drawn the attention of many researchers in the last decades. This is mainly due to the fact that bifuzzy sets are consistent with human behavior, by reflecting and modeling the hesitancy present in real-life situations. In fact, the fuzzy sets give the degree of membership of an element in a given set, while bifuzzy sets give both a degree of membership and a degree of nonmembership. As for fuzzy sets, the degree of membership is a real number between 0 and 1. This is also the case for the degree of nonmembership, and furthermore the sum of these two degrees is not greater than 1.After introducing the concept of fuzzy sets by Zadeh  in 1965,\u03b3, \u03b4)-bifuzzy Lie algebra, and discuss some of its properties. Then we introduce bifuzzy soft Lie algebras and investigate some of their properties. We use standard definitions and terminologies in this paper.In 1999, Molodtsov  initiateIn this section, we review some known basic concepts that are necessary for this paper.Lie algebra is a vector space L over a field F  on which L \u00d7 L \u2192 L denoted by \u2192 is defined satisfying the following axioms:(L1)x, y] is bilinear, = 0 for all x \u2208 L,, z] + , x] + , y] = 0 for all x, y, z \u2208 L (Jacobi identity)., z] = ]. But it is anticommutative; that is,  = \u2212. A subspace H of L closed under  will be called a Lie subalgebra.Throughout this paper, \u03bc be a fuzzy set on X; that is, a map \u03bc : X \u2192 . As an important generalization of the notion of fuzzy sets in X, Atanassov introduced the concept of a bifuzzy set defined on a nonempty set X as objects having the form\u03bc : X \u2192  and \u03bd : X \u2192  denote the degree of membership ) and the degree of nonmembership ) of each element x \u2208 X to the set A, respectively, and 0 \u2264 \u03bcA(x) + \u03bdA(x) \u2264 1 for all x \u2208 X.Let A =  be a bifuzzy set on X and let s, t \u2208  be such that s + t \u2264 1. Then the sets, t)-level subset of A. As,t) on L is called a bifuzzy Lie subalgebra if the following conditions are satisfied: \u03bcA(x + y) \u2265 min\u2061{\u03bcA(x), \u03bcA(y)},\u03bdA(x + y) \u2264 max\u2061{\u03bdA(x), \u03bdA(y)},\u03bcA(\u03b1x) \u2265 \u03bcA(x), \u03bdA(\u03b1x) \u2264 \u03bdA(x),\u03bc \u2265 max\u2061{\u03bc(x), \u03bc(y)},\u03bd \u2264 min\u2061{\u03bd(x), \u03bd(y)}A bifuzzy set x, y \u2208 L and \u03b1 \u2208 F.for all c be a point in a nonempty set X. If \u03b3 \u2208  are two real numbers such that 0 \u2264 \u03b3 + \u03b4 \u2264 1, then the bifuzzy set c = \u2329x, c\u03b3, 1 \u2212 c\u03b41\u2212\u232a is called a bifuzzy point (BP for short) in X, where \u03b3  is the degree of membership  of c and c \u2208 X is the support of c. Let c be a BP in X and let A = \u2329x, \u03bcA, \u03bdA\u232a be a bifuzzy set in X. Then c is said to belong to A, written c \u2208 A, if \u03bcA(c) \u2265 \u03b3 and \u03bdA(c) \u2264 \u03b4. We say that c is quasicoincident with A, written cqA, if \u03bcA(c) + \u03b3 > 1 and \u03bdA(c) + \u03b4 < 1. To say that c\u2208\u2228qA \u2208\u2227qA) means that c \u2208 A or cqA  \u2208 A and cqA) and c\u2208\u2228qA does not hold.Let f and g be any two bifuzzy subsets of L. Then the sum f + g is a bifuzzy subset of L defined byLet U) denote the family of all bifuzzy sets in U.Let IF is called a bifuzzy soft set over U, where f is a mapping given by f : A \u2192 IF(U). A bifuzzy soft set is a parameterized family of bifuzzy subsets of U. For any \u025b \u2208 A, f\u025b is referred to as the set of \u025b-approximate elements of the bifuzzy soft set , which is actually a bifuzzy set on U and can be\u03bcf\u025b(x) and \u03bdf\u025b(x) are the membership degree and nonmembership degree that object x holds on parameter \u025b, respectively.Let f, A) and  be two bifuzzy soft sets over U. We say that  is a bifuzzy soft subset of  and write \u22d0 if  (i)A\u2286B, (ii)\u025b \u2208 A, f(\u025b)\u2286g(\u025b).for any Let  and  are said to be bifuzzy soft equal and write  =  if \u22d0 and \u22d0. and  be two bifuzzy soft sets over U. Then their extended intersection is a bifuzzy soft set denoted by , where C = A \u222a B and\u025b \u2208 C. This is denoted by Let  and  are two bifuzzy soft sets over the same universe U then \u201c AND \u201d is a bifuzzy soft set denoted by \u2227 and is defined by \u2227 = , where, h = h(a)\u2229g(b) for all  \u2208 A \u00d7 B. Here \u2229 is the operation of a bifuzzy intersection.If  and  be two bifuzzy soft sets over U. Then their extended union is denoted by , where C = A \u222a B and\u025b \u2208 C. This is denoted by Let  and  be two fuzzy soft sets over a common universe U with A\u2229B \u2260 \u2205. Then their restricted intersection is a bifuzzy soft set  denoted by \u22d3 = , where h(\u025b) = f(\u025b)\u2229g(\u025b) for all \u025b \u2208 A\u2229B.Let  and  be two bifuzzy soft sets over a common universe U with A\u2229B \u2260 \u2205. Then their restricted union is denoted by \u22d2 and is defined as \u22d2 = , where C = A\u2229B and for all \u025b \u2208 C, h(\u025b) = f(\u025b) \u222a g(\u025b).Let  and  over U is a fuzzy soft set, denoted by , where C = A \u222a B, and defined by\u025b \u2208 C. This is denoted by The \u03b3, \u03b4)-bifuzzy Lie subalgebra, and discuss some of its properties.In this section, we introduce a new kind of generalized bifuzzy Lie algebra, an  in L is called an -bifuzzy Lie subalgebra of L if it satisfies the following conditions: x\u03b3A, \u2009y\u03b3A\u21d2(x + y), max\u2061)\u03b4A,x\u03b3A\u21d2(mx)\u03b4A,x\u03b3A, \u2009y\u03b3A\u21d2, min\u2061)\u03b4AA bifuzzy set x, y \u2208 L,\u2009\u2009m \u2208 F,\u2009\u2009s, s1, s2 \u2208 .for all x\u03b3A\u21d2(\u2212x)\u03b4A,Consider (i) x\u03b3A\u21d2(0)\u03b4A.(ii) V be a vector space over a field F such that dim\u2061(V) = 5. Let V = {e1, e2,\u2026, e5} be a basis of a vector space over a field F with Lie brackets as follows:ei, ej] = 0 for all i = j. Then V is a Lie algebra over F.Let A =  : V \u2192 \u00d7 bys = 0.4 \u2208 . By routine computations, it is easy to see that A is not an -bifuzzy Lie subalgebra of L.We define a bifuzzy set A in L, we denote L = {x \u2208 L : \u03bc(x) > 0 and \u03bd(x) < 1}.For a bifuzzy set A =  be an -bifuzzy Lie subalgebra of L; then the nonzero set L is a Lie subalgebra of L.Let x, y \u2208 L. Then \u03bcA(x) > 0 and \u03bdA(x) < 1, \u03bcA(y) > 0 and \u03bdA(y) < 1. Assume that \u03bcA(x + y) = 0 and \u03bdA(x + y) = 1. If \u03b3 \u2208 {\u2208, \u2208\u2228q}, then we can see that x(\u03bcA(x), \u03bdA(x))\u03b3A and y(\u03bcA(y), \u03bdA(y))\u03b3A, but \u03b4 \u2208 {\u2208, \u2208\u2228q, \u2208\u2227q}, a contradiction. Also, xqA and yqA, but \u03b4 \u2208 {\u2208, \u2208\u2228q, \u2208\u2227q}, a contradiction. Thus \u03bcA(x + y) > 0 and \u03bdA(x + y) < 1. Thus x + y \u2208 L. For other conditions the verification is analogous. Consequently L is a Lie subalgebra of L.Let A =  in L is called an -bifuzzy Lie algebra of L if it satisfies the following conditions: (f)x \u2208 A, y \u2208 A\u21d2(x + y), max\u2061)\u2208\u2228qA,(g)x \u2208 A\u21d2(mx)\u2208\u2228qA,(h)x \u2208 A, y \u2208 A\u21d2, min\u2061)\u2208\u2228qAA bifuzzy set x, y \u2208 L, m \u2208 F, s, s1, s2 \u2208 .for all A =  be a bifuzzy set in a Lie algebra L. Then A is an -bifuzzy Lie subalgebra of L if and only if\u03bcA(x + y)\u2a7emin\u2061(\u03bcA(x), \u03bcA(y), 0.5), \u03bdA(x + y) \u2a7d max\u2061(\u03bdA(x), \u03bdA(y), 0.5),\u03bca(mx)\u2a7emin\u2061(\u03bcA(x), 0.5), \u03bda(mx) \u2a7d max\u2061(\u03bdA(x), 0.5),\u03bcA\u2a7emax\u2061(\u03bcA(x), \u03bcA(y), 0.5), \u03bdA \u2a7d min\u2061(\u03bdA(x), \u03bdA(y), 0.5)Let x, y \u2208 L, m \u2208 F.hold for all x, y \u2208 L. We consider the following two cases:\u03bcA(x), \u03bcA(y)) < 0.5, max\u2061(\u03bdA(x), \u03bdA(y)) > 0.5,min\u2061)\u2a7e0.5, max\u2061) \u2a7d 0.5.min\u2061((f)\u21d2(i): Let 1.Case  Assume that \u03bcA(x + y) < min\u2061(\u03bcA(x), \u03bcA(y), 0.5), \u03bdA(x + y) > max\u2061(\u03bdA(x), \u03bdA(y), 0.5). Then \u03bcA(x + y) < min\u2061(\u03bcA(x), \u03bcA(y)), \u03bdA(x + y) > max\u2061(\u03bdA(x), \u03bdA(y)). Take s, t such that \u03bcA(x + y) < s < min\u2061(\u03bcA(x), \u03bcA(y)), \u03bdA(x + y) > t > max\u2061(\u03bdA(x), \u03bdA(y)). Then xs, ys \u2208 \u03bcA and xt, yt \u2208 \u03bdA, but f).2.Case  Assume that \u03bcA(x + y) < 0.5, \u03bdA(x + y) > 0.5. Then x, y \u2208 A but x, y \u2208 A; then \u03bcA(x)\u2a7es1, \u03bcA(y)\u2a7es2, \u03bdA(x) \u2a7d t1, \u03bdA(y) \u2a7d t2. Now, we haves1, s2) > 0.5, max\u2061\u2a7e0.5\u21d2\u03bcA(x + y) + min\u2061 > 1, \u03bdA(x + y) \u2a7d 0.5\u21d2\u03bdA(x + y) + max\u2061 < 1. On the other hand, if min\u2061 \u2a7d 0.5, max\u2061\u2a7e0.5, then \u03bcA(x + y)\u2a7emin\u2061, \u03bdA(x + y) \u2a7d max\u2061. Hence (x + y), max\u2061)\u2208\u2228qA. The verification of (g)\u21d4(j) and (h)\u21d4(k) is analogous and we omit the details. This completes the proof.A =  be a bifuzzy set of Lie algebra of L. Then A is an -bifuzzy Li subalgebra of L if and only if each nonempty As,t) is a Lie subalgebra of L.Let A =  is an -bifuzzy Lie subalgebra of L and let s \u2208 . If x, y \u2208 As,t) \u2265 s and \u03bcA(y) \u2265 s, \u03bdA(x) \u2264 t and \u03bdA(y) \u2264 t. Thus,x + y, mx,  \u2208 As,t) are Lie subalgebras of L. The proof of converse part is obvious. This ends the proof.Assume that A be a bifuzzy set in a Lie algebra L. Then As,t), 0.5)\u2a7emin\u2061(\u03bcA(x), \u03bcA(y)), min\u2061(\u03bdA(x + y), 0.5) \u2a7d max\u2061(\u03bdA(x), \u03bdA(y)),max\u2061(\u03bcA(mx), 0.5)\u2a7e\u03bcA(x), min\u2061(\u03bdA(mx), 0.5) \u2a7d \u03bdA(x),max\u2061, 0.5))\u2a7emin\u2061(\u03bcA(x), \u03bcA(y)), min\u2061, 0.5)) \u2a7d max\u2061(\u03bdA(x), \u03bdA(y)) for all x, y \u2208 L, m \u2208 F.max\u2061, 0.5) < min\u2061(\u03bcA(x), \u03bcA(y)) = s, min\u2061(\u03bdA(x + y), 0.5) > max\u2061(\u03bdA(x), \u03bdA(y)) = t for some x, y \u2208 L; then s \u2208 , \u03bcA(x + y) < s, \u03bdA(x + y) > t, x, y \u2208 As,t)( or \u03bcA(x + y)\u2a7es, \u03bdA(x + y) \u2a7d t, which is contradiction with \u03bcA(x + y) < s, \u03bdA(x + y) > t. Hence (1) holds. For (2), (3) the verification is analogous.Suppose that s \u2208 , x, y \u2208 As,t)\u2013(3) hold. Assume that q)-bifuzzy Lie subalgebras of L is an -bifuzzy Lie subalgebra.The intersection of any family of -bifuzzy Lie subalgebra of L and let A : = \u22c2i\u2208\u039bAi = . Let x, y \u2208 L, then by \u03bcA(x + y)\u2a7emin\u2061(\u03bcA(x), \u03bcA(y), 0.5), \u03bdA(x + y) \u2a7d max\u2061(\u03bdA(x), \u03bdA(y), 0.5), and henceA is an -bifuzzy Lie subalgebra of L.Let {L0 \u2282 L1 \u2282 \u22ef\u2282Ln = L be a strictly increasing chain of -bifuzzy Lie subalgebras of a Lie algebra L; then there exists -bifuzzy Lie subalgebra A =  of L whose level subalgebras are precisely the members of the chain with A0.5 =  = L.Let r, s \u2208  and fuzzy subset \u03bc in L, denotes \u03bc\u232ar = {x \u2208 L | xrq\u03bc}, [\u03bc]r = {x \u2208 L | xr \u2208 \u2228q\u03bc}, r, s \u2208 .For any We state here a nice characterization without proof.L be a Lie algebra and A a bifuzzy set in L. Then A is an -bifuzzy Lie subalgebra of L if and only if nonempty subsets L for all r \u2208 r and L for all r \u2208 -bifuzzy Lie subalgebra of L if and only if nonempty subsets [\u03bcA]r and L for all r, s \u2208  be a bifuzzy soft set over L. Then  is said to be a bifuzzy soft Lie subalgebra over L if f(x) is a bifuzzy Lie subalgebra of L for all x \u2208 A; that is, a bifuzzy soft set  on L is called a bifuzzy soft Lie subalgebra of L if \u03bcf\u025b(x + y)\u2a7emin\u2061{\u03bcf\u025b(x), \u03bcf\u025b(y)},\u03bdf\u025b(x + y) \u2a7d max\u2061{\u03bdf\u025b(x), \u03bdf\u025b(y)},\u03bcf\u025b(mx)\u2a7e\u03bcf\u025b(x),\u03bdf\u025b(mx) \u2a7d \u03bcf\u025b(x),\u03bcf\u025b\u2a7emin\u2061{\u03bcf\u025b(x), \u03bcf\u025b(y)},\u03bdf\u025b \u2a7d max\u2061{\u03bdf\u025b(x), \u03bdf\u025b(y)}Let x, y \u2208 L and m \u2208 K.hold for all \u211c2 = { : x, y \u2208 \u211d} be the set of all 2-dimensional real vectors. Then \u211c2 with  = x \u00d7 y is a real Lie algebra. Let \u2115 and \u2124 denote the set of all natural numbers and the set of all integers, respectively. Define f : \u2124 \u2192 \u211c2 by f(n) = fn : \u211c2 \u2192  \u00d7  for all n \u2208 \u2124,f, \u2124) is a bifuzzy soft Lie subalgebra of \u211c2.Let The following proposition is obvious.f, A) be a bifuzzy soft Lie subalgebra of L; then Let  be a bifuzzy soft set over U. For each s, t \u2208 , the set s,t) is called an -level soft set of , where f\u025bs,t)( = {x \u2208 U | \u03bcf\u025b(x) \u2265 s, \u03bdf\u025b(x) \u2264 t} for all \u025b \u2208 A.Let  be a bifuzzy soft set over L.  is a bifuzzy soft Lie subalgebra if and only if s,t) is a bifuzzy soft Lie subalgebra. For each s, t \u2208 , \u025b \u2208 A, and x1, x2 \u2208 \u025bs,t) \u2265 s, \u03bcf\u025b(x2) \u2265 s and \u03bdf\u025b(x1) \u2264 t, \u03bdf\u025b(x2) \u2264 t. From f, A)\u025bs,t) \u2265 min\u2061(\u03bcf\u025b(x1), \u03bcf\u025b(x2)), \u03bcf\u025b(x1 + x2) \u2265 s, \u03bdf\u025b(x1 + x2) \u2264 max\u2061(\u03bdf\u025b(x1), \u03bdf\u025b(x2)), \u03bdf\u025b(x1 + x2) \u2264 t. This implies that x1 + x2 \u2208 \u025bs,t)\u025bs,t)s,t)s,t), \u03bcf\u025b(x2)} and let t = max\u2061{\u03bdf\u025b(x1), \u03bdf\u025b(x2)}; then x1, x2 \u2208 \u025bs,t)\u025bs,t)\u025bs,t)(. This means that \u03bcf\u025b(x1 + x2) \u2265 min\u2061(\u03bcf\u025b(x1), \u03bcf\u025b(x2)), \u03bdf\u025b(x1 + x2) \u2264 max\u2061(\u03bdf\u025b(x1), \u03bdf\u025b(x2)). Verification for other conditions is similar. Hence we omit the details. Thus, \u025bs,t) is a bifuzzy soft Lie subalgebra over L. This completes the proof.Conversely, assume that  is called a bifuzzy soft function from X to Y.Let f, A) and  be two bifuzzy soft sets over L1 and L2, respectively, and let  be a bifuzzy soft function from L1 to L2.(1)f, A) under the bifuzzy soft function , denoted by , is the bifuzzy soft set on L2 defined by  = (\u03d5(f), \u03c8(A)), where for all k \u2208 \u03c8(A), y \u2208 L2The image of ((2)g, B) under the bifuzzy soft function , denoted by \u22121, is the bifuzzy soft set over L1 defined by \u22121 = (\u03d5\u22121(g), \u03c8\u22121(B)), where for all a \u2208 \u03c8\u22121(A), for all x \u2208 L1,The preimage of  be a bifuzzy soft function from L1 to L2. If \u03d5 is a homomorphism from L1 to L2, then  is said to be bifuzzy soft homomorphism. If \u03d5 is a isomorphism from L1 to L2 and \u03c8 is one-to-one mapping from A onto B then  is said to be bifuzzy soft isomorphism.Let  be a bifuzzy soft Lie subalgebra over L2 and let  be a bifuzzy soft homomorphism from L1 to L2. Then \u22121 is a bifuzzy soft Lie subalgebra over L1.Let \u22121 is a bifuzzy soft Lie subalgebra over L1.Let f, A) be a bifuzzy soft Lie subalgebra over L1 and let  be a bifuzzy soft homomorphism from L1 to L2. Then  may not be a bifuzzy soft Lie subalgebra over L2.Let  be a bifuzzy soft Lie subalgebra over L and let { | i \u2208 I} be a nonempty family of bifuzzy soft Lie subalgebras of . Then f, A);i\u2208I is a bifuzzy soft Lie subalgebra of \u22c0i\u2208I;\u22c0Bi\u2229Bj = \u2205 for all i, j \u2208 I, then if Let  be an -bifuzzy soft Lie subalgebra over L and let { | i \u2208 I} be a nonempty family of -bifuzzy soft K-subalgebras of ; then q)-bifuzzy soft Lie subalgebra of ;i\u2208I is an -bifuzzy soft Lie subalgebra of \u22c0i\u2208I;\u22c0Bi\u2229Bj = \u2205 for all i, j \u2208 I, then q)- bifuzzy soft Lie subalgebra of if Let  and  be two -bifuzzy soft K-subalgebras over a Lie algebra L. Then q)-bifuzzy soft Lie subalgebra over L.Let . Then h(\u03b1) = f(\u03b1) is an -bifuzzy Lie subalgebra of L since  is an -bifuzzy soft Lie subalgebra over L.2.Case  Consider (\u03b1 \u2208 B \u2212 A). Then h(\u03b1) = g(\u03b1) is an -bifuzzy Lie subalgebra of L since  is an -bifuzzy soft Lie subalgebra over L.3.Case  Consider (\u03b1 \u2208 A\u2229B). Then h(\u03b1) = f(\u03b1)\u2229g(\u03b1) is an -bifuzzy Lie subalgebra of L by the assumption. Thus, in any case, h(\u03b1) is an -bifuzzy Lie subalgebra of L. Therefore, q)-bifuzzy soft Lie subalgebra over L.f, A) and  be two -bifuzzy soft K-subalgebras over a Lie algebra L. If A and B are disjoint, then q)-bifuzzy softLie subalgebra over L.Let  and  intuitionistic bifuzzy soft sets on G. If  and  are -intuitionistic bifuzzy soft Lie subalgebra on L, then so are \u2229 and f, A) and  are an -bifuzzy soft Lie subalgebra on L and an -bifuzzy soft Lie subalgebra on L, then Let L be a Lie algebra and  and  bifuzzy soft sets on L. If  and  are -bifuzzy soft Lie subalgebra on L, then so are \u222a and Let \ud835\udd4a\ud835\udd40 the set of all -bifuzzy soft Lie subalgebras on L.Denote by f, A),  \u2208 \ud835\udd3d\ud835\udd4a\ud835\udd40, by above Lemmas, f, A)\u2229 \u2208 \ud835\udd3d\ud835\udd4a\ud835\udd40. It is obvious that f, A)\u2229 are the least upper bound and the greatest lower bound of  and , respectively. There is no difficulty in replacing {, } with an arbitrary family of \ud835\udd3d\ud835\udd4a\ud835\udd40 and so f, A), ,  \u2208 \ud835\udd3d\ud835\udd4a\ud835\udd40. Suppose that\u025b \u2208 A\u2229(B \u222a C), it follows that \u025b \u2208 A and \u025b \u2208 B \u222a C. We consider the following cases.For any . Then I(\u025b) = f(\u025b)\u2229h(\u025b) = J(\u025b).2.Case  Consider . Then I(\u025b) = f(\u025b)\u2229G(\u025b) = J(\u025b).3.Case  Consider . Then I(\u025b) = f(\u025b)\u2229(g(\u025b) \u222a h(\u025b)) = (f(\u025b)\u2229g(\u025b))\u2009\u222a\u2009(f(\u025b)\u2229h(\u025b)) = J(\u025b).I and J are the same operators, and so Therefore, f, A) and  over a Lie algebra is a bifuzzy soft set over G, denoted by , where C = A \u222a B and\u025b \u2208 C. This is denoted by  = \u2299.The product of two bifuzzy soft sets , , , and  be bifuzzy soft sets over a Lie algebra L such that \u2282 and \u2282. Then f1, A)\u2299\u2282\u2299,\u2229\u2282\u2229 and \u222a\u2282\u222a and , , and  be bifuzzy soft sets over a Lie algebra L. Then \u2299\u2299) = \u2299)\u2299.Let ,  \u2208 \ud835\udd3d\ud835\udd4aA(G).Now we consider the bifuzzy soft sets over a definite parameter set. Let f, A) and  be -bifuzzy soft Lie subalgebra over a Lie algebra L. Then so is \u2299.Let , \u2299, \u2229) is a complete lattice under the relation \u2282.Let Presently, science and technology are featured with complex processes and phenomena for which complete information is not always available. For such cases, mathematical models are developed to handle various types of systems containing elements of uncertainty. A large number of these models are based on an extension of the ordinary set theory such as bifuzzy sets and soft sets. In the current paper, we have presented the basic properties on bifuzzy soft Lie subalgebras. The most of these properties can be simply extended to bifuzzy soft Lie ideals. A Lie algebra is known algebraic structure and there are still many unsolved problems in it. In our opinion the future study of Lie algebras can be extended with the study of (i) roughness in Lie algebras and (ii) fuzzy rough Lie algebras."}
{"text": "Gm\u00d7n\u2009(m \u2265 n) be a complex random matrix and W = Gm\u00d7nHGm\u00d7n which is the complex Wishart matrix. Let \u03bb1 > \u03bb2 > \u2026>\u03bbn > 0 and \u03c31 > \u03c32 > \u2026>\u03c3n > 0 denote the eigenvalues of the W and singular values of Gm\u00d7n, respectively. The 2-norm condition number of Gm\u00d7n is Let  W = Gm\u00d7nHGm\u00d7n, where Gm\u00d7n is a complex Gaussian matrix and W is known to follow a complex Wishart distribution. Over the past decade, multiple-input and multiple-output (MIMO) systems have been at the forefront of wireless communications research and development, due to their huge potential for delivering significant capacity compared with conventional systems \u201313. The In recent years, the statistical properties of Wishart matrices have been extensively studied and applied to a large number of MIMO applications. In statistics, the random eigenvalues are used in hypothesis testing, principal component analysis, canonical correlation analysis, multiple discriminant analysis, and so forth (see ). In nucGm\u00d7n\u2009(m \u2265 n) be a complex random matrix whose elements are independent and identically distributed (i.i.d.) standard normal random variables. As we know, the n \u00d7 n complex random matrix W = Gm\u00d7nHGm\u00d7n is a complex Wishart matrix. Its distribution is denoted by W\u223cW, \u03a3 = \u03c32I. In addition, W is a positive definite Hermitian matrix with real eigenvalues; let \u03bb1 > \u03bb2 > \u22ef>\u03bbn > 0 and \u03c31 > \u03c32 > \u22ef>\u03c3n > 0 denote the eigenvalues of the W and singular values of Gm\u00d7n, respectively. The 2-norm condition number of Gm\u00d7n is \u03ba2(W) = \u03ba2(Gm\u00d7n)2.Let n matrix whose elements are independent and identically distributed standard normal real or complex random variables are given in [\u03c32I.The exact distributions of the condition number of a 2 \u00d7 given in  by Edelmgiven in  for real\u03ba2(Wn\u00d7n) for complex case, is proved.This paper is arranged as follows. W and W\u223cW, \u03a3 = \u03c32I. The determinant, trace, and norm of a square matrix B are denoted by |B|, tr\u2061(B), and ||B||, respectively. In this section, we give some results on joint density of the eigenvalues of a complex Wishart matrix p, a portion of p is a multiple , where a1 \u2265 a2 \u2265 \u22ef\u2265an \u2265 0 such that \u2211i=1nai = p, \ud835\udc9fp is the set of all portions of p, and the symbol \u2211pa\u03ba means the summation over \ud835\udc9fp; that is, \u2211pa\u03ba = \u2211\u03ba\u2208\ud835\udc9fpa\u03ba.For any nonnegative integer \u03ba be any portion of p, and let \u03bc1 > \u03bc2 > \u22ef>\u03bcn be the eigenvalues of an n \u00d7 n matrix B as follows:B are defined byLet ynomials :(3)[tr\u2061The zonal polynomial of the identity matrix is defined byW = Gm\u00d7nHGm\u00d7n, where W is positive definite Hermitian matrix with real eigenvalues, let Gm\u00d7n be a complex random matrix whose elements are independent and identically distributed (i.i.d.) standard normal random variables. Let \u03bb1 > \u03bb2 > \u22ef>\u03bbn > 0 be the eigenvalues of W and \u039b = diag\u2061. If W\u223cW, \u03a3 = \u03c32I with m \u2265 n, then the joint density of its eigenvalues is defined [For n \u00d7 n matrix B, the product of two zonal polynomials can be expressed in terms of a weighted combination of another zonal polynomial [\u03bd \u2208 \ud835\udc9fs and \u2200\u03ba \u2208 \ud835\udc9fp, one hast = s + p and g\u03bd,\u03ba\u03c4 is a constant coefficient.For an lynomial ; that ism \u00d7 n matrix B and for all \u03ba \u2208 \ud835\udc9fp, one has [t = s + p and g\u03bd,\u03ba\u03c4 is a constant coefficient.For an  one has (9)|I\u2212B|Z = diag\u2061, Z1 = diag\u2061 and D = {1 > \u03be2 > \u22ef>\u03ben > 0}, then [Let = diag\u2061\u03be,\u2026, \u03ben, W is derived by the following. The exact distribution of the 2-norm condition number of the Wishart matrix Gm\u00d7n be a complex random matrix and W = Gm\u00d7nHGm\u00d7n. \u03bb1 and \u03bbn are the maximum and minimum eigenvalues of W, \u03ba2(W) = \u03bb1/\u03bbn. Then, the exact distribution of \u03ba2(W) is given byt = s + p. Let \u03bb1 = \u03bb1, \u03b3i = 1 \u2212 \u03bbi/\u03bb1, i = 2,\u2026, n, where 1 > \u03b3n > \u22ef>\u03b32 > 0, we obtain the joint distribution of \u03bb1 and \u03b32,\u2026, \u03b3n, as follows:By making the transformation By using Taylor's formula,A = diag\u2061.By using property , we have\u03bei = \u03b3i/\u03b3n, i = 2,\u2026, n \u2212 1, using \u03ben\u22121 > \u22ef>\u03be2 > 0, we haveZ = diag\u2061 and Z1 = diag\u2061.By making the transformation n be replaced by n \u2212 1, let \u03ba be replaced by \u03c4, and let b = n + 1.By using Then,\u03bb1, \u03b3n) is given byIt follows that the distribution of (\u03ba2(W) = \u03bb1/\u03bbn = (1 \u2212 \u03b3n)\u22121; then, the distribution of \u03ba2(W) = (1 \u2212 \u03b3n)\u22121 is given byNote that  In this paper, the exact distribution of the condition number of complex Wishart matrices is derived. The distribution is expressed in terms of complex zonal polynomials. This distribution plays an important role in numerical analysis and statistical hypothesis testing."}
{"text": "C over an alphabet A is called synchronized if there exist x, y \u2208 C* such that xA*\u2229A*y\u2286C*. In this paper we describe the syntactic monoid Syn(C+) of C+ for a complete synchronized code C over A such that C+, the semigroup generated by C, is a single class of its syntactic congruence PC+. In particular, we prove that, for such a code C, either C = A or Syn(C+) is isomorphic to a special submonoid of \ud835\udcafl(I) \u00d7 \ud835\udcafr(\u039b), where \ud835\udcafl(I) and \ud835\udcafr(\u039b) are the full transformation semigroups on the nonempty sets I and \u039b, respectively.A complete code  Theory of codes is an important branch in the field of information science. Many methods, including combinatorics methods, analysis methods, and algebraic methods, are applied to study codes. As a kind of algebraic methods, it is effective to study some kinds of codes by considering syntactic monoids of the semigroups and monoids generated by these codes.C+) of C+, the semigroup generated by a maximal prefix code C for which C+ is a single class of the syntactic congruence PC+.As we have known, prefix codes have fundamental importance in theory of codes. Many authors are devoted to the investigation of prefix codes by using several methods cf. \u20134). In p. In p4])On the other hand, synchronized codes are also important both in theory and in applications. Many interesting results are obtained on this class of codes in the text of Berstel et al. . RecentlC+) of C+ for a complete synchronized code C over an alphabet A such that C+ is a single class of its syntactic congruence PC+.In this paper, by using the algebraic characterization of complete synchronized codes obtained in Liu  and someS is a left zero semigroup if ab = a for any a, b \u2208 S. Dually, we have right zero semigroups. A rectangular band is a semigroup which is isomorphic to a direct product of a left zero semigroup and a right zero semigroup. For rectangular bands, we have the following obvious result.A semigroup S be a rectangular band. Then axa = a for any a, x \u2208 S. As a consequence, aSa = {a} for any a \u2208 S. In particular, if S has an identity e, then S = {e}.Let ideal of a semigroup S is a nonempty subset I of S satisfying that the union of IS and SI is contained in I. Recall that the unique minimum ideal (with respect to set inclusion) of a semigroup S (if exists) is called the kernel of S. For the kernel of a semigroup, we have the following.An S only consists of idempotents, then this kernel is a rectangular band.If the kernel of a semigroup S be a semigroup. A function \u03bb  on S is a left translation  of S if \u03bb(xy) = (\u03bbx)y \u03c1 = x(y\u03c1)) for all x, y \u2208 S. A left translation \u03bb and a right translation \u03c1 are linked if x(\u03bby) = (x\u03c1)y in which case the pair  is a bitranslation of S. Denote the set of left translations and that of right translations on S by \u039b(S) and P(S), respectively. Clearly, \u039b(S) forms a monoid under the usual composition of functions: (\u03bb\u03bb\u2032)x = \u03bb(\u03bb\u2032x) for all x \u2208 S. Dually, P(S) forms a monoid under the usual composition of functions: x(\u03c1\u03c1\u2032) = (x\u03c1)\u03c1\u2032 for all x \u2208 S. The set of all bitranslations of S forms a submonoid of the direct product \u039b(S) \u00d7 P(S), which is called the translation hull of S, to be denoted by \u03a9(S).Let s be an element of a semigroup S. Then the function \u03bbs defined by \u03bbsx = sx for all x \u2208 S is the inner left translation induced by s. Dually, we have inner right translation \u03c1s induced by s. Finally, the pair \u03c0s =  is the inner bitranslation induced by s. The set \u03a0(S) of all inner bitranslations is the inner part of \u03a9(S). From Corollary III.1.7 in Petrich \u03c8 = (xPC*)\u03c8(yPC*)\u03c8.(b)x = 1, y \u2260 1. In this case,PC*\u03b1, yPC*\u03b1y. This implies that yPC+\u03b1y by item (1) in xPC*)(yPC*)]\u03c8 = (xPC*)\u03c8(yPC*)\u03c8.(c)x \u2260 1, y = 1. This is the dual of case (b).(d)x \u2260 1, y \u2260 1. This is obvious.On the other hand, for any M is a monoid with identity 1 and M\u2216{1} is a subsemigroup of M, then M has a kernel if and only if M\u2216{1} has a kernel. If this is the case, the two kernels are equal.If Observe thatCombining Lemmas C over A is a synchronized code if and only if the kernel of Syn(C+) is a rectangular band.A complete code I and \u039b be two nonempty sets and \u2205 \u2260 K\u2286\ud835\udcafl(I)\u2009\u00d7\u2009\ud835\udcafr(\u039b). Assume that p1 and p2 are the projections onto the first and second components of \ud835\udcafl(I) \u00d7 \ud835\udcafr(\u039b), respectively. For ,  \u2208 I \u00d7 \u039b, we denoteLet C be a complete synchronized code over an alphabet A such that C \u2260 A and C+ is a PC+-class. Then Syn(C+) is isomorphic to a submonoid K of \ud835\udcafl(I) \u00d7 \ud835\udcafr(\u039b) for some nonempty sets I and \u039b with \u2260, and the following conditions hold:\ud835\udcaf0r(I) \u00d7 \ud835\udcaf0r(\u039b)\u2286K,K\u2216{} is a subsemigroup of K,i0, \u03bb0) \u2208 I \u00d7 \u039b such that the identity \u03a9i0,\u03bb0) =  for all ,  \u2208 I \u00d7 \u039b and the identity  =  implies that \u2208{, } for all  \u2208 K.there exists -submonoid of \ud835\udcafl(I) \u00d7 \ud835\udcafr(\u039b).The above submonoid M = Syn(C+) and denote the set of PC+-classes with representatives from T by T\u2286A*. Since C+ is a PC+-class, we can also let e is an idempotent in M.Let e is disjunctive in M. Let w \u2208 A* and c \u2208 C. Then PC+-class containing 1 is {1} by C+ is a single PC+-class, we have wc, cw, c \u2208 C+. Because C* is stable, it follows that w \u2208 C*. This implies that p, q \u2208 A* and M. Then for all s, t \u2208 A*, C+ is a single PC+-class, it follows that spt \u2208 C+ if and only if sqt \u2208 C+, and so pPC+q. Thus We first assert that J = MeM. We assert that J is the kernel of M. In fact, J is an ideal of M clearly. Moreover, since C is complete, there exist u, v \u2208 A* such that uwv \u2208 C+ for all w \u2208 A*. Therefore, there exist m, n \u2208 M such that mxn = e for all x \u2208 M. Now, let I be an ideal of M and x \u2208 I. Then uxv = e for some u, v \u2208 M whence J = MeM = MuxvM\u2286I. Thus, J is the least ideal of M and so is the kernel of M. By J is a rectangular band.Now, let J = {e}, then we have em = me = e for any m \u2208 M. This implies that c \u2208 C and w \u2208 A*. Since C+ is a single PC+-class, it follows that wc, cw, c \u2208 C+. Because C* is stable, we have w \u2208 C*. Therefore, C* = A* and hence C = A. A contradiction. Thus, J \u2260 {e}. Now let \u03c1 be a congruence on M whose restriction to J is the identity relation on J. Assume that x \u2208 M and x\u03c1e. Then xe\u03c1e\u03c1ex, where e, xe, ex \u2208 J and thus e = xe = ex. Since M, it follows that y \u2208 J, we obtain y\u03c1ye, y\u03c1ey, and ye, y, ey \u2208 J, whence y = ye = ey. Thus, J is a rectangular band with the identity e. And hence, J = {e} by x = e and {e} is a \u03c1-class in M. Now, let x, y \u2208 M and x\u03c1y. Then for any u, v \u2208 M, we have uxv\u03c1uyv. Observe that {e} is a \u03c1-class, it follows that uxv = e if and only if uyv = e. Therefore, xPey in M. By the disjunctiveness of e in M, we have x = y. We conclude that \u03c1 is the equality relation on M. Thus, M is a dense extension of J.If M is isomorphic to a subsemigroup of \u03a9(J) containing the inner part \u03a0(J) as an ideal. Since J is a rectangular band, J is isomorphic to the product I \u00d7 \u039b of a left zero semigroup I and a right zero semigroup \u039b. Observe that J \u2260 {e}, |J | = |I \u00d7 \u039b | >1, whence \u2260. By \u03a9(I \u00d7 \u039b)\u2245\ud835\udcafl(I) \u00d7 \ud835\udcafr(\u039b), and in this isomorphism \u03a0(I \u00d7 \u039b)\u2245\ud835\udcaf0l(I) \u00d7 \ud835\udcaf0r(\u039b). Therefore, M is isomorphic to a subsemigroup K of \ud835\udcafl(I) \u00d7 \ud835\udcafr(\u039b) and K contains \ud835\udcaf0l(I) \u00d7 \ud835\udcaf0r(\u039b) as an ideal. Furthermore, since M is a monoid, K has an identity. Let  be the identity of K. Then for any  \u2208 \ud835\udcaf0l(I) \u00d7 \ud835\udcaf0r(\u039b),Fi, \u03bb\u03a6) = . This implies that  = . Thus, K is a submonoid of \ud835\udcafl(I) \u00d7 \ud835\udcafr(\u039b). Since M\u2216{1} is a subsemigroup of M, it follows that K\u2216{} is a subsemigroup of K. Thus, Conditions (1) and (2) hold.By J is the kernel of M and e is an idempotent in J, it follows that the image of e in K is the form a =  for some i0 \u2208 I and \u03bb0 \u2208 \u039b. Since e is disjunctive in M, a is disjunctive in K. Let ,  \u2208 I \u00d7 \u039b and \u03a9i0,\u03bb0) \u2208 p1(M) \u00d7 p2(M). This implies thatF, \u03a6) \u2208 p1(M) \u00d7 p2(M). Thus,F, \u03a8),  \u2208 K. This shows that Pa and so  =  since a is disjunctive in K. Thus  = .Since F, \u03a6) \u2208 K and  = . Then,i0\u232a, \u2329\u03bb0\u232a) = . Since e is stable in M, it follows that a is stable in K. Hence, \u2208{, }. Therefore, Condition (3) is also satisfied.Finally, let -submonoid of \ud835\udcafl(I) \u00d7 \ud835\udcafr(\u039b). In fact, observe that \u03b12 = \u23292\u232a, M1 is submonoid of \ud835\udcafl(I) \u00d7 \ud835\udcafr(\u039b), and M1\u2216{} is a subsemigroup of M1. Let \u03bb, \u03bc \u2208 \u039b. If \u03bb = 1, \u03bc = 2 or \u03bb = 1, \u03bc = 3, then there exists \u03a6 = \u03b9\u039b \u2208 p2(M1) such that \u03bb\u03a6 = 1, \u03bc\u03a6 = \u03bc. If \u03bb = 2, \u03bc = 3, then there exists \u03a6 = \u03b1 \u2208 p2(M1) such that \u03bb\u03a6 = 2, \u03bc\u03a6 = 1. Furthermore, if  \u2208 M1 with F1 = 1, 1\u03a6 = 1, then F = \u23291\u232a = \u03b9I and \u03a6 = \u23291\u232a or \u03a6 = \u03b9\u039b. Thus, M1 is a -submonoid of \ud835\udcafl(I) \u00d7 \ud835\udcafr(\u039b). It is easy to see thatC+) onto M1.On the other hand, let"}
{"text": "We discuss the existence and uniqueness of solutions for initial value problems of nonlinear singular multiterm impulsive Caputo type fractional differential equations on the half line. Our study includes the cases for a single base point fractional differential equation as well as multiple base points fractional differential equation. The asymptotic behavior of solutions for the problems is also investigated. We demonstrate the utility of our work by applying the main results to fractional-order logistic models. The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a characteristic arise naturally and are often, for example, studied in physics, chemical technology, population dynamics, biotechnology, and economics. These processes are modeled by impulsive differential equations. In 1960, Milman and Myshkis introduced the concept of impulsive differential equations , LakshmiFractional differential equations (FDEs for short), regarded as the generalizations of ordinary differential equations to an arbitrary noninteger order, find their genesis in the work of Newton and Leibniz in the seventieth century. Recent investigations indicate that many physical systems can be modeled more accurately with the help of fractional derivatives . FractioSome recent work on the existence of solutions for initial value problems of Caputo type impulsive fractional differential equations can be found in a series of papers \u201316, wherDa+\u03b1x and Db\u2212\u03b1x, a is called a left base point and b right base point. Both a and b are called base points of fractional derivatives. A fractional differential equation (FDE) containing more than one base points is called a multiple base points FDE while an FDE containing only one base point is called a single base point FDE.In the left and right fractional derivatives t1 < t2 < \u22ef<tm < b, b > 0 is a fixed real number, f :  \u00d7 R \u2192 R is continuous, Ik, Jk : R \u2192 R\u2009\u2009 are continuous functions, u(tk+) = lim\u2061t\u2192tk+\u2061u(t) and u(tk\u2212) = lim\u2061t\u2192tk\u2212\u2061u(t) and u\u2032(tk+) = lim\u2061t\u2192tk+\u2061u\u2032(t). One can see that both fractional differential equations in =t1, t2,\u2026, tm (impulse points).In , the autDc\u03b1u(t)=t1, t2,\u2026, tm (impulse points).In , the autt1 < t2 < \u22ef<tm < b, b > 0 is a fixed real number, f :  \u00d7 R \u2192 R is jointly continuous, Ik : R \u2192 R\u2009\u2009 are continuous functions, u(tk+) = lim\u2061t\u2192tk+\u2061u(t) and u(tk\u2212) = lim\u2061t\u2192tk\u2212\u2061u(t) and u\u2032(tk+) = lim\u2061t\u2192tk+\u2061u\u2032(t). Observe that the fractional differential equation in \u03b1,\u2009\u2009\u03b2 \u2264 1,\u2009\u20090 < t1 < t2 < \u22ef are fixed points, D+0 is the Riemann-Liouville fractional derivative, \u03a6 : R \u2192 R is a sup-multiplicative function, f, g, h are impulsive Caratheodory functions, m, q, n, \u03c1 : \u2192 are continuous functions, and Ik, Jk are impulse functions. In  IVPs of one base point FDEs , 29, 30 x0 \u2208 R, \u03b1 \u2208  \u2192 R satisfies that there exists l > \u2212\u03b1 such that |q(t)|\u2264tl for all t \u2208 , q may be singular at t = 0, cD\u2217 is the standard Caputo fractional derivative at the base points t = tk\u2009\u2009; that is, cD\u2217\u03b1|tk,tk+1](u(t) = \u2009cDtk+\u03b1u(t) for all t \u2208  \u00d7 R2 \u2192 R is a Caratheodory function, Ik :  \u00d7 R \u2192 R\u2009\u2009 and {Ik} is a Caratheodory function sequence, and \u0394x(tk) = lim\u2061t\u2192tk+\u2061x(t) \u2212 lim\u2061t\u2192tk\u2212\u2061x(t),\u2009\u2009k = 1,2,\u2026.In this paper, we study the following two initial value problems (IVPs for short) of nonlinear multi-term FDEs with impulses on half lines:t \u2192 \u221e is established; (iii) the method of proof relies on the Schauder fixed point theorem; (iv) our approach for dealing with impulsive problems at hand is different from the ones employed in earlier work on the topic and thus opens a new avenue for studying impulsive fractional differential equations; (v) as an application, we apply our results to fractional-order logistic models and present sufficient conditions for the existence and asymptotic behavior of solutions of these logistic models.The salient features of the present work include the following: (i) to establish sufficient conditions for the existence of solutions for the IVP  with a sThe paper is organized as follows: the auxiliary material is given in We recall some basic concepts of fractional calculus , 10 and Define the Gamma function and Beta function, respectively, as\u03b1 > 0 of a continuous function f :  \u2192 R is given byRiemann-Liouville fractional integral of order \u03b1 for a function f \u2208 ACn\u22121), R) is defined byn \u2212 1 < \u03b1 \u2264 n, n \u2208 N. If 0 < \u03b1 \u2264 1, thenCaputo's derivative of fractional-order \u03b1 > 0, the general solution of fractional differential equation cD+0\u03b1x(t) = 0 is given by x(t) = c0 + c1t + c2t2 + \u22ef+cn\u22121tn\u22121, where ci \u2208 R, i = 0,1, 2,\u2026, n \u2212 1, \u2009n \u2212 1 < \u03b1 \u2264 n.For x :  are continuous, x satisfies the differential equation cD+0\u03b1x(t) = q(t)f, cD+0px(t)) a.e. on \u2216{t1, t2, t3,\u2026}, and the limits lim\u2061t\u2192tk+\u2061x(t) and lim\u2061t\u2192tk+\u2061cD+0px(t)\u2009\u2009 exist and the following conditions are satisfied:A function  the IVP  if both x :  are continuous, x satisfies the differential equation cDtk+\u03b1x(t) = q(t)f, cDtk+px(t)) on  and lim\u2061t\u2192tk+\u2061cD+0px(t)\u2009\u2009 exist and the following conditions are satisfied:A function  the IVP  if both \u03c3 > max\u2061{0, \u03b1 + l} and \u03bc > max\u2061{\u03c3, \u03c3 \u2212 \u03b1 \u2212 l}. Letx \u2208 X, define the norm on X asX is a real Banach space.Choose f : , there exists \u03b4 > 0 such thatx1, x2 \u2208  with |x1 \u2212 x2 | <\u03b4, k = 1,2,\u2026, N \u2212 1. From  In order to show that the operator (53)Since ff is a Caratheodory function, there exists \u03b41 > 0 such thatt \u2208  and u1, u2, v1, v2 \u2208  with |u1 \u2212 u2 | <\u03b41, |v1 \u2212 v2 | <\u03b41. From , we getAs 1. From , there et \u2208 (1 + t\u03bc))\u03a9 and (tp+\u03c3\u2212\u03b1\u2212l/(1 + t\u03bc))\u2009cD+0p\u03a9 are equicontinuous on any closed subinterval  of  and equiconvergent at t = tk\u2009\u2009, and t = \u221e.(b) Let us recall that W \u2282 X be a nonempty bounded set. To prove that T is completely continuous, we need to prove that TW is bounded, TW is equicontinuous on finite closed sub-interval on , TW is equiconvergent at t = tk\u2009\u2009, and TW is equiconvergent at t = \u221e.Let W is bounded, therefore, , a, aW is berefore,  hold forTW is equicontinuous on finite closed sub-interval on .Next we show that a, b]\u2282. For \u03bc > \u03c3 > 0, we find thatTW is equiconvergent as t \u2192 0+.Now wee prove that nd thatt\u03c3\u2212\u03b1\u2212l, we haveTW is equiconvergent as t \u2192 tk+\u2009\u2009.For TW is equiconvergent as t \u2192 \u221e. Observe thatTW is equiconvergent as t \u2192 \u221e.Our next task is to show that T is completely continuous. This completes the proof.From the above steps, it follows that H1), C \u2265 0 are real numbers;where 0 < H2) is a Caratheodory sequence and there exist numbers Aki \u2265 0\u2009\u2009, Dk \u2265 0\u2009\u2009, \u03b4i \u2265 0\u2009\u2009 such that{In the sequel, we need the following assumption:Furthermore, we setH1) and (H2) hold. Then IVP (x \u2208 X ifSuppose that (Then IVP  has at lX be the Banach space as defined in T : X \u2192 X be an operator given by (H1) and (H2) that T is well defined and is completely continuous. Thus, we seek solutions of IVP  and (H2), we find thatLet us introduceThus, by , it foll\u03b4m < 1, we can choose r0 > 0 sufficiently large such that [r0 + ||\u03a8||]\u03b4mM0 < r0. Let \u03a9r0 = {x \u2208 X : ||x|| < r0}. It is easy to see that T has a fixed point (i) For n of IVP .\u03b4m = 1, we choose\u03a9r0 = {x \u2208 X : ||x|| < r0}. Then it can easily be shown that T has a fixed point (ii) In case n of IVP .\u03b4m > 1, we choose r = r0 = ||\u03a8||/(\u03b4m \u2212 1) such that\u03a9r0 = {x \u2208 X : ||x|| < r0}. As before, it is easy to show that T has a fixed point (iii) For n of IVP . This coH1) and (H2) hold with \u03b4m = 1. Then IVP  are constants, \u2211k=1\u221e|ak| is convergent, and f is a Caratheodory function; there exists l \u2208  such that |q(t)|\u2264tl for all t \u2208 .Next, consider the following IVP:H1) and (H2) hold. Then every solution of (x0 + \u2211k=1\u221eak as t \u2192 \u221e provided that (Assume that the conditions (ution of  tends toded that  is satisf is a Caratheodory function by (H1), therefore, there exists Mr > 0 such thatBy Tm on Y asIn this section, we show the existence for solutions for IVP  with mulf is a Caratheodory function and {Ik} is a Caratheodory function sequence and \u03bb0 = :inf\u2061k=1,2,3,\u2026\u2061(tk \u2212 tk\u22121) > 0. ThenTm : Y \u2192 Y is well defined;Tm coincides with the solution of IVP  such thatTmx \u2208 Y. This implies that Tm : Y \u2192 Y is well defined.(i) For Tm coincides with the solution of IVP ((ii) It follows from n of IVP .Tm is completely continuous, we split the proof into several steps.(iii) To show that Step 1.\u2009\u2009Tm is continuous.xn \u2208 Y with xn \u2192 x0 as n \u2192 \u221e. We will prove that Tmxn \u2192 Tmx0 as n \u2192 \u221e. It is easy to see that there exists r > 0 such that\u03bb0 = inf\u2061k=1,2,\u2026\u2061(tk \u2212 tk\u22121) > 0, we get tk > k\u03bb0 for all k = 0,1, 2,\u2026.Let j=K+1\u221e(1/j\u03bc+1\u2212\u03c3) is convergent, there exists K > 0 such thatSince \u2211f is a Caratheodory function, there exists \u03b41 > 0 such thatt \u2208  and u1, u2 \u2208  with |u1 \u2212 u2 | <\u03b41, |v1 \u2212 v2 | <\u03b41. From  with n > N2, we haven\u2192\u221eTmxn = Tmx0 which implies that Tm is continuous.Since 1. From , there equence,sup\u2061k=0,ow thatsup\u2061k=0,W \u2282 X be a nonempty bounded set. To prove that Tm is completely continuous, we need to prove that TmW is bounded, TmW is equicontinuous on finite closed sub-interval on , TmW is equiconvergent at t = tk\u2009\u2009, and TmW is equiconvergent at t = \u221e.Let Step 2. As in the proof of TmW is bounded.Step 3. We prove that TmW is equicontinuous on finite closed sub-interval on . For \u2282.TmW is equiconvergent as t \u2192 0+. For t \u2192 tk+, we haveTmW is equiconvergent as t \u2192 tk+\u2009\u2009.As in Step 5.\u2009\u2009TmW is equiconvergent as t \u2192 \u221e. Notice thatTmW is equiconvergent as t \u2192 \u221e. This completes the proof in which Tm is completely continuous.H1) and (H2) hold. Then IVP  and (H2), we find that\u03b4m.For \u03b4m < 1, we can choose Tm has a fixed point (i) For n of IVP .\u03b4m = 1, we select(ii) For Tm has a fixed point Let n of IVP .\u03b4m > 1, we set (iii) For Tm has a fixed point Let n of IVP . This coH1) and (H2) hold with \u03b4m = 1. Then IVP (x \u2208 Y if N0 < 1.Suppose that (Then IVP  has a unThe proof is similar to that of N\u2032(t) = rN(t), where N(t) is the population at time t and r is the proportionality constant. When the growth of the population in any environment is stopped due to the density of the population, this model modifies to a nonlinear logistic model of the form N\u2032(t) = rN(t)(1 \u2212 N(t)/\u03c0). The generalization of the nonlinear logistic model is represented by N\u2032(t) = rN(t)[1 \u2212 (N(t)/\u03c0)\u03b1]/\u03b1. For \u03b1 \u2192 0, the model is known as the Gompertz model and can be found in the literature on actuarial science and mortality analysis of elderly person , and b(t)\u2208 with a\u2217 > 0, b\u2217 > 0.In , Das et \u03b2\u22641.In , the autt1 < t2 < t3 < \u22ef, \u03b1 \u2208  \u2192 R are continuous functions, and ak \u2208 R \u2192 R\u2009\u2009 are constants.As an application of the main results established in the paper, we discuss the sufficient conditions for the existence and asymptotic behavior of solutions for the logistic models:Dk \u2208 R, Ak1, Ak2 \u2265 0 such thatSuppose thatThen IVP  has at lf = u[a(t) \u2212 b(t)u\u03b4]. ThenC = 0, A1 = a0, A1 = a1, \u03b41 = 1, \u03b42 = 2, B1 = B2 = 0. Then the conditions (H1) and (H2) hold. By Let Dk \u2208 R, Ak1, Ak2 \u2265 0 such thatSuppose thatThen IVP  has at lThe proof immediately follows from"}
{"text": "We consider an interval-valued multiobjective problem. Some necessary and sufficient optimality conditions for weak efficient solutions are established under new generalized convexities with the tool-right upper-Dini-derivative, which is an extension of directional derivative. Also some duality results are proved for Wolfe and Mond-Weir duals. Recently, Yuan and Liu  considerIn many real-life situations data suffer from inexactness. The interval-valued optimization problems are closely related to optimization problems with inexact data. Recently, Wu \u201312 derivIn this paper we consider an interval multiobjective optimization problem. Some new optimality conditions and duality results are stated under new generalized convexities with the tool-right upper-Dini-derivative. The paper is organized as follows. In n be the n-dimensional Euclidean space and let \u211d+n be its nonnegative orthant. For x =  \u2208 \u211dn and y =  \u2208 \u211dn we consider the following conventions:X \u2282 \u211dn be an arcwise connected set in Avriel and Zang , there exists z \u2208 X such that f(z)\u2266tf(x)+(1 \u2212 t)f(u).A k-dimensional vector-valued function f : X \u2192 \u211dk is called \u03c1-generalized (strong) pseudoright upper-Dini-derivative arcwise connected (with respect to H) at u, if there exists vector-valued function \u03c1 such that f(x)<(\u2266)f(u)\u21d2(d\u03c6)+) < \u03c1, for x \u2208 X; f is called \u03c1-generalized (weak) quasi-right upper-Dini-derivative arcwise connected (with respect to H) at u, if there exists vector-valued function \u03c1 such that f(x)\u2266(<)f(u)\u21d2(d\u03c6)+)\u2266\u03c1, for x \u2208 X, where \u03c1 = T.A S \u2282 \u211dn be a nonempty set and let \u03a8 : S \u2192 \u211dm be a convex-like vector-valued function on S. Then either \u03a8(x) < 0 has a solution x \u2208 S, or there exists \u03bb \u2208 \u211d+m such that the system \u03bbT\u03a8(x)\u22670 holds for all x \u2208 S, but both are never true at the same time.Let a = \u2208 CBI(\u211d), we haveaL and aU mean lower and upper bounds of a. If aL = aU, then a =  is a real number. Also, let b = . Then, by definition we haveLet CBI(\u211d) be the class of all closed and bounded intervals in \u211d. Thus if \u03b1, we haveFor a real number Using , 18, we a = \u2009\u2009and b = \u2208 CBI(\u211d). We say that a is less than b and write a\u227ab if ai\u227abi, Let a, b\u2208 CBI(\u211dn). We say that a is less than or equal to b and write a\u227cb if aL\u2266bL and aU\u2266bU.Let X be a nonempty subset of \u211dn. A function \u03c6 : X\u2192CBI(\u211d) is called an interval-valued function. In this case,\u03c6L, \u03c6U : X\u27f6\u211d, \u03c6L(x)\u2266\u03c6U(x), \u2200x \u2208 X.Let g = T, fL = T, fU = T, where fiL, fiU : \u211dn \u2192 \u211d, fi(x) =  for gj : \u211dn \u2192 \u211d, X0 = {x \u2208 X : g(x)\u22660} be the set of all feasible points of +), (dfU)+), and (dgJ(x*))+\u2009) are convex-like on X with respect to the variable x and \u2009gj is upper semicontinuous at x* for j \u2208 J\u2216J(x*), then there exist \u03beL, \u2009\u2009\u03beU \u2208 \u211d+k, \u03bc \u2208 \u211d+m,  \u2260 0 such thatAssume that tion for . If\u2009\u2009 there exists \u03b4j > 0 such thatgj, for j \u2209 J(x*), is semicontinuous at x*, then gj) is semicontinuous at t = 0. Finally we gett \u2208 , where \u03b40 = min\u2061{min\u2061i\u2009\u2009\u03b4iL, min\u2061i\u2009\u2009\u03b4iU, min\u2061j\u2009\u2009\u03b4j} > 0. These inequalities contradict that x* is a weak efficient solution for (x \u2208 X. Now we can apply dfL)+, (dfU)+, and (dgJ(x*))+) are convex-like on X, we obtain that (11)(df\u03b1)ain that  and (10)dfL)+), (dfU)+), and (dgJ(x*))+) be convex-like on X with respect to the variables x and gj, for j \u2209 J(x*), be upper semicontinuous at x*. If there exists x\u2032 \u2208 X0 such thatx* is a weak efficient solution for the problem T \u2208 \u211d+k, \u03b1 \u2208 {L, U}, and \u03bc \u2208 \u211d+m satisfying  \u2260 0.Let , a, adfL)+\u03beL, \u03beU) = 0. Then, by (\u03bcj = 0 for j \u2208 J\u2216J(x*), by (j \u2208 J(x*) such that \u03bcj > 0, by (\u03bcT(dg)+) < 0, which contradicts , by , and the > 0, by , it resutradicts  and theoIn this section we give some Karush-Kuhn-Tucker type sufficient optimality conditions under generalized convexity with upper-Dini-derivative concept.x* be a feasible solution of  \u2260 0 and \u03bc*\u22670, i \u2208 J(x*), such that fis\u2009\u2009 and gj\u2009\u2009(j \u2208 J) are  and -right upper-Dini-derivative locally arcwise connected at x* with respect to H, respectively. Also one assumes \u03b1\u22670 andx \u2208 X0 feasible. Then x* is a weak efficient solution for (Let ution of . Assume tion for .x* is not a weak efficient solution for (x\u2032 \u2208 D such that f(x\u2032)\u227af(x*). Now, since  \u2265 0, \u03bcJ(x*)*\u22670, and gJ(x*)(x\u2032)\u22660 = gJ(x*)(x*), we getWe suppose to the contrary that tion for . Then thfis\u2009\u2009 and gj\u2009\u2009(j \u2208 J) are  and -right upper-Dini-derivative locally arcwise connected at x*, by \u22670 and  we get aThe next sufficient optimality condition is given in the case of generalized pseudo- and quasi-right upper-Dini-derivative arcwise connected type, where the proof is on the line of the above theorem.x* be a feasible solution of  \u2265 0 and \u03bc*\u22670, i \u2208 J(x*), such that fs\u2009\u2009 is generalized pseudoright upper-Dini-derivative locally arcwise connected at x* with respect to H and g is \u03c1\u2032-generalized quasi-right upper-Dini-derivative locally arcwise connected at x* with respect to H, respectively. Also one assumes that there exist \u03beL*, \u03beU*,  \u2265 0 and \u03bc* \u2265 0 such that (\u03beL*)T\u03c1L + (\u03beU*)T\u03c1U + \u03bcT\u2217\u03c1\u2009\u2009\u2032\u22670 for any x \u2208 X0. Then x* is a weak efficient solution for (Let ution of . Assume tion for .f(y) =  and e =  such that  =  = 1. Let FD denote the set of all feasible solutions of  be a feasible solution for  and g are  and -right upper-Dini-derivative arcwise connected at y on X0 \u222aprXFD with respect to H, respectively. Moreover, one assumes \u03b1\u22670 and (\u03beL)T\u03c1L + (\u03beU)T\u03c1U + \u03bcT\u03c1\u2032\u22670, for all x \u2208 X0. Then the following cannot hold: f(x)\u227a\u03a8.Let tion for  and  \u2265 0 and \u03bc* \u2208 \u211dm such that  \u2208 FD and the objective values of  for  is an efficient solution of  for , then  be a weak efficient solution of  and g are  and -right upper Dini derivative arcwise connected at y on X0 \u222a \u2009prXFD with respect to H, respectively. If \u03b1\u22670 and (\u03beL*)T\u03c1L + (\u03beU*)T\u03c1U + (\u03bc*)T\u03c1\u2032\u22670, for all x \u2208 X0, then y* is a weak efficient solution of  \u2208 \u211dk, with  =  = 1. Let FD denote the set of all feasible solutions of As in x and  be a feasible solution for  and g are  and -right upper-Dini-derivative arcwise connected at y on X0 \u222aprXFD with respect to H, respectively. Also one supposes that \u03b1\u22670 and (\u03beL)T\u03c1L + (\u03beU)T\u03c1U + \u03bcT\u03c1\u2032\u22670, for all x \u2208 X0. Then the following cannot hold: f(x)\u2aaff(y).Let tion for  and  \u2265 0, and \u03bc* \u2208 \u211dm, \u03bc*\u22670, such that  is a feasible solution for (IMWVD) and the objective values of  is a weak efficient solution for  be a weak efficient solution of  and g are  and -right upper-Dini-derivative arcwise connected at y on X0 \u222aprXFD with respect to H, respectively. Further, if \u03b1\u22670 and (\u03beL*)T\u03c1L+(\u03beU*)T\u03c1U+(\u03bc*)T\u03c1\u2032\u22670, for all x \u2208 X0, then y* is a weak efficient solution of (Let (ution of . Supposeution of .In this paper we considered an interval multiobjective optimization problem. Some new optimality conditions and duality results were studied under the generalized convexity considered by Yuan and Liu . Necessa"}
{"text": "Reference is missing a book title. The complete Reference 10 is: 10. Gannushkin MS (1961) (Book in Russian). \u041e\u0431\u0449\u0430\u044f \u044d\u043f\u0438\u0437\u043e\u043e\u0442\u043e\u043b\u043e\u0433\u0438\u044f  Moscow: State Publishing House of Agricultural Literature"}
{"text": "DC0+\u03b1u(t) = f, u\u2032(t)), 0 < t < 1, u(1) = u\u2032(1) = u\u2032\u2032(0) = 0, where 2 < \u03b1 \u2264 3 is a real number, DC0+\u03b1 is the Caputo fractional derivative, and f : \u00d7[0, +\u221e) \u00d7 R \u2192 [0, +\u221e) is continuous. Firstly, by constructing a special cone, applying Guo-Krasnoselskii's fixed point theorem and Leggett-Williams fixed point theorem, some new existence criteria for fractional boundary value problem are established; secondly, by applying a new extension of Krasnoselskii's fixed point theorem, a sufficient condition is obtained for the existence of multiple positive solutions to the considered boundary value problem from its auxiliary problem. Finally, as applications, some illustrative examples are presented to support the main results.This paper is devoted to the existence of multiple positive solutions for fractional boundary value problem  Then,a, b].Let \u03b1 > \u03b2 > 0. If one assumes that u(t) \u2208 C\u2229L, then D0+\u03b2I0+\u03b1u(t) = I0+\u03b1-\u03b2u(t).Let n \u2212 1 < \u03b1 \u2264 n\u2009\u2009(n \u2208 N). The fractional differential equation D0+\u03b1x(t) = 0 has solutionci \u2208 R, i = 0,1,\u2026, n \u2212 1, n = [\u03b1] + 1.Let g \u2208 C and 2 < \u03b1 \u2264 3, the unique solution of problemFor any G is said to be the Green function of BVP  = u\u2032(1) = u\u2032\u2032(0) = 0, one hasFrom the boundary condition Therefore, by The following properties of the Green function play an important role in this paper. G defined as ((i)G \u2208 C and G > 0 for any t, s \u2208 \u00d7. For 0 \u2264 s \u2264 t < 1, we have(i) It is obvious that G/\u2202t < 0 for all s, t \u2208 \u00d7 be endowed with the normFirst, let P \u2282 X byDefine the set P is a cone in the space X.It is easy to verify that \u03b8 on the cone P be defined byT on P by the formulaLet the nonnegative continuous concave function T : P \u2192 P is completely continuous. The operator u \u2208 P, we have that Tu(t) \u2265 0 in view of nonnegativeness of G and f. It is obvious thatFor any t \u2208  and \u03c4 \u2208 , we obtain thatT(P) \u2282 P.By T : P \u2192 P is continuous in view of continuity of G and f.The operator U \u2282 P be bounded; that is, there exists a positive constant K > 0 such that ||u|| \u2264 K for all u \u2208 U. By means of the definition of ||u||, we have |u(t)| \u2264 K and |u\u2032(t)| \u2264 K for t \u2208 . Let C = max\u2061t\u2208,u\u2208U\u2061|f, u\u2032(t))| + 1. Then, for u \u2208 U, by T(U) is bounded.Let u \u2208 U, t1, t2 \u2208  with t1 < t2, thenT is completely continuous. The proof is completed. For each Now, we are in a position to state the main results.For convenience, denote\u03c32 > \u03c31 > 0 such that H1) \u2264 M\u03c32 for \u2208\u00d7\u00d7;H2) \u2265 N\u03c31 for \u2208\u00d7\u00d7.Assume that there exist two positive constants u such that \u03c31 \u2264 ||u|| \u2264 \u03c32.Then, BVP  has at lT : P \u2192 P defined by  \u2264 \u03c32, \u2212\u03c32 \u2264 u\u2032(t) \u2264 \u03c32 for all t \u2208 . It follows from condition (H1) and t \u2208 ,Tu|| \u2264 ||u||, u \u2208 P\u2229\u2202\u03a92.(i) Let \u03a91 = {u \u2208 P : ||u|| < \u03c31}. For any u \u2208 P\u2229\u2202\u03a91, we have 0 \u2264 u(t) \u2264 \u03c31, \u2212\u03c31 \u2264 u\u2032(t) \u2264 \u03c31 for all t \u2208 . It follows from condition (H2) and t \u2208 ,Tu|| \u2265 ||u||, u \u2208 P\u2229\u2202\u03a91.(ii) Let T has a fixed point in In view of n P\u2229\u03a9\u00af2\u2216\u03a9 which isa < b < c such that the following assumptions hold:A1) < Ma for \u2208\u00d7\u00d7;A2) \u2265 Nb for \u2208\u00d7\u00d7;A3) \u2264 Mc for \u2208\u00d7\u00d7.Suppose that there exist constants 0 < u1, u2, and u3 withThen, BVP  has at lWe will show that all the conditions of u|| \u2264 c. By condition (A3) and Tu|| \u2264 c for First, if A1) that ||Tu|| < a for Next, by using the analogous argument, it follows from condition (u(t) = (b + c)/2 for t \u2208 . It is easy to see thatu \u2208 P | \u03b8(u) > b} \u2260 \u2205. Hence, if u \u2208 P, then b \u2264 u(t) \u2264 c for t \u2208 . By condition (A2), we have f, u\u2032(t)) \u2265 Nb for t \u2208 . So,\u03b8(Tu) > b for u \u2208 P.Choose u1, u2, and u3 withBy L > d > \u03b7d > q > 0 such thatB1) < q/M\u2032 for \u2208\u00d7\u00d7;B2) \u2265 d/N\u2032 for \u2208\u00d7\u00d7;B3) < L/Q for \u2208\u00d7\u00d7,whereAssume that there exist constants u(t) satisfying q < \u03b1(u) < d, |u\u2032(t)| < L for t \u2208 .Then, BVP  has at lf* : \u00d7, and condition (B2) thatStep 3. In view of condition (B3), for u \u2208 P\u2229\u039b2, we have\u03b2(T*u) < L. By u(t) = (T*u)(t). Consequently, u(t) is a positive solution for the auxiliary BVP (q < \u03b1(u) < d, |u\u2032(t)| < L. In addition, by virtue of the definition of f*, we know that f*, u\u2032(t)) = f, u\u2032(t)), t \u2208 . Therefore, u is a positive solution of BVP (iary BVP  satisfyiwe have5\u03b2(T\u2217u)\u2003=mConsider the following fractional BVP:\u03b7 = 1/6, N \u2248 33.333. Choosing \u03c31 = 1/104, \u03c32 = 1/10, we havet, u, u\u2032) \u2208 \u00d7\u00d7 andt, u, u\u2032) \u2208  \u00d7 \u00d7.By a simple calculation, one can obtain that u such that 1/104 \u2264 ||u|| \u2264 1/10.With the use of Consider the following fractional BVP:a = 1/10, b = 1/50, and c = 4, then, there holdt, u, u\u2032) \u2208 \u00d7\u00d7t, u, u\u2032) \u2208 \u00d7\u00d7; andt, u, u\u2032) \u2208 \u00d7\u00d7. Hence, all the conditions of u1, u2, and u3 such thatChoosing Consider the following fractional BVP:After a simple computation, one can find thatd = 36, q = 1/3, and L = 59 = 1953125, we know thatt, u, u\u2032) \u2208 \u00d7\u00d7;t, u, u\u2032) \u2208 \u00d7\u00d7; andt, u, u\u2032) \u2208 \u00d7\u00d7. So, all the conditions of u(t) satisfying 1/3 < max\u2061t\u2208\u2061|u(t)| < 36, |u\u2032(t)| < 59.Choosing In this paper, we study the existence of multiple positive solutions for the nonlinear fractional differential equation boundary value problem  in the C"}
{"text": "The problem of absolute stability of Lur'e systems with sector and slope restricted nonlinearities is revisited. Novel time-domain and frequency-domain criteria are established by using the Lyapunov method and the well-known Kalman-Yakubovich-Popov (KYP) lemma. The criteria strengthen some existing results. Simulations are given to illustrate the efficiency of the results. Absolute stability of nonlinear systems has been investigated comprehensively for the past several decades \u201312. It iIn this paper, both time-domain criterion and frequency-domain criterion for absolute stability of Lur'e systems with sector and slope restricted nonlinearities are presented based on the Lyapunov method and the KYP lemma. Some mathematical tools are used through the derivation of the absolute stability criterion. Compared with some existing results, the proposed results are less conservative. This should be owed to the effect of the slope restricted conditions on the nonlinearities. The rest of the paper is organized as follows. In W, W > 0 (W \u2265 0) denotes that W is a positive definite (semidefinite) matrix and W < 0 denotes that W is a negative definite matrix. Re{Y} means (1/2)(Y + Y*) for any real or complex square matrix Y.Throughout this paper, the superscript \u2217 means transpose of real matrices and conjugate transpose of complex matrices. For a Hermitian matrix A \u2208 \u211dn\u00d7n, B \u2208 \u211dn\u00d7m, and C \u2208 \u211dn\u00d7m are real matrices, \u03c6(0) = 0, m, and \u03c6i(\u03c3i(t))\u2009\u2009 are assumed to satisfy\u03b3i2 \u2264 \u03b3i1, \u03b4i2 \u2265 \u03b4i1, \u03b3i2 \u2264 0, and \u03b4i2 \u2265 0. The inequalities (\u03c6(\u03c3(t)), respectively. Let \u03931 = diag\u2061, \u03941 = diag\u2061, \u03932 = diag\u2061, \u03942 = diag\u2061. Then \u03932 \u2212 \u03931 \u2264 0, \u03942 \u2212 \u03941 \u2265 0, \u03932 \u2264 0, and \u03942 \u2265 0. Setting \u03c8i(\u03c3i(t)) = d\u03c6i(\u03c3i(t))/dt, (\u03c6(\u03c3(t)) to \u2212\u03c3(t) is denoted as \u03c7(s) = C*(A \u2212 sI)\u22121B.Consider the following multi-input and multioutput Lur'e system)\u2009\u2009i = 1,,\u2026, m arei\u03c3i2(t),\u03b32i\u2264d\u03c6i(\u03c3i\u03c3i2(t),\u03b32i\u2264d\u03c6i(\u03c3x(t) = 0 is globally asymptotically stable for all nonlinear vector valued functions\u2009\u2009\u03c6(\u03c3(t)) satisfying ..x(t) = 0tisfying  are giveA \u2208 \u211dn\u00d7n, B \u2208 \u211dn\u00d7m, and symmetric matrix \u03a3 \u2208 \u211dn+m)\u00d7(n+m) \u2260 0 for \u03c9 \u2208 \u211d, and the pair  is controllable, the following two statements are equivalent.\u03c9 \u2208 \u211d.P = P* such that A, B) is not controllable.There exists a matrix Given that S22 > 0 and S11 + S12S22\u22121S12* < 0;S11 < 0 and S22 + S12*S11\u22121S12 > 0.The LMI P = P* and \u03bbi \u2208 \u211d\u2009\u2009 are necessary to be determined. It should be pointed out that P is not necessary to be positive definite and \u03bbi\u2009\u2009 are not necessary to be nonnegative.We choose the following Lur'e-Postnikov function:\u03c6(\u03c3(t)) satisfying , T1 \u2265 0, T2 > 0, and symmetric matrices P such that the LMI is feasible:System  is absoltisfying  and 3)  \u03c6(\u03c3(t)) V(x(t)).We will demonstrate that the given conditions imply the negative definiteness of V(x(t)) along the trajectory of (\u03c6i(\u03c3i(t)) are equivalent toti1 \u2265 0 and ti2 > 0, i = 1,2,\u2026, m, it followsT1 = diag\u2061 \u2265 0 and T2 = diag\u2061 > 0. ThenTaking the derivative of ctory of , we havenditions  and 4)  V(x(t)) ondition  guarante= diag\u2061t,\u2026, t1m V(x(t)) is positive definite. In ) is a little difficult and complex. Without loss of generality, letting \u03bbi < 0\u2009\u2009 and \u03bbi \u2265 0\u2009\u2009\u2009\u2009(0 \u2264 k \u2264 m), then V(x(t)) has the following form:k1 = diag\u2061 and \u0393k1 = diag\u2061. Since (\u03c3i(t)(\u03c6i(\u03c3i(t)) \u2212 \u03b3i1\u03c3i(t)) \u2265 0, \u2211i=k+1m\u03bbi\u222b0\u03c3i(t)(\u03c6i(s) \u2212 \u03b3i1s)\u2009ds \u2265 0 is satisfied. Then V(x(t)) is positive definite if P + (1/2)C\u039b\u03931C* + (1/2)C\u039b(\u0394k1 \u2212 \u0393k1)C* is positive definite, which is proved in what follows.Now we are only left to demonstrate that nite. In , P is on0\u2009\u2009i = 1,,\u2026, k andA1 = A + B\u03931C*, P1 = P + (1/2)C\u039b\u03931C*, Ak = A1 + B(\u0394k1 \u2212 \u0393k1)C*, and Pk = P1 + (1/2)C\u039b(\u0394k1 \u2212 \u0393k1)C*. Firstly, the given conditions imply that \u03b1 satisfying 0 < \u03b1 \u2264 1 can be found such that\u03c90 \u2208 \u211d. Since A1 is Hurwitzian, det\u2061(j\u03c90I \u2212 A1) \u2260 0 and \u03bd \u2260 0 such thatG(j\u03c90) = C*(j\u03c90I \u2212 A1)\u22121B. Then we deriveW1*, we haveG(j\u03c9) = C*(j\u03c9I \u2212 A1)\u22121B and j\u03c9G(j\u03c9) = C*A1(j\u03c9I \u2212 A1)\u22121B + C*B. Letting \u03c9 = \u03c90 in (Ak = A + B\u03931C* + B(\u0394k1 \u2212 \u0393k1)C* is Hurwitzian. Secondly, the given conditions imply that P + (1/2)C\u039b\u03931C* + (1/2)C\u039b(\u0394k1 \u2212 \u0393k1)C* is positive definite. Actually, pre- and postmultiplying both sides of ). This completes the proof.Denote sides of  by W1=[I we have[A1\u2217P1+P1lement, ([A1\u2217P1+P1ve that ([(j\u03c9I\u2212A1)quality (\u2212T1+12(\u03941e derive\u03bd\u2217\u0394~{\u2212T1+ows that\u03bd\u2217\u0394~{\u2212T1+ we have[A1\u2217P1+P1A + B\u03931C* is Hurwitzian if and only if P + (1/2)C\u039b\u03931C* > 0.It is found in the proof of \u03c6(\u03c3(t)) satisfying C\u039b\u03931C* > 0 and the LMI (System  is absoltisfying  and 3)  \u03c6(\u03c3(t))  the LMI  holds.The LMI  can be t\u03c6(\u03c3(t)) satisfying   \u03c6(\u03c3(t)) \u03c7(j\u03c9) = C*(A\u2212j\u03c9I)\u22121B. Substituting \u2212 lemma, ([(j\u03c9I\u2212A\u2212),where2\u03a911=[\u03a311\u03a3tituting  is deriv1 = 0, the FDI (For the case \u0393 the FDI  reduces .15.1 in . However\u03c6(\u03c3(t)) are removed, another absolute stability criterion is derived by choosing (If the slope restrictions on choosing  as the L\u03c6(\u03c3(t)) satisfying A*C\u039b + (1/2)CT(\u03931 + \u03941), \u03a922 = (1/2)\u039bC*B + (1/2)B*C\u039b \u2212 T.System  is absoltisfying  if the mThe proof is similar to that of Similar to \u03c6(\u03c3(t)) satisfying  = C*(A\u2212j\u03c9I)\u22121B and C*A(A\u2212j\u03c9I)\u22121B = C*B + j\u03c9\u03c7(j\u03c9).From the KYP lemma,  is equiv between  and 31)(32)[) = m1x1(t) + (1/2)(m0 \u2212 m1)(|x1(t) + 1| \u2212 |x1(t) \u2212 1|), \u03b1, \u03b2, \u03b3, m0, and m1 are numbers. System (\u03c3(t) = x1(t), and \u03c6(\u03c3(t)) = m1\u03c3(t) + (1/2)(m0 \u2212 m1)(|\u03c3(t) + 1| \u2212 |\u03c3(t) \u2212 1|). The nonlinearity \u03c6(\u03c3(t)) satisfies1 = \u03932 = min\u2061{m0, m1} and \u03941 = \u03942 = max\u2061{m0, m1}.Consider Chua's oscillator  with thex\u02d91(t)=\u03b1[\u03b1 = \u22120.8018, \u03b2 = 0.136, \u03b3 = 0.1097, and m0 = \u22122.96 are taken, system (m1 \u2264 2.009 by applying m1 \u2264 1.81 and m1 \u2264 1.51, respectively, by m1 = 2 at the initial value When , system  is absolWe have proposed new absolute stability criteria for Lur'e systems with sector and slope restricted nonlinearities from time-domain and frequency-domain points of view. The slope restrictions on nonlinearities improve the absolute stability conditions. We have shown that the criteria are less conservative than some existing results."}
{"text": "The exponential inequality for weighted sums of a class of linearly negative quadrant dependent random variables is established, which extends and improves the corresponding ones obtained by Ko et al. (2007) and Jabbari et al. (2009). In addition, we also give the relevant precise asymptotics. X and Y are said to be negative quadrant dependent  if P \u2264 P(X > x)P(Y > y) for all x, y \u2208 R. Based on the concept of NQD, another notion of negative dependence was formulated by Newman  denotes the integral part of x, Sn = \u2211i=1nXi, \u03c3n2 = ESn2, and u(n) = sup\u2061i\u22651\u2211j:|j\u2212i|\u2265n|Cov| and denote log\u2061n = ln\u2061(n\u2228e). This paper is organized as follows. Throughout this paper, we always suppose that (A1)Xi, i \u2265 1} be a sequence of stationary LNQD random variables with |Xi | \u2264M and let {ani : 1 \u2264 i \u2264 n, n \u2265 1} be a triangular array of numbers satisfying |ani | \u2264M1, where M and M1 are generic positive constants.Let {(A2)pn, n \u2265 1} be a positive integer sequence satisfying pn \u2264 n/2 and pn \u2192 \u221e as n \u2192 \u221e.Let {In this section, our main results will be given. For formulation of the theorems obtained, some assumptions are needed, which are listed below.\u03b5 < 1, one hasC1 and C are positive constants.Suppose that the assumption (A1) holds. Then for any 0 < Taking\u03b1 > 1. Suppose that \u03b5 is as in (ani = 1 for 1 \u2264 i \u2264 n.Assume that the assumptions (A1) and (A2) hold. Let is as in . Then oni=1n(Xi \u2212 EXi)/n is O(1)n\u22121/2log\u20611/2n, which is obviously faster than the corresponding one n\u22121/2(pnlog\u2061n)1/2 that Ko et al.  + 1 and then definej = 1,2,\u2026, rn andi=1nani(Xi \u2212 EXi) = Sn1 + Sn2 and n < 2rnpn \u2264 2n.Next, we give some notations used later. Define Now, we can obtain the following lemma.Xi, i \u2265 1} be a sequence of stationary LNQD random variables with |Xi| \u2264 M and let {ani : 1 \u2264 i \u2264 n, n \u2265 1} be a triangular array of numbers satisfying |ani| \u2264 M1, where M and M1 are generic positive constants. If 0 < \u03bbpnMM1 \u2264 1/2 for some \u03bb > 0, then on account of Definitions (Let {initions  and 10)Xi, i \u2265 ani \u2265 0. Applying EYnj = 0 and \u03bbpnMM1 \u2264 1/2, we have by |\u03bbYnj| \u2264 2\u03bbpnMM1 thatani \u2265 0 and \u03bb > 0, {\u03bbYnj, 1 \u2264 j \u2264 rn} is LNQD. Therefore, from Lemma 3.1 in Ko et al. [Sn2. The proof is completed.Without loss of generality, assume that of LNQD,\u220fj=1rnE, so that this exponent becomes equal to \u2212n\u03b52/(16C) as desired. The proof is completed.Applying Markov inequality and N is a standard normal random variable.Under the conditions of x \u2208 Rx) denotes the standard normal distribution function.Under the conditions of Based on the above lemmas, the proofs of Theorems \u03bb > 0 which satisfies \u03bbpnMM1 \u2264 1/2; then it follows from Let \u03c3 = 1 in what follows. Since\u03b2 > \u22121.Without loss of generality, set P(|N | \u2265x) = 2P(N \u2265 x) for any x > 0. It is easy to observe that\u03b2 > \u22121, which implies  = exp\u2061(M/\u03f52), where M > 4 and 0 < \u03f5 < 1/4. Obviously,I1. Set n \u2192 0 as n \u2192 \u221e. It follows that\u03f5\u21980\u03f5\u03b2+1)2(I1 = 0. Turn to I2. By \u03f5. Thus we obtain\u03f5\u21980\u03f5\u03b2+1)22(I3 \u2192 0, when M \u2192 \u221e. Combining the earlier results together yields (Next, we will prove . Let J(\u03f5ove thatlim\u2061\u03f5\u21980\u03f52"}
{"text": "By applying techniques in the theory of convex functions and Schur-geometrically convex functions, the author investigates a conjecture of Satnoianu on an algebraic inequality and generalizes some known results in recent years. For more information on the theory of means, see the monograph  \u2265 \u22ef\u2265xn][ and y[1] \u2265 \u22ef\u2265yn].The set \u03c6 : \u03a9 \u2192 \u211d+ is said to be Schur-geometrically convex on \u03a9 if ln\u2061x = \u227aln\u2061y =  implies \u03c6(x) \u2264 \u03c6(y) for every x, y \u2208 \u03a9.A function \u03c6 : \u03a9 \u2192 \u211d+ is said to be Schur-geometrically concave on \u03a9 if ln\u2061x = \u227aln\u2061y =  implies \u03c6(x) \u2265 \u03c6(y) for every x, y \u2208 \u03a9.A function Let \u03a9\u2286\u211d+n be a symmetric and geometrically convex set with inner points and \u03c6 : \u03a9 \u2192 \u211d+ a symmetric and differentiable function in \u03a9\u00b0. Then \u03c6 is a Schur-geometrically convex  function on \u03a9 if and only ifLet r \u2265 1 and t \u2265 \u22121 or for r \u2264 0 and t > \u22121. If\u2009\u20090 < r < 1 and t \u2265 \u22121, inequality r\u22651+Now we start off to demonstrate our main results.m, M be defined as in m\u03b2 \u2265 2pM\u03b1, thenp)M\u03b2 \u2264 2pm\u03b1, inequality )r is convex on ;for r \u2264 1, or for r > 1 and r \u2212 1 \u2212 2m\u2032 < 0, the function (t/(1 + t))r is concave on .for 0 < By Jensen's inequality, when r < 0, or when r > 1 and r \u2212 1 \u2212 2M\u2032 > 0, we haveu = \u2208n. When 0 < r \u2264 1, or when r > 1 and r \u2212 1 \u2212 2m\u2032 < 0, inequality (For  we have\u2211i=1n(ui1ui = \u03b1xiq/\u03b2An(xq) for i = 1,2,\u2026, n, r = 1/p, m\u2032 = \u03b1m/\u03b2M, and M\u2032 = \u03b1M/\u03b2m. Then it is easy to show that An(u) = \u03b1/\u03b2. Making use of m, if (1 \u2212 p)\u03b2 > 2p(n \u2212 1)\u03b1, thenp)(n \u2212 1)\u03b2 < 2p\u03b1, inequality (Under the conditions of )\u03b1, then\u2211i=1nn;if r < 0 and 1 + rM\u2032 > 0, the function f(u) is Schur-geometrically concave on n.if By the fact thatr > 0, or if r < 0 and 1 + rm\u2032 < 0, we haver < 0 and 1 + rM\u2032 > 0, inequality (Using  we have\u2211i=1n(1+uui = \u03b2Gn(xq)/\u03b1xiq for i = 1,2,\u2026, n leads to Gn(u) = \u03b2/\u03b1. Putting r = \u22121/p, m\u2032 = \u03b2m/\u03b1M and M\u2032 = \u03b2M/\u03b1m in inequality m, if \u03b2 \u2265 p(n \u2212 1)\u03b1, thenn \u2212 1)\u03b2 \u2264 p\u03b1, inequality (Under the conditions of )\u03b1, then\u2211i=1n[xiqn \u2265 2 and p > 1, from When n \u2265 2, \u03b1, \u03b2, xi \u2208 \u211d+ for i = 1,2,\u2026, n, and p, q \u2208 \u211d with p \u2260 0. If p \u2265 \u22121, thenp \u2264 \u22121, inequality r is a convex  function on \u211d+ for r \u2265 1 or r < 0 , by Jensen's inequality, if r \u2265 1 or r < 0, we haver \u2264 1, inequality (Since (1 +  we have\u2211i=1n(1+uui = \u03b2Hn(xq)/\u03b1xiq and r = \u22121/p shows An(u) = \u03b2/\u03b1. Further from 22) both n \u2265 2, \u03b1, \u03b2, xi \u2208 \u211d+ for i = 1,2,\u2026, n, p, q \u2208 \u211d with p \u2260 0, and m, M defined as in ((1)p < 0, one hasWhen\u2009\u2009\u22121 \u2264 (2)p < 1 and (1 \u2212 p)m\u03b2 > 2pM\u03b1, one hasWhen 0 < (3)p > 0 and m\u03b2 > pM\u03b1, one hasWhen (4)p > 0 and M\u03b2 < mp\u03b1, one hasWhen Let This follows from utilizing the well-known harmonic-geometric-arithmetic mean inequalityM = (n \u2212 1)m,(1)p < 1 and (1 \u2212 p)\u03b2 > 2p(n \u2212 1)\u03b1, one hasif\u2009\u20090 < (2)p > 0 and \u03b2 > p(n \u2212 1)\u03b1, one hasif (3)p > 0 and (n \u2212 1)\u03b2 < p\u03b1, one hasif Under the conditions of"}
{"text": "We consider fluctuation relations between the transport coefficients of a spintronic system where magnetic interactions play a crucial role. We investigate a prototypical spintronic device - a spin-diode - which consists of an interacting resonant level coupled to two ferromagnetic electrodes. We thereby obtain the cumulant generating function for the spin transport in the sequential tunnelling regime. We demonstrate the fulfilment of the nonlinear fluctuation relations when up and down spin currents are correlated in the presence of both spin-flip processes and external magnetic fields. Nonequilibrium fluctuation relations overcome the limitations of linear response theory and yield a complete set of relations that connect different transport coefficients out of equilibrium using higher-order response functions -7. Even We recently proved nonequilibrium fluctuation relations valid for spintronic systems , fully t\u03b1=L,R, as shown in Figure \u03c1\u03b1\u2191(\u03c9)\u2260\u03c1\u03b1\u2193(\u03c9) /(\u03c1\u03b1\u2191+\u03c1\u03b1\u2193). In the limit of \u03b5 is the dot level spacing, kB is the Boltzmann constant, and T is the temperature) effectively only a single energy level \u03b5\u03c3  in the dot contributes to the transport and can be occupied by 0, 1, or 2 electron charges. In the presence of an external magnetic field B, the Zeeman splitting is \u03b5\u2191\u2212\u03b5\u2193=g\u03bcBB (g is the Land\u00e9 factor and q as the electron charge). Tunneling between lead \u03b1 and the dot yields a level broadening given by \u0393\u03b1\u03c3(\u03c9)=\u03a0\u03c1\u03b1\u03c3|V\u03b1|2 (V\u03b1 is the lead-dot tunneling amplitude). Notice that the level width is then spin-dependent due to the spin asymmetry of the density of states: \u0393\u03b1\u03c3=(\u0393/2)(1+sp\u03b1), with \u0393=\u0393L=\u0393R and s=+(\u2212) for \u2191(\u2193).Consider a quantum dot coupled via tunnel barriers to two ferromagnetic leads \u0393\u226akBT, tunneling occurs sequentially, and transport is thus dominated by first-order tunnelling processes. The dynamics of the system is governed by the time evolution of the occupation probabilities calculated from the master equation P\u2261{P0,P\u2191,P\u2193,P2} denoting the probabilities associated to states with 0 electrons on the dot, 1 electron with spin \u2191 or \u2193 and 2 electrons. We also take into account spin-flip relaxation mechanisms possibly present in our system due to magnetic interactions with a spin-fluctuating environment  or spin-orbit interactions in the dot: \u03c7={\u03c7L\u2191,\u03c7L\u2193,\u03c7R\u2191,\u03c7R\u2193} the counting fields:In the limit of weak dot-lead coupling, f\u00b1(\u03b5)=1/[ exp(\u00b1\u03b5/kBT)+1]. Here, V\u03b1\u03c3 is a spin-dependent voltage bias, and \u03bci\u03c3 is the dot electrochemical potential to be determined from the electrostatic model. i=0,1 is an index that takes into account the charge state of the dot. Then, the cumulant generating function in the long time limit is given by \u03bb0(\u03c7) denotes the minimum eigenvalue of \u03c7. From the generating function, all transport cumulants are obtained [where obtained .Q\u2191 and Q\u2193: Q\u2191=Cu1(\u03d5\u2191\u2212VL\u2191)+Cu2(\u03d5\u2191\u2212VL\u2193)+Cu3(\u03d5\u2191\u2212VR\u2191)+Cu4(\u03d5\u2191\u2212VR\u2193)+C(\u03d5\u2191\u2212\u03d5\u2193) and Q\u2193=Cd1(\u03d5\u2193\u2212VL\u2191)+Cd2(\u03d5\u2193\u2212VL\u2193)+Cd3(\u03d5\u2193\u2212VR\u2191)+Cd4(\u03d5\u2193\u2212VR\u2193)+C(\u03d5\u2193\u2212\u03d5\u2191), where C\u2113i represent capacitance couplings for \u2113=u/d and i=1\u22ef4. We then find the potential energies for both spin orientations, N\u03c3 being the excess electrons in the dot. For an empty dot, i.e., N\u2191=N\u2193=0, its electrochemical potential for the spin \u2191 or \u2193 level can be written as \u03bc\u03c30=\u03b5\u03c3+U\u03c3\u2212U\u03c3. This is the energy required to add one electron into the spin \u2191 or \u2193 level when both spin levels are empty.We consider a gauge-invariant electrostatic model that treats interactions within a mean-field approach . For theN\u2191=1 or N\u2193=1, and we find Importantly, our results are gauge invariant since they depend on potential differences  between fluctuations \u0394G\u03b1,\u03b2  to the equilibrium noise G\u03b1,\u03b2\u03b3, and the noise susceptibilities, S\u03b1\u03b2,\u03b3, by means of a fluctuation relation,Small fluctuations around equilibrium and their responses are related through the fluctuation-dissipation theorem. In particular, the Kubo formula for the electrical transport relates the linear conductance  systems . Thus, ipL=p and a normal lead with polarization pR=0. We take into account the presence of spin-flip processes described by \u03b3sf. In Figure VL\u2191=V1, VL\u2193=V2, VR\u2191=V3, VR\u2193=V4. When the dot is subjected to an externally applied magnetic field, one must consider the antisymmetrized version of Equation 4 using A\u2212=A(B)\u2212A(\u2212B), where A can be G, S, or higher order correlation functions  and zero magnetic field (\u03b5\u2191=\u03b5\u2193). Then, we are able to obtain an analytical expression for the cross correlations between \u2191 and \u2193 currents in the left terminal:We now discuss the analytical expressions for the spin noises of our spin diode. We consider that the system is biased with a source-drain voltage \u03b5eff=\u03b5+e2/2C\u03a3, with p increases independently of \u03b3sf. Moreover, SL\u2191L\u2193 is always negative due to the antibunching behavior of fermions [where fermions . The shoFL\u2191L\u2191=SL\u2191L\u2191/IL\u2191,with an associated Fano factor \u03b5eff lies inside the transport window. This is due to correlations induced by Coulomb interactions [Notably, the Fano factor is always sub-Poissonian whenever ractions .Nonequilibrium fluctuation relations nicely connect nonlinear conductances with noise susceptibilities. We have derived spintronic fluctuation relations for a prototypical spintronic system: a spin diode consisting of a quantum dot attached to two ferromagnetic contacts. We have additionally investigated the fulfilment of such relations when both spin-flip processes inside the dot and an external magnetic field are present in the sample. We have also inferred exact analytical expressions for the spin noise current correlations and the Fano factor. Further extensions of our work might consider noncollinear magnetizations and energy dependent tunneling rates.The authors declare that they have no competing interests.RL and DS defined the research subject. JSL and RL performed the calculations. All authors discussed the results and co-wrote the paper. All authors read and approved the final manuscript."}
{"text": "T\u03b11 can stimulate T cell proliferation and differentiation from bone marrow stem cells, augment cell-mediated immune responses, and regulate homeostasis of immune system. In this study, we developed a novel strategy to produce T\u03b11 concatemer (T\u03b11\u2462) in Escherichia coli and compared its activity with chemically synthesized T\u03b11. Results showed that T\u03b11\u2462 can more effectively stimulate T cell proliferation and significantly upregulate IL-2 receptor expression. We concluded that the expression system for T\u03b11 concatemer was constructed successfully, which could serve as an efficient tool for the production of large quantities of the active protein.Thymosin alpha 1 (T Trypanosoma cruzi transsialidase stabilized the catalytic activity. In addition, repeats present on T. cruzi shed proteins increased trans-sialidase half-life in blood from 7 to almost 35\u2009h [The tandem repeats of proteins and peptides are studied widely and formidable progress has been made in this field. It was reported that tandem amino acid repeats have many functions of stabilizing proteins , maintaiost 35\u2009h , 5.\u03b11) is a heat-stable, acidic polypeptide composed of 28 amino acid residues blocked at the N-terminus by an acetyl group [\u03b11 could be a useful restorative therapeutic agent in the treatment of immunodeficiency diseases and immunosuppressed conditions [Thymosin alpha1  in E. coli TOP10 strain and purified by heat treatment and Q-Sepharose Fast Flow ion-exchange chromatography. Then, T\u03b11\u2462 was released by treatment with 0.5\u2009M Cyanogen bromide (CNBr) and purified by SP-Sepharose Fast Flow chromatography. In our strategy, trx acts as a chaperon to help T\u03b11\u2462 folding and CNBr treatment removed any exogenous amino acid (such as Met at the N-terminus for translation start) from T\u03b11\u2462 molecule. So we can get the \u201cnatural\u201d T\u03b11\u2462. Finally, the biological activity of T\u03b11\u2462 on T lymphocyte proliferation and IL-2R expression was assessed.In this study, TE. coli strain TOP10 (F-mcrA\u0394(mrr-hsd RMS-mcrBC) \u03c680 lacZ\u0394M15 \u0394lacX74 recA1 ara\u0394139\u0394(ara-leu)7697 galU galK rpsL (Strr)endA1 nupG) were from Invitrogen. DNA fragments were synthesized in BIOASIA. Synthetic T\u03b11 (ZADAXIN) was from Sciclone Pharmaceuticals, USA. The anti-T\u03b11 antibody (ab55635) was purchased from Abcam and FITC-anti-IL-2R\u03b2 (18344D 554452) was from BD Pharmingen. Restriction enzymes, Taq DNA polymerase, and T4 DNA ligase were purchased from TaKaRa. Expression vector pThioHisA and \u03b11\u2462 gene was cloned by gene synthesis and PCR (EcoR I site) was p1: 5\u2032-GGAATTCATGTCTGATGCAGCCGTGGAC ACCAGCAGCG-3\u2032 and the reverse primer (with an introduced Pst I site) was p2: 5\u2032-GCACTGCAGTCAGTTCTGGGCCTCCTCCACCACCT-3\u2032. The template for PCR was annealing products of 4 synthesized fragments listed in \u03b11\u2462 for identification.T and PCR . The for\u03b11\u2462 was digested with EcoR I and Pst I and cloned into expression vector pThioHisA digested with the same enzymes. The candidate plasmid pThioHisA-T\u03b11\u2462 was then confirmed by restriction enzymes digestion and DNA sequencing. The vector pGEM-T\u03b11\u2462 was transformed into E. coli TOP10 strain. A single colony was inoculated into 10\u2009mL Luria-Bertani (LB) medium supplemented with ampicillin (100\u2009\u03bcg/mL) and grown at 200\u2009rpm and 37\u00b0C overnight. Then it was inoculated into 300 mL fresh LB medium in a 500 mL shake flask and cultured until the OD600 reached 0.5. Trx-T\u03b11\u2462 expression was induced by 1\u2009mM IPTG  for 4\u2009h. Large scale fed-batch culture was performed in a 5-L fermentor as previously described [The plasmid pThioHisA-Tescribed .\u03b11\u2462 was then cleaved by CNBr (0.5\u2009M) in 70% formic acid for 24\u2009h. The cleavage reaction was stopped by addition of ten times amount of H2O [\u03b11\u2462 was purified by SP-Sepharose Fast Flow chromatography. Purified T\u03b11\u2462 was dialyzed against PBS for later use. Cell pellet was suspended in 20\u2009mM Tris/HCl buffer (pH 8.0) in proportion of 200\u2009g/L and disrupted by sonication. Then, the lysate was incubated at 80\u00b0C for 10\u2009min (shaken once every 2-3\u2009min) and cooled quickly. Samples were centrifuged at 12\u2009000\u2009g for 20\u2009min and the supernatant was loaded onto a Q-Sepharose Fast Flow chromatography column and eluted with linear NaCl gradient. The purified Trx-T\u03bcm, Invitrogen) after SDS-PAGE using a Bio-Rad Semi-Dry electrophoretic cell. Western blot analyses were carried out using a T\u03b11 specific antibody and followed by a phosphatase-conjugated goat anti-mouse IgG . Western Blue Stabilized Substrate (Promega) for alkaline phosphatase was used for visualization. Proteins were transferred to nitrocellulose membranes  and cultured in the presence of 2.5\u2009\u03bcg/mL concanavalin A (ConA) at 37\u00b0C in 5% CO2 in humid air. Six h later, 90\u2009\u03bcL of T\u03b11\u2462 diluted with RPMI 1640 media was added to all but the control wells. The synthetic T\u03b11 and media were used as positive and negative controls. After 66\u2009h incubation, 20\u2009\u03bcL of MTT (0.5\u2009mg/mL) solution was added and the plates were centrifuged  4\u2009h later. Supernatants were discarded, and 100\u2009\u03bcL of DMSO was added. After incubated at room temperature for 10\u2009min, the solubilized reduced MTT was measured at 570\u2009nm using a Bio-Rad plate reader and the optical densities were used for calculate growth rate with the formulaThe proliferation response of splenocytes was determined by MTT assay. Spleens from C57BL6 mice were dispersed through nylon mesh to generate a single-cell suspension. Then lymphocytes were separated by EZ-Sep 1\u00d7 Lymphocyte Separation Medium  and suspended at 4 \u00d7 10\u03b11\u2462 on the expression level of IL-2R on T lymphocytes, cells were isolated as before and cultured in the presence of ConA and T\u03b11\u2462. The synthetic T\u03b11 and a recombinant T\u03b11 monomer prepared in our lab were used as positive controls. Cells were collected and stained 48\u2009h later according to standard protocol. In brief, 5 \u00d7 105 cells were washed with PBS and stainedin \u201cFACS buffer\u201d  with FITC-anti-mIL-2R\u03b2 for 10\u2009min at room temperature. After washing, cells were fixed for 30 minutes on ice with 4% paraformaldehyde and analyzed on a FACSCalibur flow cytometer (BD Biosciences). To evaluate the effect of T\u03b11 has been successfully applied in clinical trials for immunodeficiency diseases therapy, the high costs is still a hard nut to crack. Fortunately, molecular biology techniques allowed us to produce recombinant T\u03b11 in E. coli. Considering that T\u03b11 is too small to be directly expressed in E. coli, it was usually assembled as concatemers. But some exogenous amino acid residues such as His6 tag or methionine (Met) introduced by the initiation codon AUG usually affects the effect of concatemers [ Although synthetic Tcatemers . \u03b11, we put forward a new strategy as showed in \u03b11 concatemers in which T\u03b11\u2462 that was assembled by three repeated copies of T\u03b11 gene owned highest proportion. After cloning into pGEM-3Zf vector, the gene was proven by enzyme digestion and DNA sequencing. The sequence of T\u03b11\u2462 gene was consistent with our design as follows: 5\u2032-atgagcgacgccgccgtggacaccagcagcgagatcaccaccaaggaccggaaggagaagaaggaggtggtggaggaggccgagaacagcgacgccgccgtggacaccagcagcgagatcaccaccaaggaccggaaggagaagaaggaggtggtggaggaggccgagaacagcgacgccgccgtggacaccagcagcgagatcaccaccaaggaccggaaggagaagaaggaggtggtggaggaggccgagaactga-3\u2032. In order to produce the real \u201cnatural\u201d concatemers of T\u03b11\u2462/TOP10 showed that a new 31\u2009kDa protein which can be specifically recognized by T\u03b11 antibody was produced. It suggested that trx-T\u03b11\u2462 was successfully expressed. Trx was used as a chaperon to guarantee the correct folding of T\u03b11\u2462 and trx-T\u03b11\u2462 was expressed as a soluble fusion protein. Both SDS-PAGE  and West\u03b11 are heat-stable proteins, so trx-T\u03b11\u2462 was easily purified by one-step Q-Sepharose Fast Flow chromatography after the lysate of recombinant bacterial cells was heated at 80\u00b0C for 10\u2009min , whereas 5\u2009\u03bcg/mL T\u03b11\u2462 could induce significant proliferation (P < 0.05). Furthermore, the effect of 10\u2009\u03bcg/mL T\u03b11\u2462 was stronger than that of 40\u2009\u03bcg/mL synthetic T\u03b11  of T\u03b11\u2462 apparently stimulated the proliferation of T lymphocytes compared with that of ZADAXIN (40\u2009\u03bcg/mL). In addition, T\u03b11\u2462 significantly upregulated IL-2R on T cell, which is very important for T cell activation and proliferation in vivo. The detailed mechanism for stronger effect of T\u03b11\u2462 and the pharmacokinetics of different tandem repeats are still under investigation. Trx-T"}
{"text": "L-fuzzy lattice is presented by means of an L-fuzzy partially ordered set. An L-fuzzy partially ordered set A is an L-fuzzy lattice if and only if one of Aa][, and Aa)( is a lattice. The concept of  L-fuzzy lattice based on a fuzzy set of a crisp lattice, so the scope is very limited, and they gave its characterizations by only one kind of its cut sets. To overcome this shortcoming, in this paper we try to present a new definition of an L-fuzzy lattice in an L-fuzzy subset of a general set in terms of an L-poset. Furthermore, its many characterizations are given using the theory of L\u03b2-sets and L\u03b1-sets proposed by Shi Fu-Gui in 1995. In this way, we obtain more generalized conclusion about L-fuzzy lattice in order to make their applications becoming comprehensive.Many concepts of fuzzy algebra were presented, since the concept of fuzzy subgroups was introduced by Rosenfeld. The concept of fuzzy lattice was also introduced by Tepavajkovski . But theajkovski  defined L denotes a completely distributive lattice, and M(L) denotes the set of all nonzero \u2228-irreducible elements in L. P(L) denotes the set of all nonunit prime elements in L. X denotes a nonempty usual sets. LX is the set of all L-fuzzy sets on X. We will not differ a crisp set from its character function. For empty set \u2205 \u2282 L, we define \u22c0\u2205 = 1 and \u22c1\u2205 = 0. According to [ \u2282 Aa)( \u2282 Aa][ and a \u2208 \u03b1(b) implies Aa][ \u2282 Ab)( = Aa) = \u22c1a\u2208M(L)(a\u2227Aa][) = \u22c0a\u2208P(L)(a\u2228Aa][ is a mapping from Aa][ into Ba][(Ca][(Ca][)\u2286f(C)a][)\u22121(Da][)\u22121(Da][)\u2286(f\u22121(D))a], where x, y, and\u2009\u2009z are different and  is an interval. We define the order in L as follow.Let e \u2208 , \u2009e \u2264 c = 1/2, \u2009c < a, \u2009c < b, \u2009a\u2270b, \u2009b\u2270a, \u2009a < 1, \u2009and\u2009\u2009b < 1. The order in  is as usual. Then L is a completely distributive lattice. Take R \u2208 LX\u00d7X such thatR is an L-fuzzy partial order on X. But it is easy to check thatR(1/2) is not a partial order on X.For all A \u2208 LX, x, y, s \u2208 X, R be an L-fuzzy partial order on A. s is called an L-fuzzy supremum of x, y if the following conditions are true:(S1)A(s) \u2265 A(x)\u2227A(y)(S2)R \u2265 R; \u2009R \u2265 R,(S3)R \u2265 R\u2227R.Let A \u2208 LX, x, y, t \u2208 X, R be an L-fuzzy partial order on A. t is called an L-fuzzy infimum of x, y if the following conditions are true:(T1)A(t) \u2265 A(x)\u2227A(y),(T2)R \u2265 R; \u2009R \u2265 R,(T3)R \u2265 R\u2227R. Let L-fuzzy partially ordered set  is called an L-fuzzy lattice on X if for any x, y \u2208 A(0), both L-fuzzy supremum and L-fuzzy infimum of x, y exist.An A, R) be an L-fuzzy partially ordered set. Then the following conditions are equivalent.A, R) is an L-fuzzy lattice on X. is a lattice.For any a \u2208 M(L),  is a lattice.For any a \u2208 \u03b1(0),  is a lattice.For any a \u2208 \u03b1*(0),  is a lattice.For any a \u2208 P(L), (Aa) is a lattice.For any Let \u21d2(2)\u21d2(3)\u21d2(6)\u21d2(1). Since above-mentioned sets have been posets, we only need to prove that supremum and infimum of a \u2208 L\u2216{0}, let x, y \u2208 A(0) and x, y \u2208 Aa][. Analogously we can prove that  \u2208 Ra][ in Aa][ in Aa][ in Aa][ and  \u2208 Ra][ = Ab]\u222a{a, b, 1X}, where a\u2270b, b\u2270a.\u2009\u2009A \u2208 LX  is a sublattice of X; that is, A is a fuzzy lattice of X. While for 1/2 \u2208 \u03b2*(1), A(1/2) = {e, f, g, 1X} is not a sublattice of X. This implies that (1)\u21cf(6).Then The following condition makes (1)\u21d2(6) be true.b, c \u2208 L, \u03b2(b\u2227c) = \u03b2(b) \u22c2 \u03b2(c), then (1)\u21d2(6) in the previous theorem is true.If for all a \u2208 L, x, y \u2208 Aa)), a \u2208 \u03b2(A(y)). Therefore, a \u2208 \u03b2(A(x)\u2227A(y)) = \u03b2(A(x))\u2229\u03b2(A(y)). From (1) we know that A(x\u2228y) \u2265 A(x)\u2227A(y) and A(x\u2227y) \u2265 A(x)\u2227A(y). So we have that a \u2208 \u03b2(A(x\u2228y)) and a \u2208 \u03b2(A(x\u2227y)). This shows that (Aa) is a sublattice of .(1)\u21d2(6). For each X be a nonempty set, let A, B be fuzzy lattices of X, and if B \u2264 A, we call B as a fuzzy sublattice of A.Let X, Y be nonempty sets, and let A, B be fuzzy sublattices of X, Y, respectively. L-fuzzy mapping f : A \u2192 B is named a fuzzy lattice homomorphism if it satisfies that for all a \u2208 P(L), fa)( : Aa)( \u2192 Ba) is a fuzzy sublattice of B;if D is a fuzzy sublattice of B, then f\u22121(D) is a fuzzy sublattice of A.if Let X, Y be nonempty sets, and let A, B be fuzzy sublattices of X, Y, respectively. f : A \u2192 B is a fuzzy mapping, then the following conditions are true:f : A \u2192 B is a fuzzy lattice homomorphism;a \u2208 M(L), fa][ is a lattice homomorphism from Aa][ to Ba][. Thereforefa][(x\u2227y) = fa][(x)\u2227fa][(y). This implies that fa][ : Aa][ \u2192 Ba][\u2286Aa)(. From (2) we know that fb][ : Ab][ \u2192 Bb][\u2286fa) \u2208 fb][\u2286fa)(. Hencefa][(x\u2227y) = fa][(x)\u2227fa][ is a lattice homomorphism from Aa][ to Ba][ is a lattice homomorphism from Aa][ to Ba][.for all Let A = X, and then the conditions are true obviously. So we can get another special case of bi-fuzzy lattice-fuzzy lattice. Now we give its definition and corresponding theorems.In Definitions L be a completely distributive lattice, X \u2260 \u2205, and let R be an L-fuzzy relation on X, if R satisfies for any x, y, z \u2208 X, there exist s, t \u2208 X such that(S1)R \u2265 R, R \u2265 R,(S2)R \u2265 R\u2227R,(T1)R \u2265 R, R \u2265 R,(T2)R \u2265 R\u2227R. then we call s, t as supremum and infimum of x, y with respect to R, respectively.Let L be a completely distributive lattice, X \u2260 \u2205, and let R be an L-fuzzy relation on X, if for any x, y \u2208 X, both supremum and infimum of x, y with respect to R exist, and then we call X as a fuzzy lattice with respect to R.Let Same to the corresponding theorems in last section,we have the following theorem.L be a completely distributive lattice, X \u2260 \u2205, and let R be an L-fuzzy relation; then (1), (2), (3), (6), (7), and (8) of the following conditions are equivalent, and (4)\u21d2(5)\u21d2(1) is true.X is a fuzzy lattice with respect to R.a \u2208 L\u2216{0},  is a lattice.For each a \u2208 M(L),  is a lattice.For each a \u2208 \u03b2(1),  is a lattice.For each a \u2208 \u03b2*(1),  is a lattice.For each a \u2208 \u03b1(0),  is a lattice.For each a \u2208 \u03b1*(0),  is a lattice.For each a \u2208 P(L),  is a lattice.For each Let"}
{"text": "D+0\u03bd1y1(t) = \u03bb1a1(t)f(y1(t), y2(t)), \u2212 D+0\u03bd2y2(t) = \u03bb2a2(t)g(y1(t), y2(t)), where D+0\u03bd is the standard Riemann-Liouville fractional derivative, \u03bd1, \u03bd2 \u2208 ((0) = 0 = y2i)((0), for 0 \u2264 i \u2264 n \u2212 2, and [D+0\u03b1y1(t)]t=1 = 0 = [D+0\u03b1y2(t)]t=1, for 1 \u2264 \u03b1 \u2264 n \u2212 2, or y1i)((0) = 0 = y2i)((0), for 0 \u2264 i \u2264 n \u2212 2, and [D+0\u03b1y1(t)]t=1 = \u03d51(y1), [D+0\u03b1y2(t)]t=1 = \u03d52(y2), for 1 \u2264 \u03b1 \u2264 n \u2212 2, \u03d51, \u03d52 \u2208 C. Our results are new and complement previously known results. As an application, we also give an example to demonstrate our result.By using Krasnoselskii's fixed point theorem, we study the existence of at least one or two positive solutions to a system of fractional boundary value problems given by \u2212 Assume that \u03a91, \u2009\u03a92 are open bounded subsets of E with 0 \u2208 \u03a91, Tu|| \u2264 ||u||, \u2200u \u2208 P\u22c2\u2009\u2202\u03a91, and ||Tu|| \u2265 ||u||, \u2200u \u2208 P\u22c2\u2009\u2202\u03a92; or||Tu|| \u2265 ||u||, \u2200u \u2208 P\u22c2\u2009\u2202\u03a91, and ||Tu|| \u2264 ||u||, \u2200u \u2208 P\u22c2\u2009\u2202\u03a92.||Let T has a fixed point in Then In this section, we apply E represent the Banach space of C when equipped with the usual supremum norm, ||\u00b7||. Then put X : = E \u00d7 E, where X is equipped with the norm |||| : = ||y1|| + ||y2|| for  \u2208 X. Observe that X is also a Banach space  is the Green function of \u03bd replaced by \u03bd1 and, likewise, G2 is the Green function of \u03bd replaced by \u03bd2. Now, we define an operator S : X \u2192 X byy1, y2) \u2208 X is a fixed point of the operator defined in (y1(t) and y2(t) solve problems . In addX \u2192 X bySy1,y2). For. ForS, bFor the sake of convenience, we setF1);F2);F3), f and a1(t).The operator \u03a9\u2286K be bounded; that is, there exists a positive constant M > 0 such that |||| \u2264 M, for all  \u2208 \u03a9. Let L = max\u2061t\u22641,\u20090\u2264||||\u2264M0\u2264 | a1(t)f(y1(t), y2(t))|+1; then, for  \u2208 \u03a9, we haveT1(\u03a9) is bounded.Let \u03b5 > 0, setting \u03b4 = min\u2061{(1/2)(\u0393(\u03bd1)\u03b5/L\u03a62)\u03bd1\u22121)1//(\u03bd1 \u2212 1)L\u03a62}, then, for each  \u2208 \u03a9, t1, t2 \u2208 , t1 < t2, and t2 \u2212 t1 < \u03b4, one has |T1(t2) \u2212 T1(t1)|<\u03b5. That is to say, T(\u03a9) is equicontinuity. In fact,On the other hand, given In the following, we divide the proof into two cases.Case 1. If \u03b4 \u2264 t1 < t2 < 1, then we havet\u03be \u2208 .Case 2. If 0 \u2264 t1 < \u03b4, t2 < 2\u03b4, then we haveT1 is completely continuous. Similarly, T2 is completely continuous. Consequently, S : K \u2192 K is a completely continuous operator. This completes the proof.In , GoodricF1)\u2013(F3) are satisfied. Then problem (Suppose that F\u2013(F3) ar problem  has at lf0 = f\u221e = g0 = g\u221e = 0\u2009\u2009or\u2009\u2009f0 = f\u221e = g0 = g\u221e = \u221e? In the rest of this paper, we give some answers to this problem.From H1)\u2009\u2009, (P1) are satisfied. Then problem , Suppose that  \u221e and H, (P1) a problem  has at lS is a completely continuous operator. At first, in view of f0 = g0 = \u221e, we have f \u2265 M(y1 + y2), for 0 < |||| < r1* < \u03c11; g \u2265 M(y1 + y2), for 0 < |||| < r2* < \u03c11, where M satisfies M \u2265 1. Set \u03c10 : = min\u2061{r1*, r2*}. So we define \u03a9\u03c10 by \u03a9\u03c10 : = { \u2208 X : |||| < \u03c10}. Then for each  \u2208 K\u22c2\u2202\u03a9\u03c10, we find thatT1|| \u2265 (1/2)|||| for  \u2208 K\u22c2\u2202\u03a9\u03c10.From T2|| \u2265 (1/2)|||| for  \u2208 K\u22c2\u2202\u03a9\u03c10. Consequently,y1, y2) \u2208 K\u22c2\u2202\u03a9\u03c10. Thus, S is cone expansion on K\u22c2\u2202\u03a9\u03c10.Similarly, we find that ||f\u221e = g\u221e = \u221e, we have f \u2265 M1(y1 + y2) for y1 + y2 \u2265 r1** > \u03c11;\u2009\u2009\u2009g \u2265 M1(y1 + y2) for y1 + y2 \u2265 r2** > \u03c11, where M1 satisfies M1 \u2265 1. Set \u03c110 : = max\u2061{r1**, r2**}. Let \u03a9\u03c10* : = { \u2208 X : |||| < \u03c10*}. Then  \u2208 K\u22c2\u2202\u03a9\u03c10* impliesT1|| \u2265 (1/2)|||| for  \u2208 K\u22c2\u2202\u03a9\u03c10*.Next, since T2|| \u2265 (1/2)|||| for  \u2208 K\u22c2\u2202\u03a9\u03c10*.Similarly, we find that ||S2|| \u2265 ||||, whenever  \u2208 K\u22c2\u2202\u03a9\u03c10*. Thus, S is cone expansion on K\u22c2\u2202\u03a9\u03c10*.Consequently, ||\u03a9\u03c11 : = { \u2208 X : |||| < \u03c11}. For  \u2208 K\u22c2\u2202\u03a9\u03c11, from (H1), (P1), we haveT2|| \u2264 (1/2)|||| for  \u2208 K\u22c2\u2202\u03a9\u03c11.Finally, let S|| \u2264 ||||, whenever  \u2208 K\u22c2\u2202\u03a9\u03c11. Thus, S is cone compression on K\u22c2\u2202\u03a9\u03c11.Consequently, ||S has a fixed point So, from d point y0,y20\u2208K\u22c2(f0 = f\u221e = g0 = g\u221e = 0 and (H2), (P2) are satisfied. Then problem , Suppose that  problem -2) has  has f0 f0 = g0 = 0, we have f < \u03b5(y1 + y2), g < \u03b5(y1 + y2), for 0 < |||| \u2264 \u03c1 < \u03c12, where \u03b5 satisfies \u03b5 \u2264 1. Let \u03a9\u03c1 : = { \u2208 X : |||| < \u03c1}.At first, in view of y1, y2) \u2208 K\u22c2\u2202\u03a9\u03c1, we find thatS|| \u2264 |||| for  \u2208 K\u22c2\u2202\u03a9\u03c1.Then for each  < \u03b51(y1 + y2), g < \u03b51(y1 + y2), for y1 + y2 \u2265 \u03c1\u2032 > \u03c12, where \u03b51 satisfies \u03b51 \u2264 1. We consider two cases.Next, in view of Case 1. Suppose that f is unbounded; there exists \u03c1* > \u03c1\u2032 such that\u03c1* > \u03c1\u2032, one has f \u2264 f < \u03b51(y1* + y2*) for 0 \u2264 |||| \u2264 \u03c1*. Then, for  \u2208 K and |||| = \u03c1*, we obtainCase 2. Suppose that f is bounded; there exists L1 such that f \u2264 L1 for all  \u2208 K. Taking \u03c1* \u2265 max\u2061{2\u03c12, L1}, for  \u2208 K and |||| = \u03c1*, we obtain\u03a9\u03c1* : = { \u2208 X : |||| < \u03c1*} such that ||T1|| \u2264 (1/2)|||| for  \u2208 K\u22c2\u2202\u03a9\u03c1*. Like S|| \u2264 ||||, for  \u2208 K\u22c2\u2202\u03a9\u03c1*.\u03a9\u03c12 : = { \u2208 X : |||| < \u03c12}. Then  \u2208 K\u22c2\u2202\u03a9\u03c12 impliesT1|| \u2265 (1/2)|||| for  \u2208 K\u22c2\u2202\u03a9\u03c12. Like S|| \u2265 |||| for  \u2208 K\u22c2\u2202\u03a9\u03c12.Finally, set S has a fixed point So, from d point y0,y20\u2208K\u22c2(\u03c11 \u2260 \u03c12 such thatH3), g \u2264 B1\u22121\u03c11, for 0 \u2264 |||| \u2264 \u03c11;H4), g \u2265 B2\u22121\u03c12, for In the following, for the sake of convenience, setH3) and (H4) are satisfied. Then problem  such that |||| between \u03c11 and \u03c12.Suppose that ( problem -2), in , in H3)\u03c11 < \u03c12.With loss of generality, we may assume that \u03a9\u03c11 : = { \u2208 X : |||| < \u03c11}. For  \u2208 K\u22c2\u2009\u2202\u03a9\u03c11, one hasS|| \u2264 |||| for  \u2208 K\u22c2\u2202\u03a9\u03c11.Let \u03a9\u03c12 : = { \u2208 X : |||| < \u03c12}. Then for  \u2208 K\u22c2\u2202\u03a9\u03c12, one hasS|| \u2265 |||| for  \u2208 K\u22c2\u2202\u03a9\u03c12. Hence, from Now, set \u03bb1 = \u03bb2 = 1 is not considered.In , problemConsider the following.y1, y2) \u2208 X is a solution of  is a fixed point of the operator U : X \u2192 X defined by\u03b21, \u03b22 : \u2192 are defined byA pair of functions y, y2 \u2208 Xution of  if and o\u03b21(t) and \u03b22(t) is strictly increasing in t and satisfies \u03b21(0) = \u03b22(0) = 0 and \u03b21(1), \u03b22(1)\u2208. Moreover, there exist constants M\u03b21 and M\u03b22 satisfying M\u03b21, M\u03b22 \u2208  such that min\u2061t\u2208\u03b21(t) \u2265 M\u03b21||\u03b21|| and min\u2061t\u2208\u03b22(t) \u2265 M\u03b22||\u03b22||.Each of K1 by\u03b30 \u2208 .Let one define a new cone U : K1 \u2192 K1 is a completely continuous operator.D1)(\u03d51(y1) \u2264 ||y1||/4, \u03d52(y2) \u2264 ||y2||/4 for each  \u2208 K1;P3), (D1), (P3) are satisfied. Then problem ,Suppose that  \u221e and H, (D1), (D1), P are satif0 = g0 = \u221e, we have f \u2265 M(y1 + y2), g \u2265 M(y1 + y2), for 0 < |||| \u2264 \u03c10 < \u03c11, where M satisfies M \u2265 1.At first, in view of \u03a9\u03c10 : = { \u2208 X : |||| < \u03c10}. Then for each  \u2208 K1\u22c2\u2202\u03a9\u03c10, we find thatU1|| \u2265 (1/2)|||| for  \u2208 K1\u22c2\u2202\u03a9\u03c10.Let U2|| \u2265 (1/2)|||| for  \u2208 K1\u22c2\u2202\u03a9\u03c10. Consequently,y1, y2) \u2208 K1\u22c2\u2202\u03a9\u03c10. Thus, U is cone expansion on K1\u22c2\u2202\u03a9\u03c10.Similarly, we find that ||f\u221e = g\u221e = \u221e, we get f \u2265 M1(y1 + y2), g \u2265 M1(y1 + y2), for y1 + y2 \u2265 \u03c110 > \u03c11, where M1 satisfies M1 \u2265 1. Let \u03c10* = max\u2061{2\u03c11, (\u03c110/\u03b30)} and \u03a9\u03c10* : = { \u2208 X : |||| < \u03c10*}; then,  \u2208 K\u22c2\u2202\u03a9\u03c10* implies y1, y2) \u2208 K1\u22c2\u2202\u03a9\u03c10*, we obtainU1|| \u2265 (1/2)|||| for  \u2208 K1\u22c2\u2202\u03a9\u03c10*.Next, since U2|| \u2265 (1/2)|||| for  \u2208 K1\u22c2\u2202\u03a9\u03c10*. Consequently, ||U|| \u2265 ||||, whenever  \u2208 K1\u22c2\u2202\u03a9\u03c10*. Thus, U is cone expansion on K1\u22c2\u2202\u03a9\u03c10*.Similarly, we find that ||\u03a9\u03c11 : = { \u2208 X : |||| < \u03c11}. For  \u2208 K\u22c2\u2202\u03a9\u03c11, from (H1), (D1), and (P3), we haveU2|| \u2264 (1/2)|||| for  \u2208 K1\u22c2\u2202\u03a9\u03c11. Consequently, ||U|| \u2264 ||||, whenever  \u2208 K1\u22c2\u2202\u03a9\u03c11. Thus, U is cone compression on K1\u22c2\u2202\u03a9\u03c11.Finally, let U has a fixed point So, from d point y0,y20\u2208K1\u22c2s of BVP  with(51f0 = f\u221e = g0 = g\u221e = 0 and (H2), (D1), (P4) are satisfied. Then problem , Suppose that (H2), D, (P4) a problem  has at lf0 = g0 = 0, we have f < \u03b5(y1 + y2), g < \u03b5(y1 + y2) for |||| \u2264 \u03c1 < \u03c12, where \u03b5 satisfies \u03b5 \u2264 1. Let \u03a9\u03c1 : = { \u2208 X : |||| < \u03c1}. Then for each  \u2208 K1\u22c2\u2009\u2202\u03a9\u03c1, we find thatU|| \u2264 |||| for  \u2208 K1\u22c2\u2009\u2202\u03a9\u03c1.At first, in view of f\u221e = g\u221e = 0, we have f < \u03b51(y1 + y2), g < \u03b51(y1 + y2), for y1 + y2 \u2265 \u03c1\u2032 > \u03c12, where \u03b51 satisfies \u03b51 \u2264 1. We consider two cases.Next, in view of Case 1. Suppose that f is unbounded and there exists \u03c1* > \u03c1\u2032 such that\u03c1* > \u03c1\u2032, one has f \u2264 f < \u03b51(y1* + y2*) for 0 \u2264 |||| \u2264 \u03c1*.y1, y2) \u2208 K1 and |||| = \u03c1*, we obtainThen, for  \u2264 L1 for all  \u2208 K1. Taking \u03c1* \u2265 max\u2061{2\u03c12, L1}, for  \u2208 K1 and |||| = \u03c1*, we obtain\u03a9\u03c1* : = { \u2208 X : |||| < \u03c1*} such that ||U1|| \u2264 (1/2)|||| for  \u2208 K1\u22c2\u2009\u2202\u03a9\u03c1*.U|| \u2264 ||||, for  \u2208 K1\u22c2\u2009\u2202\u03a9\u03c1*.Like \u03a9\u03c12 : = { \u2208 X : |||| < \u03c12}. Then  \u2208 K1\u22c2\u2009\u2202\u03a9\u03c12 impliesU1|| \u2265 (1/2)|||| for  \u2208 K1\u22c2\u2009\u2202\u03a9\u03c12.Finally, set U|| \u2265 |||| for  \u2208 K1\u22c2\u2009\u2202\u03a9\u03c12. So, from U has a fixed point Like Uy, y2|| \u2265s of BVP  with 0<|\u03bb1 = \u03bb2 = 1 having at least one positive solution. In the following, we also establish the existence of one positive solution to problem , g \u2264 B3\u22121\u03c11 for 0 \u2264 |||| \u2264 \u03c11;H6), g \u2265 B2\u22121\u03c12 for \u03b30\u03c12 \u2264 |||| \u2264 \u03c12.For the sake of convenience, setH5),\u2009\u2009(H6), and (D1) are satisfied. Then problem  such that |||| between \u03c11 and \u03c12.Suppose that , and D are sati problem , in the \u03c11 < \u03c12. Let \u03a9\u03c11 : = { \u2208 X : |||| < \u03c11}. For  \u2208 K1\u22c2\u2009\u2202\u03a9\u03c11, from (H7), (D1), one hasU|| \u2264 |||| for  \u2208 K1\u22c2\u2009\u2202\u03a9\u03c11.With loss of generality, we may assume that \u03a9\u03c12 : = { \u2208 X : |||| < \u03c12}. For  \u2208 K1\u22c2\u2009\u2202\u03a9\u03c12, one hasH8), we getU|| \u2265 |||| for  \u2208 K1\u22c2\u2009\u2202\u03a9\u03c12. Hence, from Now, set To illustrate how our main results can be used in practice, we present one example.t \u2208 :f, g : , we getH1) holds.Now, observe that \u03bb1, \u03bb2, as given by condition (P1), observe by numerical approximation, we find that\u03bb1, \u03bb2 satisfying 163530 < \u03bb1, \u03bb2 < 164547.25, condition (P1) will be satisfied.On the other hand, to calculate the admissible range of the eigenvalues Consequently, by"}
{"text": "The split equality problem (SEP) has extraordinary utility and broad applicability in many areas of applied mathematics. Recently, Byrne and Moudafi (2013)proposed a CQ algorithm for solving it. In this paper, we propose a modification for the CQ algorithm, which computes the stepsize adaptively and performs an additional projection step onto two half-spaces in each iteration. We further propose a relaxation scheme for the self-adaptive projection algorithm by using projections onto half-spaces instead of those onto the original convex sets, which is much more practical. Weak convergence results for both algorithms are analyzed. H1, H2, and H3 be real Hilbert spaces; let C \u2282 H1 and Q \u2282 H2 be two nonempty closed convex sets; let A : H1 \u2192 H3 and B : H2 \u2192 H3 be two bounded linear operators. The SEP can mathematically be formulated as the problem of finding x and y with the propertyx and y. If H2 = H3 and B = I, then the split equality problem  was introduced by Moudafi  and its CQ algorithm:\u03b3k \u2208  \u2212 \u025b) and \u03bbA and \u03bbB are the spectral radii of A*A and B*B, respectively. By studying the projected Landweber algorithm of the SEP ) \u2212 \u025b). It is easy to see that the alternating CQ algorithm  norms ||A|| and ||B|| . This means that, in order to implement the alternating CQ algorithm ,,3), the lgorithm , one hasg and He  proposedCQ algorithm  in H1 \u00d7 H2, \u03c9w stands for the set of cluster points in the weak topology. \u201cxk \u2192 x\u201d  means the strong  convergence of (xk) to x.Let xk) is said to be asymptotically regular ifA sequence (yn \u2208 T(xn) with yn strongly converging to y and xn weakly converging to x; then y \u2208 T(x).The graph of an operator is called to be weakly-strongly closed if The next lemma is well known see , 12) and and12]) H be a Hilbert space and let T : H\u21c9H be a maximal monotone mapping. If (xk) is a sequence in H bounded in norm and converging weakly to some x and (wk) is a sequence in H converging strongly to some w and wk \u2208 T(xk) for all k, then w \u2208 T(x).Let \u03a9 be a closed convex subset of real Hilbert space H. Recall that the (nearest point or metric) projection from H onto \u03a9, denoted by\u2009\u2009P\u03a9, is defined in such a way that, for each x \u2208 H, P\u03a9x is the unique point in \u03a9 such thatThe projection is an important tool for our work in this paper. Let The following two lemmas are useful characterizations of projections.x \u2208 H and z \u2208 \u03a9, then z = P\u03a9x if and only ifGiven x, y \u2208 H and z \u2208 \u03a9, it holds P\u03a9(x)\u2212P\u03a9(y)||2 \u2264 \u2329P\u03a9(x) \u2212 P\u03a9(y), x \u2212 y\u232a;||P\u03a9(x)\u2212z||2 \u2264 ||x\u2212z||2 \u2212 ||P\u03a9(x)\u2212x||2.||For any \u03b9C(x) is an indicator function of the set C defined by\u03b3 > 0 and \u03b2 > 0,I + \u03b3NC)\u22121 = PC and (I + \u03b2NQ)\u22121 = PQ, we haveThroughout this paper, assume that the split equality problem  is consi problem  can be wquations  are exacx* \u2208 H1 and y* \u2208 H2, then  solves the SEP  solves the fixed point equations ,\u2009\u2009\u03b8 \u2208 \u2009and\u2009\u03c1 \u2208 , let x0 \u2208 H1 and y0 \u2208 H2 be arbitrary. For k = 0,1, 2,\u2026, compute\u03b3k is chosen to be the largest \u03b3 \u2208 {\u03c3k, \u03c3k\u03b2, \u03c3k\u03b22,\u2026} satisfyingXk and Yk, the bounding hyperplanes of which support C and Q at uk and vk, respectively,\u03c3k = \u03c30; otherwise, set \u03c3k = \u03b3k.Given constants Xk  rather than onto the set C  and it is obvious that projections on X  are very simple. It is easy to show C \u2282 Xk and Q \u2282 Yk. The last step is used to reduce the inner iterations for searching the stepsize \u03b3k.In this algorithm,  involvesThe search rule  is well \u03b3k \u2264 \u03c30. If \u03b3k = \u03c30, then this lemma is proved; otherwise, if \u03b3k < \u03c30, by the search rule  be the sequence generated by X and Y be nonempty closed convex sets in H1 and H2 with simple structures, respectively. If (X \u00d7 Y)\u2229\u0393 is nonempty, then  converges weakly to a solution of the SEP  \u2208 \u0393; that is, x* \u2208 C, y* \u2208 Q, and Ax* = By*. Define sk = uk \u2212 \u03b3k \u2212 F); then we havePXk. Similarly, defining tk = vk \u2212 \u03b3k \u2212 G), we getsk = uk \u2212 \u03b3k \u2212 F) and tk = vk \u2212 \u03b3k \u2212 G), the second inequality follows from \u2236 = ||xk\u2212x*||2 + ||yk\u2212y*||2 is decreasing and lower bounded by 0 and thus converges to some finite limit, say, l. Moreover, (xk) and (yk) are bounded. This implies thatLet  and (ykl) of (xk) and (yk) which converge weakly to Axkl \u2212 Bykl) to Let  the SEP . The weaNC and NQ, are weakly-strongly closed and by passing to the limit in the last inclusions, we obtain, from . By passing to the limit in the relationxk, yk) weakly converges to a solution of the SEP  and (Qk) be two sequences of closed convex sets defined by\u03bek \u2208 \u2202c(xk) and\u03b7k \u2208 \u2202q(yk).In the Ck\u2283C and Qk\u2283Q for every k \u2265 0.It is easy to see that \u03c30 > 0,\u2009\u2009\u03b2 \u2208 ,\u2009\u2009\u03b8 \u2208 ,\u2009\u2009and \u03c1 \u2208 , let x0 \u2208 H1 and y0 \u2208 H2 be arbitrary. For k = 0,1, 2,\u2026, compute\u03b3k is chosen to be the largest \u03b3 \u2208 {\u03c3k, \u03c3k\u03b2, \u03c3k\u03b22,\u2026} satisfyingXk and Yk the bounding hyperplanes of which support Ck and Qk at uk and vk, respectively,\u03c3k = \u03c30; otherwise, set \u03c3k = \u03b3k.Given constants Following the proof of The search rule  is well xk, yk) be the sequence generated by X and Y be nonempty closed convex sets in H1 and H2 with simple structures, respectively. If (X \u00d7 Y)\u2229\u0393 is nonempty, then  converges weakly to a solution of the SEP  \u2208 \u0393; that is, x* \u2208 C, y* \u2208 Q, and Ax* = By*. Following the similar proof of k\u2236 = ||xk\u2212x*||2 + ||yk\u2212y*||2. Then the sequence \u0393k is decreasing and lower bounded by 0 for that \u03bc \u2208  and thus converges to some finite limit, say, l. Moreover, (xk) and (yk) are bounded. This implies thatLet  generated by xkl) and (ykl) of (xk) and (yk) which converge weakly to Axkl \u2212 Bykl) to Next we show that the sequence  weakly converges to a solution of the SEP (Following the same argument of  the SEP , which c"}
{"text": "The following errors occurred in \u03bcg\u2009mL\u22121\u2009h\u22122 instead of 37.46\u2009\u03bcg\u2009mL\u22121\u2009h\u22122.AUMC value in healthy ducks is 73.46\u2009MAT value in healthy ducks is 0.74\u2009h instead of 0.31\u2009h and MAT value in renal damaged ducks is 0.70\u2009h instead of 2.08\u2009h, with no significance in MAT between healthy and renal damaged ducks.In Page 4, last line in left column, low MAT (mean absorption time) value is 0.74\u2009h instead of 0.31\u2009h."}
{"text": "M(\u222b\u03a9\u200d | x|ap\u2212 | \u2207u|p)div(|x|ap\u2212 | \u2207u|p\u22122\u2207u) = \u03bbh(x) | u|r\u22122u, x \u2208 \u03a9, M(\u222b\u03a9\u200d | x|ap\u2212 | \u2207u|p) | x|ap\u2212 | \u2207u|p\u22122\u2009(\u2202u/\u2202\u03bd) = g(x) | u|q\u22122u, on \u2202\u03a9, where 1 < (N + 1)/2 < p < N, a < (N \u2212 p)/p. By the variational method on the Nehari manifold, we prove that the problem has at least two positive solutions when some conditions are satisfied.The paper considers the existence of multiple solutions of the singular nonlocal elliptic problem \u2212 N + 1)/2 < p < N, a < (N \u2212 p)/p, \u03bb > 0, \u03a9 is an exterior domain of \u211dN: that is, and \u03a9 = \u211dN\u2216D, where D is a bounded domain in \u211dN with the smooth boundary \u2202D(\u2009 = \u2202\u03a9), and 0 \u2208 \u03a9. g(x) and h(x) are continuous functions, M(s) = \u03b1s + \u03b2 with the parameters \u03b1, \u03b2 > 0. In this paper, we consider the existence of multiple solutions for the singular elliptic problem:\u03c1, P0, h, E, and L are all positive constants. This equation extends the classical d'Alembert's wave equation by considering the effects of changes in the length of the strings during the vibrations. Problem  in (f1)\u2009\u2009f is continuous function on h(x) and g(x) are permitted to change sign. Thus, the methods in  and decreases for t \u2208  and increases for t \u2208  and increases for t \u2208  and decreases for t \u2208 , we obtain that 1 < t0+ < t\u03b1,max\u2061(u0+) and \u03c6k(t) < 0 for t \u2208 . Then, by  on N\u03bb+. Since J\u03bb(u0+) = J\u03bb(|u0+|) and |u0+ | \u2208N\u03bb+, we may assume by u0+ is a positive solution of problem =\u03b1t2A3) and 0 < \u03bb < (p/r)\u03bb1. Then, the functional J\u03bb(u) has a minimizer u0\u2212 \u2208 N\u03bb\u2212 and J\u03bb(u0\u2212) = \u03b4\u03bb\u2212,u0\u2212 is a positive weak solution of problem ( problem . Assume (J\u03bb(u) is bounded on N\u03bb\u2212, there exists a minimizing sequence {uk} \u2208 N\u03bb\u2212 such thatuk\u21c0u0\u2212 weakly in E. For uk \u2208 N\u03bb\u2212, we deduce by \u03a9\u2202g(x) | uk|qd\u03c3 > 0; furthermore, \u222b\u03a9\u2202g(x) | u0\u2212|qd\u03c3 > 0. We also want to prove that uk \u2192 u0\u2212 strongly in E. In fact, if not, we havet0\u2212 > 0 such that Since uch thatlim\u2061k\u2192\u221e\u2009Jows thatJ\u03bb(uk)\u2265J\u03bb\u03bb* = min\u2061{\u03bb1, \u03bb0}. When 0 < \u03bb < \u03bb*, by Lemmas u0+ \u2208 N\u03bb+, u0\u2212 \u2208 N\u03bb\u2212. N\u03bb+\u2229N\u03bb\u2212 = \u2205; then u0+ and u0\u2212 are distinct. Furthermore, since J\u03bb(u0\u00b1) = J\u03bb(|u0\u00b1|) and |u0\u00b1 | \u2208N\u03bb\u00b1, we can assume that the solutions u0+ and u0\u2212 are positive. This completes the proof. We set  The object of this paper is to prove the existence of multiple solutions for the nonlinear Kirchhoff-type problem . By the"}
{"text": "Our main tool is based on a recent three-critical-point theorem obtained by Ricceri. We also give some examples to illustrate the obtained results.Existence results of three weak solutions for a Dirichlet double eigenvalue quasilinear elliptic system involving the ( Here, F, G : \u03a9 \u00d7 \u211dn \u2192 \u211d are measurable functions with respect to x \u2208 \u03a9 for every  \u2208 \u211dn and are C1 with respect to  \u2208 \u211dn for a.e. x \u2208 \u03a9, and Fui and Gui denote the partial derivatives of F and G with respect to ui, respectively.The aim of this paper is to investigate the existence of at least three weak solutions for the following Dirichlet double eigenvalue quasilinear elliptic system:F and G satisfy the following additional assumptions:1) = 0 for a.e. x \u2208 \u03a9; (G)M > 0 and every 1 \u2264 i \u2264 n, for every Moreover, X be the Cartesian product of the n Sobolev spaces W0pi1,(\u03a9) for 1 \u2264 i \u2264 n; that is, X = W0p11,(\u03a9) \u00d7 W0p21,(\u03a9) \u00d7 \u22ef\u00d7W0pn1,(\u03a9) equipped with the normi \u2264 n,pi > N for 1 \u2264 i \u2264 n, X is compactly embedded in c < +\u221e. In addition, it is known  is the Lebesgue measure of the set \u03a9, and equality occurs when \u03a9 is a ball.Here and in the following, we let x0 \u2208 \u03a9 such that B\u2286\u03a9, where B stands for the open ball in \u211dN of radius r centered at x.Moreover, leti \u2264 n.Putu =  \u2208 X such thatv =  \u2208 X.By a (weak) solution of system , we meanIn the literature many papers (a2)  the set\u03b3.In the present section we discuss the existence of multiple solutions for system . For anyWe formulate our main result as follows.\u03b8 and \u03b4 with \u2211i=1n((\u03b4\u03bai)pi/pi) > (\u03b8/\u220fi=1npi) such that 1) \u2265 0 for a.e. x \u2208 \u03a9\u2216B and all ti \u2208  for 1 \u2264 i \u2264 n; 2)(b3) \u2208 X, we let the functionals \u03a6, \u03a8 : X \u2192 \u211d be defined byX and it is known that \u03a6 and \u03a8 are well-defined and continuously G\u00e2teaux differentiable functionals whose derivatives at the point u =  \u2208 X are the functionals \u03a6\u2032(u) and \u03a8\u2032(u) given byv =  \u2208 X, and \u03a6 is sequentially weakly lower semicontinuous  of  for 1 \u2264 i \u2264 n; 5)(b6)\u2192/\u2211i=1n(|ti|pi/pi) \u2264 0.\u2009\u2009limsup\u2061 Then, settinga, b]\u2286\u039b, there exists \u03c1 > 0 with the following property: for every \u03bb \u2208 , there exists \u03b4 > 0 such that, for each \u03bc \u2208 , the systemi \u2264 n, admits at least three weak solutions in X whose norms are less than \u03c1. Let F = F for all x \u2208 \u03a9 and ti \u2208 \u211d for 1 \u2264 i \u2264 n. Since \u222bBFdx = (\u03c0N/2/\u0393(1 + N/2))(D/2)NF, Set \u03ba = \u03ba1 and p = p1. Then we have the following existence result.Let f : \u211d \u2192 \u211d be a continuous function and let g : \u03a9 \u00d7 \u211d \u2192 \u211d be an L1-Carath\u00e9odory function. Put F(t) = \u222b0tf(\u03be)d\u03be for each t \u2208 \u211d and assume that there exist two positive constants \u03b8 and \u03b4 with (\u03b4\u03ba)p > \u03b8 such that 7)(bF(t) \u2265 0 for all t \u2208 ;\u2009\u20098)(b9)(bt|\u2192+\u221e|F(t)/|t|p \u2264 0.\u2009\u2009limsup\u2061 Then, settinga, b]\u2286\u039b, there exists \u03c1 > 0 with the following property: for every \u03bb \u2208 , there exists \u03b4 > 0 such that, for each \u03bc \u2208 , the problemW0p1,(\u03a9) whose norms are less than \u03c1. Let N = 1 and p = 2. For simplicity, we fix \u03a9 =  and note that in this situation, c = (\u03b2 \u2212 \u03b1)/4 and \u03ba = ((\u03b2 \u2212 \u03b1)/D)1/2.Now, we want to point out a simple consequence of f : \u211d \u2192 \u211d be a continuous function and g :  \u00d7 \u211d \u2192 \u211d an L1-Carath\u00e9odory function. Put F(t) = \u222b0tf(\u03be)d\u03be for each t \u2208 \u211d and assume that there exist two positive constants \u03b8 and \u03b4 with (\u03b42(\u03b2 \u2212 \u03b1)/D) > \u03b8 such that assumption (b7) in 10)(b11)(bt|\u2192+\u221e|(F(t)/t2) \u2264 0.\u2009\u2009limsup\u2061 Then, settinga, b]\u2286\u039b, there exists \u03c1 > 0 with the following property: for every \u03bb \u2208 , there exists \u03b4 > 0 such that, for each \u03bc \u2208 , the problemC2 whose norms are less than \u03c1. Let  First, we present an example of the application of \u03a9 = { \u2208 \u211d2 : x2 + y2 < 4}. Consider the systemG : \u03a9 \u00d7 \u211d2 \u2192 \u211d is an arbitrary function which is measurable with respect to  \u2208 \u03a9 for every  \u2208 \u211d2 and is C1 with respect to  \u2208 \u211d2 for a.e.  \u2208 \u03a9, satisfyingM > 0 and i = 1,2. Taking into account c = 4/\u03c0, picking x0 =  andt1, t2) \u2208 \u211d2, so that D = 2 and \u03b4 = \u03b8 = 3, we have F \u2265 0 for all \u2208\u00d7 andp1 = p2 = 3, are satisfied. So, settinga, b]\u2286\u039b, there exists \u03c1 > 0 with the following property: for every \u03bb \u2208 , there exists \u03b4 > 0 such that, for each \u03bc \u2208 , system  \u00d7 W01,3(\u03a9) whose norms are less than \u03c1. Let e system\u2212\u03943u1=\u03bbe1The next example follows directly by f(t) = (1/10)t9et\u2212(10 \u2212 t) for all t \u2208 \u211d. Then one has F(t) = (1/10)t10et\u2212 for all t \u2208 \u211d. Now, consider the following two-point boundary value problemg :  \u00d7 \u211d \u2192 \u211d is an arbitrary L1-Carath\u00e9odory function. Note that, in this case, x0 = 0, D = 1, c = 1/2, and \u03b8 = 1 and \u03b4 = 2, we have F(t) \u2265 0 for all t \u2208 . Also10) is fulfilled. Furthermore, we have limsup\u2061t|\u2192+\u221e|(F(t)/t2) = 0. Thus, all hypotheses of e2/27), (5/e)\u2286\u039b, there exists \u03c1 > 0 with the following property: for every \u03bb \u2208 , there exists \u03b4 > 0 such that, for each \u03bc \u2208 , problem  whose norms are less than \u03c1.Suppose  problem\u2212u\u2032\u2032=\u03bbf whose norms are less than \u03c1.In particular, there exist two positive constants"}
{"text": "Our results in this paper are generalized and unify several results in the literature as the result of Geraghty (1973) and the Banach contraction principle.We study the generalized Ulam-Hyers stability, the well-posedness, and the limit shadowing of the fixed point problem for new type of generalized contraction mapping, the so-called  The stability problem of functional equations, first initial from a question of Ulam  in 1940,On the other hand, the notion of well-posedness and limit shadowing property of a fixed point problem have evoked much interest to many researchers, for example, De Blassi and Myjak , Reich aRecently, Samet et al.  introducX be a nonempty set and \u03b1 : X \u00d7 X \u2192 [0, \u221e) be a mapping. A mapping f : X \u2192 X is said to be \u03b1-admissible if it satisfies the following condition:Let X = . Define f : X \u2192 X and \u03b1 : X \u00d7 X \u2192 [0, \u221e) by f(x) = 2x for all x \u2208 X andf is \u03b1-admissible.Let X = [1, \u221e). Define f : X \u2192 X and \u03b1 : X \u00d7 X \u2192 [0, \u221e) byf is \u03b1-admissible.Let f is \u03b1-admissible.Every nondecreasing self-mapping \u03b1-admissible mapping. Also, they applied these results to derive fixed point theorems in partially ordered metric spaces. As application, they studied the ordinary differential equations via the main results. Several researchers studied and improved contraction mappings via the concept of \u03b1-admissible mapping in metric spaces and other spaces \u2192[0,1) which satisfies the following condition.Let tn} of nonnegative real numbers, we haveFor any sequence {This class is first introduced by Geraghty  in 1973.\u03a5.(i)\u03b21(t) = k for all t \u2208 [0, \u221e), where k \u2208 [0,1).Consider (ii)ConsiderThe following are examples of some functions in First we give the following definition as a generalization of Banach contraction mappings.X, d) be a metric space and f : X \u2192 X a given mapping. One says that f is an \u03b1-\u03b2-contraction mapping if there exist two functions \u03b1 : X \u00d7 X \u2192 [0, \u221e) and \u03b2 \u2208 \u03a5 such thatx, y \u2208 X, where 1 < \u03b4 \u2264 \u03b4\u2217.Let  = 1 for all x, y \u2208 X.It is easy to check that an Next, we introduce the transitive mapping which is useful for our main result.X be a nonempty set. A mapping \u03b1 : X \u00d7 X \u2192 [0, \u221e) is called transitive if it satisfies the following condition:Let Our first main result is the following.X, d) be a complete metric space and f : X \u2192 X an \u03b1-\u03b2-contraction mapping satisfying the following conditions: f is \u03b1-admissible;\u03b1 is transitive;x0 \u2208 X such that \u03b1) \u2265 1;there exists f is continuous. Then the fixed point problem of f has a solution; that is, there exists x* \u2208 X such that x* = f(x*).Let ) \u2265 1 (such a point exists from condition (iii)). Define the sequence {xn} in X byxn = xn\u22121 for some n \u2208 \u2115, then xn = f(xn); that is, xn is a fixed point of f and thus the proof ends. Therefore, we may assume thatf is \u03b1-admissible and \u03b1 = \u03b1) \u2265 1, we get \u03b1 = \u03b1(f(x0), f(x1)) \u2265 1. By induction, we getn \u2208 \u2115, we haven \u2208 \u2115. Therefore, the sequence {d} is strictly decreasing and so d \u2192 d as n \u2192 \u221e for some d \u2265 0. Next, we claim that d = 0. Assume on the contrary that d > 0. On taking limit as n \u2192 \u221e in  = 0, which is a contradiction. Therefore, d = 0 and thusLet ies thatd = 0 which contradicts with xn} is af by adding some condition.In the next theorem, we omit the continuity hypothesis of X, d) be a complete metric space and f : X \u2192 X an \u03b1-\u03b2-contraction mapping satisfying the following conditions:f is \u03b1-admissible;\u03b1 is transitive;x0 \u2208 X such that \u03b1) \u2265 1;there exists xn} is a sequence in X such that \u03b1 \u2265 1 for all n \u2208 \u2115 and xn \u2192 x \u2208 X as n \u2192 \u221e, then \u03b1 \u2265 1 for all n \u2208 \u2115.if { Then the fixed point problem of f has a solution; that is, there exists x* \u2208 X such that x* = f(x*).Let ) = 0; that is, x* = f(x*). Therefore, the fixed point problem of f has a solution. This completes the proof.On the other hand, from  and hypoquality, , and (25We obtain that Theorems 0) \u2265 1 for all a, b \u2208 X, where a, b are fixed points of fAdding condition 1) \u2265 1 and \u03b1 \u2265 1for all to the hypotheses of f.or x* and y* are two fixed points of f. If condition (H0) holds, then we get the uniqueness of the fixed point of f from (1) holds. From condition (H1), there exists z \u2208 X such thatx*, y* are fixed points of f and f is \u03b1-admissible, from \u22651e, from ) = 0. Assume on the contrary that,n \u2192 \u221e in }, we have x* = y*. This completes the proof.Next, we claim that lim\u2061n \u2192 \u221e in , we gety, using  and 6),,n\u2192\u221e\u2061d be a metric space and f : X \u2192 X a mapping. The fixed point problem\u03c8 : [0, \u221e)\u2192[0, \u221e) which is increasing, continuous at 0 and \u03c8(0) = 0 such that for each \u03b5 > 0 and for each w* \u2208 X which is an \u03b5-solution of the fixed point equation is problemx=f(x)is\u03c8 is defined by \u03c8(t) = ct for all t \u2265 0, where c > 0, then the fixed point equation  be a metric space and f : X \u2192 X a mapping. The fixed point problem of f is said to be well posed if it satisfies the following conditions:f has a unique fixed point x* in X;xn} in X such that lim\u2061n\u2192\u221e\u2061d) = 0, one has lim\u2061n\u2192\u221e\u2061d = 0.for any sequence {Let  be a metric space and f : X \u2192 X a mapping. We say that the fixed point problem of f has the limit shadowing property in X if, for any sequence {xn} in X satisfying lim\u2061n\u2192\u221e\u2061d) = 0, it follows that there exists z \u2208 X such that lim\u2061n\u2192\u221e\u2061d(fn(z), xn) = 0.Let  be a complete metric space. Suppose that all the hypotheses of \u03b2(0) = 0 and the function \u03be : [0, \u221e)\u2192[0, \u221e) is defined by \u03be(t): = t \u2212 \u03b2(t) which is strictly increasing and onto. Then\u03b1 \u2265 1 for all a, b which are an \u03b5-solution of the fixed point equation  \u2265 1 for all n \u2208 \u2115 such that {xn} is sequence in X in which lim\u2061n\u2192\u221e\u2061d) = 0 and x* is a fixed point of f, then the fixed point problem of f is well posed;if \u03b1 \u2265 1 for all n \u2208 \u2115 such that {xn} is sequence in X in which lim\u2061n\u2192\u221e\u2061d) = 0 and x* is a fixed point of f, then the fixed point problem of f has the limit shadowing property in X.if Let  \u2265 1. Now we have\u03be\u22121 is increasing, continuous at 0 and \u03be\u22121(0) = 0. Consequently, the fixed point problem of f is generalized Ulam-Hyers stability.From the hypothesis in (a), we claim that the fixed point problem of ution of ; that isequality . From hyf is well posed under the assumption in (b). Let {xn} be sequence in X such that lim\u2061n\u2192\u221e\u2061d) = 0. From assumption, we get \u03b1 \u2265 1 for all n \u2208 \u2115. Now, we obtain thatn \u2208 \u2115. This implies thatn \u2208 \u2115. Now we claim that lim\u2061n\u2192\u221e\u2061d = 0. Assume on the contrary thatn\u2192\u221e\u2061\u03b2) = 1. Since \u03b2 \u2208 \u03a5, we obtain that lim\u2061n\u2192\u221e\u2061d = 0 which contradicts with  = 0 and so the fixed point problem of f is well posed.Next, we prove that the fixed point problem of ies thatd\u2264ary that0<lim\u2061n\u2192\u221ef has a limit shadowing under assumption (c). Let {xn} be sequence in X such that lim\u2061n\u2192\u221e\u2061d) = 0. Similar to case (b), we get lim\u2061n\u2192\u221e\u2061d = 0. Since x* is a fixed point of f, we havef has the limit shadowing property.Finally, we prove that Some Open Problems0) and (H1) by other conditions or more general conditions?In In In b-metric space, and circular metric space?Can we extend the result in this paper to other spaces as cone metric space, complex valued metric space, partial metric space,"}
{"text": "We study the norm induced by the supremum metric on the space of fuzzy numbers. And then we propose a method for constructing a norm on the quotient space of fuzzy numbers. This norm is very natural and works well with the induced metric on the quotient space. Since the concept of fuzzy numbers was firstly introduced in the 1970s, it has been studied extensively from many different aspects of the theory and applications such as fuzzy topology, fuzzy analysis, fuzzy logic, and fuzzy decision making . The opX is a normed space with norm ||\u00b7||, it is readily checked that the formula d = ||x \u2212 y||, for x, y \u2208 X, defines a metric d on X. Thus a normed space is naturally a metric space and all metric space concepts are meaningful. However, we will show that such proposition does not hold true for the well known supremum metric on the space of fuzzy numbers. To overcome this weakness, we will consider the quotient space of fuzzy numbers up to an equivalence relation which is introduced by Mare\u0161 . For each such fuzzy set \u03bc, we denote by [\u03bc]a = {x \u2208 \u211d : \u03bc(x) \u2265 a} for any a \u2208 , \u03bcR(a)] for all a \u2208 . We denote the class of fuzzy numbers by \u2131. Notice that the real numbers \u211d can be imbedded in \u2131 by defining a fuzzy number as followsa \u2208 \u211d.A fuzzy set of \u211d is a function \u03bc, \u03bd \u2208 \u2131 and a \u2208 \u211d, owing to Zadeh's extension principle  \u2192 \u211d by assigning the midpoint of each a-level set to \u03bcM(a) for all a \u2208 , that is,For a fuzzy number \u03bc, \u03bd \u2208 \u2131, \u03bc ~ \u03bd if and only if \u03bcM = \u03bdM.For any \u2131/\ud835\udcae, +, \u00d7) is a real vector space.d\u221e, \u03bcR(a0)]\u2260{0}. Thus we have ||\u03bc|| > 0.(i)\u2009\u2009It is obvious that ||\u03bc \u2208 \u2131 and b \u2208 \u211d, we have(ii)\u2009\u2009For all \u03bc, \u03bd \u2208 \u2131, we have that\u2131.(iii)\u2009\u2009For all \u2131, the function d : \u2131 \u00d7 \u2131 \u2192 \u211d induced by ||\u00b7|| as d = ||\u03bc \u2212 \u03bd|| is not a metric on \u2131. Consequently, we get that d\u221e \u2260 ||\u03bc \u2212 \u03bd||.Although ||\u00b7|| is a norm on d has the following properties:d \u2265 0, for any \u03bc, \u03bd \u2208 \u2131;d = d, for any \u03bc, \u03bd \u2208 \u2131;d \u2264 d + d, for any \u03bc, \u03bd, \u03c9 \u2208 \u2131;d \u2260 0, for any fuzzy number \u03bc \u2209 \u211d.The function d, it is obvious that d \u2265 0, for any \u03bc, \u03bd \u2208 \u2131.(i)\u2009\u2009By the definition of \u03bc, \u03bd \u2208 \u2131, we have that(ii)\u2009\u2009For all a \u2208 , and for any \u03bc, \u03bd, \u03c9 \u2208 \u2131, we only proof the following six cases. Similarly, the others can be proved.(iii)\u2009\u2009In order to prove the triangle inequality, for any fixed Case\u2009\u20091 ( \u03bcL(a) \u2264 \u03bdL(a) \u2264 \u03bcR(a) \u2264 \u03bdR(a) and \u03c9L(a) \u2264 \u03bcL(a)). In this case we have thatCase\u2009\u20092 ( \u03bcL(a) \u2264 \u03bdL(a) \u2264 \u03bcR(a) \u2264 \u03bdR(a) and \u03bcL(a) \u2264 \u03c9L(a) \u2264 \u03c9R(a) \u2264 \u03bdR(a)). In this case we have thatCase\u2009\u20093 ( \u03bcL(a) \u2264 \u03bdL(a) \u2264 \u03bdR(a) \u2264 \u03bcR(a) and \u03c9L(a) \u2264 \u03bcL(a)). In this case we have thatCase\u2009\u20094 ( \u03bcL(a) \u2264 \u03c9L(a) \u2264 \u03c9R(a) \u2264 \u03bdL(a) \u2264 \u03bdR(a) \u2264 \u03bcR(a)). If max\u2061{|\u03bcL(a) \u2212 \u03bdR(a)|, |\u03bcR(a) \u2212 \u03bdL(a)|} = |\u03bcL(a) \u2212 \u03bdR(a)|, then we have thatCase\u2009\u20095 ( \u03bcL(a) \u2264 \u03bdL(a) \u2264 \u03c9L(a) \u2264 \u03c9R(a) \u2264 \u03bdR(a) \u2264 \u03bcR(a)). In this case we have thatCase\u2009\u20096 ( \u03bcL(a) \u2264 \u03c9L(a) \u2264 \u03bdL(a) \u2264 \u03bdR(a) \u2264 \u03c9R(a) \u2264 \u03bcR(a)). In this case we have that\u03bc \u2209 \u211d, there exists a0 \u2208  such that \u03bcR(a0) \u2212 \u03bcL(a0) > 0. Thus we have that(iv)\u2009\u2009Since \u2131/\ud835\udcae \u2192 \u211d as\u03bc\u232a\u2208\u2131/\ud835\udcae.In order to induce a metric which is compatible with the norm, we consider the quotient space of fuzzy numbers. It is very natural to define a function ||\u00b7|| : \u2131/\ud835\udcae.The function ||\u00b7|| in  is a nor\u03bc\u232a|| \u2265 0 for all \u03bc \u2208 \u2131 and ||\u2329\u03bc\u232a|| = 0 if \u2329\u03bc\u232a = \u23290\u232a. Conversely, if \u2329\u03bc\u232a\u2260\u23290\u232a, from \u03bcM \u2260 0M. Thus we have(i)\u2009\u2009It is obvious that ||\u2329\u03bc\u232a\u2208\u2131/\ud835\udcae and b \u2208 \u211d, we have(ii)\u2009\u2009For all \u2329\u03bc\u232a, \u2329\u03bd\u232a\u2208\u2131/\ud835\udcae, we have that\u2131/\ud835\udcae.(iii)\u2009\u2009For all \u2329\u03c1 : \u2131/\ud835\udcae \u00d7 \u2131/\ud835\udcae \u2192 \u211d induced by ||\u00b7|| as \u03c1 = ||\u2329\u03bc \u2212 \u03bd\u232a|| is exactly a metric on \u2131/\ud835\udcae.Now we show that the function \u03c1 is a metric on \u2131/\ud835\udcae.The function \u03c1, it is obvious that \u03c1 \u2265 0, for any \u2329\u03bc\u232a, \u2329\u03bd\u232a\u2208\u2131/\ud835\udcae. If \u03c1 = 0, then ||\u2329\u03bc \u2212 \u03bd\u232a|| = 0. Thus by \u03bc\u232a = \u2329\u03bd\u232a. In addition, by \u03c1 = ||\u2329\u03bc \u2212 \u03bc\u232a|| = 0.(i)\u2009\u2009By the definition of \u03bc\u232a, \u2329\u03bd\u232a\u2208\u2131/\ud835\udcae, we have that(ii)\u2009\u2009For all \u2329\u03bc\u232a, \u2329\u03bd\u232a, \u2329\u03c9\u232a\u2208\u2131/\ud835\udcae, we have\u03c1 is a metric on \u2131/\ud835\udcae.(iii)\u2009\u2009For any \u2329d\u221e on the space of fuzzy numbers. And then we proposed a method for constructing a norm on the quotient space of fuzzy numbers. This norm is very natural and works well with the induced metric on \u2131/\ud835\udcae. The works in this paper enable us to study the fuzzy numbers in the new environment. We hope that our results in this paper may lead to significant, new, and innovative results in other related fields.In this present investigation, we studied the norm induced by the supremum metric"}
{"text": "For the continued fraction expansion arising from \u03c4m, we solve a Gauss-Kuzmin-type problem.We consider a family { \ud835\udcaa(qn) as n \u2192 \u221e with 0 < q < 1.Chan considered some continued fraction expansions related to random Fibonacci-type sequences , 2. In , he studm \u2265 2. In this section, we show our motivation and main theorems.The purpose of this paper is to prove a Gauss-Kuzmin-type problem for the continued fraction expansions of real numbers in  whose digits are differences of consecutive nonpositive integer powers of an integer x < 1 can be written as the infinite regular continued fractionan \u2208 \u2115+\u2254{1,2, 3,\u2026}.One of the first and still one of the most important results in the metrical theory of continued fractions is the so-called Gauss-Kuzmin theorem. Any irrational 0 < an)n\u2208\u2115+. It started on October 25, 1800, with a note by Gauss in his mathematical diary (entry 113)  by\u03bb denotes the Lebesgue measure on I and \u03c4n is the nth iterate of \u03c4.Roughly speaking, the metrical theory  of continued fraction expansions is about properties of the sequence  0 < q < 1. This has been called the Gauss-Kuzmin theorem or the Kuzmin theorem.Twelve years later, in a letter dated January 30, 1812, Gauss wrote to Laplace that he did not succeed in solving satisfactorily \u201ca curious problem\u201d and that his efforts \u201cwere unfruitful.\u201d In modern notation, this problem is to estimate the erroren(x)\u2254\u03bb can be written as the forman's are nonnegative integers. Such an's are also called incomplete quotients (or continued fraction digits) of x with respect to the expansion in I, \u212cI, \u03c4m) is defined as follows: I\u2254, where \u212cI denotes the \u03c3-algebra of all Borel subsets of I and \u03c4m is the transformationIi\u2254{x \u2208 I : mi+1)\u2212I, \u212cI, \u03b3m, \u03c4m) as  with the following probability measure \u03b3m on :In addition to (i), one writes (Fix an integer quantized index map \u2009\u03b7m : I \u2192 \u2115 by\u03b7m(m\u03b1\u2212) = \u230a\u03b1\u230b. By using \u03c4m and \u03b7m, the sequence (an)n\u2208\u2115+ in (\u03c4m0(x) = x. In this way, \u03c4m gives the algorithm of Chan's continued fraction expansion (Define the )n\u2208\u2115+ in  is obtaixpansion .I, \u212cI, \u03b3m, \u03c4m) be as in I, \u212cI, \u03b3m, \u03c4m) is ergodic. = \u03b3m(\u03c4m\u22121(A)) for any A \u2208 \u212cI.The measure Let  is a \u201cdynamical system\u201d in the sense of m \u2208 . As it can be seen, \u03c7m is a constant independent of the value of x.Continued fraction expansions of almost all irrational numbers are not periodic. Nevertheless, we readily reproduce another famous probabilistic result. It is the asymptotic value given in . This is\u03bc-measure preserving operator T : \u2192 is given by the beautiful formula\u03c4m in \u2254\u222bablgiven in , 26), a, a\u03bc-measgiven in , respectm \u2265 2. Let km be as in  be as in x has the expansion in \u2212( < \u03c4mn < mi\u2212). We will show that the event \u03c4mn \u2264 x has the following asymptotic probability:\u03c4m: if a random variable X in the unit interval has the density \u03c1m, and then so does \u03c4m. The reason for this invariance is that, for 0 \u2264 x < x + h \u2264 1, \u03c4m lies between x and x + h if and only if there exists i \u2265 1, so that X lies between 1/(x + i + h) and 1/(x + i). Thush \u2192 \u221e gives that, for an arbitrary probability density function f for X, the corresponding density Gf for \u03c4m is given a.e. in I by the equationG : L1 \u2192 L1 admits the density function \u03c1m as an eigenfunction corresponding to the eigenvalue 1; that is, G\u03c1m = \u03c1m. Here L1 denotes the Banach space of all complex functions f : I \u2192 \u2102 for which \u222bI|f|d\u03bb < \u221e.We show our main theorems in this subsection. Fix an integer be as in  and let nsion in  and \u03c4m is as in , then thG is 1 and this eigenvalue is simple.The only eigenvalue of modulus 1 of \u03c4m is an ergodic operator on the unit interval [\u03c1m is the density of the invariant measure, and G is called transfer operator for \u03c4m [G has the same analytical expression as the Perron-Frobenius operator of \u03c4m under the Lebesgue measure [From another perspective, the operator interval , \u03c1m is  for \u03c4m . The tra measure .Our main result is the following theorem.\u03c4m and \u03c9m be as in  is given, define functions Fm,n (n \u2265 0) on I byx \u2208 I. Then there exists a constant 0 < qm < 1 such that Fm,n is written asLet be as in  and 31)\u03c4m and \u03c9\u03b3m is the measure defined in ((i) From , we see fined in . In factan)n\u2208\u2115 in (\u03c8-mixing under \u03b3m (and under many other probability measures including \u03bb) [A1 \u2208 \u03c3 , A2 \u2208 \u03c3, and k \u2208 \u2115+, with suitable positive constants q < 1 and C.(ii) The solution of this problem implies that (n)n\u2208\u2115 in  is exponuding \u03bb) , 12; tha\u03c8-mixing implies lots of limit theorems in both classical and functional versions. To form an idea of the results to be expected it is sufficient to look at the corresponding results for the regular continued fraction expansions [In turn, pansions .\u03c4m [\u03c4m is ergodic  In  we empha\u03c4m , 6. Thes\u2009\u2009of\u2009\u2009f\u201d .\u03c4m for m \u2265 3. For example, study the optimality of the convergence rate. Use the same strategy as in [(i) Solve the Gauss-Kuzmin-L\u00e9vy problem of gy as in .\u03c4 in ((ii) It is known that the Riemann zeta function is written by using a kind of Mellin transformation of the Gauss transformation \u03c4 in  as follo\u03c4. Then, by replacing \u03c4 with \u03c4m in  \u2264 x}, and Im,n,i = {x \u2208 Im,n : \u2009\u2009\u03b1i/(1 + (m \u2212 1)x) < \u03c4mn(x) < \u03b1i}.Let Im,n,i and  = \u03bc andFrom  and 12)12), we s\u03c4mn(x)=m\u2212i).Then  holds bep \u2208 \u2115, the derivative Fp\u2032 exists everywhere in I and is bounded. Then it is easy to see by induction that Fm,p+n\u2032 exists and is bounded for all n \u2208 \u2115+. This allows us to differentiate  is given in  byFor fm,n} in |. ThenFor {m,n} in , define We haveNow from  and by cNow  implies\u03b4i = \u03b1i \u2212 \u03b1i2 and 0 \u2264 x \u2264 1.First, we note thatx \u2208 I and i \u2208 \u2115. Hence, hm(x) \u2264 hm(0). This leads toNext, observe that the functionelations , 59), a, a(60)hmleads to\u03b1iPmi((m\u2212elations  and 54)(60)hm(x)Rm,n(x) such thatIntroduce a function Fm,n(0) = 0 and Fm,n(1) = 1, we have Rm,n(0) = Rm,n(1) = 0. To prove qm < 1 such thatBecause fm,n(x) = km + \ud835\udcaa(qmn), then its integration will show  has this desired form, it suffices to prove the following lemma.To demonstrate that x \u2208 I and n \u2208 \u2115, there exists a constant 0 < qm < 1 such thatFor any qm be as in (qm < 1. To this end, for i \u2265 2, observe thatm \u2208 \u2115, m \u2265 2.Let be as in . Using Lbe as in  it is en"}
{"text": "G1 be a group and let G2 be a metric group with the metric d. Given \u03b5 > 0, does there exist a \u03b4 > 0 such that if a function h : G1 \u2192 G2 satisfies the inequality d(h(xy), h(x)h(y)) < \u03b4, for all x, y \u2208 G1, then there exists a homomorphism H : G1 \u2192 G2 with d(h(x), H(x)) < \u03b5, for all x \u2208 G1?  Let In 1940, Ulam [G1 and G2 are Banach spaces. Indeed, he proved that each solution of the inequality ||f(x + y) \u2212 f(x) \u2212 f(y)|| \u2264 \u03b5, for all x and y, can be approximated by an exact solution, say an additive function. In this case, the Cauchy additive functional equation, f(x + y) = f(x) + f(y), is said to have the Hyers-Ulam stability.The case of approximately additive functions was solved by Hyers  under thRassias  attemptestigated \u20139.The terminologies, the generalized Hyers-Ulam stability, and the Hyers-Ulam stability can also be applied to the case of other functional equations, differential equations, and various integral equations. c > 0, the partial differential equationutt and uxx denote the second time derivative and the second space derivative of u, respectively.Given a real number \u03c6 : \u211d \u00d7 \u211d \u2192 [0, \u221e) be a function. If, for each twice continuously differentiable function u : \u211d \u00d7 \u211d \u2192 \u2102 satisfyingu0 : \u211d \u00d7 \u211d \u2192 \u2102 of the  wave equation (\u221e) such thatx, t) is independent of u and u0, then we say that the wave equation , we prove the generalized Hyers-Ulam stability of the  wave equation .\u03c6 : \u211d \u00d7 \u211d \u2192 [0, \u221e) be given such that the double integrala, b \u2208 \u211d. If a twice continuously differentiable function u : \u211d \u00d7 \u211d \u2192 \u2102 satisfies the inequalityx, t \u2208 \u211d, then there exists a solution u0 : \u211d \u00d7 \u211d \u2192 \u2102 of the wave equation  = v andx, t \u2208 \u211d. Hence, we havex, t \u2208 \u211d. Thus, it follows from inequality  is a solution of the wave equation  be given asu : \u211d \u00d7 \u211d \u2192 \u2102 satisfies inequality =\u03b1a, b \u2208 \u211d, in view of Since"}
{"text": "Due to a formatting error, the \u201cpermille\u201d symbol (\u2030) was omitted whenever it was used in the text. Below are the locations of the errors:1. On Page 3, left column, last paragraph, \u201clowest 1\u201d should be \u201clowest 1\u2030\u201d:1a. \u201cFor each target, the averages over the lowest 1\u2030 and 1% CARMSD to native are reported.\u201d1b. \u201cWe also report the average CARMSD of the lowest 1\u2030 and 1% Rosetta energies.\u201d2. On Page 3, left column, last paragraph, \u201caverage 1\u201d should be \u201caverage 1\u2030\u201d:2a. \u201cEven if Rosetta\u2019s coarse-grained energy is noisy, EdaFold still takes advantage of it and achieves a better average 1\u2030 and 1% CARMSD\u201d3. On Page 3, right column, last paragraph, \u201cbest 1\u201d should be \u201cbest 1\u2030\u201d:3a. \u201cEdaFold outperforms Rosetta on a majority of targets either in terms of best 1\u2030 and 1% average CARMSD to native\u201d4. On Page 4, table 1, \u201ctop 1\u201d should be \u201ctop 1\u2030\u201d in the row that starts with \u201cTarget\u201d.5. On Page 6, legend of figure 4, \u201cbest 1\u201d should be \u201cbest 1\u2030\u201d.There was also an error in the last sentence of the Abstract. The correct link is:http://www.riken.jp/zhangiru/software.html [^]"}
{"text": "Many known results appear as special consequences of our work. Some applications of our work to the generalized integral operator are also given.We obtain certain simple sufficiency criteria for a class of  Ap(n) denote the class of functions f(z) of the form\ud835\udd4c = {z \u2208 \u2102 : |z| < 1}. We note that A1(n) = A(n) and A1(1) = A. Also let Sp*, Cp and Mp, , denote the subclasses of Ap(n) consisting of all functions f(z), of the form  \u2261 S*, M1 \u2261 C.Let the formf(z)=zp+\u2211Mp of multivalent alpha convex functions of order \u03b2. We also consider some special cases of our results which lead to various interesting corollaries, and relevances of some of these with other known results are also mentioned.Sufficient condition was studied by various authors for different subclasses of analytic and multivalent functions; for some of the related work, see \u20139. The on \u2208 \u2115, 0 \u2264 \u03b1 \u2264 1, 0 \u2264 \u03b2 < p, and p \u2208 \u2115.We will assume throughout our discussion, unless otherwise stated, that To obtain our main results, we need the following lemmas due to Mocanu .q(z) \u2208 A(n) satisfies the conditionIf q(z) \u2208 A(n) satisfies the condition\u03b4n is the unique root of the equationIf f(z) \u2208 Ap(n) satisfiesf(z) \u2208 Mp.If q(z) byf(z) \u2208 Ap(n). Then clearly (q(z) \u2208 A(n).Let us set a function q(z) byq(z)=z(f(q(z) \u2208 S*.Differentiating  logarithus using , we haveq(z) \u2208 S*, it implies that Re(zq\u2032(z)/q(z)) > 0. Therefore, we getf(z) \u2208 Mp.From , we can \u03b1 = 0 and \u03b1 = 1 in By taking f(z) \u2208 Ap(n) satisfies\u03b2 < p, then f(z) \u2208 Sp*.If f(z) \u2208 Ap(n) satisfies\u03b2 < p, then f(z) \u2208 Cp.If n = 1 and p = 1 in Corollaries Further if we take f(z) \u2208 A satisfies\u03b2 < 1, then f(z) \u2208 S*(\u03b2).If f(z) \u2208 A satisfies\u03b2 < 1, then f(z) \u2208 C(\u03b2).If \u03b2 = 0 and p = 1 in Corollaries If we put f(z) \u2208 Ap(n) satisfies\u03b4n is the unique root of (f(z) \u2208 Mp.If  root of , then f(f(z) \u2208 Ap(n). Then clearly (q(z) \u2208 A(n).Let us set clearly  shows th\u03b4n is the root of (q(z) \u2208 S*. Now by the same arguments as in the proof of Differentiating , we haveus using , we have root of . Hence, \u03b1 = 0 in Making f(z) \u2208 Ap(n) satisfiesf(z) \u2208 Sp*.If n = 1 = p in Further if we take f(z) \u2208 A satisfies\u03b41 is the unique root of the equationf(z) belongs to the class of starlike functions of order \u03b2.If \u03b1 = 1 in Taking f(z) \u2208 Ap(n) satisfiesf(z) \u2208 Cp.If n = 1 and p = 1 in \u03b2 = 0, we get the result proved by Mocanu [If we take y Mocanu .f(z) \u2208 Ap(n), we considerG(z) \u2208 A(n), and when p = 1, \u03b3 = 1, and \u03b4 = 1, then (For considerG(z)={\u03b4\u222b0\u03b3 \u2265 \u03b4/p, \u03b4 > 0, and f(z) \u2208 Ap(n) satisfyf(z) \u2208 Sp*.If p = 1, \u03b2 = 0, and \u03b1 = 1/\u03b4, we have\u2009\u2009G(z) \u2208 M1.From , we getG\u03b4\u22121(z)G\u2032ave used . By usinf(z) \u2208 Sp*, where \u03b3 \u2265 \u03b4/p.From , we can \u03b3 \u2265 \u03b4/p, \u03b4 > 0, and f(z) \u2208 Ap(n) satisfy\u03b4n is the unique root of (f(z) \u2208 Sp*.If satisfy3|arg(f(z)p = 1, \u03b2 = 0, and \u03b1 = 1/\u03b4.The result follows on similar lines as in the last theorem using"}
{"text": "Fixed point results for a self-map satisfying locally contractive conditions on a closed ball in an ordered 0-complete quasi-partial metric space havebeen established. Instead of monotone mapping, the notion of dominated mappings is applied. We have used weaker metric, weaker contractive conditions, andweaker restrictions to obtain unique fixed points. An example is given which shows that how this result can be used when the corresponding results cannot. Our results generalize, extend, and improve several well-known conventional results. Fixed points results of mappings satisfying certain contractive conditions on the entire domain have been at the centre of vigorous research activity, see \u20133) and i and i3])Recently, many results appeared related to fixed point theorem in complete metric spaces endowed with a partial ordering in the literature. Ran and Reurings  proved aOn the other hand notion of a partial metric space was introduced by Matthews in . In partT is a contraction not on the entire space X but merely on a subset Y of X. However, if Y is closed and a sequence {xn} in X converges to some x in X, then by imposing a subtle restriction on the choice of x0, one may force the sequence to stay eventually in Y. In this case, one can establish the existence of a fixed point of T. Arshad et al. ,\u2009\u2009x, y \u2208 X is a partial metric on X. The function dpq : X \u00d7 X \u2192 R+ defined by dpq = q + q \u2212 q \u2212 q is a  metric on X. The ball x \u2208 X and \u03b5 > 0.Note that if X, q) be a quasi-partial metric. Then, we have the following; xn} in  converges to a point x \u2208 X if and only if lim\u2061n\u2192\u221e\u2061\u2009\u2009q = lim\u2061n\u2192\u221e\u2061\u2009\u2009q = q.A sequence {xn} in  is called a Cauchy sequence if the limits lim\u2061n,m\u2192\u221e\u2061\u2009\u2009q and lim\u2061n,m\u2192\u221e\u2061\u2009\u2009q exist (and are finite).A sequence {X, q) is said to be complete if every Cauchy sequence {xn} in  converges to a point x \u2208 X such that  = lim\u2061n,m\u2192\u221e\u2061\u2009\u2009q = lim\u2061n,m\u2192\u221e\u2061\u2009\u2009q.The space  be a quasi-partial metric space, let  be the corresponding partial metric space, and let  be the corresponding metric space. These statements are equivalent. (i) The sequence {xn} is Cauchy in . (ii) The sequence {xn} is Cauchy in . (iii) The sequence {xn} is Cauchy in . These statements are also equivalent. (i)  is complete. (ii)  is complete. (iii)  is complete.Let  be a partial ordered set. Then x, y \u2208 X are called comparable if x\u2aafy or y\u2aafx holds.Let  be a partially ordered set. A self-mapping f on X is called dominated if fx\u2aafx for each x in X.Let  be a complete metric space, S : X \u2192 X a mapping, r > 0, and x0 an arbitrary point in X. Suppose there exists k \u2208 [0,1) withd<(1 \u2212 k)r. Then there exists a unique point x* in x* = Sx*.Let  be a quasi-partial metric space.xn} in  is called 0-Cauchy if lim\u2061n,m\u2192\u221e\u2061\u2009\u2009q = 0 or lim\u2061n,m\u2192\u221e\u2009\u2009q = 0.A sequence {X, q) is called 0-complete if every 0-Cauchy sequence in X converges to a point x \u2208 X such that q = 0.The space  is Cauchy in  and if  is complete, then it is 0-complete but the converse assertions do not hold. For example, the space X = [0, +\u221e)\u2229Q with q = |x \u2212 y | +|x| is a 0-complete quasi-partial metric space but it is not complete  = 2 | x \u2212 y| and  is not complete).It is easy to see that every 0-Cauchy sequence in  is complete quasimetric space which is also quasi-partial, then it is 0-complete, quasi-partial metric space.Moreover, if  is called an ordered quasi-partial metric space if (i) q is a quasi-partial metric on X and (ii) \u2aaf is a partial order on X.Let X, \u2aaf, q) be a 0-complete ordered quasi-partial metric space, S : X \u2192 X a dominated map, and x0 an arbitrary point in X. Suppose that there exists k \u2208 [0,1) such thatxn} in xn} \u2192 u implies that u\u2aafxn, then there exists a point x* in x* = Sx* and q = 0. Moreover, x* is unique, if for every pair of elements x, y in X\u2009\u2009there exists a point z \u2208 X such that z\u2aafx and z\u2aafy.Let \u00af withq=Uniqueness. Let y be another point in y = Sy. If x* and y are comparable, thenx* = y. Now if x* and y are not comparable then there exists a point x* and y, that is, z\u2aafx* and z\u2aafy. Moreover by assumption z\u2aafx*\u2aafxn \u22ef \u2aafx0. Now we will prove that n \u2208 N. Now as S is dominated, it follows that Sn\u22121z\u2aafSn\u22122z\u2aaf\u22ef\u2aafz\u2aafx* and Sn\u22121z\u2aafy for all n \u2208 N, which further implies Sn\u22121z\u2aafSnx* and Sn\u22121z\u2aafSny for all n \u2208 N as Snx* = x* and Sny = y for all n \u2208 N:x* = y.X = [0, +\u221e)\u2229Q be endowed with order x\u2aafy if q \u2264 q and let q : X \u00d7 X \u2192 R+ be the 0-complete ordered quasi-partial metric on X defined by q = max\u2061{y \u2212 x, 0} + x. DefineS is dominated mapping. Take x0 = 1/2,\u2009\u2009r = 1/2; then q = 1/2,\u2009\u2009k = 4/9 withx, y \u2208 \u2229X, then,X in each case.Let S and q = 0.Now ifx, y in X.In X, \u2aaf, q) be a 0-complete ordered quasi-partial metric space, S : X \u2192 X a dominated map, and x0 an arbitrary point in X. Suppose there exists k \u2208 [0,1) withxn} in X,\u2009\u2009{xn} \u2192 u implies that u\u2aafxn, then there exists a point x* in X such that x* = Sx* and q = 0. Moreover, x* is unique, if for every pair of elements x, y in X, there exists a point z \u2208 X such that z\u2aafx and z\u2aafy.Let  be a 0-complete partial metric space, S : X \u2192 X a map, and x0 an arbitrary point in X. Suppose there exists k \u2208 [0,1) withx* in x* = Sx*. Further q = 0.Let  be a complete ordered metric space, S : X \u2192 X a dominated map, and x0 an arbitrary point in X. Suppose that there exists k \u2208 [0,1) such thatxn} in xn} \u2192 u implies that u\u2aafxn, then there exists a point x* in x* = Sx* and q = 0. Moreover, x* is unique, if for every pair of elements x, y in X there exists a point z \u2208 X such that z\u2aafx and z\u2aafy.Let  be a 0-complete ordered quasi-partial metric space, S : X \u2192 X a dominated map, and x0 an arbitrary point in X. Suppose that there exists b \u2208 [0, 1/2) such thatx, y in Let (k = b/(1 \u2212 b). If, for a nonincreasing sequence {xn} in xn} \u2192 u implies that u\u2aafxn, then there exists a point x* in x* = Sx* and q = 0. Moreover, x* is unique, if for any two points x, y in z\u2aafx and z\u2aafy, andAndxn+1 = Sxn with initial guess x0, as xn+1 = Sxn\u2aafxn for all n \u2208 {0} \u222a N. We will prove that n \u2208 N by mathematical induction. By using inequality  \u2264 q, we have q = 0. Similarly, q = 0. Now if x* and y are not comparable to then there exists a point x* and y. Moreover by assumption Sz\u2aafz\u2aafx*\u2aafxn \u22ef \u2aafx0. Now by using inequality  \u2264 q, we have q = 0. Similarly, q = 0. Hence x* = y.equality , we haveequality . Let S2zies thatq be a complete dislocated quasimetric space, S : X \u2192 X a map, and x0 an arbitrary point in X. Suppose there exists k \u2208 [0, 1/2) withx, y in \u03b8 = k/(1 \u2212 k). Then there exists a unique point x* in x* = Sx* and dq = 0.Let  be a 0-complete ordered quasi-partial metric space and let S : X \u2192 X be a dominated map. Suppose that there exists b \u2208 [0, 1/2) such thatx, y in X. If, for a nonincreasing sequence {xn} in X,\u2009\u2009{xn} \u2192 u implies that u\u2aafxn, then there exists a point x* in X such that x* = Sx* and q = 0. Moreover, x* is unique, if for any two points x, y in z\u2aafx and z\u2aafy.Let  be a 0-complete ordered quasi-partial metric space, S : X \u2192 X a dominated map, and x0 an arbitrary point in X. Suppose that there exists c \u2208 [0, 1/2) such thatx, y in Let (k = c/(1 \u2212 c). If, for a nonincreasing sequence {xn} in xn} \u2192 u implies that u\u2aafxn, then there exists a point x* in x* = Sx*. Further q = 0.Andxn+1 = Sxn with initial guess x0. As xn+1 = Sxn\u2aafxn for all n \u2208 {0} \u222a N. We will prove that n \u2208 N by mathematical induction. By using inequality (j \u2208 N. As xn+1\u2aafxn, so using inequality (n \u2208 N. Also xn+1\u2aafxn for all n \u2208 N. It implies thatxn} is a 0-Cauchy sequence in n \u2192 \u221e and using the fact that x*\u2aafxn\u2aafxn\u22121, when xn \u2192 x*, we havex* = Sx*.Consider a Picard sequence equality , we haveequality , we obtaWe can obtain the partial metric, quasi-metric, and metric version of all theorems which are still not present in the literature."}
{"text": "The strong order of the convergence of the numerical method is given, and the convergence of the numerical method is obtained. Some earlier results are generalized and improved.This paper is concerned with the convergence of stochastic  Baker and Buckwar x1, x2, x3 \u2208 \u211cn.There is a positive constant In fact, the global Lipschitz condition  implies \u03c40 = 0 and \u03c4j, j = 1,2,\u2026 are the jump times.We note for later reference that  involves\u03b8-Euler methods . Here h \u2208  is a step size, which satisfies M = T/h for some positive integer M, tk = kh\u2009\u2009. yk \u2248 x(tk), \u0394wk = w(tk+1) \u2212 w(tk), and \u0394Nk = N(tk+1) \u2212 N(tk) are the Brownian and Poisson increments, respectively.One generalisation of stochastic  methods , 21 to st \u2208 , we definey(t) immediately provides a result for yk.For ximation . So a coCi, Di, i = 1,2,\u2026 denote generic constants, independent of h. The main theorem of the paper is rather technical. We will present a number of useful lemmas in the section and then complete the proof in Throughout our analysis, h* < 1 such that for all 0 < h \u2264 h*,Under \u03b1\u03b2 \u2264 |\u03b1|2 + |\u03b2|2. Now, using E | \u0394wk|2 = mh and From , we haveAi(h), i = 1,\u2026, 5 is a constant dependent on h and A3(h) = 4K2h2(\u03b82 + (1 \u2212 \u03b8)\u03b8).For the jump, we convert to the compensated Poisson increment ombining , 14), a, a\u0394N~k\u2236=ombining  with  \u2265 1/2 and noting that (E | \u03b3(tj)|2 \u2264 B1, we obtainNow choosing h* > 0 such that, for all 0 < h \u2264 h*,Under t \u2208 \u2286. In this interval we haveC3 = 3mK2(1 + 2C1) + 3\u03bb(1 + \u03bb)K2(1 + 2C1 + B1) + 15K2.Consider irtue of \u201314) and andt \u2208 \u2286. By |2 \u2264 B1, on , z1 \u2261 yk, z2 \u2261 yk+1, C5 = 72K2(1 + 2C1) + 24K2\u03bb(2 + \u03bb)(1 + 3C1). In the following we consider C6 = 4C5; the proof is complete.Consider , T]. By , we have T]. By (3y(\u03bct)\u2212z\u00afWe can now state and prove our main result of this paper.q > 1 and Assumptions h* > 0 and C = C(q) such that, for all 0 < h < h*,Under t1 \u2264 T we haveThe analysis uses ideas from , where aM\u2032 is the smallest integer such that M\u2032h \u2265 t1.By a, b, \u025b > 0 and 1 < q < \u221e.Now the number of nonzero terms in the summation in  is a ran\u025b = hq \u2212 1)2/q(4 + \u03bbT)((q \u2212 1)\u03bb + 2q2B) + 16K1((C3 + C5)(T + 2 + 4\u03bb + \u03bb2T) + TC4 + TC6).Now, substituting , 32), , , and 4141 into  yields(By the Gronwall inequality, we have\u03b8-methods give strong convergence rate arbitrarily close to order 1/2 under appropriate moment bounds on the jump magnitude. This problem class is now widely used in mathematical finance.By Under The convergent result can be extended to the case of nonlinear coefficients that are local Lipschitz , 7, 12 bUnder the local Lipschitz condition and"}
{"text": "Conditions for the existence of locally expansive C1 solutions for such equations are given.Iterative equations which can be expressed by the following form  C be the set of all continuous self-mappings on a topological space X. For any f \u2208 C, let fm denote the mth iterate of f; that is, fm = f\u2218fm\u22121, f0 = id, \u2009m = 1,2,\u2026. Equations having iteration as their main operation, that is, including iterates of the unknown mapping, are called iterative equations. It is one of the most interesting classes of functional equations . By the choices of H and \u03d5, we have \u03bb\u03d5(0) = 0 and \u03bbi\u03d5(0) = Hi\u2032(O), where i = 0,1,\u2026, n \u2212 1.For f of (f(x) = \u03d5(c(\u03d5\u22121(x))) by the Schr\u00f6der transformation, where c is a constant to be determined, then  is a zero of the following polynomial:If function ution of , then weynomial:P(x)=xn\u2212Hlynomial .Finally, we give a basic lemma. D \u2282 Rm be a convex open set, and let a =  and a + h =  belong to D, then there exists a \u03b8 \u2208  such thatLet S(n) = {1,2,\u2026, n \u2212 1}. Let H \u2208 \u210b. Suppose that there is a neighborhood U of \u2009O \u2208 Rn that satisfies A1+) \u2265 H0\u2032(X) \u2265 0;  for all A2+), Hi\u2032(O) \u2265 Hi\u2032(X) \u2265 0. for all  Suppose that C1 solution near 0. Then,  has a lon is odd and H \u2208 \u210b. Suppose that there is a neighborhood \u2009U of O \u2208 Rn that satisfies A1\u2212) \u2264 H0\u2032(X) \u2264 0; for all A2\u00b1) \u2265 H0\u2032(X) \u2265 0 for all odd i \u2208 S(n) and Hi\u2032(O) \u2264 Hi\u2032(X) \u2264 0 for all even i \u2208 S(n). for all Suppose that C1 solution near 0. Then,  has a lon is even and H \u2208 \u210b. Suppose that there is a neighborhood \u2009U of O \u2208 Rn that satisfies A1+) \u2265 H0\u2032(X) \u2265 0; for all A2\u2213) \u2264 Hi\u2032(X) \u2264 0 for all odd i \u2208 S(n) and Hi\u2032(O) \u2265 Hi\u2032(X) \u2265 0 for all even i \u2208 S(n). for all Suppose that C1 solution near 0.Then,  has a loc > 1  and \u03c3 > 0 such that for arbitrary given \u03c4 > 0,  = 0 and \u03d5\u2032(0) = \u03c4.Under the conditions of n \u03c4 > 0,  has a \u2009Cc is real and (C1 solution \u03d5 with \u03d5(0) = 0 and \u03d5\u2032(0) \u2260 0, then by differentiating the equation, we can see that c is a root of characteristic polynomial \u2192+\u221e when x \u2192 +\u221e, and this means that P has a root c > 1. In the case of Theorems P has a root c < \u22121. Since for both of the cases c > 1 and c < \u22121, 0 < |cn+i\u2212 | <1, \u2009i = 0,1,\u2026, n \u2212 1 and c is a zero of (Hi\u2032(O), \u2009i = 0,1,\u2026, n \u2212 1. This also means that 1 \u2212 \u2211i=0n\u22121|Hi\u2032(O)||cn+2i\u22122| > 0. Now, we can choose a constant \u03c31 > 0 such that the following statements are true;A1\u03c3) holds on , where \u03c3 \u2208 {+, \u2212}; (A2\u03b4) holds on , where \u03b4 \u2208 {+, \u00b1, \u2213};  holds on , by A2\u03b4, \u03b4 \u2208 {+, \u00b1, \u2213}, we have\u03d5, we can get thatK2, we get that\ud835\udca2) \u2282 \ud835\udc9c.Moreover, for all \ud835\udca2 is continuous. Considering \u03d5, \u03c6 \u2208 \ud835\udc9c, by A2\u03b4, \u03b4 \u2208 {+, \u00b1, \u2213}, we have\ud835\udca2 is evident. By Schauder's fixed point theorem, there exists a \u03d5 \u2208 \ud835\udc9c such that \ud835\udca2\u03d5 = \u03d5. This means that  of 0 such that \u03d5\u22121 exists and is also C1 on J. Without any loss of generality, we can assume that J = \u03d5. Hence, \u03d5 :  \u2192 J is a homeomorphism. Moreover, we can choose a neighborhood I \u2282 J of 0 which is so small that ci\u03d5\u22121(x)\u2208 for all x \u2208 I, \u2009i = 1,2,\u2026, n. Let f(x) = \u03d5(c\u03d5\u22121(x)) for x \u2208 I. Clearly f is also C1 and invertible on I. Moreover, all iterates fj, \u2009j = 1,2,\u2026, n, are well defined on I, and fj(x) = \u03d5(cj\u03d5\u22121(x)), \u2009x \u2208 I. Obviously, we have f(0) = 0, f\u2032(0) = c, and f is locally expansive. Finally, for any x \u2208 I, we havef is a locally expansive C1 solution of  = 2sin(x0) + sin(x2). It is easy to verify that H satisfy the assumptions of C1 solution in a neighborhood of 0. Consider the following equation:H = \u22122sin(x0) \u2212 sin(x2). It is easy to verify that H satisfy the assumptions of C1 solution in a neighborhood of 0. Consider the following equation:H = 2sin(x0) + sin(x2). It is easy to verify that H satisfy the assumptions of C1 solution in a neighborhood of 0. Consider the following equation:"}
{"text": "We introduce the concept of Cayley bipolar fuzzy graphs and investigate some of their properties. We present some interesting properties of bipolar fuzzy graphs in terms of algebraic structures. We also discuss connectedness in Cayley bipolar fuzzy graphs. A digraph is a graph whose edges have direction and are called arcs (edges). Arrows on the arcs are used to encode the directional information: an arc from vertex (node) x to vertex y indicates that one may move from x to y but not from y to x. The Cayley graph was first considered for finite groups by Cayley in 1878. Max Dehn in his unpublished lectures on group theory from 1909 to 1910 reintroduced Cayley graphs under the name Gruppenbild (group diagram), which led to the geometric group theory of today. His most important application was the solution of the word problem for the fundamental group of surfaces with genus, which is equivalent to the topological problem of deciding which closed curves on the surface contract to a point  and \u03bcRN : X \u00d7 Y \u2192 .A bipolar fuzzy relation R be a bipolar fuzzy relation on universe X. Then R is called a bipolar fuzzy equivalence relation on X if it satisfies the following conditions: R\u2009\u2009is bipolar fuzzy reflexive; that is, R =  for each x \u2208 X;R is bipolar fuzzy symmetric; that is, R = R for any x, y \u2208 X;R is bipolar fuzzy transitive; that is, R \u2265 \u22c1y\u2227R).Let R be a bipolar fuzzy relation on universe X. Then R is called a bipolar fuzzy partial order relation on X if it satisfies the following conditions:R is bipolar fuzzy reflexive; that is, R =  for each x \u2208 X;R is bipolar fuzzy antisymmetric; that is, R \u2260 R for any x, y \u2208 X;R is bipolar fuzzy transitive; that is, R \u2265 \u22c1y\u2227R).Let R be a bipolar fuzzy relation on universe X. Then R is called a bipolar fuzzy linear order relation on X if it satisfies the following conditions:R is bipolar fuzzy partial relation;\u03bcRP\u2228\u03bcR\u22121P) > 0, (\u03bcRN\u2227\u03bcR\u22121N) < 0 for all x, y \u2208 X. where A =  is a bipolar fuzzy set in V and B =  is a bipolar fuzzy relation on E such thatxy \u2208 E. We note that B need not to be symmetric.A D be a bipolar fuzzy digraph. The indegree of a vertex x in D is defined by ind(x) = (ind\u03bcP(x), ind\u03bcN(x)), where ind\u03bcP(x) = \u2211y\u2260x\u03bcAP(xy) and ind\u03bcN(x) = \u2211y\u2260x\u03bcAN(xy). Similarly, the outdegree of a vertex x in D is defined by outd(x) = (outd\u03bcP(x), outd\u03bcN(x)), where outd\u03bcP(x) = \u2211y\u2260x\u03bcAP(xy) and outd\u03bcN(x) = \u2211y\u2260x\u03bcAN(xy). A bipolar fuzzy digraph in which each vertex has the same outdegree r is called an outregular digraph with index of outregularity r. In-regular digraphs are defined similarly.Let V, \u2217) be a group and let A =  be the bipolar fuzzy subset of V. Then the bipolar fuzzy relation R defined on V byG =  called the Cayley bipolar fuzzy graph induced by the .Let  be a group and let A =  be a bipolar fuzzy subset of V. Then the bipolar fuzzy relation R on V defined byG = , called the Cayley bipolar fuzzy graph induced by the .Let  = \u03bcAP(1) = \u03bcAP(2) = 0.5, \u03bcAN(0) = \u03bcAN(1) = \u03bcAN(2) = \u22120.4. Then the Cayley bipolar fuzzy graph G =  induced by  is given by Consider the group R in the above definition describes the strength of each directed edge. Let G denote a bipolar fuzzy graph G =  induced by the triple .We see that Cayley bipolar fuzzy graphs are actually bipolar fuzzy digraphs. Furthermore, the relation G is vertex transitive.The Cayley bipolar fuzzy graph a,\u2009\u2009b \u2208 V. Define \u03c8 : V \u2192 V by \u03c8(x) = ba\u22121x for all x \u2208 V. Clearly, \u03c8 is a bijective map. For each x, y \u2208 V,Let R(\u03c8(x), \u03c8(y)) = R. Hence \u03c8 is an automorphism on G. Also \u03c8(a) = b. Hence G is vertex transitive.Every vertex transitive bipolar fuzzy graph is regular.G =  be any vertex transitive bipolar fuzzy graph. Let u, v \u2208 V. Then there is an automorphism f on G such that f(u) = v. Note thatG is regular.Let Cayley bipolar fuzzy graphs are regular.Proof follows from Theorems G =  denote bipolar fuzzy graph. Then bipolar fuzzy relation R is reflexive if and only if \u03bcAP(1) = 1 and \u03bcAN(1) = \u22121.Let R is reflexive if and only if R =  for all x \u2208 V. NowR is reflexive if and only if \u03bcAP(1) = 1 and \u03bcAN(1) = \u22121.G =  denote bipolar fuzzy graph. Then bipolar fuzzy relation R is symmetric if and only if (\u03bcAP(x), \u03bcAN(x)) = (\u03bcAP(x\u22121), \u03bcAN(x\u22121)) for all x \u2208 V.Let R is symmetric. Then for any x \u2208 V,\u03bcAP(x), \u03bcAN(x)) = (\u03bcAP(x\u22121), \u03bcAN(x\u22121)) for all x \u2208 V. Then for all x, y \u2208 V,R is symmetric.Suppose that R is antisymmetric if and only if {x : (\u03bcAP(x), \u03bcAN(x)) = (\u03bcAP(x\u22121), \u03bcAN(x\u22121)) = .A bipolar fuzzy relation S, \u2217) be a semigroup. Let A =  be a bipolar fuzzy subset of S. Then A is said to be a bipolar fuzzy subsemigroup of S if for all x, y \u2208 S, \u03bcBP(xy) \u2265 \u03bcAP(x)\u2227\u03bcAP(y) and \u03bcBN(xy) \u2264 \u03bcAN(x)\u2228\u03bcAN(y).Let  is a bipolar fuzzy subsemigroup of .A bipolar fuzzy relation R is transitive and let x, y \u2208 V. Then R2 \u2264 R. Now for any x \u2208 V, we have R = (\u03bcAP(x), \u03bcAN(x)). This implies that {R\u2227R : z \u2208 V} = R2 \u2264 R. That is \u2228{\u03bcAP(z)\u2227\u03bcAP(z\u22121xy) : z \u2208 V} \u2264 \u03bcAP(xy) and \u2227{\u03bcAN(z)\u2228\u03bcAN(z\u22121xy) : z \u2208 V} \u2265 \u03bcAN(xy). Hence \u03bcAP(xy) \u2265 \u03bcAP(x)\u2227\u03bcAP(y) and \u03bcAN(xy) \u2264 \u03bcAP(x)\u2228\u03bcAN(y). Hence A =  is a bipolar fuzzy subsemigroup of .Suppose that A =  is a bipolar fuzzy subsemigroup of . That is, for all x, y \u2208 V\u03bcBP(xy) \u2265 \u03bcAP(x)\u2227\u03bcAP(y) and \u03bcBN(xy) \u2264 \u03bcAN(x)\u2228\u03bcAN(y). Then for any x, y \u2208 V,R\u03bcP2 \u2264 R\u03bcP and R\u03bcN2 \u2265 R\u03bcN. Hence R is transitive.Conversely, suppose that We conclude that.R is a partial order if and only if A =  is a bipolar fuzzy subsemigroup of  satisfying\u03bcAP(1) = 1 and \u03bcAN(1) = \u22121,x : (\u03bcAP(x), \u03bcAN(x)) = (\u03bcAP(x\u22121), \u03bcAN(x\u22121)) = {1, \u22121}}.{A bipolar fuzzy relation R is a linear order if and only if  is a bipolar fuzzy subsemigroup of  satisfying\u03bcAP(1) = 1 and \u03bcAN(1) = \u22121,x : (\u03bcAP(x), \u03bcAN(x)) = (\u03bcAP(x\u22121), \u03bcAN(x\u22121)) = {1, \u22121}},{x : \u03bcAP(x)\u2228\u03bcAP(x\u22121) > 0, \u03bcAN(x)\u2227\u03bcAN(x\u22121) < 0} = V. {A bipolar fuzzy relation R is a linear order. Then by x \u2208 V, (R\u2228R\u22121) > 0. This implies that R\u2228R > 0. Hence {x : \u03bcAP(x)\u2228\u03bcAP(x\u22121) > 0, \u03bcAN(x)\u2227\u03bcAN(x\u22121) < 0}.Suppose R is partial order. Now for any x, y \u2208 V, we have (x\u22121y), (y\u22121x) \u2208 V. Then by condition (iv), {x : \u03bcAP(x)\u2228\u03bcAP(x\u22121) > 0, \u03bcAN(x)\u2227\u03bcAN(x\u22121) < 0}. Therefore R is linear order.Conversely, suppose that conditions (i), (ii), and (iii) hold. Then by R is a equivalence relation if and only if  is a bipolar fuzzy subsemigroup of  satisfying\u03bcAP(1) = 1 and \u03bcAN(1) = \u22121,\u03bcAP(x), \u03bcAN(x)) = (\u03bcAP(x\u22121), \u03bcAN(x\u22121)) for all x \u2208 V. > 0, \u03bcAN(xi) < 0, for i = 1,2, 3,\u2026, n, we have, \u03bcAP(x1x2 \u22ef xn) = 0 and \u03bcAN(x1x2 \u22ef xn) = 0.G is a Hasse diagram and let x1, x2,\u2026, xn be vertices in V with n \u2265 2 and \u03bcAP(xi) > 0, \u03bcAN(xi) < 0, for i = 1,2, 3,\u2026, n. Then it is obvious that R = (\u03bcAP(xi), \u03bcAN(xi)), for i = 1,2,\u2026, n, where x0 = 1. Therefore  is a path from 1 to x1x2 \u22ef xn. Since G is a Hasse diagram, we have R = 0. This implies that \u03bcAP(x1x2 \u22ef xn) = 0 and \u03bcAN(x1x2 \u22ef xn) = 0. Conversely, suppose that for any collection x1, x2,\u2026xn of vertices in V with n \u2265 2 and \u03bcAP(xi) > 0, \u03bcAN(xi) < 0, for i = 1,2, 3,\u2026, n, we have, \u03bcAP(x1x2 \u22ef xn) = 0 and \u03bcAN(x1x2 \u22ef xn) = 0. Let  be a path in G from x0 to xn with n \u2265 2. Then R > 0, for i = 1,2,\u2026, n. Therefore \u03bcAP(xi\u22121\u22121xi) > 0, \u03bcAN(xi\u22121\u22121xi) < 0, for i = 1,2,\u2026, n. Now consider the elements x0\u22121x1, x1\u22121x2,\u2026, xn\u22121\u22121xn in V. Then by assumption \u03bcAP(x0\u22121x1x1\u22121x2 \u22ef xn\u22121\u22121xn) = 0 and \u03bcAN(x0\u22121x1x1\u22121x2 \u22ef xn\u22121\u22121xn) = 0. That is, \u03bcAP(x0\u22121xn) = 0 and \u03bcAN(x0\u22121xn) = 0. Hence, R = 0. Thus G is a Hasse diagram.Suppose G =  be any bipolar fuzzy graph; then G is connected  if and only if the induce fuzzy graph  is connected .Let S, \u2217) be a semigroup and let A =  be a bipolar fuzzy subset of S. Then the subsemigroup generated by A is the meeting of all bipolar fuzzy subsemigroups of S which contains A. It is denoted by \u2329A\u232a.Let  be a semigroup and A =  be a bipolar fuzzy subset of S. Then bipolar fuzzy subset \u2329A\u232a is precisely given by \u2329\u03bcAP\u232a(x) = \u2228{\u03bcAP(x1)\u2227\u03bcAP(x2)\u2227\u22ef\u2227\u03bcAP(xn) : x = x1x2 \u22ef xn with\u2009\u2009\u03bcAP(xi) > 0 for i = 1,2,\u2026, n}, \u2329\u03bcAN\u232a(x) = \u2227{\u03bcAN(x1)\u2228\u03bcAN(x2)\u2228\u22ef\u2228\u03bcAN(xn) : x = x1x2 \u22ef xn with \u03bcAN(xi) < 0 fori = 1,2,\u2026, n} for any x \u2208 S.Let (S defined by \u03bcAP(xi) > 0 for i = 1,2,\u2026, n}, \u03bcAN(xi) < 0 for i = 1,2,\u2026, n}, for any x \u2208 S. Let x, y \u2208 S. If \u03bcAP(x) = 0 or \u03bcAP(y) = 0, then \u03bcAP(x)\u2227\u03bcAP(y) = 0 and \u03bcAN(x) = 0 or \u03bcAN(y) = 0, and then \u03bcAN(x)\u2228\u03bcAN(y) = 0. Therefore, \u03bcAP(x) \u2260 0, \u03bcAN(x) \u2260 0, then by definition of S containing . Now let L be any bipolar fuzzy subsemigroup of S containing . Then for any x \u2208 S with x = x1x2 \u22ef xn with \u03bcAP(xi) > 0, \u03bcAN(xi) < 0, for i = 1,2,\u2026, n, we have \u03bcLP(xi) \u2265 \u03bcLP(x1)\u2227\u03bcLP(x2)\u2227\u22ef\u2227\u03bcLP(xn) \u2265 \u03bcAP(x1)\u2227\u03bcAP(x2)\u2227\u22ef\u2227\u03bcAP(xn) and \u03bcLN(xi) \u2265 \u03bcLN(x1)\u2227\u03bcLN(x2)\u2227\u22ef\u2227\u03bcLN(xn) \u2265 \u03bcAN(x1)\u2227\u03bcAN(x2)\u2227\u22ef\u2227\u03bcAN(xn). Thus \u03bcLP(x)\u2265\u2228{\u03bcAP(x1)\u2227\u03bcAP(x2)\u2227\u22ef\u2227\u03bcAP(xn) : x = x1x2 \u22ef xn with \u03bcAP(xi) > 0 for i = 1,2,\u2026, n}, \u03bcLN(x)\u2264\u2227{\u03bcAN(x1)\u2228\u03bcAN(x2)\u2228\u22ef\u2228\u03bcAN(xn) : x = x1x2 \u22ef xn with \u03bcAN(xi) < 0 for i = 1,2,\u2026, n}, for any x \u2208 S. Hence x \u2208 S. Thus \u03bcAP, \u03bcAN).Let S, \u2217) be a semigroup and A =  be a bipolar fuzzy subset of S. Then for any \u03b1 \u2208 ,  =  and (\u2329(\u03bc+)\u03b1P\u232a, \u2329(\u03bc+)\u03b1N\u232a) = , where  denotes the subsemigroup generated by  and \u2329\u232a denotes bipolar fuzzy subsemigroup generated by .Let  = . Similarly, we have (\u2329(\u03bc+)\u03b1P\u232a, \u2329(\u03bc+)\u03b1N\u232a) = .S, \u2217) be a semigroup and A =  be a bipolar fuzzy subset of S. Then by A) = A+\u232a = supp\u2061\u2329A\u232a.Let  induced by . Then we have the following results.Let A be any subset of V\u2032 and G\u2032 =  be the Cayley graph induced by . Then G\u2032 is connected if and only if \u2329A\u232a\u2287V \u2212 v1.Let G is connected if and only if\u2009\u2009supp\u2061\u2329A\u232a\u2287V \u2212 v1.A be any subset of a set V\u2032 and let G\u2032 =  be the Cayley graph induced by the triplet . Then G\u2032 is weakly connected if and only if \u2329A \u222a A\u22121\u232a\u2287V \u2212 v1, where A\u22121 = {x\u22121 : x \u2208 A}.Let S, \u2217) be a group and let A be a bipolar fuzzy subset of S. Then we define A\u22121 as bipolar fuzzy subset of S given by A\u22121(x) = A(x\u22121) for all x \u2208 S.Let (G is weakly connected if and only if supp\u2061(\u2329A \u222a A\u22121\u232a)\u2287V \u2212 v1.A be any subset of a set V\u2032 and let G\u2032 =  be the Cayley graph induced by the triplet . Then G\u2032 is semiconnected if and only if \u2329A\u232a \u222a \u2329A\u22121\u232a\u2287V \u2212 v1, where A\u22121 = {x\u22121 : x \u2208 A}.Let G is semi-connected if and only if supp\u2061(\u2329A\u232a \u222a \u2329A\u22121\u232a)\u2287V \u2212 v1.G\u2032 =  be the Cayley graph induced by the triplet . Then G\u2032 is locally connected if and only if \u2329A\u232a = \u2329A\u22121\u232a, where A\u22121 = (x\u22121 : x \u2208 A).Let G is locally connected if and only if supp\u2061(\u2329A\u232a) = supp\u2061(\u2329A\u22121\u232a).Let G\u2032 =  be the Cayley graph induced by the triplet , where V\u2032 is finite. Then G\u2032 is quasi-connected if and only if it is connected.Let G is quasi-connected if and only if it is connected.A finite Cayley bipolar fuzzy graph \u03bcP strength of a path P = v1, v2,\u2026, vn is defined as min\u2061) for all i and j and is denoted by S\u03bcP. The \u03bcN strength of a path P = v1, v2,\u2026, vn is defined as max\u2061) for all i and j and is denoted by S\u03bcN.The G =  be a bipolar fuzzy graph. Then G is said to be\u03b1-connected if for every pair of vertices x, y \u2208 G, there is a path P from x to y such that strength (P) \u2265 \u03b1,weakly \u03b1-connected if a bipolar fuzzy graph  is \u03b1-connected,semi-\u03b1-connected if for every x, y \u2208 V, there is a path of strength greater than or equal to \u03b1 from x to y or from y to x in G,locally \u03b1-connected if for every pair of vertices x and y, there is a path P of strength greater than or equal to \u03b1 from x to y whenever there is a path P\u2032 of strength greater than or equal to \u03b1 from y to x,quasi-\u03b1-connected if for every pair x, y \u2208 V, there is some z \u2208 V such that there is directed path from z to x of strength greater than or equal to \u03b1 and there is a directed path from z to y of strength greater than or equal to \u03b1.Let G =  be any bipolar fuzzy graph; then G is \u03b1-connected  if and only if the induce fuzzy graph  is connected .Let G denote the Cayley bipolar fuzzy graphs G =  induced by . Also for any \u03b1 \u2208 , we have the following results.Let G is \u03b1-connected if and only if \u2329A\u232a\u03b1\u2287V \u2212 v1.G is weakly \u03b1-connected if and only if \u2329A\u222aA\u22121\u232a\u03b1\u2287V \u2212 v1.G is semi-\u03b1-connected if and only if (\u2329A\u232a\u03b1 \u222a \u2329A\u22121\u232a\u03b1)\u2287V \u2212 v1.G be locally \u03b1-connected if and only if \u2329A\u232a\u03b1 = \u2329A\u03b1\u22121\u232a.Let G is quasi-\u03b1-connected if and only if it is \u03b1-connected.A finite Cayley bipolar fuzzy graph Fuzzy graph theory is finding an increasing number of applications in modeling real time systems where the level of information inherent in the system varies with different levels of precision. Fuzzy models are becoming useful because of their aim of reducing the differences between the traditional numerical models used in engineering and sciences and the symbolic models used in expert systems. A bipolar fuzzy set is a generalization of the notion of a fuzzy set. We have introduced the notion of Cayley bipolar fuzzy graphs in this paper. The natural extension of this research work is application of bipolar fuzzy digraphs in the area of soft computing including neural networks, decision making, and geographical information systems."}
{"text": "We consider the three-dimensional Boussinesq equations, and obtain an Osgood type regularity criterion in terms of the velocity gradient. Assume that  is the smooth solution to  for 0 \u2a7d t < T. Ifu, \u03b8) can be extended after time t = T. Here, Let , we haveThe Osgood type condition  is weakeThe rest of this paper is organized as follows. In \ud835\udcae(\u211d3) be the Schwartz class of rapidly decreasing functions. For f \u2208 \ud835\udcae(\u211d3), its Fourier transform \u03c6 \u2208 \ud835\udcae(\u211d3) such thatj \u2208 \u2124, the Littlewood-Paley projection operators Sj and f \u2208 L2(\u211d3), thenL2 sense. By telescoping the series, we thus have the following Littlewood-Paley decomposition:f \u2208 L2(\u211d3), where the summation is the L2 sense. Notice thatp \u2a7d q \u2a7d \u221e and C is an absolute constant independent of f and j.Let \u221e < s < \u221e, 1 \u2a7d p, q \u2a7d \u221e; the homogeneous Besov space \ud835\udcb5\u2032(\u211d3) is the dual space ofLet \u2212refer to  for moreH1 bounds of the solution .This section is devoted to proving 1 with \u2212\u0394u, , then  is uniformly bounded in L\u221e). This completes the proof of Taking the inner products of 1 with \u2212\u0394on as in , we getn as in (\u2207u=\u2211l<\u2212q\u0394imate asJ3\u2a7d\u2211l>q||see thatJ\u2a7d\u2211l<\u2212q\u222b\u211dy to getI1\u2a7d12||\u2207(thering (J\u2a7d[C2\u2212q/2her that12ddt||\u2207(ituting (12ddt||\u2207(mate as2J3\u2a7d\u2211l>q||"}
{"text": "F3(\u00b7) due to Marichev-Saigo-Maeda, to the product of the generalized Bessel function of the first kind due to Baricz. The results are expressed in terms of the multivariable generalized Lauricella functions. Corresponding assertions in terms of Saigo, Erd\u00e9lyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals are also presented. Some interesting special cases of our two main results are presented. We also point out that the results presented here, being of general character, are easily reducible to yield many diverse new and known integral formulas involving simpler functions.We apply generalized operators of fractional integration involving Appell's function  The fractional calculus is nowadays one of the most rapidly growing subject of mathematical analysis. It is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders. The fractional integral operator involving various special functions has found significant importance and applications in various subfields of applicable mathematical analysis. Many applications of fractional calculus can be found in turbulence and fluid dynamics, stochastic dynamical system, plasma physics and controlled thermonuclear fusion, nonlinear control theory, image processing, nonlinear biological systems, astrophysics, and in quantum mechanics. Since last four decades, a number of workers like Love , McBridew\u03bd(z) has been introduced and studied in [The computation of fractional derivatives  of special functions of one and more variables is important from the point of view of the usefulness of these results in the evaluation of generalized integrals and the solution of differential and integral equations. Motivated by these avenues of applications, a remarkably large number of fractional integral formulas involving a variety of special functions have been developed by many authors . Fraudied in \u201334. Herex > 0 and \u03b1, \u03b1\u2032, \u03b2, \u03b2\u2032, \u03b3 \u2208 \u2102 byF3(\u00b7) in the kernel. These operators were rediscovered and studied by Saigo in [Fp;\u03bc . Such fp;\u03bc (see ) were stp;\u03bc (see .F3(\u00b7) denotes so-called 3rd Appell function    2) the sw\u03bd(z), which is defined for z \u2208 \u2102\u2216{0} and b, c, \u03bd \u2208 \u2102 with \u211c(\u03bd)>\u22121 by the following series  is the familiar Gamma function   2) with e, e.g., \u201333, [34,p\u2009\u03a8q in several variables and defined by , \u03b2jk , \u03b3jk , and \u03b4jk , for all k = 1,\u2026, n, are real and positive and (ap) means the array of p-parameters a1,\u2026, ap; with similar interpretations for (bp), (\u03b3p11), (\u03b1p11), and so forth, and where (a)n is the Pochhammer symbol defined (for a \u2208 \u2102) by   2) with Our results in this Section are based on the preliminary assertions giving composition formula of fractional integral  with a p\u03b1, \u03b1\u2032, \u03b2, \u03b2\u2032, \u03b3 \u2208 \u2102 and if \u211c(\u03b3) > 0, \u211c(\u03c1) > max\u2061{0, \u211c(\u03b1 + \u03b1\u2032 + \u03b2 \u2212 \u03b3), \u211c(\u03b1\u2032 \u2212 \u03b2\u2032)}, thenLet The Marichev-Saigo-Maeda fractional integration  of produ\u03b1, \u03b1\u2032, \u03b2, \u03b2\u2032, \u03c3, \u03bb, \u03b3, \u03bdj, \u03c1j, b, c \u2208 \u2102 and satisfying the inequalitiesx > 0, |ax/b | <1, one hasA, B, C, D, E, F, G, and H are given by the followingLet \u2110. Using definition , we obtainFollowing the convergence condition of This, in accordance with , gives t\u03b1\u2032 = 0 in w\u03bd(z).On setting \u03b1, \u03b2, \u03b2\u2032, \u03c3, \u03bb, \u03b3, \u03bdj, \u03c1j, b, c \u2208 \u2102 and x > 0, |ax/b| < 1A, C, D, F, G, and H are given by , , a, a\u03b2, \u03b2\u2032,given by , respectIn the sequel, we use the following results.\u03b1, \u03b1\u2032, \u03b2, \u03b2\u2032, \u03b3 \u2208 \u2102 and if \u211c(\u03b3) > 0, \u211c(\u03c1) < 1 + min\u2061{\u211c(\u2212\u03b2), \u211c(\u03b1 + \u03b1\u2032 \u2212 \u03b3), \u211c(\u03b1 + \u03b2\u2032 \u2212 \u03b3)}, thenLet The Marichev-Saigo-Maeda fractional integration  of produ\u03b1, \u03b1\u2032, \u03b2, \u03b2\u2032, \u03c3, \u03bb, \u03b3, \u03bdj, \u03c1j, b, c \u2208 \u2102 and x > 0, |a/bx | <1 such thatA\u2032, B\u2032, C\u2032, D\u2032, E\u2032, and F\u2032 are given by the following:G, H are given by (Let given by  and 19)\u03b1, \u03b1\u2032, \u03b2,\ud835\udca5. Applying definition (b \u2212 at)\u03bb\u2212 and then changing the order of integration and summation, we findk \u2208 \u21150For convenience, let the left-hand side of the  be denotfinition  and the Now, on making use of This, in accordance with , gives t\u03b1\u2032 = 0 in w\u03bd(z) as follows.On taking \u03b1, \u03b2, \u03b2\u2032, \u03c3, \u03bb, \u03b3, \u03bdj, \u03c1j, b, c \u2208 \u2102 and x > 0, |a/bx| < 1, andA\u2032, B\u2032, C\u2032, D\u2032, G and H are given by , , ,  are oscillatory and may be regarded as generalizations of trigonometric functions. Indeed, for large argument (z \u2265 1) the function z \u2212 \u03c0\u03bd/2 \u2212 \u03c0/4). Similarly, modified Bessel functions of the first kind I\u03bd(z), which are Bessel functions of imaginary argument, may be regarded as generalization of exponentials. Exponential functions have the unique and special property that they are particularly easy to multiply and to raise to powers: eazebz = ea+b)z( and (ez)r = erz. Further, it can be easily seen that for c = 1 and b = 1, the Generalized Bessel function of the first kind (J\u03bd(z), and when c = \u22121 and b = 1 the function w\u03bd(z) becomes I\u03bd(z). Similarly, when c = 1 and b = 2, the function w\u03bd(z) reduces to c = \u22121 and b = 2, then w\u03bd(z) becomes w\u03bd(0) = 0. Therefore, the results presented in this paper are easily converted in terms of the various special Bessel functions after some suitable parametric replacement.In this section, we briefly consider another variation of the results derived in the preceding sections. Bessel functions are important special functions that appear widely in science and engineering. Bessel functions of the first kind rst kind  reduces uel from , we have\u03bd = \u2212b/2, then the generalized Bessel function w\u03bd(z) in  in  have fole, e.g., ):\u03bd = \u2212b/2 and c = \u2212c2, w\u03bd(z) in  in  in  have rel\u03bd(z) in  have fole, e.g., ):(47)w1\u03bd = 1 \u2212 b/2 and c = \u2212c2 in (w\u03bd(z) and hyperbolic sine function as (48)w1\u2212Now, by virtue of the relations  to 48),,48), one\u03b1\u2032 = 0, the results given by Malik et al. [\u03bb = 0 and b = c = 1, then our main results \u03b1\u2032 = 0, t results  and (31)Fractional integral formulas involving products of Bessel functions have been developed and play an important role in several physical problems. In fact, Bessel functions are playing the important role in studying solutions of differential equations, integral equations , and the"}
{"text": "A new kind of fuzzy module over a fuzzy ring is introduced by generalizing Yuan and Lee's definition of the fuzzy group and Akta\u015f and \u00c7a\u011fman's definition of fuzzy ring. The concepts of fuzzy submodule, and fuzzy module homomorphism are studied and some of their basic properties are presented analogous of ordinary module theory. G is fuzzy and the binary operation on G is nonfuzzy in the classical sense as Rosenfeld's definition is\u2009\u2009a\u2009\u2009mapping} andLet A = {a} and B = {b} and let R be denoted as a\u2218b; thenLet Using the notations in , we haveG be a nonempty set and let R be a fuzzy binary operation on G.\u2009\u2009 is called a fuzzy group if the following conditions are true: (G1)a, b, c, z1, z2 \u2208 G, ((a\u2218b)\u2218c)(z1) > \u03b8 and (a\u2218(b\u2218c))(z2) > \u03b8 imply z1 = z2;\u2200(G2)eo \u2208 G such that (eo\u2218a)(a) > \u03b8 and (a\u2009\u2009\u2218\u2009\u2009eo)(a) > \u03b8 for any a \u2208 G ;\u2203(G3)a \u2208 G, \u2203b \u2208 G such that (a\u2218b)(eo) > \u03b8 and (b\u2218a)(eo) > \u03b8 .\u2200Let a\u2218b)\u2218c)(d) > \u03b8\u21d4(a\u2218(b\u2218c))(d) > \u03b8.Consider (c) > \u03b8\u2009\u2009implies\u2009\u2009c \u2208 H;\u2200a \u2208 H\u2009\u2009implies\u2009\u2009a\u22121 \u2208 H.H be a fuzzy subgroup of G. LetaH\u2009\u2009(Ha) is called a left (right) coset of H.Let H be a fuzzy subgroup of G:H is called a normal fuzzy subgroup of G.Let G, R) be a fuzzy subgroup. IfG, R) is called abelian fuzzy group.Let  and  be two fuzzy groups and let f : G1 \u2192 G2 be a mapping. Iff is called a fuzzy (group) homomorphism. If f is 1-1, it is called a fuzzy monomorphism. If f is onto, it is called a fuzzy epimorphism. If f is both 1-1 and onto, it is called a fuzzy isomorphism.Let (G be a fuzzy binary operation on R. Then we have a mappingF(R) = {A | A : R \u2192 \u2009\u2009is\u2009\u2009a\u2009\u2009mapping} andA = {a} and B = {b} and let G and H be denoted as a\u2218b and a\u2217b, respectively. ThenLet Using the notations of , we haveR be a nonempty set and let G and H be two fuzzy binary operations on R. Then  is called fuzzy ring if the following conditions hold: (R1)R, G) is an abelian fuzzy group;((R2)a, b, c, z1, z2 \u2208 R, ((a\u2217b)\u2217c)(z1) > \u03b8 and (a\u2217(b\u2217c))(z2) > \u03b8 imply z1 = z2;\u2200(R3)a, b, c, z1, z2 \u2208 R, ((a\u2218b)\u2217c)(z1) > \u03b8 and ((a\u2217c)\u2218(b\u2217c))(z2) > \u03b8 imply z1 = z2; (a\u2217(b\u2218c))(z1) > \u03b8 and ((a\u2217b)\u2218(a\u2217c))(z2) > \u03b8 imply z1 = z2.\u2200Let R, G, H) be a fuzzy ring. a\u2217b)(u) > \u03b8\u21d4(b\u2217a)(u) > \u03b8, then  is said to be a commutative fuzzy ring.If (e\u2217 \u2208 R such that (a\u2217e\u2217)(a) > \u03b8 and (e\u2217\u2217a)(a) > \u03b8 for every a \u2208 R, then  is said to be fuzzy ring with identity.If \u2203R, G, H) be a fuzzy ring with identity. If (a\u2217b)(e\u2217) > \u03b8 and (b\u2217a)(e\u2217) > \u03b8, \u2200a \u2208 R, \u2203b \u2208 R, then b is said to be an inverse element of a and is denoted by a\u2217\u22121.Let  be a fuzzy ring and let S be a nonempty subset of R. Then  is a fuzzy subring of R if and only ifa\u2218b)(c) > \u03b8\u2009\u2009implies\u2009\u2009c \u2208 S\u2009\u2009and\u2009\u2009(a\u2217b)(c) > \u03b8\u2009\u2009implies\u2009\u2009c \u2208 S\u2009\u2009for all\u2009\u2009a, b \u2208 S, c \u2208 R; is called a fuzzy ideal of R if the following conditions are satisfied. x, y \u2208 I, (x\u2218y)(z) > \u03b8\u21d2z \u2208 I for all z \u2208 R;\u2200x \u2208 I, x\u22121 \u2208 I;\u2200s \u2208 I, for all r \u2208 R, (r\u2217s)(x) > \u03b8\u21d2x \u2208 I and (s\u2217r)(y) > \u03b8\u21d2y \u2208 I, x, y \u2208 R.For all A nonempty subset I be a fuzzy ideal of fuzzy ring R and let \u03a9 = {a\u2218I | a \u2208 R}. One defines a relation over \u03a9:Let R, G, H) be a fuzzy ring and  be an abelian fuzzy group and let p be fuzzy function R \u00d7 M into M. Then we have a mappingF(R) = {A | A : R \u2192 \u2009\u2009is\u2009\u2009a\u2009\u2009mapping} and F(M) = {N | N : M \u2192 is\u2009\u2009a\u2009\u2009mapping}.Let  and J be denoted as r\u2299m and a \u2295 b, respectively. ThenLet Using the notations , we haveR, G, H) be a fuzzy ring and let  be an abelian fuzzy group. M is called a (left) fuzzy module over R or (left) R-fmodule together with a fuzzy function p : R \u00d7 M \u2192 M if the following conditions hold. For all r, r1, r2 \u2208 R and for all m, m1, m2 \u2208 M,(M1)r\u2299(m1 \u2295 m2))(x) > \u03b8 and ((r\u2299m1)\u2295(r\u2009\u2299\u2009m2))(y) > \u03b8 imply x = y;((M2)r1\u2218r2)\u2299m)(x) > \u03b8 and ((r1\u2299m)\u2295(r2\u2299m))(y) > \u03b8 imply x = y;(((M3)r1\u2217r2)\u2009\u2009\u2299\u2009\u2009m)(x) > \u03b8 and (r1\u2009\u2009\u2299\u2009\u2009(r2\u2009\u2009\u2299\u2009\u2009m))(y) > \u03b8 imply x = y. be a fuzzy ring and let  be an R-fmodule; then for all r, r1, r2 \u2208 R, m, m1, m2 \u2208 M,r\u2009\u2299\u2009(m1\u2009\u2295\u2009m2))(x) > \u03b8\u21d4((r\u2009\u2009\u2299\u2009\u2009m1)\u2295(r\u2009\u2009\u2299\u2009\u2009m2))(x) > \u03b8;(r1\u2218r2)\u2299m)(x) > \u03b8\u2009\u2009\u21d4\u2009\u2009((r1\u2299m)\u2295(r2\u2299m))(x) > \u03b8;((r1\u2217r2)\u2299m)(x) > \u03b8\u2009\u2009\u21d4\u2009\u2009(r1\u2299(r2\u2299m))(x) > \u03b8.((Let (r\u2009\u2009\u2299\u2009\u2009(m1\u2009\u2009\u2295\u2009\u2009m2))(x) > \u03b8 and let x1, x2, y \u2208 M such that p > \u03b8, p > \u03b8, and J > \u03b8. Byx = y from (M1) and so ((r\u2009\u2009\u2299\u2009\u2009m1)\u2009\u2295\u2009(r\u2009\u2009\u2299\u2009\u2009m2))(x) > \u03b8. Similarly by ((r\u2299m1)\u2295(r\u2299m2))(x) > \u03b8 we have (r\u2299(m1 \u2295 m2))(x) > \u03b8.(1) Let (It is easy to prove (2) and (3) similar to the proof of (1).R, G, H) be a fuzzy ring with zero element eo and  be a left R-fmodule with identity element eJ. Then for all r \u2208 R, m \u2208 M,r\u2299eJ)(eJ) > \u03b8;(eo\u2299m)(eJ) > \u03b8;(r\u2299m)(x) > \u03b8\u21d2(r\u2299m\u22121)(x\u22121) > \u03b8;(r\u2299m)(x) > \u03b8\u21d2(r\u22121\u2299m)(x\u22121) > \u03b8.(Let (x \u2208 M such that (r\u2299eJ)(x) > \u03b8. Then\u2009\u2009(1) Let r\u2009\u2009\u2299\u2009\u2009eJ)\u2009\u2295\u2009(r\u2009\u2009\u2299\u2009\u2009eJ))(x) > \u03b8 from It follows that  > \u03b8 and x = eJ from Proposition\u2009\u20092.1 in )([x \u2295 B]) > \u03b8 and (r\u2299[a \u2295 B])([y \u2295 B]) > \u03b8. Then there exist p > \u03b8 and p > \u03b8. Since a1 \u2295 B ~ a2 \u2295 B, we have x\u2032 \u2295 B ~ y\u2032 \u2295 B from x \u2295 B] = [y \u2295 B].Let (r\u2299([a1 \u2295 B]\u2295[a2 \u2295 B]))([x \u2295 B]) > \u03b8 and ((r\u2299[a1 \u2295 B])\u2295(r\u2299[a2 \u2295 B]))([y \u2295 B]) > \u03b8 and let a \u2208 A such that J > \u03b8. Then there exist\u2009\u2009x\u2032, x1\u2032, x2\u2032 \u2208 A, b1, b2 \u2208 B such that(1) Let  > \u03b8. Then by J > \u03b8, J > \u03b8, J > \u03b8, and the proof of Theorem\u2009\u20094.2 in  = [y \u2295 B].It follows that r1\u2218r2)\u2299[a \u2295 B])([x \u2295 B]) > \u03b8 and ((r1\u2299[a \u2295 B])\u2009\u2295\u2009(r2\u2299[a \u2295 B]))([y \u2295 B]) > \u03b8 and let r \u2208 R such that G > \u03b8. Then we have x2, x3 \u2208 A, and b1, b2 \u2208 B such that(2) Let ((r1\u2299a1) \u2295 (r1\u2299b1))(x2) > \u03b8. So there exist u1 \u2208 A, v1 \u2208 B such thatThen, byJ > \u03b8 and so u1 \u2295 B ~ x2 \u2295 B.Thus r2\u2299a1)\u2295(r2\u2299b2))(x3) > \u03b8. So there exist u2 \u2208 A, v2 \u2208 B such thatSinceJ > \u03b8 and so u2 \u2295 B ~ x3 \u2295 B.Similarly, we get r1\u2299a1)\u2295(r2\u2299a1))(x1) > \u03b8. HenceSinceu1 \u2295 B ~ x2 \u2295 B, u2 \u2295 B ~ x3 \u2295 B, J > \u03b8, and J > \u03b8, we obtain x1 \u2295 B ~ y1 \u2295 B and [x \u2295 B] = [y \u2295 B].Therefore, since r1\u2217r2)\u2009\u2009\u2299\u2009\u2009[a \u2295 B])([x \u2295 B]) > \u03b8 and (r1\u2009\u2009\u2299\u2009\u2009(r2\u2009\u2009\u2299\u2009\u2009[a\u2009\u2009\u2295\u2009B]))([y\u2009\u2009\u2295\u2009\u2009B]) > \u03b8 and let r \u2208 R such that H > \u03b8. Then there exist x2, x3 \u2208 A, and b1, b2 \u2208 B such thatw \u2208 A such that p > \u03b8. Byw = y1 and p > \u03b8.(3) Finally, let  > \u03b8, and p > \u03b8, then we have x1 \u2295 B ~ y1 \u2295 B from x \u2295 B] = [y \u2295 B].Since R, G, H) be fuzzy ring and f : A \u2192 B an R-fmodule epimorphism. Then A/K is isomorphic to B where K = Ker\u2061\u2009\u2009f.Let  > \u03b8.\u2009\u20095.3 in . Then itw \u2208 A such that p > \u03b8. Since f is an R-fmodule homomorphism, we get p, f(w)) > \u03b8, so u = f(w).\u2009 \u2009By a\u2032\u2009\u2009\u2295\u2009\u2009K ~ a\u2009\u2009\u2295\u2009\u2009K, p > \u03b8, p > \u03b8, and b\u2032 \u2295 K ~ w \u2295 K and so [b \u2295 K] = [w \u2295 K]. Therefore u = f(b).Let"}
{"text": "The authors find some new inequalities of Jordan type for the sine function. These newly established inequalities are of new form and are applied to deduce some known results. The equalities in  in xm for m \u2265 2. A straightforward computation givesLet the function equality . Hence, In the final section of this paper, we will apply In order to prove Theorems f, g :  \u2192 \u211d be differentiable on . If g\u2032 \u2260 0 and f\u2032/g\u2032 are decreasing on , then the functionsa, b). Let We are now in a position to prove our theorems.\u03c0/2]. A direct calculation givesx > x on  leads to\u03c0/2) for m \u2265 2 and n \u2265 0. As a result, the function f3\u2032(x)/f4\u2032(x) is decreasing on . In virtue of \u03c0/2). SinceLet\u03c0/2]. It is easy to see that\u03c0/2]. Furthermore, we havex > x and the conditions in (f3\u2032(x)/f4\u2032(x)]\u2032 is negative on . This means that the function f3\u2032(x)/f4\u2032(x) is decreasing on . Consequently, making use of \u03c0/2). SinceH(\u03c0/2) \u2264 H(x) \u2264 H(0), the inequality (Lettions in , it is nequality  follows.After proving Theorems \u03bb \u2264 1 and A, B > 0 with A + B \u2264 \u03c0. Then,Let 0 \u2264 ce paper , many mace paper  or an evThe first application of k \u2265 2, let Ai > 0 and \u2211i=1kAi \u2264 \u03c0. If 0 \u2264 \u03bb \u2264 1, thenn \u2265 0 and m \u2265 2 are integers. For x = \u03bb\u03c0/2 in the inequality (i < j \u2264 n results in (Substituting equality  reveals e either , , if n \u2265 0 and m \u2265 2 are integers, thenFor This follows from integrating on all sides of the double inequality . n = 0 givesn = 0 and m = 2 yieldsApplying  0 gives41m+1+1\u2264\u222bished in ."}
{"text": "Adult stem cells have been widely investigated in bioengineering approaches for tissue repair therapy. We evaluated the clinical value and safety of the application of cultured bone marrow-derived allogenic mesenchymal stem cells (MSCs) for treating skin wounds in a canine model.Topical allogenic MSC transplantation can accelerate the closure of experimental full-thickness cutaneous wounds and attenuate local inflammation.n = 10; 3\u20136 years old; 7.2\u201313.1 kg) were studied.Adult healthy beagle dogs  and wound healing-related factors .Topical transplantation of MSCs results in paracrine effects on cellular proliferation and angiogenesis, as well as modulation of local mRNA expression of several factors related to cutaneous wound healing.Les cellules souches adultes ont \u00e9t\u00e9 largement \u00e9tudi\u00e9es dans les approches de bio-ing\u00e9nierie pour la th\u00e9rapie de r\u00e9paration tissulaire. Nous \u00e9valuons l'efficacit\u00e9 clinique et la s\u00e9curit\u00e9 de l'application de cellules souches m\u00e9senchymateuses allog\u00e9niques en culture d\u00e9riv\u00e9es de moelle osseuse (MSCs) pour le traitement de plaies cutan\u00e9es dans un mod\u00e8le canin.La transplantation de MSC allog\u00e9nique topique peut acc\u00e9l\u00e9rer la fermeture en toute \u00e9paisseur de plaies cutan\u00e9es exp\u00e9rimentales et att\u00e9nuer l'inflammation locale.n = 10; 3\u20136 ans; 7.2\u201313.1 kg) ont \u00e9t\u00e9 \u00e9tudi\u00e9s.Des chiens beagles adultes sains  et des facteurs li\u00e9s \u00e0 la cicatrisation .La transplantation topique de MSCs r\u00e9sulte en des effets paracrines de prolif\u00e9ration cellulaire et d'angiog\u00e9n\u00e8se ainsi qu'en une modulation de l'expression locale d'ARNm de plusieurs facteurs li\u00e9s \u00e0 la cicatrisation cutan\u00e9e.Las c\u00e9lulas madre de adultos se han estudiado extensivamente en estrategias de bioingenier\u00eda para la reparaci\u00f3n de tejidos. Hemos evaluado el valor cl\u00ednico y la seguridad de la aplicaci\u00f3n de c\u00e9lulas madre mesenquimales (MSCs) alogen\u00e9icas derivadas de la medula \u00f3sea para tratar las heridas de la piel en un modelo canino.El transplante t\u00f3pico alogen\u00e9ico de MSCs puede acelerar la cicatrizaci\u00f3n de heridas cut\u00e1neas experimentales de todo el grosor de la piel y disminuir la inflamaci\u00f3n local.Se estudiaron perros de raza Beagle sanos adultos .Se crearon heridas de todo el grosor de la piel en el dorso de perros Beagle sanos, y se inyectaron MSCs alogen\u00e9icas intrad\u00e9rmicamente. El ritmo de reparaci\u00f3n de la herida y el grado de producci\u00f3n de col\u00e1geno se analizaron histol\u00f3gicamente utilizando tinciones de hematoxilina-eosina y tricr\u00f3mico. El grado de proliferaci\u00f3n celular y la angiogenesis se evaluaron mediante inmunohistoqu\u00edmica utilizando anticuerpos espec\u00edficos para el ant\u00edgeno de proliferaci\u00f3n celular nuclear (PCNA), vimentina y \u03b1-actina de m\u00fasculo liso. La expresi\u00f3n local de mRNA para interleuquina-2 (IL-2), interfer\u00f3n- \u03b3 (IFN- \u03b3), factor b\u00e1sico de crecimiento de fibroblastos (BFGF) y metaloproteinasa de matriz-2 se evaluaron mediante RT-PCR.comparado con las heridas tratadas solo con el veh\u00edculo, las heridas tratadas con MSCs mostraron una reparaci\u00f3n m\u00e1s r\u00e1pida y un aumento en la producci\u00f3n de col\u00e1geno, proliferaci\u00f3n celular y angiogenesis. Adem\u00e1s, las heridas tratadas con MSCs mostraron una expresi\u00f3n aumentada de citoquinas proinflamatorias (IL-2 e IFN- \u03b3) y factores relacionados con la reparaci\u00f3n de la herida .El transplante t\u00f3pico de MSCs resulta en efectos paracrinos en la proliferaci\u00f3n celular y angiogenesis, as\u00ed como en la modulaci\u00f3n de la expresi\u00f3n de mRNA de diversos factores relacionados con la cicatrizaci\u00f3n de heridas.Adulte Stammzellen sind f\u00fcr ihre biotechnologische Verwendung bei der Wiederherstellung von Geweben bereits weit reichend erforscht. Wir haben die klinische Bedeutung und Sicherheit der Applikation von kultivierten allogenen mesenchymalen Stammzellen (MSCs) aus dem Knochenmark zur Behandlung von Hautwunden in einem caninen Modell evaluiert.Die topische allogene MSC Transplantation kann den Wundverschluss experimenteller Hautwunden (Brandwunden Grad III) beschleunigen und die lokale Entz\u00fcndung mildern.n=10; 3-6 Jahre alt; 7,2-13,1kg) wurden in der Studie verwendet.Erwachsene gesunde Beagles  und Wundheilungsfaktoren .Die topische Transplantation von MSCs resultiert in parakrinen Effekten auf die zellul\u00e4re Proliferation und Angiogenese sowie die Modulierung lokaler mRNA Exprimierung verschiedener Faktoren, die mit der Wundheilung der Haut im Zusammenhang stehen.\u4ee5\u6210\u4f53\u5e72\u7ec6\u80de\u7684\u751f\u7269\u5de5\u7a0b\u6cd5\u7528\u4e8e\u7ec4\u7ec7\u4fee\u590d\u6cbb\u7597\u5df2\u88ab\u5e7f\u6cdb\u7814\u7a76\u3002\u5c06\u9aa8\u9ad3\u57f9\u517b\u5f97\u5230\u7684\u5916\u6e90\u6027\u95f4\u5145\u8d28\u5e72\u7ec6\u80de(MSCs)\u7528\u4e8e\u72ac\u6a21\u578b\u4e0a\u6cbb\u7597\u76ae\u80a4\u521b\u4f24,\u6211\u4eec\u6765\u8bc4\u4f30\u5176\u4e34\u5e8a\u4ef7\u503c\u548c\u5b89\u5168\u6027\u3002\u5c40\u90e8\u5916\u6e90\u6027MSC\u79fb\u690d\u80fd\u4f7f\u8bd5\u9a8c\u7684\u76ae\u80a4\u521b\u53e3\u5168\u5c42\u6108\u5408\u52a0\u5feb,\u5e76\u4f7f\u5c40\u90e8\u708e\u75c7\u51cf\u8f7b\u3002n = 10;3\u20136 \u5c81;7.2\u201313.1 kg)\u3002\u7814\u7a76\u6210\u5e74\u5065\u5eb7\u6bd4\u683c\u72ac\u548c\u4e0e\u521b\u53e3\u6108\u5408\u76f8\u5173\u56e0\u5b50(\u78b1\u6027\u6210\u7ea4\u7ef4\u7ec6\u80de\u751f\u957f\u56e0\u5b50\u548c\u57fa\u8d28\u91d1\u5c5e\u86cb\u767d\u9176-2)\u8868\u8fbe\u51cf\u5c11\u3002\u5c40\u90e8\u79fb\u690dMSCs\u9020\u6210\u4e86\u5bf9\u7ec6\u80de\u589e\u6b96\u548c\u8840\u7ba1\u751f\u6210\u7684\u65c1\u5206\u6ccc\u4f5c\u7528,\u4e5f\u540c\u6837\u5f15\u8d77\u51e0\u4e2a\u4e0e\u76ae\u80a4\u521b\u4f24\u6108\u5408\u76f8\u5173\u56e0\u5b50\u5c40\u90e8mRNA\u8868\u8fbe\u7684\u8c03\u8282\u3002\u6210\u719f\u3057\u305f\u5e79\u7d30\u80de\u306f\u7d44\u7e54\u4fee\u5fa9\u7642\u6cd5\u306b\u5bfe\u3057\u3066\u751f\u7269\u5de5\u5b66\u7684\u306a\u30a2\u30d7\u30ed\u30fc\u30c1\u3067\u5e83\u304f\u7814\u7a76\u3055\u308c\u3066\u3044\u308b\u3002\u8457\u8005\u3089\u306f\u57f9\u990a\u3057\u305f\u9aa8\u9ac4\u7531\u6765\u306e\u540c\u7a2e\u7570\u7cfb\u306e\u9593\u8449\u7cfb\u809d\u7d30\u80de(MSCs)\u3092\u72ac\u306e\u76ae\u819a\u5275\u50b7\u306e\u6cbb\u7642\u306b\u5fdc\u7528\u3057\u3001\u81e8\u5e8a\u7684\u306a\u4fa1\u5024\u3068\u5b89\u5168\u6027\u3092\u8a55\u4fa1\u3057\u305f\u3002\u540c\u7a2e\u7570\u7cfbMSC\u306e\u5c40\u6240\u3078\u306e\u79fb\u690d\u306f\u5b9f\u9a13\u7684\u306a\u76ae\u819a\u5168\u5c64\u5275\u50b7\u306e\u9589\u9396\u3068\u708e\u75c7\u306e\u8efd\u6e1b\u3092\u52a0\u901f\u3059\u308b\u53ef\u80fd\u6027\u304c\u3042\u308b\u3002\u6210\u719f\u3057\u305f\u5065\u5e38\u30d3\u30fc\u30b0\u30eb\u72ac\u3092\u7528\u3044\u305f\u3002\u5065\u5e38\u30d3\u30fc\u30b0\u30eb\u306e\u80cc\u90e8\u306b\u76ae\u819a\u5168\u5c64\u6f70\u760d\u3092\u4f5c\u6210\u3057\u3001\u540c\u7a2e\u7570\u7cfbMSC\u3092\u76ae\u5185\u306b\u6ce8\u5c04\u3057\u305f\u3002\u5275\u50b7\u9589\u9396\u7387\u3068\u30b3\u30e9\u30fc\u30b2\u30f3\u7523\u751f\u5ea6\u3092\u30d8\u30de\u30c8\u30ad\u30b7\u30ea\u30f3\u67d3\u8272\u30c8\u30ea\u30af\u30ed\u30fc\u30e0\u67d3\u8272\u3092\u7528\u3044\u3066\u7d44\u7e54\u5b66\u7684\u306b\u89e3\u6790\u3057\u305f\u3002\u7d30\u80de\u5897\u6b96\u5ea6\u3068\u8840\u7ba1\u65b0\u751f\u306f\u6838\u5185\u5897\u6b96\u6297\u539f\u2212\u3001\u30d3\u30e1\u30f3\u30c1\u30f3\u2212\u3001\u03b1\u2212\u5e73\u6ed1\u7b4b\u30a2\u30af\u30c1\u30f3\u2212\u7279\u7570\u7684\u6297\u4f53\u3092\u7528\u3044\u3066\u514d\u75ab\u7d44\u7e54\u5316\u5b66\u67d3\u8272\u3067\u8a55\u4fa1\u3057\u305f\u3002\u5c40\u6240\u306e\u30a4\u30f3\u30bf\u30fc\u30ed\u30a4\u30ad\u30f3\u22122\u3001\u30a4\u30f3\u30bf\u30fc\u30d5\u30a7\u30ed\u30f3\u03b3\u3001\u5869\u57fa\u6027\u7dda\u7dad\u82bd\u7d30\u80de\u5897\u6b96\u56e0\u5b50\u3068\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u30e1\u30bf\u30ed\u30d7\u30ed\u30c6\u30a2\u30fc\u30bc2\u306emRNA\u767a\u73fe\u30ec\u30d9\u30eb\u3092RT-PCR\u3067\u8a55\u4fa1\u3057\u305f\u3002\u8ce6\u5f62\u5264\u306e\u307f\u3067\u6cbb\u7642\u3057\u305f\u5275\u50b7\u3068\u6bd4\u8f03\u3057\u3001MSC\u3067\u6cbb\u7642\u3057\u305f\u5275\u50b7\u306f\u3088\u308a\u65e9\u671f\u306b\u9589\u9396\u3057\u3001\u30b3\u30e9\u30fc\u30b2\u30f3\u5408\u6210\u306e\u5897\u52a0\u3001\u7d30\u80de\u5897\u6b96\u3084\u8840\u7ba1\u65b0\u751f\u3092\u793a\u3057\u305f\u3002\u3055\u3089\u306b\u3001MSC\u3067\u6cbb\u7642\u3057\u305f\u5275\u50b7\u306f\u708e\u75c7\u5f8c\u30b5\u30a4\u30c8\u30ab\u30a4\u30f3(\u30a4\u30f3\u30bf\u30fc\u30ed\u30a4\u30ad\u30f3\u22122\u3068\u30a4\u30f3\u30bf\u30fc\u30d5\u30a7\u30ed\u30f3\u2212\u03b3)\u3068\u5275\u50b7\u6cbb\u7652\u95a2\u9023\u56e0\u5b50(\u5869\u57fa\u6027\u7dda\u7dad\u82bd\u7d30\u80de\u5897\u6b96\u56e0\u5b50\u3068\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u30e1\u30bf\u30ed\u30d7\u30ed\u30c6\u30a2\u30fc\u30bc\u22122)\u306e\u767a\u73fe\u304c\u6e1b\u5c11\u3057\u3066\u3044\u305f\u3002MSC\u306e\u5c40\u6240\u3078\u306e\u79fb\u690d\u306b\u3088\u308a\u7d30\u80de\u5897\u6b96\u3068\u8840\u7ba1\u65b0\u751f\u306b\u5bfe\u3057\u3066\u306f\u30d1\u30e9\u30af\u30e9\u30a4\u30f3\u52b9\u679c\u3092\u793a\u3059\u306e\u3068\u540c\u6642\u306b,\u76ae\u819a\u5275\u50b7\u6cbb\u7652\u306b\u95a2\u9023\u3059\u308b\u69d8\u3005\u306a\u56e0\u5b50\u306e\u5c40\u6240mRNA\u767a\u73fe\u3092\u8abf\u7bc0\u3057\u305f\u3002 Normal skin depends on the pool of adult stem cells, such as epidermal skin stem cells and bone marrow (BM) stem cells, for its renewal and maintenance.18Autologous MSC transplantation is time consuming from cell collection to application and requires the patient to have normal BM function. In contrast, allogenic MSC transplantation has the advantage of prompt preparation and can be applied independently of the health status of the patient. However, allogenic cell transplantation has the risk of graft rejection, and the failure of allogenic cell transplantation has also been reported.This study received ethical approval, and all procedures dealing with animal care, handling and sampling were approved by the Institutional Animal Care and Use Committee of Konkuk University .n = 10; 3\u20136 years old; 7.2\u201313.1 kg) were used for this study. After overnight fasting, the experimental dogs were premedicated with subcutaneous atropine sulfate  and intravenous cefazolin sodium . Under general anaesthesia using intramuscular (i.m.) tiletamine\u2013zolazepam , 24 full-thickness circular wounds of 6 mm in diameter were created on the back of each dog using disposable dermal biopsy punches. The wounds were at least 2.5 cm apart . The four remaining wounds on each dog were left untreated to monitor the progress of healing. A total of 10 dogs received a set of these treatments, comprising five animals for histological analysis and the other five for cell tracing and molecular analysis , and wounds were harvested using 8 mm dermal biopsy punches. After skin harvesting, tramadol was administered. The samples were bisected along the widest line of the wound, then fixed and paraffin embedded, sectioned into 4-\u03bcm-thick slices and prepared for histology. All stained slides were examined under a light microscope , and digital photographs of the sections were taken using imaging software . The photographs were quantitatively analysed using an image analysis program . All measurements were performed by two investigators blinded to the treatment procedure.For comparison of the rate of re-epithelialization on day 7, tissue sections were stained with haematoxylin and eosin, and the epithelial gap (EG) was measured, which was defined as the distance between the advancing opposite edges of epithelial cell migration.7 MSC-treated wounds and PBS-treated wounds harvested on days 7, 14 and 21. The concentration of 1 \u00d7 107 was chosen because the 1 \u00d7 107 MSC-treated wounds showed the smallest EG and the greatest collagen deposition. Proliferating cell nuclear antigen (PCNA), vimentin and \u03b1-smooth muscle actin (\u03b1-SMA) were detected using specific monoclonal antibodies . The tissue sections were deparaffinized and hydrated, and sections for PCNA and vimentin immunostaining were pretreated for antigen retrieval using a pressure cooker in 10 mm citrate buffer. Endogenous peroxidases were quenched with 3% hydrogen peroxide for 15 min. Nonspecific antibody binding was blocked by incubation with a protein blocker (Dako). The sections were incubated with the primary antibodies or bovine serum albumin as a negative control for 30 min at 45\u00b0C, then incubated with horseradish peroxidase\u2212conjugated anti-mouse and anti-rabbit immunoglobulin (Dako) for 15 min at 45\u00b0C, followed by a 5 min incubation with 3,3\u2032-diaminobenzidine . Finally, the sections were counterstained with Gill's haematoxylin, mounted and examined under a light microscope. High-power images were used to quantify the number of DAB-positive cells or the area occupied by DAB-positive cells.To identify the effects of MSC transplantation on cellular proliferation and angiogenesis, an immunohistochemical evaluation was performed using the 1 \u00d7 10PCNA-positive cells in the epidermis was counted, and the number of these cells per 0.1 mm2 of epidermis was calculated. The number of proliferating cells in the granulation tissue was analysed in the same manner. To assess the cellularity of mesenchymal origin, the area occupied by vimentin-positive signals was measured, and the percentage of the vimentin-positive area in the granulation tissue was calculated. To analyse the degree of angiogenesis, the number of blood vessels demonstrating \u03b1-SMA-positive lumens in five random hpfs was counted. All measurements were repeated by two investigators, and the average value was used for analysis.The degree of epidermal cell proliferation was assessed by the number of proliferating cells in the newly formed epidermis. Three hpfs of the sections were selected randomly, the number of 7 MSC-treated wounds and PBS-treated wounds were harvested from the other five dogs using 8 mm biopsy punches under sedation. Tramadol was administered for pain relief. Harvested wound tissues were embedded in the optimum cutting temperature (OTC) compound, and slides of the 5-\u03bcm-thick cryosections were made. The slides were examined under a fluorescence microscope (BK51). Demonstration of transplanted MSCs was performed by detection of CFDA-SE prelabelled cells in tissue slides. Positive fluorescence signals having cell morphology were regarded as the retained MSCs.On days 3, 7 and 14 after cell transplantation, 1 \u00d7 107 MSC-treated wounds and PBS-treated wounds on days 3, 7 and 14. Harvested tissues were immediately frozen and, following homogenization, total RNA was extracted using TRIzol , according to the manufacturer's instructions. Then, RT-PCR was performed  using canine-specific primers to evaluate the expression of basic fibroblast growth factor (bFGF), keratinocyte growth factor-1 (KGF-1), interleukin-2 (IL-2), interferon-\u03b3 (IFN-\u03b3) and matrix metalloproteinase-2 (MMP-2). The PCR conditions were as follows: 30 s at 94\u00b0C for denaturation, followed by 30 cycles of denaturation for 30 s at 94\u00b0C, annealing for 30 s at 58\u00b0C, and extension for 30 s at 72\u00b0C. The primer sequences are listed in GAPDH) after testing for stable expression in skin samples. The bands obtained from the PCR products were measured by densitometry using an image analysis program . The relative mRNA expression was calculated as the ratio of the density of each mRNA to the density of GAPDH mRNA.Local expression of wound healing-related factors was evaluated in 1 \u00d7 10spss version 12.0; SPSS Inc., IBM Corporation, Endicott, NY, USA). Nonparametric Kruskal\u2013Wallis tests were used to compare the EG and collagen deposition among five different wounds. Mann\u2013Whitney U-tests were used to assess the immunohistochemistry and RT-PCR results between the 1 \u00d7 107 MSC-treated and PBS-treated wounds. Spearman rank correlations were used to analyse the relationship between the number of transplanted MSCs and the EG or collagen deposition. A value of P < 0.05 was considered statistically significant.Statistical analyses were performed using the Statistical Package for the Social Sciences , although there was no significant difference among the MSC-treated wounds  were significantly smaller than those of the PBS-treated wounds  was significantly higher than that of the PBS-treated wounds  on day 7  on days 7 and 21  on day 7. The increase in cells of mesenchymal origin in the MSC-treated wounds was maintained until day 21 (The ratio of the vimentin-positive stained area to the granulation tissue per unit area was higher in the MSC-treated wounds (23.80%) than in the PBS-treated wounds . On day 14, the number of blood vessels declined and there was no significant difference in the vascularity of the MSC-treated wounds and the PBS-treated wounds  than the PBS-treated wounds (3.7 blood vessels/hpf; d wounds .7 MSC-treated wounds and the PBS-treated wounds was measured at the mRNA level by quantitative RT-PCR. On day 7, expression of bFGF was increased in the MSC-treated wounds but was decreased in the PBS-treated wounds. The expression of bFGF in the MSC-treated wounds was decreased on day 14. Meanwhile, expression of KGF-1 was higher in the PBS-treated wounds than in the MSC-treated wounds. Expression of KGF-1 also declined on day 14 postwounding in both types of wounds. Expression of MMP-2 demonstrated a slight increase over time, and the MSC-treated wounds showed increased MMP-2 expression from day 3 to 14. Expression of IL-2 and IFN-\u03b3 showed a similar change, increasing by day 7 and decreasing by day 14. The MSC-treated wounds maintained lower IL-2 and IFN-\u03b3 expression from day 3 to 14 postwounding relative to the PBS-treated wounds (The expression of wound healing-related factors in the 1 \u00d7 10d wounds .PCNA.In this study, MSC-treated wounds showed more rapid wound closure, and this was presumed to be due to MSCs promoting re-epithelialization. Faster epithelialization, following BM-derived stem cell transplantation for cutaneous wound healing, has been demonstrated in previous studies in other species.PCNA immunohistochemistry results. Other studies in other species have shown that MSCs promote fibroblast proliferation, resulting in intensified granulation tissue formation.2In the trichrome-stained sections, increased collagen deposition in MSC-treated wounds was demonstrated, which indicated the promotion of fibroblast proliferation, as shown by the vimentin and \u03b1-SMA, which is in agreement with previous studies.\u03b1-SMA is a marker of mature blood vessels.bFGF has a promoting effect on the growth and migration of endothelial cells,MMP-2, a member of the family of matrix metalloproteinases, plays a role in degradation of the extracellular matrix, consequently facilitating endothelial cell migration.bFGF in MSC-treated wounds was increased. At the same time, increased mRNA expression of MMP-2 in MSC-treated wounds was confirmed. Thus, upregulation of bFGF and MMP-2 in MSC-treated wounds can be attributed partly to increased blood vessel formation.Apart from fibroblast proliferation, the formation of new blood vessels is necessary to maintain granulation tissue.bFGF could also partly explain the promotion of fibroblast proliferation and increased production of collagen fibres. Although accumulation of larger collagen fibres increases the wound strength, excessive accumulation of collagen may induce hypertrophic scarring.MMP in MSC-treated wounds can be favourable for improving the quality of the healed wounds. Thus, MSCs seemed to influence the degree of wound maturation. Prevention of excessive wound contraction or hypertrophic scar formation is important for maintaining normal function of the skin and its cosmetic appearance. To obtain a therapeutic strategy involving less wound contraction, dermal substitutes or biomaterial have been applied.In addition, as mentioned above, fibroblasts play a major role in wound healing.IL-2 and IFN-\u03b3, which implies that MSCs exert a suppressive effect on local inflammation. The IL-2 and IFN-\u03b3 levels do not represent the complete state of tissue inflammation. However, the results of the present study demonstrated that the MSCs exhibited a paracrine effect of decreasing postinjury inflammation. In particular, IFN-\u03b3 is known to impair re-epithelialization;IFN-\u03b3 expression may act favourably in the wound healing process.Damage of the skin inevitably induces local inflammation and recruits inflammatory cells, which act as sources of the cytokines that affect the wound healing process.in vitroin vivo.in vivo studies were inconsistent, and the persistence of the transplanted MSCs was very low in some studies.There has been no consensus explanation for the exact mechanism of action of MSCs on wound healing. One hypothesis involves transdifferentiation of MSCs into epithelial cells. The engrafted MSCs were seldom detected, and transdifferentiation of the MSCs into keratinocytes was not demonstrated in the present study, although several previous studies demonstrated transdifferentiation of MSCs both KGF, epidermal growth factor and transforming growth factor-\u03b2, in a hypoxic environment in vitro.Another hypothesis is that MSCs exert paracrine effects on wound healing, such as promoting fibroblast proliferation or exaggerating angiogenesis.6 cells/cm2 of wound was needed for a significant therapeutic effect. In contrast, the present study failed to determine the therapeutic dose of topical MSC transplantation. Confirmation of the therapeutic level of cell transplantation may be crucial in clinical cell transplantation therapy, because stem cell transplantation is associated with the underlying risks of the teratogenic and oncogenic processes; therefore, the appropriate number of cells needed for efficient wound healing must be investigated further, and the appropriate number of cells may differ according to the route of delivery.In the present study, there was no statistically significant difference in the results obtained between the different concentrations MSCs that were transplanted. This finding conflicts with a previous human clinical trial,One study reported decreased engraftment in the wound site of systemically transplanted MSCs without chemo-attractants; systemic allogenic cell transplantation is not free from the risk of a host reaction leading to transplantation failure.In conclusion, topical injection of MSCs resulted in more rapid re-epithelialization and increased collagen deposition and angiogenesis in a canine wound healing model. In addition, topically applied MSCs appeared to have a suppressive effect on local inflammation in the wounded skin, suggesting that MSCs may be applied in patients with not only large or nonhealing skin defects but also inflammatory and fibrotic skin diseases."}
{"text": "\ud835\udca9\ud835\udcab\u03a3\u03bb,\u03b4 of analytic and bi-univalent functions in the open unit disk \ud835\udd4c. For functions belonging to the class \ud835\udca9\ud835\udcab\u03a3\u03bb,\u03b4, we obtain estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3|.We introduce and investigate an interesting subclass  When \u03b4 = 1, we get S\u01cel\u01cegean's differential operator D1n = Dn, [For l-Oboudi  introducgiven by , then frz)=f(z),D\u03b41f(z)=,D\u03b4nf(z)=Dven by (1D\u03b4nf(z)=z\ud835\udd4c. In fact, the Koebe one-quarter theorem [\ud835\udd4c under every univalent function f \u2208 \ud835\udcae contains a disk of radius 1/4. Thus every function f \u2208 \ud835\udc9c has an inverse f\u22121, which is defined byf\u22121 is given bySince univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk  theorem  ensures f \u2208 \ud835\udc9c is said to be bi-univalent in \ud835\udd4c if both f and f\u22121 are univalent in \ud835\udd4c. Let \u03a3 denote the class of bi-univalent functions in \ud835\udd4c given by  if the following conditions are satisfied:\u03b2 \u2208 , \u03bb \u2265 1, the function g is given byD\u03b4n is the Al-Oboudi differential operator.Let fined by , is said\ud835\udca9\ud835\udcab\u03a3\u03bb,\u03b4 reduces to the class denoted by \ud835\udca9\ud835\udcab\u03a3\u03bb,\u03b4 which is the subclass of the functions f \u2208 \u03a3 satisfying\u03b2 \u2208 , \u03bb \u2265 1, the function g is defined by  reduces to the class denoted by \ud835\udca9\ud835\udcab\u03a3\u03bb,\u03b4 which is the subclass of the functions f \u2208 \u03a3 satisfying\u03b2 \u2208 , \u03bb \u2265 1, the function g is defined by  reduces to the class denoted by \ud835\udca9\ud835\udcab\u03a3\u03bb which is the subclass of the functions f \u2208 \u03a3 satisfying\u03b2 \u2208 , \u03bb \u2265 1, the function g is defined by  reduces to the class denoted by \ud835\udca9\ud835\udcab\u03a3\u03bb which is the subclass of the functions f \u2208 \u03a3 satisfying\u03b2 \u2208 , \u03bb \u2265 1, and the function g is defined by  reduces to the class denoted by \ud835\udca9\ud835\udcab\u03a3 which is the subclass of the functions f \u2208 \u03a3 satisfying\u03b2 \u2208  and the function g is defined by  given by\ud835\udd4c. Suppose also that the function \u03c6(z) given by\ud835\udd4c. If \u03c6(z)\u227ah(z)\u2009\u2009(z \u2208 \ud835\udd4c), thenLet the function a2| and |a3| for functions in this new subclass \ud835\udca9\ud835\udcab\u03a3\u03bb,\u03b4 of the function class \u03a3.The object of the present paper is to find estimates on the Taylor-Maclaurin coefficients |\ud835\udca9\ud835\udcab\u03a3\u03bb,\u03b4 given by In this section, we state and prove our general results involving the bi-univalent function class f(z) given by the Taylor-Maclaurin series expansion (Let the function xpansion  be in thp(z)\u227ah(z) and q(w)\u227ah(w) have the following Taylor-Maclaurin series expansions:It follows from  that(36\u03b2\u2003(z\u2208\ud835\udd4c),ei\u03b2((1\u2212\u03bb)p, q \u2208 h(\ud835\udd4c), according to p1, p2, q1, and q2, from the equalities (a2| as asserted in (From  and 42)42), we oso, from  and 43)(44)p1=\u2212qrding to |pk|=|p(k)p1=\u2212q1,2e2i\u03b2 given by the Taylor-Maclaurin series expansion (Let the function xpansion  be in thBy settingf(z) given by the Taylor-Maclaurin series expansion (Let the function xpansion  be in thBy settingf(z) given by the Taylor-Maclaurin series expansion  given by the Taylor-Maclaurin series expansion (Let the function xpansion  be in thBy settingf(z) given by the Taylor-Maclaurin series expansion  given by the Taylor-Maclaurin series expansion (Let the function xpansion  be in thBy settingf(z) given by the Taylor-Maclaurin series expansion  given by the Taylor-Maclaurin series expansion (Let the function xpansion  be in th"}
{"text": "We characterize those metrics which are Douglasian or locally projectively flat by some equations. In particular, it shows that the known fact that \u03b2 is always closed for those metrics in higher dimensions is no longer true in two-dimensional case. Further, we determine the local structures of two-dimensional -metrics which are Douglasian, and some families of examples are given for projectively flat classes with \u03b2 being not closed.We study a class of two-dimensional Finsler metrics defined by a Riemannian metric  D) is an important projective invariants in projective Finsler geometry. A Finsler metric is called  Douglasian if D = 0 and  locally projectively flat if at every point, there are local coordinate systems in which geodesics are straight. It is known that a locally projectively flat Finsler metric can be characterized by D = 0 and vanishing Weyl curvature. As we know, the locally projectively flat class of Riemannian metrics is very limited, nothing but the class of constant sectional curvature (Beltrami theorem). However, the class of locally projectively flat Finsler metrics is very rich. It is known that locally projectively flat Finsler metrics must be Douglasian, but Douglas metrics are not necessarily locally projectively flat. Therefore, it is a natural problem to study and classify Finsler metrics which are Douglasian or locally projectively flat. For this problem, we can only investigate some special classes of Finsler metrics.Projective Finsler geometry studies equivalent Finsler metrics on the same manifold with the same geodesics as points . Douglas\u03b2 = bi(x)yi on a manifold M. Such metrics are called -metrics. An -metric can be expressed in the following form:\u03d5(s) > 0 is a C\u221e function on . It is known that F is a regular Finsler metric (defined on the whole TM \u2212 {0} and positive definite) for any  with ||\u03b2||\u03b1 < bo if and only ifbo is a constant. If \u03d5 does not satisfy (F = \u03b1\u03d5(\u03b2/\u03b1) is singular.In this paper, we shall consider a special class of Finsler metrics defined by a Riemannian metric  only if\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u2009\u03d5\u03b1, \u03b2)-metrics. It is known that a Randers metric F = \u03b1 + \u03b2 is a Douglas metric if and only if \u03b2 is closed , where c \u2260 0 is a constant, then we have \u03b2 is not closed). Since every two-dimensional Riemann metric is locally conformally flat, we may put). Since  is equivproof in ), 59) h h109)\u03b1=\u03b1=(109)\u03b1=the form\u03b2=be\u03c3(x)[the form\u03b2=be\u03c3(x)[given by2\u03c4~=(\u03be2+\u03b7 only if(\u03be2+\u03b72)-metric F on R2 byk, b are constants with k \u2260 \u22121/b. Then F is a Douglas metric if and only if \u03b1 and \u03b2 can be locally defined by , s = \u03b2/\u03b1, be a two-dimensional -metric, where \u03d5(0) = 1. Let \u03d5(s) be given by   F = \u03b1\u03d5(sgiven by ). Then Ffined by  and (110 \u03c4~=0 in . There a Next we construct some singular examples for F = \u03b1\u03d5(s), s = \u03b2/\u03b1, be two-dimensional -metric, where \u03d5(0) = 1. Let \u03d5(s) be given by , s = \u03b2/\u03b1, be two-dimensional -metric, where \u03d5(0) = 1. Let \u03d5(s) be given by  is a scalar function. We can obtain the local expression of u = u(x), v = v(x) are a pair of scalar functions such thatSince  \u03b2~=\u03b2 by  and , , 120), agiven by  and 7),,\u03b1 and \u03b2 \u03b1 expressed in  is defined in  is a scalar function. Then by the result in [u = u(x), v = v(x) are a pair of scalar functions such thatNow we express esult in , we haveu, v and the fact that u, v, and \u03c3 satisfy the PDEs , where B : = b2. Firstly by , , \u03b1 and \u03b2given by ."}
{"text": "We study the local convergence properties of inexact Newton-Gauss method for singular systems of equations. Unified estimates of radius of convergence balls for one kind of singular systems of equations with constant rank derivatives are obtained. Application to the Smale point estimate theory is provided and some important known results are extended and/or improved. In the case when m = n and f\u2032(x) is invertible for each x \u2208 D, Newton's method is a classical numerical method to find an approximation solution for such system. There are a lot of results that improve, generalize, or extend the convergence of Newton's method for solving  is not invertible, we choose its Moore-Penrose inverse f\u2032(x)\u2020 instead of its classical inverse and call it Gauss-Newton's method given as follows:Consider the following system of nonlinear equations:f\u2009(x)=0,f\u2009(x)=0,A : \u211dn \u2192 \u211dm be a linear operator (or an m \u00d7 n matrix). Recall that an operator (or n \u00d7 m matrix) A\u2020 : \u211dm \u2192 \u211dn is the Moore-Penrose inverse of A, if it satisfies the following four equations:A* denotes the adjoint of A. Let ker\u2061A and im\u2009\u2009A denote the kernel and image of A, respectively. For a subspace E of \u211dn, we use \u03a0E to denote the projection onto E. Then, it is clear thatA is full row rank , AA\u2020 = Im\u211d; when A is full column rank , A\u2020A = In\u211d.Let x0 is an initial guess):rk satisfies\u03bbk} is a sequence of forcing terms such that 0 \u2264 \u03bbk < 1. In  \u2192 \u211d corresponding to  is defined by\u03bb = \u03b8 = 0,  = 0, h\u03bb,\u03b8\u2032(0) = \u2212(1 + \u03bb + \u03b8) and h\u03bb,\u03b8\u2032 is convex and strictly increasing. SetThroughout this paper, we assume that fined byh\u03bb,\u03b8(t)=\u2212For the convergence analysis, we need the following useful lemma about elementary convex analysis.R > 0. If g :  \u2192 \u211d is continuously differentiable and convex, then g(t) \u2212 g(\u03c4t))/t \u2264 (1 \u2212 \u03c4)g\u2032(t), for all t \u2208  and \u03c4 \u2208 ,(g(u) \u2212 g(\u03c4u))/u \u2264 (g(v) \u2212 g(\u03c4v))/v, for all u, v \u2208 .Let L average\u201d was used. In the case when f\u2032(\u03b6) is not surjective (see .Let \u03b3-condition for operators in Banach spaces was introduced in \u22121 exists andf\u2032(\u03b6) is full row rank, we have f\u2032(\u03b6)f\u2032(\u03b6)\u2020 = Im\u211d andf\u2032(x) is full row rank; that is, rank\u2061f\u2032(x) = rank\u2061f\u2032(\u03b6).Since xk} coincides with the sequence generated by inexact Newton-Gauss method (f\u2032(\u03b6)\u2020f(\u03b6)|| = ||\u03a0ker\u2061f\u2032(\u03b6)\u22a5|| = 1, thus, we have L-average Lipschitz condition . So, xk} converges to \u03b6 as follows. Note that f(\u00b7) = f\u2032(\u00b7)f\u2032(\u00b7)\u2020f(\u00b7); it follows that \u03b6 is a zero of f.Let fined by1f^(x)=f\u2032fore, by , we can ondition  on B = 0, f\u2032(\u03b6) is full row rank, and f\u2032 satisfies the L-average Lipschitz condition . Then, one hasSuppose that ondition  on B is full row rank, we have f\u2032(\u03b6)f\u2032(\u03b6)\u2020 = Im\u211d. It follows thatIn\u211d \u2212 f\u2032(\u03b6)\u2020(f\u2032(\u03b6) \u2212 f\u2032(x)) is invertible for any x \u2208 B. Thus, in view of the equality A\u2020A = \u03a0ker\u2061A\u22a5, for any m \u00d7 n matrix A, one has thatSince L-average condition  is defined by t0 \u2264 1/2. In fact, in view of the definition of r* given in (\u03d5(t) \u2264 0. Consequently, we get thatRecall that the majorizing sequence {fined by  and the fined by . By Lemmote that  givestk} is dndition (||f\u2032(xk)\u2020ws from (||f\u2032(xk)\u2020ombining  that(85given in , for any"}
{"text": "\ud835\udc9c be an alphabet with two elements. Considering a particular class of words (the phrases) over such an alphabet, we connect with the theory of numerical semigroups. We study the properties of the family of numerical semigroups which arise from this starting point.Let  \ud835\udc9c be a nonempty finite set called the alphabet. Elements of \ud835\udc9c are called letters or symbols. A word is a sequence of letters, which can be finite or infinite. We denote by \ud835\udc9c*  the set of all finite  words over \ud835\udc9c. The sequence of zero letters is called the empty word and is denoted by \u025b. Any subset \u2112\u2286\ud835\udc9c* is called a language over \ud835\udc9c. The length of a word u is denoted by |u|. If u, v are words, we define their product or concatenation as the word uv. We say that a word u is a factor of a word v if there exist two words x, y such that v = xuy. If u is a factor of v with x = \u025b , then u is a prefix  of v.Let We have taken these definitions from . In this\ud835\udc9c = {a, \u2323}. We say that f \u2208 \ud835\udc9c* is a phrase if it fulfills the following conditions: f,\u2323 is not a prefix or suffix of f.\u2323\u2323 is not a factor of  We denote \ud835\udc9c\u2131 = {f \u2208 \ud835\udc9c* | f\u2009is\u2009a\u2009phrase}.Let us take If we consider that \u2323 represents a gap between two words, then we have a suitable justification for the above definition.\ud835\udc9e be a language over \ud835\udc9c such that \ud835\udc9e\u2286\ud835\udc9c\u2131. We will denote by \u2113(\ud835\udc9e) = {|c| | c \u2208 \ud835\udc9e}. In this work we are going to deal with the structure of the set \u2113(\ud835\udc9e) for particular choices of \ud835\udc9e. In fact, let {w1,\u2026, wn}\u2286\ud835\udc9c* be a finite set of words such that \u2323 is not a factor of wi, \u20091 \u2264 i \u2264 n. Then \ud835\udc9e\u2286\ud835\udc9c\u2131 is the language in which each phrase f is obtained as product of factors belonging to {w1,\u2026, wn}\u222a{\u2323}. Moreover, in order to achieve the results of this paper, we assume that \u025b \u2208 \ud835\udc9e.Let aaaa, aaaaa}, then f1 = aaaaaaaa, f2 = aaaaaaaaa, and f3 = aaaaa\u2323aaaa belong to \ud835\udc9e. However, f4 = aaaa\u2323aaaaaa, f5 = aaa, and f6 = aa\u2323a do not belong to \ud835\udc9e.If we take {\u2115 be the set of nonnegative integers. A numerical semigroup is a subset S of  that is closed under addition, contains the zero element, and such that \u2115\u2216S is finite.Let \u2113) is a numerical semigroup. We will also see that there exist numerical semigroups that cannot be obtained by this procedure. This fact allows us to give the following definition.In \ud835\udc9c be the alphabet given by the set {a, \u2323}. A numerical semigroup S is the set of lengths of a language of phrases (PL-semigroup for abbreviation) if there exists \ud835\udc9e = \ud835\udc9e\u2286\ud835\udc9c\u2131 such that S = \u2113(\ud835\udc9e).Let S is a PL-semigroup if and only if x + y + 1 \u2208 S for all x, y \u2208 S\u2216{0}.The next aim of S be a numerical semigroup. Since \u2115\u2216S is a finite set, we can consider two notable invariants of S (see [Frobenius number of S is the maximum of \u2115\u2216S and is denoted by F(S). On the other hand, the genus of S is the cardinality of \u2115\u2216S and is denoted by g(S).Let f S (see ). On theFrobenius variety is a nonempty family \ud835\udcb1 of numerical semigroups that fulfills the following conditions:S, T \u2208 \ud835\udcb1, then S\u2229T \u2208 \ud835\udcb1,if S \u2208 \ud835\udcb1 and S \u2260 \u2115, then S \u222a {F(S)} \u2208 \ud835\udcb1.if A \ud835\udcaePL = {S | S is a PL-semigroup}. In \ud835\udcaePL is a Frobenius variety. This fact, together with the results of [\ud835\udcaePL in a tree with root \u2115. Moreover, we will also characterize the sons of a vertex, in order to build recursively such a tree.Let us denote sults of , will almultiplicity of a numerical semigroup S, denoted by m(S), is the minimum of S\u2216{0}. We will study the set \ud835\udcaePL(m) = {S \u2208 \ud835\udcaePL | m(S) = m} in S) | S \u2208 \ud835\udcaePL(m)} and {g(S) | S \u2208 \ud835\udcaePL(m)}. We will also see that the elements of \ud835\udcaePL(m) can be ordered in a tree with root the numerical semigroup S = {0, m, \u2192} (where the symbol \u2192 means that every integer greater than m belongs to S).The In We finish this introduction pointing out that this work admits different generalizations. Some of them are in working process and other ones have already been developed (see ).X is a nonempty subset of \u2115, we denote by \u2329X\u232a the submonoid of  generated by X; that is,X\u232a is a numerical semigroup if and only if gcd{X} = 1 . On the other hand, every numerical semigroup S is finitely generated, and therefore there exists a finite subset X of S such that S = \u2329X\u232a. In addition, if no proper subset of X generates S, then we say that X is a minimal system of generators of S. In [S has a unique (finite) minimal system of generators. The elements of such a system are called minimal generators of S.If \ud835\udc9c be an alphabet and let {w1,\u2026, wn}\u2286\ud835\udc9c* be a finite set of words. If \ud835\udc9e = \ud835\udc9e\u2286\ud835\udc9c* is the language in which each word is obtained as product of factors belonging to {w1,\u2026, wn}, then it is easy to see that \u2113(\ud835\udc9e) is a submonoid of . In addition, if gcd{|w1|,\u2026, |wn|} = 1, then \u2113(\ud835\udc9e) is a numerical semigroup. Moreover, it is a simple exercise to show that we can get any numerical semigroup in this way.Let As we indicated in the introduction we are going to study the particular case in which we consider lengths of phrases. Consequently, we will focus our attention in a particular family of numerical semigroups.\ud835\udc9c = {a, \u2323} be an alphabet. If \ud835\udc9e = \ud835\udc9e\u2286\ud835\udc9c\u2131, then \u2113(\ud835\udc9e) is a numerical semigroup.Let \u025b \u2208 \ud835\udc9e and |\u025b| = 0, we have that 0 \u2208 \u2113(\ud835\udc9e).First of all, being that l1, l2 \u2208 \u2113(\ud835\udc9e), then l1 + l2 \u2208 \u2113(\ud835\udc9e). In effect, let f1, f2 \u2208 \ud835\udc9e such that |f1| = l1 and |f2| = l2. Then the concatenation f1f2 (of f1 and f2) is an element of \ud835\udc9e with |f1f2| = l1 + l2.Now, let us see that, if f \u2208 \ud835\udc9e with |f| \u2260 0 . Since f\u2323f \u2208 \ud835\udc9e, we have that {|f|, 2|f| + 1}\u2286\u2113(\ud835\udc9e). By the previous step, we know that \ud835\udc9e is closed under addition and, consequently, \u2329|f|, 2|f| + 1\u232a\u2286\u2113(\ud835\udc9e). As gcd{|f|, 2|f| + 1} = 1, we have \u2329|f|, 2|f| + 1\u232a is a numerical semigroup. Therefore, \u2115\u2216\u2113(\ud835\udc9e) is finite.Finally, let  We conclude that \u2113(\ud835\udc9e) is a numerical semigroup.We proceed in three steps. \ud835\udc9c = {a, \u2323}. As in the introduction, we say that a numerical semigroup S is a PL-semigroup if there exists \ud835\udc9e = \ud835\udc9e\u2286\ud835\udc9c\u2131 such that S = \u2113(\ud835\udc9e). From S is a PL-semigroup and x \u2208 S\u2216{0}, then 2x + 1 \u2208 S. Consequently, there exist numerical semigroups which are not of this type. For example, S = \u23295,7, 9\u232a is not a PL-semigroup because 2 \u00b7 5 + 1 = 11 \u2209 S.From now on, unless another thing is stated, we take In the next result we give a characterization of PL-semigroups.S be a numerical semigroup. The following conditions are equivalent. S is a PL-semigroup.x, y \u2208 S\u2216{0}, then x + y + 1 \u2208 S.If Let S = \u2113) for some nonempty finite set {w1,\u2026, wn}\u2286\ud835\udc9c*. If x, y \u2208 S\u2216{0}, then there exist f, g \u2208 \ud835\udc9e\u2216{\u025b} such that |f| = x and |g| = y. It is clear that f\u2323g \u2208 \ud835\udc9e and |f\u2323g| = x + y + 1. Therefore, x + y + 1 \u2208 S.(1\u21d22) By hypothesis, n1,\u2026, np} be the minimal system of generators of S. Let us take the set {w1,\u2026, wp | |wi| = ni, \u20091 \u2264 i \u2264 p}. Our aim is to show that if \ud835\udc9e = \ud835\udc9e, then S = \u2113(\ud835\udc9e).(2\u21d21) Let {n1,\u2026, np}\u2286\u2113(\ud835\udc9e), by applying S = \u2329n1,\u2026, np\u232a\u2286\u2113(\ud835\udc9e). Now, let l \u2208 \u2113(\ud835\udc9e). In order to prove that l \u2208 S, we are going to use induction over l. If l = 0, then the result is trivially true. Let us assume that l > 0, and let f \u2208 \ud835\udc9e such that |f| = l. If \u2323 is not a factor of f, then the result follows immediately. In other case, there exist f1, f2 \u2208 \ud835\udc9e\u2216{\u025b} such that f = f1\u2323f2. By hypothesis of induction, |f1|, |f2| \u2208 S. Thereby, l = |f| = |f1| + |f2| + 1 \u2208 S.Since {numerical semigroup that admit a linear nonhomogeneous pattern. For a general study of this family of numerical semigroups see, for instance, [The previous theorem leads to the concept of nstance, , 7.S be a numerical semigroup with minimal system of generators given by {n1,\u2026, np}. Following [s \u2208 S, then we define the order of s (in S) bys).Let ollowing , if s \u2208 X is the minimal system of generators of S, then every system of generators of S contains X. Consequently, the definition of ord does not depend on the considered system of generators; that is, it only depends on s and S.From [S be a numerical semigroup with minimal system of generators given by {n1,\u2026, np} and let s \u2208 S. i \u2208 {1,\u2026, p} and s \u2212 ni \u2208 S, then ord(s \u2212 ni) \u2264 ord(s) \u2212 1.If s = \u03b11n1 + \u22ef+\u03b1pnp, with ord(s) = \u03b11 + \u22ef+\u03b1p and \u03b1i \u2260 0, then ord(s \u2212 ni) = ord(s) \u2212 1.If Let s \u2212 ni = \u03b21n1 + \u22ef+\u03b2pnp, with \u03b21 + \u22ef+\u03b2p = ord(s \u2212 ni). Then s = \u03b21n1 + \u22ef+(\u03b2i + 1)ni + \u22ef+\u03b2pnp, and thus ord(s \u2212 ni) + 1 = \u03b21 + \u22ef+(\u03b2i + 1)+\u22ef+\u03b2p \u2264 ord(s).(1) Assume that s \u2212 ni = \u03b11n1 + \u22ef+(\u03b1i \u2212 1)ni + \u22ef+\u03b1pnp, we have ord(s \u2212 ni) \u2265 \u03b11 + \u22ef+(\u03b1i \u2212 1)+\u22ef+\u03b1p = ord(s) \u2212 1. Thereby, ord(s) \u2212 1 \u2264 ord(s \u2212 ni). Now, by applying the previous item, we conclude that ord(s \u2212 ni) = ord(s) \u2212 1.(2) Since In item (2) of the next proposition, it is shown a characterization of PL-semigroups in terms of minimal systems of generators. Thus, we can decide if a numerical semigroup is a PL-semigroup in an easier way.S be a numerical semigroup with minimal system of generators given by {n1,\u2026, np}. The following conditions are equivalent. S is a PL-semigroup.i, j \u2208 {1,\u2026, p}, then ni + nj + 1 \u2208 S.If s \u2208 S\u2216{0, n1,\u2026, np}, then s + 1 \u2208 S.If s \u2208 S\u2216{0}, then s + {0,\u2026, ord(s) \u2212 1}\u2286S.If Let (1\u21d22) It is an immediate consequence of s \u2208 S\u2216{0, n1,\u2026, np}, then it is clear that there exist i, j \u2208 {1,\u2026, p} and s\u2032 \u2208 S such that s = ni + nj + s\u2032. Thus, s + 1 = (ni + nj + 1) + s\u2032 \u2208 S.(2\u21d23) If s). If ord(s) = 1, then the result is trivially true. Now, let us assume that ord(s) \u2265 2 and that \u03b11,\u2026, \u03b1p are nonnegative integers such that s = \u03b11n1 + \u22ef+\u03b1pnp, ord(s) = \u03b11 + \u22ef+\u03b1p, and ai \u2260 0 for some i \u2208 {1,\u2026, p}. By s \u2212 ni) = ord(s) \u2212 1. Then, by hypothesis of induction, we have that s \u2212 ni + {0,\u2026, ord(s) \u2212 2}\u2286S. Therefore, s + {0,\u2026, ord(s) \u2212 2}\u2286S. Moreover, (s \u2212 ni + ord(s) \u2212 2) + ni + 1 \u2208 S. Thereby, s + {0,\u2026, ord(s) \u2212 1}\u2286S.(3\u21d24) We reason by induction over ord \u2265 2. Thus, we get that x + y + 1 \u2208 S. By applying S is a PL-semigroup.(4\u21d21) If S = \u23294,5, 6\u232a; that is, let S be the numerical semigroup with minimal system of generators given by {4,5, 6}. It is obvious that 4 + 4 + 1 = 9, 4 + 5 + 1 = 10, 4 + 6 + 1 = 11, 5 + 5 + 1 = 11, 5 + 6 + 1 = 12, and 6 + 6 + 1 = 13 are elements of S. Therefore, by applying S is a PL-semigroup.Let The following result is straightforward to prove and appears in .S, T be numerical semigroups. S\u2229T is a numerical semigroup.S \u2260 \u2115, then S \u222a {F(S)} is a numerical semigroup.If Let Having in mind the definition of Frobenius variety, which was given in the introduction, we get the next result.\ud835\udcaePL = {S | S is a PL-semigroup} is a Frobenius variety.The set \u2115 \u2208 \ud835\udcaePL and, therefore, \ud835\udcaePL is a nonempty set.First of all, let us observe that S, T \u2208 \ud835\udcaePL. In order to show that S\u2229T \u2208 \ud835\udcaePL, we are going to use x, y \u2208 (S\u2229T)\u2216{0}, then x, y \u2208 S\u2216{0} and x, y \u2208 T\u2216{0}. Therefore, x + y + 1 \u2208 S\u2229T. Consequently, S\u2229T \u2208 \ud835\udcaePL.Let S \u2208 \ud835\udcaePL such that S \u2260 \u2115. By applying S \u222a F(S) \u2208 \ud835\udcaePL. Let x, y \u2208 (S \u222a F(S))\u2216{0}. If x, y \u2208 S, then x + y + 1 \u2208 S\u2286S \u222a F(S). On the other hand, if F(S)\u2208{x, y}, then x + y + 1 > F(S) and, thereby, x + y + 1 \u2208 S\u2286S \u222a F(S). We conclude that S \u222a F(S) \u2208 \ud835\udcaePL.Now, let graph\u2009\u2009G is a pair , where V is a nonempty set and E is a subset of { \u2208 V \u00d7 V | v \u2260 w}. The elements of V are called vertices of G and the elements of E are called edges of G. A path (of length n) connecting the vertices x and y of G is a sequence of different edges of the form , ,\u2026,  such that v0 = x and vn = y.A G is a tree if there exist a vertex v* (known as the root of G) such that for every other vertex x of G, there exists a unique path connecting x and v*. If  is an edge of the tree, then we say that x is a son of y.We say that a graph \ud835\udcaePL) in the following way:\ud835\udcaePL is the set of vertices of G(\ud835\udcaePL),S, S\u2032) \u2208 \ud835\udcaePL \u00d7 \ud835\udcaePL is an edge of G(\ud835\udcaePL) if S\u2032 = S \u222a {F(S)}.(We define the graph G(As a consequence of [\ud835\udcaePL) is a tree with root equal to \u2115. Moreover, the sons of a vertex S \u2208 \ud835\udcaePL are S\u2216{x1},\u2026, S\u2216{xr}, where x1,\u2026, xr are the minimal generators of S that are greater than F(S) and such that S\u2216{x1},\u2026, S\u2216{xr} \u2208 \ud835\udcaePL.The graph G the minimal system of generators of S, then msg(S) = (S\u2216{0})\u2216((S\u2216{0})+(S\u2216{0}))  \u222a (msg(S)).Let (Necessity). If x \u2212 1 \u2209 {0} \u222a (\u2115\u2216S) \u222a (msg(S)), then x \u2212 1 \u2208 S\u2216({0} \u222a (msg(S))). Accordingly, there exist y, z \u2208 S\u2216{0} such that x \u2212 1 = y + z. In fact, it is clear that y, z \u2208 S\u2216{x, 0}. Therefore, by applying S\u2216{x} is a PL-semigroup, we have x = y + z + 1 \u2208 S\u2216{x}, which is a contradiction.(Sufficiency). Let y, z \u2208 S\u2216{x, 0}. Since S is a PL-semigroup, by y + z + 1 \u2208 S. As x \u2212 1 \u2208 {0} \u222a (\u2115\u2216S) \u222a (msg(S)), we deduce that y + z + 1 \u2260 x. Thus y + z + 1 \u2208 S\u2216{x}. By applying S\u2216{x} is a PL-semigroup.As a consequence of the previous proposition, we have the next result.S be a PL-semigroup such that S \u2260 \u2115, and let x be a minimal generator of S greater than F(S). Then S\u2216{x} is a PL-semigroup if and only if x \u2212 1 \u2208 msg(S) \u222a {F(S)}.Let \ud835\udcaePL) as is shown in the following example.By applying S = \u23294,6, 7,9\u232a is a PL-semigroup with Frobenius number equal to 5. From S are S\u2216{6} = \u23294,7, 9,10\u232a and S\u2216{7} = \u23294,6, 9,11\u232a.It is clear that \ud835\udcaePL) such as it is shown in Let us observe that we can build recursively a tree, from the root, if we know the sons of each vertex. Therefore, we can build the tree G(\ud835\udcaePL), we are going to study the relation between the minimal generators of a numerical semigroup S and the minimal generators of S\u2216{x}, where x is a minimal generator of S that is greater than F(S). First of all, let us observe that if S is minimally generated by {m, m + 1,\u2026, 2m \u2212 1} , then S\u2216{m} = {0, m + 1, \u2192} is minimally generated by {m + 1, m + 2,\u2026, 2m + 1}. In other case we have the following result.In order to have an easier making of the tree G = {n1,\u2026, np}. If m(S) = n1 < np and np > F(S), then S\u2216{np} = \u2329n1,\u2026, np\u22121, np + n1\u232a.Let i \u2208 {2,\u2026, p}. Since np > F(S) and n1 < ni, we have that np + ni \u2212 n1 \u2208 S. Thus, np + ni \u2212 n1 = \u03b11n1 + \u22ef+\u03b1pnp for some \u03b11,\u2026, \u03b1p \u2208 \u2115. Thereby, np + ni = (\u03b11 + 1)n1 + \u22ef+\u03b1pnp. By applying that {n1,\u2026, np} is a minimal system of generators, we have that \u03b1p = 0. Therefore, np + ni \u2208 \u2329n1,\u2026, np\u22121\u232a. In particular, 2np \u2208 \u2329n1,\u2026, np\u22121\u232a.Let us take s \u2208 S\u2216{np}. Then s \u2208 S and, thus, there exist \u03b21,\u2026, \u03b2p \u2208 \u2115 such that s = \u03b21n1 + \u22ef+\u03b2pnp. Since 2np \u2208 \u2329n1,\u2026, np\u22121\u232a, we can assume that \u03b2p \u2208 {0,1}. On the one hand, if \u03b2p = 0, then s \u2208 \u2329n1,\u2026, np\u22121\u232a. On the other hand, if \u03b2p = 1, then there exists i \u2208 {1,\u2026, p \u2212 1} such that \u03b2i \u2260 0. If i = 1, then it is obvious that s \u2208 \u2329n1,\u2026, np\u22121, np + n1\u232a. And if i \u2260 1, since np + ni \u2208 \u2329n1,\u2026, np\u22121\u232a, we have that s \u2208 \u2329n1,\u2026, np\u22121\u232a. In any case, we conclude that S\u2216{np} = \u2329n1,\u2026, np\u22121, np + n1\u232a.Now, let S be a numerical semigroup with msg(S) = {n1,\u2026, np}. If m(S) = n1 < np and np > F(S), then Let S\u2216{np}) is {n1,\u2026, np\u22121} or {n1,\u2026, np\u22121, np + n1}. In addition, msg(S\u2216{np}) = {n1,\u2026, np\u22121} if and only if np + n1 \u2208 \u2329n1,\u2026, np\u22121\u232a.From np + n1 \u2208 \u2329n1,\u2026, np\u22121\u232a, then there exist \u03b11,\u2026, \u03b1p\u22121 \u2208 \u2115 such that np + n1 = \u03b11n1 + \u22ef+\u03b1p\u22121np\u22121. Since {n1,\u2026, np} is a minimal system of generators, we get that \u03b11 = 0. Thus, there exists i \u2208 {2,\u2026, p \u2212 1} such that \u03b1i \u2260 0. Consequently, np + n1 \u2212 ni \u2208 S.If i \u2208 {2,\u2026, p \u2212 1} such that np + n1 \u2212 ni \u2208 S, then np + n1 \u2212 ni = \u03b21n1 + \u22ef+\u03b2pnp for some \u03b21,\u2026, \u03b2p \u2208 \u2115. Thereby, np + n1 = \u03b21n1 + \u22ef+(\u03b2i + 1)ni + \u22ef+\u03b2pnp. Since {n1,\u2026, np} is a minimal system of generators, we have that \u03b2p = 0 and, therefore, np + n1 \u2208 \u2329n1,\u2026, np\u22121\u232a.Conversely, if there exists We finish this section with an illustrative example about the above corollary.S be the numerical semigroup with msg(S) = {3,5, 7}. It is obvious that F(S) = 4. By S\u2216{5} = \u23293,7, 8\u232a. In addition, 8 \u2212 7 = 1 \u2209 S. Thereby, applying S\u2216{5}) = {3,7, 8}. On the other hand, applying S\u2216{7} = \u23293,5, 10\u232a. Finally, since 10 \u2212 5 = 5 \u2208 S, we conclude that msg(S\u2216{7}) = {3,5}.Let m be a positive integer. We will denote by \u0394(m) = {0, m, \u2192}. It is clear that \u0394(m) is the greatest (with respect to set inclusion) PL-semigroup with multiplicity m. Our first aim in this section will be to show that there also exists the smallest (with respect to set inclusion) PL-semigroup with multiplicity m.Let As an immediate consequence of item (4) in S is a PL-semigroup, m \u2208 S\u2216{0}, and k \u2208 \u2115\u2216{0}, then km + i \u2208 S for all i \u2208 {0,\u2026, k \u2212 1}.If m \u2208 \u2115\u2216{0}. Then the numerical semigroup generated by {(i + 1)m + i | i \u2208 {0,\u2026, m \u2212 1}} is the smallest (with respect to set inclusion) PL-semigroup with multiplicity m.Let S = \u2329m, 2m + 1,\u2026, m2 + (m \u2212 1)\u232a. From m has to contain S. In order to conclude the proof, we will show that S is a PL-semigroup. For this purpose, since {(i + 1)m + i | i \u2208 {0,\u2026, m \u2212 1}} is a system of generators of S, it will be enough to check item (2) of i, j \u2208 {0,\u2026, m \u2212 1}, then (i + 1)m + i + (j + 1)m + j + 1 \u2208 S. We distinguish two cases. i + j + 1 \u2264 m \u2212 1, then (i + 1)m + i + (j + 1)m + j + 1 = (i + j + 2)m + (i + j + 1) \u2208 S.If i + j + 1 \u2265 m, then (i + 1)m + i + (j + 1)m + j + 1 = (i + j + 2)m + (i + j + 1) = (m + 1)m + (i + j \u2212 m + 2)m + (i + j \u2212 m + 1) \u2208 S.If Let m) = \u2329m, 2m + 1,\u2026, m2 + (m \u2212 1)\u232a and by \ud835\udcaePL(m) the set of all PL-semigroups with multiplicity equal to m. Let us recall that \u0394(m) = max\u2061(\ud835\udcaePL(m)) and \u0398(m) = min\u2061(\ud835\udcaePL(m)).We will denote by \u0398 is finite.The set S \u2208 \ud835\udcaePL(m), then \u0398(m)\u2286S\u2286\u0394(m). Since \u0394(m) and \u0398(m) are numerical semigroups, we have that \u0394(m)\u2216\u0398(m) is finite. Consequently, \ud835\udcaePL(m) is also finite.If The previous result can be considered a particular case of [m). For that, several concepts and results are introduced.Now we are interested in computing the Frobenius number and the genus of \u0398 = {s \u2208 S | s \u2212 m \u2209 S}. It is clear  = {\u03c9(0) = 0, \u03c9(1),\u2026, \u03c9(m \u2212 1)}, where \u03c9(i) is, for each i \u2208 {0,\u2026, m \u2212 1}, the least element of S that is congruent with i modulo m.If n S (see ) is Ap = max\u2061) \u2212 m,F(S) = (1/m)w) \u2212 ((m \u2212 1)/2).g = 0, \u03c9(1),\u2026, \u03c9(m \u2212 1)}\u2286\u2115 such that, for each i \u2208 {1,\u2026, m \u2212 1}, \u03c9(i) is congruent with i modulo m. Let S be the numerical semigroup generated by X \u222a {m}. The following conditions are equivalent. S, m) = X.Ap(\u03c9(i) + \u03c9(j) \u2265 \u03c9((i + j)mod\u2061m) for all i, j \u2208 {1,\u2026, m \u2212 1}.Let m \u2208 \u2115\u2216{0}, thenIf \u03c9(i) = (i + 1)m + i is congruent with i modulo m for all i \u2208 {1,\u2026, m \u2212 1}. Let us see now that, if i, j \u2208 {1,\u2026, m \u2212 1}, then \u03c9(i) + \u03c9(j) \u2265 \u03c9((i + j)mod\u2061\u2009m). Indeed, \u03c9(i) + \u03c9(j) = (i + 1)m + i + (j + 1)m + j > (i + j + 1)m + (i + j)\u2265((i + j)mod\u2061\u2009m + 1)m + (i + j)mod\u2061\u2009m = \u03c9((i + j)mod\u2061\u2009m). The proof follows from It is clear that m \u2208 \u2115\u2216{0,1}, then m)) = m \u2212 1,F(\u0394(m)) = m \u2212 1,g(\u0394(m)) = m2 \u2212 1,F(\u0398(m)) = (m \u2212 1)(m + 2)/2.g(\u0398(If Items (1) and (2) are trivial. On the other hand, items (3) and (4) are immediate consequences of m) can be rewritten asm) is a numerical semigroup generated by an arithmetic sequence with first term m and common difference m + 1 .S is a numerical semigroup, then the cardinality of the minimal system of generators of S is called the embedding dimension of S and is denoted by e(S). It is well known (see [S) \u2264 m(S). We say that a numerical semigroup S has maximal embedding dimension if e(S) = m(S). It is clear that {m, m + 1,\u2026, 2m \u2212 1} is the minimal system of generators of \u0394(m). Therefore, \u0394(m) is a numerical semigroup with maximal embedding dimension. Now we will show that \u0398(m) has also maximal embedding dimension.If The next result is [S be a numerical semigroup with multiplicity m and assume that Ap = {\u03c9(0) = 0, \u03c9(1),\u2026, \u03c9(m \u2212 1)}. Then S has maximal embedding dimension if and only if \u03c9(i) + \u03c9(j) > \u03c9((i + j)mod\u2061\u2009m) for all i, j \u2208 {1,\u2026, m \u2212 1}.Let \u03c9(i) + \u03c9(j) > \u03c9((i + j)mod\u2061\u2009m) for all i, j \u2208 {1,\u2026, m \u2212 1}. Therefore, by applying Let us observe that, in the proof of m \u2208 \u2115\u2216{0}, then \u0398(m) is a numerical semigroup with maximal embedding dimension.If m, 2m + 1,\u2026, m2 + m \u2212 1} is the minimal system of generators of \u0398(m).As a consequence of this corollary, we have that {\ud835\udcaePL(m). Thus, we define the graph G(\ud835\udcaePL(m)) in the following way:\ud835\udcaePL(m) is the set of vertices of G(\ud835\udcaePL(m));S, S\u2032) \u2208 \ud835\udcaePL(m) \u00d7 \ud835\udcaePL(m) is an edge of G(\ud835\udcaePL(m)) if S\u2032 = S \u222a {F(S)}.) is a tree with root equal to \u0394(m). Moreover, the sons of a vertex S \u2208 \ud835\udcaePL(m) are S\u2216{x1},\u2026, S\u2216{xr}, with {x1,\u2026, xr} = {x \u2208 msg(S) | x \u2260 m, x > F(S), \u2009and\u2009S\u2216{x} \u2208 \ud835\udcaePL}.The graph G(\ud835\udcaePL(m)) such as is shown in the next example.By applying \ud835\udcaePL(3)), that is, the tree of the PL-semigroups with multiplicity equal to 3.We are going to depict G is a tree, then the height of T is the maximum of the lengths of the paths that connect each vertex with the root. Let us observe that the height of G(\ud835\udcaePL(3)) is 3. In general, the height of the tree G(\ud835\udcaePL(m)) is equal toIf m.Let us study now the possible values of the Frobenius number and the genus for PL-semigroups with multiplicity m \u2208 \u2115\u2216{0}, then S) | S \u2208 \ud835\udcaePL(m)} = {x \u2208 \u2115 | m \u2212 1 \u2264 x \u2264 (m \u2212 1)(m + 2)/2};{g(S) | S \u2208 \ud835\udcaePL(m)} = (\u0394(m)\u2216\u0398(m)) \u222a {m \u2212 1}.{F(If m)\u2216\u0398(m) = {x1 > x2 > \u22ef>x\u03c1}.Let us assume that \u0394(m)\u222a{xi, \u2192} \u2208 \ud835\udcaePL(m) and that g(\u0398(m)\u222a{xi, \u2192}) = (m \u2212 1)(m + 2)/2 \u2212 i.(1) By m)\u222a{xi + 1, \u2192} \u2208 \ud835\udcaePL(m) with Frobenius number equal to xi. Thus, (\u0394(m)\u2216\u0398(m)) \u222a {m \u2212 1}\u2286{F(S) | S \u2208 \ud835\udcaePL(m)}. For the other inclusion, let us take S \u2208 \ud835\udcaePL(m) such that S \u2260 \u0394(m). Then F(S) > m and, thereby, F(S) \u2208 \u0394(m). Since \u0398(m)\u2286S, we have F(S) \u2209 \u0398(S). Therefore, we conclude that F(S) \u2208 \u0394(m)\u2216\u0398(S).(2) It is clear that \u0398(S) | S \u2208 \ud835\udcaePL(3)} = {2,3, 4,5}. Since \u0394(3) = \u23293,4, 5\u232a and \u0398(3) = \u23293,7, 11\u232a, we have that (\u0394(3)\u2216\u0398(3)) \u222a {2} = {2,4, 5,8}. Therefore, by applying S) | S \u2208 \ud835\udcaePL(3)} = {2,4, 5,8}.By \ud835\udcaePL is not a numerical semigroup. For instance, \u22c2n\u2208\u2115{0, n, \u2192} = {0}. On the other hand, it is clear that the (finite or infinite) intersection of numerical semigroups is always a submonoid of .Let us observe that, in general, the infinite intersection of elements of M be a submonoid of . We will say that M is a \ud835\udcaePL-monoid if it can be expressed like the intersection of elements of \ud835\udcaePL.Let The next lemma has an immediate proof.\ud835\udcaePL-monoids is a \ud835\udcaePL-monoid.The intersection of In view of this result, we can give the following definition.X be a subset of \u2115. The \ud835\udcaePL-monoid generated by X (denoted by \ud835\udcaePL(X)) is the intersection of all \ud835\udcaePL-monoids containing X.Let M = \ud835\udcaePL(X), then we will say that X is a \ud835\udcaePL-system of generators of M. In addition, if no proper subset of X is a \ud835\udcaePL-system of generators of M, then we will say that X is a minimal \ud835\udcaePL-system of generators of M.If \ud835\udcaePL is a Frobenius variety. Therefore, by applying [Let us recall that, by \ud835\udcaePL-monoid has a unique minimal \ud835\udcaePL-system of generators, which in addition is finite.Every The proof of the following lemma is also immediate.X\u2286\u2115, then \ud835\udcaePL(X) is the intersection of all PL-semigroups that contain X.If X is a nonempty subset of \u2115\u2216{0}, then \ud835\udcaePL(X) is a PL-semigroup.If \ud835\udcaePL(X) is a submonoid of . Therefore, in order to show that \ud835\udcaePL(X) is a numerical semigroup, it will be enough to see that \u2115\u2216\ud835\udcaePL(X) is a finite set.We know that x \u2208 X. If S is a PL-semigroup containing X, then . Since gcd{x, 2x + 1} = 1, we get that \u2329x, 2x + 1\u232a is a numerical semigroup and, thus, \u2115\u2216\u2329x, 2x + 1\u232a is finite. Consequently, \u2115\u2216\ud835\udcaePL(X) is finite.Let  then by  we know \ud835\udcaePL(X) is a PL-semigroup. Let x, y \u2208 \ud835\udcaePL(X)\u2216{0}. If S is a PL-semigroup containing X, from x, y \u2208 S\u2216{0} and from x + y + 1 \u2208 S. By applying again x + y + 1 \u2208 \ud835\udcaePL(X). Therefore, by applying \ud835\udcaePL(X) is a PL-semigroup.Now, let us see that \ud835\udcae be the set of all numerical semigroups. It is clear that \ud835\udcae is a Frobenius variety. If we take X = {2}, then the intersection of all elements of \ud835\udcae containing {2} is exactly M = \u23292\u232a, which is not a numerical semigroup.Let us observe that, in general, The next result will be key for our last purpose in this section.\ud835\udcaePL = {\ud835\udcaePL(X) | X\u2009is\u2009\u2009a\u2009\u2009nonempty\u2009\u2009finite\u2009\u2009subset\u2009\u2009of\u2009\u2009\u2115\u2216{0}}. S \u2208 \ud835\udcaePL, then (by X of \u2115\u2216{0} such that S = \ud835\udcaePL(X).By  then by  there ex\ud835\udcaePL is a Frobenius variety, by applying [Since M be a \ud835\udcaePL-monoid and let x \u2208 M. Then M\u2216{x} is a \ud835\udcaePL-monoid if and only if x belongs to the minimal \ud835\udcaePL-system of generators of M.Let As an immediate consequence of this proposition we have the following result.X be a nonempty subset of \u2115\u2216{0}. Then the minimal \ud835\udcaePL-system of generators of \ud835\udcaePL(X) is {x \u2208 X | \ud835\udcaePL(X)\u2216{x} is a PL-semigroup}.Let S = \u23293,7, 11\u232a is a PL-semigroup. By applying S = \ud835\udcaePL({3}) and {3} is the minimal \ud835\udcaePL-system of generators of S.By \ud835\udcaePL(X) when X is a fixed nonempty finite set of positive integers. Let us observe that by Now we want to describe n1,\u2026, np be positive integers. We will denote by S the set {\u03b11n1 + \u22ef+\u03b1pnp + r | r, \u03b11,\u2026, \u03b1p \u2208 \u2115, r < \u03b11 + \u22ef+\u03b1p} \u222a {0}. Our next purpose will be to show that S = \ud835\udcaePL.Let S be a numerical semigroup, let s1,\u2026, st \u2208 S\u2216{0}, and let \u03b11,\u2026, \u03b1t \u2208 \u2115. Then ord(\u03b11s1 + \u22ef+\u03b1tst) \u2265 \u03b11 + \u22ef+\u03b1t.Let n1,\u2026, np} be the minimal system of generators of S. Then, for each i \u2208 {1,\u2026, t}, there exist \u03b2i1,\u2026, \u03b2ip \u2208 \u2115 such that si = \u03b2i1n1 + \u22ef+\u03b2ipnp. Moreover, since si \u2260 0, we have that \u03b2i1 + \u22ef+\u03b2ip \u2265 1. Thus,Let {n1,\u2026, np are positive integers, then S is the smallest (with respect to set inclusion) PL-semigroup containing {n1,\u2026, np}.If x, y \u2208 S\u2216{0}, then x + y \u2208 S. In effect, we know that there exist \u03b11,\u2026, \u03b1p, \u03b21,\u2026, \u03b2p, r, r\u2032 nonnegative integers such that x = \u03b11n1 + \u22ef+\u03b1pnp + r, y = \u03b21n1 + \u22ef+\u03b2pnp + r\u2032, r < \u03b11 + \u22ef+\u03b1p, and r\u2032 < \u03b21 + \u2026+\u03b2p. Therefore, x + y = (\u03b11 + \u03b21)n1 + \u22ef+(\u03b1p + \u03b2n)np + r + r\u2032 with r + r\u2032 < (\u03b11 + \u03b21)+\u22ef+(\u03b1p + \u03b2n). Consequently, x + y \u2208 S.Let us see that if \u2115\u2216S is finite. Since n1 = 1 \u00b7 n1 + 0 \u00b7 n2 + \u22ef+0 \u00b7 np + 0 and 2n1 + 1 = 2 \u00b7 n1 + 0 \u00b7 n2 + \u22ef+0 \u00b7 np + 1, we have that n1, 2n1 \u2208 S. By applying the first step, we get that \u2329n1, 2n1\u232a\u2286S. Using the same reasoning as we did in the proof of Let us see that S is a numerical semigroup. Let us see now that S is a PL-semigroup. In order to do that, it is enough \u2216{0}, then x + y + 1 \u2208 S. Indeed, arguing as in the first step, we have that x + y + 1 = (\u03b11 + \u03b21)n1 + \u22ef+(\u03b1p + \u03b2n)np + r + r\u2032 + 1 with r + r\u2032 + 1 < (\u03b11 + \u03b21)+\u22ef+(\u03b1p + \u03b2n). Therefore, x + y + 1 \u2208 S.From the previous steps, we know that nough by  to show n1,\u2026, np}\u2286S.Following the proof of the second step, it is clear that {S is the smallest PL-semigroup that contains {n1,\u2026, np}. In fact, we will show that if T is PL-semigroup containing {n1,\u2026, np}, then S\u2286T. Thus, let x \u2208 S\u2216{0}. Then there exist \u03b11,\u2026, \u03b1p, r \u2208 \u2115 such that x = \u03b11n1 + \u22ef+\u03b1pnp + r with r < \u03b11 + \u22ef+\u03b1p. Since {n1,\u2026, np}\u2286T, by \u03b11n1 + \u22ef+\u03b1pnp + {0,\u2026, ord(\u03b11n1 + \u22ef+\u03b1pnp) \u2212 1}\u2286T. By applying r < ord(\u03b11n1 + \u22ef+\u03b1pnp) and, therefore, x \u2208 T.Finally, let us see that We divide the proof into five steps. In this way, we have proved the statement.The next result is an immediate consequence of the previous theorem.n1,\u2026, np are positive integers, then \ud835\udcaePL = S.If We finish this section with an example that illustrates its content.S = {0,4, 7,8, 9,11, \u2192} = \u23294,7, 9\u232a. Therefore, \ud835\udcaePL = \u23294,7, 9\u232a.It is clear that"}
{"text": "Using the variational approach, we prove the existence and uniqueness of variational solutions to such system. Moreover, we prove that this variational solution generates a random dynamical system. The main results are applied to a general type of nonlinear SPDEs and the stochastic generalized p-Laplacian equation.This paper deals with the following type of stochastic partial differential equations (SPDEs) perturbed by an infinite dimensional fractional Brownian motion with a suitable volatility coefficient \u03a6:  H) of classical Brownian motion. It becomes the standard Brownian motion when H equals to 1/2. However, it was proved in  \u2192 \u210b such that\u03d5 = H(2H \u2212 1) | t \u2212 s|H\u221222. From  \u2192 L2 such that the function \u03a6(\u00b7)u is contained in Lp for each u \u2208 \u210b and ||\u03a6||Lp) < \u221e. Then, we define the stochastic integral asL2. Indeed, it, from , A is (\u212c(V) \u2297 \u2131)-measurable, and \u03a6 :  \u2192 L2 such that the function \u03a6(\u00b7)u is contained in Lp for each u \u2208 \u210b and ||\u03a6||Lp) < \u221e. Furthermore, we impose some conditions on A and \u03a6 as follows.H1\u2009hemicontinuity): for all v1, v2, v \u2208 V, the map(\u2009is continuous on \u211d.H2\u2009weak monotonicity): there exists c1 \u2208 \u211d such that for all v1, v2 \u2208 V(H3\u2009coercivity): there exist c2 > 0, c3 \u2208 \u211d, \u03b1 > 1, and an (\u2131t)-adapted process f \u2208 L1 such that for all v \u2208 V,\u2009\u2009t \u2208 (H4\u2009boundedness): there exist c4 \u2265 0 and an (\u2131t)-adapted process g \u2208 L\u03b1/(\u03b1\u22121) such that for all v \u2208 V,\u2009\u2009t \u2208  = PN\u03a6(t) for all t \u2208 .such that for the corresponding orthogonal projection Let H1)\u2013(H4) are the standard monotonicity and coercivity conditions for SPDEs (cf.  is called a variational solution to \u2229L2 andt \u2208 , \u2119-a.s.A continuous ution to  if X \u2208 LNow we claim and prove the main results.H1)\u2013(H5) hold. For any given x \u2208 L2, there exists a unique variational solution to  = Y(t) + Zt(\u03c9), where Y(0) = x \u2208 L2. First, we show the existence and uniqueness of solutions to (H1 to H4. Since Zt(\u03c9) is an \u210b-valued Gaussian process, it is obvious to check the hemicontinuity and weak monotonicity for A. For the coercivity, we also have thatH3, the 2nd inequality holds using that \u2212||v+Zt(\u03c9)||V\u03b1 \u2264 \u22122\u03b11\u2212||v||V\u03b1 + ||Zt(\u03c9)||V\u03b1, and the 3rd inequality holds using Young's inequality with a small value \u03b5 > 0. Next, by H4, we estimate\u03b5 to be sufficiently small such that Zt(\u03c9)||V\u03b1 and ||Zt(\u03c9)||\u210b\u03b1 are all in L1, and g\u03b1/(\u03b1\u22121) \u2208 L1 holds due to g \u2208 L\u03b1/(\u03b1\u22121). Thus, Let us define new PDEY(t)=Y(0)ave that2V\u2217\u2329A~\u03c9(testimate||A(v+Zt(H4 we getg \u2208 L\u03b1/(\u03b1\u22121) and ||Zt(\u03c9)||V\u03b1 \u2208 L1. Thus we prove the boundedness of X(t) = Y(t) + Zt(\u03c9), we haveIndeed, by quently,  has a un\u03c6 in such a way that \u03c6 is defined by the solution of the SPDE at time t, for a noise path \u03c9, with initial point x.The next theorem shows that the unique solution of  generateX of  of  defines \u03c6 = x. Let us check then the cocycle property: for t, \u03c4 \u2208 \u211d+ and x \u2208 \u210b, we haver = s \u2212 \u03c4 leads to\u03c6 : \u211d+\u2009\u00d7\u2009\u03a9\u2009\u00d7\u2009\u210b \u2192 \u210b. Note that the maps t \u2192 \u03c6 and x \u2192 \u03c6 are continuous, thus we only need to prove the measurability of \u03c9 \u2192 \u03c6. By the proof of the existence and uniqueness of solutions to  is the weak limit of a subsequence of the Galerkin approximations \u03c6n in L\u03b1. Let \u03bek \u2208 C0\u221e(\u211d) be a Dirac sequence with supp(\u03bek)\u2286Bk1/(0), where Bk1/(0) is an open ball of radius 1/k centered at the point 0. Then )(t) is well defined for k large enough. For each such k \u2208 \u2115 and h \u2208 \u210b we haven \u2192 \u221e. Since \u03c9 \u2192 \u03c6n \u2208 L\u03b1 is measurable, so is \u03c9 \u2192 , h\u232a\u210b)(t). Moreover, it follows from , h\u232a\u210b)(t) is measurable. On the other hand, t \u2192 \u03c6 is continuous in \u210b. Therefore, , h\u232a\u210b)(t)\u2192\u2329\u03c6, h\u232a\u210b(t) and the measurability of \u03c9 \u2192 , h\u232a\u210b)(t) imply the measurability of \u03c9 \u2192 \u2329\u03c6, h\u232a\u210b(t). Since this is true for all h \u2208 \u210b, we get the measurability of \u03c9 \u2192 \u03c6. Consequently, we complete the proof that \u03c6 defines a continuous RDS.Trivially ution to , we gettions to , we know we have\u2013(H4) hold for A(X(t)) = \u2212X(t) | X(t)|p\u22122 with \u03b1 : = p. Thus, by the above results, for suitable \u03a6(t) the SPDE (L2(\u039b);, moreover, this solution defines a continuous RDS.According to [the SPDE  has a unp < \u221e, \u039b \u2282 \u211dd, \u039b is open. Denote by H0p1, the completion of C0\u221e(\u039b) with respect to the norm defined as:d \u2192 \u211dd is monotone, continuous and that for some strictly positive constants a1, a2u \u2208 \u211dd. Then we consider the following Gelfand tripleAgain let 2 \u2264 H1)\u2013(H4) hold for A(X(t)) = div(\u03a8(\u2207X(t)) \u2212 X(t) | X(t)|p\u22122) with \u03b1 : = p. Thus, by the above results, for suitable \u03a6(t), the SPDE (L2(\u039b); moreover, this solution defines a continuous RDS.According to [the SPDE  has a unp-Laplacian equation. It is useful to note that our results for SPDEs in a Hilbert space can reduce to known results for a standard infinite dimensional Wiener process if the Hurst parameter H = 1/2 though the techniques of proof are different. Furthermore, the conditions (H1)\u2013(H4) can be replaced by some much weaker assumptions  according to some recent results in [Using the variational approach we have studied a general type of fBm-driven nonlinear SPDEs with a suitable volatility coefficient in Hilbert space. We proved the existence and uniqueness of variational solutions to such system under some monotonicity and coercivity conditions. We further proved that this variational solution generates a random dynamical system. Finally, we applied the main results to two types of SPDEs including the stochastic generalized sults in . When th"}
{"text": "Rm to Rn and any continuous function, which has limit at infinite place, from limitless close subset of Rm to Rn. This extends the nonlinear approximation ability of traditional BP neural network and RBF neural network.The neural network with two weights is constructed and its approximation ability to any continuous functions is proved. For this neural network, the activation function is not confined to the odd functions. We prove that it can limitlessly approach any continuous function from limited close subset of  So far, many neural networks models have been presented and applied in pattern recognition, automatic control, signal processing, and aided decision. Among these models, BP (feedforward) neural networks and RBF  neural networks are widely used because of their nonlinear ability of approximation to any continuous function. Up to now, these two classes of neural networks have successfully been applied to approach any nonlinear continuous function defined on bounded close subsets \u20137. Howevy is the output of neurons, \u03d5 is the activation function, \u03b8 is the threshold value, \u03c9j is the direction weight value, zj is the core weight value, xj is the input, and p and s are two parameters.Wang et al.  presentezj = 0, p = 0, and s = 1, then  with one hidden layer Nn \u2192 R defined byi = 1,2,\u2026, nSuppose In this paper, one is concerned with the neural networks with two weights and a single hidden layer:s and p, the neural network with two weights can not only approximate any continuous functions defined on a bounded close subset, namely, it has the same approximation ability as BP neural network in  to denote the space of continuous real-valued functions defined on , it is equalled by the supremum norm ||\u00b7||. Let f be a real-valued function defined on . \u03b7, the modulus of continuity of f, is defined as \u03b7 = sup\u2061h\u2264\u03b40<\u2061||f(\u00b7+h) \u2212 f(\u00b7)||. Then we have lim\u2061\u03b4\u21920\u2061\u03b7 = 0, and for any real number \u03bb \u2265 0, \u2009\u03b7 \u2264 (1 + \u03bb)\u03b7.We use the following notations: the symbols f is called Lipschitz \u03b1\u2009(0 < \u03b1 \u2264 1) continuous and is written as f \u2208 LipC(f)\u03b1, if there exists a constant C(f) such that \u03b7 \u2264 C(f)\u03b4\u03b1. C(h) denotes the positive constant dependents only on h and its value may be different at different occurrence.The modulus of continuity is usually considered as the measure of the smoothness of function and the approximation error in approximation theory. The function Our main result is as follows.\u03d5 is a bounded, strictly monotonously increasing, and odd function defined on R, f \u2208 C, and n \u2208 N. Then there exists a feedforward neural network (BP) with one hidden layer Nn \u2192 R defined byi = 1,2,\u2026, n,Suppose a, b] into n equal intervals; each has length of (b \u2212 a)/n and let a = x0 < x1 < \u22ef<xn = b. For i = 1,2,\u2026, n, we set ci = (1/2m)(f(xi) \u2212 f(xi\u22121)), dn = \u03d5\u22121((m/2n) \u2212 m), m = sup\u2061x\u2208R\u2061|\u03d5(x)|, \u03b8i = dn \u2212 ((dn \u2212 dn*)n/(b \u2212 a))[a + (i \u2212 1)((b \u2212 a)/n)], dn* = \u03d5\u22121(m \u2212 (m/2n)),\u2009\u2009\u03c9i = (dn \u2212 dn*)n/(b \u2212 a).Divide , given n, we assume that x \u2208  and note thatx* \u2208  and choose f, so that Nn(xi*) = f(xi*). Let Gi(xi*) = \u03d5(\u03c9ix + \u03b8i); it follows thatx > xi*; (ii) x \u2264 xi*.For any Case (i). When x > xi*, we have, for i = 1,2,\u2026, n,Case (ii). When x \u2264 xi*, we have, for i = 1,2,\u2026, n,2n.From  and 13)Case (ii)n.From (|Gi(x)\u2212GiThe activation functions are assumed to be odd functions in , while i\u03d5 is a bounded, strictly monotonously increasing, and odd function defined on R, f \u2208 C, and n \u2208 N, z \u2209 . Then there exists a neural network with two weights and one hidden layer Nn :  \u2192 R defined byp < s, p < 0 such thatci is defined in lis is equal to \u03c9i in Suppose H : x \u2192 u, where u = (x\u2212z)s/|x\u2212z|p\u2212s, x \u2208 . Since z \u2209 , thus H is a continuous transformation. Hence f = U is also a bounded close set; moreover, 0 \u2209 U.We make fractional transformation H\u2212 : u \u2192 x, where x = z + us1/|u|s \u2212 p)/sp = f(x) = f(z + us1/|u|s \u2212 p)/sp is a continuous real-valued function defined on bounded close set U. For g(u), from Nn :  \u2192 R defined byThere exists an inverse transformation for fractional transformation p < s, p < 0, the network with two weights is not BP and RBF neural network; thus, n \u2192 \u221e, from When \u03d5 is a bounded, strictly monotonously increasing, and odd function defined on R; D is an unbounded subset of R  andf \u2208 C(D), z \u2209 D, lim\u2061x\u2192\u221e\u2061f(x) = b(constant) and then there exists a neural network with two weights and one hidden layer:p < s, p < 0 such thatSuppose H : x \u2192 u, where u = (x\u2212z)s/|x\u2212z|p\u2212s, x \u2208 D. Since z \u2209 D, thus H is a contimuous transformation on D and as x \u2192 \u221e, u \u2192 0. Let H(\u221e) = 0; H(x) is a continuous function on D \u222a {\u221e}. We prove that H(D \u222a {\u221e}) = H(D)\u222a{0} is a close set. If v is an arbitrary accumulation point of H(D)\u222a{0}, then v = 0 or v is an accumulation point of H(D). When v = 0, z \u2208 H(D)\u222a{0}. When v is an accumulation point of H(D), then there exists a sequence uk \u2208 H(D) and another sequence xk \u2208 D, such that xk \u2192 uk,\u2009\u2009uk \u2192 v \u2260 0\u2009\u2009(k \u2192 \u221e). From fractional transformation, there exists uk \u2192 v \u2260 0\u2009\u2009(k \u2192 \u221e) and inverse transformation H\u22121 : x = z + us1/|u|s \u2212 p)/sp = H(D)\u222a{0} is a close set. Letg(u) is a continuous function on H(D). Let g(0) = b = lim\u2061x\u2192\u221e\u2061f(x); it is easy to prove that g(u) is a continuous function on H(D)\u222a{0}. For a continuous function g(u) defined on H(D)\u222a{0}, by means of i=1nci\u03d5(lis((x\u2212z)s/|(x\u2212z)|s)|x\u2212z|p \u2212 \u03b8i) such thatWe make fractional transformation p < s,\u2009\u2009p < 0, the network with two weights is not BP and RBF neural network; thus, n \u2192 \u221e, from When In this paper, we construct a new BP neural network and prove that the network has the approximation ability to any nonlinear continuous function. In our result, the threshold values and direction weight values in  are diff"}
{"text": "We prove that the equation has infinitely many weak solutions by using the variant fountain theorem due to Zou (2001) and f, g do not need to satisfy the (P.S) or (P.S*) condition.We study the following  Moreover, limu\u21920g/|u|p\u22121 = 0 uniformly for\u2009\u2009x \u2208 \u03a9.There exists (F4)Assume that one of the following conditions hold:(1)u|\u2192\u221e|g/|u|p\u22122u = 0 uniformly for\u2009\u2009x \u2208 \u03a9;lim(2)u|\u2192\u221e|g/|u|p\u22122u = \u2212\u221e uniformly for\u2009\u2009x \u2208 \u03a9; furthermore, f/|u|p\u22122u and g/|u|p\u22122u are decreasing in\u2009\u2009u\u2009\u2009for\u2009\u2009u\u2009\u2009is large enough;lim(3)u|\u2192\u221e|g/|u|p\u22122u = \u221e uniformly for\u2009\u2009x \u2208 \u03a9; g/|u|p\u22122u is increasing in\u2009\u2009u\u2009\u2009for\u2009\u2009u\u2009\u2009is large enough; moreover, there exists \u03b1 > max{\u03c3, \u03b4} such thatlimwhere\u2009\u2009G = \u222b0ugdt.In this paper, we study the following\u2009\u2009p = 2.The above conditions were given in Zou  for the r < p < s < p*.A simple example which satisfies (F1)\u2013(F4) isSobolev spaceEquation  is posedu \u2208 Wp1,(\u03a9), where\u2009\u2009F = \u222b0ufdt\u2009\u2009and\u2009\u2009ds\u2009\u2009is the measure on the boundary. It is easy to see that \u03a6 \u2208 C1, \u211d) andu, v \u2208 Wp1,(\u03a9). It is well-known that the weak solution of .The corresponding energy functional of  is defin\u03bc(x) \u2261 1.Under condition , it is e\u03b7 = 0) and obtained the existence of infinitely many weak solutions. Moreover, the existence of three solutions for  was researched in , satisfies(A1)\u03bb maps bounded sets to bounded sets uniformly for\u2009\u2009\u03bb \u2208 ; furthermore, \u03a6\u03bb(\u2212u) = \u03a6\u03bb(u) for all \u2208 \u00d7 E.\u03a6(A2)B(u) \u2265 0 for all\u2009\u2009u \u2208 E; B(u) \u2192 \u221e as ||u|| \u2192 \u221e\u2009\u2009on any finite dimensional subspace of\u2009\u2009E.(A3)\u03c1k > rk > 0 such thatThere exists for all\u2009\u2009\u03bb \u2208 ,The \u03bbn \u2192 1, u(\u03bbn) \u2208 Yn, such thatThen there exist u(\u03bbn)} has a convergent subsequence for every\u2009\u2009k, then\u2009\u2009\u03a61\u2009\u2009has infinitely many nontrivial critical points {uk} \u2208 E\u2216{0} satisfying \u03a61(uk) \u2192 0\u2212 as k \u2192 \u221e.Particularly, if {u(\u03bbn)}\u2009\u2009is a\u2009\u2009(P.S*)\u2009\u2009sequence.Obviously, the sequence\u2009\u2009{E = Wp1,(\u03a9), E is a reflexive and separable Banach space; then there are\u2009\u2009ej \u2208 E\u2009\u2009and ej* \u2208 E* such thatFor our working space Xj : = span{ej}; then\u2009\u2009Yk, Zk\u2009\u2009can be defined as that in the beginning of \u03bb : E \u2192 \u211d defined byWe write\u2009\u2009B(u) \u2265 0\u2009\u2009for all\u2009\u2009u \u2208 E;\u2009\u2009B(u) \u2192 \u221e\u2009\u2009as\u2009\u2009||u|| \u2192 \u221e\u2009\u2009on any finite dimensional subspace of\u2009\u2009E;\u2009\u2009\u03a6\u03bb(\u2212u) = \u03a6\u03bb(u)\u2009\u2009for all\u2009\u2009\u03bb \u2208 , u \u2208 E.\u2009\u2009We need the following lemmas.Then\u2009\u2009q < p*, then one hasIf\u2009\u20091 \u2264 First, we check the condition of Assume (F1)\u2013(F3); then (A1)\u2013(A3) hold.n > k > 2; we assume that \u03c3 \u2264 \u03b4\u2009\u2009and define(A1) and (A2) are obvious. Let u \u2208 Zk. Note that q < p*; there exists a constant c > 0 such that Sobolev trace imbedding inequality, we haveK\u2009\u2009such that, for all\u2009\u2009\u03b7 < \u039b*,\u025b > 0, there exists\u2009\u2009c\u025b\u2009\u2009such thatp < q; we may choose \u025b > 0 and R > 0 sufficiently small thatu \u2208 Wp1,(\u03a9)\u2009\u2009with\u2009\u2009||u|| \u2264 R. So we haveu \u2208 Zk\u2009\u2009with ||u|| \u2264 R. Choosing\u03b2k(\u03c3) \u2192 0,\u2009\u2009\u03b2k(\u03b4) \u2192 0\u2009\u2009as\u2009\u2009k \u2192 \u221e, it follows that\u2009\u2009\u03c1k \u2192 0\u2009\u2009as\u2009\u2009k \u2192 \u221e, so there exists\u2009\u2009k0\u2009\u2009such that \u03c1k \u2264 R when k \u2265 k0. Thus, for k \u2265 k0, u \u2208 Zk, and ||u|| = \u03c1k, we have \u03a6\u03bb(u) \u2265 \u03c1kp/4p > 0; then ak(\u03bb) \u2265 0 for all \u03bb \u2208 .Observe that|u|\u03c3\u2264\u03b2k < 0\u2009\u2009for all\u2009\u2009\u03bb \u2208 .On the other hand, if u \u2208 Zk with ||u|| \u2264 \u03c1k, k \u2265 k0, we see thatdk(\u03bb) \u2192 0 as k \u2192 \u221e. Thus, (A3) holds.Furthermore, if By \u03bbn \u2192 1 and u(\u03bbn) \u2208 Yn such thatThere exist u(\u03bbn)} is bounded.In order to complete our proof of u(\u03bbn)} is bounded in\u2009\u2009Wp1,(\u03a9).\u2009\u2009{\u03bbn\u2032|Yn(u(\u03bbn)) = 0, thenSince \u03a6\u03b7 < \u039b such that 1 \u2212 \u03b7K > 0. If, up to a subsequence, ||u(\u03bbn)|| \u2192 \u221e as n \u2192 \u221e, then, by (F2),n is large enough. Obviously, it is a condition if (F4)(1) holds.We can choose 0 < \u039b < \u039b* and if wn = u(\u03bbn)/||u(\u03bbn)||; then, up to a subsequence,w \u2260 0 in E and limu|\u2192\u221e|g/|u|p\u22122u = \u2212\u221e in (F4)(2), then, for n is large enough, by Fatou's Lemma, we have thatu|\u2192\u221e|g/|u|p\u22122u = \u221e in (F4)(3). Thus, w = 0.Otherwise, we set tn \u2208  such thatc > 0\u2009\u2009large enough, and n is large enough, we have thatn\u2192\u221e\u03a6\u03bbn(tnu(\u03bbn)) \u2192 \u221e. Obviously,Let It follows thatp)fu \u2212 F and (1/p)gu \u2212 G are decreasing in\u2009\u2009u\u2009\u2009for u is large enough. Therefore,s > 0 and u \u2208 \u211d; it is a contradiction.If (F4)(2) holds, we have that (1/If (F4)(3) holds, then we have thatu(\u03bbn), for n is large enough, since \u03b1 > max{\u03b4, \u03c3}, we have that\u03a9((1/p)g)u(\u03bbn) \u2212 G))dx is bounded, which contradicts } is bounded.By the above arguments, we have that {c > 0 such thats < t or t < s < 0,\u2009\u2009\u2200 x \u2208 \u03a9, where H = (1/p)ft \u2212 F and In fact, our result still holds if we consider a weaker condition than (F4)(2); that is, there is"}
{"text": "We study a diffusive predator-prey model with nonconstant death rate and general nonlinear functional response. Firstly, stability analysis of the equilibrium for reduced ODE system is discussed. Secondly, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained. Furthermore, sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the system are derived by using the method of Lyapunov function. Finally, we show that there are no nontrivial steady state solutions for certain parameter configuration. One of the dominant themes in both ecology and mathematical ecology is the dynamic relationship between predators and their prey due to its universal existence and importance in population dynamics. The investigations on predator-prey models are developed during these thirty years, and more realistic models are derived in view of laboratory experiments and observations  and P represent the population density of prey and predator at x \u2208 \u03a9 and at time t, respectively. The parameter \u025b > 0 is the specific growth rate of the prey in the absence of predation and without environment limitation; in the absence of predators, the prey population grows logistically to a carrying capacity K > 0; the functional response of the predator is a Holling type II function a(PN/(\u03b2 + N)); a, \u03b2, and b are satiation coefficients or conversion rates. The specific mortality of predators in the absence of prey\u03b3 is the mortality at low density, and \u03b4 is the maximal mortality with the natural assumption \u03b3 < \u03b4. Di > 0 are constants, i = 1,2, while \u0394 denotes the Laplace operator in \u03a9 \u2282 Rn with \u03a9 bounded and connected. The advantage of the present model over the more often used models is that here the predator mortality is neither a constant nor an unbounded function, and still it is increasing with quantity.In , Duque a\u03c6(x) in Holling type I function and Holling type II function is monotonic in the first quadrant. It implies that, as the prey population increases, the consumption rate of prey per predator increases. But some experiments and observations indicate that a nonmonotonic response occurs at this level: when the nutrient concentration reaches a high level, an inhibitory effect on the specific growth rate may occur. To model such an inhibitory effect, Andrews , where K < l.there exists a Motivated by the above question a functional response may be monotonic or nonmonotonic; we consider the following diffusive predator-prey model with general nonlinear functional response:a\u03d5(N) represents Holling type II functional response or Holling type IV functional response; that is, a\u03d5(N) = aN/(\u03b2 + N) or a\u03d5(N) = aN/(\u03b2 + N2).Note that hypotheses (i)\u2013(v) are satisfied if function a\u03d5(N) = aN/(\u03b2 + N), system  and E2 =  on the boundary of \u03a9 = { : N \u2265 0, P \u2265 0} and a set of nontrivial critical points obtained as the intersection of the curvesE3 = , if \u2212\u03b3 + b\u03d5(K) > 0, whereE1 =  isJE1 has two eigenvalues \u03bb1 = \u025b, \u03bb2 = \u2212\u03b3. Hence, the equilibrium E1 =  is a saddle point and the stable and unstable manifolds lie on the P-axis and N-axis, respectively.For the reaction-diffusion predator-prey system , the reddNdt=\u025b(1\u2212rewrite (dNdt=a\u03d5(N),wheref(N)=\u025b isJE2 has two eigenvalues \u03bb1 = \u2212\u025b, \u03bb2 = \u2212\u03b3 + b\u03d5(K). Hence the equilibrium E2 =  is a saddle point, if \u2212\u03b3 + b\u03d5(K) > 0, and it is locally asymptotically stable when \u2212\u03b3 + b\u03d5(K) \u2264 0.The Jacobian matrix of system  at E2 =E3 =  isJ are the roots of the polynomialThe Jacobian matrix of system  at E3 =The following result holds.N0 \u2208 , then the equilibrium E3 =  is locally asymptotically stable with respect to system \u2013(v), and (J = J11 + J22 = a\u03d5(N0)f\u2032(N0) \u2212 P0M\u2032(P0) < 0. Moreover, since J11, J22, J12 < 0 and J21 > 0, it follows that det\u2061(J) > 0, which completes the proof.Taking into account that N0 \u2208 K/, K, the N, P) of ) of  lies in u is defined byt\u2192\u221eu = K.Assume that t > 0. Furthermore, the nonnegative solution , P) of ) of  satisfieThe nonnegativity of the solutions of  is clear\u03b7 > 0, there exists T\u03b7(>0) such that, for any t > T\u03b7,P. Taking into account the second equation of system (\u03b3 \u2264 M(P) \u2264 \u03b4, we getT = T(M0) > 0 such that\u03b7 = 1 and taking into account thatT\u2217 = max\u2061{T\u03b7, T}. Then, from (z(t) is the solution of the equationt0 \u2265 T\u2217 such thatP \u2192 0, t \u2192 \u221e, which is a contradiction.From the first equation of system , we havef system  and the uch thatP>M0get thata\u03d5(N)NP\u2265ant that2N\u2264K+ies that0<N\u2264b\u03d5(N)NN,lim\u2061N\u21920\u03d5\u2264K+uch thatb\u03d5(N)=b\u03d5 = P \u2212 M0, t > 0. We have to consider two possibilities. If there is some t0 such that \u03a6 \u2260 0 for any t > t0, we say that the zeroes of \u03a6 are bounded. If this is not true we say that the zeroes are unbounded; in this case, \u03a6 = 0 for a sequence of tn tending to +\u221e as n \u2192 \u221e.Let us define the function \u03a6 are bounded, then 0 < P \u2264 M0 for any t \u2265 t0.If the zeroes of \u03a6 is of constant sign. Let us assume that \u03a6 \u2265 0 for t \u2208 Jn, n = 1,3, 5,\u2026. Since P \u2265 M0 on the intervals Jn\u221212, arguing as in the proof that  \u2264 z(t) for t \u2208 Jn\u221212, where z is the solution of the equation z\u2032(t) = \u2212(\u03b3/2)z(t), with a suitable initial condition. Thus,n which is large enough. This contradiction completes the proof of our claim.If the zeroes are unbounded, the oof that  is not p\u03b7* > 0,An immediate consequence of the proof of the former result is that for a given r system . Hence, r system  is dissiNow, we will summarize some facts contained in Hale and Waltman studies, see , about t\u03a9 is a complete metric space with \u03a9 = \u03a90 \u222a \u2202\u03a90 for an open set \u03a90, where \u2202\u03a90 is the boundary of the set \u03a90. We will typically choose \u03a90 to be the positive cone in an ordered Banach space. A flow or semiflow on \u03a9 under which \u03a90 and \u2202\u03a90 are forward invariant is said to be permanent if it is dissipative and if there is a number \u03b7 > 0 such that any trajectory starting in \u03a90 will be at least a distance \u03b7 from \u2202\u03a90 for all sufficiently large t. To state a theorem implying permanence, we need a few definitions. An invariant set M for the flow or semiflow is said to be isolated if it has a neighborhood U such that M is the maximal invariant subset of U. Let \u03c9(\u2202\u03a90)\u2282\u2202\u03a90 denote the union of the sets \u03c9(u) over u \u2208 \u2202\u03a90. This differs from the standard definition of the \u03c9-limit set of a set but it is more convenient for our purposes; see , whereX\u039b = \u03a90 \u222a \u2202\u03a90. Assume that operator S(t) is compact for t > 0, which is defined by [S(t)\u03c6](x) = U, where U is the classical solution of system (S(t)}t\u22650 is pointwise dissipative.Let us set , system  can be rf system . MoreoveS(t) is positively invariant on \u2202\u03a90, which in turn implies that S(t) is positively invariant on \u03a90.A direct application of the maximum principle shows that \u03c9(\u2202\u03a90) = {E1, E2}. First, we consider\u03c61(x) > 0 for all \u03c61(x) \u2265 0 for all \u03c61(x)\u22620 and applying the strong maximum principle, we obtain that N > 0 for all t > 0 and t1 and choose \u03c61(x) = N.Now, let us show that u(t) and \u03c9(t) be the solutions of the equation z\u2032 = \u025b(1 \u2212 z/K)z such that z(t) = K attracts any positive solution, we conclude that N \u2192 K as t \u2192 \u221e.Let M(P) \u2265 \u03b3, we get thatP \u2192 0 as t \u2192 \u221e. Henceforth, we conclude that \u03c9(\u2202\u03a90) = {E1, E2} and it is isolated and acyclic. By choosing M1 = E1 and M2 = E2, then M = M1 \u222a M2 is the covering required by Let us now consider\u03c9(\u2202\u03a90) is positively invariant and the fact that the stable and unstable manifolds of the equilibrium E1 lie on \u03c9(\u2202\u03a90), we obtain that Ws(E1)\u2229\u03a90 = \u2205. In order to show that Ws(E2)\u2229\u03a90 = \u2205, let us linearize \u2229\u03a90 = \u2205.Taking into account that inearize  about E)P.From  and by uSince all hypotheses of Permanence implies that condition  is an imq, Q, w, and W such that q \u2264 N \u2264 Q, w \u2264 N \u2264 W as time converges to infinity. If \u03d5(N) = N/(\u03b2 + N), system  > 0, in this case system  of the system  of the operator \u2212\u0394 on \u03a9 with no-flux boundary conditions. Let us recall that the eigenvalues {\u03bbj}j=0\u221e satisfy the following inequality:Let us first briefly study the local stability of the homogeneous stationary solution U0 of  via the equation  about Ugiven by\u2202W\u2202t=D\u0394W+given by\u2202W\u2202t=D\u0394W+W = 0 is equivalent to that each matrix Bj has two eigenvalues with negative real parts for all j \u2265 0. And the eigenvalues of the matrix Bj are the roots of the polynomialHenceforth, the asymptotic stability of The following result holds.N0 \u2208 , then the equilibrium U0 is locally asymptotically stable with respect to system \u2013(v), and (Bj = A11 + A22 \u2212 \u03bbj(D1 + D2) = a\u03d5(N0)f\u2032(N0) \u2212 P0M\u2032(P0) \u2212 \u03bbj(D1 + D2) < 0. Moreover, since A11, A22, A12 < 0 and A21 > 0, it follows that det\u2061Bj = (A11 \u2212 \u03bbjD1)(A22 \u2212 \u03bbjD2) \u2212 A12A21 > 0, which completes the proof.Taking into account that N0 \u2208 K/, K, the U0.From Propositions U0. Biologically, our statement of the global stability of U0 means that no matter the two species diffuse, they will be spatially homogeneously distributed as time converges to infinity.Now, we will focus on the global stability of U0 is globally asymptotically stable.Assume thatE(t) \u2265 0 for all t \u2265 0. Differentiating E(t) along the solutions of the system (E(t) satisfies Lyapunov's asymptotic stability theorem, and the positive equilibrium U0 of system  + g) \u2265 0 for x \u2208 \u03a9, \u2202\u03c9/\u2202\u03c5 \u2264 0 for x \u2208 \u2202\u03a9, and g) \u2265 0.If \u03c9(x) + g) \u2264 0 for x \u2208 \u03a9, \u2202\u03c9/\u2202\u03c5 \u2265 0 for x \u2208 \u2202\u03a9, and g) \u2264 0.If Suppose that The following result gives an upper estimate for classical solutions of , where cK, b such that \u03b3 < b\u03d5(K) < \u03b4; then for any positive classical solution  of system  < K. This in turn implies that Let us set f system , we obta\u03c82 = D2P, we get from the second equation of system  \u2264 (b\u03d5(K) \u2212 \u03b3)/(\u03b4 \u2212 b\u03d5(K)). This in turn implies thatAnalogously, by setting f system  that \u2212 \u03b3)/(\u03b4 \u2212 b\u03d5(K)) and \u03b3 < b\u03d5(K) < \u03b4, then  is a nonconstant positive classical solution of  such that N(x), P(x) \u2264 C. Let \u03a9 by parts, we haveAssume that (ution of . By 65)N, P) is ation of  by N-N\u00af  we haveD1\u222b\u03a9|\u2207, it fol account , we obta112, in [\u03bc1\u222b\u03a9[D1 represents Holling type II functional response or Holling type IV functional response; that is, a\u03d5(N) = aN/(\u03b2 + N) or a\u03d5(N) = aN/(\u03b2 + N2). Hence, we can obtain the sufficient and necessary conditions which guaranting the predator and the prey species of the system with monotonic or nonmonotonic functional response to be permanent. If \u03d5(N) = N/(\u03b2 + N), system (Note that hypotheses (i)\u2013(v) in , system  reduces , system  that was"}
{"text": "The state transfer problem of a class of nonideal quantum systems is investigated. It is known that traditional Lyapunov methods may fail to guarantee convergence for the nonideal case. Hence, a hybrid impulsive control is proposed to accomplish a more accurate convergence. In particular, the largest invariant sets are explicitly characterized, and the convergence of quantum impulsive control systems is analyzed accordingly. Numerical simulation is also presented to demonstrate the improvement of the control performance. To avoid the confusion, we define \u2220\u2329\u03c8 | \u03c8f\u232a = 0\u00b0 if \u2329\u03c8 | \u03c8f\u232a = 0. In addition, the impulsive control matrix Bk should be chosen to satisfy \u0394V1(tk) \u2264 0; that is,\u03c1f = |\u03c8f\u232a\u2329\u03c8f| is the density matrix of target state |\u03c8f\u232a. Inequality (Bk which commutes with \u03c1f. The control law satisfying (u(t) can make the system leave the initial state even if |\u03c8(0)\u232a is an eigenstate of H0.In order that the designed control can work in the case of the initial state being orthogonal to the goal state, we rewrite  as V\u02d91=Klthat is,Bk\u2217\u03c1fBk\u2212\u03c1tisfying  will be H0 with H0|\u03c8(0)\u232a = \u03bb0|\u03c8(0)\u232a and \u2329\u03c8(0) | \u03c8f\u232a = 0, then the following conclusions hold: l \u2208 J such that \u2111\u2329\u03c8f | Hl | \u03c8(0)\u232a\u22600, then \u2329\u03c8(t) | \u03c8f\u232a \u2260 0\u2009\u2009(t > 0); if there exists \u2111\u2329\u03c8f | Hl | \u03c8(0)\u232a = 0, for all l \u2208 J, and there exists l \u2208 J such that \u211c\u2329\u03c8f | Hl | \u03c8(0)\u232a\u22600 and \u03bb0 \u2260 0, then \u2329\u03c8(t) | \u03c8f\u232a\u22600, t > t\u2032 > 0, where t\u2032 is sufficiently small; otherwise, the designed control fields cannot achieve the state steering of the closed-loop system. if For control law  and 7),,7), if tdt, as t \u2260 tk, we have dt \u2192 0, |\u03c8(dt)\u232a = (I \u2212 iHdt) | \u03c8(0)\u232a. Since \u2329\u03c8(0) | \u03c8f\u232a = 0, the inequality \u2329\u03c8f | \u03c8(dt)\u232a\u22600 is equivalent to \u2211l=1rul1\u2329\u03c8f | Hl | \u03c8(0)\u232a = \u2211l=1rul1(i\u2111\u2329\u03c8f | Hl | \u03c8(0)\u232a+\u211c\u2329\u03c8f | Hl | \u03c8(0)\u232a) \u2260 0. By (ul1\u2111\u2329\u03c8f | Hl | \u03c8(0)\u232a\u22650, for all l \u2208 J. It follows from the assumption in case (i) that there exists l \u2208 J such that ul1\u2111\u2329\u03c8f | Hl | \u03c8(0)\u232a>0, and consequently, \u2329\u03c8f | \u03c8(dt)\u232a\u22600. Since \u03c8(t) | \u03c8f\u232a\u22600, t \u2208 . By (\u03c8f | \u03c8(t1+)\u232a|\u2265|\u2329\u03c8f | \u03c8(t1)\u232a| = |\u2329\u03c8f | \u03c8(t1\u2212)\u232a|>0. Hence, we obtain that \u2329\u03c8(t) | \u03c8f\u232a\u22600, (t > 0).(i) For a sufficiently small  \u2260 0. By , we havet1). By  we have ul1(0) = 0, l \u2208 J. For a sufficiently small t* < t1, one can obtain that |\u03c8(t*)\u232a = eiH0t*\u2212 | \u03c8(0)\u232a = ei\u03bb0t*\u2212 | \u03c8(0)\u232a. Moreover, as dt \u2192 0, we have \u2329\u03c8f | \u03c8(t* + dt)\u232a = \u2329\u03c8f | (I \u2212 iHdt) | \u03c8(t*)\u232a = dt[\u2212i\u2009\u2009cos\u2061(\u03bb0t*) \u2212 sin(\u03bb0t*)]\u2211l=1r\u2329\u03c8f | Hlul1(t*) | \u03c8(0)\u232a. Noticing that there exists l \u2208 J such that \u2329\u03c8f | Hl | \u03c8(0)\u232a\u22600, we have ul1(t*) = Klfl[\u2111(ei\u03bb0t*\u2212\u2329\u03c8f | Hl | \u03c8(0)\u232a)] = Klfl[\u2212sin(\u03bb0t*)\u2329\u03c8f | Hl | \u03c8(0)\u232a] \u2260 0. Similar to the discussion in case (i), we obtain that \u2329\u03c8f | \u03c8(t* + dt)\u232a\u22600, and then \u2329\u03c8(t) | \u03c8f\u232a\u22600, t > t*. This completes the proof.(ii) Initially, the system evolves freely because For the characterization of invariant sets, properties of the states such that \u03c80 | \u03c8f\u232a = 0 and the conditions (i) or (ii) in t \u2260 tk,t \u2260 tk,\u03bbl \u2208 \u211d such that \u2329\u03c8f|(\u03bblI \u2212 Hl)|\u03c8(t)\u232a = 0, t \u2260 tk, l \u2208 J. there exists If \u2329V1 = 0. We first present the extensive LaSalle invariance principle for impulsive systems in [stems in . \ud835\udc9f:V such that V\u2032(x)fc(x) \u2264 0, t \u2260 tk and \u0394V(tk) \u2264 0, then x(t) \u2192 \u2133 as t \u2192 \u221e, where \u2133 is the largest invariant set contained in \ud835\udd3c\u225c{x : V\u2032(x)fc(x) = 0}\u2229{x : \u0394V(x) = 0}. Consider the following differential impulsive system on an open set Bk is designed such that it commutes with H0, k \u2208 \u2124+.  The following theorem presents the characterization of the invariant set for the nonideal systems under the hybrid impulsive control, by which the invariant set is smaller compared with that obtained by the conventional Lyapunov method. In the following, the unitary matrix G = \ud835\udd4an\u221212\u22c2E1\u22c2E2 with E1 = {|\u03c8\u232a : |\u03c8\u232a \u2208 Mkl, for\u2009\u2009all\u2009\u2009l \u2208 J, k = 1,2,\u2026}, E2 = {|\u03c8\u232a : |\u03c8\u232a \u2208 Nk, k = 1,2,\u2026}, andXl1, Xl2,\u2026, Xlml constitute the basis of the set {(i)s[H0s) = 0 for the whole system trajectory will be transformed to the conditions on the initial state. By the Taylor expansion and commutativity between H0 and Bk, it yields thatt = tk + dt, the free evolution of |\u03c8(t)\u232a is given bys = 0,1,\u2026, \u2211i=1kni. Noticing that the set {(i)s[H0s)ned thatD\u2212V1(tk)=random, (D\u2212V1(tk)=Mkl in , l \u2208 J, \u03c8\u232a : \u0394V1(tk) = 0, k = 1,2,\u2026}:Nk in =0 s[H0s)) = V2(t) = (1/2)\u2329\u03c8(t) \u2212 \u03c8f | \u03c8(t) \u2212 \u03c8f\u232a = 1 \u2212 \u211c\u2329\u03c8f | \u03c8(t)\u232a. Similar to the hybrid control design in \u03c9 is a new real scalar control field. For the convenience of the computation, the introduced \u03c9 may be used to adjust the global phase without changing the physical quantities regarding |\u03c8\u232a. While in practical implementation, it is not necessary to be implemented to the system. Similar conclusion can be drawn if there exists more than one control Hamiltonian Hl, l \u2265 2. The time derivative of V2 isu0 = \u03bbf + \u03c9. We design the following control to ensure t \u2260 tk:K1 > 0, and the function y1 = f1(\u00b7) is defined as that in  = 0, s = 1,\u2026, m1},X1, X2,\u2026, Xm1 are the basis of the set {I, (i)s[H0s)K = F1\u22c2F\u03c9 = \u2212\u03bbf. When the system satisfies u1 = 0, the evolution of system  = 0\u21d4\u2111(in1\u2329\u03c8f | [H0n1)\u232a) = 0. According to the similar method in the proof of t = tk, it yields thats = 0,\u2026, \u2211i=1kni. Denote the basis of the set {I, (i)s[H0s) = 0\u21d4\u2111\u2329\u03c8f | Xs\u220fj=k\u221211Bj | \u03c8(t0)\u232a = 0, s = 1,2,\u2026, m1. This equality is denoted as Uk in (Let f system  becomes system (i|\u03c8\u02d9(t)\u232a=)\u232a.From , we obta system (i|\u03c8\u02d9(t)\u232a=\u232a.From (V\u02d92(t0)=0lds thatD\u2212V2(tk)=s Uk in . \u03c8\u232a:\u0394V2(tk) = 0, k = 1,2,\u2026}. From the definition of V2, we have \u0394V2(t1) = \u211c\u2329\u03c8f | \u03c8(t1+)\u232a\u2212\u211c\u2329\u03c8f | \u03c8(t1)\u232a = \u211c\u2329\u03c8f | B1 | \u03c8(t1)\u232a\u2212\u211c\u2329\u03c8f | \u03c8(t1)\u232a. The following relations can be obtained:V2(tk) = 0\u21d4\u211c\u2329\u03c8f | \u03c8(t0)\u232a\u2212\u211c\u2329\u03c8f | \u220fj=1kBj | \u03c8(t0)\u232a = 0, which can be denoted as Wk in  = x, i = 1,2. Choose the impulsive instant to be tk = 3k \u2212 1, k \u2208 \u2124+ and K1 = K2 = 0.2. Using the hybrid impulsive control based on the state distance, simple computation yields that the invariant set G = {|3\u232a} , which implies that under the hybrid impulsive control the system converges to |\u03c8f\u232a. The populations of the controlled system are illustrated in Consider the five-level system with the internal Hamiltonian and impulsive control Hamiltonian given by Now we compare performance of the hybrid impulsive control with that of classical Lyapunov control. If the impulsive control is not applied to the system, then the hybrid impulsive control is reduced to the classical Lyapunov control, by which the performance of the controlled system is shown in \u03c0, 0}. The unitary operation is chosen as f1(x) = x, and K0 = 0.1, K1 = 0.2. Choose the impulsive instant as tk = 3k \u2212 1, k \u2208 \u2124+. Using the hybrid impulsive control based on the state error, simple computation yields that the invariant set K = {|5\u232a} , which implies that under the hybrid impulsive control the system converges to |\u03c8f\u232a. Simulation results are illustrated in Consider the five-level system with the same internal Hamiltonian as the previous example. Let the target state and the initial state be When the hybrid impulsive control is reduced to the classical Lyapunov control, the trajectory of the controlled system is plotted in H0 is not strong regular. The dynamical properties of the resulted quantum impulsive control system have been discussed to facilitate the convergence analysis. Based on two kinds of Lyapunov functions, the largest invariant sets have been characterized explicitly. Consequently, more accurate convergence of the controlled system has been achieved by the extensive LaSalle invariance principle. Compared with some existing results, the improved control performance has been shown for the nonideal case. Since the practical implementation of impulsive control has been studied in known literature, we believe that it is feasible. The optimal determination of the impulsive control Hamiltonian and impulsive instants is worth to be explored in the future work. In this paper, the coherent hybrid impulsive control for closed quantum systems has been investigated for the nonideal case that"}
{"text": "The existence of scattering operators and finite time blow-up of the solutions for the systems in higher space dimensions is also shown.The study of nonlinear Schr\u00f6dinger systems with quadratic interactions has attracted much attention in the recent years. In this paper, we summarize time decay estimates of small solutions to the systems under the mass resonance condition in 2-dimensional space. We show the existence of wave operators and modified wave operators of the systems under some mass conditions in  If we let uj = eitmjc2\u2212vj in  conservation law of solutions to  conservation law and as a result global existence in time of solutions to  is defined byH2m,s(\u211dn) = Hm,s(\u211dn) and Hm,0(\u211dn) = Hm(\u211dn) for simplicity. Lp(\u211dn) denotes the usual Lebesgue space with the normp < \u221e ands = [s] + \u03c3, 0 < \u03c3 < 1, \u03d5y(x) = \u03d5(x + y),\u2009\u20091 \u2264 p, q \u2264 \u221e, and [s] is the largest integer less than s. It is known that C be the space of continuous functions from an interval I to a Banach space E. Different positive constants might be denoted by the same letter C. The homogeneous Sobolev spaces x \u2208 \u211dn and \u03b4 \u2260 0 and define E = ei/2)t|\u03be|2\u2212( and M = ei/2t)|x|2\u2212( for t \u2260 0.We now introduce some function spaces to present exact statements of our results. For any \u211dn) (see ). We letU\u03b4(t) is written as\u2131, \u2131\u22121 are the Fourier transform and the inverse Fourier transform, respectively. We also haveJmj,k1/ = xk + i(t/mj)\u2202k = Umj1/(t)xkUmj1/(\u2212t), where j \u2208 {1,\u2026, l} and k \u2208 {1,\u2026, n} is an important tool to study time decay of solutions to nonlinear Schr\u00f6dinger equations satisfying the gauge invariant condition \u0394, we have commutation relations such thatEvolution operator tten as1(U\u03b4(t)\u03d5) \u2208 \u211d \u00d7 \u211d2, where mj is a mass of particle for j = 1,2, 3. In [The system  includes2, 3. In , the sysi\u2202tv1+12m2, 3. In , the stai\u2202tv1+12mn = 2 shown in [n = 2 based on a paper [n = 2 from a paper [This paper is organized as follows. shown in \u20137, 9. In a paper .  Sectio a paper . In the  a paper . In the n = 2, we start with time decay estimates of solutions to linear Schr\u00f6dinger systems.To state time decay of solutions to nonlinear Schr\u00f6dinger systems for j \u2264 l.For , the corvj(t) of (vj(t) = Umj1/(t)\u03d5j. It is known that vj(t) is decomposed into a main term and a remainder one asn = 2, where Rj decays rapidly in time; indeed we have the estimatet \u2260 0. By ||Umj1/(t)vj||L2(\u211d2) = ||vj||L2(\u211d2) and L\u221e(\u211d2) \u2212 L1(\u211d2) time decay estimatet \u2260 0, we have the following time decay estimates through the interpolation theorem (see [As we know the solution j(t) of  is reprerem (see ).p \u2264 \u221e, and let p, p\u2032 be conjugate indices, t \u2260 0. Then we have Umj1/(t) : Lp\u2032(\u211d2) \u2192 Lp(\u211d2) which are bounded operators and satisfyj = 1,\u2026, l.Let 2 \u2264 t, x) \u2208 \u211d \u00d7 \u211d2, where m1 and m2 are the masses of particles and \u03bb, \u03bc \u2208 \u2102. If we let u2 = (\u03bc/|\u03bb\u03bc|)v2 in the above system, then we obtain the system as belowt, x) \u2208 \u211d \u00d7 \u211d2, where \u03b3 = \u03bb\u03bc/|\u03bb\u03bc| \u2208 \u2102. Therefore we survey the results on time decay of solutions to the system i\u2202tas belowi\u2202tv1+12mm1 = m2 and \u03b3 = 1. Then there exists \u025b > 0 such that  \u2208 H2(\u211d2)\u2229H0,2(\u211d2) satisfyingt \u2208 \u211d.Assume that 2uch that  with theL\u221e(\u211d2) time decay estimates of solutions of  conservation law such thatWhen \u03b3 = 1,  satisfie\u03b3 = 1, . Under t  time decay of small solutions for (\u2131Umj1/(\u2212t)vj||L\u221e(\u211d2). The similar idea has been used for construction of H2,2(\u211d2) solutions to a single nonlinear Schr\u00f6dinger equation by a paper [m1 = m2 by using the factorization formulas of Schr\u00f6dinger evolution group stated in \u03c8j = Dmj1/\u2131Umj1/(\u2212t)vj for j = 1,2 and \u025b > 0. Asymptotic behavior in time of solutions of (O(t\u025b\u22121\u2212) which is integrable in time. This is the reason why we use the condition such that the data must be in H\u03b20,(\u211d2), \u03b2 > 1.In the case of the mass resonance condition 21 = m2,  satisfie1 = m2, . Global 1 = m2,  is obtaiions for  is prove a paper . Theoremtions of  is detera paper [i\u2202t\u03c81=\u03b3t\u2212The system  is a gen\u03d5 =  \u2208 H2,2(\u211d2) and Fj satisfies the conditions  \u2208 H2,2(\u211d2) satisfyingt \u2208 \u211d.One assumes that nditions  and 8)  \u03d5 =  by \u03d5 \u2208 H\u03b2(\u211d2)\u2229H\u03b20,(\u211d2) with 1 < \u03b2.t, x) \u2208 \u211d \u00d7 \u211d2, where m1, m2, m3 are the masses of particles and \u03bb1, \u03bb2, \u03bb3, \u03bc1, \u03bc2, \u03bc3 \u2208 \u2102\u2216{0} are constants.Now we focus on the following system:Time decay problem of solutions to  is consim1 + m2 = m3 is satisfied. One also assumes that Im\u2061\u03bbj \u2264 0 for j = 1,2, 3 and \u03ba1, \u03ba2, \u03ba3 > 0. Then there exists \u025b > 0 such that  \u2208 Hs(\u211d2)\u2229Hs0,(\u211d2) satisfyings < 2. Moreover, the time decay estimatet \u2208 \u211d.Assume that the mass resonance condition uch that  has a un\u03bbj | vj | vj acts as a dissipation one which requires logarithmic correction in time of solutions and the negative time is not considered. We have the following theorem.Ifv be the solution to the system  = \u03bc2vj; then vj,\u03bc(t) satisfies the system  = \u03bc2\u03d5j(\u03bcx). We haven = 2 in the function space \u03b1, \u03b2 satisfy 0 \u2264 \u03b2 < 1 < \u03b1 \u2264 2 and can be taken to be close to 1. Therefore our function space has relation with the invariant space and the considered data are not necessarily in L2(\u211d2).Define the scaled function by e system  with the problem . Other i problem  are givee system  for n = \u03b2 < 1 < \u03b1 \u2264 2. Then there exists \u025b > 0 such that  satisfyingt \u2260 0.Assume that  and 8)  8) hold.\u03d5=\u03d5,\u2026,\u03d5l\u2208H\u02d90Im\u2061\u03bbj < 0, where 0 \u2264 \u03b4 < 1 < \u03b1 \u2264 2. Then we have the time decay estimate such thatt \u2265 1.We note that in  we consit, x) \u2208 \u211d \u00d7 \u211d2, where m1, m2 are the masses of particles. It is obvious that the nonlinearities of (v1(0) = \u03d51 can be considered the Cauchy problem for the linear Schr\u00f6dinger equation, we find the value of v1 explicitly by v1 = Um11/(t)\u03d51. Therefore, we haveIn the final part of this subsection, we will show that the assumption  is impori\u2202tv1+12mities of . Since t we havei\u2202tv2+12m\u03d51 \u2208 H0,2(\u211d2),\u2009\u2009\u03d52 \u2208 L2(\u211d2). Lett > 1.Suppose that ution of . Then thThis fact was pointed first in , 23 in tv+ = (vi+), we assume that there exists a unique solution v = (vi) of the system (Um1/(t)v+ is the solution of linear problems with the initial data v+ and ||\u00b7||X is the norm of Banach space X. Then we define the map W+ : v+ \u21a6 v(t) and call it the wave operator. We also call v+ the final state  since it is considered the value of Um1/(\u2212t)v(t) at infinity if Um1/(t) is the unitary operator in X. The same problem can be considered for negative time.First, we briefly explain the definition of the wave operator (see ). For a e system  satisfyi\u03d51+, \u03d52+):t, x) \u2208 \u211d \u00d7 \u211d2. If there exists a nontrivial solution for the above system, then we say that there exists a usual wave operator.To study existence of wave operators for , we consm1 \u2260 m2 and m1 \u2260 m2.We consider  under thm1 \u2260 m2 and m1 \u2260 m2. Then there exists \u025b > 0 such that, for anyt \u2265 1, where 1/2 < b < 1.Let 2e system  has a unm1 \u2260 m2 and m1 \u2260 m2 are used to obtain better time decay of solutions. Oscillating properties of nonlinear terms v1+2 are different from those of solutions to linear problem which yield an additional time decay from nonlinear terms; namely, nonlinear interactions are not critical. By combining this fact and the Strichartz type estimates, the result of The mass nonresonance conditions 2m1 = m2 which is also the mass nonresonance case and the support conditions on the data.We next consider  under thm1 = m2. Assume that\u025b > 0 such that, for any \u03d5+ =  with the normt \u2265 1, where 1/2 < b < 3/4.Let e system  satisfyiFrom the result we have wave operators when the support of the Fourier transform of the Schr\u00f6dinger data is restricted. Restriction on the support of the Fourier transform of the Schr\u00f6dinger data was used to obtain an improved time decay estimate of the nonlinear term We turn to investigate existence of wave operators for . First, \u03d5 =  \u2208 H2,2(\u211d2) and let v be global in time of solutions of  of the system  from a suitable function space and define a function f by v+. Then we try to find a unique solution of nonlinear problems under the asymptotic conditionX is the norm of Banach space X. Namely, the problem is solved if we can define the function f satisfying the asymptotic condition (MW+ : v+ \u21a6 v(t) instead of the wave operator. We call MW+ the modified wave operator since we modified the final states.In onditionlim\u2061t\u2192\u221e||t, x) \u2208 \u211d \u00d7 \u211d2. In m1 = m2 and \u03b3 = 1. Since the nonlinearity is critical in this case, it is impossible to find a solution in the neighborhood of the free final state (Um11/(t)\u03d51+, Um21/(t)\u03d52+). Indeed we have the nonexistence of the usual scattering states.We consider  again whm1 = m2, \u03b3 = 1, and let\u03d5+ =  \u2208 H2(\u211d2)\u2229H0,2(\u211d2) such that \u03d51+ \u2260 0 andt \u2192 \u221e.Let 2From the result, we need to modify the final state with time dependence. We note here that the modified wave operator for nonlinear dispersive equation was first constructed in  for the m1 = m2 is determined by the solutions of the following system:In \u03b3 = \u22121 and the particular solutions of  > 0 and \u03b8(\u03be) is a real valued given function. We also find that\u03b3 = \u00b11.By calculation we find that the particular solutions of  are(75)tions of  are(76)tions of  when \u03b3 =The following theorem shows existence of the modified wave operators of .m1 = m2 and \u03b3 = \u00b11. Then there exists \u025b > 0 such that, for any \u03c9, \u03b8 \u2208 H2(\u211d2) with norm ||\u03c9||H2(\u211d2) \u2264 \u025b,  \u2264 \u025b,  has a unm1 = m2, we get the existence of modified wave operators by the contraction mapping principle. Since the identity Umj1/(t) = Mmj\u2212Dt/mj\u2131Mmj\u2212 is known for j = 1,2, we have the estimate from the above theoremt \u2265 1, where 1/2 < b < 1.Using the resonance condition, 2Existence of modified wave operator for a single nonlinear Schr\u00f6dinger equation was studied in , 27. Asyn \u2265 3, the scattering theory for i\u2202tvW+ which maps a Banach space X into itself. Namely, for any given v+ \u2208 X, we assume that there exists a unique solution v(t) \u2208 C; X) of the nonlinear system such thatv(0) which are determined by the solution v(t) in the time interval t \u2208 [0, \u221e). If the initial value problem has a unique global solution v(t) \u2208 C and we can find a unique v\u2212 \u2208 X from the solution v(t) satisfyingW\u2212\u22121 : v(0) \u2208 X \u21a6 v\u2212 \u2208 X. From this operator we can define S = W\u2212\u22121W+ : X \u2192 X. We call the operator the scattering operator.We will explain the scattering problem (See ) brieflyn \u2265 4, existence of the scattering operator was proved in the space Hn/2)\u22122((\u211dn) which is close to the invariant space n = 4, we have the results in the invariant space L2(\u211d4). In the case of n = 3, existence of the scattering operator was proved in the space H0,1/2(\u211d3), under the mass resonance condition 2m1 = m2, which is close to the invariant space In the case, n \u2265 4. Then there exist \u03b50 and C0 such that 0 < \u03b50 \u2264 1 \u2264 C0 with the following property.Let \u025b with 0 < \u025b \u2264 \u03b50 and any \u03d5 =  \u2208 B\u025b,  \u2208 C\u22122((\u211dn)). Moreover, there exist unique \u03d5\u00b1 =  \u2208 BC0\u025b such thatt \u2192 \u00b1\u221e.(1) For any ) \u2208 B\u025b,  has a un+ For any \u025b with 0 < \u025b \u2264 \u03b50 and any \u03d5+ =  \u2208 B\u025b,  \u2208 C\u22122((\u211dn)) such that v(0) = (v1(0), v2(0)) \u2208 BC0\u025b,t \u2192 +\u221e.(2)) \u2208 B\u025b,  has a un\u2212 For any \u025b with 0 < \u025b \u2264 \u03b50 and any \u03d5\u2212 =  \u2208 B\u025b,  \u2208 C\u22122((\u211dn)) such that v(0) = (v1(0), v2(0)) \u2208 BC0\u025b,t \u2192 \u2212\u221e.(2)) \u2208 B\u025b,  has a unW\u00b1 : \u03d5\u00b1 \u21a6 v(0) are defined as mappings from B\u025b to BC0\u025b\u2009\u2009for any \u025b with 0 < \u025b \u2264 \u03b50. The scattering operator S : \u03d5+ \u21a6 \u03d5\u2212 is defined as a mapping from BC0\u22121\u025b to BC0\u025b for any \u025b with 0 < \u025b \u2264 \u03b50.To state the following theorem, we introduceThe wave operators m1 = m2. Then there exist \u03b50 and C0 such that 0 < \u03b50 \u2264 1 \u2264 C0 with the following property.Let\u2009\u20092\u025b with 0 < \u025b \u2264 \u03b50 and any v with Um1/(\u2212t)v \u2208 C). Moreover, there exist unique t \u2192 \u00b1\u221e.(1) For any y \u03d5\u2208B~\u025b,  has a un+ For any \u025b with 0 < \u025b \u2264 \u03b50 and any v with Um1/(\u2212t)v \u2208 C) such that t \u2192 +\u221e.(2) \u03d5+\u2208B~\u025b,  has a un\u2212 For any \u025b with 0 < \u025b \u2264 \u03b50 and any v with Um1/(\u2212t)v \u2208 C) such that t \u2192 \u2212\u221e.(2) \u03d5-\u2208B~\u025b,  has a unW\u00b1 : \u03d5\u00b1 \u21a6 v(0) are defined as mappings from \u025b with 0 < \u025b \u2264 \u03b50. The scattering operator S : \u03d5+ \u21a6 \u03d5\u2212 is defined as a mapping from \u025b with 0 < \u025b \u2264 \u03b50.The wave operators n \u2264 6 under the mass conditions 2m1 = m2 and \u03b3 \u2208 \u211d. To state the blow-up result we need local existence in time of solutions to  > 0 such that i\u2202tvrsion of 100)\u2202t2\u2202t2(99)i\u2202 = 1 in (\u2202t2u1\u2212\u0394u1n papers , 35 with a paper  if m2 <ator for  is an opained in  for the  = 1 in (\u2202t2u1\u2212\u0394u1Small data blow-up for a system of nonlinear Schrodinger equations was studied in  under soThe asymptotic behavior of solutions to quadratic derivative nonlinear Schr\u00f6dinger systems has been considered recently in papers , 40 unde"}
{"text": "We prove the Hyers-Ulam stability for this quartic functional equation by the directed method and the fixed point method on real Banach spaces. We also investigate the Hyers-Ulam stability for the mentioned quartic functional equation in non-Archimedean spaces.We obtain the general solution of the generalized quartic functional equation  \u2131 is stable if any function f satisfying the equation \u2131\u2009\u2009approximately is near to exact solution of \u2131. Moreover, a functional equation \u2131 is hyperstable if any function f satisfying the equation \u2131\u2009\u2009approximately is a true solution of \u2131.We say a functional equation The study of stability problems for functional equations is related to a question of Ulam  concernif(2x) = 16f(x), we getThe oldest quartic functional equation was introduced by Rassias in  and thenf : \u211d \u2192 \u211d is a solution of (f(x) = Q where the function Q : \u211d4 \u2192 \u211d is symmetric and additive in each variable. The fact that every solution of +fdirect method,\u201d initiated by Hyers in 1941, had been applied by C\u0103dariu and Radu for the first time in [Bodaghi et al.  applied  time in . In othe time in  and of t time in  (for mor time in \u201320).m. In case m = 2, then , an, an3), af(x+my)+ff(x+my)+ff(x+my)+fTo achieve our aim in this section, we need the following lemma.\ud835\udcb3 and \ud835\udcb4 be real vector spaces. If a function f : \ud835\udcb3 \u2192 \ud835\udcb4 satisfies the functional equation  = m4f(x) for all integers m \u2265 2.Let equation  for all x = y = 0 in (f(0) = 0. Once more, by putting y = 0 in  = 34f(x). Now, assume that, for every k \u2264 m \u2212 1, we have f(kx) = k4f(x). If m = 2n, then f(mx) = f(2nx) = 24f(nx). Since n \u2264 m \u2212 1, we have f(nx) = n4f(x), and thus f(mx) = m4f(x). Let m = 2n + 1. Then, by substituting x,\u2009\u2009y by mx,\u2009\u2009x in  and m = 2n. Replacing these equalities in (f(mx) = m4f(x). This completes the proof.Letting y = 0 in , we get y = 0 in , we obtax,\u2009\u2009x in , respect = 0 in (f(2x)=24fx,\u2009\u2009x in , respect = 0 in (f(2x)=24ff : \ud835\udcb3 \u2192 \ud835\udcb4 satisfies the functional equation (f satisfiesIt is shown in [equation  if and o)\u2003=2m2m2\u2212f(y)\u22122(m2atisfiesf(x+my)+fequation .\ud835\udcb3 and \ud835\udcb4 be real vector spaces. Then a mapping f : \ud835\udcb3 \u2192 \ud835\udcb4 satisfies the functional equation  = 0. Let y = 0 in (f(2x) = 16f(x) for all x \u2208 \ud835\udcb3. Setting x = 0 in (f(y) = 16f(y), we have f(\u2212y) = f(y). Letting y = x in (f(3x) = 81f(x) for all x \u2208 \ud835\udcb3. By induction, we obtain f(kx) = k4f(x) for all positive integers k. Replacing x by x + y and x \u2212 y in . Hence for each x, y \u2208 \ud835\udcb3, we havef((m \u2212 1)y) = (m \u2212 1)4f(y) and sof satisfies the functional equation  and sof such thatx, y \u2208 \ud835\udcb3, where m is an integer with m \u2265 2. Then there exists a unique quartic mapping \ud835\udcaf : \ud835\udcb3 \u2192 \ud835\udcb4 such thatx \u2208 \ud835\udcb3.Let y = 0 in /2n4} is Cauchy by  = 0 for all positive integers\u2009m \u2265 2 and all x, y \u2208 \ud835\udcb3. Hence, by \ud835\udcaf : \ud835\udcb3 \u2192 \ud835\udcb4 is a quartic mapping. Now, let \ud835\udcaf\u2032 : \ud835\udcb3 \u2192 \ud835\udcb4 be another quartic mapping satisfying y = 0 in y 2x in (||f(2nx)2 so thatlim\u2061n\u2192\u221ef = \u03b2||x||\ud835\udcb3r + \u03b3||y||\ud835\udcb3s in Setting ows from  that such thatx, y \u2208 \ud835\udcb3, where m is an integer with m \u2265 2. Then there exists a unique quartic mapping \ud835\udcaf : \ud835\udcb3 \u2192 \ud835\udcb4 such thatx \u2208 \ud835\udcb3.Suppose that \u03d5 = 0. Thus from (f(0) = 0. Putting y = 0 in 2, we havex \u2208 \ud835\udcb3. If we show that the sequence {2n4f(x/2n)} is Cauchy, then it will be convergent by the completeness of \ud835\udcb4. For this, if we replace x by x/2l in \u221224\u03b2, \u03b3, r, and s be nonnegative real numbers such that r, s > 4. Suppose that f : \ud835\udcb3 \u2192 \ud835\udcb4 is a mapping fulfillingx, y \u2208 \ud835\udcb3, where m is an integer with m \u2265 2. Then there exists a unique quartic mapping \ud835\udcaf : \ud835\udcb3 \u2192 \ud835\udcb4 such thatx \u2208 \ud835\udcb3.Let x = y = 0 in (f(0) = 0. Taking \u03d5 = \u03b2||x||\ud835\udcb3r + \u03b3||y||\ud835\udcb3s in First, we note that if we put y = 0 in , we haveWe are going to investigate the hyperstability of the given quartic functional equation  by usingd) be a complete generalized metric space and let \ud835\udca5 : \u0394 \u2192 \u0394 be a mapping with Lipschitz constant L < 1. Then, for each element \u03b1 \u2208 \u0394, either d = \u221e for all n \u2265 0 or there exists a natural number n0 such thatd < \u221e for all n \u2265 n0;\ud835\udca5n\u03b1} is convergent to a fixed point \u03b2* of \ud835\udca5;the sequence {\u03b2* is the unique fixed point of \ud835\udca5 in the set \u03941 = {\u03b2 \u2208 \u0394 : d < \u221e};d\u2264(1/(1 \u2212 L))d for all \u03b2 \u2208 \u03941.Let  = 0 and let \u03c6 : \ud835\udcb3 \u00d7 \ud835\udcb3 \u2192 [0, \u221e) be a function such thatx, y \u2208 \ud835\udcb3, where m is an integer with m \u2265 2. If there exists a constant M \u2208 , such thatx, y \u2208 \ud835\udcb3, then there exists a unique quartic mapping \ud835\udcaf : \ud835\udcb3 \u2192 \ud835\udcb4 such thatx \u2208 \ud835\udcb3.Let \ud835\udd07 on \u0394 \u00d7 \u0394 as follows:C, and \ud835\udd07 = \u221e, otherwise. In a similar way to the proof of [\ud835\udd07 is a generalized metric on \u0394 and the metric space  is complete. Here, we define the mapping \ud835\udca5 : \u0394 \u2192 \u0394 byg, h \u2208 \u0394 such that \ud835\udd07 < C, by definitions of \ud835\udd07 and \ud835\udca5, we havex \u2208 \ud835\udcb3. Using  \u2264 M\ud835\udd07 for all g, h \u2208 \u0394. Hence, \ud835\udca5 is a strictly contractive mapping on \u0394 with a Lipschitz constant M. We now show that \ud835\udd07 < \u221e. Putting y = 0 in  < \u221e for all n \u2265 0, and thus in this theorem we have n0 = 0. Consequently, the parts (iii) and (iv) of \ud835\udcaf : \ud835\udcb3 \u2192 \ud835\udcb4 such that \ud835\udcaf is a fixed point of \ud835\udca5 and that \ud835\udca5nf \u2192 \ud835\udcaf as n \u2192 \u221e. Thusx \u2208 \ud835\udcb3, and sox \u2208 \ud835\udcb3. Now, it follows from  = 0 for all integers\u2009\u2009m \u2265 2\u2009\u2009and all\u2009\u2009x, y \u2208 \ud835\udcb3. It follows from \ud835\udcaf : \ud835\udcb3 \u2192 \ud835\udcb4 is a quartic mapping which is unique.We wish to make the conditions of \ud835\udcb3. Using , we gety = 0 in , we obtahow that  is true ows from  that = 0. If we put \u03c6 = \u03b1||x||\ud835\udcb3r + \u03b2||y||\ud835\udcb3r in Note that inequality  implies In the next result, we prove the hyperstability of quartic functional equations under some conditions.r, s, and \u03b1 be nonnegative real numbers with 0 < r + s \u2260 4 and let f : \ud835\udcb3 \u2192 \ud835\udcb4 be a mapping such thatx, y \u2208 \ud835\udcb3. Then f is a quartic mapping on \ud835\udcb3.Let x = y = 0 in (f(0) = 0. Again, if we put y = 0 in (f(2x) = 16f(x) for all x \u2208 \ud835\udcb3. It is easy to check that f(2nx) = 2n4f(x), and so f(x) = f(2nx)/2n4 for all x \u2208 \ud835\udcb3 and n \u2208 \u2115. Now, it follows from f is a quartic mapping when \u03c6 = \u03b1||x||\ud835\udcb3r||y||\ud835\udcb3s.Putting y = 0 in , we get y = 0 in , then we\ud835\udd42 equipped with a function  |\u00b7| from \ud835\udd42 into [0, \u221e) such that |r| = 0 if and only if r = 0, |rs| = |r||s|, and |r + s|\u2264 max{|r|, |s|} for all r, s \u2208 \ud835\udd42. Clearly |1| = |\u22121| = 1 and |n| \u2264 1 for all n \u2208 \u2115.We recall some basic facts concerning non-Archimedean spaces and some preliminary results. By a non-Archimedean field we mean a field \ud835\udcb3 be a vector space over a scalar field \ud835\udd42 with a non-Archimedean nontrivial valuation |\u00b7|. A function ||\u00b7|| : \ud835\udcb3 \u2192 \u211d is a non-Archimedean norm  if it satisfies the following conditions:(i)x|| = 0 if and only if x = 0;||(ii)rx|| = |r|||x||, ;||(iii)the strong triangle inequality (ultrametric); namely,Then  is called a non-Archimedean space. Due to the fact thatxn} is Cauchy if and only if {xn+1 \u2212 xn} converges to zero in a non-Archimedean normed space \ud835\udcb3. By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent.Let p-adic numbers as a number theoretical analogue of power series in complex analysis. The most interesting example of non-Archimedean spaces is p-adic numbers. A key property of p-adic numbers is that they do not satisfy the Archimedean axiom: for all x, y > 0, there exists an integer n such that x < ny.In , Hensel p be a prime number. For any nonzero rational number x = pr(m/n) in which m and n are coprime to the prime number p. Consider the p-adic absolute value |x|p = pr on \u211a. It is easy to check that |\u00b7| is a non-Archimedean norm on \u211a. The completion of \u211a with respect to |\u00b7| which is denoted by \u211ap is said to be the p-adic number field. One should remember that if p > 3, then |2n| = 1 for all integers n. In [Let rs n. In , the stars n. In ).\ud835\udcb3 is a normed space and \ud835\udcb4 is a complete non-Archimedean space unless otherwise stated explicitly. In the upcoming theorem, we prove the stability of the functional equation  such thatx, y \u2208 \ud835\udcb3. Suppose that f : \ud835\udcb3 \u2192 \ud835\udcb4 is a mapping satisfying the equalityx, y \u2208 \ud835\udcb3, where m is an integer with m \u2265 2. Then there exists a unique quartic mapping Q : \ud835\udcb3 \u2192 \ud835\udcb4 such thatx \u2208 \ud835\udcb3 where Let y = 0 in /16n} is Cauchy by \u22121auchy by  and 69)y = 0 in  we have||f(x)\u2212f, a, ay = 0  so thatlim\u2061n\u2192\u221ef\u2192[0, \u221e) be a function satisfying \u0393(|r|s) \u2264 \u0393(|r|)\u0393(s) for all r, s \u2208 [0, \u221e) for which \u0393(|2|) < |16|. Suppose that f : \ud835\udcb3 \u2192 \ud835\udcb4 is a mapping satisfying the inequalityx, y \u2208 \ud835\udcb3, where m is an integer with m \u2265 2. Then there exists a unique quartic mapping Q : \ud835\udcb3 \u2192 \ud835\udcb4 such thatx \u2208 \ud835\udcb3.Let \u03d5 : \ud835\udcb3 \u00d7 \ud835\udcb3 \u2192 [0, \u221e) by \u03d5 = \u03b1(\u0393(||x||) + \u0393(||y||)), we havex, y \u2208 \ud835\udcb3. We also havex \u2208 \ud835\udcb3. Now, Defining We have the following result which is analogous to \u03d5 : \ud835\udcb3 \u00d7 \ud835\udcb3 \u2192 [0, \u221e) such thatx, y \u2208 \ud835\udcb3. Suppose that f : \ud835\udcb3 \u2192 \ud835\udcb4 is a mapping satisfying the inequalityx, y \u2208 \ud835\udcb3, where m is an integer with m \u2265 2. Then there exists a unique quartic mapping Q : \ud835\udcb3 \u2192 \ud835\udcb4 such thatx \u2208 \ud835\udcb3 where Let x \u2208 \ud835\udcb3. If we replace x by x/2n+1 in the above inequality and multiply both sides of } is Cauchy. Since the non-Archimedean space \ud835\udcb4 is complete, this sequence converges in \ud835\udcb4 to the mapping Q. Indeed,x \u2208 \ud835\udcb3 and non-negative integers n. Since the right hand side of inequality \u22121ude from  and 82)81)||f(2||f(2(81)ion and (||f(x)\u221216 Indeed,Q(x)=lim\u2061applying . Now, in\u03b1 > 0, \ud835\udcb3 be a non-Archimedean space and let \u0393 : [0, \u221e)\u2192[0, \u221e) be a function satisfying \u0393(|r|s) \u2264 \u0393(|r|)\u0393(s) for all r, s \u2208 [0, \u221e) for which \u0393(|2|\u22121) < |16|\u22121. Suppose that f : \ud835\udcb3 \u2192 \ud835\udcb4 is a mapping satisfying the inequalityx, y \u2208 \ud835\udcb3, where m is an integer with m \u2265 2. Then there exists a unique quartic mapping Q : \ud835\udcb3 \u2192 \ud835\udcb4 such thatx \u2208 \ud835\udcb3.Let The proof is a direct consequence of"}
{"text": "A kth-order slant weighted Toeplitz operator on L2(\u03b2) is given by U\u03d5 = WkM\u03d5, where M\u03d5 is the multiplication on L2(\u03b2) and Wk is an operator on L2(\u03b2) given by Wkenk(z) = (\u03b2n/\u03b2nk)en(z), {en(z) = zk/\u03b2k}k\u2208\u2124 being the orthonormal basis for L2(\u03b2). In this paper, we define a kth-order slant weighted Toeplitz matrix and characterise U\u03d5 in terms of this matrix. We further prove some properties of U\u03d5 using this characterisation.Let  H2 and L2 spaces. During the past few decades, different generalisations of these spaces, like the weighted function spaces HwP and LwP and the weighted sequence spaces H2(\u03b2) and L2(\u03b2) have developed. Shields [L2(\u03b2). Lauric [H2(\u03b2). Motivated by these studies, we introduced and studied the notion of a slant weighted Toeplitz operator [L2(\u03b2). In this paper, we study a kth-order slant weighted Toeplitz operator on the space L2(\u03b2). We begin with the following preliminaries.Toeplitz operators and slant Toeplitz operators  have bee Shields  has made. Lauric  has discoperator  on L2 \u2264 1 when n \u2265 0 and 0 < (\u03b2n/\u03b2n\u22121) \u2264 1 when n \u2264 0.Let k \u2265 2, \u03b2kn/\u03b2n \u2264 M < \u221e. Consider the following spaces [Throughout the paper, we assume that for a fixed integer g spaces , 6:(1)LL2(\u03b2), ||\u00b7||\u03b2) is a Hilbert space [ek(z) = zk/\u03b2k}k\u2208\u2124 and with an inner product defined byH2(\u03b2) is a subspace of L2(\u03b2). Further, the spaceP : L2(\u03b2) \u2192 H2(\u03b2) is the orthogonal projection of L2(\u03b2) onto H2(\u03b2). For a given \u03d5 \u2208 L\u221e(\u03b2), the induced weighted multiplication operator [M\u03d5 and is given by M\u03d5 : L2(\u03b2) \u2192 L2(\u03b2) such thatThen, (rt space  with an operator  is denot\u03d51(z) = z, then M\u03d51 = Mz is the operator defined as Mzek(z) = wkek+1(z), where wk = \u03b2k+1/\u03b2k for all k \u2208 \u2124, and it is known as a weighted shift [If we put ed shift .A\u03d5 is an operator on L2(\u03b2) defined as A\u03d5 : L2(\u03b2) \u2192 L2(\u03b2) such that A\u03d5ek(z) = (1/\u03b2k)\u2211n=\u2212\u221e\u221ean\u2212k2\u03b2nen(z).The slant weighted Toeplitz operator  A\u03d5 is aW : L2(\u03b2) \u2192 L2(\u03b2) is such thatA\u03d5 can be alternately defined byIf Wk : L2(\u03b2) \u2192 L2(\u03b2) is such thatWk isWk is given bySuppose that the operator k \u2265 2, the kth order slant weighted Toeplitz operator U\u03d5 : L2(\u03b2) \u2192 L2(\u03b2) is such that U\u03d5(f) = WkM\u03d5(f) for all f \u2208 L2(\u03b2). Thus, U\u03d5ej(z) = (1/\u03b2j)\u2211n=\u2212\u221e\u221eank\u2212j\u03b2nen(z).For an integer i, j)th entry of the matrix of U\u03d5 is given byU\u03d5 with respect to this basis is as follows:The en(z) is not divisible by k. But since this is true for all n \u2208 \u2124, the second possibility is ruled out. Hence, \u03d5 \u2212 \u03c8 = 0 or \u03d5 = \u03c8.Hence,MzWk = WkMzk; (ii) MzmWk = WkMzkm, m \u2208 \u2124. Consider the following: (i) (i) It is sufficient to prove thatn is not a multiple of k. Then,Suppose that n = pk (multiple of k),Now, when On the other hand,n \u2208 \u2124,Hence from  and 18)18), we gm. (ii) We prove the result by induction on m = 1, the result is true by part (i). For m = l. Then,Suppose that the result is true for Nowm \u2208 Z+.Hence by induction, the result is true for all m = \u22121 and 1 \u2264 j \u2264 k \u2212 1,m = \u22121 and j = nk (a multiple of k),m = \u22121 also. Therefore, by using induction, we can prove it for all negative integers m. The case when m = 0 is trivially true.For \u2264 k \u2212 1,MzmWkenk+e of k),MzmWkej = \u2211n=\u2212\u221e\u221eanzn \u2208 L\u221e(\u03b2). Then,Let M\u03d5(z)U\u03c8(z) = U\u03d5(zk)\u03c8(z).Consider the following: The proof is as follows:wn = \u03b2n+1/\u03b2n for all n \u2208 \u2124. Then, the kth order slant weighted Toeplitz matrix corresponding to the weight sequence \u2329wn\u232a is a bilaterally infinite matrix \u2329\u03bbij\u232a such thatLet U is a kth order slant weighted Toeplitz operator if and only if MzU = UMzk.We have proved earlier  that U ikth order slant weighted Toeplitz operator in terms of the matrix previously defined.Next, we prove a characterisation of the U on L2(\u03b2) is a kthorder slant weighted Toeplitz operator is that its matrix with respect to the orthonormal basis {ek(z) = zk/\u03b2k}k\u2208\u2124 is a kthorder slant weighted Toeplitz matrix.A necessary and sufficient condition that an operator U is a kth order slant weighted Toeplitz operator. Then, the corresponding matrix \u2329\u03bbij\u232a is given bywn = \u03b2n+1/\u03b2n for all n \u2208 \u2124. Thus, the matrix of\u2009\u2009U is a kth order slant weighted Toeplitz matrix. Conversely, we assume that U is an operator on L2(\u03b2) whose matrix is a kth order slant weighted Toeplitz matrix. This means that for all i, j \u2208 \u2124, we haveMzU = UMzk. Therefore, we conclude that U is a kth order slant weighted Toeplitz operator.Suppose that S : L2(\u03b2) \u2192 L2(\u03b2) given by Sej = (1/wj)ej+1. Then, S*ej = (1/wj\u22121)ej\u22121.Next, consider the operator S is bounded as \u2329wn\u232a is always positive and bounded.Now, S* = Mz\u22121.Consider the following: The proof is as follows:U on L2(\u03b2) is a kthorder slant weighted Toeplitz operator if and only if\u2009\u2009U = Mz\u22121UMzk, where Mz and Mzk, are the multiplication operators induced by z and zk, respectively.A bounded operator U be a kth order slant weighted Toeplitz operator on L2(\u03b2). Then, from  satisfying U = Mz\u22121UMzk for some fixed integer k \u2265 2. Then, for all i, j \u2208 \u2124,U is a kth order slant weighted Toeplitz matrix. Hence, by U is a kth order slant weighted Toeplitz operator.For the converse part, we assume that k \u2265 2, the set of all kthorder slant weighted Toeplitz operators is weakly closed and hence strongly closed.For a fixed integer n, Un is a kth order slant weighted Toeplitz operator and Un \u2192 U weakly. Then, for f, g \u2208 L2(\u03b2), we get that \u2329Unf, g\u232a\u2192\u2329Uf, g\u232a.We assume that for each positive integer n, Un = Mz\u22121UnMzk.From the previous theorem, this implies thatU = Mz\u22121UMzk. Therefore, U is a kth order slant weighted Toeplitz operator.Hence, kth order slant weighted Toeplitz operator on H2(\u03b2) is denoted by V\u03d5 and defined as V\u03d5ej(z) = (1/\u03b2j)\u2211n=0\u221eank\u2212j\u03b2nen(z).\u2009\u2009j = 0,1, 2,\u2026.The compression of a T\u03d5 : H2(\u03b2) \u2192 H2(\u03b2) is the Toeplitz operator on H2(\u03b2) induced by \u03d5 \u2208 L\u221e(\u03b2). The matrix of V\u03d5 is unilaterally infinite and has the form\u03d5 = \u2211n=\u2212\u221e\u221eanzn.Alternatively,\u03d5 \u2192 V\u03d5 is linear and one to one. The mapping T\u03d5Wk = PM\u03d5Wk = PWkM\u03d5(zk) = PU\u03d5(zk) = V\u03d5(zk).Further, we observe the following."}
{"text": "This paper studies impulsive control systems with finite andinfinite delays. Several stability criteria are established by employingthe largest and smallest eigenvalue of matrix. Our sufficient conditionsare less restrictive than the ones in the earlier literature. Moreover,it is shown that by using impulsive control, the delay systems can bestabilized even if it contains no stable matrix. Finally, some numericalexamples are discussed to illustrate the theoretical results. As is known, a well-developed theory of impulsive control systems has come into existence x(t \u2212 u)du, where s \u2208  is a Volterra type functional. In this case, when \u03b1 = \u2212\u221e, the interval  is understood to be replaced by  \u2208 C, and \u222b0r|h(u)|du < \u221e.Consider the delay control systemtk, U)} (refer the reader to  \u2192 \u211dn is continuous.Let \u03d5 :  \u2192 \u211dn is continuous and \u222b0\u221e+|h(u)|du < \u221e.Furthermore, we can investigate the following impulsive control system with infinite delays:A \u2208 \u211dn\u00d7n, \u03bbmax\u2061(A) denotes the largest eigenvalue of A and similarly \u03bbmin\u2061(A) denotes the smallest eigenvalue of A. The norm of matrix is A \u2208 \u211dn\u00d7n, A, which is defined to be the unique positive definite matrix satisfying For any matrix A=A (see ).Let us define the following class of functions for later use:In order to prove our main results, we need the following definitions and lemmas.A \u2208 \u211dn\u00d7n is stable (or Hurwitz) if the eigenvalues of the matrix A all have negative real parts and there is a unique positive definite matrix P \u2208 \u211dn\u00d7n that solves the Lyapunov equationQ is any n \u00d7 n positive definite matrix.A matrix A \u2208 \u211dn\u00d7n is unstable if the eigenvalues of the matrix A all have positive real parts and there is a unique positive definite matrix P \u2208 \u211dn\u00d7n that solves the Lyapunov equationQ is any n \u00d7 n positive definite matrix.A matrix x(t) = x is the solution of . Then the trivial solution of (H1) > 0 such that ||\u03d5|| < \u03b4 implies ||x|| < \u025b, t \u2265 t0;stable, if for any H2)(\u03b4 in (H1) is independent of t0;uniformly stable, if the H3)(H2) holds and there exists some \u03b7 > 0 such that for any \u025b > 0 there exists some T = T > 0 such that t0 \u2265 0 and ||\u03d5|| < \u03b7 together imply ||x|| < \u025b, t \u2265 t0 + T.uniformly asymptotically stable, if (Assume that ution of  and 4)  x(t) = xution of  and (4) a, b, c \u2208 K1, p \u2208 PC, and s > 0. Suppose further that V : ;tk, \u03c8) \u2208 \u211d+ \u00d7 PC) for which \u03c8(0\u2212) = \u03c8(0);\u03c4 = inf\u2061k\u2208N{tk \u2212 tk\u22121} > 0,\u2009\u2009M2 = sup\u2061q>0\u222bqg(q)(ds/c(s)) < \u221e, and M1 = inf\u2061t\u22650\u222btt+\u03c4p(s)ds > M2.Then the trivial solution of , and g \u2208 K2 and V : ;V + I) \u2264 g)) for all  \u2208 \u211d+ \u00d7 PC) for which \u03c8(0\u2212) = \u03c8(0);\u03c4 = sup\u2061k\u2208N{tk \u2212 tk\u22121} < \u221e, M1 = sup\u2061t\u22650\u222btt+\u03c4p(s)ds < \u221e, and M2 = inf\u2061q>0\u222bg(q)q(ds/c(s)) > M1.Then the trivial solution of , and g \u2208 K2 and V : ;0, \u03d5) of , wheneveV) \u2264 g) for all k \u2208 N and all x \u2208 S(\u03c11);M1 = sup\u2061k\u2208N\u222btktk+1p(s)ds < \u221e and M2 = inf\u2061u>0\u222bg(u)u(ds/c(s)) > M1.Then the trivial solution of ..4). Our \u03c4 = inf\u2061k\u2208N{tk \u2212 tk\u22121} > 0 and assume all the eigenvalues of A have negative real parts. Suppose that there exist a constant \u03b3 \u2265 1 and two symmetrical positive definite matrices P, Q \u2208 \u211dn\u00d7n such that\ud835\udd44 = \u222b0\u03b3|h(u)|du. Then the trivial solution of  = \u03b22s and g(V(\u03c8(0))) \u2265 V(\u03c8(s)), s \u2208 , so we have \u03b22V(\u03c8(0)) \u2265 V(\u03c8(\u2212r)) and \u03b22V(\u03c8(0)) \u2265 V(\u03c8(\u2212u)). Calculating the derivative of V along solutions of ,1ln\u2061\u03b2\u03c4\u2212\u03bbmimax\u2061(P),ln\u2061\u03b2\u03c4\u2212\u03bbmiax\u2061(P),1ln\u2061\u03b2\u03c4\u2212\u03bbmiA is unstable.Our next theorem establishes conditions for uniform asymptotic stability of trivial solution of  when mat\u03c4 = sup\u2061k\u2208N{tk \u2212 tk\u22121} < \u221e and assume all the eigenvalues of A have positive real parts. Suppose that there exist a constant 0 < \u03b3 < 1 and two symmetrical positive definite matrices P, Q \u2208 \u211dn\u00d7n such that\ud835\udd44 = \u222b0\u03b3|h(u)|du. Then the trivial solution of  = \u03b32s. Then if V(\u03c8(0)) \u2265 g(V(\u03c8(s))), for s \u2208 , we have V(\u03c8(\u2212r)) \u2264 V(\u03c8(0))/\u03b32 and V(\u03c8(\u2212u)) \u2264 V(\u03c8(0))/\u03b32. Calculating the derivative of V along solutions of |du. Then the trivial solution of  = \u03b32s. Then if V(\u03c8(0)) \u2265 g(V(\u03c8(s))), for s \u2208 , we have V(\u03c8(\u2212r)) \u2264 V(\u03c8(0))/\u03b32 and V(\u03c8(\u2212u)) \u2264 V(\u03c8(0))/\u03b32. Calculating the derivative of V along solutions of =xTPxequality  on \u03b3. Thtisfies2\u03bbmin\u2061(P)|In this paper, a new technique is offered to establish the stability criteria for finite and infinite delay impulsive control systems. In the above theories, rather than employing Razumikhin techniques and Lyapunov functions, we adopt the largest and smallest eigenvalue so that not only can they be easier constructed, but also the conditions ensuring the required stability are less restrictive. Furthermore, in Consider the following impulsive control systems:h(u) = eu\u2212,\u2009\u2009u > 0 and with the following parameter matrices:\u03b3 \u2208  such thatThe trivial solution of the system  is unifoP as follows:Q = ATP + PA. So by  = 2eu\u2212, u > 0 and with the following parameter matrices:Consider the impulsive control systems  where h such thatThe trivial solution of the system  is unifoP as follows:Q = ATP + PA. So by (\u03c1 = 0.009. Also, a simple check shows that both of these conditions are satisfied by choosing \u03b3 = 0.6. Therefore, robust exponential stability of the trivial solution of (Obviously,ution of  can be o"}
{"text": "\ud835\udcae(\u211dn) and \ud835\udcae\u2032(\u211dn).Parseval's formula and inversion formula for the S-transform are given. A relation between the S-transform and pseudodifferential operators is obtained. The S-transform is studied on the spaces  The space of all rapidly decreasing functions on \u211dn is denoted by \ud835\udcae(\u211dn) or simply \ud835\udcae. Elements in the dual space \ud835\udcae\u2032 of \ud835\udcae are called tempered distribution.A function \u03c9 appears as a superposition of time-frequency shifts as follows:Some properties of S-transform can be found in \u20138 and ce\u03c9 = m(\u03be), that is, independent of t, then\u03c9 is a multiplication operator. In particular, if \u03c9 = 1, then (S\u03c9\u03d5) = (\u2131\u03d5)(\u03be).If \u03c9 = m(t), thenIf \u03c91 and \u03c92 be the window functions such that\u03d51, \u03d52 \u2208 L2(\u211dn) and let (S\u03c91\u03d51) and (S\u03c92\u03d52) be the S-transforms of \u03d51 and \u03d52, respectively. ThenLet This immediately implies the Plancherel formulaConsider\u03d5 \u2208 L2(\u211dn) and window function \u03c9 satisfy the condition (If ondition  of the pBy the previous theorem we can writeHence\u03c9 be a window function and S\u03c9 is the S-transform. Then the transform S\u03c9* defined by\u03c9. If \u03d5 \u2208 L2(\u211dn) and \u03c8 \u2208 L2(\u211dn \u00d7 \u211dn), then , thenLet ondition . If \u03c81,Consider\u03c9 satisfies the condition  function or a tempered distribution on \u211dn. Then the operatorLet The pseudodifferential operator plays an important role in the theory of partial differential equations. The pseudodifferential operator has been studied on function and distribution spaces by many authors. Details of the concept can be found in , 10.\u03d5 with respect to a window function \u03c9 is given by\u03c3 = \u03c9.Here we give a direct relation between S-transform and pseudodifferential operator which will may be very useful in the study of S-transform of distribution spaces. The continuous S-transform of a function In this section we will investigate the S-transform of tempered distribution by means of the Fourier transform.\u03c9 \u2208 \ud835\udcae(\u211dn2), then S\u03c9 maps \ud835\udcae(\u211dn) into \ud835\udcae(\u211dn2).If \u03c9\u03d5) \u2208 \ud835\udcae(\u211dn2), since the Fourier transform is continuous isomorphism from \ud835\udcae(\u211dn) to \ud835\udcae(\u211dn), and its inverse is also a continuous isomorphism from \ud835\udcae(\u211dn) to \ud835\udcae(\u211dn) (see [By  we have\u211dn) , then S\u03c9 maps \ud835\udcae\u2032(\u211dn) into \ud835\udcae\u2032(\u211dn2).If f \u2208 \ud835\udcae\u2032(\u211dn) and \u03c8 \u2208 \ud835\udcae(\u211dn2), we have\u03c9f) \u2208 \ud835\udcae\u2032(\u211dn2).For any \u03c9 \u2208 \ud835\udcae\u2032(\u211dn2), then S\u03c9 maps \ud835\udcae(\u211dn) into \ud835\udcae\u2032(\u211dn2).If \u03d5 \u2208 \ud835\udcae(\u211dn) and \u03c8 \u2208 \ud835\udcae(\u211dn2), then\u03c9 \u2208 \ud835\udcae\u2032(\u211dn2), we have\u21311(S\u03c9\u03d5) \u2208 \ud835\udcae\u2032(\u211dn2) and hence (S\u03c9\u03d5) \u2208 \ud835\udcae\u2032(\u211dn2).If"}
{"text": "The purpose of this paper is to investigate the problem of finding the approximate element of the common set of solutions of a split equilibrium problem and a hierarchical fixed point problem in a real Hilbert space. We establish the strong convergence of the proposed method under some mild conditions. Several special cases are also discussed. Our main result extends and improves some well-known results in the literature. H be a real Hilbert space, whose inner product and norm are denoted by \u2329\u00b7, \u00b7\u232a and ||\u00b7||. Let C be a nonempty closed convex subset of H. We introduce the following definitions which are useful in the following analysis.Let T:\u2009\u2009C \u2192 H is said to be (a)monotone, if(b)\u03b1 > 0 such thatstrongly monotone, if there exists (c)\u03b1-inverse strongly monotone, if there exists \u03b1 > 0 such that(d)nonexpansive, if(e)k-Lipschitz continuous, if there exists a constant k > 0 such that(f)C, if there exists a constant 0 \u2264 k < 1 such thatcontraction on It is easy to observe that every \u03b1-inverse strongly monotone T is monotone and Lipschitz continuous. It is well known that every nonexpansive operator T:\u2009\u2009H \u2192 H satisfies, for all  \u2208 H \u00d7 H, the inequalityx, y) \u2208 H \u00d7 Fix\u2061(T),The mapping tone, if\u2329Tx\u2212Ty,x\u2212uch that\u2329Tx\u2212Ty,x\u2212T is to find x \u2208 C such thatF(T) the set of solutions of (F(T) is closed and convex and PF(T) is well defined (see [The fixed point problem for the mapping uch thatTx=x.We ned (see ).x \u2208 C such thatEP\u2061(F). Numerous problems in physics, optimization, and economics reduce to finding a solution of (EP\u2061(F) is nonempty. In 2007, Plubtieng and Punpaeng [F(T)\u2229EP\u2061(F).The equilibrium problem denoted by EP is to find uch thatF\u22650,uch thatF\u22650,Punpaeng  introducH1 and H2 be two real Hilbert spaces. Given operators f:\u2009\u2009H1 \u2192 H1 and g:\u2009\u2009H2 \u2192 H2, a bounded linear operator A:\u2009\u2009H1 \u2192 H2, and nonempty, closed, and convex subsets C\u2286H1 and Q\u2286H2, the SVIP is formulated as follows: find a point x* \u2208 C such thatBi:\u2009Hi \u2192 2Hi is a set-valued mapping for i = 1,2. Later Byrne et al. [Recently, Censor et al.  introduc\u2200y\u2208Q.In , Moudafi\u2200y\u2208Q.In  for the uch that\u2329f(x\u2217),x\u2212e et al.  and Moude et al. .F1:\u2009\u2009C \u00d7 C \u2192 R and F2:\u2009Q \u00d7 Q \u2192 R be nonlinear bifunctions and let A:\u2009H1 \u2192 H2 be a bounded linear operator; then, the split equilibrium problem (SEP) is to find x* \u2208 C such thatp \u2208 EP\u2061(F1) : Ap \u2208 EP\u2061(F2)}.Very recently, Kazmi and Rizvi  studied uch thatF1\u2265uch thaty\u2217=Ax\u2217\u2208Q\u2009S:\u2009C \u2192 H be a nonexpansive mapping. The following problem is called a hierarchical fixed point problem: find x \u2208 F(T) such thatf:\u2009\u2009C \u2192 H is a contraction mapping and {\u03b1n} and {\u03b2n} are two sequences in . Under some certain restrictions on parameters, Yao et al. proved that the sequence {xn} generated by (z \u2208 F(T), which is the unique solution of the following variational inequality:U is a Lipschitzian mapping and F is a Lipschitzian and strongly monotone mapping. They proved that under some approximate assumptions on the operators and parameters, the sequence {xn} generated by  and andS:\u2009C uch that\u2329x\u2212Sx,y\u2212xuch that\u2329x\u2212Sx,y\u2212xquality:\u2329(I\u2212f)z,yated by (\u2329\u03c1U(z)\u2212\u03bcFC.In this section, we recall some basic definitions and properties, which will be frequently used in our later analysis. Some useful results proved already in the literature are also summarized. The first lemma provides some basic properties of projection onto PC denote the projection of H onto C. Then, one has the following inequalities:Let F:\u2009\u2009C \u00d7 C \u2192 \u211d be a bifunction satisfying the following assumptions: (i)F = 0, for all x \u2208 C;(ii)F is monotone; that is, F + F \u2264 0, for all x, y \u2208 C;(iii)x, y, z \u2208 C, lim\u2061t\u21920F(tz + (1 \u2212 t)x, y) \u2264 F;for each (iv)x \u2208 C, y \u2192 F is convex and lower semicontinuous;for each (v)r > 0 and z \u2208 C, there exists a bounded subset K of H1 and x \u2208 C\u2229K such thatfor fixed Let F1:\u2009C \u00d7 C \u2192 \u211d satisfies r > 0 and for all x \u2208 H1, define a mapping TrF1:\u2009H1 \u2192 C as follows:(i)TrF1 is nonempty and single-valued;(ii)TrF1 is firmly nonexpansive; that is,(iii)F(TrF1) = EP(F1);(iv)EP(F1) is closed and convex.Assume that F2:\u2009\u2009Q \u00d7 Q \u2192 \u211d satisfies s > 0 and for all u \u2208 H2, define a mapping TsF2:\u2009\u2009H2 \u2192 Q as follows:TsF2 satisfies conditions (i)\u2013(iv) of F(TsF2) = EP\u2061, where EP\u2061 is the solution set of the following equilibrium problem:Assume that F1:\u2009\u2009C \u00d7 C \u2192 \u211d satisfies TrF1 be defined as in x, y \u2208 H1 and r1, r2 > 0. ThenAssume that C be a nonempty closed convex subset of a real Hilbert space H. If\u2009\u2009T:\u2009\u2009C \u2192 C is a nonexpansive mapping with Fix\u2061(T) \u2260 \u2205, then the mapping I \u2212 T is demiclosed at 0; that is, if {xn} is a sequence in C weakly converging to x and if {(I \u2212 T)xn} converges strongly to 0, then (I \u2212 T)x = 0.Let U:\u2009\u2009C \u2192 H be \u03c4-Lipschitzian mapping and let F:\u2009\u2009C \u2192 H be a k-Lipschitzian and \u03b7-strongly monotone mapping; then for 0 \u2264 \u03c1\u03c4 < \u03bc\u03b7, \u03bcF \u2212 \u03c1U is \u03bc\u03b7 \u2212 \u03c1\u03c4-strongly monotone; that is,Let \u03bb \u2208  and \u03bc > 0. Let F:\u2009\u2009C \u2192 H be an k-Lipschitzian and \u03b7-strongly monotone operator. In association with nonexpansive mapping T:\u2009\u2009C \u2192 C, define the mapping T\u03bb:\u2009\u2009C \u2192 H byT\u03bb is a contraction provided that \u03bc < (2\u03b7/k2); that is,Suppose that an} is a sequence of nonnegative real numbers such that\u03b3n} is a sequence in  and \u03b4n is a sequence such thatn=1\u221e\u03b3n = \u221e;\u2211n\u2192\u221e\u03b4n/\u03b3n \u2264 0 or \u2211n=1\u221e|\u03b4n| < \u221e.limsup\u2061Then lim\u2061n\u2192\u221ean = 0.Assume that {C be a closed convex subset of H. Let {xn} be a bounded sequence in H. Assume that w-limit set ww(xn) \u2282 C where ww(xn) = {x:\u2009\u2009xni\u21c0x};the weak z \u2208 C, lim\u2061n\u2192\u221e||xn \u2212 z|| exists.for each  Then {xn} is weakly convergent to a point in C.Let In this section, we suggest and analyze our method and we prove a strong convergence theorem for finding the common solutions of the split equilibrium problem -16) and and16) aH1 and H2 be two real Hilbert spaces and let C\u2286H1 and Q\u2286H2 be nonempty closed convex subsets of Hilbert spaces H1 and H2, respectively. Let A:\u2009\u2009H1 \u2192 H2 be a bounded linear operator. Assume that F1:\u2009C \u00d7 C \u2192 \u211d and F2:\u2009Q \u00d7 Q \u2192 \u211d are the bifunctions satisfying F2 is upper semicontinuous in first argument. Let S, T:\u2009\u2009C \u2192 C be a nonexpansive mapping such that \u039b\u2229F(T) \u2260 \u2205. Let F:\u2009\u2009C \u2192 C be an k-Lipschitzian mapping and \u03b7-strongly monotone and let U:\u2009\u2009C \u2192 C be \u03c4-Lipschitzian mapping. Now we introduce the proposed method as follows.Let x0 \u2208 C arbitrarily, let the iterative sequences {un}, {xn}, and {yn} be generated byrn}\u2282 and \u03b3 \u2208 , L is the spectral radius of the operator A*A, and A* is the adjoint of A. Suppose that the parameters satisfy 0 < \u03bc < (2\u03b7/k2), 0 \u2264 \u03c1\u03c4 < \u03bd, where \u03b1n} and {\u03b2n} are sequences in  satisfying the following conditions: n\u2192\u221e\u03b1n = 0 and \u2211n=1\u221e\u03b1n = \u221e;lim\u2061n\u2192\u221e(\u03b2n/\u03b1n) = 0;lim\u2061n=1\u221e | \u03b1n\u22121 \u2212 \u03b1n | <\u221e and \u2211n=1\u221e|\u03b2n\u22121 \u2212 \u03b2n| < \u221e;\u2211n\u2192\u221ern < limsup\u2061n\u2192\u221ern < 2\u03c2 and \u2211n=1\u221e|rn\u22121 \u2212 rn| < \u221e.liminf\u2061For a given The proposed method is an extension and improvement of the method of Wang and Xu  for findU = f, F = I, \u03c1 = \u03bc = 1, we obtain an extension and improvement of the method of Yao et al. [If the Lipschitzian mapping o et al.  for findf with a coefficient \u03b1 \u2208 [0,1) in other papers .The contractive mapping ers see , 22, 27), 27f wit This shows that Our method can be viewed as extension and improvement for some well-known results as follows x* \u2208 \u039b\u2229F(T). Then {xn},\u2009\u2009{un}, and {yn} are bounded.Let x* \u2208 \u039b\u2229F(T); we have x* = TrnF1(x*) and Ax* = TrnF2(Ax*). ThenL, it follows that\u03b3, we getVn = \u03b1n\u03c1U(xn)+(I \u2212 \u03b1n\u03bcF)(T(yn)). Next, we prove that the sequence {xn} is bounded; without loss of generality we can assume that \u03b2n \u2264 \u03b1n for all n \u2265 1. From (Let ows from  that(36ows that\u03b32\u2329(TrnF2*). Then||un\u2212x\u2217|| 1. From , we haven, we obtain ||xn \u2212 x*|| \u2264 max\u2061{||x0 \u2212 x*||, (1/(1 \u2212 \u03c1))(||(\u03c1U \u2212 \u03bcF)x*|| + ||Sx* \u2212 x*||)}, for n \u2265 0 and x0 \u2208 C. Hence {xn} is bounded and, consequently, we deduce that {un}, {yn}, {S(xn)}, {T(xn)}, {F(T(yn))}, and {U(xn)} are bounded.By induction on x* \u2208 \u039b\u2229F(T) and {xn} the sequence generated by the n\u2192\u221e||xn+1 \u2212 xn|| = 0;lim\u2061w-limit set ww(xn) \u2282 F(T), (ww(xn) = {x : xni\u21c0x}).the weak Let un = TrnF1(xn + \u03b3A*(TrnF2 \u2212 I)Axn) and un\u22121 = Trn\u22121F1(xn\u22121 + \u03b3A*(Trn\u22121F2 \u2212 I)Axn\u22121)\u2009\u2009it follows from \u03c3n : = ||TrnF2Axn \u2212 Axn|| and \u03c7n : = ||TrnF1(xn + \u03b3A*(TrnF2 \u2212 I)Axn) \u2212 (xn + \u03b3A*(TrnF2 \u2212 I)Axn)||. Without loss of generality, let us assume that there exists a real number \u03bc such that rn > \u03bc > 0, for all positive integers n. Then we getn\u2192\u221e||un \u2212 xn|| = 0. Since x* \u2208 \u039b\u2229F(T) by using (\u03b3(1 \u2212 L\u03b3) > 0, lim\u2061n\u2192\u221e||xn+1 \u2212 xn|| = 0, \u03b1n \u2192 0, and \u03b2n \u2192 0, we obtainTrnF1 is firmly nonexpansive, we haven\u2192\u221e||xn+1 \u2212 xn|| = 0, \u03b1n \u2192 0, \u03b2n \u2192 0, and lim\u2061n\u2192\u221e||(TrnF2 \u2212 I)Axn|| = 0, we obtainz \u2208 \u039b\u2229F(T); since T(xn) \u2208 C, we haven\u2192\u221e||xn+1 \u2212 xn|| = 0,\u2009\u2009\u03b1n \u2192 0,\u2009\u2009\u03b2n \u2192 0,\u2009\u2009||\u03c1U(xn) \u2212 \u03bcF(T(yn))||, and ||Sxn \u2212 xn|| are bounded and lim\u2061n\u2192\u221e||xn \u2212 un|| = 0, we obtainxn} is bounded, without loss of generality, we can assume that xn\u21c0x* \u2208 C. It follows from x* \u2208 F(T). Therefore ww(xn) \u2282 F(T).Since n).From  and the ).From (||yn\u2212yn\u22121estimate||xn+1\u2212xnby using  and 37)un = Trows from  and (37)y using . Next, we show that w \u2208 EP\u2061(F1). Since un = TrnF1(xn + \u03b3A*(TrnF2 \u2212 I)Axn), we haveF1 thatn\u2192\u221e||un \u2212 xn|| = 0, lim\u2061n\u2192\u221e||(TrnF2 \u2212 I)Axn|| = 0, and xn\u21c0w, it easy to observe that unk \u2192 w. It follows by F1 \u2264 0, for all y \u2208 C.Since {t \u2264 1 and y \u2208 C, let yt = ty + (1 \u2212 t)w; we have yt \u2208 C. Then, from Assumptions F1 \u2265 0. From F1 \u2265 0, which implies that w \u2208 EP\u2061(F1).For any 0 < Aw \u2208 EP\u2061(F2). Since {xn} is bounded and xn\u21c0w, there exists a subsequence {xnk} of\u2009\u2009{xn} such that xnk \u2192 w and since A is a bounded linear operator, Axnk \u2192 Aw. Now set vnk = Axnk \u2212 TrnkF2Axnk. It follows from  and hence w \u2208 \u039b.Next, we show that ows from  that lim\u03c1\u03c4 < \u03bd and\u03bcF \u2212 \u03c1U is \u03bc\u03b7 \u2212 \u03c1\u03c4 strongly monotone, and we get the uniqueness of the solution of the variational inequality (z \u2208 \u039b\u2009\u2009\u2229\u2009\u2009F(T).Thus we haveequality  and denon\u2192\u221e\u2329\u03c1U(z) \u2212 \u03bcF(z), xn \u2212 z\u232a\u22640. Since {xn} is bounded, there exists a subsequence {xnk} of {xn} such thatxn \u2192 z. Consider\u03b3n = \u03b1n(\u03bd \u2212 \u03c1\u03c4) and \u03b4n = (2\u03b1n(\u03bd \u2212 \u03c1\u03c4)/(1 + \u03b1n(\u03bd \u2212 \u03c1\u03c4))){(1/(\u03bd \u2212 \u03c1\u03c4))\u2329\u03c1U(z) \u2212 \u03bcF(z), xn+1 \u2212 z\u232a + ((1 \u2212 \u03b1n\u03bd)\u03b2n/\u03b1n(\u03bd \u2212 \u03c1\u03c4))||Sz \u2212 z||||xn+1 \u2212 z||}.Next, we claim that lim\u2061sup\u2061xn \u2192 z. This completes the proof.SinceS = I \u2212 (\u03c1U \u2212 \u03bcF), then we can get the variational inequality , x \u2212 z\u232a \u2265 0, for all x \u2208 \u039b\u2229F(T), which is just the variational inequality studied by Suzuki [In hierarchical fixed point problem , if S = equality . In 58)S = I \u2212  and and16) a"}
{"text": "The aim of this paper is to give several characterizations for the property of weak exponential expansiveness for evolution families in Banach spaces. Variants for weak exponential expansiveness of some well-known results in stability theory (Datko (1973), Rolewicz (1986), Ichikawa (1984), and Megan et al. (2003)) are obtained. T(t)}t\u22650 is uniformly exponentially stable if and only if for each vector x from the Banach space X the function t \u2192 ||T(t)x|| lies in L2(\u211d+). Later, Pazy generalizes the result in )In the present paper, we introduce the concept of weak exponential expansiveness for evolution families which is an extension of classical concept of exponential expansiveness. Our main objective is to give some characterizations for weak exponential expansiveness properties of evolution families in Banach spaces, and variants for weak exponential expansiveness of some well-known results in stability theory  of all bounded linear operators on X will be denoted by ||\u00b7||.Let \ud835\udcb0 = {U}t\u2265s\u22650 of bounded linear operators is called an evolution family if the following conditions are satisfied:U = I, the identity operator on X, for all t \u2265 0;UU = U for all t \u2265 r \u2265 s \u2265 0;M \u2265 1 and \u03c9 > 0 such that ||Ux0|| \u2264 Me\u03c9(t\u2212s)||x0|| for all t \u2265 s \u2265 0 and x0 \u2208 X;there exist x0 \u2208 X and every t0 \u2265 0, the mapping r \u21a6 ||Ux0|| is continuous on  and \u03c4 \u21a6 ||U|| is continuous on  for all n \u2208 \u2115, so there exist Ln > 0 such thatL = inf\u2061n\u2061Ln. We suppose that F \u2208 \u21311, and from sup\u2061\u03bb>0\u2061((1 \u2212 e\u03bb\u2212)/\u03bb) < \u221e.Sufficiency. Let \u03b4 > 0 be such that e\u03b1\u03b4 > 1 + 3\u03b1\u03b4K, where K and \u03b1 are given by (\ud835\udcb0 is not weakly exponentially expansive. Then, by c = 3, there exists x0 \u2208 X such that for all t0 \u2265 0 and all u \u2208 (0, \u03b4] there is r \u2265 t0 witht = r + \u03b4, we havegiven by . We suppequality , the pro"}
{"text": "The Factorization Method is a noniterative method to detect the shape and position of conductivity anomalies inside an object. Themethod was introduced by Kirsch for inverse scattering problems and extended to electrical impedance tomography (EIT) by Br\u00fchl and Hanke. Since these pioneering works, substantial progress has been made on the theoretical foundations of the method. The necessary assumptions have been weakened, and the proofs have been considerably simplified. In this work, we aim to summarize this progress and present a state-of-the-art formulation of the Factorization Method for EIT with continuous data. In particular, we formulate the method for general piecewise analytic conductivities and give short and self-contained proofs. \u03a9\u2286\u211dn from current-voltage measurements on a part of its surface \u03a3\u2286\u2202\u03a9. Mathematically, this leads to the problem of recovering the coefficient \u03c3(x) in the elliptic partial differential equationu\u03c3g is the solution of  aims to reconstruct the spatial conductivity distribution inside an imaging subject equation\u2207\u00b7\u03c3\u2207u\u03c3g=0\u03c3) is compared to a reference NtD \u039b(\u03c30) in order to determine, if and where \u03c3 differs from a known background conductivity \u03c30. This problem also appears in time-difference EIT, where measurements at different times are compared to monitor temporal conductivity changes. These applications lead to the shape reconstruction problem of determining the support of \u03c3 \u2212 \u03c30 from \u039b(\u03c3) and \u039b(\u03c30).In several applications, EIT is used to determine the position of conductivity changes. This includes anomaly detection problems, where \u039b denotes the subspace of L\u221e(\u03a9)-functions with positive essential infima. H\u22c41(\u03a9) and L\u22c42(\u03a3) denote the spaces of H1- and L2-functions with vanishing integral mean on \u2202\u03a9 .We start by making the mathematical setting precise. Let \u03c3 \u2208 L+\u221e(\u03a9) and g \u2208 L\u22c42(\u03a3), there exists a unique solution u\u03c3g \u2208 H\u22c41(\u03a9) of the elliptic partial differential equationu\u03c3g \u2208 H\u22c41(\u03a9) solves (\u03c3) is a self-adjoint, compact linear operator.For equation\u2207\u00b7\u03c3\u2207u\u03c3g=0\u03c30 \u2208 L+\u221e(\u03a9) be piecewise analytic. For each point z \u2208 \u03a9 that has a neighborhood in which \u03c30 is analytic, and each unit vector d \u2208 \u211dn, ||d|| = 1, let \u03a6z,d be the solution ofz,d is called a dipole function.Let monotony lemma since it shows that a larger conductivity leads to a smaller NtD. More precisely, it shows a relation between the difference of two NtDs and the difference of the corresponding conductivities and the interior energy of an electric potential. The second lemma shows that this energy term is the image of the adjoint of an auxiliary virtual measurement operator that is defined on a subregion of \u03a9. The third lemma is a functional analytic relation between the norm of an image of an operator and the range of its adjoint. Together with the first two lemmas, it implies that the range of the auxiliary virtual measurement operator can be calculated from the NtDs. Finally, using the previous dipole functions, the last lemma shows that the range of the auxiliary virtual measurement operator determines the region on which they are defined.Our presentation of the Factorization Method in the next section relies on the following four lemmas. The first lemma is frequently called a We start with the monotony lemma.\u03c31, \u03c30 \u2208 L+\u221e(\u03a9). Then, for all g \u2208 L\u22c42(\u2202\u03a9), j : = \u039b(\u03c3j), j = 0,1, and u0 : = u\u03c30g. Let g \u2208 L\u22c42(\u2202\u03a9), we have thatThe lemma seems to go back to Ikehata, Kang, Seo, and Sheen , 23, cf.\u03c31 and \u03c30, we conclude thatBy interchanging \u03c30 \u2208 L+\u221e(\u03a9) and a measurable subset D\u2286\u03a9, we define the virtual measurement operator LD byv \u2208 H\u22c41(\u03a9) solvesu0|2 in Given a reference conductivity LD is given byu0 \u2208 H\u22c41(\u03a9) solvesThe adjoint operator of g \u2208 L\u22c42(\u03a3) and F \u2208 L2(D)n, we have thatFor all The following functional analytic lemma uses bounds on the image of an operator to characterize the range of its dual operator.X and Y be real Hilbert spaces with inner products X and Y, respectively. Let A \u2208 \u2112 and x\u2032 \u2208 X. Then, Let X, Y1, and Y2 are three real Hilbert spaces, Ai \u2208 \u2112, i = 1,2, and if there exists C > 0 with\u211b(A1)\u2286\u211b(A2).The assertion can be generalized to Banach spaces, and, in that context, it is called the \u201c14th important property of Banach spaces\u201d in Bourbaki . For thex\u2032 \u2208 \u211b(A*), then there exists y\u2032 \u2208 Y such that x\u2032 = A*y\u2032. Hence,C = ||y\u2032||.If x\u2032 \u2208 X be such that there exists C > 0 with |X | \u2264C||Ax|| for all x \u2208 X. We definef is a well-defined, continuous linear functional on \u211b(A). By setting it to zero on \u211b(A)\u22a5, we can extend f to a continuous linear functional on Y. Using the Riesz theorem, it follows that there exists y\u2032 \u2208 Y withx \u2208 X, we havex\u2032 = A*y\u2032 \u2208 \u211b(A*). Now let LD determines the region D on which it is defined. We state the lemma for a simple special case, a generalized version of the lemma will be formulated in The last lemma shows that the range of the virtual measurement operator \u03c30 = 1, D\u2286\u03a9 be open, and Let d \u2208 \u211dn, ||d|| = 1, and every point z \u2208 \u03a9\u2216\u2202D, it holds thatThen, for all unit vectors The proof is similar to the one of [z \u2208 D and \u03f5 > 0 be such that F \u2208 L2(D)n be the zero continuation of \u2207(f1 \u2212 f2) to D.First, let v \u2208 H\u22c41(\u03a9), and, for all w \u2208 H\u22c41(\u03a9),z,d|\u03a3 = v|\u03a3 = LD(F) \u2208 \u211b(LD).Then, the functionz,d|\u03a3 \u2208 \u211b(LD). Let v \u2208 H\u22c41(\u03a9) be the function from the definition of LD. Then,v = \u03a6z,d in the connected set Now let \u03a6v = \u03a6z,d in If \u03c3 differs from a reference conductivity \u03c30 by a range criterion. Before we turn to a new general formulation of the method, we first state it for a special case that is similar to the one that was treated in the original works of Br\u00fchl and Hanke [Now we will formulate the Factorization Method and characterize a region where a conductivity nd Hanke , 4.\u03c30 = 1 and \u03c3 = 1 + \u03c7D, where D\u2286\u03a9 is an open set so that z \u2208 \u03a9, z \u2209 \u2202D, and all dipole directions d \u2208 \u211dn, ||d|| = 1,Let g \u2208 L\u22c42(\u2202\u03a9),\u03c3) \u2212 \u039b(1)| = \u039b(1) \u2212 \u039b(\u03c3), and, using D and \u211b(LD) in The monotony \u03c3(x) = 1 + \u03ba(x)\u03c7D(x), when there exists a conductivity jump \u03f5 > 0 so that eitherObviously, the same arguments can be used to treat the case \u03c30 is a piecewise analytic function and that either \u03c3 \u2212 \u03c30 \u2265 0 or \u03c3\u2009\u2212\u2009\u03c30 \u2264 0. Roughly speaking, under this general assumption, the Factorization Method then characterizes the support of \u03c3 \u2212 \u03c30 up to holes in the support that have no connections to \u03a3. For a precise formulation, we use the concept of the inner and outer support from [Now we drop the assumptions that the background is constant, that there is a clear conductivity jump, and that the complement of the inclusions is connected. We will merely assume that the reference conductivity ort from  that hasort from ; see alsort from , 30.connected to \u03a3 if U\u2229\u03a9 is connected and U\u2229\u03a3 \u2260 \u2205.A relatively open set \u03ba : \u03a9 \u2192 \u211d, we define support supp\u2061(\u03ba) as the complement |>0,  the outer support out\u03a3\u2009supp\u2061\u03ba as the complement (in \u03ba|U \u2261 0.  the The interior of a set M\u2286\u03a9 is denoted by int\u2061M and its closure (with respect to \u211dn) by M is measurable, we also define(d)\u03a3\u2009M = out\u03a3supp\u2061\u03c7M. out It is easily checked that out\u03a3(supp\u2061\u03ba) = out\u03a3supp\u2061\u03ba. For a measurable function With this concept, we can extend the range characterization in \u03c30 \u2208 L+\u221e(\u03a9) be piecewise analytic. Let D\u2286\u03a9 be measurable. Let d \u2208 \u211dn, ||d|| = 1, and every point z \u2208 \u03a9 that has a neighborhood in which \u03c30 is analytic,Then, for all unit vectors z \u2208 int\u2061D, then there exists a small ball If z,d|\u03a3 \u2208 \u211b(LD), and let v \u2208 H\u22c41(\u03a9) be the function from the definition of LD, so that , and let \u03c30 \u2208 L+\u221e(\u03a9) be a piecewise analytic function. Let eitherz \u2208 \u03a9 that have a neighborhood in which \u03c30 is analytic, as well as all unit vectors d \u2208 \u211dn, ||d|| = 1, Let z \u2208 \u03a9 have a neighborhood in which \u03c30 is analytic, and let d \u2208 \u211dn be a unit vector with ||d|| = 1. We only prove the assertions for \u03c3 \u2265 \u03c30. The other case is completely analogous.Let z \u2208 inn\u2009supp\u2061\u2009\u2009(\u03c3 \u2212 \u03c30). Then there exists a small ball B\u03f5(z) and \u03b4 > 0 so that \u03c3 \u2212 \u03c30 \u2265 \u03b4 on B\u03f5(z). Using the monotony g \u2208 L\u22c42(\u2202\u03a9),D : = supp\u2061(\u03c3 \u2212 \u03c30), the monotony g \u2208 L\u22c42(\u2202\u03a9),First, let D with d \u2208 \u211dn, ||d|| = 1,A result of Hyv\u00f6nen and the author [definiteness assumption that \u03c3 \u2265 \u03c30 on \u03a9 or \u03c3 \u2264 \u03c30 on \u03a9. However, Grinberg, Kirsch, and Schmitt [exclude a region E\u2286\u03a9 from \u03a9, in such a way that the Factorization Method only requires the definiteness assumption on \u03a9\u2216E. In this subsection, we show how their idea can be incorporated into our formulation of the method.It is a long standing open theoretical problem whether the range criterion of the Factorization Method holds true without the  Schmitt , 31 show\u03c3 = 1 + \u03c7D+ \u2212 (1/2)\u03c7D\u2212, it is not known whetherD+ and D\u2212. More precisely, if we know a subset E that contains D\u2212 without intersecting D+, then we can use the Factorization Method to find D+ \u222a E (and thus D+).To point out the main idea, we first formulate the result for a simple special case. Let us stress that, for \u03c30 = 1 and \u03c3 = 1 + \u03c7D+ \u2212 (1/2)\u03c7D\u2212, where D+, D\u2212\u2286\u03a9 are open. Let E\u2286\u03a9 be an open set.(a)D+\u2286E and z \u2208 \u03a9, z \u2209 \u2202(D\u2212 \u222a E), and all dipole directions d \u2208 \u211dn, ||d|| = 1, If (b)D\u2212\u2286E and z \u2208 \u03a9, z \u2209 \u2202(D+ \u222a E), and all dipole directions d \u2208 \u211dn, ||d|| = 1, If Let g \u2208 L\u22c42(\u2202\u03a9),The monotony ince cf. (54)\u222b\u03a3g(In case (b), we obtain thatWe can also extend these ideas to the general setting of \u03c3 \u2208 L+\u221e(\u03a9) and let \u03c30 \u2208 L+\u221e(\u03a9) be a piecewise analytic function. Let E\u2286\u03a9 be a measurable set.Let \u03b1, \u03b2 \u2208 \u211d such that(a)\u03c3 \u2264 \u03c30 on \u03a9\u2216E, then for all z \u2208 \u03a9 that have a neighborhood in which \u03c30 is analytic, as well as all unit vectors d \u2208 \u211dn, ||d|| = 1, If \u2009and(b)\u03c3 \u2265 \u03c30 on \u03a9\u2216E, then for all z \u2208 \u03a9 that have a neighborhood in which \u03c30 is analytic, as well as all unit vectors d \u2208 \u211dn, ||d|| = 1, If \u2009andChoose z \u2208 inn\u2009supp\u2061(\u03c3 \u2212 \u03c30) with z \u2209 E, there exists a small ball B\u03f5(z) and \u03b4 > 0 so that \u03c30 \u2212 \u03c3 \u2265 \u03b4 on B\u03f5(z). Using the monotony g \u2208 L\u22c42(\u2202\u03a9),For every 2(\u2202\u03a9),6\u222b\u03a3g(\u039b(\u03c3)\u2212The monotony  The Factorization Method can be used to detect regions in which a conductivity differs from a known reference conductivity. In this work, we summarized the progress on the method's theoretical foundation. We formulated the method for general piecewise analytic conductivities and gave comparatively simple and self-contained proofs. We also showed how the idea of excluding a part of the imaging region can be incorporated into this formulation.\u03c30 and the existence of the dipole functions.The regularity assumptions can be weakened even further. Our proofs only require unique continuation arguments for the reference conductivity Two major open theoretical questions still exist in the context of the Factorization Method. The theoretical justification of the method requires a definiteness condition (on the whole domain or after excluding an a priori known part of the domain). It is unknown whether the method's range criterion holds without such a definiteness condition. The second open question concerns the numerical stability of the method's range criterion. So far, there are no rigorous convergence results for numerical implementations of this range criterion (see, however, Lechleiter  for a fi"}
{"text": "We have discussed some important problems about the spaces  For example, quite recently, Kayaduman and \u015eeng\u00f6n\u00fcl introduced the spaces x = (xk) such that (\u2211j=0k(xj/(j + 1))) \u2208 f, f0 and gave some important results on those spaces in , and also the inequalities q\u03c3(Ax) \u2264 L(x) (\u03c3-core of Ax\u2286K-core of x), q\u03c3(Ax) \u2264 q\u03c3(x) (\u03c3-core of Ax\u2286\u03c3-core of x), for all x \u2208 \u2113\u221e, have been studied. Here, the Knopp core, in short K-core of x, is the interval  (see , (see , and also determined necessary and sufficient conditions for a matrix A to yield K-core(Ax)\u2286st-core(x) for all x \u2208 \u2113\u221e.Let = 0 (see ). In thixn+i.In , the \u03c3-cx)] (see ). When \u03c3)], (see ). The co authors , 15\u201319 a authors  have intIn this paper, we define the spaces of Ces\u00e0ro sigma convergent and Ces\u00e0ro null sequences and give some interesting theorems.x = (xk) is said to be Ces\u00e0ro sigma convergent to the number \u03b1 if and only if lim\u2061n\u2192\u221e\u2211k=0n\u2211j=0k(x\u03c3j(p)/(n + 1)(k + 1)) = \u03b1 uniformly in p, and the set of all such sequences is denoted with \u03b1 = 0, then we write \u03b1 = \u03c3C-limx, respectively.A bounded sequence x = (xk) is said to be Ces\u00e0ro sigma bounded if and only if sup\u2061n|\u2211k=0n\u2211j=0k(x\u03c3j(p)/(n + 1)(k + 1))| < \u221e, and the set of all such sequences is denoted with A bounded sequence C denotes the Ces\u00e0ro matrix of one order. Define the sequence y = (yn), which will be frequently used as the C-transform of a sequence x = (xk); that is,\u03c3(p) = p + 1, then the spaces With the notation of , we can Now, we begin with the following theorem.V\u03c3 and V\u03c30, respectively; that is, The sequence spaces V\u03c3. In order to prove the fact V\u03c3. Consider the transformation of C defined with the notation of  whenever Cx = \u03b8 and hence C is injective. Let y \u2208 V\u03c3 and define the sequence x byC is surjective. Hence, C is linear bijection which therefore shows that the spaces V\u03c3 are linearly isomorphic, as desired. This completes the proof. The fact that the spaces V\u03c30 are linearly isomorphic can also be proved by the similar way, so we omit it.We consider only the spaces ation of  from V~\u03c3V\u03c3 and V\u03c30 are BK-spaces with the norm ||\u00b7||\u2113\u221e.The spaces The sets V\u03c3 and V\u03c30 are BK-spaces with the norm ||\u00b7||\u221e,  = {) : x \u2208 \u03bb} is subspace of \u03bb \u00d7 \u03bc and the set G(T) is normed space with the normT is a linear transformation from \u03bb to \u03bc.It is known that, if G(C) of the transformation C is closed subspace in The graph V\u03c3 are Banach spaces ) is convergent to  in G(C) for y \u2208 V\u03c3. With this supposition, and from the equalityxn \u2192 x and C(xn) \u2192 y as n \u2192 \u221e. Also, since C is continuous and from the definition of the sequential continuous, we obtain that Cx = y and this completes the proof.We know that the spaces ces see,  and the Following Ba\u015far , we starA = (ank) and B = (bnk) map the sequences x = (xk) and y = (yk) which are connected by the relation (z = (zn) and t = (tn), respectively; that is,B is applied to the C-transform of the sequence x = (xk) while the method A is directly applied to the entries of the sequence x = (xk). So, the methods A and B are essentially different. Let us assume that the matrix product BC exists which is a much weaker assumption than the conditions on the matrix B belonging to any matrix class, in general. The methods A and B in  under the application of formal summation by parts. This leads us to the fact that BC exists and is equal to A and (BC)x = B(Cx) formally holds, if one side exists. This statement is equivalent to the following relation between the entries of the matrices A = (ank) and B = (bnk):n, k \u2208 \u2115.Let us suppose that the infinite matrices relation  to the sthat is,zn=(Ax)n=A \u2208 , if and only ifA \u2208 , if and only if ( only if  and 17)A \u2208 , if and only if (I is identity matrix. only if  and 17)A \u2208  and E = (enk) are connected with the relation\u03bc be any given sequence space. Then, E \u2208 (\u03bc : V\u03c3).Suppose that the entries of the infinite matrices x = (xk) \u2208 \u03bc and consider the following equality with \u2211j=\u03c3(p) = p + 1, then If we take Now, right here, we have stated two theorem which are natural consequences of the A = (ank) be an infinite matrix real or complex numbers. Then, n\u2208\u2115\u2211k|\u2211j=k\u221e(anj/(j + 1))| < \u221e,\u2009\u2009sup\u2061\u03c3-lim\u2061n\u2192\u221e\u2211k\u2211j=k\u221e(anj/(j + 1)) = \u03b1,\u2009\u2009Let I is the identity matrix.where A = (ank) be an infinite matrix real or complex numbers. Then, (4)n\u2208\u2115\u2211k|\u2211j=k\u221e(anj/(j + 1))| < \u221e,sup\u2061(5)\u03c3-lim\u2061n\u2192\u221e\u2211j=k\u221e(anj/(j + 1)) = \u03b1k, exists for each fixed k \u2208 \u2115,(6)m\u2192\u221e\u2211k|\u2211i=0m\u2211j=k\u221e(a\u03c3i(n),j/(m + 1)(j + 1)) \u2212 \u03b1k| = 0, uniformly in n.lim\u2061Let The proof is clear from A = (ank) be an infinite matrix real or complex numbers. Then, (7)n\u2208\u2115\u2211k|\u2211j=k\u221e(anj/(j + 1))| < \u221e,sup\u2061(8)\u03c3-lim\u2061n\u2192\u221e\u2211j=k\u221e(anj/(j + 1)) = \u03b1k exists for each fixed k \u2208 \u2115,(9)\u03c3-lim\u2061n\u2192\u221e\u2211k\u2211j=k\u221e(anj/(j + 1)) = \u03b1. Let The proof is clear from Furthermore, from the Lemma 3.2 of , we haveA = (ank) and B = (bnk) are connected with the relation\u03bc be any given sequence space. Then, if (10)\u03b1k = lim\u2061n\u2211j=k\u221e(bnj/(j + 1)) exists for every k \u2208 \u2115, (11)n(\u2211k | \u2211j=k\u221e(bnj/(j + 1)) \u2212 \u03b1k|) < \u221e, sup\u2061(12)\u03b1 = \u2211k | \u03b1k | <\u221e.Suppose that the entries of the infinite matrices In this section, we use matrices classes x \u2208 \u2113\u221e. Then, \u03c3C-core of x is defined by the closed interval \u03c3C-core of x is \u03b1 if and only if \u03c3C-limx = \u03b1.Let We need the following lemma due to Das  for the C|| = ||cni(p)|| < \u221e and let lim\u2061n\u2192\u221esup\u2061p\u2208\u2115 | cni(p) | = 0. Then, there is a y = (yi) \u2208 \u2113\u221e such that ||y|| \u2264 1 andLet ||P and Q be sublinear functionals on a linear space X. Then, {X, P}\u2282{X, Q} if and only if P(x) \u2264 Q(x) for all x \u2208 X.Let x \u2208 \u2113\u221e if and only if n, i, p \u2208 \u2115.Consider the following.Sufficiency. Since x \u2208 \u2113\u221e, it is known that for any given \u03f5 > 0, there exists a positive integer i0 such that xi \u2264 L(x) + \u03f5 whenever i \u2265 i0. Now, let us writen \u2192 \u221e, uniformly in p. Therefore, we obtain by applying the operator limsup\u2061n\u2192\u221esup\u2061p\u2208\u2115 to the equality .Since  holds, wNecessity. We observe by inserting \u2212x = (\u2212xi) in place of x = (xi) in the inequality x \u2208 c, then \u2113(x) = L(x) = lim\u2061x and so \u03c3C-limAx = lim\u2061\u2009x. Hence, p. On the other hand, if we choose the sequence of matrices n, i, p \u2208 \u2115, then clearly \ud835\udc9c satisfies the conditions of y = (yi) \u2208 \u2113\u221e with ||y|| \u2264 1 andy in the hypothesis thatear thatliminf\u2061n\u2192\u03c3(p) = p + 1, we also have the following.In the special case BC-core(Ax)\u2286K-core(x) for all x \u2208 \u2113\u221e if and only if x \u2208 \u2113\u221e if and only if )reg and  holds.Consider the following.Necessity. By the similar way used in the proof of necessity of q\u03c3(x) \u2264 L(x) for any sequence x, condition  = q\u03c3(x) [)reg and  hold. Sieg, from u(x)=inf\u2061 we haveu(x)\u2265inf\u2061= q\u03c3(x) , the pro\u03c3(p) = p + 1, we also have the following theorem.In the special case BC-core(Ax)\u2286B-core(x) for all x \u2208 \u2113\u221e if and only if One can see that )reg and  holds.E\u2286\u2115 with natural density zero.c \u2282 st\u2229\u2113\u221e. Now, define a sequence t = (ti) via x \u2208 \u2113\u221e asE is any subset of \u2115 with \u03b4(E) = 0. Then, st-limti = 0 and t \u2208 st0, so we have At)n = \u2211i\u2208Eaniti, the matrix B = (bni), defined byn, must belong to the class x \u2208 st\u2229\u2113\u221e and let st-lim\u2009x = \u2113. Write E = {i : |xi \u2212 \u2113 | \u2265\u03b5} for any given \u03b5 > 0, so that \u03b4(E) = 0. Since \u03c3C-lim\u2061n\u2192\u221e\u2211iani = 1, we have\u03c3C-lim(Ax) = st-lim\u2009x; that is, Let ssity of  follows )reg and  holds. Londition  implies \u03c3(p) = p + 1, we also have the following theorem.In the special case Consider that E\u2286\u2115 with natural density zero.x \u2208 \u2113\u221e if and only if Consider that )reg and  holds.Consider the following.Necessity. Firstly, assume that x \u2208 \u2113\u221e where \u03b2(x) = st-lim\u2009sup\u2061\u2009x. Hence, since \u03b2(x) = st-lim\u2009sup\u2061\u2009x \u2264 L(x) for all x \u2208 \u2113\u221e , we havSufficiency. Let x \u2208 \u2113\u221e, then \u03b2(x) is finite. Let E be a subset of \u2115 defined by E = {i : xi > \u03b2(x) + \u03b5} for a given \u03b5 > 0. Then, it is obvious that \u03b4(E) = 0 and xi \u2264 \u03b2(x) + \u03b5 if i \u2209 E. For any real number \u03bb, we write \u03bb+ = max\u2061{\u03bb, 0} and \u03bb\u2212 = max\u2061{\u2212\u03bb, 0} whence |\u03bb | = \u03bb+ + \u03bb\u2212, \u03bb = \u03bb+ \u2212 \u03bb\u2212 and |\u03bb | \u2212\u03bb = 2\u03bb\u2212. Now, we can writen\u2192\u221esup\u2061p\u2208\u2115, we obtained from hypothesis that \u03b5 is arbitrary. and let  holds. I\u03c3(p) = p + 1, we also have the following theorem.In the special case BC-core(Ax)\u2286st-core(x) for all x \u2208 \u2113\u221e if and only if Consider that )reg and  holds."}
{"text": "\ud835\udc9c\u03b5(t) of equation x \u2208 \u03a9, converge to the global attractor \ud835\udc9c of the above-mentioned equation with \u03b5 = 0 for any t \u2208 \u211d.We will study the upper semicontinuity of pullback attractors for the 3D nonautonomouss Benjamin-Bona-Mahony equations with external force perturbation terms. Under some regular assumptions, we can prove the pullback attractors  \u03a9 \u2282 \u211d3 is a bounded domain with sufficiently smooth boundary \u2202\u03a9; u = , u2, u3) is the velocity vector field; \u03bd > 0 is the kinematic viscosity; \u025b \u2265 0 is a small nonnegative parameter; the external force g is locally square integrable in time for  \u2208 \u03a9 \u00d7 \u211d, that is, for any t \u2208 \u211d, g \u2208 Lloc\u20612, where H = (L2(\u03a9))3, V = (H01(\u03a9))3, and  and ||\u00b7|| are the inner product and norm of H, respectively.In this paper, we will consider the upper semicontinuity of pullback attractors for the following 3D Benjamin-Bona-Mahony equation:The Benjamin-Bona-Mahony (BBM) equation is a well-known model in physical applications which incorporates dispersive effects for long waves in shallow water that was introduced by Benjamin et al.  as an imFor the well-posedness of global solutions for BBM equation, we can refer to \u20137. For t2, t \u2208 \u211d. Here b \u2260 0, a \u2208 \u211d2, and p \u2265 3 is an integer. The author proved the supremum norms of the solutions with small initial data decay to zero like t\u22122/3 as t tends to infinity.Biler  investigHper2(\u03a9) and Hper1(\u03a9), respectively. Moreover, Wang et al. [By energy equation and weak continuous method, Wang  and Wangg et al.  got the By the decomposition of the semigroup, Wang  studied on, Wang  also obtH01 under certain assumptions, here w is the two-sided real-valued Wiener process on a probability space. He also proved the random attractor is invariant and attracts every pulled-back tempered random set under the forward flow. The asymptotic compactness of the random dynamical system is established by a tail-estimates method, which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity.Wang  considerH1(\u211d3) by showing the solutions are point dissipative and asymptotic compactg \u2208 L2(\u211d3) and f(u) = u + (1/2)u2. Stanislavova [Stanislavova et al.  first prislavova  investigcompactut\u2212\u0394ut\u2212\u03bdBy the method of orthogonal decomposition, Zhu , 18 obtau and Mu  deduced H01(\u03a9). Qin et al. [H02(\u03a9) by weak continuous method. Zhao et al. [J. Park and S. Park  studied n et al.  derived ut\u2212\u0394ut\u2212\u03bdo et al.  investigMoreover, \u00c7elebi et al.  deduced For the upper semicontinuity of corresponding attractors between autonomous and perturb nonautonomous systems, we can refer to Bao , Hale anTo our knowledge, there are less results on the upper semicontinuity of pullback attractors for the 3D nonautonomous BBM equations with the nonautonomous perturbation; we will pay attention to this issue in the sequel.This paper is organized as following. In \ud835\udc9c\u025b = {A\u025b(t)}t\u2208\u211d for the perturbed nonautonomous system with \u025b > 0 and global attractor \ud835\udc9c for the unperturbed autonomous system with \u025b = 0 of the following equation:\u025b = 0.In this section, we will consider the relationship between pullback attractors X be a Banach space with norm ||\u00b7||X. The Hausdorff semidistance dist\u2061X in X between B1\u2286X and B2\u2286X is defined bydX denotes the distance between two points x and y.Let S(t) : X \u2192 X (t \u2208 \u211d) is a C0-semigroup defined on X. If the global attractor \ud835\udc9c for S(t) exists, then it has the following properties: (1) \ud835\udc9c is an invariant, compact set; (2) \ud835\udc9c attracts every bounded sets in X, that is, lim\u2061t\u2192+\u221e\u2061dist\u2061(S(t)B, \ud835\udc9c) = 0 for all bounded subsets B \u2282 X.For an autonomous system, U}t\u2265\u03c4 is said to be a process in X ifU is continuous in X.For a nonautonomous system, the two-parameter mapping class {Now we will recall some definitions and framework on the existence theory of pullback attractors.\ud835\udc9c = {A(t)}t\u2208\u211d is said to be a pullback attractor for the continuous process {U} if it satisfies the following:\ud835\udc9c is invariant for all t \u2265 \u03c4.\ud835\udc9c is pullback attracting, that is, lim\u2061\u03c4\u2192+\u221e\u2061dist\u2061B, A(t)) = 0 for all bounded subsets B \u2282 X.A family of compact sets \u212c = {B(t)}t\u2208\u211d is said to be pullback absorbing for the process U, if for every t \u2208 \u211d and all bounded subsets B \u2282 X, there exists a time T > 0, such thatThe family of subsets \u212c = {B(t)}t\u2208\u211d be a family of subsets in X. A process U is said to be pullback \u212c-asymptotically compact in X if for all t \u2208 \u211d, any sequences \u03c4n \u2192 \u221e and xn \u2208 B(t \u2212 \u03c4n); the sequence {Uxn} is precompact in X.Let \u212c = {B(t)}t\u2208\u211d be pullback absorbing set for the process U and U is pullback \u212c-asymptotically compact in X. Then, the family \ud835\udc9c = {A(t)}t\u2208\u211d that is defined by A(t) = \u039b is a pullback attractor for U in X for the process {U}, whereLet the family of sets \u212c-asymptotic compactness in terms of the noncompact measure.In the following, we will characterize the pullback B \u2282 X, \u212c = {B(t)}t\u2208\u211d be a family of sets in X. A process U is said to be pullback \u212c-\u03ba contracting, if for any t \u2208 \u211d, \u025b > 0, there exists a time T\u212c > 0, such that\u03ba(B) is the Kuratowski noncompact measure defined asLet \u212c = {B(t)}t\u2208\u211d, X and satisfy that for any t \u2208 \u211d, there exists a time U is pullback \u212c-asymptotically compact, if it is pullback \u03ba contracting.Let See, for example, Wang and Qin .U is pullback \u03ba contracting and the family of sets \u212c = {B(t)}t\u2208\u211d is pullback absorbing for U, then the process U possesses a pullback attractor.Assume that the assumptions in See, for example, Wang\u2009\u2009and\u2009\u2009Qin .\u212c = {B(t)}t\u2208\u211d be a family of sets in X. Suppose U = U1 + U2 : \u211d \u00d7 \u211d \u00d7 X \u2192 X satisfies(i)t \u2208 \u211d,for any \u2009+ satisfies lim\u2061\u03c4\u2192+\u221e\u2061\u03a6 = 0 for each t \u2208 \u211d;where \u03a6 : \u211d \u00d7 \u211d \u2192 \u211d(ii)t \u2208 \u211d and T \u2265 0, \u22c3\u03c4\u2264T0\u2264U2B(t \u2212 \u03c4) is bounded and U2B(t \u2212 \u03c4) is precompact in X for any \u03c4 > 0.for any Then the process U is pullback \u212c-\u03ba contracting in X.Let See, for example, Wang and Qin .\u025b \u2208 .We now perturb the nonautonomous term with a small parameter t \u2208 \u211d,\u2009\u2009\u03c4 \u2208 \u211d, and x \u2208 X, we haveX.For each H1) holds, and for any \u025b \u2208 }t\u2208\u211d for all \u025b > 0. If there exists a compact set K \u2282 X, such that\ud835\udc9c\u025b and \ud835\udc9c have the upper semicontinuity, that is,Assume that  for the process generated by the nonautonomous dissipative system.In order to apply \u212c = {B(t)}t\u2208\u211d is pullback absorbing for U, and for each \u025b \u2208 }t\u2208\u211d is a family of compact sets in X. Suppose U\u025b = U\u025b1, + U\u025b2, : \u211d \u00d7 \u211d \u00d7 X \u2192 X satisfies(i)t \u2208 \u211d and any \u025b \u2208  = 0 for each t \u2208 \u211d;where \u03a6 : \u211d \u00d7 \u211d \u2192 \u211d(ii)t \u2208 \u211d and any T \u2265 0, \u222a\u03c4\u2264T0\u2264U\u025b2,B(t \u2212 \u03c4) is bounded, and for any t \u2208 \u211d, there exists a time T\u212c(t) > 0, which is independent of \u025b, such thatfor any \u2009K \u2282 X, such thatand there exists a compact set \u2009\u025b \u2208 }t\u2208\u211d and (H2) holds.Then for each Assume that the family See, for example, Wang and Qin .In this section, firstly, we recall some notations about the functional spaces which will be used later to discuss the regularity of pullback attracting set.A is denoted by A = \u2212\u0394 with domain D(A) = H2(\u03a9)\u22c2H01(\u03a9) and \u03bb is the first eigenvalue of A; we consider the family of Hilbert spacesD(As/2)\u21aaD(Ar/2) for any s > r and the continuous embeddings \u2208 [0, 3/2), \u210b2 = H2(\u03a9)\u22c2H01(\u03a9).The operator \ud835\udcab to the systems  = \ud835\udcabg.Then, applying the Helmholtz-Leray projector  systems \u20133), we , we \ud835\udcab to systems \u2013(3)(29u\u03c4 \u2208 H01(\u03a9), the external force g \u2208 Lloc\u20612. Also we assume that there exist constants \u03b2 > 0, 0 \u2264 \u03b1 < \u03c3/2, and \u03c3 = 2\u03bd/((2/\u03bb) + 2), such thatf is locally square integrable in time; that is, f \u2208 Lloc\u20612 and satisfies\u03b7 \u2264 min\u2061{\u03bb\u03bd, \u03bd, \u03bd/((2/\u03bb) + 2)} and any t \u2208 \u211d.Assume that es that\u222b\u2212\u221ete\u03c3s|Fi  are smooth functions satisfyings \u2208 \u211d, where C1, C2, and \u03c3 are positive constants.For the nonlinear vector function At last, we will state the main result and the proof of this paper as the following.u\u03c4 \u2208 V, then the pullback attractors \ud835\udc9c\u025b = {\ud835\udc9c\u025b(t)}t\u2208\u211d for  is defined on the Banach space V.Assume that \u201336) hol hol36) ht\u2208\u211d for  (which it\u2208\u211d for ) with \u025b or \ud835\udc9c for  with \u025b =\u025b = 0 and pullback attractors for nonautonomous system  (t \u2208 \u211d) generated by problem  hol hol36) h problem  (or prob problem \u2013(3)) witUsing similar technique as in , 17, 18,u\u03c4 \u2208 V, then problem  (\u025b \u2265 0) satisfyingU\u025b} generated by the global solutions possess pullback attractors \ud835\udc9c\u025b for all \u025b \u2265 0 in V.Assume that \u201336) hol hol36) h problem  possesseSee, for example, .u\u025b(t) = U\u025bu\u03c4 of  > 0, such thatR\u025b(t) = C\u025be\u03b7t\u2212\u222b\u221e\u2212te\u03b7s||f(s)||H2ds, and C is a positive constant independent of B, t, \u03c4.Suppose that \u201336) hol hol36) h\u03c3 = 2\u03bd/((2/\u03bb) + 2), R\u03c3 = {r : R \u2192  | lim\u2061t\u2192\u2212\u221ee\u03c3tr2(t) = 0} and denote by \ud835\udc9f\u03c3 the class of families V centered at zero with radius We choose t \u2208 \u211d,\u2009\u2009\u03c4 \u2208 \u211d, and u\u03c4 \u2208 V be fixed, and denoteu \u2208 C; V), then for all u \u2208 V, we derive that\u03c4 \u2208 \u211d.Let \u03c3, we easily getu\u03c4 \u2208 D(\u03c4),\u2009\u2009t \u2265 \u03c4.Let R\u025b(t) the nonnegative number given for each t \u2208 \u211d byV defined by\ud835\udc9f\u03c3-pullback absorbing for the process {U}.Setting \u212c\u025b = {B\u025b(t)}t\u2208\u211d is pullback absorbing in V easily. Moreover,SettingR\u025b(t), B\u025b(t) be given as above. For any t \u2208 \u211d, the solution v(t) = U\u025b1,u(t \u2212 \u03c4) of (\u03c4 \u2265 0 and ut\u2212\u03c4 \u2208 D\u025b(t \u2212 \u03c4).Let  \u2212 \u03c4) of  satisfiev and integrating over \u03a9, we deriveB(\u00b7) and \u2131i(0) = 0 asMultiplying equation in  with v a\u03b7 \u2264 min\u2061{\u03bb1\u03bd, \u03bd}.Using Poincar\u00e9's inequality, it followst \u2212 \u03c4 to t, we gett \u2265 \u03c4, which completes our proof.Integrating  from t \u2212\u212c\u025b(t) = {B\u025b(t)}t\u2208\u211d be given by  > 0 and a function I\u025b(t) > 0, such that the solution U\u025b2,u\u03c4 = w(t) of  and any ut\u2212\u03c4 \u2208 B\u025b(t \u2212 \u03c4).Let given by  and 52)\u212c\u025b(t)  w(t) of  satisfieA\u03c3w(t) in H, we deriveC depends on ||ut\u2212\u03c4||V2, \u03c3, and the first eigenvalue \u03bb of the operator A.Taking the inner product of equation in  with A derive12ddt,Arding to , we havet \u2212 \u03c4 to t, we conclude thatt > \u03c4. This completes the proof of desiring lemma.Integrating  from t \u2212t \u2208 \u211d, any \u03c4 > 0, if u0 varies in bounded sets, then the solution u\u025b(t) = U\u025bu0 of problem (u(t) = S(t)u0 of the unperturbed problem  satisfiesy\u025b(t), using  + 2)}, we knowf \u2208 Lloc\u20612, using Lemmas \u025b \u2192 0+, which implies (Denotetisfiesyt\u03f5+Ayt\u025b), using \u201336) and and(64),y\u025b|\u2202\u03a9=0,.Using \u2013(36) andhat is,ddt(||y\u025b implies .D(As/2)\u21aa(Ls)6/(3\u22122(\u03a9))3 is compact, combining Lemmas Since the embedding"}
{"text": "Some new results on existence and uniqueness of a solution are established by using fixed point theorems. Some illustrative examples are also presented. We extend previous results even in the integer case \u03b1 = 1.We study a boundary value problem for the system of nonlinear impulsive fractional differential equations of order  For the last decades, fractional calculus has received a great attention because fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various processes of science and engineering. Indeed, we can find numerous applications in viscoelasticity \u20133, dynamOn the other hand, the study of dynamical systems with impulsive effects has been an object of intensive investigations in physics, biology, engineering, and so forth. The interest in the study of them is that the impulsive differential systems can be used to model processes which are subject to abrupt changes and which cannot be described by the classical differential problems  and f : J \u00d7 R \u2192 R is a given continuous function.Fe\u010dkan et al.  investigcD0+\u03b1 is the Caputo fractional derivative of order \u03b1 \u2208  with the lower limit zero, f : J \u00d7 R \u2192 R is jointly continuous, tk satisfy 0 = t < t1 < \u22ef<tp < tp+1 = T, x(tk+) = lim\u2061\u03b5\u21920+x(tk + \u03b5) and x(tk\u2212) = lim\u2061\u03b5\u21920+x(tk \u2212 \u03b5) represent the right and left limits of x(t) at t = tk, Ik \u2208 C, and a, b, c are real constants with a + b \u2260 0.In , Guo andn-dimensional analogues of the problem \u22600. Here f, g :  \u00d7 Rn \u2192 Rn and Ii : Rn \u2192 Rn are given functions.Motivated by the papers above, in this paper, we study impulsive fractional differential equations with the two-point and integral boundary conditions in the following form:The rest of the paper is organized as follows. In C we denote the Banach space of all continuous functions from J to Rn with the normRn. We also introduce the Banach spaceA \u2208 Rn\u00d7n, then ||A|| is the norm of A.In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. By Let us recall the following known definitions and results. For more details see , 16.g \u2208 C and \u03b1 > 0, then the Riemann-Liouville fractional integral is defined byz asIf \u03b1 > 0 of a continuous function g :  \u2192 R is defined byn = [\u03b1] + 1 (the notation [\u03b1] stands for the largest integer not greater than \u03b1).The Caputo fractional derivative of order g(t), the Caputo fractional derivative becomes the conventional integer order derivative of the function g(t) as \u03b1 \u2192 n.Under natural conditions on \u03b1, \u03b2 > 0 and n = [\u03b1] + 1; then the following relations hold:Let \u03b1 > 0, g(t) \u2208 C\u22c2L1, the homogeneous fractional differential equation,ci \u2208 R, i = 0,1,\u2026, n \u2212 1, and n = [\u03b1] + 1.For g(t) \u2208 C\u22c2L1, with derivative of order n that belongs to C\u22c2L1; thenci \u2208 R, i = 0,1,\u2026, n \u2212 1, and n = [\u03b1] + 1.Assume that p, q \u2265 0, f \u2208 L1. ThenT]. Moreover, if f \u2208 C, then , then cD0+\u03b1I0+\u03b1f(t) = f(t) for all t \u2208 .If We define a solution problem  as follox \u2208 PC is said to be a solution of problem (cD0+\u03b1x(t) = f), for t \u2208 , t \u2260 ti, i = 1,2,\u2026, p, and for each i = 1,2,\u2026, p, x(ti+) \u2212 x(ti) = Ii(x(ti)), 0 = t0 < t1 < t2 < \u22ef<tp < tp+1 = T, and the boundary conditions Ax(0) + Bx(T) = \u222b0Tg)ds are satisfied.A function  problem  if cD0We have the following result which is useful in what follows.f, g \u2208 C. Then the function x is a solution of the boundary value problem for impulsive differential equationLet x is a solution of the boundary value problem  is still an arbitrary constant vector. For determining x(0) we use the boundary value condition Ax(0) + Bx(T) = \u222b0Tg(s)ds:t \u2208  = f(t), t0 < t \u2264 t1. If t \u2208  = f(t), tk < t \u2264 tk+1, and x(tk+) \u2212 x(tk) = Ik(x(tk)). The lemma is proved.Conversely, assume that atisfies . If t \u2208 X be a Banach space and W \u2282 PC. If the following conditions are satisfied,W is uniformly bounded subset of PC,W is equicontinuous in , k = 0,1, 2,\u2026, p, where t0 = 0, tp+1 = T,W(t) = {u(t) : u \u2208 W, t \u2208 J\u2032}, W(tk+) = {u(tk+) : u \u2208 W}, and W(tk\u2212) = {u(tk\u2212) : u \u2208 W} are relatively compact subsets of X,then W is a relatively compact subset of PC.Let (H1)f, g : J \u00d7 Rn \u2192 Rn are continuous functions.(H2)Lf > 0 and Lg > 0 such thatThere are constants \u2009t \u2208  and all x, y \u2208 Rn.for each (H3)li > 0, i = 1,2,\u2026, p such thatThere exist constants \u2009x, y \u2208 Rn.for all For brevity, letOur first result is based on Banach fixed point theorem. Before stating and proving the main results, we introduce the following hypotheses.J.Assume that (H1)\u2013H3) hold. If hold. IfThe proof is based on the classical Banach fixed theorem for contractions. Let us setQ be the following operator:Q maps Br into Br. It is clear that Q is well defined on PC. Moreover for x \u2208 Br and t \u2208  into itself.Let Q is a contraction. Let x, y \u2208 PC. Then, for each t \u2208  due to condition  which is a unique solution to (Next we will show that ondition  and the  we have6|(Qx)(t)(H4)Nf > 0, Ng > 0 such that |f| \u2264 Nf, |g| \u2264 Ng for each t \u2208 J and all x \u2208 Rn.There exist constants (H5)Ik \u2208 C. The second result is based on the Schaefer fixed point theorem. We introduce the following assumptions.J.Assume that (H1), (H4), and (H5) hold. Then the boundary value problem  has at lQ defined by . Then, for each k = 0,1, 2,\u2026, p and for all t \u2208 .\u03b7 > 0, there exists a positive constant l such that, for each x \u2208 B\u03b7 = {x \u2208 PC : ||x||PC \u2264 \u03b7}, we have ||Q(x)||PC \u2264 l. By (H4), (H5) we have, for each k = 1,2,\u2026, p and for all t \u2208 .\u03c41, \u03c42 \u2208  as in Step\u2009\u20092, and let x \u2208 B\u03b7. Then\u03c41 \u2192 \u03c42, the right-hand side of the above inequality tends to zero.Let X = Rn), we can conclude that the operator Q : PC \u2192 PC is completely continuous.As a consequence of Steps\u2009\u20091 to 3 together with the Arzela-Ascoli theorem (Step\u2009\u20094. One has a priori bounds.Now it remains to show that the setx = \u03bbQ(x) for some 0 < \u03bb < 1. Thus, for each t \u2208 |=|\u03bbIn this section, we give some examples to illustrate our main results.f1 = cos\u2061((1/10)x2(t)),\u2009\u2009f2 = (et\u2212/(9 + et))\u00b7(|x1|/(1 + |x1|)), and\u2009\u2009T = 1,\u2009\u2009p = 1,\u2009\u2009Consider problem  with f1\u03b1 = 0,2. Indeed,Evidently,ondition  is satisThen, by Consider\u03b1 \u2264 1,\u2009\u2009f1 = et\u2212/(1 + x22(t)),\u2009\u2009f2 = sinx1(t),\u2009\u2009T = 1,\u2009\u2009p = 1,\u2009\u2009Nf = 1, Ng = 0), and consequently boundary value problem (Here 0 <  problem  has at l"}
{"text": "RN. For time t \u2265 0, we can define a functional H(t) associated with the solution of the equations and some testing function f. When the pressure function P of the governing equations is of the form P = K\u03c1\u03b3, where \u03c1 is the density function, K is a constant, and \u03b3 > 1, we can show that the nontrivial C1 solutions with nonslip boundary condition will blow up in finite time if H(0) satisfies some initial functional conditions defined by the integrals of f. Examples of the testing functions include rN\u22121ln(r + 1), rN\u22121er, rN\u22121(r3 \u2212 3r2 + 3r + \u03b5), rN\u22121sin((\u03c0/2)(r/R)), and rN\u22121sinh\u2009r. The corresponding blowup result for the 1-dimensional nonradial symmetric case is also given.We study, in the radial symmetric case, the finite time life span of the compressible Euler or Euler-Poisson equations in  As usual, \u03c1 = \u03c1 \u2265 0 and u = u \u2208 RN are the density and the velocity, respectively. P = P(\u03c1) is the pressure function. The \u03b3-law for the pressure term P(\u03c1) can be expressed as\u03b3 \u2265 1. If K > 0, it is a system with pressure. If K = 0, it is a pressureless system.The compressible isentropic Euler 1/2.The solutions in radial symmetry are expressed by3 becomesThe Poisson equation 3 becomesThe equations in radial symmetry can be expressed in the following form:\u03b4 = 0), Makino et al.  testing function f(r) satisfying the following properties:r\u21920(f(r)/rN\u22121) exists,lim\u2061f(r)/r is increasing,then the solutions blow up in finite time.(a) For the attractive forces (\u03b4 = 0 or 1), if H0 satisfies the following initial functional condition:T = (2R\u222b0Rf(r)dr)/H0.(b) For the nonattractive forces (The key ideas in obtaining the above results are (i) to design the right form of generalized functional and find the right class of testing functions and (ii) to transform the nonlinear partial differential equations into the Riccati inequality.\u03c1) conserves its nonnegative nature.The density function 1\u03c10) \u2265 0.The mass equation 1(11)D\u03c1D\u03c10(r) \u2260 0, we havef(r) on both sides.)For the nontrivial density initial condition in radial symmetry, r from 0 to R for \u03b3 > 1 or K = 0:\u03b4 = \u22121, we havedF(r) = f(r)dr and \u03b3 > 1 or K = 0.Subsequently, we take integration with respect to o deduce\u222b0Rf(r)Vtwe have1\u222b0Rf(r)Vtf(0) = 0 by property 1 and f is increasing by property 2.Note that R > 0,f\u2032(r)\u2265(1/r)f(r) by property 2.Now, we define the assistant functional:2f(r)dr,H2(t)2R\u222b0 view of , we getT.It is well known that, with the initial conditionequality  will blo\u03b4 = 0 or 1, by a similar analysis, one can show thatT = (2R\u222b0Rf(r)dr)/H0.(b) For The proof is completed.H(t), readers may refer to Sideris' paper  and vanish at the boundaries:t \u2265 0. By considering u and \u03c1 instead, one may suppose a \u2265 0. Let f(x) be a nonnegative and nonzero C1 testing function, such that f(x)/x is increasing for x > a and the functional is given by(a)\u03b4 = 1 or \u22121, if the initial functional H0 satisfies For \u2009then the solutions blow up in finite time.(b)\u03b4 = 0, if H0 > 0, then the solutions blow up on or before the finite time T = (2b\u222babf(x)dx)/H0. For Suppose 2 becomes\u03b3 \u2260 1, one hasf(x) on both sides, taking integration with respect to x from a to b and using integration by parts, to yieldu = u = \u03c1 = \u03c1 = 0, for all t, we getf(x) and the Cauchy-Schwarz inequality  For \u03b4 = 0,(b) For"}
{"text": "N governed by a pair of parameters \u2113 and \u03b3. We study long-range percolation on the bipartite hierarchical lattice where any edge (running between vertices of unlike bipartition sets) of length k is present with probability pk = 1 \u2212 exp(\u2212\u03b1\u03b2k\u2212), independently of all other edges. The parameter \u03b1 is the percolation parameter, while \u03b2 describes the long-range nature of the model. The model exhibits a nontrivial phase transition in the sense that a critical value \u03b1c \u2208  if and only if \u2113 \u2265 1, 1 \u2264 \u03b3 \u2264 N \u2212 1, and \u03b2 \u2208 . Moreover, the infinite component is unique when \u03b1 > \u03b1c.We propose a family of bipartite hierarchical lattice of order  N \u2265 2, the hierarchical lattice of order N is defined byd on \u03a9N is defined byx, y, z \u2208 \u03a9N. This means that  is an ultrametric space. Roughly speaking, this corresponds to the leaves of an infinite N-ary tree, with metric distance half the graph distance.For an integer \u03a9N is analyzed in  on a bipartite square lattice with bipartition V1 = {v =  \u2208 \u21242 : vx \u2212 vy\u2009\u2009is\u2009\u2009odd} and V2 = {v =  \u2208 \u21242 : vx \u2212 vy\u2009\u2009is\u2009\u2009even} such that \u21242 = V1 \u222a V2. In other words, the bond percolation cannot occur on the above bipartite square lattice for occupation probability 2p(1 \u2212 p) \u2264 1/2. This is consistent with the classical result which says that bond percolation on \u21242 does not occur when occupation probability \u22641/2 . Otfound in .P(k) \u221d k\u03b3\u2212. In this model, an edge between vertices with degrees k1 and k2 is occupied with probability proportional to (k1k2)\u03b1\u2212. By using generating function method, Hooyberghs et al. [PA(k) \u221d k\u03b3A\u2212 and PB(k) \u221d k\u03b3B\u2212, respectively, has the same critical behaviors with biased percolation on a monopartite scale-free network when \u03b3A = \u03b3B = \u03b3.Another example is the biased percolation , 19 on i\u03a9N  can be viewed as a spanning subgraph of that from . The uniqueness of the infinite component holds here for the same reason as the uniqueness result for the percolation graph of \u03b2 = N2 and the graph distance  between 0 and a vertex x. It is also interesting to study the mean field percolation (N \u2192 \u221e) and compare it with that on \u03a9N [\u03a9N (other than the 2-coloring addressed in this paper) are possible.We consider t on \u03a9N . Directet on \u03a9N  and otheWe start with some notations. Then we prove x \u2208 , define Br(x) as the ball of radius r around x; that is, Br(x) = {y : d \u2264 r}. We make the following observations. Firstly, for any vertex x, Br(x) contains Nr vertices. In particular, if r < \u2113, all vertices in the ball have the same type. Secondly, Br(x) = Br(y) if d \u2264 r. Finally, for any x, y, and r, we have either Br(x) = Br(y) or Br(x)\u2229Br(y) = \u2205.For a vertex S of vertices, denote by Cn(x) be the component of vertices that are connected to x by a path using only vertices within Bn(x). For disjoint sets S1, S2\u2286\u03a9N, we denote by S1\u2194S2 the event that at least one edge joins a vertex in S1 to a vertex in S2. S1\u21aeS2 means the event that such an edge does not exist. By definition, if S1, S2\u2286\u03a9Ni for i = 1 or 2, then S1\u21aeS2 occurs with probability 1. Let Cnm(x) be the largest component in Bn(x). If there are more than one such components, just take any one of them as Cnm(x). It is clear that |Cnm(x)| = max\u2061y\u2208Bn(x) | Cn(y)| [For a set  Cn(y)| .Ak be the event that the origin 0 \u2208 \u03a9N1 connects by an edge to at least one vertex at distance k in \u03a9N2. By construction, for k < \u2113, P(Ak) = 0. For k = \u2113, there are ((N \u2212 \u03b3)/(N \u2212 1))(N \u2212 1)Nk\u22121 vertices in \u03a9N2 at distance k from 0. Hence,k > \u2113, there are ((N \u2212 \u03b3)/N)(N \u2212 1)Nk\u22121 vertices in \u03a9N2 at distance k from 0. Similarly, we obtaink > \u2113.Let by using . For k >Ak}k\u22651 are independent and\u03b2 \u2264 N, 1 \u2264 \u03b3 \u2264 N \u2212 1, and \u03b1 > 0, infinitely many of Ak occur with probability 1 by the second Borel-Cantelli lemma. Thus, \u03b8 = 1 for any \u2113 > 1, 1 \u2264 \u03b3 \u2264 N \u2212 1, \u03b1 > 0, and 0 < \u03b2 \u2264 N. The result then follows.Since all the events {\u03b1c = \u221e by virtue of the monotonicity. Note that, for j \u2265 \u2113, there are (\u03b3/N)Nj vertices in Bj(0)\u2009\u2229\u2009\u03a9N1 and ((N \u2212 \u03b3)/N)Nj vertices in Bj(0)\u2009\u2229\u2009\u03a9N2. Hence, by the comments in the proof of (i) and taking \u03b2 = N2, for any j \u2265 \u2113, we obtain\u03b1 \u2265 0.We only need to show n\u2113 = 0 and i such that \u03b8 = 0 for all \u03b1 \u2265 0. This implies \u03b1c = \u221e.Let  We have{|C(0)|=\u221e\u03b1c is a direct consequence of the proof of Theorem 1(b) in [\u03a9N1, \u03a9N2, d) can be viewed as a spanning subgraph of that of , the percolation cluster C(0) is almost surely finite; namely, \u03b8 = 0, for \u03b1 small enough.The positivity of  1(b) in . Since t\u03b1c. The main technique to be used is an iteration involving the tail probability of binomial distributions [\u03b2 < N2, we choose an integer K and a real number \u03b4 such that\u03b4 < N. For n \u2265 1, letan is the probability that the largest component of a ball of radius nK contains at least (\u03b3/N)\u03b4nK vertices in \u03a9N1 and at least ((N \u2212 \u03b3)/N)\u03b4nK vertices in \u03a9N2. Such a ball is said to be good. We set a0 = b0 = 1 by convention. It is clear that, for \u03b1 > 0, all an and bn are positive, since nK is a finite number and the connection probability in  for some c > 0.We show that there exists some \u03b1 > 0 such that liminf\u2061n\u2192\u221ebn > 0. We show that there exists some \u2115 the nonnegative integers. We can naturally label the vertices in \u03a9N via the map f : \u03a9N \u2192 \u2115 asnK is said to be very good if it is good and its largest component connects by an edge to the largest component of the first (as per the aforementioned order) good subball in the same ball of radius (n + 1)K. Clearly, the first good subball of radius nK in a ball of radius (n + 1)K is very good. Condition (NK \u2212 1)\u03b4nK \u2265 \u03b4n+1)KK((0) is good if (a) it contains NK \u2212 1 good subballs of radius nK, and (b) all these good subballs are very good.We start with ondition  implies nK in a ball of radius (n + 1)K has a binomial distribution Bin with parameters NK and an. Given the collection of good subballs, the probability that the first such good subball is very good equals to 1. Fix any of the other good subballs, say B; the probability that B is not very good is upper bounded bya and b are the number of vertices in the largest component of the first good subball in \u03a9N1 and \u03a9N2, respectively; likewise, c and d are the number of vertices in the largest component of the subball B in \u03a9N1 and \u03a9N2, respectively. By definition, we have a, c \u2265 (\u03b3/N)\u03b4nK, b, d \u2265 ((N \u2212 \u03b3)/N)\u03b4nK, and the distance between two vertices in a ball of radius (n + 1)K is at most (n + 1)K.The number of good subballs of radius B to be very good is at leat 1 \u2212 \u03b5n. Thus, the number of very good subballs is stochastically larger than a random variable obeying a binomial distribution Bin). From the above comments (a) and (b) and the definition of an, it follows that\u03ben = 1 \u2212 an, we obtainc > 0 large enough so that \u03b1 large enough so that (c) \u03b5n \u2264 exp\u2061(\u2212c(n + 1)) and (d) \u03be1 \u2264 exp\u2061(\u22122c) hold. To see (c), note that \u03b2 < \u03b42 and then\u03b1\u2192\u221e\u03b50 = 0, \u03be0 = 0 and by ), then\u03ben \u2264 exp\u2061(\u2212c(n + 1)) \u2264 exp\u2061(\u2212cn) for all n \u2208 \u2115. We then finish the proof of According to our above choice of bn, we claim thatCnK(0)\u2229\u03a9N1 | \u2265(\u03b3/N)\u03b4nK and |CnK(0)\u2229\u03a9N2 | \u2265((N \u2212 \u03b3)/N)\u03b4nK, then BnK(0) is the first good subball in the derivation above. If this component is connected to at least NK \u2212 2 other large components in Bn+1)K((0) as above, then the component containing the origin in Bn+1)K((0) has (\u03b3/N)\u03b4nK(NK \u2212 1)\u2265(\u03b3/N)\u03b4n+1)K( vertices in \u03a9N1 and ((N \u2212 \u03b3)/N)\u03b4nK(NK \u2212 1)\u2265((N \u2212 \u03b3)/N)\u03b4n+1)K converges to 1 exponentially fast. It then follows from ) > 0 for all n. Since b1 > 0, inequality (A simple coupling givesP(Bin(NK\u2212 side of  and the ows from  that(27. Hence,bn+1\u2265b1\u220fk"}
{"text": "Our results generalize some recent results from cone metric and cone rectangular metric spaces into ordered cone rectangular metric spaces. Examples are provided which illustrate the results.We prove some common fixed-point theorems for the ordered  There are a number of generalizations of metric spaces. One such generalization is obtained by replacing the real valued metric function with a vector valued metric function. In the mid-20th century (see ), the noIn , BranciaOrdered normed spaces and cones have applications in applied mathematics, for instance, in using Newton's approximation method  and in og-weak contraction is introduced by Vetro  in conend Zhang , Azam etnd Zhang , Azam annd Zhang , Malhotrnd Zhang , and thend Zhang  on orderWe need the following definitions and results, consistent with , 20.E be a real Banach space and P a subset of E. The set P is called a cone ifP\u2009\u2009is closed, nonempty, and P \u2260 {\u03b8}; here \u03b8 is the zero vector of E; a, b \u2208 \u211d, a, b \u2265 0, x, y \u2208 P\u21d2ax + by \u2208 P;x \u2208 P and \u2212x \u2208 P\u21d2x = \u03b8.Let P \u2282 E, we define a partial ordering \u201c\u2aaf\u201d with respect to P by x\u2aafy if and only if y \u2212 x \u2208 P. We write x\u227ay to indicate that x\u2aafy but x \u2260 y, while x \u226a y if and only if y \u2212 x \u2208 P0, where P0 denotes the interior of P.Given a cone P be a cone in a real Banach space E; then P is called normal, if there exists a constant K > 0 such that for all x, y \u2208 E,K satisfying the above inequality is called the normal constant of P.Let X be a nonempty set and E a real Banach space. Suppose that the mapping d : X \u00d7 X \u2192 E satisfies\u03b8\u2aafd for all x, y \u2208 X and d = \u03b8 if and only if x = y; d = d for all x, y \u2208 X; d\u2aafd + d for all x, y, z \u2208 X. Let d is called a cone metric on X, and  is called a cone metric space. In the following, we always suppose that E is a real Banach space, and P is a solid cone in E; that is, P0 \u2260 \u03d5 and \u201c\u2aaf\u201d is partial ordering with respect to P.Then, E = \u211d and P = }. Then, this cone is not normal .d : X \u00d7 X \u2192 E as follows:X, d) is nonnormal cone rectangular metric space but  is not a cone metric space because it lacks the triangular property. Define f and g be self-mappings of a nonempty set X and C = {x \u2208 X : fx = gx}. The pair  is called weakly compatible if fgx = gfx for all x \u2208 C. If w = fx = gx for some x in X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g.Let X is equipped with a partial order \u201c\u2291\u201d and mapping d : X \u00d7 X \u2192 E such that  is a cone rectangular metric space, then  is called an ordered cone rectangular metric space. Let f, g : X \u2192 X be two mappings. The mapping f is called nondecreasing with respect to \u201c\u2291\u201d, if for each x, y \u2208 X, x\u2291y implies fx\u2291fy. The mapping f is called g-nondecreasing if for each x, y \u2208 X, gx\u2291gy implies fx\u2291fy. A subset \ud835\udc9c of X is called well ordered if for all the elements of \ud835\udc9c are comparable; that is, for all x, y \u2208 \ud835\udc9c either x\u2291y or y\u2291x. \ud835\udc9c is called g-well ordered if all the elements of \ud835\udc9c are g-comparable; that is, for all x, y \u2208 \ud835\udc9c either gx\u2291gy or gy\u2291gx.If a nonempty set g = IX (the identity mapping of X), the g-well orderedness reduces into well orderedness. But, for nontrivial cases, that is, when g \u2260 IX the concepts of g-well orderedness and well orderedness are independent.  In the trivial case, that is, for X = {0,1, 2,3, 4}, let \u201c\u2291\u201d be a partial order relation on X defined by \u2291 = {, ,, , , , , , }. Let \ud835\udc9c = {0,1, 3}, \u212c = {1,4} and g : X \u2192 X be defined by g0 = 1, \u2009g1 = 2, \u2009g2 = 3, \u2009g3 = 3, \u2009g4 = 0. Then it is clear that \ud835\udc9c is not well ordered but it is g-well ordered, while \u212c is not g-well ordered but it is well ordered. Let X, \u2291, d) be an ordered cone rectangular metric space f, g : X \u2192 X two mappings. The mapping f is called ordered Reich-type contraction if for all x, y \u2208 X with x\u2291y, \u2009\u03bb, \u2009\u03bc, \u2009\u03b4 \u2208  for all x, y \u2208 X with gx\u2291gy, where \u03bb \u2208 }. Define d : X \u00d7 X \u2192 E as follows:X, d) is a complete nonnormal cone rectangular metric space but not cone metric space. Define mappings f, \u2009g : X \u2192 X and partial order \u201c\u2291\u201d on X as follows:f is an ordered g-weak contraction in  with \u03bb \u2208 [1/3,1), \u03bc = \u03b4 = 0. Indeed, we have to check the validity of  = , . Then,\u03bb \u2208 [1/3,1), \u03bc = \u03b4 = 0. Again,\u03bb, \u03b4, and \u03bc such that \u03bb + \u03bc + \u03b4 < 1.Let idity of  only forerefore,  holds foerefore,  holds fof and g. Note that f is not an ordered Reich-type contraction. Indeed, for point  =  there are no \u03bb, \u03bc, \u03b4 \u2208 [0,1) such that condition  be the cone rectangular metric space as in X, d) is a complete nonnormal cone rectangular metric space but not cone metric space. Define mappings f, g : X \u2192 X and partial order \u201c\u2291\u201d on X as follows:f is an ordered g-weak contraction in  with \u03bb = \u03b4 \u2208 [1/8,1/2), \u2009\u03bc = 0. Indeed, we have to check the validity of  = , , . Then,\u03bb = \u03b4 \u2208 [1/8,1/2), \u03bc = 0. Again,\u03bb, \u03b4, and \u03bc such that \u03bb + \u03bc + \u03b4 < 1.Let  = , and condition  are satisfied and 3 is a coincidence point of f and g. Note that f3 = g3 = 2; that is,\u2009\u20093 is a coincidence point of f and g but fg3 \u2260 gf3; therefore, f and g are not weakly compatible and have no common fixed point. All other conditions of f, \u201cnondecreasing\u201d and \u201ccompleteness of space,\u201d are replaced by another condition.  In the following theorem, the conditions on X, \u2291, d) be an ordered cone rectangular metric space and f, g : X \u2192 X is two mappings such that f(X) \u2282 g(X). Suppose that the following conditions are satisfied: f is an ordered g-weak contraction that satisfies \u2aafd for all x \u2208 X. there exists Let  = d for all x \u2208 X and gz = fu for some z \u2208 X  \u2282 g(X)); then F(u)\u2aafF(x) for all x \u2208 X. If F(u) = \u03b8, then gu = fu; that is,\u2009\u2009u is a coincidence point of f and g. If \u03b8\u227aF(u), then by assumption (B) gu\u2291fu, so gu\u2291gz, and by (A), we obtainF(u) = \u03b8; that is,\u2009\u2009gu = fu, and so u is a coincidence point of f and g.Let The existence, necessary and sufficient condition for uniqueness of common fixed point follows from a similar process as used in"}
{"text": "G\u2032/G)-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, trigonometric, and rational functions. It is shown that the new approach of generalized (G\u2032/G)-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations in mathematical physics and engineering.Exact solutions of nonlinear evolution equations (NLEEs) play a vital role to reveal the internal mechanism of complex physical phenomena. In this work, the exact traveling wave solutions of the Boussinesq equation is studied by using the new generalized (05.45.Yv, 02.30.Jr, 02.30.Ik G\u2032/G)-expansion method -expansion method -expansion method -expansion method (Naher and Abdullah \u03b7))-expansion method  expansion method to construct the exact traveling wave solutions of the Boussinesq equation.The objective of this article is to apply the new generalized (G'/G)-expansion method, we give the description of the new generalized (G'/G) expansion method. In Section Application of the method, we apply this method to the Boussinesq equation, results and discussions and graphical representation of solutions. Conclusions are given in the last section.The outline of this paper is organized as follows: In Section Description of the new generalized  is an unknown function, \u0424 is a polynomial in v and its derivatives in which highest order derivatives and nonlinear terms are involved and the subscripts stand for the partial derivatives.where Step 1: We combine the real variables x and t by a complex variable \u03b7V is the speed of the traveling wave. The traveling wave transformation (2) converts Eq. (v=v(\u03b7):where erts Eq.  into an \u03c8 is a polynomial of v and its derivatives and the superscripts indicate the ordinary derivatives with respect to \u03b7.where Step 2: According to possibility, Eq.  and \u03b2i  and d are arbitrary constants to be determined and M(\u03b7) iswhere either G=G(\u03b7) satisfies the following auxiliary nonlinear ordinary differential equation:where \u03b7; A, B, C and E are real parameters.where the prime stands for derivative with respect to Step 4: To determine the positive integer N, taking the homogeneous balance between the highest order nonlinear terms and the derivatives of the highest order appearing in Eq. N  and (d+M)-N . Subsequently, we collect each coefficient of the resulted polynomials to zero, yields a set of algebraic equations for \u03b1i  and \u03b2i , d and V.tute Eq.  and Eq. tute Eq.  includintute Eq.  with theStep 6: Suppose that the value of the constants \u03b1i , \u03b2i , d and V can be found by solving the algebraic equations obtained in Step 5. Since the general solutions of Eq. , \u03b2i , d and V into Eq. \u2009>\u20090,Family 2: When B\u2009\u2260\u20090, \u03c9=A-C and \u03a9 =B2\u2009+\u20094E(A-C)\u2009<\u20090,Family 3: When B\u2009\u2260\u20090, \u03c9\u2009=\u2009A-C and \u03a9 =\u2009B2\u2009+\u20094\u2009E(A-C)\u2009=\u20090,Family 4: When B\u2009=\u20090, \u03c9\u2009=\u2009A-C and \u0394=\u03c9E>0,Family 5: When B\u2009=\u20090, \u03c9\u2009=\u2009A-C and \u0394\u2009=\u2009\u03c9E\u2009<\u20090,G'/G) expansion method to construct many new and more general traveling wave solutions of the Boussinesq equation. Let us consider the Boussinesq equation,In this section, we will put forth the new generalized N  and (d+M)-N . We collect each coefficient of these resulted polynomials to zero yields a set of simultaneous algebraic equations  for \u03b10, \u03b11, \u03b12, \u03b21, \u03b22d, K and V. Solving these algebraic equations with the help of computer algebra, we obtain following:Substituting Eq.  togetherSet 1:n1\u2009=\u2009(\u2009-\u2009A2\u2009+\u2009V2A2\u2009-\u200912d2\u03c92\u2009+\u20098E\u03c9\u2009-\u200912Bd\u03c9\u2009-\u2009B2),\u2009n2\u2009=\u2009(\u2009-\u20092Ed\u03c9\u2009+\u20093Bd2\u03c9\u2009+\u20092d3\u03c92\u2009-\u2009EB\u2009+\u2009B2d),\u2009n3\u2009=\u2009-\u2009(\u2009-\u20092Ed2\u03c9\u2009+\u2009d4\u03c92\u2009+\u20092Bd3\u03c9\u2009+\u2009E2\u2009+\u2009B2d2\u2009-\u20092BdE),\u2009n4\u2009=\u2009-\u2009(\u2009-\u20098EB2\u03c9\u2009+\u2009V4A4\u2009-\u20092V2A4\u2009-\u200916E2\u03c92\u2009+\u2009A4\u2009-\u2009B4),\u2009\u03c9\u2009=\u2009A\u2009-\u2009C,\u2009V,\u2009d,\u2009A,\u2009B,\u2009C,\u2009E are free parameters.where Set 2: \u03c9\u2009=\u2009A-C, V, d, A, B, C, E are free parameters.Set 3: V=V, 1=0,n5\u2009=\u2009((V2\u2009-\u20091)A2\u2009+\u20098E\u03c9\u2009+\u20092B2),\u2009n6\u2009=\u2009-\u2009(16E2\u03c92\u2009+\u20098EB2\u03c9\u2009+\u2009B\u20094),\u2009n7\u2009=\u2009((V2\u2009-\u20091)2A4\u2009-\u2009256E2\u03c92\u2009-\u2009128B2E\u03c9\u2009-16B4),\u2009\u03c9\u2009=\u2009A\u2009-\u2009C,\u2009V,\u2009A,\u2009B,\u2009C,\u2009E are free parameters.where C1 = 0 but C2 \u2260 0; C2 = 0 but C1 \u2260 0 respectively:For set 1, substituting Eq.  into Eq.C1\u2009=\u20090 but C2\u2009\u2260\u20090; C2\u2009=\u20090 but C1\u2009\u2260\u20090 respectively:Substituting Eq.  into Eq.Substituting Eq.  into Eq.C1\u2009=\u20090 but C2\u2009\u2260\u20090; C2\u2009=\u20090 but C1\u2009\u2260\u20090 respectively:Substituting Eq.  into Eq.C1\u2009=\u20090 but C2\u2009\u2260\u20090; C2\u2009=\u20090 but C1\u2009\u2260\u20090 respectively:Substituting Eq.  into Eq.x-Vt.where \u03b7 = C1\u2009=\u20090 but C2\u2009\u2260\u20090; C2\u2009=\u20090 but C1\u2009\u2260\u20090 respectively:Again for set 2, substituting Eq.  into Eq.C1\u2009=\u20090 but C2\u2009\u2260\u20090; C2\u2009=\u20090 but C1\u2009\u2260\u20090 respectively:Substituting Eq.  into Eq.Substituting Eq.  into Eq.C1 = 0 but C2 \u2260 0; C2 = 0 but C1 \u2260 0 respectively:Substituting Eq.  into Eq.C1\u2009=\u20090 but C2\u2009\u2260\u20090; C2\u2009=\u20090 but C1\u2009\u2260\u20090 respectively:Substituting Eq.  into Eq.x-Vt.where \u03b7 = C1\u2009=\u20090 but C2\u2009\u2260\u20090; C2\u2009=\u20090 but C1\u2009\u2260\u20090 respectively:Similarly, for set 3, substituting Eq.  into Eq.C1\u2009=\u20090 but C2\u2009\u2260\u20090; C2\u2009=\u20090 but C1\u2009\u2260\u20090 respectively:Substituting Eq.  into Eq.Substituting Eq.  into Eq.C1\u2009=\u20090 but C2\u2009\u2260\u20090; C2\u2009=\u20090 but C1\u2009\u2260\u20090 respectively:Substituting Eq.  into Eq.C1\u2009=\u20090 but C2\u2009\u2260\u20090; C2\u2009=\u20090 but C1\u2009\u2260\u20090 respectively:Substituting Eq.  into Eq.\u03b7 = x-Vt.where It is worth declaring that some of our obtained solutions are in good agreement with already published results which are presented in the following tables Table\u00a0.Table 1Beside this table, we obtain further new exact traveling wave solutions The graphical illustrations of the solutions are given below in the figures with the aid of Maple Figures\u00a0, 4 and 5The solutions corresponding to G'/G) -expansion method. We apply the new approach of generalized (G'/G)-expansion method for the exact solution of this equation and constructed some new solutions which are not found in the previous literature. This study shows that the new generalized (G'/G)-expansion method is quite efficient and practically well suited to be used in finding exact solutions of NLEEs. Also, we observe that the new generalized (G'/G)-expansion method is straightforward and can be applied to many other nonlinear evolution equations.In this paper, we obtain the traveling wave solutions of the Boussinesq equation by using the new approach of generalized ("}
{"text": "By making use of basic hypergeometric functions, a class of complex harmonic meromorphic functions with positive coefficients is introduced. We obtain some properties such as coefficient inequality,growth theorems, and extreme points. Back in 1748, Euler considered the infinite product \u221e\u22121 = \u220fk=0\u221e(1 \u2212 qk+1)\u22121 and ever since it became very important to many areas. However, it was stated in the literature that the development of these functions was slower until Heine (1878) converted a simple observation such that lim\u2061q\u21921[(1 \u2212 qa)/(1 \u2212 q)] = a, which then returns the theory of\u2009\u20092\u03d51 basic hypergeometric series to the famous theory of Gauss\u2009\u20092F1 hypergeometric series. Various authors qzk\u22121 = kzk\u22121 = h\u2032(z), where h\u2032(z) is the ordinary derivative. For more properties of Dq, see , \u2131srf(z) : MH \u2192 MH is defined as\u2131sr, we introduce the class MH as follows.Using convolution given by  and f(z)given by , a lineaMH denote the family of harmonic meromorphic functions \u03b1\u2009\u2009(0 \u2264 \u03b1 < 1) and for all z \u2208 \ud835\udd4c\u2216{0}.Let r = 1, s = 0, a1 = q, and q \u2192 1, MH is the class of harmonic meromorphic starlike functions of order \u03b1 and for r = 1, s = 0, a1 = q, \u03b1 = 0, and q \u2192 1, the class MH gives the class of harmonic meromorphic starlike functions for all z \u2208 \ud835\udd4c\u2216{0}.For Denote \u03b1 < 1, |q | <1), then f is harmonic univalent sense preserving in \ud835\udd4c\u2216{0} and f \u2208 MH. If the form  and sati\u03b1 < 1. Consider the expressionr\u2009\u2009(0 \u2264 r < 1). Therefore, letting r \u2192 1 in . Note that f is sense preserving in \ud835\udd4c\u2216{0}. This is becauseq\u21921[|Dqh(z)|\u2265|Dqg(z)|] = [|h\u2032(z)|\u2265|g\u2032(z)|]. Hence the theorem.Suppose that  holds trpressionA(z)=|zDqpressionA(z)=|zDqpothesis , it follpothesis  holds, sNow, we prove that condition  is necesLet fined by . Then, f\u211c(z)\u2264|z| for all z, it follows from . Then,| > 1, where i = {2,\u2026, r}, j = {1,\u2026, s}, and 0 < |z | = r < 1, one hasIf fined by  is in th\u03b1 < 1 show that the bounds given in \ud835\udd4c\u2216{0}.We will only prove the right-hand inequality. The proof for the left-hand inequality is similar and we will omit it:Next, we give the following.\u03bbk \u2265 0, \u03b3k \u2265 0, and \u2211k=0\u221e(\u03bbk + \u03b3k) = 1. In particular, the extreme points of hk} and {gk}.Set\u03bbk \u2264 1\u2009 and 0 \u2264 \u03b3k \u2264 1\u2009.\u03bb0 \u2265 0. Consequently, we obtainLetq-hypergeometric functions and analytic functions can be found in [Other work related to found in , 17."}
{"text": "We establish new inequalities similar to Hardy-Pachpatte-Copson's type inequalities. These results in special cases yield some of the recent results. The classical Hardy's integral inequality is as follows.p > 1, f(x) \u2265 0 for 0 < x < \u221e, and F(x) = (1/x)\u222b0xf(t)dt, thenf \u2261 0. The constant is the best possible.If Theorem A was first proved by Hardy , in an ap > 1, m \u2260 1,f(x) \u2265 0 for 0 < x < \u221e, and F(x) is defined byf \u2261 0. The constant is the best possible.If Inequalities  and 3)  3) whichOur main results are given in the following theorems.a < b < R, c < d < R\u2032, p > 1, q < 1, and \u03b1 > 0 be constants. Let w be positive and locally absolutely continuous in \u00d7. Let h be a positive continuous function and let H = \u222bax\u222bcyhds\u2009dt, for \u2208\u00d7. Let f be nonnegative and measurable on \u00d7. Ifx, y) \u2208  \u00d7 , and if F is defined byx, y)\u2208\u00d7, thenLet f, w, h, and r reduce to f(x), w(x), h(x), and r(x), respectively, and with suitable modifications in Let This is just a new inequality established by Pachpatte .Moreover, we note that the inequality established in w(x) = r(x) = 1, \u2009H(R) = R, and \u03b1 = 1 in  cha chaw(x) This is just a new inequality established by Love .h(x) = 1, a \u2192 0, \u2009b \u2192 \u221e, and log\u2061\u2009(R/(x \u2212 a)) = 1 in (Let ) = 1 in ; then  be positive and locally absolutely continuous in \u00d7. Let h be a positive continuous function and let H = \u222bax\u222bcyhds\u2009dt, for \u2208\u00d7. Let f be nonnegative and measurable on \u00d7. Letx, y)\u2208\u00d7. If F is defined byx, y)\u2208\u00d7, thenLet f, w, h, and r reduce to f(x), w(x), h(x), and r(x), respectively, and with suitable modifications in Let This is just a new inequality established by Pachpatte .On the other hand, we note that the inequality established in w(x) = r(x) = 1, H(R) = R, and \u03b2 = 1 in  ch chw(x) =This is just a new inequality established by Love .u = wFp and in view ofx, y)\u2208\u00d7, thenIf we let H = \u222bax\u222bcyhds\u2009dt, for \u2208\u00d7, thenLetx, we haveFrom , 20), a, a20), aq < 1, then we observe thatIf p, p/(p \u2212 1) on the right side of  = wFp and in view ofx, y)\u2208\u00d7, thenIf we let H = \u222bax\u222bcyhds\u2009dt, for \u2208\u00d7, thenLety, we haveFrom , 29), a, a29), aq > 1, then we observe thatIf p, p/(p \u2212 1) on the right side of (By applying H\u00f6lder's inequality with indices  side of , we obtapth power, we obtainDividing both sides of  by the s"}
{"text": "And this system is of wide interest in different branches of science, such as physics, chemistry, biology, evolutionary game theory, and economics. We refer the reader to the book of Hofbauer and Sigmund [It is well known that  Sigmund  for its  Sigmund \u20134 and reXi(t) is the number of individuals in the ith population at time t and Xi(t) \u2265 0, \u03b2i is the intrinsic growth rate of the ith population, the \u03b1ij are interaction coefficients measuring the extent to which the jth species affects the growth rate of the ith, \u03b2i and \u03b1ij are parameters, and the values of these parameters are not very small usually.To study the bifurcation of Lotka-Volterra class, we consider three-dimensional generalized Lotka-Volterra systemsOver the last several decades, many researchers have devoted their effort to study the existence and number of isolated periodic solutions for system . There h3 with two small parameters \u03bb1 and \u03bb2 and other bounded parameters. In aijk, bijk, and cijk for i, j, k = 0,1, 2 are functions of the parameters \u03b2i and \u03b1ij in system  = O(|x|2) is C\u221e in x \u2208 \u211d3, and\u03bb1, \u03bb1, \u03bb2)x we obtainD = diag\u2061(A(\u03bb1), \u03bb2), witha1b1c1 \u2260 0 as in [S1, S2, and S3 are 2\u03c0 periodic in \u03b8, and S1, S2, and S3 = O. By a further scaling of the form\u03c0 periodic in \u03b8 but may not be well defined at p = 0. Thus, we suppose p \u226b \u03b5 > 0 for \u00d7x+x22+ on the half plane p > 0 ifB is The averaging systemAccording to Theorem 4.1.3 in , we can \u03b5 \u226a 1. Further, the periodic orbit is stable (resp. unstable) if one (resp. none) of the following conditions holds:\u0394 = 0 and \u0394 < 0 and \u0394 > 0, Suppose that  holds. Twhere \u0394 is given by .s = x3 + \u03b41 and \u03b8 \u2192 \u03b5\u22121\u03b8 and truncating the terms of order \u03b52, we have from , s(\u03b5)) withThen, by letting ave from 21)dpd\u03b8dpd\u03b8s = xd hence (dpd\u03b8=ps+\u03b5)) with2p(0)=1,\u2003\u2003ained in .0 \u2260 0. Then, for any given \u03b51 > 0 there exist an \u03b50 > 0 and a C1 function \u03d50(\u03bb1) = (2d1\u03bb1/a1) + \u03b40\u03bb12 + O(\u03bb13) and \u03d51(\u03bb1) = (2d1\u03bb1/a1) + \u03b40\u2032\u03bb12 + O(\u03bb13) such that for 0 < \u03bb12 + \u03bb22 < \u03b50, (0\u03d51(\u03bb1) \u2212 \u03b51\u03bb12 < \u03940\u03bb2 < \u03940\u03d50(\u03bb1) and has no invariant torus if \u03940\u03bb2 > \u03940\u03d50(\u03bb1). Moreover, the torus, if it exists, is stable (resp. unstable) when \u03940 < 0 (resp. >0).Suppose that  holds an2 < \u03b50,  has a un+3, where \u211d+ = {x \u2208 \u211d : x > 0}. We now look for the conditions for the existence of positive equilibria of system  = (\u03b1ij(\u03b5))3\u00d73, and we suppose M(\u03b5) is similar toNow, we shall investigate a special form of system  with a sdYidt=T, x = T, and system  = 0, and for simplicity, we write the nonlinear part of (y). By doing the following transformation:P(z) = (P1(z), P2(z), P3(z))T, which is to be determined,  denote the cubic terms in z of .ermined,  becomesrmined, \u22121=Iy noting(Dh1)\u22121=Iwherez = w + Q(w) \u2261 h2(w), where Q =  is homogeneous cubic polynomial, so that /(v(v + \u03b5)), p12 = \u2212(\u2212v2 + vu\u03b5 + v\u03b5 + u\u03b52)/(v(v + \u03b5)), p13 = 2v/(v + \u03b5), p21 = (3v + \u03b5)/(v + \u03b5), p31 = \u2212(6v2 + 6vu\u03b5 + u2\u03b52)/2v2, and p32 = \u2212u\u03b5(2v + u\u03b5)/2v2. Then, in the new variables  system  < u < 0, and then for 0 < \u03b5 \u226a \u03b50,  < u < 0. It can be checked that \u03b5 \u226a \u03b50. Then, by In this example, it is easy to see that /u. From  and 10)\u03b41 = 1,  satisfy , we need\u03b50) < u < 0 holds, where \u0394 is given by  and \u03d50(\u03bb1) are defined in \u03bb1 = \u03b5 and \u03bb2 = \u2212(24/5)\u03b5. By some easy calculations, we can obtain that for 0 < \u03b51 < A1 \u2212 A2 inequality \u03940\u03d51(\u03bb1) \u2212 \u03b51\u03bb12 < \u03940\u03bb2 < \u03940\u03d50(\u03bb1) holds. Thus, by By  and 29)29), we c"}
{"text": "In this paper, the structural characteristics of I. Schur type inequalities are exploited to generalize the corresponding inequalities by variable parameter techniques. Some novel upper and lower bounds for the I. Schur inequality have also been obtained and the upper bounds may be obtained with the help of Maple and automated proving package (Bottema). Numerical examples are employed to demonstrate the reliability of the approximation of these new upper and lower bounds, which improve some known results in the recent literature.Recently, extensive researches on estimating the value of  In fact, extensive researches for the estimated value of e have been studied )bn = (1 + (\u03b1/n))n+\u03b2 and F\u03b1,\u03b2(x) = (1 + (\u03b1/x))x+\u03b2; see xn, \u03bb \u2208 \u211d; then we have the following.In fact, the previous theorems can be viewed as the different improvements of original I. Schur inequality. The above improved inequalities follow the motivation of researching mean value sequences of the upper and lower bounds for original inequalities; the authors study the monotonicity of the sequences constructed by the mean value of upper and lower bounds for I. Schur inequality and its relationship with n \u2208 N*, the sequence Wn is monotone decreasing, and the inequality xn[1 + \u03bb/(\u03bb + 1)n] < e holds.For d(x) = ln\u2061(x + \u03bbx + \u03bb) \u2212 ln\u2061(x + \u03bbx) + xln\u2061(x + 1) \u2212 xln\u2061x\u2009\u2009(x > 0); thenIx(\u03bb) = (3x2 + 3x + x3 + 1)\u03bb2 + (2x + 2x2)\u03bb \u2212 x3. This is a quadratic function of \u03bb, and its discriminant isIx(\u03bb) areIx(\u03bb) \u2264 0 always holds for x \u2265 1 and x\u2192\u221e\u2061d\u2032(x) = 0, thus d\u2032(x) \u2265 0. It shows that sequence Wn = [1 + \u03bb/(\u03bb + 1)n]xn is monotone increasing under this condition, and the inequality in this theorem can be proved by using lim\u2061n\u2192\u221e\u2061Wn = e.We consider \u2212ln\u2061x\u22121,d\u2032\u2032(x)=\u2212)n and jn = (1 + (\u03bb/n))n+1, where \u03bb > 0, and it is easy to obtain lim\u2061n\u2192\u221ein = lim\u2061n\u2192\u221ejn = e\u03bb. Next, we consider their arithmetic mean value sequence, geometric mean value sequence, and harmonic mean value sequence, respectively; considerHere we study the monotonicity of two new sequences and their relationships with \u03bb \u2264 1 and n \u2208 N*, the sequences A1 = (1 + (\u03bb/2n))(1 + (\u03bb/n))n are monotone decreasing, and the inequality e\u03bb < (1 + (\u03bb/2n))(1 + (\u03bb/n))n holds.For 0 < M(x) = xln\u2061(x + \u03bb) \u2212 xln\u2061x + ln\u2061(2x + \u03bb) \u2212 ln\u2061(2x), x > 0; thenM\u2032\u2032(x) in (Y(x): = 4x3 \u2212 4x3\u03bb + 9x2\u03bb \u2212 4x2\u03bb2 + 6x\u03bb2 \u2212 x\u03bb3 + \u03bb3. Take \u2200 p \u2265 0; then \u03bb : = 1/(p + 1) \u2208  \u2265 0. Furthermore, it implies thatM\u2032(x) is monotone increasing, and lim\u2061x\u2192\u221e\u2061M\u2032(x) = 0 has been verified, such that M\u2032(x) < 0, and A1 = (1 + (\u03bb/2n))(1 + (\u03bb/n))n are monotone decreasing; hence e\u03bb < (1 + (\u03bb/2n))(1 + (\u03bb/n))n can be proved because of lim\u2061n\u2192\u221e\u2061A1 = e\u03bb.We consider  0; thenM\u2032(x)=ln\u2061\u03bb \u2264 1 and n \u2208 N*, the sequences G1 = (1 + (\u03bb/n))1/2(1 + (\u03bb/n))n are monotone decreasing, and the inequality (1 + (\u03bb/n))1/2(1 + (\u03bb/n))n > e\u03bb holds.For 0 < f1 = xln\u2061(1 + (\u03bb/x)) + (1/2)ln\u2061(1 + (\u03bb/x)), x \u2265 1; then\u03b4(\u03bb) = (\u22122x + 1)\u03bb2 + 2x\u03bb, which is a quadratic function of \u03bb. And its discriminant is \u03941 = 4x2 > 0. Then two roots of \u03b4(\u03bb) arex/(2x \u2212 1)}x=1\u221e is strict monotone decreasing and 1 < (2x/(2x \u2212 1)) \u2264 2, x \u2265 1. According to the characteristic of parabola curve, \u03b4(\u03bb) = (\u22122x + 1)\u03bb2 + 2x\u03bb > 0 always holds for 0 < \u03bb \u2264 1 and x \u2265 1; then from lim\u2061x\u2192\u221e\u2061f1\u2032(x) = 0, f1\u2032(x) \u2264 0. It shows that the sequences G1 are monotone decreasing.We consider +(\u03bb/x)),f1\u2032\u2032(x)=\u2212\u03bb \u2265 1, then H1 = (2injn)/(in + jn) = (1 + \u03bb/(2n + \u03bb))(1 + (\u03bb/n))n are monotone increasing, and the following inequalityn \u2208 N*.If \u03c6(x) = ln\u2061(2x + 2\u03bb) \u2212 ln\u2061(2x + \u03bb) + xln\u2061(x + \u03bb) \u2212 xln\u2061x, x > 0; thenZ(x)\u2236 = \u22124x2 \u2212 3\u03bbx + 4\u03bbx2 + 4x\u03bb2 + \u03bb3. Since \u03c4 + 1\u2236 = \u03bb \u2265 1, that is, \u03c4 \u2265 0, we have\u03b3(x) = \u2212\u03bbZ(x) \u2264 0 always holds for x > 0, \u03bb \u2265 1. Then \u03c6\u2032(x) is monotone decreasing, and according to lim\u2061x\u2192\u221e\u2061\u03c6\u2032(x) = 0, we have \u03c6\u2032(x) > 0. Hence, H1 = (1 + \u03bb/(2n + \u03bb))(1 + (\u03bb/n))n are monotone increasing sequences, and we prove thatWe consider \u2212ln\u2061x\u22121,\u03c6\u2032\u2032(x)=\u22124\u03bb < 1, are the H1 monotone increasing or monotone decreasing? With the help of the Bottema   \u2265 0 holds for 0 < \u03bb \u2264 (4/5), x \u2265 1. Moreover, it can be also shown in A remaining issue of mme; see  and refeH1, n = 1,2,\u2026, is monotone decreasing under the previous discussion in the proof of the Then we can conclude that L0\u2013L4 here mean the corresponding real continuous function for L0 = xn(1 + (1/2n)), L1 = xn(n + 1)ln\u2061(1 + (1/n)), L2 = (xn/(nln\u2061(1 + (1/n)))), L3 = xn(1 + ((4n + 1)/(8n2 + 4n))), and L4 = xn(1 + ((4n + 3)/(8n2 + 8n + 1))), respectively.In this section, we will display some new upper and lower bounds of the I. Schur inequality by using the Matlab 2011b in a personal computer. We give some figures to show their variation trend; the symbols L0 of the I. Schur inequality is not better than the other novel upper bounds L1\u2013L4; the L1 is the best upper bound among five upper bounds. Moreover, we note that the variation curves of L3, L4 are greatly similar; they are both the other alternatives. This figure also shows that the improvement of the upper bound of the I. Schur inequality is beneficial.Form T1 = xn(1 + (1/(2n + 1))),\u2009 \u2009T2 = xn(1 + (1/n))1/8, T3 = xn(1 + (1/n))1/4, \u2009 T4 = xn(1 + ((1/100)/(1 + (1/100))n)), and T5 = xn(1 + (\u22121/3)/(1 \u2212 (1/3))n), respectively. From T1 of the I. Schur inequality is better than the other novel lower bounds T2\u2013T5; the T3 is an alternative among other novel lower bounds. This figure shows that we improve the lower bound of the I. Schur inequality and it is not very successful. But we give some generalized upper and lower bounds for the e\u03bb, \u03bb \u2208 ; this work is meaningful in terms of the generalized type of the I. Schur inequality.Firstly, we define the corresponding real continuous function for H1 can be viewed as a good lower bound of e\u03bb, \u03bb \u2265 1 in terms of approximating precision, especially for the case with \u03bb = 1. It is favorable to note that the reliability of the approximation of e\u03bb based on H1 depends on the closeness between \u03bb and 1.From xn and yn, such as the arithmetic mean value, geometric mean value, logarithmic mean value, and harmonic mean value. Besides, we have given some extensions and remarks for known results in the recent literature. At the same time, we extend some ideas and conclusions in [In the previous sections, we have studied the monotonicity of the sequences and their variants based on various mean values of sions in , 30\u201332."}
{"text": "In , Toader In recent years, there have been plenty of literature, such as \u20136, dedicp \u2208 \u211d and a, b > 0, the centroidal mean pth power mean Mp are, respectively, defined byFor a, b > 0 with a \u2260 b. This conjecture was verified by Qiu and Shen .For positive numbers \u03bb and the least value \u03bc, such that the double inequality \u03b1 \u2208  and a, b > 0 with a \u2260 b. As applications, we also present new bounds for the complete elliptic integral of the second kind.The main purpose of the paper is to find the greatest value  In order to establish our main result, we need several formulas and Lemmas below.r < 1 and For 0 < fined in , 13 by/r2 is strictly increasing from  to , and the function 2\u2130 \u2212 r\u20322\ud835\udca6 is increasing from  to .The function\u2009\u2009 andfu,\u03b1 > 0, for all r \u2208  if and only if u \u2265 3(1 \u2212 \u03b1)(4/\u03c0 \u2212 1), and fu,\u03b1 < 0, for all r \u2208  if and only if u \u2264 3(1 \u2212 \u03b1)/4.  Let g(r) = (1/\u03c0)((\u2130 \u2212 r\u20322\ud835\udca6)/r2). From , one hasWe divide the proof into four cases.Case 1 (u \u2265 3(1 \u2212 \u03b1)/\u03c0).\u2009\u2009From (g(r), we clearly see that fu,\u03b1(r) is strictly increasing on . Therefore, fu,\u03b1(r) > 0, for all r \u2208 .).\u2009\u2009From  and LemmCase 2 (u \u2264 3(1 \u2212 \u03b1)/4). From (g(r), we obtain that fu,\u03b1(r) is strictly decreasing on . Therefore, fu,\u03b1(r) < 0, for all r \u2208 .4). From  and LemmCase 3 (3(1 \u2212 \u03b1)/4 < u \u2264 3(1 \u2212 \u03b1)(4/\u03c0 \u2212 1)). From (g(r), we see that there exists \u03bb \u2208 , such that fu,\u03b1(r) is strictly increasing in  andfu,\u03b1(r) leads to the conclusion that there exists 0 < \u03bb < \u03b7 < 1, such that fu,\u03b1(r) > 0 for r \u2208  and fu,\u03b1(r) < 0 for r \u2208 .)). From  and 14)Case 3  andfu,\u03b1(1\u2212)\u2264Case 4 (3(1 \u2212 \u03b1)(4/\u03c0 \u2212 1) \u2264 u < 3(1 \u2212 \u03b1)/\u03c0). Equation (Equation  leads tog(r), we clearly see that there exists \u03bb \u2208 , such that fu,\u03b1(r) is strictly increasing in . Therefore, fu,\u03b1(r) > 0 for r \u2208  follows from . From  and 14)14) togetows from  and 16)g(r), we Now, we are in a position to state and prove our main results. \u03b1 \u2208  and \u03bb, \u03bc \u2208 , then the double inequalitya, b > 0 with a \u2260 b if and only if If A, T, and a > b. Let p \u2208 , t = b/a \u2208 , and r = (1 \u2212 t)/(1 + t). Then, Since ). Then,C\u00af of the second kind in terms of elementary functions as follows.Let r \u2208  and  For r \u2208 . In the recent past, the complete elliptic integrals have attracted the attention of numerous mathematicians. In , it was r \u2208 .Guo and Qi  proved tr \u2208 .Yin and Qi  presente\u2130(r) are better than the bounds in .It was pointed out in  that theounds in  for some\u2130(r) is better than the lower bound in . The lower bound in  for \u2130(r)bound in . Indeed,x \u2208  show that the lower bound in (\u2130(r) is better than the lower bound in bound in :(25)5+6"}
{"text": "We investigate the positive solutions of the semilinear parabolicsystem with coupled nonlinear nonlocal sources subject to weighted nonlocal Dirichlet boundaryconditions. The blow-up and global existence criteria are obtained. \u03a9 is a bounded domain in \u211dN, N \u2265 1, with smooth boundary \u2202\u03a9. The exponents pi > 0, qi \u2265 0. The weighted functions \u03c6i in the boundary conditions are continuous, nonnegative on \u03a9\u03c6idy > 0 on \u2202\u03a9. The initial data \u03bd < 1, ui0(x) \u2265 0, \u22620, and satisfy the compatibility conditions.In this paper, we consider the positive solutions of the semilinear parabolic system with coupled nonlinear nonlocal sources subject to weighted nonlocal Dirichlet boundary conditions:u0(x) \u2265 0, g(t) > 0 or g(t) = (k/|\u03a9|)\u222b\u03a9utdx with k > 0. Chadam et al. [f(u) = 0 and g(t) = \u222b\u03a9\u03c8)dx and proved that the blow-up set is the whole region (including the homogeneous Neumann boundary conditions). Souplet [g(t). Pao [Many physical phenomena were formulated into nonlocal mathematical models and studied by many authors \u201313. For m et al.  studied ut\u2212\u0394u=f. Pao  discusseThe problems with both nonlocal sources and nonlocal boundary conditions have been studied as well. To motivate our study, we give a short review of examples of such parabolic equations or systems studied in the literature. For example, Lin and Liu  studied p, l > 0. And some criteria for the existence of global solution as well as for the solution to blow up in finite time were obtained.Gladkov and Kim  considerk = 2:(i)m, q < 1 and np \u2264 (1 \u2212 m)(1 \u2212 q) hold; then the solution of (Assume that ution of  exists g(ii)If one of the following conditions holds:ution of  blows up(iii)\u03a9\u03c6dy \u2265 1 and \u222b\u03a9\u03c8dy \u2265 1 for all x \u2208 \u2202\u03a9 and one of , and\u2009\u2009ST = \u2202\u03a9 \u00d7  with 0 < T < \u221e for convenience.In the following, we set pi, qi \u2265 1, i = 1,2,\u2026, k.It is known by the standard theory , 23 thatT < \u221e, thenProblem  has a popi, qi, i = 1,2,\u2026, k satisfyu1, u2,\u2026, uk) of dy < 1, i = 1,2,\u2026, k, for all x \u2208 \u2202\u03a9, then the solution of  satisfiesIf the initial data H) holds. Then the solution of  \u2265 0 on \u03a9gidy > 0 on \u2202\u03a9, i = 1,2,\u2026, k, j = 1,2,\u2026, N. If, for every i = 1,2,\u2026, k, wk+1 = w1, then wi > 0, i = 1,2, on Suppose that zi = eKt\u2212wi with zi > 0, i = 1,2,\u2026, there exists \u03b4 > 0 such that zi > 0 for zi \u2265 0 on zi vanishes at z1 \u2261 0 in \u03a9g1dy > 0 on \u2202\u03a9. This proves that z1 > 0 and consequently w1 > 0. We complete the proof.Set k}. Thenzit\u2212\u0394zi+, bi are continuous, nonnegative functions in gi \u2265 0 on \u03a9gidy < 1 on \u2202\u03a9, and there exist positive constants Ci such that \u222b\u03a9 + bi)dx \u2264 Ci. Then wi \u2265 0, i = 1,2, on Suppose that, for every wi > 0. Now we consider the general case. Set\u025b is any fixed positive constant, and K = 1 + max\u2061{\u222b\u03a9 + bi)\u2009dx, i = 1,2,\u2026, k}. By  \u2265 0 on QT. Letting \u025b \u2192 0+, we get the desired result.Suppose that the strict inequalities of  hold; by, k}. By , we get,\u03a9gidy < 1 is not necessarily valid, we have the following result. The argument of its proof can be referred to [If the boundary condition \u222bfi \u2265 0, ci, di, are nonnegative and bounded in QT, gi \u2265 0 on \u03a9gidy > 0 on \u2202\u03a9, i = 1,2,\u2026, k, j = 1,2,\u2026, N. If, for every i = 1,2,\u2026, k,\u2009\u2009wk+1 = w1, then wi \u2265 0, i = 1,2, on Suppose that By r \u2208 {1,2,\u2026, k},pr < 1,qr < 1,Let QT.then Before proving w0(x) and \u03c6 be continuous, nonnegative functions on \u03b8ij satisfy 0 < \u03b8i1 + \u03b8i2 \u2264 1. Then the solutions of the nonlocal problemLet \u03b8 > 0 be large enough such thatz = e\u03b8t2\u03c8(x) for  \u2208 \u03a9 \u00d7 , one readily checks thatThe augment is similar to the proof of , Lemma 6ution of . Clearlyai \u2208 , i = 1,2,\u2026, k, such that\u03b1 = \u2211i=1k1/ai. Let \u03a6 \u2265 max\u2061{\u03c6i, i = 1,2,\u2026, k} be a continuous function defined for z solvesz is global. Moreover, z > 1 in i = 1,2,\u2026, k. By  \u2264 C0 + M in C0 to be large enough such thatx, t) \u2208 ST, it follows thatx, t) \u2208 QT. Set Li = (1 + C0 + M)pi+qi | \u03a9| for convenience. A simple computation yieldsDefineqr > 1, no matter qr+1 > 1 or qr+1 \u2264 1, we can choose br to be small enough such that brqr1\u2212 \u2265 br+1prLr. For fixed br, there exist bi, i = 1,2,\u2026, r \u2212 1, r + 1,\u2026, k, satisfying biqi1\u2212 \u2265 bi+1piLi, i = 1,2,\u2026, k. It follows that(a) If qi \u2264 1, i = 1,2,\u2026, k and p1p2 \u22ef pk > (1 \u2212 q1)(1 \u2212 q2)\u22ef(1 \u2212 qk), we can choose b1 to be small enough such thatbi > 0, i = 2,3,\u2026, k, bk+1 = b1 satisfying biqi1\u2212 \u2265 bi+1piLi, i = 1,2,\u2026, k. Hence ((b) If k. Hence  holds toui,0(x) \u2264 bi(1 + w(x)), i = 1,2,\u2026, k for x \u2208 \u03a9.By  and 37)37), in au, v) is a positive solution of  the first eigenvalue and the corresponding eigenfunction of the linear elliptic problem:\u03d51(x) satisfies\u03b3 = min\u2061{\u03b1i(qi \u2212 1) + \u03b1i+1pi + 1, i = 1,2,\u2026, k}.We denote by qr \u2265 1, we claim that there exist positive constants \u03b1i > 1, i = 1,2,\u2026, k, such that the inequalityi = r, /pr+1}. That is,  such that ((a) If equality\u03b1i(qi\u22121)+equality\u03b1i(qi\u22121)+equality\u03b1i(qi\u22121)+equality\u03b1i(qi\u22121)+equality\u03b1i(qi\u22121)+qi < 1, i = 1,2,\u2026, k, and p1p2 \u22ef pk > (1 \u2212 q1)(1 \u2212 q2) \u22ef (1 \u2212 qk), we can choose \u03b1i > 1 such that(b) If 1.Hence  holds to\u03b3 > 1. Now let s(t) be the unique solution of the ODE probleml = min\u2061{(1/\u03b1i)\u222b\u03a9\u03d51\u03b1iqi+\u03b1i+1pi, i = 1,2,\u2026, k}. Then s(t) blows up in finite time T(s0) with s0 being large enough.Hence, for the case (a) or (b), we all have ui,0(x) \u2265 s\u03b1i(0)\u03d51\u03b1i(x), i = 1,2,\u2026, k for x \u2208 \u03a9. We complete the proof.Setui,0 > 0 in \u03a9, \u222b\u03a9\u03c6rdy > 0 on \u2202\u03a9, andui,0 > 0 on \u2202\u03a9. Denote by \u03b7 the positive constant such that ui,0 > \u03b7 on H) implies that (ui)t > 0 by the comparison principle, and in turn ui > \u03b7, i = 1,2,\u2026, k on ur satisfieszr(t) be the solution of the following Cauchy problem:zr(t) blows up under the condition\u03a9\u03c6rdy \u2265 1, by ur \u2265 zr as long as both ur and zr exist, and thus ur blows up for any positive initial data. The proof now is completed.Since"}
{"text": "We established some theorems under the aim of deriving variants of the Banach contraction principle, using the classes of inner contractions and outer contractions, on the structure of fuzzy modular spaces. X, \u03c1) is called a probabilistic modular space if X is a real vector space and \u03c1 is a mapping from X into the set of all distribution functions  is denoted by \u03c1x, and \u03c1x(t) is the value \u03c1x at t \u2208 \u211d) satisfying the following conditions: \u2009\u03c1x(0) = 0; (PM1) \u2009\u03c1x(t) = 1 for all t > 0 if and only if x = \u03b8; (PM2) \u2009\u03c1x\u2212(t) = \u03c1x(t); (PM3) \u2009\u03c1\u03b1x+\u03b2y(s + t) \u2265 \u03c1x(s)\u2227\u03c1y(t) for all x, y \u2208 X and \u03b1, \u03b2, s, t \u2208 \u211d0+, \u03b1 + \u03b2 = 1.(PM4) For every x \u2208 X, t > 0 and \u03b1 \u2208 \u211d\u2216{0}, if\u03b2 \u2208  is \u03b2-homogeneous. The concept of a modular space was introduced by Nakano . Soon afRecently, further studies have been made on the probabilistic modular spaces. Nourouzi  extendedfuzzy set A in X is a function with domain X and value in . A triangular norm is a function \u2217 : \u00d7\u2192 satisfying, for each a, b, c, d \u2208 , the following conditions: a\u22171 = a; a\u2217b \u2264 c\u2217d whenever a \u2264 c, b \u2264 d; a\u2217b = b\u2217a and (a\u2217b)\u2217c = a\u2217(b\u2217c).A V be a real or complex vector space with a zero \u03b8, \u2217 a continuous triangular norm, and \u03bc a fuzzy set on the product V \u00d7 \u211d+. Suppose that the following properties hold for x, y \u2208 V and s, t > 0: \u2009\u03bc > 0; (FM1) \u2009\u03bc = 1 for all t > 0 if and only if x = \u03b8; (FM2) \u2009\u03bc = \u03bc; (FM3) \u2009\u03bc \u2265 \u03bc\u2217\u03bc whenever z is the convex combination between x and y; (FM4) \u2009t \u21a6 \u03bc is continuous at each fixed x \u2208 V. (FM5) the mapping Then, we write  to represent the space with the pre-defined properties. In particular, we call \u03bc a fuzzy modular and the triple  a fuzzy modular space. Let t > 0.It is worth noting that every fuzzy modular is non-decreasing with respect to X be a real or complex vector space and \u03c1 be a modular on X. Take the t-norm a\u2217b = min\u2061\u2061{a, b}. For every t \u2208 , define \u03bc = t/(t + \u03c1(x)) for all x \u2208 X. Then  is a F-modular space.Let t-norm is replaced by a\u2217b = a \u00b7 b and a\u2217b = max\u2061\u2061{a + b \u2212 1,0}, respectively.Note that the above conclusion still holds even if the x \u2208 V and a non-zero real \u03bb, the equality\u03b2 \u2208  is the set of the formr \u2208  and t > 0. Now, suppose that \u03bc is \u03b2-homogeneous for some \u03b2 \u2208  in V\u2009\u03bc-converges   to its \u03bc-limit x \u2208 V if and only if \u03bc \u2192 1 as n \u2192 \u221e for all t > 0. Note here that the \u03bc-limit is unique if it does exists after all. It is then natural to say that (xn) is \u03bc-Cauchy if for any given \u025b \u2208  and t > 0, there exists N \u2208 \u2115 with \u03bc > 1 \u2212 \u025b whenever m, n > N.The and Chen , the fama\u2217b) = min\u2061\u2061{a, b}. This triangular norm has a very special property that if \u2217\u2032 is an arbitrary triangular norm, then (a\u2217\u2032b)\u2264(a\u2217b) for all a, b \u2208 . With this property, it is suitable to call this \u2217 a strongest triangular norm. As is claimed by Shen and Chen [V is a real vector space equipped with a \u03b2-homogeneous fuzzy modular \u03bc and a strongest triangular norm \u2217, then a \u03bc-convergent sequence is \u03bc-Cauchy. The authors also mentioned that if \u2217 is not the strongest one, such implementation is not always true.At this point, let us turn to a typical example of a triangular norm which is defined by  is said to be \u03bc-complete if \u03bc-Cauchy sequences actually \u03bc-converges. We note that it makes more sense if we require a complete fuzzy modular space to be equipped with the strongest triangular norm.The space  is \u03bc-complete. Suppose that at each x \u2208 V, \u03bc \u2192 1 as t \u2192 \u221e. If f\u2009:\u2009V \u2192 V is an inner contraction with constant q \u2208 , then f has a unique fixed point.Let x0 \u2208 V, we suppose that fnx0 \u2260 fn+1x0 for all n \u2208 \u2115. Let t > 0, observe thatn \u2192 \u221e, we have fnx0 \u2212 fn+1x0 \u2192 \u03b8 for every t > 0. That is, for any given t > 0 and \u025b \u2208 , there exists N \u2208 \u2115 such thatGiven a point \u03bc > 1 \u2212 \u025b for all p \u2208 \u2115. Let us assume first that \u03bc > 1 \u2212 \u025b holds at some j \u2208 \u2115. Observe thatWe now claim to show by induction that fnx0) is Cauchy. Let t > 0 and \u025b \u2208  be arbitrary, and we choose N \u2208 \u2115 according to the claim given above. For n > m > N, we may write m = N + s and n = N + s + t, for some s, t \u2208 \u2115. Now, consider thatfnx0) is Cauchy, and so the \u03bc-completeness yields that fnx0 \u2192 x\u2217 for some x\u2217 \u2208 V. It follows thatfx\u2217 = x\u2217, since \ud835\udcaf\u03bc is Hausdorff. To show that the fixed point of f is unique, assume that y\u2217 \u2208 V is a fixed point of f as well. Finally, we obtain thatx\u2217 = y\u2217, and so the conclusion is fulfilled. Next, we shall show that  is called a \u03bc-G-Cauchy sequence if for each fixed p \u2208 \u2115 and t > 0, we have lim\u2061n\u2192\u221e = 1. If every \u03bc-G-Cauchy sequence \u03bc-converges, V is said to be \u03bc-G-complete. It is to be noted that the notion of \u03bc-G-completeness is slightly stronger than the ordinary completeness. It is enough to see that every \u03bc-Cauchy sequence is also a \u03bc-G-Cauchy sequence. For our result, it is still a question whether or not the \u03bc-G-completeness assumption can be weakened.For this part, we consider a weaker form of a V be a real vector space equipped with a \u03b2-homogeneous fuzzy modular \u03bc and the strongest triangular norm \u2217 such that  is \u03bc-G-complete. If f : V \u2192 V is an outer contraction with constant q \u2208 , then f has a unique fixed point.Let x0 \u2208 V, we suppose that fnx0 \u2260 fn+1x0 for all n \u2208 \u2115. By the definition of an outer contraction, we can rewrite this notion in the following:x, y \u2208 V and t > 0.Given a point t > 0, observe thatn \u2192 \u221e, we havefnx0) is Cauchy. Let t > 0 and \u025b \u2208  be arbitrary, and we choose N \u2208 \u2115. For n > N, n \u2208 \u2115 and for each p > 0. Now, consider thatfnx0) is Cauchy, and so the \u03bc-completeness yields that fnx0 \u2192 x\u2217 for some x\u2217 \u2208 V. It follows thatn \u2192 \u221e, we havefx\u2217 = x\u2217, since \ud835\udcaf\u03bc is Hausdorff. To show that the fixed point of f is unique, assume that y\u2217 \u2208 V is a fixed point of f as well. Finally, we obtain that\u03bc(1 \u2212 q)\u2265(1 \u2212 q) which implies that \u03bc = 1. Therefore, it must be the case that x\u2217 = y\u2217, and so the conclusion is fulfilled. Let V is \u03bc-complete as in Is"}
{"text": "N-fractional calculus operator N\u03bd method, we derive the fractional solutions of the equation.This paper deals with the design fractional solution of Bessel equation. We obtain explicit solutions of the equation with the help of fractional calculus techniques. Using the  Fractional calculus has an important place in the field of math. Firstly, L'Hospital and Leibniz were interested in the topic in 1695, , NishimoN-Fractional calculus is a very interesting method because this method is applied to singular equation. Note that fractional solutions can be obtained for kinds of singular equation via this method [s method \u201312, 17. \u03bb and p are real numbers. By means of the substitution Now, consider the following the Bessel equation:xd2zdx2+dBessel equation for having the analogous singularity is given in .The differintegration operators and their generalizations , 17, 18 Rz\u03c5\u2009\u2009(\u03c5 \u2208 \u2102) and the Weyl operator Wz\u03c5\u2009\u2009(\u03c5 \u2208 \u2102), which are defined by [Two of the most commonly encountered tools in the theory and applications of fractional calculus are provided by the Riemann-Liouville operator fined by , 19.Consider\u2115 being the set of positive integers.provided that the defining integrals in  and 4)  4) existC\u2212 is a curve along the cut joining two points z and \u2212\u221e + i\u2009Im\u2061(z),\u2009\u2009C+ is a curve along the cut joining two points z and \u221e + i\u2009Im\u2061(z),\u2009\u2009D\u2212 is a domain surrounded by C\u2212, and D+ is a domain surrounded by C+. (Here D contains the points over the curve C.)Letf = f(z) be a regular function in D\u2009\u2009(z \u2208 D),t \u2260 z,f\u03bd(z)\u2009\u2009(\u03bd > 0) is said to be the fractional derivative of f(z) of order \u03bd and f\u03bd(z)\u2009\u2009(\u03bd < 0) is said to be the fractional integral of f(z) of order \u2212\u03bd, provided (in each case) that |f\u03bd(z)| < \u221e\u2009\u2009(\u03bd \u2208 \u211d).Moreover, let N\u03bd be defined by   by cf. \u201310)(8)N(8)NN\u03bd We find it to be worthwhile to recall here the following useful lemmas and properties associated with the fractional differintegration which was defined earlier .f(z) and g(z) are single-valued and analytic in some domain \u03a9\u2286\u2102, thenh1 and h2.If the functions f(z) is single-valued and analytic in some domain \u03a9\u2286\u2102, thenIf the function f(z) and g(z) are single-valued and analytic in some domain \u03a9\u2286\u2102, thengn(z) is the ordinary derivative of g(z) of order n\u2009\u2009(n \u2208 \u21150 : = \u2115 \u222a {0}), being tacitly assumed (for simplicity) that g(z) is the polynomial part (if any) of the product f(z)g(z).If the functions \u03bb,For a constant \u03bb,For a constant \u03bb,For a constant N-fractional method to nonhomogeneous Bessel equation.Now, let apply y \u2208 {y : 0 \u2260 |y\u03bd| < \u221e; \u03bd \u2208 \u211d} and f \u2208 {f : 0 \u2260 |f\u03bd| < \u221e; \u03bd \u2208 \u211d}. We consider the nonhomogeneous Bessel equation:y2 = d2y/dx2,\u2009\u2009y = y\u2009(z)\u2009\u2009(z \u2208 \u2102),\u2009\u2009f = f(z) (an arbitrary given function), and p, \u03bb are given constants.Let p = 0 of (The cases p = 0 of  and 20)p = 0 of p = 0 of  and 18)p = 0 of Sety=x\u03b7\u03c8,\u2003\u03c8=x).Thusy1=\u03b7x\u03b7\u22121\u03c8Putting , we obtaWith some rearrangement of the terms in , we have\u03b7 such thatHere, we choose \u03b7 = p + (1/2). From ((I) Let 2). From  and 24)\u03b7 = p + e\u03bcx\u03d5 twing from , we can ing from  as(32)\u03d5\u03bc such thatChoose (I) (i): For instance, taking \u2212\u03bcxfrom  and 32)\u03bc=-\u03bbi, weN\u03bd to both members of , we, we12),  Making use of the relations , rewrite\u03bd such thatChoose 1/2from .Next, writingity from :(43)\u03c91+y\u0131, we finally obtain the solution , a, a(44)\u03c9=solution . Inversely,  satisfieatisfies . Therefoatisfies  satisfieatisfies  because atisfies , 35), \u03d5=atisfies , and (45\u03d5=atisfie(I) (ii): In the case when i\u03bbxfrom  and 32)\u03bc=\u03bbi, we N\u03bd to both members of ,,50), 4646, and .\u03b7 = \u2212p + (1/2).(II) Let p by \u2212p in (I) (i) and (I) (ii), we have other solutions (p \u2260 0.With the help of the similar method in (I), replacing olutions  and 20)p by \u2212p iolutions  and 18)p by \u2212p ip \u2260 0, where k is an arbitrary constant.If p = 0, p = 0, p = 0, (5f = 0 in When stead of  and 51)f = 0 in Therefore, we get  for 58)58) and  for 59)59).\u03b7 = \u2212p + (1/2), replacing p by \u2212p in \u03b7 = \u2212p + by \u2212p in  and 57)\u03b7 = \u2212p + Let It is clear by Theorems Application 1. If we substitute p = 0, \u2009\u03bb = 1/4, and f = ix\u22121/2ei/2)x\u2212( in (i/2)x in , then weBy performing the necessary operations in , we getNow, let us show that the last equality is the solution of :(67)y2=Obviously, if  and 67)67) are pApplication \u2009\u20092. If we substitute p = \u22121 and\u2009\u2009\u03bb = 0 in y2=dt=\u03c0x24,y=kx3/2\u03c04y2=dt=\u03c0x2usly, if , it is susly, if  is givenp by ip (\u2212\u03bb instead of \u03bb) in  in ; that isp by ip in ((i) Therefore, the solutions for  are giveby ip in , 18),  In the same way, for the solutions for , substit\u03bd as follows:Choose \u03bd = pi + (1/2). From (Let 2). From  and 77)\u03bd = pi + Next, set ; then 8 is rewrielations  and 31)82)(e\u03bcx\u03c8(e\u03bcx\u03c8(82)\u03bc as follows:Choose (ii. 1) In the case when e\u03bbxfrom  and 83)\u03bc=-\u03bb, we N\u03bd to both members of , a, a10), a\u03bd such thatChoose 1/2from .Next, writingx\u22121from . This is\u22121from (u=[(fx(1/\u03c81/2\u2212pi=u/2.from  and 80)(92)\u03c81/2\u2212(ii. 2) Similarly, in the case when \u03bd = \u2212ip + (1/2). In the same way as in the procedure in (ii), replacing ip by \u2013ip\u2009\u2009(ii. 1) and (ii. 2), we can obtain y\u0131\u0131\u0131)( and y\u0131v), (, (f = 0,olutions , and (57N-fractional calculus operator N\u03bd-method is applied to the nonhomogeneous and homogeneous Bessel equation. Explicit fractional solutions of Bessel equations are obtained. Furthermore, similar solutions were obtained for the modified same equation by using the method.The"}
{"text": "Coupled nonlinear dynamical systems have been widely studied recently. However, the dynamical properties of these systems are difficult to deal with. The local activity of cellular neural network (CNN) has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice, which is composed of coupled cells. In this paper, the analytical criteria for the local activity in reaction-diffusion CNN with five state variables and one port are presented, which consists of four theorems, including a serial of inequalities involving CNN parameters. These theorems can be used for calculating the bifurcation diagram to determine or analyze the emergence of complex dynamic patterns, such as chaos. As a case study, a reaction-diffusion CNN of hepatitis B Virus (HBV) mutation-selection model is analyzed and simulated, the bifurcation diagram is calculated. Using the diagram, numerical simulations of this CNN model provide reasonable explanations of complex mutant phenomena during therapy. Therefore, it is demonstrated that the local activity of CNN provides a practical tool for the complex dynamics study of some coupled nonlinear systems. Coupled nonlinear dynamical systems have been widely studied in recent years. However, the dynamical properties of these systems are difficult to deal with. Although the research on emergence and complexity has gained much attention during the past decades, the determination, prediction, and control of the complex patterns generated from high-dimensional coupled nonlinear systems are still far from perfect. Nature abounds with complex patterns and structures emerging from homogeneous media, and the local activity is the origin of these complexities , 2. The Although Chua presents the main theorem of local activity at a cell equilibrium point , 2, it iThe remaining of this paper is organized as follows. The local activity of CNN is introduced in M \u00d7 N array of cells. Each cell is denoted by C, where i = 1,2,\u2026, M, j = 1,2,\u2026, N. The dynamics of each cell is given by the equation:The CNN architecture is composed of a two-dimensional xij, yij, uij are the state, output, and input variables of the cell, respectively. ak,l, bk,l, zij are the elements of the A-template, the B-template, and threshold, respectively. r is the radius of influence sphere. The output yij is the piece-wise linear function given byA vast majority of active homogeneous media that are known to exhibit complexity are modeled by a reaction-diffusion partial differential equation (PDE):X =  is state variables,  is spatial coordinates, fi is a coupled nonlinear vector function called the kinetic term, and D1, D2,\u2026, Dn are constants called diffusion coefficients. Replacing the Laplace in above formulation by its discrete version yieldsD = diag\u2061, From Chua and his collaborators' point, PDEs are merely mathematical abstractions of nature, and the concept of a continuum is in fact an idealization of reality. Even the collection of all electrons in a solid does not form a continuum, because much volume separating the electrons from the nucleus represents a vast empty space . Reaction state variables, but only m (m \u2264 n) state variables couple directly to their nearest neighbors via \u201creaction-diffusion\u201d. Consequently, each cell has the following state equations:Generally speaking, in a reaction-diffusion CNN, every cell has Qi = (\u2208Rn) of equation (The cell equilibrium point equation  can be dQi has the following form:The Jacobian matrix at the equilibrium point kl(Qi) are called cell parameters andQi are defined viaQi.Qi if and only if, its admittance matrix at Qi satisfies at least one of the following four conditions [YQ(s) has a pole in Re[s] > 0.\u03c9 = \u03c90, where \u03c90 is any real number.YQ(s) has a simple pole s = i\u03c9p on the imaginary axis, where its associated residue matrix:YQ(s) has a multiple pole on the imaginary axis.A reaction-diffusion CNN cell is called locally active at the equilibrium point nditions .YQ(s) Qi is called stable if and only if, all the real parts of eigenvalue \u03bbi of Jacobian matrix at the equilibrium point Qi are negative  > 0), and any one of the following conditions holds.f(s) \u2260 0.f(s) = 0, and m > n, where s is m and n orders zero point of g(s) and f(s), respectively, where f(s) = T1s3 + K1s2 + L1s + \u03941, g(s) = s4 + Ts3 + Ks2 + Ls + \u0394.A necessary and sufficient condition for Obviously proved.DenoteYQH(iw) < 0 for some w = w0 \u2208 R if any one of the following conditions holds.a11 > 0. a11 = 0, TT1 \u2212 K1 > 0. a11 = 0, TT1 \u2212 K1 = 0, Q > 0. a11 = 0, TT1 \u2212 K1 = 0, Q < 0, \u0394\u03941 > 0. a11 = 0, TT1 \u2212 K1 = 0, Q < 0, P \u2265 0, \u0394\u03941 \u2212 P2/Q/4 > 0, \u0394\u03941 \u2264 0,a11 = 0, TT1 \u2212 K1 = 0, Q = 0, P > 0. a11 = 0, TT1 \u2212 K1 = 0, Q = 0, P \u2264 0, \u0394\u03941 > 0. a11 = 0, TT1 \u2212 K1 < 0, \u0394\u03941 > 0. a11 = 0, TT1 \u2212 K1 < 0, \u0394\u03941 \u2264 0, and \u03bbj* \u2265 0, h(\u03bbj*) < 0, for j = 1 or 2.a11 < 0, D > 0, \u03a91 > 0, g(\u03a91) < 0. a11 < 0, D < 0, and \u03a9j \u2265 0, g(\u03a9j) < 0, for j = 1, 2 or 3.a11 < 0, D = 0, p = q = 0, g(\u2212F/4E) < 0. a11 < 0, D = 0, q2/4 = \u2212p3/27 \u2260 0, and \u03a9j \u2265 0, g(\u03a9j) < 0, for j = 1 or 2.Let the following parameters be defined as in YQ(i\u03c9) to satisfy condition (2) in Re[YQ(i\u03c9)] < 0,a11 > 0, then Re[YQ(i\u03c9)] < 0 when \u03c9 is large enough (See (1) of If a11 = 0, thenIf Let f(\u03bb) = \u2212Q\u03bb2 \u2212 P\u03bb \u2212 \u0394\u03941,TT1 \u2212 K1 > 0, then Re[YQ(i\u03c9)] < 0 when \u03c9 is large enough (See (2) of  If TT1 \u2212 K1 = 0, thenIf Q > 0, then Re[YQ(i\u03c9)] < 0 when \u03c9 is large enough (See (3) of If Q < 0,If 1 > 0, \u2203 \u03c90 \u2208 R, such that Re[YQ(i\u03c90)] < 0 (See (4) of If \u0394\u03941 \u2264 0, solve f\u2032(\u03bb*) = 0, we can get \u03bb* = \u22120.5P/Q, f(\u03bb*) = 0.25P2/Q \u2212 \u0394\u03941.If \u0394\u0394P \u2265 0, \u0394\u03941 \u2212 0.25P2/Q > 0, \u2203 \u03c90 > \u03bb*, such that Re[YQ(i\u03c90)] < 0 (See (5) of Then when Q = 0, then Re[YQ(i\u03c9)] = \u2212P\u03c92 \u2212 \u0394\u03941.If P > 0, then Re[YQ(i\u03c9)] < 0 when \u03c9 is large enough (See (6) of If P \u2264 0, \u0394\u03941 > 0, then \u2203\u03c90, such that Re[YQ(i\u03c90)] = \u2212\u0394\u03941 < 0 (See (7) of If TT1 \u2212 K1 < 0, let h(\u03bb) = \u2212(TT1 \u2212 K1)\u03bb3 \u2212 Q\u03bb2 \u2212 P\u03bb \u2212 \u0394\u03941,If 1 > 0, then \u2203\u03c90, such that Re[YQ(i\u03c90)] < 0 (See (8) of If \u0394\u03941 \u2264 0, solve h(\u03bb*) = 0, we can geti = 1,2, if \u03bbi* \u2265 0, h(\u03bbi*) < 0, then \u2203\u03c90, such that Re[YQ(i\u03c90)] < 0 (See (9) of If \u0394\u0394a11 < 0, let g(Q) = EQ4 + FQ3 + GQ2 + HQ + I, then g\u2032(Q) = 4EQ3 + 3FQ2 + 2GQ + H. Let x = \u03a9 + (F/4E), then the above becomes g\u2032(Q) = 4E(x3 + px + q) = 4Ef(x), then xi, i = 1,2, 3 are the roots of f(x) = 0, \u03a9i are the roots of g\u2032(\u03a9) = 0. If any one of the (10)\u2013(13) of Re[YQ(i\u03c90)] < 0.If So, if any one of conditions (1)\u2013(13) holds, Re[YQ(i\u03c90)] < 0. YQ(s) Satisfies condition (2) in j = 1, or 2, letYQ(s) satisfies condition (3) of K1w12 \u2212 \u03941 \u2260 0. K1w12 \u2212 \u03941 = 0, (L1 \u2212 T1w12)(w22 \u2212 w12) > 0. K1w22 \u2212 \u03941 \u2260 0. K1w22 \u2212 \u03941 = 0, (L1 \u2212 T1w22)(w12 \u2212 w22) > 0. K > 0, \u03941 \u2260 0, \u0394 = 0, L = KT \u2260 0, and any one of the following conditions holds.T\u03941 < 0. T(K \u2212 T1K) \u2212 \u03941 + KK1 \u2260 0. \u2009\u2009T(K \u2212 T1K) \u2212 \u03941 + KK1 = 0, T(\u03941 \u2212 KK1) + K(L1 \u2212 T1K) < 0. \u2009\u20091L > 0. \u2009\u2009\u0394 = 0, \u0394K > 0, K2 \u2212 4\u0394 > 0, and j = 1, or 2.\u0394 < 0, or\u2009\u2009AjBj1 \u2212 Aj1Bj \u2260 0. AjBj1 \u2212 Aj1Bj = 0, AjAj1 \u2212 BjBj1 > 0. \u2009\u2009For f(s) = T1s3 + K1s2 + L1s + \u03941, g(s) = s4 + Ts3 + Ks2 + Ls + \u0394, obviously, \u221e is not a single pole of YQ(s) on the imaginary axis.Let YQ(s) has a simple pole s = i\u03c9 on the imaginary axis, where its associated residuek1 \u2260 0, so f(s) \u2260 0, which implies that i\u03c9 is not a zero point of f(s) = 0, i\u03c9\u2009\u2009is not a removed pole of YQ(s).(I)YQ(s) has four poles s = \u00b1i\u03c91, \u00b1i\u03c92\u2009\u2009(\u03c91 \u2260 \u03c92 \u2260 0) on the imaginary axis. In this case,\u2009\u2009g(s) = (s2 + \u03c912)(s2 + \u03c922) = s4 + (\u03c912 + \u03c922)s2 + \u03c912\u03c922. Hence we obtain T = L = 0, K = \u03c912 + \u03c922 > 0, \u0394 = \u03c912\u03c922 > 0. Then, we can get k1 is a complex number or a negative real number. YQ(s) satisfies condition (3) in  If (II)YQ(s) has a simple pole s = 0 and two conjugate poles \u00b1i\u03c9\u2009\u2009(\u03c9 \u2260 0) on the imaginary axis, and another pole is a \u2260 0.If 1 \u2260 0, and g(s) has the form:T = \u2212a, K = \u03c92 > 0, L = \u2212a\u03c92 = KT, \u0394 = 0, \u03941 = 0, Therefore,In this case, it follows that \u0394 = 0, \u0394(1)YQ(s) at s = 0 isK > 0, \u03941 \u2260 0, \u0394 = 0, L = KT \u2260 0, T\u03941 < 0, k1 is a negative real number. YQ(s) satisfies condition (3) in The residue of (2)YQ(s) at The residue of k1 is either an imaginary number or a negative real number. YQ(s) satisfies condition (3) in (III)YQ(s) has a simple pole s = 0 on the imaginary axis, and the other poles are ai, Re[ai] \u2260 0, i = 1,2, 3, it follows that \u03941 \u2260 0, \u0394 = 0, and g(s) has the formT = \u2212(a1 + a2 + a3), K = a1a2 + a1a3 + a2a3, L = \u2212a1a2a3 \u2260 0, hence the reside of YQ(s) at s = 0 is1L > 0, k1 is a negative real number. YQ(s) satisfies condition (3) in If (IV)YQ(s) has two conjugate poles \u00b1i\u03c9(\u03c9 > 0) on the imaginary axis, and the other poles are Re[a] \u2260 0, Re[b] \u2260 0. In this case, g(s) has the formT = \u2212(a + b), K = ab + \u03c92, L = \u2212(a + b)\u03c92, \u0394 = ab\u03c92 \u2260 0. Then, ab = K \u2212 \u03c92, \u0394 = (K \u2212 \u03c92)\u03c92\u21d4\u03c94 \u2212 K\u03c92 + \u0394 = 0. Solving it, we haveK > 0, K2 \u2212 4\u0394 \u2265 0, and T = \u2212(a + b) = L/\u03c92. Then, the residue of YQ(s) at s = \u00b1\u03c9j* isk1 < 0 or it is an imaginary number. YQ(s) satisfies condition (3) in If If YQ(s) satisfies condition (3) in Therefore, when any one of conditions (I)\u2013(IV) holds, YQ(s) has a multiple pole on the imaginary axis if any one of the following conditions holds.L = 0, \u03941 \u2260 0. \u0394 = L = K = 0, and\u2009\u2009\u03941 \u2260 0\u2009\u2009or\u2009\u2009L1 \u2260 0. \u2009\u2009\u0394 = L = K = T = 0, and \u03941 \u2260 0 or L1 \u2260 0 or K1 \u2260 0.\u2009\u2009\u0394 = T = L = 0, K > 0, \u0394 = (K/2)2, and any one of the following conditions holds.1 \u2260 KK1. 2\u0394L1 \u2260 KT1. \u2009\u20092f(s) = T1s3 + K1s2 + L1s + \u03941, g(s) = s4 + Ts3 + Ks2 + Ls + \u0394. Obviously, when the conditions (I)\u2013(III) hold, 0 is the multiply poles of YQ(s).Let YQ(s) has two multiply nonzero poles \u00b1i\u03c9\u2009\u2009(\u03c9 > 0), then g(s) has the form:T = L = 0, K = 2\u03c92 > 0, \u0394 = \u03c94 = (K/2)2 > 0.If i\u03c9 are the multiply poles of YQ(s), thenf(i\u03c9) \u2260 0 whereIf \u00b1f(i\u03c9) \u2260 0, any one of \u00b1i\u03c9 is a multiply pole of YQ(s).Obviously, when any one of (1)-(2) of (IV) in YQ(s) satisfies condition (4) in So, when any one of condition (I)\u2013(IV) holds, YQ(s) satisfies When any one of Theorems These theorems can be implemented by a computer program for calculating the bifurcation diagram of the general corresponding CNN to determine emergence of complex dynamic patterns of the corresponding CNN.Life systems consist of locally coupled homogeneous media. Mostly, dynamics of life systems are suitable to be described via locally connected reaction-diffusion CNNs. It may be expected that reaction-diffusion CNN will become a promising candidate for modeling life phenomena.x, y, v, yn, vn represent the numbers of uninfected cells, infected cells infected by normal virus, normal virus, infected cells infected by mutated virus, and mutant viruses, respectively. \u03bb is the rate of reproduction of uninfected cells. Uninfected cells die at rate dx and become infected at rate bxv by normal virus and infected at rate bnxvn by mutated virus. Infected cells infected by normal and mutated virus are removed at rate ay and ayn, respectively. Normal virus is produced at rate ky and removed at rate uv, mutated virus is produced at rate kyn and removed at rate uvn. e is the rate constant describing the probability of mutation of virus , a, b, bn, d, e, k, kn, u, \u03bb are positive constants. The model was briefly analyzed in Nowak's book.In Chapter 11 \u201cTiming the emergence of resistance\u201d (Page 110) of the book \u201cVirus dynamic: mathematical principles of immunology and virology\u201d (Oxford university press), Nowak et al. proposed a mathematical model which describes the mutation selection of HBV infection during the therapy :(37)dxdThe reaction-diffusion CNN of HBV mutation selection of model has the form:2xij = xi\u22121j + xi+1j + xij\u22121 + xij+1 \u2212 4xij.D1 = 0) and solve it, we can get the two equilibrium points:Let equation  be zerosx0 = au/((1 \u2212 e)bk) and Q1, Q2 stand for the patient's complete recovery and HBV persistent infection, respectively.Qi\u2009\u2009 isConsequently, the Jacobian matrix at the equilibrium point k, u as variables, and \u03bb = 10, a = 0.5, b = 0.01, bn = 0.005, e = 0.0001, kn = 10, and d = 0.01, using Theorems Q1 and Q2 at k \u2208 , u \u2208 , see Figures Taking Q1 does not exist at the edge of chaos domain.In Figures \u03bb = 10, k = 0.01, a = 0.5, b = 0.01, bn = 0.005, kn = 10,\u2009\u2009e = 0.0001, and\u2009\u2009k = 1.0,3.0,4.9,5.1,10,24,39, u = 2,5, 9, we model the dynamic trajectories of equation , when these parameters are located in the green domain ;u, the dynamic pattern of equation  but less than a threshold value .u, the dynamic pattern of equation ,When these parameters are located in the red domain (edge of chaos);u value, the dynamic pattern of equation (regardless of the equation  is conveThe No. 2 and No. 3 variables in equation  increaseThe local activity of CNN has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice formed by coupled cells. Based on the local activity principle, the analytic criteria for the local activity in reaction-diffusion CNN with five state variables and one port are set up. The analytical criteria include four theorems, which provide the inequalities involving the parameters of the CNN. The inequalities can be used for calculating the bifurcation diagram to determine emergence of complex dynamic patterns of the reaction-diffusion CNN. As an application example, a reaction-diffusion CNN of HBV mutation-selection model is analyzed and simulated, and the bifurcation diagrams are calculated. Numerical simulations show this CNN model may explain certain complex mutant conditions during the therapy. We conclude that the local activity theory provides a practical tool for the study of the complex dynamics of certain coupled nonlinear systems."}
{"text": "The point is, the delta shock wave is not the one of transport equations, which is obviously different from cases of some other systems such as Euler equations or relativistic Euler equations. For the generalized Chaplygin gas, unlike the polytropic or isothermal gas, there exists a certain critical value \u03f52 depending only on the Riemann initial data, such that when \u03f5 drops to \u03f52, the delta shock wave appears as u\u2212 > u+, which is actually a delta solution of the same system in one critical case. Then as \u03f5 becomes smaller and goes to zero at last, the delta shock wave solution is the exact one of transport equations. Furthermore, the vacuum states and contact discontinuities can be obtained as the limit of Riemann solutions when u\u2212 < u+ and u\u2212 = u+, respectively.The limit of Riemann solutions to the nonsymmetric system of Keyfitz-Kranzer type with a scaled pressure is considered for both polytropic gas and generalized Chaplygin gas. In the former case, the delta shock wave can be obtained as the limit of shock wave and contact discontinuity when  The nonsymmetric system of Keyfitz-Kranzer type can be written as\u03c1t+ = u \u2212 p, and p = p(\u03c1), system  = \u03c1\u03b3, \u03b3 > 0 is smooth and strictly increasing withp into \u03f5p and taking p(\u03c1) = (1/\u03c1 \u2212 1/\u03c1*), \u03c1 \u2264 \u03c1*, where \u03c1* is the maximal density which corresponds to a total traffic jam and is assumed to be a fixed constant although it should depend on the velocity in practice. Then, Shen and Sun  = \u03c1+ \u2212 \u03c1\u2212. By simple calculation, we obtain\u03c1\u2212 \u2260 \u03c1+, and\u03c1\u2212 = \u03c1+.With these definitions, one can construct a We can also justify that the delta shock wave satisfies the entropy condition:In this section, we analyze some basic properties and solve the Riemann problem for .System  and 10)10) have So the 1-characteristic field is genuinely nonlinear, and the 2-characteristic field is always linearly degenerate.t \u2192 \u03b2t and x \u2192 \u03b2x, \u03b2 > 0, the solution is only connected with \u03be = x/t. Thus we should seek the self-similar solutionu, \u03c1)(\u00b1\u221e) = .Since -10) and and10) a problem -(10) and problem  can be rFor smooth solutions, system  can be r\u03be = \u03c3\u03f5, the Rankine-Hugoniot condition\u03c1] = \u03c1 \u2212 \u03c1\u2212 and \u03c3\u03f5 is the velocity of the discontinuity. From  coincid in the phase plane; that is,  bel bel\u03be = \u03c3u\u2212, \u03c1\u2212), we draw the curve u = u\u2212 for \u03c1 > 0 in the phase plane, which is parallel to the \u03c1-axis. We denote it by R when \u03c1 < \u03c1\u2212 and S when \u03c1 > \u03c1\u2212. Through the point , we draw the curve , denoted by J. Then the phase plane is divided into four regions  \u2208 I, that is, u+ > u\u2212\u2009\u2009and\u2009\u2009u+ < u\u2212 + \u03f5(\u03c1\u03b3 \u2212 \u03c1\u2212\u03b3), the solution is S + J;when  \u2208 II, that is, u+ > u\u2212\u2009\u2009and\u2009\u2009u+ > u\u2212 + \u03f5(\u03c1\u03b3 \u2212 \u03c1\u2212\u03b3), the solution is R + J;when  \u2208 III, that is, u+ < u\u2212\u2009\u2009and\u2009\u2009u+ > u\u2212 + \u03f5(\u03c1\u03b3 \u2212 \u03c1\u2212\u03b3), the solution is R + J;when  \u2208 IV, that is, u+ < u\u2212\u2009\u2009and\u2009\u2009u+ < u\u2212 + \u03f5(\u03c1\u03b3 \u2212 \u03c1\u2212\u03b3), the solution is S + J.when (Through the point (he curve  which inions see . Thus weions see -10) as  as u\u2212, \u03b1 < 1, while both the two characteristic fields are fully linearly degenerate as \u03b1 = 1.Systems  and 11)11) have \u03b1 < 1, we get rarefaction wave and shock wave which can be expressed byWhen 0 < \u03b1 < 1, through the point , we draw the curve u = u\u2212 for \u03c1 > 0 in the phase plane, denoted by R when \u03c1 < \u03c1\u2212 and S when \u03c1 > \u03c1\u2212. Through the point , we draw the curve , we draw the curve , the Riemann solution is showed as follows:u+, \u03c1+) \u2208 I, that is, u+ > u\u2212 and u+ < u\u2212 + \u03f5(\u03c1\u2212\u03b1\u2212 \u2212 \u03c1\u03b1\u2212), the solution is S + J;when  \u2208 II, that is, u+ > u\u2212 and u+ > u\u2212 + \u03f5(\u03c1\u2212\u03b1\u2212 \u2212 \u03c1\u03b1\u2212), the solution is R + J;when  \u2208 III, that is, u+ < u\u2212 and u+ > u\u2212 + \u03f5(\u03c1\u2212\u03b1\u2212 \u2212 \u03c1\u03b1\u2212), the solution is R + J;when  \u2208 IV, that is, u+ < u\u2212 and u+ < u\u2212 + \u03f5(\u03c1\u2212\u03b1\u2212 \u2212 \u03c1\u03b1\u2212), the solution is S + J.when  is given byFor any given  \u2208 V, we introduce a definition of \u03b4-measure solution, in which we introduce a definition of a generalized solution  = \u03c1+ \u2212 \u03c1\u2212, 0 < \u03b1 < 1.Suppose that f system  and 11)\u03a9 \u2282 \u211d \u00d7 \u211d\u03c1\u2212 \u2260 \u03c1+, and\u03c1\u2212 = \u03c1+.From , we obtaWe also can justify that the delta shock wave satisfies the entropy condition:\u03b1 = 1, the detailed study can be found in H(\u03be \u2212 \u03c3) and H is the Heaviside function.The first integral in  can be d{\u222b\u2212\u221e\u03c3\u03f5+\u222b\u03c3erms in .The second term in  can be cen, from 1, 84), , (85)\u222b\u03c3\u03f5e obtainlim\u2061\u03f5\u2192\u03f52\u222bWith the same reason as above, we have\u03c1\u03f5 and \u03c1\u03f5u\u03f5 as \u03f5 \u2192 \u03f52, by tracing the time-dependence of weights of the \u03b4-measure.Finally, we study the limits of \u03c6 \u2208 C0\u221e\u00d7[0, \u221e)) and set Let e obtainlim\u2061\u03f5\u2192\u03f52\u222ber hand,lim\u2061\u03f5\u2192\u03f52\u222bWith the same reason as before, we obtainu\u2212 > u+ and 0 < \u03f5 < \u03f52,  \u2208 V. So the Riemann solution of . We want to observe the behavior of strength and propagation speed of the delta shock wave when \u03f5 decreases and finally tends to zero.When ution of  and 11)u\u2212 > u+\u03c1+ \u2260 \u03c1\u2212, letting \u03f5 \u2192 0 in 11) are iu+, \u03c1+) \u2208 IV, we conclude that the shock wave and a contact discontinuity coincide as a delta shock wave when \u03f5 \u2192 \u03f52. As \u03f5 continues to drop and goes to zero eventually, the delta shock solution is nothing but the Riemann solution to transport equations (Combining the results of the above, when (quations .u\u2212 > u+, u\u2212 < u+, and u\u2212 = u+, respectively. For the polytropic gas, different from cases of some other systems such as Euler equations or relativistic Euler equations, the delta shock wave is not the one of transport equations as parameter \u03f5 tends to zero. For the generalized Chaplygin gas, the delta shock wave appears as parameter \u03f5 tends to \u03f52, depending only on the Riemann initial data. Then as \u03f5 becomes smaller and goes to zero at last, the delta shock wave solution is the exact one of transport equations.So far, the discussion for limit of Riemann solutions to the nonsymmetric system of Keyfitz-Kranzer type with both the polytropic gas and generalized Chaplygin gas has been completed. From the above analysis, as the pressure vanishes, there appear delta shock wave, vacuum state, and contact discontinuity when"}
{"text": "X, d) and for the family \ud835\udc9c of subsets of X established a new cone metric H : \ud835\udc9c \u00d7 \ud835\udc9c \u2192 E and obtained fixed point of set-valued contraction of Nadler type. Further, it is noticed in the work of Jankovi\u0107 et al., 2011 that the fixed-point problem in the setting of cone metric spaces is appropriate only in the case when the underlying cone is nonnormal. In the present paper we improve Wardowski's result by proving the same without the assumption of normality on cones.Wardowski (2011) in this paper for a normal cone metric space ( Huang and Zhang  generaliE be a Banach Space and P a subset of E. Then, P is called a cone wheneverP\u2009\u2009is closed, nonempty, and P \u2260 {\u03b8},ax + by \u2208 P for all x, y \u2208 P and nonnegative real numbers a, b,P\u2229(\u2212P) = {\u03b8}.Let P induces a partial ordering \u227c on E by x\u227cy if and only if y \u2212 x \u2208 P. So x < y will stand for x\u227cy and x \u2260 y, while x \u226a y will stand for y \u2212 x \u2208 int\u2061P, where int\u2061P denotes the interior of P. The cone P is called normal if there is a number K > 0 such that, for all x, y \u2208 E,K\u2009\u2009satisfying  for all x, y \u2208 X and d = \u03b8 if and only if x = y,2) = d for all x, y \u2208 X,3)\u227cd + d for all x, y, z \u2208 X. Then, d is called a cone metric on X, and  is called a cone metric space.  Let X, d) be a cone metric space, x \u2208 X and {xn}n\u22651 a sequence in X. Then, {xn}n\u22651 converges to x whenever for every c \u2208 E with \u03b8 \u226a c there is a natural number N such that d \u226a c for all n \u2265 N. We denote this by lim\u2061n\u2192\u221exn = x or xn \u2192 x. {xn}n\u22651 is a Cauchy sequence whenever for every c \u2208 E with \u03b8 \u226a c there is a natural number N such that d \u226a c for all n, m \u2265 N.\u2009\u2009 is called a complete cone metric space if every Cauchy sequence in X is convergent.Let  is called closed if, for any sequence {xn}\u2282A\u2009\u2009convergent to x, we have x\u2208A. Denote by N(X) the collection of all nonempty subsets of X and by C(X) a collection of all nonempty closed subsets of X. Denote by Fix\u2061T a set of all fixed points of a mapping T. In the present paper, we assume that E is a real Banach space, P is a cone in E with nonempty interior , and \u227c is a partial ordering with respect to P. In accordance with d \u2208 V, for all n \u2265 N1. Then, (\u03bbn/2/(1 \u2212 \u03bb1/2))d \u226a c, for all n \u2265 N1. Thus,m > n. Therefore, {xn}n\u22651 is a Cauchy sequence. Since X is complete, there exists u \u2208 X such that xn \u2192 u. Sincen, xn+1 \u2208 Txn, we have yn \u2208 Tu such that N2\u2009\u2009such thatn \u2265 N2,yn \u2192 u, and it implies that u \u2208 Tu.Let X = , E = CR2\u2009\u2009with the norm ||x|| = ||x||\u221e + ||x/||\u221e, P = {x \u2208 E : x \u2265 0}, x(t) = t, and y(t) = tK2. Then,\u2009\u20090 \u2264 x \u2264 y,\u2009\u2009||x|| = 2, and ||y|| = 1 + 2K. For all K \u2265 1, since K||x|| < ||y||. Therefore, P is non-normal. Define d : X \u00d7 X \u2192 E as follows:\ud835\udc9c be a family of subsets of X of the form \ud835\udc9c = { : x \u2208 X} \u222a {{x} : x \u2208 X}, and define H : \ud835\udc9c \u00d7 \ud835\udc9c \u2192 E as follows:H\u2009\u2009satisfies (H1)\u2013(H4) of T : X \u2192 \ud835\udc9c\u2009\u2009asT satisfies the conditions of \u03bb = 1/2 and 0\u2208Fix\u2061T. Suppose"}
{"text": "We investigate the moduli spaces of stable sheaves on a smooth quadric surface with linear Hilbert bipolynomial in some specialcases and describe their geometry in terms of the locally free resolution of the sheaves. \u2102, the field of complex numbers.Throughout the paper, our base field is X with a fixed Hilbert polynomial, which is itself a projective variety, and the moduli space has been studied quite intensively in the last decade for the case with linear Hilbert polynomial over projective spaces  of degree 1 such thatu, v) \u2208 \u2124\u22952.For a purely 1-dimensional sheaf mt + c is the Hilbert polynomial of \u2131 with respect to the ample line bundle \ud835\udcaaQ. Let us take any D \u2208 |\ud835\udcaaQ|, T \u2208 |\ud835\udcaaQ|, and a smooth conic C \u2208 |\ud835\udcaaQ| such that neither D, T, nor C is contained in the 1-dimensional reduced curve Supp(\u2131).Let us assume that D, T, and C induce maps jD : \u2131 \u2192 \u2131,\u2009\u2009jT : \u2131 \u2192 \u2131, and jC : \u2131 \u2192 \u2131. Since neither D nor T is contained in the 1-dimensional reduced curve Supp(\u2131), we have jD \u2260 0 and jT \u2260 0. Since \u2131 is pure, we obtain that jD, jT, and jC are injective. Thus, there are exact sequencesThe curves a : = h0 \u2297 \ud835\udcaaD) and b : = h0 \u2297 \ud835\udcaaT). The sheaves \u2131 \u2297 \ud835\udcaaD, \u2131 \u2297 \ud835\udcaaT, and \u2131 \u2297 \ud835\udcaaC have finite supports and thus the dimensions of their cohomology H0 do not change even if we twist them by a line bundle on Q. From  \u2297 \ud835\udcaaC) = m.Let us set  Q. From , we get \u03c7) = av + bu + c for all  \u2208 \u2124\u22952. If u = v, then the claim is true. Now assume that u \u2260 v, say u > v. We use u \u2212 v exact sequences like  instead of \u2131 with 0 \u2264 c < u \u2212 v to get \u03c7) = \u03c7+(u \u2212 v)b.We claim that ces like  with \u2131 \u2208 \u211a of \u2131 to be a linear bipolynomial such that\u2131 with respect to \ud835\udcaaQ is defined to be \u03c7\u2131(t) = \u03c7\u2131.One defines the Hilbert bipolynomial \u03c7\u2131 is a linear function, that is, \u03c7\u2131 = mx + ny + t for some  \u2208 \u2124\u22953.We are mainly interested in the case when \u2131 be a pure sheaf of dimension 1 on Q with \u03c7\u2131 = mx + ny + t. The p-slope of \u2131 is defined to be p(\u2131) = t/(m + n). \u2131 is called semistable (stable) with respect to the ample line bundle \ud835\udcaaQ if (1)\u2131 does not have any 0-dimensional torsion,(2)\u2131\u2032, one hasfor any proper subsheaf \u2009\u03c7\u2131\u2032 = m\u2032x + n\u2032y + t\u2032.where Let \u2131 with \u03c7\u2131 = mx + ny + t, let us define C\u2131 : = Supp(\u2131) to be its scheme-theoretic support and then we have C\u2131 \u2208 |\ud835\udcaaQ|. We often use slope stability and slope semistability instead of Gieseker stability or Gieseker semistability just to simplify the notation; they should be the same because the support is 1-dimensional, and from mt + \u03c7 and m\u2032t + \u03c7\u2032, the inequality for Hilbert and slopes \u03c7/m the same.For every semistable 1-dimensional sheaf M be the moduli space of semistable sheaves on Q with linear Hilbert bipolynomial \u03c7 = mx + ny + t.Let M in a different way as a subvariety of MQ,\u21193, the moduli space of semistable sheaves on \u21193 with linear Hilbert polynomial \u03c7(x) = mx + t, which are \ud835\udcaaQ-sheaves. To be precise, if \u2131 is \ud835\udcaaQ-sheaf, then all of its \ud835\udcaa\u21193-subsheaves are also \ud835\udcaaQ-sheaves. It implies that the notions of p-stability and \u03bc-stability of \u2131 are the same and thus MQ,\u21193 may be defined without using \u21193. Moreover, the sheaf with linear Hilbert bipolynomial \u03c7 = ax + by + c has Hilbert polynomial \u03c7(x) = (a + b)x + c with respect to \ud835\udcaaQ and thus we have a natural decompositionM is a subvariety of MQ,\u21193.We can define \u2131 be any purely 1-dimensional coherent sheaf on \u21193 with Hilbert polynomial mx + ti. Assume that \u2131 is not semistable and let\u2131  is a projective and irreducible scheme. If mn > 0, then M\u00b0 is a Zariski dense and open subset of M with dimension 2mn + 1.The moduli Q holds. But it holds, using Castelnuovo-Mumford criterion with the Serre duality\u2131 \u2208 M and j \u2265 \u22121.The first assertion follows verbatim from the proof of Proposition 2.3 and Theorem 3.1 in , only wh\u2131\u2032, \u2131\u2032\u2032) to \u2131\u2032 \u2295 \u2131\u2032\u2032, where m = m\u2032 + m\u2032\u2032 and n = n\u2032 + n\u2032\u2032. Then the dimension of the image of this map is at least 2mn \u2212 2m\u2032n\u2032 \u2212 2m\u2032\u2032n\u2032\u2032 \u2212 1 and it is at least 1 if mn > 0. In other words, general sheaf in M is stable.For the second assertion, let us consider a map\u2131 on Q with Hilbert bipolynomial \u03c7\u2131 = mx + ny + t, let us define\u2131. Since \u2131 is pure, the natural map \u03c6\u2131 : \u2131 \u2192 \u2131DD is injective. Since the support of \u2131 is 1-dimensional, \u03c6\u2131 is bijective as in Remark 4 of , where Li is a line in |\ud835\udcaaQ|.One hast = n and let us choose L \u2208 |\ud835\udcaaQ| and then it fits into\ud835\udcaaL is stable. For a line L \u2208 |\ud835\udcaaQ|, we haveL, we havef is a zero map. Thus, there exists a nontrivial extension of \ud835\udcaaL by \ud835\udcaaL and it is \ud835\udcaaL2. In particular, \ud835\udcaaL\u22952 and \ud835\udcaaL2 represent the same point in HilbQ(2y + 2). In general, \ud835\udcaaLk\u2295 and \ud835\udcaakL with k \u2265 1 represent the same point in M. Thus, \ud835\udcaaL with L \u2208 |\ud835\udcaaQ| is strictly semistable if and only if n \u2265 2. Conversely, let us choose a semistable sheaf \u2131 with \u03c7\u2131 = ny + n. In particular, the schematic support L = Supp(\u2131) of \u2131 is in |\ud835\udcaaQ|. Since \u03c7(\u2131) = n > 0, there exists a nontrivial morphism \ud835\udcaaQ \u2192 \u2131 and it induces an injection \ud835\udcaaL1 \u2192 \u2131, where L1 is a subscheme of L. Here we have L1 \u2208 |\ud835\udcaaQ| for some s \u2264 n and so \u03c7L1 = sx + s. Thus, the quotient \ud835\udca2 = \u2131/\ud835\udcaaL1 is a semistable sheaf with \u03c7\ud835\udca2 = (n \u2212 s)y + (n \u2212 s). By induction, we have [\ud835\udca2] = [\ud835\udcaaL2] with L2 \u2208 |\ud835\udcaaQ|. In particular, \u2131 is an extension of \ud835\udcaaL2 by \ud835\udcaaL1 with L1 + L2 \u2208 |\ud835\udcaaQ| and thus \u2131 is equivalent to \ud835\udcaaL1 \u2295 \ud835\udcaaL2.Let us assume that t < n and fix \u2131 \u2208 M with C : = C\u2131 \u2208 |\ud835\udcaaC|. Since \u03c7(\u2131) = t > 0, there is a nonzero map f : \ud835\udcaaQ \u2192 \u2131. Since \u2131 is an OC-sheaf, f induces a nonzero map h : \ud835\udcaaC \u2192 \u2131. Since \ud835\udcaaC has slope 1 > t/n and it is semistable, we get a contradiction. Alternatively, as in Lemma 4.10 of [T\u2286C of Im\u2061(h) and then use an injective map \ud835\udcaaT \u2192 \u2131 with \ud835\udcaaT \u2208 |\ud835\udcaaQ| with 1 \u2264 n\u2032 < n, and thus we have \u03bc(\ud835\udcaaT) = 1.Now, let us assume that 0 <  4.10 of , we may m = 1, that is, \u03c7\u2131 = x + ny + t, it is enough to check the case t = 1 since gcd = 1.For the case of M consists of \ud835\udcaaC with C \u2208 |\ud835\udcaaQ|. In particular, one has M\u2245\u2119n+12.\u03c7\ud835\udcaaC = x + ny + 1 and \ud835\udcaaC is semistable by \u2131 be a semistable sheaf with \u03c7\u2131 = x + ny + 1 and so C : = C\u2131 is a curve in |\ud835\udcaaQ|. Since we have \u03c7(\u2131) = 1, there exists a nonzero map \ud835\udcaaQ \u2192 \u2131 and it induces a nonzero map h : \ud835\udcaaC \u2192 \u2131. Note that Im\u2061(h) has no 0-dimensional torsion since \u2131 is semistable. Since \ud835\udcaaC is also semistable, we haveh factors through an injection \ud835\udcaaD\u21aa\u2131, where D is a curve contained in C. If D is properly contained in C, we have p(\ud835\udcaaD) > p(\u2131) contradicting the semistability of \u2131 and thus we have D = C; that is, h is an isomorphism from \ud835\udcaaC to its image. Since \ud835\udcaaC and \u2131 have the same Hilbert polynomial, we have \u2131\u2245\ud835\udcaaC.From the sequencex + 2y + t, it is enough to investigate the case when t = 1,2. Let us denote the moduli space M by Mt.For the moduli space of semistable sheaves with linear Hilbert bipolynomial 2M1 consists of the unique nontrivial extensions \u2131 of \ud835\udcaaP by \ud835\udcaaC for each curve C \u2208 |\ud835\udcaaQ| and a point P \u2208 C, and one also has h0(\u2131) = 1. The moduli space \u03c7(\u2131) = 1, there is a nonzero map \ud835\udcaaQ \u2192 \u2131, inducing a nonzero map h : \ud835\udcaaC \u2192 \u2131, where C : = C\u2131 \u2208 |\ud835\udcaaQ|. Since \u03c7\ud835\udcaaC = 2x + 2y, we have p(\ud835\udcaaC) = 0 < 1/4 = p(\u2131). The map h factors through an injection \ud835\udcaaD\u21aa\u2131, where D is a curve contained in C. If D is properly contained in C, we have p(\ud835\udcaaD) > p(\u2131) contradicting to the semistability of \u2131 and thus we have D = C; that is, h is an isomorphism from \ud835\udcaaC to its image, that is, we have\u03c7\ud835\udca2 = 1. In particular, we have \ud835\udca2\u2245\ud835\udcaaP, the skyscraper sheaf supported on a point P \u2208 C. Since \u2131 has no 0-dimensional torsion, the sequence does not split. Note that Ext1\u2245H1(\ud835\udcaaP)\u2228 = 0, and thus from the sequence of C we haves is the transpose of Hom, \ud835\udcaaP) \u2192 Hom which is given by the multiplication by the defining equation of C. Since P is a point on C, the map s is a zero map. In particular, the dimension of Ext1 is 1 and so \u2131 corresponds to a unique nontrivial extensionh0(\u2131) \u2264 2 and that h0(\u2131) = 1 if and only if no injective map \ud835\udcaaC \u2192 \u2131 is an isomorphism at P. This is certainly true if \u2131 is not locally free of rank 1 at P. Note that \u2131 is a line bundle at each point of C\u2216{P} and thus it is sufficient to prove h0(\u2131) = 1 when \u2131 is a line bundle on the curve C. In this case the nonexistence of a section of \u2131 that does not vanish at P is equivalent to the nonsplitting of (h0(\u2131) = 0 and so the point P is uniquely determined by \u2131.Since xtension0\u27f6\ud835\udcaaC\u27f6\u2131\u27f6\ud835\udcaaPxtension0\u27f6\ud835\udcaaC\u27f6\u2131\u27f6\ud835\udcaaP\u2131 is a nontrivial extension of \ud835\udcaaP by \ud835\udcaaC, where P is a point on C. If \u2131 is not semistable, then there exists a subsheaf \ud835\udca6 \u2282 \u2131 with p(\ud835\udca6) > p(\u2131) = 1/4 and so we have \u03c7\ud835\udca6 = m\u2032x + n\u2032y + t\u2032 with \u2264 and t\u2032 \u2265 1. If the composite s : \ud835\udca6 \u2192 \u2131 \u2192 \ud835\udcaaP is a zero map, then we have an injection \ud835\udca6\u21aa\ud835\udcaaC, contradicting the semistability of \ud835\udcaaC. Thus, the composite is surjective and so we have the following diagram:\ud835\udca6\u2032 is the kernel of the map s and \u210b is the quotient \u2131/\ud835\udca6. Since \u03c7\ud835\udca6\u2032 = m\u2032x + n\u2032y + (t\u2032 \u2212 1) and \ud835\udcaaC is semistable, we have t\u2032 = 1 and thus \u03c7\u210b = (2 \u2212 m\u2032)x + (2 \u2212 n\u2032)y with no constant term. Since \u210b is the quotient of \ud835\udcaaC, it must be \ud835\udcaaT for some curve T contained in C. But no such curves have the Hilbert polynomials with no constant term. Hence \u2131 is semistable.Conversely, let us assume that M1. Let us assume the existence of a polystable sheaf \u2131 = \u21311 \u2295 \u22ef\u2295\u2131s with s \u2265 2. We have \u03c7(\u2131) = 1 = \u03c7(\u21311)+\u22ef+\u03c7(\u2131s). If we let \u03c7\u2131i = aix + biy + ci, then we haveci > 0 for all i and thus we have s = 1, a contradiction.There is no strictly semistable sheaf in \u2131 is in M1 if and only if it admits the following resolution:\ud835\udc9c : = \ud835\udcaaQ \u2295 \ud835\udcaaQ and f : = h1l2 \u2212 h2l1 is a defining equation of C\u2131.A sheaf h0(\u2131) = 1 and so h1(\u2131) = 0. If \u2131 admits the sequence . Take any A \u2208 |\ud835\udcaaQ| which is not contained in C and with P \u2209 A. The multiplication by an equation of A gives an exact sequence\u2131|A) = deg\u2061(A\u2229C) = 4. Thus we have h1) = 0 and h0) = 5. Together with the exact sequence , we obtain that \u2131 is globally generated at P and so we have a surjection\u210b : = ker\u2061(\u03c8) and then \u210b is a torsion-free sheaf of rank 2 on Q with c1 = . By Theorem 19.9 in [\u210b is locally free. Note that \u03c7\u210b = 2xy + 3x + y + 1. Thus, we have h0) > 0 and so we have an exact sequenceZ is a 0-dimensional subscheme of Q and \u2208{, }. If  = , then we have \u03c7\u2110Z = xy + x + 1 and it is absurd since \u03c7\ud835\udcaaQ = xy + x. Thus, we have  =  and Z = \u2205. Since Ext1, \ud835\udcaaQ) = 0, we have \u210b\u2245\ud835\udcaaQ \u2295 \ud835\udcaaQ and the sequence  is given by f : = h1l2 \u2212 h2l1 is a defining equation of C = C\u2131.Note that sequence , then \u2131 sequence  tensoredjection3\u03c8:\ud835\udcaaQ\u2295\ud835\udcaaQ is globally generated and so a surjection \u03c6\u2032 : \ud835\udcaaQ \u2295 \ud835\udcaaQ \u2192 \u2131. In this case, ker\u2061(\u03c6\u2032) is no longer a direct sum of two line bundles.Using the proof of Lemma 5.3 in , we can W to beW0 \u2282 W to be the set of \u03c6 \u2208 W such that h1l2 \u2212 h2l1 \u2260 0. Then we have a surjective morphismLet us define a vector space \u03c61, \u03c62 \u2208 W0 with \u03c0(\u03c61) = \u03c0(\u03c62); that is, we have the following diagram:f is an isomorphism. Since Ext1, \ud835\udc9c) = 0, we have a map f1 \u2208 End) associated with f. Note that f1 is given by a, b \u2208 \u2102\u00d7 and z \u2208 H0). Similarly, we have a map f2 : \ud835\udc9c \u2192 \ud835\udc9c which is c1, c2 \u2208 \u2102\u00d7. In particular, we havec1 = 1. In other words, \u03c0(\u03c61) = \u03c0(\u03c62) if and only if \u03c61 and \u03c62 are in the same orbit in W0 under the action byLet us choose \u03c0 : W0 \u2192 M1 is a geometric quotient map by the action of G. In particular, one has M1\u2245W0/G and so M1 is isomorphic to \ud835\udcabic1.To get the assertion it suffices to prove that it has local sections as in Lemma 5.1 and Theorem 5.5 in .M1 is stable, M1 has a universal family \u210d1 on M1 \u00d7 \u21192  for a unique \u2131 \u2208 M1, we also have a universal family \u210d5 on M5 \u00d7 \u21192 with \u2131 as the fibre. Since h0(\u2131) = 1 and h1(\u2131) = 0 for all \u2131 \u2208 M1, the base change theorem gives that u\u2217(\u210d1) is a line bundle on M1, where u : M1 \u00d7 \u21192 \u2192 M1 is the first projection. Since h0) = 5 and h1) = 0 for all \u2131 \u2208 M1, the base change theorem gives that v\u2217(\u210d5) is a vector bundle of rank 5 on M1 by identifying M5 with M1, where v : M5 \u00d7 \u21192 \u2192 M5. For a fixed \u2131 \u2208 M1 and a matrix \u03c6 \u2208 \u03c0\u22121(\u2131), let us write f : = h1l2 \u2212 h2l1 is a defining equation of C\u2131. Take an open neighborhood U of \u2131 in M1 over which u\u2217(\u210d1) and v\u2217(\u210d5) are trivial. The matrix \u03c6 was constructed starting with a section \u03c3 of \u2131 which spans \u2131 together with the twist \u03c3\u2032 of a nonzero section of \u2131. Since u\u2217(\u210d1)|U and v\u2217(\u210d5)|U are trivial, there are maps e1 : \ud835\udcaaU \u2192 \ud835\udcaaU and e2 : \ud835\udcaaU \u2192 \ud835\udcaaU\u22955 with e1(\u2131) = \u03c3\u2032 and e2(\u2131) = \u03c3. Since \u03c3\u2032 and \u03c3 span \u2131, there is a neighborhood V of \u2131 in U such that the sections e1(\ud835\udca2) and e2(\ud835\udca2) span every \ud835\udca2 \u2208 V. The construction of \u03c6 gives that e1 and e2 induce a section of \u03c0 in a neighborhood of \u03c6 whose image by \u03c0 is V.Since every element of e 180 of ). Since M1 is irreducible and unirational. In fact, we can prove more.As an automatic consequence, we obtain that M1 is rational. Q \u00d7 Q be the diagonal and denote its ideal sheaf by \u2110\u0394. Denoting by p1 and p2 the projection from Q \u00d7 Q to each factor, let us define a sheaf \ud835\udcb0 to be p1*\ud835\udcaaQ)\u22a0\u2110\u0394 on Q \u00d7 Q. For each point P \u2208 Q, we have \ud835\udcb0|Q\u00d7{P}\u2245\u2110P. Thus, we have h1(\ud835\udcb0|Q\u00d7{P}) = 0 and so p2\u2217\ud835\udcb0 is a vector bundle of rank 8 on Q since h0(\ud835\udcb0|Q\u00d7{P}) = 8. Let us consider the projective bundle\ud835\udcb5 over a point P \u2208 Q is the set of curves of type  on Q passing through P and so there is a natural map from \ud835\udcb5 to \u2119H0)\u2245\u21198. In other words, \ud835\udcb5 is the universal curve of type  on Q and it is isomorphic to M1. Since \ud835\udcb5 is locally trivial over Q, it is rational.Let \u0394 \u2282 \u2131 \u2208 M2 admits one of the following types:\ud835\udcaaC \u2192 \u2131 \u2192 \u03b7 \u2192 0, where \u03b7 is a skyscraper \ud835\udcaaC-sheaf with degree 2,0 \u2192 \ud835\udcaaT1 \u2192 \u2131 \u2192 \ud835\udcaaT2 \u2192 0 with T1, T2 \u2208 |\ud835\udcaaQ|,0 \u2192 \ud835\udcaaT1 \u2192 \u2131 \u2192 \ud835\udcaaT2 \u2192 0, where T1 \u2208 |\ud835\udcaaQ| and T2 \u2208 |\ud835\udcaaQ| with  \u2208 {, }.0 \u2192 Any sheaf \u03c7(\u2131) = 2, we have h0(\u2131) \u2265 2. Thus, there exists a nonzero map \ud835\udcaaQ \u2192 \u2131 and it induces a nonzero map h : \ud835\udcaaC \u2192 \u2131, where C : = C\u2131 \u2208 |\ud835\udcaaQ|. The map h factors through an injection \ud835\udcaaT1\u21aa\u2131 where T1 is a curve contained in C:T1 = C, that is the map h is an isomorphism from \ud835\udcaaC to its image in \u2131, then its cokernel \u210b is the skyscraper sheaf supported on two points, say P1, P2 \u2208 C. Thus we have the sequenceSince T1 is properly contained in C and then we obtain that T1 has bidegree , , or  since p(\ud835\udcaaT1) \u2264 p(\u2131) = 1/2 and \u2131 is semistable. Let T2 \u2282 Q be the only curve such that T1 + T2 = C. Let \u210b\u2032 be the quotient of \u210b by its torsion \u03c4, that is, \u210b\u2032 : = \u210bDD.Let us assume that T1 \u2208 |\ud835\udcaaQ| and so we have \u03c7\u210b\u2032 = x + y + 1 \u2212 deg\u2061(\u03c4). Since \u2131 is semistable, we get \u03c4 = 0. Since every quotient of \u2131 has the slope at least 1/2, the same is true for \u210b. Thus, \u210b is semistable and \u210b\u2245\ud835\udcaaT2.First, assume T1 \u2208 |\ud835\udcaaQ|, that is, \u03c7\ud835\udcaaT1 = 2x + y + 1 and so we have \u03c7\u210b = y + 1. If \u210b has 0-dimensional torsion \ud835\udcaf with length k \u2265 1, then the quotient \u210b/\ud835\udcaf is a quotient of \u2131 with the p-slope 1 \u2212 k \u2264 0, contradicting the semistability of \u2131. Thus \u210b has no 0-dimensional torsion and so we have \u210b\u2245\ud835\udcaaT2 for a curve T2 \u2208 |\ud835\udcaaQ| with C = T1 + T2.Now, without loss of generality, let us assume that M2 is globally generated. Every sheaf in \u2131 \u2208 M2 and then there is no nonzero map \u2131 \u2192 \ud835\udcaaC\u2245\u03c9C since \u2131 is semistable. Thus we have h1(\u2131) = 0 and so h0(\u2131) = 2. It is clear that \u2131 of types (B) and (C) is globally generated and so we may assume that \u2131 is of type (A), but neither of (B) nor of (C).Let us take \u210b\u2286\u2131 be the image of the evaluation map H0(\u2131) \u2297 \ud835\udcaaQ \u2192 \u2131 and then \u210b is pure. Assume that \u210b \u2260 \u2131. Since \u2131 is of type (A), it is globally generated outside at most two points of Cred. In particular, we have \u03c7\u210b = 2x + 2y + c with c \u2264 1 and deg\u2061(\u2131/\u210b) = 2. Since h0(\u210b) = h0(\u2131) = 2, we have h1(\u210b) = 2 \u2212 c. Note that every nonzero section of \u210b vanishes at finitely many points since \u2131 is neither of types (B) nor (C). Since h0(\ud835\udcaaC) < h0(\u210b), we have \u210b \u2260 \ud835\udcaaC and c = 1. A nonzero section of \u210b induces an exact sequence\ud835\udca2\u2245\ud835\udcaaP for some P \u2208 Cred. Since \u210b is pure, this exact sequence does not split. As in the proof of \u210b = \u2131 and so \u2131 is globally generated.Let \u2131 is a sheaf in M2 if and only if it admits a sequencezij \u2208 H0) such that f : = z11z22 \u2212 z12z21 is a defining equation of C\u2131.\u2131 \u2208 M2 be a sheaf of type (A) and then it is globally generated by h0(\u2131) = 2, we have a surjection\u210b : = ker\u2061(\u03c8) and then it is a torsion-free sheaf of rank 2 on Q with c1 = . By Theorem 19.9 in [\u210b is locally free. Note that h0) = 2. From the sequenceH0\u22952) \u2192 H0) is an isomorphism and so h1) = 0. Similarly, we have h1) = 0 and h2(\u210b) = h1(\u2131) = 0. By Remark 2.3 in [\u210b is globally generated. Since c1) = 0 or h0) = 2, we have \u210b\u2245\ud835\udcaaQ\u22952 and the resolution  and (C), respectively. In particular, we haveLet us define a subscheme \u2131 of type (B) are strictly semistable. In particular, they are contained in \ud835\udd05. The sheaves \u2131. Let \ud835\udca6 be a subsheaf of \u2131 with p(\ud835\udca6) > 1/2 and the quotient sheaf \u210b : = \u2131/\ud835\udca6. If the composite map s : \ud835\udca6\u21aa\u2131\u21a0\ud835\udcaaT2 is a zero map, then \ud835\udca6 is a subsheaf of \ud835\udcaaT1, contradicting the semistability of \ud835\udcaaT1. The sheaf Im\u2061(s) is a subsheaf of \ud835\udcaaT2 and so we have p(Im\u2061(s)) \u2264 1/2. Similarly, the sheaf ker\u2061(s) is a subsheaf of \ud835\udcaaT1 and so p(ker\u2061(s)) \u2264 1/2. From the exact sequencep(\ud835\udca6) \u2264 1/2, a contradiction.It is enough to check the semistability of M2 the closed subscheme of M2, consisting of the strictly semistable sheaves.Let us denote by \u2202\ud835\udd162 is the permutation group of order 2. In particular, \u2202M2 is a rational variety.One has\ud835\udd05 \u2282 \u2202M2. Let \u2131 be a strictly semistable sheaf and so it has a proper quotient sheaf \u210b with p(\u210b) = 1/2. From the semistability of \u2131, \u210b has no 0-dimensional torsion. From the equality p(\u2131) = p(\u210b), we obtain that \u210b is also semistable. Since p(\u210b) = 1/2, the Hilbert bipolynomial of \u210b is either 2x + 1, 2y + 1 or x + y + 1. The first 2 cases cannot happen due to \u03c7\u210b = x + y + 1 and so \u210b\u2245\ud835\udcaaT2 with T2 \u2208 |\ud835\udcaaQ| and T2 \u2282 C\u2131 by \ud835\udca6 is the kernel of the quotient map \u210b \u2192 \u210b, then its p-slope is again 1/2 and so \ud835\udca6 is semistable. Similarly as before, we have \ud835\udca6\u2245\ud835\udcaaT1 with T1 \u2208 |\ud835\udcaaQ| and C\u2131 = T1 + T2. Hence, we have \u2131 \u2208 \ud835\udd05.Obviously, we have \u2131 be a sheaf of type (B), that is, it corresponds to a pair of two curves {T1, T2}. Let us assume that \u2131 admits another sequenceT3, T4 \u2208 |\ud835\udcaaQ|. Note that \ud835\udcaaTi is stable for all i. Thus, the composite map s : \ud835\udcaaT3 \u2192 \u2131 \u2192 \ud835\udcaaT2 is either a zero map or an isomorphism. In the former case, we have \ud835\udcaaT3\u2245\ud835\udcaaT1 and so \ud835\udcaaT4\u2245\ud835\udcaaT2. In the latter case, we have \ud835\udcaaT3\u2245\ud835\udcaaT2 and \ud835\udcaaT4\u2245\ud835\udcaaT1. Hence, the class of a strictly semistable sheaf \u2131 corresponds to a uniquely determined pair of two curves in |\ud835\udcaaQ| and we have \ud835\udd05\u2245(\u21193 \u00d7 \u21193)/\ud835\udd162. The second assertion follows from the fact that any symmetric product Sd(\u2119N) of any projective space is a rational variety .T1, T2 \u2208 |\ud835\udcaaQ|, one hasFor two curves \ud835\udcaaT2, \u2212) to the sequence of T1, we obtain\ud835\udcaaQ-sheaves, we haveT1 and T2 are reduced, and so the assertion is derived.Note that we have\u2131 of type (C), but not of type (B), are stable. In particular, the sheaves of type (C) are semistable.The sheaves \ud835\udca6 of \u2131 with p(\ud835\udca6) \u2265 1/2 and the quotient sheaf \u210b : = \u2131/\ud835\udca6. Since the composite s : \ud835\udca6\u21aa\u2131\u00b2\u21a0\ud835\udcaaT2 is not a zero map, thus we have Im\u2061(s)\u2245\ud835\udcaaT2(\u2212Z) for a 0-dimensional subscheme Z of T2 with length k. In particular, its Hilbert bipolynomial is y + 1 \u2212 k. If we let \u03c7\ud835\udca6 = m\u2032x + n\u2032y + t\u2032, then we have p(\ud835\udca6) = t\u2032/(m\u2032 + n\u2032) \u2265 1/2. In particular, we have t\u2032 \u2265 1. If we define \ud835\udca6\u2032 to be the kernel of the map s, then it is a subsheaf of \ud835\udcaaT1 and thus we have p(\ud835\udca6\u2032) = (t\u2032 \u2212 1 + k)/(m\u2032 + n\u2032 \u2212 1) \u2264 1/3. Combining the two inequalities, we have k = 0 and so the map s is surjective. Thus, we have \u210b\u2245\ud835\udcaaT1/\ud835\udca6\u2032. Note also that t\u2032 can be either 1 or 2. If t\u2032 = 2, then we have m\u2032 = n\u2032 = 2 and so \u03c7\ud835\udca6\u2032 = 2x + y + 1 = \u03c7\ud835\udcaaT1. In particular, we have \u210b = 0 and so \ud835\udca6\u2245\u2131, a contradiction. Now, assume t\u2032 = 1 and so m\u2032 + n\u2032 \u2264 2. In particular, \u210b is not a 0-dimensional sheaf. Moreover, \u210b is a quotient sheaf of \ud835\udcaaT1 with constant term 1 and so we have \u210b\u2245\ud835\udcaaT3 with T3 \u2282 T1 and T3 \u2208 |\ud835\udcaaQ|. For example, if T3 = T1, then we have \ud835\udca6\u2032 = 0 and it contradicts the nontriviality of the extension \u2131. Thus, \u2131 also admits the sequence\u03c7\ud835\udca6 = x + y + 1. Since \ud835\udca6\u2032 is a subsheaf of \ud835\udcaaT1 with \u03c7\ud835\udca6\u2032 = x, we have \ud835\udca6\u2032\u2245\ud835\udcaaT4, where T4 is a subcurve of T1 such that T1 = T3 + T4. Thus, \ud835\udca6 is an extension of \ud835\udcaaT2 by \ud835\udcaaT4. It is nontrivial, otherwise we would have \ud835\udcaaT2 as a direct factor of \u2131. Since there exists such a unique extension \ud835\udcaaT2+T4, \u2131 admits an extension of \ud835\udcaaT3 by \ud835\udcaaT2+T4:\u2131 is of type (B).As before let us assume the existence of a proper subsheaf \u2131 be a line bundle on a reduced curve C \u2208 |\ud835\udcaaQ| with degree 2. \u2131 is semistable if and only if one has:\u2131|T) \u2265 1 for all subcurves T of C in |\ud835\udcaaQ| with \u2a87\u2a87,deg\u2061(\u2131|A) \u2265 0 for each smooth subcurve A of C in |\ud835\udcaaQ| with \u2264\u2a87.deg\u2061(\u2131 is stable if and only if deg\u2061(\u2131|T) \u2265 1 for all subcurves T of C in |\ud835\udcaaQ| with \u2a87\u2a87.Let \u2131 is not stable  and take a proper subsheaf \u210b of \u2131 with p(\u210b) \u2265 p(\u2131) (resp. p(\u210b) > p(\u2131)). Taking a saturation of \u210b in \u2131, we may assume that \ud835\udca2 : = \u2131/\u210b is a pure sheaf. Call A the scheme support of \u210b and T the scheme support of \ud835\udca2. The definition of scheme support of a purely 1-dimensional sheaf gives A + T = C as effective divisors. Thus T has one of the types in the assertion. Since C is reduced and \u2131 is a line bundle on C, the support of \ud835\udca2 must be a proper subcurve T of C. If T does not have a type of \ud835\udcaaQ or \ud835\udcaaQ, then we are done. But the case of T having such types is excluded using the argument in the proof of In both parts, the \u201conly if\u201d part is obvious. Assume that \ud835\udd05\u2229\u212d \u2260 \u2205.One has B = B\u2032 + T2 with B\u2032 \u2208 |\ud835\udcaaQ| and T2 \u2208 |\ud835\udcaaQ|, and set A \u2208 |\ud835\udcaaQ| to be smooth. For any extension \u2131 \u2208 \ud835\udd05 of \ud835\udcaaB by OA, for example, \u2131 = \ud835\udcaaA \u2295 \ud835\udcaaB, let \u210b be the kernel of the composition \u2131 \u2192 \ud835\udcaaB\u2192\ud835\udcaaT2 and then \u210b is a pure sheaf with T1 : = A + B\u2032 as its scheme support and has Hilbert bipolynomial \u03c7\u210b = \u03c7\ud835\udcaaT1. Note that it has \ud835\udcaaA as its subsheaf.Let us set \u210b\u2245\ud835\udcaaT1, it is sufficient to prove that \u210b is semistable. Suppose \u210b is not semistable and take a proper saturated stable subsheaf \ud835\udca2 \u2282 \u210b with \u03c7\ud835\udca2 = ax + by + c. Its scheme support is contained in T1 and it is of type . Without loss of generality, let us assume that a \u2264 b. First, assume  = . In this case, we would have c \u2265 2 because p(\ud835\udca2) > p(\u210b) and so we have h0(\u210b) \u2265 2, contradicting the fact that h0(\u2131) = 2 and that \u2131 is globally generated. Assume a = b = 1. The map \ud835\udca2 \u2192 \u210b on A\u2216T2 must be just the inclusion \ud835\udcaaA \u2192 \u210b, because \u210b|A\u2216T2 is a line bundle. Thus either we have \ud835\udca2 = \ud835\udcaaA or \ud835\udcaaA is not saturated in \u210b. Hence the saturation \ud835\udc9c of \ud835\udcaaA in \u2131 has slope greater than 1/2, contradicting the semistability of \u2131. Now assume a = 0 and b = 1, that is, C\ud835\udca2 = B\u2032. Since B\u2032 is smooth, \ud835\udca2 is a line bundle on B\u2032. If its degree d is at least 1, then \ud835\udca2 contradicts the semistability of \u2131. If d \u2264 0, then we have p(\ud835\udca2) < p(\u210b), a contradiction. Hence \u2131 is also contained in \u212d.To prove T1 \u2208 |\ud835\udcaaQ| and T2 \u2208 |\ud835\udcaaQ|, one hasFor \ud835\udcaaT2, \u2212) to the sequence of T1, we obtain2 = 0. Note also that Ext1\u2245H0(\ud835\udcaaT2) and Ext2)\u2245H0(\ud835\udcaaT2)\u2228. Thus we have the assertion.Applying the functor Hom is the global sheaf of a sheaf with support on P1 and P2 with one copy of \u2102 on each point P1, P2,When R be a regular local ring of dimension 2 and take x, y generators of its maximal ideal. All Exti groups are with respect to R. Since R/(y) is Gorenstein, so the duality gives ExtR1(R/(y), R)\u2245R/(y) and ExtRi(R/(y), R) = 0 for all i \u2260 1. From the exact sequenceu is the multiplication by x, we get that ExtR1(R/(y), R/(x)) is the cokernel of the multiplication by x in R/(y) \u2192 R/(y); that is, we have ExtR1(R/(y), R/(x)) = \u2102. The same is true for extensions of \ud835\udcaaB by \ud835\udcaaA when A and B are transversal.Let \u2131 be a sheaf of type (A) with no 0-dimensional torsion. Then \u2131 is semistable unless it admits the sequenceT1 \u2208 |\ud835\udcaaQ| and T2 \u2208 |\ud835\udcaaQ| with \u2208{, }.Let \ud835\udca6 be a subsheaf \u2131 with maximal p-slope p(\ud835\udca6) > 1/2 and so the quotient sheaf \u210b : = \u2131/\ud835\udca6 has no 0-dimensional torsion. Let us set \u03c7\ud835\udca6 = m\u2032x + n\u2032y + t\u2032 with t\u2032 \u2265 1 and \u2a87. If the composite s : \ud835\udca6\u21aa\u2131\u00b2\u21a0\u03b7 is a zero map, then \ud835\udca6 is a subsheaf destabilizing \ud835\udcaaC, a contradiction. If s is not surjective, for instance, Im\u2061(s) = \ud835\udcaaP\u228a\u03b7, then ker\u2061(s) is a subsheaf of \ud835\udcaaC with Hilbert bipolynomial m\u2032x + n\u2032y + t\u2032 \u2212 1. Thus we have t\u2032 = 1 and the quotient \u210b\u2032 : = \ud835\udcaaC/\ud835\udca6\u2032 has Hilbert bipolynomial with zero constant term. Since \u210b\u2032 has no 0-dimensional torsion, we have \u210b\u2032\u2245\ud835\udcaaD for a curve D contained in C. But the Hilbert bipolynomial of \ud835\udcaaD has nonzero constant term, a contradiction. Thus the map s is surjective. Following the same argument before, we obtain that t\u2032 = 1 and m\u2032 + n\u2032 \u2264 1. Without loss of generality, let us assume that  = . Then we have \u03c7\u210b = 2x + y + 1 and thus we have \u210b\u2245\ud835\udcaaT1, where T1 is a curve contained in C\u2131 and T1 \u2208 |\ud835\udcaaQ|. Since \ud835\udca6 is a subsheaf of \u2131 with \u03c7\ud835\udca6 = y + 1, we have \ud835\udca6\u2245\ud835\udcaaT2 since \ud835\udca6 has no 0-dimensional torsion. Thus \u2131 fits into the sequence  to the sequence of C \u2208 |\ud835\udcaaQ|, we obtainf is the dual of the map Hom, \u03b7) \u2192 Hom given by the multiplication by the defining equation of C, the map f is a zero map and thus we have Ext1\u2245H0(\u03b7)\u2228. In particular its dimension is 2.Applying the functor Hom(\u2131 be a sheaf of type (B) fitting into an exact sequenceT1, T2 \u2208 |\ud835\udcaaQ|. Then \u2131 is of type (A) if and only if T1 and T2 have no common components; that is, C\u2131 has no multiple component.Let T1 and T2 have a common component, say T, then \u2131 has rank 2 at the general point of T and thus \u2131 is not of type (A).If T1\u2229T2 is finite. Since we have h1(\ud835\udcaaT1) = 0, the sequence (h0(\u2131) = 2 and \u2131 is globally generated. Let \u03c3 be a general section of \u2131 and then it does not vanish at the general point of any of the components of C\u2131. Since C\u2131 is reduced, \u03c3 induces an injective map \ud835\udcaaC\u2131\u21aa\u2131 and thus \u2131 has type (A).Conversely, assume that sequence  implies \u2131 be a sheaf of type (C). T2 is not a component of T1, then \u2131 is of type (A).If T2 is a double component of C\u2131, that is, T2 \u2282 T1, then it is not of type (A).If Let T2 \u2208 |\ud835\udcaaQ|.T2 is not a component of T1, \u2131 is a line bundle on C = C\u2131 outside finitely many points of C. Moreover, it is not an \ud835\udcaaT1-sheaf. Note that \u2131 is globally generated since \ud835\udcaaT1 and \ud835\udcaaT2 are globally generated with h1(\ud835\udcaaT1) = 0. Thus, a general section of \u2131 does not vanish at a general point of T2 and so it does not induce an injection \ud835\udcaaT1\u21aa\u2131. Hence, \u2131 fits into some sequence |. Let \u0393 be the projectivisation of Ext\u20611 and in particular we have \u0393\u2245\u21191 by e \u2208 \u0393 gives a semistable sheaf. Such a sheaf has rank 2 at the points of T2\u2216(T2\u2229T3) and, in particular, it is not a line bundle over its support at a general point of T2. Thus, it never fits into an exact sequence 1. Below we give a partial answer to this question in the case of \ud835\udcabic2.In general, the question whether the variety M2 is unirational with degree 4. C of bidegree  in Q and a point P \u2208 C to consider a sheaf \ud835\udcaaC(P) \u2208 M1. If \ud835\udd4bP is the tangent plane of Q at P, then we have \ud835\udd4bP\u2229C = {2P, Q1, Q2} for some points Q1, Q2 on Q since deg\u2061(C) = 4. It defines a rational map\ud835\udcaaC(P) to \ud835\udcaaC(Q1 + Q2). Note that \ud835\udcaaC(Q1 + Q2) = \ud835\udcaaC(\u22122P). We claim that the map \u03a6 is generically 4 to 1 and so the assertion follows.Let us fix a smooth curve U  be the dense open subset of M1  formed by the sheaves \u2131 such that C\u2131 is smooth. Each element of U  is uniquely determined by a smooth C \u2208 |\ud835\udcaaQ| and a degree one  line bundle on C. By Riemann-Roch, each degree one line bundle on C is associated with a unique P \u2208 C. Then the map \u03a6 sends \ud835\udcaaC(P) to \u211b : = \ud835\udcaaC(\u22122P). Fix any degree two line bundle \u2133 on C. Since we are in characteristic zero, there are exactly four line bundles \ud835\udc9c on C such that \ud835\udc9c\u22972\u2245\ud835\udcaaC. Hence, for each \u211b \u2208 Pic2(C) there are exactly 4 points P \u2208 C such that \u211b\u2245\ud835\udcaaC(\u22122P). Hence, \u03a6 is dominant and the preimage of each element of V has cardinality 4.Let M2 as of now, and we left the rationality question as a conjecture.We did not succeed in getting any smaller degree of unirationality of M2 is rational."}
{"text": "Several new theorems on the existence and multiplicity of positive radial solutions are obtained by means of fixed point index theory. Our conclusions are essential improvements of the results in Lan and Webb (1998), Lee (1997), Mao and Xue (2002), Sta\u0144czy (2000), and Han and Wang (2006).A class of elliptic boundary value problem in an exterior domain is considered under some conditions concerning the first eigenvalue of the relevant linear operator, where the variables of nonlinear term  In u(x) = z(|x|) of  = z((1 \u2212 t)n)1/(2\u2212) thus reducing  of , where zv(t) = z| for all v \u2208 E, P = {v \u2208 E : v(t)\u2a7e0\u2009\u2009for\u2009\u2009t \u2208 }, and Q = {v \u2208 P : min\u2061t\u2208v(t)\u2a7emin\u2061{a, 1 \u2212 b}||v||}, where \u2282. It is easy to show that P and Q are cones in E. Let \u03a9r = {u \u2208 E : ||u|| < r} be the open ball of radius r in E. Define a set H byLet h \u2208 H, define an operator Th byFor Then we have the following lemma.h \u2208 H,(i)Th : E \u2192 E is a completely continuous positive linear operator, and the spectral radius r(Th) \u2260 0 and Th has a positive eigenfunction \u03c6h1 corresponding to its first eigenvalue \u03bbh1 = (r(Th))\u22121;(ii)Th(P) \u2282 Q;(iii)\u03b41, \u03b42 > 0, such thatthere exist (iv)Jh by Jh(v) = \u222b01h(t)\u03c6h1(t)v(t)dt for v \u2208 E; then Jh(Thv) = \u03bbh1\u22121Jh(v) for\u2009\u2009v \u2208 E;define a functional (v)letthen P0 is a cone in E and Th(P) \u2282 P0, where \u03b41 is defined by \u2a7dTo prove E is a Banach space, Tn : E \u2192 E\u2009\u2009 are completely continuous operators, T : E \u2192 E, andT is a completely continuous operator.Suppose that T : E \u2192 E is a completely continuous linear operator and\u2009\u2009T(P)\u2282(P). If there exist \u03d5 \u2208 E\u2216(\u2212P) and a constant\u2009\u2009c > 0 such that cT\u03d5\u2a7e\u03d5, then the spectral radius\u2009\u2009r(T) \u2260 0 and T has a postive eigenfunction corresponding to its first eigenvalue \u03bb1 = (r(T))\u22121.Suppose that\u2009\u2009H that(i) It follows from the definition of Th : E \u2192 E. Obviously, Th(P) \u2282 P and Th is a linear operator; namely, Th is a positive linear operator. Next, we will show that Th is completely continuous. For any natural number n\u2009\u2009(n\u2a7e2), letHence, by Lebesgue's dominated convergence theorem, it is easy to see that hn : \u2192\u222a. Therefore, by Th : E \u2192 E is a completely continuous operator.It is clear that rding to , 19), a, aThn :t1 \u2208  such that kh(t1) > 0. Thus there is \u2282 such that t1 \u2208  and kh(s) > 0 for all t, s \u2208 . Take \u03b6 \u2208 P such that \u03b6(t1) > 0 and \u03b6(t) = 0 for all t \u2209 . Then for t \u2208 ,It is obvious that there exists c > 0 such that c(Th\u03b6)(t)\u2a7e\u03b6(t) for all t \u2208 . From r(Th) \u2260 0 and Th has a positive eigenfunction corresponding to its first eigenvalue \u03bbh1 = (r(Th))\u22121.So there exists a constant Th(P) \u2282 Q, we only need to show(ii) To prove v \u2208 P, from k \u2a7d k = s(1 \u2212 s) for t, s \u2208 , we haveIn fact, for every t \u2208 ,Notice that, for v \u2208 P,It follows from  and 26)26) that,Th maps P into Q.So  holds; t\u03c6h1 is a positive eigenfunction of Th, we know from the maximum principle (see ; we have \u03c6h1(0) = \u03c6h1(1) = 0. This implies that \u03c6h1\u2032(0) > 0 and \u03c6h1\u2032(1) < 0  Since ple (see ) that \u03c6< 0 (see ). Defineh is continuous on  and \u03a6h(s) > 0 for all s \u2208 . So, there exist \u03b41, \u03b42 > 0 such that \u03b41 \u2a7d \u03a6h(s) \u2a7d \u03b42 for all s \u2208 . Thust, s \u2208 .Then it is easy to see that \u03a6Jh(v) = \u222b01h(t)\u03c6h1(t)v(t)ds \u2a7d \u03b42\u222b01t(1 \u2212 t)h(t)v(t)dt < +\u221e for all v \u2208 E. So J : E \u2192 \u211d1 is well defined. For v \u2208 E,(iv) From , Jh(v) P0 is a cone in E. It follows from ((v) It is easy to verify that ows from  and 30)P0 is a Jh(Thv)\u2a7e\u03bbh1\u22121\u03b41||Thv|| for all v \u2208 P. The proof is completed.So Denote1) \u00d7 \u211d+, \u211d+) and for any M > 0 there exists a function hM \u2208 H such that2)(Hh \u2208 H such thatthere exists a function 3)(Hh \u2208 H such thatthere exists a function 4)/v > M1;liminf\u20615)/v > M1;liminf\u20616)(Hl > 0 such thatthere exists a number 7)(Hh \u2208 H such thatthere exists a function 8)(Hh \u2208 H with h(t)\u22620 for t \u2208  and q \u2208 C such thatthere exist We list some conditions as follows which will be useful in this section.1) holds. Then A : Q \u2192 Q is a completely continuous operator.Assume (H1), A is well defined and for every v \u2208 Q, Av is nonnegative and continuous on . Note the property of k; it is easy to see that A(Q) \u2282 Q. (H1) and Lebesgue's dominated convergence theorem ensure the continuity of A. Finally, by using Ascoli-Arzela theorem, we can prove that A is completely continuous.The proof is similar to that of Lemma 3.1 in , so we o1) holds.2) is satisfied, then i = 1 for sufficiently small positive number r;If (H3) is satisfied, then i = 1 for sufficiently large positive number R;if (H4) is satisfied, then i = 0 for sufficiently small positive number r;if (H5) is satisfied, then i = 0 for sufficiently large positive number R;if (H6) is satisfied, then i = 0;if (H7) is satisfied, then i = 0 for sufficiently small positive number r;if (H8) are satisfied, then i = 0 for sufficiently large positive number R.if (HAssume (H2), there exists r > 0 such that(i) By  \u2282 Q and the spectral radial r(Sh) = 1. For every v \u2208 Q\u2229\u2202\u03a9r, it follows from . Thus \u03c41 = \u03bc1\u22121 > 1 and \u03c41v1 = Av1 \u2a7d Shv1. By induction, we have \u03c41nv1 \u2a7d Shnv1, n = 1,2,\u2026. Then \u03c41nv1 = Shnv1 \u2a7d ||Sh||||v1|| and taking the supremum on  gives \u03c41n \u2a7d ||Shn||. By the spectral radius formula, we havei = 1.If there exist 3), there exist \u03b7 > 0 and \u03b50 \u2208  such that(ii) By (H1), there is h\u03b7 \u2208 H such that g \u2a7d h\u03b7(t) for all \u2208\u00d7. Hence,From  \u2282 Q. Let C1 = \u222b01t(1 \u2212 t)h\u03b7(t)dt < +\u221e. SetDefine W is bounded. For any v \u2208 W, from , the first eigenvalue of Sh, r(Sh)\u22121 > 1. Therefore, the inverse operator (I \u2212 Sh)\u22121 exists andSince Th(P) \u2282 Q that (I \u2212 Sh)\u22121(P) \u2282 Q. Hence, we have from  = 1.That is, (iii)\u2013(v) have been proved in , so we s7), there exists r > 0 such that(vi) By  = 0.Hence, by  we have\u03b7 > 0 and \u03b50 \u2208  such that(vii) From , there eq is bounded on , there is a constant C2 > 0 such thatSince q(v)\u2a7e\u03bbh1(1 + \u03b50)v \u2212 C2 for all v \u2208 , we have \u222babh(t)\u03c6h1(t)dt > 0. Thus from \u03c61h(t) >i = 0. The proof is completed.This is a contradiction. So, by the property of omitting a direction for fixed point index, we have Now, we are ready to state our main results of this section.1), (H2), (H3), and (H6) hold; then the singular BVP  = 1, and i = 1. From i = 0 and the additivity property of the fix point index, we obtainA has at least two fixed points, one in According to ular BVP  has at l1) and one of the following conditions are satisfied, then the singular BVP (2) and (H5) hold;(H2) and (H6) hold;(H2) and (H8) hold;(H3) and (H4) hold;(H3) and (H6) hold;(H3) and (H7) hold., (H2), and (H3), respectively. And they are somewhat easy to verify. In fact, (H1*) is a key condition in \u03c8(v); then we have g \u2a7d h(t)\u03c8(v) \u2a7d hM(t) for all \u2208\u00d7 and hM \u2208 H. Consequently, (H1) holds. From g/h(t)v \u2a7d h(t)\u03c8(v)/h(t)v = \u03c8(v)/v, it is easy to see that if (H2*) or (H3*) holds, then (H2) or (H3) holds, respectively.If (H1), the key conditions imposed on the nonlinear term g are the following ones:(A2)A \u2282  of positive measures such thatthere exists a set (A3)h \u2208 H such thatthere exists a function In , Sta\u0144czy(A2\u2032)a, b]\u2282 such thatthere exists a set \u2061g/v > M1;(HW5) liminf\u2061\u2009h \u2208 H such that(HW2) there exists a function where mh = h(s)ds)\u22121 for h \u2208 H.In a recent paper , Han and6), and\u2009h \u2208 H such that(HW3) there exists a function Additionally, they obtained a twin-solution theorem for the singular BVP  by usingmh \u2a7d \u03bbh1, (H2) is an improvement of (HW2). In fact, without loss of generality, suppose \u03c6h1, the positive eigenfunction corresponding to \u03bbh1, satisfies ||\u03c6h1|| = \u03c6h1(t0) = 1; thenSince mh \u2a7d \u03bbh1 and then (HW2) implies (H2). Therefore, Theorems 7) and (H8). These improvements allow us to deal with more singular problems.So \u03b1, \u03b2, \u03b3, and \u03b4\u2a7e0 and \u03c1 = \u03b3\u03b2 + \u03b1\u03b3 + \u03b1\u03b4 > 0, our results still hold. In fact, the Green functionk is defined in =\u2212v\u2032\u2032(t)=gFor the following singular two-point BVP whose variables of nonlinear term are separated, the hypotheses and results will be more concise:h is a fixed function, Th, \u03bbh1, and \u03c6h1 are confirmed exclusively. So we skip the subscript h in the following. Corresponding\u2009\u2009to (H1)\u2013(H8), we formulate the conditions for singular BVP (1\u2032) and h \u2208 H;2\u2032)(Hv\u21920+g(v)/v < \u03bb1;limsup\u20613\u2032)(Hv\u2192+\u221eg(v)/v < \u03bb1;limsup\u20614\u2032) > 0 and liminf\u2061v\u21920+g(v)/v > M1)\u22121;min\u20615\u2032) > 0 and liminf\u2061v\u2192+\u221eg(v)/v > M1)\u22121;min\u20616\u2032)(Hl > 0 such thatthere exists a number 7\u2032)(Hv\u21920+g(v)/v > \u03bb1;liminf\u20618\u2032)(Hv\u2192+\u221eg(v)/v > \u03bb1.liminf\u2061Since ular BVP :(H1\u2032)g t1 \u2208  with \u03c61(t1) = min\u2061t\u2208\u03c61(t). ThenTake \u03c61(t1), min\u2061t\u2208h(t), and min\u2061t\u2208\u222babkds > 0; we haveNotice that 4\u2032) implies (H7\u2032) and (H5\u2032) implies (H8\u2032).So (H8\u2032). In fact, the cone Q will be replaced by the cone P0 which is defined in (8\u2032). See the proof of Observe that the conditionfined in  as we co1\u2032), (H2\u2032), (H3\u2032), and (H6\u2032) hold; then the singular BVP (Assume that (Hular BVP  has at lThis theorem is a direct corollary of 1\u2032)\u2009\u2009and one of the following conditions are satisfied, then the singular BVP (2\u2032) and (H5\u2032) hold;(H2\u2032) and (H6\u2032) hold;(H2\u2032) and (H8\u2032) hold;(H3\u2032) and (H4\u2032) hold;(H3\u2032) and (H6\u2032) hold;(H3\u2032) and (H7\u2032) hold.(HIf (Hular BVP  has at l8), (H8\u2032) does not contain the condition that h\u22620 for t \u2208 .(i), (ii), and (iv)\u2013(vi) are direct corollaries of T(P) \u2282 P0 is completely continuous, we know that A(P) \u2282 P0 is also completely continuous. Similar to item (i) of 2\u2032), it is not difficult to prove i = 1 for sufficiently small positive number r. By (H8\u2032), there exist \u03b7 > 0 and \u03b50 \u2208  such that(iii) Since\u2009\u2009g is bounded on , there is a constant C4 > 0 such that g(v)\u2a7e\u03bb1(1 + \u03b50)v \u2212 C4 for all v \u2208 . ThusSince C5 = C4\u222b01h(t)\u03c61(t)h(s)ds)dt; then C5 > 0 is a finite constant. TakeLet v1 \u2208 P0\u2229\u2202\u03a9R and \u03bc1\u2a7e0 such that v1 = Av1 + \u03bc1\u03c61; then from 83), we hi = 0. So A has a fixed point in This contradicts with . By the ular BVP  has at lh \u2208 L1, h\u2a7e0 a.e. on  and g \u2208 C;One of the following conditions holds:(h1)v\u21920+g(v)/v < \u03b1 and \u03b2 < liminf\u2061v\u2192+\u221eg(v)/v;limsup\u2061(h2)v\u2192+\u221eg(v)/v < \u03b1 and \u03b2 < liminf\u2061v\u21920+g(v)/v, where \u03b1 = mh and \u03b2 = h(s)ds)\u22121.limsup\u2061Lan and Webb have studied BVP  in 21]..21]. The\u03b1 \u2a7d \u03bbh1 \u2a7d \u03b2. Therefore, the items (iii) and (vi) of 2\u2032), (H3\u2032), (H7\u2032), and (H8\u2032) cannot be improved anymore.Under the above condition (i), we still can prove a, b] = .At the end of this section, we present three simple examples to which our theorems can be applied, respectively. We choose [c > 0 is a constant. Obviously, g \u2a7d h(t)\u03c8(v) for all \u2208 \u00d7 \u211d+, where h(t) = t\u22121(1 \u2212 t)\u22121 andLet\u03bb = 32/3 < 11, if lim\u2061v\u21920+\u03c8(v)/v = c < \u03bbh1 and lim\u2061v\u2192+\u221e\u03c8(v)/v = c < \u03bbh1, then g satisfies all the conditions of Since c < \u03bbh1. Since lim\u2061v\u21920+\u03c8(v)/v = +\u221e > \u03bbh1 and lim\u2061v\u2192+\u221e\u03c8(v)/v = c < \u03bbh1, the item (iv) of Letp, q \u2208 , ai > 0 for i = 1,2,\u2026, n, and a1 < \u03bbh1. Since lim\u2061v\u21920+\u03c8(v)/v = a1 < \u03bbh1 and lim\u2061v\u2192+\u221e\u03c8(v)/v = +\u221e > \u03bbh1, the item (iii) of LetDefine a setc = (1 \u2212 a)n)1/(2\u2212 and d = (1 \u2212 b)n)1/ \u00d7 \u211d+, \u211d+) and for any M > 0 there exists a function pM \u2208 K such that2)(Cp \u2208 K such thatthere exists a function 3)(Cp \u2208 K such thatthere exists a function 4)/u > cn2\u22122(n \u2212 2)2M1;liminf\u20615)/u > cn2\u22122(n \u2212 2)2M1;liminf\u20616)(Cl > 0 such thatthere exists a number 7)(Cp \u2208 K such thatthere exists a function 8)(Cp \u2208 K with p(s)\u22620 for s \u2208  and q \u2208 C such thatthere exist According to , we formNow we are ready to state our main results for the elliptic BVP .1), (C2), (C3), and (C6) are satisfied, then the elliptic BVP , (1) and one of the following conditions are satisfied, then the elliptic BVP (2) and (C5) hold;(C2) and (C6) hold;(C2) and (C8) hold;(C3) and (C4) hold;(C3) and (C6) hold;(C3) and (C7) hold., \u211d+) and f \u2208 C.For the following BVP whose nonlinear terms are variable-separated, it is easy to give more concise theorems as Theorems One can see that, through variables substitution, it is easy to give some examples like in"}
{"text": "I-double gradation fuzzy topological spaces and also study these functions in relation to some other types of already known functions.In this paper, we introduce and characterize double fuzzy weakly preopen and double fuzzy weakly preclosed functions between  In the history of science, new theories have always been necessary in order for existing scientific theories to progress and this will continue to be true in the future. Two examples of essentially different mathematical theories that deal with the concept of uncertainty are probability theory and the theory of fuzzy sets. Whereas probability theory has a history of around 360 years, the theory of fuzzy sets is little more than 50 years old. Since the 1960s fuzzy methods have entered the scientific and technological world, good theoretical progress  has been made, and there have been technical advances in various areas .e\u221e Theory. Tang . The family of all fuzzy subsets on X denoted by IX. By X. For a fuzzy subset \u03bb\u2208IX, f(\u03bb) and f\u22121(\u03bb) define the direct image and the inverse image of f, defined by f\u22121(\u03bd)(x)=\u03bd(f(x)), for each \u03bb\u2208IX, \u03bd\u2208IY, and x\u2208X, respectively. For fuzzy subsets \u03bb and \u03bc in X, we write \u03bbq\u03bc to mean that \u03bb is quasi coincident (q-coincident) with \u03bc, that is, there exists at least one point x\u2208Xsuch that \u03bb(x) + \u03bc(x)>1. Negation of such a statement is denoted as Throughout this paper, let Samanta and Mondal (Garcia and Rodabaugh (I-double gradation fuzzy topology (\u03c4\u03c4\u2217) on X is a pair of maps \u03c4, \u03c4\u2217:IX\u2192I, which satisfies the following properties: \u03bb\u2208IX.(O1) \u03c4(\u03bb1\u2227\u03bb2)\u2265\u03c4(\u03bb1)\u2227\u03c4(\u03bb2) and \u03c4\u2217(\u03bb1\u2227\u03bb2)\u2264\u03c4\u2217(\u03bb1)\u2228\u03c4\u2217(\u03bb2) for each \u03bb1, \u03bb2\u2208IX.(O2) \u03bbi\u2208IX, i\u2208\u0393.(O3)  Let  \u03bb\u2264C\u03c4,\u03c4\u2217,(C2) C\u03c4,\u03c4\u2217\u2228C\u03c4,\u03c4\u2217=C\u03c4,\u03c4\u2217,(C3) C\u03c4,\u03c4\u2217\u2264C\u03c4,\u03c4\u2217 if r1\u2264r2and s1\u2265s2,(C4) C\u03c4,\u03c4\u2217,r,s)=C\u03c4,\u03c4\u2217.(C5) For  Let (X\u03c4\u03c4\u2217) be an I-dfts. Then for each r\u2208I0, s\u2208I1 and \u03bb\u2208IX, we define an operator I\u03c4,\u03c4\u2217:IX\u00d7I0\u00d7I1\u2192IXas follows: \u03bb\u03bc\u2208IX, rr1r2\u2208I0and ss1s2\u2208I1, the operator I\u03c4,\u03c4\u2217satisfies the following statements: I\u03c4,\u03c4\u2217\u2264\u03bb,I\u03c4,\u03c4\u2217\u2227I\u03c4,\u03c4\u2217=I\u03c4,\u03c4\u2217,I\u03c4,\u03c4\u2217\u2265I\u03c4,\u03c4\u2217 if r1\u2264r2and s1\u2265s2,I\u03c4,\u03c4\u2217,r,s)=I\u03c4,\u03c4\u2217,I\u03c4,\u03c4\u2217,r,s)=\u03bb, then If For X,\u03c4,\u03c4\u2217) be an I-dfts. For \u03bb\u2208IX, r\u2208I0and s\u2208I1.Let -fuzzy preopen -fpo, for short) if \u03bb\u2264I\u03c4,\u03c4\u2217,r,s). A fuzzy set \u03bb is called -fuzzy preclosed -fpc, for short) iff r,s)-fpo set. The -fuzzy preinterior of \u03bb, denoted by PI\u03c4,\u03c4\u2217 is defined by (1) r,s)-fuzzy preclosure of \u03bb, denoted by PC\u03c4,\u03c4\u2217 is defined by The -fuzzy regular open -fro, for short) if \u03bb=I\u03c4,\u03c4\u2217,r,s). A fuzzy set \u03bb is called -fuzzy regular closed -frc, for short) iff r,s)-fro set.(2) \u03bb is called -fuzzy \u03b1-open -f\u03b1o, for short) if \u03bb\u2264I\u03c4,\u03c4\u2217,r,s),r,s). A fuzzy set\u03bbis called -fuzzy \u03b1-closed -f\u03b1c, for short) iff r,s)-f\u03b1o set.(3) X,\u03c4,\u03c4\u2217) be an I-dfts. For \u03bb\u2208IX, r\u2208I0and s\u2208I1. \u03bbis -fpo -fpc) iff \u03bb=PI\u03c4,\u03c4\u2217 ),I\u03c4,\u03c4\u2217\u2264PI\u03c4,\u03c4\u2217\u2264\u03bb\u2264PC\u03c4,\u03c4\u2217\u2264C\u03c4,\u03c4\u2217,Let  is -fpc set in IYfor each \u03bb\u2208IX, r\u2208I0and s\u2208I1; double fuzzy preclosed if \u03c42(f(\u03bb))\u2265\u03c41(\u03bb) and \u03bb\u2208IX, r\u2208I0and s\u2208I1,double fuzzy open if \u03c42(f(\u03bb))\u2265rand r,s)-fro set \u03bb\u2208IX, r\u2208I0and s\u2208I1.double fuzzy almost open if Let I-dfts I-dfts f is called: \u03bb\u2208IX, r\u2208I0and s\u2208I1; \u03c41(\u03bb)\u2265rand double fuzzy weakly open if \u03b1-open if f(\u03bb) is -f\u03b1o in IYfor each \u03bb\u2208IX, r\u2208I0and s\u2208I1; \u03c41(\u03bb)\u2265rand double fuzzy Let X,\u03c4,\u03c4\u2217) be an I-dfts, \u03bc\u2208IX, xt\u2208P(X), r\u2208I0and s\u2208I1where P(X) is the family of all fuzzy points in X. \u03bcis called an -fuzzy open Q-neighborhood of xtif \u03c4(\u03bc)\u2265r, \u03c4\u2217(\u03bc)\u2264sand xtq\u03bc. We denote the set of all -fuzzy open Q-neighborhood of xtby Q\u03c4,\u03c4\u2217.Let  be an I-dfts, \u03bb\u2208IX, xt\u2208P(X), r\u2208I0and s\u2208I1. xtis called -fuzzy \u03b8-cluster point of \u03bbif for every \u03bc\u2208Q\u03c4,\u03c4\u2217, we have C\u03c4,\u03c4\u2217q\u03bb. We denote D\u03c4,\u03c4\u2217 is called -fuzzy \u03b8-closure of \u03bb.Let  an I-dfts. For \u03bb, \u03bc\u2208IXand r, s\u2208I0, we have the following: xtis -fuzzy \u03b8-cluster point of \u03bbiff xt\u2208D\u03c4,\u03c4\u2217.C\u03c4,\u03c4\u2217\u2264D\u03c4,\u03c4\u2217,\u03c4(\u03bb)\u2265rand \u03c4\u2217(\u03bb)\u2264s, then C\u03c4,\u03c4\u2217=D\u03c4,\u03c4\u2217,If \u03bbis -fpo, then C\u03c4,\u03c4\u2217=D\u03c4,\u03c4\u2217,If \u03bbis -fpo and \u03bb=C\u03c4,\u03c4\u2217,r,s), then D\u03c4,\u03c4\u2217=\u03bb.If Let -fuzzy \u03b8-closed set is called -fuzzy \u03b8-open and the -fuzzy \u03b8-interior operator denoted by T\u03c4,\u03c4\u2217 is defined by The complement of \u2264T\u03c4,\u03c4\u2217 for any \u03bb\u2208IX, r\u2208I0and s\u2208I1,T\u03c4,\u03c4\u2217=I\u03c4,\u03c4\u2217 for each \u03bb\u2208IX, r\u2208I0and s\u2208I1; \u03c4(\u03bb)\u2265rand \u03c4\u2217(\u03bb)\u2264s.From Theorem 2.4 It is easy to see that: A function \u03bb\u2208IX, r\u2208I0 and s\u2208I1; \u03c41(\u03bb)\u2265r and for each Every double fuzzy weakly open function is double fuzzy preopen and every double fuzzy preopen function is double fuzzy weakly preopen, but the converse need not be true in general.X={a,b,c} and Y={x,y,z}. Fuzzy sets \u03bb1, \u03bb2 and \u03bb3 are defined as: Let \u03c41and \u03c42 as follows: Define f(a)=z, f(b)=x and f(c)=y is double fuzzy weakly preopen but not double fuzzy preopen. Where f(\u03bb) is not Then the mapping X={a,b,c} and Y={x,y,z}. Fuzzy sets \u03bb1, \u03bb2 and \u03bb3 are defined as: Let Let f(a)=z, f(b)=xand f(c)=yis double fuzzy weakly preopen but not double fuzzy weakly open. Since Then the mapping f is double fuzzy weakly preopen,\u03bb\u2208IX, r\u2208I0and s\u2208I1,\u03bd\u2208IY, r\u2208I0and s\u2208I1,\u03bd\u2208IY, r\u2208I0and s\u2208I1.For a function \u03bb\u2208IX and (1)\u21d2(2) Let f is double fuzzy weakly preopen, Since f(\u03bb),r,s)) and so, and hence \u03bc\u2208IX; \u03c41(\u03bc)\u2265r and (2)\u21d2(1) Let f is double fuzzy weakly preopen.Hence \u03bd\u2208IY. By using (2), (2)\u21d2(3) Let (3)\u21d2(2) Trivial.\u03bd\u2208IY. Using (3), we have (3)\u21d2(4) Let r,s).Therefore, we obtain \u03bd\u2208IY, r\u2208I0 and s\u2208I1, i.e., (4)\u21d2(3) Similarly we obtain, f is double fuzzy weakly preopen,xt \u2208 P(X) and each \u03bc\u2208IX; \u03c41(\u03bc)\u2265rand xt\u2264\u03bc, there exists -fpo set \u03b3such that f(xt)\u2264\u03b3and For each For the function xt \u2208 P(X) and \u03bc\u2208IXsuch that \u03c41(\u03bc)\u2265r, xt\u2264\u03bc. Since f is double fuzzy weakly preopen, then \u03bc,r,s)), with f(xt)\u2264\u03b3.(1)\u21d2(2) Let \u03bc\u2208IX; \u03c41(\u03bc)\u2265r, ys\u2264f(\u03bc). It follows from (2) that r,s)-fpo \u03b3\u2208IYand ys\u2264\u03b3. Hence we have, f is double fuzzy weakly preopen function. \u25a1(2)\u21d2(1) Let f is double fuzzy weakly preopen;\u03bb\u2208IX, r\u2208I0and s\u2208I1; \u03c41(\u03bb)\u2265rand \u03bd\u2208IX, r\u2208I0and s\u2208I1; Let \u03bd\u2208IX; \u03c41(\u03bd)\u2265r and (1)\u21d2(2) Let and so \u03bb\u2208IX; \u03c41(\u03bb)\u2265r and r,s)-fc set and r,s) by (3) we have (2)\u21d2(3) Let (3)\u21d2(2) Trivial.(2)\u21d2(1) Trivial. \u25a1f is double fuzzy weakly preopen;\u03bd\u2208IX, r\u2208I0and s\u2208I1; \u03c41(\u03bd)\u2265rand \u03bb\u2208IX, r\u2208I0and s\u2208I1; \u03c41(\u03bb)\u2265rand r,s)-fpo set \u03bb\u2208IX;r,s)-f\u03b1o set \u03bb\u2208IX.For a function \u03bd\u2208IX, r\u2208I0 and s\u2208I1; (1)\u21d2(2) Let (2)\u21d2(3) It is clear.\u03bb be -fpo set. Hence by (3), (3)\u21d2(4) Let (4)\u21d2(5) and (5)\u21d2(1) are clear. \u25a1\u03bb\u2208IX, r\u2208I0and s\u2208I1.A function f is double fuzzy preopen.If \u03bb\u2208IX such that \u03c41(\u03bb)\u2265r and f is double fuzzy weakly preopen Let f is double fuzzy strongly continuous, then f(\u03bb) is -fpo. \u25a1However, since f(\u03bb) is -fpo for each \u03bb\u2208IX, r\u2208I0and s\u2208I1; A function f is double fuzzy weakly preopen function.If \u03bb\u2208IX; \u03c41(\u03bb)\u2265r and Let \u25a1The converse of the above theorem need not be true in general as in the following Example.X={a,b,c} and Y={x,y,z}. Define fuzzy sets \u03bb1, \u03bb2 as follows: Let Let f(a)=x, f(b)=yand f(c)=z is double fuzzy weakly preopen but it isn\u2019t double fuzzy contra-preclosed.Then the function I-dfts  is said to be -fuzzy regular space if for each \u03bb\u2208IX; \u03c4(\u03bb)\u2265rand \u03c4\u2217(\u03bb)\u2264sis a union of -fo sets \u03bci\u2208IXsuch that C\u03c4,\u03c4\u2217\u2264\u03bbfor each i\u2208J.An X,\u03c4,\u03c4\u2217) be -regular fuzzy topological space. Then, f is double fuzzy preopen.Let \u2265r and xt\u2264\u03bb, let The sufficiency is clear. For the necessity, let f is double fuzzy preopen. \u25a1Thus If \u03bb\u2208IX; \u03c41(\u03bb)\u2265r and \u03c41\u2217(\u03bb)\u2264s. Since f is double fuzzy almost open and r,s)-fro, then Let and hence f is double fuzzy weakly preopen. \u25a1This shows that X,\u03c4,\u03c4\u2217) be an I-dfts, r\u2208I0and s\u2208I1. The two fuzzy sets \u03bb, \u03bc\u2208IXare said to be -fuzzy separated iff r,s)-fuzzy separated sets is said to be -fuzzy connected.Let  an I-dfts. The fuzzy sets \u03bb, \u03bc\u2208IXsuch that r,s)-pre-separated if r,s)-fpo sets \u03bd, \u03b3such that \u03bb\u2264\u03bd, \u03bc\u2264\u03b3, I-dfts which can not be expressed as a union of two fuzzy -pre-separated sets is said to be fuzzy -pre-connected space.Let -fuzzy pre-connected space r,s)-fuzzy connected.If r,s)-fuzzy connected. Then there exist -fuzzy separated sets \u03b2, \u03b3\u2208IX such that \u03b2and \u03b3 are -fuzzy separated, there exists \u03bb, \u03bc\u2208IX; \u03c41(\u03bb)\u2265r, \u03c41(\u03bc)\u2265r and \u03b2\u2264\u03bb, \u03b3\u2264\u03bc, f(\u03b2)\u2264f(\u03bb), f(\u03b3)\u2264f(\u03bc), f is double fuzzy weakly preopen and double fuzzy strongly continuous function, from Theorem 3.10 we have f(\u03bb) and f(\u03bc) are -fpo sets. Therefore, f(\u03b2) and f(\u03b3) are -fuzzy pre-separated and Let r,s)-fuzzy pre-connected. Thus r,s)-fuzzy connected. \u25a1which is contradiction with A function \u03bb\u2208IX, r\u2208I0 and s\u2208I1; for each Clearly, every double fuzzy preclosed function is double fuzzy weakly preclosed, but the converse need not be true in general, as the next example shows.X={a,b} and Y={x,y}. Fuzzy sets \u03bb1 and \u03bb2 are defined as: Let Let f(a)=x, f(b)=y is double fuzzy weakly preclosed but is not double fuzzy preclosed.Then the function f is double fuzzy weakly preclosed;\u03bb\u2208IX, r\u2208I0and s\u2208I1; \u03c41(\u03bb)\u2265rand \u03bb\u2208IX, r\u2208I0and s\u2208I1; r,s)-fpc set \u03bb\u2208IX, r\u2208I0and s\u2208I1;r,s)-f\u03b1c \u03bb\u2208IX, r\u2208I0and s\u2208I1.For a function Straightforward. \u25a1f is double fuzzy weakly preclosed;r,s)-fro set \u03bb\u2208IX, r\u2208I0and s\u2208I1;\u03bd\u2208IY, \u03bc\u2208IX, r\u2208I0and s\u2208I1; \u03c41(\u03bc)\u2265rand f\u22121(\u03bd)\u2264\u03bc, there exists -fpo set \u03b3\u2208IYwith \u03bd\u2264\u03b3and For each ys \u2208 P(Y) and each \u03bc\u2208IX, r\u2208I0and s\u2208I1such that \u03c41(\u03bc)\u2265rand f\u22121(ys)\u2264\u03bc, there exists -fpo set \u03b3\u2208IY; ys\u2264\u03b3and For each fuzzy point \u03bb\u2208IX, r\u2208I0and s\u2208I1;\u03bb\u2208IX, r\u2208I0and s\u2208I1;r,s)-fpo set \u03bb\u2208IX, r\u2208I0and s\u2208I1.For a function We will prove (2)\u21d2(3) and (1)\u21d2(6).\u03bd\u2208IY, r\u2208I0, s\u2208I1 and let \u03bc\u2208IX; \u03c41(\u03bc)\u2265r and f\u22121(\u03bd)\u2264\u03bc. Then r,s)-fro, \u03b3 is -fpo with \u03bd\u2264\u03b3 and (2)\u21d2(3): Let \u03bd\u2208IY, r\u2208I0 and s\u2208I1; r,s)-fpo \u03b3\u2208IY with ys\u2264\u03b3 and (1)\u21d2(6): Let ys\u2208 P(Y) and each r,s)-fpo set \u03b3\u2208IY; If \u03bc(x) + s>1 and hence there exists t\u2208 such that \u03bc(x)>t>1\u2212s. Then r,s)-fpo set \u03b3\u2208IY; yt\u2264\u03b3such that \u03b3(y)>tand hence \u03b3(y)>1\u2212s. Thus \u03b3is -fpo neighborhood of ys. \u25a1Let X,\u03c4,\u03c4\u2217) be an I-dfts. A fuzzy set \u03bb\u2208IXis called -fuzzy pre-Q-neighborhood of xtif there exists -fpo set \u03bc\u2208IXsuch that xtq\u03bc\u2264\u03bb. We denote the set of all -fuzzy pre-Q-neighborhood of xtby PQ\u03c4,\u03c4\u2217.Let . A fuzzy point xt\u2208PC\u03c4,\u03c4\u2217 if and only if for every \u03bc \u2208 PQ\u03c4,\u03c4\u2217, \u03bcq\u03bbis hold.In an Straightforward. \u25a1\u03bd\u2208IX, r\u2208I0and s\u2208I1; f is double fuzzy preclosed.If \u03bd\u2208IX, r\u2208I0and s\u2208I1; f is double fuzzy weakly preclosed by using Theorem 3.12, there exists -fuzzy pre-Q-neighborhood \u03b3\u2208IYwith ys\u2264\u03b3and f(\u03bd) is -fpc and f is double fuzzy preclosed function. \u25a1Let \u03c42(f(\u03bb))\u2265rand \u03bb\u2208IX, r\u2208I0and s\u2208I1; \u03c41(\u03bb)\u2265rand A function f is double fuzzy weakly preclosed.If \u03bb\u2208IX, r\u2208I0 and s\u2208I1 such that Let \u25a1\u03bd\u2208IYand every \u03bb\u2208IX, r\u2208I0, s\u2208I1such that \u03c41(\u03bb)\u2265rand f\u22121(\u03bd)\u2264\u03bb, there exists -fpc set \u03b3\u2208IYsuch that \u03bd\u2264\u03b3and If \u03bd\u2208IYand let \u03bb\u2208IX, r\u2208I0and s\u2208I1such that \u03c41(\u03bb)\u2265rand f\u22121(\u03bd)\u2264\u03bb. Put \u03b3is -fpc set in IYsuch that \u03bd\u2264\u03b3since f is double fuzzy weakly preclosed, Let ys\u2208 P(Y) and every \u03bb\u2208IX, r\u2208I0and s\u2208I1such that \u03c41(\u03bb)\u2265rand f\u22121(ys)\u2264\u03bb, there exists -fpc set \u03b3\u2208IY; ys\u2264\u03b3such that If \u03bb\u2208IXis called -fuzzy \u03b8-compact if for each family {\u03bci\u2223i\u2208J} in x\u2208X, there exist a finite subset J0of J such that A fuzzy set r,s)-fuzzy \u03b8-closed, then f(\u03bb) is -fpc for each -fuzzy \u03b8-compact \u03bb\u2208IX, r\u2208I0and s\u2208I1.If \u03bb be -fuzzy \u03b8-compact and let xt\u2264\u03bb there is x\u2208X and since \u03bb is -fuzzy \u03b8-compact, there is f is double fuzzy weakly preclosed, by using Theorem 3.12 there exists Let ys\u2264\u03b3and f(\u03bb) is -fpc set. \u25a1Therefore X,\u03c4,\u03c4\u2217) be an I-dfts. The fuzzy sets \u03bb, \u03bc\u2208IXare -fuzzy strongly separated if there exist \u03bd, \u03b3\u2208IXsuch that \u03c4(\u03bd)\u2265rand \u03c4\u2217(\u03bd)\u2264s, \u03c4(\u03b3)\u2265rwith \u03bb\u2264\u03bd, \u03bc\u2264\u03b3and Let  is called -fuzzy pre T2if for each r,s)-fpo sets \u03bb, \u03bc\u2208IXsuch that An r,s)-fuzzy strongly separated, then r,s)-fuzzy pre-T2.If \u03b3,\u03bd\u2208IX, r\u2208I0and s\u2208I1; \u03c41(\u03b3)\u2265r, \u03c41(\u03bd)\u2265rand \u03c41(\u03bd)\u2264ssuch that r,s)-fpo sets \u03bb, \u03bc\u2208IYsuch that f is surjective. Thus r,s)-fuzzy pre-T2. \u25a1Let I-dfts  is said to be -extremally disconnected if \u03c4)\u2265rand \u03c4\u2217)\u2264sfor each \u03bb\u2208IX; \u03c4(\u03bb)\u2265rand \u03c4\u2217(\u03bb)\u2264s.an I-dfts  is said to be -fuzzy almost compact if for each -fuzzy open cover {\u03bbi\u2223i\u2208J} of X, there is a finite subset J0of J such that an \u03bbin an I-dfts  is said to be -fuzzy p-compact iff for each family of -fpo sets {\u03bci\u2223i\u2208J} satisfies x\u2208X. There exists finite subfamily J0of J such that x\u2208X.A fuzzy set r,s)-extremally disconnected I-dfts. Let f\u22121(ys) is -fuzzy almost compact for each ys \u2208 P(Y). If \u03bb\u2208IYis -fuzzy P-compact. Then f\u22121(\u03bb) is -fuzzy almost compact.Let \u03bdj\u2223\u00e6\u2208J} be -fuzzy open cover of f\u22121(\u03bb). Then for each ys\u2264\u03bb\u2227f(X), J(ys) of J. Since r,s)-extremally disconnected each r,s)-fpc set r,s)-fuzzy preclosed cover of \u03bb, K of \u03bb\u2227f(X). Hence, Let {f\u22121 (\u03bb) is -fuzzy almost compact. \u25a1so"}
{"text": "New idea and algorithm are proposed to compute asymptotic expression of limit cycles bifurcated from the isochronous center. Compared with known inverse integrating factor method, new algorithm to analytically computing shape of limit cycle proposed in this paper is simple and easy to apply. The applications of new algorithm to some examples are also given. Rewrite f(\u03b8) into the following Fourier series:\u03b8 from 0 to \u03b8, we obtain (\u03c6(\u03b8) is a periodic function with period 2\u03c0.For  series:f(\u03b8)=a02+F(\u03b8) is a periodic function with period 2\u03c0, then we get g \u00b7 (2\u03c0 \u2212 0) + \u03c6(2\u03c0) \u2212 \u03c6(0) = 0. From \u03c6(2\u03c0) = \u03c6(0), we conclude that g = 0.If The proof of the lemma is completed.\u025b, of the limit cycles bifurcated from linear isochronous center. As applications, in x\u2032 = y, y\u2032 = \u2212x + \u025b(1 \u2212 x2)y up to order o(\u03b57). In The main goal of this paper is to develop a new approach for computing analytically the global shape of the bifurcated limit cycles from an isochronous center and the paper is organized as follows. In pk and\u2009qk are both analytic functions, pk = qk = 0, k = 1,2,\u2026, and \u025b is a small real parameter. System , qk are both analytic functions, we rewrite system , R2,\u2026 are analytic functions about r, cos\u2061(\u03b8), sin(\u03b8).Firstly, we make a polar coordinates transformation o system . By elime obtaindrd\u03b8=\u025b is limit cycle of system (r(\u03b8), we get that rk(\u03b8) is periodic function with period 2\u03c0, where k = 0,1, 2,\u2026.From f system , then frkth order terms of \u025b in the obtained system, then we can obtain a series of equations:Thirdly, substituting  into 1111 and coAs to the formula of fk is analytic function about rk\u22121, rk\u22122,\u2026, r1, r0, cos\u2061(\u03b8), sin(\u03b8), k = 0,1, 2,\u2026.Functions ained in  have theAccording to  and 12)12), we gk = 0, it is easy to get that For k = 1, the term on the right hand side of . In detail, the constant in the term R1 determines r(\u03b8) = r0 + \u025br1(\u03b8) + \u03b52r2(\u03b8)+\u22ef, we get that r0. In other words, if function rn, n \u2265 1, then rn appearing in the terms in the right hand side of  of system (rk(\u03b8) in  in  in , k = 0,1k(\u03b8) in  are recudr0/d\u03b8 = 0, we get r0(\u03b8) \u2261 r0(constant). To determine the constant r0 in (r1(\u03b8).From t r0 in , the newdr1/d\u03b8 = f1, we obtainFrom f1 is a periodic function, according to \u03c61 is periodic function with period 2\u03c0.By noting that r = r(\u03b8) is a limit cycle of system (r(\u03b8) and r1(\u03b8) in  in  are perir0.So from \u025b=0.In fact, the function d system  near clod system |\u025b=0.L : x2 + y2 = r02. From [In detail, 2. From , 9, we k\u03c0-periodic differential equation. First order  averaging method to study the existence and number of periodic orbits of planar differential equation is proposed in [It should be pointed out that the function posed in , 11. Ther0 into . Thus we obtainSubstitute the value of r0 into ; we can c1, new algorithm proposed in this paper needs the expression of r2(\u03b8). From dr2/d\u03b8 = f2, we getTo determine the value of From r2(\u03b8) is a periodic function and From the fact that c1. Thus we have obtained r1(\u03b8) by  by .r0, r1(\u03b8),\u2026, rk\u22121(\u03b8), now we start to determine rk(\u03b8).Assuming that we have obtained the explicit expressions of drk/d\u03b8 = fk, we getFrom rk(\u03b8) is to determine the value of ck. According to the algorithm proposed in this paper, we resort to the expression of rk+1(\u03b8). From drk+1/d\u03b8 = fk+1, we getTo determine the expression of From rk+1(\u03b8) is a periodic function, from Because ck; thus we determine rk(\u03b8) by  by .\u025b explicitly and recursively.Thus we can compute the shape of limit cycles of system  to any go(\u03b57).In this section we will apply the method just described in the above section to compute the analytic expansion of the unique limit cycle of the Van der Pol systemx = rcos\u2061(\u03b8)y = rsin(\u03b8) to system  = \u2211k=0\u221e\u03b5krk(\u03b8) is the polar coordinates form of the limit cycles of ,\u2009\u2009k = 3,4,\u2026, 8 is omitted.Assume ycles of  and subsycles of . By compdr0/dt = 0, we get that r0 is arbitrary constant.From r0, we compute the following expression of r1(\u03b8):g1 = 0, so we get r0 = 2.To determine the value of r0 = 2 into :To determine the value of g2 = 0, so we get c1 = 0.Because r1(\u03b8) is given by (c1 = 0.Thus explicit expression of given by  with c1r1(\u03b8) and r0 = 2 into  are similar to the ones given in Section 3 of [r7(\u03b8) obtained in our method which was omitted in [\u025b = 0, (1/5), (1/2), (9/10) are plotted in x2 + y2 = 4 of system . To utilize new algorithm introduced in Consider the following perturbed system:3x\u2032=\u2212y+2xyx\u2032=\u2212y+2xyx\u2032=\u2212y+2xyx\u2032=\u2212y+2xy\u025b.In this subsection we start to compute the analytic expansion of the limit cycle of the perturbed system  to the su = rcos\u2061\u03b8, v = rsin\u03b8; then system , r2(\u03b8) are given in the following:By applying the algorithm described in f system ,(41)r is given in  is unstable and its parametric form is the following:\u025b = 1/20 is plotted by using formula 2 + y2 = 2 of unperturbed system (\u025b = 0 is drawn in sold line, and the limit cycle of the perturbed system (\u025b = 1/20 is drawn in dash line.In d system  for \u025b = d system  for \u025b ="}
{"text": "M, N)-soft intersection set, which is a generalization of soft intersection sets. We introduce the concepts of -SI filters of BL-algebras and establish some characterizations. Especially, -soft congruences in BL-algebras are concerned.The purpose of this paper is to give a foundation for providing a new soft algebraic tool in considering many problems containing uncertainties. In order to provide these new soft algebraic structures, we discuss a new soft set-( In fact, the MV-algebras, G\u00f6del algebras, and product algebras are the most known classes of BL-algebras. BL-algebras are further discussed by many researchers; see [It is well known that certain information processing, especially inferences based on certain information, is based on classical two-valued logic. In making inference levels, it is natural and necessary to attempt to establish some rational logic system as the logical foundation for uncertain information processing. BL-algebra has been introduced by H\u00e1jek as the algebraic structures for his Basic Logic . A well-ers; see \u201312.We note that the complexities of modeling uncertain data in economics, engineering, environmental science, sociology, information sciences, and many other fields cannot be successfully dealt with by classical methods. Based on this reason, Molodtsov  proposedo groups , near-rio groups , and BL-o groups , respectM, N)-SI filters of BL-algebras. In particular, some important properties of -soft congruences of BL-algebras are discussed in In this paper, we organize the recent paper as follows. In L =  is a BL-algebra [L, \u2299, 1) is a commutative monoid;;x \u2192 y)\u2228(y \u2192 x) = 1.:, 6:L, thx \u2264 y\u21d4x \u2192 y = 1;x \u2192 (y \u2192 z) = (x\u2299y) \u2192 z = y \u2192 (x \u2192 z);x\u2299y \u2264 x\u2227y;x \u2192 y \u2264 (z \u2192 x)\u2192(z \u2192 y), x \u2192 y \u2264 (y \u2192 z)\u2192(x \u2192 z);x \u2192 x\u2032 = x\u2032\u2032 \u2192 x;x\u2228x\u2032 = 1\u21d2x\u2227x\u2032 = 0;x \u2192 y)\u2299(y \u2192 z) \u2264 x \u2192 z;A,1 \u2208 (I2)x \u2208 A, \u2200y \u2208 L, x \u2192 y \u2208 A\u21d2y \u2208 A.\u2200\u2009A of L is a filter of L if and only if it satisfiesIt is easy to check that a nonempty subset (I3)x, y \u2208 L, x\u2299y \u2208 A,\u2200(I4)x \u2208 A, \u2200y \u2208 L, x \u2264 y\u21d2y \u2208 A (see [\u2200\u2208 A (see ).A nonempty subset L be a BL-algebra, U an initial universe, E a set of parameters, and P(U) the power set of U and A, B, C\u2286E.From now on, we let fA over U is a set defined by fA : E \u2192 P(U) such that fA(x) = \u2205 if x \u2209 A. Here fA is also called an approximate function. A soft set over U can be represented by the set of ordered pairs fA = {) | x \u2208 E, fA(x) \u2208 P(U)}. It is clear to see that a soft set is a parameterized family of subsets of U. Note that the set of all soft sets over U will be denoted by S(U).A soft set fA, fB \u2208 S(U).fA is said to be a soft subset of fB and denoted by fA(x)\u2286fB(x), for all x \u2208 E. fA and fB are said to be soft equal, denoted by fA = fB, if fA and fB, denoted by fA\u222aB(x) = fA(x) \u222a fB(x), for all x \u2208 E.The union of fA and fB, denoted by fA\u2229B(x) = fA(x)\u2229fB(x), for all x \u2208 E.The intersection of Let fL over U is called an SI- filter of L over U if it satisfies\u2009S1)\u2009\u2009fL(x)\u2286fL(1) for any x \u2208 L,(\u2009S2)\u2009\u2009fL(x \u2192 y)\u2229fL(x)\u2286fL(y) for all x, y \u2208 L.-SI filters in BL-algebras and investigate some characterizations. From now on, we let \u2205\u2286M \u2282 N\u2286U.In this section, we introduce the concept of -soft intersection filter -SI filter) of L over U if it satisfies1)(SIfL(x)\u2229N\u2286fL(1) \u222a M for all x \u2208 L,2)(SIfL(x \u2192 y)\u2229fL(x)\u2229N\u2286fL(y) \u222a M for all x, y \u2208 L.A soft set fL is an -SI filter of L over U, then fL is an -SI filter of L over U. Hence every SI-filter of L is an -SI filter of L, but the converse need not be true in general. See the following example.If U = S3, the symmetric 3-group is the universal set, and let L = {0, a, b, 1}, where 0 < a < b < 1. We define x\u2227y : = min\u2061{x, y}, x\u2228y : = max\u2061{x, y} and \u2299 and \u2192 as follows:L, \u2227, \u2228, \u2299, \u2192, 1) is a BL-algebra. Let M = {(13), (123)} and N = {(1), (12), (13), (123)}. Define a soft set fL over U by fL(1) = {(1), (12), (123)}, fL(b) = {(1), (12), (13), (123)} and fL(a) = fL(0) = {(1), (12)}. Then we can easily check that fL is an -SI filter of L over U, but it is not SI-filter of L over U since fL(b)\u2288fL(1).Assume that The following proposition is obvious.fL over U is an -SI filter of L over U, thenIf a soft set S(U) as follows: for any fL, gL \u2208 S(U), \u2205\u2286M \u2282 N\u2286U, we define M,N)-soft intersection filter -SI filter) of L over U if it satisfies1\u2032)-SI filter of L over U, then fL* = {x \u2208 L | (fL(x)\u2229N) \u222a M = (fL(1)\u2229N) \u222a M} is a filter of L.If fL is an -SI filter of L over U. Then it is clear that 1 \u2208 fL*. For any x, x \u2192 y \u2208 fL*, (fL(x)\u2229N) \u222a M = (fL(x \u2192 y)\u2229N) \u222a M = (fL(1)\u2229N) \u222a M. By fL(y)\u2229N) \u222a M\u2286(fL(1)\u2229N) \u222a M. Since fL is an -SI filter of L over U, we havefL(y)\u2229N) \u222a M = (fL(1)\u2229N) \u222a M, which implies y \u2208 fL*. This shows that fL* is a filter of L.Assume that fL over U is an -SI filter of L, then for any x, y, z \u2208 L,fL(x\u2299y)=M,N)(fL(x)\u2229fL(y)=M,N)(fL(x\u2227y),fL(0)=M,N)(fL(x)\u2229fL(x\u2032),If a soft set x, y \u2208 L be such that x \u2264 y. Then x \u2192 y = 1, and hence(1) Let x, y \u2208 L be such that fL(x \u2192 y) = fL(1). Then,(2) Let a3), we have x\u2299y \u2264 x\u2227y for all x, y \u2208 L. By (1), x \u2264 y \u2192 x\u2299y, we obtain 2) that fL(x\u2299y)\u2009\u2009=M,N)(\u2009\u2009fL(x)\u2229fL(y).(3) By (y \u2264 x \u2192 y and x\u2299(x \u2192 y) \u2264 x\u2227y, we have Since fL(x)\u2229fL(y)=M,N)(fL(x\u2227y). Thus fL(x\u2299y)=M,N)(fL(x)\u2229fL(y)=M,N)(fL(x\u2227y).x\u2299x\u2032 = 0.(4) It is a consequence of (3), since a4).(5) By (a7).(6) By (a8).(7) By (a9).(8) By -SI filter of L over U if and only if it satisfiesA soft set fL over U is an -SI filter of L over U if and only if it satisfies\u2009\u2009SI5)\u2009\u2009\u2200 x, y \u2208 L, fL(x\u2299y)=M,N)(fL(x)\u2229fL(y).(A soft set (\u21d2) By x, y \u2208 L. Since x \u2264 1, by (SI3), we have 1\u2032) holds. Since x\u2299(x \u2192 y) \u2264 y, by (SI3) and (SI4), we have 2\u2032) holds. Therefore, fL is an -SI filter of L over U.(\u21d0) Let M, N)-soft congruences, -soft congruences classes, and quotient soft BL-algebras.In this section, we investigate (\u03b8 from fL \u00d7 fL to P(U \u00d7 U) is called an -congruence in L over U \u00d7 U if it satisfies\u2009C1)\u2009\u2009\u03b8=M,N), \u2200x \u2208 L,(\u2009C2)\u2009\u2009\u03b8=M,N), \u2200x \u2208 L,-congruence in BL-algebra L over U \u00d7 U and x \u2208 L. Define \u03b8x in L as \u03b8x(y) = \u03b8, \u2200y \u2208 L. The set \u03b8x is called an -congruence class of x by \u03b8 in L. The set L/\u03b8 = {\u03b8x | x \u2208 L} is called a quotient soft set by \u03b8.Let \u03b8 is an -congruence in L over U \u00d7 U, then If C1) and (C3), we have By -congruence in L over U \u00d7 U, then \u03b81 is an -SI filter of L over U.If x \u2208 L, we have1\u2032) holds.For any x, y \u2208 L, by (C3) and (C5), we obtain2\u2032) holds. Thus, \u03b81 is an -SI filter of L over U.For any fL be an -SI filter of L over U. Then \u03b8 = fL(x \u2192 y)\u2229fL(y \u2192 x) is an -soft congruence in L.Let x, y, z \u2208 L, we have the following.\u2009C1) Consider(This proves that (C1) holds.\u2009C2) It is clear that (C2) holds.(\u2009C3) By (Thus (C3) holds.For any C4) Since x \u2192 y \u2264 (x\u2299z)\u2192(y\u2299z) and y \u2192 x \u2264 (y\u2299z)\u2192(x\u2299z), we haveC4) holds.(C5) Finally, we prove condition (C5):C5) holds. Therefore \u03b8f is an -soft congruence in L.-SI filter of L over U and x \u2208 L. In the following, let fx denote the -congruence class of x by \u03b8f in L and let L/f be the quotient soft set by \u03b8f.Let fL is an -SI filter of L over U, then fx=M,N)(fy if and only if fL(x \u2192 y)=M,N)(fL(y \u2192 x)=M,N)(fL(1) for all x, y \u2208 L.If fL is an -SI filter of L over U, then f\u03bc(\u03bd) = \u03b8f\u03bc(\u03bd) = \u03b8f = fL(\u03bc \u2192 \u03bd)\u2229fL(\u03bd \u2192 \u03bc); that is, f\u03bc(\u03bd) = fL(\u03bc \u2192 \u03bd)\u2229fL(\u03bd \u2192 \u03bc) for all x, y \u2208 L. If fx=M,N)=M,N)(fy(x), and hence fL(x \u2192 x) = fL(1)=M,N)(fL(y \u2192 x)\u2229fL(x \u2192 y). Thus, fL(y \u2192 x)=M,N)(fL(x \u2192 y)=M,N)(fL(1).If fL(y \u2192 x)=M,N)(fL(x \u2192 y)=M,N)(fL(1), then fL(x \u2192 z)\u2287M,N)(fL(y \u2192 z) and fL(y \u2192 z)\u2287M,N)(fL(x \u2192 z). Thus fL(x \u2192 z)=M,N)(fL(y \u2192 z). Similarly, we can prove that fL(z \u2192 x)=M,N)(fL(z \u2192 y). This implies thatz \u2208 L. Hence, fx=M,N) by ff(1) : = {x \u2208 L | f(x)=M,N)(f(1)}.We denote f is an -SI filter of L over U, then fx=M,N)(fy if and only if x~ff(1)y, where x~ff(1)y if and only if x \u2192 y \u2208 ff(1) and y \u2192 x \u2208 ff(1).If f be an -SI filter of L over U. For any fx, fy \u2208 L/f, we defineLet f is an -SI filter of L over U, then L/f =  is a BL-algebra.If L/f are well defined. In fact, if fx=M,N))(y and a~ff(1)b, and so x\u2228a~ff(1)y\u2228b. Thus fx\u2228a=M,N)(\u201d on L/f as follows:fL(x \u2192 y)=M,N)(fL(1). If fx, fy, fz \u2208 L/f, thenWe claim that the above operations on fL is an -SI filter of L over U, then L/f\u2245L/ff(1).If \u03c6 : L \u2192 L/f by \u03c6(x) = fx for all x \u2208 L. For any x, y \u2208 L, we have\u03c6 is an epic. Moreover, we haveL/f\u2245L/ff(1).Define M, N)-SI (implicative) filters of BL-algebras. We investigate their characterizations. In particular, we describe -soft congruences in BL-algebras.As a generalization of soft intersection filters of BL-algebras, we introduce the concept of -SI prime (semiprime) filters of BL-algebras. Maybe one can apply this idea to decision making, data analysis, and knowledge based systems.To extend this work, one can further investigate ("}
{"text": "We first investigate sufficient and necessary conditions of stability of nonlinear distributed order fractional system and then we generalize the integer-order Chen system into the distributed order fractional domain. Based on the asymptotic stability theory of nonlinear distributed order fractional systems, the stability of distributed order fractional Chen system is discussed. In addition, we have foundthat chaos exists in the double fractional order Chen system. Numerical solutions are used to verify the analytical results. Since then, many researchers have completed further studies on the stability of linear fractional differential systems  and satisfy \u222b01b(\u03b1)s\u03b1d\u03b1 \u2260 0 for Re(s) > 0 and f \u2208 L1 for every \u03b1 \u2208 ; then initial value problem (Let the function  problem  has a unX(t) = (x1(t), x2(t),\u2026, xn(t))T \u2208 \u211dn and b(\u03b1) = (b1(\u03b1), b2(\u03b1),\u2026, bn(\u03b1))T, 0 < \u03b1 < 1. The n-dimension distributed order fractional system is described as follows:Furthermore, the above definition in one dimension can naturally be generalized to the case of multiple dimensions; that is, let follows:Dctb(\u03b1)X(X(t) = (x1(t), x2(t),\u2026, xn(t))T \u2208 \u211dn, the matrix A \u2208 \u211dn\u00d7n, and b(\u03b1) = (b1(\u03b1), b2(\u03b1),\u2026, bn(\u03b1))T, 0 < \u03b1 < 1. Then Saberi Najafi et al. , and r = ei\u03c0.In this section, we generalize the main stability properties for systems of distributed order fractional. The linear distributed order fractional systems are expressed asi et al.  have obtDctb(\u03b1)XI \u2212 A) = 0 have negative real parts.The distributed order fractional system of  is asympB(s)I \u2212 A) = 0 is the characteristic function of the matrix A with respect to the distributed function B(s), where B(s) = \u222b01b(\u03b1)s\u03b1d\u03b1 is the distributed function with respect to the density function b(\u03b1) \u2265 0.The value of det\u2061(A), \u03c5B(s)(A), and \u03b4B(s)(A) are, respectively, the number of roots of det\u2061(B(s)I \u2212 A) = 0 with positive, negative, and zero real parts, where B(s) = (B1(s), B2(s),\u2026, Bn(s))T is the distributed function with respect to the density function b(\u03b1) = (b1(\u03b1), b2(\u03b1),\u2026, bn(\u03b1))T.The inertia of the system  is the t\u03c0nB(s)(A) = \u03b4nB(s)(A) = 0,s of the characteristic function of A with respect to B(s) = (B1(s), B2(s),\u2026, Bn(s))T satisfy |arg(s)|>\u03c0/2.all roots The linear distributed order fractional system  is asympX(0) = X0, whereX(t) = (x1(t), x2(t),\u2026, xn(t))T \u2208 \u211dn, and b(\u03b1) = (b1(\u03b1), b2(\u03b1),\u2026, bn(\u03b1))T, 0 < \u03b1 < 1.Next, we will mainly discuss the stability of a nonlinear autonomous distributed order fractional system, which can be described bys of the characteristic function of J with respect to B(s) = (B1(s), B2(s),\u2026, Bn(s))T satisfy |arg(s)|>\u03c0/2.Let f system ; that is\u03b6(t) = (\u03b61(t), \u03b62(t),\u2026, \u03b6n(t)) is a small disturbance from a fixed point. ThereforeLet hereforeDctb(\u03b1)\u03b6 = (B1(s), B2(s),\u2026, Bn(s))T satisfy |arg(s)|>\u03c0/2, which implies that the equilibrium The analytical procedure of linearization is based on the fact that if the matrix r system . Hence, l system  is as as\u03c0nB(s)(J) = \u03b4nB(s)(J) = 0.The nonlinear distributed order fractional system  in the pR3 [x, y, and z are the state variables and a, b, and c are three system parameters. The above system has a chaotic attractor when a = 35, b = 3, and c = 28 as shown in bi(\u03b1) for i = 1,2, 3 denote the nonnegative density function of order \u03b1 \u2208  = (x(t), y(t), z(t))T, b(\u03b1) = (b1(\u03b1), b2(\u03b1), b3(\u03b1))T, 0 < \u03b1 \u2264 1, and i = 1,2, 3. The Jacobian matrix of distributed order fractional Chen system  is given byThe Chen system is described by the following nonlinear differential equations on R3 , 23:(21shown in Dtb1(\u03b1)cxshown in Dtb1(\u03b1)cxbi(\u03b1) = \u03b4(\u03b1 \u2212 qi), where 0 < qi \u2264 1 for i = 1,2, 3 and \u03b4(\u03b1) is the Dirac delta function, then, we have the following fractional incommensurate-order Chen system [If n system :(25)Dt\u03b1InB(s)(J) in the case that the density function varies. The results are shown in a, b, and c.Based upon h is assumed to be very small [tk = kh\u2009\u2009 and ci\u03b1) are binomial coefficients, which can be computed as [ci\u03b1j)=quations\u2211i=1n\u03b3iDc6Dct\u03b1x(t)ry small , 43(28)Dct\u03b1x(t)=quations\u2211i=1n\u03b3iDcuted as [c0(\u03b1)=1,\u2003tion of (h\u2212\u03b1\u2211i=0kcDct\u03b1x(t)=quations\u2211i=1n\u03b3iDcT = 10, h = 0.005 with the initial conditions .To verify the efficiency of the obtained results in a, b, c) =  and  =  is asymptotically stable in the equilibrium a, b, c) =  and  =  is asymptotically stable in the equilibrium a, b, c) =  and  =  is chaotic. a, b, c) =  and  =  has chaotic attractor. a, b, c) =  and  = . From a, b, c) =  and  = .In this paper, we introduced the nonlinear distributed order fractional differential equations with respect to a nonnegative density function; hence the asymptotical stability for such systems has been investigated. In addition, we presented the distributed order fractional Chen system and then in two special cases the stability for the distributed order fractional Chen is discussed. Numerical solutions were coincident with results of"}
{"text": "Some fixed point results are given for a class of Meir-Keeler contractive maps acting on metric spaces endowed with locally transitive relations. Technical connections with the related statements due to Berzig et al. (2014) are also being discussed. X be a nonempty set. Call the subset Y of X, almost-singleton (in short: asingleton), provided y1, y2 \u2208 Y implies y1 = y2 and singleton if, in addition, Y is nonempty; note that, in this case, Y = {y}, for some y \u2208 X. Take a metric\u2009\u2009d : X \u00d7 X \u2192 R+\u2236 = .(A) Let (P0)\u211b is amorphous; that is, it has no specific properties at all;(P1)\u211b is an order; that is, it is reflexive , transitive [x\u211by and y\u211bz imply x\u211bz], and antisymmetric [x\u211by and y\u211bx imply x = y];(P2)\u211b is a quasiorder; that is, it is reflexive and transitive;(P3)\u211b is transitive (see above).Among the classes of relations to be used, the following ones  are important for us:N, \u2264); here, N = {0,1,\u2026} is the set of natural numbers and (\u2264) is defined as m \u2264 n if and only if m + p = n, for some p \u2208 N. For each n \u2208 N, let N: = {0,\u2026, n \u2212 1} stand for the initial interval (in N) induced by n. Any set P with P ~ N (in the sense: there exists a bijection from P to N) will be referred to as effectively denumerable. In addition, given some natural number n \u2265 1, any set Q with Q ~ N will be said to be n-finite; when n is generic here, we say that Q is finite. Finally, the (nonempty) set Y is called (at most) denumerable if and only if it is either effectively denumerable or finite.A basic ordered structure is x, z) \u2208 \u211b\u2218\ud835\udcae if there exists y \u2208 X with  \u2208 \u211b,  \u2208 \ud835\udcae.\u211b0 = \u2110, \u211bn+1 = \u211bn\u2218\u211b, n \u2208 N. ; x \u2208 X} is the identical relation over X). The following properties will be useful in the sequel:Given the relations k \u2208 N, let us say that \u211b is k-transitive, if \u211bk\u2286\u211b; clearly, transitive is identical with 2-transitive. We may now complete the increasing scale above as(P4)\u211b is finitely transitive; that is, \u211b is k-transitive for some k \u2265 2;(P5)\u211b is locally finitely transitive; that is, for each (effectively) denumerable subset Y of X, there exists k = k(Y) \u2265 2, such that the restriction to Y of \u211b is k-transitive;(P6)\u211b is trivial; that is, \u211b = X \u00d7 X; hence, .Given zn; n \u2265 0) in X, \u211b-ascending, if zi\u211bzi+1 for all i \u2265 0.Concerning these concepts, the following property will be useful. Call the sequence  in X and the natural number k \u2265 2 be such that (b03)\u211b is k-transitive on Z : = {zn; n \u2265 0}. Then, necessarily,Let the r. First, by the choice of our sequence,  \u2208 \u211b; whence, the case r = 0 holds. Moreover, by definition,  \u2208 \u211bk; and this, along with the k-transitive property, gives  \u2208 \u211b; hence, the case of r = 1 holds too. Suppose that this property holds for some r \u2265 1; we claim that it holds as well for r + 1. In fact, let i \u2265 0 be arbitrary fixed. Again by the choice of our sequence, (zi+1+r(k\u22121), zi+1+(r+1)(k\u22121)) \u2208 \u211bk\u22121, so that, by the inductive hypothesis :k-transitive condition, gives (k\u22121)) \u2208 \u211b. The proof is thereby complete.We will use the induction with respect to X, d) be a metric space. We introduce a d-convergence and d-Cauchy structure on X as follows. By a sequence in X, we mean any mapping x : N \u2192 X. For simplicity reasons, it will be useful to denote it as (x(n); n \u2265 0) or ; moreover, when no confusion can arise, we further simplify this notation as (x(n)) or (xn), respectively. Also, any sequence (yn : = xi(n); n \u2265 0) with i(n) \u2192 \u221e as n \u2192 \u221e will be referred to as a subsequence of . Given the sequence (xn) in X and the point x \u2208 X, we say that (xn),\u2009\u2009d-converges to x  \u2192 0 as n \u2192 \u221e; that is,x will be denoted lim\u2061n(xn); note that it is an asingleton, because d is triangular symmetric; if lim\u2061n(xn) is nonempty, then (xn) is called d-convergent. We stress that the introduced convergence concept xn),\u2009\u2009d-Cauchy when d \u2192 0 as m, n \u2192 \u221e, m < n; that is,d is triangular symmetric, any d-convergent sequence is d-Cauchy too; but, the reciprocal is not in general true. Concerning this aspect, note that any d-Cauchy sequence  is d-semi-Cauchy; that is,(B) Let  \u21a6 d is d-Lipschitz, in the sensed-continuous; that is,The mapping ; we do not give details.The proof is immediate, by the usual properties of the ambient metric X, d) be a metric space; and let \u211b\u2286X \u00d7 X be a (nonempty) relation over X; the triple  will be referred to as a relational metric space. Further, take some T \u2208 \u2131(X). Call the subset Y of X, \u211b-almost-singleton (in short: \u211b-asingleton) provided y1, y2 \u2208 Y, y1\u211by2\u2009\u2009\u21d2\u2009\u2009y1 = y2 and \u211b-singleton when, in addition, Y is nonempty. We have to determine circumstances under which Fix\u2061(T) is nonempty; and, if this holds, to establish whether T is fix-\u211b-asingleton  is \u211b-asingleton) or, equivalently, T is fix-\u211b-singleton (in the sense: Fix\u2061(T) is \u211b-singleton); to do this, we start from the working hypotheses:(b04)T is \u211b-semi-progressive: X: = {x \u2208 X; x\u211bTx} \u2260 \u2205;(b05)T is \u211b-increasing: x\u211by implies Tx\u211bTy.(C) Let ((2a)T is a Picard operator ) if, for each x \u2208 X,  is d-convergent.We say that (2b)T is a strong Picard operator ) when, for each x \u2208 X,  is d-convergent and lim\u2061n(Tnx) \u2208 Fix\u2061(T).We say that (2c)T is a globally strong Picard operator ) when it is a strong Picard operator ) and T is fix-\u211b-asingleton .We say that The basic directions under which the investigations be conducted are described by the list below, comparable with the one in Turinici :(2a)We s ascending orbital concepts (in short: (a-o)-concepts). Remember that the sequence  in X is called \u211b-ascending, if zi\u211bzi+1 for all i \u2265 0; further, let us say that  is T-orbital, when it is a subsequence of , for some x \u2208 X; the intersection of these notions is just the precise one.(2d)X, -complete, provided (for each (a-o)-sequence) d-Cauchy \u21d2\u2009\u2009d-convergent.Call (2e)T is -continuous, if ((zn)=(a-o)-sequence and We say that (2f)\u211b, -almost-self-closed, if: whenever the (a-o)-sequence  in X and the point z \u2208 X fulfill wn : = zi(n); n \u2265 0) of  with wn\u211bz, for all n \u2265 0.Call The sufficient (regularity) conditions for such properties are being founded on ascending notions (in short: a-notions). On the other hand, when the ascending properties are ignored, the same conventions give us orbital notions (in short: o-notions). The list of these is obtainable from the previous one; so, further details are not needed. Finally, when \u211b = X \u00d7 X, the list of such notions is comparable with the one in Rus  = X.When the orbital properties are ignored, these conventions give us in Rus  be a relational metric space; and let T be a self-map of X, supposed to be \u211b-semi-progressive and \u211b-increasing. The basic directions and sufficient regularity conditions under which the problem of determining the fixed points of\u2009\u2009T\u2009\u2009is to be solved were already listed. As a completion of them, we must formulate the specific metrical contractive conditions upon our data. These, essentially, consist in a \u201crelational\u201d variant of the Meir-Keeler condition ; so,  irreflexive [x \u2208 X]. Denote for x, y \u2208 X\u2009(c03)A1 = d, B1 = diam\u2061{x, Tx, y, Ty},\u2009A2 = (1/2)[d + d],\u2009A3 = max\u2061{d, d},\u2009A4 = (1/2)[d + d]. Then, let us introduce the functions \u2009(c04)B2 = max\u2061{A1, A2}, B3 = max\u2061{A1, A3}, B4 = max\u2061{A1, A4},\u2009C1 = max\u2061{A1, A2, A4}, C2 = max\u2061{A1, A3, A4},\u2009\ud835\udca2 = {A1, B2, B3, B4, C1, C2}, \ud835\udca21 = {A1, B2, B4, C1}, \ud835\udca22 = {B3, C2}.Note that, for each G \u2208 \ud835\udca2, we haveG is sufficient; note that, by the properties of d, we must haveG is diameter bounded.Let -contractive, if(c05)d < G, expressed as T is strictly -nonexpansive;(c06)\u03b5 > 0, \u2203\u03b4 > 0: [\u03b5 < G < \u03b5 + \u03b4] \u21d2\u2009\u2009d \u2264 \u03b5, expressed as T has the Meir-Keeler property ).for all  Note that, by the former of these, the Meir-Keeler property may be written as (c07)\u03b5 > 0, \u2203\u03b4 > 0: [G < \u03b5 + \u03b4] \u21d2\u2009\u2009d \u2264 \u03b5.for all Given In the following, two basic examples of such contractions will be given.\u2131(re)(R+) stand for the class of all \u03c6 \u2208 \u2131(R+) with the (strong) regressive property: . We say that \u03c6 \u2208 \u2131(re)(R+) is Meir-Keeler admissible, if(c08)\u03b3 > 0, \u2203\u03b2\u2208]0, \u03b30, \u03b30, \u03b50, \u221e(c15)\u03b5 > 0): limsup\u2061n\u03c6(tn) > \u03c8(\u03b5 + 0) \u2212 \u03c8(\u03b5), whenever tn \u2192 \u03b5 + +. Let us say that  in R and the point r \u2208 R, we denoted\u2009rn \u2192 r+ , if rn \u2192 r and\u2009rn \u2265 r , for all n \u2265 0 large enough.Here, given the sequence  of functions in \u2131(R+), let us say that T is )-contractive, provided(c16)\u03c8) \u2264 \u03c8) \u2212 \u03c6), for all x, y \u2208 X, Given T is )-contractive, for a pair  of weak generalized altering functions in \u2131(R+). Then, T is Meir-Keeler -contractive (see above).Suppose that x, y \u2208 X be such that G is sufficient), G > 0, so that (by the choice of our pair), \u03c6) > 0; wherefrom \u03c8) < \u03c8). This via (\u03c8 = increasing) yields d < G, so that T is strictly -nonexpansive.(i) Let T does not have the Meir-Keeler property ); that is, for some \u03b5 > 0,\u03b4n) in R+0, we get a couple of sequences  and  in X, so as\u03c8 = increasing), we getG \u2192 \u03b5 + +, so that passing to limsup\u2061 as n \u2192 \u221e,\u03c8, \u03c6), these relations are contradictory. This ends the argument.(ii) Assume by contradiction that X, so as(\u2200n):xn\u211b~X, d, \u211b) be a relational metric space. Further, let T be a self-map of X, supposed to be \u211b-semi-progressive and \u211b-increasing. The basic directions and regularity conditions under which the problem of determining the fixed points of T is to be solved, were already listed; and the contractive type framework was settled. It remains now to precise the regularity conditions upon \u211b. Denote, for each x \u2208 X,x), but the possibility of spec(x) = {1} cannot be removed. This fact remains valid even if x \u2208 X is orbital admissible, in the sense [i \u2260 j implies Tix \u2260 Tjx], when the associated orbit TNx : = {Tnx; n \u2265 0} is effectively denumerable. But for the developments below, it is necessary that these spectral subsets of N should have a finite Hausdorff-Pompeiu distance to N; hence, in particular, these must be infinite. Precisely, given k \u2265 1, let us say that \u211b is k-semirecurrent at the orbital admissible x \u2208 X, if \u2009n \u2208 N, there exists q \u2208 spec(x) such that q \u2264 n < q + k.for each  A global version of this convention is the following: call \u211b, finitely semirecurrent if, for each orbital admissible x \u2208 X, there exists k(x) \u2208 N, such that \u211b is k(x)-semirecurrent at x.Let ((d01)\u211b is finitely semirecurrent and nonidentical.Assume in the following thatOur main result in this exposition is the following.T is Meir-Keeler -contractive, for some G \u2208 \ud835\udca2. In addition, let X be -complete; and one of the following conditions holds:T is -continuous;\u211b is -almost-self-closed and G \u2208 \ud835\udca21;\u211b is -almost-self-closed and T is -contractive, for a certain Meir-Keeler admissible function \u03c6 \u2208 \u2131(re)(R+);\u211b is -almost-self-closed and T is )-contractive, for a certain pair  of weak generalized altering functions in \u2131(R+). Then T is a globally strong Picard operator ).Assume that \u211b-asingleton property. Let z1, z2 \u2208 Fix\u2061(T) be such that z1\u211bz2; and assume by contradiction that z1 \u2260 z2; whence G = d. This, via T being strictly -nonexpansive, yields an evaluation liked, \u211b)). The argument will be divided into several steps.First, we check the fix-Part 1. We firstly assert thatx \u2208 X be such that T is strictly -nonexpansive, one has d < G. On the other hand,Part 2. Take some x0 \u2208 X; and put . If xn = xn+1 for some n \u2265 0, we are done, so, without loss, one may assume that, for each n \u2265 0, (d02)xn \u2260 xn+1; hence, \u03c1n : = d > 0. From the preceding part, we derive\u03c1n; n \u2265 0) is strictly descending. As a consequence, \u03c1 : = lim\u2061n\u03c1n exists as an element of R+. Assume by contradiction that \u03c1 > 0; and let \u03b4 > 0 be the number given by the Meir-Keeler -contractive condition upon T. By definition, there exists a rank n(\u03b4) such that n \u2265 n(\u03b4) implies \u03c1 < \u03c1n < \u03c1 + \u03b4; hence (by a previous representation) \u03c1 < G = \u03c1n < \u03c1 + \u03b4. This, by the Meir-Keeler contractive condition we just quoted, yields (for the same n), \u03c1n+1 = d \u2264 \u03c1; contradiction. Hence, \u03c1 = 0, so thatxn; n \u2265 0) is d-semi-Cauchy.Part 3. Suppose that (d03)i, j \u2208 N such that i < j, xi = xj.there exist  Denoting p = j \u2212 i, we thus have p > 0 and xi = xi+p, so that\u03c1i = \u03c1i+np, for all n \u2265 0. This, along with \u03c1i+np \u2192 0 as n \u2192 \u221e, yields \u03c1i = 0, in contradiction with the initial choice of . Hence, our working hypothesis cannot hold; wherefromPart 4. As a consequence of this, the map i \u21a6 xi : = Tix0 is injective; hence, x0 is orbital admissible. Let k : = k(x0) \u2265 1 be the semirecurrence constant of \u211b at x0 (assured by the choice of this relation). Further, let \u03b5 > 0 be arbitrary fixed; and \u03b4 > 0 be the number associated by the Meir-Keeler -contractive property; without loss, one may assume that \u03b4 < \u03b5. By the d-semi-Cauchy property and triangular inequality, there exists a rank n(\u03b4) \u2265 0, such thatxn; n \u2265 0) is d-Cauchy. To do this, an induction argument upon s \u2265 1 will be used. The case s \u2208 {1,\u2026, 2k} is evident, by the preceding evaluation. Assume that it holds for all s \u2208 {1,\u2026, p}, where p \u2265 2k; we must establish its validity for s = p + 1. As \u211b is k-semirecurrent at x0, there exists q \u2208 spec(x0) such that q \u2264 p < q + k; note that the former of these yields (from the \u211b-increasing property of T), B1 < \u03b5 + \u03b4, so that (by the diameter boundedness property), (0<)\u2009\u2009G < \u03b5 + \u03b4. Taking the Meir-Keeler -contractive assumption imposed upon T into account givesuch that(\u2200n\u2265n(\u03b4))uch that(\u2200n\u2265n(\u03b4))Part 5. As X is -complete, z \u2208 X. If there exists a sequence of ranks (i(n); n \u2265 0) with [i(n) \u2192 \u221e as n \u2192 \u221e] such that xi(n) = z +1 = Tz) for all n, then, as (xi(n)+1; n \u2265 0) is a subsequence of , one gets z = Tz. So, in the following, we may assume that the opposite alternative is true: (d04)h \u2265 0: n \u2265 h\u2009\u2009\u21d2\u2009\u2009xn \u2260 z.\u2203 There are several cases to discuss.Case\u2009\u20095a. Suppose that T is -continuous. Then n \u2192 \u221e. On the other hand,  is a subsequence of (xn); whence d is sufficient), z = Tz.Case\u2009\u20095b. Suppose that \u211b is -almost-self-closed. Put, for simplicity reasons, b : = d. By definition, there exists a subsequence (un : = xi(n); n \u2265 0) of , such that un\u211bz, for all n. Note that, as lim\u2061n\u2009i(n) = \u221e, one may arrange for i(n) \u2265 n, for all n, so that, from (d04),Tun = xi(n)+1; n \u2265 0) being as well a subsequence of , gives (via (ves (via  and Lemmves (via (37)A1 yields a contradiction. Two alternatives must be treated.We now show that the assumption Alter 1. Suppose that G \u2208 \ud835\udca21. By the Meir-Keeler contractive condition,G \u2192 b. This, along with , , Alter 1Alter 2. Suppose that G \u2208 \ud835\udca22. The above convergence properties of  tell us that, for a certain rank n(b) \u2265 h, we must haved-Lipschitz property of d, givesb/2 < d < 3b/2,\u2009\u2009\u2200n \u2265 n(b). Combining these yieldsAlter 2a. Suppose that T is -contractive, for a certain Meir-Keeler admissible function \u03c6 \u2208 \u2131(re)(R+).  By (b \u2264 \u03c6(b); contradiction; hence, z = Tz.tep.) By  and thisives (by  above), Alter 2b. Suppose that T is )-contractive, for a certain pair  of weak generalized altering functions in \u2131(R+).  From this contractive condition,d < b, for all n \u2265 n(b). Passing to limit as n \u2192 \u221e and taking (\u03c6(b) \u2264 \u03c8(b) \u2212 \u03c8(b \u2212 0). This, however, contradicts the choice of , so that z = Tz. The proof is complete.ing with  above),d taking  into acc\u211b is transitive, this result is comparable with the one in Turinici . Then, if we take \u211b : = \ud835\udcae and G = A1, the alternative (i1) of Let (1-2)\u211b = X \u00d7 X . Then, Suppose that The following particular cases of this result are to be noted.As another consequence of T is \u211b-semiprogressive, \u211b-increasing, and -contractive, for some G \u2208 \ud835\udca2 and a certain Meir-Keeler admissible function \u03c6 \u2208 \u2131(re)(R+). In addition, let \u211b be finitely semirecurrent nonidentical,\u2009\u2009X be -complete, and one of the conditions below holds:(j1)T is -continuous;(j2)\u211b is -almost-self-closed. Then T is a globally strong Picard operator ).Assume that (2-1)\u211b = X \u00d7 X  and G = A1. Then, Suppose that (e03)T(Y)) \u2264 \u03c6(diam\u2061(Y)), for all Y \u2208 CB(X).diam\u2061 is the class of all (nonempty) closed bounded subsets of X.) Clearly, this condition is stronger than the one we already used in (2-2)\u211b = X\u2009\u00d7\u2009X; and \u03c6 \u2208 \u2131(re)(R+) is BWM-admissible . Then, if G = A1, \u03c6 is Boyd-Wong admissible; and, respectively, the Matkowski's result [\u03c6 is Matkowski admissible. Moreover, when G = C2, Suppose that s result  when \u03c6 i(2-3)\u211b is an order on X. Then, Suppose that The following particular cases of this result are to be noted.As a final consequence of T is \u211b-semiprogressive, \u211b-increasing, and )-contractive, for a certain G \u2208 \ud835\udca2 and some pair  of generalized altering functions in \u2131(R+). In addition, let \u211b be finitely semirecurrent nonidentical,\u2009\u2009X be -complete, and one of the conditions below holds:(k1)T is -continuous;(k2)\u211b is -almost-self-closed. Then T is a globally strong Picard operator ).Assume that the self-map (3-1)\u03b1(\u00b7),\u2009\u2009\u03b2(\u00b7) be a couple of functions in \u2131; and \ud835\udc9c,\u2009\u2009\u212c stand for the associated relations:Let Then, if we take \u211b : = \ud835\udc9c\u2229\u212c and G \u2208 \ud835\udca2, this result includes (cf. Lemma\u2009\u20091) the one in Berzig et al. [T is -contractive. In particular, when G = A1, this last result reduces to the one in Berzig and Karap\u0131nar [The following particular cases of this result are to be noted.g et al. , based oet al. [4\u03c8 be a metric space; and T be a self-map of Y. Given p \u2265 2, let {A1,\u2026, Ap} be a finite system of closed subsets of Y with(e04)T(Ai)\u2286Ai+1, for all i \u2208 {1,\u2026, p} (where Ap+1 = A1). Define a relation \u211b over Y as (e05)\u211b = (A1 \u00d7 A2)\u222a\u22ef\u222a(Ap \u00d7 Ap+1);then, put X = A1 \u222a \u22ef\u222aAp. Clearly, T is a self-map of X; and the relation \u211b is p-semirecurrent at each orbital admissible point of X. The corresponding version of (3-2) Let  transitive type requirements. Further aspects  may be found in Berzig [Finally, we should remark that none of these particular theorems may be viewed as a genuine extension for the fixed point statement due to Samet and Turinici ; becausen Berzig ."}
{"text": "For a smooth bivariate function defined on a general domain with arbitrary shape, it isdifficult to do Fourier approximation or wavelet approximation. In order to solve these problems, in this paper,we give an extension of the bivariate function on a general domain with arbitrary shape to a smooth, periodicfunction in the whole space or to a smooth, compactly supported function in the whole space. These smoothextensions have simple and clear representations which are determined by this bivariate function and somepolynomials. After that, we expand the smooth, periodic function into a Fourier series or a periodic waveletseries or we expand the smooth, compactly supported function into a wavelet series. Since our extensions aresmooth, the obtained Fourier coefficients or wavelet coefficients decay very fast. Since our extension tools arepolynomials, the moment theorem shows that a lot of wavelet coefficients vanish. From this, with the help ofwell-known approximation theorems, using our extension methods, the Fourier approximation and the waveletapproximation of the bivariate function on the general domain with small error are obtained. In the recent several decades, various approximation tools have been widely developed \u201314. For This paper is organized as follows. In T = 2 and the interior of T by To and always assume that \u03a9 is a simply connected domain. We say that f \u2208 Cq(\u03a9) if the derivatives (\u2202i+jf/\u2202xi\u2202yj) are continuous on \u03a9 for 0 \u2264 i + j \u2264 q. We say that f \u2208 C\u221e(\u03a9) if all derivatives (\u2202i+jf/\u2202xi\u2202yj) are continuous on \u03a9 for i, j \u2208 \u2124+. We say that a function h is a \u03b3-periodic function if h = h\u2009\u2009 \u2208 T; k, l \u2208 \u2124), where \u03b3 is an integer. We appoint that 0! = 1 and the notation [\u03b1] is the integral part of the real number \u03b1.Throughout this paper, we denote In this section, we state the main results of smooth extensions and their applications in Fourier analysis and wavelet analysis.Our main theorems are stated as follows.f \u2208 C\u221e(\u03a9), where \u03a9 \u2282 To and the boundary \u2202\u03a9 is a piecewise infinitely many time smooth curve. Then for any r \u2208 \u2124+ there is a function F \u2208 Cr(T) such that (i)F = f\u2009\u2009 \u2208 \u03a9);(ii)i+jF/\u2202xi\u2202yj) = 0 on the boundary \u2202T for 0 \u2264 i + j \u2264 r;(\u2202(iii)T\u2216\u03a9, F can be expressed locally in the formson the complement where L is a positive integer and each coefficient cij is constant.Let f \u2208 C\u221e(\u03a9), where \u03a9 is stated as in r \u2208 \u2124+, there exists a 1-periodic function Fp \u2208 Cr(\u211d2) such that Fp = f\u2009\u2009 \u2208 \u03a9).Let f \u2208 C\u221e(\u03a9), where \u03a9 is stated as in r \u2208 \u2124+, there exists a function Fc \u2208 Cr(\u211d2) with compact support T such that Fc = f\u2009\u2009 \u2208 \u03a9).Let f \u2208 C\u221e(\u03a9). If f \u2208 Cq(\u03a9) (q is a nonnegative integer), by using the similar method of arguments of Theorems In Sections Here we show some applications of these theorems.F be the smooth extension of f from \u03a9 to T which is stated as in F \u2208 Cr(T) and F = f on \u03a9. By \u0394N, denote the set of all bivariate polynomials in the form \u2211n1,n2=\u2212NNcn1,n2xn1yn2. ThenLp(D) is the norm of the space Lp(D). The right-hand side of formula  and Fp = f on \u03a9. By the well-known results 2 byFo is an odd function. By Fo \u2208 Cr and (\u2202i+jFo/\u2202xi\u2202yj) = 0 on \u2202 for 0 \u2264 i + j \u2264 r. Again, doing a 2-periodic extension, we obtain a 2-periodic odd function Fpo and Fpo \u2208 Cr(\u211d2). By the well-known results 2 byFo={ssin sum  of the FFe on 2 as follows:Fe is an even function on 2. By Fe \u2208 Cr and (\u2202i+jFe/\u2202xi\u2009\u2009\u2202yj) = 0 on \u2202 for 0 \u2264 i + j \u2264 r. Again, doing a 2-periodic extension, we obtain a 2-periodic even function Fpe and Fpe \u2208 Cr(\u211d2). By the well-known result , where \u03b1* = inf\u2061{g(x), x1 \u2264 x \u2264 x2}. Then we haveE1 = { : x1 \u2264 x \u2264 x2, 0 \u2264 y \u2264 g(x)} and g(x) \u2265 \u03b1*(x1 \u2264 x \u2264 x2). So\u03c8 chosen by us is L time smooth, then, by using the moment theorem and supp\u2061\u03c8m*,n2* \u2282 , we haveI2 = 0. Similarly, since F is bivariate polynomials on rectangles H1 and H3 \u2208\u2202\u03a9\u2009\u2009 such that T\u2216\u03a9 can be divided into the four rectanglesg \u2208 C\u221e, h \u2208 C\u221e, g* \u2208 C\u221e, and h* \u2208 C\u221e andT can be expressed into a disjoint union as follows:E\u03bd is a trapezoid with a curved side and each H\u03bd is a rectangle (see Since ngle see .f to each E\u03bd and continue to extend to each H\u03bd such that the obtained extension F satisfies the conditions of In Sections E1 with a curved side y = g(x)(x1 \u2264 x \u2264 x2) is represented asak,1}0\u221e and {bk,1}0\u221e as follows:x1 \u2264 x \u2264 x2,By , the trafollows:a0,1E1, we define a sequence of functions {S1k)}0\u221e by induction.On LetThen, by ,(30)S1, we, we(31)Sk \u2208 \u2124+, one has S1k)( \u2208 C\u221e(E1) andFor any f \u2208 C\u221e(\u03a9) and g \u2208 C\u221e, and g(x) > 0\u2009\u2009(x1 \u2264 x \u2264 x2), by the above construction, we know that S1k)( \u2208 C\u221e(E1) for any k = 0,1,\u2026.Since k = 0, sincek = l \u2212 1; that is,al,1 and bl,1 are polynomials of y whose degrees are both 2l + 1. From this and )g(x)y, holds. W))g(x)y, holds fothis and , it follthis and  holds foduction,  holds folS1k)(/\u2202yl)\u2009\u2009(0 \u2264 l \u2264 k) on the curved side \u03931 = {) : x1 \u2264 x \u2264 x2} and the bottom side \u03941 = { : x1 \u2264 x \u2264 x2} of E1.Below we compute derivatives (\u2202S1k) be stated as above. For any k \u2208 \u2124+, one hasLet k = 0, ) = 0, ) = 0. So we getl = k, note that ) = 1 and ) = 0. By ).By , we havex2, by (\u2202lS1(k)\u2202yrmula of  holds fol = 0,\u2026, k \u2212 1, by the assumption of induction and (lS1k\u22121)(/\u2202yl) = 0 and  =  = 0. Sol = k, since  = 0,  = 1, by \u2202yrmula of  holds. Brmula of  holds foS1k) on the curved side \u03931 and bottom side \u03941 of E1.Now we compute the mixed derivatives of 1 and \u03941 be the curved side and the bottom side of E1, respectively. Then, for k \u2208 \u2124+, i+jS1k)(/\u2202xi\u2202yj) = (\u2202i+jf/\u2202xi\u2202yj)\u2009\u2009 \u2208 \u03931), (\u2202i+jS1k)(/\u2202xi\u2202yj) = 0\u2009\u2009 \u2208 \u03941),(\u2202where 0 \u2264 i + j \u2264 k.Let \u0393x1 \u2264 x \u2264 x2. Then we havef by S1k)) = f); that is, (i) holds for i = j = 0. So we get (i).Now we start from the equalityent from  to 50),,(51)ddx(jS1k)(/\u2202yj) = 0\u2009\u2009(0 \u2264 j \u2264 k). From this and S1k)( \u2208 C\u221e(E1), we haveBy From this, we get the following.r, denote lr = r(r + 1)(r + 2)(r + 3). LetF \u2208 Clr(\u03a9\u22c3E1)\u2009\u2009 and \u2009\u2009F = f  \u2208 \u03a9); (ii) (\u2202i+jF/\u2202xi\u2202yj) = 0\u2208(E1\u22c2\u2202T), 0 \u2264 i + j \u2264 lr).For any positive integer f \u2208 C\u221e(\u03a9), S1k)( \u2208 C\u221e(E1), and 1 = \u03a9\u22c2E1, we get (i). By E1\u22c2\u2202T = \u03941, we get (ii). By the assumption \u03bd = 2,3, 4, by using a similar method, we define S\u03bdk) on the each trapezoid E\u03bd with a curve side. The representations of S\u03bdk) are stated in For \u03bd = 1,2, 3,4, letlr = r(r + 1)(r + 2)(r + 3). Then, for \u03bd = 1,2, 3,4, one has the following: (i)F \u2208 Clr(\u03a9\u22c3E\u03bd\u2009\u2009);(ii)i+jF/\u2202xi\u2202yj) = 0, \u2208(E\u03bd\u22c2\u2202T) for 0 \u2264 i + j \u2264 lr;(\u2202(iii)F can be expressed in the form:For any \u03bd = 2,3, 4, we haveBy The proof of (iii) is similar to the argument of f to each trapezoid E\u03bd with a curved side. In this subsection we complete the smooth extension of the obtained function F to each rectangle H\u03bd. First we consider the smooth extension of F to H1. We divide this procedure in two steps.We have completed the smooth extension of F = S4lr on E4. Now we construct the smooth extension of S4lr) from E4 to H1, where S4lr) is stated in lr = r(r + 1)(r + 2)(r + 3).In \u03c4r = r(r + 2).LetJl1,}14 be four sides of the rectangle H1:M1\u03c4r) = \u2211i,j=0lr+12di,j(1)xiyj, where di,j(1) is a constant;i+jM1\u03c4r)(/\u2202xi\u2202yj) = (\u2202i+jS4lr)(/\u2202xi\u2202yj)\u2009\u2009 \u2208 J1,1);(\u2202i+jM1\u03c4r)(/\u2202xi\u2202yj) = 0\u2009\u2009 \u2208 J1,2);(\u2202i+jM1\u03c4r)(/\u2202xi\u2202yj) = 0\u2009\u2009 \u2208 J1,3),(\u2202where 0 \u2264 i + j \u2264 \u03c4r.Let {kS4lr)(/\u2202yk) is a polynomial of degree 2lr + 1 with respect to x. Since \u03b1\u03c4r,11(y) and \u03b2\u03c4r,11(y) are both polynomials of degree 2\u03c4r + 1, (i) follows from (By ows from .Similar to the argument of y1)\u2208(E4\u22c2\u2202T), by M1(0) and (i + j \u2264 lr \u2212 (1/2)k(k + 1), we have 0 \u2264 i + j \u2264 lr \u2212 (1/2)k(k \u2212 1) and 0 \u2264 i + k \u2264 lr \u2212 (1/2)k(k \u2212 1). Again, by the assumption of induction, we getk+iS4lr)(/\u2202xi\u2202yk) = 0. From this and (\u03c4r + 1) = (lr/2) \u2265 \u03c4r, we get (iii). Since , by \u2202i+jS4(lr\u22121)).By , we get\u22c2\u2202T), by \u2202i+jS4(lr1)).By (\u2202i+jM1(k)F = S1lr on E1. We consider the difference S1lr) \u2212 M1\u03c4r). Obviously, it is infinitely many time differentiable on E1 since M1\u03c4r) is a polynomial. Now we construct its smooth extension from E1 to the rectangle H1 as follows. LetIn N1r) possesses the following properties: i+jN1r)(/\u2202xi\u2202yj) = (\u2202i+jS1lr)(/\u2202xi\u2202yj) \u2212 (\u2202i+jM1\u03c4r)(/\u2202xi\u2202yj) on J1,4;(\u2202i+jN1r)(/\u2202xi\u2202yj) = 0 on J1,2;(\u2202i+jN1r)(/\u2202xi\u2202yj) = 0 on J1,1;(\u2202i+jN1r)(/\u2202xi\u2202yj) = 0 on J1,3, where 0 \u2264 i + j \u2264 r and {J\u03bd1,}14 are stated in  = \u2211i,j=0lr+12\u03c4i,j(1)xiyj, where \u03c4i,j(1) is a constant.The arguments similar to lr \u2265 \u03c4r, we get that, for 0 \u2264 i + j \u2264 \u03c4r,By \u03c4r \u2212 (1/2)r(r + 1) \u2265 r, we get (iii). By Now we assume that\u22121)).By  and 73)(75)\u2202i+jNthis and , by usinS1lr)( \u2212 M1\u03c4r) is a polynomial of degree 2lr + 1 with respect to y. From this and , i+jF/\u2202xi\u2202yj) = 0, \u2009\u2009 \u2208 ((E1\u22c3E4\u22c3H1)\u22c2\u2202T) for 0 \u2264 i + j \u2264 r; = \u2211i,j=0lr+12cij(1)xiyj \u2208 H1), where each cij(1) is constant.LetF \u2208 Cr(\u03a9\u22c3E1\u22c3E4). Since S1lr)( \u2208 Cr(E1),F \u2208 Cr(E1\u22c3H1). Since S4lr)( \u2208 Cr(E4),F \u2208 Cr(H1\u22c3E4). So we get (i).By J1,4,by , we deduJ1,1,by , we deduH1\u22c2\u2202T = J1,2\u22c3J1,3, by  = M1\u03c4r) + N1r)  \u2208 H1), we get (iii). From \u03bd = 2,3, 4, by using a similar method, we define F = M\u03bd\u03c4r) + N\u03bdr)  \u2208 H\u03bd), where representations of M\u03bd\u03c4r) and N\u03bdr) see Section For F has been defined on the unit square T. The argument similar to i + j \u2264 r,\u03a9\u22c2\u2202T = \u2205, by (F \u2208 Cr(T) and (\u2202i+jF/\u2202xi\u2202yj) = 0 \u2009\u2009\u2208\u2202T, 0 \u2264 i + j \u2264 r). So we get (i) and (ii).Let(87)By , F has b = \u2205, by , we havecij\u03bd) can be expressed locally in the formSimilar to the argument of F satisfying the conditions of The representation of F be the smooth extension of f from \u03a9 to T which is stated as in Fp byFp is a 1-periodic function of \u211d2. By Fp \u2208 Cr(T) andTn1,n2 = \u00d7. Since Fp is 1-periodic function, we have Fp \u2208 Cr and for any n1, n2 \u2208 \u2124,2 = \u22c3n1,n2\u2208\u2124Tn1,n2, we have Fp \u2208 Cr(\u211d2). By  \u2208 Cr(\u211d2). By ={n Fc byFc={f and \u03a9 be stated as in \u03a9 be divided as in F satisfying conditions of H\u03bd}14 and the trapezoids {E\u03bd}14 with a curved side are stated in (lr = r(r + 1)(r + 2)(r + 3) and \u03c4r = r(r + 2).Let tated in  and 23)f and \u03a9 bS\u03bdk)}14, {M\u03bdk)}14, and {N\u03bdk)}14.Below we write out the representations of {S1k) by induction as follows:(i) DenoteS2k) by induction as follows:(ii) DenoteS3k) by induction as follows:(iii) DenoteS4k) by induction as follows:(iv) DenoteM1k) by induction as follows:(i) DenoteN1k) by induction as follows:DenoteM2\u03c4r) by induction as follows:(ii) DenoteN2r) by induction as follows:DenoteM3k) by induction as follows:(iii) DenoteN3k) by induction as follows:DenoteM4k) by induction as follows:(iv) DenoteN4k) by induction as follows:DenoteBy using the extension method given in \u03a9 be a trapezoid with two curved sides:L1 < \u03b7(x) < \u03be(x) < L2 (x1 \u2264 x \u2264 x2), \u03b7, \u03be \u2208 Cm. Denote the rectangle D = \u00d7. Then D = G1\u22c3\u03a9\u22c3G2, where G1 and G2 are both trapezoids with a curved side:Let f \u2208 Cq(\u03a9) (q is a nonnegative integer). We will smoothly extend f from \u03a9 to the trapezoids G1 and G2 with a curved side, respectively, as in F is smooth on the rectangle D. Moreover, we will give a precise formula. It shows that the index of smoothness of F depends on not only smoothness of f but also smoothness of \u03b7, \u03be.Suppose that a0,1 = (y \u2212 L1)/(\u03b7(x) \u2212 L1) andS1k)} on G1 as follows. Letk0 be the maximal integer satisfying 1 + 2 + \u22ef+k0 \u2264 q. For k = 1,2,\u2026, k0, we defineS1k)( \u2208 C\u03bbk(G1), where \u03bbk = min\u2061{q \u2212 1 \u2212 2 \u2212 \u22ef\u2212k, m}.Denote a0,2 = (L2 \u2212 y)/(L2 \u2212 \u03be(x)) andS2k)} on G2 as follows. Letk = 1,2,\u2026, k0, defineS2k)( \u2208 C\u03bbk(G2), where \u03bbk is stated as above.Denote \u2009\u2009k \u2264 k0 and 0 \u2264 i + j \u2264 min\u2061{k, \u03bbk},k satisfying k \u2264 \u03bbk, where [\u00b7] expresses the integral part. So \u03c4 \u2264 \u03bb\u03c4.An argument similar to Lemmas i + j \u2264 \u03c4,f \u2208 Cq(\u03a9). Now we define a function on D byFq,m \u2208 C\u03c4(D). This implies the following theorem.By , we get \u03c4,\u2202i+jS1(\u03c4(\u03a9 and the rectangle D be stated as above. If f \u2208 Cq(\u03a9), then the function Fq,m, defined in (D), where \u03c4 is stated in  = 0, and so Fm0, \u2208 C(D); for q = 2 and m \u2265 1, we have \u03c4 = 1, and so F2,1 \u2208 C1(D); for q = 5 and m \u2265 2, we have \u03c4 = 2, and so F5,2 \u2208 C2(D).Especially, for f \u2208 Cq and \u2282. In order to extend smoothly f from  to , we construct two polynomialsS0(0)(x) = f(x1)(x/x1) and for k = 1,\u2026, q,S0q)((x) is a polynomial of degree \u22642q + 1.Let Similar to the proof of f from  to , we construct two polynomialsS1(0)(x) = f(x2)((1 \u2212 x)/(1 \u2212 x2)) and for k = 1,\u2026, q,S1q)((x) is a polynomial of degree \u22642q + 1.Again extend smoothly Similar to the proof of F from  to  byS0q)((x) and S1q)((x) are polynomials of degree 2q + 1 defined as above, and F \u2208 Cq and Fl)((0) = Fl)((1) = 0 . From this, we get the following.Therefore, we obtain the smooth extension f \u2208 Cq and \u2282. Then there exists a function F \u2208 Cq satisfying F(x) = f(x) (x1 \u2264 x \u2264 x2) and Fl)((0) = Fl)((1) = 0 .Let f \u2208 Cq and \u2282, and let F be the smooth extension of f from  to  which is stated as in Fp be the 1-periodic extension satisfying Fp(x + n) = F(x) . Then Fp \u2208 Cq(\u211d) and Fp(x) = f(x) . We expand F(x) into the Fourier series which converges fast. From this, we get trigonometric approximation of f \u2208 Cq. We also may do odd extension or even extension of F from  to , and then doing periodic extension, we get the odd periodic extension Fpo \u2208 Cq(\u211d) or the even periodic extension Fpe \u2208 Cq(\u211d). We expand Fpo or Fpe into the sine series and the cosine series, respectively. From this, we get the sine polynomial approximation and the cosine polynomial approximation of f on . For F \u2208 Cq(x) , we pad zero in the outside of  and then the obtained function Fc \u2208 Cq(\u211d). We expand Fc into a wavelet series which converges fast. By the moment theorem, a lot of wavelet coefficients are equal to zero. From this, we get wavelet approximation of f \u2208 Cq.Let"}
{"text": "R denotes a ring with identity I. We study conditions under which I \u2212 A\u03bb and I \u2212 B\u03bb are left coprime or right coprime, where A, B \u2208 R. As applications, we get sufficient conditions under which the Kneading determinant of a finite rank pair of operators on an infinite dimensional space is rational.In this section,  A is a k \u00d7 k matrix with rational coefficients, one has the well-known identity between formal power series:An) denotes the trace of matrix An (matrix A raised to the nth power) and I denotes the k \u00d7 k identity matrix. This identity plays a significant role in the discussion of an important problem in dynamical systems theory. For more details see ]:For any pair (Kneading determinant was first studied by Milnor and Thurston in .\u211a[[z]]q\u00d7q be the ring of the q \u00d7 q matrices whose entries lie in \u211a[[z]]. If \u03c6 \u2208 L(\u210b), Let \u03c6, \u03c8) \u2208 L(\u210b) \u00d7 L(\u210b), I the q \u00d7 q identity matrix. Then,\u211a[[z]].Let  with finite rank . So it The following is the main result of this paper.I \u2212 \u03c6\u03bb and I \u2212 \u03c8\u03bb are left coprime or right coprime, and h = \u03c8 \u2212 \u03c6 with finite rank, then \u0394\u03c6,\u03c8), Y(\u03bb) \u2208 R[\u03bb] such thatWe say I \u2212 A\u03bb and I \u2212 B\u03bb are left coprime if and only if there exist X0, X1,\u2026, Xm \u2208 R such thatH = A \u2212 B.I \u2212 A\u03bb and I \u2212 B\u03bb are left coprime, there exist X(\u03bb) = X0 + X1\u03bb + \u22ef+Xm\u03bbm and Y(\u03bb) = Y0 + Y1\u03bb + \u22ef+Ys\u03bbs such thatIf s equals m. And we haveWe can assume that So,H = A \u2212 B, then,If we write So,X0, X1,\u2026, Xm \u2208 R such thatConversely, if there exist Then,I \u2212 A\u03bb and I \u2212 B\u03bb are right coprime if there exist polynomials X(\u03bb), Y(\u03bb) \u2208 R[\u03bb] such thatNow we give the definition of right coprime. We say I \u2212 A\u03bb and I \u2212 B\u03bb are right coprime if and only if there exist X0, X1,\u2026, Xs \u2208 R such thatH = A \u2212 B.It is similar to the proof of \u03c6, \u03c8 \u2208 L(\u210b), where \u210b is an infinite dimensional vector space over \u211a, and we denote by L(\u210b) the ring of linear transforms on \u210b.In this section, I \u2212 \u03c6\u03bb and I \u2212 \u03c8\u03bb are left coprime or right coprime, h = \u03c8 \u2212 \u03c6 is of finite rank, and W =Im(h). Then there exists a finite dimensional space W such that k = 1,2,\u2026.Suppose that I \u2212 \u03c6\u03bb and I \u2212 \u03c8\u03bb are right coprime, then by X0, X1,\u2026, Xs \u2208 L(\u210b) such thatFirst we assume that if \u210b under h. It is not very hard to check that \u03c6.Write We use induction to prove the conclusion.k = 1, then (\u03c6k \u2212 \u03c8k)(\u210b) = W.If k \u2264 l we have (\u03c6k \u2212 \u03c8k)(\u210b) \u2282 W + \u03c6W + \u22ef+\u03c6mW. Then for k = l + 1, we have Suppose that for I \u2212 \u03c6\u03bb and I \u2212 \u03c8\u03bb are left coprime; then there exist s \u2208 \u2124 and x0, x1,\u2026, xs \u2208 L(\u210b) such thatl \u2265 s + 1, \u03c6l(\u210b) \u2282 x0W + x1W + \u22ef+xsW.Now we assume that We use induction to prove the conclusion.k = 1, then (\u03c6k \u2212 \u03c8k)(\u210b) = W.If k \u2264 l we have (\u03c6k \u2212 \u03c8k)(\u210b) \u2282 W + \u03c6W + \u22ef+\u03c6mW. Then for k = l + 1, we have Suppose that for The proof is complete.Now we will give the proof of I \u2212 \u03c6\u03bb and I \u2212 \u03c8\u03bb are coprime, by If q = rank\u2061(\u03c6 \u2212 \u03c8); we denote by Iq the identity operator. Then i, j \u2208 {1,2,\u2026, q}, we have \u2211n\u22650\u03b1i\u03c6n(uj)zn = fij(z)/p(z) for some fij(z) \u2208 \u211a[z]. For more details see [Suppose ails see . So by L\u210b is the ring of countable infinite matrix with finite nonzero entries in each column.Suppose \u03b11, \u03b12, \u03b13,\u2026 are three-dimensional row vectors. We see that (\u03c6 \u2212 h)3 = 0. So I \u2212 \u03c8\u03bb and I \u2212 \u03c6\u03bb are left coprime. It is easy to check thatLet"}
{"text": "The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the form  Equation  \u226a y = \u2009\u2009if\u2009\u2009x\u2aafy\u2009\u2009with\u2009\u2009x1 \u2260 y1\u2009\u2009and\u2009\u2009x2 \u2260 y2.Consider a partial ordering\u2009\u2009\u2aaf\u2009\u2009on\u2009\u2009\u211dT on a nonempty set \u211b \u2282 \u211d2 is a continuous function T : \u211b \u2192 \u211b. The map T is monotone if x\u2aafy implies that T(x)\u2aafT(y) for all x, y \u2208 \u211b, and it is strongly monotone on \u211b if x\u227ay implies that T(x) \u226a T(y) for all x, y \u2208 \u211b. The map is strictly monotone on \u211b if x\u227ay implies that T(x)\u227aT(y) for all x, y \u2208 \u211b. Clearly, being related is invariant under iteration of a strongly monotone map.A map North-East ordering (NE) for which the positive cone is the first quadrant; that is, this partial ordering is defined by\u2009\u2009\u2009\u2aafne\u2009\u2009if\u2009\u2009x1 \u2264 x2\u2009\u2009and\u2009\u2009y1 \u2264 y2\u2009\u2009and the South-East (SE) ordering defined as\u2009\u2009\u2009\u2aafse\u2009\u2009if\u2009\u2009x1 \u2264 x2\u2009\u2009and\u2009\u2009y1 \u2265 y2.Throughout this paper, we will use the T on a nonempty set \u211b \u2282 \u211d2 which is monotone with respect to the North-East ordering is called cooperative and a map monotone with respect to the South-East ordering is called competitive.A map T is a differentiable map on a nonempty set \u211b, a sufficient condition for T to be strongly monotone with respect to the SE ordering is that the Jacobian matrix at all points x has the following sign configuration:\u211b is open and convex.If x \u2208 \u211d2, define Q\u2113(x) for \u2113 = 1,\u2026, 4 to be the usual four quadrants based at x and numbered in a counterclockwise direction; for example, Q1(x) = {y \u2208 R2 : x1 \u2264 y1, x2 \u2264 y2}. Basin of attraction of a fixed point T, denoted as x0, y0) for which the sequence of iterates Tn) converges to p. The next five results, from [For ts, from , 14, arets, from , 16.T\u2009\u2009be a competitive map on a rectangular region \u211b \u2282 \u211d2. Let T\u2009\u2009such that\u211b), and\u2009\u2009T\u2009\u2009is strongly competitive on\u2009\u2009\u0394. Suppose that the following statements are true.T\u2009\u2009has a\u2009\u2009C1\u2009\u2009extension to a neighborhood of\u2009\u2009The map\u2009\u2009T at \u03bb, \u03bc such that 0 < |\u03bb | <\u03bc, where |\u03bb | <1, and the eigenspace E\u03bb associated with \u03bb is not a coordinate axis.The Jacobian Then, there exists a curve \ud835\udc9e \u2282 \u211b through \ud835\udc9e is tangential to the eigenspace E\u03bb at \ud835\udc9e is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of \ud835\udc9e in the interior of \u211b are either fixed points or minimal period-two points. In the latter case, the set of endpoints of \ud835\udc9e is a minimal period-two orbit of T.Let\u2009\u2009\ud835\udc9e are boundary points of \u211b is of interest. The following result gives a sufficient condition for this case.We will see in \ud835\udc9e of \u211b, it is sufficient that at least one of the following conditions is satisfied.T\u2009\u2009has no fixed points nor periodic points of minimal period-two in\u2009\u2009\u0394.The map\u2009\u2009T\u2009\u2009has no fixed points in\u2009\u2009\u0394,x \u2208 \u0394.The map\u2009\u2009T\u2009\u2009has no points of minimal period-two in\u2009\u2009\u0394,x \u2208 \u0394.The map\u2009\u2009For the curve \u03bb | <1. This follows from a change of variables [For maps that are strongly competitive near the fixed point, hypothesis (b) of ariables  that allThe next result is useful for determining basins of attraction of fixed points of competitive maps.\ud835\udc9e be the curve whose existence is guaranteed by \ud835\udc9e belong to \u2202\u211b, then \ud835\udc9e separates \u211b into two connected components, namely,\ud835\udcb2\u2212\u2009\u2009is invariant, andn \u2192 \u221e\u2009\u2009for every\u2009\u2009x \u2208 \ud835\udcb2\u2212.\u2009\u2009\ud835\udcb2+\u2009\u2009is invariant, andn \u2192 \u221e\u2009\u2009for every\u2009\u2009x \u2208 \ud835\udcb2+.\u2009\u2009(B) If, in addition to the hypotheses of part (A),\u211b and\u2009\u2009T\u2009\u2009is\u2009\u2009C2\u2009\u2009and strongly competitive in a neighborhood ofT\u2009\u2009has no periodic points in the boundary of(iii)x \u2208 \ud835\udcb2\u2212, there exists\u2009\u2009n0 \u2208 \u2115such thatn \u2265 n0.For every\u2009\u2009(iv)x \u2208 \ud835\udcb2+, there exists\u2009\u2009n0 \u2208 \u2115such thatn \u2265 n0.For every\u2009\u2009(A) Assume the hypotheses of T\u2009\u2009is a map on a set \u211b and ifT, the stable setIf\u2009\u2009T\u2009\u2009is noninvertible, the set\u211b, the setsWhen\u2009\u2009\u03bc > 1 and that the eigenspace E\u03bc associated with \u03bc is not a coordinate axis. If the curve \ud835\udc9e of \u211b, then \ud835\udc9e is the stable set \u211b that is tangential to E\u03bc at \u211b are fixed points of T.In addition to the hypotheses of part (B) of f\u2009\u2009is strongly decreasing in the first argument and strongly increasing in the second argument if it is differentiable and has first partial derivative\u2009\u2009D1f\u2009\u2009negative and first partial derivative\u2009\u2009D2f\u2009\u2009positive in a considered set. The connection between the theory of monotone maps and the asymptotic behavior of  = ). The second iterate\u2009\u2009T2\u2009\u2009is given byI \u00d7 I\u2009\u2009(see [Set\u2009\u2009n\u2009\u2009in  to obtai I\u2009\u2009(see ).\u03bb, \u03bc\u2009\u2009which satisfy\u2009\u2009\u03bb < 0 < \u03bc\u2009\u2009and | \u03bb | <\u03bc, whenever\u2009\u2009f\u2009\u2009is strictly decreasing in first and increasing in second variable. Thus, the applicability of Theorems The characteristic equation of  at an eqThere are several global attractivity results for . Some off : I \u00d7 I \u2192 I\u2009\u2009is a continuous function and\u2009\u2009f\u2009\u2009is decreasing in the first argument and increasing in the second argument. Assume that\u03c6, \u03c8)\u2009\u2009and\u2009\u2009\u2009\u2009are minimal period-two solutions which are saddle points such that\ud835\udcb2s) and \ud835\udcb2s). More precisely,\u212c) = \ud835\udcb2s) and \u212c) = \ud835\udcb2s) are exactly the global stable sets of  and .Consider  where\u2009\u2009fx\u22121, x0) \u2208 \ud835\udcb2+)\u2009\u2009or\u2009\u2009 \u2208 \ud835\udcb2\u2212), then\u2009\u2009Tn)\u2009\u2009converges to the other equilibrium point or to the other minimal period-two solutions or to the boundary of the region\u2009\u2009I \u00d7 I.If\u2009\u2009\u2009\u2009is decreasing in the first variable and increasing in the second variable. By xn}n=\u22121\u221e\u2009\u2009of =It is clear that  has a una + b \u2212 c < 0, then equilibrium solutionIf\u2009\u20093a + b \u2212 c > 0, then equilibrium solutionIf\u2009\u20093a + b \u2212 c = 0, then equilibrium solution\u03bb1 = \u22121\u2009\u2009and\u2009\u2009\u03bb2 = 1/2).If\u2009\u20093Equation  has a unD = (2a + b)(6a + 5b + 4c)\u2009\u2009. It is clear that\u2009\u2009\u03bb\u2212 < 0\u2009\u2009and\u2009\u2009\u03bb+ > 0. Now, we prove that\u2009\u2009\u03bb+ \u2208 \u2009\u2009and(i)(ii)also,(iii)\u03bb\u2212 > \u22121\u21d43a + b \u2212 c < 0\u2009\u2009and\u2009\u2009\u03bb\u2212 = \u22121\u21d43a + b \u2212 c = 0.\u2009\u2009Also, if\u2009\u20093a + b \u2212 c = 0, then\u2009\u2009c = 3a + b, and we have\u2009\u2009\u03bb+ = 1/2.By , a lineaIn this section, we present results for the existence of minimal period-two solutions of .a, b, and\u2009\u2009c.(a) Equation  has the a + b \u2212 c < 0, then ((b) If 3 0, then  has the \u03c6, \u03c8, \u03c6, \u03c8,\u2026}\u2009\u2009of  follows from  = u2/(av2 + buv + cu2).\u2009\u2009The second iteration of the map\u2009\u2009T\u2009\u2009isT2\u2009\u2009is competitive by T2\u2009\u2009isThe map\u2009\u2009e system  is of thc),\u2009\u2009,\u2009\u2009, and\u2009\u2009\u2009\u2009of  The minimal period-two pointsa < c and 3a + b \u2212 c < 0, then the minimal period-two points\u03c6\u2009\u2009and\u2009\u2009\u03c8\u2009\u2009satisfy ((ii) If \u2009satisfy , are sadF = 0, for periodic point\u2009\u2009, we have\u03bb1,2 = 0, which implies that\u2009\u2009\u2009\u2009is locally asymptotically stable.(i) Since\u2009\u2009F = 1/c, for periodic solution\u2009\u2009, we have\u03bb1,2 = 0, which implies that\u2009\u2009\u2009\u2009is locally asymptotically stable.Similarly, since\u2009\u2009(ii) By , we haveF=\u03c6,\u03c8)=\u03c6,\u2202F\u2202u(\u03c6\u03c8)T2\u2009\u2009at the point\u2009\u2009\u2009\u2009is of the formNow, we obtain that Jacobian matrix of the map\u2009\u2009The corresponding characteristic equation isNotice that(i)m2 + n2 \u2265 2mn.\u2009\u2009 Consider that(ii)Notice thatWe need to show thatThis implies thatIn this section, we present global dynamics results for .T = ,\u2009\u2009u > 0, and\u2009\u2009T = ,\u2009\u2009v > 0.Notice that\u2009\u2009a + b \u2212 c < 0, then , , which is a saddle point and has the minimal period-two solution ,  which is locally asymptotically stable. The basin of attraction \ud835\udcb2s) and \ud835\udcb2s). The basins of attraction \u212c) = \ud835\udcb2s) and \u212c) = \ud835\udcb2s) are exactly the global stable sets of  and . Furthermore, the basin of attraction of the minimal period-two solution ,  is the union of the regions above \ud835\udcb2s) and below \ud835\udcb2s) in SE ordering; that is,x\u22121, x0) \u2208 \ud835\udcb2\u2212), then\u2009\u2009lim\u2061n\u2192\u221exn2 = 1/c\u2009\u2009and\u2009\u2009lim\u2061n\u2192\u221exn+12 = 0;if\u2009\u2009 \u2208 \ud835\udcb2+, then\u2009\u2009lim\u2061n\u2192\u221exn2 = 0\u2009\u2009and\u2009\u2009lim\u2061n\u2192\u221exn+12 = 1/c.if\u2009\u2009 and (T2\u2009\u2009as well. Equation (Using assumption\u2009\u20093< 0) and , it is end x\u22121, x0) \u2208 \ud835\udcb2\u2212), thenif\u2009\u2009((ii)ifThe conclusion follows from (i)Tn2) \u2192 \u2009\u2009and\u2009\u2009Tn+12) \u2192 , that is,if(ii)x\u22121, x0) \u2208 \ud835\udcb2+, then\u2009\u2009Tn+12) \u2192 \u2009\u2009and\u2009\u2009Tn2) \u2192 , that is,if\u2009\u2009,  which is locally asymptotically stable. There exists a set \ud835\udc9e which is an invariant subset of the basin of attraction of \ud835\udc9e is a graph of a strictly increasing continuous function of the first variable on an interval and separates \u211b\u2216, where \u211b = [0, \u221e) \u00d7 [0, \u221e), into two connected and invariant components n\u2192\u221exn2 = 1/c\u2009\u2009and\u2009\u2009lim\u2061n\u2192\u221exn+12 = 0;ifn\u2192\u221exn2 = 0\u2009\u2009and\u2009\u2009lim\u2061n\u2192\u221exn+12 = 1/c.ifIf 3 0, then  has a unT2\u2009\u2009as well. The existence of the set \ud835\udc9e with stated properties follows from (i)if(ii)ifConsequently,(i)Tn2) \u2192 \u2009\u2009and\u2009\u2009Tn+12) \u2192 ; that is,if(ii)Tn+12) \u2192 \u2009\u2009and\u2009\u2009Tn2) \u2192 ; that is,if,  which are locally asymptotically stable points. There exists a set \ud835\udc9e which is an invariant subset of the basin of attraction of \ud835\udc9e is a graph of a strictly increasing continuous function of the first variable on an interval and separates \u211b\u2216, where \u211b = [0, \u221e) \u00d7 [0, \u221e), into two connected and invariant components n\u2192\u221exn2 = 1/c\u2009\u2009and\u2009\u2009lim\u2061n\u2192\u221exn+12 = 0;ifn\u2192\u221exn2 = 0\u2009\u2009and\u2009\u2009lim\u2061n\u2192\u221exn+12 = 1/c.ifIf 3 0, then  has a unT\u2009\u2009at the equilibrium point\u03bb1 = \u22121\u2009\u2009and\u2009\u2009\u03bb2 = 1/2, which means that\u2009\u2009\u03bc1 = \u03bb12 = 1\u2009\u2009and\u2009\u2009\u03bc2 = \u03bb22 = 1/4\u2009\u2009are the eigenvalues of the map\u2009\u2009T2.\u2009\u2009Using (Q = (2a + b)/(a + b + c).\u2009\u2009A straightforward calculation yields that the eigenvector corresponding to the eigenvalue\u2009\u2009\u03bc2 = 1/4\u2009\u2009is of the formIn view of .\u2009\u2009Using , 52),  \u00d7 . As a consequence of this and using  \u2192 \u2009\u2009and\u2009\u2009 \u2192 ;ifun2, vn2) \u2192 \u2009\u2009and\u2009\u2009 \u2192 .ifWe see that eigenvector\u2009\u2009nd using , we have(i)Tn2) \u2192 \u2009\u2009and\u2009\u2009Tn+12) \u2192 ; that is,if(ii)Tn+12) \u2192 \u2009\u2009and\u2009\u2009Tn2) \u2192 ; that is,if.As one may notice from the figures all stable manifolds of either saddle point equilibrium points or saddle period-two solutions are asymptotic to the origin, which is the point where  is not d"}
{"text": "L2-error bound of finite element accuracy and of second order in time. Numerical examples are included to confirm our theoretical analysis.We analyze a fully discrete leapfrog/Galerkin finiteelement method for the numerical solution of the space fractional order  diffusion equation. The generalized fractional derivative spaces aredefined in a bounded interval. And some related properties are further discussed for thefollowing finite element analysis. Then the fractional diffusion equationis discretized in space by the finite element method and in time by the explicitleapfrog scheme. For the resulting fully discrete, conditionally stable scheme,we prove an  Fractional calculus and fractional partial differential equations (FPDEs) have many applications in various aspects such as in viscoelastic mechanics, power-law phenomenon in fluid and complex network, allometric scaling laws in biology and ecology, colored noise, electrode-electrolyte polarization, dielectric polarization, boundary layer effects in ducts, electromagnetic waves, quantitative finance, quantum evolution of complex systems, and fractional kinetics . And a lGenerally speaking, the finite difference method and the finite element method are the two main means to solve FPDEs. Recently, some typical fractional difference methods have been utilized to solve FPDEs numerically \u20134. On thIn this paper, we use the explicit leapfrog difference/Galerkin finite element mixed method to numerically solve the space fractional diffusion equation in order to get a higher convergence order.RLDa,x\u03b12 is commonly referred to the left (right) sided L\u00e9vy stable distribution, where the underlying stochastic process is L\u00e9vy \u03b1-stable flights; see , time T > 0. Here the spatial fractional differential operator \u0394\u03b1 is denoted by \u03ba1 \u00b7 RLDa,x\u03b12 + \u03ba2 \u00b7 RLDx,b\u03b12, where 0 \u2264 \u03ba1, \u03ba2 \u2264 1, and \u03ba1 + \u03ba2 = 1. When \u03b1 = 1, the problem models a Brownian diffusion process. And f is a source term, \u03bb is a positive constant.Here, we mainly focus on constructing and analyzing a kind of efficient numerical schemes for approximately solving space fractional diffusion equation. The considered problem reads as follows: for 1/2 < The rest of this paper is constructed as follows. In L2 sense to the Lp sense.In this section, we first give the definition of fractional derivatives. There are several definitions for the fractional derivatives, but Riemann-Liouville derivative is one of the most often used fractional derivatives, which is a reasonable generalization of the classical derivative , 19\u201322. \u03b1th order left and right Riemann-Liouville integrals of function u(x) are defined in a finite interval  as follows:\u03b1 > 0.The \u03b1th order left and right Riemann-Liouville derivatives of function u(x) defined in a finite interval  are given asn \u2212 1 < \u03b1 < n \u2208 Z+. Obviously, they are the integer derivatives of the left and right fractional integrals, respectively.The Now, we give some lemmas and corollaries which are necessary to define the generalized fractional derivative spaces.\u03a9 =  be bounded and \u03b1 > 0. Then u \u2208 Lp(\u03a9) satisfiesLet p \u2260 1, q \u2260 1 in the case 1/p + 1/q = 1 + \u03b1.The relationu(x) \u2208 RLDa,x\u03b1\u2212(Lp(\u03a9)), v(x) \u2208 \u2009RLDx,b\u03b1\u2212(Lq(\u03a9)), 1/p + 1/q \u2264 1 + \u03b1, where the function space RLDa,x\u03b1\u2212(Lp(\u03a9)) = {f(x) | f(x) = RLDa,x\u03b1\u2212\u03d5(x), \u03d5(x) \u2208 Lp(\u03a9)}, RLDx,b\u03b1\u2212(Lq(\u03a9)) = {f(x) | f(x) = RLDx,b\u03b1\u2212\u03c8(x), \u03c8(x) \u2208 Lq(\u03a9)}.The formulau(x) \u2208 RLDa,x\u03b1\u22122(Lp(\u03a9)), v(x) \u2208 RLDx,b\u03b1\u2212(Lq(\u03a9)), 1/p + 1/q \u2264 1 + \u03b1.One can further give the following corollary:u(x) \u2208 RLDa,x\u03b1\u22122(Lp(\u03a9)) implies u(x) \u2208 RLDa,x\u03b1\u2212(Lp(\u03a9)) one can prove that by using Note that the above assumption u(x) \u2208 RLDx,b\u03b1\u22122(Lp(\u03a9)), v(x) \u2208 RLDa,x\u03b1\u2212(Lq(\u03a9)), 1/p + 1/q \u2264 1 + \u03b1.ConsiderRLDa,x\u03b1\u2212(Lp(\u03a9)), we can get that if u(x) \u2208 RLDa,x\u03b1\u2212(Lp(\u03a9)), then u(x) = RLDa,x\u03b1\u2212\u03d5(x), and RLDa,x\u03b1u(x) = \u03d5(x), where \u03d5(x) \u2208 Lp(\u03a9), such that u \u2208 Lp(\u03a9), which is obtained by RLDa,x\u03b1u(x) \u2208 Lp(\u03a9) naturally holds. So, by the above idea, we define the following fractional derivative spaces from the L2 sense to the Lp sense, which will be proved to be equivalent with the fractional Sobolev spaces under some certain conditions.Note that, from the definition of the function space WL\u03b1,p) fractional derivative space and the right  fractional derivative space in a bounded interval \u03a9 =  as follows correspondingly, where 1 < p < +\u221e:Define the following norms of the left (with symbol HS\u03b1) in a bounded interval \u03a9 =  in the L2 senseDefine the symmetric fractional derivative space , WR,0\u03b1,p(\u03a9), and HS,0\u03b1(\u03a9) as the closures of C0\u221e(\u03a9) under their respective norms.Define the spaces L2 sense.From , we can WL,0\u03b1,2(\u03a9), WR,0\u03b1,2(\u03a9), HS,0\u03b1(\u03a9), and H0\u03b1(\u03a9) are equal to equivalent seminorms and norms, where H\u03b1(\u03a9) is the fractional Sobolev space in terms of the Fourier transform.The spaces H0\u03b1 when p = 2, to denote the fractional derivative space equipped with the norm ||\u00b7||\u03b1 which can be any one of (H\u03b1\u2212(\u03a9) is denoted as the dual space of H0\u03b1(\u03a9), with norm ||\u00b7||\u03b1\u2212.Therefore, in this paper we always use y one of , 15), a, aH0\u03b1 y one of , and H\u2212Lp sense.Moreover, we can present some new properties about norms for the above left and right fractional derivative spaces in the \u03b1 > 0 and \u03a9 =  \u2282 R be bounded. Then the following mapping properties hold:RLDa,x\u03b1\u2212u(x) : Lp(\u03a9) \u2192 Lp(\u03a9) is a bounded linear operator;\u2009\u2009RLDx,b\u03b1\u2212u(x) : Lp(\u03a9) \u2192 Lp(\u03a9) is a bounded linear operator;\u2009\u2009RLDa,x\u03b1u(x) : WL\u03b1,p(\u03a9) \u2192 Lp(\u03a9) is a bounded linear operator;\u2009\u2009RLDx,b\u03b1u(x) : WR\u03b1,p(\u03a9) \u2192 Lp(\u03a9) is a bounded linear operator;\u2009\u2009RLDa,x\u03b1\u2212u(x) : Lp(\u03a9) \u2192 WL\u03b1,p(\u03a9) is a bounded linear operator;\u2009\u2009RLDx,b\u03b1\u2212u(x) : Lp(\u03a9) \u2192 WR\u03b1,p(\u03a9) is a bounded linear operator.\u2009\u2009Let Properties (1) and (2) follow directly from WL\u03b1,p(\u03a9) and WR\u03b1,p(\u03a9) asProperty (3) follows directly from the definition of WL\u03b1,p(\u03a9) and the semigroup property of fractional operator,ck, k = 0,\u2026, [\u03b1] such thatProperty (5) follows from the definition of p < \u221e. And if 1 \u2264 p \u2264 q < \u221e, one hasConsiderLp(\u03a9).It is obviously true by using the norms of fractional derivative spaces and imbedding theorems for \u03a9 =  \u2282 R be bounded. Then for u \u2208 WL,0\u03b1,p(\u03a9), one hass < \u03b1, one hasLet u \u2208 WL,0\u03b1,p(\u03a9) by using Lemmas If erefore,  is true.erefore,  holds.Sh denote a uniform partition on \u03a9, with grid parameter h. For k \u2208 N, let Pk(\u03a9) denote the space of polynomials on \u03a9 with degree not greater than k. Then we define Xh as the finite element space on Sh with the basis of the piecewise polynomials of order k \u2208 Z+; that is,D is the unit of Sh.In this section, we firstly give a fully discrete scheme, where we use the leapfrog difference method in the temporal direction and the finite element method in the spatial direction and then analyze the error estimate. Let u \u2208 Hk+1(\u03a9), 0 \u2264 \u03bc \u2264 k + 1, there exists v \u2208 Xh such thatThe following property of finite element spaces is necessary for our subsequent analysis : for u \u2208t, H and an, bn, cn, \u03b3n (for integer n \u2265 0) be nonnegative numbers such thatN \u2265 0. Suppose that \u0394t\u03b3n < 1 for all n, and set \u03c3n = (1 \u2212 \u0394t\u03b3n)\u22121; thenN \u2265 0.Let \u0394t denote the step size for t so that tn = n\u0394t, n = 1,2,\u2026, N \u2212 1. For notational convenience, we denote un : = u andIn the following, we give the fully discrete scheme of . Let \u0394t uhn of (t = tn of the following fully discrete scheme:L2 inner product and (\u0394\u03b1(\u03bb \u00b7 uhn), v) = \u03ba1 \u00b7 , RLDx,b\u03b1v) + \u03ba2 \u00b7 , RLDa,x\u03b1v). For brevity, we always use (\u0394\u03b1(\u03bb \u00b7 uhn), v) instead of the right hand side of this equation.Let uhn of  be the ft > 0, there exists a unique solution uhn \u2208 Xh satisfying /2\u0394t \u2212 (\u0394\u03b1(\u03bb \u00b7 uhn), uhn) is positive, which is one of the sufficient conditions for the existence and uniqueness of uhn.Firstly, we prove that /2\u0394t \u2212 (\u0394\u03b1(\u03bb \u00b7 uhn), uhn). Hence, by using the Lax-Milgram theorem, we have that ), ut \u2208 L2), and uttt \u2208 L2), with u0 \u2208 Hk+1(\u03a9). uhn is the solution of  as \u0394t, h \u2192 0. And the approximation solution uh satisfies the following error estimates:Assume that  has a soution of , and uhximation  is convet = tn,\u2009\u2009n = 1,2,\u2026, N \u2212 1. Let un = u represent the solution of 42), we fun also satisfies\u03f5n+1 = \u039bn+1 + En+1, v = En+1 + En\u22121 into . For the second term of the right hand side, one hast, by using the Cauchy-Schwarz inequality, we obtainn = 1 to N \u2212 1, one hasWe now estimate each term in An+1 is positive and comparable to ||En||2 + ||En+1||2. To this end, we use the inverse inequality ||v||\u03b1 \u2264 C12h\u03b1\u2212||v||, v \u2208 Xh, and this yieldst \u00b7 h\u03b1\u22122 is sufficiently small such that C12\u0394t \u00b7 h\u03b1\u22122 \u2264 C13 \u2264 1, we getWe now show that, under our stability assumption , An+1 ian = ||EN||2, bn = ||En||\u03b12, H = C154||uttt||0,02).Therefore, we obtainG = hk+22||ut||k+10,2 + hk+2\u22122\u03b12||u||k+10,2 + (\u0394t)4||uttt||0,02 and using the condition  Let\u03b1 = 0.6, \u03ba1 = \u03ba2 = 1/2, \u03bb = 1, \u03a9 = , T = 1, and\u2009\u2009f is numerically obtained.(ii) LetThe experiential error results and convergence rates are displayed in t \u00b7 h\u03b1\u22122 be sufficiently small, the error analysis for the fully discrete scheme is discussed, which is an L2-error bound of finite element accuracy and of second order in time. Numerical examples are given to demonstrate the efficiency of the theoretical results.In this paper, we study the finite element method for fractional diffusion equation. We use the simple, second order accurate explicit scheme, leapfrog difference method in time, and the finite element method in space. Under the suitably accurate initial conditions and the stability requirement that \u0394"}
{"text": "We obtain some equivalent conditions of (strictly) pseudoconvex and quasiconvex fuzzy mappings. These results will be useful to present some characterizations of solutions for fuzzy mathematical programming. The occurrence of randomness and fuzziness in the real world is inevitable owing to some unexpected situations. In , Zadeh iNanda and Kar  proposedMotivated by the earlier works of Panigrahi et al. , KaramarIn this section, we quote some preliminary notations and definitions.\u03bc : \u211d \u2192  with the following properties:\u03bc\u2009\u2009is normal; that is, [\u03bc]1 = {x \u2208 \u211d : \u03bc(x) = 1} \u2260 \u2205,\u03bc is upper semicontinuous,\u03bc is convex; that is, \u03bc(\u03bbx + (1 \u2212 \u03bb)y) \u2265 min{\u03bc(x), \u03bc(y)} for all x, y \u2208 \u211d, \u03bb \u2208 ,\u03bc, supp(\u03bc) = {x \u2208 \u211d : \u03bc(x) > 0} and its closure cl(supp\u03bc) is compact.the support of Let \u211d be the set of all real numbers. A fuzzy number is a mapping \u2131 be the set of all fuzzy numbers on \u211d. The \u03b1-level set of a fuzzy number \u03bc \u2208 \u2131, 0 \u2264 \u03b1 \u2264 1, denoted by [\u03bc]\u03b1, is defined asLet \u03b1-level set of a fuzzy number is a closed and bounded interval . \u03bc\u2217(\u03b1) denotes the left-hand end point of [\u03bc]\u03b1 and \u03bc*(\u03b1) denotes the right-hand end point of [\u03bc]\u03b1. Also any m \u2208 \u211d can be regarded as a fuzzy number t = 0 and otherwise \u03bc can be identified by parameterized triples {(\u03bc\u2217(\u03b1), \u03bc*(\u03b1), \u03b1) : \u03b1 \u2208 }.It is clear that the \u03bc and \u03c5 parameterized byk, we define the addition k\u03bc as follows:r,\u03bc to be the fuzzy number \u2212\u03bc satisfying (\u2212\u03bc)(x) = \u03bc(\u2212x). In other words, if \u03bc is represented by the parametric form {(\u03bc\u2217(\u03b1), \u03bc*(\u03b1), \u03b1) : 0 \u2264 \u03b1 \u2264 1}, then \u2212\u03bc is represented by the corresponding parametric form {(\u2212\u03bc*(\u03b1), \u2212\u03bc\u2217(\u03b1), \u03b1) : 0 \u2264 \u03b1 \u2264 1}. We represent a fuzzy number \u03bc as .For fuzzy numbers \u03bc =  is said to be a triangular fuzzy number if \u03bc\u2217(1) = \u03bc*(1). Moreover, if \u03bc\u2217(\u03b1) and \u03bc*(\u03b1) are linear, then we say that\u2009\u2009\u03bc\u2009\u2009is a linear triangular fuzzy number. We denote \u2329\u03bc\u2217(0); \u03bc*(1); \u03bc*(0)\u232a.A fuzzy number u, v \u2208 \u2131, we say that u\u227cv if for each \u03b1 \u2208 , u\u2217(\u03b1) \u2264 v\u2217(\u03b1), u*(\u03b1) \u2264 v*(\u03b1). If u\u227cv, v\u227cu, then u = v. We say that u\u227av, if u\u227cv and there exists \u03b10 \u2208  such that u\u2217(\u03b10) < v\u2217(\u03b10) or u*(\u03b10) < v*(\u03b10). For u, v \u2208 \u2131, if either u\u227cv or v\u227cu, then we say that u and v are comparable, otherwise noncomparable.For F : K\u2286\u211dn \u2192 \u2131 is said to be a fuzzy mapping. For any \u03b1 \u2208  and for any x \u2208 K, we denote F(x) = .A mapping F : K\u2286\u211dn \u2192 \u2131 be a fuzzy mapping. Then, F is said to be comparable if for each pair x \u2260 y \u2208 K, F(x) and F(y) are comparable. Otherwise, F is said to be noncomparable.Let K\u2286\u211dn be an open set and assume that F : K \u2192 \u2131 is a fuzzy mapping. Let x =  \u2208 K and let Dxi, i = 1,2,\u2026, n stand for the partial differentiation with respect to the ith variable xi. Assume that for all \u03b1 \u2208 , F\u2217 and F* have continuous partial derivatives. Definei = 1,2,\u2026, n, DxiF(x)[\u03b1] defines the \u03b1-cut of a fuzzy number, then we say that F is differentiable at x, and we denoteF at x. A fuzzy mapping F is said to be differentiable at x if F\u2217 and F* for each \u03b1 \u2208  are differentiable at x.Let K\u2286\u211dn be a nonempty open convex set and let F : K \u2192 \u2131 be a differentiable fuzzy mapping. F is said to be pseudoconvex if for each x, y \u2208 K, F(x)\u227cF(y).Let K\u2286\u211dn be a nonempty convex set and let F : K \u2192 \u2131 be a fuzzy mapping. F is said to be quasiconvex if for each x, y \u2208 K and for each \u03bb \u2208 , the following implications hold:F(x) and F(y) are comparable.Let K\u2286\u211dn be a nonempty open convex set and let F : K \u2192 \u2131 be a differentiable fuzzy mapping. F is said to be strictly pseudoconvex if for each x, y \u2208 K, F(x)\u227aF(y).Let K\u2286\u211dn be a nonempty open convex set, F : K\u2286\u211dn \u2192 \u2131 be a differentiable fuzzy mapping, and F be comparable.In the following sections, we always assume that In this section, we establish the equivalent conditions of pseudoconvex and strictly pseudoconvex fuzzy mappings. We first give some lemmas which will be used in the sequel.F is a pseudoconvex fuzzy mapping. Then F is a quasiconvex fuzzy mapping.Assume that F is a quasiconvex fuzzy mapping if and only if for each x, y \u2208 K, F(x)\u227cF(y) implies that F is a pseudoconvex fuzzy mapping if and only if for each x, y \u2208 K, F is a pseudoconvex fuzzy mapping. Let x, y \u2208 K be such thatF and (Suppose that uch that0~\u227c\u2207~F(x)f F and (F(x)\u227cF(y)ary that\u2207~F(y)T\u227cF(y). Assume the contrary, that is, F(y)\u227aF(x). Hence, for each \u03b1 \u2208 ,\u03b10 \u2208  such that\u03bb \u2208 . From < we haveF\u2217\u2212uch that0~\u227c\u2207~F(x)). From ary that\u2207~F(y)T(y we haveF(x)\u227aF(y)x, y \u2208 K, x \u2260 y such thatF(x)\u227aF(y). Assume to the contrary that\u03b1 \u2208 ,\u03bb \u2208 .Conversely, let \u03b1 \u2208 ,From  and 31)31), we hnd, from , it foll0\u2264\u2207F\u2217\u227cF(x) is not possible. Otherwise, it will imply thatSuppose that uch that0~\u227a\u2207~F(x)rding to F(x)\u227aF(y)F is not a quasiconvex fuzzy mapping. Then, there exists x, y \u2208 K, \u03bb \u2208  such that F(x)\u227cF(y) and \u03b1 \u2208  such thatz1, z2 such thatx, y \u2208 K,Conversely, assume that ume thatF\u2217\u2264Fuch thatF\u2217\u2212),wherez1=x+\u03bb1,0<\u2207F\u2217(z2,"}
{"text": "Moreover, we obtain a strong convergence theorem for the sequence generated by this process in the framework Banach spaces. The results presented in this paper improve and generalize some well-known results in the literature.In this paper, we construct the hybrid block iterative algorithm for finding a common element of the set of common fixed points of an infinite family of closed and uniformly quasi -  In the theory of variational inequalities, variational inclusions, and equilibrium problems, the development of an efficient and implementable iterative algorithm is interesting and important. Equilibrium theory represents an important area of mathematical sciences such as optimization, operations research, game theory, financial mathematics and mechanics. Equilibrium problems include variational inequalities, optimization problems, Nash equilibria problems, saddle point problems, fixed point problems, and complementarity problems as special cases.C be a nonempty closed convex subset of a real Banach space E with \u2225\u00b7\u2225 and E\u2217 the dual space of E and A : C\u2192E\u2217 be an operator. The classical variational inequality problem for an operator A is to find x\u2217\u2009\u2208\u2009C such that Let VI. Recall that let A : C\u2009\u2192\u2009E\u2217 be a mapping. Then A is calledThe set of solution of (1.1) is denoted by monotone if (i) \u03b1-inverse-strongly monotone if there exists a constant \u03b1\u2009>\u20090 such that (ii) x\u2217\u2009\u2208\u2009E satisfying Ax\u2217\u2009=\u20090.Such a problem is connected with the convex minimization problem, the complementary problem, the problem of finding a point fi}i\u2208\u0393 : \u03c6i}i\u2208\u0393 : \u0393 is an arbitrary index set. The system of equilibrium problems, is to find x\u2009\u2208\u2009C such that Let {SEP. If \u0393 is a singleton, then problem (1.2) reduces to the equilibrium problem, is to find x\u2009\u2208\u2009C such that The set of solution of (1.2) is denoted by EP(f). The above formulation (1.3) was shown in (Blum and Oettli EP(f). In other words, the EP(f) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, etc. In the last two decades, many papers have appeared in the literature on the existence of solutions of EP(f); see, for example (Blum and Oettli EP(f); see, for example, (Blum and Oettli The set of solution of (1.3) is denoted by p\u2009>\u20091, the generalized duality mappingJp : For each x\u2009\u2208\u2009E. In particular, J\u2009=\u2009J2 is called the normalized duality mapping. If E is a Hilbert space, then J\u2009=\u2009I, where I is the identity mapping. Consider the functional defined by  for all C is a nonempty closed convex subset of a Hilbert space H and PC\u2009:\u2009H\u2009\u2192\u2009C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. It is obvious from the definition of function \u03d5 that As well know that if E is a Hilbert space, then \u03d5\u2009=\u2009\u2225x\u2009-\u2009y\u22252, for all x, y\u2009\u2208\u2009E. On the author hand, the generalized projection , that is, If on Alber  \u03c0C : E\u2009\u2192\u03c0C follows from the properties of the functional \u03d5\u2009=\u20090 if and only if x\u2009=\u2009y. It is sufficient to show that if \u03d5\u2009=\u20090 then x\u2009=\u2009y. From (1.4), we have \u2225x\u2225\u2009=\u2009\u2225y\u2225. This implies that \u2329x, Jy\u232a\u2009=\u2009\u2225x\u22252\u2009=\u2009\u2225Jy\u22252. From the definition of J, one has Jx\u2009=\u2009Jy. Therefore, we have x\u2009=\u2009y; see  the set of fixed points of T; that is, F(T)\u2009=\u2009{x\u2009\u2208\u2009C\u2009:\u2009Tx\u2009=\u2009x}. Recall that a point p in C is said to be an asymptotic fixed point of T\u2009\u2264\u2009\u03d5 for all x\u2009\u2208\u2009C and p\u2009\u2208\u2009F(T). The asymptotic behavior of a relatively nonexpansive mapping was studied in \u2009\u2264\u2009\u03d5 for x, y\u2009\u2208\u2009C. T is said to be relatively quasi-nonexpansive if F(T)\u2009\u2260\u2009\u2205 and \u03d5\u2009\u2264\u2009\u03d5 for all x\u2009\u2208\u2009C and p\u2009\u2208\u2009F(T). T is said to be quasi-\u03d5-asymptotically nonexpansive if F(T)\u2009\u2260\u2009\u2205 and there exists a real sequence {kn}\u2009\u2282\u2009; see E is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E.If (ii)E is a reflexive and strictly convex Banach space, then J-1 is norm-weak \u2217-continuous.If (iii)E is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping If (iv)E is uniformly smooth if and only if E\u2217 is uniformly convex.A Banach space (v)E has the Kadec-Klee property, that is, for any sequence {xn}\u2009\u2282\u2009E, if xn\u2225\u2009\u2192\u2009\u2225x\u2225, then xn\u2009\u2192\u2009x.Each uniformly convex Banach space The following basic properties can be found in Cioranescu .(i)If EWe also need the following lemmas for the proof of our main results.If E be a 2-uniformly convex Banach space. Then for all x, y \u2208 E, we have. If where J is the normalized duality mapping of E and 0\u2009<\u2009c\u2009\u2264\u20091.p-uniformly convex constant of E.The best constant IfEbe a p-uniformly convex Banach space and letpbe a given real number withp\u2009\u2265\u20092. Then for allx, y\u2009\u2208\u2009E, jx\u2009\u2208\u2009Jp(x) andjy\u2009\u2208\u2009Jp(y) \u2009\u2192\u20090 and either {xn} or {yn} is bounded, then \u2225xn\u2009-\u2009yn\u2225\u2009\u2192\u20090.(Kamimura and Takahashi ). Let E LetCbe a nonempty closed convex subset of a smooth Banach spaceEand x\u2009\u2208\u2009E. Then x0\u2009=\u2009\u03c0Cxif and only if(Alber ). LetCbeLetEbe a reflexive, strictly convex and smooth Banach space, letCbe a nonempty closed convex subset ofEand letx\u2009\u2208\u2009E. Then\u2009=\u2009\u03d5).for all LetEbe a reflexive, strictly convex smooth Banach space and letVbe as in (2.1). Then(Alber ). LetEbefor allx\u2009\u2208\u2009Eandx\u2217, y\u2217\u2009\u2208\u2009E\u2217.A be an inverse-strongly monotone mapping of C into E\u2217 which is said to be hemicontinuous if for all x, y\u2009\u2208\u2009C, the mapping F of  into E\u2217, defined by F(t)\u2009=\u2009A(tx+(1-t)y), is continuous with respect to the weak \u2217 topology of E\u2217. We define by NC(v)the normal cone for C at a point v\u2009\u2208\u2009C, that is, Let LetCbe a nonempty, closed convex subset of a Banach spaceEandAis a monotone, hemicontinuous operator ofCintoE\u2217. LetB\u2009\u2282\u2009E\u2009\u00d7\u2009E\u2217be an operator defined as follows:(Rockafellar ). LetCbeThenBis maximal monotone andB-10\u2009=\u2009VI.LetEbe a uniformly convex Banach space,r\u2009>\u20090 be a positive number andBr(0) be a closed ball ofE. Then, for any given sequenceand for any given sequenceof positive number withthere exists a continuous, strictly increasing, and convex functiong:, {rj,n} \u2282 for somea,bwith 0\u2009<\u2009a\u2009<\u2009b\u2009<\u2009c2\u03b1/2, whereis the 2-uniformly convexity constant ofE. Iffor alln\u2009\u2265\u20090, lim infn\u2192\u221e\u03b2n(1\u2009-\u2009\u03b2n)\u2009>\u20090 and lim infn\u2192\u221e\u03b1n,0\u03b1n,i\u2009>\u20090 for all i\u2009\u2265\u20091, then {xn} converges strongly top\u2009\u2208\u2009F, wherep\u2009=\u2009\u03c0Fx0.Cn+1 is closed and convex for each n\u2009\u2265\u20090. Clearly C1\u2009=\u2009C is closed and convex. Suppose that Cn is closed and convex for each z\u2009\u2208\u2009Cn, we known that We first show that  is equivalent to Cn+1 is closed and convex. Hence, F\u2009\u2282\u2009Cn for all n\u2009\u2265\u20090. Since by the convexity of \u2225\u2009\u00b7\u2009\u22252, property of \u03d5, Lemma 2.9 and by uniformly quasi- \u03d5-asymptotically nonexpansive of Sn for each q\u2009\u2208\u2009F\u2009\u2282\u2009Cn, we have Next, we show that and It follows from Lemma 2.7, that q\u2009\u2208\u2009VI and A is an \u03b1-inverse-strongly monotone mapping, we have Since Axn\u2225\u2009\u2264\u2009\u2225Axn\u2009-\u2009Aq\u2225, \u2200q\u2009\u2208\u2009VI, we also have By Lemma 2.2 and \u2225Substituting (3.5) and (3.6) into (3.4), we have Substituting (3.7) into (3.3), we also have and substituting (3.8) into (3.2), we obtain q\u2009\u2208\u2009Cn+1 implies that F\u2009\u2282\u2009Cn+1 and hence, F\u2009\u2282\u2009Cn for all n\u2009\u2265\u20090. This implies that the sequence {xn} is well defined. From definition of Cn+1 that Thus, this show that Form Lemma 2.6, it follows that \u03d5} are nondecreasing and bounded. So, we obtain that xn\u2225\u2009-\u2009\u2225x0\u2225)2} is bounded. This implies {xn} is also bounded. We denote By (3.10) and (3.11), then {\u03b8n and (3.12), it follows that Moreover, by the definition of xn} is a Cauchy sequence in C. Since m\u2009>\u2009n, by Lemma 2.6, we have Next, we show that {n\u2192\u221e\u03d5 exists and we taking m, n\u2009\u2192\u2009\u221e then, we get \u03d5\u2009\u2192\u20090. From Lemma 2.4, we have limn\u2192\u221e\u2225xm\u2009-\u2009xn\u2225\u2009=\u20090. Thus {xn} is a Cauchy sequence and by the completeness of E and there exist a point p\u2009\u2208\u2009C such that  Since limJun\u2009-\u2009Jxn\u2225\u2009\u2192\u20090, as n\u2009\u2192\u2009\u221e. By definition of Now, we claim that \u2225 Since Again form Lemma 2.4, that J is uniformly norm-to-norm continuous on bounded subsets of E, we obtain From Cn+1, we have Since  By (3.13) and (3.15), that Applying Lemma 2.4, we have Since  It follows from (3.23) and (3.19), that J is uniformly norm-to-norm continuous on bounded subsets of E, we also have Since Next, we will show that i) We show that ( By (3.13) and (3.15), that Form Lemma 2.4, that J is uniformly norm-to-norm continuous, we obtain Since From (3.45), we note that  and hence From (3.17), (3.24) and J-1 is uniformly norm-to-norm continuous on bounded sets, we have Since Using the triangle inequality, that  From (3.23) and (3.27), we have On the other hand, we observe that \u03b8n\u2009\u2192\u20090, \u2225xn\u2009-\u2009un\u2225\u2009\u2192\u20090 and \u2225Jxn\u2009-\u2009Jun\u2225\u2009\u2192\u20090, that  It follows from From (3.2), (3.3) and (3.7), we compute  and hence \u03bbn} \u2282 for some a, b with 0\u2009<\u2009a\u2009<\u2009b\u2009<\u2009c2\u03b1\u2009/\u20092, liminfn\u2192\u221e(1\u2009-\u2009\u03b2n)\u2009>\u20090 and liminfn\u2192\u221e\u03b1n,0\u03b1n,i\u2009>\u20090, for i\u2009\u2265\u20090 and kn\u2009\u2192\u20091 as n\u2009\u2192\u2009\u221e, we obtain that From (3.29), {From Lemma 2.6, Lemma 2.7 and (3.6), we compute  Applying Lemma 2.4 and (3.31) that and we also obtain i\u2009\u2265\u20091 From Again by the triangle inequality, we get  From (3.28) and (3.34), we have By using the triangle inequality, we have  That is i\u2009\u2265\u20091, Si is uniformly Li-Lipschitz continuous, hence we have. By the assumption that \u2200Si, we have Sip\u2009=\u2009p, for all i\u2009\u2265\u20091. This imply that By (3.23) and (3.36), it follows that ii) We show that j\u2009=\u20091, 2, 3, ..., m, \u03a9n0\u2009=\u2009I, for q\u2009\u2208\u2009F, we observe that , (3.21), By using triangle inequality, we have  From (3.20) and (3.39), we have Again by using triangle inequality, we have  From (3.40),we also have Since J is uniformly norm-to-norm continuous, we obtain rj,n\u2009>\u20090 we have n\u2009\u2192\u2009\u221e, \u2200j\u2009=\u20091, 2, 3, ..., m, and  From  By (A2), that f\u2009\u2264\u20090 for all y\u2009\u2208\u2009C. For 0\u2009<\u2009t\u2009<\u20091, define yt\u2009=\u2009ty\u2009+\u2009(1\u2009-\u2009t)p. Then yt\u2009\u2208\u2009C which imply that fj\u2009\u2264\u20090. From (A1), we obtain that  and fj\u2009\u2265\u20090. From (A3), we have fj\u2009\u2265\u20090 for all y\u2009\u2208\u2009C and j\u2009=\u20091, 2, 3, ..., m. Hence p\u2009\u2208\u2009EP(fj), \u2200j\u2009=\u20091, 2, 3, ..., m. This imply that  Thus iii) We show that xn\u2009\u2192\u2009p\u2009\u2208\u2009VI. Indeed, define B\u2009\u2282\u2009E\u2009\u00d7\u2009E\u2217 by . Let \u2009\u2208\u2009G(B). Since w\u2009\u2208\u2009Bv\u2009=\u2009Av\u2009+\u2009NC(v), we get w\u2009-\u2009Av\u2208NC(v). From vn\u2009\u2208\u2009C, we have By Lemma 2.8, vn\u2009=\u2009\u03c0CJ-1(Jxn\u2009-\u2009\u03bbnAxn). Then by Lemma 2.5, we have On the other hand, since  and thus A is monotone and It follows from (3.43), (3.44) and H\u2009=\u2009supn\u22651\u2225v\u2009-\u2009vn\u2225. Take the limit as n\u2009\u2192\u2009\u221e, (3.32) and (3.33), we obtain \u2329v\u2009-\u2009p, w\u232a\u2009\u2265\u20090. By the maximality of B we have p\u2009\u2208\u2009B-10, that is p\u2009\u2208\u2009VI. where p\u2009=\u2009\u03c0Fx0. From Jx0\u2009-\u2009Jxn, xn\u2009-\u2009z\u232a\u2009\u2265\u20090, \u2200z\u2009\u2208\u2009Cn. Since F\u2009\u2282\u2009Cn, we also have Finally, we show that n\u2009\u2192\u2009\u221e, we obtain  Taking limit p\u2009=\u2009\u03c0Fx0 and xn\u2009\u2192\u2009p as n\u2009\u2192\u2009\u221e. This completes the proof. \u25a1 By Lemma 2.5, we can conclude that Si\u2009=\u2009S for each If LetCbe a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach spaceE. For eachj\u2009=\u20091, 2, ..., mletfjbe a bifunction fromC\u2009\u00d7\u2009Ctowhich satisfies conditions (A1)-(A4). LetAbe an\u03b1-inverse-strongly monotone mapping ofCintoE\u2217satisfying \u2225Ay\u2225\u2009\u2264\u2009\u2225Ay\u2009-\u2009Au\u2225, \u2200y\u2208Candu\u2009\u2208\u2009VI\u2009\u2260\u2009\u2205. LetS : C\u2009\u2192\u2009Cbe a closedL-Lipschitz continuous and quasi-\u03d5-asymptotically nonexpansive mappings with a sequence {kn} \u2282 , {rj,n} \u2282 for somea, bwith 0\u2009<\u2009a\u2009<\u2009b\u2009<\u2009c2\u03b1\u2009/\u20092, where is the 2-uniformly convexity constant ofE. If lim infn\u2192\u221e(1\u2009-\u2009\u03b2n)\u2009>\u20090 and lim infn\u2192\u221e\u03b1n(1\u2009-\u2009\u03b1n)\u2009>\u20090, then {xn} converges strongly top\u2009\u2208\u2009F, wherep\u2009=\u2009\u03c0Fx0.i\u2009=\u20091, 2, we can obtain the following results on a pair of quasi- \u03d5-asymptotically nonexpansive mappings immediately from Theorem 3.1.For a special case that C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. For each j\u2009=\u20091, 2, ..., m let fj be a bifunction from C\u2009\u00d7\u2009C to A be an \u03b1-inverse-strongly monotone mapping of C into E\u2217 satisfying \u2225Ay\u2225\u2009\u2264\u2009\u2225Ay\u2009-\u2009Au\u2225, \u2200y\u2009\u2208\u2009C and u\u2009\u2208\u2009VI\u2009\u2260\u2009\u2205. Let S, T : C\u2009\u2192\u2009C be two closed quasi- \u03d5-asymptotically nonexpansive mappings and LS, LT-Lipschitz continuous, respectively with a sequence {kn}\u2009\u2282\u2009, {rj,n} \u2282 for some a, b with 0\u2009<\u2009a\u2009<\u2009b\u2009<\u2009c2\u03b1\u2009/\u20092, where E. If \u03b1n\u2009+\u2009\u03b2n\u2009+\u2009\u03b3n\u2009=\u20091 for all n\u2009\u2265\u20090 and lim infn\u2192\u221e\u03b1n\u03b2n\u2009>\u20090, lim infn\u2192\u221e\u03b1n\u03b3n\u2009>\u20090, lim infn\u2192\u221e\u03b2n\u03b3n\u2009>\u20090 and lim infn\u2192\u221e\u03b4n(1\u2009-\u2009\u03b4n)\u2009>\u20090, then {xn} converges strongly to p\u2009\u2208\u2009F, where p\u2009=\u2009\u03c0Fx0.where C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. For each j\u2009=\u20091, 2, ..., m let fj be a bifunction from C\u2009\u00d7\u2009C to A be an \u03b1-inverse-strongly monotone mapping of C into E\u2217 satisfying \u2225Ay\u2225\u2009\u2264\u2009\u2225Ay\u2009-\u2009Au\u2225, \u2200y\u2009\u2208\u2009C and u\u2009\u2208\u2009VI\u2009\u2260\u2009\u2205. Let \u03d5- nonexpansive mappings such that x0\u2208E with C1\u2009=\u2009C, we define the sequence {xn} as follows: Let J is the duality mapping on E, {\u03b1n,i} and {\u03b2n} are sequences in , {rj,n} \u2282 for some a, b with 0\u2009<\u2009a\u2009<\u2009b\u2009<\u2009c2\u03b1\u2009/\u20092, where E. If n\u2009\u2265\u20090, lim infn\u2192\u221e(1\u2009-\u2009\u03b2n)\u2009>\u20090 and lim infn\u2192\u221e\u03b1n,0\u03b1n,i\u2009>\u20090 for all i\u2009\u2265\u20091, then {xn} converges strongly to p\u2009\u2208\u2009F, where p\u2009=\u2009\u03c0Fx0.where \u03d5-nonexpansive mappings, it is an infinite family of closed and uniformly quasi- \u03d5-asymptotically nonexpansive mappings with sequence kn\u2009=\u20091. Hence the conditions appearing in Theorem 3.1F is a bounded subset in C and for each i\u2009\u2265\u20091,Si is uniformly Li-Lipschitz continuous are of no use here. By virtue of the closeness of mapping Si for each i\u2009\u2265\u20091, it yields that p\u2009\u2208\u2009F(Si) for each i\u2009\u2265\u20091, that is, Since Let Cbe a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach spaceE. Letfbe a bifunction fromC\u2009\u00d7\u2009Ctosatisfying (A1)- (A4). LetA be an \u03b1-inverse-strongly monotone mapping ofCintoE\u2217satisfying \u2225Ay\u2225\u2009\u2264\u2009\u2225Ay\u2009-\u2009Au\u2225, \u2200y\u2009\u2208\u2009Candu\u2009\u2208\u2009VI\u2009\u2260\u2009\u2205. Letbe a finite family of closed quasi-\u03d5-nonexpansive mappings such thatFor an initial pointx0\u2009\u2208\u2009EwithandC1\u2009=\u2009C, we define the sequence {xn} as follows: for somed\u2009>\u20090 and {\u03bbn} \u2282 for somea, bwith 0\u2009<\u2009a\u2009<\u2009b\u2009<\u2009c2\u03b1\u2009/\u20092, whereis the 2-uniformly convexity constant ofE. If\u03b1i\u2009\u2208\u2009 such thatthen {xn} converges strongly top\u2009\u2208\u2009F, wherep\u2009=\u2009\u03c0Fx0.LetCbe a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach spaceE. Letfbe a bifunction fromC\u2009\u00d7\u2009Ctosatisfying (A1)- (A4). Letbe an infinite family of closed and uniformly quasi-\u03d5-asymptotically nonexpansive mappings with a sequence {kn} \u2282, {rn} \u2282, {rj,n} \u2282 for somea, bwith 0\u2009<\u2009a\u2009<\u2009b\u2009<\u2009\u03b1/2. Iffor alln\u2009\u2265\u20090 and lim infn\u2192\u221e\u03b1n,0\u03b1n,i\u2009>\u20090 for alli\u2009\u2265\u20091, then {xn} converges strongly top\u2009\u2208\u2009F, wherep\u2009=\u2009\u03c0Fx0.\u03d5-asymptotically nonexpansive mappings.Theorem 4.1 improve and extend the Corollary 3.7 in Zegeye  in the aE into E\u2217. Assume that A satisfies the conditions:Next, we consider the problem of finding a zero point of an inverse-strongly monotone operator of A is \u03b1-inverse-strongly monotone,(C1)A-10\u2009=\u2009{u\u2009\u2208\u2009E : Au\u2009=\u20090}\u2009\u2260\u2009\u2205.(C2) LetCbe a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach spaceE. For eachj\u2009=\u20091, 2, ..., mletfjbe a bifunction fromC\u2009\u00d7\u2009Ctowhich satisfies conditions (A1)-(A4). LetAbe an operator ofEintoE\u2217satisfying (C1) and (C2). Letbe an infinite family of closed uniformlyLi-Lipschitz continuous and uniformly quasi-\u03d5-asymptotically nonexpansive mappings with a sequence {kn} \u2282, {rj,n}\u2282 for somea, bwith 0\u2009<\u2009a\u2009<\u2009b\u2009<\u2009c2\u03b1\u2009/\u20092, whereis the 2-uniformly convexity constant ofE. Iffor alln\u2009\u2265\u20090, lim infn\u2009\u2192\u2009\u221e(1\u2009-\u2009\u03b2n)\u2009>\u20090 and lim infn\u2192\u221e\u03b1n,0\u03b1n,i\u2009>\u20090 for alli\u2009\u2265\u20091, then {xn} converges strongly top\u2208F, wherep\u2009=\u2009\u03c0Fx0.C\u2009=\u2009E in Corollary 3.4, we also get \u03c0E\u2009=\u2009I. We also have VI\u2009=\u2009VI\u2009=\u2009{x\u2009\u2208\u2009E : Ax\u2009=\u20090}\u2009\u2260\u2009\u2205 and then the condition \u2225Ay\u2225\u2009\u2264\u2009\u2225Ay\u2009-\u2009Au\u2225 holds for all y\u2009\u2208\u2009E and u\u2009\u2208\u2009A-10. So, we obtain the result. \u25a1Setting K be a nonempty, closed convex cone in E. We define the polarK\u2217 of K as follows: Let A : K\u2009\u2192\u2009E\u2217 is an operator, then an element u\u2009\u2208\u2009K is called a solution of the complementarity problem .The set of solutions of the complementarity problem is denoted by LetKbe a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach spaceE. For eachj\u2009=\u20091, 2, ..., mletfjbe a bifunction fromC\u2009\u00d7\u2009Ctowhich satisfies conditions (A1)-(A4). LetAbe an\u03b1-inverse-strongly monotone mapping ofKintoE\u2217satisfying \u2225Ay\u2225\u2009\u2264\u2009\u2225Ay\u2009-\u2009Au\u2225, \u2200y\u2009\u2208\u2009Kandu\u2009\u2208\u2009CP\u2009\u2260\u2009\u2205. Letbe an infinite family of closed uniformlyLi-Lipschitz continuous and uniformly quasi-\u03d5-asymptotically nonexpansive mappings with a sequence {kn} \u2282, {rj,n} \u2282 for somea,bwith 0\u2009<\u2009a\u2009<\u2009b\u2009<\u2009c2\u03b1\u2009/\u20092, whereis the 2-uniformly convexity constant ofE. Iffor alln\u2009\u2265\u20090, lim infn\u2009\u2192\u2009\u221e(1\u2009-\u2009\u03b2n)\u2009>\u20090 and lim infn\u2192\u221e\u03b1n,0\u03b1n,i\u2009>\u20090 for all i\u2009\u2265\u20091, then {xn} converges strongly top\u2009\u2208\u2009F, wherep\u2009=\u2009\u03c0Fx0.VI\u2009=\u2009CP. So, we obtain the result. \u25a1As in the proof of Takahashi in (Takahashi"}
{"text": "\u2133 = (Mk). We also examine some topological properties of the resulting sequence spaces.In the present paper, we introduce some sequence spaces using ideal convergence and Musielak-Orlicz function  I-convergence of generalized sequences with respect to Musielak-Orlicz function.The notion of ideal convergence was first introduced by Kostyrko et al.  as a gen\u2110 \u2282 2X of subsets of a nonempty set X is said to be an ideal in X if\u03d5 \u2208 \u2110,A, B \u2208 \u2110 imply A \u222a B \u2208 \u2110,A \u2208 \u2110, B \u2282 A imply B \u2208 \u2110,while an admissible ideal \u2110 of X further satisfies {x} \u2208 \u2110 for each x \u2208 X; see [xn)n\u2208\u2115 in X is said to be I-convergent to x \u2208 X. If for each \u03f5 > 0, the set A(\u03f5) = {n \u2208 \u2115 : ||xn \u2212 x||\u2265\u03f5} belongs to \u2110; see [A family \u2208 X; see . A sequeo \u2110; see . For moro \u2110; see \u201310 and r\u03bb-convergent and \u03bb-bounded sequences as follows.Mursaleen and Noman  introduc\u03bb = (\u03bbk)k=1\u221e be a strictly increasing sequence of positive real numbers tending to infinity; that is,x = (xk) \u2208 w is \u03bb-convergent to the number L, called the \u03bb-limit of x, if \u039bm(x) \u2192 L, as m \u2192 \u221e, whereLet x = (xk) \u2208 w is \u03bb-bounded if sup\u2061m | \u039bm(x)|<\u221e. It is well known [m\u2009xm = a in the ordinary sense of convergence, thenThe sequence ll known  that if m\u039bm(x) = a and hence x = (xk) \u2208 w is \u03bb-convergent to a.This implies thatX be a linear metric space. A function p : X \u2192 \u211d is called paranorm ifp(x) \u2265 0, for all x \u2208 X,p(\u2212x) = p(x), for all x \u2208 X,p(x + y) \u2264 p(x) + p(y), for all x, y \u2208 X,\u03bbn) is a sequence of scalars with \u03bbn \u2192 \u03bb as n \u2192 \u221e and (xn) is a sequence of vectors with p(xn \u2212 x) \u2192 0 as n \u2192 \u221e, then p(\u03bbnxn \u2212 \u03bbx) \u2192 0 as n \u2192 \u221e.if (Let p for which p(x) = 0 implies that x = 0 is called total paranorm and the pair  is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm  = 0, M(x) > 0 for x > 0 and M(x) \u2192 \u221e as x \u2192 \u221e.An Orlicz function w be the space of all real or complex sequences x = (xk). Then,\u2113M is a Banach space with the normLindenstrauss and Tzafriri  used the\u2113M contains a subspace isomorphic to \u2113p\u2009(p \u2265 1). The \u03942-condition is equivalent to M(Lx) \u2264 kLM(x) for all values of x \u2265 0 and for L > 1.It is shown in  that eve\u2133 = (Mk) of Orlicz function is called a Musielak-Orlicz function see; [\ud835\udca9 = (Nk) defined by\u2133. For a given Musielak-Orlicz function \u2133, the Musielak-Orlicz sequence space t\u2133 and its subspace h\u2133 are defined as follows:I\u2133 is a convex modular defined byA sequence ion see; , 18. A st\u2133 equipped with the Luxemburg normWe consider \u2133 = (Mk) be a Musielak-Orlicz function and let p = (pk) be a bounded sequence of positive real numbers. We define the following sequence spaces:Let We can writep = (pk) = 1, for all k \u2208 \u2115, we haveIf we take pk \u2264 sup\u2061pk = H, D = max\u2061, thenk, and ak, bk \u2208 \u2102. Also |a|pk \u2264 max\u2061 for all a \u2208 \u2102.The following inequality will be used throughout the paper. If 0 \u2264 \u2133 = (Mk). We also make an effort to study some topological properties and prove some inclusion relations between these spaces.The main aim of this paper is to study some ideal convergent sequence spaces defined by a Musielak-Orlicz function \u2133 = (Mk) be a Musielak-Orlicz function and let p = (pk) be a bounded sequence of positive real numbers. Then, the spaces cI, c0I, mI, and m0I are linear.Let x, y \u2208 cI and let \u03b1, \u03b2 be scalars. Then, there exist positive numbers \u03c11 and \u03c12 such thatLet \u03f5 > 0, we haveFor a given \u03c13 = max\u2061{2 | \u03b1 | \u03c11, 2 | \u03b2 | \u03c12}. Since \u2133 = (Mk) is nondecreasing convex function, so by using inequality . Hence cI is a linear space. Similarly, we can prove that c0I, mI, and m0I are linear spaces.Therefore, \u2133 = (Mk) be a Musielak-Orlicz function. Then,Let x \u2208 cI. Then, there exist L \u2208 \u2102 and \u03c1 > 0 such thatLet We havek on both sides, we get x \u2208 l\u221e. The inclusion c0I \u2282 cI is obvious. Thus,Taking supremum over This completes the proof of the theorem.\u2133 = (Mk) be a Musielak-Orlicz function and let p = (pk) be a bounded sequence of positive real numbers. Then, l\u221e is a paranormed space with paranorm defined byLet g(x) = g(\u2212x). Since Mk(0) = 0, we get g(0) = 0. Let us take x, y \u2208 l\u221e\u2009. LetIt is clear that \u03c11 \u2208 B(x) and \u03c12 \u2208 B(y). If \u03c1 = \u03c11 + \u03c12, then we haveLet kMk(|\u039b(x + y)|/(\u03c11 + \u03c12))pk \u2264 1 andThus, sup\u2061\u03c3s \u2192 \u03c3, where \u03c3, \u03c3s \u2208 \u2102 and let g(xs \u2212 x) \u2192 0 as s \u2192 \u221e. We have to show that g(\u03c3sxs \u2212 \u03c3x) \u2192 0 as s \u2192 \u221e. LetLet \u03c1s \u2208 B(xs) and \u03c1s\u2032 \u2208 B(xs \u2212 x), then we observe thatIf From the above inequality, it follows thatThis completes the proof.\u2133\u2032 = (Mk\u2032) and \u2133\u2032\u2032 = (Mk\u2032\u2032) be Musielak-Orlicz functions that satisfy the \u03942-condition. Then,Let Z\u2286Z,(i)\u2009\u2009Z\u2229Z\u2286Z for Z = cI, c0I, mI, m0I.(ii)\u2009\u2009x \u2208 c0I. Then, there exists \u03c1 > 0 such that(i) Let \u03f5 > 0 and choose \u03b4 with 0 < \u03b4 < 1 such that Mk\u2032(t) < \u03f5 for 0 \u2264 t \u2264 \u03b4. Write yk = Mk\u2032\u2032(|\u039bk(x)|/\u03c1)pk and considerLet \u2133 = (Mk) satisfies \u03942-condition, we haveSince yk > \u03b4, we haveFor \u2133\u2032 = (Mk\u2032) is nondecreasing and convex, it follows thatSince \u2133\u2032 = (Mk\u2032) satisfies \u03942-condition, we haveSince x = (xk) \u2208 c0I. Thus, c0I\u2286c0I. Similarly, we can prove the other cases.Hence,k).From , 34), a, a38)lili(38)limx \u2208 c0I\u2229c0I. Then, there exists \u03c1 > 0 such that(ii) Let The rest of the proof follows from the following equality:\u2133 = (Mk) be a Musielak-Orlicz function which satisfies \u03942-condition. Then, Z\u2286Z holds for Z = cI, c0I, mI, and m0I.Let Mk\u2032\u2032(x) = x and Mk\u2032(x) = Mk(x)\u2009\u2009\u2200x \u2208 [0, \u221e).The proof follows from c0I and m0I are solid.The spaces c0I. Let x \u2208 c0I. Then, there exists \u03c1 > 0 such thatWe will prove for the space \u03b1k) be a sequence of scalars with |\u03b1k | \u22641\u2009\u2009\u2200k \u2208 \u2115. Then, the result follows from the following inequality:m0I.Let  and m0I are monotone.The spaces It is easy to prove, so we omit the details.cI and c0I are sequence algebra.The spaces x, y \u2208 c0I. Then,Let \u03c1 = \u03c11 + \u03c12. Then, we can show thatLet x \u00b7 y) \u2208 c0I. Hence, c0I is a sequence algebra. Similarly, we can prove that cI is a sequence algebra.Thus, ("}
{"text": "Firstly, an equivalent characterization of \u03f5-Henig saddle point of the Lagrangian set-valued map is obtained. Secondly, under the assumption of the generalized cone subconvexlikeness of set-valued maps, the relationship between the \u03f5-Henig saddle point of the Lagrangian set-valued map and the \u03f5-Henig properly efficient element of the set-valued optimization problem is presented. Finally, some duality theorems are given.We study  Xia and Qiu , \u03bbk \u2208 K.\u2009\u2009K is called absorbent if and only if 0 \u2208 cor(K). Let K be a nonempty subset in Y. The vector closure of K is the setLet K is vectorially closed (v-closed) if vcl(K) = K. Following Ad\u00e1n and Novo , K is veB\u2009\u2009be a nonempty convex subset in Y. B is a base of C if and only if C = cone\u2061(B) and there exists a balanced, absorbent, and convex set V such that 0 \u2209 B + V in Y. Write CV(B): = cone\u2061(V + B).Let\u2009\u2009C) \u00d7 cor(D) \u2260 \u2205 and B is a base of C. We recall a notion of\u2009\u2009\u03f5-Henig properly efficient point introduced by Zhou et al. [From now on, we suppose that cor) if and only if there exists a balanced, absorbent, and convex set V with 0 \u2209 B + V such that \u03f5-Henig properly maximal efficient point with respect to\u2009\u2009B\u2009\u2009: = \u22c3x\u2208AF(x), \u2329F(x), y*\u232a: = {\u2329y, y*\u232a\u2223y \u2208 F(x)}, and \u2329F(A), y*\u232a: = \u22c3x\u2208A\u2329F(x), y*\u232a. The meanings of G(A), \u2329G(x), z*\u232a, and \u2329G(A), z*\u232a are similar to those of F(A), \u2329F(x), y*\u232a, and \u2329F(A), y*\u232a, respectively. Let F : A\u21c9Y is called generalized C-subconvexlike on A if and only if cone\u2061(F(A)) + cor(C)\u2009\u2009is a convex set in Y. A set-valued map Z be a linear space, and let M, N\u2286Z be two nonempty sets such that M \u2212 N is a convex set in Z. If\u2009\u2009cor(M \u2212 N) \u2260 \u2205 and 0 \u2209 vcl(M \u2212 N), then there exists z* \u2208 Z*\u2216{0} such that sup\u2061z2\u2208N\u2329z2, z*\u232a<inf\u2061z1\u2208M\u2329z1, z*\u232a.Let \u03f5-Henig proper efficiency.In this section, we will establish approximate saddle point theorems of set-valued optimization problems in the sense of F : A\u21c9Y and G : A\u21c9Z be two set-valued maps on A. We consider the following vector optimization problem with set-valued maps:S : = {x \u2208 A | G(x)\u2229(\u2212D) \u2260 \u2205}.\u2009\u2009Let \u03f5 \u2208 C. \u03f5-Henig properly efficient solution of (VP) if and only if there exists Hmin\u2061(F(S), B). The pair \u03f5-Henig properly efficient element of (VP).Let L the set of all linear operators from Z to Y. A subset L+ of L is defined as L+: = {T \u2208 L | T(D)\u2286C}. The Lagrangian set-valued map of (VP) is defined byWe denote by Consider the following unconstrained vector optimization problem with set-valued maps:\u03f5 \u2208 C,\u2009\u2009\u03f5-Henig properly efficient element of \u03f5-Henig properly efficient element of (VP).Let \u03f5-Henig proper saddle point of the Lagrangian set-valued map L in linear spaces. Now, we will introduce a new notion called \u03f5-Henig proper saddle point of the Lagrangian set-valued map L if and only if\u03f5-Henig proper saddle point of the Lagrangian set-valued map L. The following proposition is an important equivalent characterization for an\u2009\u2009D be v-closed and \u03f5 \u2208 C. Then, \u03f5-Henig proper saddle point of the Lagrangian set-valued map V with 0 \u2209 B + V such that ;;;Let Necessity. Let \u03f5-Henig proper saddle point of L. Then, there exist V with 0 \u2209 B + V such thatT = 0 in  = C, it follows from  and B + V, b1 \u2260 0. Therefore,z1 \u2209 \u2212D. Similar to the above proof, there exists z2* \u2208 D+\u2216{0} such that \u2329z1, z2*\u232a>0. Taking b2 \u2208 B, we define a vector-valued map T2 : Z \u2192 Y as follows:uch thaty\u00af+T\u00af(z\u00af),T\u00af),B),y\u00af+T\u00af(z\u00af)uch thatcone\u2061\u2209\u03f5 we have\u2212T\u00af(z\u00af)\u2208Cuch thatcone\u2061,T2(z1)\u2212T\u00afSufficiency. Since r1 > 0,\u2009\u2009\u2009r2 > 0,\u2009\u2009T3 \u2208 L+,\u2009\u2009\u2009v \u2208 V,\u2009\u2009and\u2009\u2009b3 \u2208 B such that z\u2032 \u2208 \u2212D. Since T3 \u2208 L+,\u2009\u2009\u2212T3(z\u2032) \u2208 C. Therefore, there exist r3 \u2265 0 and b4 \u2208 B such thatr2/(r2 + r1r3) \u2264 1, it follows from the balance of\u2009\u2009V\u2009\u2009thatB thatV + B and r2 + r1r3 > 0, it follows from =rws from (r1(y\u2032\u2212y\u00af\u2212f B thatr2r2+r1r3e assertcone\u2061(\u22c3T\u2208ove thaty\u00af+T\u00af(z\u00af)\u03f5-strictly efficient point is equivalent to the notion of \u03f5-Henig properly efficient point in locally convex spaces. Moreover, the generalized subconvexlikeness of the set-valued map F is equivalent to ic-cone convexlikeness of the set-valued map F introduced by Sach [C) \u2260 \u2205. Therefore,  According to Theorem 1 in , the not by Sach  when theD be v-closed and \u03f5 \u2208 C. If \u03f5-Henig proper saddle point of the Lagrangian set-valued map L, then there exist Let \u03f5-Henig proper saddle point of the Lagrangian set-valued map L, it follows from V with\u2009\u20090 \u2209 B + V such that conditions (i)\u2013(iv) hold. By condition (ii), U with r1 > 0,\u2009\u2009r2 > 0,\u2009\u2009x1 \u2208 S,\u2009\u2009y1 \u2208 F(S),\u2009\u2009b1 \u2208 B, and \u2009u1 \u2208 U such thatc \u2208 C such thatB is a base of C, there exist r3 \u2265 0 and b2 \u2208 B such that c = r3b2. By ; C \u00d7 D-subconvexlike on A, where vcl(cone(G(A) + D)) = Z.  Then, there exists \u03f5-Henig properly efficient element of Let By D be v-closed, \u03f5 \u2208 C,\u2009\u2009\u03f5-Henig properly efficient element of (VP); C \u00d7 D-subconvexlike on A, where vcl(cone(G(A) + D)) = Z;  Then, there exists \u03f5-Henig proper saddle point of\u2009\u2009L.Let \u03f5-Henig proper efficiency of set-valued optimization problems in linear spaces. In this section, we will give several duality theorems characterized by \u03f5 \u2208 C and let B be a base of C. The set-valued map \u03a6 : L+\u21c9Y, defined by \u03a6(T) = \u03f5-Hmin\u2061, B), is called an \u03f5-Henig properly dual map of (VP).Let Now, we construct the following duality problem of the primal problem (VP):\u03f5 \u2208 C.\u03f5-efficient point of (VD) if and only ifLet \u03f5 \u2208 C,\u2009\u2009Let V with 0 \u2209 B + V such that C\u2216{0}\u2286CV(B)\u2216{0}. Therefore, Since \u03f5 \u2208 C and\u03f5-Henig properly efficient element of (VP) and \u03f5-efficient point of (VD). Let \u03f5-Henig properly efficient element of \u03f5-Henig properly efficient element of (VP). Because \u03f5-efficient point of (VD).  Since \u03f5-Henig properly efficient element of (VP); C \u00d7 D-subconvexlike on A, where vcl(cone(G(A) + D)) = Z.  Then,\u03f5-efficient point of (VD). Let \u03f5-Henig properly efficient element of \u03f5-efficient point of (VD).  According to \u03f5-Henig saddle point of the set-valued map in linear spaces. The relationships between the \u03f5-Henig saddle point of the set-valued map and the \u03f5-Henig properly efficient element of the set-valued optimization problem are established. Some duality theorems are obtained in the sense of \u03f5-Henig proper efficiency. When \u03f5-Henig proper efficiency is replaced by \u03f5-super efficiency in linear spaces, whether the conclusions of this paper hold is an interesting topic.Based on , we intr"}
{"text": "G be a finite group and \ud835\udca9\ud835\udc9e(G) the set of the numbers of conjugates of noncyclic proper subgroups of G. We prove that (1) if |\ud835\udca9\ud835\udc9e(G)| \u2264 2, then G is solvable, and (2) G is a nonsolvable group with |\ud835\udca9\ud835\udc9e(G)| = 3 if and only if G\u2245PSL or PSL or SL or SL.Let  G having exactly four conjugacy classes of noncyclic proper subgroups is nonsolvable if and only if G\u2245PSL or SL. As a generalization of the above result, we showed that any group having at most three conjugacy classes of nonnormal noncyclic proper subgroups is solvable, and a group G having exactly four conjugacy classes of nonnormal noncyclic proper subgroups is nonsolvable if and only if G\u2245PSL or SL  the set of the numbers of conjugates of noncyclic proper subgroups of G. It is clear that a group G with \ud835\udca9\ud835\udc9e(G) = \u2205 is either a cyclic group or a minimal noncyclic group, and a group G with \ud835\udca9\ud835\udc9e(G) = {1} is a group in which every noncyclic proper subgroup is normal. In [G in which every noncyclic proper subgroup is nonnormal; all such groups G satisfy 1 \u2209 \ud835\udca9\ud835\udc9e(G).Let rmal. In , we also\ud835\udca9\ud835\udc9e(G)| we denote the order of \ud835\udca9\ud835\udc9e(G). Note that we cannot ensure that 1 \u2208 \ud835\udca9\ud835\udc9e(G) for any solvable group G with |\ud835\udca9\ud835\udc9e(G)| = n \u2265 1. For example, let G\u2245Dpn2 be a dihedral group of order 2pn, where n \u2265 1 and p is an odd prime. Then \ud835\udca9\ud835\udc9e(Dpn2) = {p, p2,\u2026, pn}, so 1 \u2209 \ud835\udca9\ud835\udc9e(Dpn2). For the nonsolvable group of the smallest order PSL, it is easy to see that \ud835\udca9\ud835\udc9e) = {5, 6, 10}, and so |\ud835\udca9\ud835\udc9e)| = 3.By |\ud835\udca9\ud835\udc9e(G)| on the solvability of groups, we have the following result, the proof of which is given in For the influence of |G be a group.\ud835\udca9\ud835\udc9e(G)|\u22642, then G is solvable.If |G is a nonsolvable group with |\ud835\udca9\ud835\udc9e(G)| = 3 if and only if G\u2245PSL or PSL or SL or SL.  Let The following two corollaries are direct consequences of G be a group with |\ud835\udca9\ud835\udc9e(G)|\u22643. Then G is nonsolvable if and only if \ud835\udca9\ud835\udc9e(G) = {5,6, 10} or {14,78,91}.  Let G be a group and \ud835\udca9\ud835\udcaf(G) the set of the numbers of conjugates of nontrivial subgroups of G.\ud835\udca9\ud835\udcaf(G)|\u22642, then G is solvable.If |G is a nonsolvable group with |\ud835\udca9\ud835\udcaf(G)| = 3 if and only if G\u2245PSL.  Let G be a group and \ud835\udca9\ud835\udc9e*(G) the set of the numbers of conjugates of nonnormal noncyclic proper subgroups of G. Obviously \ud835\udca9\ud835\udc9e*(G)\u2286\ud835\udca9\ud835\udc9e(G).Let Arguing as in the proof of G be a group. If |\ud835\udca9\ud835\udc9e*(G)|\u22642, then G is solvable.  Let G is a nonsolvable group with |\ud835\udca9\ud835\udc9e*(G)| = 3, we cannot get that \u03a6(G) = Z(G). For example, let G\u2245PSL \u00d7 \u2124p, where p \u2265 7 is a prime. It is easy to see that |\ud835\udca9\ud835\udc9e*(G)| = 3. But \u03a6(G) = 1 and Z(G) = \u2124p.If we assume that G be a group and \ud835\udca9\ud835\udc9c(G) the set of the numbers of conjugates of nonabelian proper subgroups of G. Obviously \ud835\udca9\ud835\udc9c(G)\u2286\ud835\udca9\ud835\udc9e(G). Arguing as in the proof of Let G be a group. If |\ud835\udca9\ud835\udc9c(G)|\u22642, then G is solvable.  Let In this section, we collect some essential lemmas needed in the sequel.G be a group. If all nonnormal maximal subgroups of G have the same order, then G is solvable.  Let G be a nonsolvable group having exactly two classes of nonnormal maximal subgroups of the same order; then G/S(G)\u2245PSL, where S(G) is the largest solvable normal subgroup of G.  Let G be a group having exactly n classes of maximal subgroups of the same order, where 1 \u2264 n \u2264 3; then one of the following statements holds:G is a group with n = 1, and then G is a p-group for some prime p;suppose that G is a nonsolvable group with n = 2, and then G/\u03a6(G)\u2245) \u00d7 \u21247j, where i, j = 0, 1,\u2026, and \u21242i3\u22caPSL is a semidirect product of the normal subgroup \u21242i3 and the subgroup PSL;suppose that G is a nonsolvable group with n = 3, and then G/S(G)\u2245A6; PSL, q = 11, 13, 23, 59, 61; PSL; U3(3); PSL; PSL, and f is a prime; PSL \u00d7 PSL \u00d7 \u22ef\u00d7PSL.suppose that  Let The proof of G be a group. If |\ud835\udca9\ud835\udc9e(G)|\u22642, then G is solvable.  Let G is nonsolvable. Then by [G are noncyclic. Let \u2133\ud835\udcae(G) be the set of the numbers of conjugates of maximal subgroups of G. It follows that \u2133\ud835\udcae(G)\u2286\ud835\udca9\ud835\udc9e(G). Then |\u2133\ud835\udcae(G)|\u22642.\u2133\ud835\udcae(G). Since G is nonsolvable, G must have nonnormal maximal subgroups. Let M be any nonnormal maximal subgroup of G; one has |G : NG(M)| = |G : M|. Since |\u2133\ud835\udcae(G)|\u22642, we know that G has at most one class of nonnormal maximal subgroups of the same order. It follows that G is solvable by Suppose that 1 \u2208 \u2133\ud835\udcae(G). It follows that all maximal subgroups of G are nonnormal. By the hypothesis, G has at most two classes of maximal subgroups of the same order. Since G is nonsolvable and G has no normal maximal subgroups, one has G/\u03a6(G)\u2245\u21242i3\u22caPSL by i = 0, 1,\u2026. It is easy to see that \ud835\udca9\ud835\udc9e(G/\u03a6(G))\u2286\ud835\udca9\ud835\udc9e(G) and |\ud835\udca9\ud835\udc9e)|>2. It follows that |\ud835\udca9\ud835\udc9e(G)|>2, a contradiction.Suppose that 1 \u2209  Assume that G is solvable.Thus, our assumption is not true, so G is a nonsolvable group with |\ud835\udca9\ud835\udc9e(G)| = 3 if and only if G\u2245PSL or PSL or SL or SL.  A group The sufficiency part is evident, and we only need to prove the necessity part.\u2133\ud835\udcae(G)|\u22643. We claim thatBy the hypothesis, |\u2133\ud835\udcae(G). Then G has at most two classes of nonnormal maximal subgroups of the same order. Since G is nonsolvable, one has G/S(G)\u2245PSL by Lemmas \ud835\udca9\ud835\udc9e(G/S(G))\u2286\ud835\udca9\ud835\udc9e(G) and |\ud835\udca9\ud835\udc9e)|>3. It follows that |\ud835\udca9\ud835\udc9e(G)|>3, a contradiction. Thus, 1 \u2209 \u2133\ud835\udcae(G).Otherwise, assume that 1 \u2208 (G)\u2245PSL2, by Lemma\u2133\ud835\udcae(G)|\u22643, we have that G has at most three classes of maximal subgroups of the same order.Since |G cannot have exactly one class of maximal subgroups of the same order.By G has exactly two classes of maximal subgroups of the same order, according toG/\u03a6(G)\u2245\u21242i3\u22caPSL since G has no normal maximal subgroups, where i = 0, 1,\u2026. Since |\ud835\udca9\ud835\udc9e)|>3, it follows that |\ud835\udca9\ud835\udc9e(G)|>3, a contradiction.If G has exactly three classes of maximal subgroups of the same order. By G/S(G) might be isomorphic to A6 or PSL, q = 11, 13, 23, 59, 61 or PSL or U3(3) or PSL or PSL, and f is a prime or PSL \u00d7 PSL\u00d7\u22ef\u00d7PSL. If G/S(G) is an isomorphism to A6 or PSL, q = 11, 23, 59, 61 or PSL or U3(3) or PSL or PSL, and f is an odd prime or PSL \u00d7 PSL \u00d7 \u22ef\u00d7PSL. It is easy to see that |\ud835\udca9\ud835\udc9e(G/S(G))|>3 by [\ud835\udca9\ud835\udc9e(G)|>3, a contradiction. Thus, G/S(G)\u2245PSL\u2245PSL or PSL.Thus, ))|>3 by , 9, whic\u2133\ud835\udcae(G) and |\u2133\ud835\udcae(G)| = |\ud835\udca9\ud835\udc9e(G)| = 3. It follows that 1 \u2209 \ud835\udca9\ud835\udc9e(G), so S(G) is cyclic. We claim thatNote that 1 \u2209 G) < S(G). Let M be a maximal subgroup of G such that S(G)\u2270M. Then G = S(G)M. It is obvious that S(G)\u2229M\u22b4M. Moreover, S(G)\u2229M\u22b4S(G), since S(G) is cyclic. It follows that S(G)\u2229M\u22b4G. Therefore, G/(S(G)\u2229M) = S(G)/(S(G)\u2229M)\u22caM/(S(G)\u2229M). Let M\u22b4G; this contradicts that all maximal subgroups of G are nonnormal. Thus, our assumption is not true, so \u03a6(G) = S(G).Otherwise, assume that \u03a6(G/\u03a6(G)\u2245PSL or PSL.It follows that G) = 1, then G\u2245PSL or PSL.If \u03a6(G) \u2260 1. Let p be any prime divisor of |\u03a6(G)|. We claim that p\u226f2. Otherwise, assume that p > 2. Let T be a subgroup of \u03a6(G) such that \u03a6(G)/T\u2245\u2124p. That is, \u03a6(G/T)\u2245\u2124p. Then (G/T)/\u2124p\u2245(G/T)/\u03a6(G/T) = (G/T)/(\u03a6(G)/T)\u2245G/\u03a6(G)\u2245PSL or PSL. Since p > 2 and Schur multipliers of both PSL and PSL are \u21242, we have that G/T\u2245PSL \u00d7 \u2124p or PSL \u00d7 \u2124p. Note that |\ud835\udca9\ud835\udc9e \u00d7 \u2124p)|>3 and |\ud835\udca9\ud835\udc9e \u00d7 \u2124p)|>3. It follows that |\ud835\udca9\ud835\udc9e(G)|>3, a contradiction. Thus, p\u226f2, so \u03a6(G) is a cyclic 2-group. If |\u03a6(G)| = 2n > 2, let L be a subgroup of \u03a6(G) such that \u03a6(G)/L\u2245\u21242. Then (G/L)/\u21242\u2245(G/L)/\u03a6(G/L) = (G/L)/(\u03a6(G)/L)\u2245G/\u03a6(G)\u2245PSL or PSL. We have that G/L\u2245SL or SL. Let M be a subgroup of L such that L/M\u2245\u21242. Then (G/M)/\u21242\u2245(G/M)/(L/M)\u2245G/L\u2245SL or SL. Since Schur multipliers of both SL and SL are trivial, we have that G/M\u2245SL \u00d7 \u21242 or SL \u00d7 \u21242; this contradicts that all maximal subgroups of G are nonnormal. Thus, |\u03a6(G)| = 2. It follows that G\u2245SL or SL. Next, suppose that \u03a6( Lemmas"}
{"text": "We present a fixed-point iterative method for solving systems of nonlinear equations. The convergence theorem of the proposed method is proved under suitable conditions. In addition, some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach. One of the basic problems in mathematics is how to solve nonlinear equationsf(x)=0.Iues; see  and the The zeros of a nonlinear equation cannot in general be expressed in closed form; thus we have to use approximate methods. Nowadays, we often use iterative methods to get the approximate solution of the system ; the besIn this paper, we will present a new fixed point iterative method for solving the system  and provThis paper is organized as follows. In \u03b1* is a simple root of (f(\u03b1*) = 0. For \u03b1, \u03b1k \u2208 , using Taylor's formula, we haver = 1 in the above equality, we getf\u2032(\u03b1k + t(\u03b1 \u2212 \u03b1k)) in the interval  is replaced with its value in t = 0, that is, with f\u2032(\u03b1k), then we havef\u2032(\u03b1k + t(\u03b1 \u2212 \u03b1k)) in the interval  by its value in t = 1, that is, by f\u2032(\u03b1), then we haveR[f(\u03b1)] (still use \u201c=\u201d), we have\u03b1k+1 be the solution of (We now consider the following nonlinear equation:f(x)=0,\u2003x we have\u222b01f\u2032=f(\u03b1ky using (f(\u03b1)\u2245f(\u03b1kf(x)=0,\u2003xod from =f(\u03b1k we havef(\u03b1)=f. Choose the initial value \u03b10 \u2208 N, where \u03b1* is certain real zero of nonlinear mapping f(\u03b1) and \u03b4 > 0 is a sufficiently small constant. Take the stopping criteria \u03f51, \u03f52 > 0. Set k : = 0.Step 1 (the predictor step). Compute the predictorStep 2 (the corrector step). Computing the correctorStep 3. If |f(\u03b1k+1)|\u2264\u03f51 or |\u03b1k+1 \u2212 \u03b1k | \u2264\u03f52 then stop; otherwise, set \u03b1k : = \u03b1k+1, k : = k + 1; go to Step 1.In this section, we consider the convergence and convergent rate of \u03b1* \u2208 I be a simple zero of sufficiently differentiable function f : \u2286R \u2192 R. If \u03b10 is sufficiently close to \u03b1*, then the two-step iterative method defined by (ek = \u03b1k \u2212 \u03b1* and c1 = f\u2032\u2032\u2032(\u03b1*)/6f\u2032(\u03b1*).Let fined by -12) has has\u03b1* \u2208 f\u2032(\u03b2k), we can get\u03b1 = \u03b1* in (f(\u03b1*) = 0; we havef\u2032(\u03b1k), we can get\u03b1 = \u03b2k in (f\u2032(\u03b2k) at the point \u03b1k, we haveBy  we get( = \u03b1* in  and f(\u03b1*4).From  we get( can getf(\u03b1k)f\u2032(\u03b13).From , 18), a, a(14)ek we havef\u2032(\u03b2k)=f\u2032);thus,f(\u03b1k)=f\u2032(e obtainf(\u03b2k)=12f can getf\u2032(\u03b2k)=f\u2032 can getf\u2032(\u03b2k)ek+ituting (f\u2032(\u03b2k)ek+n-dimensional case of the method, and we also study these iterative methods' order of convergence. Consider the system of nonlinear equationsfi, i = 1,2,\u2026, n, maps a vector x = T of the n-dimensional space Rn into the real line R. The system  = (f1(x), f2(x),\u2026, fn(x))T. Thus, the system ; that is, F(x*) = 0. For any x, xk \u2208 D, we may write Taylor's expansion for F as follows ) in the interval  by its value in t = 0, that is, by F\u2032(xk), then we haveF\u2032(xk + t(x \u2212 xk)) in the interval  by its value in t = 1, that is, by F\u2032(x), then we haveR[F(x)] (still use \u201c=\u201d), we havexk+1 be the solution of :(32)F( we have\u222b01F\u2032(xk+ we haveF(x)=F(xky using (F(x)\u2245F(xk system (F(x)=0.Lod from (xk+1=xk\u2212F we have\u222b01\u2009\u200dF\u2032(x we haveF(x)=F(xk we haveF(x)=F40). Let uce from  and (40)ce from . Choose the initial value x0 \u2208 N, \u03b1 \u2208  and \u03b2 = 1 \u2212 \u03b1, where x* is certain real zero of nonlinear mapping F(x) and \u03b4 > 0 is a sufficiently small constant. Take the stopping criterions \u03f51, \u03f52 > 0. Set k : = 0.Step 1 (the predictor step). Compute the predictorStep 2 (the corrector step). Computing the correctorStep 3. If ||F(xk+1)|| \u2264 \u03f51 or ||xk+1 \u2212 xk|| \u2264 \u03f52 then stop; otherwise, set k : = k + 1; go to Step 1.\u03b2 = 0, \u03b1 = 1, our algorithms (n2 + 2n.If we take gorithms  and 43)\u03b2 = 0, \u03b1 In this section, we consider the convergence and convergent rate of F : D\u2286Rn \u2192 Rn be a sufficiently differentiable function on a convex set D\u2286Rn containing a root x* of the nonlinear system (Let r system . The iter system -43) has hasF : Dek = xk \u2212 x*, from (F\u2032(yk), we can getx = x* in (F(x*) = 0; we haveF\u2032(xk)\u22121 to the two sides of the above equation, we obtainx = yk in (yk \u2212 xk = \u2212F\u2032(xk)\u22121F(xk). It follows from the above equation thatF\u2032(yk) at the point xk, we haveDefining x*, from  and 43)ek = xk = x* in  and F.From  we get ye obtainF\u2032(xk)\u22121F|).From , 50), a, aek =  we haveF\u2032(yk)=F\u2032);thus,F(xk)=F\u2032(e obtainF(yk)=12F can getF\u2032(yk)=F\u2032 we have\u03b1I+\u03b2F\u2032(yk).From (F(xk)+F(ye obtainF(yk)=12F can getF\u2032(yk)ek+that is,ek+1=F\u2032(yp is computed approximately by the following formula:In this section we present some examples to illustrate the efficiency and the performance of the newly developed method -43) (pr (pr43) :(65)FT. We test this problem by using initial value x0 = T as a starting point. The test results are listed in The test function is as follows see , 13, 14), 1413, 1x0 = T. The test results are listed in The test function is as follows (see ):(67)x1F(x) = (f1(x), f2(x), f3(x), f4(x))T, where x = T \u2208 R4 and fi : R4 \u2192 R, i = 1,2, 3,4, such that T (the iterative sequence converges to x*) and x0 = T (the iterative sequence converges to x**) as starting point, respectively. The test results are listed in Tables The test function is hat see , 10, 13), 13F(x) x* = T and x** = T. We test this problem by using x0 = T (the iterative sequence converges to x*) and x0 = T (the iterative sequence converges to x**) as starting point, respectively. The test results are listed in Tables n = 101.The test function is as follows see , 13, 14), 1413, 10Fi(x)=xif : Rn \u2192 R is defined byF(x) = (F1(x), F2(x),\u2026, Fn(x))T,x0 = T as starting point. The test results are listed in n = 512.Consider the unrestraint optimum problem (see )(71)minA is an n \u00d7 n tridiagonal matrix defined byG(x) = (G1(x), G2(x),\u2026, Gn(x))T, and Gi(x) = sinxi \u2212 1, i = 1,2,\u2026, n. We test this problem by using x0 = T as starting point. The test results are listed in n = 1024.Consider the discrete two-point boundary value problems (see ):(75)F(p that appears in Tables From the seven examples in"}
{"text": "I-convergence of real sequences was introduced by Kostyrko et al., (2000/01) and also independently by Nuray and Ruckle (2000). In this paper, we introduce the concepts of -statistical convergence of order \u03b1 and strong -Ces\u00e0ro summability of order \u03b1 of real sequences and investigated their relationship.The idea of  The idea of statistical convergence was given by Zygmund  in the fI-convergence of real sequences was introduced by Kostyrko et al. [I-convergence was studied by Das et al. [The idea of o et al.  and alsoo et al.  -statistical convergence of order \u03b1 and strong -Ces\u00e0ro summability of order \u03b1 of real sequences and investigated their relationship. In p-Ces\u00e0ro summability, I-convergence, and difference sequences. In S\u03b1-convergent sequences and w\u03b1-summable sequences. In w\u03b1-summable sequences for different \u03b1's. In S\u03b1-convergent sequences for different \u03b1's.In this paper, we introduce the concepts of  with the usual norm ||x|| = sup\u2061|xk|, where k \u2208 \u2115 = {1,2,\u2026}, the set of positive integers. Also by bs, cs, \u21131, and \u2113p, we denote the spaces of all bounded, convergent, and absolutely and p-absolutely convergent series, respectively.Let p-Ces\u00e0ro convergence of a sequence of real numbers were introduced in the literature independently of one another and followed different lines of development since their first appearance. It turns out, however, that the two definitions can be simply related to one another in general and are equivalent for bounded sequences. The idea of statistical convergence depends on the density of subsets of the set \u2115. The density of a subset E of \u2115 is defined byThe definitions of statistical convergence and strong \u03c7E is the characteristic function of E. It is clear that any finite subset of \u2115 has zero natural density and \u03b4(Ec) = 1 \u2212 \u03b4(E).where \u03b1 and strong p-Ces\u00e0ro summability of order \u03b1 were studied by \u00c7olak [The order of statistical convergence of a sequence of numbers was given by Gadjiev and Orhan in  and thenby \u00c7olak .X = \u2113\u221e, c, or c0, where m \u2208 \u2115, \u03940x = (xk), \u0394mx = (\u0394m\u22121xk \u2212 \u0394m\u22121xk+1), and so m(X) are Banach spaces normed byX = \u2113\u221e, c, or c0. Let X be any sequence spaces, if x \u2208 \u0394m(X), then there exists one and only one y = (yk) \u2208 X such thatk; for instance, k > 2m. We use this fact in the following examples.The notion of difference sequence spaces was introduced by K\u0131zmaz  and it wRecently, the difference sequence spaces have been studied in , 36\u201339.X be nonempty set. Then a family of sets I \u2282 2X (power sets of X) is said to be an ideal if I is additive, that is, A, B \u2208 I implies A \u222a B \u2208 I, and hereditary; that is, A \u2208 I, B \u2282 A implies B \u2208 I.Let F \u2282 2X is said to be a filter of X if and only if (i) \u03d5 \u2209 F, (ii) A, B \u2208 F implies A\u2229B \u2208 F, and (iii) A \u2208 F, A \u2282 B implies B \u2208 F.A nonempty family of sets I \u2282 2X is called nontrivial if I \u2260 2X.An ideal I is said to be admissible if I\u2283{{x} : x \u2208 X}.A nontrivial ideal I is a nontrivial ideal in X(X \u2260 \u03d5), then the family of sets F(I) = {M \u2282 X : (\u2203A \u2208 I)(M = X\u2216A)} is a filter of X, called the filter associated with I.If I will stand for a nontrivial admissible ideal of \u2115.Throughout the paper We now introduce our main definitions.x \u2208 w is said to be I-convergent if there exists L \u2208 \u2102 such that, for all \u03b5 > 0, the set {n \u2208 \u2115 : |\u0394mxk \u2212 L| \u2265 \u03b5} \u2208 I. In this case, one writes  \u2212 lim\u2061xk = L. The set of all -convergent sequences will be denoted by c.A sequence \u03b1 \u2208 -statistical convergence of order \u03b1 -convergence) if there is a real number L such thatLet S\u03b1 \u2212 lim\u2061xk = L or S\u03b1(I) \u2212 lim\u2061\u0394mxk = L. The set of all -statistically convergent sequences of order \u03b1 will be denoted by S\u03b1. In the special case \u03b1 = 1, we will write S instead of S\u03b1.In this case, we write m, I)-statistical convergence of order \u03b1 is well defined for 0 < \u03b1 \u2264 1, but it is not well defined for \u03b1 > 1 in general. For this x = (xk) is defined as follows:(\u0394\u03b5 > 0 and \u03b1 > 1 we have\u03b4 > 0For every x = (xk) is -statistically convergent of order \u03b1, both to 1 and 0; that is, S\u03b1 \u2212 lim\u2061xk = 1 and S\u03b1 \u2212 lim\u2061xk = 0. But this is impossible.Therefore, the sequence m, I)-convergent sequence is -statistically convergent of order \u03b1(0 < \u03b1 \u2264 1), but converse does not hold. For this, consider a sequence x = (xk) defined byIt is easy to see that every  for \u03b1 \u2208 .It is clear that \u03b1 \u2208 -Ces\u00e0ro summable of order \u03b1 -summable) if there is a real number L such thatLet w\u03b1 \u2212 lim\u2061xk = L. The set of all strong -Ces\u00e0ro summable sequences of order \u03b1 to L will be denoted by w\u03b1. In the special case \u03b1 = 1, we will write w instead of w\u03b1.In this case, we write S\u03b1-convergent sequences and w\u03b1-summable sequences. In w\u03b1-summable sequences for different \u03b1's. In S\u03b1- convergent sequences for different \u03b1's.In this section, we give the main results of this paper. In \u03b1 \u2208  \u2212 lim\u2061xk = L1, S\u03b1\u2009\u2009lim\u2061yk = L2, and c \u2208 \u211d; thenS\u03b1 \u2212 lim\u2061cxk = cL1,S\u03b1 \u2212 lim\u2061(xk + yk) = L1 + L2. Let S\u03b1 \u2212 lim\u2061xk = L1 and c \u2208 \u211d; thenS\u03b1 \u2212 lim\u2061(cxk) = cL1.(i) Suppose that S\u03b1 \u2212 lim\u2061xk = L1 and S\u03b1 \u2212 lim\u2061yk = L2; then we haveA1\u2229A2 \u2260 \u03d5. Now for all n \u2208 A1\u2229A2, we haveS\u03b1 \u2212 lim\u2061(xk + yk) = L1 + L2.(ii) Now suppose that The proofs of the following two theorems are easy and thus omitted.\u03b1 \u2208 -convergent sequence is uniquely determined.Let x = (xk), y = (yk), and z = (zk) be real sequences such that \u0394mxk \u2264 \u0394myk \u2264 \u0394mzk. If S\u03b1 \u2212 lim\u2061xk = L = S\u03b1 \u2212 lim\u2061zk, then S\u03b1 \u2212 lim\u2061yk = L.Let The proof of the following theorem is obtained by using the same techniques of Savas and Das [\u03b1 \u2208 \u2229\u2113\u221e(\u0394m) is a closed subset of \u2113\u221e(\u0394m).Let S\u03b1-statistically convergent sequences and strong w\u03b1-summable sequences.In the following theorem we investigate the relationship between \u03b1 and \u03b2 be fixed real numbers such that 0 < \u03b1 \u2264 \u03b2 \u2264 1, and let p be a positive real number; then w\u03b1 \u2282 S\u03b2, and the inclusion is strict.Let \u03b5 > 0 and w\u03b1 \u2212 lim\u2061xk = L; then we can write\u03b4 > 0, we haveLet This completes the proof.\u03b1 = \u03b2, we show the strictness of the inclusion w\u03b1 \u2282 S\u03b2 for a special case. For this, consider the sequence x = (xk) defined byTaking \u03b5 > 0 and \u03b1 \u2208 ) for \u03b1 \u2208 , since I is admissible. So xk\u219bw\u03b1.and for any m-bounded and S\u03b1-convergent, but need not to be w\u03b1-summable. For this, consider a sequence x = (xk) defined by (x \u2208 \u2113\u221e(\u0394m) and x \u2208 S\u03b1 for \u03b1 \u2208  for \u03b1 \u2208 . Therefore, x \u2208 S\u03b1\u2216w\u03b1 for \u03b1 \u2208 .The converse of fined by . It can The following result is a consequence of w\u03b1-convergent to L, then it is S\u03b1-convergent to L.If a sequence is \u03b1 \u2264 \u03b2 \u2264 1, and let p be a positive real number; then w\u03b1 \u2282 w\u03b2 and the inclusion is strict.Let 0 < p = 1, we show the strictness of the inclusion w\u03b1 \u2282 w\u03b2 for a special case. Define the sequence x = (xk) such thatThe inclusion part of proof is trivial. Taking x \u2208 w\u03b2 for 1/2 < \u03b2 < 1 but x \u2209 w\u03b1 for 0 < \u03b1 < 1/2.So The following result is a consequence of \u03b1 \u2264 \u03b2 \u2264 1 be a positive real number. Then\u03b1 = \u03b2, then w\u03b1 = w\u03b2,if w\u03b1 \u2282 w for each \u03b1 \u2208  \u2282 S\u03b2, and the inclusion is strict.Let x \u2208 S\u03b1. Then given \u03b1 and \u03b2 such that 0 < \u03b1 \u2264 \u03b2 \u2264 1, we may writeS\u03b1 \u2282 S\u03b2.Let S\u03b1 \u2282 S\u03b2 for a special case. Define the sequence x = (xk) such thatx \u2208 S\u03b2 for 1/2 < \u03b2 \u2264 1, but x \u2209 S\u03b1 for 0 < \u03b1 \u2264 1/2.We show the strictness of the inclusion The following result is a consequence of \u03b1 \u2264 1 be a real number; then S\u03b1 \u2282 S.Let 0 <"}
{"text": "Also, we give some results about role of apoint in the existence of endpoints.We introduce  X, d) be a metric space, CB(X) the collection of all nonempty bounded and closed subsets of X, and H the Hausdorff metric with respect to d; that is, H = max\u2061{supx\u2208Ad, supy\u2208Bd} for all A, B\u2208CB(X), where d = inf\u2061y\u2208Bd. Let T : X \u2192 2X be a multifunction. An element x \u2208 X is said to be a fixed point of T whenever x \u2208 Tx. Also, an element x \u2208 X is said to be an endpoint of T whenever Tx = {x}  for all x \u2208 X. Suppose that \u03c8 :  and let d = |x \u2212 y|. Define T : X \u2192 CB(X) byx = 1 and y = 3/2, then\u03c8 :  and \u03b2 = 0 otherwise. First suppose that x \u2209  or that y \u2209 . If x, y \u2208  = 1. But, H = 0, and so \u03b2H \u2264 \u03c8). If x \u2208  = 0. Hence, \u03b2H \u2264 \u03c8). Now, suppose that x, y \u2208 . In this case, we have \u03b2 \u2265 1, H = H = (1/2)d, and N = max\u2061{d, x/2, y/2,  + d)/2}. Thus, d \u2264 N, and soT is a \u03b2-generalized weak contractive multifunction. Now, we show that T is \u03b2-admissible. If \u03b2 \u2265 1, then A, B \u2282 , and so Tx = {x/2}\u2208 and Ty = {y/2}\u2208 for all x \u2208 A and y \u2208 B. Thus, \u03b2 \u2265 1 for all x \u2208 A and y \u2208 B. Now, suppose A =  and x0 = 1/4. Then, Tx0 = {1/8}\u2208 and \u2282. Hence, \u03b2 \u2265 1. Now, we show that T satisfies the condition (H). First note that, for each \u025b > 0, there exists z \u2208 X such that sup\u2061a\u2208Tzd < \u025b. Now, we show that for each x \u2208 X there exists y \u2208 Tx such that H = sup\u2061b\u2208Tyd. If 0 \u2264 x \u2264 1, then Tx = {x/2}, T(x/2) = {x/4}, andx \u2264 3/2 we have 5/2 < 4x \u2212 (3/2) \u2264 9/2, T(4x \u2212 (3/2)) = {0}. Thus,x \u2264 9/2, then Tx = {0} and T(0) = {0}. Hence,T satisfies the conditions (R) and (K). Note that, 0 is the endpoint of T.Let X be a set and \u03b2 : 2X \u00d7 2X \u2192 [0, \u221e) a map. We say that the set X has the property (G\u03b2) whenever \u03b2 \u2265 1 for all subsets A and B of X with A\u2288B or B\u2288A.Now, we add an assumption to obtain uniqueness of endpoint. In this way, we introduce a new notion. Let X, d) be a complete metric space, \u03b2 : 2X \u00d7 2X \u2192 [0, \u221e) a mapping, and T : X \u2192 CB(X) a \u03b2-admissible, \u03b2-generalized weak contractive multifunction which has the properties (R), (K), and (H). Suppose that there exist a subset A of X and x0 \u2208 A such that \u03b2 \u2265 1. If T has the approximate endpoint property and X has the property (G\u03b2), then T has a unique endpoint.Let  = \u03b2 \u2265 1 because X has the property (G\u03b2). Hence,T has a unique endpoint.By using T has a unique endpoint, while X does has not the property (G\u03b2). Also, T has the property (R), while T is not lower semicontinuous. To see this, consider the sequence {xn} defined byk \u2265 1 and put y = 1/2 and x0 = 1. Then xn \u2192 1 and y \u2208 Tx0 = {1/2}. Let {yn} be an arbitrary sequence in X such that yn \u2208 Txn for all n \u2265 1. Then, yk\u221212 \u2208 Txk\u221212 and yk2 \u2208 Txk2 for all k. But, yk\u221212 = 4xk\u221212 \u2212 (3/2) for sufficiently large k and yk2 = xk2/2 for all k since yk\u221212 \u2192 5/2, yn\u219b1/2. This implies that T is not lower semicontinuous.In X, d) be a complete metric space, \u03b2 : 2X \u00d7 2X \u2192 [0, \u221e) a mapping, and T : X \u2192 CB(X) a \u03b2-admissible multifunction which has the properties (R), (K), and (H). Suppose that X has the property (G\u03b2), and there exist a subset A of X, x0 \u2208 A and k \u2208 [0,1) such that \u03b2 \u2265 1 and \u03b2H \u2264 kN for all x, y \u2208 X. Then T has a unique endpoint if and only if T has the approximate endpoint property.Let (\u03c8(t) = kt for all t \u2265 0. Then, It is sufficient that we define T and the property (R) are independent conditions [R) to obtain the next result. Its proof is similar to the proof of It has been proved that lower semicontinuity of the multifunction nditions . We can X, d) be a complete metric space, \u03b2 : 2X \u00d7 2X \u2192 [0, \u221e) a mapping, and T : X \u2192 CB(X) a lower semicontinuous, \u03b2-admissible, \u03b2-generalized weak contractive multifunction which has the properties (K) and (H). Suppose that there exist a subset A of X and x0 \u2208 A such that \u03b2 \u2265 1. Then T has the approximate endpoint property if and only if T has an endpoint.Let  be a complete metric space, \u03b2 : 2X \u00d7 2X \u2192 [0, \u221e) a mapping, and T : X \u2192 CB(X) a lower semicontinuous, \u03b2-admissible, \u03b2-generalized weak contractive multifunction which has the properties (K) and (H). Suppose that there exist a subset A of X and x0 \u2208 A such that \u03b2 \u2265 1. If T has the approximate endpoint property and X has the property (G\u03b2), then T has a unique endpoint.Let  be a complete metric space, \u03b2 : 2X \u00d7 2X \u2192 [0, \u221e) a mapping, and T : X \u2192 CB(X) a \u03b2-admissible multifunction which has the properties (R), (K), and (H). Suppose that X has the property (G\u03b2), and there exist a subset A of X, x0 \u2208 A and k \u2208 [0,1) such that \u03b2 \u2265 1 and \u03b2H \u2264 kN for all x, y \u2208 X. If T has the approximate endpoint property, then Fix\u2061(T) = End(T) = {x}.Let (\u03c8(t) = kt, then, by using Theorem 2.10 in [T has a fixed point. Since T has the approximate endpoint property, by using T has a unique endpoint such x. Let y \u2208 Fix\u2061(T). If Tx = Ty, then y = x. If Tx \u2260 Ty, then \u03b2 \u2265 1 because X has the property (G\u03b2). Also, we haveN = max\u2061{d, d, d,  + d)/2} = d. Thus, d = 0, and so Fix\u2061(T) = End(T) = {x}.If we put  2.10 in , T has aNext corollary shows us the role of a point in the existence of endpoints.X, d) be a complete metric space, x* \u2208 X a fixed element, and T : X \u2192 CB(X) a multifunction such that T has the property (H) and x* \u2208 Tx\u2229Ty for all subsets A and B of X with x* \u2208 A\u2229B and all x \u2208 A and y \u2208 B. Assume that H \u2264 \u03c8) for all x, y \u2208 X with x* \u2208 Tx\u2229Ty, where \u03c8 : [0, +\u221e)\u2192[0, +\u221e) is a nondecreasing upper semicontinuous function such that \u03c8(t) < t for all t > 0. Suppose that there exist a subset A0 of X and x0 \u2208 A0 such that x* \u2208 A0\u2229Tx0. Assume that for each convergent sequence {xn} in X with xn \u2192 x and x* \u2208 Txn\u22121\u2229Txn, for all n \u2265 1, one has x* \u2208 Txn\u2229Tx. Also, for each sequence {xn} in X with x* \u2208 Txn\u22121\u2229Txn for all n \u2265 1, there exists a natural number k such that x* \u2208 Txm\u2229Txn for all m > n \u2265 k. Then T has an endpoint if and only if T has the approximate endpoint property.Let  by \u03b2 = 1 whenever x* \u2208 A\u2229B and \u03b2 = 0 otherwise, and then we use It is sufficient we define X, d) be a complete metric space, x* \u2208 X a fixed element and T : X \u2192 CB(X) a lower semicontinuous multifunction such that T has the property (H) and x* \u2208 Tx\u2229Ty for all subsets A and B of X with x* \u2208 A\u2229B and all x \u2208 A and y \u2208 B. Assume thatx, y \u2208 X with x* \u2208 Tx\u2229Ty, where \u03c8 : [0, +\u221e)\u2192[0, +\u221e) is a nondecreasing upper semicontinuous function such that \u03c8(t) < t for all t > 0. Suppose that there exist a subset A0 of X and x0 \u2208 A0 such that x* \u2208 A0\u2229Tx0. Assume that for each convergent sequence {xn} in X with xn \u2192 x and x* \u2208 Txn\u22121\u2229Txn for all n \u2265 1, we have x* \u2208 Txn\u2229Tx. Then T has an endpoint if and only if T has the approximate endpoint property.Let  by \u03b2 = 1 whenever x* \u2208 A\u2229B and \u03b2 = 0 otherwise, and then we use It is sufficient to define X, d, \u2264) be an ordered metric space. Define the order \u2aaf on arbitrary subsets A and B of X by A\u2aafB if and only if for each a \u2208 A there exists b \u2208 B such that a \u2264 b. It is easy to check that (CB(X), \u2aaf) is a partially ordered set.Let  be a complete ordered metric space and T a closed and bounded valued multifunction on X such that T has the property (H) and Tx\u2aafTy for all subsets A and B of X with A\u2aafB and all x \u2208 A and y \u2208 B. Assume that H \u2264 \u03c8) for all x, y \u2208 X with Tx\u2aafTy, where \u03c8 : [0, +\u221e)\u2192[0, +\u221e) is a nondecreasing upper semicontinuous function such that \u03c8(t) < t for all t > 0. Suppose that there exist a subset A0 of X and x0 \u2208 A0 such that A0\u2aafTx0. Assume that for each convergent sequence {xn} in X with xn \u2192 x and Txn\u22121\u2aafTxn, for all n \u2265 1, one has Txn\u2aafTx. Also, for each sequence {xn} in X with Txn\u22121\u2aafTxn for all n \u2265 1, there exists a natural number k such that Txm\u2aafTxn for all m > n \u2265 k. Then T has an endpoint if and only if T has the approximate endpoint property.Let  = 1 whenever A\u2aafB and \u03b2 = 0 otherwise, and then we use Define X, d, \u2264) be a complete ordered metric space and T a closed and bounded valued multifunction on X such that T has the property (H) and Tx\u2aafTy for all subsets A and B of X with A\u2aafB, all x \u2208 A, and y \u2208 B. Assume that H \u2264 \u03c8) for all x, y \u2208 X with Tx\u2aafTy, where \u03c8 : [0, +\u221e)\u2192[0, +\u221e) is a nondecreasing upper semicontinuous function such that \u03c8(t) < t for all t > 0. Suppose that there exist a subset A0 of X and x0 \u2208 A0 such that A0\u2aafTx0. Assume that for each convergent sequence {xn} in X with xn \u2192 x and Txn\u22121\u2aafTxn, for all n \u2265 1, one has Txn\u2aafTx. Also, for each sequence {xn} in X with Txn\u22121\u2aafTxn for all n \u2265 1, there exists a natural number k such that Txm\u2aafTxn for all m > n \u2265 k. If T has the approximate endpoint property and A\u2aafB for all subsets A and B of X with A\u2288B or B\u2288A, then T has a unique endpoint.Let  = 1 whenever A\u2aafB and \u03b2 = 0 otherwise, and then we use Define X, d) be a metric space and T : X \u2192 2X a multifunction. We say that T is an HS-multifunction whenever for each x \u2208 X there exists y \u2208 Tx such that H = sup\u2061b\u2208Tyd. It is obvious that each HS-multifunction is an multifunction which has the property (H). Thus, one can conclude similar results to above ones for HS-multifunctions. Here, we provide some ones. Although by considering HS-multifunction we restrict ourselves, we obtain strange results with respect to above ones. One can prove the following by reading exactly the proofs of similar above results.Let  be a complete metric space, \u03b2 : 2X \u00d7 2X \u2192 [0, \u221e) a mapping, and T : X \u2192 CB(X) a \u03b2-admissible, \u03b2-generalized weak contractive HS-multifunction which has the properties (R) and (K). Suppose that there exist a subset A of X and x0 \u2208 A such that \u03b2 \u2265 1. Then T has an endpoint, and so T has the approximate endpoint property.Let  be a complete metric space, \u03b2 : 2X \u00d7 2X \u2192 [0, \u221e) a mapping, and T : X \u2192 CB(X) a lower semicontinuous, \u03b2-admissible, and \u03b2-generalized weak contractive HS-multifunction which has the property (K). Suppose that there exist a subset A of X and x0 \u2208 A such that \u03b2 \u2265 1. Then T has an endpoint, and so T has the approximate endpoint property.Let  be a complete metric space, x* \u2208 X a fixed element, and T : X \u2192 CB(X) an HS-multifunction such that x* \u2208 Tx\u2229Ty for all subsets A and B of X with x* \u2208 A\u2229B, all x \u2208 A, and y \u2208 B. Assume that H \u2264 \u03c8) for all x, y \u2208 X with x* \u2208 Tx\u2229Ty, where \u03c8 : [0, +\u221e)\u2192[0, +\u221e) is a nondecreasing upper semicontinuous function such that \u03c8(t) < t for all t > 0. Suppose that there exist a subset A0 of X and x0 \u2208 A0 such that x* \u2208 A0\u2229Tx0. Assume that for each convergent sequence {xn} in X with xn \u2192 x and x* \u2208 Txn\u22121\u2229Txn for all n \u2265 1 one has x* \u2208 Txn\u2229Tx. Also, for each sequence {xn} in X with x* \u2208 Txn\u22121\u2229Txn for all n \u2265 1, there exists a natural number k such that x* \u2208 Txm\u2229Txn for all m > n \u2265 k. Then T has an endpoint, and so T has the approximate endpoint property.Let  be a complete ordered metric space and T a closed and bounded valued lower semicontinuous HS-multifunction on X such that Tx\u2aafTy for all subsets A and B of X with A\u2aafB, all x \u2208 A, and y \u2208 B. Assume that H \u2264 \u03c8) for all x, y \u2208 X with Tx\u2aafTy, where \u03c8 : [0, +\u221e)\u2192[0, +\u221e) is a nondecreasing upper semicontinuous function such that \u03c8(t) < t for all t > 0. Suppose that there exist a subset A0 of X and x0 \u2208 A0 such that A0\u2aafTx0. Assume that for each sequence {xn} in X with Txn\u22121\u2aafTxn for all n \u2265 1, there exists a natural number k such that Txm\u2aafTxn for all m > n \u2265 k. Then T has an endpoint, and so T has the approximate endpoint property.Let ("}
{"text": "Mp\u2009(1 < p < \u221e) of holomorphic functions on the open unit disk \ud835\udd3b in the complex plane. These classes are in fact generalizations of the class M introduced by Kim (1986). The space Mp equipped with the topology given by the metric \u03c1p defined by \u03c1p = ||f \u2212 g||p = (\u222b0\u03c02logp(1 + M(f \u2212 g)(\u03b8))(\u03b8d/2\u03c0))p1/, with f, g\u2208Mp and Mf(\u03b8) = sup\u2a7dr<10\u2061|f(re\u03b8i)|, becomes an F-space. By a result of Stoll (1977), the Privalov space Np\u2009(1 < p < \u221e) with the topology given by the Stoll metric dp is an F-algebra. By using these two facts, we prove that the spaces Mp and Np coincide and have the same topological structure. Consequently, we describe a general form of continuous linear functionals on Mp (with respect to the metric \u03c1p). Furthermore, we give a characterization of bounded subsets of the spaces Mp. Moreover, we give the examples of bounded subsets of Mp that are not relatively compact.We consider the classes  \ud835\udd3b denote the open unit disk in the complex plane and let \ud835\udd4b denote the boundary of \ud835\udd3b. Let Lq(\ud835\udd4b)\u2009\u2009(0 < q \u2264 \u221e) be the familiar Lebesgue spaces on the unit circle \ud835\udd4b.Let M consists of all holomorphic functions f on \ud835\udd3b for which+\u2061|a| = max\u2061\u2061{log\u2061\u2061a, 0} andf. The Privalov class\u2009\u2009Np\u2009\u2009(1 < p < \u221e) consists of all holomorphic functions f on \ud835\udd3b for whichNp is denoted as Aq.Following Kim , 2), the, the2]),or whichsup\u20610<r<1p = 1, the condition  the classical Hardy space on \ud835\udd3b. It is known =lNp\u2009\u2009(1 < p < \u221e) was continued in 1977 by Stoll . Hence, f*(ei\u03b8) = g*(ei\u03b8) for almost every ei\u03b8 \u2208 \ud835\udd4b, and, by Riesz uniqueness theorem, we infer that f(z) = g(z) for all z \u2208 \ud835\udd3b. As, by . For a function f holomorphic in \ud835\udd3b and for any fixed 0 \u2264 \u03c1 < 1, denote by f\u03c1 the function defined on \ud835\udd3b as f\u03c1(z) = f(\u03c1z), z \u2208 \ud835\udd3b. Furthermore, for a given holomorphic function f on \ud835\udd3b, letFor simplicity, here as always in the sequel, we shall write f holomorphic on the unit disk \ud835\udd3b belongs to the class Mp if and only if it satisfiesA function f \u2208 Mp. Conversely, assume that f \u2208 Mp. Thenf \u2208 Mp; that is, \u222b0\u03c02(log\u2061+Mf(\u03b8))p(d\u03b8/2\u03c0) < \u221e, using (The condition  implies p. ThenMf\u03c1(\u03b8)\u27f6MfMp is closed under integration.The space f \u2208 Mp, defineF(rei\u03b8)|\u2264Mf(\u03b8), and thus MF(\u03b8) \u2264 Mf(\u03b8) for almost every \u03b8 \u2208 . Therefore F \u2208 Mp, as desired.For a given function f \u2208 Mp, f\u03c1 \u2192 f in Mp as \u03c1 \u2192 1\u2212.For each function f \u2208 Mp. Since f \u2208 N+, by Fatou's theorem, the radial limit f*(ei\u03b8) = lim\u2061r\u21921\u2212f(rei\u03b8) exists for almost every \u03b8 \u2208 . Hence, for such a fixed \u03b8, the function t \u21a6 f(tei\u03b8) is a continuous on , and thus it is uniformly continuous on . Therefore, for such a \u03b8, we havef \u2208 Mp, we obtainf\u03c1 \u2192 f in Mp as \u03c1 \u2192 1\u2212.Assume that act that  is satis we haveM(f\u2212f\u03c1) we will need the following lemmas.For the proof of completeness of the metric space \u03c1 \u2192 fn in Mp as \u03c1 \u2192 1\u2212, where this convergence is uniform with respect to n \u2208 \u2115.If {fn} is an arbitrary Cauchy sequence in Mp. Then for a given \u03b5 > 0 there is a k \u2208 \u2115 such thatn \u2265 k, we have\u03c10 < 1\u2009\u2009sufficiently near to 1, for whichSuppose that { we have\u03c1p\u2264\u03c1Mp\u2286Np is obvious, and . Let\u03c6. DefineU \u2208 Lp(\ud835\udd4b) and there is a constant Ap depending only on p such thatLp is the usual norm of the space Lp(\ud835\udd4b).Let 1 < We are now ready to state the following result.Mp = Np for each p > 1; that is, the spaces Mp and Np coincide.Mp\u2286Np for each p > 1. For the proof of the converse of this inclusion, assume that f \u2208 Np. We will show that f \u2208 Mp. As noticed in f can be factorized asI(z) is the inner function and F(z) is an outer function for the class Np; that is,\u03c9 is a constant of unit modulus. Furthermore, log\u2061+ | f* | \u2208Lp(\ud835\udd4b). As |I(z)|\u22641, for each z \u2208 \ud835\udd3b, the previous factorization and the fact that F \u2208 Mp immediately imply that f \u2208 Mp. Sincer < 1,+ | f* | \u2208Lp(\ud835\udd4b), we conclude by +MF(\u03b8) \u2208 Lp(\ud835\udd4b). This means that F \u2208 Mp and therefore f \u2208 Mp. Thus Np\u2286Mp, and therefore Mp = Np. This completes the proof.By that is,F(z)=\u03c9expf \u2208 Mp. ThenCp is a nonnegative constant depending only on p.Let F be the outer factor in the canonical factorization of f \u2208 Mp. From the proof of U(\u03b8) = log\u2061+MF(\u03b8) and \u03c6(\u03b8) = log\u2061+ | f*(ei\u03b8)| the inequality (F instead of f. Since |f(z)|\u2264|F(z)|, for each z \u2208 \ud835\udd3b, it follows that Mf(\u03b8) \u2264 MF(\u03b8) at almost every \u03b8 \u2208 ; thus <uch that\u03c1p((fn)r,Mp with the topology given by the metric \u03c1p defined by  follows from By , page 51By the Lebesgue dominated convergence theorem, we havek \u2208 \u2115 such that |c| \u2264 k. Then the triangle inequality yieldsf \u21a6 cf is a continuous map from Mp into Mp.Let The assertion (iv) is in fact the assertion of This concludes the proof.Mp, \u03c1p) and  have the same topological structure.We are now ready to prove that the (metric) spaces  and  have the same topological structure.For each j : Mp \u2192 Np. Then, by the inequality  given by Stoll .(ii)\u21d2(iii). It follows that from the obvious inequality |Np. Choose sufficiently small \u03b5 > 0 such that\u03b4, 0 < \u03b4 < \u03b5, such that (iii) holds. Choose an n \u2208 \u2115 for which 1/n < \u03b4. SetEk | = 1/n < \u03b4, and thus by (iii) we havef \u2208 Np there exists a measurable set Ef \u2282 \ud835\udd4b depending on f such that\u03b1 such that 0 < \u03b1 < \u03b5/\u03b4. Then using the inequalityf \u2208 L, we obtaindp < \u03b7, from which it follows that \u03b1L \u2282 V. Hence, L is a bounded subset of Np.(iii)\u21d2(i). Let we have\u222b02\u03c0, 0 < \u03b10 < 1, such thatf \u2208 L and |\u03b1 | \u2264\u03b10. It follows that\u03b5 > 0, choose \u03b7 > 0 satisfying\u03b10 = \u03b10(\u03b7) satisfying \u21d2(ii). Assume that uch thatows that\u222b02\u03c0(log\u2061ows that\u222b02\u03c0(log\u2061uch that\u03b4log\u2061p\u20611\u03b1+Mf(\u03b8))p : f \u2208 L} is uniformly integrable on \ud835\udd4b. The same assertion is also valid for the condition (iii). On the other hand, from the proof of +Mf(\u03b8))p : f \u2208 L} forms a bounded subset of the space L1(\ud835\udd4b); that is, there holds+|f*(ei\u03b8)|)p : f \u2208 L} is bounded in L1(\ud835\udd4b).Note that the condition (ii) from L is a subset of Mp for which the familyIf +Mf(\u03b8))p : f \u2208 L} is uniformly integrable on the circle \ud835\udd4b. This fact and the obvious inequality |f(rei\u03b8)|\u2264Mf(\u03b8), f \u2208 Mp, 0 \u2264 r < 1, for almost every \u03b8 \u2208 , imply that the family {(log\u2061+|f(rei\u03b8)|)p : f \u2208 L, 0 \u2264 r < 1} is uniformly integrable.The condition of Mp( = Np) to be bounded.The following result gives a necessary condition for a subset of L be a subset of Mp. If L is bounded in Mp, thenM\u221e = max\u2061\u03b8<2\u03c00\u2264 | f(rei\u03b8)|, K is a positive constant, and \u03c9(r), 0 \u2264 r < 1, is a positive continuous function that does not depend on f \u2208 L and for which \u03c9(r) \u2193 0 as r \u2192 1.Let f \u2208 Np, we haveL is a bounded subset of Np, by \u03b5 > 0 there exists \u03b4 = \u03b4(\u03b5) > 0, such thatE \u2282 \ud835\udd4b with the Lebesgue measure |E | <\u03b4.By the inequqlity 5.4) from the proof of Theorem 5.2 in  asfn} \u2282 Np and for each measurable set E \u2282 \ud835\udd4b we haveE| denotes the Lebesgue measure of E. From this and L = {fn} is bounded in Np.Define a sequence {E is relatively compact. This means that there exists a subsequence {fnk} of {fn} and a function f \u2208 Np such thatf(z) \u2261 e on \ud835\udd3b. On the other hand, from"}
{"text": "Genome rearrangements are studied on the basis of genome-wide analysis of gene orders and important in the evolution of species. In the last two decades, a variety of rearrangement operations, such as reversals, transpositions, block-interchanges, translocations, fusions and fissions, have been proposed to evaluate the differences between gene orders in two or more genomes. Usually, the computational studies of genome rearrangements are formulated as problems of sorting permutations by rearrangement operations.O(\u03b4n) time algorithm for solving the weighted sorting problem by CCLP operations when the weight ratio between reversals and non-reversal CCLP operations is 1:2, where n is the number of genes in the given chromosome and \u03b4 is the number of needed CCLP operations.In this article, we study a sorting problem by cut-circularize-linearize-and-paste (CCLP) operations, which aims to find a minimum number of CCLP operations to sort a signed permutation representing a chromosome. The CCLP is a genome rearrangement operation that cuts a segment out of a chromosome, circularizes the segment into a temporary circle, linearizes the temporary circle as a linear segment, and possibly inverts the linearized segment and pastes it into the remaining chromosome. The CCLP operation can model many well-known rearrangements, such as reversals, transpositions and block-interchanges, and others not reported in the biological literature. In addition, it really occurs in the immune response of higher animals. To distinguish those CCLP operations from the reversal, we call them as non-reversal CCLP operations. In this study, we use permutation groups in algebra to design an The algorithm we propose in this study is very simple so that it can be easily implemented with 1-dimensional arrays and useful in the studies of phylogenetic tree reconstruction and human immune response to tumors. Reversals, often called inversions in the biological literature, rearrange a segment of continuous integers on the chromosome by reversing the order of the integers and changing their signs [Transpositions act on two adjacent and non-overlapping segments on the chromosome by exchanging their locations [Block-interchanges function as a generalized transposition that exchanges two non-overlapping but not necessarily adjacent segments on the chromosome [Translocations affect two chromosomes by exchanging their end segments [Fusions merge two chromosomes into one chromosome and fissions split a chromosome into two chromosomes [Genome rearrangements are studied on the basis of genome-wide analysis of gene orders and important in the evolution of species -6. Sinceir signs ,7-11. Trocations ,12-15. Bromosome . Translosegments -22. Fusiomosomes ,18.inverted transposition  [Recently, great attention has been paid to the study of genome rearrangement using block-interchanges, since block-interchanges contain transpositions as a special case and, currently, the computational models involving block-interchanges are more tractable than those involving transpositions. More recently, Yancopoulos et al. defined a double cut and join (DCJ) operation that can model all the rearrangement operations described previously . The DCJsversal)  and othe\u2022 Case I \u2013 reversal:a site in Figure e site in Figure As illustrated in Figure \u2022 Case II \u2013 transposition:b site in Figure The temporary circle is cut in a new place  time algorithm for solving the problem, where n is the number of genes in the given chromosome and \u03b4 is the number of needed CCLP operations.All these seven rearrangements described above are simply called E = {1, 2, \u2026, n} be a set of n positive integers. Then a permutation of E is defined as a one-to-one function from E into itself and can simply be denoted by a product of some cycles. For example, a permutation expressed as \u03b1 =   means that \u03b1(1) = 6, \u03b1(6) = 4, \u03b1(4) = 1, \u03b1(2) = 5, \u03b1(5) = 3 and \u03b1(3) = 2. Basically, a cycle is cyclic and hence it does not matter which element in the cycle is written as the first. If the cycles in a permutation are all disjoint , then their product is called the cycle decomposition. If a cycle has k elements, then it is called a k-cycle. The element in a 1-cycle is usually called fixed. It is a convention that the 1-cycles in a permutation are not written explicitly. If all the elements in E are fixed in a permutation, then this permutation is called an identity permutation and simply denoted by 1 = (1)(2)\u22ef(n).Below, we introduce some definitions about the basics of permutation groups, as well as a couple of lemmas from Huang and Lu , that arcomposition (or product) of two given permutations \u03b1 and \u03b2 of E is a permutation, denoted by \u03b1\u03b2, such that \u03b1\u03b2(e) = \u03b1(\u03b2(e)) for all e \u2208 E. For example, suppose that \u03b1 =  and \u03b2 =  are two given permutations of E = {1,2, \u2026,6}. Then \u03b1\u03b2 = . It is not hard to see that if \u03b1 and \u03b2 are disjoint, then \u03b1\u03b2 = \u03b2\u03b1. The inverse of \u03b1, denoted by \u03b1\u20131, is a permutation such that \u03b1\u03b1\u20131 = \u03b1\u20131\u03b1 = 1. The conjugation of \u03b2 by \u03b1, denoted by \u03b1 \u22c5 \u03b2, is the permutation \u03b1\u03b2\u03b1\u20131.The \u03b1 =  be a 2-cycle and \u03b2 be an any permutation of E. If both a1 and a2 belong to the same cycle of \u03b2, then the effect of \u03b1\u03b2 (or \u03b2\u03b1) is equivalent to a fission acting on \u03b2 and hence \u03b1 is called a split operation of \u03b2. For instance, suppose that \u03b1 =  and \u03b2 =  . Then \u03b1\u03b2 =  and \u03b2\u03b1 = . On the other hand, if a1 and a2 belong to two different cycles of \u03b2, then the effect of \u03b1\u03b2 (or \u03b2\u03b1) equals to a fusion acting on \u03b2 and \u03b1 is called a join operation of \u03b2. For instance, if \u03b1 =  and \u03b2 = , then \u03b1\u03b2 =  and \u03b2\u03b1 = .As demonstrated in ,17,18, t\u03b1 of E can be written as a composition of 2-cycles in many ways [norm of \u03b1, denoted by ||\u03b1||, is the minimum number k such that \u03b1 can be expressed by a composition of k 2-cycles. The number of disjoint cycles in the cycle decomposition of \u03b1 is denoted by cn(\u03b1), which needs to count those non-expressed 1-cycles in \u03b1. For instance, if \u03b1 =  and E = {1, 2, \u2026,6}, then cn(\u03b1) = 3, rather than cn(\u03b1) = 2, because \u03b1 = (4). For any permutation \u03b1 of E, it can be shown that ||\u03b1|| = |E| \u2013 cn(\u03b1) [\u03b1 and \u03b2 of E, \u03b1 divides \u03b2, denoted by \u03b1|\u03b2, if and only if ||\u03b2\u03b1\u20131|| = ||\u03b2|| \u2013 ||\u03b1||. Actually, whether \u03b1 divides \u03b2 or not can be easily determined using the following lemma from [In fact, any permutation any ways . The normma from .Lemma 1[Let e1, e2, \u2026, ke \u2208 E and \u03b2 be any permutation of E. Then e1, e2, \u2026,k appear in the same cycle of \u03b2 in the order of ee1, e2, \u2026, k if and only ife |\u03b2.Lemma 1. Let e1,E as E = {\u00b11, \u00b12, \u2026, \u00b1n} for properly modeling reversals using the permutation groups, as described in Lemma 3 below. Let \u0393 =  \u00b7\u00b7\u00b7 . It is not difficult to verify that \u03932 = 1 and \u0393\u20131 = \u0393. If a cycle contains no e and \u2013e at the same time, where e \u2208 E, then it is called admissible and can be used to denote a DNA strand. Let \u03c0+ denote a strand of a DNA molecule \u03c0. Then \u03c0\u2013 = \u0393 \u00b7 (\u03c0+)\u20131 is the reverse complement of \u03c0+, representing another strand of \u03c0. Note that \u03c0+ and \u03c0\u2013 are disjoint. For the purpose of modeling reversals using the permutation groups, the DNA molecule \u03c0 is represented by the composition of its two strands \u03c0+ and \u03c0\u2013 , as demonstrated in [It is required to further extend the definition of rated in .Lemma 2 [Let \u03c0 and \u03c3 be two different chromosomes. Suppose that \u03b1 is a cycle in \u03c3\u03c0\u20131. Then (\u03c0\u0393) \u00b7 \u03b1\u20131is also a cycle in \u03c3\u03c0\u20131.Lemma 2 . Let \u03c0 a\u03b1 and (\u03c0\u0393) \u00b7 \u03b1\u20131 are mate cycles for each other in \u03c3\u03c0\u20131 according to Lemma 2.Actually, Lemma 3[Let u and v be in the different strands of a chromosome \u03c0, that is,  \u0142 \u03c0. Then \u03b3 = (\u03c0\u0393(v), \u03c0\u0393(u))  affects \u03c0 as a reversal.Lemma 3. Let u au, v) acts on \u03c0 as a join operation and (\u03c0\u0393(v),\u03c0\u0393(u)) acts on \u03c0 as a split operation, indicating that a reversal acting on \u03c0 can be implemented using the product of two 2-cycles and \u03c0. Actually, other non-reversal CCLP operations can be implemented by multiplying four 2-cycles (\u03c0\u0393(x),\u03c0\u0393(w))(\u03c0\u0393(v),\u03c0\u0393(u)) with the given chromosome \u03c0 if the following conditions are satisfied: (1) |\u03c0, (2)  \u0142 \u03c0 (3) w \u2260 \u0393(x) or \u0393(w) \u2260 x and (4) ) \u0142 \u03c0 or (\u0393(w), x) \u0142 \u03c0. The first condition is to make sure that  and (\u03c0\u0393(v),\u03c0\u0393(u)) respectively act on the two strands of \u03c0 as splits, which lead to two temporary circles excised from \u03c0. Note that these two temporary circles are complement to each other. The second condition is to make sure that  and (\u03c0\u0393(x), \u03c0\u0393(w)) respectively act on the two temporary circles and the cycles of the remaining \u03c0 as joins, which paste back the two temporary circles into the remaining \u03c0. It is worth mentioning that the joins also fulfill the process of linearization with possible inversion. The inversion is performed when the temporary circles are reinserted into the chromosome strands different from the ones they come from. The third and fourth conditions are to make sure that the resulting \u03c0 are admissible . Therefore, we have the following lemma.Note that in Lemma 3, (Lemma 4. Let \u03c0 be a chromosome and \u03b2 = (\u03c0\u0393(x), \u03c0\u0393(w))(\u03c0\u0393(v),\u03c0\u0393(u)). Suppose that the following four conditions are satisfied: (1) |\u03c0, (2)  \u0142 \u03c0 (3) w \u2260 \u0393(x) or \u0393(w) \u2260 x and (4) ) \u0142 \u03c0 or (\u0393(w), x) \u0142 \u03c0. Then \u03b2 affects \u03c0 as a non-reversal CCLP operation.\u03c0 into I =  using the CCLP operations when the weight ratio between reversals and non-reversal CCLP operations is 1:2. The basic idea behind this algorithm is as follows. As mentioned before, any permutation can be written as a product of 2-cycles and the effect of a reversal  acting on \u03c0 can be simulated by multiplying two  2-cycles with \u03c0. Moreover, the product of I\u03c0\u20131 and \u03c0 equals to I. All these facts indicate that one can derive a product of 2-cycles from I\u03c0\u20131 such that these 2-cycles perform as a sequence of CCLP operations to optimally transform \u03c0 into I. Below, for simplicity of describing our algorithm, x and y are said to be adjacent in a permutation \u03b1 if \u03b1(x) = y or \u03b1(y) = x.In this section, we design an efficient algorithm on the basis of the permutation groups that sorts a given chromosome Lemma 5. Let \u03c0 = \u03c0+\u03c0\u2013be a chromosome. Suppose that |I\u03c0\u20131and |\u03c0, that is, there are two elements x and y in a cycle of I\u03c0\u20131such that  acts on \u03c0 as a split. Let \u03b2 = (\u03c0\u0393(y), \u03c0\u0393(x)). Then there are two adjacent elements x\u2032 and y\u2032 in a cycle of I(\u03b2\u03c0)\u20131such that  and (\u03b2\u03c0\u0393(y\u2032),\u03b2\u03c0\u0393(x\u2032)) act on \u03b2\u03c0 as joins. Moreover, the cycles in \u03b2\u2032\u03b2\u03c0 are admissible, where \u03b2\u2032 = (\u03b2\u03c0\u0393(y\u2032), \u03b2\u03c0\u0393(x\u2032)).Proof. For convenience, let \u03c0 = \u03c0+\u2013\u03c0 = . The assumption |\u03c0 indicates that x and y are in the same cycle of \u03c0, say in \u03c0+, and hence \u03c0\u0393(x) and \u03c0\u0393(y) are in \u03c0\u2013. Hence, both  and (\u03c0\u0393(y),\u03c0\u0393(x)) act on \u03c0 as splits and \u03b2 = (\u03c0\u0393(y), \u03c0\u0393(x)) divides \u03c0 into four cycles. Let ia <ia+1 <n for 1 \u2264 i \u2264 k \u2013 2. This indicates that ka\u20131 is the maximum in ka\u20131 + 1 is not in I(\u03b2\u03c0)\u20131(a1) = I(ka\u20131) = ka\u20131 + 1, meaning that a1 and ka\u20131 + 1 are adjacent in I(\u03b2\u03c0)\u20131. In other words, there are two adjacent elements a1 and k\u2013a1 + 1 in I(\u03b2\u03c0)\u20131 such that , as well as (\u03b2\u03c0\u0393(ka\u20131 + l), \u03b2\u03c0\u0393((a1)), acts on \u03b2\u03c0 as a join. If the two cycles in (\u03b2\u03c0\u0393(ka\u20131 + 1),\u03b2\u03c0\u0393(a1))\u03b2\u03c0 are admissible , then we have completed the proof of this lemma based on Lemma 4. Now, suppose that the two cycles in (\u03b2\u03c0\u0393(ka\u20131 + 1),\u03b2\u03c0\u0393(a1))\u03b2\u03c0 are not admissible . We then show below that we can still find two other adjacent elements x\u2032 and y\u2032 in a cycle of I(\u03b2\u03c0)\u20131 such that  and (\u03b2\u03c0\u0393(y\u2032),\u03b2\u03c0\u0393(x\u2032)) can join \u03b2\u03c0 into two admissible cycles. First of all, ka\u20131 + 1 must be in \u03b2\u03c0\u0393(ka\u20131 + 1),\u03b2\u03c0\u0393(a1))\u03b2\u03c0 is an admissible chromosome), leading to that the cycle created by joining a1, ka\u20131 + 1) is not admissible. Further suppose that ja is the minimum in ja) = \u2013ja, which is the maximum in ja \u2265 ka\u20131 + 1  are adjacent in I(\u03b2\u03c0)\u20131 because I(\u03b2\u03c0)\u20131(j\u2013a\u20131) = I(\u2013ja). In the following, we consider five possibilities.Case 1.j a \u2260 \u2013n and ja \u2260 1. Then I(\u2013ja) = \u2013ja + 1, which is not in ja is the maximum in ja + 1 is in ka\u20131 cannot be the maximum in ja \u2265 ka\u20131 + 1 and hence \u2013ja + 1 >ka\u20131 which contradicts to our assumption that ka\u20131 is the maximum in I(\u2013ja) belongs to either ja\u20131, I(\u2013ja)) acts on \u03b2\u03c0 as a join and the cycles in (\u03b2\u03c0\u0393I(\u2013ja),\u03b2\u03c0\u0393(\u2013ja\u20131)))\u03b2\u03c0 are admissible.Case 2.j a = \u2013n and both 1 and \u20131 are not in I(\u2013ja) = 1 (instead of I(\u2013ja) = \u2013ja + 1 = n + 1). Because I(\u2013ja) belongs to either ja\u20131,I(\u2013ja)) acts on \u03b2\u03c0 as a join and (\u03b2\u03c0\u0393I(\u2013ja),\u03b2\u03c0\u0393(\u2013j\u2013a1)))\u03b2\u03c0 contains only admissible cycles.Case 3. ja = 1 and both n and \u2013n are not in I(\u2013ja) = \u2013n (instead of I(\u2013ja) = \u2013ja + 1 = 0 ). Clearly, I(\u2013ja) belongs to either ja\u20131,I(\u2013ja)) acts on \u03b2\u03c0 as a join and (\u03b2\u03c0\u0393I(\u2013ja), \u03b2\u03c0\u0393(\u2013ja\u20131)))\u03b2\u03c0 have two admissible cycles.Case 4.j a = \u2013n and 1 or \u20131 is in n, 1 and \u20131. Then we can exchange the roles of x\u2032 and y\u2032 in a cycle of I(\u03b2\u03c0)\u20131 such that  and (\u03b2\u03c0\u0393(y\u2032), \u03b2\u03c0\u0393(x\u2032)) can join the four cycles of \u03b2\u03c0 into two admissible cycles.Case 5. ja = 1 and n or \u2013n is in n and among them, ja is the smallest.According to the above discussion, we have completed the proof of this lemma.Theorem 1. Let \u03a6 denote a minimum weighted sequence of CCLP operations required to transform \u03c0 into I. Then the weight of \u03a6 is great than or equal toProof. Let \u03a6 contain a reversals and b non-reversal CCLP operations. It is not hard to see that a + 2b is the weight of \u03a6. Recall that the effect of a reversal can be simulated using two 2-cycles and a non-reversal CCLP operation using four 2-cycles. It indicates that \u03a6 can be written by a composition of 2a + 4b 2-cycles such that \u03a6\u03c0 = I, which equals to that I\u03c0\u20131 can be expressed as a composition of 2a + 4b 2-cycles. In other words, ||I\u03c0\u20131|| \u2264 2a + 4b. As mentioned before, we also have ||I\u03c0\u20131|| = |E| \u2013 cn(I\u03c0\u20131), which bases on the lemma proposed in [E| \u2013 cn(I\u03c0\u20131) \u2264 2a + 4b and, as a result, the weight of \u03a6 is great than or equal to posed in ,17. Therx and y in a cycle of I\u03c0\u20131 such that |\u03c0. Then, according to Lemma 5, we can always find a non-reversal CCLP operation \u03b2\u2032\u03b2 from I\u03c0\u20131 to rearrange \u03c0 into \u03b2\u2032\u03b2\u03c0, where \u03b2 = (\u03c0\u0393(y), \u03c0\u0393(x)) and \u03b2\u2032 = (\u03b2\u03c0\u0393(y\u2032),\u03b2\u03c0\u0393(x\u2032)). Assume that there are no any two adjacent elements x and y in a cycle of I\u03c0\u20131 such that |\u03c0, which implies that  \u0142 \u03c0. Then based on Lemma 3, (\u03c0\u0393(y), \u03c0\u0393(x)) can serve as a reversal to transform \u03c0 into (\u03c0\u0393(y), \u03c0\u0393(x))\u03c0. Using these properties, we design Algorithm 1 to sort \u03c0 into I by CCLP operations. It is not hard to see that a non-reversal CCLP operation derived in Algorithm 1 decreases the norm of I\u03c0\u20131 by 4 and a reversal by 2. Since non-reversal CCLP operations are weighted 2 and reversals are weighted 1, Algorithm 1 decreases the norm of I\u03c0\u20131 by 1 at the weight of Assume that there are at least two adjacent elements Theorem 2. Given a chromosome \u03c0, the weighted sorting problem by CCLP operations can be solved in O(\u03b4n) time when with weight ratio between reversals and non-reversal CCLP operations is 1:2, where \u03b4 is the number of CCLP operations needed to transform \u03c0 into I. Moreover, the weight of the optimal solution isthat can be calculated in O(n) time in advance.Proof. As discussed before, Algorithm 1 transforms \u03c0 into I by a minimum weighted sequence of \u03b4 CCLP operations, whose total weight is O(n) time. Below, the time-complexity of Algorithm 1 is analyzed. Basically, the computation in steps 1 and 2 can be done in O(n) time. As for step 3, there are \u03b4 iterations to perform. For each such iteration, it takes O(n) time to find  and  by determining every pair of adjacent elements in all the cycles of I\u03c0\u20131 and I\u03c0\u20131\u03b2, respectively, and a constant time to perform other operations in step 3.1, and also takes O(n) time to perform step 3.2. Therefore, the cost of step 3 is O(\u03b4n). Step 4 is executed in constant time. Totally, the time-complexity of Algorithm 1 is O(\u03b4n).x on a circular chromosome, a CCLP operation acting on x has an equivalent one without acting on x. Based on this property, one can further prove that the problem of sorting by CCLP operations is equivalent for circular and linear chromosomes.It is worth mentioning here that our algorithm is applicable to both circular and linear chromosomes. Actually, using similar discussion as in , one canO(\u03b4n) time algorithm for solving the weighted sorting problem by CCLP operations when the weight ratio between reversals and non-reversal CCLP operations is 1:2, where n is the number of genes and \u03b4 is the number of needed CLLP operations. As described in this article, this algorithm is very simple so that it can be easily implemented using 1-dimensional arrays and useful in the studies of phylogenetic tree reconstruction and human immune response to tumors. It would be an interesting future work to design efficient algorithms for solving the problem of sorting by CCLP operations when all the CCLP operations are weighted equally.In this article, we have introduced and studied the sorting problem by CCLP operations, where CCLP is a cut-circularize-linearize-and-paste operation that can model several known and unknown rearrangements. In addition, we have proposed an The authors declare that they have no competing interests.CLL conceived of this study, designed and analyzed its algorithm and drafted the manuscript. KHH and KTC participated in the design and analysis of the algorithm and the draft of the manuscript. All authors read and approved the final manuscript."}
{"text": "As a consequence, we deduce from it some hyperstability outcomes. Moreover, we also show how to use that result to improve some earlier stability estimations given by Isaac and Rassias.We prove a general result on Ulam's type stability of the functional equation  Its A be a nonempty set,  be a metric space, \ud835\udc9e \u2282 \u211d+A2 be nonempty, \ud835\udcaf be an operator mapping \ud835\udc9e into \u211d+A, and \u21311, \u21312 be operators mapping nonempty \ud835\udc9f \u2282 XA into XA2. We say that the equation\ud835\udcaf-stable provided for every \u03b5 \u2208 \ud835\udc9e and \u03c60 \u2208 \ud835\udc9f with\u03c6 \u2208 \ud835\udc9f of  Roughly speaking, \ud835\udcaf-stability of  solutioility of  is alwayility of ) to an eility of . The nexE1 and E2 be two normed spaces and let c \u2265 0 and p \u2260 1 be fixed real numbers. Let f : E1 \u2192 E2 be an operator such thatp \u2265 0 and E2 is complete, then there is a unique operator T : E1 \u2192 E2 that is additive  = T(x) + T(y) for x, y \u2208 E1) and such thatp < 0, then f is additive.Let p = 0 yields the result of Hyers )of Hyers  and it iof Hyers ; cf. alsof Hyers , 19) thashown in  that estp < 0, can be described as the \u03c6-hyperstability of the additive Cauchy equation for \u03c6 \u2261 c(||x||p + ||y||p) . The subsequent remark shows this.The second statement of e, e.g., , 21, 22;e, e.g., \u201325). It p < 0, a \u2265 0, I = , and f, T : I \u2192 \u211d be given by T(x) = 0 and f(x) = xp for x \u2208 I. Then clearlyx \u2264 y. Then (x + y)p \u2264 (2x)p = 2pxp \u2264 xp \u2264 xp + yp, whence |f(x + y) \u2212 f(x) \u2212 f(y)| = xp + yp \u2212 (x+y)p \u2264 xp + yp.Let In this paper we prove a quite general result that allows us to generalize and extend In the proof of the main theorem in this paper, we use the following fixed point result that can be easily derived from ). For a Z is a nonempty set,  is a complete metric space, f1, f2 : Z \u2192 Z, \ud835\udcaf : YZ \u2192 YZ is an operator satisfying the inequality+Z \u2192 \u211d+Z is an operator defined by\u03b5 : Z \u2192 \u211d+ and \u03c6 : Z \u2192 Y such thatn denotes the nth iterate of \u039b . Then there exists a unique fixed point \u03c8 of \ud835\udcaf withAssume that X, +), we denote by Aut\u2009\u2009X the family of all automorphisms of X. Moreover, for each u \u2208 XX we write ux : = u(x) for x \u2208 X and we define u\u2032 \u2208 XX by u\u2032x : = x \u2212 ux.Given a group  and  be commutative groups, d be a complete metric in E that is invariant  = d for x, y, z \u2208 X), H : (X\u2216{0})2 \u2192 \u211d+, andu \u2208 Aut\u2009\u2009X. Assume that f : X \u2192 E satisfies the inequality\ud835\udcb0 \u2282 l(X) such thatT : X \u2192 E fulfilling the inequalityLet (\ud835\udcb0 \u2282 l(X) be nonempty and let  = u\u2032x, and f2(x) = ux. Moreover,  And\u03be, \u03bc \u2208 EX0, x \u2208 X0, and u \u2208 \ud835\udcb0. Consequently, for each u \u2208 \ud835\udcb0, also ,n thatx \u2208 X0, n \u2208 \u21150 (nonnegative integers), and u \u2208 \ud835\udcb0. Hence,Z = X0, Y = E, \u03b5 : = \u03b5u, and \u03c6 = f. According to it, the limitx \u2208 X0 and u \u2208 \ud835\udcb0,Tu : X \u2192 E defined byTu\u2032 is a fixed point of \ud835\udcafu.Note that, in view of the definition of x, y \u2208 X0, x + y \u2260 0, n \u2208 \u21150, and u \u2208 \ud835\udcb0.Now we show thatn = 0 is just d. Then, by (Tu(x) + Tu(\u2212x) = 0 and consequently Tu(x \u2212 x) = Tu(0) = 0 = Tu(x) + Tu(\u2212x).In view of , it is oThen, by ,(34)Tu,tisfying  and 35)T : X \u2192 Ytions to  for all j \u2208 \u21150,j = 0 is exactly ,Tw view of ,(38)d,Twj \u2208 \u21150,d(T(x),TwTu = Tw for each u \u2208 \ud835\udcb0 , which  implies  with T :equality  means thThus we have completed the proof of X, H, E, and d be as in \ud835\udcb0 \u2282 l(X) such that (f : X \u2192 E satisfying (Let uch that  holds antisfying  is additf : X \u2192 E satisfies (T : X \u2192 E such that (H\ud835\udcb0(x) = 0 for x \u2208 X\u2216{0}, this means that f(x) = T(x) for x \u2208 X\u2216{0}, whencef is additive  proof of ).The next corollary corresponds to the results on the inhomogeneous Cauchy equation  in 30\u20133\u2013330\u201335.X, H, E, and d be as in F : X2 \u2192 E. Suppose thatF \u2260 0 for some x0, y0 \u2208 X\u2216{0}, and there exists a nonempty \ud835\udcb0 \u2282 l(X) such that (f : X \u2192 E.Let uch that  and 41)X, H, E, f : X \u2192 E is a solution to  = f(x0 + y0) \u2212 f(x0) \u2212 f(y0) = 0, which is a contradiction.Suppose that ution to . Then, with d = ||x \u2212 y||, and H : (X\u2216{0})2 \u2192 \u211d+. For each n \u2208 \u2124 define \u03bcn : X \u2192 X by \u03bcnx : = nx for x \u2208 X. Let H be given byp < 0, q < 0, c \u2265 0, and d \u2265 0. Thenx, y \u2208 X\u2216{0}, k, n \u2208 \u2124, and kn \u2260 0. Hence,M > 1 such thatf : X \u2192 E satisfying =c|H : (X\u2216{0})2 \u2192 \u211d+ has the following a bit more involved formp < 0, q < 0, c \u2265 0, and d \u2265 0 and additive injections \u03b7, \u03c7 : X \u2192 E1 . So, we have the following corollary corresponding to the hyperstability results in [Clearly, the above reasoning also works (after an easy modification) when the function sults in , 21, 24 sults in , 23, 25)H be given by . Then every f : X \u2192 E satisfying , becausex, y \u2208 X\u2216{0}, k, n \u2208 \u2124, and kn \u2260 0. So we have the following hyperstability result, as well .H be given by . Then every f : X \u2192 E satisfying , becausex, y \u2208 X\u2216{0}, k, n \u2208 \u2124, and kn \u2260 0. So, we have yet the following.It is easily seen that another example of the function tisfying  is givenH be given by . Then every f : X \u2192 E satisfying . Then, for every f : X \u2192 E satisfying  \u2264 2p\u2212. Consequently, by \ud835\udcb0 = {u0}, there is a unique additive T : X \u2192 E such thatLet  satisfy  and u0"}
{"text": "Characterizations in a uniform setting are proved, using techniques from thedomain of nonautonomous evolution equations with unbounded coefficients, and connections with the classic notion of trichotomy are given. The statements are sustained by several examples.We construct a framework for the study of dynamical systems that describe phenomena from physics and engineering in infinite dimensions and whose state evolution is set out by skew-evolution semiflows. Therefore, we introduce the concept of  The possibility of reducing the nonautonomous case in the study of associated evolution operators to the autonomous case of evolution semigroups on various Banach function spaces can be considered as an important way towards applications issued from the real world. Of great importance in the study the solution of differential equations is the approach by evolution families, as the techniques from the domain of non autonomous equations with unbounded coefficients in infinite dimensions can be extended in this direction.Appropriate for the study of evolution equations in infinite dimensions are also the skew-evolution semiflows, introduced by us in , as gene\u03c9-trichotomy for skew-evolution semiflows in Banach spaces and to give several conditions in order to describe the behavior related to the third subspace.The techniques used in the investigation of exponential stability and exponential instability were generalized for the case of exponential dichotomy in , 8 and fX, d) be a metric space, V a Banach space, and V* its topological dual. Let \u212c(V) be the space of all V-valued bounded operators defined on V. The norm of vectors on V and on V* and of operators on \u212c(V) is denoted by ||\u00b7||. Let us consider that Y = X \u00d7 V and T = { \u2208 \u211d+2 : t \u2265 t0}. I is the identity operator.Let  is called evolution cocycle over an evolution semiflow \u03c6 if it satisfies the following properties:A mapping \u03a6 : C : T \u00d7 Y \u2192 Y defined by\u03c6, is called skew-evolution semiflow on Y.The mapping f : \u211d+ \u2192 \u211d+* be a decreasing function with the property that there exists lim\u2061t\u2192\u221e\u2061f(t) = a > 0. We denote by \ud835\udc9e = \ud835\udc9e the set of all continuous functions x : \u211d+ \u2192 \u211d+, endowed with the topology of uniform convergence on compact subsets of \u211d+, metrizable by means of the distanceLet x \u2208 \ud835\udc9e, then, for all t \u2208 \u211d+, we denote that xt(s) = x(t + s), xt \u2208 \ud835\udc9e. Let X be the closure in \ud835\udc9e of the set {ft, t \u2208 \u211d+}. It follows that  is a metric space, and the mapping \u03c6 : \u0394 \u00d7 X \u2192 X, \u03c6 = \ud835\udccdt\u2212s is an evolution semiflow on X.If V = \u211d2, with the norm ||v|| = |v1| + |v2|, v =  \u2208 V. If u : \u211d+ \u2192 \u211d+*, then the mapping \u03a6u : \u0394 \u00d7 X \u2192 \u212c(V) defined by\u03c6, and C =  is a skew-evolution semiflow.We consider that A particular class of skew-evolution semiflows is emphasized in the following.C =  and a parameter \u03bb \u2208 \u211d. We define the mappingLet us consider a skew-evolution semiflow C\u03bb =  is also a skew-evolution semiflow, called \u03bb-shifted skew-evolution semiflow on Y. We will call \u03a6\u03bb the \u03bb-shifted evolution cocycle.We observe that X be the metric space defined in the first Example. We define the mapping \u03c60 : \u211d+ \u00d7 X \u2192 X, \u03c60 = xt, where xt(\u03c4) = x(t + \u03c4), for all \u03c4 \u2265 0, which is a classic semiflow on X. Let us consider for every x \u2208 X the parabolic system with Neumann's boundary conditions as follows:Let V = \u21122 be a separable Hilbert space with the orthonormal basis {en}n\u2208\u2115, e0 = 1, y \u2208 , n \u2208 \u2115.Let D(A) = {v \u2208 \u21122, v(0) = v(1) = 0}, and we define the operatorWe denote that \ud835\udc9e0-semigroup S, defined byV. For every \ud835\udccd \u2208 X, let us define an operator A(x) : D(A) \u2282 V \u2192 V, A(x) = x(0)A, which allows us to rewrite system  is a linear skew-product semiflow strongly continuous on Y. Also, for all v0 \u2208 D(A), we have obtained that v(t) = \u03a6x0, t \u2265 0, is a strongly solution of system  is a skew-product semiflow on Y, then the mapping C : T \u00d7 Y \u2192 Y, C = , \u03a6v), whereis a classic cocycle over the semiflow f system . As C0 Y. Hence, the skew-evolution semiflows generalize the notion of skew-product semiflows.is a skew-evolution semiflow on t, s, x) is the solution of the Cauchy problemMore directly, if \u03a6 is a linear skew-evolution semiflow.then Let us recall the definition of a semigroup of linear operators, and let us give an example which shows that this is generating a skew-evolution semiflow.S : \u211d+ \u2192 \u212c(V) is called semigroup of linear operators on V if the following relations hold:A mapping S : T \u00d7 X \u2192 \u212c(V), defined by \u03a6S = S(t \u2212 s), which is an evolution cocycle on V over evolution semiflows given, for example, by \u03c6 = xt\u2212s v, which are considered measurable. A particular case of skew-evolution semiflows is given by the following.Other examples of skew-evolution semiflows are given in . The asyC =  is \u2217-strongly measurable if, for every  \u2208 T \u00d7 X \u00d7 V*, the mapping given by s \u21a6 ||\u03a6)*v*|| is measurable on .A skew-evolution semiflow \u03c9-trichotomy. Some examples and connections with the classic concept of exponential trichotomy are also provided.We intend to give a new approach for the property of trichotomy for skew-evolution semiflows, the C : T \u00d7 Y \u2192 Y, C = , \u03a6v) be a skew-evolution semiflow on Y. We recall that a mapping P : X \u2192 \u212c(V) with the propertyLet projections family on V.is called P : X \u2192 \u212c(V) is said to be invariant relative to the skew-evolution semiflow C =  ifA projections family The splitting of the state space into three subspaces will be assured by the following.Pk}k\u2208{1,2,3} are said to be compatible with a skew-evolution semiflow C =  ifThree projections families {1) each of Pk, k \u2208 {1,2, 3} is invariant relative to C;(c2) \u2009for all x \u2208 X, the projections verify the relationsA\u03c8 \u2260 \u2205 and B\u03c8 \u2260 \u2205;(ii)\u03bc \u2208 A\u03c8 implies the existence of a constant M \u2208 \u211d+ such that(iii)\u03bb \u2208 Bh implies that there exists a constant m \u2208 \u211d+ such thatFor every function Let us denote thatC =  is called \u03c9-trichotomic if there exist three projections families {Pk}k\u2208{1,2,3} compatible with C and some functions \u03c8, \u03b6, \u03c7 : \u211d+ \u2192 \u211d+* with the propertiesA skew-evolution semiflow t1)P1(x)v|| \u2264 \u03c8(t \u2212 t0)||P1(x)v||;||\u03a6(t2)(\u03b6(t \u2212 t0)||P2(x)v|| \u2264 ||\u03a6P2(x)v||;t3)P3(x)v|| \u2264 \u03c7(t \u2212 t0)||P3(x)v|| and \u03c7(t \u2212 t0)||\u03a6P3(x)v|| \u2265 ||P3(x)v||,||\u03a6 \u2208 T and all  \u2208 Y.for all  be a decreasing function with the property that there exists lim\u2061t\u2192\u221e\u2061f(t) = l > 0. Let us denote that \u03bb > f(0). Let us consider the Banach space V = \u211d3 with the norm |||| = |v1| + |v2| + |v3|, v =  \u2208 V. The mappingt \u2265 t0 \u2265 0,  \u2208 Y, is an evolution cocycle over the evolution semiflow given in P1, P2, P3 : X \u2192 \u212c(\u211d3) by P1(x)v = , P2(x)v = , P3(x)v = , for all x \u2208 X and all v =  \u2208 \u211d3. The following inequalitiesLet t, s),  \u2208 T and all  \u2208 Y. The mappings \u03c8, \u03b6, \u03c7 : \u211d+ \u2192 \u211d+*, defined byhold for all ,  \u2208 T and x \u2208 X. We remark that the following relations hold:k = Pk(x), \u2200 \u2208 \u211d+ \u00d7 X;\u03a6k)\u03a6k = \u03a6k, \u2200,  \u2208 T, \u2200x \u2208 X.\u03a6for every  is \u03c9-trichotomic if and only if there exist some constants N1, N2, N3 \u2265 1, \u03bd1, \u03bd2, \u03bd3 > 0 and three projections families {Pk}k\u2208{1,2,3} compatible with C such that A skew-evolution semiflow t, t0) \u2208 T and all  \u2208 Y.for all  \u2264 Me\u03bct, for all\u2009\u2009t \u2265 0, and for \u03bb \u2208 B\u03c8 there exists a constant m \u2208 \u211d+ with the property \u03c8(t) \u2265 me\u03bbt, for all t \u2265 0.such that relations (i)\u2013(iii) of N1 \u2265 1 and \u03bd1 > 0, such that \u03c8(t) \u2264 N1e\u03bd1t\u2212, for all t \u2265 0. Hence, relation  \u2265 N2e\u03bd2t, for all t \u2265 0, which implies  is obt implies 2). From Hence, relation 3) is satSufficiency. Let us assume that there exist three projections families compatible with C and N1, N2, N3 \u2265 1, \u03bd1, \u03bd2, \u03bd3 > 0 such that relations \u2013(et3)3SufficieWe obtain the relationsC is \u03c9-trichotomic, which ends the proof.Hence, relations (i)\u2013(iii) of We obtain a characterization for the property of trichotomy, by means of the shifted skew-evolution semiflow.C =  is \u03c9-trichotomic if and only if there exist three projections families {Pk}k\u2208{1,2,3} compatible with C and1 is exponentially stable;the evolution cocycle \u03a62 is exponentially instable;the evolution cocycle \u03a6\u03bb > 0 such that the \u03bb-shifted evolution cocycle \u03a6\u03bb3 is exponentially stable and the \u2212\u03bb-shifted evolution cocycle \u03a6\u03bb\u22123 is exponentially instable.there exists a constant A skew-evolution semiflow Necessity. Statements (i) and (ii) are obtained immediately from the necessity of N3 \u2265 1 and \u03bd3 > 0 such that\u03bb = 2\u03bd3 > 0. We obtain successivelyLet us consider that t, t0) \u2208 T and all  \u2208 Y, which shows that \u03a6\u03bb3 is exponentially stable. Also, we havefor all ,  \u2208 T and all  \u2208 Y, which proves that \u03a6\u03bb\u22123 is exponentially instable.for all (Sufficiency. Relations (i) and (ii) in \u03bb > 0 such that the skew-evolution semiflow \u03a6\u03bb3 is exponentially stable, then there exists some constants N \u2265 1 and \u03bd > 0 such thatFurther, we obtaint, s),  \u2208 T and all  \u2208 Y. Denotingfor all (the first relation in 3) is obt\u03bb > 0 such that the \u2212\u03bb-shifted skew-evolution semiflow C\u03bb\u2212 is exponentially instable, then there exist some constants N \u2265 1 and \u03bd > 0 such thatIf there exists a constant t, s),  \u2208 T and all  \u2208 Y. If we consider \u03bd3 defined as in  is obt\u03c9-trichotomy is given relative to the dual space V* of the Banach space V. To this aim, let us consider three projections families {Pk}k\u2208{1,2,3} compatible with C such that the evolution cocycle \u03a61 has exponential growth and the evolution cocycle \u03a62 has exponential decay.Another characterization for the property of strongly measurable skew-evolution semiflow C =  is \u03c9-trichotomic if the and only if the following statements hold:(i)p > 1, all  \u2208 T, and all  \u2208 X \u00d7 V*;there exists (ii)p > 1, all  \u2208 T, and all  \u2208 Y;there exists (iii)\u03b1 < 0 and t0 \u2208 \u211d+ and all  \u2208 Y and there exist \u03b2 < 0 and t, t0) \u2208 T and all  \u2208 Y.there exist A \u2217-Necessity.  (i) As C is \u03c9-trichotomic, according to \u03c8 : \u211d+ \u2192 \u211d+*, with the property An equivalent relation is obtained, if we consider t, t0) \u2208 T and all  \u2208 Y. According to the properties of function \u03c8, there exist N \u2265 1 and \u03bd > 0 such that the following relations hold:for all  \u2208 T and all  \u2208 X \u00d7 V*, where we have denoted that for all  According to \u03b6 assures the existence of some constants N \u2265 1 and \u03bd > 0, such that, for all p > 0, the following relations hold:The property of function t, t0) \u2208 T and all  \u2208 Y, where we have denoted that for all ((iii) Both relations are obtained by a similar proof as in (i) and (ii), according to Sufficiency.  (i) As the evolution cocycle \u03a61 has exponential growth, there exist M \u2265 1 and \u03c1 > 0 such thatt \u2265 t0 + 1. We consider thatLet p > 1 there exists q > 1 such that 1/p + 1/q = 1. We obtainFor every Hence, we have thatt \u2208  we obtainFor which implies thatFurther, the following relations hold:t0, t], it follows thatBy integrating on ,and, if x, v) \u2208 Y. Hence,\u03b6 : \u211d+ \u2192 \u211d+* defined byfor all (t2) of with the property \u03b1-shifted evolution cocycle \u03a6\u03b13 in (i) and for the \u03b2-shifted evolution cocycle \u03a6\u03b23 in (ii) leads to the existence of two functions \u03c71, \u03c72 : \u211d+ \u2192 \u211d+*, with the properties (iii) A similar proof for the If we consider the definition of the shifted evolution cocycle, the previous relations are equivalent witht3) of We define \u03c9-trichotomy.The property described by relation  is also"}
{"text": "Three new theorems based on the generalized Carleson operators for the periodic Walsh-type wavelet packets have been established. An application of these theorems as convergence a.e. for the periodic Walsh-type wavelet packet expansion of block function with the help of summation by arithmetic means has been studied. Lp-function, 1 < p < \u221e, as the Walsh-Fourier series. Walsh-type wavelet packet expansion has been studied by the researchers Billard q1/, 1 < q < \u221e. Let \ud835\udd39q denote the space of measurable functions f on  with a unique expansion x = \u2211j=0\u221exj2j\u22121\u2212  by their associated binary sequence {xi}. For two such points x, y \u2208 , define We can carry over the operator \u2295 to the interval  by identifying those x, y \u2208 . With this definition, we havex, y for which x \u2295 y is defined, , and define {fn}n\u22652 recursively byLet n, J \u2208 \u2115, 2J \u2264 n < 2J+1.ThenUk = u1 + u2 + \u22ef+uk for k = 1,2,\u2026, n; it is also called Abel's transformation. ConsiderWn}n=0\u221e be the Walsh system. ThenC is a finite positive constant, K \u2265 1, 2K \u2264 n < 2K+1, and for all pairs x, y \u2208 \u2212log\u2061 \u2265 |a \u2212 b | /2. This completes the proof of Therefore, using f in the periodic Walsh-type wavelet packets,The dyadic arithmetic mean of partial sums for the expansion of a measurable (integrable) function ackets,4(\u03c32Nf)(x)q-block \u03b2 is associated with the dyadic interval I \u2282 [0,1). If 1 < \u03b1|I|, then |I|q1\u2212/\u03b1q \u2264 1/\u03b1, and using the fact that the operator f \u2192 \ud835\udd3ecdf(x)) is of strong type . We haveSuppose that the \u03b1|I| with I = [a, b). Put Now suppose that 1 \u2265 KN denote the operator kernel associated with the projection operators C (independent of N) such that Fix  Terence ). KN, we obtainUsing the estimate  on the k\u03b2||1 \u2264 1 and x \u2212 a | , |x \u2212 b | \u2265|I | /2, therefore,I and \u03b2 and hence Since ||\u03b1 > 0 and a q-block \u03b2 supported on the dyadic interval I \u2282 [0,1); two cases are considered.Fix \u03b1 | I|, then |I|q1\u2212/\u03b1q \u2264 1/\u03b1. Therefore, using Theorem 5.1. [If 1 < rem 5.1. , page 27\u03b1 | I| with I = [a, b). LetLet 1 \u2265 Notice thatx \u2208 [0,1), we haveFor E\u03b1l1,l2| withHence, it suffices to estimate |x \u2208 \u211d\u2216I; thenx \u2208 E\u03b1l1,l2, there is an increasing sequence Jk \u2192 \u221e for whichC > 0 and for k = 1,2,\u2026. Since Jk \u2192 \u221e, thereforeFix E\u03b1l1,l2| \u2264 1/\u03b1 and consequentlyUsing that f = \u2211k=1\u221eck\u03b2k be a function of \ud835\udd39q. ThenL1 convergence of the average sum defining f. SinceLet thereforeThis completes the proof of Following corollary can be deduced from our theorems.f \u2208 \ud835\udd39q, 1 < q < \u221e, in Let Let us write f = \u2211k=1\u221eck\u03b2k \u2208 \ud835\udd39q, let gK = \u2211k=1Kck\u03b2k, and observe that ||f\u2212gK||\ud835\udd39q \u2192 0. For each x \u2208 [0,1), writeWith ThusFrom this it follows thatThis completes the proof of the corollary."}
{"text": "We investigate the oscillation of a class of fractional differential equations with damping term. Based on a certain variable transformation, the fractional differential equations are converted into anotherdifferential equations of integer order with respect to the new variable. Then, using Riccati transformation,inequality, and integration average technique, some new oscillatory criteria for the equations are established. As for applications, oscillation for two certain fractional differential equations with damping term is investigated by the use of the presented results. In the investigations of qualitative properties for differential equations, research of oscillation has gained much attention by many authors in the last few decades . In Dt\u03b1(\u00b7) denotes the modified Riemann-Liouville derivative with respect to the variable t, the function a \u2208 C\u03b1, R+), r \u2208 C\u03b12, R+), p, q \u2208 C, R+) and C\u03b1 denotes continuous derivative of order \u03b1.In , Jumarie\u03b1 are listed as follows :(2)x(t) of  of  is callex(t) of  is calleWe organize the next as follows. In \u03be = t\u03b1/\u0393(1 + \u03b1), \u03bei = ti\u03b1/\u0393(1 + \u03b1), i = 0,1, 2,3, 4,5, R+ = , h1, h2, H \u2208 C, R) satisfyH has continuous partial derivatives \u2202H/\u2202\u03be and \u2202H/\u2202s on \u22124/3\u222b\u03be0\u03bes\u22124/3ds) = exp\u2061(3[\u0393(4/3)]\u22124/3[\u03be0\u22121/3 \u2212 \u03be\u22121/3]), which implies 1 \u2264 A(\u03be) \u2264 exp\u2061(3[\u0393(4/3)]\u22124/3\u03be0\u22121/3). On the other hand, T > \u03be2 such that T, \u221e). In \u222b\u03be0d\u03b6=\u221e.In , lettingerefore,  is oscilConsider the following fractional differential equation:t0 = 2, \u03b1 = 1/2, a(t) = t1/4, r(t) \u2261 1, q(t) \u2261 1, then we obtain (\u03be0 = 21/2/\u0393(3/2), A(\u03be) = exp\u2061([\u0393(3/2)]\u22121/2\u222b\u03be0\u03bes\u22123/2ds) = exp\u2061([\u0393(3/2)]\u22121/2[2\u03be0\u22121/2 \u2212 2\u03be\u22121/2]), which implies 1 \u2264 A(\u03be) \u2264 exp\u2061(2[\u0393(3/2)]\u22121/2\u03be0\u22121/2). On the other hand, T > \u03be2 such that T, \u221e).In , if we se obtain . So \u03be0 \u03bb = 2, for any sufficiently large l, we haveFrom the analysis above, one can see the  holds. Wds=\u221e.In ,59)\u222b\u03be0\u222b\u03be0(58)\u222bt\u03b6=\u221e.So,  hold. On\u03b6=\u221e.So, , after pds=\u221e.So  holds, ads=\u221e.So  is oscil"}
{"text": "G-metric spaces, fixed points of maps that satisfy the generalized -Chatterjea type contractive conditions are obtained. The results presented in the paper generalize and extend several well known comparable results in the literature.In the framework of ordered  P in G-metric spaces. Recently, Shatanawi \u2009\u2009for all\u2009\u2009x, y \u2208 X;ype (see ) if therChatterjea type [k \u2208  \u2264 k[d + d] for all x, y \u2208 X.jea type  if thereLet \u2192[0, \u221e) is continuous and nondecreasing with \u03c8(t) = 0\u2009\u2009if and only if t = 0} and\u03a8 = {\u2009\u03c6 | \u03c6 : [0, \u221e)5 \u2192 [0, \u221e) is lower semi-continuous with \u03c6 = 0 if and only if t1 = t2 = t3 = t4 = t5 = 0}.\u03a6 = {We define two classes of mappings as follows:G-metric space is said to have a sequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence) {xn} in\u2009\u2009X\u2009\u2009such that G \u2192 0 as n \u2192 \u221e implies that xn\u2aafx\u2009\u2009(x\u2aafxn), respectively.An ordered partial \u03c6, \u03c8)-Chatterjea type contractive conditions on partially ordered complete generalized metric space. We start with the following result.In this section, we obtain fixed point results for mappings satisfying generalized\u2009\u2009 be a partially ordered set and f be a nondecreasing self mapping on a complete G-metric space X satisfying\u03c8 \u2208 \u03a8, \u03c6 \u2208 \u03a6 withx, y \u2208 X with x\u2aafy. Suppose that there exists x0 \u2208 X with x0\u2aaffx0. If f is continuous or X a sequential limit comparison property, then f has a fixed point in X.Let  > 0 for every\u2009\u2009n \u2208 \u2115. If not, then xn = xn+1 for some n and xn becomes a fixed point of f. Using  \u2265 G for some n \u2265 0, it follows that \u03c6) = 0, a contradiction. Therefore, for all n \u2265 0,M = G. Now {G} is a decreasing sequence, so there exists L \u2265 0 such that lim\u2061n\u2192\u221e\u2061G = L. This gives lim\u2061n\u2192\u221e\u2061G\u2009 = \u2009lim\u2061n\u2192\u221e\u2061M = L. By lower semicontinuity of \u03c6,L = 0. Taking the upper limits as n \u2192 \u221e on both sides of\u03c6 = 0 and we conclude thatIf f. Using , we obtaxn} is a G-Cauchy sequence in\u2009\u2009X. If not, then there exist \u03b5 > 0 and integers nk and mk with mk > nk > k such that\u03b5 \u2264 lim\u2061k\u2192\u221e\u2061G.Next, we show that {ffect of  and 14)xn} is ak\u2192\u221e\u2061G \u2264 \u03b5 so that\u03b5 \u2264 lim\u2061k\u2192\u221e\u2061G, and\u03b5 \u2264 lim\u2061k\u2192\u221e\u2061G.On the other hand,ned with  and 16)(18)G(xnkplies by  and 19)(18)G \u2264 \u03b5. Thus,k \u2192 \u221e implies that\u03b5 > 0.But fromher with  and 16)(25)G(xnk\u03b5).From , we obtaxn} is a G-Cauchy sequence and by G-completeness of X, there exists u \u2208 X such that {xn}G-converges to u as n \u2192 \u221e. If f is continuous, then it is clear that fu = u. Next, if X has a sequential limit comparison property, then we have xn\u2aafu for all n \u2208 \u2115. From  = G. Thus, from , G, 0) = 0\u2009\u2009so that\u2009\u2009G = 0\u2009\u2009and, hence, fu = u.It follows that { \u2115. From , we have\u2115. From \u2009\u2009be a partially ordered set and\u2009\u2009f\u2009\u2009be a nondecreasing self mapping on a complete G-metric space\u2009\u2009X\u2009\u2009satisfying\u03c8 \u2208 \u03a8, \u03c6 \u2208 \u03a6 withx, y \u2208 X with x\u2aafy with ai \u2265 0 for all i = 1,2, 3,4 with a1 + a2 + a3 + 2a4 \u2264 1. Suppose that there exists x0 \u2208 X with x0\u2aaffx0. If f is continuous or X has a sequential limit comparison property, then f has a fixed point in X.Let\u2009\u2009 = max\u2061{|x \u2212 y|, |y \u2212 z|, |z \u2212 x|} be a G-metric on X. Define f : X \u2192 X by\u03c8(t) = (3/4)t and \u03c6 = (1/12)(t1 + t2 + t3 + t4 + t5) for all t, t1, t2, t3, t4, t5 \u2208 [0, \u221e).Let x, y \u2208 X with x\u2aafy, we havea1 = a2 = a3 = a4 = 1/6, where a1 + a2 + a3 + 2a4 \u2264 1. Hence, the conditions of f.Now, for all ).Thus,  is satisX, \u2aaf) be a partially ordered set and f be a nondecreasing self mapping on a complete G-metric space X satisfying\u03c6 \u2208 \u03a6,x, y \u2208 X with x\u2aafy. Suppose that there exists x0 \u2208 X with x0\u2aaffx0. If f is continuous or X has a sequential limit comparison property, then f has a fixed point in X.Let  be a partially ordered set and f be a nondecreasing self mapping on a complete G-metric space X satisfying\u03c6 \u2208 \u03a6,x, y \u2208 X with x\u2aafy, where ai > 0 for i = 1,2, 3,4 with a1 + a2 + a3 + 2a4 \u2264 1. Suppose that there exists\u2009\u2009x0 \u2208 X\u2009\u2009with\u2009\u2009x0\u2aaffx0. If\u2009\u2009f\u2009\u2009is continuous or\u2009\u2009X\u2009\u2009has a sequential limit comparison property, then f has a fixed point in\u2009\u2009X.Let ("}
{"text": "Lp spaces, 1 \u2264 p < \u221e for univariate and multivariate functions, respectively. Furthermore, we obtain these types of approximation theorems by means of \ud835\udc9c-summability which is a stronger convergence method thanordinary convergence.Korovkin-type theorem which is one of the fundamental methods inapproximation theory to describe uniform convergence of any sequence ofpositive linear operators is discussed on weighted  Lp-spaces ..Lp-spacLp-spaces was obtained by Gadjiev and Aral  and \u03c72A(t) = 1 \u2212 \u03c71A(t) for any A \u2265 0. We can choose a sufficient large A such that for every \u03b5 > 0Let k, n, and f and the linearity of the operators Lj, we getUsing the assumption of the convergence of the series  for eachK > 0 such thatBy condition , there eHence, from , we compf \u2208 Lp,\u03c9(\u211d) the inequalityLp,\u03c9 \u2282 Lp. Since the space of continuous functions is dense in Lp, given f \u2208 Lp,\u03c9(\u211d), for each \u03b5\u2032 > 0, there exists a continuous function \u03c6 on  satisfying the condition \u03c6(x) = 0 for |x| > A such thatFor every function Using the inequalities  and 18)18), we g\u03c72A1\u03c71A\u03c6 = 0 for some A1 > A, we get the inequalityOn the other hand, since M\u03c6 = max\u2061t\u2208\u211d|\u03c6(t)|\u03c71A(t), we getNow, by supposing that \u03c9 \u2208 L1(\u211d), we can choose the number A1 such thatSince Using this inequality, we havenIn\u2032:As a corollary, we get the following inequality for sup\u2061\u03c6\u03c71A is a continuous function on , for any given \u03b5\u2032 > 0, there exists a \u03b4 > 0 such thatSince t \u2208  and that x \u2208  is as follows:bk) is a sequence of positive real numbers satisfying the conditionThe Kantorovich variant of the Sz\u00e1sz-Mirakyan operators  by replaIt is known thatFurthermore by simple calculations, we obtainAlso,f \u2208 Lp,\u03c9(\u211d), we haveHence, conditions , 11) ar ar11) arLp,\u03a9(\u211dm), m \u2208 \u2115, instead of Lp,\u03a9(\u211dn) to avoid any confusion about the indices of \ud835\udc9c = {An}.Also, analogue of \ud835\udc9c = {An} be a sequence of infinite matrices with nonnegative real entries and let {Lj} be a sequence of positive linear operators from Lp,\u03a9(\u211dm) into Lp,\u03a9(\u211dm). Assume thatLet f \u2208 Lp,\u03a9(\u211dm), one hasIf\u03c71A of the ball |x| \u2264 A and \u03c72A(t) = 1 \u2212 \u03c71A(t), it is possible to choose a sufficient large A such thatConsidering the characteristic function K such that ||Akn) = 0 for |x| > A such thatOn the other hand, by condition  there exk, n, and f, and using the linearity of the operators Lj, which means the linearity of Akn)|\u03c71A(t). Furthermore, we can choose A1 such that ||\u03c72A1||p,\u03a9 < \u03b5\u2032/M\u03b8, and for sufficiently large k, we estimate sup\u2061n||Akn)((1)\u22121||p,\u03a9 < \u03b5\u2032/M\u03b8. Using these estimations in  = ex\u2212y\u2212. Note that this selection of \u03a9 satisfies condition  with an integral mean of f over the interval  \u00d7  is as follows:bk) is a sequence of positive real numbers satisfying the conditionThe Kantorovich variant of the Sz\u00e1sz-Mirakyan operators  by replaIt is known thatFurthermore we obtainAlso,f \u2208 Lp,\u03a9(\u211d2), we haveHence, conditions  and 11)11) are p"}
{"text": "Many known results appear as special consequences of our work.The aim of this paper is to study the problem of coefficient bounds for a newly defined subclass of  \ud835\udc9c(p) denote the class of functions f(z) of the form\ud835\udd4c = {z : |z| < 1}. Also let \ud835\udcaep* and \ud835\udca6p denote the well-known classes of p-valent starlike functions and p-valent convex functions, respectively.Let f(z) \u2208 \ud835\udc9c(p) given by (g(z) \u2208 \ud835\udc9c(p) given byf(z) and g(z) is given byD\u03b4+p\u22121 for p-valent analytic functions given by\u03b4 > \u2212p,x)n is a Pochhammer symbol given by\u03b4 is any integer greater than \u2212p,\ud835\udcb1\ud835\udc9fp\u03bb of p-valent analytic functions as follows.For given by  and g(z)given byg(z)=zp+\u2211p-valent function f(z) of the form , if and only if\u03b1 \u2265 0,\u2009\u2009b \u2208 \u2102\u2216{0}, \u03b4 > \u2212p, \u03bb is real with |\u03bb| < (\u03c0/2), and 0 \u2264 \u03b2 < 1.An analytic the form  belongs \u03b1, \u03b2, \u03bb, p, b, and \u03b4 in \ud835\udcb1\ud835\udc9fp\u03bb, we obtain many important subclasses studied by various authors in earlier papers; see for details [\ud835\udcb1\ud835\udc9f1\u03bb \u2261 \ud835\udcae\u03bb* and \ud835\udcb1\ud835\udc9f1\u03bb \u2261 \ud835\udca6\u03bb, studied by Spacek [y Spacek  and Robey Spacek , respecty Spacek \u201311;\ud835\udcb1\ud835\udc9f10 \u2261 \ud835\udcae\ud835\udc9f and \ud835\udcb1\ud835\udc9f10 \u2261 \ud835\udca6\ud835\udc9f, studied by both Owa et al. and Shams et al. [s et al. , 13;\ud835\udcb1\ud835\udc9f1\u03bb \u2261 \ud835\udcb0\ud835\udcae\ud835\udcab(\u03bb) and \ud835\udcb1\ud835\udc9f1\u03bb \u2261 \ud835\udcb0\ud835\udc9e\ud835\udcae\ud835\udcab(\u03bb), introduced by Ravichandran et al. [n et al. ;\ud835\udcb1\ud835\udc9f10 \u2261 \ud835\udcb1\ud835\udc9f, considered by Latha [by Latha ;\ud835\udcb1\ud835\udc9f10 \u2261 \ud835\udcae*(\u03b2) and \ud835\udcb1\ud835\udc9f10 \u2261 \ud835\udca6(\u03b2), the well-known classes of starlike and convex functions of order \u03b2.From the above special cases we note that this class provides a continuous passage from the class of starlike functions to the class of convex functions.By giving specific values to  details \u20136; we li\u03b1 \u2265 0, 0 \u2264 \u03b2 < 1, \u03b4 > \u22121, \u03bb is real with |\u03bb| < (\u03c0/2), and b \u2208 \u2102\u2216{0}.We will assume throughout our discussion, unless otherwise stated, that f(z) \u2208 \ud835\udcb1\ud835\udc9fp\u03bb with 0 \u2264 \u03b1 \u2264 \u03b2. Then f(z) \u2208 \ud835\udcb1\ud835\udc9fp\u03bb, where \u03b6 = (\u03b2 \u2212 \u03b1)/(1 \u2212 \u03b1).Let f(z) \u2208 \ud835\udcb1\ud835\udc9fp\u03bb. Then we obtain\u03b1 \u2264 \u03b2, then we can easily obtainLet f(z) \u2208 \ud835\udcb1\ud835\udc9fp\u03bb, then\u03c6n+p\u22121(\u03b4) is given by (If given by  and(15)f(z) \u2208 \ud835\udcb1\ud835\udc9fp\u03bb. Then by p(z) byp(z) is analytic in E with p(0) = 1 and Rep(z) > 0, z \u2208 E. Let\u03b7 is given by  byei\u03bb(1\u22122b+given by . Using ( E. Let1p(z)=1+\u2211nThat is,2ei\u03bb(1\u2212\u03b1) sides,22ei\u03bb(1\u2212\u03b1)nctions [|an+p\u22121|\u2264ows that  is true.nd using , we haveerefore,  holds fon = k; that is,n = k + 1, and hence by using mathematical induction,  \u2208 \ud835\udcae\ud835\udc9f, then\u03b1 = 0 in If obertson .\u03bb = 0, p = 1, b = 1, and \u03b4 = 1 in By setting f(z) \u2208 \ud835\udca6\ud835\udc9f, then\u03b1 = 0 in If f(z) \u2208 \ud835\udc9c(p) and satisfies\u03c6k(\u03b4) is given by (f(z) \u2208 \ud835\udcb1\ud835\udc9fp\u03bb.If given by , then f \u2208 \ud835\udc9c(p) and satisfiesf(z) \u2208 \ud835\udcaep*(\u03b2), the class of p-valent starlike functions of order \u03b2.If \u03bb = 0, \u03b4 = \u2212p + 2, b = 1, and \u03b1 = 0 in For f(z) \u2208 \ud835\udc9c(p) and satisfiesf(z) \u2208 \ud835\udca6p(\u03b2), the class of p-valent convex functions of order \u03b2.If p = 1 in both the last two corollaries, one obtains the results for the classes \ud835\udcae*(\u03b2) and \ud835\udca6(\u03b2) which was proved by Merkes et al. [Further for s et al.  and Silvs et al. , respect"}
{"text": "\ud835\udcab0(X) and \ud835\udca60(X) of a tvs-cone metric space , where \ud835\udcab0(X) and \ud835\udca60(X) are the space consisting of nonempty subsets of X and the space consisting of nonempty compact subsets of X, respectively. The purpose of this paper is to establish some relationships between the lower topology and the lower tvs-cone hemimetric topology  on \ud835\udcab0(X) and \ud835\udca60(X), which makes it possible to generalize some results of superspaces from metric spaces to tvs-cone metric spaces.This paper investigates superspaces  K-metric and K-normed spaces were introduced in the mid-20th century . . K-metricentury  and \ud835\udca60(X) be the space consisting of nonempty subsets of X and the space consisting of nonempty compact subsets of X, respectively. And then the following two theorems are well known .X, d) be a metric space and let \ud835\udc9e be a subset of \ud835\udcab0(X). Then the following hold.\ud835\udc9e is open in the lower topology on \ud835\udcab0(X), then \ud835\udc9e is open in the lower hemimetric topology on \ud835\udcab0(X).If \ud835\udc9e is open in the upper hemimetric topology on \ud835\udcab0(X), then \ud835\udc9e is open in the upper topology on \ud835\udcab0(X).If Let  be a metric space. Then the following hold.\ud835\udca60(X).The lower topology and the lower hemimetric topology coincide on \ud835\udca60(X).The upper topology and the upper hemimetric topology coincide on \ud835\udca60(X).The Vietoris topology and the Hausdorff hemimetric topology coincide on Let  and \ud835\udca60(X) of a tvs-cone metric space . The purpose of this paper is to establish some relationships between the lower topology and the lower tvs-cone hemimetric topology  on \ud835\udcab0(X) and \ud835\udca60(X), respectively. These results answer This paper investigates superspaces \u2115, \u211d+, and \u211d* denote the set of all natural numbers, the set of all positive real numbers, and the set of all nonnegative real numbers, respectively.Throughout this paper, E be a topological vector space with its zero vector \u03b8. A subset P of E is called a tvs-cone in E if the following are satisfied.P is a closed in E with a nonempty interior P\u00b0.\u03b1, \u03b2 \u2208 P and a, b \u2208 \u211d*\u21d2a\u03b1 + b\u03b2 \u2208 P.\u03b1, \u2212\u03b1 \u2208 P\u2009\u2009\u21d2\u2009\u2009\u03b1 = \u03b8.Let E be a topological vector space with a tvs-cone P. It is clear that \u03b8 \u2208 P from \u03b8 \u2209 P\u00b0. In fact, pick \u03b1 \u2208 E \u2212 {\u03b8}. Then {(1/n)\u03b1} \u2192 \u03b8 and {\u2212(1/n)\u03b1} \u2192 \u03b8 when n \u2192 \u221e. If \u03b8 \u2208 P\u00b0, then there is n \u2208 \u2115 such that {(1/n)\u03b1, \u2212(1/n)\u03b1}\u2286P\u00b0\u2286P. By n)\u03b1 = \u03b8. This contradicts that \u03b1 \u2260 \u03b8. So \u03b8 \u2209 P\u00b0.Let E be a topological vector space with a tvs-cone P. Some partial orderings \u2264, <, and \u226a on E with respect to P are defined as follows, respectively. Let \u03b1, \u03b2 \u2208 E.\u03b1 \u2264 \u03b2 if \u03b2 \u2212 \u03b1 \u2208 P.\u03b1 < \u03b2 if \u03b1 \u2264 \u03b2 and \u03b1 \u2260 \u03b2.\u03b1 \u226a \u03b2 if \u03b2 \u2212 \u03b1 \u2208 P\u00b0.Let E with respect to P. The meanings of these notations are clear and the following hold:\u03b1 \u2265 \u03b2\u21d4\u03b1 \u2212 \u03b2 \u2265 \u03b8\u21d4\u03b1 \u2212 \u03b2 \u2208 P,\u03b1 > \u03b2\u21d4\u03b1 \u2212 \u03b2 > \u03b8\u21d4\u03b1 \u2212 \u03b2 \u2208 P \u2212 {\u03b8},\u03b1 \u226b \u03b2\u21d4\u03b1 \u2212 \u03b2 \u226b \u03b8\u21d4\u03b1 \u2212 \u03b2 \u2208 P\u00b0,\u03b1 \u226b \u03b2\u21d2\u03b1 > \u03b2\u21d2\u03b1 \u2265 \u03b2.For the sake of convenience, we also use notations \u201c\u2265\u201d, \u201c>,\u201d and \u201c\u226b\u201d on P in a topological vector space E is called strongly minihedral if each subset of E bounded above has a supremum, equivalently, if each subset of E bounded below has an infimum.A tvs-cone P in a topological vector space E is strongly minihedral.In this paper, we always suppose that a tvs-cone E be a topological vector space with a tvs-cone P. Then the following hold.\u03b1 \u226b \u03b8, then r\u03b1 \u226b \u03b8 for each r \u2208 \u211d+.If \u03b11 \u226b \u03b21 and \u03b12 \u2265 \u03b22, then \u03b11 + \u03b12 \u226b \u03b21 + \u03b22.If \u03b1 \u226b \u03b8 and \u03b2 \u226b \u03b8, then there is \u03b3 \u226b \u03b8 such that \u03b3 \u226a \u03b1 and \u03b3 \u226a \u03b2.If Let \u03b1 \u226b \u03b8; that is, \u03b1 \u2208 P\u00b0. Then there is an open neighborhood B of \u03b1 in E such that B\u2286P. If r \u2208 \u211d+, then rB\u2286P from rB = {r\u03b2 : \u03b2 \u2208 B}. Note that r\u03b1 \u2208 rB and rB is an open subset of E. So r\u03b1 \u2208 P\u00b0; that is r\u03b1 \u226b \u03b8.(1) Let \u03b11 \u226b \u03b21 and \u03b12 \u2265 \u03b22. Then \u03b11 \u2212 \u03b21 \u226b \u03b8 and \u03b12 \u2212 \u03b22 \u2265 \u03b8; that is, \u03b11 \u2212 \u03b21 \u2208 P\u00b0 and \u03b12 \u2212 \u03b22 \u2208 P. So there is an open neighborhood B of \u03b11 \u2212 \u03b21 in E such that B\u2286P. Write (\u03b12 \u2212 \u03b22) + B = {(\u03b12 \u2212 \u03b22) + \u03b2 : \u03b2 \u2208 B}. Note that (\u03b12 \u2212 \u03b22) + B is an open subset of E, and (\u03b12 \u2212 \u03b22)+(\u03b11 \u2212 \u03b21)\u2208(\u03b12 \u2212 \u03b22) + B\u2286P from \u03b12 \u2212 \u03b22)+(\u03b11 \u2212 \u03b21) \u2208 P\u00b0; that is, (\u03b12 \u2212 \u03b22)+(\u03b11 \u2212 \u03b21) \u226b \u03b8; hence, (\u03b11 + \u03b12)\u2212(\u03b21 + \u03b22) \u226b \u03b8. It follows that \u03b11 + \u03b12 \u226b \u03b21 + \u03b22.(2) Let \u03b1 \u226b \u03b8 and \u03b2 \u226b \u03b8; that is, \u03b1, \u03b2 \u2208 P\u00b0. Then there is n1, n2 \u2208 \u2115 such that \u03b1 \u2212 ((\u03b1 + \u03b2)/n) \u2208 P\u00b0 for all n \u2265 n1 and \u03b2 \u2212 ((\u03b1 + \u03b2)/n) \u2208 P\u00b0 for all n \u2265 n2. Put \u03b3 = (\u03b1 + \u03b2)/n0, where n0 = max\u2061{n1, n2}. Then \u03b3 \u226b \u03b8 from the above (1) and (2). It is clear that \u03b1 \u2212 \u03b3 \u2208 P\u00b0 and \u03b2 \u2212 \u03b3 \u2208 P\u00b0; that is, \u03b1 \u2212 \u03b3 \u226b \u03b8 and \u03b2 \u2212 \u03b3 \u226b \u03b8. So \u03b3 \u226a \u03b1 and \u03b3 \u226a \u03b2.(3) Let We give the definition of tvs-cone metric, which is very similar to the well-known definition of metric.X be a nonempty set and let E be a topological vector space with a tvs-cone P. A mapping d : X \u00d7 X \u2192 E is called a tvs-cone metric on X, and  is called a tvs-cone metric space if the following are satisfied.d \u2265 \u03b8 for all x, y \u2208 X and d = \u03b8 if and only if x = y.d = d for all x, y \u2208 X.d \u2264 d + d for all x, y, z \u2208 X.Let \u221e as a possible value of the mapping d in the following definition, where \u221e \u2209 E and the following hold.\u221e + \u03b1 = \u221e + \u221e = \u221e for each \u03b1 \u2208 E.\u03b1 \u226a \u221e for each \u03b1 \u2208 E.Note that hemimetric takes values in the extended nonnegative real numbers . We letX be a nonempty set and let E be a topological vector space with a tvs-cone P. A mapping d : X \u00d7 X \u2192 E\u22c3{\u221e} is called a tvs-cone hemimetric on X, and  is called a tvs-cone hemimetric space if the following (1) and (2) are satisfied.d \u2264 d + d for all x, y, z \u2208 X.d = \u03b8 for all x \u2208 X.Let X, d) be a tvs-cone hemimetric space. Put B = {y \u2208 X : d \u226a \u025b} for x \u2208 X and \u025b \u226b \u03b8, and put \u212c = {B : x \u2208 X\u2009\u2009and\u2009\u2009\u025b \u226b \u03b8}. Then \u212c is a base for some topology on X.Let , B \u2208 \u212c and z \u2208 B\u22c2B. Since z \u2208 B, d \u226a \u03b1. Put \u03b31 = \u03b1 \u2212 d; then \u03b31 \u226b \u03b8. We claim that B\u2286B. In fact, if u \u2208 B, then d \u226a \u03b31; hence, d \u2264 d + d \u226a d + \u03b31 = \u03b1, and so u \u2208 B. Using the same way, we can obtain that there is \u03b32 \u226b \u03b8 such that B\u2286B. By \u03b3 \u226b \u03b8 such that \u03b3 \u226a \u03b31 and \u03b3 \u226a \u03b32. Let v \u2208 B; then d \u226a \u03b3 \u226a \u03b31 and d \u226a \u03b3 \u226a \u03b32, so v \u2208 B\u2286B and v \u2208 B\u2286B, and hence v \u2208 B\u22c2B. This has proved that B\u2286B\u22c2B. Note that z \u2208 B \u2208 \u212c. Consequently, \u212c is a base for some topology on X. In fact, put \u03c4 = {U\u2286X : there\u2009\u2009is\u2009\u2009\u212c\u2032\u2286\u212c such that U = \u22c3\u212c\u2032}; then \u03c4 is a topology on X and \u212c is a base for \u03c4.It is clear that X, d) be a tvs-cone metric space and let \u03c4 denote the topology on X induced by the tvs-cone metric d described in G of X,  and IG denote the subfamilies {P \u2208 \ud835\udcab0(X) : P\u2286G} and {P \u2208 \ud835\udcab0(X) : P\u22c2G \u2260 \u2205} of \ud835\udcab0(X), respectively.\ud835\udcaf\u2112 is called the lower topology on \ud835\udcab0(X), where \ud835\udcaf\u2112 is generated by the subbase \u2112 = {IG : G \u2208 \u03c4}.\ud835\udcaf\ud835\udcb0 is called the upper topology on \ud835\udcab0(X), where \ud835\udcaf\ud835\udcb0 is generated by the base \ud835\udcb0 = { : G \u2208 \u03c4}.\ud835\udcaf\ud835\udcb1 is called the Vietoris topology on \ud835\udcab0(X), where \ud835\udcaf\ud835\udcb1 is generated by \u2112 and \ud835\udcb0 together.Let  be a tvs-cone metric space. For P, Q \u2208 \ud835\udcab0(X), put \u03b4l = inf\u2061{\u025b \u226b \u03b8 : P\u2286S}, \u03b4u = inf\u2061{\u025b \u226b \u03b8 : Q\u2286S}, and \u03b4 = inf\u2061{\u025b \u226b \u03b8 : \u025b \u226b \u03b4l and \u025b \u226b \u03b4u}, where inf\u2061\u2205 = \u221e. Then \u03b4l, \u03b4u, and \u03b4 are tvs-cone hemimetrics on \ud835\udcab0(X). Let \ud835\udcafl, \ud835\udcafu, and \ud835\udcaf denote the topologies on \ud835\udcab0(X) induced by \u03b4l, \u03b4u, and \u03b4 described in \ud835\udcafl is called the lower tvs-cone hemimetric topology.The topology \ud835\udcafu is called the upper tvs-cone hemimetric topology.The topology \ud835\udcaf is called the Hausdorff tvs-cone hemimetric topology.The topology Let , and the relative topologies on \ud835\udca60(X) will still be denoted by \ud835\udcaf\u2112, and so forth. Also, if \u03b4u  is restricted to \ud835\udca60(X), then it does not take \u221e.It is often preferable to restrict the six topologies In this section, we need to use the following notation.Notation. Let  be a tvs-cone metric space, x \u2208 X, P \u2208 \ud835\udcab0(X), K \u2208 \ud835\udca60(X), and \u025b \u226b \u03b8. Consider the following:B = {y \u2208 X : d \u226a \u025b},S = \u22c3{B : x \u2208 P},Bl = {P\u2032 \u2208 \ud835\udcab0(X) : \u03b4l \u226a \u025b},Bu = {P\u2032 \u2208 \ud835\udcab0(X) : \u03b4u \u226a \u025b},Cl = {K\u2032 \u2208 \ud835\udca60(X) : \u03b4l \u226a \u025b},Cu = {K\u2032 \u2208 \ud835\udca60(X) : \u03b4u \u226a \u025b},C = {K\u2032 \u2208 \ud835\udca60(X) : \u03b4 \u226a \u025b}.X, d) be a tvs-cone metric space and let \ud835\udc9e be a subset of \ud835\udcab0(X). Then the following hold.\ud835\udc9e is open in the lower topology \ud835\udcaf\u2112 on \ud835\udcab0(X), then \ud835\udc9e is open in the lower tvs-cone hemimetric topology \ud835\udcafl on \ud835\udcab0(X).If \ud835\udc9e is open in the upper tvs-cone hemimetric topology \ud835\udcafu on \ud835\udcab0(X), then \ud835\udc9e is open in the upper topology \ud835\udcaf\ud835\udcb0 on \ud835\udcab0(X).If Let (\ud835\udc9e be open in the lower topology \ud835\udcaf\u2112 on \ud835\udcab0(X). Without loss of generality, we can assume that \ud835\udc9e is an element in the subbase \u2112 for the lower topology \ud835\udcaf\u2112; that is, \ud835\udc9e = IG for some G \u2208 \ud835\udcaf. Let P \u2208 \ud835\udc9e, then P\u22c2G \u2260 \u2205. Pick x \u2208 P\u22c2G; then there is \u025b \u226b \u03b8 such that B\u2286G since G is open in X. Let Q \u2208 Bl; then \u03b4l \u226a \u025b; that is, P\u2286S. Since x \u2208 P, x \u2208 S, hence x \u2208 B for some y \u2208 Q. Thus, y \u2208 B\u2286G, which means that Q\u22c2G \u2260 \u2205. It follows that Q \u2208 \ud835\udc9e. This proves that Bl\u2286\ud835\udc9e. So P is an interior point of \ud835\udc9e in the lower tvs-cone hemimetric topology \ud835\udcafl and the proof is completed.(1) Let \ud835\udc9e be open in the upper tvs-cone hemimetric topology \ud835\udcafu on \ud835\udcab0(X). Let P \u2208 \ud835\udc9e; then there is \u025b \u226b \u03b8 such that Bu\u2286\ud835\udc9e. Note that S is open in X. So  is open in the upper topology on \ud835\udcab0(X). Clearly, P \u2208 . On the other hand, if Q \u2208 , that is, Q\u2286S, then \u03b4u \u226a \u025b, and hence Q \u2208 Bu\u2286\ud835\udc9e. This has proved that \u2286\ud835\udc9e. Consequently, \ud835\udc9e is an open neighborhood of P for the upper topology \ud835\udcaf\ud835\udcb0 on \ud835\udcab0(X) and the proof is completed.(2) Let X, d) is a metric space). Moreover, there is no simple relationship between the Vietoris topology \ud835\udcaf\ud835\udcb1 and the Hausdorff tvs-cone hemimetric topology \ud835\udcaf on \ud835\udcab0(X), which is similar to (1) or (2) in (1) The converses of both (1) and (2) in \ud835\udcab0(X)\u201d in \ud835\udca60(X). Furthermore, we have the better results for the topologies on superspaces \ud835\udca60(X) (see the following).(2) It is clear that \u201cX, d) be a tvs-cone metric space. If K is a compact subset of X, then, for any \u025b \u226b \u03b8, there is a finite subset F of K such that K\u2286S.Let  : x \u2208 K} is an open cover of K; there is a finite subset F of K such that {B : x \u2208 F} covers K. It follows that K\u2286S.Let X, d) be a tvs-cone metric space. If K\u2286U with K compact in X and U open in X, then there is \u025b \u226b \u03b8 such that S\u2286U.Let \u2286U. Put \u03b5x = (1/2)\u03b7x; then \u03b5x \u226b \u03b8 from B : x \u2208 K} is an open cover of K and K is compact, there is a finite subset F of K such that {B : x \u2208 F} covers K. By \u025b \u226b \u03b8 such that \u025b \u226a \u03b5x for each x \u2208 F. We claim that S\u2286U. In fact, let u \u2208 S; then there is y \u2208 K such that u \u2208 B, that is, d \u226a \u025b. Furthermore, there is z \u2208 F such that y \u2208 B; that is, d \u226a \u03b5z. By d \u2264 d + d \u226a \u025b + \u03b5z \u226a 2\u03b5z = \u03b7z. It follows that u \u2208 B\u2286U. This has proved that S\u2286U.Let X, d) be a tvs-cone metric space. Then the following hold.\ud835\udcaf\u2112 and the lower tvs-cone hemimetric topology \ud835\udcafl coincide on \ud835\udca60(X).The lower topology \ud835\udcaf\ud835\udcb0 and the upper tvs-cone hemimetric topology \ud835\udcafu coincide on \ud835\udca60(X).The upper topology \ud835\udcaf\ud835\udcb1 and the Hausdorff tvs-cone hemimetric topology \ud835\udcaf coincide on \ud835\udca60(X).The Vietoris topology Let (\ud835\udc9e is open in the lower hemimetric topology \ud835\udcafl on \ud835\udca60(X). Let K \u2208 \ud835\udc9e; then there is \u025b \u226b \u03b8 such that Cl\u2286\ud835\udc9e. Since K is compact, by F of K such that K\u2286S. We write Gx = B for each x \u2208 F and put \ud835\udcb2 = \u22c2{IGx : x \u2208 F}. Note that B \u2208 \ud835\udcaf for each x \u2208 F. It is clear that K \u2208 \ud835\udcb2 and \ud835\udcb2 is an element of the base for the lower topology \ud835\udcaf\u2112 on \ud835\udca60(X). Let K\u2032 \u2208 \ud835\udcb2. For each x \u2208 F, we claim that Gx\u2286S. In fact, let y \u2208 Gx; then d \u226a \u025b/2. Since K\u2032 \u2208 IGx, K\u2032\u22c2Gx \u2260 \u2205. Pick z \u2208 K\u2032\u22c2Gx; then d \u226a \u025b/2; hence, d \u2264 d + d \u226a \u025b/2 + \u025b/2 = \u025b; that is, y \u2208 B\u2286S. This proves that Gx\u2286S. Furthermore, K\u2286S = \u22c3{Gx : x \u2208 F}\u2286S. Thus, \u03b4l \u226a \u025b; that is, K\u2032 \u2208 Cl\u2286\ud835\udc9e. This proves that \ud835\udcb2 \u2282 \ud835\udc9e. It follows that K is an interior point of \ud835\udc9e for the lower topology \ud835\udcaf\u2112 on \ud835\udca60(X). Consequently, \ud835\udc9e is open in the lower topology \ud835\udcaf\u2112 on \ud835\udca60(X). Combining (1) Assume that \ud835\udc9e be open in the upper topology \ud835\udcaf\ud835\udcb0 on \ud835\udca60(X). Without loss of generality, we can assume that \ud835\udc9e is an element of the base for the upper topology \ud835\udcaf\ud835\udcb0 on \ud835\udca60(X); that is, \ud835\udc9e = \u22c2\ud835\udca60(X) for some G \u2208 \ud835\udcaf. Let K \u2208 \ud835\udc9e; then K\u2286G and K is compact. By \u025b \u226b \u03b8 such that S\u2286G. If K\u2032 \u2208 Cu; then \u03b4u \u226a \u025b; hence, K\u2032\u2286S\u2286G. It follows that K\u2032 \u2208 \u22c2\ud835\udca60(X) = \ud835\udc9e. Consequently, Cu\u2286\ud835\udc9e. This has proved that \ud835\udc9e is an open neighborhood of K for the upper tvs-cone hemimetric topology \ud835\udcafu on \ud835\udca60(X). Combining (2) Let \ud835\udcaf\ud835\udcb1\u2286\ud835\udcaf and \ud835\udcaf\u2286\ud835\udcaf\ud835\udcb1. Let \ud835\udc9e be open in the Vietoris topology \ud835\udcaf\ud835\udcb1 on \ud835\udca60(X). Without loss of generality, we only need to assume that \ud835\udc9e is an element of the subbase \u2112 for the lower topology \ud835\udcaf\u2112 or an element of the base \ud835\udcb0 for the upper topology \ud835\udcaf\ud835\udcb0. If \ud835\udc9e \u2208 \u2112, then \ud835\udc9e \u2208 \ud835\udcafl by (1). So, for each K \u2208 \ud835\udc9e, there is \u025b \u226b \u03b8 such that Cl\u2286\ud835\udc9e. It is clear that C\u2286Cl. In fact, if K\u2032 \u2208 C, then \u03b4 \u226a \u025b. Note that \u03b4l \u2264 \u03b4 \u226a \u025b. So K\u2032 \u2208 Cl. It follows that K \u2208 C\u2286Cl\u2286\ud835\udc9e. So K is an interior point of \ud835\udc9e for the Hausdorff tvs-cone hemimetric topology \ud835\udcaf. Consequently, \ud835\udc9e is open in the Hausdorff tvs-cone hemimetric topology \ud835\udcaf on \ud835\udca60(X). By a similar way, if \ud835\udc9e \u2208 \ud835\udcb0, then \ud835\udc9e is open in the Hausdorff tvs-cone hemimetric topology \ud835\udcaf on \ud835\udca60(X). This has proved that \ud835\udcaf\ud835\udcb1\u2286\ud835\udcaf. Conversely, let \ud835\udc9e be open in the Hausdorff tvs-cone hemimetric topology \ud835\udcaf on \ud835\udca60(X). Then, for each K \u2208 \ud835\udc9e, there is \u025b \u226b \u03b8 such that C\u2286\ud835\udc9e. We claim that Cl\u22c2Cu\u2286C. In fact, if K\u2032 \u2208 Cl\u22c2Cu, then \u03b4l \u226a \u025b/2 and \u03b4u \u226a \u025b/2. So \u03b4 \u2264 \u03b4l + \u03b4u \u226a \u025b/2 + \u025b/2 = \u025b; hence, K\u2032 \u2208 C. This proves that Cl\u22c2Cu\u2286C. By (1) and (2), Cl and Cu are open in \ud835\udcaf\u2112 and \ud835\udcaf\ud835\udcb0, respectively. It follows that Cl\u22c2Cu is open in \ud835\udcaf\ud835\udcb1. So K is an interior point of \ud835\udc9e for the Vietoris topology \ud835\udcaf\ud835\udcb1 on \ud835\udca60(X). Consequently, \ud835\udc9e is open in the Vietoris topology \ud835\udcaf\ud835\udcb1 on \ud835\udca60(X). This has proved that \ud835\udcaf\u2286\ud835\udcaf\ud835\udcb1.We need to prove that"}
{"text": "In this paper, a Meir-Keeler contraction is introduced to propose a viscosity-projection approximation method for finding a common element of the set of solutions of a family of general equilibrium problems and the set of fixed points of asymptotically strict pseudocontractions in the intermediate sense. Strong convergence of the viscosity iterative sequences is obtained under some suitable conditions. Results presented in this paper extend and unify the previously known results announced by many other authors. H be a real Hilbert space with inner product \u2329\u00b7, \u00b7\u232a and norm ||\u00b7||, respectively. Let C be a nonempty closed convex subset of H. Let A : C \u2192 H be a nonlinear mapping and F : C \u00d7 C \u2192 \u211d be a bifunction, where \u211d denotes the set of real numbers. We consider the following generalized equilibrium problem: Find x \u2208 C such thatEP\u2061 to denote the set of solution of problem  to denote the set of solution of problem  to denote the set of solution of problem +\u2329Auch thatF+\u2329Auch thatF\u22650,uch thatF+\u2329Auch that\u2329Ax,y\u2212x\u232a\u2265uch thatF+\u2329Auch thatF+\u2329AT : C \u2192 C is said to be nonexpansive ifT is said to be uniformly L-Lipschitz continuous if there exists a constant L > 0 such thatT is said to be asymptotically nonexpansive if there exists a sequence kn \u2208 N + i(n), i = i(n) = 1,2,\u2026, N. They also obtain weak and strong convergence theorems based on the cyclic scheme above.In 2009, Qin et al.  introducRecently, Sahu et al.  considerC be a nonempty closed and convex subset of a real Hilbert space H and T : C \u2192 C be a uniformly continuous asymptotically \u03bb-strictly pseudocontractive mapping in the intermediate sense with a sequence {kn} such that Fix\u2061(T) is nonempty and bounded. Let {\u03b1n} be a sequence in  such that 0 < \u03b4 \u2264 \u03b1n \u2264 1 \u2212 \u03bb for all n \u2208 \u2115. Let {xn} \u2282 C be a sequence generated by the following (CQ) algorithm:\u03b8n = (kn \u2212 1)\u03c1n2 + en and \u03c1n = sup\u2061{||xn \u2212 p||:p \u2208 Fix\u2061(T)} < \u221e. Then, {xn} converges strongly to PT)Fix\u2061((u), where PT)Fix\u2061( is metric projection from H onto Fix\u2061(T).Let \u03b8n = (kh(n) \u2212 1)\u03c1n2 + eh(n) \u2192 0 as n \u2192 \u221e and \u03c1n = sup\u2061{||xn \u2212 p||:p \u2208 \u03a9} < \u221e. Moreover, they obtained convergence theorems under some suitable conditions.In 2011, Hu and Cai  modifiedf is a contractive mapping. He proved that the viscosity iterative sequence {xn} convergence strongly to a fixed point of T, which is the unique solution of the variational inequality:On the other hand, Moudafi  introducakahashi  and Inchakahashi  modifiedIn 2012, Kimura and Nakajo  introducC be a nonempty closed convex subset of H, and let {Tn} be a sequence of mappings of C into itself with \u03a9 = \u22c2n=1\u221eFix\u2061(Tn) \u2260 \u00d8 which satisfies the following condition: there exists {an} \u2282 \u211d with liminf\u2061n\u2192\u221ean > \u22121 such that ||Tnx \u2212 z||2 \u2264 ||x \u2212 z||2 \u2212 an||x \u2212 Tnx||2 for every n \u2208 \u2115, x \u2208 C, and z \u2208 \u03a9. Let f be a Meir-Keeler contraction of C into itself, and let {xn} be a sequence generated byn \u2208 \u2115. For every sequence {zn} \u2282 C and zn \u2192 z \u2208 C and Tnzn \u2192 z imply that z \u2208 \u03a9. Then, {xn} converges strongly to q \u2208 \u03a9, which satisfies q = P\u03a9f(q).Let \u03b8n = (kh(n) \u2212 1)\u03c1n2 + eh(n) \u2192 0 as n \u2192 \u221e and \u03c1n = sup\u2061{||xn \u2212 p||:p \u2208 \u03a9} < \u221e.In this paper, inspired and motivated by research going on in this area, we introduce a new viscosity-projection method for a family of general equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense, which is defined in the following way:Our purpose is not only to extend the viscosity-projection method with a Meir-Keeler contraction to the case of a family of general equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense, but also to obtain a strong convergence theorem by using the proposed schemes under some appropriate conditions. Results presented in this paper extend and unify the corresponding ones of \u201313, 16.C be a nonempty closed convex subset of a real Hilbert space H with inner product \u2329\u00b7, \u00b7\u232a and norm ||\u00b7||, respectively. We use notation \u21c0 for weak convergence and \u2192 for strong convergence of a sequence. For every point x \u2208 H, there exists a unique nearest point in C, denoted by PCx, such thatPC is called the metric projection of H onto C defined by PC(x) = argmin\u2061y\u2208C||x \u2212 y||. It is well known that PC is nonexpansive mapping, and u = PCx is equivalent to  the folA : C \u2192 H is said to be monotone ifA is said to be r-strongly monotone if there exists a constant r > 0 such thatA is said to be \u03b1-inverse strongly monotone if there exists a constant \u03b1 > 0 such thatA is an \u03b1-inverse strongly monotone mapping from C into H, then A is 1/\u03b1-Lipschitz continuous.Recall that a mapping F : C \u00d7 C \u2192 \u211d satisfies the following conditions: \u2009(A1)F = 0\u2009\u2009for all x \u2208 C; \u2009(A2)F is monotone, that is, F + F \u2264 0\u2009\u2009for all x, y \u2208 C; \u2009(A3)x, y, z \u2208 C, lim\u2061t\u21920F(tz + (1 \u2212 t)x, y) \u2264 F;  for each \u2009(A4)x \u2208 C, y \u21a6 F is convex and lower semi-continuous. for each To study the generalized equilibrium problem , we may F : C \u00d7 C \u2192 \u211d be a bifunction satisfying (A1)\u2013(A4). Then, for any r > 0 and x \u2208 H, there exists z \u2208 C such thatFrx = {z \u2208 C : F + (1/r)\u2329y \u2212 z, z \u2212 x\u232a\u22650, \u2200y \u2208 C}, then the following hold: Fr is single-valued; Fr is firmly nonexpansive, that is, ||Frx\u2212Fry||2 \u2264 \u2329Frx \u2212 Fry, x \u2212 y\u232a for all x, y \u2208 H; Fr) = EP(F); Fix\u2061(EP(F) is closed and convex. Let H, there hold the following identities: x+y||2 \u2264 ||x||2 + 2\u2329y, (x + y)\u232a, for all x, y \u2208 H; ||tx+(1\u2212t)y||2 = t||x||2 + (1 \u2212 t)||y||2 \u2212 t(1 \u2212 t)||x\u2212y||2, for all t \u2208 , for all x, y \u2208 H.||In a Hilbert space C be a nonempty closed convex subset of a real Hilbert space H. For any x, y, z \u2208 H and given also a real number a \u2208 \u211d, the setLet C be a nonempty closed convex subset of a real Hilbert space H and T : C \u2192 C be a uniformly L-Lipschitz continuous and asymptotically \u03bb-strict pseudocontraction in the intermediate sense. Then Fix\u2061(T) is closed and convex. Let C be a nonempty closed convex subset of a real Hilbert space H and T : C \u2192 C be a uniformly L-Lipschitz continuous and asymptotically \u03bb-strict pseudocontraction in the intermediate sense. Then I \u2212 T is demiclosed at zero, that is, if the sequence {xn} \u2282 C such that xn\u21c0x and xn \u2212 Txn \u2192 0 as n \u2192 \u221e, then x \u2208 Fix\u2061(T). Let C be a nonempty closed convex subset of a real Hilbert space H and T : C \u2192 C be an asymptotically \u03bb-strict pseudocontraction in the intermediate sense with \u03b3n = kn \u2212 1. ThenLet f of a complete metric space  into itself is called a contraction with coefficient r \u2208  if ||f(x) \u2212 f(y)||\u2264r||x \u2212 y||, for all x, y \u2208 X. It is known that f has a unique fixed point  < \u03f5 + \u03b4 implies thatRecall also that a mapping e, e.g., ). On thee, e.g.,  defined roved in . A Meir-Keeler contraction defined on a complete metric space has a unique fixed point.f be a Meir-Keeler contraction on a convex subset C of a Banach space E. Then, for every \u03f5 > 0, there exists r \u2208  such that ||x \u2212 y||\u2265\u03f5 implies that Let Cn} be a sequence of nonempty closed convex subsets of H. We define a subset s-LinCn of H as follows: x \u2208 s-LinCn if and only if there exists {xn} \u2282 H such that xn \u2192 x and xn \u2208 Cn for all n \u2208 \u2115. Similarly, a subset w-LsnCn of H is defined by y \u2208 w-LsnCn if and only if there exists a subsequence {Cni} of {Cn} and a sequence {yi} \u2282 H such that yi\u21c0y and yi \u2208 Cni for all i \u2208 \u2115. If C0 \u2282 H satisfiesCn} converges to C0 in the sense of Mosco  such that 0 < a \u2264 \u03b1n \u2264 1, 0 < b \u2264 \u03b2n \u2264 1 \u2212 \u03bb and {rm,n}\u2282 such that rm,n \u2208 \u2282, for each m = 1,2,\u2026, M and n \u2208 \u2115. Then the sequence {xn} generated by (q = P\u03a9f(q).Let rated by  convergeWe split the proof into six steps.Step\u2009\u20091. We prove that P\u03a9f exists a unique fixed point. To do this, we first show that  is nonexpansive for each m = 1,2,\u2026, M. Indeed,I \u2212 rm,nAm) is nonexpansive. By m=1MEP\u2061 is closed and convex. We also know from i=1NFix\u2061(Ti) is closed, and convex. Hence, \u03a9 = (\u22c2i=1NFix\u2061(Ti))\u2229) is a nonempty, closed and convex subset of C. Consequently, P\u03a9 is well-defined. Since P\u03a9 is nonexpansive, the composed mapping P\u03a9f of C into itself is a Meir-Keeler contraction on C; see N + i(n), where i = i(n) = 1,2,\u2026, N. By  = h(n \u2212 N) + 1 and i(n) = i(n \u2212 N), we observe thatL-Lipschitzian of Ti, we havei = 1,2,\u2026, N, we havexn \u2192 q as n \u2192 \u221e. It follows from (q \u2208 \u22c2i=1NFix\u2061(Ti).||.From  and 68)Step\u2009\u20096. \u2026, N. By  and the |).From  and (68)ows thatlim\u2061n\u2192\u221e||).From (lim\u2061n\u2192\u221e||).From . From nmxn = Frm,n\u0398nm\u22121xn, m = 1,2,\u2026, M, we havezt = ty + (1 \u2212 t)q for all t \u2208  and the monotonicity of Am, we obtaint \u2192 0, from (A3) and , m = 1,2,\u2026, M. Therefore, q \u2208 \u22c2m=1MEP\u2061. Consequently, we obtain that q \u2208 \u03a9. This completes the proof. Next, we show that n).From , we havee obtain\u2329Amzt,zt\u2212nd henceFm+We also obtain the following results by using the viscosity-hybrid projection methods, which extend and improve the hybrid method (CQ) proposed by Sahu et al.  and Hu aC be a nonempty closed convex subset of Hilbert space H. Let Fm : C \u00d7 C \u2192 \u211d be a bifunction satisfying (A1)\u2013(A4), and let Am : C \u2192 H be an \u03b1m-inverse strongly monotone mapping, for each m = 1,2,\u2026, M. Let Ti : C \u2192 C be a uniformly Li-Lipschitz continuous and asymptotically \u03bbi-strict pseudocontractive mapping in the intermediate sense with the sequences {kn,i} and {en,i} for each i = 1,2,\u2026, N. If f is a Meir-Keeler contraction of C into itself and \u03a9 = (\u22c2i=1NFix\u2061(Ti))\u2229) is nonempty and bounded. Let {xn} be a sequence defined by\u03b8n = (kh(n) \u2212 1)\u03c1n2 + eh(n) \u2192 0 as n \u2192 \u221e and \u03c1n = sup\u2061{||xn \u2212 p||:p \u2208 \u03a9} < \u221e. Assume that {\u03b1n} and {\u03b2n} are sequences in  such that 0 < a \u2264 \u03b1n \u2264 1, 0 < b \u2264 \u03b2n \u2264 1 \u2212 \u03bb and {rm,n}\u2282 such that rm,n \u2208 \u2282, for each m = 1,2,\u2026, M. Then the sequence {xn} generated by (q = P\u03a9f(q).Let fined byx1\u2208C,\u2003Q1=Cn and Qn are closed convex subsets of H and \u03a9 \u2282 Cn for every n \u2208 \u2115. We only prove that \u03a9 \u2282 Qn for every n \u2208 \u2115 and that a sequence {xn} is well-defined. We have x1 \u2208 C and \u03a9 \u2282 Q1 = C. Assume that xk \u2208 C and \u03a9 \u2282 Qk for some k \u2208 \u2115. Since \u03a9 \u2282 Ck\u2229Qk, there exists a unique element xk+1 = PCk\u2229Qkf(xk), and hence\u03a9 \u2282 Qk+1. Therefore, we prove that \u03a9 \u2282 Qn.We have that Pn=1\u221eQn\u22c2f is a Meir-Keeler contraction on C, there exists a unique element q = Pn=1\u221eQn\u22c2f(q) \u2208 \u22c2n=1\u221eQn by zn = PQnf(q) for each n \u2208 \u2115. Since \u03a9 \u2282 Qn+1 \u2282 Qn, it follows from zn \u2192 q = Pn=1\u221eQn\u22c2f(q). We also have xn = PQnf(xn\u22121) by the definition of Qn. Therefore, as in the proof of xn \u2192 q, and the desired conclusion follows immediately from On the other hand, M = N = 1, we obtain the following corollary for a general equilibrium problem and asymptotically strict pseudocontraction in the intermediate sense as a special cases.If C be a nonempty closed convex subset of Hilbert space H. Let F : C \u00d7 C \u2192 \u211d be a bifunction satisfying (A1)\u2013(A4) and A : C \u2192 H be an \u03b1-inverse strongly monotone mapping. Let T : C \u2192 C be a uniformly L-Lipschitz continuous and asymptotically \u03bb-strict pseudocontractive mapping in the intermediate sense with the sequences {kn} and {en}. If f is a Meir-Keeler contraction of C into itself and \u03a9 = Fix\u2061(T)\u2229EP\u2061 is nonempty and bounded. Let {xn} be a sequence defined by\u03b8n = (kn \u2212 1)\u03c1n2 + en \u2192 0 as n \u2192 \u221e and \u03c1n = sup\u2061{||xn \u2212 p|| : p \u2208 \u03a9} < \u221e. Assume that {\u03b1n} and {\u03b2n} are sequences in  such that 0 < a \u2264 \u03b1n \u2264 1, 0 < b \u2264 \u03b2n \u2264 1 \u2212 \u03bb and {rn}\u2282 such that rn \u2208 \u2282. Then the sequence {xn} generated by (q = P\u03a9f(q).Let fined byx1\u2208C,\u2003C1=M = N = 1 and F = 0, the general equilibrium problem \u2229VI is nonempty and bounded. Let {xn} be a sequence defined by\u03b8n = (kn \u2212 1)\u03c1n2 + en \u2192 0 as n \u2192 \u221e and \u03c1n = sup\u2061{||xn \u2212 p|| : p \u2208 \u03a9} < \u221e. Assume that {\u03b1n} is a sequence in  such that 0 < b \u2264 \u03b1n \u2264 1 \u2212 \u03bb and {rn}\u2282 such that rn \u2208 \u2282. Then the sequence {xn} generated by (q = P\u03a9f(q).Let fined byx1\u2208C,\u2003C1=F = 0, the general equilibrium problem (un = PC(xn \u2212 rnAxn). The desired conclusion follows immediately from \u03b1n = 1, \u03b2n = \u03b1n). This completes the proof. If  problem  reduces  problem , and(94"}
{"text": "As an application we prove certain fixed point results in the setup of such spaces for different types of contractive mappings. Finally, some periodic point results in b-metric-like spaces are obtained. Two examples are presented in order to verify the effectiveness and applicability of our main results.We discuss topological structure of  The number s is called the coefficient of .A partial X is a mapping \u03c3 : X \u00d7 X \u2192 \u211d+ such that for all x, y, z \u2208 X\u03c31) = 0 implies x = y,\u03c32) = \u03c3,\u03c33) \u2264 \u03c3 + \u03c3.A metric-like on a nonempty set X, \u03c3) is called a metric-like space.The pair  = x + y \u2212 xy is a metric-like on X.Let X = \u211d; then the mappings \u03c3i : X \u00d7 X \u2192 \u211d+ , defined byX, where a \u2265 0 and b \u2208 \u211d.Let X be a nonempty set and s \u2265 1 a given real number. A function \u03c3b : X \u00d7 X \u2192 \u211d+ is b-metric-like if, for all x, y, z \u2208 X, the following conditions are satisfied:\u03c3b1) = 0 implies x = y,\u03c3b2) = \u03c3b,\u03c3b3) \u2264 s[\u03c3b + \u03c3b].Let b-metric-like space is a pair  such that X is a nonempty set and \u03c3b is b-metric-like on X. The number s is called the coefficient of .A b-metric-like space  if x, y \u2208 X and \u03c3b = 0, then x = y, but the converse may not be true and \u03c3b may be positive for x \u2208 X. It is clear that every partial b-metric space is a b-metric-like space with the same coefficient s and every b-metric space is also a b-metric-like space with the same coefficient s. However, the converses of these facts need not hold.In a X = \u211d+, p > 1 a constant, and \u03c3b : X \u00d7 X \u2192 \u211d+ be defined byX, \u03c3b) is a b-metric-like space with coefficient s = 2p\u22121, but it is not a partial b-metric space. Indeed, for any 0 < y < x we have \u03c3b = (x+x)p > (x+y)p = \u03c3b, so (pb2) of Let b-metric-like spaces.The following propositions help us to construct some more examples of X, \u03c3) be a metric-like space and \u03c3b = [\u03c3]p, where p > 1 is a real number. Then \u03c3b is b-metric-like with coefficient s = 2p\u22121.Let (a+b)p \u2264 2p\u22121(ap + bp), where a, b \u2208 \u211d+.The proof follows from the fact that  = (x+y\u2212xy)p, where p > 1 is a real number, is b-metric-like on X with coefficient s = 2p\u22121.Let X = \u211d. Then the mappings \u03c3bi : X \u00d7 X \u2192 \u211d+ , defined byb-metric-like on X, where p > 1, a \u2265 0, and b \u2208 \u211d.Let X be a nonempty set such that d and pb are b-metric and partial b-metric, respectively, s > 1, and \u03c3 is a metric-like on X. Then the mappings \u03c3bi : X \u00d7 X \u2192 \u211d+ , defined byx, y \u2208 X are b-metric-like on X.Let X, pb) be a partial b-metric space and  a b-metric space with s > 1. Then conditions (\u03c3b1), (\u03c3b2), and (\u03c3b3) are obvious for the function \u03c3b5. For instance, if x, y, z \u2208 X are arbitrary then, as pb is partial b-metric and d is b-metric on X, we have\u03c3b3) is satisfied and so  is a b-metric-like space. Similarly, one can show that  and  are b-metric-like spaces.Let  = x\u2009+\u2009y \u2212 \u2009xy\u2009+\u2009|x\u2212y|p, where p > 1 is a real number, is b-metric-like on X with coefficient s = 2p\u22121.Let X = \u211d. Then the mappings \u03c3bi : X \u00d7 X \u2192 \u211d+ , defined byb-metric-like on X with coefficient s = 2p\u22121, where p > 1, a \u2265 0, and b \u2208 \u211d.Let b-metric-like \u03c3b on X generates a topology \u03c4\u03c3b on X whose base is the family of all open \u03c3b-balls {B\u03c3b : x \u2208 X, \u025b > 0}, where B\u03c3b = {y \u2208 X : |\u03c3b\u2009\u2212\u2009\u03c3b| < \u025b} for all x \u2208 X and \u025b > 0.Each b-metric-like space.Now, we define the concepts of Cauchy sequence and convergent sequence in a X, \u03c3b) be a b-metric-like space with coefficient s, and let {xn} be any sequence in X and x \u2208 X. ThenLet ((i)xn} is said to be convergent to x with respect to \u03c4\u03c3b, if lim\u2061n\u2192\u221e\u03c3b = \u03c3b;the sequence {(ii)xn} is said to be a Cauchy sequence in  if lim\u2061n,m\u2192\u221e\u03c3b exists and is finite;the sequence {(iii)X, \u03c3b) is said to be a complete b-metric-like space if for every Cauchy sequence {xn} in X there exists x \u2208 X such that.It is clear that the limit of a sequence in a We start our work by proving the following crucial lemma.X, \u03c3b) be a b-metric-like space with coefficient s > 1, and suppose that {xn} and {yn} are convergent to x and y, respectively. Then one has\u03c3b = 0, then one has lim\u2061n\u2192\u221e\u2061\u03c3b = 0.Let  = 0, thenMoreover, for each b-metric-like space it is easy to see thatn \u2192 \u221e in the first inequality and the upper limit as n \u2192 \u221e in the second inequality we obtain the first desired result. If \u03c3b = 0, then by the triangle inequality we get \u03c3b = 0 and \u03c3b = 0. Therefore, we have lim\u2061n\u2192\u221e\u03c3b = 0. Similarly, using again the triangle inequality the other assertions follow.Using the triangle inequality in a f on a metric space  is said to be a Banach contraction mapping, if there exists a number k \u2208 [0,1) such thatx, y \u2208 X. A mapping f : X \u2192 X is called a quasicontraction if for some constant \u03b1 \u2208 [0,1) and for every x, y \u2208 Xf has a unique fixed point.It is well known that a self-map  in 1974 . A resulThe existence of fixed points in partially ordered metric spaces was first investigated in 2004 by Ran and Reurings  and thenb-metric-like space. We investigate also the so-called P-property for mappings in such spaces.In this paper, we establish some fixed point theorems for quasicontractive type mappings in a partially ordered complete X, \u2aaf) be a partially ordered set, and let  be a b-metric-like space  is a partially ordered b-metric-like space). Further, let F(f) = {x \u2208 X : fx = x} be the fixed point set of f, (LF)f = {x \u2208 X : x\u2aaffx} the lower fixed point set of f, andb-metric-like space.Throughout this paper, let  is said to have the sequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence) {xn}n\u2208\u2115 in X, xn \u2192 x implies that xn\u2aafx\u2009\u2009(x\u2aafxn), for all n \u2208 \u2115.An ordered X, \u2aaf, \u03c3b) be a complete partially ordered b-metric-like space. If f : X \u2192 X is a nondecreasing map such that, for all elements x, y \u2208 X with x\u2aafy,\u03b1 \u2208 [0, 1/2s2), then F(f) \u2260 \u2205 provided that there exists an x0 \u2208 (LF)f and one of the following two conditions is satisfied:f is a continuous self-map on X,X, \u2aaf, \u03c3b) has the sequential limit comparison property.f and f is nondecreasing, therefore fnx0\u2aaffn+1x0 for each n \u2208 \u2115. Define a sequence {xn} in X with xn = fnx0 and so xn+1 = fxn for all n \u2208 \u2115. If there exists a positive integer n such that xn = xn+1, then fnx0 = fn+1x0 = ffnx0 which implies that fnx0 is a fixed point of f. Assume that xn \u2260 xn+1 for every positive integer n. Since xn\u22121\u2aafxn, therefore by replacing x by xn\u22121 and y by xn in  > \u03c3b > 0, then according to the above inequality \u03c3b \u2264 2s\u03b1\u03c3b < \u03c3b which is a contradiction.Since y xn in , we haven, \u03c3b \u2264 \u03c3b and thereforeh = 2\u03b1s. Obviously, 0 \u2264 h < 1. Repeating the above process, we getn \u2265 1, and so, for m > n, we haveh < 1/s, it follows that lim\u2061m,n\u2192\u221e\u2061\u03c3b = 0. Since X is complete, there exists an element u \u2208 X such thatf is a continuous self-map on X, then fu = u. Indeed, by the triangle inequality we haven \u2192 \u221e in the above inequality, the desired result is obtained.Hence, for all xn = fnx0\u2aafu for all n \u2208 \u2115, and it follows thatn \u2192 \u221e,  = 0) it follows that\u03c3b \u2264 \u03b1s2\u03c3b, or equivalently, fu = u.If condition (b) is fulfilled then f are comparable. Let w be another fixed point of f such that w \u2260 u. Assume, for example, that u\u2aafw. Using  is a complete b-metric-like space with coefficient s = 2 (see Let  = 2 see .f : X \u2192 X be defined by fx = ln\u2061(1 + x/4). It is easy to see that f is a nondecreasing and continuous self-map on X, and 0 \u2208 (LF)f. Using the Mean Value Theorem for any x, y \u2208 X with x \u2264 y and that fx \u2264 x/4, we have\u03b1 = 1/16 \u2208 [0, 1/8). Thus all conditions of f.Let y).Thus  is satisX, \u2aaf, \u03c3b) be a partially ordered complete b-metric-like space with coefficient s > 1. If a nondecreasing map f : X \u2192 X satisfiesx, y \u2208 X with x\u2aafy, where\u03c6 : [0, \u221e) \u2192 [0, \u221e) is a continuous and nondecreasing function such that \u03c6(t) > 0 for all t \u2208  and \u03c6(0) = 0, then F(f) \u2260 \u2205 provided that there exists an x0 \u2208 (LF)f and one of the following two conditions is satisfied:f is a continuous self-map on X.X, \u2aaf, \u03c3b) has the sequential limit comparison property. \u2264 \u03c3b and {\u03c3b} is nonincreasing and bounded from below. Hence, there exists r \u2265 0 such that lim\u2061n\u2192\u221e\u2061\u03c3b = r.Define the sequence {y xn in , we obtan \u2192 \u221e, we gets > 1, possible only if r = 0. So we haveFrom the above argument we havexn} is a Cauchy sequence. If not, then there exists \u025b > 0 for which we can find subsequences {xmk} and {xnk} of the sequence {xn} where nk is the smallest index for which nk > mk > k withk \u2192 \u221e, from .Using , we get\u221e, from (\u025b\u2264liminf\u2061).Using , we get.Using .From  is satisfied then xn = fnx0\u2aafu for all n \u2208 \u2115; it follows thatn \u2192 \u221e in  = 0, implying that u = fu.If ows thats\u03c3b = 0, which yields that u = w. The converse is trivial.Now, suppose that the fixed points of w. Using , we obtaX = [0, \u221e) be endowed with the usual order \u2264. Define \u03c3b : X \u00d7 X \u2192 \u211d+ byx, y \u2208 X. Then  is a complete ordered b-metric-like space with coefficient s = 2 f. Using the Mean Value Theorem for the function f2x = (1/16)ln\u2061(x2/3 + 1) for any x, y \u2208 X with x \u2264 y, we have\u03c6 : [0, \u221e) \u2192 [0, \u221e) by \u03c6(t) = t/8, and, for any x, y \u2208 X with x \u2264 y, we havef2x = (1/16)ln\u2061(x2/3 + 1) is an increasing function and f2x \u2264 x2/48.Let f.Thus, all conditions of f and g be two self-maps on partially ordered set X. A pair  is said to befx\u2aafgfx and gx\u2aaffgx, for all x \u2208 X [weakly increasing if ll x \u2208 X , 20,fx\u2aafgfx, for all x \u2208 X [partially weakly increasing if ll x \u2208 X .Let X be a nonempty set and f : X \u2192 X a given mapping. For every x \u2208 X, let f\u22121(x) = {u \u2208 X : fu = x}.Let X, \u2aaf) be a partially ordered set and f, g, h : X \u2192 X mappings such that fX\u2286hX and gX\u2286hX. The ordered pair  is said to beh if and only if, for all x \u2208 X, fx\u2aafgy for all y \u2208 h\u22121(fx), and gx\u2aaffy for all y \u2208 h\u22121(gx) [weakly increasing with respect to h\u22121(gx) ,h if fx\u2aafgy, for all y \u2208 h\u22121(fx) [partially weakly increasing with respect to h\u22121(fx) .Let  with respect to h and (ii) if h = I (the identity mapping on X), then the above definition reduces to the weakly increasing  mapping (see [In the above definition (i) if ing see , 24)..f = g, w\u03c8, \u03c6)-weakly contractive mappings in partially ordered b-metric-like spaces.The study of unique common fixed points of mappings satisfying weakly contractive conditions has been at the center of vigorous research activity. Motivated by the work in , 23\u201330, \u03c8 : [0, \u221e) \u2192 [0, \u221e) which satisfies that\u03c81)(\u03c8(t) is increasing and continuous,\u03c82)(\u03c8(t) = 0 if and only if t = 0.Recall  that an X, \u2aaf, \u03c3b) be an ordered b-metric-like space and f, g, R, S : X \u2192 X four self-mappings. Throughout this subsection, unless otherwise stated, for all x, y \u2208 X, letf(X)\u2286R(X) and g(X)\u2286S(X); let x0 be an arbitrary point of X. Choose x1 \u2208 X such that fx0 = Rx1 and x2 \u2208 X such that gx1 = Sx2. Continuing in this way, construct a sequence {zn} defined by zn+12 = Rxn+12 = fxn2 and zn+22 = Sxn+22 = gxn+12, for all n \u2265 0. The sequence {zn} in X is said to be a Jungck-type iterative sequence with initial guess x0.Let  be a partially ordered b-metric-like space with sequential limit comparison property, f, g, R, S : X \u2192 X four mappings such that f(X)\u2286R(X) and g(X)\u2286S(X), and RX and SX\u03c3b-complete subsets of X. Suppose that, for comparable elements Sx, Ry \u2208 X, one has\u03c8, \u03c6 : [0, \u221e) \u2192 [0, \u221e) are altering distance functions. Then, the pairs  and  have a coincidence point z* in X provided that the pairs  and  are weakly compatible, and the pairs  and  are partially weakly increasing with respect to R and S, respectively.Let  and x2 \u2208 S\u22121(gx1), and the pairs  and  are partially weakly increasing with respect to R and S, so we haveRxn+12\u2aafSxn+22, for n \u2265 0. We will complete the proof in three steps.As Step\u2009\u2009I. We will prove that lim\u2061k\u2192\u221e\u2061\u03c3b = 0.\u03c3bk = \u03c3b. Suppose \u03c3bk0 = 0, for some k0. Then, zk0 = zk0+1. If k0 = 2n, then zn2 = zn+12 gives zn+12 = zn+22. Indeed,\u03c6) = 0; that is, zn+12 = zn+22. Similarly, if k0 = 2n + 1, then zn+12 = zn+22 gives zn+22 = zn+32. Consequently, the sequence {zk} becomes constant for k \u2265 k0 and hence lim\u2061k\u2192\u221e\u2061\u03c3b = 0.Define k. We now claim that the following inequality holds:k = 1,2,\u2026.Suppose thatk = 2n, and for an n \u2265 0, \u03c3b > \u03c3b > 0. Then, as Sxn2\u2aafRxn+12, using  > \u03c3b > 0,  = 0; that is, \u03c3b = 0, a contradiction to  \u2264 \u03c3b andk = 2n.Let 1, using  we obtai, using } is a nonincreasing sequence of nonnegative real numbers. Therefore, there is an r \u2265 0 such thatk \u2192 \u221e in  \u2264 \u03c8(r) \u2212 \u03c6(r). Therefore \u03c6(r) = 0. Hence,\u03c6.Similarly, it can be shown thatk \u2192 \u221e in , we obtan \u2192 \u221e in , using 2} and {zn(k)2} of {zn2} such that n(k) > m(k) \u2265 k andn(k) is the smallest number such that the above statement holds; that is,k \u2192 \u221e in (k \u2192 \u221e in (k \u2192 \u221e in (k \u2192 \u221e and using (Sxm(k)2\u2aafRxn(k)\u221212, so from (k \u2192 \u221e in (\u03c6(liminf\u2061k\u2192\u221e\u2061M(xm(k)2, xn(k)\u221212)) = 0. Hence,zn} is a \u03c3b-Cauchy sequence.We assume on the contrary that there exists  we have\u03c3b(z2m(k) \u2265 k and\u03c3b(z2m(k)k \u2192 \u221e in  we obtai we have\u03c3b(z2m(k)that is,\u03c3b(z2m(k)k \u2192 \u221e in  we have\u025b.Also,\u03c3b(z2m(k)k \u2192 \u221e in  and 78)\u025b > 0 fornd using  and 82)\u025b > 0 for so from , we havend using , (82), ( \u2192 \u221e in (limsup\u2061k\u2192d using (limsup\u2061k\u2192so from (\u03c8(\u03c3b(z2m( \u2192 \u221e in (\u025bs\u2264limsupd using (limsup\u2061k\u2192 \u2265 k and\u03c3b(z2m(k)Step\u2009\u2009III. We will show that f, g, R, and S have a coincidence point.zn} is a \u03c3b-Cauchy sequence and R(X) and S(X) are \u03c3b-complete \u03c3b-metric spaces, there exists z* \u2208 R(X)\u2229S(X) such thatu \u2208 X such that z* = Ru andv \u2208 X such that z* = Sv andz* is a coincidence point of g and R. For this purpose, we show that Ru = gu. Since Sxn+22 \u2192 z* = Ru as n \u2192 \u221e, so Sxn+22\u2aafRu.Since {n \u2192 \u221e in  = 0, and using Ru = z* = gu.Therefore, from , we have\u03c8Similarly it can be shown that X, \u2aaf, \u03c3b) be a partially ordered \u03c3b-complete \u03c3b-metric-like space. Let f, g : X \u2192 X be two mappings. Suppose that for every two comparable elements x, y \u2208 X one has\u03c8, \u03c6 : [0, \u221e) \u2192 [0, \u221e) are altering distance functions andf and g be continuous and let the pair  be weakly increasing. Then, f and g have a common fixed point z* in X.Let ) = 0. Thus, we have fz* = gz* = z*.Let .\u2009\u2009From , we haveerefore,max\u2061{\u03c3b be a complete ordered b-metric-like space, S, T : X \u2192 X dominated maps, and x0 an arbitrary point in X. Suppose that for k \u2208 [0, 1/s) and for S \u2260 T one hasxn} in xn} \u2192 u imply that u\u2aafxn. Then there exists \u03c3b = 0 and x* = Sx* = Tx*. Also, if for any two points x, y in z\u2aafx and z\u2aafy, that is, every pair of elements has a lower bound, then x* is a unique common fixed point in Let /2 so, using inequality /2) we obtainn \u2208 N. This implies thatxn} is a Cauchy sequence in n\u2192\u221e\u2061xn = x*. Also,n \u2192 \u221e and using the fact that x*\u2aafxn when xn \u2192 x*, we havex* = Sx*. Similarly, by usingx* = Tx*. Hence S and T have a common fixed point in Let equality , we obtaquality  \u2282 F(fn). However, the converse is false. For example, the mapping f : \u211d \u2192 \u211d, defined by fx = 1/2 \u2212 x, has a unique fixed point 1/4 but every x \u2208 \u211d is a fixed point of f2. If F(f) = F(fn) for every n \u2208 \u2115, then f is said to have property P. For more details, we refer to [Clearly, a fixed point of refer to , 34 and f : X \u2192 X with the constant \u03b1 \u2208 [0, 1/2), where X is a cone metric space, has the property P [X, d) is a cone metric space and T-Hardy-Rogers contraction f : X \u2192 X satisfies some appropriate conditions, then f has property P [Recently, the study of periodic points for contraction mappings has been considered by many authors; for instance, every quasicontraction X, \u2aaf) be a partially ordered set. A mapping f is called dominating on X if x\u2aaffx for each x in X.Let  be endowed with the usual ordering. Let f : X \u2192 X be defined by x \u2208 [0,1) and fx = xn for x \u2208 [1, \u221e), for any n \u2208 \u2115. Then, for all x \u2208 X, x \u2264 fx; that is, f is a dominating map.Let We have the following results.X, \u2aaf, \u03c3b) be a partially ordered complete b-metric-like space. Let f : X \u2192 X be a nondecreasing mapping such that for all x \u2208 X with x\u2aaffx one has\u03bb \u2208 [0,1). Then f has the property P provided that F(f) is nonempty and f is dominating on F(fn).Let (u \u2208 F(fn) for some n > 1. We will show that u = fu. Since f is dominating on F(fn), therefore u\u2aaffu which implies that fn\u22121u\u2aaffnu as f is nondecreasing. Using  = 0, implying that u = fu.Let g. Using , we obtaX and f be as in f is dominating on X, then f satisfies property P.Let F(f) \u2260 \u2205. We will prove that  = 0, then it is easy to see that  > 0. Since f is dominating, we have x\u2aaffx. Also, fx\u2aaff2x as f is nondecreasing. Using  \u2264 \u03c3b. On the contrary, if \u03c3b < \u03c3b, then from the above inequality we haves\u03b1 < 2s \u00b7 (1/2s2) < 1, a contradiction.If see that  is satisg. Using , we have\u03bb = 2s\u03b1. Obviously, \u03bb \u2208 [0,1). By f has property P.So we have"}
{"text": "G =  is said to be r-regular if each vertex of G is of degree r. The vertex covering transversal domination number \u03b3vct(G) is the minimum cardinality among all vertex covering transversal dominating sets of G. In this paper, we analyse this parameter on different kinds of regular graphs especially for Qn and Hn3,. Also we provide an upper bound for \u03b3vct of a connected cubic graph of order n \u2265 8. Then we try to provide a more stronger relationship between \u03b3 and \u03b3vct.A simple graph  Kn, Km,n, Pn, Cn, and Wn and trees is dealt with in paper  and E = E1 \u222a E2 \u222a E3 \u222a E4 with \u2009\u2009U = {u1, u2,\u2026, up+12}, Vi = {vi1, vi2,\u2026, vi(q + 2p)}, Wi = {wi1, wi2,\u2026, wi(q + 2p \u2212 1)},\u2009\u2009E1 = {ujuk; 1 \u2264 j < k \u2264 2p + 1},\u2009\u2009E2 = \u22c3i=1p+12{uivik; 1 \u2264 k \u2264 q},\u2009\u2009E3 = \u22c3i=1p+12{vi(q+2k\u22121)vi(q+2k); 1 \u2264 k \u2264 p},\u2009\u2009E4 = \u22c3i=1p+12{vikwil; 1 \u2264 k \u2264 q + 2p, 1 \u2264 l \u2264 q + 2p \u2212 1}.Then(i)V(G)| = 2(2p + 1)(q + 2p),|(ii)G\u2217 is connected and (q + 2p)-regular,(iii)\u03b3vct) = 2(2p + 1).Given positive integers p = 1 and q = 2, the graph G\u2217 is as shown in (i) and (ii) are obvious. If Ij = [\u22c3i=1p+12(Wi)] \u222a {uj}, j = 1 to 2p + 1, is a maximum independent set in G\u2217. Therefore its complement Jj = V \u2212 Ij = [\u22c3i=1p+12(Vi)]\u222a[U \u2212 {uj}], j = 1 to 2p + 1, is an \u03b10 set in G\u2217. Further J1, J2,\u2026, Jp2 and Jp+12 are the only \u03b10-sets of G\u2217. Now each Sjk = \u22c3i=1p+12{vik, wij}, j = 1 to q + 2p \u2212 1, k = 1 to q is a dominating set intersecting J1, J2,\u2026, Jp2 and Jp+12 and also of minimum cardinality 2(2p + 1).It is clear that each \u03b3vct) = 2(2p + 1).Hence r \u2265 4, there exists a connected r-regular graph G of order n such that \u03b3vct(G) = n/r.For every G = G\u2217 be defined as in G is a connected r-regular graph with r = q + 2p. Also \u03b3vct(G)/n = 1/(q + 2p).Let \u03b3vct(G) = n/r.Thus \u03b3vct is replaced by \u03b3.Theorems Hn3, defined in [In this section, we provide the vertex covering transversal domination number of some regular cubic graphs especially Harary graph fined in . We alsoY3 shown in Consider the triangular prism graph Y3 has 6 vertices and 9 edges. Assume that the graph Y3 is labelled as shown in the diagram. It is clear that {uimod\u20613, vi + 1)mod\u20613mod\u20613mod\u20613mod\u20613mod\u20613(} and Si = {ui + 1)mod\u20613mod\u20613mod\u20613 = 2.Consider Peterson graph which is cubic regular shown in G is labelled as shown in Ii = {vimod\u20615, vi + 3)mod\u20615mod\u20615mod\u20615mod\u20615mod\u20615mod\u20615mod\u20615mod\u20615mod\u20615mod\u20615(} are \u03b3-sets intersecting each Ci. Hence \u03b3vct(G) = 3.Assuming that the graph Si = {vimod\u20615, vi + 3)mod\u20615mod\u20615 = \u230a(n + 1)/3\u230b.If Hn3, is a 3-regular graph and so n is even. By the definition of Hn3,, every vertex vi \u2208 Hn3, is adjacent to the vertices vi+1, vi\u22121, and vi+k where n = 2k.V = {v0, v1, v2,\u2026, vn\u22121}. The graphs H3,10 and H3,12 are shown in Let Case\u2009\u20091. Suppose n = 2k where k is odd.C1 = {v0, v2, v4,\u2026, vn\u22122} and C2 = {v1, v3, v5,\u2026, vn\u22121} are the only \u03b10-sets of Hn3,.Then Subcase\u2009\u20091. Let n \u2261 0 (mod\u20613).S = {v0, v3, v6,\u2026, vn\u22123} is a \u03b3-set which intersects C1 and C2 and |S| = n/3.Then Subcase\u2009\u20092. Suppose n \u2261 1 (mod\u20613).S = {v0, v3, v6,\u2026, vn\u22124} is a \u03b3-set which intersects C1 and C2 and |S| = (n \u2212 1)/3.Then Subcase\u2009\u20093. Suppose n \u2261 2 (mod\u20613).S = {v0, v3, v6,\u2026, vn\u22122} is a \u03b3-set which intersects C1 and C2 with |S| = (n + 1)/3.Then \u03b3vct = \u230a(n + 1)/3\u230b.Thus in all the subcases of Case\u2009\u20091, Case\u2009\u20092. Suppose n = 2k where k is even.Ii = {vimod\u2061n, vi + 2)mod\u2061nmod\u2061nmod\u2061nmod\u2061nmod\u2061nmod\u2061nmod\u2061nmod\u2061nmod\u2061nmod\u2061nmod\u2061nmod\u2061nmod\u2061nmod\u2061n.Si = {vimod\u2061n, vi + 3)mod\u2061nmod\u2061nmod\u2061n.Si = {vimod\u2061n, vi + 3)mod\u2061nmod\u2061nmod\u2061n.Si = {vimod\u2061n, vi + 3)mod\u2061nmod\u2061nmod\u2061n/3\u230b.The \u03b3vct = \u230a(n + 1)/3\u230b.Thus \u03b3vct = \u03b3.In most of the graphs considered by us, it is observed that G is a connected cubic graph of order n with n \u2265 8, then \u03b3vct(G)\u2264\u23082n/5\u2309.If I be an independent dominating set of cardinality i(G). Then I is a maximal independent set of minimum cardinality. Since I is independent, no two vertices of I are adjacent in G. Let J = V \u2212 I. Then the vertices in I are adjacent only to the vertices in J.Let Case\u2009\u20091. Suppose I itself is a \u03b20-set. Then J is an \u03b10-set. Let S = I \u222a {v} where v \u2208 J. Then S is a vertex covering transversal dominating set of G. Therefore \u03b3vct(G) \u2264 i(G) + 1. Hence \u03b3vct(G) \u2264 2n/5 + 1  \u2264 2n/5.Hence \u03b3vct(G)\u2264\u23082n/5\u2309.Thus Cases\u2009\u20091 and 2 imply that \u03b3 and \u03b3vct than that proved in [\u03b3 = \u03b3vct and \u03b3 < \u03b3vct.In this section, we prove a more stronger relationship between roved in . In viewG is a simple connected graph, then \u03b3vct(G) \u2264 \u03b3(G) + 1.If D be a minimum dominating set. If D = V(G), then obviously \u03b3vct(G) = \u03b3(G). If not, then D \u2282 V(G) and V(G) \u2212 D \u2260 \u03d5. Let u \u2208 V(G) \u2212 D. Then u is dominated by some vertex v in D. Let S = D \u222a {u}. Since uv is an edge in G, either u or v is included in every minimum vertex covering set of G. This implies that S intersects every minimum vertex covering set in G. Hence \u03b3vct(G) \u2264 \u03b3(G) + 1.Let G be a simple connected graph. If there exists a \u03b3-set which is not independent, then \u03b3vct(G) = \u03b3(G).Let D be a minimum dominating set which is not an independent set of G. Then at least two vertices, say, u, v in D, are adjacent to each other. Therefore uv is an edge in G and hence either u or v lies in every minimum vertex covering set of G. So D intersects every \u03b10-set of G. Therefore D itself is a \u03b3vct-set. Hence \u03b3vct(G) = \u03b3(G).Let \u03b3vct(G) = \u03b3(G), then there may exist a \u03b3-set which is independent also. For example, consider C6, the cycle on 6 vertices as shown in The converse is not true. If v1, v3, v5} and {v2, v4, v6} are the only \u03b10-sets of C6. Also {v1, v4} is a \u03b3-set which is independent. Further, it is a \u03b3vct-set as it intersects both the \u03b10-sets of C6. Thus there exists a \u03b3-set which is independent in C6 even though \u03b3vct(C6) = \u03b3(C6).Obviously {\u03b3vct(G) = \u03b3(G), is every \u03b3-set of G a \u03b3vct-set?\u201d The answer is \u201cnot always.\u201d The \u03b3-sets and \u03b3vct-sets in the graphs Q2 and Y3 discussed in the previous sections are the best examples for it. So it is noted that this happens if there exists a \u03b3-set which is also a \u03b2o-set. It obviously produces the result that \u201cIf \u03b3vct(G) = \u03b3(G) = \u03b2o(G), then there exists at least one \u03b3-set in G which is not a \u03b3vct-set.\u201d The next general question is that \u201cWhat happens if all the \u03b3-sets of G are \u03b20-sets?\u201d. The following theorem provides the answer to it.Now, the obvious question is \u201cIf G be a simple connected graph. If every \u03b3-set of G is a \u03b20-set, then \u03b3vct(G) = \u03b3(G) + 1.Let \u03b3-set D of G is a \u03b20-set, choose a vertex v in its complement. This is possible since D \u2260 V(G) as D is a \u03b20-set of a connected graph G. Obviously D is not a \u03b3vct-set as it does not intersect the \u03b10-set V(G) \u2212 D. Let S = D \u222a {v}. We claim that S intersects every \u03b10-set of G. Suppose that S\u2229C = \u03a6 for some \u03b10-set C in G. Then S\u2286I where I = V(G) \u2212 C is a \u03b20-set. This implies that \u03b20(G) + 1 \u2264 \u03b20(G) which is a contradiction. Hence S intersects every \u03b10-set of G. Also S is a \u03b3vct-set of G as it contains exactly one vertex more than that of the \u03b3-set D. Thus \u03b3vct(G) = \u03b3(G) + 1.Since every \u03b3vct = \u03b3, there are graphs in which \u03b3-sets do not become \u03b3vct-sets. This implies that the collection of \u03b3vct-sets in such graphs is contained in the collection of \u03b3-sets. So this may lead to consider \u03b3vct-sets in the graphs for which \u03b3vct = \u03b3 when we are in a situation to select a minimum number of \u03b3-sets in such graphs. This approach may affect a new variation in domination theory.It is easy to conclude that even though"}
{"text": "Lmx1b Causes Glaucoma and Is Semi-lethal via LDB1-Mediated Dimerisation\u201dIn the title \u201cLBD1\u201d should appear as \u201cLDB1\u201d. The correct title should read \u201cA Dominant-Negative Mutation of Mouse"}
{"text": "Rapid elevation of sodium transport through insulin is mediated by AKT in alveolar cellsCharlott Mattes, Mandy Laube & Ulrich H. ThomePhysiol Rep, 2 (3), 2014, e00269, doi: 10.1002/phy2.269\u03bcmol/L\u201d has been corrected to \u201c20 nmol/L insulin\u201d, and \u201c200 \u03bcmol/L insulin\u201d has been corrected to \u201c200 nmol/L insulin\u201d.In figure 1, the label \u201c20 Please refer to the revised figure 1 below."}
{"text": "In this paper, not only strongly lacunary ward continuity, but also some other kindsof continuities are investigated in 2-normed spaces.A function  Menger  introducN\u03b8-convergence was introduced in [N\u03b8-ward continuity and further studied in [Using the main idea in the definition of sequential continuity, many kinds of continuities were introduced and investigated, and not all but some of them are in \u201313. The duced in  and furtduced in . Stronglduced in  as namedudied in .The aim of this paper is to investigate strongly lacunary ward continuity in 2-normed spaces and prove interesting theorems.N and R will denote the set of all positive integers and the set of all real numbers, respectively. Now we recall the definition of a two-normed space. Let X be a real linear space with dim\u2061X > 1 and ||\u00b7, \u00b7|| : X \u00d7 X \u2192 R a function. Then  is called a linear 2-normed space if (i) ||x, y|| = 0\u21d4x and y are linearly dependent, (ii) ||x, y|| = ||y, x||, (iii) ||\u03b1x, y|| = |\u03b1|||x, y||, and (iv) ||x, y + z|| \u2264 ||x, y|| + ||x, z|| for \u03b1 \u2208 R and x, y, z \u2208 X. The function ||\u00b7, \u00b7|| is called a 2-norm on X. Observe that in any 2-normed space  we have that ||\u00b7, \u00b7|| is nonnegative, ||x \u2212 z, x \u2212 y|| = ||x \u2212 z, y \u2212 z||, and ||x, y + \u03b1x|| = ||x, y|| for all x, y \u2208 X, \u03b1 \u2208 R. Throughout this paper by X we will mean a 2-normed space with a two-norm ||\u00b7, \u00b7||. As an example of a 2-normed space we may take X = R2 being equipped with the 2-normx1 = , x2 = .Throughout this paper, xn) of points in X is said to converge to a point L of X in the 2-normed space X if lim\u2061n\u2192\u221e||xn \u2212 L, z|| = 0 for every z \u2208 X. This is denoted by lim\u2061n\u2192\u221e||xn, z|| = ||L, z||. A sequence (xn) of points in X is said to be a Cauchy sequence with respect to the 2-norm if lim\u2061n,m\u2192\u221e||xn \u2212 xm, z|| = 0 for every z \u2208 X. A sequence of functions (fn) is said to be uniformly convergent to a function f on a subset E of X if for each \u03b5 > 0, an integer N can be found such that ||fn(x) \u2212 f(x), z|| < \u03b5 for n \u2265 N and for all x, z \u2208 X [\u03b8 = (kr) is an increasing sequence of positive integers such that k0 = 0 and hr = kr \u2212 kr\u22121 \u2192 \u221e as r \u2192 \u221e. The intervals determined by \u03b8 will be denoted by Ir =  of points in X is called strongly lacunary convergent, or N\u03b8-convergent to an element L of X ifz \u2208 X and it is denoted by N\u03b8 \u2212 lim\u2061k\u2192\u221e||xk, z|| = ||L, z|| for every z \u2208 X [N\u03b8\u201d instead of \u201cstrongly lacunary\u201d and assume that liminf\u2061r\u2009qr > 1.A sequence  = ||x, z||, \u2200x, for each z \u2208 X, forms a locally convex topological vector space, and the topology formed by this family of seminorms gives the required topology on X. Since dim\u2061X \u2265 2, for each x \u2208 X there exists a y \u2208 X such that x and y are linearly independent, and hence by (i), py(x) = ||x, y|| \u2260 0. Thus the locally convex topological vector space induced by the set {pz : z \u2208 X} of seminorms is Hausdorff so X is a Hausdorff space [The set of seminorms {ff space . In thisE of X is called N\u03b8-sequentially compact if any sequence of points in E has an N\u03b8-convergent sequence with an N\u03b8-limit in E.A subset N\u03b8-sequentially compact subsets of X is N\u03b8-sequentially compact, intersection of any N\u03b8-sequentially compact subsets is N\u03b8-sequentially compact, any compact subset of X is N\u03b8-sequentially compact, and any finite subset of X is N\u03b8-sequentially compact. Sum of N\u03b8-sequentially compact subsets of X is N\u03b8-sequentially compact where sum of two subsets A and B is defined as A + B = {a + b : a \u2208 A, b \u2208 B}.We note that union of two f defined on a subset E of X is said to be strongly lacunary sequentially continuous or N\u03b8-sequentially continuous at a point x0 of E if (f(xk)) is an N\u03b8-convergent sequence to f(x0) whenever (xk) is an N\u03b8-convergent to x0 sequence of points in E. If f is strongly lacunary sequentially continuous at every point of E, then it is said to be strongly lacunary sequentially continuous on E.A function f defined on a subset E of X is lacunary statistically sequentially continuous at a point x0, then (f(xk)) is an N\u03b8-convergent sequence with N\u03b8 \u2212 lim\u2061||f(xk), z|| = ||f(x0), z|| for every z \u2208 X whenever (xk) is an N\u03b8-convergent sequence with N\u03b8 \u2212 lim\u2061||xk, z|| = ||x0, z|| for every z \u2208 X. We see that a function f defined on a subset E of X is strongly lacunary sequentially continuous if and only if it preserves strongly lacunary convergent sequences without stating limit of the sequence. We note that sum of two N\u03b8-sequentially continuous functions at a point x0 of X is N\u03b8-sequentially continuous at x0, and composite of two N\u03b8-sequentially continuous functions at a point x0 of X is N\u03b8-sequentially continuous at x0. In the classical case, that is in the single normed case, it is known that uniform limit of sequentially continuous function is sequentially continuous; now we see that it is also true that not only uniform limit of sequentially continuous function is sequentially continuous, but also uniform limit of N\u03b8-sequentially continuous function is N\u03b8-sequentially continuous in 2-normed spaces. Now we give the latter in the following.If a function N\u03b8-sequentially continuous functions is N\u03b8-sequentially continuous.Uniform limit of fk) be a uniformly convergent sequence of each term defined on a subset E of X with uniform limit f and let (xk) be any N\u03b8-convergent sequence of points in E with N\u03b8 \u2212 lim\u2061||xk, z|| = ||x, z|| for every z \u2208 X. Take any \u03b5 > 0. By uniform convergence of (fk), there exists an n1 \u2208 N such that ||f(x) \u2212 fk(x), z|| < \u03b5/3 for k \u2265 n1 and every x \u2208 E and z \u2208 X. Hence,r \u2265 n1 and every x \u2208 E and z \u2208 X. As fn1 is N\u03b8-sequentially continuous on E, there exists an n2 \u2208 N such that, for r \u2265 n2,z \u2208 X. Now write n0 = max\u2061{n1, n2}. Thus for r \u2265 n0 we havevk(x) = f(x) \u2212 fn1(x) for every k \u2208 N. HenceLet ) be any sequence of points in f(A) where xn \u2208 A for each positive integer n. N\u03b8-sequentially compactness of A implies that there is a subsequence (\u03b3k) = (xnk) of (xn) with N\u03b8 \u2212 lim\u2061k\u2192\u221e||\u03b3k, z|| = ||l, z|| for every z \u2208 E. Write (tk) = (f(\u03b3k)). As f is N\u03b8-sequentially continuous, (f(\u03b3k)) is N\u03b8-sequentially convergent which is a subsequence of the sequence (f(xn)) with N\u03b8 \u2212 lim\u2061k\u2192\u221e||tk, z|| = ||l, z|| for \u2200z \u2208 E. This completes the proof of the theorem.Assume that N\u03b8-quasi-Cauchy sequence.The concept of a quasi-Cauchy sequence in a 2-normed space was studied in . Now we xn) of points in a subset E of X is called N\u03b8-quasi-Cauchy if (\u0394xk) is N\u03b8-convergent to 0, that is,xk = xk+1 \u2212 xk.A sequence ) is an N\u03b8-quasi-Cauchy sequence whenever (xk) is.A function defined on a subset N\u03b8-ward continuous functions is N\u03b8-ward continuous, and sum of two N\u03b8-ward continuous functions is N\u03b8-ward continuous.We note that a composite of two N\u03b8-ward continuous image of any N\u03b8-ward compact subset of X is N\u03b8-ward compact.f is an N\u03b8-ward continuous function on a subset E of X and E is a N\u03b8-ward compact subset of A. Let (f(xn)) be any sequence of points in f(E) where xn \u2208 E for each positive integer n. N\u03b8-ward compactness of E implies that there is a subsequence (\u03b3k) = (xnk) of (xn) with N\u03b8 \u2212 lim\u2061k\u2192\u221e||\u0394\u03b3k, z|| = 0 for every z \u2208 E. Write (tk) = (f(\u03b3k)). As f is N\u03b8-ward continuous, (f(\u03b3k)) is N\u03b8-quasi-Cauchy which is a subsequence of the sequence (f(xn)) with N\u03b8 \u2212 lim\u2061k\u2192\u221e||\u0394tk, z|| = 0 for \u2200z \u2208 E. This completes the proof of the theorem.Assume that N\u03b8-ward continuous image of any compact subset of X is N\u03b8-ward compact.The proof follows from the preceding theorem.f defined on a subset E of X is sequentially continuous at x0, if for any sequence (xn) of points in E converging to x0, we have (f(xn)) converges to f(x0). f is sequentially continuous on E if it is sequentially continuous at every point of E  \u2208 \u0394N\u03b8\u21d2(f(xk)) \u2208 \u0394N\u03b8,(xk) \u2208 \u0394N\u03b8\u21d2(f(xk)) \u2208 c,(xk) \u2208 c\u21d2(f(xk)) \u2208 c,(xk) \u2208 c\u21d2(f(xk)) \u2208 \u0394N\u03b8,(xk) \u2208 N\u03b8\u21d2(f(xk)) \u2208 N\u03b8,(xk) \u2208 \u0394N\u03b8\u21d2(f(xn))\u2009\u2009is quasi-Cauchy,(xk) \u2208 \u0394N\u03b8\u21d2(f(xn)) \u2208 N\u03b8. is N\u03b8-sequentially continuity of f, and (3) is the ordinary sequential continuity of f. It is easy to see that (2) implies (1), and (1) does not imply (2); and (1) implies (4), and (4) does not imply (1); (2) implies (3), and (3) does not imply (2). (6) implies (1), but (1) does not imply (6) and (7) implies (1), but (1) does not imply (7). Now we give that the implication (1) implies (5), that is, any N\u03b8-ward continuous function is N\u03b8-sequentially continuous.We see that (1) is f is N\u03b8-ward continuous function on a subset E of X, then it is N\u03b8-sequentially continuous on E.If f is an N\u03b8-ward continuous function on a subset E of X. Let (xn) be any N\u03b8-convergent sequence in A with N\u03b8 \u2212 lim\u2061k\u2192\u221e||xk, z|| = ||x0, z|| for all z \u2208 X. Then the sequence x = (xn) defined byN\u03b8-convergent to x0. Hence it is N\u03b8-quasi-Cauchy sequence. As f is N\u03b8-ward continuous on E, the transformed sequence (yn) obtained byN\u03b8-quasi-Cauchy. Thus,z \u2208 X. It follows from this that the sequence (f(xk)) is N\u03b8 convergent to (f(x0)).Assume that f =  on the 2-normed space R2 with the usual 2-norm.The converse of this theorem is not valid in general, a counterexample can be easily constructed via the function fn) is a sequence of N\u03b8-ward continuous functions on a subset E of X and (fn) is uniformly convergent to a function f, then f is N\u03b8-ward continuous on E.If (xk) be any N\u03b8-quasi-Cauchy sequence of points in E, and let \u03b5 be any positive real number. By uniform convergence of (fk), there exists an n1 \u2208 N such that ||f(x) \u2212 fk(x), z|| < \u03b5/3 for k \u2265 n1 and every x \u2208 E and z \u2208 X. Hence,r \u2265 n1 and every x \u2208 E and z \u2208 X. As fn1 is N\u03b8-ward continuous on E, there exists an n2 \u2208 N such that, for r \u2265 n2,z \u2208 X. Now write n0 = max\u2061{n1, n2}. Thus, for r \u2265 n0, we havef preserves N\u03b8-quasi-Cauchy sequences. This completes the proof of the theorem.Let (n-normed spaces as another further study.In this paper, we investigate strongly lacunary continuity and some other kinds of continuities defined via a lacunary sequence and we prove interesting theorems related to these kinds of continuities. The results in this paper are extensively deeper than existing related results in the literature. We note that the notion of a strongly lacunary quasi-Cauchy sequence coincides with the notion of a strongly lacunary convergent sequence in a complete non-Archimedean 2-normed space, and so the set of strongly lacunary ward continuous functions coincides with the set of strongly lacunary sequentially continuous functions in a complete non-Archimedean 2-normed space (see  for the"}
{"text": "\u2133\u03c3 and \u2133\u03c3 of meromorphic bi-univalent functions defined on \u0394 = {z : z \u2208 \u2102, 1 < |z | <\u221e}. For functions belonging to these classes, estimates on the initial coefficients are obtained.We introduce and investigate two new subclasses  \ud835\udc9c denote the class of all functions of the form \ud835\udcae the class of all functions in the normalized analytic function class \ud835\udc9c which are univalent in \ud835\udd4c.Let \ud835\udd4c. In fact, the Koebe one-quarter theorem [\ud835\udd4c under every univalent function f \u2208 \ud835\udcae contains a disk of radius 1/4. Thus, every function f \u2208 \ud835\udc9c has an inverse f\u22121, which is defined by f\u22121 is given by Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk  theorem  ensures f \u2208 \ud835\udc9c is said to be bi-univalent in \ud835\udd4c if both f and f\u22121 are univalent in \ud835\udd4c. Let \u03a3 denote the class of bi-univalent functions in \ud835\udd4c given by =z+\u2211nThe coefficient problem was investigated for various interesting subclasses of the meromorphic univalent functions .b0 and b1 of functions in the newly introduced subclasses are obtained.In the present investigation, certain subclasses of meromorphic bi-univalent functions are introduced and estimates for the coefficients g given by  if the following conditions are satisfied: h = g\u22121 is given by  if the following conditions are satisfied: h = g\u22121 is given by  and \u2133\u03c3 reduce to the classes We note that, for m et al. .b0| and |b1| for functions in the above-defined classes \u2133\u03c3 and \u2133\u03c3 of the function class \u03c3\u2133.The object of the present paper is to extend the concept of bi-univalent to the class of meromorphic functions defined on \u0394 and find estimates on the coefficients |Firstly, in order to derive our main results, we need the following lemma.p \u2208 \ud835\udcab, then |ck| \u2264 2 for each k, where \ud835\udcab is the family of all functions p analytic in \ud835\udd4c for which z \u2208 \ud835\udd4c.If b0| and |b1| for functions in the class \u2133\u03c3.We begin this section by finding the estimates on the coefficients |g(z) given by the series expansion (Let the function xpansion  be in thp(z) and q(w) are functions with positive real part in \u0394 and have the forms \u211c(p(z)) > 0 and \u211c(q(z)) > 0 in \u0394, the functions p(1/z), q(1/z) \u2208 \ud835\udcab and hence the coefficients pk and qk for each k satisfy the inequality in b0| as asserted in (It follows from  that (17ents in (\u2212(1\u2212\u03bb)b0=\u22121)2p12,(1\u2212\u03bb)b0=\u03b1)b0=\u03b1p1,(1\u2212\u03bb)[(1\u2212)b0=\u03b1q1,(1\u2212\u03bb)[(1\u2212)p1=\u2212q1,2(1\u2212\u03bb)2b0o, from (2(1\u2212\u03bb)2b0erted in .b1|, we subtract 2[. Hence |b1|\u2264\u03b11\u2212\u03bb12. From |b1|\u22645\u03b121\u03bb = 0, we have the following corollary of For g(z) given by the series expansion (Let the function xpansion  be in thg(z) given by the series expansion .Next we estimate the coefficients |g(z) given by the series expansion (Let the function xpansion  be in thp(z) and q(w) are functions with positive real part in \u0394 and have the forms (It follows from  that (40he forms  and 19)(40)zg\u2032(ze forms (\u2212(1\u2212\u03bb)b0=(1\u2212\u03b2)p2,(1\u2212\u03bb)b0=((1\u2212\u03b2)p1,(1\u2212\u03bb)[(1\u2212(1\u2212\u03b2)q1,(1\u2212\u03bb)[(1\u2212\u211c(p(z)) > 0 and \u211c(q(z)) > 0 in \u0394, the functions p(1/z), q(1/z) \u2208 \ud835\udcab and hence the coefficients pk and qk for each k satisfy the inequality in b0| as asserted in b1|, we valently 4b12=(1\u2212\u03bb). Hence |b1|\u22641\u2212\u03b212 from  given by the series expansion (Let the function xpansion  be in thg(z) given by the series expansion (Let the function xpansion  be in th"}
{"text": "Ce-I functions and fuzzy completely Ce-I functions via fuzzy e-open sets. Some properties and several characterization of these types of functions are investigated. We introduced the notions of fuzzy  Ce-I functions, fuzzy Ce-continuous, fuzzy completely Ce-I functions, and fuzzy e-kernel via fuzzy e-open sets and studied their properties and several characterizations of these types of functions are investigated. In this paper, we denote fuzzy e-open, fuzzy e-closed, and fuzzy regular closed as, eo, ec, and rc, respectively.With the introduction of fuzzy sets by Zadeh  and fuzzX, \u03c4) and  (or simply X and Y) represent nonempty fuzzy topological spaces on which no separation axioms are assumed, unless otherwise mentioned.Throughout this paper, , int(\u03bc), cl\u03b4(\u03bc), and int\u03b4(\u03bc), respectively. A fuzzy set \u03bc of X is called fuzzy regular open [\u03bc = int(cl(\u03bc)) )).Let lar open  . The complement of fuzzy \u03b4-open set is called fuzzy \u03b4-closed ). A fuzzy set \u03bc of X is called fuzzy \u03b4-preopen [\u03b4-semi open [\u03bc \u2264 int(cl\u03b4(\u03bc)) )). The complement of a fuzzy \u03b4-preopen set  is called fuzzy \u03b4-preclosed .The fuzzy y \u03b4-open  if \u03bc = i-preopen   if \u03bc \u2264 \u03bc of a fuzzy topological space X is called fuzzy e-open [\u03bc \u2264 cl(int\u03b4\u03bc)\u2228int(cl\u03b4\u03bc). Fuzzy e-closed if \u03bc \u2265 cl(int\u03b4\u03bc)\u2227int(cl\u03b4\u03bc).A fuzzy set y e-open  if \u03bc \u2264 c\u03bc is called fuzzy e-closure of \u03bc and is denoted by fe-cl(\u03bc) and the union of all fuzzy e-open sets contained in \u03bc is called fuzzy e-interior of \u03bc and is denoted by fe-int(\u03bc).The intersection of all fuzzy e-closed sets containing f : X \u2192 Y is said to be fuzzy e\u2217-open [X is fuzzy e-open set in Y.A mapping  e\u2217-open  if the if : X \u2192 Y is called fuzzy e-irresolute [f\u22121(\u03bb) is fuzzy e-open in X for every fuzzy e-open set \u03bb of Y.A function resolute . f\u22121(\u03bb)\u03bc is quasicoincident [\u03bb denoted by \u03bcq\u03bb iff there exist x \u2208 X such that \u03bc(x) + \u03bb(x) > 1. If \u03bc and \u03bb are not quasicoincident, then we write A fuzzy set incident  with a fxp is quasicoincident [\u03bb denoted by xpq\u03bb iff there exist x \u2208 X such that p + \u03bb(x) > 1.A fuzzy point incident  with a fX, \u03c4) is said to be fuzzy e-T1 [x and y of X there exist fuzzy e-open sets \u03bc1and \u03bc2 such that x \u2208 \u03bc1 and y \u2208 \u03bc2 and x \u2209 \u03bc2 and y \u2209 \u03bc1.A fuzzy topological space  is said to be fuzzy e-T2 [x and y of X there exists disjoint fuzzy e-open sets \u03b7 and \u03c1 such that x \u2208 \u03b7 and y \u2208 \u03c1.A fuzzy topological space \u2227cl(\u03bb) = 0.A fuzzy topological space  Urysohn  if for eX, \u03c4) is called fuzzy S-closed [X has a finite subcover.A space  if everf : X \u2192 Y is called fuzzy completely continuous [f\u22121(\u03bb) is fuzzy regular open in X for every fuzzy open set \u03bb of Y.A function ntinuous  if f\u22121 functions is introduced and some characteristics and properties are studied.In this section, the notion of fuzzy \u03c6 : \u2192 is called fuzzy Ce-I if the inverse image of every fuzzy e-open set of Y is fuzzy e-closed in X.A mapping Ce-I and fuzzy e-irresolute are independent notions as illustrated in the following example.The concepts of fuzzy X = {a, b, c} and Y = {x, y, z} and the fuzzy sets \u03bc1, \u03bc2, \u03c5 be defined as follows: \u03c4 = {0,1, \u03bc1, \u03bc2, \u03bc1\u2228\u03bc2, \u03bc1\u2227\u03bc2} and \u03c3 = {0, \u03c5, 1}. Then, the mapping \u03c6 : \u2192 is defined by \u03c6(a) = x, \u03c6(b) = y, \u03c6(c) = z. Then, \u03c6 is fuzzy Ce-I but not fuzzy e-irresolute.Let X = {a, b, c} and Y = {x, y, z} and the fuzzy sets \u03b71, \u03b72, \u03c1 are defined as follows:Let \u03c4 = {0,1, \u03b71, \u03b72} and \u03c3 = {0, \u03c1, 1}. Then, the mapping \u03c6 : \u2192 is defined by \u03c6(a) = x, \u03c6(b) = y, \u03c6(c) = z. Then, \u03c6 is fuzzy e-irresolute but not fuzzy Ce-I.Let \u03c6 : \u2192 is called fuzzy Ce-continuous if the inverse image of every fuzzy open set of Y is fuzzy e-closed in X.A mapping Ce-I function is fuzzy Ce-continuous, but not conversely from the following example.Every fuzzy X = {a, b, c} and Y = {x, y, z} and the fuzzy sets \u03bc1, \u03bc2, \u03bd, \u03c5 are defined as follows: \u03c4 = {0,1, \u03bc1, \u03bc2, \u03bc1\u2228\u03bc2, \u03bc1\u2227\u03bc2} and \u03c3 = {0, \u03bd, 1}. Then, the mapping \u03c6 : \u2192 is defined by \u03c6(a) = x, \u03c6(b) = y, and \u03c6(c) = z. Then, \u03c6 is fuzzy Ce-continuous but not fuzzy Ce-I as the fuzzy set \u03c5 is fuzzy e-open in Y but \u03c6\u22121(\u03c5) is not fuzzy e-closed set in X.Let \u03c6 : X \u2192 Y, if \u03c6(x\u03b5)q\u03bc, the inverse image of for every fuzzy e-closed set of Y is fuzzy e-open in X iff for any x\u03b5 \u2208 X; if \u03c6(x\u03b5)q\u03bc, then x\u03b5qfe-int(\u03c6\u22121(\u03bc)).For a fuzzy function \u03bc \u2264 Y be a fuzzy e-closed set and \u03c6(x\u03b5)q\u03bc. Then, x\u03b5q\u03c6\u22121(\u03bc) and, by hypothesis, \u03c6\u22121(\u03bc) = fe-int(\u03c6\u22121(\u03bc)). We obtain, x\u03b5qfe-int(\u03c6\u22121(\u03bc)). Converse can be shown easily.Let \u03c6 : X \u2192 Y, if \u03c6(x\u03b5)q\u03bc, for any fuzzy e-closed set \u03bc \u2264 Y and for any x\u03b5 \u2208 X, x\u03b5qfe-int(\u03c6\u22121(\u03bc)) iff there exists a fuzzy e-open set \u03b4 such that x\u03b5q\u03b4 and \u03c6(\u03b4) \u2264 \u03bc.For a fuzzy function \u03bc \u2264 Y be any fuzzy e-closed set and let \u03c6(x\u03b5)q\u03bc. Then, x\u03b5qfe-int(\u03c6\u22121(\u03bc)). Take \u03b4 = fe-int(\u03c6\u22121(\u03bc)) then \u03c6(\u03b4) = \u03c6(fe-int(\u03c6\u22121(\u03bc))) \u2264 \u03c6(\u03c6\u22121(\u03bc)) \u2264 \u03bc, and \u03b4 is fuzzy e-open in X and x\u03b5q\u03b4.Let \u03bc \u2264 Y be any fuzzy e-closed set and let\u03c6(x\u03b5)q\u03bc. By hypothesis, there exists fuzzy e-open set \u03b4 such that x\u03b5q\u03b4 and \u03c6(\u03b4) \u2264 \u03bc. This implies, \u03b4 \u2264 \u03c6\u22121(\u03bc) and then x\u03b5qfe-int(\u03c6\u22121(\u03bc)).Conversely, let \u03c6 : X \u2192 Y, the following statements are equivalent:(1)f\u2009\u2009is fuzzy Ce-I.(2)\u03bc in Y, \u03c6\u22121(\u03bc) is fuzzy e-open in X.For every fuzzy e- closed set (3)\u03bc, \u03c6\u22121(fe-int(\u03bc)) is fuzzy e-closed.For every fuzzy open set (4)\u03b7, \u03c6\u22121(fe-cl\u2061(\u03b7)) is fuzzy e-open.For every fuzzy closed set (5)x\u03b5 \u2208 X and each fuzzy e-closed set \u03bc in Y containing \u03c6(x\u03b5), there exists a fuzzy e-open set \u03c1 in X containing x\u03b5 such that \u03c6(\u03c1) \u2264 \u03bc.For each (6)x\u03b5 \u2208 X and each fuzzy e-open set \u03bc in Y noncontaining \u03c6(x\u03b5), there exists a fuzzy e-closed set \u03bd in X noncontaining x\u03b5 such that \u03c6\u22121(\u03bc) \u2264 \u03bd.For each For a fuzzy function \u03c1 be a fuzzy e-open set in Y. Then, 1Y \u2212 \u03c1 is fuzzy e-closed. By (2), \u03c6\u22121(1Y \u2212 \u03c1) = 1X \u2212 \u03c6\u22121(\u03c1) is fuzzy e-open. Thus, \u03c6\u22121(\u03c1) is fuzzy e-closed. Converse can be shown easily.(1) \u21d4 (2): let \u03bc be a fuzzy open set. Since fe-int(\u03bc) is fuzzy e-open, then by (1) it follows that \u03c6\u22121(fe-int(\u03bc)) is fuzzy e-closed. The converse is easy to prove.(1) \u21d4 (3): let \u03b7 be a fuzzy closed set. Since fe-cl(\u03b7) is fuzzy e-closed set, then by (2) it follows that \u03c6\u22121(fe-cl(\u03b7)) is fuzzy e-open. The converse is easy to prove.(2) \u21d4 (4): let \u03bc be any fuzzy e-closed set in Y containing \u03c6(x\u03b5). By (2), \u03c6\u22121(\u03bc) is fuzzy e-open set in X and x\u03b5 \u2208 \u03c6\u22121(\u03bc). Take \u03c1 = \u03c6\u22121(\u03bc). Then, \u03c6(\u03c1) \u2264 \u03bc. The converse can be shown easily.(2) \u21d4 (5): let \u03bc be any fuzzy e-open set in Y noncontaining \u03c6(x\u03b5). Then, 1 \u2212 \u03bc is a fuzzy e-closed set containing \u03c6(x\u03b5). By (5), there exists a fuzzy e-open set \u03c1 in X containing x\u03b5 such that \u03c6(\u03c1) \u2264 1 \u2212 \u03bc. Hence, \u03c1 \u2264 \u03c6\u22121(1 \u2212 \u03bc) = 1 \u2212 \u03c6\u22121(\u03bc) and \u03c6\u22121(\u03bc) \u2264 1 \u2212 \u03c1. Take \u03bd = 1 \u2212 \u03c1. We obtain that \u03bd is a fuzzy e-closed set in X noncontaining x\u03b5. The converse can be shown easily.(5) \u21d4 (6): let \u03d5 : X \u2192 Y be a function and let \u03c6 : X \u2192 X \u00d7 Y be the fuzzy graph function of \u03d5, defined by \u03c6(x\u03b5) = ) for every x\u03b5 \u2208 X. If \u03c6 is fuzzy Ce-I, then \u03d5 is fuzzy Ce-I.Let \u03bc be a fuzzy e-closed set in Y; then, 1X \u00d7 \u03bc is a fuzzy e-closed set in X \u00d7 Y. Since \u03c6 is fuzzy Ce-I, then \u03d5\u22121(\u03bc) = \u03c6\u22121(1X \u00d7 \u03bc) is fuzzy e-open in X. Thus, \u03d5 is fuzzy Ce-I.Let Y\u03bb : \u03bb \u2208 \u039b} be a family of product spaces. If a function \u03c6 : X \u2192 \u220fY\u03bb is fuzzy Ce-I, then P\u03bb\u2218\u03c6 : X \u2192 Y\u03bb is fuzzy Ce-I for each \u03bb \u2208 \u039b where P\u03bb is the projection of \u220f\u2061Y\u03bb onto Y\u03bb.Let {\u03b4 be any fuzzy e-open set in Y\u03bb. Since P\u03bb is a fuzzy continuous and fuzzy open set, it is a fuzzy e-open set. Now P\u03bb : \u220fY\u03bb \u2192 Y\u03bb, P\u03bb\u22121(\u03b4) is a fuzzy e-open in \u220fY\u03bb. Therefore, P\u03bb is a fuzzy e-irresolute function. Now (P\u03bb\u2218\u03c6)\u22121(\u03b4) = \u03c6\u22121(P\u03bb\u22121(\u03b4)), since \u03c6 is fuzzy Ce-I. Hence \u03c6\u22121(P\u03bb\u22121(\u03b4)) is a fuzzy e-closed set, since P\u03bb\u22121(\u03b4) is a fuzzy e-open set. Hence, P\u03bb\u2218\u03c6 is fuzzy Ce-I.Let \u03c6 : \u220fX\u03bb \u2192 \u220fY\u03bb is fuzzy Ce-I, then \u03c6\u03bb : X\u03bb \u2192 Y\u03bb is fuzzy Ce-I for each \u03bb \u2208 \u039b.If the function \u03bb0 \u2208 \u039b be an arbitrary fixed index and let \u03c5\u03bb0 be any fuzzy e-open set of Y\u03bb0; then, \u220fY\u03bc \u00d7 \u03c5\u03bb0 is fuzzy e-open in \u220fY\u03bb, where \u03bb0 \u2260 \u03bc \u2208 \u039b. Since \u03c6 is fuzzy Ce-I function, then \u03c6\u22121(\u220fY\u03bc \u00d7 \u03c5\u03bb0) = \u220fX\u03bc \u00d7 \u03c6\u03bb0\u22121(\u03c5\u03bb0) is fuzzy e-closed in \u220fX\u03bb and hence \u03c6\u03bb0\u22121(V\u03bb0) is fuzzy e-closed in X\u03bb0. This implies \u03c6\u03bb0 is fuzzy Ce-I.Let \u03c6 : X \u2192 Y is fuzzy Ce-I and \u03b4 is fuzzy closed set of X, then \u03c6|\u03b4 : \u03b4 \u2192 Y is fuzzy Ce-I.If \u03bb be a fuzzy e-open set of Y; then, (\u03c6|\u03b4)\u22121(\u03bb) = \u03c6\u22121(\u03bb)\u2227\u03b4. Since \u03c6\u22121(\u03bb) and \u03b4 are fuzzy closed, hence (\u03c6|\u03b4)\u22121(\u03bb) is fuzzy e-closed in the relative topology of \u03b4.Let \u03b7 of a fuzzy topological space  containing \u03bc is called the fuzzy e-kernel of \u03bc , fe-K\u03bc = \u22c0{\u03b7 : \u03bc \u2264 \u03b7, \u03b7\u2009\u2009is\u2009\u2009fuzzy\u2009\u2009e-open\u2009\u2009set\u2009\u2009of\u2009\u2009X}.The intersection of all fuzzy e-open set \u03bc, \u03bb of X:(1)x \u2208 fe-K\u03bc iff \u03bc\u2227\u03b3 \u2260 0 for any fuzzy e-closed set \u03b3 containing x.(2)\u03bc \u2264 fe-K\u03bc and \u03bc = fe-K\u03bc if \u03bc is fuzzy e-open in X.(3)\u03bc \u2264 \u03bb; then, fe-K\u03bc \u2264 fe-K\u03bb.The following properties hold for fuzzy sets \u03c6 : X \u2192 Y, the following statements are equivalent:(1)\u03c6 is fuzzy Ce-I.(2)\u03c6(fe-cl(\u03bc)) \u2264 fe-K\u03c6(\u03bc) for every fuzzy set \u03bc of X.(3)fe-cl\u2061(\u03c6\u22121(\u03b7)) \u2264 \u03c6\u22121(fe-K\u03b7) for every fuzzy set \u03b7 of Y.For a fuzzy function \u03bc \u2264 X and y \u2209 fe-K\u03c6(\u03bc). There exists a fuzzy e-closed set \u03b3 in Y, such that y \u2208 \u03b3 and \u03c6(\u03bc)\u2227\u03b3 = 0. Therefore, \u03c6\u22121(\u03c6(\u03bc)\u2227\u03b3) = 0. This implies that \u03bc\u2227\u03c6\u22121(\u03b3) = 0 and fe-cl(\u03bc)\u2227\u03c6\u22121(\u03b3) = 0. Thus, \u03c6(fe-cl\u03bc)\u2227\u03b3 = 0 and y \u2209 \u03c6(fe-cl\u03bc). Hence, \u03c6(fe-cl\u03c6(\u03bc)) \u2264 fe-K\u03c6(\u03bc).(1)\u21d2(2): let \u03b7 \u2264 Y; then, \u03c6\u22121(\u03b7) \u2264 X. By hypothesis, \u03c6(fe-cl\u03c6\u22121(\u03b7)) \u2264 fe-K\u03c6(\u03c6\u22121(\u03b7)) \u2264 fe-K\u03b7. Hence, fe-cl(\u03c6\u22121(\u03b7)) \u2264 \u03c6\u22121(fe-K\u03b7).(2)\u21d2(3): let \u03b7 be any fuzzy e-open set of Y; we have fe-cl(\u03c6\u22121(\u03b7)) \u2264 \u03c6\u22121(fe-K\u03b7) = \u03c6\u22121(\u03b7), since \u03b7 is fuzzy e-open and fe-cl(\u03c6\u22121(\u03b7)) = \u03c6\u22121(\u03b7). This implies that \u03c6\u22121(\u03b7) is fuzzy e-closed in X.(3)\u21d2(1): let \u03b3 of a fuzzy topological space X is given by fe-Fr(\u03b3) = fe-cl(\u03b3)\u2227fe-cl(1X \u2212 \u03b3).The fuzzy e-Frontier of a fuzzy set x\u03b5 \u2208 X such that \u03c6 : X \u2192 Y is not fuzzy Ce-I is exactly the union of fuzzy e-Frontier if the inverse image of the fuzzy e-closed set in Y contains \u03c6(x\u03b5).The fuzzy point \u03c6 is not fuzzy Ce-I at the point x\u03b5 \u2208 X; then there exists a fuzzy e-closed set \u03b3 such that \u03c6(x\u03b5) \u2208 \u03b3 and \u03c6(\u03bc)\u2227(1Y \u2212 \u03b3) \u2260 0 for all fuzzy e-open set \u03bc such that x\u03b5 \u2208 \u03bc. It follows that \u03bc\u2227\u03c6\u22121(1Y \u2212 \u03b3) \u2260 0 and hence x\u03b5 \u2208 fe-cl\u03c6\u22121(1Y \u2212 \u03b3) = fe-cl(1X \u2212 \u03c6\u22121(\u03b3)). Thus, x\u03b5 \u2208 \u03c6\u22121(\u03b3) \u2264 fe-cl(\u03c6\u22121(\u03b3)) and hence x\u03b5 \u2208 fe-Fr(\u03c6\u22121(\u03b3)).Suppose that x\u03b5 \u2208 fe-Fr(\u03c6\u22121(\u03b3)), \u03b3 is fuzzy e-closed set of Y containing \u03c6(x\u03b5), and \u03c6 is fuzzy Ce-I at x\u03b5 \u2208 X. There exists fuzzy e-open set \u03bc such that x\u03b5 \u2208 \u03bc and \u03bc \u2264 \u03c6\u22121(\u03b3). Thus, x\u03b5 \u2208 fe-int\u03c6\u22121(\u03b3) and hence x\u03b5 \u2209 fe-Fr(\u03c6\u22121(\u03b3)) for each fuzzy e-closed set \u03b3 of Y containing \u03c6(x\u03b5), a contradiction. Therefore, \u03c6 is not fuzzy Ce-I.Conversely, suppose that \u03d5 : X \u2192 Y and \u03c6 : Y \u2192 Z:(a)\u03d5 : X \u2192 Y is fuzzy Ce-I and \u03c6 : Y \u2192 Z is fuzzy Ce-continuous then \u03c6\u2218\u03d5 : X \u2192 Z is fuzzy Ce-continuous.If (b)\u03d5 : X \u2192 Y is fuzzy Ce-I and \u03c6 : Y \u2192 Z is fuzzy e-irresolute then \u03c6\u2218\u03d5 : X \u2192 Z is fuzzy Ce-I.If The following hold for functions \u03d5 : X \u2192 Y is a fuzzy e-irresolute surjective function and \u03c6 : Y \u2192 Z is a fuzzy function such that \u03c6\u2218\u03d5 : X \u2192 Z is fuzzy Ce-I, then \u03c6 is fuzzy Ce-I.If \u03b7 be any fuzzy e-closed set in Z. Since \u03c6\u2218\u03d5 is fuzzy Ce-I, (\u03c6\u2218\u03d5)\u22121(\u03b7) is fuzzy e-open in X. Therefore, \u03d5\u22121(\u03c6\u22121(\u03b7)) = (\u03c6\u2218\u03d5)\u22121(\u03b7) is fuzzy e-open in X. Since \u03d5 is fuzzy e-irresolute, surjection implies \u03d5(\u03d5\u22121(\u03c6\u22121(\u03b7))) = \u03c6\u22121(\u03b7) is fuzzy e-open in Y. Thus, \u03c6 is fuzzy Ce-I.Let \u03d5 : X \u2192 Y is a fuzzy e\u2217 \u2212 open surjective function and \u03c6 : Y \u2192 Z is a fuzzy function such that \u03c6\u2218\u03d5 : X \u2192 Z is fuzzy Ce-continuous, then \u03c6 is fuzzy Ce-continuous.If \u03b7 be any fuzzy closed set in Z. Since \u03c6\u2218\u03d5 is fuzzy Ce-continuous, (\u03c6\u2218\u03d5)\u22121(\u03b7) is fuzzy e-open in X. Therefore, \u03d5\u22121(\u03c6\u22121(\u03b7)) = (\u03c6\u2218\u03d5)\u22121(\u03b7) is fuzzy e-open in X. Since \u03d5 is fuzzy e\u2217-open, surjection implies \u03d5(\u03d5\u22121(\u03c6\u22121(\u03b7))) = \u03c6\u22121(\u03b7) is fuzzy e-open in Y. Thus, \u03c6 is fuzzy Ce-continuous.Let Ce-I functions is introduced and the relation between other functions is studied and further some structure preservation properties are investigated.In this section, the notion of fuzzy completely \u03c6 : \u2192 is called fuzzy completely Ce-I if inverse image of every fuzzy e-open set in Y is fuzzy regular closed in X.A mapping X = {x, y, z} and the fuzzy sets \u03bc1, \u03bc2 are defined as follows: \u03c4 = {0,1, \u03bc1, \u03bc2, \u03bc1\u2228\u03bc2, \u03bc1\u2227\u03bc2} and \u03c3 = {0, \u03bc1, \u03bc1\u2227\u03bc2, 1}. Then, the mapping \u03c6 : \u2192 is defined by \u03c6(x) = 1 \u2212 x. Then, \u03c6 is fuzzy completely Ce-I.Let Ce-I function is fuzzy Ce-I and fuzzy Ce-continuous, but the converse is not true, which can be seen in the following example.Every fuzzy completely X = {a, b, c} and Y = {x, y, z} and the fuzzy sets \u03bc1, \u03bc2, and \u03bd are defined as follows: \u03c4 = {0,1, \u03bc1, \u03bc2, \u03bc1\u2228\u03bc2, \u03bc1\u2227\u03bc2} and \u03c3 = {0, \u03bd, 1}. Then, the mapping \u03c6 : \u2192 is defined by \u03c6(a) = x, \u03c6(b) = y, \u03c6(c) = z. Then, \u03c6 is fuzzy Ce-continuous and also fuzzy Ce-I but not fuzzy completely Ce-I as the fuzzy set \u03c5 is fuzzy e-open in Y but \u03c6\u22121(\u03c5) is not fuzzy regular closed set in X.Let From the above examples, we have the following implications.None of the these implications is reversible.\u03c6 : X \u2192 Y, if \u03c6(x\u03b5)q\u03bc, the inverse image of every fuzzy e-closed set of Y is fuzzy \u03b4-open in X iff for any x\u03b5 \u2208 X if \u03c6(x\u03b5)q\u03bc, then x\u03b5qint\u03b4(\u03c6\u22121(\u03bc)).For a fuzzy function \u03bc \u2264 Y be a fuzzy e-closed set and \u03c6(x\u03b5)q\u03bc. Then, x\u03b5q\u03c6\u22121(\u03bc) and, by hypothesis, \u03c6\u22121(\u03bc) = int\u03b4(\u03c6\u22121(\u03bc)). From here, x\u03b5qint\u03b4(\u03c6\u22121(\u03bc)). The converse can be shown easily.Let \u03c6 : X \u2192 Y, if \u03c6(x\u03b5)q\u03bc, for any fuzzy e-closed set \u03bc \u2264 Y and for any x\u03b5 \u2208 X, x\u03b5qint\u03b4(\u03c6\u22121(\u03bc)) iff there exists a fuzzy \u03b4-open set \u03b8 such that x\u03b5q\u03b8 and \u03c6(\u03b8) \u2264 \u03bc.For a fuzzy function \u03bc \u2264 Y be any fuzzy e-closed set and let \u03c6(x\u03b5)q\u03bc. Then, x\u03b5qint\u03b4(\u03c6\u22121(\u03bc)). Take \u03b8 = int\u03b4(\u03c6\u22121(\u03bc)); then, \u03c6(\u03b8) = \u03c6(int\u03b4(\u03c6\u22121(\u03bc))) \u2264 \u03c6(\u03c6\u22121(\u03bc)) \u2264 \u03bc; \u03b8 is fuzzy \u03b4-open in X and x\u03b5q\u03b8.Let \u03bc \u2264 Y be any fuzzy e-closed set and let \u03c6(x\u03b5)q\u03bc. By hypothesis, there exists fuzzy \u03b4-open set \u03b8 such that x\u03b5q\u03b8 and \u03c6(\u03b8) \u2264 \u03bc. This implies \u03b8 \u2264 \u03c6\u22121(\u03bc) and then x\u03b5qint\u03b4(\u03c6\u22121(\u03bc)).Conversely, let \u03c6 : X \u2192 Y, the following statements are equivalent:(1)f is fuzzy completely Ce-I.(2)\u03bc in Y, \u03c6\u22121(\u03bc) is fuzzy regular open in X.For every fuzzy e-closed set (3)\u03bc, \u03c6\u22121(fe-int(\u03bc)) is fuzzy regular closed.For every fuzzy open set (4)\u03b7, \u03c6\u22121(fe-cl\u2061(\u03b7)) is fuzzy regular open.For every fuzzy closed set (5)x\u03b5 \u2208 X and each fuzzy e-closed set \u03bc in Y containing \u03c6(x\u03b5), there exists a fuzzy regular open set \u03c1 in X containing x\u03b5 such that \u03c6(\u03c1) \u2264 \u03bc.For each (6)x\u03b5 \u2208 X and each fuzzy e-open set \u03bc in Y non containing \u03c6(x\u03b5), there exists a fuzzy regular closed set \u03bd in X noncontaining x\u03b5 such that \u03c6\u22121(\u03bc) \u2264 \u03bd.For each For a fuzzy function \u03c1 be a fuzzy e-open set in Y. Then, 1Y \u2212 \u03c1 is fuzzy e-closed. By (2), \u03c6\u22121(1Y \u2212 \u03c1) = 1X \u2212 \u03c6\u22121(\u03c1) is fuzzy regular open. Thus, \u03c6\u22121(\u03c1) is fuzzy regular closed. Thus, \u03c6 is fuzzy completely Ce-I.The converse can be shown easily.(1) \u21d4 (2): let \u03bc be a fuzzy open set. Since fe-int(\u03bc) is fuzzy e-open, then by (1) it follows that \u03c6\u22121(fe-int(\u03bc)) is fuzzy regular closed. The converse is easy to prove.(1) \u21d4 (3): let \u03b7 be a fuzzy closed set. Since fe-cl(\u03b7) is fuzzy e-closed set, then by (2) it follows that \u03c6\u22121(fe-cl(\u03b7)) is fuzzy regular open. The converse is easy to prove.(2) \u21d4 (4): let \u03bc be any fuzzy e-closed set in Y containing \u03c6(x\u03b5). By (2), \u03c6\u22121(\u03bc) is fuzzy regular open set in X and x\u03b5 \u2208 \u03c6\u22121(\u03bc). Take \u03c1 = \u03c6\u22121(\u03bc). Then, \u03c6(\u03c1) \u2264 \u03bc. The converse can be shown easily.(2) \u21d4 (5): let \u03bc be any fuzzy e-open set in Y noncontaining \u03c6(x\u03b5). Then, 1 \u2212 \u03bc is a fuzzy e-closed set containing \u03c6(x\u03b5). By (5), there exists a fuzzy regular open set \u03c1 in X containing x\u03b5 such that \u03c6(\u03c1) \u2264 1 \u2212 \u03bc. Hence, \u03c1 \u2264 \u03c6\u22121(1 \u2212 \u03bc) = 1 \u2212 \u03c6\u22121(\u03bc) and \u03c6\u22121(\u03bc) \u2264 1 \u2212 \u03c1. Take \u03bd = 1 \u2212 \u03c1. We obtain that \u03bd is a fuzzy regular closed set in X noncontaining x\u03b5. The converse can be shown easily.(5) \u21d4 (6): let \u03c61 : X \u2192 Y be a function and let \u03c62 : X \u2192 X \u00d7 Y be the fuzzy graph function of \u03c61, defined by \u03c62(x\u03b5) = ) for every x\u03b5 \u2208 X. If \u03c62 is fuzzy completely Ce-I, then \u03c61 is fuzzy completely Ce-I.Let \u03bc be a fuzzy e-closed set in Y; then, 1X \u00d7 \u03bc is a fuzzy e-closed set in X \u00d7 Y. Since \u03c62 is fuzzy completely Ce-I, then \u03c61\u22121(\u03bc) = \u03c62\u22121(1X \u00d7 \u03bc) is fuzzy regular open in X. Thus, \u03c61 is fuzzy completely Ce-I.Let \u03c61 : X \u2192 Y and \u03c62 : Y \u2192 Z: (a)\u03c61 : X \u2192 Y is fuzzy Ce-I and \u03c62 : Y \u2192 Z is fuzzy completely Ce-I, then \u03c62\u2218\u03c61 : X \u2192 Z is fuzzy e-irresolute.If (b)\u03c61 : X \u2192 Y is fuzzy completely Ce-I and \u03c62 : Y \u2192 Z is fuzzy Ce-continuous, then \u03c62\u2218\u03c61 : X \u2192 Z is fuzzy completely continuous.If The following holds for functions \u03be is said to be fuzzy e-convergent to a fuzzy point x\u03b5 in X if for any fuzzy e-open set \u03b7 in X containing x\u03b5 there exists a fuzzy set \u03c1 \u2208 \u03be such that \u03c1 \u2264 \u03b7.A fuzzy filter base \u03c6 : X \u2192 Y is fuzzy completely Ce-I for each fuzzy point x\u03b5 \u2208 X and each fuzzy filter base \u03be in X is fuzzy rc-convergent to x\u03b5, then the fuzzy filter base \u03c6(\u03be) is fuzzy e-convergent to \u03c6(x\u03b5).If a fuzzy function x\u03b5 \u2208 X and let \u03be be any fuzzy filter base in X which is fuzzy rc-converging to x\u03b5. Since \u03c6 is fuzzy completely Ce-I, then for any fuzzy e-open set \u03b7 in Y containing \u03c6(x\u03b5), there exists a fuzzy regular closed set \u03c1 in X containing x\u03b5 such that \u03c6(\u03c1) \u2264 \u03b7. Since \u03be is fuzzy rc-converging to x\u03b5, there exists a \u03b4 \u2208 \u03be such that \u03b4 \u2264 \u03c1. This means that \u03c6(\u03b4) \u2264 \u03c1 and therefore the fuzzy filter base \u03c6(\u03be) is fuzzy e-convergent to \u03c6(x\u03b5).Let \u03c6 : X \u2192 Y is a fuzzy completely Ce-I surjection and X is fuzzy S-closed, then Y is fuzzy e-compact.If \u03c6 : X \u2192 Y is a fuzzy completely Ce-I surjection and X is fuzzy S-closed. Let {\u03bdi}i\u2208I be a fuzzy e-open cover of Y. Since \u03c6 is a fuzzy completely Ce-I, then {\u03c6\u22121(\u03bdi)}i\u2208I is fuzzy regular closed cover of X and hence there exists finite set I0 of I such that X = \u22c1{\u03c6\u22121(\u03bdi); i \u2208 I0}. Therefore, we have Y = \u22c1{\u03bdi; i \u2208 I0} and Y is fuzzy e-compact.Suppose that \u03c6 : X \u2192 Y is a fuzzy completely Ce-I injection and Y is fuzzy e-T1, then X is fuzzy weakly Hausdorff.If Y is fuzzy e-T1. For any distinct fuzzy points x\u03b5 and y\u03c5 in X, there exist fuzzy e-open sets \u03b7 and \u03c1 in Y. Since \u03c6 is injective, \u03c6(x\u03b5) \u2208 \u03b7, \u03c6(y\u03c5) \u2209 \u03b7, \u03c6(x\u03b5) \u2209 \u03c1, and \u03c6(y\u03c5) \u2208 \u03c1. Since \u03c6 is fuzzy completely Ce-I, \u03c6\u22121(\u03b7) and \u03c6\u22121(\u03c1) are fuzzy regular closed sets of X such that x\u03b5 \u2208 \u03c6\u22121(\u03b7), y\u03c5 \u2209 \u03c6\u22121(\u03b7), x\u03b5 \u2209 \u03c6\u22121(\u03c1), and y\u03c5 \u2208 \u03c6\u22121(\u03c1). This shows that X is fuzzy weakly Hausdorff.Suppose \u03c6 : X \u2192 Y is a fuzzy completely Ce-I injection and Y is fuzzy e-normal, then X is fuzzy strongly normal.If \u03b7 and \u03c1 be disjoint nonempty fuzzy closed sets of X. Since \u03c6 is injective, \u03c6(\u03b7) and \u03c6(\u03c1) are disjoint fuzzy closed sets. Since Y is fuzzy e-normal, there exist fuzzy e-open sets \u03bc and \u03bb such that \u03c6(\u03b7) \u2264 \u03bc and \u03c6(\u03c1) \u2264 \u03bb and \u03bc\u2227\u03bb = 0. This implies that fe-cl(\u03bc) and fe-cl(\u03bb) are fuzzy e-closed sets in Y. Then, since \u03c6 is fuzzy completely Ce-I, \u03c6\u22121(fe-cl(\u03bc)) and \u03c6\u22121(fe-cl(\u03bb)) are fuzzy regular open sets. Then, \u03b7 \u2264 \u03c6\u22121(fe-cl(\u03bc)) and \u03c1 \u2264 \u03c6\u22121(fe-cl(\u03bb)) and \u03c6\u22121(fe-cl(\u03bc)) and \u03c6\u22121(fe-cl(\u03bb)) are disjoint; by definition X is fuzzy strongly normal.Let X, \u03c4) is said to be fuzzy e-T0 or fuzzy e-closed (fuzzy regular closed) sets of Y.A fuzzy topological space (0(r-T0 ) if for \u03c6 : X \u2192 Y is a fuzzy completely Ce-I injection and Y is fuzzy e-T0, then X is fuzzy r-T0.If \u03b7 be a any fuzzy set of X. Since Y is fuzzy e-T0, \u03c6(\u03b7) is fuzzy e-open set of Y. Then, \u03c6(\u03b7) = \u22c1i\u2208I\u22c0j\u2208J\u03bbij, where \u03bbij are fuzzy e-open set or fuzzy e-closed sets of Y. Since \u03c6 is completely Ce-I injection we have \u03b7 = \u03c6\u22121(\u03c6(\u03b7)) = \u03c6\u22121(\u22c1i\u2208I\u22c0j\u2208J\u03bbij) = \u22c1i\u2208I\u22c0j\u2208J\u03c6\u22121(\u03bbij), where \u03c6\u22121(\u03bbij) are fuzzy regular open sets or fuzzy regular closed sets of X. Thus, X is fuzzy r \u2212 T0.Let \u03c6 : X \u2192 Y is a fuzzy completely Ce-I injection and Y is fuzzy e-T2, then X is fuzzy Urysohn.If x\u03b5 and y\u03c5 be any two distinct fuzzy points in X. Since \u03c6 is injective, \u03c6(x\u03b5) \u2260 \u03c6(y\u03c5) in Y. Since Y is fuzzy e-T2, there exist fuzzy e-open sets \u03b7 and \u03c1 in Y such that \u03c6(x\u03b5) \u2208 \u03b7 and \u03c6(y\u03c5) \u2208 \u03c1 and \u03b7\u2227\u03c1 = 0. This implies that fe-cl(\u03b7) and fe-cl(\u03c1) are fuzzy e-closed sets in Y. Then, since \u03c6 is fuzzy completely Ce-I, there exists fuzzy regular open sets \u03b4 and \u03b3 in X containing x\u03b5 and y\u03c5, respectively, such that \u03c6(\u03b4) \u2264 fe-cl(\u03b7) and \u03c6(\u03b3) \u2264 fe-cl(\u03c1). This implies that \u03b4 \u2264 \u03c6\u22121(fe-cl(\u03b7)) and \u03b3 \u2264 \u03c6\u22121(fe-cl(\u03c1)); we have that \u03c6\u22121(fe-cl(\u03b7)) and \u03c6\u22121(fe-cl(\u03c1)) are disjoint and hence cl(\u03b4)\u2227cl(\u03b3) = 0; by definition, X is fuzzy Urysohn.Let"}
{"text": "We also study soft \u03b2-continuous functions and discuss their relations with soft continuous and other weaker forms of soft continuous functions.We introduce the concepts soft  The concept of soft sets was first introduced by Molodtsov  in 1999 In recent years, an increasing number of papers have been written about soft sets theory and its applications in various fields , 4. Shab\u03b2-open sets was introduced in   \u03b2-open sX be an initial universe and let E be a set of parameters. Let P(X) denote the power set of X and let A be a nonempty subset of E. A pair  is called a soft set over X, where F is a mapping given by F : A \u2192 P(X). In other words, a soft set over X is a parameterized family of subsets of the universe X. For \u03b5 \u2208 A, F(\u03b5) may be considered as the set of \u03b5-approximate elements of the soft set .Let F, A) over X is called a null soft set, denoted by \u03a6, if e \u2208 A, F(e) = \u2205.A soft set  over X is called an absolute soft set, denoted by e \u2208 A, F(e) = X.A soft set  over X for which Y(e) = Y, for all e \u2208 E.Let F, A) and  over the common universe X is the soft set , where C = A \u222a B and for all e \u2208 C,The union of two soft sets of  of two soft sets  and  over a common universe X, denoted by C = A\u2229B, and H(e) = F(e)\u2229G(e) for all e \u2208 C.The intersection  and  be two soft sets over a common universe X. A \u2282 B, and H(e) = F(e) \u2282 G(e) for all e \u2208 A.Let  is called a soft topological space over X. Let  be a soft topological space over X; then the members of \u03c4 are said to be soft open sets in X. The relative complement of a soft set  is denoted by c and is defined by c =  where Fc : A \u2192 P(X) is a mapping given by Fc(e) = X \u2212 F(e) for all e \u2208 A. Let  be a soft topological space over X. A soft set  over X is said to be a soft closed set in X, if its relative complement c belongs to \u03c4. If  is a soft topological space with \u03c4 is called the soft indiscrete topology on X and  is said to be a soft indiscrete topological space. If  is a soft topological space with \u03c4 being the collection of all soft sets which can be defined over X, then \u03c4 is called the soft discrete topology on X and  is said to be a soft discrete topological space.The triplet  be a soft topological space over X and let  be a soft set over X.F, A) is the soft set Reference : the sofF, A) is the soft set Reference : the sofLet ) is the smallest soft closed set over X which contains  and int\u2061) is the largest soft open set over X which is contained in .Clearly cl\u2061 of a soft topological space  is said to besoft open  if its c\u03b1-open [soft \u03b1-open  if soft preopen  if soft semiopen  if \u03b2-open [soft \u03b2-open  if A soft set (\u03b1-closed. (b) Every soft \u03b1-open set is soft preopen. (c) Every soft \u03b1-open set is soft semiopen. (d) Every soft semiopen set is soft \u03b2-open. (e) Every soft preclosed set is soft \u03b2-open.(a) Every soft open set is soft The proof is obvious from We have following implications; however, the converses of these implications are not true, in general, as shown in X = {x1, x2, x3}, E = {e1, e2, e3}, and F1, E), , , ... are soft sets over X, defined as follows:\u2009F1, E) = {, , }, = {, , }, = {, }, = {, , }, = {}, = {, }, = {, , }, = {, }, = {, , }, = {, , }. is a soft topological space over X. Clearly the soft closed sets are Then F, E) = {, , }; then F, E) is soft \u03b1-open set but not soft open set  is not soft open set).Then, let us take  = {, , }; then int\u2061) = {, , }, G, E) is soft semiopen set but not soft \u03b1-open set.Now, let us take  = {, }; then cl\u2061) = c, int\u2061)) = {, , }, and so L, E) is soft preopen set but not soft \u03b1-open set.Now, let us take  is soft \u03b2-open set, but it is neither soft semiopen set nor soft preopen set.Then F, A) of a soft topological space  is said to be soft \u03b2-open [\u03b2-open set is called soft \u03b2-closed. Soft \u03b2-closure and soft \u03b2-interior of a soft set are defined as follows.Recall that a soft set X, \u03c4, E) be a soft topological space and let  be a soft set over X.\u03b2-interior of a soft set  in X is denoted by Soft \u03b2-closure of a soft set  in X is denoted by Soft Let ) is the smallest soft \u03b2-closed set over X which contains  and s\u03b2int\u2061) is the largest soft \u03b2-open set over X which is contained in .Clearly \u03b2-open sets  of a soft topological space  by S\u03b2OS ).We will denote the family of all soft \u03b2-open sets is a soft \u03b2-open set. (2) Arbitrary intersection of soft \u03b2-closed sets is a soft \u03b2-closed set.(1) Arbitrary union of soft X, \u03c4, E) be a soft topological space and let  be a soft set over X; thenF, A) \u2208 S\u03b2CS\u21d4 = s\u03b2cl\u2061); \u2208 S\u03b2OS\u21d4 = s\u03b2int\u2061).(Let ((1) Let F, A) is soft \u03b2-closed set.Hence  be soft \u03b2-closed set.Conversely, let  is a soft \u03b2-closed set, Since F, E)'s.Further, (2) Similar to (1).X, \u03c4, E), the following hold for soft \u03b2-closure:s\u03b2cl\u2061(\u03a6) = \u03a6.s\u03b2cl) is soft \u03b2-closed set in  for each soft subset  of X., if In a soft space  be a soft topological space and let  and  be two soft sets over X; thens\u03b2cl\u2061))c = s\u03b2int\u2061c);))c = s\u03b2cl\u2061c);(s\u03b2cl\u2061(\u03a6) = \u03a6 and s\u03b2int\u2061(\u03a6) = \u03a6 and s\u03b2cl\u2061)) = s\u03b2cl\u2061);s\u03b2int\u2061)) = s\u03b2int\u2061).Let  and  be two soft sets over X.F, A) \u2208 S\u03b2CS})cF, A) \u2208 S\u03b2CS}F, A)c \u2208 S\u03b2OS} = s\u03b2int\u2061c).We have Similar to (1).Follows from definition.\u03b2-closed sets so s\u03b2cl\u2061(\u03a6) = \u03a6 and Since \u03a6 and \u03b2-open sets so s\u03b2int\u2061(\u03a6) = \u03a6 and Since \u03a6 and We have Similar to (6).s\u03b2cl\u2061) \u2208 S\u03b2CS, so by s\u03b2cl\u2061)) = s\u03b2cl\u2061).Since s\u03b2int\u2061) \u2208 S\u03b2OS, so by s\u03b2int\u2061)) = s\u03b2int\u2061).Since Let  the following are valid.F, A) is a soft set in X and  is a soft preopen set in X such that F, A) is a soft \u03b2-open set.If  is a soft preopen set we have (a) The proof is obvious. (b) Since  is a soft \u03b2-open set.Then X, \u03c4, E) be soft topological space and let Y be an ordinary subset of X. Then \u03c4Y = /Y :  \u2208 \u03c4) is a soft topology on Y and is called the induced or relative soft topology. The pair  is called a soft subspace of :  is called a soft open/soft closed/soft \u03b2-open soft subspace if the characteristic function of Y, namely, XY, is soft open/soft closed/soft \u03b2-open, respectively.Let  be a soft topological space. Suppose Y, \u03c4Y) is a soft \u03b2-open soft subspace of . Then Z is soft \u03b2-open soft subspace in X if and only if Z is soft \u03b2-open soft subspace in Y.Let  and  be soft classes. Let u : X \u2192 Y and p : E \u2192 K be mappings. Then a mapping f : \u2192 is defined as follows: for a soft set  in , , B),\u2009\u2009B = p(A)\u2286K is a soft set in  given by f(\u03b2) = u(\u222aF(\u03b1)\u03b1\u2208p\u22121(\u03b2)\u2229A) for \u03b2 \u2208 K. , B) is called a soft image of a soft set . If B = K, then we will write , K) as f.Let \u2192 be a mapping from a soft class  to another soft class , and let  be a soft set in soft class , where C\u2286K. Let u : X \u2192 Y and p : E \u2192 K be mappings. Then , D), D = p\u22121(C), is a soft set in the soft classes , defined as follows: f\u22121(\u03b1) = u\u22121(G(p(\u03b1))) for \u03b1 \u2208 D\u2286E.\u2009\u2009, D) is called a soft inverse image of . Hereafter we will write , E) as f\u22121.Let f : \u2192, u : X \u2192 Y, and p : E \u2192 K be mappings. Then for soft sets ,  and a family of soft sets  in the soft class , we have the following:f(\u03a6) = \u03a6,if f\u22121(\u03a6) = \u03a6,if Let X and Y stand for soft topological spaces with  and ) assumed unless otherwise stated. Moreover, throughout this paper, a soft mapping f : X \u2192 Y stands for a mapping, where f : \u2192, u : X \u2192 Y, and p : E \u2192 K are assumed mappings unless otherwise stated.Throughout the paper, the spaces f : X \u2192 Y is called soft \u03b2-continuous [\u03b1-continuous [Y is soft \u03b2-open  set in X.A soft mapping ntinuous   if the We have the following implications; however, the converses of these implications are not true, in general, as shown in X = {x1, x2, x3}, Y = {y1, y2, y3}, E = {e1, e2, e3}, and K = {k1, k2, k3} and let  and  be soft topological spaces.Let u : X \u2192 Y and p : E \u2192 K as \u2009u(x1) = {y1}, u(x2) = {y3}, u(x3) = {y2},\u2009p(e1) = {k2}, p(e2) = {k1}, p(e3) = {k3}.Define \u03c4 on X given in \u2009\u2009F, K) = {, , } and mapping;\u2192 is a soft mapping. Then  is a soft open set in Y;\u2009f\u22121) = {, , } is a soft \u03b1-open set but not soft open set in X.Let us consider the soft topology f is a soft \u03b1-continuous function but not soft continuous function.Therefore, X = {x1, x2, x3}, Y = {y1, y2, y3}, E = {e1, e2, e3}, and K = {k1, k2, k3} and let  and  be soft topological spaces.Let u : X \u2192 Y and p : E \u2192 K as mapping given in \u03c4 on X given in \u2009G, K) = {, } and mapping;\u2009f : \u2192 is a soft mapping. Then  is a soft open set in Y;\u2009f\u22121) = {, } is a soft preopen set but not soft \u03b1-open set in X. Thus, f is a soft precontinuous function but not soft \u03b1-continuous function.Let us consider the X = {x1, x2, x3}, Y = {y1, y2, y3}, E = {e1, e2, e3}, and K = {k1, k2, k3} and let  and  be soft topological spaces.Let u : X \u2192 Y and p : E \u2192 K as mapping given in \u03c4 on X given in \u2009L, K) = {, } and mapping;\u2009f : \u2192 is a soft mapping. Then  is a soft open set in Y;\u2009f\u22121) = {, , } is a soft semiopen set but not soft \u03b1-open set in X. Hence, f is a soft semicontinuous function but not soft \u03b1-continuous function.Let us consider the X = {x1, x2, x3}, Y = {y1, y2, y3}, E = {e1, e2, e3}, and K = {k1, k2, k3} and let  and  be soft topological spaces.Let u : X \u2192 Y and p : E \u2192 K as mapping given in \u03c4 on X given in \u2009\u2009f : \u2192 is a soft mapping. Then  is a soft open set in Y;\u2009\u03b2-open set, but it is neither soft semiopen set nor soft preopen set in X.Let us consider the f is a soft \u03b2-continuous function, but it is neither soft semicontinuous function nor soft precontinuous function.Therefore, f : X \u2192 Y be a function. f is called soft \u03b2-irresolute if the inverse image of soft \u03b2-open set in Y is soft \u03b2-open in X.Let f : X \u2192 Y be a function. f is called soft \u03b2-open if the image of each soft \u03b2-open set in X is soft \u03b2-open in Y.Let f : \u2192 be a soft continuous and soft open set. Then f is soft \u03b2-open set.Let F, A) be any soft \u03b2-open set. Then \u2009\u2009f) is soft \u03b2-open.\u21d2Let  is soft closed and  is soft \u03b2-open then \u03b2-open.If \u2192 be soft continuous and soft open. Then f is soft \u03b2-irresolute.Let F, A) be any soft \u03b2-open set in Y. Then f is soft continuous and soft open it follows that\u2009\u2009f\u22121))))\u2009 = cl\u2061)))). This shows that f\u22121) is soft \u03b2-open. = cl\u2061 of Y, f\u22121) is soft \u03b2-closed.A function X, \u03c4, E) the following are valid:F, A) is soft \u03b2-open \u21d4s\u03b2int\u2061) = . is soft \u03b2-closed \u21d4s\u03b2cl\u2061) = .\u2192 is soft \u03b2-irresolute if and only if for every soft set  of X, f is soft \u03b2-irresolute. Now f)) is soft \u03b2-closed set. By hypothesis f\u22121))) is soft \u03b2-closed set.Suppose that \u03b2-closure, And That is F, A) is soft \u03b2-closed set in Y. Now by hypothesis f\u22121) = s\u03b2cl\u2061)).Conversely, suppose that ) is soft \u03b2-closed set and so f is soft \u03b2-irresolute.That is f : X \u2192 Y is soft \u03b2-irresolute if and only if for all soft sets  of Y, f is soft \u03b2-irresolute. Now s\u03b2cl\u2061) is soft \u03b2-closed set so that f\u22121)) is soft \u03b2-closed set. Since \u03b2-closure that Suppose F, K) is soft \u03b2-closed set in Y. Then s\u03b2cl\u2061) = .Conversely suppose that ))\u2009 = f\u22121).Now by hypothesis s\u03b2cl\u2061)) = f\u22121)) = f\u22121).Therefore, f\u22121) is soft \u03b2-closed set and so f is soft \u03b2-irresolute.Thus, The following results are easy to establish.f : X \u2192 Y and g : Y \u2192 Z are both soft \u03b2-irresolute. Then gof : X \u2192 Z is soft \u03b2-irresolute.Suppose f : X \u2192 Y be soft continuous and soft open. Thenf is soft \u03b2-irresolute;f\u22121)) = s\u03b2cl\u2061))), with  being a soft set in Y.Let X and Y be soft topological spaces. X and Y are said to be M-soft \u03b2-homeomorphic if and only if there exists f : X \u2192 Y such that f is 1-1, onto, M soft \u03b2-continuous and soft \u03b2-open. Such an f is called soft \u03b2-homeomorphism.Let f : X \u2192 Y is soft \u03b2-homeomorphism, then f\u22121)) = s\u03b2cl\u2061)), where  is a soft set in Y.If f : X \u2192 Y is a soft \u03b2-homeomorphism, thens\u03b2cl\u2061)) = f)),f)) = s\u03b2int\u2061)),f\u22121)) = s\u03b2int\u2061)).If \u03b2-interior and soft \u03b2-closure of a soft set in topological spaces and study some of their properties. We also introduce the concept of soft \u03b2-open sets and soft \u03b2-continuous functions in topological spaces and some of their properties have been established. We hope that the findings in this paper are just the beginning of a new structure and not only will form the theoretical basis for further applications of topology on soft sets but also will lead to the development of information system and various fields in engineering.In this paper, we introduce the concept of soft"}
{"text": "In medical science, disease diagnosis is one of the difficult tasks for medical experts who are confronted with challenges in dealing with a lot of uncertain medical information. And different medical experts might express their own thought about the medical knowledge base which slightly differs from other medical experts. Thus, to solve the problems of uncertain data analysis and group decision making in disease diagnoses, we propose a new rough set model called dual hesitant fuzzy multigranulation rough set over two universes by combining the dual hesitant fuzzy set and multigranulation rough set theories. In the framework of our study, both the definition and some basic properties of the proposed model are presented. Finally, we give a general approach which is applied to a decision making problem in disease diagnoses, and the effectiveness of the approach is demonstrated by a numerical example. In real-life disease diagnoses, due to the inherent uncertainty of human's expression of preferences, and the management, storage, and extraction of various useful information available to physicians which is not always presented as crisp numbers, it is believed that fuzzy numbers own many advantages for dealing with medical information systems. Moreover, in order to seek a diagnosis for the considered patients, it is essential for physicians to take into account a number of symptoms at the same time; this process might take a long time to reach a final conclusion. What is worse, the situation of overlooking a few trivial symptoms may trigger wrong disease diagnosis. To solve this complex decision making problem, lots of efforts have been made based on combining the uncertain decision making methods with the traditional disease diagnosis study. Fuzzy set theory , proposex can be expressed as {0.7,0.8}. Ever since the establishment of the hesitant fuzzy set theory, many researchers have studied the HFS from various facets and obtained an increasing number of achievements. In the extensions of hesitant fuzzy set, Zhu et al. introduced the concept of dual hesitant fuzzy set (DHFS) in 2012 , which can be expressed as the following mathematical symbol:hF(x) is a set of some different finite values in , describing the possible membership degrees of the element x \u2208 U to the set F. For convenience, hF(x) is called a hesitant fuzzy element. The set of all hesitant fuzzy elements is called HFEs.Let Zhu et al.  further U be the universe of discourse; then a dual hesitant fuzzy set D on U is defined ash(x) and g(x) are two sets of some different finite values in , describing the possible membership degrees and nonmembership degrees of the element x \u2208 U to the set D, respectively, with the conditions 0 \u2264 \u03b3, \u03b7 \u2264 1, 0 \u2264 \u03b3+ + \u03b7+ \u2264 1, where \u03b3 \u2208 h(x), \u03b7 \u2208 g(x), \u03b3+ \u2208 h+(x) = \u222a\u03b3\u2208h(x)max\u2061{\u03b3}, and \u03b7+ \u2208 g+(x) = \u222a\u03b7\u2208g(x)max\u2061{\u03b7} for all x \u2208 U. For convenience, the pair d(x) = (h(x), g(x)) is called a dual hesitant fuzzy element. The set of all dual hesitant fuzzy elements is called DHFEs.Let U is the universe of discourse; then the set of all dual hesitant fuzzy sets on U is denoted by DHF(U).Suppose that U = {x1, x2} be a universe set; then a dual hesitant fuzzy set defined by D = {\u2329x1, {0.6,0.7}, {0.2,0.3}\u232a, \u2329x2, {0.3,0.4}, {0.5,0.6}\u232a} is a dual hesitant fuzzy set.Let (1)D\u2009\u2009is referred to as an empty dual hesitant fuzzy set , denoting the possible membership degrees and nonmembership degrees for all  \u2208 U \u00d7 V, respectively. With the conditions 0 \u2264 \u03b3, \u03b7 \u2264 1, 0 \u2264 \u03b3+ + \u03b7+ \u2264 1, where \u03b3 \u2208 hR, \u03b7 \u2208 gR, \u03b3+ \u2208 hR+ = max\u03b3\u2208hR{\u03b3}, and \u03b7+ \u2208 gR+ = max\u03b7\u2208gR{\u03b7} for all  \u2208 U \u00d7 V. Moreover, the family of all dual hesitant fuzzy relations over U \u00d7 V is denoted by DHFR(U \u00d7 V).Let U, V be two nonempty and finite universes of discourse and R \u2208 DHFR(U \u00d7 V); the pair  is called a dual hesitant fuzzy approximation space over two universes. For any A \u2208 DHF(V), the lower and upper approximations of A with respect to , denoted by Let A with respect to , respectively. The pair A with respect to  and In what follows, based on the constructive approach to dual hesitant fuzzy rough set over two universes, we extend the dual hesitant fuzzy relation into the background of multigranulation rough set. Both the definitions and some basic properties of optimistic and pessimistic DHF multigranulation rough sets over two universes will be elaborated on.U, V be two nonempty and finite universes of discourse and Ri \u2208 DHFR(U \u00d7 V)\u2009\u2009 are m dual hesitant fuzzy relations over U \u00d7 V; the pair  is called a dual hesitant fuzzy multigranulation approximation space over two universes. For any A \u2208 DHF(V), the optimistic lower and upper approximations of A with respect to  are defined as follows:i=1m\u22c1y\u2208V{hRi\u2227gA(y)}, i=1m\u22c1y\u2208V{hRi\u2227hA(y)}, and i=1m\u22c0y\u2208V{gRi\u2228gA(y)}.Let A with respect to . If A optimistic-definable in ; otherwise, A is optimistic-undefinable in . It is noted that the optimistic DHF multigranulation rough set over two universes will reduce to a DHF rough set over two universes if m = 1.We call the pair U, V be two nonempty and finite universes of discourse and Ri \u2208 DHFR(U \u00d7 V)\u2009\u2009 are m dual hesitant fuzzy relations over U \u00d7 V. For any A, A\u2032 \u2208 DHF(V), the optimistic DHF multigranulation rough set over two universes has the following properties:(1)(2)(3)(4)Let x \u2208 U, we have x, \u22c1i=1m\u22c0y\u2208V{gRi\u2228hA~(y)}, \u22c0i=1m\u22c1y\u2208V{hRi\u2227gA~(y)}\u232a\u2223x \u2208 U} = \u2009{\u2329x, \u22c1i=1m\u22c0y\u2208V{gRi\u2228gA(y)}, \u22c0i=1m\u22c1y\u2208V{hRi\u2227hA(y)}\u232a\u2223x \u2208 U} = \u2009(1) For all A\u2286A\u2032, then by hA\u03c3(k)(y) \u2264 hA\u2032\u03c3(k)(y) and gA\u03c3(k)(y) \u2265 gA\u2032\u03c3(k)(y) for all y \u2208 V. So it follows that {\u2329x, \u22c1i=1m\u22c0y\u2208V{gRi\u03c3(k)\u2228hA\u03c3(k)(y)}, \u22c0i=1m\u22c1y\u2208V{hRi\u03c3(k)\u2227gA\u03c3(k)(y)}\u232a\u2223x \u2208 U}\u2264\u2009{\u2329x, \u22c1i=1m\u22c0y\u2208V{gRi\u03c3(k)\u2228hA\u2032\u03c3(k)(y)}, \u22c0i=1m\u22c1y\u2208V{hRi\u03c3(k)\u2227gA\u2032\u03c3(k)(y)}\u232a\u2223x \u2208 U}. Hence, for each x \u2208 U, we have (2) Since x, \u22c1i=1m\u22c0y\u2208V{gRi\u2228hA\u2229A\u2032(y)}, \u22c0i=1m\u22c1y\u2208V{hRi\u2227gA\u2229A\u2032(y)}\u232a\u2223x \u2208 U} = \u2009{\u2329x, \u22c1i=1m\u22c0y\u2208V{gRi\u2228(hA(y)\u2227hA\u2032(y))}, \u22c0i=1m\u22c1y\u2208V{hRi\u2227(gA(y)\u2227gA\u2032(y))}\u232a\u2223x \u2208 U} = \u2009{\u2329x, \u22c1i=1m\u22c0y\u2208V{\u2228hA(y))\u2227\u2227hA\u2032(y))}, \u22c0i=1m\u22c1y\u2208V{\u2227gA(y))\u2227\u2227gA\u2032(y))}\u232a\u2223x \u2208 U} = \u2009{\u2329x, \u22c1i=1m\u22c0y\u2208V{gRi\u2228hA(y)}, \u22c0i=1m\u22c1y\u2208V{hRi\u2227gA(y)}\u232a\u2223x \u2208 U}\u2227\u2009{\u2329x, \u22c1i=1m\u22c0y\u2208V{gRi\u2228hA\u2032(y)}, \u22c0i=1m\u22c1y\u2208V{hRi\u2227gA\u2032(y)}\u232a\u2223x \u2208 U} = \u2009(3) Consider (4) From the discussions above, it is not difficult to prove that In the above theorem, (1) shows the complement of optimistic DHF multigranulation rough set over two universes; (2) shows the monotone of optimistic DHF multigranulation rough set over two universes with respect to the variety of dual hesitant fuzzy target; (3) and (4) show the multiplication and addition of optimistic DHF multigranulation rough set over two universes.U, V be two nonempty and finite universes of discourse and Ri, Ri\u2032 \u2208 DHFR(U \u00d7 V)\u2009\u2009 are two dual hesitant fuzzy relations over U \u00d7 V. If Ri\u2286Ri\u2032, for any A \u2208 DHF(V), one has the following properties:(1)A \u2208 DHF(V).(2)A \u2208 DHF(V).Let Ri\u2286Ri\u2032, then by Definitions hRi\u03c3(k) \u2264 hRi\u2032\u03c3(k) and gRi\u03c3(k) \u2265 gRi\u2032\u03c3(k) for any  \u2208 (U \u00d7 V). So it follows that {\u2329x, \u22c1i=1m\u22c0y\u2208V{gRi\u03c3(k)\u2228hA\u03c3(k)(y)}, \u22c0i=1m\u22c1y\u2208V{hRi\u03c3(k)\u2227gA\u03c3(k)(y)}\u232a\u2223x \u2208 U}\u2265\u2009{\u2329x, \u22c1i=1m\u22c0y\u2208V{gRi\u2032\u03c3(k)\u2228hA\u03c3(k)(y)}, \u22c0i=1m\u22c1y\u2208V{hRi\u2032\u03c3(k)\u2227gA\u03c3(k)(y)}\u232a\u2223x \u2208 U}. Hence, for each x \u2208 U, we have Since U, V be two nonempty and finite universes of discourse and Ri \u2208 DHFR(U \u00d7 V)\u2009\u2009 are m dual hesitant fuzzy relations over U \u00d7 V; the pair  is called a dual hesitant fuzzy multigranulation approximation space over two universes. For any A \u2208 DHF(V), the pessimistic lower and upper approximations of A with respect to  are defined as follows:i=1m\u22c1y\u2208V{hRi\u2227gA(y)}, i=1m\u22c1y\u2208V{hRi\u2227hA(y)}, and i=1m\u22c0y\u2208V{gRi\u2228gA(y)}.Let A with respect to . If A pessimistic-definable in ; otherwise, A is pessimistic-undefinable in . It is also noted that the pessimistic DHF multigranulation rough set over two universes will reduce to a DHF rough set over two universes if m = 1.We call the pair U, V be two nonempty and finite universes of discourse and Ri \u2208 DHFR(U \u00d7 V)\u2009\u2009 are m dual hesitant fuzzy relations over U \u00d7 V. For any A, A\u2032 \u2208 DHF(V), the pessimistic DHF multigranulation rough set over two universes has the following properties:(1)(2)(3)(4)Let In the above theorem, (1) shows the complement of pessimistic DHF multigranulation rough set over two universes; (2) shows the monotone of pessimistic DHF multigranulation rough set over two universes with respect to the variety of dual hesitant fuzzy target; (3) and (4) show the multiplication and addition of pessimistic DHF multigranulation rough set over two universes.U, V be two nonempty and finite universes of discourse and Ri, Ri\u2032 \u2208 DHFR(U \u00d7 V)\u2009\u2009 are two dual hesitant fuzzy relations over U \u00d7 V. If Ri\u2286Ri\u2032, for any A \u2208 DHF(V), one has the following properties:(1)A \u2208 DHF(V).(2)A \u2208 DHF(V).Let U, V be two nonempty and finite universes of discourse and Ri \u2208 DHFR(U \u00d7 V)\u2009\u2009 are m dual hesitant fuzzy relations over U \u00d7 V. For any A \u2208 DHF(V), the DHF multigranulation rough set over two universes has the following properties:(1)(2)Let x \u2208 U, {\u2329x, \u22c1i=1m\u22c0y\u2208V{gRi\u03c3(k)\u2228hA\u03c3(k)(y)}, \u22c0i=1m\u22c1y\u2208V{hRi\u03c3(k)\u2227gA\u03c3(k)(y)}\u232a\u2223x \u2208 U}\u2265\u2009{\u2329x, \u22c0i=1m\u22c0y\u2208V{gRi\u03c3(k)\u2228hA\u03c3(k)(y)}, \u22c1i=1m\u22c1y\u2208V{hRi\u03c3(k)\u2227gA\u03c3(k)(y)}\u232a\u2223x \u2208 U}. Hence, we have For any From In this section, we introduce a new approach to the decision making problem in medical diagnoses by utilizing the proposed model based on DHF multigranulation rough set over two universes. The main points of our model and decision making methods can be summarized as the following steps.U = {x1, x2,\u2026, xj} is a set of diagnoses and V = {y1, y2,\u2026, yk} is a set of symptoms. Let Ri \u2208 DHFR(U \u00d7 V)\u2009\u2009 be m dual hesitant fuzzy relations over U \u00d7 V, which reflects the medical knowledge base with dual hesitant fuzzy elements data given by m experts. We also let A \u2208 DHF(V) be the set of symptoms characteristic for the considered patients. Then, we obtain a dual hesitant fuzzy decision information system  in medical diagnoses.Suppose that A with respect to , respectively. That is, we obtain the set d1(x) \u2295 d2(x) = \u2009\u222a\u03b3d1(x)\u2208hd1(x),\u03b7d1(x)\u2208gd1(x),\u03b3d2(x)\u2208hd2(x),\u03b7d2(x)\u2208gd2(x){{\u03b3d1(x) + \u03b3d2(x) \u2212 \u03b3d1(x)\u03b3d2(x)}, {\u03b7d1(x)\u03b7d2(x)}}, we further obtain the set of In the following, we present an approach to the decision making for the above-mentioned problem by using DHF multigranulation rough set over two universes. At first, according to Definitions ented in , d1(x) loped in , we presT1, T2, and T3 indicate the decision making index sets which are composed of the subscripts of the largest dual hesitant fuzzy element in corresponding dual hesitant fuzzy sets T1, T2, and T3 could be obtained. Moreover, based on the risk decision making principle of classical operational research, we could present the practical meaning for the above three index sets according to their definitions. Since the optimistic multigranulation rough set is based on \u201cseeking common ground while reserving differences\u201d (SCRD) strategy, which implies that one reserves both common decisions and inconsistent decisions at the same time, thus, this opinion can be seen as a risk-seeking decision making strategy. While the pessimistic multigranulation rough set is based on \u201cseeking common ground while eliminating differences\u201d (SCED) strategy, this strategy indicates that one reserves common decisions while deleting inconsistent decisions. Hence, this opinion can be seen as a risk-averse decision making strategy. According to the above different decision making strategies, xi\u2009(i \u2208 T1) is the optimistic diagnostic result for the considered patient, xi\u2009(i \u2208 T2) is the pessimistic diagnostic result for the considered patient, and xi\u2009(i \u2208 T3) is the weighted diagnostic result for the considered patient, where T3 is the weighted decision making index set of T1 and T2 with the weighted value 0.5. Based on the above definition, the decision rules can be presented as follows:(1)T1\u2229T2\u2229T3 \u2260 \u2205, then xi\u2009(i \u2208 T1\u2229T2\u2229T3) is the determined diagnosis for the patient.If (2)T1\u2229T2\u2229T3 = \u2205 and T1\u2229T2 \u2260 \u2205, then xi\u2009(i \u2208 T1\u2229T2) is the determined diagnosis for the patient. Otherwise, if T1\u2229T2\u2229T3 = \u2205 and T1\u2229T2 = \u2205, then xi\u2009(i \u2208 T3) is the determined diagnosis for the patient.If It is noted that T1, T2, and T3 which come from optimistic and pessimistic information fusion strategies based on medical expert's risk preference, the proposed decision rules could be regarded as a multifaceted diagnostic scheme through considering multiple situations. Moreover, by utilizing the multifaceted diagnostic scheme, medical experts could obtain more reasonable and accurate diagnostic results than other approaches. Hence, the decision rules provide medical experts with a more flexible access to determine the diagnostic results for the patients.In light of the above decision rules in medical diagnoses, by virtue of the decision making index sets In what follows, we present an algorithm for the medical diagnoses model based on DHF multigranulation rough set over two universes as follows.Require. The relation between the universes U and V is provided by an expert  and a set of symptoms characteristic for the considered patients A. Ensure. The determined diagnosis for the patient:(1)calculate (2)calculate (3)determine the score function values for the sets (4)T1, T2, T3, T1\u2229T2\u2229T3, and T1\u2229T2, and confirm the determined diagnosis for the patient.compute In this section, to illustrate the efficiency of the proposed algorithm, we use a medical diagnosis problem with DHFS information which was previously studied and modeled by Farhadinia . FarhadiU = {x1, x2, x3, x4, x5} be a set of diagnoses, where xi stands for viral fever, malaria, typhoid, stomach problem, and chest problem, respectively. A patient with the given values of symptoms is denoted by V = {y1, y2, y3, y4, y5}, where yi stands for temperature, headache, cough, stomach pain, and chest pain, respectively. The medical knowledge base with DHFS data is presented in Tables Let A with all the symptoms, which is represented by the following dual hesitant fuzzy set information:In medical diagnoses, assume that we take a sample from a patient A with respect to , respectively:Following the steps of Then, we further obtain x2 > x4 > x1 > x3 > x5. Therefore, it is not difficult to obtain T1\u2229T2\u2229T3 = {2} \u2260 \u2205, which means x2 is the determined diagnosis for the patient. From the arguments of the above results, we can find that the considered patient is suffering from malaria.In what follows, according to dj\u2009\u2009 be a collection of DHFEs and we let w = T be the weight vector of dj with the equal weight. Then we have the following aggregation operators:The dual hesitant fuzzy averaging (DHFA) operator:The dual hesitant fuzzy geometric (DHFG) operator:In the following, in order to validate the effectiveness of the proposed model based on DHF multigranulation rough set over two universes, a comparison analysis is conducted by utilizing the most commonly used aggregation operators for dual hesitant fuzzy information. As presented in , we let R1, R2, and R3 in Tables R for DHFA operator and DHFG operator, respectively. Then, within the background of dual hesitant fuzzy rough set over two universes introduced in x2 > x4 > x1 > x3 > x5, which is consistent with the ranking results of DHF multigranulation rough sets over two universes. Thus, the diagnostic result also shows the considered patient is suffering from malaria. In light of the above comparison analysis, though the diagnostic outcomes for the two types of information fusion strategies are indistinguishable. It is noted that the information fusion strategies for DHFA and DHFG operators are onefold. By utilizing the optimistic and pessimistic DHF multigranulation rough sets over two universes, the proposed decision rules provide a multifaceted diagnostic scheme for medical experts, which enable them to obtain more reasonable and accurate diagnostic results than DHFA and DHFG operators.Through utilizing the above two aggregation operators, we can aggregate the DHF relation From the above analysis, the DHF multigranulation rough set over two-universe model takes full advantage of dual hesitant fuzzy set and multigranulation rough set in medical diagnoses. On one hand, compared with other generalizations of fuzzy sets, the dual hesitant fuzzy set takes into account much more information given by medical experts. That is, the nonmembership hesitancy function enables medical experts to express his or her opinions from the viewpoint of whether a patient is not suffering from a certain disease, and the hesitant information enables medical experts to hesitate among several numerical numbers when evaluating whether a patient is suffering from a certain disease or not. Thus, the dual hesitant fuzzy set provides medical experts with a more exemplary and flexible access to convey their understandings about the medical knowledge base. On the other hand, the method of multigranulation rough set is an ideal information fusion strategy which could synthesize each medical expert's view to form a final conclusion by providing optimistic and pessimistic information fusion strategies. In light of the above, the superiorities of DHF multigranulation rough set over two-universe model could decline the uncertainty to a great extent and enhance the accuracy and reliability of medical diagnoses effectively.In this paper, we have proposed a new rough set model through combining multigranulation rough set and the dual hesitant fuzzy set, called a DHF multigranulation rough set over two-universe model. In this framework, the definition and some properties of optimistic and pessimistic DHF multigranulation rough sets over two universes have been studied. Finally, we have established a general approach to the decision making problem in medical diagnoses. The outcomes of the example show that the approach proposed in this paper could deal with group decision making problems effectively. Furthermore, comparing to those theoretical results in the existing literature, the main contribution of the proposed decision making model consists in taking into account three decision making index sets based on optimistic and pessimistic information fusion strategies. By virtue of the decision making index sets, the proposed decision making model provides a multifaceted diagnostic scheme for medical experts. And with the aid of multifaceted diagnostic scheme, it is convenient for medical experts to obtain more reasonable and accurate diagnostic outcomes than other methods.This study develops a framework of DHF multigranulation rough set over two universes, in which there are still many interesting issues to be explored. In the future, we can discuss various uncertainty measures and attribute reduction approaches. It is also desirable to further apply our proposed model to other practical applications."}
{"text": "AbstractMelanopsidae. It includes nomenclaturally valid names, as well as junior homonyms, junior objective synonyms, nomina nuda, common incorrect subsequent spellings, and as far as possible discussion on the current status in taxonomy. The catalogue encompasses three family-group names, 79 genus-group names, and 1381 species-group names. All of them are given in their original combination and spelling (except mandatory corrections requested by the Code), along with their original source. For each family- and genus-group name, the original classification and the type genus and type species, respectively, are given. Data provided for species-group taxa are type locality, type horizon (for fossil taxa), and type specimens, as far as available.This nomenclator provides details on all published names in the family-, genus-, and species-group, as well as for a few infrasubspecific names introduced for, or attributed to, the family  Melanopsidae (Caenogastropoda: Cerithioidea) is one of the most diverse groups of non-marine gastropods in Earth history . To facilitate comparison, the general outline of the present work follows the excellent nomenclator of Valvatidae by A comprehensive annotated list of melanopsid names is, however, entirely missing. This catalogue presents information for all published names in the family-, genus- and species-group, as well as for a few infrasubspecific names. Discussed are nomenclaturally valid, invalid  and unavailable names , following the rules of the fourth edition of the International Code of Zoological Nomenclature , incorporating later added amendments , originally introduced as subfamily of the \u201cMelaniidae\u201d .Melanopsidae by Considered a junior synonym of the Taxon classificationAnimaliaSorbeoconchaMelanopsidaeStarobogatov in Starobogatov et al., 1992Melanopsidae.Subfamily of PageBreakFagotia Bourguignat, 1884.Melanopsidae by Considered a junior synonym of the Taxon classificationAnimaliaSorbeoconchaMelanopsidaeH. Adams & A. Adams, 1854Melaniidae.Subfamily of Melanopsis F\u00e9russac in F\u00e9russac & F\u00e9russac, 1807.In the following list, not all genus-group names are accompanied by a type species. Especially Bourguignat, who introduced the greatest number of melanopsid (sub)genera, rarely designated type species. Before 1931, a type species fixation was not a requirement for being available . Original classifications of genus-level taxa are omitted for genera that were introduced without clear family classification. Purely fossil genera are marked by a dagger.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Microcolpia.Subgenus of Melanopsisacicularis. If originally or subsequently designated as type species, Aciculariana would be an objective synonym of Microcolpia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Fagotia.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeP. Fischer, 1885[invalid]Melania.Subgenus of Melaniaholandrii Pfeiffer, 1828, by original designation.Melanella Swainson, 1840, non Bowdich, 1822. Junior objective synonym of Holandriana Bourguignat, 1884, with the same type species.Replacement name for PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeGistel, 1848[invalid]Melanopsis F\u00e9russac in F\u00e9russac & F\u00e9russac, 1807.Unnecessary substitute name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1920[invalid]Melanopsidae.Genus of Buccinumfossile Gmelin, 1791, by typification of replaced name.\u2020 Pannonia Pallary, 1916, wrongly assumed by Pannona L\u00f6renthey, 1902, and in fact a junior homonym of Pannonia Dollfus, 1912.Established as a replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Cossmann, 1909Melanopsis.Section of Melanoptychiaparadoxa Brusina, 1892, by original designation.\u2020 Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGermain, 1934[unavailable]Melanopsidae.Genus of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Stefanescu, 1896Melanopsis.Subgenus of Melanopsisbergeroni Stefanescu, 1896, by monotypy.\u2020 Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sandberger, 1870Melanopsis.Subgenus of Melanopsisgalloprovincialis Math\u00e9ron, 1843, by monotypy.\u2020 Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Swainson, 1840Melanopsis.Subgenus of Melanopsisbouei F\u00e9russac, 1823, by subsequent designation by \u2020 Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1877Melanopsis.Subgenus of Murexcariosus Linnaeus, 1767, by original designation.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGistel, 1848[invalid]Melanopsis F\u00e9russac in F\u00e9russac & F\u00e9russac, 1807.Unnecessary substitute name for Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePageBreakBourguignat, 1884 Melanopsis.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Melanopsiscoupha Bourguignat, 1884, by monotypy.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]Melanella.Subgenus of Crassiana Servain, 1882 .Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Starobogatov & Anistratenko in Anistratenko, 1993Melanopsis.Subgenus of Melanopsiscylindrica Anistratenko, 1993, by original designation.\u2020 Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1877Melanopsis.Subgenus of Melanopsisesperi F\u00e9russac, 1823, by original designation.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]Fagotia.Subgenus of Melanopsisesperi F\u00e9russac, 1823, by subsequent designation by Melanopsisesperi was the type species by monotypy, but 22 nominal species were originally included in Fagotia. Junior objective synonym of Esperiana.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanella.Subgenus of Melanellafagotiana Bourguignat, 1884, by tautonymy.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Cossmann, 1889[invalid]Melanopsispygmaea H\u00f6rnes, 1856, by original designation.\u2020 Homalia Handmann, 1887, which Homala Schumacher\u201d .Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanella.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]Fagotia.Subgenus of Letourneuxiana, one for Fagotia, one for Melanella and one for Melanopsis. All are junior homonyms of Letourneuxiana Silva e Castro, 1883 (Unionidae).Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]Melanella.Subgenus of Letourneuxiana Silva e Castro, 1883 (Unionidae).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]Melanopsis.Subgenus of Melanopsisletourneuxi Bourguignat, 1884, by monotypy.Letourneuxiana Silva e Castro, 1883 (Unionidae).Junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Fagotia.Subgenus of Fagotialocardiana Bourguignat, 1884, by tautonymy.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Melanopsislortetiana Locard, 1883, by tautonymy.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeH. Adams & A. Adams, 1854Melanopsis.Subgenus of Melanopsisdufourii F\u00e9russac, 1822, by subsequent designation by Lyrcaea\u201d as mentioned in \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sandberger, 1872[invalid]Melanopsis.Subgenus of Macrospira Guilding in Swainson, 1840. Stilospirula as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887[invalid]Melanopsis.Subgenus of Melanopsismartiniana F\u00e9russac, 1823 [objective synonym of Melanopsisfossilis ], by subsequent designation by \u2020 Martinia M\u2019Coy in M\u2019Coy & Griffith, 1844 (Brachiopoda).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Kollmann, 1984Melanopsidae.Genus of Purpuroideareussi H\u00f6rnes, 1856, by original designation.\u2020 Taxon classificationAnimaliaSorbeoconchaMelanopsidaeSwainson, 1840Melanopsis.Subgenus of Briefly described but no species included.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeSwainson, 1840[invalid]Melania.Subgenus of Melanella Bowdich, 1822. Fischer (1885: 701) introduced Amphimelania as replacement name, which is invalid, too, because it is a junior objective synonym of Holandriana, with the same type species.No originally included species. Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeSwainson, 1840Melanopsis.Subgenus of Melanopsis species but no type species was designated.Based on some of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeJ.B.L.d\u2019A. de F\u00e9russac in J.B.L. d\u2019A. de F\u00e9russac & A.E.J.P.J.F. d\u2019A. de F\u00e9russac, 1807Melaniacostata Olivier, 1804, by subsequent designation by Correct authority is denoted on p. xii of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr, 1880Melanopsidae.Genus of Melanoptychiabittneri Neumayr, 1880, by subsequent designation by \u2020 Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Andreae, 1893Melanopsidae.Genus of Melanopsisaetolica Neumayr, 1876, by subsequent designation by \u2020 The name is not available from Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Cossmann, 1909[invalid]Melanosteira Andreae, 1893.Unjustified emendation and therefore junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[unavailable]Melanopsis.Subgenus of Unavailable because no type species was designated (Art. 13.3).PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melaniidae .Genus of Melanopsisacicularis F\u00e9russac, 1823, by subsequent designation by Microcalpia\u201d as mentioned in Fischer (1886: 705) is an incorrect subsequent spelling.\u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Melanopsismingrelica Mousson, 1863, by subsequent designation by Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Melanopsismyosotidaea Bourguignat, 1884, by monotypy.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Melanopsisnodosa F\u00e9russac, 1822, by monotypy.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeFinlay, 1926Zemelanopsis.Subgenus of PageBreakMelanopsiswaitaraensis Marwick, 1926, by original designation.Coptochetuszelandicus Marshall, 1918 , in this subgenus. That species is, however, not a Melanopsidae but belongs to the marine genus Exilia Conrad, 1860 after Pakaurangia with the marine deep-water genus Calliotectum (Volutidae).Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916[invalid]Melanopsis.Subgenus of Buccinumfossile Gmelin, 1791, by original designation.\u2020 Martinia Handmann, 1887, non M\u2019Coy in M\u2019Coy & Griffith, 1844. Junior homonym of Pannonia Dollfus, 1912 (see Battistiana).Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Melanopsisparreyssi Philippi, 1847, by monotypy.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Melanopsis.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Microcolpia.Subgenus of Melanopsispotamactebia Bourguignat, 1870, by subsequent designation by PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Microcolpia.Subgenus of Microcolpiapraeclara Bourguignat, 1884, by monotypy.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1877Melanopsis.Subgenus of Buccinumpraemorsum Linnaeus, 1758, by original designation.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeNevill, 1884[invalid]Melanopsis.Subgenus of Melanopsisesperi F\u00e9russac, 1823, by original designation.Esperiana Bourguignat, 1876, with the same type species.Junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Anistratenko, 1993Fagotiinae.Genus of Pseudofagotialineata Anistratenko, 1993, by original designation.\u2020 Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Microcolpia.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Gozhik in Gozhik & Datsenko, 2007Fagotia.Subgenus of Fagotia (Sasykiana) plena Gozhik in Gozhik & Datsenko, 2007, by original designation.\u2020 PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Microcolpia.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanopsis.Subgenus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeIzzatullaev & Starobogatov, 1984Melanopsis.Subgenus of Melanopsissistanica Izzatullaev & Starobogatov, 1984, by original designation.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeTournou\u00ebr, 1882Melanopsinae.Genus of Melanopsisthomasi Tournou\u00ebr, 1877, by original designation.\u2020 PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Melanella.Subgenus of Melanellaspeciosa Bourguignat, 1884, by monotypy.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Cossmann, 1909Melanopsis.Subgenus of Melanopsisaustriaca Handmann, 1882, by original designation.\u2020 Hyphantria Handmann, 1887, non Harris, 1841 (Lepidoptera).Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Rovereto, 1899Melaniidae\u201d.Genus of \u201cMelanopsisproboscidea Deshayes, 1862, by subsequent designation by Stylospirula\u201d).\u2020 Macrospira Sandberger, 1872, non Guilding in Swainson, 1840.Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Cossmann, 1899[invalid]Stilospirula Rovereto, 1899.Unjustified emendation and therefore junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Anistratenko, 1993Fagotiinae.Genus of Turriponticaaciculina Anistratenko, 1993, by original designation.\u2020 PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Microcolpia.Subgenus of Microcolpiavilleserriana Bourguignat, 1884, by tautonymy.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeFinlay, 1926Melanopsistrifasciata Gray, 1843, by original designation.Melanopsidae. The catalogue includes taxa initially described in this family and subsequently classified in other families and vice versa. It comprises a total of 1381 names, of which 71 are unavailable (mostly nomina nuda), 252 are available but invalid (junior homonyms and junior objective synonyms) and 15 are unresolved (requiring the action of a First Reviser).The number of living melanopsid species-group taxa accepted in taxonomy today ranges between 25 and 50 ; the IUCMelanopsidae but classified therein previously. In the 19th century, many species introduced in the group that nowadays is understood as the family Melanopsidae, especially regarding varieties of Holandrianaholandrii , have been classified within the genus Melania Lamarck, 1799 , only such Melania species-group taxa attributed to the Holandrianaholandrii species-complex, or those that have been claimed to be related to that group, are included herein. Further taxa that are not included are: species introduced within the genera Faunus Montfort, 1810, Hemisinus Swainson, 1840, Dicolpus Philippi, 1887 , Loxotrema Gabb, 1868, and Boggsia Olsson, 1929 that have been occasionally classified in the Melanopsidae  species described by Bulliopsisintegra Conrad, 1862, Bulliopsismarylandica Conrad, 1862, and Bulliopsisquadrata Conrad, 1830) that he later considered melanopsids  which was considered a subfamily of Melanopsidae by Bouchet & Rocroi (2005) but probably does not belong there; species introduced in the genus Coptostylus Sandberger, 1872, which has occasionally been classified within the Melanopsidae, but is currently assigned to the Thiaridae , the Global Names Index  and the Worldwide Mollusc Species Data Base (http://www.bagniliggia.it/WMSD/WMSDhome.htm). However, many names listed in those online repositories are obvious misspellings deriving from automatic digitization procedures that have not been critically reviewed , so we can assume the same for plate 1. However, the exact wording of the original captions of 1822 needs to be seen in the future, in order to finally ascertain the availability of the associated names. A different problem appears for plate 2. While the year of publication of the species illustrated there is undoubtedly 1823, the original publication itself is uncertain \u2013 either the livraison 21 of the \u201cHistoire naturelle\u201d [27 September 1823] or the \u201cMonographie\u201d [exact date unknown]. This matter could unfortunately not be solved.To increase confusion, there is a disagreement between the original captions provided by F\u00e9russac and the ones supplied by Deshayes with livraison 29 in 1839  Antiquua\u201d [sic] (antiqua = old). If it had been introduced as a distinct taxon, the name would have been invalid as an objective junior synonym of Melanopsisfusiformis Sowerby, 1822, which F\u00e9russac listed in synonymy.The variety a\u201d [sic] : 149 posinflata, elongata) in the captions of plate 1 of the \u201cM\u00e9lanopsides fossiles\u201d, ranked below var. \u03b3 [\u201cAntiquua\u201d]. In the monograph (1823) they are clearly marked with \u201cnobis\u201d. If \u201cantiquua\u201d [= fusiformis] was considered as a distinct taxon, these names would be of infrasubspecfic rank, which is not governed by the Code. Both names would have nonetheless become PageBreakPageBreakavailable as species-group names after Art. 45.6.4.1 at least from F\u00e9russac (1822) also introduced two additional names  and Latin descriptions and sometimes marked with \u201cNob.\u201d. In his 1828-work he attributed the Latin terms minor and major to varieties of two different species. Apparently, most of these terms were meant as keywords rather than real names. This becomes obvious also from Melanopsisgibbosula for specimens treated as \u201cvar. b. minor\u201d in A similar problem as for F\u00e9russac\u2019s monograph appears in works by In order to bring stability to the problem, I propose to use only the Latin terms marked by Grateloup with \u201cNob.\u201d as available names.The works by Bourguignat and Pallary between 1853 and 1939 extended the list of melanopsid names enormously Figure. Nonetheminor and major, often several times for different species within the very same work. Pallary apparently considered some of the variety names he introduced as self-explanatory  and left them undescribed  would all become included by the Code in the species group, with the Principle of Homonymy applying throughout. Both malacologists introduced many varieties, such as PageBreakvalid names but rather as descriptive terms to fit existing species into his morphological concept , type and height of the spire (\u201cmodus\u201d) and coloration (\u201ccoloratus\u201d). None of the names used by Starm\u00fchlner are available in nomenclature because they were introduced after 1961 and are of infrasubspecific rank (Art. 10.2). They are not included in this catalogus.PageBreakfrom that person (Art. 50.1.1). In such cases, the notation is given in accordance with Recommendation 51E. Information on type locality, type horizon (for fossils only) and type specimens are indicated as far as available. The exact spelling or phrase (given in quotes) provided for the type locality in the original source is denoted, along with an English translation if required. If the localities have been indicated indirectly , the phrase is given in square brackets. Old locality names have been matched with today\u2019s geographic names as far as possible, mostly using the GeoNames geographical database (http://www.geonames.org/v3/). Places that could not be found on the map or where the matched name is uncertain are marked by a postposed question mark in the translation.The catalogue lists all names in alphabetical order in the original spelling and combination, with the necessary amendments required by the Code. The status of taxa that are invalid, unavailable or unresolved is denoted in square brackets after the taxon name; those without status declaration are available and nomenclaturally valid, irrespective of their taxonomic status. The first description, or alternatively the basis of record for unavailable names, is always indicated. Taxa solely found as fossils are marked by a dagger. Taxon authorities attributed to a person other than one of the authors in the original source are only accepted as such if there is clear evidence that the description derived Note that Information on type horizons follows the most recent age classifications found in the literature e.g., . Milo\u0161evThe following list of species-group names comprises several recurrent nomenclatural issues. In order to save space, to avoid multiple elaborate repetitions of the same rules, and to prevent that the reader needs to consult the Code constantly, I refer in text to two nomenclatural notes which are defined as follows:Note 1: Because of the Principle of Coordination (Art. 46), homonymy in the species-group does not depend on a taxon\u2019s original rank in the species-group. This also PageBreakencompasses variety and forma names published before 1961 (Art. 10.2). Many authors have been unaware that species names can constitute junior homonyms of subspecies, variety or forma names. Only in the case of simultaneously published names the taxon of higher rank takes precedence (Art. 24.1).Note 2: Several names of Melanopsidae first occurred in synonymy lists of other names  [10 minutes north of the station Djisr et-Medj\u00e2mi ], Israel.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Cernikian, Pliocene.\u201cPodvinje (\u010caplja) [\u010caplja trench near Slavonski Brod]; Novska; Farka\u0161i\u0107\u201d, Croatia.Milan et al. (1974: 89) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 3717-1357/1.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916[invalid]Middle Lutetian, Eocene.\u201cAu Nord d\u2019Albas\u201d : 204, FrMelanopsisbrevis Doncieux, 1908, non Sowerby, 1826. Junior homonym of Melanopsisabbreviata Brusina, 1874 .Replacement name for PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Calvert & Neumayr, 1880Late Sarmatian, Khersonian, late Miocene.\u201cRenki\u00f6i\u201d [north of \u0130ntepe], Turkey.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201c\u2018Ain Arouss\u201d , Syria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201cHandmann\u201d mentioned in Fischer [unavailable]Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Nomen nudum, based on an \u201cin schedis\u201d determination by Handmann .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeF\u00e9russac, 1823\u201cLa rivi\u00e8re de Laybach. [...] Les eaux thermales de Weslau, pr\u00e8s de Vienne. [...] Le Danube, \u00e0 Wissegrad et \u00e0 Bude. [...] de l\u2019\u00eele de Wight\u201d , Slovenia.Melanopsissubulata Sowerby, 1822 (regarding the specimen of the Isle of Wight) and Melanopsisdaudebartii [Prevost], 1821 , which F\u00e9russac considered varieties of Melanopsisacicularis and listed in synonymy. Currently considered as a junior synonym or subspecies of Microcolpiadaudebartii [Prevost], 1821, respectively .PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Anistratenko, 1993Duab Beds, middle to late Kimmerian, Pliocene.\u201c\u041e\u043a\u0440. \u0441. \u041c\u043e\u043a\u0432\u0438, \u041e\u0447\u0430\u043c\u0447\u0438\u0440\u0441\u043a\u0438\u0439 \u0440-\u043d\u201d , Georgia.Schmalhausen Institute of Zoology of National Academy of Sciences of Ukraine, Kiev; no number indicated.Turripontica Anistratenko, 1993.Type species of the genus Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fontannes, 1884Early Rupelian, Oligocene.\u201cBarjac, Rom\u00e9jac, Av\u00e9jan, Saint-Jean-de-Maru\u00e9jols, C\u00e9las, Issirac, Gal\u00e8s, pr\u00e8s de Montclus (Gard)\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans la rivi\u00e8re au-dessous de Krapina-Toeplitz, en Croatie\u201d [in the river below Krapinske toplice], Croatia.Note that Bourguignat denoted the authority as \u201cBourguignat, 1879\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020G\u00fcmbel, 1861Chattian, Oligocene.\u201cIn der oberen Leizach, an der Schlierach, im Sulzgraben, bei Pensberg, Rimselrain, im H\u00f6llbache, am hohen Peissenberge\u201d , Germany.G\u00fcmbel attributed the authority to Sandberger, but there is no evidence that the description really derived from that author.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882[invalid]Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.PageBreakMelanopsisacuminata G\u00fcmbel, 1861. Not included in the Fossilium Catalogus of Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]\u201cLe Jourdain, \u00e0 4 kilom\u00e8tres au-dessus de la Mer Morte\u201d , Israel/Jordan.Melanopsisacuminata G\u00fcmbel, 1861.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1901[invalid]Plio-Pleistocene.\u201cDu puits Karoubi\u201d [from the well Karoubi], Algeria.Melanopsisacuminata G\u00fcmbel, 1861. Melanopsispallaryi as replacement name .Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Seninski, 1905[invalid]Duab Beds, middle to late Kimmerian, Pliocene.\u201c\u041c\u043e\u043a\u0432\u0438\u043d\u0441\u043a\u0456\u0435 \u043f\u043b\u0430\u0441\u0442\u044b, \u0440. \u0414\u0443\u0430\u0431\u044a\u201d [Mokvi layers at Duab river], Georgia.Melanopsisacuminata G\u00fcmbel, 1861. Melanopsisseninskii as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans l\u2019aqueduc de la Palafanga, pr\u00e8s Almazora (Espagne), aux environs de Mascara (Alg\u00e9rie)\u201d .PageBreakMelanopsisdufourei [sic] graellsii sensu Rossm\u00e4ssler, 1854 and Melanopsismaroccana sensu Paladilhe, 1875 as well as a part of the material of the variety Melanopsismaroccanasubgraellsiana Bourguignat, 1864. He apparently overlooked that Melanopsismaroccana but had introduced the new variety Melanopsismaroccanazonatosubcostata. Since there was never a holotype or lectotype defined, the name acutespira comprises the syntypes of all three works; it remains nomenclaturally valid as name for the two specimens illustrated by Rossm\u00e4ssler (1854) and Bourguignat introduced this species for formerly misidentified Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1871\u201cB\u00e9lep (\u00eele Art)\u201d , New Caledonia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cDans un bassin entre le Mellah et le pont; vers Dar Mahr\u00e8s; [...] Bahlil (28 kil. au sud de F\u00e8s)\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1912\u201cDans les seguias des oasis du Touat, sp\u00e9cialement dans celles de l\u2019Adrar\u201d , Algeria.adparensis\u201d on p. 16, but \u201cadrarensis\u201d in plate captions. Since Pallary clearly denoted it from the locality Adrar, the name must be \u201cadrarensis\u201d (Art. 32.5.1).Given as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Tournou\u00ebr, 1875Phoka to Elia Formation, Plio-Pleistocene.\u201c[Prope vicum Antimaki et prope civitatem Cos], in loco Hagios-Foukas\u201d , Greece.aegaea\u201d as mentioned in The name \u201cPageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Blanckenhorn, 1901Pleistocene.\u201cIn der Fossilienbank an der Tewfik-Moschee bei Kairo. [...] Wadi Urag, Sanur, Moschasch, Raijade\u201d , Egypt.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr, 1876Gelasian, early Pleistocene.\u201cStamna, nordwestlich von Missolunghi\u201d , Greece.The name became available from Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsisaffinis F\u00e9russac\u201d and introduced Melanopsissubaffinis as replacement name. \u201cMelanopsisaffinis F\u00e9russac\u201d is, however, not an available name (see below) and Melanopsissubaffinis is thus a junior objective synonym of Melanopsisaffinis Handmann, 1882. Melanopsisbouei F\u00e9russac, 1823.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916[invalid]Late Villafranchian, early Pleistocene.PageBreak\u201cD\u2019Italie\u201d , and made the name thereby available. The illustrated specimen is, however, the holotype (by monotypy) of Melanopsisnodosa F\u00e9russac, 1822  brook near Ljubljana], Slovenia.Melaniaholandrilaevigata. It was made available at least by Introduced in synonymy of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pantanelli, 1886Tortonian\u2013early Messinian, late Miocene.\u201cS. Valentino e S. Agata\u201d [San Valentino and Sant\u2019Agata Fossili], Italy.Melanopsisnarzolina d\u2019Archiac in Viquesnel, 1846.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePfeiffer, 1828PageBreak\u201cIn der Muhr\u201d [in the river Mur], Austria or Slovenia.aequata\u201d as mentioned in Note that Bourguignat denoted the authority incorrectly as \u201cBourguignat, 1877\u201d. The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRivi\u00e8re pr\u00e8s Zenica, en Bosnie\u201d [river near Zenica], Bosnia and Herzegovina.Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in Bourguignat (1884: 83)[unavailable]Not indicated.Melanopsislaevigata, perhaps referring to an unused manuscript name. Probably the name was intended as \u201cagoraea\u201d, but the ligature was mixed up during typesetting.Nomen nudum: Taxon classificationAnimaliaSorbeoconchaMelanopsidaeAhuir Galindo, 2014\u201cAround Guefait, at the Northeastern of Morocco\u201d, Morocco.Museo Malacologico di Cupra Marittima, Italy; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Tausch, 1886Late Santonian\u2013early Campanian, late Cretaceous.\u201cCsingerthal bei Ajka\u201d [Csinger valley near Ajka], Hungary.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Wenz, 1919[invalid]Middle Lutetian, Eocene.\u201cAu Nord d\u2019Albas\u201d : 204, FrMelanopsisnodosa Doncieux, 1908, non F\u00e9russac, 1822, for which PageBreakintroduced Melanopsisdoncieuxi as replacement name. Thus, Melanopsisalbasensis is a junior objective synonym of Melanopsisdoncieuxi Pallary, 1916.Introduced as replacement name for the junior homonym Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[invalid]Melanopsisalepi Bourguignat, 1884.Unjustified emendation and therefore junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cEnvirons d\u2019Alep\u201d [surroundings of Aleppo], Syria.Melanopsiscostata .Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1904\u201cOran; [...] Mostaghanem\u201d , but Melania is feminine.Originally the gender was incorrectly given as neutrum \u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fontannes, 1887Early Cernikian, early Pliocene.\u201cCaprenu, val. Amaradii (Jud. Gorjiu)\u201d [C\u0103preni], Romania.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201cGaudry, 1862\u201d mentioned in Neumayr (1869: 372)[unavailable]Pliocene?Not indicated.Melanopsisanceps Gaudry & P. Fischer in Gaudry, 1867.Status unclear: the name was mentioned in Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLa Save \u00e0 Sissek, en Slavonie\u201d [in the Sava river near Sisak in Slavonia], Croatia.Note that Bourguignat denoted the authority as \u201cBourguignat, 1879\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020White, 1883Laramie Group, Cretaceous.\u201cValley of South Platte River, Northeastern Colorado\u201d, United States.Melanopsidae.Probably not a Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[invalid]Melanopsisammonis Tristram, 1865.Unjustified emendation and therefore junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeTristram, 1865\u201cStreams at Heshbon and Ammon, east of Jordan\u201d, Jordan.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in Azpeitia Moros [unavailable]\u201cPedroche (arroyo de), en el partido judicial de Pozoblanco (C\u00f3rdoba)\u201d , Spain.Melanopsisetrusca (see Art. 11.6).Based on a manuscript name from Hidalgo and introduced in synonymy of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Oppenheim, 1890Ronca Beds, Bartonian, Eocene.\u201cLovara di Tressino, Monte Pulli, Mussolon\u201d , Italy.Melanopsisvicentina Oppenheim, 1890.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1920Middle Pannonian, late Miocene.\u201cSulzlacke bei Margarethen n\u00e4chst Oedenburg [...]. Tinnye bei Ofen\u201d : 20 .This name, published in November 1920, is a junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRivi\u00e8res pr\u00e8s Ismidt (Anatolie)\u201d [rivers near \u0130zmit], Turkey.Fagotia [= Esperiana] gallandi Bourguignat, 1884.Note that Bourguignat denoted the authority as \u201cBourguignat, 1880\u201d. PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Gaudry & P. Fischer in Gaudry, 1867Pliocene.\u201cM\u00e9gare\u201d (p. 444), Greece.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans la Save au-dessous d\u2019Agram, dans la rivi\u00e8re de Krapina (Croatie); enfin, dans le lac Sabandja pr\u00e8s d\u2019Ismidt (Anatolie)\u201d .Fagotia [= Esperiana] esperi .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Deshayes, 1825Cuisian, late Ypresian, Eocene.\u201cLes environs de Meaux\u201d [surroundings of Meaux], France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1923\u201cDans l\u2019Ain-Touagha, \u00e0 Fatnassa dans le Nefzaoua\u201d [Ain Touagha (?) at Fatnassa in Nefzaoua], Tunisia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1885Spaniodon Beds, Karaganian, middle Miocene.\u201c\u041b\u043e\u043f\u0443\u0448\u043d\u044b\u201d : 4 , Bosnia and Herzegovina.PageBreakThe type material, with all specimens studied by Neumayr (1880), is lost. Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d : 561, AuMelanopsisfusiformis Handmann, 1882, non Sowerby, 1822. Melanopsishaueri Handmann, 1882.Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Taner, 1997Early Romanian, Pliocene.\u201cS\u00fcd\u00f6stlich vom H\u00fcgel Ba\u015falt\u0131, 2,2 km W Musak\u00f6y, 15 km NE \u00c7anakkale, W-Anatolien\u201d , Turkey.Geological-Paleontological Department, Natural History Museum Vienna, Austria, coll. no. 1996/0053/0001.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neubauer, Harzhauser, Georgopoulou, Mandic & Kroh, 2014Duab Beds, middle to late Kimmerian, Pliocene.\u201c\u041e\u043a\u0440. \u0441. \u041c\u043e\u043a\u0432\u0438, \u041e\u0447\u0430\u043c\u0447\u0438\u0440\u0441\u043a\u0438\u0439 \u0440-\u043d\u201d : 69 , France.antediluviana\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Late Villafranchian, Pleistocene.\u201cD\u2019Italie\u201d \u201d , Spain.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sch\u00fctt in Sch\u00fctt & Ortal, 1993Pleistocene, Mindel glacial epoch.\u201cGalilee, \u2018Ubeidiya [El \u2018Ubeid\u012bya], 3 km SE of the Sea of Galilee\u201d, Israel.Paleontology Collection of the Hebrew University of Jerusalem; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans les Seguia des jardins de Miliana, premier ksar au nord de l\u2019oasis d\u2019Insalah, dans le Sahara\u201d , Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Grateloup, 1838Burdigalian, early Miocene.\u201cDax; Mandillot, \u00e0 Saint-Paul\u201d, France.Melanopsismajor F\u00e9russac, 1823\u201d, which is not an available name.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Burdigalian, early Miocene.PageBreak\u201cDax, St-Paul, Mandillot\u201d .Melanopsissimulata as replacement name for the \u201cjunior homonym\u201d Melanopsisarcuata Brusina, 1878, non Matheron, 1842, which is not available name but an incorrect subsequent spelling of Melanopsisarmata Matheron, 1842.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Matheron, 1842Calcaire de Rognac, Maastrichtian, Cretaceous.\u201cRognac et St. Victoret\u201d, France.arcuata\u201d as mentioned in Melanopsisarcuata Brusina, 1897 a junior homonym and introduced the replacement name Melanopsissimulata (junior objective synonym). Paludomus Swainson, 1840 . After Cosinia Stache, 1880 .The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201c\u2018Ain Arouss\u201d , Syria.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Early Langhian, middle Miocene.\u201cZvezdanski klju\u010d\u201d [village Zvezdan], Serbia.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2503-149/1-3 .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLac Sabandja\u201d [Lake Sapanca], Turkey.Note that Bourguignat denoted the authority as \u201cBourguignat, 1880\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLac Sabandja, pr\u00e8s d\u2019Ismidt (Anatolie)\u201d [Lake Sapanca near \u0130zmit], Turkey.Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidaeIzzatullaev & Starobogatov, 1984\u201c\u0410\u0448\u0445\u0430\u0431\u0430\u0434, \u0433\u043e\u0440\u043d\u044b\u0435 \u0431\u044b\u0441\u0442\u0440\u043e \u0442\u0435\u043a\u0443\u0449\u0438\u0435 \u0440\u0443\u0447\u044c\u0438 - \u0424\u0435\u0440\u044e\u0437\u0430, \u0413\u0443\u043b\u0438 \u0438 \u0434\u0440.\u201d , Turkmenistan.Zoological Institute of Russian Academy of Sciences, St.-Petersburg; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Jodot, 1965[unavailable]Quaternary.\u201cOued Assaka\u201d [Oued Asaca], Morocco.Unavailable for two reasons: First, the original work lacks a verbal description of the taxon which is required for names published after 1930 (Art. 13.1.1). Second, the taxon was introduced after 1961 as \u201cforma\u201d which is deemed to be infrasubspecific after Art. 15.2.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1897Cernikian, Pliocene.\u201cMalino\u201d, Croatia.Milan et al. (1974: 86) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 3000-646.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1876Early Langhian, middle Miocene.Originally given as \u201cSinj\u201d and later specified as \u201c\u017dupi\u0107a potok\u201d in Milan et al. (1974: 86) stated that Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGambetta, 1929\u201cStampalia\u201d [Astypalaia Island], Greece.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Wenz, 1928Lutetian, Eocene.\u201cAu Nord d\u2019Albas\u201d : 204, FrMelanopsisdoncieuxi Wenz, 1919, non Pallary, 1916, which in turn was introduced as replacement name for Melanopsisbrevis Doncieux, 1908, non Sowerby, 1826 .Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020\u0218overth, 1953Romanian, Pliocene\u2013early Pleistocene.\u201cHurezanii-de-Sus - Hurezanii-de-jos\u201d, Romania.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans deux ou trois sources de la plaine de J\u00e9richo (Syrie)\u201d [in two or three sources of the plain of Jericho], Palestine.Melanopsissaulcyi Bourguignat, 1853.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939Not explicitly stated but probably the same as for the species .Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cD\u2019Acharn\u00e9, sur l\u2019Oronte, entre Hama et Kal\u00e2at el Moudik\u201d , Syria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 ?Sowerby in Fitton, 1836Weald Clay, early Cretaceous.\u201cPunfield\u201d, United Kingdom.Goniobasis Lea, 1862 (Pleuroceridae).After Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1912[invalid]\u201cDes bords du Chott Djerid, \u00e0 Tozeur\u201d [banks of the Chott el Dj\u00e9rid at Tozeur], Tunisia.Melanopsisattenuata Sowerby in Fitton, 1836.Junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920[invalid]\u201cT\u00e9touan\u201d, Morocco.Melanopsisattenuata Sowerby in Fitton, 1836.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1874\u201cBourail et N\u00e9k\u00e9t\u00e9\u201d [Bourail and Nak\u00e9ty], New Caledonia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsishaueri Handmann, 1882.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sandberger, 1870Maastrichtian, Cretaceous.\u201cAuzas\u201d, France.Plate 5 of Sandberger\u2019s monograph was issued in 1870, while the description on p. 110 appeared in 1871 .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fuchs, 1873[invalid]Middle Pannonian, late Miocene.\u201cSulzlacke bei Margarethen n\u00e4chst Oedenburg [...]. Tinnye bei Ofen\u201d .Melanopsisavellana Sandberger, 1870. Melanopsisampla as replacement name.Junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeAzpeitia Moros, 1929\u201cDel r\u00edo Mundo, cerca de Ayna (Albacete)\u201d , Spain.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Oppenheim, 1892Late Santonian\u2013early Campanian, late Cretaceous.\u201cCsingerthal\u201d  [near Ajka], Hungary.Melanopsisajkaensis Tausch, 1886.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Cossmann, 1909Transdanubian, Pannonian, late Miocene.\u201cV\u00f6r\u00f6s-Bereny im Hohlweg n\u00e4chst des F\u00fczf\u00f6-major und in Kenese [...] Fony\u00f3d; [...] Szt-Gy\u00f6rgy-hegy in Hegymagyos\u201d : 49 , Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Jekelius, 1944[invalid]Early Pannonian, late Miocene.\u201cTurislav-Tal bei Soceni\u201d [Turislav valley near Soceni], Romania.Melanopsisbanatica Jekelius, 1944 (same work). Melanopsisbanatica (since Melanoptychia is considered as a synonym of Melanopsis).Junior secondary homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeSch\u00fctt, 1988Unclear: Sch\u00fctt referred to two different localities: \u201cWasserfall 1 km oberhalb der Stra\u00dfenbr\u00fccke Jerash - Amman \u00fcber den Zarqa\u201d (p. 216) [waterfall 1 km above the bridge over the Zarq\u0101\u2019 along the road Jerash to Amman] and \u201cFlu\u00df Zerqa bei der alten Br\u00fccke am King Talal-See\u201d (p. 219) [Zarq\u0101\u2019 river at the old bridge at the King Talal Dam], Jordan.Natural History Museum Vienna, Austria, coll. no. 85.544.Melanopsissaulcyi Bourguignat, 1853.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Middle Pannonian, late Miocene.\u201cK\u00fap\u201d, Hungary.Milan et al. (1974: 87) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2527-173.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1911\u201cTout pr\u00e8s d\u2019Oudjda, \u00e0 4 kilom. S.-E., sourdent les belles sources de Sidi-Yahia qui alimentent une v\u00e9ritable oasis, puis la ville d\u2019Oudjda, et vont finalement se d\u00e9verser dans l\u2019oued Isly\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Bandel, 2000Late Pannonian, late Miocene.\u201cSand pit near Papkesi [Papkeszi]\u201d, Hungary.Geological-Palaeontological Institute and Museum University of Hamburg, coll. no. 4268.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Capellini, 1873Messinian, late Miocene.\u201cSterza di Laiatico\u201d, Italy.bartolini\u201d as mentioned in Ligios et al. (2012: 358) is an incorrect subsequent spelling.Illustration not on pl. 7 as indicated by Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1933\u201cL\u2019Oued Bou Regreg, au pont des Seouls\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Villatte, 1952Campanian, Cretaceous.\u201cB\u00e9lesta (Ari\u00e8ge)\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRuisseau d\u2019eau chaude \u00e0 Ouargla (prov. de Constantine) et eaux thermales du Dj\u00e9rid, au nord du chott Tiraoun (sud de la Tunisie)\u201d .belonidoea\u201d on p. 110 but \u201cbelonidaea\u201d on p. 75. Apparently, the spelling on p. 110 is based on a typesetting mistake regarding the ligature (\u201c\u0153\u201d instead of \u201c\u00e6\u201d). Letourneux & Bourguignat (1887) acted as First Reviser sensu Art. 24.2.2, giving the name as \u201cbelonidaea\u201d. The spellings \u201cbalonidaea\u201d mentioned in belonidae\u201d given in Multiple spellings occur in the original work: \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDu B\u00e9lus, pr\u00e8s de Saint-Jean-d\u2019Acre (Syrie)\u201d , Israel.PageBreakMelanopsiscostata . Melanopsislampra Bourguignat, 1884.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGermain, 1921[invalid]Melanopsisbelusi Bourguignat, 1884.Unjustified emendation and therefore junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Stefanescu, 1896Sienisian to Pelendavian, Pliocene.\u201c\u00c0 Gura-Motrului et \u00e0 Bocovatz, dans la vall\u00e9e de Jiu\u201d , Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1911\u201cPr\u00e8s du village de Berkane, \u00e0 la lisi\u00e8re Sud-Ouest de la plaine des Triffas, tout au Nord des Beni-Znassen, [...] source connue sous le nom berb\u00e8re d\u2019Ao\u00fbllout\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLe Danube \u00e0 Ibraila; la Save entre Agram et Sissek\u201d .Note that Bourguignat denoted the authority as \u201cBourguignat, 1880\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLe Danube pr\u00e8s Ibraila\u201d [Danube river at Br\u0103ila], Romania.Note that Bourguignat denoted the authority as \u201cBourguignat, 1879\u201d.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]\u201cDe B\u00e9ni Mellal; de l\u2019oued Da\u00ef\u201d , Morocco.Melanopsistextilisbicarinata Handmann, 1882 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Schr\u00e9ter, 1975[invalid]Riss/W\u00fcrm end to early W\u00fcrm Ice Age, Pleistocene.\u201cEger, az egri v\u00e1r Z\u00e1rk\u00e1ndy b\u00e1sty\u00e1j\u00e1nak \u00e1tmetsz\u00e9se\u201d , Hungary.Magyar \u00c1llami F\u00f6ldtani Int\u00e9zet , Budapest; no number indicated.Melanopsistextilisbicarinata Handmann, 1882 (see Note 1). Melanopsisdoboi to the genus Microcolpia.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Blanckenhorn, 1897Plio-Pleistocene.\u201cIn der zweiten Thonbank des linken Orontesufers und der ersten und zweiten des rechten Ufers\u201d [in the second clay bank at the left riverside of the Orontes and the first and second bank of the right riverside], Syria?Introduced as \u201cn. mut.\u201d but clearly as a binomen and hence not infrasubspecific in the sense of ICZN Art. 45.6.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903Late Pleistocene\u2013early Holocene.PageBreak\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai .Taxon classificationAnimaliaSorbeoconchaMelanopsidaePaetel, 1888\u201cW. of Shiraz\u201d : 209, IrOriginally introduced as infrasubspecific taxon (\u201csubvariety\u201d) by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1884Early Langhian, middle Miocene.\u201cStuparu\u0161a\u201d (p. 47) [near Sinj], Croatia.Milan et al. (1974: 87) stated that Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903[unresolved]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai   : 125.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903[unresolved]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai   : 125.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903[unresolved]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai   : 125.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Brusina (1903: 114)[unavailable]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai .Nomen nudum (Brusina apparently considered the term self-explanatory). Taxon classificationAnimaliaSorbeoconchaMelanopsidaeClessin, 1890\u201cIn der warmen Quelle bei Robogany im Bihargebirge in Ungarn\u201d [in the thermal spring near R\u0103b\u0103gani in the Bihar Mts], Romania.Based on an \u201cin schedis\u201d name from Hazay.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeMoricand, 1841\u201cRio de Pedra Branca, procince de Bahia\u201d , Brazil.Verena by Adams and Adams (1854) (Thiaridae), of which Melanopsiscrenocarinata is the type species  [Livanates], Greece.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr, 1880[invalid]Langhian, middle Miocene.\u201c\u017depj\u201d [D\u017eepi], Bosnia and Herzegovina.Melanoptychia as a junior synonym of Melanopsis and this species as a junior secondary homonym of Melanopsisbittneri Fuchs, 1877. They introduced Melanopsismedinae as replacement name .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902[invalid]Langhian, middle Miocene.\u201cD\u017eepe\u201d [D\u017eepi], Bosnia and Herzegovina.Melanopsisbittneri Fuchs, 1877. Melanopsiscvijici Brusina, 1902.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1892Middle Pannonian, late Miocene.\u201cMarku\u0161evec\u201d, Croatia.Milan et al. (1974: 100) indicated a holotype, but it is uncertain whether the specimen actually derives from the original type series and whether it was the only specimen Brusina had at hand. The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2539-185.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Wenz, 1929Pleistocene.\u201cDans le lac d\u2019Antioche, dans l\u2019oued Baradah pr\u00e8s A\u00efn Fidji, et dans l\u2019A\u00efn Pla\u00e7a, fontaine de la plaine du Bahr-el-Houl\u00e9\u201d : 82 Plio-Pleistocene.\u201cH\u00f6here Lage am westlichen Orontes-Ufer 12 km S \u01e6isr a\u0161-\u0160ugur\u201d , Syria.Senckenberg Forschungsinstitut und Naturmuseum Frankfurt, coll. no. SMF 307206.Melanopsisblanckenhorni Wenz, 1929. Melanopsiscostata .Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Schenk, 1969Late Santonian to early Campanian, late Cretaceous.\u201cZ\u00f6ttbachalm bei Brandenberg, Tirol\u201d , Austria.Purpuroidea Lycett, 1848 (Purpuroideidae), but considered as a junior synonym of the melanopsid Megalonodaspiniger  by The species was originally attributed to the marine genus Taxon classificationAnimaliaSorbeoconchaMelanopsidaePaladilhe, 1874\u201cEnvirons d\u2019Oran\u201d [surroundings of Oran], Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Porumbaru, 1881Early Romanian, Pliocene.\u201cBucovatzu\u201d [Bucov\u0103\u021b], Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Tausch, 1886PageBreakLate Santonian\u2013early Campanian, late Cretaceous.\u201cCsingerthal bei Ajka\u201d [Csinger valley near Ajka], Hungary.Melanopsisajkaensis Tausch, 1886.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Klika, 1891Oligocene.\u201cW\u00e4rzen; [...] Tucho\u0159ic\u201d , Czech Republic.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Oppenheim, 1891[invalid]Plio-Pleistocene.\u201cPreveza in Epirus\u201d, Greece.Melanopsisboettgeri Klika, 1891. Melanopsisconemenosianavar.turritella as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902[invalid]Transdanubian, Pannonian, late Miocene.\u201cRadmanest\u201d [R\u0103dm\u0103ne\u0219ti], Romania.The syntypes are stored in the Croatian Natural History Museum, Zagreb; no number indicated .Melanopsisboettgeri Klika, 1891. Melanopsisdelicata as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Halav\u00e1ts, 1903[invalid]Transdanubian, Pannonian, late Miocene.\u201cV\u00f6r\u00f6s-Bereny im Hohlweg n\u00e4chst des F\u00fczf\u00f6-major und in Kenese [...] Fony\u00f3d; [...] Szt-Gy\u00f6rgy-hegy in Hegymagyos\u201d , Hungary.Melanopsisboettgeri Klika, 1891. Canthidomusbalatonensis as replacement name.Junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cEnvirons de Lorca, en Espagne\u201d [surroundings of Lorca], Spain.bofolliana\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1892Middle Pannonian, late Miocene.\u201cMarku\u0161evec\u201d, Croatia.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2538-184/1-3 .bogdanovi\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cGassies\u201d mentioned in Paetel (1888: 399) and Pallary (1926a: 76)[unavailable]\u201cBond\u00e9\u201d, New Caledonia.Bulimusbondeensis\u201d from the same locality, which Melanopsis species.Nomen nudum, found only in species lists by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Manzoni, 1870Late Miocene to Pliocene.\u201cMte. Gibio nel Modanese ed a St. Agata nel Tortonese; [...] a Sogliano\u201d , Italy.Melanopsisbonellii Sismonda\u201d. The name is not available from Melanopsiscarinata non Sowerby) in the unpublished museum catalogue of Melanopsismartiniana], as well as more elongate, but they have the characteristic marginal bulge and the typical keel below the suture\u201d). Following Art. 12.1, every PageBreakname [...] must be accompanied by a description or a definition of the taxon that it denotes [...], which is not the case for the mentioning of Melanopsisbonellii in In the old literature the name appears frequently as \u201cbonelli\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeServain, 1884\u201cMigliaska\u201d [Miljacka river near Sarajevo], Bosnia and Herzegovina.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020F\u00e9russac, 1823Pannonian, late Miocene.\u201cPr\u00e8s de Bisentz, dans la vall\u00e9e de la Marsch, en Moravie; [...] pr\u00e8s de Scharditz\u201d , Czech Republic.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans la rivi\u00e8re de Krapina et dans celle entre Plaski et Ostaria, en Croatie\u201d [in the Krapina river and that between Pla\u0161ki and O\u0161tarije], Croatia.Fagotia [= Esperiana] esperi .Bourguignat denoted the authority as \u201cLetourneux, 1884\u201d, but there is no evidence that the description really derived from that author. Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRivi\u00e8re d\u2019Ogulin (Croatie); la Save, \u00e0 Sissek (Slavonie)\u201d , Croatia.PageBreakBourguignat denoted the authority as \u201cLetourneux, 1879\u201d, but there is no evidence that the description really derived from that author.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary in Germain, 1921\u201cLe Nahr ez Za\u00efr (Liban)\u201d [locality not found], Lebanon.Melanopsislampra Bourguignat, 1884.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1936\u201cA\u00efn Akseri Ifesfassen \u00e0 Imouzer d\u2019Agadir\u201d [A\u00efn Akseri Ifesfassen at Imouzzer near Agadir], Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr, 1880Pannonian, late Miocene.\u201cPosu\u0161je\u201d, Bosnia and Herzegovina.brachyptychia\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeMoricand, 1838\u201cPr\u00e8s de Villa de Barra\u201d , Brazil.Hemisinus Swainson, 1840 (Thiaridae) (Currently considered to belong in the genus iaridae) : 244.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr in Neumayr & Paul, 1875Cernikian, Pliocene.PageBreaktrench near Slavonski Brod; \u010caplja; Groma\u010dnik; roadside ditch between Groma\u010dnik and Sibinj; Ciglenik], Croatia.\u201cGraben hinter der Kirche von Podwin; [...] Graben zwischen der \u010capla und Podwin; [...] \u010capla; [...] Groma\u010dnik; [...] Strassengraben zwischen Groma\u010dnik und Sibin; [...] Cigelnik\u201d , Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1918\u201cAgoura\u00ef, A\u00efn Mahrouf\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sowerby, 1826Eocene.\u201cUpon the Hampshire coast\u201d, United Kingdom.Coptostylus Sandberger, 1872 (Thiaridae).After Taxon classificationAnimaliaSorbeoconchaMelanopsidaeParreyss in Mousson, 1854[invalid]\u201cDans les eaux de l\u2019ancien L\u00e9onthes\u201d [in the Litani river], Lebanon.Melanopsisbrevis Sowerby, 1826. Melanopsismaroccanamedia as replacement name . The authority is clear from the discussion provided in Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeMorelet, 1857[invalid]\u201c[Ad Sanctam-Mariam de Balade]\u201d [Balade], New Caledonia.PageBreakMelanopsisbrevis Sowerby, 1826. Melanopsismoreleti as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Doncieux, 1908[invalid]Middle Lutetian, Eocene.\u201cAu Nord d\u2019Albas\u201d, France.Melanopsisbrevis Sowerby, 1826. Melanopsisabbreviata as replacement name, which is itself a junior homonym of Melanopsisabbreviata Brusina, 1874 .Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1924[invalid]\u201cNavajas (Castell\u00f3n), Alcudia, Alberrique, Jativa \u201d , Spain.Melanopsisbrevis Sowerby, 1826. Not available from Rossm\u00e4ssler (1854), to whom Pallary referred to, because he did not use the term \u201cbrevis\u201d to denote a species-group taxon but only cited Melanopsisdufourii.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020[sic] Papp & Thenius, 1952Gelasian, early Pleistocene.Not stated in Museum of Palaeontology and Geology of the University of Athens; no number indicated .brevisesta\u201d is probably a lapsus calami for \u201cbrevitesta\u201d (Latin for \u201cshort shell\u201d), as used in the re-description by brevisesta\u201d is the correct name following Art. 32.5.1 and \u201cbrevitesta\u201d is an incorrect subsequent spelling.Although probably unintentionally, Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLac de Tib\u00e9riade\u201d [Sea of Galilee], Israel.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Cossmann, 1888Late Danian, Paleocene.\u201cMons\u201d : 71, BelMelaniabuccinoidea sensu Briart & Cornet, 1873, \u201cnon F\u00e9russac\u201d .Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Munier-Chalmas, 1897[invalid]Late Danian, Paleocene.\u201cMons\u201d : 71, BelMelaniabuccinoidea sensu Briart & Cornet, 1873, \u201cnon F\u00e9russac\u201d . Junior homonym, as well as junior objective synonym of Melanopsisbriarti Cossmann, 1888. Obviously, Munier-Chalmas overlooked that the name had already been introduced for the very same misidentified taxon by Cossmann.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brocchii Michelotti, 1847Michelotti 1847: 189.Late Miocene.\u201cPr\u00e8s de Tortone\u201d [near Tortona], Italy.Ptychomelaniabuccinella Sacco, 1895 (Thiaridae).Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Locard, 1883Mammal zone MN 15, Pliocene.\u201cMontgardon\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr, 1880Plio-Pleistocene.\u201cZwischen Pylle und Antimachia\u201d , Greece.PageBreakMelanopsisbroti\u201d Gassies, 1874, which was actually introduced as Melanopsisbrotiana. The replacement name Melanopsiscosiana Pallary, 1925 is thus invalid as it is a junior objective synonym  (browni\u201d as given by Currently considered to belong in the genus iaridae) : 256. ThTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020L\u00f6renthey, 1902Middle Pannonian, late Miocene.\u201cTinnye\u201d, Hungary.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Jekelius, 1944[invalid]Early Pannonian, late Miocene.\u201cTurislav-Tal bei Soceni\u201d [Turislav valley near Soceni], Romania.Melanopsisbrusinai L\u00f6renthey, 1902 , from Syria].Melanopsisbuccinoidea, the species is one of the few stable taxa in melanopsid taxonomy .PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeDrensky, 1947\u201c\u041f\u043e\u0437\u043d\u0430\u0442 \u0437\u0430 \u0441\u0435\u0433\u0430 \u0441\u0430\u043c\u043e \u043e\u0442 \u0440. \u0414\u0443\u043d\u0430\u0432, \u0441\u0435\u0432\u0435\u0440\u043d\u043e \u043e\u0442 \u0433\u0440. \u041b\u043e\u043c\u201d , Bulgaria.Amphimelaniaholandri [sic] .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeKobelt, 1879Not indicated, probably in the Middle East.Melaniacostata by Introduced in synonymy of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Magrograssi, 1928Plio-Pleistocene.\u201cRodi: colline sulla sinistra del fiume Dimilia\u201d [Rhodes island: hills on the left bank of the river Dimilia], Greece.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans l\u2019int\u00e9rieur de la grotte du Nahr-el-Kelb, pr\u00e8s de Beyrouth (Syrie)\u201d [in the cave of Nahr el-Kalb near Beirut], Lebanon.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cSadjour-Sou entre Ain-Ta\u00efb et Alep\u201d [at Sadjour-Sou between Gaziantep (Turkey) and Aleppo (Syria)].Melanopsiscostata .PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Voltz, 1852Mammal zone MN 1, early Miocene.\u201cWeisenau\u201d, Germany.Melanopsisfritzei Thom\u00e4, 1845.The name appeared first as nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1876Early Langhian, middle Miocene.Originally given as \u201cSinj\u201d and later specified as \u201c\u017dupi\u0107a potok\u201d in Milan et al. (1974: 87) stated that Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLe Danube \u00e0 Ibra\u00efla\u201d [Danube river at Br\u0103ila], Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1920Messinian, late Miocene.\u201cColognole e al casino Cubbe; [...] presso il casino Sant\u2019Andrea sotto Colognole\u201d : 397 , Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsisfossilis .The species epithet is a noun in apposition and needs not to agree in gender with the generic name (Art. 31.2.1). Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Stefanescu, 1897Pliocene.\u201cAu ravin de Tura, \u00e0 Capatzineni; [...] aussi \u00e0 Salatrucu-Mare\u201d , Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Jekelius, 1944Early Pannonian, late Miocene.\u201cTurislav-Tal bei Soceni\u201d [Turislav valley near Soceni], Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pavlovi\u0107, 1927Pannonian, zone D\u2013E, late Miocene.PageBreak\u201c\u0418\u0437 \u0420\u0430\u043c\u0430\u045b\u0435\u201d [from the village Rama\u010da (near Ripanj)], Serbia.The illustrated syntype is stored in the Natural History Museum, Belgrade, coll. no. 796 (Milo\u0161evi\u0107 1962: 23).Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sowerby, 1826Eocene.\u201cIn a well near Newport, Isle of Wight; [...] from Hampstead Cliff to Cowes, and [...] on the opposite Cliffs of Hampshire\u201d, United Kingdom.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1861[invalid]\u201cDans le Diahot, \u00e0 Balade; \u00e0 Jengen, \u00e0 Kanala, dans les marais et les petits ruisseaux\u201d , New Caledonia.Melanopsiscarinata Sowerby, 1826. Melanopsisducosi as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeIssel, 1865[invalid]\u201cNel lago di Paleaston presso Poti\u201d [Paliastomi Lake near Poti], Georgia.Melanopsiscarinata Sowerby, 1826.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGentiluomo, 1868[invalid]\u201cLago dell\u2019Accesa, Toscana\u201d, Italy.Melanopsiscarinata Sowerby, 1826.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903[invalid]PageBreakLate Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Melanopsiscarinata Sowerby, 1826. Microcolpiaparreyssiisikorai .Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Kormos, 1903[invalid]Late Pleistocene\u2013early Holocene.\u201cP\u00fcsp\u00f6kf\u00fcrd\u0151\u201d , Romania.Melanopsismucronifera\u201d as name for the taxon. Both names were simultaneously published and are junior objective synonyms. Since carinata sensu Kormos is a junior homonym of Melanopsiscarinata Sowerby, 1826, Melanopsismucronifera is the valid name of the taxon.Obviously unaware of the fact that variety names are available in nomenclature as species-group names, Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903[invalid]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Melanopsisstaubi: Melanopsiscarinata Sowerby, 1826. Microcolpiaparreyssiisikorai .Junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBiggs, 1937[invalid]\u201cFrom Jelalabad\u201d [Jalalabad], Iran.Melanopsiscarinata Sowerby, 1826.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sacco, 1889Tortonian\u2013early Messinian, late Miocene.PageBreak\u201cDelle colline tortonesi presso S. Agata\u201d [from the Torino hills near Sant\u2019Agata Fossili], Italy.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Oppenheim, 1891Gelasian, early Pleistocene.\u201cStamna\u201d, Greece.carinato-costata\u201d.Not available from Taxon classificationAnimaliaSorbeoconchaMelanopsidaePaetel, 1888\u201cKerman\u201d : 209, IrOriginally introduced as infrasubspecific taxon (\u201csubvariety\u201d) by Taxon classificationAnimaliaSorbeoconchaMelanopsidaeLinnaeus, 1767\u201cIn Aqaeductu ad Sevillam\u201d [in an aqueduct near Sevilla], Spain.Melanopsiscariosa, the species is one of the few stable taxa in melanopsid taxonomy .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in F\u00e9russac (1814: 54)[unavailable]Not indicated.Melaniabuccinoidea var. \u03b1 [= Melaniabuccinoidea ].Nomen nudum. Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRivi\u00e8re \u00e0 Ostaria et \u00e0 Ogulin (Croatie)\u201d [river at O\u0161tarije and at Ogulin], Croatia.Note that Bourguignat denoted the authority as \u201cBourguignat, 1879\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Noulet, 1854Noulet 1854: 50.Bartonian, Eocene.\u201c\u00c0 Labrugui\u00e8re; [...] \u00e0 Augmontel (Tarn)\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDe la Savina\u201d [from the Savinja river], Slovenia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928\u201cOuakda, \u00e0 4 kil. N.-E. de Colomb-B\u00e9char\u201d , Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Guppy, 1866PageBreakLate Miocene.\u201cCumana\u201d, Venezuela.capula\u201d on p. 580, but as \u201ccepula\u201d in the plate captions. cepula\u201d . Crepitacella (Rissoinidae) with Melanopsiscepula as type species.Given as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Watelet, 1851Cuisian, late Ypresian, Eocene.\u201cMercin\u201d [Mercin-et-Vaux], France.Faunus Montfort, 1810 (Pachychilidae). Semisinus [= Hemisinus] resectus .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cPlaine du Bahr-el-Houl\u00e9 (haut Jourdain) dans l\u2019A\u00efn-el-Mellaha\u201d , Israel.cerithiopis\u201d in heading of description, but as \u201ccerithiopsis\u201d throughout the rest of the work. Melanopsissaulcyi Bourguignat, 1853.Given as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cM. Mario: Farnesina\u201d : 186, ItMelanopsistransiens Cerulli-Irelli, 1914, non Blanckenhorn, 1897. Melanopsistransiens Cerulli-Irelli, 1914 a junior synonym of \u201cMelanopsisaffinis F\u00e9russac\u201d, which is not an available name.Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cBeni Abb\u00e8s\u201d [prov. B\u00e9char], Algeria.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1897Cernikian, Pliocene.\u201cKozarica\u201d [Kozarice], Croatia.Milan et al. (1974: 88) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2991-637.Melanopsisclavigera Neumayr in Neumayr & Paul, 1875.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeLocard, 1883\u201cLac d\u2019Antioche\u201d [Lake Anuk ], Turkey.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBrot, 1879\u201cSchiraz\u201d, Iran.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cDans le source de Yeni Chehir, [...] entre Antioche et Alep, \u00e0 l\u2019intersection de la route d\u2019Alexandrette\u201d , Turkey.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939Mesopotamia, beteween Ar Raqqah in the south and Tall Abya\u1e11 in the north, on the left bank of the Qarah M\u016bkh, tributary of the right side of the Nahr al Bal\u012bkh], Syria.\u201cCheragrag, en M\u00e9sopotamie, entre Rakka, au Sud et Tell Abiad au Nord, sur la rive gauche du Karamouk Sou, affluent de la rive droite du Nahr B\u00e2hlik\u201d , southern Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeNicolas, 1898\u201cAux environs de Biskra, \u00e0 Ain-Oumach [...] dans une source d\u2019eau chaude\u201d , Algeria.Melanopsissaharica Bourguignat, 1864.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Aldrich, 1886Early Eocene.\u201cHatchetigbee, Butler, Choctaw County, Alabama\u201d, United States.Bulliopsis Conrad, 1862 (Nassariidae) by Classified within the marine genus Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Early Cernikian, early Pliocene.\u201c\u010cerevi\u0107\u201d, Serbia.Milan et al. (1974: 88) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2504-150.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020PageBreakNeumayr, 1880 Plio-Pleistocene.\u201cZwischen Pylle und Antimachia\u201d , Greece.Melanopsisdelessei Tournou\u00ebr, 1875.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Merian, 1849Burdigalian, early Miocene.\u201cNahe am Gipfel des Plateaus des Randens\u201d [near the top of the plateau of Mt. Randen], Switzerland.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sandberger, 1872Badenian, middle Miocene.\u201cGrund [...], V\u00f6slau\u201d : 597, AuMelanopsisaquensis sensu H\u00f6rnes, 1856, non Grateloup, 1838.Plate 25 of Sandberger\u2019s monograph was issued in 1872, while the description on pp. 512, 521 appeared in 1875 . IntroduTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr in Neumayr & Paul, 1875Cernikian, Pliocene.\u201cCigelnik; [...] Graben zwischen Podwin und der \u010capla; [...] \u010capla; [...] An der Strasse von Sibin nach Groma\u010dnik; [...] Groma\u010dnik\u201d , Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Michelin, 1833Cretaceous?\u201c\u00c0 G\u00e9rodot, d\u00e9partement de l\u2019Aube\u201d, France.Melanopsidae.Probably not a PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.A lectotype was designated by Taxon classificationAnimaliaSorbeoconchaMelanopsidaeServain, 1884\u201cLa Migliaska \u00e0 Serajewo (Bosnie); Ostaria (Croatie)\u201d .Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1878Late Portaferrian, late Miocene\u2013early Pliocene.\u201cKarlowitz [...] G\u00f6rgetek in Syrmien\u201d : 49 , Spain.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fontannes, 1880[invalid]Mammal zone MN 11, late Miocene.Potamides Basteroti de Visan (Vaucluse)\u201d , France.\u201cLes marnes \u00e0 PageBreakMelanopsisneumayri Tournou\u00ebr, 1874.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920[invalid]\u201cA\u00eft Taleb sur le Sefrou pr\u00e8s d\u2019el Menzel, avant l\u2019oued Sebou\u201d , Morocco.Melanopsiscompacta Fontannes, 1880.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sacco, 1895Tortonian\u2013early Messinian, late Miocene.\u201cS. Agata fossili\u201d [Sant\u2019Agata Fossili], Italy.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Oppenheim, 1891Late Pliocene\u2013early Pleistocene.\u201cPreveza\u201d, Greece.M. Conemenosi Bttg. in litt.\u201d) in Appeared first as a nomen nudum  [R\u0103dm\u0103ne\u0219ti], Romania.Melanopsisfuchsi Brusina, 1884, non Handmann, 1882 and Melanopsishungarica Pallary, 1916, non Kormos, 1904, which were in turn introduced for Melaniacostata sensu Fuchs, 1870, non Olivier, 1804 , which Junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Anistratenko, 1993Duab Beds, middle to late Kimmerian, Pliocene.\u201c\u041e\u043a\u0440. \u0441. \u041c\u043e\u043a\u0432\u0438, \u041e\u0447\u0430\u043c\u0447\u0438\u0440\u0441\u043a\u0438\u0439 \u0440-\u043d\u201d , Georgia.Schmalhausen Institute of Zoology of National Academy of Sciences of Ukraine, Kiev; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Anistratenko, 1993Duab Beds, middle to late Kimmerian, Pliocene.\u201c\u041e\u043a\u0440. \u0441. \u041c\u043e\u043a\u0432\u0438, \u041e\u0447\u0430\u043c\u0447\u0438\u0440\u0441\u043a\u0438\u0439 \u0440-\u043d\u201d , Georgia.Schmalhausen Institute of Zoology of National Academy of Sciences of Ukraine, Kiev; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sacco, 1886Late Burdigalian, early Miocene.\u201cCollina di Torino\u201d [Torino hills], Italy.conjungens\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLac de Tib\u00e9riade\u201d [Sea of Galilee], Israel.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pan\u0103, 2003Parscovian\u2013Pelendavian, Pliocene.PageBreak\u201cForage Mih\u0103i\u021ba, profondeur 109 m\u201d , Romania.Laboratory of Paleontology, Bucharest, coll. no. 651.conoideus\u201d), but Melanopsis is feminine, which is why the name must be \u201cconoidea\u201d.Pan\u0103 denoted the authorship as \u201cPan\u0103, 1989\u201d. Originally the gender was indicated as masculine .Melanopsisboueimegacantha Handmann, 1887.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916[invalid]Chattian, Oligocene.\u201cDax. St-Geours, Abesse\u201d  stated that Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887[invalid]Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Melanopsisconstricta Brusina, 1878. Melanopsishandmanniana as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn\u201d, Austria.Melanopsisvindobonensis Fuchs, 1870. Note that Wenz gave the name as \u201ccostata\u201d, which is an incorrect subsequent spelling.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903[invalid]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Melanopsishazayi: Microcolpiaparreyssiisikorai .Junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020PageBreakDoncieux, 1908 Lutetian, Eocene.\u201cAu Nord d\u2019Albas\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1916[invalid]\u201cPrope Kanala [...]; insula Ouen\u201d : 148 , Hungary.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeReeve, 1860Melania, pl. 34, fig. 233.\u201cDalmatia\u201d [no locality indicated], Croatia.Based on a manuscript or \u201cin schedis\u201d name from K\u00fcster in the museum of Von dem Busch see .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeReeve, 1860[invalid]Melania, pl. 34, fig. 228.PageBreak\u201cR\u00f6merbad in Steiermark\u201d [Rimske Toplice], Slovenia after .Melaniahollandri [sic] Pfeiffer, 1828 .Based on a manuscript name by K\u00fcster and introduced in synonymy of see also . It was Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1878Cernikian, Pliocene.\u201cRepusnica, Slobodnica\u201d, Croatia.The syntype (?) illustrated in Taxon classificationAnimaliaSorbeoconchaMelanopsidaeAhuir Galindo, 2015[invalid]\u201cIn brooks and \u2018gorgos\u2019 from Anna, Valencia\u201d, Spain.Museo Malacologico di Cupra Marittima, Italy; no number indicated.Melanopsiscoronata Brusina, 1878.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePicard, 1934\u201cJarmukm\u00fcndung\u201d [Yarmouk river mouth], Jordan/Israel.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Stefanescu, 1896Pliocene.\u201c\u00c0 Mosculesti, dans la vall\u00e9e de Gilortu, \u00e0 Gura-Motrului et \u00e0 Breasta, dans la vall\u00e9e de Jiu\u201d , Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020PageBreakSch\u00fctt in Sch\u00fctt & Ortal, 1993 Pleistocene, Riss glacial epoch.\u201cGalilee, southern Hula basin, about 500 m north of the new bridge over the river Jordan at Benot Ya\u2019Aqov\u201d, Israel.Paleontology Collection of the Hebrew University of Jerusalem; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1925[invalid]Plio-Pleistocene.\u201cZwischen Pylle und Antimachia\u201d : 295  Pallary, 1916[invalid]Cernikian, Pliocene.\u201cDe Slavonie\u201d : 176, CrMelaniacostata sensu Cossmann, 1909, non Olivier, 1804. The name was corrected to \u201ccossmanni\u201d by Melaniacostata\u201d from the Pliocene of Slavonia to represent the same species and synonymized them with Melanopsisabbreviatacosmanni. Obviously, he was unaware that Melanopsiscroatica Brusina, 1884 is the first available name for them .Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916cosmanni\u201d).Cernikian, Pliocene.\u201cDe Slavonie\u201d : 176, CrMelanopsiscosmanni Pallary, 1916 by Justified emendation of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePageBreakBourguignat, 1884 \u201cEaux thermales d\u2019Ouargla et pr\u00e8s du chott Tiraoun dans le sud de la province de Constantine et de la Tunisie\u201d [in thermal waters at Ouargla (Algeria) and near chott Tiraoun (Tunisia)], Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeOlivier, 1804\u201cDe Orontes [Gesser-Chourl]\u201d , Syria.Melanopsis F\u00e9russac in F\u00e9russac & F\u00e9russac, 1807.Type species of the genus Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Ludwig, 1865[invalid]Early Rupelian, Oligocene.\u201cGrossalmerode\u201d, Germany.Melaniacostata . Melanopsisludwigi as replacement name.Junior secondary homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Botez, 1914[invalid]Viviparusstricturatus Zone, Cernikian, Pliocene.\u201cMoreni\u201d, Romania.Melanopsiscostata .Junior secondary homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Oluji\u0107, 1999[invalid]Langhian, middle Miocene.It is unclear from the original work in which of the studied localities/sections along the valleys of the Sutina, Batarelov and Vojskava rivers (4 km W of Sinj) the taxon occurred and in which not, Croatia.Melanopsiscostata . Melanopsislanzaeana.Junior secondary homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020PageBreakOluji\u0107, 1999 [invalid] Langhian, middle Miocene.It is unclear from the original work in which of the studied localities/sections along the valleys of the Sutina, Batarelov and Vojskava rivers (4 km W of Sinj) the taxon occurred and in which not, Croatia.Melanopsiscostata . Considered as a junior synonym of Melanopsislyrata Neumayr, 1869 by Junior secondary homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c[sic] Kucik\u201d mentioned in Brusina (1867: 85)[unavailable]\u201cU Savi i u potoku kod Susjeda, u \u017dirovcu, u Maksimiru kod Zagreba, u potoku Toplici kod Oroslavja i u Kutinji blizu Jastrebarskoga\u201d , Croatia.costata\u201d.Nomen nudum, based on an \u201cin schedis\u201d name from the collection of Kucik . The name was probably a typesetting mistake for \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Papp, 1953Gelasian, early Pleistocene.\u201cAghios Georgios\u201d , Greece.Museum of Palaeontology and Geology of the University of Athens; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeF\u00e9russac, 1823[invalid]\u201cDans l\u2019aqueduc de S\u00e9ville [...] et dans les ruisseaux des environs. [...] Dans les lacs et les rivi\u00e8res du royaume de Maroc\u201d .Melanopsiscariosa , which F\u00e9russac listed in synonymy.Junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020PageBreakDouvill\u00e9, 1904 [invalid] Maastrichtian, Cretaceous.\u201cDu versant oriental du Kouh Mapeul\u201d , Iran.Melanopsiscostellata F\u00e9russac, 1823. Melanopsisdouvillei as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Anistratenko, 1993Duab Beds, middle to late Kimmerian, Pliocene.\u201c\u041e\u043a\u0440. \u0441. \u041c\u043e\u043a\u0432\u0438, \u041e\u0447\u0430\u043c\u0447\u0438\u0440\u0441\u043a\u0438\u0439 \u0440-\u043d\u201d , Georgia.Schmalhausen Institute of Zoology of National Academy of Sciences of Ukraine, Kiev; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeSchmidt, 1847\u201cAus einem M\u00fchlbache bei Klinze ob Schischka\u201d [from a mill creek at Glinica near \u0160i\u0161ka], Slovenia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874[invalid]Langhian, middle Miocene.\u201cMio\u010di\u0107\u201d, Croatia.The syntypes are stored in the Croatian Natural History Museum, Zagreb; no number indicated .Melanopsisinconstans Neumayr, 1869 into three varieties, none of which he termed \u201cinconstans\u201d. The first one, \u201cvar. costulata\u201d, he referred to as the typical one, which makes it an objective synonym of the nominal subspecies and hence Melanopsisinconstans.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1897[invalid]Langhian, middle Miocene.\u201cMio\u010di\u0107\u201d, Croatia.PageBreakMilan et al. (1974: 99) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2981-627/2.Melanopsisinconstanscostulata Brusina, 1874 (see Note 1). Currently considered as a junior synonym of Melanopsisvisianiana Brusina, 1874 .Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Doncieux, 1908[invalid]Middle Lutetian, Eocene.\u201cAu Nord d\u2019Albas\u201d : 205, FrMelanopsiscostulata Brusina, 1897. Has been considered to belong to the genus Coptostylus Sandberger, 1872 (Thiaridae).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]\u201cO. Taguenout\u201d , Morocco.Melanopsiscostulata Brusina, 1897.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePageBreakPallary, 1939 [invalid] \u201cRas el \u2018Ain du Khabour\u201d [Chabur river near Ra\u2019s al \u2018Ayn], Syria.Melanopsiscostulata Brusina, 1897.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Cob\u0103lcescu, 1883Cernikian, Pliocene.\u201cCotroceni l\u0103ng\u0103 Bucure\u0219ti\u201d [Cotroceni near Bucarest], Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans les eaux chaudes du Djerid, au nord du chott Tiraoun, dans le sud de la Tunisie\u201d , Tunisia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLac Sabandja, pr\u00e8s d\u2019Ismidt, en Anatolie\u201d [Lake Sapanca near \u0130zmit], Turkey.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Cob\u0103lcescu, 1883Pliocene.\u201cBarboschi\u201d (p. 156) [Barbo\u0219i], Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBrusina, 1866[invalid]\u201cIn der Muhr\u201d : 48 , Croatia.Note that Bourguignat denoted the authority as \u201cBourguignat, 1879\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Blanckenhorn, 1897Plio-Pleistocene.Dreissena layer at Jisr Ash-Shughur], Syria.\u201cIn der plioc\u00e4nen Dreissensiaschicht von Dschisr esch-Schurr\u201d , Brazil.Verena H. Adams & A. Adams, 1854 (Thiaridae) (see Type species of the genus dae) see : 253.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1901Pleistocene.\u201cDe G\u00e9ryville\u201d [A\u00efn Sefra], Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cF\u00e8s\u201d, Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1884Cernikian, Pliocene.\u201cRepu\u0161nica\u201d : 372, CrMilan et al. (1974: 89) indicated a holotype, but it is uncertain whether the specimen actually derives from the original type series and whether it was the only specimen Brusina had at hand. The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2988-634.Melaniacostata sensu Neumayr, 1969, non Olivier, 1804. Melaniacostata\u201d from the Pliocene of Slavonia to represent the same species and synonymized them with Melanopsisabbreviatacosmanni. Obviously, he was unaware that Melanopsiscroatica Brusina, 1884 is the first available name for them .Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902[invalid]Middle Pannonian, late Miocene.\u201cMarku\u0161evec\u201d, Croatia.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2529-175/1-2 .Melanopsiscroatica Brusina, 1884. Melanopsishauerimarkusevecensis as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Dominici & Kowalke, 2014Castigaleu group, late Ypresian, Eocene.PageBreak\u201cMorillo de Lena , [...] CG-A2, sample 53\u201d, Spain.Museo di Storia Naturale, Universit\u00e0 degli Studi di Firenze, coll. no. IGF 4334E.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1870\u201cTuo\u201d [Touho], New Caledonia.Melanopsisfrustulum Morelet, 1857.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeLocard, 1883[invalid]\u201cLac d\u2019Antioche\u201d [Lake Anuk ], Turkey.Melanopsiscurta Gassies, 1870.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]\u201cPlaine du Bahr-el-Houl\u00e9 (haut Jourdain) dans l\u2019A\u00efn-el-Mellaha\u201d , Israel.Melanopsiscurta Gassies, 1870. Melanopsissaulcyi Bourguignat, 1853.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Locard, 1893[invalid]Late Burdigalian\u2013Langhian, early\u2013middle Miocene.Vermes\u201d, France.\u201cLe Locle; [...] Melanopsiscurta Gassies, 1870. Melanopsiskleini Kurr, 1856.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cPageBreak\u201d mentioned in P\u00e9r\u00e8s (1939) [unavailable] \u201cStation 119. A\u00efn Attig. Source pr\u00e8s de la route de Rabat \u00e0 Casablanca \u00e0 13 kilom\u00e8tres de Rabat\u201d , Morocco.curta\u201d self-explanatory and did not describe it. Moreover, he obviously used the name not as separate taxon but rather as descriptive term to fit existing species into his morphological concept. He even indicated Melanopsisbrevis Morelet, 1857 as its \u201ctype\u201d.First of all, the name as given by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cParreyss\u201d mentioned in Brot (1874\u20131879: 12)[unavailable]Not indicated.Melaniaholandri\u201d [sic] by Nomen nudum, \u201cin schedis\u201d name from Parreyss listed in the synonymy list of \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Langhian, middle Miocene.\u201cVatelj\u201d [Fatelj hill], Bosnia and Herzegovina.Milan et al. (1974: 89) stated that only one of the specimens illustrated by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Langhian, middle Miocene.\u201cRibari\u0107\u201d, Croatia.Melanopsislyrata Neumayr, 1869  [upper horizon of the \u017dupi\u0107a potok (near Sinj)], Croatia.Milan et al. (1974: 90) stated that Melanoptychiadalmatina Bourguignat, 1880.Junior secondary homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLe Danube \u00e0 Ibraila; la Save \u00e0 Agram; la Krapina \u00e0 Sused (Croatie)\u201d .Note that Bourguignat denoted the authority as \u201cBourguignat, 1880\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Gaudry & P. Fischer in Gaudry, 1867Late Miocene.\u201cDaphn\u00e9\u201d, Greece.daphne\u201d as mentioned in Melanopsislonga Deshayes in F\u00e9russac, 1839.The species epithet is a noun in apposition and needs not to agree in gender with the generic name (Art. 31.2.1). The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae[Prevost], 1821, Austria.PageBreakMelanopsisDaudebartii\u201d in an article in the Bulletin des Sciences, par la Soci\u00e9t\u00e9 philomatique de Paris. Constant Prevost was often considered to be the author of this article, but from the title and text it is obvious that the article is an \u201cExtrait\u201d of a talk given by Prevost earlier and summarized by an anonymous author. According to Art. 50.2 and Recommendation 51D, the correct citation should be Melanopsisdaudebartii [Prevost], 1821. The names \u201cdaudebarti\u201d, \u201caudebarti\u201d or \u201caudebardi\u201d, each occurring multiple times in the literature, are incorrect subsequent spellings. Currently, the species is classified within the genus Microcolpia \u201d [Tiout], Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cSidi Yahia, pr\u00e8s d\u2019Oudjda\u201d [Sidi Yahya near Oujda], Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[invalid]\u201cRas el \u2018Ain du Khabour\u201d [Chabur river near Ra\u2019s al \u2018Ayn], Syria.Melanopsisdebilis Pallary, 1920.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020P. Fischer, 1883Late Miocene.\u201cSmendou, province de Constantine, Alg\u00e9rie\u201d [Zighoud Youcef], Algeria.Fischer attributed the authority to Tournou\u00ebr, but from the foregoing introduction it is clear that the species was described by Fischer.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020PageBreakStoliczka, 1862 Late Transdanubian, Pannonian, late Miocene.\u201cBei Zala Apati am rechten Ufer der Zala und [...] im Gebiete des Plattensees\u201d [near Zalaap\u00e1ti at the right riverside of the Zala river and in the area around Lake Balaton], Hungary.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePaetel, 1888[invalid]\u201cPersia\u201d : 209, IrMelanopsisdecollata Stoliczka, 1862.Originally introduced as infrasubspecific taxon (\u201csubvariety\u201d) by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Late Miocene or Pliocene.Melanopsisbouei from many localities in Romania.Sabba Stefanescu, to whom Pallary referred, did not denote the localities of the figured specimens. He reported Melanopsisbouei , at K\u0101n\u012b Seip (?) and at Kars\u012b], Iraq.Pallary attributed the authority to F\u00e9russac based on a manuscript name.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020de Serres, 1829Early Campanian, Cretaceous.\u201cMartigues\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Boistel, 1898Mammal zone MN 10\u201312, late Miocene.\u201cD\u2019Ambronay [Vallon de Jurancieu]\u201d , France.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Eocene?\u201cL\u2019\u00eele de Wight\u201d [Isle of Wight], United Kingdom.Melanopsisbuccinoidea var. \u03b3) antiqua; elongata\u201d sensu F\u00e9russac, 1823 .Based on the record of \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916[invalid]Pannonian, late Miocene.\u201cDe la Moravie\u201d .Based on a part of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sacco, 1895Tortonian\u2013early Messinian, late Miocene.\u201cS. Agata fossili\u201d [Sant\u2019Agata Fossili], Italy.Melanopsisnarzolina d\u2019Archiac in Viquesnel, 1846.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sacco, 1895Tortonian\u2013early Messinian, late Miocene.\u201cS. Agata fossili\u201d [Sant\u2019Agata Fossili], Italy.dertolina\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeAnnandale & Prashad, 1919\u201cKaindak , Persian Baluchistan\u201d, Iran.Indian Museum, Calcutta, coll. no. 11535/2.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRuisseaux entre Tarsous et Mersina (Anatolie)\u201d [streams between Tarsus and Mersin], Turkey.Melanopsisbuccinoidea . In Melanopsissaulcyi Bourguignat, 1853.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1861\u201cLa Nouvelle-Cal\u00e9donie, dans l\u2019int\u00e9rieur\u201d [inland of New Caledonia], New Caledonia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020De Stefani, 1877Villafranchian, Plio-Pleistocene.\u201cSpoleto\u201d, Italy.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cKucik\u201d mentioned in Brusina (1867: 85)[unavailable]\u201cU Savi kod Zagreba\u201d [from the Sava river at Zagreb], Croatia.Melaniaholandri [sic].Nomen nudum, based on an \u201cin schedis\u201d name in the collection of Kucik . Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1915\u201cDans la source du jardin du Sultan, \u00e0 Diabet, pr\u00e8s Mogador\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884PageBreak\u201cPetit cours d\u2019eau \u00e0 Sadjour-Sou, entre A\u00efn-Ta\u00efb et Alep [...]; A\u00efn-el-Bass, dans la plaine du Bahr-el-Houl\u00e9 (Syrie)\u201d [small brook at Sadjour-Sou between Gaziantep (Turkey) and Aleppo (Syria) [...]; A\u00efn el Bass, in the plains of the Hula valley (Israel)].Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Andrusov, 1909Pontian (sensu stricto), late Miocene.\u201cBabadjan, Sundi, Meissary und Chilaalidasch\u201d , Azerbaijan.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn\u201d, Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cOgulin\u201d, Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1899\u201cDans la Souani, \u00e0 Tanger\u201d [in Souani at Tanger], Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939Unclear: given as \u201cDans l\u2019Oronte\u201d [in the Orontes river] in text but as \u201cDu lac de Homs\u201d [Lake Homs (through which the Orontes flows)] in plate captions, Syria.dircaena\u201d as mentioned in The name \u201cPageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Deshayes, 1862Lutetian, Eocene.\u201cBrasles\u201d, France.Faunus Montfort, 1810 (Pachychilidae).Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Langhian, middle Miocene.\u201cRibari\u0107\u201d : 358, CrMelanopsislyrata Neumayr, 1869 [June], non Melanopsislirata Gassies, 1869 [January]. Both names are deemed to be identical after Art. 58.2.Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cMare du moulin de la Cettina, pr\u00e8s Almissa, en Dalmatie\u201d , Croatia.Bourguignat denoted the authority as \u201cLetourneux, 1879\u201d, but there is no evidence that the description really derived from that author.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeStarobogatov, Alexenko & Levina, 1992\u201c\u0418\u0437 \u0414\u043d\u0435\u043f\u0440\u0430 \u0443 \u0425\u0435\u0440\u0441\u043e\u043d\u0430\u201d [from the Dniepr river at Kherson], Ukraine.Zoological Institute of Russian Academy of Sciences, St.-Petersburg; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Schr\u00e9ter, 1975Riss/W\u00fcrm end to early W\u00fcrm Ice Age, Pleistocene.\u201cEger, az egri v\u00e1r Z\u00e1rk\u00e1ndy b\u00e1sty\u00e1j\u00e1nak \u00e1tmetsz\u00e9se\u201d , Hungary.Magyar \u00c1llami F\u00f6ldtani Int\u00e9zet , Budapest; no number indicated.Microcolpia.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pantanelli, 1886Tortonian\u2013early Messinian, late Miocene.\u201cS. Valentino, S. Agata, Boggione\u201d , Italy.Melanopsisnarzolina d\u2019Archiac in Viquesnel, 1846.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cRas el \u2018Ain du Khabour\u201d [Chabur river near Ra\u2019s al \u2018Ayn], Syria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Graves (1847: 600)[unavailable]Eocene.\u201cCuise-Lamotte, Jaulzy, Tiverny, Saint-Vaast-de-Longmont\u201d, France.Nomen nudum. Graves attributed the authority to Defrance.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Lutetian, Eocene.\u201cAu Nord d\u2019Albas\u201d : 204, FrMelanopsisnodosa Doncieux, 1908, non F\u00e9russac, 1822.Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Wenz, 1919[invalid]Middle Lutetian, Eocene.\u201cAu Nord d\u2019Albas\u201d : 204, FrPageBreakMelanopsisbrevis Doncieux, 1908, non Sowerby, 1826. For that homonym, Melanopsisabbreviata, which is a junior homonym of Melanopsisabbreviata Brusina, 1874. (Note that Wenz 1919 was unaware of that name.) Melanopsisdoncieuxi Wenz, 1919 is, however, invalid too it is a junior homonym of Melanopsisdoncieuxi Pallary, 1916 .Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidaeIssel, 1865\u201cDi Kerman nella Persia meridionale\u201d [Kerman], Iran.Melanopsisammonis Tristram, 1865 by Melanopsisvariabilis v.d. Busch in Philippi, 1847 by Considered a junior synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Oppenheim, 1892Paleocene.\u201cDorogh, Annathal, Nagy Kovacsi\u201d , Hungary.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePreston, 1913\u201cIsland of Beilan-Beilan, to the north of the Obi Islands, Dutch East Indies\u201d [Belangbelang Island], Indonesia.Melanopsidae.Probably not a Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1901Late Miocene.\u201cSmendou\u201d [Zighoud Youcef], Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeLetourneux & Bourguignat, 1887PageBreak\u201cDe Gafsa; [...] de Tozer et de Nefta\u201d , Tunisia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1911\u201cFez\u201d [Fes], Morocco.doutte\u201d on p. 133, but as \u201cdouttei\u201d in plate caption. Since Pallary explicitly named the species after E. Doutt\u00e9, the name must read \u201cdouttei\u201d (Art. 32.5.1).Given as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Maastrichtian, Cretaceous.\u201cDu versant oriental du Kouh Mapeul\u201d , France.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020K\u00fchn, 1963[invalid]Mammal zone MN 9\u201310, late Miocene.\u201cDaphni\u201d [Dafn\u00ed valley near Athens], Greece.Museum of Palaeontology and Geology of the University of Athens, coll. no. 1963/84.Melanopsisdubiosa Matheron, 1878. Melanopsislonga Deshayes in F\u00e9russac, 1839.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1916\u201cDans le Diahot, \u00e0 Balade; \u00e0 Jengen, \u00e0 Kanala, dans les marais et les petits ruisseaux\u201d : 289 , France.Melanopsidae, the only specimen illustrated in 1822 in the \u201cHistoire naturelle\u201d was a fossil one from the Miocene of France. This fact remained widely unknown to biologists and paleontologists alike. Only few malacologists, such as Melanopsisdufourii in the biological literature actually corresponds to the real Melanopsisdufourii. Very likely, some of PageBreakthe other melanopsids from Dax . While dufouri\u201d as mentioned by numerous authors and databases and \u201cdufourei\u201d as given by The names \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Deshayes, 1825Sparnacian, early Ypresian, Eocene.\u201cLes environs de Soissons\u201d [surroundings of Soisson], France.Faunus Montfort, 1810, Melanatria Bowdich, 1822 (both Pachychilidae). Note that both authors gave the name as \u201cdufresnei\u201d, which is an incorrect subsequent spelling. Moreover, Melanatria Bowdich, 1822 [February] is not an available name because it is a replacement name for the vernacular \u201cPyrene Lamarck\u201d which was not available in nomenclature before April 1822.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeCrosse, 1869\u201cDumbea\u201d, New Caledonia.Re-described in French by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Ippolito, 1947Late Villafranchian, early Pleistocene.\u201cColle dell\u2019Oro presso Terni\u201d, Italy.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.The taxon is not included in the Fossilium Catalogus of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Deshayes, 1862Sparnacian, early Ypresian, Eocene.\u201cSaran, Cramant, Damery\u201d, France.Faunuscerithiformis  (Pachychilidae).Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat in Letourneux & Bourguignat, 1887\u201cEl-Hammam, pr\u00e8s Tozer\u201d [El-Hamma-Djerid near Tozeur], Tunisia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fontannes, 1880Miocene or Pliocene.Rh\u00f4ne Basin , France.Melanopsisnarzolina d\u2019Archiac in Viquesnel, 1846.Based on material from the Rh\u00f4ne Basin, not Italy as claimed by Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1918\u201cRas el M\u00e2 et F\u00e8s\u201d [Ras el Ma], Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans le Jourdain; [...] dans le Belus pr\u00e8s de Saint-Jean-d\u2019Acre\u201d , Israel.Melanopsisobliqua Bourguignat, 1884.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeIssel, 1866\u201cNella Maremma Toscana in un fiumicello d\u2019acqua calda detto Caldana di Ravi, presso Campiglia\u201d , Italy.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1924\u201cGandia \u201d, Spain.elatior\u201d to denote a species-group taxon but only cited Melanopsisdufourii. Pallary briefly described the taxon (by indicating its size) and therefore made the name available.Not available from Rossm\u00e4ssler (1854), to whom Pallary referred, because Rossm\u00e4ssler did not use the term \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1869\u201cNoumea\u201d, New Caledonia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in Bourguignat (1884: 15)[unavailable]\u201cDe la Save et de la Savina; [...] du Danube, pr\u00e8s de Belgrade (Serbie), de Zenica (Bosnie), de la rivi\u00e8re d\u2019Ostaria et d\u2019un ruisseau sur l\u00e0 route de Pregrada, non loin de Krapina-Toeplitz, en Croatie; [...] au pont de la Save, pr\u00e8s d\u2019Agram\u201d  Danube river near Belgrade (Serbia), Zenica (Bosnia and Herzegovina), in the river at O\u0161tarije and a stream at the road to Pregrada, near Krapinske toplice (Croatia); [...] at the Sava bridge near Zagreb].elegans\u201d as mentioned by Melanopsisholandriilegitima Rossm\u00e4ssler, 1839, under which \u201celegans\u201d was listed as a synonym by Rossm\u00e4ssler.The name \u201cVar. Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Oppenheim, 1891Pliocene.\u201cBizer\u00e8, n\u00f6rdlich von Pyrgos (Elis)\u201d , Greece.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[unavailable]Unclear: given as \u201cL\u2019Oronte \u00e0 Djishr ech Chegour\u201d [Orontes river at Jisr Ash-Shughur] in text but as \u201cL\u2019Oronte \u00e0 Homs\u201d [Orontes river at Homs (which is far to the south of the former)] in plate captions, Syria.Melanopsiscostata .Unavailable according to Art. 13.1, because it lacks a verbal description or definition. Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020F\u00e9russac, 1822Eocene.\u201cDes environs d\u2019Epernay. [...] De l\u2019\u00eele de Wight\u201d .antiquua\u201d, which was probably not intended as species-group name \u201d [at Tersanne near Hauterives], France.Melanopsisbuccinoideaelongata F\u00e9russac, 1822. Melanopsisnarzolina d\u2019Archiac in Viquesnel, 1846.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882[invalid]Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsisbuccinoideaelongata F\u00e9russac, 1822. Not included in the Fossilium Catalogus by Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cOgulin\u201d, Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887[invalid]Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Melanopsisbuccinoideaelongata F\u00e9russac, 1822.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Locard, 1893[invalid]\u201cHelicidenmergel\u201d, late Burdigalian, early Miocene.\u201cUetken, en Argovie\u201d, Switzerland.Melanopsisbuccinoideaelongata F\u00e9russac, 1822. Melanopsiscitharella Merian, 1849.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903[invalid]Late Pleistocene\u2013early Holocene.PageBreak\u201cBischofsbad\u201d , Romania.Melanopsisbuccinoideaelongata F\u00e9russac, 1822. Microcolpiaparreyssiisikorai .Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Doncieux, 1908[invalid]Middle Lutetian, Eocene.\u201cAu Nord d\u2019Albas\u201d, France.Melanopsisbuccinoideaelongata F\u00e9russac, 1822. Melanopsissublongata [sic] as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[invalid]\u201cRas el \u2018Ain du Khabour\u201d [Chabur river near Ra\u2019s al \u2018Ayn], Syria.Melanopsisbuccinoideaelongata F\u00e9russac, 1822. Melanopsiskhabourensis Pallary, 1939.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in P\u00e9r\u00e8s (1939)[unavailable]Collection stations 80 [tributary rivulet to the river Tensift] and 144  : 139, Moelongata\u201d self-explanatory and did not describe it. Moreover, he obviously used the name not as separate taxon but rather as descriptive term to fit existing species into his morphological concept. He even indicated Melanopsisolivieri Bourguignat, 1884 as its \u201ctype\u201d.First of all, the name as given by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Gillet & Marinescu, 1971[invalid]Transdanubian, Pannonian, late Miocene.\u201cR\u0103dm\u0103ne\u0219ti\u201d, Romania.Melanopsisbuccinoideaelongata F\u00e9russac, 1822. Melanopsisdefensa Fuchs, 1870.Junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1925Messinian, late Miocene.\u201cDella valle della Sterza\u201d : 420 Not indicated.Melaniabuccinoidea  by Nomen nudum, \u201cin schedis\u201d name from Tarnier listed in the synonymy list of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[unresolved]\u201cA\u00efn M\u00e9lias, pr\u00e8s de Figuig\u201d [Ain Melias near Figuig], Algeria.Melanopsisletourneuxiemaciata Pallary, 1928 (see Note 1). This case requires the action of a First Reviser.Homonym of the simultaneously published name Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[unresolved]\u201cBerguent\u201d [A\u00efn Beni Mathar], Morocco.Melanopsisfoleyiemaciata Pallary, 1928 (see Note 1). This case requires the action of a First Reviser.Homonym of the simultaneously published name Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1936[invalid]Not explicitly stated but probably the same as for the species .Melanopsisletourneuxiemaciata Pallary, 1928 and Melanopsisfoleyiemaciata Pallary, 1928 .Junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[invalid]\u201c\u00c0 J\u00e9richo, dans \u2018Ain Solthan\u201d , Palestine.Melanopsisletourneuxiemaciata Pallary, 1928 and Melanopsisfoleyiemaciata Pallary, 1928 .Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020d\u2019Orbigny, 1850Early Rupelian, Oligocene.\u201cLevit, pr\u00e8s de Castellanne\u201d [V\u00eet-de-Castellane near Castellane], France.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1897Early Langhian, middle Miocene.\u201cSinj (\u017dupi\u0107a potok)\u201d, Croatia.The syntypes are stored in the Croatian Natural History Museum, Zagreb; no number indicated .Melanopsisgeniculata Brusina, 1874, while enodota\u201d as mentioned in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cZiegler\u201d mentioned in Reeve (1860)[unavailable]Not indicated.Melaniahollandrii [sic].Perhaps a manuscript or \u201cin schedis\u201d name from Ziegler. Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Transdanubian, Pannonian, late Miocene.\u201cTihany; Kenese\u201d , Hungary.PageBreak34\u201336). This fixation is insufficient for a valid lectotype designation (Art. 74). The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2515-161/1-2.Milan et al. (1974: 90) defined a \u201clectosyntypus\u201d, a term not accepted by the Code, based on the specimen from Tihany illustrated by Melanopsisfuchsi Handmann, 1882.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Sparnacian, early Ypresian, Eocene.\u201cMont Bernon bei Epernay\u201d . The name \u201ceocaenica\u201d as mentioned in Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cPr\u00e8s de Biskra, dans un ruisseau d\u2019eau chaude \u00e0 Ouargla, dans un puits art\u00e9sien de Mazer au Ziban, ainsi qu\u2019aux environs de Boghar et dans les rivi\u00e8res du djebel Takreda entre Mogador et Maroc [sic]\u201d .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeTristram, 1865\u201cWady Um Bagkek, between Sebbeh and Jebel Usdum, at the south-west corner of the Dead Sea\u201d, Israel.Melanopsisammonis Tristram, 1865.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeRoth, 1839\u201cIn Peloponneso\u201d, Greece.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeF\u00e9russac, 1823\u201cLa Laybach\u201d [Ljubljanica river], Slovenia.Esperiana Bourguignat, 1877.Type species of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Stefanescu, 1896Pliocene.\u201c\u00c0 Bucovatz, dans la vall\u00e9e de Jiu, et \u00e0 Milcov, pr\u00e8s de Slatina, dans la vall\u00e9e de l\u2019Oltu\u201d , Romania.esperiodea\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeWesterlund, 1886\u201cItalien, Balearen [Westerlund]; aus Toscana als auch aus Jumilla (Spanien) und Merknes (Marocco) [Brot]\u201d .etrusca Villa (v. min)\u201d from the locality \u201cToscane\u201d, without description or bibliographical reference. Brot listed it under the name \u201cMelanopsisDufourii F\u00e9r.\u201d, for which a bibliographical reference (to \u201cRossm. Icon. 835\u2013839\u201d) and locality \u201cEspagne\u201d was given. Since etrusca was neither described by Brot nor by Rossm\u00e4ssler, the name etrusca is not available from this case (Art. 12.1).The name was first mentioned by Melanopsisetrusca Villa MSS.\u201d) as synonym of MelanopsisDufourii var. \u03b2, which he introduced there with a short Latin description. On p. 435 Brot even associated etrusca with an illustration of a specimen from Toscana . However, the name was explicitly referred to as synonym of the (unnamed) variety \u03b2, which is not a valid name, and thus the requirements of Art. 11.6 (and therefore 12.1) are not met (see Note 2).Melanopsislorcana] var. etrusca Villa ap. Brot (Cat. syst. Melan. 1862)\u201d and provided a description and thus made it available as species-group taxon. Since he referred to Brot\u2019s catalogue, the type locality comprises both his and Brot\u2019s records. The authority of Melanopsisetrusca remains with Westerlund. Art. 11.6.1 does not apply here as PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Middle Pannonian, late Miocene.\u201cK\u00fap\u201d, Hungary.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2494-140/1-3 .Melanopsispygmaeamucronata Handmann, 1887 by Considered a junior synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans le Jourdain, \u00e0 4 kilom\u00e8tres au-dessus de la Mer Morte\u201d , Palestine/Jordan.Melanopsislampra Bourguignat, 1884 as well as of Melanopsisobliqua Bourguignat, 1884, both of which they ranked as subspecies of Melaniacostata .Apparently by mistake, Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Cob\u0103lcescu, 1883Dacian, Pliocene.\u201cBeceni\u201d, Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr in Neumayr & Paul, 1875Cernikian, Pliocene.\u201c\u010caplathal\u201d [\u010caplja trench near Slavonski Brod], Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cA\u00eft Brahim; Tazouta; El Menzel; [...] l\u2019Oued Raha, \u00e0 Agoura\u00ef\u201d , Morocco.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cYeni Chehir\u201d [Yeni\u015fehir], Turkey.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRivi\u00e8re d\u2019Ostaria, entre Plaski et Ogulin, en Croatie; la Save, \u00e0 Sissek, en Slavonie; la Narenta, en Dalmatie\u201d , Croatia.Note that Bourguignat denoted the authority as \u201cBourguignat, 1880\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928\u201cAgoura\u00ef, \u00e0 29 kil. sud de F\u00e8s\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cAgoura\u00ef\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Magrograssi, 1928[invalid]Plio-Pleistocene.\u201cCoo: frequente nella zona centrale fossilifera\u201d [Kos island: common in the central fossiliferous zone], Greece.Melanopsismagnificaexpansa Pallary, 1920 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.PageBreak\u201cLeobersdorf\u201d, Austria.Melanopsisfossilis .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1884Portaferrian (Pannonian Basin), late Miocene\u2013Pliocene.\u201cOkrugljak\u201d [near Zagreb], Croatia.Milan et al. (1974: 90) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2966-612.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans la rivi\u00e8re de la Krapina, \u00e0 Sused, en Croatie\u201d , Croatia.Note that Bourguignat denoted the authority as \u201cBourguignat, 1879\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pantanelli, 1886Messinian, late Miocene.\u201cSterza\u201d : 550 , New Caledonia.Melanopsisfrustulum Morelet, 1857.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882[invalid]Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsisfasciata Gassies, 1874. Melanopsisvittata as replacement name. The taxon was considered as a junior synonym of Melanopsishaueri by Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBiggs, 1937[invalid]\u201cFrom a \u2018qanat\u2019 [= typical Iranian channel] at Jelalabad, near Kerman\u201d, Iran.Melanopsisfasciata Gassies, 1874.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in Brusina (1866: 106) and Brusina (1867: 86)[unavailable]\u201cCetina\u201d, Croatia.Melaniafasciata Sowerby, 1818 described from the Paleogene of the United Kingdom.Nomen nudum, \u201cin schedis\u201d name in collection of Kucik , occasionally listed with authority \u201cStentz\u201d. If available, the name would be junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1864\u201cRuisseau de Ngouca\u201d [N\u2019Goussa], Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cTaforalt. A\u00efn Sfa. F\u00e8s (Ras el M\u00e2). [...] Dar Batha, jardin public de bou Djeloud, s\u00e9guias de dar el Maghzen, s\u00e9guias en dehors de bab sidi bou Jida\u201d , Morocco.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBrot, 1868\u201cPersepolis\u201d, Iran.Brot attributed the authority to Parreyss, based on an \u201cin schedis\u201d determination by that author. However, from the following text it becomes clear that Brot is the author of the species.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neubauer, Mandic & Harzhauser, 2014Langhian, middle Miocene.\u201cNW slope of Fatelj hill near Kupres\u201d, Bosnia and Herzegovina.Geological-Paleontological Department, Natural History Museum Vienna, Austria, coll. no. 2011/0138/0107a.Melanopsismojsisovicsi Neumayr, 1880 in Introduced for formerly misidentified Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans le Jourdain, non loin de son embouchure dans la Mer Morte\u201d , Palestine/Jordan.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeRoth, 1839\u201cSmyrna\u201d [Izmir], Turkey.Melanopsisbuccinoidea .Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cA\u00eft Brahim\u201d, Morocco.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]\u201cA\u00efn M\u00e9lias, pr\u00e8s de Figuig\u201d [Ain Melias near Figuig], Algeria.Melanopsisexcoriatafestiva Pallary, 1920 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr, 1880Early Langhian, middle Miocene.\u201cWestlich von Drvar\u201d [west of Drvar], Bosnia and Herzegovina.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020De Stefani, 1877Villafranchian, Plio-Pleistocene.\u201cMonte Albuccio, [...] Valli della Pescaia, e del Riluogo [...] presso Siena; Castellacela presso Massa Marittima [...]; Colline di Piedimonte presso Terni nell\u2019Umbria [...], tra S. Gemine e Carsoli sulla strada di Narni, e fra Otricoli e le Vigne sulla strada da Roma a Foligno?\u201d , Italy.Melanopsisantiqua F\u00e9russac\u201d, which is not an available name.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cZelebor\u201d mentioned in Brot (1874\u20131879: 12)[unavailable]Not indicated.Melaniaholandri\u201d [sic] by Melaniaflava Deshayes in Deshayes & Jullien, 1876 described from Cambodia.Nomen nudum, \u201cin schedis\u201d name from Zelebor listed in synonymy of \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cA\u00efn M\u00e9lias, pr\u00e8s de Figuig\u201d [Ain Melias near Figuig], Algeria.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020G\u00fcmbel, 1861Chattian, Oligocene.\u201cWester-Buchberg, Loherfl\u00f6tz, Aubach, Rohmbach, Sulzgrabenfl\u00f6tz, Schlierachthal, Neum\u00fchle, bei Rimselrain, Pensberg, im H\u00f6llgraben, im Buchbergfl\u00f6tz, bei Schmerold, am hohen Peissenberge\u201d , Germany.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Villatte in Tambareau & Villatte, 1966Sparnacian, early Ypresian, Eocene.\u201cFontan\u00e9 (Ari\u00e8ge)\u201d, France.Laboratoire de G\u00e9ologie de la Facult\u00e9 des Sciences de Toulouse; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePaetel, 1888\u201cLake Huleh [Palestine]\u201d : 212, PaOriginally introduced as infrasubspecific taxon (\u201csubvariety\u201d) by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Tournou\u00ebr, 1879Cernikian, Pliocene.\u201cNisipulu, Rumaniae\u201d [Nisip], Romania.Amphimelania by Considered to belong in the genus PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Gmelin, 1791Gmelin 1971: 3485.Sarmatian (sensu stricto), middle Miocene.\u201cZwischen Oedinburg and Ru\u00df\u201d : 204 , Algeria.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1874\u201cOuagap\u201d [Wagap], New Caledonia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cSchmidt\u201d mentioned in Brusina (1867: 86)[unavailable]Not indicated.Melaniafragilis Lamarck, 1804 from the Paleogene (Eocene?) of the Paris Basin.Nomen nudum, \u201cin schedis\u201d name in collection of Kucik . If available, the name would be junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903Late Pleistocene\u2013Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020K\u00fchn, 1951Mammal zone MN 10, late Miocene.\u201cNordwestlich Nea Liossia\u201d [northwest of Ilion], Greece.Geological-Paleontological Department, Natural History Museum Vienna, Austria, coll. no. 1949/0004/0006.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1897Late Cernikian, late Pliocene\u2013early Pleistocene.\u201cKutina (\u0161uma Mi\u0161inka)\u201d , Croatia.Milan et al. (1974: 57) stated that PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1885Cernikian, Pliocene.\u201cKravarsko und Podvornica\u201d, Croatia.Milan et al. (1974: 91) defined a neotype based on the specimens illustrated by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Thom\u00e4, 1845Mammal zone MN 2, early Miocene.\u201cM\u00fchltal bei Wiesbaden\u201d, Germany.Fritzii\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeMorelet, 1857\u201c[Ad Sanctam-Mariam de Balade]\u201d [Balade], New Caledonia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsispygmaea H\u00f6rnes, 1856. Melanopsishandmanni, which is the objective junior synonym of Melanopsisfuchsi Handmann, 1882.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1884[invalid]Transdanubian, Pannonian, late Miocene.\u201cRadmanest\u201d (Fuchs 1870: 353) [R\u0103dm\u0103ne\u0219ti], Romania.The syntypes are stored in the Croatian Natural History Museum, Zagreb; no number indicated .PageBreakMelaniacostata sensu Fuchs, 1870, non Olivier, 1804. Junior homonym of Melanopsisfuchsi Handmann, 1882  ornata Fuchs, 1877, non Viviparaornata Neumayr, 1875, which was considered a Melanopsidae by Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1858\u201cDes petits cours d\u2019eau \u00e0 Balade; [...] \u00e0 l\u2019\u00eele des Pins\u201d , New Caledonia.Melanopsisfrustulum Morelet, 1857.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cBorn.\u201d mentioned in De Cristofori & Jan [unavailable]\u201cAustr.\u201d [Australia].Nomen nudum, found only in the species list of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBrot, 1879\u201cOuagap\u201d [Wagap], New Caledonia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020P. Fischer in Gaudry, 1867Pliocene.PageBreak\u201cM\u00e9gare\u201d (p. 444), Greece.The taxon is not included in the Fossilium Catalogus of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1870\u201cPrope Pouebo\u201d [near Pou\u00e9bo], New Caledonia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sowerby, 1822Eocene.\u201cIsle of Wight; [...] New Charlton; [...] Woolwich; [...] New Cross near Deptford; [...] Hordwell\u201d, United Kingdom.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Grateloup, 1838[invalid]Burdigalian, early Miocene.\u201cDax. [...] Mandillot\u201d, France.Melanopsisantiqua F\u00e9russac, 1823\u201d . Junior homonym of Melanopsisfusiformis Sowerby, 1822.Based on some of the illustrated syntypes of \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1870[invalid]\u201cPrope Kanala; insula Ouen\u201d , New Caledonia.Melanopsisfusiformis Sowerby, 1822. Melanopsisrossiteri as replacement name. Unaware of this, Melanopsiscookiana as replacement name, which is its junior objective synonym.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882[invalid]Pannonian, zone D, late Miocene.PageBreak\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsisfusiformis Sowerby, 1822. Melanopsisangusta as replacement name. Melanopsishaueri Handmann, 1882.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Magrograssi, 1928[invalid]Plio-Pleistocene.\u201cCoo: V. Tuvachiu, fra Antimachia e Pili\u201d , Greece.Melanopsisfusiformis Sowerby, 1822.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sacco, 1895Messinian, late Miocene.\u201cPriosa presso Narzole, Castelletto d\u2019Orba, S. Marzano Oliveto\u201d, Italy.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Vergnau-Saubade, 1968Oligocene.\u201cGaas - Lesbarritz\u201d, France.First mentioned in an unpublished thesis .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1878Middle\u2013late Cernikian, late Pliocene\u2013early Pleistocene.\u201cSibinj\u201d, Croatia.Milan et al. (1974: 57) indicated a holotype, but it is uncertain whether the specimen is part of the original type series and whether it was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2963-609.Amphimelania by caji\u201d as mentioned in gayi\u201d as given by Considered to belong in the genus PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cAffluents du lac Sabandja, pr\u00e8s d\u2019Ismidt\u201d [tributaries to the Lake Sapanca near \u0130zmit], Turkey.Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cConstantinople [...], a \u00e9t\u00e9 trouv\u00e9e dans la rivi\u00e8re d\u2019Ismidt (Anatolie)\u201d , Turkey.Microcolpiacoutagniana .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Matheron, 1842Campanian\u2013Maastrichtian, Cretaceous.\u201cLes Martigues, les Pennes, Simiane, Gardanne, Fuveau, Peynier, Trets\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Munier-Chalmas, 1884Munier-Chalmas 1884: 330, pl. 7, figs 21\u201322.Maastrichtian, Cretaceous.\u201cD\u2019Auzas\u201d, France.Munier-Chalmas (1884) indicated the authority with \u201cMun. Ch., 1870\u201d, which is probably due to the fact that issues 1\u20133 of vol. 1 of Annales de Malacologie, in which the original publication appeared, were published in 1870 .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeCrosse, 1867\u201cWagap\u201d, New Caledonia.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neubauer, Harzhauser, Georgopoulou, Mandic & Kroh, 2014Early Pleistocene (?).\u201cWeganrisse und Regenrisse 3 Km N Antirrion in Akarnanien\u201d  , Greece.Senckenberg Forschungsinstitut und Naturmuseum Frankfurt; no number indicated.Melanopsisposterior Sch\u00fctt, 1986, non Papp, 1953.Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Langhian, middle Miocene.\u201cMio\u010di\u0107\u201d, Croatia.Milan et al. (1974: 91) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 3207-853.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cSources du Nahr es Sine, au Sud de Lattaqui\u00e9, sur la route de Beyrouth\u201d , Syria.Melanopsiscostata .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Willmann, 1981Palioskala Formation, middle Tortonian, late Miocene.\u201cBachri\u00df beim H\u00f6henzug von Palioskala, ca. 700 m westlich vom Kap Phoka\u201d , Greece.Geological-Paleontological Institute, University of Kiel, Germany; no number indicated.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Grateloup, 1838Chattian\u2013Burdigalian (?), late Oligocene\u2013early Miocene.\u201cDax. [...] d\u2019Abesse et de Quillac \u00e0 Saint-Paul\u201d , France.Melanopsismajor F\u00e9russac, 1823\u201d, which is not an available name.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRivi\u00e8re pr\u00e8s d\u2019une villa sur la route de Pregrada, aux environs de Krapina, en Croatie\u201d , Croatia.Note that Bourguignat denoted the authority as \u201cBourguignat, 1879\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Robles, 1975Late Miocene.\u201cVenta del Moro \u201d, Spain.Museo Nacional de Ciencias Naturales, Madrid, coll. no. M-327.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Cernikian, Pliocene.\u201cSibinj; Farka\u0161i\u0107; Dubranjec\u201d, Croatia.Milan et al. (1974: 89) stated that the single specimen of this taxon was lost.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pavlovi\u0107, 1927[invalid]Middle Pannonian, late Miocene.\u201c\u0423 \u041a\u0430\u0440\u0430\u0433\u0430\u0447\u0443\u201d [from Karaga\u010da near Vr\u010din], Serbia.The illustrated syntype is stored in the Natural History Museum, Belgrade, coll. no. 209 (Milo\u0161evi\u0107 1962: 23).Melanopsisglabra Brusina, 1874. Melanopsisvrcinensis as replacement name.Junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Bogachev [unavailable]Miocene.\u201cTori bei Borshomi\u201d [Tori near Borjomi], Georgia.glabra\u201d on p. 53, but \u201cglarba\u201d on p. 13, 33.The name was only mentioned in a species list by Bogachev without description or illustration. Moreover, he applied multiple original spellings: the name is given as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Oluji\u0107, 1999[invalid]Langhian, middle Miocene.It is unclear from the original work in which of the studied localities/sections along the valleys of the Sutina, Batarelov and Vojskava rivers (4 km W of Sinj) the taxon occurred and in which not, Croatia.Melanopsisglabra Brusina, 1874. Melanopsislyrata Neumayr, 1869.Junior secondary homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeClessin, 1890\u201cUngarn ([...] ohne n\u00e4here Fundortangabe)\u201d , Hungary.Microcolpiaparreyssii .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sandberger, 1872Mammal zone MN 5, early\u2013middle Miocene.\u201cPontlevoy bei Blois\u201d [Pontlevoy near Blois], France.Plate 26 of Sandberger\u2019s monograph was issued in 1872, while the description on p. 520 appeared in 1875 .Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928\u201cA\u00efn M\u00e9lias, pr\u00e8s de Figuig\u201d [Ain Melias near Figuig], Algeria.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neubauer & Harzhauser in Neubauer et al., 2015Middle Pannonian, late Miocene.\u201cBrickyard near the town of Martin, Turiec Basin\u201d, Slovak Republic.M\u00fazeum Andreja Kme\u0165a in Martin, Slovak Republic, coll. no. SNM 16/2011 (PZ-703g).Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLa Glina en Hongrie\u201d [Glina river], Croatia.Microcolpiaacicularis .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBrot, 1878\u201cGlina Fl.\u201d [Glina river], Croatia.Hemisinusacicularis \u201d. Microcolpiaacicularis.\u201cIn schedis\u201d name by Parreyss, introduced by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Magrograssi, 1928Plio-Pleistocene.\u201cCoo: V. Bocasia, torrente Sefto, C. Foca, tra Antimachia e Pili\u201d , Greece.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cDans les sources de M\u00e9z\u00e9rib, au N. O. de Der\u00e2a \u201d , Syria.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Tournou\u00ebr, 1875Tafi Formation, early Pleistocene.\u201cPrope vicum Antimaki [...] et prope civitatem Cos\u201d [near near the village Antim\u00e1cheia and near the city of Kos], Greece.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1916\u201cBourail\u201d : 384, NeMelanopsiselongata Gassies, 1874, non F\u00e9russac, 1822 (see Note 1).Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1911\u201cTout pr\u00e8s d\u2019Oudjda, \u00e0 4 kilom. S.-E., sourdent les belles sources de Sidi-Yahia qui alimentent une v\u00e9ritable oasis, puis la ville d\u2019Oudjda, et vont finalement se d\u00e9verser dans l\u2019oued Isly\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Papp, 1955Early Pleistocene.\u201cBrochitza, Elis\u201d [Vrokh\u00edtsa], Greece.Museum of Palaeontology and Geology of the University of Athens; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Pannonian, zone C, late Miocene.\u201cSused (Sopot bei Goljak)\u201d , Croatia.Melanopsisbouei F\u00e9russac, 1823.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fontannes, 1880[invalid]Mammal zone MN 11, late Miocene.Potamides Basteroti de Visan (Vaucluse)\u201d , France.\u201cLes marnes \u00e0 Melanopsisboueigracilis Brusina, 1874 (see Note 1). Melanopsisneumayri Tournou\u00ebr, 1874.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeLocard, 1883[invalid]\u201cLac d\u2019Homs\u201d [Lake Homs], Syria.Melanopsisboueigracilis Brusina, 1874 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887[invalid]Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Melanopsisboueigracilis Brusina, 1874 (see Note 1). Melanopsisbouei F\u00e9russac, 1823.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Seninski, 1905Duab Beds, middle to late Kimmerian, Pliocene.\u201c\u041c\u043e\u043a\u0432\u0438\u043d\u0441\u043a\u0456\u0435 \u043f\u043b\u0430\u0441\u0442\u044b, \u0440\u0430\u0437\u0440\u0463\u0437\u044a \u0440. \u0414\u0443\u0430\u0431\u044a\u201d [Mokvi layers at Duab river], Georgia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Rolle, 1858Paleocene.\u201cHerrmanns-Stollen im Lubellina-Graben, Gemeinde Ober-Skallis, nord\u00f6stlich von Sch\u00f6nstein\u201d , Slovenia.Pyrgulifera Meek, 1877 (Pleuroceridae), but After Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fuchs, 1870[invalid]Transdanubian, Pannonian, late Miocene.\u201cTihany\u201d (p. 533), Hungary.Melanopsisgradata Rolle, 1858. Melanopsistihanyensis as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[invalid]\u201cRas el \u2018Ain du Khabour\u201d [Chabur river near Ra\u2019s al \u2018Ayn], Syria.Melanopsisgradata Rolle, 1858.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeVilla & Villa in Graells, 1846Spain [no locality indicated].graellsi\u201d as mentioned in Buccinumannulatum Chemnitz\u201d. That species, which is not available from Nassariidae?).Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939Not explicitly stated but probably the same as for the species .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Calvert & Neumayr, 1880Late Sarmatian, Khersonian, late Miocene.\u201cRenki\u00f6i\u201d [north of \u0130ntepe], Turkey.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Hoeninghaus (1831: 142)[unavailable]Burdigalian, early Miocene (?).\u201cDax\u201d, France.gratelupii\u201d) in synonymy of Melanopsisaquensis Grateloup, 1838. grateloupi\u201d) with \u201cMelanopsismajor F\u00e9russac, 1823\u201d, which is not an available name.Nomen nudum. Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRivi\u00e8res du bassin du lac Sabandja (Anatolie)\u201d [rivers of the basin of Lake Sapanca], Turkey.Fagotia [= Esperiana] gallandi Bourguignat, 1884.Note that Bourguignat denoted the authority as \u201cBourguignat, 1880\u201d. Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Sparnacian, early Ypresian, Eocene.\u201cEntre St.-Germini et Carsoli\u201d : 164 , Equatorial Guinea.Melanopsidae.Certainly not a Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cPrope Cehejin Prov. Murcica\u201d : 438 , Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLe Danube pr\u00e8s de Buda-Pesth, en Hongrie\u201d [in the Danube river near Budapest], Hungary.Hagenmulleria\u201d on p. 51 but as \u201cHagenm\u00fclleriana\u201d on p. 57. To my knowledge no later author has acted as First Reviser sensu Art. 24.2.3 and has mentioned both names and selected one as correct. Microcolpiacornea .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pavlovic, 1927Middle Pannonian, late Miocene.\u201c\u041a\u0430\u0440\u0430\u0433\u0430\u0447\u0430\u201d [Karaga\u010da near Vr\u010din], Serbia.From the writing in the original work it seems that Pavlovi\u0107 had apparently doubts about the validity of the taxon.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1856\u201cL\u2019Oued-el-Hammam [entre le Sig et Mascara]\u201d [Oued el Hammam between Sig and Mascara], Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1892[invalid]Early\u2013middle Pannonian, late Miocene.\u201cKotingbrunn; [...] Leobersdorf\u201d, Austria.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2513-159/1-4 .Melanopsisfuchsi Handmann, 1882, non Brusina, 1884. Melanopsishandmanni as well as Melanopsisfuchsi Handmann, 1882 with Melanopsispygmaea.Introduced as replacement name for the senior (!) homonym Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020W. Fischer, 1996Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Melanopsisconstricta Handmann), which is stored in the collection of the Geological Survey Austria, Vienna; no number indicated.Melanopsismartinianaconstricta Handmann, 1887, non Brusina, 1878 (see Note 1).Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Hofmann, 1870Egerian, late Oligocene\u2013early Miocene.\u201cA zsily-v\u00f6lgyi [...], tov\u00e1bb\u00e1 Valia-Aninossaban \u00e9s Paren lui Marin-ban, Lup\u00e9ny mellett. [...] gyakori Buda \u00e9s Esztergom k\u00f6z\u00f6tt [...], jelesen a Mikl\u00f3shegyen, Annav\u00f6lgyen, Mogyor\u00f3son \u00e9s Pom\u00e1zon\u201d , Hungary.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020De Laubri\u00e8re & Carez, 1881Lutetian, Eocene.\u201cBrasles\u201d, France.Faunus Montfort, 1810 (Pachychilidae).Taxon classificationAnimaliaSorbeoconchaMelanopsidaeWesterlund, 1892\u201cBei Sevilla im Guadalquivir\u201d , Spain.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr in Neumayr & Paul, 1875Cernikian, Pliocene.\u201c\u010caplathal bei Podwin\u201d [\u010caplja graben near Slavonski Brod], Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sandberger, 1873Early Rupelian, Oligocene.\u201cGrossalmerode [...]; Nordshausen\u201d : 95\u201396, Melanopsissubulata sensu Speyer, 1870, non Sowerby, 1822 and Melanopsispraerosa [= Melanopsispraemorsa] sensu Speyer, 1870, non Linnaeus, 1758.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr in Neumayr & Paul, 1875Cernikian, Pliocene.\u201cGroma\u010dnik; [...] Slobodnica; [...] Sibin [Sibinj]; [...] Malino\u201d, Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Popescu-Voite\u0219ti, 1910Lutetian, Eocene.\u201cGropile Vulpilor pr\u00e8s Tite\u0219ti\u201d [Gropile Vulpilor (?) near Tite\u0219ti], Romania.Coptostylus Sandberger, 1872 (Thiaridae).Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903Late Pleistocene\u2013Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Hantken, 1878Late Santonian\u2013early Campanian, late Cretaceous.\u201c[Ajka]\u201d , Hungary.Campylostylus  in Sandberger (1873: 341).Appeared first as a nomen nudum . Currently considered to belong in the marine genus Clea H. Adams & A. Adams, 1855 (Buccinidae)  e.g., .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans l\u2019oasis d\u2019A\u00efn-Chair, \u00e0 l\u2019extr\u00eame sud saharien du Maroc\u201d , Marocco.Note that Bourguignat denoted the authority as \u201cBourguignat, 1872\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Calvert & Neumayr, 1880Late Sarmatian, Khersonian, late Miocene.\u201cRenki\u00f6i\u201d [north of \u0130ntepe], Turkey.Amphimelania by Considered to belong in the genus Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Blanckenhorn, 1897Plio-Pleistocene.Dreissena layer], Syria.\u201cDschisr esch-Schurr, Dreissensiaschicht\u201d , Austria.Geological Survey Austria, Vienna; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fontannes, 1881Rupelian, Oligocene.\u201cBaume-Cornillane\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cSources du Jourdain; [...] d\u2019A\u00efn-el-Mellaha, dans la plaine du Bahr-el-Houl\u00e9; [...] lacs d\u2019Homs et d\u2019Antioche\u201d  A\u00efn Mallahah, in the plains of the Hula valley; [...] lakes Homs and Anuk ], Syria.Melaniacostata sensu Kobelt, 1880, non Olivier, 1804. Bourguignat attributed the authority to Letourneux, but there is no evidence that the description really derived from that author. Melanopsiscostata  as well as of Melanopsissaulcyi Bourguignat, 1884.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020De Gregorio, 1880Eocene.\u201cSan Giovanni Ilarione\u201d, Italy.Melanopsis and a separate species by Considered a Taxon classificationAnimaliaSorbeoconchaMelanopsidaeWesterlund, 1898\u201cAlbarracin\u201d, Spain.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d , AustriaMelanopsisturrita Handmann, 1887, non Rossm\u00e4ssler, 1854.Replacement name for the junior homonym Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sandberger, 1872Burdigalian, early Miocene.\u201cSt. Paul, Mandillot (Landes)\u201d, France.Melanopsisbuccinoides [sic] sensu Grateloup, 1840, non Olivier, 1801, for which Melanopsissubbuccinoides as replacement name. Hence, Melanopsishoernesi is a junior objective synonym of Melanopsissubbuccinoides. Sandberger attributed the authority to Mayer, apparently based on an \u201cin schedis\u201d determination.Plate 25 of Sandberger\u2019s monograph was issued in 1872, while the description on p. 512 appeared in 1875 . IntroduTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBlanckenhorn, 1897[invalid]Pleistocene (?)\u2013Recent.\u201cFossil [..] bei Antakije. Lebend am unteren Orontes [...], desgleichen im Kara Su, einem n\u00f6rdlichen Zufluss des Sees von Antiochia und im Sadj\u00fcr Su bei Aleppo\u201d .Melanopsishoernesi Sandberger, 1872 . Melanopsiscostata .Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePfeiffer, 1828\u201cBei Kroatisch Feistritz, am Fusse des Berges Terglou in Illyrien\u201d , Slovenia.Holandriana Bourguignat, 1884 and Amphimelania P. Fischer, 1885. The names \u201cholandri\u201d and \u201chollandri\u201d, each occurring multiple time in the literature  indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2985-631.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeAzpeitia Moros, 1929\u201cGuadalquivir en Lora del R\u00edo, Sevilla\u201d , Spain.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928\u201cF\u00e8s et Mekn\u00e8s\u201d, Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Kormos, 1904Late Pleistocene\u2013early Holocene.\u201cP\u00fcsp\u00f6kf\u00fcrd\u0151\u201d , Romania.Microcolpiaparreyssii .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916[invalid]Transdanubian, Pannonian, late Miocene.\u201cRadmanest\u201d (Fuchs 1870: 353) [R\u0103dm\u0103ne\u0219ti], Romania.Melanopsisfuchsi Brusina, 1884 non Handmann, 1882, which itself was introduced for Melaniacostata sensu Fuchs, 1870, non Olivier, 1804. It is a junior homonym of Melanopsishungarica Kormos, 1904 .Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Cernikian, Pliocene.\u201cPodvinje (\u010caplja)\u201d [\u010caplja trench near Slavonski Brod], Croatia.hibostoma\u201d and \u201chypostoma\u201d as mentioned in The names \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in Brot (1874\u20131879: 442)[unavailable]\u201cMorocco, Mogador; [...] Spanien, Andalousien\u201d .Melanopsistingitana var. \u03b2\u201d by Melanopsistingitana that matches the description by Brot is Nomen nudum, listed in synonymy of \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Krauss, 1852Ottnangian, middle Burdigalian, early Miocene.\u201cKirchberg an der Iller\u201d [Illerkirchberg], Germany.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020F\u00e9russac, 1822Sarmatian (sensu lato), middle\u2013late Miocene.\u201cDe Sestos\u201d , Turkey.Melanopsisincerta with Melanopsisdaudebartii and erroneously illustrated the latter of both species, which was already recognized by Deshayes when preparing the final explanations for the plates of the \u201cHistoire naturelle\u201d.The species was also illustrated on pl. 2 of the \u201cM\u00e9lanopsides fossiles\u201d in the \u201cHistoire naturelle\u201d  and on pTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fuchs, 1877[invalid]Pliocene.\u201cMegara\u201d, Greece.Melanopsisincerta F\u00e9russac, 1822. Melanopsisrevelata as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928\u201cSources ti\u00e8des un peu avant Kerrando, au S.-E. de Rich \u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeStarobogatov in Starobogatov et al., 1992\u201c\u0424\u0451\u0441\u043b\u0430\u0443, \u0431\u043b\u0438\u0437 \u0412\u0435\u043d\u044b\u201d [V\u00f6slau near Vienna], Austria.Zoological Institute of Russian Academy of Sciences, St.-Petersburg; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr, 1869Langhian, middle Miocene.\u201cMiocic\u201d [Mio\u010di\u0107], Croatia.Illustrated syntypes are stored at the Geological Survey Austria, Vienna, coll. no. 1869/01/3/1-10.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeWesterlund, 1898\u201cMuchalatka\u201d , Ukraine.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1897[invalid]Pannonian, zone D\u2013E, late Miocene.\u201cBegaljica\u201d, Serbia.Milan et al. (1974: 93) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 3020-666.Melanopsisinermis Handmann, 1882. Melanopsismagyari as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Jekelius, 1944[invalid]Early Pannonian, late Miocene.\u201cTurislav-Tal bei Soceni\u201d [Turislav valley near Soceni], Romania.Melanopsisinermis Handmann, 1882 \u201d , Greece.Geological-Paleontological Institute, University of Kiel, Germany; no number indicated.inexpectata\u201d as given by The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020F\u00e9russac, 1822Eocene; late Villafranchian, early Pleistocene.\u201cDes environs d\u2019Epernay. [...] Du d\u00e9p\u00f4t situ\u00e9 entre St.-Germini et Carsoli\u201d .antiquua\u201d, which was probably not intended as species-group name .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeMartens, 1874\u201cQuellen des Chabur bei Ras-el-Ain\u201d [source of Chabur river near Ra\u2019s al \u2018Ayn], Syria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928\u201cA\u00efn Chekef, pr\u00e8s F\u00e8s\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeWesterlund, 1886\u201cBei Gross-Wardein in der schnellen K\u00f6r\u00f6s\u201d [near Oradea in the Cri\u0219ul Repede], Romania.Melanopsisparreyssii figured in Microcolpiaparreyssii .Based on a specimen of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1936[invalid]Not explicitly stated but probably the same as for the species .Melanopsisparreyssiiinnodata Westerlund, 1886 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePageBreakPallary, 1939 Not explicitly stated but probably the same as for the species .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeLocard, 1883\u201cSamava\u201d [As Samawah], Iraq.MelanopsisTurcica de Mousson [...] Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Rzehak, 1883Oncophora Beds, middle Burdigalian, early Miocene.\u201cEibenschitz, [...] Oslawan\u201d , Czech Republic.Melanopsisimpressa Krauss, 1852.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887[invalid]Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Melanopsisintermedia Rzehak, 1883.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]\u201c\u00c0 B\u00e9ni Mellal\u201d [at Beni Mellal], Morocco.Melanopsisintermedia Rzehak, 1883.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a, Ziegelei c\u201d, Austria.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cTappah, \u00e0 3 km Est de Belad Sindjar et \u2018Ain Haglan\u201d , Iraq.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cM. Mario\u201d : 184, ItMelanopsisnodosa sensu Cerulli-Irelli, 1914, non F\u00e9russac, 1822. Melanopsisnodosa sensu Cerulli-Irelli, 1914 a junior synonym of \u201cMelanopsisaffinis F\u00e9russac\u201d, which is not an available name.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidaeMousson, 1861\u201cLac de Tiberias\u201d [Sea of Galilee], Israel.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882[invalid]Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsisjordanicairregularis Mousson, 1861 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeAhuir Galindo, 2016\u201cln a Za river affluent, between Guefait and Hassi Blal\u201d, Morocco.Museo Malacologico di Cupra Marittima, Italy; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884PageBreakdell\u2019Accesa near Massa Marittima, in Tuscany ; in the surroundings of Oran (Algeria); in the cave of Nahr el-Kalb near Beirut (Lebanon)].\u201cDans le lac d\u2019Accesa, pr\u00e8s de Massa, en Toscane ; aux environs d\u2019Oran; [...] dans la vall\u00e9e du Nahr-el-Kelb, pr\u00e8s de Beyrouth\u201d , Germany.Melanopsidae.Probably not a Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Stache in Sandberger, 1871Liburnian, Danian, Paleocene.\u201cZablachie bei Sebenico in Dalmatien\u201d [Zabla\u0107e near \u0160ibenik], Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans quelques sources de la plaine de J\u00e9richo (Syrie)\u201d [in a few springs in the plains of Jericho], Palestine.Bourguignat denoted the authority as \u201cLetourneux, 1882\u201d, but there is no evidence that the description really derived from that author.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Gillet & Marinescu, 1971[invalid]Transdanubian, Pannonian, late Miocene.\u201cRadmanest\u201d (Fuchs 1870: 353) [R\u0103dm\u0103ne\u0219ti], Romania.Melanopsisfuchsi Brusina, 1902, non Handmann, 1882 and Melanopsishungarica Pallary, 1916, non Kormos, 1904, which were in turn introduced for Melaniacostata sensu Fuchs, 1870, non Olivier, 1804. However, already Melanopsisconfusa as replacement name, which makes Melanopsisjekeliusi its junior objective synonym.Replacement name for the junior homonyms Taxon classificationAnimaliaSorbeoconchaMelanopsidaeRoth, 1839\u201cIn flumine Jordano; in mari Galilaeo\u201d , Israel.judaica\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeGermain, 1921[invalid]Melanopsisjordanica Roth, 1839.Unjustified emendation and therefore junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Middle Pannonian, late Miocene.\u201cMarku\u0161evec\u201d, Croatia.The syntypes are stored in the Croatian Natural History Museum, Zagreb; no number indicated .Melanopsisbouei F\u00e9russac, 1823.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pavlovi\u0107, 1927Middle Pannonian, late Miocene.PageBreak\u201c\u041a\u0430\u0440\u0430\u0433\u0430\u0447\u201d [from Karaga\u010da near Vr\u010din], Serbia.The illustrated syntype is stored in the Natural History Museum, Belgrade, coll. no. 226 (Milo\u0161evi\u0107 1962: 23).Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Mertin (1939: 254)[unavailable]Heidelberg Formation, late Santonian, late Cretaceous.\u201cFlugplatz Quedlinburg\u201d [airfield at Quedlinburg], Germany.Odostomiakaltenbachi Mertin, 1939.Nomen nudum. Obviously a lapsus calami of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pavlovi\u0107, 1903Middle Miocene.\u201c\u0418\u0437 \u0411\u0430\u0431\u0438\u043d\u043e\u0433 \u0414\u043e\u043b\u0430 \u0431\u043b\u0438\u0437\u0443 \u0421\u043a\u043e\u043f\u0459\u0430\u201d (p. 155) [Babin Dol near Skopje], Macedonia.The illustrated syntype is stored in the Natural History Museum, Belgrade, coll. no. 1448 (Milo\u0161evi\u0107 1962: 24).Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Gozhik in Gozhik & Datsenko, 2007Pliocene.\u201cO\u0437. \u0421\u0430\u0441\u044b\u043a\u201d [Lake Sasyk], Ukraine.Institute of Geological Sciences of the National Academy of Sciences of Ukraine, coll. no. 4602.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1904Langhian, middle Miocene.\u201cVarcar-Vakufa\u201d [Mrkonji\u0107 Grad], Bosnia and Herzegovina.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020K\u00fchn, 1946PageBreakLate middle Eocene\u2013early Oligocene.\u201cMonte Promina\u201d [Promina Mountains], Croatia.Geological-Paleontological Department, Natural History Museum Vienna, Austria, coll. no. 1869/0009/0013.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cSources de Khabour, dites Ras el \u2018Ain\u201d [source of Chabur river near Ra\u2019s al \u2018Ayn], Syria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cZelebor\u201d mentioned in Brot (1874\u20131879: 429)[unavailable]Not indicated.Melanopsissaulcyi by Nomen nudum, apparently based on an unpublished manuscript name from Zelebor listed in the synonymy list of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1897Langhian, middle Miocene.\u201cMio\u010di\u0107\u201d, Croatia.Milan et al. (1974: 93) indicated a holotype, but it is uncertain whether the specimen is part of the original type series and whether it was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2970-616.Melanopsislyrata Neumayr, 1869.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Melanopsisbouei F\u00e9russac, 1823.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Bourguignat, 1880Langhian, middle Miocene.\u201cVall\u00e9e de la Cettina\u201d [Cetina river valley], Croatia.The taxon is not included in the Fossilium Catalogus of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Kurr, 1856Middle\u2013late Burdigalian, early Miocene.\u201cZwiefalten und Andelfingen\u201d : 159, GeMelanopsispraerosa [= Melanopsispraemorsa] sensu Dunker, 1848 and Klein, 1852, non Linnaeus, 1758. The name \u201ckleini\u201d as mentioned in Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1893Pannonian, zone D\u2013E, late Miocene.\u201cBegaljica\u201d, Serbia.The syntypes are stored in the Croatian Natural History Museum, Zagreb; no number indicated .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeVon dem Busch in Philippi, 1847\u201cPersepolis Persiae\u201d, Iran.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsishaueri Handmann, 1882.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Bogachev [unavailable]Miocene.\u201cTori bei Borshomi\u201d [Tori near Borjomi], Georgia.The name was only mentioned in a species list by Bogachev without description or illustration.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1892Middle Pannonian, late Miocene.\u201cMarku\u0161evec\u201d, Croatia.Milan et al. (1974: 93) indicated collection numbers for \u201csyntypes\u201d illustrated in Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Cernikian, Pliocene.\u201cHrastina\u201d, Croatia.Milan et al. (1974: 58) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2481-127.Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans le canal de sortie des eaux thermales de Krapina-Toeplitz, en Croatie\u201d [in the outlet channel of the thermal waters at Krapinske toplice], Croatia.Bourguignat denoted the authority as \u201cLetourneux, 1879\u201d, but there is no evidence that the description really derived from that author.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fuchs, 1870PageBreakMiddle Pannonian, late Miocene.\u201cK\u00fap\u201d, Hungary.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Portaferrian (Pannonian Basin), late Miocene\u2013Pliocene.\u201cKurd\u201d, Hungary.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2535-181/1-4 .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1907Late Burdigalian\u2013early Langhian, early\u2013middle Miocene.Not indicated by Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Cossmann, 1888Thanetian, Paleocene.\u201cChenay [...]; Jonchery [...]; Ch\u00e2lons-sur-Vesle\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeLamarck, 1822\u201cDans les rivi\u00e8res des \u00eeles de l\u2019Archipel\u201d [it is unknown which Archipel Lamarck referred to].The species first appeared without name on a plate in the \u201cTableau encyclop\u00e9dique et m\u00e9thodique, vol. 3\u201d by PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeRossm\u00e4ssler, 1839\u201cGradaschza und Ringelsza [bei Laibach]; aus einem M\u00fchlbache bei Nassenfuss in Unterkrain\u201d  , Slovenia.Melaniaholandri in synonymy of this variety, but from the text it is clear that the list is intended to refer to the species as a whole.Note that Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020\u0141omnicki, 1886[invalid]Late Miocene.\u201cWycz\u00f3\u0142ki (wrzynka kol. trans. na zachodnio-po\u0142udniowym ko\u0144cu wsi)\u201d , Ukraine.Melanopsislaevigata Lamarck, 1822. Melanopsislomnickii as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1912[invalid]\u201cNefta\u201d, Tunisia.Melanopsislaevigata Lamarck, 1822.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1924[invalid]\u201cEaux de Mathen, Alhama de Aragon\u201d , Spain.Melanopsislaevigata Lamarck, 1822. The taxon appears at infrasubspecific rank in the plate captions of the original publication.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Stoliczka, 1860Late Turonian, late Cretaceous.\u201cNeualpe im Russbachthal\u201d [Neualm near Russbach am Pass Gsch\u00fctt], Austria.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]\u201cRuisseau de l\u2019\u00eele d\u2019Ivice, aux Bal\u00e9ares\u201d [stream on the island of Ibiza], Spain.Melanopsislaevis Stoliczka, 1860.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920[invalid]\u201cOued Masmouda; Ruisseau de Bab Hadid\u201d , Morocco.Melanopsislaevis Stoliczka, 1860.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]\u201cA\u00efn M\u00e9lias, pr\u00e8s de Figuig\u201d [Ain Melias near Figuig], Algeria.Melanopsislaevis Stoliczka, 1860.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePotiez & Michaud, 1838\u201cLes rivi\u00e8res de Madagascar?\u201d [rivers of Madagascar?].Madagasikaraspinosa  (Pachychilidae).Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Deshayes, 1862[invalid]Calcaire de Montabuzard, middle\u2013late Burdigalian.\u201cDamery\u201d, France.Melanopsislamarckii Potiez & Michaud, 1838. Faunus Montfort, 1810 (Pachychilidae). The name \u201clamarcki\u201d as mentioned in Wenz is an incorrect subsequent spelling.Junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeSouverbie, 1872Souverbie 1872: 148.\u201cBaye du Sud\u201d [Baie Sud], New Caledonia.Melanopsismariei Crosse, 1869.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cEnvirons de Constantinople\u201d [surroundings of Istanbul], Turkey.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cBelus, pr\u00e8s de Saint-Jean-d\u2019Acre, en Syrie\u201d , Israel.Melanopsiscostata . Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr in Neumayr & Paul, 1875Cernikian, Pliocene.\u201cSlobodnica; [...] Sibin [Sibinj]; [...] Malino; [...] Groma\u010dnik; [...] \u010capla [\u010caplja trench near Slavonski Brod]; [...] Cigelnik\u201d, Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pavlovi\u0107, 1927Middle Pannonian, late Miocene.\u201c\u0418\u0437 \u043a\u0430\u0440\u0430\u0433\u0430\u0447\u043a\u0438\u0445 \u043f\u0435\u0441\u043a\u043e\u0432\u0430\u201d [from the sands of Karaga\u010da near Vr\u010din], Serbia.The illustrated syntype is stored in the Natural History Museum, Belgrade, coll. no. 211 (Milo\u0161evi\u0107 1962: 23).PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Langhian, middle Miocene.\u201cRibari\u0107; Turjake [Turjaci]\u201d, Croatia.Milan et al. (1974: 94) indicated a holotype, but it is uncertain whether the specimen is part of the original type series and whether it was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2982-628.lanzae\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeLetourneux & Bourguignat, 1887\u201cDes environs de Nefta; [...] dans l\u2019Oued Gab\u00e8s et aux alentours de Tozer\u201d , Tunisia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Carez, 1879Early\u2013middle Eocene.\u201cLa Maladrerie (Brasles); Gland (Aisne)\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Lueger, 1980Pannonian, zone D, late Miocene.\u201cF\u00f6llig d1 [...] und d2\u201d [F\u00f6llig hill near Gro\u00dfh\u00f6flein], Austria.Geological-Paleontological Department, Natural History Museum Vienna, Austria. The number Lueger provided is not the collection number but the acqusition number, which could be used to trace back the object in our database.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1918\u201cTaza, Ain en nsa, source est, chaude\u201d , Morocco.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeRossm\u00e4ssler, 1839\u201cIn der Laibach; in der Save\u201d [Ljubljanica and Sava rivers] , Slovenia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in Brot (1870: 310) and Brot (1874: 370)[unavailable]\u201cLemberg, Galizien\u201d [Lviv], Ukraine.Hemisinus [now Esperiana] acicularis in Nomen nudum, apparently based on an unused manuscript name. It appears only in the synonymy list of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeReeve, 1860Melanopsis, pl. 3, figs 9a\u2013b.\u201cNew Caledonia\u201d (France) [no locality indicated].Melanopsisfrustulum Morelet, 1857.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Melanopsisfossilis .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1897Congeriarhomboidea Zone, Portaferrian, late Miocene.\u201cLepavina\u201d, Croatia.Milan et al. (1974: 94) indicated a holotype, but it is uncertain whether the specimen is part of the original type series and whether it was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 3003-649.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRivi\u00e8re entre Plaski et Ostaria, et dans la Save entre Agram et Sissek\u201d , Croatia.Fagotia [= Esperiana] acroxia Bourguignat, 1884Note that Bourguignat denoted the authority as \u201cBourguignat, 1879\u201d. Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans la rivi\u00e8re d\u2019Ogulin, en Croatie; [...] dans la Migliaska, pr\u00e8s de S\u00e9rajewo\u201d .Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans la source et la rivi\u00e8re de la Moulouiah, \u00e0 l\u2019ouest de Lalla-Maghnia\u201d , Morocco or Algeria.Bourguignat denoted the authority as \u201cBourguignat, 1872, et Letourneux\u201d, apparently referring to Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLa Save, au-dessous d\u2019Agram\u201d [Sava river below Zagreb], Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeMoricand, 1841PageBreak\u201cRio de Pedra Branca, procince de Bahia\u201d , Brazil.Verena by Adams and Adams (1854) (Thiaridae), of which Melanopsiscrenocarinata is the type species .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d : 559, AuMelanopsisscalaris Handmann, 1882, non Gassies, 1856. Melanopsishaueri Handmann, 1882.Replacement name for PageBreakPageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cDjishr ech Chegour\u201d [Jisr Ash-Shughur], Syria.unicincta\u201d in the plate captions. Melanopsismultiformis Blanckenhorn, 1897.Pallary erroneously gave the name as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Anistratenko, 1993Duab Beds, middle to late Kimmerian, Pliocene.\u201c\u041e\u043a\u0440. \u0441. \u041c\u043e\u043a\u0432\u0438, \u041e\u0447\u0430\u043c\u0447\u0438\u0440\u0441\u043a\u0438\u0439 \u0440-\u043d\u201d , Georgia.Schmalhausen Institute of Zoology of National Academy of Sciences of Ukraine, Kiev; no number indicated.Pseudofagotia Anistratenko, 1993.Type species of the genus Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1857\u201cLa rivi\u00e8re Balade\u201d [in the river Balade], New Caledonia.Melanialineolata Gray in Griffith & Pidgeon, 1833 as a Melanopsis, which resulted in secondary homonymy for his species Melanopsislineoloata Gassies, 1857. Thus, he introduced Melanopsislivida as a replacement name. Secondary homonymy, however, is not given anymore, since Melanialineolata Gray was recently shown to belong to the genus Cerithidea Swainson, 1840 (Potamididae) by Melanialineolata Gray is a junior homonym of Melanialineolata Wood, 1828). Melanopsislivida is a junior objective synonym of Melanopsislineolata Gassies . Melanopsisfrustulum Morelet, 1857.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1936\u201cGuefa\u00eft \u201d, Morocco.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1869\u201cPrope Noumea\u201d [near Noum\u00e9a], New Caledonia.Melanopsisfrustulum Morelet, 1857.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1861[invalid]\u201cLe Diahot \u00e0 Balade\u201d [in the Diahot river at Balade], New Caledonia.Melanopsislineolata Gassies, 1856, non Melanialineolata Gray in Griffith & Pidgeon, 1833. Secondary homonymy, however, is not given anymore, since Melanialineolata Gray was recently shown to belong to the genus Cerithidea Swainson, 1840 (Potamididae) by Melanialineolata Gray is a junior homonym of Melanialineolata Wood, 1828). Melanopsislivida is a junior objective synonym of Melanopsislineolata Gassies. Melanopsisfrustulum Morelet, 1857.Replacement name for the presumed secondary homonym Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBlanckenhorn, 1897Pleistocene (?)\u2013Recent.\u201cLebend im See von Antiochia. Halbfossil bei Selemije\u201d .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans le lac Sabandja, pr\u00e8s d\u2019Ismidt (Anatolie)\u201d [Lake Sapanca near \u0130zmit], Turkey.Note that Bourguignat denoted the authority as \u201cBourguignat, 1882\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Wenz, 1928Badenian, middle Miocene.PageBreak\u201cWycz\u00f3\u0142ki (wrzynka kol. trans. na zachodnio-po\u0142udniowym ko\u0144cu wsi)\u201d : 76 , Greece.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Gozhik in Gozhik & Datsenko, 2007Late Pleistocene.\u201cA\u043b\u043b\u044e\u0432\u0438\u044f V \u0442\u0435\u0440\u0440\u0430\u0441\u044b \u0440. \u0414\u0443\u043d\u0430\u0439 \u0443 \u0441. \u041d\u0430\u0433\u043e\u0440\u043d\u043e\u0435\u201d [Alluvial terrace V of the Danube river near Nagornoye], Ukraine.Institute of Geological Sciences of the National Academy of Sciences of Ukraine, coll. no. 6461.longus\u201d), but Microcolpia is feminine, which is why the name must be \u201clonga\u201d.Originally the gender was indicated as masculine , late Miocene.\u201cBabadjan\u201d \u201d, Greece.Museum of Palaeontology and Geology of the University of Athens; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sacco, 1895Early Messinian, late Miocene.\u201cS. Marzano Oliveto\u201d, Italy.Melanopsisfusulatina Sacco, 1895.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGuirao, 1854\u201cIn rivulo Rambla de Viznaga et in Pantano de Puentes non procul Lorca in Regno Murcico\u201d , Spain.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Andrusov, 1909Pontian (sensu stricto), late Miocene.\u201cBabadjan\u201d [Babadzhan], Azerbaijan.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Locard, 1893Middle\u2013late Burdigalian, early Miocene.\u201cVernier, pr\u00e8s Gen\u00e8ve\u201d, Switzerland.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeLocard, 1883\u201cLac d\u2019Antioche\u201d [Lake Anuk ], Turkey.Lorteti\u201d in Locard\u2019s remarks on p. 272, which is apparently based on a typesetting error.Appears as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1893Pannonian, zone D\u2013E, late Miocene.\u201cRipanj\u201d, Serbia.Milan et al. (1974: 94) indicated a holotype, but it is uncertain whether the specimen is part of the original type series and whether it was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 3639-1279/1.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neubauer in Neubauer et al., 2011Early Langhian, middle Miocene.\u201cSinj, Lu\u010dane [= Sutina] section\u201d, Croatia.Geological-Paleontological Department, Natural History Museum Vienna, Austria, coll. no. 2010/0042/0001.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 ?Cossmann, 1886Eocene.\u201cHoudan\u201d, France.Nozeba Iredale, 1915 (Iravadiidae) by This taxon was considered to belong to the marine genus PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in Erber (1868: 904)[unavailable]\u201cRhodus\u201d , Greece.Luciae\u201d in Nomen nudum. Erber attributed the authority to Mousson. Given as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cLac de Homs\u201d, Syria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Speyer, 1870Early Rupelian, Oligocene.\u201cGrossalmerode\u201d : 71, GerMelaniacostata Ludwig, 1865, non Olivier, 1804.Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020d\u2019Archiac in Viquesnel, 1846Pannonian, late Miocene.\u201cEntre Koulana et Lus-han\u201d , Albania.Lus-Hani\u201d; the name \u201cLushami\u201d as mentioned in Melanopsisfossilis .Originally the name was introduced as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201c\u2018Ain Haglan\u201d [not found], Iraq.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeStarobogatov & Izzatullaev, 1985\u201c\u041f\u0443\u0441\u0442\u044b\u043d\u0435 \u0414\u0435\u0448\u0442\u0435-\u041b\u0443\u0442\u201d [Dasht-e Loot desert], Iran.Zoological Institute of Russian Academy of Sciences, St.-Petersburg; no number indicated.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeGermain, 1921\u201cLac d\u2019Homs\u201d, Syria.Melanopsissaulcyi Bourguignat, 1853.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Matheron, 1842Early Campanian, Cretaceous.\u201cLes Martigues\u201d, France.Paludomus Swainson, 1840 .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr, 1869[invalid]Langhian, middle Miocene.\u201cRibaric\u201d [Ribari\u0107], Croatia.Illustrated syntype is stored at the Geological Survey Austria, Vienna, coll. no. 1869/01/6.Melanopsislyrata Neumayr, 1869 [June] is deemed to be a junior homonym of Melanopsislirata Gassies, 1869 [January] after Art. 58.2. Melanopsisdissimilis as replacement name. Melanopsiscylindracea Brusina, 1874.Unlike stated by Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLa rivi\u00e8re entre Plaski et Ostaria (Croatie)\u201d [river between Pla\u0161ki and O\u0161tarije], Croatia.Microcolpiaacicularis .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Burgerstein, 1877Late Miocene.PageBreak\u201cUeskueb\u201d [Skopje], Macedonia ], Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cZiegler\u201d mentioned in Brot (1874\u20131879: 13)[unavailable]\u201cIn der Muhr\u201d [in the river Mur], Austria.MelaniaHolandri\u201d [sic] by Nomen nudum, apparently based on an unused manuscript name from Parreyss listed in synonymy of \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Papp, 1953Gelasian, early Pleistocene.\u201cPyrgos\u201d, Greece.Museum of Palaeontology and Geology of the University of Athens; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLe Guadalquivir aux environs de S\u00e9ville\u201d , Spain.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeLea, 1837\u201cPeru\u201d [no locality indicated].Hemisinusosculati  (Thiaridae) after Junior synonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916[invalid]Burdigalian, early Miocene (?).\u201cDax\u201d, France.magna\u201d as available name attributed to Melanopsisdufourii F\u00e9russac, 1822 , but Melanopsis is feminine, which is why the name must be \u201cmagna\u201d. Junior homonym of Melanopsismagna Pallary, 1916.Originally the gender was indicated as masculine . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 3020-666.Milan et al. (1974: 93) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand when describing Melanopsisklericiinermis Brusina, 1897, non Handmann, 1882 (see Note 1).Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201cF\u00e9russac, 1823\u201d mentioned in Wenz (1929: 2773)[unavailable]Burdigalian, early Miocene.\u201cDe Mandillot, pr\u00e8s de Dax\u201d , France.Melanopsisdufourii var. \u03b5 and listed \u201cFossilis, major. F\u00e9russac\u201d in synonymy referring to plate 1 of the \u201cM\u00e9lanopsides fossiles\u201d of his \u201cHistoire naturelle\u201d . Fossilis, maxima\u201d. Obviously neither major nor maxima was intended as species-group name by F\u00e9russac. Later, the name would have been a junior homonym of Melaniabuccinoideamajor Grateloup, 1838.This name has appeared several times in the literature e.g., : 2773, bTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Grateloup, 1838Burdigalian, early Miocene.\u201cDax. [...] Mandillot; \u00e0 Saint-Paul\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeRossm\u00e4ssler, 1839[invalid]\u201cIn der Lachina bei Tschernembl, [...] in dem Bug\u201d [in the Lahinja river near \u010crnomelj (Slovenia) and in the Bug river (Ukraine)].Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1864[invalid]PageBreak\u201cMostaghanem\u201d [Mostaganem], Algeria.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]\u201cDans les aqueducs de S\u00e9ville et dans le Guadalquivir\u201d [in the aqueducts of Sevilla and in the Guadalquivir river], Spain.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]\u201cLe Guadalquivir aux environs de S\u00e9ville\u201d , Spain.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]\u201cEnvirons de Fez (Maroc)\u201d [surroundings of Fes], Morocco.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1899[invalid]\u201cEnvirons de T\u00e9touan\u201d [surrondings of T\u00e9touan], Morocco.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1899[invalid]PageBreak\u201cL\u2019O.[ued] Ida ou Guert, pr\u00e8s de Mogador (p.); O. A\u00eft Ouadel (p. 163)\u201d  , Morocco.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1901[invalid]Pleistocene.\u201cDe l\u2019oued El Biod (G\u00e9ryville)\u201d [Oude\u00ef el Biod near El Bayadh], Algeria.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1911[invalid]\u201cTout pr\u00e8s d\u2019Oudjda, \u00e0 4 kilom. S.-E., sourdent les belles sources de Sidi-Yahia qui alimentent une v\u00e9ritable oasis, puis la ville d\u2019Oudjda, et vont finalement se d\u00e9verser dans l\u2019oued Isly\u201d , Morocco.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1912[invalid]\u201cA Tozeur, sur les bords du Chott [Djerid?]\u201d [banks of the Chott el Dj\u00e9rid at Tozeur], Tunisia.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920[invalid]\u201cLa Makina\u201d, Morocco.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920[invalid]\u201cA\u00efn Allou\u201d, Morocco.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1924[invalid]\u201cEaux de Mathen, Alhama de Aragon\u201d , Spain.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1924[invalid]\u201cGuadalquivir\u201d, Spain.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Magrograssi, 1928[invalid]Plio-Pleistocene.\u201cCoo: molto frequente in tutte e due le zone fossilifere\u201d , Greece.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]Not explicitly stated but probably the same as for the species .Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]Not explicitly stated but probably the same as for the species \u201d, Morocco).Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]Not stated but probably the same or partly as for the species .Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1936[invalid]Not indicated, but probably in Morocco.Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[invalid]\u201cLac d\u2019Antioche, lac de Homs, Nahr el K\u00e9bir, Markieh, Yeni Chehir\u201d .Melanopsisbuccinoideamajor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Jodot, 1958[unavailable]Middle Miocene.\u201cRoute de Caravaca (Murcie)\u201d , Spain.Introduced as \u201cmut. nov.\u201d which is not ruled by the provisions of the Code.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Hermite [unavailable]Early Eocene.\u201cDe Binisalem et de Selva\u201d , Spain.Nomen nudum, listed by Hermite in a section called \u201cEsp\u00e8ces nouvelles cit\u00e9es et non d\u00e9crites\u201d [= \u201cnew species identified and not described\u201d], where he listed 18 new names that he intended to describe in the second volume of his \u201c\u00c9tudes g\u00e9ologiques sur les \u00eeles Bal\u00e9ares\u201d. That part, however, has never been published, probably because Hermite died in 1880.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Cossmann, 1906Ypresian, Eocene.\u201cPerauba\u201d [section Peralba near \u00c0ger], Spain.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Noulet, 1854Noulet 1854: 50.Ludian?\u2013Sannoisian, Priabonian\u2013early Rupelian, late Eocene\u2013early Oligocene.\u201cAu Mas-Saintes-Puelles (Aude); [...] de la Massale, pr\u00e8s de Castres; [...] \u00e0 Labrugui\u00e8re; [...] \u00e0 Saint-Genest-de-Contest, \u00e0 Lautrec (Tarn)\u201d , France.masensis\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Bourguignat, 1862Pleistocene.\u201cDans la daya de Habessa, ancien lac dess\u00e9ch\u00e9, situ\u00e9 \u00e0 plus de 200 lieues environ au sud d\u2019Oran\u201d , Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Robles, 1975Mammal zone MN 13\u201315, late Miocene\u2013Pliocene.PageBreak\u201cFuente del Viso (Albacete)\u201d [near Villatoya], Spain.Museo Nacional de Ciencias Naturales, Madrid, coll. no. M-435.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Braun in Walchner, 1851Braun in Walchner 1851: 1127.Early Miocene.\u201cMainzer Becken\u201d , Germany.Melanopsisfritzei Thom\u00e4, 1845.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeCrosse, 1869\u201cIn loco \u2018Baie du Sud\u2019\u201d [Baie Sud], New Caledonia.Re-described in French by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Wenz, 1930Middle Pannonian, late Miocene.\u201cMarku\u0161evec\u201d .Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidaeMorelet, 1853\u201cProv. Oranensem\u201d , Morocco.Buccina Maroccana\u201d. However, according to Opinion 184 (ICZN 1944), species names introduced in volumes 1\u201311 of Martini and Chemnitz\u2019 \u201cNeues Systematischer Conchylien Cabinet\u201d (1769\u20131795) have no status in nomenclature and the name is therefore not available from this work. The name was again listed by Melanopsisdufourii F\u00e9russac, 1822  Morelet (as well as Bourguignat) referred to Chemnitz\u2019 work \u2013 but only to figs 2080\u20132081 \u2013 and listed Melanopsisdufourei [sic] as a synonym. Morelet did not provide a description or illustrations on his own; the brief Latin description he gave below the synonymy list refers to the unnamed variety \u03b2. The reference to the illustrations in Chemnitz nonetheless suffices as indication of a new taxon; the correct name is therefore Melanopsismaroccana Morelet, 1853. Although Melanopsisdufourii as a junior synonym of Melanopsismaroccana, the two names are no objective synonyms because Morelet referred to more specimens than just the holotype of Melanopsisdufourii (fig. 16). Note, moreover, that Morelet\u2019s synonymization is likely based on a wrong concept of Melanopsisdufourii, which is a fossil species described from the Miocene of France.The first to adopt marocana\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cPr\u00e8s de Taforalt; Oued Cher\u00e2a \u00e0 Berkane\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Matheron, 1842Early Campanian, Cretaceous.\u201cLes Martigues\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020F\u00e9russac, 1823Pannonian, late Miocene.\u201cDans les environs de Bisentz et de Scharditz, en Moravie, dans la vall\u00e9e de la Marsch, affluent du Danube\u201d , Czech Republic.Melanopsisfossilis (for details see there). The name \u201cmartinii\u201d as mentioned by numerous authors , Italy.Melanopsisnarzolina d\u2019Archiac in Viquesnel, 1846.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1911\u201cBerguent, le Ras al A\u00efqun des Beni-Mattar\u201d [Ras el A\u00efn at A\u00efn Beni Mathar], Morocco.Melanopsisletourneuxi sensu Pallary, 1899 , non Bourguignat, 1884.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1922[invalid]Melanopsismauritanica Bourguignat, 1884.Unjustified emendation and therefore junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201c\u00c7\u00e0 et l\u00e0 de Maroc\u201d .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Cossmann, 1888Sparnacian, early Ypresian, Eocene.\u201cMont Bernon\u201d [near \u00c9pernay], France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeDautzenberg, 1894\u201cBir Jalo\u00fbd\u201d .PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeSch\u00fctt & Bilgin, 1974\u201cSakarya ba\u015fi, main spring of Sakarya river near village \u00c7ifteler, 60 km SE Eski\u015fehir, 160 km WSW of Ankara\u201d, Turkey.Senckenberg Forschungsinstitut und Naturmuseum Frankfurt, coll. no. SMF 232011.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBlanckenhorn, 1897\u201cDans les eaux le l\u2019ancien L\u00e9onthes\u201d : 51 \u201cDe la petite rivi\u00e8re de Guadaira qui se jette dans le Guadalquivir\u201d [in the little river Guadaira which flows into the Guadalquivir], Spain.Melanopsismaroccanamedia Blanckenhorn, 1897 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neubauer, Mandic, Harzhauser & Hrvatovi\u0107, 2013Langhian, middle Miocene.\u201c\u017depj\u201d : 480 Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Melanopsissikorai: Microcolpiaparreyssiisikorai .Nomen nudum. If available, it would be a junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeHeller & Sivan, 2000\u201cEn Haruv [...], a small spring on the Golan Heights that pours into a small cement pool\u201d [near Kefar Haruv], Syria.National mollusc collection of the Hebrew University, Jerusalem, coll. no. HUJ 7966.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeMoricand, 1841\u201cRio de Pedra Branca, procince de Bahia\u201d , Brazil.Verena by Adams and Adams (1854) (Thiaridae), of which Melanopsiscrenocarinata is the type species ], Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201c\u2018Ain Arouss (la source de la fianc\u00e9e), pr\u00e8s de Tell Abiad, d\u2019o\u00f9 na\u00eet le Nahr B\u00e2hlik, qui se jette dans Euphrate, un peu au-dessous de Rakka, la m\u00e9tropole de Haroun el Rachid\u201d , Syria.Melanopsisinfracincta Martens, 1874.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pavlovi\u0107, 1932Pontian (Dacian Basin), late Miocene\u2013Pliocene.\u201c\u0421\u0435\u043b\u0430 \u0414\u0440\u0441\u043d\u0438\u043a\u0430\u201d [village Drsnik], Kosovo.The illustrated syntype is stored in the Natural History Museum, Belgrade, coll. no. 1196 (Milo\u0161evi\u0107 1962: 24).Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cM\u00e9z\u00e9rib\u201d [Muzayrib], Syria.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Anistratenko, 1993Duab Beds, middle to late Kimmerian, Pliocene.\u201c\u041e\u043a\u0440. \u0441. \u041c\u043e\u043a\u0432\u0438, \u041e\u0447\u0430\u043c\u0447\u0438\u0440\u0441\u043a\u0438\u0439 \u0440-\u043d\u201d , Georgia.Schmalhausen Institute of Zoology of National Academy of Sciences of Ukraine, Kiev; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Late Miocene.\u201c\u00c0 St. Agata pr\u00e8s de Tortone\u201d  [at Sant\u2019Agata Fossili near Tortona], Italy.Melanopsiscarinata sensu Michelotti, 1847, non Sowerby, 1826. Melanopsisbonellii Manzoni, 1870.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cPr\u00e8s de J\u00e9richo, dans la fontaine de J\u00e9r\u00e9mie \u201d , Palestine.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRuisseau de la source de la Moulouiah, pr\u00e8s de Lalla-Maghnia sur la fronti\u00e8re marocaine (prov. d\u2019Oran)\u201d , Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Newton (1891: 203)[unavailable]Woolwich Beds, early Eocene.Woolwich, United Kingdom.Melanopsismicrostoma Bourguignat, 1884.Nomen nudum. If available, it would be a junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeMousson, 1863\u201cDe l\u2019int\u00e9rieur de la Mingr\u00e9lie [...], puis de R\u00e9duktaleh\u201d , Georgia.Mousson attributed the authority to Bayer, but there is no evidence that the description really derived from that author. Bayer only seems to have collected a part of the material.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Blanckenhorn, 1897Plio-Pleistocene.\u201cIn der tiefsten Thonbank des linken Orontesufers bei Dschisr esch-Schurr\u201d [in the lowest clay bank at the left riverside of the Orontes near Jisr Ash-Shughur], Syria.Introduced as \u201cn. mut.\u201d but clearly as a binomen and hence not infrasubspecific in the sense of ICZN Art. 45.6.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Grateloup, 1838Burdigalian, early Miocene.\u201cDax. [...] Mandillot; \u00e0 Saint-Paul\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeRossm\u00e4ssler, 1839[invalid]\u201cIn schwach schwefeligen Quellen [...] bei V\u00f6slau unweit Baden\u201d [in weakly sulfurous springs stones and sands at V\u00f6slau near Baden], Austria.Melanopsisdaudebartii [Prevost], 1821, which Rossm\u00e4ssler listed in synonymy. Moreover, the name is a junior homonym of Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeMartens, 1874[invalid]\u201cQuellen des Chabur bei Ras-el-Ain\u201d [source of Chabur river near Ra\u2019s al \u2018Ayn], Syria.PageBreakMelanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Locard, 1878[invalid]Tortonian, late Miocene.\u201c\u00c0 Tersannes pr\u00e8s de Hauterives (Dr\u00f4me)\u201d [at Tersanne near Hauterives], France.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1). Melanopsisbonellii Manzoni, 1870.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]\u201cRuisseau d\u2019eau chaude \u00e0 Ouargla (prov. de Constantine) et eaux thermales du Dj\u00e9rid, au nord du chott Tiraoun (sud de la Tunisie)\u201d .Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]\u201cLac d\u2019Antioche\u201d [Lake Anuk ], Turkey.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1). Melanopsisblanckenhorni as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]\u201cEnvirons d\u2019Alep, \u00e0 Sadjour-Sou, \u00e0 quatre kilom. en aval d\u2019A\u00efn-Ta\u00efb; ruisseaux \u00e0 Doumar, sur l\u2019oued Baradah, pr\u00e8s A\u00efn-Fidji, et \u00e0 Banias, en Syrie\u201d .Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Not indicated, but probably as for the species .Microcolpiacornea .Melania (Amphimelania) holandrif.minor Westerlund, 1886 [invalid]Not indicated.Melaniatuberculataminor Brot, 1877 from Sri Lanka (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1911[invalid]\u201cTout pr\u00e8s d\u2019Oudjda, \u00e0 4 kilom. S.-E., sourdent les belles sources de Sidi-Yahia qui alimentent une v\u00e9ritable oasis, puis la ville d\u2019Oudjda, et vont finalement se d\u00e9verser dans l\u2019oued Isly\u201d , Morocco.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1911[invalid]\u201cBerguent, le Ras al A\u00efqun des Beni-Mattar\u201d [Ras el A\u00efn at A\u00efn Beni Mathar], Morocco.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1912[invalid]\u201cDes bords du Chott Djerid, \u00e0 Tozeur\u201d [banks of the Chott el Dj\u00e9rid at Tozeur], Tunisia.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920[invalid]\u201cT\u00e9touan\u201d, Morocco.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920[unavailable]Not indicated, but probably as for the variety .Introduced as infrasubspecific taxon, which is not ruled by the provisions of the Code.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920[invalid]\u201cDar Batha\u201d [near Fes], Morocco.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920[invalid]\u201cF\u00e8s: source pr\u00e8s de la ferme proche du pont neuf, \u00e0 dar Mahr\u00e8s; Sidi Harazen (source chaude); Moulai Idriss du Zehroun, source sulfureuse chaude; El Menzel; Tazouta\u201d , Morocco.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1924[invalid]\u201cFortuna et Caravala [Caravaca?] (Murcia). Lorca, Valencia et Alicante\u201d, Spain.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of PageBreakPageBreakPageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1924[invalid]Spain [no locality indicated].Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]\u201cBel Hadi-Kenadsa (Sud Oranais)\u201d , Algeria.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]\u201cO. Taguenout\u201d , Morocco.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]\u201cDans l\u2019oued Da\u00ef\u201d [not found], Morocco.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]Not explicitly stated but probably the same as for the species .Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]\u201cMoula\u00ef Ta\u00efeb\u201d [Moulay Ta\u00efeb], Morocco.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Name appears only in the plate captions. Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]Not stated but probably the same or partly as for the species .Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]\u201cDans l\u2019oued Da\u00ef\u201d [not found], Morocco.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1930[invalid]\u201cDans l\u2019eau ti\u00e8de du bassin de Diane, \u00e0 Smyrne\u201d [in warm water basin of Halkap\u0131nar at Izmir], Turkey.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[invalid]\u201cRas el \u2018Ain du Khabour\u201d [Chabur river near Ra\u2019s al \u2018Ayn], Syria.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[invalid]\u201cDans le canal de la Butte de tir, \u00e0 7 km Sud de Baghdad\u201d , Iraq.Melanopsisnodosa F\u00e9russac in the plate captions. Junior homonym of Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Ranked as a variety of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[invalid]\u201cDans les sources de M\u00e9z\u00e9rib, au N. O. de Der\u00e2a \u201d , Syria.Melanopsisbuccinoideaminor Grateloup, 1838 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Papp, 1979Mammal zone MN 10, late Miocene.\u201cMilessi\u201d [Milesi], Greece.Institute of Paleontology, University of Vienna; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1936Not explicitly stated but probably the same as for the species .strictaminor-\u201d.Originally spelt as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Willmann, 1980Viannos Formation, early Tortonian, late Miocene.\u201cTongrube 1 km westlich von Limin Chersonisou/Kreta\u201d , Greece.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Mammal zone MN 10\u201312, late Miocene.\u201cDe Cuiseaux\u201d, France.Used as valid name by Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cFontaine froide du Hammam \u00e0 Brousse (Anatolie); Nahr-Antalies dans le Liban (Syrie); puits art\u00e9sien de Tamerna-Kedima, dans le Ziban (Alg\u00e9rie)\u201d .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Langhian, middle Miocene.\u201cVrba; Sinj (Stuparu\u0161a); Turiake [Turjaci]\u201d, Croatia.Milan et al. (1974: 94) indicated a holotype, but it is uncertain whether the specimen is part of the original type series and whether it was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2972-618.Melanopsislyrata Neumayr, 1869.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Andrusov, 1909Pontian (sensu stricto), late Miocene.\u201cBabadjan\u201d [Babadzhan], Azerbaijan.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Papp, 1955Late Pliocene to early Pleistocene.\u201cNord-Nordwestlich von der Solfatare bei Susaki\u201d [north-northwest of Solfatara Sous\u00e1ki], Greece.Museum of Palaeontology and Geology of the University of Athens; no number indicated.Amphimelaniagayi [sic] after First mentioned as nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020K\u00fchn, 1963[invalid]Mammal zone MN 10, late Miocene.\u201cPeristeri\u201d, Greece.Museum of Palaeontology and Geology of the University of Athens, coll. no. 1963/83.Melanopsis (Melanosteira) mitzopoulosi Papp, 1955 but clearly described the species as new. Neither did Melanopsislonga Deshayes in F\u00e9russac, 1839. Moreover, Papp stated that K\u00fchn\u2019s species was based on the material collected by The status of this species is unclear. First of all, K\u00fchn did not at all refer to the senior homonym Taxon classificationAnimaliaSorbeoconchaMelanopsidaeMousson, 1874Not explicitly stated but probably the same as for the species .Melanopsisnodosa F\u00e9russac, 1822, which is actually a fossil species described from the Miocene of France.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Bogachev [unavailable]Miocene.\u201cTori bei Borshomi\u201d [Tori near Borjomi], Georgia.PageBreakMelanopsisnodosamoderata Mousson, 1874 (see Note 1).The name was only mentioned in a species list by Bogachev without description or illustration. If available, it would be a junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1911\u201c\u00c0 Colomb-B\u00e9char\u201d [Bechar], Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Bogachev [unavailable]Miocene.\u201cTori bei Borshomi\u201d [Tori near Borjomi], Georgia.The name was only mentioned in a species list by Bogachev without description or illustration.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928\u201cA\u00efn Foum el An\u00e7eur et Tirboula, pr\u00e8s de Ksiba \u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Jekelius, 1944Early Pannonian, late Miocene.\u201cTurislav-Tal bei Soceni\u201d [Turislav valley near Soceni], Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Jekelius, 1944[invalid]Early Pannonian, late Miocene.\u201cTurislav-Tal bei Soceni\u201d [Turislav valley near Soceni], Romania.Melanopsismoesiensis Jekelius, 1944 (same work). Melanopsismoesiensis (since Melanoptychia is considered a synonym of Melanopsis).Junior secondary homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1911\u201cDans les sources et la rivi\u00e8re du jardin du Sultan, \u00e0 Mogador\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cD\u2019Agadir\u201d : 71, MorMelanopsispraerosa [= Melanopsispraemorsa] sensu Morelet, 1880, non Linnaeus, 1758. Note that Bourguignat denoted the authority as \u201cBourguignat, 1881\u201d.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr, 1880Langhian, middle Miocene.\u201c\u017depj\u201d [D\u017eepi], Bosnia and Herzegovina.The type material, with all specimens studied by Neumayr (1880), is lost. Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Vidal, 1917Cretaceous.\u201cLignitos de Selva y Binisal\u00e9m\u201d [lignites of Selva and Binissalem], Spain.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsisbouei F\u00e9russac, 1823.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Dollfus (1922: 118)[unavailable]Apolakkia/Monolithos Formation, Pliocene.Melanopsisbiliottii). \u201cRhodos\u201d  at the road to Marrakech], Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1916[invalid]\u201cDans les eaux de l\u2019ancien L\u00e9onthes\u201d : 51 , Romania.Melanopsissikoraicarinata that, if raised to species, he suggests \u201cMelanopsismucronifera\u201d as name for the taxon. Both names were simultaneously published and are objective PageBreaksynonyms. Since Melanopsiscarinata Kormos is a junior homonym of Melanopsiscarinata Sowerby, 1826, Melanopsismucronifera is the valid name of the taxon. Microcolpiaparreyssiisikorai .Obviously unaware of the fact that variety names are available in nomenclature as species-group names, Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.multicostellata\u201d as mentioned by Melanopsisbouei F\u00e9russac, 1823.The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai  by Considered as a junior synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Schr\u00e9ter, 1975[invalid]Riss/W\u00fcrm end to early W\u00fcrm Ice Age, Pleistocene.\u201cEger, az egri v\u00e1r Z\u00e1rk\u00e1ndy b\u00e1sty\u00e1j\u00e1nak vas\u00fati \u00e1tmetsz\u00e9se\u201d , Hungary.Magyar \u00c1llami F\u00f6ldtani Int\u00e9zet , Budapest; no number indicated.Melanopsistothivar.multifilosa Brusina, 1903. Melanopsisdoboi to the genus Microcolpia.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Blanckenhorn, 1897Plio-Pleistocene.\u201cIn der ersten Thonbank des linken Orontesufers bei Dschisr esch-Schurr\u201d [in the first clay bank at the left riverside of the Orontes near Jisr Ash-Shughur], Syria.Introduced as \u201cn. mut.\u201d  but clearly as a binomen and hence not infrasubspecific in the sense of ICZN Art. 45.6.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Roule, 1884Calcaire de Rognac, Maastrichtian, Cretaceous.\u201c\u00c0 Fuveau, Puyloubier, Olli\u00e8res\u201d, France.Melanopsidae.Probably not a Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in Brot (1874\u20131879: 427)[unavailable]Not indicated.Melanopsiscostata  by Nomen nudum; apparently based on an unpublished manuscript name from Ziegler and listed in synonymy of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cEnvirons d\u2019Agora et d\u2019Alhama (Aragon), en Espagne; Relizane, dans la plaine du Cheliff, en Alg\u00e9rie\u201d .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRuisseau du puits art\u00e9sien de Ngou\u00e7a; Chetma, pr\u00e8s de Biskra, et environs d\u2019Ouargla, dans le sud de la province de Constantine\u201d , Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928\u201cTiout, dans l\u2019Anti Atlas\u201d, Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeKobelt, 1881[invalid]\u201cTeplica\u201d [thermal springs Teplice (?)], Slovakia.nana = Latin \u201ctiny\u201d). Kobelt PageBreak(1881) referred the name to Brot\u2019s illustration, making it thereby available. However, the name is a junior homonym of Melanianana Lea & Lea, 1850.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeNevill, 1884[unavailable]\u201cPersia; [...] Shiraz\u201d, Iran.Introduced infrasubspecific taxon (\u201csubvariety\u201d), which is not ruled by the provisions of the Code.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePaetel, 1888\u201cSea of Galilee\u201d : 212, IsOriginally introduced as infrasubspecific taxon (\u201csubvariety\u201d) by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020d\u2019Archiac in Viquesnel, 1846Late Miocene.\u201cPi\u00e9mont\u201d [no locality indicated], Italy.M.[elanopsis] narsolina\u201d [sic]. However, the name Melanopsisnarzolina - probably the \u201ccorrect\u201d spelling sensu Bonelli - has been in prevailing usage since then. For reasons of stability the name narzolina should be maintained according to Art. 33.3.1 .PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1893Pannonian, zone D\u2013E, late Miocene.\u201cRipanj\u201d, Serbia.Milan et al. (1974: 95) indicated collection numbers for \u201csyntypes\u201d illustrated in nesicii\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Tournou\u00ebr, 1874Mammal zone MN 11, late Miocene.\u201cVisan\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1923\u201cGafsa, dans les eaux refroidies\u201d , Tunisia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBiggs, 1934\u201cQanat at Ginehkan near Kerman, S. Persia\u201d, Iran.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cFer.\u201d mentioned in Paetel (1888: 402)[unavailable]Mesopotamia, no locality indicated], Iraq.\u201cMesopot.\u201d , Georgia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1912[invalid]\u201cDes bords du Chott Djerid, \u00e0 Tozeur\u201d [banks of the Chott el Dj\u00e9rid at Tozeur], Tunisia.Melanopsisnobilis Seninski, 1905.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLa Save \u00e0 Agram et la rivi\u00e8re entre Plaski et Ostaria (Croatie); le Danube \u00e0 Ibra\u00efla\u201d .Fagotia [= Esperiana] esperi .Note that Bourguignat denoted the authority as \u201cBourguignat, 1879\u201d. Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsisvaricosanodescens. It is unclear whether he actually introduced a new variety or ranked the species described in 1882 as variety of Melanopsisvaricosa Handmann, 1882. Melanopsisnodescens Handmann, 1882 as a junior synonym of Melanopsisbouei F\u00e9russac, 1823.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1920Late Villafranchian, early Pleistocene.\u201cRoccantica presso Poggio Mirteto; Fra Otricoli e le Vigne\u201d : 62\u201363 , Italy.Melanopsidae, the only specimen illustrated in 1822 in the \u201cHistoire naturelle\u201d was a fossil one from the Villafranchian of Italy. This fact remained widely unknown to biologists and paleontologists alike. The consequence, however, is that probably none of the specimens referred to as Melanopsisnodosa in the biological literature . While re e.g., , 1996 orre e.g., , which rMelanopsisaffinis F\u00e9russac\u201d or \u201cMelanopsisantiqua F\u00e8russac\u201d  [\u010caplja trench near Slavonski Brod]; Kova\u010devac; Novska; Farka\u0161i\u0107; Dubranjec\u201d, Croatia.PageBreakMelanopsisnodosa F\u00e9russac, 1822. Melanopsisnodosula as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882[invalid]Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsisnodosa F\u00e9russac, 1822. Melanopsisvenusta as replacement name. Melanopsisbouei F\u00e9russac, 1823.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Doncieux, 1908[invalid]Middle Lutetian, Eocene.\u201cAu Nord d\u2019Albas\u201d, France.Melanopsisnodosa F\u00e9russac, 1822.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in Reeve (1860) and Brot (1874\u20131879: 13)[unavailable]Not indicated.Melaniahollandri [sic]. If available, it would be a junior homonym of Melanianodosa M\u00fcnster, 1841. Note that the latter species is certainly no Melanopsidae as it derives from the marine deposits of the Triassic St. Cassian Fm.Nomen nudum, apparently based on an unused manuscript name from Parreyss see  or StentTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Wenz, 1928Cernikian, Pliocene.\u201cPodvinje (\u010caplja) [\u010caplja trench near Slavonski Brod]; Kova\u010devac; Novska; Farka\u0161i\u0107; Dubranjec\u201d : 41, CroMelaniacostatanodosa Brusina, 1874, non F\u00e9russac, 1822 (see Note 1).Replacement name for PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Langhian, middle Miocene.\u201cMio\u010di\u0107\u201d, Croatia.The syntypes are stored in the Croatian Natural History Museum, Zagreb; no number indicated .Melanopsisinconstans Neumayr, 1869.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Magrograssi, 1928[invalid]Plio-Pleistocene.\u201cCoo: V. Armiri\u201d [Kos island: Armiri valley (?)], Greece.Melanopsisinconstansnodulosa Brusina, 1874 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat in Noetling, 1886\u201cAus dem Jarm\u016bk bei el Hammi [...] auch bei el-H\u0101wij\u0101n [fossil]\u201d [from the Yarmuk river at Al \u1e28ammah and as fossil near Jisr al \u1e28\u0101w\u012b], Jordan.Melanopsismultiformis Blanckenhorn, 1897. Melanopsisobliqua Bourguignat, 1884.Noetling clearly referred to \u201cBourguignat\u2019s new species\u201d, which is why the authority should read as \u201cBourguignat in Noetling, 1886\u201d. Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Wenz, 1928Cernikian, Pliocene.\u201cNovska (Bukovica)\u201d : 7, CroaM. [constricta] subcostata . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2993-639.Milan et al. (1974: 88) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand when describing PageBreakMelanopsisconstrictasubcostata Brusina, 1897, non Melanopsissubcostata d\u2019Orbigny, 1850. The name \u201cnovskaensis\u201d as mentioned in Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidaePicard, 1934\u201cObedieh\u201d [El \u2018Ubeid\u012bya], Israel.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Tausch, 1886Late Santonian\u2013early Campanian, late Cretaceous.\u201cCsingerthal bei Ajka\u201d [Csinger valley near Ajka], Hungary.Esperiana by The species was attributed to the genus Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1856\u201cDans l\u2019Oued-Lisser, sur la route de Sidi-bel-Abess, \u00e0 Tlemcen\u201d , Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBrot, 1868[invalid]\u201cPrope Cehejin Prov. Murcia\u201d , Spain.Melanopsisobesa Guirao mss.\u201d. Although Melanopsisobesa Gassies, 1856. Melanopsisguiraoi as replacement name.The species was described as \u201cPageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in Locard  and Bourguignat (1884: 141)[unavailable]\u201cLe Jourdain et le lac de Tib\u00e9riade\u201d [river Jordan and Sea of Galilee], Israel.Melanopsisobesa Gassies, 1856.Nomen nudum. If available, it would be a junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1901[invalid]Late Miocene.\u201cSmendou\u201d [Zighoud Youcef], Algeria.Melanopsisobesa Gassies, 1856. Melanopsisdoumergueiplena as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902[invalid]Middle Pannonian, late Miocene.\u201cMarku\u0161evec\u201d, Croatia.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2492-138, 2493-139 .Melanopsisobesa Gassies, 1856. It was considered as a junior synonym of Melanopsispygmaea H\u00f6rnes, 1856 by Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1904\u201cDe l\u2019A\u00efn bou Smelal pr\u00e8s de T\u00e9touan\u201d [\u2018A\u00efn bou Smelal (?) near Tetouan], Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLe B\u00e9lus, pr\u00e8s de Saint-Jean-d\u2019Acre\u201d , Israel.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cDjishr ech Chegour\u201d [Jisr Ash-Shughur], Syria.Melanopsismultiformis Blanckenhorn, 1897. Melanopsiscostata .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Blanckenhorn, 1897Plio-Pleistocene.Dreissena layers at Jisr Ash-Shughur], Syria.\u201cIn der Dreissensiaschicht von Dschisr esch-Schurr\u201d , Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeMartens, 1874[invalid]\u201cQuellen des Chabur bei Ras-el-Ain\u201d [source of Chabur river near Ra\u2019s al \u2018Ayn], Syria.Melanopsisobsoleta Fuchs, 1873.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeDautzenberg, 1894[invalid]\u201cPalmyre, dans un ruisseau et dans la rivi\u00e8re Eph\u00e9ca\u201d , Syria.Melanopsisobsoleta Fuchs, 1873.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Deshayes, 1825PageBreakCuisian, late Ypresian, Eocene.\u201cRetheuil pr\u00e8s de Pierre-Fonds\u201d , France.Coptostylus (Thiaridae), which was followed by Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920[invalid]\u201cMekn\u00e8s, pr\u00e8s de anciennes \u00e9curies\u201d , Morocco.Melanopsisobtusa Deshayes, 1824.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePaetel, 1888\u201cPersia\u201d : 209, Iracuticarinata\u201d [sic] for an infrasubspecific taxon (\u201csubvariety\u201d), which he briefly described. ocuticarinata remains the valid spelling because Nevill\u2019s \u201cacuticarinata\u201d is not available as species-group name.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Locard, 1883Mammal zone MN 9\u201312, late Miocene.\u201cPriay; [...] Niquedet\u201d (p. 150), France.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Hoeninghaus (1831: 143)[unavailable]Burdigalian, early Miocene (?).\u201cDax\u201d, France.Melanopsisoliva De Cristofori & Jan, 1832.Nomen nudum. If available, it would be a junior homonym of PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeDe Cristofori & Jan, 1832\u201cAm. mer.\u201d [South America], indicated in the previous part of the same work .Melanopsidae.Certainly not a Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cEntre Ain-Taib et Alep, \u00e0 Sadjour-Sou; [...] du Nahr-el-Kelb, pr\u00e8s Beyrouth; de divers cours d\u2019eau du Liban; de Serghaia dans l\u2019ouady Baradah pr\u00e8s de Damas; de la fontaine J\u00e9r\u00e9mie pr\u00e8s de J\u00e9richo; environs de Constantinople\u201d .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Grateloup, 1828Burdigalian, early Miocene.\u201cMandillot\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeLetourneux & Bourguignat, 1887[invalid]\u201c\u00c0 Nefta, dans les canaux d\u2019arrosement et \u00e0 El-Hammam, pr\u00e8s Tozer\u201d , Tunisia.Melanopsisolivula Grateloup, 1838. Melanopsisdoumeti Letourneux & Bourguignat, 1887.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Halav\u00e1ts, 1914Sarmatian (sensu stricto), middle Miocene.\u201cGraben unterhalb der Gemeinde-Baumschule \u00f6stlich von Oltszakad\u00e1t\u201d , Romania.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Stefanescu, 1896Pliocene.\u201c\u00c0 Plostina, \u00e0 Leurda et \u00e0 St\u00e2ngaceana, dans la vall\u00e9e de Motru; \u00e0 Valea-lui-C\u00e2ne, dans la vall\u00e9e de Gilortu; \u00e0 Breasta, \u00e0 Bocovatz et \u00e0 B\u00e2zd\u00e2na, dans la vall\u00e9e de Jiu; \u00e0 Beceni, dans la vall\u00e9e de Slanic de Buzau; \u00e0 Plaiu et \u00e0 Chiojdeni, dans la vall\u00e9e de R\u00e2mnicu-Sarat\u201d , Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Cernikian, Pliocene.\u201cBe\u010di\u0107; Podvinje (\u010caplja) [\u010caplja trench near Slavonski Brod]; Sibinj\u201d, Croatia.Milan et al. (1974: 95) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 3206-852/1.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020De Stefani, 1877Villafranchian, Plio-Pleistocene.\u201cSpoleto [...], Orciano [...], fra S. Gemini e Carsoli\u201d , Italy.Melaniabuccinoidea sensu F\u00e9russac, 1823, non Olivier, 1801 (partim) as well as for formerly misidentified Melanopsisnarzolina. Melanopsisaffinis F\u00e9russac\u201d, which is not an available name.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1926Pliocene?PageBreak\u201cD\u2019un puits, profond de 25 m\u00e8tres, situ\u00e9 dans la propri\u00e9t\u00e9 Lamur, sur la rive gauche d\u2019un ravinement creus\u00e9 par les eaux pluviales dans une d\u00e9pression de terrain, perpendiculaire au chemin d\u2019A\u00efn Be\u00efda\u201d (p. 284) , Algeria.Pallary cited in the synonymy list his paper on the fauna of the \u201cBerb\u00e9rie\u201d, which appeared in the Journal de Conchyliologie. That work, however, was not published before March 1928.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Bukowski, 1892Apolakkia/Monolithos Formation, Pliocene.\u201cRhodos\u201d [unavailable]Not indicated (except from \u201cde l\u2019Orient\u201d).Nomen nudum, listed by Mousson without any explanation.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Deshayes, 1862Sparnacian, early Ypresian, Eocene.\u201cSainceny\u201d, France.Eginea Pacaud & Harzhauser, 2012 (Pachychilidae).Type species of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fuchs, 1877Pliocene.\u201cKalamaki\u201d [near Corinth], Greece.Paludina F\u00e9russac, 1812 is a junior objective synonym of Viviparus Montfort, 1810, this species is a junior homonym of Viviparaornata Neumayr in Neumayr & Paul, 1875. Amphimelania and introduced Amphimelaniafuchsi as replacement name.Since PageBreakPageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cL\u2019Oronte \u00e0 Djisr ech Chogour\u201d [in the Orontes at Jisr Ash-Shughur], Syria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cM. Mario: Farnesina\u201d : 183, ItMelanopsispraemorsa sensu Cerulli-Irelli, 1914, non Linnaeus, 1758.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Papp, 1979Mammal zone MN 10, late Miocene.\u201cMilessi\u201d [Milesi], Greece.Institute of Paleontology, University of Vienna; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Grateloup, 1838Sparnacian, early Ypresian, Eocene.\u201c\u00c9pernay\u201d, France.Melaniabuccinoidea var. a\u201d in Introduced for \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeDunker, 1862\u201cRotoitisee\u201d [Lake Rotoiti], New Zealand.Zemelanopsistrifasciata .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Deffner & Fraas, 1877PageBreakMammal zone MN 6, middle Miocene.\u201cTrendel\u201d [in N\u00f6rdlinger Ries], Germany.Melanopsiskleinii Kurr, 1856.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLe Danube au-dessus de Routschouk (Bulgarie)\u201d [in the Danube river below Ruse], Bulgaria.Note that Bourguignat denoted the authority as \u201cBourguignat, 1879\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeCerulli-Irelli, 1914\u201cM. Mario: Farnesina; Acuatraversa\u201d, Italy.Melanopsisaffinis F\u00e9russac\u201d, which is not an available name.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Willmann, 1981Vasilios Formation, middle Tortonian, late Miocene.\u201cDermen Deressi 400 m westlich des Asklepion, ca. 3 km westlich von Kos-Ort/Kos\u201d , Greece.Geological-Paleontological Institute, University of Kiel, Germany; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLe Guadalquivir entre S\u00e9ville et Cordoue\u201d [in the Guadalquivir river between Sevilla and C\u00f3rdoba], Spain.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Watelet, 1853Cuisian, late Ypresian, Eocene.PageBreak\u201cMercin\u201d [Mercin-et-Vaux], France.Watelet attributed the authority to Deshayes, but there is no evidence that the description really derived from that author.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cSee Tiberias\u201d : 17 , Syria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLa Save pr\u00e8s d\u2019Agram\u201d [Sava river at Zagreb], Croatia.PageBreakMicrocolpiacornea .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pavlovi\u0107, 1927Middle Pannonian, late Miocene.\u201c\u0418\u0437 \u041a\u0430\u0440\u0430\u0433\u0430\u0447\u0430\u201d [from Karaga\u010da near Vr\u010din], Serbia.The illustrated syntype is stored in the Natural History Museum, Belgrade, coll. no. 213 (Milo\u0161evi\u0107 1962: 23).Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Wenz, 1919Early Pleistocene.\u201cDu puits Karoubi\u201d : 178 \u201d, Hungary.Geological-Palaeontological Institute and Museum University of Hamburg, coll. no. 4269.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1892Middle Pannonian, late Miocene.\u201cMarku\u0161evec\u201d, Croatia.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2540-186/1-4 .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBrot, 1878\u201cKrain, im Gurkfluss\u201d : 686 [inMelanopsispardalis. Mhlf. Mss.\u201d in synonymy of Hemisinusesperi  [currently in Esperiana]. Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Deshayes, 1825Cuisian, late Ypresian, Eocene.\u201cCuise-la-Mothe\u201d [Cuise-la-Motte], France.PageBreakCoptostylusalbidus  (Thiaridae).Taxon classificationAnimaliaSorbeoconchaMelanopsidaePhilippi, 1847Late Pleistocene to Recent.\u201cHungaria\u201d [no locality indicated], Hungary? .parreyssi\u201d as mentioned in Philippi attributed the authority to Johann Georg Megerle von M\u00fchlfeld, apparently based on an \u201cin schedis\u201d determination. The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Burdigalian, early Miocene (?).\u201cDax\u201d, France.parva\u201d as available name attributed to Melanopsissubbuccinoides d\u2019Orbigny, 1852.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Grateloup, 1838Burdigalian, early Miocene.\u201cDax. [...] Mandillot\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBrot, 1874\u201cKrain\u201d .PageBreakAppeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]\u201cLe Jourdain, \u00e0 4 kilom\u00e8tres au-dessus de la Mer Morte; Ain-el-Mellaha, dans la plaine du Bahr-el-Houl\u00e9\u201d , Israel.Melanopsisolivulaparvula Grateloup, 1838 (see Note 1). Melanopsislampra Bourguignat, 1884.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Bellardi & Michelotti, 1841Late Miocene.\u201cDel Tortonese\u201d [from the region of Tortona], Italy.Amphimelania by Bellardi and Michelotti attributed the authority to Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Lubenescu, 1981Middle Pannonian, late Miocene.\u201cComuna Vingard, 15 km NE de Sebe\u0219-Alba\u201d, Romania.Institute of Geology and Geophysics, University of Bucharest; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Bandel & Riedel, 1994Late Santonian\u2013early Campanian, late Cretaceous.\u201cCsingertal, near Ajka (Bakony Mountains)\u201d, Hungary.Geological-Paleontological Department, Natural History Museum Vienna, Austria, coll. no. 1994/148.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020PageBreakBourguignat, 1880 Langhian, middle Miocene.\u201cVall\u00e9e de la Cettina\u201d [Cetina river valley], Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Pannonian, zone D\u2013E, late Miocene.\u201cRipanj\u201d, Serbia.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2495-141/1-3 .Melanopsispaulovici [sic] as considered by Melanopsisripajensis as replacement name. Note that Art. 58.4. (regarding the use of u or v for the same Latin letter) does not apply here, because the names are not based on Latin words but on the names of two different persons.Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020H\u00f6rnes, 1856Badenian, middle Miocene.\u201cForchtenau in Ungarn\u201d , Austria.Amphimelania by pecchioli\u201d, which is an incorrect subsequent spelling.Considered to belong in the genus Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDe la source de la Moulouiah, au nord de Lalla-Maghnia, pr\u00e8s des fronti\u00e8res du Maroc\u201d , Morocco or Algeria.Note that Bourguignat denoted the authority as \u201cBourguignat, 1882\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sacco, 1889Late Burdigalian, early Miocene.\u201cColli torinesi, [...] presso Tetti Varetti\u201d , Italy.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1868\u201cAgora, en Aragon (Espagne)\u201d, Spain.pinchinati\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1892Middle Pannonian, late Miocene.\u201cMarku\u0161evec\u201d, Croatia.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2532-178/1-4 .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cEnvirons de Krapina-Toeplitz (Croatie)\u201d [surroundings of Krapinske toplice], Croatia.Microcolpiaacicularis .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeP\u00e9r\u00e8s, 1946[invalid]Not indicated.First mentioned as nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sandberger, 1875Middle Miocene.\u201cLocle\u201d, Switzerland.PageBreakMelanopsiskleinii Kurr, 1856.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020F\u00f6rster, 1892Early Rupelian, Oligocene.\u201cKleinkems, [...] Tagolsheim\u201d, Germany.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1918[invalid]\u201cDans les s\u00e9guias du Guers\u201d [in the irrigation channel of Guers], Morocco.Melanopsispercarinata F\u00f6rster, 1892.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Calvert & Neumayr, 1880Late Sarmatian (sensu lato), Khersonian, late Miocene.\u201cRenki\u00f6i\u201d [north of \u0130ntepe], Turkey.pergamenica\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1920Cernikian, Pliocene.\u201cBe\u010di\u0107; Podvinje (\u010caplja) [\u010caplja trench near Slavonski Brod]; Sibinj; Kova\u010devac; Novska; Moslavina\u201d : 41\u201342, Melaniacostata sensu Brusina, 1874, non Olivier, 1804, which has been considered to be the same species as Melaniacostata sensu Neumayr, 1869 . Not included in the Fossilium Catalogus of Probably a junior synonym: introduced for 69 e.g., : 41. For69 e.g., : 168 hadTaxon classificationAnimaliaSorbeoconchaMelanopsidaePageBreakPallary, 1920 \u201cOued F\u00e8s, \u00e0 1.500 m\u00e8tres en amont de la ville\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pavlovi\u0107, 1931Badenian, middle Miocene.\u201c\u0418\u0437 \u0441\u0435\u043b\u0430 \u0412\u0440\u043c\u045f\u0435\u201d [Vrmd\u017ea], Serbia.The illustrated syntype is stored in the Natural History Museum, Belgrade, coll. no. 2888 (Milo\u0161evi\u0107 1962: 24).Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1893Transdanubian, Pannonian, late Miocene.\u201cOre\u0161ac\u201d, Serbia.Lectotype designation by Milan et al. (1974: 84) invalid: it is uncertain whether the specimen illustrated by Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRivi\u00e8res de Carniole; la Save \u00e0 Steinbr\u00fcck\u201d , Slovenia.Fagotia [= Esperiana] acroxia Bourguignat, 1884.Note that Bourguignat denoted the authority as \u201cBourguignat, 1880\u201d. Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Bukowski, 1892Salakos Formation, Pliocene.\u201cRhodos\u201d . phoeneciaca\u201d) a junior synonym of Melanopsiscostata . Melanopsislampra Bourguignat, 1884.The spelling \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in (Pichler 1856: 735\u2013736)Late Cretaceous.\u201cAuf zwei Puncten der Brandenberger Ache. Ungef\u00e4hr \u00be Stunde n\u00f6rdlich von Binneck [...]. Etwa eine Viertelstunde unter Binneck [...] am linken Ufer der Ache\u201d [At two points along the Brandenberger Ache: about three quarters of an hour north of Pinegg and a quarter of an hour south of Pinegg in Tyrol], Austria.Melanopsidae there or later .After Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans la Save \u00e0 Agram et \u00e0 Sissek. Dans la rivi\u00e8re, entre Plaski et Ostaria (Croatie)\u201d [Sava river at Zagreb and Sisak. In the river between Pla\u0161ki and O\u0161tarije], Croatia.Fagotia [= Esperiana] servainiana Bourguignat, 1884.Note that Bourguignat denoted the authority as \u201cBourguignat, 1882\u201d. PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cEn Croatie, au pont de la Save \u00e0 Agram, ainsi qu\u2019\u00e0 Steinbr\u00fcck; [...] en Bosnie, dans les rivi\u00e8res de Zenica et de la Migliaska \u00e0 S\u00e9rajewo\u201d .Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Aldrich, 1895Early Eocene.\u201cGregg\u2019s Landing, Alabama\u201d, United States.Melanopsisanita Aldrich, 1886. According to Melanopsidae.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Wenz, 1928Late Miocene.\u201cSmendou\u201d [Zighoud Youcef], Algeria.Melanopsisdoumergueiobesa Pallary, 1901, non Gassies, 1856 (see Note 1).Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Gozhik in Gozhik & Datsenko, 2007Pliocene.\u201cO\u0437. \u0421\u0430\u0441\u044b\u043a\u201d [Lake Sasyk], Ukraine.Institute of Geological Sciences of the National Academy of Sciences of Ukraine, coll. no. 4603.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020PageBreakBourguignat, 1880 Langhian, middle Miocene.\u201cVall\u00e9e de la Cettina\u201d [Cetina river valley], Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]\u201cDans le Viar et le Carban\u00e8s, entre Cordoue et S\u00e9ville (Espagne)\u201d , Spain.Melanoptychiapleuroplagia Bourguignat, 1880. It was considered as a junior synonym of Melanopsissevillensis by Junior secondary homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRuisseaux de l\u2019\u00eele d\u2019Ivice, dans les Bal\u00e9ares\u201d [stream on the island of Ibiza], Spain.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pavlovi\u0107, 1927Pannonian, zone D\u2013E, late Miocene.\u201c\u0423 \u0437\u0430\u0441\u0435\u043e\u043a\u0443 \u0420\u0430\u043c\u0430\u045b\u0438 [...] \u0443 \u0420\u0438\u043f\u045a\u0443\u201d [from the village Rama\u010da and from Ripanj], Serbia.The illustrated syntype is stored in the Natural History Museum, Belgrade, coll. no. 794 (Milo\u0161evi\u0107 1962: 24).Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1911\u201cTout pr\u00e8s d\u2019Oudjda, \u00e0 4 kilom. S.-E., sourdent les belles sources de Sidi-Yahia qui alimentent une v\u00e9ritable oasis, puis la ville d\u2019Oudjda, et vont finalement se d\u00e9verser dans l\u2019oued Isly\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePageBreakPallary, 1928 [invalid] \u201cDans une source ti\u00e8de \u00e0 proximit\u00e9 de la Moulou\u00efa\u201d [in a warm source close to the river Moulouya], Morocco.Melanopsisplicata Pallary, 1911.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[invalid]\u201cRas el \u2018Ain du Khabour\u201d [Chabur river near Ra\u2019s al \u2018Ayn], Syria.Melanopsisbarbiniplicata Pallary, 1911 (see Note 1). Melanopsisinfracincta Martens, 1874.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr, 1880Pannonian, late Miocene; late Burdigalian to early Langhian, early\u2013middle Miocene.\u201cPosu\u0161je; [...] Seonica\u201d, Bosnia and Herzegovina.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Langhian, middle Miocene.\u201cMio\u010di\u0107\u201d, Croatia.Milan et al. (1974: 96) indicated a holotype, but it is uncertain whether the specimen is part of the original type series and whether it was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2976-622/1-2.Melanopsisinconstans Neumayr, 1869.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Brusina (1903: 110)[unavailable]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Melanopsisthemaki, because Melanopsisinconstansplicatula Brusina, 1874 (see Note PageBreak1). Microcolpiaparreyssiisikorai .Nomen nudum. If available, it would be a junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903[invalid]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Melanopsisinconstansplicatula Brusina, 1874 (see Note 1). Microcolpiaparreyssiisikorai .Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887[invalid]Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.plicatulus\u201d), but Melanopsis is feminine, which is why the name must be \u201cplicatula\u201d. Junior homonym of Melanopsisinconstansplicatula Brusina, 1874 (see Note 1). Melanopsissimilis as replacement name. Melanopsisbouei F\u00e9russac, 1823.Originally the gender was indicated as masculine  and Cossmann, 1910 .\u201cPomahaka, Otago\u201d [Pomahaka River], New Zealand.Melanopsis (Stilospirula?) by The species was attributed to the Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cEn Carmiole [sic], en Croatie, et en Dalmatie, notamment dans la Cettina\u201d .Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Gozhik in Gozhik & Datsenko, 2007Middle Pontian (Dacian Basin), late Miocene.\u201c\u0412\u0443\u043b\u043a\u0430\u043d\u0435\u0448\u0442\u044b\u201d [Vulc\u0103ne\u015fti], Moldova.Institute of Geological Sciences of the National Academy of Sciences of Ukraine, coll. no. 3190.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Porumbaru, 1881Early\u2013middle Romanian, Pliocene.\u201cCretzesci et Podari\u201d [Cre\u021be\u0219ti and Podari], Romania.porumbarui\u201d PageBreakas mentioned in Porumbaru attributed the authority to Brusina, but there is not clear evidence that the description really derived from that author. The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Papp, 1953Pannonian, zone B, late Miocene.\u201cLeobersdorf Sandgrube\u201d [Leobersdorf sand pit], Austria.Appeared first as nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sch\u00fctt in Symeonidis et al., 1986[invalid]Symeonidis et al. 1986: 342, pl. 4, figs 8\u20139.Gelasian, early Pleistocene.\u201cWeganrisse und Regenrisse 3 Km N Antirrion in Akarnanien\u201d [outcrops 3 km north of Antirrio], Greece.Senckenberg Forschungsinstitut und Naturmuseum Frankfurt; no number indicated.Melanopsisimpressaposterior Papp, 1953 (see Note 1). Melanopsisgearyae as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1870\u201cDans le Danube. [...] non-seulement de Brahilov, mais encore des environs de Belgrade\u201d .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Cossmann, 1907Sparnacian, early Ypresian, Eocene.\u201cPourcy\u201d, France.Coptostylus is at present considered to belong in the family Thiaridae  and near Belgrade (Serbia)].Microcolpiacornea .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sch\u00fctt in Sch\u00fctt & Ortal, 1993Early Pleistocene.\u201cThe type-locality of the Upper Pliocene sediments or the \u2018Erq el-Ahmar Formation in the Jordan Valley south of the Sea of Galilee\u201d [i.e. \u2018Erq el-Ahmar ], Israel.Paleontology Collection of the Hebrew University of Jerusalem; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeLinnaeus, 1758\u201cIn Europa australiore\u201d [southern Europe].Melanopsis [unavailable]Khersonian, late Sarmatian, late Miocene.\u201cHauptstra\u00dfe bei Avcilar, nahe der Abzweigung nach Ambarlik\u00f6y\u201d , Turkey.Nomen nudum; mentioned as \u201cnew species\u201d in PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeLea, 1837\u201cCape of Good Hope\u201d, South Africa.Faunopsis Gill, 1863. It was considered as a junior synonym of Faunusater , the type species of the genus Faunus Montfort, 1810 (Pachychilidae), by Type species of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Melanopsisbouei F\u00e9russac, 1823.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Deshayes, 1862Bartonian, Eocene.\u201cChery-Chartreuve\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Melanopsisfossilis .Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cYeni Chehir\u201d [Yeni\u015fehir], Turkey.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sacco, 1886PageBreakLate Burdigalian, early Miocene.\u201cCollina di Torino\u201d [Torino hills], Italy.Amphimelania by Considered to belong in the genus Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat in Locard, 1883\u201cDans le eaux de la fontaine de l\u2019\u00c9lys\u00e9e, \u00e0 J\u00e9richo; [...] des environs de Beyrouth; [...] dans les eaux du lac d\u2019Antioche\u201d .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Melanopsisfossilis .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Tournou\u00ebr, 1875Tafi Formation, early Pleistocene.\u201cPrope vicum Antimaki\u201d , Greece.The species epithet is a noun in apposition (named after the Greek God Proteus) and needs not to agree in gender with the generic name (Art. 31.2.1).Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Halav\u00e1ts, 1914Sarmatian (sensu stricto), middle Miocene.\u201cGraben unterhalb der Gemeinde-Baumschule \u00f6stlich von Oltszakad\u00e1t\u201d , Romania.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Gozhik, 2002Middle Pontian (Dacian Basin), late Miocene.\u201c\u0412\u0438\u043d\u043e\u0433\u0440\u0430\u0434\u0456\u0432\u043a\u0430\u201d , Ukraine.Institute of Geological Sciences of the National Academy of Sciences of Ukraine, coll. no. 3186.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sauerzopf, 1952Pannonian, zone D, late Miocene.\u201cBurgau\u201d, Austria.pseudaustriaca\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Bandel, 2000Late Pannonian, late Miocene.\u201cPapkesi [Papkeszi] near the eastern shore of Lake Balaton\u201d, Hungary.Geological-Palaeontological Institute and Museum University of Hamburg, coll. no. 4266.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Oppenheim, 1890Cernikian, Pliocene.\u201cCigelnik; [...] Graben zwischen \u010capla und der Podwiner Kirche; [...] Thal hinter der Podwiner Kirche; [...] Sibin; [...] Repu\u0161nica\u201d : 41   and Melanopsispermutabilis Pallary, 1920 . Note that Melanopsiscosmanni a replacement name of In addition, there are three other new names proposed for misidentified Melaniacostata\u201d records from Slavonia to represent the same species and synonymized them with Melanopsiscosmanni, but was unaware that Melanopsiscroatica Brusina, 1884 is the first available name for them. Although the four names  likely refer to the very same species they are based on different specimens and thus no objective synonyms.Wenz obviously considered all erroneous \u201cMelanopsissculptilis as replacement name. The misidentified Melanopsiscostata sensu Fuchs, 1877 from Megara, however, is still nameless.For the record by pseudodecorata\u201d mentioned by The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sacco, 1895Early Messinian, late Miocene.\u201cS. Marzano Oliveto\u201d, Italy.Melanopsisfusulatina Sacco, 1895.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1899\u201cEnvirons de T\u00e9touan\u201d [surrondings of T\u00e9touan], Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Papp, 1953Pannonian, zone E, late Miocene.\u201cV\u00f6sendorf (Sandriff)\u201d [V\u00f6sendorf (sandbar)], Austria.Appeared first as a nomen nudum in PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Papp, 1953Pannonian, zone B, late Miocene.\u201cLeobersdorf Sandgrube\u201d [Leobersdorf sand pit], Austria.Appeared first as nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Magrograssi, 1928Plio-Pleistocene.\u201cCoo: vicino a Cardamena\u201d [Kos island: close to Kard\u00e1maina], Greece.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sacco, 1889Late Burdigalian, early Miocene.\u201cColli torinesi\u201d [Torino hills], Italy.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Jekelius, 1944Early Pannonian, late Miocene.\u201cTurislav-Tal bei Soceni\u201d [Turislav valley near Soceni], Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sandberger, 1886Oncophora Beds, middle Burdigalian, early Miocene.Rather unprecisely given as \u201cKirchberger Schichten M\u00e4hrens\u201d [Kirchberg Formation in Moravia] by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Newton, 1891PageBreakBartonian?\u2013Rupelian, late Eocene\u2013early Oligocene.\u201cHempstead [...]. Headon Hill; Hordwell. [...] High Cliff\u201d, United Kingdom.subulatapseudo-\u201d) for a part of Sowerby\u2019s material for Melanopsisfusiformis Sowerby, 1822  [\u010caplja trench near Slavonski Brod]; Kova\u010devac\u201d, Croatia.Milan et al. (1974: 96) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 3741-1381.pterochyla\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[unresolved]\u201cDu Lac d\u2019Homs\u201d (Lorcard 1883: 288) [in Lake Homs], Syria.gracilis\u201d mentioned by pulchella its junior objective synonym. Melanopsisgracilis Locard, 1883 is a junior homonym of Melanopsisgracilis Brusina, 1874, which makes pulchella the next available name. However, Melanopsiscostatapulchella is a homonym of the simultaneously introduced Melanopsisseignettipulchella Bourguignat, 1884 (see Note 1). The action of a First Reviser is required to determine which of both pulchella is to be treated as valid.Introduced to replace - for whatever reason - the variety \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[unresolved]\u201c l\u2019oasis Sidi Yousef, \u00e0 l\u2019extr\u00eame sud de la fronti\u00e8re du Maroc; [...] ruisseau d\u2019eau chaude \u00e0 Ouargla]\u201d , Algeria.Melaniacostatapulchella Bourguignat, 1884 (see Note 1). The action of a First Reviser is required to determine which of both pulchella is to be treated as valid.Homonym of the simultaneously introduced PageBreakPageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920[invalid]\u201cDans une source entre Sidi Abdallah et Koudiat (r\u00e9gion de Taza) et dans l\u2019Innaouen, \u00e0 Sidi Abdallah\u201d [in a source between Douar Sidi Abdellah and Douar El Koudiat (Taza region) and in the Oued Abiod (?) in Douar Sidi Abdellah], Morocco.Melanopsiscostatapulchella Bourguignat, 1884 and Melanopsisseignettipulchella Bourguignat, 1884 .Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Middle Pannonian, late Miocene.\u201cMarku\u0161evec\u201d, Croatia.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2491-137/1-2 .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Stoliczka, 1860Late Turonian\u2013Coniacian, late Cretaceous.\u201cAbtenau\u201d, Austria.Megalonodareussi .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRives de la Save pr\u00e8s Sissek, en Slavonie\u201d [Sava river near Sisak in Slavonia], Croatia.PageBreakFagotia [= Esperiana] acroxia Bourguignat, 1884.Bourguignat denoted the authority as \u201cServain, 1884\u201d, but there is no evidence that the description really derived from that author. Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020H\u00f6rnes, 1856Early\u2013middle Pannonian, late Miocene.\u201cBrunn, Gumpoldskirchen, Guntramsdorf, Inzersdorf, Arsenal in Wien, Kroisbach bei Oedenburg\u201d .H\u00f6rnes attributed the authority to Partsch, apparently based on an \u201cin schedis\u201d determination.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans le Danube, o\u00f9 elle a \u00e9t\u00e9 rencontr\u00e9e \u00e7\u00e0 et l\u00e0 depuis Buda-Pesth jusqu\u2019\u00e0 Ibraila; [...] Pregrada, pr\u00e8s de Krapina, en Croatie\u201d .Melanopsispyramidalis (mihi)\u201d [nomen nudum] in a species list. Microcolpiaacicularis .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1920Early Rupelian, Oligocene.\u201cGrossalmerode, Oberzwehren, Mardorf bei Wabern, Frielendorf, Kirchhain, Dannerod, Ofleiden, Mardorf an der Ohm\u201d : 71, GerMelanopsispraerosa [= Melanopsispraemorsa] sensu Ludwig, 1865, non Linnaeus, 1758. It was regarded a junior synonym by Melanopsishassiaca as replacement name for misidentified Melanopsispraemorsa from that region and time period. Since Sandberger did not explicitly refer to Ludwig\u2019s material, pyramidalis and hassiaca are not objective synonyms.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201cPageBreak\u201d mentioned in Hermite (1879: 184) [unavailable] Early Eocene.\u201cDe Binisalem et de Selva\u201d , Spain.Melaniapyrguloeformis\u201d on p. 328. The name was probably intended as \u201cpyrgulaeformis\u201d, but the ligature was mixed up during typesetting. Hermite listed the name in a section called \u201cEsp\u00e8ces nouvelles cit\u00e9es et non d\u00e9crites\u201d [= \u201cnew species identified and not described\u201d], where he listed 18 new names that he intended to describe in the second volume of his \u201c\u00c9tudes g\u00e9ologiques sur les \u00eeles Bal\u00e9ares\u201d. That part, however, has never been published, probably because Hermite died in 1880.Nomen nudum. Given as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Melanopsisinermis Handmann, 1882.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pavlovi\u0107, 1927Middle Pannonian, late Miocene.\u201c\u0418\u0437 \u041a\u0430\u0440\u0430\u0433\u0430\u0447\u0430\u201d [from Karaga\u010da near Vr\u010din], Serbia.The illustrated syntype is stored in the Natural History Museum, Belgrade, coll. no. 222 (Milo\u0161evi\u0107 1962: 22).Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr in Neumayr & Paul, 1875Cernikian, Pliocene.\u201c\u010caplathal bei Podwin\u201d [\u010caplja trench near Slavonski Brod], Croatia.pyrum = Latin \u201cpear\u201d) and needs not to agree in gender with the generic name (Art. 31.2.1).The species epithet is a noun in apposition [unavailable]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.PageBreakMicrocolpiaparreyssiisikorai .Nomen nudum (Brusina apparently considered the term self-explanatory). Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1920[invalid]Plio-Pleistocene.\u201cDu puits Karoubi\u201d : 178 , for which Melanopsispallaryi. Thus, Melanopsisraphidia is a junior objective synonym of Melanopsispallaryi.Introduced as replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sandberger, 1871Bartonian, Eocene.\u201cCastres (Tarn)\u201d, France.Melanopsisproboscidea Desh.\u201d in plate captions.Plate 13 of Sandberger\u2019s monograph appeared in 1871, whereas the description on p. 222 was issued in 1872 . ErroneoTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Gozhik in Gozhik & Datsenko, 2007Late Pleistocene.\u201cA\u043b\u043b\u044e\u0432\u0438\u044f VII \u0442\u0435\u0440\u0440\u0430\u0441\u044b \u0440. \u0414\u043d\u0435\u0441\u0442\u0440 \u0443 \u0441. \u0420\u043e\u0433\u0438\u201d [Alluvial terrace VII of the Dniestr river near Rogi], Ukraine.Institute of Geological Sciences of the National Academy of Sciences of Ukraine, coll. no. 5243.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1892Middle Pannonian, late Miocene.\u201cMarku\u0161evec\u201d, Croatia.PageBreakThe illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2541-187/1-2 .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020L\u00f6renthey, 1902Transdanubian, Pannonian, late Miocene.\u201cBudapest-K\u00f6b\u00e1nya und Tinnye\u201d, Hungary.Melanopsisboueispinosa Handmann, 1882 and Melanopsisboueiventricosa Handmann, 1882 in synonymy of Melanopsisrarispina, although both were published earlier. The name ventricosa Handmann is a junior homonym of Melanopsisventricosa Neumayr, 1880 and thus invalid, but spinosa is available and has priority over rarispina.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1924\u201cDe la petite rivi\u00e8re de Guadaira qui se jette dans le Guadalquivir\u201d [in the little river Guadaira which flows into the Guadalquivir], Spain.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Cernikian, Pliocene.\u201cBe\u010di\u0107; Podvinje (\u010caplja) [\u010caplja trench near Slavonski Brod]; Sibinj; Nova Gradi\u0161ka; Kova\u010devac; Moslavina; Farka\u0161i\u0107\u201d, Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Willmann, 1981Upper Elia Formation, early Pleistocene.\u201cOberes Vokasia-Tal s\u00fcd\u00f6stlich von Kos-Ort/Kos (Profil K 1)\u201d , Greece.Geological-Paleontological Institute, University of Kiel, Germany; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 ?Penecke, 1885PageBreakPaleocene.\u201cIm s\u00fcdlichen Muldenfl\u00fcgel des Sonnberges (p. 334\u2013335) [on the southern mountainside of the Sonnberg ], Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1925Late Miocene.\u201cMeseta de Requena y Ayora \u201d (Royo Gomez 1922: 110) [Plateau of Requena and Ayora], Spain.Melaniacostata sensu Royo Gomez, 1922, non Olivier, 1804.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1861\u201cL\u2019int\u00e9rieur de la Nouvelle-Cal\u00e9donie\u201d [the interior of New Caledonia].Melanopsiscarinata Gassies, 1861, which is a junior homonym of Melanopsiscarinata Sowerby, 1826 .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1904Langhian, middle Miocene.\u201cVarcar-Vakufa\u201d [Mrkonji\u0107 Grad], Bosnia and Herzegovina.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020H\u00f6rnes, 1855Late Turonian, late Cretaceous.\u201cAus der Gams, nordwestlich von Hieflau in Steiermark\u201d , Austria.Megalonoda Kollmann, 1984.Type species of the melanopsid genus Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020PageBreakPallary, 1920 Pliocene.\u201cMegara\u201d : 11, GreMelanopsisincerta Fuchs, 1877, non F\u00e9russac, 1822.Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Locard, 1883Mammal zone MN 10\u201312, late Miocene.\u201cBoul\u00e9es\u201d [brook Boul\u00e9es near Miribel], France.Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1918\u201cF\u00e8s, dans les s\u00e9guias. Ras el M\u00e2, \u00e0 16 kilom. de F\u00e8s\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr in Neumayr & Paul, 1875Cernikian, Pliocene.\u201cCigelnik [Ciglenik]; [...] Novska\u201d, Croatia.Amphimelania by Neumayr mentioned uncertainties regarding the locality Ciglenik because he had lost the original collection label. The species epithet is a noun in apposition and needs not to agree in gender with the generic name (Art. 31.2.1). Considered to belong in the genus Taxon classificationAnimaliaSorbeoconchaMelanopsidaeAhuir Galindo, 2014\u201cSpring from the Southeastern Rich region\u201d, Morocco.Museo Malacologico di Cupra Marittima, Italy; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020PageBreak[sic] Pallary, 1916 [invalid] Pannonian, zone D\u2013E, late Miocene.\u201cRipanj\u201d  does not apply here, because the names are not based on Latin words but on the names of two different persons.Introduced by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neubauer, Harzhauser, Kroh, Georgopoulou & Mandic, 2014Pannonian, zone D\u2013E, late Miocene.\u201cRipanj\u201d .Melanopsishaueriserbica Brusina, 1902, non Brusina, 1893 (see Note 1).Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1870\u201cInsula Ouen\u201d [\u00cele Ouen], New Caledonia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLac Sabandja\u201d [Lake Sapanca], Turkey.Microcolpiacoutagniana .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fontannes, 1884Early Rupelian, Oligocene.PageBreak\u201cRom\u00e9jac, pr\u00e8s de Barjac (Gard)\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeIzzatullaev & Starobogatov, 1984\u201c\u0417\u0430\u043a\u0430\u0441\u043f\u0438\u0439\u0441\u043a\u0430\u044f \u043e\u0431\u043b\u0430\u0441\u0442\u044c\u201d [Transcaspian Region], Russia.Zoological Institute of Russian Academy of Sciences, St.-Petersburg; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Starobogatov in Starobogatov et al., 1992Quaternary.\u201c\u041a\u0443\u043b\u0435\u0432\u0438 (\u0431\u044b\u0432. \u0420\u0435\u0434\u0443\u0442-\u041a\u0430\u043b\u0435)\u201d , Georgia.Zoological Institute of Russian Academy of Sciences, St.-Petersburg; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1880\u201cPrope Kanala [...]; insula Ouen\u201d : 148 , Croatia.Fagotia [= Esperiana] esperi .Note that Bourguignat denoted the authority as \u201cBourguignat, 1880\u201d. Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cSpanien\u201d Not indicated.Melanopsisbuccinoidea  by Nomen nudum, apparently based on an unpublished manuscript name from Ziegler and listed in synonymy of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Magrograssi, 1928Plio-Pleistocene.\u201cCoo: V. Iracli, V. S. Giorgio\u201d , Greece.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Early\u2013middle Pannonian, late Miocene.\u201cJazvine\u201d [Jazvina], Croatia.The syntypes are stored in the Croatian Natural History Museum, Zagreb; no number indicated .Melanopsissenatoria Handmann, 1887.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Matheron, 1842Early Campanian, Cretaceous.\u201cLes Martigues\u201d, France.Melania and a junior secondary homonym of Melaniarugosa Lea, 1842 . However, the first one was published earlier .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020PageBreakHandmann, 1887 [invalid] Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.It is uncertain whether the single specimen from Wittmannsdorf near Leobersdorf stored in the Geological Survey Austria, Vienna, is really the only specimen left of the original type series , as was considered by Melanopsisrugosa Matheron, 1842. Melanopsiswolfgangfischeri as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1897[invalid]Langhian, middle Miocene.\u201cRibari\u0107\u201d, Croatia.Milan et al. (1974: 94) indicated a holotype, but it is uncertain whether the specimen is part of the original type series and whether it was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2983-629.Melanopsisrugosa Matheron, 1842. Melanopsislanzaeana Brusina, 1874.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Schr\u00e9ter, 1975[invalid]Riss/W\u00fcrm end to early W\u00fcrm Ice Age, Pleistocene.\u201cEger, az egri v\u00e1r Z\u00e1rk\u00e1ndy b\u00e1sty\u00e1j\u00e1nak vas\u00fati \u00e1tmetsz\u00e9se\u201d , Hungary.Magyar \u00c1llami F\u00f6ldtani Int\u00e9zet , Budapest; no number indicated.Melanopsisrugosa Matheron, 1842. Melanopsisdoboi to the genus Microcolpia.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fontannes, 1880Miocene or Pliocene.Rh\u00f4ne Basin? , France.PageBreakcarinatarugoso-\u201d. Based on material from the Rh\u00f4ne Basin, not Italy as claimed by Originally written as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903[invalid]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Melanopsistothi: Microcolpiaparreyssiisikorai .Junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in Brot [unavailable]\u201cBalbeck\u201d [Baalbek], Lebanon.Melaniabuccinoidea by Introduced in synonymy of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Tournou\u00ebr, 1880Early Romanian, Pliocene.\u201c\u201d , Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cKucik\u201d mentioned in Brusina (1867: 86)[unavailable]Not indicated.Nomen nudum, based on an \u201cin schedis\u201d name in the collection of Kucik .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Cernikian, Pliocene.PageBreak\u201cKova\u010devac\u201d [east of Nova Gradi\u0161ka], Croatia.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2505-151/1-2 .Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1936\u201cD\u2019Agoura\u00ef\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1864Rather unspecifically indicated in text as \u201cCours d\u2019eau du Sahara\u201d [rivers of the Sahara], but more precisely as \u201cD\u2019A\u00efn-Sidi-Taifour; Ouargla; fontaine d\u2019Oumach pr\u00e8s de Biskra\u201d  in the plate captions.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans la Save, au-dessous d\u2019Agram, en Croatie\u201d [in the Sava river below Zagreb], Croatia.Saint-Simoniana\u201d. Note that Bourguignat denoted the authority as \u201cBourguignat, 1879\u201d. Fagotia [= Esperiana] acroxia Bourguignat, 1884.Originally written as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cFoss\u00e9s d\u2019eau stagnante au camp des Pins, et \u00e7\u00e0 et l\u00e0 dans le Liban (Syrie)\u201d .Appeared first as a nomen nudum in PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cEliasbrunnen bei Jericho\u201d : 17 , Syria.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 ?Youluo, 1978Shahejie Formation (second Member), Eocene.\u201c\u5c71\u4e1c\u57a6\u5229\u201d , China.scabridus\u201d), but Melanopsis is feminine, which is why the name must be \u201cscabrida\u201d. Probably not a Melanopsidae (perhaps a Hydrobiidae or Pomatiopsidae?).Originally the gender was indicated as masculine .Appeared first as nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1856\u201cL\u2019A\u00efn-Fekan, source d\u2019eau chaude, situ\u00e9e entre Mascara et Sa\u00efda [...]; l\u2019Oued-M\u2019Ilouya, fronti\u00e8re du Maroc\u201d , Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020PageBreakHandmann, 1882 [invalid] Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsisscalaris Gassies, 1856. Melanopsislimbata as replacement name. Melanopsishaueri Handmann, 1882.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cAu pont de la Save, pr\u00e8s d\u2019Agram\u201d [at the bridge of the Sava river near Zagreb], Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeWesterlund, 1886[invalid]\u201cSiebenb\u00fcrgen b. Deva\u201d [Deva], Romania.Melanopsisscalaris Parreyss in sched.\u201d) as synonym of MelanopsisParreyssii var. \u03b2, which he introduced there with a short Latin description. Further below on p. 431, Brot even associated scalaris with an illustration of a specimen from Deva in Romania . However, the name was explicitly referred to as synonym of the (unnamed) variety \u03b2, which is not a valid name, and thus the requirements of Art. 11.6 (and therefore Art. 12.1) are not met. Melanopsisscalaris Gassies, 1856. Neubauer et al. (2014: 125) considered it as a junior synonym of Microcolpiaparreyssii.Not available from Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Papp, 1953Gelasian, early Pleistocene.\u201cR\u00f3mezi (Elis)\u201d [Rom\u00e9sion near Pyrgos], Greece.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Oppenheim, 1892[invalid]Early Campanian, Cretaceous.PageBreak\u201cPlan de Campagne bei Sept\u00eame (B.-du-Rh\u00f4ne)\u201d [Plan de Campagne near Sept\u00e8mes-les-Vallons], France.Melaniascalaroides Briart & Cornet, 1882.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928\u201cA\u00efn M\u00e9lias, pr\u00e8s de Figuig\u201d [Ain Melias near Figuig], Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Stefanescu, 1896Pliocene (Dacian?).\u201c\u00c0 Breasta, dans la vall\u00e9e de Jiu\u201d , Romania.scansorie\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeStarobogatov in Starobogatov et al., 1992\u201c\u0424\u0451\u0441\u043b\u0430\u0443 \u0431\u043b\u0438\u0437 \u0412\u0435\u043d\u044b\u201d [V\u00f6slau near Vienna], Austria.Zoological Institute of Russian Academy of Sciences, St.-Petersburg; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr, 1880Plio-Pleistocene.\u201cZwischen Pylle und Antimachia\u201d , Greece.Melanopsissporadum Tournou\u00ebr, 1876.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fuchs, 1870Middle Pannonian, late Miocene.PageBreak\u201cK\u00fap\u201d, Hungary.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1920Plio-Pleistocene.\u201cAntimaki\u201d  the taxon occurred and in which not, Croatia.Melanopsislyrata Neumayr, 1869.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeDeshayes, 1832\u201cDans l\u2019Ohio\u201d [in the Ohio river], United States.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[unresolved]Not explicitly stated but probably the same as for the species .Melanopsisseuratisemilaevigata. This case requires the action of a First Reviser.Homonym of the simultaneously published name Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[unresolved]Not explicitly stated but probably the same as for the species .Melanopsiseximiasemilaevigata. This case requires the action of a First Reviser.Homonym of the simultaneously published name Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1936[invalid]\u201cTanalt\u201d, Morocco.Melanopsisseuratisemilaevigata Pallary, 1928 and Melanopsiseximiasemilaevigata Pallary, 1928 .Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[invalid]Not explicitly stated but probably the same as for the species .Melanopsisseuratisemilaevigata Pallary, 1928 and Melanopsiseximiasemilaevigata Pallary, 1928 .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBrusina, 1870PageBreak\u201cDans la Glina pr\u00e8s de Topusko\u201d [in the Glina river near Topusko], Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr, 1880Phoka to Elia Formation, Plio-Pleistocene.\u201cPhuka\u201d [\u00c1kra \u00c1gios Fok\u00e1s], Greece.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]Not explicitly stated but probably the same as for the species .Melanopsissemiplicata Neumayr, 1880.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cA\u00eft Brahim\u201d, Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020De Stefani, 1877Villafranchian, Plio-Pleistocene.\u201cOrciano\u201d : 524 \u201cStations 117 et 117his: Oued Bou Fekrane \u00e0 13 kilom\u00e8tres de Mekn\u00e8s sur la route d\u2019El Hajeb\u201d , Morocco.Melanopsismagnifica Bourguignat, 1884.Introduced as infrasubspecific taxon (\u201cmode\u201d), which is not governed by the provisions of the Code. Moreover, the name is a nomen nudum. P\u00e9r\u00e8s referred to an earlier publication of his , claiminTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Oluji\u0107, 1999Langhian, middle Miocene.It is unclear from the original work in which of the studied localities/sections along the valleys of the Sutina, Batarelov and Vojskava rivers (4 km W of Sinj) the taxon occurred and in which not, Croatia.Melanopsislanzaeana Brusina, 1874.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Wenz, 1928Duab Beds, middle to late Kimmerian, Pliocene.\u201c\u041c\u043e\u043a\u0432\u0438\u043d\u0441\u043a\u0456\u0435 \u043f\u043b\u0430\u0441\u0442\u044b, \u0440. \u0414\u0443\u0430\u0431\u044a\u201d : 62 Pannonian, zone D\u2013E, late Miocene.\u201cRipanj\u201d, Serbia.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2530-176/1-2 .Melanopsisserbica Brusina, 1893. Melanopsishaueriripanjensis as replacement name, following Melanopsisaustriaca Handmann, 1882 with Melanopsishaueri Handmann, 1882 and listed serbica as a subspecies of Melanopsishaueri.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Vidal, 1874Maastrichtian, Cretaceous.\u201cEn Serchs y en Auzas (Alto Garona), pero principalmente en Isona\u201d .Stilospirula by Considered to belong to the subgenus Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cJardin public de bou Jeloud, oued Masmouda, avant son entr\u00e9e \u00e0 F\u00e2s el b\u00e2li\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Mayer-Eymar, 1903PageBreakSparnacian, early Ypresian, Eocene.\u201cDu village de la Serre\u201d [La Serre (?)], France.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLa Save \u00e0 Sissek (Slavonie) et dans la rivi\u00e8re de Zenica (Bosnie)\u201d [Sava river at Sisak (Croatia) and in the river at Zenica (Bosnia and Herzegovina)].Microcolpiacornea .Appeared first as a nomen nudum in Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRivi\u00e8res entre Plaski et Ostaria (Croatie)\u201d [river between Pla\u0161ki and O\u0161tarije], Croatia.Note that Bourguignat denoted the authority as \u201cBourguignat, 1882\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cPetit cours d\u2019eau \u00e0 Sadjour-Sou, entre A\u00efn-Ta\u00efb et Alep [...]; A\u00efn-el-Bass, dans la plaine du Bahr-el-Houl\u00e9 (Syrie)\u201d [small brook at Sadjour-Sou between Gaziantep (Turkey) and Aleppo (Syria) [...]; A\u00efn el Bass, in the plains of the Hula valley (Israel)].Melanopsisbuccinoidea .Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cLa Zousfana, \u00e0 la hauteur de Figuig et \u00e0 Beni Ounif\u201d , Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGrateloup, 1840PageBreak\u201cS\u00e9ville; les bords de la petite rivi\u00e8re de Guadaira, qui se jette dans le Guadalquivir\u201d , Spain.minor [...] de Grateloup\u201d as discussed by Melanopsiscariosa .The \u201cvar. Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBandel, 2000\u201cSpring and creek next to the Wadi Raiyan Plantation in the Jordan Valley near the town of Wadi Raiyan and close to the mosque of the grave of Sharhabil Ibn Hassana\u201d, Jordan.Geological-Palaeontological Institute and Museum University of Hamburg, coll. no. 4267.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1922\u201cAoullouz\u201d [Alous], Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cMrts.\u201d mentioned in Paetel (1888: 404)[unavailable]\u201cSiam\u201d, Thailand.Melampussiamensis Martens, 1866 (Ellobiidae).Nomen nudum, appears only in the species list of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Heller & Sivan, 2001[unavailable]Pleistocene.\u201cGesher Benot Ya\u2019aqov\u201d, Syria.Since Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903Late Pleistocene\u2013Holocene.PageBreak\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssii by Considered a subspecies of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d , AustriaMelanopsisplicatula Handmann, 1887, non Brusina, 1874. Melanopsisbouei F\u00e9russac, 1823.Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Brusina (1903: 112)[unavailable]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai  by Nomen nudum. Listed in synonymy of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1925[invalid]Cernikian, Pliocene.\u201cMalino\u201d : 348, CrMelanopsissimulata as replacement name for the presumed junior homonym Melanopsisarcuata Brusina, 1878, non Matheron, 1842. The alleged homonymy is, however, based on a reading error of Melanopsisarmata Matheron, 1842 by Pallary. Therefore, Melanopsissimulata is a junior objective synonym of Melanopsisarcuata Brusina, 1897.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Early Langhian, middle Miocene.\u201cSinj (Stuparu\u0161a)\u201d, Croatia.PageBreakfixation is insufficient with respect to Art. 75.3 of the Code. The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2974-620/1.Milan et al. (1974: 97) defined a \u201cneotype\u201d based on one of the specimens illustrated by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1885Spaniodon Beds, Karaganian, middle Miocene.\u201c\u041b\u043e\u043f\u0443\u0448\u043d\u044b\u201d : 5 Transdanubian, Pannonian, late Miocene.\u201cTinnye\u201d, Hungary.Melanopsissinzowi Brusina, 1885. Melanopsistinnyensis as replacement name. Melanopsisrarispina L\u00f6renthey, 1902. However, Melanopsisboueispinosa Handmann, 1882 and Melanopsisboueiventricosa Handmann, 1882 in synonymy of Melanopsisrarispina. While ventricosa Handmann is a junior homonym of Melanopsisventricosa Neumayr, 1880 and thus invalid, the name spinosa is available and has priority over rarispina.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeIzzatullaev & Starobogatov, 1984\u201c\u0412\u043e\u0441\u0442\u043e\u0447\u043d\u0430\u044f \u041f\u0435\u0440\u0441\u0438\u044f\u201d [Eastern Persia], Iran.Zoological Institute of Russian Academy of Sciences, St.-Petersburg; no number indicated.Melanopsis (Sistaniana) Izzatullaev & Starobogatov, 1984.Type species of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Papp & Psarianos, 1955Early Pleistocene.\u201cT\u00e1rapsa\u201d [Vasil\u00e1kion], Greece.Museum of Palaeontology and Geology of the University of Athens; no number indicated.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Papp, 1955Akchagylian, latest Pliocene\u2013earliest Pleistocene.\u201cSkura bei Sparta\u201d [Sko\u00fara], Greece.Museum of Palaeontology and Geology of the University of Athens; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr in Neumayr & Paul, 1875Cernikian, Pliocene.\u201cGraben zwischen Podwin und der \u010capla\u201d [\u010caplja trench near Slavonski Brod], Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Rosen, 1914Pleistocene?\u201cAm Fusse des \u010cernjajev\u2019schen Berges bei Suchum\u201d [at the foot of Mt. \u010cernjajev (?) at Sokhumi], Georgia.Melanopsissobrievskii\u201d within the subgenus Fagotia. In the plate captions it appears as \u201cFagotiasobrievskii\u201d.On p. 221 Rosen gave the species as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Jekelius, 1944Mohrensternia Zone, early Sarmatian, middle Miocene.\u201cPoli\u021bioan\u0103tal bei Soceni\u201d [Poli\u021bioan\u0103 valley near Soceni], Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Harzhauser, Kowalke & Mandic, 2002Pannonian, zone C\u2013D, late Miocene.\u201cSt. Margarethen (Burgenland)\u201d , Austria.Geological-Paleontological Department, Natural History Museum Vienna, Austria, coll. no. 2001/0126/0049.PageBreakMelanopsissoceni Jekelius, 1944.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Deshayes, 1862Thanetian, Paleocene.\u201cCh\u00e2lons-sur-Vesles, Gueux, Jonchery, Noailles\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020De Stefani in Pantanelli, 1877Pantanelli 1877: 5.Late Messinian, late Miocene.\u201cCasino, e specialmente presso le Gallozzole\u201d , Italy.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1871New Caledonia [no locality indicated].Melanopsisfrustulum Morelet, 1857.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1897Latest Burdigalian, early Miocene.\u201cDugoselo\u201d, Croatia.Milan et al. (1974: 97) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2999-645.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Porumbaru, 1881PageBreakEarly\u2013middle Romanian, Pliocene.\u201cCretzesci, Podari\u201d , Romania.soubeiranus\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1870\u201cIn Nova Caledonia\u201d .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1920[invalid]Headon Beds, Priabonian, Eocene.\u201cHordwell\u201d : 36, UniMelanopsisfusiformis Sowerby, 1846  [sic]. The name is based on an error of Melanopsispseudosubulata for that specimen and another specimen already  antiqua; elongata\u201d. Note that Melanopsissparnacensis was not included in the catalogus of Based on a specimen illustrated by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.PageBreak\u201cLeobersdorf\u201d, Austria.Melanopsisfossilis .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cRivi\u00e8re entre Ostaria et Plaski, dans la Croatie m\u00e9ridionale\u201d [river between Pla\u0161ki and O\u0161tarije], Croatia.Note that Bourguignat denoted the authority as \u201cBourguignat, 1879\u201d.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans l\u2019Oronte (Syrie)\u201d [in the Orontes river], Syria?sphoeroidoea\u201d on p. 78 but as \u201csphaeroidaea\u201d on p. 73. From the description it is clear that the name must be \u201csphaeroidaea\u201d (see Art. 33.2.1).Spelt as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Stefanescu, 1896Khersonian, late Sarmatian, late Miocene.\u201c\u00c0 Sacel, dans la vall\u00e9e de Blahnitza, district de Gorjiu\u201d , Romania.Melanopsisbouei F\u00e9russac, 1823.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Rolle, 1860Late Pliocene to early Pleistocene.\u201cSkalis prope Schoenstein\u201d [Pesje near \u0160o\u0161tanj], Slovenia.spinicosta\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sowerby in Sedgwick & Murchison, 1832Late Cretaceous.PageBreak\u201cGosau\u201d, Austria.Holotype (?) stored in the collection of the British Museum, coll. no. G 17908 : 58.Megalonoda, which was there described as new.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Seninski, 1905Duab Beds, middle to late Kimmerian, Pliocene.\u201c\u041c\u043e\u043a\u0432\u0438\u043d\u0441\u043a\u0456\u0435 \u043f\u043b\u0430\u0441\u0442\u044b, \u0440. \u0414\u0443\u0430\u0431\u044a\u201d [Mokvi layers at Duab river], Georgia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsisboueiventricosa Handmann, 1882 in synonymy of Melanopsisrarispina L\u00f6renthey, 1902, although Handmann\u2019s taxa were published earlier. The name ventricosa Handmann is a junior homonym of Melanopsisventricosa Neumayr, 1880 and thus invalid, but spinosa is available and has priority over rarispina.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Melanopsisfossilis .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeChenu, 1859Not stated; unclear if recent or fossil.Not indicated.The species is based on a single illustration, without description or any kind of explanation.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Leymerie, 1881[invalid]Maastrichtian, Cretaceous.\u201cAuzas\u201d, France.Melanopsisspriata Chenu, 1859.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Latest Burdigalian, early Miocene.\u201cDugoselo\u201d : 9, CroaMelanopsispraemorsa sensu Brusina, 1897, non Linnaeus, 1758. Melanopsispraemorsa from the late Miocene deposits of Lake Pannon and the Pliocene of Greece and Croatia under that name, but it is very unlikely that all of them belong to the same species.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Tournou\u00ebr, 1876Phoka to Elia Formation, Plio-Pleistocene.\u201cFouka\u201d (p. 449) [\u00c1kra \u00c1gios Fok\u00e1s], Greece.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Oppenheim, 1891Gelasian, early Pleistocene.\u201cStamna\u201d, Greece.Not available from Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Anistratenko, 1993Duab Beds, middle to late Kimmerian, Pliocene.\u201c\u041e\u043a\u0440. \u0441. \u041c\u043e\u043a\u0432\u0438, \u041e\u0447\u0430\u043c\u0447\u0438\u0440\u0441\u043a\u0438\u0439 \u0440-\u043d\u201d , Georgia.Schmalhausen Institute of Zoology of National Academy of Sciences of Ukraine, Kiev; no number indicated.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeIzzatullaev & Starobogatov, 1984\u201c\u0413\u0435\u0440\u043c\u043e\u0431 (\u0426\u0435\u043d\u0442\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u041a\u043e\u043f\u0435\u0442\u0434\u0430\u0433)\u201d [Germob ], Iran.Zoological Institute of Russian Academy of Sciences, St.-Petersburg; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903Late Pleistocene\u2013Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai .Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cRas el \u2018Ain du Khabour\u201d [Chabur river near Ra\u2019s al \u2018Ayn], Syria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLa Save au-dessous d\u2019Agram; rivi\u00e8res au sud de Krapina-Toeplitz, et entre Plaski et Ostaria (Croatie)\u201d , Croatia.Fagotia [= Esperiana] acroxia Bourguignat, 1884.Note that Bourguignat denoted the authority as \u201cBourguignat, 1879\u201d. Taxon classificationAnimaliaSorbeoconchaMelanopsidaeWesterlund, 1886PageBreakkurhaus of Rimske Toplice, takes its thermal water runoff, and flows over a steep gradient into the Savinja river], Slovenia.\u201cDer untere Lauf jenes kleinen Gebirgsbaches, welcher den Abh\u00e4ngen des Berges Kopitnig entspringt, am hochgelegenen Kurhause von R\u00f6merbad vorbeieilend, dessen Thermenabfluss aufnimmt, und in starkem Gef\u00e4lle der den Fuss des Berges umsp\u00fclenden Sann zufliest\u201d : 105\u2013106Melaniaholandri illustrated by Introduced for a specimen of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884Rossm\u00e4ssler, to whom Bourguignat referred, indicated the locality only as \u201cPal\u00e4stina\u201d [Palestine]. As the species was explicitly introduced for Rossm\u00e4ssler\u2019s material only this is the type locality.Melaniacostata sensu Kobelt, 1880 (figs 1899\u20131900), non Olivier, 1804.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201c\u2018Ain Arouss\u201d [\u2018Ayn al \u2018Ar\u016bs ], Syria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cCarniole\u201d .Microcolpiaacicularis .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeReeve, 1860Melanopsis, pl. 1, figs 3a\u2013b.\u201cNew Zealand\u201d [no locality indicated].Zemelanopsistrifasciata .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Cernikian, Pliocene.\u201cKova\u010devac\u201d [east of Nova Gradi\u0161ka], Croatia.PageBreakThe illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2521-167/1-5 .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Grateloup, 1838[invalid]Burdigalian, early Miocene.\u201cDax. [...] Mandillot\u201d, France.Melaniabuccinoideasubventricosa Grateloup, 1828, which Junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePantanelli, 1886\u201cAccesa\u201d, Italy.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeWesterlund, 1886[invalid]\u201cSteinbr\u00fcck\u201d [Zidani Most], Slovenia.Melaniastriata Sowerby, 1818, described from the Paleogene (Eocene?) of the United Kingdom.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887[invalid]Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Melanopsismaroccanastriata Pantanelli, 1886 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePantanelli, 1886\u201cAccesa\u201d, Italy.striata-carinata\u201d.Originally written as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaePageBreakPantanelli, 1886 \u201cAccesa\u201d, Italy.striata-sulcata\u201d.Originally written as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pavlovi\u0107, 1927Middle Pannonian, late Miocene.\u201c\u0423 \u043a\u0430\u0440\u0430\u0433\u0430\u0447\u043a\u0438\u0445 \u0436\u0443\u0442\u0438\u043c \u043f\u0435\u0441\u043a\u043e\u0432\u0438\u043c\u0430\u201d [from the yellow sands of Karaga\u010da near Vr\u010din], Serbia.The illustrated syntype is stored in the Natural History Museum, Belgrade, coll. no. 212 (Milo\u0161evi\u0107 1962: 23).Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1924Spain [no locality indicated].Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[invalid]\u201cEnvirons de Beyrouth\u201d [surroundings of Beirut], Lebanon.Melanopsisdufouriistricta Pallary, 1924 (see Note 1).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1892Middle Pannonian, late Miocene.\u201cMarku\u0161evec\u201d, Croatia.The illustrated syntype is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2496-142/1 .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Jekelius, 1944[invalid]Early Pannonian, late Miocene.\u201cTurislav-Tal bei Soceni\u201d [Turislav valley near Soceni], Romania.Melanopsisstricturata Brusina, 1892 .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fuchs, 1873Transdanubian, Pannonian, late Miocene.\u201cMoosbrunn bei Wien; Tinnye bei Ofen\u201d .sturi\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeSch\u00fctt & Bilgin, 1974\u201cRiver Pinios [Pinei\u00f3s] in Thessalia, Greece, between Larissa and Tempe [Temp\u00f3n] valley\u201d, Greece.Senckenberg Forschungsinstitut und Naturmuseum Frankfurt, coll. no. SMF 111523a.situssineri\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeAhuir Galindo, 2016\u201cSoutheastern of Guelmin, at river Seyad basin\u201d, Morocco.Museo Malacologico di Cupra Marittima, Italy; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916[invalid]Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d : 558, AuMelanopsisaffinis Handmann, 1882, \u201cnon F\u00e9russac, 1823\u201d. The latter name is, however, unavailable PageBreakfrom Melanopsisaffinis). This makes Melanopsissubaffinis an objective synonym of Melanopsisaffinis Handmann, 1882 and Melanopsisaffinis Pallary, 1916 a junior homonym of Melanopsisaffinis Handmann, 1882. Melanopsissubaffinis was considered as a junior synonym of Melanopsisbouei by Replacement name for the alleged junior homonym Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1922\u201cMarrakech\u201d, Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sandberger, 1875Kirchberg Formation, middle Burdigalian, early Miocene.\u201cKirchberg\u201d [Illerkirchberg], Germany.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020So\u00f3s in Bartha, 1955Transdanubian, Pannonian, late Miocene.\u201cV\u00e1rpalota\u201d, Hungary.Bartha clearly stated that the description was made by Lajos So\u00f3s.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Gozhik, 2002Middle Pontian (Dacian Basin), late Miocene.\u201c\u0412\u0438\u043d\u043e\u0433\u0440\u0430\u0434\u0456\u0432\u043a\u0430\u201d , Ukraine.Institute of Geological Sciences of the National Academy of Sciences of Ukraine, coll. no. 3185.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020d\u2019Orbigny, 1852Burdigalian, early Miocene.\u201cDax, St-Paul. Mandillot\u201d . According to Art. 24.1 and 57.7 the name proposed at higher rank takes precedence. This makes subcallosa an objective synonym of Melanopsistaurinensis Sacco, 1889, while the subspecies Melanopsisclavataurinensis Sacco, 1889 is still in need of a substitute name.Invalid replacement name for the homonym Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Deshayes in F\u00e9russac, 1839Late Villafranchian, Pleistocene (?).\u201cD\u2019Italie\u201d .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Morris in Forbes, 1856[invalid]Headon Beds, Priabonian, Eocene.\u201cHeadon Hill\u201d, United Kingdom.Melanopsissubcarinata Deshayes in F\u00e9russac, 1851.The description was evidently performed by Morris , France.Melaniacostata sensu Deshayes, 1825, non Olivier, 1804.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884[invalid]\u201cDans l\u2019Oronte\u201d : 168 Late Cernikian, late Pliocene\u2013early Pleistocene.\u201cNovska (Bukovica)\u201d, Croatia.Milan et al. (1974: 88) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2993-639.Melanopsissubcostata d\u2019Orbigny, 1850. Melanopsisconstrictanowskaensis as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1904\u201cSriratzel Cromfel sur la route de Rabat \u00e0 Casablanca; Temslott, dans les canaux; Agagour dans l\u2019Atlas\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1912[invalid]\u201cDe Ngoussa\u201d [N\u2019Goussa], Algeria.Melanopsissubcostulata Pallary, 1904.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePageBreakP\u00e9r\u00e8s, 1939 Several collection stations near Mekn\u00e8s, F\u00e8s, A\u00efn El Aouda and A\u00efn Chkef, Morocco : 140.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Grateloup, 1838Burdigalian, early Miocene.\u201cDax. Mandillot, \u00e0 Saint-Paul\u201d, France.Melanopsisdufourii F\u00e9russac, 1822 from Dax illustrated in Introduced for a specimen of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Morris in Forbes, 1856Headon Beds, Priabonian, Eocene.\u201cFrom the Headon series\u201d, United Kingdom.The description was evidently performed by Morris \u201d, Morocco.PageBreakPageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cAu pont de la Save, pr\u00e8s d\u2019Agram\u201d [at the bridge of the Sava river near Zagreb], Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Kormos, 1905Late Pleistocene\u2013early Holocene.\u201cP\u00fcsp\u00f6kf\u00fcrd\u0151\u201d , Romania.Microcolpiaparreyssii .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Lutetian, Eocene.\u201cAu Nord d\u2019Albas\u201d : 202, FrMelanopsiselongata Doncieux, 1908, non F\u00e9russac, 1822 (see Note 1). subelongata\u201d, which is an incorrect subsequent spelling.Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939\u201cDjishr ech Chegour\u201d [Jisr Ash-Shughur], Syria.Melanopsismultiformis Blanckenhorn, 1897. Melanopsiscostata .Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939Not explicitly stated but probably the same as for the species ], Iraq).PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Bogachev, 1908Late Sarmatian, Khersonian, late Miocene.\u201cChatma, district de Signakh, gouv. de Tiflis\u201d , Georgia.praemorsasub-\u201d.Introduced originally as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Andrusov, 1909Pontian (sensu stricto), late Miocene.\u201cAdjipirdariaki, O. von Marasy, Tscharagan, [...] Sundi, [...] Chinasty\u201d , Azerbaijan.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Penecke, 1884Cernikian, Pliocene.\u201cCapla-Graben\u201d [\u010caplja trench near Slavonski Brod], Croatia.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020d\u2019Orbigny, 1850[invalid]Early Campanian, Cretaceous.\u201cMartigues\u201d : 293, FrMelanopsisrugosa Matheron, 1842 a Melania and the species name a secondary homonym of Melaniarugosa Lea, 1842. However, Matheron\u2019s species was published earlier . Melaniasubrugosa is thus a junior objective synonym of Melanopsisrugosa Matheron, 1842.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884PageBreakkan, hot spring between Mascara and Sa\u00efda; river Moulouya, at the border to Morocco], Algeria.\u201cA\u00efn-Fekan, source d\u2019eau chaude entre Mascara et Sa\u00efda; dans Oued-M\u2019Ilouya, sur la fronti\u00e8re du Maroc\u201d .Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Jekelius, 1944Early Pannonian, late Miocene.\u201cTurislav-Tal bei Soceni\u201d [Turislav valley near Soceni], Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBrot, 1878\u201cPerna Fluss in Ungarn\u201d [not found], Hungary?HemisinusEsperi\u201d [now in Esperiana], based on a manuscript name by Parreyss. Treated as a valid name by Described and illustrated in synonymy of \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1920[invalid]Transdanubian, Pannonian, late Miocene.\u201cTihany\u201d (Fuchs 1870: 533), Hungary.Melanopsisaquensis sensu Fuchs, 1870, non Grateloup, 1838. Junior homonym of Melanopsissubtilis Brot, 1879. That name was introduced in synonymy but became available from Melanopsispetrovici Brusina, 1893.Introduced for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Magrograssi, 1928[invalid]Plio-Pleistocene.\u201cCoo: molto frequente in tutte e due le zone fossilifere\u201d , Greece.PageBreakMelanopsissubtilis Brot, 1879, which was originally introduced in synonymy but made available by Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeAnnandale, 1918\u201cBasra\u201d, Iraq.Indian Museum, Calcutta, coll. no. 11390/2M.Annandale attributed the authority to Nevill, apparently based on a manuscript of that author.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Late Villafranchian, Pleistocene.\u201cD\u2019Italie\u201d , but Melanopsis is feminine, which is why the name must be \u201csubulata\u201d.Originally the gender was indicated as masculine  [Babin Dol near Skopje], Macedonia.The illustrated syntype is stored in the Natural History Museum, Belgrade, coll. no. 1447 (Milo\u0161evi\u0107 1962: 24).Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Esu & Girotti, 2015Synania Formation, Pleistocene.\u201cS-SW of Neos Erineos\u201d [near Aigio], Greece.Senckenberg Forschungsinstitut und Naturmuseum Frankfurt, coll. no. SMF 345712.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Kormos, 1905Late Pleistocene\u2013early Holocene.PageBreak\u201cP\u00fcsp\u00f6kf\u00fcrd\u0151\u201d , Romania.Microcolpiaparreyssii .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020H\u00f6rnes, 1856Badenian, middle Miocene.\u201cGrund\u201d, Austria.Semisinus P. Fischer, 1885, which is an unjustified emendation of Hemisinus Swainson, 1840 (Thiaridae).After Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cF\u00e9r.\u201d mentioned in De Cristofori and Jan [unavailable]\u201cAustr.\u201d [Australia].Nomen nudum, found only in the species list of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans les cours d\u2019eau de la plaine du Bahr-el-Houl\u00e9, non loin d\u2019Ain-el-Mellaha\u201d , Israel.Melanopsislampra Bourguignat, 1884.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sacco, 1889[invalid]Late Burdigalian, early Miocene.\u201cColline torinesi\u201d [Torino hills], Italy.Melanopsistaurinensis Sacco, 1889. Homonym of the simultaneously described PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sacco, 1889Late Burdigalian, early Miocene.\u201cColle torinesi\u201d [Torino hills], Italy.Melanopsisclavataurinensis Sacco, 1889. Melanopsissubcallosa as replacement name for this species .Homonym of the simultaneously described Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Sacco, 1895Late Burdigalian, early Miocene.\u201cColli torinese\u201d [Torino hills], Italy.Amphimelania by taurostricta\u201d, which is an incorrect subsequent spelling.Considered to belong in the genus Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Heller & Sivan, 2002Early Pleistocene.\u201c\u2018Erq el-Ahmar\u201d [locality also known as Gesher], Israel.Paleontology Collection of the Hebrew University of Jerusalem, coll. no. HUJ 9016.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr, 1880Late Burdigalian\u2013early Langhian, early\u2013middle Miocene.\u201cSeonica\u201d, Bosnia and Herzegovina.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903[invalid]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.PageBreakMelanopsisvidovici: Microcolpiaparreyssiisikorai .Junior objective synonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Transdanubian, Pannonian, late Miocene.\u201cRadmanest\u201d [R\u0103dm\u0103ne\u0219ti], Romania.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2512-158/1-2 .tesselata\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Melanopsisinermis Handmann, 1882.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903Late Pleistocene\u2013Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai .Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBrot, 1868\u201cCarpazi [...], Miskolz [...]\u201d .Brot attributed the authority to \u201cTitius (?) (Parreyss)\u201d, probably based on an \u201cin schedis\u201d determination.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Tournou\u00ebr, 1877Late Pannonian, late Miocene.\u201cSmendou\u201d [Zighoud Youcef], Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1936\u201cCascade d\u2019Imouzer d\u2019Agadir\u201d [waterfall of Imouzzer at Agadir], Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cF\u00e9russ.\u201d mentioned in Adams and Adams (1853\u20131858: 310)[unavailable]Not indicated.Nomen nudum, appears only in Adams and Adams (1854) without explanation.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Wenz, 1928Transdanubian, Pannonian, late Miocene.\u201cTihany\u201d (Fuchs 1870: 533), Hungary.Melanopsisgradata Fuchs, 1870. Both taxa were listed as junior synonyms of Melanopsisbrusinai L\u00f6renthey, 1902 by Replacement name for the junior homonym Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020\u017divkovi\u0107, 1893Badenian, middle Miocene.\u201cVra\u017eogrnac unterhalb des Einflusses des Alapin in den Timok\u201d , Serbia.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeMorelet, 1864\u201cIn Marocco\u201d [no locality indicated].PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Wenz, 1919Middle Pannonian, late Miocene.\u201cTinnye\u201d : 214, HuMelanopsissinzowi L\u00f6renthey, 1902, non Brusina, 1885. Melanopsisrarispina L\u00f6renthey, 1902. However, Melanopsisboueispinosa Handmann, 1882 and Melanopsisboueiventricosa Handmann, 1882 in synonymy of Melanopsisrarispina. While ventricosa Handmann is a homonym of Melanopsisventricosa Neumayr, 1880 and thus invalid, the name spinosa is available and has priority over rarispina.Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Popescu-Voite\u0219ti, 1910Lutetian, Eocene.\u201cGropile Vulpilor pr\u00e8s Tite\u0219ti\u201d [Gropile Vulpilor (?) near Tite\u0219ti], Romania.Coptostylusalbidus  (Thiaridae).Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Bogachev [unavailable]Miocene.\u201cTori bei Borshomi\u201d [Tori near Borjomi], Georgia.The name was only mentioned in a species list by Bogachev without description or illustration.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Blanckenhorn in Blanckenhorn & Oppenheim, 1927Plio-Pleistocene.Dreissena layer at Jisr Ash-Shughur], Syria.\u201cAus der plioc\u00e4nen Dreissensiaschicht von Djisr esch-Schughr\u201d , Morocco.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Papp, 1953Pannonian, zone F\u2013G, late Miocene.\u201cMoosbrunn, N.-\u00d6.\u201d, Austria.Melanopsisboueitortispina) in Appeared first as nomen nudum .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1920Tafi Formation, early Pleistocene.\u201cAntimaki\u201d : 449 , Turkmenistan.Zoological Institute of Russian Academy of Sciences, St.-Petersburg; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Blanckenhorn, 1897Plio-Pleistocene.Dreissena layers at Jisr Ash-Shughur], Syria.\u201cIn der Dreissensiaschicht von Dschisr esch-Schurr\u201d  \u201cM. Mario: Farnesina\u201d, Italy.Melanopsistransiens Blanckenhorn, 1897. Melanopsiscerullii as replacement name. Melanopsisaffinis F\u00e9russac\u201d, which is, however, not an available name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Cernikian, Pliocene.\u201cBe\u010di\u0107; Podvinje (\u010caplja) [\u010caplja trench near Slavonski Brod]; Novska; Kova\u010devac; Moslavina\u201d, Croatia.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2520-166/1-2 .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Bandel, 2000Valcarga Formation, Campanian, Cretaceous.\u201cPumanous slump exposed near Torallola\u201d, Spain.Geological-Palaeontological Institute and Museum University of Hamburg, coll. no. 4270.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBrugui\u00e8re, 1789Not indicated.Melanopsistricarinata or as Melanopsisdufouriitricarinata  in Melanopsistricarinata . PageBreakPleurocerastrombiforme  (Pleuroceridae).Appeared first as nomen nudum , but Melanopsis is feminine, which is why the name must be \u201ctrifasciata\u201d. Type species of Zemelanopsis Finlay, 1926. The name \u201cbifasciata\u201d as mentioned in Originally the gender was indicated as masculine [unavailable]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai .Nomen nudum (Brusina apparently considered the term self-explanatory). Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Brusina (1903: 114)[unavailable]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai .Nomen nudum (Brusina apparently considered the term self-explanatory). Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Bourguignat, 1880Langhian, middle Miocene.\u201cVall\u00e9e de la Cettina\u201d [Cetina river valley], Croatia.Melanopsisgeniculata Brusina, 1874.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in Brot (1874\u20131879: 419)[unavailable]Not indicated.Melaniabuccinoidea  by Nomen nudum, \u201cin schedis\u201d name from Tarnier listed in synonymy of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pashko, 1968Tortonian, late Miocene.\u201cSuita e Burrelit, prerja e Zallit t\u00eb G\u00ebrmanit\u201d , Albania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Locard, 1883Mammal zone MN 14, Pliocene.\u201cTr\u00e9voux\u201d (p. 53), France.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fuchs, 1870Transdanubian, Pannonian, late Miocene.\u201cRadmanest\u201d [R\u0103dm\u0103ne\u0219ti], Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Hoernes, 1876Late Sarmatian, Khersonian, late Miocene.\u201cRenki\u00f6i\u201d [north of \u0130ntepe], Turkey.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1884Early Langhian, middle Miocene.\u201cPotravlje\u201d (p. 47), Croatia.PageBreakMilan et al. (1974: 98) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2973-619.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in De Cristofori and Jan [unavailable]\u201cAm. mer.\u201d [South America].Nomen nudum, only mentioned in a species list by Taxon classificationAnimaliaSorbeoconchaMelanopsidaeWesterlund, 1886\u201cUferstellen der Sann bei R\u00f6merbad\u201d : 102 , Serbia.Perhaps stored in the Natural History Museum, Belgrade, but not mentioned in the catalogue of Milo\u0161evi\u0107 (1962).Taxon classificationAnimaliaSorbeoconchaMelanopsidaeYen, 1939[invalid]\u201cSee von Ta-li-fu (Y\u00fcnnan)\u201d , China.Melanopsistuberculata Pavlovi\u0107, 1927. Probably not a Melanopsidae.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020De Stefani, 1880Late Villafranchian, Pleistocene.PageBreak\u201cMonticiano\u201d, Italy.Melanopsisaffinis F\u00e9russac\u201d, which is not an available name.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201c\u201d mentioned in Brot [unavailable]\u201cTaurus-Gebiete\u201d [Taurus region], Turkey.Melaniabuccinoidea by Introduced in synonymy of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1916Stated to be uncertain by Melanopsisobesa , non Gassies, 1856. Despite a valid nomenclatural act, the name is hardly useful because Philippi\u2019s species is certainly no Melanopsidae.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeMorlet, 1881\u201cTozeur, Kriz\u201d, Tunisia.tuneata\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeP\u00e9r\u00e8s, 1939\u201cStation 31. Da\u00efa Afourgagh pr\u00e8s d\u2019Annosseur. Station 144. Oued Sidi Raba \u00e0 6 kilom\u00e8tres environ en amont de son confluent avec le Bou-Regreg\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePageBreakMousson, 1874 \u201cDans le Karasu, affluent du lac d\u2019Antioche (p. 33); environs de Samava \u201d .Mousson attributed the authority to Parreyss, but there is no evidence that the description really derived from that author.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fischer-de-Waldheim, 1837Jurassic?\u201cMiatchkova\u201d [Myachikovo?], Russia.Melanopsis.Certainly not a Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBrot, 1878[invalid]\u201cGlina Fluss in Ungarn\u201d [Glina river], Croatia.HemisinusEsperi\u201d [now in Esperiana], based on a manuscript name by Parreyss. Melanopsisturgida Fischer-de-Waldheim, 1837. Fagotia [= Esperiana] esperi .Described and illustrated in synonymy of \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928[invalid]\u201cMoula\u00ef Ta\u00efeb et Taourirt du Z\u00e2 \u201d [Moulay Ta\u00efeb and in Oued Za at Taourirt (?) (eastern Morocco)], Morocco.Melanopsisturgida Fischer-de-Waldheim, 1837.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Jekelius, 1944Early Pannonian, late Miocene.\u201cTurislav-Tal bei Soceni\u201d [Turislav valley near Soceni], Romania.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePageBreakIzzatullaev & Starobogatov, 1984 \u201c\u0417\u0430\u043a\u0430\u0441\u043f\u0438\u0439\u0441\u043a\u0430\u044f \u043e\u0431\u043b\u0430\u0441\u0442\u044c\u201d [Transcaspian Region], Russia.Zoological Institute of Russian Academy of Sciences, St.-Petersburg; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Matheron, 1842Early Campanian, Cretaceous.Not stated after the description, as Matheron used to do it for the other species. Although he did not denote an exact locality, he mentioned earlier in text the occurrence of the species in deposits situated \u201cvers Vitrolles et Martigues\u201d (p. 148\u2013149) .Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1939[invalid]\u201c\u2018Ain Arouss\u201d , Syria.Melanopsisturricula Matheron, 1842.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Magrograssi, 1928Plio-Pleistocene.\u201cCoo\u201d [Kos island], Greece.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePicard, 1934\u201cJarmukm\u00fcndung\u201d [Yarmouk river mouth], Jordan/Israel.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeRossm\u00e4ssler, 1854[invalid]PageBreak\u201cIm Gebiet des unteren Guadalquivir\u201d [in the lower Guadalquivir river], Spain.Melanopsissevillensis Grateloup, 1840, arguing that \u201cGrateloup elevated this variety to species level without reason\u201d. Thus, Melanopsisturrita is an objective synonym of Melanopsissevillensis.Rossm\u00e4ssler (1854) based this new variety entirely on Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1887[invalid]Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d, Austria.Melanopsisturrita Rossm\u00e4ssler, 1854. Melanopsishispidula as replacement name.Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Jekelius, 1944[invalid]Early Pannonian, late Miocene.\u201cTurislav-Tal bei Soceni\u201d [Turislav valley near Soceni], Romania.Melanopsisturrita Rossm\u00e4ssler, 1854 and junior synonym of Melanopsisturrita Handmann, 1887, for which Melanopsishispidula as replacement name .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Blanckenhorn, 1897Plio-Pleistocene.\u201cIn der obersten Melanopsiden-Thonbank des rechten Orontesufers bei Dschisr esch-Schurr\u201d [in the uppermost clay bank at the right riverside of the Orontes near Jisr Ash-Shughur], Syria.Introduced as \u201cn. mut.\u201d but clearly as a binomen and hence not infrasubspecific in the sense of ICZN Art. 45.6.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.PageBreakMicrocolpiaparreyssiisikorai .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903[unresolved]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai   : 125.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903[unresolved]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai   : 125.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903[unresolved]Late Pleistocene\u2013early Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai   : 125.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903[unresolved]Late Pleistocene\u2013early Holocene.PageBreak\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai   : 125.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Vidal, 1874Maastrichtian, Cretaceous.\u201cIsona\u201d, Spain.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Early Cernikian, early Pliocene.\u201c\u010cerevi\u0107\u201d, Serbia.The illustrated syntypes are stored in the Croatian Natural History Museum, Zagreb, coll. no. 2511-157/1-4 .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fontannes, 1881Mammal zone MN 10\u201312, late Miocene.\u201cMontvendre\u201d, France.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Bukowski, 1892Salakos Formation, Pliocene.\u201cRhodos\u201d , in the Oued Gough (?), and Tanalt, a few kilometers south of the former in the western Anti-Atlas], Morocco.\u201cSouk el H\u00e2d des A\u00eft Souhab, dans l\u2019oued Gough, et Tanalt, \u00e0 quelques kilom\u00e8tres au sud du pr\u00e9c\u00e9dent dans l\u2019Anti Atlas occidental\u201d , Iran.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cSandri\u201d mentioned in Brusina (1866: 106)[unavailable]\u201cCetina\u201d, Croatia.Nomen nudum, listed by Brusina without any explanation.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Handmann, 1882Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d, Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeMorelet, 1857\u201c[Ad Sanctam-Mariam de Balade]\u201d [Balade], New Caledonia.Melanopsisfrustulum Morelet, 1857.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u201cGrimmer\u201d mentioned in Brot (1874\u20131879: 12) and Clessin (1890: 676)[unavailable]Not indicated.Melaniaholandri\u201d [sic] [unavailable]\u201cBei Valencia\u201d [near Valencia], Spain.ventricosior = Latin \u201cmore bulbous\u201d).Introduced as infrasubspecific taxon (forma below a variety), which is not ruled by the provisions of the Code. Moreover, the name is a nomen nudum; Westerlund apparently considered the name a descriptive term [unavailable]Miocene.\u201cBoesciwiese, \u00f6stlich von Kohldorf [...] in dem von Norden, vom Balomberge hinablaufenden Seitengraben\u201d , Hungary.Melanopsisimpressa and not a variety. Melanopsisimpressa.Nomen nudum, appears without explanation in the text. Possibly, Schr\u00e9ter wanted to denote that it is the \u201creal\u201d Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cLalla Maghnia, sur la fronti\u00e8re du Maroc\u201d , Morocco or Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 \u201c\u201d mentioned in Sandberger (1870: 88\u201389)[unavailable]Unknown.\u201cPlan d\u2019Aups (Var)\u201d .Paludomuslyravar.calva , which he described there as new.Nomen nudum, used by Matheron in correspondence with Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Oppenheim, 1890Ronca Beds, Bartonian, Eocene.\u201cLovara di Tressino, Mussolon, Monte Pulli bei Valdagno\u201d , Italy.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1903Late Pleistocene\u2013Holocene.\u201cBischofsbad\u201d , Romania.Microcolpiaparreyssiisikorai .Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Early\u2013middle Miocene.\u201cKirin (Stipan)\u201d, Croatia.Milan et al. (1974: 98) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2502-148.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cDans les cours d\u2019eau aux environs d\u2019Ismidt, en Anatolie\u201d [in rivers around \u0130zmit], Turkey.Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1936Not indicated, but probably in Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Blanckenhorn, 1897Plio-Pleistocene.\u201cIn der ersten Thonbank des linken Orontesufers bei Dschisr esch-Schurr\u201d [in the first clay bank at the left riverside of the Orontes near Jisr Ash-Shughur], Syria.Introduced as \u201cn. mut.\u201d but clearly as a binomen and hence not infrasubspecific in the sense of ICZN Art. 45.6.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Fuchs in Fuchs & Karrer, 1870Pannonian, zone E, late Miocene.\u201cZu Brunn, Inzersdorf, Rothneusiedel und Wien\u201d , Austria.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pavlovi\u0107, 1932Pontian (Dacian Basin), late Miocene\u2013Pliocene.\u201c\u0421\u0435\u043b\u0430 \u0414\u0440\u0441\u043d\u0438\u043a\u0430\u201d [village Drsnik], Kosovo.Perhaps stored in the Natural History Museum, Belgrade, but not mentioned in the catalogue of Milo\u0161evi\u0107 (1962).Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1874Langhian, middle Miocene.\u201cMio\u010di\u0107\u201d, Croatia.Milan et al. (1974: 99) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 3205-851.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Strausz, 1942Transdanubian, Pannonian, late Miocene.Canthidomusbalatonica horizon at Ny\u00e1r\u00e1d, Hungary (p. 23).The locality was indicated as number \u201c85\u201d by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1902Langhian, middle Miocene.\u201cD\u017eepe\u201d [D\u017eepi], Bosnia and Herzegovina.PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Pallary, 1916Pannonian, zone D, late Miocene.\u201cKottingbrunn [...] Ziegelei a\u201d : 560, AuMelanopsisfasciata Handmann, 1882, non Gassies, 1874. Melanopsishaueri Handmann, 1882.Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Porumbaru, 1881Early Romanian, Pliocene.\u201cBucovatzu\u201d [Bucov\u0103\u021b], Romania.vitzui\u201d as mentioned in The name \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1928\u201cBeni Mellal, dans l\u2019oued Taguenout\u201d , Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neubauer, Harzhauser, Georgopoulou, Mandic & Kroh, 2014Middle Pannonian, late Miocene.\u201c\u0423 \u041a\u0430\u0440\u0430\u0433\u0430\u0447\u0443\u201d : 60 , Turkey.Melanopsispraemorsa .PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeMarwick, 1926\u201cCoast south side Wai-iti Stream\u201d, New Zealand.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Esu & Girotti, 2015Late early Pleistocene.\u201cStirone river section, between Laurano and S. Nicomede (Emilia)\u201d, Italy.Senckenberg Forschungsinstitut und Naturmuseum Frankfurt, coll. no. SMF 345834.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neubauer, Harzhauser, Kroh, Georgopoulou & Mandic, 2014Pannonian, zone B\u2013D, late Miocene.\u201cLeobersdorf\u201d , AustriaMelanopsismartinianarugosa Handmann, 1887.See statement for Melanopsismartinianarugosa Handmann, 1887, non Matheron, 1842 (see Note 1).Replacement name for Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Mertin, 1939Mertin 193: 206, pl. 6, fig. 3.Heidelberg Formation, late Santonian, late Cretaceous.\u201cFlugplatz Quedlinburg\u201d [airfield at Quedlinburg], Germany.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Meijer, 1990Bavelian Complex, Pleistocene.\u201cClay-pit North of the village of Bavel \u201d, Netherlands.Rijks Geologische Dienst, Haarlem, The Netherlands; no number indicated.Esperiana (Microcolpia).PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaeIzzatullaev & Starobogatov, 1984\u201c\u0417\u043e\u043b\u043e\u0442\u043e\u0439 \u043a\u043b\u044e\u0447 \u043e\u043a\u043e\u043b\u043e \u0410\u0448\u0445\u0430\u0431\u0430\u0434\u0430\u201d [Zolotoy spring near Ashgabat], Turkmenistan.Zoological Institute of Russian Academy of Sciences, St.-Petersburg; no number indicated.Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020 ?Pezant, 1908Bartonian, Eocene.\u201cMonneville\u201d, France.Melanopsidae.Probably not a Taxon classificationAnimaliaSorbeoconchaMelanopsidaeDe Cristofori & Jan, 1832\u201cGuinea\u201d, indicated in the previous part of the same work .Melanopsidae.Probably not a Taxon classificationAnimaliaSorbeoconchaMelanopsidaePallary, 1920\u201cSefrou\u201d, Morocco.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGould, 1847\u201cNew Zealand\u201d [no locality indicated].Zemelanopsistrifasciata . The name \u201czealandica\u201d as mentioned in Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Neumayr, 1869Langhian, middle Miocene.PageBreak\u201cMiocic\u201d [Mio\u010di\u0107], Croatia.Illustrated syntypes are stored at the Geological Survey Austria, Vienna, coll. no. 1869/01/5/1-2.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1856\u201cL\u2019A\u00efn-Kadra, sur les hauts plateaux de l\u2019Atlas, \u00e0 deux m\u00e8tres des Chots\u201d , Algeria.Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1864[invalid]\u201cBiskra\u201d, Algeria.Melanopsismaroccanazonata Gassies, 1856 (see Note 1), which Bourguignat listed as well (p. 260\u2013261).Junior homonym of Taxon classificationAnimaliaSorbeoconchaMelanopsidaeBourguignat, 1884\u201cAus einem M\u00fchlbache bei Nassenfuss in Unterkrain\u201d [from a mill creek near Mokronog], Slovenia.Melaniaholandrivar.laevigata figured in Introduced for a specimen of Taxon classificationAnimaliaSorbeoconchaMelanopsidaePaladilhe, 1875\u201cMekn\u00e8s\u201d, Morocco.subcostatazonato-\u201d.Described originally as \u201cTaxon classificationAnimaliaSorbeoconchaMelanopsidaeGassies, 1870\u201cPrope Saint-Vincent\u201d [near Saint-Vincent], New Caledonia.Melanopsisbrevis Morelet, 1857 .PageBreakTaxon classificationAnimaliaSorbeoconchaMelanopsidaePaetel, 1888\u201cPersia\u201d : 208, IrOriginally introduced as infrasubspecific taxon (\u201csubvariety\u201d) by Taxon classificationAnimaliaSorbeoconchaMelanopsidae\u2020Brusina, 1893Pannonian, zone D\u2013E, late Miocene.\u201cRipanj\u201d, Serbia.Milan et al. (1974: 99) indicated a holotype, but it is uncertain whether the specimen was the only one Brusina had at hand . The specimen is stored in the Croatian Natural History Museum, Zagreb, coll. no. 2531-177/1."}
{"text": "Based on the light relation between a normal subgroup and a complete congruence relation of a group, we consider the homomorphism problem of rough groups and rough quotient groups and investigate their operational properties. Some new results are obtained. Rough set theory, proposed by Pawlak , is an en-ary hypergroups based on a complete residuated lattice. Xiao and Zhang \u03c1 indicates congruence class of a about \u03c1. Furthermore, if [a]\u03c1[b]\u03c1 = [ab]\u03c1, then \u03c1 is called complete.Let G  be a gro\u03c1 be a congruence relation of G. Then, N\u03c1 = {a \u2208 G |  \u2208 \u03c1} is a normal subgroup of G and  \u2208 \u03c1\u21d4a \u2208 bN\u03c1; contrarily, if N is a normal subgroup of G, one can define a congruence relation \u03c1N, where  \u2208 \u03c1N\u21d4a \u2208 bN\u03c1; then \u03c1N is a congruence relation of G, and N\u03c1 = {a \u2208 G |  \u2208 \u03c1N}.Let G and the normal subgroup of G is one to one correspondence.N be a normal subgroup of G; then \u03c1N is a complete congruence relation.Let a, b \u2208 G[a]\u03c1N[b]\u03c1N = (aN)(bN) = abN = [ab]\u03c1N. Therefore, \u03c1N is a complete congruence relation.Consider \u2200G and the normal subgroup of G is also one to one correspondence.According to \u03c1 be an equivalence relation on G and A a nonempty subset of G. Then, the sets\u03c1-lower and \u03c1-upper approximations of the set A. And \u03c1.Let \u03c1 be an equivalence relation on G and A a nonempty subset of G. ThenA.Let \u03c1 be an equivalence relation on G and, for all A, B\u2286G, one has the following.; ; If A\u2286B, then Let \u03c1 be an equivalence relation on G and \u2200A, B\u2286G one has the following.; ; If A\u2286B, then Let \u03c1 be a complete congruence relation on G and A\u2286G; if G, then A is called the lower rough group  of G and G, then A is called the upper rough group  of G and G, then A is called rough group  of G.Let \u03c1 be a complete congruence relation on G, A\u2286G, and N\u03c1\u2286A. If A is a subgroup  of G, then A is rough group  of G and Let \u03c1 be a complete congruence relation on G, A\u2286G, and N\u03c1\u2286A. Then, G\u21d4G/\u03c1.Let \u03c1 be a complete congruence relation on G, A\u2286G, and N\u03c1\u2286A. If A is a subgroup  of G, thenLet \u03c1 be a complete congruence relation on G and let A be a subgroup of G and N\u03c1\u2286A. Then, Let Based on the third isomorphism theorem of group, it is easy to prove this corollary.G and T be two groups; f : G \u2192 T is a homomorphism. If \u03c1 is a congruence relation on G and N\u03c1\u2287Ker\u2061f, then f(\u03c1) is a congruence relation on T. Further, if f is an injective, then\u2009(1)f(\u03c1) = \u03c1f(N\u03c1);\u2009(2)f(N\u03c1) = Nf(\u03c1),where f(\u03c1) = {(f(a), f(b)) |  \u2208 \u03c1}.Let f(\u03c1) is a congruence relation on T. N\u03c1 is a normal subgroup on G, then f(N\u03c1) is the normal subgroup on T. So \u2200(f(a), f(b)) \u2208 f(\u03c1)\u21d4 \u2208 \u03c1\u21d4a \u2208 bN\u03c1\u21d4f(a) \u2208 f(b)f(N\u03c1)\u21d4(f(a), f(b)) \u2208 \u03c1f(N\u03c1), so f(\u03c1) = \u03c1f(N\u03c1).Because It follows immediately from (1).It is easy to prove f : G \u2192 T be an injective homomorphism. If \u03c1 is a complete congruence relation on G and N\u03c1\u2287Ker\u2061f, then f(\u03c1) is a complete congruence relation on T.Let f(\u03c1) is a congruence relation on T. Now we prove it is complete; \u2200a, b \u2208 G, we have [f(a)]f(\u03c1)[f(b)]f(\u03c1) = (f(a)f(N\u03c1))(f(b)f(N\u03c1)) = f(aN\u03c1bN\u03c1) = f(ab)f(N\u03c1) = [f(ab)]f(\u03c1) = [f(a)f(b)]f(\u03c1).According to f(\u03c1) is complete.Therefore, f : G \u2192 T be a surjective homomorphism. If \u03c1 is a congruence relation on T, then f\u22121(\u03c1) is a congruence relation on G. Further, if f is an injective, thenf\u22121(\u03c1) = \u03c1f\u22121(N\u03c1);f\u22121(N\u03c1) = Nf\u22121(\u03c1),where f\u22121(\u03c1) = { | (f(a), f(b)) \u2208 f(\u03c1)}.Let f\u22121(\u03c1) is a congruence relation on G. N\u03c1 is a normal subgroup on T, then f\u22121(N\u03c1) is the normal subgroup on G. So \u2200 \u2208 f\u22121(\u03c1)\u21d2(f(a), f(b)) \u2208 \u03c1\u21d2f(a) \u2208 f(b)N\u03c1, and because f is injective, then a \u2208 bf\u22121(N\u03c1)\u21d2 \u2208 \u03c1f\u22121(N\u03c1); that is, f\u22121(\u03c1)\u2286\u03c1f\u22121(N\u03c1). On the contrary  \u2208 \u03c1f\u22121(N\u03c1)\u21d2a \u2208 bf\u22121(N\u03c1)\u21d2f(a) \u2208 f(b)N\u03c1\u21d2(f(a), f(b)) \u2208 \u03c1\u21d2 \u2208 f\u22121(\u03c1); that is, \u03c1f\u22121(N\u03c1)\u2286f\u22121(\u03c1); hence, f\u22121(\u03c1) = \u03c1f\u22121(N\u03c1).Because It follows immediately from (1).It is easy to prove f : G \u2192 T be an injective homomorphism. If \u03c1 is a complete congruence relation on T, then f\u22121(\u03c1) is a complete congruence relation on G.Let f\u22121(\u03c1) is a congruence relation on G. Now we prove it is complete; \u2200a, b \u2208 G, we have [a]f\u22121(\u03c1)[b]f\u22121(\u03c1) = (af\u22121(N\u03c1))(bf\u22121(N\u03c1)) = abf\u22121(N\u03c1) = [ab]f\u22121(N\u03c1).According to f\u22121(\u03c1) is complete.Therefore, f : G \u2192 T be a surjective homomorphism. If \u03c1 is a congruence relation on G, A\u2286G, and Ker\u2061f\u2286N\u03c1\u2286A, thenf is injective, then if Let f(a) = b\u21d2[a]\u03c1\u2229A \u2260 \u2205, f(a) = b\u21d2\u2203a\u2032 \u2208 [a]\u03c1, a\u2032 \u2208 A\u21d2 \u2208 \u03c1, a\u2032 \u2208 A\u21d2(f(a), f(a\u2032)) \u2208 f(\u03c1), f(a\u2032) \u2208 f(A)\u21d2f(a\u2032) \u2208 [f(a)]f(\u03c1), f(a) = b, [b]f(\u03c1)\u2229f(A) \u2260 \u2205\u21d2\u2203b\u2032 \u2208 f(A), b\u2032 \u2208 [b]f(\u03c1)\u21d2\u2203a\u2032 \u2208 A, f(a\u2032) = b\u2032, f(a\u2032) \u2208 [b]f(\u03c1)\u21d2(f(a\u2032), f(a)) \u2208 f(\u03c1), a\u2032 \u2208 A\u21d2 \u2208 \u03c1, a\u2032 \u2208 A\u21d2a\u2032 \u2208 [a]\u03c1, (1) Consider f(a) = b\u21d4[a]\u03c1 = aN\u03c1\u2286A\u21d4f([a]\u03c1) = f(a)f(N\u03c1)\u2286f(A)\u21d4bNf(\u03c1) = (2) Consider f : G \u2192 T be a surjective homomorphism, let \u03c1 be a congruence relation on G, and let A be a subgroup of G and Ker\u2061f\u2286N\u03c1\u2286A. Then,f is injective, then if Let A is a subgroup of G; then G and T; according to (1) Suppose that f is injective, according to \u2245(2) If f : G \u2192 T be a surjective homomorphism, let \u03c1 be a complete congruence relation of G based on Ker\u2061f, and let A be a subgroup of G and A\u2287Ker\u2061f. Then,f is injective, then if Let f : G \u2192 T be a surjective homomorphism and let \u03c1 be a congruence relation on T and B\u2286T. Thenf is an injective, then if Let f(a\u2032) \u2208 B, f(a\u2032) \u2208 [f(a)]\u03c1\u21d4a\u2032 \u2208 f\u22121(B), (f(a\u2032), f(a)) \u2208 \u03c1\u21d4a\u2032 \u2208 f\u22121(B), (1) Consider (2) Consider f : G \u2192 T be a surjective homomorphism and let \u03c1 be a congruence relation on T and B is a subgroup of T. Then,f is an injective, then if Let \u2009\u2009By \u03c1, \u03f1 be the congruence relations on G. Then N\u03c1\u2229\u03f1 = N\u03c1\u2229N\u03f1.Let a \u2208 N\u03c1\u2229\u03f1\u21d4 \u2208 \u03c1\u2229\u03f1\u21d4 \u2208 \u03c1,  \u2208 \u03f1\u21d4a \u2208 N\u03c1, a \u2208 N\u03f1\u21d4a \u2208 N\u03c1\u2229N\u03f1, and, therefore, N\u03c1\u2229\u03f1 = N\u03c1\u2229N\u03f1.Consider \u2200\u03c1, \u03f1 be two congruence relations on G and a \u2208 G. Then a(N\u03c1\u2229N\u03f1) = aN\u03c1\u2229aN\u03f1.Let a(N\u03c1\u2229N\u03f1)\u2286aN\u03c1\u2229aN\u03f1. On the contrary, \u2200b \u2208 aN\u03c1\u2229aN\u03f1\u21d2b \u2208 aN\u03c1, b \u2208 aN\u03f1\u21d2 \u2208 \u03c1,  \u2208 \u03f1\u21d2 \u2208 \u03c1\u2229\u03f1\u21d2b \u2208 aN\u03c1\u2229\u03f1 = a(N\u03c1\u2229N\u03f1); that is, aN\u03c1\u2229aN\u03f1\u2286a(N\u03c1\u2229N\u03f1); therefore, aN\u03c1\u2229aN\u03f1 = a(N\u03c1\u2229N\u03f1).It is easy to prove that \u03c1, \u03f1 be two congruence relations on G and a \u2208 G. Then [a]\u03c1\u2229\u03f1 = [a]\u03c1\u2229[a]\u03f1.Let b \u2208 [a]\u03c1\u2229\u03f1\u21d4b \u2208 aN\u03c1\u2229\u03f1 = a(N\u03c1\u2229N\u03f1) = aN\u03c1\u2229aN\u03f1\u21d4b \u2208 aN\u03c1, b \u2208 aN\u03f1\u21d4b \u2208 [a]\u03c1\u2229[a]\u03f1; hence [a]\u03c1\u2229\u03f1 = [a]\u03c1\u2229[a]\u03f1.Consider \u2200\u03c1, \u03f1 be two complete congruence relations on G. Then \u03c1\u2229\u03f1 is a complete congruence relation on G.Let N\u03c1, N\u03f1 are both the normal subgroups of G, then N\u03c1\u2229\u03f1 = N\u03c1\u2229N\u03f1 is a normal subgroup of G, so N\u03c1\u2229\u03f1 is a complete congruence relation.Because \u03c1, \u03f1 be two congruence relations on G and A\u2286G. ThenLet a]\u03c1\u2229\u03f1 = [a]\u03c1\u2229[a]\u03f1, then [a]\u03c1\u2229\u03f1\u2286[a]\u03c1([a]\u03f1)\u21d2[a]\u03c1\u2229A \u2260 \u2205, (1) Consider (2) Consider \u03c1, \u03f1 be two congruence relations on G. Then \u03c1\u2218\u03f1 is a congruence relation on G\u21d4\u03c1\u2218\u03f1 = \u03f1\u2218\u03c1.Let \u03c1, \u03f1 be two complete congruence relations on G and \u03c1\u2218\u03f1 = \u03f1\u2218\u03c1. Then \u03c1\u2218\u03f1 is a complete congruence relation on G.Let \u03c1\u2218\u03f1 is the congruence relation and [a]\u03c1\u2218\u03f1[b]\u03c1\u2218\u03f1 = (aN\u03c1\u2218\u03f1)(bN\u03c1\u2218\u03f1) = abN\u03c1\u2218\u03f1 = [ab]\u03c1\u2218\u03f1; hence, \u03c1\u2218\u03f1 is the complete congruence relation.By \u03c1, \u03f1 be two congruence relations on G. Then N\u03c1\u2218\u03f1 = N\u03c1N\u03f1.Let a \u2208 N\u03c1\u2229\u03f1\u21d2 \u2208 \u03c1\u2229\u03f1\u21d2\u2203b \u2208 G,  \u2208 \u03c1,  \u2208 \u03f1\u21d2a \u2208 bN\u03c1, b \u2208 N\u03f1\u21d2ab \u2208 bN\u03c1N\u03f1\u21d2a \u2208 N\u03c1N\u03f1; that is, N\u03c1\u2218\u03f1\u2286N\u03c1N\u03f1. On the contrary, \u2200a \u2208 N\u03c1N\u03f1\u21d2\u2203b \u2208 N\u03c1, c \u2208 N\u03f1, a = bc\u21d2 \u2208 \u03c1,  \u2208 \u03f1\u21d2 \u2208 \u03c1,  \u2208 \u03f1\u21d2 \u2208 \u03c1\u2218\u03f1\u21d2bc = a \u2208 N\u03c1\u2218\u03f1; that is, N\u03c1\u2218\u03f1\u2287N\u03c1N\u03f1; hence, N\u03c1\u2218\u03f1 = N\u03c1N\u03f1.Consider \u2200\u03c1, \u03f1 be two congruence relations on G and a, b \u2208 G. Then [ab]\u03c1\u2218\u03f1 = [a]\u03c1[b]\u03f1.Let ab]\u03c1\u2218\u03f1 = abN\u03c1\u2218\u03f1 = abN\u03c1N\u03f1 = (aN\u03c1)(bN\u03f1) = [a]\u03c1[b]\u03f1.Consider \u03c1\u2229A \u2260 \u2205, [c]\u03f1\u2229A \u2260 \u2205, a = bc\u21d2\u2203b\u2032 \u2208 [b]\u03c1, b\u2032 \u2208 A, c\u2032 \u2208 [c]\u03f1, c\u2032 \u2208 A\u21d2b\u2032c\u2032 \u2208 [b]\u03c1[c]\u03f1 = [bc]\u03c1\u2218\u03f1 = [a]\u03c1\u2218\u03f1, (1) Consider a]\u03c1 = [a]\u03c1[e]\u2286[a]\u03c1[e]\u03f1\u2286A, and a = bc\u21d2[b]\u03c1\u2286A, [c]\u03f1\u2286A, a = bc\u21d2[a]\u03c1\u2218\u03f1 = [bc]\u03c1\u2218\u03f1 = [b]\u03c1[c]\u03f1\u2286AA = (2) Consider \u03c1 be complete congruence relation on G and A, B\u2286G. ThenLet b \u2208 AB\u21d2a \u2208 bN\u03c1, \u2203c \u2208 A, d \u2208 B, b = cd\u21d2a \u2208 cdN\u03c1 = (cN\u03c1)(dN\u03c1) and [c]\u03c1\u2229A \u2260 \u2205, [d]\u03c1\u2229B \u2260 \u2205\u21d2\u2203c\u2032 \u2208 N\u03c1, d\u2032 \u2208 dN\u03c1, a = c\u2032d\u2032, because [c\u2032]\u03c1 = [c]\u03c1, [d\u2032]\u03c1 = [d]\u03c1\u21d2[c\u2032]\u03c1\u2229A \u2260 \u2205, [d\u2032]\u03c1\u2229B \u2260 \u2205, a = c\u2032d\u2032\u21d2a = a = bc\u21d2[b]\u03c1\u2229A \u2260 \u2205, [c]\u03c1\u2229B \u2260 \u2205\u21d2\u2203b\u2032 \u2208 [b]\u03c1, b\u2032 \u2208 A, c\u2032 \u2208 [c]\u03c1, c\u2032 \u2208 B\u21d2b\u2032c\u2032 \u2208 [b]\u03c1[c]\u03c1 = [bc]\u03c1 = [a]\u03c1, (1) Consider b\u2032 \u2208 [b]\u03c1, b\u2032\u220bA, c\u2032 \u2208 [c]\u03c1, c\u2032\u220bB, then b\u2032c\u2032 \u2208 [b]\u03c1[c]\u03c1, b\u2032c\u2032\u220bAB, Contradiction, that is, [b]\u03c1\u2286A, [c]\u03c1\u2286B, so, a = bc\u21d2[b]\u03c1\u2286A, [c]\u03c1\u2286B, (2) \u03c1 be a complete congruence relation on G and A\u2286G, n \u2208 N. ThenLet \u03c1 be a complete congruence relation on G and let A be a subgroup of G and n \u2208 N. ThenLet A is a subgroup, then G, so An = A, and \u03c1 is an equivalence relation; then \u03c1n = \u03c1; hence, we can get (1) and (2).Because \u03c1 be a complete congruence relation on G and let A, B be two subgroups of G. ThenLet A = A(e)\u2286AB, and B\u2286AB, so A \u222a B\u2286AB; hence, A \u222a B\u2286AB, so (1) Because"}
{"text": "Scientific Reports5: Article number: 1316110.1038/srep13161; published online 08122015; updated on 01122016In this Article, the images depicting the Wiring Diagarms and Numerical solutions for models 1 to 5 have been omitted. The correct \u2032M\u2009\u2192\u2009M\u2009+\u20091\u2032 and Reaction 2 \u2032M\u2009\u2192\u2009M\u2009\u2212\u20091\u2032 were incorrectly given as \u2032M\u2009\u2192\u2009M\u2009\u2212\u20091\u2032 and \u2032M\u2009\u2192\u2009M\u2009+\u20091\u2032 respectively.In Table S5, the transition values for Reaction 1"}
{"text": "Furthermore, judgement theory and discernibility matrix associated with \u03b4-confidence reduct are also introduced, from which we can obtain the approach to knowledge reduction in multilabel decision tables.Owing to the high dimensionality of multilabel data, feature selection in multilabel learning will be necessary in order to reduce the redundant features and improve the performance of multilabel classification. Rough set theory, as a valid mathematical tool for data analysis, has been widely applied to feature selection . In this study, we propose a variable precision attribute reduct for multilabel data based on rough set theory, called  Conventional supervised learning deals with the single-label data, where each instance is associated with a single class label. However, in many real-world tasks, one instance may simultaneously belong to multiple class lultilabel decision tablabels, for example, in text categorization problems, where every document may be labeled as several predefined topics, such as religion and political topics ; in imagOwing to the high dimensionality of multilabel data, feature selection in multilabel learning will be necessary in order to reduce the redundant features and improve the performance of multilabel classification. Among various feature selection approaches, rough set theory, proposed by Pawlak , has att\u03b2-reduct to improve the ability of modeling uncertain information B : x \u2208 U}, where [x]B denotes the equivalence class determined by x with respect to B; that is,A decision table is an information system X\u2286U and B\u2286A. One can define a lower approximation of X and an upper approximation of X byX and denoted alternatively as POSB(X). If X is called a rough set.Let Attribute reduct is one of the most important topics in rough set theory, which aims to delete the irrelevant or redundant attributes while retaining the discernible ability of original attributes. Among many attribute reduction methods, the positive region reduct , 8 is a S =  be a decision table and B\u2286A. B is a positive region reduct of S if and only if B satisfies the following conditions:B(D) = POSA(D),POSB\u2032(D) \u2260 POSA(D) for any B\u2032\u2286B,POSLet B(D) = \u22c3i=1rPOSB(Di) and Di\u2009\u2009 are the equivalence classes, called decision classes generated by the indiscernibility relation RD = { \u2208 U \u00d7 U : d(x) = d(y), \u2200d \u2208 D}.where POSIn this section, we first introduce the multilabel decision table and then analyze the limitations of existing attribute reduction approaches to multilabel data.S =  with A\u2229L = \u2205, where U = {x1, x2,\u2026, xn} is a nonempty finite set of objects, called universe; A = {a1, a2,\u2026, ap} is a nonempty finite set of condition attributes, called condition attribute set; F = {ak : U \u2192 Vk, k = 1,2,\u2026, p} is a set of information functions with respect to condition attributes and Vk is the domain of ak; L = {l1, l2,\u2026, lq} is a nonempty finite set of q possible labels called label set; G = {lk : U \u2192 Vk\u2032, k = 1,2,\u2026, q} is a set of information functions with respect to labels and Vk\u2032 = {0,1} is the domain of the label lk. If the object x is associated with label lk, then lk(x) = 1; otherwise lk(x) = 0. The 5-tuple  can be expressed more simply as  if F and G are understood.Multilabel data can be characterized by a multilabel decision table The object having no labels is irrelevant to multilabel learning and thus is not taken into account in the setting , 16. NotL associates with at least one object in U , one defines a \u03b4-confidence label function \u03b7B\u03b4 : U \u2192 P(L), as follows:Let \u03b4-confidence label function \u03b7B\u03b4(x) is the collection of the labels that associate with at least \u03b4% objects in [x]B. In other words, \u03b7B\u03b4(x) is the collection of the labels which associate with each object in [x]B by at least \u03b4% confidence level.The S =  given by \u03b4 = 0.6, then the 0.6-confidence label function with respect to attribute set A can be calculated thatConsider the multilabel decision table We have the following property.S =  be a multilabel decision table, B, C\u2286A. Then\u03b41 \u2264 \u03b42 \u2264 1, then \u03b7B\u03b42(x)\u2286\u03b7B\u03b41(x);if 0 \u2264 B\u2286C, then \u03b7B1(x)\u2286\u03b7C1(x)\u2286\u03b7C0(x)\u2286\u03b7B0(x);if x \u2208 U, \u03b7B0(x) \u2260 \u2205;for any x]B = [y]B, then \u03b7B\u03b4(x) = \u03b7B\u03b4(y).if [Let li \u2208 \u03b7B\u03b42(x). Then we have |[x]B\u2229Hi|/[x]B \u2265 \u03b42. Note that \u03b42 \u2265 \u03b41; thus |[x]B\u2229Hi|/[x]B \u2265 \u03b41. It means that li \u2208 \u03b7B\u03b41(x). Therefore \u03b7B\u03b42(x)\u2286\u03b7B\u03b41(x).(1) Let B\u2286C, we have [x]C\u2286[x]B.(2) Since li \u2208 \u03b7B1(x), then [x]B\u2286Hi. Since [x]C\u2286[x]B, we have [x]C\u2286Hi. Thus li \u2208 \u03b7C1(x). Therefore \u03b7B1(x)\u2286\u03b7C1(x).If \u03b7C1(x)\u2286\u03b7C0(x).According to lj \u2208 \u03b7C0(x), then [x]C\u2229Hj \u2260 \u2205. Since [x]C\u2286[x]B, we have [x]B\u2229Hj \u2260 \u2205. That means lj \u2208 \u03b7B0(x). Therefore \u03b7C0(x)\u2286\u03b7B0(x).If x \u2208 U such that \u03b7B0(x) = \u2205, then [x]B\u2229Hi = \u2205,\u2009\u2009i = 1,2,\u2026, q. Thus [x]B\u2229(H1 \u222a H2 \u222a \u22efHq) = \u2205. Since \u222ai=1qHi = U, we have [x]B\u2229(H1 \u222a H2 \u222a \u22efHq) = [x]B\u2229U \u2260 \u2205. It is a contradiction.(3) If there exists \u03b7B\u03b4(x) and [xB] = [y]B.(4) It is straightforward by the definition of \u03b4-confidence set using \u03b4-confidence label function. Furthermore, we present the definition of new attribute reduct.Now we define the consistent S =  be a multilabel decision table and B\u2286A. If \u03b7B\u03b4(x) = \u03b7A\u03b4(x), for all x \u2208 U, one says that B is a consistent \u03b4-confidence set of S. If B is a consistent \u03b4-confidence set and no proper subset of B is a consistent \u03b4-confidence set, then B is called a \u03b4-confidence reduct of S.Let \u03b4-confidence reduct is the minimal set of condition attributes that preserves the invariances of the \u03b4-confidence label function of all objects in U.A S =  given by a, b}.For the multilabel decision table a, c} and {b, c} are two positive region reducts for the same multilabel decision table. We think \u03b4-confidence reduct is more appropriate for multilabel data than positive region reduct. This is because \u03b4-confidence label function can more reasonably characterize the uncertainty implied among labels than the indiscernibility relation RL.Considering RL is also considered by the other existing attribute reduction methods. Therefore, for multilabel data, \u03b4-confidence reduct has significant advantages when compared with existing attribute reduction methods.Note that the uncertainty characterized by \u03b4-confidence reducts. Firstly, we present the judgement theorem of consistent \u03b4-confidence set.This section provides a discernibility matrice approach  to obtaiS =  be a multilabel decision table, B\u2286A and \u03b4 \u2208 . Then the following conditions are equivalent:B is a consistent \u03b4-confidence set;x, y \u2208 U, if \u03b7A\u03b4(x) \u2260 \u03b7A\u03b4(y), then [x]B\u2229[y]B = \u2205.for any Let x, y \u2208 U such that [x]B\u2229[y]B \u2260 \u2205, then [x]B = [y]B. By \u03b7B\u03b4(x) = \u03b7B\u03b4(y). Note that B is a consistent \u03b4-confidence set; we have \u03b7B\u03b4(x) = \u03b7A\u03b4(x) and \u03b7B\u03b4(y) = \u03b7A\u03b4(y). Therefore \u03b7A\u03b4(x) = \u03b7A\u03b4(y).(1)\u21d2(2). If there exist B\u2286A, it is easy to verify that I([x]B) = {[y]A : [y]A\u2286[x]B} forms a partition of [x]B.(2)\u21d2(1). Since x \u2208 U, if [y]A\u2286[x]B, then [x]B\u2229[y]B \u2260 \u2205. By the assumption we obtain \u03b7A\u03b4(x) = \u03b7A\u03b4(y).For any li \u2208 \u03b7A\u03b4(x). Then for all [y]A \u2208 I([x]B), we have li \u2208 \u03b7A\u03b4(y); that is to say, |[y]A\u2229Hi|/|[y]A| \u2265 \u03b4.Let li \u2208 \u03b7B\u03b4(x).Therefore we have thatli \u2208 \u03b7B\u03b4(x); however, li \u2209 \u03b7A\u03b4(x). For any [y]A \u2208 I([x]B), we have \u03b7A\u03b4(x) = \u03b7A\u03b4(y); hence li \u2209 \u03b7A\u03b4(y); that is to say, |[y]A\u2229Hi|/|[y]A| < \u03b4. Since [x]B = \u222a{[y]A : [y]A \u2208 I([x]B)}, we haveli \u2209 \u03b7B\u03b4(x), which is a contradiction.On the other hand, we assume that \u03b7A\u03b4(x) = \u03b7B\u03b4(x) for any x \u2208 U. According to B is a consistent \u03b4-confidence set.Thus we conclude that \u03b4-confidence set in multilabel decision tables. Now we present a method for computing all \u03b4-confidence reducts. First, we give the following notion.S =  be a multilabel decision table and U/RA = {X1, X2,\u2026, Xm}. One denotesLet ak(Xi) the value of ak with respect to the objects in Xi. DefineBy M\u03b4 is called \u03b4-confidence discernibility attribute sets. And M\u03b4 = , i, j \u2264 m) is called the \u03b4-confidence discernibility matrix.Then \u03b4-confidence discernibility matrix, we have the following property.For the M\u03b4 = , i, j \u2264 m) satisfies the following properties:M\u03b4 is a symmetric matrix; that is, for any i, j \u2264 m, M\u03b4 = M\u03b4;A; that is, for any i \u2264 m, M\u03b4 = A;elements in the main diagonals are all i, s, j \u2264 m, M\u03b4\u2286M\u03b4 \u222a M\u03b4.for any The discernibility matrix ak \u2208 A such that ak \u2208 M\u03b4 but ak \u2209 M\u03b4 \u222a M\u03b4, that is, ak \u2209 M\u03b4 and ak \u2209 M\u03b4, then according to ak(Xi) = ak(Xs) and ak(Xs) = ak(Xj). Thus ak(Xi) = ak(Xj); that is, ak \u2209 M\u03b4, a contradiction.The proofs of (1) and (2) are straightforward. We only need to prove (3). If there exists \u03b4-confidence set and discernibility matrix.In the following, we establish some connections between consistent S =  be a multilabel decision table, B\u2286A and \u03b4 \u2208 . Then, B is a consistent \u03b4-confidence set if and only if B\u2229M\u03b4 \u2260 \u2205 for all  \u2208 \u0394\u03b4.Let Xi, Xj) \u2208 \u0394\u03b4, there exist x, y \u2208 U, such that Xi = [x]A and Xj = [y]A. From the definition of \u0394\u03b4, we have \u03b7A\u03b4(x) \u2260 \u03b7A\u03b4(y). Since B is a consistent \u03b4-confidence set, we have [x]B\u2229[y]B = \u2205 from ak \u2208 B such that ak(x) \u2260 ak(y); that is, ak(Xi) \u2260 ak(Xj). Hence ak \u2208 M\u03b4; that is, B\u2229M\u03b4 \u2260 \u2205.\u201c\u21d2\u201d For any  \u2208 \u0394\u03b4. Since B\u2229M\u03b4 \u2260 \u2205, for all  \u2208 \u0394\u03b4, there exists al \u2208 B such that al \u2208 M\u03b4. Then we have al(Xi) \u2260 al(Xj); that is, al(x) \u2260 al(y) for [x]A = Xi and [y]A = Xj. It means [x]B\u2229[y]B = \u2205. We then conclude that if  \u2208 \u0394\u03b4, that is, \u03b7A\u03b4(x) \u2260 \u03b7A\u03b4(y), then [x]B\u2229[y]B = \u2205. It follows from B is a consistent \u03b4-confidence set.\u201c\u21d0\u201d Let  be a multilabel decision table, let \u03b4 \u2208 , and let M\u03b4 = , i, j \u2264 m) be the \u03b4-confidence discernibility matrix, where A = {a1,\u2026, ap}. A \u03b4-confidence discernibility function FS\u03b4 for a multilabel decision table S is a boolean function of p boolean variables a1,\u2026, ap, respectively, and is defined as follows:M\u03b4 is the disjunction of all variables a \u2208 M\u03b4.Let ai instead of In the sequel we will write S =  be a multilabel decision table. Then an attribute subset B of A is a \u03b4-confidence reduct of S if and only if \u2227B is a prime implicant of S.Let \u03b4-confidence reducts. The following example illustrates the validity of the approach.U/RA = {X1,\u2026, X5}, where\u03b7A0.6(x) in \u03b7A0.6(x1) = \u03b7A0.6(x2) = \u03b7A0.6(x3). Therefore  \u2209 \u03940.6.Consider the multilabel decision table given by \u03b4-confidence discernibility matrix shown in We can calculate the Consequently, we havea, b} is the unique 0.6-confidence reduct which accords with the results in By \u03b4-confidence reduct presented in this paper is an attribute reduction method designed for multilabel decision tables. Compared with the existing attribute reduction methods, the \u03b4-confidence reduct accurately characterizes uncertainty implied among labels; thus it is more appropriate for multilabel data. Moreover we proposed the corresponding discernibility matrix based method to compute \u03b4-confidence reduct, which is significant in both the theoretic and applied perspectives. In further research, the property of \u03b4-confidence reduct and corresponding heuristic algorithm will be considered.The"}
{"text": "By the use of the way of real analysis, we estimate the weight functions and givesome new Hilbert-type integral inequalities in the whole plane with nonhomogeneouskernels and multiparameters. The constant factors related to the hypergeometric functionand the beta function are proved to be the best possible. We also consider theequivalent forms, the reverses, and some particular cases in the homogeneous kernels. If 0 < \u03bb < 1, p > 1, and (1/p) + (1/q) = 1, Yang [In 2007, Yang  first ga 1, Yang  gave anoe et al. \u201320 also In this paper, by the use of the way of real analysis, we estimate the weight functions and give some new Hilbert-type integral inequalities in the whole plane with nonhomogeneous kernels and multiparameters, which are extensions of . The con\u03b1 > 0, we have \u0393(\u03b1) = \u222b0\u221ex\u03b1\u22121ex\u2212dx, where \u0393(\u03b1) is the \u0393 function . For \u03b2 \u03b2 > \u22121, min\u2061\u2061{\u03bc, \u03c3} > \u2212\u03b1, \u03bc + \u03c3 = \u03bb < 1 + \u03b2, and \u03b4 \u2208 {\u22121,1}, define the weight functions \u03c9\u03b4 and \u03d6\u03b4 as follows:y, x \u2208 \u222a, we haveIf \u03b4 = 1, setting u = xy, we find, for y \u2208 \u222a,(i) For rem (cf. ), in vie\u222a,\u03c91=\u222b\u03b4 = \u22121, setting y/x, we still can obtain \u03c9\u22121 = K\u03b2(\u03c3). Setting u = x\u03b4y, we find\u03b2 > \u22121,\u2009\u20090 < \u03b80 < min\u2061{\u03bc + \u03b1, \u03c3 + \u03b1},L, such that ((\u2212ln\u2061u)/(1 \u2212 u))\u03b2u\u03b80 \u2264 L;\u2009 \u2009then, by (K\u03b2(\u03c3) \u2208 R+. Hence we have ((ii) For then, by , it foll we have .F (cf. [Re(\u03b3) > Re(\u03b8) > 0, |arg(1 \u2212 z)|<\u03c0, thenz = \u22121, \u03b3 = \u03b8 + 1\u2009\u2009(\u03b8 > 0), it follows thatWe have the following formula of the hypergeometric function F (cf. ). If Re(\u03b2 = 0\u2009\u2009(\u03bb < 1), in view of +(1/q) = 1, \u03b2 > \u22121,\u2009\u2009min\u2061{\u03bc, \u03c3} > \u2212\u03b1, \u03bc + \u03c3 = \u03bb < 1 + \u03b2, \u03b4 \u2208 {\u22121,1}, K\u03b2(\u03c3) is indicated by (f(x) is a nonnegative measurable function in , then one hasIf cated by , and f, we hav1[\u222b\u2212\u221e\u221e|ln\u20611[\u222b\u2212\u221e\u221e|ln\u2061 view of  follows.p > 1,\u2009\u2009(1/p)+(1/q) = 1,\u2009\u2009\u03b2 > \u22121,\u2009\u2009min\u2061{\u03bc, \u03c3}>\u2212\u03b1,\u2009\u2009\u03bc + \u03c3 = \u03bb < 1 + \u03b2,\u2009and\u2009\u03b4 \u2208 {\u22121,1},\u2009\u2009f(x),\u2009\u2009g(y) \u2265 0, satisfying 0 < \u222b\u221e\u2212\u221e|x|p(1\u2212\u03b4\u03c3)\u22121fp(x)dx < \u221e and 0 < \u222b\u221e\u2212\u221e | y|q(1\u2212\u03c3)\u22121gq(y)dy < \u221e, then one hasK\u03b2(\u03c3) and K\u03b2p(\u03c3) are the best possible and K\u03b2(\u03c3) is defined by dy}1/q,J\u2254\u222b\u2212\u221e\u221e|y|\u03b4 = 1, we have the following equivalent inequalities:In particular, for y \u2208 \u222a, then there exist constants A and B, such that they are not all zero, andA \u2260 0 (otherwise B = A = 0). Then it follows that\u221e\u2212\u221e|x|p(1\u2212\u03b4\u03c3)\u22121fp(x)dx < \u221e. Hence (J = \u222b\u221e\u2212\u221e | y|q(1\u2212\u03c3)\u22121gq(y)dy. By  is an even function, then it follows thatv = x\u03b4 in the above integral, by Fubini theorem (cf. [K\u03b2(\u03c3) in (k with K\u03b2(\u03c3) < k, such that (K\u03b2(\u03c3) by k. Then we have k < K\u03b2(\u03c3). Hence the constant factor K\u03b2(\u03c3) in dy. By , we have 0, then  is obviothen, by , we obta we have , which i we have . We indiero, andA|x|(1\u2212\u03b4\u03c3\u03b2(\u03c3) in  is not tuch that  is valid obtain2L~\u2254{\u222b\u2212\u221e\u221e|ero, andA|x|(1\u2212\u03b4\u03c3\u03b2(\u03c3) in  is the b\u03b2(\u03c3) in  is not t. Hence (I=\u222b\u2212\u221e\u221e[|y\u03b2(\u03c3) in  is not tp > 1 by 0 < p < 1, one has the equivalent reverses of (As the assumptions of erses of  and 20)p > 1 by g(y) as  in the reverse of , such that the reverse of (K\u03b2(\u03c3) by k. By the reverse of \u03b40 > \u2212\u03b1, and then 0 < K\u03b2(\u03c3 \u2212 (\u03b40/2)) < \u221e. For 0 < \u025b < \u03b40|q|/4\u2009\u2009(q < 0), since u\u03c3+\u03b1+(2\u025b/q)\u22121 \u2264 u\u03c3+\u03b1\u2212(\u03b40/2)\u22121, u \u2208  \u2265 k, which contradicts the fact that k > K\u03b2(\u03c3). Hence, the constant factor K\u03b2(\u03c3) in the reverse of , we havg(y) as  in Theorg(y) as , we haveverse of  is obvioverse of , we obtaverse of . Hence wverse of , which iverse of . If the verse of  is not tverse of  is stillverse of , we haverem (cf. ), we finrem (cf. ), for \u025b erse of  by f(x), we obtain the following equivalent inequalities with a homogeneous kernel and the best possible constant factors:(i) For  = \u22121 in  and 20)\u03b4 = \u22121 in\u03b2 = 0\u2009\u2009(\u03bb < 1) in (K0(\u03c3) is indicated by ((ii) For  < 1) in  and 20)\u03b2 = 0\u2009\u2009 in ((iii) For  < 1) in , we findand then  follows.and then  and 19)\u03b1 = 0,\u2009\u2009\u03c3and then ."}
{"text": "Previously, we suggested prototypal models that describe some clinical states based on group postulates. Here, we demonstrate a group/category theory-like model for molecular/genetic biology as an alternative application of our previous model. Specifically, we focus on deoxyribonucleic acid (DNA) base sequences.5\u2009=\u2009{r, u, d, l, n} of operations is defined for the wallpaper pattern, with which a sequence of points can be generated corresponding to changes of a base in a DNA sequence by following the orbit of a point of the pattern under operations in group Z5. Other manipulations of DNA sequence can be treated using a vector-like notation \u2018Dj\u2019 corresponding to a DNA sequence but based on the five-letter base set; also, \u2018Dj\u2019s are expressed graphically. Insertions and deletions of a series of letters \u2018E\u2019 are admitted to assist in describing DNA recombination. Likewise, a vector-like notation Rj can be constructed for sequences of ribonucleic acid (RNA). The wallpaper group B\u2009=\u2009{Z5\u00d7\u221e, \u25cf} (an \u221e-fold Cartesian product of Z5) acts on Dj (or Rj) yielding changes to Dj (or Rj) denoted by \u2018Dj\u25e6B(j\u2192k)\u2009=\u2009Dk\u2019 (or \u2018Rj\u25e6B(j\u2192k)\u2009=\u2009Rk\u2019). Based on the operations of this group, two types of groups\u2014a modulo 5 linear group and a rotational group over the Gaussian plane, acting on the five bases\u2014are linked as parts of the wallpaper group for broader applications. As a result, changes, insertions/deletions and DNA (RNA) recombination  are described. As an exploratory study, a notation for the canonical \u201ccentral dogma\u201d via a category theory-like way is presented for future developments.We construct a wallpaper pattern based on a five-letter cruciform motif with letters C, A, T, G, and E. Whereas the first four letters represent the standard DNA bases, the fifth is introduced for ease in formulating group operations that reproduce insertions and deletions of DNA base sequences. A basic group ZDespite the large incompleteness of our methodology, there is fertile ground to consider a symmetry model for genetic coding based on our specific wallpaper group. A more integrated formulation containing \u201ccentral dogma\u201d for future molecular/genetic biology remains to be explored. Group theory is the cornerstone in classifying and studying abstract concepts involving symmetry ,2. In geDeoxyribonucleic acid (DNA) is a nucleic acid containing genetic instructions coded in ordered sequences of four bases located in genes that determine specific genetic characteristics of an organism. In the canonical Watson-Crick DNA base pairing, adenine (A) forms a base pair with thymine (T) and guanine (G) forms a base pair with cytosine (C) -26. SimiOver the latter half of the 20th century, the nature of the genetic code became fairly well established. As for the coding sequences of DNA into nucleotide units, one needs to build up more general, sophisticated, rationally functionalized systematics concerning DNA base sequences that will enable genes to be understood at the molecular biology level in more optimized form. Indeed, many approaches have been undertaken to describe gene characteristics from various viewpoints within the participating disciplines -42. In pHowever, each has its advantages and disadvantages in terms of utility and convenience in applications. To our knowledge, so far, if we intend to incorporate a sequence of bases into another sequence and/or exclude certain bases from that substitution, we need to look further afield because normally, sequencing and inserting-deleting operations cannot help in distinguishing one from the other. That means that multiple types of operations are necessary if features of DNA containing exceptional sequences are to be treated.Previously, we suggested prototypal models that describe some clinical states based on group postulates . In thisFirst, we consider a certain wallpaper pattern that helps us to visualize the operations of the present model 5);1) Associativity: x\u25cf(y\u25cfz)\u2009=\u2009(x\u25cfy)\u25cfz,  Identity: \u2018n\u2019 is an identity element such that x\u25cfn\u2009=\u2009n\u25cfx\u2009=\u2009x;\u22121 exists such that x\u25cf x\u22121\u2009=\u2009x\u22121\u25cfx\u2009=\u2009n ;3) Inverse: a unique element x4) Commutativity: x\u25cfy\u2009=\u2009y\u25cfx,5.5) Closure: any combination of operations between x\u25cfy belongs to Z5 is an Abelian group , then\u2009<\u2009Dj\u2009>\u2009changes into<Dj1\u2009>\u2009= [C1|T2(E3)A4|T5|A6(E7)C8|E9|E10|E11|E12|E13|\u2026], and subsequently into<Dj1>>\u2009= [C1|T2|A3|T4|A5|C6|E7|E8|E9|E10|E11|\u2026]. The sequence\u2009<\u2009Dj1\u2009>\u2009contains implicit \u2018E\u2019s aside from the trailing \u2018E\u2019s, and can be written as<<Dj1>}\u2009=\u2009[C1|T2|A4|T5|A6|C8|E9|E10|E11|E12|E13|\u2026]. Naturally, {<Dj1>} and\u2009<\u2009Dj1\u2009>\u2009are equivalent, but\u2009<\u2009<Dj1>\u2009>\u2009and\u2009<\u2009Dj1\u2009>\u2009differ. Moreover, as long as place numbers are recognized/traced precisely, combinations of manipulations \u2018{ }\u2019 and \u2018< >\u2019 are allowed; e.g., {<{{<Dj1>}}>}. Hence, with appropriate use, we could treat  conventional sequences of DNA via \u2018{Dj}\u2019 or \u2018<Dj>\u2019. However, below we shall focus on simple sequences \u2018Dj\u2019.{<DLooking at the beginning of a base sequence as in the following:j, N: the number bases Dj)\u2019 and \u2018(3\u2009\u2192\u20095)\u2019, is simply an additional label representing the two possible types of endings of single-stranded DNA. Nonetheless, when the number of bases is finite, two sequences can be equivalent, as for exampleandunless the prime endings <5\u2019(five prime)\u2009\u2192\u20093\u2019(three prime)\u2009>\u2009or <3\u2019\u2009\u2192\u20095\u2019\u2009>\u2009accompanies the sequence designation.j | Dj \u2208 C5\u2009\u00d7\u2009C5\u2009\u00d7\u2009C5\u2009\u00d7\u2009\u2026 } as the set of all possible sequences of recognized N-tuple single-stranded DNA bases. We can regard N to be a positive integer or infinity.In accordance with these postulates, we now can define the set D\u2009=\u2009{D5, are similarly definable because all results obtained for DNA pertain to RNA under the base substitution. Thus, set R\u2009=\u2009{Rj | Rj \u2208 C5\u2009\u00d7\u2009C5\u2009\u00d7\u2009C5\u2009\u00d7\u2009\u2026} is the set of all possible sequences of recognized N-tuple single-stranded RNA bases with C5\u2009=\u2009{C, A, U, G, E}.An analogous definition is clearly possible for the set R of RNA sequences; with \u2018T\u2019 substituted by \u2018U\u2019, operations of group Zm  | Bm \u2208 Z5\u00d7 Z5\u00d7 Z5\u2009\u00d7\u2009\u2026}\u2009=\u2009{Z5\u00d7N, \u25cf}, where elements of B act on any Dj. This means that Dj covers all possible sequences of the DNA bases, and this situation is the same for Rj of sequences of RNA bases.Next, we can consider B\u2009=\u2009{B5\u00d7N, \u25cf} is confirmed.Because B is a Cartesian product of the same Abelian group, it is also Abelian, where composition of any two elements of B is denoted by \u2018\u25cf\u2019 . Details(j\u2192k)\u2019 that effects the change Dj into Dk is definable in the following way:In a more general context, a Cartesian vector that is composed of the respective operators \u2018bHence,(j\u2192k)\u2009=\u2009Bm \u2019; despite the difference in notation, the two are identical in practice.Clearly, for arbitrary \u2018j\u2019 and \u2018k\u2019, there exists a unique \u2018m\u2019 such that \u2018B(j\u2192k)\u2019s. Consider the scenario that a certain sequence of a single strand (or one side of a double-strand) of DNA transitions from D1 to D3, in stepwise fashion,Here, we present a simple example that consists of a multiple product of \u2018B1\u2009\u2192\u2009D2\u2019. There exists an operator \u2018B(1\u21922)\u2009=\u2009[n1|n2|n3(r4|u5)l6|d7|n8|n9|\u2026]\u2019 that is able to produce this change, specifically, the insertion of two \u2018E\u2019s between \u2018C3\u2019 and \u2018G4\u2019 yields the change \u2018D1\u2009\u2192\u2009D2\u2019. However, this sort of manipulation can be troublesome. Hence, in our model, insertion/deletion of \u2018E\u2019s are instead ascribed to the way the vector Dj is interpreted. This is preferable as this avoids easier manipulations. Next, we construct the operator \u2018B(2\u21923)\u2019 that maps \u2018D2\u2009\u2192\u2009D3\u2019 (the details are shown in Appendix C). With reference to Figures\u00a0We next consider the change \u2018DIn a similar manner,4 is obtained from D1 recursively,Naturally, the final DFrom the decompositionwheredenotes the identity element of D.j irrespective of whether the \u2018E\u2019s are explicit or implicit as defined in \u00a71. Moreover, any sequence \u2018Dj\u2019 can be presented as a polygonal line; as an example, the evolution of changes \u2018D1\u2009\u2192\u2009D3\u2019 is displayed in Figure\u00a0Note that the group operations can act on D5, D, and B, another approach is possible. The five bases can be represented by five equispaced phasors with a \u20182\u03c0/5\u2019 angular phase separation located on the unit circle on the Gaussian plane, as depicted in Figure\u00a0Looking at the definitions of groups ZHerein, in the Gaussian plane, if \u2018\u03c9\u2019 is defined to be the counterclockwise rotational angle \u2018\u03c9\u2009=\u20092\u03c0/5 (rad)\u2019 and composition of \u2018\u03c9\u2019 is denoted \u2018\u25cf\u2019, then assuming \u2018\u03c9\u2019 obeys the \u2018right translation rule\u2019, we havem \u2194 Exp(m\u2009\u00b7\u2009\u03c9\u2009\u00b7\u2009i)\u2019 , m\u2009=\u2009{0, 1, 2, 3, 4, 5}). With {*} meaning one of the bases among \u2018C, A, T, G and E\u2019, we construct the following map. Denoting composition by \u2018\u25e6\u2019, \u03c9m acts on the identity trivially and hence yields the correspondencesThe general form of an arbitrary base is expressed as \u2018X1, \u03c92, \u03c93, \u2026\u2019 on bases \u2018C, A, T, G and E\u2019, we establish for instance:Expanding the operations for \u2018\u03c9\u03c9\u2009=\u2009{\u03c91, \u03c92, \u03c93, \u03c94, \u03c90 (= \u03c95)} is readily confirmed to form group {P\u03c9, \u25cf} where the identity element is \u2018\u03c90\u2019 and the inverse of \u2018\u03c9m\u2019 is \u2018\u03c9m\u22121\u2019:In continuance, the set PClosure and associativity follow from (9) and (10).Here, if we turn our attention to the wallpaper pattern, a further bijection obeying the postulates of the wallpaper group can be confirmed. Corresponding to Figures\u00a02\u22121\u2009=\u2009\u03c93) are preserved in accordance with the inverses for \u2018r, u, d, l, and n\u2019. Any right translation of the horizontal line in Figures\u00a01\u2009=\u2009T\u2009=\u2009E\u25e6d\u2019 and \u2018A\u25e6l\u2009=\u2009A\u25e6\u03c94\u2009=\u2009C\u2009=\u2009E\u25e6r\u2019 can be confirmed. All possible one-step changes between \u2018A, C, T, G and E\u2019 and \u2018\u03c91, \u03c92, \u03c93, \u03c94 and \u03c90,\u2019 are shown in Figure\u00a0Naturally, inverses e.g., \u03c92\u2212\u2009=\u2009\u03c93 are5, and group B.Therefore, this rule for \u2018E\u2019 does not break the postulates for set D, group Zm\u2019 is given; its complementary base \u2018Xm\u2020\u2019 to \u2018Xm\u2019 is defined as follows; for \u2018Xm\u2009=\u2009{Exp(m\u2009\u00b7\u2009\u03c9\u2009\u00b7\u2009i)}, m\u2009=\u2009{0, 1, 2, 3, 4, 5}, then \u2018Xm\u2020\u2019 is obtained by \u2018Xm\u2020\u2009=\u2009{Exp((5 \u2013 m)\u2009\u00b7\u2009\u03c9\u2009\u00b7\u2009i)}\u2019, where \u2018{Exp(5\u2009\u00b7\u2009\u03c9\u2009\u00b7\u2009i)}\u2009=\u2009{1}\u2009=\u2009E\u2019. In this regard,Suppose, from among \u2018C, A, T, G and E\u2019, a base \u2018X\u2020\u2009=\u2009T\u2019 and \u2018C\u2020\u2009=\u2009G\u2019.The procedure yields specifically \u2018A\u2020\u2009=\u2009E\u2019.Clearly, the complement of \u2018E\u2019 is \u2018E\u2019 itself; \u2018Em\u2019 expressed as a base can be given. We introduce the one-value function \u2018\u03c9X(m)\u2019 that provides the same results,Another notation for the \u2018XAs for \u2018m\u2019 in (14), both positive and negative integers are permissible. Thus,m\u2020\u2019 is expressible as\u2018XA simple example is illustrated below.j\u2019\u2009=\u2009[A1|T2|C3|E4|G5|T6|\u2026]\u2009=\u2009[\u03c9X(2)|\u03c9X(3)|\u03c9X(1)|\u03c9X(0)|\u03c9X(4)|\u03c9X(3)|\u2026], then,Suppose \u2018Dr that contains only {r, r2, r3, r4, re (= r0\u2009=\u2009r5\u2009=\u2009n)}. This group is isomorphic with group P\u03c9\u2009=\u2009{\u03c91, \u03c92, \u03c93, \u03c94, \u03c90 (= \u03c95)}, as is the group similarly generated over a vertical line.In accordance with the wallpaper group in Figure\u00a0\u03c9X(m)\u2019, \u2018Xm\u2019 can be expressed using another one-value function rX(m)\u2009=\u2009E\u25e6rm\u2019:Similar to \u2018m\u2020\u2019 (the complementary base of \u2018Xm\u2019) is written asHence, \u2018XExtension to vertical translations is straightforward;m\u2020\u2019 can be identified similarly although the order of letters are somewhat different.and its complementary base \u2018Xj\u2020\u2019 using \u2018rX(m)\u2019s;Consider the following simple example in identifying \u2018D\u03c9X(m)\u2019 by \u2018rX(m)\u2019 in formula (15), the same result is obtained.by replacing \u20181\u25e6b[X1\u2192 X4]\u2009=\u2009X1\u25e6r3\u2009=\u2009E\u25e6r4\u2009=\u2009G\u2019. In general, when the i-th component \u2018b[Xm1\u2192 Xm2]i\u2019 of \u2018B(m1\u2192m2)\u2019 changes Xm1 (= E\u25e6rm1\u2009=\u2009E\u25e6(m1\u03c9)) to Xm2 (= E\u25e6rm2\u2009=\u2009E\u25e6(m2\u03c9)).According to these rules, \u2018XHence, the highlighted form of the operator vector is expressed as(j\u2192k)\u2019 that changes Dj to Dk,For a further example, given the operator \u2018B(j\u2192k)\u2019 takes the formWith details shown in Appendix D, \u2018Bk is obtained through recursively applying the operations, Dj \u25e6B(j\u2192k)\u2009=\u2009Dk. (Details are presented in Appendix D).Naturally, the state Dj\u2020\u2019s might have components in reverse order in terms of sense (5\u2019 or 3\u2019), there exists however certain \u2018Dk\u2019 such that \u2018Dk\u2009=\u2009Dj\u2020, \u2019. With this, \u2018Dj\u2020\u2019 is one of the ordinal elements belonging to the same set D. Thus, the symbol \u2018\u2020\u2019 need only be present when elements are distinct.Whereas \u2018Dr, (or \u03c9 \u2208 group Pr) is \u2018a\u2019 and the number of up translations \u2018u\u2019 \u2208 group P (or 2\u03c9 (= \u03c92) \u2208 group Pr) is \u2018b\u2019 with a, b\u2009=\u2009\u2026,-2, \u22121, 0, 1, 2,\u2026. Similarly, with \u2018d \u2194 3\u03c9\u2019 \u2018l \u2194 4\u03c9\u2019, the total change can be summarized as \u2018x\u2019. We can confirm that there exists at least a pair of \u2018a, b\u2019 that satisfiesConsider Figure\u00a0a\u25cfub\u2019 constitutes a multiple composition of elements of group Z5. In addition, \u2018x\u2009=\u2009r-a\u25cfu-b\u2009=\u2009la\u25cfdb\u2019 or \u2194 \u2018(\u2212a)\u03c9\u25cf(\u2212b)(2\u03c9)\u2009=\u2009(\u2212a-2b)\u03c9\u2019. E.g., \u2018x means \u2018r\u22123\u25cfu\u22122\u2009=\u2009l3\u25cfd2\u2019 or \u2194 \u2018(\u22123)\u03c9\u25cf(\u22122\u2009\u00b7\u20092)(\u03c9)\u2009=\u2009(\u22123 -4)\u03c9\u2009=\u2009(\u22127)\u03c9\u2009=\u2009(\u22122)\u03c9\u2009=\u20093\u03c9\u2009=\u2009\u03c93\u2019.because any base in Figure\u00a0For the wallpaper group, the \u2018a\u2019 and \u2018b\u2019 should be interpreted in modulo 5 addition. The Cayley table for the wallpaper group are presented in Appendix A.\u2020\u2019 is obtained from \u2018X\u2019 using \u2018X\u2009=\u2009E\u25e6x\u2019, \u2018X\u2020\u2019 can be determined asWithin \u2018the square unit cell\u2019 in Figure\u00a0\u2020\u2019 are symmetrically disposed with respect to \u2018E\u2019 over the wallpaper pattern that would be selected as a standard for the definition of \u2018a, b\u2019. In practice, for an arbitrary \u2018X\u2019, \u2018X\u2020\u2019 can be obtained via (22) or (23) by making use of an arbitrary \u2018E\u2019 as the reference point for the symmetry.because \u2018X\u2019 and \u2018X\u2020\u2009=\u2009E\u25e6x\u22121\u2019s asm\u2019, with \u2018Dj\u2019 expressed as \u2018Dj\u2009=\u2009 i|\u2026]\u2019, one of the candidates of the appropriate \u2018B(j\u2192j\u2020)\u2019s that produces \u2018Dj\u25e6B(j\u2192j\u2020)\u2009=\u2009Dj\u2020\u2019 is identified:we can confirm descriptions (20), (22) and (23). As for the operators \u2018Bi -4bi\u2019 in (25b) are permitted to take positive or negative integer values.The exponents \u2018-2aIn these expressions, the rules for the wallpaper group (25a) can also be expressed as either for the linear group or for the rotational group (25b or 25c).More generally,j\u2009=\u2009 i|\u2026]\u2019 is changed into \u2018Dk\u2009=\u2009 i|\u2026]\u2019, and\u2018D(j\u2192k)\u2019s that provides \u2018Dj\u25e6B(j\u2192k)\u2009=\u2009Dj\u2020\u2019 is identified as\u2018B\u2020\u2019 of a certain sequence \u2018X\u2019 is reversed to <3\u2019\u2009\u2192\u20095\u2019\u2009>\u2009.As mentioned in \u00a71, if a certain sequence \u2018X\u2019 has sense <5\u2019\u2009\u2192\u20093\u2019>, the complementary sequence \u2018XTo aid understanding, we present the following examples: Givenj\u2020\u2019 is simplythen, according to (22), \u2018Dj\u25e6B(j\u2192j\u2020)\u2009=\u2009Dj\u2020\u2019 is derived. Details are given in Appendix E.Apart from these examples, additional identities for the wallpaper group can be verified using Figure\u00a0If we use the optional formula (25a\u2009\u2212\u2009c), the relation \u2018DWe develop various general formulas:Other unknown rules might underlie the wallpaper pattern.m\u2009=\u2009rX(m) in , and \u03c9X(m) in  could be regarded as a specific combination that are displayed asConcerning style in treating the wallpaper group, examples \u2018XBelow, we demonstrate, using several examples containing \u2018E\u2019s, changes and inclusion/exclusion of DNA bases using a more generalized scheme.j\u2019 be the sequence \u2018CGTAT\u2026C\u2026TA\u2019, we consider the change of its \u20181\u20133\u2019 components \u2018C1G2T3\u2019 into \u2018G1T2A3\u2019, and moreover the insertion of two bases \u2018GC\u2019 between \u2018T3\u2019 and \u2018A4\u2019 denoted \u2018\u2019:For definiteness, let \u2018Dk,We denote the result of this transformation as Dj to Dk is described recursively to find operator \u2018B(j\u2192h).The procedure from Dj\u2009\u2192\u2009Dh\u2019) in preparation for insertion of \u2018GC\u2019;First, two \u2018E\u2019s are inserted after the 3rd component (this change is denoted \u2018D(j\u2192h)Thus, \u2018Bj\u2019s dependent upon \u2018E\u2019s.This change is in accordance with those rules for vector-like \u2018D(h\u2192k) that produces the change from Dh to Dk is described as:Hence, the operator BThereby,With reference to Figure\u00a0j\u2009\u2192\u2009Dh (inserting two \u2018E\u2019s after the \u20183rd\u2019 component), and 2) Dh \u25e6B(h\u2192k)\u2009=\u2009Dk. Note that the exclusion of the \u20184\u20135\u2019 components \u2018GC\u2019 from Dk and the transformation of the \u20181\u20133\u2019 components from \u2018GTA\u2019 to \u2018CGT\u2019 constitute the recursive procedure for the inverse operatorThis indicates a code change of the \u20181\u20133\u2019 components and a \u2018GC\u2019 insertion after the 3rd as described via the two steps: 1) Dh\u2009\u2192\u2009Dj\u2019 is obtained by deleting the two \u2018E\u2019s from the \u20184\u20135\u2019 components of Dh to yield the initial state \u2018Dj\u2019 in accordance with the characteristics of the vector-like \u2018Dj\u2019s.Alternatively, \u2018D(\u2026 \u2192\u2026 ) \u2208 group B.In summary, essentially, all transitions (changes and inclusion/exclusion) of a certain sequence within the same single-stranded DNA, whether it has finite or infinite length, can be described in principle within a single operation using only the unique operator Bc1)\u2019 and \u2018ATAGCTA (= Dd1)\u2019. These have vector expressionsAs a further development, to demonstrate recombination, take two finite sequences \u2018GETAGT -th component of \u2018Dc1\u2019 and the \u2018AGCTA\u2019 of the (3\u20137)-th component of \u2018Dd1\u2019 at the same instant.To illustrate for the pair Dc1\u2019 just before the sequence to be converted, and in \u2018Dd1\u2019 just after the sequence to be converted. For example, for \u2018Dc1\u2019, five \u2018E\u2019s, \u2018EEEEE\u2019, of size equivalent to that of \u2018A3G4C5T6A7\u2019 of \u2018Dd1\u2019, are inserted just before \u2018T3\u2019 in \u2018Dc1\u2019 where \u2018A3G4C5T6A7\u2019 is to be located, that is, the interval between \u2018the 2nd \u2018E2\u2019 and 3rd \u2018T3\u2019 within \u2018Dc1\u2019. Under this procedure, Dc1 changes into Dc2:First, in the pair of sequences, a series of \u2018E\u2019s of complementary size is inserted in \u2018DEEEEE\u2019 is changed into \u2018A3G4C5T6A7\u2019 -th component of \u2018Dd1\u2019). In addition, \u2018T8A9G10T11\u2019 is transformed into the same number of \u2018E\u2019s, \u2018EEEE\u2019, at the same time. By this process, \u2018Dc2\u2019 changes in \u2018Dc3\u2019:Here, we assume that \u2018EEEE\u2019 equivalent in size to \u2018T3A4G5T6\u2019 of \u2018Dc1\u2019 would be inserted after \u2018A7\u2019 of \u2018Dd1\u2019 where \u2018T3A4G5T6\u2019 of \u2018Dc1\u2019 is to be located within \u2018Dd1\u2019. That is, \u2018T3A4G5T6\u2019 is inserted into the interval between the 7th \u2018A7\u2019 and 8th \u2018E8\u2019 within \u2018Dd1\u2019. In this procedure, Dd1 changes into Dd2:Note that bold type and underline are here merely pedagogical aids to help identify sequence changes. Meanwhile, four \u2018E\u2019s \u2018T8A9G10T11\u2019 -th components of \u2018Dc1\u2019) while \u2018A3G4C5T6A7\u2019 is transformed into the equivalent-sized \u2018EEEEE\u2019. Through this procedure, \u2018Dd2\u2019 changes in \u2018Dd3\u2019:Furthermore, we change \u2018EEEE\u2019 into the equivalent-sized \u2018c1, Dd1\u2019) with \u2018Dc1\u2019\u2009=\u2009\u2018G1E2T3A4G5T6\u2019 being transformed into \u2018Dc3\u2019\u2009=\u2009\u2018G1E2A3G4C5T6A7\u2019 and \u2018Dd1\u2019\u2009=\u2009\u2018A1T2A3G4C5T6A7\u2019 being transformed into \u2018Dd3\u2019\u2009=\u2009\u2018A1T2T3A4G5T6\u2019. We define the manipulation of the recombination  between \u2018Dc1, Dd1\u2019 in this way.As a result, if we omit the infinite series of \u2018E\u2019s from right end, we have the recombination ) and this rule depends upon the characteristics of these vectors .In the initial stage in the previous illustration, we inserted different sizes of \u2018E\u2019 sequences in each line; however, processes \u2018DAs previously explained, the operations can be performed in any of the three equivalent linear group, rotational group, and wallpaper group. Choosing the wallpaper group,Also,With respect to (42) and (44), the inverse identities are confirmed:(_\u2192_) giving transition \u2018Dc2\u2009\u2192\u2009Dc3\u2019 automatically produces an inverse change for \u2018Dd2\u2009\u2192\u2009Dd3\u2019, as stated in (46) and reduces troublesome manipulations, even if only partially.Generally, BWe next comment on other possible applications of the model. The category theory-like construction for treating DNA transcription to RNA might be conceivable, and the combination of the set and the group can comprise a category when these satisfy category theory postulates ,49. Thatj.To begin, according to our description for handling \u2018E\u2019s, it seems difficult to define inverse elements in a group theoretical way when there are deletions of \u2018E\u2019s from any place in a sequence because we cannot find sufficient numbers of \u2018E\u2019s in the target component of DThus, we consider the \u2018morphism f\u2019 that transforms the sequence of DNA bases within set D as follows ,49.j, cod(f)\u2009=\u2009Dk. Object \u2018X\u2019 is the set of \u2018Dj\u2019s. There exists a morphism \u20181X\u2019 such that \u20181X\u25cff\u2009=\u2009f\u2009=\u2009f\u25cf1X\u2019 for every \u2018morphism f\u2019, when \u20181X\u2019\u2009=\u2009[n1|n2|n3|\u2026|ni|\u2026|nN-1|nN|\u2026] (\u2208 group B). If supplemented, the \u2018morphisms f\u2019 comprise \u2018group B\u2019 \u2009=\u2009Dj) to those of its complement (Dj\u2020), and \u0399\u0399) alternation from \u2018C, A, T, and G\u2019 to \u2018C, A, U, and G\u2019 (Dj\u2020\u2009\u2192\u2009Rj\u2020)  transcription from the original DNA sequences  (for transcription to pre-messenger RNA (pre-mRNA) before splicing) can be performed via the manipulation  in \u00a73 and \u00a74. \u2018Dj\u2020\u2019 can be obtained via the linear group , the rotational group  and also the wallpaper group . Thereby, morphism \u03c1 : X\u2009\u2192\u2009Y, dom(\u03c1)\u2009=\u2009Dj, cod(\u03c1)\u2009=\u2009Dj\u2020. Object \u2018Y\u2019 is the set of \u2018Dj\u2020\u2019s .The transformation from the original DNA bases \u2018DX\u2019 and \u20181Y\u2019 such that \u20181Y\u25cf\u03c1\u2009=\u2009\u03c1\u2009=\u2009\u03c1\u25cf1X\u2019 for \u2018morphism \u03c1\u2019, whereThere exist morphisms \u20181m\u2019s \u2208 group B  [A|G|T|A|E|T|C|G|A|C|T|\u2026]\u2009\u2192\u2009(Rj\u2020=) [A|G|U|A|E|U|C|G|A|C|U|\u2026]\u2019.Next, we define manipulations that change the above \u2018DThis process can also be expressed in a similar way as transcription.j\u2020, cod(\u03c1)\u2009=\u2009Rj\u2020. Object \u2018Z\u2019 is the set of \u2018Rj\u2020\u2019s. There exist morphisms \u20181Y\u2019 and \u20181Z\u2019 such that \u20181Z\u25cf\u03c4\u2009=\u2009\u03c4\u2009=\u2009\u03c4\u25cf1Y\u2019 for every \u2018morphism \u03c4\u2019, wheremorphism \u03c4 : Y\u2009\u2192\u2009Z, dom(\u03c4)\u2009=\u2009D4\u20181Y=1Z=n1Evidently, morphism \u03c4 does not satisfy the group postulates because the source object \u2018Y\u2019 and target object \u2018Z\u2019 are different and a single set of operations cannot be defined at this stage.Additionally, as for Steps \u0406 and \u0399\u0399, the resultant process for morphisms \u03c1 and \u03c4 can be expressed as:j, cod(g)\u2009=\u2009Rj\u2020 \u2009=\u2009Dj\u2020 and Rj is the appearance \u2018T\u2019 and \u2018U\u2019 in the sequences.The only difference between DNaturally, for RNA base sequences, similar treatments are possible in the single group B:j\u2020, cod(h)\u2009=\u2009Rk\u2020 \u2009=\u2009R Figures\u00a0 and 8.Ordinarily, in prokaryotic cells, the DNA sequences are transcribed along their entire length. For eukaryotic cell, a splicing process is needed using nascent pre-messenger RNA (pre-mRNA) where introns of DNA bases are removed and exons are joined before producing a correct protein through translation, resulting in the mature messenger RNA (mRNA). Thus, the previous procedure was about the prokaryotic cell or the pre-translation of pre-mRNA in the eukaryotic cell. Therefore, to treat the products after this RNA splicing procedure in the eukaryotic cell, the following approach might be possible. The removal of introns can be regarded as changes from a certain series of bases to \u2018E\u2019s as follows.GUA\u2019 is removed from \u2018A(GUA)EUCGACU\u2026\u2019 to become \u2018AEUCGACU\u2026\u2019, this procedure can be described as; \u2018Rj\u2020\u2009\u2192\u2009Rsj\u2020\u2019,If \u2018j\u2020\u2019 form a set Rs\u2009=\u2009{Rsj\u2020 } that is a part of set R \u2009=\u2009Rj\u2020, cod(j)\u2009=\u2009Rsj\u2020. There exists a morphism \u20181Zs\u2019 such that \u20181Zs\u25cfj\u2009=\u2009j\u2009=\u2009j\u25cf1Zs\u2019.Hereon, we admit \u2018E\u2019s in the sequences of RNAs (as elements of set Rs) during the operations before morphism \u2018f\u2019 and after morphism \u2018j\u2019 to maintain theoretical consistency. Thus, if the result of a series of these maps is \u2018Rs\u2018j\u2019 changes some series of bases from \u2018C, A, U, G, E\u2019 to an equivalent-sized series of \u2018E\u2019s within the partial operations of the group B. However, morphism \u2018j\u2019 fails the group axioms, as inverse might not be definable.Finally, as in \u00a71, we apply the simultaneous deletions of all explicit \u2018E\u2019s of mRNA other than the trailing \u2018E\u2019s, the state after these deletions being denoted with \u2018< >\u2019; forj\u2020\u2009>\u2009= [A1U2C3G4A5C6U7\u2026]\u2019 is specified without explicit non-trailing \u2018E\u2019s. In this regard, as in \u00a71, if some indels (insertions/deletions) occur at certain bases of\u2009<\u2009Rsj\u2020>, as for \u2018<Rsj1\u2020\u2009>\u2009= [A1U2(E3)G4(E5)C6U7\u2026]\u2019 (with the deletion of \u2018C3\u2019 and \u2018A5\u2019), we state the result as \u2018<<Rsj1\u2020>>\u2009= [A1U2G3C4U5\u2026]\u2019. \u2018Rj\u2020\u2019s include\u2009<\u2009Rsj\u2020\u2009>\u2009s and \u2018<<Rsj\u2020>\u2009>\u2009s from the set R and both still satisfy the postulates of group B. This rule is a relative postulate and explicit \u2018E\u2019s are not absolutely forbidden in \u2018<Rsj\u2020>\u2019s or \u2018<<Rsj\u2020>>\u2019s, hence further indels of \u2018E\u2019s into \u2018<Rsj\u2020>\u2019s or \u2018<<Rsj\u2020>>\u2019s are not forbidden.the description \u2018<Rsj1\u2020>}\u2009=\u2009[A1U2G4C6U7\u2026], where place numbers \u20183\u2019 and \u20185\u2019 are absent indicating implicitly their presence in the vector.  Similarly, t-tuples of \u2018< >\u2019s are denoted \u2018<<<<Rsj\u2020>>>>\u2009(t-tuple)\u2009=\u2009<Rsj\u2020>\u2009t\u2019 representing multiple deletions of \u2018E\u2019s (t-times). Combinations of symbols \u2018{ }\u2019 and \u2018< >\u2019 are also allowed when necessary, as for example {<{{<Rsj1>}}>}, as long as the subscripted place numbers are adequately recognized/traced.Also, omissions of explicit \u2018E\u2019s are considered as in \u2018{<Rsj\u2020\u2019 should have a unique meaning with regard to protein synthesis. As a result, the subsequent reading/translation in line with \u2018codon\u2019-like \u2018[AUC|GAC|U\u2026]\u2019, \u2018[A|UCG|ACU|\u2026]\u2019 or \u2018[AU|CGA|CU\u2026]\u2019 leads in an ordinal way to a description of protein synthesis. Through the use of \u2018{ }\u2019 and/or \u2018< >\u2019, the concept \u2018E\u2019 may have benefits, although this may need to be intensely explored in future studies.Nevertheless, the multiple use of \u2018< >\u2019 to remove all \u2018E\u2019s in the vector \u2018RsThe procedure reversing transcription, found for example in retrovirus, is also describable if additional options are added to the scheme. However, these options are omitted at this stage to keep the model simple.In summary, suppose we have a \u2018category C\u2019 with objects \u2018X\u2019, \u2018Y\u2019, \u2018Z\u2019, \u2018Zs\u2019 and morphisms \u2018f\u2019\u2009, \u2018\u03c1\u2019\u2009, \u2018\u03c4\u2019\u2009, \u2018g\u2019\u2009, \u2018h\u2019 and \u2018j\u2019. We affirm that these definitions satisfy the postulates of category. A list is given in Figure\u00a0The expression \u2018hom\u2019 denotes all morphisms f: \u2018from X to X\u2019. Likewise, \u2018hom\u2019 denotes all morphisms \u03c1: \u2018from X to Y\u2019. In addition, \u2018hom\u2019 denotes all morphisms \u03c4: \u2018from Y to Z, and hom denotes all morphism g: from X to Z. Then, hom denotes all morphisms h: \u2018from Z to Z\u2019. Finally, hom denotes all morphism h: from Y to Z. (Details are displayed in Appendix F)1 and C2 are linked.As is explained in \u00a73 and \u00a74, the rotational group can be regarded as a specific bijection of the wallpaper group ,44-47, s1 and C2 with a \u2018functor F\u2019 from C1 to C2 written \u2018F: C1\u2009\u2192\u2009C2\u2019. For example, the pre-category C is denoted C1 and the product of functor F on category C1 is denoted C2\u2019 and \u2018D2\u2009=\u2009[A1C2C3(E4E5)G6T7E8E9E10\u2026]\u2019 are different over the wallpaper pattern despite being equivalent as real sequences. Although by use of electronic tools, these graphics might be of versatility for detection or identification of DNA sequences, these might produce other confusions in the present form. We hope that more appropriate devices would be performed in future study.Third, as for the graphical displays of \u2018D\u2019s in Figure\u00a0Fourth, DNA transcription to RNA and/or mRNA and translation of RNA and/or mRNA into proteins at the ribosomes are performed using a grammar rule based on a three-base set called a \u2018codon\u2019. Codons have information to synthesize twenty types of proteins; for example, \u2018CAG\u2019 codes for \u2018glutamine\u2019. As mentioned before, a number of approaches have been proposed exploit group-theoretic methods. These cover the rules for composition of triplet of bases \u2018XXX\u2019, the ways of reading codons, and models to compose geometric solids such as the tetrahedron and hexahedron, ,33,34,50Fifth, the traditional symmetry model of DNA bases often is based on the chemical types \u2018purine/pyrimidine\u2019, \u2018amino/keto\u2019, and \u2018strong/weak hydrogen binding\u2019 using biomolecular characteristics, which often have advantages for their treatments where three-dimensional graphics aid the imagination, and `matricized\u2019 expressions are possible ,35,36. ISixth, there might be too many speculative conjectures with hypothetical situations those should be used to prove scientific facts using verified methods. Thus, a more rigorous examination for a rational style with a more effective methodology is necessary.Our model is far from a complete systematization. However, we believe that it is necessary that some principal breakthrough should be pursued if we intend to systemize a descriptive model, and that if appropriate definitions are devised, that might help to systemize biomolecular/genetic biology in a more optimized manner with greater sophistication to make a significant contribution to the field.Within the large limitations of our methodology, it is considered that there is fertile ground where variants of the symmetry model for genetic coding based upon a specific wallpaper group are constructible. By integrating the linear group and rotational group over a specific wallpaper pattern, a more integrated formulation based on a group/category theory-like description is open to exploration in applications to a number of topics from molecular/genetic biology.According to Figures\u00a0Here the symbol \u2018\u2194\u2019 signifies \u2018bijection\u2019 and the meaning of \u2018x\u2019 is explained in \u00a74. Hence, operators that are regarded to effect changes from one base to another can be re-expressed as illustrated in the following examples for various types of component operations:As for B,j\u25cfBk)\u25cfBl\u2009=\u2009Bj\u25cf(Bk\u25cfBl)\u2019 holds for all positive integers j, k and l.1) Associativity: \u2018(B0\u2009=\u2009[n1|n2|n3|\u2009\u2026\u2009|ni|\u2009\u2026\u2009|n(n\u2009\u2012\u20091)|nn|nn\u2009+\u20091|nn\u2009+\u20092|nn\u2009+\u20093|\u2009\u2026]\u2019 is an identity element that satisfies \u2018\u2009B0\u25cfBm\u2009=\u2009Bm\u25cfB0\u2009=\u2009Bm\u2009\u2019.\u2009\u2009\u2019 is an element of Z5 (no movement of the point P))2) Identity: \u2018Bm\u22121\u2019 that satisfies \u2018Bm\u2012\u20091\u25cfBm\u2009=\u2009Bm\u25cfBm\u2012\u20091\u2009=\u2009B0\u2019. Actually, the components of the inverse are the inverses of each individual component.3) Inverses: there exists a unique \u2018Bj\u25cfBk\u2009=\u2009Bk\u25cfBj\u2019.4) Commutativity: \u2018Bj\u25cfBk\u2019 belongs to the set B.5) Closure law: any \u2018BAgain, with reference to Figure\u00a0k is generated through the following sequence of operations:Naturally, the series DThen,The axioms are:\u0399) A binary operation and closure law: the combination of two morphisms satisfies hom\u2009\u00d7\u2009hom\u2009\u2192\u2009hom. Moreover, hom\u2009\u00d7\u2009hom\u2009\u2192\u2009mor  and hom\u2009\u00d7\u2009hom\u2009\u2192\u2009mor  both hold.II) Associativity: If f: X\u2009\u2192\u2009X, \u03c1: X\u2009\u2192\u2009Y, \u03c4: Y\u2009\u2192\u2009Z, g: X\u2009\u2192\u2009Z, h: Z\u2009\u2192\u2009Z, and j: Z\u2009\u2192\u2009Zs. Then, \u2018f\u25cf(\u03c1\u25cf\u03c4)\u2009=\u2009(f\u25cf\u03c1)\u25cf\u03c4\u2019, \u2018\u03c1\u25cf(\u03c4\u25cfh)\u2009=\u2009(\u03c1\u25cf\u03c4)\u25cfh\u2019, \u2018f\u25cf(g\u25cfh)\u2009=\u2009(f\u25cfg)\u25cfh\u2019, and \u2018\u03c4\u25cf(h\u25cfj)\u2009=\u2009(\u03c4\u25cfh)\u25cfj\u2019 hold.X, 1Y, 1Z, 1Zs\u2019 such that \u20181X\u25cff\u2009=\u2009f\u2009=\u2009f\u25cf1X\u2019, and \u20181Y\u25cf\u03c1\u2009=\u2009\u03c1\u2009=\u2009\u03c1\u25cf1X\u2019, \u20181Z\u25cf\u03c4\u2009=\u2009\u03c4\u2009=\u2009\u03c4\u25cf1Y\u2019, \u20181Z\u25cfg\u2009=\u2009g\u2009=\u2009g\u25cf1X\u2019, \u20181Z\u25cfh\u2009=\u2009h\u2009=\u2009h\u25cf1Z\u2019. \u20181Zs\u25cfj\u2009=\u2009j\u2009=\u2009j\u25cf1Z\u2019. In practice,III) Identity: there exist morphisms \u201811,For Category C2, for each object F(X1)\u2009=\u2009X2, F(Y1)\u2009=\u2009Y2, F(Z1)\u2009=\u2009Z2, F(Zs1)\u2009=\u2009Zs2 (\u2208C2), the following relationships also hold:Similarly for category C1\u25cf\u03c11)\u2009=\u2009F(f1)\u25cfF(\u03c11), F(\u03c11\u25cf\u03c41)\u2009=\u2009F(\u03c11)\u25cfF(\u03c41), F(\u03c41\u25cfh1)\u2009=\u2009F(\u03c41)\u25cfF(h1), F(f1\u25cfg1)\u2009=\u2009F(f1)\u25cfF(g1), F(g1\u25cfh1)\u2009=\u2009F(g1)\u25cfF(h1), and F(h1\u25cfj1)\u2009=\u2009F(h1)\u25cfF(j1) are satisfied, the composition of C1 and C2 linked with \u2018functor F\u2019 is possible although the proof is omitted here.Other than these, if relationships F(f1), \u2018F(1X)\u2009=\u20091F(X) (\u2208C2)\u2019 is true, for object Y (\u2208C1), \u2018F(1Y)\u2009=\u20091F(Y) (\u2208C2)\u2019 and for object Z (\u2208C1), \u2018F(1Z)\u2009=\u20091F(Z) (\u2208C2)\u2019, is true under the condition:Furthermore, the following postulates hold: for object X (\u2208CThe authors declare that they have no competing interests.JS conceived of the main idea of this article and wrote the manuscript. SM revised the manuscript. JI gave advice on potential versatilities of the model to the biological science. In addition, all authors read and approved the final version of the manuscript."}
{"text": "AbstractPorcellanidae, which comprises 283 species, larval development remains to be described. Full development has been only described for 52 species, while part of the larval cycle has been described for 45 species. The importance of knowing the complete larval development of a species goes beyond allowing the identification of larval specimens collected in the plankton. Morphological larval data also constitute a support to cladistic techniques used in the establishment of the phylogenetic status (see Porcellanidae proposed by For most of the family atus see . Neverth Porcellanidae, commonly known as porcelain crabs, is a family of decapods belonging to the infraorder Anomura . The group comprises 283 species according to the classification proposed by Cancer pagurus, described as Cancer germanicus by Linnaeus, 1767), the morphology of a porcellanid larva was not described until 1835, when J. Vaughan Thompson published a brief description of a larva of Porcellana reared from eggs of females collected in British waters. Eight years later, Pisidialongicornis . Numerous descriptions of the larval stages have been published during more than 170 years. The number of published descriptions of the larval morphology of porcellanids, and of other groups of decapods, has grown exponentially since the 1960\u2019s , it is complicated to access it using common bibliographic search engines. As a consequence, in studies requiring the identification of planktonic organisms , or in morphological studies in which new larval stages are described, where it becomes necessary to compare results with those reported in previous publications of larval descriptions, the researcher has a difficult task in compiling the available information for the target taxon. Although this situation has yielded publication of several bibliographic compilations for brachyurans, like those of PageBreakMany larval publications first appeared more than 30 years ago; for example according to Porcellanidae.Therefore, the objective of this study is three-fold: 1) to compile the available literature on porcellanid larval morphologies; 2) to record the possible changes in the nomenclature of species, or synonymies; and 3) to describe the state-of-the-art on the larval development of species belonging to the family Google Scholar, Scopus, Science Direct and Web of Science have been used for the bibliographic compilation. The current total number of porcellanid species and the taxonomic classification used for the present checklist follow those of World Register of Marine Species (http://www.marinespecies.org).The data set of this study comprises a total of 133 entries obtained from 83 published papers (from 1835 to 2012). Search engines and scientific databases such as prezoeal stage (PR), first to fifth zoeal stage (Z1-5), and megalopal stage (M); iii) the method used to obtain the larvae, according to the following designations: from plankton samples (Pl), larvae reared under laboratory conditions from an identified ovigerous females (Lab) and larvae obtained from plankton and by instar-to-instar laboratory rearing, from unknown parentage, but often a species recognizable from its postlarval or juvenile stages (P+L). Entries marked with asterisk mean that the larval description available, in our opinion, is accurate enough to establish comparisons with other species and have all stages fully described and illustrated. In the checklist, if the taxonomical name of the species described does not match the current taxonomic name according to as\u2019 followed by the name of the species cited in the description.In the checklist, the status of current knowledge of the larval development is specified for each species as follows: i) the author(s) and the date of publication of the larval description; ii) the specific larval stages described, using the following classification: Petrochelesspinosus, which has five zoeal stages.The larval development of porcellanids usually consists of two zoeal stages and one megalopal stage, with the exception of Pisidialongicornis, referred PageBreakto as Porcellanalongicornis. The larval descriptions available were poor in number until the 1960\u2019s and 1970\u2019s, when an increasing trend in the number of publications is observed; this was possibly due to the increased number of scientists specializing in this area, to the increased facilities for cultivating larvae in laboratory conditions, and to the advances in microscope technology  and Pachycheles (44 species); however, the complete larval development has been described for only 21 species of Petrolisthes (19.8%) and only nine species of Pachycheles (20.4%).The current knowledge of larval development by genus (percentages) and the number of species in each genus are shown in Figure Porcellanidae Haworth, 1825Family Porcellana sp; Z1: LabThompson (1935) as Porcellana sp; M: PlPlAliaporcellanakikuchii Nakasone & Miyake, 1969: larvae undescribedPageBreakAliaporcellanapygmaea : larvae undescribedAliaporcellanataiwanensis Dong, Li & Chan, 2011: larvae undescribedAliaporcellanasuluensis : larvae undescribedAliaporcellanatelestophila : larvae undescribedAllopetrolisthesangulosus : full larval descriptionPR, Z1, Z2, M: Lab*Allopetrolisthespunctatus : larvae undescribedAncylochelesgravelei : larvae undescribedCapilliporcellanamurakamii : larvae undescribedCapilliporcellanawolffi Haig, 1981: larvae undescribedClastotoechusdiffractus : larvae undescribedClastotoechusgorgonensis Werding & Haig, 1983: larvae undescribedClastotoechushickmani Harvey, 1999: larvae undescribedClastotoechuslasios Harvey, 1999: larvae undescribedClastotoechusnodosus : full larval descriptionM: Lab*Enosteoideslobatus Osawa, 2009: larvae undescribedEnosteoidesmelissa : larvae undescribedEnosteoidesornatus : partial larval descriptionPorcellanaornata; PR, Z1: LabLabEnosteoidespalauensis : larvae undescribedEnosteoidesphilippinesnsis Dolorosa & Werding, 2014: larvae undescribedEuceramuspanatelus Glassell, 1938: larvae undescribedEuceramuspraelongus Stimpson, 1860: full larval descriptionM: LabPlEuceramustransversilineatus : larvae undescribedEulenaioscometes : partial larval descriptionLabHeteropolyonyxbiforma Osawa, 2001: larvae undescribedHeteroporcellanacorbicola : larvae undescribedLiopetrolisthesmitra : larvae undescribedLissoporcellanademani Dong & Li, 2014: larvae undescribedLissoporcellanaflagellicola Osawa & Fujita, 2005: larvae undescribedLissoporcellanafurcillata : larvae undescribedLissoporcellanamiyakei Haig, 1981: larvae undescribedLissoporcellanamonodi Osawa, 2007: larvae undescribedLissoporcellananakasonei : larvae undescribedLissoporcellananitida : larvae undescribedLissoporcellanapectinata Haig, 1981: larvae undescribedLissoporcellanaquadrilobata : larvae undescribedLissoporcellanaspinuligera : larvae undescribedPageBreakMegalobrachiumerosum : larvae undescribedMegalobrachiumfestae : larvae undescribedMegalobrachiumgarthi Haig, 1957: larvae undescribedMegalobrachiummortenseni Haig, 1962: partial larval descriptionPlMegalobrachiumpacificum Gore & Abele, 1974: larvae undescribedMegalobrachiumperuvianum Haig, 1960: larvae undescribedMegalobrachiumpoeyi : full larval descriptionM: Lab*Megalobrachiumroseum : full larval descriptionM: Lab*Megalobrachiumsinuimanus : larvae undescribedMegalobrachiumsmithi : larvae undescribedMegalobrachiumsoriatum : full larval descriptionM: Lab*Megalobrachiumtuberculipes : larvae undescribedMinyocerusangustus : partial larval descriptionLabMinyoceruskirki Glassell, 1938: larvae undescribedNeopetrolisthesalobatus : larvae undescribedNeopetrolisthesmaculatus : partial larval descriptionLabPlNeopetrolisthesspinatus Osawa & Fujita, 2001: partial larval descriptionLabNeopisosomaangustifrons : full larval descriptionM: Lab*Neopisosomabicapillatum Haig, 1960: larvae undescribedNeopisosomacuracaoense : larvae undescribedNeopisosomadohenyi Haig, 1960: larvae undescribedNeopisosomamexicanum : larvae undescribedNeopisosomaneglectum Werding, 1986: full larval descriptionM: Lab*Neopisosomaorientale Werding, 1986: larvae undescribedNovorostrumdecorocrus Osawa, 1998: full larval descriptionM: Lab*Novorostrumindicum : partial larval descriptionLabNovorostrumphuketense Osawa, 1998: larvae undescribedNovorostrumsecuriger : larvae undescribedOrthochelapumila Glassell, 1936: larvae undescribedPachycheles sp.Pachycheles nrs39; Z2: PlPageBreakPachychelesackleianus A. Milne-Edwards, 1880: larvae undescribedPachychelesattaragos Harvey & de Santo, 1997: larvae undescribedPachychelesbarbatus A. Milne-Edwards, 1878: larvae undescribedPachychelesbellus : larvae undescribedPachychelesbiocellatus : larvae undescribedPachychelescalculosus Haig, 1960: larvae undescribedPachycheleschacei Haig, 1956: partial larval descriptionM: PlPachycheleschubutensis Boschi, 1963: partial larval descriptionLabGonz\u00e1lez et al. (2006); Z1: Pachychelescrassus : larvae undescribedPachychelescrinimanus Haig, 1960: larvae undescribedPachychelescristobalensis Gore, 1970: larvae undescribedPachychelesgarciaensis : partial larval descriptionLabPachychelesgranti Haig, 1965: larvae undescribedPachychelesgreeleyi : larvae undescribedPachychelesgrossimanus : larvae undescribedPachycheleshertwigi Balss, 1913: partial larval descriptionLabPachychelesholosericus Schmitt, 1921: larvae undescribedPachychelesjohnsoni Haig, 1965: larvae undescribedPachycheleslaevidactylus Ortmann, 1892: full larval descriptionPachycheleshaigae; Z1, Z2, M: Lab*Pachychelesmarcortezensis Glassell, 1936: larvae undescribedPachychelesmonilifer : full larval descriptionM: Lab*Pachychelesnatalensis : full larval descriptionLabM: Lab*M: Lab*Pachychelespanamensis Faxon, 1893: larvae undescribedPachychelespectinicarpus Stimpson, 1858: larvae undescribedPachychelespilosus : full larval descriptionM: Lab*PlPachychelespisoides : larvae undescribedPachychelespubescens Holmes, 1900: full larval descriptionM: Lab*McMillan (1972); Z1, Z2, PR, Z1, Z2, M: P+L*Pachychelesriisei : partial larval descriptionPlPachychelesrudis Stimpson, 1859: full larval descriptionPageBreakLabPR, Z1, Z2, M: Lab*Pachychelesrugimanus A. Milne-Edwards, 1880: larvae undescribedPachychelessahariensis Monod, 1933: larvae undescribedPachychelessculptus : partial larval descriptionLabPachychelesserratus : full larval descriptionM: Lab*PlPachychelessetiferous Yang, 1996: larvae undescribedPachychelessetimanus : larvae undescribedPachychelesspinidactylus Haig, 1957: larvae undescribedPachychelesspinipes : larvae undescribedPachychelesstevensii Stimpson, 1858: full larval descriptionPlM: Lab*Pachychelessubsetosus Haig, 1960: larvae undescribedPachychelessusanae Gore & Abele, 1974: partial larval descriptionPlPachychelestomentosus Hendersson, 1893: full larval descriptionM: Lab* Pachychelestrichotus Haig, 1960: larvae undescribedPachychelesvelerae Haig, 1960: larvae undescribedPachychelesvicarius Nobili, 1901: larvae undescribedParapetrolisthestortugensis : partial larval descriptionPlPetrochelesaustraliensis : larvae undescribedPetrochelesspinosus : full larval descriptionPlPl; M: P+L*PR: LabPetrolisthesaegyptiacus Werding & Hiller, 2007: larvae undescribedPetrolisthesagassizii Faxon, 1893: larvae undescribedPetrolisthesamoenus : larvae undescribedPetrolisthesarmatus : full larval descriptionLabPlM: Lab*M: Lab*PlPetrolisthesartifrons Haig, 1960: larvae undescribedPetrolisthesasiaticus : partial larval descriptionLabPageBreakPetrolisthesbifidus Werding & Hiller, 2004: larvae undescribedPetrolisthesbispinosus Borradaile, 1900: larvae undescribedPetrolisthesbolivarensis Werding & Kraus, 2003: full larval descriptionM: PlPetrolisthesborradailei Kropp, 1984: larvae undescribedPetrolisthesboscii : full larval descriptionM: Lab*Petrolisthesbrachycarpus Sivertsen, 1933: larvae undescribedPetrolisthescabrilloi Glassell, 1945: larvae undescribedPetrolisthescaribensis Werding, 1983: partial larval descriptionLabPlPetrolisthescarinipes : larvae undescribedPetrolisthescelebesensis Haig, 1981: larvae undescribedPetrolisthescinctipes : full larval descriptionPR, Z1, Z2, M: Lab*Petrolisthescoccineus : partial larval descriptionLabPetrolisthescocoensis Haig, 1960: larvae undescribedPetrolisthescolumbiensis Werding, 1983: partial larval descriptionPlPetrolisthescrenulatus Lockington, 1878: larvae undescribedPetrolisthesdecacanthus Ortmann, 1897: larvae undescribedPetrolisthesdesmarestii : larvae undescribedPetrolisthesdissimulatus Gore, 1983: partial larval descriptionPlPetrolisthesdonadio Hiller & Werding, 2007: larvae undescribedPetrolisthesdonanensis Osawa, 1997: larvae undescribedPetrolisthesedwardsii : partial larval descriptionPlPetrolistheseldredgei Haig & Kropp 1987: larvae undescribedPetrolistheselegans Haig, 1981: larvae undescribedPetrolistheselegantissimus Werding & Hiller, 2015: larvae undescribedPetrolistheselongatus : full larval descriptionLab; Z2, M: P+L*PR, Z1: Lab; Z2, M: P+L*Petrolistheseriomerus Stimpson, 1871: full larval descriptionM: LabPR, Z1, Z2, M: Lab*Petrolisthesextremus Kropp & Haig, 1994: larvae undescribedPetrolisthesfimbriatus Borradaile, 1898: larvae undescribedPetrolisthesgalapagensis Haig, 1960: larvae undescribedPetrolisthesgalathinus : partial larval descriptionPageBreakPlPetrolisthesgertrudae Werding, 1996: larvae undescribedPetrolisthesglasselli Haig, 1957: larvae undescribedPetrolisthesgracilis Stimpson, 1859: larvae undescribedPetrolisthesgranulosus : full larval descriptionPR, Z1, Z2, M: Lab*Petrolistheshaigae Chace, 1962: partial larval descriptionLabPlPetrolistheshaplodactylus Haig, 1988: larvae undescribedPetrolistheshastatus Stimpson, 1858: partial larval descriptionLabPetrolistheshaswelli Miers, 1884: larvae undescribedPetrolisthesheterochrous Kropp, 1986: larvae undescribedPetrolistheshians Nobili, 1901: larvae undescribedPetrolistheshirtipes Lockington, 1878: larvae undescribedPetrolistheshirtispinosus Lockington, 1878: larvae undescribedPetrolistheshispaniolensis Werding & Hiller, 2005: larvae undescribedPetrolisthesholotrichus Nobili, 1901: larvae undescribedPetrolisthesinermis : larvae undescribedPetrolisthesjaponicus : partial larval descriptionLabLabPetrolisthesjugosus Streets, 1872: partial larval descriptionPlPetrolistheskranjiensis Johnson, 1970: larvae undescribedPetrolistheslaevigatus : full larval descriptionPR, Z1, Z2, M: Lab*Petrolistheslamarckii : full larval descriptionLabM: Lab*M: Lab*Petrolisthesleptocheles : larvae undescribedPetrolistheslewisi : partial larval descriptionPlPetrolistheslimicola Haig, 1988: larvae undescribedPetrolistheslindae Gore & Abele, 1974: larvae undescribedPetrolisthesmagdalenensis Werding, 1978: full larval descriptionM: LabM: Lab*Hern\u00e1ndez and Magan (2012); Z1, Z2, Petrolisthesmanimaculis Glassell, 1945: larvae undescribedPetrolisthesmarginatus Stimpson, 1859: partial larval descriptionPlPageBreakPetrolisthesmasakii Miyake, 1943: larvae undescribedPetrolisthesmelini Miyake & Nakasone, 1966: partial larval descriptionPetrolisthescarinipes; Z1, Z2: LabPetrolisthesmesodactylon Kropp, 1984: larvae undescribedPetrolisthesmilitaris : larvae undescribedPetrolisthesmiyakei Kropp, 1984: larvae undescribedPetrolisthesmoluccensis : partial larval descriptionLabPetrolisthesmonodi Chace, 1956: partial larval descriptionM: PlPetrolisthesnanshensis Yang, 1996: larvae undescribedPetrolisthesnigrunguiculatus Glassell, 1936: larvae undescribedPetrolisthesnobilii Haig, 1960: partial larval descriptionLabPetrolisthesnovaezelandiae Filhol, 1885: full larval descriptionM: P+L*PR, Z1: Lab; Z2, M: P+L*Petrolisthesobtusifrons Miyake, 1937: larvae undescribedPetrolisthesornatus Paulson, 1875: full larval descriptionM: Lab*Petrolisthesortmanni Nobili, 1901: partial larval descriptionPlPetrolisthesperdecorus Haig, 1981: larvae undescribedPetrolisthesplatymerus Haig, 1960: full larval descriptionM: Lab*Petrolisthespolitus : full larval descriptionM: Lab*Petrolisthespolymitus Glassell, 1937: larvae undescribedPetrolisthespubescens Stimpson, 1858: partial larval descriptionLabPetrolisthesquadratus Benedict, 1901: partial larval descriptionPlPetrolisthesrathbunae Schmitt, 1921: larvae undescribedPetrolisthesrobsonae Glassell, 1945: full larval descriptionM: Lab*Garc\u00eda-Guerrero et al. (2005); Z1, Z2, Petrolisthesrosariensis Werding, 1982: partial larval descriptionPlPetrolisthesrufescens : full larval descriptionM: Lab*Petrolisthessanfelipensis Glassell, 1936: larvae undescribedPetrolisthessanmartini Werding & Hiller, 2002: larvae undescribedPetrolisthesscabriculus : larvae undescribedPetrolisthesschmitti Glassell, 1936: larvae undescribedPageBreakPetrolisthessquamanus Osawa, 1996: larvae undescribedPetrolisthesteres Melin, 1939: larvae undescribedPetrolisthestiburonensis Glassell, 1936: larvae undescribedPetrolisthestomentosus : partial larval descriptionLabPetrolisthestonsorius Haig, 1960: full larval descriptionM: Lab*PlPetrolisthestridentatus Stimpson, 1859: full larval descriptionM: Lab*PlPetrolisthestrilobatus Osawa, 1996: partial larval descriptionLabPetrolisthestuberculatus : larvae undescribedPetrolisthestuberculosus : larvae undescribedPetrolisthestuerkayi Naderloo & Apel, 2014: larvae undescribedPetrolisthesunilobatus Henderson, 1888: full larval descriptionM: Lab*Petrolisthesuruma Osawa & Uyeno, 2013: larvae undescribedPetrolisthesviolaceus : full larval descriptionPorcellanamacrocheles; Z2: PlPR, Z1, Z2, M: Lab*Petrolisthesvirgatus Paulson, 1875: larvae undescribedPetrolistheszacae Haig, 1968: full larval descriptionM: Lab*Pisidia sp.PlPisidia sp asm10; Z1, Z2, M: PlPisidiabluteli : full larval descriptionPorcellanabluteli; Z1, Z2, M: PlPorcellanabluteli; M: PlM: PlPisidiabrasiliensis Haig, in Rodrigues da Costa, 1968: partial larval descriptionLabPlPisidiadehaanii : full larval descriptionM: Lab*Pisidiadelagoae : larvae undescribedPisidiadispar : full larval descriptionPR, Z1: Lab; Z2, M: P+LPisidiagordoni : larvae undescribedPisidiainaequalis : partial larval descriptionPorcellanainaequalis; PR, Z1, Z2: PlPageBreakPisidialongicornis : full larval descriptionPorcellanalongicornis; Z1: LabGalathea; Z2: PlPorcellanalongicornis; Z1: PlPorcellanaplatycheles; Z1: PlPorcellanalongicornis; Z1, Z2: P+LPorcellanalongicornis; Z1, M: PlPorcellanalongicornis; Z1, Z2: PlPorcellana sp; Z2: PlPorcellanalongicornis; Z1, Z2, M: P+L*Porcellanalongicornis; Z1, Z2, M: PlPorcellanalongicornis; M: PlPorcellanalongicornis; Z1, Z2, M: P+LPisidialongimana : partial larval descriptionPlPisidiamagdalenensis : larvae undescribedPisidiaserratifrons : partial larval descriptionPisidiaspinulifrons; PR, Z1: LabLabPisidiastreptocheles : larvae undescribedPisidiastreptochiroides : partial larval descriptionPR, Z1: Lab; Z2: P+LPisidiastriata Yang and Sun, 1990: larvae undescribedPisidiavanderhorsti : partial larval descriptionClastotoechusvanderhorsti; PR, Z1, Z2: Lab*Pisidiavariabilis : larvae undescribedPolyonyxbiunguiculatus : larvae undescribedPolyonyxboucheti Osawa, 2007: larvae undescribedPolyonyxbouvieri Saint Joseph, 1900: larvae undescribedPolyonyxconfinis Haig, 1960: larvae undescribedPolyonyxgibbesi Haig, 1956: partial larval descriptionPR, Z1, Z2: LabPlPolyonyxhaigae McNeil, 1968: larvae undescribedPolyonyxhendersoni Southwell, 1909: full larval descriptionPR, Z1: LabM: Lab*Polyonyxloimicola Sankolli, 1965: full larval descriptionM: Lab*Polyonyxmaccullochi Haig, 1965: larvae undescribedPolyonyxnitidus Lockington, 1878: larvae undescribedPolyonyxobesulus Miers, 1884: larvae undescribedPolyonyxpedalis Nobili, 1905: larvae undescribedPageBreakPolyonyxplumatus Yang & Xu, 1994: larvae undescribedPolyonyxquadratus Chace, 1956: larvae undescribedPolyonyxquadriungulatus Glassell, 1935: full larval descriptionM: Lab*Polyonyxsenegalensis Chace, 1956: larvae undescribedPolyonyxsinensis Stimpson, 1858: larvae undescribedPolyonyxspina Osawa, 2007: larvae undescribedPolyonyxsplendidus Sankolli, 1963: larvae undescribedPolyonyxthai Werding, 2001: larvae undescribedPolyonyxtransversus : full larval descriptionPR, Z1: Lab; Z2, M: P+LPolyonyxtriunguiculatus Zehntner, 1894: larvae undescribedPolyonyxtulearis Werding, 2001: larvae undescribedPolyonyxutinomii Miyake, 1943: larvae undescribedPolyonyxvermicola Ng & Sasekumar, 1993: larvae undescribedPorcellanaafricana Chace, 1956: larvae undescribedPorcellanacancrisocialis Glassell, 1936: full larval descriptionM: Lab*Garc\u00eda-Guerrero et al. (2006); Z1, Z2, Porcellanacaparti Chace, 1956: larvae undescribedPorcellanacorbicola Haig, 1960: larvae undescribedPorcellanacurvifrons Yang and Sun, 1990: larvae undescribedPorcellanaelegans Chace, 1956: larvae undescribedPorcellanaforesti Chace, 1956: larvae undescribedPorcellanahabei Miyake, 1961: larvae undescribedPorcellanahancocki Glassell, 1938: larvae undescribedPorcellanalillyae Lemaitre & Campos, 2000: larvae undescribedPorcellanapaguriconviva Glassell, 1936: larvae undescribedPorcellanapersica Haig, 1966: larvae undescribedPorcellanaplatycheles : full larval descriptionLabPorcellana (Polyonyx) macrocheles; Z2: PlM: PlPlM: LabPR, Z1, Z2, M: LabM: PlPlM: Lab*Porcellanapulchra Stimpson, 1858: larvae undescribedPorcellanasayana : full larval descriptionPorcellanaocellata; PR, Z1: LabM: LabPorcellanasigsbeiana A. Milne-Edwards, 1880: full larval descriptionPageBreakM: Lab*PlPorcellanellahaigae Sankarankutty, 1963: larvae undescribedPorcellanellatriloba White, 1852: larvae undescribedPseudoporcellanellamanoliensis Sankarankutty, 1961: larvae undescribedRaphidopusciliatus Stimpson, 1858: larvae undescribedRaphidopusindicus Henderson, 1893: larvae undescribedRaphidopusjohnsoni Ng & Nakasone, 1994: larvae undescribedUlloaiaperpusillia Glassell, 1938: larvae undescribed"}
{"text": "U\u2223t \u2265 \u03c4}, we introduce a new concept, called the weak \ud835\udc9f-pullback exponential attractor, which is a family of sets {\u2133(t)\u2223t \u2264 T}, for any T \u2208 \u211d, satisfying the following: (i) \u2133(t) is compact, (ii) \u2133(t) is positively invariant, that is, U\u2133(\u03c4) \u2282 \u2133(t), and (iii) there exist k, l > 0 such that distB(\u03c4), \u2133(t)) \u2264 ket\u2212\u03c4)\u2212 pullback exponential attracts B(\u03c4). Then we give a method to obtain the existence of weak \ud835\udc9f-pullback exponential attractors for a process. As an application, we obtain the existence of weak \ud835\udc9f-pullback exponential attractor for reaction diffusion equation in H01 with exponential growth of the external force.First, for a process { \ud835\udc9f-pullback attractor for nonautonomous dynamical systems and gave a general method to prove the existence of \ud835\udc9f-pullback attractor. However, pullback attractors or \ud835\udc9f-pullback attractors attract any bounded set of phase space, but the attraction to it may be arbitrarily slow. In order to describe the attracting speed, the concept of pullback exponential attractor is put forward  satisfies |f \u2212 f| \u2264 \u03be(t)|u \u2212 v|. In fact, these conditions are relatively strict; for general conditions, we can not get the result.Pullback attractor is a suitable concept to describe the long time behavior of infinite dimensional nonautonomous dynamical systems or process generated by nonautonomous partial differential equations. There are many references concerned with the existence of pullback attractors for nonautonomous PDEs see \u20135). In [. In [5])ard (see ), which ard see , 8) and  and \ud835\udc9f-puard }, for any T \u2208 \u211d, there exists a family of sets {\u2133(t)\u2223t \u2264 T} satisfying the following:(i)\u2133(t) is compact.(ii)\u2133(t) is positively invariant; that is, U\u2133(\u03c4) \u2282 \u2133(t).(iii)t \u2208 B(\u03c4), \u2133(t)) \u2264 ket\u2212\u03c4)\u2212 pullback exponential attracts B(\u03c4) for all {B(t)} \u2208 \ud835\udc9f.\u2200Motivated by these problems and some ideas in \u20136, we in\ud835\udc9f-pullback exponential attractor is not necessarily uniformly bounded or even unbounded, and the positively invariant only holds for any t \u2208  be the set of all bounded subsets of X; \ud835\udc9f is a nonempty class of parameterised setsU\u2223t \u2265 \u03c4} = {U\u2223t \u2265 \u03c4, t, \u03c4 \u2208 \u211d} act on X, that is, U : X \u2192 X, \u2200t \u2265 \u03c4.Let U} is said to be a process in X, if(1)UU = U, \u2200t \u2265 s \u2265 \u03c4,(2)U = Id is the identity operator, \u03c4 \u2208 \u211d.The pair , X) is generally referred to as a nonautonomous dynamical system, and , X)  is called a nonautonomous discrete dynamical system generated by , X). If x \u2192 Ux is continuous in X, we say that the process is continuous process; if Uxn\u21c0Ux as xn \u2192 x, we say that the process is the norm-to-weak continuous process. Obviously, continuous process is also a norm-to-weak continuous process.A two-parameter family of mappings {B(t)\u2223t \u2208 \u211d} \u2208 \ud835\udc9f is called \ud835\udc9f-pullback bounded absorbing sets for the process {U} if, for any t \u2208 \u211d and any bounded sets {D(t)\u2223t \u2208 \u211d} \u2208 \ud835\udc9f, there exists \u03c40) \u2264 t such that UD(\u03c4) \u2282 B(t) for all \u03c4 \u2264 \u03c40.A family of sets {\ud835\udc9c = {\ud835\udc9c(t)\u2223t \u2208 \u211d} \u2282 B(X) is said to be a \ud835\udc9f-pullback attractor for U if the following hold: (1)\ud835\udc9c(t) is compact for all t \u2208 \u211d;(2)\ud835\udc9c is invariant; that is, U\ud835\udc9c(\u03c4) = \ud835\udc9c(t)\u2009\u2009\u2200t \u2265 \u03c4;(3)\ud835\udc9c is \ud835\udc9f-pullback attracting; that is, lim\u03c4\u2192\u2212\u221edist\u2061\u2009B(\u03c4), \ud835\udc9c(t)) = 0, \u2200{B(t)} \u2208 \ud835\udc9f, and t \u2208 \u211d;(4)C(t)}t\u2208\u211d is another family of closed attracting sets, then \ud835\udc9c(t) \u2282 C(t)\u2009\u2009\u2200t \u2208 \u211d.if {Here dist\u2061 denotes the nonsymmetric Hausdorff distance between sets in X; that is, dist\u2061 = supa\u2208A\u2061infb\u2208B\u2061\u2016a \u2212 b\u2016.The family \u03b1(B) of B \u2282 X is defined by The Kuratowski measure of noncompactness The following summarizes some of the basic properties of the measure of noncompactness.B, B1, B2 \u2282 X. Then \u03b1(B) = 0 if, and only if,\u03b1(B1 + B2) \u2264 \u03b1(B1) + \u03b1(B2);\u03b1(B1) \u2264 \u03b1(B2) for B1 \u2282 B2;\u03b1(B1 \u222a B2) \u2264 max{\u03b1(B1), \u03b1(B2)};F1\u2283F2\u2026 are nonempty closed sets in X such that \u03b1(Fn) \u2192 0 as n \u2192 \u221e, then F = \u22c2n=1\u221eFn is nonempty and compact.if In addition, let X be an infinite dimensional Banach space with a decomposition X = X1 \u2295 X2 and let P : X \u2192 X1, Q : X \u2192 X2 be projectors with dim\u2061X1 < \u221e. Then \u03b1(B(\u03b5)) = 2\u03b5, where B(\u03b5) is a ball of radius \u03b5;\u03b1(B) < \u03b5 for any bounded subset B of X for which the diameter of QB is less than \u03b5.Let U} is called \ud835\udc9f-pullback \u03c9-limit compact for {B(t)\u2223t \u2208 \u211d} if, for any \u03b5 > 0, there exists a \u03c40, \u03b5) \u2264 t such that \u03b1B(\u03c4)) \u2264 \u03b5.A process {U\u2223t \u2265 \u03c4} is \ud835\udc9f-pullback \u03c9-limit compact for\u03c4n} \u2282 , there exists a convergence subsequence of {Uxn} whose limit lies inAssume that the process {U\u2223t \u2265 \u03c4} be a continuous or norm-to-weak continuous process and {U\u2223t \u2265 \u03c4} is \ud835\udc9f-pullback \u03c9-limit compact; let {B(t)\u2223t \u2208 \u211d} \u2282 B(X) be a family of \ud835\udc9f-pullback bounded absorbing sets for the process. Then the process {U\u2223t \u2265 \u03c4} has a \ud835\udc9f-pullback attractor \ud835\udc9c = {\ud835\udc9c(t)\u2223t \u2208 \u211d}, and Let {U\u2223n, m \u2208 \u2124, n \u2265 m}, the above conclusions also hold true.For a discrete process {X be a Banach space; \u2016\u00b7\u2016 denotes the norm of X, \ud835\udc9f is a nonempty class of parameterised setsU} is a continuous process on X.Let \ud835\udc9f-pullback exponential attractor.Now, we give our main theorems which describe the relationship between the measure of noncompactness and the weak B(n)} \u2208 \ud835\udc9f is positively invariant \ud835\udc9f-pullback bounded absorbing sets of {U}; that is, for any {D(n)} \u2208 \ud835\udc9f, N \u2208 \u2124, there exists T \u2208 \u2115, such that UD(m) \u2282 B(n) for any n \u2212 m \u2265 T, and UB(m) \u2282 B(n) for any m \u2264 n \u2264 N; then the following are equivalent: \ud835\udc9f-pullback decays exponentially for the discrete process {U}; that is, there exist k, l > 0 such thatThe measure of noncompactness U} has a weak \ud835\udc9f-pullback exponential attractor; that is, there exists a family of sets {\u2133(n)\u2223n \u2264 N} satisfying the following:The process {\u2133(n) is compact;\u2133(n) is positively invariant; that is, U\u2133(m) \u2282 \u2133(n);\u2133(n)\u2223n \u2208 \u2124} attracts {D(n)} exponentially in a \ud835\udc9f-pullback sense; more precisely, {Assume that {\ud835\udc9f-pullback decays exponentially for {U}, from U} is \ud835\udc9f-pullback \u03c9-limit compact. By \ud835\udc9f-pullback attractor of {U}. Using (3) of n \u2265 m, there exist finite points xn,im \u2208 B(n) such that UB(m) \u2282 \u22c3i=1nmB\u2212). Letting Wnm = {xn,im\u2223i = 1,2,\u2026, nm} and M(k) = \u22c3n=0\u221e+\u22c3i=0nUWk\u2212in\u2212i, we getn \u2208 \u2124, the family {M(n)\u2223n \u2264 N} is positively invariant.((I)\u21d2(II)) Since the measure of noncompactness \u2133(n) = M(n) \u222a \ud835\udc9c(n); we claim that {\u2133(n)\u2223n \u2264 N} satisfies (II).Let xk \u2208 \u2133(n), there exist mk and yk such that xk = Uyk. By (I), we get that the process {U} is pullback \ud835\udc9f-\u03c9-limit compact; we deduce from xk has subsequence convergent in \u2133(n). We get that \u2133(n) is compact.(Compactness) for any sequence UM(n) \u2282 M(n + 1), U\ud835\udc9c(n) = \ud835\udc9c(n + 1), we get (Positively invariant) since D(m)} \u2208 \ud835\udc9f, there exists T \u2208 \u2115, such that B(n)} is positively invariant, we getWnm \u2282 M(n), we get n \u2265 m. for any {x \u2208 B(m), we have x \u2208 B(m), there exists yx \u2208 \u2133(n), such that \u2133(n) is a compact set, we get that there exist y1, y2,\u2026, yl \u2208 \u2133(n) such that yx, there exists yix \u2208 {y1, y2,\u2026, yl} such that \ud835\udc9f-pullback decays exponentially.((II)\u21d2(I)) By the definition of dist\u2061, we get B(t)} \u2208 \ud835\udc9f is positively invariant \ud835\udc9f-pullback bounded absorbing sets of {U}; that is, for any {D(t)} \u2208 \ud835\udc9f, R \u2208 \u211d, there exists T \u2265 0, such that UD(\u03c4) \u2282 B(t) for any t \u2212 \u03c4 \u2265 T, and UB(\u03c4) \u2282 B(t) for any t \u2264 R, and there exists a continuous function r(t) that satisfies \u2016Ux \u2212 Uy\u2016 \u2264 r(t)\u2016x \u2212 y\u2016 for any x, y \u2208 B(\u03c4), t \u2212 \u03c4 \u2264 1; then the following are equivalent:\ud835\udc9f-pullback decays exponentially for the process {U}; that is, there exist k, l > 0 such thatThe measure of noncompactness U} has a weak \ud835\udc9f-pullback attractor; that is, there exists a family of sets {\u2133(t)\u2223t \u2264 R} satisfying the following:The process {\u2133(t) is compact;\u2133(t) is positively invariant; that is, U\u2133(\u03c4) \u2282 \ud835\udc9c(t);\u2133(t)\u2223t \u2264 R} attracts {D(t)} exponentially in \ud835\udc9f-pullback sense; more precisely,{Assume that {U} generated by {U} has a weak \ud835\udc9f-pullback exponential attractor {\u2133(n)}, that is, \u2133(n) is compact and positively invariant and \ud835\udc9f-pullback exponentially attracts {D(n)} \u2208 \ud835\udc9f. We set \u2133(t) = U\u2133(k), t \u2208 [k, k + 1), for all k \u2264 R. As proof of \u2133(t) is compact and positively invariant. Next, we will prove that \u2133(t) attracts {D(n)} \u2208 \ud835\udc9f exponentially in \ud835\udc9f-pullback sense.((I)\u21d2(II)) By D(t)} \u2208 \ud835\udc9f, there exists T \u2208 \u2115 such that UD(\u03c4) \u2282 B(t) for any t \u2212 \u03c4 \u2265 T. For discrete process {U}, by k0, l0 > 0 such that t, \u03c4 \u2208 \u211d, there exist t0, \u03c40 \u2208 [0,1) such that t = n + t0, \u03c4 = m + \u03c40; therefore\u2133(t)\u2223t \u2264 R} attracts {D(t)} exponentially in a \ud835\udc9f-pullback sense.For any {((II)\u21d2(I)) The proof is the same as that of \ud835\udc9f-pullback decays exponentially for the process {U}.We now present a method to verify that the measure of noncompactness X be a uniformly convex Banach space; that is, for all \u03b5 > 0, there exists \u03b4 > 0 such that, given x, y \u2208 X, \u2016x\u2016 \u2264 1, \u2016y\u2016 \u2264 1, \u2016x \u2212 y\u2016 > \u03b5; then \u2016x + y\u2016/2 < 1 \u2212 \u03b4. Requiring a space to be uniformly convex is not a severe restriction in application, since this property is satisfied by all Hilbert spaces, the Lp space with 1 < p < \u221e, and most Sobolev spaces Wk,p with 1 < p < \u221e.Let X be a uniformly convex Banach space; for a family of bounded sets {B(t)} \u2282 X, there exist k, l, T > 0, and for any finite dimension subspace X1 of X, such that(i)PmB(\u03c4)) is bounded;(ii)I \u2212 Pm)\u22c3t\u2212\u03c4\u2265sUx\u2016 \u2264 kel(t\u2212\u03c4)\u2212 + k, \u2200x \u2208 B(\u03c4),\u2016 is real-valued function satisfyingLet U} satisfies the enhanced flattening property; then the measure of noncompactness \ud835\udc9f-pullback decays exponentially for {U}.Assume that the process {B(t)} \u2208 \ud835\udc9f, from (2) and (7) of k \u2192 0, for \u03b50 = kel(t\u2212\u03c4)\u2212, there exists M > 0, for any m > M; we have\u03b1B(t)) \u2264 2kel(t\u2212\u03c4)\u2212; that is, the measure of noncompactness of {U}\ud835\udc9f-pullback decays exponentially.For any {\u211b be the set of all functions r(t) : R \u2192  such that limt\u2192\u2212\u221et\u03b2e\u03bbtr2(t) = 0 for some \u03b2 \u2265 0, \u03bb > 0, and denote by \ud835\udc9f the class of all families \ud835\udc9f = {D(t)\u2223t \u2208 \u211d} \u2282 B(X) such thatr(t) \u2208 \u211b,X with radius r(t).Let U} satisfies \u03b2 \u2265 0, 0 < \u03b1 < \u03bb, and t \u2212 \u03c4 \u2265 T\u2032 and for any t \u2264 R; then the process {U} has a family of positively invariant \ud835\udc9f-pullback bounded absorbing sets {B(t)\u2223t \u2264 R}; that is, for any D \u2208 \ud835\udc9f, there exists T > 0 such that UD(\u03c4) \u2282 B(t) for any t \u2212 \u03c4 \u2265 T and UB(\u03c4) \u2282 B(t).Assume that the process {D0(t)} \u2208 \ud835\udc9f, there exists T0 > 0 such that D(t)} is a family of \ud835\udc9f-pullback bounded absorbing sets. Moreover, there exists T > 0 such that t \u2208 \u211d. Let B(t) \u2282 D(t) and {B(t)} is also a family of \ud835\udc9f-pullback bounded absorbing sets. We also haveLet us define By Theorems X be a uniformly convex Banach space; {U} is a process on X, and the process {U} satisfies the following:Uu\u03c4\u20162 \u2264 K0(t \u2212 \u03c4)\u03b2e\u03bb(t\u2212\u03c4)\u2212\u2016u\u03c4\u20162 + K1 + K2e\u03b1|t| for some \u03b2 \u2265 0, 0 < \u03b1 < \u03bb, and t \u2212 \u03c4 \u2265 T and any t \u2264 R.\u2016I \u2212 Pm)x)\u2016 \u2264 kel(t\u2212\u03c4)\u2212 + k, \u2200x \u2208 B(\u03c4) = {x : \u2016x\u2016 \u2264 2K1 + K2e\u03b1|\u03c4|}, for all s \u2265 T. Here m is the dimension of subspace X1 of X, and k is real-valued function that satisfies\u2016x \u2212 Uy\u2016 \u2264 r(t)\u2016x \u2212 y\u2016, for any t \u2212 \u03c4 < 1, x, y \u2208 B(\u03c4); then the process {U} has a weak \ud835\udc9f-pullback exponential attractor; that is, for any R \u2208 \u211d, there exists a family of sets {\u2133(t)\u2223t \u2264 R} satisfying the following:\u2016\u2133(t) is compact.\u2133(t) is positively invariant; that is, U\u2133(\u03c4) \u2282 M(t).\u2133(t) attracts {D(t)} exponentially in a \ud835\udc9f-pullback sense; more precisely,Let \ud835\udc9f-pullback exponential attractor in H01(\u03a9) for the process generated by the solution of the following nonautonomous reaction diffusion equation: f \u2208 C1, g(\u00b7) \u2208 Lloc\u20612), \u03a9 is a bounded open subset of \u211dn, and there exist p \u2265 2,\u2009\u2009ci > 0,\u2009\u2009i = 1,\u2026, 5, l > 0 such thatu \u2208 \u211d.As an application of A : = \u2212\u25b5, naming \u03bb the first eigenvalue of A, and denote H = L2(\u03a9) by scalar product  and norm |\u00b7|; let ) and \u2016\u00b7\u2016 denote the scalar product and norm of H01(\u03a9) and ) = \u222b\u03a9\u2207u\u2207v\u2009dx for all u, v \u2208 H01(\u03a9). Moreover, we suppose for any t \u2208 \u211d that there exist M \u2265 0 and 0 \u2264 \u03b1 < \u03bb such thatWe set \u03c4, T \u2208 \u211d,\u2009\u2009T > \u03c4, there exists a unique solution u(\u00b7) \u2208 C\u2229L2)\u2229Lp).For this initial boundary value problem, we know from , 8 that,U}t\u2265\u03c4 defined byu(t) is the solution of  wit witU is a weak solution associated with (t > \u03c4:Assume that  satisfy \u201341) and andf andted with . Then thf and g satisfy (p < +\u221e(n \u2264 2), 2 \u2264 p \u2264 (2n \u2212 2)/(n \u2212 2) (n \u2265 3). Then the process defined by , wh, whf andfined by  has a weNext, we will prove that the process defined by  satisfy t \u2264 0, t > 0,t \u2208 \u211d, we have T > 0, for any t \u2212 \u03c4 \u2265 T; we have By , for t \u22641+e\u03b1t.By  and usin we have e\u2212\u03bbt\u222b\u2212\u221ete can get e\u2212\u03bbt\u222b\u2212\u221et\u222bR \u2208 \u211d, the process {Ut, \u03c4}} generated by (\ud835\udc9f-pullback bounded absorbing sets {B(t)\u2223t \u2264 R} and for any x \u2208 B(t), \u2016x\u2016 \u2264 2K1 + K2e\u03b1|t|.By rated by  is a fam\u211b be the set of all functions r : R \u2192  such that limt\u2192\u2212\u221e\u2061te\u03bbtr2(t) = 0 and denote by \ud835\udc9f the class of all families \ud835\udc9f = {D(t)\u2223t \u2208 R} \u2282 B(H) such thatr(t) \u2208 \u211b,H with radius r(t).Let A\u22121 is a continuous compact operator in H, by the classical spectral theorem, there exist a sequence {\u03bbj}j=1\u221e, ej}j=1\u221e of H01(\u03a9) which are orthogonal in H such thatHm = span\u2061{e1, e2,\u2026, em} in H and P : H \u2192 Hm is an orthogonal projector. For any u \u2208 H we writeSince u1(t) = Uu\u03c41 and u2(t) = Uu\u03c42 to be solutions associated with . Let w(t) = u1(t) \u2212 u2(t); by (We set ted with  with ini2(t); by  we get , 2 \u2264 p \u2264 n/(n \u2212 2) + 1(n \u2265 3), using Sobolev embedding theorem, we obtain u1(t), u2(t) \u2208 B(t), we get Taking into account  and H\u00f6ld1,2.From \u201356), we, we(54)fu \u2208 H we writeu2, we haveu(t) \u2208 B(t), hence e\u03bb(t\u2212\u03c4)\u2212\u2016u\u03c4\u20162 \u2192 0 for any u\u03c4 \u2208 B(\u03c4) and \u03bbm \u2192 +\u221e, which imply that there exists T > 0, for any t \u2212 \u03c4 \u2265 T, we have k = 1/\u03bbm + 1/(\u03bbm \u2212 \u03b1(p \u2212 1)) + 1/(\u03bbm \u2212 \u03b1) + e\u03b1(p\u22121)|t|/(\u03bbm \u2212 \u03b1(p \u2212 1)) + e\u03b1|t|/(\u03bbm \u2212 \u03b1); obviously k \u2192 0 as m \u2192 +\u221e.For any oduct of  with \u2212\u25b5ux+gt2.By , we find\u03b1p\u22121t.By , we get}t\u2265\u03c4 generated by , a, a59), arated by  satisfie"}
{"text": "Philometra Costa, 1845  are reported: Philometra inexpectata n. sp. from the mottled grouper Mycteroperca rubra and P. jordanoi  from the dusky grouper Epinephelus marginatus. Identification of both fish species was confirmed by molecular barcoding. The new species is mainly characterized by the length of equally long spicules (147\u2013165\u00a0\u03bcm), the gubernaculum (63\u201393\u00a0\u03bcm long) bearing at the tip two dorsolateral lamellar parts separated from each other by a smooth median field, a V-shaped mound on the male caudal extremity, the presence of a pair of large caudal papillae located posterior to the cloaca and by the body length of the males (1.97\u20132.43\u00a0mm). Philometra inexpectata n. sp. is the fifth known gonad-infecting philometrid species parasitizing serranid fishes in the Mediterranean region. The males of P. jordanoi were examined by scanning electron microscopy for the first time; this detailed study revealed some new taxonomically important morphological features, such as the number and arrangement of cephalic and caudal papillae, presence of amphids and phasmids and mainly the lamellate structures at the posterior end of the gubernaculum. A key to gonad-infecting species of Philometra parasitic in serranid fishes is provided.Based on light and scanning electron microscopical studies of nematode specimens  collected from the ovary of groupers  in the Mediterranean Sea off Tunisia (near Tunis and Sfax), two gonad-infecting species of  Gonad-infecting species of philometrid nematodes (Philometridae) are widely distributed in marine fishes of the Atlantic, Indian and Pacific Oceans, and sometimes occur in brackish-water environments , 26. ThePhilometra Costa, 1845 have been described from a variety of marine fishes belonging to different families and their number is quickly increasing  obtained from the market in T\u00e9touan, Morocco (probably caught in the Mediterranean Sea). Later the species was transferred to Philometra as P. jordanoi [Sanguinofilaria Yamaguti, 1941 was subsequently synonymized with Philometra [P. jordanoi [E. marginatus collected in the Mediterranean Sea near the Balearic Islands, Spain and those from wild and cultured E. marginatus in the Tyrrhenian Sea off Sicily, Italy, provided a somewhat more detailed description of philometrid gravid females, which was, more or less, in agreement with that of P. jordanoi. Nevertheless, the authors identified this material as P. lateolabracis, a species described from females collected in three species of perciform fishes off Japan [P. jordanoi to be its junior synonym.This species was originally described by L\u00f3pez-Neyra  as Sangujordanoi . The genilometra . The orijordanoi , based off Japan , and desE. marginatus in the Mediterranean Sea off Turkey (Iskenderun Bay), described the female anterior end including cephalic structures of these nematodes and the first-stage larva from the uterus. They again identified these nematodes as P. lateolabracis. From the same locality (Iskenderun Bay off Turkey) and the same host species, P. lateolabracis was also reported by Genc et al. [Subsequently, Moravec and Genc , based oc et al. .E. marginatus in waters near Majorca, Spain, which was identified as P. lateolabracis. But in their subsequent paper [P. jordanoi, to which they assigned the above-mentioned male specimen.Merella et al.  were thent paper , they reP. lateolabracis from the type-host in Japan and provided their detailed description based on LM and SEM examinations, which enabled a comparison with other gonad-infecting Philometra spp. with described males. Their study showed that P. lateolabracis is a specific parasite of Lateolabrax japonicus (Cuvier) (Lateolabracidae) and that all previous records of this parasite from many other fish species were apparently based on misidentifications. Comparison of the males of P. lateolabracis and those of P. jordanoi confirmed the validity of the latter species [However, Quiazon et al.  were the species .P. lateolabracis in E. marginatus in the Mediterranean region [P. jordanoi. Also, the nematodes designated as Philometra sp. from the ovary of E. marginatus in Iskenderun Bay off Turkey [It is apparent that all the previous records of n region , 18, 24 P. jordanoi, including the first use of SEM, made it possible to describe some new taxonomically important morphological features in this species, such as the presence of lamellate structures on the distal end of the gubernaculum, the number and distribution of cephalic and caudal papillae, the character of the male caudal mound, the structure of the male oesophagus, the location of the nerve ring and excretory pore, and the morphology of the mature female. As compared with other gonad-infecting congeneric species parasitizing serranid fishes, P. jordanoi is remarkable for its rather long spicules, which may attain up to 265\u00a0\u03bcm [P. jordanoi with other species from the gonads of serranids is more apparent from the following key.The present detailed study of the males of o 265\u00a0\u03bcm . The comSerranus cabrilla (Linnaeus); Mediterranean Sea region \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026....................\u2026\u2026\u2026\u2026\u2026.. P. serranellicabrillaeMales unknown. Body length of gravid female 40\u201360\u00a0mm. In Both males and females known \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.........................\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026.. 2Epinephelus morio ; Gulf of Mexico \u2026\u2026\u2026\u2026\u2026..............................\u2026\u2026\u2026\u2026\u2026\u2026..\u2026. P. margolisiSpicules 432\u2013468\u00a0\u03bcm long. Gravid female 65\u201385\u00a0mm long. In Spicules shorter than 300\u00a0\u03bcm \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.........................\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026. 3Spicules longer than 160\u00a0\u03bcm \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.........................\u2026\u2026\u2026\u2026\u2026\u2026...\u2026..\u2026\u2026 4Spicules shorter than 160\u00a0\u03bcm \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..........................\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 9Epinephelus bleekeri (Vaillant); Indian Ocean  \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..............\u2026\u2026\u2026\u2026...\u2026 P. tropicaSpicules conspicuously distended between second and fourth quarter of their length; spicules 168\u2013186\u00a0\u03bcm long, length of gubernaculum 120\u2013138\u00a0\u03bcm. In Spicules slender, not markedly distended \u2026\u2026\u2026\u2026\u2026..................\u2026\u2026\u2026\u2026.\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026 5Length of gubernaculum at most 87\u00a0\u03bcm \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...................\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6Gubernaculum longer than 90\u00a0\u03bcm \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.....................\u2026\u2026\u2026.\u2026\u2026\u2026\u2026. 8Epinephelus costae (Steindachner); Mediterranean Sea \u2026\u2026....\u2026\u2026\u2026\u2026\u2026... P. tunisiensisPair of large caudal papillae posterior to cloaca absent. Length of spicules 201\u2013219\u00a0\u03bcm, gubernaculum 78\u201387\u00a0\u03bcm long; length ratio of gubernaculum and spicules 1:2.52\u20132.77. In Pair of large caudal papillae posterior to cloaca present \u2026\u2026\u2026\u2026\u2026\u2026\u2026.........\u2026\u2026\u2026\u2026\u2026. 7Epinephelus merra Bloch; Indian Ocean  \u2026\u2026\u2026\u2026\u2026\u2026 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. P. indicaLength of spicules 192\u2013195\u00a0\u03bcm, gubernaculum 84\u00a0\u03bcm long; length ratio of gubernaculum and spicules 1:2.32. In Epinephelus marginatus; Mediterranean Sea \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...............................\u2026\u2026\u2026....\u2026 P. jordanoiLength of spicules 213\u2013265\u00a0\u03bcm, gubernaculum 81\u201384\u00a0\u03bcm long; length ratio of gubernaculum and spicules 1:2.61\u20133.15. In Epinephelus coioides (Hamilton); South Pacific  and Indian Ocean (Persian Gulf) \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026............................\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026... P. piscariaLength of spicules 171\u2013180\u00a0\u03bcm, representing 4\u20135% of male body length; gubernaculum 126\u2013144\u00a0\u03bcm long. Male caudal mound V-shaped. Body length of male 3.67\u20134.19\u00a0mm. In Epinephelus cyanopodus (Richardson); South Pacific  ... P. cyanopodiLength of spicules 183\u2013228\u00a0\u03bcm, representing 6\u20138% of male body length; gubernaculum 129\u2013162\u00a0\u03bcm long. Male caudal mound U-shaped. Body length of male 2.72\u20133.59\u00a0mm. In Epinephelus aeneus (Geoffroy Saint-Hilaire); Mediterranean Sea \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 P. aeneiGubernaculum with distinct dorsal barb situated postequatorially and conspicuous dorsal protuberance at posterior end in lateral view. Length of spicules 108\u2013123\u00a0\u03bcm, gubernaculum 96\u2013108\u00a0\u03bcm long. In Gubernaculum without dorsal barb situated postequatorially and dorsal protuberance at its end \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...........................\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026 10Epinephelus adscensionis (Osbeck); Gulf of Mexico \u2026\u2026\u2026\u2026\u2026\u2026...........................\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026.. P. mexicanaBody length of male 1.63\u20131.86\u00a0mm. Length of spicules 90\u2013120\u00a0\u03bcm, gubernaculum 57\u201366\u00a0\u03bcm long. Body length of gravid female 178\u2013230\u00a0mm. In Body length of male longer than 1.90\u00a0mm \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..............\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 11Epinephelus fasciatus (Forssk\u00e5l); South Pacific  \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026................................\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026. P. fasciatiSpicules 147\u2013156\u00a0\u03bcm long, length of gubernaculum 69\u201384\u00a0\u03bcm. Testis reaching anteriorly only to posterior end of oesophagus. Body length of males 2.75\u20133.32\u00a0mm, of gravid female 387\u00a0mm. In Spicules 117\u2013165\u00a0\u03bcm long. Testis reaching anteriorly at least to level of oesophageal nucleus \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...................................\u2026\u2026\u2026\u2026..\u2026.... 12Caudal mound of male simple, V-shaped, not dorsally divided \u2026\u2026\u2026\u2026\u2026.\u2026\u2026...\u2026...\u2026 13Caudal mound of male consisting of two lateral reniform parts widely separated from each other dorsally \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.........................\u2026\u2026\u2026\u2026\u2026 14Cephalopholis sonnerati ; South Pacific  \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. P. cephalopholidisLength of spicules 129\u2013147\u00a0\u03bcm, gubernaculum 96\u2013111\u00a0\u03bcm long. Testis reaching anteriorly to region of oesophageal nucleus. Body length of male 2.53\u20132.91\u00a0mm. In Mycteroperca rubra; Mediterranean Sea \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.........................\u2026\u2026P. inexpectata n. sp.Length of spicules 147\u2013165\u00a0\u03bcm, gubernaculum 63\u201393\u00a0\u03bcm long. Testis reaching anteriorly to level of nerve ring. Body length of male 1.97\u20132.43\u00a0mm. In Hyporthodus flavolimbatus (Poey); Gulf of Mexico \u2026\u2026\u2026\u2026\u2026\u2026..........\u2026\u2026.. P. hyporthodiBody length of male 3.62\u20134.07\u00a0mm, of gravid female 105\u00a0mm. Maximum body width/length ratio of gravid female 1:59. Length of spicules 135\u2013138\u00a0\u03bcm, gubernaculum 84\u00a0\u03bcm long. Caudal projections in females absent. Larvae in uterus 618\u2013648\u00a0\u03bcm long. In Mycteroperca phenax; western Atlantic Ocean (off USA) including Gulf of Mexico \u2026\u2026...\u2026. P. charlestonensisBody length of male 2.01\u20133.14\u00a0mm, of gravid females 178\u2013230\u00a0mm. Maximum body width/length ratio of gravid females 1:131\u2013163. Length of spicules 123\u2013141\u00a0\u03bcm, gubernaculum 54\u201393\u00a0\u03bcm long. Gravid and subgravid females with pair of small papilla-like caudal projections. Larvae from uterus 544\u2013648\u00a0\u03bcm long. In http://publicationethics.org), to which Parasite adheres, advises special treatment in these cases. In this case, the peer-review process was handled by an Invited Editor, Dominique Vuitton.The Editor-in-Chief of Parasite is one of the authors of this manuscript. COPE (Committee on Publication Ethics,"}
{"text": "We study the local indistinguishability problem of quantum states. By introducing an easily calculated quantity, non-commutativity, we present an criterion which is both necessary and sufficient for the local indistinguishability of a complete set of pure orthogonal product states. A constructive distinguishing procedure to obtain the concrete local measurements and classical communications is given. The non-commutativity of ensembles can be also used to characterize the quantumness for classical-quantum or quantum-classical correlated states. Nonlocality is one of the most important features in quantum mechanics. Quantum entanglement was firstly introduced to characterize and quantify the nonlocality, as it acts as a crucial resource in many quantum information processing tasksAB\u03c1. Although a set of POPS can always be distinguished globally, it may not be distinguished locally by local operations and classical communications (LOCC). This is called quantum nonlocality without entanglementAB\u03c1 is not an indicator of local indistinguishability for a set of states that constitute the pure-state decomposition of AB\u03c1491011121314A well known quantum phenomenon that exhibits quantum nonlocality is the local indistinguishability for some quantum states. State discrimination or distinguishing is essentially primitive for many quantum information tasks, such as quantum cryptography7It is natural to ask what kind of quantity or quantumness accounts for the quantum nonlocality. In this article, instead of quantum correlation measures, such as quantum discord, quantum deficit, etc., we investigate local indistinguishability for a set of POPS from the point of quantumness of ensemble. We first introduce an easily calculated quantity, non-commutativity, to quantify the quantumness of a quantum ensemble. Based on the non-commutativity, we present a necessary and sufficient criterion for the local indistinguishability. We also give a constructive distinguishing procedure to judge the local indistinguishability for any given set of POPS by using the criterion. Moreover, by proving the uniqueness of the expression of for semi-classical quantum correlated states, we show that our definition for quantumness of ensembles can be used to characterize the quantumness for semi-classical states.\u03c8\u3009 and |\u03d5\u3009 with equal probability can be viewed as a set of binary signals in some communication scheme. If |\u03c8\u3009 and |\u03d5\u3009 are orthogonal, the ensemble becomes classical since different states of classical information can be thought of merely as orthogonal quantum statesx = |\u3008\u03c8|\u03d5\u3009|, 0 < x < 1, induces quantumness for the ensemble. In fact the ensemble is most \u2018quantum\u2019 when A quantum ensemble containing only two pure states |A1, A2, \u2026, nA be a set of operators. We define the total non-commutativity for this set, A, B] = AB \u2212 BA, ||A|| is the trace norm of the operator A, \u03b5 = {ip, i\u03c1}, where i\u03c1 are density operators with probability ip. Denote iA = i\u03c1ip, then N is the measure of quantumness for the ensemble \u03b5. Here for the problem of local distinguishability of states \u03c11, \u03c12, \u2026, n\u03c1, the prior probabilities ip are irrelevant. One only needs to concern the quantity N(\u03b5) = N to judge the local distinguishability of the set of states \u03c11, \u03c12, \u2026, n\u03c1. The prior probabilities ip do make sense when one concerns the quantumness of a state given by the ensemble \u03b5 = {ip, i\u03c1} .Taking the above observation into account, we introduce non-commutativity to characterize the quantumness for a quantum ensemble. Let N has the following properties, which make it a well defined measure for quantumness of an ensemble:N is non negative; N is unitary invariant, N(\u03b5) = N(U\u03b5U\u2020), where U\u03b5U\u2020 = {ip, iUU\u03c1\u2020} with U being any unitary matrix; A1 = |\u03c8\u3009 \u3008\u03c8| and A2 = |\u03d5\u3009 \u3008\u03d5|, without considering their prior probabilities, N is zero only when the overlap x = |\u3008\u03d5|\u03c8\u3009| = 0 or 1. N gets to a maximum of 1 when For an ensemble only containing two pure states N({|ia\u3009}) + N({|ib\u3009}) \u2264 N. One can easily verify the inequality from the definition of N. The equality holds if {|ia\u3009} and {|ib\u3009} are either mutually orthogonal or identical. The sum of two sets' non-commutativity is equal to or less than the non-commutativity of the sum of the two sets: The non-commutativity \u03c81\u3009 = |1\u3009|1\u3009, st\u3009 = |s\u3009 \u2297 |t\u3009, |s\u3009 and |t\u3009 are the computational basis. Both {|i\u03c8\u3009} and {|st\u3009} are pure-state decompositions of \u03c1. However, the set {|st\u3009} can be locally distinguished while {|i\u03c8\u3009} cannot\u03c1 having zero quantum correlation is not an indicator for the local distinguishability of the states in \u03c1's pure-state decomposition. On the other hand, if we change the probability of |i\u03c8\u3009, \u03c1 is no longer an identity and can have nonzero quantum correlation, but the nine states |i\u03c8\u3009 still remain locally indistinguishable. Usually, the local indistinguishability of a set of pure states has no simple relations with the properties of the related density operatorA set of quantum states corresponds to a quantum ensemble. Nevertheless, one density operator may have many quantum ensemble decompositions. For instance, consider the density operator in a 3 \u2297 3 system, \u03b5 = {ip, |i\u03c8\u3009} classical if N(\u03b5) = 0, and quantum if N(\u03b5) > 0. If N(\u03b5) = 0, from the properties of non-commutativity, {|i\u03c8\u3009} must be either mutually orthogonal or identical and the states form a set of classical signals from informatics point of view. If N(\u03b5) > 0, among {|i\u03c8\u3009} there must be at least one pair of states that are neither orthogonal nor identical. Consider a set of bipartite POPS {|i\u03c8\u3009 = |ia\u3009 \u2297 |ib\u3009}, where |i\u03c8\u3009 are all mutually orthogonal, |ia\u3009 and |ib\u3009 are associated to partite A and B respectively. Obviously, this set forms a classical ensemble \u03b5 = {ip, |i\u03c8\u3009} since N(\u03b5) = 0, where ip is the probability with respect to |i\u03c8\u3009. Correspondingly one has ensembles A\u03b5 = {ip, |ia\u3009} and B\u03b5 = {ip, |ib\u3009} respectively. If N(A\u03b5) = N(B\u03b5) = 0, we call \u03b5 a classical-classical ensemble. If N(A\u03b5) = 0, N(B\u03b5) > 0, we call \u03b5 a classical-quantum ensemble. Analogously we can define quantum-classical and quantum-quantum ensembles.To study the local indistinguishability, and its relations with the quantumness of a quantum ensemble in terms of non-commutativity, in the following we call an ensemble finite rounds of LOCC protocols. If there are at least two states in a set which can not be distinguished by finite rounds of LOCC protocols, we say that the set can not be distinguished locally. The arguably more operational \u201casymptotic local operations and classical communications discrimination problem16A set of states is said to be reliably distinguished locally or distinguished locally if all the states in the set can be distinguished by Theorem 1 If a set of bipartite pure orthogonal product states cannot be locally distinguished, the ensemble composed of these states must be a quantum-quantum ensemble.Proof. Consider a set of bipartite POPS, not necessarily complete, {|i\u03c8\u3009 = |ia\u3009 \u2297 |ib\u3009} with \u3008i\u03c8|j\u03c8\u3009 = 0, \u2200i \u2260 j. Suppose these states form a classical-quantum or classical-classical ensemble \u03b5 = {ip, |ia\u3009 \u2297 |ib\u3009} with non zero ip. Since N({|ia\u3009}) = 0, {|ia\u3009} must be either mutually orthogonal or identical. To distinguish {|i\u03c8\u3009} locally, we can first take projective measurement on A side to distinguish those orthogonal states in {|ia\u3009}. For those identical states |ia\u3009, for example, |a1\u3009 = |a2\u3009, we can take a projective measurement on B to distinguish state |a1\u3009 from state |a2\u3009, since |b1\u3009 and |b2\u3009 must be mutually orthogonal to ensure that \u3008\u03c81|\u03c82\u3009 = 0. In this way we can distinguish the states in the set \u03b5 reliably. The analysis is similar if \u03b5 is a quantum-classical ensemble. Therefore if a set of bipartite POPS forms a classical-classical, quantum-classical or classical-quantum ensemble, the set can be locally distinguished. And if the set of POPS states cannot be locally distinguished, the ensemble composed of these states must be a quantum-quantum one. \u25a1n systems, any set of POPS is locally distinguishableSince it is impossible to form a quantum-quantum ensemble for a set of POPS in a 2 \u2297 2 system, Theorem 1 is both necessary and sufficient for 2 \u2297 2 systems. For higher dimensional cases, there are quantum-quantum ensembles whose states can be locally distinguished. In fact, for 2 \u2297 ia\u3009} and {|a\u2032j\u3009} are orthogonal if and only if \u3008ia|a\u2032j\u3009 = 0, \u2200i, j. If a set of states cannot be divided into subsets such that those subsets are mutually orthogonal, then we call the set a single set. Consider a set of states \u03b5 = {|a1\u3009 = |0\u3009, |a2\u3009 = |0 + 1\u3009, |a3\u3009 = |2\u3009}. This set can be divided into two single subsets \u03b51 = {|a1\u3009, |a2\u3009} and \u03b52 = {|a3\u3009} which are orthogonal and each subset cannot be split further. We call this partition the direct sum decomposition and denote it as \u03b5 = \u03b51 \u2295 \u03b52. A single set means that the set cannot be decomposed into such direct sums. If all the subsets are single sets in a direct sum decomposition, we say that the decomposition is a single set decomposition. It can verified that the single set decomposition of a non-single set is unique. For a single set, adding some states in the vector space spanned by itself will keep the set a single one. Nevertheless, taking away some states from a single set, the set could become a non-single one. In the following, by a set of states' decomposition we mean the single set decomposition.We say that two sets {|Lemma For a set of states \u03b5 = {|ia\u3009}, m = dim(span{|ia\u3009}), the following statements are equivalent: (a) \u03b5 is a single set. (b) There are m linear independent states \u03b5 satisfying the following relations: \u03b5.See Methods for the proof of the Lemma. From the Lemma we haveTheorem 2 For a complete set of m \u2297 n POPS, \u03b5 = {|i\u03c8\u3009 = |ia\u3009 \u2297 |ib\u3009} with \u3008i\u03c8|j\u03c8\u3009 = 0, \u2200i \u2260 j, the set \u03b5 cannot be completely locally distinguished if and only if there exist subsets Proof. Recall that a complete set of POPS can be locally distinguished if and only if the states can be distinguished by local nondestructive projective measurement and classical communication\u03b5 cannot be completely locally distinguished by local nondestructive projective measurement and classical communication, there must exist a subset \u03b5 such that From the view of accessible information, the more quantum an ensemble is, the less information one can get from the ensemble1i\u03c8\u3009 = |ia\u3009 \u2297 |ib\u3009} be a complete set of POPS, corresponding to two sets A\u03b5 = {|ia\u3009} and B\u03b5 = {|ib\u3009}. To verify the inequality (3), one needs to find all the subsets involved. This can be done in the following way.A\u03b5 and B\u03b5 into subsets, Decompose the sets s\u03c8\u3009 = |sa\u3009 \u2297 |sb\u3009}ij corresponds to two new subsets Find the overlapped states between these subsets, Theorem 2 gives a constructive distinguishing procedure to judge the local distinguishability for a complete set of POPS. Let {|n rounds for the new subsets until each of those new subsets has only one element or both A and B parts cannot be decomposed further. At last we have subset {|k\u03c8\u3009 = |ka\u3009 \u2297 |kb\u3009}stij,\u2026,  corresponding to two new sets i\u03c8\u3009} can be locally distinguished.We repeat the above process A\u03b5 = {|a1\u3009 = |1\u3009, |a2,3\u3009 = |0\u3009, |a4,5\u3009 = |2\u3009, a10\u3009 = |0\u3009, |a11\u3009 = |1\u3009, |a12\u3009 = |2\u3009} and B\u03b5 = {|b1\u3009 = |1\u3009, b6,7\u3009 = |0\u3009, |b8,9\u3009 = |2\u3009, |b10,11,12\u3009 = |3\u3009}. The distinguishing process is as follows.As an example, there is a complete set of POPS for 3 \u2297 4 system, A\u03b5) linear independent states in A\u03b5 satisfying A\u03b5 is a single set. Hence the first measurement should not be applied to A side. We denote B side, we have decomposition BM = 1(|0\u3009\u30080| + |1\u3009\u30081| + |2\u3009\u30082|) + 2|3\u3009\u30083| to distinguish Round 1: (i) There are 3 = dim(span (ii) Find the overlapped states by classical communication, a4\u3009, |a6\u3009, |a8\u3009 in \u03c81\u3009, \u2026, |\u03c89\u3009}01 is the subset described in Theorem 2. In fact they are the nine locally indistinguishable states in\u03c810\u3009, |\u03c811\u3009, |\u03c812\u3009}02 further since we have decomposition AM = 1|0\u3009\u30080| + 2|1\u3009\u30081| + 3|2\u3009\u30082|. However, Round 2: (i) Do decomposition for the above four new sets. Note (ii) Find overlapped states by classical communication, \u03c81\u3009, \u2026, |\u03c89\u3009}01, {|\u03c810\u3009}10,02, {|\u03c811\u3009}20,02, {|\u03c812\u3009}30,02, and the set cannot be completely distinguished due to the subset {|\u03c81\u3009, \u2026, |\u03c89\u3009}01.Finally we can divide the set (4) into four parts, {|\u03b5 = {ip, i\u03c1} is fixed (see Methods). Denote iX = i\u03c1ip. We have the non-commutativity describing the quantumness for the state qc\u03c1, N(qc\u03c1) satisfies the following properties, which actually makes it a candidate for quantum-correlation measures for semi-classical statesN(qc\u03c1) describes the quantumness of semi-classical states. It is different from those quantum correlation measures, such as quantum discord, quantum deficit, etc., which are based on the measurements and their estimations involve extremely complicated optimization process. Interestingly, while the quantity N(qc\u03c1) can be easily computed, it shows similar behavior to other quantum correlation measures for semi-classical states. Hence, instead of those quantum correlation measures, one can use the non-commutativity to characterize the quantum correlations of semi-classical states qc\u03c1. Moreover, non-commutativity provides a tool to explore the relation between the ensemble quantumness and those quantum correlation measures. The non-commutativity defined by (1) can be also used to describe the quantumness for bipartite semi-classical states. For a given semi-classical state, We have proposed non-commutativity as a quantumness measure for an ensemble. It has been shown that the local ensemble quantumness, instead of quantum correlation measure, like quantum discord, quantum deficit, etc., which is a function of a density operator, accounts for the local indistinguishability for a complete set of POPS. It implies that the quantumness of local ensembles must satisfy certain conditions so that the states in the ensemble cannot be locally distinguished.A constructive distinguishing procedure to obtain the concrete local measurements and classical communications has been presented to judge the local indistinguishability for a complete set of POPS. Our approach to judge the local indistinguishability can also be directly extended to distinguish a complete set of multipartite POPS or a non-complete set of POPS, when the local operations are restricted within nondestructive projective measurements.Due to that one semi-classical state corresponds to one quantum ensemble, the non-commutativity has been shown to be able to characterize the ensemble quantumness for classical-quantum or quantum-classical systems.n dimensional states A = |\u03d5\u3009\u3008\u03d5| and B = |\u03c8\u3009\u3008\u03c8|. Denote \u3008\u03d5|\u03c8\u3009 = i\u03b8xe with x \u2208 , \u03b8 \u2208  = i\u03b8xe|\u03d5\u3009\u3008\u03c8| \u2212 xei\u03b8\u2212|\u03c8\u3009\u3008\u03d5|. Under the base {|\u03d5\u3009 = |\u03d50\u3009, |\u03d51\u3009, \u2026, |n\u03d5\u22121\u3009}, one gets A, B] can then be expressed as i\u03c8j, = \u3008i\u03d5|\u03c8\u3009\u3008\u03c8|j\u03d5\u3009 and i\u03c8j,. The eigenvalues of the above matrix are \u03c8\u3009 and |\u03d5\u3009, |||| is 0 when |\u3008\u03d5|\u03c8\u3009| = 0 or 1, and 1 when Consider two pure \u03b5, say, \u03b5 such that it is not equal to or orthogonal to \u03b5 is a single set. Next choose the third vector \u03b5 such that it is independent of \u03b5 can be direct sum decomposed into two parts. Continuing with the above process, we can finally get m linear independent states \u03b5 satisfying inequality (2).(a) \u21d2 (b) First choose any vector in M, an Hermitian operator such that all |ia\u3009 in \u03b5 are the eigenvectors of M. If m linear independent M. Therefore M becomes an identity  and it cannot be used to distinguish the states in \u03b5.(b) \u21d2 (c) A nondestructive projective measurement is described by an observable, \u03b5 = \u03b51 \u2295 \u03b52. Then one can use a corresponding nondestructive projective measurement M = aI1 \u2295 bI2, where a \u2260 b, I1, I2 are identity operators, to distinguish the states in \u03b51 from \u03b52. Hence if M cannot do any in distinguishing the states in \u03b5, \u03b5 must be a single set, that is, statement (a) holds.(c) \u21d2 (a) Suppose the set is not a single one and has a decomposition i\u03b1\u3009} is an orthogonal base. Suppose there exists another orthogonal base {|i\u03b2\u3009} such that \u03c1 = \u03c3, where U|i\u03b1\u3009 = |i\u03b2\u3009, (U)ij = iju = \u3008i\u03b1|j\u03b2\u3009, \u03c3 can be reexpressed as i\u03c3iq. Set \u03c1 = \u03c3 implies that s \u2260 t, which means that both {|i\u03b1\u3009} and {|i\u03b2\u3009} are the eigenvectors of klQ, and i\u03c3iq} of state \u03c3 can be divided into degenerate part  and non-degenerate part . For non-degenerate part, taking over all k, l, one will find that the intersection of the eigenspaces belonging to the eigenvalues i\u03b1\u3009 = |i\u03b2\u3009 and then i\u03c1ip = i\u03c3iq. For degenerate part, without loss of generality, assume q1\u03c31 = q2\u03c32, then \u03c1 = \u03c3 means p1\u03c11 \u2297 |\u03b11\u3009\u3008\u03b11| + p2\u03c12 \u2297 |\u03b12\u3009\u3008\u03b12| = q1\u03c31 \u2297 (|\u03b21\u3009\u3008\u03b21| + |\u03b22\u3009\u3008\u03b22|). Therefore p1\u03c11 = p2\u03c12 = q1\u03c31 = q2\u03c32. Finally we get i\u03c1ip = i\u03c3iq, \u2200i.Let T.M., M.Z., Y.W. and S.M. wrote the main manuscript text. All authors reviewed the manuscript."}
{"text": "We consider the optimal dividends problem for a company whose cash reserves follow a general L\u00e9vy process with certain positive jumps and arbitrary negative jumps. The objective is to find a policy which maximizes the expected discounted dividends until the time of ruin. Under appropriate conditions, we use some recent results in the theory of potential analysis of subordinators to obtain the convexity properties of probability of ruin. We present conditions under which the optimal dividend strategy, among all admissible ones, takes the form of a barrier strategy. In the literatures of actuarial science and finance, the optimal dividend problem is one of the key topics. For companies paying dividends to shareholders, a commonly encountered problem is to find a dividend strategy that maximizes the expected total discounted dividends until ruin. The pioneer work can be traced to de Finetti  who consAnalysis of optimal dividends for L\u00e9vy risk processes is of particular interest which have undergone an intensive development. For example, Avram et al.  considerThe paper is organized as follows. In X = {Xt}t\u22650 be a real-valued L\u00e9vy process on a filtered probability space  where \ud835\udd3d = (\u2131t)t\u22650 is generated by the process X and satisfies the usual conditions of right continuity and completeness. Denote by Px the law of X when X0 = x. Let Ex be the expectation associated with Px. For notational convenience, we write P and E when X0 = 0. Write the L\u00e9vy triplet of X as , where a, \u03c3 \u2265 0 are real constants and \u03a0 is a positive measure on \u2216{0} which satisfies the integrability condition dx) = \u03c0(x)dx, then we call \u03c0 the L\u00e9vy density. The characteristic exponent of X is given by A is the indicator of set A. Furthermore, define the Laplace exponent of X by\u03c3 = 0 and \u222b\u221211 | x | \u03a0(dx) < \u221e. If \u03a0{} = 0, then the L\u00e9vy process X with no positive jumps is called the spectrally negative L\u00e9vy process; if \u03a0{} = 0, then the L\u00e9vy process X with no negative jumps is called the spectrally positive L\u00e9vy process. It is usual to assume that P(limt\u2192\u221eXt = +\u221e) = 1 which says nothing other than \u03a8\u2032(0+) > 0. For more information on L\u00e9vy processes we refer to the excellent book by Kyprianou , then the process Ltb can be explicitly represented by U0b = x > b, then Vb(x) the dividend value function if barrier strategy \u03beb is applied; that is,Vb is the solution to X with\u03b4 > 0, the equation \u03a8(z) = \u03b4 has a unique solution on , say \u03c1(\u03b4). A typical example is that the L\u00e9vy measure of the positive jumps has the following gamma distribution \u0393; that is, r is a positive number and \u03b3 is an even number.We denote by Vb can be expressed as\u03c8\u03c1(\u03b4) given by \u03c8\u03c1(\u03b4)(\u03b7) = \u03a8(\u03b7 + \u03c1(\u03b4)) \u2212 \u03b4. Note that the process Following similar reasoning to Yuen and Yin , Vb canY = {Yt}t\u22650, with Y0 = 0, where Yt = \u2212Xt, t \u2265 0. It is easy to see that the L\u00e9vy triplet of Y is , where \u03a0Y(dx) = \u03c0X(\u2212x)dx. Let Y, respectively. Following Kl\u00fcppelberg et al.  that L\u22121 = {Lt\u22121 : t \u2265 0} is the inverse local time such that Lt\u22121 = inf{s \u2265 0 : Ls > t}, where we take the infimum of the empty set as \u221e. Define an increasing process H by {Ht = YLt\u22121 : t \u2265 0}, that is, the process of new maxima indexed by local time at the maximum. The processes L\u22121 and H are both defective subordinators, and we call them the ascending ladder time and ladder height process of Y, respectively. It is understood that Ht = \u221e when Lt\u22121 = \u221e. Throughout the paper, we denote the nondefective versions of L, L\u22121, and H by \u2112, \u2112\u22121, and \u210b, respectively. In fact, the pair  is a bivariate subordinator. Define Y drifts to \u2212\u221e, the decreasing ladder height process is not defective. Associated with the ascending and descending ladder processes are the bivariate renewal functions U and k is its joint Laplace exponent such that q \u2265 0 is the killing rate of H so that q > 0 if and only if limt\u2192\u221eYt = \u2212\u221e, c \u2265 0 is the drift of H, and \u03a0H is its jump measure. Denote the marginal measure of U by U is called the potential/renewal measure. As for the descending ladder process, + and \u03a0\u2212 for the restrictions of \u03a0(du) and \u03a0(\u2212du) to . Furthermore, for u > 0, define In this section, we recap some basic facts about ladder processes and potential measure. Consider the dual process g et al. , we now \u03d5 : \u2192 is called a Bernstein function if it admits a representation a \u2265 0 is the killing term, b \u2265 0 is the drift, and \u03bc is the L\u00e9vy measure concentrated on  satisfying \u222b0\u221e(1\u2227x)\u03bc(dx) < \u221e. A function \u03c8 is called a special Bernstein function if the function \u03c8(\u03bb) = \u03bb/\u03d5(\u03bb) is again a Bernstein function. Let \u03d5 to be a special subordinator is that \u03bc is log-convex on . A function \u03d5 :  \u2192 \u211d is called a complete Bernstein function if there exists a Bernstein function \u03b7 such that \u2112 stands for the Laplace transform. It is known that every complete Bernstein function is a Bernstein function and that the following three conditions are equivalent: (i)\u03d5\u2009\u2009is a complete Bernstein function;(ii)\u03c8(\u03bb) = \u03bb/\u03d5(\u03bb) is a complete Bernstein function;(iii)\u03d5 is a Bernstein function whose L\u00e9vy measure \u03bc is given by where \u03bd is a measure on  satisfying We next introduce the notions of a special Bernstein function and complete Bernstein function and two useful results. Recall that a function ondra\u010dek  that  is summu et al.  and Hawku et al.  is givenH be a subordinator whose L\u00e9vy density, say \u03bc(x), x > 0, is log-convex. Then, the restriction of its potential measure to  has a nonincreasing and convex density. Furthermore, if the drift of H is strictly positive, then the density is in C1.Let H is a subordinator with Laplace exponent \u03d5 and potential measure U. Then, U has a density u which is completely monotone on  if and only if the tail of the L\u00e9vy measure is completely monotone.Suppose that \u03bc is a completely monotone function if and only if \u03bc has a completely monotone density. Thus, we have the following two equivalent statements: \u03d5 is a complete Bernstein function if and only if U has a density u which is completely monotone on ; or, equivalently, U has a density u which is completely monotone on  if and only if \u03bc has a completely monotone density.Note that the tail of the L\u00e9vy measure \u03c8(x) = \u03b1U, where \u03b1\u22121 = U = \u222b0\u221eP(Ht < \u221e)dt, with U given in \u2192 and Vb\u2217\u2032(x) = Vx\u2032(x) = 1 for x > b\u2217;(iii)\u03b4)Vb\u2217(x) = 0 for x \u2208 .(\u0393 \u2212 Suppose that x\u2192\u221eh\u2032(x) = \u221e, we have (i). For (ii), Vb\u2217\u2032(x) = h\u2032(x)/h\u2032(b\u2217) for x \u2208 ; it follows from the definition of b\u2217 that Vb\u2217(x) \u2265 1 for x \u2208 ; Vb\u2217\u2032(x) = Vx\u2032(x) = 1 for x > b\u2217 because of Vb\u2217(x) = x \u2212 b\u2217 + Vb\u2217(b\u2217); and Vx\u2032(x) = 1 since Vx(x) = h(x)/h\u2032(x). Finally, (iii) is due to (\u0393 \u2212 \u03b4)h(x) = 0 for x \u2208  and (As limb\u2217) and .h is sufficiently smooth and is convex in the interval . Then, for x > b\u2217, (i)Vb\u2217\u2032\u2032(x) = 0 \u2264 Vx\u2032\u2032(x\u2212) if \u03c3 \u2260 0;(ii)Vb\u2217\u2032(y) \u2265 Vx\u2032(y), y \u2208 ;(iii)Vb\u2217(x) \u2265 Vx(x);(iv)\u03b4)Vb\u2217(x) \u2264 0. = 0 is clear. Also, since h \u2208 C2 and is convex in the interval , we have Vx\u2032\u2032(x\u2212) = limy\u2191xVx\u2032\u2032(y) = limy\u2191xh\u2032\u2032(y)/h\u2032(x) \u2265 0. Thus, (i) is proved.If y \u2208 , by the definition of b\u2217, we have y \u2208 , by the convexity of h on , we have For Vb\u2217(b\u2217) = h(b\u2217)/h\u2032(b\u2217) \u2265 h(b\u2217)/h\u2032(x) = Vx(b\u2217) and that (Vb\u2217 \u2212 Vx) is nondecreasing on  because of (ii). Thus, Vb\u2217(x) \u2265 Vx(x); that is, (iii) holds.Note that x > b\u2217, (\u0393 \u2212 \u03b4)Vx(x\u2212) = limy\u2191x(\u0393 \u2212 \u03b4)Vx(y) = 0. For x \u2264 b\u2217, we have I1 \u2264 0, and I4 \u2265 0. For I2 + I3, we have J1 \u2264 0. For y > 0, we obtain J2 = 0. These prove (iv).For We now present the proofs of Theorems X by \u03c5 = \u03c5 = dt\u03a0(dy). Then, the L\u00e9vy decomposition [B = {Bt}t\u22650 is a standard Brownian motion and Mt is a martingale with M0 = 0.Define the jump measure of \u03bd is smooth enough for an application of the appropriate version of It\u00f4's formula and the change of variables formula. In fact, if X is of bounded variation, then \u03bd \u2208 C1 and we are allowed to use the change of variables formula [X has a Gaussian exponent, then \u03bd \u2208 C2 and we are allowed to use It\u00f4's formula [X has unbounded variation and \u03c3 = 0, then \u03bd is twice continuously differentiable almost everywhere but is not in C2 and we can use Meyer-It\u00f4's formula [tn, \u2009n \u2265 1}, we get under Px\u03bd implies that \u03bd(x) \u2212 \u03bd(y) + (x \u2212 y)\u03bd\u2032(y) \u2264 0 for any x \u2264 y. Taking expectations on both sides of Vb(x) \u2264 0 for x \u2208 \u2216{b} and Vb\u2032(x) \u2265 1 for x > 0. Similar to  = 0 and that Us\u2212\u03be + \u0394Xs \u2265 b on {\u0394Ls\u03be > 0, \u0394Xs > 0}. Consequently, Vb\u2032(Us\u2212\u03be + \u0394Xs) = 1, and hence tn, \u2009n \u2265 1}, we haven \u2192 \u221e in  is strictly convex on . Then, Vb\u2217 is concave on  because of (Vb(x) \u2265 V\u2217(x). Consequently, Vb(x) = V\u2217(x) and the proof is complete.If cause of . From Le\u03c0\u2212 is log-convex on , it follows from h(x) is strictly convex on . Then, applying \u03b4)Vb\u2217(x) \u2264 0 for all x > b\u2217. The result follows from If"}
{"text": "AX \u2212 XB = C are investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.Some complex quaternionic equations in the type  Mathematics, as with most subjects in science and engineering, has a long and varied history. In this connection one highly significant development which occurred during the nineteenth century was the quaternions, which are the elements of noncommutative algebra. Quaternions have many important applications in many applied fields, such as computer science, quantum physics, statistic, signal, and color image processing, in rigid mechanics, quantum mechanics, control theory, and field theory; see, for example, .ax + xb = c. In [ax = xb and ax = xb and \u03b1x\u03b2 + \u03b3x\u03b4 = \u03c1 linear quaternionic equation with one unknown, \u03b1x\u03b2 + \u03b3x\u03b4 = \u03c1, is solved. In [ax + xb = c is studied. In [\u03b1(x\u03b1) = (\u03b1x)\u03b1 = \u03b1x\u03b1 = \u03c1, with 0 \u2260 \u03b1 \u2208 O; second, they presented a method which allows to reduce any octonionic equation with the left and right coefficients to a real system of eight equations and finally reached the solutions of this linear octonionic equation from this real system. In [In recent years, quaternionic equations have been investigated by many authors. For example, the author of the paper  classifi = c. In , the linshed. In , the solated. In , the \u03b1x\u03b2lved. In , Bolat alved. In , the quadied. In , Bolat astem. In , Flaut astem. In \u201314.In this paper, we aim to obtain the solutions of some linear equations with two terms and one unknown by the method of matrix representations of complex quaternions over the complex quaternion field and to investigate the solutions of some complex quaternionic linear equations.The paper is organized as follows. In The following notations, definitions, propositions, lemmas, and theorems will be used to develop the proposed work. We now start the definitions of the quaternion and complex quaternion and their basic properties that will be used in the sequel.a + bi, where i is the imaginary unit with the defining property i2 = \u22121. The set of all complex numbers is usually denoted by C. From here, it can be easily said that the set of complex numbers is an extension of the set of real numbers, usually denoted by R. That is, R \u2282 C.It is well known that a complex number is a number consisting of a real and imaginary part. It can be written in the form HR is an algebra over the field R, and this algebra is called the real quaternion algebra and the set {1, i, j, k} is a basis in HR. The elements in HR take the form a = a0 + a1i + a2j + a3k, where a0, a1, a2, a3 \u2208 R, which can simply be written as a = Re\u2061a + Im\u2061\u2061a, where Re\u2061a = a0 and Im\u2061\u2061a = a1i + a2j + a3k. The conjugate of a is defined as a, b \u2208 HR. The norm of a is defined to be In the literature, firstly, the set of quaternions introduced asa, b \u2208 HR be quaternions. Then a and b are similar if and only if a0 = b0 and a12 + a22 + a32 = b12 + b22 + b32, that is, Re\u2061(a) = Re\u2061(b) and |Im\u2061\u2061a|2 = |Im\u2061\u2061b|2.Let Q = c0e0 + c1e1 + c2e2 + c3e3, where cn \u2208 C, n \u2208 {0,1, 2,3},\u03b2mn and et being uniquely determined by em and en. We denote by HC the set of the complex quaternions and HC is an algebra over the field C and this algebra is called the complex quaternion algebra. The set {1, e1, e2, e3} is a basis in HC.A complex quaternion is an element of the form Q = c0e0 + c1e1 + c2e2 + c3e3 \u2208 HC,\u2009\u2009cm \u2208 C, m \u2208 {0,1, 2,3}, can be written asam, bm \u2208 R,\u2009\u2009m \u2208 {0,1, 2,3}, and i2 = \u22121. Therefore, we can write a complex quaternion as the form Q = a + ib, where a = a0 + a1e1 + a2e2 + a3e3 and b = b0 + b1e1 + b2e2 + b3e3 are in HR. The conjugate of the complex quaternion Q is the element HR is denoted by H. For the quaternion a, b \u2208 H, if a* is defined asH, the mapa = a0e0 + a1e1 + a2e2 + a3e3 \u2208 H, is an isomorphism between H and the algebra of the matrices:\u03bb(a) \u2208 M4(R) has as columns the coefficients in R of the basis {1, e1, e2, e3} for the elements {a, ae1, ae2, ae3}. The matrix \u03bb(a) is called the left matrix representation of the element a \u2208 H.The element a \u2208 H the right matrix representation:a = a0e0 + a1e1 + a2e2 + a3e3 \u2208 H.Analogously with the left matrix representation, we have for the element \u03c1(a) \u2208 M4(R) has as columns the coefficients in R of the basis {1, e1, e2, e3} for the elements {a, e1a, e2a, e3a}.We remark that the matrix x, y \u2208 H and r \u2208 R, one has\u03bb(x + y) = \u03bb(x) + \u03bb(y),\u03bb(xy) = \u03bb(x)\u03bb(y),\u03bb(rx) = r\u03bb(x), \u03bb(1) = I4,\u03c1(x + y) = \u03c1(x) + \u03c1(y),\u03c1(xy) = \u03c1(y)\u03c1(x),\u03c1(rx) = r\u03c1(x), \u03c1(1) = I4,\u03bb(x\u22121) = (\u03bb(x))\u22121,\u03c1(x\u22121) = (\u03c1(x))\u22121, where x\u22121 is the inverse of x nonzero quaternion.For x \u2208 H, let M1\u00d74(R) be the vector representation of the element x. Therefore for all a, b, x \u2208 H the following relations are fulfilled:\u03c1(b)\u03bb(a) = \u03bb(a)\u03c1(b),\u03bb(x)) = det\u2061(\u03c1(x)) = (n(x))2, where n(x) = a02 + a12 + a22 + a32 and it is the weak norm of x. det\u2061,\u2009\u2009\u0398(Q) \u2208 M8(R); see [The matrix(R); see .a, x \u2208 H be two quaternions; then, the following relations are true.(i)a*i = ia, where i2 = \u22121. (ii)ai = ia*, where i2 = \u22121. (iii)a* = iai, where i2 = \u22121. \u2212(iv)xa)* = x*a*.((v)For X, A \u2208 HC, X = x + iy, A = a + ib,Let X, A \u2208 HC, X = x + iy, A = a + ib be given. ThenLet X, A \u2208 HC, X = x + iy, A = a + ib be given. ThenLet X \u2208 HC,\u2009\u2009X = x + iy be given. ThenX, where x, y \u2208 H and x and y.Let X, A, B \u2208 HC, X = x + iy,\u2009\u2009x, y \u2208 H be given. Then where 1 = I4 \u2208 M4(R) is the identity matrix and 0 = O4 \u2208 M4(R) is the zero matrix;; where ;\u03b12 = I4;;;, where Let In this section, the complex quaternionic equations in the typeR. In order to symbolically solve it, we need to examine some operation properties on the matrix \u0393(A) \u2212 \u03a8(B).According to (ii) and (v) cases in A = a + ib, B = c + id \u2208 HC be given, and denote \u03b4 = \u0393(A) \u2212 \u03a8(B). Then(i)\u03b4 iss = A0 \u2212 B0.the determinant of (ii)A0 \u2260 B0, or |Im\u2061\u2061A| \u2260 |Im\u2061\u2061B|, then \u03b4 is nonsingular and its inverse can be expressed asif (iii)A0 = B0 and |Im\u2061\u2061A| = |Im\u2061\u2061B|, then \u03b4 is singular and has a generalized inverse as follows:if Let A,\u2009\u2009B \u2208 HC, there are nonzero P, Q \u2208 HC such thatith case in It is a known result that, for all The results in A0 = B0 and |Im\u2061\u2061A| = |Im\u2061\u2061B|, it is easily seen thatIm\u2061\u2061A)2 = (Im\u2061\u2061B)2 = \u2212|Im\u2061\u2061A|2, we can easily deduce the following equality: \u03b43 = \u22124|Im\u2061\u2009A|2\u03b4. So, the proof of iiith case in Finally, under the conditions that Based on A = a + ib \u2208 HC be given and A \u2209 R. Then the general solution of the equationP \u2208 HC is arbitrary.Let \u03b4| = 0,  in According to (ii) and (v) cases in [\u0393(A)\u2212\u03a8(A[\u0393(A)\u2212\u03a8A and B are similar, if and only ifthe linear equation(ii)P \u2208 HC is arbitrary.in that case, the general solution of  is(36)XLet \u03b4| = 0, which is equivalent, by HC. Now substituting \u03b4\u2212 in According to (ii) and (v) cases in A = a + ib, B = c + id \u2208 HC,\u2009\u2009a, b, c, d \u2208 H, be given with A and B\u2009\u2009being not similar; that is, Re\u2061A \u2260 Re\u2061B or |Im\u2061\u2061A\u2009| \u2260 |Im\u2061\u2061B|. Then  = \u0393(A) \u2212 \u03a8(B) is nonsingular by Under the assumption of this theorem, A = a + ib \u2208 HC be given. Then the general solution of the equationP \u2208 HC is arbitrary.Let \u03b4| = 0,  in According to (ii) and (v) cases in [\u0393(A)\u2212\u03a8(A[\u0393(A)\u2212\u03a8 = \u2212(Im\u2061\u2061A)C, and then  and (v) cases in and then . In that[\u0393(A)\u2212\u03a8(B we find .Starting from known results and referring to the real matrix representations of the complex quaternions, in this paper we have investigated solutions of some linear equations with two terms and one unknown by the method of matrix representations of complex quaternions over the complex quaternion field.The methods and results developed in this paper can also extend to complex octonionic equations. We will present them in another paper."}
{"text": "H of a given group G is said to be autopermutable,if HH\u03b1 = H\u03b1H for all \u03b1 \u2208 Aut(G). We alsocall H a self-autopermutable subgroup of G, whenHH\u03b1 = H\u03b1H implies that H\u03b1 = H. Moreover, Gis said to be EAP-group, if every subgroup of G isautopermutable. One notes that if \u03b1 runs over the inner automorphisms of the group, one obtains the notions of conjugate-permutability, self-conjugate-permutability,and ECP-groups, which were studied by Foguel in 1997, Liand Meng in 2007, and Xu and Zhang in 2005, respectively. In the present paper, we determine the structure of a finite EAP-group when its centre is of index 4 in G. We also show that self-autopermutability and characteristic properties are equivalent for nilpotent groups.A subgroup  H be a subgroup of a given group G. Then we call H to be autopermutable, if HH\u03b1 = H\u03b1H for all \u03b1 \u2208 Aut(G). The subgroup H is said to be self-autopermutable, if HH\u03b1 = H\u03b1H implies that H\u03b1 = H. Moreover, we call the group G to be an EAP-group if every subgroup of G is autopermutable. Clearly, if \u03b1 runs over the inner automorphisms of the group, we obtain the notions of conjugate-permutability  =  = c3d,  = 1\u232a;\u2329a, b, c, d, e\u2223a2 = b2 = c2 = d2 = e4 = 1, a, b, c \u2208 Z(G),  = b\u232a;\u2329a, b, c, d, e\u2223a2 = b2 = c2 = d4 = e4 = 1, a, b, c \u2208 Z(G),  = a\u232a.\u2329Let G is a 2-group with cyclic centre of index 4.We remind that a nonabelian group is said to be Hamiltonian, if all of its subgroups are normal. The following result gives our claim, when G be a finite 2-group with cyclic centre of index 4. Then G is an EAP-group if and only if G\u2245Q8 or \u2329a, b\u2223an+12 = b2 = 1, ab = an+12\u232a, for all n \u2265 3.Let G to be Q8. Since Q8 is Hamiltonian group, the result follows easily. Now assume G = \u2329a, b\u2223an+12 = b2 = 1, ab = an+12\u232a, n \u2265 3. One can easily check that G contains exactly three proper subgroups of orders 2i, for 1 \u2264 i \u2264 n + 1. We also observe that the subgroups of orders 2 are autopermutable and as the subgroups of orders 2n+1 are normal, they are also autopermutable. Now, one can check that there are exactly two cyclic and one noncyclic subgroups of orders 2i, 2 \u2264 i \u2264 n, so that one of the cyclic subgroups is central and hence all the subgroups of G satisfy the required property.Consider the group G is an EAP-group, Z(G) = \u2329x\u2223xn2 = 1\u232a, G/Z(G) = {Z(G), aZ(G), bZ(G), abZ(G)}, where a2, b2 \u2208 Z(G) and so |a|, |b| \u2264 2n+1. In case n = 1, then the group G is either D8 or Q8. As explained before, D8 cannot be an EAP-group and hence G\u2245Q8. Now suppose n > 1 and the elements a and b are both of order 2. Then every element y \u2208 G has the following form (as G is nilpotent of class 2):\u03b1 given by \u03b1(y) = biajxk is an automorphism of G, which sends a into b. Thus HH\u03b1 \u2260 H\u03b1H for the subgroup H = \u2329b\u232a, which contradicts the assumption. Now, if |a | , |b | <2n+1 we may replace a and b by the elements axi and bxj, both of which are of order 2. This reduces to the previous case. Therefore we must have a or b of order 2n+1. Then G has a cyclic subgroup of order 2n+1 and so G is of order 2n+2\u2009\u2009(n > 1) with the centre of index 4. Hence, by  =  = c3d,  = 1\u232a;G = \u2329a, b, c, d, e\u2223a2 = b2 = c2 = d2 = e4 = 1, a, b, c \u2208 Z(G),  = b\u232a;G = \u2329a, b, c, d, e\u2223a2 = b2 = c2 = d4 = e4 = 1, a, b, c \u2208 Z(G),  = a\u232a.Let G be an EAP-group and G/Z(G) = {Z(G), aZ(G), bZ(G), abZ(G)}, where a2, b2 \u2208 Z(G). Assume that Z(G) is not an elementary abelian 2-group. Since Z(G) is the direct product of its cyclic subgroups, by the same argument as in Z(G) is an elementary abelian 2-group. Clearly G must be a group of order either 16 or 32. The structure of such groups is given as follows in  =  = c3d,  = 1\u232a.The sufficient condition is obvious. We only need to prove the necessity condition. Let llows in . If |G| D8 is not an EAP-group, hence the group of form (i) cannot be an EAP-group. For the group of form (ii) we can consider H = \u2329b\u232a and \u03b1 \u2208 Aut(G) which sends a and b into a and ab, respectively. Clearly, HH\u03b1 \u2260 H\u03b1H and hence G cannot be an EAP-group. Thus when |a | = |b|, then G is of the form given in either (iii) or (iv).As G| = 32. Then such groups in the list of small groups with elementary abelian centres of index 4 are only of the following forms: G = \u2329a, b, c\u2223a4 = b4 = c2 = 1,  = c,  =  = 1\u232a;G = \u2329a, b, c, d, e\u2223a2 = b2 = c2 = d4 = e4 = 1, a, b, c \u2208 Z(G),  = b\u232a;G = \u2329a, b, c, d, e\u2223a2 = b2 = c2 = d2 = e2 = 1, c, d \u2208 Z(G),  =  = ab,  = 1\u232a;G = \u2329a, b, c, d, e\u2223a2 = b2 = c2 = d2 = e4 = 1, a, b, c \u2208 Z(G),  = b\u232a;G = \u2329a, b, c, d, e\u2223a2 = b2 = c2 = d4 = e4 = 1, a, b, c \u2208 Z(G),  = a\u232a.Assume |H = \u2329b\u232a and \u03b1 \u2208 Aut(G), which sends a, b, and c into a, ab, and c, respectively. In case the group G is of form (ii), we consider H = \u2329e\u232a and \u03b1 \u2208 Aut(G) which sends a, b, c, d, and e into a, ab, ac, be, and ed, respectively. Also if the group is considered to be of form (iii), one may consider H = \u2329e\u232a and \u03b1 \u2208 Aut(G) which sends a, b, c, d, and e into e, b, c, d, and a, respectively. Now, one can easily check that in these cases HH\u03b1 \u2260 H\u03b1H and so G cannot be an EAP-group. Hence, when |G | = 32, then G is of either form (iv) or form (v). The proof is complete.For the group of form (i) we may consider the cyclic subgroup G is a 2-group. If G is not a 2-group, then we may write G = S1 \u00d7 S2 \u22ef \u00d7Sk, in such a way that S1 is a Sylow 2-subgroup and Si is an abelian Sylow pi-subgroup, where pi is an odd prime number, for 2 \u2264 i \u2264 k. Clearly, Aut(G)\u2245Aut(S1) \u00d7 Aut(S2)\u00d7\u22ef\u00d7Aut(Sk) and for any subgroup H of G, H\u2245H1 \u00d7 H2 \u00d7 \u22ef\u00d7Hk, where Hi \u2264 Si for 1 \u2264 i \u2264 k. Thus H is an autopermutable subgroup of G if H1 is an autopermutable subgroup of S1. This completes the proof.The necessity condition is obvious and Theorems H of a given group G to be weakly characteristic, when H\u03b1 \u2264 NG(H) implies that H\u03b1 = H for all \u03b1 \u2208 Aut(G). Also, given the subgroups H and K, then H satisfies the subcharacteriser condition, if H\u22b4K implies that NG)Aut((K) \u2264 NG)Aut((H), where NG)Aut((K) = {\u03b1 \u2208 Aut(G); K\u03b1 = K} \u2264 Aut(G). Clearly, if one considers the inner automorphisms of the group then weakly normal and normaliser condition properties are obtained.We call a subgroup p-subgroups of a given group.The following result of  shows thH be a p-subgroup of a group G. Then the following properties are equivalent: H is a self-conjugate-permutable subgroup;H is a weakly normal subgroup;H satisfies the subnormaliser condition.Let In this section, it is shown that self-autopermutable subgroups in nilpotent groups are always characteristic.H be a subgroup of a group G. H is self-autopermutable, then H is weakly characteristic in G.If H is weakly characteristic, then H satisfies the subcharacteriser condition in G.If Let H\u03b1 \u2264 NG(H), as H\u22b4NG(H), we have HH\u03b1 = H\u03b1H. Applying the condition that H is self-autopermutable subgroup of the group G, we get H\u03b1 = H. By definition, H is weakly characteristic.(i) If K \u2264 G, such that H\u22b4K. We have H\u03b1 \u2264 K\u03b1 = K \u2264 NG(H) for every \u03b1 \u2208 NG)Aut((K). Since H is weakly characteristic in G, we have H\u03b1 = H. Thus \u03b1 \u2208 NG)Aut((H) and the result is obtained.(ii) Let The following theorem is one of the main results in this section.H be a subgroup of a nilpotent finite group G. If H satisfies the subcharacteriser condition then H is characteristic in G.Let G\u2245P1 \u00d7 P2 \u00d7 \u22ef\u00d7Pt, where Pi is a Sylow pi-subgroup of G, for 1 \u2264 i \u2264 t. We may also write H\u2245H1 \u00d7 H2 \u00d7 \u22ef\u00d7Ht, with Hi \u2264 Pi, 1 \u2264 i \u2264 t. Since H satisfies the subcharacteriser condition in G, one can easily see that Hi satisfies the subcharacteriser condition in Pi. Therefore Hi\u22b4Pi implies that Hi is characteristic in Pi, which proves the result.Write Finally, we show that self-autopermutability, weakly characteristic, and subcharacteriser conditions are equivalent, for every subgroup of a nilpotent group.H be a subgroup of a finite nilpotent group G. Then H is a self-autopermutable;H is a weakly characteristic;H satisfies the subcharacteriser condition in G.Let The result follows by"}
{"text": "M be a lattice module over the multiplicative lattice L. A nonzero L-lattice module M is called second if for each a \u2208 L, a1M = 1M or a1M = 0M. A nonzero L-lattice module M is called secondary if for each a \u2208 L, a1M = 1M or an1M = 0M for some n > 0. Our objective is to investigative properties of second and secondary lattice modules.Let  L is a complete lattice in which there is defined as a commutative, associative multiplication which distributes over arbitrary joins and has the compact greatest element 1L (least element 0L) as a multiplicative identity (zero). An element a \u2208 L is said to be proper if a < 1L. An element p < 1L in L is said to be prime if ab \u2264 p implies either a \u2264 p or b \u2264 p. If 0L is prime, then L is said to be a domain. For a \u2208 L, we define p < 1L in L is said to be primary if ab \u2264 p implies either a \u2264 p or A multiplicative lattice a, b belong to L, (a:Lb) is the join of all c \u2208 L such that cb \u2264 a. An element e of L is called meet principal if a\u22c0be = ((a:Le)\u22c0b)e for all a, b \u2208 L. An element e of L is called join principal if ((ae\u22c1b):Le) = a\u22c1(b:Le) for all a, b \u2208 L. e \u2208 L is said to be principal if e is both meet principal and join principal. e \u2208 L is said to be weak meet (join) principal if a\u22c0e = e(a:Le)\u2009\u2009(a\u22c1(0L:Le) = (ea:Le)) for all a \u2208 L. An element a of a multiplicative lattice L is called compact if a \u2264 \u22c1b\u03b1 implies a \u2264 b\u03b11\u22c1b\u03b12\u22c1\u22ef\u22c1b\u03b1n for some subsets {\u03b11, \u03b12,\u2026, \u03b1n}. If each element of L is a join of principal (compact) elements of L, then L is called a PG-lattice (CG-lattice). If L is a CG-lattice and p is a primary element, then If M be a complete lattice. Recall that M is a lattice module over the multiplicative lattice L or simply an L-module in case there is a multiplication between elements of L and M, denoted by lB for l \u2208 L and B \u2208 M, which satisfies the following properties:lb)B = l(bB);(\u03b1l\u03b1)(\u22c1\u03b2B\u03b2) = \u22c1\u03b1,\u03b2l\u03b1B\u03b2; is the join of all a \u2208 L such that aK \u2264 N. Particularly, (0M:L1M) is denoted by ann(M). If a \u2208 L and N \u2208 M, then (N:Ma) is the join of all H \u2208 M such that aH \u2264 N. An element N of M is called meet principal if (b\u2227(B:LN))N = bN\u2227B for all b \u2208 L and for all B \u2208 M. An element N of M is called join principal if b\u2228(B:LN) = ((bN\u2228B):LN) for all b \u2208 L and for all B \u2208 M. N is said to be principal if it is both meet principal and join principal. In special cases, an element N of M is called weak meet principal  if (B:LN)N = B\u2227N\u2009\u2009((bN:LN) = b\u2228(0M:LN)) for all B \u2208 M\u2009\u2009. N is said to be weak principal if N is both weak meet principal and weak join principal.Let M be an L-module. An element N in M is called compact if N \u2264 \u22c1\u03b1B\u03b1 implies N \u2264 B\u03b11\u2228B\u03b12\u2228\u22ef\u2228B\u03b1n for some subsets {\u03b11, \u03b12,\u2026, \u03b1n}. The greatest element of M will be denoted by 1M. If each element of M is a join of principal (compact) elements of M, then M is called a PG-lattice module (CG-lattice module).Let M be an L-module. An element N \u2208 M is said to be proper if N < 1M. For all elements N of M,  is a set of all K \u2208 M such that N \u2264 K \u2264 1M and  is an L-lattice module with a \u00b7 K = aK\u2228N for all a \u2208 L and K \u2208 M such that N \u2264 K.Let For various characterizations of lattice modules, the reader is referred to \u20139.L-lattice module M is called second if for each a \u2208 L, a1M = 1M or a1M = 0M.A nonzero L-lattice module M is called secondary if for each a \u2208 L, a1M = 1M or an1M = 0M for some n > 0.A nonzero Z be the integers, let Q be the rational numbers, and let Q be Z-module. Suppose L = L(Z) is the set of all ideals of Z and M = L(Q) is the set of all submodules of Q. Thus, M as L-lattice module is a second module, since for every integer n \u2208 Z, (nZ)Q = Q or (nZ)Q = 0.Let Every second lattice module is a secondary lattice. But the converse is not true. For this, we can give the following example.Z be the integers and let Z4 be Z-module. Suppose that L = L(Z) is the set of all ideals of Z and M = L(Z4) is the set of all submodules of Z4. Thus, M as L-lattice module is a secondary lattice module, which is not a second lattice module.Let Z be the integers and L = L(Z) the set of all ideals of Z. Thus, L as L-lattice module is neither a second lattice module nor a secondary lattice module.Let L be a CG-lattice and let M be a nonzero L-lattice module. If for each compact a \u2208 L, a1M = 1M or a1M = 0M, then M is a second L-lattice module.Let r \u2208 L. Since L is a CG-lattice, then we have r = \u22c1ici such that ci's are compact elements of L. Then, we obtain r1M = (\u22c1ici)1M = \u22c1ici1M. We have two cases.Let Case 1. If ci1M = 0M for each compact ci \u2208 L, then we have r1M = (\u22c1ici)1M = \u22c1ici1M = 0M. Case 2. If ci1M = 1M for some compact ci \u2208 L, then we have r1M = (\u22c1ici)1M = \u22c1ici1M = 1M.r1M = 1M or r1M = 0M for each r \u2208 L. Consequently, M is second.Hence, M is a second L-lattice module, then ann(M) = (0M:L1M) = p is a prime element of L. In this case, M is called p-second lattice module.If M is a second L-lattice module. Clearly, ann(M) = p is a proper element of L. Let ab \u2264 p and assume that b\u2270p; that is, b1M \u2260 0M. But M is a second L-lattice module; then b1M = 1M. Since b1M = 1M and ab1M = 0M, then a1M = 0M, which implies that a \u2264 p.Suppose that M is a secondary L-lattice module, then ann(M) is a primary element of L.If M is a secondary L-lattice module. Let ab \u2264 ann(M) and a \u2264 ann(M). Since ab \u2264 ann(M) and ab1M = 0M and (b)n1M \u2260 0M for each n > 0. Since M is secondary, we have b1M = 1M. Then ab1M = a1M = 0M, which implies a \u2264 ann(M).Suppose that L be a CG-lattice. If M is a secondary L-lattice module, then L. In this case, M is called p-secondary lattice module.Let M be a secondary lattice. Then ann(M) is a primary element of L by Let L be a lattice domain and let M be a nonzero L-module. Then M is a second lattice module with ann(M) = 0L if and only if M is a secondary lattice module with Let M is a second lattice module, then M is a secondary lattice module. Since L is domain, then \u21d2: Since M is a secondary lattice module with a \u2208 L and assume that a1M \u2260 1M. Since M is a secondary lattice module, then there exists a positive integer n such that an1M = 0M; that is, a1M = 0M. Hence, we obtain M is a second lattice. Clearly, ann(M) = 0L.\u21d0: Suppose that M be a nonzero L-lattice module. An element 0M \u2260 N < 1M is said to be pure element, if aN = a1M\u22c0N for all a \u2208 L.Let L be a CG-lattice, let M be a nonzero L-lattice module, and let N be a pure element of M. If M is a p-secondary lattice module, then  and  are both p-secondary lattice modules.Let M is a p-secondary lattice module. Let s \u2208 L. Since M is a secondary lattice module, then either s1M = 1M and in this case s \u00b7 1N,1M] is a secondary lattice module.Suppose that r be compact and r is compact, there exists a positive integer n such that rn \u00b7 1N,1M] is a secondary lattice module.Let c be compact and c is compact, there exists a positive integer k such that ck \u00b7 N = ckN = 0M,N] and  are both p-second lattice modules.Let M is a p-second lattice module. Let a \u2208 L. Since N is pure, we have aN = N\u22c0a1M. As M is a second lattice module, then either a1M = 0M or a1M = 1M. This implies that either aN = 0M or aN = N. Hence,  is a second lattice module. Now, we show that ann(M) = ann. Clearly, ann(M) \u2264 ann. Let r \u2264 ann. Thus, we have rN = 0M. Now we assume that r\u2270ann(M). Then we obtain r1M = 1M, since M is a second lattice module. This implies that 0M = rN = r1M\u22c0N = 1M\u22c0N = N, a contradiction. Therefore, r \u2264 ann(M).\u21d2: Suppose that t \u2208 L. Since M is a second lattice module, either t1M = 1M and in this case t \u00b7 1N,1M] is a second lattice module. It remains to show that ann(M) = ann. Clearly ann(M) \u2264 ann. Let s \u2264 ann. Thus, s \u00b7 1N,1M] and  are both second lattice modules with ann = ann = p. Let r \u2208 L. We have two cases.\u21d0: Suppose that  is a second lattice module. Hence, we have N = rN = N\u22c0r1M, that is, N \u2264 r1M, since N is pure. Because  is a second lattice module and r\u2270ann; we obtain r \u00b7 1N,1M][, that is; s \u00b7 1M = N. Thus, s1M\u22c1N = N, and so s1M \u2264 N. Now, we assume that s\u2270ann(M). Then, we have s1M = 1M, since M is second. Hence, 1M = s1M \u2264 N, a contradiction. Consequently, we have s \u2264 ann(M).Now we show that L-module M is called a multiplication lattice module if for every element N \u2208 M, there exists an element a \u2208 L, such that N = a1M.An N of an L-module M is called prime element if N \u2260 1M and whenever r \u2208 L and X \u2208 M with rX \u2264 N, then X \u2264 N or r \u2264 (N:L1M).A element N of an L-module M is called semiprime element if N \u2260 1M and whenever r \u2208 L and X \u2208 M with r2X \u2264 N, then rX \u2264 N.A element N be a proper element of an L-module M. Then N is a semiprime element if and only if whenever r \u2208 L, X \u2208 M and k is a positive integer with rkX \u2264 N, then rX \u2264 N.Let We know that a prime element is semiprime, but the converse is not true in general. The following proposition shows that the converse is true when the module is secondary and multiplication.M be a multiplication and secondary L-lattice module. For all element N of M such that 1M \u2260 N \u2208 M, N is a semiprime element of M if and only if N is a prime element of M.Let N is a semiprime element of M and let rX \u2264 N, where r \u2208 L, X \u2208 M. Since M is a secondary lattice module, then either rn1M = 0M for some positive integer n or r1M = 1M. \u21d2: Suppose that Case\u2009\u20091. If rn1M = 0M, then rn1M \u2264 N. Since N is a semiprime element, we have r1M \u2264 N.Case\u2009\u20092. If r1M = 1M, then we have X = rX, since M is a multiplication lattice module. Then we have rX = X \u2264 N. N is a prime element of M.Therefore, \u21d0: It is obvious.M be an L-lattice module and let N be a proper element of M. N is called a primary element of M, if whenever a \u2208 L, X \u2208 M such that aX \u2264 N, then X \u2264 N or M is nonzero and 0M is primary, then M is said to be primary lattice module.Let L-lattice module M is said to be simple lattice module if M = {0M, 1M}.An Every multiplication secondary lattice module is a primary lattice module.M be a multiplication secondary module and rX = 0M for some r \u2208 L, X \u2208 M. Now, we assume that M is a secondary module, then we have r1M = 1M. Because M is a multiplication, then we have rX = X. Consequently, we obtain X = 0M.Let Every multiplication second lattice module is a simple lattice module.M be a multiplication and second module. Since M is a multiplication, for every N \u2208 M, there exists a \u2208 L such that N = a1M. Then we obtain a1M = 1M or a1M = 0M, since M is second. Thus, we have N = 1M or N = 0M for every N \u2208 M; that is, M is simple.Let L be a domain and let M be a nonzero L-lattice module. If r1M = 1M for every 0L \u2260 r \u2208 L, then M is said to be divisible.Let L-lattice module M is said to be torsion if there exists 0L \u2260 r \u2208 L such that r1M = 0M.A nonzero L be a domain. Let M be a secondary L-lattice module. Then either M is a divisible module or M is a torsion module.Let M is a secondary module over a domain L. If M is not divisible, then there exists 0L \u2260 r \u2208 L such that r1M \u2260 1M. Since M is a secondary lattice module, then there exists a positive integer n such that rn1M = 0M. Since 0L \u2260 r and L is a domain, then we have rn \u2260 0L. Consequently, there exists 0L \u2260 rn = s \u2208 L such that s1M = 0M. Therefore, M is a torsion lattice module.Suppose that"}
{"text": "C-approximating posets, the concept of countably QC-approximating posets is introduced. With the countably QC-approximating property, some characterizations of generalized completely distributive lattices and generalized countably approximating posets are given. The main results are as follows: (1) a complete lattice is generalized completely distributive if and only if it is countably QC-approximating and weakly generalized countably approximating; (2) a poset L having countably directed joins is generalized countably approximating if and only if the lattice \u03c3c(L)op of all \u03c3-Scott-closed subsets of L is weakly generalized countably approximating.As a generalization of countably  L is continuous if and only if the lattice \u03c3(L)op of all Scott-closed subsets of L is completely distributive. Gierz et al. in [C-continuous lattices. And they showed that for any poset L, \u03c3(L)op is a C-continuous lattice and that L is continuous if and only if \u03c3(L)op is continuous.The notion of continuous lattices as a model for the semantics of programming languages was introduced by Scott in . Later, t al. in  introduct al. in  introduct al. in  proved tt al. in  introduc\u03c3-Scott topology. Yang and Liu in [C-approximating posets and showed that the lattice of all \u03c3-Scott-closed subsets of a poset is a countably C-approximating lattice and that a complete lattice is completely distributive if and only if it is countably approximating and countably C-approximating.On the other hand, Lee in  introducd Liu in  introducd Liu in , 10, Maod Liu in  introducC-approximating posets to the concept of countably QC-approximating posets. With the countably QC-approximating property, we present some characterizations of GCD lattices and generalized countably approximating posets.In this paper, we generalize the concept of countably L, \u2264) be a poset. Then L with the dual order is also a poset and denoted by Lop. A principal ideal  is a set of the form \u2193x = {y \u2208 L\u2223y \u2264 x} . For X\u2286L, we write \u2193X = {y \u2208 L\u2223\u2203 \u2009\u2009x \u2208 X, y \u2264 x}, \u2191X = {y \u2208 L\u2223\u2203 \u2009\u2009x \u2208 X, x \u2264 y}. A subset X is a(n) lower set  if X = \u2009\u2193X . The supremum of X is denoted by \u2228X or sup\u2061X. A subset D of L is directed if every finite subset of D has an upper bound in D. A subset D of L is countably directed if every countable subset of D has an upper bound in D. Clearly every (countably) directed set is nonempty, and every countably directed set is directed but not vice versa. A poset L is a directed complete partially ordered set  if every directed subset of L has a supremum. A poset is said to have countably directed joins if every countably directed subset has a supremum.We quickly recall some basic notions and results . Let  to denote the power set of X and Pfin(X) to denote the set of all nonempty finite subsets of X. For a poset L, define a preorder \u2264  on P(L)\u2216{\u2205} by G \u2264 H if and only if \u2191H\u2286\u2191G for all G, H\u2286L. That is, G \u2264 H if and only if for each y \u2208 H there is an element x \u2208 G with x \u2264 y. We say that a nonempty family F of subsets of L is (countably) directed if it is (countably) directed in the Smyth preorder. More precisely, F is directed if for all F1, F2 \u2208 F, there exists F \u2208 F such that F1, F2 \u2264 F; that is, F\u2286\u2191F1\u2229\u2191F2.For a set c on points of L to the nonempty subsets of L, one obtains the concept of weakly generalized countably approximating posets.Generalizing the relation \u226aL be a poset having countably directed joins. A binary relation \u226ac on P(L)\u2216{\u2205} is defined as follows. A\u226acB if and only if for any countably directed set D\u2286L, \u2228\u2009\u2009D \u2208 \u2191B implies D\u2009\u2229\u2191A \u2260 \u2205. We write F\u226acx for F\u226ac{x} and y\u226acH for {y}\u226acH. If for each x \u2208 L, \u2191x = \u2229{\u2191F\u2223F \u2208 \u03c9(x)}, where \u03c9(x) = {F\u2223F \u2208 Pfin(L) and F\u226acx}, then L is called a weakly generalized countably approximating poset. A weakly generalized countably approximating poset which is also a complete lattice is called a weakly generalized countably approximating lattice.Let L with the condition that for each x \u2208 L, \u03c9(x) is countably directed is called a generalized countably approximating poset (lattice) in [A weakly generalized countably approximating poset (lattice) tice) in .As a generalization of completely distributive lattice, the following concept of GCD lattices was introduced in .L be a poset. A binary relation \u22b2 on P(L) is defined as follows. A\u22b2B if and only if whenever S is a subset of L for which \u2228S exists, \u2228S \u2208 \u2191B implies S\u2009\u2229\u2191A \u2260 \u2205. A complete lattice L is called a generalized completely distributive lattice or shortly a GCD lattice, if and only if for all x \u2208 L, \u2191x = \u2229{\u2191F\u2223F \u2208 Pfin(L) and F\u22b2x}.Let U of a poset L is Scott-open if \u2191U = U and for any directed set D\u2286L, sup\u2061D \u2208 U implies U\u2229D \u2260 \u2205. All the Scott-open sets of L form a topology, called the Scott topology and denoted by \u03c3(L). The complement of a Scott-open set is called a Scott-closed set. The collection of all Scott-closed sets of L is denoted by \u03c3(L)op. The topology on L generated by {L\u2216\u2193x\u2223x \u2208 L} as a subbase is called the upper topology and denoted by \u03bd(L).A subset \u03c3-Scott-open sets.Replacing directed sets with countably directed sets in L be a poset. A subset U of L is called \u03c3-Scott-open if \u2191U = U and for any countably directed set D\u2286L, sup\u2061D \u2208 U implies U\u2229D \u2260 \u2205. All the \u03c3-Scott-open sets of L form a topology, called the \u03c3-Scott topology and denoted by \u03c3c(L). The complement of a \u03c3-Scott-open set is called a \u03c3-Scott-closed set. The collection of all \u03c3-Scott-closed sets of L is denoted by \u03c3c(L)op.Let L, the \u03c3-Scott topology \u03c3c(L) is closed under countably intersections and the Scott topology \u03c3(L) is coarser than \u03c3c(L); that is, \u03c3(L)\u2286\u03c3c(L).(1) For a poset \u03c3-Scott-closed if and only if it is a lower set and closed under countably directed joins.(2) A subset of a poset is \u03c3-Scott-closed subsets for a poset, Mao and Xu in [C-approximating posets.To study the order structure of the lattice of all nd Xu in  introducL be a poset and x, y \u2208 L. We say that x is \u03c3-beneath y, denoted by x\u227a\u03c3y, if for any nonempty \u03c3-Scott-closed set F\u2286L for which \u2228F exists, \u2228F \u2265 y always implies that x \u2208 F. Poset L is said to be countably C-approximating if for each x \u2208 L, x = \u2228\u2193\u03c3\u227ax, where \u2193\u03c3\u227ax = {y \u2208 L\u2223y\u227a\u03c3x}. A complete lattice which is also countably C-approximating is called a countably C-approximating lattice.Let L, the lattice \u03c3c(L)op is countably C-approximating.For a poset L be a poset and C \u2208 \u03c3c(\u03c3c(L)op)op. It is straightforward to check that \u22c1\u03c3c(L)opC = \u222aC. For each F \u2208 \u03c3c(L)op, we have that F = \u22c1\u03c3c(L)op{\u2193x\u2223x \u2208 F}. Suppose C \u2208 \u03c3c(\u03c3c(L)op)op with \u22c1\u03c3c(L)opC\u2287F. Then for each x \u2208 F, since \u22c1\u03c3c(L)opC = \u222aC\u2287F, there exists A \u2208 C such that x \u2208 A. Noticing that A \u2208 \u03c3c(L)op is a lower set, we have \u2193x\u2286A \u2208 C. It follows from C \u2208 \u03c3c(\u03c3c(L)op)op being a lower set that \u2193x \u2208 C. Thus by x\u227a\u03c3F holds in \u03c3c(L)op. Hence, F = \u22c1\u03c3c(L)op{\u2193x\u2223x \u2208 F}\u2286\u2228\u2193\u03c3\u227aF\u2286F. So, F = \u2228\u2193\u03c3\u227aF and by the arbitrariness of F \u2208 \u03c3c(L)op, we conclude that \u03c3c(L)op is countably C-approximating.Let QC-approximating posets. Firstly, we generalize the relation \u227a\u03c3 on points of a poset L to the nonempty subsets of L.In this section, we introduce the concept of countably L, the \u03c3-beneath relation \u227a\u03c3 on nonempty subsets of L is defined as follows: A\u227a\u03c3B if and only if whenever S is a nonempty \u03c3-Scott-closed subset of L for which \u2228S exists, \u2228S \u2208 \u2191B implies S\u2009\u2229\u2191A \u2260 \u2205. We write F\u227a\u03c3x for F\u227a\u03c3{x}. Set c(x) = {F\u2223F \u2208 Pfin(L)\u2009\u2009and\u2009\u2009F\u227a\u03c3x}.For a poset The next proposition is basic and the proof is omitted.L be a poset. ThenG, H\u2286L, G\u227a\u03c3H\u21d2G \u2264 H;\u2200G, H\u2286L, G\u227a\u03c3H\u21d4\u2200h \u2208 H, G\u227a\u03c3h;\u2200E, F, G, H\u2286L, E \u2264 G\u227a\u03c3H \u2264 F\u21d2E\u227a\u03c3F;\u2200x, y \u2208 L, {x}\u227a\u03c3{y}\u21d4x\u227a\u03c3y.\u2200Let \u03c3, we have the concept of countably QC-approximating posets.With the relation \u227aL is said to be countably quasi-C-approximating, shortly countably QC-approximating, if for all x \u2208 L, \u2191x = \u2229{\u2191F\u2223F \u2208 c(x)}. A countably QC-approximating poset which is also a complete lattice is called a countably QC-approximating lattice.A poset C-approximating posets are countably QC-approximating.Countably L be a countably C-approximating poset. Then for all x \u2208 L,F\u2223F \u2208 c(x)} = \u2191x. By L is countably QC-approximating.Let By L, the lattice \u03c3c(L)op is countably QC-approximating.For any poset QC-approximating lattices and GCD lattices.In the sequel, we explore relationships between countably Every GCD lattice is weakly generalized countably approximating.L be a GCD lattice. For all x \u2208 L and F \u2208 Pfin(L), F\u22b2x implies F\u226acx. Then \u2191x\u2286\u2229{\u2191F\u2223F \u2208 \u03c9(x)}\u2286\u2229{\u2191F\u2223F \u2208 Pfin(L) and F\u22b2x} = \u2191x. So \u2191x = \u2229{\u2191F\u2223F \u2208 \u03c9(x)}. By L is weakly generalized countably approximating.Let QC-approximating.Every GCD lattice is countably L be a GCD lattice. For each x \u2208 L and F \u2208 Pfin(L), F\u22b2x implies F\u227a\u03c3x. Then \u2191x\u2286\u2229{\u2191F\u2223F \u2208 c(x)}\u2286\u2229{\u2191F\u2223F \u2208 Pfin(L) and F\u22b2x} = \u2191x. Thus \u2191x = \u2229{\u2191F\u2223F \u2208 c(x)}. By L is countably QC-approximating.Let The following theorem characterizes GCD lattices.L be a complete lattice. Then the following statements are equivalent:L is a GCD lattice;L is countably QC-approximating and weakly generalized countably approximating.Let (1)\u21d2(2): follows from Propositions L is countably QC-approximating and weakly generalized countably approximating. Then for each x \u2208 L, by the weakly generalized countably approximating property of L, we have \u2191x = \u2229{\u2191F\u2223F \u2208 \u03c9(x)}. Now for each F \u2208 \u03c9(x), we show that \u2191F = \u2229{\u2191F\u2032\u2223F\u2032 \u2208 Pfin(L) and F\u2032\u227a\u03c3F}. To this end, it suffices to show that \u2229{\u2191F\u2032\u2223F\u2032 \u2208 Pfin(L) and F\u2032\u227a\u03c3F}\u2286\u2191F. Suppose t \u2208 \u2229{\u2191F\u2032\u2223F\u2032 \u2208 Pfin(L) and F\u2032\u227a\u03c3F} and t \u2209 \u2191F. Then for any yF \u2208 F, t \u2209 \u2191yF. By the countably QC-approximating property of L, there exists FyF \u2208 c(yF) such that FyF\u227a\u03c3yF and t \u2209 \u2191FyF. Let t \u2208 \u2229{\u2191F\u2032\u2223F\u2032 \u2208 Pfin(L) and F\u2032\u227a\u03c3F}. Thus \u2191x = \u2229{\u2191F\u2223F \u2208 \u03c9(x)} = \u2229{\u2191F\u2032\u2223F\u2032 \u2208 Pfin(L), \u2203F \u2208 Pfin(L) such that F\u2032\u227a\u03c3F\u226acx}.(2)\u21d2(1): suppose that F\u2032\u227a\u03c3F\u226acx, we will show that F\u2032\u22b2x. For any A\u2286L with \u2228A \u2265 x, let G = {\u2228E\u2223E is a countable subset of A}. Then G is a countably directed set and \u2228G = \u2228A \u2208 \u2191x. Since F\u226acx, there exists a countable subset E\u2286A such that \u2228E = \u2228\u2193E \u2208 \u2191F. By E is \u03c3-Scott-closed. It follows from F\u2032\u227a\u03c3F that \u2193E\u2229\u2191F\u2032 \u2260 \u2205. This implies A\u2229\u2191F\u2032 \u2260 \u2205, showing that F\u2032\u22b2x. Thus, \u2191x\u2286\u2229{\u2191W\u2223W \u2208 Pfin(L), W\u22b2x}\u2286\u2229{\u2191F\u2032\u2223F\u2032 \u2208 Pfin(L), \u2203 F \u2208 Pfin(L), F\u2032\u227a\u03c3F\u226acx} = \u2191x. So, \u2191x = \u2229{\u2191W\u2223W \u2208 Pfin(L) and W\u22b2x}. Therefore, L is a GCD lattice.Suppose L is called a hypercontinuous poset x} is directed and x = sup\u2061{y \u2208 L\u2223y\u227a\u03bd(L)x}, where y\u227a\u03bd(L)x\u21d4x \u2208 int\u03bd(L)\u2191y. A hypercontinuous poset which is also a complete lattice is called a hypercontinuous lattice.Recall that a poset set (see ) if for L be a complete lattice. Then L is a GCD lattice if and only if Lop is a hypercontinuous lattice.Let L, both L and Lop are continuous, and \u03bd(L) = \u03c3(L). It follows from  is hypercontinuous.Let \u03c3c(L)op is a GCD lattice. The following theorem gives comprehensive characterizations of generalized countably approximating posets.So, in view of L be a poset having countably directed joins. Then the following statements are equivalent:L is a generalized countably approximating poset;\u03c3c(L) is a hypercontinuous lattice;\u03c3c(L)op is a GCD lattice;\u03c3c(L)op is a weakly generalized countably approximating lattice.Let (i)\u21d4(ii) by (ii)\u21d4(iii) by (iii)\u21d4(iv) follows from"}
{"text": "Z with its two crisp boundaries named upper-bound set and lower-bound set. In this paper, the concept of similarity degree between two interval sets is defined at first, and then the similarity degrees between an interval set and its two approximations  and lower approximation set Z)) are presented, respectively. The disadvantages of using upper-approximation set Z) or lower-approximation set Z) as approximation sets of the uncertain set (uncertain concept) Z are analyzed, and a new method for looking for a better approximation set of the interval set Z is proposed. The conclusion that the approximation set R0.5(Z) is an optimal approximation set of interval set Z is drawn and proved successfully. The change rules of R0.5(Z) with different binary relations are analyzed in detail. Finally, a kind of crisp approximation set of the interval set Z is constructed. We hope this research work will promote the development of both the interval set model and granular computing theory.The interval set is a special set, which describes uncertainty of an uncertain concept or set  Since the twenty-first century, researchers have done more and more research on uncertain problems . It is aU and is induced by an equivalence relation. In the knowledge space, two certain sets named upper approximation set and lower approximation set are used to describe the target concept X as its two boundaries. If knowledge granularity in knowledge space is coarser, then the border region of described target concept is wider and approximate accuracy is relatively lower. On the contrary, if knowledge granularity in knowledge space is finer, then the border region is narrower and approximate accuracy is relatively higher.Rough set theory is a mathematical tool to handle the uncertain information, which is imprecise, inconsistent, or incomplete. The basic thought of rough set is to obtain concepts and rules through classification of relational database and discover knowledge by the classification induced by equivalence relations; then approximation sets of the target concept are obtained with many equivalence classes. Rough set is a useful tool to handle uncertain problems, as well as fuzzy set theory, probability theory, and evidence theory. Because rough set theory has novel ideas and its calculation is easy and simple, it has been an important technology in intelligent information processing \u20139. The kThe interval set theory is an effective method for describing ambiguous information \u201312 and cZ is built in a certain knowledge space induced by many conditional attributes, and we find that this approximation set may have better similarity degree with the target concept Z than that of Z is obtained by cut-set with some threshold. And then, the decision-making rules can be obtained through the approximation set instead of Z in current knowledge granularity space. In addition, the change rules of similarity between a target concept Z and its approximation sets are analyzed in detail.In this paper, an approximation set of the target concept The method used is getting the approximation of interval sets with a special approximation degree. With this method, we can use certain sets to describe an interval set in Pawlak's space. Our motivation is to get a mathematical theory model, which can be helpful to promote interval sets development in knowledge acquisition.Z with the different knowledge granularity spaces are discussed in The rest of this paper is organized as follows. In In order to introduce the approximation set of interval set more easily, many basic concepts will be reviewed at first.U be a finite set which is called universal set, and then let 2U be the power set of \u2009U and let interval set Z be a subset of 2U. In mathematical form, interval set Z is defined as Z =  = {Z \u2208 2U\u2223Zl\u2286Z\u2286Zu}. If Zl = Zu, Z is a usual classical set.An interval set is a new collection, and it is described by two sets named upper bound and lower bound. The interval set can be defined as follows. Let U be all papers submitted to a conference. After being reviewed, there are 3 kinds of results. The first kind of results is the set of papers certainly accepted and represented by Zl. The second kind of results is the set of papers that need to be further reviewed and represented by Zu \u2212 Zl. The last kind of results is the set of papers rejected and represented by U \u2212 Zu. Although every paper just can be rejected or accepted, no one knows the final result before further evaluation. Through reviewing, the set of papers accepted by the conference is described as .In order to better explain the interval set, there is an example , 18 as fR\u2286A, let us define one unclear binary relationship IND(R) = {\u2223 \u2208 U2, \u2009\u2200b \u2208 R \u2192 b(x) = b(y)}.For any attribute set S = \u2329U, A, V, f\u232a. U is the domain, and A = C \u222a D is the set of all attributes. Subset C is a set of conditional attributes, and D is a set of decision-making attributes. V = \u222ar\u2208AVr is the set of attribute values. Vr describes the range of attribute values r where r \u2208 A. f : U \u00d7 A \u2192 V is an information function which describes attribute values of object x in U.A knowledge expression system can be described as S = \u2329U, A, V, f\u232a. For any X\u2286U and R\u2286A, upper approximation set X on R are defined as follows:U/IND(R) = {X\u2223 = b(y)))} is the classification of equivalence relation R on U. Upper approximation set and lower approximation set of rough set X on R can be defined in another form as follows:x]R \u2208 U/IND(R) and [x]R is an equivalence class of x on relation R. U according to knowledge R; U according to knowledge R. Let  boundary region of target concept X on relation R. Let  positive region of target concept X on relation R. Let  negative region of target concept X on relation R. BNR(X) is a set of objects which just possibly belong to target concept X.A knowledge-expression system is described as A and B be two subsets of domain U, which means A\u2286U, \u2009B\u2286U. Defining a mapping S : U \u00d7 U \u2192 , that is,  \u2192 S, S is the similarity degree between A and B, if S satisfies the following conditions.A, B\u2286U, \u20090 \u2a7d S \u2a7d 1 (boundedness).For any A, B\u2286U, \u2009S = S (symmetry).For any A, B\u2286U, \u2009S = 1; \u2009S = 0 if and only if A\u2229B = \u03d5.For any Let Any formula satisfying (1), (2), and (3) is a similarity degree formula between two sets. Zhang et al.  gave outZ =  = {Z \u2208 2U\u2223Zl\u2286Z\u2286Zu} be an interval set and let N =  = {N \u2208 2U\u2223Nl\u2286N\u2286Nu} be also an interval set. Similarity degree between two interval sets can be defined as follows:S accords with Let Z =  = {Z \u2208 2U\u2223Zl\u2286Z\u2286Zu} be an interval set. Let R be an equivalence relation on domain U. Upper approximation set of this interval set Z is defined as Z is defined as Let Zu and inner circle standing for a set Zl represent an interval set Z, and each block represents an equivalence class. The black region represents Zu and inner circle standing for a set Zl represent an interval set Z, and each block represents an equivalence class. The black region represents Figures Z, then the similarity degree between Z and Z, then the similarity degree between Z and If Z? In this paper, the better approximation sets of target concept will be proposed. Let U be a nonempty set of objects. Let Z\u2286U, x \u2208 Z, and the membership degree of x belonging to set Z is defined as\u03bcZR(x) \u2a7d 1.If the knowledge space keeps unchanged, is there a better approximation set of the target concept U be a nonempty set of objects, and let knowledge space be U/IND(R). Let Z\u2286U, \u2009x \u2208 Z, and the membership degree belonging to set Z isR\u03bb(Z) = {x \u2208 Z\u2223\u03bcZR(x)\u2a7e\u03bb, \u20091\u2a7e\u03bb > 0}, then R\u03bb(Z) is called \u03bb-approximation set of set Z.Let Z =  = {Z \u2208 2U\u2223Zl\u2286Z\u2286Zu} and R\u03bb(Z) = ; then R\u03bb(Z) is called \u03bb-approximation set of the interval set Z.Let Zu and inner circle standing for a set Zl represent an interval set Z, and each block represents an equivalence class. The black region represents R0.5(Zl), and the whole colored region (black and gray region) represents R0.5(Zu).a, b, c, and d be all real numbers. If 0 < a < b, \u20090 < c < d, then a/b < (a + d)/(b + c).Let a, b, c, and d be all real numbers. In the numbers, 0 < a < b, \u20090 < c < d. If a/b\u2a7ec/d, then a/b \u2a7d (a \u2212 c)/(b \u2212 d). If \u2009a/b \u2a7d c/d, then a/b\u2a7e(a \u2212 c)/(b \u2212 d).Let R0.5(Z) and Z, Theorems In order to better understand the similarity degree between U be a finite domain, let Z be an interval set on U, and let R be an equivalence relation on U. Then, Let U/R = {{x1, x2}, {x3, x4}, {x5, x6}}, Zu = {x1, x2, x3, x4, x5}, Zl = {x2, x3, x4}. Then, R0.5(Zu) = {x1, x2, x3, x4, x5, x6}, R0.5(Zl) = {x1, x2, x3, x4}.For example, let S) = (1/2) + (3/(2 \u00d7 4)) = 7/8, And then we can have According to x \u2208 R0.5(Zl), we have \u03bcZlR(x)\u2a7e0.5. That is,(1) There we first proveR is an equivalence relation on U, the classifications induced by R can be denoted as [x1]R, [x2]R,\u2026, [xn]R. Then, R0.5(Zl) = {x\u2223\u03bcZlR(x)\u2a7e0.5} = {x\u2223\u03bcZlR(x) = 1}\u222a{x\u22230.5 \u2a7d \u03bcXR(x) < 1}. Obviously, x\u22230.5 \u2a7d \u03bcZlR(x) < 1} = [xi1]R \u222a [xi2]R \u222a \u22ef\u222a[xik]R. So, Zl \u222a R0.5(Zl) = Zl \u222a ([xi1]R \u2212 Zl)\u222a([xi2]R \u2212 Zl)\u222a\u22ef\u222a([xik]R \u2212 Zl) and the intersection set between any two elements in\u2009\u2009Zl, ([xi1]R \u2212 Zl), ([xi2]R \u2212 Zl),\u2026, ([xik]R \u2212 Zl) is empty, we have that |Zl \u222a R0.5(Zl)| = |Zl | +|([xi1]R \u2212 Zl)|+|([xi2]R \u2212 Zl)|+\u22ef + |([xik]R \u2212 Zl)|. So,Because xi1]R\u2229Zl | \u2a7e|[xi1]R \u2212 Zl|. In the same way, according to |[xi2]R\u2229Zl | \u2a7e|[xi2]R \u2212 Zl | ,\u2026, |[xik]R\u2229Zl | \u2a7e|[xik]R \u2212 Zl| and Because(2) In a similar way with (1), we can have the inequalityFrom (1) and (2), we have Z and its approximation set R0.5(Z) is better than the similarity degree between Z and its lower approximation set U be a finite domain, let Z be an interval set on U, and let R be an equivalence relation on U. IfLet U/R = {{x1}, {x2, x3, x4, x5, x6}, {x7, x8, x9, x10}},\u2009\u2009Zu = {x1, x2, x7}, Zl = {x1, x2}. Then, R0.5(Zu) = {x1}, R0.5(Zl) = {x1},For example, let S) = (1/(2 \u00d7 2)) + (1/(2 \u00d7 3)) = 5/12,And then we can have According to xj1]R, [xj2]R,\u2026, [xjs]R are empty sets. Becausexj1]R\u2229Zl \u2260 \u03d5. In the same way, we have [xj2]R\u2229Zl \u2260 \u03d5,\u2026, [xjs]R\u2229Zl \u2260 \u03d5. Then we have Zl\u2229R0.5(Zl) = Zl \u2212 ([xj1]R\u2009\u2229\u2009Zl)\u2212([xj2]R\u2009\u2229\u2009Zl)\u2212\u22ef \u2212 ([xjs]R\u2009\u2229\u2009Zl) = Zl \u2212 (Zl \u2212 R0.5(Zl)). Because the intersection sets between any two elements in [xj1]R\u2229Zl, [xj2]R\u2229Zl,\u2026, [xjs]R\u2229Zl are empty sets, we have Zl\u2229R0.5(Zl) = Zl \u2212 (Zl \u2212 R0.5(Zl)), |Zl\u2229R0.5(Zl)| = |Zl | \u2212|Zl \u2212 R0.5(Zl)|, and \u222a\u22ef\u222a([xjs]R \u2212 Zl)). Because the intersection sets between any two elements in ([xj1]R \u2212 Zl), ([xj2]R \u2212 Zl),\u2026, ([xjs]R \u2212 Zl) are empty sets, (1) Let For(2) In a similar way with (1), we can easily obtain the conclusion thatAccording to (1) and (2), the inequality Z and its approximation set R0.5(Z) is better than the similarity degree between Z and its lower approximation set U be a finite domain, Z an interval set on U, and R an equivalence relation on U. If 1\u2a7e\u03bb > 0.5, then Let U/R = {{x1}, {x2, x3}, {x4, x5, x6}},\u2009\u2009Zu = {x1, x2, x3, x4, x5}, Zl = {x1, x2, x3}. Then, R0.5(Zu) = {x1, x2, x3, x4, x5, x6}, R0.75(Zu) = {x1, x2, x3},R0.5(Zl) = {x1, x2, x3}, R0.75(Zl) = {x1, x2, x3}.For example, let S) = (3/(2 \u00d7 3)) + (5/(2 \u00d7 6)) = 11/12, S) = (3/(2 \u00d7 3)) + (3/(2 \u00d7 5)) = 4/5, And then we can have x \u2208 R0.5(Zl), then \u03bcZlR(x)\u2a7e0.5, which meansR0.5(Zl) = {x\u2223\u03bcZlR(x)\u2a7e0.5} = {x\u2223\u03bcZlR(x) = 1}\u2009\u222a\u2009{x\u22230.5 \u2a7d \u03bcZlR(x) < 1}, we can easily get x\u22230.5 \u2a7d \u03bcZlR(x) < 1} = [xi1]R \u222a [xi2]R \u222a \u22ef\u222a[xik]R and R\u03bb(Zl) = {x\u2223\u03bcZlR(x)\u2a7e\u03bb > 0.5} = {x\u2223\u03bcZlR(x) = 1}\u222a{x\u22230.5 < \u03bb \u2a7d \u03bcZlR(x) < 1}, and then we can get R\u03bb(Zl)\u2286R0.5(Zl). To simplify the proof, let {x\u22230.5 < \u03bb \u2a7d \u03bcZlR(x) < 1} = [xi1]R \u222a [xi2]R \u222a \u22ef\u222a[xiq]R and q \u2a7d k in this paper. So, Zl\u2229[xi2]R | +\u22ef+|Zl\u2229[xik]R|. And(1) For all Zl\u2009\u2229\u2009[xiq+1]R | +\u22ef+|Zl\u2009\u2229\u2009[xik]R | \u2a7e|([xiq+1]R \u2212 Zl)|+\u22ef + |([xik]R \u2212 Zl)|, according to And because |(2) The inequalityAccording to (1) and (2), the inequality Based on Theorems U be a finite domain, Z an interval set on U, and R an equivalence relation on U. Z\u22862U. IfLet U be a finite domain, Z an interval set on U, and R an equivalence relation on U. if 0.5 \u2a7d \u03bb1 < \u03bb2 \u2a7d 1, then S)\u2a7eS).Let U/R = {{x1}, {x2, x3}, {x4, x5, x6}, {x7, x8, x9, x10}}, Zu = {x1, x2, x3, x5, x6, x7}, Zl = {x1, x2, x3, x5, x6}. Then, R0.6(Zu) = {x1, x2, x3, x4, x5, x6}, R0.8(Zu) = {x1, x2, x3}, R0.6(Zl) = {x1, x2, x3, x4, x5, x6}, R0.8(Zl) = {x1, x2, x3}.For example, S) = (5/(2 \u00d7 6)) + (5/(2 \u00d7 7)) = 65/84, S) = (3/(2 \u00d7 5)) + (3/(2 \u00d7 7)) = 18/35, S) > S). This example is in accordance with the theorem.And then we can have x \u2208 R\u03bb1(Zl), we haveR is an equivalence relation on U, all the classifications induced by R can be denoted by [x1]R, [x2]R,\u2026, [xn]R. We have R\u03bb1(Zl) = {x\u2223\u03bcZlR(x)\u2a7e\u03bb1} = {x\u2223\u03bcZlR(x) = 1}\u222a{x\u2223\u03bb1 \u2a7d \u03bcZlR(x) < 1} and x\u2223\u03bb1 \u2a7d \u03bcZlR(x) < 1} = [xi1]R \u222a [xi2]R \u222a \u22ef\u222a[xiu]R. We can also get R\u03bb2(Zl) = {x\u2223\u03bcZlR(x)\u2a7e\u03bb2} = {x\u2223\u03bcZlR(x) = 1}\u222a{x\u2223\u03bb2 \u2a7d \u03bcZlR(x) < 1}. Let {x\u2223\u03bb2 \u2a7d \u03bcZlR(x) < 1} = [xi1]R \u222a [xi2]R \u222a \u22ef\u222a[xiv]R, where 0.5 \u2a7d \u03bb1 < \u03bb2 \u2a7d 1 and v \u2a7d u. So, \u222a[xiu]R), xi1]R, [xi2]R,\u2026, [xiu]R are empty sets, we haveZl \u222a R\u03bb1(Zl) = Zl \u222a ([xi1]R \u2212 Zl)\u222a([xi2]R \u2212 Zl)\u222a\u22ef\u222a([xiv]R \u2212 Zl)\u222a\u22ef\u222a([xiu]R \u2212 Zl) and because the intersection sets between any two elements in Zl, ([xi1]R \u2212 Zl), ([xi2]R \u2212 Zl),\u2026, ([xiv]R \u2212 Zl),\u2026, ([xiu]R \u2212 Zl) are empty sets, we easily havexiv+1]R\u2229Zl | \u2a7e|[xiv+1]R \u2212 Zl|. According to |[xiv+2]R\u2229Zl | \u2a7e|[xiv+2]R \u2212 Zl | ,\u2026, |[xiu]R\u2229Zl | \u2a7e|[xiu]R \u2212 Zl|, we have |Zl\u2229[xiv+1]R| + \u22ef+|Zl\u2229[xiu]R | \u2a7e|([xiv+1]R \u2212 Zl)|+\u22ef+|([xiu]R \u2212 Zl)|. And based on (1) For any (2) In the same way as \u2009(1), the inequalityS)\u2a7eS) is held. So, the proof of According to (1) and (2), the inequality Z and its approximation set R\u03bb(Z) is a monotonically decreasing function with the parameter \u03bb, and the similarity degree reaches its maximum value when \u03bb = 0.5.Theorems S) in Pawlak's approximation spaces with different knowledge granularities. In this paper, we focus on discussing how the similarity degree between Z and R0.5(Z) changes when the granules are divided into more subgranules in Pawlak's approximation space. In other words, it is an important issue concerning how S) changes with different knowledge granularities in Pawlak's approximation space.In different Pawlak's approximation spaces with different knowledge granularities, the change rules of the uncertainty of rough set are a key issue , 22. Manx1]R, [x2]R,\u2026, [xn]R be classifications of U under equivalence relation R. Let [x1]R\u2032, [x2]R\u2032,\u2026, [xn]R\u2032 be classifications of U under equivalence relation R\u2032. If R\u2032\u2286R, then [xi]R\u2032\u2286[xi]R (1 \u2264 i \u2264 n). And then, U/R\u2032 is called a refinement of U/R, which is written as xj \u2208 U, then [xj]R\u2032 \u2282 [xj]R. And then, U/R\u2032 is called a strict refinement of U/R, which is written as U/R\u2032\u227aU/R.Let R\u2032\u2286[x]R is always satisfied, and \u2203y \u2208 U, \u2009[y]R\u2032 \u2282 [y]R. And then, there must be two or more granules in U/R\u2032 whose union is [y]R. To simplify the proof, we suppose that there is just only one granule which is divided into two subgranules, denoted by [xit1]R\u2032 and [xit2]R\u2032 in U/R\u2032, and other granules keep unchanged.Next, we will analyze the relationship between There are 9 cases, and only 6 cases are possible.U be a finite domain, Z an interval set on U, and R and R\u2032 two equivalence relations on U. Let [xit]R be one granule which is divided into two subgranules marked as [xit1]R\u2032 and [xit2]R\u2032. IfS) \u2a7d S).Let (1)xit]R is contained in both positive region of Zl and positive region of Zu. In this case, obviously S) = S) is held.[(2)xit]R is contained in both positive region of Zl and negative region of Zu. In this case, obviously, S) = S) is held.[(3)xit]R is contained in both negative region of Zl and negative region of Zu. In this case, obviously, S) = S) is held.[(4)xit]R is contained in both negative region of Zl and boundary region of Zu. In this case,R \u222a [xi2]R \u222a \u22ef\u222a[xim]R where m\u2a7ek. When [xit]R is in boundary region of Zu, we should further discuss this situation. To simplify the proof, we suppose that there is just only one granule marked as [xit]R in U/R which is divided into two subgranules marked as [xit1]R\u2032 and [xit2]R\u2032 in U/R\u2032. And the other granules keep unchanged.(a)k < t \u2a7d m, then [xit]R\u2284R0.5(Zu).If (1)xit1]R\u2032\u2286R0.5\u2032(Zu), [xit2]R\u2032\u2284R0.5\u2032(Zu). From the proof of If [Because [xit1]R\u2032 \u222a [xit2]R\u2032 = [xit]R, [xit1]R\u2032\u2286R0.5\u2032(Zu), and [xit2]R\u2032\u2284R0.5\u2032(Zu), we havexi11]R\u2032\u2286R0.5\u2032(Zu), |[xit1]R\u2032\u2229Zu | /(|[xit1]R\u2032\u2229Zu | +|[xit1]R\u2032 \u2212 Zu|)\u2a7e0.5, which means |[xit1]R\u2032\u2229Zu | \u2a7e|[xit1]R\u2032 \u2212 Zu|. According to (2)xit1]R\u2032\u2284R0.5\u2032(Zu), \u2009[xit2]R\u2032\u2284R0.5\u2032(Zu), thenIf [Because [xit]R\u2284R0.5(Zu), the case that [xi11]R\u2032\u2286R0.5\u2032(Zu) and [xi12]R\u2032\u2286R0.5\u2032(Zu) is impossible.(b)t \u2a7d k, then [xit]R\u2286R0.5(Zu).If 1 \u2a7d (1)xit1]R\u2032\u2286R0.5\u2032(Zu) and [xit2]R\u2032\u2286R0.5\u2032(Zu), then we can easily haveIf [(2)xit1]R\u2032\u2286R0.5\u2032(Zu) and [xit2]R\u2032\u2284R0.5\u2032(Zu), thenIf [Because [xit2]R\u2032\u2284R0.5\u2032(Zu), we have the following.(i)xit2]R\u2032\u2229Zu = \u03d5, then we have |Zu\u2229[xit1]R | = |Zu\u2229[xit]R| and |([xit1]R\u2032 \u2212 Zu)|<|([xit]R \u2212 Zu)|. Therefore,If [(ii)xit1]R\u2032\u2286Zu, then |[xit1]R\u2032\u2229Zu | = |[xit1]R\u2032|. Therefore,If [Because(iii)xit1]R\u2032\u2286BNR\u2032(Zu) and [xit2]R\u2032\u2286BNR\u2032(Zu), because [xit1]R\u2032\u2286R0.5\u2032(Zu) and [xit2]R\u2032\u2284R0.5\u2032(Zu), we haveIf [BecauseLet S) \u2a7d S) when(5)xit]R is contained in boundary region of Zl and positive region of Zu. In this case,R is contained in boundary region of Zl and boundary region of Zu.\u2229Zl|/|[xit2] \u2212 Zl| and S)\u2a7e|[xit2]\u2229Zu|/|[xit2] \u2212 Zu|, we easily have S) \u2a7d S).According to the proofs of (4) and (5), if \u2009From (1), (2), (3), (4), (5), and (6), Z and its approximation set R0.5(Z) is a monotonically increasing function when the knowledge granules in U/R are divided into many finer subgranules in U/R\u2032, where U/R\u2032 is a refinement of U/R.Z. In this paper, the approximation set R0.5(Z) of target concept Z in current knowledge space is proposed from a new viewpoint and related properties are analyzed in detail.With the development of uncertain artificial intelligence, the interval set theory attracts more and more researchers and gradually develops into a complete theory system. The interval set theory has been successfully applied to many fields, such as machine learning, knowledge acquisition, decision-making analysis, expert system, decision support system, inductive inference, conflict resolution, pattern recognition, fuzzy control, and medical diagnostics systems. It is an important tool of granular computing as well as the rough set which is one of the three main tools of granular computing , 26. In R0.5(Z) of the interval set Z is defined and the change rules of S) in different knowledge granularity spaces are analyzed. These researches show that R0.5(Z) is a better approximation set of Z than both In this paper, the interval set is transformed into a fuzzy set at first, and then the uncertain elements in boundary region are classified by cut-set with some threshold. Next, the approximation set ormation . The fuzormation \u201332. Receormation , the casormation , and incormation \u201338. In t"}
{"text": "There are errors in Q. variabilis: 2\u03c7 \u200a=\u200a 0.125, df \u200a=\u200a 1, P \u200a=\u200a 0.724; Q. aliena: 2\u03c7 \u200a=\u200a 1.865, df \u200a=\u200a 1, P \u200a=\u200a 0.172; Q. serrata var. brevipetiolata: 2\u03c7 \u200a=\u200a 0, df \u200a=\u200a 1, P \u200a=\u200a 1.The authors have provided the corrected versions of"}
{"text": "We consider the emerging problem of comparing the similarity between (unlabeled) pedigrees. More specifically, we focus on the simplest pedigrees, namely, the 2-generation pedigrees. We show that the isomorphism testing for two 2-generation pedigrees is GI-hard. If the 2-generation pedigrees are monogamous  then the isomorphism testing problem can be solved in polynomial time. We then consider the problem by relaxing it into an NP-complete decomposition problem which can be formulated as the Minimum Common Integer Pair Partition (MCIPP) problem, which we show to be FPT by exploiting a property of the optimal solution. While there is still some difficulty to overcome, this lays down a solid foundation for this research. Pedigrees, or commonly known as family trees, are important tools in evolutionary and computational biology. They are important for geneticists, as with a valid pedigree the recombination events can be deduced more accurately , or diseThere have been many practical methods for reconstructing pedigrees ,4,5,17. It is known that a lot of computations on pedigree graphs are NP-hard ,20,16, set al. [On the other hand, methods for comparing pedigrees are rare. The brute-force method will not work when the data set has size in the thousands ,14. Peopet al. , where tet al. [In this paper, we follow the work by Kirkpatrick et al.  to consiP = (I(P), E(P)) with vertices I(P) and edges E(P), together with a gender function s : I(P) \u2192 {male, female} such that:An (unlabeled) pedigree is a directed graph P is acyclic.1 v \u2208 I(P), the in-degree of v is either two or zero.2 For all nodes a, c),  \u2208 E(P), we have s(a) \u2260 s(b).3 For two edges  nodes to represent males . See Figure u \u2208 I(P) is monogamous if it mates with exactly one partner, i.e., the number of individuals u', u' \u2260 u, such that ,  \u2208 E(P) for some x \u2208 I(P) is exactly one. A pedigree is monogamous if all the individuals are monogamous. In Figure P = (I(P), E(P)) is generational if there is a function Let g(v) = 1 for all v \u2208 I(P) with in-degree zero.1 u, v) \u2208 E(P), we have g(v) = g(u) + 1.2 For all (g(v) is called the generation of v. For a generational pedigree P, we use g I(P) to represent the individuals of P whose generation is g. The pedigree on Figure The number P = (I(P), E(P)), P' = (I(P'),E(P')) with the associated gender functions s(\u2212), s'(\u2212) respectively, a bijection \u03d5: I(P) \u2192 I(P') is a pedigree isomorphism between P and P' if:Given two pedigrees u \u2208 I(P), s(u) = s'(\u03d5(u)), and1 For every u, v) \u2208 E(P) if and only if (\u03d5(u), \u03d5(v)) \u2208 E(P').2 (Graph Isomorphism (GI) is one of the most famous problems in computational complexity whose precise complexity has been open since 1972 ,12. It iIn , it was Theorem 1 Testing the isomorphism between two 2-generation pedigrees is GI-hard.Proof. We reduce bipartite graph isomorphism problem to our problem. Let B1 = , B2 =  be two bipartite graphs (with no isolated nodes). For our construction, we perform the following:U1,U2 are marked male.1 All nodes in V1,V2 are marked female.2 All nodes in B1 is converted into a pedigree P1 = (I(P1), E(P1)) as follows: (3.1) I(P1) = U1 \u222a V1 are the generation-1 nodes; (3.2) E(P1) is initially set as empty; (3.3) for  \u2208 E1 we create a new generation-2 node uv such that s(uv) = female, E(P1) \u2190 E(P1) \u222a {, }.3 B2 is converted into a pedigree P2 identically as in step (3).4 B1 and B2 are isomorphic iff P1 and P2 , both 2-generational, are isomorphic. We only show the necessary direction here as the other one is easy. If P1 and P2 are isomorphic, the first property we make use of is that all the generation-2 nodes are female. So, in the isomorphism between P1 and P2, if a generation-2 node uv \u2208 I(P1) is mapped to a generation-2 node xy \u2208 I(P2), we can simultaneously contract uv, xy to their corresponding male parents in P1 and P2. Consequently, we obtain the isomorphism between B1 and B2.\u00a0\u00a0\u00a0\u25a1We claim that Note that in our construction, all generation-2 individuals are female; moreover, a pair of generation-1 individuals mate with exactly one female child. A simple example on this reduction is shown on Figure Although the isomorphism testing problem is GI-hard even for 2-generation pedigrees, in some situations the problem is not hard to solve. In fact, when both of the 2-generation pedigrees are monogamous then the problem can be solved in linear time.i females and j males, we say that these two parents and the i + j children form an \u2329i, j\u232a-family. In Figure When a pair of generation-1 couple mate to have generation-2 children, for instance Theorem 2 Testing the isomorphism between two 2-generation monogamous pedigrees is polynomial time solvable.Proof. It is easily seen that when a 2-generation pedigree Q1 is monogamous then it is composed of a set of disjoint \u2329i, j\u232a-families. So to test the isomorphism between two monogamous 2-generation pedigrees Q1,Q2 it suffices to check whether two sets of integral pairs are identical, which can be done in O(n log n) time using the standard optimal sorting algorithms in two passes similar to the radix sort. In the first pass, we sort all the pairs according to their first components, and in the second, for each contiguous list of pairs with the same first component, we sort them according to the second components.\u00a0\u00a0\u00a0\u25a1The hardness result in the previous section implies that it might be too much if we use the standard isomorphism to measure the similarity of 2-generation pedigrees. In practice, ambiguities exist in pedigree-related datasets. In fact, it is estimated that 2-10% of people do not know their biological father . For 2-gP, it is not difficult to identify all (not necessarily disjoint) \u2329i, j\u232a-families .  = {n1, n2,..., tn} such that n = 9, {1, 2, 2, 4} is a partition of n. It should be noted that while it is simple to partition an integer, the number of such partitions is usually (counter-intuitively) huge. For instance, the integer 10 has 190569292 distinct partitions [Throughout this paper, for MCIP, we focus on integers in X = {x1, x2,...,px} is a multiset union of all the partitions \u03c4(ix), i.e., \u222ai\u2264p1\u2264\u03c4(ix). A multiset Z is a common partition of two multisets X = {x1, x2,..., px}, Y = {y1, y2,..., qy} if there are partitions \u03c41, \u03c42 with \u222ai\u2264p1\u2264\u03c41(ix) = \u222aj\u2264q1\u2264\u03c42(jy) = Z. The size of the partition Z is denoted as |Z|. For example, given X = {5, 8}, Y = {3,10}, a common partition of X, Y is Z = {1, 2, 2, 4, 4}, and the size of this partition is 5. It is easily seen that the necessary condition for X and Y to admit a common partition is that the sums of the integers in X and Y are equal. Throughout this paper, whenever we talk about a common partition for sets of integers X and Y, we always assume that this condition is met.A partition of a multiset A and B, and an integer k.Instance: Two multiple sets of integers A, B admit a common partition of size k?Question: Does For the ease of presentation, we use MCIP to represent this instance.a, b\u232a, the projection S be a set of 2-tuples of integers, Given a 2-tuple of integers, \u2329S, T, a common partition of S and T is a set of 2-tuples H = {\u2329g1, h1\u232a, \u2329g2, h2\u232a, \u22ef, \u2329kg, kh\u232a} such that k is the size of the partition H. Again, it is easily seen that the necessary condition for S and T to admit a common partition is that the sums of the integers in S, T, we always assume that this condition is met.Given two sets of 2-tuples S and T, and an integer k.Instance: Two multiple sets of 2-tuples of integers S, T admit a common partition of size k?Question: Does i, j\u232a represents the pedigree of a couple which has i female and j male chilren. Again, we use MCIPP to represent this instance. As MCIPP is a generalization for MCIP, all the known negative results regarding MCIP hold for MCIPP; i.e., MCIP and MCIPP are both NP-complete and APX- hard, following [d-MCIP has also been considered, where the input is d multisets with the same sum. Efficient asymptotic approximation algorithms have been obtained for large d [d + O(1) [d = 2 in this paper.) Also, note that the integer 0 in a solution for MCIP is meaningless while it is possible that 0 can appear either in the input or in the solution for MCIPP. So for MCIPP, we focus on integers in Recall that a 2-tuple \u2329ollowing . cn) = O*(f(k)), where f (\u2212) is any computable function on k and c is a constant. FPT algorithms are efficient tools for handling some NP-complete problems, especially when k is small in practical datasets [Finally, a Fixed-Parameter Tractable (FPT) algorithm is an algorithm for a decision problem with input size datasets ,11.a, c, we say a dominates c if a >c. Given a pair of 2-tuples of integers \u2329a, b\u232a and \u2329c, d\u232a, we say \u2329a, b\u232a dominates \u2329c, d\u232a if a \u2265 c and b \u2265 d. To simplify the writing, we say that \u2329a, b\u232a and \u2329c, d\u232a form a dominating pair if either \u2329a, b\u232a dominates \u2329c, d\u232a or vice versa. Likewise, \u2329a, b\u232a and \u2329c, d\u232a form a non-dominating pair if either a >c, b <d or a <c, b >d.Given a pair of integers We first describe some optimality properties for both the optimization versions of MCIP and MCIPP. When the context is clear, we still use MCIP and MCIPP to denote the corresponding optimization versions of the instances.Lemma 1 Let A, B be the input for MCIP. In any feasible solution, if a partition for some \u00d7 \u2208 A, \u03c4 (x) = {x1, x2,...,px}, and a partition for some y \u2208 B, tau(y) = {y1, y2,...,qy}, satisfies that |\u03c4(x) \u2229 \u03c4(y)| > 1 then this solution for MCIP is not optimal.Proof. Suppose to the contrary that |\u03c4(x) \u2229 \u03c4(y)| > 1, and the corresponding partition for A, B is optimal. WLOG, suppose \u03c4(x) = {x1, x2,..., px} and \u03c4(y) = {y1, y2,..., qy} contain r common elements {z1, z2,..., rz} then we can update \u03c4(x) \u2190 \u03c4(x) \u2212 {z1, z2,..., rz}\u222a{z1 + z2 + ... + rz} and \u03c4(y) \u2190 \u03c4(y) \u2212 {z1, z2,..., rz}\u222a{z1 + z2 + ... + rz}. Then the solution size for MCIP on A, B is reduced by r \u2212 1, contradicting the optimality of the assumption.\u00a0\u00a0\u00a0\u25a1x \u2208 A and some y \u2208 B, they share at most one common element. Notice that this lemma also holds for MCIPP, i.e., in an optimal partition of \u2329s1, s2\u232a \u2208 S and \u2329t1, t2\u232a \u2208 T, \u03c4 and \u03c4 share at most one common 2-tuple. Similarly, we can assume that in the input for MCIP  ) there is no common pair of integers in A and B (resp. no common pair of 2-tuples in S and T), as it must be put in the optimal solution.With the above lemma, we can now assume that for any optimal partition for some The following property is trivial and holds for both MCIP and MCIPP.Lemma 2 Let |MCIPP*| be the optimal solution size for MCIPP. Then |MCIPP*| >max{|S|, |T|}.a, b\u232a and \u2329c, d\u232a, we can use subtraction to partition them into two common pairs. For example, if a \u2264 c and b \u2264 d, then we can obtain the common partition {\u2329a, b\u232a, \u2329c \u2212 a, d \u2212 b\u232a. So, given \u23292, 4\u232a and \u23294, 5\u232a we can obtain a partition {\u23292,4\u232a, \u23292,1\u232a} for \u23294, 5\u232a. We also say that this is a dominating - partition operation. Apparently, for MCIP, this gives a way to partition a pair of integers as well. For instance, given 2 and 6, we can subtract 2 from 6 to obtain a partition {2, 4} for 6.For a pair of dominating 2-tuples \u2329a, b has the property that either a dominates b or vice versa. This is not the case for a non-dominating pairs of 2-tuples, e.g. \u23291, 4\u232a and \u23292, 3\u232a. We start with this fundamental lemma.We next describe some properties on non-dominating pairs of 2-tuples which are unique for MCIPP -- for MCIP, a pair of integers Lemma 3 Let A', B' be a set of positive integers with the same total sum, moreover, let us suppose |A'| = m + 1, |B'| = m. Then there must exist elements a \u2208 A', b \u2208 B' such that a <b.Proof. As |A'| > |B'|, we can arbitrarily select m elements from A' and match up them with those in B in an one-to-one fashion. As the sum of integers in A and B are the same, in this matching at least one of elements a \u2208 A must be smaller than its matched counterpart b \u2208 B -- otherwise, the sum of integers in A would be larger than that of B'.\u00a0\u00a0\u00a0\u25a1Corollary 1 Let A, B be two sets of n > 1 positive integers with the same total sum. WLOG, let A = {a1, a2, \u22ef, na}, B = {b1, b2, \u22ef, nb}. Then there must exist an element b \u2208 B' = B \u2212 {jb} which is greater than some element a \u2208 A' = {a1, a2, \u22ef, ai\u22121, i a\u2212 jb, ai+1, \u22ef, na}, i where a>jb.Proof. Obviously we have |A'| = n and |B'| = n \u2212 1. Then this corollary follows directly from Lemma 3.\u00a0\u00a0\u00a0\u25a1A, B is obvious -- we can successively find pairs of dominating integers. In fact, in the proof of Corollary 1, once we obtain a' = k a\u2208 A' and b' = \u2113 b\u2208 B' such that a' <b', we can repeatedly use the above argument to A'' = A' \u2212 {a'} = {a1, a2, \u22ef, ak\u22121, ak+1, \u22ef, na} and B'' = {b1, b2, \u22ef, b\u2113\u22121, b\u2113 \u2212 a', b\u2113+1, \u22ef, bn}, where |A''| = |B''| = n \u2212 1 and the two sets A, B have the same sum.The implication of Corollary 1 for MCIP with input S, T} is each composed of m non-dominating 2-tuples, then we can find a pair s = \u2329s1, s2\u232a \u2208 S and t = \u2329t1, t2\u232a \u2208 T (assuming s1 >t1 and s2 <t2) such that we can put \u2329t1, s2\u232a in some solution set while the resulting instance S' =  \u222a {\u2329s1 \u2212 t1, 0\u232a},T' =  \u222a {\u23290, t2 \u2212 s2\u232a} is still a valid instance for MCIPP.  Then, following Corollary 1, there exists a pair of dominating 2-tuples s' \u2208 S', t' \u2208 T'. Moreover, if we apply the dominating-partition process on these two tuples s', t', following Corollary 1, we can repeatedly find dominating tuples until all tuples in S, T are all commonly partitioned. This is because after we apply the dominating-partition on s', t' (say s' <t') to obtain an MCIPP instance S'',T'', we have Now let us see how this can be applied to MCIPP. When we have an instance of MCIPP whose input {Algorithm Heuristic-MCIPPS, TInput: \u03c4 for S, T, initially empty.Output: A common partition S| \u2265 2 and |T| \u2265 21\u00a0\u00a0\u00a0While |2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Repeats \u2208 S and t \u2208 T,2.1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0select a pair of dominating 2-tuples, 2.2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0compute two decomposing 2-tuples by subtraction,S \u2190 S \u2212 {s}, T \u2190 (T \u2212 {t}) \u222a{t \u2212 s} if s <t,2.3\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0update S \u2190 (S \u2212 {s}) \u222a{s \u2212 t}, T \u2190 T \u2212 {t} if s >t,2.4\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0update \u03c4 \u2190\u03c4 \u222a {min},2.5\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0update 3\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Until no dominating 2-tuples can be found.S and T4\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0If there are at least two pairs of non-dominating 2-tuples in 5\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Thens' \u2208 S, t' \u2208 T which leads to successive dominating pairs.5.1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0use a brute-force method to select two non-dominating 2-tuples S| = 1, |T| = 1, then find the smaller tuple in S and T, x.6\u00a0\u00a0\u00a0If |\u03c4 \u2190 \u03c4 \u222a {x}.7\u00a0\u00a0\u00a0Return s \u2208 S, t \u2208 T which can make the process of repeatedly processing dominating pairs possible. Let us show an example, S = {\u23299, 4\u232a, \u23291, 11\u232a, \u23296, 3\u232a} and T = {\u23292, 8\u232a, \u232912, 1\u232a, \u23292, 9\u232a}. In this example, among the 9 non-dominating pairs between S and T, there are 4 solutions enabling us to successively find dominating pairs. One of them is s = \u23296, 3\u232a and t = \u23292, 9\u232a, which gives us a common partition of size 6. The other 5 solutions all lead to a common partition of size 7.Of course, due to the 'existence' constraint in Lemma 3 and Corollary 1, we would have to use a brute-force method to find a pair The above discussion enables us to design an algorithm Heuristic-MCIPP to prove the next lemma.Lemma 4 Let |MCIPP| be the size of the solution returned by Heuristic-MCIPP. Then |MCIPP| \u2264 |S| + |T|.Proof. When there is no non-dominating pairs in the input, with the running of the algorithm Heuristic-MCIPP, we have |MCIPP| \u2264 |S| + |T| \u2212 1. The reason is that when each of S and T has at least two 2-tuples, we can use the dominating-partition procedure to obtain two 2-tuples in the solution set for each pair of dominating 2-tuples from S, T. When there are a total of three elements in S, T, say, one in S and two in T, we just need to return the two elements in T as their sum matches the one in S already. (This is certainly true for MCIP as pointed out in [d out in .)p \u2265 2 pairs of non-dominating pairs at Step 4-5 of the Heuristic-MCIPP algorithm, following Corollary 1 and the subsequent arguments, there exists a non-dominating pair s = \u2329s1, s2\u232a \u2208 S, t = \u2329t1, t2\u232a \u2208 T which leads to successive dominating pairs. (We can use the brute-force method to find this in O(p2(|S| + |T|)) time.) In this case, the solution obtained by Heuristic-MCIPP has size at most 1 + (|S| + |T| \u2212 1) = |S| + |T|, where the first one corresponds to  (if s1 <t1) or  (if s1 >t1).\u00a0\u00a0\u00a0\u25a1When there are S| + |T| \u2264 2max{|S|, |T|} \u2264 2|MCIPP*|. On the other hand, designing approximation algorithms is not our focus for this paper; in fact, by a simple modification for the Maximum Packing method in [In fact, the above three lemmas imply that Heuristic-MCIPP provides a factor-2 approximation for MCIPP, as we have |ethod in  we can oMCIP| < |A| + |B| \u2212 1. The latter in fact immediately implies that for MCIP there is always an optimal solution which does not partition at least an integer from either A or B. That further implies that there is a simple FPT algorithm for MCIP based on the bounded-degree search. We will show a stronger property in the next section to improve the FPT algorithm for MCIP, and subsequently, an FPT algorithm for MCIPP can be obtained.Note that Lemma 4 is different from its counterpart for MCIP, which, according to Lemma 2.2 in , states We first give the following lemma for MCIP.Lemma 5 Let A, B be the input for MCIP and let a be the smallest element in A or B. Then there is an optimal solution for MCIP which contains a, i.e., there is an optimal solution which does not partition the smallest element in A and B.Proof. We first show the following claim: in an optimal solution \u03c4 for MCIP with input A, B, let a1, a2 be a pair of elements in \u03c4(z), z \u2208 A \u222a B, with the condition that (1) a1 + a2 <a, and (2) a2 is the minimum among all pairs of elements in \u03c4 satisfying the condition (1), then there is an optimal solution \u03c4' which partitions some element z \u2208 A \u222a B with \u03c4'(z) = \u03c4(z) \u2212 {a1, a2} \u222a {a1 + a2}.z \u2208 A and let a1 \u2208 \u03c4(y1) and a2 \u2208 \u03c4(y2) for two distinct integers y1, y2 \u2208 B. Following the definition of \u2329a1, a2\u232a, \u03c4(y1) contains at least one more element other than a1, say a3; and following (2) we have a3 >a2. Suppose that a3 \u2208 \u03c4(x) for some x \u2208 A. By Lemma 1, x \u2260 z. We replace a3 by a1 by \u03c4' with \u03c4'(x) = \u03c4(x) \u2212 {a3} \u222a {a2}, \u03c4'(y1) = \u03c4(y1) \u2212 {a1, a3} \u222a {\u03c4'(z) = \u03c4(z) \u2212 {a1, a2} \u222a {\u03c4' has the same size as \u03c4, so it is also a minimum size common partition for A, B.The proof for the above claim is as follows. WLOG, let a is partitioned in some optimal partition \u03c4 with \u03c4' with t \u2212 1 such steps, we obtain an optimal partition which contains the minimum element a in A \u222a B.\u00a0\u00a0\u00a0\u25a1It is obvious that, as long as the smallest element A = {2, 5, 5}, B = {6, 6}, and an optimal partition \u03c4 = {1,1, 5, 5} where \u03c4(2) = {a1 = 1, a2 = 1}. By the construction in the proof of Lemma 5, y1 = 6, y2 = 6, a3 = 5, \u03c4' = {1, 2, 4, 5}, where a = 2 is kept.An example of the above proof is given as follows. We have et al. [MCIP| < |A| + |B| \u2212 1, there must be an optimal solution whose corresponding matching graph between the partitioned elements in A, B contains no cycle, which means there is at least one leaf node. Then this leaf node corresponds to an unpartitioned integer in A or B. The above lemma in fact implies a faster FPT algorithm for MCIP. Pick the smallest element a \u2208 A \u222a B (say a \u2208 A), we try to partition some other integer z \u2208 B by subtracting a from it. Then we repeat over the new problem instance involving z \u2212 a. This process is repeated k times when either a solution is founded or we have to report that there is no solution of size k. The running time is O*k) = O*(kk).We comment that we can use Lemma 2.2 by Chen et al.  directlyTo obtain an FPT algorithm for MCIPP, we also need a similar lemma.Lemma 6 Let S, T be the input for MCIPP. Then there is an optimal solution for MCIPP which either contains \u2329a, b\u232a \u2208 S \u222a T or \u2329c, d\u232a \u2208 S \u222a T, or contains \u2329a, d\u232a, where a is the minimum element in and d is the minimum element in Proof. Again, we first show the following claim: in an optimal solution \u03c4 for MCIPP with input S, T, let \u2329a1, a2\u232a, \u2329b1, b2\u232a be two 2-tuples in \u03c4(z), z \u2208 S \u222a T, such that (1) a1 + b1 \u2264 a, and (2) b1 is the minimum among all pairs of 2-tuples in \u03c4 satisfying (1), then there is an optimal solution \u03c4' which partitions some 2-tuple z \u2208 S \u222a T with \u03c4'(z) = \u03c4(z) \u2212 {\u2329a1, a2\u232a, \u2329b1, b2\u232a} \u222a { a1 + b1, a2}. z = \u2329z1, z2\u232a \u2208 S and let \u2329a1, a2\u232a \u2208 \u03c4(y1) and \u2329b1, b2\u232a \u2208 \u03c4(y2) for two distinct 2-tuples y1, y2 \u2208 T. Following the definition of , tau(y1) contains at least one more pair \u2329c1, c2\u232a, with c1 \u2265 b1. Suppose that \u2329c1, c2\u232a \u2208 \u03c4(x) for some x \u2208 S. Again, by Lemma 1, x \u2260 z. We replace \u2329c1, c2\u232a by \u2329c1 \u2212 b1, c2\u232a, and \u2329a1, a2\u232a by \u2329a1 + b1, a2\u232a. Subsequently, we obtain another optimal partition \u03c4' with \u03c4'(x) = \u03c4(x) \u2212 {\u2329c1, c2\u232a}\u222a{\u2329c1 \u2212 b1, c2\u232a, \u2329a2, b2\u232a}, \u03c4'(y1) = \u03c4(y1) \u2212 {\u2329a1, a2\u232a, \u2329c1, c2\u232a \u222a {\u2329a1 + b1, a2\u232a}, \u2329c1 \u2212 b1, c2\u232a, and \u03c4'(z) = \u03c4(z) \u2212 {\u2329a1, a2\u232a, \u2329b1, b2\u232a \u222a {\u2329a1 + b1, a2\u232a}. Again, \u03c4' is also a minimum size common partition for S, T.WLOG, let Similar to Lemma 5, it is obvious that we can repeatedly apply the above steps to obtain an optimal solution with does not partition the smallest element in a, b\u232a, \u2329c, d\u232a \u2208 S \u222a T or \u2329a, d\u232a \u2208 S \u222a T, where a is the minimum element in d is the minimum element in a, b\u232a, \u2329c, d\u232a or \u2329a, d\u232a. For one step, the running time for the former would be O(k1 + k2) for the first two cases and for the latter would also be O(k1 + k2) -- as \u2329a, d\u232a could be subtracted from O(k1 + k2) pairs, where k1 = |S|, k2 = |T|. As k1, k2 \u2264 k, the running time of this step is bounded by O(2k). Running this for k steps, the running time of the whole algorithm is O*(2kkk). Hence, we have the following theorem.With the above lemma, it is again possible to have an FPT algorithm, Exact-MCIPP, for MCIPP using bounded degree search. At each step, we search for \u2329Exact-MCIPPAlgorithm S, T, kInput: \u03c4 for S, T, initially empty.Output: A common partition k \u2265 11\u00a0\u00a0\u00a0While 2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Repeata be the minimum element in 2.1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0let d be the minimum element in 2.2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0let a, d\u232a \u2208 S \u222a T then \u03c4 \u2190 \u03c4 \u222a {\u2329a, d\u232a}, delete \u2329a, d\u232a from S \u222a T, and update S, T and k \u2190 k \u2212 1,2.3\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if \u2329a, b\u232a \u2208 S \u222a T then \u03c4 \u2190 \u03c4 \u222a {\u2329a, b\u232a}, delete \u2329a, b\u232a from S \u222a T, and update S, T and k \u2190 k \u2212 1,2.4\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if \u2329c, d\u232a \u2208 S \u222a T then \u03c4 \u2190 \u03c4 \u222a {\u2329c, d\u232a}, delete \u2329c, d\u232a from S \u222a T, and update S, T and k \u2190 k \u2212 1,2.5\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if \u2329S = \u2205 or T = \u2205 or k = 0.3\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Until S = \u2205 and T = \u22054\u00a0\u00a0\u00a0If both \u03c4,4.1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Then return 4.2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Else return 'no solution'.Theorem 3 Minimum Common Integer Pair Partition is FPT.k is large. In [k is relatively small.The running time of the above FPT algorithm is still too high to be applied alone to the similarity comparison for arbitrary 2-generation pedigrees, i.e., when arge. In , the salk values, we suggest a combination of the FPT algorithm and approximation algorithms [k is not too large, we can run this FPT algorithm; when k is too large for the FPT algorithm to handle, we can then use the approximation algorithms.  problem, which generalizes the NP-complete problem of Minimum Common Integer Partition (MCIP) problem. We show that MCIPP (hence MCIP) is FPT (Fixed-Parameter Tractable). It would be interesting to significantly improve the running time of the FPT algorithms presented in this paper.We consider the problem of testing the isomorphism and similarity of the simplest possible unlabeled pedigrees. We show that the isomorphism testing is GI-hard, excluding any chance for a polynomial time algorithm . We define a new similarity measure based on \u2329The authors declare that they have no competing interests.BZ conceived the study. All authors contributed to the algorithm design and analysis, read and approved the final manuscript."}
{"text": "Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of \u03a9-algebras. A complduced in . In 5]   discrim\u03a9-algebras .\u03a9 is an ordered triple  where F\u03a9 and C\u03a9 are disjoint sets (possibly empty) and d\u03a9 is a function from F\u03a9 into the set of positive integers. F\u03a9 is the set of function symbols of \u03a9, C\u03a9 is the set of constant symbols of \u03a9, and d\u03a9 is the degree function or arity function of \u03a9. Given an operator domain \u03a9, an \u03a9-algebra \ud835\udc9c is an ordered triple A is a nonempty set (the domain or carrier or universe of \ud835\udc9c);c \u2208 C\u03a9, cA \u2208 A;for each f \u2208 F\u03a9 with d\u03a9(f) = n, fA : An \u2192 A is an n-ary operation defined on A.for each An operator domain \u03a9-algebra whose carrier is a singleton is called trivial. All other \u03a9-algebras are nontrivial. If \ud835\udc9c and \u212c are \u03a9-algebras, then \ud835\udc9c is a subalgebra of \u212c provided thatA\u2286B;c \u2208 C\u03a9, cA = cB;for each f \u2208 F\u03a9 with d\u03a9(f) = n, fA = fBAn|.for each An \ud835\udc9c and \u212c be \u03a9-algebras. A function \u03d5 : A \u2192 B is a homomorphism provided thatc \u2208 C\u03a9, \u03d5(cA) = cB;for each f \u2208 F\u03a9 with d\u03a9(f) = n and all  \u2208 An, \u03d5) = fB(\u03d5(a1),\u2026, \u03d5(an)).for each Let \ud835\udc9c be an \u03a9-algebra. An equivalence relation R on A is a congruence on \ud835\udc9c provided for all f \u2208 F\u03a9 with d\u03a9(f) = n and all xiRyi for i = 1,\u2026, n, one has Epimorphism, monomorphism, isomorphism, endomorphism, and automorphism are defined in the obvious way. Let R is a congruence on \ud835\udc9c, then the quotient set A/R = {[a]R; a \u2208 A} can be made into a \u03a9-algebra \u212c = \ud835\udc9c/R by setting cB = [cA]R for all c \u2208 C\u03a9 and defining f \u2208 F\u03a9 with d\u03a9(f) = n and all  \u2208 An. The function A \u2192 A/R given by a \u21a6 [a]R is an epimorphism \ud835\udc9c \u2192 \ud835\udc9c/R.If \u03d5 : \ud835\udc9c \u2192 \u212c is a homomorphism, then the image of \u03d5 is a subalgebra of \u212c; moreover, the relation Ker(\u03d5) is defined by xKer(\u03d5)y if and only if \u03d5(x) = \u03d5(y) is a congruence on \ud835\udc9c and A/Ker(\u03d5) is isomorphic to the image of \u03d5. Let (Ai)i \u2208 I be an indexed family of \u03a9-algebras. Let i \u2208 I. Let P = \u220fi\u2208IAi. We make P into an \u03a9-algebra cP(i) = ci for all i \u2208 I and definingf \u2208 F\u03a9 with d\u03a9(f) = n, all  \u2208 Pn, and all i \u2208 I. \ud835\udcab is the direct product of the family (\ud835\udc9ci)i\u2208I. For each fixed i0 \u2208 I the projection \u03c0io : \ud835\udcab \u2192 \ud835\udc9ci0 given by \u03b1 \u21a6 \u03b1(i0) is an epimorphism. If all the \ud835\udc9ci are the same algebra \ud835\udc9c, then \ud835\udcab = \ud835\udc9cI is a direct power of \ud835\udc9c. In that event the diagonal map \u03b4 : \ud835\udc9c \u2192 \ud835\udc9cI given by a \u21a6 \u03b4(a) defined by \u03b4(a)(i) = a for all a \u2208 A and all i \u2208 I is a monomorphism.If \u03a9-algebras closed under taking subalgebras, homomorphic images, and direct products is a variety of \u03a9-algebras. Note that since the trivial \u03a9-algebra is a homomorphic image of any \u03a9-algebra whatsoever, every variety contains the trivial algebra.A nonempty class of F\u03a9 = {\u00b7,\u22121} with d\u03a9(\u00b7) = 2 and d\u03a9(\u22121) = 1 and C\u03a9 = {1}, then the class of all groups is a variety of \u03a9-algebras.(1) If R be an integral domain. If F\u03a9 = R \u222a {+, \u2212, [xx]} with d\u03a9(\u2212) = d\u03a9(\u03b1) = 1 for all \u03b1 \u2208 R and d\u03a9(+) = d\u03a9([x]) = 2 and C\u03a9 = {0}, then the class of all Lie algebras over R is a variety of \u03a9-algebras.(2) Let P(I) be the power set of a set I. A subset M0\u2286P(I) will be called an ideal in the ring P(I) if \u2205 \u2208 M0 and M0 is closed under finite unions and also closed under the formation of subsets. M0 will be a proper ideal in P(I) if is an ideal in P(I) and I \u2209 M0. The dual of a proper ideal in a Boolean algebra is a filter. Specifically, if I is a nonempty set and \ud835\udc9f is a family of subsets of I, then \ud835\udc9f is a filter on I provided that\u2205 \u2209 \ud835\udc9f;I \u2208 \ud835\udc9f;A, B) \u2208 \ud835\udc9f2 implies A\u2229B \u2208 \ud835\udc9f;(A\u2286B\u2286I and A \u2208 \ud835\udc9f implies B \u2208 \ud835\udc9f.Let \ud835\udcab = \u220f\ud835\udc9ci is the direct product of the indexed family (\ud835\udc9ci)i\u2208I of \u03a9-algebras and \ud835\udc9f is a filter on I, then we may define a congruence R(\ud835\udc9f) on \ud835\udcab by \u03b1\u2009\u2245R\ud835\udc9f\u2009\u03b2 provided {i \u2208 I : \u03b1(i) = \u03b2(i)} \u2208 \ud835\udc9f. The quotient algebra \ud835\udcab/R\ud835\udc9f is the reduced product of the family (\ud835\udc9ci)i\u2208I modulo the filter \ud835\udc9f on I. If \ud835\udcab = \ud835\udc9cI is a direct power, then \ud835\udcab/R\ud835\udc9f, written \ud835\udc9cI/\ud835\udc9f, may be called the reduced power of \ud835\udc9c modulo the filter \ud835\udc9f on I. In that event one can prove that the mapping d : \ud835\udc9c \u2192 \ud835\udc9cI/\ud835\udc9f defined by a \u21a6 [\u03b4(a)]R(\ud835\udc9f) where \u03b4 is the diagonal embedding \u03b4 : \ud835\udc9c \u21a6 \ud835\udc9cI, \u03b4(a)(i) = a for all a \u2208 A, i \u2208 I, is an algebra monomorphism.If d : \ud835\udc9c \u2192 \ud835\udc9cI/D is called the canonical embedding of \ud835\udc9c into the reduced power \ud835\udc9cI/\ud835\udc9f. We give as examples two extreme cases.The map \ud835\udc9f = {I} be the trivial filter on I. Then \ud835\udcab/R(\ud835\udc9f) is isomorphic to \ud835\udcab so that direct products (powers) may be viewed as special cases of reduced products (powers).(1) Let \ud835\udc9f on I is called an ultrafilter on I. If D is an ultrafilter on I, then \ud835\udcab/R(\ud835\udc9f)(\ud835\udc9cI/\ud835\udc9f) is called the ultraproduct of the family (\ud835\udc9ci)i\u2208I  modulo the ultrafilter \ud835\udc9f on I. If \ud835\udcab = \ud835\udc9cI is a direct power then the ultraproduct is called an ultrapower.(2) A maximal filter \u03c9 of nonnegative integers with its natural order as the first limit ordinal. Suppose \ud835\udc9c0 is an \u03a9-algebra and, for each n < \u03c9, \ud835\udc9fn is an ultrafilter on an index set In. Let \ud835\udc9cn+1 = \ud835\udc9cnIn/\ud835\udc9fn if \ud835\udc9cn has already been defined and let dn : \ud835\udc9cn \u2192 \ud835\udc9cn+1 be the canonical embedding. Then the direct limit \ud835\udc9c\u03c9 of the system \ud835\udc9co with respect to the sequence (\ud835\udc9fn)n<\u03c9 of ultrafilters. Note that in this event the limit map d\u03c9 : \ud835\udc9c0 \u2192 A\u03c9 is a monomorphism.Now view the set \u03a9 there corresponds the first order language with equality L\u03a9. Besides \u03a9 we need a countably infinite set {xn : n < \u03c9} of distinct variables. The terms or polynomials or words of L\u03a9 are defined recursively as follows.Constant symbols and variables are terms.f \u2208 F\u03a9 and d\u03a9(f) = n and  is a tuple of terms already defined, then f is a term.If To each operator domain t1 = t2 where  is an ordered pair of terms of L\u03a9 is an atomic formula of L\u03a9. The negation ~(t1 = t2) of an atomic formula of L\u03a9 is a negated atomic formula of L\u03a9. If \u03b1 is either an atomic formula or a negated atomic formula of L\u03a9, then \u03b1 is a literal of L\u03a9. We omit the recursive definition of  formula of L\u03a9 but appeal to the classical result that every formula of L\u03a9 is logically equivalent to one in prenex normal form. Such a formula is of the form Q1y1 \u2026 Qnyn\u03d5 where each yi = xni is a variable, each Qi is a quantifier, \u2200, \u2203, and \u03d5, the matrix of the formula, is a Boolean combination of atomic formulas. We do not exclude the possibility n = 0. Such formulas are quantifier free. Each quantifier free formula of L\u03a9 is equivalent to a quantifier free formula of L\u03a9 in disjunctive normal form  as well as to a quantifier free formula of L\u03a9 in conjunctive normal form . A formula of L\u03a9 containing no unquantified variables is a sentence of L\u03a9. We omit the recursive definition of a formula \u03a8 of L\u03a9 holding in an \u03a9-algebra \ud835\udc9c under an interpretation of the variables and trust the reader to understand intuitively what it means for a sentence a of L\u03a9 to hold in an \u03a9-algebra \ud835\udc9c. If in Q1y1 \u2026 Qnyn all the Qi are \u2200, then the above formula is a universal formula of L\u03a9. Similarly, if all the Qi are \u2203, then the above formula is an existential formula of L\u03a9. A universal  formula of L\u03a9 containing no unquantified variables is a universal  sentence of L\u03a9. Clearly, the negation of a universal sentence is logically equivalent to an existential sentence and vice versa. Vacuous quantifications are permitted. Thus, a quantifier free formula of L\u03a9 is considered a universal formula of L\u03a9 as well as an existential formula of L\u03a9. For example, the formula 1 \u00b7 1\u22121 = 1 in the language of group theory is considered both a universal sentence and an existential sentence.An expression of the form L\u03a9 containing at most the variables in L\u03a9. Clearly a universal sentence of L\u03a9 of the form L\u03a9 containing at most the variables in y, is equivalent to the negation of a primitive sentence of L\u03a9. We will find it convenient to call such sentences negated primitive. Given an \u03a9-algebra \ud835\udc9c, we let Th\u2200(\ud835\udc9c) be the set of all universal sentences of L\u03a9 true in \ud835\udc9c, Th\u2203(\ud835\udc9c) the set of all existential sentences of L\u03a9 true in \ud835\udc9c, and Th(\ud835\udc9c) the set of all sentences of L\u03a9 true in \ud835\udc9c. We call Th\u2200(\ud835\udc9c) the universal theory of \ud835\udc9c, Th\u2203(\ud835\udc9c) is the existential theory of \ud835\udc9c and Th(\ud835\udc9c) the theory of A. Clearly Th\u2200(\ud835\udc9c) = Th\u2200(\u212c) if and only if Th\u2203(\ud835\udc9c) = Th\u2203(\u212c). In that event we say that \ud835\udc9c and \u212c are universally equivalent and write \ud835\udc9c\u2009\u2261\u2200\u2009\u212c. If Th(\ud835\udc9c) = Th(\u212c) we say that \ud835\udc9c and \u212c are elementarily equivalent and write \ud835\udc9c \u2261 \u212c. Being elementarily equivalent is a sufficient but not necessarily necessary condition for being universally equivalent.An existential sentence of the form L\u03a9 whose matrix L\u03a9. Similarly, if L\u03a9 whose matrix L\u03a9.Suppose \u03a9-algebra \ud835\udc9c is a subalgebra of the \u03a9-algebra \u212c. Then every existential sentence of L\u03a9 holding in \ud835\udc9c holds also in \u212c and every universal sentence of L\u03a9 holding in \u212c holds also in \ud835\udc9c. The next result is rather obvious. We omit a proof.Suppose that the \u03a9-algebra \ud835\udc9c be a subalgebra of the \u03a9-algebra \u212c. Then the following three statements are equivalent in pairs:\ud835\udc9c\u2009\u2261\u2200\u2009\u212c;L\u03a9 true in \u212c is also true in \ud835\udc9c;every primitive sentence of L\u03a9 true in \ud835\udc9c is also true in \u212c.every negated primitive sentence of Let the L\u03a9 whose matrix L\u03a9.Now let L\u03a9 has the form si = ti) as (si \u2260 ti)) i(si \u2260 ti)\u2227(s = t) so that A universal Horn sentence of \ud835\udc9c is an \u03a9-algebra, then we have the inclusions I(\ud835\udc9c) is the set of identities of L\u03a9 true in \ud835\udc9c, Q(\ud835\udc9c) is the set of quasi-identities of L\u03a9 true in \ud835\udc9c, and H(\ud835\udc9c) is the set of universal Horn sentences of L\u03a9 true in \ud835\udc9c.  An \u03a9-algebra \ud835\udc9c is a model of a set S of sentences of L\u03a9 provided every sentence s \u2208 S holds in \ud835\udc9c. Appealing to the classical Godel-Henkin Completeness Theorem, we see that a set S of sentences of L\u03a9 is consistent if and only if it has a model. Let S be a consistent set of sentences of L\u03a9. Then M(S) will be the class of all models or the model class of S.It follows from this discussion that if \u03a9-algebras is axiomatic provided that it is the model class M(S) for at least one consistent set S of sentences of L\u03a9. Every axiomatic class of \u03a9-algebras is nonempty and closed under isomorphism. Note that every set of quasi-identities  of L\u03a9 holds in the trivial \u03a9-algebra, hence it is consistent. A celebrated theorem of Garrett Birkhoff asserts that a class of \u03a9-algebras is a variety if and only if it is the model class of a set of identities of L\u03a9. If we define a quasivariety of \u03a9-algebras to be an axiomatic class of \u03a9-algebras containing the trivial \u03a9-algebra and closed under taking subalgebras and direct products, then a well-known characterization, due to Mal'cev, along the lines of Birkhoff's Theorem asserts that a class of \u03a9-algebras is a quasivariety if and only if it is the model class of a set of quasi-identities of L\u03a9. Note that the model class operator M applied to sets of sentences reverses inclusions.A class of \ud835\udc9c be an \u03a9-algebra. Recall that I(\ud835\udc9c) is the set of identities of L\u03a9 true in \ud835\udc9c, Q(\ud835\udc9c) is the set of quasi-identities of L\u03a9 true in \ud835\udc9c and H(\ud835\udc9c) is the set of universal Horn sentences of L\u03a9 true in \ud835\udc9c, and Th\u2200(\ud835\udc9c) is the set of universal sentences of L\u03a9 true in \ud835\udc9c. (All of these sets are consistent since they have A as a model.) Applying the model class operator, we get M(I(\ud835\udc9c)) = var(\ud835\udc9c) is the least variety of \u03a9-algebras containing \ud835\udc9c. The set M(Q(\ud835\udc9c)) = qvar(\ud835\udc9c) is the least quasivariety of \u03a9-algebras containing \ud835\udc9c. The set M(H(\ud835\udc9c)) = uhc(\ud835\udc9c) is the least universally axiomatizable Horn class containing \ud835\udc9c. Finally, M(Th\u2200(\ud835\udc9c)) ) is the least universally axiomatizable class containing \ud835\udc9c. With the above notation, we have Now let \u03f5 : \ud835\udc9c \u2192 \u212c is an elementary embedding provided for each formula \u03a8 and each tuple  from \ud835\udc9c it is the case that \u03a8(\u03f5(a1),\u2026, \u03f5(an)) holds in \u212c if and only if \u03a8 holds in \ud835\udc9c. The existence of an elementary embedding is a sufficient, but in general unnecessary, condition for elementary equivalence. If \ud835\udc9c is a subalgebra of \u212c and the inclusion map embeds \ud835\udc9c elementarily into \u212c, then \u212c is said to be an elementary extension of \ud835\udc9c. Obviously isomorphic \u03a9-algebras are elementarily equivalent. Thus, necessary conditions for a nonempty class \ud835\udcb3 of \u03a9-algebras to be axiomatic are that \ud835\udcb3 be closed under elementary equivalence and taking ultraproducts. These conditions are also sufficient. That is the content of Theorem 3, Section 42 of [A monomorphism on 42 of  (see 8]\u03f5 : \ud835\udc9c \u2192 \u212cS be a consistent set of sentences of L\u03a9. A consequence of a theorem of Los is that if every member of the family (\ud835\udc9ci)i\u2208I of \u03a9-algebras lies in M(S), then so does every ultraproduct of the family. From that it is easy to deduce that every ultrapower and every ultralimit of an \u03a9-algebra \ud835\udc9c must be elementarily equivalent to \ud835\udc9c.  The canonical embedding is in fact an elementary embedding .\u03c3 be a sentence of L\u03a9.\u03c3 is equivalent to a universal sentence of L\u03a9 if and only if \u03c3 is preserved under taking subalgebras.\u03c3 is equivalent to a Horn sentence of L\u03a9 if and only if \u03c3 is preserved under taking reduced products.\u03c3 is a universal sentence of L\u03a9, then \u03c3 is equivalent to a universal Horn sentence of L\u03a9 if and only if \u03c3 is preserved under taking direct products.If Let For proofs see, for example, .\u03a9-algebras under a restriction on the operator domain \u03a9. Specifically, we consider those \u03a9 whose set C\u03a9 of constant symbols is a singleton C\u03a9 = {\u03b8}. We then consider those varieties of \u03a9-algebras satisfying, at least, the laws f = \u03b8 as f varies over the set F\u03a9 of function symbols of \u03a9. Recall that quantifier free sentences of L\u03a9 are special cases of universal sentences of L\u03a9. We say that such varieties contain a zero. For example, the varieties of groups, monoids, and Lie algebras contain a zero but the variety of semigroups does not. We conclude this section with several observations.We will have need in the next section to consider special varieties of \u03a9-algebra \ud835\udc9c is a subalgebra of the \u03a9-algebra \u212c. Since \ud835\udc9c\u2009\u2261\u2200\u2009\u212c if and only every primitive sentence of L\u03a9 true in \u212c is also true in \ud835\udc9c it follows that necessary and sufficient conditions for \ud835\udc9c\u2009\u2261\u2200\u2009\u212c are that every finite system (in finitely many variables) of equations and inequations Pi, pi, Qj, and qj are terms of L\u03a9, which has a solution in \u212c must already have a solution in \ud835\udc9c.Suppose the i(si \u2260 ti) contains no atomic formula is false in the trivial \u03a9-algebra. Hence, if such a sentence is true in an \u03a9-algebra \ud835\udc9c, then uhc(\ud835\udc9c), the universal Horn class of \ud835\udc9c, must be a proper subclass of qvar(\ud835\udc9c), the quasivariety generated by \ud835\udc9c. On the other hand, if \ud835\udc9c lies in a variety containing a zero \u03b8, then \ud835\udc9c contains the trivial subalgebra {\u03b8A} so such a sentence could not hold in \ud835\udc9c.A negated primitive Horn sentence \u03a9-algebra \u212c is a direct power of the \u03a9-algebra \ud835\udc9c. For some special \ud835\udc9c we will want to show that \ud835\udc9c and \u212c are universally equivalent. Since the diagonal map \u03b4 : \ud835\udc9c \u2192 \u212c embeds \ud835\udc9c isomorphically into \u212c every universal sentence of L\u03a9 true in \u212c must also be true in \ud835\udc9c. Thus, in the above situation it will suffice for our purposes to show that every universal sentence of L\u03a9 true in \ud835\udc9c is also true in \u212c.Suppose the \u03a9 be an operator domain and let \ud835\udcb1 be a variety of \u03a9-algebras. Throughout this section we will assume that all algebras lie in \ud835\udcb1. We will sometimes  assume that \ud835\udcb1 contains a zero, which in that event implies, among other things, a restriction on \u03a9.Let \ud835\udc9c and \u212c be elements of \ud835\udcb1.\ud835\udc9c separates \u212c provided to every pair x \u2260 y of unequal elements of \u212c there is a homomorphism \u03d5 : \u212c \u2192 \ud835\udc9c such that \u03d5(x) \u2260 \u03d5(y).\ud835\udc9c discriminates \u212c provided given finitely many pairs xi \u2260 yi, i = 1,\u2026, n, of unequal elements of \u212c there is a homomorphism \u03d5 : \u212c \u2192 \ud835\udc9c such that \u03d5(xi) \u2260 \u03d5(yi) for all i = 1,\u2026, n.\ud835\udc9c is discriminating provided it discriminates every element of \ud835\udcb1 which it separates.Let \ud835\udc9c is discriminating if and only if it discriminates its direct square \ud835\udc9c2.The proof is identical to that for groups. See, for example, .\ud835\udc9c is discriminating, then \ud835\udc9c2\u2009\u2261\u2200\u2009\ud835\udc9c; that is, \ud835\udc9c has the same universal theory as its direct square \ud835\udc9c2.If \ud835\udc9c with its image in \ud835\udc9c2 under the diagonal embedding \u03b4 : A \u2192 \ud835\udc9c2 given by \u03b4(a) = . Viewing \ud835\udc9c as a subalgebra of \ud835\udc9c2, it will suffice to show that every finite system \ud835\udc9c2 already has a solution in \ud835\udc9c.We identify x1,\u2026, xn) =  is a solution in \ud835\udc9c2 to the above system. Since \ud835\udc9c discriminates \ud835\udc9c2 there is a homomorphism \u03d5 : \ud835\udc9c2 \u2192 \ud835\udc9c such that Suppose that  = (\u03d5(b1),\u2026, \u03d5(bn)) is a solution to the system in \ud835\udc9c. Hence, \ud835\udc9c2\u2009\u2261\u2200\u2009\ud835\udc9c.But then  = uhc(\ud835\udc9c);\u212c in \ud835\udcb1 such that \ud835\udc9c\u2009\u2261\u2200\u2009\u212c.there is a discriminating algebra Let Momentarily assuming the theorem, we have the following consequence.\ud835\udcb1 contains a zero and let \ud835\udc9c be an algebra in \ud835\udcb1. Then the following three conditions are equivalent in pairs:\ud835\udc9c is squarelike;ucl(\ud835\udc9c) = qvar(\ud835\udc9c);\u212c in \ud835\udcb1 such that \ud835\udc9c\u2009\u2261\u2200\u2009\u212c.there is a discriminating algebra Suppose the variety \ud835\udc9c) = qvar(\ud835\udc9c). Since uhc(\ud835\udc9c) is axiomatizable by universal Horn sentences it is closed under taking subalgebras and direct products. Since \ud835\udc9c \u2208 uhc(\ud835\udc9c) and the trivial algebra {\u03b8A} is a subalgebra of \ud835\udc9c, uhc(\ud835\udc9c) contains the trivial algebra. Thus the axiomatic class uhc(\ud835\udc9c) is closed under taking subalgebras and direct products and contains the trivial algebra. Hence, it is a quasivariety. Therefore, uhc(\ud835\udc9c) = qvar(\ud835\udc9c).Assuming the theorem it will suffice to show that uhc is axiomatizable by universal Horn sentences. Assume deducing a contradiction that u is a universal sentence of L\u03a9 true in \ud835\udc9c but not a consequence of any set of universal Horn sentences of L\u03a9. Of course then u itself cannot be a Horn sentence. We may assume that the matrix of u is written in conjunctive normal form and hence u has the form u is equivalent to the conjunction u would be a Horn sentence. We claim that at least one of the negated primitive sentences L\u03a9 true in \ud835\udc9c. Suppose not. For each i such that Hi be a set of universal Horn sentences of L\u03a9 true in \ud835\udc9c such that Hi. For each i for which i we take H = \u222aiHi. It follows then that u would be a consequence of the set H of universal Horn sentences of L\u03a9 true in \ud835\udc9c, contrary to hypothesis. The contradiction shows that at least one negated primitive sentence L\u03a9 true in \ud835\udc9c.(1) \u21d2 (2) Assume i notationally). Let n \u2265 2. For each fixed \u03bd0 = 1,\u2026, n let \u03a8\u03bd0 be the quasi-identity Fix such a conjunct \u03bd0 is true in \ud835\udc9c. Then for every tuple \ud835\udc9c, \ud835\udc9c. Equivalently, for every tuple \ud835\udc9c, \ud835\udc9c. Since a disjunction is true if at least one of its disjuncts is true we would have in the above event that \ud835\udc9c for every tuple \ud835\udc9c. This implies that \u03bd0. But quasi-identities are special cases of universal Horn sentences and L\u03a9 true in \ud835\udc9c. The contradiction shows that none of the quasi-identities \u03a8\u03bd can hold in \ud835\udc9c. Thus, for each \u03bd = 1,\u2026, n, the existential sentence \ud835\udc9c.Suppose deducing a contradiction that \u03a8\ud835\udc9c such that simultaneously \u03bc = 1,\u2026, m and \ud835\udc9cn so that \ud835\udc9cn. By \ud835\udc9cn\u2009\u2261\u2200\u2009A. It follows that the existential sentence \ud835\udc9c. But that contradicts the fact that its negation  \ud835\udc9c. The contradiction shows that ucl(\ud835\udc9c) must have at least one set of universal Horn axioms. Hence, ucl(\ud835\udc9c) = uhc(\ud835\udc9c) if \ud835\udc9c is squarelike.Now let \ud835\udc9c) = uhc(\ud835\udc9c). Now A \u2208 uhc(\ud835\udc9c) and uhc(\ud835\udc9c) is closed under taking direct products. Let I be an infinite index set and let \u212c = \ud835\udc9cI. Then \u212c \u2208 ucl(\ud835\udc9c) so \u212c is a model of Th\u2200(\ud835\udc9c) and every universal sentence of L\u03a9 true in \ud835\udc9c is also true in \u212c. Thus, \ud835\udc9c\u2009\u2261\u2200\u2009\u212c by \u212c2 is isomorphic to \u212c so that \u212c is discriminating by (2) \u21d2 (3) Suppose ucl2 which is elementarily equivalent to \u212c2 by \u22c6\u212c)2\u2009\u2261\u2200\u2009\u212c2. But \u212c is discriminating so \u212c2\u2009\u2261\u2200\u2009\u212c by \ud835\udc9c\u2009\u2261\u2200\u2009\u212c so ultimately (\u22c6\u212c)2\u2009\u2261\u2200\u2009\ud835\udc9c. Thus, every universal sentence of L\u03a9 true in \ud835\udc9c is also true in (\u22c6\u212c)2. But \ud835\udc9c2 embeds in (\u22c6\u212c)2 so every universal sentence of L\u03a9 true in \ud835\udc9c must also be true in \ud835\udc9c2. Then \ud835\udc9c2\u2009\u2261\u2200\u2009\ud835\udc9c by \ud835\udc9c is squarelike.(3) \u21d2 (1) Suppose that Exactly as for groups  we have\ud835\udc9ci)i\u2208I be an indexed family of \u03a9-algebras and let \ud835\udc9f be a filter on I. Let \ud835\udc9c be the reduced product \ud835\udcab/R(\ud835\udc9f) where \ud835\udcab = \u220fi\u2208I\ud835\udc9ci. Then \ud835\udc9c2 is isomorphic to the reduced product of the family (\ud835\udc9ci2)i\u2208I modulo the filter \ud835\udc9f on I.Let i\u2208I be a family of \u03a9-algebras and let \ud835\udc9f be an ultrafilter on I. Let \ud835\udc9c be the ultraproduct constructed from this data. If \u03d5 is a sentence of L\u03a9 let the support of \u03d5 be the set Supp(\u03d5) of all i \u2208 I such that \u03d5 holds in \ud835\udc9ci. A consequence of Los' Theorem  \u2208 \ud835\udc9f. Now consider \ud835\udc9c2. By \ud835\udc9c2 is isomorphic to the ultraproduct of the family (\ud835\udc9c2)i\u2208I modulo the filter \ud835\udc9f on I. Suppose that each \ud835\udc9ci lies in \ud835\udcb3 so that (\ud835\udc9c2)i\u2009\u2261\u2200\u2009\ud835\udc9ci for all i \u2208 I. Let \u03d5 be a universal sentence of L\u03a9 holding in \ud835\udc9c. Then Supp(\u03d5) \u2208 \ud835\udc9f. Thus, for each i \u2208 Supp(\u03d5), \u03d5 holds in \ud835\udc9ci. But Ai\u2009\u2261\u2200\u2009(\ud835\udc9c2)i since Ai \u2208 \ud835\udcb3. But then \u03d5 holds in \ud835\udc9c2 since Supp(\u03d5) \u2208 \ud835\udc9f. It follows that every universal sentence of L\u03a9 true in \ud835\udc9c must also be true in \ud835\udc9c2 and hence \ud835\udc9c2\u2009\u2261\u2200\u2009\ud835\udc9c by \ud835\udcb3 is closed under taking ultraproducts as well as elementary equivalence. Hence, \ud835\udcb3 is axiomatic.Let e, e.g., ) is that\ud835\udc9c be a squarelike algebra in \ud835\udcb1. Then \ud835\udc9c is elementarily equivalent to a discriminating member of \ud835\udcb1. Consequently, the class of squarelike algebras in \ud835\udcb1 is the least axiomatic class containing the discriminating members of \ud835\udcb1.Let A proof which uses the ultralimit construction is exactly the same as that for groups and may be found, for example, in . See als\ud835\udcb1 has a set of Horn axioms.The class of squarelike algebras in \ud835\udcb1 is closed under taking direct unions. Thus, that class, in addition to having a set of Horn axioms, has a set of universal-existential axioms.A theorem of Los and Suszko asserts that a model class has a set of universal-existential axioms if and only if it is closed under direct unions. For definitions of the relevant terms and a proof of the Los-Suszko Theorem see, for example, . It is e\ud835\udc9ci)i\u2208I be a family of squarelike algebras in \ud835\udcb1 and let \ud835\udc9f be a filter on I. By the theorem there is, for each i \u2208 I, a discriminating algebra \u212ci in \ud835\udcb1 such that \ud835\udc9ci \u2261 \u212ci. By \ud835\udc9ci)i\u2208I modulo the filter \ud835\udc9f on I is elementarily equivalent to the reduced product of the family (\u212ci)i\u2208I modulo the filter \ud835\udc9f on I. In particular, the reduced product of the family (\ud835\udc9ci)i\u2208I modulo the filter \ud835\udc9f on I is universally equivalent to the reduced product of the family (\u212ci)i\u2208I modulo the filter \ud835\udc9f on I. Hence, the reduced product of the family (\ud835\udc9ci)i\u2208I is squarelike by \ud835\udcb1 is preserved under taking reduced products. Now, as mentioned in page 225 of [Exactly as for groups  one proe 225 of , an axio\ud835\udcb1 contains a zero \u03b8 we can explicitly describe a set of axioms for the class of squarelike algebras in \ud835\udcb1 by mimicking the situation for groups. For the remainder of this section we will restrict ourselves to those varieties \ud835\udcb1 which contain a zero \u03b8. The class of squarelike algebras in the variety \ud835\udcb1 containing a zero is the model class of the laws of \ud835\udcb1 together with the sentences Pi, pi, Qj, and qj vary over terms of L\u03a9 containing at most the variables in In the special case when example, .\ud835\udcb1 be a variety containing a zero. Let \ud835\udcac be a subquasivariety of \ud835\udcb1. An algebra \ud835\udc9c \u2208 \ud835\udcacq-discriminates \ud835\udcac provided that given finitely many quasi-identities \ud835\udc9c such that simultaneously i, j. An algebra \ud835\udc9c in \ud835\udcb1 is q-discriminating provided it q-discriminates qvar(\ud835\udc9c).Let \ud835\udcb1 be a variety containing a zero. An algebra \ud835\udc9c in \ud835\udcb1 is q-algebraically closed if and only if whenever a finite system \u212c\u2208 qvar(\ud835\udc9c) it also has a solution in \ud835\udc9c.Let Exactly as for groups we have the following.\ud835\udcb1 be a variety containing a zero. Let \ud835\udc9c be an algebra in \ud835\udcb1. The following conditions are equivalent in pairs.\ud835\udc9c is q-discriminating.\ud835\udc9c is q-algebraically closed.\ud835\udc9c is squarelike.Let See, for example, .\u03a9-algebras which contains the trivial algebra and is closed under taking subalgebras and direct products must be the model class of at least one set of quasi-identities of L\u03a9. Granting us the result that an axiomatic class of \u03a9-algebra is closed under taking subalgebras if and only if it is the model class of at least one set of universal sentences of L\u03a9 (Theorem 5.2.4 of [A2\u2009\u2261\u2200\u2009A implies ucl(A) = uhc(A) can be adapted to provide a proof of Malcev's Theorem.A classical theorem of Malcev asserts that an axiomatic class of 5.2.4 of ) we obse\u03a9-algebras which contains the trivial algebra and is closed under taking subalgebras and direct products must be the model class of at least one set of quasi-identities of L\u03a9.An axiomatic class of \u03a9-algebra is closed under taking subalgebras if and only if it is the model class of at least one set of universal sentences of L\u03a9 (Theorem 5.2.4 of [We assume that an axiomatic class of 5.2.4 of ).\u03c4 be an axiomatic class of \u03a9-algebras containing the trivial algebra and closed under taking subalgebras and direct products. Then \u03c4 is the model class of a set of universal sentences of L\u03a9. Assume deducing a contradiction that u is a universal sentence of L\u03a9 holding in every algebra \ud835\udc9c \u2208 \u03c4 but that u is not a consequence of any set of quasi-identities of L\u03a9 holding in every algebra \ud835\udc9c \u2208 \u03c4. We may assume that the matrix of u is written in conjunctive normal form. Hence u has the form u is equivalent to the conjunction u would be equivalent to a conjunction of quasi-identities true in every algebra \ud835\udc9c \u2208 \u03c4 contrary to hypothesis.Now let \u03c4. Therefore at least one We claim that it is impossible to have a conjunct L\u03a9 true in every algebra \ud835\udc9c \u2208 \u03c4. Suppose not. For each i such that Hi be the set of quasi-identities of L\u03c9 true in every algebra \ud835\udc9c \u2208 \u03c4 such that Hi. For each i for which i we take H = \u222aiHi. Then u would be a consequence of the set H of quasi-identities  of L\u03a9 true in every algebra \ud835\udc9c \u2208 \u03c4, contrary to hypothesis. The contradiction shows that at least one negated primitive sentence L\u03a9 true in every algebra \ud835\udc9c \u2208 \u03c4. Fix such a conjunct i notationally). Let n \u2265 2. For each fixed \u03bd0 = 1,\u2026, n let \u03c8\u03bd0 be the quasi-identity We now claim that at least one of the negated primitive sentences \u03c8\u03bd0 is true in every algebra \ud835\udc9c \u2208 \u03c4. Fix an \ud835\udc9c \u2208 \u03c4. Then for every tuple \ud835\udc9c\ud835\udc9c.Suppose deducing a contradiction that \ud835\udc9c, \ud835\udc9c. Since a disjunction is true if at least one of its disjuncts is true we have in that event that \ud835\udc9c for every tuple \ud835\udc9c. Since \ud835\udc9c was arbitrary, \u03c8\u03bd0 true in every algebra in \u03c4. But \u03bd = 1,\u2026, n, there is an algebra \ud835\udc9c\u03bd \u2208 \u03c4 such that the existential sentence \ud835\udc9c\u03bd. Now let \ud835\udc9c\u03bd such that simultaneously \u03bc = 1,\u2026, m and Equivalently, for every tuple \u03bd=1n\ud835\udc9c\u03bd. It follows that the existential sentence \u03bd=1n\ud835\udc9c\u03bd \u2208 \u03c4.Let \ud835\udc9c \u2208 \u03c4. The contradiction shows that \u03c4 must be axiomatizable by at least one set of quasi-identities of L\u03a9 completing the proof.But that contradicts the fact that its negation"}
{"text": "The aim of this paper is to introduce power idealization filter topologies with respect to filter topologies and power ideals of lattice implication algebras, and to investigate some properties of power idealization filter topological spaces and their quotient spaces. By generalizing Boolean algebras and Lukasiewicz implication algebras , Xu 2]  2] definNow we recall some definitions and notions of lattice implication algebras and topological spaces.L, \u2227, \u2228, 0,1) be a bounded lattice with the greatest 1 and the smallest 0. A system  is called a quasi-lattice implication algebra if \u2009\u2032 : L \u2192 L is an order-reserving involution and \u2192:L \u00d7 L \u2192 L is a map  satisfying the following conditions for any x, y, z \u2208 L:x \u2192 (y \u2192 z) = y \u2192 (x \u2192 z),x \u2192 x = 1,x \u2192 y = y\u2032 \u2192 x\u2032,x \u2192 y = y \u2192 x = 1 implies x = y,x \u2192 y) \u2192 y = (y \u2192 x) \u2192 x. is called a lattice implication algebra if the implication operator \u2192 further fulfils the following conditions:(6)x\u2228y) \u2192 z = (x \u2192 z)\u2227(y \u2192 z),((7)x\u2227y) \u2192 z = (x \u2192 z)\u2228(y \u2192 z). will be simply denoted by L.A lattice implication algebra  is called a topological space. A subset B of \u03c4 is called a base of \u03c4, if for each A \u2208 \u03c4 and each x \u2208 A, there exists B \u2208 B such that x \u2208 B\u2286A.Elements of L be an implication algebra. A subset F of L is called a filer, if F satisfies the following: (1) 1 \u2208 F; (2) x, x \u2192 y \u2208 F implies y \u2208 F. The collection of all filters in L is denoted by F(L), or F briefly. Clearly, F consists a base of some topology TF(L), briefly TF. Usually, TF is called the filter topology generated by F. And the pair  is called the filter topological space. A subset U\u2286L is called TF-neighborhood of x \u2208 L, or neighborhood of x in TF if x \u2208 U \u2208 TF. The set of all TF-neighborhoods of x is denoted by NTF(x). Since F\u2286TF and [x) = \u2229{F : x \u2208 F \u2208 F} \u2208 F, [x) is the smallest element of NTF(x).Let TF are denoted by c and i. Clearly, for every A\u2286L, c(A) = \u2229{L\u2216[x) : x \u2208 L, [x)\u2229A = \u2205} and i(A) = \u222a{[x) : x \u2208 L, [x)\u2286A}. The following proposition describes c(A).The closure operator and interior operator of L, TF) be the filer topology generated by F(L). Then for A\u2286L, c(A) = {x \u2208 L : [x)\u2229A \u2260 \u2205}.Let (x) is the smallest TF-neighborhood of x.The proof is trivial since [L be a lattice implication algebra and let 2L be the power set of L. A nonempty subset I of 2L is called a power ideal of L if I satisfies the following: (1) A, B \u2208 2L and A\u2286B \u2208 I imply A \u2208 I; (2) A, B \u2208 I implies A \u222a B \u2208 I. The collection of all power ideals in 2L is denoted by I(L), or briefly I. Note that I\u2205 = {\u2205} is the smallest power ideal and IL = 2L is the greatest power ideal. Moreover, if I, J \u2208 I, then (1) I\u2229J \u2208 I; (2) I\u2228J = {I \u222a J : I \u2208 I, J \u2208 J} \u2208 I.Let L be a lattice implication algebra, let TF be the filter topology, and let I be a power ideal. An operator *on\u2061 \u20092L is defined as follows:A\u2286L.Let TF and I. A* is called local function of A. We usually write A*(I) or A* instead of A*.The operator *is called the local function with respect to x \u2208 A* if and only if [x)\u2229A \u2209 I. Thus A* = {x \u2208 L : [x)\u2229A \u2209 I}. The following proposition gives some further details of A*.Clearly, L, TF) be the filter topological space and I, J \u2208 I. ThenA*(I\u2205) = c(A) and A*(IL) = \u2205;A\u2286B, then A*(I)\u2286B*(I);if I\u2286J, then A*(J)\u2286A*(I);if A*(I) = c(A*(I))\u2286c(A);A*)*(I)\u2286A*(I); = \u2205;if A \u2208 TFc, then A*(I)\u2286A;if B \u2208 I, then (A \u222a B)*(I) = A*(I) = (A\u2216B)*(I);if A \u222a A*(I))*(I) = A*(I);(A \u222a B)*(I) = A*(I) \u222a B*(I);(A*(I)\u2216B*(I) = (A\u2216B)*(I)\u2216B*(I)\u2286(A\u2216B)*(I);I, then [x)*(I) = L for each x \u2208 L;if {1} \u2209 I and 1 \u2208 A\u2286L, then [A)*(I) = [A*(I)) = L.if {1} \u2209 Let (x \u2208 A*(I\u2205) if and only if [x)\u2229A \u2260 \u2205 if and only if x \u2208 c(A). Thus A*(I\u2205) = c(A). Since [x)\u2229A \u2208 2L = IL for each x \u2208 L, A*(L) = \u2205.(1) By A\u2286B and x \u2208 A*(I). Then [x)\u2229A \u2209 I. Since I is a power ideal and [x)\u2229A\u2286[x)\u2229B, [x)\u2229B \u2209 I and so x \u2208 B*(I). Thus A*(I)\u2286B*(I).(2) Let I\u2286J and x \u2208 A*(I). Then [x)\u2229A \u2209 J. It follows that [x)\u2229A \u2209 I and so x \u2208 A*(I). Thus A*(J)\u2286A*(I).(3) Let x \u2209 c(A), then x \u2208 L\u2216c(A) \u2208 TF and so [x)\u2286L\u2216c(A). Thus [x)\u2229A\u2286(L\u2216c(A))\u2229A = \u2205 \u2208 I. This implies x \u2209 A*(I) and so A*(I)\u2286c(A). Then c(A*(I))\u2286c(c(A)) = c(A).(4) If A*(I)\u2286c(A*(I)). Next, we prove c(A*(I))\u2286A*(I).It is clear that x \u2208 c(A*(I)). By x)\u2229A*(I) \u2260 \u2205. Then there exists y \u2208 [x)\u2229A*(I). By y \u2208 A*(I), [y)\u2229A \u2209 I. By y \u2208 [x), [y)\u2286[x). Thus [x)\u2229A \u2209 I and so x \u2208 A*(I). Therefore c(A*(I))\u2286A*(I).Let A*(I))*(I)\u2286c(A*(I)) = A*(I).(5) By (4), (x)\u2229A\u2286A \u2208 I for each x \u2208 L, A*(I) = \u2205.(6) Since [x \u2208 A*(I)\u2216A. Then x \u2208 L\u2216A \u2208 TF. Thus [x)\u2286L\u2216A and so [x)\u2229A\u2286(L\u2216A) = \u2205 \u2208 I. Hence x \u2209 A*(I) which is a contradiction. Therefore A*(I)\u2286A.(7) Suppose that A\u2216B)*(I)\u2286A*(I)\u2286(A \u222a B)*(I). Next, we prove the inverse inclusions.(8) By (2), (x \u2209 (A\u2216B)*(I), then ([x)\u2229A)\u2216B = [x)\u2229(A\u2216B) \u2208 I. Thus [x)\u2229A\u2286I \u222a B \u2208 I which follows from I is a power ideal. This implies x \u2209 A*(I). Thus A*(I)\u2286(A\u2216B)*(I) and so A*(I) = (A\u2216B)*(I).If x \u2209 A*(I), then [x)\u2229A \u2208 I. Since B \u2208 I,x \u2209 (A \u222a B)*(I). This implies (A \u222a B)*(I)\u2286A*(I) and so (A \u222a B)*(I) = A*(I).If A*(I)\u2286(A \u222a A*(I))*(I). Conversely, if x \u2209 A*(I), then [x)\u2229A \u2208 I. Let [x)\u2229A = I. Then A\u2286I \u222a (L\u2216[x)). By (2), (7), (8), and [x) \u2208 TF,A \u222a A*(I)\u2286(L\u2216[x)) \u222a A and sox \u2209 (A \u222a A*(I))*(I) and so (A \u222a A*(I))*(I)\u2286A*(I).(9) Clearly, A*(I) \u222a B*(I)\u2286(A \u222a B)*(I) is clear. Conversely, if x \u2209 A*(I) \u222a B*(I), then [x)\u2229A, [x)\u2229B \u2208 I. Thus [x)\u2229(A \u222a B) = ([x)\u2229A)\u222a([x) \u222a B) \u2208 I. This implies x \u2209 (A \u222a B)*(I). Therefore (A \u222a B)*(I)\u2286A*(I) \u222a B*(I).(10) A*(I)\u2216B*(I)\u2286(A\u2216B)*(I). Assume that x \u2208 (A*(I)\u2216B*(I))\u2216(A\u2216B)*(I). Then [x)\u2229(A\u2216B) \u2208 I and [x)\u2229B \u2208 I. Thusx \u2209 A*(I) which is a contradiction. Thus A*(I)\u2216B*(I)\u2286(A\u2216B)*(I) and so A*(I)\u2216B*(I)\u2286(A\u2216B)*(I)\u2216B*(I). Finally, (A\u2216B)*(I)\u2216B*(I)\u2286A*(I)\u2216B*(I) follows from (2). Therefore A*(I)\u2216B*(I) = (A\u2216B)*(I)\u2216B*(I).(11) We firstly prove I, 1 \u2209 I for each I \u2208 I. Assume that there exists x \u2208 L such that [x) \u2260 L. Let y \u2209 L\u2216[x)*(I). Thus [y)\u2229[x) \u2208 I. Since [x\u2228y) = [y)\u2229[x), 1 \u2209 [x\u2228y) which is a contradiction. Therefore [x)*(I) = L for each x \u2208 L.(12) Since {1} \u2209 y \u2208 L\u2216A*(I). Then 1 \u2208 [y)\u2229A \u2208 I which is a contradiction. Thus A*(I) = L and [A*(I)) = L. Since 1 \u2208 A, L = [1)*(I)\u2286[A)*(I) follows from (12). Therefore [A)*(I) = L.(13) Assume that there exists L, TF) be the filter topological space and I \u2208 I. The operator cI*  on 2L, defined by c*(A) = A \u222a A* for A\u2286L, satisfies the following statements:c*(\u2205) = \u2205; c*(L) = L,c*(c*(A)) = c*(A)\u2286c(A),c*(A))* = c*(A*) = A*,(c*(A \u222a B) = c*(A) \u222a c*(B),A \u2208 TFc or A \u2208 I implies c*(A) = A.Let (c*(\u2205) = \u2205 follows from \u2205* = \u2205. c*(L) = L is clear.(1) (2) By (4) and (9) of c*(A*) = A* \u222a (A*)* = A*.(3) By (5) and (9) of (4) By (10) of (5) The result follows from (6) and (7) of L, TF) be the filter topological space and I \u2208 I. The operator c* stated in TF and the topology generated by Ic . Such a topology is called a power idealization filter topology and often denoted by TF*, TF*(I), or TF*.Let (TF* = {A\u2286L : c*(L\u2216A) = L\u2216A}. We prove that TF* is a topology on L.Let \u2205, L \u2208 TF* follows from (1) of (1) A, B \u2208 TF*, then c*(L\u2216A) = L\u2216A and c*(L\u2216B) = L\u2216B. ThusA\u2229B \u2208 TF*.(2) If At \u2208 TF* for t \u2208 T, where T is an index set. Then c*(L\u2216At) = L\u2216At andt\u2208TAt \u2208 TF*.(3) Let TF, Ic\u2286TF*.Finally, by (6) and (7) of L = {0, a, b, c, d, 1}, 0\u2032 = 1, a\u2032 = c, b\u2032 = d, c\u2032 = a, d\u2032 = b, 1\u2032 = 0, and the implication operator \u2192 be defined by a \u2192 b = a\u2032\u2228b for a, b \u2208 L. Then  is the Hasse lattice implication algebra  be the filter topological space and I, J \u2208 I. ThenTF*(I\u2205) = TF and TF*(IL) = 2L,I\u2286J, then TF*(I)\u2286TF*(J).if Let (A*(I\u2205) = c(A) and A*(IL) = \u2205. Thus cI\u2205*(A) = A if and only if c(A) = A. Similarly, cIL*(A) = A for each A\u2286L. Therefore (1) holds.(1) By (1) of I\u2286J. By (3) of cJ*(A)\u2286cI*(A) for A\u2286L. Thus if A \u2208 TF*(I), then A \u2208 TF*(J). Therefore TF*(I)\u2286TF*(J).(2) Let I \u2208 I satisfies TFc\u2286I, then L \u2208 I and so I = 2L = IL. Thus by (1) of TF* = Ic = 2L. If TFc\u2216{L}\u2286I, we have the following proposition.Clearly, if L, TF) be the filter topological space. If I \u2208 I satisfies TFc\u2216{L}\u2286I, then TF* = Ic \u222a {\u2205}.Let (Ic \u222a {\u2205}\u2286TF* follows from TF*\u2288Ic \u222a {\u2205}. Then there exists B \u2208 TF*\u2216(Ic \u222a {\u2205}) such that (L\u2216B)*\u2286L\u2216B \u2260 L. Let y \u2208 (L\u2216B)\u2216(L\u2216B)*. Then y \u2209 (L\u2216B)*. Thus [y)\u2229(L\u2216B) \u2208 I. Put [y)\u2229(L\u2216B) = I. We have L\u2216B\u2286I \u222a (L\u2216[y)). Since TFc\u2216{L}\u2286I, L\u2216[y) \u2208 I. Thus L\u2216B\u2286I \u222a (L\u2216B) \u2208 I and so L\u2216B \u2208 I. Hence B \u2208 Ic. It is a contradiction. Therefore TF*\u2286Ic \u222a {\u2205}.L, TF) be the filter topological space and I \u2208 I. If A\u2286L satisfies A\u2229I = \u2205 for each I \u2208 I, then c*(A) = c(A).Let (c*(A)\u2286c(A) is clear. Conversely, if x \u2209 c*(A), then x \u2209 A and x \u2209 A*. Thus I = [x)\u2229A \u2208 I and A\u2286I \u222a (L\u2216[x)). Since A\u2229I = \u2205, A\u2286L\u2216[x). Observe that x \u2209 L\u2216[x) and L\u2216[x) is TF-closed. We have x \u2209 c(A). Thus c(A)\u2286c*(A). Therefore c*(A) = c(A).I \u2208 TFc, then TF* = TF.If TF\u2286TF*. Conversely, let IM be the greatest element of I and A\u2286L. Thus (A\u2216IM)\u2229I = \u2205 for each I \u2208 I. By c*(A\u2216IM) = c(A\u2216IM). By (8) of A\u2216IM)* = A*. Now, notice that A\u2229IM \u2208 I\u2286TFc and thus c(A\u2229IM) = A\u2229IM. We havec* = c. Therefore TF* = TF.It is clear that L, TF) be the filter topological space and I \u2208 I. If I \u2208 I and U \u2208 TF, then U\u2216I \u2208 TF*.Let (P = L\u2216U. Then P \u2208 TFc. By (7) and (8) of L\u2216(P \u222a I) = U\u2229(L\u2216I) = U\u2216I \u2208 TF*.Let L, TF) be the filter topological space and I \u2208 I. ThenTF*. Moreover,NTF*(x) for each x \u2208 L, where NTF*(x) is the set of all TF*-neighborhoods of x in . Clearly, [x)\u2216Ix is the smallest TF*-neighborhoods of x, where Ix is the greatest element of I satisfying x \u2209 Ix.Let (BTF*\u2286TF*. Let B\u2286L. Then B \u2208 TF*\u21d4L\u2216B is TF*-closed \u21d4(L\u2216B)*\u2286(L\u2216B)\u21d4B\u2286L\u2216(L\u2216B)*. Thus x \u2208 B\u21d2x \u2209 (L\u2216B)*\u21d2 there exists U \u2208 NTF(x) such that (L\u2216B)\u2229U \u2208 I. Let I = (L\u2216B)\u2229U. Then L\u2216B\u2286I \u222a (L\u2216U) andBTF* is a base of TF*.By BTF*(x)\u2286NTF*(x). Let A \u2208 NTF*(x) and y \u2208 A. Since BTF* is a base of TF*, there are U, V \u2208 TF and I, J \u2208 I such that x \u2208 U\u2216I\u2286A and y \u2208 V\u2216J\u2286A. We can assume y \u2209 I and x \u2209 J . Theny \u2208 (U \u222a V)\u2216(I \u222a J) \u2208 BTF*(x) andBTF*(x) is a base of NTF*(x).Clearly, Ix is the greatest element of I satisfying x \u2209 Ix, then [x)\u2216Ix \u2208 NTF*(x) is the smallest TF*-neighborhoods of x.Clearly, if L, \u03c4) be a topological space and I \u2208 I. The topology that was generated by B = {U\u2216I : U \u2208 \u03c4, I \u2208 I} is denoted by T* [T* = TF*(I).Let  . Clearly\u03c8 = {\u2205, L} be the indiscrete topology on L and I \u2208 I. Then T* = {\u2205} \u222a Ic.Let Tc\u2216{L} = {\u2205} \u2208 I and By L, \u03c4) be a topological space and I \u2208 I. Then T* = \u03c4\u2228T*, where \u03c4\u2228T* is the topology generated by the base {U\u2229V : U \u2208 \u03c4, V \u2208 T*}.Let . Since BT* = {U\u2229V : U \u2208 \u03c4, V \u2208 T*}, BT* is also a base of \u03c4\u2228T*. Therefore T* = \u03c4\u2228T*.Clearly, L, TF) be the filter topological space and I \u2208 I. Then TF* = TF\u2228T*.Let  be the filter topological space and I, J \u2208 I. ThenT* = T*\u2228T*,TF*(I\u2228J) = T*) = T*),TF*(I\u2228J) = TF*(I)\u2228TF*(J),T*) = TF*(I).Let (T*(I\u2228J)\u2287T*(I)\u2228T*(J). Conversely, let \u2205 \u2260 A \u2208 T*(I\u2228J). By I \u2208 I and J \u2208 J such thatT*(I\u2228J)\u2286T*(I)\u2228T*(J).(1) By (2) of (2) By (1), TF*(I\u2228J) = T*(TF*(J), I).Similarly, TF*(I\u2228J) = TF*(I)\u2228TF*(J).(3) By (1) and I = J. Then the proof follows from (2).(4) Let L, TF) be the filter topological space, I, J \u2208 I and A\u2286L. ThenA* = A* \u222a A*,A* = A*)\u2229A*).Let  \u222a A*\u2286A*. Conversely, x \u2209 A* \u222a A*. Then [x)\u2229A \u2208 I and [x)\u2229A \u2208 J. Let [x)\u2229A = I and [x)\u2229A = J. Then A\u2286I \u222a (L\u2216[x)) and A\u2286J \u222a (L\u2216[x)). Thusx)\u2229A\u2286I\u2229J which implies x \u2209 A*. Therefore A*\u2286A* \u222a A*.(1) By (3) of x \u2209 A*. Then [x)\u2229A \u2208 I\u2228J. Then there exist I \u2208 I and J \u2208 J such that [x)\u2229A = I \u222a J. We can assume I\u2229J = \u2205, ). Thus x \u2209 I or x \u2209 J, . Now, we take x \u2209 I for example. Thenx) \u2208 TF and x \u2208 [x)\u2216I, [x)\u2216I \u2208 BTF*(I). Thus x \u2209 A*) and so x \u2209 A*)\u2229A*). Hence(3) Let x \u2209 A*(TF*(I), J). Then there exists I \u2208 I such that ([x)\u2216I)\u2229A \u2208 J. Let ([x)\u2216I)\u2229A = J. Then [x)\u2229A = I \u222a J which implies x \u2209 A*. Similarly, if x \u2209 A*(TF*(J), I), then x \u2209 A*. Therefore A*\u2286A*)\u2229A*).Conversely, let A* = A*).Consider I = J. The proof follows from (2) of Let TF*(I\u2229J) = TF*(I)\u2229TF*(J).Consider TF*(I\u2229J)\u2286TF*(I)\u2229TF*(J). Conversely, if A \u2209 TF*(I\u2229J), thenL\u2216A)*\u2288(L\u2216A) or (L\u2216A)*\u2288(L\u2216A). Thus A \u2209 TF*(I) or A \u2209 TF*(J). Therefore TF*(I)\u2229TF*(J)\u2286TF*(I\u2229J).By (2) of L1, \u2227, \u2228, \u2009\u2032, \u21921, 01, 11) and  be two lattice implication algebras. A mapping f from L1 to L2 is called lattice implication homomorphism, if f(x\u21921y) = f(x)\u21922f(y) for any x, y \u2208 L1. The set of all lattice implication homomorphisms from L1 to L2 is denoted by hom.Let . Then, clearly,Let pologies .L1 and L2 be two implication algebras and let f \u2208 hom and I \u2208 2L1, J \u2208 2L2 be power ideals.f is injective, then f\u22121(J) = {f\u22121(J) : J \u2208 J} \u2208 I(L1).If f is surjective, then f(I) = {f(I) : I \u2208 I} \u2208 I(L2).If Let \u2205 \u2208 J, then \u2205 = f\u22121(\u2205) \u2208 f\u22121(J).(1) Since I2 \u2208 f\u22121(J) and I1\u2286I2, then there exist J2 \u2208 J such that I2 = f\u22121(J2). Thus f(I1)\u2286f(I2) = J2 and f(I1) \u2208 J. Since f is injective, I1 = f\u22121(f(I1)) \u2208 f\u22121(J).If I1, I2 \u2208 f\u22121(J), then there exist J1, J2 \u2208 J such that I1 = f\u22121(J1) and I2 = f\u22121(J2). One has I1 \u222a I2 = f\u22121(J1) \u222a f\u22121(J2) = f\u22121(J1 \u222a J2). Since J1 \u222a J2 \u2208 J, I1 \u222a I2 \u2208 f\u22121(J). Therefore f\u22121(J) is a power ideal.If \u2205 \u2208 I, \u2205 = f(\u2205) \u2208 f(I).(2) By J2 \u2208 f(J) and J1\u2286J2, then there exists I2 \u2208 I such that J2 = f(I2). Let I1 = f\u22121(J1). Then I1\u2286I2 and I1 \u2208 I. Since f is surjective, J1 = f(I1) \u2208 f(I).If J1, J2 \u2208 f(J), then there exist I1, I2 \u2208 I such that J1 = f(I1) and J2 = f(I2). Thus I1 \u222a I2 \u2208 I. Hence J1 \u222a J2 = f(I1) \u222a f(I2) = f(I1 \u222a I2) \u2208 f(I). Therefore f(I) is a power ideal.If L1 and L2 be two implication algebras and f \u2208 hom. Thenf is injective, then for each x \u2208 L1, f\u22121([f(x))) \u2208 Tl(TF(L2), f) is the smallest Tl-neighborhood of x;if f is bijective, then for each y \u2208 L2, f([f\u22121(y))) \u2208 Tr(TF(L1), f) is the smallest Tr-neighborhood of y.if Let f(x)) is the smallest TF(L2)-neighborhood of f(x). Then x \u2208 f\u22121([f(x))) \u2208 Tl(TF(L2), f). Let x \u2208 V \u2208 Tl(TF(L2), f). Then f(x) \u2208 f(V) \u2208 TF(L2) and [f(x))\u2286f(V). Thus f\u22121([f(x)))\u2286f\u22121(f(V)) = V. Therefore f\u22121([f(x))) is the smallest one.(1) Clearly, [f\u22121(y)) is the smallest TF(L1)-neighborhood of f\u22121(y). Thus y \u2208 f([f\u22121(y))) \u2208 Tr(TF(L1), f). Now, let y \u2208 U \u2208 Tr(TF(L1), f). Then f\u22121(y) \u2208 f\u22121(U) \u2208 TF(L1). Thus [f\u22121(y))\u2286f\u22121(U) and so f([f\u22121(y)))\u2286f(f\u22121(U)) = U. Therefore (2) holds.(2) Clearly, [L1 and L2 be two implication algebras and f \u2208 hom.f is injective and J \u2208 I(L2), then for each x \u2208 L1, [f(x))\u2216JM \u2208 TF*) is the smallest neighborhood of f(x), where JM is the greatest element of J satisfying f(x) \u2209 JM.If f is bijective and I \u2208 I(L1), then for each y \u2208 L2, [f\u22121(y))\u2216IM \u2208 TF*) is the smallest neighborhood of f\u22121(y), where IM is the greatest element of I satisfying f\u22121(y) \u2209 IM.If Let L1 and L2 be two implication algebras and f \u2208 hom.(1)f is injective and J \u2208 I(L2), thenIf (2)f is bijective and I \u2208 I(L1), thenIf Let c2* and cl be the closure operators of the left side and the right side of the equation. We only need to prove c1* = cl.(1) Let A\u2286L1 and x \u2209 c1*(A). Then x \u2209 A and x \u2209 A*(f\u22121(J), Tl(TF(L2), f)). By (1) of f\u22121([f(x)))\u2229A \u2208 f\u22121(J). Thus there exists J \u2208 J such that f\u22121([f(x)))\u2229A = f\u22121(J). Since x \u2209 A and f is injective, f(x) \u2209 J. Let JM \u2208 J be the greatest one satisfying f(x) \u2209 JM. Then f\u22121([f(x)))\u2229A\u2286f\u22121(JM). Thusx \u2209 cl(A). Therefore cl(A)\u2286c1*(A).Let y \u2209 cl(A). By x \u2209 A, f\u22121([f(x)))\u2229A\u2286f\u22121(JM) and [f(x))\u2229f(A)\u2286JM. Then J1 = [f(x))\u2229f(A) \u2208 J. Since f is injective, f\u22121([f(x)))\u2229A = f\u22121(J1). By (1) of x \u2209 A*(f\u22121(J), Tl(TF(L2), f)). Therefore x \u2209 c1*(A) and c1*(A)\u2286cl(A).Conversely, let c2* and cr be the closure operators of the left side and the right side of the equation. We only need to prove c2* = cr.(2) Let y \u2209 c2*(A). Then y \u2209 A and y \u2209 A*((f(I), Tr(TF(L1), f))). By (2) of f([f\u22121(y)))\u2229A \u2208 f(I). Thus there exists I \u2208 I such that f([f\u22121(y)))\u2229A = f(I). Since f is bijective, [f\u22121(y))\u2229f\u22121(A) = I. By y \u2209 A, f\u22121(y) \u2209 f\u22121(A) and so f\u22121(y) \u2209 I. Since IM \u2208 I is the greatest element of I satisfying f\u22121(y) \u2209 IM, [f\u22121(y))\u2229f\u22121(A)\u2286IM and [f\u22121(y))\u2229(L1\u2216IM)\u2229f\u22121(A) = \u2205. By f being bijective again, we havey \u2209 cr(A). Hence c2*(A)\u2286cr(A).Let z \u2209 cr(A). Since IM is the greatest element of I satisfying f\u22121(z) \u2209 IM, by (2) of f\u22121(z))\u2216IM \u2208 TF*) is the smallest neighborhood of f\u22121(z). Thus f([f\u22121(z))\u2216IM)\u2229A = \u2205. Since f is bijective,f\u22121(z))\u2229f\u22121(A)\u2286IM and [f\u22121(z))\u2229f\u22121(A) \u2208 I. Thus f([f\u22121(y)))\u2229A \u2208 f(I) which implies y \u2208 A*((f(I), Tr(TF(L1), f))). Since z \u2209 cr(A), z \u2209 A. Therefore z \u2209 c2*(A) and so c2*(A)\u2286cr(A).Conversely, let f \u2208 hom is surjective but not bijective, then (2) of Generally, if L1 = {01, a, b, c, d, 11} be the Hasse lattice implication algebra of L2 = {02, e, h, 12} and 02\u2032 = 12, e\u2032 = h, h\u2032 = e, and 12\u2032 = 02. The Hasse diagram and the implication operator of L2 are shown by Let f from L1 to L2 is defined asf \u2208 hom and f is surjective. Let I = {\u2205, {c}}. Then I \u2208 I(L1) and f(I) = {\u2205, {h}} \u2208 I(L2).A mapping h} \u2208 f(I), by b, c}*) = {01, b, c, d}\u2288{b, c}. We havef\u22121 = {02, a, d, 12}, {02, e, 12} \u2209 Tr), f).Since {Tr), f) \u2260 T*(f(I), Tr(TF(L1), f)).In fact, we haveL1 and L2 be two implication algebras and let f \u2208 hom be bijective.(1)J \u2208 I(L2), thenIf (2)I \u2208 I(L1), thenIf Let The proof follows from L1 and L2 be two implication algebras and f \u2208 hom.(1)f is injective and J1, J2 \u2208 I(L2), thenIf (2)f is bijective and I1, I2 \u2208 I(L1), thenIf Let f\u22121(J1\u2228J2) = f\u22121(J1)\u2228f\u22121(J2). By (3) of (1) Clearly, (2) Is similar to (1).L1 and L2 be two implication algebras and let f \u2208 hom be bijective. If J \u2208 I(L2) and I \u2208 I(L1), thenLet The proof follows from (4) of"}
{"text": "Unfortunately, the original version of this article  containeThe results subsection of the Abstract also contains a mistake as a result of the mislabel. The first sentence \u201cThe 422 participants  had mean age (\u00b1 standard deviation) of 38.3\u2009\u00b1\u200920.5 and 42.9\u2009\u00b1\u200920.7 years, respectively\u201d should read \u201cThe 422 participants  had mean age (\u00b1 standard deviation) of 38.3\u2009\u00b1\u200920.5 and 42.9\u2009\u00b1\u200920.7 years, respectively\u201d. This error has been corrected in the original article.The \u2018Baseline Characteristics of Participants\u2019 subsection in the results contains an error. Reading from line 4, the sentence \u201cThe mean age amongst participant was 42.9\u2009\u00b1\u200920.7 and 38.3\u2009\u00b1\u200920.5 for females and males, respectively\u201d should be corrected to \u201cThe mean age amongst participant was 42.9\u2009\u00b1\u200920.7 and 38.3\u2009\u00b1\u200920.5 for males and females, respectively\u201d. We regret these errors which has now been corrected."}
{"text": "\u2110-regular pre-\u2110-open sets in which the notion of pre-\u2110-open set is involved. We characterize these sets and study some of their fundamental properties. We also present other notions called extremally pre-\u2110-disconnectedness, locally pre-\u2110-indiscreetness, and pre-\u2110-regular sets by utilizing the notion of pre-\u2110-open and pre-\u2110-closed sets by which we obtain some equivalence relation for pre-\u2110-regular pre-\u2110-open sets.We deal with the new class of pre- \u2110 on a topological space  is a nonempty collection of subsets of X which satisfies (i) A \u2208 \u2110 and B \u2282 A implies B \u2208 \u2110 and (ii) A \u2208 \u2110 and B \u2208 \u2110 implies A \u222a B \u2208 \u2110. Given a topological space  with an ideal \u2110 on X and if \u2118(X) is the set of all subsets of X, a set operator (\u00b7)\u22c6 : \u2118(X) \u2192 \u2118(X), called a local function [A with respect to \u03c4 and \u2110, is defined as follows: for A \u2282 X, A\u22c6 = {x \u2208 X\u2223U\u2229\ud835\udc9c \u2209 \u2110}, for every {U \u2208 \u03c4(x)} where \u03c4(x) = {U \u2208 \u03c4\u2223x \u2208 U}. A Kuratowski closure operator cl\u22c6 for a topology \u03c4\u22c6, called the \u22c6-topology, finer than \u03c4 is defined by cl\u22c6(A) = A \u222a A\u22c6 [A\u22c6 for A\u22c6 and \u03c4\u22c6 for \u03c4\u22c6. If \u2110 is an ideal on X, then  is called an ideal space. A subset A of an ideal space  is said to be \u2110-open [A \u2282 int\u2061(A\u22c6). A subset A of an ideal space  is said to be pre-\u2110-open [A \u2282 int\u2061(cl\u22c6(A)). The complement of pre-\u2110-open set is called pre-\u2110-closed. The family of all pre-\u2110-open sets in  is denoted by P\u2110O or simply P\u2110O(X). Clearly \u03c4 \u2282 P\u2110O(X). The largest pre-\u2110-open set contained in A, denoted by p\u2110int\u2061(A), is called the pre-\u2110-interior of A. The smallest pre-\u2110-closed set containing A, denoted by p\u2110cl(A), is called the pre-\u2110-closure of A. A subset A of an ideal space  is said to be \u03b1-\u2110-open [A \u2282 int\u2061(cl\u22c6(int\u2061(\ud835\udc9c))). The family of all \u03b1-\u2110-open sets is a topology finer than \u03c4. We will denote the \u03b1-\u2110-interior subset of A of X by \u03b1\u2110int\u2061(A). A subset A of an ideal space  is \u03c4\u22c6-dense [cl\u22c6(A) = X. A space  is \u2110-submaximal [\u03c4\u22c6-dense subset of X is open.The subject of ideals in topological space has been studied by Kuratowski  and Vaidfunction  of A wit\u22c6 ; when the \u2110-open  if A \u2282 ie-\u2110-open  if A \u2282 i\u03b1-\u2110-open  if A \u2282 i\u22c6-dense  if cl\u22c6 be an ideal space and let A be a subset of X. Then p\u2110int\u2061(A) = A\u2229int\u2061(cl\u22c6(A)) [Let (cl\u22c6(A)) .\u2110-open set A of a space  is said to be pre-\u2110-regular pre-\u2110-open if A = p\u2110int\u2061(p\u2110cl(A)). The complement of a pre-\u2110-regular pre-\u2110-open set is called pre-\u2110-regular pre-\u2110-closed set, equivalently A = p\u2110cl(p\u2110int\u2061(A)). A subset A of a space  with an ideal \u2110 is said to be pre-\u2110-regular if it is pre-\u2110-open and pre-\u2110-closed. Clearly, X and \u2205 are pre-\u2110-regular pre-\u2110-open.A pre-\u2110-regular set is pre-\u2110-regular pre-\u2110-open.Also every pre-A is pre-\u2110-regular that implies A is pre-\u2110-open and pre-\u2110-closed. A is pre-\u2110-open which implies A = p\u2110int\u2061(A) = p\u2110int\u2061(p\u2110cl(A)). Hence A is pre-\u2110-regular pre-\u2110-open.Assume But the converse is not true as shown by X, \u03c4, \u2110) where X = {a, b, c}, \u03c4 = {\u2205, {a}, {b}, {a, b}, X}, and \u2110 = {\u2205, {a}}. Here {a} is pre-\u2110-regular pre-\u2110-open but not pre-\u2110-closed.Consider the ideal space  where X = {a, b, c}, \u03c4 = {\u2205, {b, c}, X}, and \u2110 = {\u2205, {a}}. Here P\u2110O(X) = {\u2205, {b}, {c}, {a, b}, {a, c}, {b, c}, X}. Thus, A = {a, b}, B = {a, c} are both pre-\u2110-regular pre-\u2110-open. But A\u2229B is not, since it is not even pre-\u2110-open.Consider the ideal space  with an ideal \u2110 as in b, c} is \u2110-open but not pre-\u2110-regular pre-\u2110-open. Consider the space X = {a, b, c} and \u03c4 = {\u2205, {a}, {b}, {a, b}, X}, \u2110 = {\u2205, {a}}. Here {a} is pre-\u2110-regular pre-\u2110-open but not \u2110-open. Observe that every pre-\u2110-regular pre-\u2110-open set is pre-\u2110-open but the converse need not be true. Here {b, c} is pre-\u2110-open but not pre-\u2110-regular pre-\u2110-open.The notions of pre-X, \u03c4, \u2110) be an ideal space and let A, B be any subsets of X. Then the following hold.(a)A \u2282 B, then p\u2110int\u2061(p\u2110cl(A)) \u2282 p\u2110int\u2061(p\u2110cl(B)).If (b)A \u2208 PIO(X), then A \u2282 p\u2110int\u2061(p\u2110cl(A)).If (c)A \u2208 P\u2110O(X), p\u2110int\u2061(p\u2110cl(p\u2110int\u2061(p\u2110cl(A)))) = p\u2110int\u2061(p\u2110cl(A)).For every (d)A and B are disjoint pre-\u2110-open sets, then p\u2110int\u2061(p\u2110cl(A)) and p\u2110int\u2061(p\u2110cl(B)) are disjoint.If (e)A is a pre-\u2110-regular pre-\u2110-open, then p\u2110cl(X \u2212 A) is pre-\u2110-regular pre-\u2110-closed.If (f)A is pre-\u2110-regular pre-\u2110-open, then p\u2110int\u2061(A) is pre-\u2110-regular pre-\u2110-open.If Let ((a)A \u2282 B\u21d2p\u2110cl(A) \u2282 p\u2110cl(B). Therefore p\u2110int\u2061(p\u2110cl(A)) \u2282 p\u2110int\u2061(p\u2110cl(B)).Suppose (b)A \u2208 P\u2110O\u21d2A = p\u2110int\u2061(A) \u2282 p\u2110int\u2061(p\u2110cl(A)).Suppose that (c)p\u2110int\u2061(p\u2110cl(A)) \u2208 P\u2110O, so by (b) we have p\u2110int\u2061(p\u2110cl(A))\u2282p\u2110int\u2061(p\u2110cl(p\u2110int\u2061(p\u2110cl(A)))). On the other hand p\u2110int\u2061(p\u2110cl(A)) \u2282 p\u2110cl(A) which implies p\u2110cl(p\u2110int\u2061(p\u2110cl(A)))\u2282p\u2110cl(p\u2110cl(A)) = p\u2110cl(A). Therefore, p\u2110int\u2061(p\u2110cl(p\u2110int\u2061(p\u2110cl(A))))\u2282p\u2110int\u2061(p\u2110cl(A)). Hence p\u2110int\u2061(p\u2110cl(p\u2110int\u2061(p\u2110cl(A)))) = p\u2110int\u2061(p\u2110cl(A)).It is obvious that (d)A and B are disjoint pre-\u2110-open sets, we have A\u2229B = \u2205 which implies A\u2229p\u2110cl(B) = \u2205\u21d2A\u2229p\u2110int\u2061(p\u2110cl(B)) = \u2205. Since p\u2110int\u2061(p\u2110cl(B)) is pre-\u2110-open, p\u2110cl(A)\u2229p\u2110int\u2061(p\u2110cl(B)) = \u2205. Hencep\u2110int\u2061(p\u2110cl(A))\u2229p\u2110int\u2061(p\u2110cl(B)) = \u2205.Since (e)A is pre-\u2110-regular pre-\u2110-open, A = p\u2110int\u2061(p\u2110cl(A)) implies X \u2212 A = X \u2212 p\u2110int\u2061(p\u2110cl(A)) = p\u2110cl(p\u2110int\u2061(X \u2212 A)). Therefore, p\u2110cl(X \u2212 A) = p\u2110cl(p\u2110int\u2061(p\u2110cl(X\u2009\u2212\u2009A))). Hence p\u2110cl(X \u2212 A) is pre-\u2110-regular pre-\u2110-closed.Given that (f)A is pre-\u2110-regular pre-\u2110-open, then p\u2110cl(X \u2212 A) is pre-\u2110-regular pre-\u2110-closed. Hence X \u2212 (p\u2110cl(X \u2212 A)) is pre-\u2110-regular pre-\u2110-open that implies p\u2110int\u2061(A) is pre-\u2110-regular pre-\u2110-open.By (e) if X, \u03c4, \u2110) the following are equivalent. (a)\u2110-open set is open.Every pre-(b)\u03c4\u22c6-dense set is open.Every For an ideal topological space (A be a \u03c4\u22c6-dense set that implies cl\u22c6(A) = X which implies int\u2061(cl\u22c6(A)) = X, so that A \u2282 int\u2061(cl\u22c6(A)) = X. By (a) every pre-\u2110-open set is open and hence A is open.(a) \u21d2 (b): let B be a pre-\u2110-open subset of X, so that B \u2282 int\u2061(cl\u22c6(B)) = G, say. Then cl\u22c6(B) = cl\u22c6(G), so that cl\u22c6((X \u2212 G) \u222a B) = cl\u22c6(X \u2212 G) \u222a cl\u22c6(B) = (X \u2212 G) \u222a cl\u22c6(B) = X, and thus (X \u2212 G) \u222a B is \u03c4\u22c6-dense in X. Thus (X \u2212 G) \u222a B is open. Now B = ((X \u2212 G) \u222a B)\u2229G is the intersection of two open sets, so that B is open.(b) \u21d2 (a): let X, \u03c4) with an ideal \u2110 is \u2110-submaximal, then any finite intersection of pre-\u2110-open set is pre-\u2110-open.If a space  with an ideal \u2110 is \u2110-submaximal, then any finite intersection of pre-\u2110-regular p-\u2110-open set is pre-\u2110-regular pre-\u2110-open.If a space ). Also, for each i = 1,2,\u2026, n, \u2229Oi \u2282 Oi which implies p\u2110int\u2061(p\u2110cl(\u2229Oi))\u2282p\u2110int\u2061(p\u2110cl(Oi)). Also, each Oi is pre-\u2110-regular pre-\u2110-open that implies Oi = p\u2110int\u2061(p\u2110cl(Oi)) which implies p\u2110int\u2061(p\u2110cl(\u2229Oi))\u2282\u2229Oi and so \u2229Oi = p\u2110int\u2061(p\u2110cl(\u2229Oi)). Hence \u2229Oi is pre-\u2110-regular pre-\u2110-open.Let {\u2110-regular pre-\u2110-open set is pre-\u2110-regular pre-\u2110-open. But the intersection of two pre-\u2110-regular pre-\u2110-closed sets fails to be pre-\u2110-regular pre-\u2110-closed as shown by It should be noted that an arbitrary union of pre-X, \u03c4, \u2110) as in a, c}, {a, b} are pre-\u2110-regular pre-\u2110-closed but their intersection is not pre-\u2110-regular pre-\u2110-closed.Consider the ideal space .(a)A is pre-\u2110-closed, then p\u2110int\u2061(A) is pre-\u2110-regular pre-\u2110-open.If (b)A = p\u2110int\u2061(A), then p\u2110cl(A) is pre-\u2110-regular pre-\u2110-closed.If (c)A and B are pre-\u2110-regular pre-\u2110-closed sets, then A \u2282 B if and only if p\u2110int\u2061(A) \u2282 p\u2110int\u2061(B).If (d)A and B are pre-\u2110-regular pre-\u2110-open sets, then A \u2282 B if and only if p\u2110cl(A) \u2282 p\u2110cl(B).If The following hold for a subset A is pre-\u2110-closed, A = p\u2110cl(A).(a) Since p\u2110int\u2061(p\u2110cl(p\u2110int\u2061(A))) = p\u2110int\u2061(p\u2110int\u2061(A)) = p\u2110int\u2061(A). Hence p\u2110int\u2061(A) is pre-\u2110-regular pre-\u2110-open.Now, p\u2110cl(p\u2110int\u2061(p\u2110cl(A))) = p\u2110cl(p\u2110cl(A)) = p\u2110cl(A). Hence p\u2110cl(A) is pre-\u2110-regular pre-\u2110-closed.(b) Now A and B are pre-\u2110-regular pre-\u2110-closed sets, therefore, A = p\u2110cl(p\u2110int\u2061(A)) and B = p\u2110cl(p\u2110int\u2061(B)). Clearly, p\u2110int\u2061(A) \u2282 p\u2110int\u2061(B) if A \u2282 B.(c) Given that p\u2110int\u2061(A) \u2282 p\u2110int\u2061(B). Now A = p\u2110cl(p\u2110int\u2061(A)) \u2282 p\u2110cl(p\u2110int\u2061(B)) \u2282 B. Hence A \u2282 B.Conversely, A and B are pre-\u2110-regular pre-\u2110-open, therefore, A = p\u2110int\u2061(p\u2110cl(A)) and B = p\u2110int\u2061(p\u2110cl(B)). Suppose A \u2282 B, p\u2110cl(A) = p\u2110cl(p\u2110int\u2061(p\u2110cl(A)))\u2282p\u2110cl(p\u2110int\u2061(p\u2110cl(B)))\u2282p\u2110cl(B). Therefore, p\u2110cl(A) \u2282 p\u2110cl(B).(d) Given that p\u2110cl(A) \u2282 p\u2110cl(B). Now A = p\u2110int\u2061(p\u2110cl(A)) \u2282 p\u2110int\u2061(p\u2110cl(B)) \u2282 B.Conversely, A of an ideal topological space  is said to be \u2110-rare if it has no interior points in \u03c4\u22c6.A subset X, \u03c4, \u2110) be an ideal space. Then the following hold.(a)\u2110-regular pre-\u2110-open.The empty set is the only subset which is nowhere dense and pre-(b)A is pre-\u2110-regular pre-\u2110-closed, then every \u2110-rare set is pre-\u2110-open.If Let (A is nowhere dense and A is pre-\u2110-regular pre-\u2110-open. Then A = p\u2110int\u2061(p\u2110cl(A)) = p\u2110cl(A)\u2229int\u2061(cl\u22c6(p\u2110cl(A))), by A \u2282 p\u2110cl(A)\u2229int\u2061(cl\u22c6(cl(A)))\u2282p\u2110cl(A)\u2229int\u2061(cl(A)) = p\u2110cl(A)\u2229\u2205 = \u2205.(a) Suppose A is pre-\u2110-regular pre-\u2110-closed. Then A = p\u2110cl(p\u2110int\u2061(A)) = p\u2110int\u2061(A)\u222acl(int\u2061\u22c6(p\u2110int\u2061(A)))\u2282p\u2110int\u2061(A) \u222a cl(int\u22c6(A)) = p\u2110int\u2061(A) \u222a \u2205 = p\u2110int\u2061(A). Therefore, A = p\u2110int\u2061(A). Hence A is pre-\u2110-open.(b) Suppose X, \u03c4, \u2110) is called extremally pre-\u2110-disconnected if the pre-\u2110-closure of every pre-\u2110-open set is pre-\u2110-open.An ideal space  the following are equivalent.(a)X, \u03c4, \u2110) is extremally pre-\u2110-disconnected.((b)\u2110-regular pre-\u2110-open subset is pre-\u2110-regular.Every pre-For a topological space  is extremally pre-\u2110-disconnected. Suppose A is pre-\u2110-regular pre-\u2110-open. Then A is pre-\u2110-open and so p\u2110cl(A) is a pre-\u2110-open set. Hence A = p\u2110int\u2061(p\u2110cl(A)) = p\u2110cl(A). Hence A is pre-\u2110-closed which implies A is pre-\u2110-regular.(a) \u21d2 (b): assume (A is pre-\u2110-open. Then p\u2110cl(A) is pre-\u2110-regular pre-\u2110-closed which implies X \u2212 p\u2110cl(A) is pre-\u2110-regular pre-\u2110-open. Hence X \u2212 p\u2110cl(A) is pre-\u2110-regular. Therefore, X \u2212 p\u2110cl(A) is pre-\u2110-closed and so p\u2110cl(A) is pre-\u2110-open. Hence  is extremally pre-\u2110-disconnected.(b) \u21d2 (a): suppose X, \u03c4, \u2110) be an extremally pre-\u2110-disconnected space and A \u2282 X. Then the following are equivalent:(a)A is pre-\u2110-regular,(b)A = p\u2110cl(p\u2110int\u2061(A)),(c)X \u2212 A is pre-\u2110-regular pre-\u2110-open,(d)A is pre-\u2110-regular pre-\u2110-open.Let (A is pre-\u2110-regular. Then A is pre-\u2110-open and pre-\u2110-closed and so A = p\u2110int\u2061(A) and A = p\u2110cl(A). Hence A = p\u2110cl(p\u2110int\u2061(A)).(a) \u21d2 (b): suppose A = p\u2110cl(p\u2110int\u2061(A)). Then X \u2212 A = X \u2212 p\u2110cl(p\u2110int\u2061(A)) = p\u2110int\u2061(p\u2110cl(A)) so X \u2212 A is pre-\u2110-regular pre-\u2110-open.(b) \u21d2 (c): let (c) \u21d2 (d) is clear.(d) \u21d2 (a) follows from X, \u03c4, \u2110) is called locally pre-\u2110-indiscrete if every pre-\u2110-open subset of X is pre-\u2110-closed (or) if every pre-\u2110-closed subset of X is pre-\u2110-open.An ideal space  be an ideal space. Then the following are equivalent.(a)X, \u03c4, \u2110) is locally pre-\u2110-indiscrete.((b)\u2110-open subset is pre-\u2110-regular.Every pre-(c)\u2110-open subset is pre-\u2110-regular pre-\u2110-open.Every pre-(d)p\u2110int\u2061(p\u2110cl({x})) \u2260 \u2205, for every x \u2208 X.(e)X.The empty set is the only nowhere dense subset of Let  is locally pre-\u2110-indiscrete. Let A be a pre-\u2110-open subset of X. By hypothesis, A is pre-\u2110-closed. Hence A is pre-\u2110-regular.(a) \u21d2 (b): assume that . Also by hypothesis, A is pre-\u2110-closed. Therefore, A = p\u2110int\u2061(p\u2110cl(A)). Hence A is pre-\u2110-regular pre-\u2110-open.(b) \u21d2 (c): if x} is preopen, {x} is pre-\u2110-open. By (c), {x} is a pre-\u2110-regular pre-\u2110-open set. Therefore, {x} = p\u2110int\u2061(p\u2110cl({x})).(c) \u21d2 (d): since {\u2110-regular pre-\u2110-open.(d) \u21d2 (e): by A is a pre-\u2110-closed set. Now int\u2061(cl(A \u2212 p\u2110int\u2061(A))) = int\u2061(cl(A \u2212 (A\u2229int\u2061(cl\u22c6(A))))) = int\u2061(cl(A\u2229(X \u2212 A)\u222a(int\u2061(cl\u22c6(X \u2212 A))))) = int\u2061(cl(A\u2229(X \u2212 A)\u222a(A \u2212 int\u2061(cl\u22c6(A))))) = int\u2061(cl(A \u2212 int\u2061(cl\u22c6(A))))\u2282int\u2061(cl(A) \u2212 int\u2061(cl\u22c6(A))) = int\u2061(cl(A) \u2212 cl(int\u2061(cl\u22c6(A))))\u2282int\u2061(cl(A)\u2009\u2212\u2009(int\u2061(cl\u22c6(A)))) = \u2205. Therefore A \u2212 p\u2110int\u2061(A) is nowhere dense which implies A = p\u2110int\u2061(A), and so A is pre-\u2110-open. Hence is A locally pre-\u2110-indiscrete.(e) \u21d2 (a): suppose that X, \u03c4, \u2110) is said to be p\u2110R-door if every subset of X is either pre-\u2110-regular pre-\u2110-open or pre-\u2110-regular pre-\u2110-closed.An ideal space  be a p\u2110R-door space; then every pre-\u2110-open set in the space is pre-\u2110-regular pre-\u2110-open.Let ) which implies that p\u2110int\u2061A = p\u2110int\u2061(p\u2110int\u2061(p\u2110cl(A))). Since A is pre-\u2110-open, A = p\u2110int\u2061(p\u2110cl(A)). Hence A is pre-\u2110-regular pre-\u2110-open.Let X, \u03c4, \u2110) be an ideal space. A subset A of X is both \u03b1-\u2110-open and \u03b1-\u2110-closed; then A is a pre-\u2110-regular pre-\u2110-open set.Let  be an ideal space. A subset A of X is \u03b1-\u2110-open and pre-\u2110-regular pre-\u2110-open; then A = int\u2061(cl\u22c6(int\u2061(A))).Let (A is \u03b1-\u2110-open. Then A \u2282 int\u2061(cl\u22c6(int\u2061(A))). And A is pre-\u2110-regular pre-\u2110-open which implies A = p\u2110int\u2061(p\u2110cl(A))\u2283p\u2110int\u2061(cl\u22c6(int\u2061(p\u2110cl(A))))\u2283p\u2110int\u2061(cl\u22c6(int\u2061(A)))\u2283int\u2061(cl\u22c6(int\u2061(A))). Therefore A = int\u2061(cl\u22c6(A)).Suppose"}
{"text": "Based on our decomposition of stochastic processes and our asymptotic representations of Fourier cosine coefficients, we deduce an asymptotic formula of approximation errors of hyperbolic cross truncations for bivariate stochastic Fourier cosine series. Moreover we propose a kind of Fourier cosine expansions with polynomials factors such that the corresponding Fourier cosine coefficients decay very fast. Although our research is in the setting of stochastic processes, our results are also new for deterministic functions. From this, we see that hyperbolic cross approximation with polynomial factors can reconstruct the stochastic process on 2 by using the least Fourier cosine coefficients.Finally, based on our decomposition of stochastic processes, we propose Fourier cosine expansions with polynomial factors (see ) whose hThis paper is organized as follows. In We recall some concepts in calculus of stochastic processes and stochastic Fourier cosine series.\u03be, we denote its expectation, second-order moment, and variance by E[\u03be], E[\u03be2], and Var\u2061(\u03be), respectively. If \u03be(t) is a stochastic variable for each t \u2208 d, then we say \u03be(t) is a stochastic process on d. In this paper, we always assume that a stochastic process \u03be(t) is real-valued and satisfies E[\u03be2(t)] < \u221e for each t. This ensures that its expectation, variance, and second-order moment always exist. Calculus of stochastic processes is a generalization of classical calculus. Let {\u03ben}1\u221e be a sequence of stochastic variables and let \u03be be a stochastic variable. If lim\u2061n\u2192\u221e\u2061E[|\u03ben\u2212\u03be|2] = 0, we say that \u03be is the limit of {\u03ben}1\u221e. Starting from the concept of the limit, one defines continuity, derivatives, partial derivatives, integrals, and double integrals of stochastic processes  and the derivative \u03be\u2032 is continuous on , then\u03be(t) be a continuously differentiable stochastic process on  and let f(t) be a continuously differentiable deterministic function on . Then [For a stochastic variable rocesses . Moreove1]. Then (6)(\u03be(t)tions in . Xiu revtions in .\u03be(t) is a univariate stochastic process on  and E[\u222b01\u03be2(t)dt] < \u221e, then it can be expanded into the Fourier cosine seriesIf \u03be is a bivariate stochastic process on 2 and E\u03be2dt1dt2] < \u221e, then it can be expanded into the Fourier cosine seriesIf \u03be be a stochastic process on 2. Partial sums of its Fourier cosine series areSNr)((\u03be) is N2.Let SNh)((\u03be) is of order Nlog\u2061N. The hyperbolic cross approximations have been widely used in multivariate function approximation \u03be, denote its mixed derivative \u2202l1+l2\u03be/\u2202t1l1\u2202t2l2 by \u03bel1,l2) represents the set of continuous stochastic processes on 2. If \u03bel,l), then we say \u03be has the smoothness index l on 2.For a bivariate stochastic process an} and {bn} be two sequences. If K1 | an | \u2264|bn | \u2264K2 | an| for any n, then we say an ~ bn; if |an | \u2264K3 | bn| for any n, then we say an = O(bn); here Ki are constants independent of n. If an \u2192 0, bn \u2192 0, and an/bn \u2192 0 as n \u2192 \u221e, then we say an = o(bn). If an1,n2 \u2192 0 and bn1,n2 \u2192 0, and an1,n2/bn1,n2 \u2192 0 as an1,n2 = o.Let {In order to study stochastic Fourier cosine series, we give a decomposition of stochastic processes. Although this decomposition is given in the setting of stochastic processes, it is also new for deterministic functions.\u03be be a bivariate stochastic process on 2 and \u03bel,l) for some l \u2265 1. First, based on a fundamental polynomial p(t) = t2/2 \u2212 1/6, we construct a stochastic polynomial P\u03be as follows:\u03be at vertexes {0,1}2.Let Denote\u03be be a bivariate stochastic process on 2 with smoothness index l \u2265 1 and \u03be1 be stated in  = \u03be1p\u2032(1) = \u03be1\u2009\u2009(0 \u2264 t1 \u2264 1). Therefore, by \u03be3 = \u03be1 \u2212 \u03be2, we get \u03be3 = 0\u2009\u2009. Similarly, \u03be3 vanishes on other sides of the square 2.Consider the bottom side 0 \u2264 2 as follows.From  and 21)21), we g\u03be be a stochastic process on 2 and \u03bel,l) for some l \u2265 1. Then the decompositionP\u03be, \u03be2, and \u03be3 are stated in , a, a\u03be be atated in , respect\u03be is a stochastic process on  and the derivative \u03bel) for some l \u2265 2. Its Fourier cosine coefficients are as follows:E[\u03be\u2032\u2032(t)] \u2208 C, using the Riemann-Lebesgue lemma [rn) \u2264 E[rn2], we have Var\u2061(rn) = o(1/n4). By  = \u2211k=0N\u22121ck(\u03be)cos\u2061(\u03c0nt) satisfyE[||SN(\u03be)\u2212\u03be||22] = o(1/N3) if and only if \u03be\u2032(0) = \u03be\u2032(1) = 0.Suppose that ue lemma \u201312, we gn,wheregn=\u22122(n\u03c0)follows:cn(\u03be)=2\u222b0, we getE[rn2]\u2003=4\u03be is a bivariate stochastic process on 2 and \u03bel,l) for some l \u2265 2. We expand \u03be into the Fourier cosine series\u03bbn1,n2 is stated in , we will precisely compute the first two terms and estimate the expectation and variance of the last term on the right-hand side of cn1,n2(P\u03be), by the representation  at {0,1}2 and signs for addition and subtraction are determined by odevity of n1 and n2. Since \u222b01p(t)dt = 0, we get c0,0(P\u03be) = cn1,0(P\u03be) = cn20,(P\u03be) = 0.(i) For the first term entation  of stochcn1,n2(\u03be2), by (\u03be2 is the sum of products of separated variables.(ii) For the second term \u03be2), by , we know\u03be2 are equal tocn(\u03c6i) is the Fourier cosine coefficient of the univariate stochastic process \u03c6i and each cn(p) and each cn(p(1 \u2212 \u00b7)) are both Fourier cosine coefficients of univariate deterministic functions p and p(1 \u2212 \u00b7). A direct computation shows that, for n1 \u2260 0 and n2 \u2260 0,Denotecn1,n2(\u03be3), using the integration by parts, we deduce that\u03be3 = \u03be3 = 0\u2009\u2009(0 \u2264 t2 \u2264 1), the interior integral is equal to\u03be3 = 0\u2009\u2009(0 \u2264 t1 \u2264 1), we have \u03be3 = 0\u2009\u2009(0 \u2264 t1 \u2264 1). This implies that\u03be3 \u2208 Cs that E[\u03be3] is a continuous function on 2 and E[\u03be3\u03be3] is a continuous on 4. By the Riemann-Lebesgue lemma, we havecn1,n2(\u03be3)) \u2264 E.(iii) For the last term cn1,n2(\u03be) satisfies , (ii), and (iii), we getatisfies , where \u03b7tated in  and eachtated in .cn(\u03c6i).Now we further estimate these four univariate Fourier cosine coefficients \u03c6i be stated as in (cn(\u03c6i) satisfyLet each ed as in . Then Foi = 1, since \u03c61 = \u03be1. By \u03be1 = \u03be \u2212 P\u03be, we have \u03c61\u2032(0) = \u03c61\u2032(1) = 0. From this, we deduce that Fourier cosine coefficients satisfyBy similarity, we only prove the case We will deduce these asymptotic representations of Fourier cosine coefficients.\u03be be a stochastic process on 2 and \u03bel,l) for some l \u2265 2, and let \u03b7n1,n2 and \u03c6i be stated in ((i) for n1 \u2192 \u221e, cn1,n2(\u03be) = bn1,n2 + \u03c4n1,n2, where\u2009\u03c4n1,n2 satisfiesand (ii) for n2 \u2192 \u221e, \u2009and (iii) for n1 \u2192 \u221e and n2 \u2192 \u221e, cn1,n2(\u03be) = (4/\u03c02n12n22)\u03b7n1,n2 + \u03c4n1,n2* and \u03c4n1,n2* satisfiesLet tated in  and 38)\u03be be a stn1 \u2192 \u221e, denotecn1,n2(\u03be) = bn1,n2 + \u03c4n1,n2. Using E = o(1/n12)(1/n22) andWhen (\u03be3).By , cn1,n2From this, we can give asymptotic representations of expectation, second-order moment, and variance of Fourier cosine coefficients.(i) for n1 \u2192 \u221e and n2 \u2260 0,\u2009bn1,n2 is stated in  for n2 \u2192 \u221e and n1 \u2260 0,\u2009o\u201d is uniform for n1.where tated in  and \u201co\u201d Under conditions of By similarity, we only prove .n1 \u2192 \u221e, \u2009\u2009n2 \u2260 0,From o we get .l increases.From \u03be be a stochastic process on 2 and \u03bel,l) for some l \u2265 2. We expand \u03be into the Fourier cosine series:Let SNr)((\u03be) of Fourier cosine series ((i)PN. The first term PN on the right-hand side of the formula  and  and , we haveK1 > 0 such thatn2=1\u221eE[(bn2(1))2 + (bn2(2))2] converges, and denote its sum by A. SoBy , we deduPk21 = (A/12\u03c04k3) + o(1/k3). Notice thatE = O(1/n14n24) (see (Pk+121 = Pk21 + o(1/k3). This implies thatbn2(1) and bn2(2) are stated in  (see ). We havtated in .o\u201d is uniform for n1 andE = O(1/n14n24), we have PN3 = o(1/N3). Finally, by QN and RN in cn1By (01p(t)dt = 0, we have cn1,0(P\u03be) = 0. By (78)\u2003PN2=(\u03be3).By  and \u222b01 = 0. By , we haveSimilar to , we havegain, by , we haveplies by  that for some l \u2265 2. Then the partial sums SN(\u03be) of its Fourier cosine series satisfyM = (1/6\u03c04)(A + B + 4C + 4D) and A, B, C, D are stated in ,  for some l \u2265 2. We consider hyperbolic cross truncations of its Fourier cosine series:JN = \u2211n2=1N\u22121\u2211n1=[(N\u22121)/n2]\u221eE. We rewrite JN in the formSuppose that tity and , 86), a, a\u03be is atity and , we haveJN1. By o\u201d is uniform for n2 and\u03bcn2 = \u2212(1/\u03c02n22)\u2009\u2009E[\u03c91(cn2(\u03c61) \u2212 cn2(\u03c62))] + E[(cn2(\u03c61) \u2212 cn2(\u03c62))2].We first consider E[(cn2(\u03c61) \u2212 cn2(\u03c62))2] \u2264 2E[cn22(\u03c61) + cn22(\u03c62)] = o(1/n24). Therefore, \u03bcn2 = o(1/n24). From this, we haven2 \u2264 N,By using the Schwarz inequality in Probability theory,tituting  and 97)(96)|E ~ (log\u2061N/N3). This result cannot be improved as the smoothness index l increases.From SNh)((\u03be) isNc ~ log\u2061N. From this, we getBy , the numSNr)((\u03be) is Nc = N2, by Since the number of Fourier cosine coefficients in the partial sum \u03be is a stochastic process on 2 and \u03bel,l) for some l \u2265 3. By using notation  into bivariate Fourier cosine series. Finally, we obtain the Fourier cosine expansion of \u03be with polynomial factors:p(t) = (t2/2) \u2212 (1/6). Notice that \u03c61(t2) = \u03be1 and \u03be1 \u2208 Cs, where \u03be1 = \u03be \u2212 P\u03be. Using \u03c61\u2032\u2032\u2032 \u2208 Cs and \u03c61\u2032(0) = \u03c61\u2032(1) = 0, and soE[cn(\u03c61)] = o(1/n3) andcn(\u03c62),cn(\u03c63), and cn(\u03c64), we getIn order to use the least Fourier cosine coefficients to reconstruct stochastic processes, we introduce the Fourier cosine expansion with polynomial factors. Suppose that notation , the decnotation  can be rcn1,n2(h3) decay fast. By \u03bel,l)\u2009\u2009(l \u2265 3) and Now we show that bivariate Fourier cosine coefficients SN(\u03c6i) is the first N terms partial sums of Fourier cosine series of \u03c6i and SNh)((\u03be3) is the hyperbolic cross truncation of Fourier cosine series of bivariate stochastic process of \u03be3 which is stated in (TNh)((\u03be) is a combination of stochastic polynomials and cosine polynomials. By (n=1kan)2 \u2264 k2\u2211n=1kan2, we get\u03be3 = \u03be3 = \u03be3 = \u03be3 = 0. Based on (E[||TNh)((\u03be)\u2212\u03be||22] = o((log\u2061N)/N5). Noticing that the number of Fourier cosine coefficients in TN : Nc ~ Nlog\u2061N, we get the following.Take the hyperbolic cross truncations of the expansion ,117)TNtated in . The hypials. By  and  for some l \u2265 3. ThenNc is the number of Fourier cosine coefficients in TNh)((\u03be).Suppose that Comparing \u03be be a stochastic process on 2 and \u03bel,l) for some l \u2265 3. ThenSNr)((\u03be) and SNh)((\u03be) are partial sums and hyperbolic cross truncations of Fourier cosine series of \u03be, respectively, TNh)((\u03be) is the hyperbolic cross truncations of Fourier expansion of \u03be with polynomial factors, and Nc is the number of Fourier cosine coefficients in each sum.Let \u03be by TNh)((\u03be), we need the least Fourier cosine coefficients.From this, we see that if we reconstruct"}
{"text": "E-inversive semigroups. We prove that group fuzzy congruences and normal fuzzy subsemigroups determined each other in E-inversive semigroups. Moreover, we show that the set of group t-fuzzy congruences and the set of normal subsemigroups with tip t in a given E-inversive semigroup form two mutually isomorphic modular lattices for every t\u2208 .The aim of this paper is to investigate the lattices of group fuzzy congruences and normal fuzzy subsemigroups on  The investigation of fuzzy sets is initiated by Zadeh in . As specSeveral authors investigated fuzzy congruences for some special classes of semigroups. In 2000, Zhang  characteE-inversive semigroups is a very wide class of semigroups which contains groups, inverse semigroups, and regular semigroups as proper subclasses and some kinds of crisp congruences on this class of semigroups have been investigated extensively; see  the set of group t-fuzzy congruences and the set of normal fuzzy subsemigroups with tip t on S form two mutually isomorphic modular lattices. Our results generalize and enrich several results obtained in . A fuzzy set \u03bc in a set S is said to be contained in a fuzzy set \u03b7 if \u03bc(x) \u2264 \u03b7(x) for all x in S and this is denoted by \u03bc\u2286\u03b7. The union \u03bc \u222a \u03b7 and the intersection \u03bc\u2229\u03b7 of two fuzzy sets \u03bc and \u03b7 in a set S are defined byx in S. Further, if \u03bci is a fuzzy subset in S for i \u2208 I where I is an index set, then \u22c2i\u2208I\u03bci is defined byx \u2208 S.Zadeh  defined  semigroup is a nonempty set with an associative binary operation. A semigroup S is called E-inversive if for all a \u2208 S there exists a\u2032 \u2208 S such that a\u2032aa\u2032 = a. In this case, we denote W(a) = {a\u2032 \u2208 S | a\u2032aa\u2032 = a\u2032} and call the elements in W(a) the weak inverses of a for any a \u2208 S. It is easy to see that groups, regular semigroups, and semigroups with zeros are all E-inversive semigroups. For more details on E-inversive semigroups, see . A t-fuzzy equivalence on S is a fuzzy subset in S \u00d7 S which satisfies the following conditions: a \u2208 S)\u2009\u2009\u03c1 = t,\u2009\u2009\u03c1 = \u03c1 \u2264 t,\u2009\u2009\u03c1 \u2265 \u03c1\u2227\u03c1.a, b, c \u2208 S)\u2009\u2009\u03c1 \u2264 \u03c1\u2227\u03c1,a, b, c, d \u2208 S)\u2009\u2009\u03c1 \u2265 \u03c1\u2227\u03c1. = \u03c1 for all x \u2208 S.\u03c1a = \u03c1b if and only if \u03c1 = t for all a, b \u2208 S.S/\u03c1 = {\u03c1a | a \u2208 S} is a semigroup with the multiplication \u03c1a\u03c1b = \u03c1ab for any a, b \u2208 S.Let S. In particular, we show that the set of group t-fuzzy congruences on S forms a modular lattice. Firstly, we give the concept of group t-fuzzy congruences which is parallel to that of usual group fuzzy congruences defined in Kuroki GFCt(S).A t-fuzzy congruences on S.The following result provides a characterization of group t-fuzzy congruence \u03c1 on S is a group t-fuzzy congruence if and only if e, f \u2208 E(S))\u2009\u2009\u03c1 = t;(\u2200a \u2208 S)(\u2200a\u2032 \u2208 W(a))\u2009\u2009\u03c1 = t. . This implies that \u03c1 = t for all e, f \u2208 E(S) by a\u2032a \u2208 E(S) for any a \u2208 S and a\u2032 \u2208 W(a), it follows that \u03c1a\u2032a is the identity of S/\u03c1 and so \u03c1a = \u03c1a\u03c1a\u2032a = \u03c1aa\u2032a. This yields that \u03c1 = t by If a \u2208 S, a\u2032 \u2208 W(a), and e \u2208 E(S). Then by condition (1) and \u03c1aa\u2032 = \u03c1a\u2032a = \u03c1e. By condition (2) and S/\u03c1 is a group.Conversely, let \u03c1 \u2208 GFCt(S), a, b \u2208 S, and e \u2208 E(S). ThenLet \u03c1 \u2208 GFCt(S), by a\u2032 \u2208 W(a) and b\u2032 \u2208 W(b). This implies that\u03c1 = \u03c1. By similar arguments, we can showa, b \u2208 S and e \u2208 E(S).Since \u03c3, \u03c4 \u2208 GFCt(S), we define \u03c3\u2218\u03c4 as follows:a, b \u2208 S. Then we have the following.As usual, for \u03c3, \u03c4 \u2208 GFCt(S). \u03c3\u2218\u03c4 = \u03c4\u2218\u03c3.\u03c3\u2218\u03c4 is the least group t-fuzzy congruence of\u2009\u2009S containing \u03c3 and \u03c4. \u03c3\u2229\u03c4 is the greatest group t-fuzzy congruence of S contained in \u03c3 and \u03c4.Let a, b \u2208 S and c\u2032 \u2208 W(c), by \u03c3\u2218\u03c4 = \u03c4\u2218\u03c3 for all a, b \u2208 S.(1) For all a, b, c \u2208 S. Since\u03c3\u2218\u03c4 = t. Similarly, we have\u03c3\u2218\u03c4 is the least t-fuzzy congruence on S containing \u03c3 and \u03c4. Finally, we can easily show that \u03c3\u2218\u03c4 \u2208 GFCt(S) by \u03c3\u2218\u03c4 is the least group t-fuzzy congruence of S containing \u03c3 and \u03c4.(2) Let (3) This is clear.By GFCt(S), \u2286) forms a modular lattice for any t in . and a, b \u2208 S. \u03c1 = \u03c1 for all a\u2032, a* \u2208 W(a). \u03c1(a\u2032b(a\u2032b)\u2032, a\u2032b) = \u03c1 for all a\u2032 \u2208 W(a) and (a\u2032b)\u2032 \u2208 W(a\u2032b). Let \u03c1 \u2208 GFCt(S), by \u03c1 \u2265 \u03c1 = t whence \u03c1 = t. Thus,\u03c1 = \u03c1.(1) Since a\u2032b(a\u2032b)\u2032, aa\u2032 \u2208 E(S), by \u03c1(a\u2032b(a\u2032b)\u2032, a\u2032b) = \u03c1.(2) In view of the fact that E-inverse semigroups.In this section, we consider some basic properties of normal fuzzy subsemigroups of t \u2208 . A fuzzy subset \u03bc in S is called a normal\u2009\u2009fuzzy\u2009\u2009subsemigroup\u2009\u2009with\u2009\u2009tip\u2009\u2009t in S if x, y \u2208 S)\u2009\u2009\u03bc(xy) \u2265 \u03bc(x)\u2227\u03bc(y),\u2009\u2009t \u2265 \u03bc(a) \u2265 \u03bc(xay)\u2227\u03bc(xy),(\u2200e \u2208 E(S))\u2009\u2009\u03bc(e) = t.(\u2200Let t in S by NFSt(S) and let NFS(S) = \u22c3t\u2208NFSt(S).We denote the set of normal fuzzy subsemigroups with tip In fact, normal fuzzy subsemigroups with tip 1 are introduced in Zhang  where thG be a group with identity e, t \u2208  and let \u03bc be a fuzzy set in G. From Ajmal and Thomas  and let \u03bc be a fuzzy subset in G. Then \u03bc is a normal fuzzy subsemigroup with tip t of G if and only if \u03bc is a normal fuzzy subgroup of G with tip t.Let e is the unique idempotent in G and the inverse a\u22121 of a is certainly the unique weak inverse of a for all a \u2208 G. If \u03bc is a normal fuzzy subsemigroup with tip t of G, then by \u03bc is a normal fuzzy subgroup of G with tip t. Conversely, let \u03bc be a normal fuzzy subgroup of G with tip t and a, x, y \u2208 S. Then\u03bc is a normal fuzzy subsemigroup of G with tip t.Observe that ondition . This imi\u2208I\u03c3i \u2208 NFSt(S) for \u03c3i \u2208 NFSt(S), i \u2208 I, and the elementNFSt(S), we have the following theorem by Since \u22c2NFSt(S), \u2286) forms a lattice., e, f \u2208 E(S), a, b \u2208 S, and a\u2032, a* \u2208 W(a), b\u2032 \u2208 W(b).\u03bc(ef) = t. \u03bc(a\u2032b) = \u03bc(b\u2032a). \u03bc(a\u2032b) = \u03bc(a*b). Let e, f \u2208 E(S), we have \u03bc(e) = \u03bc(f) = t. This implies that \u03bc(ef) \u2265 \u03bc(e)\u2227\u03bc(f) = t whence \u03bc(ef) = t.(2)The result follows from the facts that(3)This follows from the proof of Theorem 2.12 in Zhang .(1) Since GFCt(S) is isomorphic to NFSt(S) as lattices whence NFSt(S) is modular for all t in . We first give some useful propositions.In this section, we show that \u03bc \u2208 NFSt(S) and\u03c1\u03bc \u2208 GFCt(S), where a\u2032 \u2208 W(a).Let \u03c1\u03bc is well defined. Now, let a, b, c \u2208 S and a\u2032 \u2208 W(a), b\u2032 \u2208 W(b), c\u2032 \u2208 W(c). Then we have the following facts:(1)a\u2032a \u2208 E(S), \u03c1\u03bc = \u03bc(a\u2032a) = t.Since (2)By In view of b\u2032b \u2208 E(S), it follows that \u03bc(b\u2032b) = t and(3) Since ac)\u2032 \u2208 W(ac), we have ac(ac)\u2032 \u2208 E(S) and\u03bc(a\u2032b) \u2264 \u03bc((ac)\u2032bc). Dually, we have \u03bc(a\u2032b) \u2264 \u03bc((ca)\u2032cb). This implies that\u03c1\u03bc \u2264 \u03c1\u03bc\u2227\u03c1\u03bc.(5)e \u2208 W(e), we have \u03c1\u03bc = \u03bc(ef) = t.By (6)\u03c1\u03bc = \u03bc(a\u2032aa\u2032a) = \u03bc(aa\u2032) = t.From the above six items, we can see that \u03c1\u03bc \u2208 GFCt(S) by (4) For any (\u03c1 \u2208 GFCt(S) and define\u03bc\u03c1 \u2208 NFSt(S), where a\u2032 \u2208 W(a).Let \u03bc\u03c1 is well defined. Now, let\u03bc\u03c1(xy) \u2265 \u03bc\u03c1(x)\u2227\u03bc\u03c1(y). On the other hand, also by t \u2265 \u03c1 = \u03bc\u03c1(a) \u2265 \u03bc\u03c1(xay)\u2227\u03bc\u03c1(xy). Finally, since e \u2208 W(e) for all e \u2208 E(S), we have \u03bc\u03c1(e) = \u03c1 = \u03c1 = t for all e \u2208 E(S). Therefore, \u03bc\u03c1 \u2208 NFSt(S).By At this stage, we can give the main result of this paper.\u03bc\u03c1 and \u03c1\u03bc are defined as in Propositions The mappings\u03bc \u2208 NFSt(S), a \u2208 S, and a\u2032 \u2208 W(a). Then aa\u2032 \u2208 W(aa\u2032). This implies thata, b \u2208 S and a\u2032 \u2208 W(a), (a\u2032b)\u2032 \u2208 W(a\u2032b), we havet and \u03a6t are mutually inverse bijections. Obviously, \u03a8t and \u03a6t preserve the inclusion relations.From Propositions GFCt(S), \u2286) and (NFSt(S), \u2286) are isomorphic. As a consequence, the lattice (NFSt(S), \u2286) is also modular.The lattices (GFCt(S), \u2286) is isomorphic to the lattice (NFSt(S), \u2286). Thus, the lattice (NFSt(S), \u2264) is also modular by By GFC(S), \u2286) and (NFS(S), \u2286) are two mutually isomorphic lattices.(\u03c1i \u2208 GFCti(S), i \u2208 I, and \u03c3j \u2208 NFStj(S), j \u2208 J, where I and J are index sets. Moreover,GFC(S), \u2286) and (NFS(S), \u2286), respectively. By GFC(S), \u2286) and (NFS(S), \u2286) are two lattices. Moreover, if we letGFC(S), \u2286) and (NFS(S), \u2286) are isomorphic from It is routine to check thatWe end this section by giving an example to illustrate our previous results.S be a semigroup with the following multiplication table:S is an E-inversive semigroup which is nonregular. Moreover,Let t in the interval . For every s \u2208  with s \u2264 t, define a fuzzy set \u03bcs in S as follows:\u03bcs is a normal fuzzy subsemigroup with tip t in S. Furthermore, in view of the fact that\u03bcs \u2208 NFSt(S). By \u03c1\u03bcs satisfies that\u03c1\u03bcs \u2208 GFCt(S). By GFCt(S) = {\u03c1\u03bcs | s \u2264 t, s \u2208 }. By virtue of GFCt(S), \u2286) and (NFSt(S), \u2286) are isomorphically modular lattices.Fix an element E-inversive semigroups. Our results generalize the corresponding results of groups and regular semigroups. From the results presented in the paper, the lattices of group fuzzy congruences and normal fuzzy subsemigroups on E-inversive semigroups can be regarded as a source of possibly new modular lattices. On the other hand, this paper also leaves some questions which can be considered as future works. For example, from Ajmal and Thomas [S is a group, then (NFS(S), \u2286) is a modular lattice. Thus, the following question would be interesting: is (NFS(S), \u2286) also modular for an E-inversive semigroup S?In this paper, we have introduced and investigated the lattices of group fuzzy congruences and normal fuzzy subsemigroups on d Thomas , if S is"}
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{"text": "R0 algebras and pseudo-R0 algebras are obtained, and the mutually independence of axioms is proved. We introduce the notions of filters and normal filters in pseudo-weak-R0 algebras. The structures and properties of the generated filters and generated normal filters in pseudo-weak-R0 algebras are obtained. These can be seen as noncommutative generalizations of the corresponding ones in weak-R0 algebras.The most simplified axiom systems of pseudo-weak- Flondor et al.  = 0.97\u2228(0.97\u21dd0.9) \u2264 0.97\u22280.96 = 0.97 < 1.(v)A is a bounded lattice given by Suppose that Then \u2212, \u2009~, \u2192, and \u21dd on A are defined by the following:The operations \u2009A satisfies (P1)\u2013(P3) and (P5), but not (P4): (a \u2192 b)\u2228(a \u2192 c) = (a\u2212\u2228b)\u2228(a\u2212\u2228c) = (d\u2228b)\u2228(d\u2228c) = b\u2228a = e, but a \u2192 b\u2228c = a \u2192 e = 1.Then R0 algebras and investigate the structures and properties of the generated filters and generated normal filters in pseudo-weak-R0 algebras.We introduce the notions of filters and normal filters in pseudo-weak-F of a pseudo-weak-R0 algebra A is said to be a filter of A if it satisfies(F1)x, y \u2208 F\u21d2x\u2299y \u2208 F,(F2)x \u2208 F, x \u2264 y\u21d2y \u2208 F.A nonempty subset F of a pseudo-weak-R0 algebra A, the following are equivalent:F is a filter,F and x, x\u21ddy \u2208 F\u21d2y \u2208 F,1 \u2208 F and x, x \u2192 y \u2208 F\u21d2y \u2208 F.1 \u2208 For a subset F. By (F1), x, x\u21ddy \u2208 F\u21d2x\u2299(x\u21ddy) \u2208 F. By (38) and (F2), x\u2227y \u2208 F, and so y \u2208 F.(i)\u21d2(ii). By (F2), we have 1 \u2208 x, x \u2192 y \u2208 F, by (19), x \u2264 (x \u2192 y)\u21ddy. By (4), x\u21dd((x \u2192 y)\u21ddy) = 1 \u2208 F. By (ii), y \u2208 F.(ii)\u21d2(iii). If x \u2208 F, x \u2264 y, then x \u2192 y = 1 \u2208 F, so y \u2208 F; that is, (F2) holds; if x, y \u2208 F, by (41), x \u2192 (y \u2192 (x\u2299y)) = (x\u2299y)\u2192(x\u2299y) = 1 \u2208 F, and so x\u2299y \u2208 F, which means (F1) holds.(iii)\u21d2(i). If A are both filters of a pseudo-weak-R0 algebra A.Clearly, {1} and For a subset F of a pseudo-weak-R0 algebra A, the following are equivalent:F is a filter,x, y \u2208 F, y \u2264 x \u2192 z\u21d2z \u2208 F,x, y \u2208 F, y \u2264 x\u21ddz\u21d2z \u2208 F.x, y \u2208 F, y \u2264 x \u2192 z, by (F2) and z \u2208 F. Conversely, if x \u2208 F, by x \u2264 x \u2192 1, we have 1 \u2208 F; suppose that x, x \u2192 y \u2208 F, by x \u2192 y \u2264 x \u2192 y, we have y \u2208 F. By F is a filter.(i)\u21d4(ii). If (i)\u21d4(iii). Similarly.A is also a filter. If S\u2286A, the least filter containing S; that is, the intersection of all filters of A containing S is called the filter generated by S and denoted by [S). If S = {a}, [{a}) is written [a). ClearlyNext, we consider filter generated by a set. It is easy to verify that the intersection of filters of A be a pseudo-weak-R0 algebra and let S be a nonempty subset of A. ThenLet B denote the right side of the first equality. If x, y \u2208 B, then there are a1, a2,\u2026, an, b1, b2,\u2026, bm \u2208 S such that a1\u2299\u22ef\u2299an \u2264 x and b1\u2299\u22ef\u2299bm \u2264 y. By (37), a1\u2299\u22ef\u2299an\u2299b1\u2299\u22ef\u2299bm \u2264 x\u2299y, so x\u2299y \u2208 B. If x \u2208 B and x \u2264 y, we have a1\u2299\u22ef\u2299an \u2264 x \u2264 y, so y \u2208 B. Hence B is a filter. If C is a filter and S\u2286C, for any x \u2208 B, there are a1, a2,\u2026, an \u2208 S such that a1\u2299\u22ef\u2299an \u2264 x. By (F2), x \u2208 C, hence B\u2286C.Only prove the first equality. Using (34) to the first equality, we can get the rest of the two equalities. Let a0 : = 1; For convenience, we shall write A is a pseudo-weak-R0 algebra and a \u2208 A, thenIf Let F be a filter of a pseudo-weak-R0 algebra A and a \u2208 A; thenF be a filter of a pseudo-weak-R0 algebra A and a, b \u2208 A; thenLet x \u2208 [F \u222a {a})\u2229[F \u222a {b}), by n1,\u2026, nm, l1,\u2026, lk \u2265 0, s1,\u2026, sm, t1,\u2026, tk \u2208 F such thatp = s1\u2299\u22ef\u2299sm\u2299t1\u2299\u22ef\u2299tk and q = max\u2061{n1,\u2026, nm, l1,\u2026, lk}, and thenx \u2265 (p\u2299aq)m\u2228(p\u2299bq)k \u2265 ((p\u2299aq)m\u2228(p\u2299bq))k \u2265 ((p\u2299aq)\u2228(p\u2299bq))mk = (p\u2299(aq\u2228bq))mk \u2265 (p\u2299(a\u2228b)q2)mk. x \u2208 [F \u222a {a\u2228b}). Hence [F \u222a {a})\u2229[F \u222a {b})\u2286[F \u222a {a\u2228b}). Inverse contains is obvious.Assume that F be a filter of a pseudo-weak-R0 algebra A and a, b \u2208 A. If a\u2228b \u2208 F, thenLet A be a pseudo-weak-R0 algebra and a, b \u2208 A; then [a)\u2229[b) = [a\u2228b).Let F = {1} in Taking R0 algebra.Next we introduce the notion of normal filters in a pseudo-weak-F of a pseudo-weak-R0 algebra A is called normal if x, y \u2208 A, x \u2192 y \u2208 F if and only if x\u21ddy \u2208 F.A filter F be a normal filter of a pseudo-weak-R0 algebra A. Then there is s \u2208 F such that b\u2299s \u2264 c if and only if there is t \u2208 F such that t\u2299b \u2264 c.Let s \u2208 F such that b\u2299s \u2264 c, by (34), s \u2264 b\u21ddc. By s \u2208 F, we have b\u21ddc \u2208 F, and so b \u2192 c \u2208 F. Put b \u2192 c = t \u2208 F, and then t\u2299b \u2264 c. Converse is similar.If there is If F is a normal filter of a pseudo-weak-R0 algebra A and a \u2208 A, thent2 \u2208 F such thatt2,\u2026, tm \u2208 F such thats = tm + \u22ef+t2 + s1 \u2208 F and n = n1 + \u22ef+nm, we have s\u2299an \u2264 x. That is that the first equality holds.We show the first equality. By By the first equality and If F is a normal filter of a pseudo-weak-R0 algebra A and a \u2208 A, thenBy s \u2208 F such that s\u2299an \u2264 x, if and only if there is s \u2208 F such that s \u2208 F such that Since there is s \u2208 F such that an\u2299s \u2264 x, if and only if there is s \u2208 F such that s \u2208 F such that Similarly, byIf F is a normal filter of a pseudo-weak-R0 algebra A and a \u2208 A, thens \u2208 F such that s\u2299an \u2264 x, if and only if there is s \u2208 F such that s \u2264 an \u2192 x, if and only if an \u2192 x \u2208 F. There is s \u2208 F such that an\u2299s \u2264 x, if and only if there is s \u2208 F such that s \u2264 an\u21ddx, if and only if an\u21ddx \u2208 F. By There is R0 algebras and pseudo-R0 algebras and proved the mutually independence of axioms. We introduced the notions of filters and normal filters in pseudo-weak-R0 algebras and gave the structures and properties of the generated filters and generated normal filters in pseudo-weak-R0 algebras. These will be conducive to further study pseudo-weak-R0 algebras  and pseudo-R0 algebras . In the future, we will investigate relations between various kinds of filters of pseudo-logic algebras. We may also study fuzzy type of filters of pseudo-weak-R0 algebras and pseudo-R0 algebras.We obtained the most simplified axiom systems of pseudo-weak-"}
{"text": "In this paper, generalized synchronization (GS) is extended from real space to complex space, resulting in a new synchronization scheme, complex generalized synchronization (CGS). Based on Lyapunov stability theory, an adaptive controller and parameter update laws are designed to realize CGS and parameter identification of two nonidentical chaotic (hyperchaotic) complex systems with respect to a given complex map vector. This scheme is applied to synchronize a memristor-based hyperchaotic complex L\u00fc system and a memristor-based chaotic complex Lorenz system, a chaotic complex Chen system and a memristor-based chaotic complex Lorenz system, as well as a memristor-based hyperchaotic complex L\u00fc system and a chaotic complex L\u00fc system with fully unknown parameters. The corresponding numerical simulations illustrate the feasibility and effectiveness of the proposed scheme. By taking the derivative of J(\u03d5)\u2208Cm\u00d7n is the Jacobian matrix of \u03d5(x), and J(\u03d5) = Jr(\u03d5) + jJi(\u03d5). By substituting Eqs Define the complex CGS error vector asere e = e, e2, \u22efemTherefore, the problem of CGS for two nonidentical complex systems  and 2)  2) is trTheorem 1 For a given complex map vector \u03d5(x), CGS and parameter identification of the response system , K\u03b8 = diag and K\u03b4 = diag are error feedback control strength whose elements are all positive constants, which can adjust converging velocity.e system  and the Proof We introduce a positive Lyapunov function asV(t) along the trajectories of the error dynamical system  and Based on Lyapunov stability theory, since x1, x2\u2208C, x3, x4\u2208R, x1, x2. a1, a2, a3 and a4 are unknown real parameters, \u03b11 and \u03b21 are considered as known positive constants. When \u03b11 = 4, \u03b21 = 0.01, a1 = 36, a2 = 20, a3 = 3.2, a4 = 3 and x(0) = T, the Lyapunov exponents of system  = T, the system +u4where y1\u03b8 = T, \u03b4 = T, u = T, andThe drive system  and respThe complex map vector is given byThe Jacobian matrix of the complex map vector is calculated ase1,r = y1,r \u2212 x1,r+x2,i, e1,i = y1,i \u2212 x1,i \u2212 x2,r, e2,r = y2,r \u2212 2x2,r \u2212 x2,i, e2,i = y2,i \u2212 2x2,i+x2,r, e3 = y3 \u2212 x3 \u2212 x4, According to Eqs \u03b11 = 4, \u03b21 = 0.01, \u03b12 = 0.67 \u00d7 10\u22123, \u03b22 = 0.02 \u00d7 10\u22123, the true values of unknown parameters are \u03b8 = T, \u03b4 = T. The initial conditions of system (x(0) = T, y(0) = T. The initial values of all unknown parameters are randomly chosen as zero, and the control strength is set as K = diag, K\u03b8 = diag, K\u03b4 = diag. The corresponding simulation results are shown in Figs In order to verify the validity and effectiveness of CGS between system  and 13)13) with f system  and (13)= 8, 11, 0, 8/3T. x1, x2\u2208C, x3\u2208R, c1, c2 and c3 are unknown real parameters. When c1 = 27, c2 = 23, c3 = 1, and x(0) = T, the complex Chen system \u2212c3x3where x1n system , is also\u03b8 = T, \u03b4 = T, u = T, andThe drive system  and respThe complex map vector is given byThe Jacobian matrix of the complex map vector is calculated ase1,r = y1,r \u2212 x1,i, e1,i = y1,i+x1,r, e2,r = y2,r \u2212 x2,i, e2,i = y2,i+x2,r, e3 = y3+x3, e3 = y4 \u2212 x3.According to Eqs \u03b12 = 0.67 \u00d7 10\u22123, \u03b22 = 0.02 \u00d7 10\u22123, the true values of unknown parameters \u03b8 = T, the initial conditions of system (x(o) = T, y(0) = T, the initial values of unknown parameters K = diag, K\u03b8 = diag, K\u03b4 = diag. The corresponding simulation results are shown in Figs y1,r, y2,r, y4 are synchronized with x1,i, x2,i, x3, and y1,i, y2,i, y3 are antisynchronized with x1,r, x2,r, x3. The synchronization errors, as shown in Numerical simulations are presented to verify the validity and effectiveness of CGS between systems  and 13)13) with ters \u03b8 = ,23,1T, \u03b4f system  and (13)y1, y2\u2208C, y3\u2208R, d1, d2 and d3 are unknown real parameters, u1, u2 and u3 are controllers. When d1 = 29, d2 = 21, d3 = 2, and y(0) = T, the complex L\u00fc system \u2212d3y3+u3where y1\u03b8 = T, \u03b4 = T, u = T, andThe drive system  and respThe complex map vector is given byThe Jacobian matrix of the complex map vector is calculated asAccording to Eqs \u03b11 = 4, \u03b21 = 0.01, \u03b8 = T, \u03b4 = T, y(0) = ,K = diag, K\u03b8 = diag, K\u03b4 = diag. The CGS process is plotted in y1,r, y2,r are antisynchronized with x2,i, x1,i, y1,i, y2,i are synchronized with x2,r, x1,r, and y3 is synchronized with In order to verify the validity and effectiveness of CGS between Systems  and 25)25) with 3]T, \u03b4 = ,21,2T, xThis paper investigates a novel synchronization scheme named complex generalized synchronization, and its application to synchronization and parameter identification of two nonidentical complex nonlinear systems with fully unknown parameters. An adaptive controller and a parameter estimator are proposed and proved theoretically based on Lyapunov stability theory. Three illustrative examples are presented to verify the correctness and effectiveness of the proposed scheme, namely, CGS of a memristor-based hyperchaotic complex L\u00fc system and a memristor-based chaotic complex Lorenz system, CGS of a chaotic complex Chen system and a memristor-based chaotic complex Lorenz system, as well as CGS of a memristor-based hyperchaotic complex L\u00fc system and a chaotic complex L\u00fc system. The proposed CGS scheme has some advantages, for instance, it can be applied to synchronize complex systems with different orders , can be transformed to other types of synchronization with different given complex map vectors (feasibility), can be achieved in a short time with the appropriate control strength (timelines), and can be almost impossibly predicted with the complex map vector (security). So, CGS has extensively potential applications to secure communication, digital cryptography, and so on, which will be involved in our future works."}
{"text": "There are errors in the published paper. \u201cSection 2.4.1\u201d should be labelled as subsection \u201cDisturbance patterns\u201d and \u201citem 2.2\u201d should be labelled as subsection \u201cForest inventory.\u201d"}
{"text": "Under some hypotheses, it is proved that the boundary value problem has a unique solution.We mainly deal with the boundary value problem for triharmonic function with value in a universal Clifford algebra: \u0394 It iA and Aj are invertible constants; we denote the inverse elements as A\u22121 and Aj\u22121.\u2009\u2009u(x), (Dju)(x), g(x), fj(x) \u2208 H\u03b2), j = 1,\u2026, 5, 0 < \u03b2 \u2264 1. The explicit solutions for (In  and 2),,2), A anVn,s (0 \u2264 s \u2264 n) be an n-dimensional (n \u2265 1) real linear space with basis {e1, e2,\u2026, en}, and Cl the universal Clifford algebra over Vn,s. For more information on Cl (0 \u2264 s \u2264 n), we refer to  = 0 in Rn; then, for any x \u2208 Rn,\u03c9n denotes the area of the unit sphere in Rn.Suppose \u03943[u] = 0 in Rn and |u(x)| = O(|x|) (|x | \u2192\u221e); then \u0394[u] = 0 and \u03942[u] = 0 in Rn.Suppose \u03943[u] = 0 in Rn and u(x) is bounded in Rn; then u(x) \u2261 C.Suppose \u03943[u] = 0 in Rn\u2216\u2202\u03a9, and for x \u2208 \u2202\u03a9, u \u2208 C6), [u]+(x) = [u]\u2212(x) \u2208 C5), and, moreover, Dj[u]+(x) = Dj[u]\u2212(x) \u2208 H\u03b1j), 0 < \u03b1j \u2264 1, where j = 0,1,\u2026, 5. Then \u03943[u] = 0 in Rn.Suppose \u03943[u] = 0 in \u03a9\u2212, Dj[u](x) \u2208 H\u03b1j), 0 < \u03b1j \u2264 1 , and |u(x)| = O(1) (|x| \u2192 \u221e); then for x \u2208 \u03a9\u2212Hj(y \u2212 x) is as in  = \u2212\u03a8j(y), . For x \u2208 Rn\u2216\u2202\u03a9, denoting3[u*] = 0 in Rn\u2216\u2202\u03a9. By using Plemelj formula, combining with weak singularity of Hj(x) (j \u2265 2), we obtain the following:\u03b1j\u2032 \u2264 1 . By using 3[u*] = 0 in Rn. It is clear that we have |u*(x)| = O(1) (|x | \u2192\u221e). In view of For 3[u] = 0 in \u03a9\u2212, Dj[u](x) \u2208 H\u03b1j), 0 < \u03b1j \u2264 1 , and |u(x)| = O(1) (|x | \u2192\u221e), thenLet u(x)| = O(1) (|x | \u2192\u221e) in u(\u221e) = 0, then the results in When the condition |In this section, we will consider the Riemann boundary value problem ; the expThe Riemann boundary value problem  is solvaG(x) \u2208 H\u03b1)\u2009\u20090 < \u03b1 < 1 and G(x) satisfies the following condition:M is the positive constant mentioned in Suppose w(x) = D5[u](x) then w+(x) = w\u2212(x)A5 + f5(x), x \u2208 \u2202\u03a9. Moreover, by D5[u](\u221e) = 0. we get the following:1(x) being as in (2u(x) \u2212 \u03942\u03a81(x)\u2236 = \u03c61(x), x \u2208 Rn\u2216\u2202\u03a9, and using (\u03942u)+(x) = (\u03942u)\u2212(x)A4 + f4(x), x \u2208 \u2202\u03a9, we conclude that\u03c61(\u221e) = 0, and we then get the following representation formula:2(x) from (2u(x) \u2212 \u03942\u03a81(x) \u2212 \u03942\u03a82(x) = 0, x \u2208 Rn\u2216\u2202\u03a9. DenoteD3u)+(x) = (D3u)\u2212(x)A3 + f3(x), x \u2208 \u2202\u03a9, we obtain that\u03c62(\u221e) = 0, and we then get the following representation formula:Denoting ng as in , it is eng as in . By Coro(x) from  that \u03942ng as in . By Coro3(x) being as in (D3u(x) \u2212 D3\u03a81(x) \u2212 D3\u03a82(x) \u2212 D3\u03a83(x) = 0, x \u2208 Rn\u2216\u2202\u03a9. We denoteu)+(x) = (\u0394u)\u2212(x)A2 + f2(x), x \u2208 \u2202\u03a9. We conclude that\u03c63(\u221e) = 0; then4(x) being as in (u(x) \u2212 \u0394\u03a81(x) \u2212 \u0394\u03a82(x) \u2212 \u0394\u03a83(x) \u2212 \u0394\u03a84(x) = 0, x \u2208 Rn\u2216\u2202\u03a9. DenotingDu)+(x) = (Du)\u2212(x)A1 + f1(x), x \u2208 \u2202\u03a9, we get\u03c64(\u221e) = 0, and we then get the following representation formula:Here \u03a8ng as in , then weng as in . Corollang as in , that \u0394uken from . By Coro5(x) as defined in (Du(x) \u2212 \u2211i=14D\u03a8i(x) = 0, x \u2208 Rn\u2216\u2202\u03a9. Defineu+(x) = u\u2212(x)G(x) + g(x) we arrive at\u03c65(\u221e) = 0. We obtain thatFinally, we use \u03a8fined in  and get ken from . It is c\u03c66(y) is a H\u00f6lder continuous function to be determined on \u2202\u03a9. Then, by using Plemelj formula, ), we haveT is a contraction operator mapping the Banach space H\u03b1) into itself, which has a unique fixed point for the operator T. Thus, there exists a unique solution to (Letting  side of , we getondition , the intution to . The pro"}
{"text": "Res. Natl. Inst. Stand. Technol. Volume 100, Number 3, May\u2013June 1995, p. Page 211, above Eq. (3).Replace \u201cthe magnetic field within the balance chamber is vertical with the form\u2026\u201d by \u201cthe vertical magnetic field strength within the balance chamber has the form\u2026\u201d.Page 217, Eq. (8).Ia\u201d by \u201cIa\u201d.Replace \u201c\u03bcmHmax\u201d by \u201c\u03bc0Hmax\u201d.Page 218, two paragraphs above Sec. 5.1. Replace \u201cPage 223, first paragraph of Sec. 11.1.Replace \u201c: it must be less than that based on a sum\u2026; it must be greater than that based on a sum\u2026\u201d by \u201c: it must be greater than that based on a sum\u2026; it must be less than that based on a sum\u2026\u201d.Page 224, Table 5, column 4, row 2.Replace \u201c0.1116\u201d by \u201c\u22120.1116\u201d.Page 224, Eq. (10) should be:\u03c7\u201d by \u201cThis again underestimates \u03c7\u201d.Page 225, two paragraphs above the Acknowledgment. Replace \u201cThis again overestimates"}
{"text": "G = (V(G), E(G)) be a simple connected graph with vertex set V(G) = {v1, v2,\u2026, vn} and edge set E(G). Its adjacency matrix A(G) = (aij) is defined as n \u00d7 n matrix (aij), where aij = 1 if vi is adjacent to vj and aij = 0, otherwise. Denote by d(vi) or dG(vi) the degree of the vertex vi. It is well known that A(G) is a real symmetric matrix. Hence, the eigenvalues of A(G) can be ordered asA(G) is called the spectral radius of G, denoted by \u03c1(G). It is easy to see that if G is connected, then A(G) is nonnegative irreducible matrix. By the Perron-Frobenius theory, \u03c1(G) has multiplicity one and exists a unique positive unit eigenvector corresponding to \u03c1(G). We refer to such an eigenvector corresponding to \u03c1(G) as the Perron vector of G.Let Pn and Cn the path and the cycle on n vertices, respectively. The characteristic polynomial of A(G) is det\u2061(xI \u2212 A(G)), which is denoted by \u03a6(G) or \u03a6. Let X be an eigenvector of G corresponding to \u03c1(G). It will be convenient to associate with X a labelling of G in which vertex vi is labelled xi (or xvi). Such labellings are sometimes called \u201cvaluation\u201d [Denote by luation\u201d .\u03c9 are given. And, in [\u03c9 the minimum value of the spectral radius is attained for a kite graph PKn\u2212\u03c9,\u03c9, where PKn\u2212\u03c9,\u03c9 is a graph on n vertices obtained from the path Pn\u2212\u03c9 and the complete graph K\u03c9 by adding an edge between an end vertex of Pn\u2212\u03c9 and a vertex of K\u03c9  = \u03c9 \u2212 1. If G \u2208 I\u03c9+1,\u03c9, then, from the Perron-Frobenius theorem, the first \u03c9 \u2212 1 smallest values of the spectral radius of I\u03c9+1,\u03c9 are PK\u03c9;i1, (0 \u2264 i \u2264 \u03c9 \u2212 2), respectively, where PK\u03c9;i1, is the graph obtained from PK\u03c91, by adding i (0 \u2264 i \u2264 \u03c9 \u2212 2) edges. So in the following, we consider that n \u2265 \u03c9 + 2.Let In order to complete the proof of our main result, we need the following lemmas.v be a vertex of the graph G. Then the inequalitiesG is connected, then \u03bb1(G) > \u03bb1(G \u2212 v).Let For the spectral radius of a graph, by the well-known Perron-Frobenius theory, we have the following.G be a connected graph and H a proper subgraph of G. Then \u03c1(H) < \u03c1(G).Let G be a graph on n vertices, thenG is a regular graph.Let v be a vertex of a graph G and suppose that two new paths P = v(vk+1)vk \u22ef v2v1 and Q = v(ul+1)ul \u22ef u2u1 of lengths k and l (k \u2265 l \u2265 1) are attached to G at v( = vk+1 = ul+1), respectively, to form a new graph Gk,l  be the graphs as defined above. Then \u03c1 > \u03c1.Let v be a vertex of the graph G and N(v) the set of vertices adjacent to v.Let G be a connected graph, and let u, v be two vertices of G. Suppose that v1, v2,\u2026, vs \u2208 N(v)\u2216(N(u)\u22c3{u}) (1 \u2264 s \u2264 d(v)) and x =  is the Perron vector of G, where xi corresponds to the vertex vi (1 \u2264 i \u2264 n). Let G* be the graph obtained from G by deleting the edges vvi and adding the edges uvi (1 \u2264 i \u2264 s). If xu \u2265 xv, then \u03c1(G) < \u03c1(G*).Let v be a vertex of G, let \u03c6(v) be the collection of circuits containing v, and let V(Z) denote the set of vertices in the circuit Z. Then the characteristic polynomial \u03a6(G) satisfiesw adjacent to v, and the second summation extends over all Z \u2208 \u03c6(v).Let G is a sequence of vertices v1, v2,\u2026, vk with k \u2265 2 such thatv1 = vk);the vertices in the sequence are distinct ;d(vi) satisfy d(v1) \u2265 3, d(v2) = \u22ef = d(vk\u22121) = 2 (unless k = 2) and d(vk) \u2265 3.the vertex degrees An internal path of a graph Wn be the tree on n vertices obtained from Pn\u22124 by attaching two new pendant edges to each end vertex of Pn\u22124, respectively.Let G \u2260 Wn is a connected graph and uv is an edge on an internal path of G. Let Guv be the graph obtained from G by subdivision of the edge uv. Then \u03c1(Guv) < \u03c1(G).Suppose that H1 be the graph obtained from K\u03c9 and a path P4 : v1v2v3v4 by joining a vertex of K\u03c9 and a nonpendant vertex, say, v2, of P4 by a path with length 2 and let H2 be the graph obtained from K\u03c9 by attaching two pendant edges at two different vertices of K\u03c9  > \u03c1(H1).Let bove see . If \u03c9 \u2265 \u03c9 \u2265 3, by direct computations, we have \u03c1(H2) > \u03c1(H1). In the following, we suppose that \u03c9 \u2265 6. From For 5 \u2265 \u03c9 > \u03c1(H1) \u2265 \u03c1(K\u03c9) = \u03c9 \u2212 1 \u2265 \u03bb2(H1) and \u03c9 > \u03c1(H2) \u2265 \u03c1(K\u03c9) = \u03c9 \u2212 1. Then from (\u03c1(H2) > \u03c9 \u2212 1 + (2/\u03c92) > \u03c1(H1).By direct calculation, we haveg1  see . It is eLet -1vn see .PKn\u2212\u03c9,\u03c9i be the graphs defined as above  \u2265 \u03c1 \u2248 2.00659 > \u03c1(Cn).Clearly, G1 = PKn\u22123,3n\u22123 \u2212 vn\u22121vn\u22122 + vn\u22123vn\u22121, let G2 = PKn\u22123,3n\u22123 + vn\u22121vn, and let Cn\u22121,1 be the graph obtained from Cn\u22121 and an isolated vertex by adding an edge between some vertex of Cn\u22121 and the isolated vertex  < \u03c1 < \u03c1(Wn) = \u03c1(Cn) < \u03c1. Thus, we only need to prove that \u03c1(G) > \u03c1 if G \u2260 Pn, PKn\u22122,2n\u22122, Wn, Cn, PKn\u22122,2n\u22123. If G is a tree, note that G \u2260 Pn, PKn\u22122,2n\u22122, Wn, PKn\u22122,2n\u22123, then, from \u03c1(G) > \u03c1. If G contains some cycle as a subgraph, then, from Lemmas \u03c1(G) \u2265 \u03c1 > \u03c1.Let PKn\u2212\u03c9,\u03c9i, G1 and G2 be the graphs defined as above  < \u03c1(G1). If n \u2265 12, from Lemmas \u03c1 < \u03c1 < \u03c1 < 2.23808. From x < 2.23808, we have\u03c1 < \u03c1 and 2.23601 < \u03c1 < 2.23808. Then, we have\u03c1 < \u03c1(G1). By similar method, we have for n \u2265 8For 8 \u2264 H3 be the graph obtained from K\u03c9 by attaching two pendant edges at some vertex of K\u03c9; let H4 be the graph obtained from K\u03c9 and P2 by adding two edges between two vertices of K\u03c9 and two end vertices of P2  > \u03c1 if G \u2260 PKn\u22123,3, PKn\u22123,3n\u22122, PKn\u22123,3n\u22123, PKn\u22123,3n\u22124.Let We distinguish the following three cases.Case\u2009\u20091. If there exist at least two vertices outside of K3 that are adjacent to some vertices of K3, then we have that G contains either H2 (\u03c9 = 3) or H3 (\u03c9 = 3) as a proper subgraph. If G contains H2 (\u03c9 = 3) as a proper subgraph, from Lemmas G contains H3 (\u03c9 = 3) as a proper subgraph, from Lemmas  Lemmas 1\u03c1(G)>\u03c1 is a tree. If there exist two vertices u, r \u2208 V(G\u2009\u2212\u2009V(K3)) such that d(u) \u2265 3 and d(r) \u2265 3, then, from Lemmas \u03c1(G) > \u03c1. If there exists only one vertex u \u2208 V(G\u2009\u2212\u2009V(K3)) such that d(u) \u2265 4, then, from Lemmas \u03c1(G) \u2265 \u03c1(G1) > \u03c1. If there exists exactly one vertex u \u2208 V(G\u2009\u2212\u2009V(K3)) such that d(u) = 3, note that G \u2260 PKn\u22123,3n\u22122, PKn\u22123,3n\u22123, PKn\u22123,3n\u22124, then from Lemmas \u03c1(G) > \u03c1.Subcase 2. Suppose that G \u2212 V(K3) contains cycle Cg as a subgraph. If g = 3,4, then, from Lemmas \u03c1(G) \u2265 \u03c1(G2) > \u03c1. If g \u2265 5, then, from Fg from G by deleting vertices such that \u03c1(G) \u2265 \u03c1(Fg), where Fg is the graph obtained from K3 and a cycle Cg by joining a vertex of K3 and a vertex of Cg with a path and |V(Fg)|\u2264n , (1 \u2264 i \u2264 g \u2212 1), v1vg \u2208 E(Cg), and d(v1) = 3. Then, from Lemmas \u03c1(Fg \u2212 v2v3) > \u03c1. Thus, we have \u03c1(G) > \u03c1.)|\u2264n see . SupposePKn\u2212\u03c9,\u03c9i and \u03c9 \u2265 4).Let X = T be the Perron vector of xi corresponds to vi. It is easy to prove that xn = xn\u22121. From \u03c9 \u2265 4 we have Let M\u03c92 (\u03c9 \u2265 4) be the graph as shown in Let n vertices with maximum clique size \u03c9 \u2265 4 and n \u2265 \u03c9 + 5, the first four smallest spectral radii are exactly obtained for PKn\u2212\u03c9,\u03c9, PKn\u2212\u03c9,\u03c9n\u22122, PKn\u2212\u03c9,\u03c9n\u22123, respectively.Among all connected graphs on G be a connected graph with maximum clique size \u03c9 \u2265 4 and n \u2265 \u03c9 + 5 vertices. Suppose that K\u03c9 is a maximum clique of G. From Lemmas \u03c1(G) > \u03c1 if G \u2260 PKn\u2212\u03c9,\u03c9, PKn\u2212\u03c9,\u03c9n\u22122, PKn\u2212\u03c9,\u03c9n\u22123. We distinguish the following three cases.Let Case\u2009\u20091. If there exist at least two vertices outside of K\u03c9 that are adjacent to some vertices of K\u03c9, then G contains either H2 or H3 as a proper subgraph. If G contains H2 as a proper subgraph, from Lemmas G contains H3 as a proper subgraph, from Lemmas Case\u2009\u20092. Suppose that there exists a vertex, say, u, which does not belong to K\u03c9, such that u is adjacent to at least two vertices of K\u03c9. From Lemmas Case\u2009\u20093. Suppose that there uniquely exists a vertex u which does not belong to K\u03c9 such that u is adjacent to a vertex of K\u03c9. If G \u2212 V(K\u03c9) is a tree, note that G \u2260 PKn\u2212\u03c9,\u03c9, PKn\u2212\u03c9,\u03c9n\u22122, PKn\u2212\u03c9,\u03c9n\u22123, then, from Lemmas \u03c1(G) > \u03c1. Suppose that G \u2212 V(K\u03c9) contains cycle Cg as a subgraph. If g = 3, note that \u03c1(G) > \u03c1(G*) > \u03c1, where G* = PKn\u2212\u03c9,\u03c9n\u22123 + vn\u22121vn. If g \u2265 4, then by the similar reasoning as that of Subcase 2 of Case 3 of \u03c1(G) > \u03c1.H3 and H4 be the graphs defined as above T be the Perron vector of H3, where xi corresponds to vi. From AX = \u03c1(H3)X, we havex > \u03c9 \u2212 1 and \u03c9 \u2265 3, we have\u03c1(H3) > \u03c1(K\u03c9) = \u03c9 \u2212 1. From (\u03c1(H3) which is the largest root of equation r1(x) = 0. Similarly, we have \u03c1(H4) which is the largest root of equationx > \u03c9 \u2212 1,\u03c1(H3) < \u03c1(H4).Let \u22124.Thenr1(\u03c9\u22121)=\u2212 we haver1\u2032(x)=3xG be a graph on n vertices with maximum clique size \u03c9 \u2265 3 and n = \u03c9 + 2. Let PK\u03c92,, H2, H3, and H4 be the graphs defined as above  > \u03c1(H4). We distinguish the following two cases.From Lemmas Case\u2009\u20091. Suppose that there exists exactly one vertex outside of K\u03c9 that is adjacent to at least two vertices of K\u03c9. Then G contains M\u03c92 (see \u03c1(M\u03c92) > \u03c1(H4). \u03c92 see  as a subCase\u2009\u20092. Suppose that the two vertices outside of K\u03c9 that are all adjacent to some vertices of K\u03c9. Note that G \u2260 H2, H3, H4. Then G contains one of graphs M\u03c92 as a subgraph, where H3 by adding an edge between two pendant vertices. From \u03c1(G) > \u03c1(M\u03c92) > \u03c1(H4).H5 be the graph obtained from H2 and an isolated vertex by adding an edge between a pendant vertex of H2 and the isolated vertex; let H6 be the graphs as shown in Let H5 be the graphs defined as above T be the Perron vector of xi corresponds to vi. It is easy to see that x1 = x5. From \u03c9 \u2265 4, we haveLet H5 and H6 be the graphs defined as above  > \u03c1(H5). In the following, we suppose that \u03c9 \u2265 5. Then, from Lemmas \u03c9 > \u03c1(H5) > \u03c1(K\u03c9) = \u03c9 \u2212 1 \u2265 4. Let X = T be the Perron vector of H5, where xi corresponds to vi. From AX = \u03c1(H5)X, we have\u03c9 > \u03c1(H5) > \u03c9 \u2212 1 \u2265 4,\u03c1(H6) = \u03c1(H5 \u2212 v1v2 + v1v6) > \u03c1(H5).For H7 be the graph obtained from H3 and an isolated vertex by adding an edge between v\u03c9 and the isolated vertex; let H8 be the graph obtained from H3 and an isolated vertex by adding an edge between v2 and the isolated vertex; let H9 be the graph obtained from H3 and an isolated vertex by adding an edge between one pendant vertex and the isolated vertex; and let H10 be the graph obtained from PK\u03c93,\u03c9+1 and an isolated vertex by adding an edge between v\u03c9+1 and the isolated vertex , the first four smallest spectral radii are obtained for PK\u03c93,, PK\u03c93,\u03c9+1, H5, respectively.Let \u03c1(G) > \u03c1(H5) if G \u2260 PK\u03c93,, PK\u03c93,\u03c9+1, H5. We distinguish the following four cases.From Lemmas Case\u2009\u20091. There exists exactly one vertex outside of K\u03c9 that is adjacent to only one vertex of K\u03c9. Then G must be one of graphs PK\u03c93,, PK\u03c93,\u03c9+1, and Case\u2009\u20092. There exists one vertex outside of K\u03c9 that is adjacent to at least two vertices of K\u03c9. Then G contains M\u03c92 (see \u03c1(G) > \u03c1(M\u03c92) > \u03c1(H5).\u03c92 see  as a proCase\u2009\u20093. If there exactly exist two vertices outside of K\u03c9 that are adjacent to some vertices of K\u03c9, then G contains H5 or H9  \u2265 \u03c1(H9) > \u03c1(H5). If G contains H5 as a subgraph, note that G \u2260 H5, then, from \u03c1(G) > \u03c1(H5).Case\u2009\u20094. If there exist three vertices outside of K\u03c9 that are adjacent to some vertices of K\u03c9, then G contains one of graphs H6, H7, and H8  > \u03c1(H7) > \u03c1(H6) > \u03c1(H5). Then, from \u03c1(G) > \u03c1(H5).PKn\u2212\u03c9,\u03c9i and Let x > \u03c9 \u2212 1 (\u03c9 \u2265 4), we have\u03c1 > \u03c1(K\u03c9) = \u03c9 \u2212 1. Thus, for x > \u03c9 \u2212 1 (\u03c9 \u2265 4), we have \u03c9 \u2265 4).From PKn\u2212\u03c9,\u03c9i and H2 be the graphs defined as above  > \u03c1. In the following, we suppose that \u03c9 \u2265 6. From \u03c9 \u2265 6, we have\u03c9 > \u03c1 \u2265 \u03c1(K\u03c9) = \u03c9 \u2212 1 \u2265 \u03bb2. Then from  > \u03c9 \u2212 1 + 1/\u03c92. From the proof of \u03c1(H2) > \u03c9 \u2212 1 + 2/\u03c92 (\u03c9 \u2265 6). The result follows.For PK4,\u03c9\u03c9+. In the  we haveg3(\u03c9\u22121+1\u03c9n vertices with maximum clique size \u03c9 and n = \u03c9 + 4 (\u03c9 \u2265 4), the first four smallest spectral radii are obtained for PK\u03c94,, PK\u03c94,\u03c9+2, PK\u03c94,\u03c9+1  > \u03c1 if G \u2260 PK\u03c94,, PK\u03c94,\u03c9+2, PK\u03c94,\u03c9+1. We distinguish the following three cases.Let Case\u2009\u20091. There exists exactly one vertex outside of K\u03c9 that is adjacent to one vertex of K\u03c9.Subcase 1. Suppose that G \u2212 V(K\u03c9) is a tree. If G contains exactly one pendant vertex, then G = PK\u03c94,. If G contains exactly two pendant vertices, then G = PK\u03c94,\u03c9+1 or G = PK\u03c94,\u03c9+2. If G contains three pendant vertices, then G = H10 (see \u03c1(H10) > \u03c1.H10 see . From LeSubcase 2. Suppose that G \u2212 V(K\u03c9) contains a cycle. If G \u2212 V(K\u03c9) contains C4, then G contains H11 as a subgraph, where H11 is obtained from PK\u03c94,\u03c9+1 by adding an edge between two pendant vertices. From \u03c1(H11) > \u03c1. If G \u2212 V(K\u03c9) does not contain C4, then G contains \u03c1(G) > \u03c1.Case\u2009\u20092. There exists at least one vertex outside of K\u03c9 that is adjacent to at least two vertices of K\u03c9. Then G contains M\u03c92 (see \u03c1(G) > \u03c1(M\u03c92) > \u03c1(H2) > \u03c1.\u03c92 see  as a subCase\u2009\u20093. There exist at least two vertices outside of K\u03c9 that are adjacent to some vertices of K\u03c9. Then G contains H2 or H3 as a subgraph  > \u03c1(H2) > \u03c1. Thus, from \u03c1(G) > \u03c1.n with maximum clique size \u03c9 \u2265 2, are determined.In this paper, the first four graphs, which have the smallest values of the spectral radius among all connected graphs of order"}
{"text": "Particularly, two coverings with the same reductions are proved to generate the same f-lower and f-upper approximations. Finally, we discuss the relationships between the new model and some other variable precision rough set models.Classical rough set theory is a technique of granular computing for handling the uncertainty, vagueness, and granularity in information systems. Covering-based rough sets are proposed to generalize this theory for dealing with covering data. By introducing a concept of misclassification rate functions, an extended variable precision covering-based rough set model is proposed in this paper. In addition, we define the  In the era of big data, it is difficult to obtain useful information in huge data. Many researchers have proposed lots of efficient means of dealing with the difficulty. As one of these efficient means, classical rough set theory based on equivalence relations is proposed by Pawlak , 2 in thIn variable precision rough sets, the misclassification rate of equivalence classes of all elements in a universe is identical. Similarly, in variable precision covering-based rough sets, the misclassification rate of neighborhoods of all elements in a universe is identical too. However, in practical applications, since there are different understanding or demands about equivalence classes or neighborhoods of different elements, the misclassification rate usually varies. Hence, it is necessary to propose misclassification rate functions.f-lower and f-upper approximations in terms of neighborhoods and investigate their properties.To address the above issue, we propose a variable precision covering-based rough set model based on functions by introducing misclassification rate functions in this paper. We present the concepts of the f-lower and f-upper approximations. Moreover, the relationships between this model and some other variable precision rough set models are exhibited. The rest of this paper is arranged as follows. U is a non-empty finite set.In this section, we present some fundamental concepts and existing results of classical rough sets, covering-based rough sets, and variable precision covering-based rough sets. Throughout this paper, the universe U be a universe and R an equivalence relation on U. R will generate a partition U/R. The elements in U/R are called equivalence classes. Let [x]R denote an equivalence class which includes x \u2208 U. For any X\u2286U, we can describe X by the two sets,Let X with respect to R, respectively.They are called the lower and upper approximations of Covering-based rough sets are presented as the extension of classical rough sets by extending partitions to coverings on a universe.U be a universe of discourse and C a family of subsets of U. If none subsets in C is empty and \u22c3C = U, then C is called a covering of U. The pair  is called a covering approximation space.Let Neighborhoods are important concepts in covering-based rough sets.C be a covering of U and x \u2208 U. NC(x) = \u22c2{K \u2208 C\u2223x \u2208 K} is called the neighborhood of x with respect to C. When there is no confusion, we omit the subscript C.Let C be a covering of U and x \u2208 U. For any X\u2286U, the lower and upper approximations of X with respect to  are defined as follows, respectively:\u2009\u2009Let C is a partition of\u2009\u2009U, then covering-based rough sets degenerate into classical rough sets.Clearly, if C be a covering of U and K \u2208 C. If K is a union of some sets in C \u2212 {K}, we say K is a reducible element of C; otherwise K is an irreducible element of C. The family of all irreducible elements of C is called the reduct of C, denoted as Reduct\u2009\u2009(C).Let \u03b2\u2009\u2009(0 \u2264 \u03b2 < 0.5) in classical rough sets, namely, some degrees of misclassification are allowed, Ziarko proposed variable precision rough sets.By introducing the parameter X and Y be two non-empty subsets of\u2009\u2009U. We say that X is included in Y, if every element of X is an element of Y. In this case, the inclusion relation is certain. That is to say, there does not exist the slightest misclassification between X and Y. However, slight misclassification is allowed in practical applications. Therefore, we present the majority inclusion relation which is a generalized definition of inclusion relation. Before the majority inclusion relation is presented, we introduce the measure of the relative degree of misclassification of one set with respect to others.Let X and Y be two subsets of a universe U. The measure c of the relative degree of misclassification of the set X with respect to set Y is defined asX| denotes the cardinality of X.Let X and Y be two subsets of a universe U, and 0 \u2264 \u03b2 < 0.5. The majority inclusion relation is defined asLet X\u2286Y if and only if c = 0.By the definition, it is clear that \u03b21 < \u03b22 < 0.5, then If 0 \u2264 U be a universe of discourse, R an equivalence relation, and 0 \u2264 \u03b2 < 0.5. For any X\u2286U, the \u03b2-lower and \u03b2-upper approximations of X with respect to R are defined as follows, respectively:\u2009\u2009Let \u03b2 = 0, it is easy to see that variable precision rough sets become classical rough sets. By extending partitions to coverings, variable precision rough sets are generalized to variable precision covering-based rough sets. We present the definition of variable precision covering-based rough sets as follows.When C be a covering of U and 0 \u2264 \u03b2 < 0.5. For any X\u2286U, the \u03b2-lower and \u03b2-upper approximations of X with respect to  are defined as follows, respectively:\u2009\u2009Let \u03b2 = 0, then variable precision covering-based rough sets are covering-based rough sets; if C is a partition of U, then variable precision covering-based rough sets are variable precision rough sets; if C is a partition of U and \u03b2 = 0, then variable precision covering-based rough sets become classical rough sets.Clearly, if In variable precision rough sets, the misclassification rate of equivalence classes of all elements in a universe is identical. Similarly, in variable precision covering-based rough sets, the misclassification rate of neighborhoods of all elements in a universe is identical too. However, in practical applications, we have different understanding or demands about equivalence classes or neighborhoods of different elements. That means the misclassification rate usually varies. Therefore, we present variable precision covering-based rough sets based on functions by introducing a concept of misclassification rate functions.C be a covering of U and NC(x) the neighborhood of x. It is obvious that \u222ax\u2208UNC(x) = U. That is, the collection of neighborhoods of all elements in U is still a covering of U. It is called neighborhood covering of U with respect to C and denoted by Cov(C) [Let y Cov(C) .C be a covering of U. A function f defined from Cov(C) to [0,0.5) is called misclassification rate function.Let C be a covering of U and f a misclassification rate function. For any X\u2286U, the f-lower and f-upper approximations of X with respect to  are defined as follows, respectively:\u2009\u2009Let U = {x1, x2, x3, x4, x5, x6, x7, x8} and C = {K1, K2, K3}, where K1 = {x1, x2, x3, x4, x5}, K2 = {x4, x5, x6, x7} and K3 = {x6, x8}. Suppose that f(N(x)) = |N(x)|/2|U| and X = {x2, x3, x4, x5, x6}. Through the definition of neighborhoods, we have N(x1) = N(x2) = N(x3) = {x1, x2, x3, x4, x5}, N(x4) = N(x5) = {x4, x5}, N(x6) = {x6}, N(x7) = {x4, x5, x6, x7} and N(x8) = {x6, x8}. Therefore, f(N(x1)) = f(N(x2)) = f(N(x3)) = 5/16, f(N(x4)) = f(N(x5)) = f(N(x8)) = 2/16, f(N(x6)) = 1/16 and f(N(x7)) = 4/16. By c(N(xi), X) = 1/5\u2009\u2009, c(N(xj), X) = 0\u2009\u2009, c(N(x7), X) = 1/4 and c(N(x8), X) = 1/2. Hence f \u2261 0, then f \u2261 \u03b2 = 1/5, then Let In practical applications, according to various needs, the different misclassification rate functions can be given by workers or researchers.f(N(xi)) = 0.1\u2009\u2009 and f(N(x7)) = f(N(x8)) = 0.2. By Suppose that f-positive region Pos\u2009cf(X), f-boundary region Bn\u2009cf(X) and f-negative region Neg\u2009cf(X) of X with respect to  by the following definition.Similar to classical rough sets or variable precision rough sets, we present the concepts of U be a universe of discourse and f a misclassification rate function. For any X\u2286U, the f-positive region Pos\u2009cf(X), f-boundary region Bn\u2009cf(X) and f-negative region Neg\u2009cf(X) of X with respect to  are defined as follows, respectively:Let U be a universe of discourse. For any X\u2286U, the following conclusions are true,cf(X) = {x \u2208 U\u2223c(N(x), X) \u2265 1 \u2212 f(N(x))},Neg\u2009cf(X) = {x \u2208 U\u2223f(N(x)) < c(N(x), X) < 1 \u2212 f(N(x))},Bn\u2009cf(\u2212X) =Negcf(X).Pos\u2009Let x \u2208 U\u2223c(N(x), X) \u2265 1 \u2212 f(N(x))}.(1) x \u2208 U\u2223f(N(x)) < c(N(x), X)<1 \u2212 f(N(x))}.(2) x \u2208 U, c(N(x), \u2212X) = 1 \u2212 c(N(x), X). x \u2208 U\u22231 \u2212 c(N(x), X) \u2264 f(N(x))}\u2009\u2009=\u2009\u2009{x \u2208 U\u2223c(N(x), X) \u2265 1 \u2212 f(N(x))} = Neg\u2009cf(X).(3) For any f-accuracy and f-roughness, which are important numerical characteristics of this type of rough sets.In the following definition, we present the C be a covering of U. For any X\u2286U, the f-accuracy and f-roughness are defined as follows, respectively:\u2009\u2009\u03c1f(X) = 1 \u2212 \u03bbf(X).Let In this subsection, we present the properties and some significant results concerning the new model.C be a covering of\u2009\u2009U and f a misclassification rate function. For all X, Y\u2286U, the following conclusions are true,X\u2286Y, then If Let x \u2208 U, f(N(x)) < 0.5 < 1 \u2212 f(N(x)). By (1) For any x \u2208 U, since c(N(x), \u2205) = 1, it follows that c(N(x), U) = 0, it follows that (2) For any X\u2286Y, then c(N(x), Y) \u2264 c(N(x), X). Hence, (3) If X, Y\u2286U, since c(N(x), X \u222a Y) \u2264 c(N(x), X) and c(N(x), X \u222a Y) \u2264 c(N(x), Y), it follows that (4) For all (5) Similar to the proof of (4).X, Y\u2286U, since c(N(x), X) \u2264 c(N(x), X\u2229Y) and c(N(x), Y) \u2264 c(N(x), X\u2229Y), it follows that (6) For all (7) Similar to the proof of (6).x \u2208 U, c(N(x), \u2212X) = 1 \u2212 c(N(x), X). Hence, (8) For any C be a covering of U and f, g two misclassification rate functions. Suppose that f(N(x)) \u2264 g(N(x)) for any x \u2208 U. For every X\u2286U, the following relationships are true,Let According to C be a covering of U and f a misclassification rate function. Denote \u03b11 = min\u2061\u2061{f(N(x))\u2223N(x) \u2208Cov(C)} and \u03b12 = max\u2061\u2061{f(N(x))\u2223N(x) \u2208Cov(C)}. For any X\u2286U, thenLet C, D be two different coverings of U and f a misclassification rate function. If Reduct\u2009\u2009(C) = Reduct\u2009\u2009(D), then X\u2286U.Let C) = Cov(D). Suppose that K1 is a reducible element of C. For any x \u2208 U, x \u2208 K1 or x \u2209 K1. If x \u2209 K1, it is straightforward that NC(x) = NC\u2212{K1}(x). Otherwise, there exists K\u2032 \u2282 K1 such that x \u2208 K\u2032 and K\u2032 is an irreducible element of C. Thus, NC(x) = \u22c2x\u2208K\u2208CK = \u22c2K\u2032\u22c2K1 = \u22c2K\u2032 = NC\u2212{K1}(x). Hence NC(x) = NC\u2212{K}(x) for any reducible element K \u2208 C. According to this method, it is easy to see that NC(x) = NC)Reduct\u2009\u2009((x). From the arbitrariness of x, Cov(C) = Cov(Reduct\u2009\u2009(C)). Since Reduct\u2009\u2009(C) = Reduct\u2009\u2009(D), it follows that Cov(C) = Cov(D). This completes the proof.From U = {x1, x2, x3, x4, x5, x6, x7, x8}. C = {K1, K2, K3, K4}, where K1 = {x1, x2, x3}, K2 = {x3, x4, x5}, K3 = {x4, x5, x6, x7} and K4 = {x6, x8}. D = {K1, K2, K3, K4}, where K1 = {x1, x2, x3, x4, x5}, K2 = {x4, x5, x6, x7}, K3 = {x6, x8} and K4 = {x4, x5, x6, x7, x8}. Clearly, Reduct\u2009\u2009(C) = Reduct\u2009\u2009(D). Suppose X = {x2, x3, x4, x5, x6}. Hence Let However, the converse proposition of C be the covering in D = {K1, K2, K3, K4}, where K1 = {x1, x2, x3, x4, x5}, K2 = {x4, x5, x6, x7}, K3 = {x6, x8} and K4 = {x1, x2, x3, x4, x5, x6}. Suppose that X = {x2, x3, x4, x5, x6}. It is easy to know that C) \u2260 Reduct\u2009\u2009(D).Let f be a misclassification rate function and \u03b2 a constant, where 0 \u2264 \u03b2 < 0.5.\u201c1\u201d:C) is a partition of U;Cov is a partition of U and f \u2261 0;Cov(\u201c4\u201d:C) is a partition of U and f \u2261 \u03b2;Cov(\u201c5\u201d:C) is a partition of U;Cov(\u201c6\u201d:\u03b2 = 0;\u201c7\u201d:f \u2261 0;\u201c8\u201d:f \u2261 \u03b2.In this subsection, we will use a figure  to explaf-lower and f-upper approximations. Moreover, we exhibited the relationships between this model and some other variable precision rough set models. In future work, we will seek more specific misclassification rate functions for dealing with more types of data.In this paper, we proposed the variable precision covering-based rough set model based on functions as a generalization of a variable precision covering-based rough set model and studied its properties. Through the concept of reductions, we obtained that two coverings with the same reductions generate the same"}
{"text": "The main result is illustrated with the aid of an example.By employing a nonlinear alternative for contractive maps, we investigate the existence of solutions for a boundary value problem of fractional  Here, we emphasize that the nonlocal conditions are regarded as more plausible than the standard initial conditions for the description of some physical phenomena. In (g(x) may be understood as g(x) = \u2211j=1p\u03b1jx(tj), where \u03b1j, j = 1,\u2026, p, are given constants and 0 < t1 < \u22ef<tp \u2264 1. For more details, we refer to the work by Byszewski and Lakshmikantham q is defined byFor a real parameter q-analogue of the Pochhammer symbol  is defined asq-analogue of the exponent (x \u2212 y)k isq-gamma function \u0393q(y) is defined asy \u2208 R\u2216{0, \u22121, \u22122,\u2026}. Observe that \u0393q(y + 1) = [y]q\u0393q(y).The f be a function defined on . The fractional q-integral of the Riemann-Liouville type of order \u03b2 \u2265 0 is (Iq0f)(t) = f(t) andLet \u03b2 = 1 in q-integral:q-integrals and fractional q-integrals, see Sections 1.3 and 4.2, respectively, in  \u00d7 R \u2192 P(R) is said to be a Carath\u00e9odory function if t \u21a6 F is measurable for each x \u2208 R;x \u21a6 F is upper semicontinuous for almost all t \u2208 .Further, a Carath\u00e9odory function F is called L1-Carath\u00e9odory if (iii)\u03b1 > 0, there exists \u03c6\u03b1 \u2208 L1 such thatfor each \u2009x\u2016 \u2264 \u03b1 and for a.e. t \u2208 .for all \u2016A multivalued map G to be the set Gr(G) = { \u2208 X \u00d7 Y, \u2009y \u2208 G(x)} and recall two results for closed graphs and upper-semicontinuity.We define the graph of G : X \u2192 Pcl(Y) is u.s.c., then Gr(G) is a closed subset of X \u00d7 Y, that is, for every sequence {xn}n\u2208N \u2282 X and {yn}n\u2208N \u2282 Y, if n \u2192 \u221e, xn \u2192 x\u2217, yn \u2192 y\u2217, and yn \u2208 G(xn), then y\u2217 \u2208 G(x\u2217). Conversely, if G is completely continuous and has a closed graph, then it is upper semicontinuous.If X be a separable Banach space. Let F :  \u00d7 X \u2192 Pcp,c(X) be an L1-Carath\u00e9odory function. Then, the operatorC \u00d7 C.Let x \u2208 AC1 is called a solution of problem  with f(t) \u2208 F), a.e. on  such that Dq\u03c5x(t) = f(t), a.e. on  and x(0) = g(x) and x(\u03c9) = b\u222b\u03b41x(s)dqs.A function To prove our main result in this section we will use the following form of the nonlinear alternative for contractive maps , we haveStep 2. F2 is compact, convex valued, and completely continuous. This will be established in several claims.Claim 1. F2 maps bounded sets into bounded sets in C. For that, let B\u03c1 = {x \u2208 C : \u2016x\u2016 \u2264 \u03c1} be a bounded set in C. Then, for each h \u2208 F2(x), x \u2208 B\u03c1, we haveh \u2208 F2(B\u03c1), we haveClaim 2. F2 maps bounded sets into equicontinuous sets. As before, let B\u03c1 be a bounded set and let h \u2208 F2(x) for x \u2208 B\u03c1. Let t1, t2 \u2208  with t1 < t2 and x \u2208 B\u03c1. Then, for each h \u2208 F2(x), we obtainx and tends to zero as t2 \u2212 t1 \u2192 0. Therefore, it follows by the Arzel\u00e1-Ascoli theorem that F2 : C \u2192 P is completely continuous.Claim 3. F2 has a closed graph. Let xn \u2192 x\u2217, hn \u2208 F2(xn), and hn \u2192 h\u2217. Then, we need to show that h\u2217 \u2208 F2(x\u2217). Associated with hn \u2208 F2(xn), there exists fn \u2208 SF,xn such that, for each t \u2208 ,f\u2217 \u2208 SF,x\u2217 such that, for each t \u2208 ,L1 \u2192 C defined byLet us consider the continuous linear operator \u0398 : n \u2192 \u221e. Thus, it follows by SF is a closed graph operator. Further, we have hn(t) \u2208 \u0398. Since xn \u2192 x\u2217, therefore, we havef\u2217 \u2208 SF,x\u2217. Hence, F2 has a closed graph . In consequence, the operator F2 is compact valued.Observe thatF1 and F2 satisfy hypotheses of x \u2208 \u03bbF1(x) + \u03bbF2(x) for \u03bb \u2208 , then there exists f \u2208 SF,x such that x = \u03bbF(x), that is,\u03bb \u2208  and x \u2208 \u2202BM with x = \u03bbF(x), where BM = {x \u2208 C : \u2016x\u2016 \u2264 M}. Then, x is a solution of x = \u03bbF(x) with \u2016x\u2016 = M. Now, by the last inequality, we getF has a fixed point on  by Thus, the operators tradicts . Hence, q-fractional boundary value problem:Consider the following \u03c5 = 3/2, q = 1/2, b = 1/5, \u03c9 = 1/4, \u03b4 = 3/4, l = 1/15, and\u03d1 = 0.191667, \u03bc0 \u2248 1.3990, k0 = 3.9564, ||p|| = 43/48, \u03c8(||x||) = 1, and M > 1.7026 by (H3). Thus, all the conditions of Here,"}
{"text": "We establish the stabilities and blowup results for the nonisentropic Euler-Poisson equations by the energy method. By analysing the second inertia, we show that the classical solutions of the system with attractive forces blow up in finite time in some special dimensions when the energy is negative. Moreover, we obtain the stabilities results for the system in the cases of attractive and repulsive forces. As usual, \u03c1 = \u03c1 \u2265 0, u = u \u2208 \u211dN, and S are the density, the velocity, and the entropy, respectively. P is the pressure function, for which the constants K \u2265 0 and \u03b3 \u2265 1.The compressible nonisentropic Euler  are C1 solutions with compact support \u03a9 = \u03a9(t) for each fixed time t. We also denote the total mass by M, where \u03c10 = \u03c10(x): = \u03c1.Lastly, we will denote \u03b3 > 1; namely, In this section, we establish some lemmas for the proof of the main results. The following lemma will be used to derive the energy functional for \u03c1, u, S) of system  of system \u2009\u2003\u2003\u2009=\u2212\u03b4\u03c1\u2207Appendix:\u222b\u03a912u2\u2207\u00b7 of system \u03b3 = 1. Co\u03c1, u, S) of system  of system  with respect to t.For the classical solution (f system  with \u03b3 >E(t) is a decreasing function and is conserved if the system is not damped.Thus, By \u03c1, u, S) of system  with respect to t.For the classical solution (f system  with \u03b3 =E(t) is a decreasing function and is conserved if the system is not damped.Thus, By \u03c1, u, S) is a classical solution of system =\u222b\u03a9\u03c1xx is the gradient operator with respect to the spatial variable x.Firstly,N = 2, N = 2 is established.Note thatN \u2265 3, N \u2265 3 are also established.For Now, we are ready to present the stability results.Considering the classical solutions of system , we haveCase 1. For \u03b4 = 1, \u03b2 = 0, N = 3 or 4, \u03b3 \u2265 2(N \u2212 1)/N, and E(0) \u2265 0, we have Case 2. For \u03b4 = \u22121, \u03b2 = 0, N \u2265 4, and \u03b3 \u2265 (N + 2)/N, we haveCase 3. For \u03b4 = \u22121, \u03b2 = 0, N = 2, and \u03b3 > 1, we have First of all, by definitions , 6), an, an6), aCase 1. By Propositions equality , we haveCase 2. By Propositions N \u2265 3 by /N, and E(0) < 0, then the classical solutions of (If \u2009 \u2009Case 1 (\u03b2 = 0). As before, we have, from Propositions t, we see that H(t) is negative as the leading coefficient of the right hand side of (H(t) is nonnegative by definition (ows that H(t)\u2264(N\u22122finition . This isCase 2 (\u03b2 \u2260 0). Now (e\u03b2t on both sides and taking integration that A1 and A2. Note that this implies that H(t) is negative for sufficient large t as \u03b2 > 0 and (N \u2212 2)E(0) < 0. Therefore, the solutions blow up in finite time. 0). Now  becomes("}
{"text": "By using the concept of exceptional family, we proposea sufficient condition of a solution to general order complementarity problems (denoted by GOCP) in Banach space, which is weaker than that in N\u00e9meth, 2010 (Theorem 3.1). Thenwe study some sufficient conditions for the nonexistence of exceptional family forGOCP in Hilbert space. Moreover, we prove that without exceptional family isa sufficient and necessary condition for the solvability of pseudomonotone generalorder complementarity problems. There are several types of order complementarity problems in real world applications. Among them, the linear order complementarity problem was systematically studied (see ). The prRn to general Hilbert space  weakly proper to CCP . Smi. Smi15])ace (see ). Using ace (see ) and Kalace (see ), some eace see , 21). In. InRn tace (see ). In 201CCP (see ). In 201CCP (see ).Motivated and inspired by the works mentioned above, in this paper, by using the concept of exceptional family in , we propThe remainder of this paper is organized as follows. The preliminary results which will be used in this paper are stated in In this section, we recall some background materials and preliminary results used in the subsequent sections. Firstly, we give some concepts from .X be a Banach space whose norm is denoted by ||\u00b7||. Let K \u2282 H be a closed set. K is called a wedge, if for any \u03bb \u2265 0 and x, y \u2208 K, \u03bbx \u2208 K and x + y \u2208 K. A wedge K is called a cone if K\u2229(\u2212K) = {0}.Let X is called an order if it meetsx\u2aafx for all x \u2208 X;reflexivity; that is, x\u2aafy and y\u2aafx, then x = y;antisymmetry; that is, if x\u2aafy and y\u2aafz, then x\u2aafz.transitivity; that is, if A relation \u2aaf on X is induced by a cone K \u2282 X; that is, x\u2aafy if and only if y \u2212 x \u2208 K. Hence K = {x \u2208 X : 0\u2aafx} by using the relation \u2aaf on X. Then we denote an ordered Banach space by .We say a relation \u2aaf on X is induced by a cone K \u2282 X if and only if it isx\u2aafy, then x + z\u2aafy + z for all z \u2208 X;translation invariant; that is, if x\u2aafy, then \u03bbx\u2aaf\u03bby for any \u03bb > 0;scale invariant; that is, if xn}n>0 and {yn}n>0 in X with xn\u2aafyn for all n > 0, then x*\u2aafy*, where x* and y* are the limits of {xn}n>0 and {yn}n>0, respectively.continuous; that is, if for any two convergent sequences {A relation \u2aaf on X, ||\u00b7||, K) is called a vector lattice if for every x, y \u2208 X there exists x\u2227y : = inf\u2061{x, y} with respect to the order induced by K. In this case we say that the cone K is latticial. By the above concepts, we give the following property from  \u00d7 \u03a9 \u2192 X be a completely continuous mapping. If y \u2209 ht(\u2202\u03a9), then deg\u2061 is a constant for 0 \u2264 t \u2264 1, where ht(x) = x \u2212 H.Let \u03a9\u2286X be an open bounded subset, I : X \u2192 X an identity mapping, and f\u2009\u2009: I \u2212 f, \u03a9, y) \u2260 0, then equation x \u2212 f(x) = y has at least one solution in \u03a9.Let First we recall the definition of general order complementarity problems see , 5, 6) a, 6 a5, 6X, ||\u00b7||) be a Banach space ordered by the latticial cone K \u2282 X and D \u2282 X a nonempty closed convex set. Consider m mappings f1, f2,\u2026, fm : X \u2192 X. The general order complementarity problem defined by the family of mappings {fi}i=1m and the set D isLet  be a Banach space ordered by the latticial cone K \u2282 X and D \u2282 X a nonempty closed convex set. Consider m mappings f1, f2,\u2026, fm : X \u2192 X. A sequence {xr}r>0\u2286D is said to be an exceptional family for GOCP if the following conditions are satisfied:xr|| \u2192 +\u221e as r \u2192 +\u221e,||r > 0, there exists a real number \u03bcr > 0 such that ur1\u2227\u22ef\u2227urm = 0, with uri = \u03bcrxr + fi(xr), for i = 1,2,\u2026, m.for every real number Let (The following lemma comes from the property proved in . If there exists an r > 0 such thatht(x), Ur, 0) is constant for t \u2208 . This together with x \u2212 \u03d5(x), Ur, 0) = deg\u2061 = 1. Therefore, we know that problem \u03d5(x) = x is solvable from From r > 0, there exist a vector xr \u2208 Br and a scalar tr \u2208  such that htr(xr) = 0; that is, xr \u2212 (1 \u2212 tr)\u03d5(xr) = 0.On the other hand, for every tr = 0, then xr \u2212 \u03d5(xr) = 0, which again implies solvability of the problem GOCP. If tr = 1, then xr = 0, which contradict with the fact xr \u2208 Br. Hence tr \u2260 1. If 0 < tr < 1, then from the definition of \u03d5(x) we gettr, we obtain\u03bcr = tr/(1 \u2212 tr). Let uri = \u03bcrxr + fi(xr), i = 1,2,\u2026, m; thenxr}r>0 is an exceptional family for GOCP. The proof is complete.If fi and Si = I \u2212 fi being completely continuous operators for all i = 1,2,\u2026, m. Moreover, [Sm(D) + K \u2282 D holds, which does not need this condition in Notice that, in , they usH whose inner product and norm are denoted by \u2329\u00b7, \u00b7\u232a and ||\u00b7||, respectively. We propose some sufficient conditions and prove that they guarantee existence of solutions to the general order complementarity problem. Firstly, we give the condition as follows.In this section, we consider the general order complementarity problems in Hilbert space H, \u2329\u00b7, \u00b7\u232a) be a Hilbert space ordered by the latticial cone K \u2282 H and D \u2282 H a nonempty set. f1, f2,\u2026, fm : H \u2192 H satisfy the following condition: there exists \u03c1 > 0 such that for all x \u2208 D with ||x|| > \u03c1, there exists y \u2208 H with ||y|| < ||x|| such thatLet  be a Hilbert space ordered by the latticial cone K \u2282 H, D \u2282 H an unbounded closed convex set and fi, Si = I \u2212 fi are completely continuous for all i = 1,2,\u2026, m. If fi}i=1m, D) and hence, GOCP is solvable.Let  has an exceptional family {xr}r>0 \u2282 D. By Suppose that GOCP\u2227\u22ef\u2227fm(xr)\u232a\u22650. We havefi}i=1m, D). Then the problem is solvable.Take H, \u2329\u00b7, \u00b7\u232a) be a Hilbert space ordered by the latticial cone K \u2282 H and D \u2282 H a nonempty set. f1, f2,\u2026, fm : H \u2192 H satisfy the following condition: there exists a nonempty bounded subset C \u2282 D such that for every x \u2208 D\u2216C, there exists y \u2208 C such thatLet  be a Hilbert space ordered by the latticial cone K \u2282 H and D \u2282 H an unbounded closed convex set and fi, Si = I \u2212 fi are completely continuous for all i = 1,2,\u2026, m. If fi}i=1m, D) and hence, GOCP is solvable.Let  such that \u2329x \u2212 y, f1(x)\u2227\u22ef\u2227fm(x)\u232a\u22650. Hence Let p-order coercivity condition  to  to p-ordH, \u2329\u00b7, \u00b7\u232a) be a Hilbert space ordered by the latticial cone K \u2282 H and D \u2282 H an unbounded set. Consider m mappings f1, f2,\u2026, fm : H \u2192 H. f1(x)\u2227\u22ef\u2227fm(x) is said to be p-order coercive with respect to D, if there exists p \u2208  be a Hilbert space ordered by the latticial cone K \u2282 H and D \u2282 H an unbounded closed convex set and fi, Si = I \u2212 fi are completely continuous for all i = 1,2,\u2026, m. If there exists some p \u2208 \u2227\u22ef\u2227fm(x) is p-order coercive with respect to D, then there exists no exceptional family for GOCP and hence, GOCP is solvable.Let (f1(x)\u2227\u22ef\u2227fm(x) is p-order coercive with respect to D for some p \u2208  weakly proper to GOCP.H, \u2329\u00b7, \u00b7\u232a) be a Hilbert space ordered by the latticial cone K \u2282 H and D \u2282 H a nonempty set. Consider m mappings f1, f2,\u2026, fm : H \u2192 H. f1(x)\u2227\u22ef\u2227fm(x) is said to be(a)D if, for every x, y \u2208 D, x \u2260 y, one haspseudomonotone on (b)D if, for every x, y \u2208 D, x \u2260 y, one hasquasiomonotone on Let  be a Hilbert space ordered by the latticial cone K \u2282 H and D \u2282 H an unbounded set. Consider m mappings f1, f2,\u2026, fm : H \u2192 H. f1(x)\u2227\u22ef\u2227fm(x) is said to be(a)D, if for every sequence {xr} \u2282 D with lim\u2061r\u2192+\u221e||xr|| = +\u221e, there exists a y \u2208 H and some r such thatweakly proper on (b)D, if for every sequence {xr} \u2282 D with lim\u2061r\u2192+\u221e||xr|| = +\u221e, there exists a y \u2208 H and some r such thatstrictly weakly proper on Let  be a Hilbert space ordered by the latticial cone K \u2282 H and D \u2282 H an unbounded closed convex set and fi, Si = I \u2212 fi are completely continuous for all i = 1,2,\u2026, m. If f1(x)\u2227\u22ef\u2227fm(x) is pseudomonotone on D, then the following conditions are equivalent: fi}i=1m, D) has no exceptional family;GOCP has at least a solution;GOCP({f1(x)\u2227\u22ef\u2227fm(x) is weakly proper on D.Let ((1)\u21d2(2) follows from fi}i=1m, D) has at least a solution, there exists x* \u2208 D such that f1(x*)\u2227\u22ef\u2227fm(x*) = 0. Then for every sequence {xr} \u2282 D with lim\u2061r\u2192+\u221e||xr|| = +\u221e, we havef1(x)\u2227\u22ef\u2227fm(x) is weakly proper on D.(2)\u21d2(3). Since GOCP has an exceptional family. Then there exists {xr}r>0\u2286D, ||xr|| \u2192 +\u221e\u2009\u2009(r \u2192 +\u221e), and \u03bcr > 0 such thatf1(x)\u2227\u22ef\u2227fm(x) is weakly proper on D, then there exists a y \u2208 H and some r such thatf1(x)\u2227\u22ef\u2227fm(x) is pseudomonotone on D yieldsfi}i=1m, D) has no exceptional family. From the above, we complete the proof.(3)\u21d2(1). Suppose that GOCP\u2227\u22ef\u2227f1(x)\u2227\u22ef\u2227fm(x) is pseudomonotone on D, GOCP has no exceptional family \u21d4 GOCP which has at least a solution.The above theorem shows that if H, \u2329\u00b7, \u00b7\u232a) be a Hilbert space ordered by the latticial cone K \u2282 H and D \u2282 H an unbounded closed convex set and fi, Si = I \u2212 fi are completely continuous for all i = 1,2,\u2026, m. If f1(x)\u2227\u22ef\u2227fm(x) is quasimonotone on D, then there exists no exceptional family for GOCP and hence, GOCP is solvable.Let  has an exceptional family. Then from the proof of f1(x)\u2227\u22ef\u2227fm(x) is strictly weakly proper on D, then there exists a y \u2208 H and some r such thatf1(x)\u2227\u22ef\u2227fm(x) is quasimonotone on D yieldsfi}i=1m, D) has no exceptional family. From the above, we complete the proof.Suppose that GOCP({)\u232a\u22650.By  we get(In this paper, by using the concept of exceptional family in , we prop"}
{"text": "We show that these ranks generalize some known rank functions over semirings such as the determinantal rank. We also show that this family of ranks satisfies the rank-sum and Sylvester inequalities. We classify all bijective linear maps which preserve these ranks.We use the  For matrices over semirings, all of these definitions are no longer equivalent and each of these generalizes to a distinct rank function for matrices over semirings. There are many different rank functions for matrices over semirings and their properties and the relationships between them have been much studied . In tt of Tan , 5 to de\u03f5-determinant of Tan first introduced in  and B = [bij] be m by n matrices over a semiring  and let C = [cij] be an n by p matrix over the same semiring. Then A + B = [aij \u2295 bij] and AC = [\u2a01k=1naik \u2297 ckj]. The set of n by n matrices over a semiring is itself a semiring. For any k \u2208 S, the matrix kA = [k \u2297 aij].Let A = [aij] is an n by n matrix over a commutative ring, then the standard determinant expression of A is Sn is the symmetric group of order n and sgn\u2061(\u03c3) = +1 if \u03c3 is even permutation and sgn\u2061(\u03c3) = \u22121 if \u03c3 is odd permutation. Here sgn\u2061(\u03c3)a\u03c3(1)1a\u03c3(2)2 \u22ef an\u03c3(n) is called a term of the determinant.If Since we do not have subtraction in a semiring, we cannot write the determinant of a matrix over a semiring in this form. We split the determinant into two parts, the positive determinant and the negative determinant.A be an n by n matrix over a commutative semiring S. We define the positive and the negative determinant as An is the alternating group of order n, that is, the set of all even permutations of order n and Sn\u2216An is the set of all odd permutations of order n.Let A over a commutative ring takes the form As such we note that the determinant of a matrix In the semiring case, one cannot subtract the negative determinant from the positive determinant and so the positive determinant and the negative determinant are listed as a pair. This pair is called the bideterminant.A be an n by n matrix over a commutative semiring. The bideterminant of A is (det\u2061+(A), det\u2061\u2212(A)).Let The definition of the permanent involves no subtractions; hence it carries over to the semiring case unchanged.A = [aij] be an n by n matrix over a semiring; then the permanent of A is Let A) = det\u2061+(A) \u2295 det\u2061\u2212(A).The permanent of a square matrix is the sum of its positive and negative determinants: per called the difference preorder on semirings.S be a semiring. We define the difference preorder \u2265 on S as follows: if x, y \u2208 S then x \u2265 y if there exists z \u2208 S such that x = y \u2295 z.Let The difference preorder may not be a partial order. However for many semirings such as the nonnegative semiring, max-plus semiring, and any Boolean algebra or distributive lattice, the difference semiring corresponds to the natural order on the set.In , 5, Tan S be a semiring. A bijection \u03c4 : S \u2192 S is called a symmetry if \u03c4(\u03c4(a)) = a for all a \u2208 S and \u03c4(a \u2297 b) = a \u2297 \u03c4(b) = \u03c4(a) \u2297 b for all a, b \u2208 S.Let \u03f5-function in  denote the k by k submatrix of A whose th entry is a\u03b1i,\u03b2j. We define \u03c0(\u03b1) = \u2211j=1k\u03b1j.Tan has shown that the S be a commutative semiring and let \u03f5 \u2208 S satisfy \u03f52 = 1. Let A \u2208 Mm,n(S), B \u2208 Mn,p(S), 1 \u2264 k \u2264 min\u2061. Let \u03b1 and \u03b2 be two subsets of {1,2,\u2026, n} of cardinality k. Then there exists a \u03b4 \u2208 S such that det\u2061\u03f5((AB)) =  \u2297 det\u2061\u03f5)\u2295[\u03b4 \u2297 (1\u2295\u03f5)].Let S be a commutative semiring and let \u03f5 \u2208 S satisfy \u03f52 = 1. If A \u2208 Mn(S) and \u03b1\u2286{1,2,\u2026, n}, then det\u2061\u03f5(A) = \u2a01\u03b2\u2286{1,2,\u2026,n};|\u03b2|=|\u03b1|(\u03f5)\u03c0(\u03b1)+\u03c0(\u03b2) \u2297 det\u2061\u03f5(A[\u03b1\u2223\u03b2]) \u2297 det\u2061\u03f5(A[\u03b1c\u2223\u03b2c]).Let S is a commutative ring and \u03f5 = \u22121, the \u03f5-determinant reduces to the regular determinant and the two theorems above reduce to the usual Binet-Cauchy theorem and the Laplace expansion.In the case where One corollary of the generalized Binet-Cauchy theorem is the following difference preorder inequality for square matrices.S be a commutative semiring and let \u03f5 \u2208 S satisfy \u03f52 = 1. The inequality det\u2061\u03f5(AB) \u2265 det\u2061\u03f5(A) \u2297 det\u2061\u03f5(B) holds for all A, B \u2208 Mn(S).Let S is a Boolean algebra (and by necessity \u03f5 = 1 which means det\u2061\u03f5 is the permanent) has appeared in ) \u2295 \u03b4 \u2297 [1 \u2295 \u03f5]. Since the first summand is 0, so Ir+1\u03f5 is contained in the ideal generated by 1 \u2295 \u03f5. Hence rank\u2061det\u2061\u03f5(A) \u2264 r.Let \u03f5, I)-rank.We can also prove a version of Sylvester's inequality for the  \u2264 min\u2061, rank\u2061det\u2061\u03f5,I(B)) holds for all A \u2208 Mm,n(S) and B \u2208 Mn,p(S).Let r = min\u2061, rank\u2061det\u2061\u03f5,I(B)). If r \u2265 min\u2061 we are done so suppose r < min\u2061 and let \u03b1 and \u03b2 be both arbitrary subsets of {1,2,\u2026, n} of cardinality r + 1. Then either Ir+1\u03f5(A) or Ir+1\u03f5(B) is contained in I. It follows from the Binet-Cauchy theorem that det\u2061\u03f5 \u2208 I.Let \u03f5 \u2208 I is required for our version of Sylvester's inequality to hold; this is largely our motivation for insisting on this condition.It should be noted that the condition 1 \u2295 We also have the following rank-sum inequality.S be a commutative semiring and let \u03f5 be an element of S such that \u03f52 = 1 and that 1 \u2295 \u03f5 is not a unit. Let I be an proper ideal of S which contains 1 \u2295 \u03f5. The inequality rank\u2061det\u2061\u03f5,I(A + B) \u2264 rank\u2061det\u2061\u03f5,I(A) + rank\u2061det\u2061\u03f5,I(B) holds for all A, B \u2208 Mm,n(S).Let m = n and rank\u2061det\u2061\u03f5,I(A) + rank\u2061det\u2061\u03f5,I(B) = n \u2212 1. Hence A, B \u2208 Mn(S). Let us suppose that r = rank\u2061det\u2061\u03f5,I(A). This implies that n \u2212 r \u2212 1 = rank\u2061det\u2061\u03f5,I(B). We can use \u03f5(A + B) \u2208 I. Note that every term in the expansion of det\u2061\u03f5(A + B) is of a power of \u03f5 times det\u2061\u03f5(A[\u03b1\u2223\u03b2]) \u2297 det\u2061\u03f5(B[\u03b1c\u2223\u03b2c]) where \u03b1 and \u03b2 are subsets of {1,2,\u2026, n} satisfying |\u03b1 | = |\u03b2|. Let k = |\u03b1 | = |\u03b2|. If k \u2264 r, then n \u2212 k \u2265 n \u2212 r > rank\u2061det\u2061\u03f5,I(B) and since |\u03b1c | = |\u03b2c | = n \u2212 k we must have det\u2061\u03f5(B[\u03b1c\u2223\u03b2c]) \u2208 I. Similarly, if k > r, then det\u2061\u03f5(A[\u03b1\u2223\u03b2]) \u2208 I. Therefore every term in the expansion of det\u2061\u03f5(A + B) is in I and hence rank\u2061det\u2061\u03f5,I(A + B) \u2264 n \u2212 1 = rank\u2061det\u2061\u03f5,I(A) + rank\u2061det\u2061\u03f5,I(B).We begin by proving the inequality in the special case where r = rank\u2061det\u2061\u03f5,I(A) + rank\u2061det\u2061\u03f5,I(B). If r \u2265 min\u2061 then we are done so suppose r < min\u2061. Now let \u03b1 and \u03b2 be subsets of {1,2,\u2026, m} and {1,2,\u2026, n}, respectively, both of cardinality r + 1. Then rank\u2061det\u2061\u03f5,I((A + B)[\u03b1\u2223\u03b2]) \u2264 rank\u2061det\u2061\u03f5,I(A[\u03b1\u2223\u03b2]) + rank\u2061det\u2061\u03f5,I(B[\u03b1\u2223\u03b2])\u2264rank\u2061det\u2061\u03f5,I(A) + rank\u2061det\u2061\u03f5,I(B) = r. Hence det\u2061\u03f5((A + B)[\u03b1\u2223\u03b2]) \u2208 I and since (A + B)[\u03b1\u2223\u03b2] is an arbitrary r + 1 by r + 1 submatrix of A + B, we have rank\u2061det\u2061\u03f5,I(A + B) \u2264 r = rank\u2061det\u2061\u03f5,I(A) + rank\u2061det\u2061\u03f5,I(B).Now we prove the general case. Let \u03f5, I)-rank of matrices over antinegative commutative semiring.In this section, we look at bijective linear operators which preserve (S be a commutative semiring and A be an m by n matrix over S. The term rank of A is the minimum number of lines (rows and columns) needed to include all nonzero entries of A. The term rank of a matrix A is denoted by t(A).Let f(A) \u2264 t(A) whenever A is an m by n matrix over S.For any commutative semiring, one has S be a semiring and A, B \u2208 Mm,n(S). We write A \u2264 B if there exists C \u2208 Mm,n(S) such that A \u2295 C = B. We note that the relation (\u2264) is a reflexive and transitive relation but not antisymmetric in general. Therefore it is a preorder. It is easy to check that any linear operator T : Mm,n(S) \u2192 Mm,n(S) preserves this preorder. Further, if S is an antinegative semiring then the term rank is a monotone function; that is, if A \u2264 B then t(A) \u2264 t(B).Let S be a commutative semiring and A, B \u2208 Mm,n(S). The Schur product of A and B, denoted as A\u2218B, is an m by n matrix whose th entry is aij \u2297 bij.Let S be a commutative semiring. A matrix A \u2208 Mm,n(S) is called a submonomial matrix if every line (row or column) of A contains at most one nonzero entry. A matrix A \u2208 Mn(S) is called a monomial matrix if every line (row or column) of A contains exactly one nonzero entry.Let P, Q, B) operator is a fundamental concept in the theory of linear preservers over semirings.The concept of  to itself. One says that T is a strong  operator if there exist P \u2208 Mm(S), Q \u2208 Mn(S), and B \u2208 Mm,n(S) such that P and Q are permutation matrices, and all of the entries of B are units and either T(X) = P(X\u2218B)Q or m = n and T(X) = P(XT\u2218B)Q.Let We also use a theorem from the same reference. We note though there is an error earlier in  for the S be a commutative antinegative semiring. If rank\u2061:Mm,n(S) \u2192 \u2124 is a function which satisfies 0 \u2264 rank\u2061(A) \u2264 t(A) for all A \u2208 Mm,n(S) with equality whenever A is a submonomial matrix, then any bijective linear operator which preserves this rank function must be a strong  operator.Let \u03f5, I)-rank satisfies the hypotheses of the above theorem, we now have the following corollary which classifies all bijective linear operators which preserve the -rank.Since the -rank preserver on Mm,n(S) must be a strong  operator.Let \u03f5-rank.In this section, we explore connections between the sign pattern matrices and  sign pattern matrix. If A = [aij] is a real matrix, then the sign pattern of A is obtained from A, by replacing each entry by its signs [A is denoted by Sg(A) = [sg(aij)], where A matrix whose entries are from the set {+1, \u22121,0} is called ats signs , 15. Then by n sign pattern matrices by Qn. Sometimes we may not know the signs of certain entries, so a new symbol, #, has been introduced to denote such entries.Thus in a sign pattern matrix all we know is the sign of each entry. We do not know the exact values of the entries. We denote the set of all The generalized sign pattern matrices are the matrices over the set {+1, \u22121,0, #}, where # corresponds to entries where the sign is unknown.S = {+1, \u22121,0, #}, then  is a commutative semiring with identity, where the operations of addition and multiplication are defined as follows: The set {+1, \u22121,0, #} can be viewed as a semiring. If S. More about the sign pattern semiring can be found in [Clearly all the properties of a semiring are satisfied where 0 is the additive identity and +1 is multiplicative identity. Here +1 and \u22121 are the units of found in .A be a real matrix. The qualitative class of A is Q(A), the set of all real matrices with the same sign pattern as A.Let A is called sign-nonsingular (SNS) if every matrix in its qualitative class is nonsingular.A sign pattern matrix \u03f5-rank. The sign pattern semiring has only two elements whose square is the identity, namely, 1 and \u22121. The ideal generated by 1 = 1 + 1 is the entire semiring but # = 1 + \u22121 generates the unique proper ideal {#, 0}. Therefore \u22121 is the only available choice for \u03f5 and we have a unique \u03f5-rank. Hence det\u2061\u03f5(A) = det\u2061+(A)\u2295(\u22121 \u2297 det\u2061\u2212(A)). It is easy to show that an n by n sign pattern matrix has \u03f5-rank n if and only if it is an SNS matrix. Hence the \u03f5-rank of a sign pattern matrix A is the largest integer k for which there exists a k by k SNS submatrix of A.For matrices over the sign pattern semiring, we can give a more concrete interpretation of the det\u2061\u03f5,I(A) \u2264 rank\u2061det\u2061\u03f5(A). In other words amongst the family of -ranks, choosing I to be the ideal generated by 1 \u2295 \u03f5 gives us the largest possible rank function from this family. The minimal rank functions from this family arise from the choice of I to be a maximal ideal which contains 1 \u2295 \u03f5. In general, there may be many maximal ideals. In this section, we will look at semirings which have a unique maximal ideal. We will use the term sublocal semiring to denote a semiring which has a unique maximal ideal. Sublocality in semirings is essentially the straightforward generalization of the very useful concept of locality in rings. We use the term sublocal semiring because the term local semiring has been used to define a slightly different concept.It was remarked earlier that rank\u2061I of a commutative semiring S is called a k-ideal if for any y \u2208 S with x, x + y \u2208 I one has y \u2208 I.An ideal R to be semiring, the semiring ideals of R are exactly the semiring k-ideals of R which are also exactly the ring ideals of R.We note that if we consider a commutative ring I of a commutative semiring S is called a maximal  ideal if there exists no other proper ideal \u2009\u2009J such that I \u2282 J.A proper ideal S be a commutative semiring. One says that S is a local semiring if S has only one k-maximal ideal.Let k-maximal ideals. This is useful as some semirings do not have proper k-ideals. For example, the sign pattern semiring has only one proper ideal {0, #} and this is not a k-ideal.Now we will define sublocal semirings using maximal ideals instead of S be a commutative semiring. One says that S is a sublocal semiring if S has only one maximal ideal.Let I = {0, #}. We note that this maximal ideal is contained in the proper subsemiring P = {0, +1, #}; P however fails to be an ideal in the sign pattern semiring. The set of all natural numbers, \u2115 = {0,1, 2,\u2026}, forms a sublocal semiring whose only one maximal ideal I = {\u2115/{1}}. All chain semirings are sublocal semirings with S/{1} as a unique maximal ideal. A semifield is a commutative semiring in which all elements except 0 have a multiplicative inverse. (The Boolean and max-plus semirings are examples of semifields.) All semifields are sublocal semirings as the zero ideal is the unique maximal ideal.We note that both local and sublocal semirings are direct but different semiring generalizations of the concept of a local ring. Local semirings have been useful in semiring theory; see  for examWe begin with the following elementary lemma whose proof is identical to the corresponding result for rings.a of a commutative semiring S is a unit of S if and only if a lies outside all maximal ideals of S.An element a be a unit of S; then the ideal generated by a must be S itself and hence a lies outside all maximal ideals of S. If a is not a unit of S, then 1 \u2209 aS and hence there exits a maximal ideal M of S such that a \u2208 aS\u2286M.Let k-ideal. The analog for sublocal semirings is an easy consequence of One of the key results of  is that S is a sublocal semiring if and only if the set of all nonunits of S forms an ideal.A commutative semiring k-ideal is an ideal, it follows that every local semiring is a sublocal semiring. The converse is false. Consider the nonnegative integers \u2115 with the usual addition and multiplication. The unique maximal ideal is \u2115\u2216{1}; the maximal k-ideals are of the form p\u2115 for any prime p.Since every Since the set of nonunits in any sublocal semiring is an ideal, the nonunits are closed under addition. Hence we have the following observation which will prove useful later on.S be a sublocal semiring. Let a1, a2 \u2208 S. If a1 \u2295 a2 is a unit of S, then either a1 or a2 is a unit of S.Let \u03f5 satisfying the condition that \u03f52 = 1 and 1 \u2295 \u03f5 is not a unit. Such an element may not exist in a given semiring; the max-min and max-plus semirings are examples of semiring which lack an \u03f5. Fortunately, there is a known construction which allows us to append such an element. This construction is from [The ranks introduced in the previous sections all require an element  is from , in whicS, \u2295, \u2297) is a commutative semiring then S2 = {\u2223a, b \u2208 S} is also a commutative semiring with addition and multiplication defined as follows: for all a, b, c and d \u2208 S, S2. Essentially this construction allows us to append an element \u03f5 =  with the property \u03f52 = 1 to the semiring S in a natural way giving us a way to apply the \u03f5-determinant theory to semirings which do not have nontrivial self inversive elements. The ideal in S2 generated by  =  + \u03f5 is \u0394 = { : x \u2208 S} which we will call the diagonal ideal. The \u03f5-determinant in this case is the standard bideterminant and the \u03f5-rank is the standard determinantal rank defined as follows.If  \u2260 det\u2061\u2212(B).Let \u03f5, I)-rank of matrices over general semirings.The determinantal rank has been much studied , , , } is also a semiring with the addition and the multiplication defined above for S2. Moreover it is isomorphic to the sign pattern semiring as  = 0,  = +1,  = \u22121, and  = #.Recall that S2 inherits some important properties from S.We complete our paper by showing that the symmetrized semiring S be a commutative semiring. If S is antinegative and has no zero divisors then S2 is also antinegative and has no zero divisors.Let S is antinegative, only 0 has an additive inverse. Let us suppose that  \u2208 S2 has an additive inverse, so there exists  \u2208 S2, such that \u2295 = . Consequently a1 \u2295 a2 = 0 and b1 \u2295 b2 = 0. Since S is antinegative so a1 = a2 = b1 = b2 = 0. Hence  = . Thus only the additive identity has an additive inverse in S2 which means S2 is an antinegative semiring. Now suppose that  \u2208 S2 is a zero divisor, so there exists a nonzero  \u2208 S2, such that \u2297 = . It follows that ((a1 \u2297 a2)\u2295(b1 \u2297 b2)) = 0 and ((a1 \u2297 b2)\u2295(a2 \u2297 b1)) = 0. Since S is antinegative, a1 \u2297 a2 = 0 and b1 \u2297 b2 = 0. Also S has no zero divisors so either a1 = 0 or a2 = 0 and either b1 = 0 or b2 = 0; combining this with ((a1 \u2297 b2)\u2295(a2 \u2297 b1)) = 0, we get either  =  or  = . Hence S2 has no zero divisors.Since S is a semiring, we let U(S) denote the set of units of S. There is a very close relation between the units of S and the units of S2.If U(S2) = { : x \u2208 U(S)}\u22c3{ : x \u2208 U(S)}.If S is a commutative antinegative semiring with no zero divisors then x be a unit in S. Then there exists a nonzero element a in S such that x \u2297 a = 1. Consider \u2297 =  =  and \u2297 =  = . Consequently  and  are units of S2. Now we have to prove that these are the only units for S2. Suppose  \u2208 S2 is a unit in S2. Then there exists a nonzero element  \u2208 S2 such that \u2297 = . Thus ((a \u2297 x)\u2295(b \u2297 y), (b \u2297 x)\u2295(a \u2297 y)) = . Consequently (a \u2297 x)\u2295(b \u2297 y) = 1 and (b \u2297 x)\u2295(a \u2297 y) = 0. Since (b \u2297 x)\u2295(a \u2297 y) = 0 and S is an antinegative semiring, so b \u2297 x = 0 and a \u2297 y = 0. Also given that S has no zero divisors it follows that either b = 0 or x = 0 (note that both b and x cannot be zero because (a \u2297 x)\u2295(b \u2297 y) = 1) and either a = 0 or y = 0 \u2295(b \u2297 y) = 1). Since  and  are nonzero elements of S2 so the units of S2, , have only two choices which are  and . Putting  =  in (a \u2297 x)\u2295(b \u2297 y) = 1, we get a \u2297 x = 1, and this means that x is a unit of S. Putting  =  in (a \u2297 x)\u2295(b \u2297 y) = 1, we get b \u2297 y = 1, and this means that y is a unit of S. Thus all the units in S2 are of the type  and  where x is a unit in S.Let S is a sublocal antinegative semiring with no zero divisors then so is S2.We can now prove that if S is a sublocal antinegative semiring with no zero divisors then S2 is also a sublocal antinegative semiring with no zero divisors.If S is a sublocal antinegative semiring with no zero divisors; then by S2 is an antinegative semiring with no zero divisors. Hence we only need to prove that S2 is sublocal; we show this by proving that the set of all nonunits in S2 forms an ideal of S2. Let M = { : where\u2009\u2009\u2009\u2009is\u2009\u2009not\u2009\u2009a\u2009\u2009unit\u2009\u2009of\u2009\u2009S2} be the set of all nonunits of S2. Let  and  \u2208 M such that \u2295 = , where x is a unit in S. Hence a1 \u2295 a2 = x and b1 \u2295 b2 = 0. Since S is antinegative so b1 = 0 and b2 = 0, and also a1 \u2295 a2 = x, where x is a unit in S so  = , where a1 is a unit in S, or  = , where a2 is a unit in S. It follows that either  is a unit or  is a unit in S2, which is a contradiction to the fact that both  and  \u2208 M. Thus the sum of nonunits in S2 is a nonunit. A similar argument works if \u2295 = , where x is a unit in S. Now suppose that for  \u2208 M and  \u2208 S2 we have \u2297 = , where x is a unit in S. Then (a \u2297 s1)\u2295(b \u2297 s2) = x and (a \u2297 s2)\u2295(b \u2297 s1) = 0. Since S is antinegative so a \u2297 s2 = 0 and b \u2297 s1 = 0, and also S has no zero divisors so either a = 0 or s2 = 0 and either b = 0 or s1 = 0. Clearly  or  cannot be  since (a \u2297 s1)\u2295(b \u2297 s2) = x, so we have  =  or . Further x is a unit in S and (a \u2297 s1)\u2295(b \u2297 s2) = x, so either a \u2297 s1 is a unit in S or b \u2297 s2 is a unit in S. Hence either a is a unit in S or b is a unit in S. We get  =  or  is a unit in S2, which is a contradiction to the fact that  \u2208 M. Thus \u2297 \u2208 M for all  \u2208 M and  \u2208 S2. Hence the set of all nonunits in S2 forms an ideal of S2. Consequently S2 is a sublocal semiring. A similar argument works if \u2297 = , where x is a unit in S.Suppose so using  either a"}
{"text": "G is called a fractional -deleted graph if G \u2212 {e} admits a fractional -factor for any e \u2208 E(G). A graph G is called a fractional -critical deleted graph if, after deleting any n vertices from G, the resulting graph is still a fractional -deleted graph. The toughness, as the parameter for measuring the vulnerability of communication networks, has received significant attention in computer science. In this paper, we present the relationship between toughness and fractional -critical deleted graphs. It is determined that G is fractional -critical deleted if t(G) \u2265 ((b2 \u2212 1 + bn)/a).A graph  G be a graph with the vertex set V(G) and the edge set E(G). For a vertex x \u2208 V(G), we use dG(x) and NG(x) to denote the degree and the neighborhood of x in G, respectively. Let \u03b4(G) denote the minimum degree of G. For any S\u2286V(G), the subgraph of G induced by S is denoted by G[S].All graphs considered in this paper are finite, are loopless, and are without multiple edges. The notation and terminology used but undefined in this paper can be found in . Let G bThe problem of fractional factor can be considered as a relaxation of the well-known cardinality matching problem. It has wide-ranging applications in areas such as scheduling, network design, and the combinatorial polyhedron. For instance, several large data packets are to be sent to various destinations through several channels in a communication network. The efficiency of this work can be improved if large data packets are to be partitioned into small parcels. The feasible assignment of data packets can be seen as a fractional flow problem and it becomes a fractional factor problem when the destinations and sources of a network are disjoint.g and f are two integer-valued functions on V(G) such that 0 \u2264 g(x) \u2264 f(x) for all x \u2208 V(G). A fractional -factor is a function h that assigns to each edge of a graph G a number in  so that for each vertex x we have g(x) \u2264 \u2211e\u2208E(x)h(e) \u2264 f(x). If g(x) = a and f(x) = b for all x \u2208 V(G), then a fractional -factor is a fractional -factor. Moreover, if g(x) = f(x) = k for all x \u2208 V(G), then a fractional -factor is just a fractional k-factor. Throughout this paper, k \u2265 1 is an integer, and we will not reiterate it again.Suppose that G is called a fractional -critical graph if, after deleting any n vertices from G, the resulting graph still has a fractional -factor. A graph G is called a fractional -deleted graph if, after deleting any m edges, the resulting graph still has a -factor. Fractional deleted graph and fractional critical graph, as extensions of the concept of fractional factor, describe the existence of fractional factor in communication networks when certain channels or certain sites are damaged.A graph G is called a fractional -critical deleted graph if, after deleting any n vertices from G, the resulting graph is still a fractional -deleted graph. In particular, the fractional -critical deleted graph is just fractional -critical deleted graph if m = 1.Gao  proposedg, f, n)-critical deleted graph.LetG be a graph and let g, f be two nonnegative integer-valued functions defined on V(G) satisfying g(x) \u2264 f(x) for all x \u2208 V(G). Let n be a nonnegative integer. Then G is a fractional -critical deleted graph if and only ifS and T of V(G) with |S | \u2265n.Let  toughness was first introduced by Chv\u00e1tal in  such that \u03b4(H) \u2265 1 and 1 \u2264 dG(x) \u2264 k \u2212 1 for every x \u2208 V(H), where T\u2286V(G) and k \u2265 2. Let T1,\u2026, Tk\u22121 be a partition of the vertices of H satisfying dG(x) = j for each x \u2208 Tj, where one allows some Tj to be empty. If each component of H has a vertex of degree at most k \u2212 2 in G, then H has a maximal independent set I and a covering set C = V(H) \u2212 I such thatcj = |C\u2229Tj| and ij = |I\u2229Tj| for every j = 1,\u2026, k \u2212 1.Let The lemma below can be deduced from Lemma 2.2 in .G be a graph and let H = G[T] such that dG(x) = k \u2212 1 for every x \u2208 V(H) and no component of H is isomorphic to Kk, where T\u2286V(G) and k \u2265 2. Then there exist an independent set I and the covering set C = V(H) \u2212 I of H satisfyingIi) = k \u2212 i}, 1 \u2264 i \u2264 k, and \u2211i=1k | Ii)-critical deleted graph due to |V(G)|\u2265n + b + 2. In what follows, we assume that G is not complete.If G satisfies the conditions of g, f, n)-critical graph. By \u03b5 \u2264 2, there exist subsets S and T of V(G) such thatS and T such that |T| is minimum. Obviously, we deduce T \u2260 \u2205 and dG\u2212S(x) \u2264 g(x) \u2212 1 \u2264 b \u2212 1 for any x \u2208 T.Suppose that l be the number of the components of H\u2032 = G[T] which are isomorphic to Kb and let T0 = {x \u2208 V(H\u2032)\u2223dG\u2212S(x) = 0}. Let H be the subgraph obtained from H\u2032 \u2212 T0 by deleting those l components isomorphic to Kb.Let V(H)| = 0, then by virtue of (\u03c9(G \u2212 S) = |T0 | +l > 1, then t(G) \u2264 (|S|/\u03c9(G \u2212 S)) \u2264 ((b(|T0 | +l) + bn + 1)/a(|T0 | +l)) < ((b + bn + 1)/a), which contradicts t(G) \u2265 ((b2 \u2212 1 + bn)/a) and b \u2265 2. If \u03c9(G \u2212 S) = |T0 | +l = 1, then |T0 | +l = 1. Since dG\u2212S(x)+|S | \u2265dG(x) \u2265 \u03b4(G) \u2265 2t(G), we have 2t(G) \u2264 b \u2212 1 + |S | \u2264b \u2212 1 + ((b(n + 1) + 1)/a), which contradicts t(G) \u2265 ((b2 \u2212 1 + bn)/a).If |irtue of  we inferV(H)|>0. Let H = H1 \u222a H2, where H1 is the union of components of H which satisfies that dG\u2212S(x) = b \u2212 1 for every vertex x \u2208 V(H1) and H2 = H \u2212 H1. In terms of H1 has a maximum independent set I1 and the covering set C1 = V(H1) \u2212 I1 such thatIi) = b \u2212 i}, 1 \u2264 i \u2264 b, and \u2211i=1b | Ii)( | = |I1|. Let Tj = {x \u2208 V(H2)\u2223dG\u2212S(x) = j} for 1 \u2264 j \u2264 b \u2212 1. Each component of H2 has a vertex of degree at most b \u2212 2 in G \u2212 S by the definitions of H and H2. According to H2 has a maximal independent set I2 and the covering set C2 = V(H2) \u2212 I2 such thatcj = |C2\u2229Tj| and ij = |I2\u2229Tj| for every j = 1,\u2026, b \u2212 1. Set W = V(G) \u2212 S \u2212 T and U = S \u222a C1 \u222a (NG(I1)\u2229W) \u222a C2 \u222a (NG(I2)\u2229W). We infert0 = |T0|. Let t(G) = t. Then when \u03c9(G \u2212 U) > 1, we have\u03c9(G \u2212 U) = 1. In terms of (b | T | \u2212dG\u2212S(T) \u2265 a | S | \u2212bn \u2212 1, we obtainNow we consider that |We infer|U|\u2264|S|+| we have|U|\u2265t\u03c9(G\u2212erms of (|S|+|C1|\u2003uch that|V(H1)|\u2264\u2211uch that\u2211j=1b\u22121|+\u2211rtue of (|V(H1)|+at0 + l.The following proof splits into two cases by the value of at \u2265 b2 \u2212 1 + bn, we have (at \u2212 b)(t0 + l) \u2212 bn \u2212 1 \u2265 b2 \u2212 b \u2212 2 \u2265 0. Hence,  + j + a \u2264 a2 \u2212 1. By t(G) \u2265 ((b2 \u2212 1 + bn)/a), we get n = 0 and \u2211j=1b\u22122ij = 0, which contradicts the definition of H2 and the choice of I2 (see j=1b\u22122ij \u2260 0).There is at least one a < b, then at \u2264 b(b \u2212 2)+(a \u2212 b + 1) + b = (b2 \u2212 1)+(a \u2212 b)+(2 \u2212 b) < b2 \u2212 1, which contradicts t(G) \u2265 ((b2 \u2212 1 + bn)/a).If t0 + l \u2265 2 or b \u2265 3, then by (bt \u2212 a)(t0 + l) \u2212 bn \u2212 1 \u2265 1 we haveb \u2265 a and h1(2) < 0, if a = 1, we deduceab + b \u2212 a \u2212 i + 2 \u2212 b2 \u2264 \u2212b2 + (a + 1)b \u2212 a due to i \u2265 2. Letb \u2265 a, we inferIf n \u2265 1, we obtainIf n = 0, t0 + l = 1, and  =   = , then h1 < 0 and h2 < 0, a contradiction). Then the result follows from the main result in [G is fractional 2-deleted graph if t(G) \u2265 (3/2).In conclusion, we have esult in  which deIn this case,  becomesSubcase 1 (|I1 | = 0). In this subcase,  = , \u2211j=2b\u22121ij = 0, |C2 | \u2264|I2|, and |T | \u22642 | I2|. Thus,I2 | = 1, then |S | \u2264((2(b \u2212 1) + bn + 1)/a) = 3 + 2n and \u03b4(G)\u2264|S | +1 \u2264 4 + 2n. This contradicts \u03b4(G) \u2265 2t(G) \u2265 6 + 4n. Hence, |I2 | \u22652.(i) If Z = {x\u2223x \u2208 C2, dG\u2212S(x) = 1} and z = |Z|. Thus, 0 \u2264 z \u2264 |I2| and NG\u2212S(v) \u2208 I2 if v \u2208 Z. We obtainZ\u2032 = {x\u2223x \u2208 NG(I2)\u2229W, dG\u2212S(x) = 1}, we infera, b) =  and |C2 | \u2264|I2|, we get 2n(1 \u2212 (1/|I2|)) \u2264 (1/|I2|) \u2212 1. By |I2 | \u22652, we derive the contradiction.Let a = b, then max\u2061{hj} = hb\u22121 = \u2212bn and the second largest value of hj is hb\u22122 = \u2212bn \u2212 1. In terms of the analysis of H2, choose a vertex with the smallest degree and add it to I2. Hence, by the definition of H2, we confirm that H2 is connected; each vertex in I2 has degree b \u2212 1 in G \u2212 S except that one vertex has degree b \u2212 2 in G \u2212 S. This fact impliesI2 | = 1, then |S | \u22641 + n and \u03b4(G)\u2264|S | +(b \u2212 1) \u2264 b + n, which contradicts \u03b4(G) \u2265 2t(G) > b + n. Hence, |I2 | \u22652 andn(1 \u2212 (1/|I2|))\u2264((1/b|I2|) \u2212 1), which contradicts b \u2265 2 and |I2 | \u22652.(ii) If Subcase 2 (|I2 | = 0). In this subcase,  | \u22641, and |I(1) | \u22642. Now, we consider the following three subcases.subcase,  becomesSubcase 2.1 (|I(1) | = 1). In this subcase, we have \u2211i=3b | Ii) = k \u2212 1, there exists a vertex y \u2208 In such that NHn(x)\u2229NHn(y) \u2260 \u2205,\u201d we obtain |I1 | \u22652,bn(1 \u2212 (1/|I1|))\u2264(b \u2212 a)(1 \u2212 b) + ((1 \u2212 a)/|I1|), a contradiction.a 2.2 in : \u201cfor eaSubcase 2.2 (|I(1) | = 2). In this subcase, \u2211i=3b | Ii)( | = 0. We can get a contradiction via a similar discussion as in Subcase 2.1.Subcase 2.3(|I(1)| = 0). In this subcase, we have \u2211i=4b | Ii)( | = 0 and |I(3) | \u22641. If |I1 | = 1, then |S | \u2264(((b \u2212 1) + bn + 1)/a). Thus, we inferI1 | \u22652. Let Y = NG(I1)\u2229W.y \u2208 Y such that y only adjacent to one vertex in I1. ResetI1 | \u22652, we obtainbn(1 \u2212 (1/|I1|))\u2264(b \u2212 a)(1 \u2212 b) + ((1 \u2212 a)/|I1|), a contradiction.If there is a vertex Y is adjacent to at least two vertices in I1, we getU = S \u222a C1 \u222a (NG(I1)\u2229W). Due to |I1 | \u22652, we deducebn(1 \u2212 (1/|I1|))\u2264(b \u2212 a)(1 \u2212 b)+((1/|I1|) \u2212 (a/2)), which contradicts a \u2265 1 and |I1 | \u22652.If each vertex in Subcase 3 (|I1 | \u22600 and |I2 | \u22600). From what we have discussed in Subcase 1, we get \u2211j=1b\u22121(b \u2212 2)(b \u2212 j)ij \u2264 \u2211j=1b\u22121(at \u2212 aj \u2212 b + j)ij + bn + 1. Then, we deducei=4b | Ii) | \u22641, |I(1) | \u22642, and n = 0, by what we have discussed in Subcase 2. It is enough to discuss the situation of |I(1) | = 0; other two cases for |I(1) | = 1 and |I(1) | = 2 can be considered in a similar way.I(1) | = 0, we get \u2211i=4b | Ii) | \u22641,I1 | +|I2 | \u22652, we getb \u2212 a)(b \u2212 1)(|I1 | +|I2|) \u2264 1 \u2212 a | I2|, a contradiction.Under the condition of |We complete the proof of the theorem."}
{"text": "The Acknowledgments contain several errors in the \u201c2014 iGEM Interlab Study Contributors\u201d section. In the \u201cETH_Zurich\u201d subsection, the correct spelling of \u201cVerena Jagger\u201d is \u201cVerena J\u00e4ggin\u201d. In the \u201cHUST-China\u201d subsection, the correct spelling of \u201cYunjun Yang\u201d is \u201cYunjun Yan\u201d. In the \u201cLeicester\u201d subsection, the name \u201cRoss Campbell\u201d should be listed after \u201cRichard Badge\u201d. In the \u201cMarburg\u201d subsection, the name \u201cDaniel F. Hurtgen\u201d should be listed after \u201cMatthias Franz\u201d."}
{"text": "Scientific Reports5: Article number: 1422910.1038/srep14229; published online: 09182015; updated: 01072016.This Article contains typographical errors in Table 2.\u22121/2) for samples \u2018HPGA-50 (4.9\u2009\u00d7\u2009102)\u2019, \u2018HPGA-20 (2.1\u2009\u00d7\u2009103)\u2019 and \u2018GA (4.2\u2009\u00d7\u2009103)\u2019 were incorrectly given as \u2018HPGA-50 (4.9\u2009\u00d7\u200910\u22122)\u2019, \u2018HPGA-20 (2.1\u2009\u00d7\u200910\u22123)\u2019 and \u2018GA (4.2\u2009\u00d7\u200910\u22123)\u2019 respectively.Under the column \u2018Before Cycling\u2019, the Warburg coefficient (ohm\u00b7s\u22121/2) for samples \u2018HPGA-50 (5.4\u00d7102)\u2019, \u2018HPGA-20 (2.3\u00d7103)\u2019 and \u2018GA (7.4 \u00d7 103)\u2019 were incorrectly given as \u2018HPGA-50 (5.4\u00d710\u22122)\u2018, \u2018HPGA-20 (2.3\u00d710\u22123)\u2019 and \u2018GA (7.4\u00d710\u22123)\u2019 respectively.In addition, under the column \u2018After Cycling\u2019, the Warburg coefficient (ohm\u00b7s"}
{"text": "Moreover, a comparison of \u2228-rules, \u2228-\u2227 mixed rules, and \u2227-rules ismade from the perspectives of inclusion and inference relationships. Finally,some real examples and numerical experiments are conducted to comparethe proposed rule acquisition algorithms with the existing one in terms ofthe running efficiency.Rule acquisition is one of the main purposes in the analysis of formaldecision contexts. Up to now, there have been several types of rules informal decision contexts such as decision rules, decision implications, andgranular rules, which can be viewed as  Formal concept analysis (FCA) is a field of applied mathematics based on the mathematization of formal concepts and conceptual hierarchy [ formal context  consisting of an object set G, an attribute set M, and an incidence relation I between G and M [ierarchy . This th G and M . Its key G and M , machine G and M , knowled G and M \u201315, and  G and M , 17. Wha implications [ association rules [In FCA, a basic way of describing dependencies between the attributes of a formal context is viaications , 19 or aon rules . Consideon rules \u201323 discuon rules . formal decision context [ training context [ decision rules, in formal decision contexts through combining conditional formal concepts with decision formal concepts. In order to eliminate superfluous decision rules, Li et al. [ granular computing into decision rules for decreasing computation time. Moreover, the notion of a decision rule was extended into the cases of incomplete formal decision contexts [ real formal decision contexts [ decision implications, whose premises and conclusions are taken from conditional attributes and decision attributes, respectively. It should be pointed out that decision rules are special decision implications. Following the above discussion, Zhai et al. [ granular rules, which are special decision rules, with their premises and conclusions being the intents of conditional formal concepts and decision formal concepts, respectively. A detailed investigation on relation between granular rules and decision rules can also be found in [In the real world, a formal context often contains target attributes for the purpose of making decision analysis. A formal context equipped with additional target attributes is called a context   including an object set G, an attribute set M, and an incidence relation I\u2286G \u00d7 M, in which  \u2208 I indicates that the object x has the attribute a and  \u2209 I means the opposite.A formal context is a triple  is said to be regular if, for any  \u2208 G \u00d7 M, the following conditions hold: a1, a2 \u2208 M such that  \u2208 I and  \u2209 I;there exist x1, x2 \u2208 G such that  \u2208 I and  \u2209 I.there exist A formal context  forms an antitone Galois connection, while the pairs of operators  and  form isotone Galois connections . More prG, M, I) be a formal context. For X, X1, X2\u2286G and B, B1, B2\u2286M, the following properties hold:X1\u2286X2\u21d2X2\u2191\u2286X1\u2191, X1\u25a1\u2286X2\u25a1, X1\u22c4\u2286X2\u22c4;B1\u2286B2\u21d2B2\u2193\u2286B1\u2193, B1\u25a1\u2286B2\u25a1, B1\u22c4\u2286B2\u22c4;X\u2286X\u2191\u2193, X\u25a1\u22c4\u2286X\u2286X\u22c4\u25a1;B\u2286B\u2193\u2191, B\u25a1\u22c4\u2286B\u2286B\u22c4\u25a1;X\u2191\u2193\u2191 = X\u2191, X\u25a1\u22c4\u25a1 = X\u25a1, X\u22c4\u25a1\u22c4 = X\u22c4;B\u2193\u2191\u2193 = B\u2193, B\u25a1\u22c4\u25a1 = B\u25a1, B\u22c4\u25a1\u22c4 = B\u22c4.Let  be a formal context, X\u2286G, and B\u2286M. If X\u2191 = B and B\u2193 = X, then  is called a formal concept; if X\u25a1 = B and B\u22c4 = X, then  is called an object-oriented concept; if X\u22c4 = B and B\u25a1 = X, then  is called a property-oriented concept. For each of the cases, X and B are called the extent and intent of , respectively.Let  are, respectively, ordered byG, M, I), respectively. Hereinafter, we denote formal concept lattice by When the formal, object-oriented, and property-oriented concepts of a formal context (d by(2)X,B1\u2264F and  are, respectively, defined byIn the concept lattices It should be pointed out that the relation among formal, object-oriented, and property-oriented concept lattices was discussed in , 39, andG, M, I, N, J), where  and  with M\u2229N = \u2205 are two formal contexts. Here, M and N are called the conditional attribute set and the decision attribute set of , respectively.A formal decision context is a quintuple  is also said to be regular [G, M, I) and  are regular. Hereinafter, the concerned formal decision contexts are all assumed to be regular. Moreover, for convenience, Like the formal context, a formal decision context \u03a0 =  formal concept lattices only, and the derived rules ironment .In order to widen the domain of application of rule acquisition, now we continue to put forward two new types of rules, called \u2228-rules and \u2228-\u2227 mixed rules, based on formal, object-oriented, and property-oriented concept lattices.In this subsection, we propose the notion of a \u2228-rule in formal decision contexts based on formal and object-oriented concept lattices.G, M, I, N, J) be a formal decision context, let G, M, I), and let G, N, J). For any X \u2260 \u2205, Y \u2260 G, and X\u2286Y, then the expression B\u2192\u2228\u2009\u2009C is called a \u2228-rule generated between the object-oriented concept  and formal concept . Here, B and C are called the premise and conclusion of the \u2228-rule B \u2192 C, respectively. The set of all the \u2228-rules generated between the object-oriented concepts in RO(\u03a0).Let \u03a0 = (B\u2192\u2228\u2009\u2009C \u2208 RO(\u03a0), we conclude that each x \u2208 G having at least one conditional attribute in B has all the decision attributes in C. More specifically, if B = {b1, b2,\u2026, bs} and C = {c1, c2,\u2026, ct}, then B\u2192\u2228\u2009\u2009C means that \u201cif b1\u2228b2\u2228\u22ef\u2228bs, then c1\u2227c2\u2227\u22ef\u2227ct,\u201d where \u2228 and \u2227 denote logical disjunction and conjunction operators.Thus, for any B\u2192\u2228\u2009\u2009C with B = {b1, b2,\u2026, bs} and C = {c1, c2,\u2026, ct} can be integrated by the following attribute implication rules (or association rules with their confidences being one):It should be pointed out that the \u2228-rules have something to do with both the attribute implication rules and the association rules  be a formal decision context. For B1\u2192\u2228\u2009\u2009C1, B2\u2192\u2228\u2009\u2009C2 \u2208 RO(\u03a0), if B2\u2286B1 and C2\u2286C1, one says that B2\u2192\u2228\u2009\u2009C2 can be implied by B1\u2192\u2228\u2009\u2009C1. One denotes this implication relationship by B1\u2192\u2228\u2009\u2009C1\u21d2B2\u2192\u2228\u2009\u2009C2. For any B\u2192\u2228\u2009\u2009C \u2208 RO(\u03a0), if there exists B0\u2192\u2228\u2009\u2009C0 \u2208 RO(\u03a0)\u2216{B\u2192\u2228\u2009\u2009C} such that B0\u2192\u2228\u2009\u2009C0\u21d2B\u2192\u2228\u2009\u2009C, then B\u2192\u2228\u2009\u2009C is said to be redundant in RO(\u03a0); otherwise, B\u2192\u2228\u2009\u2009C is said to be nonredundant in RO(\u03a0). We denote by RO*(\u03a0) the set of all of the nonredundant \u2228-rules in RO(\u03a0).Let \u03a0 =  be a formal decision context. DenoteLet \u03a0 = , B\u2192\u2228\u2009\u2009C \u2208 RO(\u03a0). Then, B\u2192\u2228\u2009\u2009C is redundant in RO(\u03a0) if and only if \u03b1O = 0 or \u03b2O = 0, or, equivalently, B\u2192\u2228\u2009\u2009C is nonredundant in RO(\u03a0) if and only if \u03b1O = 1 and \u03b2O = 1.For a formal decision context \u03a0 = (B\u2192\u2228\u2009\u2009C is redundant in RO(\u03a0), there exists B0\u2192\u2228\u2009\u2009C0 \u2208 RO(\u03a0)\u2216{B\u2192\u2228\u2009\u2009C} such that B0\u2192\u2228\u2009\u2009C0\u21d2B\u2192\u2228\u2009\u2009C. By B\u2286B0 and C\u2286C0, which implies that X\u2286X0 and Y0\u2286Y. Noting that X0\u2286Y0 and B0\u2192\u2228\u2009\u2009C0 is different from B\u2192\u2228\u2009\u2009C, we conclude that X \u2282 X0\u2286Y0\u2286Y or X\u2286X0\u2286Y0 \u2282 Y. Consequently, \u03b1O = 0 or \u03b2O = 0.Necessity. If \u03b1O = 0 or \u03b2O = 0, we can prove that B\u2192\u2228\u2009\u2009C is redundant in RO(\u03a0). In fact, when \u03b1O = 0, there exists X \u2282 X0\u2286Y. Suppose B0\u2192\u2228\u2009\u2009C\u21d2B\u2192\u2228\u2009\u2009C. As a result, B\u2192\u2228\u2009\u2009C is redundant in RO(\u03a0). When \u03b2O = 0, there exists X\u2286Y0 \u2282 Y. Suppose B\u2192\u2228\u2009\u2009C0\u21d2B\u2192\u2228\u2009\u2009C. Consequently, B\u2192\u2228\u2009\u2009C is redundant in RO(\u03a0).Sufficiency. If Now, we are ready to put forward a method to derive the nonredundant \u2228-rules from a formal decision context. The method can briefly be described as in G, M, I) can be derived from the formal concept lattice of the complementary formal context of  [LO| denotes the cardinality of the object-oriented concept lattice of  and |LF| denotes that of the formal concept lattice of .Note that the object-oriented concept lattice of  . Then, iG, M, I, N, J), where G = {1,2, 3,4, 5}, M = {a, b, c, d, e, f}, and N = {d1, d2, d3}. The object-oriented concept lattice of  is shown in G, N, J) is shown in \u2009r1: if c, then d1 and d2;\u2009r2: if e, then d2 and d3;\u2009r3: if b, c, d, e, or f, then d2.According to r3 can be divided into the following attribute implication rules:\u2009r3(1): if b, then d2;\u2009r3(2): if c, then d2;\u2009r3(3): if d, then d2;\u2009r3(4): if e, then d2;\u2009r3(5): if f, then d2.It should be pointed out that In this subsection, we continue to put forward the notion of a \u2228-\u2227 mixed rule in formal decision contexts based on formal and property-oriented concept lattices.G, M, I, N, J) be a formal decision context, let G, M, I), and let G, N, J). For any X \u2260 \u2205, Y \u2260 G, and X\u2286Y, then the expression B\u2192\u2228\u2227\u2009\u2009C is called a \u2228-\u2227 mixed rule generated between the property-oriented concept  and the formal concept . Here, B and C are called the premise and conclusion of the \u2228-\u2227 mixed rule B \u2192 C, respectively. The set of all of the \u2228-\u2227 mixed rules generated between the property-oriented concepts in RP(\u03a0).Let \u03a0 = (B\u2192\u2228\u2227\u2009\u2009C \u2208 RP(\u03a0), we conclude that each object having at least one conditional attribute in B and no conditional attribute in M\u2216B has all the decision attributes in C. More specifically, if B = {b1, b2,\u2026, bs}, M\u2216B = {bs+1, bs+2,\u2026, bn}, and C = {c1, c2,\u2026, ct}, then B\u2192\u2228\u2227\u2009\u2009C means the following: \u201cif b1\u2009\u2009\u2228b2\u2009\u2009\u2228\u22ef\u2228\u2009\u2009bs and \u00acbs+1\u2009\u2009\u2227\u2009\u2009\u00acbs+2\u2227\u2009\u2009\u22ef\u2009\u2009\u2227\u2009\u2009\u00acbn, then c1\u2009\u2009\u2227\u2009\u2009c2\u2009\u2009\u2227\u2009\u2009\u22ef\u2009\u2009\u2227\u2009\u2009ct,\u201d where \u2228, \u2227, and \u00ac denote logical disjunction, conjunction, and negation operators, respectively.Thus, for any B\u2192\u2228\u2227C with B = {b1, b2,\u2026, bs}, M\u2216B = {bs+1, bs+2,\u2026, bn}, and C = {c1, c2,\u2026, ct} can be integrated by the following attribute implication rules (or association rules with their confidences being one):It should be pointed out that the \u2228-\u2227 mixed rules have something to do with both the attribute implication rules and the association rules. Concretely, a \u2228-\u2227 mixed rule G, M, I, N, J) be a formal decision context. For any B1\u2192\u2228\u2227\u2009\u2009C1, B2\u2192\u2228\u2227\u2009\u2009C2 \u2208 RP(\u03a0), if B2\u2286B1 and C2\u2286C1, one says that B2\u2192\u2228\u2227\u2009\u2009C2 can be implied by B1\u2192\u2228\u2227\u2009\u2009C1. One denotes this implication relationship by B1\u2192\u2228\u2227\u2009\u2009C1\u21d2B2\u2192\u2228\u2227\u2009\u2009C2. For any B\u2192\u2228\u2227\u2009\u2009C \u2208 RP(\u03a0), if there exists B0\u2192\u2228\u2227\u2009\u2009C0 \u2208 RP(\u03a0)\u2216{B\u2192\u2228\u2227\u2009\u2009C} such that B0\u2192\u2228\u2227\u2009\u2009C0\u21d2B\u2192\u2228\u2227\u2009\u2009C, then B\u2192\u2228\u2227\u2009\u2009C is said to be redundant in RP(\u03a0); otherwise, B\u2192\u2228\u2227\u2009\u2009C is said to be nonredundant in RP(\u03a0). One denotes by RP*(\u03a0) the set of all the nonredundant \u2228-\u2227 mixed rules in RP(\u03a0).Let \u03a0 =  be a formal decision context. DenoteLet \u03a0 = , B\u2192\u2228\u2227\u2009\u2009C \u2208 RP(\u03a0). Then, B\u2192\u2228\u2227C is redundant in RP(\u03a0) if and only if \u03b1P = 0 or \u03b2P = 0, or, equivalently, B\u2192\u2228\u2227C is nonredundant in RP(\u03a0) if and only if \u03b1P = 1 and \u03b2P = 1.For a formal decision context \u03a0 =  can be derived from the formal concept lattice of the complementary formal context of  [LP| denotes the cardinality of the property-oriented concept lattice of  and |LF| denotes that of the formal concept lattice of .Note that the property-oriented concept lattice of  . Then, iG, M, I, N, J) be the formal decision context in G = {1,2, 3,4, 5}, M = {a, b, c, d, e, f}, and N = {d1, d2, d3}. The property-oriented concept lattice of  is shown in G, N, J) can be found in Let \u03a0 =  be a formal decision context, let G, M, J), and let G, N, J). For any X, B, Y, and C are all nonempty and X\u2286Y, then the expression B\u2192\u2227\u2009\u2009C is called a decision rule generated between the formal concepts  and . Here, B and C are called the premise and conclusion of the decision rule B\u2192\u2227\u2009\u2009C, respectively. The set of all the decision rules generated between the formal concepts in RF(\u03a0).Let \u03a0 =  be a formal decision context. For B1\u2192\u2227\u2009\u2009C1, B2\u2192\u2227\u2009\u2009C2 \u2208 RF(\u03a0), if B1\u2286B2 and C2\u2286C1, one says that B2\u2192\u2227\u2009\u2009C2 can be implied by B1\u2192\u2227\u2009\u2009C1. One denotes this implication relationship by B1\u2192\u2227\u2009\u2009C1\u21d2B2\u2192\u2227\u2009\u2009C2. For any B\u2192\u2227\u2009\u2009C \u2208 RF(\u03a0), if there exists B0\u2192\u2227\u2009\u2009C0 \u2208 RF(\u03a0)\u2216{B\u2192\u2227\u2009\u2009C} such that B0\u2192\u2227\u2009\u2009C0\u21d2B\u2192\u2227\u2009\u2009C, then B\u2192\u2227\u2009\u2009C is said to be redundant in RF(\u03a0); otherwise, B\u2192\u2227\u2009\u2009C is said to be nonredundant in RF(\u03a0). One denotes by RF*(\u03a0) the set of all the nonredundant decision rules in RF(\u03a0).Let \u03a0 = (B\u2192\u2227\u2009\u2009C \u2208 RF(\u03a0), one concludes that each object having all the conditional attributes in B also has all the decision attributes in C. More specifically, if B = {b1, b2,\u2026, bs} and C = {c1, c2,\u2026, ct}, then B\u2192\u2227\u2009\u2009C means the following: \u201cif b1\u2227b2\u2227\u22ef\u2227bs, then c1\u2227c2\u2227\u22ef\u2227ct,\u201d where \u2227 denotes logical conjunction operator. Moreover, it is easy to observe that decision rules, \u2228-rules, and \u2228-\u2227 mixed rules are different from each other in terms of their logical reasoning methodologies.Thus, for any The following example is used to show that there does not exist inclusion relationship among decision rules, \u2228-rules, and \u2228-\u2227 mixed rules. That is, we need to confirm three statements: (1) a decision rule may not be a \u2228-rule or \u2228-\u2227 mixed rule; (2) a \u2228-rule may not be a decision rule or \u2228-\u2227 mixed rule; (3) a \u2228-\u2227 mixed rule may not be a decision rule or \u2228-rule.G, M, I, N, J) be the formal decision context in G = {1,2, 3,4, 5}, M = {a, b, c, d, e, f}, and N = {d1, d2, d3}. The formal concept lattice of  is shown in G, N, J) can be found in Let \u03a0 = \u03a9:\u2009G, M, I, N, J) be the formal decision context in G = {1,2, 3,4, 5}, M = {a, b, c, d, e, f}, and N = {d1, d2, d3}. Then, according to the discussion in Examples r1\u2032\u2032 cannot be implied by the \u2228-rules r1, r2, r3(1), r3(2), r3(3), r3(4), and r3(5) based on the inference rule \u03a9, neither can the \u2228-\u2227 mixed rules r1\u2032, r2\u2032, r3\u2032, and r4\u2032;the decision rule r1, r2, r3(1), r3(2), r3(3), r3(4), and r3(5) cannot be implied by the decision rules r1\u2032\u2032, r2\u2032\u2032, and r3\u2032\u2032 based on the inference rule \u03a9, neither can the \u2228-\u2227 mixed rules r1\u2032, r2\u2032, r3\u2032, and r4\u2032;each of the \u2228-rules r1\u2032, r2\u2032, and r4\u2032 cannot be implied by the decision rules r1\u2032\u2032, r2\u2032\u2032, and r3\u2032\u2032 based on the inference rule \u03a9, neither can the \u2228-rules r1, r2, r3(1), r3(2), r3(3), r3(4), and r3(5).each of the \u2228-\u2227 mixed rules Let \u03a0 =  as the decision attribute(s) and other attributes as the conditional attributes. Then, the scaling approach  was usedRO*(\u03a0), RP*(\u03a0), and RF*(\u03a0) are the nonredundant \u2228-rules, \u2228-\u2227 mixed rules, and decision rules, respectively. It can be seen from In the experiments, we still denote the proposed rule acquisition algorithms by Algorithms m hold, then decisions hold.\u201d In order to enrich the existing rule acquisition theory in formal decision contexts, we have proposed two new types of rules, called \u2228-rules and \u2228-\u2227 mixed rules, based on formal, object-oriented, and property-oriented concept lattices. Moreover, a comparison of \u2228-rules, \u2228-\u2227 mixed rules, and \u2227-rules has been made from the perspectives of inclusion and inference relationships. Finally, some numerical experiments have been conducted to compare the proposed rule acquisition algorithms with the existing one in terms of the running efficiency.Rule acquisition is one of the main purposes in the analysis of formal decision contexts. Although there have been several types of rules in formal decision contexts such as decision rules, decision implications, and granular rules, these rules are all \u2227-ones since they have the following form: \u201cif conditions 1,2,\u2026, and From the point of view of real applications, the results obtained in this paper need to be further extended to the cases of fuzzy formal decision contexts , incompl"}
{"text": "If the limit A \u2208 R exists in eitherr\u2192\u221e(log\u2061p][Mh\u22121(Mf(r))/log\u2061p][Mh\u22121(r)) = A orlim\u2061sup\u2061r\u2192\u221e(log\u2061p][Mh\u22121(Mf(r))/log\u2061p][Mh\u22121(r)) = A,lim\u2061inf\u2061 thenLet Mh\u22121(r) is an increasing function of r, it follows from Lemmas \u03b4 > 0, for all sufficiently large values of r, that(i) Since r thatr thatr thatTherefore from  we get frly from  it follohereforelog\u2061pMh\u22121ows thatlim\u2061sup\u2061rr.Hencelim\u2061inf\u2061r (r thatlim\u2061sup\u2061r(ii) follows with a similar argument.f, g, and h be any three transcendental entire functions with g(0) = 0. If p, q, and m are any three positive integers with m > q, then \u03c1hq) = \u221e under any of the following conditions: \u03c1hp, q)((g) = \u221e;\u03c1hp, q)((f), \u03bbg) > 0;min\u2061, \u03bbhp, q)((f)) > 0.min\u2061((g) = \u221e, since Mh\u22121(r) is an increasing function of r, it follows from r, that(i) If \u03c1hp, q)((f) > 0 and \u03bbg > 0.(ii) Suppose Mh\u22121(r) is an increasing function of r, we get from \u03b4 > 0 and any \u025b > 0, for all sufficiently large values of r,As In the line of f, g, and h be any three transcendental entire functions with g(0) = 0. If p, q, and m are any three positive integers with m > q, then \u03bbhp, q)((f\u2218g) = \u221e if any of the following facts happens:\u03bbhp, q)((g) = \u221e;\u03bbhp, q)((f), \u03bbg) > 0.min\u2061 = 0. If p, q, and m are any three positive integers with m > q and any of the following two facts happens\u03c1hp, q)((f), \u03bbg) > 0 ormin\u2061((f), \u03bbg) > 0,min\u2061( thenLet (i) Since(ii) The proof can be carried out in the line of (i) and p, q)th order and relative th lower order. In this process, After modifying the notion of relative order of higher dimensions in case of entire functions in , where a"}
{"text": "We show that 1-quasiconformal mappings on Goursat groups are CR or anti-CR mappings. This can reduce the determination of 1-quasiconformal mappings to the determination of CR automorphisms of CR manifolds, which is a fundamental problem in the theory of several complex variables. Rn are elements of group SO. Quasiconformal mappings on the Heisenberg group were introduced by Mostow  \u2208 C\u221e, if Z1, Z2 \u2208 C\u221e. The number n is called CR dimension of M and k = dim\u2061\u2009M \u2212 2n is called the codimension of M. A smooth mapping f : M1 \u2192 M2 is called CR ifH is then defined as H = TM\u2229iTM and J is the restriction of the complex structure in the ambient complex manifold to H. We recall that, for a manifold M = {\u03c11 = 0, \u03c12 = 0,\u2026, \u03c1k = 0} with M is given by T1,0M = T1,0Cn\u2229\u2009CTM = \u2009{Z \u2208 T1,0Cn | Z\u03c11 = 0, Z\u03c12 = 0,\u2026, Z\u03c1k = 0}, and dim\u2061T1,0M = n \u2212 k.AGn can be realized as the real submanifold of Cn:z = x + iy, wj = uj + ivj, j = 2,\u2026, n, by the diffeomorphism T : Sn \u2192 Gn defined byz = (1/2)((\u2202/\u2202x) \u2212 i(\u2202/\u2202y)), \u2202/\u2202wj = (1/2)((\u2202/\u2202uj) \u2212 i(\u2202/\u2202vj)), j = 2,\u2026, n, is the complex tangent vector of Sn, and we can define the standard CR structure on Sn by T1,0Sn = CZ and dim\u2061T1,0Sn = 1. We can also define quasiconformal mappings on Sn by  : \u03a9 \u2192 f(\u03a9) is a 1-quasiconformal mapping with P; then f is CR or anti-CR; that is, eitherp is some P-differential point.Let Cn. According to the extension theorem , locallP in In the cases of Heisenberg group and Engel group, the condition of Gn, given byd, given byp, q \u2208 Gn. The metric is left-invariant and is related to the dilations by the formulaGn.There is a natural homogeneous norm ||\u00b7|| on  horizontal tangent space at p \u2208 Gn is a subspace HpGn of the tangent space TpGn, and it is spanned by the vector fields X0(p) and X1(p). An absolutely continuous curve \u03b3 :  \u2192 Gn is horizontal if its tangent vectors t \u2208 , lie in the horizontal tangent space H\u03b3(t)Gn. By Chow ,,Df(p) isB on g byX0, X1] = X2, the corresponding matrix isDefine the bilinear form f be a 1-quasiconformal mapping between domains in Gn; since Df\u2217(p) is grading-preserving, then Df\u2217(p) maps Vj\u2009\u2009 onto itself, respectively. In particular V1 = span\u2061{X0, X1}, so for any X \u2208 V1,S. Since V2 = RX2, then\u03bb.Let S above can be written as the following form:\u03b8 \u2208 [0,2\u03c0), where c = |Df\u2217(p)X|G for any X \u2208 V1 with |X|G = 1, \u03bb2/c4 = 1; \u03bb, \u03b8, c depend on p \u2208 Gn.The matrix Df\u2217(p)X = SX, X \u2208 V1, now supposeSince Df\u2217(p) is Lie algebra homomorphism, that is,Df\u2217(p)X2 = \u03bbX2, we find thatBecause f, we haveG is defined by S is an orthogonal matrix. Together with (\u03bb2/c4 = 1. Note that ((1/c)S)T((1/c)S) = I, where I is the 2 \u00d7 2 unit matrix; hences11 = ccos\u2061\u03b8, \u03b8 \u2208 [0,2\u03c0), we have .Thuss21=\u2212\u03bbc2sx; hences112+s122 we have . The resP-differential and the usual derivatives. For the corresponding result on Heisenberg group see p.31 of [C will be used to denote various constants, and the various uses of the letter do not, however, denote the same constant.We get the following relation between f is a quasiconformal mapping between domains in Gn with f(0) = 0, and if f is P-differentiable at 0 and Df(0) preserves the plan P in (0)\u2202xk and Df\u2217(0) is the P-derivative of f at 0.Suppose lan P in , then f f is P-differentiable at 0, by definition, there exists 0 < \u03b4 < 1 such that, for c < \u03b4 and |||| \u2264 1,Df(0) commutes with \u03b4c; then for points  in particular, we haveSince Df(0) preserves the plane\u2009\u2009P, that is,x0\u2032, x1\u2032, x2\u2032,\u2026, xn\u2032) = f; we getNow compare this with Euclidean distances. Since Df(0) is Lipschitz; then by the equivalence of dc and d, we haveC > 0, namely,From Lemma 3.3 in , Df(0) if is continuous, then when |z| < \u03b4 < 1, we haveM2 is a positive constant. Now substituting f is conttituting  and thenThen by , for anyz = , thenIf we let P-differential and the usual tangential mapping. See Proposition 3.2 in [We need the following relation between n 3.2 in  for the f : \u03a91 \u2192 \u03a92 be a quasiconformal mapping between domains in the Goursat group Gn. Suppose f is P-differentiable at p and Df(p) preserves the plan P in X(p) = \u2211k=0nXfk(p)\u2202xk and \u03c4q\u2217 are the tangential mappings of \u03c4q for q \u2208 Gn.Let lan P in ; then\u22121\u2218f\u2218\u03c4p; then F is also a quasiconformal mapping and F(0) = 0. Moreover, by Chain rule  preserves the plan P.Suppose a 3.7 in ), F is PX0(0) = \u2202x0 and X1(0) = \u2202x1. Therefore,By ,1.From  we can sWe need the following proposition due to Capogna f be a 1-has form , and \u03bb =CV1, the corresponding complex basis is ZG, ZG is as Df~\u2217(p)ZGf in the formk = 2,3,\u2026, n. ThenFrom \u03c6 \u2208 Ck(Sn),In fact, for any q = T\u22121(p) \u2208 Sn; by ,  is unknown now. It is also very interesting to determine it, since if this can be solved, we can get the 1-quasiconformal mappings on these groups.The CR automorphism of"}
{"text": "We show with examples that our new class of mappings is a real generalization of several known classes of mappings. We also establish fixed point results for such mappings in metric spaces. Applying our new results, we obtain fixed point results on ordinary metric spaces, metric spaces endowed with an arbitrary binary relation, and metric spaces endowed with graph.We extend the notion of generalized weakly contraction mappings due to Choudhury et al. (2011) to generalized  The well-known Banach's contraction principle has been generalized in many ways over the years \u20136. One oOn the other hand, the concept of the altering distance function was introduced by Khan et al. . In 2011\u03b1-\u03c8-contraction mappings and \u03b1-admissible mappings and established various fixed point theorems for such mappings in complete metric spaces. Afterwards, many fixed point results via the concepts of \u03b1-admissible mappings occupied a prominent place in many aspects \u2192\u222a{2,3, 4,\u2026}. From , we get that T satisfies condition , d}) = 2x \u2212 1 and soy < x, we have \u03d5, d}) = 2x \u2212 1, and hencey > x, we have \u03d5, d}) = 2y \u2212 1, and henceNow we obtain thatT satisfies condition ) \u2265 1 and soIf T satisfies condition . A self-mapping T\u2009:\u2009X \u2192 X is said to be \u03b20-subadmissible if the following condition holds:Let X be a nonempty set and \u03b1\u2009:\u2009X \u00d7 X \u2192 [0, \u221e) a mapping.\u03b1 is said to be forward transitive  if for each x, y, z \u2208 X for which \u03b1 \u2265 1 and \u03b1 \u2265 1 one has \u03b1 \u2265 1;\u03b1 is said to be 0-backward transitive  if for each x, y, z \u2208 X for which 0 < \u03b1 \u2264 1 and 0 < \u03b1 \u2264 1 one has 0 < \u03b1 \u2264 1.Let \u03b1-\u03b2-weakly contraction mappings type A.In this subsection, we give the fixed point results for generalized X, d) be a complete metric space, \u03b1, \u03b2\u2009:\u2009X \u00d7 X \u2192 [0, \u221e) two given mappings, and T\u2009:\u2009X \u2192 X a generalized \u03b1-\u03b2-weakly contraction type A; then the following conditions hold: (a)T is continuous;(b)T is \u03b1-admissible and \u03b20-subadmissible;(c)\u03b1 is forward transitive and \u03b2 is 0-backward transitive;(d)x0 \u2208 X such that 0 < \u03b2 \u2264 1 \u2264 \u03b1.there exists Then T has a fixed point in X.Let (x0 \u2208 X in assumption (d) and letting xn+1 = Txn for all n \u2208 N \u222a {0}, if there exists n0 \u2208 N \u222a {0} such that xn0 = xn0+1, then xn0 is a fixed point of T. This finishes the proof. Therefore, we may assume that xn \u2260 xn+1 for all n \u2208 N \u222a {0}. Since 0 < \u03b2 \u2264 1 \u2264 \u03b1, we getT is \u03b1-admissible and \u03b20-subadmissible thatxn} is a sequence in X such that xn+1 = Txn andn \u2208 N \u222a {0}. By using the generalized \u03b1-\u03b2-weakly contractive condition type A of T, we haven \u2208 N \u222a {0}. Now we obtain thatn \u2208 N \u222a {0}. From  \u2264 d for some n \u2208 N \u222a {0}. Then we haved < d for all n \u2208 N \u222a {0}. This means that {d} is a monotone decreasing sequence. Since {d} is bounded below, there exists r \u2265 0 such thatn \u2208 N \u222a {0}. Taking n \u2192 \u221e in the above inequality, we haver = 0; that is,Suppose that r.Using , we getxn} is a Cauchy sequence. Suppose that {xn} is not a Cauchy sequence. Then there exists \u03b5 > 0 such thatnk > mk \u2265 k, where k \u2208 N. Further, corresponding to mk, we can choose nk in such a way that it is the smallest integer with nk > mk \u2265 k satisfying .From  and 44)xn} is a).From .From (lim\u2061k\u2192\u221ed(.Using (lim\u2061k\u2192\u221ed(.Using (lim\u2061k\u2192\u221ed(an provelim\u2061k\u2192\u221ed(\u03b1-\u03b2-weakly contraction mapping type A in C) \u2264 1 \u2264 \u03b1 for all n \u2208 N and xn \u2192 x as n \u2192 \u221e, then 0 < \u03b2 \u2264 1 \u2264 \u03b1 for all n \u2208 N.if {In the next theorem, the continuity of a generalized X, d) be a complete metric space, \u03b1, \u03b2 : X \u00d7 X \u2192 [0, \u221e) two given mappings, and T\u2009:\u2009X \u2192 X a generalized \u03b1-\u03b2-weakly contraction type A; then the following conditions hold:(a)C) holds;condition ((b)T is \u03b1-admissible and \u03b20-subadmissible;(c)\u03b1 is forward transitive and \u03b2 is 0-backward transitive;(d)x0 \u2208 X such that 0 < \u03b2 \u2264 1 \u2264 \u03b1.there exists Then T has a fixed point in X.Let , we obtain thatn \u2208 N. Now, let us claim that Tx* = x*. Supposing the contrary, from the fact that T is a generalized \u03b1-\u03b2-weakly contractive type A and =C\u2032) and defined byTn+1x0 = T(Tnx0) and T0x0 = x0.Let C\u2032) to the hypotheses of O exists, where z is an element in X satisfying (T has a unique fixed point.By adding condition (tisfying . Then T x and x* are two fixed points of T. By condition (C\u2032) there exists z \u2208 X such thatT is \u03b1-admissible and \u03b20-subadmissible thatn \u2208 N. Since the limit of O exists, we get that {Tnz} converges to some element in X. Let us claim that Tnz \u2192 x as n \u2192 \u221e. Suppose the contrary; that is\u03b1-\u03b2-weakly contractive condition type A of T, we haven \u2208 N. On the other hand, we haven \u2208 N. From =\u03b1-\u03b2-weakly contraction mappings type B.In this subsection, we obtain the existence and uniqueness of fixed point theorems for generalized X, d) be a complete metric space, \u03b1, \u03b2\u2009:\u2009X \u00d7 X \u2192 [0, \u221e) two given mappings, and T\u2009:\u2009X \u2192 X a generalized \u03b1-\u03b2-weakly contraction type B; then the following conditions hold:(a)T is continuous;(b)T is \u03b1-admissible and \u03b20-subadmissible;(c)\u03b1 is forward transitive and \u03b2 is 0-backward transitive;(d)x0 \u2208 X such that 0 < \u03b2 \u2264 1 \u2264 \u03b1.there exists Then T has a fixed point in X.Let  \u2264 d for some n \u2208 N \u222a {0}. From  < d for all n \u2208 N \u222a {0}. This means that {d} is a monotone decreasing sequence. It follows from a sequence {d} bounded below that there exists r \u2265 0 such thatn \u2208 N \u222a {0}. Taking n \u2192 \u221e in the above inequality, we haver = 0; that is,As in the proof of , we get\u03c8 be a complete metric space, \u03b1, \u03b2\u2009:\u2009X \u00d7 X \u2192 [0, \u221e) two given mappings, and T\u2009:\u2009X \u2192 X a generalized \u03b1-\u03b2-weakly contraction type B; then the following conditions hold:(a)C) holds;condition ((b)T is \u03b1-admissible and \u03b20-subadmissible;(c)\u03b1 is forward transitive and \u03b2 is 0-backward transitive;(d)x0 \u2208 X such that 0 < \u03b2 \u2264 1 \u2264 \u03b1.there exists Then T has a fixed point in X.Let  \u2260 0. By using  to the hypotheses of O exists, where z is an element in A satisfying  = \u03b2 = 1 for all x, y \u2208 X in Setting X, d) be a complete metric space and T\u2009:\u2009X \u2192 X a continuous generalized weakly contraction mapping. Then T has a fixed point in X.Let  be a complete metric space and T\u2009:\u2009X \u2192 X a continuous mapping andx, y \u2208 X, where k \u2208 [0,1) andT has a fixed point in X.Let  be a complete metric space and T\u2009:\u2009X \u2192 X a continuous mapping andx, y \u2208 X, whereT has a fixed point in X.Let  be a complete metric space and let T\u2009:\u2009X \u2192 X be a continuous mapping andx, y \u2208 X, whereT has a fixed point in X.Let  be a metric space and R a binary relation over X. A mapping T\u2009:\u2009X \u2192 X is said to be a generalized weakly contraction with respect to R if for each x, y \u2208 X for which xRy one has\u03c8\u2009:\u2009[0, \u221e)\u2192[0, \u221e) is altering distance function, and \u03d5\u2009:\u2009[0, \u221e)\u2192[0, \u221e) is a continuous function with \u03d5(t) = 0 if and only if t = 0.Let  be a metric space and R a binary relation over X and T\u2009:\u2009X \u2192 X a generalized weakly contraction with respect to R; then the following conditions hold: (A)T is continuous;(B)X has a transitive property with respect to R;(C)T is comparative mapping with respect to R;(D)x0 \u2208 X such that x0R(Tx0).there exists Then T has a fixed point in X.Let  defined by\u03b1 = \u03b2 = 1. Since T is comparative mapping with respect to R, we get T is \u03b1-admissible and \u03b20-subadmissible. Also, \u03b1 is forward transitive and \u03b2 is 0-backward transitive since X has a transitive property with respect to R. Since T is a generalized weakly contraction with respect to R, we have, for all x, y \u2208 X,T is generalized \u03b1-\u03b2-weakly contraction mapping types A and B. Now all the hypotheses of T follows from Consider two mappings T, we need the following condition:CR) be a metric space and R a binary relation over X and T\u2009:\u2009X \u2192 X a generalized weakly contraction with respect to R; then the following conditions hold: (A)CR) holds on X;the condition ((B)X has a transitive property with respect to R;(C)T is comparative mapping with respect to R;(D)x0 \u2208 X such that x0R(Tx0).there exists Then T has a fixed point in X.Let (\u03b1 and \u03b2 given by (CR) implies property (C).The result follows from given by  and by oCR\u2032) to the hypotheses of O exists, where z is an element in X satisfying (T has a unique fixed point.By adding condition (tisfying . Then T \u03b1 and \u03b2 given by (CR\u2032) implies property (C\u2032).The result follows from given by  and by oX, d) be a metric space. A set {\u2009:\u2009x \u2208 X} is called a diagonal of the Cartesian product X \u00d7 X and is denoted by \u0394. Consider a graph G such that the set V(G) of its vertices coincides with X and the set E(G) of its edges contains all loops; that is, \u0394\u2286E(G). We assume G has no parallel edges, so we can identify G with the pair (V(G), E(G)). Moreover, we may treat G as a weighted graph by assigning to each edge the distance between its vertices. A graph G is connected if there is a path between any two vertices.Throughout this section, let  be a metric space endowed with a graph G and T\u2009:\u2009X \u2192 X mapping. One says that T preserves edges of G ifLet  be a metric space endowed with a graph G and T\u2009:\u2009X \u2192 X mapping. One says that X has a transitive property with respect to graph G ifLet  be a metric space endowed with a graph G. A mapping T\u2009:\u2009X \u2192 X is said to be a generalized weakly contraction with respect to graph G if for each x, y \u2208 X for which  \u2208 E(G) one has\u03c8\u2009:\u2009[0, \u221e)\u2192[0, \u221e) is altering distance function, and \u03d5\u2009:\u2009[0, \u221e)\u2192[0, \u221e) is a continuous function with \u03d5(t) = 0 if and only if t = 0.Let  be a metric space, T\u2009:\u2009X \u2192 X a given mapping, \u03c8\u2009:\u2009[0, \u221e)\u2192[0, \u221e) an arbitrary altering distance function, and \u03d5\u2009:\u2009[0, \u221e)\u2192[0, \u221e) an arbitrary continuous function with \u03d5(t) = 0 if and only if t = 0. If \u03d5) \u2264 \u03c8) for all x \u2208 X, then T is trivially generalized weakly contraction with respect to graph G, where G = (V(G), E(G)) = .Let  be a metric space endowed with a graph G and T\u2009:\u2009X \u2192 X a generalized weakly contraction with respect to graph G; then the following conditions hold: (A)T is continuous;(B)X has a transitive property with respect to graph G;(C)T preserves edges of G;(D)x0 \u2208 X such that  \u2208 E(G).there exists Then T has a fixed point in X.Let  defined by\u03b1 = \u03b2 = 1. Since T preserves edges of G, we get T is \u03b1-admissible and \u03b20-subadmissible. Also, \u03b1 is forward transitive property and \u03b2 is 0-backward transitive since X has transitive property with respect to graph G. Since T is a generalized weakly contraction with respect to graph G, we getx, y \u2208 X. This implies that T is generalized \u03b1-\u03b2-weakly contraction mapping types A and B. Therefore, all the hypotheses of T follows from Consider two mappings T, we need the following condition.In order to remove the continuity of X, d) be a metric space endowed with a graph G. One says that X has G-regular property if {xn} is the sequence in X such that  \u2208 E(G) for all n \u2208 N and it converges to the point x \u2208 X; then  \u2208 E(G) for all n \u2208 N.Let  be a metric space endowed with a graph G and T\u2009:\u2009X \u2192 X a generalized \u03b1-\u03b2-weakly contraction with respect to graph G; then the following conditions hold: (A)X has G-regular property;(B)X has a transitive property with respect to graph G;(C)T preserves edges of G;(D)x0 \u2208 X such that  \u2208 E(G).there exists Then T has a fixed point in X.Let (\u03b1 and \u03b2 given by (G-regular property implies property (C).The result follows from given by  and by oCG\u2032) to the hypotheses of O exists, where z is an element in X satisfying (T has a unique fixed point.By adding condition (tisfying . Then T \u03b1 and \u03b2 given by (CG\u2032) implies property (C\u2032).The result follows from given by  and by oBy using X, d) be a metric space endowed with a graph G and T\u2009:\u2009X \u2192 X a generalized weakly contraction with respect to graph G; then the following conditions hold:(A)T is continuous;(B)G is connected graph;(C)T preserves edges of G;(D)x0 \u2208 X such that  \u2208 E(G).there exists Then T has a fixed point in X.Let  be a metric space endowed with a graph G and T\u2009:\u2009X \u2192 X a generalized \u03b1-\u03b2-weakly contraction with respect to graph G; then the following conditions hold:(A)X has G-regular property;(B)G is connected graph;(C)T preserves edges of G;(D)x0 \u2208 X such that  \u2208 E(G).there exists Then T has a fixed point in X.Let (CG\u2032) to the hypotheses of O exists, where z is an element in X satisfying (T has a unique fixed point.By adding condition (tisfying . Then T"}
{"text": "G-metric, metric-like, and b-metric to define new notion of generalized b-metric-like space and discuss its topological and structural properties. In addition, certain fixed point theorems for two classes of G-\u03b1-admissible contractive mappings in such spaces are obtained and some new fixed point results are derived in corresponding partially ordered space. Moreover, some examples and an application to the existence of a solution for the first-order periodic boundary value problem are provided here to illustrate the usability of the obtained results.We unify the concepts of  A function \u03c3b : X \u00d7 X \u2192 R+ is a b-metric-like if, for all x, y, z \u2208 X, the following conditions are satisfied:\u2009\u03c3b1)\u2009\u2009\u03c3b = 0 implies x = y;(\u2009\u03c3b2)\u2009\u2009\u03c3b = \u03c3b;(\u2009\u03c3b3)\u2009\u2009\u03c3b \u2264 s[\u03c3b + \u03c3b]. such that X is a nonempty set and \u03c3b is a b-metric-like on X. The number s is called the coefficient of .A b-metric-like space  if x, y \u2208 X and \u03c3b = 0, then x = y, but the converse may not be true and \u03c3b may be positive for all x \u2208 X. It is clear that every b-metric space is a b-metric-like space with the same coefficient s but not conversely in general.In a X = R+, let p > 1 be a constant, and let \u03c3b : X \u00d7 X \u2192 R+ be defined byLet X, \u03c3b) is a b-metric-like space with coefficient s = 2p\u22121.Then,  be a metric-like space and \u03c3b = [\u03c3]p, where p > 1 is a real number. Then, \u03c3b is a b-metric-like with coefficient s = 2p\u22121.Let  = (x + y \u2212 xy)p, where p > 1 is a real number, is a b-metric-like on X with coefficient s = 2p\u22121.Let X = R. Then, the mappings \u03c3bi : X \u00d7 X \u2192 R+ defined byb-metric-like on X, where p > 1, a \u2265 0, and b \u2208 R.Let b-metric-like \u03c3b on X generates a topology \u03c4\u03c3b on X whose base is the family of all open \u03c3b-balls {B\u03c3b : x \u2208 X, \u025b > 0}, where B\u03c3b = {y \u2208 X : |\u03c3b \u2212 \u03c3b| < \u025b} for all x \u2208 X and \u025b > 0.Each b-metric-like space, or G\u03c3b-metric space, as a proper generalization of both of the concepts of b-metric-like spaces and G-metric spaces.Now, we introduce the concept of generalized X be a nonempty set. Suppose that a mapping G\u03c3b : X \u00d7 X \u00d7 X \u2192 R+ satisfies the following:G\u03c3b1) = 0 implies x = y = z;G\u03c3b2) = G\u03c3b, where p is any permutation of x, y, z ;G\u03c3b3) \u2264 s[G\u03c3b + G\u03c3b] for all x, y, z, a \u2208 X .Let G\u03c3b is called a G\u03c3b-metric and  is called a generalized b-metric-like space.Then, b-metric-like space.The following proposition will be useful in constructing examples of a generalized X, \u03c3b) be a b-metric-like space with coefficient s. Then,b-metric-like functions on X.Let  and  are generalized b-metric-like spaces with coefficient s = 2p\u22121. Note that, for x = y = z > 0,\u2009G\u03c3bm = (2x)p > 0 and G\u03c3bs = 3(2x)p > 0.Let X = . Then, the mappings G\u03c3b1m, G\u03c3b1s : X \u00d7 X \u00d7 X \u2192 R+ defined byp > 1 is a real number, are generalized b-metric-like spaces with coefficient s = 2p\u22121.Let By some straight forward calculations, we can establish the following.X be a G\u03c3b-metric space. Then, for each x, y, z, a \u2208 X, it follows that:G\u03c3b > 0 for x \u2260 y;G\u03c3b \u2264 s + G\u03c3b);G\u03c3b \u2264 2sG\u03c3b;G\u03c3b \u2264 sG\u03c3b + s2G\u03c3b + s2G\u03c3b.Let X, G\u03c3b) be a G\u03c3b-metric space. Then, for any x \u2208 X and r > 0, the G\u03c3b-ball with center x and radius r isLet  on X, which we call it G\u03c3b-metric topology.The family of all X, G\u03c3b) be a G\u03c3b-metric space. Let {xn} be a sequence in X. Consider the following.(1)x \u2208 X is said to be a limit of the sequence {xn}, denoted by xn \u2192 x, if lim\u2061n,m\u2192\u221eG\u03c3b = G\u03c3b.A point (2)xn} is said to be a G\u03c3b-Cauchy sequence, if lim\u2061n,m\u2192\u221eG\u03c3b exists (and is finite).{(3)X, G\u03c3b) is said to be G\u03c3b-complete if every G\u03c3b-Cauchy sequence in X is G\u03c3b-convergent to an x \u2208 X such that be a G\u03c3b-metric space. Then, for any sequence {xn} in X and a point x \u2208 X, the following are equivalent:xn} is G\u03c3b-convergent to x;{G\u03c3b \u2192 G\u03c3b, as n \u2192 \u221e;G\u03c3b \u2192 G\u03c3b, as n \u2192 \u221e;Let  and  be two generalized b-metric like spaces and let f : \u2192 be a mapping. Then, f is said to be G\u03c3b-continuous at a point a \u2208 X if, for a given \u025b > 0, there exists \u03b4 > 0 such that x \u2208 X and |G\u03c3b \u2212 G\u03c3b|<\u03b4 imply that |G\u03c3b\u2032(f(a), f(a), f(x)) \u2212 G\u03c3b\u2032(f(a), f(a), f(a))|<\u025b. The mapping f is G\u03c3b-continuous on X if it is G\u03c3b-continuous at all a \u2208 X. For simplicity, we say that f is continuous.Let  and  be two generalized b-metric like spaces. Then, a mapping f : X \u2192 X\u2032 is G\u03c3b-continuous at a point x \u2208 X if and only if it is G\u03c3b-sequentially continuous at x; that is, whenever {xn} is G\u03c3b-convergent to x, {f(xn)} is G\u03c3b-convergent to f(x).Let  be a G\u03c3b-metric space and suppose that {xn}, {yn}, and {zn} are G\u03c3b-convergent to x, y, and z, respectively. Then, we haveyn} = {zn} = a are constant, thenLet  a function. We say that T is an \u03b1-admissible mapping ifLet \u03c8 :  with the usual ordering on R. Define the generalized b-metric-like function G\u03c3b given by G\u03c3b = max\u2061{(x + y)2, (y + z)2, (z + x)2} with s = 2. Consider the mapping f : X \u2192 X defined by f(x) = ln\u2061(1 + xex\u2212/4) and the function \u03c8 \u2208 \u03a8b given by \u03c8(t) = (1/8)t, \u2009t \u2265 0. It is easy to see that f is increasing function. Now, we show that f is a G\u03c3b-continuous function on X.Let xn} be a sequence in X such that lim\u2061n\u2192\u221eG\u03c3b = G\u03c3b, so we have max\u2061{(lim\u2061n\u2192\u221exn + x)2, 4x2} = 4x2\u2009\u2009and, equally, lim\u2061n\u2192\u221exn = \u03b1 \u2264 x. On the other hand, we havef is G\u03c3b-sequentially continuous on X.Let {x, y, z \u2208 X, we havef satisfies all the assumptions of u = 0).For all elements In this section, we present an application of our results to establish the existence of a solution to a periodic boundary value problem see , 31)..31]).X = C be the set of all real continuous functions on . We first endow X with the b-metric-likeu, v \u2208 X where p > 1 and then we endow it with the generalized b-metric-like G\u03c3b defined byX, G\u03c3b) is a complete generalized b-metric-like space with parameter K = 2p\u22121. We equip C with a partial order given byC, \u2aaf) is regular; that is, whenever {xn} in X is an increasing sequence such that xn \u2192 x as n \u2192 \u221e, we have xn\u2aafx for all n \u2208 N \u222a {0}.Let r, as in , it is pt \u2208 I = , with T > 0, and f : I \u00d7 R \u2192 R is a continuous function.Consider the first-order periodic boundary value problem\u03b1 \u2208 C1 such thatt \u2208 I = .A lower solution for  is a fun\u03bb > 0 such that, for all x, y \u2208 C, we havea < 1. Then, the existence of a lower solution for (Assume that there exists tion for  providestion for .t \u2208 I.Problem  can be rx(0) = x(T), we getx(0) in =x(0)creasing . Note thution of .x, y, z \u2208 C. Then, we have\u03b1 is a lower solution for (\u03b1\u2aafS(\u03b1) [Let tion for , so it i (\u03b1\u2aafS(\u03b1) .\u03c8(t) = ln\u2061((a/2p\u22121)t + 1) where 0 \u2264 a < 1. Hence, there exists a fixed point Hence, the hypotheses of  problem ."}
{"text": "Nature Communications6: Article number: 8903 10.1038/ncomms8903 (2015); Published: 07302015; Updated: 03102016y axis should be \u2018cm2V\u22121s\u22121' not \u2018V\u22121s\u22121cm\u22121'. Similarly, the second and third sentences of the second paragraph of the \u2018Quantum yield calculation' section should read \u2018From Fig. 2 and Table 1, we obtain \u03d5\u03bc(1+c1)=11\u2009cm2V\u22121s\u22121 and \u03d5k2=11 \u00d7 10\u221210\u2009cm3s\u22121. These data compare well with the corresponding values of 8.2\u2009cm2V\u22121s\u22121 and 9.2 \u00d7 10\u221210\u2009cm3s\u22121 in Wehrenfennig et al.8 where the support was mesoporous alumina.'This article contains errors in the units used for carrier mobility. In Fig. 2a\u2013d, the units on the"}
{"text": "In recent years, with the wide application of computers, difference system has become one of the important theoretical bases for computer, information system, engineering control, ecological balance, and so forth. As typical nonlinear difference equations, rational difference equations have become a research hot spot in mathematical modelling. The behavior of solutions of the system for rational difference equation has received extensive attention.In , Ozban hIn , KurbanlThe periodicity of the positive solutions of the rational difference systemxn+1=1yn,In , Ozban seader to , 6 and tA, B \u2208 [0, +\u221e) and x\u22121, x0, y\u22121, and y0 \u2208 [0, +\u221e).In this paper, we investigate the behavior of positive solutions for the system of rational difference equationsBefore stating our main results, we state some definitions used in this paper.xn, yn)}n=0\u221e that satisfies }n=0\u221e of  = , n = 0,1,\u2026, and T is called a period.A solution {}n=0\u221e be a solution of system , all solutions of system }n=0\u221e be a solution of system xA, B > 0, it can be seen easily that all solutions of }n=0\u221e be an arbitrary positive solution of ((i)A < B2, then, for each integer n \u2265 0, the subsequence {xn+3l}l=0\u221e \u2192 0, and the subsequence {yn+3l}l=0\u221e \u2192 \u221e.If (ii)A > B2, then, for each integer n \u2265 0, the subsequence {xn+3l}l=0\u221e \u2192 \u221e, and the subsequence {yn+3l}l=0\u221e \u2192 0.If (iii)xn+3yn+3 = xnyn, n = 0,1,\u2026.Let {(ution of .(i)If A n, we will show thatl = 0, xn = (A/B2)0xn. Assume that  For every fixed how thatxn+3l=AB2how thatxn+3l=AB2ion thatyn+3l=B2Ahow thatxn+3l=AB2ion thatyn+3l=B2Ax\u22121 = 8, x0 = 1, y\u22121 = 3, and y0 = 7.(i)A = 1, B = 2, and n = 2, xn+3l}l=0\u221e and {yn+3l}l=0\u221e (shown by the black spot at the top of every peak).For (ii)A = 2, B = 1, and n = 2, xn+3l}l=0\u221e and {yn+3l}l=0\u221e (shown by the black spot at the top of every peak).For Let"}
{"text": "There is an error in Table 4. The value listed for \u201cWork + Regular Saving\u201d under \u201cModel 2\u201d should be \u201c0.382\u20136.623\u201d. Please see the corrected"}
{"text": "Reconciliation methods explain topology differences between a species tree and a gene tree by evolutionary events other than speciations. However, not all phylogenies are trees: hybridization can occur and create new species and this results into reticulate phylogenies. Here, we consider the problem of reconciling a gene tree with a species network via duplication and loss events. Two variants are proposed and solved with effcient algorithms: the first one finds the best tree in the network with which to reconcile the gene tree, and the second one finds the best reconciliation between the gene tree and the whole network. Moreover, we propose a faster algorithm solving the problem described in [Reconciliations explain topology incompatibilities between a species tree and a gene tree by evolutionary events - other than speciation - affecting genes , for a r and thisribed in  when resWe start by giving some basic definitions that will be useful in the paper.Definition 1 (Rooted phylogenetic network)A rooted phylogenetic network N on a label set (N), has indegree 0 and outdegree 2. All other internal nodes have either indegree 1 and outdegree 2 (speciation node), or indegree 2 and outdegree 1 (hybridization node).V(N), I(N), E(N), L(N) and N. The size of N, denoted by |N|, is equal to |V(N)| + |E(N)|. Given x in V(N), we denote by x Nthe subnetwork of N rooted at x, i.e. the subgraph of N consisting of all edges and nodes reachable from x. If x is a leaf of N, we denote by s(x) the label of x in x is a speciation node, we denote by p(x) the only parent of x.Denote by x and y of N, we say that x is lower or equal to y in N, denoted by x \u2264N y , if and only if there exists a path (possible reduced to a single node) in N from y to x (resp. and x \u2260 y). If x \u2264N y, then, for every path p from y to x, denote by length(p) the number of speciation nodes in N such that x <N z \u2264N y. If N is a tree, then, for every two nodes u, v of N, N LCA [N that is above or equal to both u, v.Given two nodes N   is the lx in N, two paths of N starting from x are said to be separated if each path contains a different child of x. Let x, y be two nodes of N. Denote by z of N such that there exist two separated paths in N from z to x and y. For example, in Figure N is a tree and x, y are not comparable, N M contains exactly one node, which coincides with N LCA.Given a speciation node N has at most k hybridization nodes, we say that N is of level-k [networks and trees, respectively. In this paper, we allow trees to contain artificial nodes, i.e. nodes with indegree and outdegree 1.If every biconnected component of  level-k . A rooteB be a biconnected component of a network N. Then B contains exactly one node r(B) without ancestors in B [[r(B) the root of B. If B is not trivial, i.e. B consists of more than one node, we can contract it by removing all nodes of B other than r(B), then connect r(B) to every node with indegree 0 created by this removal. Then the following definitions are well-posed.Let rs in B [, Lemma 5Definition 2 (Tree bc(N)) Given a network N, the tree bc(N) is obtained from N by contracting all its biconnected components.N and its associated tree bc(N).For example, Figure bc(N) that corresponds to a biconnected component B in N. Given two biconnected component i, Bj B, we say that i B\u2264 N Bj (resp. i <N Bj B) if and only if i Bis the parent (resp. a child) of Bj if bc(N). We also denote by NLCA the biconnected component corresponding to N.Let denote by Definition 3 (Elementary network) Given a network N, each biconnected component B that is not a leaf of N defines an elementary network, denoted by N(B), consisting of B and all cut-edges coming out from B.Definition 4 (Switchings of a network [) Given a network N, a switching S of N is obtained from N by choosing, for each hybridization node, an incoming edge to switch on and the other to switch off. Once this is done, we also switch off all switched-on edges with the target node having only switched-off outgoing edges  for an example). For each bi-connected component B of N, we also denote by S(B) the subgraph of S restricted to N(B). network ) Given ac) a switching of the level-2 network in (a). We denote by onV(S) the set of nodes of S that are not an endpoint of any switched-off edge. A path of S is a path of N that uses only switched-on edges.Switchings will be used in the next section to model gene histories for genes evolving in a species network. For example, Figure G will denote a tree and N a network such that there is a bijection between L(N) and G represents a gene tree such that each leaf corresponds to a contemporary gene and is labeled by the species containing this gene, while N is a species network such that each leaf represents an extant species. In the cophylogeny problem, G represents a parasite tree such that each leaf corresponds to a parasite species that is labeled by the species that hosts it, while N is a host network such that each leaf represents an extant species.Hereafter, G is associated to a node of S and an event - a speciation u of G with a leaf x of S such that s(u) = s(x). A speciation in a node u of G is constrained to the existence of two separated paths from the mapping of u to the mappings of its two children, while the only constraint given by a duplication event is that evolution of G cannot go back in time. More formally:We will now extend the definition of ation in  to netwoDefinition 5 (Reconciliation) Given a tree G and a network N such that a reconciliation between G and N is a function \u03b1 that maps each node u of G to a pair (r\u03b1(u), \u03b1e(u)) where \u03b1r(u) is a node of V(N) eand \u03b1(u) is an event of type  or orsuch that:\u00a0if and only if u \u2208 L(G), r\u03b1(u)\u2208L(N) and s(u) = s(r\u03b1(u));\u2022 for every u \u2208 I(G) with child nodes {u1, u2}, if then \u2022 for any two nodes u, v of V(G) G u, if such that v <r then \u03b1(v) \u2264N \u03b1r(u). r Otherwise, \u03b1(v) N \u03b1r<(u).\u2022 N is a tree, this definition coincides with the one given in [Note that, if given in .\u03b1, denoted by d(\u03b1) is the number of nodes u of G such that x, y of N, the distance between x and y, denoted by dist, is defined as follows:The number of duplications of  done in . In morey \u2264N x, then dist = minp length(p) over all possible paths p from x to y;\u2022 If dist = +\u221e.\u2022 otherwise, u \u2208 I(G) with child nodes {u1, u2}, the number of losses associated with u in a reconciliation \u03b1, denoted by \u03b1l(u), is defined as follows:Then, for every \u03b1l(u) = min{dist) + dist), dist)+dist)} where x1, x2 are the two children of r\u03b1(u);\u2022 if \u03b1l(u) = dist(r\u03b1(u), r\u03b1(u1)) + dist(r\u03b1(u), r\u03b1(u2)).\u2022 if \u03b1, denoted by l(\u03b1), is the sum of \u03b1l(\u00b7) for all internal nodes of G. Thus, the cost of \u03b1, denoted by cost(\u03b1), is d(\u03b1)\u00b7\u03b4 + l(\u03b1)\u00b7\u03bb, where \u03b4 and \u03bb are respectively the cost of a duplication and a loss event. We use cost to denote the cost of the minimum reconciliations between G and N. A reconciliation having the minimum cost is called a most parsimonious reconciliation.The number of losses of a reconciliation S be a switching of N, then a reconciliation between G and S is defined similarly to Definition 5 except that only switched-on edges are considered when defining paths, and only nodes in onV(S) are counted for calculating the length of the shortest path in the definition of dist. This is done to model the fact that, since each gene of an hybridization species is inherited from one of its two parents, we should not count as a loss the fact that the other parent does not contribute. Moreover, note that, for every two nodes x, y of S such that S yx \u2264, there is a unique path from y to x in S.Let N is a tree, there is a unique reconciliation (the LCA reconciliation) between G and N which has minimum cost and this reconciliation can be found in O(|G|) time [When G|) time -18 as fou in L(G), r\u03b1(u) is defined as the only node x of S such that s(u) = s(x), and \u2022 for each node u in I(G) with child nodes {u1, u2}, r\u03b1(u) = N LCA(r\u03b1(u1), r\u03b1(u2)); if r\u03b1(u1) N \u2264r\u03b1(u2) or r\u03b1(u2) N \u03b1r\u2264(u1) then \u2022 for each node e \u03b1is totally defined by r\u03b1, hence it can be omitted, and we will use \u03b1 to refer to r\u03b1. Note that the algorithms used on trees to find the LCA reconciliation can also be used on switchings, which - when only switched-on edges are considered - are actually trees. Hereafter, when we refer to the reconciliation between a tree and a switching, we refer to the LCA reconciliation between them. The problems in which we are interested in here are defined as follows:In the LCA reconciliation, the mapping Problem 1 BEST SWITCHING FOR THEODELInput A tree G, a network N such that and positive costs \u03b4 and \u03bb for respectively  and  events.Output A switching S of N such that the cost is minimum over all switchings of N.Problem 2 MINIMUMECONCILIATION ON NETWORKSInput A tree G, a network N such that and positive costs \u03b4 and \u03bb for respectively  and  events.Output A minimum reconciliation between G and N.Remark 1 For the sake of simplicity, we suppose that G does not contain any internal node u such that u are mapped to a leaf of N). If it is not the case, we can simplify the instance by replacing in G each such subtree Gu by a leaf labeled by the unique label in   For every u \u2208 V(G), B(u) is defined as the lowest biconnected component B of N such that \u2112(Nr(B)) contains \u2112(uG).Then the following remark holds:Remark 2 If u \u2208 L(G), then B(u) is the only leaf of N such that s(u) = s(B(u)). If u \u2208 L(G), then B(u) = N LCA(B(u1), B(u2)) where u1, u2 are child nodes of u in G.N Gthe tree obtained from G by adding some artificial nodes on the edges of G and label each node of N Gby a biconnected component of N via an extension of the labeling function B(\u00b7) as follows:We define by Definition 7 (Tree N G) N is obtained from G as follows: for each internal node u in G with child nodes uThe tree G1, u2 such that there exist k biconnected components (u) and strictly above B(u1), we add k artificial nodes v1 > ... >k on the edge v, and B(j v) is fixed to 2.N G. We have the following lemma:See Figure Lemma 1 Let u be a node of I(N G), and u' be one of the children of u. If u is an artificial node, then B(u) is the parent of B(u') and a child of B(p(u)); otherwise, B(u') is either equal to B(u) or one of its children.Proof: Suppose that u is an artificial node. Then, by Definition 7, B(p(u)), B(u), B(u') are three consecutive nodes of N G, thus B(u) is the parent of B(u') and a child of B(p(u)). Suppose now that u is not an artificial node, and let u\" be the child of u in G such that u\" \u2264Gn u' \u2264Gn u. If B(u\") = B(u'), then by definition, u\" = u' because no artificial node is added between u and u\", and thus the claim holds. If B(u') \u2260 B(u\"), then by Definition 7, B(u') is the highest biconnected component of N that is below B(u) and above B(u\"), which is then a child of B(u).Definition 8 (B G) B is the set of all maximal connected subgraph H of GN such that BLet B be a biconnected component of N different from a leaf, then G(u) = B for every u \u2208 I(H).For example, in Figure Lemma 2 Let B be a biconnected component of N different from a leaf, then we have the following:(i) for every H \u2208 B, H is either a binary tree or an edge whose upper extremity is an artificial node. Moreover, for every leaf u of H, BG(u) is a child of B.(ii) if B = B(r(G)), B consists of one binary treethen G.Proof: (i) First, suppose that I(H) contains an artificial node u. Then this artificial node is the only internal node of H; indeed, by Lemma 1, the value of B(\u00b7) for the parent and the child of u are both different from B. Thus, H consists of only one edge whose upper extremity is u. If I(H) does not contain any artificial node, it follows that H is a binary tree. Moreover, by Lemma 1 and Definition 8, B(u) is a child of B for every leaf u of H.(ii) Suppose now that B = B(r(G)), and B Gcontains at least two components H1, H2, rooted at two different nodes r1, r2 where B(r1) = B(r2) = B. Let B(v) = B for every node v on the two paths from r to r1 and to r2, because B(r(G)) = B(r1) = B(r2) = B and H1 and H2 since they can both be extended. Hence B Gcontains only one component. Suppose that this component is an edge; thus, its upper extremity is an artificial node that has been added on the path from a node u to a node v of G where u is strictly higher than r(G). But this is not possible, because r(G) is the highest node of G. Therefore, B Gcontains one binary tree.i Bdifferent from a leaf, denote by i Sbe a switching of N(iB), and let H be a tree in u \u2208 L(H), B(u) is a child of i Band thus r(B(u)) is a leaf of N(iB), which is also a leaf of iS. Hence, we can define a reconciliation between H and iS, denoted by u of H is mapped to the leaf r(B(u)) of iS.Given a biconnected component Lemma 3 Let S be a switching of N, and let \u03b1 be the reconciliation between G and S. For every u \u2208 I(G), there exists such that u \u2208 I(H), and The proof of this lemma is deferred to the appendix. The following definition will be useful later on.Definition 9 ) i be a biconnected component of N different from a leaf, and Si a switching of NLet B(iB); then cost is defined as follows:i if B= B(r(G)), and otherwise;\u2022 where u is the only leaf of H.\u2022 H to a reconciliation between G and any switching of N that contains iS. For example, in Figure H is the edge  and, if i Sis the switching on the left, then cost = \u03bb.As we will see later, this cost corresponds to the contribution of N :The next lemma is a central one, and it permits to solve Problem 1 independently per each biconnected component of Lemma 4 Let {B1,..., p} be the biconnected components of N that are not leaf nodes, and let S be a switching of N where each elementary network N(Bi) has Si as switching. Then BProof: Let \u03b1 be the reconciliation between G and S. Denote by \u03b1d(iS) the number of nodes in I(iS) associated to a duplication by \u03b1. By Remark 1, no duplication happens at leaves of S. Additionally, u of H (Lemma 3).i Sif and only if it is also on (resp. off) in S. Let x, y be two nodes of S such that S xy \u2264. Then we define z in onV(iS) \\ L(iS) such that S z \u2264S xy <. Then, for each internal node u of G, we define the number of losses associated with u in iS, denoted by \u03b1l, similarly to \u03b1l(u) but using the function dist. Then, u1, u2 be two children of u in G, then \u03b1l >0 only if the path from \u03b1(u) to either \u03b1(u1) or \u03b1(u2) in S contains at least a node of iB. Therefore, we have three possible cases: (1) \u03b1(u) is in is in iB, (3) \u03b1(u) is above r(iB) while either \u03b1(u1) or \u03b1(u2) is in a biconnected component below iB. In case (1), by Lemma 3, there exists a binary tree H of u\u2208I(H), and v\u2208I(iS), thus i Sthat case (2) holds for u1, then u1 must be the root of a binary tree H1 of t G, and the contribution of u1 to \u03b1l is u1 \u2260 r(G). Finally, let suppose that case (3) holds for u1 and let a, ub ube the artificial nodes added on the edge  of G such that B(au) = i Band B(bu) is a child of iB. Hence, u1 to \u03b1l is dist(r(iS), r(B(bu))). Let call u in the first, second, and third case. By construction, V1 is disjoint from u has two children u1, u2 such that u1 is in iB, and u2 is below iB, then u must be in iB, i.e. cannot be above r(iB). Thus,Let us now consider the loss count. Note that a node/edge is on (resp. off) in Therefore,G are considered. Hence, the second term between squared brackets corresponds to As proved above (in case (3)), the first term between squared brackets is equal to Therefore, N for which its reconciliation with G has the smallest cost, by analyzing each biconnected component of N independently. See Figure Algorithm 1 computes the switching of Algorithm 1 Solving Problem 1Input: A species network N and a gene tree G such that \u03b4, \u03bb for duplication and loss events, respectively.1: Output: A switching S of N that is optimal in the sense of Problem 1.2: N Gand its labeling function B(\u00b7);3: Compute the tree i Bof N that is not a leaf;4: Compute for each biconnected component i Bof N do5: for each switching N(iB) do6:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a07:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0N(iB) with the lowest value of j costover all j;8:\u00a0\u00a0\u00a0\u00a0\u00a0S of N in which each elementary network N(iB) has 9: Return the switching Theorem 2 Let N be a level-k network with p biconnected components. Algorithm 1 runs in O(|N| + 2k\u00b7p\u00b7|G|) time and returns a switching S of N such that cost is minimum.Proof: Complexity: It is well-known that all biconnected components of N can be computed in linear time, i.e. O(|N|), by using depth-first-search [LCA operations on a tree can be performed in constant time [B(\u00b7) can be computed in O(|NG|). Hence, the tree N Gcan be constructed in times O(|N | G+ |N|).t-search . After aant time ,22. ThusN Gwhich takes time O(|N G|), we can compute iB. Each biconnected component i Bof N has at most k hybridization nodes, then N(iB) has at most 2k switchings. For each switching N(iB), it takes O(|B G|) to compute j cost[Bi Gis the size of N G. Therefore, the total complexity of the loop at lines 5 - 8 is O(2k \u00b7|N G|). Each edge of G can have at most p artificial nodes added to it. Hence in the worst case O(|N G|) = O(|G|\u00b7p), i.e. the total complexity of Algorithm 1 is O(|N| + 2k\u00b7p\u00b7|G|).By a simple traversal of te costj ,24. The Correctness: Let S be a switching of N where each elementary network N(iB) has i Sas the switching. By Lemma 4, cost = cost is minimum if and only if iS. Lines 5-8 in Algorithm 1 search, for each network N(iB), the tree G and a network N, in [G on N can be solved in polynomial time, when host switchings  are also accounted for. In their model, each node of N is dated by a single date while each node of G is dated by a set of dates, and they search for a parsimonious reconciliation between G and N, i.e. one that has minimum cost, under the constraint that an event associated to a node u of G can only happen at a node/edge of N whose date is contained in the set of possible dates of u. Although the algorithm complexity stays polynomial, it is very high: O(\u03c43\u00b7|G|\u00b7|N|5) for a binary tree and a binary network, where \u03c4 is the number of possible dates of the nodes of G and N, which is at most O(|G| + |N|). A drawback of this model is that it requires information on the dates that is often not available. Moreover, transfers are not always possible in all parts of Tree of Life. Here, we take into account only speciation, duplication and loss events, and consider G and N as undated (see Problem 2 and Definition 5 for more details). Using a similar dynamic algorithm on this simpler model, and by some further analyses, we provide an algorithm that is a generalization of the LCA algorithm to networks that has a much smaller complexity than the algorithm in [O(h2\u00b7|G|\u00b7|N|), where h is the number of hybridization nodes of N.Given a tree rk N, in  the authrithm in , namely x, y be two nodes of N. Denote by z to x and to y passes through z'. For example, in Figure m1, m2, m3 are in m1 and m2 are in m3 to x, y must pass m2.Let For the sake of simplicity, we consider only reconciliations without duplications on hybridization nodes: indeed, since losses are not counted at hybridization nodes, a duplication on such nodes can be moved to its only child without changing the total cost of the reconciliation.Lemma 5 Let \u03b1 be a reconciliation of minimum cost between G and N, then for every node u of G that has two children u1, u2, we have:(i) if  then (ii) if r then either \u03b1(u1) \u2264N r\u03b1(u2) rand \u03b1(u) = r\u03b1(u2), ror \u03b1(u2) \u2264N r\u03b1(u1) rand \u03b1(u) = r\u03b1(u1).Proof: (i) Suppose that r\u03b1(u) must be a node of x1, x2 be two children of r\u03b1(u) such that \u03b1l(u) = dist)+dist). Suppose that y in r\u03b1(u) to r\u03b1(u1) (resp. to r\u03b1(u2)) passes for y. Let y1, y2 be the two children of y.x1 to r\u03b1(u1) passes through y, y1, while the one from x2 to r\u03b1(u2) passes y, y2. Consider the reconciliation \u03b1' such that v \u2260 u, while \u03b1' respects Definition 5, and that d(\u03b1) = d(\u03b1'). Denote by f = dist(r\u03b1(u), y), then f >0. The numbers of losses in \u03b1 and \u03b1' are different on those associated with u and p(u) (if u is not the root of G). We thus have \u03b1l(p(u)) \u2265 \u03b1'l(p(u)) - f if u \u2260 r(G). Moreover, \u03b1l(u) = dist) + dist) = dist + 1 + dist) + dist + 1 +dist) \u2265 dist(r\u03b1(u), y) + dist + dist(r\u03b1(u), y)+dist)) \u2265 2\u00b7f +\u03b1' l(u). Hence, whether u coincides with r(G) or not, l(\u03b1) > l(\u03b1'), i.e. cost(\u03b1) > cost(\u03b1'), a contradiction.First, we suppose that the shortest path from x1 to r\u03b1(u1) and to r\u03b1(u2) pass through y, and then pass through y1 (or y2). This means that y1 is above both r\u03b1(u1) and r\u03b1(u2). Let y' be one of the lowest nodes below or equal to y1 that is above both r\u03b1(u1) and r\u03b1(u2). Let y', and sup-1 2 pose, without loss of generality, that r\u03b1(u1) and r\u03b1(u2). Hence, the shortest path from x1 to r\u03b1(u1) must pass, in the order, through y, y1, y', and y', while the shortest path from x2 to r\u03b1(u2) must pass through y, y1, y', and \u03b1'as done above apart for \u03b1'(u), which is fixed to y', we arrive at a contradiction by a similar argument.Suppose now that both the shortest paths from r\u03b1(u1), r\u03b1(u2) are not comparable. Hence, r\u03b1(u) to r\u03b1(u1) and to r\u03b1(u2) must pass through y. Similarly to case (i), we have that the reconciliation \u03b1' that coincides with \u03b1 apart for the fact that \u03b1, a contradiction. Hence, r\u03b1(u) N 2 \u2133in(r\u03b1(u1), r\u03b1(u2)). Considering now the reconciliation \u03b1' that coincides with \u03b1 but for \u03b1' (u), which is fixed to d(\u03b1) = d(\u03b1')+1, and u \u2260 r(G). Let x1, x2 be the two children of r\u03b1(u). If the shortest path from r\u03b1(u) to r\u03b1(u1) passes through x1 while the shortest path from r\u03b1(u) to r\u03b1(u2) passes through x2, then cost(\u03b1) > cost(\u03b1'), a contradiction. Thus, both the two shortest paths from r\u03b1(u) to r\u03b1(u1) and r\u03b1(u2) must pass through x1 (or x2). Let y be one of the lowest nodes located on both of these paths. Then, y \u2208 N \u2133in(r\u03b1(u1), r\u03b1(u2)). By the same argument as in the previous case, the reconciliation \u03b1\" coinciding with \u03b1 but for \u03b1, a contradiction.(ii) Suppose that r\u03b1(u1) N \u03b1r \u2264(u2) or r\u03b1(u2) N \u03b1r \u2264(u1). Suppose that the first case holds (the second case is similar), but r\u03b1(u) \u2260 r\u03b1(u2), i.e. r\u03b1(u2) N \u03b1r <(u). Let x1, x2 be two children of r\u03b1(u). If the shortest path from r\u03b1(u) to r\u03b1(u1) passes through x1 while the shortest path from r\u03b1(u) to r\u03b1(u2) passes through x2, then by replacing e\u03b1(u) by an r\u03b1(u) to r\u03b1(u1) and r\u03b1(u2) must pass through x1 (or x2). Let y be a node that is located on both paths such that there is no other node below y on these two paths. Then, y \u2260 N \u2133in(r\u03b1(u1), r\u03b1(u2)). By the same argument as in the case (i), the reconciliation \u03b1' such that v \u2260 u, \u03b1, a contradiction.Hence, either G and N. Let \u03b1 be a reconciliation between G and N. For every u \u2208 V(G), denote by \u03b1cost(u) the cost of the reconciliation of \u03b1 restricted to uG. Hence, if \u03b1 is a most parsimonious reconciliation, then \u03b1cost(u) is the minimum cost among all reconciliations between u Gand N that maps u to r\u03b1(u). Algorithm 2 aims at computing, for each u, the set u) containing all pairs  such that c is the minimum cost among all reconciliations between u Gand N mapping u to x. It is straightforward to see that the cost of a most parsimonious reconciliation between G and N is the minimum cost involved in a pair in r(G)).Now, we are ready to describe a dynamic algorithm to compute a reconciliation of minimum cost between merge used in Algorithm 2 takes as input two lists of pairs  - where x is a node of N and c is a real number - and merges them keeping, for each x, the pair  with the smallest value of c. The method computeMin used in Algorithm 2 is detailed in Algorithm 3. This method computes, for two nodes y, z of N, the set y and z up to the root of N . For this, it labels each node v in such a way that, if v is not strictly above y and z, then label (v) = \u2205. Otherwise, label(v) is the lowest node such that every path from v to y and z passes through it. This method also computes the value of the function dist between y (resp. z) and each node visited in the corresponding breath-first-search.The function Algorithm 2 Solving Problem 2Input: A network N and a tree G such that \u03b4, \u03bb for duplication and loss events, respectively.1: Output: The set u) of pairs  for every 2: for each node u of G in post-order do3: u) \u2190 \u2205;4:\u00a0\u00a0\u00a0if u is a leaf then5:\u00a0\u00a0\u00a0x be the leaf of S such that s(x) = s(u);6:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Let u) \u2190 {};7:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0else8:\u00a0\u00a0\u00a0u1, u2 be the two children of u;9:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Let for each  \u2208 u1) and each  \u2208 u2) do10:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0computeMin;11:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0C \u2190 \u2205;12:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0for each do13:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x1, x2 be the two children of x in N ;14:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Let c = c1 + c2 + \u03bb\u00b7min{dist + dist, dist + dist};15:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0C \u2190 C \u222a {};16:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if N z y \u2264then17:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0c = \u03b4 + \u03bb\u00b7dist+ c1 + c2;18:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0C \u2190 C \u222a {};19:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0else if N y z \u2264then20:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0c = \u03b4 + \u03bb\u00b7dist + c1 + c2;21:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0C \u2190 C \u222a {};22:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0u) = merge23:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a024: Return The following theorem proves the correctness of Algorithm 2:Theorem 3 Algorithm 2 returns a matrix  such that, for every u \u2208 V(G),  is contained in u) u and N mapping u to x with cost cif and only if there exists a most parsimonious reconciliation between G.Algorithm 3 computeMiny, store the ordered list of visited nodes in BFS(y) and compute, for each u in BFS(y), dist;1: Proceed a BFS from z;2: Do the same from for each node v \u2208 BFS(y) do3: if v = y or v = z or v is not in BFS(z) then label(v) = \u22054:\u00a0\u00a0\u00a0else if v has only one child v1 that is in BFS(z) or BFS(y) then5:\u00a0\u00a0\u00a0label(v) = label(v1);6:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0else7:\u00a0\u00a0\u00a0v1, v2 be the two children of v;8:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Let if label(v1) = label(v2) \u2260 \u2205 then9:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0label(v) = label(v1);10:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0else11:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0label(v) = {v}; Add v into 12:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Proof: For each u \u2208 V(G), we need to prove that, if \u03b1' is a reconciliation of minimum cost between u Gon N, then (\u03b1'(u), \u03b1'cost) is contained in u). This is obviously true for every leaf of G (by lines 5-7). Let now u be an internal node having two children u1, u2. Then, following Lemma 5, u) can be computed from u1) and u2) by using the information contained in dist. Lines 10-23 in Algorithm 2 computes u) following this lemma.v that is above y, z, denote by low(v) the lowest node such that every path from v to y, z must pass through this node. There is only one such node. Indeed, suppose that there are two such nodes m, m', then every path from v to y must pass through both m, m', i.e. either m <N m' or m' <N m, contradicting the lowest property of m, m'. To prove the claim, we need to show that if v is a node of BFS(y), then:It remains now to prove that Algorithm 3 correctly computes v is not strictly above y and z, then label(v) = \u2205;(i) if label(v) = low(v) and line 12 of Algorithm 3 is performed if and only if v is actually in (ii) otherwise, v be a node of BFS(y), then (i) holds by line 4. Now consider the case where v is strictly above y and z. We will prove (ii) by recursion on v following the order of the nodes in BFS(y). The recursion begins from the set of lowest nodes that are strictly above y and z, i.e. the set of nodes of v in v. Let v1, v2 be two children of v, then by hypothesis v1, v2 are not strictly above y, z, i.e. label(v1) = label(v2) \u2260 \u2205; and low(v) = v. Thus, due to line 12, label(v) = v = low(v). Now let v be a node strictly above y, z such that (ii) is correct for each node below v which is strictly above or equal to y, z. If v is a hybridization node, then it is evident that low(v) = low(v1) where v1 is the only child of v. Moreover, since label(v1) = low(v1) (by the hypothesis of recurrence), then label(v) = label(v1) = low(v1) = low(v). If v is a speciation node having two children v1, v2 such that v2 is not in either BFS(y) or BFS(z), then we also have low(v) = low(v1). Hence, due to lines 5 - 6, we have label(v) = label(v1) = low(v1) = low(v). Now consider the last case, i.e. v is a speciation node having two children v1, v2 that are both in BFS(y) and BFS(z). If there exists a node q = low(v1) = low(v2), then low(v) = q because every from v to y, z must pass either through v1 or v2, i.e. always pass through q. Following line 10, we fix label(v) = label(v1) = q = low(v). Moreover, in this case v can not be in v to y, z passes through a node q below v. In the last case, we have v1, v2 are above y, z, then there exists a node q1 = label(v1), and a different node q2 = label(v2). We will prove that q such that every path from v to y, z must pass through q. Hence, every path from v1 (resp. v2) to y, z must also pass through q, so q1 N q \u2264(resp. q2 N q\u2264). It means that there is a path from v1 to q to q2 and then to y, z that does not pass through q1, a contradiction. Hence v is in v to y, z must pass through is v itself (line 12).Let We now present some intermediate results that will be useful to prove the complexity of Algorithm 2.L be a subset of L(N ). If |L| = 1, then m of N such that m is above all leaves in L and there exist at least two separated paths from m to two distinct leaves of L.We extend the definition of u of G, N L(u) is defined as the set of leaves of N to which \u03b1 maps the leaf set of uG, i.e. s(u) = s(x)}.Given a node Lemma 6 Let \u03b1 be a most parsimonious reconciliation between G and N, then, for every node u of Proof: It is true for every leaf u of G. Let u now be an internal node having two children u1, u2.u1, u2 \u2208 L(G), then following Remark 1, N L(u) consists of two distinct nodes r\u03b1(u1), r\u03b1(u2). By Lemma 5, r\u03b1(u) to r\u03b1(u1) and r\u03b1(u2). It means that If u1 \u2209 L(G), then following Remark 1, |N L(u1)| \u2265 2, then there always exist two distinct leaves x, y of N L(u) such that x \u2208 NL(u1), y \u2208 NL(u2), i.e. x is below r\u03b1(u1) and y is below r\u03b1(u2). If r\u03b1(u) to r\u03b1(u1), r\u03b1(u2). By extending these two paths from r\u03b1(u1) to x, and from r\u03b1(u2) to y, we have to two separated paths from r\u03b1(u) to x, y. In other words, r\u03b1(u) \u2208 N \u2133(N L(u)). If e\u03b1(u) = r\u03b1(u2) N \u03b1r \u2264(u1), then r\u03b1(u) = r\u03b1(u1) following Lemma 5, and r\u03b1(u) is not a leaf of N following Remark 1. Let u' be the highest node such that u' \u2264G u1 and r\u03b1(u'), i.e. from r\u03b1(u), to two distinct leaves of NL(u'), i.e. two distinct leaves of N L(u). We can prove the claim similarly for the case when r\u03b1(u1) N \u03b1r \u2264(u2).Let Lemma 7 If N is a network that contains h hybridization nodes, then for every subset L of .The proof of this lemma is deferred to the appendix. We are now ready to state the complexity of Algorithm 2.Theorem 4 The time complexity of Algorithm 2 is O(h2\u00b7|G|\u00b7|N|) where h is the number of hybridization nodes of N.Proof: For every u \u2208 V(G), N that u can be mapped to, which is bounded by O(h) following Lemma 7.for loop at lines 3 - 23 is performed |G| times, and, at each iteration, the for loop at lines 10-23 is performed O(h2) times. In each iteration of the second loop, the operation computeMin, as detailed in Algorithm 3, requires two breath-first-search traversals, which can be performed in time O(|N|). Moreover, for every node x of x to y, z, which can be extended to be two separated paths from x to two distinct leaves l1, l2 of N L(u) where l1 is a leaf below y of and l2 is a leaf below z. This is always possible because, by Remark 1, |N L(u)| > 1. Hence, x must be in O(h). The operation merge at lines 23 for two lists of size O(h) can be implemented in times O(h), if we know that the resulting list is also of size O(h). Hence, it takes O(|N| + h) = O(|N|) times for each iteration of the loop 10 - 23. Therefore, the total complexity is O(h2\u00b7 |G|\u00b7|N|).The G and N can be then obtained by a standard back-tracking of the matrix x,c) of r(G)) such that c is the minimum value over all pairs in r(G)).Finally, a reconciliation of minimum cost between In this paper, we have studied two variants of the reconciliation problem between a gene tree and a species network. In particular, for the problem of finding the \"most parsimonious\" switching of the network, even though the number of switchings can be exponential with respect to the number of hybridization nodes, we proposed an algorithm that is exponential only with respect to the level of the network, which is often low for biological data. Moreover, the problem of finding a reconciliation between a gene tree and a network, which was solved in  for a mou \u2208 I(G), there must exist H \u2208 GB(u) such that u \u2208 I(H). If H is an edge, then u is the only internal node of H, which must be an artificial node by Lemma 2. But this is not possible because nodes of G cannot be artificial. Hence, H must be a binary tree. Let denote i B= B(u), and i S= S(B(u)). We will prove now that u.By Definition 8, for every u be an internal node of G that has two children u1, u2 in G, and let H be the binary tree of u \u2208 I(H). Denote j = 1, 2, if j uis a leaf, let j Hbe equal to j u, otherwise j His the binary tree of j u\u2208 I(jH). For the sake of convenience, if j uis a leaf, we also denote by j uto the only leaf x of N such that s(x) = s(j u). Note that s(x) = \u03b1(j u).Let j = 1, 2 (which is evidently true if j uis a leaf), and we will show that this implies that the claim is true for u.We now suppose that u in N Gsuch that Let H = H1 = H2. This implies that H will contain an artificial node. The same holds for u2. Thus, (i) If H = H2. As in point (i), this implies H that is mapped to ri Bare above all nodes of (ii) If ii)). Then, similarly as in point (ii), H mapped respectively to (iii) Suppose now that u \u2208 I(G) by recursion.Therefore, in all cases we always have We will prove this lemma by recursion on the number of hybridization nodes. See the figure N is a tree, and it is evident that If there is no hybridization node, then h hybridization nodes. Let N now be a network that has h + 1 hybridization nodes. Let  be an edge of N having as target a hybridization node  such that it does not exist any hybridization node above a . Let N' be the network obtained by removing  from N , then N' has exactly h hybridization nodes. Let q in Q must be above a. Indeed, if q is not above a, then every path from q to every node of L does not contains , thus q is in a, hence all nodes of Q must be contained in a path leading a, and this path does not contain any hybridization to node. Let enumerate the nodes in Q as q1,... m qfrom the lowest to the highest one.Suppose that the claim is correct for every network having If Q| = m >1. In the following, we will define m - 1 edges of N having as target a hybridization node such that if N\" is the network obtained from N' by removing these edges, then Suppose now that |* Lthe set of nodes in L that are below m qin N'. Hence * L \\ Lis not empty since otherwise m qwould be in iq, i = 1,..., m, let i qsuch that a. Hence, a since there is no hybridization node above a, thus every path from i qto * L \\ Lmust pass through a, b. By definition, there must exist at least two separated paths from i qto two leaves of L. Hence, for every i, there exists always a path from i qto a node of * Lthat passes through l(iq).Denote by m Lthe set of nodes in * Lsuch that, for every l \u2208 mL, there is a path from q to l that passes through m Lis not empty. Recursively, for every i< m, let i Lbe the set of nodes in L* \\ Lm \\... Li+1 such that, for every l \u2208 iL, there is a path from qi to l that passes i< m, and for every i \u2260 j, i L\u2229 jL = \u2205.Denote by i < m, an index c(i) that is strictly greater than i such that Lc(i) \u2260 \u2205, together with two paths i p(resp. i q(resp. qc(i)) to a node of Lc(i) as follows: if l(qi) is not in iL, then by the definition, there exists a unique j such that j Lcontains l(qi), and j > i. We fix c(i) = j. Next, we define i p(resp. i q(resp. qc(i)) to l(iq) that passes l(iq) is in iL, then let c(i) be the smallest number that is greater than i and Lc(i) \u2260 \u2205 . Let l' be a node of Lc(i). Since i qis in N M(L), then there must exist a path from i qto l', and we define i pas this path. The path qc(i) to l' that passes through ip, L. Denote by i hthe highest common hybridization node of i pand ihs are distinct since each i pstarts at a different node iq, and each qc(i) that is strictly greater than iq. We define  recursively in increasing order of i from 1 to m - 1 as follows. If i = 1, then i bis the highest hybridizatition node on ip. If i >1, then i bis the highest hybridization node on i pand different from all k bfor every k < i. There exists always such a node ib, for example ih. Therefore, all ibs are distinct. Denote by i athe parent of i bon the path ip.We will define, for each N\" the network obtained from N' by removing all edges . For every node x in x is also in x', x\" the two children of x. By definition, for every l \u2208 L, there exists a path, denoted by f'(l), in N' from x to l such that at least one path among them passes through x' and one other passes through x\". To prove that x is in N\" ) from x to each leaf l of L, denoted by f\"(l), such that at least one path among them passes through x'and one other passes through x\".Denote by x is above m q contains mq, then l \u2208 L* and f'(l) must pass through x' because there is no hybridization node above mq. Suppose that k be the index such that l \u2208 kL, then we can choose a path f\"(l) in N' from x to l that does not contain any  as follows. This path starts from x, passes through x', goes down to k q, then takes the path from k qto l that contains q\". Note that this path does not include any i psince, by construction, every path i pstarts from i qand goes to a node in Lc(i) that is different from iL, while this path passes through k qand goes to a node in k L. Moreover, this path and i pcan not have any common hybridization node above i abecause i bis the highest hybridization node on ip. Hence, it can not pass through  for any i. If l is not in m qto l must pass through x, going down to l. It is evident that this path does not include any ip, and it cannot have with i pany common hybridization node above i abecause i bis the highest hybridization node on ip. Hence it does not pass through any . If f'(l) does not contain mq, then we fix f\"(l) = f'(l). It is easy to see that f'(l) does not contain any edge  because otherwise f'(l) and i pmust have at least a common hybridization node above i a(since there is no hybridization node above iq). But this is not possible because i bis the highest hybridization node on ip. Remark that at least one of the paths f'(l) in this case must pass through x\" since all paths in the first case must pass through x'. Hence at least two of the paths f\"(l) are separated, thus x is in Consider first the case that x is not above mq, then similarly as in the previous case where f'(l) does not contain mq, we deduce that f'(l) does not contain any  for every l. Hence, by choosing f\"(l) = f'(l) for every l, we are done.Consider now the case that Therefore, we have N\" contains h - |Q| + 1 hybridization nodes, then following the hypothesis of recurrence, The network The authors declare that they have no competing interests.Both authors contributed to design the models, algorithms and to write the paper."}
{"text": "Denote \u03c8t(x) = tn\u2212\u03c8(x/t) with t > 0 and x \u2208 Rn. Given a function f \u2208 Lloc\u20611(Rn), the Lusin area integral of f is defined bya(x) denote the usual cone of aperture onea = 1, we denote S\u03c8,a(f) as S\u03c8(f).Let g-functions and the Littlewood-Paley g\u03bc*-functions besides the Lusin area integrals. The Littlewood-Paley g-functions, which can be viewed as a \u201czero-aperture\u201d version of S\u03c8, and g\u03bc*-functions, which can be viewed as an \u201cinfinite-aperture\u201d version of S\u03c8, are, respectively, defined by\u03c8 to be the Poisson kernel, then the functions defined above are the classical Littlewood-Paley operators.Now let us turn to the introduction of the other two Littlewood-Paley operators. It is well known that the Littlewood-Paley operators include also the Littlewood-Paley b \u2208 Lloc\u20611(Rn),\u2009\u2009m \u2265 1, the corresponding m-order commutators of Littlewood-Paley operators above generated by a function b are defined by\u03bc > 0.Letting p\u2009\u2009(Rn) in  is bounded on Lp(\u03c9) in )(Rn) in . In 2005p(\u03c9) in ). There p(\u03c9) in \u20139 and soLp(\u00b7)(Rn) become one class of important research subject in analysis filed due to the fundamental paper ,\u2009\u2009, and , we have the following results.For commutators  is defined by (b \u2208 BMO(Rn), p(\u00b7) \u2208 B(Rn), m \u2265 1,\u2009\u20091 < r < \u221e. If, for any x \u2208 Rn and r0 > 0, function u satisfiesC > 0 independent of f such that, for any function sequences {fh}h=1\u221e with ||{\u2211h|fh|r}r1/||Mp(\u00b7),u(Rn) < \u221e, the following inequality holds:Suppose that function fined by . Let b \u2208bm, g\u03c8] is defined by ,u(Rn) < \u221e, the following inequality holds:Suppose that  is defined by (\u03bc > 3 + 2(\u025b + \u03b3)/n, and 0 < \u03b3 < \u025b. Then under the same condition as the one in C > 0 independent of f such that, for any function sequences {fh}h=1\u221e with ||{\u2211h|fh|r}r1/||Mp(\u00b7),u(Rn) < \u221e, the following inequality holds:Suppose that (fj)|q)q1/||Lp(\u00b7)(Rn) \u2264 C||(\u2211j|fj|q)q1/||Lp(\u00b7)(Rn),|||q)q1/||Lp(\u00b7)(Rn) \u2264 C||(\u2211j|fj|q)q1/||Lp(\u00b7)(Rn),|||q)q1/||Lp(\u00b7)(Rn) \u2264 C||(\u2211j|fj|q)q1/||Lp(\u00b7)(Rn),||n/2, \u025b}.Let Next, let us show the proofs of Theorems fh}h||lr \u2208 Mp(\u00b7),u(Rn); for any x0 \u2208 Rn, r0 > 0, denotefh0 = fh\u03c7B,\u2009\u2009fhj = fh\u03c7B\u2216B, j \u2208 N\u2216{0}.Let ||{D1, notice that supp fh0 \u2282 B; using D2. To do this, we need to consider S\u03c8,a first. Without loss of generality, we may assume that a \u2265 1. Letx \u2208 B,\u2009\u2009z \u2208 B\u2216B,\u2009\u2009j \u2265 1, theny > 2j+1r0, then t > |x \u2212 y|/a \u2265 (|x| \u2212 |y|)/a > (2j+1r0 \u2212 r0)/a \u2265 2jr0/a. Thus, by condition (iii), we getI, we obtainD1, D2, we obtainNoting that, in order to prove Now let us prove Theorems g\u03c8, similar to the estimate of S\u03c8,a, via a simple calculation, we get that , supp fhj \u2282 B\u2216B,\u2009\u2009j \u2265 1, thenFor hat (see ) if x \u2208 g\u03bc*, by the definitions of S\u03c8,a and g\u03bc*, we haveS\u03c8,a in the proof of x \u2208 B, supp fhj \u2282 B\u2216B, j \u2265 1, then\u03bc > 3 + 2(\u025b + \u03b3)/n, we obtainFor b \u2208 BMO, ||{fh}h||lr \u2208 Mp(\u00b7),u(Rn). For any x0 \u2208 Rn, r0 > 0, denotefh0 = fh\u03c7B and fhj = fh\u03c7B\u2216B, j \u2208 N\u2216{0}.Let E1, notice that supp fh0 \u2282 B; by E2. According to the estimate of S\u03c8,a in the proof of x \u2208 B, z \u2208 B\u2216B, j \u2265 1, thenTo finish the proof of E1, E2, we haveThus,bm, g\u03c8], according to the estimate of g\u03c8 in the proof of x \u2208 B, supp fhj \u2282 B\u2216B, j \u2265 1, thenFor , according to the estimate of  in the proof of x \u2208 B, supp fhj \u2282 B\u2216B, j \u2265 1, thenFor ["}
{"text": "After publication of  we becamIn the original publication, the images of Figures 1, 3, 5\u20139 are shuffled. In addition, images of Figures 1, 3, 4, 6\u20139 contain incorrectly encoded symbols. They should be replaced with Figures , 8, 9 prn of 11\u201d in the main text should be removed.All four occurrences of \u201cYasuda and Miyano Page O(|E|2 log |V | log |E|)\u201d should read \u201cO(|E|2 log |V | log |E|)\u201dAll three occurrences of \u201cE+)\u201d should read \u201cCC(E+)\u201dIn the Results, both of two \u201cCC.\u201d in subsection Circular chromosome graphThe line with the sentence \u201cTherefore, In the Methods, a Q.E.D. symbol \u201c\u2610(white box)\u201d should be inserted at the end of the following lines:In the Background, \u201cBCRABL\u201d should read \u201cBCR-ABL\u201dLemma 1, the expression \u201cc = p1e1p2e2 . . . etcptc+1\u201d should read \u201c In the Methods, in the proof of Lemma 2, the expression \u201c2|V |(nN+nT )+(4|V |+1)Pe \u2217 \u2217 \u2217ES n(e) \u2264U(4|V|+1)(|E|+1)\u201d should read \u201c In the same proof, just above Lemma 3, the following expressionsIn the Methods, in the proof of should readLemma 4, in the paragraph that begins with \u201cAll of these steps\u201d, the expression \u201cm = f(e) + f(\u0113)(\u2208ES).\u201d should read \u201cm = f(e) + f(\u0113) (e\u2208ES).\u201d (Insert a white space before \u201c(e\u2208ES)\u201d.) In the Methods, in the proof of C satisfies (6).\u201d, the expression \u201cw)=0(e\u2208EL\u222aER).\u201d should read \u201cw)=0 (e\u2208EL\u222aER).\u201d (Insert a white space before \u201c(e\u2208EL\u222aER)\u201d.) In the same proof, just before \u201cTherefore, Incorrectly formatted descriptions in  should bFormulation of the problem, the phrase \u201cits computational complexity was not analyzed\u201d should read \u201cits computational complexity was not intensively analyzed\u201dIn the Results, in subsection Polynomial-time solvable variation, both of two \u201cEL\u222aER\u201d should read \u201cE\u201dIn the Results, in the first paragraph of subsection Definition 2, the phrase \u201cif all g\u2208CC are good\u201d should read \u201cif all g\u2208CC are good and n(e)\u2009=\u20090 for e\u2208E\u2009\u2212\u2009EW \u201dIn the Results, in Definition 2, the expression \u201cEW\u2009=\u2009{e\u2208ES |n(e)\u2009\u2265\u20091}\u222a{e\u2208EL\u222aER |e is required}\u201d should read \u201cEW\u2009=\u2009{e\u2208E |e is required}\u201dIn the Results, in the paragraph just after Definition 3 should read as follows:In the Results, the last sentence that begins with \u201cFinally, if some\u201d just before g\u2208CC that are not good still remain, edges e in g are forcibly removed from EW by changing e not required and setting n(e) to 0. Finally, if n(e)\u2009>\u20090 for some e\u2208E\u2212 EW, e is changed to be required and added to EW by confirming its existence, or n(e) is forcibly set to 0.\u201d\u201cIn addition, if some Definition 3 should read \u201cLet G\u2009=\u2009 be a chromosome graph that satisfies WCC with respect to given VW\u2009\u2282\u2009V and EW\u2009\u2282\u2009E. Then, find a set C of chromosomes on G that minimizes W(C) when (3) is satisfied, each v \u2208 VW is at an end of some c\u2208C, and each e\u2208EW appears in C.\u201dIn the Results, Lemma 4, the sentences \u201cFor e\u2208ES \u222a {eN, eT }, we set l(e)\u2009=\u2009n(e), l(\u0113)\u2009=\u20090, and u(\u0113)\u2009=\u2009n(e). For e\u2208EL\u222aER, we set l(e)\u2009=\u20091.\u201d should read \u201cFor e\u2208ES\u222a{eN, eT}, we set l(e)\u2009=\u2009n(e), l(\u0113)\u2009=\u20090, and u(\u0113)\u2009=\u2009n(e) if e is not required, whereas l(e)\u2009=\u2009max{n(e), 1}, l(\u0113)\u2009=\u20090, and u(\u0113)\u2009=\u2009max{n(e)\u2009\u2212\u20091, 0} if e is required. We assume that eN is required because nN \u22651. We also assume that eT is required if |VW|\u2009\u2265\u20091. For e\u2208EL\u222aER, we set l(e)\u2009=\u20091 if e is required, or l(e)\u2009=\u20090 otherwise.\u201dIn the Methods, in the paragraph just above Lemma 4, the description \u201cor n(e)\u2009\u2265\u20091\u201d should be removed.In the Methods, in the paragraph just after Proof of Theorem 3, the phrase \u201cby making all edges in EL\u222aER required\u201d should read \u201cby making all edges required\u201dIn the Methods, in subsection The following items correct inaccurate descriptions in the original manuscript. We regret any inconvenience that they might have caused."}
{"text": "Based on linguistic term sets and hesitant fuzzy sets, the concept of hesitant fuzzy linguistic sets was introduced. The focus of this paper is the multicriteria decision-making (MCDM) problems in which the criteria are in different priority levels and the criteria values take the form of hesitant fuzzy linguistic numbers (HFLNs). A new approach to solving these problems is proposed, which is based on the generalized prioritized aggregation operator of HFLNs. Firstly, the new operations and comparison method for HFLNs are provided and some linguistic scale functions are applied. Subsequently, two prioritized aggregation operators and a generalized prioritized aggregation operator of HFLNs are developed and applied to MCDM problems. Finally, an illustrative example is given to illustrate the effectiveness and feasibility of the proposed method, which are then compared to the existing approach. Since the fuzzy set was proposed by Zadeh in 1965 \u20135. Due tHesitant fuzzy sets (HFSs), an extension of traditional fuzzy sets, can address this problem. HFSs were first introduced by Torra and Narukawa , 7, and When faced with problems that are too complex or ill-defined to be solved by quantitative expressions, linguistic variables can be an effective tool because the use of linguistic information enhances the reliability and flexibility of classical decision models . Linguiss2, \u232a is an HFLN and 0.3, 0.4, and 0.5 are the possible membership degrees to the linguistic term s2. HFLSs have enabled great progress in describing linguistic information and to some extent may be considered an innovative construct. The main advantage of HFLSs is that they can describe two fuzzy attributes of an object: a linguistic term and a hesitant fuzzy element (HFE). The former provides an evaluation value, such as \u201cexcellent\u201d or \u201cgood.\u201d The latter describes the hesitancy for the given evaluation value and denotes the membership degrees associated with the specific linguistic term. However, the operations proposed by Lin et al.  in the case of it being applied to X.Let hE(x) is a set of values in , denoting the possible membership degrees of the element x \u2208 X to the set E. hE(x) is called a hesitant fuzzy element (HFE)  and denotes the possible membership degrees that x belongs to s\u03b8(x).Let X has only one element, the HFLS A is reduced to \u2329s\u03b8(x), hA(x)\u232a. For computational convenience, we call \u03b1 = \u2329s\u03b8(\u03b1), h\u03b1\u232a as an HFLN.When hA(x) = {r} has only one element, it indicates that the degree that x belongs to s\u03b8(x) is r. For example, \u2329s2, 0.3\u232a is called a fuzzy linguistic number, which is a special case of HFLN.When X = {x1, x2} be a universal set. If an HFLS A = {\u2329x, s\u03b8(x), hA(x)\u232a\u2223x \u2208 X} = {\u2329x1, s5, {0.4,0.6,0.7}\u232a, \u2329x2, s6, {0.1,0.5,0.7}\u232a} is divided into two subsets that contain only one object, respectively, then \u2329s5, {0.4,0.6,0.7}\u232a and \u2329s6, {0.1,0.5,0.7}\u232a are HFLNs. 0.4, 0.6, and 0.7 are the possible membership degrees that x1 belongs to s5; 0.1, 0.5, and 0.7 are the possible membership degrees that x2 belongs to s6.Let An HFLN is an extension of a linguistic term and an HFE. Compared to linguistic terms, HFLNs embody the possible membership degrees that an evaluation object attaches to the linguistic term, and they can depict the fuzziness more accurately than an uncertain linguistic variable does. When compared to HFEs, HFLNs add linguistic terms and assign the membership function to a specific linguistic evaluation value, which make the membership degrees no longer relative to a fuzzy concept, but to linguistic terms, such as \u201cpoor\u201d or \u201cgood.\u201ds5)\u201d is an acceptable evaluation result for the car and is given by three decision makers. Then, each decision maker uses a value to express his/her opinion about the car under the evaluation of \u201cgood (s5).\u201d Decision maker A may give the value 0.4 for \u201cgood,\u201d whilst decision maker B may give 0.6 and decision maker C may give 0.7. In this case, HFLNs may be a better choice, and the evaluation result can be denoted by \u2329s5, {0.4,0.6,0.7}\u232a.In fuzzy set theory, the hesitant values in HFLNs are called possible membership degrees, which are caused by the hesitancy and uncertainty of decision makers. In the example of the performance evaluation of a car, suppose that \u201cgood   are diffTo use data more efficiently and to express the semantics more flexibly, linguistic scale functions assign different semantic values to linguistic terms under different situations . They ar\u03b8i \u2208  is a numeric value, then the linguistic scale function f that conducts the mapping from si to \u03b8i\u2009\u2009 is defined as follows:\u03b80 < \u03b81 < \u22ef<\u03b8t2.If \u03b8i\u2009\u2009 reflects the preference of the decision makers when they are using the linguistic term si \u2208 S\u2009\u2009. Therefore, the function/value in fact denotes the semantics of the linguistic terms.\u2009(1) ConsiderClearly, the symbol \u2009(2) ConsiderThe evaluation scale of the linguistic information given above is divided on average.\u2009(3) ConsiderWith the extension from the middle of the given linguistic term set to both ends, the absolute deviation between adjacent linguistic subscripts also increases.With the extension from the middle of the given linguistic term set to both ends, the absolute deviation between adjacent linguistic subscripts will decrease.f*(si) = \u03b8i, and is a strictly monotonically increasing and continuous function. Therefore, the mapping from R+ is one-to-one because of its monotonicity, and the inverse function of f* exists and is denoted by f\u2217\u22121.To preserve all the given information and facilitate the calculation, the above function can be expanded to \u03b1 = \u2329s\u03b8(\u03b1), h\u03b1\u232a and \u03b2 = \u2329s\u03b8(\u03b2), h\u03b2\u232a be two HFLNs. Some operations of \u03b1 and \u03b2 are defined as follows:\u03b1 \u2295 \u03b2 = \u2329s\u03b8(\u03b1)+\u03b8(\u03b2), \u222ar1\u2208h\u03b1,r2\u2208h\u03b2{r1 + r2 \u2212 r1r2}\u232a;\u03bb\u03b1 = \u2329s\u03bb\u03b8(\u03b1), \u222ar\u2208h\u03b1{1 \u2212 (1 \u2212 r)\u03bb}\u232a;\u03b1 \u2297 \u03b2 = \u2329s\u03b8(\u03b1)\u00d7\u03b8(\u03b2), \u222ar1\u2208h\u03b1,r2\u2208h\u03b2{r1r2}\u232a;\u03b1\u03bb = \u2329s\u03b8(\u03b1)\u03bb, \u222ar\u2208h\u03b1{r\u03bb}\u232a.Let \u03b1 = \u2329s2, {0.3,0.4}\u232a; for example, 0.3 and 0.4 are the possible membership degrees that the object belongs to s2; that is, {0.3,0.4} is the explanatory part of s2 and should be closely related to s2 in the additive operation.The operations proposed in  have somIn order to overcome the existing limitations given above, new operations of HFLNs based on linguistic scale functions are defined as follows.\u03b1 = \u2329s\u03b8(\u03b1), h\u03b1\u232a and \u03b2 = \u2329s\u03b8(\u03b2), h\u03b2\u232a be two HFLNs. Some operations of \u03b1 and \u03b2 are defined as follows:(1)(2)(3)(4)(5)Let f* is a mapping from the linguistic term si to the numeric value \u03b8i and f\u2217\u22121 is a mapping from \u03b8i to si. So, the first part of (1)\u2013(5) is a linguistic term. In addition, it is obvious that the second part of (1)\u2013(5) is an HFE. In summary, according to According to f* is utilized. Thus, decision makers can flexibly select the linguistic scale function f* depending on their preferences and the actual semantic situations. In addition, the new addition operation of HFLNs is more reasonable and reliable, because the final hesitant fuzzy membership has closely combined each element of the original HFLNs.The operations defined above are based on linguistic scale functions, which can get different results when a different linguistic scale function \u03b1 \u2295 \u03b2, \u03bb\u03b1, \u03b1 \u2297 \u03b2, and \u03b1\u03bb necessarily appear in defining basic operations, but their results have no practical meaning. In the aggregation process, for example, using weighted operators, \u03b1 \u2295 \u03b2 is combined with \u03bb\u03b1 and \u03b1 \u2297 \u03b2 is combined with \u03b1\u03bb; therefore, the calculation results are interpretable in practice.S = {s0, s1, s2, s3, s4, s5, s6} = {very poor, poor, slightly poor, fair, slightly good, good, very good}, \u03b1 = \u2329s2, {0.1,0.3}\u232a, \u03b2 = \u2329s3, {0.2}\u232a, and \u03bb = 2.Let f1*(si) = i/6\u2009\u2009(0 \u2264 i \u2264 6), then\u03b1) = \u2329s4, {0.7,0.9}\u232a;neg = \u2329s4, {0.7,0.9}\u232a;neg = \u2329s4, {0.7,0.9}\u232a;neg, h\u03b1i\u232a\u2009\u2009 be any three HFLNs; thus the following properties are true.\u03b11 \u2295 \u03b12 = \u03b12 \u2295 \u03b11;\u03b11 \u2297 \u03b12 = \u03b12 \u2297 \u03b11;\u03b11 \u2295 \u03b12) \u2295 \u03b13 = \u03b11 \u2295 (\u03b12 \u2295 \u03b13);(\u03b11 \u2297 \u03b12) \u2297 \u03b13 = \u03b11 \u2297 (\u03b12 \u2297 \u03b13);(\u03bb(\u03b11 \u2295 \u03b12) = \u03bb\u03b11 \u2295 \u03bb\u03b12, (\u03bb \u2265 0);\u03b11 \u2297 \u03b12)\u03bb = \u03b11\u03bb \u2297 \u03b12\u03bb, (\u03bb \u2265 0).(Let \u2009(3) ConsiderAccording to \u03b11 \u2295 \u03b12) \u2295 \u03b13 = \u03b11 \u2295 (\u03b12 \u2295 \u03b13).So,  and (6) can be easily proven.\u03b1 = \u2329s\u03b8(\u03b1), h\u03b1\u232a be an HFLN. The score function E(\u03b1) of \u03b1 can be represented as follows:s(h\u03b1) is the score function of h\u03b1.Let \u03b1 = \u2329s3, {0.2,0.4,0.5,0.7}\u232a. If t = 3 and f1*(si) = i/2t, by applying , h\u03b1\u232a = \u2329s\u03b8(\u03b1), \u222ar\u2208h\u03b1{r}\u232a be an HFLN. A variance function h\u03b1 can be denoted by D(\u03b1) of \u03b1 can be represented as follows:l(h\u03b1) is the number of the values in h\u03b1.Let \u03b1 = \u2329s3, {0.3,0.6}\u232a. If t = 3 and f1*(si) = i/2t, by applying , h\u03b11\u232a and \u03b12 = \u2329s\u03b8(\u03b12), h\u03b12\u232a be any two HFLNs.E(\u03b11) > E(\u03b12), then \u03b11 > \u03b12.If E(\u03b11) = E(\u03b12), thenIf \u2009D(\u03b11) > D(\u03b12), then \u03b11 > \u03b12;if \u2009D(\u03b11) = D(\u03b12), then \u03b11 = \u03b12.if Let \u03b1 = \u2329s3, {0.3,0.6}\u232a and \u03b2 = \u2329s6, {0.15,0.3}\u232a. If t = 3 and f1*(si) = i/2t, then E(\u03b1) = E(\u03b2) = 0.225, D(\u03b1) = 0.489, D(\u03b2) = 0.994, and thus \u03b1 < \u03b2.Let In this section, two prioritized aggregation operators for HFLNs are proposed based on the PA operator, and some desirable properties are also analyzed. Subsequently, these operators are extended to a generalized form. Finally, a method for solving MCDM problems with HFLNs, where the criteria are in different priority levels, is developed.The PA operator was originally introduced by Yager  and is sG = {G1, G2,\u2026, Gn} be a collection of criteria and ensure that there is a prioritization between the criteria expressed by the linear ordering G1\u227bG2\u227bG3\u227b\u22ef\u227bGn, which indicates that the criteria Gj has a higher priority than Gk, if j < k. Gj(x) is an evaluation value denoting the performance of the alternative x under the criteria Gj and satisfies Gj(x)\u2208. Ifwj = Tj/\u2211i=1nTi, T1 = 1 and Tj = \u220fk=1j\u22121Gk(x)\u2009\u2009, then PA is called the PA operator.Let \u03b11, \u03b12,\u2026, \u03b1n} and {\u03b21, \u03b22,\u2026, \u03b2n} are two sets of criteria values under criteria {G1, G2,\u2026, Gn}, where G1\u227bG2\u227bG3\u227b\u22ef\u227bGn. Now the PA operators under a hesitant fuzzy linguistic environment will be analyzed in the following subsections.PA operators have usually been used in situations where input arguments are exact values. Therefore, PA operators could be extended to accommodate situations where the input arguments are hesitant fuzzy linguistic information. Based on In this subsection, the prioritized weighted average operator under a hesitant fuzzy linguistic environment is investigated. The definition of the HFLPWA operator and its relevant theorems are given as follows.\u03b1j\u2009\u2009 be a collection of HFLNs, and then the HFLPWA operator can be defined as follows:Tj = \u220fk=1j\u22121E(\u03b1k)\u2009\u2009, T1 = 1, and E(\u03b1k) is the score function of \u03b1k.Let Based on the operations of HFLNs described in \u03b1j = Gj(x) = \u2329s\u03b8(\u03b1j), h\u03b1j\u232a\u2009\u2009 be a collection of HFLNs. Then the aggregated value, obtained by using the HFLPWA operator, is also an HFLN, andTj = \u220fk=1j\u22121E(\u03b1k)\u2009\u2009, T1 = 1, and E(\u03b1k) is the score function of \u03b1k.Let n.(1)n = 2, sinceFor (2)n = k, thenIf  holds foWhen n = k + 1, by the operations described in n = k + 1. Thus,  = \u2329s\u03b8(\u03b1j), h\u03b1j\u232a\u2009\u2009 be a collection of HFLNs. If \u03b1\u2212 = \u2329min\u2061j{s\u03b8(\u03b1j)}, r\u2212\u232a, \u03b1+ = \u2329max\u2061j{s\u03b8(\u03b1j)}, r+\u232a, where r\u2212 = min\u2061rj\u2208\u222ah\u03b11,h\u03b12,\u2026,h\u03b1n{rj} and r+ = max\u2061rj\u2208\u222ah\u03b11,h\u03b12,\u2026,h\u03b1n{rj}, thenLet \u03b11, \u03b12,\u2026, \u03b1n) = \u03b1 = \u2329s\u03b8(\u03b1), h\u03b1\u232a, and then E(\u03b1) = f*(s\u03b8(\u03b1)) \u00b7 s(h\u03b1).Let HFLPWA(j{s\u03b8(\u03b1j)} \u2264 s\u03b8(\u03b1j) \u2264 max\u2061j{s\u03b8(\u03b1j)} for all j, we haveSince min\u2061\u03b1j = Gj(x) = \u2329s\u03b8(\u03b1j), h\u03b1j\u232a\u2009\u2009 be a collection of HFLNs and \u03b11, \u03b12,\u2026, \u03b1n). ThenLet \u03b1j is decided by the priority and value of \u03b1j and will not be influenced by its position in the permutation. So, The weight of \u03b1 = \u2329s2, {0.2,0.4}\u232a, for example, and if \u03b11 = \u03b12 = \u03b1, then HFLPWA = \u2329s2, {0.2,0.218,0.382,0.4}\u232a \u2260 \u03b1. In addition, we do not consider the monotonicity of HFLPWA because the weights will be recalculated and vary if the values used in the HFLPWA operator change. It is difficult to consider the monotonic property when the parameters are irregularly variable.It should be noted that the HFLPWA operator cannot satisfy idempotency. Take In this subsection, the prioritized weighted geometric operator under a hesitant fuzzy linguistic environment is investigated. The definition of the HFLPWG operator and its relevant theorems are given as follows.\u03b1j\u2009\u2009 be a collection of HFLNs, and then the HFLPWG operator can be defined as follows:Tj = \u220fk=1j\u22121E(\u03b1k)\u2009\u2009, T1 = 1, and E(\u03b1k) is the score function of \u03b1k.Let Similar to the HFLPWA operator, the HFLPWG operator satisfies the following properties.\u03b1j = \u2329s\u03b8(\u03b1j), h\u03b1j\u232a\u2009\u2009 be a collection of HFLNs. Then the aggregated value, obtained by using the HFLPWG operator, is also an HFLN, andTj = \u220fk=1j\u22121E(\u03b1k)\u2009\u2009, T1 = 1, and E(\u03b1k) is the score function of \u03b1k.Let n.(1)n = 2, sinceFor we have(2)n = k, thenIf  holds foWhen n = k + 1, by the operations described in n = k + 1. Thus,  = \u2329s\u03b8(\u03b1j), h\u03b1j\u232a\u2009\u2009 be a collection of HFLNs. If \u03b1\u2212 = \u2329min\u2061j{s\u03b8(\u03b1j)}, r\u2212\u232a, \u03b1+ = \u2329max\u2061j{s\u03b8(\u03b1j)}, r+\u232a, where r\u2212 = min\u2061rj\u2208\u222ah\u03b11,h\u03b12,\u2026,h\u03b1n{rj} and r+ = max\u2061rj\u2208\u222ah\u03b11,h\u03b12,\u2026,h\u03b1n{rj}, thenLet \u03b11, \u03b12,\u2026, \u03b1n) = \u03b1 = \u2329s\u03b8(\u03b1), h\u03b1\u232a, and then E(\u03b1) = f*(s\u03b8(\u03b1)) \u00b7 s(h\u03b1).Let HFLPWG(j{s\u03b8(\u03b1j)} \u2264 s\u03b8(\u03b1j) \u2264 max\u2061j{s\u03b8(\u03b1j)} for all j, we haveSince min\u2061\u03b1j = Gj(x) = \u2329s\u03b8(\u03b1j), h\u03b1j\u232a\u2009\u2009 be a collection of HFLNs and \u03b11, \u03b12,\u2026, \u03b1n). ThenLet Similarly to \u03b1 = \u2329s2, {0.2,0.4}\u232a, for example; if \u03b11 = \u03b12 = \u03b1, then HFLPWG = \u2329s2, {0.2,0.213,0.376,0.4}\u232a \u2260 \u03b1. Similarly, we do not consider the monotonicity of HFLPWG because of the variability of weights.Besides, the HFLPWG operator cannot satisfy idempotency. Take In general, the HFLPWA operator emphasizes the impact of the overall evaluation data and the compensation between different evaluation results, while the HFLPWG operator emphasizes the balance in the system and the coordination between different evaluation results. In this section, the generalized form of the HFLPWA and HFLPWG operators will be proposed, that is, the hesitant fuzzy linguistic generalized prioritized weighted aggregation (HFLGPWA) operator.\u03b1j\u2009\u2009 be a collection of HFLNs, and then the HFLGPWA operator can be defined as follows:\u03bb > 0, Tj = \u220fk=1j\u22121E(\u03b1k)\u2009\u2009, T1 = 1, and E(\u03b1k) is the score function of \u03b1k.Let (1)\u03bb \u2192 0, then the HFLGPWA operator degenerates into the HFLPWG operator:If (2)\u03bb = 1, then the HFLGPWA operator degenerates into the HFLPWA operator:If Obviously, the HFLPWA and HFLPWG operators are the special cases of the HFLGPWA operator.In this subsection, the HFLGPWA operator will be applied to MCDM problems with hesitant fuzzy linguistic information.C1, C2,\u2026, Cn}, and the prioritization relationships that exist among them are C1\u227bC2\u227b\u22ef\u227bCn. Every criterion in Ci has a higher priority than every criterion in Cj if i < j. Under these criteria, there is a set of alternatives {x1, x2,\u2026, xm} and the criteria values of the alternatives are expressed as HFLNs Cj(xi)\u2009\u2009. Suppose that R = (Cj(xi))m\u00d7n is the decision matrix. Subsequently, a ranking of alternatives is required.For MCDM problems with hesitant fuzzy linguistic information, assume that there is a set of criteria {In the following paragraphs, the HFLGPWA operator is applied to MCDM problems with hesitant fuzzy linguistic information.This method involves the following steps.The common types of criteria in MCDM problems are maximizing criteria and minimizing criteria. For the minimizing criteria the negation operator in xi\u2009\u2009 with respect to Cj\u2009\u2009 are also denoted by Cj(xi) = \u2329s\u03b8(Cij), hCij\u232a.For convenience, the normalized criteria values of Xi\u2009\u2009 of xi by applying the formula as follows:Obtain the comprehensive evaluation values E(Xi)\u2009\u2009 of the comprehensive values Xi of the alternatives xi\u2009\u2009, in order to rank all alternatives xi\u2009\u2009 and then select the best one(s). If the score values are E(Xi) = E(Xj)\u2009\u2009(i \u2260 j), it is necessary to calculate the accuracy values D(Xi) and D(Xj) and then rank the alternatives xi and xj in accordance with these accuracy values.Use E(Xi) and D(Xi)\u2009\u2009.Use The following case is adapted from .ABC Nonferrous Metals Co. Ltd. is a large state-owned company whose main business is producing and selling nonferrous metals. It is also the largest manufacturer of multispecies nonferrous metals in China, with the exception of aluminum. To expand its main business, the company is always engaged in overseas investment, and a department which consists of executive managers and several experts in the field has been established specifically to make decisions on global mineral investment.x1, x2, x3, x4, x5}. There are many factors that affect the investment environment and four factors are considered based on the experience of the department personnel, including C1: resources ; C2: politics and policy ; C3: economy ; C4: infrastructure (such as railway and highway facilities).Recently, the overseas investment department decided to select a pool of alternatives from several foreign countries based on preliminary surveys. After thorough investigation, five countries  are taken into consideration, that is, {S = {s0, s1, s2, s3, s4, s5, s6} = {very poor, poor, slightly poor, fair, slightly good, good, very good} is used. The evaluation information is given in the form of HFLNs, where Cj(xi) is the evaluation value of the alternative xi on the criterion Cj. In Cj(xi) there is a consensus on the chosen linguistic term and each decision maker can use a value to express his/her opinion; in other words, the value denotes to what degree xi matches this given linguistic term under Cj. Each decision maker gave his/her own evaluations (membership degrees) based on the surveys of the five countries as well as his/her knowledge and experience. Then all the possible membership degrees under each given linguistic term are gathered together. The same membership degrees for a given linguistic term will appear only once in an HFLN. Consequently, following a heated discussion, they came to a consensus on the final evaluations which are expressed by HFLNs as shown in The decision makers, including the experts and executive managers, have gathered to determine the decision information. The linguistic term set C1\u227bC2\u227bC3\u227bC4.Assume that the prioritization relationship for the criteria is f1*(si) = i/2t and adopt the following steps.To get the optimal alternative(s), let xi\u2009\u2009 do not need to be normalized.Considering that all the criteria are of maximizing type, the performance values of the alternatives Use E(Xi)\u2009\u2009 of the comprehensive evaluation values Xi of the alternatives xi\u2009\u2009 can be calculated and are shown in The score values \u03bb, the score values E(Xi) would increase slightly, where i = 1,2, 3,4, 5.It has been ascertained that, with the increase of E(Xi)\u2009\u2009, the rankings of the alternatives are shown in In accordance with the score values x5 is the best choice. This result can reveal the stability of the proposed method.From f* and parameter \u03bb on the decision-making process in this example, different f* and \u03bb are used in Steps In order to illustrate the influence of the linguistic scale function f* or parameter \u03bb is utilized. If \u03bb \u2264 5, the best alternative is x5; if \u03bb \u2265 8, the best one is x3. Moreover, the worst alternative is always x4 except for one situation.We can conclude that the rankings of the alternatives may be a little different when a different linguistic scale function To further illustrate the advantages of the proposed MCDM approach under a hesitant fuzzy linguistic environment, the method in  is used Lin et al.  utilizedS(x5) > S(x1) > S(x3) > S(x2) > S(x4), the ranking is x5\u227bx1\u227bx3\u227bx2\u227bx4, and the most desirable car is x5.Since x1 and x3, and this may be caused by the different operations and comparison method for HFLNs. The operations and comparison method for HFLNs in [f* used in this paper are applicable and effective under different semantic environment. In addition, the HFLWA operator in [Obviously, the rankings obtained by the proposed method in this paper may be a little different from that obtained by the method in . The onlHFLNs in  considerrator in  emphasizAccording to the above comparison analyses, the proposed method for MCDM problems with HFLNs has the following advantages.First, HFLNs used in this paper can express the evaluation information more flexibly. They can depict fuzzy linguistic information more accurately and retain the completeness of the original data or the inherent thoughts of decision makers, which is the prerequisite of guaranteeing accuracy of final outcomes.f* is used. Thus, decision makers can flexibly select the linguistic scale function f* depending on their preferences and the actual semantic situations.Second, the operations of HFLNs in this paper are defined based on linguistic scale functions, which can achieve different results when a different linguistic scale function Third, the proposed hesitant fuzzy linguistic prioritized aggregation operators can deal with MCDM problems under the hesitant fuzzy linguistic environment in which the criteria are in different priority levels. What is more, the criteria weights, which are calculated by the prioritized aggregation operator according to the criteria priority levels, are more objective and reasonable than a set of known criteria weights.To address situations where decision-making problems use qualitative variables rather than numerical ones and to reflect the uncertainty, hesitancy, and inconsistency of decision makers, HFLSs have been introduced and used in this paper. Considering the limitations in the existing literature, new operations of HFLNs were introduced. Then, on the basis of the PA operator, two prioritized aggregation operators for HFLNs were proposed and extended to a generalized form. Furthermore, an MCDM method based on the generalized prioritized aggregation operator under a hesitant fuzzy linguistic environment was developed. Finally, an illustrative example demonstrated the application of the proposed method and comparison analysis was made with the representative method. The results indicated that the method proposed in this paper is feasible and effective in solving MCDM problems with HFLSs.\u03bb may also influence the results. Decision makers can select the most appropriate linguistic scale function f* according to their interests and actual semantic situations. In a word, the main advantages of the proposed method are not only that the proposed operators accommodate a hesitant fuzzy linguistic environment, but also its consideration of the priority among criteria, which is more feasible and practical. In the future research, the linguistic scale function can be applied in other linguistic sets, such as ILSs, LHFSs, and HFLSs. Moreover, we believe that the study of information measures and outranking relations for HFLSs does make great sense.It is well known that the PA operator has many advantages over other operators as it does not need to provide weight vectors and, when using this approach, it is only necessary to know the priority among criteria. The foremost characteristic of these proposed operators is that they take into account the priority among criteria. Although traditional prioritized aggregation operators are generally suitable for aggregating information which is in the form of numerical values or simple fuzzy values, they are unable to deal with hesitant fuzzy linguistic information. The proposed HFLGPWA operator can accommodate situations where the input arguments consist of hesitant fuzzy linguistic information. In addition, the results may change using different linguistic scale functions, and the parameter"}
{"text": "L is strongly semicontinuous if and only if L is semicontinuous and meet semicontinuous. It is proved that semi-FS domains are strongly semicontinuous. Some interpolation properties of semiway-below relations in (strongly) semicontinuous bc-domains are given. In terms of these properties, it is proved that strongly semicontinuous bc-domains, in particular strongly semicontinuous lattices, are all semi-FS domains.We are mainly concerned with some special kinds of semicontinuous domains and relationships between them. New concepts of strongly semicontinuous domains, meet semicontinuous domains and semi-FS domains are introduced. It is shown that a dcpo  A logic-oriented approach to domain theory to formalize the properties of computation is provided in  to denote all order-preserving functions from L to M, use [L\u21aaM] to denote all the functions preserving suprema of prime ideals from L to M and use [L \u2192 M] to denote all the Scott-continuous functions from L to M. All of them are under the pointwise order. It is easy to see that [L \u2192 M]\u2286[L\u21aaM]\u2286.Let L, M be dcpos. Then [L\u21aaM] is a dcpo.Let G be a directed subset of [L\u21aaM] and f(x) = \u2228g\u2208Gg(x) for all x \u2208 L. Then it is easy to see that f is order-preserving. For any prime ideal P \u2208 PI(L), we havef \u2208 [L\u21aaM], showing that [L\u21aaM] is a dcpo.Let L be a dcpo. If D\u2286 is directed and sup\u2061\u03b4\u2208D\u03b4 = IdL, then we say that D is an approximate identity for L.Let L be a dcpo. If L has an approximate identity D\u2286[L\u21aaL] such that \u03b4(x) \u21d0 x for all \u03b4 \u2208 D and for all x \u2208 L, then L is a strongly semicontinuous domain.Let x \u2208 L, let A = {\u03b4(x) : \u03b4 \u2208 D}. Then A\u2286\u21d3x\u2009\u2229\u2193x. Note that D is directed and x \u2208 \u2193(\u2228A) = A\u2191\u2193. By the L is a strongly semicontinuous domain.For any L be a dcpo. A function \u03b4 : L \u2192 L on L is finitely separating if there is a finite set F\u03b4 such that, for each x \u2208 L, there exists y \u2208 F\u03b4 such that \u03b4(x) \u2264 y \u2264 x.Let L is called a semi-FS domain if there is an approximate identity D\u2286[L\u21aaL] consisting of finitely separating functions.A dcpo L, there is an approximate identity D\u2286[L \u2192 L]\u2286[L\u21aaL] for L consisting of finitely separating functions. So, an FS-domain is a semi-FS domain.For an FS-domain L be a dcpo. If \u03b4 \u2208 [L\u21aaL] is finitely separating, then, for all x \u2208 L, \u03b4(x) \u21d0 x. Thus a semi-FS domain is a strongly semicontinuous domain.Let x \u2208 L and P \u2208 PI(L) with x \u2264 \u2228P. Since \u03b4 is finitely separating, there exists a finite set F\u03b4 such that for each d \u2208 P there exists yd \u2208 F\u03b4 with \u03b4(d) \u2264 yd \u2264 d. Let F\u03b4\u2032 = {yd \u2208 F\u03b4 : d \u2208 P}, a nonempty finite subset of F\u03b4. Then for each y \u2208 F\u03b4\u2032, we can get dy \u2208 P such that \u03b4(dy) \u2264 y \u2264 dy. As P is a prime ideal, there exists d0 \u2208 P such that y \u2264 dy \u2264 d0 for all y \u2208 F\u03b4\u2032. Hence for all d \u2208 P, \u03b4(d) \u2264 yd \u2264 d0 and \u03b4(x) \u2264 \u03b4(\u2228P) = \u2228d\u2208P\u03b4(d) \u2264 d0. Therefore, \u03b4(x) \u2208 P. It follows from \u03b4(x) \u21d0 x.Suppose that L is a strongly semicontinuous domain.By The next example gives a strongly semicontinuous domain which is not a semi-FS domain, showing that the reverse of L = {0, b1, b2}\u222a{aj : j = 1,2,\u2026} be the domain showing in bi < aj for i = 1,2 and j = 1,2,\u2026. It is clear that L is an L-domain but not compact in the Lawson topology. By \u2286[L\u21aaL]\u2286 = [L \u2192 L]. So, [L \u2192 L] = [L\u21aaL]. Since L is not an FS-domain, L is not a semi-FS domain either.Let Every strongly semicontinuous bc-domain is a semi-FS domain.L be a strongly semicontinuous bc-domain. For each x \u2208 L, S \u2208 Pfin(L), define \u03b4S : L \u2192 L by \u03b4S(x) = \u2228{y \u2208 S\u2229\u2193x : y \u21d0 x}. If {y \u2208 S\u2229\u2193x : y \u21d0 x} = \u2205, then \u03b4S(x) = \u22a5, the least element of L. So, \u03b4S(x) is well-defined. It is easy to see that \u03b4S is order-preserving with \u03b4S(x) \u2264 x for all x \u2208 L. Next we show that \u03b4S preserves suprema of prime ideals. For each P \u2208 PI(L), it suffices to show that \u03b4S(\u2228P) = \u03b4S(k) \u2264 \u2228p\u2208P\u03b4S(p), where k = \u2228P. If {y \u2208 S\u2229\u2193k : y \u21d0 k} = \u2205, then, by the definition of \u03b4S, we see that \u03b4S(k) = \u22a5\u2009\u2264\u2228p\u2208P\u03b4S(p). Let {y1,\u2026, yl} = {y \u2208 S\u2229\u2193k : y \u21d0 k} \u2260 \u2205, then m = \u2228i=1lyi exists in L. By the definition of \u03b4S, \u03b4S(k) = m \u2264 k. Since yi \u21d0 k for i = 1,2,\u2026, l, we have that m \u21d0 k. By (SI\u2264) in m* \u2265 m such that m \u21d0 m* \u21d0 k. It follows from m* \u21d0 k = \u2228P that m, m* \u2208 P. Noticing that m \u2264 m* and yi \u2264 m \u21d0 m* \u2208 P, we have {y1,\u2026, yl}\u2286{y \u2208 S\u2229\u2193m* : y \u21d0 m*} which yields that \u2228p\u2208P\u03b4S(p) \u2265 \u03b4S(m*) \u2265 \u03b4S(k). So, \u03b4S preserves suprema of prime ideals, and \u03b4 \u2208 [L\u21aaL].Let S, T \u2208 Pfin(L) with S\u2286T. It is easy to see that \u03b4S \u2264 \u03b4T and D = {\u03b4S}S\u2208Pfin(L) is directed. Let F\u03b4S = {\u03b4S(x) : x \u2208 L}. Then F\u03b4S is finite by the finiteness of S. So, \u03b4S is a finitely separating function. Since L is strongly semicontinuous, for each x \u2208 L and \u03b4S \u2208 D, \u03b4S(x) = \u2228{y \u2208 S\u2229\u2193x : y \u21d0 x} \u2264 x andSuppose that D = {\u03b4S}S\u2208Pfin(L)\u2286[L\u21aaL] is an approximate identity for L consisting of finitely separating functions, and L is a semi-FS domain.Therefore, Every strongly semicontinuous lattice is a semi-FS domain."}
{"text": "The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties. The theory of rough sets was originally proposed by Pawlak  in 1982 The relationships between lattice theory and rough sets are another topic receiving much attention in recent years. Cattaneo and Ciucci  focus onThe remaining part of the paper is organized as follows. In this section, we introduce some basic definitions see , 19, 20), 2019, 2L, \u2264) is said to be a lattice if inf\u2061\u2061{x, y} and sup\u2061\u2061{x, y}, denoted by \u2227 and \u2228, respectively, exist, for all x, y \u2208 L. A lattice L is said to be complete if, for every A\u2286L, \u22c0\u03b1\u2208A\u03b1 and \u22c1\u03b1\u2208A\u03b1 exist.A partially ordered set  be a complete lattice with the maximum element 1 and minimum element 0 and \u201c\u227a\u201d a binary relation on L. If the following conditions hold: for all \u03b1, \u03b2, \u03bc, \u03b7, \u03b3 \u2208 L,\u03b1\u227a\u03b2\u21d2\u03b1 \u2264 \u03b2,\u03bc \u2264 \u03b1\u227a\u03b2 \u2264 \u03b7\u21d2\u03bc\u227a\u03b7,\u03b1\u227a\u03b3, \u03b2\u227a\u03b3\u21d2\u03b1\u2228\u03b2\u227a\u03b3,\u03b1 \u2208 L, 0\u227a\u03b1\u227a1,\u2200then the relation \u201c\u227a\u201d is called an auxiliary ordering on L. If the relation \u201c\u227a\u201d satisfies conditions (i), (ii), and (iv), it is called a weak auxiliary ordering on L. The weak auxiliary ordering \u201c\u227a\u201d is called completely approximate if(iii)\u2032\u03b1t\u227a\u03b2(t \u2208 T)\u21d2\u22c1t\u2208T\u03b1t\u227a\u03b2,then \u201c\u227a\u201d is called a strong auxiliary ordering on L.Let  be a completely distributive lattice. If the mapping c : L \u2192 L satisfies the following conditions:\u03b1, \u03b2 \u2208 L, if \u03b1 \u2264 \u03b2, then \u03b2c \u2264 \u03b1c,reverse law: \u2200\u03b1 \u2208 L, (\u03b1c)c = \u03b1,recovery law: \u2200then  is called a fuzzy lattice.Let  be a fuzzy lattice and \u03c0\u2286L. If the subset \u03c0 satisfies the following conditions:\u03c0,0 \u2209 \u03b1, \u03b2 \u2208 \u03c0, \u03b1\u2227\u03b2 \u2260 0\u21d2\u03b1 \u2264 \u03b2\u2009\u2009or\u2009\u2009\u03b2 \u2264 \u03b1,\u03b1, \u03b2, \u03b3 \u2208 \u03c0, \u03b1 \u2264 \u03b3, \u03b2 \u2264 \u03b3\u21d2\u03b1\u2227\u03b2 \u2260 0,\u03b7 = \u2228{\u03b1\u2223\u03b1 \u2208 \u03c0, \u03b1 \u2264 \u03b7} for all \u03b7 \u2208 L,\u03c00\u2286\u03c0 and \u03c00 is linear order subset, then \u2228\u03c00 \u2208 \u03c0,if then \u03c0 is called a molecular set, and the element of \u03c0 is called a molecule. The fuzzy lattice with molecule (L(\u03c0), \u2228, \u2227,c, 0,1) is called a molecular lattice.Let  be a fuzzy lattice and \u03b4\u2286L. If the subset \u03b4 satisfies the following conditions:\u03b4,0, 1 \u2208 \u03b1t \u2208 \u03b4(t \u2208 T)\u21d2\u22c1t\u2208T\u03b1t \u2208 \u03b4,\u03b1, \u03b2 \u2208 \u03b4\u21d2\u03b1\u2227\u03b2 \u2208 \u03b4,then  is called a lattice topology space.Let  be a fuzzy lattice. If the dual mappings \u2009(P1) \u2009(P2) \u2009(P3) \u2009(P4) then L. If \u03b1 is called a definable element. If \u03b1 is called a rough element.Let (\u2009(P3)\u2217then L, and the system If (P3) is replaced by the following condition (P3)\u2217:L, \u2228, \u2227,c, 0,1) be a fuzzy lattice. Denote by APR(L) ) the set of all dual approximation operators  on L, respectively, and by \u03c30apr(L) the set of all definable elements in the Pawlak algebra Let  be a fuzzy lattice and  rougher than Let (APR(L)\u2286APRw(L).Let \u03b1, \u03b2 \u2208 L with \u03b1 \u2264 \u03b2, we have \u03b1 = \u03b1\u2227\u03b2, and soLet I, O \u2208 APR(L) and Define two dual approximation operators APRW(L) according to the following result.It is possible to characterize L, \u2228, \u2227,c, 0,1) be a fuzzy lattice. Then (APRW(L), \u227a) is a complete distributive lattice with maximum element I and minimum element O.Let  Consider \u2009(P1) \u2009(P2) \u2009\u03b1 \u2264 \u03b2, then t \u2208 T, implying that (P3) if \u2009t \u2208 T.(P4) \u2009(2) Consider \u2009\u227at \u2208 T. In fact, let t \u2208 T. Then t \u2208 T and \u03b1 \u2208 L by definition. It follows that(3) Consider On the other hand, it is obvious that t \u2208 T. Hence, Suppose that APRW(L), \u227a) is a complete lattice. It is obvious that I is the maximum element and O is the minimum element and that (APRW(L), \u227a) is distributive. This completes the proof.Summing up the above analysis,  be a fuzzy lattice and  a lattice topology space. Define the operators \u03b1 \u2208 L. Then the system Let ) is a zero-dimensional lattice topology space.Let T is an index set. Now, for any s \u2208 T, we have \u03b1s \u2264 \u22c1t\u2208T\u03b1t, and so t\u2208T\u03b1t \u2265 T is an index set. For any s \u2208 T, we have \u03b1s \u2265 \u22c0t\u2208T\u03b1t, and so \u2229\u03b1c\u2208\u2229It is evident that L, \u03c30apr(L)) is a zero-dimensional lattice topology space.Summing up the above analysis,  be a fuzzy lattice. Define the operator \u201c\u2218\u201d on APR(L) as follows:k, define  transitive closure of Let  be a fuzzy lattice. ThenLet ((1) It is straightforward and omitted.\u03b1, \u03b2 \u2208 L, we have\u2009(P1) \u2009(P2) \u2009(P3) (P4)\u03b1 \u2208 L, \u2200(2) It suffices to prove that (3) From the proof of L.If L, \u2228, \u2227,c, 0,1) be a fuzzy lattice and \u03b1 \u2208 L, one hasLet  be a fuzzy lattice and \u201c\u227a\u201d a strong auxiliary ordering on L. Define two operators L \u2192 L as follows:Let ((1)The strong auxiliary ordering \u227a implies that 1\u227a1 is true, and so we have(2)\u03b1, \u03b2, \u03b3 \u2208 L be such that \u03b3\u227a\u03b1\u2227\u03b2. Then \u03b3 \u2264 \u03b3\u227a\u03b1\u2227\u03b2 \u2264 \u03b1 and \u03b3 \u2264 \u03b3\u227a\u03b1\u2227\u03b2 \u2264 \u03b2. By condition (ii) in \u03b3\u227a\u03b1, \u03b3\u227a\u03b2, and soLet ThereforeIt is obvious that condition (P1) holds. In what follows, we prove that conditions (P2)\u2013(P4) are satisfied.(3)By the duality of approximation operators, it suffices to prove that On the other hand, by condition (iii) in Summing up the above statements, L.Let \u03b1, \u03b2 \u2208 Lbe such that \u03b1\u227a\u03b2. Then \u03b1 \u2264 \u03b2 since (1) Let \u03b1, \u03b2, \u03b7, \u03bb \u2208 L be such that \u03b7 \u2264 \u03b1\u227a\u03b2 \u2264 \u03bb. Then \u03b7\u227a\u03bb.(2) Let \u03b1, \u03b2, \u03bb \u2208 L be such that \u03b1\u227a\u03b3 and \u03b2\u227a\u03b3. Then \u03b1\u2228\u03b2\u227a\u03b3.(4)\u03b1 for all \u03b1 \u2208 L.It follows from (3) Let L.Summing up the above statements, \u201c\u227a\u201d is an auxiliary ordering on In this section, we focus on the approximate structure on fuzzy lattices. This structure can be regarded as abstract system of rough set.D, \u2265) a directed set, and {S(d)}d\u2208D a molecular net and \u03b1 \u2208 \u03c0. If there exists d0 \u2208 D such that d \u2265 d0, then \u03b1 is called a rough limit of S, denoted by S is denoted by lim\u2061S.Let In the sequel, we provide two examples of molecular Pawlak algebra in topology space.U, \u03c1) be a metric space, L = \u03a6(U), and \u03c0 = {{x}\u2223x \u2208 U}, where \u03a6(U) denotes the set of all subsets of U. Definex0 be the apr-limit of the molecular sequence {xn}n\u2208N  \u2264 1 for n \u2265 no. Obviously, Let  fuzzy power set on U, \u03c0 = {{x\u03bb}\u2223x \u2208 U, \u03bb \u2208 , define two fuzzy sets FA and FA on U/R asA, B \u2208 F(U),FA\u222aB = FA \u222a FB,FA\u2229B = FA\u2229FB,FA\u2229B\u2286FA\u2229FB,FA\u222aB\u2287FA \u222a FB,FA\u2286FA.Moreover, for any A \u2208 F(U), define x \u2208 U. Then Let x\u03bb is represented asd0 \u2208 D such that xRyd and \u03bbd \u2264 \u03bb for d \u2265 d0, where S(d) = y\u03bbdd and Xi is poly-point set.In the system U/R = {{x}\u2223x \u2208 U}, namely, R = 1 if and only if x = y, then we havex\u03bb and it is know that d0 \u2208 D such that S(d) = x\u03bb for d \u2265 d0.If R-fuzzy-rough-convergent if and only if it is apr-convergent in the molecular Pawlak algebra A molecular net is called formulas  and 32)R-fuzzy-rThe following is now straightforward.R-fuzzy-rough-convergent classes satisfy the Moore-Smith conditions can induce a topology, called R-rough fuzzy topology, which is a nullity topology with square members.The L(\u03c0), \u2228, \u2227,c, 0,1) be a molecular lattice. Then any mapping h : \u03c0 \u2192 L satisfying \u03b1 \u2264 h(\u03b1) can at least induce a molecular Pawlak algebra.Let (R2)(R3)(R4)(R5)(R6)Let U) denotes the set of all subsets of U.R\u2286U \u00d7 U be a binary relation on U. ThenR is reflexive if and only if R is symmetric if and only if R is transitive if and only if\u2009\u2009the reflexive relation Let Let It is straightforward and omitted.(1)Rp defined asU;the binary relation (2)U is a finite universe or the cover U is finite, then Let  It follows from X\u2286U, we have(2) For any U is finite and let X = {y1, y2,\u2026, ym}\u2286U. Then we haveSuppose that universe U is finite; that is, there exist xk \u2208 U, k = 1,2,\u2026, m such that U. Analogous to the above proof, to prove that X\u2286U, U that Now suppose that the cover R the set of all similar relations on U. Then it is evident that R is infinite-intersection-closed. And the following result holds.In the sequel, denote by R be the set of all similar relations on U. Then we havex]t\u2208TRt\u22c2 = \u22c2t\u2208T[x]Rt for {Rt}t\u2208T\u2286R, where T is an index set;t\u2208TRt\u22c2 = {y\u2223 \u2208 \u22c2t\u2208TRt} = {y\u2223\u2200 t \u2208 T,  \u2208 Rt} = \u22c2t\u2208T{y\u2223 \u2208 Rt} = \u22c2t\u2208T[x]Rt.(1) Consider R1\u2286[x]R2. Now, let X\u2286U. If x]R2\u2286X, implying that [x]R1 = [x]R1\u2229R2 = [x]R1\u2229[x]R2\u2286[x]R2\u2286X; that is, (2) Let x, y) \u2208 R1. It follows that X\u2286U by the assumption, we know that x, y) \u2208 R2. Therefore, R1\u2286R2.Conversely, suppose that U be a finite set and R a similar relation on U. Then there exists an equivalent relation R.Let U as follows:U and R, which also implies that U.It follows from the proof of R* is an equivalent relation such that R\u2286R*. By Suppose that In this paper, we have investigated the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. The relationships between the Pawlak approximation structures and these mathematic structures are established, and some related properties are presented. These works would provide a new direction for the study of rough set theory and information systems. As for future research, it will be interesting to continue the study of molecular Pawlak algebra and general partial approximation spaces."}
{"text": "We put forward a new general iterative process. We prove a convergence result as well as a stability result regarding this new iterative process for weak contraction operators. N, we denote the set of all positive integers. In this paper, we obtain results on the stability and strong convergence for a new iteration process  be complete metric space and T : E \u2192 E a self-map on E; and the set of fixed points of T in E is defined by FT = {p \u2208 E : Tp = p}. Let {xn}n\u2208N \u2282 E be the sequence generated by an iteration involving T which is defined byx0 \u2208 E is the initial point and f is a proper function. Suppose that sequence {xn}n\u2208N converges to a fixed point p of T. Let {yn}n\u2208N \u2282 E and setLet  and \u2211n=0\u221e\u03bcn = \u221e, then lim\u2061n\u2192\u221esn = 0.If {T : C \u2192 E is said to be contraction if there is a fixed real number a \u2208 ,d \u2264 c[d + d].Let  such thatx, y \u2208 E.A mapping E, ||\u00b7||) be Banach space. Assume that C\u2286E is a nonempty closed convex subset and T : C \u2192 C is a mapping satisfying (F(T) \u2260 \u2205.Let  \u2260 \u2205.From Theorems From , we haveIn addition,Substituting  in 13),,13), we p \u2212 Tp|| = 0, we haven \u2208 N.Since ||\u03b4 < 1,\u2009\u2009\u2118n \u2208 , and \u2211n=0\u221e\u2118n = \u221e, we haveSince 0 < n\u2192\u221e||xn \u2212 p|| = 0 yields xn \u2192 p \u2208 F(T). This completes the proof of theorem.So lim\u2061E, ||\u00b7||) be Banach space and T : E \u2192 E a self-mapping with fixed point p with respect to weak contraction condition in the sense of Berinde yn + \u03b6nTyn. Suppose that lim\u2061n\u2192\u221e\u03f5n = 0. Then, we shall prove that lim\u2061n\u2192\u221eyn = p. Using contraction condition ,,19), we p \u2212 Tp|| = 0, we haveSince ||\u2118n(1 \u2212 \u03b4) < 1 and using n\u2192\u221eyn = p.Since 0 < 1 \u2212 n\u2192\u221eyn = p, we show that lim\u2061n\u2192\u221e\u03b5n = 0 as follows:Conversely, letting lim\u2061n\u2192\u221e||yn \u2212 p|| = 0, it follows that lim\u2061n\u2192\u221e\u03b5n = 0. Therefore the iteration scheme is T stable.Since lim\u2061T :  \u2192 , Tx = x/2, \u2118n, \u03ben, \u03b6n, \u03d1n = 0, n = 1,2,\u2026, 15, and n \u2265 16.\u2009\u2009It is easy to show that T is a weak contraction operator satisfying  \u2212 y(t)| denotes the space of all continuous functions. It is well known that C is a real Banach space with respect to ||\u00b7||\u221e norm; more details can be found in  such that Kg > 0,there exists the following inequality:5)(CKg(b \u2212 t0) < 1, and according to a solution of problem \u2229C1. The problem can be reconstituted as follows:2 problem -29) we  we Kg(b6) \u2192 C is defined by the following form:Assume that the following conditions are satisfied:Using weak-contraction mapping, we obtain the following.C1)\u2013(C5) are performed. Then the problem \u2229C1.We suppose that conditions ( problem -29) has hasC1)\u2013T. The fixed point of T is shown via p such that Tp = p.We consider iterative process  for the t \u2208 , it is clear that lim\u2061n\u2192\u221exn = p. Therefore, letting t \u2208 , we obtainFor the first part, that is, for Hence, we obtainBy continuing this way, we haveHence, we obtainSubstituting  into 3434, we obKg(b \u2212 t0)) < 1, we haveSince (1 \u2212 2\u03b6n(1 \u2212 2Kg(b \u2212 t0)) = \u03bcn < 1 and ||xn\u2212p||\u221e = sn, and then the conditions of n\u2192\u221e||xn\u2212p||\u221e = 0.We take"}
{"text": "Biliary atresia (BA) is a human infant disease with inflammatory fibrous obstructions in the bile ducts and is the most common cause for pediatric liver transplantation. In contrast, the sea lamprey undergoes developmental BA with transient cholestasis and fibrosis during metamorphosis, but emerges as a fecund adult. Therefore, sea lamprey liver metamorphosis may serve as an etiological model for human BA and provide pivotal information for hepatobiliary transformation and possible therapeutics.hmgcr), and bile acid biosynthesis, cyp7a1. Injection of hsp90 siRNA for 4\u00a0days altered gene expressions of met, hmgcr, cyp27a1, and slc10a1. Bile acid concentrations were increased while bile duct and gall bladder degeneration was facilitated and synchronized after hsp90 siRNA injection.We hypothesized that liver metamorphosis in sea lamprey is due to transcriptional reprogramming that dictates cellular remodeling during metamorphosis. We determined global gene expressions in liver at several metamorphic landmark stages by integrating mRNA-Seq and gene ontology analyses, and validated the results with real-time quantitative PCR, histological and immunohistochemical staining. These analyses revealed that gene expressions of protein folding chaperones, membrane transporters and extracellular matrices were altered and shifted during liver metamorphosis. HSP90, important in protein folding and invertebrate metamorphosis, was identified as a candidate key factor during liver metamorphosis in sea lamprey. Blocking HSP90 with geldanamycin facilitated liver metamorphosis and decreased the gene expressions of the rate limiting enzyme for cholesterol biosynthesis, HMGCoA reductase  contains supplementary material, which is available to authorized users. Manduca sexta and the single burst of triiodothyronine (T3) for metamorphosis of Xenopus larvae to the froglet stage 3keto-petromyzonol sulfate; Bridge Organic Inc., Vicksburg, MI, USA) was added to the whole liver (24.9\u2009\u00b1\u20091.7\u00a0mg). Liver tissues were homogenized and incubated in a shaker with 70\u00a0rpm at room temperature overnight. The homogenized tissues were then centrifuged at 13,000 \u00d7 g for 10\u00a0min. The supernatant was transferred to a new tube, freeze-dried overnight, and stored at \u221220\u00a0\u00b0C until analyses. Samples were reconstituted in 1\u00a0mL of methanol:water (1:1) and placed in an autosampler for LC-MS/MS analysis.Bile acid analyses followed the method developed by Li et al.  with minRTQ-PCR was performed using the TaqMan MGB or SYBR Green system (Life Technologies) as described previously . Gene se40S ribosomal protein: 5\u2032acctacgcaggaacagctatgaccATCTCGAGCAGCTGAAgctccaatgtggtggaattcgtcg3\u2032. 60S ribosomal protein: 5\u2032cgcatccgcgcaatgaAGACCATCCAGAGCAAtcagatcgtggacatacccgac3\u2032. Bsep: 5\u2032gtgtctcaggagccggtgttgTTCGACTGCAGCATTGccgacaacattcgctacggtgcc3\u2032. Slc10a1: 5\u2032ctgtcccggagggaacctctccaacgtgttcgcgctggcgcTCGACGGAGACATGAAcctcagcatcctcatgaccacgtg3\u2032. Col2a1: 5\u2032ttcacttactctgtgctggaggatgggTGCACTACGCACACCGgcgtgtggggcaagacggtgatcgagtacagg3\u2032For each gene, the following information is shown: gene name and synthetic oligo used as the standard for RTQ-PCR. 5\u2032 and 3\u2032 primer sequences are underlined. (Note: the 3\u2032 primer is complementary to the sequence shown). TaqMan MGB probes are shown in uppercase. Hgf : 5\u2032cggcattgcttggaaggaaaaggggaaaattaccgcggccttgtgaacaaaacagccaccgac3\u2032.Hsp90:cgtgctgcacctgaaggaggaCCAATCTGAGTACCTGGAGgagaagcgcatcaaagacatcg3\u2032. Met: 5\u2032ctgcagacgcagaggttcaccACCAAGTCGGATGTGTGgtcgtttggcgttctgctg3\u2032.5\u2032"}
{"text": "Meanwhile, on the basis of functional analysis skill, the existence of the smooth solution and the uniform validity of the asymptotic expansion are proved.We study a kind of vector singular perturbed delay-differential equations. By using the methods of boundary function and fractional steps, we construct the formula of asymptotic expansion and confirm the interior layer at  Singular perturbed differential equations are often used as mathematical models describing processes in biological sciences and physics, such as genetic engineering and the El Nino phenomenon of atmospheric physics . In ordeIn addition, in the study of population models and propagation of epidemic virus, we sometimes require the construction of models. The models are often expressed by singular perturbed delay-differential equations. We can get the equilibrium points of singular perturbed delay-differential equations and confirm the laws of processes. Therefore, using the research methods and theoretical results of singular perturbed delay-differential problems to solve natural and social processes is essential.In recent years, more and more attention was paid to the study of singular perturbed delay-differential problems , especiaIn this paper, we will discuss the interior layer for a class of nonlinear singularly perturbed differential-difference equations and construct its asymptotic expansion formula. Then, the existence of the smooth interior layer solution and the uniform validity of the asymptotic expansion are proved. The results of this paper are new and complement the previous ones.Fi\u2009\u2009 are sufficiently smooth on the domain D = {(\u03bc y\u2032(t), y(t), t) | ||\u03bc y\u2032(t)|| \u2264 l1, ||y(t)|| \u2264 l2, 0 \u2264 t \u2264 2\u03c3}, and l1, l2 are given positive real numbers. The restriction on 2\u03c3 will not influence the essence of the problems.We consider the following nonlinear singularly perturbed differential-difference equations\u03bc = 0; then we can obtain the degenerate equations   \u03bc = 0; tquations 3)F = T and using the method of boundary function [\u03c3]:t = 0, andt = \u03c3, and lim\u2061\u03c40\u2192+\u221e\u2061\u03a0kx(\u03c40) = 0, lim\u2061\u03c4\u2192\u2212\u221e\u2061Qk(\u2212)x(\u03c4) = 0 hold. The system  are undetermined n-dimensional vector functions.In , where Re\u03bbi(t) < 0, i = 1,\u2026, n, Re\u03bbi(t) > 0, i = n + 1,\u2026, 2n, and I is the n \u00d7 n identity matrix.Suppose that the characteristic equation of the systems  given byBy (H1) and , y\u00afk(t),0x(\u03c40), we haveFor \u03a0 we have14)LetLet0x, 0) on the phase plane G1 of vector function \u03a00y.(H3)n-dimensional stable manifold is \u03a00z = \u03a6L(\u03a00y), and \u03b1(0) \u2212 \u03c6(0) \u2208 G1.Suppose that the By (H2), there exists an 0y, \u03a00z, which are both satisfied with exponential decay.By (H2) and (H3), systems  and 14)14) have kx, we have0z, \u03c6(0) + \u03a00y, \u03b1(\u2212\u03c3), 0) and Gk(\u03c40) is a known vector function about \u03a01y(\u03c40),\u2026, \u03a0k\u22121y(\u03c40).For \u03a0In fact, the homogeneous system of 18)d\u03a0kyation of . Thus, id\u03a0kyationtituting , we haveky = \u03a61(\u03c40)C be the general solution of C be the general solution of kx(\u03c40) is completely determined. Obviously, \u03a0kx(\u03c40) decays exponentially as \u03c4 \u2192 \u221e.Let \u03a0ution of , under tution of , we obtaution of  as21)\u2003irtue of  and 17)ky = \u03a61(lx(\u03c40) satisfy the following inequality:C1, k1 are all positive constants.Under conditions (H1)\u2013(H3), the boundary functions \u03a0Q0(\u2212)x\u03c6(\u03c3), 0) is a hyperbolical singular point on the plane n-dimensional stable manifold passing through (\u03c6(\u03c3), 0).(H4)n-dimensional stable manifold is G2 is a domain of Suppose that this We first consider the system of 0(\u2212)xdQ0(\u2212)yd\u03c4Q0(\u2212)x can be completely determined, but it contains the unknown vector p0.By (H4) and , Q0(\u2212)Qk(\u2212)x is determined by the following system:Hk(\u2212)(\u03c4) is a known vector; elements of matrix Q0(\u2212)z, \u03c6(\u03c3) + Q0(\u2212)y, \u03b1(0), \u03c3). Because Qk(\u2212)x(\u03c4) can be completely determined. We can easily obtain the exponential decay of Qk(\u2212)x(\u03c4). system:dQk(\u2212)yd\u03c4Qlx(\u03c4) satisfy the following inequality:C2, k2 are positive constants.Under the condition (H4), the boundary functions \u03c3, 2\u03c3], let \u03bc y\u2032 = z; the system (x(+) = T and using the method of boundary function :t = \u03c3, andt = 2\u03c3, and lim\u2061\u03c4\u2192+\u221e\u2061Qk(+)x(\u03c4) = 0, lim\u2061\u03c4\u2217\u2192\u2212\u221e\u2061Rkx(\u03c4\u2217) = 0 hold.In , where Re\u03bbi(t) < 0, i = 1,\u2026, 2n, Re\u03bbi(t) > 0, i = 2n + 1,\u2026, 4n.Suppose that the characteristic equations of the systems  have 4n For the zeroth approximation of the left boundary layer  we have(40)whe\u03a00y and  and 14)Q0(+)xn-dimensional stable manifold passing the equilibrium point of (\u03c8(\u03c3), 0, \u03c6(0), 0).(H7)n-dimensional stable manifold isSuppose that this 2where (+) = (\u03a61(+), \u03a62(+))T, and (\u03b1(0) \u2212 \u03c6(0), p0 \u2212 \u03c8(\u03c3)) \u2208 G3.By condition (H6), there exists a 2Q0(+)x(\u03c4) can be determined.By (H7) and , 40), Q, Q40), QQk(+)x(\u03c4) satisfies the following boundary value problem:2(\u03c4) and \u03a82(\u03c4) is similar to that of \u03a6 and \u03a8, respectively, but Qk+x(\u03c4) contains the unknown vector functions p1,\u2026, pk.Ql(+)x(\u03c4) satisfy the following inequality:C3, k3 are all positive constants.Under conditions (H5)\u2013(H7), the boundary functions R0x(\u03c4\u2217), we haveFor n-dimensional stable manifold, which is in some region G4 of vector function R0y.(H8)n-dimensional stable manifold is R0z = \u03a6R(R0y), and Suppose that the Consider the first approximate system of (48), the system  has a soRkx, we haveGk(\u03c40) is a known vector function.For Rkx(\u03c4\u2217) is treated in the same way as \u03a0kx(\u03c40) and is omitted here. Obviously, Rkx(\u03c4\u2217) satisfy the following Lemma.The determination of Rlx(\u03c4\u2217) satisfy the following inequality:C4, k4 are all positive constants.Under the condition (H8), the boundary functions Qk(\u2212)x(\u03c4), Qk(+)x(\u03c4) contain the unknown vector functions p0,\u2026, pk. Next, we will use the continuous conditions to determine them.Obviously, t = \u03c3, the solution x of equivalent system needs to satisfyp0. (H4), (H7), and ((H9)Suppose that  has a soAs the solution of the original problem is continuous at  satisfy(52)Firpk can be completely determined.Next, bringing  and 39)39) into Nk(pk)=y\u00afLet\u03bc0 > 0, c > 0, such that, for 0 < \u03bc \u2264 \u03bc0, the solution x of the systems \u2013(H9) and \u2013(H9) andy = T. For 0 \u2264 t \u2264 1, the degenerate solution of (0y(\u03c40), R0y(\u03c4\u2217), Q0(\u2212)y(\u03c4), and Q0(+)y(\u03c4) are given by the following systems, respectively,0y = ((1/2)e\u03c40\u2212, (1/2)e\u03c40\u2212)T, R0y = ((1/2)e\u03c4\u2217, (1/2)e\u03c4\u2217)T, Q0(\u2212)y(\u03c4) = T, Q0(+)y(\u03c4) = (p01e\u03c4\u2212 + (1/2)e\u03c4\u2212 + (1/4)\u03c4e\u03c4\u2212, p02e\u03c4\u2212 + (1/2)e\u03c4\u2212 + (1/4)\u03c4e\u03c4\u2212)T. By the smooth connection (dQ0(\u2212)y/d\u03c4)|\u03c4=0 = (dQ0(+)y/d\u03c4)|\u03c4=0, we have H(p0) = (2p01 + (1/2) \u2212 (1/4), 2p02 + (1/2) \u2212 (1/4))T = 0. Obviously, p0 = T and |H(p0)/d p0|p0=\u22121/8 = 4 \u2260 0. Thus, we obtain that the zero order asymptotic solution of (Let us consider the systems\u03bc2y\u2032\u2032=y(t\u03bc2y\u2032\u2032=y(tWe can see that the zero order asymptotic solution is close to the reduced solution.n-dimensional singularly perturbed differential equations with time delay. Under some assumptions, we obtain the asymptotic solution of the system (Using the boundary layer function method, we consider a class of e system . In compe system  in this e system  can be u"}
{"text": "Multiscale information system is a new knowledge representation system for expressing the knowledge with different levels of granulations. In this paper, by considering the unknown values, which can be seen everywhere in real world applications, the incomplete multiscale information system is firstly investigated. The descriptor technique is employed to construct rough sets at different scales for analyzing the hierarchically structured data. The problem of unravelling decision rules at different scales is also addressed. Finally, the reduct descriptors are formulated to simplify decision rules, which can be derived from different scales. Some numerical examples are employed to substantiate the conceptual arguments. As one of the important mathematical tools for granular computing , 2, the Pawlak's rough set was proposed on the basis of an indiscernibility relation, which can generate a granulation space on the universe of discourse. Such granulation space is actually a partition since the indiscernibility relation is an equivalence relation. With respect to different requirements, a variety of the expanded rough sets models have been proposed. For example, the tolerance relation \u20137, similObviously, the above rough sets are constructed on the basis of one and only one set of the information granules, which can be generated from a binary relation or a covering. From this point of view, we may call these rough sets the single-granulation rough sets. In single-granulation rough sets, a partition is a granulation space, a binary neighborhood system induced by a binary relation is a granulation space, and a covering is also a granulation space. Nevertheless, it should be noticed that, in , the autPresently, the development of multigranulation rough sets approaches is progressing rapidly. For instance, Qian et al. classified their multigranulation rough sets into two categories: one is the optimistic case and the other is the pessimistic case. Yang et al.  generaliIt must be noticed that the multiscale information system is a very important knowledge representation approach; it can help us to analyze data from the viewpoint of different levels of granulations. For example, maps can be hierarchically organized into different scales, from large to small and vice versa. The smaller the scale, the finer the partition that can be obtained; conversely, the bigger the scale, the coarser the partition that can be obtained. However, what Wu and Leung investigated is complete multiscale information systems. Gore, in his influential book Earth in the Balance , notes tIn the next section, we first introduce some basic notions related to Pawlak's rough set and multiscale information system. The incomplete multiscale information system and rule induction problem are explored in S = , in whichU is a nonempty finite set of objects; it is called the universe;a \u2208 AT, Va is the domain of attribute a.AT is a nonempty finite set of attributes, such that, \u2200Formally, an information system  can be cx \u2208 U, let us denote by a(x) the value that x holds on a (a \u2208 AT). For an information system S, one then can describe the relationship between objects through their attributes values. With respect to a subset of attributes such that A\u2286AT, an indiscernibility relation A = {y \u2208 U :  \u2208 IND(A)} is the A-equivalence class containing x. The pair X with respect to the set of attributes A.The relation IND, whereU = {x1,\u2026, xn} is a nonempty, finite set of objects called the universe of discourse;a1,\u2026, am} is a nonempty, finite set of attributes, and each aj \u2208 A is a multiscale attribute; that is, for the same object in U, attribute aj can take on different values at different scales.AT = {A multiscale information system  is a tupI of levels of granulations. Therefore, a multiscale information system can be rewritten as a system such that , where ajk : U \u2192 Vajk is a surjective function and Vajk is the domain of the kth scale attribute ajk. For the set of the kth scale attributes ATk = {a1k, a2k,\u2026, amk}, one then denotes an equivalence relation such thatIn multiscale information system, Wu and Leung  assumed RkAT is denoted by U/RkAT such thatx]RkAT is the equivalence class that includes x at scale k.The partition induced by I-scales, such hierarchical structure can be expressed by the inclusion relation among m equivalence relations; that is,It should be noticed that, in multiscale information system, since different scales represent different levels of granulations and then there is a hierarchical structure among S =  =  is referred to as a multiscale decision system, where  =  is a multiscale information system and d \u2209 {ajk : k = 1,2,\u2026, I, j = 1,2,\u2026, m}, d : U \u2192 Vd is a special attribute called decision. Such multiscale decision system can be decomposed into I decision systems Sk =  = , k = 1,2,\u2026, I, with the same decision d. Each decomposed decision system Sk represents information on a special level of granulation, that is, scale.A system S1 =  = , is consistent; otherwise it is referred to as inconsistent. Following such work, we will propose a more generalized definition of the concept of consistency in multiscale decision system.In , Wu and S =  =  be a multiscale decision system; then S is referred to as k-scale consistent if and only if k-scale decision system, that is, Sk =  = , is consistent; otherwise S is referred to as k-scale inconsistent.Let k = 1, 1-scale consistency is same to what have been proposed by Wu and Leung. In such case, we have R1AT\u2286Rd}{. Moreover, since R1AT\u2286R2AT\u2286\u22ef\u2286RIAT, then we do not always have R2AT\u2286Rd}{,\u2026, RIAT\u2286Rd}{; it follows that, in I-scale levels of granulations, we need only at least one of the levels of granulations to satisfy the condition of consistent; then this type of the consistent, that is, 1-scale consistent, can be referred to as the optimistic consistent in multiscale decision system.By I-scale consistent. In such case, we have RIAT\u2286Rd}{. Moreover, since R1AT\u2286R2AT\u2286\u22ef\u2286RIAT, then we also have R2AT\u2286Rd}{,\u2026, RIAT\u2286Rd}{; it follows that, in I-scale levels of granulations, we need all the levels of granulations to satisfy the condition of consistent; then this type of the consistent, that is, I-scale consistent, can be referred to as the pessimistic consistent in multiscale decision system.On the other hand, let us consider the k-scale consistent in multiscale decision system.From discussions above, it is not difficult to observe that the optimistic and pessimistic consistent are all special cases of S =  =  be a multiscale decision system; if S is k-scale consistent, then S is also k\u2032-scale consistent, where k\u2032 \u2208 {1,2,\u2026, I} and k\u2032 \u2264 k.Let The above proposition tells us that if a multiscale decision system is consistent in a given scale, then such multiscale decision system is also consistent in the scale, which is smaller than the given scale.It should be noticed that the inverse of S =  =  be a multiscale decision system; then \u2200X\u2286U, the k-scale lower and upper approximations of X are denoted by Let k-scale rough set of X in multiscale decision system. Since the k-scale rough set shown in k-scale rough set. We omit these properties in this paper.The pair x]R1AT\u2286X. Since R1AT\u2286R2AT\u2286\u22ef\u2286RmAT, then we do not have that [x]R2AT\u2286X,\u2026, [x]RIAT\u2286X; it follows that, in I levels of granulations, we need only at least one of the levels of granulations to satisfy the inclusion condition between equivalence class and target concept; such explanation is compatible with that in Qian et al.'s optimistic multigranulation lower approximation [x]R1AT\u2229X \u2260 \u2205. Since R1AT\u2286R2AT\u2286\u22ef\u2286RmAT, then we always have [x]R2AT\u2229X \u2260 \u2205 \u22ef [x]RIAT\u2229X \u2260 \u2205; it follows that, in I levels of granulations, we need all the levels of granulations to satisfy the intersection condition between equivalence class and target concept; such explanation is also compatible with that in Qian et al.'s optimistic multigranulation upper approximation. From this point of view, 1-scale rough set is also referred to as the optimistic multiscale rough set in multiscale decision system.By ximation , 33. MorI-scale lower approximation; we have [x]RIAT\u2286X. Since R1AT\u2286R2AT\u2286\u22ef\u2286RIAT, then we also have [x]R1AT\u2286X,\u2026, [x]RI\u22121AT\u2286X; it follows that, in I levels of granulations, we need all the levels of granulations to satisfy the inclusion condition between equivalence class and target concept; such explanation is compatible with that in Qian et al.'s pessimistic multigranulation lower approximation RIAT\u2229X \u2260 \u2205. Since R1AT\u2286R2AT\u2286\u22ef\u2286RmAT, then we do not always have [x]R1AT\u2229X \u2260 \u2205 \u22ef [x]RI\u22121AT\u2229X \u2260 \u2205; it follows that, in I levels of granulations, we need at least one of the levels of granulations to satisfy the intersection condition between equivalence class and target concept; such explanation is also compatible with that in Qian et al.'s pessimistic multigranulation upper approximation. From this point of view, I-scale rough set is also referred to as the pessimistic multiscale rough set in multiscale decision system.On the other hand, let us consider ximation . MoreoveS =  =  be a multiscale decision system; then, \u2200X\u2286U, one hasLet k-scale lower approximations become smaller while the k-scale upper approximations become bigger. In other words, we can obtain a string of rough sets through different levels of granulations in multiscale decision system.The above proposition tells us that, with the monotonous increasing of levels of granulations, the k-scale rough approximation is defined byX| denotes the cardinal number of set X. Obviously, 0 \u2264 \u03b1k(X) \u2264 1 holds.The accuracy of S =  =  be a multiscale decision system; then, \u2200X\u2286U, one hasLet S =  in this paper. Given an incomplete multiscale information system S, if ajk(x) = \u2217, then we say that the value of object x is unknown on the attribute aj in terms of the k-scale. Moreover, we assume that the unknown value \u2217 can be compared with any other values in the domain of the corresponding attributes [An incomplete multiscale information system is still denoted by tributes , 5. TherS, if Ak\u2286ATk, then any attribute-value pair  is called an Ak-atomic property where ajk \u2208 Ak and vjk \u2208 Vajk. Any Ak-atomic property or conjunction of different Ak-atomic properties is called the Ak-descriptor. If  is the atomic property occurring in Ak-descriptor tk, we simply say that  \u2208 tk. Obviously, tk is constructed at scale k; it can also be called a k-scale descriptor.In the discussion to follow, the symbols \u2227 and \u2228 denote the logical connectives \u201cand\u201d (conjunction) and \u201cor\u201d (disjunction), respectively . Given atk be an Ak-descriptor; if, for all  \u2208 tk, we have  \u2208 sk, that is, tk is constructed from a subset of atomic properties occurring in sk, then we say tk is coarser than sk or sk is finer than tk and is denoted by tk\u2ab0sk or sk\u2aaftk. If tk is constructed from a proper subset of atomic properties occurring in sk, then we say tk is properly coarser than sk and is denoted by tk\u227bsk or sk\u227atk.Let tk be an Ak-descriptor; the attributes set occurring in tk is denoted by Ak(tk). Moreover, if tk is an Ak-descriptor and Ak(tk) = Ak, then tk is called full Ak-descriptor. Here, suppose that \u2227ajk\u2208Ak is a full Ak-descriptor; we denoteajk\u2208Ak|| is referred to as the support of \u2227ajk\u2208Ak.Let Here, let us denoteU could be partitioned into several subsets that may overlap at scale k, and the result is denoted by U/Ak such thatBy the descriptor technique, the universe k, k\u2032 \u2208 {1,2,\u2026, I} and k\u2032 \u2264 k, U/ATk\u2032\u2291U/ATk means that the following two conditions hold: tk|| \u2208 U/ATk, there must be ||tk\u2032|| \u2208 U/ATk\u2032 such that ||tk\u2032||\u2286||tk||;\u2200||tk\u2032|| \u2208 U/ATk\u2032, there must be ||tk|| \u2208 U/ATk such that ||tk\u2032||\u2286||tk||.\u2200||In complete multiscale decision system, the hierarchical structure is represented by a partial relation among different equivalence relations or among different partitions. In incomplete multiscale decision system, since, for each level of granulation, we can obtain a family of the supports of the descriptors, which form coverings on the universe of discourse, and then we can use those supports of the descriptors to represent the hierarchical structure such thatU/ATk\u2032 and U/ATk are all partitions, then condition (1) implies condition (2) or condition (2) implies condition (1); it follows that only one of the above conditions is needed. However, since, in incomplete multiscale information system, U/ATk\u2032 and U/ATk may be the coverings instead of the partitions, then the above two conditions are needed simultaneously.It should be noticed that if S =  = , \u2200k, k\u2032 \u2208 {1,2,\u2026, I} and k\u2032 \u2264 k, a surjective function can be defined asThe above two conditions for expressing the hierarchical structure in incomplete multiscale information system are consistent with the basic thinking of surjective function. In other words, in an incomplete multiscale information system S =  = , where U = {x1, x2,\u2026, x20}, AT = {a1, a2, a3, a4} is the set of the condition attributes, and d is the decision attribute. The system has three levels of granulations, where \u201cG,\u201d \u201cF,\u201d \u201cB,\u201d \u201cL,\u201d \u201cM,\u201d \u201cH,\u201d \u201cY,\u201d and \u201cN\u201d stand for, respectively, \u201cgood,\u201d \u201cfair,\u201d \u201cbad,\u201d \u201clow,\u201d \u201cmedium,\u201d \u201chigh,\u201d \u201cyes,\u201d and \u201cno.\u201d1, AT2, and AT3 descriptors are shown in Tables By the descriptor technique we mentioned above, it is not difficult to obtain the descriptors and their supports in each level of granulation. The results of full ATS =  =  be an incomplete multiscale decision system; then S is referred to as k-scale consistent if and only if k-scale decision system Sk =  =  is consistent; that is, U/ATk\u2291U/IND({d}); otherwise S is referred to as k-scale inconsistent.Let I-scale consistent incomplete multiscale decision system is referred to as the pessimistic consistent incomplete multiscale decision system.Similar to the complete case, the 1-scale consistent incomplete multiscale decision system is referred to as the optimistic consistent incomplete multiscale decision system while the S =  =  be an incomplete multiscale decision system; if S is k-scale consistent, then S is also k\u2032-scale consistent, where k\u2032 \u2208 {1,2,\u2026, I} and k\u2032 \u2264 k.Let It should be noticed that the inverse of S =  =  be an incomplete multiscale decision system; then, \u2200X\u2286U, the k-scale lower and upper approximations of X are denoted by k-scale boundary region of X is thenLet X1 = {x1, x2, x3, x13, x14, x15}, X2 = {x4, x5, x6, x7, x8, x9, x10, x11, x12}, X3 = {x16, x17, x18, x19, x20}, then by  1-scale lower and upper approximations:(1)t31||, ||t51||, ||t71||,||t81||, ||t91||, ||t101||, ||t141||, ||t151||, ||t191||, ||t231||, ||t261||},t41||, ||t61||, ||t111||, ||t121||,||t131||, ||t161||, ||t171||, ||t181||, ||t201||, ||t211||, ||t221||, ||t251||},t41||, ||t61||, ||t91||, ||t111||, ||t121||, ||t131||, ||t141||, ||t161||, ||t171||, ||t181||, ||t191||, ||t201||, ||t211||, ||t221||, ||t251||}, 2-scale lower and upper approximations:(2) 3-scale lower and upper approximations:(3)Take, for instance, By the above computations, we can see that S =  =  be an incomplete multiscale decision system; then, \u2200X\u2286U, one hasLet The results in S =  =  be an incomplete multiscale decision system; then a k-scale decision rule is represented bytk \u2208 FDES(ATk), s = , w \u2208 Vd, and tk and s are, respectively, called the condition and decision parts of the rule rk.The end result of rough set is a representation of the information contained in the data system considered in terms of \u201cif\u2026 then\u2026\u201d decision rules , 44. Sink-scale decision rule rk : tk \u2192 s, we associate a quantitative measure, called the certainty, of rk and it is defined byFor each k-scale decision rule rk : tk \u2192 s is referred to as certain if and only if Cer(rk) = 1; a k-scale decision rule rk : tk \u2192 s is referred to as possible if and only if 0 < Cer(rk) < 1. Similar to the traditional rough set approach, the certain decision rules are supported by the descriptors in lower approximation while the possible rules are supported by the descriptors in boundary region. In other words, if tk \u2192 s is a certain decision rule; if ||tk|| \u2208 BNDESkAT||), then tk \u2192 s is a possible decision rule.A S =  = ,U/AT1\u2291U/AT2\u2291\u22ef\u2291U/ATI, then, \u2200||tk|| \u2208 U/ATk, there must be ||tk\u2032|| \u2208 U/ATk\u2032 such that ||tk\u2032||\u2286||tk||; it tells us that if we have a certain k-scale decision rule such that rk : tk \u2192 s, then we can also obtain a certain k\u2032-scale decision rule; that is, rk\u2032 : tk\u2032 \u2192 s;since U/AT1\u2291U/AT2\u2291\u22ef\u2291U/ATI, then, \u2200||tk\u2032|| \u2208 U/ATk\u2032, there must be ||tk|| \u2208 U/ATk such that ||tk\u2032||\u2286||tk||; it tells us that if we have a possible k\u2032-scale decision rule such that rk\u2032 : tk\u2032 \u2192 s, then we can also obtain a possible k-scale decision rule; that is, rk : tk \u2192 s.since In an incomplete multiscale decision system  1-scale decision rules:(1)(a) 1-scale certain decision rules:r11 : a11(x) = 1\u2227a21(x) = 3\u2227a31(x) = 5\u2227a41(x) = 2 \u2192 d(x) = 1 // supported by t31,r21 : a11(x) = 3\u2227a21(x) = 3\u2227a31(x) = 2\u2227a41(x) = 2 \u2192 d(x) = 1 // supported by t81,r31 : a11(x) = 4\u2227a21(x) = 3\u2227a31(x) = 5\u2227a41(x) = 2 \u2192 d(x) = 1 // supported by t101,r41 : a11(x) = 5\u2227a21(x) = 3\u2227a31(x) = 5\u2227a41(x) = 2 \u2192 d(x) = 1 // supported by t151,r51 : a11(x) = 7\u2227a21(x) = 3\u2227a31(x) = 5\u2227a41(x) = 2 \u2192 d(x) = 1 // supported by t231,r61 : a11(x) = 8\u2227a21(x) = 3\u2227a31(x) = 5\u2227a41(x) = 2 \u2192 d(x) = 1 // supported by t261,r71 : a11(x) = 1\u2227a21(x) = 1\u2227a31(x) = 3\u2227a41(x) = 1 \u2192 d(x) = 2 // supported by t21,r81 : a11(x) = 2\u2227a21(x) = 1\u2227a31(x) = 3\u2227a41(x) = 1 \u2192 d(x) = 2 // supported by t41,r91 : a11(x) = 3\u2227a21(x) = 1\u2227a31(x) = 3\u2227a41(x) = 1 \u2192 d(x) = 2 // supported by t61,r101 : a11(x) = 4\u2227a21(x) = 4\u2227a31(x) = 1\u2227a41(x) = 1 \u2192 d(x) = 2 // supported by t111,r111 : a11(x) = 4\u2227a21(x) = 4\u2227a31(x) = 3\u2227a41(x) = 3 \u2192 d(x) = 2 // supported by t121r121 : a11(x) = 5\u2227a21(x) = 1\u2227a31(x) = 3\u2227a41(x) = 1 \u2192 d(x) = 2 // supported by t131,r131 : a11(x) = 5\u2227a21(x) = 5\u2227a31(x) = 4\u2227a41(x) = 1 \u2192 d(x) = 2 // supported by t161,r141 : a11(x) = 5\u2227a21(x) = 5\u2227a31(x) = 4\u2227a41(x) = 4 \u2192 d(x) = 2 // supported by t171,r151 : a11(x) = 6\u2227a21(x) = 1\u2227a31(x) = 3\u2227a41(x) = 1 \u2192 d(x) = 2 // supported by t181,r161 : a11(x) = 6\u2227a21(x) = 6\u2227a31(x) = 5\u2227a41(x) = 4 \u2192 d(x) = 2 // supported by t201,r171 : a11(x) = 6\u2227a21(x) = 6\u2227a31(x) = 5\u2227a41(x) = 2 \u2192 d(x) = 2 // supported by t211,r181 : a11(x) = 7\u2227a21(x) = 1\u2227a31(x) = 3\u2227a41(x) = 1 \u2192 d(x) = 2 // supported by t221,r191 : a11(x) = 8\u2227a21(x) = 1\u2227a31(x) = 3\u2227a41(x) = 1 \u2192 d(x) = 2 // supported by t251,r201 : a11(x) = 7\u2227a21(x) = 7\u2227a31(x) = 6\u2227a41(x) = 5 \u2192 d(x) = 3 // supported by t241,r211 : a11(x) = 8\u2227a21(x) = 7\u2227a31(x) = 6\u2227a41(x) = 6 \u2192 d(x) = 3 // supported by t271;(b) 1-scale possible decision rules:r1\u20321 : a11(x) = 1\u2227a21(x) = 1\u2227a31(x) = 1\u2227a41(x) = 1 \u2192 d(x) = 1, Cer (r1\u20321) = 0.5 // supported by t11,r2\u20321 : a11(x) = 2\u2227a21(x) = 2\u2227a31(x) = 1\u2227a41(x) = 1 \u2192 d(x) = 1, Cer (r2\u20321) = 0.5 // supported by t51,r3\u20321 : a11(x) = 3\u2227a21(x) = 3\u2227a31(x) = 2\u2227a41(x) = 2 \u2192 d(x) = 1, Cer (r3\u20321) = 0.5 // supported by t71,r4\u20321 : a11(x) = 4\u2227a21(x) = 1\u2227a31(x) = 3\u2227a41(x) = 1 \u2192 d(x) = 1, Cer (r4\u20321) = 0.5 // supported by t91,r5\u20321 : a11(x) = 5\u2227a21(x) = 2\u2227a31(x) = 4\u2227a41(x) = 1 \u2192 d(x) = 1, Cer (r5\u20321) = 0.5 // supported by t141,r6\u20321 : a11(x) = 6\u2227a21(x) = 3\u2227a31(x) = 5\u2227a41(x) = 2 \u2192 d(x) = 1, Cer (r6\u20321) = 0.5 // supported by t191,r7\u20321 : a11(x) = 4\u2227a21(x) = 1\u2227a31(x) = 3\u2227a41(x) = 1 \u2192 d(x) = 2, Cer (r7\u20321) = 0.5 // supported by t91,r8\u20321 : a11(x) = 5\u2227a21(x) = 2\u2227a31(x) = 4\u2227a41(x) = 1 \u2192 d(x) = 2, Cer (r8\u20321) = 0.5 // supported by t141,r9\u20321 : a11(x) = 6\u2227a21(x) = 3\u2227a31(x) = 5\u2227a41(x) = 2 \u2192 d(x) = 2, Cer (r9\u20321) = 0.5 // supported by t191,r10\u20321 : a11(x) = 1\u2227a21(x) = 1\u2227a31(x) = 1\u2227a41(x) = 1 \u2192 d(x) = 3, Cer (r10\u20321) = 0.5 // supported by t11,r11\u20321 : a11(x) = 2\u2227a21(x) = 2\u2227a31(x) = 1\u2227a41(x) = 1 \u2192 d(x) = 3, Cer (r11\u20321) = 0.5 // supported by t51,r12\u20321 : a11(x) = 3\u2227a21(x) = 3\u2227a31(x) = 2\u2227a41(x) = 2 \u2192 d(x) = 1, Cer (r12\u20321) = 0.5 // supported by t71; 2-scale decision rules:(2)(a) 2-scale certain decision rules:r12 : a12(x) = F\u2227a22(x) = G\u2227a32(x) = M\u2227a42(x) = L \u2192 d(x) = 1 // supported by t72,r22 : a12(x) = F\u2227a22(x) = G\u2227a32(x) = L\u2227a42(x) = L \u2192 d(x) = 1 // supported by t82,r32 : a12(x) = F\u2227a22(x) = G\u2227a32(x) = H\u2227a42(x) = L \u2192 d(x) = 1 // supported by t92,r42 : a12(x) = F\u2227a22(x) = G\u2227a32(x) = M\u2227a42(x) = M \u2192 d(x) = 2 // supported by t22,r52 : a12(x) = F\u2227a22(x) = F\u2227a32(x) = M\u2227a42(x) = M \u2192 d(x) = 2 // supported by t42,r62 : a12(x) = F\u2227a22(x) = F\u2227a32(x) = M\u2227a42(x) = L \u2192 d(x) = 2 // supported by t52,r72 : a12(x) = F\u2227a22(x) = B\u2227a32(x) = M\u2227a42(x) = M \u2192 d(x) = 2 // supported by t62,r82 : a12(x) = B\u2227a22(x) = B\u2227a32(x) = H\u2227a42(x) = H \u2192 d(x) = 3 // supported by t102,r92 : a12(x) = B\u2227a22(x) = F\u2227a32(x) = H\u2227a42(x) = H \u2192 d(x) = 3 // supported by t112,r102 : a12(x) = B\u2227a22(x) = G\u2227a32(x) = H\u2227a42(x) = H \u2192 d(x) = 3 // supported by t122;(b) 2-scale possible decision rules:r1\u20322 : a12(x) = G\u2227a22(x) = G\u2227a32(x) = L\u2227a42(x) = L \u2192 d(x) = 1, Cer (r1\u20322) = 0.5 // supported by t12,r2\u20322 : a12(x) = F\u2227a22(x) = G\u2227a32(x) = M\u2227a42(x) = L \u2192 d(x) = 1, Cer (r2\u20322) = 0.5 // supported by t32,r3\u20322 : a12(x) = F\u2227a22(x) = G\u2227a32(x) = M\u2227a42(x) = L \u2192 d(x) = 2, Cer (r3\u20322) = 0.5 // supported by t32,r4\u20322 : a12(x) = G\u2227a22(x) = G\u2227a32(x) = L\u2227a42(x) = L \u2192 d(x) = 3, Cer (r4\u20322) = 0.5 // supported by t12; 3-scale decision rules:(3)(a) 3-scale certain decision rules:r13 : a13(x) = N\u2227a23(x) = N\u2227a33(x) = N\u2227a43(x) = Y \u2192 d(x) = 2 // supported by t43;(b) 3-scale possible decision rules:r1\u20323 : a13(x) = Y\u2227a23(x) = Y\u2227a33(x) = Y\u2227a43(x) = Y \u2192 d(x) = 1, Cer (r1\u20323) = 0.5 // supported by t13,r2\u20323 : a13(x) = N\u2227a23(x) = Y\u2227a33(x) = N\u2227a43(x) = Y \u2192 d(x) = 1, Cer (r2\u20323) = 0.5 // supported by t23,r3\u20323 : a13(x) = N\u2227a23(x) = Y\u2227a33(x) = N\u2227a43(x) = Y \u2192 d(x) = 2, Cer (r3\u20323) = 0.5 // supported by t23,r4\u20323 : a13(x) = N\u2227a23(x) = N\u2227a33(x) = N\u2227a43(x) = N \u2192 d(x) = 2, Cer (r4\u20323) = 0.667 // supported by t33,r5\u20323 : a13(x) = Y\u2227a23(x) = Y\u2227a33(x) = Y\u2227a43(x) = Y \u2192 d(x) = 3, Cer (r5\u20323) = 0.5 // supported by t13,r6\u20323 : a13(x) = N\u2227a23(x) = N\u2227a33(x) = N\u2227a43(x) = N \u2192 d(x) = 3, Cer (r6\u20323) = 0.333 // supported by t33.Following the results of approximations we obtained in S =  =  be an incomplete multiscale decision system, tk \u2208 FDES(ATk) is a full ATk-descriptor, tk\u2032 \u2208 DES(ATk) is an ATk-descriptor, and tk\u2032\u2ab0tk; then tk\u2032 is referred to as the reduct descriptor of tk if and only if the following two conditions hold: tk\u2032|| = ||tk||;||tk\u2032\u2032|| \u2260 ||tk|| for each tk\u2032\u2032\u227btk\u2032.||Let tk is a conjunction of the atomic properties in tk, which preserves the support of tk. The reduct descriptor allows us to classify objects with the smallest number of required atomic properties.By tk, let us definetk discernibility matrix.To compute the reduct descriptor of S =  =  be an incomplete multiscale decision system in which tk \u2208 FDES(ATk), tk\u2032 \u2208 DES(ATk), and tk\u2032\u2ab0tk; then\u2009tk|| = ||tk\u2032||\u21d4AT(tk\u2032)\u2229DIS \u2260 \u2205 for each DIS \u2208 Mtk.||Let tk, y) \u2208 Mtk such that AT(tk\u2032)\u2229DIS = \u2205; then by the definition of DIS, we have y \u2208 ||tk\u2032|| because tk\u2032\u2ab0tk. Moreover, since ||tk\u2032|| = ||tk||, then y \u2208 ||tk||; such result is contradictive to the condition DIS \u2208 Mtk\u21d2y \u2209 ||tk||.\u21d2: Suppose \u2203DIS \u2208 Mtk, since AT(tk\u2032)\u2229DIS \u2260 \u2205; then there must be ajk \u2208 ATk(tk\u2032) such that y \u2209 ||||; it follows that y \u2209 ||tk\u2032||, from which we can conclude that y \u2209 ||tk||\u21d2y \u2209 ||tk\u2032||; that is, ||tk\u2032||\u2286||tk||. That completes the proof.\u21d0: Since S =  =  be an incomplete multiscale decision system in which tk \u2208 FDES(ATk); then the tk discernibility function is defined asLet S =  =  be a multiscale decision system in which tk \u2208 FDES(ATk), tk\u2032 \u2208 DES(ATk), and tk\u2032\u2ab0tk; then\u2009tk\u2032 is a reduct descriptor of tk if and only if ATk(tk\u2032) is the prime implicant of the tk discernibility function \u0394tk.Let tk\u2032)\u2229DIS \u2260 \u2205 for each DIS \u2208 Mtk. We claim that, for each ajk \u2208 ATk(tk\u2032), there must be DIS \u2208 Mtk such that AT(tk\u2032)\u2229DIS = {ajk}. If fact, if \u2200DIS \u2208 Mtk, there exist ajk \u2208 DIS such that |AT(tk\u2032)\u2229DIS| > 2, where ajk \u2208 ATk(tk\u2032)\u2229DIS; let tk\u2032\u2032\u227btk\u2032, where ATk(tk\u2032\u2032) = ATk(tk\u2032)\u2212{ajk}; then by tk\u2032\u2032|| = ||tk||, which contradicts the fact that tk\u2032 is a reduct descriptor of tk; it follows that ATk(tk\u2032) is the prime implicant of the \u0394tk.\u21d2: By k(tk\u2032) is the prime implicant of the \u0394tk, then by k(t\u2032)\u2229DIS \u2260 \u2205 for each DIS \u2208 Mtk. Moreover, for each ajk \u2208 ATk(tk\u2032), there exists DIS \u2208 Mtk such that ATk(tk\u2032)\u2229DIS = {ajk}. Consequently, \u2200tk\u2032\u2032\u227btk\u2032 and ATk(tk\u2032\u2032) = ATk(tk\u2032)\u2212{ajk}, ||tk\u2032\u2032|| \u2260 ||tk||; we then conclude that tk\u2032 is a reduct descriptor of tk. That completes the proof.\u21d0: If ATIn S =  =  be an incomplete multiscale decision system in which tk \u2208 FDES(ATk), tk\u2032 \u2208 DES(ATk), and tk\u2032\u2ab0tk; definetk\u2032 is referred to as the lower approximation reduct descriptor of tk if and only if the following two conditions hold: tLk = tLk\u2032;tLk\u2032\u2032 \u2260 tLk for each tk\u2032\u2032\u227btk\u2032;Let tk\u2032 is referred to as the boundary region reduct descriptor of tk if and only if the following two conditions hold: tBk = tBk\u2032;tBk\u2032\u2032 \u2260 tBk for each tk\u2032\u2032\u227btk\u2032.tk is a minimal conjunction of the atomic properties in tk, which preserves the inclusion relation between support of tk and the decision classes; a boundary region reduct descriptor of tk is a minimal conjunction of the atomic properties in tk, which preserves the intersection relation between support of tk and the decision classes.By tLk = \u2205, then ||tk||\u2288Xi for each Xi \u2208 U/IND({d}); it follows that no certain decision rules are supported by the k-scale descriptor tk. Therefore, it is meaningless to compute the lower approximation reduct descriptor of tk if tLk = \u2205.It should be noticed that if S =  = , if tLk = {Xi}, then ||tk||\u2286Xi; we can obtain a certain decision rule such that rk : tk \u2192 Xi. Moreover, suppose that tk\u2032 is a lower approximation reduct descriptor of tk; then tk\u2032 is a minimal conjunction of the atomic properties in tk, which preserves the inclusion property; therefore, the decision rule rk : tk \u2192 Xi can be simplified to be rk\u2032 : tk\u2032 \u2192 Xi. The condition part of rk : tk \u2192 Xi is shortened because tk\u2032\u2ab0tk. Similar to what have been analyzed, by the boundary region reduct descriptor, the possible rules supported by tk can also be simplified.Given an incomplete multiscale decision system tk, let us definetk lower approximation and tk boundary region discernibility matrixes, respectively.To compute the lower approximation and boundary region reduct descriptor of S =  =  be an incomplete multiscale decision system in which tk \u2208 FDES(ATk), tk\u2032 \u2208 DES(ATk), and tk\u2032\u2ab0tk: tLk = tLk\u2032\u21d4ATk(tk\u2032)\u2229DISL \u2260 \u2205 for each DISL \u2208 MLtk;tBk = tBk\u2032\u21d4ATk(tk\u2032)\u2229DISB \u2260 \u2205 for each DISB \u2208 MBtk.Let We only prove (1); the proof of (2) is similar to the proof of (1).L \u2208 MLtk such that ATk(tk\u2032)\u2229DISL = \u2205; then we have y \u2208 ||tk\u2032|| because tk\u2032\u2ab0tk. Moreover, by condition we have tLk = tLk\u2032; that is, ||tk||\u2286Xi\u21d4||tk\u2032||\u2286Xi; we know y \u2208 Xi; such result is contradictive to the assumption because DISL \u2208 MLtk\u21d2y \u2209 Xi.\u21d2: Suppose \u2203DIStk\u2032\u2ab0tk, then ||tk\u2032||\u2287||tk||, from which we obtain tLk\u2287tLk\u2032. Thus, it must be proved that tLk\u2286tLk\u2032. \u2200Xi \u2208 tLk, we have ||tk||\u2286Xi. Then \u2200y \u2209 Xi, we have DISL \u2208 MLtk; by condition we know that ATk(tk\u2032)\u2229DISL \u2260 \u2205; it follows that y \u2209 ||t\u2032||; then there must be ajk \u2208 ATk(tk\u2032) such that y \u2209 ||ajk, vjk||; it follows that y \u2209 ||tk\u2032||, from which we can conclude that ||tk\u2032||\u2286Xi; that is, Xi \u2208 tLk\u2032, tLk\u2287tLk\u2032. That completes the proof.\u21d0: Since S =  =  be an incomplete multiscale decision system in which tk \u2208 FDES(ATk); thentk lower approximation discernibility function;tk boundary region discernibility function.Let S =  =  be an incomplete multiscale decision system in which tk \u2208 FDES(ATk), tk\u2032 \u2208 DES(ATk), and tk\u2032\u2ab0tk; then tk\u2032 is a lower approximation reduct descriptor of tk if and only if ATk(tk\u2032) is the prime implicant of the tk lower approximation discernibility function \u0394Ltk;tk\u2032 is a boundary region reduct descriptor of tk if and only if ATk(tk\u2032) is the prime implicant of the tk boundary region discernibility function \u0394Btk.Let The proof of In By the discernibility matrixes we mentioned above, it is not difficult to obtain the lower approximation and boundary region reduct descriptors as Tables  1-scale certain decision rules:a11(x) = 1,4, 5,7, 8\u2227a21(x) = 3 \u2192 d(x) = 1,a11(x) = 1,3, 4,5, 7,8\u2227a31(x) = 5 \u2192 d(x) = 1,a11(x) = 1,4, 5,7, 8\u2227a41(x) = 2 \u2192 d(x) = 1,a11(x) = 1,2, 3,5, 6,7, 8\u2227a31(x) = 3 \u2192 d(x) = 2,a11(x) = 2,3, 5,6, 7,8\u2227a21(x) = 1 \u2192 d(x) = 2,a11(x) = 3,6, 7,8\u2227a41(x) = 1 \u2192 d(x) = 2,a21(x) = 4,5, 6 \u2192 d(x) = 2,a11 = 4\u2227a31 = 1 \u2192 d(x) = 2,a41(x) = 3,4 \u2192 d(x) = 2,a21(x) = 7\u2228a31(x) = 6\u2228a41(x) = 5,6 \u2192 d(x) = 3; 2-scale certain decision rules:a12(x) = F\u2227a32(x) = L, H \u2192 d(x) = 1,a32(x) = H\u2227a42(x) = L \u2192 d(x) = 1,a42(x) = M \u2192 d(x) = 2,a12(x) = F\u2227a22(x) = F, B \u2192 d(x) = 2,a22(x) = F, B\u2227a32(x) = M \u2192 d(x) = 2,a22(x) = F\u2227a42(x) = L \u2192 d(x) = 2,a12(x) = B\u2228a42(x) = H \u2192 d(x) = 3,a22(x) = B, F\u2227a32(x) = H \u2192 d(x) = 3; 3-scale certain decision rules:a23(x) = N\u2227a43(x) = Y \u2192 d(x) = 2.By the lower approximation reduct descriptors shown in  1-scale possible decision rules:a11(x) = 1\u2227a31(x) = 1 \u2192 d(x) = 1\u22283 with certainty 0.5,a21(x) = 1\u2227a31(x) = 1 \u2192 d(x) = 1\u22283 with certainty 0.5,a11(x) = 2\u2227a21(x) = 2 \u2192 d(x) = 1\u22283 with certainty 0.5,a11(x) = 2\u2227a31(x) = 1 \u2192 d(x) = 1\u22283 with certainty 0.5,a11(x) = 3\u2227a21(x) = 3 \u2192 d(x) = 1 with certainty 0.67,a11(x) = 3\u2227a21(x) = 3 \u2192 d(x) = 3 with certainty 0.33,a31(x) = 2 \u2192 d(x) = 1\u22283 with certainty 0.5,a11(x) = 3\u2227a41(x) = 2 \u2192 d(x) = 1 with certainty 0.67,a11(x) = 3\u2227a41(x) = 2 \u2192 d(x) = 3 with certainty 0.33,a11(x) = 4\u2227a31(x) = 3 \u2192 d(x) = 2 with certainty 0.67,a11(x) = 4\u2227a31(x) = 3 \u2192 d(x) = 1 with certainty 0.33,a11(x) = 5\u2227a31(x) = 4 \u2192 d(x) = 2 with certainty 0.75,a11(x) = 5\u2227a31(x) = 4 \u2192 d(x) = 1 with certainty 0.25,a11(x) = 6\u2227a31(x) = 5 \u2192 d(x) = 2 with certainty 0.75,a11(x) = 6\u2227a31(x) = 5 \u2192 d(x) = 1 with certainty 0.25; 2-scale possible decision rules:a12 = G\u2227a32 = L \u2192 d(x) = 1\u22283 with certainty 0.5,a12 = F\u2227a32 = M \u2192 d(x) = 2 with certainty 0.75,a12 = F\u2227a32 = M \u2192 d(x) = 1 with certainty 0.25; 3-scale possible decision rules:a13 = Y\u2228a33 = Y \u2192 d(x) = 1\u22283 with certainty 0.5,a13 = N\u2228a23 = Y \u2192 d(x) = 1\u22282 with certainty 0.5,a13 = N\u2228a43 = Y \u2192 d(x) = 2 with certainty 0.67,a13 = N\u2228a43 = Y \u2192 d(x) = 1 with certainty 0.33,a23 = Y\u2228a33 = N \u2192 d(x) = 1\u22282 with certainty 0.5,a33 = N\u2228a43 = Y \u2192 d(x) = 2 with certainty 0.67,a33 = N\u2228a43 = Y \u2192 d(x) = 1 with certainty 0.33,a23 = N \u2192 d(x) = 2 with certainty 0.75,a23 = N \u2192 d(x) = 3 with certainty 0.25,a43 = N \u2192 d(x) = 2 with certainty 0.67,a43 = N \u2192 d(x) = 3 with certainty 0.33.By the boundary region reduct descriptors shown in In this paper, the incomplete multiscale information system is explored through the well-known rough set approach. The incomplete multiscale information system is different from the traditional incomplete information system since it reflects data represented at different levels of granulations, which are transformed from smaller to bigger scales. At each level of granulation, that is, scale, we can construct the corresponding rough set and then derive decision rules. Obviously, the decision rules derived from a coarser scale are more general than that from a finer scale. The incomplete multiscale information system is closely related to multigranulation rough set approach. The rough set at the smallest scale is optimistic multigranulation rough set while the rough set at the biggest scale is pessimistic multigranulation rough set.Obviously, the study of multiscale information system will give new life to the development of rough set theory. Furthermore, by considering the preference-ordered \u201347 domai"}
{"text": "M be a 2\u2010torsion free prime \u0393\u2010ring satisfying the condition a\u03b1b\u03b2c=a\u03b2b\u03b1c,\u2200a,b,c\u2208M and \u03b1,\u03b2\u2208\u0393, U be an admissible Lie ideal of M and F=(fi)i\u2208N be a generalized higher \u2010derivation of M with an associated higher \u2010derivation D=(di)i\u2208N of M. Then for all n\u2208N we prove that Let Mathematics Subject Classification (2010): 13N15; 16W10; 17C50 M and \u0393 be additive abelian groups. If there is a mapping M\u00d7\u0393\u00d7M\u2192M  into x\u03b1y) such that (i) (x+y)\u03b1z=x\u03b1z+y\u03b1z,x(\u03b1+\u03b2)y=x\u03b1y+x\u03b2y,x\u03b1(y+z)=x\u03b1y+x\u03b1z, (ii) (x\u03b1y)\u03b2z=x\u03b1(y\u03b2z), for all x,y,z\u2208M and \u03b1,\u03b2\u2208\u0393, then M is called a \u0393\u2010ring. This concept is more general than a ring and was introduced by Barnes , where Z(M) denotes the center of M, then U is called an admissible Lie ideal of M. In \u2010derivations in rings have been introduced by Faraj et al. \u2010derivation extends the concept given in \u22602, U a square closed Lie ideal of R and d a \u2010 derivation of R, then d(ur)=d(u)r+ud(r),\u2200,u\u2208U,r\u2208R. This result is a generalization of a result in \u2010derivation, generalized \u2010derivation and generalized higher \u2010derivation, where U is a Lie ideal of a \u0393\u2010ring M. Examples of a Lie ideal of a \u0393\u2010ring, \u2010derivation, generalized \u2010derivation, higher \u2010derivation and generalized higher \u2010derivation are given here. A result in \u2010derivation. Throughout the article, we use the condition a\u03b1b\u03b2c=a\u03b2b\u03b1c,\u2200a,b,c\u2208M and \u03b1,\u03b2\u2208\u0393 and this is represented by (*). We make the basic commutator identities \u03b2= \u03b2\u03b1y+xzy+x\u03b1\u03b2, \u03b2= \u03b2\u03b1z+yxz+y\u03b1\u03b2, \u2200x,y,z\u2208M,\u2200\u03b1,\u03b2\u2208\u0393. According to the condition (*), the above two identities reduces to \u03b2= \u03b2\u03b1y+x\u03b1\u03b2,\u03b2= \u03b2\u03b1z+y\u03b1\u03b2,\u2200x,y,z\u2208M,\u2200\u03b1,\u03b2\u2208\u0393.The notion of a \u0393\u2010ring has been developed by Nobusawa , as a gey Barnes . A \u0393\u2010rinHerstein , HersteiHerstein , Awtar ej et al. , as a gein Awter . In thisin Awter , they prin Awter , Theoremand Paul , TheoremU,R)\u2010derivation of an ordinary ring developed by Faraj et al. \u2010derivation of \u0393\u2010rings as defined below.In view of the concept of \u2010derivation of M if d(u\u03b1m + s\u03b1u) = d(u)\u03b1m + u\u03b1d(m) + d(s)\u03b1u + s\u03b1d(u),\u2200u\u2208U,m,s\u2208M and \u03b1\u2208\u0393.\u2010 derivation of M if there exists a \u2010derivation d of M such that f(u\u03b1m+s\u03b1u)=f(u)\u03b1m+u\u03b1d(m)+f(s)\u03b1u+s\u03b1d(u),\u2200u\u2208U,m,s\u2208M and \u03b1\u2208\u0393.\u2010derivation and a generalized \u2010derivation are confirmed by the following examples.The existence of a Lie ideal of a \u0393\u2010ring,  and M is a \u0393\u2010ring.Let N={:x\u2208R}\u2286M and U1={:u\u2208U} then N is a sub \u0393\u2010ring of M and U1 is a Lie ideal of N. Let f:R\u2192R be a generalized \u2010derivation. Then there exists a \u2010derivation d:R\u2192R such that f(u\u03b1x+s\u03b1u)=f(u)\u03b1x+u\u03b1d(x)+f(s)\u03b1u+s\u03b1d(u).If D :N\u2192N by D)=(d(x),d(x)), then we have If we define a mapping D(u1\u03b1x1+y1\u03b1u1)=D(u1)\u03b1x1+u1\u03b1D(x1)+D(y1)\u03b1u1+y1\u03b1D(u1), where After calculation we have D is a \u2212 derivation on N.Hence F :N\u2192N be the additive mapping defined by F)=(f(x),f(x)), then considering Let F is a generalized \u2212derivation on N.Hence ), Lemma 2.4) Let M be a 2\u2010torsion free \u0393\u2010ring satisfying the condition (*). U be a Lie ideal of M and f be a generalized \u2010derivation of M. Thenf(u\u03b1m\u03b2u)=f(u)\u03b1m\u03b2u+u\u03b1d(m)\u03b2u+u\u03b1m\u03b2d(u),\u2200u\u2208U,m\u2208M and \u03b1,\u03b2\u2208\u0393.(ii)f(u\u03b1m\u03b2v+v\u03b1m\u03b2u)=f(u)\u03b1m\u03b2v+u\u03b1d(m)\u03b2v+u\u03b1m\u03b2d(v)+f(v)\u03b1m\u03b2u+v\u03b1d(m)\u03b2u+v\u03b1m\u03b2d(u),\u2200u,v\u2208U,m\u2208M and \u03b1,\u03b2\u2208\u0393.d be a \u2010derivation of M, then we define \u03a6\u03b1=d(u\u03b1m)\u2212d(u)\u03b1m\u2212u\u03b1d(m),\u2200u\u2208U,m\u2208M and \u03b1\u2208\u0393.\u2010derivation of M, then(i)\u03b1 =\u2212\u03a6\u03b1, \u2200u\u2208U,m\u2208M and \u03b1\u2208\u0393.\u03a6(ii)\u03b1=\u03a6\u03b1+\u03a6\u03b1,\u2200u,v\u2208U,m\u2208Mand \u03b1\u2208\u0393.\u03a6(iii)\u03b1=\u03a6\u03b1+\u03a6\u03b1,\u2200u\u2208U,m,n\u2208M and \u03b1\u2208\u0393.\u03a6(iv)\u03b1+\u03b2=\u03a6\u03b1+\u03a6\u03b2,\u2200u\u2208U,m\u2208M and \u03b1,\u03b2\u2208\u0393.\u03a6The proofs are obvious by using the Definition 3.f is a generalized \u2010derivation of M and d is a \u2010derivation of M, then we define \u03a8\u03b1=f(u\u03b1m)\u2212f(u)\u03b1m\u2212u\u03b1d(m),\u2200u\u2208U,m\u2208M and \u03b1\u2208\u0393.\u2010derivation of M, then(i)\u03a8\u03b1=\u2212\u03a8\u03b1,\u2200u\u2208U,m\u2208Mand \u03b1\u2208\u0393.(ii)\u03a8\u03b1=\u03a8\u03b1+\u03a8\u03b1,\u2200u,v\u2208U,m\u2208Mand \u03b1\u2208\u0393.(iii)\u03a8\u03b1=\u03a8\u03b1+\u03a8\u03b1,\u2200u\u2208U,m,n\u2208M and \u03b1\u2208\u0393.(iv)\u03a8\u03b1+\u03b2=\u03a8\u03b1+\u03a8\u03b2,\u2200u\u2208U,m\u2208M and \u03b1,\u03b2\u2208\u0393.The proofs are obvious by using the Definition 4.), Lemma 2.11) Let U be a Lie ideal of a 2\u2010torsion free prime \u0393\u2010ring M satisfying the condition (*) and U is not contained in Z(M). If a,b\u2208M (resp. b\u2208U and a\u2208M) such that a \u03b1U\u03b2b=0,\u2200\u03b1,\u03b2\u2208\u0393, then a=0 or b=0., Theorem 2.13) Let M be a 2\u2010torsion free prime \u0393\u2010ring satisfying the condition (*), U be an admissible Lie ideal of M and f be a generalized \u2010derivation of M, then \u03a8\u03b1=0,\u2200u,v\u2208U and \u03b1\u2208\u0393., Theorem 2.14) Let M be a 2\u2010torsion free prime \u0393\u2010ring satisfying the condition (*), U a square closed Lie ideal of M and f be a generalized \u2010derivation of M, then f(u\u03b1m)=f(u)\u03b1m+u\u03b1d(m),\u2200u\u2208Um\u2208M and \u03b1\u2208\u0393.\u2010derivations in \u0393\u2010rings.In this section, we introduce generalized higher \u2010derivation of M if there exists an higher \u2010derivation D=(di)i\u2208N of M such that for each \u03b1,\u03b2\u2208\u0393.Let N and U1 are as in Example 1. If fn :R\u2192R be a generalized higher \u2010derivation. Then there exists a higher  derivation dn :R\u2192R such that Let Dn :N\u2192N by Dn)=(dn(x),dn(x)). Then Dn is a higher \u2010derivation on N.If we define a mapping Fn :N\u2192N be the additive mapping defined by Fn)=(fn(x),fn(x)). Then by the similar calculation as in Example 1, we can show that, Fn is a generalized higher \u2010derivation on N.Let Let M be a 2\u2010torsion free \u0393\u2010ring satisfying the condition (*), U be a Lie ideal of M and F=(fi)i\u2208Nbe a generalized higher \u2010derivation of M. Thenand \u03b1,\u03b2\u2208\u0393.x=u\u03b1((2u)\u03b2m+m\u03b2(2u))+((2u)\u03b2m+m\u03b2(2u))\u03b1u.Let m and s by (2u)\u03b2m+m\u03b2(2u) and (2u)\u03b1m+m\u03b1(2u) respectively in Replacing Thus we haveU,M)\u2010 derivation and using the condition (*) On the other hand by the definition of higher  and (2) we get Let M be a 2\u2010torsion free \u0393\u2010ring satisfying the condition (*), U be a Lie ideal of M and F=(fi)i\u2208Nbe a generalized higher \u2010derivation of M. Thenand \u03b1,\u03b2\u2208\u0393.u gives us Linearizing of On the other hand \u03b1,\u03b2\u2208\u0393.Now comparing above two expressions, we get M be a 2\u2010torsion free \u0393\u2010ring satisfying the condition (*) and U be a Lie ideal of M. Let F=(fi)i\u2208N be a generalized higher \u2010derivation of M. For every fixed n\u2208N, we define D=(di)i\u2208N be a higher \u2010derivation of M. For every fixed n\u2208N, we define Let n\u2208N if and only if n\u2208N. Also n\u2208N if and only if n\u2208N.Let M be a 2\u2010torsion free \u0393\u2010ring satisfying the condition (*) and U be a Lie ideal of M. For every u\u2208U,m\u2208M,\u03b1\u2208\u0393 and n\u2208N, thenandU,M)\u2010derivation of M and generalized higher \u2010derivation of M.The proofs are obvious by the Definition 6, higher (Let M be a 2\u2010torsion free prime \u0393\u2010ring satisfying the condition (*), U be an admissible Lie ideal of M and F=(fi)i\u2208Nbe a generalized higher \u2010derivation of M. Thenand n\u2208N.We have n\u2208N, that m\u2208N and m<n.Now we assume, by induction on x=4(u\u03b1v\u03b2w\u03b3v\u03b1u+v\u03b1u\u03b2w\u03b3u\u03b1v).Let Then by using Lemma 6, we have D=(di)i\u2208N is a higher \u2010derivation of M. On the other hand, by Lemma 5 and fn(x) and using Now comparing the two expressions of M we get Using Lemma 7 and 2\u2010torsion freeness of D=(di)i\u2208N is a higher \u2010derivation of M, thus we have U is noncentral, thus we get n\u2208N.Since Now we prove the main result.Let M be a 2\u2010torsion free prime \u0393\u2010ring satisfying the condition (*), U be an admissible Lie ideal of M and F=(fi)i\u2208Nbe a generalized higher \u2010derivation of M. Thenand n\u2208N.We have n\u2208N, that m<n.Now we assume, by induction on F=(fi)i\u2208N is a generalized higher \u2010derivation of M, we have Now since D=(di)i\u2208N is a higher \u2010derivation of M, thus we haveSince F=(fi)i\u2208N is a generalized higher \u2010derivation of M, thus we have Since Since On the other hand, by using Equation  and LemmBy comparing (5) and (6) and using the condition (*), we getu, gives usLinearizing of (7) with respect to v by v\u03b1v in (8) and since Replacing n\u2208N.Hence by Lemma 4 and since n\u2208N.Thus by the Remark 2, we have"}
{"text": "The amount of functional genomic information has been growing rapidly but remains largely unused in genomic selection. Genomic prediction and estimation using haplotypes in genome regions with functional elements such as all genes of the genome can be an approach to integrate functional and structural genomic information for genomic selection. Towards this goal, this article develops a new haplotype approach for genomic prediction and estimation.A multi-allelic haplotype model treating each haplotype as an \u2018allele\u2019 was developed for genomic prediction and estimation based on the partition of a multi-allelic genotypic value into additive and dominance values. Each additive value is expressed as a function of h\u2009\u2212\u20091 additive effects, where h\u2009=\u2009number of alleles or haplotypes, and each dominance value is expressed as a function of h(h\u2009\u2212\u20091)/2 dominance effects. For a sample of q individuals, the limit number of effects is 2q\u2009\u2212\u20091 for additive effects and is the number of heterozygous genotypes for dominance effects. Additive values are factorized as a product between the additive model matrix and the h\u2009\u2212\u20091 additive effects, and dominance values are factorized as a product between the dominance model matrix and the h(h\u2009\u2212\u20091)/2 dominance effects. Genomic additive relationship matrix is defined as a function of the haplotype model matrix for additive effects, and genomic dominance relationship matrix is defined as a function of the haplotype model matrix for dominance effects. Based on these results, a mixed model implementation for genomic prediction and variance component estimation that jointly use haplotypes and single markers is established, including two computing strategies for genomic prediction and variance component estimation with identical results.The multi-allelic genetic partition fills a theoretical gap in genetic partition by providing general formulations for partitioning multi-allelic genotypic values and provides a haplotype method based on the quantitative genetics model towards the utilization of functional and structural genomic information for genomic prediction and estimation. With the factorization of a\u2009=\u2009W\u03b1\u03b1 and d\u2009=\u2009W\u03b4\u03b4, genomic additive relationship is a function of W\u03b1W\u03b1\u2009' and genomic dominance relationship is a function of W\u03b4W\u03b4\u2009' \u2009\u00d7\u20091 column vector of 1\u2019s, ah\u2009=\u2009W\u03b1h\u03b1h\u2009=\u2009[h(h\u2009+\u20091)/2]\u2009\u00d7\u20091 column vector of additive values , dh\u2009=\u2009W\u03b4h\u03b4h\u2009=\u2009[h(h\u2009+\u20091)/2]\u2009\u00d7\u20091 column vector of dominance values (dominance deviations), W\u03b1h\u2009=\u2009[h(h\u2009+\u20091)/2]\u2009\u00d7\u2009(h\u2009\u2212\u20091) model matrix of \u03b1hwith w\u03b1ij,k defined by Eqs. h\u2009=\u2009[h(h\u2009+\u20091)/2]\u2009\u00d7\u20091 column vector of dominance values (dominance deviations), W\u03b4h\u2009=\u2009[h(h\u2009+\u20091)/2]\u2009\u00d7\u2009[h(h\u2009\u2212\u20091)/2] matrix of \u03b4h with w\u03b4ij,kf defined by Eqs. \u03b1h\u2009=\u2009(h\u2009\u2212\u20091)\u2009\u00d7\u20091 column vector with \u03b1lk defined by Eq. \u03b4h\u2009=\u2009[h(h\u2009\u2212\u20091)/2]\u2009\u00d7\u20091 column vector with \u03b4kf defined by Eq. For convenience of computer programming, Eqs. lele Eq. , or 1 colele Eq.  or 1 comlele Eq. . SimilarA hypothetical numerical example is used to illustrate the genetic partition of multi-allelic genotypic values described by Eqs. Using Eqs. Using Eqs. The genotypic values calculated as the summation of the additive and dominance values are:g\u2009=\u20091\u03bc\u2009+\u2009ah\u2009+\u2009dh\u2009=\u20091\u03bc\u2009+\u2009W\u03b1h\u03b1h\u2009+\u2009W\u03b4h\u03b4h described by Eqs. ij\u2009=\u2009gji,\u2009aij\u2009=\u2009aji and dij\u2009=\u2009dji, the genotypic variance (\u03c3g2), additive variance (\u03c3a2) and dominance variance (\u03c3d2) are:By comparing with the genotypic values in Table\u00a0g2\u2009=\u2009\u03c3a2\u2009+\u2009\u03c3d2.It is readily seen that \u03c3W\u03b1h,\u2002W\u03b4h,\u2002ah,\u2002dh,\u2002\u03b1h and \u03b4h in Eq. th haplotype block has hi haplotypes, hi\u22121 additive effects, and n\u03b4i dominance effects or heterozygous genotypes. Let n\u03b1\u2009=\u2009total number of additive effects of all r haplotype blocks, n\u03b4\u2009=\u2009total number of dominance effects (or heterozygous genotypes) of all r haplotype blocks. Then, n\u03b1\u2009=\u2009\u2211i\u2009=\u20091rhi\u2009\u2212\u2009r, and n\u03b4\u2009=\u2009\u2211i\u2009=\u20091rn\u03b4i. For a given sample of q individuals, the limit number of effects is 2q-1 for additive effects and is the number of heterozygous genotypes for dominance effects. For a sample with N observations on q individuals, the mixed model to implement the multi-allelic haplotype model of Eq. Z\u2009=\u2009N\u2009\u00d7\u2009q incidence matrix allocating phenotypic observations to each individual\u2009=\u2009identity matrix for one observation per individual (N\u2009=\u2009q),\u2002\u03b1h\u2009=\u2009n\u03b1\u2009\u00d7\u20091 column vector of haplotype additive effects, W\u03b1h\u2009=\u2009q\u2009\u00d7\u2009n\u03b1 model matrix of \u03b1h,\u2002\u03b4h\u2009=\u2009n\u03b4\u2009\u00d7\u20091 column vector for dominance effects of haplotype genotypes, W\u03b4h\u2009=\u2009q\u2009\u00d7\u2009n\u03b4 model matrix of \u03b4h,\u2002\u03b1s\u2009=\u2009m\u00a0\u00d7\u00a01 column vector of single-SNP additive effects, b\u2009=\u2009c\u2009\u00d7\u20091 column vector of fixed effects such as heard-year-season in dairy cattle (c\u2009=\u2009number of fixed effects), and X\u2009=\u2009N\u2009\u00d7\u2009c model matrix of b. To define two equivalent models with complementary computing advantages and identical GBLUP and GREML results, the mixed model of Eq. ah\u2009=\u2009T\u03b1h\u03b1h = multi-allelic genomic breeding values, dh\u2009=\u2009T\u03b4h\u03b4h = multi-allelic genomic dominance values, and each T matrix can be defined by any of the six definitions of genomic relationships we previously discussed and implemented [T matrices are defined as: T\u03b1h\u2009=\u2009W\u03b1h/k\u03b1h1/2,\u2002T\u03b4h\u2009=\u2009W\u03b4h/k\u03b4h1/2, where k\u03b1h\u2009=\u2009the average of diagonal elements of W\u03b1hW\u03b1h\u2009', and k\u03b4h = the average of diagonal elements of W\u03b4hW\u03b4h\u2009'. The genomic relationship matrices of Eq. A mixed model to implement the multi-allelic haplotype model of Eq. The multi-allelic genomic relationships of Eqs. It is important to distinguish between single-locus multi-allelic markers such as microsatellite markers from haplotypes where each haplotype is treated as an \u2018allele\u2019 and each haplotype block is treated as a \u2018locus\u2019, because recombination between loci within a haplotype block generally exists, leading to lowered haplotype similarity than single-locus similarity among relatives. As the number of loci increases in each haplotype block, genomic relationships using haplotypes are expected to decrease from those using single-locus markers. Therefore, the utility of haplotype genomic relationships using Eqs. y)\u2009=\u2009Xb, E(\u03b1h)\u2009=\u2009E(\u03b4h)\u2009=\u2009E(\u03b1s)\u2009=\u2009E(\u03b4s)\u2009=\u20090, Var(\u03b1h)\u2009=\u2009\u03c3\u03b1h2In\u03b1, Var(ah)\u2009=\u2009G\u03b1h\u2009=\u2009\u03c3\u03b1h2Ah, Var(\u03b4h)\u2009=\u2009\u03c3\u03b4h2In\u03b4, Var(dh)\u2009=\u2009G\u03b4h\u2009=\u2009\u03c3\u03b4h2Dh, and Var(e)\u2009=\u2009R\u2009=\u2009\u03c3e2IN, where \u03c3\u03b1h2 = variance of multi-allelic additive effects, \u03c3\u03b4h2 = variance of multi-allelic dominance effects, \u03c3e2 = residual variance, and In\u03b1, In\u03b4, Im and IN are identity matrices of orders n\u03b1, n\u03b4, m and N, respectively. All random effects are assumed to be uncorrelated so that the phenotypic variance-covariance matrix is:To establish mixed models using multi-allelic markers or haplotypes, assumptions for the first and second moments of the mixed model of Eq. \u03b1h\u2009=\u2009\u03c41, \u03b4h\u2009=\u2009\u03c42; T\u03b1h\u2009=\u2009T1, T\u03b4h\u2009=\u2009T2; ui\u2009=\u2009Ti\u03c4i, i\u2009=\u20091,2; Ah\u2009=\u2009S1,\u2002Dh\u2009=\u2009S2; and \u03c3\u03b1h2\u2009=\u2009\u03c312,\u2002\u03c3\u03b4h2\u2009=\u00a0\u03c322. Then, Eqs. To simply notations for the two equivalent mixed models, terms in Eqs. Zi\u2009=\u2009ZTi, an equivalent model of Eqs. By defining y) and V, but these two models have different computational advantages that can be complementary to each other. For each model, two methods can be established for genomic prediction and estimation: the method of conditional expectation (CE) and the method of mixed model equations (MME), yielding a total of four methods for the two equivalent models. Model-I using CE is the best method for large numbers of SNP markers and multiple genetic factors, Model-II using MME is the best method for large numbers of individuals, and Model-I using MME and Model-II using CE have no computing advantage. Therefore, Model-I using CE and Model-II using MME will be used for genomic prediction and estimation. Using our previous naming of these two methods, GBLUP and GREML of Model-I using CE will be referred to as the CE set of formulations, and GBLUP and GREML of Model-II using MME as the QM set of formulation, where QM means \u2018q\u2009>\u2009m\u2019. These two methods yield identical results of prediction and estimation and are applicable to singular genomic relationship matrices. Assuming one observation per individual, CE based on Eqs. \u03b1\u2009+\u2002n\u03b4 according to the size of the largest matrix to invert for each method \u2212X\u2009'\u2009V\u2212\u20091, and th type of genetic values on the phenotypic values in the training population. Two equivalent methods with identical results can be used to predict genetic values of individuals without phenotypic observations : placing all individuals with or without records in the same mixed model by setting to zero the Z matrix for the validation population, or calculate predictions separately based on the regressed phenotypic values of the training population [th type of genetic values for individuals in the validation population is calculated as:Si01\u2009=\u2009q0\u2009\u00d7\u2009q genomic relationship matrix between the training and validation populations for the ith type of genetic values .Using the CE method of Model-I Eqs.  and 37, pulation , 39. UsiZg\u2009=\u2009, \u03bbi\u2009=\u2009\u03c3e2/\u03c3i2, t\u2009=\u2009n\u03b1, n\u03b4, m and N for i\u2009=\u20091,2, respectively, and \u2295 denotes direct sum that defines a block diagonal matrix. With haplotype and SNP effects from Eq. th type of genetic values for individuals in the training and validation populations are obtained as:Ti0\u2009=\u2009the Ti matrix calculated using SNPs of the validation population. Equations \u011d\u2009=\u2009\u2211i\u2009=\u200912\u00fbi = predicted genotypic values of all individuals, and \u011d0\u2009=\u2009\u2211i\u2009=\u200912\u00fbi0 = predicted genotypic values of the validation population. Prediction reliabilities of additive, dominance and genotypic predictions as the squared correlations between the genomic and true values has the same formulations as the Rai2, Rdi2 and Rgi2 formulae in [Using the QM method \u2212X\u2009', and ti\u2009=\u2009n\u03b1 for i\u2009=\u20091 and ti\u2009=\u2009n\u03b4 for i\u2009=\u20092.Using the CE method of Model-I Eqs.  and 37, hod Eqs.  and 39, The EM-REML of Eqs. \u03b1s\u2009=\u2009m\u2009\u00d7\u20091 column vector of SNP additive effects, T\u03b1s\u2009=\u2009q\u2009\u00d7\u2009m model matrix of \u03b1s, \u03b4s\u2009=\u2009m\u2009\u00d7\u20091 column vector of SNP dominance effects, T\u03b4s\u2009=\u2009q\u2009\u00d7\u2009m model matrix of \u03b4s, Var(\u03b1s)\u2009=\u2009\u03c3\u03b1s2Im, Var(as)\u2009=\u2009G\u03b1s\u2009=\u2009\u03c3\u03b1s2As, Var(\u03b4s)\u2009=\u2009\u03c3\u03b4s2Im, Var(ds)\u2009=\u2009G\u03b4s\u2009=\u2009\u03c3\u03b4s2Ds, As\u2009=\u2009genomic additive relationship matrix, and Ds\u2009=\u2009SNP genomic dominance relationship matrix, and where As\u2009=\u2009T\u03b1sT\u03b1s\u2009' and Ds\u2009=\u2009T\u03b4sT\u03b4s\u2009'. Let \u03b1s\u2009=\u2009\u03c43, \u03b4\u2009=\u2009\u03c44; ui\u2009=\u2009Ti\u03c4i, i\u2009=\u20091,\u2026,4; As\u2009=\u2009S3, Dh\u2009=\u2009S4; and \u03c3\u03b1s2\u2009=\u2009\u03c332, \u03c3\u03b4s2\u2009=\u2009\u03c342. The GBLUP and GREML formulations to jointly include haplotype and single SNP additive and dominance effects essentially entails to extending the range of the subscript i from 2 to 4 for Eqs. Haplotype analysis and single SNP analysis can be analyzed jointly for genomic prediction in the same mixed model by adding single SNP effects from our previous work  to the mi2\u2009=\u2009\u03c3i2/\u03c3y2, where \u03c3y2\u2009=\u2009\u2211i\u2009=\u200914\u03c3i2\u2009+\u2009\u03c3e2 = phenotypic variance. The total heritability of all types of genetic effects is the summation of all effect heritabilities, i.e., H2\u2009=\u2009\u2211i\u2009=\u200914hi2. Genomic heritability estimation has flexibility unavailable from heritability estimation using pedigree relationships: the heritability estimation for a single SNP, a chromosome region, or a set of selected SNPs. Using the GREML formulae of Eqs. GREML estimation using the joint mixed model with haplotype and SNP effects offer flexibility to estimate the heritability for various types of functional genomic information in any given autosome regions based on formulations we implemented in GVCBLUP , e.g., tThe multi-allelic haplotype model can be used for the integration of functional genomic information with genomic prediction and estimation. This integration defines haplotype blocks using functional genomic information under the hypothesis that a chromosome region with functional information required more than a single point to affect a phenotype, followed by genomic prediction and estimation using a haplotype analysis such as the methods developed in this article. Each gene could be a \u2018natural haplotype block\u2019 and the use of gene blocks improved the prediction accuracy for some human phenotypes in our preliminary results . Other tThe mixed model approach outlined above allows rare haplotypes. In the extreme case of rare haplotypes with one observation per haplotype or haplotype frequency of 1/h when h is large, the multi-allelic model with the mixed model implementation still is applicable for additive effects and values. Missing genotypic values is a problem for dominance effects and values. The dominance effect defined by Eq. A multi-allelic haplotype model for genomic prediction and estimation is established using the quantitative genetics model that partitions a multi-allelic genotypic value into additive and dominance values, factorizes each additive value into a product between a function of allele frequencies and additive effect, and factorizes each dominance value into a product between a function of allele frequencies and dominance effect. Haplotype genomic additive and dominance relationship matrices and formulations are then derived for GBLUP and GREML utilizing haplotypes in haplotype blocks. These results fill a gap in the theory of quantitative genetics for multi-allelic genetic partition and provide a haplotype approach within the theory of quantitative genetics towards the integration of functional and structural genomic information for genomic selection.The only data set used in this article is shown in Tables"}
{"text": "The concept of \u0393-semihyperrings was introduced by Dehkordi and Davvaz as a generalization of semirings, semihyperrings, and \u0393-semiring. In this paper, by using the notion of triangular norms, we define the concept of triangular fuzzy sub-\u0393-semihyperrings as well as triangular fuzzy \u0393-hyperideals of a \u0393-semihyperring, and we study a few results in this respect. M be an additive group whose elements are denoted by a, b, c,\u2026 and \u0393 another additive group whose elements are \u03b3, \u03b2, \u03b1,\u2026. Suppose that a\u03b3b is defined to be an element of M and that \u03b3a\u03b2 is defined to be an element of \u0393 for every a, b, \u03b3, and \u03b2. If the products satisfy the following three conditions: (1)(a1 + a2)\u03b3b = a1\u03b3b + a2\u03b3b, a(\u03b31 + \u03b32)b = a\u03b31b + a\u03b32b, a\u03b3(b1 + b2) = a\u03b3b1 + a\u03b3b2; (2)(a\u03b3b)\u03b2c = a\u03b3(b\u03b2c) = a(\u03b3b\u03b2)c; (3) if a\u03b3b = 0 for any a and b in M, then \u03b3 = 0; then M is called a \u0393-ring in the sense of Nobusawa  which associates with each point x \u2208 X its grade or degree of membership \u03bc\u03bc(x) \u2208 . Fuzzy sets generalize classical sets since the characteristic functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. After the introduction of fuzzy sets by Zadeh, reconsideration of the concept of classical mathematics began. In 1971, Rosenfeld  \u00d7  \u2192  satisfying the following conditions: (T1) T = T; (T2) T) = T, z); (T3) T \u2264 T if y \u2264 z; (T4) T = x, for all x, y, z \u2208 . For every t-norm T, we set \u0394T = {x \u2208 , T = x}. A t-norm on  is called a continuous t-norm if T is a continuous function from  \u00d7  to  with respect to the usual topology. Note that the function \u201cMin\u201d is a continuous t-norm. A triangular conorm (t-norm for short) is a binary operation S on the unit interval , that is, a function S : \u00d7\u2192, which for all x, y, z \u2208  satisfies (T1)\u2013(T3) and (S4) S = x. From an axiomatic point of view, t-norms and t-conorms differ only with respect to their respective boundary conditions. In fact, the concepts of t-norms and t-conorms are dual in some sense. Anthony and Sherwood . Then, for any x, y \u2208 R and \u03b3 \u2208 \u0393, we havex, y \u2208 \u0393,x \u2209 \u0393 and y \u2208 \u0393\u2009,x, y \u2209 \u0393. Regarding the above cases, we have T = 9/17, T = 15/40, and T = 25/81. Thus, in every case, we obtain\u03bc is a T-fuzzy sub-\u0393-semihyperring of R.Suppose that R be a \u0393-semihyperring, T be a t-norm, and \u03bc be a T-fuzzy sub-\u0393-semihyperring of S. Thenx1,\u2026, xn \u2208 R and \u03b3 \u2208 \u0393, whereLet The proof is straightforward by mathematical induction.R be a \u0393-semihyperring, T be a t-norm, and \u03bc be a T-fuzzy sub-\u0393-semihyperring of S. Let A and B be nonempty subsets of R. Then\u03b3 \u2208 \u0393.Let The proof is straightforward.R be a \u0393-semihyperring, T be a t-norm, and \u03bc be a fuzzy subset of S with imaginable property and b the maximum of Im\u2009\u03bc. Then, the following two statements are equivalent:\u03bc is a T-fuzzy sub-\u0393-semihyperring of S,\u03bc\u22121 is a sub-\u0393-semihyperring of S whenever a \u2208 \u0394T and 0 < a \u2264 b.Let a \u2208 \u0394T and 0 < a \u2264 b. If x, y \u2208 \u03bc\u22121, then inf\u2061z\u2208x+y\u2061{\u03bc(z)} \u2265 T(\u03bc(x), \u03bc(y)) \u2265 T = a, which implies that x + y\u2286\u03bc\u22121. Similarly, assume that a \u2208 \u0394T and 0 < a \u2264 b. If x, y \u2208 \u03bc\u22121 and \u03b3 \u2208 \u0393, then inf\u2061z\u2208x\u03b3y\u2061{\u03bc(z)} \u2265 T(\u03bc(x), \u03bc(y)) \u2265 T = a. Then, we have x\u03b3y\u2286\u03bc\u22121, and so \u03bc\u22121 is a sub-\u0393-semihyperring of R.(1)\u21d2(2): Suppose that x, y \u2208 S and \u03b3 \u2208 \u0393. Since Im\u2009\u03bc\u2286\u0394T, both \u03bc(x) and \u03bc(y) are in \u0394T. Now, we haveT(\u03bc(x), \u03bc(y)) \u2208 \u0394T. Assume that a = T(\u03bc(x), \u03bc(y)). If a = 0, thena = T(\u03bc(x), \u03bc(y)) \u2264 \u03bc(x)\u2227\u03bc(y) \u2264 \u03bc(x) \u2264 b. Hence x, y \u2208 \u03bc\u22121, which implies that x + y\u2286\u03bc\u22121 and x\u03b3y\u2286\u03bc\u22121. Therefore T(\u03bc(x), \u03bc(y)) \u2264 inf\u2061z\u2208x+y\u2061{\u03bc(z)} and T(\u03bc(x), \u03bc(y)) \u2264 inf\u2061z\u2208x\u03b3y\u2061{\u03bc(z)}.(2)\u21d2(1): Suppose that R be a \u0393-semihyperring, T be a t-norm, and \u03bc be a fuzzy subset of R. Then(1)\u03bc is called a T-fuzzy left \u0393-hyperideal of R if(2)\u03bc is called a T-fuzzy right \u0393-hyperideal of R if(3)\u03bc is called a T-fuzzy \u0393-hyperideal of R if it is both a T-fuzzy left \u0393-hyperideal and a T-fuzzy right \u0393-hyperideal of R.Let R be a \u0393-semihyperring, T be a t-norm, and \u03bc be a fuzzy subset of S with imaginable property and b the maximum of Im\u2009\u03bc. Then, the following two statements are equivalent:\u03bc is a T-fuzzy \u0393-hyperideal of R,\u03bc\u22121 is a \u0393-hyperideal of R whenever a \u2208 \u0394T and 0 < a \u2264 b.Let The proof is similar to the proof of \u03bc be a fuzzy subset of R and t \u2208 . The set U = {x \u2208 R | \u03bc(x) \u2265 t} is called a level subset of \u03bc. So, we obtain the following corollary.Let R be a \u0393-semihyperring and \u03bc be a fuzzy subset of R. Then\u03bc is a Min-fuzzy sub-\u0393-semihyperring of R if and only if every nonempty level subset is a sub-\u0393-semihyperring of R;\u03bc is a Min-fuzzy \u0393-hyperideal of R if and only if every nonempty level subset is a \u0393-hyperideal of R.Let A be a subset of R. Then\u03c7A is a T-fuzzy sub-\u0393-semihyperring of R if and only if A is a sub-\u0393-semihyperring of R;the characteristic function \u03c7A is a T-fuzzy \u0393-hyperideal of R if and only if A is a \u0393-hyperideal of R.the characteristic function Let R be a \u0393-semihyperring and K be a sub-\u0393-semihyperring of R. Let T be the t-norm defined by T = max\u2061\u2061{0, a + b \u2212 1} and \u03bc be a fuzzy subset of R defined bya, b \u2208  and x \u2208 R, where r, s \u2208  such that s < r. Then, \u03bc is a T-fuzzy sub-\u0393-semihyperring of R. In particular, if r = 1 and s = 0, then \u03bc is imaginable.Let The proof is similar to the proof of Theorem\u2009\u20092.6 in .R1 and R2 be \u03931 and \u03932-semihyperrings, respectively. If there exists a map \u03c6 : R1 \u2192 R2 and a bijection f : \u03931 \u2192 \u03932 such thatx, y \u2208 R1 and \u03b3 \u2208 \u0393, then we say  is a homomorphism from R1 to R2. Also, if \u03c6 is a bijection then  is called an isomorphism and R1 and R2 are isomorphic.Let R1 and R2 be \u03931 and \u03932-semihyperrings, respectively. Let  be a homomorphism from R1 to R2. If \u03bb is a T-fuzzy sub-\u0393-semihyperring of R2, then \u03c6\u22121(\u03bb) is a T-fuzzy sub-\u0393-semihyperring of R1 too.Let x, y \u2208 R1 and \u03b3 \u2208 \u0393. Then, we have\u03c6\u22121(\u03bb) is a T-fuzzy sub-\u0393-semihyperring of R1.Suppose that R1 and R2 be \u03931 and \u03932-semihyperrings, respectively. Let  be a homomorphism from R1 to R2. If \u03bb is a T-fuzzy \u0393-hyperideal of R2, then \u03c6\u22121(\u03bb) is a T-fuzzy \u0393-hyperideal of R1 too.Let The proof is similar to the proof of ai}i\u2208I and {bj}j\u2208J be two sets of real numbers in . Then, we say T is infinitely distributive ifT is continuous, then T is infinitely distributive , we have \u03c6(\u03bc)t = \u22c2t>\u03f5>0\u03c6(\u03bct\u2212\u03f5).Let The proof is similar to the proof of Lemma\u2009\u20093.5 in .R1 and R2 be \u03931 and \u03932-semihyperrings, respectively, and let \u03bc be a fuzzy subset of R1. Let  be an onto homomorphism from R1 to R2. If \u03bc is a Min-fuzzy sub-\u0393-semihyperring of R1, then \u03c6(\u03bc) is a Min-fuzzy sub-\u0393-semihyperring of R2 too.Let \u03bc is a Max-fuzzy sub-\u0393-semihyperring of R1. By \u03c6(\u03bc) is a Max-fuzzy sub-\u0393-semihyperring of R2 if every nonempty level subset U(\u03c6((\u03bc); t)) is a sub-\u0393-semihyperring of R2. Thus, assume that U(\u03c6((\u03bc); t)) is any nonempty level subset. If t = 0, then U(\u03c6((\u03bc); t)) = R2, and if 0 < t \u2264 1, then by U(\u03c6((\u03bc); t)) = \u22c2t>\u03f5>0\u03c6). By U for each t > \u03f5 > 0 is a sub-\u0393-semihyperring of R1. Hence, \u03c6) is a sub-\u0393-semihyperring of R2. By \u03c6) being an intersection of a family of sub-\u0393-semihyperrings is also a sub-\u0393-semihyperring of R2 and the proof is completed.Suppose that R1 and R2 be two \u0393-semihyperrings and let \u03bc and \u03bb be fuzzy subsets of R1 and R2, respectively. The product of \u03bc and \u03bb is defined to be the fuzzy subset \u03bc \u00d7 \u03bb of R1 \u00d7 R2 with (\u03bc \u00d7 \u03bb) = T(\u03bc(x), \u03bb(x)), for all  \u2208 R1 \u00d7 R2.Let R1 and R2 be two \u0393-semihyperrings and let \u03bc and \u03bb be fuzzy subsets of R1 and R2, respectively. Thenif \u03bc and \u03bb are T-fuzzy sub-\u0393-semihyperrings of R1 and R2, respectively, then \u03bc \u00d7 \u03bb is a T-fuzzy sub-\u0393-semihyperring of R1 \u00d7 R2; if \u03bc and \u03bb are T-fuzzy \u0393-hyperideals of R1 and R2, respectively, then \u03bc \u00d7 \u03bb is a T-fuzzy \u0393-hyperideal of R1 \u00d7 R2. Let It is straightforward.A of subsets of a \u0393-semihyperring R satisfies the ascending chain condition (or Acc) if there does not exist a properly ascending infinite chain A1 \u2282 A2 \u2282 \u22ef of subsets from A. Recall that a subset B \u2208 A is a maximal element of A if there does not exist a subset in A that properly contains B. Similar to  for every n \u2208 Z, R = \u22c3n\u2208ZAn, and \u0393 = Z. Then, R is a Noetherian \u0393-semihyperring with respect to the following hyperoperations:x \u2208 An and y \u2208 Am.Let Ak | k \u2208 N} be a family of \u0393-hyperideals of a \u0393-semihyperring R, where A1\u2283A2\u2283A3\u22ef. Let \u03bc be a fuzzy subset of R defined byx \u2208 R, where A0 stands for R. Let T be a t-norm with Im\u2009\u03bc\u2286\u0394T. Then, \u03bc is a T-fuzzy \u0393-hyperideal of R.Let {x, y \u2208 R. Suppose that x \u2208 Ak\u2216Ak+1 and y \u2208 Ar\u2216Ar+1 for k = 0,1, 2,\u2026 and r = 0,1, 2,\u2026. Without loss of generality we may assume that k \u2264 r. Then, obviously y \u2208 Ak. Since Ak is a \u0393-hyperideal of R, it follows that x + y\u2286Ak and x\u03b1y\u2286Ak which imply that inf\u2061z\u2208x+y\u03bc(z) \u2265 k/(k + 1) = T(\u03bc(x), \u03bc(y)) and inf\u2061z\u2208x\u03b1y\u03bc(z) \u2265 k/(k + 1) = \u03bc(y) for all \u03b1 \u2208 \u0393. If x \u2208 \u22c2k=0\u221eAk and y \u2208 \u22c2k=0\u221eAk, then x + y\u2286\u22c2k=0\u221eAk. Hence, inf\u2061z\u2208x+y\u03bc(z) = 1 = T(\u03bc(x), \u03bc(y)) and inf\u2061z\u2208x\u03b1y\u03bc(z) = 1 \u2265 \u03bc(y). If x \u2209 \u22c2k=0\u221eAk and y \u2208 \u22c2k=0\u221eAk, then there exists n \u2208 N such that x \u2208 An\u2216An+1. It follows that x + y\u2286An which implies that inf\u2061z\u2208x+y\u03bc(z) \u2265 n/(n + 1) = T(\u03bc(x), \u03bc(y)) and inf\u2061z\u2208x\u03b1y\u03bc(z) \u2265 n/(n + 1) = \u03bc(y) for all \u03b1 \u2208 \u0393. Finally, assume that x \u2208 \u22c2k=0\u221eAk and y \u2209 \u22c2k=0\u221eAk, then y \u2208 An\u2216An+1 for some n \u2208 N. Therefore, x + y\u2286An, and thus inf\u2061z\u2208x+y\u03bc(z) \u2265 n/(n + 1) = T(\u03bc(x), \u03bc(y)) and inf\u2061z\u2208x\u03b1y\u03bc(z) \u2265 n/(n + 1) = \u03bc(y), for all \u03b1 \u2208 \u0393. Hence, \u03bc is a T-fuzzy \u0393-hyperideal of R.Let R be a \u0393-semiring satisfying descending chain condition, let \u03bc be a fuzzy subset of R, and let T be a t-norm with Im\u2009\u03bc\u2286\u0394T. Let \u03bc be a T-fuzzy \u0393-hyperideal of R. If a sequence of elements of Im\u2009\u03bc is strictly increasing, then \u03bc has finite number of values.Let tk} be a strictly increasing sequence of elements of Im\u2061\u03bc. Then 0 \u2264 t1 \u2264 t2 \u2264 \u22ef\u22641. Then, U is an ideal of M for all r = 2,3,\u2026 Let x \u2208 U. Then \u03bc(x) \u2265 tr \u2265 tr\u22121, and so x \u2208 U. Hence U\u2286U. Since tr\u22121 \u2208 Im\u2061\u03bc, there exists xr\u22121 \u2208 M such that \u03bc(xr\u22121) = tr\u22121. It follows that xr\u22121 \u2208 U but xr\u22121 \u2209 U. Thus, U \u2282 U and so we obtain a strictly descending sequence U\u2283U\u2283U\u2283\u22ef of \u0393-hyperideals of R which is not terminating. This contradicts the assumption that M satisfies the descending chain condition. Consequently, \u03bc has finite number of values.Let {R be a \u0393-semiring, \u03bc be a fuzzy subset of R, and T be a t-norm with Im\u2009\u03bc\u2286\u0394T. Then, the following conditions are equivalent: R is a Noetherian \u0393-semihyperring,R is a well-ordered subset of .the set of values of any \u0393-hyperideal of Let \u03bc be a T-fuzzy \u0393-hyperideal of R. Suppose that the set of values of \u03bc is not a well-ordered subset of . Then, there exists a strictly decreasing sequence {tk} such that \u03bc(xk) = tk. It follows that U \u2282 U \u2282 \u22ef is a strictly ascending chain of \u0393-hyperideals of M, where U = {x \u2208 M | \u03bc(x) \u2265 tr}, for every r = 1,2,\u2026. This contradicts the assumption that R is a Noetherian \u0393-semihyperring.(1)\u21d2(2): Let R is not a Noetherian \u0393-hyperring. There exists a strictly ascending chainR. Note that A = \u22c2k=0\u221eAk is a \u0393-hyperideal of R. Define a fuzzy subset in R by\u03bc is a T-fuzzy \u0393-hyperideal of R. Let x, y \u2208 R. If x \u2208 Ak\u2216Ak\u22121 and y \u2208 Ak\u2216Ak\u22121, then x \u2212 y \u2282 Ak and x\u03b1y \u2282 Ak. It follows that inf\u2061z\u2208x+y\u03bc(z) = 1/k = T(\u03bc(x), \u03bc(y)) and inf\u2061z\u2208x\u03b1y\u03bc(z) \u2265 1/k \u2265 \u03bc(y), for all \u03b2 \u2208 \u0393. Suppose that x \u2208 Ak and y \u2208 Ak\u2216Ar for all r < k. Since Ak is a \u0393-hyperideal of R, it follows that x + y \u2282 Ak. Hence, inf\u2061z\u2208x+y\u03bc(z) \u2265 1/k \u2265 1/(k + 1) \u2265 \u03bc(y) and so inf\u2061z\u2208x+y\u03bc(z) \u2265 min\u2061{\u03bc(x), \u03bc(y)} \u2265 T(\u03bc(x), \u03bc(y)). Similarly, for the case x \u2208 Ak\u2216Ar and y \u2208 Ak, we have inf\u2061z\u2208x+y\u03bc(z) \u2265 min\u2061{\u03bc(x), \u03bc(y)} \u2265 T(\u03bc(x), \u03bc(y)) and inf\u2061z\u2208x\u03b1y\u03bc(z) \u2265 \u03bc(y), for all \u03b2 \u2208 \u0393. Thus, \u03bc is a T-fuzzy \u0393-hyperideal of R. Since the chain is not terminating, \u03bc has a strictly descending sequence of values. This contradicts the assumption that the value set of any ideal is well-ordered. Therefore, R is a Noetherian \u0393-semihyperring.(2)\u21d2(1): Suppose that the condition (2) is satisfied and \u03bc\u03b1 | \u03b1 \u2208 \u039b} of fuzzy subsets in R, we define the join \u2228\u03b1\u2208\u039b\u03bc\u03b1 and the meet \u2228\u03b1\u2208\u039b\u03bc\u03b1 as follows:x \u2208 R, where \u039b is any index set.For a family {T-fuzzy \u0393-hyperideals in R is a completely distributive lattice with respect to meet \u201c\u2227\u201d and join \u201c\u2228\u201d.The family of \u03b1\u2208\u039b\u03bc\u03b1 and \u2227\u03b1\u2208\u039b\u03bc\u03b1 are T-fuzzy \u0393-hyperideals of R for family {\u03bc\u03b1 | \u03b1 \u2208 \u039b} of T-fuzzy \u0393-hyperideals of R. For any x, y \u2208 R, we haveSince  is a completely distributive lattice with respect to the usual ordering in , it is sufficient to show that \u2228x, y \u2208 M and \u03b2 \u2208 \u0393. Then\u03b1\u2208\u039b\u03bc\u03b1 and \u2227\u03b1\u2208\u039b\u03bc\u03b1 are T-fuzzy \u0393-hyperideals of R. This completes the proof.Now, let"}
{"text": "The proof of the main result is based upon a fixed point theorem of a sum operator. It is expected in this paper not only to establish existence and uniqueness of positive solution, but also to show a way to construct a series to approximate it by iteration.It is expected in this paper to investigate the existence and uniqueness of positive solution for the following difference equation: \u2212\u0394 T > 1 be an integer; \u2124a,b : = {a, a + 1,\u2026, b}, where a, b are positive integers. Difference equation appears as natural descriptions of observed evolution phenomena because most measurements of time evolving variables are discrete and so it arises in many physical problems, as nonlinear elasticity theory or mechanics, and engineering topics. In recent years, the study of positive solutions for discrete boundary value problems has attracted considerable attention, but most research dealt with two-point boundary value problem; see [Let lem; see  and the lem; see \u20137. Howev\u03b1 < 1, \u03b2 > 0, and \u03b7 \u2208 \u2124T\u221212,.In this paper, we consider the existence uniqueness and positive solutions for difference equation E = C denote the class of real valued functions \u03c9 on \u2124T+10, with norm ||\u03c9|| = max\u2061k\u2208\u2124T+10, | \u03c9(k)|. Observe that E is a Banach space. Set P = {u \u2208 E : u(t) \u2265 0, t \u2208 \u2124T+10,} to be the normal cone in E with the normality constant 1. For u, v \u2208 E, the notation u ~ v means that there exist \u03bb > 0 and \u03bc > 0 such that \u03bbv \u2264 u \u2264 \u03bcv. Clearly, ~ is an equivalence relation. Given h > \u03b8, we denote by Ph the set Ph = {x \u2208 E\u2223x ~ h}.Let \u2124T+10, with the discrete topology, every \u03c9 \u2208 C is continuous.As suggested by the notation, by equipping D = P or D = P0 and let \u03b3 be a real number with 0 \u2264 \u03b3 < 1. An operator A : P \u2192 P is said to be \u03b3-concave if it satisfies Let A : E \u2192 E is said to be homogeneous if it satisfies A : P \u2192 P is said to be subhomogeneous if it satisfies An operator  The main tool of this paper is the following fixed point theorem.P be a normal cone in a real Banach space E, A : P \u2192 P an increasing \u03b3-concave operator, and B : P \u2192 P an increasing subhomogeneous operator. Assume that\u2009(i)h > 0 such that Ah \u2208 Ph and Bh \u2208 Ph;there is \u2009(ii)\u03b40 > 0 such that, for any x \u2208 P, Ax \u2265 \u03b40Bx.there exists a constant Then the operator equation Ax + Bx = x has a unique solution x* \u2208 Ph. Moreover, constructing successively the sequence yn = Ayn\u22121 + Byn\u22121, n = 1,2,\u2026, for any initial value y0 \u2208 Ph, one has y \u2192 x* as n \u2192 \u221e.Let B is a null operator, When In this section, we will apply T \u2208 {4, 5,\u2026}, \u03b7 \u2208 \u2124T\u221212,, and \u03b1, \u03b2 \u2208 \u211d are real numbers with \u03b2 \u2260 \u22121 and (T + 1 \u2212 \u03b1\u03b7) + \u03b2(1 \u2212 \u03b1) \u2260 0, for any y defined in \u2124T+10,, the nonlocal boundary value problem If t \u2208 \u2124T+10,, s \u2208 \u2124T1,, the Green function G in \u2009(i)G > 0, t \u2208 \u2124T+10,, s \u2208 \u2124T1,;\u2009(ii)G \u2264 r(s)h(t), t \u2208 \u2124T+10,, s \u2208 \u2124T1,, where \u2009(iii)s \u2265 t, G = p(s)h(t), t \u2208 \u2124T+10,, s \u2208 \u2124T1,, where for For s < t, s < \u03b7, notice that t \u2212 \u03b7 < t \u2212 s, \u03b7 \u2264 s < t, t \u2264 s < \u03b7, t \u2264 s, \u03b7 \u2264 s, t \u2208 \u2124T+10, and s \u2208 \u2124T1,, G \u2264 r(s)h(t).Since (i) and (iii) are obvious, here we just prove (ii). For \u2009(A1)f, g : \u2124T+10, \u00d7 [0, \u221e)\u2192[0, \u221e) are continuous and increasing with respect to the second variable, g \u2260 0;\u2009(A2)g \u2265 \u03bbg for \u03bb \u2208 , \u2009t \u2208 \u2124T+10,, x \u2208 [0, \u221e) and there exists a constant \u03b3 \u2208  such that f \u2265 \u03bb\u03b3f for \u03bb \u2208 , t \u2208 \u2124T+10,, x \u2208 [0, \u221e);\u2009(A3)\u03b40 > 0 such that f \u2265 \u03b40g, t \u2208 \u2124T+10,, x \u2208 [0, \u221e).there exists a constant The problem  = t + \u03b2, t \u2208 \u2124T+10,. Moreover, for any initial value u0 \u2208 Ph, constructing successively the sequence un(t) \u2192 u*(t) as n \u2192 \u221e, where G is given as  \u2260 0;\u2009Aggiven as .A : P \u2192 E and B : P \u2192 E by u is a solution of (u = Au + Bu. From (A2) and A : P \u2192 P and B : P \u2192 P. In the sequel we check that A, B satisfy all assumptions of Define two operators P \u2192 E by 7(Au)(t)=ution of  if and oA, B are two increasing operators. In fact, by (A1) and u, v \u2208 P with u \u2265 v, we know that u(t) \u2265 v(t), t \u2208 \u2124T+10,, and obtain Bu \u2265 Bv.Firstly, we prove that A is a \u03b3-concave operator and B is a subhomogeneous operator. In fact, for any \u03bb \u2208  and u \u2208 P, from (A2), we know that A is a \u03b3-concave operator. At the same time, for any \u03bb \u2208  and u \u2208 P, from (A2), we get B is subhomogeneous.Next we show that Ah \u2208 Ph and Bh \u2208 Ph. From (A3) and l1 = \u03b40p(T)g > 0 and l2 = \u2211s=1Tr(s)f. Hence we have l1h(t)\u2264(Ah)(t) \u2264 l2h(t), t \u2208 \u2124T+10,; that is, Ah \u2208 Ph. We can similarly prove that Bh \u2208 Ph. Thus condition (i) of Now we show that Au \u2265 \u03b40Bu, u \u2208 P. By applying Au + Bu = u has a unique solution u* \u2208 Ph. Moreover, constructing successively the sequence un = Aun\u22121 + Bun\u22121, n = 1,2,\u2026, for any initial value u0 \u2208 Ph, we have un \u2192 u* as n \u2192 \u221e. That is, problem  \u2192 u*(t) as n \u2192 \u221e.In the following we show that condition (ii) of 2(Au)(t)=The following result can be obtained by \u2009(A1)\u2032f : \u2124T+10, \u00d7 [0, \u221e)\u2192[0, \u221e) is continuous and increasing with respect to the second variable, f \u2260 0;\u2009(A2)\u2032\u03b3 \u2208  such that f \u2265 \u03bb\u03b3f for \u03bb \u2208 , t \u2208 \u2124T+10,, x \u2208 [0, \u221e).there exists a constant Then problem u* \u2208 Ph, where h(t) = t + \u03b2, t \u2208 \u2124T+10,. Moreover, for any initial value u0 \u2208 Ph, constructing successively the sequence un(t) \u2192 u*(t) as n \u2192 \u221e, where G is given as =\u03b1u\u2192[0, \u221e) are continuous and increasing with respect to the second variable, g \u2260 0, and for \u03bb \u2208 , t \u2208 \u21240,10, y \u2208 [0, \u221e), Pt+2.Consider the following nonlinear discrete problem: \u2212\u03942y(t\u22121)\u03b1 \u2260 1, the nonlocal boundary value problem If t \u2208 \u2124T+10,, s \u2208 \u2124T1,, the Green function G in \u2009(i)G > 0, t \u2208 \u2124T+10,, s \u2208 \u2124T1,;\u2009(ii)t \u2208 \u2124T+10,, s \u2208 \u2124T1,, where For We omit it since it is obvious.h(t) = 1, t \u2208 \u2124T+10,. Moreover, for any initial value n \u2192 \u221e, where Assume that A1), (A2), and A3) are satisfied; then the problem , a are satigiven as .It is similar to the proof of The following corollary can be obtained by h(t) = 1, \u2009t \u2208 \u2124T+10,. Moreover, for any initial value n \u2192 \u221e, where Assume that (A1)\u2032 and (A2)\u2032 are satisfied; then the problem given as .In a similar way, we can get the corresponding results for the difference equation  subject 9u(0)=\u03b1u\u2192[0, \u221e) are continuous and increasing with respect to the second variable, g \u2260 0, and for \u03bb \u2208 , t \u2208 \u21240,5, y \u2208 [0, \u221e), P1.Consider the following nonlinear discrete problem: \u2212\u03942y(t\u22121)"}
{"text": "Rough set theory is a suitable tool for dealing with the imprecision, uncertainty, incompleteness, and vagueness of knowledge. In this paper, new lower and upper approximation operators for generalized fuzzy rough sets are constructed, and their definitions are expanded to the interval-valued environment. Furthermore, the properties of this type of rough sets are analyzed. These operators are shown to be equivalent to the generalized interval fuzzy rough approximation operators introduced by Dubois, which are determined by any interval-valued fuzzy binary relation expressed in a generalized approximation space. Main properties of these operators are discussed under different interval-valued fuzzy binary relations, and the illustrative examples are given to demonstrate the main features of the proposed operators. Rough set theory proposed by Pawlak  is an exA rough set model is composed of two parts: the approximation space and the approximated object. Rough set theory comes with a lot of extensions and generalizations. Yao et al. researched the generalized rough sets by considering sets and relations of the approximation space and the approximated object , 16. In Most researches on the fuzzy rough set theory focus on point-valued fuzzy sets and point-valued fuzzy binary relations. But the fuzzy notion described by using point values may lose some available information in the real-life information systems sometimes. If the description is done by interval values, it may acquire a better effectiveness than that by using point ones, for example, a self-evolving interval type-2 fuzzy neural network with online structure and parameter learning , encodinIn this paper, we further study the generalized fuzzy rough approximation operators defined in . In partThe rest of the paper is organized as follows. In In this section, we introduce some basic notions and properties related to interval-valued fuzzy sets which will be used in this paper. We first review an interval-valued subset originated by . We firsI be a closed unit interval; that is, I = . [I] = { : a\u2212 \u2264 a+, a\u2212, a+ \u2208 I} is the set of all interval-valued subsets of I. a = \u2208[I] is an interval value. When a\u2212 = a+, the interval-valued a =  becomes a real number in [I]. In particular, real numbers return intervals of zero length, say 1 =  and 0 = .Let a, b \u2208 [I]. a \u2264 b if and only if a\u2212 \u2264 b\u2212, a+ \u2264 b+; a = b if and only if a\u2212 = b\u2212, a+ = b+; a < b if and only if a \u2264 b and a \u2260 b.Let a, b \u2208 [I]. a\u2270b indicates that a is not less than or equal to b; a\u2009\u226e\u2009b indicates that a is not less than b; a\u2271b indicates that a is not greater than or equal to b; a\u226fb indicates that a is not greater than b.Let I] may not exhibit order relations, so According to the order relation defined in ai \u2208 [I], bi \u2208 I, i \u2208 J, J = {1,2, \u22ef, m}; one definesLet I], \u2264) is a complete lattice, and the triple  is an algebraic system, which is derived by  with the maximal element  and the minimum element .Obviously, . In particular, when A = U, A(x) = , for all x \u2208 U, and when A = \u2205, A(x) = , for all x \u2208 U.Let A, B \u2208 FI(U), A\u2286B means A(x) \u2264 B(x) and for all x \u2208 U, (A\u2229B)(x) = A(x)\u2227B(x), (A \u222a B)(x) = A(x)\u2228B(x), and (~A)(x) = 1 \u2212 A(x).Similar to fuzzy sets, the operators \u2286, \u2229, \u222a, and complement of interval-valued fuzzy sets are defined as follows. For all \u03b1 \u2208 [I], A \u2208 FI(U). \u03b1A is called numerical product of \u03b1 and A and is defined as (\u03b1A)(x) = \u03b1\u2227A(x), for all x \u2208 U.Let \u03b1 \u2208 [I], A \u2208 FI(U). A\u03b1 = {x \u2208 U : A(x) \u2265 \u03b1} is called \u03b1-cut set of A and \u03b1-cut set of A.Let A \u2208 FI(U); thenLet x \u2208 U,For all Then Similarly, one can show that U and W be two finite and nonempty universes of discourse. Then the mapping IR : U \u00d7 W \u2192 [I] is called an interval-valued fuzzy relation from U to W, where U \u00d7 W = { : x \u2208 U, y \u2208 W}. When U = W, IR is called an interval-valued fuzzy relation on U.Let IR from U to W is an interval-valued fuzzy set denoted by IR \u2208 FI(U \u00d7 W). So Definitions IR \u2208 FI(U \u00d7 W) is an interval-valued fuzzy relation. If we see it as an interval-valued fuzzy set, then IR\u03b1 = { \u2208 U \u00d7 W : R \u2265 \u03b1}.Obviously, an interval-valued fuzzy relation IR be an interval-valued fuzzy relation from U to W; then IR is said to be serial if and only if for all x \u2208 U, there exists y \u2208 W such that IR = .Let IR be an interval-valued fuzzy relation on U; then IR is reflexive if and only if IR = , for all x \u2208 U; IR is symmetric if and only if IR = IR, for all x, y \u2208 U; IR is transitive if and only if x, z \u2208 U; IR is Euclidean if and only if y, z \u2208 U.Let \u03b1-cut set or strong \u03b1-cut set to an interval-valued fuzzy relation, for all \u03b1 \u2208 [I], still satisfies the corresponding definition of IR is, respectively, reflexive, symmetric, and transitive, then One can prove that the binary relation obtained by calculating U and W be two finite universes of discourse. If R is an arbitrary binary fuzzy relation from U to W, then the triple  is called a generalized fuzzy approximation space.Let U, W, R) be a generalized fuzzy approximation space, for all x \u2208 U; one defines R(x) = {) : y \u2208 W}.Let (R(x) is the row of the fuzzy relation which includes x, and obviously R\u03b1(x) = (R(x))\u03b1.U, W, R) be a generalized fuzzy approximation space, for all \u03b1, \u03b2 \u2208 , A \u2208 F(W),Let  lower and upper approximations of A with respect to .U, W, R) be a generalized fuzzy approximation space, for all A \u2208 F(W). One definesLet , and the operators The pair The dual properties are quite useful in proving the properties of the approximation operators. When one intends to prove two dual properties, it suffices to prove one of them, which simplifies the proof procedure. The properties of the lower and upper approximation operators are characterized as follows.U, W, R) be a generalized fuzzy approximation space. Then for all A \u2208 F(W), Let (x \u2208 U.(1) Note that x \u2208 U, y \u2208 W if R \u2265 1 \u2212 \u03b1, then A(y) > \u03b1; that is, for all y \u2208 W, R < 1 \u2212 \u03b1 or A(y) > \u03b1.For all Hence we have \u03b1 \u2208 I, so that For the second case, Hence,Similarly,Therefore, (2) Now, we prove the validity of the relationship Hence, Similarly, x is a certain value, the variables y1 and y2 are functions of the variable \u03b1. Refer to Suppose that y1 is equal to the maximum of function y2, such that y2 equal to zero at the point \u03b2, which makes the maximum of function y2 approach \u03b2, but it does not exist. In this paper, the lower and upper approximation operators in In the proof of olds. In , the lowU and W be two finite universes of discourse. If IR is an arbitrary binary interval-valued fuzzy relation from U to W, then the triple  is called a generalized interval-valued fuzzy approximation space. In particular, when U = W, the space is denoted by .Let U, W, IR) be a generalized interval-valued fuzzy approximation space, for all x \u2208 U,Let  be a generalized interval-valued fuzzy approximation space, A \u2208 FI(W), for all x \u2208 U; one definesLet . The operators The pair U, W, IR) be a generalized interval-valued fuzzy approximation space, for all \u03b1, \u03b2 \u2208 [I], A \u2208 FI(W); one definesLet , respectively.U, W, IR) be a generalized interval-valued fuzzy approximation space, A \u2208 FI(W); one definesLet . The operators The pair The approximation operators introduced in U, W, IR) be a generalized interval-valued fuzzy approximation space, A \u2208 FI(W); then for all Let \u2229A\u03b1 \u2260 \u2205. This means that there exist y \u2208 W, IR \u2265 \u03b1, and A(y) \u2265 \u03b1.We observe that, for all By the interval-valued operations of Therefore, y \u2208 W, IR\u2227A(y) \u2265 \u03b1; that is, there exists y \u2208 W, so that IR \u2265 \u03b1 and A(y) \u2265 \u03b1 cannot be deduced by Now, we prove that the reverse of Next, we give an example illustrating that the relationship U, W, IR) is a generalized interval-valued fuzzy approximation space,Suppose that (IR\u03b1(x1) = A\u03b1 = \u2205, we have IR\u03b1(x1)\u2229A\u03b1 = \u2205 and we get On the other hand, since U, W, IR) be a generalized interval-valued fuzzy approximation space, A \u2208 FI(W); then, for all \u03b1 \u2208 [I], Let  > 1 \u2212 \u03b1 then A(y) \u2265 \u03b1; that is, for all y \u2208 W, IR \u2264 1 \u2212 \u03b1 or A(y) \u2265 \u03b1. By the interval-valued operations as in Note that, for all IR\u2228A(y) \u2265 \u03b1 means that )\u2212\u2228A(y)\u2212 \u2265 \u03b1\u2212 and )+\u2228A(y)+ \u2265 \u03b1+, which cannot deduce that 1 \u2212 IR \u2265 \u03b1 or A(y) \u2265 \u03b1 in y \u2208 W, ~IR\u2228A(y) \u2265 \u03b1, we cannot deduce that IR \u2264 1 \u2212 \u03b1 or A(y) \u2265 \u03b1, and note that, for all y \u2208 W, ~IR\u2228A(y) \u2265 \u03b1 if and only if y \u2208 W, IR \u2264 1 \u2212 \u03b1 or A(y) \u2265 \u03b1 cannot hold.Now we prove that the reverse of Next, we show that U, W, IR) is a generalized interval-valued fuzzy approximation space,Suppose that  be a generalized interval-valued fuzzy approximation space, A \u2208 FI(W); then Let \u2227A(y)\u2208[I], such that y \u2208 IR\u03b1(x)\u2229A\u03b1. We observe that y \u2208 IR\u03b1(x)\u2229A\u03b1 means that IR\u03b1(x)\u2229A\u03b1 \u2260 \u2205, which can deduce that In fact, for all y, So, for arbitrary value of Therefore, U, W, IR) be a generalized interval-valued fuzzy approximation space, A \u2208 FI(W); then Let  We verify that y0 \u2208 W, IR > 1 \u2212 \u03b11, and from (1\u2212\u03b11)+ = 1, we have IR\u2212 > \u2228y\u2208W\u2212\u2227(1 \u2212 A(y)+)).Note that, for all y0 \u2208 W, we have IR\u2212 > IR\u2212\u2227(1 \u2212 A(y0)+)\u2009\u2009.Further from IR\u2212 > 1 \u2212 A(y0)+\u2009\u2009and\u2009\u20091 \u2212 A(y0)+ = IR\u2212\u2227(1 \u2212 A(y0)+).Therefore we obtain that \u03b11\u2212 = 0, we get A(y0) \u2265 \u03b11; that is, y0 \u2208 A\u03b11.Because y0, So, for arbitrary value of (2) Similar to the proof shown in (1), we have x, For any Therefore, According to Theorems U, W, IR) be a generalized interval-valued fuzzy approximation space, A \u2208 FI(W); then Let  be a generalized interval-valued fuzzy approximation space; then the lower approximation operator Let , a \u2208 [I],For all x \u2208 U and x \u2208 W.Here (1) We prove that x \u2208 U, letx is \u201cfor all x\u201d and \u2203x is \u201cthere exists x,\u201d which are the same as follows.For all D3 = D2 \u2212 D1, for all \u03b2 \u2208 D3; two cases appear:Obviously, a, b \u2208 D1; we have For the first case, suppose \u03b2, it is easy to see that For arbitrary Hence, Similarly, (2) We verify x \u2208 U, letFor all We observe thatSimilarly, A\u2286B then A\u2286B, then A\u03b1\u2286B\u03b1. According to (3) We prove that if A\u2286B, then Similarly, if (4) From (3), one immediately obtains (4).From U, W, IR) be a generalized interval-valued fuzzy approximation space; then the following conditions are equivalent: IR is serial;A\u2286FI(W);Let  be a generalized interval-valued fuzzy approximation space; then the following conditions are equivalent: IR is reflexive;A\u2286FI(W);A\u2286FI(W).Let  be a generalized interval-valued fuzzy approximation space; then the following properties hold: x, y) \u2208 U \u00d7 W;x, y) \u2208 U \u00d7 W.Let  be a generalized interval-valued fuzzy approximation space and A is an interval-valued fuzzy set on U; then \u2200\u03b1 \u2208 [I], Let  is a generalized interval-valued fuzzy approximation space, where U = {x1, x2}, IR(x1) = /x1 + /x2, A = /x1 + /x2, \u2009\u2009\u03b1 = , \u03b21 = , and \u03b22 = . Since IR\u03b1(x1) = \u2205, A\u03b1 = {x1, x2}, we have IR\u03b1(x1)\u2229A\u03b1 = \u2205. Hence Suppose that (IR\u03b21(x1) = {x1}, A\u03b21 = {x1, x2}, IR\u03b22(x1) = {x2}, A\u03b22 = {x1, x2}, we see that On the other hand, since Note that U, IR) is a generalized interval-valued fuzzy approximation space, where U = {x1, x2, x3}, IR(x1) = /x1 + /x2 + /x3, A = /x1 + /x2 + /x3,\u03b1 = , \u03b21 = , \u03b22 = .Suppose that  be a generalized interval-valued fuzzy approximation space; then the following conditions are equivalent: IR is transitive;A \u2208 FI(U);A \u2208 FI(U).Let (A \u2208 FI(U), from (1)\u21d2(2) For all Hence x, y, z \u2208 U, let(3)\u21d2(1) For all y \u2208 U, suppose that \u03b1 = D7(y) = IR\u2227IR; then IR \u2265 \u03b1, IR \u2265 \u03b1; hence \u03b1 \u2208 D6 and by the arbitrary y, For all \u03b1 \u2208 D6, there exists y \u2208 U, such that \u03b1 \u2264 IR\u2227IR. For arbitrary \u03b1, For all IR is transitive.Hence, by (2)\u21d4(3) This conclusion follows immediately from the duality.IR is symmetric, then the approximation operators satisfy A\u2286U; if IR is Euclidean, then the approximation operators satisfy A\u2286U. These properties do not hold in the interval-valued fuzzy rough sets. Next, we give a counterexample to show it.In , if IR iU, IR) is a generalized interval-valued fuzzy approximation space, U = {x1, x2, x3}, andSuppose that  be a generalized interval-valued fuzzy approximation space; then the following conditions are equivalent: Let \u21d2(1) For all By Hence, At the same time, we haveTherefore (1) has been proven.(1)\u21d2(3) First we prove thatx \u2208 U, if IR\u03b1(x)\u2229A\u03b1 \u2260 \u2205; that is, there exists z \u2208 IR\u03b1(x)\u2229A\u03b1. So we have IR \u2265 \u03b1 and A(z) \u2265 \u03b1.Note that, for all Sincewe havey \u2208 U, IR+ \u2264 1 \u2212 \u03b1\u2212 or IR\u2212 \u2265 \u03b1\u2212and IR\u2212 \u2264 1 \u2212 \u03b1+ or IR+ \u2265 \u03b1+ imply that if IR+ > 1 \u2212 \u03b1\u2212, then IR\u2212 \u2265 \u03b1\u2212, and if IR\u2212 > 1 \u2212 \u03b1+, then IR+ \u2265 \u03b1+. It follows that IR+ > 1 \u2212 \u03b1\u2212 and IR\u2212 > 1 \u2212 \u03b1+ imply that IR\u2212 \u2265 \u03b1\u2212 and IR+ \u2265 \u03b1+; therefore, IR > 1 \u2212 \u03b1 implies that IR \u2265 \u03b1, because IR > 1 \u2212 \u03b1 and IR \u2265 \u03b1 are equivalent to z \u2208 IR\u03b1(y), respectively. If z \u2208 IR\u03b1(y), and since A(z) \u2265 \u03b1, z \u2208 IR\u03b1(y)\u2229A\u03b1; that is, IR\u03b1(y)\u2229A\u03b1 \u2260 \u2205. So Hence, for all y, it follows that x, we obtain For arbitrary By Therefore (2)\u21d4(3) This conclusion follows immediately from the duality.U, IR) be a generalized interval-valued fuzzy approximation space; then the following conditions are equivalent: IR = 0\u2009\u2009or\u2009\u2009IR = 1, x, y \u2208 U;Let +)\u2228IR\u2212) = 1 and \u2212)\u2228IR+) = 1; namely, for all x \u2208 U, IR+ = 0\u2009\u2009or\u2009\u2009IR\u2212 = 1 and IR\u2212 = 0 or IR+ = 1.(1)\u21d4(2) We observe that, for all IR+ = 0, then IR\u2212 = 0; we have IR = 0. If IR+ \u2260 0, then IR\u2212 = 1 and IR+ = 1; we have IR = 1.On the one hand, if y \u2208 U, IR = 0 or IR = 1.Hence, (2)\u21d2(4) We first prove that x \u2208 A\u03b1, for all y \u2208 U, IR = 0 or IR = 1 if and only if IR \u2260 0 deduces IR = 1. It follows that if IR > 1 \u2212 \u03b1, then IR \u2265 \u03b1; that is, x \u2208 IR\u03b1(y). Further since x \u2208 A\u03b1, IR > 1 \u2212 \u03b1 implies that x \u2208 IR\u03b1(y)\u2229A\u03b1. So IR\u03b1(y)\u2229A\u03b1 \u2260 \u2205. Note that IR > 1 \u2212 \u03b1 and IR\u03b1(y)\u2229A\u03b1 \u2260 \u2205 are equivalent to IR > 1 \u2212 \u03b1, then x \u2208 IR\u03b1(y)\u2229A\u03b1. It shows that if y, x, Suppose that On the other hand, in view of x, z \u2208 U, from the proof of \u201c(2)\u21d2(1)\u201d in (3)\u21d2(1) For all Furthermore, since z = x, we have I], we have When (3)\u21d4(4) This conclusion follows immediately from the duality.U, IR) be a generalized interval-valued fuzzy approximation space.IR is reflexive and transitive, then A \u2208 FI(U).If IR is reflexive and x, z \u2208 U, then A \u2208 FI(U).If Let  is a generalized interval-valued fuzzy approximation space. (1)IR is reflexive and transitive, thenIf (2)IR is reflexive and x, z \u2208 U, thenIf Suppose that  is reflexive, symmetric, and transitive, respectively, under the classical binary relation. Thus, if IR can satisfy the above functions, this technology can be applied in reasoning, learning, and decision-making. In Sections In this paper, one can prove that the binary relation obtained by calculating"}
{"text": "R be a prime ring of characteristic different from 2, with extended centroid C, U its two-sided Utumi quotient ring, F a nonzero generalized derivation of R, f a noncentral multilinear polynomial over C in n noncommuting variables, and a, b \u2208 R such that ab = 0 for any r1,\u2026, rn \u2208 R. Then one of the following holds: (1) a = 0; (2) b = 0; (3) there exists \u03bb \u2208 C such that F(x) = \u03bbx, for all x \u2208 R; (4) there exist q \u2208 U and \u03bb \u2208 C such that F(x) = (q + \u03bb)x + xq, for all x \u2208 R, and f2 is central valued on R; (5) there exist q \u2208 U and \u03bb, \u03bc \u2208 C such that F(x) = (q + \u03bb)x + xq, for all x \u2208 R, and aq = \u03bca, qb = \u03bcb.Let  R be a prime ring with center Z(R). We denote by  = ab \u2212 ba the simple commutator of the elements a, b \u2208 R and by k = k\u22121, b], for k > 1, the kth commutator of a, \u2009b. Throught this paper we will use the following notation: U will be the (two-sided) Utumi quotient ring of a ring R . The definition, the axiomatic formulation, and the properties of this quotient ring U can be found in  \u2208 Z(R), for all x \u2208 R, then R is commutative. In k = 0 for all x \u2208 L and k \u2265 1 a fixed integer; then char(R) = 2 and R satisfies s4, the standard identity of degree 4.In any case, when f Posner  says thative. In  Lanski gf be a multilinear polynomial over C in n noncommuting variables and denote by f(X) the set of all evaluations of f in X\u2286R. In case f is not central valued on R, it is well known that the additive subgroup generated by f(R) contains a noncentral Lie ideal of R. Moreover any noncentral Lie ideal of R contains all the commutators  for x, y in some nonzero ideal of R, unless char(R) = 2 and dimCRC = 4.Let d(x), x]k = 0, in case x \u2208 f(I), where I is a two-sided ideal of R. They show that either f is central valued in R or char(R) = 2 and R satisfies s4.In light of this and following the line of investigation of the previous cited papers, in  P. H. LeR is often tightly connected to the behaviour of additive mappings defined on R, which act on suitable subsets of the whole ring. In k = 0, where d is a derivation of R and u \u2208 f(I), where I is a one-sided ideal of R. In particular, for I = R, if a \u2260 0 and f is not central valued on R, then char(R) = 2 and R satisfies s4.More recently, Liu  and Wangd is replaced by a generalized derivation F. An additive map F : R \u2192 R is said to be a generalized derivation if there is a derivation d of R such that, for all x, y \u2208 R, F(xy) = F(x)y + xd(y). A significative example is a map of the form F(x) = ax + xb, for some a, b \u2208 R; such generalized derivations are called inner. Generalized derivations have been primarily studied on operator algebras. Therefore any investigation from the algebraic point of view might be interesting .The main result in  is the fR be a prime ring of characteristic different from 2, with extended centroid C, U its two-sided Utumi quotient ring, F \u2260 0 a nonzero generalized derivation of R, f a noncentral multilinear polynomial over C in n noncommuting variables, and a \u2208 R such that r1,\u2026, rn \u2208 R. Then either a = 0 or one of the following holds: (1)\u03bb \u2208 C such that F(x) = \u03bbx, for all x \u2208 R;there exists (2)q \u2208 U and \u03bb \u2208 C such that F(x) = (q + \u03bb)x + xq, for all x \u2208 R, and f2 is central valued on R.there exist Let We would like to remark that the same conclusions hold in case we consider the right annihilator, more precisely.R be a prime ring of characteristic different from 2, with extended centroid C, U its two-sided Utumi quotient ring, F \u2260 0 a nonzero generalized derivation of R, f a noncentral multilinear polynomial over C in n noncommuting variables, and a \u2208 R such that r1,\u2026, rn \u2208 R. Then either a = 0 or one of the following holds: (1)\u03bb \u2208 C such that F(x) = \u03bbx, for all x \u2208 R;there exists (2)q \u2208 U and \u03bb \u2208 C such that F(x) = (q + \u03bb)x + xq, for all x \u2208 R, and f2 is central valued on R.there exist Let F(x), x] : x \u2208 f(R)} and prove the following.Here we will consider a more general situation, involving a two-sided annihilating condition. More specifically, we study simultaneously left and right annihilators of the set {. Let a, b, c \u2208 R, such that c \u2209 Z(R) and ab = 0 for all x \u2208 S. Then there exists \u03bb \u2208 Z(R) such that ac = \u03bba and cb = \u03bbb.Let In order to prove K be an infinite field and t \u2265 2. If A1,\u2026, Ak are not scalar matrices in Mt(K), then there exists some invertible matrix B \u2208 Mt(K) such that each matrix BA1B\u22121,\u2026, BAkB\u22121 has all nonzero entries.Let See Lemma\u2009\u20091.5 in .R be a prime ring with extended centroid C. Suppose \u2211i=1naixbi + \u2211j=1mcjxdj = 0, for all x \u2208 R, where ai, bi, cj, dj \u2208 R, for i = 1,\u2026, n and j = 1,\u2026, m. If a1,\u2026, an are C-independent then each bi is C-dependent on d1,\u2026, dm. Analogously if b1,\u2026, bn are C-independent then each ai is C-dependent on c1,\u2026, cm.Let It is Martindale's result contained in .R be a prime ring with extended centroid C. Suppose a+b = 0, for all x, y \u2208 R, where a, b \u2208 R. Then a = \u2212b \u2208 C.Let It is an easy consequence of K be an infinite field, R = Mm(K) the algebra of m \u00d7 m matrices over K, Z(R) the center of R, and S = . Assume that there exist a, \u2009b, \u2009c, \u2009q nonzero elements of R such that axq + cxb = 0 for all x \u2208 S. If q \u2208 Z(R) then one of the following holds: (1)a, \u2009b, \u2009c are central matrices and aq + bc = 0;(2)b is a central matrix and aq + bc = 0.Let q \u2208 Z(R), by the assumption, we have that aqx + cxb = 0 for all x \u2208 S. Clearly if c \u2208 Z(R) then aqx + xbc = 0 for all x \u2208 S, and by aq = \u2212bc \u2208 Z(R); that is, a, b, c \u2208 Z(R). On the other hand, if b \u2208 Z(R), then (aq + bc)x = 0 for all x \u2208 S and it follows easily that aq + bc = 0.Since c and b both nonscalar matrices. We will prove that in this case we get a contradiction.In light of this, we consider eij the usual matrix unit with 1 in the -entry and zero elsewhere.Here we denote by c and b have all nonzero entries, say c = \u2211klcklekl and b = \u2211klbklekl, for 0 \u2260 ckl, 0 \u2260 bkl \u2208 K.By eji \u2208 S for all i \u2260 j, then, for any i \u2260 j, i, j)-entry of X is cijbij = 0, a contradiction.Since For sake of clearness, we may write the previous lemma as follows.K be an infinite field, R = Mm(K) the algebra of m \u00d7 m matrices over K, Z(R) the center of R, and S = . Let a1, \u2009a2, \u2009a3, \u2009a4 be nonzero elements of R such that a1xa2 + a3xa4 = 0 for all x \u2208 S. Assume there exists i \u2208 {1,2, 3,4} such that ai \u2208 Z(R). Then a1 = \u03b1a3 and a2 = \u2212\u03b1a4, for a suitable \u03b1 \u2208 Z(R).Let K be an infinite field, R = Mm(K) the algebra of m \u00d7 m matrices over K, and Z(R) the center of R. Assume that there exist a, \u2009b, \u2009c, \u2009q nonzero elements of R such that axq + cxb = 0 for all x \u2208 S = . If q \u2209 Z(R) and b \u2212 \u03b1q \u2208 Z(R), for a suitable \u03b1 \u2208 K, then b \u2212 \u03b1q = a + \u03b1c = 0.Let a + \u03b1c is not a scalar matrix. By a + \u03b1c and q have all nonzero entries, say a + \u03b1c = \u2211kltklekl and q = \u2211klqklekl, for 0 \u2260 tkl, 0 \u2260 qkl \u2208 K.Assume that b = \u03b2I + \u03b1q, for a suitable \u03b2 \u2208 K, by our assumption we have that x \u2208 S. In particular for x =  = eij, with i \u2260 j, j, i)-entry of X is 0 = tjiqji, a contradiction.Since a + \u03b1c must be a central matrix. In light of this, there exist \u03b2, \u03b3 \u2208 K such that b = \u03b1q + \u03b2 and a = \u2212\u03b1c + \u03b3, so that 0 = (\u2212\u03b1c + \u03b3)xq + cx(\u03b1q + \u03b2) = (\u03b2c)x + x(\u03b3q), for all x \u2208 S = . Once again by q \u2209 Z(R), it follows that \u03b2 = \u03b3 = 0; that is, b = \u03b1q and a = \u2212\u03b1c.Therefore K be an infinite field, R = Mm(K) the algebra of m \u00d7 m matrices over K, and S = . Suppose there exist a, \u2009b, \u2009c, \u2009q \u2208 R such that axq + cxb = 0 for all x \u2208 S. Denote akl, \u2009bkl, \u2009ckl, and qkl elements of K. If there are i \u2260 j such that qji \u2260 0, cji \u2260 0, and bji = 0, then ari = 0 and brk = 0 for all r \u2260 i and k \u2260 r .Let x = eij, we have r \u2260 i, the -entry of the matrix X is 0 = ariqji + cribji = ariqji. Since qji \u2260 0, one has ari = 0 for all r \u2260 i, in particular aji = 0. Thus, in case m = 2 we are done (since bji = aji = 0).Consider the assumption m \u2265 3, and choose x = eit, with t \u2260 i, j. Hence we also have (1)s \u2260 i, the -entry of the matrix X is ajiqjs + cjibjs = 0;for all (2)s \u2260 i, j, the -entry of the matrix Y is ajiqts + cjibts = 0;for all (3)j, i)-entry of the matrix Y is ajiqti + cjibti = 0;the ((4)k \u2260 i, t, the -entry of the matrix Y is ajiqtk + cjibtk = 0 .for all By (1) and (2) and since aji = 0 and cji \u2260 0, we have both bjs = 0, for all s \u2260 i, and bts = 0 for all t \u2260 i, j and s \u2260 i, j. So by (3)\u2009\u2009bti = 0 for all t \u2260 i. Finally by (4), btk = 0 for all t \u2260 i, j and k \u2260 t.Assume in what follows that K be an infinite field, R = Mm(K) the algebra of m \u00d7 m matrices over K, and S = . Suppose there exist a, b, c, q \u2208 R such that axq + cxb = 0 for all x \u2208 S. Denote bkl, ckl, and qkl elements of K. Assume there are i \u2260 j such that bji = 0. If qrs \u2260 0, crs \u2260 0 for all r \u2260 s, then one of the following holds: (1)a = b = 0;(2)m = 2, cq = 0, and there exists 0 \u2260 \u03bb \u2208 K such that Let m \u2265 3. The first step is to apply twice b to be a diagonal matrix. In fact bji = 0, qji \u2260 0, and cji \u2260 0 imply that brk = 0 for all r \u2260 i and k \u2260 r; in particular, since m \u2265 3, there exists t \u2260 i such that blt = 0, for all l \u2260 t, i. Since qlt \u2260 0, clt \u2260 0, we have brk = 0 for all r \u2260 t and k \u2260 r, so bik = 0 for all k \u2260 i, as required. Say b = \u2211kbkkekk.Firstly we consider the case R induced by the invertible matrix P = I + erj, for r \u2260 i, j: \u03c6(x) = P\u22121xP. Of course \u03c6(a)x\u03c6(q) + \u03c6(c)x\u03c6(b) = 0, for all x \u2208 S. Moreover the -entries of \u03c6(q), \u03c6(c), and \u03c6(b) are, respectively, qji \u2260 0, cji \u2260 0, and bji = 0. Therefore, again by r, j)-entry of \u03c6(b) is zero, for all r \u2260 i. By calculations 0 = (\u03c6(b))rj = bjj \u2212 brr; that is, bjj = brr.Consider now the inner automorphism of \u03c7 is the inner automorphisms induced by the invertible matrix Q = I + eri, as above \u03c7(a)x\u03c7(q) + \u03c7(c)x\u03c7(b) = 0, for all x \u2208 R. Since the -entries of \u03c7(q), \u03c7(c), and \u03c7(b) are, respectively, qij \u2260 0, cij \u2260 0, and bij = 0, and again any -entry of \u03c7(b) is zero, for all r \u2260 j; that is, 0 = (\u03c6(b))ri = bii \u2212 brr and bii = brr = bjj = \u03b2, for all r \u2260 i, j. Thus b = \u03b2I is a central matrix in R. By b = \u03b1q for some \u03b1 \u2208 K or b = 0. Since the first case cannot occur, we get b = 0 and also a = 0 which follows from aq = 0 and q \u2260 0.On the other hand, if m = 2; that is, R = M2(K). In this case it is well known that for any element x \u2208  there exist \u03b1, \u03b2, \u03b3 \u2208 K such that b21 = 0. In case b12 = 0, then by the same above argument we show that b \u2208 Z(R) and we are done again. Thus we consider the case b12 \u2260 0. Moreover, by applying a21 = 0. Hence we may write x = e12 \u2208  we have X is 0 = c21b22; that is, b22 = 0 and the -entry of the matrix X is 0 = a11q21; that is, a11 = 0. On the other hand, for x = e21 \u2208 , we have Y is 0 = a12q12 + c12b12; that is, a12 \u2260 0 and b12/q12 = \u2212a12/c12. Moreover the -entry of the matrix Y is 0 = a22q12 + c22b12. Therefore, if denoted \u03bb = \u2212b12/q12, one has a22 = \u03bbc22 and a12 = \u03bbc12.Let now Y is 0 = a12q11 + c12b11. Thus b11 = \u2212\u03bbq11 and b12 = \u2212\u03bbq12.Analogously, the -entry of the matrix \u03b1 \u2260 0, we also have cq = 0.Finally, by our assumption and for K be an infinite field, R = M2(K) the algebra of m \u00d7 m matrices over K, and S = . Let a, b, c \u2208 R and denote akl, bkl, ckl, and pkl elements of K. Suppose c \u2209 Z(R) and ab = 0 for all x \u2208 S. Assume there are i \u2260 j such that pji = 0. If ars \u2260 0, brs \u2260 0, for all r \u2260 s, then ac = cb = 0.Let acxb \u2212 axcb = 0 for all x \u2208 S. By ac = cb = 0 or ab = 0 and there exists 0 \u2260 \u03bb \u2208 K such that ab = 0 implies that the following holds: ac we get cb we also have a11 = 0, by  = 0 and using (b21(c22a11 + c11a11) = 0. Since b21 \u2260 0 and a11 \u2260 0, then c11 = \u2212c22 = \u03bc.By our hypothesis, we have lso have c21b11+c2\u03bc \u2260 0, denoted by I the identity matrix in R, and let Assume c and c\u2032 induce the same inner derivation, then by our assumptions we have that ab = 0 for all x \u2208 S. By applying again ac\u2032 = c\u2032b = 0 or ab = 0 and there exists 0 \u2260 \u03bd \u2208 K such that c\u2032 satisfies the equalities (30)ac'=c0 and by  also c2S: Z is T is c \u2260 0, then c12 \u2260 0. Therefore the sum of  the algebra of t \u00d7 t matrices over K, and S = . Let a, b, c \u2208 R and denote akl, bkl, pkl, and qkl elements of K. Suppose c \u2209 Z(R) and ab = 0 for all x \u2208 S. Then there exists \u03bb \u2208 Z(R) such that ac = \u03bba and cb = \u03bbb.Let a, b, ac, or cb is a scalar matrix we are done by a, b, ac, and cb are noncentral matrices.Clearly if one of Q \u2208 Mt(K) such that QaQ\u22121 = a\u2032, QbQ\u22121 = b\u2032, Q(ac)Q\u22121 = (ac)\u2032, and Q(cb)Q\u22121 = (cb)\u2032 have all nonzero entries.By ac, a} are linearly Z(R)-dependent if and only if {(ac)\u2032, a\u2032} are linearly Z(R)-dependent; analogously {cb, b} are linearly Z(R)-dependent if and only if {(cb)\u2032, b\u2032} are linearly Z(R)-dependent. Moreover ac = cb = 0 if and only if (ac)\u2032 = (cb)\u2032 = 0. Therefore, in order to prove our result, we may replace a, \u2009b, \u2009ac, \u2009cb, respectively, by a\u2032, \u2009b\u2032, \u2009(ac)\u2032, \u2009(cb)\u2032, so that a, \u2009b, \u2009ac, and cb have all nonzero entries.Notice that {x = eij \u2208 S we have j, i)-entry of X is qjibji \u2212 ajipji = 0. Denote 0 \u2260 \u03b7 = qji/aji, so that pji = \u03b7bji. Let I be the identity matrix in R and u = c \u2212 \u03b7I. Since u and c induce the same inner derivation in R, then ab = 0; that is, a(c \u2212 \u03b7I)xb \u2212 ax(c \u2212 \u03b7I)b = 0, for all x \u2208 S. Moreover a and b have all nonzero entries, and the -entry of (c \u2212 \u03b7I)b is zero. Thus we may apply Lemmas ac = \u03b7a and cb = \u03b7b, as required.For C is infinite, the conclusion follows from If one assumes that K be an infinite field which is an extension of the field C and let R. Since P is a multilinear generalized polynomial in the indeterminates x1, \u2009x2, then it is a generalized polynomial identity for Now let f(R), the set of all evaluations of the noncentral multilinear polynomial f over C, and assume that F is an inner generalized derivation, so that there exist c, q \u2208 U such that F(x) = cx + xq, for all x \u2208 R, and f(R) satisfies a, \u2009b are nonzero elements of R.In this section we consider In order to prove the first result we premit the following.R = Mt(C) be the algebra of t \u00d7 t matrices over C of characteristic different from 2. Notice that the set f(R) = {f : r1,\u2026, rn \u2208 R} is invariant under the action of all inner automorphisms of R. Hence if denoted by r =  \u2208 R \u00d7 R \u00d7 R \u00d7 \u22ef\u00d7R = Rn, then for any inner automorphism \u03c6 of Mt(C), we have that Let f is not central then, by b = 0, for all r1,\u2026, rn \u2208 R.This means that R, generated by the set S = {x2 : x \u2208 f(R)}. By  of R is contained in S. In the first case we conclude that f2 is central valued in R and we are done. In either case we have a]b = 0, for all r1, r2 \u2208 R, and by Consider the additive subgroup of (R)}. By , either K be an infinite field which is an extension of the field C and let f is central-valued on R if and only if it is central-valued on R. Moreover it is multihomogeneous of multidegree  in the indeterminates x1,\u2026, xn.Now let P is a multilinear generalized polynomial \u0398 in 2n indeterminates; moreover x1,\u2026, xn, y1,\u2026, yn) is a generalized polynomial identity for R and C) \u2260 2 we obtain P = 0, for all Hence the complete linearization of a \u2208 R, 0 \u2260 b \u2208 R, c, q \u2208 U such that ab = 0, for all x \u2208 f(R), then R satisfies a nontrivial generalized polynomial identity, unless when one of the following holds: (1)c, q \u2208 C;(2)c \u2212 q \u2208 C and there exists \u03bb \u2208 C such that ac = \u03bba, \u2009qc = \u03bbb.If there exist 0 \u2260 R does not satisfy any nontrivial generalized polynomial identity with coefficients in U. Therefore, R. By calculations x1,\u2026, xn \u2208 R. If c \u2208 C and q \u2208 C, the proof is completed; hence we suppose that c and q are not simultaneously central. By b, qb} are linearly C-independent then R satisfies the trivial generalized polynomial identity af2qb = 0. It means, since a \u2260 0, qb = 0, a contradiction. Analogously, if we suppose {a, ac} linearly C-independent, we get ac = 0, a contradiction.Assume that ulations \u03a6x1,\u2026,xn=\u03b1, \u03b2 \u2208 C such that qb = \u03b2b and ac = \u03b1a; now c, q \u2208 Z(R);(2)\u03bb \u2208 C such that c \u2212 q = \u03bb, and f2 is central valued on R;there exists (3)\u03bb, \u03bc \u2208 C such that c \u2212 q = \u03bb, ac = \u03bca, and cb = \u03bcb.there exist Let 0 \u2260 R is not a domain.By R satisfies the nontrivial generalized polynomial identity: U. In case C is infinite, we have P = 0 for all C. Since both U and R by U or C being finite or infinite. Thus we may assume that R is centrally closed over C which is either finite or algebraically closed. By Martindale's theorem b. We remark that H satisfies P(x) = a(cx2 \u2212 x2q + x(q \u2212 c)x)b = 0 , for any nontrivial idempotent element e = e2 \u2208 H, and obtain R, it follows that either ae = 0 or (1 \u2212 e)b = 0 or (1 \u2212 e)(q \u2212 c)e = 0. Here our aim is to prove that in any case (1 \u2212 e)(q \u2212 c)e = 0. To do this, we firstly assume that ae = 0. In 2b = 0, which implies ace = 0.Assume next that ma\u2009\u20092 in , the set  = 0, for any idempotent element e \u2208 H. Since H is not a domain, then H is generated by its minimal idempotent elements; therefore  = (0); that is, q \u2212 c \u2208 C. Let \u03bb \u2208 C such that q = c + \u03bb. By our assumption it follows that H satisfies ab that is H satisfies ab. In this last replace x by x + 1 and obtain that H satisfies ab. Since char(H) \u2260 2, then acrb \u2212 arcb = 0, for all r \u2208 H. By 2b = 0, for all r1,\u2026, rn \u2208 R, then either a = 0 or b = 0.Let U and thus all generalized derivations of R will be implicitly assumed to be defined on the whole U and obtained the following result.In  Lee provF on a dense right ideal of R can be uniquely extended to U and assumes the form F(x) = cx + d(x), for some c \u2208 U and a derivation d on U.Every generalized derivation fd the polynomial obtained from f by replacing each coefficient \u03b1\u03c3 with d(\u03b1\u03c3). Thus we write d) = fd + \u2211if,\u2026, rn), for all r1,\u2026, rn in R.In this section we denote by In light of this, we finally prove our main result.a \u2260 0 and b \u2260 0. Since R satisfies the generalized differential identity R satisfies d is an inner derivation induced by an element q \u2208 U, then R satisfies the generalized polynomial identity: Suppose both d be an outer derivation of R. In this case R satisfies the differential identity:R satisfies the generalized polynomial identity: i = 1,\u2026, n, R satisfies the blended component q \u2208 R \u2212 Z(R) and replace any yi by . Thus R satisfies q \u2208 Z(R).Hence let rem see , 22), R , R d be"}
{"text": "An objective measure of the uncertainty, regarding the interval constraint, accounted for by using the HSGM is proposed for the Bayesian inference. For this purpose, we drive a maximum entropy prior of the normal mean, eliciting the uncertainty regarding the interval constraint, and then obtain the uncertainty measure by considering the relationship between the maximum entropy prior and the marginal prior of the normal mean in HSGM. Bayesian estimation procedure of HSGM is developed and two numerical illustrations pertaining to the properties of the uncertainty measure are provided.This paper considers a  Specifically, we assign a normal prior distribution for \u03b8 and an inverse gamma (IG) prior for \u03c42, that is, \u03b8 ~ N and \u03c42 ~ IG, which are commonly used in a normal model as conjugate priors, where all the hyperparameters \u03bc and \u03c32,\u2009\u2009c, and d are assumed to be known in the first place.Consider the following model for normally distributed data:yi=\u03b8+\u03f5i,\u2003\u03b8 a priori; a suitable restriction on the parameter space \u0398 = R, such as using a truncated normal distribution, is expected. However, it is often the case that the actual observations of  and thun  the degree of uncertainty regarding the interval constraint of  Leonard . Thus, s\u03b8 in . In \u03b8 by using Boltzmann's maximum entropy theorem  = tj, j = 1,\u2026, k, takes the k-parameter exponential family form\u03bb1, \u03bb2,\u2026, \u03bbk can be determined, via the k-constraints, in terms of t1,\u2026, tk. See Leonard and Hsu [Assume now that we can specify the partial information concerning a location parameter the formE[tj(\u03b8)]= and Hsu  for the \u03b8 of our interest be the normal mean in , has different formula according to the degree of uncertainty regarding the interval constraint.Let the location parameter  mean in . Then thR, and \u03b8 \u2208 \u0398 with certainty.\u0398 = \u03b8 that we can specify values for both mean \u03bc and variance \u03c32. Then the N prior specification is the maximum entropy prior for \u03b8 . Thus ta, b], and \u03b8 \u2208 \u0398 with certainty.\u0398 =  = \u03bc and E[(\u03b8 \u2212 \u03bc)2] = \u03c32 on the space \u03b8 \u2208 R by a priori information. Further suppose that we have certain prior information that the parameter \u03b8 is concentrated on the region , that is,\u03c0(\u03b8). The last condition is equivalent to E[I(a \u2264 \u03b8 \u2264 b)] = 1. Therefore, by Bolzmann's maximum entropy theorem, the prior density of \u03b8 for the t1(\u03b8) = \u03b8, t1 = \u03bc, t2(\u03b8) = (\u03b8 \u2212 \u03bc)2, t2 = \u03c32, and t3(\u03b8) = I\u2009\u2009(a \u2264 \u03b8 \u2264 b). Since t3 = 1, the maximum entropy prior, subject to these three restrictions, is thus\u03bb1 = 0, \u03bb2 = \u22121/2\u03c32, \u03bb3 = \u2212ln\u2061{\u03a6(v(b)) \u2212 \u03a6(v(a))}, where v(a) = (a \u2212 \u03bc)/\u03c3 and v(b) = (b \u2212 \u03bc)/\u03c3.On the other hand, when we are completely sure about the priori interval constraint of \u03b8\u2264b)}by , becauseNa,b) distribution in Johnson et al. , and \u03b8 \u2208 \u0398 with (1 \u2212 \u03b1)\u2009\u00d7\u2009100% uncertain.\u0398 = . Then along with the priori moment conditions E[\u03b8] = \u03bc and E[(\u03b8 \u2212 \u03bc)2] = \u03c32, the additional uncertain prior information about the interval constraint can be expressed by\u03b1 (or (1 \u2212 \u03b1) \u00d7 100%) denotes the degree of uncertainty. Thus the priori uncertain interval constraint is equivalent toNow suppose that we have partial priori information that we can specify values for both mean \u03b8, reflecting the uncertain interval constraint in  of X1 and X2 whose joint distribution is a bivariate normal N2,If raint in  is(17)\u03b8t1(\u03b8) = \u03b8, t1 = \u03bc, t2(\u03b8) = (\u03b8 \u2212 \u03bc)2, t2 = \u03c32, t3(\u03b8) = I\u2009\u2009(a \u2264 \u03b8 \u2264 b), and t3 = \u03b1, and setting \u03bb1 = 0 and \u03bb2 = \u22121/2\u03c32, the maximum entropy prior \u03c0(\u03b8) in (Taking  \u03c0(\u03b8) in  reduces \u03c0(\u03b8) in (\u03c03(\u03b8)\u221dexp side of  as follo\u03c0(\u03b8) in (\u03c03(\u03b8)\u221dexpondition , because\u03be* = (1 \u2212 \u03b4)\u22121/2 and Thus the maximum entropy prior density for the \u03c01(\u03b8) = \u03c02(\u03b8) = \u03c03(\u03b8) for a = \u2212\u221e and b = \u221e. This is consistent with our partial priori information that E[\u03b8] = \u03bc and Var\u2061(\u03b8) = \u03c32 for \u03b8 \u2208 R. Some algebra using the moments of the WNa,b) distribution given in Kim . Also note, from the conditional property of N2, that \u03b4 \u2192 1 for finite vales of \u03bc,\u03c3,a, and b with b > a. Applying these two results to \u03c03(\u03b8), we see that it approximates to \u03c02(\u03b8) as \u03b4 \u2192 1. It is straightforward to see from \u03c03(\u03b8) = \u03c01(\u03b8)\u2009\u2009for \u03b4 = 0, because the WNa,b) distribution is equivalent to N for \u03b4 = 0.Note from  that \u03bev*\u03c03(\u03b8)) and Ent(\u03c02(\u03b8)) as a function of \u03b4 \u2208  for three values of \u03c32, two cases of \u0398 = , and \u03bc = 0. In Ent. Since each graph coincides with the results of \u03c01(\u03b8)) > Ent(\u03c03(\u03b8)) > Ent(\u03c02(\u03b8)) for \u03b4 \u2208 . (ii) The entropy of \u03c03(\u03b8) is a monotone decreasing function of \u03b4. (iii) Each entropy of the three priors increases as \u03c32 becomes large. (iv) DiffEnt is closely related with degree of uncertainty  \u00d7 100%) for it is a monotone decreasing function of \u03b4 \u2208  and \u03b1 is a function of \u03b4. (v) DiffEnt is a monotone increasing function of \u03c32 for the case where a value of \u03b4 is given.Each graph of From a relationship between Lemmas \u03c302 = (1 \u2212 \u03b4)\u03c32, \u03c312 = \u03b4\u03c32, and \u03b4 \u2208 . Then the two-stage prior of \u03b8 defined by HSGM in (\u03c03(\u03b8) and the degree of uncertainty regarding the interval constraint, \u03b8 \u2208 , accounted for by the HSGM isLet \u03c302 = (1 \u2212 \u03b4)\u03c32, \u03c312 = \u03b4\u03c32, and \u03b4 \u2208 , the marginal prior distribution of \u03b8 in a,b). Thus the prior density p(\u03b8) in (\u03c03(\u03b8) in (\u03c03(\u03b8) reflects uncertainty about the interval constraint \u03b8 \u2208  by the degree of (1 \u2212 \u03b1) \u00d7 100% with If  p(\u03b8) in  is equal3(\u03b8) in . Now Lem\u03b8 accounted for by HSGM, and it shows that the uncertainty measure is different from that of O'Hagan and Leonald }, with (1 \u2212 \u03b1) \u00d7 100% degree of uncertainty, where \u03bc and \u03c32 are true prior mean and variance of the parameter \u03b8 when the uncertain priori interval condition does not exist.Let us revisit the HSGM with the following two stages of a prior hierarchy of \u03b8 in \u03b8 and \u03c42 proportional to the product of likelihood and the prior distribution,Dn = {y1,\u2026, yn}, IG is the inverse-gamma density with parameters c and d, and \u03c03(\u03b8) is the density . Note that the joint posterior is not simplified in an analytic form of the known density and thus intractable for posterior inference. Instead, we derive each of the conditional posterior distributions of \u03b8 and \u03c42 in an explicit form, which will be useful for posterior inference such as Gibbs sampling .(i) \u03b8 is given byThe full conditional posterior distribution of where\u03c9\u22172 = \u03c32\u03c42/(n\u03c32 + \u03c42).(ii)\u03c42 is the inverse-Gamma distributionThe full conditional posterior distribution of Given the joint posterior distribution , we have\u03b8 given that \u03c42 and Dn is proportional to(i) The unnormalized conditional density of \u03c42 isc + 2/n, d + \u2211i=1n(yi \u2212 \u03b8)2) distribution.(ii) It is straightforward to see from  that the\u03b8, \u03c42) because their full conditional distributions are given in \u03b8, \u03c42). The only difficulty would lie in generating random samples from WN distribution, that is, \u03b8\u2009\u2009\u2223\u2009\u2009 ~ WNa,b) as given in ,\u03b3*(b)) with \u03b3*(a) = (a \u2212 \u03b81*)/\u03c911*, \u03b3*(b) = (b \u2212 \u03b81*)/\u03c911*, \u03b81* = (1 \u2212 \u03b4)\u03bc + \u03b4\u03b8*, and \u03c911\u22172 = \u03b4(1 \u2212 \u03b4)\u03c32 + \u03b42\u03c9\u22172. Johnson et al. [\u03d5(\u00b7) denotes the density of N. It is seen that each of the first terms in  is put on \u03b8 (e.g. [\u03c03(\u03b8). That is each of the second terms in the posterior mean and covariance vanishes when \u03b8 is assumed to be N a priori without any interval constraint. In this sense, the second terms in , in, in\u03b8, \u03c4iven in . The HSGM\u2009\u2009is based on the two stages of a prior hierarch advocated by O'Hagan and Leonard [This paper considered the normal models which include the two-stage prior of the normal mean, referred to as Leonard  and elic Leonard . By expl"}
{"text": "X), where Bin(X) is the collection of all groupoids. Finally, we define a fuzzy-d-subset of a groupoid and investigate some of its properties.The notion of a fuzzy upper bound over a groupoid is introduced and some properties of it are investigated. We also define the notions of an either-or subset of a groupoid and a strong either-or subset of a groupoid and study some of their related properties. In particular, we consider fuzzy upper bounds in Bin( Is\u00e9ki and Tanaka introduced two classes of abstract algebras: BCK-algebras and BCI-algebras , 2.d-algebras which is a useful generalization of BCK-algebras and then investigated several relations between d-algebras and BCK-algebras as well as several other relations between d-algebras and oriented digraphs. Han et al. ) is a BCK-algebra.The notion of the semigroup x\u2217x = 0,(D2)x = 0,0\u2217(D3)x\u2217y = 0 and y\u2217x = 0 imply x = y for all x, y \u2208 X.A algebra   is a nond-algebra X satisfying the following additional axioms:(D4)x\u2217y)\u2217(x\u2217z))\u2217(z\u2217y) = 0,\u2009(((D5)x\u2217(x\u2217y))\u2217y = 0 for all x, y, z \u2208 X.\u2009 has the multiplicationx\u2217x = e; e\u2217x = e; x\u2217y = y\u2217x = e yield (x \u2212 y)(x \u2212 e) = 0, (y \u2212 x)(y \u2212 e) = 0 and x = y, or x = e = y; that is, x = y; that is,  is a d-algebra.Consider the real numbers X, we let Bin(X) the collection of all groupoids , where \u2217 : X \u00d7 X \u2192 X is a map and where \u2217 is written in the usual product form. Given elements  and  of Bin(X), define a product \u201c\u25a1\u201d on these groupoids as follows:x, y \u2208 X. Using the notion, Kim and Neggers . Then \u03bc(0) \u2265 \u03bc(x) for all x \u2208 X; that is, \u03bc is a fub over .Let , we consider the following conditions: for any x, y \u2208 X, A)\u2009\u2009(\u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(0) implies \u03bc(x) = \u03bc(y),B)\u2009\u2009(\u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(0) implies x = y,C)\u2009\u2009(\u03bc(x\u2217y) = \u03bc(y\u2217x) implies \u03bc(x) = \u03bc(y),D)\u2009\u2009(\u03bc(x\u2217y) = \u03bc(y\u2217x) implies x = y. We have a diagram of implications as follows:A)\u21d2(B), (B)\u21d2(D), (A)\u21d2(C), and (C)\u21d2(D).Given a fub X, \u2217, 0) be a groupoid with |X| \u2265 2 and let \u03bc : X \u2192  be a constant map; that is, \u03bc(x) = \u03b1 for some \u03b1 \u2208  for all x \u2208 X. Then (A) holds as does (C). On the other hand, x \u2260 y implies (B) fails as does (D). Thus we have countered examples to (A)\u21d2(B) and (C)\u21d2(D).(a) Let  having the condition (B). But \u03bc does not have the condition (D), since \u03bc(1\u22172) = \u03bc(2) = \u03bc(2\u22171) and 1 \u2260 2. Hence (B)\u21d2(D) is false.(b) Let X = 2A, where A = {1,2, 3,4}. Define a binary operation \u201c\u2217\u201d on X by x\u2217y : = {t | t \u2208 x, t \u2209 y}, \u2009\u2200x, y \u2208 X. Thus, if x\u2286y, then x\u2217y = \u2205. Let \u03bc(x): = 1/(1 + |x|)\u2009\u2009, where |x| is the cardinality of x. Assume \u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(\u2205). Since \u03bc(\u2205) = 1, we have |x\u2217y| = 0 = |y\u2217x| and so x\u2217y = \u2205 = y\u2217x. Therefore x\u2286y, y\u2286x. Hence x = y. On the other hand, if x = {1,2, 3}, y = {3,4, 5}, then x\u2217y = {1,2}, y\u2217x = {4,5}, and \u03bc(x\u2217y) = 1/3 = \u03bc(y\u2217x), but x \u2260 y. This shows that (B) is true, but (D) is false.(c) Let X, \u2217, 0) be a left-zero-semigroup; that is, for any x, y \u2208 X, x\u2217y = x, and let \u03bc : X \u2192  be a fub over . Assume that \u03bc(x\u2217y) = \u03bc(y\u2217x) for any x, y \u2208 X. Since  is a left-zero-semigroup, we have \u03bc(x) = \u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(y). Hence condition (C) holds for any fub \u03bc over .(d) Let  = , where R is the set of all real numbers and \u201c\u2212\u201d is the usual subtraction on R. Also, if \u03bc(x) = ex2\u2212, then \u03bc(0) = 1 \u2265 \u03bc(x) for all x \u2208 R; that is, \u03bc is a fub over . Note that \u03bc(x \u2212 y) = \u03bc(y \u2212 x) = \u03bc(0) implies x \u2212 y = y \u2212 x = 0 and x = y, so that \u03bc(x) = \u03bc(y) as well; that is, the conditions (B) and (A) hold. Now if we let y : = \u22121 and x : = 2 + y, then \u03bc(x \u2212 y) = \u03bc(2 + y \u2212 y) = \u03bc(2) = e\u22124 and \u03bc(y \u2212 x) = \u03bc(y \u2212 2 \u2212 y) = \u03bc(\u22122) = e\u22124, so that \u03bc(x \u2212 y) = \u03bc(y \u2212 x). Since ey+1)\u22124 = \u03bc(2 + y) = ey)2\u2212(2+ = ey2\u2212ey+1)\u22124( < ey2\u2212 = \u03bc(y). Thus (C) fails to hold.(e) Let (A), we give two conditions (A)* and (A)\u03b1*, 0 \u2264 \u03b1 \u2264 1 as follows:A)*\u2009\u2009(\u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(0) implies \u03bc(x) = \u03bc(y) = \u03bc(0),A)\u03b1*\u2009\u2009(\u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(0) implies \u03bc(x) = \u03bc(y) \u2265 \u03b1.As a refinement of the condition  = \u03b1 for all x \u2208 X, then for any groupoid \u2009\u2009\u03bc satisfies the condition (A)*. Since (A)*\u21d2(A), \u03bc satisfies the condition (A).(a) If X, \u2217, 0): = , where R is the set of all real numbers and \u201c\u2212\u201d is the usual subtraction on R, and let \u03bc(x): = ex2\u2212. Then \u03bc(0) = 1 \u2265 \u03bc(x) > 0 for any x \u2208 R, and \u03bc(x) = 1 if and only if x = 0. Assume that \u03bc(x \u2212 y) = \u03bc(y \u2212 x) = \u03bc(0). Then ex\u2212y)2\u2212( = ey\u2212x)2\u2212( = 1 and so x \u2212 y = 0. Hence x = y. Therefore \u03bc(x) = \u03bc(y); that is, (A) holds. Let x = y \u2260 0 imply \u03bc(y) = \u03bc(x) = ex2\u2212 < 1 = \u03bc(0). Thus (A)* does not hold. Since lim\u2061x\u2192\u221e\u03bc(x) = 0, given \u03b1 > 0, there is an x such that \u03bc(x) < \u03b1; that is, (A)\u03b1* does not hold for \u03b1 > 0.(b) Let , let \u03bc\u03b2(x): = (ex2\u2212 + \u03b2)/(1 + \u03b2), where \u03b2 \u2265 0. Then \u03bc\u03b2(0) = 1 \u2265 \u03bc\u03b2(x) for all x \u2208 R and lim\u2061x\u2192\u221e\u03bc\u03b2(x) = \u03b2/(1 + \u03b2) \u2264 \u03bc\u03b2(x) for all x \u2208 X. If we take \u03b1 : = \u03b2/(1 + \u03b2), then \u03bc\u03b2(x) \u2265 \u03b1 for all x \u2208 X. If we assume \u03bc\u03b2(x \u2212 y) = \u03bc\u03b2(y \u2212 x) = \u03bc\u03b2(0), then (x \u2212 y)2 = (y \u2212 x)2 = 0 and hence \u03bc\u03b2(x) = \u03bc\u03b2(y); that is, the condition (A)\u03b1* holds. If \u03b11 < \u03b12, then (A)\u03b12*\u21d2(A)\u03b11* holds clearly. If we set \u03b21 : = \u03b11/(1 \u2212 \u03b11) and \u03b22 : = \u03b12/(1 \u2212 \u03b12), then \u03b21 < \u03b22 and lim\u2061x\u2192\u221e\u03bc\u03b21(x) = lim\u2061x\u2192\u221e((ex2\u2212 + \u03b21)/(1 + \u03b21)) = (\u03b21)/(1 + \u03b21) = \u03b1 < \u03b12. Take an x \u2208 R so that \u03b11 < \u03bc\u03b21(x) < \u03b12. Then ex2\u2212 + \u03b21 < \u03b12(1 + \u03b21). It follows that ex2\u2212 < (\u03b22/(1 + \u03b22))(1 + \u03b21) \u2212 \u03b21 < (\u03b22(1 + \u03b21) \u2212 \u03b21(1 + \u03b22))/(1 + \u03b22) = (\u03b22 \u2212 \u03b21)(1 + \u03b21)/(1 + \u03b22); that is, x2 > ln\u2061[(1 + \u03b22)/(\u03b22 \u2212 \u03b21)(1 + \u03b21)]. Hence, if A)\u03b11*\u21d2(A)\u03b12* does not hold.(c) For the groupoid  and that they both satisfy the condition (A). If \u03bc : = \u03bb\u03bc1 + (1 \u2212 \u03bb)\u03bc2, where 0 \u2264 \u03bb \u2264 1, then \u03bc is also a fub over  having the condition (A).Suppose that \u03bb < 1. For any x \u2208 X, we have \u03bc(0) = \u03bb\u03bc1(0)+(1 \u2212 \u03bb)\u03bc2(0) \u2265 \u03bb\u03bc1(x)+(1 \u2212 \u03bb)\u03bc2(x) = \u03bc(x). Hence \u03bc is a fub over .Let 0 < \u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(0), but \u03bc1(x\u2217y) < \u03bc1(0) or \u03bc2(x\u2217y) < \u03bc2(0). If \u03bc1(x\u2217y) < \u03bc1(0), then \u03bb\u03bc1(x\u2217y) < \u03bb\u03bc1(0) and thus (1 \u2212 \u03bb)\u03bc2(x\u2217y) = \u03bc(x\u2217y) \u2212 \u03bb\u03bc1(x\u2217y) > \u03bc(x\u2217y) \u2212 \u03bb\u03bc1(0) = \u2009\u03bc(0) \u2212 \u03bb\u03bc1(0) = (1 \u2212 \u03bb)\u03bc2(0). This shows that \u03bc2(x\u2217y) > \u03bc2(0), which is a contradiction. Similarly, \u03bc2(x\u2217y) > \u03bc2(0) implies \u03bc1(x\u2217y) > \u03bc1(0), which is also a contradiction. Hence we obtain \u03bc1(x\u2217y) = \u03bc1(0), \u03bc2(x\u2217y) = \u03bc2(0). Similarly, we obtain \u03bc1(y\u2217x) = \u03bc1(0), \u03bc2(y\u2217x) = \u03bc2(0). Since \u03bci has the condition (A), we have \u03bci(x) = \u03bci(y)\u2009\u2009. Hence \u03bc(x) = \u03bb\u03bc1(x)+(1 \u2212 \u03bb)\u03bc2(x) = \u03bb\u03bc1(y)+(1 \u2212 \u03bb)\u03bc2(y) = \u03bc(y). Thus \u03bc has the condition (A).Assume that \u03bc1 and \u03bc2 are fubs over a groupoid  and that \u03bc2 has the condition (A)* with \u03bc1(0) < \u03bc2(0). If \u03bc : = \u03bc1\u2228\u03bc2, where (\u03bc1\u2228\u03bc2)(x): = max\u2061{\u03bc1(x), \u03bc2(x)}, then \u03bc is a fub over  having the condition (A)*.Suppose that \u03bc = \u03bc1\u2228\u03bc2, then \u03bc(0) = \u03bc1(0)\u2228\u03bc2(0), \u03bc(x) = \u03bc1(x)\u2228\u03bc2(x). If \u03bc1(x) \u2265 \u03bc2(x), then \u03bc(x) = \u03bc1(x) and \u03bc(0) \u2265 \u03bc1(0) \u2265 \u03bc1(x) = \u03bc(x) implies \u03bc(0) \u2265 \u03bc(x) for any x \u2208 X. Similarly, if \u03bc2(x) \u2265 \u03bc1(x), then \u03bc(0) \u2265 \u03bc(x) for any x \u2208 X. Hence \u03bc is a fub over If \u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(0). Then \u03bc1(x\u2217y)\u2228\u03bc2(x\u2217y) = \u03bc(x\u2217y) = \u03bc(0) = \u2009\u03bc1(0)\u2228\u03bc2(0) = \u03bc2(0). If \u03bc2(x\u2217y) < \u03bc2(0), then \u03bc1(x\u2217y) = \u03bc2(0) and \u03bc1(0) \u2265 \u03bc1(x\u2217y) = \u03bc2(0), which is a contradiction. Hence \u03bc2(x\u2217y) \u2265 \u03bc2(0); that is, \u03bc2(x\u2217y) = \u03bc2(0). Similarly, \u03bc(y\u2217x) = \u03bc(0) implies \u03bc2(0) = \u03bc2(y\u2217x). Hence \u03bc2(x\u2217y) = \u03bc2(y\u2217x) = \u03bc2(0). Since \u03bc2 has the condition (A)*, we have \u03bc2(x) = \u03bc2(y) = \u03bc2(0) and so \u03bc(x) = \u03bc(0). Similarly, we obtain \u03bc(y) = \u03bc(0). Thus \u03bc has the condition (A)*.Assume that A and B are subsets of X such that A\u2216B, A\u2229B, and B\u2216A are nonempty. Let 0 \u2208 A\u2229B and let \u03bc1 : = \u03c7A, \u03bc2 : = \u03c7B, where x \u2208 A\u2216B, y \u2208 B\u2216A, and  is the left-zero-semigroup. It follows that \u03bc(x\u2217y) = \u03bc1(x\u2217y)\u2228\u03bc2(x\u2217y) = \u03bc(x) = \u03bc1(x)\u2228\u03bc2(x) = 1\u22280 = 1 and that \u03bc(y\u2217x) = 1 = \u03bc(y), \u03bc(0) = 1 = \u03bc1(1) = \u03bc2(1). Hence \u03bc(x) = \u03bc(y) = \u03bc(0); that is, the condition (A)* holds.(a) Suppose R, \u2212, 0) be the set of all real numbers equipped with subtraction and the zero element 0. Let \u03bc1(x): = 1 + x \u2212 \u2308x\u2309 and \u03bc2(x): = 1 + \u230ax\u230b \u2212 x. If x \u2208 Z, then x = \u2308x\u2309 = \u230ax\u230b and hence \u03bc1(x) = \u03bc2(x) = 1 = \u03bc(0). If x = n + \u03b1 where n \u2208 Z, 0 < \u03b1 < 1/2, then \u03bc1(x) = \u03b1 and \u03bc2(x) = 1 \u2212 \u03b1. Hence \u03bc(x) = \u03bc1(x)\u2228\u03bc2(x) = 1 \u2212 \u03b1. If x = n + \u03b2 where n \u2208 Z, 1/2 < \u03b2 < 1, then \u03bc1(x) = \u03b2 and \u03bc2(x) = 1 \u2212 \u03b2. Hence \u03bc(x) = \u03bc1(x)\u2228\u03bc2(x) = \u03b2. Assume \u03bc(x \u2212 y) = \u03bc(y \u2212 x) = \u03bc(0). Since \u03bc(0) = 1, we have x \u2212 y, y \u2212 x \u2208 Z, say y \u2212 x = n \u2208 Z. If we let x : = m + \u03b3 where m \u2208 Z, 0 < \u03b3 < 1, then y = x + n = (n + m) + \u03b3. It follows that \u03bc(x) = \u03bc(y) has the value either \u03b3 or 1 \u2212 \u03b3. Hence \u03bc satisfies the condition (A), but not the condition (A)*.(b) Let (\u03bc1(0) = \u03bc2(0) = 1. This shows that the condition \u03bc1(0) < \u03bc2(0) is a very necessary condition in Note that, in X, \u2217) be a groupoid and let \u2205 \u2260 U\u2286X. Then U is said to be either-or if x\u2217y \u2208 U, y\u2217x \u2208 U implies either {x, y}\u2286U or {x, y}\u2286Uc for any x, y \u2208 X.Let  and that \u03bc satisfies the condition (A). Let Ker\u2061\u03bc : = {x \u2208 X | \u03bc(x) = \u03bc(0)}. Then Ker\u2061\u03bc is an either-or subset of X.Suppose that x\u2217y, y\u2217x \u2208 Ker\u2061\u03bc for any x, y \u2208 X. Then \u03bc(x\u2217y) = \u03bc(0) = \u03bc(y\u2217x). Since \u03bc satisfies the condition (A), we have \u03bc(x) = \u03bc(y). Thus, also, if x \u2208 Ker\u2061\u03bc, then y \u2208 Ker\u2061\u03bc, and if x \u2209 Ker\u2061\u03bc, then y \u2209 Ker\u2061\u03bc. Therefore Ker\u2061\u03bc is an either-or subset of X.Let \u03bc in Note that 0 \u2208 Ker\u2061U is said to be with alternative if there are elements {x, y}\u2286U, {u, v}\u2286Uc such that {x\u2217y, y\u2217x}\u2286U, {u\u2217v, v\u2217u}\u2286U.An either-or subset Z, +) is the groupoid of all integers with respect to the usual addition and if U : = 2Z is the set of all even integers, then U is an either-or subset of X. In fact, let x + y = y + x = 2u \u2208 U and x = 2v, where u, v \u2208 Z. Then y = 2u \u2212 2v = 2(u \u2212 v) \u2208 U. Thus, U is an either-or subset of . On the other hand, Uc consists of all odd integers. Now if x + y = y + x is odd for any x, y \u2208 Z and if x is odd, then y = (x + y) \u2212 x is even. If x is even, then y is odd. Hence Uc fails to be an either-or subset of . The subset U = 2Z of  is an either-or subset with alternative. Both Z and Zc = \u2205 are either-or subsets, but without alternative.If . Suppose that U is an either-or subset of the groupoid  and that \u03bc = \u03c7U is the characteristic function of U; that is, \u03bc(x) = 1 if x \u2208 U and \u03bc(x) = 0 otherwise. Then \u03bc satisfies the condition (A).Let 0 \u2208 U be a \u201cselected\u201d element; that is, \u03bc(0) = 1. Then \u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(0) implies x\u2217y, y\u2217x \u2208 U. Since U is an either-or subset of the groupoid , we have either {x, y}\u2286U or {x, y}\u2286Uc; that is, \u03bc(x) = \u03bc(y) = 1 or \u03bc(x) = \u03bc(y) = 0. In any case \u03bc(x) = \u03bc(y) and the condition (A) holds.Let 0 \u2208 X, \u2217) be a groupoid and let \u2205 \u2260 S\u2286X. Let EO(S) be the collection of all either-or subsets of the groupoid , sometimes denoted as EO), which contain the subset S of X. Then, since  is an either-or subset of , it follows that EOR(S): = \u2229i{Ui | Ui \u2208 EO)}\u2287S.Let  be a groupoid and let \u2205 \u2260 S\u2286U. Then EOR) is an either-or subset of  containing S.Let ). If {x, y}\u2286Uc for some either-or subset U of  which contains S, then EOR)\u2286U implies Uc\u2286EOR)c and {x, y}\u2286EOR)c. If this is not the case, then {x, y}\u2009\u2288\u2009Uc for any either-or subset U of  containing S, whence {x, y}\u2286U. Thus {x, y}\u2286EOR). This shows that EOR) is an either-or subset of .Let We have the following corollaries.X, \u2217) be a groupoid. If \u2205 \u2260 S\u2286T, then EOR)\u2286EOR).Let  be a groupoid. Then we haveLet  be the left-zero-semigroup; that is, x\u2217y = x for all x, y \u2208 X, and let 0 be a fixed element for which \u03bc(0) \u2265 \u03bc(x) for all x \u2208 X, where \u03bc : X \u2192  is a fuzzy subset of X. Then \u03bc(x\u2217y) = \u03bc(x) = \u03bc(y\u2217x) = \u03bc(y) = \u03bc(0) implies \u03bc(x) = \u03bc(y) = \u03bc(0) and \u03bc satisfies the condition (A)*. If U is any nonempty subset of X, then x\u2217y, y\u2217x \u2208 U implies {x, y}\u2286U, so that U is an either-or subset without alternative. Hence EOR) = U for any \u2205 \u2260 U\u2286X whatever. Of course, if  is the right-zero-semigroup, that is, x\u2217y = y for any x, y \u2208 X, then the same conclusions hold.(a) Let  be the group of all integers with respect to addition. We claim that nZ = {nx | x \u2208 Z} is an either-or subset of . Given n \u2208 Z, if n = 0, and if x + y, y + x \u2208 {0} and x = 0, then y = 0 and hence {x, y} = {0}\u2286{0}. On the other hand, if x \u2260 0, then y = \u2212x implies {x, y} = {x, \u2212x}\u2286{0}c = Z \u2212 {0}. Hence {0} = 0Z is an either-or subset of the groupoid . Let n \u2260 0. If x + y = y + x = nu, and if x = nv, then y = (x + y) \u2212 x = nu \u2212 nv = n(u \u2212 v); that is, y \u2208 nZ as well. Thus {x, y}\u2286nZ. Assume that x \u2209 nZ; that is, x = nu\u2032 + \u03b1 for some u\u2032 \u2208 Z and \u03b1 \u2208 Z such that 0 < \u03b1 < n. Then y = nu \u2212 x = n(u \u2212 u\u2032) \u2212 \u03b1 \u2209 nZ. Hence {x, y}\u2286(nZ)c.(b) Let ) = nZ for all n \u2208 Z. Since {0} is the smallest either-or subset of , EOR) = {0} = 0Z. Let n \u2260 0 in Z. Then 0 \u2208 EOR). In fact, if Ui is an either-or subset of  containing n, then either {n, 0}\u2286Ui or {n, 0}\u2286Uic, since n + 0 = 0 + n = n \u2208 Ui. It follows that 0 \u2208 Ui for all either-or subset Ui of ; that is, 0 \u2208 EOR).We claim that EOR). If Ui is an either-or subset of , either {n, \u2212n}\u2286Ui or {n, \u2212n}\u2286Uic for any either-or subset Ui of , since n + (\u2212n) = (\u2212n) + n = 0 \u2208 Ui. Since n \u2208 Ui, we have \u2212n \u2208 Ui for all either-or subset Ui. This proves that \u2212n \u2208 EOR). Hence {\u2212n, 0, n}\u2286EOR).We claim that \u2212B is an either-or subset of  containing n. Then it is easy to see that 0, \u2212n \u2208 B. If we let x : = n, y : = \u22122n, then x + y = y + x = \u2212n \u2208 B. Since x = n \u2208 B and B is an either-or subset of , we obtain \u22122n = y \u2208 B. Similarly, if we let x = \u2212n and y : = 2n, then we obtain 2n = y \u2208 B. Similarly, we obtain \u00b13n, \u00b14n,\u2026, \u2208B. Thus nZ\u2286B; that is, B is an either-or subset of  containing nZ. It follows that nZ = \u2229{Ui | Ui : \u2009either-or\u2009\u2009subset\u2009\u2009containing\u2009\u2009n} = EOR).Assume that N : = {1,2, 3,\u2026} and let a \u2208 N. If x + y = a, then x < a, y < a and thus {a} is an either-or subset of N without alternative, since {x, y}\u2286{a}c. If Uak : = {a, 2a,\u2026, ka} for any a \u2208 N, let x + y = y + x \u2208 Uak. Then there exists h \u2208 N such that x + y = ha, h \u2264 k. If x = ja, j < h, then y = ha \u2212 x = (h \u2212 j)a \u2208 Uak. Hence {x, y}\u2286Uak. If x \u2209 Uak, then y \u2209 Uak and {x, y}\u2286(Uak)c, so that Uak is an either-or subset of N with alternative if k \u2265 2.(c) Let X, \u2217) be a groupoid and let U\u2286X such that x\u2217y, y\u2217x \u2208 U implies x = y. Then U is an either-or subset of .Let .We call such a subset X, \u2217, f) be a leftoid, and let U = {a} be a singleton. If x\u2217y = f(x) \u2208 U and y\u2217x = f(y) \u2208 U, then f(y) = f(x) = a. Hence, if |f\u22121(a)| = 1, then x = y, and U is a strong either-or subset of .(a) Let  Let X, \u2217, 0) be a groupoid and let U be a strong either-or subset of X with 0 \u2208 U. If \u03bc : = \u03c7U is the characteristic function of U, then \u03bc is a fub over . Furthermore, \u03bc satisfies the condition (B) and Ker\u2061\u03bc = U.Let  = 1 \u2265 \u03bc(x) for all x \u2208 X and \u03bc is a fub over . Let \u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(0) for any x, y \u2208 X. Since \u03bc(0) = 1, we have x\u2217y, y\u2217x \u2208 U. Hence x = y, since U is a strong either-or subset of X. Therefore \u03bc satisfies the condition (B) and in that case Ker\u2061\u03bc = \u03bc\u22121(\u03bc(0)) = U.Since 0 \u2208 X, \u2217, e) be a group all of whose elements have finite order and let \u03bc : X \u2192  be a fuzzy subgroup. Then \u03bc has the condition (A) and Ker\u2061\u03bc is an either-or subset of X.Let (\u03bc be a fuzzy subgroup of X. Then \u03bc(x\u2217y) \u2265 min\u2061{\u03bc(x), \u03bc(y)} for any x, y \u2208 X. Thus \u03bc(x2) = \u03bc(x\u2217x) \u2265 \u03bc(x) and \u03bc(xn) \u2265 min\u2061{\u03bc(xn\u22121), \u03bc(x)}. By induction \u03bc(xn\u22121) \u2265 \u03bc(x) implies \u03bc(xn) \u2265 \u03bc(x), so that \u03bc(e) = \u03bc(xn) \u2265 \u03bc(x) implies \u03bc(e) \u2265 \u03bc(x) for all x \u2208 X; that is, \u03bc is a fub over .Let \u03bc = \u03bc\u22121(\u03bc(e)) = U, then \u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(e) implies \u03bc((x\u2217y)\u2217y\u22121) = \u03bc(x) \u2265 min\u2061{\u03bc(x\u2217y), \u03bc(y\u22121)) and hence \u03bc(x) \u2265 \u03bc(y\u22121) \u2265 \u03bc(y), so that \u03bc(x) = \u03bc(y); the condition (A) holds for \u03bc, and Ker\u2061\u03bc is an either-or subset of .If Ker\u2061\u03bc be a fub over  with the condition (A) and a fub over  with the condition (A)*. If : = \u25a1, then \u03bc is a fub over  with the condition (A)*.Let \u03bc(x\u25a1y) = \u03bc(y\u25a1x) = \u03bc(0) for any x, y \u2208 X. Then \u03bc((x\u2217y)\u2022(y\u2217x)) = \u03bc((y\u2217x)\u2022(x\u2217y)) = \u03bc(0) implies \u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(0), so that \u03bc(x) = \u03bc(y) = \u03bc(0). Hence \u03bc satisfies the condition (A)* over .Assume that \u03bc be a fub over both  and  with the condition (A)*. If : = \u25a1, then \u03bc is an lub over  with the condition (A)*.Let Straightforward.X be a nonempty set. If \u2329\u03bc; (A)\u232a and \u2329\u03bc; (A)*\u232a are the collections of groupoids  such that \u03bc satisfies the condition (A) *), then \u2329\u03bc; (A)*\u232a\u2286\u2329\u03bc; (A)\u232a and\u03bc; (A)*\u232a, \u25a1) is a subsemigroup of (Bin(X), \u25a1).Let \u03bc be a fub over  with the condition (B), and let \u03bc be a fub over  with the condition (A)*. If : = \u25a1, then \u03bc is a fub over  with the condition (B).Let \u03bc(x\u25a1y) = \u03bc(y\u25a1x) = \u03bc(0) for any x, y \u2208 X. Then \u03bc((x\u2217y)\u2022(y\u2217x)) = \u03bc((y\u2217x)\u2022(x\u2217y)) = \u03bc(0). Since \u03bc satisfies the condition (A)* over , we have \u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(0). Since \u03bc has the condition (B) over , we obtain x = y, proving the theorem.Assume that X, \u2217, 0), the following are interesting properties in fuzzy subgroupoids \u03bc : X \u2192 :\u03bc(x\u2217x) = \u03bc(0) for all x \u2208 X,\u03bc(0\u2217x) = \u03bc(0) for all x \u2208 X,\u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(0), then \u03bc(x) = \u03bc(y) for all x, y \u2208 X.if Given a groupoid  is called a fuzzy-d-subset of the groupoid  if it satisfies conditions (1), (2), and (3).A fub\u2009\u2009X : = R be the set of all real numbers and let \u201c+\u201d be the usual addition on R. Then every fuzzy-d-subset \u03bc of  is constant. In fact, for all x \u2208 X, we have \u03bc(0) = \u03bc(x + x) = \u03bc(2x). If we let y : = 2x, then \u03bc(y) = \u03bc(0) for all y \u2208 X; that is, \u03bc is a constant function on X.(a) Let X, \u2217, 0) be a group with identity 0. Let \u03bc be a fuzzy subset of X with (2); then \u03bc(x) = \u03bc(0\u2217x) = \u03bc(0) for all x \u2208 X, whence \u03bc is a constant function. Therefore \u03bc is a fuzzy-d-subset of .(b) Let  be a d-algebra. If \u03bc = \u03c7{0} is the characteristic function of {0}; that is, \u03bc(0) = 1 and \u03bc(x) = 0 otherwise, then \u03bc is a fuzzy-d-subset of X and Ker\u2061\u03bc = {0}.Let  = \u03bc(0), \u03bc(0\u2217x) = \u03bc(0). If \u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(0), then x\u2217y = y\u2217x = 0. Since  is a d-algebra, we obtain x = y. Hence \u03bc(x) = \u03bc(y). This shows that \u03bc is a fuzzy-d-subset of  and Ker\u2061\u03bc = \u03bc\u22121(\u03bc(0)) = \u03bc\u22121(1) = {0}.Let S be an either-or subset of a groupoid  such that \u0394 : = {x\u2217x | x \u2208 X} and \u03940: = {0\u2217x | x \u2208 X} are subsets of S. Let \u03bc : = \u03c7S be the characteristic function of S and 0 \u2208 S. Then \u03bc is a fuzzy-d-subset of .Let S, we have \u03bc(0) = 1 \u2265 \u03bc(x) for all x \u2208 X. Since \u0394, \u03940\u2286S, we obtain \u03bc(x\u2217x) = \u03bc(0), \u03bc(0\u2217x) = \u03bc(0). Assume that \u03bc(x\u2217y) = \u03bc(y\u2217x) = \u03bc(0). Then x\u2217y, y\u2217x \u2208 S. Since S is an either-or set over , either {x, y}\u2286S or {x, y}\u2286Sc. It follows that either \u03bc(x) = \u03bc(y) = 1 or \u03bc(x) = \u03bc(y) = 0. Hence \u03bc is a fuzzy-d-subset of .Since 0 \u2208"}
{"text": "It has been shown that a normal S-iterative method converges to the solution of a mixed type Volterra-Fredholm functional nonlinear integral equation. Furthermore, a data dependence result for the solution of this integral equation has been proven. The scientists working in almost every field of science are faced with nonlinear problems, because nature itself is intrinsically nonlinear. Such problems can be modelled as nonlinear mathematical equations. Solving nonlinear equations is, of course, considered to be a matter of the uttermost importance in mathematics and its manifold applications. There are numerous systematic approaches which are classified as direct and iterative methods to solve such equations in the existing literature. Indeed, by using direct methods, finding solutions to a complicated nonlinear equation can be an almost insurmountable challenge. In this context, iterative methods have become very important mathematical tools for finding solutions to a nonlinear equation. For a comprehensive review and references to the extensive literature on the iterative methods, the interested reader may refer to some recent works \u20138.Recently, Sahu  and KhanX be an ambient space and let T be a self-map of X. A normal S-iterative method is defined by\u03ben}n=0\u221e is a real sequence in  satisfying certain control condition(s).Let It has been shown both analytically and numerically in , 10 thatThis iterative method, due to its simplicity and fastness, has attracted the attention of many researchers and has been examined in various aspects; see \u201320.a1; b1] \u00d7 \u22ef\u00d7 is an interval in Rm, K, H :  \u00d7 \u22ef\u00d7 \u00d7  \u00d7 \u22ef\u00d7 \u00d7 R \u2192 R continuous functions, and F :  \u00d7 \u22ef\u00d7 \u00d7 R3 \u2192 R.In this paper, inspired by the performance and achievements of normal S-iterative method , we willAlso we give a data dependence result for the solution of integral equation  with theWe end this section with some known results which will be useful in proving our main results.A1);A2);A3)(\u03b1 + (\u03b2LK + \u03b3LH)(b1 \u2212 a1) \u22ef (bm \u2212 am) < 1.We suppose that the following conditions are satisfied:x* \u2208 C.Then  has a un\u03b2n}n=0\u221e be a nonnegative sequence for which one assumes there exists n0 \u2208 N, such that for all n \u2265 n0 one has satisfied the inequality\u03bcn \u2208 , for all n \u2208 N, \u2211n=0\u221e\u03bcn = \u221e, and \u03b3n \u2265 0, for all n \u2208 N. Then the following inequality holds:Let {A1)\u2013(A5) in \u03ben}n=0\u221e \u2282  be a real sequence satisfying \u2211n=0\u221e\u03ben = \u221e. Then  and normal S-iterative method , where ||\u00b7||C is Chebyshev's norm. Let {xn}n=0\u221e be an iterative sequence generated by normal S-iterative method \u2013(A4), we have thatA5),\u03bek \u2208  for all k \u2208 N, assumption (A5) yieldsex \u2265 1 \u2212 x for all x \u2208 , we can rewrite , an, an2), aave that||xn+1\u2212x\u2217yn\u2212x\u2217||,||yn\u2212x\u2217||, we get||xn+1\u2212x|We now prove the data dependence of the solution for integral equation  with theB be as in the proof of T, K, H, Let F, K, and H be defined as in xn}n=0\u221e be an iterative sequence defined by normal S-iterative method  1/2 \u2264 \u03ben, for all n \u2208 N, and (ii) \u2211n=0\u221e\u03ben = \u221e. One supposes further that (iii) there exist nonnegative constants \u03b51 and \u03b52 such that u \u2208 R and for all t, s \u2208  \u00d7 \u22ef\u00d7.Let e method  associatx* and If quations  and 16)x* and x~A1)\u2013(A4) and (iii), we obtainA5) and 1/2 \u2264 \u03ben in the resulting inequality, we getUsing , 15), , , 17), a, a17), ae obtain||xn+1\u2212x~(bi\u2212ai),||yn\u2212y~n|, we get||xn+1\u2212x~"}
{"text": "The notions of int-soft semigroups and int-soft left  ideals are introduced, and several properties are investigated. Using these notions and the notion of inclusive set, characterizations of subsemigroups and left  ideals are considered. Using the notion of int-soft products, characterizations of int-soft semigroups and int-soft left  ideals are discussed. We prove that the soft intersection of int-soft left  ideals  is also int-soft left  ideals . The concept of int-soft quasi-ideals is also introduced, and characterization of a regular semigroup is discussed. Molodtsov  introduc\u015f and \u00c7a\u011fman [\u00c7a\u011fman and Engino\u011flu  introducd \u00c7a\u011fman  studied d \u00c7a\u011fman \u201326). Recd \u00c7a\u011fman  investigS be a semigroup. Let A and B be subsets of S. Then the multiplication of A and B is defined as follows:Let S is said to be regular if for every x \u2208 S there exists a \u2208 S such that xax = x.A semigroup A of S is calledS if AA\u2286A, that is, ab \u2208 A for all a, b \u2208 A,a subsemigroup of S if SA\u2286A , that is, xa \u2208 A  for all x \u2208 S and a \u2208 A,a left  ideal of S if it is both a left and a right ideal of S,a two-sided ideal of S if AS\u2229SA\u2286A.a quasi-ideal of A nonempty subset A soft set theory was introduced by Molodtsov , and \u00c7a\u011fU be an initial universe set and let E be a set of parameters. Let P(U) denote the power set of U and A, B, C,\u2026\u2286E.In what follows, let \u03b1, A) over U is defined to be the set of ordered pairs\u03b1 : E \u2192 P(U) such that \u03b1(x) = \u2205 if x \u2209 A.A soft set . The subscript A in the notation \u03b1 indicates that \u03b1 is the approximate function of .The function E has a binary operation \u21aa. For any nonempty subset A of E, a soft set  over U is said to be intersectional over U if it satisfiesAssume that \u03b1, A) over U and a subset \u03b3 of U, the \u03b3-inclusive set of , denoted by iA, is defined to be the setFor a soft set  over U is called an intersectional soft semigroup  over U if it satisfiesA soft set  over U is called an intersectional soft left  ideal  ideal) over U if it satisfiesA soft set  over U is both an int-soft left ideal and an int-soft right ideal over U, we say that  is an intersectional soft two-sided ideal  over U.If a soft set  be a soft set over U defined as follows:\u03b31, \u03b32, \u03b33, and \u03b34 are subsets of U with \u03b31\u228b\u03b32\u228b\u03b33\u228b\u03b34. Then  is an int-soft two-sided ideal over U.Let U is an int-soft semigroup over U. But the converse is not true as seen in the following example.Obviously, every int-soft left  ideal over S = {0,1, 2,3, 4,5} be a semigroup with the following Cayley table:(1)\u03b1, S) be a soft set over U defined as follows:\u03b31, \u03b32, \u03b33, \u03b34, and \u03b35 are subsets of U with \u03b31\u228b\u03b32\u228b\u03b33\u228b\u03b34\u228b\u03b35. Then  is an int-soft semigroup over U. But it is not an int-soft left ideal over U since \u03b1(3 \u00b7 5) = \u03b1(3) = \u03b34\u2289\u03b33 = \u03b1(5).Let ((2)\u03b2, S) be a soft set over U defined as follows:\u03b31, \u03b32, \u03b33, and \u03b34 are subsets of U with \u03b31\u228b\u03b32\u228b\u03b33\u228b\u03b34. Then  is an int-soft semigroup over U. But it is not an int-soft right ideal over U since \u03b2(3 \u00b7 4) = \u03b2(2) = \u03b33\u2289\u03b32 = \u03b2(3).Let  is a soft set over U, which is called the characteristic soft set. The soft set  is called the identity soft set over U.For a nonempty subset A of\u2009\u2009S, the following are equivalent.A is a left  ideal of\u2009\u2009S.\u03c7A, S) over\u2009\u2009U is an int-soft left  ideal over\u2009\u2009U.The characteristic soft set \u2287\u2205 = \u03c7A(y). If y \u2208 A, then xy \u2208 A since A is a left ideal of S. Hence \u03c7A(xy) = U = \u03c7A(y). Therefore  is an int-soft left ideal over U. Similarly,  is an int-soft right ideal over U when A is a right ideal of S.Assume that \u03c7A, S) is an int-soft left ideal over U. Let x \u2208 S and y \u2208 A. Then \u03c7A(y) = U, and so \u03c7A(xy)\u2287\u03c7A(y) = U; that is, \u03c7A(xy) = U. Thus xy \u2208 A and therefore A is a left ideal of S. Similarly, we can show that if  is an int-soft right ideal over U, then A is a right ideal of S.Conversely suppose that  over\u2009\u2009U\u2009\u2009is an int-soft two-sided ideal over\u2009\u2009U.The characteristic soft set  over\u2009\u2009U is an int-soft semigroup over U if and only if the nonempty \u03b3-inclusive set of  is a subsemigroup of\u2009\u2009S for all \u03b3\u2286U.A soft set  over U is an int-soft semigroup over U. Let \u03b3\u2286U be such that iS \u2260 \u2205. Let x, y \u2208 iS. Then \u03b1(x)\u2287\u03b3 and \u03b1(y)\u2287\u03b3. It follows from . Thus iS is a subsemigroup of S.Assume that  is a subsemigroup of S for all \u03b3\u2286U. Let x, y \u2208 S be such that \u03b1(x) = \u03b3x and \u03b1(y) = \u03b3y. Taking \u03b3 = \u03b3x\u2229\u03b3y implies that x, y \u2208 iS. Hence xy \u2208 iS, and so \u03b1(xy)\u2287\u03b3 = \u03b3x\u2229\u03b3y = \u03b1(x)\u2229\u03b1(y). Therefore  is an int-soft semigroup over U.Conversely, suppose that the nonempty \u03b1, S) over U is an int-soft left  ideal over U if and only if the nonempty \u03b3-inclusive set of\u2009\u2009 is a left  ideal of\u2009\u2009S for all \u03b3\u2286U.A soft set  over\u2009\u2009U is an int-soft two-sided ideal over U if and only if the nonempty \u03b3-inclusive set of\u2009\u2009 is a two-sided ideal of\u2009\u2009S for all \u03b3\u2286U.A soft set  and  over U, we define\u03b1, S) and  is defined to be the soft set U in which  soft intersection of  and  is defined to be the soft set U in which \u03b1, S) and  is defined to be the soft set U in which S to P(U) given byFor any soft sets , , , and\u2009\u2009 be soft sets over U. IfLet\u2009\u2009 and  be characteristic soft sets over\u2009\u2009U where A and B are nonempty subsets of\u2009\u2009S. Then the following properties hold:Let  Let x \u2208 S, suppose x \u2208 AB. Then there exist a \u2208 A and b \u2208 B such that x = ab. Thus we havex \u2208 AB, we get \u03c7AB(x) = U. Suppose x \u2209 AB. Then x \u2260 ab for all a \u2208 A and b \u2208 B. If x = yz for some y, z \u2208 S, then y \u2209 A or z \u2209 B. Hencex \u2260 yz for all x, y \u2208 S, then(2) For any \u03b1, S) over U is an int-soft semigroup over U if and only if A soft set  is an int-soft semigroup over U.Assume that \u03b1, S) is an int-soft semigroup over U. Then \u03b1(x)\u2287\u03b1(y)\u2229\u03b1(z) for all x \u2208 S with x = yz. Thusx \u2208 S. Hence Conversely, suppose that  and a soft set  over\u2009\u2009U, the following are equivalent: \u03b2, S) is an int-soft left ideal over U, is an int-soft left ideal over U. Let x \u2208 S. If x = yz for some y, z \u2208 S, thenSuppose that  is an int-soft left ideal over U.Conversely, assume that Similarly, we have the following theorem.\u03c7S, S) over U and a soft set  over U, the following assertions are equivalent: \u03b2, S) is an int-soft right ideal over U, over U and a soft set  over U, the following assertions are equivalent:\u03b2, S) is an int-soft two-sided ideal over U, and  are int-soft semigroups over\u2009\u2009U, then so is the soft intersection If  and  are int-soft left ideals  over U, then so is the soft intersection If\u2009\u2009 and  are int-soft two-sided ideals over U, then so is the soft intersection If\u2009\u2009 and  be soft sets over U. If  is an int-soft left ideal over U, then so is the int-soft product Let\u2009\u2009 = (xa)b andy is not expressible as y = ab for a, b \u2208 S, then x, y \u2208 S, and so U.Let\u2009\u2009Similarly, we have the following theorem.\u03b1, S) and\u2009\u2009 be soft sets over U. If  is an int-soft right ideal over U, then so is the int-soft product Let  be a soft set over U. For a subset \u03b3 of U with iS \u2260 \u2205, define a soft set  over U by\u03b4 is a subset of U with \u03b4\u228a\u03b1(x).Let  is an int-soft semigroup over U, then so is .If\u2009\u2009, then xy \u2208 iS since iS is a subsemigroup of S by x \u2209 iS or y \u2209 iS, then \u03b1*(x) = \u03b4 or \u03b1*(y) = \u03b4. Thus\u03b1*, S) is an int-soft semigroup over U.Let By similar manner, we can prove the following theorem.\u03b1, S) is an int-soft left ideal  over U, then so is .If\u2009\u2009 is an int-soft two-sided ideal over U, then so is .If\u2009\u2009 is an int-soft right ideal over U and  is an int-soft left ideal over U, then If\u2009\u2009 is an int-soft right ideal over\u2009\u2009U, then \u03b2, S) over\u2009\u2009U.Let x \u2208 S. Then there exists a \u2208 S such that xax = x since S is regular. Thus\u03b1, S) is an int-soft right ideal over U. Since xax = x, we obtainLet In a similar way we prove the following.S be a regular semigroup. If\u2009\u2009 is an int-soft left ideal over\u2009\u2009U, then \u03b1, S) over\u2009\u2009U.Let S is regular, then \u03b1, S) and int-soft left ideal  over\u2009\u2009U.If a semigroup S is a regular semigroup and let  and  be an int-soft right ideal and an int-soft left ideal, respectively, over U. By Assume that \u03b1, S) over U is called an int-soft quasi-ideal over U ifA soft set  ideal is an int-soft quasi-ideal over In fact, we have the following example.S = {0, a, b, c} be a semigroup with the following Cayley table:\u03b1, S) be a soft set over U defined as follows:\u03b3 is a subset of U. Then  is an int-soft quasi-ideal over U and is not an int-soft left  ideal over U.Let G be a nonempty subset of\u2009\u2009S. Then G is a quasi-ideal of S if and only if the characteristic soft set  is an int-soft quasi-ideal over\u2009\u2009U.Let G is a quasi-ideal of S. Let a be any element of S. If a \u2208 G, thena \u2209 G, then \u03c7G(a) = \u2205. On the other hand, assume thatb, c, d, and e of S with a = bc = de such that \u03c7G(b) = U and \u03c7G(e) = U. Hence a = bc = de \u2208 GS\u2229SG\u2286G, which contradicts that a \u2209 G. Thus we have \u03c7G, S) is an int-soft quasi-ideal over U.We first assume that \u03c7G, S) is an int-soft quasi-ideal over U. Let a be any element of GS\u2229SG. Then bx = a = yc for some b, c \u2208 G and x, y \u2208 S. It follows from  over\u2009\u2009U.For a semigroup S is regular and let a \u2208 S. Then a = axa for some x \u2208 S. Hence\u03b1, S) is an int-soft quasi-ideal over U,Assume that A be a quasi-ideal of S. Then ASA\u2286AS\u2229SA\u2286A and  is an int-soft quasi-ideal over U. For any a \u2208 A, we haveb, c \u2208 S such that \u03c7A(c) = U. Thenb = st, \u2009\u03c7A(s) = U = \u03c7S(t) for some s, t \u2208 S. It follows that c, s \u2208 A and t \u2208 S so that a = bc = (st)c \u2208 ASA. Hence A\u2286ASA, and thus A = ASA. Therefore S is regular.Conversely, suppose that (2) is valid and let"}
{"text": "Also, the necessary and sufficient conditions of efficient solution for fractional programming are established and a parameterization technique is used to establish duality results under generalized second order  A fractional programming problem arises in many types of optimization problems such as portfolio selection, production, information theory, and numerous decision making problems in management science. More specifically, it can be used in engineering and economics to minimize a ratio of physical or economical function or both, such as cost/time, cost/volume, and cost/benefit, in order to measure the efficiency or productivity of the system. Many economic, noneconomic, and indirect applications of fractional programming problem have also been given by Bector , Bector The central concept in optimization is known as the duality theory which asserts that, given a  minimization problem, the infimum value of the primal problem cannot be smaller than the supermom value of the associated  maximization problem and the optimal values of primal and dual problems are equal. Duality in fractional programming is an important class of duality theory and several contributions have been made in the past , 10\u201314. V, \u03c1) invexity assumption.Multiobjective fractional programming duality has been of much interest in the recent past. Schaible  and Bect\u03c1-univexity.Duality for various forms of mathematical problems involving square roots of positive semidefinite quadratic forms has been discussed by many authors , 23\u201325. z et al.  to charaz et al.  introducz et al.  and Jagaz et al.  for both\u03c1-univexity assumption.To relax convexity assumption imposed on the function in theorems on optimality and duality, various generalized convexity concepts have been proposed. Hanson  introduc\u03c1-univexity assumption.Motivated by the earlier authors in this paper, we have introduced three approaches given by Dinklebaeh  and Jaga\u211dn be the n-dimensional Euclidean space and \u211d+n its nonnegative orthant. The following conventions for inequality will be used throughout this paper. For any x = , y = , we denote the following.(i)x > y\u21d4xi > yi, for all i = 1,2,\u2026, n.(ii)x \u2265 y\u21d4xi \u2265 yi, for all i = 1,2,\u2026, n.Throughout the paper, let X be a nonempty open subset of \u211dn.Let Consider the following nondifferentiable multiobjective fractional programming problem.(i) MFP0. Minimize \u2009where (ii) MFP1. Minimize \u2009where \u2009\u03bdi are fixed parameters.(iii)MFP\u03bb. Minimize \u03bbF(x); \u03bb is k-dimensional strictly positive vector,\u2009all subject to same constraintfi : \u211dn \u2192 \u211d, gi : \u211dn \u2192 \u211d, i = 1,2,\u2026, k and h = ; hj : \u211dn \u2192 \u211d, j = 1,2,\u2026, m, are differentiable functions, Bi and Ci, i = 1,2,\u2026, k are positive semidefinite matrices of order n. In the sequel, we assume that fi(x) \u2265 0 and gi(x) > 0 on \u211dn for i = 1,2,\u2026, k.X0 = {x \u2208 X\u2286\u211dn : hj(x) \u2264 0, j = 1,2,\u2026, m} for all feasible solutions of MFP0, MFP1, and MFP\u03bb and denote I = {1,2, 3,\u2026, k}, M = {1,2, 3,\u2026, m}, J1 = {j \u2208 M : hj(x) = 0}, and J2 = {j \u2208 M : hj(x) < 0}. It is obvious that J1 \u222a J2 = M.Let fi : X \u2192 \u211d, \u03b7 : X \u00d7 X \u2192 \u211dn, p \u2208 \u211dn, \u03c1 \u2208 \u211d.Throughout the paper, consider \u03c8 : \u211d \u2192 \u211d satisfying a \u2264 0\u21d2\u03c8(a) \u2264 0 or \u03c8(a) \u2264 0\u21d2a \u2264 0 and \u03c8(\u2212a) = \u2212\u03c8(a), K : X \u00d7 X \u2192 \u211d+. For Assume that fi is said to be second order \u03c1-univex at \u03b7, \u03c8, and K, if The real differentiable function fi is said to be second order \u03c1-pseudounivex at \u03b7, \u03c8, and K, if The real differentiable function fi is said to be second order \u03c1-quasiunivex at \u03b7, \u03c8, and K, if The real differentiable function p = 0, the above definitions reduce to the definitions of \u03c1-univex, \u03c1-pseudounivex, and \u03c1-quasiunivex as introduced in  + \u2211j=1myjhj(x) is second order \u03c1-pseudounivex with respect to \u03b7, \u2009\u03c8, and K at 2P(x))p = 0, where fi : X \u2192 \u211d, gi : X \u2192 \u211d, hj : X \u2192 \u211d, i = 1,2,\u2026, k; j = 1,2,\u2026, m, p \u2208 \u211dn, \u03b7 : X \u00d7 X \u2192 \u211dn, K : X \u00d7 X \u2192 \u211d+, and \u03c8 : \u211d \u2192 \u211d satisfying \u03c8(a) \u2264 0\u21d2a \u2264 0.(ii)\u03c1 \u2265 0. Let Then Suppose that the hypothesis holds.Since the conditions of So we can write Now for \u03c1 \u2265 0, we have For P(x) is second order \u03c1-pseudounivex with respect to \u03b7, \u2009\u03c8, and K at K, \u2009\u03c8, it gives t \u2208 {1,2,\u2026, k}.Since \u03bbi > 0, implies that The above relation, together with the relation elations , 11), a, a\u03bbi > elations , we get ies that \u2211i=1k\u03bbi[flations Q(x) = \u2211i=1k\u03bbi[fi(x) + (xTBix)1/2 \u2212 vi{gi(x) \u2212 (xTCix)1/2}] is second order \u03c1-pseudounivex with respect to \u03b7, \u03c80, and K at H(x) = \u2211j=1myjhj(x) is second order \u03c3-quasiunivex with respect to \u03b7, \u03c81, and K at fi : X \u2192 \u211d, gi : X \u2192 \u211d, hj : X \u2192 \u211d, i = 1,2,\u2026, k; j = 1,2,\u2026, m, p \u2208 \u211dn, \u03b7 : X \u00d7 X \u2192 \u211dn, K : X \u00d7 X \u2192 \u211d+, and \u03c80, \u03c81 : \u211d \u2192 \u211d satisfying \u03c80(a) \u2265 0\u21d2a \u2265 0 and \u03c81(a) \u2264 0\u21d2a \u2264 0.(ii)\u03c1 + \u03c3 \u2265 0. Then MFP1.Let Suppose hypothesis holds.From the relations , 11), a, a11), a\u03c3-quasiunivexity of H(x) with respect to \u03b7, \u2009\u03c81, and K implies the following: \u03c1 + \u03c3 \u2265 0, we get Q(x) is \u03c1-pseudounivex with respect to \u03b7, \u2009\u03c80, and K, we obtained K and \u03c80, we get So we have the following: \u22640. From , we get llowing: \u03b7T{0. From \u2212Q MMFD0. Maximize \u2009where (ii) MMFD1. Maximize\u2009Gi(u) = fi(u) + yJ1ThJ1(u) + uTBiw \u2212 \u03bdi{gi(u) \u2212 uTCiz}, i = 1,2,\u2026, k; \u03bdi are fixed parameters.where (iii)MMFD\u03bb. Maximize \u03bbG(u); \u03bb is k-dimensional strictly positive vector,\u2009all subject to same constraintsfi : X \u2192 \u211d, gi : X \u2192 \u211d, hj : X \u2192 \u211d, i = 1,2,\u2026, k; j = 1,2,\u2026, m are differentiable functions, w, z \u2208 \u211dn, p \u2208 \u211dn. Bi, and Ci, i = 1,2,\u2026, k are positive semidefinite matrices of order n.\u03b7 : X \u00d7 X \u2192 \u211dn, K : X \u00d7 X \u2192 \u211d+, and \u03c80, \u03c81 : \u211d \u2192 \u211d satisfying \u03c80(a) \u2265 0\u21d2a \u2265 0 and b \u2264 0\u21d2\u03c81(b) \u2264 0 and \u03c1, \u03c3 \u2208 \u211d.For the following theorems, we assume that x be a feasible solution for the primal MFP\u03bb and let  be feasible for dual SMMFD\u03bb. If (i)i=1k\u03bbiGi(\u00b7) is second order \u03c1-pseudounivex with respect to \u03b7, \u03c80, K, and for yJ2 \u2208 \u211dm\u2212|J1|, yJ2ThJ2(\u00b7) is second order \u03c3-quasiunivex with respect to \u03b7, \u03c81, and K along with\u2211(ii)\u03c1 + \u03c3 \u2265 0, then Inf\u2061\u2061(\u03bbF(x)) \u2265 Sup\u2061(\u03bbG(u)).Let yJ2ThJ2 is second order \u03c3-quasiunivex with respect to \u03b7, \u2009\u03c81, and K and in view of p\u22650. So yJ2ThJ2 \u2208 \u211dn, we have \u03b7T{\u2211i=1k\u03bbi[\u2207Gi(u) + \u22072Gi(u)p]} \u2212 \u03c3||x\u2212u||2 \u2265 0.Again from the dual constraint , we have0. Using  in above\u03c1 + \u03c3 \u2265 0, we get \u03c1||x\u2212u||2 \u2265 \u2212\u03c3||x\u2212u||2.Since i=1k\u03bbiGi(u) is second order \u03c1-pseudounivex with respect to \u03b7, \u03c80, and K, by K\u03c80{\u2211i=1k\u03bbiGi(x) \u2212 \u2211i=1k\u03bbiGi(u) + (1/2)pT\u2211i=1k\u03bbiGi(u)p} \u2265 0.So, we have \u03b7T{\u2211\u03c80 and K, we get h(x) \u2264 0\u21d2yJ1ThJ1(x) \u2264 0, for yJ1 \u2265 0.Using the property of Equation  gives h is second order \u03c1-pseudounivex with respect to \u03b7, \u2009\u03c80, and K and for yJ2 \u2208 \u211dm\u2212|J1|, yJ1ThJ2(\u00b7) is second order \u03c3-quasiunivex with respect to \u03b7, \u2009\u03c81, and K along with (iii) \u03c1 + \u03c3 \u2265 0, then Let \u03bb), by \u03bbi \u2208 \u211d+; w, z \u2208 \u211dn, vi \u2208 \u211d+, and y \u2208 \u211dm such that Since So \u03bb and SMMFD\u03bb are equal to zero. It follows from And the objective values of MFP\u03bb. Then applying Lemmas So p = 0, Ci = 0, i = 1,2,\u2026, k, then our dual programming reduces to the dual programming proposed by Tripathy [If Tripathy .\u03c1-univexity assumption. The results developed in this paper can be further extended to higher order mixed type fractional problem containing square root term. Also the present work can be further extended to a class of nondifferentiable minimax mixed fractional programming problems.In this paper, three approaches given by Dinklebaeh  and Jaga"}
{"text": "L in the reproducing kernel space W2m(D). Then we show spectral analysis of L and derive several property theorems.We first introduce some related definitions of the bounded linear operator  It is well known that spectral analysis of linear operators  is an imSo far, the spectral decomposition method  has becoW2m(D) and derive some useful conclusions.To our knowledge, reproducing kernel space has been applied in many fields, such as linear systems \u20138, nonliL in the reproducing kernel space W2m(D). In L in W2m(D) are given. In L and also establish several theorems. The paper is organized as follows. In D be an abstract set, W2m(D) the reproducing kernel space, and BL[W2m(D) \u2192 W2n(D)] the bounded linear operator space. \u2200L \u2208 BL[W2m(D) \u2192 W2n(D)] with m, n \u2208 N, if there exists nonvanishing vector u \u2208 W2m(D), such that\u03bb is called an eigenvalue of L and u is called the eigenvector of L according to \u03bb, where I denotes the identity operator.Let L \u2208 BL[W2m(D) \u2192 W2n(D)] and all eigenvectors and zero vector of L compose the eigenvector space which is denoted by E\u03bb.\u2200E\u03bb is a linear closed subspace of W2m(D).Obviously, E\u03bb by dim\u2061\u2061E\u03bb; it is called the multiplicity of eigenvalue \u03bb. That is, dim\u2061\u2061E\u03bb is the number of vectors of maximum linear independence.Denote the dimension of K be a binary function on D = {\u2223a \u2264 s \u2264 b, a \u2264 t \u2264 b}, L \u2208 BL[W2m(D) \u2192 W2n(D)], m, n \u2208 N with\u03bb is the eigenvalue of L if and only if the following integral equation has nonzero solution:K = \u2211i=1nfi(s)gi(t), {fi}i=1n is a linear independence vector system, then ((a)\u03bb = 0, then (u \u2208 W2m(D) and \u222babgi(t)u(t)dt = 0, i = 1,2,\u2026, n. It follows that, for the eigenvalue \u03bb = 0 of L, the eigenvector space is infinite-dimensional.If  0, then  has nonz(b)\u03bb \u2260 0, then the solution of  are constants.If ution of  can be dLet olution:\u03bbu(s)\u2212\u222babfi}i=1n in (Ci must satisfy the following linear equation system:Combine  with 4)4) and, ii=1n in , Ci musCi  are undetermined coefficients. In addition, in order to solve the eigenvector, we just need to solve Ci  in   5) are eE\u03bb of L for the homogeneous equation  \u2192 W2n(D)], m, n \u2208 N, D(L)\u2286W2m(D), and R(L)\u2286W2n(D), where D(L) denotes the domain of L and R(L) denotes the range of values of L. If the inverse operator L\u22121 of L exists and is linearly bounded, then L is called a regular operator.Let L be a linear operator; \u2009D(L)\u2286W2m(D) and R(L)\u2286W2n(D); if L\u22121 exists, then L\u22121L = ID(L) and LL\u22121 = IR(L), where ID(L) and IR(L) are, respectively, identity operators of subspace D(L) and R(L). Inversely, if there exists a linear operator C : W2n(D) \u2192 W2m(D), such that CL = ID(L) and LC = ID(C), then L\u22121 exists and I\u22121 = C. In fact, \u2200u1, u2 \u2208 D(L); if Lu1 = Lu2, then u1 = CLu1 = CLu2 = u2. Hence, L is invertible. Since LC = ID(C), then \u2200v \u2208 D(C); we have u = Cv such that Lu = v. That is, D(C)\u2286R(L).Let R(L) = D(C). Hence, we have L\u22121 = (CL)L\u22121 = CLL\u22121 = C. Particularly, when D(C) = W2n(D) and C is a bounded linear operator, we can derive the following results.Summing up the above disscusion, L \u2208 BL[W2m(D) \u2192 W2n(D)], m, n \u2208 N; then L is a regular operator if and only if \u2203C \u2208 BL[W2n(D) \u2192 W2m(D)], such that CL = ID(L) and LC = ID(C).Let L \u2208 BL[W2m(D) \u2192 W2n(D)], m, n \u2208 N; if L is a regular operator, then L* is also a regular operator and (L*)\u22121 = (L\u22121)*.Let L\u22121 \u2208 BL[W2n(D) \u2192 W2m(D)], m, n \u2208 N, andL* is a regular operator and (L*)\u22121 = (L\u22121)*. The proof is complete.Since D(L)\u2286W2m(D), m \u2208 N, \u03bb \u2208 C; C denotes the complex number field.\u03bbI \u2212 L is a regular operator, that is, \u03bbI \u2212 L is a one-to-one linear operator from D(L) to W2m(D). In addition, the inverse operator (\u03bbI \u2212 L)\u22121 is a linear bounded operator. Then \u03bb is called a regular point of L. All regular points compose the regular set of L, which is denoted by \u03c1(L).\u03bb is not a regular point, then \u03bb is called a spectral point of L. All spectral points compose the spectral set of L, which is denoted by \u03c3(L).If Let \u03c1(L)\u22c3\u03c3(L) = C. Then we have the following property results.In view of L be a bounded linear operator in reproducing kernel space W2m(D), m \u2208 N; then \u03bb is a regular point of L if and only if \u2200f \u2208 W2m(D), there exists a solution g of (\u03bbI \u2212 L)g = f, which satisfies ||g|| \u2264 m||f||, where m is a positive constant.Let R(\u03bbI \u2212 L) = W2m(D), then \u2200f \u2208 W2m(D), \u2203g \u2208 W2m(D) such that (\u03bbI \u2212 L)g = f. In addition, in view of the boundedness of (\u03bbI \u2212 L)\u22121 and the Cauchy-Schwartz inequality, we havem = ||(\u03bbI\u2212L)\u22121|| > 0; then ||g|| \u2264 m||f||.\u21d2 Since \u03bbI \u2212 L)g = f, we have R(\u03bbI \u2212 L) = W2m(D). Next, we will prove that \u03bbI \u2212 L is one-to-one. In fact, \u2200f \u2208 W2m(D); if (\u03bbI \u2212 L)g1 = f, (\u03bbI \u2212 L)g2 = f, theng1 \u2212 g2 is 0. Hence, ||g1 \u2212 g2|| \u2264 m||0||; that is, g1 = g2. Therefore, \u03bbI \u2212 L is one-to-one and (\u03bbI \u2212 L)\u22121 exists. Furthermore, since ||g|| \u2264 m||f||, we have\u03bbI \u2212 L)\u22121 exists and is a bounded linear operator. Summing up the above, \u03bbI \u2212 L is a regular operator, where \u03bb is a regular point of L.\u21d0 Since (f \u2208 W2m(D); when L is a continuous linear operator and \u03bb is the regular point of L, (\u03bbI \u2212 L)g = f has a unique solution g. Furthermore, the continuity of g depends on the right term. In other words, if {fi}i=1n are column vectors and fn \u2192 f, then gn \u2192 g.L be a bounded linear operator in the reproducing kernel space W2m(D), m \u2208 N. If \u03bb is not the eigenvalue of L and (\u03bbI \u2212 L)g1 = (\u03bbI \u2212 L)g2, one has L(g1 \u2212 g2) = \u03bb(g1 \u2212 g2), g1 = g2. That is, \u03bbI \u2212 L is invertible.Let g \u2260 0 such that (\u03bbI \u2212 L)g = 0. That is, \u03bb is not the eigenvalue of L.Otherwise, the invertible operator can convert the nonvanishing vector to nonvanishing vector. Hence, there exists W2m(D) is a finite dimension space and \u03bb is not the eigenvalue of L, we can derive that C = \u03bbI \u2212 L is an invertible mapping. Obviously, R(C) = W2m(D). In fact, let {ei}i=1n be the basis of W2m(D); then {(\u03bbI \u2212 L)ei}i=1n is a linear independent system in W2n(D) and also a basis of W2n(D). Therefore, R(L) = W2n(D). In view of the inverse operator Theorem, (\u03bbI \u2212 L)\u22121 is bounded. It follows that \u03bb \u2208 \u03c1(L).When So, the proof of the theorem is complete.L can only be an eigenvalue in finite dimension normed space. This is entirely consistent with the conclusion of the theory of linear algebra. But if W2m(D) is an infinite-dimensional space and \u03bb is not the eigenvalue of L, then \u03bb may not be a regular point of L, so far as \u03bbI \u2212 L is not a map from W2m(D) to W2n(D).\u03bb \u2208 C, \u222babu(t)dt = \u03bbu(t) has only zero solutions. Hence, L has not eigenvalue. That is, zero is not the eigenvalue. However, the range of values is all functions of the from \u222babu(t)dt for (0I \u2212 L). This shows that the spectral point is complex in infinite-dimensional space for the operator L.For example, let\u03bbI \u2212 L is not one-to-one, then \u03bb is called point spectral of L; the set of point spectral is denoted by \u03c3p(L).If \u03bbI \u2212 L is one-to-one and R(\u03bbI \u2212 L) is dense in W2m(D), then \u03bb is called continuous spectral of L; the set of continuous spectral is denoted by \u03c3c(L).If \u03bbI \u2212 L is one-to-one and R(\u03bbI \u2212 L) is not dense in W2m(D), then \u03bb is called residual spectral of L; the set of residual spectral is denoted by \u03c3r(L).If Now, we will classify the spectral set by three situations.\u03c3p(L), \u03c3c(L), and \u03c3r(L) are mutually disjoint sets and \u03c3(L) = \u03c3p(L) \u222a \u03c3c(L) \u222a \u03c3r(L).Obviously, L \u2208 BL[W2m(D) \u2192 W2n(D)], m, n \u2208 N, \u03b5 > 0, \u2203N \u2208 N*, \u2200n > N, such that Ln||<(r + \u03b5)n. In view of the completeness of W2m(D), there exists m > N, such thatn=0\u221eLn converges in the sense of ||\u00b7|| and the limit is denoted by C = \u2211n=0\u221eLn.Let Cm = \u2211n=0mLn; thenCm \u2212 C|| \u2192 0, m \u2265 N, we havem \u2192 \u221e, then\u03c1(L) and (I\u2212L)\u22121 = \u2211n=0\u221eLn.Let L|| < 1, one obtainsFor ||Summing up the above parts, we have the following theorems.L \u2208 BL[W2m(D) \u2192 W2n(D)], m, n \u2208 N, then one has the following.\u03c1(L).Consider 1 \u2208 I \u2212 L)\u22121 = \u2211n=0\u221eLn.Consider \u22121|| \u2264 1/(1 \u2212 ||L||).When ||Let L \u2208 BL[W2m(D) \u2192 W2n(D)], m, n \u2208 N; if \u03bb| > r if and only if \u03bb is a regular point of L.|\u03bb| > r, (\u03bbI\u2212L)\u22121 = \u2211n=0\u221e(Ln/\u03bbn+1).When |\u03bb| > ||L||, ||(\u03bbI\u2212L)\u22121|| \u2264 (|\u03bb|\u2212||L||)\u22121.When |Let \u03bb \u2260 0, since (\u03bbI \u2212 L) = \u03bb(I \u2212 L/\u03bb), \u03bb \u2208 \u03c1(L) if and only if 1 \u2208 \u03c1(L/\u03bb). Replacing L by L/\u03bb in \u03bb is a regular point of L and ||(\u03bbI\u2212L)\u22121|| \u2264 (|\u03bb|\u2212||L||)\u22121 with |\u03bb| > r.\u2200\u03bb| > ||L||, we have ||L/\u03bb|| < 1. In view of (2) of In addition, when |L \u2208 BL[W2m(D) \u2192 W2n(D)], m, n \u2208 N; then one has the following.\u03c1(L) is an open set.\u03c1(L) is nonempty, \u2200\u03bb0 \u2208 \u03c1(L); if \u03bb is a regular point of L and (\u03bbI\u2212L)\u22121 = \u2211n=0\u221e(\u22121)n(\u03bb0I\u2212L)n+1)\u2212((\u03bb\u2212\u03bb0)n, where |\u03bb \u2212 \u03bb0| < 1/r\u03bb0.When \u03c3(L) is a closed set.Consider Let \u03c1(L) = \u2205, the conclusion is obvious. If \u03c1(L) \u2260 \u2205, then\u03bb0I\u2212L)\u22121 is a bounded linear operator in the reproducing kernel space W2m(D). We use (\u03bb \u2212 \u03bb0)(\u03bb0I\u2212L)\u22121 instead of L in \u03bb \u2212 \u03bb0| < 1/r\u03bb0, [I+(\u03bb\u2212\u03bb0)(\u03bb0I\u2212L)\u22121]\u22121 exists and is bounded. Hence, when |\u03bb \u2212 \u03bb0| < 1/r\u03bb0, \u03bb \u2208 \u03c1(L); that is, \u03c1(L) is an open set.(1) If \u03c1(L) is nonempty, \u2200\u03bb0 \u2208 \u03c1(L), let (2) If \u03c1(L)\u22c3\u03c3(L) = C and (1), \u03c3(L) is a closed set.(3) Since \u03c3(L)\u2286{\u03bb\u2223|\u03bb| \u2264 r}, which means that (4) In view of (1) of The proof is complete.L \u2208 BL[W2m(D) \u2192 W2n(D)], m, n \u2208 N, r(L) = max\u2061\u03bb\u2208\u03c3(L)\u2061|\u03bb|; r(L) is called the spectral radius of L.Let \u03bb| > r(L), due to the fact that \u03bb is a regular point of L, then for any f \u2208 W2m(D), (\u03bbI \u2212 L)g = f has a unique solution g.For |\u03bb| \u2264 r(L), it cannot guarantee this equation has a solution for any f \u2208 W2m(D). In many practical problems, in order to calculate the spectral range, one needs to estimate the spectral radius. In terms of (4) of r(L) \u2264 ||L||. In practical terms, this estimate is convenient, but it is imprecise.For |From the purpose of solving equations, spectral radius has the following meanings.L \u2208 BL[W2m(D) \u2192 W2n(D)], m, n \u2208 N; then Let \u03bb| > ||L||, one obtainsf \u2208 W2m(D), f((\u03bbI\u2212L)\u22121) = \u2211n=0\u221e(f(Ln)/\u03bbn+1). If |\u03bb| > r(L), then \u03bb is a regular point of L. In addition, since {\u03bb\u2223|\u03bb| > r(L)}, then Laurent expansions of f((\u03bbI\u2212L)\u22121) are established, where |\u03bb| > r(L).In terms of (4) of a = r(L), \u2200\u03b5 > 0; we haveBn = Ln/(a+\u03b5)n, \u2200f \u2208 W2m*(D); thenLet Bn} must be bounded. It follows that there exists a positive constant M, such that ||Bn|| \u2264 M and ||Ln|| \u2264 (a+\u03b5)n||Bn|| \u2264 (a+\u03b5)nM. Namely,\u03b5 \u2192 0; thenIn terms of the resonance Theorem, {L \u2208 BL[W2m(D) \u2192 W2n(D)], m, n \u2208 N; then \u03c3(L) \u2260 \u2205.Let \u03c3(L) = \u2205, in view of the properties of the reproducing kernel space, W2m(D)\u2260{0}; hence, I \u2260 {0}, where the unit element is denoted by I. In terms of the functional extension Theorem, \u2203f \u2208 W2m*(D), such that f(I) \u2260 0. In addition, \u2200\u03bb0 \u2208 \u03c1(L) and \u2203r\u03bb0 \u2208 R+; when |\u03bb \u2212 \u03bb0| < 1/r\u03bb0, we have\u03c3(L) = \u2205 and \u03bb| > ||L||, one obtains\u03bb| \u2265 ||L|| + 1, we havef((\u03bbI\u2212L)\u22121) is bounded. In terms of the Liouville Theorem, f((\u03bbI\u2212L)\u22121) must be a constant, so we have \u03c3(L) \u2260 \u2205.If The proof is complete.L \u2208 BL[W2m(D) \u2192 W2n(D)], m, n \u2208 N, L is called a generalized nilpotent operator.If L \u2208 BL[W2m(D) \u2192 W2n(D)], m, n \u2208 N, \u2286D,L, one obtainsL is a generalized nilpotent operator; spectral point \u03bb = 0 is not the eigenvalue of L.In terms of L \u2208 BL[W2m(D) \u2192 W2n(D)], m, n \u2208 N, \u03bb \u2208 C; if there exists {un}n=1\u221e \u2208 W2m(D), such that (\u03bbI \u2212 L)un \u2192 0, then \u03bb is called an approximate spectral point. All approximate spectral points are denoted by \u03c3a(L); the other spectral point is called remainder spectral point. All the remainder spectral points are denoted by \u03c3r(L).Let L \u2208 BL[W2m(D) \u2192 W2n(D)], m, n \u2208 N; then\u03c3p(L)\u2286\u03c3a(L),\u03c3a(L)\u22c2\u03c3r(L) = \u2205, and \u03c3a(L)\u22c3\u03c3r(L) = \u03c3(L),\u03c3r(L) is an open set,\u03c3(L)\u2286\u03c3a(L), where \u2202\u03c3(L) denotes the boundary of \u03c3(L),\u2202\u03c3a(L) is a nonempty closed set.Let \u03bb \u2208 \u03c3p(L), then there exists nonzero element u of W2m(D), such thatu|| = 1; we choose un = u, n = 1,2,\u2026, and then ||un|| = 1 and (L \u2212 \u03bbI)un \u2192 0; namely, \u03bb \u2208 \u03c3a(L); this shows that \u03c3p(L)\u2286\u03c3a(L).(1) If \u03c3a(L)\u22c2\u03c3r(L) = \u2205 and \u03c3a(L)\u22c3\u03c3r(L) = \u03c3(L).(2) In terms of \u03bb \u2208 \u03c3r(L), then \u03bb \u2209 \u03c3a(L). Hence, \u2203\u03b1 \u2208 N*, such that\u03bb\u2032 \u2212 \u03bb| < \u03b1/2, \u2200u \u2208 W2m(D), we have\u03bb\u2032 which satisfies |\u03bb\u2032 \u2212 \u03bb| < \u03b1/2 it is impossible to be an approximate spectral point of L. Hence, if one can prove that when |\u03bb\u2032 \u2212 \u03bb| < \u03b1/2, \u03bb\u2032 is not a regular point of L, then \u03bb\u2032 \u2208 \u03c3r(L). That is, \u03bb is an inner point of \u03c3r(L), so \u03c3r(L) is an open set.(3) If \u03bb\u2032 \u2212 \u03bb| < \u03b1/2, \u03bb\u2032 \u2209 \u03c1(L). But not vice versa, \u2203\u03bb0 \u2208 C, |\u03bb0 \u2212 \u03bb| < \u03b1/2; then \u03bb0 \u2208 \u03c1(L). Note that ||(L \u2212 \u03bb\u2032I)u|| \u2265 ||(L \u2212 \u03bbI)u|| \u2212 |\u03bb \u2212 \u03bb\u2032|||u|| \u2265 (\u03b1/2)||u||; if \u03bb\u2032 = \u03bb0, then\u03bc be a regular point of L; if |\u03bc \u2212 \u03bb0| < 1/r\u03bb0, then\u03bc = \u03bb; note that\u03bb \u2208 \u03c1(L). This is a contradiction with \u03bb \u2208 \u03c3r(L). It follows that \u03c3r(L) is an open set.Now, we prove that when |\u03c3(L) is a closed set, when \u03bb \u2208 \u2202\u03c3(L), \u03bb \u2208 \u03c3(L). In addition, \u03bb \u2209 \u03c3r(L), \u2202\u03c3(L)\u2288\u03c3r(L), so we have \u2202\u03c3(L)\u2286\u03c3a(L).(4) Since \u03c3a(L) = \u03c3(L) \u2212 \u03c3r(L), then \u03c3a(L) is a closed set. Furthermore, since \u2202\u03c3(L)\u2286\u03c3a(L), \u03c3a(L) \u2260 \u2205, this shows that \u2202\u03c3(L) and \u03c3a(L) are all nonempty sets.(5) Since The proof is complete.E\u03bb of the bounded linear operator L in the reproducing kernel space W2m(D). Then we show some definitions and properties of the regular operator. The regular set and spectral set of bounded linear operator are also introduced. From the solvability of the equation, we show the spectral classification and give three conditions. Finally, we introduce the spectral analysis of the bounded linear operator L. It includes the definitions of spectral radius, nilpotent operator, approximate spectral point, and remainder spectral point. We also establish some property theorems of the bounded linear operator in the reproducing kernel space W2m(D).This paper first introduces the eigenvalue, eigenvector, eigenvector space, and dim\u2061\u2061"}
{"text": "In particular, we give so far the smallest number of locally indistinguishable states of a completable orthogonal product basis in arbitrary quantum systems. Furthermore, we construct a series of small and locally indistinguishable orthogonal product bases in m\u2009\u2297\u2009n (m\u2009\u2265\u20093 and n\u2009\u2265\u20093). All the results lead to a better understanding of the structures of locally indistinguishable product bases in arbitrary bipartite quantum system.As we know, unextendible product basis (UPB) is an incomplete basis whose members cannot be perfectly distinguished by local operations and classical communication. However, very little is known about those incomplete and locally indistinguishable product bases that are not UPBs. In this paper, we first construct a series of orthogonal product bases that are completable but not locally distinguishable in a general  Furthermore, they proved that a UPB cannot be perfectly distinguished by LOCC. DiVincenzo et al.et al.et al.d\u2009\u2212\u20091 orthogonal states that are locally indistinguishable in d\u2009\u2297\u2009d (d\u2009\u2265\u20093) and conjectured that any set of no more than 2(d\u2009\u2212\u20091) product states is locally distinguishable in a d\u2009\u2297\u2009d (d\u2009\u2265\u20093) quantum system. Wang et al.m\u2009+\u2009n)\u2009\u2212\u20099 orthogonal product states and proved the local indistinguishability of these states in an m\u2009\u2297\u2009n quantum system, where m\u2009\u2265\u20093 and n\u2009\u2265\u20093. Recently, Zhang et al.n\u2009+\u2009m\u2009\u2212\u20094 locally indistinguishable orthogonal product states that do not constitute a UPB and presented a smaller set with 2n\u2009\u2212\u20091 orthogonal product states that cannot be perfectly distinguished by LOCC in m\u2009\u2297\u2009n (3\u2009\u2264\u2009m\u2009\u2264\u2009n). All the results show it is a meaningful work to research the structure of the locally indistinguishable product basis and the smallest number of locally indistinguishable orthogonal product states in arbitrary high-dimensional quantum systems.Unextendible product basis (UPB) is a set of orthogonal product states that spans a subspace whose complementary subspace contains no product state. As an object with rich mathematical structure, it was introduced by Bennett m, n)\u2009\u2212\u20094 members, respectively, in a general m\u2009\u2297\u2009n (m\u2009\u2265\u20093 and n\u2009\u2265\u20093) quantum system. Our results show that Yu et al.\u2019s conjectured\u2009\u2212\u20091) product states is locally distinguishable in a d\u2009\u2297\u2009d (d\u2009\u2265\u20093) quantum system, is not true. In fact, eight is so far the smallest number of locally indistinguishable states of a completable orthogonal product basis2223m, n)\u2009\u2212\u20091 members respectively, in m\u2009\u2297\u2009n (m\u2009\u2265\u20093 and n\u2009\u2265\u20093). It should be pointed out that five is so far the smallest number of locally indistinguishable states of an orthogonal product basis by Refs. In this paper, we construct a series of completable and locally indistinguishable orthogonal product bases, which have eight members, twelve members, \u00b7\u00b7\u00b7, 4\u2009min is a set S of pure orthogonal product states spanning a subspace HSof H. An uncompletable orthogonal product basis is a PB whose complementary subspace, i.e., the subspace in H spanned by vectors that are orthogonal to all the vectors in HS, contains fewer mutually orthogonal product states than its dimension. An unextendible product basis (UPB) is an uncompletable product basis for whichcontains no product state20. We call a PB is completable if it is not an uncompletable orthogonal product basis.Definition 2Consider a multipartite quantum systemwith q parties. A strongly uncompletable product basis (SUCPB) is a PB spanning a subspace Hsin a locally extended Hilbert space Hextsuch that for all Hextthe subspacecontains fewer mutually orthogonal product states than its dimension.Definition 3. Suppose that {Mt} is a set of measurement operators, which can act on the measured Hilbert space. And t denote one of the possible measurement outcomes. If the measured state is |\u03d5\u232a before it is measured, the probability of the measurement outcome t is given byand the postmeasurement state is. Furthermore, the measurement operators {Mt} satisfy the completeness, i.e.,. If we denoteas Et, it is easy to see that Etis a positive semidefinite operator. We will say that the measurement is a positive operator-valued measure (POVM) and the objects Etare the POVM elements corresponding to each measurement outcome tDIt is easy to see that POVM is a general measurement to a measured quantum state according to the definition 3. As we mentioned in the preceding part, LOCC denote local operations and classical communication. When it comes to identify a given state that is chosen from a known set of orthogonal states by LOCC, the local operations are local POVMs or local unitary operations. That is, if a measured state is a bipartite (or multipartite) quantum system, each party that holds one particle of the bipartite (or multipartite) quantum system can only perform POVM or unitary operations on his (or her) own particle. For simplicity, we usually say a set of orthogonal states is locally distinguishable if it can be distinguished by LOCC.efinition 4We will say that a POVM is trivial if all the POVM elements are proportional to the identity operator since such a measurement yields no information about the measured state. Any measurement not of this type will be called nontrivial.DDifferent from Definition 4, we give a new definition about trivial measurement here. It should be noted that we say a measurement is trivial if it satisfies our new definition.efinition 5. A POVM is trivial to a set of orthogonal states,, if and only if we cannot get any useful information about the measured state that is arbitrarily selected from the set by the POVM, i.e., for each of the POVM elements,, we haveDefinition 6Alice goes first if Alice is the first person to perform a nontrivial measurement upon the system.Demma 1Given a PBon a Hilbert spaceof total dimension D. If the set S is completable in H or a locally extended Hilbert space Hext, then the density matrixis separable, where I is the identity matrix of rank D.Lp\u2009\u2212\u20094 members that cannot be locally distinguished in a general m\u2009\u2297\u2009n (m\u2009\u2265\u20093 and n\u2009\u2265\u20093) quantum system and give a proof for its indistinguishability.Now we construct a completable product basis with 4theorem 1. In an m\u2009\u2297\u2009n quantum system, the 4p\u2009\u2212\u20094 orthogonal product statescannot be perfectly distinguished by LOCC, where m\u2009\u2265\u20093, n\u2009\u2265\u20093, p is an arbitrary integer from 3 to min, j\u2009=\u2009i\u2009+\u20091 when i\u2009=\u20091, \u00b7\u00b7\u00b7, p\u2009\u2212\u20092 and j\u2009=\u20091 while i\u2009=\u2009p\u2009\u2212\u20091.Proof. Many proof techniques are borrowed from Ref. m \u00d7 m POVM elements Mt\u2009\u2297\u2009IB|\u03c8i\u232a: i\u2009=\u20091, \u00b7\u00b7\u00b7, 4p\u2009\u2212\u20094} should be mutually orthogonal. For the states |\u03c8i\u232a and |\u03c8j\u232a, where 1\u2009\u2264\u2009i\u2009\u2264\u2009p\u2009\u2212\u20091, 1\u2009\u2264\u2009j\u2009\u2264\u2009p\u2009\u2212\u20091 and i\u2009\u2260\u2009j, we have i\u2009\u2264\u2009p\u2009\u2212\u20091, 1\u2009\u2264\u2009j\u2009\u2264\u2009p\u2009\u2212\u20091 and i\u2009\u2260\u2009j. For the states j\u2009=\u2009i\u2009+\u20091 when i\u2009=\u20091, \u00b7\u00b7\u00b7, p\u2009\u2212\u20092 and j\u2009=\u20091 when i\u2009=\u2009p\u2009\u2212\u20091, we can get j\u2009=\u20091, 2, \u00b7\u00b7\u00b7, p\u2009\u2212\u20091. For the states j\u2009=\u2009i\u2009+\u20091 when i\u2009=\u20091, \u00b7\u00b7\u00b7, p\u2009\u2212\u20092 and j\u2009=\u20091 while i\u2009=\u2009p\u2009\u2212\u20091, we have i\u2009=\u20091, \u00b7\u00b7\u00b7, p\u2009\u2212\u20091.The post measurement states {Therefore, we haveMt for each of the 4p\u2009\u2212\u20094 states. It is easy to see that t\u2009=\u20091, 2, \u00b7\u00b7\u00b7, l, where p\u2009\u2212\u20094 states can lead to the outcome that is corresponding to Mt with the same probability Mt\u2009\u2297\u2009IB\u2009|\u2009t\u2009=\u20091, 2, \u00b7\u00b7\u00b7, l} is trivial to the 4p\u2009\u2212\u20094 states. In other words, Alice cannot get any information about which the measured state will be by the measurement {Mt}. Thus Alice cannot go first. In fact, a similar argument can be used to exhibit that Bob faces the same dilemma, i.e., he cannot gain any useful information by a nondisturbing measurement, either. Therefore, they cannot perfectly distinguish these states by LOCC. In other words, the 4p\u2009\u2212\u20094 states cannot be perfectly distinguished by LOCC. This completes the proof.Now we consider the probability of the measurement outcome corresponding to the measurement operator In general, the local indistinguishability of an incomplete PB are proved by Definition 4422p\u2009\u2212\u20094 states of (1) can become a completed orthogonal product basis in m\u2009\u2297\u2009n (m\u2009\u2265\u20093 and n\u2009\u2265\u20093) by adding the following mn\u2009\u2212\u20094p\u2009+\u20094 states:It is noted that the product basis (1) is completable since the 4p can be an arbitrary integer from 3 to min. That is, we actually construct a series of orthogonal product bases that are locally indistinguishable in m\u2009\u2297\u2009n (m\u2009\u2265\u20093 and n\u2009\u2265\u20093). In particular, we have the following corollary by Theorem 1 when p\u2009=\u20093.From Theorem 1, we know that the parameter corollary 1. In an m\u2009\u2297\u2009n quantum system, the eight orthogonal product statescannot be perfectly distinguished by LOCC, where m\u2009\u2265\u20093 and n\u2009\u2265\u20093.d\u2009\u2297\u2009d quantum system when m\u2009=\u2009n\u2009=\u2009d\u2009\u2265\u20093. This fact shows that the conjectured\u2009\u2212\u20091) product states is locally distinguishable in a d\u2009\u2297\u2009d (d\u2009\u2265\u20093) quantum system, is not true. By Refs m\u2009\u2297\u2009n (m\u2009\u2265\u20093 and n\u2009\u2265\u20093).As a special case, we can get eight states (2) that cannot be perfectly distinguished by LOCC in a p\u2009\u2212\u20091 members that cannot be perfectly distinguished by LOCC in an m\u2009\u2297\u2009n quantum system, where m\u2009\u2265\u20093, n\u2009\u2265\u20093 and 3\u2009\u2264\u2009p\u2009\u2264\u2009min. Then we give a simple proof for its local indistinguishability.Now we give a small orthogonal product basis with 2theorem 2. In an m\u2009\u2297\u2009n quantum system, the 2p\u2009\u2212\u20091 orthogonal product statescannot be perfectly distinguished by LOCC, where m\u2009\u2265\u20093, n\u2009\u2265\u20093, p is an arbitrary integer from 3 to min, j\u2009=\u2009i\u2009+\u20091 when i\u2009=\u20091, \u00b7\u00b7\u00b7, p\u2009\u2212\u20092 and j\u2009=\u20091 while i\u2009=\u2009p\u2009\u2212\u20091.Proof. Similar to the proof of Theorem 1, one of the two parties (Alice and Bob) has to start with a nondisturbing measurement to distinguish these states, i.e., the postmeasurement states should be mutually orthogonal. Without loss of generality, suppose that Alice goes first with a set of general m \u00d7 m POVM elements t\u2009=\u20091, \u00b7\u00b7\u00b7, l), wherei\u2009\u2264\u2009p\u2009\u2212\u20091, 1\u2009\u2264\u2009j\u2009\u2264\u2009p\u2009\u2212\u20091 and i\u2009\u2260\u2009j, and j\u2009=\u20091, 2, \u00b7\u00b7\u00b7, p\u2009\u2212\u20091 by the same way as the proof of Theorem 1 since the postmeasurement states should be mutually orthogonal. For the states \u03c8p\u221212\u232a, where j\u2009=\u2009i\u2009+\u20091 when i\u2009=\u20091, \u00b7\u00b7\u00b7, p\u2009\u2212\u20092 and j\u2009=\u20091 while i\u2009=\u2009p\u2009\u2212\u20091, we have i\u2009=\u20091, 2, \u00b7\u00b7\u00b7, p\u2009\u2212\u20091. Therefore, we haveWe can get Mt for each of the 2p\u2009\u2212\u20091 states. It is easy to see that t\u2009=\u20091, 2, \u00b7\u00b7\u00b7, l, where p\u2009\u2212\u20091 states can lead to the outcome that is corresponding to Mt with the same probability Mt\u2009\u2297\u2009IB\u2009|\u2009t\u2009=\u20091, 2, \u00b7\u00b7\u00b7, l} is trivial to the 2p\u2009\u2212\u20091 states. That is, Alice cannot get any useful information about which the measured state will be by the measurement {Mt}. In fact, if Bob goes first with a nondisturbing measurement, they cannot distinguish the 2p\u2009\u2212\u20091 states, either. Therefore, the 2p\u2009\u2212\u20091 states cannot be perfectly distinguished by LOCC. This completes the proof. \u25a1Now we consider the probability of the measurement outcome corresponding to the measurement operator p can be an arbitrary integer from 3 to min in Theorem 2. That means that we construct a series of orthogonal product bases that are locally indistinguishable in m\u2009\u2297\u2009n (m\u2009\u2265\u20093 and n\u2009\u2265\u20093). We have the following corollary directly by Theorem 2 when p\u2009=\u20093.The parameter corollary 2. In an m\u2009\u2297\u2009n quantum system, the five orthogonal product statesare locally indistinguishable, where m\u2009\u2265\u20093 and n\u2009\u2265\u20093.m\u2009=\u2009n\u2009=\u20093, the five states of (4) form a UPB. In Ref. et al. exhibits two results. One is that a UPB is not completable even in a locally extended Hilbert space. The other is that if a set of orthogonal product states is exactly measurable by LOCC, then the set can be completed in some extended space. Thus it is obvious that the five states of (4) in m\u2009\u2297\u2009n are locally indistinguishable by the two results, which is coincident with Corollary 2. Since any four orthogonal product states are shown to be locally distinguishableWhen p\u2009\u2212\u20091 states of Theorem 2 are uncompletable in the m\u2009\u2297\u2009n quantum system, where m\u2009\u2265\u20093, n\u2009\u2265\u20093 and p is an arbitrary integer from 3 to min. By the analysis of the last paragraph, we know that the 2p\u2009\u2212\u20091 states of Theorem 2 are uncompletable in the m\u2009\u2297\u2009n (m\u2009\u2265\u20093 and n\u2009\u2265\u20093) quantum system when p\u2009=\u20093. Then we prove that the 2p\u2009\u2212\u20091 states of Theorem 2 are uncompletable in the m\u2009\u2297\u2009n (m\u2009\u2265\u20094 and n\u2009\u2265\u20094) quantum system when p\u2009=\u20094. It is noted that some proof techniques are borrowed from Ref. Now we consider whether or not the 2S\u2009=\u2009{|\u03c81\u232a, |\u03c82\u232a, \u00b7\u00b7\u00b7, |\u03c87\u232a}. The density matrix S, which are not all mutually orthogonal:Let S is a SUCPB because p\u2009\u2212\u20091 states of Theorem 2 are uncompletable in the m\u2009\u2297\u2009n (m\u2009\u2265\u20094 and n\u2009\u2265\u20094) quantum system when p\u2009=\u20094. On the other hand, we can prove that the 2p\u2009\u2212\u20091 states of Theorem 2 are uncompletable in the m\u2009\u2297\u2009n (m\u2009\u2265\u20095 and n\u2009\u2265\u20095) quantum system by the same method, where 5\u2009\u2264\u2009p\u2009\u2264\u2009min. By Theorem 2, we get the following 2p\u2009\u2212\u20091 states in the p\u2009\u2297\u2009p (p\u2009\u2265\u20095) quantum system.These six vectors are not enough to span the full Hilbert space j\u2009=\u2009i\u2009+\u20091 when i\u2009=\u20091, \u00b7\u00b7\u00b7, p\u2009\u2212\u20092 and j\u2009=\u20091 while i\u2009=\u2009p\u2009\u2212\u20091. Let S\u2032\u2009=\u2009{|\u03c81\u232a, |\u03c82\u232a, \u00b7\u00b7\u00b7, |\u03c8p\u221212\u232a}. The density matrix p2\u2009\u2212\u2009(2p\u2009\u2212\u20091)\u2009=\u2009(p\u2009\u2212\u20091)2. We can enumerate the product states that are orthogonal to the members of S\u2032, which are not all mutually orthogonal:where i\u2009=\u20091, 2, \u00b7\u00b7\u00b7, p\u2009\u2212\u20091; j\u2009=\u2009i\u2009+\u20091 for i\u2009=\u20091, 2, \u00b7\u00b7\u00b7, p\u2009\u2212\u20092 while j\u2009=\u20091 for i\u2009=\u2009p\u2009\u2212\u20091; and j\u2009+\u20091\u2009=\u2009i\u2009+\u20092 for i\u2009=\u20091, 2, \u00b7\u00b7\u00b7, p\u2009\u2212\u20093 while j\u2009+\u20091\u2009=\u20091 for i\u2009=\u2009p\u2009\u2212\u20092 and j\u2009+\u20091\u2009=\u20092 for i\u2009=\u2009p\u2009\u2212\u20091. These 2p\u2009\u2212\u20092 vectors are not enough to span the full Hilbert space p\u2009\u2212\u20092 product states, whereas p\u2009\u2212\u20091)2. Therefore S\u2032 is a SUCPB because p\u2009\u2212\u20091 states of Theorem 2 are uncompletable in the m\u2009\u2297\u2009n (m\u2009\u2265\u20095 and n\u2009\u2265\u20095) quantum system when 5\u2009\u2264\u2009p\u2009\u2264\u2009min. Therefore, the 2p\u2009\u2212\u20091 orthogonal product states of (3) are uncompletable in the m\u2009\u2297\u2009n quantum system, where m\u2009\u2265\u20093, n\u2009\u2265\u20093 and p is an arbitrary integer from 3 to min.where p\u2009\u2212\u20094 members that cannot be perfectly distinguished by LOCC in an m\u2009\u2297\u2009n quantum system, where m\u2009\u2265\u20093, n\u2009\u2265\u20093 and p is an arbitrary integer from 3 to min, and give a simple but quite effective proof. As a special case, we get eight orthogonal product states that can be completable but cannot be locally distinguished in m\u2009\u2297\u2009n (m\u2009\u2265\u20093 and n\u2009\u2265\u20093). On the other hand, we construct a samll locally indistinguishable orthogonal product basis with 2p\u2009\u2212\u20091 members in m\u2009\u2297\u2009n, which are uncompletable, where m\u2009\u2265\u20093, n\u2009\u2265\u20093 and p is an arbitrary integer from 3 to min. Our work is useful to understand the structures both of completable and uncompletable product bases that cannot be distinguished by LOCC in arbitrary bipartite quantum system.In this paper, we construct a completable orthogonal product basis with 4How to cite this article: Xu, G.-B. et al. Locally indistinguishable orthogonal product bases in arbitrary bipartite quantum system. Sci. Rep.6, 31048; doi: 10.1038/srep31048 (2016)."}
{"text": "By using Tsuji's characteristic, we investigate uniqueness of meromorphic functions in an angular domain dealing with the shared set, which is different from the set of the paper  and obtain a series of results about the unique range set of meromorphic functions in angular domain. The purpose of this paper is to deal with the uniqueness problem of meromorphic functions sharing one set in an angular domain by using Tsuji's characteristic. Thus, the notation and theory of Nevanlinna see , 2) abou abou2]) C to denote the open complex plane, \u03a9(\u2282C) to denote an angular domain.We use IM theorem of two meromorphic functions sharing five distinct values.In 1929, Nevanlinna (see ) first if and g are two nonconstant meromorphic functions that share five distinct values a1, a2, a3, a4, and a5\u2009\u2009IM in C, then f(z) \u2261 g(z).If C. In recent years, there are many results on the uniqueness of meromorphic function in an angular domain sharing values and sets . In 200ets see \u201316). Zha. ZhaC. Iets ..C. In reS be a set of distinct elements in \u03a9\u2286C. Definefa(z) = f(z) \u2212 a if a \u2208 C and f\u221e(z) = 1/f(z). We also defineLet f and g be two nonconstant meromorphic functions in C. If E = E, we say f and g share the set S\u2009\u2009CM (counting multiplicities) in \u03a9. If f and g share the set S\u2009\u2009IM (ignoring multiplicities) in \u03a9. In particular, when S = {a}, where f and g share the value a\u2009\u2009CM in \u03a9 if E = E, and we say f and g share the value a\u2009\u2009IM in \u03a9 if \u03a9 = C, we give the simple notation as before, Let  on (see ).In 2006, Lin et al.  dealt wiS1 = {\u221e}, S2 = {\u03c9 | \u03c9n\u22121(\u03c9 + a) \u2212 b = 0}, and S3 = {0}, where n(\u22654) is an integer and a, b are two nonzero constants, such that the algebraic equation \u03c9n\u22121(\u03c9 + a) \u2212 b = 0 has no multiple roots. Assume that f is a meromorphic function of lower order \u03bc(f) \u2208  in \u03b4 : = \u03b4 > 0 for some \u03c3 < \u221e with \u03bc(f) \u2264 \u03c3 \u2264 \u03bb(f), there exists an angular domain \u03a9 = \u03a9 : = {z : \u03b1 < arg\u2061z < \u03b2} with 0 \u2264 \u03b1 < \u03b2 \u2264 2\u03c0 andE = E and E = E\u2009\u2009 hold for a meromorphic function g of finite order or, more generally, with the growth satisfying either log\u2061T = O) orE1 is a set of finite linear measures, then f \u2261 g.Let In 2011, Chen and Lin  further S1 and S2 be defined as in n \u2265 8 be an integer. Assume that f is a meromorphic function of lower order \u03bc(f) \u2208  in C and \u0398 > 2/(n \u2212 1) and that g is a meromorphic function of finite order or, more generally, with the growth satisfying either log\u2061T = O) or condition (\u03c3 < \u221e with \u03bc(f) \u2264 \u03c3 \u2264 \u03bb(f), there exists an angular domain \u03a9 = \u03a9 with 0 \u2264 \u03b1 < \u03b2 \u2264 2\u03c0 and condition  = E\u2009\u2009, then f \u2261 g.Let ondition . Then, fondition , such thIM theorem of Nevanlinna's to an angular domain. Tsuji's characteristic will be introduced in In 2010, Zheng  proved tf(z) and g(z) be both meromorphic functions in an angular domain \u03a9 = {z : \u03b1 < arg\u2061z < \u03b2} with 0 \u2264 \u03b1 < \u03b2 \u2264 2\u03c0, and let f(z) be transcendental in Tsuji's sense. Assume that aj\u2009\u2009 are 5 distinct complex numbers. If f(z) \u2261 g(z).Let S = {w \u2208 A : P1(w) = 0}, wherec be a complex number satisfying c \u2260 0,1, and obtain the following results.In this paper, we will focus on the uniqueness problem of shared set of meromorphic functions in an angular domain by using Tsuji's characteristic. In fact, we will study the uniqueness of meromorphic functions in an angular domain sharing one set of the form f(z) and g(z) be both meromorphic functions in an angular domain \u03a9 = {z : \u03b1 < arg\u2009\u2009z < \u03b2} with 0 \u2264 \u03b1 < \u03b2 \u2264 2\u03c0, and let f(z) be transcendental in Tsuji's sense. If E = E and n is an integer \u226511, then f \u2261 g.Let S is called a unique range set for meromorphic functions in an angular domain \u03a9, if for any two nonconstant meromorphic functions f and g the condition E = E implies f \u2261 g. We denote by \u266fS the cardinality of a set S. Thus, from A set S with \u266fS = 11, such that any two meromorphic functions f and g in an angular domain \u03a9 which are transcendental in Tsuji's sense must be identical if E = E.There exists one finite set f(z) and g(z) be both meromorphic functions in an angular domain \u03a9 = {z : \u03b1 < arg\u2009\u2009z < \u03b2} with 0 \u2264 \u03b1 < \u03b2 \u2264 2\u03c0, and let f(z) be transcendental in Tsuji's sense. If E = E, \u0398T > 3/4, \u0398T > 3/4, and n is an integer \u22657, then f \u2261 g.Let S with \u266fS = 7, such that any two analytic functions f and g in \u03a9 which are transcendental in Tsuji sense must be identical if E1 = E1.There exists one finite set f(z) and g(z) be both meromorphic functions in an angular domain \u03a9 = {z : \u03b1 < arg\u2061z < \u03b2} with 0 \u2264 \u03b1 < \u03b2 \u2264 2\u03c0, and let f(z) be transcendental in Tsuji's sense. If E1 = E1 and n is an integer \u226515, then f \u2261 g.Let S is called a unique range set with weight 1 for meromorphic functions in \u03a9, if for any two nonconstant meromorphic functions f and g the condition E1 = E1 implies f \u2261 g. Thus, from A set S with \u266fS = 15, such that any two meromorphic functions f and g in an angular domain \u03a9 which are transcendental in Tsuji's sense must be identical if E1 = E1.There exists one finite set f(z) and g(z) be both meromorphic functions in an angular domain \u03a9 = {z : \u03b1 < arg\u2061z < \u03b2} with 0 \u2264 \u03b1 < \u03b2 \u2264 2\u03c0, and let f(z) be transcendental in Tsuji's sense. If E1 = E1, \u0398T > 5/6, \u0398T > 5/6, and n is an integer \u22659, then f \u2261 g.Let From S with \u266fS = 9, such that any two analytic functions f and g in \u03a9 which are transcendental in Tsuji sense must be identical if E1 = E1.There exists one finite set f in Tsuji sense.We found that the conclusions of Theorems Thus, a question arises naturally, whether the conclusions of these theorems and corollaries hold for a general function in an angular domain.For the above question, we can get the following theorem.f(z) is transcendental in Tsuji sense. Assume that, for some \u03b5 > 0,\u03c9 = \u03c0/(\u03b2 \u2212 \u03b1), N = \u222b1r/t)dt, and n is the number of poles of f(z) in \u03a9\u2229{z : 1 < |z| \u2264 t}. Then f(z) \u2261 g(z).Let the assumptions of Theorems f in an angular domain \u03a9 and \u03c9 = \u03c0/(\u03b2 \u2212 \u03b1), we definebn = |bn|ei\u03b2n are the poles of f(z) in \u039e = {z = rei\u03b8 : \u03b1 < \u03b8 < \u03b2, 1 < t \u2264 r(sin\u2061(\u03c9(\u03b2n \u2212 \u03b1)))\u03c9\u22121} appearing often according to their multiplicities and then Tsuji characteristic of f isn\u03b1,\u03b2 the number of poles of f(z) in \u039e, thenbn occurs in the sum \u2211bn|<r(sin(\u03c9(\u03b2n\u2212\u03b1)))\u03c9\u221211<| only once, and we denote it by f in \u03a9 and for all complex numbers a, iff is called transcendental with respect to the Tsuji characteristic ).teristic , and we on, from , we haveq distinct points Q\u03b1,\u03b2 satisfies  is the counting function of the zeros of f\u2032 in \u03a9 where f does not take any of the values aj\u2009\u2009.In fact, from  and 16]16], we catisfies  and N\u03b1,f(z) is a meromorphic function in \u03a9. Then for 0 < r < R, one hasK is a constant independent of r and R.Assume that M, N, Q, T, and N0 instead of M\u03b1,\u03b2, N\u03b1,\u03b2, Q\u03b1,\u03b2, T\u03b1,\u03b2, and N\u03b1,\u03b20, respectively.For sake of simplicity, we omit the subscript in all notations and use \u03a9 as follows.By a similar discussion as in , one canf(z) be a meromorphic function in \u03a9. Then for all irreducible rational function R in f with coefficients meromorphic and small with respect to f in \u03a9, one hasQ is stated as in  in f.Let ed as in  and d isf is a nonconstant meromorphic function in \u03a9. ThenQ is stated as in , we haveSincethat is,N\u2264).SinceT=MNext, we will give two main lemmas of this paper as follows.F and G be transcendental meromorphic functions in \u03a9 in Tsuji sence satisfying E = E, and let c1, c2,\u2026, cq be q(\u22652) distinct nonzero complex numbers. IfN1) is the counting function which only counts simple zeros of the function \u00b7 in \u039e, and I is some set of r of infinite linear measure, thena, b, c, d \u2208 C are constants with ad \u2212 bc \u2260 0.Let SetH\u22620; from Q(r) : = o{T(r)} and T(r) = max\u2061{T, T}. Since E = E and by an elementary calculation, we can conclude that if z0 is a common simple zero of F and G in \u03a9, then H(z0) = 0. Thus, from (N1)(r) = N1) = N1). The poles of H in \u03a9 can only occur at zeros of F\u2032 and G\u2032 in \u03a9 or poles of F and G in \u03a9. Moreover, H only has simple zeros in \u03a9. Hence, from (r)\u2264N \u2261 0; that is,a, b, c, d \u2208 C and ad \u2212 bc \u2260 0.Sincehen from  and 28)(28)N\u00af+r).From qT\u2264N\u2209 E thatq{T+),since\u2211j=1qN\u00af(2r).From  and 33)(28)N\u00af = E1, and let c1, c2,\u2026, cq be q(\u22652) distinct nonzero complex numbers. IfI are stated as in a, b, c, d \u2208 C are constants with ad \u2212 bc \u2260 0.Let H be stated as in the proof of E1 = E1, it follows thatr \u2209 E,f, g are transcendental in Tsuji sense in \u03a9, it follows thatH(z) \u2261 0; that is,a, b, c, d \u2208 C and ad \u2212 bc \u2260 0.Let r).From  and 40)H be statThe following result can be derived from the proof of Frank-Reinders' theorem in .n \u2265 6 andP(w) is a unique polynomial for transcendental meromorphic functions; that is, for any two transcendental meromorphic functions f and g in Tsuji sense, P(f) \u2261 P(g) implies f \u2261 g.Let f(z) be a meromorphic function in \u03a9, for any real number \u03b5 > 0, \u03a9\u03b5 = \u03a9. Then for \u03b5 > 0, one hasc < 1 is a constant depending on \u03b5, \u03c9 = (\u03c0/(\u03b2 \u2212 \u03b1)), N = \u222b1r/t)dt, and n is the number of poles of f(z) in \u03a9\u2229{z : 1 < |z| \u2264 t}.Let P1(w), we have P1(1) = 1 \u2212 c : = c1 \u2260 0, P1(0) = \u2212c : = c2 \u2260 0, andQ1, Q2 are polynomials of degrees n \u2212 3 and 2, respectively. We also see that Qi\u2009\u2009 and P1 have only simple zeros.From the definition of F and G be defined as F = P1(f) and G = P1(g). Since E = E, we have E = E. From  and bj\u2009\u2009 are the zeros of Q1(w) and Q2(w) in \u03a9, respectively.Let G). From  and 48)F and G bT = nT + Q(r). Thus, combining  is nonempty and E = E, we have b = 0, a \u2260 0. HenceA = c/a, B = d/a \u2260 0.From , we haveombining  and 50)(50)N\u00af and (P1(g) + B/A in \u03a9 has a multiplicity of at least n. Here, three following subcases will be discussed.From the definition of 1(w) and , we can ai \u2260 0,1 are distinct values. It follows thatn \u2212 2 values satisfying the above inequality. Thus, from . It follows that every zero of g in \u03a9 has a multiplicity of at least 2 and every zero of g \u2212 bi\u2009\u2009 in \u03a9 has a multiplicity of at least n. Then, by g is transcendental in Tsuji sense in \u03a9 and n \u2265 11, we can get a contradiction.From , we haveBy using the same argument as in B \u2260 1, from  \u2212 c2/B has at least n \u2212 2 distinct zeros\u2009\u2009e1, e2,\u2026, en\u22122. Then, by If  1, from , we haveg).From  and 60)B \u2260 1, fr2, from , it folln \u2265 11 and f is transcendental in Tsuji sense in \u03a9, we can get a contradiction.By applying A = 0 and B = 1; that is, P1(f) = P1(g). Notting the form of P1(w), we can get that P(f) = P(g). Then, by f \u2261 g.Thus, we have Therefore, the proof of T > 3/4 and \u0398T > 3/4, it follows thatn \u2265 7, we getF \u2261 (aG + b)/(cG + d), where a, b, c, d \u2208 C and ad \u2212 bc \u2260 0. Thus, by using the same argument as in Since \u0398applying , from (5applying  and 52)T >E1 = E1, we have E1 = E1. From  are the distinct zeros of P1(w). And from  = nT + Q(r) and n \u2265 15, we havea, b, c, d \u2208 C and ad \u2212 bc \u2260 0. By using arguments similar to that in proof of f \u2261 g.Since G). From \u201348), we, weE1(SAnd from  and 65)E1E1 > 5/6 and \u0398T > 5/6, it follows thatn \u2265 9, we getF \u2261 (aG + b)/(cG + d), where a, b, c, d \u2208 C and ad \u2212 bc \u2260 0. Thus, by using the same argument as in Since \u0398ows thatlimsup\u2061r\u2192applying  and 68)T >Hence, the proof of f is transcendental in Tsuji sense, then the conclusions of Since condition  implies"}
{"text": "X2 constructed as finite  linear combinations of functions denoted as , where  is a groupoid (binary system) and \u03bc is a fuzzy subset of X and where : = \u03bc(x\u2217y) \u2212 min\u2061{\u03bc(x), \u03bc(y)}. Many properties, for example, \u03bc being a fuzzy subgroupoid of X, \u2217), can be restated as some properties of . Thus, the context provided opens up ways to consider well-known concepts in a new light, with new ways to prove known results as well as to provide new questions and new results. Among these are identifications of many subsemigroups and left ideals of (Bin(X), \u25a1) for example.We discuss properties of a class of real-valued functions on a set  Fuzzy Commutative Algebra, presented a fuzzy ideal theory of commutative rings, and applied the results to the solution of fuzzy intersection equations. The book included all the important work that has been done on L-subspaces of a vector space and on L-subfields of a field.The notion of a fuzzy subset of a set was introduced by Zadeh . His semX) and obtained a semigroup structure. Fayoumi , where  is a groupoid (binary system) and \u03bc is a fuzzy subset of X and where : = \u03bc(x\u2217y) \u2212 min\u2061{\u03bc(x), \u03bc(y)}. Many properties, for example, \u03bc being a fuzzy subgroupoid of , can be restated as some properties of . Thus, the context provided opens up ways to consider well-known concepts in a new light, with new ways to prove known results as well as to provide new questions and new results. Among these are identifications of many subsemigroups and left ideals of (Bin(X), \u25a1), for example.In this paper, we discuss properties of a class of real valued functions on a set X, we let Bin(X) denote the collection of all groupoids , where \u2217 : X \u00d7 X \u2192 X is a map and where \u2217 is written in the usual product form. Given elements  and  of Bin(X), define a product \u201c\u25a1\u201d on these groupoids as follows:x, y \u2208 X. Using that notion, Kim and Neggers proved the following theorem.Given a nonempty set X), \u25a1) is a semigroup; that is, operation \u201c\u25a1\u201d as defined in general is associative. Furthermore, the left zero semigroup is the identity for this operation. a fuzzy subset \u03bc of X, that is, a mapping \u03bc : X \u2192 , and (ii) a groupoid (binary system) , where \u2217 : X2 \u2192 X is a mapping. It is thus possible to consider a fuzzy subgroupoid of a groupoid as a composition :\u2192, where \u03bc and  interact to satisfy the condition:x, y \u2208 X.Given a set  the intersection between a fuzzy subset \u03bc and a groupoid  as given by the function denoted by  : X2 \u2192 R, wherex, y \u2208 X.One might easily conceive related notions such as \u201can almost fuzzy subgroupoid of a fuzzy groupoid,\u201d which though here unspecified will be considered in what follows. Since \u201calmost\u201d itself is a typical \u201cfuzzy math\u201d notion, we will considerX, \u2217); \u03bc] need not be a fuzzy subset of .Note that intersection  = \u03bc(x + y) \u2212 min\u2061{\u03bc(x), \u03bc(y)} = (x + y) \u2212 \u230ax + y\u230b \u2212 min\u2061{x \u2212 \u230ax\u230b, y \u2212 \u230ay\u230b}. If we let x = y = 1/2, then  = \u22121/2. Hence,  is not a fuzzy subset of .Let X be set R of all real numbers. Define a binary operation \u2217 on R by x\u2217y : = x \u2212 \u230ax\u230b. Then \u03bc(x\u2217y) = \u03bc(x \u2212 \u230ax\u230b) = (x \u2212 \u230ax\u230b) \u2212 \u230ax \u2212 \u230ax\u230b\u230b = x \u2212 \u230ax\u230b = \u03bc(x) for all x, y \u2208 X. It follows that  = \u03bc(x\u2217y) \u2212 min\u2061{\u03bc(x), \u03bc(y)} = \u03bc(x) \u2212 min\u2061{\u03bc(x), \u03bc(y)} \u2265 0. Hence,  is a fuzzy subset of .Let X be set R of all real numbers, and let + be the usual addition on R and let x\u2217y : = x \u2212 \u230ax\u230b for all x, y \u2208 R. Define : = \u25a1. Then x\u25a1y = (x\u2217y)+(y\u2217x) = x \u2212 \u230ax\u230b + y \u2212 \u230ay\u230b. It follows that \u03bc(x\u25a1y) = x + y \u2212 (\u230ax\u230b + \u230ay\u230b) \u2212 \u230ax + y \u2212 (\u230ax\u230b + \u230ay\u230b)\u230b for all x, y \u2208 X. Assume that x \u2212 \u230ax\u230b = y \u2212 \u230ay\u230b = 0.6. Then \u03bc(x\u25a1y) = 1.2 \u2212 1 = 0.2 and hence  = \u03bc(x\u25a1y) \u2212 min\u2061{\u03bc(x), \u03bc(y)} = \u22120.4. Hence,  is not a fuzzy subgroupoid of .Let X, \u2217); \u03bc], note that the smallest value obtainable is when \u03bc(x\u2217y) = 0 and \u03bc(x) = \u03bc(y) = 1. If such a pair  exists, then  = \u03bc(x\u2217y) \u2212 min\u2061\u2061{\u03bc(x), \u03bc(y)} = \u22121. We call such a pair  a \u03bc-orthogonal pair on .Given an intersection  be the characteristic function of the rationals; that is, if x is rational, then \u03bc(x) = \u03c7Q(x) = 1 and if x is irrational, then \u03bc(x) = 0. Suppose that x and y \u2260 0 are both rational. Then X, \u2217); \u03bc] = \u03bc(x\u2217y) \u2212 min\u2061{\u03bc(x), \u03bc(y)} = 0 \u2212 1 = \u22121. Thus,  is an \u03c7Q-orthogonal pair on .Let X, \u2217) \u2208Bin(X), if \u03bc is a fuzzy subalgebra of , then there is no pair  \u2208 X2 such that  is a \u03bc-orthogonal pair on .Given , then \u03bc(x\u2217y) \u2265 min\u2061{\u03bc(x), \u03bc(y)} for all x, y \u2208 X. Assume that  is a \u03bc-orthogonal pair on . Then \u03bc(x\u2217y) \u2212 min\u2061{\u03bc(x), \u03bc(y)} = \u22121. It follows that 0 \u2264 \u03bc(x\u2217y) = min\u2061{\u03bc(x), \u03bc(y)} \u2212 1 \u2264 0, which shows that min\u2061{\u03bc(x), \u03bc(y)} = 1. Since \u03bc is a fuzzy subalgebra of , we obtain \u03bc(x\u2217y) = \u03bc(x) = \u03bc(y) = 1. Hence, 0 = \u03bc(x\u2217y) \u2212 min\u2061{\u03bc(x), \u03bc(y)} = \u22121, a contradiction.If X, \u2217); \u03bc], a pair  \u2208 X2 is said to be a \u03bc-parallel pair on  if  = 1. In \u03bc-parallel pair on . In fact, Given an intersection , we define \u03b1 byx, y \u2208 X. Using these notions, we will consider that a generating set for a vector space of groupoids  \u2208 Bin(X) can be expressed as finite sums \u2211i=1n\u03bbi, where \u03bbi \u2208 R for i = 1,\u2026, n, and \u03bci : \u2192 is a fuzzy subset of  and where ; \u03bci]): = \u03b1 as usual for real-valued functions. If we let S : = \u2211i=1n\u03bbi, then \u2212\u2211i=1n | \u03bbi | \u2264S \u2264 \u2211i=1n | \u03bbi| for all x, y \u2208 X. Given a fuzzy subset \u03bc of , we define a map 1 \u2212 \u03bc : X \u2192  by (1 \u2212 \u03bc)(x): = 1 \u2212 \u03bc(x) for all x \u2208 X.Given an element X, \u2217) \u2208Bin(X), if \u03bc is a fuzzy subset of , thenx, y \u2208 X.Given ; \u03bc]+. Then, for all x, y \u2208 X, we haveLet S in X, \u2217) \u2208 Bin(X).Note that such an X, \u2217),  \u2208Bin(X), we haveGiven  \u2208Bin(X) and a fuzzy subset \u03bc : X \u2192 , there exist a groupoid  \u2208Bin(X) and a fuzzy subset \u03bd : X \u2192  such that  = max\u2061{\u03bc(x), \u03bc(y)} \u2212 min\u2061{\u03bd(x), \u03bd(y)} for all x, y \u2208 X.Given  \u2208 Bin(X) and a fuzzy subset \u03bc : X \u2192 , we define a surjective map \u03bd : X \u2192 Im\u2061(\u03bc). Since max\u2061{\u03bc(x), \u03bc(y)} \u2208 Im\u2061(\u03bc), \u03bd\u22121(max\u2061{\u03bc(x), \u03bc(y)}) \u2260 \u2205. Define a binary operation \u22c6 on X as follows: for any x, y \u2208 X, x\u22c6y : = z for some z \u2208 \u03bd\u22121(max\u2061{\u03bc(x), \u03bc(y)}. It follows that \u03bd(x\u22c6y) = max\u2061{\u03bc(x), \u03bc(y)}. This means that  = \u03bd(x\u22c6y) \u2212 min\u2061{\u03bd(x), \u03bd(y)} = max\u2061{\u03bc(x), \u03bc(y)} \u2212 min\u2061{\u03bd(x), \u03bd(y)}.Given  \u2208Bin(X) and a fuzzy subset \u03bc : X \u2192 , there exists a groupoid  \u2208Bin(X) such that  = S.Given ; \u03bd] = max\u2061{\u03bc(x), \u03bc(y)} \u2212 min\u2061{\u03bc(x), \u03bc(y)} = |\u03bc(x) \u2212 \u03bc(y)| = S by In the proof of X, \u2217) \u2208 Bin(X), we define a set F(X): = {\u03bc | \u03bc : X \u2192 :a\u2009\u2009map}. We consider the composition \u03bc\u25a1\u03bd of fuzzy subsets \u03bc, \u03bd \u2208 F(X) as a function \u03c6(x) whose variables are \u03bc(x) and \u03bd(x); that is, (\u03bc\u25a1\u03bd)(x): = \u03c6(\u03bc(x), \u03bd(x)). For example, (\u03bc\u25a1\u03bd)(x) = (1/2)(\u03bc(x) + \u03bd(x)),(\u03bc\u03bd)(x) = \u03bb\u03bc(x) + (1 \u2212 \u03bb)\u03bd(x), \u03bb \u2208 , or Given a groupoid (\u03bc\u25a1\u03bd)\u25a1\u03c1 \u2260 \u03bc\u25a1(\u03bd\u25a1\u03c1). In fact, if we define \u03bc\u25a1(\u03bd\u25a1\u03c1)](x)\u03bc \u2208 F(X) is said to be \u25a1-idempotent if \u03bc\u25a1\u03bc = \u03bc.In general, ; \u03bc], , we define a new interaction as follows:X, \u25a1) = \u25a1 and (\u03bc\u25a1\u03bd)(x): = \u03c6(\u03bc(x), \u03bd(x)) for all x \u2208 X.Given interactions \u25a1)\u25a1 = \u25a1; \u03bd]\u25a1 for any ,,  \u2208 Bin(X).It is easy to see that if  \u2208 Bin(X) and \u03bc, \u03bd \u2208 F(X), we define \u03bc\u25a1\u03bd : = (1/2)(\u03bc + \u03bd). It follows thatGiven a groupoid ; \u03bc], we have already seen that \u22121 \u2264  \u2264 1. If 0 \u2264  \u2264 1, then \u03bc is a fuzzy subgroupoid of  and conversely. Thus, we will be interested in bounds \u03b1 \u2264  \u2264 \u03b2 as classification parameters of the . Hence, we usually take \u03b1 : = inf\u2061{ | x, y \u2208 X} and \u03b2 : = sup\u2061{ | x, y \u2208 X} for the \u201cnarrowest\u201d fit.Given an interaction ,  have the following bounds: \u03b1 \u2264  \u2264 \u03b2, \u03b3 \u2264  \u2264 \u03b4, then the interaction  has the bound \u03b1 + \u03b3 \u2264  \u2264 \u03b2 + \u03b4.Given x, y \u2208 X, we let p : = min\u2061{\u03bc(x\u2217y), \u03bc(y\u2217x)} and q : = min\u2061{\u03bc(x), \u03bc(y)}. If \u03b1 \u2264  \u2264 \u03b2, \u03b3 \u2264  \u2264 \u03b4, then \u03bc((x\u2217y) \u00b7 (y\u2217x)) \u2212 min\u2061{\u03bc(x\u2217y), \u03bc(y\u2217x)}\u2208. Since \u03bc(x\u2217y) \u2212 min\u2061{\u03bc(x), \u03bc(y)}\u2208 and \u03bc(y\u2217x) \u2212 min\u2061{\u03bc(x), \u03bc(y)}\u2208, we have \u03bc(x\u2217y), \u03bc(y\u2217x)\u2208. It follows that p = min\u2061{\u03bc(x\u2217y), \u03bc(y\u2217x)}\u2208; that is, p \u2212 q \u2208 . Thus, \u03bc(x\u25a1y) \u2212 min\u2061{\u03bc(x), \u03bc(y)} = \u03bc((x\u2217y) \u00b7 (y\u2217x)) \u2212 q = \u03bc((x\u2217y) \u00b7 (y\u2217x)) \u2212 p + p \u2212 q \u2208 + = .Given \u03bc in X, \u2217)\u25a1; \u03bc) if \u03b2 + \u03b4 > 1.Note that map \u03b1 \u2264  \u2264 \u03b2 and \u03b3 \u2264  \u2264 \u03b4. If there exist \u03c3, \u03c4 \u2208 R such that |\u03bc(x\u2217y) \u2212 \u03bc(y\u2217x)|\u2208 for all x, y \u2208 X, thenLet \u03b3 \u2264  \u2264 \u03b4, thenx, y \u2208 X. If \u03b1 \u2264  \u2264 \u03b2, then\u03c3, \u03c4 \u2208 R such that |\u03bc(x\u2217y) \u2212 \u03bc(y\u2217x)|\u2208 for all x, y \u2208 X. Let \u03bc(x\u2217y) + \u03c1 = \u03bc(y\u2217x) for some \u03c1 \u2265 0. Then \u03bc(x\u2217y) \u2212 min\u2061{\u03bc(x), \u03bc(y)} + \u03c1 = \u03bc(y\u2217x) \u2212 min\u2061{\u03bc(x), \u03bc(y)} \u2264 \u03b2. It follows that \u03bc(x\u2217y) \u2212 min\u2061{\u03bc(x), \u03bc(y)} \u2264 \u03b2 \u2212 \u03c1 \u2264 \u03b2 \u2212 \u03c3. By applying \u2212mi\u03bc(y\u2217x) \u2212 min\u2061{\u03bc(x), \u03bc(y)} = \u03bc(x\u2217y) \u2212 min\u2061{\u03bc(x), \u03bc(y)} + \u03c1, we obtainSimilarly, since \u2264applying , we obta}.Using , 17), a, a\u03bc(y\u2217x)}.Using , we obta\u03bc \u2208 F(X). A groupoid  \u2208 Bin(X) is said to be \u03bc-commutative if \u03bc(x\u2217y) = \u03bc(y\u2217x) for all x, y \u2208 X.Let X, \u00b7) \u2208Bin(X) is \u03bc-commutative, then \u25a1; \u03bc) is also \u03bc-commutative for all  \u2208Bin(X).If  = \u03bc((x\u2217y) \u00b7 (y\u2217x)) = \u03bc((y\u2217x) \u00b7 (x\u2217y)) = \u03bc(y\u25a1x).For all \u03bc-commutative groupoids forms a left ideal of the semigroup (Bin(X), \u25a1).Note that X, \u2217) \u2208Bin(X), if \u03bci : X \u2192  are fuzzy subsets of  and \u03bb \u2208 , thenGiven (\u03bc : = \u03bb\u03bc1 + (1 \u2212 \u03bb)\u03bc2. Given a, b, c, d \u2208 , we have min\u2061{a + b, c + d} \u2265 min\u2061{a, c} + min\u2061{c, d}. It follows that, for all x, y \u2208 X,Let f : X2 \u2192  is said to be representable if f can be represented as  for some groupoid  \u2208 Bin(X) and a fuzzy subset \u03bc : X \u2192 . We denote it by f = .A function an}n=1\u221e be a sequence of real numbers such that, for any n and k,A, B > 0. Then {an}n=1\u221e is called a special sequence of type .Let {A\u2032 \u2265 A and B\u2032 \u2265 B, then a special sequence of type  is also a special sequence of type .If an}n=1\u221e is a special sequence of type  and {bn}n=1\u221e is a special sequence of type , then, for real numbers \u03bb, \u03b4 > 0, {\u03bban + \u03b4bn}n=1\u221e is a special sequence of type . If A = B = C = D = 1 and \u03bb + \u03b4 = 1, then \u03bbA + \u03b4C = 1 and \u03bbB + \u03b4D = 1. Such a special sequence of type  is called a standard special sequence.If {bn = an+t, t \u2265 0 and if {an}n=1\u221e is a special sequence of type , then {bn}n=1\u221e is also a special sequence of type . It follows that if {an}n=1\u221e is a special sequence, then, for all n \u2265 1, lim\u2061k\u2192\u221e(an+1 + \u22ef+an+k)/k = 0.If an = \u03b1}n=1\u221e, then lim\u2061k\u2192\u221e(an+1 + \u22ef+an+k)/k = lim\u2061k\u2192\u221e(k\u03b1/k) = \u03b1, whence, \u03b1 \u2260 0 implies that {an = \u03b1}n=1\u221e is not a special sequence.Notice that if {k\u2192\u221e[1/(n + 1) + \u22ef+1/(n + k)]/k = 0, since lim\u2061k\u2192\u221e(1/k)\u222bnn+k(dx/x) = lim\u2061k\u2192\u221e[ln\u2061(1 + k/n)/1/k] = 0. However, there is no B such that, for any n and k, [1/(n + 1) + \u22ef+1/(n + k)]/k \u2264 B/k; that is, 1/(n + 1) + \u22ef+1/(n + k) \u2264 B since \u2211n=1\u221e(1/n) diverges. Hence, the sequence {an = 1/n}n=1\u221e is not special for any pair , with A, B > 0, a pair of real numbers.Notice that lim\u2061We have the following observations.X, \u2217) be a groupoid. For x1 \u2208 X, determine a sequence d(x1): = {x1, x2,\u2026, xn,\u2026} as follows: x2 = x1\u2217x1, \u2009x3 = x2\u2217x2,\u2026, xn+1 = xn\u2217xn,\u2026.. We consider d(x1) to be a doubling sequence for x1 relative to .Let ; \u03bc], then the doubling sequence must be a standard special sequence.If f =  for some groupoid  and a fuzzy subset \u03bc : X \u2192 . If d(x1) is a doubling sequence in , then\u03bc(xn+1) \u2212 \u03bc(x1) \u2264 1, we obtain \u22121/n \u2264 (1/n)[\u2211i=1nf] \u2264 1/n. Replace x1 by xk+1,\u2026, xn by xn+k, the same inequalities are obtained. Hence, we conclude that {d(x1)} is a standard special sequence.Let f \u2265 \u03b1 > 0 or f \u2264 \u03b2 < 0, then f is not representable as f = .If f \u2265 \u03b1 > 0, then, for any expression  for some groupoid  \u2208 Bin(X) and a fuzzy subset \u03bc : X \u2192 , then we may deduce many properties of the representation. Note that \u03bc(\u03c0/4) = 0 and \u03bc((\u03c0/4)\u2217(\u03c0/4)) = 1. In fact, Sin = sin(\u03c0/2) = \u03bc((\u03c0/4)\u2217(\u03c0/4)) \u2212 \u03bc(\u03c0/4) = 1. Much other information is available via the use of trigonometric properties. Thus, for example, Sin = sin(x + y + \u03c0) = \u2212sin(x + y) = \u2212Sin = \u2212 = \u03bc(x\u2217(y + \u03c0)) \u2212 min\u2061{\u03bc(x), \u03bc(y + \u03c0)] so that \u03bc(x\u2217(y + \u03c0)) + \u03bc(x\u2217y) = min\u2061{\u03bc(x), \u03bc(y + \u03c0)} + min\u2061{\u03bc(x), \u03bc(y)}. Thus, for example, \u03bc(x) = 1 implies \u03bc(x\u2217(y + \u03c0)) + \u03bc(x\u2217y) = \u03bc(y + \u03c0) + \u03bc(y) and \u03bc(x) = 0 implies \u03bc(x\u2217(y + \u03c0)) = \u2212\u03bc(x\u2217y), whence \u03bc(x\u2217y) = 0 for any y whatsoever. Hence, if Ker\u2061\u03bc = \u03bc\u22121(0), then Ker\u2061\u03bc is a right ideal of .If we assume that f+\u22ef+f]/k must behave properly to yield at least the possibility of a representation.In the sine function case, doubling sequences may generate sequences with positive and negative values whose average over sections  has a sign pattern on {f}n=1\u221e of the following type {+, \u2212, +, +, \u2212, \u2212, +, +, +, \u2212, \u2212, \u2212, +, +, +, +,\u2026} and that f \u2265 \u03b1 > 0 when it is positive. If f is representable, that is, f = , and if {f}n=1\u221e is a doubling sequence, then for n = k(k + 1), there are k slots of positive value between i = n and i = n + k. Thus, summing over all these slots, one obtains an upper bound of value k (the number of slots). Another associated value is the average of the functional values of that segment from i = n to i = n + k, and thus at least \u03b1 > 0. This violates the condition that the doubling sequence obtained from f =  must be a standard special sequence. Hence, f is not representable.Here is another example of a negative solution to the representation problem. Suppose that {, if it is believed/known that f = , develop a method of finding a groupoid  and a fuzzy subset \u03bc such that f = . So far, the best results have been of the negative kind, but interesting nevertheless.Given f : X2 \u2192  is quite enormous with common properties to be examined. The doubling sequence technique developed above is an example of such a common property which can be applied to the existence problem of representations for function of the type addressed here.As a closing comment, we observe that the class of representable functions"}
{"text": "Res. Natl. Inst. Stand. Technol. Volume 100, Number 3, May\u2013June 1995, p. Page 211, above Eq. (3).Replace \u201cthe magnetic field within the balance chamber is vertical with the form\u2026\u201d by \u201cthe vertical magnetic field strength within the balance chamber has the form\u2026\u201d.Page 217, Eq. (8).Ia\u201d by \u201cIa\u201d.Replace \u201cPage 218, two paragraphs above Sec. 5.1.\u00b5mHmax\u201d by \u201c\u00b50Hmax\u201d.Replace \u201cPage 223, first paragraph of Sec. 11.1.Replace \u201c: it must be less than that based on a sum\u2026; it must be greater than that based on a sum\u2026\u201d by \u201c: it must be greater than that based on a sum\u2026; it must be less than that based on a sum\u2026\u201d.Page 224, Table 5, column 4, row 2.Replace \u201c0.1116\u201d by \u201c\u22120.1116\u201d.Page 224, Eq. (10) should be:Page 225, two paragraphs above the Acknowledgment.Replace \u201cThis again overestimates \u03c7\u201d by \u201cThis again underestimates \u03c7\u201d."}
{"text": "In this paper, wegeneralize the results of Azam et al. (2011), and Bhatt et al. (2011), by improving the conditionsof contraction to establish the existence and uniqueness of common fixed point for twoself-mappings on complex valued b-metric spaces. Some examples are given to illustratethe main results.Azam et al. (2011), introduce the notion of complex valued metric spaces and obtainedcommon fixed point result for mappings in the context of complex valued metric spaces. Rao et al. (2013) introduce the notion of complex valued  The concept of complex valued b-metric spaces was introduced in 2013 by Rao et al. .The pair  is called a b-metric space. The number s \u2265 1 is called the coefficient of .Let X, d) be a metric space and \u03c1 = )p, where p > 1 is a real number. Then  is a b-metric space with s = 2p\u22121.Let  = Re(z2) and Im\u2061(z1) = Im\u2061(z2),Re(z1) < Re(z2) and Im\u2061(z1) = Im\u2061(z2),Re(z1) = Re(z2) and Im\u2061(z1) < Im\u2061(z2),Re(z1) < Re(z2) and Im\u2061(z1) < Im\u2061(z2).We will write z1\u22e8\u2009\u2009z2 if z1 \u2260 z2 and one of (2), (3), and (4) is satisfied; also we will write z1\u227az2 if only (4) is satisfied.Let a, b \u2208 R and a \u2264 b, then az\u227ebz for all z \u2208 C;if z1\u22e8z2, then |z1 | <|z2|;if 0\u227ez1\u227ez2 and z2\u227az3, then z1\u227az3.if We can easily check that the following statements are held:X be a nonempty set. A function d : X \u00d7 X \u2192 C is called a complex valued metric on X if for all x, y, z \u2208 X the following conditions are satisfied:d and d = 0 if and only if x = y;0\u227ed = d;d\u227ed + d.The pair  is called a complex valued metric space.Let X = C. Define the mapping d : X \u00d7 X \u2192 C byX, d) is a complex valued metric space.Let X = C. Define the mapping d : X \u00d7 X \u2192 C byX, d) is a complex valued metric space.Let X be a nonempty set and let s \u2265 1 be a given real number. A function d : X \u00d7 X \u2192 C is called a complex valued b-metric on X if for all x, y, z \u2208 X the following conditions are satisfied:d and d = 0 if and only if x = y;0\u227ed = d;d\u227es[d + d].The pair  is called a complex valued b-metric space.Let X = . Define the mapping d : X \u00d7 X \u2192 C byX, d) is a complex valued b-metric space with s = 2.Let X, d) be a complex valued b-metric space. Consider the following.x \u2208 X is called interior point of a set A\u2286X whenever there exists 0\u227ar \u2208 C such that B: = {y \u2208 X : d\u227ar}\u2286A.A point x \u2208 X is called a limit point of a set A whenever, for every 0\u227ar \u2208 C,\u2009\u2009B\u2229(A \u2212 X) \u2260 \u2205.A point A\u2286X is called open whenever each element of A is an interior point of A.A subset A\u2286X is called closed whenever each element of A belongs to A.A subset \u03c4 on X is a family F = {B : x \u2208 X and 0\u227ar}.A subbasis for a Hausdorff topology Let  be a complex valued b-metric space and {xn} a sequence in X and x \u2208 X. Consider the following.c \u2208 C, with 0\u227ar, there is N \u2208 N such that, for all n > N,\u2009\u2009d\u227ac, then {xn} is said to be convergent, {xn} converges to x, and x is the limit point of {xn}. We denote this by lim\u2061n\u2192\u221exn = x or {xn} \u2192 x\u2009\u2009as\u2009\u2009n\u2009\u2009 \u2192 \u221e.If for every c \u2208 C, with 0\u227ar, there is N \u2208 N such that, for all n > N,\u2009\u2009d\u227ac, where m \u2208 N, then {xn} is said to be Cauchy sequence.If for every X is convergent, then  is said to be a complete complex valued b-metric space.If every Cauchy sequence in Let  be a complex valued b-metric space and let {xn} be a sequence in X. Then {xn} converges to x if and only if |d|\u21920\u2009\u2009as\u2009\u2009n \u2192 \u221e.Let  be a complex valued b-metric space and let {xn} be a sequence in X. Then {xn} is a Cauchy sequence if and only if |d|\u21920\u2009\u2009as\u2009\u2009n \u2192 \u221e, where m \u2208 N.Let  be a complete complex valued metric space and let \u03bb, \u03bc be nonnegative real numbers such that \u03bb + \u03bc < 1. Suppose that S, T : X \u2192 X are mappings satisfyingx, y \u2208 X. Then S, T have a unique common fixed point in X.Let  be a complete complex valued metric space and let S, T : X \u2192 X be mappings satisfyingx, y \u2208 X, where a \u2208 [0,1). Then S, T have a unique common fixed point in X.Let  be a complete complex valued b-metric space with the coefficient s \u2265 1 and let S, T : X \u2192 X be mappings satisfyingx, y \u2208 X, where \u03bb, \u03bc are nonnegative reals with s\u03bb + \u03bc < 1. Then S, T have a unique common fixed point in X.Let |>|d|, we gets\u03bb + \u03bc < 1 and s \u2265 1, we get \u03bb + \u03bc < 1.For any arbitrary point, 2n+1 in ; we have\u03b4 = \u03bb/(1 \u2212 \u03bc) < 1, and for all n \u2265 0, consequently, we havem > n,\u2009\u2009m,\u2009\u2009n \u2208 N, and since s\u03b4 = s\u03bb/(1 \u2212 \u03bc) < 1, we getxn} is a Cauchy sequence in X.Therefore, with  we have|d|\u22640, a contradiction with |S and T have unique common fixed point of S and T. To show this, assume that u* is another common fixed point of S and T. Thend|<|d|, a contradiction. So u = u* which proves the uniqueness of common fixed point in X. This completes the proof.Now we show that X, d) be a complete complex valued b-metric space with the coefficient s \u2265 1 and let T : X \u2192 X be a mapping satisfyingx, y \u2208 X, where \u03bb, \u03bc are nonnegative reals with s\u03bb + \u03bc < 1. Then T has a unique fixed point in X.Let  be a complete complex valued b-metric space with the coefficient s \u2265 1 and let T : X \u2192 X be a mapping satisfyingx, y \u2208 X, where \u03bb, \u03bc are nonnegative reals with s\u03bb + \u03bc < 1. Then T has a unique fixed point in X.Let |\u2264\u03bb | d|<|d|, a contradiction. So, Tu = u. HenceT is unique. This completes the proof.From ows fromd=dX = C. Define a function d : X \u00d7 X \u2192 C such thatz1 = x1 + iy1 and z2 = x2 + iy2.Let X, d) is a complete complex valued b-metric space with s = 2, it is enough to verify the triangular inequality condition.To verify that , we haveTnz = 0 for n > 1, sox, y \u2208 X and \u03bb, \u03bc \u2265 0 with 2\u03bb + \u03bc < 1. So all conditions of T.Now, define two self-mappings b-metric spaces.Our next theorem is a generalization of X, d) be a complete complex valued b-metric space with the coefficient s \u2265 1 and let S, T : X \u2192 X be mappings satisfyingx, y \u2208 X, where sa \u2208 [0,1). Then S, T have a unique common fixed point in X.Let |\u22640, a contradiction with |S and T have unique common fixed point of S and T. To show this, assume that u* is another common fixed point of S and T. Thend|\u22640, and then u = u* which proves the uniqueness of common fixed point in X. This completes the proof.Now we show that X, d) be a complete complex valued b-metric space with the coefficient s \u2265 1 and let T : X \u2192 X be a mapping satisfyingx, y \u2208 X, where sa \u2208 [0,1). Then T has a unique fixed point in X.Let  be a complete complex valued b-metric space with the coefficient s \u2265 1 and let T : X \u2192 X be a mapping satisfyingx, y \u2208 X, where sa \u2208 [0,1) and n \u2208 N. Then T has a unique fixed point in X.Let ("}
{"text": "The authors wish to make the following corrections to this paper :Mar. Drugs [We have found eight inadvertent errors in our paper published in r. Drugs . \u03946,12,15\u201d should be \u201c18:3\u03946,9,12\u201d. Line 8 and column 8 of Table 2, \u201c20:4\u03945,8,11,14,17\u201d should be \u201c20:4\u03948,11,14,17\u201d. Line 9 and column 7, \u201c\u03945/cis\u201d should be \u201c\u03944/cis\u201d. Therefore replace this incorrect table:On page 1327, Line 6 and column 9 of Table 2, \u201c18:3With the following corrected table:\u03949,12 \u2192 18:3\u03946,9,12 and 18:3\u03949,12,15 \u2192 18:3\u03946,9,12,15)-Desaturation\u201d should be \u201cThe -Desaturation\u201d. On page 1331, line 6, in the title of Section 2.4.11, \u201cThe Question of the -Desaturation\u201d should be \u201cThe Question of the -Desaturation\u201d. On page 1332, line 1, \u201cE \u03946FAD\u201d should be \u201cER\u03946FAD\u201d. On page 1335, lines 25\u201326, in the Conclusions Section, \u201cERD6FAD, and ERD5FAD\u201d should be \u201cER\u03946FAD, and ER\u03945FAD\u201d.On page 1330, line 9, \u201c18:3 and 14\u201d should be \u201c18:3 and 18:4\u201d. Line 29, in the title of Section 2.4.10, \u201cThe (18:2These changes have no material impact on the conclusions of our paper. We apologize to our readers for any inconvenience caused."}
{"text": "Based on He's variational iteration method idea, we modified the fractional variational iteration method and applied it to construct some approximate solutions of the generalized time-space fractional Schr\u00f6dinger equation (GFNLS). The fractional derivatives are described in the sense of Caputo. With the help of symbolic computation, some approximate solutions and their iterative structure of the GFNLS are investigated. Furthermore, the approximate iterative series and numerical results show that the modified fractional variational iteration method is powerful, reliable, and effective when compared with some classic traditional methods such as homotopy analysis method, homotopy perturbation method, adomian decomposition method, and variational iteration method in searching for approximate solutions of the Schr\u00f6dinger equations. In the past decades, due to the numerous applications of fractional differential equations (FDEs) in the areas of nonlinear science , many imThe variational iteration method (VIM) established in 1999 by He in  is thoroWe give some basic definitions and properties of the fractional calculus theory which are used further in this paper; we define the following fractional integral and derivatives , 21.f(x) is said to be in the space C\u03bc, where \u03bc \u2208 R, x > 0, if there exists a real number p(>\u03bc) such that f(x) = xpf1(x), where f1(x) \u2208 C[0, \u221e) and it is said to be in the space C\u03bcm if and only if fm) \u2208 C\u03bc, \u03bc \u2265 \u22121, is defined as follows:The Riemann-Liouville fractional integral operator of order \u03b1 > 0, x > 0, and f(x) \u2208 C\u22121n, the Caputo fractional derivative operator of order \u03b1 on the whole space is defined as follows:For Also one has the following properties:u = u, \u2202\u03b1u/\u2202t\u03b1 = Dt\u03b1u, \u2202\u03b22u/\u2202x\u03b22 = Dx\u03b2(Dx\u03b2u), v(x) is the trapping potential, and a, \u03b3 are the slowly increasing dispersion coefficient and nonlinear coefficient, respectively. If we select \u03b1 = \u03b2 = 1, v(x) = 0, this equation turns to the famous nonlinear Schr\u00f6dinger equations in optical fiber [Consider the following generalized time and space fractional nonlinear Schr\u00f6dinger equation with variable coefficients , 23:(5)al fiber \u201326.u0 = u = f(x), where \u03bb is a general Lagrange's multiplier which can be identified optimally with the variational theory. The function \u03bb that will be identified optimally via integration by parts [un+1, n \u2265 0, of the solution u will be readily obtained through \u03bb and any selective function u0. The initial values are usually used for choosing the zeroth approximation u0. With \u03bb determined, then several approximations uk, k = 1,2,\u2026 follow immediately. Consequently, the exact solution may be procured by using u = lim\u2061n\u2192\u221e\u2061un. The convergence of FVIM has been proved in [u|2, as we all know, a complex function u(\u03be) can be written as c(\u03be)ei\u03b8(\u03be), where c(\u03be) and \u03b8(\u03be) are real functions, noticed that |u(\u03be)|2 = |c(\u03be)|2, we can give some modification for the iteration formulation  = Asec\u2009hx; using the iteration (E\u03b1(ait\u03b1) is the Mittag-Leffler function. If we let \u03b1 = 1 in i\u2202\u03b1ui\u2202\u03b1u(7)i\u2202follows:un+1=un+ition of  = sinx, using the iteration (ck(x) = ck,0sinx + ck,1sin(x + \u03c0\u03b2) + ck,2sin(x + 2\u03c0\u03b2)+\u22ef+ck,ksin(x + k\u03c0\u03b2), ck,0 = (\u22121)k, ck,1 = (1/2)ck\u22121,1 \u2212 ck\u22121,0,\u2026, ck,k\u22121 = (1/2)ck\u22121,k\u22122 \u2212 ck\u22121,k\u22121, and ck,k = (1/2)ck\u22121,k\u22121, k \u2265 2, c0,0 = 1; c1,0 = \u22121, c1,1 = (1/2); c2,0 = 1, c2,1 = \u22121, c2,2 = (1/4); \u2026.Consider the following time-space fractional NLS equation :(20)i\u2202\u03b1follows:un+1=un+i\u03b1 = 1 and let \u03b2 = 1 in  = cos\u2061x, with the same process, we can also obtain the following exact solution of (The exact solution of  is23)uu(23)u=li\u03b2 = 1 in  can be oution of :(25)u=i\u03b2 = 1, the solution (If one selects solution  is more solution  and 25)\u03b2 = 1, thsolution  to our ku4, uabs, and u are plotted in Figures Comparisons between the real part of some numerical results and the exact solution  are summIn this paper, the MFVIM is used for finding approximate and exact solutions of the GFNLS equation with Caputo derivative. The obtained results indicate that the MFVIM is effective, convenient, and powerful method for solving nonlinear fractional complex differential equations when comparing it with some other traditional asymptotic decomposition methods such as HAM, VIM, and ADM. We believe that these methods should play an important role for finding exact and approximate solutions in the mathematical physics."}
{"text": "This review on recent research advances of the lipid peroxidation product 4-hydroxy-nonenal (HNE) has four major topics: I. the formation of HNE in various organs and tissues, II. the diverse biochemical reactions with Michael adduct formation as the most prominent one, III. the endogenous targets of HNE, primarily peptides and proteins  physiological consequences discussed), and IV. the metabolism of HNE leading to a great number of degradation products, some of which are excreted in urine and may serve as non-invasive biomarkers of oxidative stress. Table of ContentsPreface\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202622511. Lipid Peroxidation as a Free Radical Amplification Process\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202622522. Structure, Properties and Generation of HNE\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202622553. Major Reaction Mechanisms\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262257\u20033.1. Reactions of the C=C Double Bond\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262257\u2003\u20033.1.1. Michael Additions\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262257\u2003\u20033.1.2. Reduction\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262258\u2003\u20033.1.3. Epoxidation\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262259\u20033.2. Reactions of the Carbonyl Group\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262259\u2003\u20033.2.1. Acetal and Thio-Acetal Formation\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262259\u2003\u20033.2.2. Schiff-Base Formation\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262259\u2003\u20033.2.3. Oxidation\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262259\u2003\u20033.2.4. Reduction\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262260\u20033.3. Reactions of the Hydroxy Group\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202622624. Biophysical Effects\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202622625. Biochemical Targets of HNE\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262262\u20035.1. Reactions with Peptides and Proteins\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262263\u2003\u20035.1.1. Substrates\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262265\u2003\u20035.1.1.1. Glutathione\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262265\u2003\u20035.1.1.2. Carnosine\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262267\u2003\u20035.1.1.3. Thioredoxin\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262267\u2003\u20035.1.1.4. Cytochrome c\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262268\u2003\u20035.1.2. Enzymes\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262268\u2003\u20035.1.2.1. Oxidoreductases\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262269\u2003\u20035.1.2.1.1. Lactate Dehydrogenase\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262269\u2003\u20035.1.2.1.2. Glyceraldehyde-3-Phosphate Dehydrogenase (GAPDH)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262269\u2003\u20035.1.2.2. Transferases\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262270\u2003\u20035.1.2.2.1. Glutathione-S-Transferase (GST)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262270\u2003\u20035.1.2.2.2. Liver Kinase B1 (LKB1)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262270\u2003\u20035.1.2.2.3. 5'-AMP-Activated Protein Kinase (AMPK)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262271\u2003\u20035.1.2.2.4. ZAK Kinase (Sterile Alpha Motif and Leucine Zipper Containing Kinase AZK)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262271\u2003\u20035.1.2.2.5. Serine/Threonine-Protein Kinase AKT2 (Proteinkinase B2)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262271\u2003\u20035.1.2.3. Hydrolases\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262271\u2003\u20035.1.2.3.1. ATP Synthase\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262271\u2003\u20035.1.2.3.2. Phosphatase and Tensin Homolog Deleted on Chromosome 10 (PTEN)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262272\u2003\u20035.1.2.3.3. Sirtuin 3 (SIRT3)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262272\u2003\u20035.1.2.3.4. Cathepsins\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262273\u2003\u20035.1.2.3.5. Neprilysin (NEP)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262273\u2003\u20035.1.2.4. Lyases\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262273\u2003\u20035.1.2.4.1. Mitochondrial Aconitase (ACO2)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262273\u2003\u20035.1.2.4.2. \u03b1-Enolase\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262273\u2003\u20035.1.2.5. Isomerases\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262274\u2003\u20035.1.2.5.1. Protein Disulfide Isomerase (PDI)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262274Cis/Trans-Isomerase A1 (Pin1)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2003\u20035.1.2.5.2. Peptidyl-Prolyl 2274\u2003\u20035.1.2.6. Ligases: Glutamine Synthetase\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262274\u2003\u20035.1.3. Carriers\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262274\u2003\u20035.1.3.1. Albumin\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262274\u2003\u20035.1.3.2. Hemoglobin and Myoglobin\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262275\u2003\u20035.1.3.3. Liver and Adipocyte Fatty Acid-Binding Protein (FABP)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262275\u2003\u20035.1.3.4. Apolipoprotein B-100 (ApoB)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262275\u2003\u20035.1.3.5. \u03b2-Lactoglobulin\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262276\u2003\u20035.1.4. Transporters and Channels\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262276\u2003\u20035.1.4.1. Glutamate Transport Protein\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262276\u2003\u20035.1.4.2. \u03b1-Synuclein (\u03b1-Syn)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202622762+-ATPase (SERCA1a)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2003\u20035.1.4.3. Sarco/Endoplasmic Reticulum Ca2277\u2003\u20035.1.4.4. Transient Receptor Potential Vanilloid 1 (TRPV1)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262277\u2003\u20035.1.4.5. Dopamine Transporter\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262278\u2003\u20035.1.5. Receptors\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262278\u2003\u20035.1.5.1. Platelet-Derived Growth Factor Receptor-\u03b2 (PDGFR-\u03b2)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262278\u2003\u20035.1.5.2. Lectin-Like Oxidized Low-Density Lipoprotein Receptor-1 (LOX-1)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262278\u2003\u20035.1.5.3. Toll-Like Receptor 4 (TLR4)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262278\u2003\u20035.1.6. Cytoskeletal Proteins\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262279\u2003\u20035.1.6.1. Tau Proteins\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262279\u2003\u20035.1.6.2. Ankyrin\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262279\u2003\u20035.1.6.3. Spectrins\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262280\u2003\u20035.1.7. Chaperones: Heat Shock Proteins 70 and 90\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262280\u2003\u20035.1.8. Uncoupling Proteins 2 and 3 (UCP2 and UCP3)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262282\u2003\u20035.1.9. Growth Factors: Platelet-Derived Growth Factor (PDGF)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262283\u2003\u20035.1.10. Peptide Hormones\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262283\u2003\u20035.1.10.1. Insulin\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262283\u2003\u20035.1.10.2. Angiotensin II\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262283\u2003\u20035.1.11. Extracellular Matrix Proteins: Collagen\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262283\u2003\u20035.1.12. Histones: Histone-H2A\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262284\u20035.2. Reactions with Lipids\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262284\u20035.3. Reactions with Cofactors and Vitamins\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262284\u2003\u20035.3.1. Vitamin C (Ascorbic Acid)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262284\u2003\u20035.3.2. Pyridoxamine\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262285\u2003\u20035.3.3. Lipoic Acid\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262285\u20035.4. Reactions with Nucleic Acids\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202622856. Formation of HNE in Mammalian Cells and Tissues\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262287\u20036.1. HNE Formation in Cellular and Organ Systems\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262287\u20036.2. HNE in the Whole Healthy Organism\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262289\u20036.3. Influence of Nutrition\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202622907. Metabolism of HNE\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262291\u20037.1. HNE Metabolism in Mammalian Cells and Organs\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262293\u20037.2. HNE Metabolism in Subcellular Organelles\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20032294\u20037.3. HNE Metabolism in Whole Animals and Interorgan Relationships\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262295\u20037.4. Primary HNE Intermediates\u2014Enzymatic Reactions and Quantitative Results\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262295\u20037.5. Secondary HNE Intermediates\u2014Enzymatic Reactions and Quantitative Results\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262301\u20037.6. HNE Metabolism as a Component of the Antioxidative Defense System\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262306\u20037.7. HNE Intermediates as Potential Biomarkers of LPO\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262307\u20037.8. Further Medical Applications of HNE Metabolism\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202623078. Conclusions\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262309Conflicts of Interest\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262309Abbreviations\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262309References\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262313Tables and FiguresTable 1. HNE concentrations in cells, tissues and organs\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262289Table 2. Maximal velocity of total HNE degradation in cells, subcellular organelles, and perfused organs\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262292Table 3. Primary HNE metabolites in different cells and tissues after the addition of 100 \u03bcM HNE to the biological system\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262301Figure 1. Idealized representation of the initiation and propagation reactions of lipid peroxidation2253Figure 2. Formation of lipid hydroperoxides and cyclic peroxides from arachidonic acid.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262254Figure 3. Chemical structure of 4-hydroxy-2-trans-nonenal (HNE)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262255Figure 4. Overview of the reactions of 4-hydroxy-nonenal with different biomolecules\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262257Figure 5. HNE plasma concentration in dependence on age of the blood donor (5 to 90 years)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262288Figure 6. Degradation/metabolism of 4-HNE in rat hepatocytes\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262292Figure 7. Identification of HNE and 4-hydroxynonenoic acid (HNA) by isocratic HPLC separation\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262297Figure 8. Mass spectrum of dihydroxynonene urethane (HPLC plus MS) with fluorimetric detection\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262298Figure 9. HNE metabolites.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262299Figure 10. HPLC chromatogram of the isoindol derivative of the HNE-GSH conjugate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262306Due to its diverse biochemical reactions and far-reaching biological effects 4-hydroxy-nonenal (HNE) is a fascinating compound, the discovery of which is hidden deep in the scientific literature. When Hermann Esterbauer isolated this aldehyde for the first time a quantitative analysis of its chemical elements was made resulting in values in between those for 4-hydroxy-octenal and 4-hydroxy-nonenal . Since these values were closer to those of 4-hydroxy-octenal Esterbauer and his supervisor Schauenstein published it with this chemical formula  to produce phosphatidylinositol-4,5-bisphosphate.473 and Thr308 -HNE was used ,297,298. minutes ,302,303 Originally it was assumed that HNE metabolism is especially fast in hepatocytes ,305,306 + and lactate/pyruvate ratio of resting cells. The consequence of the high demand of reduced NADH by partially uncoupled mitochondria for ATP production is a lower capacity of resting cells to use NADH for HNE reduction (DHN or GSH-DHN formation). In support, addition of NADH to the homogenates of rat hepatoma cells (MH1C1) resulted in a 1.5-fold increase of aldehyde consumption HNE into rats, the majority of the dose appeared in urine (67.1% after 48 h) HNE to rats, besides DHN-MA 15 polar urinary metabolites accounting for about 50% of the urinary radioactivity were separated by HPLC by the same group HNE. Five metabolites were present in the bile, two of them corresponded to GSH-HNE and to GSH-DHN adducts. Two further metabolites were identified as DHN and HNA-lactone mercapturic acid conjugates, whereas the fifth metabolite in the bile remained unidentified . The kinet al., 2005 [in vitro studies using rat liver cytosolic incubations and HNE-glutathione conjugates as substrate and found large differences in the reduction and retro-Michael conversion steps of the metabolism between the conjugates originating from the two enantiomers [The metabolism of the two enantiomers of HNE, the (R)- and the (S)-enantiomer of its racemic mixture, was the aim of a study by Gueraud l., 2005 . The patntiomers . Studiesntiomers  , and the formation of the products was simply calculated using the radioactivity of the spot or peak and dividing it by the specific radioactivity of the added HNE which can be taken as constant taking into account the overshoot of added radioactive HNE in comparison with the \u201clow\u201d formation of new HNE. After addition of labelled HNE aliquots of suspensions were taken. The HNE metabolizing reactions were stopped by the addition of e.g., 0.5 mL suspension to an equal volume of acetonitrile/acetic acid mixture . After centrifugation two aliquots of the supernatant were eluted on TLC plates. The elution with hexane/diethylether  allows the separation of HNE, HNA, and DHN. The elution with butanol/acetic acid/water  allows the determination of GSH-adducts of HNE . For quaSometimes acetonitrile/acetic acid extracts were used for the determination of HNE and HNA by an HPLC-method ,293. TheThe GSH-HNE-adduct was measured as an isoindol derivative after the reaction of the adduct existing in neutralized perchloric acid extracts with o-phthaldialdehyde in the presence of mercaptoethanol .3H-labeled HNE [The reaction is started by addition of a sample aliquot to the o-phthaldiadehyde solution. The reaction mixture is transferred into a Hamilton syringe and the reaction is stopped by injection of the mixture into the HPLC system after 2 min. HPLC separation is carried out in an isocratic manner with 65% methanol-0.1 M sodium acetate buffer, pH 6.5 (v/v) in the presence of an ion-pair reagent (tetraethylammoniumhydroxide). Isoindol is detected by means of a fluorescence detector at wavelengths of 345 nm and 445 nm for excitation and emission, respectively. The formation of enantiomer products in the reaction of GSH-HNE adduct with o-phthaldialdehyde  led to teled HNE ,297.et al., 1998 was combined with different detection methods: electrospray ionization MS for the detection of HNE-GSH and DHN-GSH, GC-chemical ionization MS for HNA [Different HPLC separations were applied in tracer kinetic experiments and in measurements of non-labelled HNE intermediates ,312,319.Various enzyme inhibitors were useful in studies on HNE metabolism. In investigations of HNE metabolic pathways of vascular smooth muscle cells it was shown that the formation of HNA was inhibited by cyanamide; indicating that the acid is derived from an ALDH-catalyzed pathway; while the overall rate of HNE metabolism was insensitive to inhibition of aldose reductase or aldehyde dehydrogenase. Nevetheless; inhibition of HNA formation by cyanamide led to a corresponding increase in the fraction of HNE metabolized by the GSH-linked pathway; indicating that ALDH-catalyzed oxidation competes with glutathione conjugation .etc.The catabolism of HNE may be regarded as a very important part of the antioxidative defense system of cells and organisms. This is due to the fact, that HNE like other aldehydic products of LPO is able to exert cytotoxic, mutagenic, signal, and carcinogenic effects as described above. The degradation of HNE to less toxic intermediates diminished the reactions between HNE and biomolecules and therefore effectively contributes to the antioxidative protection of cells and organisms. This protective effect is the more effective the more rapidly the metabolism of HNE leads to stable HNE products, which can be easily excreted. The primary importance of HNE-degrading pathways which function both at physiological and pathophysiological HNE levels, seems to be the protection of proteins from modification by aldehydic LPO products. This should especially be valid in regions with high HNE formation rate such as in postischemic or reperfused tissues or in synovial tissue of joints during rheumatoid arthritis and other inflammatory diseases or in lymphedematous tissue, et al., 2005 favor the 1,4-dihydroxynonene mercapturic acid (DHN-MA) [3per os and LPO was estimated every day for a 4-day-period. All three parameters significantly increased in response to oxidative stress. DHN-MA showed the highest increase during the 24\u201348 h period after treatment.Two groups of HNE derivatives seem to be useful as possible biomarkers in clinical chemistry and laboratory medicine. The first are the mercapturic acid conjugates including the derivatives of HNE, HNA, and DHN. These conjugates are excreted with the urine and, therefore can be analyzed from urine. The second are HNE-protein adducts of blood serum or plasma. The HNE-modified plasma proteins are already used for evaluation of rheumatic and autoimmune diseases . Both bo(DHN-MA) . The autet al., 2006 applying the specific and very sensitive method of electrospray ionization triple quadrupole mass spectrometry (ESI-MS-MS) [The quantitation of the glutathione conjugate of HNE (HNE-GSH) in the human post-mortem brain was reported by Volkel I-MS-MS) . Levels I-MS-MS) ,9,350.Several studies indicate that the glutathione conjugates of HNE metabolism are mediators of cell signalling and growth. A modification of the metabolic pattern, possibly by pharmacological interventions, could be of therapeutical importance . Additioet al., 2006 also demonstrated a contribution of aldose reductase to diabetic hyperproliferation of vascular smooth muscle cells [The demonstration that aldose reductase mediates endotoxin-induced inflammation and cardiomyopathy suggests that inhibition of this enzyme may be useful to attenuate maladaptive host responses and to treat acute cardiovascular dysfunction associated with endotoxic shock ,352. Srile cells . The autet al. [Tammali et al.  showed tet al. .Another oncological application of modulation of HNE metabolism was suggested in relation to the inhibition of cytosolic class 3 aldehyde dehydrogenase (ALDH3) by antisense oligonucleotides in hepatoma cells . SeveralN-acetylcysteine (NAC) is a pharmaceutical drug and nutritional supplement used primarily as a mucolytic agent and in the management of paracetamol (acetaminophen) overdose. It was used in various investigations to attenuate HNE-mediated organ damage, such as in alcoholic liver damage [The carbonyl scavenger r damage . In thesr damage . An impor damage .Drosophila) slows the age-related and the oxidative damage-related HNE accumulation [Further studies showed that caloric restriction in mammals and in flies (mulation . Additiomulation . Additiomulation , and a dmulation .This review tries to demonstrate that there is a multitude of possible targets of HNE within a cell. Several factors can be envisaged which determine the actual reactants within a cell: The concentration of HNE which can differentially modulate cell death, growth and differentiation , the con"}
{"text": "We show that if p = 1 (or p \u2265 2 and k is odd), then every well-defined solution of this equation is eventually periodic with period k, which generalizes the results of (Elsayed and Stevip \u2265 2 and k being even which has a well-defined solution that is not eventually periodic.We study the following max-type difference equation  The max operator arises naturally in certain models in automatic control theory (see ). In recAn}n=1\u221e+ is a periodic sequence with period p and k, r \u2208 {1,2,\u2026} with gcd = 1 and k \u2260 r, and the initial conditions xd1\u2212, xd2\u2212,\u2026, x0 are real numbers with In this paper, we consider the following max-type equation:k = 4, r = 1, and p = 1. Elsayed and Stevik = 3, r = 1, and p = 1. In [k = 2, r = 1, and p = 1, then every well-defined solution of  = 1 and k \u2260 r.p \u2265 2 and k is odd, then every well-defined solution of kr < 0 eventually for all 1 \u2264 i \u2264 k \u2212 1 and every {xnk\u2212ir}n=r\u221e+ (1 \u2264 i \u2264 k \u2212 1) is not constant sequence eventually.We claim that {p \u2265 2 and k is odd, then we have xnkxnk\u2212kr\u220fi=1k\u22121xnk\u2212irxnk\u2212ir < 0 eventually. This is a contradiction.If p = 1, then we write An = A for all n \u2265 1 and choose m0 \u2265 m1 \u2265 \u22ef\u2265mk2 such that xmjk\u2212jr < xmj+1)k\u2212jr( and xmj+1)k\u2212(j+1)r( = xmj+1+1)k\u2212(j+1)rk+rk+rk+r. By induction, we can show that xnk+jr is a constant sequence eventually for every 1 \u2264 j \u2264 k. Note {jr : 1 \u2264 j \u2264 k}kmod\u2061 = {0,1, 2,\u2026, k \u2212 1} since gcd = 1. Then xnk+i is a constant sequence eventually for every i \u2208 {0,1,\u2026, k \u2212 1}, which implies that {xn}n=1\u2212d\u221e+ is eventually periodic with period k.By the above claim we may choose an From the proof of k, r \u2208 {1,2,\u2026} with k \u2260 r. If {An}n=1\u221e+ is a periodic sequence, then every positive (or negative) solution of  = 1 and k \u2260 r. If As \u2265 0 for some s \u2208 {1,2,\u2026, p}, then every well-defined solution of k = 1. Then xnk+i is a constant sequence eventually for every i \u2208 {0,1,\u2026, k \u2212 1}, which implies that {xn}n=1\u2212d\u221e+ is eventually periodic with period k.Using arguments similar to the ones developed in the proof of p \u2265 2 and k being even which has a well-defined solution that is not eventually periodic.Now we construct an example with k, r \u2208 {1,2,\u2026} and k is even with gcd = 1 and k \u2260 r and An is a periodic sequence with Ai2 = A < Ai\u221212 = B < 0 for all i \u2265 1. Choose the initial conditions xi\u2212 = B for odd i \u2208 {0,1,\u2026, d} and xi\u2212 = 1 for even i \u2208 {0,1,\u2026, d} with d = max\u2061{r, k}; we can obtain a solution {xn}n=\u22121\u221e of  If  It is easy to verify that lim\u2061n\u2192\u221exn2 = \u221e and lim\u2061n\u2192\u221exn\u221212 = 0.\u2009r > k, then(2) If It is easy to verify that lim\u2061n\u2192\u221exn2 = \u221e and lim\u2061i\u2192\u221exn\u221212 = 0.Consider the max-type equationxn=max\u2061{AAn}n=1\u221e+ is a periodic sequence with period ps and s, k, r \u2208 {1,2,\u2026} with gcd = 1 and k \u2260 r, and p = 1 (or p \u2265 2), and the initial conditions xd1\u2212, xd2\u2212,\u2026, x0 are real numbers with yni = xns+i for every 1 \u2264 i \u2264 s and n = 0,1, 2,\u2026. Then ((1)p = 1 (or p \u2265 2 and k is odd), then it follows from i \u2264 s, every well-defined solution of equation k. Thus every well-defined solution of (sk.If ution of  is event(2)p \u2265 2 and, for every 1 \u2264 i \u2264 s, there exists some ji such that Ajis+i \u2265 0, then it follows from i \u2264 s, every well-defined solution of equation k. Thus every well-defined solution of (sk.If ution of  is event(3)p \u2265 2 and k is even, then it follows from i \u2264 s, we can construct an equation If such that it has a well-defined solution which is not eventually periodic.Consider the max-type equation,\u2026. Then  reduces"}
{"text": "Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we initiate the study of soft congruence relations by using the soft set theory. The notions of soft quotient rings, generalized soft ideals and generalized soft quotient rings, are introduced, and several related properties are investigated. Also, we obtain a one-to-one correspondence between soft congruence relations and idealistic soft rings and a one-to-one correspondence between soft congruence relations and soft ideals. In particular, the first, second, and third soft isomorphism theorems are established, respectively. To solve complicated problems in economics, engineering, environmental science, medical science, and social science, methods in classical mathematics are not always successfully used because various uncertainties are typical for these problems. Therefore, there has been a great deal of alternative research and applications in the literature concerning some special tools such as probability theory, fuzzy set theory , 2, rougCurrently, works on soft set theory are progressing rapidly. Maji et al.  discusseThe rest of the paper is organized as follows. In In this section, we recall some notions and definitions see , 22, 23), 2322, 2U be an initial universe and let E be a set of parameters. Let \u2118(U) denote the power set of U and let A and B be nonempty subsets of E.From now on, let F, A) is called a soft set over U, where F is a mapping given by F : A \u2192 \u2118(U).A pair  and  over a common universe U, we say that  is a soft subset of  if it satisfiesA\u2286B;F(\u03b1)\u2286G(\u03b1) for all \u03b1 \u2208 A.For two soft sets  is said to be a soft super set of .In this case,  and  over a common universe U is the soft set  where C = A \u00d7 B and for all  \u2208 C, H = F(\u03b1)\u2229G(\u03b2). In this case, we write The AND-operation of two soft sets  and  over a common universe U is the soft set  where C = A \u00d7 B and for all  \u2208 C, H = F(\u03b1) \u222a G(\u03b2). In this case, we write G, B) = .The OR-operation of two soft sets  and  over a common universe U is the soft set  where C = A \u222a B and for all \u03b1 \u2208 C,The extended intersection of two soft sets \u2229\u025b = .In this case, we write  and  over a common universe U is the soft set  where C = A\u2229B \u2260 \u2205 and for all \u03b1 \u2208 C, H(\u03b1) = F(\u03b1)\u2229G(\u03b1). In this case, we write \u2229R = .The restricted intersection of two soft sets \u2229R = \u2205\u2205, where \u2205\u2205 is the unique soft set over U with an empty parameter set.If F, A) and  over a common universe U is the soft set  where C = A \u222a B and for all \u03b1 \u2208 C,The extended union of two soft sets \u222a\u025b = .In this case, we write  and  over a common universe U is the soft set  where C = A\u2229B \u2260 \u2205 and for all \u03b1 \u2208 C, H(\u03b1) = F(\u03b1) \u222a G(\u03b1). In this case, we write \u222aR = .The restricted union of two soft sets \u222aR = \u2205\u2205.If Fi, Ai) | i \u2208 I} of soft sets over U as follows in . Then RE2 and \u2205\u22052 are the greatest element and the least element of SC(R)E, respectively.Let (R)A the set of all those soft congruence relations defined over R with a fixed parameter set A. Then RA2 and \u2205A2 are the greatest element and the least element of SC(R)A, respectively.In a similar way, denote by SC = {(f(x), f(y)) \u2208 R\u2032 \u00d7 R\u2032 | x\u03b7y}; then f(\u03b7) is a congruence relation on R\u2032.Let x\u2032, y\u2032 \u2208 R, and  \u2208 f(\u03b7), since f is a ring epimorphism, there exists x, y \u2208 R, such that x\u2032 = f(x), y\u2032 = f(y), and x\u03b7y. It is easy to verify that f(\u03b7) is a congruence relation on R\u2032.For all f : R \u2192 R\u2032 be a mapping of rings and let  be a soft set over R. Then we can define a soft set (f(F), A) over R\u2032 where f(F) : A \u2192 \u2118(R\u2032) is defined as f(F)(\u03b1) = f(F(\u03b1)) for all \u03b1 \u2208 A. Here, by definition, we see that Supp\u2061(f(F), A) = Supp\u2061.Let f : R \u2192 R\u2032 be a ring epimorphism. If  is a soft congruence relation over R, then (f(\u03c1), A) is a soft congruence relation over R\u2032, where f(\u03c1)(\u03b1) = {(f(x), f(y)) \u2208 R\u2032 \u00d7 R\u2032 |  \u2208 \u03c1(\u03b1)} for all \u03b1 \u2208 A.Let f(\u03c1), A) is a nonnull soft set since  is a soft congruence relation over R, which is a nonnull soft set by \u03b1 \u2208 Supp\u2061(f(\u03c1), A), we have f(\u03c1)(\u03b1) = f(\u03c1(\u03b1)) \u2260 \u2205. Then the nonempty set \u03c1(\u03b1) is a congruence relation on R, and so we deduce that its onto homomorphic image f(\u03c1(\u03b1)) is a congruence on R\u2032. Hence f(\u03c1)(\u03b1) is a congruence on R\u2032 for all \u2208Supp\u2061(f(\u03c1), A). That is, (f(\u03c1), A) is a soft congruence relation over R\u2032.Note first that  be a soft congruence relation over R, where f(\u03c1)(\u03b1) = {(f(x), f(y)) \u2208 R\u2032 \u00d7 R\u2032 |  \u2208 \u03c1(\u03b1)} for all \u03b1 \u2208 A.\u03c1, A) is trivial, then (f(\u03c1), A) is the trivial soft congruence relation over R\u2032.If  is whole, then (f(\u03c1), A) is the whole soft congruence relation over R\u2032.If  = { | x \u2208 R}. Since f is a ring epimorphism, we have that f(\u03c1)(\u03b1) = f(\u03c1(\u03b1)) = { | x\u2032 = f(x), x \u2208 R} for all \u03b1 \u2208 A. So, (f(\u03c1), A) is the trivial soft congruence relation over R\u2032 by (1) By \u03b1 \u2208 A, \u03c1(\u03b1) = { | x, y \u2208 R}. Since f is a ring epimorphism, we have that f(\u03c1)(\u03b1) = { | x\u2032 = f(x), y\u2032 = f(y),  \u2208 \u03c1(\u03b1), x, y \u2208 R} for all \u03b1 \u2208 A. It follows from f(\u03c1), A) is the whole soft congruence relation over R\u2032.(2) By This completes the proof.Let us define now some definitions about soft quotient rings that we will use in the following paragraphs.\u03c1, A) be a soft congruence relation over R; we have that  = 0 is an idealistic soft ring over R by \u03b1 \u2208 A, x\u03c1(\u03b1) = x + F(\u03b1). Let R/ = {x + F(\u03b1) | x \u2208 R, \u03b1 \u2208 A} be an initial universe set and consider the set-valued function F* : A \u2192 \u2118) given by F*(\u03b1) = {x + F(\u03b1) | x \u2208 R} for all \u03b1 \u2208 A. We say that  is a soft quotient ring of R with respect to the idealistic soft ring , where \u2329F*(\u03b1), +*, \u00b7*\u232a is a quotient ring of \u2329R, +, \u00b7\u232a with respect to an ideal F(\u03b1).Let  be an idealistic soft ring over \u2329Z, +, \u00b7\u232a and A = N+. For all \u03b1 \u2208 A, we have F(\u03b1) = {\u03b1x | x \u2208 Z}\u22b2Z. Let Z/ = {x + F(\u03b1) | x \u2208 Z, \u03b1 \u2208 A} be an initial universe set and consider the set-valued function F* : A \u2192 \u2118) given by F*(\u03b1) = Z/F(\u03b1) = {0 + F(\u03b1), 1 + F(\u03b1),\u2026, (\u03b1 \u2212 1) + F(\u03b1)} = Z/\u2329\u03b1\u232a for all \u03b1 \u2208 A; it is clear that F*(\u03b1) is a quotient ring of Z with respect to F(\u03b1). Then  is a soft quotient ring of Z with respect to .Let  be a soft quotient ring of R with respect to an idealistic soft ring  and let  be a soft quotient ring of R\u2032 with respect to an idealistic soft ring , respectively. We say that  is soft homomorphic to , denoted by ~Q, if, for all \u03b1 \u2208 A, there exists \u03b2 \u2208 B such that a mapping f : F*(\u03b1) \u2192 G*(\u03b2) is a homomorphism; that is, F*(\u03b1) ~ G*(\u03b2).Let  is soft isomorphic to , which is denoted by \u2243Q.If S is a subring of R, we write S \u2264 R; if H is an ideal of R, we write H\u22b2R.Finally, we establish the first soft isomorphism theorem. In order to do this, we need to introduce some relative notions and results. Note that, if F, A) and  be soft sets over a common universe U.  is said to be contained by , denoted by \u03b1 \u2208 A, there exists \u03b2 \u2208 B such that F(\u03b1)\u2286G(\u03b2).Let  and  over a common universe U are said to be soft set equal, denoted by  = , if it satisfies the following:\u03b1 \u2208 A, \u2203\u03b2 \u2208 B such that F(\u03b1) = G(\u03b2);for all b \u2208 B, \u2203a \u2208 A such that G(b) = F(a).for all Two soft sets  and G : B \u2192 \u2118(R) be set-valued functions defined as follows: F(\u03b1) = {x, z}, F(\u03b1\u2032) = {u, v}, G(\u03b2) = {u, v}, and G(\u03b2\u2032) = {x, z}. Therefore,  and  are soft set equal.Let f : R \u2192 R\u2032 be a ring epimorphism.\u03c1 : A\u2192\u2118(R \u00d7 R) be a set-valued function given by \u03c1(\u03b1) = { | x, y \u2208 R, f(x) = f(y)} for all \u03b1 \u2208 A. Then  is a soft congruence relation over R.Let F : A \u2192 \u2118(R) be a set-valued function given by F(\u03b1) = Kerf = {x \u2208 R | f(x) = 0} for all \u03b1 \u2208 A. Then  is an idealistic soft ring over R.Let Let f : R \u2192 R\u2032 be a ring epimorphism. Let S = {H | H \u2264 R, H\u2287K = Kerf} and T = {H\u2032 | H\u2032 \u2264 R\u2032}. Then, for H \u2208 S, we make H correspond to Hf = {f(x) | x \u2208 H}, which is a bijective mapping between S and T. And H\u22b2R\u21d4Hf\u22b2R\u2032; here, R/H\u2245R\u2032/Hf.Let f : R \u2192 R\u2032 be a ring epimorphism and consider an idealistic soft ring  over R defined as G, B) \u2208 SR(R), let f(G) : B \u2192 \u2118(R\u2032) be a set-valued function given by f(G)(\u03b1) = f(G(\u03b1)) = {f(x) | x \u2208 G(\u03b1)} for all \u03b1 \u2208 B. We make  correspond to (f(G), B), which is a bijective mapping between SR(R) and SR(R\u2032).For  is an idealistic soft ring over R if and only if (f(G), B) is an idealistic soft ring over R\u2032. And \u2243Q(f(G)*, B), where  is a soft quotient ring of R with respect to , and (f(G)*, B) is a soft quotient ring of R\u2032 with respect to (f(G), B).,  \u2208 SR(R) if (f(G1), B1) = (f(G2), B2); that is, for all \u03b1 \u2208 B1, \u2203\u03b2 \u2208 B2 such that f(G1)(\u03b1) = f(G2)(\u03b2) and for all b \u2208 B2, \u2203a \u2208 B1 such that f(G2)(b) = f(G1)(a), but by G1(\u03b1) = G2(\u03b2) and G2(b) = G1(a), which means that  = . Thus it is injection. Let  \u2208 SR(R); then we have, for all \u03b1 \u2208 B, G(\u03b1) \u2264 R and its homomorphism image f(G)(\u03b1) = f(G(\u03b1)) \u2264 R\u2032, so f(G), B) \u2208 SR(R\u2032). Let (f(G), B) \u2208 SR(R\u2032); then we have for all \u03b1 \u2208 B, f(G)(\u03b1) \u2264 R\u2032. Define G(\u03b1) = {x \u2208 R | f(x) \u2208 f(G)(\u03b1)} for all \u03b1 \u2208 B; then G(\u03b1) \u2264 R. And for all a \u2208 A, \u2203\u03b1 \u2208 B, Kerf = F(a)\u2286G(\u03b1). So G, B) \u2208 SR(R). Thus it is surjection. Therefore, we make  correspond to (f(G), B), which is a bijective mapping between SR(R) and SR(R\u2032).(1) Let  is an idealistic soft ring over R\u21d4G(\u03b1)\u22b2R \u21d4 (by f(G(\u03b1))\u22b2R\u2032\u2009\u2009\u21d4(f(G), B) is an idealistic soft ring over R\u2032. From G*(\u03b1)\u2245f(G)*(\u03b1) for all \u03b1 \u2208 B, which means that \u2243Q(f(G)*, B) by (2) (\u2208 B)\u21d4 by  f(G(\u03b1))\u22b2f : Z \u2192 Z/\u23296\u232a be a ring epimorphism, A = Z and B = C = {\u03b1 | \u03b1\u2009\u2009is\u2009\u2009divisor\u2009\u2009of\u2009\u20096}. Consider the set-valued function F : A \u2192 \u2118(Z) given by F(\u03b1) = 6Z\u22b2Z. Then  is an idealistic soft ring over Z. Let G, B) \u2208 SR(Z), we have G(\u03b2) = \u03b2Z. And let H, C) \u2208 SR(Z/\u23296\u232a), we have H(c) = cZ/\u23296\u232a.B = C = {1,2, 3,6} and let f(G) : B \u2192 \u2118(R\u2032) be a set-valued function given by f(G)(\u03b2) = \u03b2Z/\u23296\u232a for all \u03b2 \u2208 B. Then, (f(G), B) \u2208 SR(Z/\u23296\u232a). It is clear that we make  correspond to (f(G), B), which is a bijective mapping between SR(Z) and SR(Z/\u23296\u232a).Let B = {2,3} and G*(\u03b2) = {x + G(\u03b2) | x \u2208 Z} = Z/\u2329\u03b2\u232a. Then  is an idealistic soft ring over R iff G(\u03b2) = \u03b2Z\u22b2Z , f(G(\u03b2)) = \u03b2Z/\u23296\u232a\u22b2Z/\u23296\u232a , and (f(G), B) is an idealistic soft ring over Z/\u23296\u232a. It is clear that  is a soft quotient ring of Z with respect to , and (f(G)*, B) is a soft quotient ring of Z/\u23296\u232a with respect to (f(G), B). Let \u03b2 \u2208 B; we have G*(\u03b2) = {x + G(\u03b2) | x \u2208 Z} = Z/\u2329\u03b2\u232a and f(G)*(\u03b2) = {x + f(G)(\u03b2) | x \u2208 Z/\u23296\u232a} = (Z/\u23296\u232a)/(\u03b2Z/\u23296\u232a). Note that Z/\u23292\u232a\u2245(Z/\u23296\u232a)/(2Z/\u23296\u232a) and Z/\u23293\u232a\u2245(Z/\u23296\u232a)/(3Z/\u23296\u232a); we have \u2243Q(f(G)*, B) by Let Let In this section, we study internal connections between soft congruence relations and soft ideals of soft rings and obtain the second and third soft isomorphism theorems. In order to do this, we recall the following notions.F, A) and  be soft rings over R. Then  is called a soft subring of , denoted by B\u2286A;G(\u03b1) is a subring of F(\u03b1) for all \u03b1\u2208Supp\u2061.Let  be a soft ring over R. A nonnull soft set  over R is called a soft ideal of , denoted by B\u2286A;G(\u03b1) is an ideal of F(\u03b1) for all \u03b1\u2208Supp\u2061.Let  be a soft ring over R. A nonnull soft set  over R \u00d7 R is called a soft binary relation of  if it satisfies the following:B\u2286A;\u03c1(\u03b1) is a binary relation of F(\u03b1) for all \u03b1 \u2208 B.Let  be a soft ring over R. A soft binary relation  of  is called a soft congruence relation of  if \u03c1(\u03b1) is a congruence relation on F(\u03b1) for all \u2208Supp\u2061.Let  be a soft ring over R. Then we have the following.\u03c1, B) be a soft congruence relation of . If  = 0, then  is a soft ideal of , and we have \u03c1(\u03b1) = { | x, y \u2208 F(\u03b1), x \u2212 y \u2208 G(\u03b1)} for all \u03b1 \u2208 B.Let  be a soft ideal of  and consider a soft binary relation  of  defined by \u03c1(\u03b1) = { | x, y \u2208 F(\u03b1), x \u2212 y \u2208 G(\u03b1)} for all \u03b1 \u2208 B. Then  is a soft congruence relation of , and 0 = .Let , \u03c1(\u03b1) is a congruence relation on a ring F(\u03b1). Since  = 0, we have G(\u03b1) = 0\u03c1(\u03b1)\u22b2F(\u03b1), and \u03c1(\u03b1) = { | x, y \u2208 F(\u03b1), x \u2212 y \u2208 G(\u03b1)} for all \u03b1 \u2208 B by G, B) is a non-null soft set, we have (1) By G, B), G(\u03b1)\u22b2F(\u03b1). By the hypothesis, we know that \u03c1(\u03b1) is a congruence relation on F(\u03b1) and 0\u03c1(\u03b1) = G(\u03b1) by \u03c1, B) is a soft congruence relation of  and 0 = .(2) By \u03c1, B) of a soft ring  can be represented by the soft ideal generated by . Also, we observe that any soft ideal  of  is the soft congruence class of 0 with respect to the soft congruence relation  generated by .Here, we obtain that any soft congruence relation )E the set of all soft congruence relations and by SI)E the set of all soft ideals of a soft ring . We can establish the following two mappings:\u03c8 : SC)E \u2192 SI)E, \u03c8) = , where  = 0;\u03c6 : SI)E \u2192 SC)E, \u03c6) = , where, for all \u03b1 \u2208 B, we define  \u2208 \u03c1(\u03b1) in F(\u03b1), which is equivalent to x \u2212 y \u2208 G(\u03b1).Denote by SC)E and SI)E of .The above two mappings F, A) and  be soft rings over R. We say that  is a generalized soft subring of , denoted by \u03b2 \u2208 B, there exists \u03b1 \u2208 A such that G(\u03b2) is a subring of F(\u03b1).Let  and  be soft rings over R. We say that  is a generalized soft ideal of , denoted by \u03b2 \u2208 B, there exists \u03b1 \u2208 A such that G(\u03b2) is an ideal of F(\u03b1).Let  be a generalized soft ideal of a soft ring  which is over \u2329R, +, \u00b7\u232a and let / = {x + F(\u03b1) |  \u2208 A\u00d7\u2217B, x \u2208 G(\u03b2)} be an initial universe set, where A\u00d7\u2217B = { \u2208 A \u00d7 B | F(\u03b1)\u22b2G(\u03b2)}. Let us consider the set-valued function F\u2217 : A\u00d7\u2217B \u2192 \u2118/) given by F\u2217 = {x + F(\u03b1) | x \u2208 G(\u03b2)} for all  \u2208 A\u00d7\u2217B. We say that  is a generalized soft quotient ring of soft ring  with respect to the generalized soft ideal , where \u2329F\u2217, +\u2217, \u00b7\u2217\u232a is a quotient ring of ring \u2329G(\u03b2), +, \u00b7\u232a with respect to F(\u03b1).Let  given by G(\u03b2) = {x \u2208 R | x \u00b7 \u03b2 = {0,2}}. Then G(0) = R, G(1) = {0,2}, G(2) = Z/\u23294\u232a, and G(3) = {0,2}. As we see, all these sets are subrings of R. Hence,  is a soft ring over R. On the other hand, consider the function F : A \u2192 \u2118(R) given by F(\u03b1) = {x \u2208 R | x \u00b7 \u03b1 = 0}. As we see,Let F, A) is a generalized soft ideal of . For all  \u2208 A \u00d7 B, where F(\u03b1)\u22b2G(\u03b2), F\u2217 = {x + F(\u03b1) | x \u2208 G(\u03b2)}; we haveHence,  are quotient rings of ring \u2329G(\u03b2), +, \u00b7\u232a with respect to F(\u03b1). Thus  is a generalized soft quotient ring of soft ring  with respect to the generalized soft ideal .As we see, F, A) and  be generalized soft ideals of soft rings  and , respectively. Suppose  is a generalized soft quotient ring of  with respect to , and  is a generalized soft quotient ring of  with respect to . We say that  is soft homomorphic to , denoted by \u223cG, if, for all  \u2208 A\u00d7\u2217A\u2032, there exists  \u2208 B\u00d7\u2217B\u2032 such that a mapping f : F\u2217 \u2192 G\u2217 is a homomorphism; that is, F\u2217 ~ G\u2217.Let  is soft isomorphic to , which is denoted by \u2243G.If F, A) and  be soft rings over the rings R and R\u2032, respectively. Let f and g be two mappings. The pair  is called a soft ring homomorphism if the following conditions are satisfied:f is a ring epimorphism;g is surjective;f(F(\u03b1)) = G(g(\u03b1)) for all \u03b1 \u2208 A.Let  and ,  is said to be soft homomorphic to , denoted by ~. In addition, if f is a ring isomorphism and g is a bijective mapping, then  is called a soft ring isomorphism. In this case, we say that  is soft isomorphic to , denoted by \u2243.If we have a soft ring homomorphism between  and  over R is the soft set  where C = A \u00d7 B and for all  \u2208 C, H = F(\u03b1) + G(\u03b2). In this case, we write + = .The basic sum of two soft rings  and  be soft rings over rings R and R\u2032, respectively. Let  be a soft ring homomorphism between  and , and A\u2032\u2286A.\u03c1, A\u2032) of  defined by \u03c1(\u03b1) = { | x, y \u2208 F(\u03b1), f(x) = f(y)} for all \u03b1 \u2208 A\u2032. Then  is a soft congruence relation of .Let us consider a soft binary relation (F\u2032 : A\u2032 \u2192 \u2118(R) be a set-valued function given by F\u2032(\u03b1) = Kerf(F(\u03b1)) = {x \u2208 F(\u03b1) | f(x) = 0} for all \u03b1 \u2208 A\u2032. Then  is a soft ideal of .Let Let  and  be soft rings over rings R and R\u2032, respectively. Let  be a soft ring epimorphism between  and , and consider a soft ideal  of  defined as F1, A1) \u2208 SS) and consider the set-valued function f(F1) : A1 \u2192 \u2118(R\u2032) given by f(F1)(\u03b1) = f(F1(\u03b1)) = {f(x) | x \u2208 F1(\u03b1)} for all \u03b1 \u2208 A1. Then we make  correspond to (f(F1), A1), which is a bijective mapping between SS) and SS).Let  is a soft ideal of  if and only if (f(F1), A1) is a generalized soft ideal of . And \u2243G(f(F1)\u2217, A1\u00d7\u2217B), where  is a generalized soft quotient ring of  with respect to , and (f(F1)\u2217, A1\u00d7\u2217B) is a generalized soft quotient ring of  with respect to (f(F1), A1).,  \u2208 SS). If (f(F1), A1) = (f(F2), A2), that is, for all \u03b1 \u2208 A1, \u2203\u03b2 \u2208 A2 such that f(F1)(\u03b1) = f(F2)(\u03b2) and for all b \u2208 A2, \u2203a \u2208 A1 such that f(F2)(b) = f(F1)(a), but by F1(\u03b1) = F2(\u03b2) and F2(b) = F1(a), which means that  = . Thus it is injection. Let  \u2208 SS); we have that, for all \u03b1 \u2208 A1, F1(\u03b1) \u2264 F(\u03b1); then its homomorphism image f(F1(\u03b1)) \u2264 f(F(\u03b1)) = G(g(\u03b1)), and so f(F1), A1) \u2208 SS). Let (f(F1), A1) \u2208 SS), we have that, for all \u03b1 \u2208 A1, \u2203g(\u03b1) \u2208 B such that f(F1(\u03b1)) \u2264 G(g(\u03b1)). So we can define F1(\u03b1) = {x \u2208 R | f(x) \u2208 f(F1(\u03b1))}; then for all \u03b1 \u2208 A1, F1(\u03b1) \u2264 F(\u03b1) and for all \u03b2 \u2208 A\u2032, \u2203\u03b1 \u2208 A1, Ker\u2061f(F(\u03b2)) = F\u2032(\u03b2)\u2286F1(\u03b1). So F1, A1) \u2208 SS). Thus it is surjection. Therefore, we make  correspond to (f(F1), A1), which is a bijective mapping between SS) and SS).(1) Let (\u03b1 \u2208 A1)\u21d4 (by f(F1(\u03b1))\u22b2f(F(\u03b1)) = G(g(\u03b1))  \u2208 B) \u03b1, \u03b1) \u2208 A1\u00d7\u2217A, \u2203) \u2208 A1\u00d7\u2217B such that F1\u2217\u2245f(F1)\u2217), which means that \u2243G(f(F1)\u2217, A1\u00d7\u2217B).(2) A1)\u21d4 by  f(F1(\u03b1)S be a subring and let H be an ideal of R, respectively. Then S + H = {x + y | x \u2208 S, y \u2208 H} is a subring of R, S\u2229H is an ideal of S, and a mapping f : S/(S\u2229H)\u2192(S + H)/H, for x + (S\u2229H) \u2208 S/(S\u2229H), f(x + (S\u2229H)) = x + H, is an isomorphism. That is, S/(S\u2229H)\u2245(S + H)/H.Let F, A) be a soft ring and  be an idealistic soft ring over R.H, C) = + is a soft ring over R, and  is a generalized soft ideal of ; = \u2229R is a generalized soft ideal of ; \u2208 B\u00d7\u2217C, there exists  \u2208 D\u00d7\u2217A such that a mapping f : G\u2217 \u2192 K\u2217, with respect to x + K(d) \u2208 K\u2217 and f(x + K(d)) = x + G(b), is an isomorphism, where x \u2208 F(a). That is, \u2243G, where  is a generalized soft quotient ring of  with respect to , and  is a generalized soft quotient ring of  with respect to .For all  \u2264 R and for all \u03b2 \u2208 B, G(\u03b2)\u22b2R. By \u03b1, \u03b2) \u2208 C, F(\u03b1) + G(\u03b2) = H \u2264 R; then \u03b2 \u2208 B, \u2203 \u2208 C such that G(\u03b2)\u22b2H. Hence, (1) By given conditions, we observe that, for all \u03b1 \u2208 D = A\u2229B \u2260 \u2205, K(\u03b1) = F(\u03b1)\u2229G(\u03b1). Since F(\u03b1) \u2264 R, G(\u03b1)\u22b2R, then by \u03b1 \u2208 D, F(\u03b1)\u2229G(\u03b1) = K(\u03b1)\u22b2F(\u03b1). Therefore  is a generalized soft ideal of .(2) By \u03b2 \u2208 B, \u2203 \u2208 C such that G\u2217 = {x + G(\u03b2) | x \u2208 H}, and we observe that, for all \u03b1 \u2208 D = A\u2229B, \u2203\u03b1 \u2208 A such that K\u2217 = {y + F(\u03b1) | y \u2208 K(\u03b1)}, where G(\u03b2)\u22b2H and K(\u03b1)\u22b2F(\u03b1). By \u03b1, \u03b2) \u2208 B\u00d7\u2217C, \u2203 \u2208 D\u00d7\u2217A, such that a mapping f, with respect to x + K(\u03b1) \u2208 K\u2217 and f(x + K(\u03b1)) = x + G(\u03b1), is an isomorphism, where x \u2208 F(\u03b1). That is, G\u2217\u2245K\u2217, which means that \u2243G by (3) By In this paper, concepts of soft congruence relations are presented. Two types of soft congruence relations that are soft congruence relation over a ring and soft congruence relation of a soft ring are defined in a natural way. Then we study relations among soft congruence relations and homomorphisms, soft ring homomorphisms. Also, we obtain a one-to-one correspondence between soft congruence relations and idealistic soft rings and a one-to-one correspondence between soft congruence relations and soft ideals. In particular, we established the first, second, and third soft isomorphism theorems, respectively. In the light of these results, our future work on this topic will be focused on applying soft congruence relations to fuzzy information over rings."}
{"text": "By these properties, the following important results are proved: (1) a fuzzy filter  of a pseudo-BCI algebra is a fuzzy associative filter if and only if it is a fuzzy a-filter; (2) a filter  of a pseudo-BCI algebra is associative if and only if it is an a-filter ; (3) a fuzzy filter of a pseudo-BCI algebra is fuzzy a-filter if and only if it is both a fuzzy p-filter and a fuzzy q-filter.Some new properties of fuzzy associative filters , fuzzy  BCI algebra, which originated from BCI-logic; it is a kind of nonclassical logic and inspired by the calculus of combinators  is a fuzzy filter of X if and only if the level set \u03bct = {x \u2208 X | \u03bc(x) \u2265 t} is filter of X for all t \u2208 Im\u2061(\u03bc).Let X be a pseudo-BCI algebra. Then a fuzzy set \u03bc : X \u2192  is a fuzzy filter of X if and only if it satisfiesx, y, z \u2208 X, x \u2264 y \u2192 z\u21d2\u03bc(z) \u2265 min\u2061{\u03bc(x), \u03bc(y)};for all x, y, z \u2208 X, x \u2264 y\u21ddz\u21d2\u03bc(z) \u2265 min\u2061{\u03bc(x), \u03bc(y)}.for all Let \u03bc : X \u2192  is called a fuzzy p-filter of a pseudo-BCI algebra X if it satisfies (FF1) and(FPF1)x, y, z \u2208 X, \u03bc(z) \u2265 min\u2061{\u03bc((x \u2192 y)\u21dd(x \u2192 z)), \u03bc(y)};for all (FPF2)x, y, z \u2208 X, \u03bc(z) \u2265 min\u2061{\u03bc((x\u21ddy)\u2192(x\u21ddz)), \u03bc(y)}.for all A fuzzy set \u03bc : X \u2192  is called a fuzzy a-filter of a pseudo-BCI algebra X if it satisfies (FF1) and(FaF1)x, y, z \u2208 X, \u03bc(z \u2192 x) \u2265 min\u2061{\u03bc((x\u21dd1)\u2192(y\u21ddz)), \u03bc(y)};for all (FaF2)x, y, z \u2208 X, \u03bc(z\u21ddx) \u2265 min\u2061{\u03bc((x \u2192 1)\u21dd(y \u2192 z)), \u03bc(y)}.for all A fuzzy set \u03bc : X \u2192  is called a fuzzy associative filter of a pseudo-BCI algebra X if it satisfies (FF1) and(FAF1)x, y, z \u2208 X, \u03bc(z) \u2265 min\u2061{\u03bc(x \u2192 (y\u21ddz)), \u03bc(x \u2192 y)};for all (FAF2)x, y, z \u2208 X, \u03bc(z) \u2265 min\u2061{\u03bc(x\u21dd(y \u2192 z)), \u03bc(x\u21ddy)}.for all A fuzzy set \u03bc : X \u2192  is called a fuzzy q-filter of a pseudo-BCI algebra X if it satisfies (FF1) and(FqF1)x, y, z \u2208 X, \u03bc(x \u2192 z) \u2265 min\u2061{\u03bc((x\u21ddy) \u2192 z), \u03bc(y)};for all (FqF2)x, y, z \u2208 X, \u03bc(x\u21ddz) \u2265 min\u2061{\u03bc((x \u2192 y)\u21ddz), \u03bc(y)}.for all A fuzzy set \u03bc be a fuzzy a-filter of a pseudo-BCI algebra X. Then \u03bc satisfiesLet \u03bc be a fuzzy a-filter of a pseudo-BCI algebra X. Then \u03bc satisfiesLet \u03bc be a fuzzy a-filter of a pseudo-BCI algebra X. Then the following statements hold for all x, y, z \u2208 X:x, y \u2208 X, \u03bc(x) \u2265 min\u2061{\u03bc(y), \u03bc(x \u2192 y)}, \u03bc(x) \u2265 min\u2061{\u03bc(y), \u03bc(x\u21ddy)};for all x, y \u2208 X, \u03bc(x \u2192 y) = \u03bc(x\u21ddy);for all x \u2208 X, \u03bc((x \u2192 1) \u2192 x) = \u03bc((x\u21dd1)\u21ddx) = \u03bc(1);for all x, y, z \u2208 X, \u03bc(x \u2192 z) \u2265 min\u2061{\u03bc((x\u21ddy) \u2192 z), \u03bc(y)};for all x, y, z \u2208 X, \u03bc(x\u21ddz) \u2265 min\u2061{\u03bc((x \u2192 y)\u21ddz), \u03bc(y)};for all x \u2208 X, \u03bc(x \u2192 (x\u21dd1)) = \u03bc(x\u21dd(x \u2192 1)) = \u03bc(1);for all x, y, z \u2208 X, \u03bc((x \u2192 y) \u2192 y) \u2265 \u03bc(x), \u03bc((x\u21ddy)\u21ddy) \u2265 \u03bc(x).for all Let x, y \u2208 X, by y \u2192 1 \u2264 (x \u2192 y)\u2192(x \u2192 1). From this, applying \u03bc(x \u2192 1) = \u03bc(x) and \u03bc(y \u2192 1) = \u03bc(y). Therefore, \u03bc(x) \u2265 min\u2061{\u03bc(y), \u03bc(x \u2192 y)}.(1) For any \u03bc(x) \u2265 min\u2061{\u03bc(y), \u03bc(x\u21ddy)}.Similarly, we have x, y \u2208 X, by x \u2192 y) \u2192 1 = (x \u2192 1)\u21dd(y \u2192 1). By \u03bc(x \u2192 y) \u2264 \u03bc(x\u21ddy). Similarly, we can get \u03bc(x\u21ddy) \u2264 \u03bc(x \u2192 y). By \u03bc(x \u2192 y) = \u03bc(x\u21ddy).(2) For any x \u2208 X, applying \u03bc((x \u2192 1) \u2192 x) = \u03bc(1).(3) For any \u03bc((x\u21dd1)\u21ddx) = \u03bc(1).Similarly, we have x, y \u2208 X, by \u03bc((x\u21ddy)\u21ddx) \u2265 \u03bc(y). From this and (2), we get \u03bc((x\u21ddy) \u2192 x) \u2265 \u03bc(y).(4) For any x\u21ddy) \u2192 x \u2264 (x \u2192 z)\u21dd((x\u21ddy) \u2192 z). Using On the other hand, by (5) The proof is similar to (4).x \u2208 X, by (4), we have\u03bc(x \u2192 (x\u21dd1)) = \u03bc(1). Similarly, \u03bc(x\u21dd(x \u2192 1)) = \u03bc(1).(6) For any x \u2208 X, by x \u2264 (x \u2192 y)\u21ddy. From this, (2), and (7) For any \u03bc((x\u21ddy)\u21ddy) \u2265 \u03bc(x).Similarly, we have \u03bc be a fuzzy associative filter of a pseudo-BCI algebra X. Then \u03bc satisfiesLet It is easily proved by \u03bc be a fuzzy associative filter of a pseudo-BCI algebra X. Then the following statements hold:x \u2208 X, \u03bc(x \u2192 (x\u21dd1)) = \u03bc(x\u21dd(x \u2192 1)) = \u03bc(1);for all x, y \u2208 X, \u03bc(y) \u2265 \u03bc((x \u2192 1)\u21dd(x \u2192 y));for all x, y \u2208 X, \u03bc(y) \u2265 \u03bc((x\u21dd1)\u2192(x\u21ddy));for all x, y \u2208 X, \u03bc(x \u2192 y) = \u03bc(x\u21ddy);for all x, y \u2208 X, \u03bc(x \u2192 (x \u2192 y)) \u2265 \u03bc(y), \u03bc(x\u21dd(x\u21ddy)) \u2265 \u03bc(y);for all x \u2208 X, \u03bc(x) \u2265 \u03bc((x \u2192 1) \u2192 1)), \u03bc(x) \u2265 \u03bc((x\u21dd1)\u21dd1));for all x, y \u2208 X, \u03bc(x \u2192 (y \u2192 1)) \u2265 \u03bc(x \u2192 y), \u03bc(x\u21dd(y\u21dd1)) \u2265 \u03bc(x\u21ddy);for all x, y \u2208 X, \u03bc(y \u2192 x)) \u2265 \u03bc(x \u2192 y), \u03bc(y\u21ddx)) \u2265 \u03bc(x\u21ddy);for all x \u2208 X, \u03bc((x\u21dd1) \u2192 x) = \u03bc((x \u2192 1)\u21ddx) = \u03bc(1);for all x, y \u2208 X, \u03bc((x \u2192 y) \u2192 y) \u2265 \u03bc(x), \u03bc((x\u21ddy)\u21ddy) \u2265 \u03bc(x);for all x \u2208 X, \u03bc(x) = \u03bc(x \u2192 1) = \u03bc(x\u21dd1);for all x, y \u2208 X, \u03bc(x) \u2265 min\u2061{\u03bc(y), \u03bc(x \u2192 y)}, \u03bc(x) \u2265 min\u2061{\u03bc(y), \u03bc(x\u21ddy)}.for all Let x \u2208 X (by \u03bc(x \u2192 (x\u21dd1)) = \u03bc(1). Similarly, \u03bc(x\u21dd(x \u2192 1)) = \u03bc(1).(1) For any x \u2208 X by ,(16)) = \u03bc(1), \u03bc(y) \u2265 \u03bc(x\u21dd(x \u2192 y)). Therefore,(2) For any (3) It is similar to (2).x, y \u2208 X, by (2), we have \u03bc(x\u21ddy) \u2265 \u03bc((y \u2192 1)\u21dd(y \u2192 (x\u21ddy))). On the other hand, applying \u03bc((y \u2192 1)\u21dd(y \u2192 (x\u21ddy))) \u2265 \u03bc(x \u2192 y). Thus, \u03bc(x\u21ddy) \u2265 \u03bc(x \u2192 y). Similarly, we can get \u03bc(x \u2192 y) \u2265 \u03bc(x\u21ddy). Therefore, \u03bc(x \u2192 y) = \u03bc(x\u21ddy).(4) For any x, y \u2208 X, since (by \u03bc(y\u21dd(x \u2192 (x \u2192 y))) = \u03bc(x \u2192 (x\u21dd1)). By (1), \u03bc(x \u2192 (x\u21dd1)) = \u03bc(1); hence, \u03bc(y\u21dd(x \u2192 (x \u2192 y))) = \u03bc(1). Moreover, by (5) For any since by (22)y\u21dd(x\u03bc(x\u21dd(x\u21ddy)) \u2265 \u03bc(y).Similarly, (6) By (2) and (3), we can get (6).x, y \u2208 X, by y \u2192 (y \u2192 1))\u2264(x \u2192 y)\u2192(x \u2192 (y \u2192 1)).(7) For any Applying (1) and \u03bc(x\u21dd(y\u21dd1)) \u2265 \u03bc(x\u21ddy).Similarly, x, y \u2208 X, by \u03bc(y \u2192 x) \u2265 \u03bc(x\u21dd(x \u2192 (y \u2192 x)))\u2009\u2009. And, using \u03bc(y \u2192 x) \u2265 \u03bc(x \u2192 (y \u2192 1)). From this and (7), we get(8) For any \u03bc(y\u21ddx) \u2265 \u03bc(x\u21ddy).Similarly, (9) By (1) and (8), we can get (8).(10) It is similar to the proof of x \u2208 X, by \u03bc(x) \u2265 \u03bc(x \u2192 (x\u21ddx)) = \u03bc(x \u2192 1). On the other hand, using (8), \u03bc(x \u2192 1) \u2265 \u03bc(1 \u2192 x) = \u03bc(x). Hence, \u03bc(x) = \u03bc(x \u2192 1) = \u03bc(x\u21dd1).(11) For any (12) It is similar to the proof of Checking the proof of \u03bc be a fuzzy filter of a pseudo-BCI algebra X. If \u03bc satisfies(C1)x, y \u2208 X, \u03bc(y) \u2265 \u03bc(x \u2192 (x\u21ddy));for all (C2)x, y \u2208 X, \u03bc(y) \u2265 \u03bc(x\u21dd(x \u2192 y)),for all \u2009then the following statements hold:(C3)x \u2208 X, \u03bc(x \u2192 (x\u21dd1)) = \u03bc(x\u21dd(x \u2192 1)) = \u03bc(1);for all (C4)x, y \u2208 X, \u03bc(x \u2192 y) = \u03bc(x\u21ddy);for all (C5)x \u2208 X, \u03bc(x) = \u03bc(x \u2192 1) = \u03bc(x\u21dd1);for all (C6)x, y \u2208 X, \u03bc(x) \u2265 min\u2061{\u03bc(y), \u03bc(x \u2192 y)}, \u03bc(x) \u2265 min\u2061{\u03bc(y), \u03bc(x\u21ddy)}.for all Let It is similar to \u03bc be a fuzzy filter of a pseudo-BCI algebra X. Then \u03bc is a fuzzy associative filter of X if and only if it satisfies(C1)x, y \u2208 X, \u03bc(y) \u2265 \u03bc(x \u2192 (x\u21ddy));for all (C2)x, y \u2208 X, \u03bc(y) \u2265 \u03bc(x\u21dd(x \u2192 y)).for all Let \u03bc is a fuzzy associative filter of X; by Assume that \u03bc satisfies conditions (C1) and (C2). For any x, y, z \u2208 X, by y \u2192 (x \u2192 z)\u2264(x \u2192 y)\u2192(x \u2192 (x \u2192 z)). Using \u03bc(y \u2192 (x \u2192 z)) = \u03bc(y\u21dd(x \u2192 z)) = \u03bc(x \u2192 (y\u21ddz)).Conversely, assume that Thus,\u03bc(x \u2192 (x \u2192 z)) \u2265 min\u2061{\u03bc(x \u2192 (y\u21ddz)), \u03bc(x \u2192 y)}.(P1) On the other hand, applying \u03bc(z) \u2265 min\u2061{\u03bc(1), \u03bc(x \u2192 (x \u2192 z))} = \u03bc(x \u2192 (x \u2192 z)).(P2) \u03bc(z) \u2265 min\u2061{\u03bc(x \u2192 (y\u21ddz)), \u03bc(x \u2192 y)}. That is, (FAF1) holds. Similarly, condition (FAF2) holds. Therefore, by \u03bc is a fuzzy associative filter of X.Combining (P1) and (P2), we get that \u03bc be a fuzzy filter of a pseudo-BCI algebra X. Then \u03bc is a fuzzy a-filter of X if and only if it satisfies(a1)x, y \u2208 X, \u03bc(x \u2192 y) \u2265 \u03bc((y\u21dd1) \u2192 x);for all (a2)x, y \u2208 X, \u03bc(x\u21ddy) \u2265 \u03bc((y \u2192 1)\u21ddx).for all Let \u03bc be a fuzzy filter of a pseudo-BCI algebra X. Then the following conditions are equivalent:\u03bc is a fuzzy a-filter of X;\u03bc is a fuzzy associative filter of X.Let \u03bc is a fuzzy a-filter of X. For any x, y \u2208 X, by \u03bc((x\u21dd1)\u21dd(x\u21ddy)) = \u03bc((x\u21dd1)\u2192(x\u21ddy)). Thus,y \u2264 1 \u2192 y \u2264 (x\u21dd1)\u21dd(x\u21ddy). From this and y \u2264 (x\u21dd1)\u21dd(x\u21ddy), we have y \u2192 ((x\u21dd1)\u21dd(x\u21ddy)) = 1; it follows that\u03bc(y) \u2265 \u03bc(x\u21dd(x \u2192 y)). By \u03bc is a fuzzy associative filter of X.(i) \u21d2 (ii). Suppose that \u03bc is a fuzzy associative filter of X. For any x, y \u2208 X, by y\u21dd1) \u2192 x \u2264 (x \u2192 y)\u21dd(y\u21dd1). Applying \u03bc((y\u21dd1) \u2192 y) = \u03bc(1). It follows that(ii) \u21d2 (i). Suppose that \u03bc(x\u21ddy) \u2265 \u03bc((y \u2192 1)\u21ddx). By \u03bc is a fuzzy a-filter of X.Similarly, we can get a-filters of pseudo-BCI algebras. At first, we give the following results (the proofs are omitted).Now, we discuss the relationship between associative filters and F of pseudo-BCI algebra X is a filter  of X if and only if the characteristic function \u03c7F of F is a fuzzy filter  of X.A nonempty subset X be a pseudo-BCI algebra. Then a fuzzy set \u03bc : X \u2192  is a fuzzy associative filter (fuzzy a-filter) of X if and only if the level set \u03bct = {x \u2208 X | \u03bc(x) \u2265 t} is associative filter (a-filter) of X for all t \u2208 Im\u2061(\u03bc).Let  Transfer Principle for Fuzzy Sets in [In fact, the above proposition is a consequence of the so-called Sets in .Combining Propositions F be a filter of a pseudo-BCI algebra X. Then the following conditions are equivalent:F is an a-filter of X;F is an associative filter of X.Let In , the auta-filters, and fuzzy q-filters in pseudo-BCI algebras.Finally, we discuss the relationship among fuzzy associative filters, fuzzy \u03bc be a fuzzy p-filter of a pseudo-BCI algebra X. Then \u03bc satisfiesx \u2208 X, \u03bc(x) = \u03bc((x \u2192 1)\u21dd1);for all x \u2208 X, \u03bc(((x \u2192 1)\u21dd1)\u21ddx) = \u03bc(1).for all Let x \u2208 X, by \u03bc((x \u2192 1)\u21dd1) \u2265 \u03bc(x). It follows that \u03bc(x) = \u03bc((x \u2192 1)\u21dd1).(1) For any x \u2208 X, by (2) For any \u03bc be a fuzzy filter of a pseudo-BCI algebra X. Then the following conditions are equivalent:\u03bc is a fuzzy a-filter of X;\u03bc is both a fuzzy p-filter and a fuzzy q-filter of X.Let \u03bc is a fuzzy a-filter of X. It is easy to prove that \u03bc is both a fuzzy p-filter and a fuzzy q-filter of X.Assume that \u03bc be both a fuzzy p-filter and fuzzy q-filter of X. For any x, y \u2208 X, by \u03bc is a fuzzy a-filter of X.Conversely, let"}
{"text": "Particle size versus ground time clearly shows the existence of a size-induced regime transition . Thermoelectric properties of \u03b2-MnO2 powders as a function of electrical resistance in the range of RP\u2009=\u200910 - 80\u03a9 were measured. Based on the data presented, we propose a model for the \u03b2-MnO2 system in which nanometer-scale MnO2 crystallites bond together through weak van der Waals forces to form larger conglomerates that span in size from nanometer to micrometer scale.Particle sizes of manganese oxide (\u03b2-MnO One of the most known and commercially available TE material is bismuth telluride (Bi2Te3), which exhibits one of the highest ZT values at room temperature . Thermoelectric devices (TED) made out of Bi2Te3 are already commercially available and are used for small-scale energy harvesting , [101], [200], [111], [210], [211], [220], [002], and [310], respectively. The position and orientation of these peaks confirmed that our particles have \u03b2-MnO2 crystal structure   with different properties, and from a TE standpoint, \u03b2-MnO2 is preferred since the electrical conductivity of \u03b1-phase MnO2 is approximately six orders of magnitude lower [Shown in Fig.\u00a0te) Ref. . This crde lower . Althoug2 system, the group of Wallia et al. [2 nanopowder by using ball-milling method with crystals ranging from 400 to 700\u00a0nm in size; however, their SEM micrograph shows particle conglomerates that are ranging in size from a few hundreds of nm up to 2\u00a0\u03bcm. They present no XRD results that would support their crystalline size claim. The groups of Xia et al. [2 powders, but no supporting SEM/TEM or XRD measurements are presented. Finally, the group of Song et al. [Extracting information about the crystalline size based on the broadening of the XRD peaks is not a trivial task. It involves understanding diffraction peak broadening and also choosing the right method of analysis . For exaa et al.  claimed a et al.  and Preia et al.  obtainedg et al.  reports D\u2009=\u2009(k\u2009\u22c5\u2009\u03bb)/(\u03b2\u2009\u22c5\u2009cos(\u03b8)), where D is the particle size in nanometers, \u03bb is the wavelength of X-ray radiation , k is a constant equal to 0.94, and \u03b2 is the measured FWHM of the peak, and \u03b8 is the peak position shown in Fig.\u00a0D4 = 306.30\u00a0nm smaller than the particle size for sample S6  D6 = 546.26\u00a0nm. Therefore, it is important that the data should be obtained from more than one peak in order to get consistent and reliable results.Just to solidify our findings, we also used Scherrer formula to estimate the particle size. The Scherrer formula is RP), for samples S4 through S8, varies exponentially with tube length (L). We believe that this response induces further exponential behavior observed in \u03c3 and power factor (\u03c3\u2009\u22c5\u2009S2) versus RP  versus electrical resistance. The S measured varies linearly with RP, and it has the tendency to have larger values at smaller resistances  at values of \u2212\u2009316\u03bcV/K\u2002(\u2212288\u03bcV/K) and at resistances of 9.8\u2002\u03a9\u2002(69\u03a9)  (Table\u00a0S observed by Song and co-workers [Rp\u2009=\u200910 - 80\u2002\u03a9) as compared to Song\u2019s resistance values . Also, our method of particle modification was quite different. For example, we used a simple mortar and pestle  while Song et al. used a ball-milling method. Furthermore, our data also suggests that the S increases with decreasing RP, which is opposite to the work of Song et al.  and   9\u03a9) Fig.\u00a0. Our absof Refs. , 13, 14 -workers . This isRP for samples S4 through S8. We find that the largest  \u03c3 data were measured for samples S4 (S8) at values of (0.058 S/cm) and at resistances of Rp\u2009=\u20099.8\u2002\u03a9 (Rp\u2009=\u200979\u2002\u03a9).Figure\u00a0\u03c3\u2009\u22c5\u2009S2) versus RP. The largest  power factors we had obtained came for samples S4 (S8) 5.8\u2009\u00d7\u200910\u2212\u20097\u2002W/(m\u2009\u22c5\u2009K2) 5.7\u2009\u00d7\u200910\u2212\u20098\u2002W/(m\u2009\u22c5\u2009K2) at resistances of Rp\u2009=\u20099.8\u2002\u03a9\u2009\u22c5\u2009(Rp\u2009=\u200979\u03a9).In Figure\u00a0, we present the literature data compared to our data for power factor versus electrical conductivity. Our power factor values are lower than other works by Walia et al. ) (Ref. [\u2212\u20095\u2002W/(m\u2009\u22c5\u2009K2)) (Ref. [\u2212\u20096\u2002W/(m\u2009\u22c5\u2009K2)) (Ref. [\u2212\u20096\u2002W/(m\u2009\u22c5\u2009K2)) (Ref. [\u2212\u20096\u2002W/(m\u2009\u22c5\u2009K2) and 5.09\u2009\u00d7\u200910\u2212\u20096\u2002W/(m\u2009\u22c5\u2009K2)) ) (Ref. ), Islam )) (Ref. ), Xia et)) (Ref. ), and So)) (Ref. ). It is l. (Ref. ).Fig. 6LZ (1/K) and the unitless figure of merit ZT that were reported in the literature. It appears that the highest Z and ZT values were obtained by the work of Walia et al. . If we were to assume a thermal conductivity value of k\u2009=\u20090.2096  for the other works that did not report thermal conductivity  [Z\u2009=\u20093.66\u2009\u00d7\u200910\u2212\u200951/K and ZT\u2009=\u20091.1\u2009\u00d7\u200910\u2212\u20092) [Z\u2009=\u20093.42\u2009\u00d7\u200910\u2212\u200951/K and ZT\u2009=\u20091.02\u2009\u00d7\u200910\u2212\u20092) [Z\u2009=\u2009(7.29 - 9.84)\u2009\u00d7\u200910\u2212\u200931/K and ZT\u2009=\u2009(2.43 - 3.28)\u2009\u00d7\u200910\u2212\u20095) [Z\u2009=\u2009(0.254 - 1.59)\u2009\u00d7 10\u2212\u2009131/K and ZT\u2009=\u2009(3.0 - 8.3)\u2009\u00d7\u200910\u2212\u200911) [S values of |S|\u2009=\u200920,000 - 40,000\u2002\u03bcV/V, their reported electrical conductivity was \u03c3\u2009=\u2009(3.18 - 12.7)\u2009\u00d7\u200910\u2212\u20095\u2002S/cm, which has a strong contribution toward lowering their ZT values.In Table\u00a0ty Table\u00a0. Still ta et al.  followed\u00d7\u200910\u2212\u20092) , Song et\u00d7\u200910\u2212\u20095) , and Bha\u200910\u2212\u200911) , 9, 10).\u200910\u2212\u200911)  reported2 system behaves differently as compared to other works. For example, the work of Song et al. [2 crystallites at nanometer scale bond together through weak van der waals forces to form larger conglomerates that span anywhere from mesoscopic to microscopic scale.All our data lead us to believe that our \u03b2\u2212MnOg et al.  might ha2 powders as a function of electrical resistance in the range of R\u2009=\u200910 - 80\u03a9. We found two distinct particle size regimes , which were further confirmed by our thermoelectric measurements. According to SEM and TEM data, most of the MnO2 show a wide range of crystallite sizes that span from nanometer all the way to micrometer sizes. The data presented suggest that the thermoelectric properties of \u03b2-MnO2 depend heavily on particle size distribution and particle morphology. The details of particle agglomeration is presented as a model in which smaller MnO2 crystallites  bond together through weak van der Waals forces to form larger conglomerates with sizes ranging from mesoscopic to microscopic scale. Future research consists in a systematic study of thermoelectric properties as a function of high-energy ball-milling process parameters such as ground time, and angular speed.We have investigated thermoelectric properties of \u03b2-MnO"}
{"text": "Using certain results of exponential Diophantine equations, we prove that (i) if p \u2261 \u00b13(mod\u2009\u20098), then the equation 8x + py = z2 has no positive integer solutions ; (ii) if p \u2261 7(mod\u2009\u20098), then the equation has only the solutions  = (q + 2), 2, 2q + 1), where q is an odd prime with q \u2261 1(mod\u2009\u20093); (iii) if p \u2261 1(mod\u2009\u20098) and p \u2260 17, then the equation has at most two positive integer solutions .Let  Z, N be the sets of all integers and positive integers, respectively. Let p be a fixed odd prime. Recently, the solutions  of the equationp = 19, then ((Sroysang ) if p = p = 13, then ((Sroysang ) if p = p = 17, then  = , , and .(Rabago ) if p = Let In this paper, using certain results of exponential Diophantine equations, we prove a general result as follows.p \u2261 \u00b13(mod\u2061\u2009\u20098), then . If p \u2261 7(mod\u2061\u2009\u20098), then (q is an odd prime with q \u2261 1(mod\u2061\u2009\u20093).If 8), then  has no s8), then  has onlyp \u2261 1(mod\u2061\u2009\u20098) and p \u2260 17, then .If Obviously, the above theorem contains the results of , 2. Finap \u2260 17, then .If n \u2212 1 is a prime, where n is a positive integer, then n must be a prime.If 2See Theorem 1.10.1 of .p is an odd prime with p \u2261 1(mod\u2061\u2009\u20094), then the equationu, v).If See Section 8.1 of .X, Y, m, n) = .The equationSee Theorem 8.4 of .D be a fixed odd positive integer. If the equationu, v), then the equationX, n), except the following cases:D = 2r2 \u2212 3 \u00b7 2r+1 + 1,  = , , , and , where r is a positive integer with r \u2265 3;D = ((1/3)(2r+12 \u2212 17))2 \u2212 32,  = ((1/3)(2r+12 \u2212 17), 5), (1/3), and ((1/3)(17 \u00b7 2r+12 \u2212 1), 4r + 7), where r is a positive integer with r \u2265 3;D = 2r12 + 2r22 \u2212 2r1+r2+1 \u2212 2r1+1 \u2212 2r2+1 + 1,  = , , and , where r1, r2 are positive integers with r2 > r1 + 1 > 2.Let See .D is an odd prime and D belongs to the exceptional case (i) of D = 17.If D is an odd prime with D = 2r2 \u2212 3 \u00b7 2r+1 + 1. Then we haver, since r \u2265 3, then r \u2265 4, and by (r \u2212 1) \u2212 2r/2+1 \u2261 \u22121(mod\u2061\u2009\u20098), a contradiction.We now assume that  we have2r\u221212\u22122r+ and by /2 of D = 17.If Using the same method as in the proof of D belongs to the exceptional case (ii), then  with 3\u2223n.If i), then  has at mr, there exists at most one number of 5, 2r + 3, and 4r + 7 which is a multiple of 3. Thus, by Notice that, for any positive integer X, Y, m, n) = .The equationSee .x, y, z) is a solution of  = 1.We now assume that  = 2, then from .If 2\u2223hen from  we get, a, a15), ap satisfies (p \u2261 7(mod\u2061\u2009\u20098). Otherwise, since 2\u2224y, we see from (p \u2261 py \u2261 z2 \u2212 8x \u2261 1(mod\u2061\u2009\u20098). It implies that if p \u2261 \u00b13(mod\u2061\u2009\u20098), then . If p \u2261 7(mod\u2061\u2009\u20098), then . By the above analysis, we have 2\u2224y. If y > 1, then y \u2265 3 and  =  with 3\u2223m. But, by X, n) =  with 3\u2223n. Since p \u2261 1(mod\u2061\u2009\u20098), by u, v). Therefore, by Lemmas x, y, z). Thus, the theorem is proved.Here and below, we consider the remaining cases that  \u2265 3 and  has the  But, by y=1.Subs"}
{"text": "BE-algebras.We study several degrees in defining a fuzzy positive implicative filter, which is a generalization of a fuzzy filter in  BE-algebra. Ahn and So  by\u03bc is a fuzzy implicative filter of X with degree  and a fuzzy filter of X with degree , but it is neither a fuzzy filter of X nor a fuzzy positive implicative filter of X with degree  sinceConsider a self-distributive X : = {1, a, b, c} be a BE-algebra . But it is neither a fuzzy filter of X nor a fuzzy positive implicative filter of X with degree  sinceLet algebra  with thX with degree  sinceAlso, it is not a fuzzy implicative filter of \u03bc is a fuzzy positive implicative filter of a BE-algebra X with degree , then \u03bc is a fuzzy filter of X with degree .If z : = y in (e4), we havex, y \u2208 X. Thus, \u03bc is a fuzzy filter of X with degree .By putting The converse of \u03bb, \u03ba) is a fuzzy filter if and only if  = .Note that a fuzzy filter with degree . Then, the following holds:Let \u03bc is a fuzzy positive implicative filter of a BE-algebra X with degree  and let x, y \u2208 X. Using (e4) and (e1), we haveAssume that \u03bc be a fuzzy filter of a BE-algebra X with degree  satisfying\u03bc is a positive implicative filter of X with degree .Let x, y, z \u2208 X. Using (e2), we haveLet \u03bc is a positive implicative filter of a BE-algebra X with degree .Thus, \u03bc be a fuzzy filter of X. Then, \u03bc is a fuzzy positive implicative filter of X, if and only ifLet It follows from Propositions BE-algebra X with degree  satisfies the following assertions:x, y \u2208 X)(\u03bc(x\u2217y) \u2265 \u03bb\u03ba\u03bc(y));(x \u2264 y\u21d2\u03bc(y) \u2265 \u03bb\u03ba\u03bc(x)).. If \u03bb = \u03ba, thenx, y \u2208 X)(\u03bc(x\u2217y) \u2265 \u03bb2\u03bc(y)),(x \u2264 y\u21d2\u03bc(y) \u2265 \u03bb2\u03bc(x)).. Then,Let x, y \u2208 X. Let \u03bc be a fuzzy positive implicative filter of X with degree . By \u03bc is a fuzzy filter of X with degree . Since x \u2264 (y\u2217x)\u2217x, using y\u2217x)\u2217x)\u2217y \u2264 x\u2217y. Hence,Let X be a BE-algebra. X is said to be commutative if the following identity holds:(C)x\u2217y)\u2217y = (y\u2217x)\u2217x; that is, x\u2228y = y\u2228x, where x\u2228y = (y\u2217x)\u2217x, for all x, y \u2208 X. is a fuzzy implicative filter of X with degree .Let \u03bc be a fuzzy positive implicative filter of X with degree . By \u03bc is a fuzzy filter of X with degree . Using (BE4) and x\u2217(y\u2217z))\u2217((x\u2217y)\u2217(x\u2217(x\u2217z))) = 1, for any x, y, z \u2208 X. Hence, by \u03bc(x\u2217(x\u2217z)) \u2265 min\u2061{\u03ba\u03bc(x\u2217(y\u2217z)), \u03bb\u03ba2\u03bc(x\u2217y)}. On the other hand, using (BE4) and (C), we obtainLet FPI(X) the set of all positive implicative filters of a BE-algebra X. Note that a fuzzy subset \u03bc of a BE-algebra X is a fuzzy positive implicative filter of X, if and only if\u03bc of a BE-algebra X, there exist \u03bb, \u03ba \u2208  and t \u2208  such that\u03bc is a fuzzy positive implicative filter of X with degree ,U \u2209 FPI(X)\u222a{\u2205}. Denote by BE-algebra X = {1, a, b, c, d} which is given in \u03bc : X \u2192  byt \u2208  = {1, b} is not a positive implicative filter of X, since b\u2217((a\u2217d)\u2217a) = 1, b \u2208 {1, b}, and a \u2209 {1, b}. But \u03bc is a fuzzy positive implicative filter of X with degree .Consider a \u03bc be a fuzzy subset of a BE-algebra X. For any t \u2208  with t \u2264 max\u2061{\u03bb, \u03ba}, if U is an enlarged positive implicative filter of X related to U, then \u03bc is a fuzzy positive implicative filter of X with degree .Let \u03bc(1) < t \u2264 \u03bb\u03bc(x), for some x \u2208 X and t \u2208  \u2265 t/\u03bb \u2265 t/max\u2061{\u03bb, \u03ba} and so x \u2208 U; that is, U \u2260 \u2205. Since U is an enlarged filter of X related to U, we have 1 \u2208 U; that is, \u03bc(1) \u2265 t. This is a contradiction, and thus \u03bc(1) \u2265 \u03bb\u03bc(x), for all x \u2208 X.Assume that a, b, c \u2208 X such that \u03bc(b) < \u03bamin\u2061{\u03bc(a\u2217((b\u2217c)\u2217b)), \u03bc(a)}. If we take t : = \u03bamin\u2061{\u03bc(a\u2217((b\u2217c)\u2217b)), \u03bc(a)}, then t \u2208 \u2217b) \u2208 U\u2286U and a \u2208 U\u2286U. It follows from b \u2208 U so that \u03bc(b) \u2265 t, which is impossible. Therefore,x, y, z \u2208 X. Thus, \u03bc is a fuzzy positive implicative filter of X with degree .Now suppose that there exist \u03bc be a fuzzy subset of a BE-algebra X. For any t \u2208  with t \u2264 k/n, if U is an enlarged positive implicative filter of X related to Ut), then \u03bc is a fuzzy positive implicative filter of X with degree .Let t \u2208  be such that U\u2009\u2009(\u2260\u2205) is not necessarily a positive implicative filter of a BE-algebra X. If \u03bc is a fuzzy positive implicative filter of X with degree , then U is an enlarged positive implicative filter of X related to U.Let tmin\u2061{\u03bb, \u03ba} \u2264 t, we have U\u2286U. Since U \u2260 \u2205, there exists x \u2208 U and so \u03bc(x) \u2265 t. By (e1), we obtain \u03bc(1) \u2265 \u03bb\u03bc(x) \u2265 \u03bbt \u2265 tmin\u2061{\u03bb, \u03ba}. Therefore, 1 \u2208 U.Since x, y, z \u2208 X be such that x\u2217((y\u2217z)\u2217y) \u2208 U and x \u2208 U. Then \u03bc(x\u2217((y\u2217z)\u2217y)) \u2265 t and \u03bc(x) \u2265 t. It follows from (e4) thaty \u2208 U. Thus, U is an enlarged positive implicative filter of X related to U.Let"}
{"text": "Some numerical examples are given to verify our theoretical results.We study the boundedness and persistence, existence, and uniqueness of positive equilibrium, local and global behavior of positive equilibrium point, and rate of convergence of positive solutions of the following system of rational difference equations:  Systems of nonlinear difference equations of higher order are of paramount importance in applications. Such equations also appear naturally as discrete analogues and as numerical solutions of systems differential and delay differential equations which model diverse phenomena in biology, ecology, physiology, physics, engineering, and economics. For applications and basic theory of rational difference equations, we refer to \u20133. In 44\u201310, app\u03b1i, \u03b2i, ai, and bi for i \u2208 {1, 2} and initial conditions x0, x\u22121, y0, and y\u22121 are positive real numbers.Gibbons et al.  investigMore precisely, we investigate the boundedness character, persistence, existence, and uniqueness of positive steady state, local asymptotic stability, and global behavior of unique positive equilibrium point and rate of convergence of positive solutions of system  which coThe following theorem shows the boundedness and persistence of every positive solution of system .\u03b21 < a1 and \u03b22 < a2; then every positive solution {} of system } of system } be a positive solution of system }n=\u22121\u221e of system  \u2208 I \u00d7 J for i \u2208 {\u22121,0}. Along with system . An equilibrium point of \u03b5 > 0 there exists \u03b4 > 0 such that, for every initial condition , i \u2208 {\u22121,0} if n > 0, where ||\u00b7|| is usual Euclidian norm in R2.An equilibrium point (ii)An equilibrium point (iii)\u03b7 > 0 such thatAn equilibrium point (iv)n \u2192 \u221e.An equilibrium point (v)An equilibrium point Let e system .(i)An eqF = , where f and g are continuously differentiable functions at FJ is Jacobian matrix of system , n = 0,1,\u2026, is a system of difference equations such that F. If all eigenvalues of the Jacobian matrix JF about \u03bb| < 1, then Assume that The following theorem shows the existence and uniqueness of positive equilibrium point of system .\u03b21 < a1 and \u03b22 < a2; then there exists unique positive equilibrium point of system  \u2208  \u00d7 ; then it follows from (f(x) = \u03b12/(a2 \u2212 \u03b22 + b2x) and x \u2208 . Then, we obtain thatF(x) = 0 has at least one positive solution in .Consider the following system of equations:x=\u03b11+\u03b21xF(x) = 0 has a unique positive solution in . The proof is therefore completed.Furthermore, assume that condition  is satisb1b2U1U2 + \u03b21\u03b22 + \u03b21(a2 + b2L1) + \u03b22(a1 + b1L2) < (a1 + b1L2)(a2 + b2L1).The unique positive equilibrium point f system  is local\u03bb) = \u03bb4 and b1b2U1U2 + \u03b21\u03b22 + \u03b21(a2 + b2L1) + \u03b22(a1 + b1L2) < (a1 + b1L2)(a2 + b2L1) and |\u03bb| = 1; then one has\u03bb) and \u03a6(\u03bb) \u2212 \u03a8(\u03bb) have the same number of zeroes in an open unit disk |\u03bb| < 1. Hence, all the roots of \u2003=\u03bb4iven by4P(\u03bb)\u2003=\u03bbArguing as in , we havef : \u00d7\u2192 and g : \u00d7\u2192 are continuous functions and a, b, c, and d are positive real numbers with a < b, c < d. Moreover, suppose that f : \u00d7\u2192 and g : \u00d7\u2192 such that following conditions are satisfied: f is increasing in x and decreasing in y, and g is decreasing in x and increasing in y;m1,\u2009\u2009M1,\u2009\u2009m2,\u2009and\u2009M2 be real numbers such that m1 = f, M1 = f, m2 = g, and M2 = g; then m1 = M1 and m2 = M2.let Assume that xn+1 = f, yn+1 = g has a unique positive equilibrium point Then, the system of difference equations a1 \u2212 \u03b21 + b1L2)2(a2 \u2212 \u03b22 + b1L1)2 > \u03b11\u03b12b1b2.The unique positive equilibrium point of system  is globaf = (\u03b11 + \u03b21x)/(a1 + b1y) and g = (\u03b12 + \u03b22y)/(a2 + b2x). Then, it is easy to see that f is increasing in x and decreasing in y. Moreover, g is decreasing in x and increasing in y. Let  be a solution of the systemK = (a1 \u2212 \u03b21 + b1L2)2(a2 \u2212 \u03b22 + b1L1)2. Finally, from } be an arbitrary solution of the system } is a positive solution of the system } of } of  the folla1 < \u03b21; then it follows from xn \u2264 \u03b11/(a1 \u2212 \u03b21) = U1, n = 1,2,\u2026. Furthermore, from system  > 1; then we obtain that {wn} is divergent. Hence, by comparison, we have yn \u2192 \u221e as n \u2192 \u221e.(i) Suppose that uation:wn+1=c2+a2 < \u03b22; then from yn \u2264 \u03b12/(a2 \u2212 \u03b22) = U2,\u2009\u2009n = 1,2,\u2026. Moreover, from system  > 1; then one has {zn} that is divergent. Hence, by comparison we have xn \u2192 \u221e as n \u2192 \u221e.(ii) Assume that uation:zn+1=c1+a1 > \u03b21 and a2 > \u03b22; then system (\u03b21 \u2212 a1) + b2\u03b11 \u2212 b1\u03b12 and \u03bd = (a2 \u2212 \u03b22)(\u03b21 \u2212 a1) \u2212 b2\u03b11 + b1\u03b12. From ,p,q2,)2,q1+q2=\u03bd+. From ((p1+p2)2, from ((q1+q2)2\u03b11 = 0.5, \u03b21 = 12, a1 = 13, b1 = 0.2, \u03b12 = 0.1, \u03b22 = 17, a2 = 17.5, and b2 = 0.3. Then, system (x0 = 0.46, x\u22121 = 0.5, y\u22121 = 0.11, and y0 = 0.14.Let xn is shown in yn is shown in In this case, the unique positive equilibrium point of the system  is given\u03b11 = 10,\u2009\u2009\u03b21 = 1.5,\u2009\u2009a1 = 1.6,\u2009\u2009b1 = 0.003,\u2009\u2009\u03b12 = 12,\u2009\u2009\u03b22 = 23,\u2009\u2009a2 = 23.1, and\u2009\u2009b2 = 0.02.\u2009\u2009Then, system (x\u22121 = 82,\u2009\u2009x0 = 89,\u2009\u2009y\u22121 = 5.9, and\u2009\u2009y0 = 6.Let\u2009\u2009xn is shown in yn is shown in In this case, the unique positive equilibrium point of the system  is given\u03b11 = 3.2,\u2009\u2009\u03b21 = 8,\u2009\u2009a1 = 8.1,\u2009\u2009b1 = 5.5,\u2009\u2009\u03b12 = 4.2,\u2009\u2009\u03b22 = 16,\u2009\u2009a2 = 16.1,\u2009\u2009and b2 = 8.5. Then, system (x\u22121 = 3.9,\u2009\u2009x0 = 3.5,\u2009\u2009y\u22121 = 0.1, and\u2009\u2009y0 = 0.12.Let xn is shown in yn is shown in In this case, the unique positive equilibrium point of the system  is givenIn literature, several articles are related to qualitative behavior of competitive system of planar rational difference equations . It is v"}
{"text": "SS,\u03a3* and \u00a3S,\u03a3 of biunivalent functions denoted by subordination. The results presented in this paper improve the recent work of Crisan (2013).We obtain the Fekete-Szeg\u00f6 inequalities for the classes  A denote the class of analytic functions in the unit disk S we will denote the class of all functions in A which are univalent in U.Let U under every function f from S contains a disk of radius 1/4. Thus every such univalent function has an inverse f\u22121 which satisfies The Koebe one-quarter theorem  states tU if both U. Let \u03a3 denote the class of biunivalent functions defined in the unit disk U.A function f and g are analytic in U, then f is said to be subordinate to g, written as w(z), analytic in U, with If the functions a2|. Subsequently, Brannan and Clunie [f \u2208 \u03a3. Netanyahu [a2| = 4/3 if f(z) \u2208 \u03a3. Brannan and Taha [an| for n \u2208 \u2115\u2216{1,2}; \u2115 = {1,2, 3,\u2026} is presumably still an open problem.Lewin  studied d Clunie  conjectuetanyahu  showed tand Taha  introducand Taha \u201312. The \u03d5 be an analytic and univalent function with positive real part in U with \u03d5(0) = 1, \u03d5\u2032(0) > 0, and \u03d5 maps the unit disk U onto a region starlike with respect to 1 and symmetric with respect to the real axis. Taylor's series expansion of such function is of the form B1 > 0.Let S*(\u03d5) and C(\u03d5) we denote the following classes of functions: By S*(\u03d5) and C(\u03d5) are the extensions of classical sets of starlike and convex functions and in such a form were defined and studied by Ma and Minda [S*(\u03d5) and C(\u03d5) as well as Fekete-Szeg\u00f6 inequalities for S*(\u03d5) and C(\u03d5). Their proof of Fekete-Szeg\u00f6 inequalities requires the univalence of \u03d5. Ali et al. [\u03d5. So in this paper, we assume that \u03d5 has series expansion \u03d5(z) = 1 + B1z + B2z2 + \u22ef, B1, B2 are real, and B1 > 0. A function f is bistarlike of Ma-Minda type or biconvex of Ma-Minda type if both f and f\u22121 are, respectively, Ma-Minda starlike or convex. These classes are denoted, respectively, by S\u03a3*(\u03d5) and C\u03a3(\u03d5)  (see ).SS* of starlike functions with respect to symmetric points in U, consisting of functions f \u2208 A that satisfy the condition Re(zf\u2032(z)/(f(z) \u2212 f(\u2212z))) > 0, z \u2208 U. Similarly, in [CS of convex functions with respect to symmetric points in U, consisting of functions f \u2208 A that satisfy the condition Re((zf\u2032(z))\u2032/(f\u2032(z) + f\u2032(\u2212z))) > 0, z \u2208 U. In the style of Ma and Minda, Ravichandran (see [SS*(\u03d5) and CS(\u03d5).In , Sakagucarly, in , Wang etran (see ) definedf \u2208 A is in the class SS*(\u03d5) if CS(\u03d5) if A function SS,\u03a3* and \u00a3S,\u03a3. These inequalities will result in bounds of the third coefficient which are, in some cases, better than these obtained in [In this paper, motivated by the earlier work of Zaprawa , we obtaained in .In order to derive our main results, we require the following lemma.p(z) = 1 + p1z + p2z2 + p3z3 + \u22ef is an analytic function in U with positive real part, then If f \u2208 \u03a3 is said to be in the class SS,\u03a3* if the following subordination holds: g(w) = f\u22121(w).A function \u03b1 = 0, the class SS,\u03a3* reduces to the class SS*(\u03d5) introduced by Ravichandran [We note that, for chandran .f given by  and \u03bc \u2208 \u211d. Then Let given by  be in thf \u2208 SS,\u03a3* and g be the analytic extension of f\u22121 to U. Then there exist two functions u and v, analytic in U with u(0) = v(0) = 0, |u(z)| < 1, |v(w)| < 1, z, w \u2208 U, such that p and q by Rep(z) > 0 and Req(w) > 0. From , a, af \u2208 S0. From  then If f \u2208 SS,\u03a3* then If If ualities  and 31)(32)\u03d5z=1+If ualities  and 31)(34)\u03d5z=1+a3| obtained by Crisan [Corollaries y Crisan .f \u2208 \u03a3 is said to be \u00a3S,\u03a3 if the following subordination holds: g(w) = f\u22121(w).A function \u03b1 = 0, the class \u00a3S,\u03a3 reduces to the class CS(\u03d5) introduced by Ravichandran [We note that, for chandran .f given by  and \u03bc \u2208 \u211d. Then Let given by  be in thf \u2208 \u00a3S,\u03a3 and g be the analytic extension of f\u22121 to U. Then there exist two functions u and v, analytic in U with u(0) = v(0) = 0, |u(z)| < 1, |v(w)| < 1, z, w \u2208 U, such that \u03bc = 1 or \u03bc = 0 we get the following.Let uch that 2zf\u2032z\u2032f\u2032zw. From  then If f \u2208 \u00a3S,\u03a3 then If If ualities  and 51)(52)\u03d5z=1+If ualities  and 51)(54)\u03d5z=1+a3| obtained by Crisan [Corollaries y Crisan ."}
{"text": "Using three critical points theorems, we prove the existence of at least three solutions for a quasilinear biharmonic equation. \u03a9 \u2282 \u211dN\u2009\u2009(4 > N \u2265 1) is a nonempty bounded open set with a sufficient smooth boundary \u2202\u03a9, \u03bb > 0, f : \u03a9 \u00d7 \u211d \u2192 \u211d is an L1-Carath\u00e9odory function, and g : \u211d \u2192 \u211d is a Lipschitz continuous function with Lipschitz constant L > 0; that is, t1, t2 \u2208 \u211d and g(0) = 0.In this paper, we show the existence of at least three weak solutions for the Navier boundary value problem N = 1, in  \u00d7 \u211d \u2192 \u211d is a L2-Carath\u00e9odory function. Later, some authors generalized this type of equation  is a nonempty bounded open set with a sufficient smooth boundary \u2202\u03a9, p > max\u2061\u2061{1, N/2}, \u03bb > 0, and f : \u03a9 \u00d7 \u211d \u2192 \u211d is an L1-Carath\u00e9odory function. After that some authors used different critical point theorems to get one nontrivial, at least three, and infinitely many solutions , 19N = 1u and Su  also useu and Su  to estabons see , 16, 18), 18N = 1ons \u2229W01,2(\u03a9).The goal of the present paper is to establish some new criteria for  to have or which  admits aA special case of our main results is the following theorem.g : \u211d \u2192 \u211d be a Lipschitz continuous function with the Lipschitz constant L > 0 and g(0) = 0 such that L < 1/K2m(\u03a9), where K is defined by (f : \u211d \u2192 \u211d be a continuous function and put F(t) = \u222b0tf(\u03be)d\u03be for each t \u2208 \u211d. Assume that F(d) > 0 for some d > 0 and F(\u03be) \u2265 0 in  and \u03bb* > 0 such that for each \u03bb > \u03bb* the problem Let fined by . Let f :First we here recall for the reader's convenience our main tools to prove the results. The first result has been obtained in  and the X be a separable and reflexive real Banach space, \u03a6 : X \u2192 \u211d a nonnegative continuously G\u00e2teaux differentiable and sequentially weakly lower semicontinuous functional whose G\u00e2teaux derivative admits a continuous inverse on X*, and \u03a8 : X \u2192 \u211d a continuously G\u00e2teaux differentiable functional whose G\u00e2teaux derivative is compact. Assume that there exists x0 \u2208 X such that \u03a6(x0) = \u03a8(x0) = 0 and that r > 0, x1 \u2208 X, such that r < \u03a6(x1) and \u22121 in the weak topology. Then, for each X and, moreover, for each h > 1, there exists an open interval \u03c3 such that, for each \u03bb \u2208 \u039b2, (X whose norms are less than \u03c3.Let equation \u03a6\u2032(u)\u2212\u03bb\u03a8\u2032X be a reflexive real Banach space; let \u03a6 : X \u2192 \u211d be a sequentially weakly lower semicontinuous, coercive, and continuously G\u00e2teaux differentiable whose G\u00e2teaux derivative admits a continuous inverse on X*, and let \u03a8 : X \u2192 \u211d be a sequentially weakly upper semicontinuous and continuously G\u00e2teaux differentiable functional whose G\u00e2teaux derivative is compact. Assume that there exist r \u2208 \u211d and u1 \u2208 X with 0 < r < \u03a6(u1), such that (A1)u\u2208\u03a6\u22121\u03a8(u) < r(\u03a8(u1)/\u03a6(u1));sup\u2061(A2)\u03bb \u2208 \u039br\u2236 = ]\u03a6(u1)/\u03a8(u1), r/sup\u2061u\u2208\u03a6\u22121\u03a8(u);(A5)\u03a9sup\u2061t\u2208Fdx < (1 \u2212 K2Lm(\u03a9))c2Fdx/((1 \u2212 K2Lm(\u03a9))c2+(1 + K2Lm(\u03a9))(\u03b8d)2)).\u222bThen, for each \u03bb in X and, moreover, for each h > 1, there exist an open interval \u03c3 such that, for each \u03bb \u2208 \u039b2\u2032, problem (X whose norms are less than \u03c3.Assume that there exist a positive function w given by r\u2236 = ((1 \u2212 K2Lm(\u03a9))/2)(c/K)2. It is easy to verify that w \u2208 W2,2(\u03a9)\u2229W01,2(\u03a9), and, in particular, one has w(x) \u2264 d for each x \u2208 \u03a9, the condition (A4) ensures that 1\u2032\u2286\u039b1 and \u039b2\u2286\u039b2\u2032, we have the desired conclusion directly from We claim that all the assumptions of tly from  we see tc and d with c < \u03b8d such that the assumption (A4) in (B4)\u03a9sup\u2061t\u2208Fdx/(1 \u2212 K2Lm(\u03a9))c2 < \u222bBFdx/(1 + K2Lm(\u03a9))(\u03b8d)2;\u222b(B5)t|\u2192+\u221e|F/t2 < \u222b\u03a9sup\u2061t\u2208Fdx/c2.limsup\u2061Then, for each Assume that there exist two positive constants w as given in (r\u2236 = ((1 \u2212 K2Lm(\u03a9))/2)(c/K)2 and bearing in mind that All the assumptions of given in  and r\u2236 =\u03bb > \u03bb*\u2236 = m(\u03a9)(1 + K2Lm(\u03a9))(\u03b8d)2/m)F(d) for some d > 0. Since cm}m\u2208\u2115\u2286\u2009\u2009]0, +\u221e[ such that lim\u2061m\u2192+\u221ecm = 0 and F(\u03becm) = sup\u2061\u03be|\u2264cm|F(\u03be). Hence, there is Fix u \u2208 X, we let the functionals \u03a6, \u03a8 : X \u2192 \u211d be defined by u \u2208 X is the functional \u03a6\u2032(u) \u2208 X*, given by v \u2208 X. Furthermore, the differential \u03a6\u2032 : X \u2192 X* is a Lipschitzian operator. Indeed, for any u, v \u2208 X, there holds g is Lipschitz continuous and the embedding X\u21aaL2(\u03a9) is compact, the claim is true. In particular, we derive that \u03a6 is continuously differentiable. The inequality , it turns out that \u03a6\u2032 is a strongly monotone operator. So, by applying Minty-Browder theorem [X \u2192 X* admits a Lipschitz continuous inverse. On the other hand, the fact that X is compactly embedded into C0(\u03a9) implies that the functional \u03a8 is well defined, continuously G\u00e2teaux differentiable, and with compact derivative, whose Gateaux derivative at the point u \u2208 X is given by v \u2208 X. Note that the weak solutions of  = 0, we have from  > r. From the definition of \u03a6 and by using )dx \u2265 0  = 0 so x0 = 0 and x1 = w from \u03bb \u2208 \u039b1, the problem  for any fixed [, using , taking [, using  into accby using  we have (A2) and , we haveX \u2192 \u211d as given in the proof of w) > r. From the definition of \u03a6 we have \u03b7, \u03d1 \u2208 \u211d with x \u2208 \u03a9 and all t \u2208 \u211d. Fix u \u2208 X. Then x \u2208 \u03a9. Now, to prove the coercivity of the functional \u03a6 \u2212 \u03bb\u03a8, first we assume that \u03b7 > 0. So, for any fixed \u03b7 \u2264 0, clearly we obtain lim\u2061u||\u2192+\u221e||(\u03a6(u) \u2212 \u03bb\u03a8(u)) = +\u221e. Both cases lead to the coercivity of functional \u03a6 \u2212 \u03bb\u03a8.To apply  X. Then F)X.So, assumptions A1) and (A2) in  and (A2)"}
{"text": "A stochastic SIS-type epidemic model with general nonlinear incidence and disease-induced mortality is investigated. It is proved that the dynamical behaviors of the model are determined by a certain threshold value  Our real life is full of randomness and stochasticity. Therefore, using stochastic dynamical models can gain more real benefits. Particularly, stochastic dynamical models can provide us with some additional degrees of realism in comparison to their deterministic counterparts. There are different possible approaches which result in different effects on the epidemic dynamical systems to include random perturbations in the models. In particular, the following three approaches are seen most often. The first one is parameters perturbation; the second one is the environmental noise that is proportional to the variables; and the last one is the robustness of the positive equilibrium of the deterministic models.In recent years, various types of stochastic epidemic dynamical models are established and investigated widely. The main research subjects include the existence and uniqueness of positive solution with any positive initial value in probability mean, the persistence and extinction of the disease in probability mean, the asymptotical behaviors around the disease-free equilibrium and the endemic equilibrium of the deterministic models, and the existence of the stationary distribution as well as ergodicity. Many important results have been established in many literatures, for example, \u201316 and tThe stochastic epidemic models with general nonlinear incidence are not investigated. Up to now, only some special cases of nonlinear incidence, for example, saturated incidence rate, are considered. But, we all know that the nonlinear incidence rate in the theory of mathematical epidemiology is very important.For the stochastic epidemic models with the standard incidence, up to now, we do not find any interesting researches.The conditions obtained on the existence of unique stationary distribution are very rigorous. Whether there is a unique stationary distribution only when the model is permanent in the mean with probability one is still an open problem.However, from articles \u201316 and tS and I denote the susceptible and infectious individuals, \u039b denotes the recruitment rate of the susceptible, \u03bc is the natural death rate of S and I, \u03b1 is the disease-related death rate, the transmission of the infection is governed by a nonlinear incidence rate \u03b2f, where \u03b2 denotes the transmission coefficient between compartments S and I, f is a continuously differentiable function of S and I, and \u03b3 denotes the per capita disease contact rate.Motivated by the above work, in this paper, we consider the following deterministic SIS epidemic model with nonlinear incidence rate and disease-induced mortality:dStdt=\u039b\u2212\u03b2\u03b2 of disease in deterministic model (B(t) is a one-dimensional standard Brownian motion defined on some probability space. Thus, model , for example, f = SI/N (standard incidence) and f = h(S)g(I), many idiographic criteria on the extinction, permanence, and stationary distribution are established. Lastly, some affirmative answers for the open problems which are proposed in this paper also are given by the numerical examples .The organization of this paper is as follows. In R+2 = { : x1 > 0, x2 > 0}, R+0 = [0, \u221e), and R+ = . Throughout this paper, we assume that model  with a filtration {Ft}t\u22650 satisfying the usual conditions; that is, {Ft}t\u22650 is right continuous and F0 contains all P-null sets.Denote S and I denote the susceptible and infected fractions of the population, respectively, and N = S + I is the total size of the population among whom the disease is spreading; the parameters \u039b, \u03bc, \u03b2, and \u03b3 are given as in model (\u03b2Sg(I); B(t) denotes one-dimensional standard Brownian motion defined on the above probability space; and \u03c3 represents the intensity of the Brownian motion B(t). Throughout this paper, we always assume the following. is two-order continuously differentiable for any S \u2265 0, I \u2265 0, and S + I > 0. For each fixed I \u2265 0, f is increasing for S > 0 and for each fixed S \u2265 0, f/I is decreasing for I > 0. f = f = 0 for any S > 0 and I > 0, and \u2202f/\u2202I > 0, where S0 = \u039b/\u03bc.In model , S and Iin model ; the traf = h(S)g(I), then assumption (H) becomes in the following form:\u2009(h(S) and g(I) are continuously differentiable for S \u2265 0 and I \u2265 0, h(S) is increasing for S \u2265 0, and g(I)/I is decreasing for I > 0.Particularly, when H), by simple calculating, we can obtain that for any S > 0 and I > 0, 0 \u2264 f \u2264 /\u2202I)I, and for any S2 > S1 > 0, \u2202f/\u2202I \u2265 \u2202f/\u2202I.From  = SI/N (standard incidence), where N = S + I, f = SI/(1 + \u03c91I + \u03c92S) (Beddington-DeAngelis incidence) with constants \u03c91 \u2265 0 and \u03c92 \u2265 0, and f = SI/(1 + \u03c9I2) with constant \u03c9 \u2265 0, then (H) is satisfied.When f.Now, we give the following result for function p > q > 0, let D = { : S > 0, I > 0, q \u2264 S + I \u2264 p}. Then,For any constants H), we have H), we haveH and G are continuous for  \u2208 D. Therefore, conclusion , I(0)) \u2208 R+2, model (S(t), I(t)) defined on t \u2208 R+0 satisfying (S(t), I(t)) \u2208 R+2 for all t \u2265 0 with probability one. Furthermore, when \u03b1 > 0 then S0 \u2264 lim\u2009inft\u2192\u221eN(t) \u2264 lim\u2009supt\u2192\u221eN(t) \u2264 S0, and when \u03b1 = 0 then limt\u2192\u221eN(t) = S0, where N(t) = S(t) + I(t) and S0 = \u039b/(\u03bc + \u03b1).For any initial value (R0 is the basic reproduction number of deterministic model (Define the constantsic model . On the (a)\u03c32 \u2264 \u03b2//\u2202I) and (b)\u03c32 > \u03b22/2(\u03bc + \u03b3 + \u03b1).Then disease I in model (S(0), I(0)) \u2208 R+2, solution (S(t), I(t)) of model (t\u2192\u221eI(t) = 0 a.s.Assume that one of the following conditions holds:S(t), I(t)) \u2208 R+2 a.s. for all t \u2265 0 and lim\u2009supt\u2192\u221e(S(t) + I(t)) \u2264 S0. For any \u03b7 > 0 there is T0 > 0 such that S(t) + I(t) < S0 + \u03b7 for all t \u2265 T0. Hence, for any t \u2265 T0,\u03b5 > 0, \u03c3 = 0, g(u) is monotone increasing for u \u2208 R+, and when \u03c3 > 0, g(u) is monotone increasing for u \u2208 [0, (\u03b2 + \u03b5)/\u03c32) and monotone decreasing for u \u2208 [(\u03b2 + \u03b5)/\u03c32, \u221e).By ula see , 18), we, weS(t),\u03c3 = 0, from /\u2202I \u2264 \u03b2/\u03c32, we can choose \u03b7 > 0 such that \u03b7 \u2264 \u03b5 and \u2202f/\u2202I < (\u03b2 + \u03b5)/\u03c32. From  holds, then when  0, from , we dire2. From  we also 0, from , we obt\u03c3 > 0, g(u) has maximum value (\u03b2 + \u03b5)2/2\u03c32 \u2212 (\u03bc + \u03b3 + \u03b1) at u = (\u03b2 + \u03b5)/\u03c32, and for any t \u2265 0, we have \u03b5, we obtaint\u2192\u221eI(t) = 0 a.s. This completes the proof.If condition (b) holds, then, since a.s.From  and 19)\u03c3 > 0, g and (b) of ic model .R0 \u2264 1, then, for any \u03c3 > 0, \u03c3 > 0, the conclusions of R0 \u2264 2. From \u03c31 \u2264 \u03c32, we easily prove that when R0 > 2, we have \u03c31 > \u03c32 and \u03c3 > \u03c31 the conclusions of When (a)R0 \u2264 1 and \u03c3 > 0;(b)R0 \u2264 2 and 1 < (c)R0 > 2 and \u03c3 > \u03c31.Then disease I in model  = SI/N (standard incidence). Assume that one of the following conditions holds:(a)\u03c32 \u2264 \u03b2 and (b)\u03c32 > \u03b22/2(\u03bc + \u03b3 + \u03b1).Then disease I in model  = h(S)g(I). Assume that (H\u2217) holds and one of the following conditions holds:(a)\u03c32 \u2264 \u03b2/h(S0)g\u2032(0) and (b)\u03c32 > \u03b22/2(\u03bc + \u03b3 + \u03b1).Then disease I in model , I(0)) \u2208 R+2, solution (S(t), I(t)) of model (If in model  is perma\u03b5 > 0 such thatS(0), I(0)) \u2208 R+2, solution (S(t), I(t)) of model (t\u2192\u221e(1/t)\u222b0tI(s)ds \u2264 S0 and for above \u03b5 > 0 there is T0 > 0 such that S0 \u2212 \u03b5 \u2264 S(t) + I(t) \u2264 S0 + \u03b5 a.s. for all t \u2265 T0. Denote the set D\u03b5 = { : S0 \u2212 \u03b5 \u2264 S + I \u2264 S0 + \u03b5}. Since dN(t) = (\u039b \u2212 \u03bcN(t) \u2212 \u03b1I(t))dt, we obtain for any t > T0t \u2265 T0,f/I for S > 0 and I > 0 is continuously differentiable, limI\u21920/I) = \u2202f/\u2202I exists for any S > 0, and set D\u03b5 is convex and connected, by the Lagrange mean value theorem when t \u2265 T0 we have \u03be(t), \u03d5(t)) \u2208 D\u03b5. Let constantsM1, M2 < \u221e. For any t \u2265 T0, we havet\u2192\u221e(H(t)/t) = 0 a.s. Therefore, from Lemma\u2009\u20095.2 given in [t\u2192\u221e(1/t)\u222b0tI(s)ds \u2265 \u03b8/\u03b80 a.s. This completes the proof.From such that3\u03b2\u2202fS0,0\u2202Nt\u03bc.From , for any, I(0)) \u2208 R+2, solution (S(t), I(t)) of model , I(0)) \u2208 R+2, solution (S(t), I(t)) of model (t\u2192\u221e(1/t)\u222b0tS(s)ds \u2264 S0 and for any small enough constant \u03b5 > 0 there is T0 > 0 such that S0 \u2212 \u03b5 \u2264 S(t) + I(t) \u2264 S0 + \u03b5 for all t \u2265 T0. Hence, by t \u2265 T0 we have f(S(t), I(t)) \u2264 MSS(t), where MS = maxD\u03b5{f/S} < \u221e. Integrating the first equation of model  = SI/N (standard incidence). If Let  R~0=\u03b2-1/\u03c32/ = h(S)g(I). Assume that (H\u2217) holds and Let We further have the result on the weak permanence of model  in proba\u03be > 0 such that, for any initial value (S(0), I(0)) \u2208 R+2, solution (S(t), I(t)) of model , I(0)) \u2208 R+2, solution (S(t), I(t)) of model (\u03be > 0 satisfies the equationLet in model . If R~0>t\u2192\u221e(S(t) + I(t)) = S0. Without loss of generality, we assume that S(t) + I(t) \u2261 S0 for all t \u2265 0. From  = f/I. Then, for any t \u2265 0,g(u) = \u03b2u \u2212 (\u03c32/2)u2 \u2212 (\u03bc + \u03b3). With condition g(0) = \u2212(\u03bc + \u03b3) < 0 andg(u) = 0 has a positive root \u03b7 in /\u2202I) which isu(I) is monotone decreasing for I \u2208 , u(S0) = 0, and\u03be \u2208  such that u(\u03be) = f/\u03be = \u03b7 and g(u(\u03be)) = g(\u03b7) = 0.From  0. From , for any\u03c3 > 0 and \u03b2/\u03c32 < \u2202f/\u2202I, since function g(u) has maximum value g(\u03b2/\u03c32) at u = \u03b2/\u03c32 and g(\u03b2/\u03c32) > g/\u2202I), there is a unique \u03b7 \u2208 /\u2202I) and g(\u03b7) = 0 we have \u03b7 < \u03b2/\u03c32. Hence, When g(u(I)) > 0 is strictly increasing on g(u(I)) > 0 is strictly decreasing on g(u(I)) < 0 is strictly decreasing on I \u2208 .From the above discussion we obtain that \u03c32 \u2264 \u03b2//\u2202I), similarly to the above discussion, we can obtain that g(u(I)) > 0 is strictly decreasing on I \u2208  and g(u(I)) < 0 is strictly decreasing on I \u2208 .When \u03b50 \u2208  such that P(\u03a91) > \u03b50, where \u03a91 = {lim\u2009supt\u2192\u221eI(t) < \u03be}. Hence, for every \u03c9 \u2208 \u03a91, there is a constant T1 = T1(\u03c9) \u2265 T0 such thatg(u(I(t))) \u2265 g(u(\u03be \u2212 \u03b50)) > 0 for all t \u2265 T1. From (t \u2265 T1t\u2192\u221e(log\u2061I(t)/t) \u2264 g(u(\u03be \u2212 \u03b50)) > 0, which implies I(t) \u2192 \u221e as t \u2192 \u221e. This leads to a contradiction with  such that P(\u03a92) > \u03b51, where \u03a92 = {lim\u2009inft\u2192\u221eI(t) > \u03be}. Hence, for every \u03c9 \u2208 \u03a92, there is T2 = T2(\u03c9) \u2265 T0 such thatg(u(I(t))) \u2264 g(u(\u03be + \u03b51)) < 0 for all t \u2265 T2. Together with (t \u2265 T2t\u2192\u221e(log\u2061I(t)/t) \u2264 g(u(\u03be + \u03b51)) < 0, which implies I(t) \u2192 0 as t \u2192 \u221e. This leads to a contradiction with , I(t)) of model (S(0), I(0)) \u2208 R+2 oscillates about a positive number \u03be. Therefore, an interesting open problem is whether there is a more less positive m than number \u03be such that any solution (S(t), I(t)) of model (S(0), I(0)) \u2208 R+2, lim\u2009inft\u2192\u221eI(t) \u2265 m a.s. In of model  with ini\u03be will arise from the change when the noise intensity \u03c3 changes. Therefore, it is very interesting and important to discuss how number \u03be changes along with the change of \u03c3. We have the following result.From \u03b1 = 0 in model /\u2202I)/(\u03bc + \u03b3). Then one has the following.(a)\u03be as the function of \u03c3 is defined for (b)\u03be is monotone decreasing for (c)\u03c3\u21920\u03be = I\u2217, where  is the endemic equilibrium of deterministic model (limic model .(d)R0 \u2264 2, then R0 > 2, then \u03be2 satisfies If 1 \u2264 Assume that in model  and R~0>\u03be as the function of \u03b7 is defined for \u03b7 \u2208 /\u2202I). From\u03b7 \u2208 /\u2202I) when \u03be as a function of \u03c3 is defined for Since\u03b7 with respect to \u03c3, we haved\u03b7/d\u03c3 > 0. From the definition of \u03be, we easily see that \u03be is monotone decreasing for \u03b7. From (H), we obtain that d\u03be/d\u03b7 exists and is continuous for \u03b7. Since (\u2202/\u2202\u03be)/\u03be) < 0, we have d\u03be/d\u03b7 < 0. Therefore, d\u03be/d\u03c3 = (d\u03be/d\u03b7)(d\u03b7/d\u03c3) < 0. It follows that \u03be is monotone decreasing as \u03c3 increases. Thus, both lim\u03c3\u21920\u03be and \u03c3\u21920\u03be = \u03be1 and \u03c3\u21920/\u03be) = lim\u03c3\u21920\u03b7 = (\u03bc + \u03b3)/\u03b2. This shows that f/\u03be1 = (\u03bc + \u03b3)/\u03b2. Let  be the endemic equilibrium of deterministic model /I\u2217 = (\u03bc + \u03b3)/\u03b2. Hence, \u03be1 = I\u2217. This shows that lim\u03c3\u21920\u03be = I\u2217.Computing the derivative of  \u03b7. From  and (H), we have5d\u03b7d\u03c3\u22122\u03b2\u03c33R0 \u2264 2, then from . Therefore, 0 < \u03be < I\u2217 when \u03c3 > 0. If 1 \u2264 R0 \u2264 2, then when \u03be has a minimum value 0 and if R0 > 2 then when \u03be has a minimum value \u03be2 > 0 by Conclusion (d).Conclusion (b) of in model , number \u03b1 = 0 then R0 > 2 then when \u03c3 > \u03c31, where \u03c31 is given in (t\u2192\u221eI(t) = 0 a.s. for any solution (S(t), I(t)) of model (S(0), I(0)) \u2208 R+2, which implies that \u03be = 0. Therefore, when R0 > 2, we can propose an interesting open problem: whether there is a critical value \u03c3 \u2208  we have the fact that \u03be is monotonically decreasing and \u03be > 0 and when \u03c3 > \u03c3\u2217 we have \u03be = 0.It is clear that when in model  \u03b1 = 0 th=\u03c3\u00af from . On the given in , we haveR0 > 2, then from  is the endemic equilibrium of deterministic model  = SI, we easily validate that Theorems When \u03b1 > 0 in model , I(t)) of model (S(0), I(0)) \u2208 R+2, one has limt\u2192\u221eI(t) \u2265 m, a.s. In For the case in model , an inteof model  with iniFrom Theorems >1 model  is perma>1 model  also hasIf en model  is positS, I) : S \u2265 0, I \u2265 0, S0 \u2264 S + I \u2264 S0}. Let (S(t), I(t)) be any solution of model (S(0), I(0)) \u2208 \u0393 a.s. for all t \u2265 0. Let a > 0 be a large enough constant, and letS, I) \u2208 \u0393\u2216D, then either 0 < S < 1/a or 0 < I < 1/a. The diffusion matrix for model  \u2265 \u03c32/(aS0 \u2212 1))2.Here, the method given in the proof of Theorem\u2009\u20095.1 in  is improor model  is(63)ASv < 1 is a constant. Computing L\u03a81, by \u03be, \u03d5) \u2208 \u0393 andM1; M2 < \u221e. We hence haveL\u03a82, by L\u03a83, we haveM0 = max\u0393{f/S} < \u221e. From the above calculations, we obtain that for any  \u2208 \u0393\u2216Dv > 0 is small enough, it follows thata > 0 is large enough\u03be such thatChoose a Lyapunov function as follows:given in , we knowComparing Since en model  is positf, we have the following idiographic results on the stationary distribution as the consequences of Particularly, for some special cases of nonlinear incidence f = SI/N (standard incidence). If Let f = h(S)g(I). Assume that (H\u2217) holds and Let \u03c3 of stochastic perturbation and basic reproduction number R0 of deterministic model (Combining ic model .R0 \u2264 1. Then for any \u03c3 > 0 the disease in model ((a) Let in model  is extinR0 \u2264 2. Then for any (b) Let 1 < R0 > 2. Then for any \u03c3 > \u03c31, where \u03c31 is given in ((c) Let given in , the disf = SI/(1 + \u03c9I), where \u03c9 > 0 is a constant. The corresponding discretization system of model  are the Gaussian random variables which follow standard normal distribution N.In this section we analyze the stochastic behavior of model  by meansof model  is given\u03b2 = 0.60, \u03bc = 11, \u03b3 = 13, \u03c3 = 0.075, and \u03b1 = 2.In model  we choosR0 = 4.195 > 2, \u03b2/S0 \u2212 \u03c32 = \u22120.0023 < 0, and \u03c32 \u2212 \u03b22/2(\u03bc + \u03b3) = \u22120.0019 < 0 which is the case of By computing we have \u03b2 = 0.9, \u03bc = 30, \u03b3 = 12, and \u03c3 = 0.09.In model , choose By computing we have \u03b2 = 0.5, \u03bc = 30, \u03b3 = 20, \u03c3 = 0.02, and \u03b1 = 2.In model  choose \u039bR0 = 1.2500, and \u03be = 0.1037. The numerical simulations are found in I(t) of model g(I). A series of criteria in the probability mean on the extinction of the disease, the persistence and permanence in the mean of the disease, and the existence of the stationary distribution are established. Furthermore, the numerical examples are carried out to illustrate the proposed open problems in this paper.In this paper we investigated a class of stochastic SIS epidemic models with nonlinear incidence rate, which include the standard incidence, Beddington-DeAngelis incidence, and nonlinear incidence It is easily seen that the research given in  for the The researches given in this paper show that stochastic model  has more\u03c3 > 0 of the stochastic perturbation, then R0 > 1 we still can have In addition, we easily see that when intensity ic model  the dise"}
{"text": "There are 2 errors in the first paragraph of the Results subsection titled \u201cHIV tat and gp 120 and Cell-free HIV Virions Induce Activation of MAPK Signaling in Polarized Oral Epithelial Cells.\u201d In the first sentence, \u03b1\u03d6 should be \u03b1v. In the third sentence, \u03d6 should be \u03b1v.In the Discussion section, there are 2 errors in the fourth sentence of the third paragraph. In the fourth sentence, \u201c\u03c9\u03b7\u03b9\u03c7\u03b7\u201d should be \u201cwhich\u201d and \u201c\u03bb\u03b5\u03b1\u03b4ing\u201d should be \u201cleading.\u201d"}
{"text": "T be a singular integral operator with its kernel satisfying |K(x \u2212 y) \u2212 \u2211k=1\u2113\u200dBk(x)\u03d5k(y)| \u2264 C | y|\u03b3/|x \u2212 y|n+\u03b3, |x | > 2 | y | > 0, where Bk and \u03d5k\u2009\u2009 are appropriate functions and \u03b3 and C are positive constants. For bj \u2208 BMO(\u211dn), the multilinear commutator T and Lp-boundedness and the weighted weak type Llog\u2061L estimate for the multilinear commutator Let  In the classical Calder\u00f3n-Zygmund theory, the H\u00f6rmander's condition\u222b|x|>2|y|Lp-boundedness of certain singular integral operators, Grubb and Moore |2 \u2208 RH\u221e(Rnl), where yi \u2208 Rn and i, k = 1,\u2026, l;there exist functions K4)\u2009, we defined the convolution operator associated to the kernel K byFor (1)p < \u221e and \u03c9 \u2208 Ap. Then there exists a constant C > 0 such thatLet 1 < (2)\u03c9 \u2208 A1. Then there exists a constant C > 0 such that for all \u03bb > 0Let Let l = 1, B1(x) = K(x), and \u03d51(y) \u2261 1, then condition (K(x) = sinx/x satisfies conditions (K1)\u2013(K4), but does not satisfy the H\u00f6rmander's condition \u2009\u2009. The generalized commutator, the so-called the multilinear commutator, For In 2002, P\u00e9rez and Trujillo-Gonz\u00e1lez  studied In 1993, Alvarez et al.  establisK be a linear operator and 1 < p < \u221e. Suppose that for all \u03c9 \u2208 Ap(Rn), the linear operator K satisfies the following weighted estimateC depends only on n, p, and the Ap constant of \u03c9. Then for b \u2208 BMO(Rn) and any weight function \u03bd \u2208 Ap, the commutator  is bounded from Lp(\u03bd) to Lp(\u03bd) with bound depending on n, p, and the Ap constant of \u03c9.Let T defined by (K1)\u2013(K4).The goal of this paper is to study the weighted norm inequalities for multilinear commutator of the convolution operator fined by  with itsm-times, we can easily get the following weighted Lp inequalities for the multilinear commutator By T be the singular integral operator defined by (K1)\u2013(K4). If 1 < p < \u221e, \u03c9 \u2208 Ap, and bj \u2208 BMO(Rn)\u2009\u2009, then there exists a positive constant C such thatLet fined by  with itsH1(Rn) into L1(Rn) when b \u2208 BMO(Rn); see ).).3]). By\u03d51,\u2026, \u03d5l} is a finite family of bounded functions on Rn such that |det\u2061[\u03d5k(yi)]|2 \u2208 RH\u221e(Rnl). Then, for any cube Q centered at the origin and any f \u2208 L1(Q), there exists the projection PQf of f onto span\u2061{\u03d51,\u2026, \u03d5l} \u2282 L1(Q) and satisfiesC depends only on n, l, and the RH\u221e constant of |det\u2061[\u03d5k(yi)]|2.Suppose {\u221e)\u2192[0, \u221e) is said to be a Young function, if \u03a6 is continuous, convex, and increasing with \u03a6(0) = 0 and lim\u2061t\u2192\u221e\u03a6(t) = \u221e. We use A function \u03a6 : [0, f over a cube Q \u2282 Rn is defined byThe \u03a6-average of a locally integrable function ies  = t(1 + log\u2061+t)\u03b1\u2009\u2009(\u03b1 > 0) with its complementary Young function The Young function that we are going to use is \u03a6\u03b1 = 1, we simply write \u03a6(t) = t(1 + log\u2061+t) and f||L(log\u2061L),\u2009Q = ||f||Q\u03a6,\u2009 and When The following generalized H\u00f6lder's inequality holds (see (2.5) in ):(24)1|\u03c9 \u2208 A\u221e and a cube Q \u2282 Rn, denoteWe also need the following notations . For 0 \u2264 t < \u221e, define \u03c6j\u22121(t) = inf\u2061{s : \u03c6j(s) > t}\u2009\u2009. If for all 0 \u2264 t < \u221et1, t2,\u2026, tk < \u221e, there existk(t) = t(1 + log\u2061+t)k\u2009\u2009 and \u03a8(t) = et \u2212 1, we have \u03a6k\u22121(t) \u2248 t/(log\u2061t)k and \u03a8\u22121(t) \u2248 log\u2061t A(t) = \u03a6j(C\u22121t) since A\u22121(t) = C\u03a6j\u22121(t), then it follows from s, t1, t2,\u2026, tm\u2212j < \u221e, we haveNoting that f and a cube Q, denoteFor a locally integrable function \u03c9 \u2208 A\u221e and b \u2208 BMO(Rn). Then, for any cube Q \u2282 Rn,C0 and C are positive constants independent of b and Q, and ||b||\u2217 is the BMO-norm of b.Let p < \u221e, \u03c9p \u2208 A1, bj \u2208 BMO(Rn)\u2009\u2009, and Q be a cube. Then for any positive integer m and k = 0,1,\u2026,Let 1 \u2264 m = 1, we write b = b1 and \u03c9 \u2208 A1 and b \u2208 BMO(Rn), there exists constant C > 0 such that, for all \u03bb > 0,When \u03bb > 0, we consider the Calder\u00f3n-Zygmund decomposition of f at height \u03bb and get a sequence of nonoverlapping cubes {Qi}, where Qi = Q is a cube centered at yi with radius ri, such thatFor any fixed f|Qi the restriction of f to Qi. Let gi(x) be the projection of f|Qi onto Yi = span\u2061{\u03d51(\u00b7\u2212yi), \u2009\u03d52(\u00b7\u2212yi),\u2026, \u03d5l(\u00b7\u2212yi)}. We decompose f into two parts, f = g + h, whereh(x) = f(x) \u2212 g(x) = \u2211ihi(x) with hi(x) = f(x) \u2212 gi(x) for x \u2208 Qi.Denote by hi is supported on Qi and it follows from ),g(x)|\u2264\u03bb, for a.e. x \u2208 Rn\u2216\u222aiQi. On the other hand, for any x \u2208 \u222aiQi there exists an i so that x \u2208 Qi, and noting that gi(x) is the projection of f|Qi onto Yi, then it follows from Furthermore, we havedeed, by  and 38)40)|g(x)|g(x)(40)\u03c9 \u2208 A1, then by , a, a\u03c9 \u2208 A then by , we haveQi, by Qi, by  first. Applying (K4), and Let us consider Applying , conditiows from  that(50J(2). By the weak type  estimate of T , .of T see , and LemNote that  impliesThen by  we haveJ(1) and J(2), we haveCombining the estimates for ong with  and 46)J(1) andong with , which iIn this section, we will use an induction argument to prove m and j\u2009\u2009(1 \u2264 j \u2264 m), we denote by Cjm the family of all finite subsets \u03c3 = {\u03c3(1), \u03c3(2),\u2026, \u03c3(j)} of {1,2,\u2026, m} of j different elements. For any \u03c3 \u2208 Cjm, we write \u03c3\u2032 = {1,2,\u2026, m}\u2216\u03c3.As in , given pbj \u2208 BMO(Rn) and \u03c3 = {\u03c3(1), \u03c3(2),\u2026, \u03c3(j)} \u2208 Cjm\u2009\u2009(1 \u2264 j \u2264 m), we denote by Q is a cube in Rn and For m = 1 in j < m; namely, for all 1 \u2264 j < m and any \u03c3 \u2208 Cjm, we haveWe have proved that \u03bb > 0, we consider the Calder\u00f3n-Zygmund decomposition of f at height \u03bb as in Qi}, Qi*, g, h, hi, and \u03a9 as there.For any fixed For the same reason as in , we haveSimilar to , we haveReasoning as the proof of Lemma 3.1 in  , and J(1) in Applying , conditiJ2, by the weak type  estimate for T (see J(2) in For or T see , 27), 43) we haJ3 by applying the induction hypothesis.Now, let us consider hi(x) = (f(x) \u2212 gi(x))\u03c7Qi(x)\u2009\u2009, we can split J3 into two partsNoting that \u03c3 \u2208 Cjm, we denote by \u03c3\u2032 = {\u03c3\u2032(1), \u03c3\u2032(2),\u2026\u03c3\u2032(m \u2212 j)}, so thatFor Cs,0 and Cs such that for s = 1 \u22ef m \u2212 jFrom \u03b3s = ||\u2217)\u22121\u2009\u2009; then it follows from the induction hypothesis and (Set esis and  that(72By  and 43)43), we hm(ab) \u2264 C\u03a6m(a)\u03a6m(b) for a, b > 0, we haveNoting that \u03a6J3(2). By Jensen's inequality,Finally, we consider J3(1), we haveBy the induction hypothesis, , and 7575, similApplying , we haveong with  and 74)(77)\u222bQi{1J1, J2, and J3, we obtainBy , 65), a, a65), aThe proof of the general case of"}
{"text": "R0 has been worked out. The local and global asymptoticalstability analysis of the equilibria are performed, respectively. Furthermore, if we take the treatedrate \u03c4 as the bifurcation parameter, periodic orbits will bifurcate from endemic equilibrium when\u03c4 passes through a critical value. Finally, some numerical simulations are given to support ouranalytic results.An SEIV epidemic model for childhood disease with partial permanent immunity isstudied. The basic reproduction number  S(t), I(t), and V(t), respectively, represent the number of susceptible individuals at time t, infective individuals at time t, and vaccinated individuals at time t. At the earliest, most researches on these types of models assume that the disease incubation is negligible, so that each susceptible individual, once infected, instantaneously turns into infectious and later recovers obtaining a permanent immunity. Soon afterwards, the models become more general. Researchers assume that a susceptible individual first goes through a latent period after infection before becoming infectious .It is primarily important for health administrators to protect children from disease that can be prevented by vaccination. Although preventive vaccines have reduced the incidence of infectious diseases among children, childhood disease is an important public health problem. We often use mathematical models to realize the transmission dynamics of childhood diseases and to estimate control programs \u20134. RecenA represents the number of additional populations of childhood; \u03c9 represents the rate at which vaccine wanes; \u03bc represents the natural death rate; \u03b2 represents the rate at which susceptible individuals become infected by those who are infectious; p represents the fraction of recruited individuals who are vaccinated; \u03c4 represents the rate at which infected individuals are treated; and \u03c3 represents the rate at which exposed individuals become infectious.In , the aut\u03b2SI is called incidence rate which plays an important role in the transmission dynamics. In addition, incidence rate can determine the tendency of epidemics. At the earliest, in the classical epidemic disease model, scholars made much focus on the bilinear incidence /\u03bc(\u03bc + \u03c9), pA/(\u03bc + \u03c9), 0,0); that is, E = I = 0. And {\u2223S > 0, E \u2265 0, I \u2265 0, V > 0} is a positively invariant set of system (t\u2192\u221e\u2061(S + E + I + V) \u2264 A/\u03bc. Therefore, the set \u03a9. Set x = T; then system (FV\u22121 develops a meaningful definition of R0 and is the expected number of new infections for system (\u03c1(FV\u22121) = \u03c3\u03b2A[\u03bc(1 \u2212 p) + \u03c9]/\u03bc(\u03bc + \u03c9)(\u03bc + \u03c3)(\u03bc + \u03c4 + \u03f5) is the spectral radius of matrix FV\u22121. Thus by /\u03bc(\u03bc + \u03c9), pA/(\u03bc + \u03c9), 0,0) when R0 = 1. In other words, we will discuss under what conditions system  is the linearization matrix of system  right eigenvector V and a left eigenvector W with respect to the zero eigenvalue.Matrix Suppose the following.fk as the kth component of f, and a and b totally decide the local dynamic of system . And the disease-free equilibrium changes its local stability at the point  /\u03bc(\u03bc + \u03c9), pA/(\u03bc + \u03c9), 0,0) and notice that the condition R0 = 1 can be seen as \u03b2 = \u03b2* = \u03bc(\u03bc + \u03c4 + \u03f5)(\u03bc + \u03c3)(\u03bc + \u03c9)/\u03c3A[(1 \u2212 p)\u03bc + \u03c9] in terms of the parameter \u03b2.We will show that system  may exhi\u03bb1 = \u2212\u03bc, \u03bb2 = \u2212\u03bc \u2212 \u03c9, \u03bb3 = \u22122\u03bc \u2212 \u03c3 \u2212 \u03c4, and \u03bb4 = 0.Calculate the eigenvalues of the following matrix:J has a simple eigenvalue of 0; and all others have negative real parts. Thus, we can make use of the center manifold theory. The disease-free equilibrium P0 is a nonhyperbolic equilibrium when \u03b2 = \u03b2* . This completes the verification with respect to (A1) of The matrix V = T as a right eigenvector associated with the zero eigenvalue \u03bb4 = 0. It is calculated by W =  satisfying W \u00b7 V = 1 is obtained by W turns out to be Now we set lated by \u2212\u03bc\u03c90\u03c4\u2212\u03b2\u2217Apanding  and (\u22022fk/\u2202xi\u2202\u03b2) which are nonzero, the following are deduced: So, we evaluate m system , and the0,\u03b2\u2217).By  and 39)a and b ab is always positive. Therefore the sign of the coefficient a determines the local dynamics around the disease-free equilibrium for \u03b2 = \u03b2* by Obviously \u03b10 = (M \u2212 \u03c3\u03c4(\u03bc + \u03c9))/\u03c3A[\u03bc(1 \u2212 p) + \u03c9]. The coefficient a is positive if and only if \u03b1 > \u03b10. Under this circumstance, the direction of the bifurcation for system \u03c4 = \u03c4*, \u03c4 = \u03c4*, we have \u03c4, the roots are all in the following general forms: \u03bbj(\u03c4) = \u03b11(\u03c4) + i\u03b12(\u03c4) into (For =c3.From  and 46)\u03c4 = \u03c4*, c3.From ((\u03bb2+c2) < 1. If R1* > 1 the disease-free equilibrium P0 is globally asymptotically stable in \u03a9; if R1* < 1 the disease-free equilibrium P0 is globally asymptotically stable in \u03a9 when R0 < R0*.Let R0(1 + \u03b1A/\u03bc) < 1, R0 < 1. If R1* > 1, P0 is the only equilibrium of\u2009\u2009(\u03a9. From the first equation of\u2009\u2009(dS/dt \u2264 (1 \u2212 p)A + (A/\u03bc)(\u03c9 + \u03c4)\u2212(\u03bc + \u03c9)S. A solution of the equation dy/dt = (1 \u2212 p)A + (A/\u03bc)(\u03c9 + \u03c4)\u2212(\u03bc + \u03c9)y is a upper solution of S(t). Due to that y \u2192 A[\u03bc(1 \u2212 p)+(\u03c9 + \u03c4)]/\u03bc(\u03bc + \u03c9) when t \u2192 \u221e, we can easily get that, for a small enough \u03b5 > 0 which is sufficiently small, there exists a t0 such that S(t) \u2264 y(t) \u2264 A[\u03bc(1 \u2212 p)+(\u03c9 + \u03c4)]/\u03bc(\u03bc + \u03c9) + \u03b5 as t > t0.When rium of\u2009\u2009 which istion of\u2009\u2009, we obtaL = \u03c3E + (\u03bc + \u03c3)I, thus Considering the Lyapunov function R0(1 + \u03b1A/\u03bc) < 1, we can choose \u03b5 small enough such that (1 + \u03b1A/\u03bc)R0 \u2212 1 + (1 + \u03b1A/\u03bc)\u03b5\u03c3\u03b2/(\u03bc + \u03c3)(\u03bc + \u03c4 + \u03f5) < 0. Thus,L\u2032 = 0 if and only if I = 0. The singleton P0 is the maximum positive invariant set in { \u2208 \u03a9, L\u2032 = 0}. The global stability of P0 for every solution follows from LaSalle's Invariance Principle.For R1* < 1, system  so that the disease can be eliminated. Thus, when R1* < 1 the disease-free equilibrium P0 is globally asymptotically stable in \u03a9 when R0 < R0*.If , system  has two , system  shows a P* for R0 > 1. Due to S + V + E + I \u2192 A/\u03bc when t \u2192 \u221e, we can determine V(t) by S(t), E(t), and I(t). So system  \u2192 g(x) when t \u2192 \u221e locally uniformly for x \u2208 Rn, then system ,dydt=g(y)P is a locally asymptotically stable equilibrium of (x is the \u03c9-limit set of a forward bounded solution x(t) of (x includes a point y0 such that the solution of (y(0) = y0 converges to P when t \u2192 \u221e, then \u03c9 = P; that is, x(t) \u2192 P when t \u2192 \u221e.Set brium of  and x is x(t) of . If x inution of  with y(0P of the limit system (x(t) of system (x(t) \u2192 P when t \u2192 \u221e.If solutions of system  are bount system  is globaf system  satisfieP* is globally asymptotically stable for R0 > 1 by the geometrical approach :Next we will look for conditions which satisfied (H3). Consider the Jacobian matrix  and get p(x) = P = diag{E/I, E/I, E/I}. Then PfP\u22121 = diag{E\u2032/E \u2212 I\u2032/I, E\u2032/E \u2212 I\u2032/I, E\u2032/E \u2212 I\u2032/I}. Thus, the matrix B = PfP\u22121 + PJ[2]P\u22121 can be written in block form as Let R3. We choose a standard in R3 as \u03bc be the LozinskiSet nique in , the folc \u2264 S, I \u2264 A/\u03bc, where c is the constant of uniform persistence; it is obvious that d > 0.From the second and third equations of system , we can S(t), E(t), I(t)) \u2208 \u03a9, we obtain For each  = 0.4800 and R0 = 0.00031987 < 1 which satisfies R0(1 + \u03b1A/\u03bc) = 0.0011 < 1 which satisfies P0 and it is globally asymptotically stable  = 63.8400, R0 = 48.1293, R1* = 267.6519, and R0* = 0.0032. Therefore R0* < 1 < R0 < R1*, due to P0 and an endemic equilibrium P*. And we can calculate d = 45.53 > 0 which guarantees \u03c4 = 40 > \u03c4*, then the endemic equilibrium P* becomes unstable and a periodic orbits bifurcates from P*, which is demonstrated by We take parameters R0 and found that when R0 = 1 and R1* < 1 system  < 1, respectively. Also we have studied the local and global asymptotic stability of the endemic equilibrium. Moreover, taking the disease-caused death rate \u03c4 as bifurcation parameter, we discussed the Hopf bifurcation of system  < 1, if R1* > 1 holds or R1* < 1, R0 < R0* hold; system  < 0, which means that vaccinating more susceptible populations decreases the likelihood of the occurrence of backward bifurcation [From the sense of epidemiology, when ; system  has one equality  or 78)  R0(1 + equality  has a un, system  has a unurcation ."}
{"text": "On PDF page\u00a05, first column, the GenBank accession number \u201c1910160\u201d should be updated to \u201cKX174310.\u201d The correct URL is http://www.ncbi.nlm.nih.gov/nuccore/KX174310.Volume 7, no 3, doi:mon, +700/\u2212 less than 5,062\u00a0bp but greater than 700\u00a0bp)\u201d should be updated to \u201cRD1mon, +309\u00a0bp/\u2212513\u00a0bp).\u201dOn PDF page\u00a011, second column, \u201cRD1"}
{"text": "We derive some simple relations that demonstrate how the posterior convergence rate is related to two driving factors: a \u201cpenalized divergence\u201d of the prior, which measures the ability of the prior distribution to propose a nonnegligible set of working models to approximate the true model and a \u201cnorm complexity\u201d of the prior, which measures the complexity of the prior support, weighted by the prior probability masses. These formulas are explicit and involve no essential assumptions and are easy to apply. We apply this approach to the case with model averaging and derive some useful oracle inequalities that can optimize the performance adaptively without knowing the true model. In Jiang , there adu2-divergence for any u \u2208  (defined later), which is more general than the squared Hellinger distances d\u22121/22 (corresponding to u = \u22121/2). A recent work by Norets  and \ud835\udcab is the support of the prior \u03c0.Consider a \u201cposterior sequence\u201d of densities penalized divergence\u201d dt2 \u2261 inf\u2061K\u2282\ud835\udcab[sup\u2061q\u2208Kd2 + n\u22121ln\u2061(1/\u03c0(K))], for some divergence d of a prior \u03c0 from the true density p0.Here we define a \u201cu \u2208 . This is because the dt divergence is monotonically increasing in t. For any du\u2212 so that 1/u is not an integer, we can use a more stringent divergence du\u2032\u2212 with the \u2212u\u2032 being the next larger value from the integer range {\u22121/2, \u22121/3, \u22121/4,\u2026} to bound the convergence rate in du\u2212.The result can be extended to the continuously valued penalized divergence\u201d of the form dt2 \u2261 inf\u2061K[sup\u2061q\u2208Kdt2 + n\u22121ln\u2061(1/\u03c0(K))], related to a prior \u03c0. This first part sup\u2061q\u2208Kdt2 describes the maximal divergence of a set K (proposed by a prior \u03c0) from p0. We can understand this part as the approximation error of the prior \u03c0 when it is used to propose densities to approximate a true density p0. The second part penalizes an unlikely set K with a small prior \u03c0(K). Combining the two parts, we can perhaps try to interpret d2 as the approximation error (away from p0) by a not-too-unlikely set proposed by a prior \u03c0. This \u201cpenalized divergence\u201d is a critically important driving factor for determining the convergence rates in the previous results. It is noted that although this factor corresponds to the approximation ability of \u03c0, it already has a complexity penalty built in it implicitly. This is from the penalty against a small prior; the second part is n\u22121ln\u2061(1/\u03c0(K)), which is, roughly speaking, about d/n, where d is the number of parameters proposed by the prior \u03c0  \u221d \u03b4d).In this and other works, we notice that we often encounter in the convergence rate results a quantity similar to the \u201cd/n, up to some logarithm factors, where d is the dimension of the parameters involved in the prior. It is noted, however, that with model averaging the higher dimensional model can be downweighted by the model prior, so that effectively one can make d to be of order 1 for this complexity factor, so that the convergence rate will be controlled by the first factor  alone.The other factor behind the convergence rate is related to the complexity of the model, which is proportional to \u03c0 = \u03c0m\u03c0(df\u2223m), jointly over a model index m \u2208 M in a set of nonoverlapping models M, and density f \u2208 \ud835\udcabm (the support of prior given model m)  \u221d \u2211m\u2208M\u222bf\u2208\ud835\udcabmI \u2208 A)\u03a0i=1nf(Yi)\u03c0m\u03c0(df\u2223m) for an event A. In this case, let L1 balls of radius (1/n)u1/ needed to cover the prior-support \ud835\udcabm under model m. Then we have the following, under the iid assumption.The convergence rate result in p for iid data satisfying u \u2208 {1/2,1/3,1/4,\u2026} and any 0 < t < \u221e, with probability 1, for almost all large sample size n, one has \ud835\udcab and \ud835\udcabm are supports of the mixing prior \u03c0 = \u2211m\u03c0m\u03c0(df\u2223m) and the model-m prior \u03c0(df\u2223m), respectively.Consider a \u201cposterior sequence\u201d of densities m for the bound on the right hand side. Again, the convergence rate is displayed explicitly, and we will try to explain the driving factors of the convergence rate later. This is unlike the previous works where one has to conjecture a rate \u03f5 and check that it satisfies many conditions.This is an oracle inequality that achieves the best performance of all models \u03c0, developing an idea pioneered by Walker u1/ over all such covers \u222ajBj of \ud835\udcab, where each Bj is an L1 ball of radius nu\u22121/. We may name it as the \u201c\u2113u-norm prior complexity\u201d for covering the prior support. An unbounded prior support may still be coverable by infinitely many Bj's, so that Nu(\u03c0) is finite, even with an infinite covering number Nu(\u03c0) is the \u201c\u2113u-norm complexity\u201d of this prior \u03c0 defined in this remark.The complexity in Nu(\u03c0) defined in the previous remark in parametric models, where densities p \u2208 \ud835\udcab are parameterized by a d dimensional parameter \u03b8, and a prior on \u03b8 induces a prior \u03c0 on the densities in \ud835\udcab. A more rigorous treatment is given in the example of L1 distance between two densities p\u03b81 and p\u03b82 by the maximal norm |\u00b7|\u221e: \u222b | p1 \u2212 p2 | \u2264cd|\u03b81\u2212\u03b82|\u221e for some constant c > 1. Then, to cover the parameter space, we can use \u2113\u221e ball Aj's in the parameter space with radius h = (cdnu1/)\u22121, so that the corresponding densities cover the L1-ball Bi with the required radius nu\u22121/. These sets Aj, with small volumes vol(Aj) = (2h)d, can be used to form a fine partition of the parameter space, so that the norm \u2211j\u2208N\u03c0(Bj)u = \u2211j\u2208N\u03c0(\u03b8 \u2208 Aj)u = [\u2211j\u2208N\u03c0(\u03b8j)uvol(Aj)]vol(Aj)u\u22121 \u2248 [\u222b\u03c0(\u03b8)ud\u03b8]((2h)d)u\u22121, where \u03c0(\u03b8j) is the prior density function evaluated at some intermediate point in the set Aj. The sum in the square bracket is a Riemann sum over a fine grid, which we will assume to be approximated by an integral under some regularity conditions, even if the domain may be unbounded. Therefore, we have an upper bound of the norm complexity as n. Assume that the prior density is Lu integrable in the parameter space, and the norm |\u03c0|u scales as (const)d as in the case of an iid prior \u03c0(\u03b8) = \u220fj=1d\u03c01(\u03b8j). Then the complexity term in the bound of d.We now describe heuristically how to bound the \u201cnorm complexity\u201d Nu(\u03c0) is the \u201c\u2113u-norm complexity\u201d of this prior \u03c0, which in this case should be the infimum of the \u2113u norm [\u2211m\u2208M\u2211j\u2208\ud835\udca9m{\u03c0m\u03c0(Bj\u2223m)}u]u1/ over all such covers \u222am,jBmj of \ud835\udcab, where each Bmj is an L1 ball of radius nu\u22121/, and under each model m, \u222aj\u2208\ud835\udca9mBmj represents a cover of its prior support \ud835\udcabm using possibly infinitely many balls. The defining expression of Nu(\u03c0) can also be related to the norm complexities of all the conditional priors \u03c0 given the model choices: Nu(\u03c0) = [\u2211m\u2208M{\u03c0mNu(\u03c0(\u00b7\u2223m))}u]u1/. With model averaging using some suitable weights \u03c0m, this term Nu(\u03c0) and its effect on the convergence rate no longer diverge with the complexity of the model, in contrast to the conclusion of dt2. An example below (in its second part) is used to illustrate this.Similar to This is a simple binary regression example intended for illustration. We will see that model averaging can be used to derive nearly optimal convergence rates that are adaptive to the assumptions on the true model. In the first part, we will illustrate how to bound the penalized divergence with a uniform prior with a bounded support. In the second part, we will illustrate how to bound the norm complexity when the prior has an unbounded support.Z\u2223X ~ Bin), X \u2208 Unif, where the true conditional mean function \u03bc(X) is denoted as \u03bc0(X)\u2208 , which is bounded away from 0 and 1, for any value of X. We consider an m-piecewise constant working model for the mean function \u03bc. Suppose the prior is \u03c0 = \u03c0m\u03c0 where m indicates the m-piecewise constant model \u03bc(x) = \u2211j=1m\u03b8jI. We consider an independent uniform prior \u03c0 = \u220fj=1mI. Consider a binary regression model We will consider two different setups of the true model.\u03bc0 has continuous derivative bounded by D. We call this a \u201cdense\u201d setup since we may need a large piecewise constant model (with large m increasing with sample size n) to approximate this quite arbitrary true mean function \u03bc0.In the first setup, the true u = \u22121/2 and t = 1. In the present case, we have d12 = \u222b01dx(\u03bc1 \u2212 \u03bc2)2/(\u03bc2(1 \u2212 \u03bc2)), \u2264(\u03b4(1 \u2212 \u03b4))\u22121\u222b01dx(\u03bc1 \u2212 \u03bc2)2, if p1 = \u03bc1z(1 \u2212 \u03bc1)z1\u2212 and p2 = \u03bc2y(1 \u2212 \u03bc2)z1\u2212. We will sometimes use the mean function \u03bc's to denote the corresponding distances as d12 and so on.To apply \u03bc0(x), there exists a \u03bc* in the support of \u03c0(\u00b7\u2223m) so that |\u03bc0 \u2212 \u03bc* | <D/m everywhere. In fact, we can take \u03bc* = \u2211j=1m\u03b8j*I where \u03b8j* = \u03bc0((j \u2212 1)/m)\u2208. Now let \u03bc be close to \u03bc*, in the sense that \u03bc = \u2211j=1m\u03b8jI and \u03b8j \u2208 \u03b8j* \u00b1 \u0394, j = 1, ..., m, for some small \u0394 \u2208 . Take Km to be the set of all such densities of Bin. Then \u03bc \u2208  everywhere, since \u03b8j \u2208 \u03b8j* \u00b1 \u0394 and \u03b8j* \u2208  for all j.The approximation property of the piecewise constants implies that, for any true p0 and p be the densities corresponding to \u03bc0 and \u03bc, respectively. Then sup\u2061p\u2208Kmd12 = sup\u2061p\u2208Km\u222b01dx(\u03bc0 \u2212 \u03bc)2/(\u03bc(1 \u2212 \u03bc))\u2264sup\u2061p\u2208Km\u222b01dx(|\u03bc0 \u2212 \u03bc* | +|\u03bc \u2212 \u03bc* | )2/{min\u2061x(\u03bc(1 \u2212 \u03bc))}\u2264(D/m + \u0394)2/((\u03b4 \u2212 \u0394)(1 \u2212 \u03b4 + \u0394)) due to the triangle inequality. The prior probability over  is \u03c0m(2\u0394)m. Therefore, the \u201cpenalized divergence\u201d \u03c0m \u221d eK(m\u22121)ln\u2061n\u2212 for some large enough constant K > 0 and apply d2  and the resulting convergence rate d\u22121/22 are both of order O(n\u22122/3ln\u2061n), which is within a ln\u2061n factor to the minimax optimal result. It is noted that the model averaging automatically achieves this near optimal rate.We can take m0-piecewise constant, where we do not know the value m0. We call this a sparse case since we only need an m-piecewise constant model \u03bc* to approximate the true mean function \u03bc0 perfectly, where m = m0 can be much smaller than the choice of m ~ n1/3 in Consider a second setup, where we assume that the true model is a d12 by taking m = m0 and \u0394 = 1/n and obtain (using |\u03bc* \u2212 \u03bc0 | = 0) the following: d\u22121/22 are, therefore, both close to the parametric rate O(m0/n), as if we knew m0 beforehand.Then modifying the above reasoning, we can bound the infimum in the \u201cpenalized divergence\u201d \u03c0 is adaptive in the sense that in either the dense or the sparse case, the resulting posterior distribution works nearly optimally, even if we do not really know whether the true model is dense or sparse.In summary, the prior \u03c0(\u00b7\u2223m) with a bounded support. In this subsection, we will consider a parametrization in the log-odds scale, with an unbounded prior support, for illustrating how to calculate the norm complexity described in Remarks Z\u2223X ~ Bin, X \u2208 Unif, where the true \u03bc, denoted as \u03bc0, is bounded away from 0 and 1. We consider an m-piecewise constant working model for the mean function \u03bc = \u2211j=1me\u03b8j/(1 + e\u03b8j)I. Suppose the prior is iid for each \u201clog-odds\u201d parameter \u03b8j supported on \u211c; then In the example above, we have considered a uniform prior for Z, X): p1,2 = \u03bc1,2zz1\u2212, with parameters \u03b8j1's and \u03b8j2's, respectively. Then one can easily derive a following relationship for the L1 distance: \u222b | p1 \u2212 p2 | = 2\u222b | \u03bc1 \u2212 \u03bc2 | \u2264(1/2)sup\u2061j=1m | \u03b8j1 \u2212 \u03b8j2|. For covering all the densities in this working model with L1 balls Bmj with radius nu\u22121/ ), we can use the densities with parameters in \u2113\u221e balls  : s = 1,\u2026, m], with radius h/2 = 2nu\u22121/.Consider two densities of (Nu(\u03c0(\u00b7\u2223m))\u2264[\u2211j1=\u2212\u221e\u221e\u22ef\u2211jm=\u2212\u221e\u221e\u03c0h],\u2026, \u03b8m \u2208 \u2223m)u]u1/ = [\u220fs=1m\u2211js=\u2212\u221e\u221e\u03c0h]\u2223m)u]u1/ = [\u2211j=\u2212\u221e\u221e\u03c0h]\u2223m)u]m/u, since the priors of \u03b81,\u2026, \u03b8m\u2223m are iid.Then \u03b8j (given model m) is continuous, symmetric, and decreasing from the origin \u03c0(\u03b8j\u2223m) = fm(|\u03b8j|) for some decreasing functions fm \u2208 Lu  priors or double-exponential densities).Now assume that the prior density for each j=\u2212\u221e\u221e\u03c0h]\u2223m)u = 2\u2211j=0\u221e\u03c0h]\u2223m)u \u2264 2\u2211j=0\u221efm(jh)uhu\u22121\u2264[2hfm(0)u + \u222b\u221e\u2212\u221efm(|w | )udw]hu\u22121, where we used the decreasingness of fm in the last two steps. The integration exists since fm \u2208 Lu.Then \u2211Nu(\u03c0(\u00b7\u2223m)) \u2264 O(hm(u\u22121)/u) = O(nm(1\u2212u)/u2), which is finite despite the unbounded priors support. Then according to Nu(\u03c0) = [\u2211m(\u03c0mNu(\u03c0(\u00b7\u2223m)))u]u1/ will be O(nu)/u2(1\u2212) if \u03c0m \u221d eK(m\u22121)ln\u2061n\u2212 for some large enough constant K > 0. So the norm complexity term in O(ln\u2061n/n), which, when compared with the last formula in O(1) by model averaging. Therefore, the norm complexity term does not affect the convergence rate significantly due to model averaging, and the convergence rate is mainly determined by the penalized divergence d12. The bounding of the penalized divergence is similar to the example discussed in the previous subsection and we omit the details. The resulting convergence rates are essentially the same as when the uniform priors (with bounded supports) are used, despite the fact that we now allow priors with unbounded supports .Then we have P0\u03a0(A) \u2264 P0\u03d5 + P0\u03a0(A)(1 \u2212 \u03d5) for any \u201ctest\u201d \u03d5 as a function of D valued in . Consider P0 represents the expectation under the true density p0.P0\u222bf\u2208A(f(D)/p0(D))(1 \u2212 \u03d5)\u03c0(df) = \u222bf\u2208APf(D)(1 \u2212 \u03d5)\u03c0(df) due to Fubini's Theorem, where Pf represents the expectation under density f.t > 0, we have Using Markov's theorem, for any All these combine to (\u2217) Bj, there exist \u03d5j such that for any \u03b1, \u03b2 > 0, u \u2208 , sup\u2061f\u2208Bj(\u03b1P0\u03d5j + \u03b2Pf(1 \u2212 \u03d5j)) = sup\u2061f\u2208Bj\u222bmin\u2061{(\u03b1p0), (\u03b2f)} \u2264 sup\u2061f\u2208Bj\u222b{(\u03b1p0)u1\u2212(\u03b2f)u} = \u03b1u1\u2212\u03b2usup\u2061f\u2208BjP0(p0/f)u\u2212.Now apply a result that is a straightforward extension of Ghosal et al.  so that \u03b1/\u03b2 = \u03bbj in the above statement.Given any j\u2208NBj to be a convex cover of A and define a combined test \u03d5 = max\u2061j\u2208N\u03d5j and plug it into (\u2217), we have If we take \u222aTherefore, we have \u03bbj = (1 \u2212 u)\u22121ua\u03c0(Bj)\u22121 and Take c = (t/(t + u))t/(t+u)\u2212(u/(t + u))u/(t+u)\u2212(uu(1 \u2212 u)u1\u2212)t/(u+t)\u2212 \u2264 4. Then notice that P0(p0/f)t = 1 + tdt2; we obtain the proof of p0 and f, are the densities for the entire data set p0D and fD, resp., and we do not assume iid.)We get t \u2208 \u222a, Use the fact that, for any This leads to the proof by applying \u03c0m focuses on only one model.This is a special case of \ud835\udcab = \u222am\u2208M\ud835\udcabm, where \ud835\udcabm is the support of \u03c0(\u00b7\u2223m).Repeat the proofs of Propositions A\u2229\ud835\udcab is doubly indexed as \u222am\u2208M\u222aj\u2208\ud835\udca9mBmj, where \ud835\udca9m has cardinality at most j\u2208\ud835\udca9mBmj is a convex cover of the support A\u2229\ud835\udcabm. Then the result in Suppose the convex cover of A = [f : du\u22122 > \u03f5].In Bmj are such that inf\u2061j,minf\u2061f\u2208Bmjdu\u22122 > \u03b4. Then we have (\u2020) Bmj needed to cover A\u2229\ud835\udcabm.Suppose all the convex sets Bmj in more detail. They are used to cover A, so without generality, each Bmj contains a point in A, say p1, which is not close to p0 since du\u22122 > \u03f5. If Bmj is small so that any two points are close together, then any point p2 in Bmj  can be made to be also not close to p0, so that du\u22122 > \u03b4 for some \u03b4 > 0 related to \u03f5. This would be easy to establish by a triangular inequality, were it not for the difficulty that the divergence du\u2212 is not a true distance for u \u2260 1/2. So we would not be able to say, for example, that Bmj should be a small du\u2212-ball.Now we try to define the convex sets u = 1/k for any k = 2,3, 4,\u2026, we have |p1u \u2212 p2u|u1/ = |p1k1/ \u2212 p2k1/ | |p1k1/ \u2212 p2k1/|k\u22121 \u2264 |p1k1/ \u2212 p2k1/ | \u2211i=0k\u22121 | p1i/kp2k\u22121\u2212i)/k( | 2k\u22121 = |(p1k1/)k \u2212 (p2k1/)k | 2k\u22121, so that To resolve this difficulty, we derive the following inequalities: Bmj is an L1 ball with small enough radius \u03bb, then any point p2 in Bmj has a not too small divergence du\u22122 \u2265 \u03f5 \u2212 |du\u22122 \u2212 du\u22122|\u2265\u03f5 \u2212 u\u221212(2\u03bb)u \u2265 \u03f5 \u2212 u\u221214\u03bbu = \u03b4. We can take \u03bb = [(u/4)(\u03f5 \u2212 \u03b4)]u1/. The L1 balls are also convex as required. We will take \u03f5 \u2212 \u03b4 = 4/(nu). Then the radius \u03bb = (1/n)u1/. Therefore, we conclude that we will take Bmj's to be small L1 balls with radius nu\u22121/.Therefore, if A \u222a \ud835\udcabm. We can use this upper bound L1 balls of radius (1/n)u1/ needed to cover the entire prior-support \ud835\udcabm under model m.Now we try to find \u03f5 > 0, (\u2021), dt2(p0\u03c0) \u2261 inf\u2061K\u2208\ud835\udcab[sup\u2061q\u2208Kdt2 \u2212 n\u22121ln\u2061\u03c0(K)] and \ud835\udcab is the support of the prior.Then (\u2020) implies, for any eu/t)\u22122/(1+n)2 under a choice of \u03f5 = \u03f5n for the right-hand side of (\u2021). Then the event [du\u22122 \u2264 \u03f5n] will happen for all large enough n, almost surely, due to the Borel-Cantelli lemma.Let the probability in (\u2021) be bounded by 4/ = inf\u2061K\u2208\ud835\udcab[sup\u2061q\u2208Kdt2 \u2212 n\u22121ln\u2061\u03c0(K)], under a mixture prior \u03c0(K) = \u2211s\u2208M\u03c0s\u03c0(K\u2223s), can be bounded by taking K = Km \u2282 \ud835\udcabm, for any m \u2208 M, as inf\u2061K[sup\u2061q\u2208Kdt2 \u2212 n\u22121ln\u2061\u03c0(K) \u2264 inf\u2061minf\u2061Km{sup\u2061q\u2208Kmdt2 \u2212 n\u22121ln\u2061[\u03c0m\u03c0(Km\u2223m)]}\u2261inf\u2061m[dt2) \u2212 n\u22121ln\u2061\u03c0m].The quantity"}
{"text": "The exact closed-form expressions of the numbers of spanning trees for 4.8.8 lattice, hexagonal lattice, and 3 G = (V(G), E(G)) denote a graph with no multiple edges and no loops and with vertex set V(G) = {v1, v2,\u2026, vn} and edge set E(G). The degree ks of a vertex vs is the number of edges attached to it. A k-regular graph is a graph with the property that each of its vertices has the same degree k. The adjacency matrix A(G) of G is the n \u00d7 n matrix with elements A(G)sj = 1 if vs and vj are connected by an edge and zero otherwise. The Laplacian matrix Q(G) is the n \u00d7 n matrix with the element Q(G)sj = ks\u03b4sj \u2212 A(G)sj, where \u03b4sj is the Kronecker delta, equal to 1 if s = j, and zero otherwise. Denote by t(G) the number of spanning trees of a graph G. Enumeration of spanning trees on the graph is a problem of fundamental interest in mathematics and physics. This number can be calculated in several ways. A basic result is \u201cthe Matrix-Tree Theorem.\u201dLet G be a graph with vertex set {v1, v2,\u2026, vn} and let Q(G) be its Laplacian matrix. Then,Q(G)s}{ is the submatrix of Q(G) by deleting the sth row and the sth column from Q(G) for 1 \u2264 s \u2264 n.Let Q(G) is always zero. We can express t(G) that can be expressed by the nonzero eigenvalue of Q(G) as follows.Note that one of the eigenvalues of \u03bc1 \u2264 \u03bc2 \u2264 \u22ef\u2264\u03bcn\u22121 be the Laplacian eigenvalues of a connected graph G with n vertices. Then, t(G) = \u03bc1\u03bc2 \u2026 \u03bcn\u22121/n.Let 0 < G is said to be n-rotational symmetric if the cyclic group of order n is a subgroup of the automorphism group of G. Yan and Zhang [n-rotational symmetric graph. As applications, they got explicit expressions for the numbers of spanning trees and the asymptotic tree number entropy for some lattices with cylindrical boundary condition.By two methods, Ciucu et al.  obtainednd Zhang  also obtt(G) has asymptotically exponential growth; one defines the quantity z(G) byLattices are of special interest for their structures. In particular, the number of spanning trees in a lattice was studied extensively. It turns out that t(G) have been obtained for many lattices. Wu [d dimensions under free, periodic, and a combination of free and periodic boundary conditions and a quartic lattice embedded on a M\u00f6bius strip and the Klein bottle. Shrock and Wu [d-dimensional body-centred cubic lattice and thermodynamical limit. They also gave an exact integral expression for thermodynamical limit on the face-centred cubic lattice and 4.8.8 lattice. Chang and Wang [Closed-form expressions for ices. Wu  evaluateices. Wu  obtainedk and Wu  got a gek and Wu  got closand Wang  considerand Wang , 11.3 \u00b7 42 lattice. The number of spanning trees of 4.8.8 lattice is gotten in 3 \u00b7 42 lattice in Sections In this paper, we present an exact closed-form result for the asymptotic growth constant for spanning trees of lattices embedded on Klein bottle, exactly for 4.8.8 lattice, hexagonal lattice, and 3B\u22121 and BT be the inverse and the transpose of a matrix B. And let Im denote the m \u00d7 m identity matrix. SetU be an m \u00d7 m matrices with entriesm \u00d7 m matrices U\u22121 arem \u00d7 m matrices U\u22121K1U, U\u22121(K1T)U and U\u22121K2U are\u03b8t = 2t\u03c0/m for t, j = 1,2,\u2026, m.Introduce some notation firstly. Let L4.8.8 is shown in bj, bj*), for 1 \u2264 j \u2264 n in L4.8.8, we obtain a graph with cylindrical boundary condition, denoted by L4.8.8c. Adding edges , for 1 \u2264 s \u2264 m in L4.8.8c, a 4.8.8 lattice with toroidal boundary condition, denoted by L4.8.8t, can be gotten.The 4.8.8 lattice L4.8.8c:a = 1 \u2212 cos\u2061(2j\u03c0/m), b = 10 \u2212 2cos\u2061(2j\u03c0/m), and c = 14 \u2212 6cos\u2061(2j\u03c0/m).Yan and Zhang  got the L4.8.8t can be expressed as\u03b81 = 2s\u03c0/n and \u03b82 = 2j\u03c0/m. Chang and Shrock [L4.8.8t by an exact closed-form evaluation of the integral given in [Shrock and Wu  showed tas, as*), for 1 \u2264 s \u2264 m in L4.8.8c, 4.8.8 lattice L4.8.8K with Klein bottle boundary condition can be gotten. By a suitable labelling of vertices of L4.8.8K, the adjacency matrix X of it can be written in terms of a linear combination of direct products of smaller ones:By adding edges , the Laplacian characteristic polynomial of L4.8.8K can be expressed asj = 1,\u2026, m/2 \u2212 1, \u03d5j(x) = det\u2061(xIn4 \u2212 Aj\u2032 \u2212 Cj\u2032) and for j = m/2, m. Note that\u03bc1, \u03bc2,\u2026, \u03bcmn\u221214 are the nonzero Laplacian eigenvalues of L4.8.8K.For an even value of dIn4 \u2212 A \u2212 B \u2212 BT \u2212 C also is a Laplacian matrix of a graph, denoted by L4.8.80 (see \u03d5m(0) = det\u2061(\u2212Am\u2032 \u2212 Cm\u2032) = det\u2061(\u2212dIn4 + A + B + BT + C) = 0 and \u03d5m\u2032(0) = (\u22121)n\u2212144nt(L4.8.80). So, we haveL4.8.80 by Lh0 . Let Y be a subset of the row/column index set of P. For convenience, let PY denote the determinant of the matrix obtained from P by deleting all rows and columns whose indices are in Y. For j = 1,\u2026, m/2 \u2212 1, noticing that \u03b8j = \u2212\u03b8m\u2212j and D1(\u03b8j) = D1(\u2212\u03b8j), expanding the determinant In the following, we turn to calculate \u03d5j(0){1}, \u03d5j(0){1,2,3,4}, \u03d5j(0){1,2,3,4,5}, and \u03d5j(0)n}{1,8.Now, we turn to calculate Fn2 = \u03d5j(0){1}, Ln\u221212 = \u03d5j(0){1,2,3,4}, Fn2\u2032 = \u03d5j(0)n}{1,8, and Ln\u221212\u2032 = \u03d5j(0)n}{1,2,3,4,8. Also set \u0393j\u22121 = \u0393j{1,2,3,4}, j = 2,\u2026, n, \u0393n \u2208 {Fn, Ln, Fn\u2032, Ln\u2032}.Let By the Laplace expansion theorem, we obtain several expansions. First, an expansion by rows 1,2, and 3:The recursion relations  and 21)21) give\u0393n=(28\u221212By combining  and 24)24), we oExpanding the determinant along the first row and then expanding the resulting determinants along the first column, we haveThe number of spanning trees of 4.8.8 lattice can be expressed asLh is shown in a1 and b1*, am* and bn2*, and bs and bs* for f or s = 2,\u2026, 2n \u2212 1 in Lh, we obtain a graph with cylindrical boundary condition, denoted by Lhc. Adding edges  for 1 \u2264 s \u2264 m, in Lhc, a hexagonal lattice with toroidal boundary condition, denoted by Lht, can be gotten.The hexagonal lattice Lhc:a = 1 \u2212 cos\u2061(2j\u03c0/m), b = 7 \u2212 cos\u2061(2j\u03c0/m), and c = 3 \u2212 cos\u2061(2j\u03c0/m).Yan and Zhang  got the Lht can be expressed as\u03b81 = 2s\u03c0/n and \u03b82 = 2j\u03c0/m.Shrock and Wu  showed tas, as*) for 1 \u2264 s \u2264 m, in Lhc, a hexagonal lattice LhK with Klein bottle boundary condition can be gotten. For the number of spanning trees of LhK, we have the following result.By adding edges n\u00d7n, in which asj1 = ajs1 = 1 if s is odd and j = s + 1; else, asj1 = 0; A2 = (asj2)n\u00d7n, in which asj2 = ajs2 = 1, if s is even, and j = s + 1; else, asj2 = 0; C1 = (csj)n\u00d7n, in which cn1 = cn1 = 1, otherwise 0.By suitable labelling of vertices of A3(x) = (asj3)n\u00d72n2, in which asj3 = aj+2,s+23 = ei/2x(\u22121), if s is even, and j = s + 1, an\u22121,2n23 = 1 + eix\u2212, and an,2n\u2212123 = 1 + eix; else, asj3 = 0; C3(x) = (csj3)n\u00d72n2, in which cn1,23 = \u22121 \u2212 eix, cn,123 = \u22121 \u2212 eix\u2212, otherwise 0. Expanding the determinant T along the first row and then expanding the resulting determinants along the first column, we haveFn\u221214 = T{1}, Ln\u22121)22 + c, b = 3 \u2212 cos\u2061(2j\u03c0/m) \u2212 c and The recursion relations  and 37)37) give\u0393n=(6\u22122coombining  and 40)(38)\u0393n= for 1 \u2264 s \u2264 m, in L3\u00b7423c, a 33 \u00b7 42 lattice with toroidal boundary condition, denoted by L3\u00b7423t, can be gotten.The 3L3\u00b7423c:a = 7 \u2212 9cos\u2061(2j\u03c0/m) + 2cos\u20612(2j\u03c0/m), b = 17 \u2212 13cos\u2061(2j\u03c0/m) + 2cos\u20612(2j\u03c0/m), and c = 11 \u2212 11cos\u2061(2j\u03c0/m) + 2cos\u20612(2j\u03c0/m).Yan and Zhang  got the L3\u00b7423t can be expressed as\u03b81 = 2s\u03c0/m and \u03b82 = 2j\u03c0/n.Chang and Wang  showed tas, as*) for 1 \u2264 s \u2264 m, in L3\u00b7423c, a 33 \u00b7 42 lattice L3\u00b7423K with Klein bottle boundary condition can be gotten. For the number of spanning trees of L3\u00b7423K, we have the following theorem.By adding edges n\u00d72n2, where bsj = 1, if s is odd, and j = s + 1; else, bsj = \u03b4sj, C = (csj)n\u00d72n2, in which cn1,2 = cn,12 = 1, otherwise 0. Using the same notations as By a suitable lebelling of vertices of Fn2 = \u03d5j(0){1}, Ln\u221212 = \u03d5j(0){1,2}, Fn\u221212\u2032 = \u03d5j(0)n}{1,2, and Ln\u221212\u2032 = \u03d5j(0)n}{1,2,2. Also set \u0393j\u22121 = \u0393j{1,2}, j = 2,\u2026, n, \u0393n \u2208 {Fn, Ln, Fn\u2032, Ln\u2032}.Let By the Laplace expansion theorem, we obtain several expansions. First, an expansion by rows 1 and 2 is as follows:c = (11 \u2212 11cos\u2061(2j\u03c0/m) + 2cos\u2061(2j\u03c0/m))2 \u2212 2 \u2212 2cos\u2061(2j\u03c0/m). By combining (The recursion relations  and 50)50) give\u0393n=(22\u221222ombining  and 53)(51)\u0393n=/|V(Gn)|) = h. If {Gn\u2032} is a sequence of connected subgraph of {Gn}, such that lim\u2061n\u2192\u221e(|{v \u2208 V(Gn\u2032); dGn(v) = dGn\u2032(v)}|/|V(Gn)|) = 1, then lim\u2061n\u2192\u221e(log\u2061t(Gn\u2032)/|V(Gn\u2032)|) = h.Let {3 \u00b7 42 lattice have the same property.By"}
{"text": "On page\u00a01, Importance section, line\u00a05, \u201ctrichloroacetic acid cycle\u201d should be \u201ctricarboxylic acid cycle.\u201d This error was due to a copyediting mistake.Volume 1, no. 2, doi:"}
{"text": "C\u2135-fibration property. We give and prove the relationship between the C\u2135-fibration property and an approximate fibration property. Furthermore, we study the pullback maps for C\u2135-fibrations.We extend the path lifting property in homotopy theory for topological spaces to bitopological semigroups and we show and prove its role in the  S\u2135-fibrations as extension of Hurewicz fibrations. In  into X\u03c4 with the compact-open topology. Recall  be two given S-maps withy \u2208 Y, s \u2208 I, where \u03c1c is a compact-open topology on P(O) which is induced by \u03c1. Define an S-homotopy H :  \u2192 P byy \u2208 Y, s, t \u2208 I. That is, s \u2208 I. Hence C\u2135-fibration.Since c-lifting function \u03bbf is called regular if for every x \u2208 X\u03c42, X\u03c42 \u2208(X \u00d7 O)\u03c42,t \u2208 I.In The following theorem is an analogue of results of Fadell in Hurewicz fibration theory .f12 : \u2192 be a regular C\u2135-fibration and let S-map defined by Mi(\u03b1) = (\u03b1(0), fi\u2218\u03b1) for all \u03b1 \u2208 P(X\u03c4i) where i = 1,2. Then (1)M1\u2218\u03bbf = X \u00d7 id|f2)\u0394 S-homotopy S-maps \u03bbf\u2218M2 and f1[(H(\u03b1)(s))(t)] = f2(\u03b1(t)) for all t, s \u2208 I, \u03b1 \u2208 P(X\u03c42).Let x, \u03b1) \u2208 \u0394(f2),M1\u2218\u03bbf = X \u00d7 id|f2)\u0394 and s \u2208 I, define a path \u03b2s1\u2212 \u2208 P(O\u03c1) by\u03bbf, we can define an S-homotopy H : P \u2192 P bys \u2208 I, \u03b1 \u2208 P(X\u03c42). Then\u03b1 \u2208 P(X\u03c42), t \u2208 I. That is, s, t \u2208 I, \u03b1 \u2208 P(X\u03c42). Hence For the second part, for f : X\u03c4 \u2192 O\u03c1 of compact metrizable spaces X\u03c4 and O\u03c1 is called an approximate fibration if, for every space Y\u03c9 and for given \u03f5 > 0, there exists \u03b4 > 0 such that whenever g : Y\u03c9 \u2192 X\u03c4 and H : Y\u03c9 \u00d7 I \u2192 O\u03c1 are maps with d < \u03b4, then there is homotopy G : Y\u03c9 \u00d7 I \u2192 X\u03c4 such that G0 = g andCoram and Duvall  introducCN\u03c0-fibration property in inducing an approximate fibration property.The following theorem shows the role of the  S-map f : \u2192 with metrizable spaces X\u03c4 and O\u03c1, by d\u03c4 and d\u03c1, we mean the metric functions on X and O, respectively; by X \u00d7 O we mean the product metrizable space of X\u03c4 and O\u03c1 with a metric functionGf we mean the graph of f ) : x \u2208 X}) which is an S-subspace of ((X \u00d7 O)\u03c4\u00d7\u03c1, \u03c0); for a positive integer n > 0, by Gn(f), we mean the (1/n)-neighborhood of Gf in a metrizable space X \u00d7 O which is also S-subspace of ((X \u00d7 O)\u03c4\u00d7\u03c1, \u03c0).For anf : X\u03c4 \u2192 O\u03c1 be a map with compact metrizable spaces X\u03c4 and O\u03c1. Then f is an approximate fibration if and only if, for every positive integer n > 0, there exists a positive integer m \u2265 n such that the S-map fn : (Gn(f), \u03c0)\u2192 has the CN\u03c0-fibration property by the inclusion S-map Imn : (Gm(f), \u03c0)\u2192(Gn(f), \u03c0), where fn = b for all  \u2208 Gn(f).Let n be any positive integer. For \u03f5 = 1/n > 0, let \u03b4 be given in the definition of approximate fibration. Since \u03b4/2 > 0 and f is a continuous function, then let \u03b4\u2032 be chosen such that if x, x\u2032 \u2208 X and d\u03c4 < \u03b4\u2032, then d\u03c1(f(x), f(x\u2032)) < \u03f5/2. Choose a positive integer m\u2a7en, such that 1/m \u2264 \u03b4\u2032, \u03b4/2.Let Y\u03c9, \u03c0) \u2208 N\u03c0 and let g : \u2192(Gm(f), \u03c0) and G :  \u2192 P be two given S-maps with G0 = fn\u2218(Imn\u2218g). Define a map g\u2032 : Y\u03c9 \u2192 X\u03c4 by g\u2032(y) = J1[g(y)] and a homotopy G\u2032 : Y\u03c9 \u00d7 I \u2192 O\u03c1 by G\u2032 = G(y)(t) for all y \u2208 Y and t \u2208 I. We get that g(y) = (g\u2032(y), G\u2032) for all y \u2208 Y. Since g(y) \u2208 Gm(f), then there exists x \u2208 X such thaty \u2208 Y. This impliesy \u2208 Y. Hence, since f is an approximate fibration, there exists a homotopy H\u2032 : Y\u03c9 \u00d7 I \u2192 X\u03c4 such that H0\u2032 = g\u2032 andy \u2208 Y, t \u2208 I. Define an S-homotopy H :  \u2192 P(Gn(f), \u03c0) byy \u2208 Y and fn\u2218Ht = Gt for all t \u2208 I. Hence fn has the CN\u03c0-fibration property by Imn.Now let  < \u03b4\u2032, then d\u03c1(f(x), f(x\u2032)) < \u03f5/2. Choose a positive integer n > 0 such that 1/n \u2264 \u03b4\u2032, \u03f5/2. By hypothesis, there exists a positive integer m \u2265 n such that fn has the CN\u03c0-fibration property by Imn.Conversely, let \u03b4 = 1/m. Let Y\u03c9 be any space and let g : Y\u03c9 \u2192 X\u03c4 and G : Y\u03c9 \u00d7 I \u2192 O\u03c1 be two given maps withy \u2208 Y. Define an S-map g\u2032 : \u2192(Gm(f), \u03c0) by g\u2032(y) = (g(y), G) and an S-homotopy G\u2032 :  \u2192 P by G\u2032(y)(t) = G for all y \u2208 Y and t \u2208 I. Since G0\u2032 = fn\u2218(Imn\u2218g\u2032), then there exists an S-homotopy F :  \u2192 P(Gn(f), \u03c0) such that F0 = Imn\u2218g\u2032 and fn\u2218Ft = Gt\u2032 for all t \u2208 I. By the last part, we can define a homotopy H : Y\u03c9 \u00d7 I \u2192 X\u03c4 byF(y)(t) = , G). Since F(y)(t) \u2208 Gn(f), then there exists x \u2208 X such thaty \u2208 Y, t \u2208 I. Hence f is an approximate fibration.Take"}
{"text": "A periodic mathematical model of cancer treatment by radiotherapy is presented and studied in this paper. Conditions on the coexistence of the healthy and cancer cells are obtained. Furthermore, sufficient conditions on the existence and globally asymptotic stability of the positive periodic solution, the cancer eradication periodic solution, and the cancer win periodic solution are established. Some numerical examples are shownto verify the validity of the results. A discussion is presented for further study. Cancer is a well-known killer of humans worldwide, and its treatments are varied and sporadically successful. There are four main types of cancer treatments, which are surgery, chemotherapy, radiotherapy, and immunotherapy. In this paper, we only consider cancer treatment by radiotherapy.Radiotherapy, as a primary treatment strategy, has been proven to be an effective tool in combating with cancer , 2. RadiIt is an important and effective way to deeply understand the real-world problems by establishing mathematical models and analyzing their dynamical behaviors  be the concentration of healthy cells, and let x2(t) be the concentration of cancer cells; then our model takes the formD(t) is the strategy of the radiotherapy. We assume that D(t) \u2261 \u03b3 > 0 when t \u2208 /[(\u03b11 \u2212 K2\u03b21) + \u025b(\u03b12 \u2212 K1\u03b22)]. Furthermore, if [\u03b12(\u03b11 \u2212 K2\u03b21) + \u03b11(\u03b12 \u2212 K1\u03b22)]/[(\u03b11 \u2212 K2\u03b21) + \u025b(\u03b12 \u2212 K1\u03b22)] < min\u2061{\u03b11/\u025b, \u03b12}, then\u03b12(\u03b11 \u2212 K2\u03b21) + \u03b11(\u03b12 \u2212 K1\u03b22)]/[(\u03b11 \u2212 K2\u03b21) + \u025b(\u03b12 \u2212 K1\u03b22)]\u2a7emin\u2061{\u03b11/\u025b, \u03b12}, \u03b3 is inexistent.(c) Consider  for any positive initial values. However, if we treat it all the time, by condition . Hence, if the treatment is periodic, the solution of system  during the treatment stage, and it will tend to the cancer win state  during the no treatment stage. This will lead to the appearance of a periodic solution.Under conditions of ondition  we know ondition  has a glondition  and Lemmondition  has a unf system  will ten\u03c9-periodic solution is guaranteed by The existence of a positive x2(t) \u2261 0:t \u2208 } \u2192 0 as n \u2192 \u221e. Together with (V(t) \u2192 0 as t \u2192 \u221e. Hence, x1(t) \u2192 x1*(t) and x2(t) \u2192 0 as t \u2192 \u221e. The cancer eradication periodic solution is unique and global stable. This completes the proof.By a simple calculation and from the last inequality of condition , it can her with , we fina\u03b3L < \u03b12\u03c9 holds. Further, if\u03c32 = min\u2061t\u2208{x2*(t)}. Then system , x2(t)) be any solution of system  + \u025b(\u03b12 \u2212 K1\u03b22) = 0.007625 > 0 and \u03bc2 = \u03b12(\u03b11 \u2212 K2\u03b21) + \u03b11(\u03b12 \u2212 K1\u03b22) = 0.03075 > 0. The range of \u03b3 is determined by case (a) in \u03b3 < 0.45. We choose \u03b3 = 0.35 here. It is easy to verity that all the conditions of Theorems L = 15 hours , 0) exists by L = 8 hours) and L = 9 hours). From \u03c31 = min\u2061t\u2208{x1*(t)}\u2a7e0.44, and, \u03c31\u2a7e0.44. It is easy to verify that condition (L = 8 hours) and Figures L = 9 hours).To show the existence and global stability of the cancer eradication periodic solution, we illustrate ondition  is satisIt can be seen from Figures \u03b3 to the cancer eradication periodic solution with \u03c9 = 10 hours and L = 8 hours. Here, we also choose \u025b = 0.3. For the sake of getting a cancer eradication periodic solution, we need the condition \u025b\u03b3L < \u03b11\u03c9. Then we have \u03b3 < \u03b11\u03c9/(\u025bL) = 0.83. However, the true \u03b3(\u03c9 \u2212 L)+(\u2212\u03b7)\u03c9 < 0. From the fact that \u2212\u03b7 = max\u2061{\u03b11 \u2212 \u025b\u03b3 \u2212 \u03b11\u03c31/K1, \u03b12 \u2212 \u03b3}, then we have \u03b3(\u03c9 \u2212 L)+(\u03b12 \u2212 \u03b3)\u03c9 < \u03b3(\u03c9 \u2212 L)+(\u2212\u03b7)\u03c9 < 0, which implies \u03b3 > 0.625. Hence, here we choose \u03b3 = 0.65,0.75 and 0.8 separately to investigate the influence of the variance of the \u03b3 to the cancer eradication periodic solution. The dynamical behavior of the system with \u03b3 = 0.65 can be seen in Figures As we all know, larger dosage radiation can kill cancer cells more effectively, but it also may increase the rate to the healthy cells from the radiation. We now investigate effects of the variance of the parameter \u03b3 = 0.75, it is easy to verify that \u025b\u03b3L < \u03b11\u03c9 and system }\u2a7e0.43; then condition }\u2a7e0.42, then it is easy to verify that condition (\u03b3 = 0.65), Figures \u03b3 = 0.75), and Figures \u03b3 = 0.8) that under the circumstance that the system has a cancer eradication periodic solution and all the other conditions are not changed, the increase of the radiation dosage will quicken the extinction of the cancer cells but will also decrease the concentration of the healthy cells at the same time. One can refer to the three figures; in \u03b3 = 0.65, 0.1 \u2a7d x1, in \u03b3 = 0.75, 0.05 \u2a7d x1 \u2a7d 0.1, and in \u03b3 = 0.8, x1 \u2a7d 0.05.If we take ondition  is satisondition  is satis\u03b11 = 0.2, \u03b12 = 0.5, \u03b21 = 0.48, \u03b22 = 0.05, K1 = 0.65, and K2 = 1. It is easy to verify that condition ) by L = 2 hours. From \u03c32 = min\u2061t\u2208{x2*(t)}\u2a7e0.55. It is easy to verify that condition ) by L = 1 hour. From \u03c32 = min\u2061t\u2208{x2*(t)}\u2a7e0.6. It is easy to verify that condition  = 0.5 \u2a7d K1 and x2(0) = 0.8 \u2a7d K2. Also note that the choice of the values in the theorems are based on the range determined by [Throughout Figures mined by , but the\u03b3x2 to the cancer cells and \u025b\u03b3x1 to the healthy cells. However, during the recovery stage, the model has taken the most basic Lotka-Volterra competition type. We gave the range of radiation dosage \u03b3 under the following three results: the healthy cells and cancer cells are coexist, the cancer eradication periodic solution is globally stable, and the cancer win periodic solution is globally stable. Note that, in the discussion on the global stability of the cancer eradication periodic solution, we showed the relationship between the radiation dosage \u03b3 and the treatment time L; that is, \u025b\u03b3L < \u03b11\u03c9 and \u03b3(\u03c9 \u2212 L) < \u03b7\u03c9, which are also given during the discussion on the cancer win periodic solution (\u03b3L < \u03b12\u03c9). However, during the discussion on the coexistence of the healthy cells and cancer cells, we only give the range of the radiation dosage \u03b3, which shows that it fits all the treatment time L \u2208 . This is a flaw, which is caused by the analysis methods in this paper. We will improve our analysis techniques and make a further study in the future.In this paper, we employed a pair of ordinary differential equations to model the dynamics between the healthy cells and cancer cells for the cancer treatment by radiotherapy. We separated the treatment into two stages: treatment stage and recovery stage (no treatment stage). During the treatment stage, the radiation harvesting amount is The cancer treatment model discussed in this paper is only based on one treatment measure. It may be more effective for the cancer treatment if we add a medication during the recovery stage, which is still an open problem and we will carry out the research in the further work."}
{"text": "Apocrita has a special structure that its first abdominal segment has been incorporated into the thorax as the propodeum. The remaining abdomen, metasoma, is connected to this hybrid region via a narrow propodeal-metasomal articulation forming a \u201cwasp waist\u201d, which serves an important function of providing maneuverability, flexibility and posture for oviposition. However, the origin and transformation of the propodeal-metasomal articulation are still vague. Ephialtitidae, as the basal group of Apocrita from the Early Jurassic to the Early Cretaceous, have shown various types of propodeal-metasomal articulations.Acephialtitia colossa gen. et sp. nov., Proephialtitia acanthi gen. et sp. nov. and P. tenuata sp. nov., collected respectively from the Early Cretaceous Yixian Formation at Liutiaogou and the Middle Jurassic Jiulongshan Formation at Daohugou, both in Inner Mongolia, China. These genera are assigned to the Ephialtitidae based on their complete wing venation, e.g. 2r-rs, 2r-m, 3r-m and 2\u00a0m-cu always present in the forewings and Rs, M and Cu in the hind wings. These new fossil ephialtitids have well-preserved propodeal-metasomal articulations indicating metasoma is broadly attached to propodeum.This study describes and illustrates two new genera with three new species, The broad articulation between the propodeum and metasoma in basal Ephialtitidae, likely passed on from a still more basal family Karatavitidae, suggests three separate pathways of the transformation of the \u201cwasp waist\u201d in three different derived lineages leading from Ephialtitidae to: (i) Kuafuidae and further to the remaining Apocrita, (ii) Stephanidae, and (iii) Evanioidea. In addition, the demise of ephialtitid wasps lagging behind the flourishing of angiosperms suggests that ephialtitid extinction might have been mainly driven by competition with numerous new taxa (eg. the abundant Cretaceous xylophilous Baissinae and Ichneumonoidea) appeared just before or/and soon after the J/K boundary.The online version of this article (doi:10.1186/s12862-015-0317-1) contains supplementary material, which is available to authorized users. The extinct family Ephialtitidae was considered as the second most stem group in the suborder Vespina of Hymenoptera. Together with the most stem group of Karatavitidae, they jointly constituted the superfamily Ephialtitoidea . In receAcephialtitia colossa gen. et sp. nov., Proephialtitia acantha gen. et sp. nov. and Proephialtitia tenuata sp. nov., based on three well-preserved, nearly complete female specimens. These specimens were collected respectively from the Early Cretaceous Yixian Formation at Liutiaogou and the late Middle Jurassic Jiulongshan Formation at Daohugou, both are in Inner Mongolia, China. According to the accurate Ar\u2013Ar and SHRIMP U\u2013Pb dating, the age of the Yixian Formation is considered as the Early Cretaceous  - Cell 3rm much longer than 1mcu\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026..8Cells 2-3rm short, 3rm shorter than 1mcu\u2026\u2026\u2026\u2026\u2026\u2026 Stephanogaster Rasnitsyn, 1975 [distinctly bent\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026.- Forewing 4 mm or shorter. Hind wing when known with r cell closed and 1-M shorter andAsiephialtites Rasnitsyn, 1975 [straight (or nearly so)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026..\u2026\u2026\u2026\u2026Forewing 5 mm or longer. Hind wing when known with r cell closed and 1-M longer and- 3r-m present\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202611(2). 3r-m and 1a-2a lost\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.10\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Liadobracona Zessin, 1981 [in, 1981 Sessiliventer Rasnitsyn, 1975 [- 2 m-cu present, 2-RS almost straight with no sign of 1r-rs \u2026\u2026yn, 1975 2 m-cu lost, 2-RS angular at junction with rudimentary 1r-rs\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Montsecephialtites Rasnitsyn & Mart\u00ednez-Delcl\u00f2s, 2000 [- 2r-m present \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.12(9). 2r-m lost, 1a-2a lost\u2026\u2026.. \u00f2s, 2000 - 2r-m pr- 2r-m subvertical, not sinuate, often distant from 3r-m for much more than its length\u2026\u2026.152r-m oblique, sinuate, distant from 3r-m for about its own length or less\u2026\u2026\u2026\u2026\u2026\u2026..13Tuphephialtites Zhang et al. 2002 [- 2r-rs and 2r-m clearly distant, 1a-2a lost \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026142r-rs and 2r-m practically coincide, 1a-2a present\u2026..al. 2002 - 2r-rs aCratephialtites Rasnitsyn, 1999 [Cretephialtites Rasnitsyn & Ansorge, 2000 [- Cell 3rm receiving 2cu-a \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026ge, 2000 Cell 2rm receiving 2cu-a\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026Ephialtites Meunier, 1903 [- If (rarely) cell 2rm as above, hind wing with 1-RS oblique. Ovipositor never bent upward\u2026...16(12). Cell 2rm long, surpassing level of pterostigmal base basally. Hind wing with 1-RS vertical to both R and 2-RS. Ovipositor evenly bent upward\u2026......er, 1903 - If \u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.Antenna 12-segmented, 1a-2a present, metasoma widest near midlength\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026. Brigittepterus Rasnitsyn, Ansorge & Zessin, 2003 [in, 2003 - 1-RS reclivous, 1r-rs rarely present\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202622(1). 1-RS proclivous, short 1r-rs present\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026Symphyogaster Rasnitsyn, 1975 [Cephenopsis might run here but differs in having complete 1r-rs reaching- 3r-m lost, female metasoma either longer or not subcylindrical\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026.233r-m lost, female metasoma subcylindrical, as long as head and mesosoma combined\u2026\u2026..- 1a-2a lost\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026261a-2a present\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202624\u2026\u2026\u2026\u2026\u2026\u2026..Micrephialtites Rasnitsyn, 1975 [distinctly angled with 2nd in side view\u2026- Cell 3rm usually shorter than 2rm and/or 1mcu. 1st metasomal segment narrowing basal\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.25Cells 2-3rm and 1mcu of about same length. 1st metasomal segment big, cylindrical,Karataus Rasnitsyn, 1977 [Hind femur very thick\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026 yn, 1977 - Cells 2rm and 3rm not simultaneously as long as above. Hind femur ordinary\u2026\u2026\u2026\u2026\u2026Symphytopterus Rasnitsyn, 1975 [\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Cell 2rm along M much longer than cell 1mcu, and 3rm along M about as long as 1mcu.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026Karataviola Rasnitsyn, 1975 [Body more elongate- 2r-m distant from 2r-rs for about its length. Antenna filiform .Trigonalopterus Rasnitsyn, 1975 [Body more robust\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026.(23). 2r-m oblique, almost reaching 2r-rs. Antenna setaceous (narrowing towards apex).Symphyogaster, Pracproapocritus, Acephialtitia and Karataviola form a branch Ephialtitidae, which as the sister group of the remaining big clade (Kuafuidae\u2009+\u2009(Orussoidae\u2009+\u2009((Stephanidae\u2009+\u2009Evanioidae)\u2009+\u2009(Ceraphronmorpha\u2009+\u2009(Proctotrupomorpha\u2009+\u2009(Ichneumonomorpha\u2009+\u2009Vespomorpha))))). This first big clade supported by the following characters: 1-Rs short, proclined or subvertical, reclined , 1r-rs  longer than 2r-rs, or lost traceless , hind wing m-cu lost , hind wing jugal lobe  not delimited , first abdominal segment/propodeum fused with metapleuron . In the second big clade, Kuafuidae as the sister group to the remain groups is supported by 2r-rs entirely lost . In the third big clade, Stephanidae is sister group to Evanioidea, and together they are sister group to the remaining (Ceraphronomorpha\u2009+\u2009(Proctotrupomorpha\u2009+\u2009(Ichneumonomorpha\u2009+\u2009Vespomorpha))), which supported hind wing cell r open or very small .An analyses using NONA resulted in forty-five most parsimonious trees, each consisting of 29 steps, consistency index\u2009=\u20090.82; retention index\u2009=\u20090.83. . The family Ephialtitidae was interpreted as a stem group of Apocrita demonstrating the origin and transformation of the wasp waist characters of the Apocrita [A key character of Ephialtitidae, as well as of more primitive taxa , is that the first abdominal segment (propodeum) is more or less convex (bent) transversally but even  longitudinally Figure\u00a0A. The fiThe second pathway of transformation is the pattern Figure\u00a0A basicalSome other Ephialtitidae Figure\u00a0D demonstAfter studying the wasp waists of described fossil species of Ephialtitidae, we identified three typical but different propodeal-metasomal articulations: the shape of the first metasomal segment varies from narrow and greatly elongate with distal end broader than the proximal end , to sli& Zhang, ), then, Apocrita ,5,28. OuApocrita ,30. ThisThe above hypothesis suggests three separate pathways of the wasp waist transformation. Concerning our considerations about various ovipositing postures of Apocrita, this poses limitation only on the most sophisticated \u201c=\u201d-posture for both ephialtitid-stephanid and evanioid lineages, as well as on the \u201cL\u201d-posture for only the most basal versions of the ephialtitid mechanism Figure\u00a0A. The ovBased on the information of Table\u00a0Based on the information and data summarized in Table\u00a0Archaefructus liaoningensis, Archaefructus sinensis and Archaefructus eoflora from the Early Cretaceous Yixian Formation, are now widely accepted as important fossil angiosperm plants [Sinocarpus decussates, from the same Formation. These fossil angiosperms provided important information about early angiosperms which co-existed with many gymnosperm and other plants in the same ecosystems [The above considerations suggest addressing to profound changes in vegetation occurred in the Cretaceous while seeking for a possible cause of demise of the Ephialtitidae. It is well known that the non-marine Earth transformed from the gymnosperm- to angiosperm-forested that time. Early angiosperms, vascular flowering plants with seeds enclosed in an ovary, have many primitive features which are considerably different from extant angiosperms . Primitim plants -37 Leng m plants  describeosystems ,40. Duriosystems , and theosystems ,32 . At the same time, Ephialtitidae appeared as a rare group since the very beginning of Cretaceous and lost the fossil record after the Aptian Table\u00a0. This paA thorough review of the various types of the propodeal-metasomal articulation of Apocrita suggests that the wide articulation between the propodeum and metasoma in basal Ephialtitidae was passed on from a still more basal family of Karatavitidae and provided three separate pathways of transformation of the wasp waist. In addition, the demise of ephialtitid wasps lagging behind the flourishing of angiosperms suggests that ephialtitid extinction driven by competition with numerous new taxa (e.g. the abundant Cretaceous xylophilous Baissinae and Ichneumonoidea) appeared just before or/and soon after the J/K boundary, rather than the transformed from the gymnosperm to angiosperm-forested led to shortage of food sources for hosts of the larvae of ephialtitids.The authors declare that the study makes no uses of human, clinical tools and procedures, vertebrate and regulated invertebrate animal subjects and/or tissue, and plants."}
{"text": "R0 algebras and R0-algebras to the noncommutative forms is given. We show that pseudo-weak-R0 algebras are categorically isomorphic to pseudo-IMTL algebras and that pseudo-R0 algebras are categorically isomorphic to pseudo-NM algebras. Some properties, the noncommutative forms of the properties in weak-R0 algebras and R0-algebras, are investigated. The simplified axiom systems of pseudo-weak-R0 algebras and pseudo-R0 algebras are obtained.A positive answer to the open problem of Iorgulescu on extending weak- It is well known that certain information processing, especially inferences based on certain information, is based on classical two-valued logic. Due to strict and complete logical foundation , making inferences about certain information can be done with high confidence levels. Thus, it is natural and necessary to attempt to establish some rational logic system as the logical foundation for uncertain information processing. It is evident that this kind of logic cannot be two-valued logic itself but might form a certain extension of two-valued logic. Various kinds of nonclassical logic systems have therefore been extensively researched in order to construct natural and efficient inference systems to deal with uncertainty.In recent years, motivated by both theory and application, the study of t-norm-based logic systems and the corresponding pseudo-logic systems has become of greater focus in the field of logic. Here, t-norm-based logical investigations preceded the corresponding algebraic investigations, and in the case of pseudo-logic systems, algebraic development preceded the corresponding logical development.A noncommutative generalization of reasoning can be found, for example, in psychological processes. In clinical medicine on behalf of transplantation of human organs, an experiment was performed in which the same two questions have been posed to two groups of interviewed people as follows. (1) Do you agree to donate your organs for medical transplantation after your death? (2) Do you agree to accept organs of a donor if you need them? When the order of questions was changed in the second group, the number of positive answers here was much higher than in the first group.The following reviews some situation concerning some important logic algebras and the corresponding pseudo-logic algebras. BCK- and BCI-algebras were introduced by Imai and Iseki  and haveR0-algebras were introduced by Wang R0-algebProblem. Recall that the IMTL-algebras, introduced in 2001 by Esteva and Godo, are categorically isomorphic to weak-R0 algebras, introduced in 1997 by Wang, and that NM-algebras are categorically isomorphic to R0-algebras, introduced also in 1997 by Wang. Extend weak-R0 algebras and R0-algebras to the noncommutative case.R0 algebras and R0-algebras to the noncommutative forms, called pseudo-weak-R0 algebras and pseudo-R0 algebras. We show that pseudo-weak-R0 algebras are categorically isomorphic to pseudo-IMTL-algebras and that pseudo-R0 algebras are categorically isomorphic to pseudo-NM algebras. Some properties, the noncommutative forms of the properties in weak-R0 algebras and R0-algebras, are investigated. Furthermore, we discuss the simplified axiom systems of pseudo-weak-R0 algebras and pseudo-R0 algebras.In this paper, we extend weak-We recall some definitions and results which will be used in the sequel.A, \u2228, \u2227, \u2299, \u2192, 0,1) of type  such that for all x, y, z \u2208 A(B1)A, \u2228, \u2227, 0,1) is a bounded lattice,((B2)A, \u2299, 1) is a monoid,((B3)x\u2299y \u2264 z if and only if x \u2264 y \u2192 z,(B4)x \u2192 y)\u2228(y \u2192 x) = 1. is a structure ((B5)x\u2212\u2212 = x,where x\u2212 = x \u2192 0.An IMTL- (involutive MTL-) algebra is an MTL-algebra satisfying the following additional condition:(B6)x\u2299y)\u2212\u2228((x\u2227y)\u2192(x\u2299y)) = 1.(A WNM- (weak nilpotent minimum-) algebra is an MTL-algebra satisfying the following additional condition:An NM- (nilpotent minimum-) algebra is an IMTL-algebra satisfying condition (B6).A, \u2228, \u2227, \u2299, \u2192, \u21dd, 0,1) of type  such that for all x, y, z \u2208 A(pB1)A, \u2228, \u2227, 0,1) is a bounded lattice,((pB2)A, \u2299, 1) is a monoid,((pB3)x\u2299y \u2264 z if and only if x \u2264 y \u2192 z if and only if y \u2264 x\u21ddz,(pB4)x \u2192 y)\u2228(y \u2192 x) = (x\u21ddy)\u2228(y\u21ddx) = 1. is a structure ((pB5)x~\u2212 = x\u2212~ = x,where x\u2212 = x \u2192 0 and x~ = x\u21dd0,A pseudo-IMTL (pseudo-involutive MTL) algebra is a pseudo-MTL algebra satisfying the following additional condition:(pB6)x\u2299y)\u2212\u2228((x\u2227y)\u2192(x\u2299y)) = (x\u2299y)~\u2228((x\u2227y)\u21dd(x\u2299y)) = 1,(where x\u2212 = x \u2192 0 and x~ = x\u21dd0.A pseudo-WNM (pseudo-weak nilpotent minimum) algebra is a pseudo-MTL algebra satisfying the following additional condition:A pseudo-NM (pseudo-nilpotent minimum) algebra is a pseudo-IMTL algebra satisfying condition (pB6).M be a -type algebra, where \u00ac is a unary operation and \u2227, \u2228, and \u2192 are binary operations. If there is a partial ordering \u2264 on M, such that  is a bounded distributive lattice, \u2227, \u2228 are infimum and supremum operations with respect to \u2264, \u00ac is an order-reversing involution with respect to \u2264, and the following conditions hold for any a, b, c \u2208 M:(R1)a \u2192 \u00acb = b \u2192 a,\u00ac(R2)a = a, a \u2192 a = 1,1 \u2192 (R3)b \u2192 c \u2264 (a \u2192 b)\u2192(a \u2192 c),(R4)a \u2192 (b \u2192 c) = b \u2192 (a \u2192 c),(R5)a \u2192 (b\u2228c) = (a \u2192 b)\u2228(a \u2192 c), a \u2192 (b\u2227c) = (a \u2192 b)\u2227(a \u2192 c),where 1 is the largest element of M, then one calls M a weak-R0 algebra.Let R0 algebra M is a weak-R0 algebra satisfying the following additional condition:(R6)a \u2192 b)\u2228((a \u2192 b)\u2192(\u00aca\u2228b)) = 1. such that  is a bounded distributive lattice, \u2212 and ~ are order-reversing pseudo-involution , and the following axioms hold for any x, y, z \u2208 A:\u2009x \u2192 y = y\u2212\u21ddx\u2212,\u2009\u2009x\u21ddy = y~ \u2192 x~,(pR1) \u2009x = 1\u21ddx = x; x \u2192 x = x\u21ddx = 1,(pR2*) 1 \u2192 \u2009x \u2192 y \u2264 (z \u2192 x)\u2192(z \u2192 y), x\u21ddy \u2264 (z\u21ddx)\u21dd(z\u21ddy),(pR3) \u2009x \u2192 (y\u21ddz) = y\u21dd(x \u2192 z),(pR4) \u2009x \u2192 (y\u2228z) = (x \u2192 y)\u2228(x \u2192 z), x\u21dd(y\u2228z) = (x\u21ddy)\u2228(x\u21ddz);\u2009\u2009x \u2192 (y\u2227z) = (x \u2192 y)\u2227(x \u2192 z), x\u21dd(y\u2227z) = (x\u21ddy)\u2227(x\u21ddz).(pR5*) A pseudo-weak-R0 algebra A is a pseudo-weak-R0 algebra satisfying the following additional axiom:\u2009x \u2192 y)\u2228((x \u2192 y)\u21dd(x\u2212\u2228y)) = (x\u21ddy)\u2228((x\u21ddy)\u2192(x~\u2228y)) = 1.(pR6)  Comparing \u2212 and ~ in (ii) The operations A, \u2227, \u2228, 0,1),the distributivity of bounded lattice  = (x \u2192 y)\u2227(x \u2192 z), x\u21dd(y\u2227z) = (x\u21ddy)\u2227(x\u21ddz).Hence, we have the following simplified definition. In what follows, we will use the following definition.(iii) As we will see in R0 algebra is a structure  satisfying the following:(pL1)A, \u2227, \u2228, 0,1) is a bounded lattice,((pL2)x \u2264 y, then y\u2212 \u2264 x\u2212 and y~ \u2264 x~,if (pL3)x~\u2212 = x\u2212~ = x,(pR1)x \u2192 y = y\u2212\u21ddx\u2212, x\u21ddy = y~ \u2192 x~,(pR2)x = 1\u21ddx = x,1 \u2192 (pR3)x \u2192 y \u2264 (z \u2192 x)\u2192(z \u2192 y), x\u21ddy \u2264 (z\u21ddx)\u21dd(z\u21ddy),(pR4)x \u2192 (y\u21ddz) = y\u21dd(x \u2192 z),(pR5)x \u2192 (y\u2228z) = (x \u2192 y)\u2228(x \u2192 z), x\u21dd(y\u2228z) = (x\u21ddy)\u2228(x\u21ddz).A pseudo-weak-R0 algebra A is a pseudo-weak-R0 algebra satisfying the following additional axiom:(pR6)x \u2192 y)\u2228((x \u2192 y)\u21dd(x\u2212\u2228y)) = (x\u21ddy)\u2228((x\u21ddy)\u2192(x~\u2228y)) = 1. is a bounded lattice satisfying x\u2212~ = x~\u2212 = x for any x \u2208 A. Define operations \u2192 and \u21dd as pseudo-Lukasiewicz implication on A:A, \u2227, \u2228, \u2192, \u21dd, \u2212, ~, 0,1) is a pseudo-weak-R0 algebra.Let R0 algebra, the following properties hold:~ = 0\u2212 = 1, 1~ = 1\u2212 = 0,0x\u2212 = x \u2192 0, x~ = x\u21dd0,x\u21ddx = x \u2192 x = 1,x \u2264 y if and only if x\u21ddy = 1 if and only if x \u2192 y = 1,i\u2208Ixi)~ = \u22c1i\u2208Ixi~, (\u22c0i\u2208Ixi)\u2212 = \u22c1i\u2208Ixi\u2212, whenever the arbitrary meets and unions exist,(\u22c0i\u2208Ixi)~ = \u22c0i\u2208Ixi~, (\u22c1i\u2208Ixi)\u2212 = \u22c0i\u2208Ixi\u2212, whenever the arbitrary meets and unions exist,\u21ddz = (x\u21ddz)\u2228(y\u21ddz), (x\u2227y) \u2192 z = (x \u2192 z)\u2228(y \u2192 z),(x\u21dd(y\u2227z) = (x\u21ddy)\u2227(x\u21ddz), x \u2192 (y\u2227z) = (x \u2192 y)\u2227(x \u2192 z),x\u2228y)\u21ddz = (x\u21ddz)\u2227(y\u21ddz), (x\u2228y) \u2192 z = (x \u2192 z)\u2227(y \u2192 z),\u2192(x\u21ddz), x \u2192 y \u2264 (y \u2192 z)\u21dd(x \u2192 z),A, \u2227, \u2228, 0,1) is a bounded distributive lattice. and (L2), 1 = 1~\u2212 \u2264 0\u2212 and 1 = 1\u2212~ \u2264 0~, and hence 0~ = 0\u2212 = 1. 1~ = 1\u2212 = 0 follows similarly.(1) Since 0 \u2264 1x\u2212 = 1\u21ddx\u2212 = x\u2212~ \u2192 1~ = x \u2192 0, x~ = 1 \u2192 x~ = x~\u2212\u21dd1\u2212 = x\u21dd0.(2) By (1), (pR1), and (pR2), ~ \u2192 0~ = 1 \u2192 1 = 1. By (pR3) and (L3), 0\u21dd0 \u2264 (x\u2212\u21dd0)\u21dd(x\u2212\u21dd0); that is, 1 \u2264 x\u21ddx, and so x\u21ddx = 1. The second equality follows similarly.(3) By (pR1) and (pR2), 0\u21dd0 = 0x \u2192 y = 1, by (pR1), (pR3), and (pR2), x = 1 \u2192 x = x\u2212\u21dd0\u2264(y\u2212\u21ddx\u2212)\u21dd(y\u2212\u21dd0) = (x \u2192 y)\u21dd(1 \u2192 y) = 1\u21ddy = y. If x \u2264 y, by (pR5) and (3), x \u2192 y = x \u2192 (x\u2228y) = (x \u2192 x)\u2228(x \u2192 y) = 1\u2228(x \u2192 y) = 1. Similarly, we have x \u2264 y if and only if x\u21ddy = 1.(4) If i\u2208Ixi \u2264 xi for each i \u2208 I, by (L2), xi~ \u2264 (\u22c0i\u2208Ixi)~ for each i \u2208 I, and so \u22c1i\u2208Ixi~ \u2264 (\u22c0i\u2208Ixi)~. Since xi~ \u2264 \u22c1i\u2208Ixi~ for each i \u2208 I, by (L2) and (L3), (\u22c1i\u2208Ixi~)\u2212 \u2264 xi for each i \u2208 I, and so (\u22c1i\u2208Ixi~)\u2212 \u2264 \u22c0i\u2208Ixi, (\u22c0i\u2208Ixi)~ \u2264 \u22c1i\u2208Ixi~. The second equality follows similarly.(5) Since \u22c0(6) Similarly.x \u2264 y, by (pR5), z\u21ddy = z\u21dd(x\u2228y) = (z\u21ddx)\u2228(z\u21ddy), and so z\u21ddx \u2264 z\u21ddy. Similarly, z \u2192 x \u2264 z \u2192 y.(7) If x \u2264 y, then y\u2212 \u2264 x\u2212. By (pR1), y \u2192 z = z\u2212\u21ddy\u2212 \u2264 z\u2212\u21ddx\u2212 = x \u2192 z. Similarly, y\u21ddz \u2264 x\u21ddz.(8) If x\u2227y) \u2192 z = z\u2212\u21dd(x\u2227y)\u2212 = z\u2212\u21dd(x\u2212\u2228y\u2212) = (z\u2212\u21ddx\u2212)\u2228(z\u2212\u21ddy\u2212) = (x \u2192 z)\u2228(y \u2192 z). Similarly, (x\u2227y)\u21ddz = (x\u21ddz)\u2228(y\u21ddz).(9) By (pR1), (5), and (pR5),  if and only if u \u2192 (x \u2192 (y\u2227z)) = 1 if and only if u \u2192 ((y\u2227z)\u2212\u21ddx\u2212) = 1 if and only if (y\u2227z)\u2212\u21dd(u \u2192 x\u2212) = 1 if and only if (u \u2192 x\u2212)~ \u2192 (y\u2227z) = 1 if and only if (u \u2192 x\u2212)~ \u2264 y\u2227z if and only if (u \u2192 x\u2212)~ \u2264 y and (u \u2192 x\u2212)~ \u2264 z if and only if (u \u2192 x\u2212)~ \u2192 y = 1 and (u \u2192 x\u2212)~ \u2192 z = 1 if and only if u \u2264 x \u2192 y and u \u2264 x \u2192 z if and only if u \u2264 (x \u2192 y)\u2227(x \u2192 z). The second equality follows similarly.(10) By (4), (pR1), and (pR4), for any x\u2228y) \u2192 z = z\u2212\u21dd(x\u2228y)\u2212 = z\u2212\u21dd(x\u2212\u2227y\u2212) = (z\u2212\u21ddx\u2212)\u2227(z\u2212\u21ddy\u2212) = (x \u2192 z)\u2227(y \u2192 z). Similarly, (x\u2228y)\u21ddz = (x\u21ddz)\u2227(y\u21ddz).(11) By (10),  Since x \u2192 y)\u2192((y \u2192 z)\u21dd(x \u2192 z)) = (y \u2192 z)\u21dd((x \u2192 y)\u2192(x \u2192 z)) = 1. Hence, x \u2192 y \u2264 (y \u2192 z)\u21dd(x \u2192 z). Similarly, we have the second inequality.(13) By (pR4) and (pR3), (x\u2227y)\u2228(x\u2227z) \u2264 x\u2227(y\u2228z); by (pR5), (10), and (4),x\u2227(y\u2228z))\u2264(x\u2227y)\u2228(x\u2227z).(14) Obviously, x\u21ddy \u2264 x\u2228z\u21ddy\u2228z, x \u2192 y \u2264 x\u2228z \u2192 y\u2228z,(16)x\u21ddy \u2264 x\u2227z\u21ddy\u2227z, x \u2192 y \u2264 x\u2227z \u2192 y\u2227z,(17)x\u21ddy)\u2264(x\u21ddz)\u2228(z\u21ddy), (x \u2192 y)\u2264(x \u2192 z)\u2228(z \u2192 y),((18)x\u21ddy)\u2228(y\u21ddx) = (x \u2192 y)\u2228(y \u2192 x) = 1,((19)x \u2264 (x \u2192 y)\u21ddy, x \u2264 (x\u21ddy) \u2192 y,(20)x \u2192 y = ((x \u2192 y)\u21ddy) \u2192 y, x\u21ddy = ((x\u21ddy) \u2192 y)\u21ddy,(21)x \u2192 (y \u2192 x) = x\u21dd(y\u21ddx) = x\u21dd(y \u2192 x) = x \u2192 (y\u21ddx) = 1,(22)x\u2212 \u2192 (x \u2192 y) = x~\u21dd(x\u21ddy) = x\u2212\u21dd(x \u2192 y) = x~ \u2192 (x\u21ddy) = 1,(23)y \u2264 (x\u21ddy)\u2227(x \u2192 y),(24)x\u2228y = ((x\u21ddy) \u2192 y)\u2227((y\u21ddx) \u2192 x) = ((x \u2192 y)\u21ddy)\u2227((y \u2192 x)\u21ddx),(25)x\u2228y) \u2192 x = y \u2192 x, (x\u2228y)\u21ddx = y\u21ddx,((26)x \u2192 (x\u2227y) = x \u2192 y, x\u21dd(x\u2227y) = x\u21ddy,(27)x \u2264 y\u2212 if and only if y \u2264 x~,(28)x \u2192 y~ = y\u21ddx\u2212, x\u21ddy\u2212 = y \u2192 x~,(29)x \u2192 y\u2212)~ = (y\u21ddx~)\u2212.(In a pseudo-weak-z\u21dd(y\u2228z) = 1, x\u21ddy \u2264 x\u21dd(y\u2228z) = (x\u21dd(y\u2228z))\u2227(z\u21dd(y\u2228z)) = (x\u2228z)\u21dd(y\u2228z). The second inequality follows similarly.(15) Since x\u2227z)\u21ddz = 1, x\u21ddy \u2264 (x\u2227z)\u21ddy = ((x\u2227z)\u21ddy)\u2227((x\u2227z)\u21ddz) = (x\u2227z)\u21dd(y\u2227z). The second inequality follows similarly.(16) Since (x\u21ddy \u2264 (x\u2228z)\u21dd(y\u2228z) = ((x\u2228z)\u21ddy)\u2228((x\u2228z)\u21ddz) = ((x\u21ddy)\u2227(z\u21ddy))\u2228\u2009((x\u21ddz)\u2227(z\u21ddz))\u2264(x\u21ddz)\u2228(z\u21ddy). The second inequality follows similarly.(17) Consider x\u2228y)\u21dd(x\u2228y) = ((x\u2228y)\u21ddx)\u2228((x\u2228y)\u21ddy) = ((x\u21ddx)\u2227(y\u21ddx))\u2228((x\u21ddy)\u2227(y\u21ddy)) = (x\u21ddy)\u2228(y\u21ddx). The second equality follows similarly.(18) Consider 1 = (x \u2192 ((x \u2192 y)\u21ddy) = (x \u2192 y)\u21dd(x \u2192 y) = 1 and x\u21dd((x\u21ddy) \u2192 y) = (x\u21ddy)\u2192(x\u21ddy) = 1.(19) Consider x \u2192 y)\u21ddy) \u2192 y)\u21dd(x \u2192 y) \u2265 x \u2192 ((x \u2192 y)\u21ddy) = (x \u2192 y)\u21dd(x \u2192 y) = 1 and (x \u2192 y)\u21dd(((x \u2192 y)\u21ddy) \u2192 y) = ((x \u2192 y)\u21ddy)\u2192((x \u2192 y)\u21ddy) = 1, then ((x \u2192 y)\u21ddy) \u2192 y = (x \u2192 y). Similarly, we have ((x\u21ddy) \u2192 y)\u21ddy = x\u21ddy.(20) Since (((x \u2192 (y \u2192 x) = 1 if and only if x\u21dd(y \u2192 x) = 1 if and only if y \u2192 (x\u21ddx) = 1. The rest follow similarly.(21) x\u2212 \u2192 (x \u2192 y) \u2265 0 \u2192 y = 1. The rest follow similarly.(22) Consider y = 1\u21ddy \u2264 x\u21ddy and y = 1 \u2192 y \u2264 x \u2192 y.(23) Consider x\u21dd((x\u21ddy) \u2192 y) = (x\u21ddy)\u2192(x\u21ddy) = 1 and y\u21dd((x\u21ddy) \u2192 y) = (x\u21ddy)\u2192(y\u21ddy) = 1, then x, y \u2264 (x\u21ddy) \u2192 y, and so x\u2228y \u2264 (x\u21ddy) \u2192 y. Similarly, x\u2228y \u2264 (y\u21ddx) \u2192 x. Hence, x\u2228y \u2264 ((x\u21ddy) \u2192 y)\u2227((y\u21ddx) \u2192 x). On the other hand, by (9), (pR5), (20), and (18),(24) Since x\u2228y) \u2192 x = (x \u2192 x)\u2227(y \u2192 x) = y \u2192 x, (x\u2228y)\u21ddx = (x\u21ddx)\u2227(y\u21ddx) = y\u21ddx.(25) Consider (x \u2192 (x\u2227y) = (x \u2192 x)\u2227(x \u2192 y) = x \u2192 y, x\u21dd(x\u2227y) = (x\u21ddx)\u2227(x\u21ddy) = x\u21ddy.(26) Consider \u2009x \u2264 y\u2212 if and only if x\u21ddy\u2212 if and only if y \u2192 x~ = 1 if and only if y \u2264 x~.(27) x \u2192 y~ = y~\u2212\u21ddx\u2212 = y\u21ddx\u2212, x\u21ddy\u2212 = y\u2212~ \u2192 x~ = y \u2192 x~.(28) Consideru, (x \u2192 y\u2212)~ \u2264 u if and only if u\u2212 \u2264 x \u2192 y\u2212 if and only if u\u2212\u21dd(x \u2192 y\u2212) = 1 if and only if x \u2192 (u\u2212\u21ddy\u2212) = 1 if and only if x \u2192 (y \u2192 u) = 1 if and only if x\u21dd(y \u2192 u) = 1 if and only if y \u2192 (x\u21ddu) = 1 if and only if y\u21dd(x\u21ddu) = 1 if and only if y\u21dd(u~ \u2192 x~) = 1 if and only if u~ \u2192 (y\u21ddx~) = 1 if and only if u~ \u2264 y\u21ddx~ if and only if (y\u21ddx~)\u2212 \u2264 u.(29) For each R0 algebra  A, we define a binary operation \u2299 as follows, for any x, y \u2208 A:In a pseudo-weak-x\u2299y = (x\u2192y\u2212)~ = (y\u21ddx~)\u2212.(30) R0 algebra, the following properties hold:(31)x \u2192 y = (x\u2299y~)\u2212, x\u21ddy = (y\u2212\u2299x)~,(32)x\u2299y)\u2299z = x\u2299(y\u2299z),((33)x = x\u22991 = x,1\u2299(34)x\u2299y \u2264 z if and only if x \u2264 y \u2192 z if and only if y \u2264 x\u21ddz,(35)x\u2299(x\u21ddy) \u2264 y \u2264 x\u21dd(x\u2299y), (x \u2192 y)\u2299x \u2264 y \u2264 x \u2192 (y\u2299x),(36)x\u2299(x\u21ddy) \u2264 x \u2264 y\u21dd(y\u2299x), (x \u2192 y)\u2299x \u2264 x \u2264 y \u2192 (x\u2299y),(37) if x \u2264 y, then x\u2299z \u2264 y\u2299z and z\u2299x \u2264 z\u2299y,(38)x\u2299(x\u21ddy) \u2264 x\u2227y, (x \u2192 y)\u2299x \u2264 x\u2227y,(39)x\u22990 = 0\u2299x = 0,(40)x\u2299(\u22c1i\u2208Ixi) = \u22c1i\u2208I(x\u2299xi), (\u22c1i\u2208Ixi)\u2299x = \u22c1i\u2208I(xi\u2299x), whenever the arbitrary unions exist,(41)x\u2299y) \u2192 z = x \u2192 (y \u2192 z), (y\u2299x)\u21ddz = x\u21dd(y\u21ddz),((42)y\u21dd(\u22c0i\u2208Ixi) = \u22c0i\u2208I(y\u21ddxi), y \u2192 (\u22c0i\u2208Ixi) = \u22c0i\u2208I(y \u2192 xi), whenever the arbitrary meets exist,(43)i\u2208Ixi)\u21ddy = \u22c0i\u2208I(xi\u21ddy), (\u22c1i\u2208Ixi) \u2192 y = \u22c0i\u2208I(xi \u2192 y), whenever the arbitrary unions and meets exist,(\u22c1(44)x\u2299x~ = x\u2212\u2299x = 0,(45)x\u2299y \u2264 x\u2227y \u2264 x, y,(46)x\u2228(y\u2299z)\u2265(x\u2228y)\u2299(x\u2228z),(47)x \u2192 y \u2264 (x\u2299z)\u2192(y\u2299z), x\u21ddy \u2264 (z\u2299x)\u21dd(z\u2299y),(48)x\u2299(y \u2192 z) \u2264 y \u2192 (x\u2299z), (y\u21ddz)\u2299x \u2264 y\u21dd(z\u2299x).In a pseudo-weak-(31) Straightforward.x\u2299y)\u2299z = (x \u2192 y\u2212)~\u2299z = ((x \u2192 y\u2212)~ \u2192 z\u2212)~ = (z\u2212\u2212\u21dd(x \u2192 y\u2212))~ = (x \u2192 (z\u2212\u2212\u21ddy\u2212))~ = (x \u2192 (y \u2192 z\u2212))~ and x\u2299(y\u2299z) = x\u2299(y \u2192 z\u2212)~ = (x \u2192 (y \u2192 z\u2212))~.(32) By (pR1) and (pR4), (x = (1 \u2192 x\u2212)~ = x\u2212~ = x, x\u22991 = (x \u2192 1\u2212)~ = (x \u2192 0)~ = x\u2212~ = x.(33) Consider1\u2299x\u2299y \u2264 z\u21d4(x \u2192 y\u2212)~ \u2264 z\u21d4z\u2212 \u2264 x \u2192 y\u2212\u21d4z\u2212\u21dd(x \u2192 y\u2212) = 1\u21d4x \u2192 (z\u2212\u21ddy\u2212) = 1\u21d4x \u2192 (y \u2192 z) = 1\u21d4x \u2264 y \u2192 z. And(34) Considerx\u2299y \u2264 z\u21d4(y\u21ddx~)\u2212 \u2264 z\u21d4z~ \u2264 y\u21ddx~\u21d4z~ \u2192 (y\u21ddx~) = 1\u21d4y\u21dd(z~ \u2192 x~) = 1\u21d4y\u21dd(x\u21ddz) = 1\u21d4y \u2264 x\u21ddz.Consider x\u21ddy \u2264 x\u21ddy, by (34), x\u2299(x\u21ddy) \u2264 y. Since x\u2299y \u2264 x\u2299y, by (34), y \u2264 x\u21dd(x\u2299y). The second inequality follows similarly.(35) Since x \u2192 y \u2264 1 = x \u2192 x, we have (x \u2192 y)\u2299x \u2264 x. Since x\u2299y \u2264 x\u2299y, we have x \u2264 y \u2192 (x\u2299y). The second inequality follows similarly.(36) Since x \u2264 y\u21d2y \u2192 z\u2212 \u2264 x \u2192 z\u2212\u21d2(x \u2192 z\u2212)~ \u2264 (y \u2192 z\u2212)~; that is, x\u2299z \u2264 y\u2299z. Similarly, we have that x \u2264 y implies that z\u2299x \u2264 z\u2299y.(37) x\u2299(x\u21ddy) \u2264 x. By (35), x\u2299(x\u21ddy) \u2264 y. So, x\u2299(x\u21ddy) \u2264 x\u2227y. The second inequality follows similarly.(38) By (36), x\u22990 = (x \u2192 0\u2212)~ = (x \u2192 1)~ = 1~ = 0,0\u2299x = (0 \u2192 x\u2212)~ = 1~ = 0.(39) Consider u, x\u2299(\u22c1i\u2208Ixi) \u2264 u\u21d4\u22c1i\u2208Ixi \u2264 x\u21ddu\u21d4xi \u2264 x\u21ddu for each i \u2208 I\u21d4x\u2299xi \u2264 u for each i \u2208 I\u21d4\u22c1i\u2208I(x\u2299xi) \u2264 u for each i \u2208 I. The second equality follows similarly.(40) For any x\u2299y) \u2192 z = (x \u2192 y\u2212)~ \u2192 z = z\u2212\u21dd(x \u2192 y\u2212) = x \u2192 (z\u2212\u21ddy\u2212) = x \u2192 (y \u2192 z). The second equality follows similarly.(41) Consider \u21d4y\u2299z \u2264 \u22c0i\u2208Ixi\u21d4y\u2299z \u2264 xi for each i \u2208 I\u21d4z \u2264 y\u21ddxi for each i \u2208 I\u21d4z \u2264 \u22c0i\u2208I(y\u21ddxi). The second equality follows similarly.(42) For any (43) Similarly.x\u2299x~ = (x \u2192 x~\u2212)~ = 1~ = 0, x\u2212\u2299x = (x\u2212 \u2192 x\u2212)~ = 0.(44) Consider x\u2299y\u21ddx = y\u21dd(x\u21ddx) = 1 and x\u2299y \u2192 y = x \u2192 (y \u2192 y) = 1. Hence, x\u2299y \u2264 x\u2227y.(45) By (41), x\u2228y)\u2299(x\u2228z) = (x\u2299x)\u2228(x\u2299z)\u2228(y\u2299x)\u2228(y\u2299z) \u2264 x\u2228x\u2228x\u2228(y\u2299z) = x\u2228(y\u2299z).(46) By (40), (x\u2299z)\u2192(y\u2299z) = (x \u2192 z\u2212)~ \u2192 (y \u2192 z\u2212)~ = (y \u2192 z\u2212)\u21dd(x \u2192 z\u2212) \u2265 x \u2192 y. Similarly, we have the other one.(47) By (13), (y\u2299(y\u21ddz) \u2264 z. By (37), (y\u2299(y\u21ddz))\u2299x \u2264 z\u2299x; that is, y\u2299((y\u21ddz)\u2299x) \u2264 z\u2299x, and so (y\u21ddz)\u2299x \u2264 y\u21dd(z\u2299x). The second inequality follows similarly.(48) By (35), R0 algebra, the following property holds:In a pseudo-x\u2299y)\u2212\u2228((x\u2227y)\u2192(x\u2299y)) = (y\u2299x)~\u2228((x\u2227y)\u21dd(y\u2299x)) = 1.(49)\u2009\u2009(By (pR1), (5), and (pR6),R0 algebra. If A is a chain, then the operations \u2192 and \u21dd can be written by the following:Let A be a pseudo-x \u2264 y, by (4), x \u2192 y = x\u21ddy = 1. Now, we show that if x\u2270y, then x \u2192 y = x\u2212\u2228y and x\u21ddy = x~\u2228y. By (12), we have x\u2212\u2228y \u2264 x \u2192 y and x~\u2228y \u2264 x\u21ddy. On the other hand, by (pR6), we havex\u2270y, x \u2192 y \u2260 1 and x\u21ddy \u2260 1. By A being a chain, we have (x \u2192 y)\u21dd(x\u2212\u2228y) = 1 and (x\u21ddy)\u2192(x~\u2228y) = 1, and so x \u2192 y \u2264 x\u2212\u2228y and x\u21ddy \u2264 x~\u2228y.If R0 algebras coincide with pseudo-IMTL algebras and that pseudo-R0 algebras coincide with pseudo-NM algebras. Hence, as the corollaries of commutative cases, weak-R0 algebras coincide with IMTL-algebras and R0-algebras coincide with NM-algebras, which are obtained in [We show that pseudo-weak-ained in .\ud835\udc9c =  be a pseudo-weak-R0 algebra. Define \u03a6(\ud835\udc9c) =  byxR\u2212 = x\u2192R0, xR~ = x\u21ddR0. Then, \u03a6(\ud835\udc9c) is a pseudo-IMTL algebra.(i) Let \ud835\udc9c =  be a pseudo-IMTL algebra. Define \u03a8(\ud835\udc9c) =  by\ud835\udc9c) is a pseudo-weak-R0 algebra.(ii) Conversely, let (iii) The above defined maps \u03a6 and \u03a8 are mutually inverse.(i) (pB1) is verified by (pL1).(pB2) is verified by (32) and (33).(pB3) is verified by (34).(pB4) is verified by (18).(pB5) is verified by (pL3).\ud835\udc9c) is a pseudo-IMTL algebra by Thus, \u03a6(R\u201d and \u201cI\u201d of the operations.(ii) For convenience, in the following proof, we omit the indices \u201c(pL1): By (pB1).x \u2264 y, by (pB3), x \u2264 y \u2264 z \u2192 (y\u2299z) and x \u2264 y \u2264 z\u21dd(z\u2299y), and so(pL2): If x \u2264 y, by y \u2192 z)\u2299x \u2264 (y \u2192 z)\u2299y and x\u2299(y\u21ddz) \u2264 y\u2299(y\u21ddz) \u2264 z, and soIf x \u2264 y, by y \u2192 0 \u2264 x \u2192 0 and y\u21dd0 \u2264 x\u21dd0; that is, y\u2212 \u2264 x\u2212 and y~ \u2264 x~, which means that (pL2) holds.Thus, if (pL3): By (pB5).x\u2299((x\u21ddy)\u2299(y\u21ddz)) = (x\u2299(x\u21ddy))\u2299(y\u21ddz) \u2264 y\u2299(y\u21ddz) \u2264 z and ((y \u2192 z)\u2299(x \u2192 y))\u2299x = (y \u2192 z)\u2299((x \u2192 y)\u2299x)\u2264(y \u2192 z)\u2299y \u2264 z, and so(pR1): By (pB2) and (pB3), x \u2192 y \u2264 (y \u2192 0)\u21dd(x \u2192 0) = y\u2212\u21ddx\u2212 \u2264 x\u2212~ \u2192 y\u2212~ = x \u2192 y and x\u21ddy \u2264 (y\u21dd0)\u2192(x\u21dd0) = y~ \u2192 x~ \u2264 x~\u2212\u21ddy~\u2212 = x\u21ddy, which means that x \u2192 y = y\u2212\u21ddx\u2212 and x\u21ddy = y~ \u2192 x~.Thus, by x\u22991 = x \u2264 x and 1\u2299x = x \u2264 x, and, by (pB3), x \u2264 1 \u2192 x and x \u2264 1\u21ddx. On the other hand, 1 \u2192 x = (1 \u2192 x)\u22991 \u2264 x and 1\u21ddx = 1\u2299(1\u21ddx) \u2264 x. Hence, 1 \u2192 x = x and 1\u21ddx = x.(pR2): By (pB2), z\u2299(z\u21ddx))\u2299(x\u21ddy) \u2264 x\u2299(x\u21ddy) \u2264 y; that is, z\u2299((z\u21ddx)\u2299(x\u21ddy)) \u2264 y, so (z\u21ddx)\u2299(x\u21ddy) \u2264 z\u21ddy; hence, x\u21ddy \u2264 (z\u21ddx)\u21dd(z\u21ddy). Similarly, (x \u2192 y)\u2299((z \u2192 x)\u2299z)\u2264(x \u2192 y)\u2299x \u2264 y; that is, ((x \u2192 y)\u2299(z \u2192 x))\u2299z \u2264 y, so (x \u2192 y)\u2299(z \u2192 x) \u2264 z \u2192 y; hence, x \u2192 y \u2264 (z \u2192 x)\u2192(z \u2192 y).(pR3): By (pB2), (pB3), and y\u2299(x \u2192 (y\u21ddz)))\u2299x = y\u2299((x \u2192 (y\u21ddz))\u2299x) \u2264 y\u2299(y\u21ddz) \u2264 z, and so y\u2299(x \u2192 (y\u21ddz)) \u2264 x \u2192 z; it follows that x \u2192 (y\u21ddz) \u2264 y\u21dd(x \u2192 z). Similarly, we have y\u21dd(x \u2192 z) \u2264 x \u2192 (y\u21ddz). Hence, y\u21dd(x \u2192 z) = x \u2192 (y\u21ddz).(pR4): By (pB2), (pB3), and x \u2264 y, by (pB3), z\u2299(z\u21ddx) \u2264 x \u2264 y and (z \u2192 x)\u2299z \u2264 x \u2264 y, and so(pR5): If y \u2192 z \u2264 y \u2192 z and (y \u2192 z)\u2299z \u2264 z, by (pB3), y \u2264 (y \u2192 z)\u21ddz and z \u2264 (y \u2192 z)\u21ddz, and so y\u2228z \u2264 (y \u2192 z)\u21ddz. Hence, by x \u2192 (y\u2228z) \u2264 x \u2192 ((y \u2192 z)\u21ddz) = (y \u2192 z)\u21dd(x \u2192 z), so (y \u2192 z)\u2299(x \u2192 (y\u2228z)) \u2264 x \u2192 z. Similarly, (x\u21dd(y\u2228z))\u2299(y\u21ddz) \u2264 x\u21ddz. Namely,Since By (pB3) and the same proof of (40), we havex \u2192 (y\u2228z) = 1\u2299(x \u2192 (y\u2228z)) = ((y \u2192 z)\u2228(z \u2192 y))\u2299(x \u2192 (y\u2228z)) = ((y \u2192 z)\u2299(x \u2192 (y\u2228z))\u2228((z \u2192 y))\u2299(x \u2192 (y\u2228z))\u2264(x \u2192 z)\u2228(x \u2192 y). On the other hand, by x \u2192 (y\u2228z) \u2265 x \u2192 y and x \u2192 (y\u2228z) \u2265 x \u2192 z, and so x \u2192 (y\u2228z)\u2265(x \u2192 y)\u2228(x \u2192 z). Hence, x \u2192 (y\u2228z) = (x \u2192 y)\u2228(x \u2192 z). Similarly, we have x\u21dd(y\u2228z) = (x\u21ddy)\u2228(x\u21ddz). Namely, (pR5) holds.By (pB2), (pB4), \ud835\udc9c) is a pseudo-weak-R0 algebra by Thus, \u03a8((iii) We put the index \u201cc\u201d to the operations of the structure obtained by composition of \u03a6 and \u03a8.\ud835\udc9c =  be a pseudo-weak-R0 algebra. We prove that \u03a8(\u03a6(\ud835\udc9c)) = \ud835\udc9c. Indeed,x\u2228cy = x\u2228Iy = x\u2228Ry, x\u2227cy = x\u2227Iy = x\u2227Ry, xc\u2212 = x\u2192I0 = x\u2192R0, xc~ = x\u21ddI0 = x\u21ddR0, x\u2192cy = x\u2192Iy = x\u2192Ry, and x\u21ddcy = x\u21ddIy = x\u21ddRy.Let \ud835\udc9c =  be a pseudo-IMTL algebra. We prove that \u03a6(\u03a8(\ud835\udc9c)) = \ud835\udc9c. Indeed, x\u2228cy = x\u2228Ry = x\u2228Iy, x\u2227cy = x\u2227Ry = x\u2227Iy, x\u2299cy = (x\u2192RyR\u2212)R~ = (x\u2192IyI\u2212)I~ = x\u2299Iy, x\u2192cy = x\u2192Ry = x\u2192Iy, andx\u21ddcy = x\u21ddRy = x\u21ddIy.Let \ud835\udc9c =  be a pseudo-R0 algebra. Define \u03a6(\ud835\udc9c) =  byxR\u2212 = x\u2192R0, xR~ = x\u21ddR0. Then, \u03a6(\ud835\udc9c) is a pseudo-NM algebra.(i) Let \ud835\udc9c =  be a pseudo-NM algebra. Define \u03a8(\ud835\udc9c) =  by\ud835\udc9c) is a pseudo-R0 algebra.(ii) Conversely, let (iii) The above defined maps \u03a6 and \u03a8 are mutually inverse.\ud835\udc9c). (pB6) is verified by (49).(i) By \ud835\udc9c). Now, we prove that (pR6) also holds in \u03a8(\ud835\udc9c). For convenience, in the following proof, we omit the indices \u201cR\" and \u201cI\u201d of the operations.(ii) By u \u2208 A, u \u2264 x \u2192 (y \u2192 z)\u21d4u\u2299x \u2264 y \u2192 z\u21d4u\u2299(x\u2299y) = (u\u2299x)\u2299y \u2264 z\u21d4u \u2264 (x\u2299y) \u2192 z and u \u2264 x\u21dd(y\u21ddz)\u21d4x\u2299u \u2264 y\u21ddz\u21d4(y\u2299x)\u2299u = y\u2299(x\u2299u) \u2264 z\u21d4u \u2264 (y\u2299x)\u21ddz. Hence,By (pB2) and (pB3), for any x\u2299y)~ = (x\u2299y)\u21dd0 = y\u21dd(x\u21dd0) = y\u21dd(1 \u2192 x~) = y\u21ddx~. Similarly, (x\u2299y)\u2212 = (x\u2299y) \u2192 0 = x \u2192 (y \u2192 0) = x \u2192 (1\u21ddy\u2212) = x \u2192 y\u2212. Hence,By By (pL2), (pL3), and the same proof of (5) and (6), we haveThus, by (iii) By R = \u21ddR, R\u2212=R~, and \u2192I = \u21ddI, then we have the main results in [In Theorems sults in .R0 algebras coincide with IMTL-algebras.Weak-R0-algebras coincide with NM-algebras.R0 algebras and R0-algebras to the noncommutative forms, called pseudo-weak-R0 algebras and pseudo-R0 algebras. The properties of pseudo-weak-R0 algebras and pseudo-R0 algebras were investigated, and the simplified axiom systems of pseudo-weak-R0 algebras and pseudo-R0 algebras were discussed. Finally, we showed that pseudo-weak-R0 algebras are categorically isomorphic to pseudo-IMTL algebras and that pseudo-R0 algebras are categorically isomorphic to pseudo-NM algebras. Based on these results, we will study filter theory of pseudo-weak-R0 algebras and pseudo-R0 algebras and investigate relations between various kinds of filters of pseudo-logic algebras. We may also study fuzzy type of filters of pseudo-weak-R0 algebras and pseudo-R0 algebras.We gave a positive answer to Iorgulescu's open problem. We extended weak-"}
{"text": "The present study investigated the effects of \u03b5\u2010viniferin and resveratrol on epithelial secretory and barrier functions in isolated rat small and large intestinal mucosa. Mucosa\u2013submucosa tissue preparations of various segments of the rat large and small intestines were mounted on Ussing chambers, and short\u2010circuit current (Isc) and tissue conductance (Gt) were continuously measured. The mucosal addition of \u03b5\u2010viniferin (>10\u22125\u00a0mol/L) and resveratrol (>10\u22124\u00a0mol/L) to the cecal mucosa, which was the most sensitive region, induced an increase in Isc and a rapid phase decrease (P\u20101) followed by rapid (P\u20102) and broad (P\u20103) peak increases in Gt in concentration\u2010dependent manners. Mucosal \u03b5\u2010viniferin (10\u22124\u00a0mol/L), but not resveratrol (10\u22124\u00a0mol/L), increased the permeability of FITC\u2010conjugated dextran (4\u00a0kDa). The mucosal \u03b5\u2010viniferin\u2013evoked changes in Isc (Cl\u2212 secretion), but not in Gt, were attenuated by a selective cyclooxygenase (COX)\u20101 inhibitor and a selective EP4 prostaglandin receptor. The mucosal \u03b5\u2010viniferin\u2013evoked increase in Isc was partially attenuated, and P\u20102, but not P\u20101 or P\u20103, change in Gt was abolished by a transient receptor potential cation channel, subfamily A, member 1 (TRPA1) inhibitor. Moreover, the mucosal \u03b5\u2010viniferin concentration\u2010dependently attenuated the mucosal propionate (1\u00a0mmol/L)\u2010evoked increases in Isc and Gt. Immunohistochemical studies revealed COX\u20101\u2013immunoreactive epithelial cells in the cecal crypt. The present study showed that mucosal \u03b5\u2010viniferin modulated transepithelial ion transport and permeability, possibly by activating sensory epithelial cells expressing COX\u20101 and TRPA1. Moreover, mucosal \u03b5\u2010viniferin decreased mucosal sensitivity to other luminal molecules such as short\u2010chain fatty acids. In conclusion, these results suggest that \u03b5\u2010viniferin modifies intestinal mucosal transport and barrier functions. Resveratrol has a variety of beneficial effects, including antioxidant  in a colonic epithelium\u2010like monolayer\u2010cultured cell line, T84 cells, and in the mouse jejunum  Cl\u2212 secretion  were purchased from Japan SLC, Inc. , and were allowed food and water ad\u00a0libitum before all experiments. The animals were handled and euthanized in accordance with the Guidelines for the Care and Use of Laboratory Animals of the University of Shizuoka, and the study was approved by the University of Shizuoka Animal Usage Ethics Committee. SD rats were anesthetized with isoflurane and decapitated with a guillotine. The terminal ileum, cecum, proximal colon, middle colon, distal colon, and rectum were removed and placed in ice\u2010cold Krebs\u2013Ringer solution saturated with O2 and CO2  rats  maintained at 37\u00b0C and gassed with 95% O2/5% CO2. The transepithelial potential difference  was measured using paired Ag\u2013AgCl electrodes  through Krebs\u2013agar bridges, and was clamped at 0\u00a0mV by applying a short\u2010circuit current (Isc) with another pair of Ag\u2013AgCl electrodes  connected to a voltage\u2010clamp apparatus . The lumen negative electrogenic current represents a positive Isc per unit area of tissue (\u03bcA/cm2). To record tissue conductance (Gt [mS/cm2]), voltage command pulses  were applied at 1\u00a0min intervals, and Gt was calculated using Ohm's law as the current needed to alter the clamped voltage. The current output was continuously recorded on a data acquisition and analog\u2010to\u2010digital conversion system . Before the experiments, the tissues were stabilized for 1\u00a0h and tissue viability was checked by electrical field stimulation  using a pair of aluminum foil ribbon electrodes.The cross\u2010sectional area of the mucosa\u2013submucosa preparations in the Ussing flux chamber was 0.64\u00a0cm\u22124\u00a0mol/L) was added to the mucosal bathing solution 30\u00a0min before the addition of EtOH , \u03b5\u2010viniferin (10\u22124\u00a0mol/L), and resveratrol (10\u22124\u00a0mol/L). Samples (100\u00a0\u03bcL) were taken from the serosal bathing solution, and fresh Krebs\u2013Ringer (100\u00a0\u03bcL) was added at \u221215, 0 15, 30, 45, and 60\u00a0min (6 samples). The concentrations of FD4 were determined by measuring the fluorescence intensity of samples and standard solutions using a microplate reader  at emission and excitation wavelengths of 520\u00a0nm and 490\u00a0nm, respectively. Based on a volume of the serosal bathing solution (Vm) of 10\u00a0mL and sample volume (Vs) of 100\u00a0\u03bcL, the total FD4 mass that crossed the tissue at each time (Mn) was:C1, C2, \u2026 Cn are the sample concentrations at sampling times \u221215 \u2026 60\u00a0min. The FD4 flux (Jm\u2192s) per unit area (0.64\u00a0cm2) was calculated as:Jnm\u2192s=Mn+1\u2212Mn15min\u00d710.64cm2where J1m\u2192s, J2m\u2192s, \u2026Jn\u20101m\u2192s are the FD4 fluxes in each period . Because the decrease in the mucosal concentration of FD4 during experiments is considered to be negligible, the permeability (Pnm\u2192s) of FD4 in each period was calculated by Fick's law as follows:Pnm\u2192s=Jnm\u2192sCm\u2212Cs\u2245Jnm\u2192sCm=Jnm\u2192s10\u22124mol/Lwhere Cm and Cs (Cm\u00a0\u226b\u00a0Cs) are the concentrations of FD4 in the mucosal and serosal bathing solutions, respectively. This method was built in reference to a chapter in the textbook, Molecular Biopharmaceutics \u2010conjugated dextran  in the mucosa\u2013submucosal tissue preparation was measured under short\u2010circuit conditions in the Ussing chamber. In this experiment, FD4  117 NaCl, 4.7 KCl, 1.2 MgCl\u03b5\u2010Viniferin, resveratrol, and sodium propionate were purchased from Wako Pure Chemical Industries, Ltd. . Carbachol (CCh), FD4, bumetanide, 5\u2010nitro\u20102\u2010(3\u2010phenylpropylamino) benzoic acid (NPPB), atropine, hexamethonium, SC\u2010560, and NS\u2010398 were purchased from Sigma . TTX and HC030031 were purchased from Tocris . Piroxicam was purchased from Biomol Research Laboratories . 4,4\u2032\u2010Diisothiocyanatostilbene\u20102,2\u2032\u2010disulfonic acid (DIDS) was purchased from ANA SPEC . AH\u20106809 was purchased from Cayman Chemical . ONO\u20108713, ONO\u2010AE3\u2010208, and ONO\u2010AE3\u2010240 were kind gifts from ONO Pharmaceutical Co., Ltd. .\u03b5\u2010Viniferin and resveratrol were dissolved in ethanol (EtOH) just before use. TTX was dissolved in citrate buffer (pH 4.8) and stored at \u221220\u00b0C until use. CCh, FD4, propionate, atropine, and HEX were dissolved in distilled water, and stored at 4\u00b0C until use. All of the other chemicals were dissolved in DMSO and stored at \u221220\u00b0C until use.\u03bcm\u2010thick sections on a cryostat . The sections were placed on glass slides, fixed in 100% methanol at \u221220\u00b0C for 10\u00a0min, and air\u2010dried under a cold blower for 30\u00a0min. The sections were incubated with 10% normal donkey serum and 1% Triton X\u2010100 in phosphate\u2010buffered saline (PBS) at room temperature for 1\u00a0h to block nonspecific antibody binding. The sections were incubated with goat anticyclooxygenase (COX)\u20101 antibody  in PBS containing 0.3% Triton X\u2010100 at 4\u00b0C overnight. After washing in PBS (3\u00a0\u00d7\u00a010\u00a0min), the sections were incubated with donkey anti\u2010goat IgG conjugated with Alexa594  at room temperature for 1\u00a0h. The sections were washed in PBS (3\u00a0\u00d7\u00a010\u00a0min), and coverslips were mounted on the glass slides with mounting medium . Immunoreactivity for COX\u20101 was visualized using a fluorescence microscope , and the images were captured using a cooled charge\u2010coupled device digital camera and digital imaging software .The isolated cecal tissues were immediately frozen with Tissue\u2010Tek optimal cutting temperature compound  in a \u221275\u00b0C acetone bath . The tissues were cut into 4\u2010t tests were used to determine the statistical significance between two groups because some tissue preparations from the same animals were used in both experimental groups. In all analyses, Ps\u00a0<\u00a00.05 were considered statistically significant.All physiologic data in the present study are expressed as the mean\u00a0\u00b1\u00a0SEM. Multiple comparisons were made using Dunnett's test and Tukey's test for each group relative to the control group and among all combinations of groups, respectively. Paired \u03b5\u2010viniferin  evoked a monophasic increase in Isc  and triphasic changes in Gt . In contrast, mucosal addition of 10\u22124\u00a0mol/L resveratrol evoked very small changes in Isc and Gt  and Gt, likely via \u03b5\u2010viniferin\u2013evoked P\u20101  and P\u20103  , \u03b5\u2010viniferin evoked a small, but long\u2010lasting decrease in Gt, as shown in Figure\u00a0\u03bcL), did not affect Isc or Gt.In the rat cecal mucosa\u2013submucosal preparations, mucosal, but not serosal, addition of \u03b5\u2010viniferin (3\u00a0\u00d7\u00a010\u22126 to 3\u00a0\u00d7\u00a010\u22124\u00a0mol/L) and resveratrol (10\u22125 to 3\u00a0\u00d7\u00a010\u22124\u00a0mol/L) were added to the mucosal bathing solution, and the mean changes in Isc  of the mucosal \u03b5\u2010viniferin\u2013evoked \u2206Isc was 4.81\u00a0\u00d7\u00a010\u22125\u00a0mol/L, the maximum effect (Emax) was 87.19\u00a0\u03bcA/cm2, and the Hill coefficient value (nH) was 2.37 (coefficient of determination [R2]\u00a0=\u00a01.000). At 10\u22125\u00a0mol/L or lower concentrations, mucosal \u03b5\u2010viniferin only evoked a decrease in Gt, which was long\u2010lasting. The EC50 of the mucosal \u03b5\u2010viniferin\u2013evoked P\u20101 \u2206Gt was 2.53\u00a0\u00d7\u00a010\u22126\u00a0mol/L, Emax was \u22120.69\u00a0mS/cm2, and nH was 1.22 (R2\u00a0=\u00a00.887). Although it is unclear whether the P\u20102 \u2206Gt evoked by 3\u00a0\u00d7\u00a010\u22124\u00a0mol/L \u03b5\u2010viniferin was at the maximum effective concentration, the EC50 of \u2206Gt was calculated to be 9.44\u00a0\u00d7\u00a010\u22125\u00a0mol/L, when we assumed that the Emax was the mean value for \u2206Gt at 3\u00a0\u00d7\u00a010\u22124\u00a0mol/L \u03b5\u2010viniferin (6.92\u00a0mS/cm2) and that the nH was same as the that for P\u20103 described below (R2\u00a0=\u00a01.000). Therefore, the EC50 for mucosal \u03b5\u2010viniferin\u2013evoked P\u20103 \u2206Gt was calculated to be 2.35\u00a0\u00d7\u00a010\u22125\u00a0mol/L, the Emax of mucosal \u03b5\u2010viniferin\u2013evoked P\u20103 \u2206Gt was 4.49\u00a0mS/cm2, and nH was 4.53 (R2\u00a0=\u00a00.999).Various concentrations of Isc did not seem to reach the maximum concentrations, but the EC50 of P\u20102 \u2206Isc was 2.52\u00a0\u00d7\u00a010\u22124\u00a0mol/L (R2\u00a0=\u00a00.975) when we assumed that Emax and nH were the same as those for \u03b5\u2010viniferin\u2013evoked \u2206Isc. Likewise, the resveratrol\u2010evoked increase in Gt did seem to reach the maximum concentration, but the EC50 was 1.82\u00a0\u00d7\u00a010\u22124\u00a0mol/L (R2\u00a0=\u00a00.912) when we assumed the Emax and nH were the same as those for \u03b5\u2010viniferin\u2013evoked P\u20103 \u2206Gt.The resveratrol\u2010evoked P\u20101 and P\u20102 \u2206\u03b5\u2010viniferin may evoke more potent responses at concentrations of >3\u00a0\u00d7\u00a010\u22124\u00a0mol/L, \u03b5\u2010viniferin at these concentrations evoked unlimited continuous increases in Gt. This indicates that \u03b5\u2010viniferin at concentrations >3\u00a0\u00d7\u00a010\u22124\u00a0mol/L may have cytotoxic effects because the epithelial barrier function was thought to have failed. Thus, \u03b5\u2010viniferin was used at 10\u22124\u00a0mol/L in the following experiments.Although \u03b5\u2010viniferin (10\u22124\u00a0mol/L) on Isc and Gt were measured and compared among the rat terminal ileum, cecum, proximal colon, middle colon, and distal colon. Mucosal addition of \u03b5\u2010viniferin (10\u22124\u00a0mol/L) had weak effects on Isc in the intestinal tissues, except for the cecum \u2013evoked increase was most potent in the cecum and the magnitude of the increase gradually decreased from the proximal to distal segments of the large intestine  were measured in terms of Gt. Basal Isc was 18.07\u00a0\u00b1\u00a01.62\u00a0\u03bcA/cm2, basal Gt was 11.15\u00a0\u00b1\u00a00.41\u00a0mS/cm2, and basal FD4 Pm\u2192s was 1.26\u00a0\u00b1\u00a00.20\u00a0\u00d7\u00a010\u22127\u00a0cm/s (n\u00a0=\u00a020).To investigate the effects of mucosal \u03b5\u2010viniferin evoked an increase in Isc, and the mean Isc reached its maximum at 7\u00a0min . The Gt first decreased at 3\u00a0min , the P\u20102 peak occurred at 8\u00a0min , and the P\u20103 peak occurred at 26\u00a0min . FD4 Pm\u2192s increased, and reached a maximum  at 30\u201345\u00a0min after mucosal addition of \u03b5\u2010viniferin. This value was significantly greater (P\u00a0<\u00a00.05 by Dunnett's test) than the FD4 Pm\u2192s after the addition of the vehicle control  at 30\u201345\u00a0min.Mucosal addition of \u22124\u00a0mol/L), Isc decreased initially , and then slowly increased . Gt slowly increased . However, FD4 permeability did not significantly change.After the mucosal addition of resveratrol . All preparations were bathed with normal solution at first. After the bathing solutions of the serosal, mucosal, or both sides were replaced with Cl\u2212\u2010free solution, and the compensations of fluid resistance were reset after changing the solutions. Basal Isc significantly increased in the serosal Cl\u2212\u2010free solution, but decreased in the mucosal Cl\u2212\u2010free solution \u2013evoked increases in Isc were almost completely abolished for Cl\u2212\u2010free solution on the serosal and both sides, but not the mucosal side . These results indicate that the mucosal \u03b5\u2010viniferin\u2013evoked increase in Isc is due to electrogenic Cl\u2212 secretion. Moreover, with serosal Cl\u2212\u2010free solution, mucosal \u03b5\u2010viniferin evoked a decrease in Isc . Serosal and both side Cl\u2212\u2010free solutions led to unlimited increases in Gt, which made it difficult or impossible to measure the mucosal \u03b5\u2010viniferin\u2013evoked changes in Gt.The \u03b5\u2010viniferin evoked Cl\u2212 secretion, a Na+\u2010K+\u20102Cl\u2212 (NKCC) transporter inhibitor, bumetanide (10\u22124\u00a0mol/L), was added to the serosal bathing solution. Alternatively, a Ca2+\u2010dependent channel blocker, DIDS (10\u22124\u00a0mol/L), or a cAMP\u2010dependent Cl\u2212 channel  blocker, NPPB (10\u22125 to 10\u22124\u00a0mol/L), were added to the mucosal bathing solution before the mucosal addition of \u03b5\u2010viniferin. Bumetanide (10\u22124\u00a0mol/L) and NPPB (\u22655\u00a0\u00d7\u00a010\u22125\u00a0mol/L) significantly attenuated the \u03b5\u2010viniferin\u2013evoked increase in Isc, but not DIDS . Although bumetanide attenuated the \u03b5\u2010viniferin\u2013evoked increase in Isc, it did not attenuate the Gt responses . In contrast, NPPB (5\u00a0\u00d7\u00a010\u22125\u00a0mol/L) enhanced the \u03b5\u2010viniferin\u2013evoked P\u20101 Gt reduction and nearly abolished the P\u20103 Gt change . The \u03b5\u2010viniferin\u2013evoked changes in Gt in the presence of 10\u22124\u00a0mol/L NPPB were not measured because Gt gradually increased after the addition of 10\u22124\u00a0mol/L NPPB.To confirm that mucosal \u20134\u00a0mol/L resveratrol evoked only a weak decrease in P\u20101 \u0394Isc (\u22125.66\u00a0\u00b1\u00a01.61\u00a0\u03bcA/cm2) and an increase in \u0394Gt (0.63\u00a0\u00b1\u00a00.30\u00a0mS/cm2) , resveratrol at 3\u00a0\u00d7\u00a010\u22124\u00a0mol/L elicited Isc and Gt responses, which were similar to those induced by 10\u22124\u00a0mol/L \u03b5\u2010viniferin  was added to the mucosal bathing solution 30\u00a0min after mucosal addition of resveratrol (at 0 to 3\u00a0\u00d7\u00a010\u22124\u00a0mol/L), and the changes in Isc and Gt were measured. In this experiment, resveratrol significantly attenuated the \u03b5\u2010viniferin\u2013evoked changes in Isc and Gt in concentration\u2010dependent manners . Mucosal resveratrol at 10\u22124\u00a0mol/L significantly attenuated the mucosal \u03b5\u2010viniferin (10\u22124\u00a0mol/L)\u2013evoked increase in Isc to 26.07\u00a0\u00b1\u00a04.12% of the control value  and P\u20103 \u2206Gt to 29.31\u00a0\u00b1\u00a08.20% of the control value . When the mucosal bathing solution contained 3\u00a0\u00d7\u00a010\u22124\u00a0mol/L resveratrol, the administration of \u03b5\u2010viniferin (10\u22124\u00a0mol/L) decreased Isc  and abolished the P\u20102 response. However, there were no statistically significant differences because the P\u20102 response in the control conditions was weak and varied considerably between tests. Moreover, in the presence of 3\u00a0\u00d7\u00a010\u22124\u00a0mol/L resveratrol, \u03b5\u2010viniferin (10\u22124\u00a0mol/L) caused a continuous increase in Gt, as shown in Figure\u00a06C, so P\u20103 \u2206Gt could not be determined.Although mucosal addition of 10\u03b5\u2010viniferin\u2013evoked changes in Isc and Gt were mediated via TRPA1 channels, like those evoked by AITC. In this setting, a TRPA1 inhibitor, HC030031 (10\u22124\u00a0mol/L), significantly reduced the \u03b5\u2010viniferin\u2013evoked increase in Isc from 60.95\u00a0\u00b1\u00a010.70\u00a0\u03bcA/cm2 to 38.04\u00a0\u00b1\u00a08.02\u00a0\u03bcA/cm2 , and the peak time after the addition of \u03b5\u2010viniferin was significantly delayed from 6.71\u00a0\u00b1\u00a00.93\u00a0min to 14.68\u00a0\u00b1\u00a02.05\u00a0min . For Gt, the \u03b5\u2010viniferin\u2013evoked P\u20101 and P\u20103 responses were not affected, but the P\u20102 response was almost completely abolished by HC030031, as shown in Figure\u00a07A and B .In our previous study, we reported that mucosal administration of a pungent principle, allyl isothiocyanate (AITC), evoked anion secretion by activating TRPA1 channels , HEX , or atropine (a muscarinic ACh receptor antagonist) to the serosal bathing solution 30\u00a0min before the addition of \u03b5\u2010viniferin (10\u22124\u00a0mol/L). However, TTX, HEX, and atropine did not affect the \u03b5\u2010viniferin\u2013evoked changes in Isc and Gt .The neural reflex pathways in the submucosal plexus in the gut wall play important roles in luminal stimuli\u2010evoked secretory responses in the intestinal mucosa Cooke . Thus, wn\u2010propyl\u20102\u2010thiouracil) , a selective COX\u20101 inhibitor , a selective COX\u20102 inhibitor , or a combination of SC\u2010560 and NS\u2010398 were added to the serosal bathing solution 30\u00a0min before the mucosal addition of \u03b5\u2010viniferin (10\u22124\u00a0mol/L). Piroxicam, SC\u2010560, and the combination of SC\u2010560 and NS\u2010398, but not NS\u2010398 alone, significantly attenuated the \u03b5\u2010viniferin\u2013evoked increase in Isc from the control value of 59.40\u00a0\u00b1\u00a010.53\u00a0\u03bcA/cm2 (n\u00a0=\u00a06) to 13.33\u00a0\u00b1\u00a02.79\u00a0\u03bcA/cm2 with piroxicam , to 23.10\u00a0\u00b1\u00a06.36\u00a0\u03bcA/cm2 with SC\u2010560 , and 22.92\u00a0\u00b1\u00a07.70\u00a0\u03bcA/cm2 with the combination of SC\u2010560 and NS\u2010398 . These results indicate that the \u03b5\u2010viniferin\u2013evoked increase in Isc is mediated by COX\u20101 production of prostaglandin (P)G. In contrast, piroxicam, SC\u2010560, and NS\u2010398 did not significantly affect the \u03b5\u2010viniferin\u2013evoked changes in Gt .In our previous studies, some mucosal stimulants including a bitter tastant , an EP1 and EP2 receptor antagonist , a selective EP3 receptor antagonist , or a selective EP4 receptor antagonist  were added to the serosal bathing solution 30\u00a0min before the mucosal addition of \u03b5\u2010viniferin (10\u22124\u00a0mol/L). In this experiment, neither ONO\u20108713 nor AH\u20106809 affected the \u03b5\u2010viniferin\u2013evoked increase in Isc, but ONO\u2010AE3\u2010208 at concentrations \u226510\u22126\u00a0mol/L significantly attenuated the \u03b5\u2010viniferin\u2013evoked increase in Isc . These results indicate that the \u03b5\u2010viniferin\u2013evoked increase in Isc is mediated by EP4 receptors.To determine which PG receptor subtype mediates the \u03b5\u2010viniferin would attenuate the mucosal propionate\u2010evoked secretory responses. To confirm this hypothesis, mucosal propionate (10\u22123\u00a0mol/L)\u2010evoked Isc responses were measured 30\u00a0min after the addition of \u03b5\u2010viniferin at a variety of concentrations, as described earlier.Short\u2010chain fatty acids, including acetate (two carbons), propionate (three carbons), and butyrate (four carbons), are the predominant anions in the large intestine and exist at concentrations of \u2265100\u00a0mmol/L. They are produced by bacterial fermentation of indigestible dietary fibers and oligosaccharides. Mucosal SCFAs (with a potency order of propionate\u00a0\u2265\u00a0butyrate\u00a0\u226b\u00a0acetate) are known to stimulate the large intestinal mucosa to secrete transepithelial anions Yajima . In our \u03b5\u2010viniferin, propionate (10\u22123\u00a0mol/L) transiently increased Isc  and Gt  (n\u00a0=\u00a06). However, the mucosal propionate (10\u22123\u00a0mol/L)\u2010evoked increases in Isc and Gt were attenuated by mucosal \u03b5\u2010viniferin in a concentration\u2010dependent manner at \u03b5\u2010viniferin concentrations ranging from 3\u00a0\u00d7\u00a010\u22126 to 3\u00a0\u00d7\u00a010\u22124\u00a0mol/L . Pretreatment with 3\u00a0\u00d7\u00a010\u22124\u00a0mol/L \u03b5\u2010viniferin nearly abolished the propionate response. The concentration\u2013response curves of \u03b5\u2010viniferin for inhibiting the propionate\u2010evoked Isc and Gt responses were drawn by fitting the data to the inhibitory Hill equation, as described in Figure\u00a011. In this analysis, the half\u2010maximal inhibitory concentration (IC50) of \u03b5\u2010viniferin for \u0394Isc was 4.12\u00a0\u00d7\u00a010\u22125\u00a0mol/L and the nH was 2.11 (R2\u00a0=\u00a00.998). For Gt, the IC50 was 2.68\u00a0\u00d7\u00a010\u22125\u00a0mol/L and nH was 1.68 (R2\u00a0=\u00a00.985), where 119.4%, which was the mean value of the propionate\u2010evoked Gt in 3\u00a0\u00d7\u00a010\u22126\u00a0mol/L \u03b5\u2010viniferin, was used as the value for the maximal effect.In the absence of \u03b5\u2010viniferin might stimulate some epithelial sensory cells expressing COX\u20101, because luminal \u03b5\u2010viniferin evoked Isc and Gt responses, but serosal \u03b5\u2010viniferin did not. An earlier study revealed that COX\u20101\u2013expressing epithelial cells are scattered throughout the crypts in the rat large intestine  induce transepithelial electrogenic Cl\u2212 secretion by activating EP4 receptors via PGs produced by COX\u20101; (2) elicit a rapid decrease and a sustained increase in transepithelial ion permeability; and (3) increase transepithelial permeability of a nonionic macromolecule, FD4. In addition, mucosal \u03b5\u2010viniferin inhibited mucosal propionate\u2010evoked Cl\u2212 secretion. These results indicate that \u03b5\u2010viniferin stimulates the intestinal epithelium from the luminal side and effects mucosal barrier functions.The results of the present study show that \u03b5\u2010viniferin at concentrations \u226510\u22125\u00a0mol/L elicited a concentration\u2010dependent monophasic positive change in Isc and triphasic changes in Gt, which comprised an abrupt decrease (P\u20101) followed by fast (P\u20102) and sustained (P\u20103) increases in the rat cecal mucosa, whereas serosal administration of 10\u22124\u00a0mol/L \u03b5\u2010viniferin did not  and a sustained positive \u2206Isc (P\u20102), and a single sustained increase in Gt. In contrast, serosal administration of 3\u00a0\u00d7\u00a010\u22124\u00a0mol/L resveratrol did not affect Isc and a small decrease (P\u20101) and increase (P\u20102) in Gt , \u03b5\u2010viniferin evoked the long\u2010lasting decreases in P\u20101 Gt, as shown in Figure\u00a0\u22125\u00a0mol/L), the \u03b5\u2010viniferin\u2013evoked changes in Gt consisted of an abrupt and long\u2010lasting decrease in ion permeability and subsequent increases in ion permeability. At lower concentration, \u03b5\u2010viniferin may enhance epithelial barrier function, and at higher concentrations \u03b5\u2010viniferin may further enhance ionic and nonionic permeability (as mentioned below) to induce secretory and inflammatory functions as part of a host defense mechanism.At lower concentrations \u2264100\u22125\u00a0mol/L\u03b5\u2010viniferin in terms of the Isc and Gt responses occurred in the cecal mucosa , but not resveratrol (10\u22124\u00a0mol/L), elicited a transient (15\u201345\u00a0min after addition) increase in FD4 Pm\u2192s  also increased FD4 Pm\u2192s in the rat cecum. These findings suggest that luminal \u03b5\u2010viniferin directly and/or indirectly affects epithelial tight junctions, and increases paracellular permeability of nonionic macromolecules. However, the mechanism underlying the \u03b5\u2010viniferin\u2013evoked increase in FD4 Pm\u2192s needs to be examined in future studies.Mucosal m\u2192s Fig.\u00a0. In our \u03b5\u2010viniferin\u2013induced increase in Isc was due to transepithelial Cl\u2212 secretion, because these responses were attenuated by Cl\u2212\u2010free solution on the serosal and both sides, but not on the mucosal side  blocker  and apical DIDS (10\u22124\u00a0mol/L), the apical administration of NPPB at concentrations \u22655\u00a0\u00d7\u00a010\u22125\u00a0mol/L significantly enhanced P\u20101 and significantly attenuated P\u20103 \u2206Gt  antagonist, significantly attenuated \u03b5\u2010viniferin\u2013evoked Cl\u2212 secretion \u2010evoked increase in Isc and Gt in a concentration\u2010dependent manner  and/or FFA3 (GPR41) (Karaki et\u00a0al. \u03b5\u2010viniferin may allosterically bind to these receptors, with a possible stoichiometry of 2:1 because the nH was nearly 2. Nevertheless, further studies are necessary to confirm this hypothesis. Moreover, at the lower concentration of 3\u00a0\u00d7\u00a010\u22126\u00a0mol/L, \u03b5\u2010viniferin very weakly enhanced the propionate\u2010evoked increase in Gt (Fig.\u00a0\u03b5\u2010viniferin on transepithelial ion permeability need to be investigated in future studies.Bacterial fermentation in the lumen of the large intestine produces numerous metabolites. The predominant molecules are SCFAs, particularly acetate, propionate, and butyrate. Propionate and butyrate, but not acetate, were reported to induce anion secretion in the rat Yajima  and guinner Fig.\u00a0. This inGt Fig.\u00a0C. Theref\u03b5\u2010viniferin may activate COX\u2010expressing cecal crypt cells, increase the tissue PG level as part of a tissue alarm\u2010response system, and enhance the tissue's host defense functions.In the gastrointestinal lumen, especially in the large intestine, the luminal microbiota can synthesize a variety of compounds and some of these compounds may have cytotoxic effects. Thus, the intestinal mucosa has protective roles, including enhancing the integrity of the epithelial barrier and fluid secretion when the mucosa senses potentially cytotoxic chemicals. We hypothesized that the tissue concentrations of PGs may constitute an alarm\u2010response system, in which an increase in the PG concentration might enhance the protective functions of the mucosa in host defense (Karaki and Kuwahara \u03b5\u2010viniferin to the mucosal side of the rat intestine modulates transepithelial ion transport, ion permeability, and the permeability of nonionic macromolecule as well as the effects of the other luminal molecules, such as SCFAs. These results also imply that \u03b5\u2010viniferin has beneficial effects on intestinal functions by enhancing the mucosal host defense mechanism.The present study shows that administration of A. Kuwahara have received research fund from FANCL Corporation. I. Ishikawa is an employee of FANCL Corporation."}
{"text": "In this paper, we essentially deal withK\u00f6the-Toeplitz duals of fuzzy level sets defined using a partial metric. Since the utilization of Zadeh'sextension principle is quite difficult in practice, we prefer the idea of level sets in order to constructsome classical notions. In this paper, we present the sets of bounded, convergent, and null series and theset of sequences of bounded variation of fuzzy level sets, based on the partial metric. We examine therelationships between these sets and their classical forms and give some properties including definitions,propositions, and various kinds of partial metric spaces of fuzzy level sets. Furthermore, we study some oftheir properties like completeness and duality. Finally, we obtain the K\u00f6the-Toeplitz duals of fuzzy levelsets with respect to the partial metric based on a partial ordering. We define the classical sets bs(H), cs(H), and cs0(H) consisting of the sets of all bounded, convergent, and null series, respectively; that isbs(H), cs(H), and cs0(H) are complete metric spaces with the partial metric Hs defined byu = (uk) and v = (vk) are the elements of the sets bs(H), cs(H), or cs0(H).By bv(H), bvq(H), and bv\u221e(H) consisting of sequences of q-bounded variation by using the partial metric Hs with respect to the partial ordering \u2291H, as follows:Hs denotes the partial metric of fuzzy level sets defined byu, v \u2208 E1 with the partial ordering \u2291H. One can conclude that the sets bv(H), bvq(H), and bv\u221e(H) are complete metric spaces with the following partial metrics:u = (uk) and v = (vk) are the elements of the sets bv(H), bvq(H), or bv\u221e(H) and (\u0394u)k = uk \u2212 uk+1 for all k \u2208 N.Secondly, we introduce the sets A some of which reduced to the Maddox's spaces l\u221e, c, c0, and l of sequences of fuzzy numbers for the special cases of that matrix A. Quite recently, Talo and Ba\u015far \u2291p if and only if  is a subset of .For the partial metric max\u2061{X, p) be a partial metric space and (xn) a sequence in . Then, we say the following:xn) converges to a point x \u2208 X if and only if p = lim\u2061n\u2192\u221e\u2009p.A sequence (xn) is a Cauchy sequence if there exists (and is finite) lim\u2061m,n\u2192\u221e\u2009p.A sequence  is said to be complete if every Cauchy sequence (xn) in X converges, with respect to the topology \u03c4p, to a point x \u2208 X such that p = lim\u2061m,n\u2192\u221e\u2009p. It is easy to see that every closed subset of a complete partial metric space is complete.A partial metric space ) \u2282 Bp(f(x0), \u025b).A mapping xn) in a partial metric space  converges to a point x \u2208 X, for any \u03f5 > 0 such that x \u2208 Bp, there exists n0 \u2265 1 so that for any n \u2265 n0, xn \u2208 Bp.A sequence  be a partial metric space. Then, (i)xn) is a Cauchy sequence in  if and only if it is a Cauchy sequence in the metric space ,( (ii)X, p) is complete if and only if the metric space  is complete. Furthermore, lim\u2061n\u2192\u221e\u2009ps = 0 if and only if p = lim\u2061n\u2192\u221e\u2009p = lim\u2061m,n\u2192\u221e\u2009p.a partial metric space , the limit of the sequence (\u22121/n) is 0 since one has lim\u2061n\u2192\u221e\u2009ps, where ps is the usual metric induced by p on R\u2212.In the partial metric space u is normal; that is, there exists an x0 \u2208 R such that u(x0) = 1. (ii)u is fuzzy convex; that is, u[\u03bbx + (1 \u2212 \u03bb)y] \u2265 min\u2061{u(x), u(y)} for all x, y \u2208 R and for all \u03bb \u2208 . (iii)u is upper semicontinuous. (iv)x \u2208 R : u(x) > 0} in the usual topology of R.The set f. Zadeh ), where AR by E1 and call it as the space of fuzzy numbers. \u03bb-level set [u]\u03bb of u \u2208 E1 is defined byu]\u03bb is closed, bounded, and nonempty interval for each \u03bb \u2208  which is defined by [u]\u03bb = . R can be embedded in E1, since each r \u2208 R can be regarded as a fuzzy number We denote the set of all fuzzy numbers on  Let [u]\u03bb =  for u \u2208 E1 and for each \u03bb \u2208 . Then the following statements hold. (i)u\u2212 is a bounded and nondecreasing left continuous function on ]0,1]. (ii)u+ is a bounded and nonincreasing left continuous function on ]0,1]. (iii)The functions u\u2212 and u+ are right continuous at the point \u03bb = 0. (iv)u\u2212(1) \u2264 u+(1).Representation Theorem 1 (see ). Let \u03bb =  for each \u03bb \u2208 . The fuzzy number u corresponding to the pair of functions u\u2212 and u+ is defined by u : R \u2192 , u(x) = sup\u2061{\u03bb : u\u2212(\u03bb) \u2264 x \u2264 u+(\u03bb)}.\u03bb-level set.Now we give the definitions of triangular fuzzy numbers with the \u03bcu) is interpreted, as follows:u]\u03bb : =  =  holds for each \u03bb \u2208 .The membership function u, v, w \u2208 E1 and \u03b1 \u2208 R. Then the operations addition, scalar multiplication, and product defined on E1 by u + v = w\u21d4[w]\u03bb = [u]\u03bb + [v]\u03bb\u2009for\u2009all\u2009\u2009\u03bb \u2208  then w\u2212(\u03bb) = u\u2212(\u03bb) + v\u2212(\u03bb)\u2009\u2009and\u2009\u2009w+(\u03bb) = u+(\u03bb) + v+(\u03bb)\u2009\u2009for\u2009all\u2009\u2009\u03bb \u2208 .Let W be the set of all closed bounded intervals A of real numbers with endpoints d on W by d is a metric on W  is a complete metric space, (cf. Nanda .A sequence (The following statements hold.un) \u2208 \u03c9(F) is equivalent to the fact thatuk) \u2208 \u03c9(F) is bounded then the sequences of functions {uk\u2212(\u03bb)} and {uk+(\u03bb)} are uniformly bounded in .The boundedness of the sequence (l\u221e(H), c(H), c0(H), and lp(H) consisting of the bounded, convergent, null, and p-summable sequences of fuzzy level sets with the partial metric Hs, as follows:l\u221e(H), c(H), and c0(H) are complete metric spaces with the partial metric H\u221e defined byu = (uk) and v = (vk) are the elements of the sets c(H), c0(H), or l\u221e(H). Also, the space lp(H) is complete metric space with the partial metric Hp defined byu = (uk) and v = (vk) are the points of lp(H).Following Kadak and Ozluk , we give\u03bc(H) denote any of the spaces bs(H), cs(H), and cs0(H), and u = (uk), v = (vk) \u2208 \u03bc(H). Define the partial distance function H\u221ep on \u03bc(H) by\u03bc(H), H\u221ep) is a complete metric space.Let cs(H) and cs0(H), we prove the theorem only for the space bs(H). Let u = (uk), v = (vk), and w = (wk) \u2208 bs(H). Then, (i)by using the axiom (P1) in  (ii)By using the axiom (P2) in  (iii)By using the axiom (P3) in  (iv)H \u2264 H + H \u2212 H and Hs = 0, we haveBy using the axiom (P4) in Therefore, one can conclude that (bs(H), H\u221ep) is a partial metric space on bs(H). It remains to prove the completeness of the space bs(H). \u2009Let (um) be any Cauchy sequence on bs(H), where um = {u1m) in Theorem 3.1 [ukm) = {uk(1), uk(2),\u2026} is a Cauchy sequence and is uniformly convergent. Now, we suppose that lim\u2061m\u2192\u221e\u2009ukm). We must show thatN \u2208 N for all m > N, taking the limit for r \u2192 \u221e in (k \u2208 N. Since (ukm) \u2208 bs(H), there exists a number k \u2208 N. Thus,  is bounded sequence of fuzzy numbers hence u \u2208 bs(H). Also, from  = 0. Since (um) is an arbitrary Cauchy sequence, bs(H) is complete.Since the proof is similar for the spaces orem 3.1 , we say m, r > NHs\u2211k=0nuk \u2192 \u221e in , Hq\u0394, and H\u221e\u0394 byu = (uk), v = (vk) are the element of the spaces bv(H), bvq(H), or bv\u221e(H), respectively. Then, (bv(H), H\u0394), (bvq(H), Hq\u0394), and (bv\u221e(H), H\u221e\u0394) are complete metric spaces.Define the distance functions bv(H) and bv\u221e(H), we prove the theorem only for the space bvq(H). One can easily establish that Hq\u0394 defines a metric on bvq(H). Let xi = {x0i). Then for every \u03f5 > 0, there exists a positive integer n0(\u03f5) \u2208 N for all i, j > n0, such thatx)n = xn \u2212 xn\u22121 and n \u2208 N from , which leads us to the fact that the sequence {(\u0394x)ni} is a Cauchy sequence and is convergent. Now, we suppose that (\u0394x)ni \u2192 (\u0394x)n as n \u2192 \u221e. We have from , thati > n0(\u03f5). Let firstly j \u2192 \u221e and nextly m \u2192 \u221e in  \u2264 \u03f5. Finally, by using Minkowski's inequality for each m \u2208 Nx \u2208 bvq(H). Since Hq\u0394 \u2264 \u03f5 for all i > n0(\u03f5), it follows that xi \u2192 x as i \u2192 \u221e. Since (xi) is an arbitrary Cauchy sequence, the space bvq(H) is complete. This step concludes the proof.Since the proof is similar for the spaces uch thatHq\u0394xi,xj: N from , that\u2211k=0mHs\u0394x\u03b1-duals. An account of the duals of sequence spaces can be found in K\u00f6the [The idea of dual sequence space, which plays an important role in the representation of linear functionals and the characterization of matrix transformations between sequence spaces, was introduced by K\u00f6the and Toeplitz , whose min K\u00f6the . One canin K\u00f6the .\u03bb(H), \u03bc(H), and S(\u03bb(H), \u03bc(H)) of sequences defined by\u03bb(H) and \u03bc(H) for all k \u2208 N. One can easily observe for a sequence set \u03bd(H) of fuzzy level sets that the inclusions\u03bb(H)}\u03b1, {\u03bb(H)}\u03b2, and {\u03bb(H)}\u03b3 of a set \u03bb(H) \u2282 \u03c9(F) are, respectively, defined bywkzk) the coordinatewise product of the sequences w and z of level sets for all k \u2208 N. Then {\u03bb(H)}\u03b2 is called \u03b2-dual of \u03bb(H) or the set of all factor sequences of \u03bb(H) are in cs(H). Firstly, we give a remark concerning with the convergence factor sequences of fuzzy level sets with partial metric.In this section, we focus on the alpha-, beta- and gamma-duals of the classical sets of sequences of fuzzy numbers with partial metric. For the sets \u2205 \u2260 \u03bb(H) \u2282 \u03c9(F). Then the following statements are valid.(a)\u03bb(H)}\u03b2 is a set of sequence and \u03c6(F)<{\u03bb(H)}\u03b2 < \u03c9(F) (\u201c<\u201d stands for \u201cis a linear subset of \u201d) where{(b)\u03bb(H) \u2282 \u03bc(H) \u2282 \u03c9(F) then {\u03bc(H)}\u03b2 < {\u03bb(H)}\u03b2.If (c)\u03bb(H)\u2282{\u03bb(H)}\u03b2\u03b2 : = ({\u03bb(H)}\u03b2)\u03b2.(d)\u03c6(F)}\u03b2 = \u03c9(F) and {\u03c9(F)}\u03b2 = \u03c6(F).{Let m = (mk) and n = (nk)\u2208{\u03bb(H)}\u03b2.(a)l \u2208 \u03bb(H). Then we get (mklk) \u2208 cs(H); (nklk) \u2208 cs(H) and (mk + nk)lk = (mklk)+(nklk) \u2208 cs(H). Since l is arbitrary, m + n \u2208 {\u03bb(H)}\u03b2. For any \u03b1 \u2208 R and w = (wk)\u2208{\u03bb(H)}\u03b2 we have\u03b1w \u2208 {\u03bb(H)}\u03b2. Therefore, {\u03bb(H)}\u03b2 is a linear subset of \u03c9(F).Let (d)\u03c9(F)}\u03b2 \u2282 \u03c6(F). Suppose that w = (wn)\u2208{\u03c9(F)}\u03b2 and z = (zn) be given with geometric division by zn : = (1/wn) if \u03c6(F) from the case (a), then there exists an integer N \u2208 N for all n \u2265 N such that wnzn) \u2208 cs(H) implies that w \u2208 \u03c6(F). The rest is an immediate consequence of this part; we omit the detail.Using (a) we need only show {Since the proof is trivial for conditions (b) and (c), we prove only (a) and (d). Let c0(H)}\u03b2 = {c(H)}\u03b2 = {l\u221e(H)}\u03b2 = l1(H).{l1(H)}\u03b2 = l\u221e(H).{The following statements hold. l\u221e(H)}\u03b2 \u2282 {c(H)}\u03b2 \u2282 {c0(H)}\u03b2 by l1(H)\u2282{l\u221e(H)}\u03b2 and {c0(H)}\u03b2 \u2282 l1(H). Now, consider w = (wk) \u2208 l1(H) and z = (zk) \u2208 l\u221e(H) are given. Thenwz \u2208 cs(H). So the condition l1(H)\u2282{l\u221e(H)}\u03b2 holds.(a) Obviously {y = (yk) \u2208 \u03c9(F)\u2216l1(H) we prove the existence of an x \u2208 c0(H) with yx \u2209 cs(H). According to y \u2209 l1(H) we may take an index sequence (np) which is a strictly increasing real valued sequence with n0 = 0 and x = (xk) \u2208 c0(H) by xk : = ((sgn\u2061yk)/p), where the real signum function defined byu = (uk) \u2208 E1, thus, we getnp\u22121 \u2264 k < np. Therefore yx \u2209 cs(H) and thus y \u2209 {c0(H)}\u03b2. Hence {c0(H)}\u03b2 \u2282 l1(H).Conversely, for a given l\u221e(H)\u2282({l\u221e(H)}\u03b2)\u03b2 = {l1(H)}\u03b2 since {l\u221e(H)}\u03b2 = l1(H). Now we assume the existence of a w = (wn)\u2208{l1(H)}\u03b2\u2216l\u221e(H). Since w is an unbounded sequence there exists a subsequence (wnk) of (wn) such that k \u2208 N1. The sequence (xn) is defined by xn : = (sgn\u2061(wnk)/(k + 1)2) if n = nk and x \u2208 l1(H). Howeverw \u2209 {l1(H)}\u03b2, which contradicts our assumption and {l1(H)}\u03b2 \u2282 l\u221e(H). This step completes the proof.(b) From the condition (c) of \u03b6-duals .Further to the statements in \u2205 \u2260 \u03bb(H) \u2282 \u03c9(F). Then the following statements are valid.(a)\u03c6(F)<{\u03bb(H)}\u03b1 < {\u03bb(H)}\u03b2 < {\u03bb(H)}\u03b3 < \u03c9(F); in particular, {\u03bb(H)}\u03b6 is a set of sequence.(b)\u03bb(H) < \u03bc(H) < \u03c9(F) then {\u03bc(H)}\u03b6 < {\u03bb(H)}\u03b6.If (c)I is an index set, if \u03bb(H)i are sets of sequences and if \u03bb(H): = \u22c3i\u2208I\u03bb(H)i, thenR.If (d)\u03bb(H)<{\u03bb(H)}\u03b6\u03b6 : = ({\u03bb(H)}\u03b6)\u03b6. If Let l\u221e(H) < cs(H) < bs(H). We only show conditions (c) and (d) taking \u03b6 = \u03b1. Other parts can be obtained in a similar way.(c)\u03bb(H)i \u2282 \u2329\u03bb(H)\u232a that the following conditionsNow, as an immediate consequence \u2009y \u2208 \u22c2i\u2208I{\u03bb(H)i}\u03b1, that is y \u2208 {\u03bb(H)i}\u03b1, then xy \u2208 l1(H) for all x \u2208 \u03bb(H) and therefore y \u2208 {\u03bb(H)}\u03b1 \u2282 \u2329\u03bb(H)\u232a\u03b1.hold by (b). On the other hand, if (d)\u03bb(H)<{\u03bb(H)}\u03b1\u03b1. Let w \u2208 \u03bb(H); then, wz \u2208 l1(H) for all z \u2208 {\u03bb(H)}\u03b1; thus, w \u2208 {\u03bb(H)}\u03b1\u03b1 and \u03bb(H)<{\u03bb(H)}\u03b1\u03b1 by (a).We prove Condition (b) is obviously true, and (a) follows from \u03bb(H)\u2260{\u03bb(H)}\u03b6\u03b6 as we get from \u03b6 = \u03b2 and \u03bb(H): = c0(H). We have {c0(H)}\u03b2\u03b2 = l\u221e(H) \u2260 c0(H). This remark gives rise to the following definition.In general \u03b6 \u2208 {\u03b1, \u03b2, \u03b3}, and let \u03bb(H) be a set of sequence. \u03bb(H) is called \u03b6-space if \u03bb(H) = {\u03bb(H)}\u03b6\u03b6. Further, an \u03b1-space is also called a K\u00f6the space or perfect sequence space.Let From Remarks 4.3(d) and (b) we obtain immediately the following remark.\u03bb(H) is a set of sequence over real field and , then {\u03bb(H)}\u03b6 is a \u03b6-space; that is, {\u03bb(H)}\u03b6 = {\u03bb(H)}\u03b6\u03b6\u03b6.If \u03bb(H)}\u03b1 = {\u03bb(H)}\u03b2 = {\u03bb(H)}\u03b3. This gives rise to the notion of solidity.Now we look for sufficient conditions for {\u03bb(H) be a set of sequence over the field R. Then \u03bb(H) is called solid ifLet \u03bb(H) < \u03c9(F) is any set of sequence over the field R; then, the following statements hold.\u03bb(H) is a K\u00f6the space, then \u03bb(H) is solid.If \u03bb(H) is solid, then {\u03bb(H)}\u03b1 = {\u03bb(H)}\u03b2 = {\u03bb(H)}\u03b3.If \u03bb(H) is a K\u00f6the space, then \u03bb(H) is a \u03b6-space.If Consider \u03bb(H) < \u03c9(H) be a set of sequence over the field R.(a)\u03bb(H) is a K\u00f6the space and u \u2208 \u03c9(F), then u \u2208 {\u03bb(H)}\u03b1\u03b1 if and only if the condition uz \u2208 l1(H) holds for all z \u2208 {\u03bb(H)}\u03b1. Besides this we obtain u = (uk) \u2208 \u03bb(H) and v = (vk) \u2208 \u03c9(F) and the statementz \u2208 {\u03bb(H)}\u03b1. Therefore vz \u2208 l1(H). Hence v \u2208 \u03bb(H) and \u03bb(H) is solid over the real field.If (b)\u03bb(H) is solid. To show {\u03bb(H)}\u03b1 = {\u03bb(H)}\u03b2 = {\u03bb(H)}\u03b3, it suffices to verify {\u03bb(H)}\u03b3 < {\u03bb(H)}\u03b1 as we have v = (vk)\u2208{\u03bb(H)}\u03b3; that is,\u03bb(H), for z = (zk) \u2208 \u03bb(H), where zk = uksgn\u2061(ukvk) and the condition u = (uk) \u2208 \u03bb(H) for all k \u2208 N. Therefore by combining this with the inclusion (uv \u2208 l1(H). Hence v \u2208 {\u03bb(H)}\u03b1 and {\u03bb(H)}\u03b3 < {\u03bb(H)}\u03b1.Consider that is,sup\u2061n\u2208N\u2211k(c)This is an obvious consequence of Let \u03c6(F), \u03c9(F), lp(H), c0(H), and l\u221e(H) of sequences are solid.The sets c(H) and bv(H) of sequences are not solid; therefore, none of them is a K\u00f6the space.The sets \u03b6 \u2208 {\u03b1, \u03b2, \u03b3}, thenFor each l1(H)}\u03b6 = l\u221e(H) and {l\u221e(H)}\u03b6 = l1(H),{\u03c9(F)}\u03b6 = \u03c6(F) and {\u03c6(F)}\u03b6 = \u03c9(F).{\u03b6 \u2208 {\u03b1, \u03b2, \u03b3} and c0(H) < \u03bc(H) < l\u221e(H), then {\u03bc(H)}\u03b6 = l1(H) and \u03bc(H)\u2282{\u03bc(H)}\u03b6\u03b6 = l\u221e(H). In particular {c0(H)}\u03b6 = {c(H)}\u03b6 = l1(H), and each of c(H), c0(H) is not a \u03b6-space.If The following statements hold.c0(H) and l\u221e(H) are solid, we know that {c0(H)}\u03b6 = {l\u221e(H)}\u03b6 = l1(H). So the statements in (d) obtain from Given specified sets are solid in (a) and (b) is an immediate consequence of their definition. Additionally, the parts (i) and (ii) of (c) can be obtained \u03b6-duals of the spaces cs(H), bs(H), bv(H), and bv0(H). We will find that none of these sets is solid; in particular, none of them is a K\u00f6the space.Next, we determine the cs(H)}\u03b1 = {bv(H)}\u03b1 = {bv0(H)}\u03b1 = {bs(H)}\u03b1 = l1(H).{cs(H)}\u03b2 = bv(H), {bv(H)}\u03b2 = cs(H), {bv0(H)}\u03b2 = bs(H), {bs(H)}\u03b2 = bv0(H).{cs(H)}\u03b3 = bv(H), {bv(H)}\u03b3 = bs(H), {bv0(H)}\u03b3 = bs(H), {bs(H)}\u03b3 = bv(H).{The following statements hold.cs(H), bs(H), bv(H), and bv0(H) of sequences are \u03b2-spaces, but they are not K\u00f6the spaces. Moreover, the sets bs(H) and bv(H) of sequences are \u03b3-spaces, whereas both cs(H) and bv0(H) are not \u03b3-spaces. None of the spaces cs(H), bs(H), bv(H), and bv0(H) is solid.In particular the sets cs(H)}\u03b6, \u03b6 \u2208 {\u03b1, \u03b2, \u03b3} and the proofs of all other cases are quite similar.(a)x = (xk) \u2208 cs(H) and y = (yk) \u2208 l1(H). Then,Let \u2009y \u2208 {cs(H)}\u03b1 which gives that l1(H)\u2282{cs(H)}\u03b1.Therefore, \u2009y = (yk)\u2208{cs(H)}\u03b1\u2216l1(H). Then we can construct an index sequence (np) with np < np+1 and x = (xk) byConversely, suppose that \u2009x = (xk) \u2208 cs(H). According to the choice of np the inequalitiesThen \u2009xy \u2209 l1(H), which implies y \u2209 {cs(H)}\u03b1. This contradicts that y \u2208 {cs(H)}\u03b1. Therefore {cs(H)}\u03b1 \u2282 l1(H).hold. Thus \u2009xk) byAs well if we take the sequence (We prove the cases for the spaces {\u2009bv0(H)}\u03b1 \u2282 l1(H) holds.the condition {(b)u = (uk)\u2208{cs(H)}\u03b2 and w = (wk) \u2208 c0(H). Define the sequence v = (vk) \u2208 cs(H) by vk = (wk \u2212 wk+1) for all k \u2208 N. Therefore, \u2211kukvk converges, butLet \u2009l1(H) \u2282 cs(H) yields that (uk)\u2208{cs(H)}\u03b2 \u2282 {l1(H)}\u03b2 = l\u221e(H). Then we derive by passing to the limit in (n \u2192 \u221e which implies thatand the inclusion limit in  as n \u2192 \u221e\u2009k \u2208 N. Hence (uk \u2212 uk\u22121)\u2208{c0(H)}\u03b2 = {c0(H)}\u03b1 = l1(H); that is, u \u2208 bv(H). Therefore, {cs(H)}\u03b2\u2286bv(H).for every \u2009u = (uk) \u2208 bv(H). Then, (uk \u2212 uk\u22121) \u2208 l1(H). Further, if v = (vk) \u2208 cs(H), the sequence (wn) defined by wn = \u2211k=0nvk for all k \u2208 N, is an element of the space c(H). Since {c(H)}\u03b1 = l1(H), the series \u2211kwk(uk \u2212 uk+1) is convergent. Also, we haveConversely, suppose that \u2009wn) \u2208 c(H) and (uk) \u2208 bv(H) \u2282 c(H), the right-hand side of inequality uk or \u2211k=0\u221eukvk converges and bv(H)\u2286{cs(H)}\u03b2.Since bv(H)\u2286{cs(H)}\u03b2 and since {cs(H)}\u03b2 \u2282 {cs(H)}\u03b3, so bv(H)\u2282{cs(H)}\u03b3. We need to show that {cs(H)}\u03b3 \u2282 bv(H). Let u = (un)\u2208{cs(H)}\u03b3 and v = (vn) \u2208 c0(H). Then, for the sequence (wn) \u2208 cs(H) defined by wn = (vn \u2212 vn+1) for all n \u2208 N, we can find a number K > 0 such that n \u2208 N. Since (vn) \u2208 c0(H) and (un)\u2208{cs(H)}\u03b3 \u2282 l\u221e(H), there exists a real number M > 0 such that n \u2208 N. Therefore,By using (a), it is known that \u2009uk \u2212 uk\u22121)\u2208{c0(H)}\u03b3 = {c0(H)}\u03b1 = l1(H); that is, (un) \u2208 bv(H). Therefore, since the inclusion {cs(H)}\u03b3 \u2282 bv(H) holds, we conclude that {cs(H)}\u03b3 = bv(H), as desired.Hence (Partial metrics are more flexible than metrics; they generate partial orders and their topological properties are more general than the one for metrics, argued by the fact that the self-distance of each point need not be zero. They are useful in partially defined information for the study of domains and semantics in computer science.The concept of level sets associated with a fuzzy set was originally introduced by Zadeh. With the aid of level sets we are able to provide a formulation for a fuzzy set in terms of crisp subsets via the representation theorem. The importance of having such a representation is that it can allow us to extend operations defined on crisp sets to the case of fuzzy sets. Our focus here is on using the idea of level sets to construct the sets of sequences of fuzzy numbers within partial metric spaces.This work presents the alpha-, beta-, and gamma-duals of the sets of bounded, convergent, and null series and the set of sequences of bounded variation of fuzzy level sets, based on the partial metric. The potential applications of the obtained results include the characterization of matrix transformations between these sets of sequences."}
{"text": "We shall explore a nonlinear discrete dynamical system that naturally occursin population systems to describe a transmission of a trait from parents to their offspring. We considera Mendelian inheritance for a single gene with three alleles and assume that to form a new generation,each gene has a possibility to mutate, that is, to change into a gene of the other kind. We investigatethe derived models and observe chaotic behaviors of such models. Recently, chaotic dynamical systems become very popular in science and engineering. Besides the original definition of the Li-Yorke chaos , there hIn this paper, we introduce and examine a family of nonlinear discrete dynamical systems that naturally occurs to describe a transmission of a trait from parents to their offspring. Here, we shall present some essential analytic and numerical results on dynamics of such models.In , it was A and a  represent a gene pool for a population; that is, x1, x2 are the percentage of the population which carries the alleles A and a, respectively. For the convenience, we express it as a linear combination of the alleles A and ax1, \u2009x2 \u2264 1 and x1 + x2 = 1. The rules of the Mendelian inheritance indicate that the next generation has the following form:As the first example, we consider a Mendelian inheritance of a single gene with two alleles d a (see ). Let anPAA,A  is the probability that the child receives the allele A  from parents with the allele A; PAa,A  is the probability that the child receives the allele A  from parents with the alleles A and a, respectively; and Paa,A  is the probability that the child receives the allele A  from parents with allele a. It is evident thatHere, A and a, respectively, if the distribution of the current generation is known.Thus, the evolution  is a nonPAA,A = Paa,a = 1 and PAA,a = Paa,A = 0, the dynamical system ( = V(xn\u22121), is called a trajectory of V starting from an initial point x0. Recall that a point x is called a fixed point of V if V(x) = x. We denote a set of all fixed points by Fix\u2061(V). A dynamical system V is called regular if a trajectory {xn) were deeply studied in \u201316 and qV : X \u2192 X is said to be ergodic if the limitx \u2208 X.A dynamical system V0 : S2 \u2192 S2 is not ergodic:Based on some numerical calculations, Ulam has conjectured  that anyIn , Zakharekth order Cesaro mean by the following formula:k \u2265 1 and Ces0n) = Vn(x). It is clear that the first order Cesaro mean Ces1n) is nothing but (1/n)\u2211i=0n\u22121Vi(x). Based on these notations, Zakharevich's result says that the first order Cesaro mean {Ces1n)}n=0\u221e of the trajectory of the operator V0 given by (kn)}n=0\u221e, for any k \u2208 N, of the trajectory of the operator V0 diverges for any initial point except fixed points. This leads to a conclusion that the operator V0 might have unpredictable behavior. In fact, in [V0 exhibits the Li-Yorke chaos. It is worth pointing out that some strange properties of Volterra QSO were studied in [We define the given by  divergesfact, in , it was udied in , 25. any non Volterra QSO acting on the finite dimensional simplex is ergodic, that is, operators having chaotic behavior can be only found among Volterra QSO. However, in this paper, we are aiming to present the continual family of nonergodic and chaotic QSO which are non Volterra QSO.In the literature, all examples of nonergodic QSO have been found in the class of Volterra QSO see , 20, 21), 2120, 2 a single gene with two alleles always exhibits an ergodic behavior . It is of independent interest to study the evolution of a mutation in population system having a single gene with three alleles. In the next section, we consider an inheritance of a single gene with three alleles a1, a2, and a3 and show that a nonlinear dynamical system corresponding to the mutation exhibits a nonergodic and Li-Yorke chaotic behavior.Note that if QSO is regular, then it is ergodic. However, the reverse implication is not always true. It is known that the dynamical system  is eitheIn this section, we shall derive a mathematical model of an inheritance of a single gene with three alleles.x represents a linear combination x = x1a1 + x2a2 + x3a3 of the alleles a1, a2, and a3 in which the following conditions are satisfied 0 \u2264 x1, x2, x3 \u2264 1 and x1 + x2 + x3 = 1, that is, x1, x2, x3 are the percentages of the population which carry the alleles a1, a2, and a3 respectively.As it was in the previous section, an element  both parents have the same alleles. Specifically, we will consider two types of the simplest mutations; assume thata1 \u2192 a2, a2 \u2192 a3, and a3 \u2192 a1 occur with probability \u03b1;mutations a1 \u2192 a3, a3 \u2192 a2, and a2 \u2192 a1 occur with probability \u03b1.mutations We assume that prior to a formation of a new generation each gene has a possibility to mutate, that is, to change into a gene of the other kind. We assume that the mutation occurs ifIn this case, the corresponding dynamical systems are defined on the two-dimensional simplexLet us first recall the definition of the Li-Yorke chaos , 3, 4.X, d) be a metric space. A continuous map V : X \u2192 X is called Li-Yorke chaotic if there exists an uncountable subset S \u2282 X such that for every pair  \u2208 S \u00d7 S of distinct points, we have thatS is a scrambled set and  \u2208 S \u00d7 S is a Li-Yorke pair.Let V\u03b1, W\u03b1  systems  and (13) systems . As we a\u03b1 = 1. In the first case, the operator V1 is a permutation of Zakharevich's operator V0 + \u03b1V1 and W\u03b1 = (1 \u2212 \u03b1)W0 + \u03b1W1.It is easy to check that V\u03b1 is a convex combination of two Li-Yorke chaotic operators V0, V1, meanwhile, in the second case, the evolution operator W\u03b1 is a convex combination of the Li-Yorke chaotic and regular operators W0, W1. These operators V\u03b1, W\u03b1 were not studied in [V\u03b1 and W\u03b1. The reason is that, in the first case, the convex combination presents a transition from one chaotic biological system to another chaotic biological system ; meanwhile, in the second case, the convex combination presents a transition from the ordered biological system to the chaotic biological system. In the next section, we are going to present some essential analytic and numerical results on dynamics of the operators V\u03b1 and W\u03b1 given by V\u03b1 is a V\u03b1 : S2 \u2192 S2:V\u03b1(x) = x\u2032 =  and 0 < \u03b1 < 1. As we already mentioned, this operator can be written in the following form: V\u03b1 = (1 \u2212 \u03b1)V0 + \u03b1V1 for any 0 < \u03b1 < 1, whereWe are aiming to present some analytic results on dynamics of LetV\u03b1 : S2 \u2192 S2 be the evolution operator given by . Let Fix(V\u03b1) and \u03c9(x0) be sets of fixed points and omega limiting points of\u2009\u2009V\u03b1, respectively. Then the following statements hold true.P and V\u03b1 are commutative, that is, P\u2218V\u03b1 = V\u03b1\u2218P.Operators x \u2208 Fix\u2061(V\u03b1) then Px \u2208 Fix\u2061(V\u03b1).If V\u03b1) is a finite set then |Fix\u2061(V\u03b1)| \u2261 1(mod\u20613).If Fix\u2061(P(\u03c9(x0)) = \u03c9(Px0), for any x0 \u2208 S2.One has that Let given by , where \u03b1V\u03b1), where \u03b1 \u2208 . It is worth mentioning that Fix\u2061(V0) = {e1, e2, e3, C} and Fix\u2061(V1) = {C}, where e1 = , e2 = , and e3 =  are vertices of the simplex S2 and C =  is a center of the simplex S2.We are aiming to study the fixed point set Fix\u2061(x0 \u2208 Fix\u2061(V\u03b1) is nondegenerate [x0:Recall that a fixed point generate  if and oV\u03b1 : S2 \u2192 S2 be the evolution operator given by . Let C =  be a center of the simplex S2. Then the following statements hold true. All fixed points are nondegenerate.Fix(V\u03b1) = {C} for any \u03b1 \u2208 .One has that Let given by , where \u03b1x \u2208 Fix\u2061(V\u03b1) be a fixed point. One can easily check that(i) Let \u03b1 < 1, then the above expression is positive. Therefore, all fixed points are nondegenerate.If 1/2 \u2264 \u03b1 < 1/2. In this case, the above expression is equal to zero if and only if x1x2 + x1x3 + x2x3 = (1 \u2212 2\u03b1)/(4(1 \u2212 \u03b1 + \u03b12)). Since x1 + x2 + x3 = 1, we have that x12 + x22 + x32 = (1 + 2\u03b12)/(2(1 \u2212 \u03b1 + \u03b12)).Let 0 < x1 \u2265 max\u2061{x2, x3} (See x2 \u2265 x3. Since x \u2208 Fix\u2061(V\u03b1), we have that x2 = (1 \u2212 \u03b1)x22 + 2x2x3 + \u03b1x32. We then obtain thatx3 \u2265 x2. This shows that, in the case 0 < \u03b1 < 1/2, all fixed points are nondegenerate.Without loss of generality, we may assume that V\u03b1) = {C}. The simple calculation shows that C \u2208 Fix\u2061(V\u03b1). It is clear that V\u03b1(\u2202S2) \u2282 int\u2061S2. This means that the operator V\u03b1 does not have any fixed point on the boundary \u2202S2 of the simplex S2, that is, Fix\u2061(V\u03b1)\u2229\u2202S2 = \u2205. Moreover, all fixed points are nondegenerate. Due to Theorem 8.1.4 in [V\u03b1)| should be odd. On the other hand, due to Corollary 8.1.7 in [V\u03b1)| \u2264 4. In V\u03b1)| = 1. Therefore, we get that Fix\u2061(V\u03b1) = {C}.(ii) We shall show that Fix\u2061 is as follows.A local behavior of the fixed point V\u03b1 : S2 \u2192 S2 be the evolution operator given by . Then the following statements hold true.\u03b1 \u2260 1/2, then the fixed point C =  is repelling.If \u03b1 = 1/2, then the fixed point C =  is nonhyperbolic.If Let given by , where \u03b1x1 + x2 + x3 = 1, the spectrum of the Jacobian matrix of the operator V\u03b1 : S2 \u2192 S2 at the fixed point C =  must be calculated as follows:\u03b1 \u2260 1/2, then the fixed point C =  is repelling and if \u03b1 = 1/2, then the fixed point C =  is nonhyperbolic. This completes the proof.It is worth mentioning that, since \u03b1 \u2260 1/2 and \u03b1 = 1/2.We shall separately study two cases V\u03b1 : S2 \u2192 S2 be the evolution operator given by (\u03b1 \u2260 1/2. Then \u03c9V\u03b1(x0) \u2282 intS2 is an infinite compact subset, for any x0 \u2260 C.Let given by , where \u03b1\u03b1 \u2260 1/2. Since V\u03b1 is continuous and V\u03b1(S2) \u2282 int\u2061S2, an omega limiting set \u03c9(x0) is a nonempty compact set and \u03c9(x0) \u2282 int\u2061S2, for any x0 \u2260 C. We want to show that \u03c9(x0) is infinite, for any x0 \u2260 C. Since C is repelling, we have that C \u2209 \u03c9(x0). Let us pick up any point x* \u2208 \u03c9(x0) from the set \u03c9(x0). Since the operator V\u03b1 does not have any periodic point, the trajectory {V\u03b1n)((x*)}n=1\u221e of the point x* is infinite. Since V\u03b1 is continuous, we have that {V\u03b1n)((x*)}n=1\u221e \u2282 \u03c9(x0). This shows that \u03c9(x0) is infinite for any x0 \u2260 C.Let \u03c9V0(x0) and \u03c9V1(x0) of the operators V0 and V1 are infinite. However, unlike the operator V\u03b1, we have inclusions \u03c9V0(x0)\u2282\u2202S2 and \u03c9V1(x0)\u2282\u2202S2. Moreover, both operators V0 and V1 are nonergodic [It is worth mentioning that the sets of omega limiting points nergodic , 27.V\u03b1 : S2 \u2192 S2 given by  is nonhyperbolic and the spectrum of the Jacobian matrix of the operator V1/2 at the fixed point C, calculated by  = |x1 \u2212 x2||x1 \u2212 x3||x2 \u2212 x3| is a Lyapunov function.C = .Every trajectory converges to the fixed point Let given by . The folV1/2 be an operator given by (\u03d5(V1/2(x)) \u2264 \u03d5(x), for any x \u2208 S2. This means that \u03d5 is decreasing a long the trajectory of V1/2. Consequently, \u03d5 is a Lyapunov function.(i) Let given by . It follgiven by  that(25\u03d5(V1/2n)((x))}n=1\u221e is a decreasing bounded sequence. Therefore, the limit lim\u2061n\u2192\u221e\u2061\u03d5(V1/2n)((x)) = \u03bb exists. We want to show that \u03bb = 0. Suppose that \u03bb \u2260 0. It means that \u03bb \u2260 0, we get that(ii) We know that {\u025b0 such that for any n one has thatOn the other hand, since \u03bb = 0.This is a contradiction. It shows that \u03c9(x0) \u2282 l1 \u222a l2 \u222a l3. We want to show that \u03c9(x0) = l1\u2229l2\u2229l3.Therefore, We know that ows from  that(29V\u03b1 : S2 \u2192 S2 given by (We are going to present some pictures of attractors (an omega limiting set) of the operator given by .\u03b1 = 0 and \u03b1 = 1, the corresponding operators V0, V1 have similar spiral behaviors which reel along the boundary of the simplex [\u03c9V0(x0)\u2282\u2202S2 and \u03c9V1(x0)\u2282\u2202S2.In the cases  simplex , 20. HowV\u03b1 while \u03b1 approaches to 1/2 from both left and right sides. In order to see some antisymmetry, we shall provide attractors of V\u03b1 and V\u03b11\u2212 at the same time.We are interested in the dynamics of the evolution operator \u03b1 is an enough small number, then we can see that the omega limiting sets of operators V\u03b1 and V\u03b11\u2212 are separated from the boundary \u2202S2 W0 + \u03b1W1, for any 0 < \u03b1 < 1, whereW0 = V0 is Zakharevich's operator  which is attracting.The operator e1, e2, e3 are 3-periodic points.The vertexes of the simplex \u03d5(x) = x12 + x22 + x32 \u2212 1/3 is a Lyupanov function.W1 is regular in the set intS2.The operator Let V\u03b1, we can prove the following results.By means of the same methods and techniques which are used for the operator W\u03b1 : S2 \u2192 S2 be the evolution operator given by ; that is, Fix(W\u03b1) = {C}. Moreover, one has thatif if if Let given by . Then itW\u03b1 : S2 \u2192 S2 be the evolution operator given by (\u03c9(x0) \u2282 intS2 is an infinite compact set, for any x0 \u2260 C.If \u03c9(x0) = {C}, for any x0 \u2208 S2.If Let given by . Then thW\u03b1 : S2 \u2192 S2 given by  of the operator given by .\u03b1 = 0 and \u03b1 = 1, the operator W0 is chaotic and the operator W1 is regular. Since W\u03b1 = (1 \u2212 \u03b1)W0 + \u03b1W1, the evolution operator W\u03b1 gives the transition from the regular behavior to the chaotic behavior. Consequently, we are aiming to find the bifurcation point in which we can see the transition from the regular behavior to the chaotic behavior.In the cases \u03b1 is a very small number then attractors of the operator W\u03b1 are separated from the boundary of the simplex  in the system can be considered as a transition between two different types of systems having Mendelian inheritances. Namely, the first mutation presents the transition between two chaotic biological systems; meanwhile the second mutation presents the transition between regular and chaotic systems.V\u03b1 : S2 \u2192 S2 given by \u2282\u2202S2 and \u03c9V1(x0)\u2282\u2202S2. If \u03b1 is an enough small number then we observed that the omega limiting sets of operators V\u03b1 and V\u03b11\u2212 are separated from the boundary \u2202S2  chaotic systems, at some point of the time, the system should become stable.In the first mutation, we have presented some pictures of attractors of the operator given by . In the \u2202S2 see . If \u03b1 beW\u03b1 : S2 \u2192 S2 given by  and regular transformations, it is natural to expect the bifurcation scenarios in this evolution. Namely, in order to have a transition from regular to chaotic behavior we have to cross from the bifurcation point. Numerical result \u03b1\u22cd0.13397 also confirms the theoretical result about the exact value of bifurcation point. However, the biological plausiblity of this value is unknown for the authors.In the second mutation, we have presented some pictures of attractors of the operator given by . In the plex see . HoweverIn this paper, we have considered two types of mutations of three alleles which occurred with the same probability. It is natural to consider mutations with different probabilities among alleles. In this case, it is expected to have more complicated dynamics in the biological system. The future research is to study the dynamics of the mutated biological system having a single gene with a finite number of alleles."}
{"text": "This paper presents new structural and algorithmic results around the scaffolding problem, which occurs prominently in next generation sequencing. The problem can be formalized as an optimization problem on a special graph, the \"scaffold graph\". We prove that the problem is polynomial if this graph is a tree by providing a dynamic programming algorithm for this case. This algorithm serves as a basis to deduce an exact algorithm for general graphs using a tree decomposition of the input. We explore other structural parameters, proving a linear-size problem kernel with respect to the size of a feedback-edge set on a restricted version of Scaffolding. Finally, we examine some parameters of scaffold graphs, which are based on real-world genomes, revealing that the feedback edge set is significantly smaller than the input size. During the last decade, a huge amount of new genomes have been sequenced, leading to an abundance of available DNA resources. Nevertheless, most of recent genome projects stay unfinished, in the sense that databases contain much more incompletely assembled genomes than whole stable reference genomes . One reamulation , nearly mulation , howeverscaffold graph, representing the link between already assembled contigs. The main idea is to represent each contig by two vertices linked by an edge (these \"contig-edges\" form a perfect matching on the scaffold graph). Other edges are constructed and weighted using complementary information on the contigs. The weight of a non-contig edge uv, with uu\u2032, vv\u2032 being contig-edges, corresponds to a confidence measure that the uu\u2032 contig is succeeded by the vv\u2032 contig (oriented as u\u2032 \u2212 u \u2212 v \u2212 v\u2032). The scaffold graph is a flexible tool to study the scaffolding issues. Indeed, the graph is a syntactical data-structure which may represent several semantics. For instance, the scaffold graphs used for our previous experiments have been built using Next-Generation Sequencing data, namely paired-end reads. However, we also could provide other type of information to compute the weight on the edges, like for instance ancestral support in a phylogenetic context, or comparison to other extent genomes which could be used as multiple references. The way to define the weight on the edges does not change the main goal of our method, which is to determine the optimal ordering and orientation of the contigs, given a specific criterion. It is also possible to mix two or more criteria in order to take several sources of information into account.The approach presented here relies on a combinatorial problem on a dedicated graph, called \u03c3p and \u03c3c representing the respective numbers of linear and circular chromosomes sought in the genome. Those parameters seems quite artificial at a first sight, but they are well-motivated as follows. First, in a number of species, genomes are hard to assemble with classical methods because of an heterogeneous chromosomal structure or difficulties to perform classical assembly. This is the case, for instance, with the micro-chromosomes in the chicken genome  denote the subgraph of G induced by V\u2032 and let G \u2212 V\u2032 := G[V \\ V\u2032]. For S \u2286 E, let G \u2212 S :=  and, for any edge or vertex x, we abbreviate G \u2212 {x} =: G \u2212 x. For a set of pairs S , we let Gr(S ) denote the graph having S as edgeset, that is, Gr(S ) := . For a function \u03c9 : E \u2192 \u2115 and a set S \u2286 E, we abbreviate \u03a3e\u2208S \u03c9(e) =: \u03c9(S ). Let G =  be a graph with a matching M and let S be a matching in G \u2212 M. The number of paths (resp. cycles) in G[S \u222a M] is denoted by G[S \u222a M] are alternating with respect to M, then, we call S a ||S||p-||S||c-cover (or simply cover) for G with respect to M (we will omit M if its clear from context). If ||S||c = 0, we may also refer to S as a ||S||p-path cover (or simply path cover). The general scaffold problem is expressed as follows see also ,17): SCAInput: A scaffold graph , M), \u03c3p \u2208 \u2115, \u03c3c \u2208 \u2115, k \u2208 \u2115Question: Is there a \u03c3p-\u03c3c-cover S for G with respect to M with \u03c9(S ) \u2265 k?Tree Decompositions. A tree decomposition of a graph G =  is a pair , X : T V\u2192 2V ) such that (1) for all uv \u2208 E, there is some i \u2208 T Vwith uv \u2286 X(i) and (2) for all v \u2208 V, the subtree v T:= T[{X(i) | v \u2208 X(i)}] is connected. We call the images of X \"bags\" and the size of the largest bag minus one is the width of the decomposition. A decomposition of minimum width for G is called the optimal for G and its width is called the treewidth of G. It is a folklore theorem that each graph G has an optimal tree decomposition  that is nice, that is, each bag X(i) is one of the following types:Leaf bag: i is a leaf of T and X(i) = \u2205,Introduce vertex v bag: i is internal with child j and X(i) = X( j) \u222a {v} with v \u2209 X(j),Forget v bag: i is internal with child j and X(i) = X(j) \u2212 v with v \u2208 X(i),Introduce edge uv bag: i is internal with child j and uv \u2208 E and uv \u2286 X(i) = X(j),Join bag: i is internal with children j and \u2113 and X(i) = X(j) = X(\u2113).e \u2208 E is introduced exactly once. Given a width-tw tree decomposition, we can obtain a nice tree decomposition of width tw in n \u00b7 poly(tw) time  and u T+v := T [V(uT)\u222a{v}]. Let v \u2208 V and 1 \u2264 j \u2264 |C(v)|. Then, we define vs := vs[1..0] := \u22a5.In the following, for a vertex Algorithm 1: Algorithm to fill the dynamic programming table.1I \u2190 leaves of T;2 while \u2203v \u2208 V \\ I s.t. C(v) \u2286 I do3 \u00a0\u00a0\u00a0\u00a0foreach 1 \u2264 j \u2264 |C(v)| with u := vs[j] do4 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0foreach i \u2264 \u03c3p do5 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0if uv \u2208 M then6 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0v \u2190 max\u03b1\u2208{0,1} maxi\u2212(1\u2212\u03b1)\u2113\u2264u + v;7 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0v \u2190 max\u03b1\u2208{0,1} maxi\u2113\u2264u + v;8 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0else9 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0v \u2190 maxi \u2113\u2264max\u03b1\u2208{0,1}u + v;10\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a011 if v = r then return maxc\u2208{0,1}relse I \u2190 I \u222a {v};Semantics: For  \u2208 {0, 1} \u00d7 [\u03c3p] \u00d7 [|C(v)|] \u00d7 V we let v denote the maximum weight of an i-0-cover S for v := v.I of initialized vertices and, as soon as r is initialized, the algorithm stops. Thus, we assume r \u2209 I. Finally, the maximum weight of a \u03c3p-0-cover for T can be computed by maxc\u2208{0,1}r.We maintain a set Lemma 1 Algorithm 1 is correct, that is, for all  \u2208 [\u03c3p] \u00d7 [|C(v)|] \u00d7 {0, 1} \u00d7 V and for any maximum-weight i-0-cover S for v = \u03c9(S).\u03c3p times per vertex. As they compute the maximum over \u03c3p values, the whole algorithm runs in Concerning the running time, we note that the body of the loop in line 3 is executed exactly once per vertex and, hence, lines 6,7,9, and 10 are executed Corollary 1 SCAFFOLDINGon trees can be solved intime.SCAFFOLDING in O(twtw) poly time. It is based on using a dynamic programming table to keep track of solutions that interact with the bags of a tree decomposition in a certain way  :=Gr(P2). For permutations P and Q, we define P \u25a1 Q as the set of pairs uv such that P(x) = \u22a5 \u2295 Q(x) = \u22a5 for all x \u2208 uv and there is a u-v-path in Gr(P \u222a Q). Furthermore, for a function d : A \u2192 B and  \u2208 A \u00d7 B, we define d[x \u2192 y] as the result of setting d(x) := y (that is (d \\ ({x} \u00d7 B)) \u222a {}). Here, d[x \u2192 \u22a5] means to remove x from the domain of d. Let T =  be a tree decomposition of G with root X(r) \u2208 \u03c7. For a bag X(i), let iGdenote the subgraph of G that contains exactly those edges of G that are introduced in a bag of the subtree of T that is rooted at X(i) and let To present our algorithm, we use special permutations (involutions) to model matchings that allow reflexive pairing . Thus, slightly abusing notation, we will consider permutations as sets of pairs. We denote the subset of reflexive pairs of a permutation X(i) will be indexed by (i) a function d : X(i) \u2192 {0, 1, 2}, (ii) a permutation P with Uuv\u2208P uv = d\u22121(1) and(iii) integers p \u2264 \u03c3p and c \u2264 \u03c3c. See Figure d and P. An entry will have the following semantics:A table entry for the bag Definition 2 Let i \u2208 T. We call a set Si V\u2286 E(iG) \\ M eligible with respect to a tuple  if1 each vertex v \u2208 X(i) has degree d(v) in 2 for each uv \u2208 P, if u \u2260 v, then there is a u-v-path in and, if u = v, then there is a path q of non-zero-length in \u22121(1) \u2212 u (we say that q is dangling from u).3 contains p paths and c cycles that do not intersect d\u22121(1),Semantics: A table entry i is the maximum weight of any set that is eligible with respect to .S for G from r.Then, we can read the maximum weight of a solution X(i) with children X(j) and X(\u2113) (possibly j = \u2113), we compute i depending on the type of the bag X(i) (entries that are not mentioned explicitly are set to \u2212\u221e):Given a nice tree decomposition and a bag Leaf bag: Set i := 0.Introduce vertex v: Newly introduced vertices do not have introduced edges yet. Thus, we force the degree of v in d, P, p, c]i := [d[v \u2192 \u22a5], P, p, c] jif d(v) = 0 and \u221e, otherwise.Forget vertex v: A vertex v that we forget in bag X(i) can have degree 0,1, or 2 in q dangling from it. If q ends in some other vertex u \u2208 X(i) \\ {v} = X( j), then, the permutation for X(i) contains uu and the permutation for X(j) contains uv. Otherwise, q is dangling from v in X(j) contains vv. Formally, letIntroduce edge uv: Let d(u), d(v) \u2265 1 and, by symmetry, let d(u) \u2265 d(v) \u2265 1. We define a value z (representing the assumption that uv is in S ) as follows. Let d\u2032 := d, that is, we let d\u2032 be the result of decreasing both d(u) and d(v) by one.Case 1 d(u) = d(v) = 2. Then, P avoids u and v. Since we assume uv \u2208 S , this means that u and v have dangling paths u qand v qin d\u2032\u22121(1). If u q= vq, then adding uv to S closes a cycle in X(j) contains uv ). Otherwise, uq\u2260 vq. Then, if u qintersects d\u2032\u22121(1) \\ {u, v} in a vertex x, then the permutation for X(j) contains ux  and Figure 3(d)), otherwise, it contains uu ). Likewise, if v qintersects d\u2032\u22121(1) \\ {u, v} in a vertex y, then the permutation for X(j) contains vy, otherwise, it contains vv. Note that, if both u qand v qintersect d\u2032\u22121(1) \\ {u, v} ), then we have xy \u2208 P. Note further that, if neither u qnor v qintersects d\u2032\u22121(1) \\ {u, v} ), then u q\u222a v q\u222a {uv} is a path in d\u22121(1) and, thus, we have to decrease the number p of such paths we are looking for in Case 2: d(u) = d(v) = 1. Then, both u and v are not incident to any edges in uv incident to both. Thus, we set z only if uv \u2208 P. Formally, let z := j if uv \u2208 P.Case 3: d(u) = 2, d(v) = 1. Then, there is a path q containing uv and ending in v in q ends in a vertex x in d\u22121(1) \u2212 v, we have vx \u2208 P and the permutation for X(j) contains ux. Otherwise, we have vv \u2208 P and the permutation for X(j) contains uu. Note that, since v \u2208 d\u22121(1), we know that P(v) \u2260 \u22a5. Formally, for vx \u2208 P, let z := j, if v = x and z := j, otherwise. Finally, let i := z if uv \u2208 M and i := max{z + \u03c9(uv), j}, otherwise.Join: The join bag X(i) \"glues\" the (disjoint) partial solutions of its children together at the vertices of X(i) = X( j) = X(\u2113). In particular, the degrees in d. Furthermore, the permutations P1 and P2 for X(j) and X(\u2113), respectively, have to \"fit\" P: For example, let uv \u2208 P1 and vw \u2208 P2, implying that there are paths j qand \u2113 qin u and v and v and w, respectively. Then, in j q\u222a q\u2113 connecting u and w and containing v (with d(v) = 2). Finally, the numbers of paths and cycles have to \"fit\" p and c: For example, if the permutations for both X(j) and X(\u2113) contain uu 1), then u that is neither in Lemma 2 The described algorithm is correct, that is, the computed value i corresponds to the semantics.Theorem 1 SCAFFOLDING can be solved in O(twtw \u00b7\u03c3p \u00b7 \u03c3c \u00b7 n) time, given a width-tw tree decomposition of the input instance.SCAFFOLDING, we consider a more restricted problem variant, where all paths and cycles of the solution have to be of certain, respective lengths. SCAFFOLDING. Let G\u2217 = G[V\u2217] denote the convex hull of \u25cbG, that is, V\u2217 is the set of vertices on some shortest path between some vertices u, v \u2208 \u25cbV. For each v \u2208 V\u2217, the tree rooted at v that is incident with v in G \u2212 E\u2217 is called the \"pendant tree\" vT of v.For a p. Since the correctness of this is trivial, we omit the proof.First, we remove isolated paths by cutting them into pieces of length \u2113Tree Rule 1  by ). Otherwise, return \"NO\".p that is furthest from the root of the pendant tree.The next rule cuts branching edges in pendant trees. To this end, it finds an occurrence of a path of length \u2113Tree Rule 2 v. Then, delete from G all edges that are incident with the least common ancestor of u and w but are not on the unique u-w-path in T.Lemma 3 Let I =  be a yes-instance of RESTRICTED SCAFFOLDING such that G is reduced with respect to Tree Rule 2 and let v \u2208 V\u2217v is a path p that is alternating with respect to M . Then, T\u2229 E(vT) and |p| < \u2113p and v is an endpoint of p.G is reduced with respect to Tree Rule 2 and, thus, we can reject all instances for which Lemma 3 does not hold. Hence, in the following, we assume that Lemma 3 holds for the input instance. The next reduction rule helps unify the way in which pendant trees (which are now paths) attach to G\u2217, simplifying the rest of the presentation.In the following, we assume that Tree Rule 3  \\ M. Then, delete from G all edges incident with v that are not in M + e.G is reduced with respect to the tree reduction rules presented in the previous section. For i \u2208 \u2115, let i Vdenote the set of vertices of G that have degree i in G. The goal in this subsection is to reduce the length of chains of degree-two vertices in G. Thus, we consider paths whose inner vertices are all in V2. We call these paths deg-2 path. If the solution has a cycle running through this path, then we cannot modify its length. Therefore, we focus on paths that are guaranteed to not be in a cycle in the solution:In the following, we assume that p in G is split into by the solution are recurring, that is, if the solution contains the third edge of p, then it also contains the 3 + (\u2113p + 1)th edge of p. This gets slightly more complicated if there are pendants in p, but first, let us consider paths without any pendants.The path reduction rules are based on the idea that the path segments that a deg-2 path Path Rule 1  be a deg-2 path with |p| > max{\u2113p, \u2113c} + \u2113p + 1 i and let e:= iui+1 for all i. Then, for all ei with i u\u2264 \u2113p, add \u03c9(ie) to \u03c9(ei+1\u2113p +) i. Finally, decrease \u03c3and contract ep by 1.The remaining two rules deal with deg-2 paths containing pendants. Note that any path in the solution that contains the pendant can only \"continue\" in two directions.Path Rule 2  and q =  w contain be deg-2 paths and let T\u03b3 > 0 edges. For all i \u2264 \u2113pi , let e:= iui+1, let fi u:= ivi+1 and let iv+ := (i \u2212 \u03b3) mod (\u2113p + 1). Then, for all i \u2264 \u2113p, add \u03c9(ie) to \u03c9( i+f) i. Finally, decrease \u03c3and contract ep by 1.In case of two pendants, the solution is restricted by the distance between the pendants. If we can infer what the solution does by this length, we implement this right away, otherwise, we can represent the choices that a solution can take with a single pendant.Path Rule 3  be a u-v-deg-2 path such that u, v \u2208 V\u2217. Let \u03b3u > 0 and \u03b3v > 0 u and Tv, respectively, and let w be the vertex at maximum distance to u in Tu. Let be the number of edges in T\u03b3 := |p| mod (\u2113p + 1) and let G\u2032 be the result of replacing ux1 by wx1 of the same weight in G.1) If \u03b3 + \u03b3u\u2260 f \u2113p + 1 = \u03b3 + \u03b3v, then delete ux1  If \u03b3 + \u03b3u = \u2113p + 1 \u2260 f \u03b3 + \u03b3v, then delete vxp|\u22121|.(3) If \u03b3 + \u03b3u = \u2113p + 1 = \u03b3 + \u03b3v, then return G\u2032  If \u03b3 = 1 and \u03b3u + \u03b3v + 1 = \u2113p, then return G\u2032  If \u03b3 = 1 and \u03b3u + \u03b3v + 1 \u2260 \u2113p, then delete ux1  If \u03b3 \u2260 1 and \u03b3 + \u03b3u + \u03b3v \u2261 \u2113p mod (\u2113p + 1), then delete all edges e \u2208 E(G) \\ (M \u222a {ux1}).(7) In all other cases, return \"NO\".(p \u00b7 FES(G) or 11\u2113c \u00b7 FES(G). To prove this, let G\u2020 =  be the result of contracting all degree-2 vertices in G\u2217.Finally, we can show that an input graph that is reduced with respect to these rules cannot be larger than 11\u2113Lemma 4 Let G be reduced with respect to all presented reduction rules. Then, |V| \u2264 \u2113 \u00b7 (|V\u2020| + 3|E\u2020|) with \u2113 := max{\u2113c, \u2113p}.Proof By Lemma 3, we know that \u2200v\u2208V\u2020|E(vT)| < \u2113p. Therefore, if Lemma 4 is false, there is an edge uv \u2208 E\u2020 such that there are > 3\u2113 vertices between u and v (i.e. uv is a contraction of more than 3\u2113 edges of G). Nevertheless, by irreducibility with respect to Path Rule 3, there is at most one vertex w between u and v such that wTis not empty and the distance between u and v cannot be greater than 2\u2113 + 1 (by irreducibility with respect to Path Rule 1 and 2). So, |E(wT)| \u2265 \u2113, contradicting Lemma 3.Theorem 2 RESTRICTED SCAFFOLDING admits a kernel containing at most 11\u2113 \u00b7 FES(G) vertices and (11\u2113 + 1) \u00b7 FES(G) edges where \u2113 := max{\u2113p, \u2113c}.nucleotide database (http://www.ncbi.nlm.nih.gov). Then, for each of them, a set of simulated paired-end reads was generated with the tool wgsim  is the size of a smallest set of edges whose deletion leaves an acyclic graph. The degeneracy is the smallest value d for which every subgraph has a vertex of degree at most d. It is a kind of measure of sparsity of the graph. We notice that scaffold graphs look quite sparse, with few vertices of high degrees and a feedback edge set number that is usually significantly lower than the number of vertices. While degree-based graph parameters like the degeneracy d are tiny in all instances, we recall that our problem generalizes Hamiltonian Cycle, which is already N P-hard on 3-regular graphs. However, Table Table Staphylococcus Aureus genome is no longer available in the NCBI Nucleotide database  and the Escherichia Coli genome has been updated since the presented version. Errors in assemblies are quite frequent  =  for the ebola and monarch genomes and  =  for the more complicated inputs. The tests were run on an AMD Opteron(tm) Processor 6376 at 2300 MHz.We implemented the dynamic programming algorithm presented in Section 11 in C++ using boost and the treewidth optimization library TOL . We ran Figure Concerning the rice chloroplast, among the 84 contigs, only three were misplaced. All three of them are small (< 130 bp), two of them strongly overlap and the third has two occurrences in the reference genome, one complete and one partial. The seven remaining scaffolds follow exactly the right relative order and orientation of the contigs on the reference genome. Chloroplast genomes have a particularity which make them interesting as data for scaffolding. They present an inverted repeat region of approximately 20 kbp . Figure SCAFFOLDING problem which is an integral part of genome sequencing. We showed that it can be solved in polynomial time on trees or graphs that are close to being trees (constant treewidth) by a dynamic programming algorithm. We proved a linear-size problem kernel for the parameter \"feedback edge set\" for a restricted version in which the lengths of paths and cycles are fixed. We implemented an exact algorithm solving SCAFFOLDING in f(tw) \u00b7 poly(n) time and evaluated our implementation experimentally, supporting the claim that this method produces high-quality scaffolds. Our experiments are run on data sets that are based on specific real-world genomes, which we also examined to identify a number of interesting parameters that may help design parameterized algorithms and random scaffold graph generators that produce more realistic instances. We are currently transferring the preprocessing rules to the general problem variant. We are highly interested in further graph classes that are closer to real-world instances than trees and on which the problem might be polynomial-time solvable. From an algorithmic point of view, we remark that only few bags of the used tree decompositions are small  and, in case of ties, j (ascending).Induction Base: The statement holds for all vertices v if j = 0 since Induction Step (\u2265): First, we show that v \u2265 \u03c9(S). To this end, let u := vs[j] and, noting that all vertices except maybe v of M, let q denote the path in G[S \u222a M] containing u. Let w be the vertex paired with v by M. Let \u03b1 denote the number of edges in q \u2229 E(u T+ v) \\ M incident with u and let \u03b2 \u2264 c denote the number of edges in vs[1..(j \u2212 1)]. Let u S:= Up\u2208S p \u2229 E(uT) and let u S and j\u22121 S are path covers of u Tand Case 1: uv \u2208 M and c = 0. Then, Case 2: uv \u2208 M and c = 1. Then, \u03b2 = 1. Furthermore, the path containing u is split over u S + uv and j-1S , implying Case 3: uv \u2209 M and c = 0. Then, ||S||p = ||uS||p+ ||j\u22121S ||p and we haveCase 4: uv \u2209 M and c = 1. If uv \u2209 S , then \u03b2 = 1. Furthermore, ||S||p = ||uS||p + ||j\u22121S||p and we haveuv \u2208 S , implying \u03b1 = \u03b2 = 0. If w \u2209 vs[1..j], then Otherwise, w \u2208 vs[1..j] and the path containing u is split over u S and j\u22121S . Thus, Otherwise, Induction Step (\u2264): Next, we show that v \u2264 \u03c9(S ) by proving that a i-path cover S\u2032 for c, i, j]v exists and, thus, v = \u03c9(S\u2032) \u2264 \u03c9(S ) by optimality of S. To this end, let u := vs[j] and let w := M(v).Case 1: w = u and c = 0. By line 6, there are \u2113 and \u03b1 such that v= u +v. By induction hypothesis, there are path covers u S and j\u22121 S corresponding to u and v, respectively. Then, Case 2: u = w and c = 1. By line 7, there are \u2113 and \u03b1 such that v= u+v. By induction hypothesis, there are path covers u S and j\u22121S corresponding to u and v, respectively. Then, S\u2032 := uS \u222a j\u22121 S is a path cover for j\u22121S contains an edge incident to v. Thus, Case 3: w \u2260 u and c = 0. By line 9, there are \u2113 and \u03b1 such that v = u + v. By induction hypothesis, there are path covers u S and j\u22121 S corresponding to u and v, respectively. Since c = 0 we have uv \u2209 u S and, thus, Case 4: w \u2260 u and c = 1.Case 4a: There are \u2113 and \u03b1 such that v = u + v (see line 10). By induction hypothesis, there are path covers u S and j\u22121 S corresponding to u and v, respectively. Since uv \u2209 M, some edge incident to u in u Tis in M. Then, S\u2032||p = ||uS ||p+ ||j\u22121S ||p= i. Furthermore,Case 4b: There is some \u2113 such that v = \u03c9(uv) + u + v (see line 10). Then, w \u2208 vs[1..j] and, by induction hypothesis, there are path covers u S and j\u22121S  corresponding to u and v, respectively. Then, S\u2032 := (uS +uv) \u222a j\u22121S is a path cover for w,v and u are on the same path p in u is incident to an edge of M in uT, we know that p does not end in u. Thus, ||S\u2032||p = ||uS ||p+ ||j\u22121S ||p\u2212 1 = i. Furthermore,Case 4c: There is some \u2113 such that v = \u03c9(uv) + u + v (see line 10). Then, w \u2209 vs[1..j] and, by induction hypothesis, there are path covers u S and j\u22121 S corresponding to u and v, respectively. Thus, S\u2032 := (u S + uv) \u222a j\u22121 S is a path cover for u is incident to an edge of M in uT, we have ||S\u2032||p = ||uS ||p + ||j\u22121S ||p = i. Furthermore,Proof of Section 11Proof of Lemma 2 The proof is by induction on the distance of i to r (descending). In the induction base, i is a leaf of T and X(i) = \u2205 and i Gis empty. Thus, the domains of d and P are empty. Thus, i = 0 and all other entries are \u2212\u221e.X(i) with children X( j) and X(\u2113) (possibly j = \u2113):For the induction step, we distinguish the possible bag types of Introduce vertex v: Since i Gdoes not contain edges incident to v, only tuples with d(v) = 0 and P(v) = \u22a5 are valid.Forget vertex v: Let iS be a maximum weight set that is eligible for . We show that i = \u03c9(iS).\"\u2264\":Case 1: i = [d[v \u2192 1], P + vv, p \u2212 1, c] j. By induction hypothesis, there is a set j Scorresponding[d[v \u2192 1], P + vv, p \u2212 1, c] j. We show that j Sis eligible for  and, thus, \u03c9(iS ) \u2265 \u03c9(jS ) = i. Since X(i) \u2282 X( j) and all paths between vertices in d\u22121(1) that are represented by P + vv are also represented by P, the first two conditions are satisfied by j Sfor i. Since q in v. But since v \u2209 d\u22121(1), we know that d\u22121(1). Thus, p paths that do not intersect d\u22121(1) and j Ssatisfies the third condition.Case 2: i = [d[v \u2192 1], (P \u2212 uu) + uv, p, c] j for some uu \u2208 P. By induction hypothesis, there is a set j Scorresponding to [d[v \u2192 1], (P\u2212uu)+uv, p, c] j. Since uv \u2208 (P \u2212 uu) + uv, there is a u-v-path q in q intersects d\u22121(1) in u. Thus, p paths that do not intersect d\u22121(1) and, thus, j Sis eligible for .Case 3: i = [d[v \u2192 x], P, p, c] j for some x \u2208 {0, 2}. By induction hypothesis, there is a set j Scorresponding to [d[v \u2192 x], P, p, c] j. Then, j Sis also eligible for .\"\u2265\": Let Case 1: x \u2208 {0, 2}. Then, iS is eligible for  and, by induction hypothesis, \u03c9(iS ) \u2264 [d[v \u2192 x], P, p, c] j \u2264 i.Case 2: x = 1. Then, by Definition 2(1), there is a path q in v. If q has another end u in d\u22121(1), then i Sis eligible for  + uv, p \u2212 1, c, j) and, thus, \u03c9(iS ) \u2264 [d[v \u2192 1], (P \u2212 uu) + uv, p \u2212 1, c]. Otherwise, iS is eligible for . In both cases, \u03c9(iS ) \u2264 i.Introduce edge uv: Let iS correspond to i. We show that i= \u03c9(iS ). Let d\u2032 and z be as described in the dynamic programming and note that d, P, p, c]i =  j, then, by induction hypothesis, there is a set jS corresponding to  j and uv \u2209 M. Then, j Sis eligible for  and, thus, \u03c9(iS ) \u2265 \u03c9(jS ) = i.d, P, p, c]i = z + \u03c9(uv): To show i \u2264 \u03c9(iS ), we consider a set j Sthat corresponds to the entry for X( j) from which i is computed and whose existence is granted by induction hypothesis. Then, we show that jS + uv is eligible for , implying \u03c9(iS ) \u2265 \u03c9(j S+ uv) = z + \u03c9(uv) if uv \u2209 M and \u03c9(iS ) \u2265 \u03c9(jS ) = z if uv \u2208 M. Note that, in each of the cases, the degrees of u and v in j Ssatisfies Definition 2(1) for d\u2032.In the following, we proceed in a similar manner for the case that i if S is eligible for  and its weight \u03c9(S ) = iis maximum among all sets that are.We say a Case 1: d(u) = d(v) = 2. Then, both u and v have degree 1 in Case 1a: z = j. Then, there is a u-v-path in d\u22121(1) = d\u2032\u22121(1) \\ {u, v}, adding uv does not touch any paths intersecting d\u22121(1). Thus, Definition 2(2) is satisfied. Further, adding uv closes a cycle in d\u22121(1) than Case 1b: z =  j. Then, both u and v have dangling paths in uv connects two paths such that the resulting path does not intersect d\u22121(1). Thus, there are one more such paths in Case 1c: z =  j for some x \u2208 d\u22121(1) = d\u2032\u22121(1) \\ {u, v} with xx \u2208 P. Then, v and a u-x-path ). Thus, adding uv connects two paths such that the resulting path intersects d\u22121(1) exclusively in x. Hence, there is a path dangling from x in j S+ uv satisfies Definition 2(2). Since no paths or cycles avoiding d\u2032\u22121(1) are affected by adding uv, Definition 2(3) is satisfied. The case that z =  j is analogous.Case 1d: z =  j for some x, y \u2208 d\u22121(1) with xy \u2208 P. Then, u qand v qthat start in u and v, respectively, and end in x and y, respectively ). Thus, adding uv connects u qand v qto a single path q that starts in x and ends in y, thus intersecting d\u22121(1). Hence both Definition 2(2) and (3) are satisfied.Case 2: d(u) = d(v) = 1. Thus, u and v are not incident to any edges in uv forms a new path connecting u and v in z = jand uv \u2208 P, we conclude that both Definition 2(2) and (3) are satisfied.Case 3: d(u) = 2, d(v) = 1. Then, v has no incident edges in u in u has degree 1 in q in vx \u2208 P for some x \u2208 d\u22121(1) and q\u2032 := q + uv.Case 3a: x = v. Then, z =  j. Since q\u2032 which is a path dangling from u, Definition 2(2) is satisfied. Since no other paths are touched, Definition 2(3) is satisfied.Case 3b: x \u2260 v. Then, z = j. Since q\u2032 which is a u-x-path, Definition 2(2) is satisfied. Since no other paths are touched, Definition 2(3) is satisfied.uv \u2209 iS \u2229 M, then i Sis eligible for . By induction hypothesis,  j \u2265 \u03c9(iS ) and, thus, i \u2265  j} \u2265 \u03c9(iS ). Otherwise, uv \u2208 i S\u222a M\"\u2265\": If iS \u2212 uv is eligible for a tuple corresponding to one of the entries over which we maximize to compute z. Thus, \u03c9(iS ) \u2264 \u03c9(i S\u2212 uv) + \u03c9(uv) \u2264 z + \u03c9(uv) if uv tt M and \u03c9(iS ) \u2264 z if uv \u2208 M. Note that, in each case, i S\u2212 uv satisfies Definition 2(1) since the degrees of u and v decrease by one when removing uv.In the following, we show that Case 1: d(u) = d(v) = 2. Then, uv is part of a path q in u nor v is an endpoint of q.Case 1a: q is closed  ). Then, q does not intersect d\u22121(1), implying that q \u2212 uv is a u-v-path in d\u2032\u22121(1) only in u and v. Thus, i S\u2212 uv is eligible for .Case 1b: q is open and does not intersect d\u22121(1) ). Then, q \u2212 uv decomposes into paths u qand vqintersecting d\u2032\u22121(1) only in u and v, respectively. Thus, i S\u2212 uv is eligible for .Case 1c: q is open and intersects d\u22121(1) in a single vertex x ). Then, q \u2212 uv decomposes into paths u qand v qin d\u2032\u22121(1) in x. Since no path avoiding d\u22121(1) is touched, i S\u2212 uv is eligible for either  \u222a {xu, vv}, p, c, j) or  \u222a {uu, vx}, p, c, j).Case 1d: q is open and intersects d\u22121(1) in 2 distinct vertices x and y ). Then, q \u2212 uv decomposes into a u-x-path u qand a v-y-path v qin i S- uv is eligible for  \u222a {ux, vy}, p, c, j).Case 2: d(u) = d(v) = 1. Then, q = {uv} is a path in u and v are isolated in uv \u2208 P and i S\u2212 uv is eligible for .Case 3: d(u) = 2 and d(v) = 1. Thus, q = . If q intersects d\u22121(1) \u2212 v in a vertex x, then q \u2212 uv intersects d\u2032\u22121(1) \u2212 u and, thus, i S\u2212 uv is eligible for  + ux, p, c, j). Otherwise, q is a dangling from v in q \u2212 uv is a dangling from u in i S\u2212 uv is eligible for  + uu, p, c, j).Join: \"\u2264\": Let d1, d2, P1, P2, p1, p2, c1, c2 be such that i =  j + \u2113. By induction hypothesis, there are sets j Sand S \u2113 corresponding to  j and \u2113, respectively, such thati S:= jS \u222a S t is eligible for  and, thus, \u03c9(S ) \u2265 \u03c9(iS ) =  j + \u2113 = i. First, by (2), we have that Definition 2(1) is satisfied. Second, to show that Definition 2(2) is satisfied, consider some uv \u2208 P. Then, there is a path q =  in Gr(P1 \u222a P2). Note that u-x1-path, an x1-x2-path, . . . . The concatenation of these paths forms a u-v path in uu \u2208 (P1)1, there is a path u in u. Analogously, a similar path uu \u2208 (P2)1 in uu \u2208 (P1 \u2229 P2)1, there is a path u and avoiding d\u22121(1) and u qis neither in p1 paths avoiding p2 paths avoiding p1 + p2 + |(P1 \u2229 P2)1| such paths. Similarly, c1 + c2 + |(P1 \u2229 P2)2| cycles avoiding d\u22121(1).We show that j S:= i S\u2229E(jG ) and let \u2113 S:= i S\u2229E(\u2113G). We show that j Sand \u2113S are eligible for tuples  and , respectively, such that (2)-(5) hold. First, for all u \u2208 X(i) = X( j) = X(\u2113) let d1 (d2) be the number of edges of j G (\u2113G) incident with u. Since P1 be the set of pairs uv with u, u \u2260 v, then there is a u-v path in u = v, then u. Let P2 be defined analogously for d2 and uv \u2208 P, since d(u) = 1 \u21d0\u21d2 d1(u) = 1 \u2295 d2(u) = 1, we have P1(u) = \u22a5 \u2295 P2(u) = \u22a5 and there is a u-v-path q in P1 \u222a P2) contains a u-v-path and we conclude that (3) holds. Third, let p1 and c1 be the number of paths and cycles, respectively, in p2 and c2 in d2. Then Definition 2(3) is satisfied. Further, let p \u2032 denote the number of pairs uu \u2208 P1 \u2229 P2, that is, p \u2032 := |(P1 \u2229 P2)1|. Then, since uu, a different path consisting of the concatenation of the two paths dangling from u in p1 + p2 + p\u2032 paths avoiding d\u22121(1). Thus, (4) holds and, in complete analogy, (5) holds. \u00a0\u00a0\u00a0\u25a1\"\u2265\": Let Proof of Theorem 1, sketch The bottleneck in the computation are the join nodes so we focus on computing their dynamic programming table. To calculate the number of entries that have to be considered in order to compute i, assume that P1 and P2 are fixed. Then so is P and u is incident to an edge in M that has been introduced in the subtree rooted at j, then d1(u) > 0 and, thus, d1(u) = 2. However, if u is not incident with any matching edges in d1(u) < 2, since otherwise, u would be incident to two non-matching edges. Thus, u is incident to a matching edge in P1 and P2 also fixes d1 and, by extension, d2. Finally, we can choose p1 and c1 in order to compute p2 and c2. Since we need to consider only permutations that are also involutions, there are less than twtw ways to choose P1 and P2. Thus, the maximum is over at most twtw \u00b7\u03c3p \u00b7 \u03c3c elements. Since there are O(n) bags in the tree decomposition, the algorithm can be executed in the claimed running time. \u25a1Lemma 5 Tree Rule 2 is correct, that is, the instance I =  is yes if and only if the result I\u2032 =  of applying Tree Rule 2 to I is yes.Proof Clearly, since all vertices of the graph G must be covered, then the only way to pack the vertex u is to include it into a path of length p lincluded the vertex v\u2032 = LCA and then decrease the number of paths by one. \u25a1Proof of Lemma 3 We show that v Tdoes not contain branching vertices (vertices with at least two children) since it is straightforward that, if v Tis a path, it has to be alternating for I to be a yes-instance and, if its length exceeds \u2113p, then Tree Rule 2 applies. Towards a contradiction, assume that v Thas branching vertices and let z denote such a vertex in v Tsuch that, among all branching vertices, z is furthest from v. Let u be a leaf of z Tthat has maximum distance to z and let d denote this distance. Since z is the only branching vertex in z Tand I is a yes-instance, the unique u-z path in z Tis alternating. Thus, by irreducibility with respect to Tree Rule 2, we know that d < \u2113p. However, since z is branching, there is another leaf w at distance d\u2032 \u2264 d < \u2113p to z in zT. Thus, if the unique u-w-path in z Tis not alternating or its length is not \u2113p, then I is not a yes-instance. But otherwise, Tree Rule 2 is applicable to u and w, contradicting irreducibility. \u25a1Lemma 6 Tree Rule 3 is correct, that is, the instance I =  is a yes-instance if and only if the result of applying Tree Rule 3 to I is.Proof Let v and e be as defined in Tree Rule 3 and let e = {u, v}. To show correctness of Tree Rule 3, we prove that all optimal solutions for I contain e. To this end, let S be an optimal solution with e \u2209 S . By Lemma 3, v Tis an alternating path p ending in v with |p| < \u2113p. By definition, M(u) is on p, so |p| \u2265 2. But then, G \u2212 e contains an isolated path of length strictly less than \u2113p, implying that S is not a solution for I. \u25a1Lemma 7 Path Rule 1 is correct, that is, the instance I =  is a yes-instance if and only if the result I\u2032 =  of applying Path Rule 1 to I is.Proof Let G\u2032.S be a solution for I, let G[S \u222a M] contains p since |p| > \u2113c. We show that S \u2032 := S \\ p \u2032 is a solution of for I\u2032 with \u03c9\u2032(S \u2032) = \u03c9(S ). First, note that all vertices of G\u2032 are covered by S \u2032. Second, note that p \\ p\u2032 is alternating since |p| = \u2113p + 1. Finally, G\u2032[S \u2032 \u222a M\u2032] contains one path less than G[S \u222a M].\"\u21d2\": Let S \u2032 be a solution for I\u2032 and let u denote the vertex onto which I\u2032 and let S for I with \u03c9(S ) = \u03c9\u2032(S \u2032). First, since |p| > \u2113c + \u2113p + 1, no cycle in G[S \u222a M] contains e. If e \u2209 S \u2032 \u222a M\u2032, then S := S \u2032 \u222a (p\u2032 \u2212 e0). If e \u2208 S \u2032 \u222a M\u2032, then je \u2209 S \u2032 \u222a M\u2032 for some \u2113p <j \u2264 2\u2113p. Then, \"\u21d0\": Let Lemma 8 Path Rule 2 is correct, that is, the instance I =  is a yes-instance if and only if the result I\u2032 =  of applying Path Rule 2 to I is.Proof First, note that no solution for I or I\u2032 covers w in a cycle since w Tis necessarily covered by a path. Also note that q exists in I\u2032.S be a solution for I and note that there is some i \u2264 \u2113p such that ie\u2209 S \u222aM . Then, however, i\u2212\u03b3f\u2209 S and S \u2032 := S \\ p is a solution for I\u2032 and \u03c9\u2032(S \u2032) = \u03c9(S ).\"\u21d2\": Let S \u2032 be a solution for I\u2032 and note that there is some i \u2264 \u2113p \u2212 \u03b3 such that i f\u2209 S \u2032 \u222a M\u2032 . But then, S \u2032 \u222a (p \u2212 i+\u03b3e) is a solution for I and \u03c9(S ) = \u03c9\u2032(S \u2032). \u25a1\"\u21d0\": Let Lemma 9 Path Rule 3 is correct, that is, the instance I =  is a yes-instance if and only if the result I\u2032 =  of applying Path Rule 3 to I is.Proof \"\u21d2\": Let S be solution for I and let u qand v qdenote the paths containing u and v, respectively, in G[S \u222a M].Case 1: Neither u qnor v qcontain edges of p. Then, \u03b3 = 1 and ux1 \u2209 S \u222a M. If \u03b3u +\u03b3v +1 = \u2113p, then Case (4) applies and otherwise, Case (5) applies. In both cases, S is also a solution for I\u2032.Case 2: v qcontains edges of p, but u qdoes not. Then, \u03b3v+ \u03b3 = \u2113p + 1 and ux1 \u2209 S \u222a M. If \u03b3u + \u03b3 = \u2113p + 1, then Case (3) applies and otherwise, Case (1) applies. In both cases, S is also a solution for I\u2032.Case 3: u qcontains edges of p, but v qdoes not. Then, \u03b3u + \u03b3 = \u2113p + 1 and vxp|\u22121| \u2209 S \u222a M and ux1 \u2208 S \\ M. If \u03b3u + \u03b3 = \u2113p + 1, then Case (3) applies and switching ux1 for wx1 in S gives a solution of same weight for I\u2032. Otherwise, Case (2) applies and S is also a solutionI\u2032.for Case 4: Both u qand v qcontain edges of p. Then, \u03b3u + \u03b3v + \u03b3 \u2261 \u2113p mod (\u2113p + 1) and ux1 \u2208 S \\ M. If \u03b3 \u2260 1, then Case (6) applies and S is also a solution for I\u2032. Otherwise, Case (3) applies and switching ux1 for wx1 in S gives a solution of same weight for I\u2032.S \u2032 be a solution for I\u2032. Note that, if G\u2032 does not contain wx1, then S is clearly also a solution for I. Thus, assume that G\u2032 contains wx1 and, thus, either Case (3) or (4) applies to I. We show that the result S of switching wx1 for ux1 in S \u2032 is a solution of the same weight for I. To show this, it suffices to show that, if a path q contains wx1, then q ends at u. Assume this is false, that is, q contains wx1 and some edge e \u2208 E(G\u2032) \\ M incident with u. Since w Tis not empty, no cycle in G\u2032[S \u2032 \u222a M] contains u. Then, the length of the u-v-path containing p \u2212 ux1 in G\u2032 is equivalent to \u03b3 + \u03b3u modulo \u2113p + 1.\"\u21d0\": Let Case 1: vxp|\u22121 |\u2208 S \u222a M. Then, since q does not end in u, we have I, implying that Case (3) applies to I. But then, \u03b3 + \u03b3v = \u2113p + 1 and, hence, wx1 \u2209 S \u222a M.Case 2: vxp|\u22121| \u2209 S \u222a M. Then, since q does not end in u, we have u + \u03b3 \u2260 \u2113p + 1 since 0 < \u03b3 \u2264 \u2113p and 0 < \u03b3u < \u2113p. Thus, Case (3) does not apply to I, implying that Case (4) applies to I. But then, \u03b3 = 1 and, since vxp|\u22121| \u2209 S \u222a M, we have wx1 \u2209 S \u222a M. \u25a1Observation 1 Let G be connected. Then, |V\u2020| \u2264 2 FES(G) since |E\u2020| \u2264 |V\u2020| + FES(G\u2020) \u2264 |V\u2020| + FES(G) and 2|E\u2020| \u2265 3|V\u2020|.Proof of Theorem 2 By Lemma 4, we have The authors declare that they have no competing interests.MW, AC and RG conceived the method and the proofs. MW implemented and tested the algorithms on all datasets. MW, AC and RG wrote the paper."}
{"text": "An upper bound method for the process of plane strain extrusion through a wedge-shaped die is derived. A technique for constructing a kinematically admissible velocity field satisfying the exact asymptotic singular behavior of real velocity fields in the vicinity of maximum friction surfaces  is described. Two specific upper bound solutions are found using the method derived. The solutions are compared to an accurate slip-line solution and it is shown that the accuracy of the new method is very high. The approximate methods for analysis of plane strain and axisymmetric extrusion/drawing can be conveniently divided into three groups: (i) the slab method, (ii) semianalytical solutions for flow through infinite channels, and (iii) the upper bound method. The slab method has been used, for example, in . In gene\u03b1) through which a sheet of plastic material is being pushed by force P  be taken relative to the axis of symmetry and the virtual apex O of the die. In addition, introduce Cartesian coordinates  whose origin is situated at O and whose x-axis coincides with the axis \u03b8 = 0. In the polar coordinates, the surfaces of the die are determined by the equation \u03b8 = \u00b1\u03b1. Since \u03b8 = 0 is an axis of symmetry for the flow, it is sufficient to find the solution in the region \u03b8 \u2265 0. It is assumed that the sheet is rigid perfectly/plastic and obeys Mises yield criterion. The tensile yield stress is denoted by \u03c30.Consider a wedge-shaped die \u03beeq=\u03beeq=\u03berr,\u03b8 = \u03b1 is automatically satisfied. The equation of incompressibility in the polar coordinates reduces to \u2202u/\u2202r + u/r = 0. Here u is the radial velocity. The general solution of the incompressibility equation isf(\u03b8) is an arbitrary function of \u03b8 and R0 is the radial coordinate of point A  must tend to a finite limit as \u03b8 \u2192 0. A necessary condition for that is C = 0. Then, it follows from  passes through point A and the line \u03c1 = \u03c1d1(\u03b8) passes through point B (\u03c1d0(\u03b1) = 1 and \u03c1d1(\u03b1) = h. Substituting these conditions into (f(\u03b8):\u03c1 = \u03c1d(\u03b8) on the velocity discontinuity lines yieldd\u03c1d/d\u03b8 by means of (\u03c1d(\u03b8) should be replaced with \u03c1d0(\u03b8) for the velocity discontinuity line between the plastic zone and rigid zone 1 and with \u03c1d1(\u03b8) for the velocity discontinuity line between the plastic zone and rigid zone 2.Assume that the circumferential velocity vanishes everywhere. Then, the velocity boundary condition at  point A . The nonation isu=\u2212UR0fi=eri=er\u03b8 = \u03b1ng form:ur=\u2212vi.wation isu=\u2212UR0fsin\u03c6ation is\u03c1=\u03c1d(\u03b8)=1ws from (\u03c1=\u03c1d(\u03b8)=t point B . Therefoed from (\u03c1=\u03c1d0(\u03b8)=ituting (ur=\u2212v(erc zone isup=uer.Tation isu=\u2212UR0f(\u03b8account ([u]=[(u+vituting (d\u03c1d\u03b8sin\u03b8+f(\u03b8) is independent of \u03c1, integrating with respect to \u03c1 and using (F(\u03b8) = \u222b0\u03b8f(z)dz. It is evident from this equation that the value of Wd is the same for each of the velocity discontinuity lines. The radial velocity at the friction surface is found from (u|\u03b8=\u03b1 = \u2212UR0f(\u03b1)r\u22121. Then, using , a, a(25)WVmeans of  as28)WmponentsWi=WV+2Wdtheorem [PuU2B=Wi,d using  is\u03c4r\u03b8 = ksin2\u03c8, where \u03c4r\u03b8 is the shear stress in the polar coordinates. Then, it follows from  and G1(\u03b1) are determined by numerical integration. Then, pu is readily found from (G0(\u03b1) and G1(\u03b1) with \u03b1 at several m-values is illustrated in Figures G1(\u03b1) is practically a linear function of \u03b1. The variation of pu with \u03b1 at several h-values and m = 1 is depicted in pu has been compared to an accurate slip-line solution derived in [h at several values of \u03b1 (in the nomenclature of the present paper) is depicted in Figure 7.26 in [pu with 1 \u2212 h at \u03b1 = \u03c0/12 and \u03b1 = \u03c0/6 has been calculated by means of pu by meion (see ) that \u03c40r\u03b2c\u2212cos\u2061\u03c8.Here \u03b2 \u03c8(\u03b8) isd\u03c8d\u03b8=c\u2212cows from (\u03c8\u03b1=12arcstion of (\u03b8=\u222b0\u03c8cos\u2061eans of (u=\u2212UR0r\u03b2( \u03c8(\u03b8) isd\u03c8d\u03b8=c\u2212co R0sin\u03b1 , 40) is ispu bythe form\u03b1=\u222b0\u03c8\u03b1cosR0sin\u03b1 (\u03b2=sin\u03b1[\u222b0eans of (u=\u2212UR0r\u03b2( \u03c8(\u03b8) isd\u03c8d\u03b8=c\u2212coR0sin\u03b1 (\u03b2=sin\u03b1[\u222b0cos\u20612\u03c8),F(\u03b8)\u2003=\u222b0\u03b8erefore,  is satised from (f(\u03b8)=\u03b2(c\u2212cos\u20612\u03c8),F(\u03b8)\u2003=\u222b0\u03b8und from . The varrived in  by the m 7.26 in . The varmeans of , (34), (ed from (f(\u03b8)=\u03b2(c\u2212cos\u20612\u03c8),F(\u03b8)\u2003=\u222b0\u03b8a and b are arbitrary constants. It is evident that  is  is 44) if(\u03b8)=a+2bent that . It follf(\u03b8)=a+2bnt that (F(\u03b8)=a\u03b8+b= \u03b1 into  gives(4f(\u03b8)=a+2b \u03b1 into (46)f(\u03b8)=e use of . In partA general kinematically admissible velocity field has been built up for the process of extrusion through a wedge-shaped die. This velocity field satisfies the asymptotic behavior of the real velocity field in the vicinity of maximum friction surfaces.Equations for the power dissipation within the plastic zone, at the velocity discontinuities and at the friction surface, have been developed in terms of ordinary integrals. These allow the calculation of the power required for the extrusion process.A semianalytic solution for flow through an infinite converging channel has been adopted to calculate an upper bound on the extrusion force by the method developed. The very close agreement between this upper bound solution and a slip-line solution found by the method of Riemann has been found.A very simple kinematically admissible velocity field has been proposed to calculate an upper bound on the extrusion force by the method developed. The very close agreement between the two upper bound solutions has been found. This suggests that simple kinematically admissible velocity fields satisfying  can be uThe method can be extended to axisymmetric extrusion. To this end, it is just necessary to account for  in the g"}
{"text": "The quotient hemirings via fuzzy strong h-ideals are investigated. The relationships between fuzzy congruences and fuzzy strong h-ideals of hemirings are discussed. Pay attention to an open question on fuzzy congruences. Finally, the normal fuzzy strong h-ideals of hemirings are explored.The aim of this paper is to introduce the concepts of fuzzy strong  The concept of fuzzy sets was formulated by Zadeh , and sinSemirings, regarded as a generalization of rings, have been recently found particularly useful in solving problems in different disciplines of applied mathematics and information sciences because semirings provide an algebraic framework for modelling. A special semiring with a zero and endowed with the commutative addition is said to be a hemiring. Nowadays, semirings (hemirings) are useful in optimization theory, graph theory of discrete event dynamical systems, matrices, determinants, generalized fuzzy computation, automata, theory, formal language theory, coding theory, and analysis of computer programs.h-ideals and k-ideals of hemirings were thoroughly investigated by la Torre . Let F(S) denote the set of all fuzzy sets of S. A fuzzy set \u03bc in S of the formy and value r and is denoted by yr. In particular, if r = 1, y1 denotes the fuzzy point with support y and value 1.A fuzzy set \u03bc of S is called a fuzzy left ideal if for all x, y \u2208 S, we have1)(F\u03bc(x + y) \u2265 \u03bc(x)\u2227\u03bc(y),2)(F\u03bc(xy) \u2265 \u03bc(y).A fuzzy left ideal \u03bc of S is called a fuzzy left h-ideal if for all a, b, x, z \u2208 S, x + a + z = b + z \u2192 \u03bc(x) \u2265 \u03bc(a)\u2227\u03bc(b). A fuzzy right h-ideal is defined similarly.A fuzzy set \u03bc and \u03bd be fuzzy sets of S. The sum \u03bc + \u03bd of \u03bc and \u03bd is defined byLet In particular,x \u2208 S and \u03bc \u2208 F(S), then y + \u03bc = y1 + \u03bc.Note that if \u03bc of S is called a fuzzy strong left(right) h-ideal of S if for all x, y, z, a, b \u2208 S,\u03bc(x + y) \u2265 \u03bc(x)\u2227\u03bc(y),\u03bc(xy) \u2265 \u03bc(y)(\u03bc(xy) \u2265 \u03bc(x)),x + a + z = y + b + z \u2192 (y1 + \u03bc)(x) \u2265 \u03bc(a)\u2227\u03bc(b). Note that if \u03bc is a fuzzy strong h-ideal of S, then \u03bc(0) \u2265 \u03bc(x).A fuzzy set S = {0, a, b, c} be a set with an addition operation (+) and a multiplication operation (\u00b7) as follows:Let \u03bc in S by \u03bc(0) = \u03bc(a) = 0.6, \u03bc(b) = \u03bc(c) = 0.5. Then \u03bc is a fuzzy strong h-ideal of S.Define a fuzzy set \u03bc and \u03bd be fuzzy sets of S. Then h-sum of \u03bc and \u03bd is defined bya\u2032, a\u2032\u2032, b\u2032, b\u2032\u2032, z \u2208 S such that x + a\u2032 + b\u2032 + z = a\u2032\u2032 + b\u2032\u2032 + z; otherwise, (\u03bc+h\u03bd)(x) = 0.Let \u03bc be a fuzzy set of S and r \u2208 . Then the sets \u03bcr = {x\u2223\u03bc(x) \u2265 r} and \u03bcrs = {x\u2223\u03bc(x) > r} are called an r-level subset and an r-strong level of \u03bc, respectively. We now characterize the fuzzy strong h-ideal of S by using their (strong) level subsets.Let \u03bc of S is a fuzzy strong h-ideal of S if and only if nonempty \u03bcr is a strong h-ideal of S for all r \u2208 .A fuzzy set \u03bc be a fuzzy strong h-ideal of S, x, y \u2208 \u03bcr, and a \u2208 S. Then \u03bc(x + y) \u2265 \u03bc(x)\u2227\u03bc(y) \u2265 r, \u03bc(ax) \u2265 \u03bc(a)\u2228\u03bc(x) \u2265 r, and so x + y, ax \u2208 \u03bcr. Similarly, we get xa \u2208 \u03bcr, hence \u03bcr is an ideal of S.Let x, y, z \u2208 S and a, b \u2208 \u03bcr be such that x + a + z = y + b + z. Hence, \u03bc(a) \u2265 r, \u03bc(b) \u2265 r. Thus we haveNow, let b\u2032 \u2208 S such that x = y + b\u2032 and \u03bc(b\u2032) \u2265 r; that is, b\u2032 \u2208 \u03bcr, and so x \u2208 y + \u03bcr. Therefore, \u03bcr is a strong h-ideal of S.This implies that there exists x\u2032, y\u2032 \u2208 S. If possible, let \u03bc(x\u2032 + y\u2032) < \u03bc(x\u2032)\u2009\u2227\u2009\u03bc(y\u2032). Choose r such that \u03bc(x\u2032 + y\u2032) < r < \u03bc(x\u2032)\u2009\u2227\u2009\u03bc(y\u2032). Then x\u2032, y\u2032 \u2208 \u03bcr, but x\u2032 + y\u2032 \u2209 \u03bcr, a contradiction. Hence, \u03bc(x + y) \u2265 \u03bc(x)\u2227\u03bc(y) for all x, y \u2208 S. Similarly, we have \u03bc(xy) \u2265 \u03bc(x)\u2228\u03bc(y) for all x, y \u2208 S.Conversely, assume that the given conditions hold. Let x\u2032, y\u2032, z\u2032, a\u2032, b\u2032 \u2208 S such that x\u2032 + a\u2032 + z\u2032 = y\u2032 + b\u2032 + z\u2032 and (y1\u2032 + \u03bc)(x\u2032) < \u03bc(a\u2032)\u2227\u03bc(b\u2032); choose r such that (y1\u2032 + \u03bc)(x\u2032) < r < \u03bc(a\u2032)\u2227\u03bc(b\u2032). Then a\u2032, b\u2032 \u2208 \u03bcr, but (y1\u2032 + \u03bc)(x\u2032) = \u22c1x\u2032=y\u2032+c\u03bc(c) < r. That is, x\u2032 \u2209 y\u2032 + \u03bcr, a contradiction. Hence, (y1 + \u03bc)(x) \u2265 \u03bc(a)\u2227\u03bc(b) for all x, y, z, a, b \u2208 S with x + a + z = y + b + z. This implies that \u03bc is a fuzzy strong h-ideal of S.Now assume there exist h-ideals are investigated. Finally, we give an isomorphism theorem of hemirings.In this section, the quotient hemirings via fuzzy strong \u03b8 be an equivalence relation on S. Recall that \u03b8 is called a congruence relation on S if  \u2208 \u03b8 and  \u2208 \u03b8 imply  \u2208 \u03b8 and  \u2208 \u03b8.Let I be a strong h-ideal of S, x, y \u2208 S. We call x congruent to y mod I, if and only if there exist a, b \u2208 Iand z \u2208 S be such that x + a + z = y + b + z I if and only if x \u2208 y + I,x]I + [y]I = [x + y]I,[ab\u2223a \u2208 [x]I, b \u2208 [y]I}\u2286[xy]I. {Let \u03bc be a fuzzy strong h-ideal of S and \u03bcr an r-level subset of S. We denote by \u03bcx = {y \u2208 S\u2223y \u2208 [x]\u03bcr} the equivalence class containing x and by S/\u03bc = {\u03bcx\u2223x \u2208 S} the set of all equivalence classes of S.Let \u03bc is a fuzzy strong h-ideal of S and \u03bcr2 = \u03bcr, then S/\u03bc is a hemiring under the binary operations:x, y \u2208 S.If \u03bcx = \u03bcu and \u03bcy = \u03bcv. Since \u03bcx = {y \u2208 S\u2223y \u2208 [x]\u03bcr}, we have \u03bcx = [x]\u03bcr, and so [x]\u03bcr = [u]\u03bcr, [y]\u03bcr = [v]\u03bcr. Then [x]\u03bcr + [y]\u03bcr = [u]\u03bcr + [v]\u03bcr. By x]\u03bcr + [y]\u03bcr = [x + y]\u03bcr, and so [x + y]\u03bcr = [u + v]\u03bcr; that is, \u03bcx+y = \u03bcu+v. Hence the addition is well defined.Firstly, we show that the above binary operations are well defined. In fact, if x]\u03bcr \u00b7 [y]\u03bcr\u2286[xy]\u03bcr. Next we show [xy]\u03bcr\u2286[x]\u03bcr \u00b7 [y]\u03bcr. Let z = ab \u2208 [xy]\u03bcr; since \u03bcr2 = \u03bcr, then z = ab \u2208 [xy]\u03bcr = xy + \u03bcr = xy + x\u03bcr + y\u03bcr + \u03bcr2 = (x + \u03bcr)(y + \u03bcr) = [x]\u03bcr \u00b7 [y]\u03bcr. This implies that z = ab \u2208 [x]\u03bcr \u00b7 [y]\u03bcr, and so [xy]\u03bcr\u2286[x]\u03bcr \u00b7 [y]\u03bcr; that is, [x]\u03bcr \u00b7 [y]\u03bcr = [xy]\u03bcr. This means that \u03bcx \u00b7 \u03bcy = \u03bcxy. Hence the multiplication is well defined. Now it is easy to verify that S/\u03bc is a hemiring.By f be a homomorphism from S to S\u2032, \u03bc a fuzzy subset of S, and \u03bc\u2032 a fuzzy subset of S\u2032. Then the image f(\u03bc) of \u03bc and the preimage f\u22121(\u03bc) of \u03bc\u2032 are both fuzzy sets defined, respectively, as follows:x \u2208 S.Let f : S \u2192 S\u2032 be a homomorphism of hemirings. A strong h-ideal I of S is called f-compatible, if for all x, y, z, a, b \u2208 S, f(x + a + z) = f(y + b + z) implies x \u2208 y + I. A fuzzy strong h-ideal \u03bc of S is called f-compatible, if for all x, y, z, a, b \u2208 S, f(x + a + z) = f(y + b + z) implies (y + \u03bc)(x) \u2265 \u03bc(a)\u2227\u03bc(b).Let f is a monomorphism, then every fuzzy strong h-ideal is f-compatible.If the above f : S \u2192 S\u2032 be an epimorphism of hemirings. If \u03bc is a f-compatible fuzzy strong h-ideal of S, then f(\u03bc) is a fuzzy strong h-ideal of S\u2032.Let x\u2032, y\u2032 \u2208 S\u2032; then(1) Let x\u2032, y\u2032 \u2208 S\u2032; then(2) Let x, y, z, a, b \u2208 S and x\u2032, y\u2032, z\u2032, a\u2032, b\u2032 \u2208 S be such that x\u2032 + a\u2032 + z\u2032 = y\u2032 + b\u2032 + z\u2032, f(x) = x\u2032, f(y) = y\u2032, f(z) = z\u2032, f(a) = a\u2032, f(b) = b\u2032; then we have f(x + a + z) = f(y + b + z). Since \u03bc is f-compatible, we have (y1 + \u03bc)(x) \u2265 \u03bc(a)\u2227\u03bc(b); thus(3) Let f(\u03bc) is a fuzzy strong h-ideal of S\u2032.Therefore, Similarly, we can obtain the following result.f: S \u2192 S\u2032 be a monomorphism of hemirings. If \u03bc\u2032 is a fuzzy strong h-ideal of S\u2032, then f\u22121(\u03bc\u2032) is an f-compatible fuzzy strong h-ideal of S.Let The proof is similar to \u03bc and \u03bd be two fuzzy sets of S. If they are fuzzy strong left  h-ideals of S, then so are \u03bc\u2229\u03bd and \u03bc+h\u03bd.Let \u03bc\u2229\u03bd is a fuzzy strong left h-ideal of S. For any x, y \u2208 S,We first show that \u03bc(xy) \u2265 \u03bc(y) and \u03bd(xy) \u2265 \u03bd(y), it follows thatSince \u03bc\u2229\u03bd is a fuzzy left ideal of S.Therefore, a, b, x, y, z be such that x + a + z = y + b + z. ThenLet \u03bc\u2229\u03bd is a fuzzy strong left h-ideal of S.Hence \u03bc+h\u03bd is a fuzzy strong left h-ideal. In fact,Now we show that x, y \u2208 S, we have(1) for any x, y \u2208 S, we have(2) For any a, b, x, y, z\u2032 be any elements of S such that x + a + z\u2032 = y + b + z\u2032. If there exist c\u2032, c\u2032\u2032, d\u2032, d\u2032\u2032, e\u2032, e\u2032\u2032, f\u2032, f\u2032\u2032, z\u2032\u2032, z\u2032\u2032\u2032 \u2208 S, such thatz\u2032\u2032\u2032\u2032 = z\u2032 + z\u2032\u2032 + z\u2032\u2032\u2032 + a + b, and so(3) Let \u03bc+h\u03bd)(a) = 0 or (\u03bc+h\u03bd)(b) = 0, and so [y1 + (\u03bc+h\u03bd)](x) \u2265 0 = (\u03bc+h\u03bd)(a)\u2227(\u03bc+h\u03bd)(b). Summing up the above statements, \u03bc+h\u03bd is a fuzzy strong left h-ideal of S. The case for fuzzy strong right h-ideals can be similarly proved.This givesS) the set of all fuzzy strong h-ideals of S with the same tip t; that is, \u03bc(0) = \u03bd(0) for all \u03bc, \u03bd \u2208 FSI(S). Then we have the following result.Now denote by FSI(S), +h, \u2229) is a bounded complete lattice under the relation \u201c\u2286\u201d.The . It follows from \u03bc\u2229\u03bd \u2208 FSI(S) and \u03bc+h\u03bd \u2208 FSI(S). It is clear that \u03bc\u2229\u03bd is the greatest lower bound of \u03bc and \u03bd. We now show that \u03bc+h\u03bd is the least upper bound of \u03bc and \u03bd, since \u03bc(0) = \u03bd(0); for any x \u2208 S, we haveLet \u03bc\u2286\u03bc+h\u03bd. Similarly, we have \u03bd\u2286\u03bc+h\u03bd. Now, let \u03c9 \u2208 FSI(S) be such that \u03bc, \u03bd\u2286\u03c9; then we have \u03bc+h\u03bd\u2286\u03c9+h\u03c9\u2286\u03c9. Hence, \u03bc\u2228\u03bd = \u03bc+h\u03bd. It is clear that replace the {\u03bc, \u03bd} with arbitrary family of FSI(S) and so (FSI(S), +h, \u2229) is a complete lattice under the relation \u201c\u2286\u201d. \u2205 and \u03c7S are the minimal and the maximal elements in (FSI(S), +h, \u2229), respectively. Therefore, (FSI(S), +h, \u2229) is a bounded complete lattice.Hence, Finally, we give an isomorphism theorem of hemirings.f: S \u2192 S\u2032 be an isomorphism of hemirings and \u03bd a fuzzy strong h-ideal of S\u2032. If \u03bdr2 = \u03bdr and f\u22121(\u03bd)r2 = f\u22121(\u03bd)r, thenLet S/f\u22121(\u03bd) and S\u2032/\u03bd are both hemirings. Define \u03be : S/f\u22121(\u03bd) \u2192 S/\u03bd byIt follows from Theorems \u03be is well defined as follows: f\u22121(\u03bd)x = f\u22121(v)y\u21d2[x]f\u22121(v)r = [y]f\u22121(\u03bd)r\u21d2x + f\u22121(\u03bd)r = y + f\u22121(\u03bd)r\u21d2f(x + f\u22121(\u03bd)r) = f(y + f\u22121(\u03bd)r). Since f is a homomorphism, then f(x) + f(f\u22121(\u03bd)r) = f(y) + f(f\u22121(\u03bd)r)\u21d2f(x) + vr = f(y) + \u03bdr\u21d2[f(x)]\u03bdr = [f(y)]\u03bdr\u21d2\u03bdf(x) = \u03bdf(y).(1) \u03be is a homomorphism:(2) \u03be is an epimorphism; for any \u03bdy \u2208 S\u2032/\u03bd, since f is epimorphism, then there exists x \u2208 S, such that f(x) = y, so \u03be((f\u22121(\u03bd))x) = \u03bdf(x) = \u03bdy.(3) \u03be is monomorphism: \u03bdf(x) = \u03bdf(y)\u21d2[f(x)]\u03bdr = [f(y)]\u03bdr\u21d2f(x) + \u03bdr = f(y) + \u03bdr\u21d2f(x) + f(f\u22121(\u03bd)r) = f(x) + f(f\u22121(\u03bd)r)\u21d2f(x + f\u22121(\u03bd)r) = f(y + f\u22121(\u03bd)r); since f is monomorphism, so x + f\u22121(\u03bd)r = y + f\u22121(\u03bd)r; thus [x]f\u22121(\u03bd)r = [y]f\u22121(\u03bd)r, so we have (f\u22121(\u03bd))x = (f\u22121(\u03bd))y; hence S/f\u22121(\u03bd)\u2245S\u2032/\u03bd, and the proof is completed.(4) h-ideals of hemirings. The following concepts can be seen in \u2192 be an increasing function. Then a fuzzy set \u03bcf : S \u2192  defined by \u03bcf(x) = f(\u03bc(x)) is a fuzzy strong left h ideal of S. In particular, if f(\u03bc(0)) = 1, then \u03bcf is normal; if f(t) \u2265 t for all t \u2208 , \u03bc\u2286\u03bcf.Let x, y \u2208 S, then we haveF1). Similarly, \u03bcf(xy) = f(\u03bc(xy)) \u2265 f(\u03bc(y)) = \u03bcf(y), which proves (F2) holds. Hence \u03bcf is a fuzzy left ideal of S.For all a, b, x, y, z \u2208 S be such that x + a + z = y + b + z. ThenNow, let \u03bc is a fuzzy strong left h-ideal, we haveSince so\u03bcf is a fuzzy strong left h-ideal of S; if f(\u03bc(0)) = 1, then \u03bcf is normal; suppose that f(t) = f(\u03bc(x)) \u2265 \u03bc(x) for all x \u2208 S, which proves \u03bc\u2286\u03bcf.Therefore, N(S) denote the set of all normal fuzzy strong left h-ideal of S. Note that N(S) is a poset under the set inclusion.Let \u03bc \u2208 N(S) be nonconstant such that it is a maximal element of (N(S), \u2286). Then \u03bc takes only two values 0 and 1.Let \u03bc is normal, we have \u03bc(0) = 1. For some x \u2208 S, let \u03bc(x) \u2260 1; we know that \u03bc(x) = 0; otherwise, there exists x0 \u2208 S, such that 0 < \u03bc(x0) < 1. Now we define on S a fuzzy set \u03bd by putting \u03bd(x) = (\u03bc(x) + \u03bc(x0))/2 for some x \u2208 S. Then \u03bd is well defined.Since x, y \u2208 S, we haveF1) holds.For all F2) holds. Hence v is a fuzzy left ideal of S. Now, let a, b, x, y, z \u2208 S be such that x + a + z = y + b + z; since \u03bc is a normal fuzzy strong left h-ideal, we have (y1 + \u03bc)(x) = \u22c1x=y+e\u2009\u03bc(e) \u2265 \u03bc(a)\u2227\u03bc(b), which implies\u03bd is a fuzzy strong left h-ideal of S. By \u03bd+ is a normal fuzzy strong left h-ideal of S. Note that\u03bd+(x0) < 1 = \u03bd+(0); this means that \u03bd+ is nonconstant and \u03bc is not a maximal of N(S). This is a contradiction.Similarly, we can obtainh-ideal \u03bc of S is called maximal if \u03bc+ is a maximal element of N(S).A nonconstant fuzzy strong left h-ideal \u03bc of S is maximal, then\u03bc is normal,\u03bc takes only the values 0 and 1,\u03c7\u03bc0 = \u03bc,\u03bc0 is a maximal strong left h-ideal of S.If a fuzzy strong left \u03bc be a maximal fuzzy strong left h-ideal of S. Then \u03bc+ is a nonconstant maximal element of the poset (N(S), \u2286). It follows from \u03bc+ takes only the values 0 and 1. Note that \u03bc+(x) = 1 if and only if \u03bc(x) = \u03bc(0), and \u03bc+(x) = 0 if and only if \u03bc(x) = \u03bc(0) \u2212 1. By \u03bc(x) = 0, and so \u03bc(0) = 1. Hence \u03bc is normal and \u03bc+ = \u03bc. This proves that (1) and (2) hold.Let (3) Obviously.\u03bc0 = {x \u2208 S\u2223\u03bc(x) = 1} is a strong left h-ideal. Obviously \u03bc0 \u2260 S since \u03bc takes two values. Now let A be a strong h-ideal that contains \u03bc0. Then \u03bc\u03bc0\u2286\u03bcA, and in consequence, \u03bc = \u03bc\u03bc0\u2286\u03bcA. It follows from \u03bcbeing normal that \u03bcA is also normal and takes only two values: 0 and 1. By the assumption, \u03bc is maximal, so \u03bc = \u03bcA or \u03bc = \u03c9, where \u03c9(x) = 1 for all x \u2208 S. In the last case \u03bc0 = S, which is impossible. So, \u03bc = \u03bcA; that is, \u03bcA = \u03c7A. Hence \u03bc0 = A.(4) It is clear that h-ideals of hemirings. We also discuss some concept of fuzzy strong h-ideals of hemrings and then we consider quotient hemirings and their isomorphism theorem. Finally, we introduce normal fuzzy strong h-ideals of hemirings. In the future study of hemirings, we can apply fuzzy congruences of hemirings to some applied fields, such as decision making, data analysis, and forecasting.In this paper, we consider the relationships between fuzzy congruences and fuzzy strong"}
{"text": "In this paper, we extend this result and put forward a generalized method which can completely solve the family of systems of Diophantine equations x2 \u2212 6y2 = \u22125 and x = az2 \u2212 b for each pair of integral parameters a, b. The proof utilizes algebraic number theory and p-adic analysis which successfully avoid discussing the class number and factoring the ideals.Mignotte and Peth\u00f6 used the Siegel-Baker method to find all the integral solutions ( Z, N, and Q be the sets of all integers, positive integers, and rational numbers, and let a, b be the integers. The system of Diophantine equationsa = 2 and b = 1; however, their method was complicated as a combination of algebraic and transcendental number theory. In 1998, Cohn [a = 2 and b = 1. In 2004, Le [D \u2212 1 is the power of an odd prime. As an example, solutions of the equations for D = 6 and D = 8 are given in the paper so as to show the effectiveness of the method.Let quationsx2\u22126y2=\u22125quationsx2\u22126y2=\u221252004, Le  used a sp-adic method [In this paper, we use algebraic number theory and Skolem's c method  to solvec method  to compuIn order to well interpret the main result, the symbol notation used in this paper is defined as below.m \u2265 0 is an integer and A denote \u03b22 = \u00b1\u03b7\u03b50 or \u03b22 = \u00b1\u03b7, with \u03b22 in Here, we assume that The main result of this paper is as follows.x, y, z) be an integral solution of (z exists only when it satisfies one of the following four equations for k \u2208 Z: N1)(az + \u03b8A = \u00b1\u03b1\u03b5k with 2N2)(az + \u03b8A = \u00b1\u03b1\u03b5k with 2N3)(az + \u03b8A = \u00b1\u03b1\u03b5k with 2N4), \u03b1 is the nonassociated factor such that \u03b1\u2212 denotes the relative conjugate of \u03b1, and A is referred to above.Let  = , , , , , and .The system of  for a = Before the proof of the theorem, m, \u03b50, \u03b2, and If A, we haveRewriting A is an algebraic number. Thus, the lemma is proven.Since There are four separate cases in consideration during the proof. Since the process is very similar in each case, some details will be omitted for simplicity. Now we prove the theorem.After rewriting the first equation of , factoriAdding , we get2x=\u00b1(\u03b7\u03b50nCase 1. Assume that n = 2m + 1 is odd and a to obtainQ(\u03b8), we haveCase 2. In another case, when\u2009\u2009n = 2m is even and \u00afn. Then2x=\u03b7\u03b502m+nserting , we getserting  denotes the standard p-adic valuation and p is an odd prime.For any integer i > 1, the p-adic valuation of the term of p-adic valuation of the term of We knowQ(\u03b8) is computed. Furthermore, nonassociated factor of Q(\u03b8) is also calculated. The idea of computation of fundamental unit and nonassociated factorization stems from Zhu and Chen [To complete the proof of the corollary, the fundamental unit in the totally complex quartic field and Chen  and Buchand Chen , which oCase 1. Substituting a = 2 and b = 1 into  is reduced toEquation\u03b5 = 11 + 2\u03b8 \u2212 6\u03b82 \u2212 4\u03b83 and \u03b1 = 6 + 4\u03b8 \u2212 \u03b83. From A is an element in the field z + \u03b8A in the basis 1, \u03b8, \u03b82, and \u03b83 of Q(\u03b8), we obtain 2z + \u03b8A = a + b\u03b8 + c\u03b82 + d\u03b83\u2009\u2009; then, from c = 0. In short, we denote this fact by (2z + \u03b8A)2 = 0.From p-adic analysis, a suitable prime p is needed. Here, we take p = 7. A straightforward computation shows that\u03b1\u03b50)2 \u2261 0(mod\u2061\u20097), (\u03b1\u03b57)2 \u2261 0(mod\u2061\u20097), and (\u03b1\u03b5i)2\u22620(mod\u2061\u20097)\u2009, we get k \u2261 0,7(mod\u2061\u20098).To use the k \u2261 0(mod\u2061\u20098); since \u03b58 \u2261 1(mod\u2061\u20097), we obtain \u03b58 = 1 + 7\u03be; thenk = 8m. So we have\u03b1\u03be)2 = (\u221230 + 30\u03b8 + 60\u03b82 + 30\u03b83)2\u22620(mod\u2061\u20097), we get m = 0 and z = \u00b13 by working modulo 7r+2 on formula (r||m.(i) Let  formula , where 7k \u2261 7(mod\u2061\u20098), we havek = \u22121 + 8m. So we have(ii) When m = 0 and z = \u00b168.Similar deduction shows that Case 2. Secondly, we similarly consider the following equation:\u03b5 = \u221249 \u2212 20\u03b8 + 3\u03b82 + (7/2)\u03b83 and \u03b1 = 2 + \u03b8. By the same argument we choose p = 23 and \u03b511 \u2261 1(mod\u2061\u200923). Similarly we have k \u2261 0,1(mod\u2061\u200911); then we can deduce z = \u00b11 and z = \u00b191, respectively.Case 3. Consider the following equation:\u03b5 = \u221285 + 34\u03b8 + 30\u03b82 + 38\u03b83, \u03b1 = 4 + \u03b83, p = 5, and \u03b54 \u2261 1(mod\u2061\u20095). Direct deductions show that z = \u00b12 and z = \u00b111798.Case 4. The final equation is\u03b5 = 1 \u2212 2\u03b8 \u2212 \u03b82 \u2212 (1/2)\u03b83, \u03b1 = 12 \u2212 \u03b8, p = 13, and \u03b55 \u2261 1(mod\u2061\u200913). A similar deduction yields z = 0 and z = \u00b16.x, y, z) is an integral solution of  = , , , , , and .All in all, if \u00b1z=tituting  into theThis completes the proof of the corollary.a \u2264 10 and 1 \u2264 b \u2264 10 are solved and results are listed in z-value of solutions .Like before, we can solve a family of systems of Diophantine equations . As a di"}
{"text": "Mm1 of meromorphic spirallike functions. Such results as integral representations, convolution properties, and coefficient estimates are proved. The results presented here would provide extensions of those given in earlier works. Several other results are also obtained.We introduce and investigate a new subclass  M denote the class of functions f of the formLet f, g \u2208 M, where f is given by  and MK* the subclasses of f \u2208 M which are defined, respectively, by\u03b8 = 0 in (f \u2208 M consisting of meromorphic functions which are starlike and convex of order \u03b3\u2009\u2009(0 \u2264 \u03b3 < 1), respectively. For some recent investigations on meromorphic spirallike functions and related topics, see, for example, the earlier works f, we introduce the following class of meromorphic functions.Using the operator \u03b8| < \u03c0/2, 0 \u2264 \u03bb < 1/2, and \u03b7 > 1, let Mml denote a subclass of M consisting of functions satisfying the condition thatHmlf is given by  becomes the class M(\u03b7).We note that, for Mml.In the present paper, we aim at proving some interesting properties such as integral representations, convolution properties, and coefficient estimates for the class The following lemma will be required in our investigation.An}n=1\u221e is defined bySuppose that the sequence {n \u2265 2, we deduce from .We begin by proving the following integral representation for the class f \u2208 Mml. Then\u03c9 is analytic in U with \u03c9(0) = 0 and |\u03c9(z)| < 1.Let f \u2208 Mml and\u03c4 \u2208 P, which implies\u03c9 is analytic in U with \u03c9(0) = 0 and |\u03c9(z)| < 1. We find from .Next, we derive a convolution property for the class \u03be \u2208 C and |\u03be| = 1. Then f \u2208 Mml if and only ifLet f \u2208 Mml if and only ifFrom the definition , we knowind from  thatz.Now, we discuss the coefficient estimates for functions in the class f \u2208 Mml. ThenSuppose that f \u2208 Mml. Then there exists \u03c4 \u2208 P such thatzn in both sides of \u22122\u03bbf \u2208 Mws from (ei\u03b8(1\u22122\u03bb)mbining (ei\u03b8(1\u22122\u03bb)ides of (2ei\u03b8(1\u2212\u03bb),\u03b2;l;m),|an|\u22642(\u03b7\u2212follows:A1=(1\u22122\u03bb)follows:A1=(1\u22122\u03bb) we have|an|\u2264An\u2003.In what follows, we present some sufficient conditions for functions belonging to the class \u03b6 be a real number with 0 \u2264 \u03b6 < 1. If f \u2208 M satisfies the conditionf \u2208 Mml provided thatLet \u03c9 is analytic in U with \u03c9(0) = 0 and |\u03c9(z)| < 1. Thus, we have\u03b8 \u2265 (1 \u2212 \u03b6)/(\u03b7 \u2212 1). This completes the proof of From , it foll\u03b6 = 1 \u2212 (\u03b7 \u2212 1)cos\u2061\u03b8 in If we take f \u2208 M satisfies the inequalityf \u2208 Mml.If f \u2208 M given by .If a function given by  satisfie\u03b7 \u2212 1)cos\u2061\u03b8, ifIn virtue of"}
{"text": "Solitons are of the important significant in many fields of nonlinear science such as nonlinear optics, Bose-Einstein condensates, plamas physics, biology, fluid mechanics, and etc. The stable solitons have been captured not only theoretically and experimentally in both linear and nonlinear Schr\u00f6dinger (NLS) equations in the presence of non-Hermitian potentials since the concept of the parity-time  In Section of Results, we introduce the NLS equation with third-order dispersion in the presence of complex We focus on the generalized form of the third-order NLS equation5253\u03c8\u2009\u2261\u2009\u03c8 is a complex wave function of x, z, z denotes the propagation distance, the real parameter \u03b2 stands for the coefficient of TOD, the V(x)\u2009=\u2009V(\u2212x) and W(x)\u2009=\u2009\u2212W(\u2212x) describing the real-valued external potential and gain-and-loss distribution, respectively, and g\u2009>\u20090 (or <0) is real-valued inhomogeneous self-focusing (or defocusing) nonlinearity. The power of \u03b2\u2009=\u20090 becomes the \u03b2\u2009\u2260\u20090) and gain-and-loss distribution. Here our following results are also suitable for the case x\u2009\u2192\u2009t in where Raman effect, nonlinear dispersion terms , and higher-order dispersion terms are neglected495055We start to study the physically interesting potential in V0 and TOD parameter \u03b2 can be used to modulate the amplitudes of the reflectionless potential V(x)W(x), respectively. Moreover, V(x) and W(x) are both bounded |\u2009\u2264\u2009|V0|, V(x), W(x)\u2009\u2192\u20090 as |x|\u2009\u2192\u2009\u221e (see W(x)\u2009~\u20090 as |x|\u2009>\u2009M\u2009>\u20090. The sole difference between the potential (2) and the usual Scraff-II potential\u03b2\u2009sec\u2009h\u2009x\u2009tan\u2009h\u2009x) in Scarff-II potential for the same amplitudes.where the real constant \u2009\u2192\u2009\u221e see . It is eg\u2009=\u20090) in the \u03c8\u2009=\u2009\u03a6(x)ei\u03bbz\u2212 to yieldWe firstly consider the linear spectrum problem  are the corresponding eigenvalue and eigenfunction, respectively, and limx|\u2192\u221e|\u03a6(x)\u2009=\u20090. Since the discrete spectrum of a complex V0, \u03b2 for which the complex where V0\u2009<\u20090 such that the shape of potential V(x) seems to be V-shaped with zero boundary conditions (see L (see Methods). V0, \u03b2) space. Two almost parallel straight lines (\u03b2\u2009\u2248\u2009\u00b10.12) separate the limited space {|\u22120.02\u2009\u2264\u2009V0\u2009\u2264\u2009\u22123, |\u03b2|\u2009\u2264\u20090.5}. The regions of broken and unbroken \u03b2\u2009=\u20090.1 and varying V0, the spontaneous symmetry breaking does not occur from the two lowest states since the maximum absolute value of imaginary parts of \u03bb is less than 6\u2009\u00d7\u200910\u221214 and they can be regarded as zero /6, and the existent condition g(V0\u2009\u2212\u2009\u03ba\u03b2\u2009+\u20091)\u2009>\u20090 is required. For the signs of parameter v and nonlinearity g, we find the following four cases for the existent conditions of bright solitons (4) -space); (ii) v\u2009=\u2009g\u2009=\u2009\u22121 and V0\u2009<\u2009\u2212\u03b1 -space); (iii) v\u2009=\u2009g\u2009=\u20091 and V0\u2009>\u2009\u03b1 -space); (iv) v\u2009=\u2009\u2212g\u2009=\u20091 and V0\u2009<\u2009\u03b1 -space).where the phase wavenumber is defined by TOD coefficient g\u2009=\u2009\u00b11) via the direct propagation of the initially stationary state (4) with a noise perturbation of order about 2% in v\u2009=\u2009\u2212g\u2009=\u2009\u22121 and different parameters V0 and \u03b2. For V0\u2009=\u2009\u22120.8, \u03b2\u2009=\u20090.1 belonging to the domain of the unbroken linear V0\u2009=\u2009\u22120.8, if we change \u03b2\u2009=\u20091.1 corresponding to the domain of the broken linear \u03b2 increases a little bit , then the nonlinear mode becomes unstable  for this case, \u03d50 can be excited to a stable and weakly oscillatory (breather-like behavior) situation  has no discrete spectra, but we still find the stable nonlinear modes /g\u2009>\u20090 and \u03b2\u2009>\u20090 . For v\u2009=\u20091 (or \u22121), we have S\u2009>\u20090 (or <0), which implies that the pamaeter v change the directions of power flows from gain to loss regions. The power of the solutions (4) is For the above-obtained nonlinear modes (4), we have the corresponding transverse power-flow or Poynting vector given by g\u2009=\u2009\u22121, if we choose V0\u2009=\u20091.1, \u03b2\u2009=\u2009v\u2009=\u20091 and consider the initial condition \u03d5 given by V0\u2009=\u2009\u22121.5, \u03b2\u2009=\u20090.1, v\u2009=\u2009\u22121 and consider the initial condition \u03d5 given by g\u2009=\u20091, if we choose V0\u2009=\u20091.2, \u03b2\u2009=\u2009v\u2009=\u20091 and consider the initial condition \u03d5 given by V0\u2009=\u2009\u22120.8, \u03b2\u2009=\u20090.1, v\u2009=\u2009\u22121 and consider the initial condition \u03d5 given by We here study the interactions of two bright solitons in the \u03b2\u2009\u2192\u2009\u03b2(z) in z, that is, we focus on simultaneous adiabatic switching on the TOD and gain-and-loss distribution, modeled byNowadays we turn to the excitation of nonlinear modes by means of a slow change of the control TOD parameter V(x), W(x) are given by \u03b2\u2009\u2192\u2009\u03b2(z), and \u03b2(z) is chosen aswhere \u03b21,2 being real constants. It is easy to see that the solutions (4) with \u03b2\u2009\u2192\u2009\u03b2(z) do not satisfy z\u2009=\u20090 and z\u2009\u2265\u20091000 solutions (4) indeed satisfy with \u03c8 of \u03b2\u2009\u2192\u2009\u03b2(z) given by v\u2009=\u20091 and different parameters g,\u2009V0,\u2009\u03b21,2, \u03c8|2 given by z\u2009=\u20090, \u03b2\u2009=\u2009\u03b21 are all of the higher amplitudes and then the amplitudes decrease slowly as z increases such that they reach the alternative stable sates beginning from about z\u2009=\u20091000. For v\u2009=\u2009\u22121 and different parameters g,\u2009V0,\u2009\u03b21,2, z increases such that they reach the stable and weakly oscillatory (breather-like behavior) situations beginning from about z\u2009=\u20091000, but z\u2009=\u20090 to z\u2009=\u20091100 and then the wave slowly increases a little bit to reach another stable and weakly oscillatory (breather-like behavior) feature. In particular, in v\u2009=\u2009\u2212g\u2009=\u20091, V0\u2009=\u20090.8, \u03b2\u2009=\u20090.7  to the stable states subject to exact solitons (4) of v\u2009=\u2009\u2212g\u2009=\u20091, V0\u2009=\u20090.8, \u03b2\u2009=\u20091 , that is,We here consider nonlinear modes of the generalized form of \u03b2(x) of a Gaussian functionWe here are interested in the TOD coefficient \u03b20\u2009\u2260\u20090 being a real amplitude of the Gaussian profile and another complex with \u03b2\u2009=\u2009\u03b2(x) is given by \u03c3\u2009\u2260\u20090 being a real constant. In fact, we can also consider the general case \u03b2(x). We know that the potential V(x) approaches to the harmonic potential x2/2  and W(x)\u2009\u2192\u20090 as |x|\u2009\u2192\u2009\u221e . For the given amplitude \u03b20\u2009=\u20090.1 of TOD coefficient, we give the first six lowest eigenvalues  is an error function. In the following we take g\u2009=\u20091 without loss of generality.where g\u2009=\u20091) via the direct propagation of the initially stationary state (10) with a noise perturbation of order about 2% in \u03c3\u2009=\u20090. For \u03c3\u2009=\u2009\u03b20\u2009=\u20090.1, in which the potential becomes almost a harmonic potential x2/2  of nonlinear localized modes (10) for self-focusing case \u2009=\u2009sgn(\u03c3). Since the gain-and-loss distribution W(x) given by \u03b20 and \u03c3, which generate that there are more one intervals for gain (or loss) distribution, thus though the power flows along the positive (negative) direction for the x axis as \u03c3\u2009>\u20090 (<0), it is of the complicated structures from the gain-and-loss view.For the above-obtained nonlinear modes (10), we have the corresponding transverse power-flow or Poynting vector given by g\u2009=\u20091, if we choose \u03c3\u2009=\u2009\u22120.1, \u03b20\u2009=\u20090.1 and consider the initial condition \u03d5 given by \u03c3\u2009=\u20090.2, \u03b20\u2009=\u20090.1 and consider the initial condition \u03d5 given by Moreover, we also study the interactions of two bright solitons in the \u03b2(x)\u2009\u2192\u2009\u03b2 in \u03b20 is regarded as a function of propagation distance z, that is, we focus on simultaneous adiabatic switching on the TOD, potential, and gain-and-loss distribution.Nowadays we turn to the excitation of nonlinear modes by means of a slow change of the control TOD parameter \u03c3\u2009=\u20090.1, we consider the varying amplitude \u03b20\u2009\u2192\u2009\u03b20(z) in For the given V(x) and gain-or-loss distribution W(x) given by \u03c8 satisfied by which makes the TOD coefficient given by \u03b20\u2009=\u20090.1, and consider the effect of varying amplitude \u03c3\u2009\u2192\u2009\u03c3(z) on nonlinear modes:Now we fix the amplitude of TOD, V(x) and gain-or-loss distribution W(x) given by \u03c8 satisfied by \u03b20(z) and \u03c3(z) given by which makes the potential In conclusions, we have introduced some non-Hermitian , we assume that \u03c8\u2009=\u2009\u03a6(x)ei\u03bbz\u2212, then we have x|\u2009\u2192\u2009\u221e, reduces to \u03bb\u2009=\u20091/(3\u03b22) and \u03b2\u2009>\u20090 , we have its three roots \u039b1\u2009=\u2009i/\u03b2, \u03b2 depends on the space x, e.g., \u03b2(x)\u2009=\u2009\u03b20\u2009exp(\u2212x2), and V(x), W(x) are given by \u03b2(x), W(x)\u2009\u2192\u20090 and V(x)\u2009\u2192\u2009x2/2 as |x|\u2009\u2192\u2009\u221e. Thus for this case x|\u2009\u2192\u2009\u221e, where the condition For \u03c8\u2009=\u2009\u03d5(x)ei\u03bcz\u2212, where \u03d5(x) is a complex field function and \u03bc the corresponding propagation constant. We have \u03c1(x) and \u03c6(x) being real functions and separate the real and imaginary parts to yieldWe consider the stationary solutions of For the given \u03c8\u2009=\u2009\u03d5(x)ei\u03bcz\u2212 of 59\u03b4 and F(x) and G(x) are the eigenvalue and eigenfunctions of the linearized eigenvalue problem. We substitute the expression into \u03f5 to yield the following linear eigenvalue problemTo further study the linear stability of the above-obtained nonlinear stationary solutions \u03b4 has no imaginary component, otherwise they are linearly unstable.where How to cite this article: Chen, Y. and Yan, Z. Solitonic dynamics and excitations of the nonlinear Schr\u00d6edinger equation with third-order dispersion in non-Hermitian Sci. Rep. 6, 23478; doi: 10.1038/srep23478 (2016)."}
{"text": "As applications, we derive some new fixed point theorems in partially ordered modular metric spaces, Suzuki type fixed point theorems in modular metric spaces and new fixed point theorems for integral contractions. In last section, we develop an important relation between fuzzy metricand modular metric and deduce certain new fixed point results in triangular fuzzy metric spaces. Moreover, some examples are provided here to illustrate the usability of the obtained results.The notion of modular metric spaces being a natural generalization of classical modulars over linear spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, and Calderon-Lozanovskii spaces was recently introduced. In this paper we investigate theexistence of fixed points of generalized  But the way we approached the concept of modular metric spaces is different. Indeed we look at these spaces as the nonlinear version of the classical modular spaces introduced by Nakano , we will write\u03bb > 0 and x, y \u2208 X.Let \u03c9 :  \u00d7 X \u00d7 X \u2192  is said to be modular metric on X if it satisfies the following axioms:x = y if and only if \u03c9\u03bb = 0, for all \u03bb > 0;\u03c9\u03bb = \u03c9\u03bb, for all \u03bb > 0, and x, y \u2208 X;\u03c9\u03bb+\u03bc \u2264 \u03c9\u03bb + \u03c9\u03bc, for all \u03bb, \u03bc > 0 and x, y, z \u2208 X.If instead of (i), we have only the condition (i\u2032)\u03c9 is said to be a pseudomodular (metric) on X. A modular metric \u03c9 on X is said to be regular if the following weaker version of (i) is satisfied:\u03c9 is said to be convex if for \u03bb, \u03bc > 0 and x, y, z \u2208 X, it satisfies the inequality\u03c9 on a set X and any x, y \u2208 X, the function \u03bb \u2192 \u03c9\u03bb is nonincreasing on . Indeed, if 0 < \u03bc < \u03bb, thenA function Following example presented by Abdou and Khamsi  is an imX be a nonempty set and \u03a3 a nontrivial \u03c3-algebra of subsets of X. Let P be a \u03b4-ring of subsets of X, such that E\u2229A \u2208 P for any E \u2208 P and A \u2208 \u03a3. Let us assume that there exists an increasing sequence of sets Kn \u2208 P such that X = \u22c3Kn. By E we denote the linear space of all simple functions with supports from P. By M\u221e we will denote the space of all extended measurable functions; that is, all functions f : X \u2192  such that there exists a sequence {gn} \u2282 E, |gn| \u2264 |f|, and gn(x) \u2192 f(x) for all x \u2208 X. By 1A we denote the characteristic function of the set A. Let \u03c1 : M\u221e \u2192  be a nontrivial, convex, and even function. We say that \u03c1 is a regular convex function pseudomodular if\u03c1(0) = 0;\u03c1 is monotone; that is, |f(x)| \u2264 |g(x)| for all x \u2208 X implies \u03c1(f) \u2264 \u03c1(g), where f, g \u2208 M\u221e;\u03c1 is orthogonally subadditive; that is, \u03c1(f1A\u222aB) \u2264 \u03c1(f1A) + \u03c1(f1B) for any A, B \u2208 \u03a3 such that A\u2229B \u2260 \u2205, f \u2208 M;\u03c1 has the Fatou property; that is, |fn(x)|\u2191|f(x)| for all x \u2208 X implies \u03c1(fn)\u2191\u03c1(f), where f \u2208 M\u221e;\u03c1 is order continuous in E; that is, gn \u2208 E and |gn(x)| \u2193 0 implies \u03c1(gn) \u2193 0.Similarly, as in the case of measure spaces, we say that a set A \u2208 \u03a3 is \u03c1-null if \u03c1(g1A) = 0 for every g \u2208 E. We say that a property holds \u03c1-almost everywhere if the exceptional set is \u03c1-null. As usual we identify any pair of measurable sets whose symmetric difference is \u03c1-null as well as any pair of measurable functions differing only on a \u03c1-null set. With this in mind we definef \u2208 M is actually an equivalence class of functions equal to \u03c1-a.e. rather than an individual function. Where no confusion exists we will write M instead of M. Let \u03c1 be a regular function pseudomodular.\u03c1 is a regular function semimodular if \u03c1(\u03b1f) = 0 for every \u03b1 > 0 implies f = 0\u2009\u2009\u03c1-a.e.We say that \u03c1 is a regular function modular if \u03c1(f) = 0 implies f = 0\u2009\u2009\u03c1-a.e.We say that The class of all nonzero regular convex function modulars defined on X will be denoted by R. Let us denote \u03c1 = \u03c1(f1E) for f \u2208 M, E \u2208 \u03a3. It is easy to prove that \u03c1 is a function pseudomodular in the sense of Definition 2.1.1 in . On the other hand, Tw \u2208  for all w \u2208 . Then, \u03b1 \u2265 \u03b7. That is, T is an \u03b1-admissible mapping with respect to \u03b7. If {xn} is a sequence in X such that \u03b1 \u2265 \u03b7 with xn \u2192 x as n \u2192 \u221e, then Txn, T2xn, T3xn \u2208  for all n \u2208 N. That is,n \u2208 N. Clearly, \u03b1 \u2265 \u03b7. Let \u03b1 \u2265 \u03b7. Now, if x \u2209  or y \u2209 , then 1/8 \u2265 1/4, which is a contradiction. So, x, y \u2208 . Therefore,T has a unique fixed point.Let X\u03c9, \u2aaf) be a partially ordered modular metric space. Recall that T : X\u03c9 \u2192 X\u03c9 is nondecreasing if for all x, y \u2208 X, x\u2aafy\u21d2T(x)\u2aafT(y). Fixed point theorems for monotone operators in ordered metric spaces are widely investigated and have found various applications in differential and integral equations  be a complete partially ordered modular metric space and T : X\u03c9 \u2192 X\u03c9 self-mapping satisfying the following assertions:(i)T is nondecreasing;(ii)x0 \u2208 X such that x0\u2aafTx0;there exists (iii)T is continuous function;(iv)H \u2208 \u0394H such that for all x, y \u2208 X\u03c9 and \u03bb > 0 with x\u2aafy we haveassume that there exists \u2009c.where 0 < l < Then T has a fixed point. Moreover, if for all x, y \u2208 Fix\u2061\u2061(T) we have x\u2aafy and H > 0 for all u > 0, then T has a unique fixed point.Let  be a complete partially ordered modular metric space and T : X\u03c9 \u2192 X\u03c9 self-mapping satisfying the following assertions:(i)T is nondecreasing;(ii)x0 \u2208 X such that x0\u2aafTx0;there exists (iii)xn} is a sequence in X such that xn\u2aafxn+1 with xn \u2192 x as n \u2192 \u221e, then eitherif {\u2009n \u2208 N;holds for all (iv)H \u2208 \u0394H such that for all x, y \u2208 X\u03c9 and \u03bb > 0 with x\u2aafy we haveassume that there exists \u2009l < c.where 0 < Then T has a fixed point. Moreover, if for all x, y \u2208 Fix\u2061\u2061(T) we have x\u2aafy and H > 0 for all u > 0, then T has a unique fixed point.Let  be a complete partially ordered modular metric space and T : X\u03c9 \u2192 X\u03c9 self-mapping satisfying the following assertions:(i)T is nondecreasing;(ii)x0 \u2208 X such that x0\u2aafTx0;there exists (iii)xn} is a sequence in X such that xn\u2aafxn+1 with xn \u2192 x as n \u2192 \u221e, then eitherif {\u2009n \u2208 N;holds for all (iv)x, y \u2208 X\u03c9 and \u03bb > 0 with x\u2aafy we haveassume that for all \u2009\u03c8 \u2208 \u03a8 and 0 < l < c.where Then T has a fixed point. Moreover, if for all x, y \u2208 Fix\u2061\u2061(T) we have x\u2aafy, then T has a unique fixed point.Let  be a complete partially ordered modular metric space and T : X\u03c9 \u2192 X\u03c9 self-mapping satisfying the following assertions:(i)T is nondecreasing;(ii)x0 \u2208 X such that x0\u2aafTx0;there exists (iii)xn} is a sequence in X such that xn\u2aafxn+1 with xn \u2192 x as n \u2192 \u221e, then eitherif {\u2009n \u2208 N;holds for all (iv)x, y \u2208 X\u03c9 and \u03bb > 0 with x\u2aafy we haveassume that for all \u2009\u03c8 \u2208 \u03a8 and 0 < l < c.where Then T has a fixed point. Moreover, if for all x, y \u2208 Fix\u2061\u2061(T) we have x\u2aafy, then T has a unique fixed point.Let  be a complete partially ordered modular metric space and T : X\u03c9 \u2192 X\u03c9 self-mapping satisfying the following assertions:(i)T is nondecreasing;(ii)x0 \u2208 X such that x0\u2aafTx0;there exists (iii)xn} is a sequence in X such that xn\u2aafxn+1 with xn \u2192 x as n \u2192 \u221e, then eitherif {\u2009n \u2208 N;holds for all (iv)x, y \u2208 X\u03c9 and \u03bb > 0 with x\u2aafy we haveassume that for all \u2009a + b < 1 and 0 < l < c.where Then T has a fixed point. Moreover, if for all x, y \u2208 Fix\u2061\u2061(T) we have x\u2aafy, then T has a unique fixed point.Let  > 0 for all u > 0, then T has a unique fixed point.Let \u03b1, \u03b7 :  \u00d7 X\u03c9 \u00d7 X\u03c9 \u2192  is continuous.The function M denotes the degree of nearness between x and y with respect to t.A 3-tuple  be a fuzzy metric space. The fuzzy metric M is called triangular wheneverx, y, z \u2208 X and all t > 0.Let  is nondecreasing on .For all X induces a modular metric.As an application of X, M, \u2217) be a triangular fuzzy metric space. Definex, y \u2208 X and all \u03bb > 0. Then \u03c9\u03bb is a modular metric on X.Let  is triangular, then we getLet As an application of X, M, \u2217) be a complete triangular fuzzy metric space and T : X \u2192 X self-mapping satisfying the following assertions:(i)T is an \u03b1-admissible mapping with respect to \u03b7;(ii)x0 \u2208 X such that \u03b1 \u2265 \u03b7;there exists (iii)T is an \u03b1-\u03b7-continuous function;(iv)H \u2208 \u0394H such that for all x, y \u2208 X and \u03bb > 0 with \u03b7 \u2264 \u03b1 we haveassume that there exists \u2009l < c.where 0 < Then T has a fixed point. Moreover, if for all x, y \u2208 Fix\u2061\u2061(T) we have \u03b7 \u2264 \u03b1 and H > 0 for all u > 0, then T has a unique fixed point.Let  be a complete triangular fuzzy metric space and T : X \u2192 X self-mapping satisfying the following assertions:(i)T is an \u03b1-admissible mapping with respect to \u03b7;(ii)x0 \u2208 X such that \u03b1 \u2265 \u03b7;there exists (iii)xn} is a sequence in X such that \u03b1 \u2265 \u03b7 with xn \u2192 x as n \u2192 \u221e, then eitherif {\u2009n \u2208 N;holds for all (iv)condition (iv) of Then T has a fixed point. Moreover, if for all x, y \u2208 Fix\u2061\u2061(T) we have \u03b7 \u2264 \u03b1 and H > 0 for all u > 0, then T has a unique fixed point.Let  be a partially ordered complete triangular fuzzy metric space and T : X \u2192 X self-mapping satisfying the following assertions:(i)T is nondecreasing;(ii)x0 \u2208 X such that x0\u2aafTx0;there exists (iii)T is continuous function;(iv)H \u2208 \u0394H such that for all x, y \u2208 X and \u03bb > 0 with x\u2aafy we haveassume that there exists \u2009l < c.where 0 < Then T has a fixed point. Moreover, if for all x, y \u2208 Fix\u2061\u2061(T) we have x\u2aafy and H > 0 for all u > 0, then T has a unique fixed point.Let  be a partially ordered complete triangular fuzzy metric space and T : X \u2192 X self-mapping satisfying the following assertions:(i)T is nondecreasing;(ii)x0 \u2208 X such that x0\u2aafTx0;there exists (iii)xn} is a sequence in X such that xn\u2aafxn+1 with xn \u2192 x as n \u2192 \u221e, then eitherif {\u2009n \u2208 N;holds for all (iv)H \u2208 \u0394H such that for all x, y \u2208 X\u03c9 and 0\u03bb > 0 with x\u2aafy we haveassume that there exists \u2009l < c.where 0 < Then T has a fixed point. Moreover, if for all x, y \u2208 Fix\u2061\u2061(T) we have x\u2aafy and H > 0 for all u > 0, then T has a unique fixed point.Let  be a complete triangular fuzzy metric space and T continuous self-mapping on X. Assume thatx, y \u2208 X and \u03bb > 0, where 0 < l < c. Then T has a unique fixed point.Let  be a complete triangular fuzzy metric space and T continuous self-mapping on X. Assume thatx, y \u2208 X\u03c9 and \u03bb > 0, where 0 < l < c and a + b < 1. Then T has a unique fixed point.Let ("}
{"text": "A. We find explicit expressions for the generalized Drazin inverse of the sum a + b, under new conditions on a, b \u2208 A. As an application we give some new representations for the generalized Drazin inverse of an operator matrix.We investigate additive properties of the generalized Drazin inverse in a Banach algebra  A be a complex Banach algebra with unite 1. We use \u03c3(a) to denote the spectrum of a \u2208 A. The sets of all nilpotent and quasinilpotent elements (\u03c3(a) = {0}) of A will be denoted by Anil and Aqnil, respectively.Let a \u2208 A  is the erse see . For a cp = p2 \u2208 A is an idempotent, we denote a \u2208 A asa11 = pap, If a \u2208 Ad and a\u03c0 = 1 \u2212 aad be the spectral idempotent of a corresponding to {0}. It is well known that a \u2208 A can be represented in the following matrix form d under the conditions ab = bab\u03c0 and ab = a\u03c0bab\u03c0, respectively. Then we will apply these formulas to provide some representations for the generalized Drazin inverse of the operator matrix In this paper, we first give the formulas of a \u2208 (pAp)d and x and y are generalized Drazin invertible, andIf (ii)x \u2208 Ad and a \u2208 (pAp)d, then xd and yd are given by (If given by  and 7)..x \u2208 Ad Let a, b \u2208 Aqnil. If ab = ba or ab = 0, then a + b \u2208 Aqnil.Let The following result is a generalization of [a \u2208 Aqnil, b \u2208 Ad, and ab = bab\u03c0, then a + b \u2208 Ad andIf b \u2208 Aqnil. Therefore, b\u03c0 = 1 and from ab = bab\u03c0 we obtain ab = ba. Using a + b \u2208 Aqnil and \u22121, Now we assume ab = bab\u03c0 anda1b1 = 0 and a3b1 = 0. Since b1 is invertible, we have a1 = 0 and a3 = 0.Let us representHence we haveab = bab\u03c0 implies that a4b2 = b2a4. Hence, using a4 + b2 \u2208 Aqnil. By a + b \u2208 Ad andbd, a, and a + b, we easily obtain formula \u22121, then a1\u03c0 = 0.From u reduces toa2b4 = b4a2b4\u03c0 we get = a2b4(b4d)2 = b4a2b4\u03c0(b4d)2 = 0.Hence, the expression of a2b4d = 0, we haveHence, from formula  and a2tituting  and 24)a2b4dtituting , we getu by the above expression and considering matrix representations of a and b, after direct computations, we obtain the formula (a + b)d.Now, replacing  formula  for  and D \u2208 B(Y) are generalized Drazin invertible.This section is devoted to the generalized Drazin inverse of 2 \u00d7 2 operator matrix:Next we will state some auxiliary lemmas.A and D be generalized Drazin invertible and let M be matrix of form d = 0, (A2Ad)d = Ad, (A2Ad)\u03c0 = A\u03c0, and Ind(AA\u03c0) = Ind(A) and Ind(A2Ad) = 1.Let X permits us to write A = S(C \u2295 N)S\u22121, where S and C are nonsingular, and N is nilpotent with index Ind(A). Thus Ad = S(C\u22121 \u2295 0)S\u22121. Now, it is evident that A2Ad = S(C \u2295 0)S\u22121 and AA\u03c0 = S(0 \u2295 N)S\u22121, which lead to the affirmations of this lemma.The Jordan canonical form of Md under conditions BC = 0, DC = 0, and BD = 0. Here we replace the last two conditions by the two weaker conditions DC = D\u03c0CAA\u03c0 and BD = AA\u03c0B.In [A and D be generalized Drazin invertible and let M be matrix of form n+2.From fined in . From DAA\u03c0B = BD and DC = D\u03c0CAA\u03c0, we obtain PQ = P\u03c0QPQ\u03c0. Applying Since AdBD = 0, we haveAdBD = 0, we haveFrom nce from , we obtaBC = 0 and BD = AA\u03c0B\u2009\u2009imply that BDnC = 0. From BDnC = 0 and AdBD = 0, we getThe conditions From , 38), a, a38), aThe proof is finished.SinceA and D be generalized Drazin invertible and let M be matrix of form 47) it foThe proof is finished.A and D be generalized Drazin invertible and let M be matrix of form nC = BDnD\u03c0C = BDnC = 0 for any nonnegative integer n; we can apply Q with D replaced by DD\u03c0; we haveBDnC = 0 implies thatNote that rve that  and 51)DD\u03c0 is qAA\u03c0B = BD2Dd and D2DdC = D\u03c0CAA\u03c0, we obtain PQ = P\u03c0QPQ\u03c0. Applying From Applying P+Qd=Q\u03c0Pdo we getQdP=Ad\u0393\u0394\u0394BDnC = 0, we haveBDnC = 0 yieldBy direct computation we verify thatrve that  and BDnQ\u03c0Pd=Pd,\u2003 0 yieldQ\u03c0\u2211n=0\u221eP++2.From .The proof is finished.A and D be generalized Drazin invertible and let M be matrix of form d = 0, so we getXn is defined in 72) it foThe proof is finished.Using  and TheoCA = D\u03c0DCA\u03c0, BDd = 0, CB = 0, and BCA\u03c0 = 0. ThenIf Using the case of A and D be generalized Drazin invertible and let M be matrix of form , a, a80), aUsing  and TheoA and D be generalized Drazin invertible and let M be matrix of form (ABD\u03c0 = BD, DCA = 0, BC = ABCAd, and CB = 0. ThenLet  of form . If ABD"}
{"text": "In this paper, we present protocols which use entanglement more efficiently than teleportation to distinguish some classes of orthogonal product states in m\u2009\u2297\u2009n, which are not UPB. For the open question, our results offer rather general insight into why entanglement is useful for such tasks, and present a better understanding of the relationship between entanglement and nonlocality.It is known that there are many sets of orthogonal product states which cannot be distinguished perfectly by local operations and classical communication (LOCC). However, these discussions have left the following open question: What entanglement resources are necessary and/or sufficient for this task to be possible with LOCC? In  That is, with enough entanglement, LOCC can be used to teleportHowever, in recent years, entanglement has been shown to be a valuable resource4144m\u2009\u2297\u2009n(m\u2009\u2264\u2009n) can be distinguished perfectly with a m\u2009\u2297\u2009n, which are not UPB.In 2008, Cohen presented that certain classes of unextendible product bases (UPB) in m\u2009\u2297\u2009n, we present that only needing a 2\u2009\u2297\u20092 maximally entangled state, these states are also distinguishable by LOCC. Each of the locally indistinguishable orthogonal product states that we consider can be extended to a complete \u201cnonlocality without entanglement\u201d basis, and the latter can also be distinguished by the same  protocols. Our results show that the locally indistinguishable quantum states may become distinguishable with a small amount of entanglement resources. And these results also present a better understanding of the relationship between entanglement and nonlocality.In this paper, we will consider the locally indistinguishable orthogonal product states in the general bipartite quantum systems and present the LOCC protocols which, using entanglement as a resource, distinguish these states considerably more efficiently than teleportation. Specifically, in 5\u2009\u2297\u20095, we show a set of locally indistinguishable orthogonal product states can be distinguished by LOCC with a 2\u2009\u2297\u20092 maximally entangled state. Furthermore, for several classes of locally indistinguishable orthogonal product states on a higher-dimensional \u03d5\u232a\u2009=\u2009|0\u232aA|0\u2009\u00b1\u20091\u232aB, but in this paper |\u03d5\u232a\u2009=\u2009|0\u2009\u00b1\u20091\u232aA|0\u232aB.In this section, we present that a set of locally indistinguishable orthogonal product states in 5\u2009\u2297\u20095, can be perfectly distinguished by LOCC with a 2\u2009\u2297\u20092 maximally entangled state. In 5\u2009\u2297\u20095, the following 21 orthogonal product states cannot be distinguished by LOCCIn the following, applying the proof method which was presented by Cohenheorem 1 In 5\u2009\u2297\u20095, a 2\u2009\u2297\u20092 maximally entangled state is sufficient to perfectly distinguish the above 21 states by LOCC.TProof. First of all, Alice and Bob share a 2\u2009\u2297\u20092 maximally entangled state Then, Bob performs a two-outcome measurement, each outcome corresponding to a rank-5 projector:ab, and then operating with B1 on systems bB, is that each of the initial states is transformed as:To be precise, the result of bringing in the ancillary systems in state |\u03a8\u232aA1\u2009=\u2009|1\u232aa\u23291|\u2009\u2297\u2009|3\u2009+\u20094\u232aA\u23293\u2009+\u20094|. The only remaining possibility is A2\u2009=\u2009|1\u232aa\u23291|\u2009\u2297\u2009|3\u2009\u2212\u20094\u232aA\u23293\u2009\u2212\u20094|, A3\u2009=\u2009|1\u232aa\u23291|\u2009\u2297\u2009|1\u2009+\u20092\u232aA\u23291\u2009+\u20092|, A4\u2009=\u2009|1\u232aa\u23291|\u2009\u2297\u2009|1\u2009\u2212\u20092\u232aA\u23291\u2009\u2212\u20092|, respectively.Let us now describe how the parties can proceed from here to distinguish the states. Alice makes a seven-outcome projective measurement, and we begin by considering the first outcome, A5\u2009=\u2009|0\u232aa\u23290|\u2009\u2297\u2009|4\u232aA\u23294| onto the Alice\u2019s Hilbert space, it leaves B and |2\u2009\u00b1\u20093\u232aB.Using a rank-1 projector A6\u2009=\u2009|0\u232aa\u23290|\u2009\u2297\u2009|2\u232aA\u23292|\u2009+\u2009|0\u232aa\u23290|\u2009\u2297\u2009|3\u232aA\u23293| onto the Alice\u2019s Hilbert space, it leaves B61\u2009=\u2009|0\u232ab\u23290|\u2009\u2297\u2009|1\u232aB\u23291|, B62\u2009=\u2009|0\u232ab\u23290|\u2009\u2297\u2009|3\u232aB\u23293|, respectively. When Bob uses a projector B63\u2009=\u2009|0\u232ab\u23290|\u2009\u2297\u2009|0\u232aB\u23290|, it can leave A. In the same way, we can easily distinguish Using a rank-2 projector A7\u2009=\u2009|0\u232aa\u23290|\u2009\u2297\u2009|0\u232aA\u23290|\u2009+\u2009|0\u232aa\u23290|\u2009\u2297\u2009|1\u232aA\u23291|\u2009+\u2009|1\u232aa\u23291|\u2009\u2297\u2009|0\u232aA\u23290|. This leaves B71\u2009=\u2009|0\u232ab\u23290|\u2009\u2297\u2009|0\u232aB\u23290|, it leaves A. When Bob uses a projector B72\u2009=\u2009|0\u232ab\u23290|\u2009\u2297\u2009(|1\u232aB\u23291|\u2009+\u2009|2\u232aB\u23292|), it can leave A, and leave B73\u2009=\u2009|0\u232ab\u23290|\u2009\u2297\u2009|3\u232aB\u23293|\u2009+\u2009|1\u232ab\u23291|\u2009\u2297\u2009|4\u232aB\u23294|, it leaves the last two states Alice\u2019s last outcome is a rank-3 projector onto the remaining part of Alice\u2019s Hilbert space B2 on systems bB, it creates new states which differ from the states (3) only by ancillary systems |00\u232aab\u2009\u2192\u2009|11\u232aab and |11\u232aab\u2009\u2192\u2009|00\u232aab. Thus, the latter can be handled using the exact same method as described above for B1.For operating with That is, we have succeeded in designing a protocol that perfectly distinguishes the states (1) using LOCC with an additional resource of a two-qubit maximally entangled state. This completes the proof.ab\u2009=\u2009m|00\u232a\u2009+\u2009n|11\u232a(m\u2009\u2260\u2009n), is shared, the states |\u03d513,14\u232a will be not orthogonal after Bob performs a two-outcome measurement. This means that 1 ebit is necessary for the protocol. Then, it is interesting to design a protocol to distinguish these states with less than one ebit of entanglement, or to prove that there is no any protocol to accomplish it.In addition, a 2\u2009\u2297\u20092 maximally entangled state is necessary to distinguish the set of product states for the above protocol. When a partially entangled state, |\u03a8\u232am\u2009\u2297\u2009nk\u2009+\u20091)\u2009\u2297\u2009(2l), (2k)\u2009\u2297\u2009(2l) and (2k\u2009+\u20091)\u2009\u2297\u2009(2l\u2009+\u20091).In this section, we consider the set of locally indistinguishable orthogonal product states in k\u2009+\u20091)\u2009\u2297\u2009(2l)In this subsection, we first present the following locally indistinguishable orthogonal product states in (2heorem 2 In (2k\u2009+\u20091)\u2009\u2297\u2009(2l), a 2\u2009\u2297\u20092 maximally entangled state is sufficient to perfectly distinguish the above 6(k\u2009+\u2009l)\u2212\u20096 states by LOCC.TProof. Similarly to the proof of Theorem 1, Alice and Bob first share a 2\u2009\u2297\u20092 maximally entangled state Then, Bob performs a two-outcome measurement, each outcome corresponding to a rank-2l projector:B1. To be precise, the result of bringing in the ancillary systems in state |\u03a8\u232aab, and then operating with B1 on systems bB, is that each of the initial states is transformed as:As Theorem 1, we only need to discuss the projector i\u2009=\u20091, \u2026, 2k by 2k projectors Ai\u2009=\u2009|1\u232aa\u23291|\u2009\u2297\u2009|i\u2009+\u2009(i\u2009+\u20091)\u232aA\u2329i\u2009+\u2009(i\u2009+\u20091)|, Ai+1\u2009=\u2009|1\u232aa\u23291|\u2009\u2297\u2009|i\u2009\u2212\u2009(i\u2009+\u20091)\u232aA\u2329i\u2009\u2212\u2009(i\u2009+\u20091)|, i\u2009=\u20091, 3, \u2026, 2k\u2009\u2212\u20091, respectively.Then, Alice makes a (2k\u2009+\u20093)-outcome projective measurement. Similarly to the proof of Theorem 1, Alice can identify Ak+12\u2009=\u2009|0\u232aa\u23290|\u2009\u2297\u2009|2k\u232aA\u23292k| onto the Alice\u2019s Hilbert space, it leaves i\u2009\u00b1\u2009(i\u2009+\u20091)\u232aB, i\u2009=\u20090, 2, \u2026, 2l\u2009\u2212\u20094.Using a rank-1 projector Ak+22\u2009=\u2009|0\u232aa\u23290|\u2009\u2297\u2009|2\u232aA\u23292|\u2009+\u2009|0\u232aa\u23290|\u2009\u2297\u2009|3\u232aA\u23293|\u2009+\u2009\u2026\u2009+\u2009|0\u232aa\u23290|\u2009\u2297\u2009|2k\u2009\u2212\u20091\u232aA\u23292k\u2009\u2212\u20091|, onto the Alice\u2019s Hilbert space, it leaves i\u2009=\u20092k\u2009+\u20092l\u2009\u2212\u20091, \u2026, 6k\u2009+\u20094l\u2009\u2212\u20099 and annihilates other states in (6). Then, Bob can identify i\u2009=\u20094k\u2009+\u20092l\u2009\u2212\u20092, \u2026, 4k\u2009+\u20094l\u2009\u2212\u20096 by 2l\u2009\u2212\u20093 projectors Bk+2)i1 projectors Ak+32\u2009=\u2009|0\u232aa\u23290|\u2009\u2297\u2009|0\u232aA\u23290|\u2009+\u2009|0\u232aa\u23290|\u2009\u2297\u2009|1\u232aA\u23291|\u2009+\u2009|1\u232aa\u23291|\u2009\u2297\u2009|0\u232aA\u23290|. This leaves Bk+3)12(2\u2009=\u2009|0\u232ab\u23290|\u2009\u2297\u2009(|1\u232aB\u23291|\u2009+\u2009|2\u232aB\u23292|\u2009+\u2009\u2026\u2009+\u2009|(2l\u2009\u2212\u20094)\u232aB\u2329(2l\u2009\u2212\u20094)|), it can leave i\u2009=\u20096k\u2009+\u20094l\u2009\u2212\u20096, \u2026, 6k\u2009+\u20096l\u2009\u2212\u200910. Then, Alice can easily identify A, and leave Bk+3)3(2\u2009=\u2009|0\u232ab\u23290|\u2009\u2297\u2009|(2l\u2009\u2212\u20093)\u232aB\u2329(2l\u2009\u2212\u20093)|\u2009+\u2009|0\u232ab\u23290|\u2009\u2297\u2009|(2l\u2009\u2212\u20092)\u232aB\u2329(2l\u2009\u2212\u20092)|\u2009+\u2009|1\u232ab\u23291|\u2009\u2297\u2009|(2l\u2009\u2212\u20091)\u232aB\u2329(2l\u2009\u2212\u20091)|, it leaves the last four states A31\u2009=\u2009|0\u232aa\u23290|\u2009\u2297\u2009|1\u232aA\u23291| and A32\u2009=\u2009|0\u232aa\u23290|\u2009\u2297\u2009|0\u232aA\u23290|\u2009+\u2009|1\u232aa\u23291|\u2009\u2297\u2009|0\u232aA\u23290| which leave two sets of orthogonal states Alice\u2019s last outcome is a rank-3 projector onto the remaining part of Alice\u2019s Hilbert space Therefore, a 2\u2009\u2297\u20092 maximally entangled state is sufficient to perfectly distinguish the states (4) by LOCC. This completes the proof.k)\u2009\u2297\u2009(2l)In this subsection, we consider the following locally indistinguishable orthogonal product states in \u2009\u2297\u2009(2l), a 2\u2009\u2297\u20092 maximally entangled state is sufficient to perfectly distinguish the above 6(k\u2009+\u2009l)\u22129 states by LOCC.TProof. Similarly to the proof of Theorem 2, Alice and Bob first share a 2\u2009\u2297\u20092 maximally entangled state B1. In the same way, the initial states are transformed as:As Theorem 1, we only need to discuss the projector i\u2009=\u20091, \u2026, 2k\u2009\u2212\u20092 by 2k\u2009\u2212\u20092 projectors Aj\u2009=\u2009|1\u232aa\u23291|\u2009\u2297\u2009|i\u2009+\u2009(i\u2009+\u20091)\u232aA\u2329i\u2009+\u2009(i\u2009+\u20091)|, Aj+1\u2009=\u2009|1\u232aa\u23291|\u2009\u2297\u2009|i\u2009\u2212\u2009(i\u2009+\u20091)\u232aA\u2329i\u2009\u2212\u2009(i\u2009+\u20091)|, i\u2009=\u20091, 3, \u2026, 2k\u2009\u2212\u20095, j\u2009=\u2009i, i\u2009=\u20092k\u2009\u2212\u20092, j\u2009=\u2009i\u2009\u2212\u20091, respectively.Then, Alice makes a (2k\u2009+\u20091)-outcome projective measurement. Similarly to the proof of Theorem 2, Alice can identify Ak2\u2009\u2212\u20091\u2009=\u2009|0\u232aa\u23290|\u2009\u2297\u2009|2k\u232aA\u23292k| onto the Alice\u2019s Hilbert space, it leaves i\u2009\u00b1\u2009(i\u2009+\u20091)\u232aB, i\u2009=\u20090, 2, \u2026, 2l\u2009\u2212\u20094.Using a rank-1 projector Ak2\u2009=\u2009|0\u232aa\u23290|\u2009\u2297\u2009|2\u232aA\u23292|\u2009+\u2009|0\u232aa\u23290|\u2009\u2297\u2009|3\u232aA\u23293|\u2009+\u2009\u2026\u2009+\u2009|0\u232aa\u23290|\u2009\u2297\u2009|2k\u2009\u2212\u20092\u232aA\u23292k\u2009\u2212\u20092|, it leaves i\u2009=\u20092k\u2009+\u20092l\u2009\u2212\u20093,\u2026, 6k\u2009+\u20094l\u2009\u2212\u200912. Similarly to the proof of Theorem 2, these states can also be distinguished by LOCC.When Alice uses a rank-(2k-3) projector Ak+12\u2009=\u2009|0\u232aa\u23290|\u2009\u2297\u2009|0\u232aA\u23290|\u2009+\u2009|0\u232aa\u23290|\u2009\u2297\u2009|1\u232aA\u23291|\u2009+\u2009|1\u232aa\u23291|\u2009\u2297\u2009|0\u232aA\u23290|, it leaves Bk+1)12(2\u2009=\u2009|0\u232ab\u23290|\u2009\u2297\u2009(|1\u232aB\u23291|\u2009+\u2009|2\u232aB\u23292|\u2009+\u2009\u2026\u2009+\u2009|(2l\u2009\u2212\u20094)\u232aB\u2329(2l\u2009\u2212\u20094)|), it can leave i\u2009=\u20096k\u2009+\u20094l\u2009\u2212\u20099, \u2026, 6k\u2009+\u20096l\u2009\u2212\u200913. Then, Alice can identify A, and leave Bk+1)3(2\u2009=\u2009|0\u232ab\u23290|\u2009\u2297\u2009|(2l\u2009\u2212\u20093)\u232aB\u2329(2l\u2009\u2212\u20093)|\u2009+\u2009|0\u232ab\u23290|\u2009\u2297\u2009|(2l\u2009\u2212\u20092)\u232aB\u2329(2l\u2009\u2212\u20092)|\u2009+\u2009|1\u232ab\u23291|\u2009\u2297\u2009|(2l\u2009\u2212\u20091)\u232aB\u2329(2l\u2009\u2212\u20091)|, it leaves the last four states For the last rank-3 projector Thus, the states (7) can be perfectly distinguished using LOCC with an additional resource of a 2\u2009\u2297\u20092 maximally entangled state. This completes the proof.k\u2009+\u20091)\u2009\u2297\u2009(2l\u2009+\u20091)In this subsection, we consider a generalization of (1) to higher-dimensional systems (2heorem 4 In (2k\u2009+\u20091)\u2009\u2297\u2009(2l\u2009+\u20091), a 2\u2009\u2297\u20092 maximally entangled state is sufficient to perfectly distinguish the above 6(k\u2009+\u2009l)\u22123 states by LOCC.Tk\u2009+\u20091)\u2009\u2297\u2009(2l\u2009+\u20091), the states (10) are a generalization of the states (1). Thus, the proof is similar to the proof of Theorem 1. In the following, we only give a simple proof.In \u232abB\u23290(2l\u2009\u2212\u20091)|\u2009+\u2009|1(2l)\u232abB\u23291(2l)| and B2\u2009=\u2009|10\u232abB\u232910|\u2009+\u2009|11\u232abB\u232911|\u2009+\u2009\u2026\u2009+\u2009|1(2l\u2009\u2212\u20091)\u232abB\u23291(2l\u2009\u2212\u20091)|\u2009+\u2009|0(2l)\u232abB\u23290(2l)|.B1, the initial states are transformed as:For i\u2009=\u20091, \u2026, 4k\u2009+\u20092l\u2009\u2212\u20092, 4k\u2009+\u20094l\u2009+\u20092, \u2026, 6k\u2009+\u20096l\u2009\u2212\u20093.In the same way, it is easily to distinguish Ak+32\u2009=\u2009|0\u232aa\u23290|\u2009\u2297\u2009|0\u232aA\u23290|\u2009+\u2009|0\u232aa\u23290|\u2009\u2297\u2009|1\u232aA\u23291|\u2009+\u2009|1\u232aa\u23291|\u2009\u2297\u2009|0\u232aA\u23290|, which leaves Bk+3)12(2\u2009=\u2009|0\u232ab\u23290|\u2009\u2297\u2009(|1\u232aB\u23291|\u2009+\u2009|2\u232aB\u23292|\u2009+\u2009\u2026\u2009+\u2009|(2l\u2009\u2212\u20092)\u232aB\u2329(2l\u2009\u2212\u20092)|), it can leave i\u2009=\u20094k\u2009+\u20092l\u2009+\u20091, \u2026, 4k\u2009+\u20094l\u2009\u2212\u20092, 4k\u2009+\u20094l\u2009+\u20091. Then, Alice can easily identify A, and leave Bk+3)3(2\u2009=\u2009|0\u232ab\u23290|\u2009\u2297\u2009|(2l\u2009\u2212\u20091)\u232aB\u2329(2l\u2009\u2212\u20091)|\u2009+\u2009|1\u232ab\u23291|\u2009\u2297\u2009|2l\u232aB\u23292l|, which leaves the last two orthogonal states Alice\u2019s last outcome is a rank-3 projector That is, we have succeeded in designing a protocol to distinguish the states (10) by LOCC with a two-qubit maximally entangled state. This completes the proof.In the proof, the important idea is that entanglement provides multiple Hilbert space, and that the parties can, independently of one another, act on these subspaces in ways that differ from one subspace to the next. This allows an apart of Hilbert space such that initially nonorthogonal pairs of local states end up being orthogonal, aiding the process of distinguishing the set of states. It should be noted that our protocols do not rely on details of the individual states, but only on the general way they are distributed through the space.m\u2009\u2297\u2009n for the above protocol. Furthermore, it will be good to do the analysis using n2, which is the optimal probability of converting such a state into a maximally entangled state, where m\u2265n. However, we have not succeeded in doing so. Hence, it remains an open question whether it is possible to distinguish these states with less than one ebit of entanglement.As Theorem 1, a 2\u2009\u2297\u20092 maximally entangled state is also necessary to distinguish these product states in In addition, each of the locally indistinguishable orthogonal product states that we consider can be extended to a complete \u201cnonlocality without entanglement\u201d basism\u2009\u2297\u2009n, which are constructed by Wang et al.et al.In this paper, we present that the locally indistinguishable orthogonal product states in How to cite this article: Zhang, Z.-C. et al. Entanglement as a resource to distinguish orthogonal product states. Sci. Rep.6, 30493; doi: 10.1038/srep30493 (2016)."}
{"text": "The objective is to determine the optimal resource allocation and the optimal schedule to minimize a total cost function that dependents on the total completion (waiting) time, the total machine load, the total absolute differences in completion (waiting) times on all machines, and total resource cost. If the number of machines is a given constant number, we propose a polynomial time algorithm to solve the problem. Yang and Kuo  denote the jth job on machine Mi and Ci[j](Wi[j]) denote the completion (waiting) time of job Ji[j]; pi[j], ai[j], bi[j], \u03b8i[j], ui[j], and Wi[j] = Ci[j] \u2212 pi[j]. Let Li = max\u2061{Cij | j = 1,2,\u2026, ni}, TCi = \u2211j=1niCij (TWi = \u2211j=1niWij), and TADCi = \u2211j=1ni\u2211l=jni | Cij \u2212 Cil| (TADWi = \u2211j=1ni\u2211l=jni | Wij \u2212 Wil|) be the load, the total completion (waiting) times, and the total absolute differences in completion (waiting) times of machine Mi. Then, the total machine load, the total completion (waiting) time, and the total absolute deviation of job completion (waiting) time on all machines are \u2211i=1mLi, \u2211i=1mTCi (\u2211i=1mTWi), and \u2211i=1mTADCi (\u2211i=1mTADWi), respectively. The objective is to determine the optimal resource allocations and the optimal schedule on the machines so that the corresponding value of the following cost functions is optimal:\u03b11 \u2265 0, \u03b12 \u2265 0, \u03b13 \u2265 0, and \u03b14 \u2265 0 are given constants and Gij is the per time unit cost associated with the resource allocation. Using the three-field notation introduced by Graham et al. ; in addition, Ci[j] = \u2211l=1jpi[l], Li = \u2211j=1nipi[j], and TCi = \u2211j=1niCi[j] = \u2211j=1ni(ni \u2212 j + 1)pi[j]; hence, we have\u03c9ij = \u03b11 + \u03b12(ni + 1 \u2212 j) + \u03b13(j \u2212 1)(ni \u2212 j + 1) andFrom Kanet , we haveFrom , for anyRm\u2009|\u2009pij = aijrbij + \u03b1t \u2212 \u03b8ijuij\u2009|\u2009\u03b11\u2211i=1mLi + \u03b12\u2211i=1mTCi + \u03b13\u2211i=1mTADCi + \u03b14\u2211i=1m\u2211j=1niGijuij can be determined byui[j]*, i = 1,2,\u2026, m; j = 1,2,\u2026, ni, represents the optimal resource allocation of the job in position j on machine Mi.For a given sequence, the optimal resource allocation of the problem Rm\u2009|\u2009pij = aijrbij + \u03b1t \u2212 \u03b8ijuij\u2009|\u2009\u03b11\u2211i=1mLi + \u03b12\u2211i=1mTCi + \u03b13\u2211i=1mTADCi + \u03b14\u2211i=1m\u2211j=1niGijuij, taking the derivative by ui[j] to /ui[j] = \u03b14Gi[j] \u2212 \u03b2i[j]\u03a9ij for i = 1,2,\u2026, m; j = 1,2,\u2026, ni. Hence, if \u03b14Gi[j] \u2212 \u03b2i[j]\u03a9ij > 0, we should not allocate any resource to job Ji[j]; if \u03b14Gi[j] \u2212 \u03b2i[j]\u03a9ij < 0, we will allocate the maximal feasible amount of resource to job Ji[j]; and if \u03b14Gi[j] \u2212 \u03b2i[j]\u03a9ij = 0, any feasible resource allocation can be optimal.For the problem i[j] to , we havexijr = 1 if job Jj is scheduled in position r on machine Mi, and xijr = 0 otherwise. If the number of jobs on machine Mi is known in advance, then we formulate the Rm\u2009|\u2009pij = aijrbij + \u03b1t \u2212 \u03b8ijuij\u2009|\u2009\u03b11\u2211i=1mLi + \u03b12\u2211i=1mTCi + \u03b13\u2211i=1mTADCi + \u03b14\u2211i=1m\u2211j=1niGijuij problem as the following assignment problem:We define bject to\u2211i=1m\u2009\u2211r=1,2,\u2026,n,\u2211j=1nxijrP =  vectors exist. Note that ni may be 0,1, 2,\u2026, n for i = 1,2,\u2026, m. So, if we get the numbers of jobs on the first m \u2212 1 machines, the number of jobs processed on the last machine is then determined uniquely due to n1 + n2 + \u22ef+nm = n. Therefore, the upper bound of the number of P vectors is (n + 1)m\u22121. Based on the above analysis, we have the following result.Next, the question is how many Rm\u2009|\u2009pij = aijrbij + \u03b1t \u2212 \u03b8ijuij\u2009|\u2009\u03b11\u2211i=1mLi + \u03b12\u2211i=1mTCi + \u03b13\u2211i=1mTADCi + \u03b14\u2211i=1m\u2211j=1niGijuij can be solved in O(nm+2) time; that is, the problem is polynomially solvable because m is a constant.The problem Rm\u2009|\u2009pij = aijrbij + \u03b1t \u2212 \u03b8ijuij\u2009|\u2009\u03b11\u2211i=1mLi + \u03b12\u2211i=1mTCi + \u03b13\u2211i=1mTADCi + \u03b14\u2211i=1m\u2211j=1niGijuij via the following algorithm.Base on the above analysis, we can determine the optimal solution for the problem Consider the following. Step\u2009\u20091. For all the possible vectors , solve the assignment problems .roblems (\u201316)). T). TStep\u2009Step\u2009\u20092. The optimal solution for the problem is the one with the minimum value of the total cost \u03b11\u2211i=1mLi + \u03b12\u2211i=1mTCi + \u03b13\u2211i=1mTADCi + \u03b14\u2211i=1m\u2211j=1niGijuij.Step\u2009\u20093. Calculate the optimal resources allocation by using pi[j] (Bagchi [\u03bdij = \u03b11 + \u03b12(ni \u2212 j) + \u03b13j(ni \u2212 j),Similar to the analysis of the problem  (Bagchi ) into  =  is given, the optimal sequence can be determined by solving the following assignment problem:For the problem Rm\u2009|\u2009pij = aijrbij + \u03b1t \u2212 \u03b8ijuij\u2009|\u2009\u03b11\u2211i=1mLi + \u03b12TWi + \u03b13TADWi + \u03b14\u2211i=1m\u2211j=1niGijuij can be solved in O(nm+2) time; that is, the problem is polynomially solvable because m is a constant.The problem Rm\u2009|\u2009pij = aijrbij + \u03b1t \u2212 \u03b8ijuij\u2009|\u2009\u03b11\u2211i=1mLi + \u03b12TWi + \u03b13TADWi + \u03b14\u2211i=1m\u2211j=1niGijuij can be obtained by the following algorithm.The optimal solution for the problem Consider the following. Step\u2009\u20091. For all the possible vectors , solve the assignment problems . roblems \u2013). ). Step\u2009\u2009Step\u2009\u20092. The optimal solution for the problem is the one with the minimum value of the total cost \u03b11\u2211i=1mLi + \u03b12\u2211i=1mTWi + \u03b13\u2211i=1mTADWi + \u03b14\u2211i=1m\u2211j=1niGijuij. Step\u2009\u20093. Calculate the optimal resources allocation by using  time algorithm for the problems Rm\u2009|\u2009pij = aijrbij + \u03b1t \u2212 \u03b8ijuij\u2009|\u2009\u03b11\u2211i=1mLi + \u03b12\u2211i=1mTCi + \u03b13\u2211i=1mTADCi + \u03b14\u2211i=1m\u2211j=1niGijuij and Rm\u2009|\u2009pij = aijrbij + \u03b1t \u2212 \u03b8ijuij\u2009|\u2009\u03b11\u2211i=1mLi + \u03b12\u2211i=1mTWi + \u03b13\u2211i=1mTADWi + \u03b14\u2211i=1m\u2211j=1niGijuij, respectively. The algorithms can also be easily applied to the cases bij > 0 (aging effect) and \u03b1 < 0. Future research may focus on similar problems with more general processing time model and extend the problems to flow shop machine settings.In this paper, we have studied the problem of scheduling"}
{"text": "A scheme is presented and software is documented for representing as integers input decimal numbers that have been stored in a computer as double precision floating point numbers and for carrying out multiplications, additions and subtractions based on these numbers in an exact manner. The input decimal numbers must not have more than nine digits to the left of the decimal point. The decimal fractions of their floating point representations are all first rounded off at a prespecified location, a location no more than nine digits away from the decimal point. The number of digits to the left of the decimal point for each input number besides not being allowed to exceed nine must then be such that the total number of digits from the leftmost digit of the number to the location where round-off is to occur does not exceed fourteen. Mainstream computers base integer and floating point arithmetic on fixed word lengths. As a consequence, only values with a limited number of significant digits can be represented directly, so that the results of arithmetic operations may have to be rounded off or truncated. Such errors can be avoided or, at least, mitigated, by implementing special algorithms for the execution of arithmetic operations. A fully \u201cexact arithmetic\u201d, however, would have to be based on quotients of integers for representing numerical values. In any case, a final limitation is due to finite memory.The need for exact arithmetic became apparent during the development of software for generating triangular and tetrahedral nets from very large point sets. Typically, this need is not due to high accuracy requirements for results\u2014the input data are often noisy or given up to only a few significant digits\u2014but is rather due to the need to maintain the consistency of a combinatorial structure. Building or manipulating such geometry-based combinatorial structures requires the calculation of indicators such as determinants in order to evaluate their sign and to check for zero values: round-off may lead to a false sign or zero value. An example considered later in this paper is to decide whether four given spatial points are coplanar. The approach of using exact computations for implementing computational geometry algorithms in a robust manner has been addressed in , 3], , , 6], , , [3], 4) reduce In this paper, we document software for exact integer arithmetic, accommodating an indeterminate number of digits, for multiplication, addition, subtraction, but excluding division. We found that, in many computational geometry applications, decision variables such as determinants can be calculated without division. Also the sign of a decision variable stated as a quotient but not evaluated is readily derived from the signs of the numerator and denominator.We also describe a preprocessing step, called \u201ctwo-integer decomposition\u201d, which leads from floating point input to one composed of integers only. At the root of this step lies the concept of space as an integer grid of points, all of which have integer coordinates in some shared unit. After completing the transition from floating point numbers to intermediate representations as pairs of integers\u2014prompted by the fact that Fortran 77 does not provide a double precision integer format\u2014a polynomial decomposition creates the number representations to be used in the exact arithmetic calculations. Software for this preprocessing step together with software for exact integer arithmetic has been successfully incorporated into several computational geometry related programs such as REGTET .In what follows, a \u201cstandard computer\u201d is a computer that uses 64 bits of storage for a double precision number and 32 bits for an integer. Given a standard computer, even though it may not store exactly an input decimal number as a double precision floating point number, it is safe to assume that the number will be represented as accurately as possible by a double precision floating point number up to its fourteenth significant digit.x(i), i = 1, \u2026, n, be a double precision array into which input numbers ix, i = 1, \u2026, n, have been read. The two-integer decomposition process is a preprocessing step that takes place before any computations based on the input data are carried out. It rounds off the numbers in the array at a prespecified location of their decimal fractions and decomposes each rounded off number into two integers that are saved in integer arrays, say ix(i), ix2(i), i = 1, \u2026, n. The rounded off numbers are then saved in array x(i), i = 1, \u2026, n.Let k, l, 1 \u2264 k \u2264 9, 0 \u2264 l \u2264 9, k + l \u2264 14, and assuming each input number ix, i = 1, \u2026, n, has no more that k digits to the left of the decimal point, each number x(i), i = 1, \u2026, n, is rounded off at the lth digit of its decimal fraction and decomposed into two integers in one of two ways according to its size. If the absolute value of x(i) times (10.0d0)l is less than 230(=1073741824), x(i) is multiplied by (10.0d0)l and rounded off at the decimal point. The resulting integer is then placed in ix(i) while ix2(i) is set to zero. Finally, x(i) is redefined to be the double precision value of integer ix(i) divided by (10.0d0)l. On the other hand, if the absolute value of x(i) times (10.0d0)l exceeds or equals 230, x(i) is truncated at the decimal point. The resulting integer  is placed in ix(i). In addition, the signed decimal fraction obtained by subtracting the double precision value of this integer from the initial value of x(i) is multiplied by (10.0d0)l and rounded off at the decimal point. The resulting integer  is placed in ix2(i). Next, x(i) is redefined to be the double precision value of integer ix(i) plus the value obtained by dividing the double precision value of integer ix2(i) by (10.0d0)l. Finally, if the integer ix2(i) is zero then ix2(i) is set to 230 so that ix2(i) is zero if and only if the initial absolute value of x(i) (before the two-integer decomposition process) times (10.0d0)l is less than 230.Given integers d0, when the signed decimal fraction of a number is obtained by truncating the number at its decimal point and subtracting the result from the initial value of the number, and when a number is rounded off with the two-integer decomposition process. However once the two-integer decomposition process is completed all computations that follow, exact and otherwise, are carried out in terms of the arrays x(i), ix(i), ix2(i), i = 1, \u2026, n, under the assumption that for the purposes of the user for each i, i = 1, \u2026, n, x(i) represents closely enough the input number ix rounded off at the lth digit of its decimal fraction, and an integer (not necessarily stored by the computer) in terms of ix(i) and ix2(i) represents closely enough the input number ix times 10l rounded off at the decimal point.The following is Fortran code for carrying out the two-integer decomposition process. Variables are either integer or double precision following convention. It is noted that while some loss in precision may occur at the time the input numbers are read and transformed into double precision floating point numbers, some additional loss in precision may occur here as well when the decimal point in a number is shifted by dividing or multiplying it by a multiple of 10.0\u2003\u2003mfull=1073741824\u2003\u2003if(l.lt.0 .or. l.gt.9) stop 10\u2003\u2003isclu = 1\u2003\u2003dscle = 1.0d0\u2003\u2003if(l.eq.0) go to 200\u2003\u2003do 100 i = 1, l\u2003\u2003isclu = 10*isclu\u2003\u2003dscle = 10.0d0*dscle100 continue200 continue\u2003 dfull = dble(mfull)\u2003 dfill=dfull/dscle\u2003 do 300 i = 1, n\u2003\u2003ix2(i) = 0\u2003\u2003if(dabs(x(i)).lt.dfill) then\u2003\u2003\u2003 ix(i) = idnint(dscle*x(i))\u2003\u2003\u2003if(iabs(ix(i)).lt.mfull) then\u2003\u2003\u2003\u2003 x(i) = dble(ix(i))/dscle\u2003\u2003\u2003\u2003go to 300\u2003\u2003\u2003endif\u2003\u2003endif\u2003\u2003if(dabs(x(i)).ge.dfull) stop 20\u2003\u2003ix(i) = idint(x(i))\u2003\u2003if(iabs(ix(i)).ge.mfull) stop 30\u2003\u2003decml = (x(i) - dint(x(i)))*dscle\u2003\u2003ix2(i) = idnint(decml)\u2003\u2003if(iabs(ix2(i)).eq.0) then\u2003\u2003\u2003x(i) = dble(ix(i))\u2003\u2003\u2003ix2(i) = mfull\u2003\u2003else\u2003\u2003\u2003x(i) = dble(ix(i)) + (dble(ix2(i))/dscle)\u2003\u2003endif300 continuel, 0 \u2264 l \u2264 9, let ix, i = 1, \u2026, n, be input numbers whose double precision floating point representations have been rounded off at the lth digit of their decimal fractions through the two-integer decomposition process. Let x(i), ix(i), ix2(i), i = 1, \u2026, n be the arrays produced by the two-integer decomposition process that contain the rounded off numbers and the two-integer decompositions. For each i, i = 1, \u2026, n, an integer J is symbolically defined as follows . If ix2(i) equals zero then J is set equal to ix(i). If ix2(i) equals 230 then J is set equal to ix(i) \u00b7 10l. Finally, if ix2(i) is neither zero nor 230 then J is set to ix(i) \u00b7 10l + ix2(i). In all cases for each i, i = 1, \u2026, n, J is considered to approximate closely enough (for the purposes of the user) the input number ix times 10l rounded off at the decimal point.Given an integer M to 215. Given J, 1 \u2264 i \u2264 n, the polynomial decomposition process is a procedure (presented below in the form of Fortran subroutine decmp2) that decomposes the integer J into a unique collection of integers isga, isga in {\u22121, 0, 1}, ika, ika > 0, ka, 0 \u2264 ka < M, k = 1, \u2026, ika, such that J equals isgaisga the sign of J. Integers ka, k = 1, \u2026, ika, are saved in an integer array, say ia(k), k = 1, \u2026, ika, and the collection of integers isga, ika, ia(k), k = 1, \u2026, ika, and the symbolic expression isgaJ, with isga as the sign of the representation. For each i, i = 1, \u2026, n, the polynomial decomposition of the integer J is identified each time an exact computation involving additions, subtractions, or multiplications is required that references the input number ix. During one such computation, for each i, 1 \u2264 i \u2264 n, if the number ix is referenced in the computation, once the polynomial decomposition of the corresponding integer J is identified, each reference of ix in the computation is replaced by the symbolic polynomial representation of J. The computation then takes effect as a sequence of additions, subtractions, or multiplications of symbolic polynomial representations with the final result being itself the symbolic polynomial representation of some integer. This final result can usually be used in only one of two ways. If it is known that for some positive integer m the integer that is equal to the final symbolic polynomial representation is approximately equal to the product of (10l)m and the true value of the computation, then this integer is computed approximately as a double precision floating point number from its symbolic polynomial representation and the true value of the computation is then approximately obtained by dividing it by ((10.0d0)l)m. On the other hand, if the purpose of the computation is simply that of obtaining the sign of the true result then the sign of the final symbolic polynomial representation is a satisfactory answer.Set J, 1 \u2264 i \u2264 n, can also be defined for any integer K (not necessarily stored by the computer) in the same manner. Accordingly, the following is a Fortran subroutine called decomp for finding the polynomial decomposition isga, ia(k), k = 1, 2,  of an integer iwi (stored by the computer) with absolute value less than 230. Here mhalf equals 215(=32768).The concepts of polynomial decomposition and symbolic polynomial representation defined above for subroutine decompinteger ia(*), isga, iwi, mhalf, iviif(iwi.gt.0) then\u2003isga = 1\u2003ivi = iwielseif(iwi.lt.0) then\u2003isga =-1\u2003ivi = -iwielse\u2003isga = 0\u2003ia(1) = 0\u2003ia(2) = 0\u2003returnendifia(2) = ivi/mhalfia(1) = ivi - ia(2)*mhalfreturnendisclu is set to 10l then isclu is less than 230 (since l \u2264 9) so that the polynomial decomposition isgu , iu(i), i = 1, 2,  of isclu can be obtained by calling subroutine decomp with a Fortran instruction as follows.In particular if call decompisga, ika, ia(k), k = 1, \u2026, ika, of the integer J, 1 \u2264 i \u2264 n. Here iwi equals ix(i), iwi2 equals ix2(i), mhalf equals 215, mfull equals 230, and iu(k), k = 1, 2, is an array such that the polynomial decomposition of 10l is iu(1), iu(2) . In addition, it is assumed that subroutines mulmul and muldif (presented below) exist for multiplying and subtracting, respectively, two symbolic polynomial representations.Finally, the following is a Fortran subroutine called decmp2 for finding the polynomial decomposition \u2003\u2003subroutine decmp2\u2003\u2003integer nkmax\u2003\u2003parameter (nkmax=5)\u2003\u2003integer ia(*), isga, ika, iwi, iwi2, mhalf, mfull, iu(*)\u2003\u2003integer ie(nkmax), io(nkmax), isge, isgo, ike, iko, isgu, iku\u2003\u2003call decomp\u2003\u2003ika = 2\u2003\u2003if(iwi2.ne.0) then\u2003\u2003\u2003isgu = 1\u2003\u2003\u2003iku = 2\u2003\u2003\u2003call mulmul\u2003\u2003\u2003if(iwi2.eq.mfull) iwi2 = 0\u2003\u2003\u2003call decomp\u2003\u2003\u2003isgo = -isgo\u2003\u2003\u2003iko = 2\u2003\u2003\u2003call muldif\u2003\u2003endif\u2003\u2003return\u2003\u2003endisga, ika, ia(k), k = 1, \u2026, ika, and isgb, ikb, ib(k), k = 1, \u2026, ikb, of two integers K1 and K2, respectively, the following is a Fortran subroutine called mulmul that produces the polynomial decomposition isgo, iko, io(k), k = 1, \u2026, iko, of the integer K1 \u00b7 K2 by multiplying the symbolic polynomial representation of K1 by that of K2  to produce a symbolic polynomial representation of K1 \u00b7 K2 from which the polynomial decomposition of K1 \u00b7 K2 can be obtained. Here nkmax is the dimension of the arrays ia, ib, io in the calling routine and mhalf equals 215. It is noted that the value of mhalf is of importance here since given integers i, j, 1 \u2264 i \u2264 ika, 1 \u2264 j \u2264 ikb, then 0 \u2264 ia(i) < 215, 0 \u2264 ib(j) < 215, so that the product ia(i) \u00b7 ib(j) is less than 230 and therefore can be stored in a 32 bit integer word.Given the polynomial decompositions \u2003\u2003subroutine mulmul\u2003\u2003integer ia(*), ib(*), io(*)\u2003\u2003integer isga, isgb, isgo, ika, ikb, iko, nkmax, mhalf\u2003\u2003integer i, ipt, ipr, iko1, k, j\u2003\u2003if(isga.eq.0.or.isgb.eq.0)then\u2003\u2003\u2003isgo=0\u2003\u2003\u2003iko = 2\u2003\u2003\u2003io(1) = 0\u2003\u2003\u2003io(2) = 0\u2003\u2003\u2003return\u2003\u2003endif\u2003\u2003iko = ika + ikb\u2003\u2003if(iko.gt.nkmax) stop 110\u2003\u2003if(isga.gt.0)then\u2003\u2003\u2003if(isgb.gt.0)then\u2003\u2003\u2003 \u2003isgo = 1\u2003\u2003\u2003else\u2003\u2003\u2003 \u2003isgo =-1\u2003\u2003\u2003endif\u2003\u2003else\u2003\u2003\u2003if(isgb.gt.0)then\u2003\u2003\u2003\u2003isgo =-1\u2003\u2003\u2003else\u2003\u2003\u2003\u2003isgo = 1\u2003\u2003\u2003endif\u2003\u2003endif\u2003\u2003\u2003iko1 = iko - 1\u2003\u2003\u2003ipr = 0\u2003\u2003\u2003do 200 i = 1, iko1\u2003\u2003\u2003ipt = ipr\u2003\u2003\u2003k = i\u2003\u2003\u2003do 180 j = 1, ikb\u2003\u2003\u2003if(k .lt. 1) go to 190\u2003\u2003\u2003if(k .gt. ika) go to 150\u2003\u2003\u2003ipt = ipt + ia(k)*ib(j)150\u2003\u00a0\u00a0continue\u2003\u2003\u2003k = k - 1180\u2003\u00a0\u00a0continue190\u2003\u00a0\u00a0continue\u2003\u2003\u2003ipr = ipt/mhalf\u2003\u2003\u2003io(i) = ipt - ipr*mhalf200\u2003\u00a0\u00a0continue\u2003\u2003\u2003io(iko) = ipr\u2003\u2003\u2003if stop 120\u2003\u2003\u2003iko1 = iko\u2003\u2003\u2003do 300 i = iko1, ika+1, -1\u2003\u2003\u2003if(io(i) .ne. 0) go to 400\u2003\u2003\u2003iko = iko - 1300\u2003\u00a0\u00a0continue400\u2003\u00a0\u00a0continue\u2003\u2003\u2003return\u2003\u2003\u2003endisga, ika, ia(k), k = 1, \u2026, ika, and isgb, ikb, ib(k), k = 1, \u2026, ikb, of two integers K1 and K2, respectively, the following is a Fortran subroutine called muldif that produces the polynomial decomposition isgo, iko, io(k), k = 1, \u2026, iko, of the integer K1 \u2212 K2 by subtracting the symbolic polynomial representation of K2 from that of K1  to produce a symbolic polynomial representation of K1 \u2212 K2 from which the polynomial decomposition of K1 \u2212 K2 can be obtained. Here nkmax is the dimension of the arrays ia, ib, io in the calling routine and mhalf equals 215. It is noted that by setting isgb equal to \u2212isgb the polynomial decomposition of K1 + K2 can also be obtained with this subroutine.Given the polynomial decompositions \u2003\u2003subroutine muldif\u2003\u2003integer ia(*), ib(*), io(*)\u2003\u2003integer isga, isgb, isgo, ika, ikb, iko, nkmax, mhalf\u2003\u2003integer i, iko1, irel\u2003\u2003if(isgb.eq.0)then\u2003\u2003\u2003if(isga.eq.0)then\u2003\u2003\u2003\u2003isgo=0\u2003\u2003\u2003\u2003iko = 2\u2003\u2003\u2003\u2003io(1) = 0\u2003\u2003\u2003\u2003io(2) = 0\u2003\u2003\u2003\u2003return\u2003\u2003\u2003endif\u2003\u2003\u2003isgo = isga\u2003\u2003\u2003iko = ika\u2003\u2003\u2003do 100 i=1,iko\u2003\u2003\u2003\u2003io(i) = ia(i)100\u2003\u00a0\u00a0continue\u2003\u2003elseif(isga.eq.0)then\u2003\u2003\u2003isgo =-isgb\u2003\u2003\u2003iko = ikb\u2003\u2003\u2003do 200 i=1,iko\u2003\u2003\u2003\u2003io(i) = ib(i)200\u2003\u00a0\u00a0continue\u2003\u2003else\u2003\u2003\u2003iko = ika\u2003\u2003\u2003if(ikb.lt.ika) then\u2003\u2003\u2003\u2003do 300 i=ikb+1,ika\u2003\u2003\u2003\u2003\u2003ib(i) = 0300\u2003\u2003continue\u2003\u2003\u2003elseif(ika.lt.ikb) then\u2003\u2003\u2003\u2003iko = ikb\u2003\u2003\u2003\u2003do 400 i=ika+1,ikb\u2003\u2003\u2003\u2003\u2003ia(i) = 0400\u2003\u2003continue\u2003\u2003\u2003endif\u2003\u2003\u2003if(isga*isgb.gt.0)then\u2003\u2003\u2003\u2003irel = 0\u2003\u2003\u2003\u2003do 500 i = iko, 1, -1\u2003\u2003\u2003\u2003\u2003if(ia(i).gt.ib(i))then\u2003\u2003\u2003\u2003\u2003\u2003irel = 1\u2003\u2003\u2003\u2003\u2003\u2003go to 600\u2003\u2003\u2003\u2003elseif(ia(i).lt.ib(i))then\u2003\u2003\u2003\u2003\u2003\u2003irel = -1\u2003\u2003\u2003\u2003\u2003\u2003go to 600\u2003\u2003\u2003\u2003endif500\u2003\u2003continue600\u2003\u2003continue\u2003\u2003\u2003\u2003if(irel.eq.0)then\u2003\u2003\u2003\u2003\u2003isgo = 0\u2003\u2003\u2003\u2003\u2003do 700 i=1,iko\u2003\u2003\u2003\u2003\u2003\u2003io(i) = 0700\u2003\u2003\u2003\u00a0\u00a0continue\u2003\u2003\u2003\u2003else\u2003\u2003\u2003\u2003\u2003isgo=isga*irel\u2003\u2003\u2003\u2003\u2003io(1) = (ia(1)-ib(1))*irel\u2003\u2003\u2003\u2003\u2003do 800 i=2,iko\u2003\u2003\u2003\u2003\u2003\u2003if(io(i-1).lt.0) then\u2003\u2003\u2003\u2003\u2003\u2003\u2003io(i) =-1\u2003\u2003\u2003\u2003\u2003\u2003\u2003io(i-1) = io(i-1) + mhalf\u2003\u2003\u2003\u2003\u2003\u2003else\u2003\u2003\u2003\u2003\u2003\u2003\u2003io(i) = 0\u2003\u2003\u2003\u2003\u2003\u2003endif\u2003\u2003\u2003\u2003\u2003\u2003io(i) = io(i) + (ia(i)-ib(i))*irel800\u2003\u2003\u2003\u00a0\u00a0continue\u2003\u2003\u2003\u2003\u2003if(io(iko).lt.0) stop 210\u2003\u2003\u2003\u2003endif\u2003\u2003else\u2003\u2003\u2003\u2003isgo=isga\u2003\u2003\u2003\u2003io(1) = ia(1)+ib(1)\u2003\u2003\u2003\u2003do 900 i=2,iko\u2003\u2003\u2003\u2003\u2003if(io(i-1).ge.mhalf) then\u2003\u2003\u2003\u2003\u2003\u2003io(i) = 1\u2003\u2003\u2003\u2003\u2003\u2003io(i-1) = io(i-1) - mhalf\u2003\u2003\u2003\u2003\u2003else\u2003\u2003\u2003\u2003\u2003io(i) = 0\u2003\u2003\u2003\u2003\u2003endif\u2003\u2003\u2003\u2003\u2003io(i) = io(i) + ia(i)+ib(i)900\u2003\u2003continue\u2003\u2003\u2003\u00a0\u00a0if(io(iko).ge.mhalf) then\u2003\u2003\u2003\u2003\u2003iko = iko+1\u2003\u2003\u2003\u2003\u2003if(iko.gt.nkmax) stop 220\u2003\u2003\u2003\u2003\u2003io(iko) = 1\u2003\u2003\u2003\u2003\u2003io(iko-1) = io(iko-1) - mhalf\u2003\u2003\u2003endif\u2003\u2003endif\u2003endif\u2003if(iko .eq. 2) go to 990\u2003iko1 = iko\u2003do 950 i = iko1, 3, -1\u2003\u2003if(io(i) .ne. 0) go to 990\u2003\u2003iko = iko - 1950continue990continue\u2003\u00a0\u00a0return\u2003\u00a0\u00a0endn, n \u2265 4, let S be a set of n points in 3-dimensional space. Given an integer l, 0 \u2264 l \u2264 9, let ix, iy, iz, i = 1, \u2026, n, be the input decimal coordinates of the points in S, and assume that their double precision floating point representations have been rounded off at the lth digit of their decimal fractions through applications, one per coordinate, of the two-integer decomposition process. Accordingly, let x(i), ix(i), ix2(i), y(i), iy(i), iy2(i), z(i), iz(i), iz2(i), i = 1, \u2026, n, be the arrays produced by the three applications of the two-integer decomposition process that contain the rounded off x-, y-, z-coordinates and their two-integer decompositions.Given an integer p1, p2, p3 in S that are vertices of a non-degenerate triangle, a fundamental problem in computational geometry is that of finding the location of a point p4 in S relative to the plane H that contains the triangle. Let H+ be the open half-space defined by H for which p1, p2, p3 appear in a counterclockwise direction around the boundary of the triangle when looking at the triangle from H+. Let H\u2212 be the other half-space defined by H. Determining in which of H, H+, H\u2212, the point p4 is located may not on occasion be satisfactorily done using floating point arithmetic. Accordingly, the following is a Fortran subroutine called crsinn for doing this using polynomial decompositions. On output the sign isgo  of some polynomial decomposition determines the location of p4 .Given points isgox, ikox, iox(k), k = 1, \u2026, ikox, isgoy, ikoy, ioy(k), k = 1, \u2026, ikoy, isgoz, ikoz, ioz(k), k = 1, \u2026, ikoz, of integers that are the coordinates of a vector v pointing into H+ and perpendicular to H. It also produces the polynomial decomposition isgo, iko, io(k), k = 1, \u2026, iko, of an integer whose sign isgo determines the location of p4 and whose value when divided by both 10l and the length of v is the perpendicular distance from p4 to H. Here mhalf equals 215, mfull equals 230, and if ir, isec, ithi, ifou are locations in the arrays ix, ix2, etc. corresponding to the points p1, p2, p3, p4, respectively. In addition, isclp(k), k = 1, 2, is an array such that the polynomial decomposition of 10l is isclp(1), isclp(2) .This routine actually does more. It produces polynomial decompositions \u2003\u2003subroutine crsinn\u2003\u2003integer ix(*), iy(*), iz(*), ix2(*), iy2(*), iz2(*)\u2003\u2003integer io(*), iox(*),ioy(*), ioz(*)\u2003\u2003integer ifir, isec, ithi, ifou\u2003\u2003integer isclp(*), mhalf, mfull, nkmax\u2003\u2003parameter (nkmax = 30)\u2003\u2003integer iu(nkmax), iv(nkmax), iw(nkmax)\u2003\u2003integer ixt(nkmax), iyt(nkmax), izt(nkmax)\u2003\u2003integer ix3(nkmax), iy3(nkmax), iz3(nkmax)\u2003\u2003integer ix4(nkmax), iy4(nkmax), iz4(nkmax)\u2003\u2003integer ixf(nkmax), iyf(nkmax), izf(nkmax)\u2003\u2003integer ixfiw, iyfiw, izfiw, ixsew, iysew, izsew\u2003\u2003integer ixthw, iythw, izthw, ixfow, iyfow, izfow\u2003\u2003integer ixfi2, iyfi2, izfi2, ixse2, iyse2, izse2\u2003\u2003integer ixth2, iyth2, izth2, ixfo2, iyfo2, izfo2\u2003\u2003integer isgxf, isgyf, isgzf, ikxf, ikyf, ikzf\u2003\u2003integer isgx2, isgy2, isgz2, ikx2, iky2, ikz2\u2003\u2003integer isgx3, isgy3, isgz3, ikx3, iky3, ikz3\u2003\u2003integer isgx4, isgy4, isgz4, ikx4, iky4, ikz4\u2003\u2003integer isgo, iko, isgox, ikox, isgoy, ikoy, isgoz, ikoz\u2003\u2003integer isgu, isgv, isgw, iku, ikv, ikw\u2003\u2003ixfiw = ix(ifir)\u2003\u2003iyfiw = iy(ifir)\u2003\u2003izfiw = iz(ifir)\u2003\u2003ixsew = ix(isec)\u2003\u2003iysew = iy(isec)\u2003\u2003izsew = iz(isec)\u2003\u2003ixthw = ix(ithi)\u2003\u2003iythw = iy(ithi)\u2003\u2003izthw = iz(ithi)\u2003\u2003ixfow = ix(ifou)\u2003\u2003iyfow = iy(ifou)\u2003\u2003izfow = iz(ifou)\u2003\u2003ixfi2 = ix2(ifir)\u2003\u2003iyfi2 = iy2(ifir)\u2003\u2003izfi2 = iz2(ifir)\u2003\u2003ixse2 = ix2(isec)\u2003\u2003iyse2 = iy2(isec)\u2003\u2003izse2 = iz2(isec)\u2003\u2003ixth2 = ix2(ithi)\u2003\u2003iyth2 = iy2(ithi)\u2003\u2003izth2 = iz2(ithi)\u2003\u2003ixfo2 = ix2(ifou)\u2003\u2003iyfo2 = iy2(ifou)\u2003\u2003izfo2 = iz2(ifou)\u2003\u2003call decmp2\u2003\u2003call decmp2\u2003\u2003call decmp2\u2003\u2003call decmp2\u2003\u2003call muldif\u2003\u2003call decmp2\u2003\u2003call muldif\u2003\u2003call decmp2\u2003\u2003call muldif\u2003\u2003call decmp2\u2003\u2003call muldif\u2003\u2003call decmp2\u2003\u2003call muldif\u2003\u2003call decmp2\u2003\u2003call muldif\u2003\u2003call decmp2\u2003\u2003call muldif\u2003\u2003call decmp2\u2003\u2003call muldif\u2003\u2003call decmp2\u2003\u2003call muldif\u2003\u2003call mulmul\u2003\u2003call mulmul\u2003\u2003call muldif\u2003\u2003call mulmul\u2003\u2003call mulmul\u2003\u2003call mulmul\u2003\u2003call muldif\u2003\u2003call mulmul\u2003\u2003isgu =-isgu\u2003\u2003call muldif\u2003\u2003call mulmul\u2003\u2003call mulmul\u2003\u2003call muldif\u2003\u2003call mulmul\u2003\u2003isgu =-isgu\u2003\u2003call muldif\u2003\u2003return\u2003\u2003endp4 relative to the plane H it may be desirable to know the perpendicular distance from p4 to H. The following is Fortran code for this purpose. It uses the polynomial decompositions that are part of the output of subroutine crsinn. Variables here are either integer or double precision following convention. Here r215 equals (2.0d0)15, dscle equals (10.0d0)l, and dist is the resulting signed perpendicular distance. In addition, it is assumed that subroutine doubnm (presented below) exists for transforming the polynomial decomposition of an integer into the double precision floating point value of the integer.Sometimes besides knowing the location of the point \u2003\u2003call crsinn\u2003\u2003call doubnm\u2003\u2003call doubnm\u2003\u2003call doubnm\u2003\u2003call doubnm\u2003\u2003dnux = dmax1(dabs(xnum),dabs(ynum),dabs(znum))\u2003\u2003xnum = xnum/dnux\u2003\u2003ynum = ynum/dnux\u2003\u2003znum = znum/dnux\u2003\u2003dnom = dsqrt(xnum**2+ynum**2+znum**2)\u2003\u2003dist = ((dnum/dnux)/dnom)/dscleThe following is subroutine doubnm that was called above.\u2003\u2003subroutine doubnm\u2003\u2003integer io(*)\u2003\u2003double precision dnum, r215, rpwr\u2003\u2003integer isgo, iko, i\u2003\u2003if(isgo.eq.0) then\u2003\u2003\u2003\u2003dnum = 0.0d0\u2003\u2003\u2003\u2003go to 900\u2003\u2003else\u2003\u2003\u2003\u2003if(iko .lt. 2) stop 310\u2003\u2003\u2003\u2003if(iko .gt. 68) stop 320\u2003\u2003\u2003\u2003rpwr = 1.0d0\u2003\u2003\u2003\u2003dnum = dble(io(1))\u2003\u2003\u2003\u2003do 100 i = 2, iko\u2003\u2003\u2003\u2003\u2003\u00a0\u00a0rpwr = rpwr*r215\u2003\u2003\u2003\u2003\u2003\u00a0\u00a0dnum = dnum + dble(io(i))*rpwr100\u2003\u2003\u00a0\u00a0continue\u2003\u2003endif\u2003\u2003if(isgo.lt.0) dnum = -dnum900 continue\u2003\u2003return\u2003\u2003endx-, y-, z-coordinates of the point, in that order. Given i, 1 \u2264 i \u2264 12, point i is the point corresponding to the ith line. It is assumed that the coordinates of the twelve points are read into double precision arrays x(i), y(i), z(i), i = 1, \u2026, 12, so that x(i), y(i), z(i) contain the x-, y-, z-coordinates, respectively, of point i.Twelve lines follow, each line containing three numbers. Each line corresponds to a point in 3-dimensional space, and the three numbers in the line correspond to the l equal to 8, through the two-integer decomposition process, the numbers above are rounded off at the lth = 8th digit of their decimal fractions and saved in x(i), y(i), z(i), i = 1, \u2026, 12, so that then they appear as follows.Given i, i = 1, \u2026, 12, the first two integers in the ith line are the two integers into which the x-coordinate of point i is decomposed. Similarly the next two integers correspond to the y-coordinate, and the last two to the z-coordinate. The two-integer decompositions of the twelve points are then saved into ix(i), ix2(i), iy(i), iy2(i), iz(i), iz2(i), i = 1, \u2026, 12, in the obvious manner. It is noted that when mfull = 230 = 1073741824 appears as the second integer corresponding to a coordinate it is to be interpreted as a zero.Each rounded off coordinate is also decomposed into two integers. Twelve lines follow, each line containing six integers. For each l as above equal to 8 and setting isclu to 10l = 108, by calling subroutine decomp the polynomial decomposition of isclu is found to be isgcl, isclp(1), isclp(2), with isgcl equal to 1, isclp(1) equal to 24832 and isclp(2) equal to 3051 .Given x-coordinate of point 2 and the y-coordinate of point 4 minus the product of the y-coordinate of point 2 and the x-coordinate of point 4 is described. Using rounded-off numbers the result of the computation should equal x(2) \u00b7 y(4) \u2212 y(2) \u00b7 x(4), i. e.,As an example of how exact computations are carried out that reference coordinates of the input points given above, the computation that is the product of the (38.000000000) \u00b7 (\u221249.840807040) \u2212 (7.049967880) \u00b7 (85.557213020).8 rounded off at the decimal point. Since the resulting integer may be too big to be saved into a 32 bit integer word, its polynomial decomposition in the form of two or more 32 bit integer words is obtained instead. Thus, for example, the polynomial decomposition of the integer which is the x-coordinate of point 4 times 108 rounded off at the decimal point can be obtained by calling subroutine decmp2 using the two-integer decomposition of the coordinate, i.e. the integers 85 and 55721302, and the polynomial decomposition of 108 as obtained above. The resulting polynomial decomposition is then found to be isgox4, ikox4, iox4(k), k = 1, \u2026, ikox4, with isgox4 equal to 1, ikox4 equal to 3, iox4(k), k = 1, 2, 3, equal to 29270, 31723, and 7  + 7 \u00b7 (215)2). In the same manner the polynomial decompositions associated with the x-, y-coordinates of point 2, and the y-coordinate of point 4 are found to be, respectively, isgox2, ikox2, iox2(k), k = 1, \u2026, ikox2, isgoy2, ikoy2, ioy2(k), k = 1, \u2026, ikoy2, isgoy4, ikoy4, ioy4(k), k = 1, \u2026, ikoy4, with isgox2 equal to 1, ikox2 equal to 3, iox2(k), k = 1, 2, 3, equal to 26112, 17662, 3, isgoy2 equal to 1, ikoy2 equal to 2, ioy2(k), k = 1, 2, equal to 26036, 21514, isgoy4 equal to \u22121, ikoy4 equal to 3, ioy4(k), k = 1, 2, 3, equal to 2368, 21030, 4. Finally, using these polynomial decompositions as input the desired result is obtained by calling subroutines mulmul and muldif as follows. Here nkmax is the dimension of all of the arrays (input and output).On the other hand, if exact computations are required then each of the four numbers involved must first be converted into an integer which is the number times 10call mulmulcall mulmulcall muldif8)2 rounded off at the decimal point is then found to be isgo, iko, io(k), k = 1, \u2026, iko, with isgo equal to \u22121, iko equal to 5, io(k), k = 1, \u2026, 5, equal to 21112, 15183, 31880, 21597, 21. The desired result is then approximately equal to this integer, i.e. 21112 + 15183 \u00b7 (215) + 31880 \u00b7 (215)2 + 21597 \u00b7 (215)3 + 21 \u00b7 (215)4, divided by (108)2. By calling subroutine doubnm the integer can be approximated by a double precision number which when divided by (108)2 is approximately \u22122497.1262712133, an approximation of the desired result.The polynomial decomposition of an integer that approximates the desired result times isgo equals \u22121 so that point 12 must be in H\u2212. In addition iko equals 7, io(k), k = 1, \u2026, 7, equals 21844, 27853, 3870, 5372, 13630, 9887, 213, isgox equals \u22121, ikox equals 5, iox(k), k = 1, \u2026, 5, equals 11868, 15341, 2677, 15631, 62, isgoy equals 1, ikoy equals 5, ioy(k), k = 1, \u2026, 5, equals 11577, 8756, 364, 27887, 63, isgoz equals \u22121, ikoz equals 5, ioz(k), k = 1, \u2026, 5, equals 5921, 23919, 26934, 16812, 19. By calling subroutine doubnm the integer whose polynomial decomposition is isgo, iko, io(k), k = 1, \u2026, iko, can be approximated by a double precision number dnum. Similarly, the three integers, say ix, iy, iz, whose polynomial decompositions are isgox, ikox, iox(k), k = 1, \u2026, ikox, isgoy, ikoy, ioy(k), k = 1, \u2026, ikoy, isgoz, ikoz, ioz(k), k = 1, \u2026, ikoz, can be approximated, respectively, by double precision numbers xnum, ynum, znum. The vector , which is perpendicular to the plane H and points into H+, can then be approximated by the vector . Finally, by dividing dnum by both the length of the vector  and 108 the number \u221225.047402554921 is approximately obtained, an approximation of the signed perpendicular distance from point 12 to plane H, the negative sign indicating that point 12 is in H\u2212.On output http://math.nist.gov/~JBernal/JBernal_Sft.html. Besides being used for locating a point relative to a plane, in these programs the scheme has also been used for locating a point relative to a sphere, for computing the intersection of a line and a plane, for computing the center of a sphere, etc.A scheme has been presented and software has been documented for transforming into a series of integers input decimal numbers that have been read into a computer as double precision floating point numbers and for carrying out multiplication, addition and subtraction operations based on these numbers using their integer representations. The total number of significant digits of each input number must not be greater than 14, and the number of digits to the left of the decimal point must not exceed 9. Through a preprocessing step the double precision floating point representation of each input decimal number is rounded off at a prespecified location of its decimal fraction, a location no more than 9 digits to the right of the decimal point, and the rounded off number is decomposed into two integers. All operations that follow involving the input number are then carried out in terms of the rounded off double precision floating point number and when this is not satisfactory in terms of the two integers. This scheme has been successfully incorporated into several computational geometry related programs such as REGTET ["}
{"text": "Scientific Reports5: Article number: 1061310.1038/srep10613; published online: 06032015; updated: 01122016This Article contains typographical errors in the keys of Figure 4c, 4d, 4e, and 4f.6 \u2018\u03b7or\u2019 have been incorrectly given as \u2018\u03b1\u2019.In Figure 4c, the values of the exponents for power-law decay of GK \u2018\u03b7p\u2019 have been incorrectly given as \u2018\u03b1\u2019.In Figure 4e the values of the exponents for power-law decay of Gp\u2019 and \u2018\u03beor\u2019 which were incorrectly given as \u2018[p\u2019 and \u2018[or\u2019 respectively.Lastly in Figures 4d and 4f, there are typographical errors in the symbols \u2018\u03beThe correct Figure 4 appears below as"}
{"text": "In Volume 93, Issue 10, October 2015, throughout pages 674\u221283 \u201cRivaroxavan\u201d should read: \u201cRivaroxaban\u201d."}
{"text": "T\u2227-fuzzy (implicative) ideal is proposed in MV-algebras. The relationships between falling fuzzy (implicative) ideals and T-fuzzy (implicative) ideals are discussed, and conditions for a falling fuzzy (implicative) ideal to be a T\u2227-fuzzy (implicative) ideal are provided. Some characterizations of falling fuzzy (implicative) ideals are presented by studying proprieties of them. The product \u229b and the up product \u229a operations on falling shadows and the upset of a falling shadow are established, by which T-fuzzy ideals are investigated based on probability spaces.Based on the falling shadow theory, the concept of falling fuzzy (implicative) ideals as a generalization of that of a  Then there is a unique operation x \u2192 y satisfying the condition T \u2264 y if and only if z \u2264 x \u2192 y for all x, y, z \u2208 ; namely, x \u2192 y = max\u2061{z\u2223T \u2264 y}.Let x \u2192 y from t-norm.The operation A, \u2295, \u00ac, 0) of type  satisfying the following equations: for any x, y, z \u2208 A, (MV1)x \u2295 (y \u2295 z) = (x \u2295 y) \u2295 z;(MV2)x \u2295 y = y \u2295 x;(MV3)x \u2295 0 = x;(MV4)x = x;\u00ac\u00ac(MV5)x \u2295 \u00ac0 = \u00ac0;(MV6)x \u2295 y) \u2295 y = \u00ac(\u00acy \u2295 x) \u2295 x.\u00ac be an MV-algebra. For any x, y \u2208 A, we define 1 = \u00ac0, x \u2297 y = \u00ac(\u00acx \u2295 \u00acy), x \u2192 y = \u00acx \u2295 y, x \u2296 y = x \u2297 \u00acy, x\u2228y = \u00ac(\u00acx \u2295 y) \u2295 y = (x \u2296 y) \u2295 y, x\u2227y = \u00ac(\u00acx\u2228\u00acy), 0x = 0, (n + 1)x = nx \u2295 x, x0 = 1, and xn+1) is a bounded distributive lattice 2. There are many types of probability distributions, but only three types are the most classic ones \u2009\u2009x, y \u2208 I implies x \u2295 y \u2208 I; (2)\u2009\u2009x \u2264 y and y \u2208 I imply x \u2208 I. For purpose of convenience, let \u2205 be an ideal of A in the the rest of the sections. If an ideal I satisfies the following condition: (x \u2296 y) \u2296 z \u2208 I and y \u2296 z \u2208 I imply x \u2296 z \u2208 I for any x, y, z \u2208 A, then I is called an implicative ideal of A.A nonempty set I be a nonempty set of A. Then the following statements are equivalent: I is an ideal of A;I; \u2200x, y \u2208 A, x \u2296 y \u2208 I, and y \u2208 I imply x \u2208 I;0 \u2208 x, y, z \u2208 A, if z \u2296 x \u2264 y and x, y \u2208 I, then z \u2208 I.\u2200Let T-fuzzy ideals and falling fuzzy ideals of MV-algebras. The relationships between T-fuzzy ideals and falling fuzzy ideals are provided, and some characterizations of falling fuzzy ideals are displayed.In this section, we will introduce the notions of \u03bc of A is called a T-fuzzy ideal of A, if \u03bc satisfies \u2200x, y \u2208 A, \u03bc(x \u2295 y) \u2265 T(\u03bc(x), \u03bc(y));x \u2264 y implies \u03bc(x) \u2265 \u03bc(y).A fuzzy set T-fuzzy ideal of A is a T\u2227-fuzzy ideal when T = T\u2227 and a T\u2227-fuzzy ideal is also called a fuzzy ideal , L is an ideal of A, where L = {x \u2208 A\u2223\u03bc(x) \u2265 t}.\u2200Let T-fuzzy ideals for the further discussion.In the following, we give some properties of \u03bc1, \u03bc2 be fuzzy sets of A. If \u03bc1, \u03bc2 are T-fuzzy ideals of A, then \u03bc1\u2293\u03bc2 is a T-fuzzy ideal of A.Let The proof is straightforward.As a direct consequence of \u03bci\u2009\u2009(i \u2208 \u039b) be fuzzy sets of A, where \u039b is an index set. If \u03bci is a T-fuzzy ideal of A, then \u2293i\u2208\u039b\u03bci is a T-fuzzy ideal of A.Let A = {0, a, b, 1} where 0 < a < 1, 0 < b < 1. Define the operations \u2295 and \u00ac on A as follows:Let A, \u2295, \u00ac, 0) is an MV-algebra. Define fuzzy sets \u03bc1 and \u03bc2 of A as follows:It is clear that (a \u2295 b) = 0.3 < (\u03bc1\u2294\u03bc2)(a)\u2227(\u03bc1\u2294\u03bc2)(b) = 0.6\u22270.5 = 0.5.Then T-fuzzy ideals is not a T-fuzzy ideal. In order to investigate the algebraic properties of the set of all T\u2227-fuzzy ideals in BL-algebras,  and m is the usual Lebesgue measure. Denote A1 = { \u2208 R2\u2223y > 0} and A2 = { \u2208 R2\u2223y < 0}. The mapping \u03be : \u03a9 \u2192 P(A) is defined by\u03be(t) is an ideal of A for any t \u2208 . Thus A, where Let  be a probability space and F(A): = {f\u2223f : \u03a9 \u2192 A}. Define the operations \u229e and ~ on F(A) as follows: \u2200\u03c9 \u2208 \u03a9, f, g \u2208 F(A),Let (F(A) be defined by 0(\u03c9) = 0 for any \u03c9 \u2208 \u03a9. Then it is easy to check that (F(A), \u229e, ~, 0) is an MV-algebra. Define the operation \u229f on F(A) by (f\u229fg)(\u03c9) = f(\u03c9) \u2296 g(\u03c9), for any \u03c9 \u2208 \u03a9, f, g \u2208 F(A).Let 0 \u2208 S of A and f \u2208 F(A), let Sf : = {\u03c9 \u2208 \u03a9\u2223f(\u03c9) \u2208 S} andSf \u2208 A.For any subset \u03a9, A, P) be a probability space, S a nonempty subset of A, and \u03c9 \u2208 \u03a9. If S is an ideal of A, then \u03be(\u03c9) = {f \u2208 F(A)\u2223f(\u03c9) \u2208 S} is an ideal of F(A).Let  = 0 \u2208 S for any \u03c9 \u2208 \u03a9, we have that 0 \u2208 \u03be(\u03c9). Let f, g \u2208 \u03be(\u03c9) be such that f\u229fg \u2208 \u03be(\u03c9) and g \u2208 \u03be(\u03c9). For any \u03c9 \u2208 \u03a9, we have f(\u03c9) \u2296 g(\u03c9) = (f\u229fg)(\u03c9) \u2208 A and g(\u03c9) \u2208 S. Hence f(\u03c9) \u2208 S; that is, f \u2208 \u03be(\u03c9). Thus \u03be(\u03c9) is an ideal of F(A).Suppose that \u03be\u22121(f) = {\u03c9 \u2208 \u03a9\u2223f \u2208 \u03be(\u03c9)} = {\u03c9 \u2208 \u03a9\u2223f(\u03c9) \u2208 S} = Sf \u2208 A and \u03be is a random set of F(A), we get that F(A), where Noticing that \u03bc be a T\u2227-fuzzy ideal of A. Then \u03bc is a falling fuzzy ideal of A.Let \u03a9, A, P) = , where A is a Borel field on  and m is the usual Lebesgue measure. Since \u03bc is a T\u2227-fuzzy ideal of A, then L is an ideal of A for any t \u2208 , by \u03be :  \u2192 P(A) by \u03be(t) = L for any t \u2208 ; then \u03bc is a falling fuzzy ideal of A.Consider the probability space  be the power set of E. Let \u2295, \u00ac, and 0 denote, respectively, the join, the complement, and the smallest element in A : = P(E). It is clear that  is an MV-algebra.Let \u03a9, A, P) = , where A is a Borel field on  and m is the usual Lebesgue measure. The mapping \u03be : \u03a9 \u2192 P(A) is defined byLet (\u03be(t) is an ideal of A for any t \u2208 . Thus A, where Then T\u2227-fuzzy ideal of A since But \u03a9, A, P) be a probability space and \u03be : \u03a9 \u2192 P(A). For any x \u2208 A, let \u03a9 = {\u03c9 \u2208 \u03a9\u2223x \u2208 \u03be(\u03c9)}; then \u03a9 \u2208 A. In what follows, we give a number of equivalent conditions of falling fuzzy ideals for further discussion.Let (\u03be : \u03a9 \u2192 P(A). Then A if and only if (1)\u2009\u20090 \u2208 \u03be(\u03c9), (2)\u2009\u2009y \u2208 \u03be(\u03c9) and x \u2208 A\u2216\u03be(\u03c9) imply x \u2296 y \u2208 A\u2216\u03be(\u03c9) for any \u03c9 \u2208 \u03a9, x, y \u2208 A.Let A, then \u03be(\u03c9) is an ideal of A for any \u03c9 \u2208 \u03a9, and it follows that 0 \u2208 \u03be(\u03c9). Let x, y \u2208 A be such that y \u2208 \u03be(\u03c9) and x \u2208 A\u2216\u03be(\u03c9). Supposing that x \u2296 y \u2208 \u03be(\u03c9), then x \u2208 \u03be(\u03c9), a contradiction, and thus (2) is valid.Assuming that \u03c9 \u2208 \u03a9, let x, y \u2208 A be such that x \u2296 y \u2208 \u03be(\u03c9) and y \u2208 \u03be(\u03c9). Supposing that x \u2208 A\u2216\u03be(\u03c9), it follows that x \u2296 y \u2208 A\u2216\u03be(\u03c9), which is a contradiction, and thus x \u2208 \u03be(\u03c9). Hence A.Conversely, assume that (1) and (2) are valid. For any \u03be : \u03a9 \u2192 P(A) be a random set and \u03be. Then A if and only if z \u2296 x \u2264 y implies \u03a9\u2229\u03a9\u2286\u03a9 for any x, y, z \u2208 A.Let x, y, z \u2208 A be such that z \u2296 x \u2264 y. For any \u03c9 \u2208 \u03a9\u2229\u03a9, we get x \u2208 \u03be(\u03c9), y \u2208 \u03be(\u03c9). Considering that A, we have that \u03be(\u03c9) is an ideal of A. By z \u2208 \u03be(\u03c9); that is, \u03c9 \u2208 \u03a9. Hence \u03a9\u2229\u03a9\u2286\u03a9.Let z \u2296 x \u2264 y implies \u03a9\u2229\u03a9\u2286\u03a9 for any x, y, z \u2208 A. Let \u03c9 \u2208 \u03a9 be such that \u03be(\u03c9) \u2260 \u2205. Then there exists x \u2208 \u03be(\u03c9); that is, \u03c9 \u2208 \u03a9. Since 0 \u2296 x = 0 \u2264 x, then \u03a9\u2229\u03a9 = \u03a9\u2286\u03a9. It follows that \u03c9 \u2208 \u03a9; that is, 0 \u2208 \u03be(\u03c9). Let x, y \u2208 A be such that x \u2296 y \u2208 \u03be(\u03c9) and y \u2208 \u03be(\u03c9). Then \u03c9 \u2208 \u03a9\u2229\u03a9. Since x \u2296 (x \u2296 y) \u2264 y, it follows that \u03a9\u2229\u03a9\u2286\u03a9. Thus \u03c9 \u2208 \u03a9; that is, x \u2208 \u03be(\u03c9). Hence \u03be(\u03c9) is an ideal of A, and so A.Conversely, assume that \u03be : \u03a9 \u2192 P(A) be a random set and \u03be. If A, then the following statements hold: for any x, y \u2208 A, \u03a9\u2286\u03a9;\u03a9\u2229\u03a9\u2286\u03a9.Let A, we have that \u03be(\u03c9) is an ideal of A for any \u03c9 \u2208 \u03a9. It follows that 0 \u2208 \u03be(\u03c9); that is, \u03c9 \u2208 \u03a9. Hence \u03a9\u2286\u03a9.Noticing that \u03c9 \u2208 \u03a9\u2229\u03a9, we get x \u2296 y \u2208 \u03be(\u03c9) and y \u2208 \u03be(\u03c9). Since \u03be(\u03c9) is an ideal of A, then x \u2208 \u03be(\u03c9); that is, \u03c9 \u2208 \u03a9. Thus \u03a9\u2229\u03a9\u2286\u03a9.For any \u03be : \u03a9 \u2192 P(A). Then A if and only if, for any x, y \u2208 A, \u03a9\u2229\u03a9\u2286\u03a9;x \u2264 y implies \u03a9\u2286\u03a9.Let A, then \u03be(\u03c9) is an ideal of A for any \u03c9 \u2208 \u03a9. For any x, y \u2208 A, if x \u2208 \u03be(\u03c9), y \u2208 \u03be(\u03c9), then x \u2295 y \u2208 \u03be(\u03c9). That is, \u03c9 \u2208 \u03a9 and \u03c9 \u2208 \u03a9 imply \u03c9 \u2208 \u03a9. Hence \u03a9\u2229\u03a9\u2286\u03a9. If x \u2264 y, that is, x \u2296 0 \u2264 y, it follows that \u03a9\u2286\u03a9 by Assuming that \u03c9 \u2208 \u03a9, let x, y \u2208 A be such that x \u2208 \u03be(\u03c9) and y \u2208 \u03be(\u03c9). It follows that \u03c9 \u2208 \u03a9\u2229\u03a9. Thus \u03c9 \u2208 \u03a9; that is, x \u2295 y \u2208 \u03be(\u03c9). If x \u2264 y and y \u2208 \u03be(\u03c9) for any x, y \u2208 A, then \u03c9 \u2208 \u03a9\u2286\u03a9. Thus x \u2208 \u03be(\u03c9). Hence \u03be(\u03c9) is an ideal of A, and so A.Conversely, assume that (1) and (2) are valid. For any \u03be : \u03a9 \u2192 P(A). Then A if and only if, for any x, y \u2208 A, \u03a9\u2229\u03a9\u2286\u03a9;\u03a9\u2286\u03a9.Let x \u2264 y if and only if x\u2227y = x for any x, y \u2208 A.The proof is obvious since \u03be : \u03a9 \u2192 P(A). Then A if and only if, for any x, y \u2208 A, \u03a9\u2229\u03a9\u2286\u03a9;\u03a9\u2286\u03a9.Let x \u2297 y \u2264 y. Conversely, assume that conditions (1) and (2) hold. By hypothesis, we have \u03a9\u2286\u03a9(y \u2297 (\u00acy \u2295 x); \u03be) for any x, y \u2208 A. Now let x \u2264 y and \u03c9 \u2208 \u03a9. Then y \u2208 \u03be(\u03c9) and y \u2297 (\u00acy \u2295 x) = x\u2227y = x \u2208 \u03be(\u03c9). It means that x \u2264 y implies \u03a9\u2286\u03a9, and so A by One direction is clear since \u03be : \u03a9 \u2192 P(A). If the following conditions are valid: \u03a9 = \u03a9;\u03a9\u2229\u03a9\u2286\u03a9, for any x, y \u2208 A,then A.Let \u03c9 \u2208 \u03a9, we have \u03c9 \u2208 \u03a9; that is, 0 \u2208 \u03be(\u03c9). For any x, y \u2208 A, if x \u2296 y \u2208 \u03be(\u03c9) and y \u2208 \u03be(\u03c9), then \u03c9 \u2208 \u03a9, \u03c9 \u2208 \u03a9. It follows that \u03c9 \u2208 \u03a9\u2229\u03a9\u2286\u03a9. Hence \u03c9 \u2208 \u03a9; that is, x \u2208 \u03be(\u03c9). Thus \u03be(\u03c9) is an ideal of A, and so A.Assume that conditions (1) and (2) are valid. For any \u03be : \u03a9 \u2192 P(A). If A, then the following relationships hold: for any x, y, z \u2208 A, \u03a9\u2229\u03a9\u2286\u03a9;\u03a9 = \u03a9, then \u03a9\u2286\u03a9;if \u03a9((x \u2296 y) \u2296 z; \u03be)\u2229\u03a9\u2286\u03a9((x \u2296 z) \u2296 z; \u03be);\u03a9\u2229\u03a9 = \u03a9\u2229\u03a9;\u03a9\u2229\u03a9 = \u03a9;x \u2264 y, then \u03a9\u2229\u03a9 = \u03a9.if Let x, y, z \u2208 A, we have \u03a9((x \u2296 z) \u2296 z; \u03be)\u2287\u03a9((x \u2296 z) \u2296 y; \u03be)\u2229\u03a9 = \u03a9((x \u2296 y) \u2296 z; \u03be)\u2229\u03a9 by (1).Here we only prove (3), and the other cases directly follow from According to \u03be : \u03a9 \u2192 P(A). If the following conditions are valid: \u03a9 = \u03a9;\u03a9((x \u2296 y) \u2296 z; \u03be)\u2229\u03a9\u2286\u03a9((x \u2296 z) \u2296 z; \u03be), for any x, y, z \u2208 A,then A.Let \u03a9\u2229\u03a9 = \u03a9((x \u2296 y) \u2296 0; \u03be)\u2229\u03a9\u2286\u03a9((x \u2296 0) \u2296 0; \u03be) = \u03a9, for any x, y \u2208 A. It follows from A.By hypothesis, we get T\u2227-fuzzy ideals. Under what conditions a falling fuzzy ideal becomes a T-fuzzy ideal, we will give some answers to the questions in the following.From \u03be : \u03a9 \u2192 P(A). If A, then Tm-fuzzy ideal of A.Let A, we have that \u03a9\u2286\u03a9 for any x \u2208 A, by \u03a9\u2229\u03a9\u2286\u03a9 for any x, y \u2208 A by \u03c9 \u2208 \u03a9\u2223x \u2296 y \u2208 \u03be(\u03c9)}\u2229{\u03c9 \u2208 \u03a9\u2223y \u2208 \u03be(\u03c9)}\u2286{\u03c9 \u2208 \u03a9\u2223x \u2208 \u03be(\u03c9)}. Thus P(\u03c9\u2223x \u2296 y \u2208 \u03be(\u03c9) or Tm-fuzzy ideal of A.Noticing that \u03be : \u03a9 \u2192 P(A). If A, then the following statements hold: \u2200x, y \u2208 A, \u03a9\u2286\u03a9 or \u03a9\u2286\u03a9, then T\u2227-fuzzy ideal of A;if \u03a9 and \u03a9 are independent random events, then Tp-fuzzy ideal of A.if Let x \u2208 A. According to hypothesis and \u03a9\u2286\u03a9 or \u03a9\u2286\u03a9 for any x, y \u2208 A. If \u03a9\u2286\u03a9, then \u03a9\u2286\u03a9, then T\u2227-fuzzy ideal of A.By x \u2208 A. Since A, then {\u03c9 \u2208 \u03a9\u2223x \u2296 y \u2208 \u03be(\u03c9)}\u2229{\u03c9 \u2208 \u03a9\u2223y \u2208 \u03be(\u03c9)}\u2286{\u03c9 \u2208 \u03a9\u2223x \u2208 \u03be(\u03c9)}, for any x, y \u2208 A. It follows that Tp-fuzzy ideal of A.It is directly obtained that \u03a9, A, P) be a probability space, and let \u03be, \u03be1, \u03be2 : \u03a9 \u2192 P(A), respectively. Then the product of Let  be a probability space and \u03be : \u03a9 \u2192 P(A). Then x \u2264 y implies x, y \u2208 A.Let  be a probability space and \u03be : \u03a9 \u2192 P(A). If the probability distribution P of two-dimensional random variables is diagonal , then T-fuzzy ideal of A if and only if where T \u2208 {T\u2227, Tm, Tp}.Let  be a probability space and \u03be : \u03a9 \u2192 P(A). If the probability distribution P of two-dimensional random variables is diagonal, then T\u2227-fuzzy ideal of A if and only if Let , and then T\u2227-fuzzy ideal of A.Conversely, assume that \u03a9, A, P) be a probability space and \u03be : \u03a9 \u2192 P(A). If P of two-dimensional random variables is antidiagonal (or independent), then Tm-fuzzy ideal  of A if and only if Let ; then Tm-fuzzy ideal of A.We only consider that Tm-fuzzy ideal of A. The proof of Conversely, suppose that \u03a9, A, P) be a probability space, where the probability distribution P of two-dimensional random variables is diagonal. Let \u03be1, \u03be2 : \u03a9 \u2192 P(A), respectively. If T\u2227-fuzzy ideals of A, then T\u2227-fuzzy ideal of A.Let  become falling fuzzy implicative ideals.In the section, we introduce the notion of falling fuzzy implicative ideals as a generalization of \u03bc in A is called a T-fuzzy implicative ideal of A if it satisfies \u2200x, y, z \u2208 A, \u03bc(0) \u2265 \u03bc(x);\u03bc(x \u2296 z) \u2265 T(\u03bc((x \u2296 y) \u2296 z), \u03bc(y \u2296 z)).A fuzzy set \u03a9, A, P) be a probability space and \u03be : \u03a9 \u2192 P(A) be a random set. If \u03be(\u03c9) is an implicative ideal of A for any \u03c9 \u2208 \u03a9, then the falling shadow \u03be, that is, \u2200u \u2208 A,A.Let  is an MV-algebra.Routine computations prove that  = , where A is a Borel field on  and m is the usual Lebesgue measure. The mapping \u03be : \u03a9 \u2192 P(A) is defined byLet (\u03be(t) is an implicative ideal of A for any t \u2208 . Thus A, where Then \u03a9, A, P) be a probability space, S a nonempty subset of A, and \u03c9 \u2208 \u03a9. If S is an implicative ideal of A, then \u03be(\u03c9) = {f \u2208 F(A)\u2223f(\u03c9) \u2208 S} is an implicative ideal of F(A).Let . Then A if and only if (1)\u2009\u20090 \u2208 \u03be(\u03c9), (2)\u2009\u2009y \u2296 z \u2208 \u03be(\u03c9) and x \u2296 z \u2208 A\u2216\u03be(\u03c9) imply (x \u2296 y) \u2296 z \u2208 A\u2216\u03be(\u03c9) for any \u03c9 \u2208 \u03a9, x, y, z \u2208 A.Let A, it follows that \u03be(\u03c9) is an implicative ideal of A for any \u03c9 \u2208 \u03a9, and thus 0 \u2208 \u03be(\u03c9). Let x, y, z \u2208 A be such that y \u2296 z \u2208 \u03be(\u03c9) and x \u2296 z \u2208 A\u2216\u03be(\u03c9). If (x \u2296 y) \u2296 z \u2208 \u03be(\u03c9) hold, then x \u2296 z \u2208 \u03be(\u03c9), which is a contradiction, and so (2) is valid.Suppose that x, y, z \u2208 A be such that (x \u2296 y) \u2296 z \u2208 \u03be(\u03c9) and y \u2296 z \u2208 \u03be(\u03c9) for any \u03c9 \u2208 \u03a9. If x \u2296 z \u2208 A\u2216\u03be(\u03c9), it follows from hypothesis that (x \u2296 y) \u2296 z \u2208 A\u2216\u03be(\u03c9), a contradiction, and so x \u2296 z \u2208 \u03be(\u03c9). Therefore A.Conversely, let \u03be : \u03a9 \u2192 P(A). If A, then we have, for any x, y \u2208 A, \u03a9((x \u2296 y) \u2296 z; \u03be)\u2229\u03a9\u2286\u03a9;\u03a9 = \u03a9, for any n \u2265 1;\u03a9\u2229\u03a9 = \u03a9 = \u03a9;\u03a9 = \u03a9, for any n \u2265 1;\u03a9\u2229\u03a9 = \u03a9;\u03a9((x \u2296 y) \u2296 y; \u03be)\u2286\u03a9.Let \u03c9 \u2208 \u03a9((x \u2296 y) \u2296 z; \u03be)\u2229\u03a9, we have (x \u2296 y) \u2296 z \u2208 \u03be(\u03c9) and y \u2296 z \u2208 \u03be(\u03c9). Since \u03be(\u03c9) is an ideal of A, it follows that x \u2296 z \u2208 \u03be(\u03c9); that is, \u03c9 \u2208 \u03a9, and so \u03a9((x \u2296 y) \u2296 z; \u03be)\u2009\u2009\u2229\u2009\u2009\u03a9\u2009\u2009\u2286\u2009\u2009\u03a9.For any n = 1. For the case of n = 2, we have \u03a9 = \u03a9\u2009\u2009\u2287\u2009\u2009\u03a9((1 \u2296 \u00acx)\u2296\u00acx; \u03be)\u2009\u2009\u2229\u2009\u2009\u03a9 = \u03a9 by (1). On the other hand, since x2 \u2264 x, then \u03a9\u2286\u03a9 by \u03a9 = \u03a9. Suppose that n \u2265 3 and \u03a9 = \u03a9, then we get \u03a9 = \u03a9\u2009\u2009=\u2009\u2009\u03a9\u2009\u2009\u2287\u2009\u2009\u03a9((xn\u22122 \u2296 \u00acx)\u2009\u2009\u2296\u2009\u2009\u00acx; \u03be)\u2009\u2009\u2229\u2009\u2009\u03a9\u2009\u2009=\u2009\u2009\u03a9. The other direction is that \u03a9\u2286\u03a9 since xn \u2264 x, which shows that (2) is valid.It is true for x, y \u2208 A, we have x, y \u2264 x\u2228y \u2264 x \u2295 y, and so \u03a9\u2009\u2009\u2286\u2009\u2009\u03a9\u2009\u2009\u2286\u2009\u2009\u03a9\u2229\u03a9. On the other hand, since (x \u2295 y) \u2296 x = \u00acx\u2227y, then \u03a9\u2009\u2009\u2287\u2009\u2009\u03a9((x \u2295 y) \u2296 x; \u03be)\u2009\u2009\u2229\u2009\u2009\u03a9\u2009\u2009\u2287\u2009\u2009\u2009\u2009\u222a\u2009\u2009\u03a9)\u2009\u2009\u2229\u2009\u2009\u03a9\u2009\u2009=\u2009\u2009\u2229\u03a9)\u2009\u2009\u222a\u2009\u2009\u2229\u03a9)\u2009\u2009=\u2009\u2009\u03a9\u2009\u2009\u222a\u2009\u2009\u2229\u03a9)\u2009\u2009=\u2009\u2009\u03a9\u2229\u03a9, and so it proves (3).For any It is straightforward by (3).\u03a9\u2009\u2009=\u2009\u2009\u03a9\u2009\u2009\u2287\u2009\u2009\u03a9((1 \u2296 (1 \u2296 y))\u2296\u00acx; \u03be)\u2009\u2009\u2229\u2009\u2009\u03a9((1 \u2296 y)\u2296\u00acx; \u03be)\u2009\u2009=\u2009\u2009\u03a9\u2229\u03a9. On the other hand, since x \u2297 y \u2264 x and x \u2296 y \u2264 x, we have \u03a9\u2286\u03a9\u2009\u2009\u2229\u2009\u2009\u03a9; that means (5) holds.On the one hand, \u03c9 \u2208 \u03a9((x \u2296 y) \u2296 y; \u03be), we have (x \u2296 y) \u2296 y \u2208 \u03be(\u03c9). Since y \u2296 y = 0 \u2208 \u03be(\u03c9) and \u03be(\u03c9) is an implicative ideal of A, it follows that x \u2296 y \u2208 \u03be(\u03c9), that is \u03c9 \u2208 \u03a9, and thus \u03a9((x \u2296 y) \u2296 y; \u03be)\u2286\u03a9.For any Some of equivalent conditions of falling fuzzy implicative ideals are given in the next theorem.\u03be : \u03a9 \u2192 P(A). Then A if and only if for any x, y, z \u2208 A, \u03a9\u2286\u03a9;\u03a9((x \u2296 y) \u2296 z; \u03be)\u2229\u03a9\u2286\u03a9.Let A, it follows that \u03a9\u2286\u03a9 by Assume that \u03c9 \u2208 \u03a9 and x \u2208 \u03be(\u03c9). Then \u03c9 \u2208 \u03a9\u2286\u03a9; that is, 0 \u2208 \u03be(\u03c9). For any x, y, z \u2208 A, if (x \u2296 y) \u2296 z \u2208 \u03be(\u03c9) and y \u2296 z \u2208 \u03be(\u03c9), then \u03c9 \u2208 \u03a9((x \u2296 y) \u2296 z; \u03be)\u2229\u03a9\u2286\u03a9, and so x \u2296 z \u2208 \u03be(\u03c9). It follows that \u03be(\u03c9) is an implicative ideal of A. Thus A.Conversely, assume that (1) and (2) hold. Let As a direct consequence of \u03be : \u03a9 \u2192 P(A). If \u03a9 = \u03a9;\u03a9((x \u2296 y) \u2296 z; \u03be)\u2229\u03a9\u2286\u03a9, for any x, y \u2208 A,then A.Let The following result provides a condition for a falling shadow to be a falling fuzzy implicative ideal.\u03be : \u03a9 \u2192 P(A) be a random set and \u03be. Then A if and only if it satisfies \u03a9\u2286\u03a9 for any x \u2208 A;\u03a9((x \u2296 (y \u2296 x)) \u2296 z; \u03be)\u2229\u03a9\u2286\u03a9 for any x, y, z \u2208 A.Let A. (1) is directly from \u03c9 \u2208 \u03a9((x \u2296 (y \u2296 x)) \u2296 z; \u03be)\u2229\u03a9, we have (x \u2296 (y \u2296 x)) \u2296 z \u2208 \u03be(\u03c9) and z \u2208 \u03be(\u03c9); therefore x \u2296 (y \u2296 x) \u2208 \u03be(\u03c9). Since (y \u2296 (y \u2296 x))\u2296(y \u2296 x)\u2296(x \u2296 (y \u2296 x))\u2264(y \u2296 (y \u2296 x)) \u2296 x = 0 \u2208 \u03be(\u03c9), we obtain (y \u2296 (y \u2296 x))\u2296(y \u2296 x) \u2208 \u03be(\u03c9), and so y \u2296 (y \u2296 x) = x \u2296 (x \u2296 y) \u2208 \u03be(\u03c9) by \u03c9 \u2208 \u03a9(x \u2296 (x \u2296 y); \u03be). Noticing that ((x \u2296 y) \u2296 z)\u2296(x \u2296 (y \u2296 x)) = (x \u2296 (x \u2296 (y \u2296 x))) \u2296 y \u2296 z = (x\u2227(y \u2296 x)) \u2296 y \u2296 z = 0 \u2208 \u03be(\u03c9), we get (x \u2296 y) \u2296 z \u2208 \u03be(\u03c9). Since z \u2208 \u03be(\u03c9), it follows that x \u2296 y \u2208 \u03be(\u03c9); that is, \u03c9 \u2208 \u03a9. Considering that \u03be(\u03c9) is an implicative ideal of A, we have \u03c9 \u2208 \u03a9\u2229\u03a9(x \u2296 (x \u2296 y); \u03be)\u2286\u03a9 by Suppose that x, y, z \u2208 A, we get \u03a9(x \u2296 (y \u2296 x); \u03be)\u2286\u03a9 and \u03a9\u2229\u03a9 = \u03a9((x \u2296 (x \u2296 x)) \u2296 z; \u03be)\u2229\u03a9\u2286\u03a9, and so A. For any \u03c9 \u2208 \u03a9((x \u2296 y) \u2296 z; \u03be)\u2229\u03a9, we obtain (x \u2296 y) \u2296 z \u2208 \u03be(\u03c9) and y \u2296 z \u2208 \u03be(\u03c9). Since (x \u2296 z) \u2296 (x \u2296 (x \u2296 z)) \u2296 (y \u2296 z) \u2264 (x \u2296 y) \u2296 (x \u2296 (x \u2296 z)) = (x\u2227(x \u2296 z)) \u2296 y \u2264 (x \u2296 z) \u2296 y = (x \u2296 y) \u2296 z, and \u03be(\u03c9) is an ideal of A, then (x \u2296 z)\u2296(x \u2296 (x \u2296 z)) \u2208 \u03be(\u03c9); that is, \u03c9 \u2208 \u03a9((x \u2296 z)\u2296(x \u2296 (x \u2296 z)); \u03be)\u2286\u03a9. Thus \u03a9((x \u2296 y) \u2296 z; \u03be)\u2229\u03a9\u2286\u03a9, it follows that A.Conversely, for any We can also give some conditions for a falling fuzzy ideal to be a falling fuzzy implicative ideal.\u03be : \u03a9 \u2192 P(A). If A, then the following conditions are equivalent: A,\u03a9((y \u2296 \u00acx) \u2296 z; \u03be)\u2229\u03a9\u2286\u03a9, for any x, y, z \u2208 A;\u03a9 = \u03a9((x \u2295 x) \u2296 nx; \u03be) for any x \u2208 A and n \u2265 1;\u03a9(x \u2296 (y \u2296 x); \u03be)\u2286\u03a9, for any x, y \u2208 A;\u03a9((x \u2296 y) \u2296 y; \u03be)\u2286\u03a9, for any x, y \u2208 A.Let A. For any x, y, z \u2208 A, we have \u03a9((y \u2296 \u00acx) \u2296 z; \u03be)\u2229\u03a9 = \u03a9((\u00acz \u2296 \u00acy)\u2296\u00acx; \u03be)\u2229\u03a9\u2286\u03a9 = \u03a9, by (1)\u21d4(2) Suppose that x, y, z \u2208 A be such that (x \u2296 y) \u2296 z \u2208 \u03be(\u03c9) and y \u2296 z \u2208 \u03be(\u03c9). It follows that \u03c9 \u2208 \u03a9((x \u2296 y) \u2296 z; \u03be)\u2229\u03a9 = \u03a9((\u00acy \u2296 z)\u2296\u00acx; \u03be)\u2229\u03a9\u2286\u03a9 = \u03a9, and so x \u2296 z \u2208 \u03be(\u03c9). Noticing that \u03be(\u03c9) is an ideal of A, thus \u03be(\u03c9) is an implicative ideal, and therefore A.Conversely, let \u03c9 \u2208 \u03a9, we have 0 \u2208 \u03be(\u03c9). Since x \u2296 x = 0 \u2208 \u03be(\u03c9) and ((x \u2295 x) \u2296 x) \u2296 x = (x \u2295 x)\u2296(x \u2295 x) = 0 \u2208 \u03be(\u03c9), taking into consideration that \u03be(\u03c9) is an implicative ideal, we have (x \u2295 x) \u2296 x \u2208 \u03be(\u03c9); that is, \u03c9 \u2208 \u03a9((x \u2295 x) \u2296 x; \u03be), and thus \u03a9\u2286\u03a9((x \u2295 x) \u2296 x; \u03be). Since (x \u2295 x) \u2296 nx \u2264 (x \u2295 x) \u2296 x, then \u03a9\u2286\u03a9((x \u2295 x) \u2296 x; \u03be)\u2286\u03a9((x \u2295 x) \u2296 nx; \u03be) by \u03a9((x \u2295 x) \u2296 nx; \u03be)\u2286\u03a9; therefore, (2) holds.(1)\u21d4(3) For any n = 1, we get \u03a9 = \u03a9((x \u2295 x) \u2296 x; \u03be) for any x \u2208 A. It follows that (x \u2295 x) \u2296 x \u2208 \u03be(\u03c9) for any \u03c9 \u2208 \u03a9. Let x, y, z \u2208 A be such that (x \u2296 y) \u2296 z \u2208 \u03be(\u03c9) and y \u2296 z \u2208 \u03be(\u03c9). By x \u2296 (z \u2295 z))\u2296((x \u2296 y) \u2296 z)\u2296(y \u2296 z) = ((x \u2296 z)\u2296(y \u2296 z)) \u2296 z \u2296 ((x \u2296 y) \u2296 z)\u2264(x \u2296 y) \u2296 z \u2296 ((x \u2296 y) \u2296 z) = 0. Noticing that \u03be(\u03c9) is an ideal of A, we obtain x \u2296 (z \u2295 z) \u2208 \u03be(\u03c9). Since (z \u2295 z) \u2296 z \u2208 \u03be(\u03c9) and x \u2296 z \u2264 ((z \u2295 z) \u2296 z)\u2295(x \u2296 (z \u2295 z)), then x \u2296 z \u2208 \u03be(\u03c9). Hence \u03be(\u03c9) is an implicative ideal, and thus A.Conversely, given (1)\u21d2(4) It is immediate from the proof of x \u2296 y)\u2296(x \u2296 (x \u2296 y)) = (x \u2296 y) \u2296 y for any x, y \u2208 A, then \u03a9((x \u2296 y) \u2296 y; \u03be)\u2286\u03a9((x \u2296 y)\u2296(y \u2296 (x \u2296 y)); \u03be)\u2286\u03a9, and thus (5) is valid.(4)\u21d2(5) Since (x \u2296 (y \u2296 z) \u2296 z) \u2296 z = (x \u2296 z)\u2296(y \u2296 z) \u2296 z \u2264 (x \u2296 y) \u2296 z, we have \u03a9((x \u2296 y) \u2296 z; \u03be)\u2286\u03a9((x \u2296 (y \u2296 z) \u2296 z) \u2296 z; \u03be) = \u03a9(((x \u2296 z) \u2296 z)\u2296(y \u2296 z); \u03be). It follows that \u03a9((x \u2296 y) \u2296 z; \u03be)\u2229\u03a9\u2286\u03a9(((x \u2296 z) \u2296 z)\u2296(y \u2296 z); \u03be)\u2229\u03a9\u2286\u03a9((x \u2296 z) \u2296 z; \u03be)\u2286\u03a9, and thus A.(5)\u21d2(1) Noticing that (\u03be : \u03a9 \u2192 P(A). If A, then one has Tm-fuzzy implicative ideal of A;\u03a9\u2286\u03a9 or \u03a9\u2286\u03a9 for any x, y \u2208 A, then T\u2227-fuzzy implicative ideal of A;if x, y \u2208 A, if \u03a9 and \u03a9 are independent random events, then Tp-fuzzy implicative ideal of A.for any Let The proof is similar to that of"}
{"text": "G be a group. Denote by \u03c0(G) the set of prime divisors of |G|. Let GK(G) be the graph with vertex set \u03c0(G) such that two primes p and q in \u03c0(G) are joined by an edge if G has an element of order p \u00b7 q. We set s(G) to denote the number of connected components of the prime graph GK(G). Denote by N(G) the set of nonidentity orders of conjugacy classes of elements in G. Alavi and Daneshkhah proved that the groups, An where n = p, p + 1, p + 2 with s(G) \u2265 2, are characterized by N(G). As a development of these topics, we will prove that if G is a finite group with trivial center and N(G) = N(Ap+3) with p + 2 composite, then G is isomorphic to Ap+3.Let  G be a finite group and let Z(G) be its center. For any 1 \u2260 x \u2208 G, suppose that xG denotes the conjugacy classes in G containing x and CG(x) denotes the centralizer of x in G. We will use N(G) to denote the set {n\u2009:\u2009G has a conjugacy class of size n}. Thompson in 1987 gave the following conjecture with respect to N(G).In this paper, all groups considered are finite and simple groups are nonabelian simple groups. Let Thompson's Conjecture  = N(L), then G\u2245L.ure  denote the set of all prime divisors of |G|. Let GK(G) be the graph with vertex set \u03c0(G) such that two primes p and q in \u03c0(G) are joined by an edge if G has an element of order p \u00b7 q. We set s(G) to denote the number of connected components of the prime graph GK(G). A classification of all finite simple groups with disconnected prime graph was obtained in  \u2212 1; in particular, exp\u2061 \u2264 p + 2.exp\u2061 = \u2211i=1\u221e[(p + 3)/ri]\u2009\u2009for each r \u2208 \u03c0(Ap+3)\u2216{2}. Furthermore, exp\u2061 < (p \u2212 1)/2, where 3 \u2264 r \u2208 \u03c0(Ap+3). In particular, if r > [(p + 3)/2], then exp\u2061 = 1.exp\u2061(|Let (1) By the definition of Gaussian integer function, we have thatr \u2208 \u03c0(Ap+3).(2) Similarly, as (1), we have thatr > [(p + 3)/2], exp\u2061 = 1.If The proof is completed.Sn be the symmetric group of degree n. Assume that the cycle has c1 1-cycles, c2 2-cycles, and so on, up to ck\u2009\u2009k-cycles, where 1c1 + 2c2 + \u22ef+kck = n. Then the number of conjugacy classes in Sn isAn be the alternating group of degree n.Let x \u2208 An. Then, for the size of the conjugacy class xG of x in An, one has the following. i, ci = 0 and, for all odd i, i \u2208 {0,1}, then |xG | = z/2.If, for all even xG | = z.In all other cases, |In particular, |xG | \u2265z/2.Let See .n \u2265 6 is a natural number, then there are at least s(n) prime numbers pi such that (n + 1)/2 < pi < n. Heres(n) = 6 for n \u2265 48;s(n) = 5 for 42 \u2264 n \u2264 47;s(n) = 4 for 38 \u2264 n \u2264 41;s(n) = 3 for 18 \u2264 n \u2264 37;s(n) = 2 for 14 \u2264 n \u2264 17;s(n) = 1 for 6 \u2264 n \u2264 13.In particular, for every natural number n > 6, there exists a prime p such that (n + 1)/2 < p < n \u2212 1, and, for every natural number n > 3, there exists an odd prime number p such that n \u2212 p < p < n.If See Lemma 1 of .a, b, and n be positive integers such that  = 1. Then there exists a prime p with the following properties:p divides an \u2212 bn,p does not divide ak \u2212 bk for all k < n,with the following exceptions: a = 2,\u2009\u2009b = 1; n = 6 and a + b = 2k; n = 2.Let See .2 \u2212 2(13)4 = \u22121 and (3)5 \u2212 2(11)2 = 1 every solution of the equationm = n = 2; that is, it comes from a unit p \u2212 q \u00b7 21/2 of the quadratic field Q(21/2) for which the coefficients p and q are primes.With the exceptions of the relations (239)See , 21.L be a nonabelian simple group and let O denote the order of the outer automorphism group of L.Let L be a nonabelian simple group. Then the orders and their outer automorphism of L are as listed in Tables Let See .In this section, we give the main theorem and its proof.G be a finite group with trivial center and N(G) = N(Ap+3) with p + 2 composite. Then G is isomorphic to Ap+3.Let Ap+3 are characterized by N(G) if p = 7,13,19 . The. TheAp+We divide the proof into the following lemmas.L = Ap+3. Then the following hold.(1)r \u2264 [(p + 3)/2], then we can write p + 3 = kr + m with 0 \u2264 m < r and conjugacy class sizes of r-elements of L areIf 2 \u2260 \u2009i with 1 \u2264 i \u2264 k = [(p + 3)/r].for possible \u2009r is an odd prime divisor of |G|, then conjugacy class sizes of r-element of L areIn particular, if \u2009p + 3 = 2r + k and 0 \u2264 k < r.where (2)r = 2, then one can write p + 3 = 2k + m with 0 \u2264 m \u2264 1 and conjugacy class sizes of 2-elements of L areIf \u2009i with 1 \u2264 i \u2264 k = [(p + 3)/2].for possible (3)r > [(p + 3)/2], then one can write p + 3 = r + m with 0 \u2264 m < r and conjugacy class sizes of r-elements of L areIf \u2009r = p, then the conjugacy class size of p-elements of L isIn particular, if (4)N(G) are maximality with respect to divisibility.The following numbers from (a)One of the following holds:\u2009\u2009p + 3)!/2r2 if 2 \u00b7 r = p + 3;(\u2009\u2009p + 3)!/4r2 if 2 \u00b7 r + 2 = p + 3;(\u2009\u2009p + 3)!/2(k \u2212 1)r2 if 2 \u00b7 r + k = p + 3 and k = 2n with n \u2265 2;((b)p + 3)!/6p. = N(L). Then |L|\u2223|G| and \u03c0(G) = \u03c0(L).Let L | = \u220fn\u2208N(L)n. Since |xG||CG(x)| = |G|, every member from N(G) divides the order of G and |L|\u2223|G|. So by \u03c0(G) = \u03c0(L).Note that |G is a finite group with trivial center and N(G) = N(L). Then the following hold.r1, r2, p from \u03c0(L) such that r1, r2, p > [(p + 3)/2]. In particular, the Sylow r-subgroup S of G is a cyclic group of order r where r \u2208 {r1, r2, p}. There does not exist an element of order r1 \u00b7 r2, r1 \u00b7 p, or r2 \u00b7 p.There exist different primes n \u2208 N(G), if n is divisible at most by ra, then the Sylow r-subgroup S of G is of order ra.For all Suppose that r1, r2, p from \u03c0(G) such that r1, r2, p > [(p + 3)/2].(1) By r1, r2, p are prime divisors of |G| and r12, r22, p2 do not divide |xG| for all x \u2208 G. Then by S is elementary abelian. Therefore if |x | = r, then |xG| is an r\u2032-number.From Lemmas S | \u2265p2. Consider an element y of G withLet |p\u2224|y|. Let x be an element of CG(y) having order p. Then CG(xy) = CG(x)\u2229CG(y), |xG|\u2223|(xy)G|, and |yG|\u2223|(xy)G| by S is abelian, S \u2264 CG(x). Hence, p\u2224|xG|. It follows that |xG| equals (p + 3)!/6p or (p + 1)(p + 2)(p + 3)/3 by Assume that xG| equals (p + 3)!/6p, then (p + 3)!/6\u2223|(xy)G|. On the other hand, |yG|\u2223|(xy)G|; then we have thatN(G) such that |xG | \u2223 | (xy)G| and |yG | \u2223 | (xy)G|.If |xG| equals (p + 1)(p + 2)(p + 3)/3. In the following, we will consider the following three cases.Therefore |Case 1. |yG | = (p + 3)!/2 \u00b7 r2 if 2r = p + 3.r\u2223(p + 1)(p + 2)(p + 3)/3. Therefore (p + 3)!/r\u2223|(xy)G|, a contradiction, since |xG|\u2223|(xy)G|, |yG|\u2223|(xy)G|, and the maximality of |yG| = (p + 3)!/2 \u00b7 r2.Obviously,\u2009\u2009Case 2. |yG | = (p + 3)!/4 \u00b7 r2 if 2r + 2 = p + 3.r\u2223(p + 1)(p + 2)(p + 3)/3. Therefore (p + 3)!/r\u2223|(xy)G|. Also we get a contradiction as in Case 1.Obviously, Case 3. |yG | = (p + 3)!/2(k \u2212 1)r2 if 2r + k = p + 3 and k = 2n with n \u2265 2.r\u2224(p + 1)(p + 2)(p + 3)/3. It follows that |yG | = |(xy)G|. By CG(y) \u2264 CG(x) and so |xG | \u2223 | yG|, a contradiction.In this case, p\u2223|y|. Let |y | = p \u00b7 t. Since S is elementary abelian, the numbers p and t are coprime. Lety = uv and CG(uv) = CG(u)\u2229CG(v). Therefore,Assume that v of G is of order p. Since the Sylow p-subgroup of G is elementary abelian, then p\u2224|vG|. It follows thatOn the other hand, the element vG | = (p + 3)!/6p, then |vG|\u2223|yG|, a contradiction. Hence |vG | = (p + 1)(p + 2)(p + 3)/3. We consider the following three cases.If |Case 1. |yG | = (p + 3)!/2 \u00b7 r2 if 2r = p + 3.r\u2223(p + 1)(p + 2)(p + 3)/3. But r\u2224(p + 3)!/2r2, a contradiction, since |xG|\u2223|(xy)G|, |yG|\u2223|(xy)G|, and the maximality of |yG | = (p + 3)!/2 \u00b7 r2.Obviously,\u2009\u2009Case 2. |yG | = (p + 3)!/4 \u00b7 r2 if 2r + 2 = p + 3.r\u2223(p + 1)(p + 2)(p + 3)/3. But r\u2224(p + 3)!/4r2, a contradiction, since |xG|\u2223|(xy)G|, |yG|\u2223|(xy)G|, and the maximality of |yG| = (p + 3)!/4 \u00b7 r2.Obviously,\u2009\u2009Case 3. |yG | = (p + 3)!/2(k \u2212 1)r2 if 2r + k = p + 3 and k = 2n with n \u2265 2.r\u2224(p + 1)(p + 2)(p + 3)/3. It follows that |yG | = |(xy)G| since the maximality of |vG|. By CG(y) \u2264 CG(x) and so |xG|\u2223|yG|, a contradiction.In this case,\u2009\u2009p-subgroup of G is of order p.Therefore the Sylow Similarly we can prove the other two cases.r1 \u00b7 r2, r1 \u00b7 p, or r2 \u00b7 p.There does not exist an element of order n is divisible at most by r2.(2) Without loss of generality, we assume that S | \u2265r3. Consider an element x of G such thatAssume that |r\u2224|x|. Then there is an element y of G of order r. By CG(xy) = CG(x)\u2229CG(y), |xG|\u2223|(xy)G|, and |yG|\u2223|(xy)G|.Let y is an r-central element, then S \u2264 CG(y) and |yG| is an r\u2032-number. We consider the following three cases.If Case 1. 2r = p + 3.yG | = (p + 3)!/2 \u00b7 r2. Then we have that |xG | = |yG | = |(xy)G| and so r2\u2223\u2223 | CG(x)| = |CG(y)| since y is an r-central element |, then there is a conjugacy class size which is a multiple of the number (p + 3)!/r3 \u00b7 3! contradicting 2r = p + 3). Thus r\u2223|xG| = |yG| since r3\u2223|G|, a contradiction. In this case |Case 2. 2r + 2 = p + 3.yG | = (p + 3)!/4 \u00b7 r2. We similarly can rule out this case as \u201cCase 1: 2r = p + 3\u201d. We have that |Case 3. 2r + k = p + 3 and k = 2n with n \u2265 2.yG | = (p + 3)!/2(k \u2212 1)r2 or (2r + 1)(2r + 2) \u22ef (2r + k)/2(k \u2212 2) ). If the former, then |xG | = |yG | = |(xy)G| and so r2\u2223\u2223 | CG(x)| = |CG(y)| since y is an r-central element |, then there is a conjugacy class size which is a multiple of the number (p + 3)!/r3 \u00b7 3! contradicting 2r + k = p + 3). Thus r\u2223|xG| = |yG| since r3\u2223|G|, a contradiction. Then |yG | = (2r + 1)(2r + 2)\u22ef(2r + k)/2(k \u2212 2). But there is no member from N(G) such that |xG | \u2223 | (xy)G| and |yG | \u2223 | (xy)G|.In this case, |y is a noncentral r-element, then we choose an element z of order p such that r2\u2223\u2223 | zG|. By hypothesis, r\u2223 | CG(z)| and, obviously, r\u2224|z|. Then by Lemma 1.2 of  and r \u2265 [(p + 3)/2]\u201d.r > [(p + 3)/2] and r \u2260 r1, r2, p. In this case, by r-subgroup of G is of order r. Hence there is a Hall {r, p}-subgroup H. Since H must be cyclic, then there is an element of order r \u00b7 p, a contradiction by the proof of r \u2264 [(p + 3)/2] and r \u2260 2,3. If 2r = p + 3, 2r + 2 = p + 3, or 2r + k = p + 3 with k = 2n and n \u2265 2, then the index of A is at most r2. By m of \u03d5(p) such that p divides rm \u2212 1, and the subgroup Z(P)\u2229A, then the order of p. By y lies in the center of a Sylow p-subgroup of G. This contradicts It follows that r \u2208 {5,7,\u2026, p}.Let O\u03c0,\u03c0\u2032(G) = O\u03c0(G). In particular, G is insoluble.Therefore K \u2264 H \u2264 G such that H/K\u2245Ap+3.There is a normal series 1 \u2264 G | = (p + 3)!/2.By Lemmas M : H/K = S1 \u00d7 S2 \u00d7 \u22ef\u00d7Sk is a direct product of nonabelian simple groups S1, S2,\u2026, Sk. Since G cannot contain a Hall {r1, r2, p}-subgroup, numbers r1, r2, and p divide the order of exactly one of these groups that is listed as in S1. Since G* and M* denote the factor groups M/S1, respectively. If k > 1, then a Sylow 2-subgroup of G* is nontrivial and its center Z has a nontrivial intersection with M*. Consider a nontrivial element y of T = S2 \u00d7 \u22ef\u00d7Sk such that its image in Z. Since y centralizes S1, it lies in the center of a Sylow 2-subgroup of p, a contradiction. Thus M = S1, and r1, r2, p\u2223 | H/K| |, a contradiction, from r1, r2, p\u2223 | K|, then there is an element of order r \u00b7 p with r \u2208 {r1, r2} contradicting r \u2208 \u03c0(G) = \u03c0(L). In the following, we consider S1 which is listed as in Tables By erefore2HK\u2264G\u00af\u2264Aut(i) Case 1. H/K\u2245An with n \u2265 6.n = p, p + 1,\u2026, p + k with p + 2, p + 4,\u2026 composite and p + k + 1 prime. If k \u2265 4, then (p + k)!/2\u2223(p + 3)!, a contradiction. Therefore H/K is isomorphic to Ap,\u2009\u2009Ap+1, Ap+2, or Ap+3.Then x be an element of order p in H. Then |xH| is p\u2032-number since |H|p = pH/K\u2245Ap.Let Ap | \u2223(p + 3)!, then 3\u2223 | K|. We have |xH | = (p \u2212 1)!/2. On the other hand, |xG | = (p + 3)!/6p. It follows that |xK | \u2223(p + 1)(p + 2)(p + 3)/3 and so there is an element of r \u00b7 p or of order r\u2032 \u00b7 p with 3 < r\u2032 < r < p and r and r\u2032 divide one of the prime divisors of the numbers p + 1, p + 2, or p + 3, which contradicts H/K\u2245Ap+1 and H/K\u2245Ap+2\u201d.Since |H/K\u2245Ap+3.Therefore (ii) Case 2. H/K is not isomorphic to a sporadic simple group according to (iii) Case 3. H/K is isomorphic to a simple group of Lie type.q be a prime power.(1)H/K\u2245Bn(q) with n \u2265 2.\u2009\u03c0(G) = {2,3, 5,7,\u2026, p} and soIn this situation, by \u2009p\u2223q or p\u2223\u220fi=1n(qi2 \u2212 1). If p\u2223q, then q is a power of p. Since |Gp | = p by n \u2265 2. Therefore p\u2223\u220fi=1n(qi2 \u2212 1). It follows that p\u2223qt2 \u2212 1 for some 1 \u2264 t \u2264 n as p is prime. On the other hand, qn2\u2223r or qn2\u2223rm. If the former, then q = r and n = 1, a contradiction. It follows that qn2\u2223rm. Hence r\u2223q and m = kn2 for some integer k \u2265 1. By kn2 < p/2 < (qn2 \u2212 1)/2, but the equation has no solution in N. Furthermore, since Cn(q) has the same order as Bn(q), we also can rule out.It follows that (2)H/K\u2245Dn(q) with n \u2265 4.\u2009qn \u2212 1))qn(n\u22121)(qn \u2212 1)\u220fi=1n\u22121(qi2 \u2212 1)\u2223(p + 3)!. Since the Sylow p-subgroup of G is of order p, p\u2224q as, otherwise, q = p and thus n = 1, a contradiction. It follows that p\u2223qn \u2212 1 or p\u2223qt2 \u2212 1 for some integer 1 \u2264 t \u2264 n \u2212 1.Therefore we have that \u2223r < p with r prime, and so qn(n\u22121) < pn \u2212 1. Thus n = 1, a contradiction.Let \u2009p\u2223qt2 \u2212 1 for some integer 1 \u2264 t \u2264 n \u2212 1. Then qn(n\u22121)\u2223r < p with r prime, and so qn(n\u22121) \u2212 1 \u2264 qt2 \u2212 1 < qn2 \u2212 1. It follows that n = 1,2, 3, a contradiction.Let (3)H/K\u22452An(q) with n \u2265 2.\u2009In this situation,\u2009p-subgroup of G is of order p and n \u2265 2, we obtain that p\u2223qn(n+1)(1/2) or p\u2223qt+1 \u2212 (\u22121)t+1 for some integer 1 \u2264 t \u2264 n.Since the Sylow \u2009q = p and n = 1, a contradiction.If the former, then we have that \u2009p\u2223qt+1 \u2212 (\u22121)t+1 for some integer 1 \u2264 t \u2264 n. Then qn(n+1)(1/2)\u2223rm for some m. If m = 1, then q = r and n = 1, a contradiction. It follows that qn(n+1)(1/2)\u2223rm for some m > 1 and r\u2223q. By r \u2264 [p/2] < p/2 and n(n + 1)/2 \u2264 m \u2264 p/2.Let \u2009t is odd, then p\u2223rt+1 \u2212 1. By Lemmas q = 2, t = 5, and r = 2. Hence p = 3 or 7. If p = 3, we can rule out this case since p \u2265 5. If p = 7, then n(n + 1) \u2264 3 and so n = 1,2, a contradiction, since t = 5 < n.If \u2009t is even, then p\u2223rt+1 + 1. So p\u2223r + 1 or p\u2223rt \u2212 rt\u22121 + \u22ef+1. If the former, then p\u2223r + 1 < [p/2] + 1, a contradiction. If the latter, then p < rt + 1. It follows that p \u2264 rt \u2212 1. Let t = 2k with 1 < k < [p/2]. Then p \u2264 rk + 1 or p \u2264 rk \u2212 1 and so p\u2223rk + 1 or p\u2223rk \u2212 1. If p\u2223rk \u2212 1, then, by r = 2 and t = 5; we also can rule out this case as above. So p\u2223rk + 1. It follows that p\u2223rt \u2212 1. Similarly, as p\u2223rk \u2212 1, we can rule out this case.If (4)H/K\u2245E8(q).\u2009Therefore we have that\u2009It follows that\u2009p\u2223q120, then we can rule out this case since the Sylow p-subgroup of G is of order p. Hence p\u2223qt \u2212 1, where t \u2208 {2,8, 12,14,18,20,24,30}. On the other hand, rm\u2223q120. If m = 1, then q = r and 1 > 120, a contradiction. If m > 1, then q = r and m \u2264 120. By p and so p \u2208 {5,7, 11,13,\u2026, 101,103,107,109,113}. It is easy to rule out this case by considering the orders of G. Similarly, we can exclude that H/K\u2245E6(q), E7(q), and F4(q).If (5)H/K\u2245G2(q).\u2009q6(q6 \u2212 1)(q2 \u2212 1)\u2223(p + 3)!. It follows that q6\u2223p, p\u2223q6 \u2212 1, or p\u2223q2 \u2212 1.Then we have \u2009q6\u2223p, we rule out this case.If \u2009p\u2223p6 \u2212 1, then there exists a prime r such that q6\u2223rm for some integer m. Therefore q = r and 6 \u2264 m < p by p = 5 and so we have a contradiction by considering the order of G. Similarly, we also can rule out this case \u201cp\u2223q2 \u2212 1\u201d.If (6)H/K\u22452E6(q).\u2009q + 1))q36(q12 \u2212 1)(q9 + 1)(q8 \u2212 1)(q6 \u2212 1)(q5 + 1)(q2 \u2212 1)\u2223(p + 3)!. It follows that p\u2223qt \u2212 1 with t = 12,8, 6,2, p\u2223qk + 1 with k = 9,5, or p\u2223q36. If p\u2223q36, then we rule out this case since the Sylow p-subgroup of G is of order p. So p\u2223qt \u2212 1 with t = 12,8, 6,2, p\u2223qk + 1 with k = 9,5, and so there exists a prime r such that q36\u2223rm for some integer m. It means that 36 \u2264 m \u2264 p. Therefore p = 31,29,23,19,17,13,11,7. We also can rule out this case by order consideration.It is easy to see that H/K\u22452B2(q) with q = 2m+12.\u2009q2(q2 + 1)(q \u2212 1)\u2223p!. Thus q2\u2223p, p\u2223q2 + 1, or p\u2223q \u2212 1.It follows that \u2009q2\u2223p, then we rule out this case.If \u2009p\u2223q2 + 1, then there is a prime r such that q2\u2223rm and so 2 \u2264 m \u2264 p by p = 2, a contradiction. Similarly we can rule out this case \u201cp\u2223q + 1\u201d.If \u2009H/K\u22472F4(2m+12).Similarly (8)H/K\u22452G2(q), q = 3n+12 with n \u2265 1.\u2009q3(q3 + 1)(q \u2212 1)\u2223p!. Since the Sylow p-subgroup of G is of order p, then p\u2224q3. It follows that p\u2223q3 + 1 or p\u2223q \u2212 1. If p\u2223q3 + 1, then there exists a prime r such that q3\u2223rm for some integer m. If m = 1, then 1 > 3, a contradiction. Hence 3 \u2264 m \u2264 p by p = 3, a contradiction. If p\u2223q \u2212 1 and r\u2223q, then there exists a Frobenius group of r \u00b7 p with a kernel of order r and a complement of order p, respectively, and so there is an element of order r \u00b7 p, which contradicts We see that (9)H/K\u22453D4(q).\u2009q12(q8 + q4 + 1)(q6 \u2212 1)(q2 \u2212 1)\u2223p!. In this case, since G has a Sylow p-subgroup of order p, then p\u2223q8 + q4 + 1, q\u2223q6 \u2212 1, or p\u2223q2 \u2212 1. If p\u2223q8 + q4 + 1, then there exists a prime r such that rm\u2223q12 and so m \u2264 12. By p \u2264 12. It follows that p = 5,7, 11. Order consideration rules out these cases \u201cp = 5,7, 11\u201d. Similarly we can rule out this case \u201cp\u2223q2 \u2212 1\u201d.We have that (10)H/K\u2245An(q) with n \u2265 1.\u2009It is easy to get that\u2009p\u2223qn(n+1)/2 or p\u2223\u220fi=1n(qi+1 \u2212 1). If p\u2223qn(n+1)/2, then n = 1 since the Sylow p-subgroup of G is of order p, a contradiction. Hence p\u2223\u220fi=1n(qi+1 \u2212 1) and so p\u2223qt+1 \u2212 1 for some integer 1 \u2264 t \u2264 n. It follows that there exists a prime r such that qn(n+1)/2\u2223rm and so n(n + 1)/2 \u2264 m \u2264 p/2 by p-subgroup of G is of order p, then n(n + 1)\u2223p and so n = 1, a contradiction.It follows that This completes the proof of the lemma.Let Consider the following.G\u2245Ap+3.By K \u2264 G into the chief ones. We prove that K = 1. By \u03c0(K)\u2286{2,3}.So K is a 2-group, in this case, let x centralizes K. It follows that there is an element of 2 \u00b7 p which contradicts If K is a 3-group. Then similarly as the case \u201cK is a 2-group\u201d, we have that N(G) and CG(x) is abelian. So by Lemma 1.12 of [K \u2264 Z(G) = 1.If  1.12 of , K \u2264 Z = N(Ap+3) and |G | = |Ap+3|. Then G\u2245Ap+3.Let An with n = 10,16,22,26 are characterized by N(G). Then by [One knows that the alternating groups  Then by , 15, 21,G be a finite group with trivial center. Assume that N(G) = N(An) with n = p, p + 1, p + 2, p + 3. Then G\u2245An.Let"}
{"text": "H1-Galerkin mixed finite element (H1-GMFE) method to look for the numerical solution of time fractional telegraph equation. We introduce an auxiliary variable to reduce the original equation into lower-order coupled equations and then formulate an H1-GMFE scheme with two important variables. We discretize the Caputo time fractional derivatives using the finite difference methods and approximate the spatial direction by applying the H1-GMFE method. Based on the discussion on the theoretical error analysis in L2-norm for the scalar unknown and its gradient in one dimensional case, we obtain the optimal order of convergence in space-time direction. Further, we also derive the optimal error results for the scalar unknown in H1-norm. Moreover, we derive and analyze the stability of H1-GMFE scheme and give the results of a priori error estimates in two- or three-dimensional cases. In order to verify our theoretical analysis, we give some results of numerical calculation by using the Matlab procedure.We discuss and analyze an  \u03a9 is a bounded domain with boundary \u2202\u03a9 and J =  is a given source function, u0(x) and u1(x) are two given initial functions, and the time Caputo fractional-order derivatives \u2202t0,\u03b1u and \u2202t0,\u03b12u are defined, respectively, by\u03b1 < 1.In this paper, our purpose is to present and discuss a mixed finite element method for the time fractional telegraph equationIn the current literatures, we can see that some numerical methods for solving fractional partial differential equations (PDEs), which include finite element methods \u20138, mixedH1-GMFE method. This method includes some advantages, such as avoiding the LBB consistency condition, allowing different polynomial degrees of the finite element spaces, and obtaining the optimal a priori estimates in both H1 and L2-norms. In 1/2. When m = 0, we simply write the norm ||\u00b7||0 as ||\u00b7||.For the need of study, we denote the natural inner product as  in /\u2202x and split /\u2202t = \u2202u/\u2202t = 0 and \u22022u/\u2202t2 = \u22022u/\u2202t2 = 0, we easily getv/\u2202x, v \u2208 H01, and integrate with respect to space from xL to xR to obtainVh \u2282 H01 and Wh \u2282 H1, which satisfy the following approximation properties: for 1 \u2264 p \u2264 \u221e and k, r positive integers  with step length \u0394t = T/M and nodes tn = n\u0394t, for some positive integer M. For a smooth function \u03d5 on , define \u03d5n = \u03d5(tn). In the following analysis, for deriving the convenience of theoretical process, we now denoteFor formulating fully discrete scheme, let 0 = t0,\u03b1\u03c3 at t = tn is approximated by, for 0 < \u03b1 < 1,The time fractional derivative \u2202t0,\u03b12\u03c3 at t = tn is estimated by, for 1 < 2\u03b1 < 2,The time fractional order derivative \u2202u and \u03c3.In the next analysis, we will derive and prove some a priori error results for E0n = E\u03b1n + E\u03b12n.Based on the approximation formulas  and 22)22) of tiuhn, \u03c3hn) \u2208 Vh \u00d7 Wh such thatNow, we formulate a fully discrete procedure: find (Making a combination of -25) wit wit25) wIn the following discussion, we will derive the proof for the fully discrete a priori error estimates.\u03c30 = ux0 and \u03c3t(0) = ux1, suppose that un \u2208 H01\u2229Hk+1, \u03c3n \u2208 Hr+1, \u03c3h0 = Rh\u03c3(0), and \u03c3ht(0) = Lh\u03c3t(0), where Lh is the L2 projection defined by  = 0, wh \u2208 Wh. Then there exists a positive constant C free of space-time discrete parameters h and \u0394t such that, for 1/2 < \u03b1 < 1,\u03b1 \u2192 1With L2-norm ||\u03d1n|| and the H1-norm ||\u03d1n||1. Taking vh = \u03d1n in \u2211k=sily get||\u03d1n||\u2264||quality:\u0393(2\u2212\u03b1)\u2211k=tion to (\u2212\u0393(2\u2212\u03b1)\u2211kNoting (\u03b2\u0393\u03b1(\u03d5n+\u03d1nrrive at\u0393(2\u2212\u03b1)\u2211k=Bk\u03b12 \u2212 Bk+1\u03b12 > 0, we arrive atBk\u03b1 \u2212 Bk+1\u03b1 > 0, we use the similar method to haveB0\u03b1 = B0\u03b12 = 1, we arrive at\u03bc0||\u03b4n||1 \u2264 B/||\u03b4n||1 \u2264 B/||\u03b4n|| to getn = 1 in /Bj\u22121\u03b12)(hk+1 + hr+1 + \u0394t2 + \u0394t\u03b11+2) hold for j = 1,2,\u2026, n \u2212 1, we now prove that the inequality  \u21a6 Vh \u00d7 Wh such thatIn order to get fully discrete mixed finite element scheme, we now choose the finite element spaces integers ,(59)infIn the following discussion, we will give the stability for the system  and 61)61). Firsq0 = \u2207u0 and qt(0) = \u2207u1, the following inequality holds:With By a similar discussion as in , we have||qh\u22121||\u2264The following stable inequality for the system  and 61)61) holdsvh = \u2207uhn and use Cauchy-Schwarz inequality and Poincar\u00e9 inequality to arrive atwh = qhn and use Cauchy-Schwarz inequality, Young inequality, and proof of ||qhn||+\u03barrive at||uhn||\u2264|q.For deriving the a priori error analysis, we define the Ritz projection Let u~h; see , 44; we q0 = \u2207u0, qt(0) = \u2207u1, un \u2208 H01\u2229Hk+1, and qn \u2208 Hr+1, there exists a positive constant C free of space-time meshes h and \u0394t such that, for 1/2 < \u03b1 < 1,\u03b1 \u2192 1With We can use a similar proof as in \u03ba = 1/2 and \u03b2 = 0, the exact solution u = t\u03b12+sin(2\u03c0x), for all \u2208\u00d7, and the determined source term f by the exact solution in , b = \u0394t\u03b1\u2212/\u0393(2 \u2212 \u03b1), Here, in order to show the numerical performances on the rate of convergence and a priori error estimates, we now choose ution in  and then= 1. Letuhn=\u2211k=1Nressions :(78)[. From the calculated data, we can find that the rates of convergence, which are higher than the results O(\u0394t\u03b13\u22122 + h2) of theory, gradually decrease with the increased \u03b1 . In u and \u03c3 in H1-norm, which are unchanged with the changed \u03b1 = 0.6,0.7,0.8,0.9, are given.In H1-GMFE method.In view of the above analysis on the numerical results, we now announce that the time fractional telegraph equation can be well solved by the H1-GMFE method for time fractional telegraph equation, especially, has not been made and discussed. In this paper, we give the detailed proof's process of the error analysis on H1-GMFE method for time fractional telegraph equation. Further, we provide a numerical procedure to verify the theoretical results of the studied method.As far as we know, more and more people have proposed and analyzed a lot of numerical methods for fractional partial differential equations. However, the discussions on mixed finite element methods for solving fractional partial differential equations are fairly limited. The a priori error analysis of H1-Galerkin moving mixed finite element method, which is based on a combination of H1-GMFE methods and moving finite element methods.In the near future, our aim is to study an"}
{"text": "\u03a9 \u2282 \u211dN be a bounded domain, let \u03c3 be a nonnegative locally integrable function or, more generally, a locally finite measure on \u03a9, and let \u03c9 be a nonnegative Borel measure. In this paper, we consider the following nonlinear partial differential equations with measure data:P(u) = P\u2113,\u03b1,\u03b2(u) \u2208 L\u03c3,loc1(\u03a9) is defined, following (x), which includes fractional Laplacian (\u2212\u0394)\u03b3. Therefore, we also can obtain similar results of these integral equations.The four previous theorems are particular case of the more general class of nonlinear Wolff integral equations: p-superharmonic. Theorems \u03c3 and \u03c9 provided that there exist solutions of . We also denote by B = {x \u2208 \u211dn : |x \u2212 x0| < r} the open ball with center x0 and radius r > 0; when it is not important or clear from the context, we shall omit denoting the center as Br = B.In this section, we first recall some notations and definitions. In the following, we denote by p > 0 and \u03c3 be a nonnegative Borel measures in \u03a9 which are finite on compact subsets of \u03a9. The \u03c3-measure of a measurable set E \u2282 \u03a9 is denoted by \u03c3(E)\u2254\u222bEd\u03c3. We denote by Lp\u2009\u2009, resp.) the space of measurable functions f such that |f|p is integrable  with respect to \u03c3. When d\u03c3 = dx, we write Lp(\u03a9) (Llocp(\u03a9), resp.).Let \u03b1 > 0,\u2009\u2009p > 1, such that \u03b1p < n, the R-truncated Wolff's potential Wp1,R\u03bc(x) of a nonnegative Borel measure \u03bc on \u211dN is defined by Wp1,[\u03bc](x) the \u221e-truncated Wolff's potential.For u : \u03a9 \u2192 , p-harmonic in D, the implication holds: h \u2264 u on \u2202D implies h \u2264 u in D. Note that p-superharmonic function u does not necessarily belong to Wlocp1,(\u03a9), but its truncation Tk(u) = min\u2061{u, k} does for every integer k; therefore, we will need the generalized gradient of a p-superharmonic function u defined by Du = limk\u2192\u221e\u2207(Tk(u)). For more properties of p-superharmonic, see (x) is the Wolff potential of \u03bc.Let r > 0, we consider a ball B \u2282 \u03a9 and shrinking balls Bj\u2254Brj(x0), where rj = r2j\u2212 with j \u2265 0 is an integer.Given \u03bc be locally finite nonnegative measures on \u03a9. Then, there exists a constant C = C > 0 such that for any s > 0 we have\u03d5 = \u2211j=0\u221ecj\u03c7Bj with cj = C[\u03c3(Bj)/(r2j\u2212)N\u2212p]p\u22121)1/ > 0 such that if \u03bc = Fk[\u2212u] thenBR3 \u2282 \u03a9.Let p-superharmonic supersolution, then \u03c9 is absolutely continuous with respect to \u03c3. The fact can be used to obtain a characterization of removable singularities for the homogeneous quasilinear equation: In this section, we will give the proof of our main theorem. Firstly, we prove the following integral estimate for solutions of quasilinear equations , which s\u20092.18 in  and Theo\u20092.18 in .\u03c3 and \u03c9 be locally finite nonnegative measures on \u03a9 and p > 1. There exists a constant C = C > 0 such that if u(x) is a solution to  \u2282 \u03a9. Suppose that u is a positive solution of (x \u2208 BR(x0),C depends on N, p.For fixed ution of . In viewution of , we find\u03c3 on BR(x0) and let d\u03c3\u2032 = \u03c7BR(x0)d\u03c3; thus, taking into account (x \u2208 BR(x0),M is defined as\ud835\udd10 is a nonlinear integral operator defined by \ud835\udd10f = Wp1,R(f\u2113\u03b2d\u03c3\u2032). The iterates of \ud835\udd10 are denoted by \ud835\udd10if = \ud835\udd10(\ud835\udd10i\u22121f). It is then easy to see from s > 0, \u03d5(y) appears in Restrict the integration  account , we obtaaccount (ux\u2265C\u03b1ll!1account (ux\u2265C\u03b1ll!1BR(x0),uxC1+l\u03b2/pIn the following, we will divide the proof into two cases.Case\u2009\u20091 (u(x0) < \u221e). In this case,  = \u221e). According to  is the sum of the k \u00d7 k principal minors of D2u, which coincides with the Laplacian F1[u] = \u0394pu if k = 1, and the Monge-Amp\u00e8re operator Fn[u] = det\u2061(D2u) if k = n.We now move to and Wang \u201317:(54)FWk/(k+1),k+12R in place of Wp1,R. Therefore, the proof is omitted.The proof of the following theorems is completely analogous to that of . One onlu(x) be a solution of (\u03a9 with p > 1 and \u2113\u03b2 > k. Suppose that BR4(x0) \u2282 \u03a9. Then, there exists a constant M = M such that r \u2264 R/32.Let ution of  in \u03a9 witu(x) be a solution of (\u03a9 with p > 1 and P(u) = \u03c3up\u22121. Suppose that BR4(x0) \u2282 \u03a9. Then, there exists a constant M = M such that r \u2264 R/32.Let ution of  in \u03a9 wit"}
{"text": "EQ-algebras. First,we present several characterizations of fuzzy positiveimplicative prefilters (filters), fuzzy implicative prefilters(filters), and fuzzy fantastic prefilters (filters). Next,using their characterizations, we mainly consider therelationships among these special fuzzy filters. Particularly,we find some conditions under which a fuzzyimplicative prefilter (filter) is equivalent to a fuzzy positiveimplicative prefilter (filter). As applications, weobtain some new results about classical filters in EQ-algebras and some related results about fuzzy filters inresiduated lattices.We introduce and study sometypes of fuzzy prefilters (filters) in  BL-algebras, MTL-algebras, MV-algebras, and so forth are the best known classes of residuated lattices .A\u03bc be a fuzzy set in L. For all t \u2208 , the set \u03bct = {x \u2208 L | \u03bc(x) \u2265 t} is called a level subset of \u03bc.Let F be a nonempty subset, we denote the characteristic function of F by \u03c7F.Let a, b \u2208 , we denote max\u2061{a, b} and min\u2061{a, b} by a\u2228b and a\u2227b, respectively.For convenience, for any \u03bc, \u03bd in L, we define \u03bc\u2286\u03bd if and only if \u03bc(x) \u2264 \u03bd(x) for all x \u2208 L.For any fuzzy sets EQ-algebras and give some of their properties that will be used in the sequel.In this section, we introduce the notion of fuzzy prefilters (filters) in L denote an EQ-algebra unless otherwise is specified.In what follows, let \u03bc be a fuzzy set in L. \u03bc is called a fuzzy prefilter of L if it satisfies(FF1)\u03bc(1) \u2265 \u03bc(x) for all x \u2208 L,(FF2)\u03bc(y) \u2265 \u03bc(x)\u2227\u03bc(x \u2192 y) for all x, y \u2208 L.\u2009\u03bc is called a fuzzy filter if it satisfiesA fuzzy prefilter (FF3)\u03bc((x\u2299z)\u2192(y\u2299z)) \u2265 \u03bc(x \u2192 y) for all x, y, z \u2208 L.Let EQ-algebras exist.The following examples show that fuzzy prefilters (filters) in L = {0, a, b, 1} be a chain with Cayley tables as Let L, \u2227, \u2299, ~, 1) is an EQ-algebra in , \u03bct is either empty or a prefilter (filter) of L.Let The\u2009\u2009proof\u2009\u2009is straightforward.F be a nonempty subset of L. F is a prefilter (filter) if and only if \u03c7F is a fuzzy prefilter (filter) of L.Let EQ-algebra, which will be used in the sequel.Next, we discuss some properties of fuzzy filters in an L be an EQ-algebra and let \u03bc be a fuzzy filter of L; for all x, y, z \u2208 L, the following hold: \u03bc(x\u2299y) = \u03bc(x)\u2227\u03bc(y),\u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 y)\u2227\u03bc(y \u2192 z).Let x\u2299y \u2264 x\u2227y \u2264 x, y, then \u03bc(x\u2299y) \u2264 \u03bc(x)\u2227\u03bc(y) by y \u2264 1 \u2192 y, it follows that \u03bc(y) \u2264 \u03bc(1 \u2192 y). Since \u03bc is a fuzzy filter of L, then \u03bc((x\u22991)\u2192(x\u2299y)) \u2265 \u03bc(1 \u2192 y) by (FF3); that is, \u03bc(x \u2192 (x\u2299y)) \u2265 \u03bc(1 \u2192 y). By (FF2), we get \u03bc(x\u2299y) \u2265 \u03bc(x)\u2227\u03bc(x \u2192 (x\u2299y)). Hence, we have \u03bc(x\u2299y) \u2265 \u03bc(x)\u2227\u03bc(x \u2192 (x\u2299y)) \u2265 \u03bc(x)\u2227\u03bc(1 \u2192 y) \u2265 \u03bc(x)\u2227\u03bc(y). Therefore, \u03bc(x\u2299y) = \u03bc(x)\u2227\u03bc(y).(1) Since x \u2192 y)\u2299(y \u2192 z) \u2264 x \u2192 z, we have \u03bc(x \u2192 z) \u2265 \u03bc((x \u2192 y)\u2299(y \u2192 z)) by \u03bc((x \u2192 y)\u2299(y \u2192 z)) = \u03bc(x \u2192 y)\u2227\u03bc(y \u2192 z). Therefore, \u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 y)\u2227\u03bc(y \u2192 z).(2) Since  in \u03bc be a fuzzy prefilter in L. \u03bc is called a fuzzy positive implicative prefilter of L if it satisfies(FF4)\u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 (y \u2192 z))\u2227\u03bc(x \u2192 y) for all x, y, z \u2208 L.Let \u03bc of L is called a fuzzy positive implicative filter if it satisfies (FF4).A fuzzy filter For better understanding of the above definition, we illustrate it by the following example.L be the EQ-algebra and let \u03bc be the fuzzy set of L defined in \u03bc is a fuzzy positive implicative filter of L.Let The following result gives a characterization of fuzzy positive implicative prefilters by fuzzy prefilters.\u03bc be a fuzzy prefilter in L. \u03bc is a fuzzy positive implicative prefilter of L if and only if \u03bca : L \u2192  is a fuzzy prefilter in L, where \u03bca(x) = \u03bc(a \u2192 x) for all a, x \u2208 L.Let \u03bc is a fuzzy positive implicative prefilter of L. Since a \u2192 1 = 1, then \u03bc(a \u2192 1) = \u03bc(1). It follows that \u03bca(1) = \u03bc(a \u2192 1) = \u03bc(1). From a \u2192 x \u2264 1, we have \u03bc(a \u2192 x) \u2264 \u03bc(1); that is, \u03bca(x) \u2264 \u03bc(1). Then, \u03bca(x) \u2264 \u03bca(1). On the other hand, since \u03bc is a fuzzy positive implicative prefilter of L, then \u03bc(a \u2192 y) \u2265 \u03bc(a \u2192 (x \u2192 y))\u2227\u03bc(a \u2192 x); that is, \u03bca(y) \u2265 \u03bca(x \u2192 y)\u2227\u03bca(x). Therefore, \u03bca is a fuzzy prefilter in L.Suppose\u2009\u2009that\u2009\u2009a \u2208 L, since \u03bca is a fuzzy prefilter in L, then \u03bcx is a fuzzy prefilter in L. It follows that \u03bcx(z) \u2265 \u03bcx(y \u2192 z)\u2227\u03bcx(y). By the definition of \u03bcx, we have \u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 (y \u2192 z))\u2227\u03bc(x \u2192 y). Therefore, \u03bc is a fuzzy positive implicative prefilter of L.Conversely, for any \u03bca.As a consequence of \u03bc be a fuzzy positive implicative prefilter of L. Then, for any a \u2208 L, \u03bca is the fuzzy prefilter containing \u03bc.Let \u03bc is a fuzzy\u2009\u2009positive implicative prefilter of L. By \u03bca is a fuzzy prefilter. For any x \u2208 L, x \u2264 a \u2192 x. It follows from \u03bc(x) \u2264 \u03bc(a \u2192 x); that is, \u03bc(x) \u2264 \u03bca(x). And so \u03bc\u2286\u03bca. Therefore, \u03bca is the fuzzy prefilter containing \u03bc.Assume that \u03bc, \u03bd be two fuzzy prefilters of L. Then, for any a, b \u2208 L, the following statements hold: \u03bca = \u03bc if and only if \u03bc(a) = \u03bc(1),a \u2264 b implies that \u03bcb\u2286\u03bca,\u03bc\u2286\u03bd implies that \u03bca\u2286\u03bda,\u03bc\u2229\u03bd)a = \u03bca\u2229\u03bda, (\u03bc \u222a \u03bd)a = \u03bca \u222a \u03bda. = \u03bca(a) = \u03bc(a \u2192 a) = \u03bc(1); hence, \u03bc(a) = \u03bc(1).(1) Suppose that \u03bc(a) = \u03bc(1). For any x \u2208 I, since x \u2264 a \u2192 x and \u03bc is a fuzzy prefilter, we have \u03bc(x) \u2264 \u03bc(a \u2192 x) = \u03bca(x). Hence, \u03bc\u2286\u03bca. On the other hand, since \u03bc is a fuzzy prefilter, then \u03bc(x) \u2265 \u03bc(a \u2192 x)\u2227\u03bc(a) = \u03bc(a \u2192 x)\u2227\u03bc(1) = \u03bc(a \u2192 x) for all x \u2208 L. It follows that \u03bc(x) \u2265 \u03bca(x); that is, \u03bc\u2287\u03bca. Consequently, we obtain \u03bca = \u03bc.Conversely, assume that a \u2264 b. For any x \u2208 L, then b \u2192 x \u2264 a \u2192 x. Since \u03bc is a fuzzy prefilter, we have \u03bc(b \u2192 x) \u2264 \u03bc(a \u2192 x). It follows that \u03bcb(x) \u2264 \u03bca(x). So \u03bcb\u2286\u03bca.(2) Suppose that (3) and (4) It is easy to prove them, and we hence omit the details.In the following, we give some equivalent conditions of fuzzy positive implicative prefilters (filters) for further discussions.\u03bc be a fuzzy prefilter (filter) in L. The following are equivalent:\u03bc is a fuzzy positive implicative prefilter (filter) of L,\u03bc(x \u2192 y) \u2265 \u03bc(x \u2192 (x \u2192 y)) for all x, y \u2208 L,\u03bc(x \u2192 y) = \u03bc(x \u2192 (x \u2192 y)) for all x, y \u2208 L.Let \u03bc is a fuzzy positive implicative prefilter of L. Then, \u03bc(x \u2192 y) \u2265 \u03bc(x \u2192 (x \u2192 y))\u2227\u03bc(x \u2192 x) follows from \u03bc(x \u2192 y) \u2265 \u03bc(x \u2192 (x \u2192 y)).\u2009\u2009(1)\u21d2(2)\u2009\u2009Suppose\u2009\u2009that x \u2192 y \u2264 x \u2192 (x \u2192 y), it follows that \u03bc(x \u2192 y) \u2264 \u03bc(x \u2192 (x \u2192 y)), which together with (2) leads to \u03bc(x \u2192 y) = \u03bc(x \u2192 (x \u2192 y)).(2)\u21d2(3) From x \u2192 (y \u2192 z)\u2264((y \u2192 z)\u2192(x \u2192 z))\u2192(x \u2192 (x \u2192 z)) and x \u2192 y \u2264 (y \u2192 z)\u2192(x \u2192 z). Since \u03bc is a fuzzy prefilter in L, then \u03bc(x \u2192 (y \u2192 z)) \u2264 \u03bc(((y \u2192 z)\u2192(x \u2192 z))\u2192(x \u2192 (x \u2192 z))) and \u03bc(x \u2192 y) \u2264 \u03bc((y \u2192 z)\u2192(x \u2192 z)). From \u03bc(x \u2192 (x \u2192 z)) \u2265 \u03bc((y \u2192 z)\u2192(x \u2192 z))\u2227\u03bc((y \u2192 z)\u2192(x \u2192 z))\u2192(x \u2192 (x \u2192 z))) \u2265 \u03bc(x \u2192 y)\u2227\u03bc(x \u2192 (y \u2192 z)). Combining (3), we get \u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 y)\u2227\u03bc(x \u2192 (y \u2192 z)). Therefore, \u03bc is a fuzzy positive implicative prefilter of L.(3)\u21d2(1) By \u03bc be a fuzzy set in L. \u03bc is a fuzzy positive implicative prefilter (filter) of L if and only if, for all t \u2208 , \u03bct is either empty or a positive implicative prefilter (filter) of L.Let The proof\u2009\u2009is easy, and we hence omit the details.F be a nonempty subset of L. F is a positive implicative prefilter (filter) if and only if \u03c7F is a fuzzy positive implicative prefilter (filter).Let Now, we continue to characterize fuzzy positive implicative filters.L be an EQ-algebra and let \u03bc be a fuzzy filter. Then, \u03bc is a fuzzy positive implicative filter of L if and only if \u03bc(x\u2227(x \u2192 y) \u2192 y) = \u03bc(1) for all x, y \u2208 L.Let \u03bc is a fuzzy positive implicative filter of L, then \u03bc(x\u2227\u2009\u2009(x \u2192 y) \u2192 y) \u2265 \u03bc(x\u2227(x \u2192 y)\u2192(x \u2192 y))\u2009\u2009\u2227\u2009\u2009\u03bc(x\u2227(x \u2192 y) \u2192 x). For any x, y \u2208 L, since x\u2227(x \u2192 y) \u2264 x \u2192 y and x\u2227(x \u2192 y) \u2264 x, then x\u2227(x \u2192 y)\u2192(x \u2192 y) = 1 and x\u2227(x \u2192 y) \u2192 x = 1. Hence, \u03bc(x\u2227(x \u2192 y)\u2192(x \u2192 y)) = \u03bc(1) and \u03bc(x\u2227(x \u2192 y) \u2192 x) = \u03bc(1). Consequently, we have \u03bc(x\u2227(x \u2192 y) \u2192 y) = \u03bc(1).Since \u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 (y\u2227(y \u2192 z)))\u2227\u03bc((y\u2227(y \u2192 z)) \u2192 z) \u2265 \u03bc(x \u2192 (x\u2227y))\u2227\u2009\u2009\u03bc((x\u2227y)\u2192(y\u2227\u2009\u2009(y \u2192 z)))\u2009\u2009\u2227\u2009\u2009\u03bc(y\u2227\u2009\u2009(y \u2192 z) \u2192 z). It follows from x \u2192 (y \u2192 z)\u2264(x\u2227y)\u2192(y\u2227(y \u2192 z)) and x \u2192 y = x \u2192 (x\u2227\u2009\u2009y). Then, \u03bc((x\u2227y)\u2192(y\u2227(y \u2192 z))) \u2265 \u03bc(x \u2192 (y \u2192 z)) and \u03bc(x \u2192 (x\u2227y)) = \u03bc(x \u2192 y). Combining \u03bc(y\u2227(y \u2192 z) \u2192 z) = \u03bc(1), we obtain that \u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 (y \u2192 z))\u2227\u03bc(x \u2192 y). Therefore, \u03bc is a fuzzy positive implicative filter of L.Conversely, by EQ-algebra.In order to get some related results about fuzzy filters in residuated lattices see , we charL be a residuated EQ-algebra and \u03bc be a fuzzy filter. The following are equivalent:\u03bc is a fuzzy positive implicative filter of L,\u03bc(x \u2192 (x\u2299x)) = \u03bc(1) for all x \u2208 L,\u03bc(x\u2227y \u2192 (x\u2299y)) = \u03bc(1) for all x, y \u2208 L,\u03bc(x\u2227(x \u2192 y)\u2192(x\u2299y)) = \u03bc(1) for all x, y \u2208 L.Let L is a residuated EQ-algebra, we have 1 = (x\u2299x)\u2192(x\u2299x) \u2264 x \u2192 (x \u2192 (x\u2299x)) by \u03bc(1) \u2264 \u03bc(x \u2192 (x \u2192 (x\u2299x))), which implies that \u03bc(x \u2192 (x \u2192 (x\u2299x))) = \u03bc(1). From \u03bc(x \u2192 (x\u2299x)) \u2265 \u03bc(x \u2192 (x \u2192 (x\u2299x)))\u2227\u03bc(x \u2192 x) = \u03bc(x \u2192 (x \u2192 (x\u2299x))) = \u03bc(1). Consequently, we have \u03bc(x \u2192 (x\u2299x)) = \u03bc(1).\u2009\u2009(1)\u21d4(2) Since \u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 (x\u2299x))\u2227\u03bc((x\u2299x) \u2192 z) = \u03bc(1)\u2227\u03bc((x\u2299x) \u2192 z) = \u03bc((x\u2299\u2009\u2009x) \u2192 z) \u2265 \u03bc((x\u2299\u2009\u2009x)\u2192(y\u2299\u2009\u2009(y \u2192 z)))\u2227\u03bc((y\u2299\u2009\u2009(y \u2192 z)) \u2192 z). By y\u2299(y \u2192 z) \u2264 z. Then, \u03bc((y\u2299(y \u2192 z)) \u2192 z) = \u03bc(1). It follows that \u03bc(x \u2192 z) \u2265 \u03bc((x\u2299x)\u2192(y\u2299(y \u2192 z))). Hence, we obtain that \u03bc(x \u2192 z) \u2265 \u03bc((x\u2299x)\u2192(y\u2299(y \u2192 z))) \u2265 \u03bc((x\u2299\u2009\u2009x)\u2192(x\u2299\u2009\u2009y))\u2009\u2009\u2227\u2009\u2009\u03bc((x\u2299\u2009\u2009y)\u2192(y\u2299\u2009\u2009(y \u2192 z))) by \u03bc is a fuzzy filter, we have \u03bc((x\u2299x)\u2192(x\u2299y)) \u2265 \u03bc(x \u2192 y) and \u03bc((x\u2299y)\u2192(y\u2299(y \u2192 z))) \u2265 \u03bc(x \u2192 (y \u2192 z)) by (FF3). Combining them, we get \u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 (y \u2192 z))\u2227\u03bc(x \u2192 y). Therefore, \u03bc is a fuzzy positive implicative filter of L.Conversely, by \u03bc is a fuzzy positive implicative filter of L. Then, \u03bc(x\u2227(x \u2192 (x\u2299y))\u2192(x\u2299y)) = \u03bc(1) by L is a residuated EQ-algebra, we have y \u2264 x \u2192 (x\u2299y) by x\u2227y \u2264 x\u2227(x \u2192 (x\u2299y)), which implies that x\u2227(x \u2192 (x\u2299y))\u2192(x\u2299y) \u2264 x\u2227y \u2192 (x\u2299y). From \u03bc(x\u2227y \u2192 (x\u2299y)) \u2265 \u03bc(x\u2227(x \u2192 (x\u2299y))\u2192(x\u2299y)) = \u03bc(1). Therefore, we have \u03bc(x\u2227y \u2192 (x\u2299y)) = \u03bc(1).(1)\u21d2(3) Suppose that \u03bc(x\u2227(x \u2192 y)\u2192(x\u2299(x \u2192 y))) = \u03bc(1). By x\u2299(x \u2192 y) \u2264 x\u2227y, which implies that x\u2227(x \u2192 y)\u2192(x\u2299(x \u2192 y)) \u2264 x\u2227(x \u2192 y)\u2192(x\u2227y). Then, \u03bc(x\u2227(x \u2192 y)\u2192(x\u2299(x \u2192 y))) \u2264 \u03bc(x\u2227(x \u2192 y)\u2192(x\u2227y)). Hence, \u03bc(x\u2227(x \u2192 y)\u2192(x\u2227y)) = \u03bc(1). By \u03bc(x\u2227(x \u2192 y)\u2192(x\u2299y)) \u2265 \u03bc(x\u2227(x \u2192 y)\u2192(x\u2227y))\u2227\u03bc(x\u2227y \u2192 (x\u2299y)) = \u03bc(1)\u2227\u03bc(1) = \u03bc(1). Therefore, we have \u03bc(x\u2227(x \u2192 y)\u2192(x\u2299y)) = \u03bc(1).(3)\u21d2(4) From (3), it follows that x\u2299(x \u2192 y) \u2264 x\u2227(x \u2192 y), it follows that x\u2299\u2009\u2009(x \u2192 y)\u2192(x\u2299\u2009\u2009y) \u2265 x\u2227(x \u2192 y)\u2192(x\u2009\u2299\u2009\u2009y). Then, we have \u03bc(x\u2009\u2299\u2009(x \u2192 y)\u2192(x\u2009\u2299\u2009y)) \u2265 \u03bc(x\u2009\u2227(x \u2192 y)\u2192(x\u2299y)) = \u03bc(1), which implies that \u03bc(x\u2299(x \u2192 y)\u2192(x\u2009\u2299\u2009y)) = \u03bc(1). Taking x = y, we have \u03bc(x \u2192 (x\u2299x)) = \u03bc(1). Therefore, \u03bc is a fuzzy positive implicative filter by (2).(4)\u21d2(1) From L be a residuated EQ-algebra and \u03bc be a fuzzy filter. The following are equivalent:\u03bc is a fuzzy positive implicative filter of L,\u03bc((x\u2299y) \u2192 z) = \u03bc((x\u2227y) \u2192 z) for all x, y, z \u2208 L,\u03bc(x\u2227(x \u2192 y)\u2192(x\u2227y)) = \u03bc(1) for all x, y \u2208 L,\u03bc(x\u2227(x \u2192 y)\u2192(y\u2227(y \u2192 x))) = \u03bc(1) for all x, y \u2208 L.Let \u03bc is a fuzzy positive implicative filter. We have \u03bc(y\u2227(y \u2192 z) \u2192 z) = \u03bc(1) by L is a residuated EQ-algebra, then (x\u2299y) \u2192 z \u2264 x \u2192 (y \u2192 z) for all x, y, z \u2208 L by \u03bc((x\u2299y) \u2192 z) \u2264 \u03bc(x \u2192 (y \u2192 z)). By x \u2192 (y \u2192 z)\u2264(x\u2227y)\u2192(y\u2227(y \u2192 z)). Then, \u03bc(x \u2192 (y \u2192 z)) \u2264 \u03bc((x\u2227y)\u2192(y\u2227(y \u2192 z))). By \u03bc((x\u2227y) \u2192 z) \u2265 \u03bc((x\u2227y)\u2192(y\u2227(y \u2192 z)))\u2227\u03bc((y\u2227(y \u2192 z)) \u2192 z) \u2265 \u03bc((x\u2299y) \u2192 z)\u2227\u03bc(1) = \u03bc((x\u2299y) \u2192 z). On the other hand, since x\u2299y \u2264 x\u2227y, then (x\u2227y) \u2192 z \u2264 (x\u2299y) \u2192 z. It follows that \u03bc((x\u2227y) \u2192 z) \u2264 \u03bc((x\u2299y) \u2192 z). Consequently, we have \u03bc((x\u2299y) \u2192 z) = \u03bc((x\u2227y) \u2192 z).\u2009\u2009(1)\u21d4(2) Suppose that L is a residuated EQ-algebra, then it is a good EQ-algebra. It follows from x\u2299(x \u2192 y) \u2264 y. Then, (x\u2299(x \u2192 y)) \u2192 y = 1. So, \u03bc((x\u2299(x \u2192 y)) \u2192 y) = \u03bc(1). Combining (2), we obtain \u03bc((x\u2299(x \u2192 y)) \u2192 y) = \u03bc(x\u2227(x \u2192 y) \u2192 y) = \u03bc(1). By \u03bc is a fuzzy positive implicative filter of L.Conversely, since \u03bc is a fuzzy positive implicative filter. Then, we have \u03bc(x\u2227(x \u2192 y)\u2192(x\u2009\u2299y)) = \u03bc(1) for all x, y \u2208 L by y \u2264 x \u2192 y, it follows that x\u2009\u2299y \u2264 x\u2009\u2299(x \u2192 y). Then, we have x\u2227(x \u2192 y)\u2192(x\u2009\u2299y) \u2264 x\u2227(x \u2192 y) \u2192 x\u2299(x \u2192 y). Hence, we get \u03bc(1) = \u03bc(x\u2227(x \u2192 y)\u2192(x\u2299y)) \u2264 \u03bc(x\u2227(x \u2192 y) \u2192 x\u2299(x \u2192 y)), which implies that \u03bc(x\u2227(x \u2192 y) \u2192 x\u2299(x \u2192 y)) = \u03bc(1). Moreover, by x\u2299(x \u2192 y) \u2264 x\u2227y. Then, x\u2227(x \u2192 y)\u2192(x\u2009\u2299(x \u2192 y)) \u2264 x\u2009\u2227(x \u2192 y)\u2192(x\u2227y). It follows that \u03bc(x\u2009\u2227(x \u2192 y)\u2192(x\u2009\u2299(x \u2192 y))) \u2264 \u03bc(x\u2009\u2227(x \u2192 y)\u2192(x\u2227y)). Therefore, \u03bc(x\u2227(x \u2192 y)\u2192(x\u2227y)) = \u03bc(1).(1)\u21d2(3) Suppose that x \u2264 y \u2192 x, it follows that x\u2227y \u2264 (y \u2192 x)\u2227y. Then, x\u2227(x \u2192 y)\u2192(x\u2227y) \u2264 x\u2227(x \u2192 y)\u2192(y\u2227(y \u2192 x)). Hence, we have \u03bc(x\u2227(x \u2192 y)\u2192(x\u2227y)) \u2264 \u03bc(x\u2227(x \u2192 y)\u2192(y\u2227(y \u2192 x))). Combining (3), we get \u03bc(x\u2227(x \u2192 y)\u2192(y\u2227(y \u2192 x))) = \u03bc(1).(3)\u21d2(4) From \u03bc(x\u2227(x \u2192 (x\u2299y))\u2192((x\u2299y)\u2227((x\u2299y) \u2192 x))) = \u03bc(1). Then, \u03bc(x\u2227(x \u2192 (x\u2299y))\u2192(x\u2299y)) = \u03bc(1). Since L is a residuated EQ-algebra, we get y \u2264 x \u2192 (x\u2299y) by x\u2227(x \u2192 (x\u2299y))\u2192(x\u2299y) \u2264 x\u2227y \u2192 (x\u2299y). Then, \u03bc(1) = \u03bc(x\u2009\u2227(x \u2192 (x\u2009\u2299y))\u2192(x\u2009\u2299y)) \u2264 \u03bc(x\u2009\u2227y \u2192 (x\u2009\u2299\u2009\u2009y)), which implies that \u03bc(x\u2227y \u2192 (x\u2299y)) = \u03bc(1). Therefore, \u03bc is a fuzzy positive implicative filter of L by (4)\u21d2(1) By (4), we have EQ-algebras.As an application of Theorems L be a residuated EQ-algebra and let F be a filter of L. The following are equivalent:F is a positive implicative filter of L,x \u2192 (x\u2299x) \u2208 F for all x \u2208 L,x\u2227y \u2192 (x\u2299y) \u2208 F for all x, y \u2208 L,x\u2227(x \u2192 y)\u2192(x\u2299y) \u2208 F for all x, y \u2208 L,x\u2227(x \u2192 y) \u2192 y \u2208 F for all x, y \u2208 L,x\u2227(x \u2192 y)\u2192(x\u2227y) \u2208 F for all x, y \u2208 L,x\u2227(x \u2192 y)\u2192(y\u2227(y \u2192 x)) \u2208 F for all x, y \u2208 L,x\u2227(x \u2192 y) \u2192 y \u2208 F for all x, y \u2208 L.Let The extension property for fuzzy positive implicative prefilters is obtained from the following proposition.L be a good EQ-algebra and let \u03bc and \u03bd be two fuzzy prefilters which satisfy \u03bc\u2286\u03bd, \u03bc(1) = \u03bd(1). If \u03bc is a fuzzy positive implicative prefilter, then so is \u03bd.Let t = x \u2192 (x \u2192 y); then, x \u2192 (x \u2192 (t \u2192 y)) = t \u2192 (x \u2192 (x \u2192 y)) = 1. Since \u03bc is a fuzzy positive implicative prefilter, we have \u03bc(x \u2192 (t \u2192 y)) = \u03bc(x \u2192 (x \u2192 (t \u2192 y))) = \u03bc(1) by \u03bc(x \u2192 (t \u2192 y)) = \u03bc(1) = \u03bd(1). From \u03bc\u2286\u03bd, it follows that \u03bd(x \u2192 (t \u2192 y)) \u2265 \u03bc(x \u2192 (t \u2192 y)) = \u03bd(1). So, \u03bd(x \u2192 (t \u2192 y)) = \u03bd(1). Since \u03bd is a fuzzy prefilter, then \u03bd(x \u2192 y) \u2265 \u03bd(t \u2192 (x \u2192 y))\u2227\u03bd(t). It follows that \u03bd(x \u2192 y) \u2265 \u03bd(1)\u2227\u03bd(t) = \u03bd(t) = \u03bd(x \u2192 (x \u2192 y)). By \u03bd is a fuzzy positive implicative prefilter.We set L be an EQ-algebra and let \u03bc and \u03bd be two fuzzy filters which satisfy \u03bc\u2286\u03bd, \u03bc(1) = \u03bd(1). If \u03bc is a fuzzy positive implicative filter, then so is \u03bd.Let It follows from EQ-algebras and discuss some of their properties.In this section, we introduce the notion of fuzzy implicative prefilters (filters) in \u03bc be a fuzzy set in L. \u03bc is called a fuzzy implicative prefilter of L if it satisfies(FF1)\u03bc(1) \u2265 \u03bc(x) for all x \u2208 L,(FF5)\u03bc(x) \u2265 \u03bc(z \u2192 ((x \u2192 y) \u2192 x))\u2227\u03bc(z) for all x, y, z \u2208 L.Let \u03bc of L is called a fuzzy implicative filter if it satisfies (FF3).A fuzzy implicative prefilter EQ-algebras exist.The following example shows that fuzzy implicative prefilters in L = {0, a, b, c, 1} be a chain with Cayley tables as Let L, \u2227, \u2299, ~, 1) is an EQ-algebra in , \u03bct is either empty or an implicative prefilter (filter) of L.Let The proof is easy, and we hence omit the details.F be a nonempty subset of L. F is an implicative prefilter (filter) if and only if \u03c7F is a fuzzy implicative prefilter (filter).Let \u03bc be a fuzzy prefilter in L. We call that \u03bc has weak exchange principle if it satisfies for all x, y, z \u2208 L\u2009\u2009\u03bc(x \u2192 (y \u2192 z)) = \u03bc(y \u2192 (x \u2192 z)).Let \u03bc of a good EQ-algebra L has weak exchange principle.From L be the EQ-algebra and \u03bc be the fuzzy set of L defined in L is not a good EQ-algebra and \u03bc is a fuzzy prefilter of L, which has weak exchange principle.Let EQ-algebra with a bottom element 0.We have the following characterizations of fuzzy implicative prefilters in an L be an EQ-algebra with a bottom element 0 and let \u03bc be a fuzzy prefilter with weak exchange principle. The following are equivalent:\u03bc is a fuzzy implicative prefilter of L,\u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 (\u00acz \u2192 y))\u2227\u03bc(y \u2192 z) for all x, y, z \u2208 L,\u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 (\u00acz \u2192 z)) for all x, z \u2208 L,\u03bc(x \u2192 z) = \u03bc(x \u2192 (\u00acz \u2192 z)) for all x, z \u2208 L.Let y \u2192 z \u2264 (x \u2192 y)\u2192(x \u2192 z) and \u00acz \u2192 (x \u2192 y)\u2264((x \u2192 y)\u2192(x \u2192 z))\u2192(\u00acz \u2192 (x \u2192 z)), then \u03bc(y \u2192 z) \u2264 \u03bc((x \u2192 y)\u2192(x \u2192 z)) and \u03bc(\u00acz \u2192 (x \u2192 y)) \u2264 \u03bc(((x \u2192 y)\u2192(x \u2192 z))\u2192(\u00acz \u2192 (x \u2192 z))). From \u03bc(\u00acz \u2192 (x \u2192 z)) \u2265 \u03bc((x \u2192 y)\u2192(x \u2192 z))\u2227\u03bc(((x \u2192 y)\u2192(x \u2192 z))\u2192(\u00acz \u2192 (x \u2192 z))) \u2265 \u03bc(y \u2192 z)\u2227\u03bc(\u00acz \u2192 (x \u2192 y)) \u2265 \u03bc(y \u2192 z)\u2227\u03bc(x \u2192 (\u00acz \u2192 y)). Notice that \u00ac(x \u2192 z)\u2264\u00acz, and we have \u00acz \u2192 (x \u2192 z)\u2264\u00ac(x \u2192 z)\u2192(x \u2192 z). Then, \u03bc(\u00acz \u2192 (x \u2192 z)) \u2264 \u03bc(\u00ac(x \u2192 z)\u2192(x \u2192 z)). Since \u03bc is a fuzzy implicative prefilter of L, we have \u03bc(x \u2192 z) \u2265 \u03bc(\u00ac(x \u2192 z)\u2192(x \u2192 z)) by \u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 (\u00acz \u2192 y))\u2227\u03bc(y \u2192 z).\u2009\u2009(1)\u21d4(2) Since \u03bc satisfies \u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 (\u00acz \u2192 y))\u2227\u03bc(y \u2192 z) for all x, y, z \u2208 L. Then, \u03bc(1 \u2192 x) \u2265 \u03bc(1 \u2192 (\u00acx \u2192 x))\u2227\u03bc(x \u2192 x) \u2265 \u03bc(\u00acx \u2192 x)\u2227\u03bc(1) = \u03bc(\u00acx \u2192 x). Since \u03bc is a fuzzy prefilter, then \u03bc(x) \u2265 \u03bc(1)\u2227\u03bc(1 \u2192 x) = \u03bc(1 \u2192 x). It follows that \u03bc(x) \u2265 \u03bc(\u00acx \u2192 x). By \u03bc is a fuzzy implicative prefilter of L.Conversely, suppose that \u03bc is a fuzzy implicative prefilter of L. Then, we have \u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 (\u00acz \u2192 z))\u2227\u03bc(z \u2192 z) = \u03bc(x \u2192 (\u00acz \u2192 z))\u2227\u03bc(1) by (2). Therefore, we obtain \u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 (\u00acz \u2192 z).(1)\u21d2(3) Suppose that z \u2264 \u00acz \u2192 z, it follows that x \u2192 z \u2264 x \u2192 (\u00acz \u2192 z). Then, \u03bc(x \u2192 z) \u2264 \u03bc(x \u2192 (\u00acz \u2192 z)) as \u03bc is a fuzzy prefilter. Combining (3), we get \u03bc(x \u2192 z) = \u03bc(x \u2192 (\u00acz \u2192 z)).(3)\u21d2(4) From \u03bc be a fuzzy prefilter which satisfies \u03bc(x \u2192 z) = \u03bc(x \u2192 (\u00acz \u2192 z)) for all x, z \u2208 L. Then, \u03bc(1 \u2192 x) = \u03bc(1 \u2192 (\u00acx \u2192 x)). Since \u03bc is a fuzzy prefilter, then \u03bc(x) \u2265 \u03bc(1 \u2192 x)\u2227\u03bc(1) = \u03bc(1 \u2192 x). From \u00acx \u2192 x \u2264 1 \u2192 (\u00acx \u2192 x), it follows that \u03bc(1 \u2192 (\u00acx \u2192 x)) \u2265 \u03bc(\u00acx \u2192 x). Consequently, we obtain \u03bc(x) \u2265 \u03bc(\u00acx \u2192 x). Using \u03bc is a fuzzy implicative prefilter of L.(4)\u21d2(1) Let EQ-algebras and discuss the relations among various fuzzy prefilters (filters). Particularly, we find some conditions under which a fuzzy implicative prefilter (filter) is equivalent to a fuzzy positive implicative prefilter (filter). Moreover, as applications, we obtain some relations among corresponding classical filters in EQ-algebras and some related results about fuzzy filters in residuated lattices.In this section, we introduce fuzzy fantastic prefilters (filters) in F be a prefilter in L. F is called a fantastic prefilter of L if it satisfies the following:(F6)y \u2192 x \u2208 F implies that ((x \u2192 y) \u2192 y) \u2192 x \u2208 F for all x, y \u2208 L.Let F is a filter and it satisfies (F6), then it is called a fantastic filter of L.If It is time to give some examples of fantastic prefilters.L = {0, a, b, 1} be a chain with Cayley tables as Let L, \u2227, \u2299, ~, 1) is an EQ-algebra. One can check that F = {1} is a fantastic prefilter.Then, \u03bc(((x \u2192 y) \u2192 y) \u2192 x) \u2265 \u03bc(y \u2192 x) for all x, y \u2208 L.Let \u03bc of L is called a fuzzy fantastic filter if it satisfies (FF6).A fuzzy filter L, \u2227, \u2299, ~, 1) be an EQ-algebra in \u03bc in L as follows: \u03bc(1) = 0.7 and \u03bc(0) = \u03bc(a) = \u03bc(b) = 0.4. One can check that \u03bc is a fuzzy fantastic prefilter of L.Let .\u03bc be a fuzzy prefilter (filter) in L. Then, \u03bc is a fuzzy fantastic prefilter (filter) of L if and only if \u03bc(((x \u2192 y) \u2192 y) \u2192 x) \u2265 \u03bc(z \u2192 (y \u2192 x))\u2009\u2009\u2227\u2009\u2009\u03bc(z) for all x, y, z \u2208 L.Let \u03bc is\u2009\u2009a fuzzy fantastic prefilter of L. From \u03bc(((x \u2192 y) \u2192 y) \u2192 x) \u2265 \u03bc(y \u2192 x) for all x, y \u2208 L. Since \u03bc is a fuzzy prefilter in L, then \u03bc(y \u2192 x) \u2265 \u03bc(z \u2192 (y \u2192 x))\u2227\u03bc(z). Therefore, we obtain \u03bc(((x \u2192 y) \u2192 y) \u2192 x) \u2265 \u03bc(z \u2192 (y \u2192 x))\u2227\u03bc(z).Suppose that \u03bc satisfies \u03bc(((x \u2192 y) \u2192 y) \u2192 x) \u2265 \u03bc(z \u2192 (y \u2192 x))\u2227\u03bc(z) for all x, y, z \u2208 L. Taking z = 1, we obtain \u03bc(((x \u2192 y) \u2192 y) \u2192 x)) \u2265 \u03bc(1 \u2192 (y \u2192 x))\u2227\u03bc(1) = \u03bc(1 \u2192 (y \u2192 x)). From y \u2192 x \u2264 1 \u2192 (y \u2192 x), it follows that \u03bc(y \u2192 x) \u2264 \u03bc(1 \u2192 (y \u2192 x)). Consequently, we obtain \u03bc(((x \u2192 y) \u2192 y) \u2192 x) \u2265 \u03bc(y \u2192 x). Therefore, \u03bc is a fuzzy fantastic prefilter of L.Conversely, suppose that \u03bc be a fuzzy set in L. \u03bc is a fuzzy fantastic prefilter (filter) of L if and only if, for all t \u2208 , \u03bct is either empty or a fantastic prefilter (filter) of L.Let The proof is easy, and we hence omit the details.F be a nonempty subset of L. F is a fantastic prefilter (filter) if and only if \u03c7F is a fuzzy fantastic prefilter (filter).Let In what follows, we pay attention to the relations among various special fuzzy prefilters (filters).First, the relationship between fuzzy implicative prefilters and fuzzy fantastic prefilters can be described by the following theorem.EQ-algebra.Each fuzzy implicative prefilter (filter) with weak exchange principle is a fuzzy fantastic prefilter (filter) in an \u03bc\u2009\u2009is a fuzzy implicative prefilter with weak exchange principle. From x \u2264 ((x \u2192 y) \u2192 y) \u2192 x, it follows that (((x \u2192 y) \u2192 y) \u2192 x) \u2192 y \u2264 x \u2192 y. This implies that ((((x \u2192 y) \u2192 y) \u2192 x) \u2192 y)\u2192(((x \u2192 y) \u2192 y) \u2192 x)\u2265(x \u2192 y)\u2192(((x \u2192 y) \u2192 y) \u2192 x). Since \u03bc is a fuzzy implicative prefilter, then \u03bc(((((x \u2192 y) \u2192 y) \u2192 x) \u2192 y)\u2192(((x \u2192 y) \u2192 y) \u2192 x)) \u2265 \u03bc((x \u2192 y)\u2192(((x \u2192 y) \u2192 y) \u2192 x)) by \u03bc((x \u2192 y)\u2192(((x \u2192 y) \u2192 y) \u2192 x)) = \u03bc(((x \u2192 y) \u2192 y)\u2192((x \u2192 y) \u2192 x)) as \u03bc has weak exchange principle. From y \u2192 x \u2264 ((x \u2192 y) \u2192 y)\u2192((x \u2192 y) \u2192 x), it follows that \u03bc(y \u2192 x) \u2264 \u03bc(((x \u2192 y) \u2192 y)\u2192((x \u2192 y) \u2192 x)). Together with them, we obtain \u03bc(((((x \u2192 y) \u2192 y) \u2192 x) \u2192 y)\u2192(((x \u2192 y) \u2192 y) \u2192 x)) \u2265 \u03bc(y \u2192 x). On the other hand, since \u03bc is a fuzzy implicative prefilter, then \u03bc(((x \u2192 y) \u2192 y) \u2192 x) \u2265 \u03bc(1 \u2192 ((((x \u2192 y) \u2192 y) \u2192 x) \u2192 y)\u2192(((x \u2192 y) \u2192 y) \u2192 x))\u2227\u03bc(1) = \u03bc(1 \u2192 ((((x \u2192 y) \u2192 y) \u2192 x) \u2192 y)\u2192(((x \u2192 y) \u2192 y) \u2192 x)) \u2265 \u03bc(((((x \u2192 y) \u2192 y) \u2192 x) \u2192 y)\u2192(((x \u2192 y) \u2192 y) \u2192 x)). Consequently, we obtain \u03bc(((x \u2192 y) \u2192 y) \u2192 x) \u2265 \u03bc(y \u2192 x). Therefore, we get that \u03bc is a fuzzy fantastic prefilter by Suppose that From the following example, we can see that the converse of L be the EQ-algebra and let \u03bc be the fuzzy set of L defined in \u03bc is a fuzzy fantastic prefilter of L. But it is not a fuzzy implicative prefilter of L, since \u03bc(b) < \u03bc((b \u2192 0) \u2192 b).Let Next, the following theorems show the relationship between fuzzy implicative prefilters (filters) and fuzzy positive implicative prefilters (filters).EQ-algebra.Each fuzzy implicative prefilter with weak exchange principle is a fuzzy positive implicative prefilter in an \u03bc is a fuzzy implicative prefilter of L. Then, \u03bc is a fuzzy prefilter of L by \u03bc((x \u2192 (x \u2192 z)) \u2265 \u03bc(x \u2192 y)\u2227\u03bc((x \u2192 y)\u2192(x \u2192 (x \u2192 z))). Since y \u2192 (x \u2192 z)\u2264(x \u2192 y)\u2192(x \u2192 (x \u2192 z)), then \u03bc(y \u2192 (x \u2192 z)) \u2264 \u03bc((x \u2192 y)\u2192(x \u2192 (x \u2192 z))). In a similar way, we get that \u03bc(x \u2192 (x \u2192 z)) \u2264 \u03bc(((x \u2192 x) \u2192 z)\u2192(x \u2192 z)). Hence, we obtain \u03bc(((x \u2192 x) \u2192 z)\u2192(x \u2192 z)) \u2265 \u03bc(x \u2192 y)\u2227\u03bc(y \u2192 (x \u2192 z)) = \u03bc(x \u2192 y)\u2227\u03bc(x \u2192 (y \u2192 z)) as \u03bc has weak exchange principle. From ((x \u2192 z) \u2192 z)\u2192(x \u2192 z) \u2264 1 \u2192 (((x \u2192 z) \u2192 z)\u2192(x \u2192 z)), it follows that \u03bc(((x \u2192 z) \u2192 z)\u2192(x \u2192 z)) \u2264 \u03bc(1 \u2192 (((x \u2192 z) \u2192 z)\u2192(x \u2192 z))). Since \u03bc is a fuzzy implicative prefilter of L, then \u03bc(x \u2192 z) \u2265 \u03bc(1 \u2192 (((x \u2192 z) \u2192 z)\u2192(x \u2192 z)))\u2227\u03bc(1) = \u03bc(1 \u2192 (((x \u2192 z) \u2192 z)\u2192(x \u2192 z))). Consequently, we obtain \u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 y)\u2227\u03bc(x \u2192 (y \u2192 z)). Therefore, \u03bc is a fuzzy positive implicative prefilter.Suppose that\u2009\u2009EQ-algebra.Each fuzzy implicative filter is a fuzzy positive implicative filter in an x\u2227(x \u2192 y) \u2264 x and x\u2227(x \u2192 y) \u2264 x \u2192 y, it follows that x\u2227(x \u2192 y) \u2264 x \u2192 y \u2264 (x\u2227(x \u2192 y)) \u2192 y. Then, we have ((x\u2227(x \u2192 y)) \u2192 y) \u2192 y \u2264 (x\u2227(x \u2192 y)) \u2192 y, which implies that ((x\u2227(x \u2192 y)) \u2192 y) \u2192 y)\u2192((x\u2227(x \u2192 y)) \u2192 y) = 1. Hence, we obtain \u03bc(((x\u2227(x \u2192 y)) \u2192 y) \u2192 y)\u2192((x\u2227(x \u2192 y)) \u2192 y)) = \u03bc(1). Since \u03bc is a fuzzy implicative filter of L, then \u03bc(((x\u2227(x \u2192 y)) \u2192 y) \u2192 y)\u2192((x\u2227(x \u2192 y)) \u2192 y)) = \u03bc((x\u2227(x \u2192 y)) \u2192 y) by \u03bc((x\u2227(x \u2192 y)) \u2192 y) = \u03bc(1). Therefore, we get that \u03bc is a fuzzy positive implicative filter by From The following example shows that the converse of the above theorem may not be true.L be the EQ-algebra and let \u03bc be the fuzzy set of L defined in \u03bc is a fuzzy positive implicative filter of L. But it is not a fuzzy implicative filter of L, since \u03bc(a) < \u03bc((a \u2192 0) \u2192 a).Let The following result displays the relations among fuzzy positive implicative prefilters (filters), fuzzy implicative prefilters (filters), and fuzzy fantastic prefilters (filters). Also, it provides a condition under which the converse of Theorems L be a good EQ-algebra. Then, \u03bc is a fuzzy implicative prefilter (filter) if and only if \u03bc is both a fuzzy positive implicative prefilter (filter) and a fuzzy fantastic prefilter (filter) of L.Let \u03bc is a fuzzy implicative prefilter. Since L is a good EQ-algebra, then \u03bc has weak exchange principle. Hence, we have that \u03bc is a fuzzy fantastic prefilter of L by \u03bc is a fuzzy positive implicative prefilter.Suppose\u2009\u2009that \u03bc is both a fuzzy positive implicative prefilter and a fuzzy fantastic prefilter of L. Since \u03bc is a fuzzy positive implicative prefilter, then \u03bc((x \u2192 y) \u2192 y) \u2265 \u03bc((x \u2192 y)\u2192((x \u2192 y) \u2192 y)) by L is a good EQ-algebra; we obtain x \u2264 (x \u2192 y) \u2192 y by x \u2192 y) \u2192 x \u2264 (x \u2192 y)\u2192((x \u2192 y) \u2192 y). It follows that \u03bc((x \u2192 y) \u2192 x) \u2264 \u03bc((x \u2192 y)\u2192((x \u2192 y) \u2192 y)). Consequently, we obtain \u03bc((x \u2192 y) \u2192 y) \u2265 \u03bc((x \u2192 y) \u2192 x). On the other hand, since \u03bc is a fuzzy fantastic prefilter of L, we get \u03bc(((x \u2192 y) \u2192 y) \u2192 x)) \u2265 \u03bc(y \u2192 x) by x \u2192 y) \u2192 x \u2264 y \u2192 x, it follows that \u03bc(y \u2192 x) \u2265 \u03bc((x \u2192 y) \u2192 x), which implies that \u03bc(((x \u2192 y) \u2192 y) \u2192 x) \u2265 \u03bc((x \u2192 y) \u2192 x). Moreover, since \u03bc is a fuzzy prefilter of L, we have \u03bc(x) \u2265 \u03bc(((x \u2192 y) \u2192 y) \u2192 x)\u2227\u03bc((x \u2192 y) \u2192 y) and \u03bc((x \u2192 y) \u2192 x) \u2265 \u03bc(z \u2192 ((x \u2192 y) \u2192 x))\u2227\u03bc(z). Consequently, we obtain \u03bc(x) \u2265 \u03bc((x \u2192 y) \u2192 x)\u2227\u03bc((x \u2192 y) \u2192 x) = \u03bc((x \u2192 y) \u2192 x) \u2265 \u03bc((z \u2192 ((x \u2192 y) \u2192 x))\u2227\u03bc(z). Therefore, \u03bc is a fuzzy implicative prefilter of L by Conversely, suppose that Combining Corollaries L be a good EQ-algebra and let F be a nonempty subset of L. Then, F is an implicative prefilter (filter) if and only if F is both a positive implicative prefilter (filter) and a fantastic prefilter (filter) of L.Let EQ-algebras are good EQ-algebras, in view of Since residuated L be a residuated EQ-algebra and let F be a fantastic filter of L. Then, F is an implicative filter if and only if F is a positive implicative filter of L.Let L be a residuated EQ-algebra and let F be a positive implicative filter of L. Then, F is an implicative filter if and only if F is a fantastic filter of L.Let EQ-algebras.The above results sufficiently show that fuzzy filters are a useful tool to obtain results on classical filters. Moreover, the above results further develop the classical filter theory in In what follows, we continue to find the conditions under which a fuzzy positive implicative prefilter (filter) is a fuzzy implicative prefilter (filter).F be a positive implicative prefilter of an EQ-algebra L. Then, for all a \u2208 L, Fa = {x \u2208 L | a \u2192 x \u2208 F} is the least prefilter containing F and a.Let F is a positive implicative prefilter of L. Since a \u2192 1 = 1 \u2208 F, then 1 \u2208 Fa. For all x, y \u2208 L, if x, x \u2192 y \u2208 Fa, then a \u2192 x \u2208 F and a \u2192 (x \u2192 y) \u2208 F. Since F is a positive implicative prefilter, we have a \u2192 y \u2208 F; that is, y \u2208 Fa. Therefore, Fa is a prefilter in L.Suppose that x \u2208 L, if x \u2208 F, since x \u2264 a \u2192 x, and F is a prefilter, we have a \u2192 x \u2208 F; that is, x \u2208 Fa. Hence, F\u2286Fa. Moreover, since a \u2192 a = 1 \u2208 F, then a \u2208 Fa. Therefore, F \u222a {a}\u2286Fa. Now, let G be a prefilter of L such that F \u222a {a}\u2286G; then, for any x \u2208 Fa, we have a \u2192 x \u2208 F\u2286G. Since a \u2208 G and G is a prefilter in L, we have x \u2208 G; that is, Fa\u2286G. Therefore, Fa is the least prefilter containing F and a.For any F of an EQ-algebra L is called maximal if and only if it is proper and no proper prefilter of L strictly contains F; that is, for each prefilter, G \u2260 F; if F\u2286G then G = L.A prefilter \u03bc be a fuzzy prefilter with weak exchange principle in an EQ-algebra L. The following are equivalent:\u03bc is a fuzzy implicative prefilter and \u03bc\u03bc(1) is a maximal prefilter in L,\u03bc is a fuzzy positive implicative prefilter and \u03bc\u03bc(1) is a maximal prefilter in L,\u03bc satisfies that \u03bc(x) \u2260 \u03bc(1) and \u03bc(y) \u2260 \u03bc(1) can imply that \u03bc(x \u2192 y) = \u03bc(1) and \u03bc(y \u2192 x) = \u03bc(1) for all x, y \u2208 L.Let \u2009\u2009(1)\u21d2(2) It follows from \u03bc(x) \u2260 \u03bc(1) and \u03bc(y) \u2260 \u03bc(1); thus, x \u2209 \u03bc\u03bc(1) and y \u2209 \u03bc\u03bc(1). Since \u03bc is a fuzzy positive implicative prefilter of L, we can easily prove that \u03bc\u03bc(1) is a positive implicative prefilter in L. It follows from Fy = {t \u2208 L | y \u2192 t \u2208 \u03bc\u03bc(1)} is the least prefilter containing \u03bc\u03bc(1) and y. Notice that \u03bc\u03bc(1) is a maximal prefilter in L; we get Fy = L. Hence, x \u2208 Fy; that is, y \u2192 x \u2208 \u03bc\u03bc(1), which implies that \u03bc(y \u2192 x) = \u03bc(1). In a similar way, we can get \u03bc(x \u2192 y) = \u03bc(1).(2)\u21d2(3) Suppose that \u03bc is not a fuzzy implicative prefilter of L. Then, by x, y \u2208 L such that \u03bc(x) < \u03bc((x \u2192 y) \u2192 x). Hence, \u03bc(x) \u2260 \u03bc(1). Now, we consider two cases: either \u03bc(y) = \u03bc(1) or \u03bc(y) \u2260 \u03bc(1).(3)\u21d2(1) Assume on the contrary that \u03bc(y) = \u03bc(1), since \u03bc(y) \u2264 \u03bc(x \u2192 y), then \u03bc(x \u2192 y) = \u03bc(1). Since \u03bc is a fuzzy prefilter in L, then \u03bc(x) \u2265 \u03bc((x \u2192 y) \u2192 x)\u2227\u03bc(x \u2192 y) = \u03bc((x \u2192 y) \u2192 x).If \u03bc(y) \u2260 \u03bc(1), combining \u03bc(x) \u2260 \u03bc(1), we have \u03bc(x \u2192 y) = \u03bc(1) by assumption. Similarly, we have \u03bc(x) \u2265 \u03bc((x \u2192 y) \u2192 x).If \u03bc is a fuzzy implicative prefilter of L. It follows from \u03bc is a fuzzy positive implicative prefilter of L. Hence, \u03bc\u03bc(1) is a positive implicative prefilter in L. By Fa = {x \u2208 L | a \u2192 x \u2208 \u03bc\u03bc(1)} is the least prefilter containing \u03bc\u03bc(1) and a.In any case, we have a contradiction. Therefore, \u03bc\u03bc(1) is a maximal prefilter in L, it is sufficient to show that, for all a \u2208 L \u2212 \u03bc\u03bc(1), Fa = {x \u2208 L | a \u2192 x \u2208 \u03bc\u03bc(1)} = L. For all t \u2208 L, if t \u2208 \u03bc\u03bc(1), then t \u2208 Fa = {x \u2208 L | a \u2192 x \u2208 \u03bc\u03bc(1)}. If t \u2209 \u03bc\u03bc(1), then \u03bc(t) \u2260 \u03bc(1). Since a \u2209 \u03bc\u03bc(1), then \u03bc(a) \u2260 \u03bc(1). It follows from (3) that \u03bc(a \u2192 t) = \u03bc(1); that is, a \u2192 t \u2208 \u03bc\u03bc(1), which implies that t \u2208 Fa = {x \u2208 L | a \u2192 x \u2208 \u03bc\u03bc(1)}. In any case, we have Fa = L. Therefore, \u03bc\u03bc(1) is a maximal prefilter in L. This completes the proof.In order to prove that L be a good EQ-algebra. Suppose that \u03bc is a fuzzy prefilter and \u03bc\u03bc(1) is a maximal prefilter in L. Then, \u03bc is a fuzzy implicative prefilter of L if and only if it is a fuzzy positive implicative prefilter of L.Let Next, we further find the conditions under which a fuzzy positive implicative filter is equivalent to a fuzzy implicative filter.L be a good EQ-algebra with a bottom element 0 and let \u03bc be a fuzzy positive implicative filter. Then, \u03bc is a fuzzy implicative filter if and only if it satisfies \u03bc(\u00ac\u00acx) = \u03bc(x) for all x \u2208 L.Let \u03bc is a fuzzy implicative filter. From \u00ac\u00acx = \u00acx \u2192 0 \u2264 \u00acx \u2192 x, it follows that \u03bc(\u00acx \u2192 x) \u2265 \u03bc(\u00ac\u00acx). By \u03bc(x) = \u03bc(\u00acx \u2192 x). Consequently, we obtain \u03bc(x) \u2265 \u03bc(\u00ac\u00acx). Since L is a good EQ-algebra with a bottom element 0, then x \u2264 (x \u2192 0) \u2192 0 by x \u2264 \u00ac\u00acx. It follows that \u03bc(x) \u2264 \u03bc(\u00ac\u00acx). Therefore, we get \u03bc(\u00ac\u00acx) = \u03bc(x).Suppose that \u03bc is a fuzzy positive implicative filter and satisfies \u03bc(\u00ac\u00acx) = \u03bc(x) for all x \u2208 L. Since \u00acx \u2192 x \u2264 (x \u2192 0)\u2192(\u00acx \u2192 0) = \u00acx \u2192 (\u00acx \u2192 0), then \u03bc(\u00acx \u2192 (\u00acx \u2192 0)) \u2265 \u03bc(\u00acx \u2192 x). It follows from \u03bc(\u00acx \u2192 0) = \u03bc(\u00acx \u2192 (\u00acx \u2192 0)). So we have \u03bc(\u00ac\u00acx) \u2265 \u03bc(\u00acx \u2192 x). Combining \u03bc(\u00ac\u00acx) = \u03bc(x), we obtain \u03bc(x) \u2265 \u03bc(\u00acx \u2192 x). Therefore, \u03bc is a fuzzy implicative filter of L by Conversely, suppose that EQ-algebra L satisfies \u00ac\u00acx = x for all x \u2208 L. We have the following.Notice that an involutive L be a good involutive EQ-algebra. Then, \u03bc is a fuzzy implicative filter of L if and only if it is a fuzzy positive implicative filter of L.Let EQ-algebras, which are good involutive EQ-algebras, we can obtain some related results about fuzzy filters in residuated lattices.Since involutive residuated lattices are involutive residuated L be an involutive residuated lattice and let \u03bc be a fuzzy filter of L. Then, the following are equivalent:\u03bc is a fuzzy implicative filter of L,\u03bc is a fuzzy positive implicative filter of L,\u03bca is a fuzzy filter for all a \u2208 L,\u03bc(x \u2192 y) \u2265 \u03bc(x \u2192 (x \u2192 y)) for all x, y \u2208 L,\u03bc(x \u2192 y) = \u03bc(x \u2192 (x \u2192 y)) for all x, y \u2208 L,\u03bc(x\u2227(x \u2192 y) \u2192 y) = \u03bc(1) for all x, y \u2208 L,\u03bc(x \u2192 (x\u2299x)) = \u03bc(1) for all x \u2208 L,\u03bc(x\u2227y \u2192 (x\u2299y)) = \u03bc(1) for all x, y \u2208 L,\u03bc(x\u2227(x \u2192 y)\u2192(x\u2299y)) = \u03bc(1) for all x, y \u2208 L,\u03bc((x\u2299y) \u2192 z) = \u03bc((x\u2227y) \u2192 z) for all x, y, z \u2208 L,\u03bc(x\u2227(x \u2192 y)\u2192(x\u2227y)) = \u03bc(1) for all x, y \u2208 L,\u03bc(x\u2227(x \u2192 y)\u2192(y\u2227(y \u2192 x))) = \u03bc(1) for all x, y \u2208 L,\u03bc(x) \u2265 \u03bc((x \u2192 y) \u2192 x) for all x, y \u2208 L,\u03bc(x) = \u03bc((x \u2192 y) \u2192 x) for all x, y \u2208 L,\u03bc(x) \u2265 \u03bc(\u00acx \u2192 x) for all x \u2208 L,\u03bc(x) = \u03bc(\u00acx \u2192 x) for all x \u2208 L,\u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 (\u00acz \u2192 y))\u2227\u03bc(y \u2192 z) for all x, y, z \u2208 L,\u03bc(x \u2192 z) \u2265 \u03bc(x \u2192 (\u00acz \u2192 z)) for all x, z \u2208 L,\u03bc(x \u2192 z) = \u03bc(x \u2192 (\u00acz \u2192 z)) for all x, z \u2208 L.Let EQ-algebras. Using characterizations of these fuzzy prefilters (filters), we mainly consider the relations among special fuzzy prefilters (filters). We find some conditions under which a fuzzy positive implicative prefilter (filter) is equivalent to a fuzzy implicative prefilter (filter). As applications of our obtained results, we give some new characterizations about classical filters in EQ-algebras and some related results about fuzzy filters in residuated lattices. In our future work, we will introduce the notion of states on EQ-algebras and discuss the relations between fantastic filters and states on EQ-algebras.In this paper, we present several characterizations of some fuzzy prefilters (filters) in"}
{"text": "G2 . All these four kinds of mappings form a group G5. And all the groups Gi, i = 2,3, 4 are normal subgroups of G5. Moreover, for G5, a generator set is given, which consists of all the involutive negations of the second kind and the standard negation of the first kind. As a subset of the unit square, the interval-valued set is also studied. Two groups are found: one group consists of all the isomorphisms on LI, and the other group contains all the isomorphisms and all the strict negations on LI, which keep the diagonal. Moreover, the former is a normal subgroup of the latter. And all the involutive negations on the interval-valued set form a generator set of the latter group.The main results are about the groups of the negations on the unit square, which is considered as a bilattice. It is proven that all the automorphisms on it form a group; the set, containing the monotonic isomorphisms and the strict negations of the first (or the second or the third) kind, with the operator \u201ccomposition,\u201d is a group  In , the gro2 is a very special bilattice, which is a subset of the real plane. Thus it is of particular interest. In 2\u2223x1 \u2264 x2}, with the natural order \u2264k if and only if x1 \u2264 y1, x2 \u2264 y2. Their properties could be found in 2\u2223x1 + x2 \u2264 1}, with the order \u2009\u2264t\u2009 if and only if x1 \u2264 y1, x2 \u2265 y2. Both LI and L* are sublattices of the bilattice 2, with orders \u2264k and \u2264t. Therefore, similar to 2, \u2264t and \u2264k, defined as, for any x = , y = \u22082,k is the natural order on the unit square.In , 7, threN\u00ac, called reflection, is a unary operator on the square, satisfying the following properties: \u2200x = , y = \u22082,x\u2264ky, then N\u00ac(x)\u2009\u2265k\u2009N\u00ac(y);if x\u2264ty, then N\u00ac(x)\u2009\u2264t\u2009N\u00ac(y);if N\u00ac =  and N\u00ac = .The first kind negation N\u2212, named conflation, is a unary operator satisfying(1\u2032)x\u2264ky, then N\u2212(x)\u2265kN\u2212(y);if (2\u2032)x\u2009\u2264t\u2009y, then N\u2212(x)\u2009\u2265t\u2009N\u2212(y);if (3\u2032)N\u2212 =  and N\u2212 = .The second kind negation N~ is a unary operator satisfying(1\u2032\u2032)x\u2264ky, then N~(x)\u2264kN~(y);if (2\u2032\u2032)x\u2009\u2264t\u2009y, then N~(x)\u2009\u2265t\u2009N~(y);if (3\u2032\u2032)N~ =  and N~ = .The last kind negation N on the unit square, as a bilattice .For convenience, in this paper, these three kinds of negations are collectedly called negations N, N\u2032 is defined as (N\u2218N\u2032) = N). Then, it is not hard to check that N\u00ac\u2218N\u2212 and N\u2212\u2218N\u00ac are negations of the third kind; N\u00ac\u2218N~ and N~\u2218N\u00ac are the second kind negations, and N~\u2218N\u2212 and N\u2212\u2218N~ are of the first kind.The composition of two negations Similar to the definition of strict interval-valued negations in , the strk and \u2264t are strict, that is, in k and \u2264t are replaced by <k and <t in the premise, then the conclusion will be changed to <k and <t correspondingly.A negation on the unit square is called strict, if it is continuous and both \u2264N1 = (n1(x1), n2(x2)) and N2 = (n1(x2), n2(x1)), withci, i = 1,2 are constants in . Then both N1 and N2 are negations, but neither of them is strict.Let N satisfies N(N(x)) = x, then it is called an involutive negation.If a negation Ns,1 defined as Ns,1 = , for all \u22082, is the standard negation of the first kind and named as the first standard negation. The mapping Ns,2 defined as Ns,2 = , for all \u22082, is called the second standard negation. And Ns,3 =  is called the third standard kind negation. It is obvious that Ns,1, Ns,2, and Ns,3 are involutive.The mapping Obviously, each involutive negation is strict, but the converse is not valid.The following negation is strict but not involutive:N\u00ac = (n1(x2), n2(x1)), with n1, n2 being strict negations on , and every second kind strict negation could be characterized as N\u2212 = (n1(x1), n2(x2)), with n1 and n2 being strict negations on .Each of the first kind strict negations could be characterized as N\u00ac = (n1(x2), n2(x1)), with n1 = n2\u22121, negations on , and every second kind involutive negation could be characterized as N\u2212 = (n1(x1), n2(x2)), with n1 and n2 being involutive negations on .Each of the first kind involutive negations could be characterized as Similarly, each of the third kind negations could be characterized as follows.N~ is a strict negation of the third kind, if and only if there are two isomorphisms \u03d51 and \u03d52 on , s.t. N~ could be represented as N~ = N\u00ac\u2218N\u2212, in which N\u00ac is the first kind and N\u2212 is the second kind. Then from Since each ni and \u03d5i, i = 1,2 are bijections on , we can know each strict negation N is a bijection on 2.From Lemmas Note that, the composition of two involutive negations of different kinds is still a negation, but it may not be involutive. The following example shows it.Ns2,,2 \u2192 2 is a monotonic isomorphism on the unit square, if it is bijective and preserves both the orders \u2264k and \u2264t.A continuous mapping \u03a6 : N\u2218\u03a6 and \u03a6\u2032\u2218\u03a6 as (N\u2218\u03a6) = N) and (\u03a6\u2032\u2218\u03a6) = \u03a6\u2032).Obviously, each monotonic isomorphism is continuous. Define \u03d51 and \u03d52 on the unit interval  such thatIf \u03a6 is a monotonic isomorphism on the unit square, then there are isomorphisms x1, x2) \u2208 2, \u2264k. Since \u03a6 preserves \u2264k, we can know \u03a6\u2264k\u03a6. Thus, the greatest element of , \u2264k) is \u03a6. Because \u03a6 is a bijection, we have \u03a6 = . Similarly, \u03a6 = , \u03a6 = , and \u03a6 =  can be proven.Firstly, let us show \u03a6 = , \u03a6 = , \u03a6 = , and \u03a6 = . For any \u2264k \u2208 2, then Ns,1\u2265kNs,1. Because \u03a6 preserves \u2264k, we haveNs,1 reverses the order \u2264k. Similarly, since both \u03a6 and Ns,1 keep the order \u2264t, we can get that \u03a6\u2218Ns,1 preserves the order \u2264t. Since \u03a6 is a bijection, \u03a6 strictly preserves the orders \u2264k and \u2264t. Similarly, it can be proven that \u03a6\u2218Ns,1 strictly preserves the order \u2264t and strictly reverses the order \u2264k.Suppose  = , Ns,1 = , the following hold:Ns,1 is continuous, strictly reverses the order \u2264k, and strictly preserves the order \u2264t, show that \u03a6\u2218Ns,1 is a strict negation of the first kind, denoted by N\u00ac = \u03a6\u2218Ns,1.Since \u03a6 = , \u03a6 = , and x1, x2) \u2208 2,  =  = \u03a6)) = \u03a6, that is, \u03a6 = N\u00ac\u2218Ns,1. From N\u00ac = (n1(x2), n2(x1)), with n1, n2 being negations on the unit interval . Since Ns,1 = , we could get \u03a6 = (n1(1 \u2212 x1), n2(1 \u2212 x2)). Because n1(1 \u2212 x1) and n2(1 \u2212 x2) are isomorphisms on the unit interval, \u03a6 can be characterized as \u03a6 = (\u03d51(x1), \u2009\u03d52(x2)), in which \u03d51(x1) = n1(1 \u2212 x1), \u03d52(x2) = n2(1 \u2212 x2) are isomorphisms on the unit interval.Then for any  = (1 \u2212 \u03d5(x2), 1 \u2212 \u03d5(x1)). It shows that Ns,1\u2218\u03a6 is a continuous bijection on 2, which strictly preserves the order \u2264k and strictly reverses \u2264t and satisfies  =  and  = . Therefore, Ns,1\u2218\u03a6 is a strict negation of the first kind.If Ns = Ns,2 or Ns = Ns,3. Therefore, \u03a6\u2218Ns and Ns\u2218\u03a6 are strict negations and of the same kind as Ns, for Ns = Ns,i, \u2009i = 1, \u20092, \u20093.Similar for N is a negation and Ns is the standard negation of the same kind as N. Then, N\u2218Ns is a monotonic isomorphism \u03a6. As a result, for any  \u2208 2,N = \u03a6\u2218Ns.Next, let us show the second part. Suppose Based on the notions in the above section, the groups of the negations on the unit square will be discussed.In , the fol1) For every strict negation n on the unit interval, there exist three involutive negations n1, n2 and n3 s.t. n = n1\u2218n2\u2218n3.(2) For every isomophism \u03d5 on the unit interval, there exist four involutive negations n1, n2, n3 and n4 s.t. \u03d5 = n1\u2218n2\u2218n3\u2218n4. For every strict negation N of the second kind on the unit square, there exist three involutive negations N1, N2, and N3 of the second kind, s.t. N = N1\u2218N2\u2218N3.(2) For every monotonic isomorphism \u03a6 on the unit square, there exist four involutive negations N1, N2, N3, and N4 of the second, s.t. \u03a6 = N1\u2218N2\u2218N3\u2218N4., i = 1,2, 3,4, 5 are groups, with the same unit element id. Moreover, G1\u25c3G2,3,4,5\u25c3G5, that is, G1 is a normal subgroup of Gi, i = 2,3, 4,5, and G2,3,4,5 are normal subgroups of G5.f in Gi, i = 1,\u2026, 5, id\u2218f = f\u2218id = f, that is, id is the unit element of Gi, i = 1,\u2026, 5.Obviously, for any element 1, \u03a62, the composition of them is still a monotonic isomorphism. Thus G1 is closed under the operator \u201c\u2218\u201d. For two strict negations N1, N2 of the same kind, the composition of them is also a monotonic isomorphism. The composition of a strict negation N and a monotonic isomorphism \u03a6 is still a strict negation of the same kind with N. Therefore, G2, G3, and G4 are closed under the operator \u201c\u2218\u201d. For any two strict negations N1, N2 of different kind, the composition is a strict negation of the other kind. So G5 is closed.For any two monotonic isomorphisms \u03a6G5. This proof also shows the associativity of Gi, i = 1,2, 3,4.Now, let us prove the associativity of F = (f1(x1), f2(x2)) or F = (f1(x2), f2(x1)), in which both of f1 and f2 are isomorphisms on  or both are strict negations on . Let F1, F2, F3 be negations or monotonic isomorphisms on 2. Then,F1\u2218[F2\u2218F3] = [F1\u2218F2]\u2218F3, which shows the associativity of G5.From Lemmas N, obviously \u03a6\u22121 and N\u22121 exist. Moreover, \u03a6\u22121 is also a monotonic isomorphism and N\u22121 is a negation of the same kind as N, that is, the inverse about the composition operator \u201c\u2218\u201d exists. Therefore, Gi, i = 1,\u2026, 5 are groups.For any monotonic isomorphism \u03a6 and any strict negation S1 \u2282 S2,3,4 \u2282 S5, G1 < G2,3,4 < G5, we only need to prove that they are normal subgroups.Since G1\u25c3G2. For any \u03a6 \u2208 S1, obviously, \u03a6\u2218G1\u2218\u03a6\u22121 \u2282 G1, because G1 is a group. For N \u2208 S2\u2216S1, from N = (n1(x2), n2(x1)) and N\u22121 = (n1\u22121(x2), n2\u22121(x1)). For any \u03a6 = (\u03d51(x1), \u03d52(x2)) \u2208 S1,n1\u2218\u03d52\u2218n2\u22121 and n2\u2218\u03d51\u2218n1\u22121 are isomorphisms on the unit interval , N\u2218\u03a6\u2218N\u22121 is in S1. Thus, N\u2218G1\u2218N\u22121 \u2282 G1, that is, G1 is a normal subgroup of G2.Firstly, let us show G1\u25c3G3,4. Since, S5 = S2 \u222a S3 \u222a S4, we could know G1\u25c3G5.Similarly, we could prove G2\u25c3G5. Because G2 is a group, for all f \u2208 G2 , we have f\u2218G2\u2218f\u22121 \u2282 G2. For any N \u2208 G5\u2216G2, N is a negation of the second kind or the third kind. If N is of the second kind, then N\u22121 is also of the second kind. Thus for all \u03a6 \u2208 G2, N\u2218\u03a6\u2218N\u22121 \u2208 G2, and for all N\u2032 \u2208 G2, N\u2218N\u2032\u2218N\u22121 is of the first kind, since N\u2032\u2218N\u22121 is of the third kind. If N is of the third kind, then N\u22121 is also of the third kind. Thus for all \u03a6 \u2208 G2, N\u2218\u03a6\u2218N\u22121 \u2208 G2, and for all N\u2032 \u2208 G2, N\u2218N\u2032\u2218N\u22121 is of the first kind, since N\u2032\u2218N\u22121 is of the second kind. Therefore, for all f \u2208 G5, f\u2218G2\u2218f\u22121 \u2282 G2, that is, G2 is a normal subgroup of G5. Similarly, G3\u25c3G5, G4\u25c3G5 could be proven.Now, let us show From SI,2 of all the involutive negations of the second kind could generate the group G3, that is, for any f \u2208 G3, there is some involutive negations Ni, i = 1,\u2026, m of the second kind, such that f = N1\u2218N2\u2218\u22ef\u2218Nm. And if f is a negation, m could be 3; if f is an isomorphism, m could be 4.The set G3. However, the following problem is still unproven.This theorem shows the relation between the involutive negations of the second kind and the group G2 (or G4) be generated by the set of all the involutive negations of the first (or the third) kind?Could G5 is given in the following theorem.A generator set of SI,2 \u222a {Ns,1} is a generator set of the group G5. And SI,2 \u222a {Ns,3} is also a the generator set of G5.The set N of the first or the third kind, there exists a monotonic isomorphism \u03a6, s.t. N = Ns,1\u2218\u03a6 or N = Ns,2\u2218\u03a6. Since Ns,1\u2218Ns,2 = Ns,3, from SI,2 \u222a {Ns,1} generates the group G5.By Ns,2\u2218Ns,3 = Ns,1, SI,2 \u222a {Ns,3} is also a the generator set of G5.Because LI is a sublattice of the unit square 2, so the negations of the interval-valued set could be defined as the restriction of the negations on the unit square.The interval-valued set NI on the interval-valued set is the restriction of some negation N of the first kind on the unit square, and satisfies thatA negation An interval-valued negation is strict, if it is the restriction of some strict negations of the first kind on the unit square and satisfies .NI is involutive, if it satisfiesAn interval-valued negation NI, it keeps the diagonal \u0394 = { \u2208 LI : x1 = x2}, that is,From this definition, each strict interval-valued negation is an injection and each involutive negation is strict . Also, iI on the interval-valued set is an isomorphism, if it is a bijection and keeps the natural order.A mapping \u03a6I is an isomorphism on the interval-valued set, then \u03a6I keeps both the orders \u2264k and \u2264t , because \u03a6I keeps the diagonal. Define the mapping (\u03a6I)\u22121 as (\u03a6I)\u22121 = (\u03d5\u22121(x1), \u03d5\u22121(x2)). Then (\u03a6I)\u22121 is also an isomorphism on LI, that is, (\u03a6I)\u22121 \u2208 SI, andI)\u22121 is the inverse of \u03a6I. Thus, G1I is a group.Suppose \u03a6NI be a strict negation on LI, which keeps the diagonal \u0394. From n1, n2 on , s.t. NI = (n1(x2), n2(x1)). From (n1 = n2. Then (NI)\u22121 = (n1\u22121(x2), n1\u22121(x1)) is also a strict negation on LI and keeps the diagonal. Also we can check thatNI)\u22121 is the inverse of NI. Thus, G2I is a group.Let )). From , n1 = nG1I\u25c3G2I could be proven.Similar to the proof of This theorem could be extended to the unit square.1) All the monotonic isomorphisms on the unit square, which keep the diagonal, form a group, called G6.(2) All the strict negations of the first kind and the monotonic isomorphisms on the unit square, which keep the diagonal, form a group, called G7.(The proof is similar to From Theorems 1) The set of all the involutive negations on LI generates the group G2I.(2) The set of all the involutive negations of the first kind on the unit square, which keep the diagonal, generates the group G7.(LI, which keep the diagonal, form a group. Moreover, some generator sets of the groups are given.In this paper, we firstly study the negations on the unit square. The main results are Theorems"}
{"text": "The concepts of -fuzzy soft (invertible) n-ary subhypergroups over a commutative n-ary hypergroup are introduced and some related properties and characterizations are obtained. The homomorphism properties of -fuzzy soft (invertible) n-ary subhypergroups are also derived.Maji et al. introduced the concept of fuzzy soft sets as a generalizationof the standard soft sets and presented an application of fuzzy soft sets in a decisionmaking problem. The aim of this paper is to apply the concept of fuzzy soft sets to  To solve complicated problems in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics, which we can consider as mathematical tools for dealing with uncertainties. However, all of these have their advantages as well as inherent limitations in dealing with uncertainties. One major problem shared by those theories is their incompatibility with the parameterization tools. To overcome these difficulties, Molodtsov  introducd-algebras, respectively. Acar et al. r = {x \u2208 H\u2223xr\u2208\u03b3\u2228q\u03b4F(\u03b5)}. The next theorem presents the relationships between -fuzzy soft (invertible) n-ary subhypergroups over H and crisp (invertible) n-ary subhypergroups of H.For any fuzzy soft set \u2329F, A\u232a\u2208FS. Then,F, A\u232a is an -fuzzy soft (invertible) n-ary subhypergroup over H if and only if nonempty subset F(\u03b5)r is an (invertible) n-ary subhypergroup of H for all \u03b5 \u2208 A and r \u2208 -fuzzy soft (invertible) n-ary subhypergroup over H if and only if nonempty subset \u2329F(\u03b5)\u232ar is an (invertible) n-ary subhypergroup of H for all \u03b5 \u2208 A and r \u2208 -fuzzy soft (invertible) n-ary subhypergroup over H if and only if nonempty subset [F(\u03b5)]r is an (invertible) n-ary subhypergroup of H for all \u03b5 \u2208 A and r \u2208  and (3). (1) can be easily proved.\u03b4 = 1 + \u03b3. Let \u2329F, A\u232a be an -fuzzy soft n-ary subhypergroup over H and assume that \u2329F(\u03b5)\u232ar \u2260 \u2205 for some \u03b5 \u2208 A and r \u2208 \u232ar. Then rx1q\u03b4F(\u03b5),\u2026, rxnq\u03b4F(\u03b5); that is, F(\u03b5)(x1) + r > 2\u03b4,\u2026, F(\u03b5)(xn) + r > 2\u03b4. Since \u2329F, A\u232a is an -fuzzy n-ary subhypergroup over H, we have max\u2061{F(\u03b5)(x), \u03b3} \u2265 min\u2061{F(\u03b5)x1xn, \u03b4} for all x \u2208 f(x1n). Hence, by r > \u03b4,r \u2264 1 = 2\u03b4 \u2212 \u03b3, that is, r + \u03b3 \u2264 2\u03b4, we have F(\u03b5)(x) + r > 2\u03b4 and so x \u2208 \u2329F(\u03b5)\u232ar. Similarly we can show that e \u2208 \u2329F(\u03b5)\u232ar and, for all y, y1n\u22121 \u2208 \u2329F(\u03b5)\u232ar, there exists x \u2208 \u2329F(\u03b5)\u232ar such that y \u2208 f. Therefore, \u2329F(\u03b5)\u232ar is an n-ary hypergroup of H.(2) Assume that 2x1n \u2208 H and \u03b5 \u2208 A. If there exists x \u2208 f(x1n) such that max\u2061{F(\u03b5)(x), \u03b3} < min\u2061{F(\u03b5)x1xn, \u03b4}, take r = 2\u03b4 \u2212 max\u2061{F(\u03b5)(x), \u03b3}. Then r \u2208 (x) \u2264 2\u03b4 \u2212 r, F(\u03b5)(x1) > max\u2061{G(\u03b5)(x), \u03b3} = 2\u03b4 \u2212 r,\u2026, F(\u03b5)(xn) > max\u2061{G(\u03b5)(x), \u03b3} = 2\u03b4 \u2212 r, that is, x1n \u2208 \u2329F(\u03b5)\u232ar, but\u2009\u2009x \u2209 \u2329F(\u03b5)\u232ar, a contradiction. Hence max\u2061{F(\u03b5)(x), \u03b3} \u2265 min\u2061{F(\u03b5)x1xn, \u03b4} and so max\u2061{inf\u2061x\u2208f(x1n)\u2061F(\u03b5)(x), \u03b3} \u2265 min\u2061{F(\u03b5)x1xn, \u03b4}. Similarly we can show that conditions (F1a) and (F3a) hold. Therefore, \u2329F, A\u232a is an -fuzzy soft n-ary subhypergroup over H.Conversely, assume that the given condition holds. Let F, A\u232a be an -fuzzy soft n-ary subhypergroup over H and assume that [F(\u03b5)]r \u2260 \u2205 for some \u03b5 \u2208 A and r \u2208 ]r. Then rx1\u2208\u03b3\u2228q\u03b4F(\u03b5),\u2026, rxn\u2208\u03b3\u2228q\u03b4F(\u03b5); that is, F(\u03b5)(x1) \u2265 r > \u03b3 or F(\u03b5)(x1) > 2\u03b4 \u2212 r \u2265 2\u03b4 \u2212 (2\u03b4 \u2212 \u03b3) = \u03b3,\u2026, F(\u03b5)(xn) \u2265 r > \u03b3 or F(\u03b5)(xn) > 2\u03b4 \u2212 r \u2265 2\u03b4 \u2212 (2\u03b4 \u2212 \u03b3) = \u03b3. Since \u2329F, A\u232a is an -fuzzy n-ary subhypergroup over H, we have max\u2061{F(\u03b5)(x), \u03b3} \u2265 min\u2061{F(\u03b5)x1xn, \u03b4} for all x \u2208 f(x1n) and so F(\u03b5)(x) \u2265 min\u2061{F(\u03b5)x1xn, \u03b4} since \u03b3 < min\u2061{F(\u03b5)x1xn, \u03b4} in any case. Now we consider the following cases.(3) Let \u2329Case\u2009\u20091 . In this case, 2\u03b4 \u2212 r \u2265 \u03b4 \u2265 r. F(\u03b5)(x1) \u2265 r or \u22ef or F(\u03b5)(xn) \u2265 r, then F(\u03b5)(x) \u2265 min\u2061{F(\u03b5)x1xn, \u03b4} \u2265 r. Hence zr\u2208\u03b3F(\u03b5).If F(\u03b5)(xi) + r > 2\u03b4, for all 1 \u2264 i \u2264 n, then F(\u03b5)(x) \u2265 min\u2061{F(\u03b5)x1xn, \u03b4} = \u03b4 \u2265 r. Hence zr\u2208\u03b3F(\u03b5).If Case\u2009\u20092 . In this case, r > \u03b4 > 2\u03b4 \u2212 r. F(\u03b5)(xi) \u2265 r, for all 1 \u2264 i \u2264 n, then F(\u03b5)(x) \u2265 min\u2061{F(\u03b5)x1xn, \u03b4} = \u03b4 > 2\u03b4 \u2212 r. Hence zrq\u03b4F(\u03b5).If F(\u03b5)(x1) + r > 2\u03b4 or \u22ef or F(\u03b5)(xn) + r > 2\u03b4, then F(\u03b5)(x) \u2265 min\u2061{F(\u03b5)x1xn, \u03b4} > 2\u03b4 \u2212 r. Hence zrq\u03b4F(\u03b5).If xr\u2208\u03b3\u2228q\u03b4F(\u03b5); that is, x \u2208 [F(\u03b5)]r. Similarly, we can show that e \u2208 [F(\u03b5)]r and, for all y, y1n\u22121 \u2208 [F(\u03b5)]r, there exists x \u2208 [F(\u03b5)]r such that y \u2208 f. Therefore, [F(\u03b5)]r is an n-ary hypergroup of H.Thus, in any case, x1n \u2208 H and \u03b5 \u2208 A. If there exists x \u2208 f(x1n) such that max\u2061{F(\u03b5)(x), \u03b3} < r = min\u2061{F(\u03b5)x1xn, \u03b4}, then F(\u03b5)(x1) \u2265 r > \u03b3,\u2026, F(\u03b5)(xn) \u2265 r > \u03b3, F(\u03b5)(x) < r, and F(\u03b5)(x) + r < 2r \u2264 2\u03b4, that is, (x1)r\u2208\u03b3F(\u03b5),\u2026, (xn)r\u2208\u03b3F(\u03b5), but x1n \u2208 [F(\u03b5)]r, but x \u2209 [F(\u03b5)]r, a contradiction. Hence max\u2061{F(\u03b5)(x), \u03b3} \u2265 min\u2061{F(\u03b5)x1xn, \u03b4} and so max\u2061{inf\u2061x\u2208f(x1n)\u2061F(\u03b5)(x), \u03b3} \u2265 min\u2061{F(\u03b5)x1xn, \u03b4}. Similarly, we can show that conditions (F1a) and (F3a) hold. Therefore, \u2329F, A\u232a is an -fuzzy soft n-ary subhypergroup over H.Conversely, assume that the given condition holds. Let As a direct consequence of \u03b3, \u03b3\u2032, \u03b4, \u03b4\u2032 \u2208  be such that \u03b3 < \u03b4, \u03b3\u2032 < \u03b4\u2032, \u03b3 < \u03b3\u2032, and\u2009\u2009\u03b4\u2032 < \u03b4. Then every -fuzzy (invertible) n-ary soft subhypergroup over H is an -fuzzy soft (invertible) n-ary subhypergroup over H.Let P \u2208 P*(H). Then P is an (invertible) n-ary subhypergroup of H if and only if \u03a3 is an -fuzzy soft (invertible) n-ary subhypergroup over H for any A\u2286E and r \u2265 \u03b4.Let F, A\u232a \u2208 FS. Theni \u2264 n.Let \u2329It is straightforward.F, A\u232a be an -fuzzy soft n-ary subhypergroup over H. Theni \u2264 n;x \u2208 H and 1 \u2264 i \u2264 n.Let \u2329F, A\u232a be an -fuzzy soft n-ary subhypergroup over H, \u03b5 \u2208 A, and 1 \u2264 i \u2264 n. Since max\u2061{F(\u03b5)(e), \u03b3} \u2265 min\u2061{F(\u03b5)(x), \u03b4}, for all x \u2208 H, we have(1) Let \u2329x \u2208 H and 1 \u2264 i \u2264 n. By (1), we have(2) Let F, A\u232a be an -fuzzy soft n-ary subhypergroup over H. Then \u2329F, A\u232a is invertible if and only ifx, y \u2208 H, \u03b5 \u2208 A, and 1 \u2264 i \u2264 n. ProofLet \u2329It is straightforward by Lemmas F, A\u232a,\u2329F1, A\u232a,\u2026, \u2329Fi, A\u232a\u2009\u2009(2 \u2264 i \u2264 n) be -fuzzy soft invertible n-ary subhypergroups over H. ThenF, e\u2217)(y), \u03b4}, \u03b3} = max\u2061\u2061{min\u2061\u2061{F, e\u2217)(x),\u03b4},\u03b3}, for all x, y \u2208 H and \u03b5 \u2208 A;max\u2061{min\u2061{x, y, y1n\u22121 \u2208 H such that y \u2208 f and \u03b5 \u2208 A, one has max\u2061{F(\u03b5)(x), \u03b3} \u2265 min\u2061{F(\u03b5)y1yn\u22121, F(\u03b5)(y), \u03b4}.for all\u2009\u2009Let \u2329x, y \u2208 H and \u03b5 \u2208 A. Then, by F, e\u2217)(y), \u03b4}, \u03b3} \u2265 max\u2061{min\u2061\u2061{F, e\u2217)(x), \u03b4}, \u03b3}. In a similar way, we have max\u2061\u2061{min\u2061\u2061{F, e\u2217)(x), \u03b4}, \u03b3} \u2265 max\u2061\u2061{min\u2061\u2061{F, e\u2217)(y), \u03b4}, \u03b3}. Hence max\u2061\u2061{min\u2061\u2061{F, e\u2217)(y), \u03b4}, \u03b3} = max\u2061\u2061{min\u2061\u2061{F1i, e\u2217)(x), \u03b4}, \u03b3}.(1) Let x, y, y1n\u22121 \u2208 H be such that y \u2208 f and \u03b5 \u2208 A. Then (2) Let Fk, A\u232a be -fuzzy soft invertible n-ary subhypergroup over H for all 1 \u2264 k \u2264 n and 2 \u2264 i \u2264 n. Then \u03b3, \u2208\u03b3\u2228q\u03b4)-fuzzy soft invertible n-ary subhypergroup over H.Let \u2329Fk, A\u232a is an -fuzzy soft invertible n-ary subhypergroup over H for all\u2009\u20091 \u2264 k \u2264 n, let \u03b5 \u2208 A. It is clear that max\u2061\u2061{F(F1i(\u03b5),e\u2217)(e),\u03b3} \u2265 min\u2061\u2061{F(F1i(\u03b5),e\u2217)(x),\u03b4} for all x \u2208 H. From j \u2264 n, we havex, y \u2208 H; that is, F(F1i(\u03b5), e\u2217) is invertible. In fact, by Lemmas Since each \u2329x, y, y1n\u22121 \u2208 H be such that y \u2208 f. Then by F(F1i(\u03b5), e\u2217)(x), \u03b3} \u2265 min\u2061{F(F1i(\u03b5),e\u2217)y1yn\u22121, F(F1i(\u03b5), e\u2217)(y), \u03b4}.Next, let \u03b3, \u2208\u03b3\u2228q\u03b4)-fuzzy soft invertible n-ary subhypergroup over H.Summing up the above arguments, F, A\u232a and \u2329G, B\u232a be two -fuzzy soft invertible n-ary subhypergroups over H. Then both F, A\u232a\u22d2\u2329G, B\u232a are -fuzzy soft n-ary subhypergroups over H.Let \u2329F, A\u232a and \u2329G, B\u232a be -fuzzy soft invertible n-ary subhypergroups over H and \u03b5 \u2208 A \u222a B, we consider the following cases.Let \u2329Case\u2009\u20091 (\u03b5 \u2208 A \u2212 B). In this case, L(\u03b5) = F(\u03b5). It follows that L(\u03b5) satisfies conditions (F1a), (F2a), and (F3a) since \u2329F, A\u232a is an -fuzzy soft invertible n-ary subhypergroup over S. Case\u2009\u20092 (\u03b5 \u2208 B \u2212 A). In this case, L(\u03b5) = G(\u03b5). It follows that L(\u03b5) satisfies conditions (F1a), (F2a), and (F3a) since \u2329G, B\u232a is an -fuzzy soft invertible n-ary subhypergroup over S. Case\u2009\u20093 (\u03b5 \u2208 A\u2229B). In this case, L(\u03b5) = F(\u03b5)\u2229G(\u03b5). Now we show that F(\u03b5)\u2229G(\u03b5) satisfies conditions (F1a), (F2a), and (F3a). (1)x \u2208 H, we haveFor any (2)x, x1n \u2208 H be such that x \u2208 f(x1n). ThenLet It follows that max\u2061{inf\u2061x\u2208f(x1n)\u2061(F(\u03b5)\u2229G(\u03b5))(x), \u03b3} \u2265 min\u2061{(F(\u03b5)\u2229G(\u03b5))x1xn, \u03b4}.(3)x, y, y1n\u22121 \u2208 H be such that y \u2208 f. Analogous to (2), we haveLet \u03b3, \u2208\u03b3\u2228q\u03b4)-fuzzy soft n-ary subhypergroup over H. The case for \u2329F, A\u232a\u22d2\u2329G, B\u232a can be similarly proved.Therefore, \u03b3, \u2208\u03b3\u2228q\u03b4)-fuzzy soft (invertible) subhypergroups. Let us first introduce the following definition.In this section, we consider the homomorphism properties of  and FS be two fuzzy soft classes, and let \u03c6 : U \u2192 U\u2032 and \u03d5 : E \u2192 E\u2032 be mappings. Define a mapping  : FS \u2192 FS by the following: for \u2329F, A\u232a \u2208 FS, the image of \u2329F, A\u232a under , denoted by \u2329F, A\u232a = \u2329\u03c6(F), \u03d5(A)\u232a, is a fuzzy soft set in FS given by\u03b5\u2032 \u2208 \u03d5(A) and x\u2032 \u2208 U\u2032. For \u2329F\u2032, A\u2032\u232a \u2208 FS, the inverse image of \u2329F\u2032, A\u2032\u232a under , denoted by \u22121\u2329F\u2032, A\u2032\u232a = \u2329\u03c6\u22121(F\u2032), \u03d5\u22121(A\u2032)\u232a, is a fuzzy soft set in FS given by\u03b5 \u2208 \u03d5\u22121(B) and x \u2208 U.Let The following results can be easily deduced.F, A\u232a, \u2329G, B\u232a\u2208FS and \u03c6 : U \u2192 U\u2032 and \u03d5 : E \u2192 E\u2032 be two mappings. ThenF, A\u232a\u22d0\u22121\u2329F, A\u232a) and \u2329F, A\u232a = \u22121\u2329F, A\u232a) if both \u03c6 and \u03d5 are injective;\u2329\u03c6, \u03d5)\u22d0\u2329F, A\u232a\u22d2\u2329G, B\u232a;\u2329F, A\u232a\u22d0\u03b3,\u03b4)\u2329G, B\u232a.if \u2329Let \u2329F\u2032, A\u2032\u232a, \u2329G\u2032, B\u2032\u232a \u2208 FS and \u03c6 : U \u2192 U\u2032 and \u03d5 : E \u2192 E\u2032 be two mappings. Then\u03c6, \u03d5)\u22121\u2329F\u2032, A\u2032\u232a)\u22d0\u2329F\u2032, A\u2032\u232a and \u22121\u2329F\u2032, A\u2032\u232a) = \u2329F\u2032, A\u2032\u232a if both \u03c6 and \u03d5 are surjective;\u22121 = \u22121\u2329F\u2032, A\u2032\u232a\u22d2\u22121\u2329G\u2032, B\u2032\u232a;\u22121\u2329F\u2032, A\u2032\u232a\u22d0\u03b3,\u03b4)\u22121\u2329G\u2032, B\u2032\u232a.if \u2329Let \u2329H, f) and  be two n-ary hypergroups with identities e and e\u2032, respectively, and \u03c6 a mapping from H to H\u2032. Then, \u03c6 is called a homomorphism if \u03c6(e) = e\u2032 and \u03c6(f(x1n)) = f\u2032(\u03c6(x1),\u2026, \u03c6(xn)) for all x1n \u2208 H. If such a homomorphism is surjective, injective, or bijective, it is called an epimorphism, a monomorphism, or an isomorphism.Let  and  be two n-ary hypergroups with identities e and e\u2032, respectively, \u03c6 : H \u2192 H\u2032 a homomorphism, and \u03d5 : E \u2192 E\u2032 a mapping. Then )\u22d0F\u2032\u2329F1, A1\u232a,\u2026, \u2329Fn, An\u232a) for all \u2329F,A\u232a1n \u2208 FS such that \u22c2i=1nAi \u2260 \u2205 and ) = F\u2032\u2329F1, A1\u232a,\u2026, \u2329Fn, An\u232a) if \u03d5 is injective.Let , \u03b5\u2032 \u2208 \u03d5(\u22c2i=1nAi)\u2286\u22c2i=1n\u03d5(Ai), and x\u2032 \u2208 H\u2032. If \u03c6\u22121(x\u2032) = \u2205, then F\u2032(\u03c6(F1),\u2026, \u03c6(Fn))(\u03b5\u2032)(x\u2032) = 0 = \u03c6(F(F1n)(\u03b5\u2032))(x\u2032). Otherwise, we have\u03c6, \u03d5))\u22d0F\u2032\u2329F1, A1\u232a,\u2026, \u2329Fn, An\u232a). If \u03d5 is injective, then the symbol \u201c\u2265\u201d in the above proof can be replaced with \u201c=\u201d; hence ) = F\u2032\u2329F1, A1\u232a,\u2026, \u2329Fn, An\u232a).Let \u2329H, f) and  be two commutative n-ary hypergroups with identities e and e\u2032, respectively, \u03c6 : H \u2192 H\u2032 a homomorphism, and \u03d5 : E \u2192 E\u2032 an injective mapping. Let \u2329F, A\u232a be an -fuzzy n-ary soft subhypergroup over H which has the sup-property. Then, \u2329F, A\u232a is an -fuzzy n-ary subhypergroup over H\u2032. If \u03c6 is an epimorphism from H onto H\u2032 and \u2329F, A\u232a is invertible, then \u2329F, A\u232a is also invertible.Let -fuzzy n-ary soft subhypergroup over H which has the sup-property and \u03b5\u2032 \u2208 \u03d5(A). Then there exists a unique \u03b5 \u2208 A such that \u03d5(\u03b5) = \u03b5\u2032 and we have the following.(1)\u03c6(F)(\u03b5\u2032)(e\u2032), \u03b3} \u2265 min\u2061{\u03c6(F)(\u03b5\u2032)(x\u2032), \u03b4} for all x\u2032 \u2208 H.It is clear that max\u2061{(2)By (3)y\u2032, y1\u2032,\u2026, yn\u22121\u2032 \u2208 H\u2032. If \u03c6\u22121(yj\u2032) = \u2205\u2009\u2009(1 \u2264 j \u2264 n \u2212 1) or \u03c6\u22121(y\u2032) = \u2205, thenLet for y\u2032 \u2208 f\u2032. Otherwise, there exist y, y1n\u22121 \u2208 H such that \u03c6(y) = y\u2032, \u03c6(yi) = yi\u2032, \u03c6(F)(\u03b5\u2032)(y\u2032) = F(\u03b5)(y), and \u03c6(F)(\u03b5\u2032)(yi\u2032) = F(\u03b5)(yi) for all 1 \u2264 i \u2264 n \u2212 1 since \u2329F, A\u232a has the sup-property. Now, for y, y1n\u22121 \u2208 H, since \u2329F, A\u232a is an -fuzzy soft n-ary subhypergroup over H, there exists x \u2208 H such that y \u2208 f and max\u2061{F(\u03b5)(x), \u03b3} \u2265 min\u2061{F(\u03b5)y1yn\u22121, F(\u03b5)(y), \u03b4}. Hence we have\u03c6, \u03d5) is an -fuzzy soft n-ary subhypergroup over H\u2032.Assume that \u2329\u03c6 is an epimorphism from H onto H\u2032 and that \u2329F, A\u232a is an invertible -fuzzy soft n-ary subhypergroup over H. Then, for any x\u2032, y\u2032 \u2208 H\u2032, we haveNext assume that \u03c6, \u03d5)\u2329F, A\u232a is invertible.Therefore,  and  be two commutative n-ary hypergroups with identities e and e\u2032, respectively, \u03c6 : H \u2192 H\u2032 a homomorphism, and\u2009\u2009\u03d5 : E \u2192 E\u2032 a mapping. Let \u2329F\u2032, A\u2032\u232a be an -fuzzy soft n-ary subhypergroup over H\u2032. Then \u22121\u2329F\u2032, A\u2032\u232a is an -fuzzy soft n-ary subhypergroup over H. If both \u03c6 and \u03d5 are injective and \u2329F\u2032, A\u2032\u232a is an -fuzzy soft (invertible) n-ary subhypergroup over H\u2032, then \u22121\u2329F\u2032, A\u2032\u232a is an -fuzzy soft (invertible) n-ary subhypergroup over H.Let -fuzzy soft n-ary subhypergroup over H\u2032. Let \u03b5 \u2208 \u03d5\u22121(A\u2032). Then we have the following.(1)x \u2208 H, max\u2061{\u03c6\u22121(F\u2032)(\u03b5)(e), \u03b3} = max\u2061{F\u2032(\u03d5(\u03b5))(\u03c6(e)), \u03b3} \u2265 min\u2061{F\u2032(\u03d5(\u03b5))(\u03c6(x)), \u03b4} = min\u2061{\u03c6\u22121(F\u2032)(\u03b5)(x), \u03b4}.For any (2)x, x1n \u2208 H be such that x \u2208 f(x1n). Then \u03c6(x) \u2208 \u03c6(f(x1n)) = f\u2032(\u03c6(x)1n) and soLet It follows that max\u2061{inf\u2061x\u2208f(x1n)\u2061\u03c6\u22121(F\u2032)(\u03b5)(z), \u03b3} \u2265 min\u2061{\u03c6\u22121(F\u2032)(\u03b5)x1xn, \u03b4}.(3)x, y, y1n\u22121 \u2208 H be such that y \u2208 f. Then \u03c6(y) \u2208 f\u2032(\u03c6(x), \u03c6(y1),\u2026, \u03c6(yn\u22121)). Since \u2329F\u2032, A\u2032\u232a is invertible, by Let Summing up the above analysis, \u22121\u2329F\u2032, A\u2032\u232a is an -fuzzy n-ary soft subhypergroup over H. If both \u03c6 and \u03d5 are injective and \u2329F\u2032, A\u2032\u232a is an -fuzzy soft invertible n-ary subhypergroup over H\u2032, it is easy to check that \u22121\u2329F\u2032, A\u2032\u232a is an -fuzzy soft invertible n-ary subhypergroup over H.Assume that \u2329n-ary hypergroups which generalize some algebra structures: soft groups, fuzzy n-ary hypergroups, fuzzy hypergroups, and so on. The goal is to explain new methodological development in n-ary hypergroups which will also be of growing importance in the future. The results presented in this paper can hopefully provide more insight into and a full understanding of fuzzy soft set theory and algebraic hyperstructures. Our future work on this topic will focus on studying some other classes of fuzzy soft n-ary hypergroups.In this paper, our aim is to promote the research and development of fuzzy soft technology by studying fuzzy soft"}
{"text": "The concept of soft translations of soft subalgebras and soft ideals over BCI/BCK-algebras is introduced and some related properties are studied. Notions of Soft extensions of soft subalgebras and soft ideals over BCI/BCK-algebras are also initiated. Relationships between soft translations and soft extensions are explored. Recently soft set theory has emerged as a new mathematical tool to deal with uncertainty. Due to its applications in various fields of study researchers and practitioners are showing keen interest in it. As enough number of parameters is available here, so it is free from the difficulties associated with other contemporary theories dealing with uncertainty. Prior to soft set theory, probability theory, fuzzy set theory, rough set theory, and interval mathematics were common mathematical tools for dealing with uncertainties, but all these theories have their own difficulties. These difficulties may be due to lack of parametrization tools , 2. To oAt present, work on the soft set theory is progressing rapidly. Maji et al.  describeThe study of BCI/BCK-algebras was initiated by Imai and Iseki  as the gTranslations play a vital role in reducing the complexity of a problem. In geometry it is a common practice to translate a system to some new position to study its properties. In linear algebra translations help find solution to many practical problems. In this paper idea of translations is being extended to soft BCI/BCK algebras.This paper is arranged as follows: in First of all some basic concepts about BCI/BCK-algebra are given. For a comprehensive study on BCI/BCK-algebras  is a verX, \u2217, 0) is called a BCI-algebra if it satisfies the following conditions:x, y, z \u2208 X)\u2009\u2009(((x\u2217y)\u2217(x\u2217z))\u2217(z\u2217y) = 0),\u2009\u2009((x\u2217(x\u2217y))\u2217y = 0),(\u2200x \u2208 X)\u2009\u2009(x\u2217x = 0),\u2009\u2009.x \u2208 X)\u2009\u2009(0\u2217x = 0),\u2009\u2009(x\u22170 = x),\u2009\u2009(x\u2217y = 0\u21d2(x\u2217z)\u2217(y\u2217z) = 0, (z\u2217y)\u2217(z\u2217x) = 0),\u2009\u2009((x\u2217y)\u2217z = (x\u2217z)\u2217y),\u2009\u2009(((x\u2217z)\u2217(y\u2217z))\u2217(x\u2217y) = 0).\u2009\u2009.(\u2200A subset U be a universe and E be a set of parameters. Let P(U) denote the power set of U and let A, B be nonempty subsets of E.Now we recall some basic notions in soft set theory. Let F, A) is called a soft set over U, where F is a mapping given by F : A \u2192 P(U).A pair  is called a relative null soft set (with respect to the parameters set A), denoted by \u2205A, if F(a) = \u2205, for all a \u2208 A. is called a relative whole soft set (with respect to the parameters set A), denoted by UA, if G(e) = U, for all e \u2208 A. is denoted by c and is defined by c = , where Fc : A \u2192 P(U) is a mapping given by Fc(a) = U \u2212 F(a), \u2200a \u2208 A. Clearly, c)c = .The complement of a soft set  over U is called a full soft set if \u22c3a\u2208AF(a) = U.A soft set  be set valued map defined asA\u2286X. Then FA also denotes a soft set over a BCI/BCK algebra X. From here onward a soft set will be denoted by symbols like FA, unless stated otherwise.Let FA over a BCI/BCK-algebra X is called a soft subalgebra of X if it satisfiesX =  denote a BCI/BCK-algebra, and for any soft set FA over X, we denote T : = X \u2212 \u222a{FA(x)\u2223x \u2208 X} unless otherwise specified.A soft set T = (\u22c3x\u2208XFA(x))c = \u22c2x\u2208XFAc(x).That is T\u2229FA(x) = \u2205 for all x \u2208 X. If FA is a full soft set then T is an empty set. Therefore throughout this paper only those soft set are considered which are not full.It is easy to see that FA be a soft set over X and let U1\u2286T. A mapping FU1T : X \u2192 P(X) is called a soft U1-translation of FA if, for all x \u2208 X,Let U1\u2286T and FA be a soft set over X, then FA(x) \u222a U1\u2287FA(y) \u222a U1 implies FA(x)\u2287FA(y), for all x, y \u2208 X.Let U1\u2286T, FA(x)\u2229U1 = \u2205 and FA(y)\u2229U1 = \u2205. Let a \u2208 FA(y) then a \u2208 FA(y) \u222a U1\u2286FA(x) \u222a U1 this implies a \u2208 FA(x) or a \u2208 U1 but a \u2209 U1 because FA(y)\u2229U1 = \u2205. So a \u2208 FA(x) that is FA(x)\u2287FA(y), for all x, y \u2208 X.Since FA be a soft subalgebra of X and U1\u2286T. Then the soft U1-translation FU1T of FA is a soft subalgebra of X.Let x, y \u2208 X. ThenFU1T is a soft subalgebra of X.Let FA be a soft set over X such that the U1-translation FU1T of FA is a soft subalgebra of X for some U1\u2286T. Then FA is a soft subalgebra of X.Let FU1T is a soft subalgebra of X for some U1\u2286T. Let x, y \u2208 X, we havex, y \u2208 X. Hence FA is a soft subalgebra of X.Assume From Propositions FA of X is a soft subalgebra of X if and only if U1-translation FU1T of FA is a soft subalgebra of X for some U1\u2286T.A soft set FA and GB be two soft sets over X. If FA(x)\u2286GB(x) for all x \u2208 X, then we say that GB is a soft extension of FA.Let X = {0,1, 2,3} presented as follows:FA and GB of X as in Consider a BCI/BCK-algebra FA(0)\u2286GB(0), FA(1)\u2286GB(1), FA(2)\u2286GB(2), and FA(3)\u2286GB(3), which implies that GB is a soft extension of FA.Here S-extension is being introduced.Next the concept of soft FA and GB be two soft sets over X. Then GB is called a soft S-extension of FA, if the following conditions hold:GB is a soft extension of FA.FA is a soft subalgebra of X, then GB is a soft subalgebra of X.If Let FU1T(x)\u2287FA(x) for all x \u2208 X. As a consequence of As we know FA be a soft subalgebra of X and U1\u2286T. Then the soft U1-translation FU1T of FA is a soft S-extension of FA.Let The converse of X = {0,1, 2,3} given as follows:FA of X by Consider a BCI/BCK-algebra FA is a soft subalgebra of X. For soft set FA, T = {3}. Let GB be a soft set over X given by Then GB is a soft S-extension of X. But it is not a soft U1-translation of FA for any nonempty U1\u2286T.Then FA of X, U1\u2286T and U2 \u2208 P(X) with U2\u2287U1, letFA is a soft subalgebra of X, then it is clear that UU1 is a subalgebra of X for all U2 \u2208 P(X) with U2\u2287U1. But, if we do not give condition that FA is a soft subalgebra of X, then UU1 may not be a subalgebra of X as seen in the following example.For a soft set X = {0,1, 2,3, 4} be a BCI/BCK-algebra presented as follows:Let FA of X by Define a soft subset FA is not a soft subalgebra of X with T = {1}. Since FA(3\u22174) = {0}\u2289{0,4} = FA(3)\u2229FA(4) For U2 = {1,4} and U1 = {1}, we obtain UU1 = {3,4} which is not a subalgebra of X since 3\u22173 = 0 \u2209 UU1.Then U1-translations and UU1 is studied in case of soft subalgebra.In the following theorem, relationship between FA be a soft set over X and U1\u2286T. Then the soft U1-translation FU1T of FA is a soft subalgebra of X if and only if UU1 is a subalgebra of X for all U2 \u2208 P(U) with U2\u2287U1.Let U1-translation FU1T of FA is a soft subalgebra of X. Then by FA is a soft subalgebra of X if FU1T is a soft subalgebra of X. Further let a, b \u2208 UU1, then FA(a)\u2287U2 \u2212 U1 and FA(b)\u2287U2 \u2212 U1 are subalgebras of X for all U2 \u2208 P(U) with U2\u2287U1. Considera\u2217b \u2208 UU1, which shows that UU1 is a subalgebra of X, for all U2\u2286P(U), with U2\u2287U1.Assume that the soft UU1 is a subalgebra of X for all U2\u2286P(U) with U2\u2287U1. Now assume that there exist a, b \u2208 X such thatFA(a)\u2287U2 \u2212 U1 and FA(b)\u2287U2 \u2212 U1 but FA(a\u2217b) \u2282 U2 \u2212 U1. This shows that a, b \u2208 UU1 and a\u2217b \u2209 UU1, which is a contradiction and so FU1T(a\u2217b)\u2287FU1T(a)\u2229FU1T(b) for all a, b \u2208 X. Hence FU1T is a soft subalgebra of X.Conversely, suppose that FA be a soft subalgebra of X and let U1, U2\u2286T. If U1\u2287U2, then the soft U1-translation FU1T of FA is a soft S-extension of the soft U2-translation FU2T of FA.Let U1\u2287U2, this implies FU1T(x)\u2287FU2T(x), for all x \u2208 X. So U1-translation is an extension of U2-translation, and from FU1T and FU2T are soft subalgebras of FA. Hence soft U1-translation FU1T of FA is a soft S-extension of the soft U2-translation FU2T of FA.Since FA of X and U2\u2286T, the soft U2-translation FU2T of FA is a soft subalgebra of X. If GB is a soft S-extension of FU2T and then there exists U1\u2286T such that U1\u2287U2 and GB(x)\u2287FU1T(x), for all x \u2208 X. Thus, we have the following theorem.For every soft subalgebra FA be a soft subalgebra of X and U2\u2286T. For every soft S-extension GB of soft U2-translation FU2T of FA, there exists a U1\u2286T such that U1\u2287U2 and GB are a soft S-extension of U1-translation of FA.Let FA of X and U2\u2286T, the soft U2-translation FU2T of FA is a soft subalgebra of X. If GB is a soft S-extension of FU2T and then there exists U1\u2286T such that U1\u2287U2 and GB(x)\u2287FU1T(x), for all x \u2208 X. Then by GB is a soft S-extension of U1-translation of FA.For every soft subalgebra S-extension GB of a soft subalgebra FA of X is said to be normalized if there exists x0 \u2208 X such that GB(x0) = X.A soft FA be a soft subalgebra of X. A soft set GB of X is called a maximal soft S-extension of FA if it satisfies the following conditions:GB is a soft S-extension of FA,X which is a soft extension of GB.there does not exist another soft subalgebra of Let Z+ be a set of positive integers and let \u201c\u2217\u201d be a binary operation on Z+ defined byx, y \u2208 Z+, where  is the greatest common divisor of x and y. Then  is a BCK-algebra. Let FA and GB be soft sets of Z+ which are defined by FA(x) = {1,2, 3} and GB(x) = Z+ for all x \u2208 Z+. Clearly, FA and GB are soft subalgebras of Z+. By using definition of maximal soft S-extension, then it is easy to see that GB is a maximal soft S-extension of FA.Let GB of X is a normalized soft S-extension of a soft subalgebra FA of X, then GB(0) = X.If a soft set GB is a normalized soft S-extension of a soft subalgebra FA of X then there exists x0 \u2208 X such that GB(x0) = X, for some x0 \u2208 X. ConsiderGB(0) = X.Assume that FA be a soft subalgebra of X. Then every maximal soft S-extension of FA is normalized.Let S-extensions.This follows from the definitions of the maximal and normalized soft Now concept of translation of a soft ideal of a BCI/BCK-algebra is introduced.FA of a BCI/BCK-algebra is called a soft ideal of X, denoted by FA\u22b2SX, if it satisfies:x \u2208 X)\u2009(FA(0)\u2287FA(x)),\u2009(F(x)\u2287(FA(x\u2217y)\u2229FA(y))). = FA(0) \u222a U1\u2287FA(x) \u222a U1 = FU1T(x) andFU1T is a soft ideal of X for some U1\u2286T. Let x, y \u2208 X. ThenFA(0)\u2287FA(x). NextFA(x)\u2287FA(x\u2217y)\u2229FA(y)  by . Hence FIn this section concept of soft ideal extension is being introduced and some of its properties are studied.FA and GB be the soft subsets of X. Then GB is called the soft ideal extension of FA, if the following conditions hold:GB is a soft extension of FA.FA\u22b2SX\u21d2GB\u22b2SX.Let FA of X, U1\u2286T and U2 \u2208 P(X) with U2\u2287U1, define EU1: = {x \u2208 X\u2223FA(x) \u222a U1\u2287U2}.For a soft subset FA\u22b2SX, then UU1\u22b2X for all U2 \u2208 P(U) with U2\u2287U1.It is clear that if U1\u2286T, let FU1T be the soft U1-translation of FA. Then the following are equivalent:FU1T\u22b2SX.U2 \u2208 P(U))\u2009\u22b2X).(\u2200For FU1T\u22b2SX and let U2 \u2208 P(U) be such that U2\u2283U1. Since FU1T(0)\u2287FU1T(x) for all x \u2208 X, we havex \u2208 EU1.x, y \u2208 X be such that x\u2217y \u2208 EU1 and y \u2208 EU1. Then FA(x\u2217y) \u222a U1\u2287U2 and FA(y) \u222a U1\u2287U2, that is, FU1T(x\u2217y) = FA(x\u2217y) \u222a U1\u2287U2 and FU1T(y) = FA(y) \u222a U1\u2287U2. Since FU1T\u22b2SX, it follows thatFA(x) \u222a U1\u2287U2 so that x \u2208 EU1. Therefore EU1\u22b2X.(1)\u21d2(2) Consider EU1\u22b2X for every U2 \u2208 P(U) with U2\u2287U1. If there exists x \u2208 X with U3\u2287U1 such that FU1T(0) \u2282 U3\u2286FU1T(x) and then FA(x) \u222a U1\u2287U3 but FA(0) \u222a U1 \u2282 U3. This shows that x \u2208 EU1 and 0 \u2209 EU1. This is a contradiction, and so FU1T(0)\u2287FU1T(x), for all x \u2208 X.(2)\u21d2(1) Suppose that a, b \u2208 X such that FU1T(a) \u2282 U4\u2286FU1T(a\u2217b)\u2229FU1T(b). Then FA(a\u2217b) \u222a U1\u2287U4 and FA(b) \u222a U1\u2287U4, but FA(a) \u222a U1 \u2282 U4. Hence a\u2217b \u2208 EU1 and b \u2208 EU1, but a \u2209 EU1. This is impossible and therefore FU1T(x)\u2287FU1T(x\u2217y)\u2229FU1T(y), for all x, y \u2208 X. Consequently FU1T\u22b2SX.Now assume that there exist FA\u22b2SX and U1, U2\u2286T. If\u2009\u2009U1\u2287U2, then the soft U1-translation FU1T of FA is a soft ideal extension of the soft U2-translation FU2T of FA.Let U1\u2287U2, this implies that (FU1T(x)\u2287FU2T(x))\u2009\u2009(\u2200x \u2208 X). This shows that FU1T is a soft extension of FU2T.SinceFU2T is a soft ideal of X, then FU1T(0) = FA(0) \u222a U1\u2287FA(x) \u222a U1 = FU1T(x) for all x \u2208 X, so we have (FU1T(0)\u2287FU1T(x)). ConsiderFU1T(x)\u2287FU1T(x\u2217y)\u2229FU1T(y))\u2009\u2009 so FU1T is a soft ideal of X. Hence FU1T is a soft ideal extension of FU2T.Now, let Soft set theory is a mathematical tool to deal with uncertainties. Translation and extension are very useful concepts in mathematics to reduce the complexity of a problem. These concepts are frequently employed in geometry and algebra. In this papers, we presented some new notions such as soft translations and soft extensions for BCI/BCK-algebras. We also examined some relationships between soft translations and soft extensions. Moreover, soft ideal extensions and translations have been introduced and investigated as well. It is hoped that these results may be helpful in other soft structures as well."}
{"text": "To prove the existence of traveling wave solutions, an invariant cone is constructed by upper and lower solutions and Schauder's fixed point theorem is applied. The nonexistence of traveling wave solutions is proved by two-sided Laplace transform. However, to apply two-sided Laplace transform, the prior estimate of exponential decrease of traveling wave solutions is needed. For this aim, a new method is proposed, which can be applied to reaction-diffusion systems consisting of more than three equations.Based on Code\u00e7o's cholera model (2001), an epidemic cholera model that incorporates the pathogen diffusion and disease-related death is proposed. The formula for minimal wave speed  Vibrio cholera. An estimated 3\u20135 million cases and over 100,000 deaths occur each year around the world /c, a1 = \u2212(1 + \u03ba) < 0, and a2 = (\u03ba \u2212 c2)/c. To investigate distribution of roots of =\u03bb3+ang lemma .0 > 0, ((a) If \u03940 > 0,  has one 0 = 0, ((b) If \u03940 = 0,  has a mu0 < 0, ((c) If \u03940 < 0,  has threThen, we get the following lemma about the distribution of eigenvalues.U0 > 1/g\u2032(0). Then, there exists a constant c* > 0 which is the only positive root of  < 0 if 0 < \u03b5 < \u03bb2 \u2212 \u03bb1.one can assume roots of ; then H = 0 has only one positive root c* such that \u0394 < 0 for c > c* and \u0394 > 0 for 0 < c < c*. Direct calculations show that \u03940 = \u0394/(108c4). Since a0 > 0 and a1 < 0, Descartes' rule of signs shows that (H(\u03bb) is a cubic polynomial.Obviously, ows that  has onlyows that  has rootU0 > 1/g\u2032(0) and c > c* unless other conditions are specified. Denote \u03bb1 < \u03bb2 to be the two positive roots of .In this section, we always suppose roots of  and defi\u03be \u2260 ln\u2061V0/\u03bb1.The function \u03be < ln\u2061V0/\u03bb1 and, therefore, g\u2032\u2032(V) \u2264 0 for any V \u2265 0, we have\u03b8 < 1.Firstly, assume atisfies  and g\u2032\u2032.For \u03c3 > 0 be sufficiently large to ensure 1/\u03b1ln\u2061(U0/\u03c3) < ln\u2061V0/\u03bb1. When \u03be > 1/\u03b1ln\u2061(U0/\u03c3), then \u03be < 1/\u03b1ln\u2061(U0/\u03c3). Then, \u03b8 < 1. Let \u03c3 = 1/\u03b1. Since\u03b1 > 0 sufficiently small and \u03c3 > 0 sufficiently large such thatLet \u03b5 = min\u2061{\u03b1, \u03bb1, \u03bb2 \u2212 \u03bb1}/2. Then, for M > 0 large enough, the function \u03be \u2260 1/\u03b5ln\u2061(1/M).Let \u03be = 1/\u03b1ln\u2061(U0/\u03c3), that \u03be = 1/\u03b5ln\u2061(1/M), and that 1/\u03b5ln\u2061(1/M) < 1/\u03b1ln\u2061(U0/\u03c3) if and only if M > (\u03c3/U0)\u03b5/\u03b1)(. Let M > (\u03c3/U0)\u03b5/\u03b1)(. When \u03be > 1/\u03b5ln\u2061(1/M), then e\u03bb1\u03be(1 \u2212 Me\u03b5\u03be) < 0, It is clear that \u03be < 1/\u03b5ln\u2061(1/M). Then, \u03be < 1/\u03b1ln\u2061(U0/\u03c3), \u03b8 < 1. Since\u03be < 1/\u03b1ln\u2061(U0/\u03c3), thenH(\u03bb1 + \u03b5) < 0, inequality . Let \u03bc be a positive constant which will be specified in the following. For \u03a6(\u03be) = (\u03d51(\u03be), \u03d52(\u03be)), defineC. Firstly, we change system (\u03b21 > lim\u2061V\u2192+\u221eg(V),1 < 0 < \u039b2 are the two roots of equation \u039b2 \u2212 c\u039b \u2212 1 = 0. Furthermore, define F =  : \u0393 \u2192 C by\u03bc < min\u2061{\u03b21/c, \u03ba/c, \u2212\u039b1, \u039b2}.To apply Schauder's fixed point theorem, we will introduce a topology in e system  into theF(\u0393) \u2282 \u0393.Consider U(\u00b7), V(\u00b7)) \u2208 \u0393; that is Let (\u03b21 > lim\u2061V\u2192+\u221eg(V). Then,First of all, we haveF2(U(\u00b7), V(\u00b7))(\u03be). If \u03be \u2265 \u03be0\u225c1/\u03b5ln\u2061(1/M), then \u03be < \u03be0. From \u03be \u2208 R.Now, we study Similarly, we can show F =  : \u0393 \u2192 C is continuous with respect to the norm |\u00b7|\u03bc in B\u03bc.Map 1(\u00b7) = (U1(\u00b7), V1(\u00b7)), \u03a62(\u00b7) = (U2(\u00b7), V2(\u00b7)) \u2208 \u0393, we haveM1 = \u03b21 + J + U0g\u2032(0), and V* is between V1(\u03be) and V2(\u03be). Therefore, we have\u03be \u2264 0, we getFor \u03a6\u03be > 0, it follows that\u03be \u2208 R, whereF1 : \u0393 \u2192 C is continuous with respect to the norm |\u00b7|\u03bc in B\u03bc.When F2(U(\u03be), V(\u03be)). Firstly, we haveM3 = (J + U0g\u2032(0))\u03ba/(\u03ba \u2212 c\u03bc). Therefore,\u03be < 0, it holds that\u03be \u2265 0, we haveF2 : \u0393 \u2192 C is continuous with respect to the norm |\u00b7|\u03bc in B\u03bc. The proof is completed.In addition, consider F =  : \u0393 \u2192 \u0393 is compact with respect to the norm |\u00b7|\u03bc in B\u03bc.Map \u03be) = (U(\u03be), V(\u03be)) \u2208 \u0393, it is clear thatH1(\u03be)| = |[\u03b21 \u2212 g(V(\u03be))]U(\u03be)| \u2264 \u03b21U0, we haved/d\u03be)F1(\u03a6(\u00b7))(\u00b7)|\u03bc < 2\u03b21U0/c. Since \u03a6(\u00b7) = (U(\u00b7), V(\u00b7)) \u2208 \u0393, we getd/d\u03be)F2(\u03a6(\u00b7))(\u00b7)|\u03bc < 2(\u03b21U0 + V0)/(\u039b2 \u2212 \u039b1). Consequently, |(d/d\u03be)F1(\u03a6(\u00b7))(\u00b7)|\u03bc and |(d/d\u03be)F2(\u03a6(\u00b7))(\u00b7)|\u03bc are bounded, which shows that F(\u0393) is uniformly bounded and equicontinuous with respect to the norm |\u00b7|\u03bc in B\u03bc.For any \u03a6 is uniformly bounded and equicontinuous with respect to the norm |\u00b7|\u03bc in B\u03bc, which implies that Fn : \u0393 \u2192 \u0393 is a compact operator. SinceF2n(U(\u00b7),V(\u00b7))(\u00b7)\u2212F2(U(\u00b7),V(\u00b7))(\u00b7)|\u03bc \u2192 0 when n \u2192 +\u221e. Thus, |Fn(U(\u00b7),V(\u00b7))(\u00b7)\u2212F(U(\u00b7),V(\u00b7))(\u00b7)|\u03bc \u2192 0 when n \u2192 +\u221e. By Proposition 2.1 in Zeilder (\u03bb) is increasing in (\u03bb) = +\u221e or \u03bb* = +\u221e.Assume  has a trondition . From thondition , it follG(V): = g\u2032(0)V \u2212 g(V), from which it follows that G\u2032(0) = 0. Since g(V) satisfies hypothesis (A1), we have G(V) \u2265 0 for any V \u2265 0.Furthermore, we haveality of  can be r\u03bb* < +\u221e, it is clear that L[V(\u00b7)](\u03bb*\u2212) = +\u221e. SinceG\u2032(0) = 0 imply L[Q(\u00b7)](\u03bb*) < +\u221e. However, \u03bb*)<+\u221e, which contradicts  = +\u221e, it is impossible that the above equality holds, again a contradiction.Taking two-sided Laplace transform of , we have\u0398(\u03bb)L[V(\u00b7\u0398(\u03bb)L[V(\u00b7U(y + c\u03c4), W(y + c\u03c4), V(y + c\u03c4)) is a traveling wave solution of system (\u221e\u2212\u221e+V(\u03be)d\u03be = \u222b\u221e\u2212\u221e+H2(\u03be)d\u03be. Furthermore,Suppose (f system  satisfyif system . Then, b"}
{"text": "Scientific Reports6: Article number: 28657; 10.1038/srep28657 published online: 06282016; updated: 08032016.st\u20133rd Qu\u2019 and \u2018Range\u2019 \u201c\u22123.07\u20131.80\u201d and \u201c\u22125.06\u20130.42\u201d should read: \u201cof \u201c\u22123.07\u2013\u2009\u22121.80\u201d and \u201c\u22125.06\u2013\u2009\u22120.42\u201d respectively. In addition, the Girls-pre values for \u20181st\u20133rd Qu\u2019 and \u2018Range\u2019 \u201c\u22122.54\u20130.85\u201d and \u201c\u22126.01\u20130.42\u201d should read: \u201c\u22122.54\u2013\u2009\u22120.85\u201d and \u201c\u22126.01\u2013\u2009\u22120.42\u201d respectively.This Article contains typographical errors in Table 1. In the column \u2018Pubertal age\u2019 the values for Boys-pre \u20181"}
{"text": "This paper mainly studies the Laplacian-energy-like invariants of the modified hexagonal lattice, modified Union Jack lattice, and honeycomb lattice. By utilizing the tensor product of matrices and the diagonalization of block circulant matrices, we derive closed-form formulas expressing the Laplacian-energy-like invariants of these lattices. In addition, we obtain explicit asymptotic values of these invariants with software-aided computations of some integrals. G, which is studied in chemistry and used to approximate the total electron energy of a molecule [E\u03c0 is closely related to the energy releasing from the formation progress of the conjugated carbon oxides and could be approximately calculated by H\u00fcckel molecular orbital theory. And in the method of HMO, the calculation of E\u03c0 can be attributed to the sum of the absolute values of all the eigenvalues of its molecular graph [Molecular structure descriptors or topological indices are used for modelling information of molecules, including toxicologic, chemical, and other properties of chemical compounds in theoretical chemistry. Topological indices play a very important role in mathematical chemistry, especially in the quantitative structure-property relationship (QSPR) and quantitative structure activity relationship (QSAR). Many topological indices have been introduced and investigated by mathematicians, chemists, and biologists, which contain energy , the Lapmolecule . During ar graph \u201320.Compared with adjacency matrix, the definition of Laplacian matrix added to all vertices degrees. As Mohar said, the Laplacian eigenvalues can reflect more the combination properties of graphs. Cvetkovi\u0107 and Simi\u0107 \u201323 pointG, the vertex set and edge set of G will be denoted by V(G) = {v1, v2,\u2026, vn} and E(G) = {e1, e2,\u2026, em}, respectively [G are, respectively, A(G) and D(G); then the matrix L(G) = D(G) \u2212 A(G) is called the Laplacian matrix of the graph G [G are \u03c7G(\u03bb) = det\u2061(\u03bbI \u2212 A(G)) and \u03bcG(\u03bb) = det\u2061(\u03bbI \u2212 L(G)) [A(G) and L(G) are symmetric matrices; their eigenvalues are real numbers [G as \u03bb1 \u2265 \u03bb2 \u2265 \u22ef\u2265\u03bbn, and the Laplacian eigenvalues are \u03bc1 \u2265 \u03bc2 \u2265 \u22ef\u2265\u03bcn [G is a connected graph, then \u03bci > 0, i = 1,2,\u2026, n \u2212 1, \u03bcn = 0\u2009\u2009[All the graphs discussed in this paper are simple, finite, and undirected. For a graph ectively . The adj graph G , 28. The \u2212 L(G)) . Both A, is defined asLet A = m\u00d7n, B = s\u00d7t, the tensor product of A and B, denoted by A \u2297 B, is defined asFor two matrices Gn} be a sequence of finite simple graphs with bounded average degree such that Let {Hn} be a sequence of spanning subgraphs of {Gn} such that Let {Gn and Hn have the same asymptotic Laplacian-energy-like invariant.That is, In what follows, we will explore the Laplacian-energy-like invariants formulas of the modified hexagonal lattice, modified Union Jack lattice, and honeycomb lattice.t.The modified hexagon lattice with toroidal boundary condition is denoted by MH\u03b1i = 2\u03c0i/n1, \u03b2j = 2\u03c0j/n2. ThenLet In1, In2 are the unit matrices and M \u2297 N is tensor product of matrices M and N. ConsiderWith the proper labelling of the vertices of the modified hexagonal lattice, its Laplacian matrix isL) can be defined as follows: The matrix g0, g1,\u2026, gn\u22121} be a cyclic group of order n. Obviously, \u03c1 : gi \u2192 Bni can express the group. The cyclic group of order n has linear values of n\u2009\u2009\u03c7i\u2009\u2009,\u2009\u2009\u03c7i(g) = \u03c9ni, where \u03c9n are said n-times unit roots.Let {1 = Therefore, there is a reversible matrix In fact, So In2 \u2297 (6In1 \u2212 Dn1 \u2212 Dn1\u22121) \u2212 Dn2 \u2297 (In1 + Dn1\u22121) \u2212 Dn2\u22121 \u2297 (In1 + Dn1) is a diagonal matrix whose diagonal elements are i \u2264 n1 \u2212 1 and\u2009\u20090 \u2264 j \u2264 n2 \u2212 1.It is not difficult to find that L are \u03bc = 6 \u2212 2cos\u2061\u03b1i \u2212 2cos\u2061\u03b2j \u2212 2cos\u2061(\u03b1i \u2212 \u03b2j), 0 \u2264 i \u2264 n1 \u2212 1, and\u2009\u20090 \u2264 j \u2264 n2 \u2212 1, where \u03b1i = 2\u03c0i/n1 and \u03b2j = 2\u03c0j/n2.This means that the eigenvalues of the matrix By formula , the LapSo The numerical integration value in last line is calculated with the software MATLAB . As suchBy Theorems MHt, MHc, and MHf with toroidal, cylindrical, and free boundary conditions, then,For the modified hexagonal lattices St.The modified Union Jack lattice with toroidal boundary condition is denoted by \u03b1i = 2\u03c0i/n1;\u2009\u2009\u03b2j = 2\u03c0j/n2. Then Let With a proper labelling of the vertices of the modified Union Jack lattice, its Laplacian matrix can be represented as Based on LetActually, So i \u2264 n1 \u2212 1 and\u2009\u20090 \u2264 j \u2264 n2 \u2212 1.It is not difficult to find that L are \u03bc = 8 \u2212 2cos\u2061\u03b1i \u2212 2cos\u2061\u03b2j \u2212 4cos\u2061\u03b1i\u03b2j, 0 \u2264 i \u2264 n1 \u2212 1, and\u2009\u20090 \u2264 j \u2264 n2 \u2212 1, where \u03b1i = 2\u03c0i/n1 and \u03b2j = 2\u03c0j/n2.This means that the eigenvalues of the matrix By formula , the LapSo By Theorems St, Sc, and Sf with toroidal, cylindrical, and free boundary conditions, then, For the modified Union Jack lattices t, can be constructed by starting with an m \u00d7 n square lattice and adding two diagonal edges to each square.The honeycomb lattice with toroidal boundary condition, denoted by HC\u03b1i = 2\u03c0i/n1 and \u03b2j = 2\u03c0j/n2. Then Let M = n1n2 and F is an M \u00d7 M matrix. The matrix F can be written in the following form: I represents the unit matrix of n1 \u00d7 n1 and IM represents the unit matrix of M \u00d7 M, respectively.Similarly, the Laplacian matrix of the honeycomb lattice is F can be written as Based on Let Similarly, hence, So In2 \u2297 (In1 + Dn1\u22121) + Dn2\u22121 \u2297 In1 is a diagonal matrix whose diagonal elements are 1 + \u03c9n1i\u2212 + \u03c9n2j\u2212, so matrix L) can be reduced to the following form: It is not difficult to find that \u03bcI \u2212 L)) = 0, we can get By det\u2061) characteristic eigenvalues are i \u2264 n1 \u2212 1 and 0 \u2264 j \u2264 n2 \u2212 1.Therefore, the \u03b1i = 2\u03c0i/n1 and \u03b2j = 2\u03c0j/n2. By formula , HCc, and HCf with toroidal, cylindrical, and free boundary conditions, then, For the honeycomb lattices HCIn this paper, we mainly studied the Laplacian-energy-like invariants of the modified hexagonal lattice, modified Jack lattice, and honeycomb lattice. The Laplacian-energy-like invariants formulas of these lattices are obtained. The proposed results imply that the asymptotic Laplacian-energy-like invariants of those lattices are independent of the three boundary conditions.The problems on the various topological indices of lattices have much important significance in the mathematical theory, chemical energy, statistical physics, and networks science. This paper investigated the Laplacian-energy-like invariants of some lattices. However, the other topological indices of the general lattices remain to be studied."}
{"text": "The \u201cHydraulically weighted diameter (mm)\u2014average\u201d and \u201cHydraulically weighted diameter (mm) \u2013standard deviation\u201d columns in S2 Table(XLSX)Click here for additional data file."}
{"text": "GL(3) and discuss the analytic behavior of the test functions on both sides. Applications to Weyl\u2019s law, exceptional eigenvalues, a large sieve and L-functions are given.We develop a fairly explicit Kuznetsov formula on  GL(2) and the cornerstone for the investigation of moments of families of L-functions, including striking applications to subconvexity and non-vanishing. It can be viewed as a relative trace formula for the group G=GL(2) and the abelian subgroup U2\u00d7U2\u2286G\u00d7G where U2 is the group of unipotent upper triangular matrices. The Kuznetsov formula in the simplest case is an equality of the shape n,m\u2208\u2124\u2216{0},\u03b4n,m is the Kronecker symbol,uj for the group SL2(\u2124) having spectral parameter tj and Hecke eigenvalues \u03bbj(n) for n\u2208\u2115 (and \u03bbj(\u2212n):=\u00b1\u03bbj(n) depending on whether uj is even or odd),the sum on the left-hand side runs over an orthogonal basis of Hecke\u2013Maa\u00df cusp forms \u03c3t(n) is the Fourier coefficient of an Eisenstein series defined by dspect=\u03c0\u22122ttanh(\u03c0t)\u2009dt is the spectral measure,h is some sufficiently nice, even test function, andh\u00b1 is a certain integral transform of h, the sign being sgn(nm), described in\u00a0, SO and SU; see also\u00a0 with compact support. Let\u03b5>0 and letIfX1=X2=1, thenMoreover, there is a constantcdepending onf1,f2such that the following holds: ifthenand if\u03bd1,\u03bd2are given by  an anLetA,given by  and in at1,t2,X1,X2 as above, I as a function of \u03bd1,\u03bd2\u2208{z\u2208\u2102:|\u2111z|\u22641/2} is under some technical assumptions a function with a bump at \u2111\u03bd1=\u03c41 and \u2111\u03bd2=\u03c42 of size X1=X2=1. Only if we need a test function that blows up at exceptional eigenvalues we will choose X2 to be large. The asymmetry in X1,X2 in\u00a0>1/50. It is now easy to see that the remaining u1-integral for A\u2192\u221e vanishes, and we are left with the sum over the residues. Next we shift in the same way the u2-integral to \u2212\u221e, and express\u00a0 whenever j=3, r\u2208{1,2}  \u03b1j,\u03b1r\u22643, \u03c41,\u03c42\u22650,\u2111\u03bd2\u2265\u2111\u03bd1\u22650 and show that they all satisfy the bound\u00a0 or the term j=3, r=2 (if \u03c1<0) satisfies . By Stilarge by  +O(1) and can therefore be bounded by  and can therefore be bounded by X1=X2=1. It follows from  \\documnated by . Similarnated by  the 4 tenated by . We procnated by  for X1=ows from  and  into three pairs \u03bd1|<\u03b5 the second line can be bounded by the mean value theorem. Then we use the functional equation s\u0393(s)=\u0393(s+1) of the Gamma-function in connection with Stirling\u2019s formula as before and bound the preceding display by \u03bd1, this can again be estimated by the mean value theorem giving the crude bound \u03b11, \u03b12, \u03b13 are pairwise distinct, that is \u03bd1=0 follows by continuity.\u2003\u25a1Finally we need to treat the case ondition  is emptyondition  is also ondition  for X1=eturn to  and partible for . This co\u03c41, \u03c42 sufficiently large one has \u03bd1,\u03bd2\u2208i\u211d in a neighborhood of i\u03c41, i\u03c42, respectively, and it is very small outside this region.An inspection of the proof, in particular \u20133.13), , 3.13), \u0393=SL3(\u2124). We denote by Let \u03bd1,\u03bd2  in the notation of\u00a03 is contained in a fundamental domain for \u03bd1,\u03bd2 is sufficiently large.\u2003\u25a1We conclude from Lemma\u00a01 that as in\u00a0 the uppesee e.g.\u00a0), we obtSL2(\u2124) and spectral parameter \u03bd\u2208i\u211d ). A cusp form u has a Fourier expansion W\u03bd was defined in\u00a0. Again we see that an arithmetically normalized cusp form u is essentially L2-normalized, and W\u03bd has roughly norm one with respect to the inner product We briefly discuss cusp forms fined in\u00a0. Similarhence by\u00a0  the evows from , 6.576.4There are three types of Eisenstein series on the space minimal parabolic Eisenstein series: for \u03bd1,\u211c\u03bd2 sufficiently large we define the minimal parabolic Eisenstein series \u03bd1 and \u03bd2, and its non-zero Fourier coefficients are given by I-function.The first term gives rise to fined in\u00a0. It has 5.1 . Then we define s, and as the minimal parabolic Eisenstein series it is an eigenform of all Hecke operators; in particular for s=1/2+\u03bc it is an eigenform of T with eigenvalue c. This can be seen by setting m1=m2=1 and comparing with  denotes the least common multiple. In particular, ifAll implied constants depend only on\u03b5.It remains to show  which isD1\u2223D2, we put m1,n1\u2009(mod D1) and n2\u2009(mod D2/D1), and for p1q1\u22611\u2009(mod D1), p2q2\u22611\u2009(mod D2) we have . Analytic properties of We choose now With the notation developed so far, we haveLetC1,C2\u22650, \u03b5>0. ThenandIn the special case whenA1,A2\u22641, X1=1, X2=X\u22651, R1+R2\u224d\u03c41+\u03c42this can be improved toLetgbe a fixed smooth function with compact support in . Then forR1,R2\u226b1 sufficiently large we haveOn the left hand side we have suppressed the dependence ofon\u03c41, \u03c42.A1,A2 from below, and therefore D1,D2 in\u00a0(X1=X2=1. It is instructive to compare this with the GL(2) situation: one can construct a sufficiently nice test function h on the spectral side with essential support on  such that the integral transforms h\u00b1 in\u00a0, , 8.8),  are not 1,D2 in\u00a0 from abos h\u00b1 in\u00a0 are negl\u03c41,\u03c42 can be performed at almost no cost, in other words, we save a factor (R1R2)\u03b51\u2212 compared to trivial integration.The bound  shows thy-integrals in\u00a02. Hence the second part of  defined by  instead of \u03b5. To this end, we combine as before \u22121/2\u224d(R1+R2)\u22121/2. Our present assumption X1=1, X2=X\u22651, A1,A2\u22641 together with the size constraints ((y)\u226aj1 uniformly in all other variables. We can assume that \u03f51=sgn(x1) and |x1|\u224d\u03c41+\u03c42, otherwise we can save as many powers of \u03c41+\u03c42 as we wish by repeated partial integration. In that case we make another change of variables and re-write (\u03c41+\u03c42)\u22121/2: we cut out smoothly the region y1=1+O((\u03c41+\u03c42)\u22121/2) which we estimate trivially. For the rest we apply integration by parts. The treatment of the y2 integral is very similar. Here our assumptions imply that the integral is of the form \u03c41+\u03c42)\u22121/2.The proof of  is a smae C1 in\u00a0 by C1+\u03b7he bound  follows s before  and the egral in\u00a0 are bothstraints \u20138.16) i iR1+R2\u03c41,\u03c42. In the situation (\u03b5+\u03b7(C1+C2) instead of \u03b5. In order to show , this condition is empty unless A\u2265B\u22650. Since we are assuming that \u039e1,\u039e2 are sufficiently large, we can deduce from (x3-region is x2) and hence the total contribution under the assumption \u2212100) by the first part of (T1T2(T1+T2). On the spectral side, we drop the Eisenstein spectrum and large parts of the cuspidal spectrum to conclude by  balls. To prove the lower bound, we choose  K so large that O((T1+T2)\u03b5) by known bounds for the zeta function on the line \u211cs=1, the third line contributes similarly O((T1+T2)\u03b51+) by Weyl\u2019s law for SL2(\u2124) and lower bounds for the L-functions in the denominator\u00a0[T1,T2 sufficiently large.\u2003\u25a1For the  combine , 8.2), , proof oX2=1 in\u00a0, fix a f part of  and  is O(T2). In order to prove Theorem\u00a02, it is therefore enough to consider those Maa\u00df forms with |\u03c1j|\u2265\u03b5. Moreover, by symmetry it is enough to bound only Maa\u00df forms satisfying  The long Weyl element contributes at most X=T2 completes the proof of Theorem\u00a02.\u2003\u25a1The tisfying . In . The left hand side of  and using  by an approximate functional equation. As we are summing over the archimedean parameters of the L-functions, we need an approximate functional equation whose weight function is essentially independent of the underlying family. This has been obtained in [\u03d5j as in Theorem\u00a04 put n|=n(1)+\u22ef+n(6) and Finally we prove LetG0:\u2192\u211d be a smooth function with functional equationG0(x)+G0(1/x)=1 and derivatives decaying faster than any negative power ofxasx\u2192\u221e. LetM\u2208\u2115 and fix a Maa\u00df form\u03d5as above. There are explicitly computable rational constantscn,\u2113\u2208\u211a depending only onn, \u2113, Msuch that the following holds forFor any\u03b5>0 one haswhere |\u03baj|=1 and the implied constant depends at most on\u03b5, M, and the functionG0.T is sufficiently large. Let s=\u03c3>0. By\u00a0(Cj\u226a\u03b7jT2\u226aT3. This is at most O(T\u03b55+). By Theorem\u00a01 or (O(T\u03b55+). This completes the proof of Theorem\u00a04.\u2003\u25a1It is now a simple matter to prove Theorem\u00a04. We can assume that =\u03c3>0. By\u00a0 and Mellem\u00a03 and  the firsrem\u00a01 or  it is ea"}
{"text": "Then we study some properties about approximation of lattice ordered effect algebras.Many authors have studied roughness on various algebraic systems. In this paper, we consider a lattice ordered effect algebra and discuss its roughness in this context. Moreover, we introduce the notions of the interior and the closure of a subset and give some of their properties in effect algebras. Finally, we use a Riesz ideal induced congruence and define a function  Quantum effects play a basic role in the foundations of quantum mechanics. In 1994, Foulis and Bennett introduced effect algebras for modeling unsharp measurement in a quantum mechanical system . It is aThe theory of rough sets was first introduced by Pawlak  as a tooE can be regarded as (possibly) unsharp experimentally testable propositions about a physical system. A subset X of E can be regarded as (possibly) some unsharp experimentally testable propositions about a physical system. We think about a rough approximation of X to be the set of propositions which is indiscernible with some propositions in X. It means that two propositions p and q are observable simultaneously in terms of quantum measurement theory that p, q \u2208 E are indiscernible.Events of quantum logics do not describe \u201cunsharp measurements\u201d since unsharp measurements do not have a \u201cyes-no\u201d character. To include such events another algebraic structure was introduced by Foulis and Bennett (1994), called an effect algebra. Hence elements of an effect algebra In this paper, we discuss the roughness of lattice ordered effect algebra and introduce the notions of the interior and the closure of a subset. We give some properties of the interior and the closure of a subset in lattice ordered effect algebras.In this section, we recall some definitions and results which will be used in the sequel.E with partial binary operation \u2295 and two nullary operations 0 and 1 satisfying the following axioms.(E1)a \u2295 b = b \u2295 a if a \u2295 b is defined.(E2)a \u2295 b) \u2295 c = a \u2295 (b \u2295 c) if one side is defined.((E3)a \u2208 E there exists a unique b \u2208 E such that a \u2295 b = 1.For every (E4)a is defined then a = 0.If 1 \u2295 An effect algebra is a partial algebra R, where a\u22a5b is defined by a + b \u2264 1, in which case a \u2295 b = a + b.(1) A simple example of an effect algebra is \u2286n-chain, Cn = {0, a, 2a,\u2026, na = 1}, where ia\u22a5ja if and only if i + j \u2264 n for i, j = 0,1, 2,\u2026, n, in which case ia \u2295 ja = (i + j)a.(2) Another example is an H be a complex Hilbert space and let \u03a6(H) be the set of all bounded self-adjoint operators on H. The positive cone \u03a6(H)+ in \u03a6(H) is the set of all A \u2208 \u03a6(H) that satisfy \u2329Ax, x\u232a \u2265 0 for all x \u2208 H. We then write A \u2264 B if B \u2212 A \u2208 \u03a6(H)+. Letting 0 and 1 be the zero and identity operators, respectively, we have that 1 \u2208 \u03a6(H)+ and (\u03a6(H), +, 0, \u2264) is a partially ordered Abelian group. It can be checked that \u2118(H) = \u03a6(H)+ is an effect algebra but not a Boolean effect algebra.(3) Let E, we can introduce a partial order \u2264 on E: a \u2264 b iff there exists c : a \u2295 c = b. We denote b \u2296 a = c iff a \u2295 c = b. It is easy to check that \u2296 is a well defined partial operation. In \u03b8\u2286X}. Here X0 is called the interior (or the lower rough approximation) of X, X, and X. Apr(X) is called the rough approximation of X (or the rough set) in  (see ,\u2200\u03b8 is a congruence and [0]\u03b8 = [p]\u03b8 = [q]\u03b8 = I and [p\u2032]\u03b8 = [q\u2032]\u03b8 = [1]\u03b8 = {p\u2032, q\u2032, 1}. Let X = {0,1}; then X0 = \u2205 and Clearly U, \u03b8) be an approximation space and let X be a subset of U.X is called a crisp set.If X0 = \u2205, then X is called having an empty interior with respect to \u03b8.If X. If X is called having an empty exterior with respect to \u03b8.We define Let  be an approximation space. Clearly, \u2205 and U are definable with respect to \u03b8. The family of all crisp sets is denoted by Cri(Apr).Let  instead of the approximation space . Also, in this case we use the symbols XI0 instead of X0.Let E be an effect algebra, let I be a Riesz ideal of E, and let X be a subset of E. Then the following statements are equivalent:X is a crisp set;XI0 = X iff X is a union of some equivalence classes.Let The proof is trivial.The following properties of approximation spaces are well known and obvious. They are similar to Proposition\u2009\u20093.1 in  and PropE be an effect algebra and let I be a Riesz ideal of E. For the approximation space , for arbitrary subsets X, Y\u2286E, and for each x \u2208 X, one hasE and \u2205 are crisp sets,XI0, X]I are crisp sets with respect to I,X\u2286Y, then XI0\u2286YI0 and if X\u2229Y)I0 = XI0\u2229YI0,(X \u222a Y)I0\u2287XI0 \u222a YI0, denoted by Aprc(X) is defined by The rough complement of Apr(The proof is similar to the I = [a]I = {0, a}, [b]I = [1]I = {b, 1}. Therefore, we have XI0 = \u2205, YI0 = {b, 1}, X \u222a Y)I0 = E, X \u222a Y)I0\u2009\u2288\u2009XI0 \u222a YI0, Then . We also can check that I = {0, a} is a Riesz ideal. Let X = {a, 1} and Y = {0, b, c} be subsets of E. Then [0]I = [a]I = {0, a}, [b]I = [c]I = {b, c}, [d]I = [1]I = {d, 1}, [e]I = {e}. Therefore, we have YI0 = {b, c}, X \u222a Y)I0 = {0, a, b, c}, X \u222a Y)I0\u2009\u2288\u2009XI0 \u222a YI0, Let X be a subset of an effect algebra E and let X\u2227 be the annihilator of X in E defined by X\u2227 = {a \u2208 E : a\u2227x = 0, \u2200x \u2208 X}. Denote X\u22a5 = (X\u2032)\u2227, where X\u2032 = {x\u2032 \u2208 E : x \u2208 X}.Let I be a Riesz ideal of effect algebra E and let X be a subset of E. ThenX\u22a5)I0\u2009\u2286\u2009(XI0)\u22a5,\u22a5 and again by X\u22a5)I0\u2286(XI0)\u22a5. Similarly, we can prove (2), (3), and (4).(1) By The following example shows that inclusion symbols \u2286 in E; \u2295, 0,1) be a lattice ordered effect algebra in X = {a}, Y = {0, a} be subsets of E and let I = {0, a} be a Riesz ideal of E. It is easy to check that X\u22a5 = {0, a}, Y\u22a5 = {0}, [0]I = [a]I = I, [1]I = [b]I = {b, 1}. So we have (Y\u22a5)I0 = \u2205, (YI0)\u22a5 = {0}, \u2009YI0)\u22a5\u2009\u2288\u2009(Y\u22a5)I0, in a lattice ordered effect algebra E and build a relationship between it and congruence classes. Then we study some properties about approximation in lattice ordered effect algebras.In this section, we define a function We give some examples of lattice ordered effect algebras.R is also a lattice ordered effect algebra.(1) In G be a partially ordered abelian group with an operator \u201c+\u201d. Let u \u2208 G with u > 0, and let L : = G+ = {g \u2208 G\u22230 \u2264 g \u2264 u}. Then L can be organized into a lattice ordered effect algebra  by defining p \u2295 q iff p + q \u2264 u, in which case p \u2295 q = p + q. In the effect algebra L we have a\u2032 = u \u2212 a and the effect algebra partial order on L coincides with the restriction to L of the partial order on G. An effect algebra of the form G+  is called an interval effect algebra or, for short, an interval algebra.(2) Let E in (3) Effect algebra E, we define e = (a \u2296 (a\u2227b))\u2228(b \u2296 (a\u2227b)).In a lattice ordered effect algebra E be a lattice ordered effect algebra. Then the following properties hold for every a, b, c \u2208 E:e = e,e = 0 if and only if a = b,e = e,e \u2264 e\u2228e, if c \u2264 a\u2227b.Let e = (a \u2296 (a\u2227b))\u2228(b \u2296 (a \u2296 b)) = (a\u2228b)\u2296(a\u2227b), e = (a\u2032\u2228b\u2032) \u2296 (a\u2228b)\u2032 = (a\u2227b)\u2032 \u2296 (a\u2228b)\u2032 = (1 \u2296 (a\u2227b))\u2296(1 \u2296 (a\u2228b)), by e = (a\u2228b)\u2296(a\u2227b). (iv): e = (a \u2296 (a\u2227b))\u2228(b \u2296 (a\u2227b)) = (a\u2228b)\u2296(a\u2227b), e = (a \u2296 (a\u2227c))\u2228(c \u2296 (a\u2227c)) = (a\u2228c)\u2296(a\u2227c), e = (b \u2296 (b\u2227c))\u2228(c \u2296 (b\u2227c)) = (b\u2228c)\u2296(b\u2227c); if c \u2264 a\u2227b, we have e\u2228e \u2265 e.(i) and (ii) are obvious. (iii): I be a Riesz ideal in a lattice ordered effect algebra E. Then I is a lattice ideal and ~I is a lattice congruence. Moreover for all a, b \u2208 E, a\u2009~\u2009Ib if and only if (a \u2296 (a\u2227b))\u2228(b \u2296 (a\u2227b)) \u2208 I.Let I is a Riesz ideal in a lattice ordered effect algebra E, then for all a, b \u2208 E, a\u2009~\u2009Ib if and only if e \u2208 I.From the lemma, we have that if E be an effect algebra and let X be a subset of E. Then X is called convex if for every x, y \u2208 X and z \u2208 E, x \u2264 z \u2264 y implies z \u2208 X.Let I be a Riesz ideal of effect algebra E. Then [a]I is convex for each a \u2208 E.Let a \u2208 E, take t, s \u2208 [a]I with t \u2264 s and let t \u2264 z \u2264 s. Since z \u2296 t \u2264 s \u2296 t\u2009~\u2009I0, we have z \u2296 t\u2009~\u2009I0, too. Hence, z\u2009~\u2009It. That is z \u2208 [a]I. So [a]I is convex for each a \u2208 E.Given I be a Riesz ideal of linear ordered effect algebra E. If a \u2264 b and [a]I \u2260 [b]I, then for each t \u2208 [a]I and s \u2208 [b]I, t \u2264 s.Let t \u2208 [a]I and s \u2208 [b]I such that s < t. We show that it is a contradiction. First, let t \u2264 b. So we obtain s \u2264 t \u2264 b and by t \u2208 [b]I which is a contradiction. Now let b \u2264 t. Thus a \u2264 b \u2264 t and by b \u2208 [a]I which is a contradiction again. Therefore for each t \u2208 [a]I and s \u2208 [b]I, t \u2264 s.Suppose that there exist X is a subset of lattice ordered effect algebra E, it is easy to check that for every X, Y\u2286E if we have X\u2286Y, then X\u2032\u2286Y\u2032.If E be a lattice ordered effect algebra and let I be a Riesz ideal of E. Let X be a subset of E. ThenXI0)\u2032 = (X\u2032)I0. \u2208 I and y \u2208 X. By e \u2208 I and y\u2032 \u2208 X\u2032, so y\u2032 \u2208 [x]I\u2229X\u2032 and this implies that x]I\u2229X\u2032 \u2260 \u2205. Suppose that y \u2208 [x]I\u2229X\u2032; then y\u2032 \u2208 [x\u2032]I\u2229X. Similarly, (i) Let x \u2208 (XI0)\u2032, so x\u2032 \u2208 XI0 by x\u2032]I\u2286X. Hence for each y \u2208 [x\u2032]I, we have e \u2208 I and hence e \u2208 I by y\u2032 \u2208 [x]I\u2286X\u2032. So x \u2208 (X\u2032)I0. Conversely, let x \u2208 (X\u2032)I0, so [x]I\u2286X\u2032. Suppose that y \u2208 [x]I\u2286X\u2032; then y\u2032 \u2208 [x\u2032]I\u2286X by x\u2032 \u2208 XI0. By X\u2032 = {x\u2032 : x \u2208 X} again, we have x \u2208 (XI0)\u2032 and this proves part (ii).(ii) Let E be a lattice ordered effect algebra and let X be a subeffect algebra of E. ThenE,X is an ideal and I\u2286X, then XI0 is a subeffect algebra of E.if Let X, then x \u2208 X such that x \u2208 [a]I; then x\u2032\u2009~\u2009Ia\u2032, so [a\u2032]I\u2229X \u2260 \u2205, therefore a\u22a5b, then there are x \u2208 [a]I, y \u2208 [b]I, such that x, y \u2208 X and x\u22a5y. Since X is subeffect algebra of E, we have x \u2295 y \u2208 X. Note that x\u2009~\u2009Ia and y\u2009~\u2009Ib; we have x \u2295 y\u2009~\u2009Ia \u2295 b and hence x \u2295 y \u2208 [a \u2295 b]I\u2229X. That is, (i) Since 0,1 \u2208 I\u2286X. First we prove 0 \u2208 XI0 and 1 \u2208 XI0. For all x \u2208 [0]I, by x \u2296 i = 0 \u2296 0, i \u2264 x, i \u2208 I, thus x = i \u2208 I\u2286X, so [0]I\u2286X, therefore 0 \u2208 XI0. Similarly, 1 \u2208 XI0 can be proved. Then we prove that x \u2208 XI0 implies x\u2032 \u2208 XI0. For all y \u2208 [x\u2032]I, we have y\u2032\u2009~\u2009Ix. It follows that y\u2032 \u2208 X, then y \u2208 X. Therefore x\u2032 \u2208 XI0. Let x, y \u2208 XI0 and x\u22a5y. For all z \u2208 [x \u2295 y]I, by z \u2296 i = (x \u2295 y) \u2296 j \u2264 x \u2295 y \u2208 X, i \u2264 z, j \u2264 x \u2295 y, i \u2208 I. This yields that z \u2296 i \u2208 X, thus z \u2208 X. Therefore x \u2295 y \u2208 XI0.(ii) Let II0 are subeffect algebras of E.By the above theorem, we note that E be a lattice ordered effect algebra. ThenE,E.Let E such that E = C(E) and an orthomodular effect algebra to be a lattice ordered effect algebra in which every element is principal.We define a Boolean effect algebra to be an effect algebra I be a Riesz ideal of a linear ordered effect algebra E and let X be a convex subset of E. Then XI0 and Let x, y \u2208 XI0 and x \u2264 z \u2264 y. We divide the proof into three cases. First let x~Iy, we have x, y \u2208 [a]I for some a \u2208 E. By a]I is convex, so that z \u2208 [a]I. It follows that [z]I = [a]I = [x]I\u2286X; therefore z \u2208 XI0. Next let [x]I \u2260 [y]I and z \u2208 [x]I. Then [z]I = [x]I. Since x \u2208 XI0, [z]I = [x]I\u2286X. So z \u2208 XI0. Similarly for [x]I \u2260 [y]I and z \u2208 [y]I, we can prove that z \u2208 XI0 and we omit it. Finally let [x]I \u2260 [y]I, z]I\u2286X. So let t \u2208 [z]I. By u \u2208 [x]I and v \u2208 [y]I we have u \u2264 t \u2264 v. Since [x]I, [y]I\u2286X, we have u, v \u2208 X. Note that X is convex; we obtain t \u2208 X and hence [z]I\u2286X. It follows that z \u2208 XI0. This shows that XI0 is convex.Let x \u2264 z \u2264 y. We divide the proof into three cases. First let x~Iy; we have x, y \u2208 [a]I for some a \u2208 E. By a]I is convex, so that z \u2208 [a]I. It follows that [z]I = [a]I = [x]I and [x]I\u2229X \u2260 \u2205; therefore [z]I\u2229X \u2260 \u2205. Hence x]I \u2260 [y]I and z \u2208 [x]I. Then [z]I = [x]I. Since x]I\u2229X \u2260 \u2205 and hence [z]I\u2229X \u2260 \u2205. Therefore x]I \u2260 [y]I and z \u2208 [y]I, we can prove that x]I \u2260 [y]I, z]I\u2229X \u2260 \u2205. Since x]I\u2229X \u2260 \u2205 and [y]I\u2229X \u2260 \u2205. So there exist u \u2208 [x]I\u2229X and v \u2208 [y]I\u2229X. By u \u2264 z \u2264 v. So z \u2208 X since X is convex. This implies that [z]I\u2229X \u2260 \u2205 and hence Now, let E be a lattice ordered effect algebra, I and J two Riesz ideals of E, and X a subset of E.X is an ideal of E and I, J\u2286X, then XI0\u2229XJ0 = XI\u2229J0,If X is crisp with respect to I or J, then If Let XI0\u2229XJ0\u2286XI\u2229J0. Conversely, assume x \u2208 XI\u2229J0. Then [x]I\u2229J\u2286X and hence x \u2208 X. Now, let y \u2208 [x]I, so e \u2208 I\u2286X. Because X is an ideal and x \u2208 X, so x\u2227y \u2208 X. Hence we have e\u2295(x\u2227y) \u2208 X. On the other hand, by e = (x\u2228y)\u2296(x\u2227y) and so y \u2264 x\u2228y = e\u2295(x\u2227y). This means that y \u2208 X, which implies x \u2208 XI0. Similarly, we can obtain x \u2208 XJ0 and so x \u2208 XI0\u2229XJ0. Therefore XI0\u2229XJ0 = XI\u2229J0.(i) By X is crisp with respect to I, so (ii) First, assume that E such that E = C(E) and an orthomodular effect algebra to be a lattice ordered effect algebra in which every element is principal.We define a Boolean effect algebra to be an effect algebra X and Y be subsets of E. We define X\u2228Y = {a\u2228b : a \u2208 X, b \u2208 Y} and define the downset X\u2193 = {c \u2208 E : \u2203a \u2208 X, c \u2264 a}.Let E be an orthomodular effect algebra and let I be a Riesz ideal of E, and let X, Y be subsets of E. Then Let a]I\u2229(X\u2228Y)\u2193 \u2260 \u2205. Thus there exists b \u2208 [a]I\u2229(X\u2228Y)\u2193 and consequently there exist x \u2208 X and y \u2208 Y such that b \u2264 x\u2228y. On the other hand, since [a]I = [b]I, by i, j \u2208 I, i \u2264 a, j \u2264 b such that a = (b \u2296 j) \u2295 i \u2264 b\u2228i \u2264 x\u2228y\u2228i = (x\u2228i)\u2228y. By x\u2228i]I = [x]I, Let E be an orthomodular effect algebra and let I be a Riesz ideal of E, and let X, Y be subsets of E. Then (XI0\u2228YI0)\u2193\u2286((X\u2228Y)\u2193)I0.Let a \u2208 (XI0\u2228YI0)\u2193, so there exist x \u2208 XI0 and y \u2208 YI0 such that a \u2264 x\u2228y. Now, let b \u2208 [a]I; thus by by i, j \u2208 I, i \u2264 a, j \u2264 b such that b = (a \u2296 i) \u2295 j \u2264 a\u2228j \u2264 x\u2228y\u2228j = (x\u2228j)\u2228y. By x\u2228j \u2208 [x\u2228j]I = [x]I\u2286X and y \u2208 Y. Hence b \u2208 X\u2228Y, this implies that a \u2208 ((X\u2228Y)\u2193)I0.Suppose that The following example shows that we can not replace the inclusion symbol \u2286 by equal sign in E = {0, a, b, c, d, e, 1}, as the lattice-ordered effect algebra in X = {0}, Y = {0, b, c, 1} be the subsets of E and let I = {0, a} be the Riesz ideal of E. We have (X\u2228Y)\u2193 = E, XI0 = \u2205 and YI0 = {b, c}. Therefore ((X\u2228Y)\u2193)I0 = E and (XI0\u2228YI0)\u2193 = \u2205. This shows that (XI0\u2228YI0)\u2193 \u2260 ((X\u2228Y)\u2193)I0.Consider E is a Boolean effect algebra.In what follows, that is, Propositions E be a Boolean effect algebra and let I, J be two Riesz ideals of E. Then (I\u2228J)\u2193 is also a Riesz ideal of E.Let a, b \u2208 (I\u2228J)\u2193 and a\u22a5b; then a \u2264 x1\u2228y1, b \u2264 x2\u2228y2, where xi \u2208 I, yi \u2208 J, i = 1,2. Since a \u2295 b = a\u2228b, then a \u2295 b \u2264 (x1\u2228y1)\u2228(x2\u2228y2) = (x1\u2228x2)\u2228(y1\u2228y2)\u2208(I\u2228J)\u2193. So (I\u2228J)\u2193 is an ideal. Let x \u2208 (I\u2228J)\u2193; then x \u2264 i\u2228j, where i \u2208 I, j \u2208 J. For a, b \u2208 E, a\u22a5b and x \u2264 a \u2295 b, we have x \u2264 (a\u2228b)\u2227(i\u2228j). Taking a1 = a\u2227(i\u2228j) = (a\u2227i)\u2228(a\u2227j), b1 = (b\u2227i)\u2228(b\u2227j), we have a1, b1 \u2208 (I\u2228J)\u2193 and a1 \u2264 a, b1 \u2264 b. Since E is a distributive lattice ordered effect algebra and  is an orthomodular lattice, it follows that a1 \u2295 b1 = (a\u2227(i\u2228j))\u2295(b\u2227(i\u2228j)) = (a\u2227(i\u2228j))\u2228(b\u2227(i\u2228j)) = (a\u2228b)\u2227(i\u2228j). Therefore, x \u2264 a1 \u2295 b1. Hence, (I\u2228J)\u2193 is a Riesz ideal.Let E be a Boolean effect algebra, and let I, J be two Riesz ideals of E and let X be a subset of E. Then Let X is a subset of E. Let x]I\u2228J)\u2193(\u2229X \u2260 \u2205. Thus there exists s \u2208 [x]I\u2228J)\u2193\u2208(I\u2228J)\u2193 and s \u2208 X; hence by h, k \u2208 (I\u2228J)\u2193, h \u2264 s, k \u2264 x, such that x = (s \u2296 h) \u2295 k and also there exist i \u2208 I, j \u2208 J such that k \u2264 i\u2228j. Then x = (s \u2296 h) \u2295 k \u2264 s\u2228k \u2264 s\u2228(i\u2228j) = (s\u2228i)\u2228(s\u2228j). On the other hand by s\u2228i]I = [s]I and [s\u2228j]J = [s]J. Therefore, Assume that The following example shows that the symbol inclusion in E = {0, a, b, 1} as the Boolean effect algebra in I = {0, a}, J = {0} are two ideals of E. And obviously, (I\u2228J)\u2193 = {0, a} is an ideal of E too. We have [0]I = I, [a]I = I, [b]I = {1, b}, [1]I = {b, 1}, [0]J = {0}, [a]J = {a}, [b]J = {b}, [1]J = {1}. Assume X = {b}; then Consider E be a Boolean effect algebra, I, J two Riesz ideals of E, and X a subset of E. Then XI\u2228J)\u2193(0\u2286(XI0\u2228XJ0)\u2193. Furthermore, if a \u2208 (XI0\u2228XJ0)\u2193, then one obtains [a]I\u2228J)\u2193(\u2286(XI0\u2228XJ0)\u2193.Let a \u2208 XI\u2228J)\u2193\u2193\u2193\u2193.Let a \u2208 (XI0\u2228XJ0)\u2193. There exist x1 \u2208 XI0 and x2 \u2208 XJ0 such that a \u2264 x1\u2228x2. Suppose that b \u2208 [a]I\u2228J)\u2193\u2193, i \u2264 a, j \u2264 b such that b = (a \u2296 i) \u2295 j. Also, there are h \u2208 I and k \u2208 J such that j \u2264 h\u2228k. Then b \u2264 a\u2228j \u2264 x1\u2228x2\u2228h\u2228k \u2264 (x1\u2228h)\u2228(x2\u2228k). By x1\u2228h]I = [x1]I and [x2\u2228k]I = [x2]I. This implies that [a]I\u2228J)\u2193(\u2286(XI0\u2228XJ0)\u2193.Now, suppose that From the above proposition, we have the following corollary.E be a Boolean effect algebra, and let I, J be two Riesz ideals of E and let X be a lattice ideal of E. Then one has XI\u2228J)\u2193(0 = (XI0\u2228XJ0)\u2193.Let XI\u2228J)\u2193(0\u2286(XI0\u2228XJ0)\u2193. Since X is an ideal, we can show easily, (XI0\u2228XJ0)\u2193\u2286X. Assume a \u2208 (XI0\u2228XJ0)\u2193, so [a]I\u2228J)\u2193(\u2286(XI0\u2228XJ0)\u2193\u2286X from a \u2208 XI\u2228J)\u2193(0. It yields that (XI0\u2228XJ0)\u2193\u2286XI\u2228J)\u2193\u2193(0 = (XI0\u2228XJ0)\u2193.By E be a lattice ordered effect algebra and F a Boolean effect algebra. If I is a Riesz ideal of E and h : E \u2192 F is a monomorphism, then h(I) is a Riesz ideal of F.Let h(I) is an ideal. Let h(x), h(y) \u2208 h(I) and h(x)\u22a5h(y), where x, y \u2208 I. Since h is a monomorphism, then h(x) \u2295 h(y) = h(x \u2295 y) \u2208 h(I). So h(I) is an ideal of F. Next, we prove that h(I) is a Riesz ideal. For any h(x) \u2208 h(I), let h(x) \u2264 a \u2295 b, a, b \u2208 F, a\u22a5b. Take a1 = h(x)\u2227a, b1 = h(x)\u2227b; then a1, b1 \u2208 h(I) and a1 \u2264 a, b1 \u2264 b. Since F is a distributive lattice ordered effect algebra, and  is an orthomodular lattice, it follows that a1 \u2295 b1 = (h(x)\u2227a)\u2295(h(x)\u2227b) = h(x)\u2227(a\u2228b). Therefore, h(x) \u2264 a1 \u2295 b1. Hence, h(I) is a Riesz ideal of F.First, we prove that E, F be lattice ordered effect algebras, let h : E \u2192 F be a morphism, and let Ih : = {a \u2208 E : h(a) = 0} be the kernel of h.X is a subset of E, then If h is a monomorphism and  is a Boolean effect algebra. If X is a subset of E and I is a Riesz ideal of E containing Ih, then Assume that Let h(x) = y. So we have [x]Ih\u2229X \u2260 \u2205. Let a \u2208 [x]Ih\u2229X. We have a \u2208 [x]Ih and a \u2208 X, so there exist i, j \u2208 Ih, i \u2264 a, j \u2264 x such that a = (x \u2296 j) \u2295 i. Since h is a morphism, we get that h(a) = h(x \u2296 j) \u2295 h(i) = h(x) \u2296 h(j) \u2295 h(i) = h(x). It follows that h(a) = y \u2208 h(X), which proves the theorem.(1) Since y = h(x). Since [x]I\u2229X \u2260 \u2205, suppose z \u2208 [x]I\u2229X, so we have [z]I = [x]I and z \u2208 X. Hence there exist i, j \u2208 I, i \u2264 z, j \u2264 x such that z = (x \u2296 j) \u2295 i. Since h a monomorphism, then [h(z)]h(I) = [h(x)]h(I) and h(z) \u2208 h(X). Therefore, (2) Let y]h(I)\u2229h(X) \u2260 \u2205. Suppose z \u2208 [y]h(I)\u2229h(X); it implies that [y]h(I) = [z]h(I) and z \u2208 h(X). Since h is a monomorphism, there exists x \u2208 E such that y = h(x) and there exist t \u2208 X and s \u2208 I such that z = h(t) and e(h(t), h(x)) = h(s), so h) = h(s) and it implies that h \u2296 s) = 0. Hence e \u2296 s \u2208 I. So e \u2208 I and t \u2208 [x]I by x]I\u2229X \u2260 \u2205, so Conversely, let In order to well illustrate the above theorem, we give the following example.E = {0, a, b, 1}, as the lattice ordered effect algebra in F = {0, y1, y2, 1}. For all y \u2208 F, define y1 \u2295 y2 = y2 \u2295 y1 = 1, y \u2295 0 = 0 \u2295 y = y; otherwise \u2295 is undefined. Then  is a Boolean effect algebra.Consider \u03d5 : E \u2192 F is given by \u03d5(1) = 1, \u03d5(a) = y1, \u03d5(b) = y2, \u03d5(0) = 0. We can easily check that \u03d5 is a monomorphism. Since ker\u2061\u03d5 = {0}, so [0]ker\u2061\u03d5 = {1}, [a]ker\u2061\u03d5 = {a}, [b]ker\u2061\u03d5 = {b}, [1]ker\u2061\u03d5 = {1}. Suppose that X = {a, b}; then \u03d5(X) = {y1, y2} and The map The earliest research on rough set theory of algebraic structures are on semigroups, groups, and modules. We set up the rough structure on partial algebraic system. The core foundation of rough set theory and application is a pair of approximation operators induced from the approximation space, namely, the closure operator and interior operator . We try to define rough approximation operators by a congruence relation on effect algebras and thus induce rough structure of effect algebras. Because effect algebras are incomplete algebras, not any two elements can compute, so we give a full operation on an effect algebra to construct a distance function. Moreover by use of Riesz ideals and distance functions we induced a congruence relation and then we obtain rough approximation operators and rough effect algebra system."}
{"text": "S = \u2212\u0394V/\u0394T, the determination of the temperature plays an important role. We present a method for the determination of the temperature difference using a combination of a finite element simulation, which reveals the temperature distribution of the sample, and the measurement of the resistance change due to laser heating of sensor leads on both sides next to the junction. Our results for the measured thermopower are in agreement with recent reports in the literature.We report the development of a novel method to determine the thermopower of atomic-sized gold contacts at low temperature. For these measurements a mechanically controllable break junction (MCBJ) system is used and a laser source generates a temperature difference of a few kelvins across the junction to create a thermo-voltage. Since the temperature difference enters directly into the Seebeck coefficient  S = \u2212\u0394V/\u0394T, where \u0394V is the thermo-voltage and \u0394T is the temperature difference. In general S is a function of energy and temperature:The energy and heat management in electronic devices has become a challenge in recent years due to the down-scaling of electronic components to the nanoscale, where the transport is governed by quantum-mechanical properties, which are partially not explored thoroughly yet. This includes solid-state semiconducting devices  and orgaEF is the Fermi energy, \u03c4(E) is the transmission function, e is the electron charge, kB is the Boltzmann constant and T the temperature of the system  + c, where b = 36 \u00b5m and s = 7.6 \u00b5m. The formula to calculate the spot size leads to  All other parameters, A, b0, and c are not relevant for the determination.To estimate the diameter of the laser spot we used the knife edge technique. Thereby the laser spot is moved across a 36 \u00b5m wide  stripe of gold while the reflected intensity is detected with a photodiode. Due to the different reflectivity of gold and Kapton Cirlex there is an intensity profile corresponding to the convolution of a box function for the stripe and a 2D Gauss profile for the laser. It can be easily shown that the direction parallel to the stripe does not play any role, so the convolution reduces to a 1D integral, which leads to a difference of two error functions, which is used to fit the dataset. \u22121 K\u22121, specific heat cp = 128 J kg\u22121 K\u22121, mass density \u03c1m = 19300 kg m\u22123, electrical resistivity \u03c1(T) = 8.73575 \u00d7 10\u22129 \u03a9m + 0.08325 \u00d7 10\u22129 \u03a9m K\u22121\u00b7T; polyimide: \u03ba = 0.15 W m\u22121 K\u22121, cp = 1100 J kg\u22121 K\u22121, \u03c1m = 1300 kg m\u22123; Kapton: \u03ba = (\u22121.372384 \u00d7 10\u22123 + 5.601653 \u00d7 10\u22123\u00b7T + 2.082966 \u00d7 10\u22126\u00b7T2 \u2212 5.05445 \u00d7 10\u22129\u00b7T3) W m\u22121 K\u22121 (for 5 K \u2264 T < 140 K), \u03ba = (\u22127.707532 \u00d7 10\u22123 + 5.769136 \u00d7 10\u22123\u00b7T + 5.622796 \u00d7 10\u22127\u00b7T2 \u2212 4.329984 \u00d7 10\u221210\u00b7T3) W m\u22121 K\u22121 (for 140 K \u2264 T \u2264 300 K) [cp = (2.809666 \u2212 1.394349\u00b7T\u22121 + 0.2106639\u00b7T\u22122 + 4.752016 \u00d7 10\u22123\u00b7T\u22123 \u2212 3.279998 \u00d7 10\u22124\u00b7T\u22124 + 4.282249 \u00d7 10\u22126\u00b7T\u22125) J kg\u22121 K\u22121 (for 4 K \u2264 T < 30 K), cp = \u221286.86946 + 7.816917\u00b7T \u2212 0.03664788 \u00d7 10\u22122\u00b7T\u22122 + 9.9128 \u00d7 10\u22125\u00b7T\u22123 \u2212 1.08441 \u00d7 10\u22127\u00b7T\u22124 J kg\u22121 K\u22121 (for 30 K \u2264 T \u2264 300 K) [m = 1420 kg m\u22123.For the simulations we used the following material parameters: Gold: Thermal conductivity \u03ba = 320/1.85 W m"}
{"text": "Let {Xn;n \u2265 1} be a sequence of independent copies of a real-valued random variable X and set Sn = X1+\u22ef+Xn, n \u2265 1. We say X satisfies the -Chover-type law of the iterated logarithm ) if X \u2208CTLIL. We obtain sets of necessary and sufficient conditions for X\u2208CTLIL for the five cases: \u03b1 = 2 and 0 < \u03b2 <\u221e, \u03b1 = 2 and \u03b2 = 0, 1<\u03b1<2 and \u2212\u221e<\u03b2<\u221e, \u03b1 = 1 and \u2212\u221e <\u03b2 <\u221e, and 0 < \u03b1 <1 and \u2212\u221e <\u03b2 <\u221e. As for the case where \u03b1 = 2 and \u2212\u221e <\u03b2 <0, it is shown that X\u2209CTLIL for any real-valued random variable X. As a special case of our results, a simple and precise characterization of the classical Chover law of the iterated logarithm ) is given; that is, X\u2208CTLIL if and only if \u03b1 \u2264 2.Let 0 < Primary: 60F15; Secondary: 60G50 Xn; n \u2265 1} is a sequence of independent copies of a real-valued random variable X. As usual, the partial sums of independent identically distributed (i.i.d.) random variables Xn, n \u2265 1 will be denoted by Lx= log(e\u2228x), x \u2265 0.Throughout, {X has a symmetric stable distribution with exponent \u03b1\u2208, i.e., t\u2208, Chover When X is in the domain of normal attraction of a nonnormal stable law, Pakshirajan and Vasudeva X is in the domain of attraction of a nonnormal stable law, Scheffler X is in the generalized domain of operator semistable attraction of some nonnormal law, Chen and Hu This is what we call the classical Chover law of iterated logarithm (LIL). Since then, several papers have been devoted to develop the classical Chover LIL. See, for example, Hedye Motivated by the previous study of the classical Chover LIL, we introduce a general Chover-type LIL as follows.\u03b1 \u2264 2 and \u2212\u221e <\u03b2 <\u221e. Let {X,Xn; n \u2265 1} be a sequence of real-valued i.i.d. random variables. We say X satisfies the -Chover-type law of the iterated logarithm ) ifLet 0 < X\u2208CTLIL  holds with \u03b2=1/\u03b1) when X has a symmetric stable distribution with exponent \u03b1\u2208.From the classical Chover LIL and Definition 1.1, we see that X\u2208CTLIL. The main results are stated in Section 2. We obtain sets of necessary and sufficient conditions for X\u2208CTLIL for the five cases: \u03b1 = 2 and 0 < \u03b2 <\u221e (see Theorem 2.1), \u03b1=2 and \u03b2 = 0 (see Theorem 2.2), 1 < \u03b1 < 2 and \u2212\u221e<\u03b2 <\u221e (see Theorem 2.3), \u03b1 = 1 and \u2212\u221e < \u03b2 < \u221e (see Theorem 2.4), and 0 < \u03b1 < 1 and \u2212\u221e < \u03b2 < \u221e (see Theorem 2.5). The proofs of Theorems 2.1-2.5 are given in Section 4. For proving Theorems 2.1-2.5, three preliminary lemmas are stated in Section 3. Some llustrative examples are provided in Section 5This paper is devoted to a characterization of \u03b1 = 2 and 0 < \u03b2 <\u221e.The main results of this paper are the following five theorems. We begin with the case where \u03b2 <\u221e. Let {X,Xn;n \u2265 1} be a sequence of i.i.d. real-valued random variables. ThenLet 0 < if and only if\u03b1 = 2 and \u03b2 = 0, we have the following result.For the case where X,Xn;n \u2265 1} be a sequence of i.i.d. non-degenerate real-valued random variables. ThenLet {if and only ifIn either case, we havec be a constant. Note thatLet \u221e <\u03b2 <0, X \u2209CTLIL for any real-valued random variable X.Thus, from Theorem 2.2, we conclude that, for any \u2212X \u2208CTLIL for the three cases where 1 < \u03b1 < 2 and \u2212\u221e < \u03b2 < \u221e, \u03b1 = 1 and \u2212\u221e <\u03b2 <\u221e, and 0 < \u03b1 <1 and \u2212\u221e <\u03b2 <\u221e respectively.In the next three theorems, we provide necessary and sufficient conditions for \u03b1<2 and \u2212\u221e <\u03b2 <\u221e. Let {X,Xn;n \u2265 1} be a sequence of i.i.d. real-valued random variables. ThenLet 1<if and only if\u221e <\u03b2 <\u221e. Let {X,Xn;n \u2265 1} be a sequence of i.i.d. real-valued random variables. ThenLet \u2212if and only ifIn particular, \u03b1 <1 and \u2212\u221e <\u03b2 <\u221e. Let {X,Xn;n \u2265 1} be a sequence of i.i.d. real-valued random variables. ThenLet 0 < if and only ifX\u2208CTLIL) is obtained as follows. For 0 < \u03b1\u2264 2, we haveFrom our Theorems 2.1, 2.3, 2.4, and 2.5, a simple and precise characterization of the classical Chover LIL \u221e<\u03b2<\u221e such that There exists a constant \u2212Then we haveif and only if\u221e<\u03b2<\u221e such that(ii) There exists a constant \u2212if and only if\u221e<\u03b2<\u221e such that(iii) There exists a constant \u2212if and only ifProof We prove the sufficiency part of Part (i) first. It follows from (3.3) that the setNote thatWe thus conclude thatwhich ensures (3.2).b\u2260\u03b2, let h=(b+\u03b2)/2. ThenWe now prove the necessity part of Part (i). For all It follows from (3.2) that the setNote thatWe thus have thatNote that (3.1) implies thatThus it follows from (3.4) thati.e., (3.3) holds.We leave the proofs of Parts (ii) and (iii) to the reader since they are similar to the proof of Part (i). This completes the proof of Lemma 3.1. \u25a1The following result is a special case of Corollary 2 of Einmahl and Li b>0. Let {X,Xn;n \u2265 1} be a sequence of i.i.d. random variables. ThenLet if and only ifx \u2265 0.where The following result is a generalization of Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers and follows easily from Theorems 1 and 2 of Feller \u03b1<2 and \u2212\u221e<b<\u221e. Let {X,Xn;n \u2265 1} be a sequence of i.i.d. random variables. ThenLet 0 < if and only if\u03b1<2 or \u03b1=1 and \u2212\u221e<b\u2264 0.where In this section, we only give the proofs of Theorems 2.1-2.2. By applying Lemmas 3.1 and 3.3, the proofs of Theorems 2.3-2.5 involves only minor modifications of the proof of Theorem 2.1 and will be omitted.Proof of Theorem 2.1 We prove the sufficiency part first. Note that the second part of (2.2) implies thatb>\u03b2,We thus see that for all h=(b+\u03b2)/2. Since \u03b2<h<b, it follows from (4.1) thatwhere We thus conclude from (2.2) thatwhich, by applying Lemma 3.2, ensures that\u03b2>0, the second part of (2.2) implies thatSince which ensures thatcn=LLn, n \u2265 1. It then follows from (4.2) and (4.3) thatLet By Lemma 3.1, we see that (4.4) is equivalent toi.e., (2.1) holds.We now prove the necessity part. By Lemma 3.1, (2.1) is equivalent to (4.4) which ensures that (4.2) holds. By Lemma 3.2, we conclude from (4.2) that\u03b2<\u221e, it follows from (4.5) thatSince 0 < \u03b21<\u03b2 then, using the argument in the proof of the sufficiency part, we have thatIf Hence, by Lemma 3.1, (4.6) implies thatwhich is in contradiction to (2.1). Thus (2.2) holds. The proof of Theorem 2.1 is complete. \u25a1Proof of Theorem 2.2 Using the same argument used in the proof of the sufficiency part of Theorem 2.1, we have from (2.4) thatX is a non-degenerate random variable, by the classical Hartman-Wintner-Strassen LIL, we have thatSince which implies thatcn=LLn, n \u2265 1. It then follows from (4.7) and (4.8) thatLet By Lemma 3.1, we see that (4.9) is equivalent toi.e., (2.5) holds, so does (2.3).Using the same argument used in the proof of the necessity part of Theorem 2.1, we conclude from (2.3) thatClearlyThus (2.4) holds. The proof of Theorem 2.2 is therefore complete. \u25a1In this section, we provide the following examples to illustrate our main results. By applying Theorems 2.3-2.5, we rededuce the classical Chover LIL in the first example.\u03b1\u2264 2. Let X be a symmetric real-valued stable random variable with exponent \u03b1. Clearly, \u03b1\u2264 2.Let 0 < \u03b1<2, we haveFor 0 < it follows thatand hence thatX\u2208CTLIL .Thus, by Theorems 2.3-2.5, \u03b1=2, we have X\u2209CTLIL but X\u2208CTLIL.However, for X\u2208CTLIL for some certain \u03b1 and \u03b2 even if the distribution of X is not in the domain of attraction of the stable distribution with exponent \u03b1.From our second example, we will see that \u03b1\u2264 2. Let dn= exp(2n), n \u2265 1. Given \u2212\u221e<\u03bb<\u221e. Let X be a symmetric i.i.d. real-valued random variable such thatLet 0 < whereThenX is not in the domain of attraction of the stable distribution with exponent \u03b1. Also \u03b1\u2264 2 or \u03b1=1 and \u03bb<0. It is easy to see thatThus the distribution of and hence that\u03b1 = 2 and 0 < \u03bb<\u221e, then X\u2208CTLIL.If \u03b1=2 and \u2212\u221e<\u03bb\u2264 0, then X\u2208CTLIL.If \u03b1<2, then X\u2208CTLIL.If 0 < Thus, by Theorems 2.1-2.5, we haveX may satisfy the other Chover-type LIL studied by Chen and Hu X\u2209CTLIL.Our third example shows that f(x) of X byDefine the density function p\u22600, 0 < \u03b3<1, c=c is a positive constant such that \u221e<b<\u221ewhere \u03b1 = 2 and p < 0, then X\u2208CTLIL.If \u03b1 = 2 and p > 0, then X\u2209CTLIL for any 0\u2264 \u03b2 <\u221e.If \u03b1 < 2, then X \u2209CTLIL for any \u2212\u221e <\u03b2 <\u221e. However, for 0 < \u03b1 < 2, by Theorem 3.1 in Chen and Hu If 0 < From Theorem 2.1-2.5, we haveB(x) is the inverse function of x\u03b1/ exp(p(logx)\u03b3), x \u2265e.where"}
{"text": "We propose a quite concrete approach based on the notion of K\u00f6the-Toeplitz duals with respect to the non-Newtonian calculus. Finally, we derive some inclusion relationships between K\u00f6the space and solidness.The important point to note is that the non-Newtonian calculus is a self-contained system independent of any other system of calculus. Therefore the reader may be surprised to learn that there is a uniform relationship between the corresponding operators of this calculus and the classical calculus. Several basic concepts based on non-Newtonian calculus are presented by Grossman (1983), Grossman and Katz (1978), and Grossman (1979). Following Grossman and Katz, in the present paper, we introduce the sets of bounded, convergent, null series and  It is certainly not unusual to measure deviations by ratios rather than differences. For instance, during the Renaissance, many scholars, including Galileo, discussed the following problem. Two estimates, 10 and 1000, are proposed as the value of a horse, which estimates, if any, deviates more from the true value of 100? The scholars who maintained that deviations should be measured by differences concluded that the estimate of 10 was closer to the true value. However, Galileo eventually maintained that the deviations should be measured by ratios, and he concluded that two estimates deviated equally from the true value. From the story, the question comes out this way, what if we measure by ratios? The answer is the main idea of non-Newtonian calculus which consists of many calculuses such as the classical, geometric, anageometric, and bigeometric calculus.Bashirov et al. , 2 have bs*, cs*, and cs0* consisting of the sets of all bounded, convergent, null series based on the non-Newtonian calculus, as follows:\u03c9* = {x = (xk) : xk \u2208 C*for\u2009all\u2009\u2009k \u2208 N}. One can conclude that the sets bs*, cs*, and cs0* are complete non-Newtonian metric spaces with the metric d\u221eN defined byFollowing Tekin and Ba\u015far  we can cbv*, bvp*, and bv\u221e* of bounded variation sequences in the sense of non-Newtonian calculus, as follows:bv*, bvp*, and bv\u221e* are complete with corresponding metrics on the right-hand side with (\u0394x)k = xk \u2296 xk\u22121, x)k\u2032 = xk \u2296 xk+1 for all k \u2208 N.Secondly, we introduce several sets  generator is a one-to-one function whose domain is R and whose range is a subset of B\u2286R, the set of real numbers. Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one generator. As a generator, we choose the function \u03b2 such that its basic algebraic operations are defined as follows:x, y \u2208 R(N), where the non-Newtonian real field R(N)\u2254{\u03b2{x} : x \u2208 R} as in [AR} as in .\u03b2-positive real numbers, denoted by R+(N), are the numbers x in R such that \u03b2-negative real numbers, denoted by R\u2212(N), are those for which beta-zero, beta-one, \u03b2(0) and \u03b2(1). Further, \u03b2-integers turns out to be the following:The X be a nonempty set and let d* : X \u00d7 X \u2192 R(N) be a function such that, for all x, y, z \u2208 X, the following axioms hold: \u2009x = y,(NM1)\u2009\u2009\u2009d* = d*,(NM2)\u2009\u2009\u2009(NM3)\u2009\u2009 Then, the pair  and d* are called a non-Newtonian metric space and a non-Newtonian metric on X, respectively.Let X =  be a non-Newtonian metric space. Then the basic notions can be defined as follows. x = (xk) is a function from the set N into the set R(N). The \u03b2-real number xk denotes the value of the function at k \u2208 N and is called the kth term of the sequence.A sequence xn) in a metric space X =  is said to be \u2217-convergent if for every given n0 = n0(\u03b5) \u2208 N and x \u2208 X such that n > n0 and is denoted by *lim\u2061n\u2192\u221exn = x or n \u2192 \u221e.A sequence (xn) in X =  is said to be non-Newtonian Cauchy (\u2217-Cauchy) if for every n0 = n0(\u03b5) \u2208 N such that m, n > n0.A sequence  is defined as \u03b2(|\u03b2\u22121(x)|) and is denoted by |x|\u03b2. For each number x in B \u2282 R(N), x1 and x2 is defined by alpha-generator in  also are the ordered pair of arithmetics (\u03b2-arithmetic and \u03b1-arithmetic). The sets beta-generator generates beta-arithmetics, respectively. Definitions given for \u03b2-arithmetic are also valid for \u03b1-arithmetic.Suppose that \u03b1-arithmetic is used for arguments and \u03b2-arithmetic is used for values; in particular, changes in arguments and values are measured by \u03b1-differences and \u03b2-differences, respectively. The operators of this calculus type are applied only to functions with arguments in A and values in B. The \u2217-limit of a function with two generators \u03b1 and \u03b2 is defined byf is \u2217-continuous at a point a in A if and only if a is an argument of f and *lim\u2061x\u2192af(x) = f(a). When \u03b1 and \u03b2 are the identity function I, the concepts of \u2217-limit and \u2217-continuity are identical with those of classical limit and classical continuity.The important point to note here is that \u03b1-arithmetic to \u03b2-arithmetic is the unique function \u03b9 (iota) that possesses the following three properties:(i)\u03b9 is one to one.(ii)\u03b9 is from A onto B.(iii)u and v in A,For any numbers,  It turns out that \u03b9(x) = \u03b2{\u03b1\u22121(x)} for every x in A and that n. Since, for example, \u03b1-arithmetic can readily be transformed into a statement in \u03b2-arithmetic.The isomorphism from C*; that is,Let C*, \u2295, \u2299) is a field. of \u2217-complex numbers, the formal notation,N-series. Also, for integers, n \u2208 N, the finite \u2217-sums sn* = \u2217\u2211\u2211k=0nzk* are called the partial sums of complex N-series. If the sequence \u2217-converges to a complex number s*, then we say that the series \u2217-converges and write s* = \u2217\u2211\u2211k=0nzn*. The number s* is then called the \u2217-sum of this series. If (sn)\u2009\u2009\u2217-diverges, we say that the series \u2217-diverges or it is \u2217-divergent.Given a sequence -triangle inequality).(ii)(iii)Let  and zk*, tk* \u2208 C* for k \u2208 {0,1, 2,3,\u2026, n}. Then, For any Following Tekin and Ba\u015far , we can z* \u2208 C* be an arbitrary element. The distance function d* is called \u2217-norm of z* and is denoted by z1*, z2* \u2208 C* we have d* is the induced metric from Let z* by z* but leaves the real part the same. ThusLet (i) Let \u03b1 and \u03b2 be the same generator functions and z* \u2208 C*. Then the following condition holds(ii) Let C*, d*) is a complete metric space, where d* is defined by l\u221e*, c*, c0*, and lp*, p \u2265 1, are sequence spaces.The sets (b)\u03bb* denote any of the spaces l\u221e*, c*, and c0* and z = (zk*), t = (tk*) \u2208 \u03bb*. Define d\u221e* on the space \u03bb* by \u03bb*, d\u221e*) is a complete metric space.Let (c)l\u221e*, c*, and c0* are Banach spaces with the norm ||z||\u221e* defined byThe spaces (d)lp* is Banach spaces with the norm ||z||p* defined byThe space The following statements hold.bs*, cs*, cs0* and bv*, bvp*, bv\u221e* consisting of all bounded, convergent, null series and the sets of bounded variation sequences in the sense of non-Newtonian calculus which correspond to the sets bs, cs, cs0 and bv, bvp, bv\u221e over the complex field C, respectively.In the present section, we introduce the sets \u03bc* denote any of the spaces bs*, cs*, and cs0* and z = (zk*), t = (tk*) \u2208 \u03bc*. Define d\u221eN on the space \u03bc* by\u03b1, \u03b2 operators and corresponding function \u03b9 = \u03b2\u03b1\u22121. Then,  is a complete metric space.Let cs* and cs0*, we prove the theorem only for the space bs*. Let the \u2217-sums \u2217\u2211\u2211k=0nzk*, \u2217\u2211\u2211k=0ntk* \u2208 C*, where z = (zk*), t = (tk*) \u2208 C*. Then the following metric axioms in \u2009(NM1) From  it can b\u2009d\u221eN = d\u221eN holds.(NM2) It is trivial that the condition \u2009z = (zk*), t = (tk*), w = (wk*) \u2208 C*. In fact by taking into account (NM3) We show that \u2217triangle inequality in  Since the axioms (NM1)\u2013(NM3) are satisfied,  is a non-Newtonian metric space. It remains to prove the completeness of the space bs*.Since the proof is similar to the spaces xm) be a \u2217-Cauchy sequence in bs*, where xm = {x1m) = {xk(1), xk(2),\u2026, xkm) and show that x \u2208 bs*. From (r \u2192 \u221e and m > n0 we havexm) \u2208 bs*, there exists k \u2208 N. Thus, (m > n0k \u2208 N whose right-hand side does not involve k. Hence (xk) is a bounded sequence of geometric complex numbers; that is, x = (xk) \u2208 bs*. Also from (m > n0m \u2192 \u221e for an arbitrary \u2217-Cauchy sequence (xm). Hence bs* is complete.Let , y = (yk) \u2208 bv*, and (\u0394x)k\u2032 = xk \u2296 xk+1. Then,  is a complete metric space. Similarly, one can conclude that the other bounded variation sets  and  are also complete.Let bv* is a Banach space with the norm ||x||bv* defined byThe space The idea of dual sequence space which plays an important role in the representation of linear functionals and the characterization of matrix transformations between sequence spaces was introduced by K\u00f6the and Toeplitz , whose m\u03bb*, \u03bc* \u2208 C*, the set S, defined by\u03bb* and \u03bc* for all k \u2208 N. One can easily observe for a sequence space \u03bd* of \u2217-complex numbers that the inclusions S \u2282 S if \u03bd* \u2282 \u03bb* and S \u2282 S if \u03bc* \u2282 \u03bd* hold.In this section, we focus on the alpha-, beta-, and gamma-duals of the classical sequence spaces over non-Newtonian complex field. For \u03bb* \u2282 \u03c9* which are, respectively, denoted by {\u03bb*}\u03b1, {\u03bb*}\u03b2, and {\u03bb*}\u03b3, as follows:wk\u2299zk) is the coordinatewise product of \u2217-complex numbers w and z for all k \u2208 N. Then {\u03bb*}\u03b2 is called beta-dual of \u03bb* or the set of all convergence factor sequences of \u03bb* in cs*. Firstly, we give a remark concerning the \u2217-convergence factor sequences.Firstly, we define the alpha-, beta-, and gamma-duals of a set Throughout the text, we also use the notation \u201c<\u201d for a non-Newtonian linear subspace which was created in .\u2205 \u2260 \u03bb* \u2282 \u03c9*. Then the following statements are valid. \u03bb*}\u03b2 is a sequence space if \u03c6* < {\u03bb*}\u03b2 < \u03c9*, where {\u03bb* \u2282 \u03bc* \u2282 \u03c9*, then {\u03bc*}\u03b2 < {\u03bb*}\u03b2.If \u03bb* \u2282 {\u03bb*}\u03b2\u03b2 : = ({\u03bb*}\u03b2)\u03b2.\u03c6*}\u03b2 = \u03c9* and {\u03c9*}\u03b2 = \u03c6*.{Let w = (wk), m = (mk), and n = (nk)\u2208{\u03bb*}\u03b2. (a)\u03bb*}\u03b2 < \u03c9* holds from the hypothesis. We show that m \u2295 n \u2208 {\u03bb*}\u03b2 for m, n \u2208 {\u03bb*}\u03b2. Suppose that l \u2208 \u03bb*. Then (mk\u2299lk) \u2208 cs* and (nk\u2299lk) \u2208 cs* for all l \u2208 \u03bb*. We can deduce thatIt is trivial that {\u2009m \u2295 n \u2208 {\u03bb*}\u03b2. Now, we show that t\u2299w \u2208 {\u03bb*}\u03b2 for any t \u2208 C* and w = (wk)\u2208{\u03bb*}\u03b2, since (wk\u2299lk) \u2208 cs* for all l \u2208 \u03bb*. Combining this with ((tk\u2299wk)\u2299lk) = tk\u2299(wk\u2299lk) \u2208 cs* for\u2009all l \u2208 \u03bb* we get t\u2299w \u2208 {\u03bb*}\u03b2. Therefore, we have proved that {\u03bb*}\u03b2 is a subspace of the space \u03c9*.Hence, (d)\u03c9*}\u03b2 \u2282 \u03c6*. Suppose that w = (wn)\u2208{\u03c9*}\u03b2 and z = (zn) are given with \u2217-division by zn\u2299wn = 1* if wn \u2260 0* and zn : = 1* otherwise. By taking into account the set \u03c6* from inclusion (a), then there exists an integer N \u2208 N for all n \u2265 N such that Using (a) we need only to show {\u2009w\u2299z \u2208 cs* implies that w \u2208 \u03c6*. The rest is an immediate consequence of this part and we omitted the details.Further, Since the proof is trivial for the conditions (b) and (c), we prove only (a) and (d). Let c0*}\u03b2 = {c*}\u03b2 = {l\u221e*}\u03b2 = l1*.{l1*}\u03b2 = l\u221e*.{The following statements hold: l\u221e*}\u03b2 \u2282 {c*}\u03b2 \u2282 {c0*}\u03b2 by l1* \u2282 {l\u221e*}\u03b2 and {c0*}\u03b2 \u2282 l1*. Now, consider that w = (wk) \u2208 l1* and z = (zk) \u2208 l\u221e* are given. By taking into account the cases (c)-(d) of w\u2299z \u2208 cs*. So the condition l1* \u2282 {l\u221e*}\u03b2 holds.(a) Obviously {y = (yk) \u2208 \u03c9*\u2216l1*, we prove the existence of an x \u2208 c0* with y\u2299x \u2209 cs*. According to y \u2209 l1* we may confirm an index sequence (np) which is strictly increasing with x = (xk) \u2208 c0* by xk : = (sgn*yk\u2298p), the non-Newtonian complex signum function is defined byy = (yk) \u2208 C*. Finally, by using \u03b1 = \u03b2, we getnp\u22121 \u2264 k < np. Therefore y\u2299x \u2209 cs* and thus y \u2209 {c0*}\u03b2. Hence {c0*}\u03b2 \u2282 l1*.Conversely, for a given l\u221e* \u2282 ({l\u221e*}\u03b2)\u03b2 = {l1*}\u03b2 since {l\u221e*}\u03b2 = l1*. Now we assume the existence of a w = (wn)\u2208{l1*}\u03b2\u2216l\u221e*. Since w is unbounded there exists a subsequence (wnk) of (wn) and we can find a real number (k + 1)2 such that k \u2208 N1. The sequence (xn) is defined by xn : = (sgn*(wnk)\u2298(k + 1)2) if n = nk and x \u2208 l1*. However,w \u2209 {l1*}\u03b2, which contradicts our assumption and {l1*}\u03b2 \u2282 l\u221e*. This step completes the proof.(b) From the condition (c) of \u03b6-duals .In addition to the statements in \u2205 \u2260 \u03bb* \u2282 \u03c9*. Then the following statements are valid:\u03c6* < {\u03bb*}\u03b1 < {\u03bb*}\u03b2 < {\u03bb*}\u03b3 < \u03c9*; in particular, {\u03bb*}\u03b6 is a sequence space over C*;\u03bb* < \u03bc* < \u03c9*, then {\u03bc*}\u03b6 < {\u03bb*}\u03b6;if I is an index set, if \u03bbi* are sequence spaces, and if \u03bb* : = \u22c3i\u2208I\u03bbi*, then \u2329\u03bb*\u232a\u03b6 = \u22c2i\u2208I{\u03bbi*}\u03b6, where the notation \"\u2329\u232a\" stands for the span of linear subspace over C*;\u03bb* \u2282 {\u03bb*}\u03b6\u03b6 : = ({\u03bb*}\u03b6)\u03b6.Let l\u221e* < cs* < bs*. We only show the cases (c) and (d) taking \u03b6 = alpha. The rest of the parts can be obtained in a similar way.(c)\u03bbi* \u2282 \u2329\u03bb*\u232a the following \u2329\u03bb*\u232a\u03b1 \u2282 {\u03bbi*}\u03b1 and \u2329\u03bb*\u232a\u03b1 \u2282 \u22c2i\u2208I{\u03bbi*}\u03b1 hold by (b). On the other hand, if y \u2208 \u22c2i\u2208I{\u03bbi*}\u03b1, that is, y \u2208 {\u03bbi*}\u03b1, then x\u2299y \u2208 l1* for all x \u2208 \u03bb* and therefore y \u2208 {\u03bb*}\u03b1 \u2282 \u2329\u03bb*\u232a\u03b1.Now, as an immediate consequence (d)\u03bb* \u2282 {\u03bb*}\u03b6\u03b6. Let w \u2208 \u03bb*; then w\u2299z \u2208 l1* for all z \u2208 {\u03bb*}\u03b1; thus w \u2208 {\u03bb*}\u03b6\u03b6 and \u03bb* \u2282 {\u03bb*}\u03b6\u03b6 by (a).We can deduce The case (b) obviously is true, and (a) follows from \u03bb* \u2260 {\u03bb*}\u03b6\u03b6 as we get from \u03b6 = \u03b2 and \u03bb* : = c0*. We have {c0*}\u03b2\u03b2 = l\u221e* \u2260 c0*. This remark gives rise to the following definition.Here \u03b6 \u2208 {\u03b1, \u03b2, \u03b3} and let \u03bb* be a sequence space over the field C*. \u03bb* is called \u03b6-space if \u03bb* = {\u03bb*}\u03b6\u03b6. Further, an \u03b1-space is also called a K\u00f6the space or perfect sequence space.Let From \u03bb* is a sequence space over the field C* and , then {\u03bb*}\u03b6 is a \u03b6-space; that is, {\u03bb*}\u03b6 = {\u03bb*}\u03b6\u03b6\u03b6.If \u03bb*}\u03b1 = {\u03bb*}\u03b2 = {\u03bb*}\u03b3. This gives rise to the notion of solidity.Now we look for sufficient conditions for {X be a sequence space. Then X is called solid ifLet \u03bb* be a sequence space over the field C*. Then \u03bb* is solid if and only ifLet u = (uk) \u2208 \u03c9* and x = (xk) \u2208 \u03bb* with \u03bek, \u03b7k \u2208 R. Obviously the condition \u03c8k, \u03b4k \u2208 R. We obtain \u03c8k2 + \u03b4k2 \u2264 \u03bek2 + \u03b7k2 since k \u2208 N. We may choose a real number \u03b3k \u2208 R with \u03c8k2 + \u03b3k2 = \u03bek2 + \u03b7k2 and consider w = (wk) defined by w \u2208 \u03bb* and \u03b6k \u2208 R such that \u03b6k2 + \u03b4k2 = \u03bek2 + \u03b7k2 it follows similarly that Let \u03bb* < \u03c9* is any sequence space over the field C*; then the following statements hold. \u03bb* is a K\u00f6the space, then \u03bb* is solid.If \u03bb* is solid, then {\u03bb*}\u03b1 = {\u03bb*}\u03b2 = {\u03bb*}\u03b3.If \u03bb* is a K\u00f6the space, then \u03bb* is a \u03b6-space.If Consider that \u03bb* < \u03c9* be a sequence space over the field C*.(a)\u03bb* is a K\u00f6the space and u \u2208 \u03c9*, then u \u2208 {\u03bb*}\u03b1\u03b1 if and only if the condition u\u2299z \u2208 l1* holds for all z \u2208 {\u03bb*}\u03b1. Besides this we obtain u = (uk) \u2208 \u03bb* and v = (vk) \u2208 \u03c9* and the statementIf \u2009z \u2208 {\u03bb*}\u03b1. Therefore v\u2299z \u2208 l1*. Hence w \u2208 \u03bb* and \u03bb* is solid over the field C*.holds for each (b)\u03bb* is a solid sequence space over C*. To show {\u03bb*}\u03b1 = {\u03bb*}\u03b2 = {\u03bb*}\u03b3, it is sufficient condition to verify {\u03bb*}\u03b3 < {\u03bb*}\u03b1 in v = (vk) \u2208 {\u03bb*}\u03b3; that is,Consider that \u2009\u03bb*, for z = (zk) \u2208 \u03bb*, the condition u = (uk) \u2208 \u03bb* for all k \u2208 N. Therefore by combining this with the inclusion (By taking into account solidness of nclusion  we deduc\u2009u\u2299v \u2208 l1* < cs*. Hence v \u2208 {\u03bb*}\u03b1 and {\u03bb*}\u03b3 < {\u03bb*}\u03b1.holds and (c)This is an obvious consequence of Let \u03c6*, \u03c9*, lp*, c0*, and l\u221e* of sequences are solid.The sets c* and bv* of sequences are not solid; therefore none of them is a K\u00f6the space.The sets \u03b6 \u2208 {\u03b1, \u03b2, \u03b3}, thenFor each (i)l1*}\u03b6 = l\u221e* and {l\u221e*}\u03b6 = l1*;{(ii)\u03c9*}\u03b6 = \u03c6* and {\u03c6*}\u03b6 = \u03c9*.{\u03b6 \u2208 {\u03b1, \u03b2, \u03b3} and c0* < \u03bc* < l\u221e*, then {\u03bc*}\u03b6 = l1* and \u03bc* \u2282 {\u03bc*}\u03b6\u03b6 = l\u221e*. In particular, {c0*}\u03b6 = {c*}\u03b6 = l1*, and each of c*, c0* is not a \u03b6-space.If The following statements hold. c0* and l\u221e* are solid, we know that {c0*}\u03b6 = {l\u221e*}\u03b6 = l1*. So the statements in (d) are obtained from That the specified spaces in cases (a-b) are solid is an immediate consequence of their definition. Additionally, the cases (i-ii) of (c) can be obtained \u03b6-duals of the spaces cs*, bs*, bv*, and bv0*. We will find that none of these sequence spaces is solid; in particular, none of them is a K\u00f6the space.Next, we determine the cs*}\u03b1 = {bv*}\u03b1 = {bv0*}\u03b1 = l1*; {bv0*} = bv*\u2229c0*,{cs*}\u03b2 = bv*, {bv*}\u03b2 = cs*, {bv0*}\u03b2 = bs*, {bs*}\u03b2 = bv0*;{cs*}\u03b3 = bv*, {bv*}\u03b3 = bs*, {bv0*}\u03b3 = bs*, {bs*}\u03b3 = bv*.{The following statements hold: cs*, bs*, bv*, and bv0* of sequences are \u03b2-spaces (\u03b6 = beta), but they are not K\u00f6the spaces. Moreover, the sets bs* and bv* of sequences are \u03b3-spaces (\u03b6 = gamma), whereas both cs* and bv0* are not \u03b3-spaces. None of the spaces cs*, bs*, bv*, and bv0* is solid.In particular, the sets cs*}\u03b6, \u03b6 \u2208 {\u03b1, \u03b2, \u03b3}, and the proofs of all other cases are quite similar.We prove the cases for the spaces {x = (xk) \u2208 cs* and y = (yk) \u2208 l1*. Then,x||cs* is defined by (y \u2208 {cs*}\u03b1 which gives the fact that l1* \u2282 {cs*}\u03b1.(a) Let fined by . Therefoy = (yk) \u2208 {cs*}\u03b1\u2216l1*. Then to every natural number p we can construct an index sequence (np) with np < np+1 and p \u2208 N. Define x = (xk) by\u03b2-division \u03b2{(\u22121)k2p\u2212} for np < k \u2264 np+1. Then x = (xk) \u2208 cs*. According to the choice of np, the inequalitiesp2p diverges. Thus, x\u2299y \u2209 l1*, which implies y \u2209 {cs*}\u03b1. This contradicts the fact that y \u2208 {cs*}\u03b1. Therefore {cs*}\u03b1 \u2282 l1*.Conversely, suppose that bv0*}\u03b1 \u2282 l1* holds as well if we take the sequence (xk) byThe condition {u = (uk) \u2208 {cs*}\u03b2 and w = (wk) \u2208 c0*. Define the sequence v = (vk) \u2208 cs* by vk = (wk \u2296 wk+1) for all k \u2208 N. Therefore, \u2217\u2211\u2211k\u2299\u2009\u2009vk\u2009\u2009\u2217-converges, butl1* \u2282 cs* yields (uk) \u2208 {cs*}\u03b2 \u2282 {l1*}\u03b2 = l\u221e*. Then we derive by passing to the \u2217-limit in  \u2208 {c0*}\u03b2 = {c0*}\u03b1 = l1*; that is, u \u2208 bv*. Therefore, {cs*}\u03b2\u2286bv*.(b) Let ges, but\u2211\u2217\u2002k=0\u2002n\u2009u = (uk) \u2208 bv*. Then, (uk \u2296 uk\u22121) \u2208 l1*. Further, if v = (vk) \u2208 cs*, the sequence (wn) defined by wn = \u2217\u2211\u2211k=0nvk for all k \u2208 N is an element of the space c*. Since {c*}\u03b1 = l1*, the N-series \u2217\u2211\u2211kwk\u2299(uk \u2296 uk\u22121) is \u2217-convergent. Also, we havewn) \u2208 c* and (uk) \u2208 bv* \u2282 c*, the right-hand side of inequality \u2299uk or \u2217\u2211\u2211k=0\u221euk\u2299vk\u2009\u2009\u2217-converges. Hence bv*\u2286{cs*}\u03b2.Conversely, suppose that  we have\u2211\u2217\u2002k=m\u2002n \u2208 {cs*}\u03b3 and v = (vn) \u2208 c0*. Then, for the sequence (wn) \u2208 cs* defined by wn = (vn \u2296 vn+1) for all n \u2208 N, we can find a non-Newtonian real number n \u2208 N. Since (vn) \u2208 c0* and (un)\u2208{cs*}\u03b3 \u2282 l\u221e*, there exists a non-Newtonian real number n \u2208 N. Therefore,uk \u2296 uk\u22121) \u2208 {c0*}\u03b3 = {c0*}\u03b1 = l1*; that is, (un) \u2208 bv*. Therefore, since the inclusion {cs*}\u03b3 \u2282 bv* holds, we conclude that {cs*}\u03b3 = bv*, as desired.(c) By using (a), it is known that Although all arithmetics are isomorphic, only by distinguishing among them, we do obtain suitable tools for constructing all the non-Newtonian calculi. But the usefulness of arithmetics is not limited to the construction of calculi; we believe there is a more fundamental reason for considering alternative arithmetics; they may also be helpful in developing and understanding new systems of measurement that could yield simpler physical laws. average speed. The definition \u201cdistance traveled per unit time\u201d is incomplete because it fails to provide a method of determining the average speed of an accelerated particle. The definition \u201cdistance divided by time,\u201d though not incorrect, is a gross oversimplification that fails to reveal the underlying issues. Fortunately there is a completely satisfactory definition, which undoubtedly was known to Galileo. Then we isolate a constant in each given uniform motion by defining speed to be the distance traveled in any unit time-interval. Finally, for a particle that moves nonuniformly in a distance d in time t, we define the average speed to be the speed that a particle in uniform motion must have in order to travel a distance d in time t. In our opinion, neither the simplicity nor the obviousness of the answer, d/t, justifies its use as the definition of average speed .Of course, we can only speculate as to future applications of the non-Newtonian calculi. Perhaps they can be used to define new scientific concepts, to yield new or simpler scientific laws, to solve heretofore unsolved problems, or to formulate and solve new problems. For example, one constructs non-Newtonian calculi of functions of two or more real variables by choosing an arithmetic for each axis. It might even be profitable to seek deeper connections among the corresponding operators of the calculi. Like as, some of them are postponed to our future works."}
{"text": "We present a local convergence analysis of inexact Newton method for solving singular systems of equations. Under the hypothesis that the derivative of the function associated with the singular systems satisfies a majorant condition, we obtain that the method is well defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the singular systems. It also allows us to obtain an estimate of convergence ball for inexact Newton method and some important special cases. F : \u03a9 \u2282 Rn \u2192 Rm is a nonlinear operator with its Fr\u00e9chet derivative denoted by DF; \u03a9 is open and convex. In the case m = n, the inexact Newton method was introduced in  \u2192 R is continuously differentiable and convex, theng(t) \u2212 g(\u03c4t))/t \u2264 (1 \u2212 \u03c4)g\u2032(t) for all t \u2208  and \u03c4 \u2208 ,(g(u) \u2212 g(\u03c4u))/u \u2264 (g(v) \u2212 g(\u03c4v))/v for all u, v \u2208 .Let DF(\u03b6) is not surjective (see .Let DF(x) is of full rank for each x \u2208 \u03a9. Note that, for surjective-underdetermined systems, the fixed points of the Newton operator NF(x)\u2236 = x \u2212 DF(x)\u2020F(x) are the zeros of F, while for injective-overdetermined systems, the fixed points of NF are the least square solutions of  = 0, DF(\u03b6) \u2260 0 and that DF satisfies the modified majorant condition , where r is given in (DF(x) \u2264rank\u2009DF(\u03b6) for any x \u2208 B and that\u03b8 satisfies 0 \u2264 \u03b8 < 1. Let {xk} be sequence generated by inexact Newton's method with any initial point x0 \u2208 B\u2216{\u03b6} and the conditions for the residual rk and the forcing term \u03bbk:\u03ba(A): = ||A\u2020||||A|| denotes the condition number of A \u2208 Rm\u00d7n. Then, {xk} converges to a zero \u03b6 of DF(\u00b7)\u2020F(\u00b7) in tk} is defined by  is given by\u03bb = 0 (in this case \u03bbk = 0 and rk = 0) in If DF(x) is full column rank for every x \u2208 B, then we have DF(x)\u2020DF(x) = IRn. Thus,\u03b8 = 0. We immediately have the following corollary.If DF(x) \u2264rank\u2009DF(\u03b6) and thatx \u2208 B. Suppose that F(\u03b6) = 0, DF(\u03b6) \u2260 0 and that DF satisfies the modifed majorant condition \u2216{\u03b6} and the condition \u2020F(\u00b7) in tk} is defined by (\u03b8 = 0.Suppose thatrank\u2009ondition . Let {xondition  for the fined by  for \u03b8 = DF(\u03b6) is full row rank, the modified majorant condition (In the case when ondition  can be rondition .F(\u03b6) = 0 and DF(\u03b6) is full row rank and that DF satisfies the majorant condition , where r is given in (DF(x) \u2264rank\u2009DF(\u03b6) for any x \u2208 B and that condition \u2216{\u03b6} and the conditions for the residual rk and the forcing term \u03bbk:xk} converges to a zero \u03b6 of F(\u00b7) in tk} is defined by  is given by  = 0, DF(\u03b6) is full row rank and that DF satisfies the majorant condition , where r is given in (DF(x) \u2264rank\u2009DF(\u03b6) for any x \u2208 B and that condition \u2216{\u03b6} and the conditions for the control residual rk and the forcing term \u03bbk:xk} converges to a zero \u03b6 of f(\u00b7) in tk} is defined by  is invertible in h\u03bb,\u03b8 is given by u\u2009R2 be endowed with the l1-norm. Consider the operator F : R2 \u2192 R2 defined byF is analytic on R2 and its derivative is given byDF(x)) = 1 and the Moore-Penrose inverse isk = 1,2,\u2026, we can obtain thatx1, x2)T \u2208 \u03a9, we have\u03b6 = T. Then, \u03b6 \u2208 \u03a9 satisfies F(\u03b6) = 0 and\u03bbk} such that 0 \u2264 \u03bbk \u2264 \u03bbmax\u2061 = 1/2; see \u22121 exists andDF(\u03b6) is full row rank, we have DF(\u03b6)DF(\u03b6)\u2020 = IRm andDF(x) is full row rank; that is, rank\u2061\u2009DF(x) = rank\u2061\u2009DF(\u03b6). The proof is complete.Since xk} coincides with the sequence generated by inexact Newton's method (DF(\u03b6)\u2020F(\u03b6)|| = ||\u03a0ker\u2061DF(\u03b6)\u22a5|| = 1, we haveB. So, xk} converges to \u03b6 follows. Note that F(\u00b7) = DF(\u00b7)DF(\u00b7)\u2020F(\u00b7); it follows that \u03b6 is a zero of F.Let s method  for F^. fore, by , we can ondition  on B = 0, DF(\u03b6) is full row rank, and DF satisfies the majorant condition . Then we haveSuppose that ondition  on B is full row rank, we have DF(\u03b6)DF(\u03b6)\u2020 = IRm. It follows thatIRn \u2212 DF(\u03b6)\u2020(DF(\u03b6) \u2212 DF(x)) is invertible for any x \u2208 B. Thus, in view of the equality A\u2020A = \u03a0ker\u2061A\u22a5 for any m \u00d7 n matrix A, one has thatSince Using"}
{"text": "C1-almost periodic solutions for this class of networks with time-varying delays are established. Two examples are given to show the effectiveness of the proposed method and results.On a new type of almost periodic time scales, a class of BAM neural networks is considered. By employing a fixed point theorem and differential inequality techniques,some sufficient conditions ensuring the existence and global exponential stability of  C1-almost periodic function is an important subclass of almost periodic functions. However, to the best of our knowledge, few authors have studied problems of C1-almost periodic solutions of BAM neural networks.It is well known that bidirectional associative memory (BAM) neural networks have been extensively applied within various engineering and scientific fields such as pattern recognition, signal and image processing, artificial intelligence, and combinatorial optimization \u20133. SinceC1-almost periodic solutions for continuous and discrete systems, respectively. Motivated by the above, our purpose of this paper is to study the existence and stability of C1-almost periodic solutions for the following BAM neural networks on time scales:T is an almost periodic time scale which will be defined in the next section; xi(t) and yj(t) correspond to the activation of the ith neurons and the jth neurons at the time t, respectively; ai(t), bj(t) are positive functions and they denote the rates with which the cells i and j reset their potential to the resting state when isolated from the other cells and inputs at time t; pji(t) and qij(t) are the connection weights at time t; \u03b3ji(t), \u03c1ij(t) are nonnegative, which correspond to the finite speed of the axonal signal transmission at time t; Ii(t), Jj(t) denote the external inputs at time t; and fj and gi are the activation functions of signal transmission. For each interval J of R, we denote JT = J\u2229T.On the other hand, the theory of calculus on time scales  and there exist positive constants \u03b1j, \u03b2i such that\u2009u | , |v | \u2208 R, i = 1,2,\u2026, n, j = 1,2,\u2026, m;where |H2)(ai(t) > 0, bj(t) > 0, \u03b3ji(t) \u2265 0, \u03c1ij(t) \u2265 0, pji(t), qij(t), Ii(t), Jj(t) are bounded almost periodic functions on T, i = 1,2,\u2026, n, j = 1,2,\u2026, m.Throughout this paper, we assume the following:\u03c6k(\u00b7) denotes a real-valued bounded rd-continuous function defined on T, andSystem  is supplIn this section, we will first recall some basic definitions and lemmas which are used in what follows.T be a nonempty closed subset  of R. The forward and backward jump operators \u03c3, \u03c1 : T \u2192 T and the graininess \u03bc : T \u2192 R+ are defined, respectively, byLet t \u2208 T is called left-dense if t > inf\u2061T and \u03c1(t) = t, left-scattered if \u03c1(t) < t, right-dense if t < sup\u2061T and \u03c3(t) = t, and right-scattered if \u03c3(t) > t. If T has a left-scattered maximum m, then Tk = T\u2216{m}; otherwise, Tk = T. If T has a right-scattered minimum m, then Tk = T\u2216{m}; otherwise, Tk = T.A point f : T \u2192 R is right-dense continuous provided it is continuous at right-dense point in T and its left-side limits exist at left-dense points in T. If f is continuous at each right-dense point and each left-dense point, then f is said to be continuous function on T.A function y : T \u2192 R and t \u2208 Tk, we define the delta derivative of y(t), y\u0394(t), to be the number (if it exists) with the property that for a given \u03b5 > 0 there exists a neighborhood U of t such thats \u2208 U.For y is continuous, then y is right-dense continuous, and if y is delta differentiable at t, then y is continuous at t.If y be right-dense continuous. If Y\u0394(t) = y(t), then we define the delta integral byLet r : T \u2192 R is called regressive ift \u2208 Tk. The set of all regressive and rd-continuous functions r : T \u2192 R will be denoted by R = R(T) = R. We define the set R+ = R+ = {r \u2208 R : 1 + \u03bc(t)r(t) > 0, \u2200t \u2208 T}.A function r is a regressive function, then the generalized exponential function er is defined byp, q : T \u2192 R be two regressive functions; we defineIf p, q : T \u2192 R are two regressive functions; then,e0 \u2261 1 and ep \u2261 1;ep(\u03c3(t), s) = (1 + \u03bc(t)p(t))ep;ep) = ep/(1 + \u03bc(s)p(s));ep = ep\u2296;1/ep\u2296)\u0394 = (\u2296p)(t)ep\u2296. is called an almost periodic function in t \u2208 T uniformly for x \u2208 D if the \u03b5-translation set of fT for all \u03b5 > 0 and for each compact subset S of D; that is, for any given \u03b5 > 0 and each compact subset S of D, there exists a constant l > 0 such that each interval of length l contains a \u03c4 \u2208 E{\u03b5, f, S} such that\u03c4 is called the \u03b5-translation number of f and and l is called the inclusion length of E{\u03b5, f, S}.Let \u03b1 = {\u03b1n} and \u03b2 = {\u03b2n} be two sequences. Then \u03b2 \u2282 \u03b1 means that \u03b2 is a subsequence of \u03b1. We introduce the translation operator T, and T\u03b1f = g means that For convenience, we introduce some notations. Let f \u2208 C, and if for any given sequence \u03b1\u2032 \u2282 \u03a0 and each compact subset S of D there exists a subsequence \u03b1 \u2282 \u03b1\u2032 such that T\u03b1f exists uniformly on T \u00d7 S, then f is called an almost periodic function in t uniformly for x \u2208 D.Let f \u2208 C1 is said to be a C1-almost periodic function, if f, f\u0394 are two almost periodic functions on T.A function x \u2208 Rn, and let A(t) be an n \u00d7 n rd-continuous matrix on T; the linear systemT if there exist positive constants k and \u03b1, projection P, and the fundamental solution matrix X(t) of (0 is a matrix norm on T.Let  X(t) of , satisfyA(t) is an almost periodic matrix function and f(t) is an almost periodic vector function.Consider the following linear almost periodic system:x(t):X(t) is the fundamental solution matrix of (If the linear system  admits eatrix of .ci(t) be an almost periodic function on T, where ci(t) > 0, \u2212ci(t) \u2208 R+, \u2200t \u2208 T, andT.Let t0 \u2208 T, then F defined byf.Every rd-continuous function has an antiderivative. In particular, if p \u2208 R and a, b, c \u2208 T, thenIf By Lemmas f(t) is an rd-continuous function and c(t) is a positive rd-continuous function which satisfies that \u2212c(t) \u2208 R+. Lett0 \u2208 T; then,Suppose that g is a real-valued almost periodic function on T and f : R \u2192 R is a Lipschitz function, then t \u2192 f(g(t)) is an almost periodic function on T.If f, g : T \u2192 R are almost periodic functions, then the following hold:f + g is almost periodic function;fg is almost periodic function.If x = T \u2208 Rn+m to denote a column vector, in which the symbol T denotes the transpose of vector. We let |x| denote the absolute-value vector given by |x | = T and define \u2016x\u2016 = max\u2061i\u2264n+m1\u2264|xi|.First, for convenience, we introduce some notations. We will use AP(T) = {c(t) : c(t) be a bounded real-valued, almost periodic function on T}, AP1(T) = {c(t) : c(t), c\u0394(t) \u2208 AP(T)}, and\u03c6 \u2208 B, if we define induced modulusB is a Banach space.Let H1), (H2), and the following hold:H3), t \u2212 \u03c1ij(t) \u2208 T, \u2200t \u2208 T, i = 1,2,\u2026, n, j = 1,2,\u2026, m;\u2212H4), \u03c6 \u2208 B, we consider the following almost periodic differential equation:i\u2264n,1\u2264j\u2264m1\u2264{inf\u2061t\u2208Tai(t), inf\u2061t\u2208Tbj(t)} > 0, it follows from T. Thus, by T; that is,  \u2282 E. For any given \u03c6 \u2208 E, it suffices to prove that \u2016\u03a6(\u03c6)\u2016B \u2264 r0. By conditions (H1)\u2013(H4), we haveE) \u2282 E.For any given quation:xi\u0394t\u2212aitxThus, by x\u03c6itW=\u222b\u2212\u221equation:xi\u0394t\u2212aitxquation:xi\u0394t\u2212aitx we havesup\u2061t\u2208Tx\u03c6\u03c6, \u03c8 \u2208 E and combining conditions (H1) and (H4), we obtain thatTaking ain thatsup\u2061t\u2208Tx\u03c6E to E. Since E is a closed subset of B, \u03a6 has a fixed point in E, which means that  = (x1*(t), x2*(t),\u2026, xn*(t), y1*(t),\u2026, ym*(t))T of system (\u03c6*(t) = (\u03c61*(t), \u03c62*(t),\u2026, \u03c6n*(t), \u03c6n+1*(t),\u2026, \u03c6n+m*(t))T is said to be globally exponentially stable. There exist a positive constant \u03bb with \u2296\u03bb \u2208 R+ and M > 1 such that every solutionThe H1)\u2013(H4) hold and sup\u2061t\u2208T\u03bc(t) < +\u221e; then, system  hoC1-almost periodic solution\u03c6*(t) = (\u03c61*(t), \u03c62*(t),\u2026, \u03c6n*(t), \u03c6n+1*(t),\u2026, \u03c6n+m*(t))T. Suppose thatui(s) = xi(s) \u2212 xi*(s), vj(s) = yj(s) \u2212 yj*(s) and i = 1,2,\u2026, n, j = 1,2,\u2026, m, and the initial conditions of , we get\u221e) and Hi(\u03f5) \u2192 \u2212\u221e, \u03f5 \u2192 +\u221e, there exist Hi(\u03f5i) = 0, Hi(\u03f5) > 0 for \u03f5 \u2208  and eai\u2212) and integrating on T, by ebj\u2212) and integrating on T, we haveH4) we have M > 1. Thus, there exists \u03bb \u2264 \u03bb0,\u03bb \u2208 R+ andt1 \u2208 T, p > 1 and some k such thatH2)\u2013(H4), we obtainC1-almost periodic solution of system  : R \u2192 R be arbitrary almost periodic functions; then, (H2)-(H3) hold. Let \u03b11 = \u03b12 = \u03b21 = \u03b22 = 1/4; then, (H1) holds. Next, let us check (H4); if we take r0 = 1, thenH4) holds for r0 = 1. By Theorems C1-almost periodic solution in the regionIn , take T T = Z:\u03b3ji, \u03c1ij\u2009\u2009 : Z \u2192 Z be arbitrary almost periodic functions; then, (H2)-(H3) hold. Let \u03b11 = \u03b12 = \u03b21 = \u03b22 = 1/4; then, (H1) holds. Next, let us check (H4); if we take r0 = 1, thenH4) holds for r0 = 1. By Theorems C1-almost periodic solution in the regionIn , take T C1-almost periodic solutions for a class of neural networks with time-varying delays on a new type of almost periodic time scales are established. To the best of our knowledge, this is the first time to study the existence of C1-almost periodic solutions of BAM neural networks on time scales. Our methods that are used in this paper can be used to study other types of neural networks, such as Cohen-Grossberg neural networks and fuzzy cellular neural networks.In this paper, by using calculus theory on time scales, a fixed point theorem, and differential inequality techniques, some sufficient conditions ensuring the existence and global exponential stability of"}
{"text": "We consider a new iterative method due to Kadioglu and Yildirim (2014) for further investigation. We study convergence analysis of this iterative method when applied to class of contraction mappings. Furthermore, we give a data dependence result for fixed point of contraction mappings with the help of the new iteration method. Recent progress in nonlinear science reveals that iterative methods are most powerful tools which are used to approximate solutions of nonlinear problems whose solutions are inaccessible analytically. Therefore, in recent years, an intensive interest has been devoted to developing faster and more effective iterative methods for solving nonlinear problems arising from diverse branches in science and engineering.D is a nonempty convex subset of a Banach space B, T is a self map of D, and {\u03b1n}n=0\u221e, {\u03b2n}n=0\u221e are real sequences in .Very recently the following iterative methods are introduced in  and 2],,2], respWhile the iterative method  fails toS  for all n \u2208 Nk=0\u221e\u03b1k\u03b2k = \u221e leads ton\u2192\u221e||xn \u2212 un|| = 0. SinceDefineUsing the same argument as above one can easily show the implication (ii)\u21d2(i); thus it is omitted here.D, B, and T with fixed point x\u2217 be as in \u03b1n}n=0\u221e, {\u03b2n}n=0\u221e be real sequences in  satisfyingn\u2192\u221e\u03b1n = lim\u2061n\u2192\u221e\u03b2n = 0. lim\u2061Let x0 = u0 \u2208 D, consider iterative sequences {xn}n=0\u221e and {un}n=0\u221e defined by (un}n=0\u221e converges to x\u2217 faster than {xn}n=0\u221e does.For given fined by  and 2),,x0 = uThe following inequality comes from inequality  of TheorThe following inequality is due to . Let T : D \u2192 D be a mapping and for all T is a contraction with contractivity factor x\u2217 = 3; see B = R andes S [S* , Noor [1es S [S* , and NorS [S* [S  iteratioS [S* [S .We are now able to establish the following data dependence result.T satisfying condition  1/2 \u2264 \u03b1n, (ii) \u03b2n \u2264 \u03b1n for all n \u2208 N, and (iii) \u2211n=0\u221e\u03b1n = \u221e. If Tx\u2217 = x\u2217 and n \u2192 \u221e, then we have\u03b5 > 0 is a fixed number and \u03b4 \u2208 .Let ondition . Let {xrated by  for T anIt follows from , 3), , , and assFrom assumption (i) we haveequality  becomesn\u2192\u221exn = x\u2217. Thus, using this fact together with the assumption Denote that"}
{"text": "We define a new class of multivalent meromorphic functions using the generalised hypergeometric function. We derived this class related to conic domain. It is also shown that this new class of functions, under certain conditions, becomes a class of starlike functions. Some results on inclusion and closure properties are also derived. Mp denote the class of functions of the form p-valent in the punctured unit disc centred at origin E\u2217 = {z : 0 < |z| < 1} = E\u2216{0}. Also by f(z)\u227ag(z) we mean f(z) is subordinate to g(z) which implies the existence of an analytic function, called Schwartz function w(z) with |w(z)| < 1, for z \u2208 E\u2217 such that f(z) = g(w(z)), where f(z) and g(z) are multivalent meromorphic functions. Note that if g is univalent in E then the above subordination is equivalent to f(0) = g(0) and f(E) \u2282 g(E).Let \u03b3 < 1 and k \u2208 [0, \u221e, qk,\u03b3(z) for conic regions are convex univalent and given by R(t) is Legendre's complete elliptic integral of the first kind with t \u2208 , and z \u2208 E is chosen in such a way that k = cosh\u2061(\u03c0R\u2032(t)/R(t)).The set of points, for 0 < qFs for complex parameters \u03b11,\u2026, \u03b1q and \u03b21,\u2026, \u03b2s with \u03b2j \u2260 0, \u22121, \u22122, \u22123,\u2026, for j = 1,2, 3,\u2026, s, is defined as q \u2264 s + 1, q, s \u2208 \u21150 = \u2115 \u222a {0}, and (\u03b1)n is the well-known Pochhammer symbol related to the factorial and the Gamma function by the relation The generalised hypergeometric function fined as Fsq\u03b11,\u2026,\u03b1Hp : Mp \u2192 Mp as follows: Hp,q,s(\u03b11) = Hp then the following identity holds for this operator:  Liu and Srivastava  defined MQp of meromorphic function associated with conic domain, for k \u2265 0, 0 \u2264 \u03bb < 1, and p \u2265 1, as follows: Shareef  defined MQp of meromorphic function associated with conic domain, for k \u2265 0, 0 \u2264 \u03bb < 1, p \u2265 1, and b \u2265 1, as follows: qk,\u03b3 is a convex and univalent function, for h(z)\u227aqk,\u03b3(z) it means h(E\u2217) is contained in qk,\u03b3(E\u2217), where Hp,q,s(\u03b11).We now define a new subclass h2(z) be convex in E and \u211c(\u03bbh2(z) + \u03bc) > 0, where \u03bc \u2208 \u2102, \u03bb \u2208 \u2102\u2216{0}, and z \u2208 E. If h1(z) is analytic in E, with h1(0) = h2(0), then Let h(z) = 1 + \u2211n=1\u221ecnzn and H(z) = 1 + \u2211n=1\u221ednzn and h\u227aH. If H(z) is univalent and convex in E, then n \u2265 1.Let qk,\u03b3(z) = 1 + q1z + q2z2 + \u22ef then If MQp.In this section we explore some of the geometric properties exhibited by the class MQp.We begin by discussing an inclusion property for the class \u211c(\u03b11) > p\u211c \u2212 1) then If f \u2208 MQp and set z we get f \u2208 MQq, therefore \u211c + \u03b11 + p) > 0 or equivalently \u211c(\u03b11) > p\u211c \u2212 1). Hence, f \u2208 MQp.Let ntiating  with resz we get zH\u03b11fz\u2032=\u03b1 and set \u22121p1+1bzHPutting  is closed under a certain integral.We now show that the class f(z) \u2208 MQp, then the integral f(z) into MQp.If g(z)\u227aqk,\u03b3(z) which implies From  we have z\u03b7B\u03b7H\u03b1fz=tiating .Now we get coefficient estimates of the class f(z) \u2208 MQp and f(z) is given by (k \u2264 n.If f(z) \u2208 MQp; then, by definition, we have Let H(\u03b11)f(z) = f(z); then h(z) = 1 + \u2211n=1\u221ecnzn, then  is univalent and qk,\u03b3(E) is convex, applying Rogosinski's theorem we have q1 is given in  giv giv(44)A"}
{"text": "The eigenvalue complementarity problem (EiCP) is a kind of very useful model, which is widely used in the study of many problems in mechanics, engineering, and economics. The EiCP was shown to be equivalent to a special nonlinear complementarity problem or a mathematical programming problem with complementarity constraints. The existing methods for solving the EiCP are all nonsmooth methods, including nonsmooth or semismooth Newton type methods. In this paper, we reformulate the EiCP as a system of continuously differentiable equations and give the Levenberg-Marquardt method to solve them. Under mild assumptions, the method is proved globally convergent. Finally, some numerical results and the extensions of the method are also given. The numerical experiments highlight the efficiency of the method. A \u2208 Rn\u00d7n and the matrix B \u2208 Rn\u00d7n, which are positive definite matrix, then we consider to find a scalar \u03bb \u2208 R and a vector x \u2208 Rn\u2216{0}, such thatEigenvalue complementarity problem (EiCP) is proposed in the study of the problems in mechanics, engineering, and economics. The EiCP is also called cone-constrained eigenvalue problem in \u20134. The EThis paper is organized as follows. In Mn(R) denotes a real matrix of order n and Mn,m(R) denotes a real matrix of order n \u00d7 m. 0n =  and en = .Throughout the paper, In this section, firstly, we reformulate  as a sysf : Rn+1 \u2192 Rn is a continuously differentiable function and the matrix A \u2208 Rn\u00d7n. Using the nonlinear complementarity problem (NCP) function \u03d5 : R2 \u2192 R, which satisfied the following basic property\u03c9 solves the EiCP operty, . Thus, xample, [F(\u03c9)=\u03a8(\u03c9)Now, we give the Levenberg-Marquardt method for solving . The glo\u03b1 < 1, 0 < \u03b2 < 1, p > 2, \u03c1 > 0, 0 < \u03b4 \u2264 2. P > 0, \u03f5 > 0, \u03c90 \u2208 Rn+1, \u03bc0 = ||F(\u03c90)||\u03b4, and k : = 0.Given 0 < \u03c9)||\u2264\u03f5, then stop. Otherwise, compute dk byIf ||\u2207\u03a6(\u03c9k)Tdk + \u03c1||dk||P > 0, let dk = \u2212\u2207\u03a6(\u03c9k); otherwise, dk is computed by ||\u03b4 and k : = k + 1, and go to Let Now, we give the global convergence of \u03c9k}, k = 1,2,\u2026 generated by Suppose that {\u03c9k}K \u2192 \u03c9\u22c6, {\u03c9k}K is a subsequence of {\u03c9k}, and k = 1,2,\u2026. When there are infinitely many k \u2208 K such that dk = \u2212\u2207\u03a6(\u03c9k), by Proposition\u2009\u20091.16 in [\u03c9k}K is a convergent subsequence of {\u03c9k}, then dk is always computed by (\u03c9k}K for which\u03c9k \u2192 \u03c9\u22c6. Suppose that \u03c9\u22c6 is not a stationary point of \u03a6. By (\u03c9k)|| = 0. Then, the algorithm has stopped. On the other hand, we know that there exists a constant \u03b6 > 0 such that\u03c9k)Tdk \u2264 \u2212\u03c1||dk||P and the fact that the gradient \u2207\u03a6(\u03c9k) is bounded on the convergent sequence {\u03c9k}, we get or whichlim\u2061k\u2208Kk\u2192k\u2192\u221edk<\u221e,lim\u2061\u2009sup\u2061uch thatF'\u03c9kTF'\u03c9km such thatm(0) = 0, m(k) = min\u2061{M0, m(k \u2212 1) + 1}, and M0 > 0 is a integer.In dk in dk \u2208 Rn of the equationrk is the residuals and satisfies\u03b1k \u2264 a < 1 for every k.In \u03c1 = 10, P = 3, \u03b1 = 0.1, and \u03f5 = 10\u22124.We give some numerical experiments for the method. And we compare We considerBy , we knowWe considerFrom , we alsoWe considerFrom , we alsoDiscussion. In this section, we study the numerical behaviors of \u03bb, \u03bc) \u2208 R \u00d7 R and  \u2208 Rn\u2216{0} \u00d7 Rm\u2216{0} such thatA \u2208 Mn(R), B \u2208 Mn,m(R), B \u2208 Mm,n(R), and D \u2208 Mm(R).We also can use f(\u03c9) = x, g(\u03c9) = y, P(\u03c9) = \u03bbx \u2212 Ax \u2212 By, and Q(\u03c9) = \u03bcy \u2212 Cx \u2212 Dy. We can write the above bieigenvalue complementarity problems as f(\u03c9) \u2265 0, g(\u03c9) \u2265 0, P(\u03c9) \u2265 0, Q(\u03c9) \u2265 0, fT(\u03c9)P(\u03c9) = 0, gT(\u03c9)Q(\u03c9) = 0, \u03c9 = 1, and \u03c9 = 1. By using F-B function, similar to rewriting the EiCP, we can rewrite the above bieigenvalue complementarity problems as the following equations:\u03c1 = 10, P = 3, \u03b1 = 0.1, and \u03f5 = 10\u22124.Let A = 0, D = 0,We consider the BECP, where We use M standing for the pencil associated to a finite collection {A0, A1,\u2026, Ar} of real matrices of order n andWe consider using We consider the quadratic pencil model, whereIn this paper, we reformulate the EiCP as a system of continuously differentiable equations and use the Levenberg-Marquardt method to solve them. The numerical experiments show that our method is a promising method for solving the EiCP. The numerical experiments of the extensions confirm the efficiency of our method."}
{"text": "Melanopus is a morphological group of Polyporus which contains species with a black cuticle on the stipe. In this article, taxonomic and phylogenetic studies on Melanopus group were carried out on the basis of morphological characters and phylogenetic evidence of DNA sequences of multiple loci including the internal transcribed spacer (ITS) regions, the large subunit nuclear ribosomal RNA gene (nLSU), the small subunit nuclear ribosomal RNA gene (nSSU), the small subunit mitochondrial rRNA gene sequences (mtSSU), the translation elongation factor 1-\u03b1 gene (EF1-\u03b1), the largest subunit of RNA polymerase II (RPB1), the second largest subunit of RNA polymerase II (RPB2), and \u03b2-tubulin gene sequences (\u03b2-tubulin). The phylogenetic result confirmed that the previously so-called Melanopus group is not a monophyletic assemblage, and species in this group distribute into two distinct clades: the Picipes clade and the Squamosus clade. Four new species of Picipes are described, and nine new combinations are proposed. A key to species of Picipes is provided. Melanopus Pat. was established by Patouillard Polyporus admirabilis Peck, Bulletin of the Torrey Botanical Club 26: 69 (1899). [MB 157762]Basionym: Cerioporus admirabilis (Peck) Zmitr. et Kovalenko, International Journal of Medicinal Mushrooms 18 (1): 33 (2016). [MB 812034]\u2261 Picipes americanus (Vlas\u00e1k & Y.C. Dai) J.L. Zhou & B.K. Cui, comb. nov. [MB 817137]Polyporus americanus Vlas\u00e1k & Y.C. Dai, Fungal Diversity 64: 136 (2014). [MB 803796]Basionym: Picipes austroandinus (Rajchenb. & Y.C. Dai) J.L. Zhou & B.K. Cui, comb. nov. [MB 817138]Polyporus austroandinus Rajchenb. & Y.C. Dai, Fungal Diversity 64: 138 (2014). [MB 803797]Basionym: Picipes conifericola (H.J. Xue & L.W. Zhou) J.L. Zhou & B.K. Cui, comb. nov. [MB 817139]Polyporus conifericola H.J. Xue & L.W. Zhou, Mycological Progress 13(1): 139 (2014). [MB 801216]Basionym: Picipes fraxinicola (L.W. Zhou & Y.C. Dai) J.L. Zhou & B.K. Cui, comb. nov. [MB 817140]Piptoporus fraxineus Bondartsev & Ljub., Novosti Sistematiki Nizshikh Rastenii 2: 135 (1965). [MB 337052]Basionym: Polyporus fraxinicola L.W. Zhou & Y.C. Dai, Fungal Diversity 64: 141 (2014). [MB 803799]\u2261 Picipes rhizophilus (Pat.) J.L. Zhou & B.K. Cui, comb. nov. [MB 817141]Polyporus rhizophilus Pat., Journal de Botanique 8: 219 (1894). [MB 150169]Basionym: Cerioporus rhizophilus (Pat.) Zmitr. et Kovalenko, International Journal of Medicinal Mushrooms 18 (1): 33 (2016). [MB 812040]\u2261 Picipes submelanopus (H.J. Xue & L.W. Zhou) J.L. Zhou & B.K. Cui, comb. nov. [MB 817142]Polyporus submelanopus H.J. Xue & L.W. Zhou, Mycotaxon 122: 436 (2013). [MB 800237]Basionym: Picipes taibaiensis (Y.C. Dai) J.L. Zhou & B.K. Cui, comb. nov. [MB 817143]Polyporus rhododendri Y.C. Dai & H.S. Yuan, Annales Botanici Fennici 46 (1): 58 (2009). [MB 540894]Basionym: Polyporus taibaiensis Y.C. Dai, Fungal Diversity 64: 142 (2014). [MB 803798]\u2261 Picipes virgatus  J.L. Zhou & B.K. Cui comb. nov. [MB 817144]Polyporus virgatus Berk. & M.A. Curtis, Botanical Journal of the Linnean Society 10: 304 (1869). [MB 202513]Basionym: Leucoporus virgatus (Berk. & M.A. Curtis) Pat., \u00c9num\u00e9ration M\u00e9thodique des Champignons Recueillis \u00e0 la Guadeloupe et \u00e0 la Martinique: 25 (1903). [MB 102236]\u2261 Picipes baishanzuensis was collected from subtropical area of China. It is characterized by its radially striped infundibuliform pileus with a slender black stipe. In phylogenetic analysis (Pi. virgatus (100/100/1.00). Morphologically, Pi. virgatus and Pi. baishanzuensis share infundibuliform pileus, similar pore size, decurrent tubes and wrinkled dark stipe; however, the basidiospores of Pi. virgatus are much larger (9\u201312.5 \u00d7 4\u20135 \u03bcm for Pi. virgatus and 6.6\u20137.9 \u00d7 2.5\u20133.1 \u03bcm for Pi. baishanzuensis) [Polyporus tuberaster also has depressed pileus and decurrent pores, but its pileus is covered with dark brown flecks, pores (0.5\u20132 per mm) and basidiospores (12\u201314.5 \u00d7 4.8\u20136 \u03bcm) [Pi. baishanzuensis; besides, P. tuberaster usually grows on the ground, arising from a black underground sclerotium [analysis , it stro.8\u20136 \u03bcm)  are muchlerotium .Picipes subtropicus was found in subtropical areas of China. It can be identified by a continuous variation in pore size, bright pileal surface color, short black stipe-like base and medium cylindrical basidiospores (5.1\u20136.2 \u00d7 2.2\u20132.7 \u03bcm). In phylogenetic analysis , larger basidiospores (7\u20138.5 \u00d7 2.5\u20134 \u03bcm) and pantropical distribution [Picipes badius share similar basidiocarps and pore size with P. subtropicus; but it differs in its larger basidiospores (7.5\u20139.5 \u00d7 3\u20133.5 \u03bcm), simple-septate generative hyphae and absence of cystidioles [Picipes baishanzuensis was also found in subtropical areas of China, but its infundibuliform pilei, slender stipe and lager basidiospores (6.6\u20137.9 \u00d7 2.5\u20133.1 \u03bcm) are quite different from Pi. subtropicus.analysis , it did Picipes subtubaeformis was described from temperate zone of China. It can be distinguished by the irregularly semicircular or elliptical pileus, terra-brown to black stipe, and oblong to cylindrical basidiospores (5.7\u20136.8 \u00d7 2.7\u20133.1 \u03bcm). In the phylogenetic analysis (Pi. subtubaeformis grouped together with Pi. tubaeformis (88/88/1.00); morphologically, both of them have orange to reddish-brown pileus and dark stipe, but Pi. tubaeformis differs in its slender stipe and basidiospores  [Pi. virgatus and Pi. subtubaeformis have reddish-brown or chestnut basidiocarps with centrally to laterally dark stipe, but the former one has both larger pores (3\u20134 per mm) and basidiospores [Pi. virgatus is absence of cystidioles [Picipes taibaiensis is another temperate species described from China. It has similar upper pileal surface color with Pi. subtubaeformis, but the flabelliform or spathulate pileus, larger basidiospores (7.5\u201310.5 \u00d7 3.2\u20133.8 \u03bcm) and fusoid cystidioles make it different from Pi. subtubaeformis [analysis , Pi. sub2.75 \u03bcm) . Both Piaeformis .Picipes tibeticus is a special species found from eastern Tibetan Plateau. it can be identified by its reddish-brown to blackish-brown fan-shaped or semicircular basidiocarps, small angular pores (6\u20139 per mm), oblong basidiospores (5\u20135.9 \u00d7 2.8\u20133.3 \u03bcm) and growth on coniferous trees. Phylogenetically, it grouped together with Pi. conifericola  and basidiospores (8\u201310 \u00d7 3\u20133.9 \u03bcm). In addition, Pi. submelanopus has both simple septate and clamped generative hyphae [e hyphae .Picipes admirabilis was was initially collected on wood of apple trees in northeastern United States [Pi. admirabilis is a variety of P. varius or belongs to group Melanopus for its black stipe [Pi. admirabilis, P. gayanus L\u00e9v. and P. pseudobetulinus (Murashk. ex Pil\u00e1t) Thorn, Kotir. & Niemel\u00e4 as members of Polyporus group Admirabilis [P. pseudobetulinus has recently been combined into Favolus as F. pseudobetulinus (Murashk. ex Pil\u00e1t) Sotome & T. Hatt. [Pi. admirabilis as a member of genus Cerioporus Qu\u00e9l. But according to our phylogenetic analysis, Pi. admirabilis strongly clusters in the picipes clade (Pi. admirabilis has a long (up to 8 cm) and pale buff to black stipe, firm-corky basidiocarps and uninflated hyphae [Picipes.d States . Lloyd cck stipe ,40. N\u00fa\u00f1eirabilis . Among tT. Hatt.  regardedes clade . Morphold hyphae . Based oPicipes rhizophilus was treated as a member of group Polyporellus, and it is a special polypore merely grows on the grass roots [Pi. rhizophilus into Cerioporus as C. rhizophilus (Pat.) Zmitr. et Kovalenko. In our current phylogenetic analysis, Pi. rhizophilus strongly groups into the picipes clade . These above-mentioned features fit Picipes well, so we consider this species as a member of Picipes.ss roots ,7,41,42.ss roots  transferes clade . AccordiPicipes is showed to be a monophyletic group based on the 8-gen-squences data analysis, and sixteen species are included in this clade , cylindrical basidiospores and lignicolous habit. According to our study with more samples, we find several species with large pores, oblong basidiospores and terrestrial habit are also members of Picipes.is clade . Among tDe by De  for its . Corner  considerovalenko  considerPolyporus umbellatus is a particular species that merely grows on the ground from a sclerotium close to stumps of hardwoods, and characterized by numerous stipitate pilei arising from a common, strongly branched stipe [Polyporus group Dendropolyporus [P. umbellatus was reported as a distinctive species that could not cluster with any other species [P. umbellatus into genus Cladomeris Qu\u00e9l. But in our analysis, P. umbellatus strongly clusters with P. tuberaster and P. hapalopus in the core polyporus clade  Zmitr. et Kovalenko, but this name is illegitimate because of its earlier homonym N. suavissimus (Fr.) J.S. Seelan.Previous phylogenetic analyses showed that favolus clade and neofavolus clade did not gather together and favolus clade has closer relationships with beraster ,8. But on et al.  estimateovalenko  also treLentinus Fr. and Polyporus have been suspected for a long time. Both Pegler [Lentinus divides from polypores, and this assumption had been evidenced by Hibbett & Vilgalys [Polyporellus has a much closer relationship with Lentinus compared with other Polyporus spp. [Lentinus s. str. and group Polyporellus which have the inflated generative hyphae. Seelan et al. [Lentinus and Polyporellus group is circular pores, with independent transitions to angular pores and lamellae. Zmitrovich [Lentinus as L. arcularius (Batsch) Zmitr., L. brumalis (Pers.) Zmitr., L. crinitus (L.) Fr. and L. tricholoma (Mont.) Zmitr. But the last name is illegitimate because of its earlier homonym L. tricholoma Berk. & Cooke. Then Zmitrovich & Kovalenko [P. tricholoma to L. flexipes (Fr.) Zmitr. et Kovalenko. In this article, we prefer to treat species in polyporellus clade as members of Lentinus.The relationships of h Pegler  and Singh Pegler  believedVilgalys  and HibbVilgalys . Moleculrus spp. ,46,51\u201356rus spp.  proposeditrovich  combinedovalenko  renamed Melanopus distribute into two different clades: picipes clade and squamosus clade. This conclusion verified the view that Melanopus group is not a monophyletic assemblage of dark-stiped Polyporus species, and whether having black cuticle on stipe or not, is not a sufficient feature to define the natural Melanopus group. In our study, sixteen species including four new species of Picipes are recognized. A key to species of Picipes is provided.Our phylogenetic analysis based on multiple gene sequences data of ITS, nLSU, EF1-\u03b1, mtSSU, \u03b2-tubulin, RPB1, RPB2 and nSSU suggested that species of group 1aGrowing on woods or ground\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 21bPi. rhizophilusGrowing on grass roots\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 2aGenerative hyphae bearing simple septa\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 32bGenerative hyphae only with clamps\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 43aPi. submelanopusPores 2\u20133 per mm, generative hyphae bearing both simple septa and clamp connections, growing on ground or hardwoods, basidiospores 8\u201310 \u00d7 3\u20133.9 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 3bPi. badiusPores 5\u20136 per mm, generative hyphae only with simple septa, growing on hardwoods, basidiospores 6.5\u20138 \u00d7 3\u20133.8 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 4aStipe very short or attach to the substrate with a flattened base\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 54bStipe usually more than 1 cm long\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 65aPi. subtropicusPores 8\u20139 per mm when young and becoming 5\u20137 per mm when mature, basidiospores 5.1\u20136.2 \u00d7 2.2\u20132.7 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5bRhododendron, basidiospores 7.5\u201310.5\u00d73.2\u20133.8 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Pi. taibaiensisPores 3\u20135 per mm, growing on 6aBasidiospores more than 9 \u03bcm long\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 76bBasidiospores less than 9 \u03bcm long\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 87aAustrocedrus or Lomatia woods, pores 4\u20135 per mm, basidiospores 9\u201311.5 \u00d7 3\u20133.8 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.Pi. austroandinusGrowing on 7bPi. virgatusGrowing on hardwoods, pores 3\u20134 per mm, basidiospores 9\u201312.5 \u00d7 4\u20135 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 8aCystidioles present\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..98bPi. fraxinicolaCystidioles absent\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 9aPores 3\u20136 per mm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 109bPores 6\u201310 per mm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 1410aBasidiospores usually more than 3 \u03bcm wide\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 1110bBasidiospores usually less than 3 \u03bcm wide\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 1211aPi. admirabilisCystidioles subulate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 11bPi. melanopusCystidioles fusoid\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 12aPilei not infundibuliform in shape\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 1312bPi. baishanzuensisPilei infundibuliform in shape\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 13aPi. americanusPilei nearly circular, basidiospores 7\u20139 \u00d7 2.5\u20133.1 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 13bPi. subtubaeformisPilei irregularly semicircular or elliptical, basidiospores 5.7\u20136.8 \u00d7 2.7\u20133.1 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.14aBasidiospores more than 6 \u03bcm in length\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 1514bPi. tibeticusBasidiospores 5\u20135.9 \u00d7 2.8\u20133.3 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 15aPi. conifericolaGrowing on coniferous trees\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 15bPi. tubaeformisGrowing on hardwoods\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026."}
{"text": "The graph with the largest signless Laplacian spectral radius among all bicyclic graphs with perfect matchings is determined. G =  be a simple connected graph with vertex set V = {v1, v2,\u2026, vn} and edge set E. Its adjacency matrix A(G) = (aij) is defined as n \u00d7 n matrix (aij), where aij = 1 if vi is adjacent to vj, and aij = 0, otherwise. Denote by d(vi) or dG(vi) the degree of the vertex vi. Let Q(G) = D(G) + A(G) be the signless Laplacian matrix of graph G, where D(G) = diag\u2061(d(v1), d(v2),\u2026, d(vn)) denotes the diagonal matrix of vertex degrees of G. It is well known that A(G) is a real symmetric matrix and Q(G) is a positive semidefinite matrix. The largest eigenvalues of A(G) and Q(G) are called the spectral radius and the signless Laplacian spectral radius of G, denoted by \u03c1(G) and q(G), respectively. When G is connected, A(G) and Q(G) are a nonnegative irreducible matrix. By the well-known Perron-Frobenius theory, \u03c1(G) is simple and has a unique positive unit eigenvector and so does q(G). We refer to such an eigenvector corresponding to q(G) as the Perron vector of G.Let G are independent if they are not adjacent in G. A set of mutually independent edges of G is called a matching of G. A matching of maximum cardinality is a maximum matching in G. A matching M that satisfies 2 | M | = n = |V(G)| is called a perfect matching of the graph G. Denote by Cn and Pn the cycle and the path on n vertices, respectively.Two distinct edges in a graph A(G) is det\u2061(xI \u2212 A(G)), which is denoted by \u03a6(G) or \u03a6. The characteristic polynomial of Q(G) is det\u2061(xI \u2212 Q(G)), which is denoted by \u03a8(G) or \u03a8.The characteristic polynomial of Cp and Cq be two vertex-disjoint cycles. Suppose that v1 is a vertex of Cp and vl is a vertex of Cq. Joining v1 and vl by a path v1v2 \u22ef vl of length l \u2212 1, where l \u2265 1 and l = 1 means identifying v1 with vl, denoted by B, is called an \u221e-graph , is called a \u03b8-graph  be the set of all bicyclic graphs on n = 2\u03bc\u2009\u2009(\u03bc \u2265 2) vertices with perfect matchings. Obviously Bn(2\u03bc) consists of two types of graphs: one type, denoted by Bn+(2\u03bc), is a set of graphs each of which is an \u221e-graph with trees attached; the other type, denoted by Bn++(2\u03bc), is a set of graphs each of which is \u03b8- graph with trees attached. Then we have Bn(2\u03bc) = Bn+(2\u03bc) \u222a Bn++(2\u03bc).Let The investigation on the spectral radius of graphs is an important topic in the theory of graph spectra, in which some early results can go back to the very beginnings (see ). The reIn this paper, we deal with the extremal signless Laplacian spectral radius problems for the bicyclic graphs with perfect matchings. The graph with the largest signless Laplacian spectral radius among all bicyclic graphs with perfect matchings is determined.G \u2212 u or G \u2212 uv denote the graph obtained from G by deleting the vertex u \u2208 V(G) or the edge uv \u2208 E(G). A pendant vertex of G is a vertex with degree 1. A path P : vv1v2 \u22ef vk in G is called a pendant path if d(v1) = d(v2) = \u22ef = d(vk\u22121) = 2 and d(vk) = 1. If k = 1, then we say vv1 is a pendant edge of the graph G.Let In order to complete the proof of our main result, we need the following lemmas.G be a connected graph and u, v two vertices of G. Suppose that v1, v2,\u2026, vs \u2208 N(v)\u2216{N(u) \u222a u}\u2009\u2009(1 \u2264 s \u2264 d(v)) and x =  is the Perron vector of G, where xi corresponds to the vertex vi\u2009\u2009(1 \u2264 i \u2264 n). Let G* be the graph obtained from G by deleting the edges vvi and adding the edges uvi\u2009\u2009(1 \u2264 i \u2264 s). If xu \u2265 xv, then q(G) < q(G*).Let G is commonly known as its matching number, denoted by \u03bc(G).The cardinality of a maximum matching of From w and v be two vertices in a connected graph G and suppose that s paths {ww1w1\u2032, ww2w2\u2032,\u2026, wwsws\u2032} of length 2 are attached to G at w and t paths {vv1v1\u2032, vv2v2\u2032,\u2026, vvtvt\u2032} of length 2 are attached to G at v to form Gs,t. Then either q > q\u2009\u2009(1 \u2264 i \u2264 t) or q > q\u2009\u2009(1 \u2264 i \u2264 s)or \u03bc = \u03bc = \u03bc.Let u is a vertex of graph G with d(u) \u2265 2. Let G : uv be a graph obtained by attaching a pendant edge uv to G at u. Suppose t paths {vv1v1\u2032,\u2026, vvtvt\u2032} of length 2 are attached to G : uv at v to form Lt0,. LetLt0, has a perfect matching, then we have that Mt1, has a perfect matching andSuppose G is a sequence of vertices v1, v2,\u2026, vm with m \u2265 2 such thatv1 = vm);the vertices in the sequences are distinct ;d(vi) satisfy d(v1) \u2265 3, d(v2) = \u22ef = d(vm\u22121) = 2 (unless m = 2) and d(vm) \u2265 3.the vertex degrees An internal path of a graph G be a connected graph, and uv \u2208 E(G). The graph Guv is obtained from G by subdividing the edge uv, that is, adding a new vertex w and edges uw, wv in G \u2212 uv. By similar reasoning as that of Theorem 3.1 of [Let m 3.1 of , we haveP : v1v2 \u22ef vk\u2009\u2009(k \u2265 2) be an internal path of a connected graph G. Let G\u2032 be a graph obtained from G by subdividing some edge of P. Then we have q(G\u2032) < q(G).Let v1v2 \u22ef vk\u2009\u2009(k \u2265 3) is an internal path of the graph G and v1vk \u2209 E(G) for k = 3. Let G* be the graph obtained from G \u2212 vivi+1 \u2212 vi+1vi+2\u2009\u2009(1 \u2264 i \u2264 k \u2212 2) by amalgamating vi, vi+1, and vi+2 to form a new vertex w1 together with attaching a new pendant path w1w2w3 of length 2 at w1. Then q(G*) > q(G) and \u03bc(G*) \u2265 \u03bc(G).Suppose that q(G*) > q(G). Next, we prove that \u03bc(G*) \u2265 \u03bc(G). Let M be a maximum matching of G. If vivi+1 \u2208 M or vi+1vi+2 \u2208 M, then {M \u2212 {vivi+1}}\u222a{w2w3} or {M \u2212 {vi+1vi+2}}\u222a{w2w3} is a matching of G*. Thus, \u03bc(G*) \u2265 \u03bc(G); If vivi+1 \u2209 M and vi+1vi+2 \u2209 M, then there exist two edges viu and vi+2v \u2208 M. Thus, {M \u2212 {viu}}\u222a{w2w3} is a matching of G*. Hence, \u03bc(G*) \u2265 \u03bc(G), completing the proof.From S(G) be the subdivision graph of G obtained by subdividing every edge of G.Let G be a graph on n vertices and m edges, \u03a6(G) = det\u2061(xI \u2212 A(G)), \u03a8(G) = det\u2061(xI \u2212 Q(G)). Then \u03a6(S(G)) = xm\u2212n\u03a8.Let u be a vertex of a connected graph G. Let Gk,l\u2009\u2009 be the graph obtained from G by attaching two pendant paths of lengths k and l at u, respectively. If k \u2265 l \u2265 1, then q > q.Let v1v2 \u22ef vk\u2009\u2009(k \u2265 3) is a pendant path of the graph G with d(v1) \u2265 3. Let G* be the graph obtained from G \u2212 v1v2 \u2212 v2v3 by amalgamating v1, v2, and v3 to form a new vertex w1 together with attaching a new pendant path w1w2w3 of length 2 at w1. Then q(G*) > q(G) and \u03bc(G*) \u2265 \u03bc(G).Suppose that q(G*) > q(G). By the proof as that of \u03bc(G*) \u2265 \u03bc(G).By e = uv be an edge of G, and let C(e) be the set of all circuits containing e. Then \u03a6(G) satisfiesZ \u2208 C(e).Let v be a vertex of G, and let \u03c6(v) be the collection of circuits containing v, and let V(Z) denote the set of vertices in the circuit Z. Then the characteristic polynomial \u03a6(G) satisfiesw adjacent to v, and the second summation extends over all Z \u2208 \u03c6(v).Let G be a connected graph, and let G\u2032 be a proper spanning subgraph of G. Then \u03c1(G) > \u03c1(G\u2032), and, for x \u2265 \u03c1(G), \u03a6(G\u2032) > \u03a6(G).Let G) denote the maximum degree of G. From Let \u0394 > q(G\u2032).Let G =  be a connected graph with vertex set V = {v1, v2,\u2026, vn}. Suppose that v1v2 \u2208 E(G), v1v3 \u2208 E(G), v1v4 \u2208 E(G), d(v3) \u2265 2, d(v4) \u2265 2, d(v1) = 3, and d(v2) = 1. Let Gv1v3(Gv1v4) be the graph obtained from G\u2009\u2212\u2009v1v3(G \u2212 v1v4) by amalgamating v1 and v3(v4) to form a new vertex w1(w3) together with subdivising the edge w1v2(w3v2) with a new vertex w2(w4). If q(Gv1v3) > q(G) or q(Gv1v4) > q(G);either \u03bc(Gv1v3) \u2265 \u03bc(G) and \u03bc(Gv1v4) \u2265 \u03bc(G).Let u is a vertex of the bicyclic graph G with dG(u) \u2265 2. Let G : uv be a graph obtained by attaching a pendant edge uv to G at u. Suppose that a pendant edge vw1 and t paths {vv1v1\u2032,\u2026, vvtvt\u2032} of length 2 are attached to G : uv at v to form Lt1,. Let Mt+10, = Lt1, \u2212 vv1 \u2212 \u22ef\u2212vvt + uv1 + \u22ef+uvt. Then we haveq > q, (t \u2265 1);\u03bc \u2264 \u03bc.Suppose G1, G2,\u2026, G6 be the graphs as \u03bc \u2265 3, we have q(G1) > q(Gi), .Let \u03bc \u2265 12, for S(G2)) \u2212 \u03a6(S(G1)) > 0. Hence, \u03c1(S(G1)) > \u03c1(S(G2)) for \u03bc \u2265 12. When \u03bc = 4,5,\u2026, 11, by direct calculation, we also get \u03c1(S(G1)) > \u03c1(S(G2)), respectively. So, \u03c1(S(G1)) > \u03c1(S(G2)) for \u03bc \u2265 4. By q(G1) > q(G2)\u2009\u2009(\u03bc \u2265 4). By similar method, the result is as follows.From \u03a6(S(G1))\u2003G \u2208 Bn(2\u03bc)\u2009\u2009(n \u2265 6), then q(G) \u2264 q(G1), with equality if and only if G = G1.If X = T be the Perron vector of G. From \u03bc \u2265 3, Let G* \u2208 Bn(2\u03bc) such that q(G*) is as large as possible. Then G* consists of a subgraph H which is one of graphs B, B, and P | is the number of vertices of T including the vertex z. In the following, we prove that tree T is formed by attaching at most one path of length 1 and other paths of length 2 at z.Let P : v0v1 \u22ef vk is a pendant path of G* and vk is a pendant vertex. If k \u2265 3, let H1 = G* \u2212 v2v3 + v0v3. From H1 \u2208 Bn(2\u03bc) and q(H1) > q(G*), which is a contradiction.Suppose u \u2208 V(T \u2212 z), we prove that d(u) \u2264 2. Otherwise, there must exist some vertex u0 of T \u2212 z such that d = max\u2061{d | v \u2208 V(T) \u2212 z, d(v) \u2265 3}. From the above proof, we have the pendant paths attached u0 which have length of at most 2. Obviously, there exists an internal path between u0 and some vertex w of G*, denoted by m \u2265 2, let H2 be the graph obtained from G*\u2009\u2212\u2009u0w1\u2009\u2212\u2009w1w2 by amalgamating u0, w1, and w2 to form a new vertex s1 together with attaching a new pendant path s1s2s3 of length 2 at s1. From H2 \u2208 Bn(2\u03bc) and q(H2) > q(G*), which is a contradiction. If m = 1, by H3 such that H3 \u2208 Bn(2\u03bc) and q(H3) > q(G*), which is a contradiction.For each vertex T which is obtained by attaching some pendant paths of length 2 and at most one pendant path of length 1 at z.From the proof as above, we have the tree G* which must be attached at the same vertex of H.From G* is isomorphic to one of graphs G1, G2,\u2026, G6 ). We prove that G* is isomorphic to one of graphs G1, G2, and G3.Cp of G* with length of at least 4. From G*, which is not a triangle, has length 1. Note that all the pendant paths of length 2 in G* must be attached at the same vertex, then there must exist edges v1v2 \u2208 E(G*), v1v3 \u2208 E(Cp), and v1v4 \u2208 E(Cp) and d(v1) = 3, d(v2) = 1, d(v3) \u2265 3, and d(v4) \u2265 3. Let H4\u2009\u2009(H5) be the graph obtained from G* \u2212 v1v3\u2009\u2009(G* \u2212 v1v4) by amalgamating v1 and v3(v4) to form a new vertex y1(y3) together with subdividing the edge y1v2\u2009\u2009(y3v2) with a new vertex y2\u2009\u2009(y4). From Hi \u2208 Bn+(2\u03bc)\u2009\u2009 and either q(H4) > q(G*) or q(H5) > q(G*), which is a contradiction. Then for each cycle Cg of G*, we have g = 3.Assume that there exists some cycle l \u2265 4. If there exists an internal path G*. Then, by H6 such that q(H6) > q(G*) and H6 \u2208 Bn+(2\u03bc), which is a contradiction. Thus, d(vi) \u2265 3\u2009\u2009 and either d(v2) = 3 or d(v3) = 3. By H7 such that q(H7) > q(G*) and H7 \u2208 Bn+(2\u03bc), which is a contradiction. Hence, l \u2264 3.Assume that We distinguish the following three subcases:Subcase\u2009\u20091.1 (l = 1). Then G* is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of ,G\u00af5 see .xu \u2265 xv, let H8 = G* \u2212 rv \u2212 sv + ru + su; if xv \u2265 xu, let H9 = G* \u2212 ut + tv. Obviously, Hi \u2208 Bn+(2\u03bc)\u2009\u2009 and either q(H8) > q(G*) or q(H9) > q(G*) by Assume that Subcase\u2009\u20091.2 (l = 2). Then G* is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of G\u00af14 see .xv1 \u2265 xv2, let H10 = G* \u2212 v2u + v1u; if xv2 \u2265 xv1, let H11 = G* \u2212 v1r + v2r. Obviously, Hi \u2208 Bn+(2\u03bc)\u2009\u2009 and either q(H10) > q(G*) or q(H11) > q(G*) by Assume that Subcase\u2009\u20091.3 (l = 3). Then G* is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of G\u00af20 see .xv1 \u2265 xv2, let H12 = G* \u2212 v2v3 + v1v3; if xv2 \u2265 xv1, let H13 = G* \u2212 v1z1 + v2z1. Obviously, Hi \u2208 Bn+(2\u03bc)\u2009\u2009 and either q(H12) > q(G*) or q(H13) > q(G*) by Assume that G* is isomorphic to one of graphs G1, G2 and G3.Thus, G* is obtained by attaching all the pendant paths of length 2 at vertex y4 of xv1 \u2265 xy4, let H14 be the graph obtained from \u03bc \u2212 3 pendant paths of length 2 at v1. If xy4 \u2265 xv1, let H15 = G* \u2212 v1y3 \u2212 v1y1 \u2212 v1y2 + y4y3 + y4y1 + y4y2. Obviously, H14 = H15 = G1 and q(G1) > q(G*) by G* = G1. By similar reasoning, the result follows.Assume that Case\u2009\u20092 (G* \u2208 Bn++(2\u03bc)). By similar reasoning as that of Case\u2009\u20091, we have G* is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of G\u00af24 see .G* is isomorphic to one of graphs G4, G5 and G6  > q(Gi), . Thus, G* = G1.So,"}
{"text": "The aim of the present paper is to investigate coefficient estimates, Fekete-Szeg\u0151 inequality, and upper bound of third Hankel determinant for some families of starlike and convex functions of reciprocal order. A denote the class of functions f(z) which are analytic in the open unit disk U = {z \u2208 C : |z| < 1} and normalized byS*(\u03b1) and K(\u03b1) denote the usual classes of starlike and convex functions of order \u03b1, 0 \u2264 \u03b1 < 1, respectively. In 1975, Silverman , if it satisfies the inequalityA function F\u03bb(z) byqth Hankel determinant Hq(n), q \u2265 1, n \u2265 1, for a function f(z) \u2208 A is studied by Noonan and Thomas  unless otherwise stated.Throughout in this paper we assume that For our results we will need the following Lemmas.q(z) is a function with Req(z) > 0 and is of the formIf q(z) is of the form (If the form  with posq(z) is of the form  \u2208 L. Thenn = 3,4, 5,\u2026Let q(z) byF\u03bb(z) is given by (q(z) is analytic in U with q(0) = 1, Req(z) > 0.Let us define the function given by  with12), we hlity and 1\u2212nAn\u22642\u03b31lity and 1\u2212nAn\u22642\u03b31k,whereAk=1+\u03bbk\u22121\u03bb = 0 and \u03b3 = 1 \u2212 \u03b1, we get the following result.If we take f(z) \u2208 S\u2217(\u03b1). Then, for n = 3,4, 5,\u2026, one hasa2| \u2264 2(1 \u2212 \u03b1).Let \u03bb = 1 and \u03b3 = 1 \u2212 \u03b1, we get the following result.Making f(z) \u2208 K\u2217(\u03b1). Then, for n = 3,4, 5,\u2026, one hasa2| \u2264 (1 \u2212 \u03b1).Let f(z) \u2208 L and be of the form (Let the form . Then(3f(z) \u2208 L. Then from  \u2208 L. ThenLet f(z) \u2208 L and be of the form (Let the form . Then(3f(z) \u2208 L. Then, from /\u2202\u03c1 > 0 for c \u2208  and \u03c1 \u2208 , maximum of F will exist at \u03c1 = 1 and let F = G(c). Thenc, we obtainG(c)/\u2202c > 0 for c \u2208 , G(c) has a maximum value at c = 2 and henceLet en, from , we havef(z) \u2208 L and be of the form  \u2208 L and is of the form (If the form , then(4Since"}
{"text": "We give further improvements of the Jensen inequality and its converse on time scales, allowing also negative weights. These results generalize the Jensen inequality and its converse for both discrete and continuous cases. Further, we investigate the exponential and logarithmic convexity of the differences between the left-hand side and the right-hand side of these inequalities and present several families of functions for which these results can be applied. By a time scale \ud835\udd4b we mean any nonempty closed subset of real numbers. Using the delta and nabla derivatives, the notions of delta and nabla integrals were defined , was inf : \ud835\udd4b \u2192 \u211d be continuous function and let a, b \u2208 \ud835\udd4b. Then, the diamond-\u03b1 integral of f from a to b is defined by Let \u03b1 = 1 the diamond-\u03b1 integral reduces to the standard delta integral and for \u03b1 = 0 the diamond-\u03b1 integral reduces to the standard nabla integral.From the above definition it is clear that for \ud835\udd4b = \u211d, then \ud835\udd4b = \u2124, then \ud835\udd4b = h\u2124, where h > 0, then \ud835\udd4b = q\u21150, where q > 1, then Moreover, if Recently, the Jensen inequality, the improvement of the Jensen inequality, and their converses are given for time scale integrals see \u20134)..4]).\ud835\udd4b be a time scale, and let a, b \u2208 \ud835\udd4b. Then the time scale interval is denoted and defined by \ud835\udd4b = \u2229\ud835\udd4b.Let c, d \u2208 \u211d. If g \u2208 C), w \u2208 C with \u222bab | w(t) | \u22c4\u03b1t > 0 and \u03a6 \u2208 C, \u211d) is convex, then Let \u03b1-Steffensen-Popoviciu (\u03b1-SP) weight.Note that in this Jensen inequality we have nonnegative weights. In order to give a better version of this Jensen inequality on time scales, Dinu  gives thg \u2208 C. Then, w \u2208 C is an \u03b1-Steffensen-Popoviciu (\u03b1-SP) weight for g on \ud835\udd4b if C, where Let \u03b1-SP weight for a nondecreasing function g on time scales.In the following lemma he gives a characterization for w \u2208 C such that \u222babw(t)\u22c4\u03b1t > 0. Then, w is an \u03b1-SP weight for a nondecreasing function g \u2208 C if and only if it verifies the following condition: s \u2208 \ud835\udd4b.Let w is also an \u03b1-SP weight for the nondecreasing continuous function g.If the following stronger (but more suitable) condition holds \u03b1-SP weights, for any continuous function g and every \u03b1 \u2208 . But there are some \u03b1-SP weights that are allowed to take the negative values. The Jensen inequality on time scales, where it is allowed that the weight function takes some negative values, is given in the following theorem.As given in , all posg \u2208 C and let w \u2208 C such that \u222babw(t)\u22c4\u03b1t > 0. Then, the following two statements are equivalent: (i)w is an \u03b1-SP weight for g on \ud835\udd4b;\u2009(ii)\u2009for every convex function \u03a6 \u2208 C, it holds Let g be nondecreasing function. If \ud835\udd4b = \u2115, then \ud835\udd4b = \u211d in Let \u03b1-Hermite-Hadamard (\u03b1-HH) weight, its characterization for a nondecreasing function g on time scales, and the improvement of the converse of the Jensen inequality for some negative weights.Considering the converse of the Jensen inequality, Dinu  gives thg \u2208 C. Then w \u2208 C is an \u03b1-Hermite-Hadamard (\u03b1-HH) weight for g on \ud835\udd4b if C, where Let w \u2208 C be such that \u222babw(t)\u22c4\u03b1t > 0. Then w is an \u03b1-HH weight for a nondecreasing function g \u2208 C if and only if it verifies the following condition: s \u2208 \ud835\udd4b.Let In the next result Dinu  gives thg \u2208 C. Then every \u03b1-SP weight for g on \ud835\udd4b is an \u03b1-HH weight for g on \ud835\udd4b, for all \u03b1 \u2208 .Let In the following two sections of our paper we give some further generalizations of the Jensen-type inequalities on time scales allowing negative weights, and we also give the mean-value theorems of the Lagrange and Cauchy type for the functionals obtained by taking the difference of the left-hand side and right-hand side of these new inequalities. These results also generalize the results given in  for contm, M \u2208 \u211d, where m \u2260 M. Consider the Green function G : \u00d7 \u2192 \u211d defined by G is convex and continuous with respect to both x and y.Let C2 can be represented by G is defined in  any fined in . Using (ented by \u03a6(x)=M\u2212xMIn the following theorem, we give the generalization of the Jensen inequality on time scales, where negative weights are also allowed.g \u2208 C be such that g\u2286. Let w \u2208 C be such that \u222babw(t)\u22c4\u03b1t \u2260 0 and \u222babg(t)w(t)\u22c4\u03b1t/\u222babw(t)\u22c4\u03b1t \u2208 . Then, the following two statements are equivalent: (i)Cfor every convex function \u03a6 \u2208 (ii)y \u2208 G : \u00d7 \u2192 \u211d is defined in  and (ii) are also equivalent if we change the sign of inequality in both \u03a6(\u222babg(t) \u2208 G(\u222babg(t)G  is continuous and convex, it follows that ((i) \u21d2 (ii): let (i) hold. As the function ows that  holds.C2. Then by using (y) \u2265 0 for all y \u2208 , and hence it follows that for every convex function \u03a6 \u2208 C2 inequality ((ii) \u21d2 (i): let (ii) hold. Let \u03a6 \u2208 by using  we get  the reverse inequality in  and (ii\u2032) are also equivalent if we change the sign of inequality in both statements (i\u2032) and (ii\u2032).Let the conditions of g is nondecreasing and that it has the first derivative. Let m = g(a) and M = g(b), and make the substitution y = g(s). Then we get g is nondecreasing, we have that g\u2032(s) \u2265 0 for all s \u2208 \ud835\udd4b. If \u03a6 \u2208 C2 is convex, then \u03a6\u2032\u2032(g(s)) \u2265 0 for all s \u2208 \ud835\udd4b. Hence, if and only if s \u2208 \ud835\udd4b, then for every continuous convex function \u03a6 inequality  and let w \u2208 C be such that \u222babw(t)\u22c4\u03b1t > 0. Then w is an \u03b1-SP weight for g on \ud835\udd4b if and only if y \u2208 , where G is defined in  be nondecreasing function and w \u2208 C such that \u222babw(t)\u22c4\u03b1t > 0. Then s \u2208 \ud835\udd4b, if and only if y \u2208 , where G is defined in : m, M]. Clearly, if \u03a6 is continuous and convex, then \ud835\udca51 is nonnegative.Under the assumptions of g \u2208 C and \u03a6 \u2208 C2. Let \ud835\udca51 be defined as in  = x2/2.Let ed as in . Then thm, M], applying the integral mean-value theorem on \u03d51 and \u03d52 are continuous and convex, we have \u03be \u2208  such that  and \u03a6, \u03a8 \u2208 C2. Let \ud835\udca51 be defined as in J1\u03c7 be defined as the linear combination of functions \u03a6 and \u03a8 by \u03c7 \u2208 C2. By applying \u03c7, it follows that there exists \u03be \u2208  such that \ud835\udca51 = 0. By hypothesis \ud835\udca51 \u2260 0  \u2260 0), so it follows that Let alent to .In x) = x2/2 in Note that setting the function \u03a8 as \u03a8, \u03a6, \u03a8 :  \u2192 \u211d, and let w \u2208 C be an \u03b1-SP weight for g. Let \ud835\udca51 be defined as in ((i)C2, then there exists \u03be \u2208  such that (If \u03a6 \u2208 uch that  holds.(ii)C2, then there exists \u03be \u2208  such that  (statement (ii), resp.,) directly follows from g \u2208 C be monotone function, \u03a6, \u03a8 :  \u2192 \u211d, and w \u2208 C such that \u222babw(t)\u22c4\u03b1t > 0. Let C2, then there exists \u03be \u2208  such that (If \u03a6 \u2208 uch that  holds.(ii)C2, then there exists \u03be \u2208  such that  (statement (ii), resp.,) directly follows from Using the similar method as in previous section, in the following theorem we obtain the generalization of the converse of the Jensen inequality on time scales, where negative weights are also allowed.g \u2208 C be such that g\u2286 and let c, d \u2208 \u2009\u2009(c \u2260 d) be such that c \u2264 g(t) \u2264 d for all t \u2208 \ud835\udd4b. Let w \u2208 C be such that \u222babw(t)\u22c4\u03b1t \u2260 0. Then, the following two statements are equivalent. (i)CFor every convex function \u03a6 \u2208 (ii)y \u2208 G : \u00d7 \u2192 \u211d is defined in  and (ii) are also equivalent if we change the sign of inequality in both \u222bab\u03a6(g(t) \u2208 \u222babG(g(t)The idea of the proof is very similar to the proof of G  is continuous and convex, it follows that ((i) \u21d2 (ii): let (i) hold. As the function ows that  holds.C2. Then by using (y) \u2265 0 for all y \u2208 , and hence it follows that for every convex function \u03a6 \u2208 C2 the inequality ((ii) \u21d2 (i): let (ii) hold. Let \u03a6 \u2208 by using  we get , the reverse inequality in  and (ii\u2032) are also equivalent if we change the sign of inequality in both statements (i\u2032) and (ii\u2032).Let the conditions of m, M], while in the results from the previous section we demanded that Note that in all the results in this section we allow that the mean value c = m and d = M in Setting g \u2208 C be such that g\u2286. Let w \u2208 C be such that \u222babw(t)\u22c4\u03b1t \u2260 0. Then, the following two statements are equivalent. (i)CFor every convex function \u03a6 \u2208 (ii)y \u2208 G : \u00d7 \u2192 \u211d is defined as in  and (ii) are also equivalent if we change the sign of inequality in both \u222bab\u03a6(g(t) \u2208 \u222babG(g(t)As a consequence of c = m and d = M. Then (Let  M. Then  transforabw(t)\u22c4\u03b1t > 0, and suppose that g is nondecreasing and that it has the first derivative. Now, similarly as in \ud835\udd4b. If \u03a6 \u2208 C2 is convex, then \u03a6\u2032\u2032(g(s)) \u2265 0 for all s \u2208 \ud835\udd4b. Hence, if and only if s \u2208 \ud835\udd4b, then for every continuous convex function \u03a6 inequality equality .g \u2208 C and let w \u2208 C be an \u03b1-SP weight for g on \ud835\udd4b. Then y \u2208 .Let The proof follows directly from \ud835\udca52: m, M]. Clearly, if \u03a6 is continuous and convex, then \ud835\udca52 is nonnegative.Under the assumptions of g \u2208 C and \u03a6 \u2208 C2. Let \ud835\udca52 be defined as in  = x2/2.Let ed as in . Then thThe idea of the proof is very similar to the proof of m, M], applying the integral mean-value theorem on  and \u03a6, \u03a8 \u2208 C2. Let \ud835\udca52 be defined as in J2The proof is very similar to the proof of First we recall some definitions and facts about exponentially convex and logarithmically convex functions  which wf : I \u2192 \u211d(I\u2286\u211d) is n-exponentially convex in the Jensen sense on I, if \u03bei \u2208 \u211d and xi \u2208 I, i = 1,\u2026, n.A function f : I \u2192 \u211d is n-exponentially convex if it is n-exponentially convex in the Jensen sense and continuous on I.A function n-exponentially convex functions in the Jensen sense are k-exponentially convex in the Jensen sense for every k \u2208 \u2115, k \u2264 n.It is clear from the definition that 1-exponentially convex functions in the Jensen sense are in fact nonnegative functions. Also, By definition of positive semidefinite matrices and some basic linear algebra, we have the following proposition.f is an n-exponentially convex function in the Jensen sense, then the matrix [f((xi + xj)/2)]i,j=1k is positive, semidefinite for all k \u2208 \u2115, k \u2264 n. Particularly, det\u2061[f((xi + xj)/2)]i,j=1k \u2265 0 for all k \u2208 \u2115, k \u2264 n.If f : I \u2192 \u211d is exponentially convex in the Jensen sense on I, if it is n-exponentially convex in the Jensen sense for all n \u2208 \u2115.A function f : I \u2192 \u211d is exponentially convex if it is exponentially convex in the Jensen sense and continuous.A function (i)f : I \u2192 \u211d defined by f(x) = cekx, where c \u2265 0 and k \u2208 \u211d;(ii)f : \u211d+ \u2192 \u211d defined by f(x) = xk\u2212, where k > 0;(iii)f : \u211d+ \u2192 \u211d+ defined by k > 0.Some examples of exponentially convex functions are (see ) as follf : I \u2192 \u211d+ is log-convex in the Jensen sense on I, if and only if \u03c1, \u03c3 \u2208 \u211d and x, y \u2208 I. It follows that a positive function is log-convex in the Jensen sense if and only if it is 2-exponentially convex in the Jensen sense. Also, using basic convexity theory, it follows that a positive function is log-convex if and only if it is 2-exponentially convex.It is known (and easy to show) that The following lemma is equivalent to the definition of convex function x1, x2, x3 \u2208 I are such that x1 < x2 < x3, then the function f : I \u2192 \u211d is convex if and only if the following inequality holds: If We will also need the following result .f : I \u2192 \u211d is a convex function and x1, x2, y1, y2 \u2208 I are such that x1 \u2264 y1, x2 \u2264 y2, x1 \u2260 x2, and y1 \u2260 y2, then the following inequality is valid: If When dealing with functions with different degree of smoothness, divided differences are found to be very useful.f : I \u2192 \u211d at mutually different points x0, \u2009\u2009x1, \u2009\u2009x2 \u2208 I is defined recursively by The second order divided difference of a function x0, x1, x2; f] is independent of the order of the points x0, x1, and x2. This definition may be extended to include the case in which some or all the points coincide  such that the function \u03c1 \u21a6  is n-exponentially convex in the Jensen sense on J for every choice of three mutually different points x0, x1, x2 \u2208 . Then \u03c1 \u21a6 \ud835\udca5i is an n-exponentially convex function in the Jensen sense on J. If the function \u03c1 \u21a6 \ud835\udca5i is also continuous on J, then it is n-exponentially convex on J.Let fined in  and 50)\ud835\udca5i, i = \u03bd : I \u2192 \u211d by \u03bej, \u03bek \u2208 \u211d, rj, rk \u2208 J, 1 \u2264 j, k \u2264 n, rjk = (rj + rk)/2, and \u03a6rjk \u2208 \u03a9. Using the assumption that for every choice of three mutually different points x0, x1, x2 \u2208 \u03c1 \u21a6  is n-exponentially convex in the Jensen sense on J, we obtain \u03bd is convex (and continuous) function on I. Hence \ud835\udca5i \u2265 0, i = 1,2, which implies that \u03c1 \u21a6 \ud835\udca5i is n-exponentially convex on J in the Jensen sense.Define the function \u03c1 \u21a6 \ud835\udca5i is continuous on J, then \u03c1 \u21a6 \ud835\udca5i is n-exponentially convex by definition.If The following corollary is an immediate consequence of \ud835\udca5i, i = 1,2, be linear functionals defined in  such that the function \u03c1 \u21a6  is exponentially convex in the Jensen sense on J for every choice of three mutually different points x0, x1, x2 \u2208 . Then \u03c1 \u21a6 \ud835\udca5i is an exponentially convex function in the Jensen sense on J. If the function \u03c1 \u21a6 \ud835\udca5i is also continuous on J, then it is exponentially convex on J.Let fined in  and 50)\ud835\udca5i, i = \ud835\udca5i, i = 1,2, be linear functionals defined in  such that the function \u03c1 \u21a6  is 2-exponentially convex in the Jensen sense on J for every choice of three mutually different points x0, x1, x2 \u2208 . Then, the following statements hold. (i)\u03c1 \u21a6 \ud835\udca5i is a 2-exponentially convex function in the Jensen sense on J.(ii)\u03c1 \u21a6 \ud835\udca5i is continuous on J, then it is also 2-exponentially convex on J. If \u03c1 \u21a6 \ud835\udca5i is additionally strictly positive, then it is also log-convex on J, and for r, s, t \u2208 J such that r < s < t, we have If (iii)\u03c1 \u21a6 \ud835\udca5i is strictly positive and differentiable function on J, then for every p, q, u, v \u2208 J such that p \u2264 u, q \u2264 v, we have p, \u03a6q \u2208 \u03a9.If Let fined in  and 50)\ud835\udca5i, i = \u03c1 \u21a6 \ud835\udca5i is continuous and strictly positive, its log-convexity is an immediate consequence of f(x) = log\u2061\ud835\udca5i and r, s, t \u2208 J (r < s < t), we get (i) and the first part of (ii) are immediate consequences of equality .\u03c1 \u21a6 \ud835\udca5i be strictly positive and differentiable and therefore continuous too. By (ii), the function \u03c1 \u21a6 \ud835\udca5i is log-convex on J; that is, the function \u03c1 \u21a6 log\u2061\ud835\udca5i is convex on J, and by p \u2264 u, q \u2264 v, p \u2260 q, and u \u2260 v, concluding that p = q and u = v follow from (To prove (iii), let , and by log\u2061Ji, and furthermore, they still hold when all three points coincide for a family of twice differentiable functions with the same property. The proofs are obtained by recalling Note that the results from \u03a9 = {\u03a6\u03c1 : \u03c1 \u2208 J} in order to construct different examples of exponentially convex functions and construct some means.In this section we will vary on choice of a family d2/dx2)\u03ba\u03c1(x) = e\u03c1x > 0 which shows that \u03ba\u03c1 is convex on \u211d for every \u03c1 \u2208 \u211d. From \u03c1 \u21a6 (d2/dx2)\u03ba\u03c1(x) is exponentially convex. Therefore, \u03c1 \u21a6  is exponentially convex ,  are exponentially convex in the Jensen sense. It is easy to verify that these mappings are continuous, so they are exponentially convex.Consider a family of functions vex (see )  from  from  becomes nd using  we have \ud835\udca5i\u2009\u2009 are positive, using Theorems \u03bap \u2208 \u03a91 and \u03a8 = \u03baq \u2208 \u03a91, it follows that \u2135p,q\u2208. If we set g =  then we have that \u2135p,q are means (of the function g). Note that by (If  that by  they ared2/dx2)\u03b2\u03c1(x) = x\u03c1\u22122 = e\u03c1\u22122)log\u2061x\u03b2\u03c1(x) is exponentially convex. Therefore \u03c1 \u21a6  is exponentially convex . Here we assume that \u2282, so our family \u03a92 of \u03b2\u03c1 fulfills the conditions of \u2133p,q from  are positive, by applying Theorems \u03b2p \u2208 \u03a92 and \u03a8 = \u03b2q \u2208 \u03a92, it follows that for i = 1,2 there exist \u03bei \u2208  such that \u03bei \u21a6 \u03beip\u2212q is invertible for p \u2260 q, we have Consider a family of functions  \u03a9) from  becomes \u2133p,q is continuous, symmetric, and monotonous  = , then we have that \u2133p,q are means (of the function g).Also, nous (by ). If we r in case g = . For r \u2260 0 by substituting g \u21a6 gr, p \u21a6 p/r, and q \u21a6 q/r in \u03b3\u03c1(x) = \u03c1x\u2212 > 0, which shows that \u03b3\u03c1 is convex function for \u03c1 > 0. Also, from \u03c1 \u21a6 (d2/dx2)\u03b3\u03c1(x) is exponentially convex. Therefore \u03c1 \u21a6  is exponentially convex . Here we assume that \u2282, so our family \u03a93 of \u03b3\u03c1 fulfills the conditions of \u2133p,q from (p and q.Consider a family of functions  \u03a9) from  becomes , and by  it is mo\ud835\udca5i\u2009\u2009 are positive, using Theorems \u03b3p \u2208 \u03a93 and \u03a8 = \u03b3q \u2208 \u03a93, it follows that \u2135p,q\u2208. Here L is the logarithmic mean defined by L = (p \u2212 q)/(log\u2061p \u2212 log\u2061q) for p \u2260 q, L = p.If \u03b4\u03c1 is convex function for \u03c1 > 0. Also, from \u03c1 \u21a6 (d2/dx2)\u03b4\u03c1(x) is exponentially convex. Therefore \u03c1 \u21a6  is exponentially convex . Here we assume that \u2282, so our family \u03a94 of \u03b4\u03c1 fulfills the conditions of \u2133p,q from (p and q by (Consider a family of functions  \u03a9) from  becomes and q by .\ud835\udca5i\u2009\u2009 are positive, using Theorems \u03b4p \u2208 \u03a94 and \u03a8 = \u03b4q \u2208 \u03a94, it follows that \u2135p,q\u2208.If"}
{"text": "Under the new H\u00f6lder conditions, we consider the convergence analysis of the inverse-free Jarratt method in Banach space which is used to solve the nonlinear operator equation. We establish a new semilocal convergence theorem for the inverse-free Jarratt method and present an error estimate. Finally, three examples are provided to show the application of the theorem. C is the set of all continuous functions in ; k is the Green function:We consider the following boundary value problem:x\u2032\u2032=\u2212\u03bbG(xion (see ):(2)x(sF(s) = 0, whereF is defined on an open convex \u03a9 of a Banach space X with values in a Banach space Y.Instead of conomics , chemist0, whereF:\u03a9\u2282Ca,b\u27f6conomics \u20138 are soF(x) = 0. Particularly iterative methods are often used to solve this problem ., 10.F(x)To improve the convergence order, many modified methods have been presented. The famous Halley's method and the supper-Halley method are the third-order convergence. References \u201322 give \u03c9 :  and z \u2208 (1/(1 \u2212 f(a0)g)); hence, x1, y0 \u2208 S. ConsiderF\u2032(x1)\u22121 exists, andBy (A1)\u2013(A6), F\u2032(x2)\u22121F\u2032(x0) exists, and \u2016F\u2032(x2)\u22121F\u2032(x0)\u2016 \u2264 f(a1)\u2016F\u2032(x1)\u22121F\u2032(x0)\u2016. By induction, we can prove that the following By n \u2265 1:F\u2032(xn)\u22121F\u2032(x0) exists and \u2016F\u2032(xn)\u22121F\u2032(x0)\u2016\u2009\u2009\u2264f(an\u22121)\u2016F\u2032(xn\u22121)\u22121F\u2032(x0)\u2016;yn \u2212 xn\u2016 \u2264 f(an\u22121)g\u2016yn\u22121 \u2212 xn\u22121\u2016;\u2016H \u2264 M\u2016F\u2032(xn)\u22121F\u2032(x0)\u2016\u2016yn \u2212 xn\u2016 \u2264 an;N\u2016F\u2032(xn)\u22121F\u2032(x0)\u2016\u2016yn \u2212 xn\u20162 \u2264 bn;F\u2032(xn)\u22121F\u2032(x0)\u2016\u03c9(\u2016yn \u2212 xn\u2016)\u2016yn \u2212 xn\u20162 \u2264 cn;\u2016xn+1 \u2212 yn\u2016 \u2264 (an/2)(1 + an)\u2016yn \u2212 xn\u2016;\u2016xn+1 \u2212 xn\u2016 \u2264 [1 + (an/2)(1 + an)]\u2016yn \u2212 xn\u2016;\u2016xn+1 \u2212 x0\u2016 \u2264 R\u03b7, where R = [1 + (a0/2)(1 + a0)]\u2009(1/(1 \u2212 f(a0)g)).\u2016Under the hypotheses of X and Y be two Banach spaces and F : \u03a9 \u2282 X \u2192 Y has continuous Fr\u00e9chet derivative of the third-order on a nonempty open convex \u03a9. One supposes that \u03930 = F\u2032(x0)\u22121 \u2208 L exists for some x0 \u2208 \u03a9 and conditions (A1)\u2013(A6) and xn} generated by x* of S \u2212 R\u03b7)\u2229\u03a9. Furthermore, the following error estimate is obtained:\u03b3 = f2(a0)g = a1/a0 and \u0394 = 1/f(a0), R = (1 + (a0/2)(1 + a0))\u2009(1/(1 \u2212 \u03b3\u0394)).Let xn} is a Cauchy one. From (II) and by n \u2265 0, m \u2265 1,x)k \u2212 1 > kx, so (3 + p)k \u2212 1 > k(2 + p). Hence, we havexn} is a Cauchy sequence and x* = lim\u2061n\u2192\u221exn. Obviously, xm \u2208 B, for all m \u2265 1, as if n = 0 in yn \u2208 B, for all n \u2265 0.Firstly, we prove that the sequence { we havexn+m\u2212xn\u2003<n \u2192 \u221e in F\u2032(xn)\u22121F(xn)\u2016 \u2192 0. Besides \u2016F(xn)\u2016 \u2192 0, since \u2016F(xn)\u2016 \u2264 \u2016F\u2032(xn)\u2016\u2016F\u2032(xn)\u22121F(xn)\u2016 and {\u2016F\u2032(xn)\u2016} is a bounded sequence, therefore F(x*) = 0 by the continuity of F in Now, let m \u2192 \u221e in By letting y* of S \u2212 R\u03b7)\u2229\u03a9. Then01F\u2032(x* + t(y* \u2212 x*))dt exists andx* = y*.To show uniqueness, let us assume that there exists a second solution This completes the proof of In this section, we apply the convergence theorem and show three numerical examples.F(x) = x10/3 + x7/2 \u2212 x \u2212 1 = 0 on x \u2208 . Then, we easily get thatK, p) H\u00f6lder conditionp \u2208  \u2264 t1/3\u03c9(z) for t \u2208  and z \u2208  \u2192 C,K, p) H\u00f6lder condition; for instance, the max-norm is considered. In this case,\u03c9(z) = \u03a3i=1mLizpi satisfy \u03c9(tz) \u2264 tq\u03c9(z), where q = min\u2061{pi, p2,\u2026, pm}.Let us consider a particular case of F\u2032\u2032\u2032 is Lipschitz continuous in \u03a9, we can choose \u03c9(z) = Kz, K > 0, so that Jarratt's method is of R-order, at least four order. If F\u2032\u2032\u2032 is  H\u00f6lder continuous in \u03a9, then we can choose \u03c9(z) = Lzp, L < 0, p \u2208  = 1 and we look for a domain in the forma0 = M\u03b7 = 0.00123353 < 1/2, b0 = 6.52127 \u00d7 10\u22126, c0 = 1.47132 \u00d7 10\u22126, \u03b3 = f2(a0)g = 3.48167 \u00d7 10\u22127 < 1, \u0394 = 0.998766\u22ef, and R = 1.00062\u22ef. This means that the hypothesis of n = 1,2, 3,4, we getThis equation arises in the theory of the radiative transfer and neutron transport and in the kinetic theory of gasses. Let us define the operator F = 0, wherex0 =  =  and \u03a9 = {x\u2223\u2016x \u2212 x0\u2016 \u2264 1.5}. We take the max-norm in R2 and the norm \u2016A\u2016 = max{|a11| + |a12|, |a21| + |a22|} for B on R2 byu = T and Let us consider the system of equations \u03b7 = \u2016F\u2032(x0)\u22121F(x0)\u2016 = 0.09598\u22ef, M = 9.20456\u22ef, N = 10.7635\u22ef, and p = 1/3.Then, we get the following results: We get that the hypotheses of"}
{"text": "Sharp radius of Janowski starlikeness is obtained for functions f whose nth coefficient satisfies |an | \u2264 cn + d\u2009\u2009 or |an | \u2264 c/n\u2009\u2009(c > 0\u2009\u2009and\u2009\u2009n \u2265 3). Other radius constants are also obtained for these functions, and connections with earlier results are made.Let  A denote the class of analytic functions f defined in the open unit disk D\u2254{z \u2208 C : |z | <1}, normalized by f(0) = 0 = f\u2032(0) \u2212 1, and let S denote its subclass consisting of univalent functions. If f(z) = z + \u2211n=2\u221eanzn \u2208 S, de Branges . O. OS poss\u2009(n \u2265 2) \u20137. The n 1)cos\u2061\u03b2 , and thecos\u2061\u03b2  of Janowski starlike functions  for appropriate choices of the parameters A and B. For example, for 0 \u2264 \u03b2 < 1, ST(\u03b2)\u2254ST is the familiar class of starlike functions of order \u03b2. Denote by ST\u03b2 the class ST\u03b2\u2254L0 = ST. Janowski .The class unctions  consistsJanowski  obtainedAb consisting of functions f(z) = z + \u2211n=2\u221eanzn,\u2009\u2009, in the disk D. The subclass of univalent functions in Ab have been studied in -radius are derived for these classes. Several known radius constants are shown to be special cases of the results obtained.This paper finds radius constants for functions f \u2208 A to belong to the class L is given in the following lemma.A sufficient condition for functions \u03b2 \u2208 R\u2216{1} and \u03b1 \u2265 0. If f(z) = z + \u2211n=2\u221eanzn \u2208 A satisfies the inequalityf \u2208 L.Let L-radius is obtained for f \u2208 Ab satisfying the coefficient inequality |an | \u2264cn + d.Making use of this lemma, the sharp \u03b2 \u2208 R\u2216{1}, 6\u03b1 + 3 \u2212 \u03b2 \u2265 0, and \u03b1 \u2265 0. The L-radius for f(z) = z + \u2211n=2\u221eanzn \u2208 Ab satisfying the coefficient inequality |an | \u2264cn + d, c, d \u2265 0, n \u2265 3, is the real root in  of the equation\u03b2 < 1, this number is also the L0-radius of f \u2208 Ab. The results are sharp.Let r0 is the L-radius for f \u2208 Ab if and only if f(r0z)/r0 \u2208 L. Therefore, by r0 is the real root in  of  of . Using t\u03b2 < 1, consider the functionz = r0 in  of -radius for f \u2208 Ab. For \u03b2 < 1, (N(r0)/D(r0) is positive, and therefore the equalityr0 is the sharp L0-radius for f \u2208 Ab when \u03b2 < 1.For  of , f0 satatisfiesRe-radius of f(z) = z + \u2211n=2\u221eanzn \u2208 Ab satisfying the coefficient inequality |an | \u2264c/n for n \u2265 3 and c > 0 is the real root in  of the equation\u03b2 < 1, this number is also the L0-radius of f \u2208 Ab. The results are sharp.Let r0 is the L-radius of functions f \u2208 Ab when inequality  (r0 of  and 9)  r0 is t\u03b2 < 1, consider the functionz = r0 in  of -radius for f \u2208 Ab. For \u03b2 < 1, the rational expression in -radius for f \u2208 Ab. For \u03b2 > 1, sharpness of the result is demonstrated by the function f0 given byTo verify sharpness for  of , f0 satatisfiesRe = z + \u2211n=2\u221eanzn \u2208 A satisfies the inequalityf \u2208 ST.Let \u22121 \u2264 ST-radius for f \u2208 Ab satisfying the coefficient inequality |an | \u2264cn + d.The next result finds the sharp B < A \u2264 1. The ST-radius for f(z) = z + \u2211n=2\u221eanzn \u2208 Ab satisfying the coefficient inequality |an | \u2264cn + d, n \u2265 3 and c, d \u2265 0, is the real root in  of the equationLet \u22121 \u2264 r0 is the ST-radius of f \u2208 Ab if and only if f(r0z)/r0 \u2208 ST. Hence, by r0 is the root in  of  of . From  of , and  of  of , the fun1.Then,  yields = z + \u2211n=2\u221eanzn \u2208 Ab satisfying the coefficient inequality |an | \u2264c/n, n \u2265 3 and c > 0, is the real root in  of the equationLet \u22121 \u2264 r0 is the ST-radius of f \u2208 Ab where r0 is the real root of  of (f0 satisfies (By  root of . Therefo root of  and 19)r0 is thgiven by . Indeed, of . Evidentatisfies , and hen"}
{"text": "We focus our discussion on the uncertainty measures of vague soft sets. Wepropose axiomatic definitions of similarity measure and entropy for vague soft sets. Furthermore, wepresent a new category of similarity measures and entropies for vague soft sets. The basic properties ofthese measures are discussed and the relationships among these measures are analyzed. In 1999, Molodtsov \u201311.Accordingly, works on soft set theory are progressing rapidly. Maji et al.  defined The measurement of uncertainty is an important topic for the theories dealing with uncertainty. Majumdar and Samanta  initiateIn this section, we recall some fundamental notions of soft sets and vague soft sets. See especially , 33, 34 U be a nonempty set, called universe. A fuzzy set \u03bc on U is defined by a membership function \u03bc : U \u2192 . For x \u2208 U, the membership value \u03bc(x) essentially specifies the degree to which x belongs to the fuzzy set \u03bc. We denote by F(U) the set of all fuzzy sets on U.The theory of fuzzy sets initiated by Zadeh  providesAmong the extensions of the classic fuzzy set, vague set is one of the most popular sets treating imprecision and uncertainty. It was proposed by Gau and Buehrer .A over the universe U can be expressed by the notion A = {, 1 \u2212 fA(x)]); x \u2208 U}; that is, A(x) = , and the condition 0 \u2264 tA(x) \u2264 1 \u2212 fA(x) should hold for any x \u2208 U, where tA(x) is called the membership degree (true membership) of element x to the vague set A, while fA(x) is the degree of nonmembership  of the element x to the vague set A.A vague set tA(x) is a lower bound on the grade of membership of x to A derived from the evidence for x and fA(x) is a lower bound on the negation of x derived from the evidence against x. The vague value  indicates that the exact grade of membership of x to A may be unknown, but it is bounded by tA(x) and 1 \u2212 fA(x).In this definition, \u03bc corresponds to the following vague set:Every fuzzy set A, B be two vague sets over the universe U. The union, intersection, and complement of vague sets are defined as follows:Let A, B be two vague sets over the universe U. If, for\u2009\u2009all\u2009\u2009x \u2208 U, tA(x) \u2264 tB(x), 1 \u2212 fA(x) \u2264 1 \u2212 fB(x), then A is called a vague subset of B, denoted by A\u2286B.Let The operations on vague sets are natural generalizations of the corresponding operations on fuzzy sets. Also, the notion of vague subset is a generalization of the notion of fuzzy subset.U be the universe set and E the set of all possible parameters under consideration with respect to U. Usually, parameters are attributes, characteristics, or properties of objects in U.  will be called a soft space. Molodtsov defined the notion of a soft set in the following way.In 1999, Molodtsov  be a mapping. For  \u2208 VSS(U),\u2009H is called the entropy of  if it satisfies the following conditions:H1) = 0\u21d4for\u2009\u2009all\u2009\u2009e \u2208 E, x \u2208 U, tF(e)(x) = 0, fF(e)(x) = 1, or tF(e)(x) = 1, fF(e)(x) = 0;H2) = 1\u21d4for\u2009\u2009all\u2009\u2009e \u2208 E, x \u2208 U, tF(e)(x) = fF(e)(x);H3) = Hc);H4)\u2286 and tG(e)(x) \u2264 fG(e)(x), or \u2287 and tG(e)(x) \u2265 fG(e)(x), then H \u2264 H.for\u2009\u2009all\u2009\u2009Let H1) means the entropy of a soft set is minimal. By (H2), the entropy of the most vague soft set is maximal. (H3) means the entropies of a vague soft set and its complement are equal. (H1), (H2), and (H3) are natural generalizations of the corresponding conditions needed for the entropy of vague set. Now we focus our discussion on the condition (H4). If \u2286, tG(e)(x) \u2264 fG(e)(x) for any e \u2208 E, x \u2208 U, then tG(e)(x) \u2264 0.5 by tG(e)(x) \u2264 1 \u2212 fG(e)(x). In this case, \u2286 means F(e) is crisper than G(e) for any e \u2208 E. Similarly, if \u2287, tG(e)(x) \u2265 fG(e)(x) for any e \u2208 E, x \u2208 U, then F(e) is crisper than G(e) for any e \u2208 E. Thus (H4) is reasonable, but it is not complete. In fact, let  be crisper that . There may exist parameters e1, e2 \u2208 E such that F(e1)\u2286G(e1) and F(e2)\u2287G(e2). Also, for a specific parameter e \u2208 E, there may exist some elements x \u2208 U1 \u2282 U such that tG(e)(x) \u2264 fG(e)(x), tF(e)(x) \u2264 tG(e)(x), and fF(e)(x) \u2265 fG(e)(x), and tG(e)(x) \u2265 fG(e)(x), tF(e)(x) \u2265 tG(e)(x), and fF(e)(x) \u2264 fG(e)(x) for any x \u2208 U \u2212 U1. In these cases, (H4) cannot guarantee H \u2264 H because \u2288 and \u2288. As illustration, we consider the following example.In this definition,  and  are defined by(1) Let F(e1) and F(e2) are crisper than G(e1) and G(e2), respectively. We noticed that \u2288 and \u2288. Thus (H4) can not guarantee H \u2264 H.By the interpretation of true membership and false membership, E = {e}, U = {x1, x2}. The vague soft sets  and  are defined by(2) Let F(e) is crisper than G(e). Also, \u2288 and \u2288. (H4) can not guarantee H \u2264 H.Similarly, U) is the set of all vague soft sets over U, but (H1)~(H4) are all talking about the vague soft sets with the whole parameter set E. Thus, the entropy derived from this definition is actually a partial entropy.By the way, VSS\u2192 be a mapping. For  \u2208 VSS(U), H is called the entropy of  if it satisfies the following conditions:H1) = 0\u21d4for\u2009\u2009all\u2009\u2009e \u2208 A, x \u2208 U, tF(e)(x) = 0, fF(e)(x) = 1, or tF(e)(x) = 1, fF(e)(x) = 0;H2) = 1\u21d4for\u2009\u2009all\u2009\u2009e \u2208 A, x \u2208 U, tF(e)(x) = fF(e)(x);H3) = Hc);H4),  \u2208 VSS(U). If for\u2009\u2009all\u2009\u2009e \u2208 A, x \u2208 U, tF(e)(x) \u2264 tG(e)(x), fF(e)(x) \u2265 fG(e)(x) whenever tG(e)(x) \u2264 fG(e)(x), and tF(e)(x) \u2265 tG(e)(x), fF(e)(x) \u2264 fG(e)(x) whenever tG(e)(x) \u2265 fG(e)(x), then H \u2264 H.Let  \u00d7 VSS(U)\u2192 be a mapping. For  \u2208 VSS(U) and  \u2208 VSS(U), M, ) is called the degree of similarity between  and  if it satisfies the following conditions:M1), ) = M, );\u2009\u2009M2), )\u2208;\u2009\u2009M3), ) = 1\u21d4 = ;\u2009\u2009M4), ) = 0\u21d4for\u2009\u2009all\u2009\u2009e \u2208 E, x \u2208 U, tF(e)(x) = 0, fF(e)(x) = 1, tG(e)(x) = 1, fG(e)(x) = 0, or tF(e)(x) = 1, fF(e)(x) = 0, tG(e)(x) = 0, fG(e)(x) = 1;\u2009\u2009M5)\u2286\u2286 implies M, ) \u2264 min\u2061, ), M, )).\u2009\u2009(Let d : VSS(U) \u00d7 VSS(U)\u2192 be a mapping. For  \u2208 VSS(U) and  \u2208 VSS(U), d, ) is called the degree of distance between  and  if it satisfies the following conditions:d1), ) = d, );d2), )\u2208;d3), ) = 1\u21d4for\u2009\u2009all\u2009\u2009e \u2208 E, x \u2208 U, tF(e)(x) = 0, fF(e)(x) = 1, tG(e)(x) = 1, fG(e)(x) = 0, or tF(e)(x) = 1, fF(e)(x) = 0, tG(e)(x) = 0, fG(e)(x) = 1;d4), ) = 0\u21d4 = ;d5)\u2286\u2286 implies d, ) \u2265 max\u2061, ), d, ))., ) and d, ) be the similarity measure and distance measure between two vague soft sets  and  as defined in Definitions M, ) and d, ) can be given as follows:Let M, ) + d, ) = 1 just by the definitions of similarity measure and distance measure. Furthermore, in Definitions E are compared. Thus the similarity measure and distance measure are all partial measures. By the way, conditions (M2) and (d2) are clearly not necessary because the codomain of M and d has already been restricted to .We notice that this theorem holds for the special similarity measure and distance measure presented in Theorems 3.2 and 3.3 . But it Similarity measure and distance measure are closely related. In what follows, we focus our discussion on entropy and similarity measure. Taking the above observations into account, we propose the following definition of similarity measure for vague soft sets.M : VSS(U) \u00d7 VSS(U)\u2192 be a mapping. For  \u2208 VSS(U) and  \u2208 VSS(U), M, ) is called the degree of similarity between  and  if it satisfies the following conditions:M1), ) = M, );M2), ) = 1\u21d4 = ;M3), ) = 0\u21d4for\u2009\u2009all\u2009\u2009e \u2208 A\u2229B\u2009(A\u2229B \u2260 \u2205), x \u2208 U, tF(e)(x) = 0, fF(e)(x) = 1, tG(e)(x) = 1, fG(e)(x) = 0, or tF(e)(x) = 1, fF(e)(x) = 0, tG(e)(x) = 0, fG(e)(x) = 1;M4)\u2286\u2286 implies M, ) \u2264 min\u2061, ), M, )). bya \u2265 0, b \u2265 0, and a + b > 0; then, N\u03b8 and N\u03b4 are similarity measures for fuzzy sets in the sense thatN\u03b8 = 0, N\u03b4 = 0 and N\u03b8 = 1, N\u03b4 = 1 whenever A \u2208 F(U);N\u03b8 = N\u03b8, N\u03b4 = N\u03b4 whenever A, B \u2208 F(U);A, B, C \u2208 F(U), N\u03b8 \u2264 min\u2061, N\u03b8), N\u03b4 \u2264 min\u2061, N\u03b4) whenever A\u2286B\u2286C.for all Suppose a and b, one can obtain some typical similarity measures for fuzzy sets  is a similarity measure, where, for any ,  \u2208 VSS(U),M1) is trivial.(M2) If  = , then A = B and tF(e) = tG(e), fF(e) = fG(e) for any e \u2208 A. It follows that N\u03b8(tF(e), tG(e)) = 1, N\u03b8(fF(e), fG(e)) = 1 and hence N\u03b8(F(e), G(e)) = 1. Thus M\u03b8, ) = (1/|A|)\u2211e\u2208AN\u03b8(F(e), G(e)) = |A|/|A| = 1., ) = 1. By N\u03b8(F(e), G(e)) \u2264 1 we have 1 = M\u03b8, ) \u2264 |A\u2229B|/|A \u222a B|. It follows that A = B and N\u03b8(tF(e), tG(e)) = 1 and N\u03b8(fF(e), fG(e)) = 1 for any e \u2208 A. If there exist e \u2208 A, x \u2208 U such that tF(e)(x) \u2260 tG(e)(x), thenN\u03b8(tF(e), tG(e)) < 1. This is a contradiction. Thus tF(e)(x) = tG(e)(x) for each e \u2208 A, x \u2208 U. Similarly, we have fF(e)(x) = fG(e)(x) for each e \u2208 A, x \u2208 U. Consequently, we have  = .Conversely, we assume that M3) Let M, ) = 0. It follows that, for each e \u2208 A\u2229B, N\u03b8(F(e), G(e)) = 0 and hence N\u03b8(tF(e), tG(e)) = 0, N\u03b8(fF(e), fG(e)) = 0. By N\u03b8(tF(e), tG(e)) = 0 we have, for each x \u2208 U, tF(e)(x) = 0, tG(e)(x) = 1 or tF(e)(x) = 1, tG(e)(x) = 0.tF(e)(x) = 0, tG(e)(x) = 1, then fG(e)(x) = 0 by tG(e)(x) + fG(e)(x) \u2264 1. Thus tG(e)(x) = 1 by N\u03b8(fF(e), fG(e)) = 0.If tF(e)(x) = 1, tG(e)(x) = 0, then fF(e)(x) = 0 by tF(e)(x) + fF(e)(x) \u2264 1. Therefore fG(e)(x) = 1 by N\u03b8(fF(e), fG(e)) = 0.If  Let , ,  \u2208 VSS(U) and \u2286\u2286. It follows that A\u2286B\u2286C and F(e)\u2286G(e)\u2286P(e) for each e \u2208 A. Consequently, tF(e)\u2286tG(e)\u2286tP(e), fP(e)\u2286fG(e)\u2286fF(e). Thus we have N\u03b8(tF(e), tP(e)) \u2264 N\u03b8(tF(e), tG(e)), N\u03b8(tF(e), tP(e)) \u2264 N\u03b8(tG(e), tP(e)), N\u03b8(fF(e), fP(e)) \u2264 N\u03b8(fF(e), fG(e)), N\u03b8(fF(e), fP(e)) \u2264 N\u03b8(fG(e), fP(e)), and hence N\u03b8(F(e), P(e)) \u2264 N\u03b8(F(e), G(e)), N\u03b8(F(e), P(e)) \u2264 N\u03b8(G(e), P(e)). Therefore we obtain \u00d7 VSS(U)\u2192 is a similarity measure, where, for any ,  \u2208 VSS(U),It can be proved in the same manner with a = 0, b = 1; then, we havea = 0, b = 2; then, we haveIn , let a =U = {x1, x2}, E = {e1, e2, e3, e4}, A = {e1, e2, e3}, and B = {e1, e2, e4}. Suppose that  and  are vague soft sets over U given byLet By the definition, we haveIt follows thatSimilarly, we haveA, B \u2208 F(U). We consider N\u03b8. For any x \u2208 U, by 1 \u2212 |A(x) \u2212 B(x)| \u2265 0, min\u2061(A(x), B(x)) \u2265 0, a \u2212 a|A(x) \u2212 B(x)| + b \u00b7 min\u2061(A(x), B(x)) = a(1 \u2212 |A(x) \u2212 B(x)|) + b \u00b7 min\u2061(A(x), B(x)) we conclude that \u2211x\u2208U(a \u2212 a | A(x) \u2212 B(x)|+b \u00b7 min\u2061(A(x), B(x))) is increasing with respect to a and b. Thereforea and b. ByN\u03b8 is increasing with respect to a and b. Thus we have the following corollary.Let F, A),  \u2208 VSS(U). Then M1, ) \u2264 M2, ), M1, ) \u2264 M6, );M3, ) \u2264 M4, ) \u2264 M5, );M4, ) \u2264 M6, ).,  \u2208 VSS(U). Then M7, ) \u2264 M8, ).Let  is an entropy, whereF, A) \u2208 VSS(U).H1) We note that M\u03b8 is a similarity measure. For any  \u2208 VSS(U), we have H\u03b8 = 0\u21d4M\u03b8, ) = 0\u21d4for\u2009\u2009all\u2009\u2009e \u2208 A, x \u2208 U, tF(e)(x) = 0, fF(e)(x) = 1, tFc(e)(x) = 1, fFC(e)(x) = 0, or tF(e)(x) = 1, fF(e)(x) = 0, tFc(e)(x) = 0, fFc(e)(x) = 1\u21d4for\u2009\u2009all\u2009\u2009e \u2208 A, x \u2208 U, tF(e)(x) = 0, fF(e)(x) = 1, or tF(e)(x) = 1, fF(e)(x) = 0.(H2) For any  \u2208 VSS(U), we have H\u03b8 = 1\u21d4M\u03b8, ) = 1\u21d4 = \u21d4for\u2009\u2009all\u2009\u2009e \u2208 A, x \u2208 U, tF(e)(x) = tFc(e)(x), fF(e)(x) = fFc(e)(x)\u21d4for\u2009\u2009all\u2009\u2009e \u2208 A, x \u2208 U, tF(e)(x) = fF(e)(x).(H3) is trivial.(H4) Let ,  \u2208 VSS(U), and for\u2009\u2009all\u2009\u2009e \u2208 A, x \u2208 U, tF(e)(x) \u2264 tG(e)(x), fF(e)(x) \u2265 fG(e)(x) if tG(e)(x) \u2264 fG(e)(x), and tF(e)(x) \u2265 tG(e)(x), fF(e)(x) \u2264 fG(e)(x) if tG(e)(x) \u2265 fG(e)(x). We note that N\u03b8(tF(e), tFc(e)) = N\u03b8(fF(e), fFc(e)) = N\u03b8(tF(e), fF(e)) and hence H\u03b8 = (1/|A|)\u2211e\u2208AN\u03b8(tF(e), fF(e))., fF(e)) \u2264 N\u03b8(tG(e), fG(e)) and consequently, H\u03b8 \u2264 H\u03b8.By the definition, for each H\u03b4 is an entropy, whereF, A) \u2208 VSS(U).The proof is similar to that of Mi\u2009\u2009(1 \u2264 i \u2264 8), we can obtain the corresponding entropies Hi\u2009\u2009(1 \u2264 i \u2264 8) as follows:Using similarity measures F, A) given in We consider the vague soft set (Soft set theory was originally proposed as a general mathematical tool for dealing with uncertainties. Wang and Qu  introduc"}
{"text": "Later, we obtain the initial fuzzy soft proximity determined by a family of fuzzy soft proximities. Finally, we investigate relationship between fuzzy soft proximities and proximities.We study the fuzzy soft proximity spaces in Katsaras's sense. First, we show how a fuzzy soft topology is derived from a fuzzy soft proximity. Also, we define the notion of fuzzy soft  In 1999, Molodtsov \u20139.Fuzzy soft set which is a combination of fuzzy set and soft set was introduced by Maji et al. . Roy andX as a map \u03b4 : LX \u00d7 LX \u2192 {0,1} satisfying certain conditions, where L is a completely distributive lattice with an order reversing involution. In connection with a fuzzy topology in [Proximity structure was introduced by Efremovic in 1951 , 26. It ology in , the difology in . In 2005ology in  presenteExtensions of proximity structures to the soft sets and also fuzzy soft sets have been studied by some authors. More recently, Hazra et al.  defined \u03b4 on X induces a fuzzy soft topology \u03c4(\u03b4) on the same set. Also, we define the notion of fuzzy soft \u03b4-neighborhood in a fuzzy soft proximity space and obtain a few results analogous to the ones that hold for \u03b4-neighborhood in proximity spaces. We prove the existences of initial fuzzy soft proximity structures. Based on this fact, we introduce products of fuzzy soft proximity spaces. The relation between a fuzzy soft proximity and a proximity is also investigated.Motivated by their works, we continue investigating the properties of fuzzy soft proximity spaces in Katsaras's sense. We show that each fuzzy soft proximity X be an initial universe, IX be the set of all fuzzy subsets of X and E be the set of all parameters for X.In this section, we recall some basic notions regarding fuzzy soft sets which will be used in the sequel. Throughout this work, let f on the universe X with the set E of parameters is defined by the set of ordered pairsf is a mapping given by f : E \u2192 IX.A fuzzy soft set X is denoted by FS.Throughout this paper, the family of all fuzzy soft sets over f, g \u2208 FS. Then, we have the following.f is called null fuzzy soft set, denoted by f(e) = 0X for every e \u2208 E.The fuzzy soft set f(e) = 1X, for all e \u2208 E, then f is called absolute fuzzy soft set, denoted by If f is a fuzzy soft subset of g if f(e) \u2264 g(e) for each e \u2208 E. It is denoted by f\u2291g.f and g are equal if f\u2291g and g\u2291f. It is denoted by f = g.f is denoted by fc, where fc : E \u2192 IX is a mapping defined by fc(e) = 1X \u2212 f(e) for all e \u2208 E. Clearly, (fc)c = f.The complement of f and g is a fuzzy soft set h defined by h(e) = f(e)\u2228g(e) for all e \u2208 E. h is denoted by f\u2294g.The union of f and g is a fuzzy soft set h defined by h(e) = f(e)\u2227g(e) for all e \u2208 E. h is denoted by f\u2293g.The intersection of Let J be an arbitrary index set and let {fi}i\u2208J be a family of fuzzy soft sets over X. Then,h defined by h(e) = \u2228i\u2208Jfi(e) for every e \u2208 E and this fuzzy soft set is denoted by \u2294i\u2208Jfi;the union of these fuzzy soft sets is the fuzzy soft set h defined by h(e) = \u2227i\u2208Jfi(e) for every e \u2208 E and this fuzzy soft set is denoted by \u2293i\u2208Jfi.the intersection of these fuzzy soft sets is the fuzzy soft set Let J be an index set and f, g, fi, gi \u2208 FS, for all i \u2208 J. Then, the following statements are satisfied:f\u2293(\u2294i\u2208J\u2061gi) = \u2294i\u2208J(f\u2293gi);f\u2294(\u2293i\u2208Jgi) = \u2293i\u2208J(f\u2294gi);i\u2208Jfi)c = \u2294i\u2208Jfic;(\u2293i\u2208Jfi)c = \u2293i\u2208Jfic; is a fuzzy point in X (x) = \u03b1 \u2208 (x\u2032) = 0 for all x\u2032 \u2208 X \u2212 {x}) and f(e\u2032) = 0X for every e\u2032 \u2208 E\u2216{e}. It will be denoted by ex\u03b1.A fuzzy soft set ex\u03b1 is said to belong to a fuzzy soft set f, denoted by \u03b1 \u2264 f(e)(x).The fuzzy soft point FS and FS be the families of all fuzzy soft sets over X and Y, respectively. Let \u03c6 : X \u2192 Y and \u03c8 : E \u2192 K be two mappings. Then, the mapping \u03c6\u03c8 is called a fuzzy soft mapping from X to Y, denoted by \u03c6\u03c8 : FS \u2192 FS.(1)f \u2208 FS. Then \u03c6\u03c8(f) is the fuzzy soft set over Y defined as follows:Let \u2009k \u2208 K and all y \u2208 Y.for all \u2009\u03c6\u03c8(f) is called an image of a fuzzy soft set f.(2)g \u2208 FS. Then \u03c6\u03c8\u22121(g) is the soft set over X defined as follows:Let \u2009e \u2208 E and all x \u2208 X.for all \u2009\u03c6\u03c8\u22121(g) is called a preimage of a fuzzy soft set g.The fuzzy soft mapping \u03c6\u03c8 is called injective, if \u03c6 and \u03c8 are injective. The fuzzy soft mapping \u03c6\u03c8 is called surjective, if \u03c6 and \u03c8 are surjective.Let fi \u2208 FS and gi \u2208 FS for all i \u2208 J, where J is an index set. Then, for a fuzzy soft mapping \u03c6\u03c8 : FS \u2192 FS, the following conditions are satisfied:f1\u2291f2, then \u03c6\u03c8(f1)\u2291\u03c6\u03c8(f2);if g1\u2291g2, then \u03c6\u03c8\u22121(g1)\u2291\u03c6\u03c8\u22121(g2);if \u03c6\u03c8(\u2294i\u2208Jfi) = \u2294i\u2208J\u03c6\u03c8(fi);\u03c6\u03c8(\u2293i\u2208Jfi)\u2291\u2293i\u2208J\u03c6\u03c8(fi);\u03c6\u03c8\u22121(\u2294i\u2208Jgi) = \u2294i\u2208J\u03c6\u03c8\u22121(gi);\u03c6\u03c8\u22121(\u2293i\u2208Jgi) = \u2293i\u2208J\u03c6\u03c8\u22121(gi);Let f, fi \u2208 FS for all i \u2208 J, where J is an index set, and let g \u2208 FS. Then, for a fuzzy soft mapping \u03c6\u03c8 : FS \u2192 FS, the following conditions are satisfied:f\u2291\u03c6\u03c8\u22121(\u03c6\u03c8(f)), and the equality holds if \u03c6\u03c8 is injective;\u03c6\u03c8(\u03c6\u03c8\u22121(g))\u2291g, and the equality holds if \u03c6\u03c8 is surjective;\u03c6\u03c8(\u2293i\u2208Jfi) = \u2293i\u2208J\u03c6\u03c8(fi) if \u03c6\u03c8 is injective;\u03c6\u03c8 is surjective.Let f \u2208 FS and g \u2208 FS. The fuzzy soft product f \u00d7 g is defined by the fuzzy soft set h where h : E \u00d7 K \u2192 IX\u00d7Y and h = f(e) \u00d7 g(k) for all  \u2208 E \u00d7 K.Let f \u2208 FS and g \u2208 FS and let pX : X \u00d7 Y \u2192 X, qE : E \u00d7 K \u2192 E and pY : X \u00d7 Y \u2192 Y, qK : E \u00d7 K \u2192 K be the projection mappings in classical meaning. The fuzzy soft mappings (pX)qE and (pY)qK are called fuzzy soft projection mappings from X \u00d7 Y to X and from X \u00d7 Y to Y, respectively, where (pX)qE(f \u00d7 g) = f and (pY)qK(f \u00d7 g) = g.Let X is a fuzzy soft set. Also, every fuzzy soft set is a parameterized collection of fuzzy subsets in some universe.Every parameterized collection of fuzzy subsets in \u03bc\u03b1 : \u03b1 \u2208 \u0394} of fuzzy subsets in X. Then, f : \u0394 \u2192 IX defined by f(\u03b1) = \u03bc\u03b1 is a fuzzy soft set over X.Consider any parameterized collection {\u03c4 be the collection of fuzzy soft sets over X; then \u03c4 is said to be a fuzzy soft topology on X iffst1) is called a fuzzy soft topological space. The members of \u03c4 are called fuzzy soft open sets in X. A fuzzy soft set f over X is called a fuzzy soft closed in X if fc \u2208 \u03c4.. Let us consider the following fuzzy soft sets on X with the set E of parameters:X.Let X, \u03c4) be a fuzzy soft topological space and f \u2208 FS. The fuzzy soft interior of f is the fuzzy soft set fo = \u2294{g : g\u2009\u2009is\u2009\u2009a\u2009\u2009fuzzy\u2009\u2009soft\u2009\u2009open\u2009\u2009set\u2009\u2009and\u2009\u2009g\u2291f}.Let (fst2) for fuzzy soft open sets, fo is fuzzy soft open. It is the largest fuzzy soft open set contained in f.By property  be a fuzzy soft topological space and f \u2208 FS. The fuzzy soft closure of f is the fuzzy soft set Let (X which contains f.Clearly f on X another fuzzy soft set fo1)(fo2)(fo3)(fo4), the fuzzy soft set f in the fuzzy soft topological space .Then, the familyThis operator is called the fuzzy soft closure operator.X, \u03c41) and  be two fuzzy soft topological spaces. A fuzzy soft mapping \u03c6\u03c8 : \u2192 is called fuzzy soft continuous if \u03c6\u03c8\u22121(g) \u2208 \u03c41 for every g \u2208 \u03c42.Let  be a fuzzy soft topological space, where \u03c4 = {f\u03b1 : \u03b1 \u2208 \u0394}. Then, the collection \u03c4e = {f\u03b1(e)\u2223\u03b1 \u2208 \u0394} for every e \u2208 E defines a fuzzy topology on X.Let  : e \u2208 E} be a parameterized family of fuzzy topological spaces. Let us define a fuzzy soft topological space  as the following: let \u03c4 be the collection of all the mappings f, where f : E \u2192 IX such that f(e) \u2208 \u03c4e for each e \u2208 E. Then, \u03c4 is a fuzzy soft topology on X. Indeed,fst1) \u2208 \u03c4e for each e \u2208 E; therefore, \u2294i\u2208Jfi is a mapping \u2294i\u2208Jfi : E \u2192 IX such that (\u2294i\u2208Jfi)(e) = \u2228i\u2208Jfi(e) \u2208 \u03c4e for each e \u2208 E; consequently, \u2294i\u2208Jfi \u2208 \u03c4;let {fst3), g(e) \u2208 \u03c4e for each e \u2208 E and hence f\u2293g is a mapping f\u2293g : E \u2192 IX such that (f\u2293g)(e) = f(e)\u2227g(e) \u2208 \u03c4e for each e \u2208 E; thus, f\u2293g \u2208 \u03c4.let Let {(p2)(A\u03b4(B \u222a C) if and only if A\u03b4B or A\u03b4C;p5):(p1)\u2205\u03b4\u03b4.where X, \u03b4) is called a proximity space; two subsets A and B of the set X are close with respect to \u03b4 if A\u03b4B; otherwise, they are remote with respect to \u03b4.The pair (fp2)(\u03bc\u03b4(\u03c1\u2228\u03c3) if and only if \u03bc\u03b4\u03c1 or \u03bc\u03b4\u03c3;fp5) comprising a set X and a fuzzy proximity \u03b4 on the set X.A fuzzy proximity space is a pair \u00d7FS is called a Katsaras -soft fuzzy proximity on a set X, where E and K are arbitrary nonempty sets viewed on the sets of parameters, if, for any f, g, h \u2208 FS and k \u2208 K, the following conditions are satisfied:sfp1)(sfp2)(if sfp3)(f\u03b4k(g\u2294h) if and only if f\u03b4kg or f\u03b4kh;sfp5) such that if A mapping X, \u03b4) is called a Katsaras -soft fuzzy proximity space, where, for every k \u2208 K, \u03b4k \u2282 FS \u00d7 FS is a relation on FS.The pair  \u00d7 FS is called a fuzzy soft proximity on X if \u03b4 satisfies the following conditions:fsp1)(fsp2)(f\u03b4(g\u2294h) if and only if f\u03b4g or f\u03b4h;fsp5) such that if A binary relation X, \u03b4) is called a fuzzy soft proximity space.The pair  is a fuzzy soft proximity space, then it satisfies the following properties:f\u03b4g and k\u2292f, h\u2292g, then k\u03b4h;if f\u03b4f for each If  be a fuzzy soft proximity space. For every f \u2208 FS, we defineLet  be a fuzzy soft proximity space. Then, the mapping fo1)\u2013(fo4). Therefore, the collectionX.Let (fo1)\u2013(fo4).We will show that the mapping fo1) Suppose that e \u2208 E and x \u2208 X. Take any g \u2208 FS such that fsp2), g(e)(x) = 0 or f(e)(x) = 0. In both cases, we obtain g(e)(x) \u2264 1 \u2212 f(e)(x). Therefore, (fo2) It is enough to show that g\u03b4f if and only if fsp5), there is an h \u2208 FS such that e \u2208 E and an x \u2208 X such that a, where exa1\u2212 \u2208 FS. Since 1 \u2212 a \u2264 1 \u2212 h(e)(x), we have exa1\u2212\u03b4f, since, otherwise, we would have exa1\u2212\u03b4f and (fo3) It is easy to verify that e \u2208 E and an x \u2208 X such that \u03f5 > 0 satisfyingh \u2208 FS such that h(e)(x) < a \u2212 \u03f5. By the inequality k \u2208 FS such that h(e)(x) \u2212 \u03f5/2 < k(e)(x). Now, since fsp2), it follows that h(e)(x) \u2212 \u03f5/2 < (k\u2293h)(e)(x). Thus,(fo4) Because of  be a fuzzy soft proximity space. For f, g \u2208 IX, the fuzzy soft set g is said to be a fuzzy soft \u03b4-neighborhood of f if f\u22d0g.Let  be a fuzzy soft proximity space. Then the relation \u22d0 satisfies the following properties:fspn1)(fspn2)(f\u22d0g implies fspn3)(f\u22d0g implies fspn4)(f\u22d0(g\u2293h) if and only if f\u22d0g and f\u22d0h;fspn5) such that f\u22d0h\u22d0g.Let (fspn1) is obvious.(fspn2) If f\u22d0g, then fsp3), (fspn3) Let f\u22d0g. Then, from (fsp2), it follows that (fspn4) Consider f\u22d0g\u2293hf\u22d0g and f\u22d0h.(fspn5) If f\u2291g and (fspn6) Consider that f\u22d0g implies fsp5), there exists an h \u2208 FS such that f\u22d0h\u22d0g. be a fuzzy soft proximity space and f, g \u2208 FS. Then, the following statements are satisfied:f\u22d0g if and only if f\u22d0g, then there is a k \u2208 \u03c4(\u03b4) such that f\u2291k\u2291g;if h, k such that f\u22d0h, g\u22d0k, and if Let (g\u03b4f if and only if (i) It is clear by the fact that f\u22d0g. Then, k \u2208 \u03c4(\u03b4) and f\u2291k\u2291g.(ii) Let fsp5), there is a fuzzy soft set k such that h such that h and k such that f\u22d0h, g\u22d0k, and (iii) If FS satisfying (fspn1)\u2212(fspn6). Then, \u03b4 is a fuzzy soft proximity on X defined as follows:g is a fuzzy soft \u03b4-neighbourhood of f if and only if f\u22d0g.Let \u22d0 be a relation on fsp1)\u2013(fsp5).fsp1). By (fspn1), we have Let fsp2)(f\u22d0gc and from (fspn3) it follows thatLet fsp3)(f\u22d0gc. By (fspn2), g\u22d0fc and hence If fsp4)(Consider fsp5)(fspn6), there is a fuzzy soft set h such that Let We first need to verify axioms  is a fuzzy soft proximity space and f \u2208 FS, thenIf  we obtain e \u2208 E and an x \u2208 X such that g(e)(x) : f\u22d0g} = a. Then there exists an \u03f5 > 0 such thatk such that k(e)(x) < a \u2212 \u03f5. Because Let us take a fuzzy soft set X, \u03b41) and  be two fuzzy soft proximity spaces. A fuzzy soft mapping \u03c6\u03c8 : \u2192 is a fuzzy soft proximity mapping if it satisfiesf, g \u2208 FS.Let  and  be two fuzzy soft proximity spaces. A fuzzy soft mapping \u03c6\u03c8 : \u2192 is a fuzzy soft proximity mapping if and only ifh, k \u2208 FS.Let \u2192 is fuzzy soft continuous with respect to \u03c4(\u03b41) and \u03c4(\u03b42).A fuzzy soft proximity mapping f \u2208 \u03c4(\u03b42). Now let us take any h \u2208 FS such that \u03c6\u03c8 is a fuzzy soft proximity mapping, we obtain fsp2), it follows that e \u2208 E and every x \u2208 X, we have\u03c6\u03c8\u22121(f) \u2208 \u03c4(\u03b41).Let \u03b41 and \u03b42 are two fuzzy soft proximities on X, we define\u03b42 is finer than \u03b41, or \u03b41 is coarser than \u03b42.If We prove the existences of initial fuzzy soft proximity structure. Based on this fact, we define the product of fuzzy soft proximity spaces.X be a set and { : \u03b1 \u2208 \u0394} a family of fuzzy soft proximity spaces, and, for each \u03b1 \u2208 \u0394, let (\u03c6\u03c8)\u03b1 : FS\u2192 be a fuzzy soft mapping. The initial structure \u03b4 is the coarsest fuzzy soft proximity on X for which all mappings (\u03c6\u03c8)\u03b1 : \u2192\u2009\u2009(\u03b1 \u2208 \u0394) are fuzzy soft proximity mapping.Let X be a set { : \u03b1 \u2208 \u0394} a family of fuzzy soft proximity spaces, and, for each \u03b1 \u2208 \u0394, let (\u03c6\u03c8)\u03b1 : FS\u2192 be a fuzzy soft mapping. For any f, g \u2208 FS, define f\u03b4g if and only if, for every finite families {fi : i = 1,\u2026, n} and {gj : j = 1,\u2026, m}, where f = \u2294i=1nfi and g = \u2294j=1mgj, there exist an fi and a gj such that\u03b4 is the coarsest fuzzy soft proximity on X for which all mappings (\u03c6\u03c8)\u03b1 : \u2192\u2009\u2009(\u03b1 \u2208 \u0394) are fuzzy soft proximity mapping.Let \u03b4 is a fuzzy soft proximity on X.fsp1)(is obvious.fsp2). Conversely, assume that f = \u2294i=1nfi and g = \u2294j=1mgj of f and g, respectively, such that \u03b1 = sij \u2208 \u0394, where i = 1,\u2026, n and j = 1,\u2026, m. In the same way, there are finite covers f = \u2294p=1qkp and h = \u2294j=m+1m+lgj of f and h, respectively, such that \u03b1 = tpj \u2208 \u0394, where p = 1,\u2026, q and j = m + 1,\u2026, m + l. Now, f = \u2294{fi\u2293kp : i = 1,\u2026, n; p = 1,\u2026, q} and g\u2294h = \u2294{gj : j = 1,\u2026, m + l} are finite covers of f and g\u2294h, respectively. Hence, from the fact that \u03b1 = sij or \u03b1 = tpj, it follows that It is easy to verify that if fsp5) such that f\u03b4h or h \u2208 FS. The validity of (fsp5) will follow from the fact that \u03a9 is empty. Suppose, on the contrary, that  \u2208 \u03a9. Then, (\u03c6\u03c8)\u03b1(f)\u03b4\u03b1(\u03c6\u03c8)\u03b1(g) for each \u03b1 \u2208 \u0394. Indeed, let h \u2208 FS and k = (\u03c6\u03c8)\u03b1\u22121(h). If f\u03b4k, then (\u03c6\u03c8)\u03b1(f)\u03b4\u03b1(\u03c6\u03c8)\u03b1(k). Because (\u03c6\u03c8)\u03b1(k)\u2291h, we have (\u03c6\u03c8)\u03b1(f)\u03b4\u03b1h. Similarly, if \u03b4\u03b1 is a fuzzy soft proximity on X\u03b1, we obtain (\u03c6\u03c8)\u03b1(f)\u03b4\u03b1(\u03c6\u03c8)\u03b1(g). Also, we observe that for each  \u2208 \u03a9 there are positive integers n, m and covers f = \u2294i=1nfi and g = \u2294j=1mgj such that, for every pair \u2208{1,\u2026, n}\u00d7{1,\u2026, m}, there exists an \u03b1 \u2208 \u0394 satisfying l = n + m. It easy to see that l > 2. Then, for each  \u2208 \u03a9, let us choose such an integer l. But l is not uniquely determined by . Let \u03ba be the set of all integers corresponding to members of \u03a9 and let l be the smallest member of \u03ba. Take a  \u2208 \u03a9 such that l is the integer corresponding to it. Then, there are covers f = \u2294i=1nfi and g = \u2294j=1mgj such that l = n + m and for every pair \u2208{1,\u2026, n}\u00d7{1,\u2026, m} and there exists an \u03b1 \u2208 \u0394 satisfying n, m is greater than 1. Consider n > 1 and let f\u2032 = f1\u2294\u22ef\u2294fn\u22121. In this case, one of the following conditions should be true:Let us define the set (i)h \u2208 FS, either f\u2032\u03b4h or for every (ii)h \u2208 FS, either fn\u03b4h or for every We first prove that h1, h2 \u2208 FS such that h = h1\u2293h2, we obtain f, g) \u2208 \u03a9.In fact, suppose that neither (i) nor (ii) holds. Then, there are f\u2032\u2291f and f\u2032, g) \u2208 \u03a9. But this is now a contradiction since (n \u2212 1) + m = l \u2212 1 \u2208 \u03ba, contrary to the choice of l. If (ii) holds, we get a contradiction in a similar way. Therefore, the set \u03a9 is empty. Thus, \u03b4 is a fuzzy soft proximity on X.Suppose that (i) holds. Because \u03c6\u03c8)\u03b1 : \u2192 are fuzzy soft proximity mapping. Let \u03b4* be another fuzzy soft proximity on X making each of the mappings (\u03c6\u03c8)\u03b1 : \u2192 fuzzy soft proximity mapping. We will show that \u03b4 < \u03b4*, which will complete the proof. Let f\u03b4*g and consider any covers f = \u2294i=1nfi and g = \u2294j=1mgj of f and g, respectively. Since f = (f1\u2294\u22ef\u2294fn)\u03b4*g, by (fsp4), there is an i \u2208 {1,\u2026, n} such that fi\u03b4g. In the same way, since fi\u03b4*g = (g1\u2294\u22ef\u2294gm), by (fsp4), there is a j \u2208 {1,\u2026, m} such that fi\u03b4gj. From the fact that all mappings (\u03c6\u03c8)\u03b1 : \u2192 are fuzzy soft proximity mapping, it follows that (\u03c6\u03c8)\u03b1(fi)\u03b4\u03b1(\u03c6\u03c8)\u03b1(gj) for each \u03b1 \u2208 \u0394. Thus, we get f\u03b4g.It is easy to see that all mappings \u2192 is a fuzzy soft proximity mapping if and only if (\u03c6\u03c8)\u03b1\u2218\u03c6\u03c8 : \u2192 is a fuzzy soft proximity mapping for every \u03b1 \u2208 \u0394.A fuzzy soft mapping \u03c6\u03c8)\u03b1\u2218\u03c6\u03c8 is a fuzzy soft proximity mapping for every \u03b1 \u2208 \u0394. Let f\u03b4*g and let \u03c6\u03c8(f) = \u2294i=1nfi and \u03c6\u03c8(g) = \u2294j=1mgj. Then, we havef\u03b4*g, by (fsp4), there exist i, j such that \u03c6\u03c8\u22121(fi)\u03b4*\u03c6\u03c8\u22121(gj). Because\u03c6\u03c8)\u03b1(fi)\u03b4\u03b1(\u03c6\u03c8)\u03b1(gj) for every \u03b1 \u2208 \u0394. This shows that \u03c6\u03c8(f)\u03b4\u03c6\u03c8(g).The necessity is easy. We prove the sufficiency. Suppose that  : \u03b1 \u2208 \u0394} be a family of fuzzy soft proximity spaces and let X = \u220f\u03b1\u2208\u0394X\u03b1 and E = \u220f\u03b1\u2208\u0394E\u03b1 be product sets. An initial fuzzy soft proximity structure \u03b4 = \u220f\u03b1\u2208\u0394\u03b4\u03b1 on X with respect to all the fuzzy soft projection mappings (pX\u03b1)qE\u03b1, where pX\u03b1 : X \u2192 X\u03b1 and qE\u03b1 : E \u2192 E\u03b1, is called the product fuzzy soft proximity structure.Let { is said to be a product fuzzy soft proximity space. : \u03b1 \u2208 \u0394} be a family of fuzzy soft proximity spaces. Let X = \u220f\u03b1\u2208\u0394X\u03b1 and E = \u220f\u03b1\u2208\u0394E\u03b1 be sets and for each \u03b1 \u2208 \u0394 let (pX\u03b1)qE\u03b1 be a fuzzy soft mapping. For any f, g \u2208 FS, define f\u03b4g if and only if, for every finite families {fi : i = 1,\u2026, n} and {gj : j = 1,\u2026, m}, where f = \u2294i=1nfi and g = \u2294j=1mgj, there exist an fi and a gj such that (pX\u03b1)qE\u03b1(fi)\u03b4\u03b1(pX\u03b1)qE\u03b1(gj) for each \u03b1 \u2208 \u0394. Then,\u03b4 = \u220f\u03b1\u2208\u0394\u03b4\u03b1 is the coarsest fuzzy soft proximity on X for which all mappings (pX\u03b1)qE\u03b1\u2009\u2009(\u03b1 \u2208 \u0394) are fuzzy soft proximity mapping;\u03c6\u03c8 : \u2192 is a fuzzy soft proximity mapping if and only if (pX\u03b1)qE\u03b1\u2218\u03c6\u03c8 : \u2192 is a fuzzy soft proximity mapping for every \u03b1 \u2208 \u0394.a fuzzy soft mapping Consider { be a proximity space. By letting, for f, g \u2208 FS,Let \u2013(fsp5).fsp1)(From fsp2)(It is clear because fsp4)(A and B of X such that C and D of X such that It is easy to see that if fsp5)(A and B of X such that p5), there is a C\u2286X such that If We will show that X, \u03b4*) be a fuzzy soft proximity space.\u03b4 on X such that \u03b4* = \u03b4i.There is a proximity relation A and B of X such that If A\u03b4B if and only if X.The relation Let (a) and (b) are equivalent and they imply (c).Then, (The implication (a)\u21d2(b) is obvious.\u03b4 satisfies (p5). Let A, B of X. Because f \u2208 FS such that C, D, G, H of X such that \u03b4 has the axiom (p5).To prove that (b)\u21d2(c), since the other axioms are readily verified, it is enough to show that \u03b4 on the power set of X as follows:\u03b4 is a proximity on X. To complete the proof, it will suffice to prove that \u03b4* = \u03b4i, that is, that f\u03b4ig if and only if f\u03b4*g. Assume that A and B of X such that \u03b4i, we obtain subsets A, B of X such that In order to prove (b)\u21d2(a), let us define the binary relation \u03b4-neighborhood. We believe that these results will help the researchers to advance and promote the further study on fuzzy soft topology to carry out a general framework for their applications in practical life.Each proximity space determines in a natural way a topological space with beneficial properties. Also, this theory possesses deep results, rich machinery, and tools. With the development of topology, the theory of proximity makes a great progress. Hence, the concept of proximity has been studied by many authors in both the fuzzy setting and the soft setting. In the present work, we mainly establish some properties of fuzzy soft proximity spaces in Katsaras's sense. We have shown that each fuzzy soft proximity determines a fuzzy soft topology by using fuzzy soft closure operator. Also, we present an alternative description of the concept of fuzzy soft proximity, which is called fuzzy soft"}
{"text": "As an important tool for data analysis and knowledge processing, formal concept analysis (FCA) has been applied to many fields. In this paper, we introduce a new method to find all formal concepts based on formal contexts. The amount of intents calculation is reduced by the method. And the corresponding algorithm of our approach is proposed. The main theorems and the corresponding algorithm are examined by examples, respectively. At last, several real-life databases are analyzed to demonstrate the application of the proposed approach. Experimental results show that the proposed approach is simple and effective. Formal concept analysis (FCA), proposed by Wille in 1982 [Most of the researches on FCA concentrate on the following topics: construction and pruning algorithm of concept lattices , 4; relan objects, then we should calculate 2n times to obtain all intents. Obviously, the computational costing is very huge. To solve this problem, we give a new method to obtain all intents. And correspondingly, the formal concepts are determined.Formal concepts are very important notions of FCA. And intents and extents are also very important elements of formal concepts. The set of intents (extents) is isomorphic to the corresponding concept lattice under the order relationship \u201c\u2287\u201d (\u201c\u2286\u201d). So, if the set of intents is determined, the corresponding concept lattice is identified. Thus, obtaining all intents or extents is very important. Generally, the basic way to obtain all intents or extents is via their definitions. If there are This paper is organized as follows. In In this section, we recall some basic notions and properties in FCA.G, M, I) consists of two sets G and M and a relation I between G and M. The elements of G are called the objects and the elements of M are called the attributes of the context. In order to express that an object g is in a relation I with an attribute m, we write gIm or  \u2208 I and read it as \u201cthe object g has the attribute m.\u201dA formal context , Ganter and Wille [A\u2286G and B\u2286M byWith respect to a formal context  be a formal context. \u2200A1, A2, A\u2286G, \u2200B1, B2, B\u2286M; the following properties hold.A1\u2286A2\u21d2A2*\u2286A1*,\u2009 \u2009B1\u2286B2\u21d2B2\u2032\u2286B1\u2032.A\u2286A\u2217\u2032,\u2009\u2009B\u2286B\u2032\u2217.A* = A\u2217\u2032\u2217, B\u2032 = B\u2032\u2217\u2032.A\u2286B\u2032\u21d4B\u2286A*.A1\u222aA2)* = A1*\u2229A2*, (B1\u222aB2)\u2032 = B1\u2032\u2229B2\u2032.(A1\u2229A2)*\u2287A1* \u222a A2*, (B1\u2229B2)\u2032\u2287B1\u2032 \u222a B2\u2032. is called a formal concept, where A is called the extent of the formal concept and B is called the intent of the formal concept. For any g \u2208 G, a pair  is a formal concept and is called an object concept. Similarly, for any m \u2208 M, a pair  is a formal concept and is called an attribute concept. The family of all formal concepts of  forms a complete lattice that is called the concept lattice and is denoted by L. For any ,  \u2208 L, the partial order is defined byA1, B1) and  are defined byIf respectively.G, M, I) be a formal context. |G| < \u221e and |M| < \u221e. Denote LM = {Y\u2223Y\u2286M and Y is an intent of }.Let . G = {1,2, 3,4, 5,6} is an object set and M = {a, b, c, d, e} is an attribute set. The corresponding concept lattice L is shown in G, M, and \u2205.n objects, then we should calculate 2n times to get all intents. Obviously, the amount of computation is very large. So our paper presents a new approach to solve the problem. In this section, we give this new method and some theorems to explain its rationality and validity.The basic way to obtain all intents or extents is via their definitions. If there are Before giving the method, we firstly propose a related definition.G, M, I) be a formal context. |G | <\u221e, |M | <\u221e. Denote \u03b1n = {X\u2223X\u2286G and |X | = n}, \u03b2n = {Y\u2223Y\u2286M, Y = X* and X \u2208 \u03b1n}, n = 1,2,\u2026, |G|, where |\u00b7| presents the cardinal of a set.Let  such that \u03b2n+1\u2286\u03b2n, then \u03b2n+2\u2286\u03b2n+1.If there exists Y \u2208 \u03b2n+2. By Xn+2 \u2208 \u03b1n+2 such that Y = Xn+2*.Suppose Xn+2 \u2260 \u2205, there exists a \u2208 Xn+2 such that Y = Xn+2* = (Xn+2\u2216{a}\u222a{a})* = (Xn+2\u2216{a})*\u2229{a}*. Noting that |Xn+2\u2216{a}| = n + 1, we have Xn+2\u2216{a} \u2208 \u03b1n+1. Moreover, from \u03b2n+1\u2286\u03b2n, we know that there exists Xn \u2208 \u03b1n satisfying (Xn+2\u2216{a})* = Xn*; that is, Y = Xn*\u2229{a}* = (Xn\u222a{a})*.Since Y \u2208 \u03b2n+1.Now, we discuss two cases to prove a \u2209 Xn. In this case, |Xn \u222a {a}| = n + 1. Thus, Y = Xn*\u2229{a}* = (Xn\u222a{a})* \u2208 \u03b2n+1.The one case is that a \u2208 Xn. In this case, Y = Xn*\u2229{a}* = Xn*. Because Xn+2\u2216Xn \u2260 \u2205, there exists b \u2208 Xn+2\u2216Xn such that Xn\u2286Xn \u222a {b}\u2286Xn \u222a Xn+2. Therefore, we have Xn*\u2287(Xn\u222a{b})*\u2287(Xn\u222aXn+2)*; that is, Xn*\u2287(Xn\u222a{b})*\u2287Xn*\u2229Xn+2*. That means, Y\u2287(Xn\u222a{b})*\u2287Y. Therefore, we have Y = (Xn\u222a{b})*. Thereby, we can obtain Y \u2208 \u03b2n+1.The other one is that \u03b2n+2\u2286\u03b2n+1 holds.To sum up the above two cases, n  such that \u03b2n+1\u2286\u03b2n, then for any m, m \u2265 n, we have \u03b2m+1\u2286\u03b2m\u2286\u03b2n.If there exists \u03b2n+1\u2286\u03b2n, we have \u03b2n+2\u2286\u03b2n+1 by \u03b2m+1\u2286\u03b2m\u2286\u03b2m\u22121\u2286\u03b2m\u22122\u2286\u22ef\u2286\u03b2n+1\u2286\u03b2n.According to the condition Y \u2208 \u03b2k, 1 \u2264 k \u2264 |G|, 1 < m < k; if Y \u2209 \u03b2m, then we have Y \u2209 \u03b2m\u22121.Suppose We will adopt the proof by contradiction.Y \u2208 \u03b2m\u22121; there is Xm\u22121 \u2208 \u03b1m\u22121 satisfying Y = Xm\u22121*; according to the condition and Xk \u2208 \u03b1k satisfying Y = Xk*.Suppose Xk\u2216Xm\u22121 \u2260 \u2205, there exists a \u2208 Xk\u2216Xm\u22121 such that |Xm\u22121 \u222a {a}| = m; that is, Xm\u22121 \u222a {a} \u2208 \u03b1m. Obviously, Xm\u22121\u2286Xm\u22121 \u222a {a}\u2286Xm\u22121 \u222a Xk, and thus, Xm\u22121*\u2287(Xm\u22121\u222a{a})*\u2287(Xm\u22121\u222aXn)*. That is, Xm\u22121*\u2287(Xm\u22121\u222a{a})*\u2287Xm\u22121*\u2229Xk*. That means Y\u2287(Xm\u22121\u222a{a})*\u2287Y. Therefore, we have Y = (Xm\u22121\u222a{a})*. From Y \u2208 \u03b2m. It is a contradiction with Y \u2209 \u03b2m.Because Y \u2209 \u03b2m\u22121 holds.Therefore, Y \u2208 \u03b2k, 1 \u2264 k \u2264 |G|, and Y \u2209 \u03b2k\u22121,\u03b2k\u2286\u03b2k\u22121; then for any m, 1 \u2264 m < k, Y \u2209 \u03b2m.Suppose Y \u2208 \u03b2k, and Y \u2209 \u03b2k\u22121, we have Y \u2209 \u03b2k\u22122 by Y \u2209 \u03b2m.Because G, M, I) is canonical; then \u03b2n+1\u2286\u03b2n if and only if\u2009\u2009\u222ai=1n\u03b2i \u222a {M} = LM.Suppose . If Y \u2260 {M}, then there exists n0 such that Y \u2208 \u03b2n0. By Xn0 \u2208 \u03b1n0 such that Y = Xn0*. Obviously, Y \u2208 LM. Since Y is arbitrary and M \u2208 LM, we have \u222ai=1n\u2009\u03b2i \u222a {M}\u2286LM.For any Y \u2208 LM, by Y = (Y*)*, Y*\u2286G. Without loss of the generality, we can suppose |Y* | = m. If m \u2264 n, then Y \u2208 \u03b2m\u2286\u222ai=1n\u03b2i by m > n, from above Y \u2208 \u03b2m\u2286\u03b2n\u2286\u222ai=1n\u03b2i. Since M \u2208 \u222ai=1n\u03b2i \u222a {M} and Y is arbitrary, we obtain \u222ai=1n\u03b2i \u222a {M}\u2287LM.For any i=1n\u03b2i \u222a {M} = LM.Therefore, \u222aSufficiency. We assume \u03b2n+1\u2288\u03b2n and prove \u222ai=1n\u03b2i \u222a {M} \u2260 LM.\u03b2n+1\u2288\u03b2n, then there is Y \u2208 \u03b2n+1, but Y \u2209 \u03b2n. From m, 1 \u2264 m < n, we have Y \u2209 \u03b2m. So, Y \u2209 \u222ai=1n\u03b2i.  is canonical and Y \u2208 \u03b2n+1, so Y \u2260 M. Thus, Y \u2209 \u222ai=1n\u03b2i \u222a {M}.If Y \u2208 \u03b2n+1, by Xn+1 \u2208 \u03b1n+1, such that Y = Xn+1*. By the definition of LM, Y \u2208 LM.On the other hand, since Y such that Y \u2209 \u222ai=1n\u03b2i \u222a {M}, but Y \u2208 LM. Therefor, \u222ain\u03b2i \u222a {M} \u2260 LM.That means there exists one set LM. Now, the process to calculate all intents is summarized as follows. Step\u2009\u20091. Calculate \u03b11 and \u03b21 by  Step\u2009\u20092. Calculate \u03b12 and \u03b22 by \u03b22\u2286\u03b21, then the set of intents is \u03b21 \u222a {M}. Otherwise, we proceed Step\u2009\u20093. Step\u2009\u20093. Calculate \u03b13 and \u03b23 by \u03b23\u2286\u03b22, then the set of intents is \u03b21 \u222a \u03b22 \u222a {M}. Otherwise, calculate \u03b2i (1 \u2264 i \u2264 |G|) continuously. The computation needs to stop at \u03b2n+1 which exactly meets \u03b2n+1\u2286\u03b2n. Meanwhile, the set of intents is \u222ai=1n\u03b2i \u222a {M}.\u03b2i, 1 \u2264 i \u2264 |G| and the computation needs only to stop at \u03b2n+1 which exactly meets \u03b2n+1\u2286\u03b2n. Now all the intents have been found and there is no extra computing.The merit of our method is that we do not need to calculate all In the following, we use an example in the literature  to examia (water) from the original formal context. The objects are living beings mentioned in the film and are denoted by G = {1,2, 3,4, 5,6, 7,8}, where 1 is leech, 2 is bream, 3 is frog, 4 is dog, 5 is spike-weed, 6 is reed, 7 is bean, and 8 is maize. And the attributes in M = {b, c, d, e, f, g, h, i} are the properties which the film emphasizes: b: lives in water, c: lives on land, d: needs chlorophyll to produce food, e: two seed leaves, f: one seed leaf, g: can move around, h: has limbs, and i: suckles its offspring.The formal context in L of this formal context is shown in The corresponding concept lattice \u03b1i and \u03b2i  firstly:We calculate \u03b24, \u03b25,\u2026, \u03b28 and find \u03b22\u2287\u03b23\u2287\u03b24\u2287\u03b25\u2287\u22ef\u2287\u03b28. And we can also know \u03b22\u2288\u03b21. In fact, we only need to calculate \u03b21, \u03b22, \u03b23. Once we have \u03b23\u2286\u03b22, but \u03b22\u2288\u03b21, the computation can be stopped.Similarly, we can calculate \u03b21 \u222a \u03b22 \u222a {M}; that is, LM = {{b, g}, {g}, {b}, \u2205, {b, g, h}, {g, h}, {c, g, h}, {b, c}, {c}, {b, d, f}, {d}, {d, f}, {c, d}, {c, d, f}, {b, g, c, h}, {c, g, h, i}, {b, c, d, f}, {c, d, e}, M}. These results are easily examined from According to The time complexity of N = min\u2061{|G | , |M|}; by O(CNi). So we can get two matters as follows.O(2N).The time complexity of algorithm is kth step; then the time complexity of O(\u2211i=1i=k(CNi)) by O(\u2211i=1i=k(CNi)) \u2264 O(2N).Suppose that Denote a,\u2026, h, where a is headache, b is fever, c stands for painful limbs, d represents swollen glands in neck, e is cold, f is stiff neck, g is rash, and h is vomiting. Input the formal context and run the program; we obtain the set of all intents when n = 2:\u2205, {b}, {a, b}, {b, e}, {g}, {b, e, g}, {a, b, f, h}, {a, b, c, e}, {a, b, c, d, e, f, g, h}}\u2009{.We present an example demonstrating performance of Living beings and water  introducPatients and ill symptoms  introducBacterial Taxonomy [ Escherichia coli (ecoli), Salmonella typhi (styphi), Klebsiella pneumoniae (kpneu), Proteus vulgaris (pvul), Proteus morganii (pmor), and Serratia marcesens (smar). The phenotypic characters are H2S, MAN, LYS, IND, ORN, CIT, URE, ONP, VPT, INO, LIP, PHE, MAL, ADO, ARA, and RHA.Taxonomy . Data arMembership of Developing Countries in Supranational Group [al Group . In thisIn this section, we conduct some experiments to compare I| presents the number of intents and the efficiency is equivalent to (Time 1 \u2212 Time 2)/Time 1. It can be seen that I|.The results are shown in To find new methods to solve the difficult problems of the concept lattice construction is a hot problem. Constructing concept lattices is a novel research branch for data processing and data analysis. Different methods play essential roles in different problems. This paper first defines some basic notions. Based on the basic notion of intents, we obtain a new judgment method of finding all intents of formal concepts. Moreover, an example is given to explain the feasibility of this method. At last, we give the corresponding algorithm of this method and do the experiments to illustrate the effectiveness of this method.G| with |M| of a formal context. If |G | \u2264|M|, then we use subsets of G to determine subsets of M and output the set of intents. Otherwise, according to the duality principle, the subsets of M can be used to determine subsets of G and output the set of extents. We will improve the corresponding algorithm of this method in the future.For"}
{"text": "Finally to illustrate easy application and rich behavior of EP method, several examples are given.Firstly in this paper we introduce a new concept of the 2nd power of a fuzzy number. It is exponent to production (EP) method that provides an analytical and approximate solution for fully fuzzy quadratic equation (FFQE) :  The problem of finding the roots of equations like the quadratic equation has many applications in applied sciences like finance , 2, econThe rest of the paper is set out as follows. In the second section some related basic definitions of fuzzy mathematics for the analysis are recalled. In The basic definitions are given as follows.R;x0 \u2208 R with A.A fuzzy number is a function x0 \u2208 R with Note: in this paper we consider fuzzy numbers which have a unique u~(x0)=1 .F. Obviously, R\u2286F. For 0 < r \u2264 1, we define r-cut of fuzzy number r \u2208 . We denote the r-cut of fuzzy number The set of all these fuzzy numbers is denoted by \u22650}\u00af. In  from 4)F. Obvioux < 0\u2009\u2009(x > 0).A fuzzy number r \u2264 1, which satisfy the following requirements:r \u2264 1.A fuzzy number r \u2264 1, and scalar k, we define addition, subtraction, and scalar product by k and multiplication is, respectively, as follows.\u2009Addition: \u2009Subtraction: \u2009Scalar product:\u2009Multiplication:For two important cases multiplication of two fuzzy numbers is defined by the following terms.\u2009If \u2009If \u2009If \u2009If For arbitrary r-cuts is similar to arithmetics of the parametric form recalled previously .Two fuzzy numbers \u03b1 in parametric form is r \u2264 1. A triangular fuzzy number is popular and represented by \u03b1 > 0 and \u03b2 > 0, which has the parametric form as follows:A crisp number D : F \u00d7 F \u2192 R \u222a {0} and let k \u2208 R and Therefore  is a complete metric space.Let \u03b1, as a real root of crisp 1-cut equation and then we get \u03b11(r)\u2a7e0, \u03b12(r)\u2a7e0, \u03b11\u2032(r) < 0, and \u03b12\u2032(r) < 0. To find fi(\u03b11(r), \u03b12(r)), for i = 1,2, we consider two cases:FFQE has analytical solution;FFQE does not have analytical solution.Let (r)) andF(X~)=D~,fi(\u03b11(1), \u03b12(1)) = 0, for i = 1,2;To find fi(\u03b11(r), \u03b12(r)), i = 1,2, at first we substitute \u03b1 \u2212 f1(\u03b11(r), \u03b12(r)), \u03b1 + f2(\u03b11(r), \u03b12(r))) and \u2245(\u03b1 \u2212 \u03b11(r), \u03b1 + \u03b11(r))(\u03b1 \u2212 \u03b12(r), \u03b1 + \u03b12(r)), in parametric form of (r = 1. Using condition (1) and \u03b1 \u2208 R; that means\u03bb, \u03bc, and \u03c7(0) = \u03b11(0) + \u03b12(0). The third equation comes from the equality below, for r = 0,\u03bb, \u03bc, and \u03c7(0). To construct solution of the new method, in the above parametric form, let \u03c7(r) = \u03b11(r) + \u03b12(r) and \u03bd(r) = \u03b11(r) \u00d7 \u03b12(r); then, by solving a 2 \u00d7 2 system via \u03c7(r) and \u03bd(r), we obtain \u03c7(r) and we set\u03b11(r) + \u03b12(r) \u2208 R, because \u03b11(r) and \u03b12(r) are the roots of Z2 \u2212 \u03c7(r)Z + \u03bd(r) = 0.In this case we have  form of , and theution of . Notice \u03c7(r).Using the proposed method we obtain the following set of expressions for Case (1). (1)if (2)if (3)if (4)if Case (2). (1)if (2)if (3)if (4)if where\u03be \u2208 {\u03bb, \u03bc}.\u03bb = \u03bc = 0.5 because of In this case we do not have analytical solution, and we do not have \u03b1 is real core point of \u03bb = \u03bc = 0.5 is the best choice in EP method.If Without loss of generality suppose that \u03bb and \u03bc, we must solve the minimization problem as follows:To find the optimum parameters \u03b1 \u2208 R, we choose \u03bb = \u03bc = 0.5. This completes the proof.To delimitate maximum error for any \u03bb = \u03bc = 0.5 isNecessary condition for existence of EP solution with \u03c7(r) is sum of \u03b11(r) and \u03b12(r) and we want \u03b11(r) and \u03b12(r) to be nonnegative, then necessary condition for existence of EP solution with \u03bb = \u03bc = 0.5 is \u03c7(r) \u2265 0 and since \u2286, for all 0 \u2264 r \u2264 1, it is sufficient to have \u03c7(0) \u2265 0. Considering denominators of (\u03c7(0) \u2265 0, and this completes the proof.Since ators of \u201320) we  we \u03c7(r) ators of  and 25)\u03c7(r) is s\u03bb = \u03bc = 0.5 does not have solution,The EP method with This lemma is conclusion of \u03bb = \u03bc = 0.5 in EP method and \u03c7(0) \u2265 0, then sufficient condition for existence of solution is \u03bd(0) \u2265 0.Suppose \u03b11(r) and \u03b12(r) are nonnegative if \u03c7(r) \u2265 0 and \u03bd(r) \u2265 0.We know \u03bb = \u03bc = 0.5 and \u03bd(0) \u2265 0.Because of construction of EP method for \u03c7(0) \u2265 0 and \u03bb = \u03bc = 0.5 in EP method, then we have solution with (1)if (2)if (3)if (4)if with \u2009(1)if (2)if (3)if (4)if Suppose \u03c7(r) and \u03bd(r) in r = 0; we obtain\u03bd(0) \u2265 0, if the numerator is nonnegative and this completes the proof.We consider only one case to discuss. We consider nd solve  via \u03c7(r)\u22124.In the next examples we use round numbers with approximation less than 10Let =(3/2/2) .We will look for a solution where Equation Therefore we have analytical solution and EP solution by  and 29)29).\u03bb = 0.7067, \u03bc = 0.3296, andBy  and 14)14), we fLetting \u03bb = \u03bc = 0.5 too.Therefore this example does not have analytical solution. We look for EP solution. By  we find Now we consider an example with Letting \u03bb = \u03bc = 0.5 as follows: Therefore, this example does not have analytical solution. We look for EP solution. By , 30), a, a30), aNotice that the numerical methods needed In this paper we introduced a new method to solve a fully fuzzy quadratic equation. To this purpose we found the optimum spreads to decrease maximum error. One of the advantages of this method is that complications do not depend on the sign of the coefficients and variable. It is possible that these equations do not have any analytical solution, but the proposed method gives us an approximate analytical solution."}
{"text": "In Gao et\u00a0al. \u22123\u00a0mmol/kg.\u201d The correct dosage was given to animals.In paragraph 2, letter (d) indicated that the animals were given the dosage \u201c \u2026 2.46 mmol/kg once a day at P7, P11 \u2026. This should be corrected to \u201c2.46\u00a0x\u00a010The authors apologise for the typographical error."}
{"text": "Scientific Reports6: Article number: 29863; 10.1038/srep29863published online: 07202016; updated: 08252016This Article contains typographical errors.In the Results and Discussion section under subheading \u2018Properties of second-harmonic generation imaging with Bessel beam excitation\u2019,\u03c1z\u2009\u00d7\u20091/\u03c1z2, i.e. should not exhibit the same drop as 2PEF from an extended sample (Table 1).\u201d\u201cInterestingly, in the case of SHG, if all contributions scattered by the sample along the optical axis interfere in a constructive manner, the signal should scale as should read:\u03c1z2\u2009\u00d7\u20091/\u03c1z2, i.e. should not exhibit the same drop as 2PEF from an extended sample (Table 1).\u201d\u201cInterestingly, in the case of SHG, if all contributions scattered by the sample along the optical axis interfere in a constructive manner, the signal should scale as \u03c1z2\u2009\u00d7\u20091/\u03c1z2\u2009=\u20091\u2019 is incorrectly given as \u2018\u03c1z\u2009\u00d7\u20091/\u03c1z2\u2019.In Table 1, \u2018"}
{"text": "The notions of double-framed soft subfields, double-framed soft algebras over double-framedsoft subfields, and double-framed soft hypervector spaces are introduced, and theirproperties and characterizations are considered. The concept of soft set has been introduced by Molodtsov in 1999 [g\u02d8man  studied ginog\u02d8lu  introducginog\u02d8lu  considerginog\u02d8lu \u201310).The hyperstructure theory was introduced by Marty  at the 8In this paper, using the notion of DFS sets which is introduced in , we intrH \u00d7 H \u2192 P\u2217(H) is called a hyperoperation or join operation, where P\u2217(H) is the set of all nonempty subsets of H. The join operation is extended to subsets of H in natural way, so that A\u2218B is given bya\u2218A and A\u2218a are used for {a}\u2218A and A\u2218{a}, respectively. Generally, the singleton {a} is identified by its element a.A map \u2218:F be a field and  an abelian group. A hypervector space over F is defined to be the quadruplet , where \u201c\u2218\u201d is a mappinga, b \u2208 F and x, y \u2208 V the following conditions hold:(H1)a\u2218(x + y)\u2286a\u2218x + a\u2218y,(H2)a + b)\u2218x\u2286a\u2218x + b\u2218x,((H3)a\u2218(b\u2218x) = (ab)\u2218x,(H4)a\u2218(\u2212x) = (\u2212a)\u2218x = \u2212(a\u2218x),(H5)x \u2208 1\u2218x. Let V, +, \u2218, F) over a field F is said to be strongly left distributive (see [A hypervector space (ive (see ) if it sU be an initial universe set and let E be a set of parameters. We say that the pair  is a soft universe. Let P(U) denote the power set of U and A, B, C,\u2026\u2286E.Molodtsov  defined  soft set over U, where A pair U is a parameterized family of subsets of the universe U. For \u03b5 \u2208 A, \u03b5-approximate elements of the soft set In other words, a soft set over mples in .E be a set of parameters, and A, B, C,\u2026 are subsets of E.Let  double-framed soft set  over U where \u03b1 are mappings from A to P(U).A double-framed pair U and two subsets \u03b3 and \u03b4 of U, the \u03b3-inclusive set and the \u03b4-exclusive set of eA, respectively, are defined as follows: double-framed including set of For a DFS-set F be a field unless otherwise specified.In what follows, let U is called a double-framed soft subfield of F if the following conditions are satisfied: A double-framed soft set F, then  and \u03b1(1)\u2287\u03b1(0). If a \u2208 F, we have(1) For all a \u2208 F be such that a \u2260 0. Then(2) Let (3) It is by (1).It is easy to show that the following theorem holds.U is a double-framed soft subfield of F if and only if the nonempty \u03b3-inclusive set\u03b4-exclusive setF for all \u03b3, \u03b4 \u2208 P(U).A soft set V be algebra over F and let U. A double-framed soft set  double-framed soft algebra over Let V be an algebra over F and let U. If \u03b1(0)\u2286\u03b2(x) for all x \u2208 V.Let x \u2208 V, we have \u03b1(0)\u2286\u03b1(1)\u2286\u03b2(x).For any V be an algebra over F and let U. Then, a double-framed soft set Let a, b \u2208 F and x, y \u2208 V. Also conditions (2) and (3) are hold by Assume that \u03b1(0)\u2286\u03b1(1)\u2286\u03b2(x) for all x \u2208 V. Thus,a \u2208 F and x \u2208 V. Therefore, Conversely, suppose that V be a hypervector space over F and F. A soft set V is called a double-framed soft hypervector space of V related to \u03b1(1)\u2286\u03b2(\u03b8) where \u03b8 is the zero of .Let V be a hypervector space over F and F. If V related to  and \u03b2(\u03b8)\u2287\u03b1(0),Let It is an immediate consequence of V be a hypervector space over F. If V related to a double-framed soft subfield F, thenLet x \u2208 V. Since x \u2208 1\u2218x by (H5), we have \u03b2(x)\u2286\u22c2y\u22081\u2218x\u03b2(y).Let \u03b2(x) = \u22c3y\u22081\u2218x\u03b2(y) for all x \u2208 V.By using V over F is strongly left distributive. Let F. Then, a DFS-set V is a double-framed soft hypervector space of V related to  and \u22c3z\u2208a\u2218x+b\u2218y\u03b2(z)\u2286(\u03b1(a) \u222a \u03b2(x)) \u222a (\u03b1(b) \u222a \u03b2(y)), and \u03b2(x)\u2287\u03b1(1),for all a, b \u2208 F and all x, y \u2208 V.Assume that a hypervector space V related to a, b \u2208 F and x, y \u2208 V. Then,Assume that x, y \u2208 V, we haveF, we have \u03b2(a)\u2287\u03b1(1)\u2287\u03b1(0), and \u03b2(a)\u2287\u03b1(1)\u2287\u03b1(\u22121). Note that 0 \u2208 0\u2218x for all x \u2208 V. It follows thatx \u2208 V. Let a \u2208 F and x \u2208 V. Then,\u03b1(1)\u2286\u03b2(\u03b8). Therefore, V related to Conversely suppose the conditions (1) and (2) are true. For all V be a hypervector space over F and F. If a DFS-set V is a double-framed soft hypervector space of V related to \u03b3-inclusive set,\u03b4-exclusive set,V over the fields eF, respectively, for all \u03b3, \u03b4 \u2208 P(U).Let F. Let y \u2208 a\u2218x. Then,\u03b3 \u2208 P(U). Let x, y \u2208 eV. Then, \u03b2(x)\u2286\u03b4 and \u03b2(y)\u2286\u03b4. It follows thatx \u2212 y \u2208 iV. Note that eF is a subfield of F. Let a \u2208 eF, x \u2208 iV, and y \u2208 a\u2218x. Then,y \u2208 iV which shows that a\u2218x\u2286iV. Therefore, iV is a hypervector space over the field eF for all \u03b4 \u2208 P(U).Let V be a hypervector space over F and F. If a DFS-set V is a double-framed soft hypervector space of V related to V over the field \u03b3, \u03b4 \u2208 P(U).Let"}
{"text": "We give some results concerning the existence oftripled fixed points for a class of condensing operators in Banach spaces. Further, as an application, we study the existence of solutions for a generalsystem of nonlinear integral equations. Then, the intersection set A\u221e = \u22c2n=1\u221eAn is nonempty and A\u221e is precompact.Let  \u2192 1 implies tn \u2192 0. Then, T has at least one fixed point.Let The following result is a corollary of the previous theorem.C\u2009\u2009be a nonempty closed, bounded, and convex subset of X and T : C \u2192 C a continuous mapping such that for any subset A of C\u03c6 : \u211b+ \u2192 \u211b+ is a nondecreasing and upper semicontinuous functions; that is, for every t > 0, \u03c6(t) < t. Then, T has at least one fixed point.Let G : X \u00d7 X \u2192 X is an element  \u2208 X \u00d7 X such that G = x and G = y.A coupled fixed point of a mapping \u03bc1, \u03bc2,\u2026, \u03bcn be measures of noncompactness in Banach spaces E1, E2,\u2026, En, (respectively).Let E1 \u00d7 E2 \u00d7 \u22ef\u00d7En, where Xi is the natural projection of X on Ei, for i = 1,2,\u2026, n, and F is a convex function defined byThen, the functionF be as follows:E1 \u00d7 E2, where, for i = 1,2, \u03bci are measure of noncompactness in Ei\u2009\u2009and Xi, i = 1,2 are the natural projections of X on Ei.Aghajani and Sabzali  illustra\u03a9 be a nonempty, bounded, closed, and convex subset of a Banach space E and let \u03c6 : \u211b+ \u2192 \u211b+. Assume that \u03c6 is a nondecreasing and upper semicontinuous function. Let G : \u03a9 \u00d7 \u03a9 \u2192 \u03a9 be a continuous operator satisfying\u03bc. Then, G has at least a coupled fixed point.Let x, y, z) of a mapping G : X \u00d7 X \u00d7 X \u2192 X is called a tripled fixed point ifA tripled , we have thatE \u00d7 E \u00d7 E where Xi, i = 1,2, 3 are the natural projections of X on Ei.We can notice that by takingSo, we obtain the following theorem.\u03a9 be a nonempty, bounded, closed, and convex subset of a Banach space E and let \u03c6 : \u211b+ \u2192 \u211b+ be a nondecreasing and upper semicontinuous function such that \u03c6(t) < t for all t > 0. Then, for any measure of noncompactness \u03bc, the continuous operator G : \u03a9 \u00d7 \u03a9 \u00d7 \u03a9 \u2192 \u03a9 satifyingLet G has at least a tripled fixed point.To prove this theorem, let us define the measure of noncompactness \u211b+) endowed with the norm\u211b+) for a positive fixed t on \u212c\u211b+)BC( is defined as follows:\u03c90(X), we need first to introduce the modulus of continuity.We can see an application of x \u2208 X and \u03f5 > 0;x on  and letLet (i)\u03be, \u03b7, q : \u211b+ \u2192 \u211b+ are continuous and \u03be(t) \u2192 \u221e as t \u2192 \u221e;(ii)\u03c8 : \u211b \u2192 \u211b is continuous and there exist positive \u03b4, \u03b1 such thatt1, t2 \u2208 \u211b+;the function (iii)f : \u211b+ \u00d7 \u211b \u00d7 \u211b \u00d7 \u211b \u00d7 \u211b \u00d7 \u211b \u2192 \u211b is continuous and there exists a nondecareasing continuous function \u03a6 : \u211b \u2192 \u211b with \u03a6(0) = 0, so that(iv)f| is bounded on \u211b+; that is,the function defined by |(v)h : \u211b+ \u00d7 \u211b+ \u00d7 \u211b \u00d7 \u211b \u2192 \u211b is a continuous function and there exists a positive solution r0 of the inequalityD is positive constant defined by the equalityx, y, z, u, v, w \u2208 BC(\u211b+).Assume thatBC(\u211b+) \u00d7 BC(\u211b+) \u00d7 BC(\u211b+).Suppose that (i)\u2013(v) hold; then the system of integral equationsG : BC(\u211b+) \u00d7 BC(\u211b+) \u00d7 BC(\u211b+) \u2192 BC(\u211b+) be an operator defined byx, y, z) \u2208 BC(\u211b+) \u00d7 BC(\u211b+) \u00d7 BC(\u211b+), let\u211b+) \u00d7 BC(\u211b+) \u00d7 BC(\u211b+) is equivalent to the tripled fixed point of G.Let ution of  in BC(t) is continuous for any  \u2208 BC(\u211b+) \u00d7 BC(\u211b+) \u00d7 BC(\u211b+). Hence, we havex, y, z), \u025b > 0,T > 0 such that if t > T, thenx,y, z \u2208 BC(\u211b+). We notice two cases.Obviously, Then, by  and 30)G and , there eCase 1. If t > T, then from (hen from  and 40)Case 1. ICase 2. Similarly, for t \u2208 , we haveqT = sup\u2061{q(t) : t \u2208 }, and\u03b2 is continuous on  \u00d7  \u00d7  \u00d7 , we have \u03b2(\u03f5) \u2192 0 and \u03f5 \u2192 0. Thus, using (iii), we getG is a continuous function from  we haveGx,y,zt\u2212Gly, from , we concG satisfies all the conditions of T > 0 and \u03f5 > 0, assume that X1, X2, and X3 are nonempty chosen subsets of t1, t2 \u2208 , with |t2 \u2212 t1| \u2a7d \u03f5. Without loss of generality, letx, y, z) \u2208 X1 \u00d7 X2 \u00d7 X3,f and h on the compact sets \u2009\u2009\u00d7\u2009\u2009\u2009\u2009\u00d7\u2009\u2009\u2009\u2009\u2009\u00d7\u2009\u2009\u2009\u2009\u00d7\u2009\u2009 and  \u00d7  \u00d7  \u00d7  \u00d7 , respectively, we get \u03c9r0,D1T \u2192 0 and \u03c9r0T \u2192 0 as \u03f5 \u2192 0.Now, we show that the map T \u2192 \u221e in ,  \u2208 X1 \u00d7 X2 \u00d7 X3, and t \u2208 \u211b+, we havex, y, z), , and t are arbitrary in  \u00d7 BC(\u211b+) \u00d7 BC(\u211b+).Moreover, \u03a6 is a nondecreasing continuous function with \u03a6(0) = 0 and (iii), and we obtain,\u03f5\u27f60.By , we get\u03f5\u27f60.By \u2013(v).\u03be, \u03b7, q : \u211b+ \u2192 \u211b+ are continuous and \u03be \u2192 \u221e as t \u2192 \u221e. Further, the function \u03c8 : \u211b \u2192 \u211b is continuous for \u03b4 = \u03b1 = 1, and we havet1, t2 \u2208 \u211b+. Conditions (i) and (ii) hold.Obviously, Now, letMoreover,x, y, z \u2208 BC(\u211b+) \u00d7 BC(\u211b+) \u00d7 BC(\u211b+),r > 0, we have thatLet us verify the last condition (v). First,\u211b+) \u00d7 BC(\u211b+) \u00d7 BC(\u211b+).Consequently, the system has at least one solution in BC("}
{"text": "After publication of the original article the authors realised that Douglas. R. Call\u2019s name was displayed incorrectly. The correct spelling of his name is included in this erratum; the correct spelling has also been updated in the original article. The authors realised that in Line 347\u2014\u201cAAT\u201d should have been \u201cATA\u201d, and \u201cOAA\u201d should have been \u201cAAO\u201d. In Line 348\u2014\u201cOIO\u201d should have been \u201cIOO\u201d, \u201cCRD\u201d should have been \u201cDRC\u201d and \u201cAAT\u201d should have been \u201cATA\u201d. In Line 349\u2014\u201cOIO\u201d should have been \u201cIOO\u201d and \u201cCRD\u201d should have been \u201cDRC\u201d. In Line 325\u2014\u201c47\u201d should have been \u201c41\u201d. In Line 329\u2014\u201c48\u201d should have been \u201c47\u201d. In Line 340\u2014\u201c49\u201d should have been \u201c48\u201d. In Line 503 reference \u201847\u2019 should be removed. In Line 506\u2014\u201848\u2019 should have been \u201847\u2019 and in Line 510\u2014reference \u201849\u2019 should have been \u201848\u2019."}
{"text": "This paper studies the existence of solutions for a system of coupled hybridfractional differential equations with Dirichlet boundary conditions. We makeuse of the standard tools of the fixed point theory to establish the main results. The existence and uniqueness result is elaborated with the aid of an example. Fractional calculus is the study of theory and applications of integrals and derivatives of an arbitrary (noninteger) order. This branch of mathematical analysis, extensively investigated in the recent years, has emerged as an effective and powerful tool for the mathematical modeling of several engineering and scientific phenomena. One of the key factors for the popularity of the subject is the nonlocal nature of fractional-order operators. Due to this reason, fractional-order operators are used for describing the hereditary properties of many materials and processes. It clearly reflects from the related literature that the focus of investigation has shifted from classical integer-order models to fractional-order models. For applications in applied and biomedical sciences and engineering, we refer the reader to the books \u20134. For sHybrid fractional differential equations have also been studied by several researchers. This class of equations involves the fractional derivative of an unknown function hybrid with the nonlinearity depending on it. Some recent results on hybrid differential equations can be found in a series of papers see \u201337)..37]).cD\u03b4, \u2009cD\u03c9 denote the Caputo fractional derivative of orders \u03b4, \u03c9, respectively, fi \u2208 C and hi \u2208 C, i = 1,2.Motivated by some recent studies on hybrid fractional differential equations, we consider the following Dirichlet boundary value problem of coupled hybrid fractional differential equations:The aim of this paper is to obtain some existence results for the given problem. Our first theorem describes the uniqueness of solutions for the problem  by meansIn this section, some basic definitions on fractional calculus and an auxiliary lemma are presented , 2.q for a continuous function g is defined asThe Riemann-Liouville fractional integral of order n-times continuously differentiable function g :  denotes the integer part of the real number q.For at least \u03d5 \u2208 C, the integral solution of the problemGiven c0, c1 \u2208 R are arbitrary constants. Alternatively, we havex(0) = 0 = x(1) in =f=f\u2223w(t) \u2208 C1} denote a Banach space equipped with the norm ||w|| = max\u2061\u2061{|w(t)|, t \u2208 }, where W = U, V. Notice that the product space ||) with the norm |||| = ||x|| + ||y||,  \u2208 U \u00d7 V is also a Banach space.Let U \u00d7 V \u2192 U \u00d7 V byA1fi\u2009\u2009 are continuous and bounded; that is, there exist positive numbers \u03bcfi such that |fi| \u2264 \u03bcfi, \u2200 \u2208  \u00d7 R \u00d7 R.The functions A2) such that |h1| \u2264 \u03c10 + \u03c11|x| + \u03c12|y| and |h2| \u2264 \u03c30 + \u03c31|x| + \u03c32|y|, \u2200x, y \u2208 R, i = 1,2.There exist real constants For brevity, let us setIn view of Now we are in a position to present our first result that deals with the existence and uniqueness of solutions for the problem . This reA1) holds and that h1, h2 :  \u00d7 R2 \u2192 R are continuous functions. In addition, there exist positive constants \u03b7i, \u03b6i, i = 1,2 such that\u03bd1(\u03b71 + \u03b72) + \u03bd2(\u03b61 + \u03b62) < 1\u03bd1 and \u03bd2 are given by  = \u03ba1 < \u221e, sup\u2061t\u2208h2 = \u03ba2 < \u221e and define a closed ball: x, y)|| \u2264 r.Let us set sup\u2061].Hence||\u03981ind that||\u03982x1, y1),  \u2208 U \u00d7 V and for any t \u2208 , we have\u03bd1(\u03b71 + \u03b72) + \u03bd2(\u03b61 + \u03b62) < 1, it follows that \u0398 is a contraction. So Banach's fixed point theorem applies and hence the operator \u0398 has a unique fixed point. This, in turn, implies that the problem  = (1/2)(|sin\u2061u(t)| + 1), f2 = (1/2)(|cos\u2061\u2061u(t)| + 1), h1 = (1/4(t+2)2)(|x|/(1 + |x|)) + 1 + (1/32)\u2009\u2009sin2\u2061y\u2009\u2009, and h2 = (1/32\u03c0)sin\u2061(2\u03c0x) + |y|/16(1 + |y|) + 1/2. Note thatConsider the following coupled system of hybrid fractional differential equations:Dc3/2. Let P(F) = {x \u2208 G : x = \u03bbFx\u2009forsome\u20090 < \u03bb < 1}. Then either the set P(F) is unbounded or F has at least one fixed point.Let A1) and (A2) hold. Furthermore, it is assumed that \u03bd1\u03c11 + \u03bd2\u03c31 < 1 and \u03bd1\u03c12 + \u03bd2\u03c32 < 1, where \u03bd1 and \u03bd2 are given by how that||\u03982\u03c41, \u03c42 \u2208  with \u03c41 < \u03c42 and obtainx, y). This implies that the operator \u0398 is equicontinuous. Thus, by the above findings, the operator \u0398 is completely continuous.Now we show that the operator \u0398 is equicontinuous. For that, we take P = { \u2208 U \u00d7 V\u2223 = \u03bb\u0398, 0 \u2264 \u03bb \u2264 1} is bounded. Let  \u2208 P; then we have  = \u03bb\u0398. Thus, for any t \u2208 , we can writeP is bounded. Hence all the conditions of In the next step, it will be established that the set  view of , can be"}
{"text": "P-orders and R-orders are introduced and some related properties are investigated. An example is presented to show that the R-union of two internal cubic soft sets might not be internal. A sufficient condition is provided, which ensure that the R-union of two internal cubic soft sets is also internal. Moreover, some properties of cubic soft subalgebras of BCK/BCI-algebras based on a given parameter are discussed.Operations of cubic soft sets including \u201cAND\u201d operation and \u201cOR\u201d operation based on  P-union, P-intersection, R-union, and R-intersection of cubic sets and investigated several related properties. Later on, Jun et al.  the set of all interval numbers. Let us define what is known as refined minimum and refined maximum  of two elements in [I]. We also define the symbols \u201c\u2ab0,\u201d \u201c\u2aaf,\u201d and \u201c = \u201d in case of two elements in [I]. Consider two interval numbers i \u2208 \u039b. We define complement, denoted by By anX be a nonempty set. A function A : X \u2192 [I] is called an interval-valued fuzzy set  in X. Let [I]X stand for the set of all IVF sets in X. For every A \u2208 [I]X and x \u2208 X, A(x) =  is called the degree of membership of an element x to A, where A\u2212 : X \u2192 I and A+ : X \u2192 I are fuzzy sets in X which are called a lower fuzzy set and an upper fuzzy set in X, respectively. For simplicity, we denote A = . For every A, B \u2208 [I]X, we defineAc of A \u2208 [I]X is defined as follows: Ac(x) = A(x)c for all x \u2208 X; that is,Ai | i \u2208 \u039b} of IVF sets in X where \u039b is an index set, the union G = \u22c3i\u2208\u039bAi and the intersection F = \u22c2i\u2208\u039bAi are defined as follows:x \u2208 X, respectively.Let U be an initial universe set and let E be a set of parameters. Let P(U) denote the power set of U and A \u2282 E.Molodtsov  defined F, A) is called a soft set over U, where F is a mapping given byA pair  may be considered as the set of \u025b-approximate elements of the soft set . Clearly, a soft set is not a set. For illustration, Molodtsov considered several examples in \u2208[I] and t \u2208 .Given a parameter F, A) over U is an \u025b-cubic soft subalgebra over U and let x, y \u2208 U. If \u03b41, \u03b42]\u2208[I], then x, y \u2208 \u03bbF(\u025b)\u2192(t) for all t \u2208 , then \u03bbF(\u025b)(x) \u2264 t and \u03bbF(\u025b)(y) \u2264 t. Using (\u03bbF(\u025b)(x\u2217y) \u2264 max\u2061{\u03bbF(\u025b)(x), \u03bbF(\u025b)(y)} \u2264 t, and so x\u2217y \u2208 \u03bbF(\u025b)\u2192(t). Therefore \u03bbF(\u025b)\u2192(t) are subalgebras of U.Assume that a cubic soft set \u2192(t) are subalgebras of U for all \u2208[I] and t \u2208 . Assume that there exists a, b \u2208 U such that\u03b41 < min\u2061{\u03b31, \u03b33} and \u03b42 < min\u2061{\u03b32, \u03b34}. Takingx, y \u2208 U. Now, assume that \u03bbF(\u025b)(a\u2217b) > max\u2061{\u03bbF(\u025b)(a), \u03bbF(\u025b)(b)} for some a, b \u2208 U. Then there exists t0 \u2208  such thata, b \u2208 \u03bbF(\u025b)\u2192(t0) but a\u2217b \u2209 \u03bbF(\u025b)\u2192(t0). This is a contradiction, and thereforex, y \u2208 U. Consequently,  is an \u025b-cubic soft subalgebra over U.Conversely, suppose that \u025b \u2208 A, if a cubic soft set  over U is an \u025b-cubic soft subalgebra over U, then \u03bbF(\u025b)(0) \u2264 \u03bbF(\u025b)(x), for all x \u2208 U.Given a parameter x \u2208 U, we have\u03bbF(\u025b)(0) = \u03bbF(\u025b)(x\u2217x) \u2264 max\u2061{\u03bbF(\u025b)(x), \u03bbF(\u025b)(x)} = \u03bbF(\u025b)(x).For every F, A) be an \u025b-cubic soft subalgebra over U for a parameter \u025b \u2208 A. If there is a sequence in U such that n\u2192\u221e\u03bbF(\u03b5)(xn) = 0, then \u03bbF(\u025b)(0) = 0.Let (\u03bbF(\u025b)(0) \u2264 \u03bbF(\u025b)(x), for all x \u2208 U, we haven. Note that \u03bbF(\u025b)(0) \u2264 \u03bbF(\u025b)(xn) = 0. Hence \u03bbF(\u025b)(0) = 0.Since \u025b \u2208 A, if a cubic soft set  over U is an \u025b-cubic soft subalgebra over U, then the sets U\u03bbF(\u025b)\u2236 = {x \u2208 U | \u03bbF(\u025b)(x) = \u03bbF(\u025b)(0)} are subalgebras of U.Given a parameter x, y \u2208 U. If \u03bbF(\u025b)(x\u2217y) \u2264 max\u2061{\u03bbF(\u025b)(x), \u03bbF(\u025b)(y)} = max\u2061{\u03bbF(\u025b)(0), \u03bbF(\u025b)(0)} = \u03bbF(\u025b)(0). Combining this and \u03bbF(\u025b)(x\u2217y) = \u03bbF(\u025b)(0). This shows that x\u2217y \u2208 U\u03bbF(\u025b). Therefore U\u03bbF(\u025b) are subalgebras of U.Let \u025b \u2208 A, if a cubic soft set  over U is an \u025b-cubic soft subalgebra over U, then the set U.Given a parameter The proof is straightforward.P-order and the R-order. In [R-union of two internal cubic soft sets an internal cubic soft set? We have given an example to show that the answer to this question is negative, and then we have provided a condition for the R-union of two internal cubic soft sets to be an internal cubic soft set. We also have investigated several properties of cubic soft subalgebras of BCK/BCI-algebras based on any given parameter. Some important issues to be explored in the future includedeveloping strategies for obtaining more valuable results,applying these notions and results for studying related notions in other (soft) algebraic structures.In this paper, we first have considered operations of cubic soft sets, that is, \u201cAND\u201d operation and \u201cOR\u201d operation based on the rder. In , Muhiudd"}
{"text": "In this paper, further properties and characterizations of int-soft left (right) ideals are studied, and the notion of int-soft  bi-ideals is introduced. Relations between int-soft generalized bi-ideals and int-soft semigroups are discussed, and characterizations of (int-soft) generalized bi-ideals and int-soft bi-ideals are considered. Given a soft set  As a new mathematical tool for dealing with uncertainties, the notion of soft sets is introduced by Molodtsov . The aut\u03b1, S) over U, we establish int-soft  bi-ideals generated by .Our aim in this paper is to apply the soft sets to one of abstract algebraic structures, the so-called semigroup. So, we take a semigroup as the parameter set for combining soft sets with semigroups. This paper is a continuation of . We firsS be a semigroup. Let A and B be subsets of S. Then the multiplication of A and B is defined as follows:Let S is said to be regular if for every x \u2208 S there exists a \u2208 S such that xax = x.A semigroup S is said to be left  zero if xy = x  for all x, y \u2208 S.A semigroup S is said to be left (right) simple if it contains no proper left (right) ideal.A semigroup S is said to be simple if it contains no proper two-sided ideal.A semigroup A of S is calledS if AA\u2286A, that is, ab \u2208 A for all a, b \u2208 A,a subsemigroup of S if SA\u2286A , that is, xa \u2208 A  for all x \u2208 S and a \u2208 A,a left  ideal of S if it is both a left and a right ideal of S,a two-sided ideal of S if ASA\u2286A,a generalized bi-ideal of S if it is both a semigroup and a generalized bi-ideal of S.a bi-ideal of A nonempty subset A soft set theory is introduced by Molodtsov , and \u00c7a\u011fU be an initial universe set and E be a set of parameters. Let P(U) denote the power set of U and A, B, C,\u2026, \u2286E.In what follows, let \u03b1, A) over U is defined to be the set of ordered pairs\u03b1 : E \u2192 P(U) such that \u03b1(x) = \u2205 if x \u2209 A.A soft set . The subscript A in the notation \u03b1 indicates that \u03b1 is the approximate function of .The function \u03b1, A) over U and a subset \u03b3 of U, the \u03b3-inclusive set of , denoted by iA, is defined to be the setFor a soft set  and  over U, we define\u03b1, S) and  is defined to be the soft set U in which \u03b1, S) and  is defined to be the soft set U in which \u03b1, S) and  is defined to be the soft set U in which S to P(U) given byFor any soft sets  over U is called an int-soft semigroup over U if it satisfiesA soft set  over U is called an int-soft left  ideal over U if it satisfiesA soft set  over U is both an int-soft left ideal and an int-soft right ideal over U, we say that  is an int-soft two-sided ideal over U.If a soft set  ideal over ral (see ).\u03b1, S) be an int-soft left ideal over U. If G is a left zero subsemigroup of S, then the restriction of  to G is constant; that is, \u03b1(x) = \u03b1(y) for all x, y \u2208 G.Let  = \u03b1(y) for all x, y \u2208 G.Let Similarly, we have the following proposition.\u03b1, S) be an int-soft right ideal over U. If G is a right zero subsemigroup of S, then the restriction of  to G is constant; that is, \u03b1(x) = \u03b1(y) for all x, y \u2208 G.Let  be an int-soft left ideal over U. If the set of all idempotent elements of S forms a left zero subsemigroup of S, then \u03b1(x) = \u03b1(y) for all idempotent elements x and y of S.Let , we have uv = u and vu = v. Hence\u03b1(u) = \u03b1(v) for all idempotent elements u and v of S.Assume that the setSimilarly, we have the following proposition.\u03b1, S) be an int-soft right ideal over U. If the set of all idempotent elements of S forms a right zero subsemigroup of S, then \u03b1(x) = \u03b1(y) for all idempotent elements x and y of S.Let  with \u03a6\u228b\u03a8, define a map \u03c7A as follows:\u03c7A, S) is a soft set over U, which is called the -characteristic soft set. The soft set , S) is called the -identity soft set over U. The -characteristic soft set with \u03a6 = U and \u03a8 = \u2205 is called the characteristic soft set and is denoted by . The -identity soft set with \u03a6 = U and \u03a8 = \u2205 is called the identity soft set and is denoted by .For a nonempty subset \u03c7A, S) and , S) be -characteristic soft sets over U where A and B are nonempty subsets of S. Then the following properties hold: Let  Let x \u2208 S, suppose x \u2208 AB. Then there exist a \u2208 A and b \u2208 B such that x = ab. Thus we havex \u2208 AB, we get \u03c7AB(x) = \u03a6. Suppose x \u2209 AB. Then x \u2260 ab for all a \u2208 A and b \u2208 B. If x = yz for some y, z \u2208 S, then y \u2209 A or z \u2209 B. Hencex \u2260 yz for all x, y \u2208 S, then(2) For any \u03c7S, S), let  be a soft set over U such that \u03b1(x)\u2286\u03a6 for all x \u2208 S. Then the following assertions are equivalent:\u03b1, S) is an int-soft left ideal over U,-identity soft set  is an int-soft left ideal over U. Let x \u2208 S. If x = yz for some y, z \u2208 S, thenSuppose that  is an int-soft left ideal over U.Conversely, assume that Similarly, we have the following theorem.\u03c7S, S), let  be a soft set over U such that \u03b1(x)\u2286\u03a6 for all x \u2208 S. Then the following assertions are equivalent: \u03b1, S) is an int-soft right ideal over U,-identity soft set , S), let  be a soft set over U such that \u03b1(x)\u2286\u03a6 for all x \u2208 S. Then the following assertions are equivalent:\u03b1, S) is an int-soft two-sided ideal over U,-identity soft set  ideal over U. In fact, the soft intersection of int-soft left  ideals containing a soft set  over U is the smallest int-soft left  ideal over U.Note that the soft intersection of int-soft left  ideals over \u03b1, S) over U, the smallest int-soft left  ideal over U containing  is called the int-soft left  ideal over U generated by  and is denoted by l\u2009\u2009r, 2).For any soft set l = , wherea \u2208 S.Let a \u2208 S. Since a = ea, we havex, y \u2208 S, we have\u03b2, S) is an int-soft left ideal over U. Now let  be an int-soft left ideal over U such that \u03b1(a)\u2286\u03b3(a) for all a \u2208 S and\u03b1, S)l = .Let Similarly, we have the following theorem.S be a monoid with identity e. Then r = , wherea \u2208 S.Let \u03b1, S) over U is called an int-soft generalized bi-ideal over U if it satisfiesA soft set  be a soft set over U = Z defined as follows:\u03b1, S) is an int-soft generalized bi-ideal over U = Z. But it is not an int-soft semigroup over U = Z since \u03b1(c)\u2229\u03b1(c) = 4Z\u22884N = \u03b1(b) = \u03b1(cc).Let \u03b1, S) over U is both an int-soft semigroup and an int-soft generalized bi-ideal over U, then we say that  is an int-soft bi-ideal over U.If a soft set  be an int-soft generalized bi-ideal over U and let x and y be any elements of S. Then there exists a \u2208 S such that y = yay, and so\u03b1, S) is an int-soft semigroup over U.Let  bi-ideal.A of S, the following are equivalent: A is a generalized bi-ideal of S,\u03c7A, S) over U is an int-soft generalized bi-ideal over U for any \u03a6, \u03a8 \u2208 P(U) with \u03a6\u228b\u03a8.the -characteristic soft set  with \u03a6\u228b\u03a8 and x, y, z \u2208 S. If x, z \u2208 A, then \u03c7A(x) = \u03a6 = \u03c7A(z) and xyz \u2208 ASA\u2286A. Hencex \u2209 A or z \u2209 A, then \u03c7A(x) = \u03a8 or \u03c7A(z) = \u03a8. Hence\u03c7A, S) is an int-soft generalized bi-ideal over U for any \u03a6, \u03a8 \u2208 P(U) with \u03a6\u228b\u03a8.Assume that \u03c7A, S) over U is an int-soft generalized bi-ideal over U for any \u03a6, \u03a8 \u2208 P(U) with \u03a6\u228b\u03a8. Let a be any element of ASA. Then a = xyz for some x, z \u2208 A and y \u2208 S. Then\u03c7A(a) = \u03a6. Thus a \u2208 A, which shows that ASA\u2286A. Therefore A is a generalized bi-ideal of S.Conversely, suppose that the -characteristic soft set  over U is an int-soft generalized bi-ideal over U if and only if the nonempty \u03b3-inclusive set of  is a generalized bi-ideal of S for all \u03b3\u2286U.A soft set  is an int-soft semigroup over U. Let \u03b3\u2286U be such that iS \u2260 \u2205. Let a \u2208 S and x, y \u2208 iS. Then \u03b1(x)\u2287\u03b3 and \u03b1(y)\u2287\u03b3. It follows from . Thus iS is a generalized bi-ideal of S.Assume that  is a generalized bi-ideal of S for all \u03b3\u2286U. Let x, y, z \u2208 S be such that \u03b1(x) = \u03b3x and \u03b1(z) = \u03b3z. Taking \u03b3 = \u03b3x\u2229\u03b3z implies that x, z \u2208 iS. Hence xyz \u2208 iS, and so\u03b1, S) is an int-soft generalized bi-ideal over U.Conversely, suppose that the nonempty \u03b1, S) over U is an int-soft semigroup over U if and only if the nonempty \u03b3-inclusive set of  is a subsemigroup of S for all \u03b3\u2286U.A soft set  over U is an int-soft bi-ideal over U if and only if the nonempty \u03b3-inclusive set of  is a bi-ideal of S for all \u03b3\u2286U.A soft set  and a soft set  over U, the following are equivalent: \u03b1, S) is an int-soft generalized bi-ideal over U, is an int-soft generalized bi-ideal over U. Let a be any element of S. If x, y, u, v \u2208 S such that a = xy and x = uv. Since  is an int-soft generalized bi-ideal over U, it follows from  is an int-soft generalized bi-ideal over U.Therefore  be an int-soft semigroup over U. Then  is an int-soft bi-ideal over U if and only if \u03c7S, S) is the identity soft set over U.Let  and  are int-soft generalized bi-ideals over U, then so is the soft intersection If  and  are int-soft bi-ideals over U, then so is the soft intersection If  be an int-soft generalized bi-ideal over U. For any x \u2208 S, we have\u03b1(x) = \u03b1(e). Therefore \u03b1 is a constant function.Let \u03b1, S) over U, the smallest int-soft  bi-ideal over U containing  is called an int-soft  bi-ideal over U generated by  and is denoted by b.For any soft set  be a soft set over U such that \u03b1(x)\u2286\u03b1(e) for all x \u2208 S. Then b = , wherea \u2208 S.Let a \u2208 S. Since a = eea it follows from hypothesis thatx, y, z \u2208 S, we get\u03b2(x)\u2229\u03b2(z)\u2286\u03b2(xyz). Let y = e. Then \u03b2(x)\u2229\u03b2(z)\u2286\u03b2(xz) for all x, z \u2208 S. Hence  is an int-soft  bi-ideal over U. Let  be an int-soft  bi-ideal over U such that a \u2208 S, we have\u03b1, S)b = .Let"}
{"text": "Activated G\u03b1q subunits have been shown to interact directly with Fhit, up-regulate Fhit expression and enhance its suppressive effect on cell growth and migration. Other signaling molecules may be involved in modulating G\u03b1q/Fhit interaction.G proteins are known to modulate various growth signals and are implicated in the regulation of tumorigenesis. The tumor suppressor Fhit is a newly identified interaction partner of Gq and Fhit, co-immunoprecipication assay was performed on HEK293 cells co-transfected with different combinations of Flag-Fhit, G\u03b116, G\u03b116QL, pcDNA3 vector, and PLC\u03b2 isoforms. Possible associations of Fhit with other effectors of G\u03b1q were also demonstrated by co-immunoprecipitation. The regions of G\u03b1q for Fhit interaction and PLC\u03b2 stimulation were further evaluated by inositol phosphates accumulation assay using a series of G\u03b116/z chimeras with discrete regions of G\u03b116 replaced by those of G\u03b1z.To test the relationship of PLC\u03b2 with the interaction between G\u03b1q. Expression of PLC\u03b2 increased the affinities of Fhit for both wild-type and activated G\u03b1q. Swapping of the Fhit-interacting \u03b12-\u03b24 region of G\u03b1q with G\u03b1i eliminated the association of G\u03b1q with Fhit without affecting the ability of the mutant to stimulate PLC\u03b2. Other effectors of G\u03b1q including RGS2 and p63RhoGEF were unable to interact with Fhit.PLC\u03b21, 2 and 3 interacted with Fhit regardless of the expression of G\u03b1q in a unique way. PLC\u03b2 interacts with Fhit and increases the interaction between G\u03b1q and Fhit. The G\u03b1q/PLC\u03b2/Fhit complex formation points to a novel signaling pathway that may negatively regulate tumor cell growth.PLC\u03b2 may participate in the regulation of Fhit by GThe online version of this article (doi:10.1186/s12885-015-1802-z) contains supplementary material, which is available to authorized users. FHIT gene at the chromosomal fragile site FRA3B is often regarded as an early target of DNA damage in precancerous cells. Its gene product, the ubiquitously expressed Fhit (Fragile Histidine Triad) protein, is a member of the HIT (histidine triad) superfamily with three signature histidines in the conserved nucleoside binding motif. Fhit binds and hydrolyzes various dinucleoside polyphosphates  into two nucleotides where one is a nucleoside monophosphate inositol overnight. The labeled cells were then washed once with IP3 assay medium  and then incubated with 500\u00a0\u03bcl IP3 assay medium at 37\u00a0\u00b0C for 1\u00a0h. Reactions were stopped by replacing the assay medium with 750\u00a0\u03bcL ice-cold 20\u00a0mM formic acid and the lysates were kept in 4\u00a0\u00b0C for 30\u00a0min before the separation of [3H]inositol phosphates from other labeled species by sequential ion-exchange chromatography as described previously [HEK293 cells were seeded at a density of 2\u00d710eviously .t test. Differences at values of P\u2009<\u20090.05 were considered significant (* P\u2009<\u20090.05).Data were expressed as the mean\u2009\u00b1\u2009S.E. of at least three independent sets of experiments. The probability of an observed difference being a coincidence was evaluated by Dunnett q family  via their \u03b12-\u03b24 region without affecting G\u03b1q-induced PLC\u03b2 activation [q [q in co-immunoprecipitation assays. HEK293 cells were co-transfected with Flag-tagged Fhit and wild-type or activated G\u03b116 (G\u03b116QL) with or without PLC\u03b21, PLC\u03b22 or PLC\u03b23. Because activated G\u03b1q signaling always increase the expression levels of Fhit [q subunits, we have opted for using G\u03b116 as a representative G\u03b1q member. In vector transfected cells, Fhit pulled down detectably more G\u03b116QL than wild-type G\u03b116 . The reduction of G\u03b116QL in the PLC\u03b23-immunoprecipitates was not due to variations in the expression levels or pull down efficiency, as these parameters were essentially similar in the different samples , we have previously identified the \u03b12-\u03b24 region of G\u03b116 as critical for Fhit interaction [q is seemingly involved in binding to PLC\u03b2 [q, PLC\u03b2 and Fhit, we constructed two new chimeras named z\u03b12\u03b24 (G\u03b116 backbone with \u03b12\u03b24 region from G\u03b1z) and 16\u03b12\u03b24 (G\u03b1z backbone with \u03b12\u03b24 region from G\u03b116), wherein the \u03b12-\u03b24 region of G\u03b116 or G\u03b1z was swapped with each other . Because the G\u03b1-specific antibodies are N-terminal targeting, the z\u03b12\u03b24 and 16\u03b12\u03b24 chimeras were recognized by anti-G\u03b116 and anti-G\u03b1z antisera, respectively. When examined for their ability to stimulate PLC\u03b2, activated z\u03b12\u03b24QL efficiently stimulated the formation of inositol phosphates to an extent similar to that of G\u03b116QL  tested as well as the endogenously expressed PLC\u03b21 could be detected in the immunoprecipitates of Fhit in the absence of G\u03b1q overexpression  [q could be detected   but thisted Fig.\u00a0. The inc\u03b2s Figs.\u00a0 may resuq is essential for the binding of Fhit [q-PLC\u03b23 complex, PLC\u03b23 interacts with the \u03b12 and \u03b13 region of G\u03b1q by a helix-turn-helix domain [q/p63RhoGEF complex [q was present  for PLC\u03b23 interaction are the same with members of G\u03b1i subfamily while the exceptional residue on G\u03b1i subunits corresponding to R214 of G\u03b1q is identical to that of G\u03b116 [16 (a G\u03b1q member) with the corresponding region of G\u03b1z (a G\u03b1i subfamily member) would still allow the z\u03b12\u03b24QL chimera to interact productively with PLC\u03b2  of cell growth inhibition by G\u03b1q and PLC\u03b2.PLC\u03b2 is a key molecule in transducing activated Gt cancer  and it pt cancer . On the t cancer . PLC\u03b23-dt cancer . The intq binds to Fhit through the \u03b12-\u03b24 region of G\u03b1q (Fig.\u00a0q including the \u03b12 region (Fig.\u00a0q is the \u03b12 region (Fig.\u00a016 with the corresponding region of G\u03b1z did not affect the activated G\u03b116-induced PLC\u03b2 activation (Fig.\u00a0q and change the binding interface between G\u03b1q and PLC\u03b2 without affecting the PLC\u03b2 activity. The altered binding interfaces of activated G\u03b1q and PLC\u03b2 may trigger G\u03b1q and PLC\u03b2 to form a \u2018clamp\u2019 around Fhit. Beside protein interactions that were found between two protein pairs among G\u03b1q, PLC\u03b2 and Fhit, PLC\u03b2s increased the interaction between G\u03b1q and Fhit. But Fhit or G\u03b1q did not enhance the interaction between PLC\u03b2 and G\u03b1q or Fhit, respectively (Fig.\u00a0q by PLC\u03b2 is that PLC\u03b2 may interact with and stabilize the complex of Fhit and G\u03b1q. Another possibility is that when binding to G\u03b1q, PLC\u03b2 may provide direct binding sites on itself for Fhit which also leads to the formation of a heterotrimeric protein complex. In both possibilities, the activated G\u03b1q recruits PLC\u03b2 which acts as a positive regulator for the association of Fhit with activated G\u03b1q. In the future, the involvement of PLC\u03b2 in the regulation of Fhit by G\u03b1q and their possible roles on cancer therapy should be demonstrated.Activated G\u03b1ion Fig.\u00a0. As subsion Fig.\u00a0, Fhit maely Fig.\u00a0. One posq and Fhit. This regulatory effect appears to be unique to PLC\u03b2 because other G\u03b1q effectors such as RGS2 and p63RhoGEF could not interact with Fhit. Substitution of the \u03b12-\u03b24 region of G\u03b1q with G\u03b1i did not affect G\u03b1q-induced PLC\u03b2 activation but eliminated the interaction between G\u03b1q with Fhit. This new G\u03b1q/PLC\u03b2/Fhit signaling complex represents a novel pathway of G\u03b1q regulation on tumor suppression.We showed that PLC\u03b2 could interact with Fhit, and the expression of PLC\u03b2 increased the interaction between G\u03b1"}
{"text": "Following up on this idea we obtain several results and conclusions of interest. We also discuss the notion of a couplet  on X, consisting of a two-step derivation d and its square D = d \u2218 d, for example, whose defining property leads to further observations on the underlying ranked trigroupoids also.We define a ranked trigroupoid as a natural followup on the idea of a ranked bigroupoid. We consider the idea of a derivation on such a trigroupoid as representing a two-step process on a pair of ranked bigroupoids where the mapping  The notion of derivations arising in analytic theory is extremely helpful in exploring the structures and properties of algebraic systems. Several authors , 2 studid-algebras which is another useful generalization of BCK-algebras and then investigated several relations between d-algebras and BCK-algebras as well as several other relations between d-algebras and oriented digraphs +[xd(1) \u2212 d(x)] = D(x) + xd(1) \u2212 d(x). It follows that, for any x \u2208 R,xd(y) \u2212 d(x)y = xyd(1) \u2212 xd(1)y = 0. Hence, by (D(xy) = D(x)y \u2212 xD(y) + xd(y) \u2212 d(x)y = D(x)y \u2212 xD(y); that is,Thus : = 1 in , then D = D(x)\u00b7(\u22121) \u2212 xD(\u22121). Hence D(x) = \u2212xD(\u22121) for all x < 0.If we let  = \u22121 in , then wed is a two-step derivation on , then d(xy) = d(x)y + xd(y) and d(x + y) = (d(x) + y)\u2212(x + d(y)) for any x, y \u2208 R. It follows that d(2x) = d(x + x) = (d(x) + x)\u2212(x + d(x)) = 0, which proves that d(x) = d(2 \u00b7 (x/2)) = 0 for all x \u2208 R.In X : = ) 1 and 02 are both zeros, then x \u2260 01, 02 yields 01 = x\u2217x = 02.The collection of frame algebras includes the collection of BCK-algebras and it is a variety. In a frame algebra the element 0 is unique. Indeed, if 0X, \u2217i, 0) forms a subsemigroup of the semigroup (Bin(X), \u25a1).The collection of all frame algebras , , if we let : = \u2009\u2009\u25a1\u2009\u2009, then x\u2009\u2009\u25a1\u2009\u2009y = (x\u2217y)\u2022(y\u2217x) for all x, y \u2208 X. It follows that x\u2009\u2009\u25a1\u2009\u2009x = (x\u2217x)\u2022(x\u2217x) = 0\u20220 = 0, 0\u2009\u2009x = (0\u2217x)\u2022(x\u22170) = 0\u2009\u2009\u2022\u2009\u2009x = 0, and x\u2009\u2009\u25a1\u2009\u20090 = (x\u22170)\u2022(0\u2217x) = x\u20220 = x, proving that  is a frame algebra. This proves the proposition.Given frame algebras ,  \u2208 Bin(X), we defineGiven groupoids  and  be frame algebras. If \u2208S, then  is a frame algebra.Let \u2208S, then x\u2009\u2009\u2207\u2009\u2009y \u2208 {x\u2217y, x\u2022y} for all x, y \u2208 X. It follows that x\u2009\u2009\u2207\u2009\u2009x \u2208 {x\u2217x, x\u2022x} = {0} implies that x\u2009\u2009\u2207\u2009\u2009x = 0. Moreover, 0\u2009\u2009\u2207\u2009\u2009x \u2208 {0\u2217x, 0\u2022x} = {0} shows that 0\u2009\u2009\u2207\u2009\u2009x = 0, and x\u2009\u2009\u2207\u2009\u20090 \u2208 {x\u22170, x\u20220} = {x} shows that x\u2009\u2009\u2207\u2009\u20090 = x, proving that  is a frame algebra.If  is called an fr(3)-algebra if(G)X, \u2217, 01), ,  are frame algebras,((H)1 = 02 = 03.0A ranked trigroupoid  be an fr(3)-algebra. If d : X \u2192 X is a two-step derivation on X, then d(0) = 0,d(x) = d(x)\u2022x = d(x)\u2009\u2009\u25ca\u2009\u2009x for all x \u2208 X.Let , then for any x, y \u2208 x, we haved(0) = d(0\u2217y) = (d(0)\u2217y)\u2022(0\u2217d(y)) = (d(0)\u2217y)\u20220 = d(0)\u2217y; that is, d(0) = d(0)\u2217y, for all y \u2208 X. If we let y : = d(0), then by applying (A) we obtain d(0) = d(0)\u2217d(0) = 0.(i) If x \u2208 X, we have d(x) = d(x\u2217\u2009\u20090) = (d(x)\u22170)\u2009\u2009\u2022\u2009\u2009(x\u2217d(0)) = d(x)\u2022x and d(x) = d(x\u2022\u2009\u20090) = (d(x)\u20220)\u2009\u2009\u25ca\u2009\u2009(x\u2022d(0)) = d(x)\u2009\u2009\u25ca\u2009\u2009x.(ii) Given d on a trigroupoid , we denote its kernel by Ker\u2061(d): = {x \u2208 X\u2223d(x) = 0}.Given a two-step derivation X, \u2217, \u2022, \u25ca) be an fr(3)-algebra. If d : X \u2192 X is a two-step derivation on X, then x\u2217d(x), x\u2022d(x) \u2208 Ker(d),x \u2208 Ker(d) implies that x\u2217y, x\u2022y \u2208 Ker(d),Ker(d)\u2286Ker(d2),for all x, y \u2208 X.Let (y : = d(x) in (d(x\u2217d(x)) = (d(x)\u2217d(x))\u2009\u2009\u2022\u2009\u2009(x\u2217d(d(x))) = 0\u2009\u2009\u2022\u2009\u2009(x\u2217d2(x)) = 0 an d(x\u2022\u2009\u2009d(x)) = (d(x)\u2009\u2009\u2022\u2009\u2009d(x))\u2009\u2009\u25ca\u2009\u2009(x\u2022\u2009\u2009d(d(x))) = 0\u2009\u2009\u25ca\u2009\u2009(x\u2022\u2009\u2009d2(x)) = 0, proving that x\u2217d(x), x\u2022d(x) \u2208 Ker\u2061(d) for any x \u2208 X.(i) If we let  d(x) in  and 25)y : = d(xx \u2208 Ker\u2061(d), then d(x\u2217y) = (d(x)\u2217y)\u2022(x\u2217d(y)) = (0\u2217y)\u2022(x\u2217d(y)) = 0 and d(x\u2022y) = (d(x)\u2022y)\u2009\u2009\u25ca\u2009\u2009(x\u2022\u2009\u2009d(y)) = 0 for any y \u2208 X, proving that x\u2217y,\u2009 \u2009x\u2022y \u2208 Ker\u2061(d).(ii) If x \u2208 Ker\u2061(d), then d2(x) = d(d(x)) = d(0) = 0 by x \u2208 Ker\u2061(d2).(iii) If X, \u2217, \u2022, \u25ca) be an fr(3)-algebra. If d : X \u2192 X is a two-step derivation on X, theny \u2208 X.Let (x \u2208 Ker\u2061(d2), then d(d(x)) = 0 and hencey \u2208 X, which proves that x\u2217y \u2208 Ker\u2061(d2).If d2(x\u2022y) may not be computable unless the behavior of d(u\u2009\u2009\u25ca\u2009\u2009v) is specified, since d2(x\u2022y) = d((d(x)\u2022y)\u2009\u2009\u25ca\u2009\u2009(x\u2022d(y))) = d(u\u2009\u2009\u25ca\u2009\u2009v) for some u, v \u2208 X.Note that X, \u2217, \u2022, \u25ca) be an fr(3)-algebra. If  is a couplet of , then Ker(D)\u2217X\u2286Ker(D).Let  is a couplet of  and if x \u2208 Ker\u2061(D), then D(x)\u2217y = 0\u2217y = 0 for any y \u2208 X. It follows from (D(x\u2217y) = 0\u2009\u2009\u25ca\u2009\u2009[(d(x)\u2217y)\u2022[(d(x)\u2217d(y))\u2022(x\u2217D(y))]] = 0, proving that x\u2217y \u2208 Ker\u2061(D).If  be a poset with minimal element 0. Define a binary operation \u201c\u2217\u201d on X byX, \u2217, 0) is a BCK-algebra, called a standard BCK-algebra inherited from the poset .Let  be a standard BCK-algebra. Let  be an fr(3)-algebra and let  be a couplet of . If D(x\u2217y) \u2260 0 for some x, y \u2208 X, then D(x\u2217y) = D(x)\u2217y = D(x) and x\u2217y = x.Let (D(x\u2217y) \u2260 0. We claim that D(x)\u2217y \u2260 0. Suppose that D(x)\u2217y = 0. Since  is a fr(3)-algebra, by applying (D(x\u2217y) = 0 is a contradiction. Since  is a standard BCK-algebra, we obtain D(x)\u2217y = D(x). We claim that x\u2217y = x. If x\u2217y = 0, then D(x\u2217y) = D(0) = 0 is a contradiction. It follows that D(x\u2217y) = D(x) = D(x)\u2217y, proving the proposition.Let applying , we obtaX : = R be the real field and let  be a ranked trigroupoid, where , , and\u2009\u2009 are linear groupoids; that is, x\u2217y : = A + Bx + Cy, x\u2022y : = a + bx + cy, and x\u2009\u2009\u25ca\u2009\u2009y : = \u03b1 + \u03b2x + \u03b3y for any x, y \u2208 X, where A, B, C, a, b, c, \u03b1, \u03b2, \u03b3 \u2208 X (fixed).Let d : X \u2192 X is a two-step derivation on , then for any x, y \u2208 X, we haveThus, if d(x) = 0 for all x \u2208 X, then A = 0,\u2009\u2009\u03b1 + (\u03b2 + \u03b3)a = 0. Hence, d(x) = 0 X, then (a+(b+c)A+X, then (a+(b+c)A+ Hence,  be a ranked trigroupoid, where , ,  are linear groupoids; that is, x\u2217y : = A + Bx + Cy, x\u2022y : = a + bx + cy, and x\u2009\u2009\u25ca\u2009\u2009y : = \u03b1 + \u03b2x + \u03b3y for any x, y \u2208 X, where A, B, C, a, b, c, \u03b1, \u03b2, \u03b3 \u2208 X (fixed). Let d : X \u2192 X be a two-step derivation such that d(x) = 0 for all x \u2208 X. bc \u2260 0 and b + c \u2260 0, then x\u2217y = \u2212a/(b + c), x\u2022y : = a + bx + cy, x\u2009\u2009\u25ca\u2009\u2009y = 0;If bc \u2260 0 and b + c = 0, then x\u2217y = A, x\u2022y = b(x \u2212 y), x\u2009\u2009\u25ca\u2009\u2009y = 0.If Let bc \u2260 0, then it follows from  and x\u2009\u2009\u25ca\u2009\u2009y = 0.(i) If ows from  that we bc \u2260 0 and b + c = 0, then 0 = a + (b + c)A = a and A is arbitrary, and hence we obtain the result.(ii) If X : = R be the real field and let  be a ranked trigroupoid, where , ,  are linear groupoids; that is, x\u2217y : = A + Bx + Cy, x\u2022y : = a + bx + cy, and x\u2009\u2009\u2009\u25ca\u2009\u2009y : = \u03b1 + \u03b2x + \u03b3y for any x, y \u2208 X, where A, B, C, a, b, c, \u03b1, \u03b2, \u03b3 \u2208 X (fixed). Let d : X \u2192 X be a two-step derivation such that d(x) = 0 for all x \u2208 X:b = c = 0 and b + c \u2260 0, then x\u2217y = A + Bx + Cy, x\u2022y : = 0, x\u2009\u2009\u25ca\u2009\u2009y = \u03b2x + \u03b3y;if b \u2260 0, c = \u03b2 = 0, then x\u2217y = \u2212a/b + Bx, x\u2022y = a + bx, x\u2009\u2009\u25ca\u2009\u2009y = 0;if b \u2260 0, c = 0, \u03b2 \u2260 0, then x\u2217y = \u03b1/b\u03b2 + Bx, x\u2022y = \u2212\u03b1/\u03b2 + bx, x\u2009\u2009\u25ca\u2009\u2009y = \u03b1 + \u03b2y;if b = 0, c \u2260 0, \u03b3 = 0, then x\u2217y = \u2212a/c + Cy, x\u2022y = a + cy, x\u2009\u2009\u25ca\u2009\u2009y = 0;if b = 0, c \u2260 0, \u03b3 \u2260 0, then x\u2217y = \u03b1/c\u03b3 + Cy, x\u2022y = \u2212\u03b1/\u03b3 + cy, x\u2009\u2009\u25ca\u2009\u2009y = \u03b1 + \u03b3y.if Let The proof is similar to d(x) = 0 for all x \u2208 X, and most of them are classified by the properties of b, c in x\u2022y = a + bx + cy.In Propositions The notion of two-step derivations is a generalization of derivations. This leads to the study of trigroupoids, and we explore some relations with several algebras, for example, BCK-algebras, frame algebras, and so forth. The classification of linear ranked trigroupoids then explains a number of concrete algebraic structures with derivations."}
{"text": "In several advanced fields like control engineering, computer science, fuzzy automata, finite state machine, and error correcting codes, the use of fuzzified algebraic structures especially ordered semigroups plays a central role. In this paper, we introduced a new and advanced generalization of fuzzy generalized bi-ideals of ordered semigroups. These new concepts are supported by suitable examples. These new notions are the generalizations of ordinary fuzzy generalized bi-ideals of ordered semigroups. Several fundamental theorems of ordered semigroups are investigated by the properties of these newly defined fuzzy generalized bi-ideals. Further, using level sets, ordinary fuzzy generalized bi-ideals are linked with these newly defined ideals which is the most significant part of this paper. The concept that belongs to relation (\u2208) and quasicoincident with relation (q) relation of a fuzzy point to fuzzy set was introduced by Pu and Liu  belongs to  a fuzzy set \u03bc, if \u03bc(x) \u2265 t  + t > 1) and is denoted by  \u2208 \u03bc , where t \u2208 , where \u03b1, \u03b2 \u2208 {\u2208, q, \u2208\u2228q, \u2208\u2227q} and \u03b1 \u2260 \u2208\u2227q. The idea of generalized fuzzy subgroups has increased the importance of algebraic structures by attracting the attention of many researchers and opened ways for future researchers in this field. Furthermore, Jun  of real numbers.AAfter the introduction of fuzzy set theory , Rosenfe\u03bc of S is called a fuzzy subsemigroup if for all x, y \u2208 S,A fuzzy subset \u03bc of S is called a fuzzy generalized bi-ideal of S if for all x, y, z \u2208 S the following conditions hold:x \u2264 y \u2192 \u03bc(x) \u2265 \u03bc(y),\u03bc(xyz) \u2265 min\u2061{\u03bc(x), \u03bc(z)}.A fuzzy subset \u03bb is called a fuzzy bi-ideal of S if the following conditions hold for all x, y, z \u2208 S:x \u2264 y \u2192 \u03bc(x) \u2265 \u03bc(y),\u03bc(xyz) \u2265 min\u2061{\u03bc(x), \u03bc(z)}.A fuzzy subsemigroup S. But the converse is not true, as given in  \u2208 \u03bb  if \u03bb(x) \u2265 t  + t > 1). If  \u2208 \u03bb or q\u03bb, then we write \u2208\u2228q\u03bb. The symbol q does not hold.is called a\u03bc of S is called an -fuzzy generalized bi-ideal of S if it satisfies the following conditions:x, a, y \u2208 S) \u2009\u2009, \u2009\u2009.-fuzzy generalized bi-ideal of S if and only if it satisfies the following conditions:x, a, y \u2208 S)\u2009\u2009(\u03bc(xay) \u2265 min\u2061{\u03bc(x), \u03bc(y), 0.5}),\u2009\u2009 \u2265 min\u2061{\u03bc(y), 0.5}).\u2009\u2009x \u2264 y),\u2009\u2009x, y \u2208 S)\u2009\u2009(max\u2061{\u03bc(x), 0.5} \u2265 \u03bc(y) with x \u2264 y),(\u2200(4)x, a, y \u2208 S)\u2009\u2009(max\u2061{\u03bc(xay), 0.5} \u2265 min\u2061{\u03bc(x), \u03bc(y)})., we have \u03bc(y) \u2265 t and r + \u03bc(y) \u2264 1, which implies that t \u2264 0.5, a contradiction. Hence (3) is valid.(1)\u21d2(3). If there exists x, y \u2208 S with x \u2264 y and r \u2208  < r. If \u03bc(x) \u2265 \u03bc(y), then \u03bc(y) \u2264 \u03bc(x) < r. It follows that \u03bc(x) < \u03bc(y), then by (3), we have 0.5 \u2265 \u03bc(y). Let \u03bc(y) < r and \u03bc(y) \u2264 0.5. It follows that (3)\u21d2(1). Let x, a, y \u2208 S such thatt \u2264 1, x; t] \u2208 \u03bc and  \u2208 \u03bc. By (2), we have \u03bc(x) \u2265 t and t + \u03bc(x) \u2264 1) or (\u03bc(y) \u2265 t and t + \u03bc(y) \u2264 1), which implies that t \u2264 0.5, a contradiction.(2)\u21d2(4). If there exists x, a, y \u2208 S and r, t \u2208  < min\u2061{r, t}.(4)\u21d2(2). Let \u03bc(xay) \u2265 min\u2061{\u03bc(x), \u03bc(y)}, then min\u2061{\u03bc(x), \u03bc(y)} < min\u2061{r, t} and consequently \u03bc(x) < r or \u03bc(y) < t. It follows that (a) If \u03bc(xay) < min\u2061{\u03bc(x), \u03bc(y)}, then by (4), we have 0.5 \u2265 min\u2061{\u03bc(x), \u03bc(y)}. Let \u03bc(x) < r and \u03bc(x) \u2264 0.5 or \u03bc(y) < t and \u03bc(y) \u2264 0.5. It follows (b) If \u03b3, \u2208\u03b3\u2228q\u03b4)-fuzzy filters, -fuzzy ideals, and -fuzzy generalized bi-ideals of ordered semigroups.From the time that fuzzy subgroups gained general acceptance over the decades, it has provided a central trunk to investigate similar type of generalization of the existing fuzzy subsystems of other algebraic structures. A contributing factor for the growth of fuzzy subgroups is increased by the reports from Yin and Zhan  and Ma e\u03b3, \u03b4 \u2208  be such that \u03b3 < \u03b4. For a fuzzy point  and a fuzzy subset \u03bc of X, we say thatx; t]\u2208\u03b3\u03bc if \u03bc(x) \u2265 t > \u03b3.q\u03b4\u03bc if \u03bc(x) + t > 2\u03b4.\u2208\u03b3\u2228q\u03b4\u03bc if \u2208\u03b3\u03bc or q\u03b4\u03bc.\u2208\u03b3\u2227q\u03b4\u03bc if \u2208\u03b3\u03bc and q\u03b4\u03bc.\u03b1\u03bc does not hold for \u03b1 \u2208 {\u2208\u03b3, q\u03b4, \u2208\u03b3\u2228q\u03b4, \u2208\u03b3\u2227q\u03b4}.In what follows let \u03b1 = \u2208\u03b3\u2227q\u03b4 is omitted. Because the set { | \u2208\u03b3\u2227q\u03b4} is empty for \u03bc(x) \u2264 \u03b4 that is, if x \u2208 S and r \u2208  \u2265 t > \u03b3 and \u03bc(x) + t > 2\u03b4. It follows that 2(x) = (x) + (x) \u2265 (x) + t > 2\u03b4 so that \u03bc(x) > \u03b4 which is contradiction to \u03bc(x) \u2264 \u03b4. This means that { | \u2208\u03b3\u2227q\u03b4} = \u2205.Note that, the case \u03bc of S is called an -fuzzy generalized bi-ideal of S if it satisfies the following conditions:(B1)x, y \u2208 S)\u2009\u2009.(\u2200(B2)x, a, y \u2208 S)\u2009\u2009\u2009\u2009.-fuzzy generalized bi-ideal of S.Then by \u03bc of S is an -fuzzy generalized bi-ideal of S if and only if the following conditions hold for all x, a, y \u2208 S:(B3)x \u2264 y)\u2192(max\u2061{\u03bc(x), \u03b3} \u2265 min\u2061{\u03bc(y), \u03b4}),((B4)\u03bc(xay), \u03b3} \u2265 min\u2061{\u03bc(x), \u03bc(y), \u03b4}). \u2265 t > \u03b3, \u03bc(x) < t, and \u03bc(x) + t < 2t \u2264 2\u03b4; that is, \u2208\u03b3\u03bc but (B1)\u21d2(B3). If there exists x, y \u2208 S with x \u2264 y and t \u2208  \u2265 t > \u03b3, \u03bc(x) < t and \u03bc(x) + t < 2\u03b4. It follows that \u03bc(x) < \u03b4 and so max\u2061{\u03bc(x), \u03b3} < min\u2061{t, \u03b4} \u2264 min\u2061{\u03bc(y), \u03b4}, a contradiction. Hence (B1) is valid.(B3)\u21d2(B1). Assume that there exists x, a, y \u2208 S such that\u03bc(xay) + r < 2r \u2264 2\u03b4; that is, \u2208\u03b3\u03bc, \u2208\u03b3\u03bc but \u03bc(xay), \u03b3} \u2265 min\u2061{\u03bc(x), \u03bc(y), \u03b4} for all x, y \u2208 S.(B2)\u21d2(B4). If there exists x, a, y \u2208 S and r, t \u2208  + min\u2061{r, t} \u2264 2\u03b4. It follows that \u03bc(xay) < \u03b4 and so(B4)\u21d2(B2). Assume that there exist S, whenever \u03bc is an -fuzzy generalized bi-ideal of S (\u03b1 \u2260 \u2208\u03b3\u2227q\u03b4) and 2\u03b4 = 1 + \u03b3.The set \u03bc is an -fuzzy generalized bi-ideal of S. Let a \u2208 S, \u03bc(x) > \u03b3, \u03bc(y) > \u03b3.Assume that Case\u2009\u20091. Then \u03b1\u03bc and \u03b1\u03bc, where \u03b1 \u2208 {\u2208\u03b3, \u2208\u03b3\u2228\u2009\u2009q\u03b4}. By (B2),\u03bc(xay) \u2265 min\u2061{\u03bc(x), \u03bc(y)} > \u03b3 or \u03bc(xay) + {\u03bc(x), \u03bc(y)} > 2\u03b4, and so \u03bc(xay)\u2265\u2009\u2009min{\u03bc(x), \u03bc(y)} > \u03b3 or \u03bc(xay) > 2\u03b4 \u2212 min\u2061{\u03bc(x), \u03bc(y)} \u2265 2\u03b4 \u2212 1 = \u03b3. Hence It follows that Case\u2009\u20092. Then \u03b1\u03bc and \u03b1\u03bc, where \u03b1 = q\u03b4, since 2\u03b4 = 1 + \u03b3. Analogous to the proof of Case\u20091, we have x, y \u2208 S and x \u2264 y if S.\u03bc of S defined as follows:A is a nonempty subset of S. Then\u2009\u2009A is a generalized bi-ideal of S if and only if \u03bc is an -fuzzy generalized bi-ideal of S.Consider a fuzzy subset A is a generalized bi-ideal of S and x, y \u2208 S with x \u2264 y, t \u2208  \u2265 t > \u03b3\u2009\u2009follows that y \u2208 A.Case\u2009\u20092. If q\u03b4\u03bc, then \u03bc(y) + t > 2\u03b4 and so \u03bc(y) > 2\u03b4 \u2212 t \u2265 2\u03b4 \u2212 1 = \u03b3\u2009\u2009follow that y \u2208 A.y \u2208 A and hence x \u2208 A. By definition of \u03bc we have \u03bc(x) \u2265 \u03b4. If t \u2264 \u03b4, then \u03bc(x) \u2265 \u03b4 \u2265 t > \u03b3 and hence \u2208\u03b3\u03bc but if t > \u03b4, then \u03bc(x) + t \u2265 \u03b4 + t > 2\u03b4; that is, q\u03b4\u03bc. Thus for \u2208\u03b3\u03bc we have \u2208\u03b3\u2228q\u03b4\u03bc.In the above two cases x, y, a \u2208 S and r, t \u2208  \u2265 r > \u03b3 and \u03bc(y) \u2265 t > \u03b3 follow that x, y \u2208 A.Case\u2009\u20092. If q\u03b4\u03bc and q\u03b4\u03bc, then \u03bc(x) + r > 2\u03b4 and \u03bc(y) + t > 2\u03b4\u2009\u2009and so \u03bc(x) > 2\u03b4 \u2212 r \u2265 2\u03b4 \u2212 1 = \u03b3 and \u03bc(y) > 2\u03b4 \u2212 t \u2265 2\u03b4 \u2212 1 = \u03b3 follow that x, y \u2208 A.Case\u2009\u20093. Similarly, if \u2208\u03b3\u03bc and q\u03b4\u03bc, then x, y \u2208 A.Case\u2009\u20094. If q\u03b4\u03bc and \u2208\u03b3\u03bc, then x, y \u2208 A.x, y \u2208 A. Hence xay \u2208 A and by definition we have that \u03bc(xay) \u2265 \u03b4. If min{r, t} \u2264 \u03b4, then \u03bc(xay) \u2265 \u03b4 \u2265 min\u2061{r, t} > \u03b3; that is,\u2009\u2009\u2208\u03b3\u03bc. If min{r, t} > \u03b4, then \u03bc(xay) + min\u2061{r, t} > \u03b4 + \u03b4 = 2\u03b4; that is, q\u03b4\u03bc. Therefore \u2208\u03b3\u2228q\u03b4\u03bc.Thus, in any case, \u03bc is an -fuzzy generalized bi-ideal of S. It is easy to see that A is a generalized bi-ideal of S.Conversely, assume that \u03b3\u2228q\u03b4, \u2208\u03b3\u2228q\u03b4)-fuzzy generalized bi-ideal of S is an -fuzzy generalized bi-ideal of S.Every -fuzzy generalized bi-ideal of S is an -fuzzy generalized bi-ideal of S.Every -fuzzy generalized bi-ideal of S;by \u03bc is not an -fuzzy generalized bi-ideal of S, since \u22080.30\u03bc and \u22080.30\u03bc but \u03bc is not an -fuzzy generalized bi-ideal of S, since \u22080.30\u2228q0.60\u03bc and \u22080.30\u2228q0.60\u03bc but Then,\u03bc of S, we define the following sets for all t \u2208 -fuzzy generalized bi-ideal and crisp generalized bi-ideal of S.The next theorem provides the relationship between -fuzzy generalized bi-ideal of S,\u03bct(\u2260\u2205) is a generalized bi-ideal of S for all t \u2208 -fuzzy generalized bi-ideal of S. Let x, y \u2208 S with x \u2264 y and t \u2208 -fuzzy generalized bi-ideal of S, therefore \u2208\u03b3\u2228q\u03b4\u03bc. If \u2208\u03b3\u03bc, then x \u2208 \u03bct and if q\u03b4\u03bc, then \u03bc(x) > 2\u03b4 \u2212 t > t > \u03b3; that is,\u2009\u2009x \u2208 \u03bct.(1)\u21d2(2). Let x, y, a \u2208 S be such that x, y \u2208 \u03bct for some t \u2208 -fuzzy generalized bi-ideal of S, therefore \u2208\u03b3\u2228q\u03b4\u03bc. If \u2208\u03b3\u03bc, then xay \u2208 \u03bct and if q\u03b4\u03bc, then \u03bc(xay) > 2\u03b4 \u2212 t > t > \u03b3; that is,\u2009\u2009xay \u2208 \u03bct. Therefore \u03bct is a generalized bi-ideal of S.Let \u03bct(\u2260\u2205) is a generalized bi-ideal of S for all t \u2208 , \u03b3} < min\u2061{\u03bc(y), \u03b4}; then there exists t \u2208 , \u03b3} < t \u2264 min\u2061{\u03bc(y), \u03b4}; this follows that \u2208\u03b3\u03bc; that is, y \u2208 \u03bct but \u03bc(x), \u03b3} \u2265 min\u2061{\u03bc(y), \u03b4} for all x, y \u2208 S with x \u2264 y. Let x, y, a \u2208 S and max\u2061{\u03bc(xay), \u03b3} < min\u2061{\u03bc(x), \u03bc(y), \u03b4}; then max\u2061{\u03bc(xay), \u03b3} < t \u2264 min\u2061{\u03bc(x), \u03bc(y), \u03b4} for some t \u2208 \u21d2(1). Assume that \u03bc is an -fuzzy generalized bi-ideal.Consequently, \u03bc\u2009\u2009of an ordered semigroup S, then \u03bc is an -fuzzy generalized bi-ideal of S if and only if \u03bcr\u03b4(\u2260\u2205) is a generalized bi-ideal of S for all t \u2208 -fuzzy generalized bi-ideal of S. Let x, y \u2208 S with x \u2264 y and t \u2208  > 2\u03b4 \u2212 t. Since t \u2208 -fuzzy generalized bi-ideal of S; therefore\u03bc(x) \u2265 2\u03b4 \u2212 t. Hence x \u2208 \u03bct\u03b4.Let x, y, a \u2208 S be such that x, y \u2208 \u03bct\u03b4 for some t \u2208  > 2\u03b4 \u2212 t, that is, \u03bc(y) > 2\u03b4 \u2212 t. Since \u03bc is an -fuzzy generalized bi-ideal of S, therefore\u03bc(xay) \u2265 2\u03b4 \u2212 t. Hence xay \u2208 \u03bct\u03b4. Consequently, \u03bct\u03b4 is generalized bi-ideal.Let \u03bct\u03b4(\u2260\u2205) be a generalized bi-ideal of S for all t \u2208 , \u03b3} < t = min\u2061{\u03bc(y), \u03b4}; this follows that y \u2208 \u03bct\u03b4 but \u03bc(x), \u03b3} \u2265 min\u2061{\u03bc(y), \u03b4} for all x, y \u2208 S with x \u2264 y. Let x, y, a \u2208 S and max\u2061{\u03bc(xay), \u03b3} < t = min\u2061{\u03bc(x), \u03bc(y), \u03b4}; this implies that x \u2208 \u03bct\u03b4 and y \u2208 \u03bct\u03b4 but \u03bc(xay), \u03b3} \u2265 min\u2061{\u03bc(x), \u03bc(y), \u03b4}. Consequently, \u03bc is an -fuzzy generalized bi-ideal.Conversely, let \u03bc\u2009\u2009of an ordered semigroup S is an -fuzzy generalized bi-ideal of S if and only if [\u03bc]t\u03b4(\u2260\u2205) is a generalized bi-ideal of S for all t \u2208 -fuzzy subsemigroup if it satisfies the following condition:(B5)x, y \u2208 S)\u2009\u2009\u2009\u2009.-fuzzy bi-ideal of S if it is -fuzzy subsemigroup and -fuzzy generalized bi-ideal of S.A fuzzy subset \u03bc of S is an -fuzzy bi-ideal of S if and only if the following conditions hold for all x, a, y \u2208 S:(B6)x \u2264 y)\u2192(max\u2061{\u03bc(x), \u03b3} \u2265 min\u2061{\u03bc(y), \u03b4}),((B7)\u03bc(xy), \u03b3} \u2265 min\u2061{\u03bc(x), \u03bc(y), \u03b4}),(max\u2061{(B8)\u03bc(xay), \u03b3} \u2265 min\u2061{\u03bc(x), \u03bc(y), \u03b4}). + r < 2r \u2264 2\u03b4; that is, \u2208\u03b3\u03bc, \u2208\u03b3\u03bc but \u03bc(xy), \u03b3} \u2265 min\u2061{\u03bc(x), \u03bc(y), \u03b4} for all x, y \u2208 S.(B5)\u21d2(B7). If there exists x, y \u2208 S and r, t \u2208  + min\u2061{r, t} \u2264 2\u03b4. It follows that \u03bc(xy) < \u03b4 and so(B7)\u21d2(B5). Assume that there exist The remaining proof follows from \u03b3, \u2208\u03b3\u2228q\u03b4)-fuzzy bi-ideals and -fuzzy generalized bi-ideals is provided.Next, the relationship between -fuzzy bi-ideal \u03bc of S is -fuzzy generalized bi-ideal.Every -fuzzy generalized bi-ideal of S but not an -fuzzy bi-ideal, since \u22080.28\u03bc but Then by A of an ordered semigroup S is a generalized bi-ideal of S if and only if\u03b3, \u2208\u03b3\u2228q\u03b4)-fuzzy generalized bi-ideal of S.A nonempty subset The proof is straightforward and is omitted.In the last couple of decades, the importance of fuzzification of ordered semigroups and related structures is increased due to the pioneering role of aforementioned structures in advanced fields like computer science, error correcting codes, and fuzzy automata. In contribution to this fact, we define and investigate x \u2264 y),(B10)A fuzzy subset \u03bc of S such that \u03bc(x) \u2265 \u03b4 for any x \u2208 S in the case \u03bc(x) < r and \u03bc(x) + r < 2\u03b4. Thus \u03bc(x) + \u03bc(x) < \u03bc(x) + r \u2264 2\u03b4, which implies \u03bc(x) < \u03b4. This means that The case when S = {a, b, c, d, e} with the following multiplication table and order relation:Consider an ordered semigroup \u03bc : S \u2192  as follows:\u03bc is an (S.Define a fuzzy subset \u03bc of S is an (S if and only if the following conditions hold:(B11)x, y \u2208 S)\u2009\u2009(max\u2061{\u03bc(x), \u03b4} \u2265 \u03bc(y) with x \u2264 y),(\u2200(B12)x, a, y \u2208 S)\u2009\u2009(max\u2061{\u03bc(xay), \u03b4} \u2265 min\u2061{\u03bc(x), \u03bc(y)}). , \u03b4} < \u03bc(y). Then max\u2061{\u03bc(x), \u03b4} < t \u2264 \u03bc(y) for some t \u2208 , \u03b4} \u2265 \u03bc(y) for all x, y \u2208 S with x \u2264 y.(B9)\u21d2(B11). Assume that there exists x, y \u2208 S with x \u2264 y and t \u2208  < t, \u03bc(y) \u2265 t, and \u03bc(y) + t > 2\u03b4 and hence \u03bc(y) > \u03b4. This follows that(B11)\u21d2(B9). Assume that there exists x, a, y \u2208 S such that max\u2061{\u03bc(xay), \u03b4} < min\u2061{\u03bc(x), \u03bc(y)}, then there exists t \u2208 , \u03b4} < t \u2264 min\u2061{\u03bc(x), \u03bc(y)}. It follows that \u03bc(xay), \u03b4} \u2265 min\u2061{\u03bc(x), \u03bc(y)} for all x, a, y \u2208 S.(B10)\u21d2(B12). If x, a, y \u2208 S and r, t \u2208  < min\u2061{r, t}, \u03bc(x) \u2265 r, \u03bc(y) \u2265 t, \u03bc(x) + r > 2\u03b4, and \u03bc(y) + r > 2\u03b4. It follows that \u03bc(x) > \u03b4 and \u03bc(y) > \u03b4, and so(B12)\u21d2(B10). Assume that there exist \u03bc of S is called S if it satisfies the following conditions:(B13)A fuzzy subset \u03bc of S is called S if it is S.A fuzzy subset \u03bc of S is an (S if and only if the following conditions hold:(B14)x, y \u2208 S)\u2009\u2009(max\u2061{\u03bc(x), \u03b4} \u2265 \u03bc(y) with x \u2264 y),(\u2200(B15)x, y \u2208 S)\u2009\u2009(max\u2061{\u03bc(xy), \u03b4} \u2265 min\u2061{\u03bc(x), \u03bc(y)})(\u2200(B16)x, a, y \u2208 S)\u2009\u2009(max\u2061{\u03bc(xay), \u03b4} \u2265 min\u2061{\u03bc(x), \u03bc(y)}). , \u03b4} < t \u2264 min\u2061{\u03bc(x), \u03bc(y)}. It follows that \u03bc(xy), \u03b4} \u2265 min\u2061{\u03bc(x), \u03bc(y)} for all x, y \u2208 S.(B13)\u21d2(B15). Suppose x, y \u2208 S and r, t \u2208  < min\u2061{r, t}, \u03bc(x) \u2265 r, \u03bc(y) \u2265 t, \u03bc(x) + r > 2\u03b4, and \u03bc(y) + r > 2\u03b4. It follows that \u03bc(x) > \u03b4 and \u03bc(y) > \u03b4, and so(B15)\u21d2(B13). If there exist \u03bc of S is  > \u03b4. As \u03bc is an S, therefore\u03bc(x) > \u03b4 and hence Assume that x, a, y \u2208 S be such that \u03bc(x) > \u03b4, \u03bc(y) > \u03b4, and by (B8)Let S.that is, \u03bc of S defined asConsider a fuzzy subset A is a generalized bi-ideal of S if and only if \u03bc is an S.Then A is a generalized bi-ideal of S. Let x, a, y \u2208 S and r, t \u2208  < min\u2061{r, t} \u2264 1 and so \u03bc(xay) = \u03b4 < min\u2061{r, t}; that is, xay \u2209 A. It follows that x \u2209 A or y \u2209 A, and so \u03bc(x) = \u03b4 < r or \u03bc(y) = \u03b4 < t. Hence Case\u2009\u20092. \u03bc(xay) + min\u2061{r, t} \u2264 2\u03b4. If \u03bc(xay) = \u03b4, analogous to the proof of Case\u20091, we have \u03bc(xay) = 1; then max\u2061{\u03bc(x), \u03bc(y)} + min\u2061{r, t}\u2264\u20091 + min\u2061{r, t} = \u2009\u03bc(xay) + min\u2061{r, t} \u2264 2\u03b4. It follows that \u03bc(x) + r \u2264 2\u03b4 or \u03bc(y) + t \u2264 2\u03b4. Hence Case\u2009\u20093. x, y \u2208 S with x \u2264 y.In a similar way we can show that \u03bc is a -fuzzy generalized bi-ideals and  is also constructed.Due to the significant role of ordered semigroups and their different characterizations in several applied fields such as control engineering, fuzzy automata, coding theory, and computer science, the latest research has been carried out in the last few decades by considering various characterizations of ordered semigroups in terms of different types of fuzzy ideals. In this paper, we determined a more generalized form of Davvaz and Khan  approach"}
{"text": "Here, we aim to present some new fractional integral inequalities involving generalized Erd\u00e9lyi-Kober fractional q-integral operator due to Gaulu\u00e9, whose special cases are shown to yield corresponding inequalities associated with Kober type fractional q-integral operators. The cases of synchronous functions as well as of functions bounded by integrable functions are considered.In recent years, a remarkably large number of inequalities involving the fractional  If f and g are synchronous on , that is,x, y \u2208 , then we have  = q(x) for any x, y \u2208 , we get the Chebyshev inequality :1)T\u2212f(3)T\u2212f\u2212f|\u2265m and |g\u2032(x)|\u2265r for x \u2208 , then we haveIf f and g are asynchronous on , then we havef and g are two differentiable functions on  with |f\u2032(x)|\u2264M and |g\u2032(x)|\u2264R for x \u2208  and p is a positive integrable function on , then we haveIf nctional  has attrnctional  has alsoq-integral inequalities has been of great importance due to the fundamental role in the theory of differential equations. In recent years, a number of researchers have done deep study, that is, the properties, applications, and different extensions of various fractional q-integral operators .q-calculus analogs of some classical integral inequalities. In particular, we will find q-generalizations of the Chebyshev integral inequalities by using the generalized Erd\u00e9lyi-Kober fractional q-integral operator introduced by Galu\u00e9  is defined byDqf(0) = lim\u2061t\u21920Dqf(t). It is noted thatf(t) is differentiable.We begin by noting that F. J. Jackson was the first to develop F(t) is a q-antiderivative of f(t) if DqF(t) = f(t). It is denoted byThe function  Jackson integral of f(t) is thus defined, formally, byThea < b. The definite q-integral is defined as follows:Suppose that 0 < A more general version of  is givenz) (n \u2208 N0) to real numbers. The q-factorial function [n]q!\u2009\u2009(n \u2208 N0) of n! defined byn by a \u2212 1 in (q-Gamma function \u0393q(a) byThe classical Gamma function \u0393(follows:(1\u2212q)\u2212n\u220fkq-analogue of (t \u2212 a)n is defined by the polynomial\u03b3 \u2208 R, thenThe R(\u03b2), R(\u03bc) > 0 and \u03b7 \u2208 C. Then a generalized Erd\u00e9lyi-Kober fractional integral Iq\u03b1,\u03b2,\u03b7 for a real-valued continuous function f(t) is defined by (26)Iq\u03b7q-analogue of the Kober fractional integral operator is given by (27)Iq\u03b7\u03bc > 0 and k \u2208 N0. If f : ,Iq\u03b7,\u03bc,\u03b2{ny adding .q < 1, let f and g be two continuous and synchronous functions on [0, \u221e), and let l, m, n : [0, \u221e)\u2192[0, \u221e) be continuous functions. Then, the following inequality holds true:\u03bc, \u03bd, \u03b2, \u03b4 > 0 and \u03b7, \u03b6 \u2208 C.Let 0 < u = m and v = n in , after a little simplification, we getu, v by l, n and u, v by l, m in  and Iq\u03b7,\u03bc,\u03b2{n}(t), respectively, we get the following two inequalities:Setting v = n in , we have = n in . The special case of 42) in Thequality  in Theor\u03b2 = 1 in , and let l, m, n : [0, \u221e)\u2192[0, \u221e) be continuous functions. Then, the following inequality holds true:\u03bc > 0 and \u03b7 \u2208 C.Let 0 < q < 1, let f and g be two continuous and synchronous functions on [0, \u221e), and let l, m, n : [0, \u221e)\u2192[0, \u221e) be continuous functions. Then, the following inequality holds true:\u03bc, \u03bd > 0 and \u03b7, \u03b6 \u2208 C.Let 0 < \u03b7 = 0 and \u03b2 = 1 in \u03b7 = \u03b6 = 0 and \u03b2 = \u03b4 = 1 in If we take q-integral operator in the case where the functions are bounded by integrable functions and are not necessary increasing or decreasing as are the synchronous functions.In this section we obtain some new inequalities involving Erd\u00e9lyi-Kober fractional q < 1, let f be an integrable function on [0, \u221e), and let u, v : [0, \u221e)\u2192[0, \u221e) be continuous functions. Assume the following.H1)\u2009 such thatThere exist two integrable functions Then, for t > 0, \u03bc, \u03b2 > 0, and \u03b7 \u2208 C, we haveLet 0 < H1), for all \u03c4 \u2265 0 and \u03c1 \u2265 0, we have\u03b2t\u03b2(\u03b7+\u03bc)\u2212/\u0393q(\u03bc))(t\u03b2 \u2212 \u03c4\u03b2q)\u03bc\u22121)(\u03c4\u03b2(\u03b7+1)\u22121u(\u03c4), \u03c4 \u2208 , and integrating both sides with respect to \u03c4 on , we obtain\u03b2t\u03b2(\u03b7+\u03bc)\u2212/\u0393q(\u03bc))(t\u03b2 \u2212 \u03c1\u03b2q)\u03bc\u22121)(\u03c1\u03b2(\u03b7+1)\u22121v(\u03c1), \u03c1 \u2208 , and integrating both sides with respect to \u03c1 on , we get inequality f(\u03c1)e obtainIq\u03b7,\u03bc,\u03b2{uequality  as requeAs special cases of Theorems q < 1, let f be an integrable function on [0, \u221e) satisfying m \u2264 f(t) \u2264 M, for all t \u2208 [0, \u221e), let u, v : [0, \u221e)\u2192[0, \u221e) be continuous functions, and let\u2009\u2009m, M \u2208 R. Then, for t > 0, \u03bc, \u03b2 > 0, and \u03b7 \u2208 C, we haveLet 0 < q < 1, let f be an integrable function on [1, \u221e), and let u, v : [0, \u221e)\u2192[0, \u221e) be continuous functions. Assume that there exists an integrable function \u03c6(t) on [0, \u221e) and a constant M > 0 such thatt > 0, \u03bc, \u03b2 > 0, and \u03b7 \u2208 C; we haveLet 0 < q < 1, let f be an integrable function on [0, \u221e), let u, v : [0, \u221e)\u2192[0, \u221e) be continuous functions, and let \u03b81, \u03b82 > 0 satisfying 1/\u03b81 + 1/\u03b82 = 1. Suppose that (H1) holds. Then, for t > 0, \u03bc, \u03b2 > 0, and \u03b7 \u2208 C, we haveLet 0 < x = \u03c62(\u03c4) \u2212 f(\u03c4) and y = f(\u03c1) \u2212 \u03c61(\u03c1), \u03c4, \u03c1 \u2265 0, we have\u03c4, \u03c1 \u2208 , and integrating with respect to \u03c4 and \u03c1 from 0 to t, we deduce the desired result in 1\u03b81x we have1\u03b81(\u03c62(\u03c4)esult in .q < 1, let f be an integrable function on [0, \u221e) satisfying m \u2264 f(t) \u2264 M, for all t \u2208 [0, \u221e), let u, v : [0, \u221e)\u2192[0, \u221e) be continuous functions, and let m, M \u2208 R. Then, for t > 0, \u03bc, \u03b2 > 0, and \u03b7 \u2208 C, we haveLet 0 < q < 1, let f be an integrable function on [0, \u221e), let u, v : [0, \u221e)\u2192[0, \u221e) be continuous functions, and let \u03b81, \u03b82 > 0 satisfying \u03b81 + \u03b82 = 1. In addition, suppose that (H1) holds. Then, for t > 0, \u03bc, \u03b2 > 0, and \u03b7 \u2208 C, we haveLet 0 < x = \u03c62(\u03c4) \u2212 f(\u03c4) and y = f(\u03c1) \u2212 \u03c61(\u03c1), \u03c4, \u03c1 > 1, we have\u03c4, \u03c1 \u2208 , and integrating with respect to \u03c4 and \u03c1 from 0 to t, we deduce inequality \u03b81x+ we have\u03b81(\u03c62(\u03c4)\u2212equality .q < 1, let f be an integrable function on [0, \u221e) satisfying m \u2264 f(t) \u2264 M, for all t \u2208 [0, \u221e), let u, v : [0, \u221e)\u2192[0, \u221e) be continuous functions, and let m, M \u2208 R. Then, for t > 0, \u03bc, \u03b2 > 0, and \u03b7 \u2208 C, we haveLet 0 < a \u2265 0, p \u2265 q \u2265 0, and p \u2260 0. Then,Assume that q < 1, let f be an integrable function on [0, \u221e), let u : [0, \u221e)\u2192[0, \u221e) be a continuous function, and let constants p \u2265 q \u2265 0, p \u2260 0. In addition, assume that (H1) holds. Then, for any k > 0, t > 0, \u03bc, \u03b2 > 0, and \u03b7 \u2208 C, the following two inequalities hold:Let 0 < H1) and p \u2265 q \u2265 0, p \u2260 0, it follows thatk > 0. Multiplying both sides of (\u03b2t\u03b2(\u03b7+\u03bc)\u2212/\u0393q(\u03bc))(t\u03b2 \u2212 \u03c4\u03b2q)\u03bc\u22121)(\u03c4\u03b2(\u03b7+1)\u22121u(\u03c4), \u03c4 \u2208 , and integrating the resulting identity with respect to \u03c4 from 0 to t, one has inequality (i). Inequality (ii) is proved by setting a = f(\u03c4) \u2212 \u03c61(\u03c4) in By condition (ows that(\u03c62(\u03c4)\u2212f satisfying m \u2264 f(t) \u2264 M, for all t \u2208 [0, \u221e), let u, v : [0, \u221e)\u2192[0, \u221e) be continuous functions, and let m, M \u2208 R. Then, for t > 0, \u03bc, \u03b2 > 0, and \u03b7 \u2208 C, we haveLet 0 < q < 1, let f and g be two integrable functions on [0, \u221e), and let u, v : [0, \u221e)\u2192[0, \u221e) be continuous functions. Suppose that (H1) holds and moreover we assume the following.\u2009H2) There exist \u03c81 and \u03c82 integrable functions on [0, \u221e) such that, from (H1) and (H2), we have for t \u2208 [0, \u221e) that\u03b2t\u03b2(\u03b7+\u03bc)\u2212/\u0393q(\u03bc))(t\u03b2 \u2212 \u03c4\u03b2q)\u03bc\u22121)(\u03c4\u03b2(\u03b7+1)\u22121u(\u03c4), \u03c4 \u2208 , and integrating both sides with respect to \u03c4 on , we obtain\u03b2t\u03b2(\u03b7+\u03bc)\u2212/\u0393q(\u03bc))(t\u03b2 \u2212 \u03c1\u03b2q)\u03bc\u22121)(\u03c1\u03b2(\u03b7+1)\u22121v(\u03c1), \u03c1 \u2208 , and integrating both sides with respect to \u03c1 on , we get the desired inequality (i).To prove g(\u03c1)e obtaing(\u03c1)Iq\u03b7,\u03bcii)\u2013(iv), we use the following inequalities:To prove , let u, v : [0, \u221e)\u2192[0, \u221e) be continuous functions, and let \u03b81, \u03b82 > 0 satisfying 1/\u03b81 + 1/\u03b82 = 1. Suppose that (H1) and (H2) hold. Then, for t > 0, \u03bc, \u03b2 > 0, and \u03b7 \u2208 C, the following inequalities hold:Let i)\u2013(iv) can be proved by choosing the parameters in the Young inequality [The inequalities (i)f and g be two integrable functions on [0, \u221e), let u, v : [0, \u221e)\u2192[0, \u221e) be continuous functions, and let \u03b81, \u03b82 > 0 satisfying \u03b81 + \u03b82 = 1. Suppose that (H1) and (H2) hold. Then, for t > 0, \u03bc, \u03b2 > 0, and \u03b7 \u2208 C, the following inequalities hold:Let i)\u2013(iv) can be proved by choosing the parameters in the Weighted AM-GM [The inequalities (ed AM-GM :(80)(i)f and g be two integrable functions on [0, \u221e), let u, v : [0, \u221e)\u2192[0, \u221e) be continuous functions, and let constants p \u2265 q \u2265 0, p \u2260 0. Assume that (H1) and (H2) hold. Then, for any k > 0, t > 0, \u03bc, \u03b2 > 0, and \u03b7 \u2208 C, the following inequalities hold:Let i)\u2013(iv) can be proved by choosing the parameters in The inequalities (q-integral inequalities and fractional calculus. Moreover, they are expected to find some applications for establishing uniqueness of solutions in fractional boundary value problems and in fractional partial differential equations. In last, use of the generalized Erd\u00e9lyi-Kober fractional q-integral operator due to Gaulu\u00e9 is the advantage of our results because after setting suitable parameter values in our main results, we get known results established by number of authors.We conclude our present investigation with the remark that the results derived in this paper are general in character and give some contributions to the theory of"}
{"text": "Scientific Reports5: Article number: 1076810.1038/srep10768; published online: 06112015; updated: 09212015LF\u2009=\u2009\u03b2\u00b7L\u2019 should read \u2018LF\u2009=\u2009(\u03b1\u2009+\u2009\u03b2)\u2009\u00b7\u2009L\u2019. The correct Fig. 3a appears below as This Article contains a typographical error in Fig. 3a, where \u2018"}
{"text": "In the description of the test for cell viability, \u201c4\u00a0\u03bcM ethidium bromide\u201d should be replaced by \u201c4\u00a0\u03bcM ethidium homodimer-1 (EthD-1)\u201d.In the Results and discussion section, the sentence \u201cThe surface morphology of gelatin matrix is visualized using AFM and SEM, respectively \u201d should read \u201cThe surface morphology of gelatin matrix is visualized using SEM and AFM, respectively .\u201dIn the Results and discussion section, the sentence \u201cAFM images also depicted a relatively smooth gelatin surface with an RMS Rq of 155.6\u2009\u00b1\u200918\u00a0nm \u201d should read \u201cAFM images also depicted a relatively smooth gelatin surface with an RMS Rq of 155.6\u2009\u00b1\u200912.3\u00a0nm \u201d.In the original version of this article , there w"}
{"text": "F*(K) be the set of all fuzzy complex numbers. In this paper some classical and measure-theoretical notions are extended to the case of complex fuzzy sets. They are fuzzy complex number-valued distance on F*(K), fuzzy complex number-valued measure on F*(K), and some related notions, such as null-additivity, pseudo-null-additivity, null-subtraction, pseudo-null-subtraction, autocontionuous from above, autocontionuous from below, and autocontinuity of the defined fuzzy complex number-valued measures. Properties of fuzzy complex number-valued measures are studied in detail.Let  It is well known that additivity of a classical measure primly depicted measure problems under error-free condition. But when measure error was unavoidable, additivity could not fully depict the measure problems under certain condition. To overcome such difficulties, fuzzy measure has been developed. Research on fuzzy measures was very deep in those aspects: research based on a certain number of subsets of a classic set and a real value nonaddable measure  \u201337 34\u201337R is the set of all real numbers set, K is the set of all complex numbers, X is an ordinary set, F*(R) is the set of all real fuzzy numbers on R, \u0394(R) is the set of all interval numbers,  is a measurable space , and F*(K) is the set of all fuzzy complex numbers on K. Let F+*(K) = In this paper, K is denoted by F*(K).Let A, B of R, write \u225cA + iB = {x + iy\u2223x \u2208 A, y \u2208 B}. The operation \u2218\u2208{+, \u2212, \u00b7} is described as follows: c1\u2032\u2218c2\u2032 =  for any c1\u2032, c2\u2032 \u2208 F*(K);c \u00b7 c1\u2032 =  for any c1\u2032 \u2208 F*(K) and c =  \u2208 K (c\u2032 \u2208 F*(K) is said to be a fuzzy infinity [c1\u2032, c2\u2032 \u2208 F*(K), one makes the following appointments:c1\u2032 \u2264 c2\u2032\u2009\u2009Rec1\u2032 \u2264 Rec2\u2032 and Im\u2061c1\u2032 \u2264 Im\u2061c2\u2032;if and only if \u2009c1\u2032 = c2\u2032 if and only if c1\u2032 \u2264 c2\u2032 and c2\u2032 \u2264 c1\u2032;\u2009c1\u2032 < c2\u2032 if and only if c1\u2032 \u2264 c2\u2032 and Rec1\u2032 \u2264 Rec2\u2032 or Im\u2061c1\u2032 < Im\u2061c2\u2032;\u2009c\u2032 \u2265 0 if and only if Rec\u2032 \u2265 0, Im\u2061c\u2032 \u2265 0.One uses A* to denote a family (which is obviously nonempty) of subsets of F*(K) that satisfies the following conditions:for each for each For any subsets infinity  (writtenF*(R):for any for any for any A mapping F*(R).The mapping F*(K):for any for any for any Analogously, a mapping F*(R). Then the mapping \u03c1\u2032 : F*(K) \u00d7 F*(K) \u2192 F*(R) defined byF*(K).Let cn\u2032} \u2282 F*(K) and let c\u2032 \u2208 F*(K). {cn\u2032} is said to converge to c\u2032 according to a fuzzy metric \u03c1\u2032 on F*(K) (written as lim\u2061n\u2192\u221ecn\u2032 = c\u2032) if, for each \u03b5 > 0, there exists a positive integer N such that \u03c1\u2032 < \u03b5 for all n \u2265 N.Let {The notion of complex fuzzy measure on family of classical sets was given in .\u03c3-algebra A composed of subsets of X is a mapping \u03bc(\u03d5) = 0;A \u2282 B, then |\u03bc(A)| \u2264 |\u03bc(B)|;if An}1\u221e\u2191, then \u03bc(\u22c3n=1\u221eAn) = lim\u2061n\u2192\u221e\u03bc(An);if {An}1\u221e\u2193 and |\u03bc(An0)| < +\u221e for some n0, then \u03bc(\u22c2n=1\u221eAn) = lim\u2061n\u2192\u221e\u03bc(An).if {Let In this paper we need an expansion of this notion. First we defined the concept of fuzzy complex value distance.F*(K):for any for any for any A mapping F*(K) if and only if F*(K).It can be easily seen that a mapping \u03b5 > 0, there exists a positive integer N such that n \u2265 N hold, then Let Z be a nonempty complex number set, let F(Z) be the set of all complex fuzzy sets on Z, and let F(Z). A complex fuzzy set-value complex fuzzy measure is a mapping for any n = 1,2,\u2026), then (lower semicontinuous) if n = 1,2,\u2026) and n0, then (upper semicontinuous) if Let Apparently, a complex fuzzy set-value complex fuzzy measure is also a kind of special generalized fuzzy measures.0-add if null-additive  if autocontinuous from above  if autocontinuous from below  if autocontinuous if it is both autoc. \u2193 and autoc. \u2191.A mapping F(Z) is said to be pseudo-null-additive  if, for each \u03b5 = \u03b51 + i\u03b52 > 0, there exists a \u03b4 = \u03b41 + i\u03b42 > 0 such that A complex fuzzy set-value complex fuzzy measure X be a set and F(X) the set of all fuzzy sets on X. Then a subfamily F \u2282 F(X) is addable if and only if it satisfies the following conditions :X \u2208 F;x \u2208 X).where We first have the following result.F \u2282 F(Z) is a complex fuzzy value fuzzy complex measure on F.Every fuzzy complex measure n0. By monotonicity of n \u2265 n0. Since We only prove the upper continuity and lower continuity of Assume \u03c3-algebra F \u2282 F(Z) is exhaustive.Every complex fuzzy set-value complex fuzzy measure z0 \u2208 Z or z0 \u2208 Z; that is, n \u2265 1 or n \u2265 1. Without loss of generality, we assume the first. Then there are two distinct indexes kn1 and kn2 such that Suppose \u03c3-algebra F, On the other hand, since If k, Since n \u2265 k, As Similarly, From properties of upper limits and lower limits we can see the following theorem holds.F(Z), then, for any Let Let Similar to F(Z), then, for any Let F(Z) which is pseudo-zero addable about Assume that Let Similarly, we have the following.F(Z) which is pseudo-zero subtractable about Suppose that \u03c3-algebra F \u2282 F(Z) and autoc.\u2009\u2193, then Let \u03b50 = \u03b50\u2032 + i\u03b50\u2032\u2032  such that, for any natural numbers n, m, there exist n0 \u2265 1. This conflicts with the hypothesis.Suppose that \u03b4n} of real numbers satisfying \u03b4n > 0  and \u03b4n\u21980 and a subsequence k \u2265 1). Furthermore, Suppose that \u03b5\u2032, \u03b5\u2032\u2032, \u03b41\u2032, \u03b41\u2032\u2032 > 0, let \u03b5 = \u03b5\u2032 + i\u03b5\u2032\u2032, \u03b41 = \u03b41\u2032 + i\u03b41\u2032\u2032, \u03b41\u2032 \u2208 , and \u03b41\u2032\u2032 \u2208 . Since \u03b41 \u2208  such that n1 such that \u03b42 = \u03b42\u2032 + i\u03b42\u2032\u2032 such that \u03b42\u2032 \u2208 , \u03b42\u2032\u2032 \u2208 , \u03b42 = \u03b42\u2032 + i\u03b42\u2032\u2032 > 0, there exists an n2 > n1 such that For any real numbers \u03b42 = \u03b42\u2032 + i\u03b42\u2032\u2032 > 0 and there exists \u03b43 = \u03b43\u2032 + i\u03b43\u2032\u2032 > 0, \u03b43\u2032 \u2208 , and \u03b43\u2032\u2032 \u2208  such that \u03b43 = \u03b43\u2032 + i\u03b43\u2032\u2032 > 0, \u03b43\u2032 \u2208 , \u03b43\u2032\u2032 \u2208  since n3 > n2, such that For nk+1 > nk > nk\u22121 > \u22ef>n1, and \u03b4k < \u03b4k\u22121\u2227\u03b5/2k\u22121, such that k = 1,2, 3,\u2026, r + 1, r \u2265 1).Generally we can get Let Hence"}
{"text": "As an application of our results, periodic points of weakly contractive mappings are obtained. We also derive certain new coincidence point and common fixed point theorems in partially ordered G-metric spaces. Moreover, some examples are provided here to illustrate the usability of the obtained results.The aim of this paper is to present some coincidence and common fixed point results for generalized ( G-metric space, was introduced by Mustafa and Sims . Let f : X \u2192 X be defined by fx = x1/3 and let \u03b1 : X \u00d7 X \u00d7 X \u2192 [0, \u221e) be defined by \u03b1 = y + z \u2212 2x + 1. Then, x \u2264 x1/3 = fx for all x \u2208 X. That is, 2x1/3 \u2212 2x + 1 \u2265 1. Thus, f is an \u03b1-dominating map.Let X, G) be a G-metric space. We say that X is \u03b1-regular if and only if the following hypothesis holds.Let  \u2265 1 such that xn \u2192 z as n \u2192 \u221e, it follows that \u03b1 \u2265 1 or \u03b1 \u2265 1 or \u03b1 \u2265 1, for all n \u2208 N.For any sequence {X be a set and let f, g : X \u2192 X be given mappings. We say that the pair  is partially weakly G-\u03b1-admissible if and only if \u03b1 \u2265 1 for all x \u2208 X.Let X be a nonempty set and f : X \u2192 X a given mapping. For every x \u2208 X, let f\u22121(x) = {u \u2208 X\u2223fu = x}.Let X be a set and let f, g, R : X \u2192 X be given mappings. We say that the pair  is partially weakly G-\u03b1-admissible with respect to R if and only if for all x \u2208 X, \u03b1 \u2265 1, where y \u2208 R\u22121(fx).Let f = g, we say that f is partially weakly G-\u03b1-admissible with respect to R.If R = IX (the identity mapping on X), then the previous definition reduces to the partially weakly G-\u03b1-admissible pair.If f, g, h, R, S, and T for which ordered pairs , , and  are partially weakly G-\u03b1-admissible with respect to R, S, and T, respectively.Following is an example of mappings X = [0, \u221e). We define functions f, g, h, R, S, T : X \u2192 X byLet Jungck in  introducX, G) be a G-metric space and let f, g : X \u2192 X. The pair  is said to be compatible if and only if lim\u2061n\u2192\u221e\u2061G = 0, whenever {xn} is a sequence in X such that lim\u2061n\u2192\u221e\u2061fxn = lim\u2061n\u2192\u221e\u2061gxn = t for some t \u2208 X.Let -contractive mappings , , and  which are partially weakly \u03b1-admissible with respect to R, S, and T, respectively, in a G-metric space.The aim of this paper is to prove some coincidence and common fixed point theorems for nonlinear weakly  be a metric space and let f, g, h, R, S, T : X \u2192 X be six self-mappings. In the rest of this paper, unless otherwise stated, for all x, y, z \u2208 X, let\u03b1 : X3 \u2192 [0, \u221e) be a function having the following property:\u03b1 : [0, \u221e)3 \u2192 [0, \u221e) by \u03b1 = ex\u2212y\u2212z2.Let  be a G-complete G-metric space. Let f, g, h, R, S, T : X \u2192 X be six mappings such that f(X)\u2286R(X), g(X)\u2286S(X), and h(X)\u2286T(X). Suppose that, for every three elements x, y, and z with \u03b1 \u2265 1, one has\u03c8, \u03c6 : [0, \u221e)\u2192[0, \u221e) are altering distance functions. Let f, g, h, R, S, and T be continuous, the pairs , , and  compatible, and the pairs , , and  partially weakly \u03b1-admissible with respect to R, S, and T, respectively. Then, the pairs , , and  have a coincidence point z in X. Moreover, if \u03b1 \u2265 1, then z is a coincidence point of f, g, h, R, S, and T.Let (x0 \u2208 X be an arbitrary point. Since f(X)\u2286R(X), we can choose x1 \u2208 X such that fx0 = Rx1. Since g(X)\u2286S(X), we can choose x2 \u2208 X such that gx1 = Sx2. Also, as h(X)\u2286T(X), we can choose x3 \u2208 X such that hx2 = Tx3.Let zn} defined byn \u2265 0.Continuing this process, we can construct a sequence {x1 \u2208 R\u22121(fx0), x2 \u2208 S\u22121(gx1), and x3 \u2208 T\u22121(hx2) and , , and  are partially weakly \u03b1-admissible with respect to R, S, and T, respectively, we obtain thatNow, since n \u2208 N.Continuing this process, from , we getGk = G. Suppose Gk0 = 0, for some k0. Then, zk0 = zk0+1 = zk0+2. In the case that k0 = 3n, then zn3 = zn+13 = zn+23 gives zn+13 = zn+23 = zn+33. Indeed,Define zn+13 = zn+23 = zn+33.Thus,M, we can get this result.Analogously, for other values of k0 = 3n + 1, then zn+13 = zn+23 = zn+33 gives zn+23 = zn+33 = zn+43. Also, if k0 = 3n + 2, then zn+23 = zn+33 = zn+43 implies that zn+33 = zn+43 = zn+53. Consequently, zk0 is a coincidence point of the pairs , , and . Indeed, let k0 = 3n. Then, we know that zn3 = zn+13 = zn+23 = zn+33.Similarly, if So,T(xn3) = f(xn3), R(xn+13) = g(xn+13), and S(xn+23) = h(xn+23).This means that f, T), , and  are compatible. So, they are weakly compatible. Hence, fT(xn3) = Tf(xn3), gR(xn+13) = Rg(xn+13), and hS(xn+23) = Sh(xn+23) or, equivalently, fzn3 = Tzn+13, gzn+13 = Rzn+23, and hzn+23 = Szn+33.On the other hand, the pairs , similarly, one can show that zn+13 (zn+23) is a coincidence point of the pairs , , and .In the other cases, when k; that is, zk \u2260 zk+1 for each k.So, suppose thatWe complete the proof in three steps as follows.Step\u2009\u20091. We will prove that lim\u2061k\u2192\u221e\u2061G = 0.\u03b1 \u2265 1, using  = G, then (If 2), then  becomesIfand then  will beIfand then  becomesIfand then  becomesFinally, ifand then  becomesSimilarly it can be shown thatG} is a nondecreasing sequence of nonnegative real numbers. Thus, there is an r \u2265 0 such thatHence, we conclude that {Reviewing the above argument, from , we haveIn general, we can show thatk \u2192 \u221e in  \u2264 \u03c8(r) \u2212 \u03c6(r), and hence \u03c6(r) = 0. This gives us that\u03c6. Also, from Letting nd using , 35), a, an \u2192 \u221e Step\u2009\u20092. We will show that {zn} is a G-Cauchy sequence in X. So, we will show that, for every \u025b > 0, there exists k \u2208 N such that, for all m, n \u2265 k, G < \u025b.\u025b > 0 for which we can find subsequences {zm(k)3} and {zn(k)3} of {zn3} such that n(k) > m(k) \u2265 k satisfying thatn(k) is the smallest number such that ,quality,G(z3m(k),\u221e, using  and 39)\u025b > 0 fork \u2192 \u221e, using ,\u221e, using  and 38)(42)G, \u221e, from , we havek \u2192 \u221e, using ,\u221e, using  and 38)(46)G+\u221e, using  and 45)(48)G3, Rxn(k)+13, Sxn(k)+23) \u2265 1, putting x = xm(k)3, y = xn(k)+13, and z = xn(k)+23 in ,\u221e, using  and 38)(53)G(z3m(k)+2 in , for allM(xm(k)3, xn(k)+13, xn(k)+23) = G(zm(k)3, zn(k)+13, zn(k)+23), from (k \u2192 \u221e in (If 2), from  and 41)M(x3m(k)k \u2192 \u221e in , we havection to .k \u2192 \u221e in , M, a, a(60)M(62)M(64)M65) yieldStep\u2009\u20093. We will show that f, g, h, R, S, and T have a coincidence point.zn} is a G-Cauchy sequence in the complete G-metric space X, there exists z \u2208 X such thatSince {f, T) is compatible, soAs  = 0, lim\u2061n\u2192\u221eG = 0, and the continuity of T and f, we obtainMoreover, from lim\u2061By the rectangle inequality, we haven \u2192 \u221e in n \u2192 \u221e in gz = Rz and hz = Sz.Similarly, we can obtain that \u03b1 \u2265 1. By  > 0; that is, fz \u2260 gz = hz or fz = gz \u2260 hz.Let M = G = G, from  = 0, a contradiction.If z), from , we haveM =  + G + G)/6 and fz \u2260 gz = hz, thenfz = gz = hz, a contradiction to G(fz = gz = hz) > 0.If so, from , we havefz = gz = hz = Tz = Rz = Sz.In the other cases, by a similar manner, we can show that In the following theorem, we will omit the compatibility and continuity assumptions.X, G) be an \u03b1-regular G-metric space and f, g, h, R, S, T : X \u2192 X six mappings such that f(X)\u2286R(X), g(X)\u2286S(X), and h(X)\u2286T(X) and RX, SX, and TX are G-complete subsets of X. Suppose that, for elements x, y, and z with \u03b1 \u2265 1, we have\u03c8, \u03c6 : [0, \u221e)\u2192[0, \u221e) are altering distance functions. Then, the pairs , , and  have a coincidence point z in X provided that the pairs , , and  are weakly compatible and the pairs , , and  are partially weakly \u03b1-admissible with respect to R, S, and T, respectively. Moreover, if \u03b1 \u2265 1, then z \u2208 X is a coincidence point of f, g, h, R, S, and T.Let (z \u2208 X such thatR(X) is G-complete and {zn+13}\u2286R(X), therefore z \u2208 R(X), so there exists u \u2208 X such that z = Ru andv, w \u2208 X such that z = Sv = Tw andw is a coincidence point of f and T.Following the proof of Txn3 \u2192 z = Tw = Ru = Sv as n \u2192 \u221e, \u03b1-regularity of X implies that \u03b1 \u2265 1. Therefore, from  = G.Let n \u2192 \u221e in , it can be shown that z is a coincidence point of the pairs  and .Similarly, in other cases for The rest of the proof is similar to the proof of Assume thatR = S = T = IX (the identity mapping on X) in the previous theorems, we obtain the following common fixed point result.Taking X, G) be a G-complete G-metric space. Let f, g, h : X \u2192 X be three mappings. Suppose that, for every three elements x, y, and z with \u03b1 \u2265 1, we have\u03c8, \u03c6 : [0, \u221e)\u2192[0, \u221e) are altering distance functions. Let the pairs , , and  be partially weakly \u03b1-admissible. Then, the triple  has a common fixed point z in X provided that (a) f, g, and h are continuous or (b) X is \u03b1-regular.Let  be a G-metric space. Let f, R : X \u2192 X be two mappings such that f(X)\u2286R(X). Suppose that, for every three elements x, y, and z with \u03b1 \u2265 1, we have\u03c8, \u03c6 : [0, \u221e)\u2192[0, \u221e) are altering distance functions. Let the pair  be compatible and f is partially weakly \u03b1-admissible w.r.t. R. Then,  has a coincidence point z in X provided that (a) f and R are continuous and  is a G-complete G-metric space or (b) X is \u03b1-regular and R(X) is G-complete.Let  \u2265 1 or \u03b1 \u2265 1, where u and v are common fixed points of f and R.Under the hypotheses of z \u2208 X such that fz = Rz. Since f and R are weakly compatible  is compatible), we have fRz = Rfz. Let w = Rz = fz. Therefore, we haveR is an \u03b1-dominating map,Since z = w, then z is a common fixed point of f and R. If z \u2260 w, then, from  \u2265 1, from  =  + G + G)/6. Then, from \u03c8 + G + G)/6) = 0. So, fz = fw. Now, since w = Rz = fz and fw = Rw, we have w = Rw = fw.Therefore, \u03b1 \u2265 1 or \u03b1 \u2265 1, where u and v are common fixed points of f and R. We claim that common fixed point of f and R is unique. Assume on the contrary that fu = Ru = u and fv = Rv = v and u \u2260 v. Without any loss of generality, we may assume that \u03b1 = \u03b1 \u2265 1. Using  =  + G + G)/6. Then we haveLet u = v, a contradiction.Therefore, In the other cases the proof will be done in a similar way.X = [0, \u221e), G on X given by G = |x \u2212 y| + |y \u2212 z| + |z \u2212 x|, for all x, y, z \u2208 X, and \u03b1 : X3 \u2192 [0, \u221e) given by \u03b1 = ex\u2212y\u2212z2. Define self-maps f, g, h, R, S, and T on X byf, g) is partially weakly \u03b1-admissible with respect to R, let x \u2208 X and y \u2208 R\u22121fx; that is, Ry = fx. By the definition of f and R, we have ln\u2061(1 + x) = ey3 \u2212 1. So, y = ln\u2061(ln\u2061(1 + x) + 1)/3 and hence\u03b1 \u2265 1.Let g, h) is partially weakly \u03b1-admissible with respect to S, let x \u2208 X and y \u2208 S\u22121gx; that is, Sy = gx. By the definition of g and S, we have ln\u2061(1 + (x/2)) = ey2 \u2212 1. So,\u03b1 \u2265 1.To prove that  is partially weakly \u03b1-admissible with respect to T, let x \u2208 X and y \u2208 T\u22121hx; that is, Ty = hx. By the definition of h and T, we have ln\u2061(1 + (x/3)) = ey6 \u2212 1. So,\u03b1 \u2265 1.To prove that .Furthermore, \u03c8, \u03c6 : [0, \u221e)\u2192[0, \u221e) as \u03c8(t) = bt and \u03c6(t) = (b \u2212 1)t, for all t \u2208 [0, \u221e), where 1 < b \u2264 36.Define x, y, and z with \u03b1 \u2265 1 we haveM = G. Therefore, all the conditions of Using the mean value theorem for all F(f) = {x \u2208 X : fx = x} be the fixed point set of f.Let f is also a fixed point of fn, for every n \u2208 N; that is, F(f) \u2282 F(fn). However, the converse is false. For example, the mapping f : R \u2192 R, defined by fx = (1/2) \u2212 x, has the unique fixed point 1/4, but every x \u2208 R is a fixed point of f2. If F(f) = F(fn), for every n \u2208 N, then f is said to have property P. For more details, we refer the reader to [Clearly, a fixed point of eader to , 42\u201345 aAssume thatR = IX (the identity mapping on X) in Taking X, G) be a G-complete G-metric space. Let f : X \u2192 X be a mapping such that f is partially weakly \u03b1-admissible and, for every x, y, z \u2208 X such that \u03b1 \u2265 1,\u03c6 : [0, \u221e)\u2192[0, \u221e) is an altering distance function. Then, f has a fixed point z in X provided that (a) f is continuous or (b) X is \u03b1-regular.Let (X and f be as in f has property P if f is an \u03b1-dominating map.Let F(f) \u2260 \u2205. Let u \u2208 F(fn) for some n > 1. We will show that u = fu. We have \u03b1 \u2265 1, as f is \u03b1-dominating. Using  = G, then, from  and repeating the above process, we get\u03c6 implies thati \u2264 n \u2212 1. Now, taking i = n \u2212 1, we have u = fu.Starting from Now, letSo, we haveRepeating the above process, we getFrom the above inequalities, we have\u03c6 implies thati \u2264 n \u2212 1. Now, taking i = n \u2212 1, we have u = fu.Therefore,In other three cases, the proof will be done in a similar way.Fixed point theorems for monotone operators in ordered metric spaces are widely investigated and have found various applications in differential and integral equations  be a partially ordered G-metric space. We say that X is regular if and only if the following hypothesis holds.Let  be a partially ordered set and f, g, h : X \u2192 X given mappings such that fX\u2286hX and gX\u2286hX. We say that f and g are weakly increasing with respect to h if and only if for all x \u2208 X, fx\u2aafgy, for all y \u2208 h\u22121(fx), and gx\u2aaffy, for all y \u2208 h\u22121(gx).Let  be a partially ordered set and f and g two self-maps on X. An ordered pair  is said to be partially weakly increasing with respect to h if fx\u2aafgy, for all y \u2208 h\u22121(fx).Let (h = I (the identity mapping on X), then the previous definition reduces to the weakly increasing mapping [If  mapping  .f, g) is weakly increasing with respect to h if and only if ordered pairs  and  are partially weakly increasing with respect to it.Note that a pair  be a partially ordered set and letLet  be a partially ordered G-complete G-metric space. Let f, g, h, R, S, T : X \u2192 X be six mappings such that f(X)\u2286R(X), g(X)\u2286S(X), and h(X)\u2286T(X). Suppose that, for every three elements Tx\u2aafRy\u2aafSz, one has\u03c8, \u03c6 : [0, \u221e)\u2192[0, \u221e) are altering distance functions. Let f, g, h, R, S, and T be continuous, the pairs , , and  compatible, and the pairs , , and  partially weakly increasing with respect to R, S, and T, respectively. Then, the pairs , , and  have a coincidence point z in X. Moreover, if Tz\u2aafRz\u2aafSz, then z is a coincidence point of f, g, h, R, S, and T.Let  be a regular partially ordered G-metric space, f, g, h, R, S, T : X \u2192 X six mappings such that f(X)\u2286R(X), g(X)\u2286S(X), and h(X)\u2286T(X), and RX, SX, and TXG-complete subsets of X. Suppose that, for elements Tx\u2aafRy\u2aafSz, one has\u03c8, \u03c6 : [0, \u221e)\u2192[0, \u221e) are altering distance functions. Then, the pairs , , and  have a coincidence point z in X provided that the pairs , , and  are weakly compatible and the pairs , , and  are partially weakly increasing with respect to R, S, and T, respectively. Moreover, if Tz\u2aafRz\u2aafSz, then z \u2208 X is a coincidence point of f, g, h, R, S, and T.Let ("}
{"text": "BL-algebra. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy G\u00f6del ideals of a BL-algebra are introduced and their several properties are investigated. We give a procedure to generate a fuzzy ideal by a fuzzy set. We prove that every fuzzy irreducible ideal is a fuzzy prime ideal but a fuzzy prime ideal may not be a fuzzy irreducible ideal and prove that a fuzzy prime ideal \u03c9 is a fuzzy irreducible ideal if and only if \u03c9(0) = 1 and |Im\u2061(\u03c9)| = 2. We give the Krull-Stone representation theorem of fuzzy ideals in BL-algebras. Furthermore, we prove that the lattice of all fuzzy ideals of a BL-algebra is a complete distributive lattice. Finally, it is proved that every fuzzy Boolean ideal is a fuzzy G\u00f6del ideal, but the converse implication is not true.In this paper we investigate further properties of fuzzy ideals of a  It is well-known that an important task of the artificial intelligence is to make computer simulate human being in dealing with certainty and uncertainty in information. Logic gives a technique for laying the foundations of this task. Information processing dealing with certain information is based on the classical logic. Nonclassical logic includes many valued logic and fuzzy logic which takes the advantage of the classical logic to handle information with various facets of uncertainty , such asBL, in short). A well known example of a BL-algebra is the interval  endowed with the structure induced by a continuous t-norm. MV-algebras \u2227[(y \u2192 x) \u2192 x],x \u2264 y implies y\u2212 \u2264 x\u2212,x = x,x \u2192 x = 1,x \u2192 1 = 1,1 \u2192 x \u2264 y \u2192 x, or equivalently, x \u2192 (y \u2192 x) = 1,x \u2192 y) \u2192 y) \u2192 y = x \u2192 y,\u2212 = x\u2212\u2227y\u2212, (x\u2227y)\u2212 = x\u2212\u2228y\u2212,\u2212\u2212 = y \u2192 x\u2212 for any x, y \u2208 A (see . Let \u03bc be a fuzzy set in A and t \u2208 , the set \u03bct = {x \u2208 A | \u03bc(x) \u2265 t} is called a level subset of \u03bc.A fuzzy set in A and 0A represent two special fuzzy sets in A satisfying 1A(x) = 1 for any x \u2208 A and 0A(x) = 0 for any x \u2208 A, respectively.The notations 1\u03bc, \u03bd, \u03bc\u03bb (\u03bb \u2208 \u039b) in A where \u039b is an index set, we define \u03bc\u2228\u03bd, \u03bc\u2227\u03bd, \u2228{\u03bc\u03bb : \u03bb \u2208 \u039b} and \u2227{\u03bc\u03bb : \u03bb \u2208 \u039b} as follows: for all x \u2208 A,For any fuzzy sets \u03bc \u2264 \u03bd we mean that \u03bc(x) \u2264 \u03bd(x) for all x \u2208 A.By A be a BL-algebra. A fuzzy set \u03bc in A is called a fuzzy ideal of A if, for all x, y \u2208 A,(FI1)\u03bc(0) \u2265 \u03bc(x),(FI2)\u03bc(y) \u2265 \u03bc(x)\u2227\u03bc((x\u2212 \u2192 y\u2212)\u2212).Let \u03bc be a fuzzy set in A. Then \u03bc is a fuzzy ideal if and only if, for each t \u2208 , \u03bct is either empty or an ideal of A.Let BL-algebras. The set of all fuzzy sets in BL-algebra A is denoted by F(A) and the set of all fuzzy ideals in A is denoted by FI(A).In this section, we give a procedure to construct the fuzzy ideal generated by a fuzzy set. First of all we give some further properties of fuzzy ideals in \u03bc \u2208FI(A) and \u03bc((x\u2212 \u2192 y\u2212)\u2212) = \u03bc(0), then \u03bc(x) \u2264 \u03bc(y) for any x, y \u2208 A. In particular, \u03bc(x) \u2264 \u03bc(y) where x\u2212 \u2264 y\u2212 for any x, y \u2208 A.If \u03bc(y) \u2265 \u03bc((x\u2212 \u2192 y\u2212)\u2212)\u2227\u03bc(x) = \u03bc(0)\u2227\u03bc(x) = \u03bc(x), we have \u03bc(x) \u2264 \u03bc(y).Since As an immediate consequence of the proposition we have the following.\u03bc \u2208FI(A) and y \u2264 x then \u03bc(x) \u2264 \u03bc(y) for any x, y \u2208 A.If \u03bc \u2208FI(A). Then for any x \u2208 A, \u03bc(x) = \u03bc(0) if and only if \u03bc(x\u2212\u2212) = \u03bc(0).Let \u03bc(x) = \u03bc(0). By \u03bc \u2208 FI(A), we have\u03bc(x\u2212\u2212) = \u03bc(0).(\u21d2) Suppose that \u03bc(x\u2212\u2212) = \u03bc(0). Since for any x \u2208 A, x \u2264 x\u2212\u2212, by \u03bc(0) = \u03bc(x\u2212\u2212) \u2264 \u03bc(x), thus \u03bc(x) = \u03bc(0).(\u21d0) Assume that \u03bc in A is a fuzzy ideal if and only if for all x, y, z \u2208 A, z\u2212 \u2192 (y\u2212 \u2192 x\u2212) = 1 implies \u03bc(x) \u2265 \u03bc(y)\u2227\u03bc(z).A fuzzy set \u03bc is a fuzzy ideal of A and z\u2212 \u2192 (y\u2212 \u2192 x\u2212) = 1, then z\u2212 \u2264 y\u2212 \u2192 x\u2212 = (y\u2212 \u2192 x\u2212)\u2212\u2212, and by \u03bc(z) \u2264 \u03bc((y\u2212 \u2192 x\u2212)\u2212), so \u03bc(x) \u2265 \u03bc((y\u2212 \u2192 x\u2212)\u2212)\u2227\u03bc(y) \u2265 \u03bc(y)\u2227\u03bc(z).Assuming that z\u2212 \u2192 (y\u2212 \u2192 x\u2212) = 1 implies \u03bc(x) \u2265 \u03bc(y)\u2227\u03bc(z) for all x, y, z \u2208 A. Since x\u2212 \u2192 (x\u2212 \u2192 0\u2212) = x\u2212 \u2192 (x\u2212 \u2192 1) = 1, so \u03bc(o) \u2265 \u03bc(x)\u2227\u03bc(x) = \u03bc(x); (FI1) holds. By 1 = (x\u2212 \u2192 y\u2212) \u2192 (x\u2212 \u2192 y\u2212) = (x\u2212\u2192y\u2212)\u2212\u2212 \u2192 (x\u2212 \u2192 y\u2212), then \u03bc(y) \u2265 \u03bc((x\u2212\u2192y\u2212)\u2212)\u2227\u03bc(x), (FI2) holds.Conversely, suppose that By induction and \u03bc be a fuzzy set in A. \u03bc is a fuzzy ideal if and only if, for any x, y1,\u2026, yn \u2208 A, yn\u2212 \u2192 (yn\u22121\u2212 \u2192 (\u22ef\u2192(y1\u2212 \u2192 x\u2212)\u22ef)) = 1 implies \u03bc(x) \u2265 \u03bc(y1)\u2227\u22ef\u2227\u03bc(yn).Let A be a BL-algebra and x, a, b, a1,\u2026, an, b1,\u2026, bm \u2208 A, if b\u2217a \u2192 x = 1, a1\u2217 \u22ef \u2217an \u2192 a = 1, b1\u2217 \u22ef \u2217bm \u2192 b = 1, thenLetting a1\u2217 \u22ef \u2217an \u2192 a = 1, b1\u2217 \u22ef \u2217bm \u2192 b = 1 it follows that a1\u2217 \u22ef \u2217an \u2264 a, b1\u2217 \u22ef \u2217bm \u2264 b. Since the operation \u201c\u2217\u201d is isotone, so we have a1\u2217 \u22ef \u2217an\u2217b1\u2217 \u22ef \u2217bm \u2264 a\u2217b. While b\u2217a \u2192 x = 1 implies b\u2217a \u2264 x, henceFrom f be a fuzzy set in A. A fuzzy ideal \u03bc is called to be generated by f if f \u2264 \u03bc and f \u2264 \u03bd implies \u03bc \u2264 \u03bd for any fuzzy ideal \u03bd in A. The fuzzy ideal generated by f will be denoted by  we have f \u2264 1A and the intersection of any family of fuzzy ideals in A is a fuzzy ideal in A.It is worth noticing that this definition is well-defined because 1A = {0, a, b, 1}. Define \u2217,\u2192, \u2228, and \u2227 as follows:A; \u2227, \u2228, \u2217, \u2192, 0,1) is a BL-algebra. Define a fuzzy set \u03bc in A by \u03bc(a) = \u03bc(b) = 0.5, \u03bc(1) = 0, \u03bc(0) = 0.8. It is easy to check that (\u03bc](0) = 0.8, (\u03bc](a) = (\u03bc](b) = (\u03bc](1) = 0.5.Let \u03bc and \u03bd be fuzzy sets in A.\u03bc is a fuzzy ideal in A then  = 1  for any x, y, z \u2208 A. Given any arbitrary small \u025b > 0, there exists a1,\u2026, an; b1,\u2026, bm \u2208 A, such that\u03bc(z) \u2265 \u03bc(x)\u2227\u03bc(y), and \u03bc is a fuzzy ideal in A by First, we prove x\u2212 \u2192 x\u2212 = 1 for any x \u2208 A, it follows that f(x) \u2264 \u03bc(x), so f \u2264 \u03bc.Next, since \u03bd is any fuzzy ideal in A with f \u2264 \u03bd, then f(x) \u2264 \u03bd(x) for any x \u2208 A,Finally, supposing that \u03bc \u2264 \u03bd.by \u03bc = p \u2192 x = 1.Let \u03bc be a fuzzy ideal in A. If s, t \u2208  satisfies s \u2265 \u03bc(a), t \u2265 \u03bc(b), s\u2227t \u2264 \u03bc(a\u2227b) where a, b \u2208 A, thenLet \u03bc \u2264  \u2212 \u025b < (\u03bc\u2228as)(a1)\u2227\u22ef\u2227(\u03bc\u2228as)(an).(\u03bc\u2228as)(ai) = \u03bc(ai)\u2009\u2009, we obtainSince  \u2212 \u025b < (\u03bc\u2228bt)(b1)\u2227\u22ef\u2227(\u03bc\u2228bt)(bm).l\u2217a1\u2212\u2217 \u22ef \u2217an\u2212 \u2192 x\u2212 = 1,(\u03bc\u2228as](x) \u2212 \u025b < (\u03bc\u2228as)(a1)\u2227\u22ef\u2227(\u03bc\u2228as)(an)\u2227(\u03bc\u2228as)(a) = \u03bc(a1\u2227\u22ef\u2227\u03bc(an)\u2227s.b\u2212)k\u2217b1\u2212\u2217 \u22ef \u2217bm\u2212 \u2192 x\u2212 = 1,((iv) \u03bc\u2228bt](x) \u2212 \u025b < (\u03bc\u2228bt)(b1)\u2227\u22ef\u2227(\u03bc\u2228bt)(bm)\u2227(\u03bc\u2228bt)(b) = \u03bc(b1)\u2227\u22ef\u2227\u03bc(bm)\u2227t.a\u2212)l \u2192 (a1\u2212\u2217 \u22ef \u2217an\u2212\u2217b1\u2212\u2217 \u22ef \u2217bm\u2212 \u2192 x\u2212) = 1.(Because (iii\u2032)b\u2212)k \u2192 (a1\u2212\u2217 \u22ef \u2217an\u2212\u2217b1\u2212\u2217 \u22ef \u2217bm\u2212 \u2192 x\u2212) = 1.(By the similar argument and (iii) we can getp \u2208 N such thatBy (i\u2032), (iii\u2032), and \u03bc\u2228as]\u2227 \u2264 \u03bc(x)\u2228\u03bc(y).A nonconstant fuzzy ideal \u03bc in A is a fuzzy prime ideal in A if and only if \u03bct = \u2205 where t > \u03bc(0); \u03bct is a prime ideal of A where inf\u2061{\u03bc(x) | x \u2208 A} < t \u2264 \u03bc(0); \u03bct = A where 0 \u2264 t \u2264 inf\u2061{\u03bc(x) | x \u2208 A}.A nonconstant fuzzy set It is easy and omitted.a, b \u2208  and a < b. If I is a prime ideal of A and J is an proper ideal of A with I \u2282 J, then the function fIJab : A \u2192  is a fuzzy prime ideal in A whereLet I is a prime ideal of A, then the characteristic function \u03c7I of I is a fuzzy prime ideal in A whereIf \u03bc be a fuzzy ideal in A, then \u03bc is a fuzzy prime ideal in A if and only if \u03bc\u03bc(0) is a prime ideal of A.Letting \u03bc\u03bc(0) is a prime ideal of A. \u03bct = \u2205 where t > \u03bc(0); \u03bc\u03bc(0)\u2286\u03bct \u2260 A where inf\u2061{\u03bc(x) | x \u2208 A} < t \u2264 \u03bc(0), so \u03bct is a prime ideal of A; \u03bct = A where 0 \u2264 t \u2264 inf\u2061{\u03bc(x) | x \u2208 A}. Thus \u03bc is a fuzzy prime ideal in A by The \u201conly if\u201d part is easy. We now prove the part \u201cif\u201d as follows. Suppose that \u03bc in A is a fuzzy prime ideal in A if and only if \u03bc(x\u2227y) = \u03bc(0) implies \u03bc(x) = \u03bc(0) or \u03bc(y) = \u03bc(0).A nonconstant fuzzy ideal \u03bc is a fuzzy prime ideal in A; then \u03bc\u03bc(0) is a prime ideal of A by \u03bc(x\u2227y) = \u03bc(0), then x\u2227y \u2208 \u03bc\u03bc(0), and so x \u2208 \u03bc\u03bc(0) or y \u2208 \u03bc\u03bc(0). Hence \u03bc(x) = \u03bc(0) or \u03bc(y) = \u03bc(0).Suppose that x, y \u2208 A, \u03bc(x\u2227y) = \u03bc(0) implies \u03bc(x) = \u03bc(0) or \u03bc(y) = \u03bc(0). That is, x\u2227y \u2208 \u03bc\u03bc(0) implies x \u2208 \u03bc\u03bc(0) or y \u2208 \u03bc\u03bc(0), and thus \u03bc\u03bc(0) is a prime ideal of A. Therefore \u03bc is a fuzzy prime ideal in A by Conversely, suppose, for any Note. The above theorem shows that the definition on fuzzy prime ideals in this paper and one in  + \u03c9(a\u2227b)} and \u03bc = \u2228\u03c9(b) < s < \u03c9(a\u2227b), by \u03bc\u2227\u03bd = \u03c9, but \u03bc \u2260 \u03c9, \u03bd \u2260 \u03c9, a contradiction.Suppose But the converse of the above theorem is not true.A = {0, a, b, 1}. Define \u2217, \u2192, \u2228, and \u2227 as follows:A; \u2227, \u2228, \u2217, \u2192, 0,1) is a BL-algebra. It is easy to check that I1 = {0, a}, I2 = {0, b} are prime ideals of A.Let \u03c9 in A by \u03c9(0) = \u03c9(a) = 1/2, \u03c9(1) = \u03c9(b) = 1/4, and then \u03c9 is a fuzzy prime ideal in A. Indeed, \u03c9t = \u2205 where t > 1/2; \u03c9t = {0, a} where 1/4 < t \u2264 1/2; \u03c9t = A where 0 \u2264 t \u2264 1/4. Let \u03bc be a fuzzy ideal in A defined by \u03bc(0) = 3/4, \u03bc(a) = 3/4, \u03bc(b) = \u03bc(1) = 1/4. Let \u03bd be a fuzzy ideal in A defined by \u03bd(0) = \u03bd(a) = 1/2, \u03bd(b) = \u03bd(1) = 3/8. It is easy to verify that \u03bc\u2227\u03bd = \u03c9 but \u03bc \u2260 \u03c9, \u03bd \u2260 \u03c9. Therefore \u03c9 is not a fuzzy irreducible ideal in A.Define a fuzzy set This example shows the converse of \u03be be a fuzzy set in A defined by \u03be(0) = \u03be(a) = 1, \u03be(b) = \u03be(1) = t(0 < t < 1), it is easy to check that \u03be is a fuzzy irreducible ideal in A.Letting BL-algebras, an ideal is prime if and only if it is irreducible . we get the fuzzy set f\u2228g:Let f, g are fuzzy ideals in A, but f\u2228g is not a fuzzy ideal in A since (f\u2228g)s = {0, a, b} is not an ideal of A.It is easy to see that \u03bc, \u03bd \u2208 FI(A), denote \u03bc\u2294\u03bd = :(\u03bb \u2208 \u039b \u2260 \u2205),Now we give the following definition: for any A be a BL-algebra, then (FI(A); \u2294, \u2227, 0A, 1A) is a complete distributive lattice where 0A and 1A are the least lower bound and the largest upper bound ofFI(A), respectively, and satisfies the following infinitely distributive law:(DL)\u03bc\u2227\u2294{\u03bd\u03b1 | \u03b1 \u2208 \u039b} = \u2294{\u03bc\u2227\u03bd\u03b1 | \u03b1 \u2208 \u039b}, for all \u03bc, \u03bd\u03b1 \u2208FI(A)(\u03b1 \u2208 \u039b \u2260 \u2205).Letting \u03bc, \u03bd \u2208 FI(A),\u2009\u2009\u03bc\u2294\u03bd and \u03bc\u2227\u03bd are the supremum and the infimum in FI(A) of \u03bc and \u03bd, and 0A \u2264 \u03bc \u2264 1A, so (FI(A); \u2294, \u2227, 0A, 1A) is a bounded lattice. This lattice is obviously complete.It is easy to verify that for any \u03bd\u03bb \u2208 FI(A)(\u03bb \u2208 \u039b \u2260 \u2205),x \u2208 A and arbitrary small \u025b > 0, there exist a1, a2,\u2026, an \u2208 A such that\u03bd\u03b1 | \u03b1 \u2208 \u039b}(ai)\u2009\u2009, there are \u03b11,\u2026, \u03b1n \u2208 \u039b such thatIn order to check (DL) it suffices to prove, for any Thus by \u03bc(x) \u2264 \u03bc(b1), \u03bc(b2),\u2026, \u03bc(bn).(i) a1\u2212 \u2264 b1\u2212. By a similar way we may prove that a2\u2212 \u2264 b2\u2212,\u2026, an\u2212 \u2264 bn\u2212. ThereforeSinceBy \u03bd\u03b11(a1) \u2264 \u03bd\u03b11(b1),\u2026, \u03bd\u03b1n(an) \u2264 \u03bd\u03b1n(bn).(ii) By (i) and (ii) we haveand so\u03bc\u2227\u03bd\u03b1 | \u03b1 \u2208 \u039b} is a fuzzy ideal of A, and by \u025b is arbitrary small, we haveIt is obviousThe proof is completed.BL-algebra and investigate some of their properties.In this section, we introduce the notion of fuzzy G\u00f6del ideals of \u03bc be a fuzzy ideal of A. \u03bc is called a fuzzy G\u00f6del ideal if \u03bc((x\u2212 \u2192 (x\u2212)2)\u2212) = \u03bc(0) for all x \u2208 A.Let It is obvious that each fuzzy ideal in G\u00f6del algebra is a fuzzy G\u00f6del ideal. In \u03bc be a fuzzy subset of A. \u03bc is a fuzzy G\u00f6del ideal if and only if, for each t \u2208 , \u03bct is a G\u00f6del ideal of A where \u03bct \u2260 \u2205.Let Next theorem is called the extension theorem of fuzzy G\u00f6del ideals.\u03bc and \u03bd be fuzzy ideal of A with \u03bc \u2264 \u03bd and \u03bc(0) = \u03bd(0). If \u03bc is a fuzzy G\u00f6del ideal, then so is \u03bd.Let \u03bc be a fuzzy ideal of A. Define a fuzzy set \u03c7\u03bc in A byLet \u03c7\u03bc is also a fuzzy ideal in A.Obviously, \u03bc of A is a fuzzy G\u00f6del ideal if and only if \u03c7\u03bc is a fuzzy G\u00f6del ideal of A.A fuzzy subset \u03bc is a fuzzy G\u00f6del ideal of A, then by t \u2208 , \u03bct is a G\u00f6del ideal of A. In particular, \u03bc\u03bc(0) = {x \u2208 A | \u03bc(x) = \u03bc(0)} is a G\u00f6del ideal of A. We notice for any t \u2208 t \u2208 , (\u03c7\u03bc)t is a G\u00f6del ideal of A where (\u03c7\u03bc)t \u2260 \u2205. By \u03c7\u03bc is a fuzzy G\u00f6del ideal of A.If \u03c7\u03bc is a fuzzy G\u00f6del ideal of A. It is clear that \u03c7\u03bc \u2264 \u03bc and \u03c7\u03bc(0) = \u03bc(0). By \u03bc is a fuzzy G\u00f6del ideal of A.Conversely, suppose \u03bc be a fuzzy ideal of A. The following conditions are equivalent:\u03bc is a fuzzy G\u00f6del ideal,\u03bc(((x\u2212)2 \u2192 y\u2212)\u2212) = \u03bc(0) implies \u03bc((x\u2212 \u2192 y\u2212)\u2212) = \u03bc(0),\u03bc(((x\u2212\u2217y\u2212) \u2192 z\u2212)\u2212) = \u03bc(0) implies \u03bc(((x\u2212 \u2192 y\u2212)\u2192(x\u2212 \u2192 z\u2212))\u2212) = \u03bc(0).Let \u03bc is a fuzzy G\u00f6del ideal and \u03bc(((x\u2212)2 \u2192 y\u2212)\u2212) = \u03bc(0). Then we have\u03bc((x\u2212 \u2192 y\u2212)\u2212) = \u03bc(0).(i)\u21d2(ii) Suppose that \u03bc(((x\u2212\u2217y\u2212) \u2192 z\u2212)\u2212) = \u03bc(0). Since y\u2212 \u2192 z\u2212 \u2264 (x\u2212 \u2192 y\u2212)\u2192(x\u2212 \u2192 z\u2212), then we havex\u2212)2 \u2192 ((x\u2212 \u2192 y\u2212) \u2192 z\u2212))\u2212 \u2264 ((x\u2212\u2217y\u2212) \u2192 z\u2212)\u2212. Hence\u03bc(((x\u2212)2 \u2192 ((x\u2212 \u2192 y\u2212) \u2192 z\u2212))\u2212) = \u03bc(0). From (ii) we get that\u03bc(((x\u2212)2 \u2192 (x\u2212)2)\u2212) = \u03bc(0), by (iii) we have\u03bc is a fuzzy G\u00f6del ideal.(ii)\u21d2(iii) Suppose that \u03bc be a fuzzy ideal. \u03bc is a fuzzy G\u00f6del ideal if and only if \u03bc((x\u2212 \u2192 (y\u2212 \u2192 z\u2212))\u2212) = \u03bc(0) and \u03bc((x\u2212 \u2192 y\u2212)\u2212) = \u03bc(0) imply \u03bc((x\u2212 \u2192 z\u2212)\u2212) = \u03bc(0).Let \u03bc is a fuzzy G\u00f6del ideal. Let\u03bc((x\u2212 \u2192 z\u2212)\u2212) = \u03bc(0).Suppose that \u03bc((x\u2212 \u2192 (x\u2212 \u2192 (x\u2212)2))\u2212) = \u03bc(0) and \u03bc((x\u2212 \u2192 x\u2212)\u2212) = \u03bc(0), then by the assumption we have \u03bc((x\u2212 \u2192 (x\u2212)2)\u2212) = \u03bc(0), so \u03bc is a fuzzy G\u00f6del ideal.Conversely, since \u03bc be a fuzzy ideal. Then \u03bc is a fuzzy G\u00f6del ideal if and only if the following condition holds:(\u2217)\u03bc(x\u2212 \u2192 ((y\u2212)2 \u2192 z\u2212)\u2212) = \u03bc(0) and \u03bc(x) = \u03bc(0) imply \u03bc((y\u2212 \u2192 z\u2212)\u2212) = \u03bc(0) for any x, y, z \u2208 A.Let \u03bc is a fuzzy G\u00f6del ideal of A. Letting \u03bc((x\u2212 \u2192 ((y\u2212)2 \u2192 z\u2212))\u2212) = \u03bc(0) and \u03bc(x) = \u03bc(0), then\u03bc(((y\u2212)2 \u2192 z\u2212)\u2212) = \u03bc(0). Since \u03bc is a fuzzy G\u00f6del ideal, by \u03bc((y\u2212 \u2192 z\u2212)\u2212) = \u03bc(0). (\u2217) holds.Suppose that \u03bc((x\u2212 \u2192 (y\u2212 \u2192 z\u2212))\u2212) = \u03bc(0) and \u03bc((x\u2212 \u2192 y\u2212)\u2212) = \u03bc(0). Since\u03bc((x\u2212 \u2192 (y\u2212 \u2192 z\u2212))\u2212) = \u03bc(0), we get that\u03bc((x\u2212 \u2192 y\u2212)\u2212) = \u03bc(0) we get \u03bc((x\u2212 \u2192 z\u2212)\u2212) = \u03bc(0). Thus \u03bc is a fuzzy G\u00f6del ideal by Conversely, suppose that (\u2217) is true. Let BL-algebra and proved that fuzzy Boolean ideals are equivalent to fuzzy implicative ideals. In the following, we investigate the relation between fuzzy Boolean ideals and fuzzy G\u00f6del ideals.Zhang et al. . introdu\u03bc in A is called a fuzzy Boolean ideal if \u03bc(x\u2227x\u2212) = \u03bc(0) for all x \u2208 A.A fuzzy ideal A is a fuzzy G\u00f6del ideal in A.Each fuzzy Boolean ideal in \u03bc is a fuzzy Boolean ideal in A and \u03bc(((x\u2212)2 \u2192 y\u2212)\u2212) = \u03bc(0). Then\u03bc((x\u2212 \u2192 y\u2212)\u2212) = \u03bc(0). By \u03bc is a fuzzy G\u00f6del ideal in A.Suppose that But the converse of the above theorem is not true.A = {0, a, b, c, d, 1}. Define \u2217, \u2192, \u2228, and \u2227 as follows:A; \u2227, \u2228, \u2217, \u2192, 0,1) is a BL-algebra. Define a fuzzy set \u03bc in A by \u03bc(a) = \u03bc(b) = \u03bc(c) = \u03bc(1) = 0.5, \u03bc(0) = \u03bc(d) = 0.8; one can easily check that \u03bc is a fuzzy G\u00f6del ideal in A, but it is not a fuzzy Boolean ideal, since \u03bc(a\u2227a\u2212) = \u03bc(a\u2227c) = \u03bc(c) = 0.5 \u2260 0.8 = \u03bc(0).Let BL-algebras is technically more difficult, so far little research literature. Zhang et al. [BL-algebras. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy G\u00f6del ideals are introduced and studied. We give a procedure to generate a fuzzy ideal by a fuzzy set. Using this result we prove that any fuzzy irreducible ideal is a fuzzy prime ideal and meanwhile we give an example to show that a fuzzy prime ideal may not be a fuzzy irreducible ideal; we also give the Krull-Stone representation theorem of fuzzy ideals in BL-algebras. Furthermore we prove that the set of all fuzzy ideals forms a complete distributive lattice. In addition, we prove that any fuzzy Boolean ideal in BL-algebras is a fuzzy G\u00f6del ideal, but the converse is not true.Study of fuzzy ideal theory in g et al.  initiateBL-algebras should be related to (1) several special types of fuzzy ideals; (2) decomposition properties of fuzzy ideals. Our obtained results can be applied in information science, engineering, computer science, and medical diagnosis.In our opinion, the future study of fuzzy ideals in"}
{"text": "Choose an orthonormal frames field {e1,\u2026, en, en+1} along M such that {e1,\u2026, en} are tangent to M and {en+1} is normal to M. Their dual frames are {\u03b81,\u2026, \u03b8n} and {\u03b8n+1}, respectively; obviously, \u03b8n+1 = 0 when it is restricted over M. Let h denote the second fundamental form of the immersion \u03c6 : M \u2192 Sn+1(1). WriteH, S, Pk to be mean curvature, square of the length, and kth power polynomial of second fundamental form, respectively. It is well known that hypersurface \u03c6 : Mn \u2192 Sn+1(1) is minimal when H \u2261 0 and is totally geodesic when S \u2261 0. For invariant \u03c1, it is obvious that \u03c1(Q) \u2265 0 for all Q \u2208 M, \u03c1(Q) = 0 if and only if Q is an umbilical point of M, and 0 \u2264 \u03c1(Q) \u2264 C for all Q \u2208 M and some positive constant C since M is compact.Let Wn,n/2)(\u03c6) \u2265 4\u03c02 holds for all immersed tori \u03c6 : M \u2192 S3; recently, it has been completely solved by Marques and Neves in article [n \u2265 3, the functional Wn,n/2)((\u03c6) was studied extensively in [In differential geometry, there is a famous classical Willmore functional of hypersurface which is defined as2dv.See \u20134 for th article  using Miively in \u201311.Due to the importance of Willmore conjecture, many geometric experts generalized the classic Willmore functional to a wide range and some interesting results have been obtained.Wp):k1 and k2 denote the principal curvatures of M and k1 \u2265 k2. Under some proper conditions, Cai obtained some interesting inequalities.In , Cai stuWn,1):In , Guo andenon. In , Xu and Wn,r):In , Wu reseWn,F)((\u03c6) is called Wn,F) be an n-dimensional hypersurface in an (n + 1)-dimensional unit sphere Sn+1(1); then M is a Wn,F) hypersurface, then M is a Wn,F)(-Willmore hypersurface if and only ifLet F(\u03c1) = \u03c1n/2, \u03c1, \u03c1r, When M is an n-dimensional compact minimal hypersurface in (n + 1)-dimensional unit sphere Sn+1(1), thenIt is well known that Simons' integral inequality plays an important role in the study of minimal hypersurface. It says that if M is an n-dimensional compact classical Willmore hypersurface in unit sphere Sn+1(1), thenWn,1), , 15 obtaM be an n-dimensional compact Wn,F)(-Willmore hypersurface in sphere Sn+1(1); then we have Simons' type equality and can give a discussion according to the sign of F, F\u2032, and F\u2032\u2032:\u03c1 \u2261 n, there holds\u03c1 \u2261 0, there holdsF\u2032 \u2261 0, F \u2261 c \u2260 0, there holdsnF \u2212 2uF\u2032 \u2265 0, F\u2032 \u2265 0, F\u2032\u2032 \u2265 0, there holdsnF \u2212 2uF\u2032 \u2264 0, F\u2032 \u2265 0, F\u2032\u2032 \u2265 0, there holdsLet F(\u03c1) = \u03c1n/2, \u03c1, \u03c1r, F, F\u2032, and F\u2032\u2032.When Using the integral equalities in M be an n-dimensional closed Wn,F)(-Willmore hypersurface in unit sphere Sn+1(1), we have the following.F\u2032 \u2261 0, F \u2261 c \u2260 0 over , if 0 \u2264 \u03c1 \u2264 n, then \u03c1 = 0 or \u03c1 = n.When \u03c1 = 0, then H = 0, \u03c1 = 0, S = 0, and M is totally geodesic;For \u03c1 = n, then H = 0, \u03c1 = n, S = n, and m with 1 \u2264 m \u2264 n \u2212 1.For nF \u2212 2uF\u2032 > 0,\u2009\u2009F\u2032 > 0,\u2009\u2009F\u2032\u2032 \u2265 0 over , if 0 \u2264 \u03c1 \u2264 n, then \u03c1 = 0 or \u03c1 = n.When \u03c1 = 0, then M is totally umbilical;For \u03c1 = n, then n \u2261 0(mod\u2061\u2009\u20092), H = 0, \u03c1 = n, S = n, and For nF \u2212 2uF\u2032 = 0, F\u2032 > 0, F\u2032\u2032 \u2265 0 over , if 0 \u2264 \u03c1 \u2264 n, then F(u) = cun/2, c > 0, \u03c1 = 0, or \u03c1 = n.When \u03c1 = 0, M is totally umbilical;For \u03c1 = n, m with 1 \u2264 m \u2264 n \u2212 1.For nF \u2212 2uF\u2032 < 0, F\u2032 > 0, F\u2032\u2032 \u2265 0 over , if H = 0, 0 \u2264 \u03c1 \u2264 n, then H = 0, \u03c1 = 0 or H = 0, \u03c1 = n.When H = 0, \u03c1 = 0, then M is totally geodesic;For H = 0, \u03c1 = n, then n \u2261 0(mod\u2061\u2009\u20092), H = 0, \u03c1 = n, S = n, For Let F(\u03c1) = \u03c1n/2, \u03c1, \u03c1r, F, F\u2032, and F\u2032\u2032. The examples appeared in When \u03c6 : Mn \u2192 Sn+1(1) be an n-dimensional closed hypersurface in an (n + 1)-dimensional unit sphere and let \u03a6 : M \u00d7  \u2192 Sn+1(1) be a variation of \u03c6; it means that\u03c60 \u2261 \u03c6.Let s =  and \u03c3 =  be the orthonormal local frames of TSn+1(1) and T*Sn+1(1), respectively; then e =  = \u03c6t\u22121s is the orthonormal local frames of the pullback vector bundle \u03c6t\u22121TSn+1(1) over M \u00d7 {t}, such that {e1,\u2026, en} are tangent to M and {en+1} is normal to M due toLet \u03c9 to denote the connection form over TSn+1(1); by the pullback operation, we have the following decomposition:\u03b8i} are the orthonormal frames of T*M and \u03b8n+1 \u2261 0 when it is restricted over M, {Vi, Vn+1 = :f} are the variation vector fields of \u03a6, {\u03d5ij} is the connection form of TM, and {\u03d5in+1 = hij\u03b8j} is the second fundamental form. In particular, we haveUse \u03a9 and \u03a9\u22a4 to denote curvature forms of TSn+1(1) and TM, respectively and write their components:Use \u03d5ij}, the covariant derivatives of hij, H, \u03c1, Vi, f, and bi can be defined asUsing connection form {dM,\u2009\u2009dM\u00d7 = dM + dt(\u2202/\u2202t) and d to denote the differential operators on M, M \u00d7 , and Sn+1(1), respectively. By calculating directly and comparing both sides of the following equations:We use  article . In fact=d\u03c3\u2212\u03c3\u2227\u03c9,\u03a6\u2217(\u03a9)=dM\u00d7=d\u03c3\u2212\u03c3\u2227\u03c9,\u03a6\u2217(\u03a9)=dM\u00d7article [\u03a6\u2217(\u03a9AB)=\u03a6side of )\u2227\u03a6\u2217(\u03c9),0=dM\u00d7 be a hypersurface; one has structure equations:Let \u03c6 : Mn \u2192 Sn+1(1) be a hypersurface; one has Ricci identity:Let \u03c6 : Mn \u2192 Sn+1(1) be a hypersurface and let V = V\u22a4 + V\u22a5 = Viei + fen+1 be a variation vector field; one hasLet With the same notations as above, one hasWn,F) be a hypersurface and let V = V\u22a4 + V\u22a5 = Viei + fen+1 be a variation vector field. Suppose that dv = \u03b81\u2227\u22ef\u2227\u03b8n denotes the volume element; one hasLet t)(\u03b8i) = \u03b8j. Hence,By \u03c6 : Mn \u2192 Sn+1(1) be a hypersurface and let V = V\u22a4 + V\u22a5 = Viei + fen+1 be a variation vector field; one hasLet \u03c1,\u2009\u2009S and H, one hasBy the definition of By Lemmas Sn+1(1), in particular, the isoparametric hypersurfaces. Since all principal curvatures k1,\u2026, kn are constant, then \u03c1, H, S are constant and H = (1/n)P1, S = P2, \u03c1 = P2 \u2212 (1/n)(P1)2. Thus, Wn,F)(-Willmore if and only if F(0) = 0.Obviously, totally geodesic hypersurfaces are n \u2261 0(mod\u20612),P1,\u2009\u2009P2,\u2009\u2009P3,\u2009\u2009\u03c1, respectively, P1 = 0,\u2009\u2009P2 = n,\u2009\u2009P3 = 0,\u2009\u2009\u03c1 = n. Obviously Cn/2,n/2 always is a Wn,F)(-Willmore hypersurface of Sn+1(1) for any function F.For a particular hypersurface with \u03bb,\u2009\u2009\u03bc,\u2009\u20090 < \u03bb, \u03bc < 1, \u03bb2 + \u03bc2 = 1. P1, P2, P3, \u03c1 are, respectively,\u03bc/\u03bb = x > 0, Wn,F), m = n/2.Consider thatSome lemmas are needed for the establishment of Simons' type integral equalities and the discussion of gap phenomenon.\u03c1, one hash|2 = \u2211ijkhij,k2.For \u03c1 and Laplacian operator, we haveT1 = \u2211ijkhijhij,kk; by Lemmas T2 = \u2211ijkhij; by By the definition of ,kj); by T2=\u2211ijkhiy Lemmas T1=\u2211ijkhiituting , one hasFor We know thatM, we haveIntegrating the equality in hij = 0, for all\u2009\u2009i \u2260 j,\u2009\u2009hi = hii.In order to prove x : Mn \u2192 Sn+1(1) be a compact hypersurface with \u2207h \u2261 0; then there are two cases.h1 = \u22ef = hn = \u03bb = constant, and M is a totally umbilic (\u03bb > 0) or totally geodesic (\u03bb = 0);h1 = \u22efhm = \u03bb = constant > 0, hm+1 = \u22ef = hn = \u2212(1/\u03bb), 1 \u2264 m \u2264 n \u2212 1, and M is a Riemannian product of M1 \u00d7 M2, where Let Cm,n\u2212m, 1 \u2264 m \u2264 n \u2212 1 are the only compact minimal (H = 0) hypersurfaces of dimension n in unit sphere Sn+1(1) satisfying S = n.Clifford torus Obviously, case (1) of nF \u2212 2uF\u2032 > 0,\u2009\u2009F\u2032 > 0,\u2009\u2009F\u2032\u2032 \u2265 0 over , by the case (4) of \u03c1 \u2264 n, the right hand side of identity , H = 0, \u03c1 = n, S = n, and M = Cn/2,n/2.(2) When e (4) of \u222bM2\u03c1(\u03c1\u2212n)e (4) of \u222bM2\u03c1(\u03c1\u2212n)nF \u2212 2uF\u2032 = 0, F\u2032 > 0, F\u2032\u2032 \u2265 0\u2009\u2009, we know that F(u) = cun/2, c = const > 0. By the case (4) of \u03c1 \u2264 n, then \u03c1 = 0 or \u03c1 = n. For \u03c1 = 0, M is totally umbilical; for \u03c1 = n, substituting it into the above equality together with h = 0; by M is a torus; then, by M is a classic Willmore torus Wm,n\u2212m for some m.(3) When"}
{"text": "We also introduce common (EA) property for two hybrid pairs F, G : X \u2192 2X and f, g : X \u2192 X. We establish some common coupled fixed point theorems for two hybrid pairs of mappings under \u03c6-\u03c8 contraction on noncomplete metric spaces. An example is also given to validate our results. We improve, extend and generalize several known results. The results of this paper generalize the common fixed point theorems for hybrid pairs of mappings and essentially contain fixed point theorems for hybrid pair of mappings.We introduce the concept of (EA) property and occasional  X, d) be a metric space and let CB(X) be the set of all nonempty closed bounded subsets of X. Let D denote the distance from x to A \u2282 X and let H denote the Hausdorff metric induced by d; that is,Let , and let g be a self-mapping on X. An element  \u2208 X \u00d7 X is called(1)F, g} if g(x) \u2208 F and g(y) \u2208 F,a coupled coincidence point of hybrid pair {(2)F, g} if x = g(x) \u2208 F and y = g(y) \u2208 F.a common coupled fixed point of hybrid pair {Let F and g by C{F,g}. Note that if  \u2208 C{F, g}, then  is also in C{F, g}.We denote the set of coupled coincidence points of mappings F : X \u00d7 X \u2192 2X be a multivalued mapping and let g be a self-mapping on X. The hybrid pair {F, g} is called w-compatible if g)\u2286F whenever  \u2208 C{F, g}.Let F : X \u00d7 X \u2192 2X be a multivalued mapping and let g be a self-mapping on X. The mapping g is called F-weakly commuting at some point  \u2208 X \u00d7 X if g2x \u2208 F and g2y \u2208 F.Let f : X \u2192 X and T : X \u2192 2X. Liu et al. [f : X \u2192 X and T : X \u2192 2X.Aamri and El Moutawakil  defined u et al.  introducu et al.  extendedw-compatibility for hybrid pair F : X \u00d7 X \u2192 2X and f : X \u2192 X. We also introduce common (EA) property for two hybrid pairs F, G : X \u00d7 X \u2192 2X and f, g : X \u2192 X. We establish some common coupled fixed point theorems for two hybrid pairs of mappings under \u03c6-\u03c8 contraction on noncomplete metric spaces. The \u03c6-\u03c8 contraction is weaker contraction than the contraction defined in Gnana Bhaskar and Lakshmikantham [In this paper, we introduce the concept of (EA) property and occasional ikantham  and Luonikantham . We imprikantham , Gnana Bikantham , Jain etikantham , Lakshmiikantham , Liu et ikantham , and Luoikantham . The resWe first define the following.f : X \u2192 X and F : X \u00d7 X \u2192 CB(X) are said to satisfy the (EA) property if there exist sequences {xn}, {yn} in X, some u, v in X, and A,\u2009\u2009B in CB(X) such thatMappings f, g : X \u2192 X and F, G : X \u00d7 X \u2192 CB(X). The pairs {F, f} and {G, g} are said to satisfy the common (EA) property if there exist sequences {xn}, {yn}, {un}, and {vn} in X, some u, v in X, and A, B, C, D in CB(X) such thatLet X = [1, +\u221e) with the usual metric. Define f, g : X \u2192 X and F, G : X \u00d7 X \u2192 CB(X) byF, f} and {G, g} are said to satisfy the common (EA) property.Let F : X \u00d7 X \u2192 2X and f : X \u2192 X are said to be occasionally w-compatible if and only if there exists some point  \u2208 X \u00d7 X such that fx \u2208 F, fy \u2208 F, and fF\u2286F.Mappings X = [0, +\u221e) with usual metric. Define f : X \u2192 X, F : X \u00d7 X \u2192 CB(X) byf and F, but fF\u2286F and f and F are not w-compatible. However, the pair {F, f} is occasionally w-compatible.Let \u03c6 : [0, +\u221e)\u2192[0, +\u221e) satisfying the following:\u2009\u03c6)\u2009\u2009\u03c6 is continuous and strictly increasing,(i\u2009\u03c6)\u2009\u2009\u03c6(t) < t for all t > 0,(ii\u2009\u03c6)\u2009\u2009\u03c6(t + s) \u2264 \u03c6(t) + \u03c6(s) for all t, s > 0.\u2192[0, +\u221e) which satisfies\u2009\u03c8)\u2009\u2009lim\u2061t\u2192r\u03c8(t) > 0 for all r > 0 and lim\u2061t\u21920+\u03c8(t) = 0,(i\u2009\u03c8)\u2009\u2009\u03c8(t) > 0 for all t > 0 and \u03c8(0) = 0. and (ii\u03c6), we have that \u03c6(t) = 0 if and only if t = 0. For example, functions \u03c61(t) = kt where k > 0, \u03c62(t) = t/(t + 1), \u03c63(t) = ln\u2061(t + 1), and \u03c64(t) = min\u2061{t, 1} are in \u03a6,\u2009\u2009\u03c81(t) = kt where k > 0,\u2009\u2009\u03c82(t) = (ln\u2061(2t + 1))/2, and Note that, by  be a metric space. Assume F, G : X \u00d7 X \u2192 CB(X) and f, g : X \u2192 X to be mappings satisfying the following.(1)F, f} and {G, g} satisfy the common (EA) property.{(2)x, y, u, v \u2208 X, there exist some \u03c6 \u2208 \u03a6 and some \u03c8 \u2208 \u03a8 such thatFor all (3)f(X) and g(X) are closed subsets of X. Then(a)F and f have a coupled coincidence point,(b)G and g have a coupled coincidence point,(c)F and f have a common coupled fixed point, if f is F-weakly commuting at  and f2x = fx and f2y = fy for  \u2208 C{F, f},(d)G and g have a common coupled fixed point, if g is G-weakly commuting at (e)F, G, f,\u2009\u2009and g have common coupled fixed point provided that both (c) and (d) are true.Let  property, there exist sequences {xn}, {yn}, {un}, and {vn} in X, some u, v in X, and A, B, C, D in CB(X) such thatf(X) and g(X) are closed subsets of X, then there exist n \u2192 \u221e in the above inequality, by using (\u03c6),\u2009\u2009(ii\u03c6), and (i\u03c8), we obtain\u03c6) and (ii\u03c6), impliesfx \u2208 B and fy \u2208 D, it follows thatx, y) is a coupled coincidence point of F and f. This proves (a). Again, by using condition (2) of n \u2192 \u221e in the above inequality, by using (\u03c6),\u2009\u2009(ii\u03c6), and (i\u03c8), we obtain\u03c6) and (ii\u03c6), impliesG and g. This proves (b).Since {uch thatlim\u2061n\u2192\u221eFxx~,y~\u2208X,u=fx=gx~,uch thatlim\u2061n\u2192\u221eFxx~,y~\u2208X,u=fx=gx~,f which is F-weakly commuting at ; that is, f2x \u2208 F, f2y \u2208 F and f2x = fx, f2y = fy. Thus, fx = f2x \u2208 F and fy = f2y \u2208 F; that is, u = fu \u2208 F and v = fv \u2208 F. This proves (c). A similar argument proves (d). Then (e) holds immediately.Furthermore, from condition (c), we have f = g in Put X, d) be a metric space. Assume F, G : X \u00d7 X \u2192 CB(X) and g : X \u2192 X to be mappings such that(1)F, g} and {G, g} satisfy the common (EA) property,{(2)x,y,u,v \u2208 X, there exist some \u03c6 \u2208 \u03a6 and some \u03c8 \u2208 \u03a8 such thatfor all (3)g(X) is a closed subset of X. Then(a)F and g have a coupled coincidence point,(b)G and g have a coupled coincidence point,(c)F and g have a common coupled fixed point, if g is F-weakly commuting at  and g2x = gx and g2y = gy for  \u2208 C{F, g},(d)G and g have a common coupled fixed point, if g is G-weakly commuting at (e)F, G,\u2009\u2009and g have common coupled fixed point provided that both (c) and (d) are true.Let  be a metric space. Assume F : X \u00d7 X \u2192 CB(X) and g : X \u2192 X to be mappings such that(1)F, g} satisfies the (EA) property,{(2)x, y, u, v \u2208 X, there exist some \u03c6 \u2208 \u03a6 and some \u03c8 \u2208 \u03a8 such thatfor all  If (3) of (a)F and g have a coupled coincidence point,(b)F and g have a common coupled fixed point, if g is F-weakly commuting at  and g2x = gx and g2y = gy for  \u2208 C{F, g}.Let  be a metric space. Assume F, G : X \u00d7 X \u2192 CB(X) and f, g : X \u2192 X to be mappings satisfying (1) of (1)x,y,u,v \u2208 X, there exists some \u03c8 \u2208 \u03a8 such thatfor all  If (3) of (a)F and f have a coupled coincidence point,(b)G and g have a coupled coincidence point,(c)F and f have a common coupled fixed point, if f is F-weakly commuting at  and f2x = fx and f2y = fy for  \u2208 C{F, f},(d)G and g have a common coupled fixed point, if g is G-weakly commuting at (e)F, G, f, and\u2009\u2009g have common coupled fixed point provided that both (c) and (d) are true.Let  of \u03c6(t) = (1/2)t, t \u2208 [0, +\u221e), and then the above condition reduces to condition (2) of \u03c81 = (1/2)\u03c8 and hence by If f = g in Put X, d) be a metric space. Assume F, G : X \u00d7 X \u2192 CB(X) and g : X \u2192 X to be mappings satisfying (1) of (1)x, y, u, v \u2208 X, there exists some \u03c8 \u2208 \u03a8 such thatfor all  If (3) of (a)F and g have a coupled coincidence point,(b)G and g have a coupled coincidence point,(c)F and g have a common coupled fixed point, if g is F-weakly commuting at  and g2x = gx and g2y = gy for  \u2208 C{F, g},(d)G and g have a common coupled fixed point, if g is G-weakly commuting at (e)F, G, and\u2009\u2009g have common coupled fixed point provided that both (c) and (d) are true.Let  be a metric space. Assume F : X \u00d7 X \u2192 CB(X) and g : X \u2192 X to be mappings satisfying (1) of (1)x, y, u, v \u2208 X, there exists some \u03c8 \u2208 \u03a8 such thatfor all  If (3) of (a)F and g have a coupled coincidence point,(b)F and g have a common coupled fixed point, if g is F-weakly commuting at  and g2x = gx and g2y = gy for  \u2208 C{F, g}.Let  be a metric space. Assume F, G : X \u00d7 X \u2192 CB(X) and f, g : X \u2192 X to be mappings satisfying (1) of (1)F, f} and {G, g} are w-compatible.{(2)Suppose that either(a)g(X) is a closed subset of X and G(X \u00d7 X)\u2286f(X) or(b)f(X) is a closed subset of X and F(X \u00d7 X)\u2286g(X).Then F, G, f, and\u2009\u2009g have a common coupled fixed point.Let  property, there exist sequences {xn}, {yn}, {un}, and {vn} in X, some u, v in X, and A, B, C, D in CB(X) satisfying (g(X) is a closed subset of X, and then there exist G and g. Hence, w-compatibility of {G, g}, we have gu \u2208 G and gv \u2208 G. Now, we shall show that u = gu and v = gv. Suppose, not. Then, by condition (2) of n \u2192 \u221e in the above inequality, by using (\u03c6), we obtainu \u2208 A, v \u2208 C, gu \u2208 G, and gv \u2208 G, therefore, by (i\u03c8), we getu = gu and v = gv. Hence, we haveG(X \u00d7 X)\u2286f(X), then there exist x, y \u2208 X such that fx = u = gu \u2208 G and fy = v = gv \u2208 G. Now, by condition (2) of \u03c6),\u2009\u2009(ii\u03c6), and (ii\u03c8), we get\u03c6) and (ii\u03c6), impliesx, y) is a coupled coincidence point of F and f. Hence,  \u2208 C{F, f}. From w-compatibility of {F, f}, we have fF\u2286F; hence f2x \u2208 F and f2y \u2208 F; that is, fu \u2208 F and fv \u2208 F. Now, we shall show that fu = u and fv = v. Suppose, not. Then, by condition (2) of \u03c8), we getfu = u and fv = v. Hence, we haveu, v) is a common coupled fixed point of the pairs {F, f} and {G, g}. The proof is similar when (b) holds.Since {tisfying . Supposeby using  and (i\u03c6)f = g in If we put X, d) be a metric space. Assume F, G : X \u00d7 X \u2192 CB(X) and g : X \u2192 X to be mappings satisfying (1) of (1)F, g} and {G, g} are w-compatible;{(2)suppose that either(a)g(X) is a closed subset of X and G(X \u00d7 X)\u2286g(X) or(b)g(X) is a closed subset of X and F(X \u00d7 X)\u2286g(X).Then F, G, and\u2009\u2009g have a common coupled fixed point.Let  be a metric space. Assume F : X \u00d7 X \u2192 CB(X) and g : X \u2192 X to be mappings satisfying (1) of (1)F, g} is w-compatible;{(2)g(X) is a closed subset of X and F(X \u00d7 X)\u2286g(X).Then F and g have a common coupled fixed point.Let  be a metric space. Assume F, G : X \u00d7 X \u2192 CB(X) and f, g : X \u2192 X to be mappings satisfying (1) of F, G, f,\u2009\u2009and g have a common coupled fixed point.Let  of \u03c6(t) = (1/2)t, t \u2208 [0, +\u221e), then it reduces to condition (2) of \u03c81 = (1/2)\u03c8 and hence by If f = g in If we put X, d) be a metric space. Assume F, G : X \u00d7 X \u2192 CB(X) and g : X \u2192 X to be mappings satisfying (1) of F, G, and g have a common coupled fixed point.Let  be a metric space. Assume F : X \u00d7 X \u2192 CB(X) and g : X \u2192 X to be mappings satisfying (1) of F and g have a common coupled fixed point.Let  be a metric space. Assume F, G : X \u00d7 X \u2192 CB(X) and f, g : X \u2192 X to be mappings satisfying (2) of (1)F, f} and {G, g} are occasionally w-compatible.{Then F, G, f,\u2009\u2009and g have a common coupled fixed point.Let ,\u03c8), we haveu = fu = gu and v = fv = gv. Suppose, not. Then, by condition (2) of \u03c8), we haveu, v) is a common coupled fixed point of F, G, f,\u2009\u2009and g.Since the pairs {ows thatf2x\u2208Ffx,fhus, by  be a metric space. Assume F, G : X \u00d7 X \u2192 CB(X) and g : X \u2192 X to be mappings satisfying (2) of (1)F, g} and {G, g} are occasionally w-compatible.{Then F, G,\u2009\u2009and g have a common coupled fixed point.Let  be a metric space. Assume F : X \u00d7 X \u2192 CB(X) and g : X \u2192 X to be mappings satisfying (2) of (1)F, g} is occasionally w-compatible.{Then F and g have a common coupled fixed point.Let  be a metric space. Assume F, G : X \u00d7 X \u2192 CB(X) and f, g : X \u2192 X to be mappings satisfying (1) of F, G, f, and\u2009\u2009g have a common coupled fixed point.Let  of \u03c6(t) = (1/2)t, t \u2208 [0, +\u221e), then it reduces to condition (2) of \u03c81 = (1/2)\u03c8 and hence by If f = g in Put X, d) be a metric space. Assume F, G : X \u00d7 X \u2192 CB(X) and g : X \u2192 X to be mappings satisfying (1) of F, G, and g have a common coupled fixed point.Let  be a metric space. Assume F : X \u00d7 X \u2192 CB(X) and g : X \u2192 X to be mappings satisfying (1) of F and g have a common coupled fixed point.Let  defined as d = max\u2061{x, y} and d = 0 for all x, y \u2208 X. Let F, G : X \u00d7 X \u2192 CB(X) be defined asf, g : X \u2192 X be defined as\u03c6 : [0, +\u221e)\u2192[0, +\u221e) by\u03c8 : [0, +\u221e)\u2192[0, +\u221e) byx, y, u, v \u2208 X with x, y, u, v \u2208 [0,1), we haveSuppose that Case (a). If (x2 + y2)/4 = (u + v)/8, thenCase (b). If (x2 + y2)/4 \u2260 (u + v)/8 with (x2 + y2)/4 < (u + v)/8, thenu + v)/8<(x2 + y2)/4. Thus, the contractive condition (2) of x, y, u, v \u2208 X with x, y, u, v \u2208 [0,1). Again, for all x, y, u, v \u2208 X with x, y \u2208 [0,1) and u, v = 1, we havex, y, u, v \u2208 X with x, y \u2208 [0,1) and u, v = 1. Similarly, we can see that the contractive condition (2) of x, y, u, v \u2208 X with x, y, u, v = 1. Hence, the hybrid pairs {F, f} and {G, g} satisfy condition (2) of x, y, u, v \u2208 X. In addition, all the other conditions of z =  is a common coupled fixed point of F, G, f, and\u2009\u2009g."}
{"text": "U, let k such that U can be decomposed into a product of k rotations about either U. Here, a rotation means an element D of the special orthogonal group SO(3) or an element of the special unitary group SU(2) that corresponds to D. Decompositions of U attaining the minimum number For any pair of three-dimensional real unit vectors  However, we follow the custom, in quantum physics, to call not only an element of SO(3) but also that of the special unitary group SU(2) a rotation. This is justified by the well-known homomorphism from SU(2) onto SO(3)  are presented explicitly.Then, a natural question arises: What is the least value, r angles\u00a0,4. MoreoIn this work, not only explicit constructions but also simple inequalities on geometric quantities, which directly show lower bounds on the number of constituent rotations, will be presented. Remarkably, the proposed explicit constructions meet the obtained lower bounds, which shows both the optimality of the constructions and the tightness of the bounds.D\u2208SO(3). That interesting result\u00a0. The Hermitian conjugate of a matrix U is denoted by U\u2020.The notation to be used includes the following: I denotes the 2\u00d72 identity matrix; X, Y and Z denote the following Pauli matrices:\u03b8 , where For F from SU(2) onto SO(3) to be defined in \u00a7D\u2208SO(3) andUsing the homomorphism 3.2This work's results lead to an elementary self-contained proof of the following known theorem (appendix\u00a0F).For anywith3.3The following lemma presents a well-known parametrization of SU(2) elements.For any elementU\u2208SU(2), there exist someand\u03b2\u2208 such that\u03b1,\u03b2 and \u03b3 in this lemma are often called Euler angles.2 The lemma can be rephrased as follows: any matrix in SU(2) can be written asa and b such that |a|2+|b|2=1\u00a0 such thatF is onto SO(3), there exists an element U\u2208SU(2) such that 4 With this element U, some \u03b2\u2208, write U\u2020VU=Rz(\u03b1)Ry(\u03b2)Rz(\u03b3) in terms of the parametrization (3.6). Then, since As We also have the following lemma, which is easy but worth recognizing.Let arbitraryandU\u2208SU(2) be given. PutandThen, for anyif and only if (iff)This readily follows from 4.Here, we present the result establishing U uniquely, For Functions For anywithand \u03b2\u2208, ifthenwhereand\u03b1,\u03b2 and \u03b3 vary, in general, if Note that there is no loss of generality in assuming We give two constructions or decompositions, which will turn out to attain the minimum number Given arbitrarywithand\u03b2\u2208, putandThen, for anyand\u03b21,\u2026,\u03b2k\u2208 holds for some \u03b21,\u2026,\u03b2k\u2208 \u2309.5 Hence, this proposition gives a decomposition of an arbitrary element \u03b2/(2\u03b4)\u2309+1 rotations.6The least value of \u03b2/2\u2264\u03b4\u2264\u03c0/2, \u03b4\u22600, and \u03b1j, \u03b3j and \u03b8j for which (4.5) holds is given by T=\u03c3tj, wherej=1,\u2026,k. (These make (4.6) hold.)For Given anywithputandFor an arbitraryU\u2208SU(2), choose parametersand\u03b2\u2032\u2208 such thatThen,Furthermore, for anyand\u03b2\u20321,\u2026,\u03b2\u2032k\u2032\u2208 holds for some \u03b2\u20321,\u2026,\u03b2\u2032k\u2032\u2208 /(2\u03b4)\u2309=\u2308\u03b2\u2032/(2\u03b4)+1/2\u2309. Moreover, if \u03b2\u2032\u2265\u03b4 and k\u2032=\u2308\u03b2\u2032/(2\u03b4)+1/2\u2309, the parameter \u03b1\u20321 can be chosen so that it satisfies \u03b1\u20321=0 as well as (4.11) and (4.12). Hence, when \u03b2\u2032\u2265\u03b4, this proposition and the fact just mentioned give a decomposition of an arbitrary element \u03b2\u2032<\u03b4, a decomposition of U into the product of four rotations.The least value of \u03b1\u2032j,\u03b3\u2032j and \u03b8\u2032j, j=1,\u2026,k\u2032, for which (4.11) and (4.12) hold is given by T= \u03c3tj, where j=1,\u2026,k\u2032.An explicit instance of the set of parameters 5.S2, which is denoted by d. Specifically,S2. We have the following lemma.  and (5.4) will be used in the following forms:D and D\u2032\u2208SO(3) equal the product of 2k\u22121 rotations and that of 2k rotations, respectively, in lemma\u00a0k is an integer). It will turn out that these bounds are tight.This can be shown easily by induction on 6.6.1Here, the structure of the whole proof of the results in this work is described. Theorem\u00a06.2The following lemma is fundamental to the results in this work.For anyand for anysuch thatthe following two conditions are equivalent.There exist somesuch thatI.\u2003II.\u2003U\u2208SU(2) such thatvx,vy and vz such that(1) Take an element (2) A direct calculation shows(3) We shall prove I \u21d2 II. On each side of (6.7) and (6.8), squaring and summing the resultant pair, we have(4) Next, we shall prove II \u21d2 I.\u03b1,\u03b2) into , where the two pairs are related byTransforming (\u03b7 satisfying (6.12) and (6.15). From II, however, we have (6.10), and hence, \u03b7.Now suppose \u03b6 satisfying (6.13) and (6.14). But (6.17) follows again from II or (6.10) since \u03b6 similarly.\u2003\u25aaIn a similar way, if \u03b8j such that \u03b2j\u2208Rz(\u03b1\u2032)Ry(\u2212\u03b4)=Rv(\u03b1\u2032), where \u03b1,\u03b2,\u03b3) replaced by , it readily follows that there exist some \u03b1\u2032j,\u03b3\u2032j and j=1,\u2026,k\u2032, that satisfy the following: j=1,\u2026,k\u2032, andNote \u03b1\u20321 in remark\u00a0\u03b2\u20321=2\u03b4 and t1=\u03c0/2) or, more directly, from an equation Ry(2\u03b4)=Rv(\u03c0)Rz(\u2212\u03c0), where Remarks\u00a06.3U\u2208SU(2);D\u2208SO(3). The following lemma largely solves the issue of determining the optimal number Let Letand\u03b4be as in theorem\u00a0Then, for anyand\u03b2\u2208,andwhereis as defined in theorem\u00a0Letand\u03b4be as in theorem\u00a0Then, for anyand\u03b2\u2208,\u03b2=0, since \u03b2>0.In the case where To establish (6.19), we shall show the first and third inequalities inj=2k\u22121 with \u03bd is odd and Note first that remark\u00a0\u03b2\u22642(k\u22121)\u03b4 by (5.2) of lemma\u00a0\u03b2/(2\u03b4)\u2309\u2264k\u22121, and therefore,We shall evaluate f\u2265\u03b4. Recalling that To establish (6.20), we shall first treat the major case where \u03b2\u2032 in proposition\u00a0\u03b2\u2032=f when \u03b2\u2032=f, rewrite (4.8), using lemma\u00a0\u03b2\u2032=f in view of (3.6).Note that remark\u00a0j=2k with \u03bd is odd and \u03b2\u2032\u2264(2k\u22121)\u03b4 by (5.4) of lemma\u00a0\u03b2\u2032+\u03b4)/(2\u03b4)\u2309\u2264k, and, therefore,f\u2265\u03b4. The proof of (6.20) in the other case is given in appendix\u00a0C. This completes the proof of the lemma. The proved lemma immediately implies the corollary.\u2003\u25aaTo prove the first inequality in (6.25), assume (6.23) holds for some U\u2208SU(2),U in terms of three parametric expressions:b. (This is because writing U in (4.1) as Note that for any U\u2208SU,Nm^,n^ showsU\u2208SU(2), where we assume U\u2020 in place of U [note 7 See appendix\u00a0G for a detailed description of the above construction method.From the viewpoint of construction, we summarize the (most directly) suggested way to obtain an optimal construction of a given element 7.k such that U can be decomposed into the product of k rotations about either U in SU(2), or in SO(3), where U attaining the minimum number This work has established the least value 8.U is an arbitrary fixed rotation and A\u2282S2 with |A|=2  is considered. In the series of Brezov et al.\u00a0[Naturally, the present author could not find any (explicit or implicit) indication that Brezov et al. \u201312 suggev et al. \u201312 may bv et al.\u00a0\u201312, theyet al. [Despite such differences in essence and background, note in the proof of this paper's formula 6.20) for the minimum even number of factors in lemma\u00a00 for theet al. \u201312."}
{"text": "In order to solve the large scale linear systems, backward and Jacobi iteration algorithms are employed. The convergence is the most important issue. In this paper, a unified backward iterative matrix is proposed. It shows that some well-known iterative algorithms can be deduced with it. The most important result is that the convergence results have been proved. Firstly, the spectral radius of the Jacobi iterative matrix is positive and the one of backward iterative matrix is strongly positive (lager than a positive constant). Secondly, the mentioned two iterations have the same convergence results (convergence or divergence simultaneously). Finally, some numerical experiments show that the proposed algorithms are correct and have the merit of backward methods. A is a given n \u00d7 n complex or real matrix.The primal goal of this paper is to study the iterative methods of the linear systems:A in  and ordinary differential equations (ODE) \u20133 it is A in . In the A in . In the A in . TherefoA in  is very The methods to solve linear systems can be roughly divided into two categories: direct methods and iterative methods 1. Iterative methods are more suitable than direct methods for large linear systems , 12. TheIn this paper, we do some research with the iterative algorithm. To this end, the paper is organized as follows. In D = diag\u2061(A) is a diagonal matrix obtained from A and nonsingular and CL and CU are strictly lower and upper triangular matrices obtained from A, (D\u22121A = (I \u2212 L \u2212 U).The basic idea to solve  is matriThe Jacobi iterative matrix isudied in \u201317.\u03c4 > 0 is a real constant, obviously . Matrix A, vector b, \u03c4 > 0, \u03c91, \u03c92, algorithm stop cutoff \u03f5. Step\u2009\u20091 . Compute D = diag\u2061(A), CL, CU, L = D\u22121CL, U = D\u22121CU, x(0) = 0, and set i\u22540. Step\u2009\u20092. Compute matrix rding to . Step\u2009\u20093. Compute xi+1)( with (+1) with . Step\u2009\u20094. If ||xi+1)(\u2212xi)||2\u03c91, \u2009\u2009\u03c92, and \u03c4, we have the following.\u03c91 = 0, \u03c92 = 0, and \u03c4 = 1, we obtain the Jacobi iterative method.When \u03c91 = 0, \u03c92 = 0, and \u03c4 = \u03c9, we obtain the backward JOR iterative method.When \u03c91 = 1, \u03c92 = 0, and \u03c4 = 1, we obtain the backward G-S iterative method.When \u03c91 = \u03c9, \u03c92 = 0, and \u03c4 = \u03c9, we obtain the backward SOR iterative method.When \u03c91 = \u03c9, \u03c92 = 0, and \u03c4 = \u03b1, we obtain the backward AOR iterative method.When \u03c91 = \u03c9, \u03c92 = \u03c9, and \u03c4 = \u03c9(2 \u2212 \u03c9), we obtain the backward SSOR iterative method.When \u03c91 = \u03c9, \u03c92 = \u03c9, and \u03c4 = \u03c9, we obtain the backward EMA iterative method.When \u03c91 = \u03c9, \u03c92 = \u03c9, and \u03c4 = \u03b1, we obtain the backward PSD iterative method.When \u03c91 = \u03c9, \u03c92 = \u03c9, and \u03c4 = 1, we obtain the backward PJ iterative method.When With special values of The convergence relationship between the Gauss-Seidel iterative matrix and the Jacobi iterative matrix is studied in , and theIn order to obtain the convergence results, we give some well-known results which will be used in the proof of A = M \u2212 N with A and M nonsingular is called a regular splitting if M\u22121 \u2265 0 and N \u2265 0. It is called a weak regular splitting if M\u22121 \u2265 0 and M\u22121N \u2265 0.The splitting It is obvious that a regular splitting is a weak regular splitting.T \u2208 Rn\u00d7n is convergent; that is, \u03c1(T) < 1 if and only if (I \u2212 T)\u22121 exists and (I \u2212 T)\u22121 = \u2211k=1\u221eTk \u2265 0.The nonnegative matrix A = M \u2212 N be a weak regular splitting of A, H = M\u22121N. Then the following statements are equivalent.(1)\u2009A\u22121 \u2265 0; that is, A is inverse-positive.(2)\u2009A\u22121N \u2265 0.(3)\u2009\u03c1(H) = \u03c1(A\u22121N)/(1 + \u03c1(A\u22121N)) so that \u03c1(H) < 1.Let A \u2265 0 be an irreducible n \u00d7 n matrix. Then(1)\u2009A has a positive real eigenvalue equal to its spectral radius;(2)\u2009\u03c1(A), there corresponds an eigenvector x > 0,to (3)\u2009\u03c1(A) increases when any entry of A increases,(4)\u2009\u03c1(A) is a simple eigenvalue of A.Let A = (aij) \u2265 0 be an irreducible n \u00d7 n matrix. Then for any x > 0, eitherLet By the lemmas above, we give the convergence theorem in the following.A of \u2009\u03c1(B) > 0, \u2009(2)One and only one of the following mutually exclusive relations is valid.(i)(ii)(iii)Let the coefficient matrix A of  be irredThus, The Jacobi iterative method and the backward MPSD iterative method are either both convergent or both divergent.\u03c1(\u03c92L) = \u03c1(\u03c91U) = 0 with I \u2212 \u03c92L)\u22121 \u2265 0, (I \u2212 \u03c91U)\u22121 \u2265 0, andCombining aii \u2260 0 and A is irreducible, I \u2212 L \u2212 U = D\u22121A and B = L + U are irreducible. By 0 \u2264 \u03c9k < \u03c4 \u2264 1, k = 1,2, we have (1 \u2212 \u03c4)I + (\u03c4 \u2212 \u03c91)U + (\u03c4 \u2212 \u03c92)L \u2265 0 and irreducible. Thus, by T > 0, such that \u03b7k = \u03c4 \u2212 \u03c9k + \u03bb\u03c9k, k = 1,2; by calculation,Since Thus, by , S~\u03c4,\u03c91,B \u2265 0 is irreducible, by \u03c1(B) > 0. If \u03c9k < \u03c4 \u2264 1, k = 1,2, \u03bb \u2212 (1 \u2212 \u03c4) \u2265 0 because the left side of  Since  side of  is nonne \u2212 \u03c4. By ,16)\u03b71U\u03b71UB \u2265 0 \u03b7k > 0, k = 1,2, B = (bij) \u2265 0 and x > 0, we obtain that B = 0. Thus, \u03c1(B) = 0. This contradicts \u03c1(B) > 0. So, Since (2) For mutually exclusive relations, consider the following.\u03c1(B) < 1, let(i) If 0 < M\u22121 = (I \u2212 \u03c92L)\u22121(I \u2212 \u03c91U)\u22121 \u2265 0 and N \u2265 0, T = M \u2212 N is a regular splitting:Since B \u2265 0, 0 < \u03c1(B) < 1, and 0 < \u03c4 \u2264 1, we know that T\u22121 = (1/\u03c4)(I \u2212 B)\u22121 \u2265 0. By By \u03b7k > 0, k = 1,2, bii = 0, \u2200 i,If 2)<1, by , we have\u03bb < 1 and 1 \u2212 \u03c9k > 0\u2009\u2009, there isBy \u03c1(B) < 1.Combining  with 2929, we haBx = x. Since x > 0, we have\u03c1(B) = 1.(ii) If 2)=1, by , we have\u03b7k > 0, k = 1,2, bii = 0, \u2200 i,(iii) If 2)>1, by , we have\u03bb > 1 and 1 \u2212 \u03c9k > 0\u2009\u2009, there isBy \u03c1(B) > 1.Combining  with 3333, we ha\u03c1(B) = 1 and \u03c1(B) < 1 or \u03c1(B) > 1. This contradicts \u03c1(B) = 1. So, If \u03c1(B) > 1 and \u03c1(B) \u2264 1. This contradicts \u03c1(B) > 1. So, If \u03c91, \u03c92, and \u03c4, we have the following corollaries.With special values of A of \u2009\u03c1(B) > 0, (2)\u2009One and only one of the following mutually exclusive relations is valid.\u2009(i)(ii)(iii)Let the coefficient matrix A of  be irredThus, The Jacobi iterative method and the backward JOR iterative method are either both convergent or both divergent.A of \u2009\u03c1(B) > 0, (2)\u2009One and only one of the following mutually exclusive relations is valid.Thus, The Jacobi iterative method and the backward Gauss-Seidel iterative method are either both convergent or both divergent.A of \u2009\u03c1(B) > 0, (2)\u2009One and only one of the following mutually exclusive relations is valid. \u2009Thus, The Jacobi iterative method and the backward SOR iterative method are either both convergent or both divergent.A of \u2009\u03c1(B) > 0, (2)\u2009One and only one of the following mutually exclusive relations is valid.(i)(ii)(iii)Let the coefficient matrix A of  be irredThus, The Jacobi iterative method and the backward AOR iterative method are either both convergent or both divergent.A of \u2009\u03c1(B) > 0, (2)\u2009One and only one of the following mutually exclusive relations is valid. \u2009Thus, The Jacobi iterative method and the backward SSOR iterative method are either both convergent or both divergent.A of \u2009\u03c1(B) > 0, (2)\u2009One and only one of the following mutually exclusive relations is valid.\u2009\u2009(i)(ii)(iii)Let the coefficient matrix A of  be irredThus, The Jacobi iterative method and the backward EMA iterative method are either both convergent or both divergent.A of \u2009\u03c1(B) > 0, (2)\u2009One and only one of the following mutually exclusive relations is valid. (i)(ii)(iii)Let the coefficient matrix A of  be irredThus, The Jacobi iterative method and the backward PSD iterative method are either both convergent or both divergent.A of \u2009\u03c1(B) > 0, (2)\u2009One and only one of the following mutually exclusive relations is valid. \u2009\u2009\u2009(i)(ii)(iii)Let the coefficient matrix A of  be irredThus, The Jacobi iterative method and the backward PJ iterative method are either both convergent or both divergent.The convergence results between the backward MPSD and Jacobi iterative matrix are proposed, and The convergence results between some special cases of backward MPSD  and Jacobi iterative matrix are obtained. These results involve some special iterative methods which are proposed in the references.In this section, we show five examples. The first three examples are used to show the convergence of the proposed iterative methods. A and the vector b of (Let the coefficient matrix tor b of  be(35)AThe Jacobi iterative matrix is\u03c1(B) = 1/2 < 1.(1)\u03c4 = \u03b1 = 1/2, \u03c91 = \u03c92 = \u03c9 = 1/4. We obtain the backward PSD iterative matrix Let (2)\u03c4 = 1, \u03c91 = \u03c92 = \u03c9 = 1/2. We obtain the backward PJ iterative matrix Let (3)\u03c4 = \u03c9 = 1/2, \u03c91 = \u03c92 = 0. We obtain the backward JOR iterative matrix Let (4)\u03c4 = \u03c91 = \u03c92 = \u03c9 = 1/2. We obtain the backward EMA iterative matrix Let  With these iterative methods and the presented algorithm, the solution is x = T.By caculation, we obtain 0 < A and the vector b of \u22024\u22022u\u2202\u03c1(B) = 1159/1601 < 1.(1)\u03c4 = \u03b1 = 1/2, \u03c91 = \u03c92 = \u03c9 = 1/4. We obtain the backward PSD iterative matrix Let (2)\u03c4 = 1, \u03c91 = \u03c92 = \u03c9 = 1/2. We obtain the backward PJ iterative matrix Let (3)\u03c4 = \u03c9 = 1/2, \u03c91 = \u03c92 = 0. We obtain the backward JOR iterative matrix Let (4)\u03c4 = \u03c91 = \u03c92 = \u03c9 = 1/2. We obtain the backward EMA iterative matrix Let The Jacobi iterative matrix is\u03c91, \u03c92, and \u03c4 in this example.From Figures A of , and s = \u2212p/(n + 2) [n = 100 and p = 1. By caculation, we obtain 0 < \u03c1(B) = 199/203 < 1.(1)\u03c4 = \u03b1 = 1/2, \u03c91 = \u03c92 = \u03c9 = 1/4. We obtain the backward PSD iterative matrix Let (2)\u03c4 = 1, \u03c91 = \u03c92 = \u03c9 = 1/2. We obtain the backward PJ iterative matrix Let (3)\u03c4 = \u03c9 = 1/2, \u03c91 = \u03c92 = 0. We obtain the backward JOR iterative matrix Let (4)\u03c4 = \u03c91 = \u03c92 = \u03c9 = 1/2. We obtain the backward EMA iterative matrix Let Let the coefficient matrix A of  be(48)A/(n + 2) . Here, wA and the vector b of (\u03c1(B) = 987/305 > 1.(1)\u03c4 = \u03b1 = 1/2, \u03c91 = \u03c92 = \u03c9 = 1/4. We obtain the backward PSD iterative matrix Let (2)\u03c4 = 1, \u03c91 = \u03c92 = \u03c9 = 1/2. We obtain the backward PJ iterative matrix Let (3)\u03c4 = \u03c9 = 1/2, \u03c91 = \u03c92 = 0. We obtain the backward JOR iterative matrix Let (4)\u03c4 = \u03c91 = \u03c92 = \u03c9 = 1/2. We obtain the backward EMA iterative matrix Let  It shows that the backward MPSD iteration is invalid for this example.Let the coefficient matrix tor b of  be(53)AA of (Let the coefficient matrix A of  be(59)A=A in [\u03c1(B) = 811/822 < 1.We can see the analogous matrix A in . By cacu\u03c4 = 1, \u03c91 = 1, \u03c92 = 0. We obtain the backward Gauss-Seidel iterative matrix Let From The Jacobi iteration is the basic iteration for linear systems and easier to the analysis of the convergence than other iterations. In the paper, we proposed the backward MPSD iteration and obtained the convergence result between backward MPSD iteration  and Jacobi iteration. We pointed out that the backward MPSD iteration and the Jacobi iteration are either both convergent or both divergent under the assumptions in"}
{"text": "In this work, nonparametric nonsymmetric measure of divergence, a particular part of Vajda generalized divergence at m = 4, is taken and characterized. Its bounds are studied in terms of some well-known symmetric and nonsymmetric divergence measures of Csiszar's class by using well-known information inequalities. Comparison of this divergence with others is done. Numerical illustrations (verification) regarding bounds of this divergence are presented as well.Vajda (1972) studied a generalized divergence measure of Csiszar's class, so called \u201cChi- If we take pi \u2265 0 for some i = 1, 2, 3,\u2026, n, then we have to suppose that 0f(0) = 0f(0/0) = 0.Let \u0393f-divergence measure, which is given byf :  \u2192 R  is a real, continuous, and convex function and P = , Q =  \u2208 \u0393n, where pi and qi are probability mass functions. Many known divergences can be obtained from this generalized measure by suitably defining the convex function f. Some of those are as follows.Csiszar  introducP, Q \u2208 \u0393n. These measures are as follows (see , \u0394 = 2[1 \u2212 W], h = 1 \u2212 B, and I = (1/2)[F + F], where W = 2\u2211i=1n(piqi/(pi + qi)) is the harmonic mean divergence, F = \u2211i=1npilog\u2061(2pi/(pi + qi)) is the relative JS divergence [Nonsymmetric measures are those that are not symmetric with respect to probability distributions ows see \u201310):(7):(7)P, Qvergence . EquatioIn this section, we obtain nonsymmetric divergence measure for convex function and further define the properties of function and divergence. Firstly, f is convex and normalized, that is, f(1) = 0, then Cf and its adjoint Cf are both nonnegative and convex in the pair of probability distribution  \u2208 \u0393n \u00d7 \u0393n.If the function f :  \u2192 R be a function defined as(a)f1(1) = 0, f1(t) is a normalized function.Since (b)f1\u2032\u2032(t) \u2265 0 for all t \u2208 \u21d2f1(t) is a convex function as well.Since (c)f1\u2032(t) < 0 at  and f1\u2032(t) > 0 at \u21d2f1(t) is monotonically decreasing in  and monotonically increasing in  and f1\u2032(1) = 0, f1\u2032\u2032(1) = 0.Since Now, let fined asft=f1t=t\u2212f1(t) in  is called adjoint of \u03c74 = \u2211i=1n((pi \u2212 qi)4/qi3)  in  = \u2211i=1n(|pi \u2212 qi|m/qim\u22121), m \u2265 1.Now put 1(t) in \u03c74 is convex and nonnegative in the pair of probability distribution  \u2208 \u0393n \u00d7 \u0393n.In view of (b)\u03c74 = 0 if P = Q or pi = qi .(c)\u03c74 \u2260 \u03c74\u21d2\u03c74 is a nonsymmetric divergence measure.Since Properties of the divergence measure defined in  are as fCf, which is given by f-divergence measures.In this section, we are taking well-known information inequalities on f1, f2 : I \u2282 R+ \u2192 R be two convex and normalized functions, that is, f1(1) = f2(1) = 0, and suppose the following assumptions.(a)f1 and f2 are twice differentiable on  where 0 < \u03b1 \u2264 1 \u2264 \u03b2 < \u221e, \u03b1 \u2260 \u03b2.(b)m, M such that m < M andf2\u2032\u2032(t) > 0 for all t \u2208 .There exist the real constants Let P, Q \u2208 \u0393n and satisfying the assumption 0 < \u03b1 \u2264 pi/qi \u2264 \u03b2 < \u221e, then one has the following inequalities:Cf is given by  and \u03c74 be defined as in \u03b1 < 1, thenIf 0 < (b)\u03b1 = 1, thenIf Let \u0394(ed as in  and 11)P, Q) andf2\u2032\u2032(t) > 0 for all t > 0 and f2(1) = 0, f2(t) is a convex and normalized function, respectively. Now put f2(t) in (g(t) = f1\u2032\u2032(t)/f2\u2032\u2032(t) = 3(t \u2212 1)2(t + 1)3/2t5, where f1\u2032\u2032(t) and f2\u2032\u2032(t) are given by (g\u2032(t) = 0\u21d2t = 1, t = \u22121, and t = 5.Let us consider6f2t=t\u221212given by  and 16)(16)f2t=tg(t) is decreasing in  and [5, \u221e) but increasing in [1,5).It is clear that g(t) has a minimum and maximum value at t = 1 and t = 5, respectively, because g\u2032\u2032(1) = 24 > 0 and g\u2032\u2032(5) = \u2212216/15625 < 0, so(a)\u03b1 < 1, thenif 0 < (b)\u03b1 = 1, thenif Results (P and Q.Also Results  and 15)g(t) has Results , 17),  h < 0, som=inf\u2061t\u22080 1, thenM=sup\u2061t\u2208\u03b1h and \u03c74 be defined as in \u03b1 < 1, thenIf 0 < (b)\u03b1 = 1, thenIf Let ed as in  and 11)h af2\u2032\u2032(t) > 0 for all t > 0 and f2(1) = 0, f2(t) is a convex and normalized function, respectively. Now put f2(t) in (g(t) = f1\u2032\u2032(t)/f2\u2032\u2032(t) = 48(t \u2212 1)2/t7/2, where f1\u2032\u2032(t) and f2\u2032\u2032(t) are given by (g\u2032(t) = 0\u21d2t = 1 and t = 7/3.Let us considergiven by  and 24)(24)f2t=1g(t) is decreasing in  and [7/3, \u221e) but increasing in [1,7/3).It is clear that g(t) has a minimum and maximum value at t = 1 and t = 7/3, respectively, because g\u2032\u2032(1) = 96 > 0 and g\u2032\u2032(2.33) = \u22121951/913 < 0, so(a)\u03b1 < 1, thenif 0 < (b)\u03b1 = 1, thenif Results (P and Q.Also Results  and 23)g(t) has Results , 25),  h < 0, som=inf\u2061t\u22080 1, thenM=sup\u2061t\u2208\u03b1 1, thenM=sup\u2061t\u22081I and \u03c74 be defined as in \u03b1 < 1, thenIf 0 < (b)\u03b1 = 1, thenIf Let ed as in , respectf2\u2032\u2032(t) > 0 for all t > 0 and f2(1) = 0, f2(t) is a convex and normalized function, respectively. Now put f2(t) in (g(t) = f1\u2032\u2032(t)/f2\u2032\u2032(t) = 24(t \u2212 1)2(t + 1)/t4, where f1\u2032\u2032(t) and f2\u2032\u2032(t) are given by (g\u2032(t) = 0\u2009\u2009\u21d2\u2009\u2009t = 1, Let us considergiven by  and 32)(32)f2t=tg(t) is decreasing in  and It is clear that g(t) has a minimum and maximum value at t = 1 and g\u2032\u2032(1) = 96 > 0 and (a)\u03b1 < 1, thenif 0 < (b)\u03b1 = 1, thenif Results (P and Q.Also Results  and 31)g(t) has Results , 33),  h65<0, som=inf\u2061t\u22080 1, thenM=sup\u2061t\u2208\u03b1 1, thenM=sup\u2061t\u22081Results , after iP* and \u03c74 be defined as in (Let ed as in  and 11)P*P, Q \u2208 \u0393n, one hasFor f2\u2032\u2032(t) \u2265 0 for all t > 0 and f2(1) = 0, f2(t) is a convex and normalized function, respectively. Now put f2(t) in (g(t) = f1\u2032\u2032(t)/f2\u2032\u2032(t) = 2/(2t5 + t4 + t3 + t2 + t + 2), where f1\u2032\u2032(t) and f2\u2032\u2032(t) are given by (g(t) is always decreasing in , soP and Q.Let us considergiven by  and 39)(39)f2t=t.Result  is obtai.Result , 40), a, a(39)f2, \u221e), som=inf\u2061t\u2208\u03b1E* and \u03c74 be defined as in E*f2\u2032\u2032(t) \u2265 0 for all t > 0 and f2(1) = 0, f2(t) is a convex and normalized function, respectively. Now put f2(t) in (g(t) = f1\u2032\u2032(t)/f2\u2032\u2032(t) = 16/t3/2(5t2 + 6t + 5), where f1\u2032\u2032(t) and f2\u2032\u2032(t) are given by (g(t) is always decreasing in , soP and Q.Let us considergiven by  and 44)(44)f2t=t.Result  is obtai.Result , 45), a, a(44)f2, \u221e), som=inf\u2061t\u2208\u03b1Now in this section, we obtain bounds of divergence measure  in termsJR and \u03c74 be defined as in \u03b1 < 1, thenIf 0 < (b)\u03b1 = 1, thenIf Let ed as in  and 11)JRf2\u2032\u2032(t) > 0 for all t > 0 and f2(1) = 0, f2(t) is a convex and normalized function, respectively. Now put f2(t) in (g(t) = f1\u2032\u2032(t)/f2\u2032\u2032(t) = 12(t2 \u2212 1)2/(t + 3)t5, where f1\u2032\u2032(t) and f2\u2032\u2032(t) are given by (g\u2032(t) = 0\u21d2t = 1,\u2009\u2009t = \u22121, and t \u2248 116/59.Let us considergiven by  and 50)(50)f2t=tg(t) is decreasing in  and [116/59, \u221e) but increasing in [1,116/59).It is clear that g(t) has a minimum and maximum value at t = 1 and t = 116/59, respectively, because g\u2032\u2032(1) = 24 > 0 and g\u2032\u2032(116/59) \u2248 \u22127/10 < 0, so(a)\u03b1 < 1, thenif 0 < (b)\u03b1 = 1, thenif Results (P and Q.Also Results  and 49)g(t) has Results , 53),  h 1, thenM=sup\u2061t\u2208\u03b1 1, thenM=sup\u2061t\u22081Results , after iK and \u03c74 be defined as in \u03b1 < 1, thenIf 0 < (b)\u03b1 = 1, thenIf Let ed as in  and 11)K af2\u2032\u2032(t) > 0 for all t > 0 and f2(1) = 0, f2(t) is a convex and normalized function, respectively. Now put f2(t) in (g(t) = f1\u2032\u2032(t)/f2\u2032\u2032(t) = 12(t \u2212 1)2/t4, where f1\u2032\u2032(t) and f2\u2032\u2032(t) are given by (g\u2032(t) = 0\u21d2t = 1 and t = 2.Let us considergiven by  and 58)(58)f2t=tg(t) is decreasing in  and [2, \u221e) but increasing in [1,2).It is clear that g(t) has a minimum and maximum value at t = 1 and t = 2, respectively, because g\u2032\u2032(1) = 24 > 0 and g\u2032\u2032(2) = \u22123/4 < 0, so(a)\u03b1 < 1, thenif 0 < (b)\u03b1 = 1, thenif Results (P and Q.Also Results  and 57)g(t) has Results , 59),  h < 0, som=inf\u2061t\u22080 1, thenM=sup\u2061t\u2208\u03b1 1, thenM=sup\u2061t\u22081Results , after iG and \u03c74 be defined as in \u03b1 < 1, thenIf 0 < (b)\u03b1 = 1, thenIf Let ed as in  and 11)G af2\u2032\u2032(t) > 0 for all t > 0 and f2(1) = 0, f2(t) is a convex and normalized function, respectively. Now put f2(t) in (g(t) = f1\u2032\u2032(t)/f2\u2032\u2032(t) = 24(t \u2212 1)2(t + 1)/t3, where f1\u2032\u2032(t) and f2\u2032\u2032(t) are given by (g\u2032(t) = 0\u21d2t = 1 and t = \u22123.Let us considergiven by  and 66)(66)f2t=tg(t) is decreasing in  and increasing in [1, \u221e).It is clear that g(t) has a minimum value at t = 1, because g\u2032\u2032(1) = 120 > 0, so(a)\u03b1 < 1, thenif 0 < (b)\u03b1 = 1, thenif Results (P and Q.Also Results  and 65)g(t) has Results , 67),  h > 0, som=inf\u2061t\u22080 1, thenM=sup\u2061t\u2208\u03b1 1, thenM=sup\u2061t\u22081Results , after iP, Q), h, JR, K, and \u03c74 and verify inequalities .In this section, we give two examples for calculating the divergences \u0394, ,ualities , and 56P, Q, h and Q its approximated Poisson probability distribution with parameter (\u03bb = np = 5), for the random variable X; then we have Let By using p = 1/2.Put the approximated numerical values from  in 14),,14), 2222, 48),,48), andP be the binomial probability distribution with parameters  and Q its approximated Poisson probability distribution with parameter (\u03bb = np = 7), for the random variable X; then we have Let p = 7/10.By using \u03b1=5772\u2264piues from , and ) ) , 38), ) , and (64\u221e). Function is decreasing in  and increasing in . \u03c74, E*, K, \u0394, and G. We have considered pi = , qi = , where a \u2208 . It is clear from \u03c74 has a steeper slope than E*, K, \u0394, and G.Many research papers have been studied by Taneja, Kumar, Dragomir, Jain, and others, who gave the idea of divergence measures, their properties, their bounds, and relations with other measures. Taneja especially did a lot of quality work in this field: for instance, in  he derivDivergence measures have been demonstrated to be very useful in a variety of disciplines such as anthropology, genetics, finance, economics and political science, biology, analysis of contingency tables, approximation of probability distributions, signal processing, pattern recognition, sensor networks, testing of the order in a Markov chain, risk for binary experiments, region segmentation, and estimation.This paper also defines the properties and bounds of Vajda's divergence and derives new relations with other symmetric and nonsymmetric well-known divergence measures."}
{"text": "The adipate and oxalate ions are located on centres of inversion. The Dy3+ cation has a distorted tricapped trigonal\u2013prismatic geometry and is coordinated by nine O atoms, four belonging to three adipate anions, four to two oxalate anions and one from an aqua ligand. The cations are bridged by adipate ligands, generating a two-dimensional network parallel to (010). This network is further extended into three dimensions by coordination of the rigid oxalate ligands and is further consolidated by O\u2014H\u22efO hydrogen bonds. A part of the adipate anion is disordered over two positions in a 0.75:0.25 ratio.In the title coordination polymer, [Dy DOI: 10.1107/S1600536814024544/gk2622Isup2.hklStructure factors: contains datablock(s) I. DOI: Click here for additional data file.10.1107/S1600536814024544/gk2622fig1.tifx y z x y z x y - z x y z x y z . DOI: x, 1\u00a0\u2212\u00a0y, 1\u00a0\u2212\u00a0z; (ii) 1\u00a0\u2212\u00a0x, 1\u00a0\u2212\u00a0y, 1\u00a0\u2212\u00a0z; (iii) \u2212x, \u2212 y, 1- z; (iv) \u2212 x, \u2212 y, \u2212z; (v) 1\u00a0\u2212\u00a0x, 1\u00a0\u2212\u00a0y, 2\u00a0\u2212\u00a0z.The fragment of the structure of the title compounds, with the atom-numbering scheme. Displacement ellipsoids are drawn at the 30% probability level and H atoms are shown as small spheres of arbitrary radii. Symmetry code: (i) \u22121033325CCDC reference:  crystallographic information; 3D view; checkCIF reportAdditional supporting information:"}
{"text": "Scientific Reports5: Article number: 1342010.1038/srep13420; published online: 08262015; updated: 10302015In the HTML version of this Article, the legend of Figure 1 contains typographical errors.d) Hex\u2009\u2248\u2009HC\u2009H*\u2009\u2248\u2009Hsat and e Hex\u2009\u2265\u2009H*\u2009\u2248\u2009Hsat\u2009HC\u201d\u201c(should read:d) Hex\u2009\u2248\u2009HC\u2009\u226aH*\u2009\u2248\u2009Hsat and (e) Hex\u2009\u2265\u2009H*\u2009\u2248\u2009Hsat\u2009\u226b\u2009HC\u201d\u201c("}
{"text": "The Rosenzweig-MacArthur (1963) criterion is a graphical criterion that has been widely used for elucidating the local stability properties of the Gause (1934) type predator-prey systems. It has not been stated whether a similar criterion holds for models with explicit resource dynamics ), like the chemostat model. In this paper we use the implicit function theorem and implicit derivatives for proving that a similar graphical criterion holds under chemostat conditions, too. The parameters C > 0, D > 0, a > 0, b > 0, m > 0, A > 0, B > 0, and M > 0 stand for concentration, dilution rate, search rate for the prey, handling time for the prey , conver\u210b1 = 0 is asymptotically invariant for  < 1, 0 \u2264 \u03b8 < 2\u03c0, x3(0) \u2264 0 and positive constant parameters. In this case, consider the function\u210b2 = 0 is asymptotically invariant for (\u2217-marks in x3 = 0 has some equilibrium on the unit circle as its limit set. But if x3(0) > 0, then the limit set is the whole unit circle. We see, however, that the chain recurrent set [A reduction into two dimensions  is often Waltman  gave the r\u2009sin\u2009\u03b8:r\u02d9=r1\u2212r,\u03b8 r\u2009sin\u2009\u03b8:r\u02d9=r1\u2212r,\u03b8 r\u2009sin\u2009\u03b8:r\u02d9=r1\u2212r,\u03b8 functionH2=x32.Alb \u2260 0, then the growth function of f, \u03c8, and F are C1),(A-II)f(0) = 0, f(x) > 0 for x > 0,(A-III)x \u2212 1)F(x) < 0 for x \u2260 1,((A-IV)x \u2212 \u03bb1)\u03c8(x) > 0, x \u2260 \u03bb1 > 0,(We start this study by relating  to a wid a Gause  type prend Cheng . In gene\u03c8(x) = 0 (the predator isocline) in this case gives\u03bb1 < 1, then solutions of system ) that has the JacobianJ) = \u03bb1f(\u03bb1)\u03c8\u2032(\u03bb1) > 0 and Tr\u2009J) = f(\u03bb1)F\u2032(\u03bb1) with f(\u03bb1) > 0, so the Trace-determinant criterion [x = \u03bb1) intersects the prey isocline (y = F(x)) at point where the prey isocline decreases F\u2032(\u03bb1) < 0 and is unstable when F\u2032(\u03bb1) > 0. In fact, all the topological properties including results of global stability and uniqueness of limit cycles for all parameters of  and finally the chemostat estimate\u03ba = 0 corresponds to the known case b = 0 . We remb/m.Now,  takes thm.Now,  in Kuang [x in f, \u03c8, \u03c1, and H are C1),(C-II)f(0) = 0, f(\u03be) > 0 for \u03be > 0,(C-III)H\u2032(s) < 0, H(1) = 0,(C-IV)\u03be \u2212 \u03bb2)\u03c8(\u03be) > 0, \u03be \u2260 \u03bb2 > 0,((C-V)\u03c1(\u03be) > 0, \u03c1\u2032(\u03be) > 0, \u03be > 0,(C-VI)f(\u03be) + \u03c8(\u03be) < 0, \u03bb2 < \u03be \u2264 1.\u2212We rewrite the system on a form allowing isoclines to be analyzable. It is far from clear how this should be done in the chemostat case. We decided to work with the following form:\u03b1 > \u03bc. Before going further, we prove a basic theorem. We note that system  correspo\u03be \u2265 0, \u03b7 \u2265 0, \u03be + \u03b7 \u2264 1. Solutions of  = , and  = . Thus, solutions remain positive. To prove that solutions remain bounded, we assume 1 \u2264 \u03be + \u03b7 \u2264 \u03b3/\u03ba and consider the series of inequalitiesBy uniqueness of solutions no solutions can intersect the four solutions H decreasing and concave down as far as possible, too. We have one equilibrium at the origin, one at the carrying capacity , and one equilibrium at , where \u03b7\u2217 satisfies the condition \u03b7\u2217 = \u03c1(\u03bb2)H(\u03bb2 + \u03b7\u2217). We prove first, that the first two equilibria are saddles. The corresponding Jacobians are given byJ also contains the information that the eigenvector corresponding to the positive eigenvalue points into the triangle \u03be \u2265 0, \u03b7 \u2265 0, and \u03be + \u03b7 \u2264 1. This criterion can be formulated as\u03b1 > \u03bc and f(1) > \u03c8(1) > 0.We further conclude that\u03bb2, \u03b7\u2217) we start by doing some estimates concerning its location and define an implicit function \u03be \u2264 1. The sign of the derivative of the implicit function \u03bb2, \u03b7\u2217) and get For the interior equilibrium \u2013(C-VI). The interior fixed point of  is local\u03be > 0, \u03b7 > 0, \u03be + \u03b7 < 1 is invariant and its boundary is not approached by any of the solutions. We illustrate the graphical conclusions of The last assertion is due to the Poincar\u00e9-Bendixson theorem  because\u03b7(0) = 0 and \u03b7(0) = 1, resp., in the above equation and remember the chemostat estimate (\u03b2\u03be2 and \u2212\u03ba\u03b1\u03b72 are of different sign). The level curve consists therefore of either two intersecting lines or a hyperbola. One branch of this hyperbola is the predator isocline curve Finally, we return to the specific expressions that we used as our prototype example  in orderestimate ). We arecause of . We contcause of  can be wen as in , the isoThe prey isocline t system  with (22"}
{"text": "Escherichia coli RuvB hexameric ring motor proteins, together with RuvAs, promote branch migration of Holliday junction DNA. Zero mode waveguides (ZMWs) constitute of nanosized holes and enable the visualization of a single fluorescent molecule under micromolar order of the molecules, which is applicable to characterize the formation of RuvA\u2013RuvB\u2013Holliday junction DNA complex. In this study, we used ZMWs and counted the number of RuvBs binding to RuvA\u2013Holliday junction DNA complex. Our data demonstrated that different nucleotide analogs increased the amount of Cy5-RuvBs binding to RuvA\u2013Holliday junction DNA complex in the following order: no nucleotide, ADP, ATP\u03b3S, and mixture of ADP and ATP\u03b3S. These results suggest that not only ATP binding to RuvB but also ATP hydrolysis by RuvB facilitates a stable RuvA\u2013RuvB\u2013Holliday junction DNA complex formation.The  Escherichia coli, RuvA, RuvB, and RuvC are involved in the processing of Holliday junction DNA into mature recombinant DNA molecules12379Homologous recombination is a crucial biological process not only for the repair of damaged chromosomes, but also for generating genetic diversity. Holliday junction DNA is an important intermediate of the homologous recombination that consists of two homologous duplex DNA molecules linked by a single-stranded crossover. In Thermus themophilus and Thermatoga maritime RuvBs have a crescent\u2013like structure with three domains N, M, and C111311181920Crystallographic studies showed that \u221215\u2009L), it is very difficult to visualize single fluorescent molecule of interest under submicromolar order concentrations of the fluorescent molecules. To overcome this limitation, zero mode waveguides (ZMWs) have been developed and applied to single molecule real time DNA sequencing2425\u221219\u201310\u221220\u2009L. Thus, the ZMWs is said to enable the visualization of a single fluorescent molecule under micromolar order of the molecules272829The single molecule fluorescence imaging technique using total internal reflection fluorescence (TIRF) microscope is a conventional and powerful method for the characterization of biomolecule interactions in real time22In case of RuvB, approximately submicromolar concentrations of RuvB were required for RuvB binding to a Holliday junction DNA. Thus, using TIRF microscope, the observation of fluorescently labeled RuvB binding to a Holliday junction DNA was very difficult. In this study, to characterize RuvB binding to RuvA\u2013Holliday junction DNA complex, we fabricated ZMWs and labeled RuvB with Cy5. Then, we succeeded in visualizing Cy5-RuvBs binding to a RuvA\u2013Holliday junction DNA complex immobilized on the nanohole. We counted the number of Cy5 photobreaching steps under various nucleotide conditions and determined the most probable numbers of RuvBs binding to the complex. Our data shows that, in the presence of ATP\u03b3S and ADP, a more stable RuvA\u2013RuvB\u2013Holliday junction complex was formed, suggesting that ATP synergistically facilitates both RuvB hexameric ring formation and RuvA\u2013RuvB\u2013Holliday junction DNA complex formation, which is crucial for Holliday junction DNA branch migration.E. coli RuvB has no Cys residues.To characterize RuvB binding to Holliday junction DNA using the single molecule fluorescence imaging technique, we constructed and purified a RuvB mutant, RuvB-S39C, and then label the purified RuvB protein with Cy5-maleimide as described in Materials and Methods. We used RuvB-S39C to label RuvB protein by the highly specific conventional reaction between the sulfhydryl group and maleimide group because wild type To determine the effect of Ser-Cys mutation, we measured branch migration activity of the purified RuvB mutant in the presence of RuvA using a stopped-flow system . We usedThe labeling ratio of Cy5-labeled RuvB-S39C was 42%. The Holliday junction DNA branch migration activity of Cy5-labeled RuvB-S39C was comparable with that of unlabeled RuvB-S39C , indicatAs described in the methods, 400\u2009nM of Cy5-labeled RuvB-S39C was used to visualize RuvB binding to the RuvA\u2013Holliday junction DNA complex. Because ZMWs enable us to visualize single fluorescently labeled biomolecules at a high concentration of them, we fabricated ZMWs for single molecule Cy5-RuvB observation. Two methods have primarily been reported for ZMW fabrication31To confirm that RuvA\u2013RuvB complexes is capable of promoting branch migration of Holliday junction DNA in the nanoholes, Cy3-labeled Holliday junction DNA was immobilized on a streptavidin coated glass surface, as described in Materials and Methods. The ratio of fluorescent spots of Holliday junction DNA to nanoholes was about 90% before the addition of RuvB proteins to the nanoholes . After tAs described above, we demonstrated that the RuvA\u2013RuvB complex promoted branch migration of Cy3 labeled Holliday junction DNA immobilized on the nanohole, indicating that the RuvA\u2013RuvB\u2013Holliday junction DNA complex was formed in the nanohole. Next, we performed single molecule characterization of the RuvA\u2013RuvB\u2013Holliday junction DNA complex formation using Cy5-labeled RuvB. In the presence of ATP, the RuvA\u2013RuvB protein complex promotes Holliday junction DNA branch migration, resulting in the disassembly of Holliday junction DNA and the formation of Y-form DNA. Here, we used ATP\u03b3S and ADP as nucleotide cofactors. It was impossible for us to visualize Cy5-RuvB binding to the junction in the presence of ATP. We observed bright spots that emitted stable Cy3 and Cy5 fluorescence from nanoholes on which Cy3-Holliday junction DNA was immobilized. Most of the Cy3 and Cy5 fluorescence intensity decreased in a stepwise manner due to photobreaching . The num2+, RuvBs exist as a monomer and/or dimer233We characterized RuvB binding to the complex under conditions without nucleotides, with ADP, ATP\u03b3S or both ADP and ATP\u03b3S . BecauseIn the case that the RuvB protomer is a dimer, we considered that the distribution of Cy5-RuvB in dimer was random, and as shown in To date, RuvB properties on RuvA\u2013RuvB\u2013Holliday junction complex formation or DNA\u2013binding activity have been largely characterized by an electrophoresis mobility shift assay (EMSA) with glutaraldehyde cross-linking3536Single fluorescence imaging techniques enabled us to characterize the protein\u2013protein or protein\u2013DNA complex in more detail. We could visualize the assembly or disassembly processes of the complex and count the number of molecules constituting the complex in real time. In this study, to characterize the single molecule formation of RuvA\u2013RuvB\u2013Holliday junction DNA complex, we labeled RuvB with Cy5 and fabricated ZMWs. We measured the number of RuvBs binding to a RuvA\u2013Holliday junction DNA complex under various nucleotide conditions . Interes1337381314In the presence of ATP\u03b3S and ADP, 97%\u201398% of Holliday junction DNAs interacted with RuvBs and approximately 3% of the complexes contained six RuvBs  and 2. T40412+. Thus, in this study, we determined the distribution of the number of RuvBs binding to a RuvA\u2013Holliday junction DNA based on two models. One model is based on the model that the RuvB protomer is a monomer -\u0394ruvABC100::kan) was used for protein overexpressionThe expression plasmid for RuvB-S39C was constructed by PCR mediated site-directed mutagenesis as described previouslyRuvA and RuvB proteins were purified as previously described3N, N-dimethylformamide (DMF). RuvB-S39C and Cy5-maleimide was mixed at the ratio of 1:5 in 1.0\u2009mL of mixture containing 10 \u03bcM RuvB-S39C, 50\u2009\u03bcM Cy5-maleimide, 20\u2009mM HEPES-KOH (pH 7.0), and 8% Glycerol. The mixture was incubated for 16\u2009h at 4\u2009\u00b0C. After the coupling reaction, purification of Cy5-RuvB was performed using Resource Q (GE) in a purification buffer containing 30\u2009mM Tris-HCl (pH 7.5), 1 mM EDTA and 15% Glycerol. The protein was eluted with a 20\u2009mL linear gradient from 0 M to 1 M NaCl in a purification buffer.Using RuvB-S39C and Cy5-maleimide, Cy5-RuvB was prepared as below. Approximately 1.0 mg of Cy5-maleimide was dissolved in 10\u2009\u03bcL of 2, and Holliday junction DNA was constructed as described previously and used for the branch migration assayTwo Holliday junction DNAs were prepared as below. Four oligonucleotides  were mixed in a buffer containing 50\u2009mM Tris-HCl (pH 7.5), 100\u2009mM NaCl, and 10\u2009mM MgCl2 and 1 mM DTT. Solution II contained 400\u2009nM RuvB, 2\u2009mM ATP, 20\u2009mM Tris-HCl (pH 8.0), 10\u2009mM MgCl2 and 1 mM DTT. Branch migration starts by mixing Solution I and Solution II at 25\u2009\u00b0C. All data curves represent the average of at least four experiments.Branch migration activity of the RuvA\u2013RuvB complex was carried out using a stopped-flow spectrafluorometer  equipped with a photomultiplier tube. The filter used with the photomultiplier was Semrock FF01-567/15-25. Excitation was at 520\u2009nm for Cy3. Solution I and Solution II were prepared as below. Solution I contained 100\u2009nM RuvA, 10\u2009nM Holliday junction DNA, 20\u2009mM Tris-HCl (pH 8.0), 10\u2009mM MgClFused silica coverslips were immersed in 4% ammonium and 4.3% hydrogen peroxide for 10\u2009min at 75\u2009\u00b0C and then washed thoroughly with deionized water. The coverslips were dried with an air blower, and then baked at 200\u2009\u00b0C for 10\u2009min. The coverslips were then cleaned by air plasma at 18\u2009W for 10\u2009min. Before coating with Hexamethyldisilazane , the coverslips were immersed in 2-butanone and cleaned by sonication for 5\u2009min. A resist film of ma-N 2403 (Micro resist technology) was then coated on the HMDS coated coverslip with a spin coater. The ESPACER (Showa Denko) was then coated on the ma-N 2403 coated coverslip. Electron beam (EB) lithography (Elionix Inc.) was performed with an accelerating voltage of 80\u2009kV, and a beam current of 100\u2009pA. After EB patterning, the coverslips were immersed in deionized water for 30\u2009s, and the pattern was then developed by immersing it in ma-D 525 (Micro resist technology) for 2\u2009min. After development, the coverslips were washed thoroughly with deionized water for 5\u2009min and dried with an air blower. Aluminum coating was performed in a thermal evaporator using a BN composite boat. The thickness of the coated aluminum was monitored using a thickness monitor (Eiko Engineering Co. Ltd.). After aluminum coating, the remaining photoresist was lifted off by immersing it in 2-butanone with sonication for 5\u2009min.The ZMWs were observed by scanning electron microscopy  and the diameter of each nanohole was measured.Samples were observed at 25\u2009\u00b1\u20092\u2009\u00b0C on an Olympus IX71 inverted microscope with a 100X oil-immersion objective as described previouslyN-hydroxy-succinimidyl (NHS) group  and 0.1\u2009mg NHS-Bio-PEG  dissolved in 50\u2009mM MOPS buffer (pH 7.5). After the coating, the ZMWs were washed with MilliQ water and dried on a clean bench with an air blower.ZMWs were washed with acetone under sonication for 5\u2009min and then washed with 2-propanol under sonication for 5\u2009min. The ZMWs were dried with an air blower and cleaned with air plasma at 18\u2009W for 5\u2009min. The ZMWs were immersed in a preheated 0.6% (vol/vol) aqueous solution of poly(vinylphosphonic acid) (Funakoshi) for 10 min at 90\u2009\u00b0C. They were washed briefly with deionized water, dried with an air blower, and annealed on a hot plate at 80\u2009\u00b0C for 10\u2009min. The Polyethylene glycol (PEG) coating was then performed as described previously2). PEG-coated ZMWs were incubated for 5\u2009min at room temperature with a drop containing 0.06\u2009mg/ml streptavidin in Buffer A. The ZMWs were then rinsed with Buffer A. A 20\u2009\u03bcl mixture containing 400\u2009nM RuvA and 40\u2009nM Holliday junction DNA in Buffer A was added to the ZMWs and incubated for 5\u2009min. The ZMWs were rinced with Buffer A containing 2.5\u2009mM Protocatechuic acid , 250\u2009nM Protocatechuate-3,4-dioxygenase , and 2\u2009mM Trolox (Sigma) to wash out the unbound RuvA and Holliday junction DNA. The ZMW was set on the microscope. A mixture containing 400\u2009nM Cy5-RuvBs and the indicated amount of nucleotides in Buffer A with 2.5\u2009mM PCA, 250\u2009nM PCD, and 2\u2009mM Trolox was added to the ZMW and the fluorescence imaging from Cy3 labeled Holliday junction DNA and Cy5-RuvB were recorded.To observe RuvB binding to RuvA\u2013Hollliday junction DNAs, the DNA was immobilized in the nanoholes. PEG-coated ZMWs were incubated for 5\u2009min at room temperature with a drop containing 1% F127 in Water. The ZMWs were then rinsed with Buffer A (20 mM HEPES-KOH (pH 8.0), 10\u2009mM MgClno-nucleotide, FADP, FATP\u03b3S, and FADP and ATP\u03b3S, respectively.The calculated data were obtained by a binominal distribution between 42% of Cy5-RuvB and 58% of nonlabeled RuvB as shown in no-nucleotide\u2009=\u2009(a/100\u2009+\u20090.58\u2009\u00d7\u2009b/100\u2009+\u20090.3364\u2009\u00d7\u2009c/100\u2009+\u20090.1551\u2009\u00d7\u2009d/100\u2009+\u20090.1132\u2009\u00d7\u2009e/100\u2009+\u20090.0656\u2009\u00d7\u2009f/100\u2009+\u20090.0381\u2009\u00d7\u2009g/100\u2009\u2212\u20090.58)2\u2009+\u2009(0.42\u2009\u00d7\u2009b/100\u2009+\u20090.4872\u2009\u00d7\u2009c/100\u2009+\u20090.4239\u2009\u00d7\u2009d/100\u2009+\u20090.3278\u2009\u00d7\u2009e/100\u2009+\u20090.2377\u2009\u00d7\u2009f/100\u2009+\u20090.1654\u2009\u00d7\u2009g/100\u2009\u2212\u20090.35)2\u2009+\u2009(0.1764\u2009\u00d7\u2009c/100\u2009+\u20090.3069\u2009\u00d7\u2009d/100\u2009+\u20090.3560\u2009\u00d7\u2009e/100\u2009+\u20090.3442\u2009\u00d7\u2009f/100\u2009+\u20090.2994\u2009\u00d7\u2009g/100\u2009\u2212\u20090.07)2\u2009+\u2009(0.0741\u2009\u00d7\u2009d/100\u2009+\u20090.1719\u2009\u00d7\u2009e/100\u2009+\u20090.2492\u2009\u00d7\u2009f/100\u2009+\u20090.2891\u2009\u00d7\u2009g/100)2\u2009+\u2009(0.0311\u2009\u00d7\u2009e/100\u2009+\u20090.0902\u2009\u00d7\u2009f/100\u2009+\u20090.1570\u2009\u00d7\u2009g/100)2\u2009+\u2009(0.0131\u2009\u00d7\u2009f/100\u2009+\u20090.0455\u2009\u00d7\u2009g/100)2\u2009+\u2009(0.0055\u2009\u00d7\u2009g/100)2.FADP\u2009=\u2009(a/100\u2009+\u20090.58\u2009\u00d7\u2009b/100\u2009+\u20090.3364\u2009\u00d7\u2009c/100\u2009+\u20090.1551\u2009\u00d7\u2009d/100\u2009+\u20090.1132\u2009\u00d7\u2009e/100\u2009+\u20090.0656\u2009\u00d7\u2009f/100\u2009+\u20090.0381\u2009\u00d7\u2009g/100\u2009\u2212\u20090.32)2\u2009+\u2009(0.42\u2009\u00d7\u2009b/100\u2009+\u20090.4872\u2009\u00d7\u2009c/100\u2009+\u20090.4239\u2009\u00d7\u2009d/100\u2009+\u20090.3278\u2009\u00d7\u2009e/100\u2009+\u20090.2377\u2009\u00d7\u2009f/100\u2009+\u20090.1654\u2009\u00d7\u2009g/100\u2009\u2212\u20090.37)2\u2009+\u2009(0.1764\u2009\u00d7\u2009c/100\u2009+\u20090.3069\u2009\u00d7\u2009d/100\u2009+\u20090.3560\u2009\u00d7\u2009e/100\u2009+\u20090.3442\u2009\u00d7\u2009f/100\u2009+\u20090.2994\u2009\u00d7\u2009g/100\u2009\u2212\u20090.23)2\u2009+\u2009(0.0741\u2009\u00d7\u2009d/100\u2009+\u20090.1719\u2009\u00d7\u2009e/100\u2009+\u20090.2492\u2009\u00d7\u2009f/100\u2009+\u20090.2891\u2009\u00d7\u2009g/100 \u2009\u2212\u20090.08)2\u2009+\u2009(0.0311\u2009\u00d7\u2009e/100\u2009+\u20090.0902\u2009\u00d7\u2009f/100\u2009+\u20090.1570\u2009\u00d7\u2009g/100)2\u2009+\u2009(0.0131\u2009\u00d7\u2009f/100\u2009+\u20090.0455\u2009\u00d7\u2009g/100)2\u2009+\u2009(0.0055\u2009\u00d7\u2009g/100)2.FATP\u03b3S\u2009=\u2009(a/100\u2009+\u20090.58\u2009\u00d7\u2009b/100\u2009+\u20090.3364\u2009\u00d7\u2009c/100\u2009+\u20090.1551\u2009\u00d7\u2009d/100\u2009+\u20090.1132\u2009\u00d7\u2009e/100\u2009+\u20090.0656\u2009\u00d7\u2009f/100\u2009+\u20090.0381\u2009\u00d7\u2009g/100\u2009\u2212\u20090.29)2\u2009+\u2009(0.42\u2009\u00d7\u2009b/100\u2009+\u20090.4872\u2009\u00d7\u2009c/100\u2009+\u20090.4239\u2009\u00d7\u2009d/100\u2009+\u20090.3278\u2009\u00d7\u2009e/100\u2009+\u20090.2377\u2009\u00d7\u2009f/100\u2009+\u20090.1654\u2009\u00d7\u2009g/100\u2009\u2212\u20090.35)2\u2009+\u2009(0.1764\u2009\u00d7\u2009c/100\u2009+\u20090.3069\u2009\u00d7\u2009d/100\u2009+\u20090.3560\u2009\u00d7\u2009e/100\u2009+\u20090.3442\u2009\u00d7\u2009f/100\u2009+\u20090.2994\u2009\u00d7\u2009g/100\u2009\u2212\u20090.24)2\u2009+\u2009(0.0741\u2009\u00d7\u2009d/100\u2009+\u20090.1719\u2009\u00d7\u2009e/100\u2009+\u20090.2492\u2009\u00d7\u2009f/100\u2009+\u20090.2891\u2009\u00d7\u2009g/100\u2009\u2212\u20090.10)2\u2009+\u2009(0.0311\u2009\u00d7\u2009e/100\u2009+\u20090.0902\u2009\u00d7\u2009f/100\u2009+\u20090.1570\u2009\u00d7\u2009g/100\u2009\u2212\u20090.02)2\u2009+\u2009(0.0131\u2009\u00d7\u2009f/100\u2009+\u20090.0455\u2009\u00d7\u2009g/100)2\u2009+\u2009(0.0055\u2009\u00d7\u2009g/100)2.FADP and ATP\u03b3S\u2009=\u2009(a/100\u2009+\u20090.58\u2009\u00d7\u2009b/100\u2009+\u20090.3364\u2009\u00d7\u2009c/100\u2009+\u20090.1551\u2009\u00d7\u2009d/100\u2009+\u20090.1132\u2009\u00d7\u2009e/100\u2009+\u20090.0656\u2009\u00d7\u2009f/100\u2009+\u20090.0381\u2009\u00d7\u2009g/100\u2009\u2212\u20090.19)2\u2009+\u2009(0.42\u2009\u00d7\u2009b/100\u2009+\u20090.4872\u2009\u00d7\u2009c/100\u2009+\u20090.4239\u2009\u00d7\u2009d/100\u2009+\u20090.3278\u2009\u00d7\u2009e/100\u2009+\u20090.2377\u2009\u00d7\u2009f/100\u2009+\u20090.1654\u2009\u00d7\u2009g/100\u2009\u2212\u20090.35)2\u2009+\u2009(0.1764\u2009\u00d7\u2009c/100\u2009+\u20090.3069\u2009\u00d7\u2009d/100\u2009+\u20090.3560\u2009\u00d7\u2009e/100\u2009+\u20090.3442\u2009\u00d7\u2009f/100\u2009+\u20090.2994\u2009\u00d7\u2009g/100\u2009\u2212\u20090.30)2\u2009+\u2009(0.0741\u2009\u00d7\u2009d/100\u2009+\u20090.1719\u2009\u00d7\u2009e/100\u2009+\u20090.2492\u2009\u00d7\u2009f/100\u2009+\u20090.2891\u2009\u00d7\u2009g/100\u2009\u2212\u20090.13)2\u2009+\u2009(0.0311\u2009\u00d7\u2009e/100\u2009+\u20090.0902\u2009\u00d7\u2009f/100\u2009+\u20090.1570\u2009\u00d7\u2009g/100\u2009\u2212\u20090.02)2\u2009+\u2009(0.0131\u2009\u00d7\u2009f/100\u2009+\u20090.0455\u2009\u00d7\u2009g/100\u2009\u2212\u20090.01)2\u2009+\u2009(0.0055\u2009\u00d7\u2009g/100)2.Fa, b, c, d, e, f, and g indicate percentages of 0, 1, 2, 3, 4, 5, and 6 RuvBs binding to a RuvA-Holliday junction DNA. The a, b, c, d, e, f, and g satisfied the following conditions.a, b, c, d, e, f, and g were nonnegative integers.The inequalities as below were not allowed.no-nucleotide, GADP, GATP\u03b3S, and GADP and ATP\u03b3S, respectively.In case that the RuvB protomer is a dimer containing Cy5-RuvB at random, the calculated data were also obtained by a binominal distribution between 42% of Cy5-RuvB and 58% of nonlabeled RuvB as shown in no-nucleotide\u2009=\u2009(h/100\u2009+\u20090.3364\u2009\u00d7\u2009i/100\u2009+\u20090.1132\u2009\u00d7\u2009j/100\u2009+\u20090.0381\u2009\u00d7\u2009k/100\u2009\u2212\u20090.58)2\u2009+\u2009(0.4872\u2009\u00d7\u2009i/100\u2009+\u20090.3278\u2009\u00d7\u2009j/100\u2009+\u20090.1654\u2009\u00d7\u2009k/100\u2009\u2212\u20090.35)2\u2009+\u2009(0.1764\u2009\u00d7\u2009i/100\u2009+\u20090.3560\u2009\u00d7\u2009j/100\u2009+\u20090.2994\u2009\u00d7\u2009k/100\u2009\u2212\u20090.07)2\u2009+\u2009(0.1719\u2009\u00d7\u2009j/100\u2009+\u20090.2891\u2009\u00d7\u2009k/100)2\u2009+\u2009(0.0311\u2009\u00d7\u2009j/100\u2009+\u20090.1570\u2009\u00d7\u2009k/100)2\u2009+\u2009(0.0455\u2009\u00d7\u2009k/100)2\u2009+\u2009(0.0055\u2009\u00d7\u2009k/100)2.GADP\u2009=\u2009(h/100\u2009+\u20090.3364\u2009\u00d7\u2009i/100\u2009+\u20090.1132\u2009\u00d7\u2009j/100\u2009+\u20090.0381\u2009\u00d7\u2009k/100 \u2009\u2212\u20090.32)2\u2009+\u2009(0.4872\u2009\u00d7\u2009i/100\u2009+\u20090.3278\u2009\u00d7\u2009j/100\u2009+\u20090.1654\u2009\u00d7\u2009k/100\u2009\u2212\u20090.37)2\u2009+\u2009(0.1764\u2009\u00d7\u2009i/100\u2009+\u20090.3560\u2009\u00d7\u2009j/100\u2009+\u20090.2994\u2009\u00d7\u2009k/100 \u2009\u2212\u20090.23)2\u2009+\u2009(0.1719\u2009\u00d7\u2009j/100\u2009+\u20090.2891\u2009\u00d7\u2009k/100 \u2009\u2212\u20090.08)2\u2009+\u2009(0.0311\u2009\u00d7\u2009j/100\u2009+\u20090.1570\u2009\u00d7\u2009k/100)2\u2009+\u2009(0.0455\u2009\u00d7\u2009k/100)2\u2009+\u2009(0.0055\u2009\u00d7\u2009k/100)2.GATP\u03b3S\u2009=\u2009(h/100\u2009+\u20090.3364\u2009\u00d7\u2009i/100\u2009+\u20090.1132\u2009\u00d7\u2009j/100\u2009+\u20090.0381\u2009\u00d7\u2009k/100\u2009\u2212\u20090.29)2\u2009+\u2009(0.4872\u2009\u00d7\u2009i/100\u2009+\u20090.3278\u2009\u00d7\u2009j/100\u2009+\u20090.1654\u2009\u00d7\u2009k/100\u2009\u2212\u20090.35)2\u2009+\u2009(0.1764\u2009\u00d7\u2009i/100\u2009+\u20090.3560\u2009\u00d7\u2009j/100\u2009+\u20090.2994\u2009\u00d7\u2009k/100\u2009\u2212\u20090.24)2\u2009+\u2009(0.1719\u2009\u00d7\u2009j/100\u2009+\u20090.2891\u2009\u00d7\u2009k/100\u2009\u2212\u20090.10)2\u2009+\u2009(0.0311\u2009\u00d7\u2009j/100\u2009+\u20090.1570\u2009\u00d7\u2009k/100 \u2013 0.02)2\u2009+\u2009(0.0455\u2009\u00d7\u2009k/100)2\u2009+\u2009(0.0055\u2009\u00d7\u2009k/100)2.GADP and ATP\u03b3S\u2009=\u2009(h/100\u2009+\u20090.3364\u2009\u00d7\u2009i/100\u2009+\u20090.1132\u2009\u00d7\u2009j/100\u2009+\u20090.0381\u2009\u00d7\u2009k/100\u2009\u2212\u20090.19)2\u2009+\u2009(0.4872\u2009\u00d7\u2009i/100\u2009+\u20090.3278\u2009\u00d7\u2009j/100\u2009+\u20090.1654\u2009\u00d7\u2009k/100\u2009\u2212\u20090.35)2\u2009+\u2009(0.1764\u2009\u00d7\u2009i/100\u2009+\u20090.3560\u2009\u00d7\u2009j/100\u2009+\u20090.2994\u2009\u00d7\u2009k/100\u2009\u2212\u20090.30)2\u2009+\u2009(0.1719\u2009\u00d7\u2009j/100\u2009+\u20090.2891\u2009\u00d7\u2009k/100\u2009\u2212\u20090.13)2\u2009+\u2009(0.0311\u2009\u00d7\u2009j/100\u2009+\u20090.1570\u2009\u00d7\u2009k/100 \u2013 0.02)2\u2009+\u2009(0.0455\u2009\u00d7\u2009k/100 \u2013 0.01)2\u2009+\u2009(0.0055\u2009\u00d7\u2009k/100)2.Gh, i, j, and k indicate percentages of 0, 2, 4, and 6 RuvB dimers binding to a RuvA-Holliday junction DNA. The h, i, j, and k satisfied the following conditions.h, i, j, and k were nonnegative integers.The inequalities as below were not allowed.no-nucleotide, HADP, HATP\u03b3S, and HADP and ATP\u03b3S, respectively.In case that all of Cy5 labeled RuvB dimers contain a Cy5-RuvB and a non-labeled RuvB, the calculated data were obtained by a binominal distribution between 84% of RuvB dimer containing single Cy5-RuvB and 16% of nonlabeled RuvB dimer as shown in no-nucleotide\u2009=\u2009(l/100\u2009+\u20090.16\u2009\u00d7\u2009m/100\u2009+\u20090.0256\u2009\u00d7\u2009n/100\u2009+\u20090.0041\u2009\u00d7\u2009p/100\u2009+\u20090.0007\u2009\u00d7\u2009q/100\u2009+\u20090.0001\u2009\u00d7\u2009r/100\u2009\u2212\u20090.58)2\u2009+\u2009(0.84\u2009\u00d7\u2009m/100\u2009+\u20090.2688\u2009\u00d7\u2009n/100\u2009+\u20090.0645\u2009\u00d7\u2009p/100\u2009+\u20090.0138\u2009\u00d7\u2009q/100\u2009+\u20090.0028\u2009\u00d7\u2009r/100\u2009+\u20090.0005\u2009\u00d7\u2009s/100\u2009\u2212\u20090.35)2\u2009+\u2009(0.7056\u2009\u00d7\u2009n/100\u2009+\u20090.3387\u2009\u00d7\u2009p/100\u2009+\u20090.1084\u2009\u00d7\u2009q/100\u2009+\u20090.0289\u2009\u00d7\u2009r/100\u2009+\u20090.0069\u2009\u00d7\u2009s/100\u2009\u2212\u20090.07)2\u2009+\u2009(0.5927\u2009\u00d7\u2009p/100\u2009+\u20090.3793\u2009\u00d7\u2009q/100\u2009+\u20090.1517\u2009\u00d7\u2009r/100\u2009+\u20090.0486\u2009\u00d7\u2009s/100)2\u2009+\u2009(0.4978\u2009\u00d7\u2009q/100\u2009+\u20090.3983\u2009\u00d7\u2009r/100\u2009+\u20090.1912\u2009\u00d7\u2009s/100)2\u2009+\u2009(0.4182\u2009\u00d7\u2009r/100\u2009+\u20090.4015\u2009\u00d7\u2009s/100)2\u2009+\u2009(0.3513\u2009\u00d7\u2009s/100)2.HADP\u2009=\u2009(l/100\u2009+\u20090.16\u2009\u00d7\u2009m/100\u2009+\u20090.0256\u2009\u00d7\u2009n/100\u2009+\u20090.0041\u2009\u00d7\u2009p/100\u2009+\u20090.0007\u2009\u00d7\u2009q/100\u2009+\u20090.0001\u2009\u00d7\u2009r/100\u2009\u2212\u20090.32)2\u2009+\u2009(0.84\u2009\u00d7\u2009m/100\u2009+\u20090.2688\u2009\u00d7\u2009n/100\u2009+\u20090.0645\u2009\u00d7\u2009p/100\u2009+\u20090.0138\u2009\u00d7\u2009q/100\u2009+\u20090.0028\u2009\u00d7\u2009r/100\u2009+\u20090.0005\u2009\u00d7\u2009s/100\u2009\u2212\u20090.37)2\u2009+\u2009(0.7056\u2009\u00d7\u2009n/100\u2009+\u20090.3387\u2009\u00d7\u2009p/100\u2009+\u20090.1084\u2009\u00d7\u2009q/100\u2009+\u20090.0289\u2009\u00d7\u2009r/100\u2009+\u20090.0069\u2009\u00d7\u2009s/100\u2009\u2212\u20090.23)2\u2009+\u2009(0.5927\u2009\u00d7\u2009p/100\u2009+\u20090.3793\u2009\u00d7\u2009q/100\u2009+\u20090.1517\u2009\u00d7\u2009r/100\u2009+\u20090.0486\u2009\u00d7\u2009s/100 \u2013 0.08)2\u2009+\u2009(0.4978\u2009\u00d7\u2009q/100\u2009+\u20090.3983\u2009\u00d7\u2009r/100\u2009+\u20090.1912\u2009\u00d7\u2009s/100)2\u2009+\u2009(0.4182\u2009\u00d7\u2009r/100\u2009+\u20090.4015\u2009\u00d7\u2009s/100)2\u2009+\u2009(0.3513\u2009\u00d7\u2009s/100)2.HATP\u03b3S\u2009=\u2009(l/100\u2009+\u20090.16\u2009\u00d7\u2009m/100\u2009+\u20090.0256\u2009\u00d7\u2009n/100\u2009+\u20090.0041\u2009\u00d7\u2009p/100\u2009+\u20090.0007\u2009\u00d7\u2009q/100\u2009+\u20090.0001\u2009\u00d7\u2009r/100\u2009\u2212\u20090.29)2\u2009+\u2009(0.84\u2009\u00d7\u2009m/100\u2009+\u20090.2688\u2009\u00d7\u2009n/100\u2009+\u20090.0645\u2009\u00d7\u2009p/100\u2009+\u20090.0138\u2009\u00d7\u2009q/100\u2009+\u20090.0028\u2009\u00d7\u2009r/100\u2009+\u20090.0005\u2009\u00d7\u2009s/100\u2009\u2212\u20090.35)2\u2009+\u2009(0.7056\u2009\u00d7\u2009n/100\u2009+\u20090.3387\u2009\u00d7\u2009p/100\u2009+\u20090.1084\u2009\u00d7\u2009q/100\u2009+\u20090.0289\u2009\u00d7\u2009r/100\u2009+\u20090.0069\u2009\u00d7\u2009s/100\u2009\u2212\u20090.24)2\u2009+\u2009(0.5927\u2009\u00d7\u2009p/100\u2009+\u20090.3793\u2009\u00d7\u2009q/100\u2009+\u20090.1517\u2009\u00d7\u2009r/100\u2009+\u20090.0486\u2009\u00d7\u2009s/100 \u2013 0.10)2\u2009+\u2009(0.4978\u2009\u00d7\u2009q/100\u2009+\u20090.3983\u2009\u00d7\u2009r/100\u2009+\u20090.1912\u2009\u00d7\u2009s/100 \u2013 0.02)2\u2009+\u2009(0.4182\u2009\u00d7\u2009r/100\u2009+\u20090.4015\u2009\u00d7\u2009s/100)2\u2009+\u2009(0.3513\u2009\u00d7\u2009s/100)2.HADP and ATP\u03b3S\u2009=\u2009(l/100\u2009+\u20090.16\u2009\u00d7\u2009m/100\u2009+\u20090.0256\u2009\u00d7\u2009n/100\u2009+\u20090.0041\u2009\u00d7\u2009p/100\u2009+\u20090.0007\u2009\u00d7\u2009q/100\u2009+\u20090.0001\u2009\u00d7\u2009r/100\u2009\u2212\u20090.19)2\u2009+\u2009(0.84\u2009\u00d7\u2009m/100\u2009+\u20090.2688\u2009\u00d7\u2009n/100\u2009+\u20090.0645\u2009\u00d7\u2009p/100\u2009+\u20090.0138\u2009\u00d7\u2009q/100\u2009+\u20090.0028\u2009\u00d7\u2009r/100\u2009+\u20090.0005\u2009\u00d7\u2009s/100\u2009\u2212\u20090.35)2\u2009+\u2009(0.7056\u2009\u00d7\u2009n/100\u2009+\u20090.3387\u2009\u00d7\u2009p/100\u2009+\u20090.1084\u2009\u00d7\u2009q/100\u2009+\u20090.0289\u2009\u00d7\u2009r/100\u2009+\u20090.0069\u2009\u00d7\u2009s/100\u2009\u2212\u20090.30)2\u2009+\u2009(0.5927\u2009\u00d7\u2009p/100\u2009+\u20090.3793\u2009\u00d7\u2009q/100\u2009+\u20090.1517\u2009\u00d7\u2009r/100\u2009+\u20090.0486\u2009\u00d7\u2009s/100 \u2013 0.13)2\u2009+\u2009(0.4978\u2009\u00d7\u2009q/100\u2009+\u20090.3983\u2009\u00d7\u2009r/100\u2009+\u20090.1912\u2009\u00d7\u2009s/100 \u2013 0.02)2\u2009+\u2009(0.4182\u2009\u00d7\u2009r/100\u2009+\u20090.4015\u2009\u00d7\u2009s/100 \u2013 0.01)2\u2009+\u2009(0.3513\u2009\u00d7\u2009s/100)2.Hl, m, n, p, q, r, and s indicate percentages of 0, 1, 2, 3, 4, 5, and 6 RuvB dimers binding to a RuvA-Holliday junction DNA. The l, m, n, p, q, r, and s satisfied the following conditions.l, m, n, p, q, r, and s were nonnegative integers.The inequalities as below were not allowed.How to cite this article: Iwasa, T. et al. Synergistic effect of ATP for RuvA\u2014RuvB\u2014Holliday junction DNA complex formation. Sci. Rep.5, 18177; doi: 10.1038/srep18177 (2015)."}
{"text": "N-dimensional Euler equations with spherical symmetry. We first show that there are only trivial solutions when the velocity is of the form c(t)|x|\u03b1\u22121x + b(t)(x/|x|) for any value of \u03b1 \u2260 1 or any positive integer N \u2260 1. Then, we show that blowup phenomenon occurs when \u03b1 = N = 1 and \u03b3 > 1) and the isothermal case (\u03b3 = 1). The blowup phenomenon of solutions is investigated for the initial-boundary value problem (IBVP) of the  N-dimensional Euler equations for compressible fluid:\u03c1, u, and p represent the density, velocity, and pressure of the fluid, respectively. n is the unit normal vector on the unit sphere. The \u03b3-law for p is given by  and b(t) is a time-dependent drifting function.In , the autf system  in spher\u03c1 = \u03c1 and u = u(x/r), system r\u03b1 + b(t) for any real \u03b1 and integer N in the isentropic and isothermal cases except the case \u03b3 = \u03b1 = N = 1. For \u03b3 = \u03b1 = N = 1, one has the following two cases.(1)t,c(t) \u2192 0, \u2192 as t \u2192 \u221e.If (2)t,c(t)\u2192\u2212\u221e and \u2192 as t \u2192 \u221e.If There are only trivial solutions to the r system  of the fAs a corollary, we also obtain the following.\u03c1, u) be a solution for (a0\u2254a(0) > 0, and \u03b3 = N = 1. Then a(t) satisfies\u03bb \u2208 \u211d. Furthermore, one has the following five cases.(1)\u03bb < 0, then the solution \u2192 as t \u2192 T\u2217 for some finite T\u2217 > 0.If (2)\u03bb > 0, then \u03c1 is bounded above and the solution \u2192 as t \u2192 \u221e.If (3)\u03bb = 0 and a1 > 0, \u03c1 is bounded above and the solution \u2192 as t \u2192 \u221e.If (4)\u03bb = 0 and a1 = 0, then the solution is trivial.If (5)\u03bb = 0 and a1 < 0, then the solution \u2192 as t \u2192 \u2212a0/a1.If Let  is set to be positive. Thus, we suppose \u03c1 > 0 in the following to avoid the trivial solutions \u03c1 \u2261 0.It is well-known that \u03b3 > 1, one hasFor 1, one has\u03c1\u03b3\u22122. Then, the result follows.From 1, one ha\u03b3 > 1, one hasFor 2, one hasFrom 2, one ha\u03b3 = 1.Similarly, we have the following two lemmas for \u03b3 = 1, one hasFor \u03b3 = 1, one hasu = c(t)r + b(t) and \u03b3 > 1.For A1 \u2260 A2 or B1 \u2260 B2. If A \u2260 0, B \u2260 0, andConsider the following dynamical systemc3+A1cc\u02d9+c3+A1cc\u02d9+A1B2 \u2212 A2B1 = 0, then it is clear that c = 0 is the only solution. So we suppose A1B2 \u2212 A2B1 \u2260 0. One has from (If has from  that(21)B = 0 and A \u2260 0, then c = 0 is the only solution.If A = 0 and B \u2260 0, then c = 0 is the only solution.If A and B are not zero.So, we suppose both \u03be.From , one has1, one hasA(A1B2 \u2212 A2B1)/2B2 \u2260 0, then c = 0 is the only solution. So we suppose it is zero; that is, 22), one us, from 1, one hathat is,  holds. T one has  and 19)(23)c\u00a8=\u2212A one has  is satis\u03b3 > 1. Then there are only trivial solutions to the N-dimensional Euler system (u = c(t)r\u03b1 + b(t) with \u03b1 \u2260 1.Assume r system  of the f4, we haveN > 1, if \u03b1 = 0, then u = 0 from  is independent of t and r, respectively. Thus, \u03c1 is a constant.First, we set+bt.From 4, we hav, we haveu=ctr\u03b1\u22121. we have2u=ctr\u03b1\u22121.\u03b1\u22121\u2212c\u02d9,2ut+uurt=c\u03b1 \u2260 0 and \u22121, after substituting , a, a\u03b1 \u2260 0 tituting  into (9)tituting , and 7)\u03b1 \u2260 0 and\u03b1 \u2209 {0,1/3,1/2,2/3,1, 2}, then the powers 3\u03b1 \u2212 1 and 2\u03b1 \u2212 1 are different and unique among the powers in (\u03b1 + (\u03b3 \u2212 1)(\u03b1 + N \u2212 1) and (\u03b3 \u2212 1)(2\u03b1 + 3N \u2212 3) + 4\u03b1 cannot be both zero for N \u2260 1, we conclude that c = 0. Hence, u = 0 and \u03c1 is a constant.If owers in . In this\u03b1 \u2208 {1/3,1/2,2/3,2}, we have For \u03b1 = 1/3 or 2/3, as 2\u03b1 \u2212 1 is unique among other powers, one has\u03b3 \u2212 1)(2\u03b1 + 3N \u2212 3) + 4\u03b1 \u2260 0, we conclude that c = 0. Thus, u = 0 and \u03c1 is a constant.For \u03b1 = 1/2, as \u03b1 \u2212 1 and \u22121 are different and unique among other powers, one hasN \u2260 1. As \u03b1 + (\u03b3 \u2212 1)(N \u2212 1) \u2260 0, we conclude that c = 0. Thus, u = 0 and \u03c1 is a constant.For \u03b1 = 2, as 3\u03b1 \u2212 1 is unique among other powers, one has\u03b1 + (\u03b3 \u2212 1)(\u03b1 + N \u2212 1) \u2260 0, we conclude that c = 0. Thus, u = 0 and \u03c1 is a constant.For \u03b1 = \u22121.Next, we consider the case \u03b1 = \u22121, the corresponding equation of (3N \u2212 5) and 2 \u2212 (N \u2212 2)(\u03b3 \u2212 1) cannot be both zero for N \u2260 1, we conclude that c = 0 and the solutions are trivial.For ation of  is(39)E1N = 1 and \u03b1 \u2260 1, one can show that there are only trivial solutions with similar procedures. The proof is complete.For \u03b3 > 1 in \u03b3 = 1.Using similar analysis as that given for the case \u03b3 = 1. Then there are only trivial solutions to the N-dimensional Euler system (u = c(t)r\u03b1 + b(t) with \u03b1 \u2260 1.Assume r system  of the f\u03b1 = 1 will be analyzed as follows.Next, the crucial case \u03b3 > 1. Then there are only trivial solutions to the N-dimensional Euler system (u = c(t)r\u03b1 + b(t) with \u03b1 = 1.Assume r system  of the f\u03b3 > 1 and \u03b1 = 1, one has1 and r\u03b1 + b(t) with \u03b1 = 1 and N > 1. For N = \u03b1 = 1, one has the following two cases.(1)t, c(t) \u2192 0, \u2192 as t \u2192 \u221e.If (2)t, c(t)\u2192\u2212\u221e and \u2192 as t \u2192 \u221e.If Assume r system  of the f\u03b3 = \u03b1 = 1, the corresponding system of  in [N in (For ystem of  is46)G1G1\u03b3 = \u03b1 =here47) It is cl1. Then (2c3+4cc\u02d9+n (7) in  when we  It is c2 and  must be zero. More precisely, if T\u2217 is finite and limt\u2192T\u2217a(t) > 0, then we can extend the solution by solving (a(T\u2217)\u2254limt\u2192T\u2217a(t) > 0. This contradicts the definition of T\u2217.Let =a\u02d9/a in 1. One haatisfies . Consideatisfies  with ini solving  with ini\u03bb < 0, and then\u03bc\u2254(1/2)a12 \u2212 \u03bbln\u2061a0. It follows thatt2 is negative, we see that a(t) will be negative if t is sufficiently large. This implies that Next, suppose  and then(56)wheru \u2192 \u2212\u221e and \u03c1 \u2192 \u221e as t \u2192 T\u2217.On the other hand, from , one has\u03bb > 0, one has(i)a \u2265 e\u03bc/\u03bb\u2212 > 0,(ii)T\u2217 = \u221e,(iii)From (i), (ii), and (iii) above, we see that \u03c1 is bounded above byt\u2192\u221ea(t) = \u221e. This is because if limt\u2192\u221ea(t) is finite, then a(t) is bounded by some positive number M > 0. But, from 1\u03c1t,r=a0\u03c1\u03bb > 0, \u2192 as t \u2192 \u221e.Next, we show thatlimt\u2192\u221e\u2061a\u02d9\u03bb = 0 can be verified trivially, the proof is complete.As the cases for"}
{"text": "Besides giving characterizations of these spaces, we study some of their properties. Also, we investigate the behavior of intuitionistic fuzzy \u03b2-compactness, intuitionistic fuzzy\u03b2-almost compactness, and intuitionistic fuzzy \u03b2-nearly compactness under several types ofintuitionistic fuzzy continuous mappings.The concept of intuitionistic fuzzy  In this paper some properties of intuitionistic fuzzy \u03b2-compactness were investigated. We use the finite intersection property to give characterization of the intuitionistic fuzzy \u03b2-compact spaces. Also we introduce intuitionistic fuzzy \u03b2-almost compactness and intuitionistic fuzzy \u03b2-nearly compactness and established the relationships between these types of compactness.As a generalization of fuzzy sets introduced by Zadeh , the conX be a nonempty fixed set and I the closed interval . An intuitionistic fuzzy set (IFS) A is an object of the following form: \u03bcA(x) : X \u2192 I and \u03bdA(x) : X \u2192 I denote the degree of membership, namely, \u03bcA(x), and the degree of nonmembership, namely, \u03bdA(x), for each element x \u2208 X to the set A, respectively, and 0 \u2264 \u03bcA(x) + \u03bdA(x) \u2264 1 for each x \u2208 X.Let A and B be IFSs of the forms A = {\u2329x, \u03bcA(x), \u03bdA(x)\u232a; x \u2208 X} and B = {\u2329x, \u03bcB(x), \u03bdB(x)\u232a; x \u2208 X}. ThenA\u2286B if and only if \u03bcA(x) \u2264 \u03bcB(x) and \u03bdA(x) \u2265 \u03bdB(x);A\u2229B = {\u2329x, \u03bcA(x)\u2227\u03bcB(x), \u03bdA(x)\u2228\u03bdB(x)\u232a; x \u2208 X};A \u222a B = {\u2329x, \u03bcA(x)\u2228\u03bcB(x), \u03bdA(x)\u2227\u03bdB(x)\u232a; x \u2208 X}.Let A = {\u2329x, \u03bcA, \u03bdA\u232a; x \u2208 X} instead of A = {\u2329x, \u03bcA(x), \u03bdA(x)\u232a; x \u2208 X}.We will use the notation ~ = {\u2329x, 0,1\u232a; x \u2208 X} and 1~ = {\u2329x, 1,0\u232a; x \u2208 X}.Consider 0\u03b1, \u03b2 \u2208  such that \u03b1 + \u03b2 \u2264 1. An intuitionistic fuzzy point (IFP) p\u03b1, \u03b2) and  or simply by X and Y, we will denote the intuitionistic fuzzy topological spaces (IFTSs). Each IFS which belongs to \u03c4 is called an intuitionistic fuzzy open set (IFOS) in X. The complement X is called an intuitionistic fuzzy closed set (IFCS) in X.An intuitionistic fuzzy topology (IFT) in Coker's sense on a nonempty set X and Y be two nonempty sets and let f : \u2192 be a function.Let B = {\u2329y, \u03bcB(y), \u03bdB(y)\u232a; y \u2208 Y} is an IFS in Y, then the preimage of B under f is denoted and defined by f\u22121(B) = {\u2329x, f\u22121(\u03bcB(x)), f\u22121(\u03bdB(x))\u232a; x \u2208 X}. Since \u03bcB(x), \u03bdB(x) are fuzzy sets, we explain that f\u22121(\u03bcB(x)) = \u03bcB(x)(f(x)) and f\u22121(\u03bdB(x)) = \u03bdB(x)(f(x)).If p\u03b1,\u03b2) of p\u03b1,\u03b2) be an IFTS and let A = {\u2329x, \u03bcA(x), \u03bdA(x)\u232a; x \u2208 X} be an IFS in X. Then the intuitionistic fuzzy closure and intuitionistic fuzzy interior of A are defined byA) = \u2229{C : C\u2009\u2009is\u2009\u2009an\u2009\u2009IFCS\u2009\u2009in\u2009\u2009X\u2009\u2009and\u2009\u2009C\u2287A};cl\u2009(A) = \u222a{D : D\u2009\u2009is\u2009\u2009an\u2009\u2009IFOS\u2009\u2009in\u2009\u2009X\u2009\u2009and\u2009\u2009D\u2286A}.int\u2009 is an IFCS, int\u2009(A) is an IFOS in X, and A is an IFCS in X if and only if cl\u2009(A) = A; A is an IFOS in X if and only if int\u2009(A) = A.Let  be an IFTS and let A and B be IFSs in X. Then the following properties hold:A)\u2286A\u2286cl\u2009(A).int\u2009 if A\u2286cl(int(clA)). The complement of an IF\u03b2OS A in IFTS X is called an intuitionistic fuzzy \u03b2-closed set (IF\u03b2CS) in X.An IFS f be a mapping from an IFTS X into an IFTS Y. The mapping f is calledf\u22121(B) is an IFOS in X, for each IFOS B in Y;intuitionistic fuzzy continuous if and only if \u03b2-continuous if and only if f\u22121(B) is an IF\u03b2OS in X, for each IFOS B in Y.intuitionistic fuzzy Let X, \u03c4) be an IFTS and let A = {\u2329x, \u03bcA(x), \u03bdA(x)\u232a; x \u2208 X} be an IFS in X. Then the intuitionistic fuzzy \u03b2-closure and intuitionistic fuzzy \u03b2-interior of A are defined by\u03b2cl\u2061(A) = \u2229{C : C\u2009\u2009is\u2009\u2009an\u2009\u2009IF\u03b2CS\u2009\u2009in\u2009\u2009X\u2009\u2009and\u2009\u2009C\u2287A};\u03b2int\u2061(A) = \u222a{D : D\u2009\u2009is\u2009\u2009an\u2009\u2009IF\u03b2OS\u2009\u2009in\u2009\u2009X\u2009\u2009and\u2009\u2009D\u2286A}.Let \u2192 from an intuitionistic fuzzy topological space  to another intuitionistic fuzzy topological space  is said to be intuitionistic fuzzy \u03b2-irresolute if f\u22121(B) is an IF\u03b2OS in  for each IF\u03b2OS B in .A function X be an IFTS. A family of {\u2329x, \u03bcGi(x), \u03bdGi(x)\u232a; i \u2208 J} intuitionistic fuzzy open sets (intuitionistic fuzzy \u03b2-open sets) in X satisfies the condition 1~ = \u222a{\u2329x, \u03bcGi(x), \u03bdGi(x)\u232a; i \u2208 J} which is called an intuitionistic fuzzy open (intuitionistic fuzzy \u03b2-open) cover of X. A finite subfamily of an intuitionistic fuzzy open (intuitionistic fuzzy \u03b2-open) cover {\u2329x, \u03bcGi(x), \u03bdGi(x)\u232a; i \u2208 J} of X which is also an intuitionistic fuzzy open (intuitionistic fuzzy \u03b2-open) cover of X is called a finite subcover of {\u2329x, \u03bcGi(x), \u03bdGi(x)\u232a; i \u2208 j}.Let X is called intuitionistic fuzzy compact if each fuzzy open cover of X has a finite subcover for X.An IFTS X is called intuitionistic fuzzy almost compact  if, for every IF open cover {Uj : j \u2208 J} of X, there exists a finite subfamily J0 \u2282 J such that X = \u222a{cl\u2061(Uj) : j \u2208 J0}.An IFTS X is said to be intuitionistic fuzzy \u03b2-compact (IF \u03b2-compact) if every IF \u03b2-open cover of X has a finite subcover.An IFTS X, \u03c4) and  be two IFTSs. A function f : X \u2192 Y is said to be intuitionistic fuzzy weakly continuous (IF weakly continuous) if, for each IFOS V in Y, f\u22121(V)\u2286int\u2009(f\u22121(cl\u2061V)).Let  is said to be IF \u03b2-compact relative to X if, for every collection {Ai; i \u2208 I} of IF \u03b2-open subsets of X such that B\u2286\u222a{Ai; i \u2208 I}, there exists a finite subset I0 of I such that B\u2286\u222a{Ai; i \u2208 I0}.An IFTS B of  is said to be IF \u03b2-compact if it is IF \u03b2-compact as a subspace of X.An IFTS \u03b2-closed sets {Ai; i \u2208 I} has the finite intersection property (in short FIP) if, for any subset I0 of I, \u2229i\u2208I0Ai \u2260 0~.A family of IF X the following statements are equivalent. X is IF \u03b2-compact.\u03b2-closed subsets of X satisfying the FIP has a nonempty intersection.Any family of IF For an IFTS X be IF \u03b2-compact space and let {Ai; i \u2208 I} be a family of IF \u03b2-closed subsets of X satisfying the FIP. Suppose \u2229i\u2208IAi = 0~. Then \u03b2-open sets cover X, then from IF \u03b2-compactness of X there exists a finite subset I0 of I such that i\u2208I0Ai = 0~ which gives a contradiction and therefore \u2229i\u2208IAi \u2260 0~. Thus (i)\u21d2(ii).Let Ai; i \u2208 I} be a family of IF \u03b2-open sets cover X. Suppose that for any finite subset I0 of I we have \u222ai\u2208I0Ai \u2260 1~. Then i\u2208IAi \u2260 1~ and contradicts that {Ai; i \u2208 I} is an IF \u03b2-open cover of X. Hence our assumption \u222ai\u2208I0Ai \u2260 1~ is wrong. Thus \u222ai\u2208I0Ai = 1~ which implies that X is IF \u03b2-compact. Thus (ii)\u21d2(i).Let {\u03b2-closed subset of an intuitionistic fuzzy \u03b2-compact space is intuitionistic fuzzy \u03b2-compact relative to X.An intuitionistic fuzzy A be an IF \u03b2-closed subset of X. Let {Gi; i \u2208 I} be cover of A by IF \u03b2-open sets in X. Then the family \u03b2-open cover of X. Since X is IF \u03b2-compact, there is a finite subfamily {G1, G2,\u2026, Gn} of IF \u03b2-open cover, which also covers X. If this cover contains \u03b2-open subcover of A. So A is IF \u03b2-compact relative to X.Let X, \u03c4) and  be intuitionistic fuzzy topological spaces and let f : \u2192 be intuitionistic fuzzy \u03b2-irresolute, surjective mapping. If  is IF \u03b2-compact space then so is .Let \u2192 be intuitionistic fuzzy \u03b2-irresolute mapping of an intuitionistic fuzzy \u03b2-compact space  onto an IFTS . Let {Ai : i \u2208 I} be any intuitionistic fuzzy \u03b2-open cover of . Then {f\u22121(Ai) : i \u2208 I} is collection of intuitionistic fuzzy \u03b2-open sets which covers X. Since X is intuitionistic fuzzy \u03b2-compact, there exists a finite subset I0 of I such that subfamily {f\u22121(Ai); i \u2208 I0} of {f\u22121(Ai):\u2009\u2009i \u2208 I} covers X. It follows that {Ai; i \u2208 I0} is a finite subfamily of {Ai : i \u2208 I} which covers Y. Hence Y is intuitionistic fuzzy \u03b2-compact.Let X, \u03c4) and  be intuitionistic fuzzy topological spaces and let f : \u2192 be intuitionistic fuzzy \u03b2-irresolute mapping. If A is IF \u03b2-compact relative to X then f(A) is IF \u03b2-compact relative to Y.Let (Ai : i \u2208 I} be a family of IF \u03b2-open cover of Y such that f(A)\u2286\u222ai\u2208IAi. Then A\u2286f\u22121(f(A))\u2286f\u22121(\u222ai\u2208IAi) = \u222ai\u2208If\u22121(Ai). Since f is IF \u03b2-irresolute, f\u22121(Ai) is IF \u03b2-open cover of X. And A is IF \u03b2-compact in ; there exists a finite subset I0 of I such that A\u2286\u222ai\u2208I0f\u22121(Ai). Hence f(A)\u2286f(\u222ai\u2208I0f\u22121(Ai)) = \u222ai\u2208I0f(f\u22121(Ai))\u2286\u222ai\u2208I0(Ai). Thus f(A) is IF \u03b2-compact relative to Y.Let {\u03b2-continuous image of IF \u03b2-compact space is IF compact.An IF f : \u2192 be an IF \u03b2-continuous from an IF \u03b2-compact space X onto IFTS Y. Let {Ai : i \u2208 I} be IF open cover of Y. Then {f\u22121(Ai); i \u2208 I} is IF \u03b2-open cover of X. Since X is IF \u03b2-compact, there exists finite subset I0 of I such that finite family {f\u22121(Ai); i \u2208 I0} covers X. Since f is onto, {Ai : i \u2208 I0} is a finite cover of Y. Hence Y is IF compact.Let X, \u03c4) and  be two intuitionistic fuzzy topological spaces. A mapping f : X \u2192 Y is said to be intuitionistic fuzzy strongly \u03b2-open if f(V) is IF\u03b2OS of Y for every IF\u03b2OS V of X.Let \u2192 be an IF strongly \u03b2-open, bijective function and Y is IF \u03b2-compact; then X is IF \u03b2-compact.Let Ai : i \u2208 I} be IF \u03b2-open cover of X, and then {f(Ai) : i \u2208 I} is IF \u03b2-open cover of Y. Since Y is IF \u03b2-compact, there is a finite subset I0 of I such that finite family {f(Ai) : i \u2208 I0} covers Y. But 1X~ = f\u22121(1Y~) = f\u22121f(\u222ai\u2208I0(Ai)) = \u222ai\u2208I0Ai and therefore X is IF \u03b2-compact.Let {\u03b2-compactness, IF \u03b2-almost compactness, and IF \u03b2-nearly compactness.In this section we investigate the relationships between IF X, \u03c4) is said to be IF \u03b2-almost compactness if and only if, for every family of IF \u03b2-open cover {Ai : i \u2208 I} of X, there exists a finite subset I0 of I such that \u222ai\u2208I0\u03b2cl\u2061Ai = 1~.An IFTS  is said to be IF \u03b2-nearly compactness if and only if, for every family of IF \u03b2-open cover {Ai : i \u2208 I} of X, there exists a finite subset I0 of I such that \u222ai\u2208I0\u03b2int\u2061(\u03b2cl\u2061Ai) = 1~.An IFTS  is said to be IF \u03b2-regular if, for each IF \u03b2-open set A \u2208 X, A = \u222a{Ai \u2208 IX\u2223Ai\u2009\u2009is\u2009\u2009IF\u2009\u2009\u03b2-open, \u03b2cl\u2061Ai\u2286A}.An IFTS  be IFTS. Then IF \u03b2-compactness implies IF \u03b2-nearly compactness which implies IF \u03b2-almost compactness.Let  be IF \u03b2-compact. Then for every IF \u03b2-open cover {Ai : i \u2208 I} of X, there exists a finite subset \u222ai\u2208I0Ai = 1~. Since Ai is an IF\u03b2OS, for each i \u2208 I, Ai = \u03b2int\u2061(Ai) for each i \u2208 I. Ai = \u03b2int\u2061Ai\u2286\u03b2int\u2061(\u03b2\u2009cl\u2009\u2061Ai) for each i \u2208 I. Here 1~ = \u222ai\u2208I0Ai = \u222ai\u2208I0\u03b2int\u2061Ai\u2286\u222ai\u2208I0\u03b2int\u2061(\u03b2\u2009cl\u2009\u2061Ai). Thus \u222ai\u2208I0\u03b2int\u2061(\u03b2cl\u2061Ai) = 1~ which implies that  is IF \u03b2-nearly compactness. Now let  be IF \u03b2-nearly compact. Then for every IF \u03b2-open cover {Ai : i \u2208 I} of X, there exists a finite subset \u222ai\u2208I0\u03b2int\u2061(\u03b2cl\u2061Ai) = 1~. Since \u03b2int\u2061(\u03b2cl\u2061Ai)\u2286\u03b2cl\u2061Ai for each i \u2208 I0, 1~ = \u222ai\u2208I0\u03b2int\u2061(\u03b2cl\u2061Ai)\u2286\u222ai\u2208I0\u03b2cl\u2061Ai. Thus \u222ai\u2208I0\u03b2cl\u2061Ai = 1~. Hence  is IF \u03b2-almost compact.Let  is IFTS, where \u03c4 = {0~, 1~, An}, n \u2208 N, where An : X \u2192  is defined by An = {\u2329x, 1 \u2212 1/n, 1/n\u232a; x \u2208 X}, n \u2208 N. The collection {An : n \u2208 N} is IF \u03b2-open cover of X. But no finite subset of {An : n \u2208 N} covers X. Hence X is not IF \u03b2-compact. But \u03b2cl\u2061An = 1~ for n \u2265 3. Thus there exists a finite subfamily {An : n \u2208 N0} for N0\u2286N such that \u222an\u2208N0\u03b2cl\u2061An = 1~. Thus X is IF \u03b2-almost compactness. Also \u03b2int\u2061(\u03b2cl\u2061An) = \u03b2int\u2061\u2009(1~) = 1~ for n \u2265 3. Thus there exists a finite subfamily {An : n \u2208 N0} for N0\u2286N such that \u222an\u2208N0\u03b2int\u2061\u03b2cl\u2061An = 1~. Thus X is IF \u03b2-nearly compactness.Let X, \u03c4) be IFTS. If  is IF \u03b2-almost compact and IF \u03b2-regular then  is IF \u03b2-compact.Let  is IF \u03b2-regular, Ai = \u222a{Bi \u2208 IX\u2223Bi\u2009\u2009is\u2009\u2009IF\u2009\u2009\u03b2-open, \u03b2cl\u2061\u2009Bi\u2286Ai} for each i \u2208 I. Since 1~ = \u222ai\u2208I(\u222ai\u2208IBi) and  is IF \u03b2-almost compact there exists a finite set I0 of I such that \u222ai\u2208I0\u03b2cl\u2061Bi = 1~. But \u03b2cl\u2061(Bi)\u2286Ai\u2009\u2009(\u03b2int\u2061(\u03b2cl\u2061Bi)\u2286\u03b2cl\u2061(Bi)). We have \u222ai\u2208I0Ai\u2287\u222ai\u2208I0\u03b2cl\u2061Bi = 1~. Thus, \u222ai\u2208I0Ai = 1~. Hence  is IF \u03b2-compact.Let {X, \u03c4) be IFTS. If  is IF \u03b2-nearly compact and IF \u03b2-regular then  is IF \u03b2-compact.Let  is IF \u03b2-regular, Ai = \u222a{Bi \u2208 IX\u2223Bi\u2009\u2009is\u2009\u2009IF\u2009\u2009\u03b2-open, \u03b2\u2009cl\u2061\u2009Bi\u2286Ai} for each i \u2208 I. Since 1~ = \u222ai\u2208I(\u222ai\u2208IBi) and  is IF \u03b2-nearly compact there exists a finite set I0 of I such that \u222ai\u2208I0\u03b2int\u2061(\u03b2cl\u2061\u2061Bi) = 1~. But \u03b2int\u2009\u2061(\u03b2cl\u2061Bi)\u2286\u03b2cl\u2061(Bi)\u2286Ai. We have \u222ai\u2208I0Ai\u2287\u222ai\u2208I0\u03b2cl\u2061Bi\u2287\u222ai\u2208I0\u03b2int\u2061(\u03b2cl\u2061Bi) = 1~. Thus, \u222ai\u2208I0Ai = 1~. Hence  is IF \u03b2-compact.Let {X, \u03c4) is IF \u03b2-almost compact, if and only if, for every family {Ai : i \u2208 I} of IF \u03b2-open sets having the FIP, \u2229i\u2208I\u03b2cl\u2061Ai \u2260 0~.An IFTS  is IF \u03b2-almost compact, there exists a finite subset I0 of I such that i\u2208I0\u03b2int\u2061(\u03b2\u2009cl\u2009Ai) = 0~. But Ai = \u03b2\u2061int\u2061Ai\u2286\u03b2int\u2009\u2061(\u03b2cl\u2061Ai). This implies that \u2229i\u2208I0Ai = 0~ which is in contradiction with FIP of the family. Conversely, let {Ai : i \u2208 I} be a family of IF \u03b2-open sets such that \u222ai\u2208IAi = 1~. Suppose that there exists no finite subset I0 of I such that \u222ai\u2208I0\u03b2cl\u2061Ai = 1~. Since Ai\u2286\u03b2int\u2061\u2009(\u03b2cl\u2061Ai) for each i \u2208 I, \u222ai\u2208IAi \u2260 1~ which is in contradiction with \u222ai\u2208IAi = 1~.Let {X, \u03c4) and  be IFTS and let f : \u2192 be intuitionistic fuzzy \u03b2-irresolute, surjective mapping. If  is IF \u03b2-almost compact space then so is .Let \u2192 be intuitionistic fuzzy \u03b2-irresolute mapping of an intuitionistic fuzzy \u03b2-compact space  onto an IFTS . Let {Ai : i \u2208 I} be any intuitionistic fuzzy \u03b2-open cover of . Then {f\u22121(Ai) : i \u2208 I} is an intuitionistic fuzzy \u03b2-open cover of X. Since X is intuitionistic fuzzy \u03b2-almost compact, there exists a finite subset I0 of I such that \u222ai\u2208I0\u03b2cl\u2061\u2009(f\u22121(Ai)) = 1X~. And f(1X~) = f(\u222ai\u2208I0\u03b2cl\u2061(f\u22121(Ai))) = \u222ai\u2208I0f(\u03b2cl\u2061(f\u22121(Ai))) = 1Y~. But \u03b2cl\u2061\u2009(f\u22121(Ai))\u2286f\u22121(\u03b2cl\u2009Ai) and from the surjectivity of f, f(\u03b2cl\u2009(f\u22121(Ai)))\u2286f(f\u22121(\u03b2cl\u2061Ai)) = \u03b2cl\u2061Ai. So \u222ai\u2208I0\u03b2cl\u2061Ai\u2287\u222ai\u2208I0f(\u03b2cl\u2061(f\u22121(Ai))) = 1Y~. Thus \u222ai\u2208I0\u2009\u2009\u03b2cl\u2061Ai = 1Y~. Hence  is IF \u03b2-almost compact.Let X, \u03c4) and  be intuitionistic fuzzy topological spaces and let f : \u2192 be intuitionistic fuzzy \u03b2-continuous, surjective mapping. If  is IF \u03b2-almost compact space then  is IF almost compact.Let . Then {f\u22121(Ai) : i \u2208 I} is an intuitionistic fuzzy \u03b2-open cover of X. Since X is intuitionistic fuzzy \u03b2-almost compact, there exists a finite subset I0 of I such that \u222ai\u2208I0\u03b2cl\u2061(f\u22121(Ai)) = 1X~. And from the surjectivity of f, 1Y~ = f(1X~) = f(\u222ai\u2208I0\u03b2cl\u2061(f\u22121(Ai)))\u2286\u222ai\u2208I0f(\u03b2cl\u2061(f\u22121(Ai)))\u222ai\u2208I0\u03b2cl\u2061f(f\u22121(Ai))\u2286\u222ai\u2208I0cl\u2061f(f\u22121(Ai))\u2286\u222ai\u2208I0cl\u2009\u2061Ai which implies that \u222ai\u2208I0cl\u2061Ai = 1Y~. Hence  is IF almost compact.Let {X, \u03c4) and  be two intuitionistic fuzzy topological spaces. A function f : X \u2192 Y is said to be intuitionistic fuzzy \u03b2-weakly continuous (IF \u03b2-weakly continuous) if, for each IF\u03b2OS V in Y, f\u22121(V)\u2286\u03b2int\u2061(f\u22121(\u03b2cl\u2061V)).Let  to an IFTS  is IF strongly \u03b2-open if and only if f(\u03b2int\u2061V)\u2286\u03b2int\u2061f(V).A mapping f is IF strongly \u03b2-open mapping then f(\u03b2int\u2061V) is an IF\u03b2OS in Y for IF\u03b2OS V in X. Hence f(\u03b2int\u2061V) = \u03b2\u2009intf(\u03b2int\u2061V) = \u03b2int\u2061f(V). Thus f(\u03b2int\u2061V)\u2286\u03b2int\u2061f(V).If V be IF\u03b2OS in X and then V = \u03b2int\u2061V. Then by hypothesis, f(V) = f(\u03b2int\u2061V)\u2286\u03b2int\u2061f(V). This implies that f(V) is IF\u03b2OS in Y.Conversely, let X, \u03c4) and  be IFTS and let f : \u2192 be intuitionistic fuzzy \u03b2-weakly continuous, surjective mapping. If  is IF \u03b2-compact space then  is IF \u03b2-almost compact.Let (Ai : i \u2208 I} be IF \u03b2-open cover of Y such that \u222ai\u2208IAi = 1Y~. Then \u222ai\u2208If\u22121(Ai) = f\u22121(\u222ai\u2208IAi) = f\u22121(1Y~) = 1X~.  is IF \u03b2-compact, and there exists a finite subset I0 of I such that \u222ai\u2208I0f\u22121(Ai) = 1X~. Since f is IF \u03b2-weakly continuous, f\u22121(Ai)\u2286\u03b2int\u2061(f\u22121(\u03b2cl\u2061Ai))\u2286f\u22121(\u03b2cl\u2061Ai). This implies that \u222ai\u2208I0f\u22121(\u03b2cl\u2061Ai)\u2287\u222ai\u2208I0f\u22121(Ai) = 1X~. Thus \u222ai\u2208I0f\u22121(\u03b2cl\u2061Ai) = 1X~. Since f is surjective, 1Y~ = f(1X~) = f(\u222ai\u2208I0f\u22121(\u03b2cl\u2061Ai)) = \u222ai\u2208I0f(f\u22121(\u03b2cl\u2061Ai)) = \u222ai\u2208I0\u03b2cl\u2061Ai. Hence \u222ai\u2208I0\u03b2cl\u2061Ai = 1Y~. Hence  is IF \u03b2-almost compact.Let {X, \u03c4) and  be intuitionistic fuzzy topological spaces and let f : \u2192 be intuitionistic fuzzy \u03b2-irresolute, surjective, and strongly \u03b2-open mapping. If  is IF \u03b2-nearly compact space then so is .Let . Since f is IF \u03b2-irresolute, then {f\u22121(Ai) : i \u2208 I} is an intuitionistic fuzzy \u03b2-open cover of X. Since  is IF \u03b2-nearly compact, there exists a finite subset I0 of I such that \u222ai\u2208I0\u03b2int\u2061(\u03b2cl\u2061f\u22121(Ai)) = 1X~. Since f is surjective, 1Y~ = f(1X~) = f(\u222ai\u2208I0\u03b2int\u2009\u2061(\u03b2cl\u2061f\u22121(Ai))) = \u222ai\u2208I0f(\u03b2\u2009int\u2061\u2061\u2061\u2009(\u03b2cl\u2061f\u22121(A))). Since f is IF strongly \u03b2-open, f(\u03b2int\u2061(\u03b2cl\u2061f\u22121(Ai)))\u2286\u03b2int\u2061f(\u03b2\u2009clf\u22121(Ai)) for each i \u2208 I. Since f is IF \u03b2-irresolute, then f(\u03b2cl\u2061f\u22121(Ai))\u2286\u03b2cl\u2061f(f\u22121(Ai)). Hence we have 1Y~ = \u222ai\u2208I0f(\u03b2int\u2061(\u03b2cl\u2061f\u22121(Ai)))\u2009\u2009\u2286\u2009\u2009\u222ai\u2208I0\u03b2int\u2061f(\u03b2cl\u2061f\u22121(Ai))\u2009\u2009\u2286\u2009\u2009\u222ai\u2208I0\u03b2int\u2061(\u03b2cl\u2061f(f\u22121(Ai))) = \u222ai\u2208I0\u03b2int\u2061(\u03b2cl\u2061(Ai)). Thus 1Y~ = \u222ai\u2208I0\u03b2int\u2061(\u03b2cl\u2061(Ai)). Hence  is IF \u03b2-nearly compact.Let {"}
{"text": "The notions of an ideal and a fuzzy ideal in BN-algebras are introduced. The properties and characterizations of them are investigated. The concepts of normal ideals and normal congruences of a BN-algebra are also studied, the properties of them are displayed, and a one-to-one correspondence between them is presented. Conditions for a fuzzyset to be a fuzzy ideal are given. The relationships between ideals and fuzzy ideals of a BN-algebra are established. The homomorphic properties of fuzzy ideals of a BN-algebra are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals are obtained. The class of BM-algebras contains Coxeter algebras . Some oras  the class of all BCK-algebras . The interrelationships between some classes of algebras mentioned before are visualized in X and Y are classes of algebras, then X \u2192 Y means X \u2282 Y).We will denote by \u03bc, we make the least fuzzy ideal containing \u03bc. This leads us to show that the set of fuzzy ideals of a BN-algebra is a complete lattice. Moreover, the homomorphic properties of fuzzy ideals are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras in terms of fuzzy ideals are given in In this paper we consider ideals and fuzzy ideals in BN-algebras. In A; \u2217, 0) of type  is called a Coxeter algebra \u03b8 we denote the congruence class containing x; that is, [x]\u03b8 = {y \u2208 A : x\u03b8y}.Let \u03b8 be a congruence on \ud835\udc9c. Then \u03b8 \u2208 CN(\ud835\udc9c) if and only if [0]\u03b8 is a normal ideal of \ud835\udc9c.Let I = [0]\u03b8 and let \u03b8 \u2208 CN(\ud835\udc9c). It follows easily that 0 \u2208 I. Let x and y be elements of A such that x\u2217y, y \u2208 I. Then, (x\u2217y)\u03b80 and y\u03b80. Since y\u03b80, we obtain (x\u2217y)\u03b8(x\u22170) and hence x = x\u22170 \u2208 I. Consequently, I is an ideal. Now, suppose that x\u2217y, a\u2217b \u2208 I, where x, y, a, b \u2208 A. Then, (x\u2217y)\u03b80 and (a\u2217b)\u03b80. By the definition of a normal congruence, ((x\u2217a)\u2217(y\u2217b))\u03b80; that is, (x\u2217a)\u2217(y\u2217b) \u2208 I. Thus, I = [0]\u03b8 is normal.Set The converse is obvious.There is a bijection between the set of normal ideals and the set of normal congruences of a BN-algebra.\ud835\udc9c be a BN-algebra. We consider functions f : IdN(\ud835\udc9c) \u2192 CN(\ud835\udc9c) and g : CN(\ud835\udc9c) \u2192 IdN(\ud835\udc9c) given as follows:I \u2208 CN(\ud835\udc9c) and [0]\u03b8 \u2208 IdN(\ud835\udc9c), we conclude that f and g are well-defined. Next, note that I = [0]I~. Indeed,g\u2218f = idN(\ud835\udc9c)Id and f\u2218g = idCN(\ud835\udc9c). Thus, f and g are inverse bijections between IdN(\ud835\udc9c) and CN(\ud835\udc9c).Let .Hence,g\u2218fI=g~I=e obtainf\u2218g\u03b8=f0\u03b8=\ud835\udc9c =  and \u212c =  be BN-algebras. A mapping f : A \u2192 B is called a homomorphism from \ud835\udc9c into \u212c if f(x\u2217y) = f(x)\u2217f(y) for any x, y \u2208 A.Let f(0A) = 0B. Indeed, f(0A) = f(0A\u22170A) = f(0A)\u2217f(0A) = 0B. We denote by ker\u2061f the subset {x \u2208 A : f(x) = 0B} of A (it is the kernel of the homomorphism f).Observe that f : A \u2192 B be a homomorphism from \ud835\udc9c into \u212c. Then ker\u2061f is an ideal of \ud835\udc9c.Let A \u2208 ker\u2061f; that is, (I1) holds. Let x\u2217y \u2208 ker\u2061f and y \u2208 ker\u2061f. Then 0B = f(x\u2217y) = f(x)\u2217f(y) = f(x)\u22170B = f(x). Consequently, x \u2208 ker\u2061f. Therefore, (I2) is satisfied. Thus I is an ideal of \ud835\udc9c.Obviously, 0A; \u2217, 0) be the algebra given in A : A \u2192 A is a homomorphism and the ideal ker\u2061(idA) = {0} is not normal.The kernel of a homomorphism is not always a normal ideal. Let  be a BN-algebra and let \u212c =  be a BM-algebra. Let f : A \u2192 B be a homomorphism from \ud835\udc9c into \u212c. Then ker\u2061f is a normal ideal.Let ker\u2061f is an ideal of \ud835\udc9c. Let x, y, a, b \u2208 A and x\u2217y, a\u2217b \u2208 ker\u2061f. Then 0B = f(x\u2217y) = f(x)\u2217f(y). From f(x) = f(y). Similarly, f(a) = f(b). Consequently, f((x\u2217a)\u2217(y\u2217b)) = (f(x)\u2217f(a))\u2217(f(y)\u2217f(b)) = (f(x)\u2217f(a))\u2217(f(x)\u2217f(a)) = 0B, and hence, (x\u2217a)\u2217(y\u2217b) \u2208 ker\u2061f.By I be a normal ideal of \ud835\udc9c. For x \u2208 A, we write x/I = {y \u2208 A : x~Iy}; that is, x/I = [x]I~. We note thatA/I = {x/I : x \u2208 A} and set x/I\u2217\u2032y/I = x\u2217y/I. The operation \u2217\u2032 is well-defined, since ~I is a congruence of \ud835\udc9c. It is easy to see that \ud835\udc9c/I =  is a BN-algebra. The algebra \ud835\udc9c/I is called the quotient BN-algebra of \ud835\udc9c modulo I.Let \ud835\udc9c and \u03b8 be given as in \ud835\udc9c is a BN-algebra and \u03b8 is a normal congruence of \ud835\udc9c. Since [0]\u03b8 = {0,1}, from I = {0,1} is a normal ideal of \ud835\udc9c. We have 0/I = 1/I = {0,1} and 2/I = 3/I = {2,3}. Then A/I = {0/I, 2/I}. Clearly, 0/I\u2217\u20320/I = 0/I = 2/I\u2217\u20322/I and 0/I\u2217\u20322/I = 2/I = 2/I\u2217\u20320/I.Let \ud835\udc9c be a BN-algebra and let \u212c be a BM-algebra. Let f : A \u2192 B be a homomorphism from \ud835\udc9c onto \u212c. Then \ud835\udc9c/ker\u2061f is isomorphic to \u212c.Let I = ker\u2061f is a normal ideal of \ud835\udc9c. Define a mapping g : A/I \u2192 B by g(x/I) = f(x) for all x \u2208 I. Let x/I = y/I. Then, x~Iy; that is, x\u2217y \u2208 I. Hence, f(x)\u2217f(y) = 0B. By f(x) = f(y). Consequently, g(x/I) = g(y/I). This means that g is well-defined. It is easy to check that g is a homomorphism from \ud835\udc9c/I onto \u212c. Observe that g is one-to-one. Let g(x/I) = g(y/I). Then f(x) = f(y) and hence f(x\u2217y) = 0B; that is, x\u2217y \u2208 I. Therefore, x~Iy and consequently, x/I = y/I. Thus g is an isomorphism from \ud835\udc9c/I onto \u212c.By \u03b1, \u03b2}, then \u03b1\u2227\u03b2 = min\u2061{\u03b1, \u03b2} and \u03b1\u2228\u03b2 = max\u2061{\u03b1, \u03b2}. Recall that a fuzzy set in A is a function \u03bc : A \u2192 .We now review some fuzzy logic concepts. First, for \u0393\u2286 we define \u22c0\u0393 = inf\u2061\u0393 and \u22c1\u0393 = sup\u2061\u0393. Obviously, if \u0393 = {\u03bc and \u03bd in A, we defineA.For any fuzzy sets A and B be any two sets, \u03bc any fuzzy set in A, and f : A \u2192 B any function. Set f\u2190(y) = {x \u2208 A : f(x) = y} for y \u2208 B. The fuzzy set \u03bd in B defined byy \u2208 B, is called the image of \u03bc under f and is denoted by f(\u03bc).Let A and B be any two sets, f : A \u2192 B any function, and \u03bd any fuzzy set in f(A). The fuzzy set \u03bc in A defined by preimage of \u03bd under f and is denoted by f\u2190(\u03bd).Let Now, we give the definition of a fuzzy ideal in a BN-algebra.\u03bc in A is called a fuzzy ideal of a BN-algebra \ud835\udc9c if it satisfies, for all x, y \u2208 A, \u03bc(0) \u2265 \u03bc(x),\u03bc(x) \u2265 \u03bc(x\u2217y)\u2227\u03bc(y).A fuzzy set \u03bc be a fuzzy ideal of \ud835\udc9c. Then, for any x, y \u2208 A, if x \u2264 y, then \u03bc(x) \u2264 \u03bc(y).Let x \u2264 y, then x\u2217y = 0. From y\u2217x = 0. Hence, by (d2) and (d1), we have \u03bc(y) \u2265 \u03bc(y\u2217x)\u2227\u03bc(x) = \u03bc(0)\u2227\u03bc(x) = \u03bc(x).If \u2131Id(\ud835\udc9c) the set of all fuzzy ideals of a BN-algebra \ud835\udc9c.Denote by \ud835\udc9c =  be the BN-algebra given in \u03b13 < \u03b12 < \u03b11 \u2264 1. Define a fuzzy set \u03bc in A by\u03bc satisfies (d1) and (d2). Thus \u03bc \u2208 \u2131Id(\ud835\udc9c).Let I be an ideal of a BN-algebra \ud835\udc9c and let \u03b1, \u03b2 \u2208  with \u03b1 \u2265 \u03b2. Define \u03bcI\u03b1,\u03b2 as follows:\u03bcI\u03b1,\u03b2 = \u03bc. Since 0 \u2208 I, \u03bc(0) = \u03b1 \u2265 \u03bc(x) for all x \u2208 A. To prove (d2), let x, y \u2208 A. If x \u2208 I, then \u03bc(x) = \u03b1 \u2265 \u03bc(x\u2217y)\u2227\u03bc(y). Suppose now that x \u2209 I. By the definition of an ideal, x\u2217y \u2209 I or y \u2209 I. Therefore, \u03bc(x\u2217y)\u2227\u03bc(y) = \u03b2 = \u03bc(x). Thus \u03bc is a fuzzy ideal of \ud835\udc9c.Let \u03c7I of I:\ud835\udc9c.In particular, the characteristic function \u03bc in A is a fuzzy ideal of \ud835\udc9c if and only if it satisfies (d1) and x, y, z \u2208 A, if (z\u2217y)\u2217x = 0, then \u03bc(z) \u2265 \u03bc(x)\u2227\u03bc(y).for all A fuzzy set \u03bc \u2208 \u2131Id(\ud835\udc9c) and let x, y, z \u2208 A. Suppose that (z\u2217y)\u2217x = 0. Since \u03bc is a fuzzy ideal, we have \u03bc(z\u2217y) \u2265 \u03bc((z\u2217y)\u2217x)\u2227\u03bc(x) = \u03bc(0)\u2227\u03bc(x) = \u03bc(x) and \u03bc(z) \u2265 \u03bc(z\u2217y)\u2227\u03bc(y). Therefore, \u03bc(z) \u2265 \u03bc(x)\u2227\u03bc(y).Let \u03bc satisfy (d3). From (B1) we have (x\u2217y)\u2217(x\u2217y) = 0. By (d3), \u03bc(x) \u2265 \u03bc(x\u2217y)\u2227\u03bc(y). Then \u03bc satisfies (d2) and hence \u03bc \u2208 \u2131Id(\ud835\udc9c).Conversely, let \u03bc be a fuzzy set in A. Then \u03bc \u2208 \u2131Id(\ud835\udc9c) if and only if its nonempty level subset\ud835\udc9c for all \u03b1 \u2208 .Let \u03bc \u2208 \u2131Id(\ud835\udc9c). Let \u03b1 \u2208  and U \u2260 \u2205. Then \u03bc(x0) \u2265 \u03b1 for some x0 \u2208 A. Since \u03bc(0) \u2265 \u03bc(x0), we have 0 \u2208 U. Let x, y \u2208 A such that x\u2217y, y \u2208 U. Then \u03bc(x\u2217y) \u2265 \u03b1 and \u03bc(y) \u2265 \u03b1. It follows from (d2) thatx \u2208 U. Therefore U is an ideal of \ud835\udc9c.Assume that \u03b1 \u2208 , U = \u2205 or U is an ideal of \ud835\udc9c. If (d1) is not valid, then there exists x0 \u2208 A such that \u03bc(0) < \u03bc(x0) = \u03b2. Then U \u2260 \u2205 and, by assumption, U is an ideal of \ud835\udc9c. Hence 0 \u2208 U and consequently \u03bc(0) \u2265 \u03b2. This is a contradiction and (d1) is valid. Now assume that (d2) does not hold. Then there are a, b \u2208 A such that \u03bc(a) < \u03bc(a\u2217b)\u2227\u03bc(b). Taking\u03bc(a) < \u03b3 < \u03bc(a\u2217b)\u2227\u03bc(b) \u2264 \u03bc(a\u2217b) and \u03b3 < \u03bc(b). Therefore a\u2217b, b \u2208 U but a \u2209 U. This is impossible, and \u03bc is a fuzzy ideal of \ud835\udc9c.Conversely, suppose that, for each By \u03bc is a fuzzy ideal of a BN-algebra \ud835\udc9c, then the set\ud835\udc9c.If The following example shows that the converse of \ud835\udc9c be a BN-algebra. Define a fuzzy set \u03bc in A byA\u03bc = {0} is the ideal of \ud835\udc9c but \u03bc \u2209 \u2131Id(\ud835\udc9c) (because \u03bc does not satisfy (d1)).Let \ud835\udc9c =  given in \u03bc be defined as in \u03b1 \u2208  we haveA are ideals of \ud835\udc9c, this is another proof  be a strictly decreasing sequence in . Let \u03bc be the fuzzy set in A defined byI0 = \u2205. Then \u03bc is a fuzzy ideal of \ud835\udc9c.Let I = \u22c3n\u2208\u2115In. By I is an ideal of \ud835\udc9c. Obviously, \u03bc(0) = t1 \u2265 \u03bc(x) for all x \u2208 A; that is, (d1) holds. Now we show that \u03bc satisfies (d2). Let x, y \u2208 A. We have two cases.Let Case 1\u2009\u2009(x \u2209 I). Then x\u2217y \u2209 I or y \u2209 I. Therefore \u03bc(x\u2217y)\u2227\u03bc(y) = 0 = \u03bc(x). Case 2\u2009\u2009(x \u2208 In \u2212 In\u22121 for some n \u2208 \u2115). Then x\u2217y \u2209 In\u22121 or y \u2209 In\u22121. Hence \u03bc(x\u2217y) \u2264 tn or \u03bc(y) \u2264 tn. Therefore \u03bc(x\u2217y)\u2227\u03bc(y) \u2264 tn = \u03bc(x).\u03bc is a fuzzy ideal of \ud835\udc9c.Thus (d2) is also satisfied and consequently T be a nonempty set of indexes. Let \u03bct \u2208 \u2131Id(\ud835\udc9c) for t \u2208 T. The meet \u22c0t\u2208T\u03bct of fuzzy ideals \u03bct of \ud835\udc9c is defined as follows:Let \u03bct \u2208 \u2131Id(\ud835\udc9c) for t \u2208 T. Then \u22c0t\u2208T\u03bct \u2208 \u2131Id(\ud835\udc9c).Let \u03bc = \u22c0t\u2208T\u03bct. Then, by (d1),x \u2208 A. Let x, y \u2208 A. Since \u03bct \u2208 \u2131Id(\ud835\udc9c), we have \u03bct(x) \u2265 \u03bct(x\u2217y)\u2227\u03bct(y). Hence\u03bc(x) \u2265 \u03bc(x\u2217y)\u2227\u03bc(y) and therefore \u03bc \u2208 \u2131Id(\ud835\udc9c).Let \u03bd be a fuzzy set in A. A fuzzy ideal \u03bc of \ud835\udc9c is said to be generated by \u03bd if \u03bd \u2264 \u03bc and, for any fuzzy ideal \u03c1 of \ud835\udc9c, \u03bd \u2264 \u03c1 implies \u03bc \u2264 \u03c1. The fuzzy ideal generated by \u03bd will be denoted by  let \u03bc\u2228\u03bd denote the join of \u03bc and \u03bd; that is, \u03bc\u2228\u03bd =  = \u03bc(x)\u2228\u03bd(x) for all x \u2208 A.For From \ud835\udc9c be a BN-algebra. Then (\u2131Id(\ud835\udc9c); \u2227, \u2228) is a complete lattice.Let The following two theorems give the homomorphic properties of fuzzy ideals.\ud835\udc9c and \u212c be BN-algebras and let f : A \u2192 B be a homomorphism and \u03bd \u2208 \u2131Id(\u212c). Then f\u2190(\u03bd) \u2208 \u2131Id(\ud835\udc9c).Let x \u2208 A. Since f(x) \u2208 B and \u03bd \u2208 \u2131Id(\u212c), we have \u03bd(0) \u2265 \u03bd(f(x)) = (f\u2190(\u03bd))(x), but \u03bd(0) = \u03bd(f(0)) = (f\u2190(\u03bd))(0). Thus we get (f\u2190(\u03bd))(0)\u2265(f\u2190(\u03bd))(x) for any x \u2208 A; that is, f\u2190(\u03bd) satisfies (d1).Let x, y \u2208 A. Since \u03bd \u2208 \u2131Id(\u212c), we obtainf\u2190(\u03bd))(x)\u2265(f\u2190(\u03bd))(x\u2217y)\u2227(f\u2190(\u03bd))(y). Consequently, f\u2190(\u03bd) \u2208 \u2131Id(\ud835\udc9c).Now let \ud835\udc9c and \u212c be BN-algebras and let f : A \u2192 B be a homomorphism and \u03bc \u2208 \u2131Id(\ud835\udc9c). Then, if \u03bc is constant on ker\u2061f = f\u2190(0), then f\u2190(f(\u03bc)) = \u03bc.Let x \u2208 A and f(x) = y. Hencea \u2208 f\u2190(y), we have f(x) = f(a). Hence f(a\u2217x) = 0; that is, a\u2217x \u2208 ker\u2061f. Thus \u03bc(a\u2217x) = \u03bc(0). Therefore, \u03bc(a) \u2265 \u03bc(a\u2217x)\u2227\u03bc(x) = \u03bc(0)\u2227\u03bc(x) = \u03bc(x). Similarly, \u03bc(x) \u2265 \u03bc(a). Hence \u03bc(x) = \u03bc(a). Thusf\u2190(f(\u03bc)) = \u03bc.Let \ud835\udc9c and \u212c be BN-algebras and let f : A \u2192 B be a surjective homomorphism and \u03bc \u2208 \u2131Id(\ud835\udc9c) such that A\u03bc\u2287ker\u2061f. Then f(\u03bc) \u2208 \u2131Id(\u212c).Let \u03bc is a fuzzy ideal of \ud835\udc9c and 0 \u2208 f\u2190(0), we havex \u2208 A. Hencey \u2208 B. Thus f(\u03bc) satisfies (d1). Suppose thatxB, yB \u2208 B. Since f is surjective, there are xA, yA \u2208 A such that f(xA) = xB and f(yA) = yB. HenceA\u03bc\u2287ker\u2061f, \u03bc is constant on ker\u2061f. Then, by \u03bc is a fuzzy ideal. Thus, we obtain f(\u03bc) \u2208 \u2131Id(\u212c).Since In this section we characterize Noetherian BN-algebras and Artinian BN-algebras using some fuzzy concepts, in particular, fuzzy ideals.\ud835\udc9c is called Noetherian if for every ascending sequence I1\u2286I2\u2286\u22ef of ideals of \ud835\udc9c there exists k \u2208 \u2115 such that In = Ik for all n \u2265 k. A BN-algebra \ud835\udc9c is called Artinian if for every descending sequence I1\u2287I2\u2287\u22ef of ideals of \ud835\udc9c there exists k \u2208 \u2115 such that In = Ik for all n \u2265 k.A BN-algebra \ud835\udc9c be a BN-algebra. The following statements are equivalent: \ud835\udc9c is Noetherian,\u03bc of \ud835\udc9c, Im\u2061(\u03bc) = {\u03bc(x) : x \u2208 A} is a well-ordered set.for each fuzzy ideal Let \ud835\udc9c is Noetherian and \u03bc is a fuzzy ideal of \ud835\udc9c such that Im\u2061(\u03bc) is not a well-ordered subset of . Then there exists a strictly decreasing sequence (\u03bc(xn)), where xn \u2208 A. Let tn = \u03bc(xn) and Un = U = {x \u2208 A : \u03bc(x) \u2265 tn}. Then, by Un is an ideal of \ud835\udc9c for every n \u2208 \u2115. So U1 \u2282 U2 \u2282 \u22ef is a strictly ascending sequence of ideals of \ud835\udc9c. This is a contradiction with the assumption that \ud835\udc9c is Noetherian. Therefore Im\u2061(\u03bc) is a well-ordered set for each fuzzy ideal \u03bc of \ud835\udc9c.(a) \u21d2 (b): Assume that \ud835\udc9c is not Noetherian. Then there exists a strictly ascending sequence I1 \u2282 I2 \u2282 \u22ef\u2282In \u2282 \u22ef of ideals of \ud835\udc9c. Let \u03bc be a fuzzy set in A such thatI0 = \u2205. By \u03bc \u2208 \u2131Id(\ud835\udc9c), but Im\u2061(\u03bc) is not a well-ordered set, which is impossible. Therefore \ud835\udc9c is Noetherian.(b) \u21d2 (a): Assume that (b) is true. Suppose that \ud835\udc9c be a BN-algebra. If, for every fuzzy ideal \u03bc of \ud835\udc9c, Im\u2061(\u03bc) is a finite set, then \ud835\udc9c is Noetherian.Let \ud835\udc9c be a BN-algebra and let T = {t1, t2,\u2026} \u222a {0}, where (tn) is a strictly decreasing sequence in . Then the following conditions are equivalent: \ud835\udc9c is Noetherian,\u03bc of \ud835\udc9c, if Im\u2061(\u03bc)\u2286T, then there exists k \u2208 \u2115 such that Im\u2061(\u03bc)\u2286{t1, t2,\u2026, tk} \u222a {0}.for each fuzzy ideal Let \ud835\udc9c is Noetherian. Let \u03bc be a fuzzy ideal of \ud835\udc9c such that Im\u2061(\u03bc)\u2286T. From Im\u2061(\u03bc) is a well-ordered subset of . Then, since 1 > t1 > t2 > \u22ef>tn > \u22ef>0 and Im\u2061(\u03bc)\u2286{t1, t2,\u2026} \u222a {0}, there exists k \u2208 \u2115 such that Im\u2061(\u03bc)\u2286{t1, t2,\u2026, tk} \u222a {0}.(a) \u21d2 (b): Assume that \ud835\udc9c is not Noetherian. Then there exists a strictly ascending sequence I1 \u2282 I2 \u2282 \u22ef of ideals of \ud835\udc9c. Define a fuzzy set \u03bc in A byI0 = \u2205. By \u03bc is a fuzzy ideal of \ud835\udc9c. This is a contradiction with our assumption. Thus \ud835\udc9c is Noetherian.(b) \u21d2 (a): Assume that (b) is true. Suppose that \ud835\udc9c be a BN-algebra and let T = {t1, t2,\u2026} \u222a {0,1}, where (tn) is a strictly increasing sequence in . Then the following conditions are equivalent: \ud835\udc9c is Artinian,\u03bc of \ud835\udc9c, if Im\u2061(\u03bc)\u2286T, then there exists k \u2208 \u2115 such that Im\u2061(\u03bc)\u2286{t1, t2,\u2026, tk} \u222a {0,1}.for each fuzzy ideal Let ti1 < ti2 < \u22ef<tim < \u22ef is a strictly increasing sequence of elements of Im\u2061(\u03bc). Let Um = U for m \u2208 \u2115. It is immediately seen that U1\u2283U2\u2283\u22ef\u2283Um\u2283\u22ef is a strictly descending sequence of ideals of \ud835\udc9c. This contradicts the assumption that \ud835\udc9c is Artinian.(a) \u21d2 (b): Suppose that \ud835\udc9c is not Artinian. Then there exists a strictly descending sequence I1\u2283I2\u2283\u22ef\u2283In\u2283\u22ef of ideals of \ud835\udc9c. Define a fuzzy set \u03bc in A by\u03bc(0) = 1 \u2265 \u03bc(x) for all x \u2208 A; that is, (d1) holds. Now we show that \u03bc satisfies (d2). Let x, y \u2208 A. We have three cases.(b) \u21d2 (a): Assume that (b) is true. Suppose that Case 1\u2009\u2009(x \u2209 I1). Then x\u2217y \u2209 I1 or y \u2209 I1. Therefore \u03bc(x\u2217y)\u2227\u03bc(y) = 0 = \u03bc(x).Case 2\u2009\u2009(x \u2208 In \u2212 In+1 for some n \u2208 \u2115). Then x\u2217y \u2209 In+1 or y \u2209 In+1. Hence \u03bc(x\u2217y) \u2264 tn or \u03bc(y) \u2264 tn. Therefore \u03bc(x\u2217y)\u2227\u03bc(y) \u2264 tn = \u03bc(x). Case 3\u2009\u2009(x \u2208 \u22c2{In : n \u2208 \u2115}). It is obvious.\u03bc is a fuzzy ideal of \ud835\udc9c. This contradicts our assumption. Thus \ud835\udc9c is Artinian.Thus \ud835\udc9c be a BN-algebra. If, for every fuzzy ideal \u03bc of \ud835\udc9c, Im\u2061(\u03bc) is a finite set, then \ud835\udc9c is Artinian.Let The following example shows that the converse of p be a prime number. Set A = {z \u2208 \u2102 : zpn = 1 for some n \u2265 0}. It is known that  is the p-quasicyclic group. Define x\u2217y = xy\u22121 for all x, y \u2208 A. By \ud835\udc9c =  is a BN-algebra. Let In = {z \u2208 \u2102 : zpn = 1} for n \u2208 \u2115 \u222a {0}. It follows easily that I is an ideal of \ud835\udc9c if and only if I = A or I = In for some n \u2265 0. We have I0 = {1} \u2282 I1 \u2282 \u22ef\u2282In \u2282 \u22ef\u2282A and hence \ud835\udc9c is Artinian. Define \u03bc byI\u22121 = \u2205. Since A = \u22c3n\u2208\u2115In, \u03bc is a fuzzy set in A. By the proof of \u03bc \u2208 \u2131Id(\ud835\udc9c). However, Im\u2061(\u03bc) = {1/n : n \u2208 \u2115} is not a finite set.Let This paper begins by considering the notion of ideals in BN-algebras. We give its characterizations and introduce the concept of normal ideals investigating its properties. We also define the notion of normal congruences proving that there is a one-to-one correspondence between normal ideals and normal congruences of a BN-algebra. Moreover we obtain the isomorphism theorem for BN-algebras. Next, we define the notion of fuzzy ideals of BN-algebras giving its characterizations and providing conditions for a fuzzy set to be a fuzzy ideal. We give the relationships between ideals and fuzzy ideals of a BN-algebra and also provide the homomorphic properties of fuzzy ideals. Finally, we display characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals.The next step in studying fuzzy ideals in BN-algebras may be introducing and investigating the notions of fuzzy maximal ideals and fuzzy prime ideals of BN-algebras."}
{"text": "Application for the sequence { The sinc function is defined to beThis function plays a key role in many areas of mathematics and its applications \u20136.x = \u03c0/2.The following result that provides a lower bound for the sinc is well known as Jordan inequality :(2)sincThis inequality has been further refined by many authors in the past few years \u201335.In , it was We noticed that the lower bound in  is the fTo the best of the authors' knowledge, few results have been obtained on fractional lower bound for the sinc function. It is the first aim of the present paper to establish the following fractional lower bound for the sinc function.x \u2208 , one hasFor any m, the following Carleman type inequality holds:an\u2a7e0, n = 1,2, 3,\u2026, with 0 < \u2211n=1\u221ean < \u221e, whereIn , Yang prbn}n=1\u221e has very interesting properties. Yang [n,From a mathematical point of view, the sequence {es. Yang  and Gylles. Yang  have proIn , the autbn}n=1\u221e.As an application of n\u2a7e2, one hasFor any positive integer The proof is not based on . We firsx \u2208 \u2a7e0. Elementary calculations reveal thatt < 1/3, we haveSet equality  is equivquality =t2(\u2212x) and 16\u03c04/(3\u03c02+x2)2, we find thatx2/\u03c02 = t, 0 \u2a7d t \u2a7d 1. Consider the function f(t) defined byf(0) = 2/9 and f(1) = 0. n\u2a7e4,n\u2a7e4, we havec with  such that g(c) = 0. Using Maple, we find that on the open interval  the equation g(t) = 0 has a unique real root t1 \u2248 0.89\u22ef.By using the power series expansions of sin(ind that1+sinc(x)Hence, from  we get, a, a31)f(f((31)f = 2/\u03c0, H(1) = 1, and H is concave up, it follows thatTaking the natural log givesln\u2061H(x)=(1.Using  from (42ows thatH(x)\u2a7d2(1\u2212"}
{"text": "The inverse of (x + 1) enters into  = (1 + x)\u2192(1 + x)\u22121;expansion, (1 + x)\u22121 = 1 \u2212 x + x2 \u2212 x3 + x4 \u2212 \u22ef (|x | <1); integration, \u21d2x \u2212 x2/2 + x3/3 \u2212 x4/4 + \u22ef;term-by-term limits x = 1 minus x = 0, \u21d2(1 \u2212 1/2)+(1/3 \u2212 1/4)+(1/5 \u2212 1/6)\u22ef.taking theFor ers into  and we pm = 2:f(n)\u2236 = (\u22121)n+1/n. Note that after the integration, the limit x = 1 can be taken. Equation (x2 + 1), separating the roots \u00b11 and \u00b1i; now (1 + x2)\u22121 enters into .m = 2,4) for any integer m \u2208 N (in principle) and include (when possible) the appropriate Dirichlet character.In this paper, as said, we will generalize the constructions above  \u2261 1 + 1/2 + 1/3 + \u22ef+1/N + \u22ef+1/(3N \u2212 2) + 1/(3N \u2212 1) + 1/3N; so these sums diverge, for N arbitrarily large, and indeed as  for i = 1,2,\u2026, m as displaced sums:Also, we definemm = 1/m + 1/2m + \u22ef+1/mN = (1/m)(1 + 1/2 + \u22ef+1/N) soAnd finally, \u2211m(N) (in (mi(N) with i : {1,2,\u2026, m}:Therefore, \u2211m(N) (in ) is the Note that these sums make sense, spite diverging, because all have only positive terms and they are finite sums.N): divergences are no higher than logarithmic. We will try to be careful in subtracting two divergent series. We use a philosophy close to physics : we first truncate the series, taking a fixed upper bound N \u226b 1 . Then we subtract other series, also divergent but with a similar type of divergence : the result should be convergent (radiative corrections); see, for example, the book by Schwinger [The convergence or divergence of the above series is usually self-explained: convergence occurs always for the decreasing alternative series (Leibniz), but as the convergence is not absolute, but conditional, the ordering in the series should be maintained. The higher divergence behaves, if at all, with a factor \u221dlog\u2061 result for the case m = 2\u2009\u2009 by four (1) Redundance (Direct) Proof. \u22112(N) and \u221122(N), defined above, diverge in a known way, for fixed (and large) N and, as said, )((1) ReSo(2) Integration Proof. Trivial here, as it is already \u222b01(3) Summation Proof. We have the convergent series (so now N = \u221e) digamma function \u03a8(z)  is the ordinary, Euler's gamma function), with help of the expression [z = \u22121/2. In total we get, as expected,(4) Use of Hansen Formula. This formula, again depending on the Digamma function \u03a8(z), reads  N = \u221e)1\u221121\u2212\u221122=1n\u2032 above in \u22111(1/n\u2032(n\u2032 \u2212 1/2)) as (n\u2032 = n + 1), because n runs now from zero; hence this is equivalent to  = \u2212\u03b3 and \u03a8(1/2) = \u2212\u03b3 \u2212 2log\u2061(2))We put our alent to  with the Mathematica Computer Programs give many integrals and double sums also directly.Of the four methods, the most common and \u201ceasy\u201d is the second (integration); it can, in principle , always be used. Them = 2 case that what appears as result is the sign function f(n)\u2236 = \u2211n=1\u221e((\u22121)n+1/n); later, for some m > 2, (including m = 4) this will become a Dirichlet character: here we have, as r\u00e9sum\u00e9 (to repeat),Please note in this  m = 3 first. Now (x3 \u2212 1) = (x \u2212 1)(1 + x + x2) as the roots are +1, \u03c9 = exp\u2061\u2061(2\u03c0i/3) and conjugate \u03c9 is equivalent to a plane rotation by 120\u00b0. By direct integration, we obtain at onceWe discuss  positive integer m we obtain, after an easy calculation (\u03c9 is root of xm \u2212 1 = 0),For a generalm = 3,4, and 6, yielding (This formula  can be uyielding  and(29)I = \u222b01dx(1/(1 \u2212 Tr\u2061\u03c9x + x2)) for \u03c9 = exp\u2061\u2061(2\u03c0ir/m), for r natural between 1 and m \u2212 1. The integral is nowlater, we shall need also m = 3, the specific series is 1 \u2212 S + S2 \u2212 S3 + S4 \u2212 \u22ef, with S \u2261 x + x2; after developing, we get Coming back to 31 \u2212 \u221132 finishes the calculation for m = 3 .Hence \u2211\u03c72(3), we have\u03c72(3) =  (so 3-periodic). We can even use Dirichlet's L-functions, which will include the 1/n factors, but we refrain from doing that, as it does not illuminate the matter any further.In terms of the  Dirichle recapitulation, the series are obtained from the polynomial of roots \u22601:x = 1. In principle, the result of the series summation can be also obtained from the Hansen formula : complete solution for the m = 3 case. We quote the following four divergent series for later use, still in this m = 3 case (the limit \u2192 meaning just N \u226b 1):Finally for this m = 4, we have first the natural cyclotomic expressionm = 3, our first final result is here:\u03c72m = 4) =  being periodic mod 4 (and restricted multiplicative). Note also why do we get i = 1 and 3 in \u22114i (not 2!) mod 4: the expansion is for 1/(1 + x2) \u2248 1 \u2212 x2 + x4 \u2212 x6, and so forth, so it is with even powers only, so with only odd powers after integration!Now, for 4, \u221140, and \u221142 are automatically obtainable.We are done with this, as \u2211 second factorization of x4 \u2212 1 is obtained by separating only the x = 1 root:TheB\u2236 = (1 + x + x2 + x3) contains the x = \u22121 root, one writes B = (x + 1)(1 + x2), where the integral can be computed at once . The final result for factorization (II) is redundant result, as defines \u221141 \u2212 \u221142 computable from the above calculation (I). In fact, we end up this m = 4 case by writing the analogy to \u221144m implies only a difference calculation, as it was also the case for m = 3, prime.The nonprime structure of m = 4 case has one redundancy (two factorizations), and it is done (solved) once a single calculation is made; for example, \u222b01dx(1/(1 + x2)) = \u03c0/4.So the full solution for the m = 6. But here there are also several factorizations, as for the m = 4 case above. The simplest is (perhaps)Now it is the turn of x3) = (1 + x)(1 \u2212 x + x2), the integral is easy, with this factoring:As  = 1 + x3 yield the infinite but convergent sumThe operations INV, EXP, INT, and TAK  jump, now by three: it is due to the cubic xn3 terms in 1/(1 + x3).Notice again the redundant; we just write them:All other factorizations are therefore61, as \u221164 is directly computable , as \u221163 is again directly computable.Redundancy arises because from  one obtamputable ; so, in x6 \u2212 1) = (x3 \u2212 1)(x3 + 1) the simplest factorization (x3 + 1) = (x + 1)(1 \u2212 x + x2) yields, for P2(x) = 1 \u2212 x + x2, the double series expansion redundant calculation, as included in the (summed) \u222b01dx(1/(1 + x3)).It is remarkable here that in the original expansion  = Rotation by 72\u00b0 and \u03c92 = exp\u2061\u2061(4\u03c0i/5) = Rotation by 144\u00b0.Here the cyclotomic equation is, for We obtain easilyT \u2261 \u222b01(dx/(1 + x + x2 + x3 + x4)) can be done as the real denominator splits in two real ones and quadratic, but we abstain to write explicitly (\u03c9 = e\u03c0i/52)The integral x + x2 + x3 + x4) = \u22ef = (1 \u2212 1/2)+(1/6 \u2212 1/7)+\u22ef = \u2211n=1\u221e(1/(5n \u2212 4) \u2212 1/(5n \u2212 3)) still computable  but still too long. The final result will beThe expression for the integral is too long to be written. In summation terms, it is 1/ function Mathematica) isThe simplest summation /2 pairs of complex conjugate roots; \u201ca priori,\u201d the only integral versus series is the simplest case, generalizing the above result:This is the general trend for prime numbers p \u2212 1)/2 coincides also with the Euler number.The number  = 8 \u2212 4 = 4:We deal now with some composite numbers. Compound numbers are easier; we just add a calculation for x8 \u2212 1) = (x2 \u2212 1)(1 + x2 + x4 + x6). It is equivalent toAnother factorization is (x8 \u2212 1) = (x \u2212 1)(1 + x + x2 + \u22ef+x7), which yields \u221181 \u2212 \u221182, so the remaining \u221187 is obtained by difference . The case m = 8 is potentially resolved. Again, we refrain from elaborating.Still, a third factorization is  in the literature (like \u222bdx(1/(1 + x)) = log\u2061(2)) to very complicated cases, still feasible: the integrals have denominators factoring in quadratic ones, and the series are of the type \u2211(1/(an2 + bn + c)), computable, in principle, by means of the Hansen's formula.Asx + x2 + x3 + \u22ef+xq) we identify it with the series \u2211q+11; this is correct, but we have checked it \u201ccase by case,\u201d offering no general proof, and so forth. Also we feel that some new series might perhaps appear, whenever the quadratic components offer an integer expansion: those are two questions for the future.There are, however, some questions left in our work: for example, the series for 1/(1 +"}
{"text": "Tolypocladium are herein revised based on the most comprehensive dataset to date. Two species-level phylogenies of Tolypocladium were constructed: a single-gene phylogeny (ITS) of 35 accepted species and a multigene phylogeny  of 27 accepted species. Three new species, Tolypocladium pseudoalbum sp. nov., Tolypocladium subparadoxum sp. nov., and Tolypocladium yunnanense sp. nov., are described in the present study. The genetic divergences of four markers  among Tolypocladium species are also reported. The results indicated that species of Tolypocladium were best delimited by rpb1 sequence data, followed by the sequence data for the rpb2, tef-1\u03b1, and ITS provided regions. Finally, a key to the 48 accepted species of Tolypocladium worldwide is provided.The taxonomy and phylogeny of the genus  Tolypocladium was originally described as an anamorph genus by Gams in 1971 to accommodate three species collected from soil: T. cylindrosporum W. Gams, T. geodes W. Gams, and T. inflatum W. Gams [T. lignicola G.L. Barron, T. parasiticum G.L. Barron, and T. trigonosporum G.L. Barron, all of which were isolated from bdelloid rotifers, were added to this genus [T. nubicola and T. tundrense from soil in 1983 [Tolypocladium: T. balanoide , T. microsporum (basionym: Verticillium microsporum) and T. niveum (basionym: Pachybasium niveum). Additionally, Bissett [T. niveum were similar to those of T. inflatum. Because T. niveum precedes T. inflatum, Bissett proposed that T. inflatum be synonymized with T. niveum [T. inflatum produces cyclosporine and is the type species of the genus Tolypocladium. The name T. inflatum is also commonly accepted [T. inflatum with T. niveum [Tolypocladium is morphologically characterized by sparingly branched conidiophores, swollen phialides, and one-celled conidia borne in slimy heads. Approximately 20 species have been included in the Tolypocladium based on morphological characteristics.is genus ,3,4. Bis in 1983  and reas Bissett  noted th. niveum . Howeveraccepted . Therefo. niveum . The genTolypocladium has been discussed extensively for decades. Cordyceps sensu lato was recently reclassified into three families  and four genera  based on multigene phylogeny [Tolypocladium species fall within the Ophicordycipitaceae [Elaphocordyceps Sung and Spatafora 2007 was proposed for 23 species of the Cordyceps Fr. (1818: 316); these species parasitize the fungal genus Elaphomyces and some species of arthropods  [Elaphocordyceps species within the Ophiocordycipitaceae form a clade sister to those of the genus Ophiocordyceps. Gams established the Chaunopycnis to accommodate C. alba, which morphologically resembles Tolypocladium species in its conidiogenesis [Tolypocladium was chosen over Elaphocordyceps and Chaunopycnis as Tolypocladium is the oldest and most commonly used name [Chaunopycnis was integrated into the genus Tolypocladium. Accordingly, C. alba, C. ovalispora, and C. pustulata were renamed T. album, T. ovalisporum, and T. pustulatum, respectively [The taxonomy of hylogeny . Moleculpitaceae ,8. The gogenesis . With thTolypocladium records, including 5 varieties, are listed in the Index Fungorum . Tolypocladium balanoides, which was reassigned to Drechmeria , and Tolypocladium parasiticum, which was reassigned to Metapochonia (as Tolypocladium parasiticum), should be excluded from the Tolypocladium. However, some of these records are doubtful, because the original identifications were presumptive based on host associations or based on the morphology of only one or two ascospore stages of the asexual or sexual morph. For 16 species, no molecular data are available in the GenBank database [Tolypocladium species have a cosmopolitan distribution and a broad host range that includes bdelloid rotifers, mosquito larvae, nematodes, fireflies, beetles, cicada nymphs, batmoth larvae, macrocystic fungi, Ophiocordyceps sinensis, and even plants (as endophytes) [At present, 53 database . Tolypocophytes) ,18,19.Tolypocladium species have been widely studied due to their importance in the medicinal domain. These species can produce cyclosporine A, tolypoalbin, tolypin, cyclosporine D hydroperoxide, cylindromicin, and tolyprolinol [T. inflatum, is widely used in autoimmune disease treatment and to prevent allograft rejection [T. album [Tolypocladium sp. FKI-7981, contains a rare moiety prolinol and was the first natural product isolated from Tolypocladium species. Tolyprolinol exhibits moderate antimalarial activity without cytotoxicity or any other antimicrobial properties [prolinol ,21, all prolinol . Cyclospejection ,24,25. TT. album . TolypinT. album . Like koT. album . Tolyprooperties .Tolypocladium. Therefore, the diversity of Tolypocladium may be underestimated. In the present study, we aimed first to investigate and document the worldwide diversity of Tolypocladium fungi using our current collection of specimens and data collected over the last several years. We used comprehensive morphological and molecular phylogenetic reconstructions to identify and reevaluate our specimens. Based on these reconstructions, we herein describe and illustrate three new taxa. We then clarify the phylogenetic affinities of these new taxa using rDNA sequence analyses.Recent investigations and phylogenetic analyses have ascribed many new taxa to Tolypocladium species were collected in Kunming, Pu\u2019er, Yunnan, China. Voucher specimens and the corresponding isolated strains were deposited in the Yunnan Herbal Herbarium (YHH) and the Yunnan Fungal Culture Collection (YFCC), respectively, of Yunnan University, Kunming, China.Tolypocladium strains were isolated from soil samples, as described in our previous publication [lication . In brieMorphological studies were performed as described in our previous study . MicromoSSU-CoF and nrSSU-CoR [SSU, the primer pair LR5 and LR0R [LSU, and the primer pair EF1\u03b1-EF and EF1\u03b1-ER [tef-1\u03b1). The primer pair RPB1-5\u2032F and RPB1-5\u2032R and the primer pair RPB2-5\u2032F and RPB2-5\u2032R [rpb1 and rpb2), respectively. The ITS fragment was amplified using the primer pair ITS5 and ITS4 [Total DNA was extracted from the fungal mycelia on PDA plates or from herbarium materials using the modified CTAB procedure . The prirSSU-CoR  was usedand LR0R ,35 was u EF1\u03b1-ER ,36 was uRPB2-5\u2032R ,36 were and ITS4 .2+) , 1 \u03bcL forward primer (10 \u00b5mol/L), 1 \u03bcL reverse primer (10 \u00b5mol/L), 0.25 \u03bcL Taq DNA polymerase , 2 \u03bcL dNTP (2.5 mmol/L), 1 \u03bcL DNA template (500 ng/\u03bcL), and 17.25 \u03bcL sterile ddH2O. Amplification reactions were performed in a Bio-Rad T100 thermal cycler . The PCR cycling conditions for the amplification of nrSSU were as follows: 95 \u00b0C for 4 min; eight cycles of 94 \u00b0C for 50 s, 56 \u00b0C for 50 s, and 72 \u00b0C for 2 min, with the annealing temperature decreasing 0.5 \u00b0C/cycle; 25 cycles of 94 \u00b0C for 50 s, 52 \u00b0C for 50 s, and 72 \u00b0C for 2 min; and 72 \u00b0C for 10 min. The nucleotide sequences of ITS, nrLSU, tef-1\u03b1, rpb1, and rpb2 were amplified using the following cycling conditions: 95 \u00b0C for 4 min; eight cycles of 94 \u00b0C for 50 s, 56 \u00b0C for 50 s, and 72 \u00b0C for 70 s, with the annealing temperature decreasing 0.5 \u00b0C/cycle; 25 cycles of 94 \u00b0C for 50 s, 52 \u00b0C for 50 s, and 72 \u00b0C for 70 s; and 72 \u00b0C for 10 min. PCR products were purified using a gel extraction and PCR purification combo kit  and sequenced on an automatic sequence analyzer  using the amplification primers.The matrix for the polymerase chain reaction (PCR) was comprised of 2.5 \u03bcL PCR 10\u00d7 buffer  were aligned and manually checked using Bioedit v7.0.9 [http://paup.phylosolutions.com, accessed on 28 August 2022) [To investigate the placement of our samples within  GenBank . Individt v7.0.9 . To idenst 2022) . The resSSU, nrLSU, tef-1\u03b1, rpb1, rpb2, and ITS sequences were retrieved from GenBank, and combined with those generated in this study. Taxon information and GenBank accession numbers are given in SSU sequences of some species were excluded from the phylogenetic analyses, and gaps were treated as missing data. After alignment of the five genes individually, the alignments were concatenated. A partition homogeneity test was conducted in PAUP* 4.0a166 [tef-1\u03b1, rpb1, and rpb2) and one each for nrLSU and nrSSU [p = 0.02).Phylogenetic analyses were based on a concatenated five-gene dataset and the ITS sequences alone. nr 4.0a166 , and thend nrSSU ,43. The Maximum likelihood (ML) phylogenetic analyses were conducted using RaxML 7.0.3  with theDrechmeria W. Gams and H.-B. Jansson, Harposporium Lohde, Ophiocordyceps Petch, Purpureocillium Luangsa-Ard, Hywel-Jones, Houbraken and Samson, and Tolypocladium. Two species of Polycephalomyces Kobayasi were used as outgroups. ITS analysis was performed on Tolypocladium taxa only. Phylogenetic trees were visualized with FigTree v1.4.0 [The following taxa were included in the five-gene concatenated dataset: e v1.4.0 , edited tef-1\u03b1, rpb1, and rpb2 to assess the species boundaries of the 10 Tolypocladium species , Tolypocladium , Purpureocillium , Drechmeria , Harposporium , and Polycephalomyces   were recognized in Tolypocladium   within Osp. nov. , while Taradoxum . T. yunnsp. nov. .Purpureocillium lilacinum CBS 284.36 and Purpureocillium lilacinum NHJ 3497 were chosen as outgroup sequences. The three phylogenetic algorithms  recovered trees with similar topologies  formed an independent lineage with Tolypocladium  the minimum thresholds (p-distances) required to distinguish species within the ectively ; and (2) and ITS .Tolypocladium W. Gams, Persoonia 6(2): 185 (1971). emend. C. A. Quandt et al. IMA Fungus 5: 125 (2014).Synonyms: Chaunopycnis W. Gams, Persoonia 11: 75 (1980).Elaphocordyceps G. H. Sung and Spatafora, Stud. Mycol. 57: 36 (2007).Sexual morph: Stromata are solitary or several, simple or branched. The stipe is tough, dark-brownish to greenish, cylindrical, and abruptly to enlarging in the fertile part. The fertile part is cylindrical to clavate. Perithecia are superficial, wholly or partially immersed, ordinal or oblique in arrangement. Asci are cylindrical with a thickened ascus apex. Ascospores are usually cylindrical, multiseptate, disarticulate into part spores, and are occasionally non-disarticulating. Part spores are cylindrical.Asexual morph:\u00a0Tolypocladium-like, Chaunopycnis-like, or Verticillium-like. Conidiophores typically are short and bear whorls of phialides. Phialides often have bent necks and are usually swollen at the base. Conidia are ellipsoidal, globose, or reniform, and aggregate in small heads at the tips of the phialides.Tolypocladium pseudoalbum, H. Yu, Y. Wang and Q.Y. Dong, sp. nov., MycoBank: MB 845430.Etymology: Referring to the morphological resemblance of this species to Tolypocladium album, despite its phylogenetic dissimilarity.Type: China, Yunnan Province, Kunming City, Wild Duck Forest Park , from the soil on the forest floor, 10 August 2019, Yao Wang .Teleomorph: Unknown.Anamorph: Colonies on PDA are moderately fast-growing, attaining a diameter of 42\u201344 mm in 21 days at 22 \u00b0C. Colonies pulvinate, with high mycelial density, white or pale yellow, reverse deep yellow. Hyphae branched, smooth-walled, septate, hyaline, 1.1\u20132.7 \u03bcm wide. Cultures readily produce phialides and conidia on PDA after two weeks at room temperature. Phialides arising from aerial hyphae, solitary, 12.3\u201348.5 \u00d7 1.0\u20132.0 \u03bcm, cylindrical, tapering gradually toward the apex, neck 1.4\u20134.6 \u00d7 0.8\u20131.8 \u00b5m. Conidia hyaline, one-celled, globose to broadly ellipsoidal 1.8\u20133.4 \u00d7 1.3\u20131.9 \u03bcm. Chlamydospores present.Habitat: Soil.Known distribution: China.Additional specimens examined: China, Yunnan Province, Kunming City, Songming County, Dashao Village , from the soil on the forest floor, 12 August 2018, Yao Wang (living culture: YFCC 876).Comments: Five species are closely related to T. pseudoalbum sp. nov., i.e., T. pustulatum, T. tropicale, T. endophyticum, T. amazonense, and T. yunnanense sp. nov. This clade is characterized by cylindrical to lageniform phialides, globose to broadly ellipsoidal conidia, and primarily white colonies. The phialides of T. pseudoalbum sp. nov. (12.3\u201348.5 \u00d7 1.0\u20132.0 \u03bcm) are longer than those of T. album (3.5\u201310 \u00d7 1.0\u20131.5 \u00b5m).Tolypocladium subparadoxum H. Yu, Y. Wang and Q.Y. Dong, sp. nov., MycoBank: MB 845431.Etymology: Referring to the phylogenetic placement is closely related to T. paradoxum.Holotype: China, Yunnan Province, Pu\u2019er City, Simao District , from soil on the forest floor, 27 August 2021, Yao Wang .Teleomorph: Not observed.Anamorph: Colonies on PDA are moderately fast-growing, attaining a diameter of 36\u201338 mm in 21 days at 22 \u00b0C. Colonies flocculent, fluffy, with low mycelial density, white or pale yellow, reverse deep yellow. Hyphae smooth-walled, branched, septate, hyaline, 0.8\u20132.2 \u03bcm wide. Cultures produce phialides and conidia on PDA after two weeks at room temperature. Phialides arising from aerial hyphae, solitary, or in verticils of two to four, 5.4\u201340.1 \u00d7 0.9\u20131.8 \u03bcm, cylindrical, tapering gradually toward the apex, neck 3.2\u20135 \u00d7 0.7\u20131.2 \u00b5m. Conidia hyaline, one-celled, ellipsoidal or globose, single or aggregating in heads at the apex of phialides, 2.6\u20136.5 \u00d7 1.0\u20132.9 \u03bcm. Chlamydospores not observed.Habitat: Soil, larvae of cicada.Known distribution: China, Japan.Additional specimens examined: NBRC 106958, Niryo, Takatsuki-shi, Osaka Prefecture.Comments: Our phylogenetic analysis indicates that Tolypocladium subparadoxum sp. nov. is closely related to Tolypocladium sp. and T. paradoxum. The two strains (YFCC 879 and NBRC 106958) formed a distinct lineage. NBRC 106958 was firstly isolated from cicada in Japan by S. Ban  and subsequently isolated from soil in China (YFCC 879). Since no significant morphological differences were found between the Chinese collections and that of Japan , and minor conidia (2.3\u20134.8 \u00d7 1.9\u20135.2 \u00b5m vs 2.6\u20136.5 \u00d7 1.0\u20132.9 \u03bcm)  .Tolypocladium subparadoxum similar to T. dujiaolongae and sharing cicada host, solitary, or verticillate, cylindrical or conical phialides, globose to ovoid conidia, and conidia aggregating mostly in small heads, but the latter differs by its relatively shorter phialides (11\u201335 \u00d7 1.0\u20132.7 \u03bcm vs 5.4\u201340.1 \u00d7 0.9\u20131.8 \u03bcm) [\u20131.8 \u03bcm) . Our phyTolypocladium geodes is also similar to T. subparadoxum in their soil habitats and ellipsoidal or globose conidia. However, T. geodes has relatively shorter phialides (5.6\u201312.4 \u00d7 1.4\u20132.4 \u00b5m) and somewhat minor conidia (1.9\u20132.4 \u00d7 1.6\u20132.0 \u00b5m) [Tolypocladium yunnanense H. Yu, Y. Wang and Q.Y. Dong, sp. nov., Figure 5MycoBank: MB 845432.Etymology:Yunnanense (Lat.) refers to the type locality .Holotype: China, Yunnan Province, Kunming City, Wild Duck Forest Park , from soil on the forest floor, 12 August 2018, Yao Wang .Teleomorph: Unknown.Anamorph: Colonies on PDA are moderately fast-growing, attaining a diameter of 44\u201346 mm in 21 days at 22 \u00b0C. Colonies pulvinate, with high mycelial density, whitish to orange-yellow, reverse deep yellow. Hyphae smooth-walled, branched, septate, hyaline, 1.0\u20132.4 \u03bcm wide. Cultures produce phialides and conidia on PDA after two weeks at room temperature. Phialides are usually curved, solitary, 7.6\u201362.6 \u00d7 0.9\u20132.3 \u03bcm, cylindrical, narrowing slightly or abruptly into a neck, 3\u20134.2 \u00d7 0.5\u20131 \u00b5m. Conidia hyaline, one-celled, elliptical to subglobose, 1.2\u20132.4 \u00d7 0.9\u20131.9 \u03bcm. Chlamydospores present.Habitat: Soil.Known distribution: China.Additional specimens examined: China, Yunnan Province, Pu\u2019er City, Simao District , from soil on the forest floor, 7 October 2019, Yao Wang (living culture: YFCC 878).Comments:Tolypocladium yunnanense sp. nov. is characterized by its solitary cylindrical phialides (7.6\u201362.6 \u00d7 0.9\u20132.3 \u03bcm), elliptical to subglobose conidia (1.2\u20132.4 \u00d7 0.9\u20131.9 \u03bcm), and white colonies. The five-gene phylogenetic analysis suggested that T. yunnanense sp. nov. was closely related to five other species . Phylogenetic analyses of this clade using ITS sequences, for which more complete data were available, showed that T. yunnanense sp. nov. formed clade with T. album, T. pseudoalbum sp. nov., T. tropicale, T. amazonense, and T. endophyticum. Morphologically, Tolypocladium yunnanense sp. nov. has longer phialides than other species in this clade: Tolypocladium yunnanense sp. nov., 7.6\u201362.6 \u00d7 0.9\u20132.3 \u03bcm; T. pustulatum, 4\u201310 \u00d7 2\u20134 \u00b5m, T. tropicale, 4.6 \u00d7 1.5 \u00b5m; T. endophyticum, 4.1 \u00d7 1.6 \u00b5m; T. amazonense, 4.1 \u00d7 1.6 \u00b5m; T. pseudoalbum sp. nov., 12.3\u201348.5 \u00d7 1.0\u20132.0 \u03bcm, and T. album, 3.5\u201310 \u00d7 1.0\u20131.5 \u00b5m.Tolypocladium species worldwideKey to Sexual state observed\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..1Sexual state not observed\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..27\u00a01a. Perithecia superficial or half-immersed \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..2\u00a01b. Perithecia completely immersed\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263\u00a0T. inegoense2a. Perithecia pyriform, relatively larger, 520\u2013550 \u00d7 260\u2013280 \u00b5m, asci relatively larger, 400\u2013450 \u00d7 7\u20137.5 \u00b5m, part spores 2.5\u20133.0 \u00d7 3.0 \u00b5m, on cicada nymphs, stromata relatively longer, 14 cm long\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Elaphomyces, stromata shorter, 3.5\u20134.5 cm long\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026T. ramosum2b. Perithecia ovoid, relatively smaller, 320\u2013380 \u00d7 220\u2013280 \u00b5m, asci cylindrical, smaller, 240\u2013250 \u00d7 6 \u00b5m, not dissociate into part spores, on \u00a03a. Perithecia ellipsoid, subglobose to ovoid\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.4\u00a03b. Perithecia ampullaceous\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..26\u00a0Scarabaeidae ; soil, humus, Picea glauca, roots of Picea mariana, the surface of Mycobates sp. , the sclerotium of Ophiocordyceps gracilis \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026..T. inflatum4a. From multiple substrate/host \u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0T. flavonigrum16a. Perithecia relatively narrower, 567\u2013697 \u00d7 206\u2013248 \u00b5m, part spores smaller, 2\u20135 \u00d7 1.5\u20132 \u00b5m, stromata 1.5\u20133 cm long\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u00a0T. japonicum16b. Perithecia relatively wider, 500\u2013700 \u00d7 250\u2013350 \u00b5m, part spores larger, 10\u201318 \u00d7 2.5\u20134 \u00b5m, stromata 2.5\u20137 cm long\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u00a017a. Perithecia larger\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202618\u00a0T. virens17b. Perithecia smaller, 400 \u00d7 250 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u00a0T. longisegmentatum18a. Stromata 12 cm long, part spores very long, 40\u201365 \u00b5m long.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u00a018b. Stromata shorter than 12 cm, part spores < 40 \u00b5m long\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202619\u00a019a. Part spores \u2264 8 \u00b5m long\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.20\u00a019b. Part spores > 8 \u00b5m long\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u202622\u00a0T. intermedium20a. Asci shorter than 300 \u00b5m (240\u2013300 \u00d7 7\u20138 \u00b5m), perithecia relatively smaller (450\u2013540 \u00d7 230\u2013260 \u00b5m).\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u00a020b. Asci longer than 300 \u00b5m, perithecia larger\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u202621\u00a0T. fractum21a. Stipe slender, 0.5\u20131.0 mm thick, yellowish green to olivaceous, stromata shorter, 1.5\u20132.5 cm long, part spores, 2\u20135 \u00d7 1.5\u20132 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0T. valliforme21b. Stipe 1\u20135 mm thick, dark brown, smooth or furfuraceous, stromata 5\u20137 cm long, part spores longer, 3\u20138 \u00d7 2 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0T. delicatistipitatum22a. Perithecia < 550 \u00b5m long (480\u2013540 \u00b5m)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u00a022b. Perithecia > 550 \u00b5m long\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..23\u00a0T. miomoteanum23a. Part spores < 15 \u00b5m long (8\u201311 \u00b5m)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u00a023b. Part spores \u2265 15 \u00b5m long\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026.24\u00a024a. Part spores < 3 \u00b5m wide\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..25\u00a0T. inusitaticapitatum24b. Part spores \u2265 3 \u00b5m wide (3.0\u20134.5 \u00b5m)\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0T. capitatum25a. Fertile part olive-brown to olive-black, perithecia relatively larger, 650\u2013950 \u00d7 250\u2013420 \u00b5m, asci wider, 350\u2013540 \u00d7 10\u201312 \u03bcm, part spores cylindrical or somewhat fusoid, 8\u201325 \u00d7 2.5\u20133 \u00b5m\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u00a0T. rouxii25b. Fertile part purple-brown, blacker when older, perithecia smaller, 600\u2013750 \u00d7 200\u2013300 \u00b5m, asci slender, 350\u2013500 \u00d7 8\u201310 \u00b5m, part spores filiform, spindle-shaped, 15\u201320 \u00d7 2\u20133 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u00a0T. dujiaolongae26a. Perithecia relatively shorter, 520\u2013740 \u00d7 300\u2013330 \u03bcm, part spores cylindrical, 3\u20137 \u00d7 2\u20133 \u03bcm, on cicada nymphs\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Elaphomyces\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026T. minazukiense26b. Perithecia relatively longer, 900\u2013930 \u00d7 220\u2013250 \u00b5m, part spores fusoid, 16\u201318 \u00d7 3 \u00b5m, on \u00a027a. From multiple substrate/host\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u202628\u00a027b. From only a type of substrate/host\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.31\u00a028a. Phialides cylindrical\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202629\u00a0T. cylindrosporum28b. Phialides ellipsoidal to subglobos\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u00a0T. album29a. Colonies white, conidia globose to ovoid \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u00a029b. Colonies white to pale yellow, conidia ellipsoidal, globose or broadly ellipsoidal\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202630\u00a0T. pustulatum30a. Phialides 4\u201310 \u00d7 2\u20134 \u00b5m, conidia 2\u20133 \u00d7 1.5\u20132.5 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0T. subparadoxum30b. Phialides 5.4\u201340.1 \u00d7 0.9\u20131.8 \u03bcm, conidia larger, 2.6\u20136.5 \u00d7 1\u20132.9 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a031a. From substrate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026.32\u00a031b. On insects\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..45\u00a032a. Substrate is not fungus\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202633\u00a032b. Substrate is fungus\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202643\u00a033a. From plant tissue\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u202634 \u00a033b. From soil\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202637\u00a0T. endophyticum34a. Conidia relatively more minor \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u00a034b. Conidia larger, diam > 1.3 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u202635\u00a0T. ovalisporum35a. Conidia > 4 \u00b5m long (4.5\u20139.0 \u00d7 2.5\u20133.5 \u00b5m)\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a035b. Conidia < 4 \u00b5m long\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u202636\u00a0T. amazonense36a. Phialides 4.6 \u00b1 1.2 \u00d7 1.5 \u00b1 0.3\u00b5m, conidia spherical, larger, 2.1\u20132.2 \u00b5m diam\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u00a0T. tropicale36b. Phialide 4.6 \u00d7 1.5 \u00b5m, conidia spherical, relatively smaller, 1.5 \u00b1 0.1 \u00b5m diam\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a037a. Phialides cylindrical\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.38\u00a037b. Phialides subglobose or ellipsoidal\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202641\u00a038a. Conidia ellipsoidal, globose or broadly ellipsoidal\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u202639\u00a0T. microsporum38b. Conidia asymmetrically flattened, with a minute apical\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u00a039a. Colonies white\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..40\u00a0T. pseudoalbum39b. Colonies white or pale yellow \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u00a0T. geodes40a. Phialides shorter, 5.6\u201312.4 \u00d7 1.4\u20132.4 \u00b5m, conidia 1.9\u20132.4 \u00d7 1.6\u20132.0 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u00a0T. yunnanense40b. Phialides longer, 7.6\u201362.6 \u00d7 0.9\u20132.3 \u03bcm, conidia 1.2\u20132.4 \u00d7 0.9\u20131.9 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u00a041a. Conidia only one type\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202642\u00a0T. tundrense41b. Conidia two types  \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0T. nubicola42a. Phialides relatively longer, 4.4\u20137.8 \u00d7 1.5\u20132.7 \u00b5m, conidia cylindrical, 2.6\u20134.1 \u00d7 0.8\u20131.3 \u00b5m, colonies white to pale cream\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u00a0T. terricola42b. Phialides shorter, 2.8\u20133.5 \u00d7 2.0\u20133.0 \u00b5m, conidia broadly oval, 2.5\u20133 \u00d7 2.0\u20132.5 \u00b5m, colonies white\u2026.\u00a0Elaphomyces\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026T. guangdongense43a. On \u00a0Ophiocordyceps sinensis\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20264443b. From \u00a0T. reniformisporum44a. Conidia reniform, 1.0\u20133.2 \u00d7 0.7\u20131.6 \u00b5m, phialides 3.4\u201310.6 \u00d7 1.1\u20133.8 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u00a0T. sinense44b. Conidia spherical, 1.4\u20133.6 \u00b5m diam, phialides 7.6\u201319.4 \u00d7 2.9\u20133.6 \u00b5m.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0T. extinguens45a. On mosquito larvae, conidia two types \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. \u00a045b. On bdelloid rotifers, conidia only one type\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u202646\u00a0T. lignicola46a. Phialides thicker, 4\u20138 \u00d7 3\u20134.5 \u00b5m, conidia circular, 2.5\u20133.2 \u00d7 1.5\u20132.0 \u00b5m, colonies pure white\u2026\u2026\u2026T. trigonosporum46b. Phialides slender, 4.8\u20139.8 \u00d7 1.4\u20133.5 \u00b5m, conidia like an equilateral triangle or less ellipsoidal, 2\u20133 \u00d7 1.3\u20131.7 \u00b5m, colonies white or pale yellow\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Tolypocladium is one of the most diverse fungal groups in terms of shape, substrate or host, and habitat range. Many new species have recently been added to Tolypocladium [T. pseudoalbum sp. nov., T. subparadoxum sp. nov., and T. yunnanense sp. nov.) based on phylogenetic analyses and morphological characteristics. Phylogenetically, these three species fell within the Tolypocladium clade, while morphologically all three species possessed cylindrical phialides and ellipsoidal or globose conidia. It is challenging to distinguish species of Tolypocladium based only on morphological characteristics, because several species in this genus are morphologically cryptic [ocladium ,13,14,73 cryptic ,11. Sexu cryptic . HoweverTolypocladium play a significant role in a variety of artificial and wild ecosystems and may participate in antifungal, host\u2013fungi, and insecticidal interactions [Tolypocladium based on host associations or morphology [Tolypocladium was developed based on morphological characteristics. However, the advent of molecular biology, which was an important scientific milestone, revolutionized the taxonomic characterization of this genus. Over the last few decades, the number of accepted species in Tolypocladium has doubled.Species of ractions ,77. Manyrphology ,12. Overrphology . In mostTolypocladium were included in the key developed in this study. However, because the sequence loci for many of these taxa were incomplete, only 27 species were included in the multigene phylogenetic analyses , Tolypocladium species are mainly known from insects [Tolypocladium species in the soil and in plant roots. Recently, Tolypocladium species in Chinese soils were surveyed, but no new species were identified.athogens ,12. Beca insects , and few"}
{"text": "We also introduce and study a new type of generalized open sets in GTSs, called \u03b2-open sets, and apply them to obtain more properties of \u03bc-\u03b2-Lindel\u00f6f sets.We introduce and study  The primary purpose of this article is to introduce and study \u03bc on a non-empty set X is a collection of subsets of X such that \u03bc is closed under non-empty arbitrary unions. For a GT \u03bc on X, the pair \u03bc will be called \u03bc-open sets, and a subset A of \u03bc-closed if \u03bc-open. A space A is a subset of a space \u03bc-closure of A\u03bc-closed sets containing A, and the \u03bc-interior of A\u03bc-open sets contained in A. A GT \u03bc on X is called a strong GT \u03bc-space \u03bc is a strong GT. Strong GTs were introduced by Lugojan A generalized topology (GT) A will stand, respectively, for the closure of A in X and the interior of A in X. A subset A of a topological space \u03b2-open \u03b1-open \u03b2-open, \u03b1-open) subsets of a topological space \u03bc is a strong GT on X.If A, A.Throughout this article, the set-theoretic framework is ZFC, that is the Zermelo-Fraenkel system of axioms (ZF) together with the Axiom of Choice (AC); for a set 2We recall the following definitions and facts for their importance in the material of our article.Definition 2.1A be a subset of a space A is called\u03bc-semi-open if (i) \u03bc-preopen if (ii) \u03bc-\u03b2-open if (iii) \u03bc-\u03b1-open if (iv) \u03bc-semi-open sets by \u03c3, the class of \u03bc-preopen sets by \u03c0, the class of \u03bc-\u03b2-open sets by \u03b2, and the class of \u03bc-\u03b1-open sets by \u03b1. It was pointed out in \u03c3, \u03c0, \u03b1, and \u03b2 is a GT. However, it is easy to see that for a space \u03c3 and \u03b2 is a strong GT, while \u03c0 and \u03b1 need not.For a space Proposition 2.2Letbe a space. Then(i);(ii);(iii);(iv).Definition 2.3A of a \u03bc-space \u03bc-Lindel\u00f6f if any cover of A by \u03bc-open subsets of X has a countable subcover.\u03bc-space \u03bc-Lindel\u00f6f if any cover of X by \u03bc-open sets has a countable subcover.(ii) A Definition 2.4A of a topological space A by preopen subsets of X has a countable subcover.(i) A subset X by preopen subsets of X has a countable subcover.(ii) A topological space Definition 2.5A of a \u03bc-space \u03bc-Lindel\u00f6f if any cover of A by \u03bc-preopen subsets of X has a countable subcover, that is A is \u03bc-space \u03bc-Lindel\u00f6f if any cover of X by \u03bc-preopen sets has a countable subcover, that is (ii) A Definition 2.6A of a topological space XA by semi-open subsets of X has a countable subcover.(i) A subset X by semi-open subsets of X has a countable subcover.(ii) A topological space Definition 2.7A of a space \u03bc-semi-Lindel\u00f6f relative to XA by \u03bc-semi-open subsets of X has a countable subcover, or equivalently A is A is \u03bc-semi Lindel\u00f6f to mean A is \u03bc-semi Lindel\u00f6f relative to X.(i) A subset \u03bc-semi-Lindel\u00f6f X by \u03bc-semi-open sets has a countable subcover, or equivalently (ii) A space Proposition 2.8Let A be a subset of a topological space X. Then(i)A is semi-open if and only if there exists an open set U such that.(ii)A is preopen if and only if, where U is open and D is dense.(iii)A is \u03b2-open if and only if, where S is semi-open and D is dense.3\u03bc-\u03b2-Lindel\u00f6f sets in GTSs.This section is mainly devoted to introducing and studying Definition 3.1A of a topological space \u03b2-Lindel\u00f6f relative to A by \u03b2-open subsets of X has a countable subcover.\u03b2-Lindel\u00f6f if any cover of X by \u03b2-open subsets of X has a countable subcover.(ii) A topological space Definition 3.2A of a space \u03bc-\u03b2-Lindel\u00f6f if any cover of A by \u03bc-\u03b2-open subsets of X has a countable subcover, that is A is (i) A subset \u03bc-\u03b2-Lindel\u00f6f if any cover of X by \u03bc-\u03b2-open sets has a countable subcover, that is (ii) A space Remark 3.3\u03bc-\u03b2-Lindel\u00f6f subsets of a space \u03bc-\u03b2-Lindel\u00f6f.It is easy to see that the countable union of The following two remarks follow from Remark 3.4A be a subset of a space Let A is \u03bc-\u03b2-Lindel\u00f6f if and only if A is \u03b2-Lindel\u00f6f;(i) A is \u03bc-\u03b2-Lindel\u00f6f if and only if A is \u03b2-Lindel\u00f6f;(ii) A is \u03bc-\u03b2-Lindel\u00f6f if and only if A is \u03b2-Lindel\u00f6f.(iii) Remark 3.5A is a subset of a topological space If The next remark follows from Remark 3.6A be a subset of a space Let A is \u03bc-\u03b2-Lindel\u00f6f if and only if A is strongly (i) A is \u03bc-\u03b2-Lindel\u00f6f if and only if A is strongly (ii) \u03bc-\u03b2-Lindel\u00f6f sets and some other related types.Now, we will study the relationships between Remark 3.7A of a space \u03bc-\u03b2-Lindel\u00f6f, then A is \u03bc-semi-Lindel\u00f6f; in particular, if a space \u03bc-\u03b2-Lindel\u00f6f, then \u03bc-semi-Lindel\u00f6f. However, the converse need not be true even for topological spaces as it can be easily seen from each of the following two examples.Clearly, if a subset Example 3.8X be an uncountable set equipped with the co-countable topology. Then it follows from X is open, but X is Lindel\u00f6f, so X is semi-Lindel\u00f6f. On the other hand, since the dense subsets of X are the uncountable subsets of X, it follows from X are the uncountable subsets of X. As X is uncountable, there exists a disjoint family X by preopen sets and X is not strongly Lindel\u00f6f, and thus, X is not \u03b2-Lindel\u00f6f.Let Example 3.9X be an uncountable set equipped with the indiscrete topology. Then it follows from X is open, but X is Lindel\u00f6f, so X is semi-Lindel\u00f6f. On the other hand, since every singleton subset of X is preopen, it is clear that X is not strongly Lindel\u00f6f, and thus, X is not \u03b2-Lindel\u00f6f.Let Remark 3.10A of a \u03bc-space \u03bc-\u03b2-Lindel\u00f6f, then A is strongly \u03bc-Lindel\u00f6f, in particular, if a \u03bc-space \u03bc-\u03b2-Lindel\u00f6f, then \u03bc-Lindel\u00f6f. However, the converse need not true even for topological spaces as it can be easily seen from the following example.Clearly, if a subset Example 3.11X be an uncountable discrete space, X are unchanged and a basic open neighborhood of p has the form A is a countable subset of X. Then it is easy to see that X and A is an uncountable subset of X, is semi-open in X is uncountable, there exists a disjoint family \u03b2-Lindel\u00f6f.Let A of a space \u03bc-\u03b2-closed \u03bc-\u03b2-open. The following proposition tells that a \u03bc-\u03b2-closed subset of a \u03bc-\u03b2-Lindel\u00f6f space is \u03bc-\u03b2-Lindel\u00f6f; the straightforward proof is omitted.Recall that a subset Proposition 3.12Let A be a \u03bc-\u03b2-Lindel\u00f6f subset of a spaceand B be a \u03bc-\u03b2-closed subset of X. Thenis \u03bc-\u03b2-Lindel\u00f6f. In particular, a \u03bc-\u03b2-closed subset A of a \u03bc-\u03b2-Lindel\u00f6f spaceis \u03bc-\u03b2-Lindel\u00f6f.Proposition 3.13If every proper \u03bc-\u03b2-closed subset of a spaceis \u03bc-\u03b2-Lindel\u00f6f, thenis \u03bc-\u03b2-Lindel\u00f6f.ProofX by \u03bc-\u03b2-open subsets of X. Choose \u03bc-\u03b2-closed subset of X, thus by assumption, \u03bc-\u03b2-Lindel\u00f6f, so there exist X is \u03bc-\u03b2-Lindel\u00f6f.\u00a0\u25a1Let \u03bc-\u03b2-Lindel\u00f6f sets in a subspace; to proceed, we recall the following definition.As for the next issue, we characterize Definition 3.14A be a non-empty subset of a space A is the collection A is the GTS Proposition 3.15Let B be a non-empty subset of a spaceand. Then A is \u03bc-\u03b2-Lindel\u00f6f if and only if A is-Lindel\u00f6f.ProofNecessity. Suppose that A by A by \u03bc-\u03b2-open sets, but A is \u03bc-\u03b2-Lindel\u00f6f, so there exist A is Sufficiency. Suppose that A by \u03bc-\u03b2-open sets. Then A, but A is A is \u03bc-\u03b2-Lindel\u00f6f.\u00a0\u25a1Corollary 3.16Let A be a non-empty subset of a space. Then A is \u03bc-\u03b2-Lindel\u00f6f if and only if A is-Lindel\u00f6f.The proof of the next proposition is straightforward, and thus omitted.Proposition 3.17A non-empty subset A of a spaceis \u03bc-\u03b2-Lindel\u00f6f if and only if for every familyof \u03bc-\u03b2-closed sets having the property that for every non-empty countable subfamilyof,, then.\u03bc-\u03b2-Lindel\u00f6f sets using filter bases; to proceed, we recall the following definition.Now, we will characterize Definition 3.18X is called a strong filter base on X if whenever X is called a maximal strong filter base on X if whenever X with (ii) A strong filter base X is contained in a maximal strong filter base on X.It was pointed out in Definition 3.19\u03bc-\u03b2-open subset U of X such that \u03bc-\u03b2-open subset U of X such that A filter base Remark 3.20Let x, then x;(i) If x if and only if x.(ii) If \u03bc-\u03b2-Lindel\u00f6f sets using filter bases, the straightforward proof is omitted.The next proposition meets our goal for characterizing Proposition 3.21For a non-empty subset A of a space, the following are equivalent:(i) A is \u03bc-\u03b2-Lindel\u00f6f;(ii) Every maximal strong filter base on X, each of whose members meet A,-converges to some point of A;(iii) Every strong filter base on X, each of whose members meet A,-accumulates at some point of A.\u03bc-\u03b2-Lindel\u00f6f spaces; to proceed, we recall the following.At the end of this section, we study a sum property of Definition 3.22\u03bc of subsets of Remark 3.23\u03bc be as in \u03bc is a GT on Remark 3.24\u03bc-open and \u03bc-closed for each Proposition 3.25Letbe a-space for each, and letbe the generalized topological sum of. Thenis \u03bc-\u03b2-Lindel\u00f6f if and only ifis \u03bc-\u03b2-Lindel\u00f6f for eachand \u039b is countable.ProofNecessity. By \u03bc-\u03b2-closed for each \u03bc-\u03b2-Lindel\u00f6f, it follows by \u03bc-\u03b2-Lindel\u00f6f for each \u03bc-\u03b2-open for each \u03bc-\u03b2-open sets, but \u03bc-\u03b2-Lindel\u00f6f, so, \u039b is countable.Sufficiency. Follows from 4\u03b2-open sets. This study will be useful later to obtain further properties of \u03bc-\u03b2-Lindel\u00f6f sets in GTSs.In this section, we introduce and study a new type of generalized open sets in GTSs, called Definition 4.1A be a subset of X. Then A is called \u03b2-open if whenever \u03bc-\u03b2-open set x such that A is called \u03b2-closed if \u03b2-open. The collection of all \u03b2-open subsets of X will be denoted by Let Proposition 4.2(i) If A is a \u03bc-\u03b2-open subset of a space, then A is-\u03b2-open, that is.(ii) Ifis a space, thenis an-space.(iii) Let A be a subset of a space. Then A is-\u03b2-open if and only if A is-\u03b2-open.ProofA be \u03bc-\u03b2-open and A is \u03b2-open.(i) Let \u03b2-open, that is \u03b2-open for each \u03b2-open, there exists a \u03bc-\u03b2-open set x such that \u03b2-open, that is X is \u03bc-\u03b2-open, it follows by (i) that X is \u03b2-open, that is (ii) Clearly, \u2205 is A is \u03b2-open and let \u03bc-\u03b2-open set U containing x such that U is \u03b2-open. By U is \u03b2-open. Thus, A is \u03b2-open. Conversely, suppose that A is \u03b2-open and let \u03b2-open set U containing x such that U is \u03b2-open. Thus, there exists a \u03bc-\u03b2-open set V containing x such that A is \u03b2-open.\u00a0\u25a1(iii) Suppose that The following example shows that if Example 4.3X is an uncountable set and \u03bc is the co-countable topology on X. Since the dense subsets of X are the uncountable subsets of X and since every semi-open subset of X is open, it follows from \u03bc-\u03b2-open subsets of X are the uncountable subsets of X. We will show now that X. Let A and B be any uncountable subsets of X such that A and B are \u03bc-\u03b2-open, and thus \u03b2-open by \u03b2-open because if X such that X.Let Remark 4.4A is a countable subset of a space A is \u03b2-closed. However, the converse need not be true, in general, even for topological spaces as the following example tells.It is easy to see that if Example 4.5X be an uncountable set equipped with the discrete topology \u03bc. Then clearly every subset of X is \u03bc-\u03b2-open; thus by X is \u03b2-open. Now let A be a subset of X such that \u03b2-closed which is uncountable.Let The next example shows that the converse of Example 4.6X be an uncountable set and A is a non-empty countable subset of X. Then by \u03b2-open; however, \u03bc-\u03b2-open becauseLet \u03b2-open sets in terms of \u03bc-\u03b2-open sets and countable sets.The following proposition characterizes Proposition 4.7Letbe a space and A be a subset of X. Then A is-\u03b2-open if and only if whenever, there exists a \u03bc-\u03b2-open setcontaining x and a countable subsetof X such that.ProofNecessity. Let A be \u03b2-open and \u03bc-\u03b2-open set x such that Sufficiency. Let \u03bc-\u03b2-open set x and a countable subset X such that A is \u03b2-open.\u00a0\u25a1Corollary 4.8Letbe a space and A be an-\u03b2-closed subset of X. Thenfor some \u03bc-\u03b2-closed subset B of X and some countable subset C of X.ProofA be an \u03b2-closed subset of X. Then \u03b2-open. If \u03bc-\u03b2-open set x and a countable subset X such that B is \u03bc-\u03b2-closed, and Let \u03b2-open \u03bc-\u03b2-Lindel\u00f6f set.The next proposition describes the shape of an Proposition 4.9Let A be an-\u03b2-open subset of a space. If A is \u03bc-\u03b2-Lindel\u00f6f, thenfor some \u03bc-\u03b2-open subset B of X and some countable subset C of X.ProofA is \u03b2-open, it follows that for each \u03bc-\u03b2-open set x such that A is \u03bc-\u03b2-Lindel\u00f6f, so there exist B is \u03bc-\u03b2-open and C is countable.\u00a0\u25a1Since 5\u03bc-\u03b2-Lindel\u00f6f sets in GTSs.The primary purpose of this section is to provide more properties and mapping properties of Proposition 5.1Let A be a subset of a space. Then A is \u03bc-\u03b2-Lindel\u00f6f if and only if A is-Lindel\u00f6f.ProofNecessity. Suppose that A by \u03bc-\u03b2-open set x such that A by \u03bc-\u03b2-open sets, but A is \u03bc-\u03b2-Lindel\u00f6f, so there exist A, it is covered by a countable subcollection A is Sufficiency. Follows from Proposition 5.2Let A be a \u03bc-\u03b2-Lindel\u00f6f subset of a spaceand B be an-\u03b2-closed subset of X. Thenis \u03bc-\u03b2-Lindel\u00f6f. In particular, an-\u03b2-closed subset A of a \u03bc-\u03b2-Lindel\u00f6f spaceis \u03bc-\u03b2-Lindel\u00f6f.Proof\u03bc-\u03b2-open sets. Then A by \u03b2-open sets, but A is \u03bc-\u03b2-Lindel\u00f6f, so it follows from \u03bc-\u03b2-Lindel\u00f6f.\u00a0\u25a1Suppose that \u03bc-\u03b2-Lindel\u00f6f sets in GTSs.The next issue is to study several mapping properties of \u03ba-open set is \u03bc-open, and called \u03bc-closed set is \u03ba-closed.Recall that a function Proposition 5.3Letbe a-continuous function, whereandare spaces. If A is \u03bc-\u03b2-Lindel\u00f6f, thenis \u03ba-\u03b2-Lindel\u00f6f.Proof\u03ba-\u03b2-open sets. Then A, but f is \u03b2-open for each A is \u03bc-\u03b2-Lindel\u00f6f, it follows from \u03ba-\u03b2-Lindel\u00f6f.\u00a0\u25a1Suppose that Corollary 5.4Letbe a-continuous function, whereandare spaces. If A is \u03bc-\u03b2-Lindel\u00f6f, thenis \u03ba-\u03b2-Lindel\u00f6f.\u03bc-\u03b2-Lindel\u00f6f spaces; to proceed, we recall the following.Now, we are ready to study a product property of Proposition 5.5Letandbe GTSs, and let. Thenis a GT on.Remark 5.6\u03bb from (i) (ii) Lemma 5.7Letandbe spaces, and \u03bb be the generalized product topology on. Then the projection(resp.) is-continuous (resp.-continuous).ProofA be a \u03bc-\u03b2-open subset of X. Then \u03bb-\u03b2-open. Now by \u03bb-\u03b2-open.\u00a0\u25a1We will show that the projection Corollary 5.8Letandbe spaces, and \u03bb be the generalized product topology on. Ifis \u03bb-\u03b2-Lindel\u00f6f, thenis \u03bc-\u03b2-Lindel\u00f6f andis \u03ba-\u03b2-Lindel\u00f6f.ProofFollows from Proposition 5.9Letbe a-closed function, whereandare spaces. If for each,is \u03bc-\u03b2-Lindel\u00f6f, thenis \u03bc-\u03b2-Lindel\u00f6f whenever A is \u03ba-\u03b2-Lindel\u00f6f.Proof\u03bc-\u03b2-open sets. Then it follows by assumption that for each \u03bc-\u03b2-open. Let \u03b2-open as f is A by \u03b2-open sets, but A is \u03ba-\u03b2-Lindel\u00f6f, so it follows from \u03bc-\u03b2-Lindel\u00f6f.\u00a0\u25a1Suppose that Corollary 5.10Letbe a-closed function, whereandare spaces. If for each,is \u03bc-\u03b2-Lindel\u00f6f, thenis \u03bc-\u03b2-Lindel\u00f6f whenever A is \u03ba-\u03b2-Lindel\u00f6f.This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.Mohammad Shawqi Sarsak: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.The authors declare no conflict of interest."}
{"text": "Correction to: Microchimica Acta (2021) 189:14https://doi.org/10.1007/s00604-021-05113-4\u2212\u20091 instead of 100\u00a0\u03bcg\u22c5mL\u2212\u20091. The corrected line should read as: \"Briefly, using the lateral flow reagent dispenser, control lines were drawn using 200\u00a0\u03bcg\u22c5mL\u2212\u20091 of mouse IgG and test lines were drawn using a solution consisting of Ab01680 antibodies (1000\u00a0\u03bcg\u22c5mL\u2212\u20091) and ACE2 protein (100\u00a0\u03bcg\u22c5mL\u2212\u20091) at a rate of 0.125\u00a0mL\u22c5min\u2212\u20091 on nitrocellulose membrane\".In this article, there is an error in the concentration of Ab01680 antibodies used. It should be 1000\u2009\u03bcg\u22c5mLThe original article has been corrected."}
{"text": "At the end, an example is given to illustrate the theoretical result.This paper describes the existence and uniqueness of the solution,  Many physical problems can be expressed in mathematical models using differential equations. Differential equations enable us to study the rapid changes in physical problems, for example, blood flows, river flows, biological systems, control theory, and mechanical systems with impact. A system of differential equations with impulses can be used to model several above-listed problems. A few existing results for a general class of impulsive systems were discussed by Ahmad . The theH1 the group and by H2 the metric group with a metric \u03b4 and a constant \u03bd > 0. The problem is to study if there exists \u03bb > 0 satisfies for every h : H1\u27f6H2 such thatf : H1\u27f6H2 that satisfiesAt the University of Wisconsin, Ulam  proposedf(x+y)=f(x)+f(y), and their solutions have been discussed in several spaces. A linear transformation is a solution of a linear functional equation. By considering the H1 and H2 as Banach spaces, Hyers [The linear functional equations, of the form s, Hyers  discusses, Hyers  and Rasss, Hyers  extendedIn 2012, Wang et al.  studied \u03b2-Hyers\u2013Ulam\u2013Rassias stability and generalized \u03b2-Hyers\u2013Ulam\u2013Rassias stability of the impulsive difference system of the formH, B \u2208 \u211dn\u00d7n, f \u2208 \u2102 and \u0398n \u2208 B space of bounded and convergent sequences, \u2124+={0,1,2,\u2026} and \ud835\udd4f=\u211dn, I={0,1,2,\u2026, n}. In fact, we are presenting a discrete version of the work given in [\u03b2-Hyers\u2013Ulam\u2013Rassias stability was discussed for differential equations. With the help of [\u03b2-Hyers\u2013Ulam\u2013Rassias stability of the difference equation.In this paper, we will explain the given in , in whic help of , 18, we n-dimensional Euclidean space will be denoted by \u211dn along with the vector norm \u2016\u00b7\u2016, and n \u00d7 n matrices with real-valued entries will be denoted by \u211dn\u00d7n. The vector infinite norm is defined as \u2016v\u2016=maxi\u2264n1\u2264|vi|, and the matrix infinite-norm is given as \u2016A\u2016=maxi\u2264n1\u2264\u2211j=1n|aij| where v \u2208 \u211dn and A \u2208 \u211dn\u00d7n, also vi and aij are the elements of the vector v and the matrix A, respectively. \u2102 will be the space of all convergent sequences from I to \ud835\udd4f with norm \u2016v\u2016=supn\u2208I\u2016vn\u2016. We will use \u211d, \u2124, and \u2124+ for the set of all real, integer, and nonnegative integer numbers, respectively. The next lemma is a basic result about the solution of the difference system  is called \u03b2-norm, with 0 < \u03b2 \u2264 1, where \ud835\udd4d is a vector space over the field K, if the function satisfied the following properties:\u03b2=0 if and only \u210b=0\u2016\u210b\u2016\u03ba\u210b\u2016\u03b2=|\u03ba|\u03b2\u2016\u210b\u2016\u03b2, \u2009foreachk\u2009\u03ba \u2208 K\u2009and\u2009\u210b \u2208 \ud835\udd4d\u20161\u2016\u03b2 \u2264 \u2016\u210b\u2016\u03b2+\u2016\u210b1\u2016\u03b2, \u2009forall\u2009\u210b, \u2009\u210b \u2208 \ud835\udd4d\u2016\u210b+\u210bA function \u2016\u00b7\u2016\ud835\udd4d, \u2016\u00b7\u2016\u03b2) is said to be \u03b2-norm space.And . A sequence \u0398n will be an \u03f5-approximate solution of \u0398n\u03b2-Hyers\u2013Ulam\u2013Rassias stable if for every \u03f5-approximate solution Yn of system  satisfies  and a sequence hk, k \u2208 M satisfyingFrom , it is catisfies  if and oThe solution of n \u2265 0 withfor any \u03b3k by \u03b3kn, thenIf we replace G1: for f, Ik : \u2124X\u00d7X+\u00d7\u27f6\ud835\udd4f, k \u2208 I, there exist constants \u2112f > 0 and \u2112Ik > 0, such thatG2:G2\u2217:To describe the uniqueness and existence of the solution of system , we willG1, G2, and G2\u2217 are held, then system .If assumptions \ud835\udc9c : \u2102\u27f6\u2102 byDefine \u2102, we haveNow, for \u0398, \u0398\u2032 \u2208 This implies that\ud835\udc9c is a contraction map using the Banach contraction principle, we say that system  such that\u03b2-Hyers\u2013Ulam\u2013Rassias stable over discrete bounded interval, if G1, G2 and G3 are satisfied.System  is \u03b2-HyeThe solution of system  is as foYn be the solution of inequality r \u2264 (xr+yr), x, y \u2265 0, \u2009for\u2009any\u2009r \u2208 {0,1}, whereNow,\u03b2-Hyers\u2013Ulam\u2013Rassaias stable. \u25a1Hence, system  is \u03b2-Hye\u03b2-Hyers\u2013Ulam\u2013Rassias stability on an unbounded discrete interval, we must need the following assumptions:To explain G4: the operators family \u2016Hn\u2212i\u2016 \u2264 Me\u03c9(n \u2212 i) and \u2016T(n \u2212 nk)\u2016 \u2264 Me\u03c9(n \u2212 nk).G1 and G3 \u2212 G8 are holds, then system \u0398nYn be the solution of inequality  is 1/2\u2212 Hyers\u2013Ulam\u2013Rassias stable with respect to Z+ and Also, \u00f9\u03b2-Hyers\u2013Ulam\u2013Rassias stability of difference equations has been considered as one of the important topics of the literature, in which different types of conditions have been used in the form of inequalities, and most results have been obtained through discrete Gronwall inequality. In this paper, we have investigated the existence and uniqueness of the solution through the Banach contraction principle and \u03b2-Hyers\u2013Ulam\u2013Rassias stability of the impulsive difference system with the help of Gronwall inequality.Nowadays, studies on the qualitative behavior of impulsive difference equations have a significant contribution to the literature. In particular, the discussion regarding the"}
{"text": "Scientific Reportshttps://doi.org/10.1038/s41598-022-18138-3, published online 30 September 2022Correction to: The original version of this Article contained an error in Figure 2, where the y-axis label \u201cRange mm\u201d in panel (A) was incorrectly described as \u201cRange nm\u201d, and the y-axis units in panel (B) \u201c2.5\u201d, \u201c2.0\u201d, \u201c1.5\u201d, \u201c1.0\u201d, \u201c0.5\u201d and \u201c0.0\u201d were incorrectly described as \u201c25\u201d, \u201c20\u201d, \u201c15\u201d, \u201c10\u201d, \u201c5\u201d and \u201c0\u201d, respectively. The original Figure\u00a0The original Article has been corrected."}
{"text": "Perenniporia is an important genus of Polyporaceae. In its common acceptation, however, the genus is polyphyletic. In this study, phylogenetic analyses on a set of Perenniporia species and related genera were carried out using DNA sequences of multiple loci, including the internal transcribed spacer (ITS) regions, the large subunit nuclear ribosomal RNA gene (nLSU), the small subunit mitochondrial rRNA gene (mtSSU), the translation elongation factor 1-\u03b1 gene (TEF1) and the b-tubulin gene (TBB1). Based on morphology and phylogeny, 15 new genera, viz., Aurantioporia, Citrinoporia, Cystidioporia, Dendroporia, Luteoperenniporia, Macroporia, Macrosporia, Minoporus, Neoporia, Niveoporia, Rhizoperenniporia, Tropicoporia, Truncatoporia, Vanderbyliella, and Xanthoperenniporia, are proposed; 2 new species, Luteoperenniporia australiensis and Niveoporia subrusseimarginata, are described; and 37 new combinations are proposed. Illustrated descriptions of the new species are provided. Identification keys to Perenniporia and its related genera and keys to the species of these genera are provided. Perenniporia Murrill, typified by P. medulla-panis (Jacq.) Donk, is a large and cosmopolitan genus. The genus has been redefined by Decock and Stalpers [Stalpers , who desStalpers ,3.Perenniporia has been intensively studied and the number of species has increased significantly. Many new species were published based on morphological characters [In the last few decades, aracters ,12,13,14aracters ,27,28,29Perenniporia. A recent trend is to make the morphologically distinct species or morphologically homogeneous alliance separate from Perenniporia. For instance, Perenniporiella Decock & Ryvarden [Perenniporia based on non-truncate basidiospores. This was later confirmed by Robledo et al. [Hornodermoporus, Truncospora, and Vanderbylia are usually treated as synonyms of Perenniporia [Perenniporia s. s. clade, each morphologically homogenous [These multiple additions considerably enlarged the morphological concept of Ryvarden  was sepao et al.  by phylonniporia ,32,33,34mogenous ,20,26,31Perenniporia is polyphyletic, and some monophyletic clades have been separated from Perenniporia s. l. and recognized as independent genera [P. narymica (Pil\u00e1t) Pouzar was segregated into Yuchengia B.K. Cui & K.T. Steffen [Perenniporia minutissima (Yasuda) T. Hatt. & Ryvarden from Perenniporia as Perenniporiopsis minutissima (Yasuda) C.L. Zhao based on the phylogenetic analyses, as well as the morphological differences of rigidly osseous basidiocarps when there are dry and large basidiospores. Cui et al. [Perenniporia hattorii Y.C. Dai and B.K. Cui from Perenniporia into Amylosporia B.K. Cui, C.L. Zhao & Y.C. Dai because of its amyloid skeletal hyphae and basidiospores, and P. subadusta (Z.S. Bi & G.Y. Zheng) Y.C. Dai from Perenniporia into Murinicarpus B.K. Cui & Y.C. Dai on account of its stipitate basidiocarps and cystidia in the hymenium. Chen et al. [Perenniporia subacida (Peck) Donk formed a well-supported lineage that is distinct from the Perenniporia s. s. clade and proposed that P. subacida should be treated in a new genus named Poriella C.L. Zhao.The current phylogenetic analysis confirmed that t genera ,35,36,37Amylosporia, Murinicarpus, Perenniporiopsis, Poriella, and Yuchengia [Perenniporia species are still doubtful, and some species should be separated from Perenniporia. In order to clarify the taxonomy and phylogeny of Perenniporia s. l., reliable specimens and sequences were studied using morphological methods and phylogenetic analyses of ITS, nLSU, mtSSU, TEF1, and TBB1.Although some monophyletic lineages have been assigned a new genus name  and the Institute of Applied Ecology, Chinese Academy of Sciences, China (IFP). Macro-morphological descriptions were based on field notes. Special color terms followed the Petersen protocol . Micro-mhttp://www.biology.duke.edu/fungi/mycolab/primers.htm, accessed on 12 October 2021). The mtSSU regions were amplified with the primer pairs MS1 and MS2 [A CTAB rapid plant genome extraction kit-DN14  was used to extract total genomic DNA from dried specimens and to perform the polymerase chain reaction (PCR) according to the manufacturer\u2019s instructions, with some modifications ,41. The  and MS2 . Part of and MS2 . TBB1 wa and MS2 . The PCR and MS2 . The PCRHeterobasidion annosum (Fr.) Bref. and Stereum hirsutum (Willd.) Pers. were used as outgroups for the ITS + nLSU analysis [Fomitopsis pinicola (Sw.) P. Karst. and Daedalea quercina (L.) Pers. were selected as outgroups for the ITS + nLSU + mtSSU + TEF1 + TBB1 analysis. All sequences were aligned in MAFFT 7 [http://mafft.cbrc.jp/alignment/server/, accessed on 11 June 2022) and manually adjusted in BioEdit [analysis , Fomitop MAFFT 7   and involved 1000 ML searches under the GTRGAMMA model. Only the maximum likelihood best tree from all searches was kept. In addition, 1000 rapid bootstrap replicates were run with the GTRCAT model to assess the reliability of the nodes.ML studies were conducted with RAxML-HPC through the Cipres Science Gateway  . Four Mahttp://tree.bio.ed.ac.uk/software/figtree/, accessed on 12 June 2022). Branches that received bootstrap support for maximum likelihood (BS) and Bayesian posterior probabilities (BPP) \u2265 50% (BS) and \u22650.90 (BPP) were considered significantly supported. The final concatenated sequence alignment and the retrieved topology were deposited in TreeBase .Trees were viewed in FigTree v1.4.2 . The topology from the ML analysis with a maximum likelihood bootstrap (BS) \u2265 50% and Bayesian posterior probabilities (BPP) \u2265 0.90 labeled on branches is shown .The combined five-gene  sequence dataset included sequences from 119 fungal samples representing 70 taxa. The dataset had an aligned length of 3498 characters, of which 2285 characters were constant, 226 were variable and parsimony-uninformative, and 987 were parsimony-informative. BI analysis generated topologies similar to those of ML analysis, with an average standard deviation of split frequencies = 0.007253 (BI). The topology from the ML analysis with a maximum likelihood bootstrap (BS) \u2265 50% and Bayesian posterior probabilities (BPP) \u2265 0.90 labeled on branches is shown .Aurantioporia B.K. Cui & Xing Ji, gen. nov.MycoBank: MB 847338Differs from other genera by its resupinate, rhizomorphic basidiocarps with an orange pore surface, a dimitic hyphal system with arboriform skeletal hyphae, tissues becoming violet in KOH, ellipsoid, truncate, and slightly dextrinoid basidiospores.Aurantioporia bambusicola  B.K. Cui & Xing JiType species: Aurantioporia (Lat.) refers to the orange pore surface of the genus.Etymology: Basidiocarps are annual to perennial and resupinate with rhizomorphs. The pore surface is yellow to orange when fresh, grayish orange, and orange-brown to dark orange when dry; pores are round to angular; dissepiments thin, entire. The subiculum is extremely thin and cream to pale orange. The tubes are concolorous with pore surface. The hyphal system is dimitic; generative hyphae with clamp connections; skeletal hyphae arboriform, IKI\u2212, CB+; tissues become violet to dark in KOH. Basidiospores are ellipsoid, truncate, hyaline, thick-walled, smooth, slightly dextrinoid, and CB+.P. aurantiaca (A. David & Rajchenb.) Decock & Ryvarden and P. bambusicola Choeyklin, T. Hatt. & E.B.G. Jones formed a single well-supported clade  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847362Pyrofomes aurantiacus A. David & Rajchenb., Mycotaxon 22(2): 312 (1985).Basionym: Perenniporia aurantiaca (A. David & Rajchenb.) Decock & Ryvarden, Mycol. Res. 103(9): 1140 (1999).\u2261 Perenniporia aurantiaca, see David and Rajchenberg [For a detailed description of chenberg  and Decochenberg .Aurantioporia aurantiaca was originally described in Pyrofomes by David and Rajchenberg [Perenniporia by Decock and Ryvarden [Aurantioporia aurantiaca from French Guyana  fell into the Aurantioporia clade in our phylogeny.Notes: chenberg  and lateRyvarden . The seqAurantioporia bambusicola  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847363Perenniporia bambusicola Choeyklin, T. Hatt. & E.B.G. Jones, Fungal Diversity 36: 122 (2009).Basionym: Perenniporia bambusicola, see Choeyklin et al. [For a detailed description of n et al.  and Cui n et al. .Aurantioporia bambusicola was first described in Perenniporia from Thailand [Aurantioporia aurantiaca also shares an orange pore surface, but Aurantioporia aurantiaca grows on hardwood trees [Notes: Thailand . It is cod trees .Specimen examined: CHINA. Yunnan, Cangyuan County, Banlao, on bamboo, 11 July 2013, Cui 11050 (BJFC).AurantioporiaKey to species of A. bambusicola1. Growing on bamboo; distributed in Southeast Asia\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0A. aurantiaca1. Growing on other hardwoods; distributed in neotropical areas\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Citrinoporia B.K. Cui & Xing Ji, gen. nov.MycoBank: MB 847346Differs from other genera by its slightly cushion shape, yellow pore surface, a dimitic hyphal system with dextrinoid, and cyanophilous shortly arboriform vegetative hyphae and ellipsoid, truncate, thick-walled, dextrinoid, and cyanophilous basidiospores.Type species:Citrinoporia (Lat.) refers to the yellowish pore surface of the genus.Etymology: Basidiocarps are annual to perennial and resupinate. The pore surface is white to yellow; pores are round. Subiculum is cream to buff, corky. Tubes are buff to pale brown and corky to hard corky. The hyphal system is dimitic: generative hyphae with clamp connections; skeletal hyphae arboriform, IKI\u2212, CB+; tissues becoming pale brown to black in KOH. Cystidia is absent, cystidioles are present. Basidiospores are ellipsoid, truncate, hyaline, thick-walled, smooth, dextrinoid, and CB+.P. citrinoalba B.K. Cui, C.L. Zhao & Y.C. Dai and P. corticola (Corner) Decock clustered together and formed a clade distinct from the Perenniporia s. s clade with full support  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847364Perenniporia citrinoalba B.K. Cui, C.L. Zhao & Y.C. Dai, Fungal Diversity 97: 270 (2019)Basionym: Perenniporia citrinoalba, see Cui et al. [For a detailed description of i et al. .Citrinoporia citrinoalba was newly described in Perenniporia from tropical China [Notes: al China . It is cSpecimens examined: CHINA. Hainan, Qiongzhong County, Limushan Forest Park, on fallen trunk of Castanopsis, 15 June 2014, Dai 13643 , on fallen angiosperm trunk, 18 November 2015, Cui 13615 (BJFC).Citrinoporia corticola (Corner) B.K. Cui & Xing Ji, comb. Nov.MycoBank: MB 847365Parmastomyces corticola Corner, Beih. Nova Hedwigia 96: 96 (1989).Basionym: Perenniporia corticola (Corner) Decock, Mycologia 93(4): 776 (2001).\u2261 Perenniporia dipterocarpicola T. Hatt. & S.S. Lee, Mycologia 91(3): 525 (1999)= Citrinoporia corticola was originally described in Parmastomyces Kotl. & Pouzar from Malaysia by Corner [Parmastomyces corticola and confirmed that this species has a dimitic hyphal system with clamped generative hyphae and transferred the species to Perenniporia. Citrinoporia corticola and Citrinoporia citrinoalba share yellow pore surfaces, dimitic hyphal structures, and truncate basidiospores, but the basidiospores of C. citrinoalba (5.5\u20136 \u00d7 4.7\u20135.2 \u00b5m) [Citrinoporia corticola (4.4\u20135 \u00d7 3.4\u20134 \u03bcm) [Notes: y Corner  as haviny Corner  studied \u20135.2 \u00b5m)  are larg.4\u20134 \u03bcm) .Specimens examined: MALAYSIA. Selangor, Kota Damansara, Community Forest Reserve, on angiosperm stump, 17 April 2018 Dai 18633, 18641 (BJFC).CitrinoporiaKey to species of C. corticola1. Basidiospores 4.4\u20135 \u03bcm; growing mainly on trees of Dipterocarpaceae\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0C. citrinoalba1. Basidiospores 5.5\u20136 \u00b5m; growing on trees of Fagaceae\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Cystidioporia B.K. Cui & Xing Ji, gen. nov.MycoBank: MB 847348Differs from other genera by its resupinate basidiocarps, slightly dextrinoid and cyanophilous skeletal hyphae, presence of thick-walled cystidia, and thick-walled, oblong-ellipsoid, truncate, slightly dextrinoid, and cyanophilous basidiospores.Cystidioporia piceicola (Y.C. Dai) B.K. Cui & Xing JiType species: Cystidioporia (Lat.) refers to resembling Perenniporia but with cystidia.Etymology: Basidiocarps are annual to biennial, resupinate, soft corky when fresh, and hard corky when dry. Pore surface is cream to buff when fresh and pale yellowish upon drying. Pores are round and large; dissepiments are thin, entire. Subiculum is yellowish ochraceous and corky. Tubes are yellowish ochraceous or straw yellow and corky. Hyphal system is dimitic to trimitic; generative hyphae with clamp connections; skeletal hyphae is slightly dextrinoid, CB+; tissues unchanged in KOH. Cystidia present, thick-walled, strongly CB+. Basidiospores are ellipsoid, truncate, hyaline, thick-walled, smooth, slightly dextrinoid, and CB+.Perenniporia piceicola Y.C. Dai formed a single clade distant from the Perenniporia s. s. clade. Moreover, this species has thick-walled cystidia, large pores, and basidiospores  [Perenniporia. Thus, the new genus is set up, and the following combination is proposed.Notes: In our present phylogenetic analyses , two spe\u20137.5 \u00b5m)  which arCystidioporia piceicola (Y.C. Dai) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847366Perenniporia piceicola Y.C. Dai, Ann. Bot. Fenn. 39(3): 173 (2002).Basionym: Perenniporia piceicola, see Dai et al. [For a detailed description of i et al. .Cystidioporia piceicola was originally described in Perenniporia by Dai et al. [Picea and Abies.Notes: i et al. ; it is cAbies, 16 September 2018, Cui 17062 (BJFC), and on fallen trunk of Picea, 16 September 2018, Cui 17069 (BJFC).Specimens examined: CHINA. Yunnan, Lijiang, Yunshanping, on fallen trunk of Picea likiangensis, 18 June 1999, Dai 3089 , on fallen trunk of Dendroporia B.K. Cui & Xing Ji, gen. nov.MycoBank: MB 847349Differs from other genera by annual and resupinate basidiocarps with gray to pale brown pore surface, a dimitic hyphal system with weakly dextrinoid skeletal hyphae, tissues darkening in KOH, presence of dendrohyphidia and large rhomboid crystals, and hyaline to pale yellowish, thick-walled, ellipsoid, truncate, non-dextrinoid, and cyanophilous basidiospores.Dendroporia cinereofusca (B.K. Cui & C.L. Zhao) B.K. Cui & Xing JiType species: Dendroporia (Lat.) refers to the presence of dendrohyphidia.Etymology: Basidiocarps are annual, resupinate, adnate, and corky. Pore surface is gray to pale brown. Subiculum is thin and clay buff to pale brown. Tubes are concolorous with pore surface and corky. Hyphal system is dimitic; generative hyphae with clamp connections; skeletal hyphae weakly dextrinoid and CB+; tissues are brown to black in KOH. Dendrohyphidia present at dissepimental edges; cystidia are absent; cystidioles are present. Large rhomboid crystals are present. Basidiospores ellipsoid, truncate, hyaline to pale yellowish, thick-walled, smooth, IKI\u2212, and CB+.Perenniporia cinereofusca B.K. Cui & C.L. Zhao formed a strongly supported clade distinct from the Perenniporia s. s. clade  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847367Perenniporia cinereofusca B.K. Cui & C.L. Zhao, Mycoscience 55: 419 (2014).Basionym: Perenniporia cinereofusca, see Zhao et al. [For a detailed description of o et al. .Dendroporia cinereofusca was first described in Perenniporia from tropical China [Notes: al China  and is cSpecimens examined: CHINA. Hainan, Ledong County, Jianfengling Nature Reserve, on fallen angiosperm trunk, 18 November 2007, Dai 9289 ; Lingshui County, Diaoluoshan Forest Park, on fallen angiosperm trunk, 20 November 2007, Cui 5280 .Luteoperenniporia B.K. Cui & Xing Ji, gen. nov.MycoBank: MB 847350Differs from other genera by its resupinate basidiocarps with buff-yellow to cinnamon-buff pore surface, a dimitic hyphal system with weak to strong dextrinoid skeletal hyphae, the presence of cystidioles, and thick-walled, ellipsoid, and non-truncate, dextrinoid, and cyanophilous basidiospores.Luteoperenniporia bannaensis (B.K. Cui & C.L. Zhao) B.K. Cui & Xing JiType species: Luteoperenniporia (Lat.) refers to resembling Perenniporia but with a buff-yellow pore surface when dry.Etymology: Basidiocarps are annual to perennial and resupinate. Pore surface iscream, buff to pale cinnamon buff when fresh, and becoming buff, buff-yellow to cinnamon-buff upon drying; pores are round to angular; dissepiments thin, entire to lacerate. Subiculum is thin and buff to cinnamon-buff. Tubes are concolorous with pore surface and corky. Hyphal system is dimitic, generative hyphae with clamp connections; skeletal hyphae weakly to strongly dextrinoid, CB+; tissues are unchanged in KOH. Cystidia is absent; cystidioles are present. Basidiospores are ellipsoid, non-truncate, hyaline, thick-walled, smooth, dextrinoid, and CB+.Luteoperenniporia formed a single clade with high support  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847368Perenniporia bannaensis B.K. Cui & C.L. Zhao, Fungal Diversity 58: 52 (2013).Basionym: Perenniporia bannaensis, see Zhao et al. [For a detailed description of o et al. .Luteoperenniporia bannaensis was recently described in Perenniporia from China by Zhao et al. [L. yinggelingensis in morphology and phylogeny; they share annual and resupinate basidiocarps, cream to buff pore surface and a dimitic hyphal system with dextrinoid and cyanophilous skeletal hyphae, and both species are distributed in the tropics. However, L. yinggelingensis is distinguished from L. bannaensis by its larger pores and basidiospores  [Notes: o et al. ; it is c\u20135.5 \u03bcm) .Specimens examined: CHINA. Yunnan, Xishuangbanna, Mengla County, Wangtianshu Nature Reserve, on fallen angiosperm trunk, 2 November 2009, Cui 8560 , Cui 8562 .Luteoperenniporia mopanshanensis (C.L. Zhao) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847369Perenniporia mopanshanensis C.L. Zhao, Mycotaxon 134(1): 132 (2019).Basionym: Perenniporia mopanshanensis, see Zhao and Ma [For a detailed description of o and Ma .Luteoperenniporia mopanshanensis was recently described in Perenniporia by Zhao and Ma [L. mopanshanensis and L. bannaensis are both reported from Yunnan Province in southern China. They share resupinate basidiocarps, a dimitic hyphal system with strongly dextrinoid skeletal hyphae, non-truncate, and strongly dextrinoid and similar sized basidiospores. However, L. bannaensis differs by having an annual growth habit and smaller pores (6\u20138 per mm) [Notes: o and Ma . L. mopa per mm) .Luteoperenniporia yinggelingensis (B.K. Cui & Y.C. Dai) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847370Perenniporia yinggelingensis B.K. Cui & Y.C. Dai, Fungal Diversity 97: 300 (2019).Basionym: Perenniporia yinggelingensis, see Cui et al. [For a detailed description of i et al. .Luteoperenniporia yinggelingensis was newly described from a tropical area of China [L. yinggelingensis is close to L. mopanshanensis, but the latter species has perennial basidiocarps, larger pores (3\u20135 per mm), indistinct sterile margins, and strongly dextrinoid skeletal hyphae [Notes: of China . It is cl hyphae .Specimens examined: CHINA. Hainan, Baisha County, Yinggeling Nature Reserve, on fallen angiosperm trunk, 17 November 2015, Cui 13625 ; on fallen angiosperm branch, 17 June 2016, Cui 13856 (BJFC).Luteoperenniporia australiensis B.K. Cui & Xing Ji, sp. nov.; Figure 4MycoBank: MB 847371Luteoperenniporia by its annual to perennial growth habit, resupinate basidiocarps with buff to cinnamon-buff pore surfaces, a dimitic hyphal system with dextrinoid skeletal hyphae, and ellipsoid, non-truncate, dextrinoid basidiospores (6.2\u20137.5 \u00d7 4\u20135.2 \u03bcm).Differs from other species of Eucalyptus, 15 May 2018, Cui 16743 (BJFC).Holotype: AUSTRALIA. Tasmania, Keogh\u2019s Creek Walk, on fallen trunk of australiensis (Lat.) refers to the country where the new species was found.Etymology: Fruitbody: Basidiocarps are annual to perennial, resupinate, without odor or taste when fresh, corky to hard corky when dry, and up to 14 cm long, 8.5 cm wide, and 7 mm thick at center. Pore surface is buff-yellow, pinkish buff to pale cinnamon-buff when fresh, and buff, pale yellowish-brown to cinnamon-buff upon drying; pores are round to angular, 4\u20136 per mm, dissepiments thin, entire to lacerate. Sterile margin is distinct to indistinct, buff, and up to 2 mm wide. Subiculum is thin, buff, and up to 1 mm thick. Tubes are concolorous with pore surface, corky to hard corky when dry, and up to 6 mm long.Hyphal structure: Hyphal system is dimitic; generative hyphae bearing clamp connections; skeletal hyphae dextrinoid, and CB+; tissues are unchanged in KOH.Subiculum: Generative hyphae is infrequent, hyaline, thin-walled, occasionally branched, and 1.5\u20132.5 \u03bcm in diameter; skeletal hyphae is dominant, hyaline, thick-walled with a wide to narrow lumen, rarely branched, interwoven, and 2\u20135 \u03bcm in diameter.Tubes: Generative hyphae is infrequent, hyaline, thin-walled, occasionally branched, and 1\u20132.4 \u03bcm in diameter; skeletal hyphae is dominant, hyaline, thick-walled with a wide to narrow lumen, occasionally branched, interwoven, and 1\u20133 \u03bcm. Cystidia are absent; fusoid cystidioles are present, hyaline, thin-walled, and 15.5\u201322 \u00d7 5.8\u20138 \u03bcm. Basidia clavate, with four sterigmata and a basal clamp connection, 15\u201324.5 \u00d7 6.5\u20139.7 \u03bcm; basidioles are dominant and in shape similar to basidia, but slightly smaller.Spores: Basidiospores are ellipsoid, non-truncate, hyaline, thick-walled, smooth, dextrinoid, CB+, (6\u2013) 6.2\u20137.4 (\u20137.7) \u00d7 (3.9\u2013) 4\u20135.2 (\u20135.4) \u03bcm, L = 6.77 \u03bcm, W = 4.58 \u03bcm, Q = 1.44\u20131.54 (n = 90/3).Luteoperenniporia australiensis is characterized by its annual to perennial and resupinate basidiocarps, buff to cinnamon-buff pore surface, entire to lacerate pores, a dimitic hyphal system with dextrinoid skeletal hyphae, presence of cystidioles, and ellipsoid, non-truncate, dextrinoid, and cyanophilous basidiospores. Luteoperenniporia bannaensis is similar to L. australiensis by sharing a buff-yellow to pinkish buff pore surface, a dimitic hyphal system with dextrinoid, and cyanophilous skeletal hyphae. However, L. bannaensis differs from L. australiensis by its smaller pores (6\u20138 per mm) and smaller basidiospores (5.2\u20136 \u00d7 4\u20134.6 \u03bcm) [Luteoperenniporia yinggelingensis may be confused with L. australiensis by having resupinate basidiocarps, and similar pore size and basidiospores. However, L. yinggelingensis is distinguished from L. australiensis mainly by its annual growth habit and cream to buff pore surface [Perenniporia subaurantiaca (Rodway & Cleland) P.K. Buchanan & Ryvarden is also described from Tasmania, it is similar to Luteoperenniporia australiensis in its resupinate basidiocarps, dimitic hyphal system with dextrinoid and cyanophilous skeletal hyphae, and presence of cystidioles, but P. subaurantiaca has greyish cream to greyish orange pore surfaces and larger basidiospores (7.2\u20139.5 \u00d7 4.2\u20135.5 \u03bcm) [Notes: \u20134.6 \u03bcm) . Luteope surface . Perenni\u20135.5 \u03bcm) Eucalyptus, 9 May 2018, Cui 16524 (BJFC), on fallen trunk of Eucalyptus, 9 May 2018, Cui 16525 (BJFC), 10 May 2018, Cui 16535 (BJFC), on living tree of Eucalyptus, 10 May 2018, Cui 16533, Cui 16534 (BJFC); Tasmania, Keogh\u2019s Creek Walk, on fallen trunk of Eucalyptus, 15 May 2018, Cui 16742 (BJFC).Additional specimens (paratypes) examined: AUSTRALIA. Victoria, Yarra Ranges National Park, at the base of living LuteoperenniporiaKey to species of L. bannaensis1. Pores 6\u20138 per mm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20261. Pores 3\u20136 per mm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262L. mopanshanensis2. Skeletal hyphae unbranched\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262. Skeletal hyphae branched\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a03L. yinggelingensis3. Basidiocarps annual, distributed in tropical areas\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. australiensis3. Basidiocarps annual to perennial, distributed in temperate to subtropical areas\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Macroporia B.K. Cui & Xing Ji, gen. nov.MycoBank: MB 847352Differs from other genera by its annual and resupinate basidiocarps, a dimitic hyphal system with dextrinoid skeletal hyphae, the presence of thin-walled cystidioles, and hyaline, thick-walled, ellipsoid, truncate, dextrinoid, and cyanophilous basidiospores.Macroporia macropora (B.K. Cui & C.L. Zhao) B.K. Cui & Xing JiType species: Macroporia (Lat.) refers to the species with relatively large pores in Perenniporia.Etymology: Basidiocarps are annual, resupinate, adnate. Pore surface is white, cream to buff when fresh, and becoming buff, pinkish buff to yellowish buff upon drying; pores are angular; dissepiments are thin, entire to lacerate. Subiculum is thin, cream. Tubes are concolorous with pore surface, corky to hard corky. Hyphal system is dimitic, generative hyphae with clamp connections; skeletal hyphae branched, dextrinoid, CB+; tissues are unchanged in KOH. Cystidia is absent; cystidioles are usually present. Basidiospores are ellipsoid, truncate, hyaline, thick-walled, smooth, dextrinoid, and CB+.Perenniporia lacerata B.K. Cui & C.L. Zhao, P. macropora B.K. Cui & C.L. Zhao, P. subrhizomorpha Xue W. Wang, L.W. Zhou & X.M. Tian and P. tibetica B.K. Cui & C.L. Zhao grouped together and formed a well-supported clade  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847372Perenniporia lacerata B.K. Cui & C.L. Zhao, Mycoscience 54: 232 (2013).Basionym: Perenniporia lacerata, see Zhao and Cui [For a detailed description of  and Cui .Macroporia lacerata was originally described in Perenniporia from China by Zhao and Cui [Notes:  and Cui . It is cSpecimens examined: CHINA. Henan, Xiuwu County, Yuntaishan Park, on fallen angiosperm trunk, 3 September 2009, Cui 7220 ; Neixiang County, Baotianman Nature Reserve, on rotten angiosperm wood, 22 September 2009, Dai 11268 .Macroporia macropora (B.K. Cui & C.L. Zhao) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847373Perenniporia macropora B.K. Cui & C.L. Zhao, Mycologia 105: 947 (2013).Basionym: Perenniporia macropora, see Zhao and Cui [For a detailed description of  and Cui .Macroporia macropora was first described in Perenniporia from southern China by Zhao and Cui [M. macropora is very close to M. lacerata in the current phylogeny, but M. lacerata differs from M. macropora by its smaller (3\u20135 per mm) and lacerate pores, the absence of dendrohyphidia, and smaller basidiospores (6.1\u20137 \u00d7 5\u20135.7 \u03bcm) [Notes:  and Cui . It is d\u20135.7 \u03bcm) .Specimens examined: CHINA. Guangxi, Ningming County, Nonggang Nature Reserve, on fallen angiosperm branch, 8 July 2007, Zhou 407 , 7 July 2007, Zhou 297 .Macroporia subrhizomorpha  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847374Perenniporia subrhizomorpha Xue W. Wang, L.W. Zhou & X.M. Tian, Phytotaxa 528(2): 129 (2021).Basionym: Perenniporia subrhizomorpha, see Tian et al. [For a detailed description of n et al. .Macroporia subrhizomorpha was recently described in Perenniporia as P. subrhizomorpha by Tian et al. [Perenniporia rhizomorpha may be confused with P. subrhizomorpha by sharing cream rhizomorphs and similar pore sizes, but the former species differs by its non-truncate basidiospores [Perenniporia tibetica also has rhizomorphs but differs from P. subrhizomorpha by larger pores (2\u20133 per mm) [Notes: n et al. . Perenniiospores . Perenni per mm) .Macroporia tibetica (B.K. Cui & C.L. Zhao) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847375Perenniporia tibetica B.K. Cui & C.L. Zhao, Mycoscience 53: 366 (2012).Basionym: Perenniporia tibetica, see Cui and Zhao [For a detailed description of and Zhao .Macroporia tibetica is characterized by resupinate basidiocarps with white to cream rhizomorphs, a dimitic hyphal system with slightly dextrinoid skeletal hyphae, and ellipsoid, truncate or not, dextrinoid, and cyanophilous basidiospores. M. macropora and M. tibetica share resupinate basidiocarps and similar pores, but the former differs in having dendrohyphidia and lacking rhizomorphs [Notes: zomorphs .Specimens examined: CHINA. Xizang, Linzhi County, Tongmai, on fallen angiosperm trunk, 16 September 2010, Cui 9457 , Cui 9459 .MacroporiaKey to species of M. lacerata1. Pores lacerate\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0\u00a01. Pores entire\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0\u00a02M. macropora2. Basidiocarps without rhizomorphs\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a02. Basidiocarps with rhizomorphs\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263M. tibetica3. Pores 2\u20133 per mm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026M. subrhizomorpha3. Pores 4\u20136 per mm\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Macrosporia B.K. Cui & Xing Ji, gen. nov.MycoBank: MB 847353Differs from other genera by its annual and resupinate basidiocarps, cinnamon-buff pore surface, a trimitic hyphal system with weakly dextrinoid and cyanophilous skeletal hyphae, and hyaline, thick-walled, ellipsoid, truncate, strongly dextrinoid, and cyanophilous basidiospores.Macrosporia nanlingensis (B.K. Cui & C.L. Zhao) B.K. Cui & Xing JiType species: Macrosporia (Lat.) refers to the large basidiospores.Etymology: Basidiocarps are annual, resupinate, adnate, corky when fresh, and becoming hard corky upon drying. Pore surface is cream-buff to yellowish buff when fresh, becoming cinnamon-buff upon drying; pores are round; dissepiments are thick, entire. Subiculum is cream to buff. Tubes are concolorous with the pore surface and hard corky. Hyphal system is trimitic; generative hyphae with clamp connections; skeletal and binding hyphae are weakly dextrinoid and CB+. Cystidia is absent; cystidioles are present. Basidiospores areellipsoid, truncate, hyaline, thick-walled, smooth, strongly dextrinoid, and CB+.Perenniporia nanlingensis B.K. Cui & C.L. Zhao formed a single clade that was distant from the Perenniporia s. s. clade. Morphologically, it differs from Perenniporia s. s. by its annual and resupinate basidiocarps with cinnamon-buff pore surfaces and larger basidiospores. Therefore, Macrosporia gen. nov. is proposed to accommodate P. nanlingensis.Notes: In our study, Macrosporia nanlingensis (B.K. Cui & C.L. Zhao) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847376Perenniporia nanlingensis B.K. Cui & C.L. Zhao, Mycol. Prog. 11: 556 (2012).Basionym: Perenniporia nanlingensis, see Zhao and Cui [For a detailed description of  and Cui .Macrosporia nanlingensis was first described in Perenniporia from southern China [Notes: rn China ; it is cSpecimens examined: CHINA. Guangdong Province, Ruyang County, Nanling Nature Reserve, on dead angiosperm tree, 16 September 2009, Cui 7589 , Cui 7620 .Minoporus B.K. Cui & Xing Ji, gen. nov.MycoBank: MB 847354Differs from other genera by its annual and pileate basidiocarps, cream to pale buff pileal surface when fresh, a dimitic hyphal system with weakly amyloid and cyanophilous skeletal hyphae, and hyaline, thick-walled, ellipsoid, truncate, dextrinoid, and cyanophilous basidiospores.Minoporus minor (Y.C. Dai & H.X. Xiong) B.K. Cui & Xing JiType species: Minoporus (Lat.) refers to the small pilei.Etymology: Basidiocarps are annual, pileate, solitary, and soft corky when fresh, becoming hard corky upon drying. Pilei are semicircular to spathulate. Pileal surface is cream to pale buff when fresh, becoming cinnamon-buff when dry. Pore surface is cream when fresh, becoming cinnamon-buff when dry; pores are round. Context is white to cream, corky. Tubes are concolorous with pore surface and hard corky. Hyphal system dimitic; generative hyphae with clamp connections; skeletal hyphae is weakly amyloid and CB+. Cystidia and cystidioles are absent. Basidiospores are ellipsoid, truncate, hyaline, thick-walled, smooth, dextrinoid, and CB+.Perenniporia minor Y.C. Dai & H.X. Xiong formed a highly supported single clade  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847377Perenniporia minor Y.C. Dai & H.X. Xiong, Mycotaxon 105: 60 (2008).Basionym: Perenniporia minor, see Xiong et al. [For a detailed description of g et al. .Notes: This species was described from northeastern China by Xiong et al.  and is cAcer, 14 September 2007, Dai 9198 ; Liaoning, Huanren County, Laotudingzi Nature Reserve, on fallen branch of Quercus, 31 July 2008, Cui 5738 (BJFC).Specimens examined: CHINA. Jilin, Antu County, Changbaishan Nature Reserve, Huangsongpu, on fallen branch of Neoporia B.K. Cui & Xing Ji, gen. nov.MycoBank: MB 847355Differs from other genera by its annual and resupinate basidiocarps with a buff-yellow pore surface, a dimitic hyphal system with dextrinoid and cyanophilous skeletal hyphae, and ellipsoid, non-truncate, dextrinoid, and cyanophilous basidiospores.Neoporia rhizomorpha  B.K. Cui & Xing JiType species: Neoporia (Lat.) refers to the genus resembling Perenniporia.Etymology: Basidiocarps are annual, resupinate and corky when dry. Pore surface is cream to buff when fresh, buff-yellow to grayish orange upon drying; pores are round to angular. Subiculum is cream to buff. Tubes are concolorous with the pore surface and corky. Hyphal system is dimitic; generative hyphae with clamp connections; skeletal hyphae dextrinoid, CB+; tissues unchanged in KOH. Basidiospores are ellipsoid, non-truncate, hyaline, thick-walled, smooth, dextrinoid, and CB+.Perenniporia, P. bostonensis C.L. Zhao, P. koreana Y. Jang & J.J. Kim, and P. rhizomorpha B.K. Cui, Y.C. Dai & Decock grouped together and formed a highly supported clade  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847378Perenniporia bostonensis C.L. Zhao, Phytotaxa 351(1): 67 (2018).Basionym: Perenniporia bostonensis, see Shen et al. [For a detailed description of n et al. .Neoporia bostonensis was recently described in Perenniporia from North America by Shen et al. [Notes: n et al. . It is cNeoporia koreana (Y. Jang & J.J. Kim) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847379Perenniporia koreana Y. Jang & J.J. Kim, Mycotaxon 130(1): 174 (2015).Basionym: Perenniporia bannaensis, see Jang et al. [For a detailed description of g et al. .Neoporia koreana was originally described from Republic of Korea as Perenniporia koreana by Jang et al. [Neoporia koreana from type specimens fell into Neoporia in our phylogeny. Thus, P. koreana is transferred to Neoporia. N. koreana is similar to N. bostonensis in resupinate basidiocarps, similar sized pores, dextrinoid skeletal hyphae, and non-truncate basidiospores, but the former has larger basidiospores (6\u20137 \u00d7 3.9\u20135.2 \u03bcm) [Notes: g et al. ; it has \u20135.2 \u03bcm)  and the \u20135.2 \u03bcm) .Neoporia rhizomorpha  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847380Perenniporia rhizomorpha B.K. Cui, Y.C. Dai & Decock, Mycotaxon 99: 176 (2007).Basionym: Perenniporia rhizomorpha, see Cui et al. [For a detailed description of i et al. .Neoporia rhizomorpha was first described in Perenniporia based on morphological characters from China [Neoporia due to its resupinate basidiocarps with rhizomorphs.Notes: om China . It is uSpecimens examined: CHINA. Anhui, Huangshan, Yellow Mountain, on fallen angiosperm trunk, 13 October 2004, Dai 6165 ; Fujian, Wuyishan County, Wuyishan Nature Reserve, on fallen angiosperm branch, 19 October 2005, Cui 7248 .NeoporiaKey to species of N. rhizomorpha1. Basidiocarps with rhizomorphs\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20261. Basidiocarps without rhizomorphs\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a02N. bostonensis2. Basidiospores 3.5\u20134.5 \u00d7 3\u20134 \u03bcm\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0N. koreana2. Basidiospores 6\u20137 \u00d7 3.9\u20135.2 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Niveoporia B.K. Cui & Xing Ji, gen. nov.MycoBank: MB 847356Differs from other genera by perennial basidiocarps with white pore surface when fresh, distinct rusty red to reddish brown sterile margin, a dimitic hyphal system with dextrinoid and cyanophilous skeletal hyphae, the presence of cystidioles, hyaline, and thick-walled, ellipsoid, and truncate basidiospores.Niveoporia russeimarginata (B.K. Cui & C.L. Zhao) B.K. Cui & Xing JiType species: Niveoporia (Lat.) refers to the white pore surface.Etymology: Basidiocarps are perennial, resupinate to pileate, corky to woody hard when dry. Pilei dimidiate to fan shaped. Pileal surface is clay-buff to reddish brown when fresh, grayish brown to umber brown when dry, glabrous, and concentrically sulcate. Pore surface is white when fresh and white to cream upon drying; pores are round. The sterile margin is sometimes distinct, rusty red to reddish brown. Context is buff to fawn. Tubes are cream to pale cinnamon. Hyphal system is dimitic; generative hyphae with clamp connections; skeletal hyphae dextrinoid, CB+; tissues unchanged in KOH. Cystidia is absent; cystidioles are present. Basidiospores are ellipsoid, truncate, hyaline, thick-walled, smooth, dextrinoid or not, and CB+.P. decurrata F. Wu & X.H. Ji, P. russeimarginata B.K. Cui & C.L. Zhao and an undescribed species grouped together and formed a well-supported clade  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847381Perenniporia decurrata Corner, Beih. Nova Hedwigia 96: 105 (1989)Basionym: Perenniporia chiangraiensis F. Wu & X.H. Ji, Mycosphere 8(8): 1103 (2017).= Perenniporia decurrata, see Corner [For a detailed description of e Corner .Perenniporia decurrata was first described from Malaysia [Perenniporia chiangraiensis was recently described from Northern Thailand based on morphological characters and molecular data by Ji et al. [P. decurrata, which is a priority name for this species. The species is characterized by pileate basidiocarps with concentrically sulcate pileal surfaces, white pore surfaces, the presence of dendrohyphidia and cystidioles, and ellipsoid, truncate, thick-walled, and non-dextrinoid basidiospores.Notes: Malaysia . Perennii et al. . HoweverSpecimens examined: CHINA. Yunnan, Xishuangbanna, Menglun, on fallen angiosperm trunk, 12 September 2006, Yuan 2334 (IFP). THAILAND. Chiang Rai, Doi Mae Salong, on angiosperm tree root, 22 July 2016, Dai 16637 (BJFC).Niveoporia russeimarginata (B.K. Cui & C.L. Zhao) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847382Perenniporia russeimarginata B.K. Cui & C.L. Zhao, Mycologia 105(4): 947 (2013).Basionym: Perenniporia russeimarginata, see Zhao and Cui [For a detailed description of  and Cui .Niveoporia russeimarginata was first described in Perenniporia from southern China [Perenniporia alboferruginea Decock, described from Cameroon in Africa, is similar to N. russeimarginata in having resupinate basidiocarps with ferruginous red upper margins and a dimitic hyphal system. However, P. alboferruginea differs by having an annual growth habit, larger pores (5\u20136 per mm), the absence of cystidioles, and non-dextrinoid basidiospores [Notes: rn China . Perenniiospores .Specimens examined: CHINA. Yunnan, Chuxiong, Zixishan Nature Reserve, on fallen angiosperm trunk, 1 August 2005, Yuan 1225 , Yuan 1262 .Niveoporia subrusseimarginata B.K. Cui & Xing Ji, sp. nov., Figure 6MycoBank: MB 847383Niveoporia species by its resupinate to pileate basidiocarps with rusty reddish brown sterile margins and pores measuring 5\u20136 per mm.Differs from other Holotype: CHINA. Yunnan, Binchuan County, Jizushan Park, on angiosperm stump, 14 September 2018, Cui 16991 (BJFC).Subrusseimarginata (Lat.) refers to the species resembling Niveoporia russeimarginata.Etymology: Fruitbody: Basidiocarps are perennial, resupinate, sometimes pileate, corky, without odor or taste when fresh, becoming woody hard upon drying. Pilei are irregular, projecting up to 2 cm, 5 cm wide, and 3 cm thick at the base; with resupinate up to 9 cm long, 5 cm wide, and 1.3 cm thick at center. Pileal surface is orange-brown to reddish brown when fresh, umber brown when dry, glabrous, and concentrically sulcate. The pore surface is white when fresh, cream upon drying; pores are round to angular, 5\u20136 per mm, dissepiments thick, entire. Sterile margin is distinct to indistinct, cinnamon to rusty reddish brown, and up to 2 mm wide. Subiculum is buff, thin, up to 0.5 mm thick. Tubes are buff, woody hard when dry, and up to 11.5 mm long.Hyphal structure: Hyphal system is dimitic; generative hyphae bearing clamp connections; skeletal hyphae is weakly dextrinoid and CB+; tissues are unchanged in KOH.Subiculum: Generative hyphae are infrequent, hyaline, thin-walled, unbranched, and 1\u20132.2 \u03bcm in diameter; skeletal hyphae are dominant, hyaline, thick-walled, occasionally branched, interwoven, and 1\u20132.5 \u03bcm in diameter.Tubes: Generative hyphae are infrequent, hyaline, thin-walled, occasionally branched, and 1\u20132 \u03bcm in diameter; skeletal hyphae are dominant, hyaline, thick-walled, moderately branched, and interwoven, 1\u20132.3 \u03bcm. Cystidia are absent; fusoid cystidioles are present, hyaline, thin-walled, and 12.5\u201314 \u00d7 4.8\u20136 \u03bcm. Basidia are clavate, with four sterigmata and a basal clamp connection, 12.5\u201317.5 \u00d7 6.3\u20139 \u03bcm; basidioles are dominant and in shape similar to basidia, but slightly smaller.Spores: Basidiospores are broadly ellipsoid, truncate or not, hyaline, thick-walled, smooth, weakly dextrinoid, CB+, (4\u2013)4.2\u20134.8 (\u20135) \u00d7 (3\u2013)3.2\u20133.8(\u20134) \u03bcm, L = 4.48 \u03bcm, W = 3.51 \u03bcm, Q = 1.27\u20131.3 (n = 90/3).Niveoporia subrusseimarginata is characterized by resupinate to pileate basidiocarps with white to cream pore surface, cinnamon to rust sterile margin, dimitic hyphal system with weakly dextrinoid skeletal hyphae, the presence of cystidioles, and broadly ellipsoid, truncate or not basidiospores.Notes: Quercus, 14 September 2018, Cui 16973 (BJFC), on fallen angiosperm branch, 14 September 2018, Cui 16980 (BJFC), on angiosperm stump, 14 September 2018, Cui 16988, 16990 (BJFC).Additional specimens (paratypes) examined: CHINA. Yunnan, Binchuan County, Jizushan Park, on the stump of NiveoporiaKey to species of N. decurrata1. Sterile margin indistinct\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a01. Sterile margin distinct, reddish brown\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262N. subrusseimarginata2. Pores 5\u20136 per mm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0N. russeimarginata2. Pores 6\u20138 per mm\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Perenniporia Murrill, Mycologia 34(5): 595 (1942).MycoBank: MB 18204Perenniporia medulla-panis (Jacq.) Donk, Persoonia 5(1): 76 (1967).Type species: Basidiocarps are annual to perennial and resupinate. Pore surface is white to cream when fresh, cream to buff when dry; pores are round and small. Subiculum is thin, cream, and corky. Tubes are concolorous with pore surface and corky. Hyphal system is dimitic to trimitic; generative hyphae with clamp connections; skeletal hyphae are non-dextrinoid to dextrinoid or amyloid, cyanophilous; tissues are unchanged in KOH. Cystidia are absent, cystidioles are present. Basidiospores are ellipsoid, truncate, hyaline, thick-walled, smooth, dextrinoid, and CB+.Perenniporia s. l. clustered into several clades, Perenniporia medulla-panis is grouped with P. substraminea B.K. Cui & C.L. Zhao and P. hainaniana B.K. Cui & C.L. Zhao. These three species share similar morphological characteristics and form the core clade of Perenniporia. The above concept of Perenniporia s. s. is determined from P. medulla-panis, P. hainaniana, and P. substraminea.Notes: In our phylogenetic analyses, the species of Perenniporia hainaniana. China. Hainan, Changjiang County, Bawangling Nature Reserve, on angiosperm stump, 8 May 2009, Cui 6364 . Perenniporia medulla-panis. China. Guangxi, Jinxiu County, Dayaoshan Nature Reserve, on living angiosperm tree, 15 July 2017, Cui 14515 (BJFC). Perenniporia substraminea. China. Zhejiang, Taishun County, Wuyanling Nature Reserve, on angiosperm stump, 22 August 2011, Cui 10177 .Specimens examined: Perenniporia s. s.Key to species of P. substraminea1. Pores > 8 per mm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20261. Pores < 7 per mm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262P. hainaniana2. Dendrohyphidia present at dissepimental edges\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026P. medulla-panis2. Dendrohyphidia absent at dissepimental edges\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Poriella C.L. Zhao, Agronomy 11: 5 (2021).MycoBank: MB 840061Poriella subacida (Peck) C.L. Zhao, Agronomy 11: 6 (2021).Type species: Basidiocarps are annual to perennial, resupinate to effused-reflexed, and corky when dry. Pore surface is dingy yellowish, cinnamon to ochraceous; pores are round to angular. Context is thin, cream, buff to pale ochraceous. Tubes are concolorous with pore surface, corky. Hyphal system is dimitic to trimitic; generative hyphae with clamp connections; skeletal hyphae is unbranched, strongly dextrinoid, and cyanophilous. Basidiospores are ellipsoid to subglobose, non-truncate, hyaline, thick-walled, smooth, non-dextrinoid to dextrinoid, and CB+.Poriella was newly set up by Chen et al. [Poriella subacida, originally described as Polyporus subacidus Peck [Perenniporia in current studies [Perenniporia, P. africana Ipulet & Ryvarden, P. ellipsospora Ryvarden & Gilb., and P. valliculorum Spirin et Zmitr. grouped together with Poriella subacida and formed a highly supported clade  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847384Perenniporia africana Ipulet & Ryvarden, Syn. Fung. 20: 94 (2005).Basionym: Perenniporia africana, see Ipulet and Ryvarden [For a detailed description of Ryvarden  and Cui Ryvarden .Poriella africana was originally described from Uganda as Perenniporia africana [Notes: africana . It has Specimens examined: CHINA. Anhui, She County, Qingliangfeng Nature Reserve, on fallen angiosperm trunk, 14 December 2009, Cui 8674, 8676 (BJFC).Poriella ellipsospora (Ryvarden & Gilb.) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847385Perenniporia ellipsospora Ryvarden & Gilb., Mycotaxon 19: 140 (1984).Basionym: Perenniporia ellipsospora, see Cui et al. [For a detailed description of i et al. .Poriella ellipsospora and P. subacida both have resupinate basidiocarps, strongly dextrinoid skeletal hyphae, and ellipsoid and non-truncate basidiospores, but the former has larger pores (3\u20134 per mm) and dextrinoid basidiospores, and the latter has smaller pores (4\u20136 per mm) and non-dextrinoid basidiospores [Notes: iospores .Specimens examined: CHINA. Yunnan, Lanping County, Changyanshan Nature Reserve, on fallen angiosperm trunk, 18 September 2011, Cui 10276, 10284 (BJFC).Poriella valliculorum (Spirin & Zmitr.) B.K. Cui & Xing Ji, comb. nov.Mycobank: MB 847386Perenniporia valliculorum Spirin & Zmitr., Folia Cryptogamica Petropolitana (Sankt-Peterburg) 6: 51 (2005).Basionym: Perenniporia valliculorum, see Spirin et al. [For a detailed description of n et al. .Poriella valliculorum was originally described from Russia as Perenniporia valliculorum by Spirin et al. [Poriella well. In addition, the sequence of P. valliculorum from the type specimen fell into Poriella in our phylogeny  B.K. Cui & Xing JiType species: Rhizoperenniporia (Lat.) refers to resembling Perenniporia but with rhizomorphs.Etymology: Basidiocarps are annual to perennial, resupinate, and corky when dry. Rhizomorphs are present. Pore surface is grayish white to pale buff when dry; pores are round; dissepiments thick, entire. Subiculum is thin, cream. Tubes are concolorous with pore surface and corky. Hyphal system is dimitic; generative hyphae with clamp connections; skeletal hyphae are dextrinoid and CB+; tissues are unchanged in KOH. Cystidia are absent; cystidioles are present. Basidiospores are ellipsoid, truncate, hyaline, thick-walled, smooth, dextrinoid, and CB+.Perenniporia japonica (Yasuda) T. Hatt. & Ryvarden formed a single clade distant from the Perenniporia s. s. clade. In addition, P. japonica has basidiocarps with rhizomorphs, which are different from other species of Perenniporia s. s. Therefore, the new genus Rhizoperenniporia is proposed to include P. japonica.Notes: In our phylogenetic analyses , two speRhizoperenniporia japonica (Yasuda) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847387Trametes japonica Yasuda, Bot. Mag., Tokyo 32: 356 (1918).Basionym: Perenniporia japonica (Yasuda) T. Hatt. & Ryvarden, Mycotaxon 50: 36 (1994).= Perenniporia japonica, see N\u00fa\u00f1ez and Ryvarden [For a detailed description of Ryvarden  and Cui Ryvarden .Perenniporia s. l., P. aurantiaca, P. bambusicola, P. rhizomorpha, P. subrhizomorpha, and P. tibetica also have resupinate basidiocarps with rhizomorphs. However, P. aurantiaca differs from P. japonica by an orange pore surface, arboriform vegetative hyphae, and tissues becoming violet in KOH [Perenniporia bambusicola is distinguished by an orange pore surface turning dark violet to black in KOH, arboriform vegetative hyphae, and growing only on bamboo [Perenniporia rhizomorpha differs by its non-truncate basidiospores [Perenniporia subrhizomorpha differs by the absence of cystidioles and larger basidiospores (5.7\u20136.5 \u00d7 4.3\u20135.5 \u03bcm) [Perenniporia tibetica differs by having larger pores and basidiospores  [Notes: In t in KOH . Perennin bamboo . Perenniiospores . Perenni\u20135.5 \u03bcm) . Perenni\u20136.8 \u03bcm) .Vitex, 25 August 2016, Dai 17035 (BJFC); Huguan County, Baquan Gorge, on fallen trunk of Lonicera, 27 August 2016, Dai 17068, 17080 (BJFC).Specimens examined: CHINA. Shanxi, Yangcheng County, Manghe Nature Reserve, on rotten wood of Tropicoporia B.K. Cui & Xing Ji, gen. nov.MycoBank: MB 847358Differs from other genera by its dimitic to trimitic hyphal system with usually non-dextrinoid and inamyloid skeletal hyphae, broadly ellipsoid to subglobose, truncate, dextrinoid, and cyanophilous basidiospores.Tropicoporia aridula (B.K. Cui & C.L. Zhao) B.K. Cui & Xing JiType species: Tropicoporia (Lat.) refers to the distribution of the genus in tropical areas.Etymology: Basidiocarps are annual to perennial, mostly resupinate, and pseudopileate to rarely pileate. Pore surface is cream, buff-yellow to grayish orange; pores are round to angular. Hyphal system is dimitic to trimitic; generative hyphae with clamp connections; skeletal hyphae are non-dextrinoid to slightly dextrinoid and CB+. Basidiospores are broadly ellipsoid to subglobose, truncate, hyaline, thick-walled, smooth, dextrinoid, and CB+.Perenniporia aridula B.K. Cui & C.L. Zhao, P. vanhulleae Decock & Ryvarden, P. centrali-africana Decock & Mossebo, and P. brasiliensis C.R.S. de Lira et al. grouped together and formed a clade distinct from the Perenniporia s. s. clade, although the branch support was low  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847388Perenniporia aridula B.K. Cui & C.L. Zhao, Fungal Diversity 58: 48 (2013)Basionym: Perenniporia aridula, see Zhao et al. [For a detailed description of o et al. .Tropicoporia aridula was first described in Perenniporia by Zhao et al. [Tropicoporia vanhulleae is closely related to T. aridula in morphology and phylogeny, but the former has smaller basidiospores (5.5\u20136.0 \u00d7 4.5\u20135.5 \u00b5m) [Notes: o et al. . It is t\u20135.5 \u00b5m) .Specimens examined: CHINA. Yunnan, Yuanjiang County, on fallen angiosperm trunk, 9 June 2011, Dai 12396 , on fallen bamboo, 9 June 2011, Dai 12398 .Tropicoporia brasiliensis  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847389Perenniporia brasiliensis Lira, A.M.S. Soares, Ryvarden & Gibertoni, Persoonia 38: 355 (2017).Basionym: Perenniporia brasiliensis, see Lira et al in Crous et al. [For a detailed description of s et al. .Tropicoporia brasiliensis was originally described from Brazil as Perenniporia brasiliensis by Lira et al ; it hass et al. .Tropicoporia centrali-africana (Decock & Mossebo) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847390Perenniporia centrali-africana Decock & Mossebo, Systematics and Geography of Plants 71(2): 608 (2002).Basionym: Perenniporia centrali-africana, see Decock and Mossebo [For a detailed description of  Mossebo .Tropicoporia centrali-africana was originally described from Cameroon as Perenniporia centrali-africana [Notes: africana . Lira etafricana ) also reTropicoporia vanhulleae (Decock & Ryvarden) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847391Perenniporia vanhulleae Decock & Ryvarden, Index Fungorum 234: 1 (2015)Basionym: Perenniporia vanhulleae, see Decock and Ryvarden [For a detailed description of Ryvarden .Tropicoporia vanhulleae was originally described from Africa as Perenniporia vanhulleae by Decock and Ryvarden [T. vanhulleae from the type specimens fell into Tropicoporia in our phylogeny. Therefore, Perenniporia vanhulleae is transferred to Tropicoporia.Notes: Ryvarden . The seqTropicoporiaKey to species of T. centrali-africana1. Basidiocarps resupinate to pileate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a01. Basidiocarps resupinate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262T. brasiliensis2. Basidiospores < 5 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262. Basidiospores > 5 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263T. aridula3. Basidiospores 6\u20137 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0T. vanhulleae3. Basidiospores 5.5\u20136.0 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Truncatoporia B.K. Cui & Xing Ji, gen. nov.MycoBank: MB 847359Differs from other genera by its resupinate to pileate basidiocarps, a dimitic to trimitic hyphal system with dextrinoid and cyanophilous skeletal hyphae, and thick-walled, ellipsoid, truncate, and cyanophilous basidiospores.Truncatoporia truncatospora (Lloyd) B.K. Cui & Xing JiType species: Truncatoporia (Lat.) refers to the truncate basidiospores of the genus.Etymology: Basidiocarps are annual to perennial, resupinate to pileate, and corky. Pileal surface is brown to ochraceous. Pore surface is buff to pale yellowish buff upon drying; pores are round to angular; dissepiments thin, entire. Context cream buff to pale brown. Tubes are concolorous with pore surface. Hyphal system is dimitic to trimitic; generative hyphae with clamp connections; skeletal hyphae are dextrinoid and CB+; tissues are unchanged in KOH. Basidiospores are ellipsoid, truncate, hyaline, thick-walled, smooth, dextrinoid or not, and CB+.Perenniporia, P. pyricola Y.C. Dai & B.K. Cui and P. truncatospora (Lloyd) Ryvarden, grouped together and formed a well-supported clade  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847392Perenniporia pyricola Y.C. Dai & B.K. Cui, Mycosystema 29(6): 815 (2010).Basionym: Perenniporia pyricola, see Dai [For a detailed description of  see Dai .Truncatoporia pyricola was first described in Perenniporia from Northeast China [Pyrus and Prunus. The species has perennial and resupinate basidiocarps with cream to pale cinnamon, a dimitic hyphal system with dextrinoid and cyanophilous skeletal hyphae, thick-walled and truncate, dextrinoid, and cyanophilous basidiospores.Notes: st China . It has Pyrus, 2 August 2008, Dai 10265 ; Tianjin, Ji County, Panshan Mountain, Living tree of Crataegus, 6 August 2015, Dai 15496 (BJFC), Dai 15498 (BJFC).Specimens examined: CHINA. Liaoning, Anshan, Qianshan Park, on living tree of Truncatoporia truncatospora (Lloyd) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847393Trametes truncatospora Lloyd, Mycol. Writ. 6: 853 (1919).Basionym: Perenniporia truncatospora (Lloyd) Ryvarden, Acta Mycol. Sin. 5: 228 (1986).= Perenniporia truncatospora, see Cui et al. [For a detailed description of i et al. .Truncatoporia truncatospora and T. pyricola both have dextrinoid skeletal hyphae and truncate basidiospores, but the former has pileate basidiocarps, smaller pores (6\u20138 per mm), and non-dextrinoid basidiospores, and the latter has resupinate basidiocarps, larger pores (3\u20135 per mm) [Notes:  per mm) , and dexQuercus, 14 September 2019, Cui 17770 (BJFC).Specimen examined: CHINA. Sichuan, Ganluo County, Shengli, Gaoqiao Village, on living tree of TruncatoporiaKey to species of T. pyricola1. Basidiocarps resupinate, 3\u20135 per mm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026T. truncatospora1. Basidiocarps pileate, 6\u20138 per mm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Vanderbyliella B.K. Cui & Xing Ji, gen. nov.MycoBank: MB 847360Differs from other genera by its pileate basidiocarps with an orange brown pileal surface, a dimitic hyphal system with strongly dextrinoid skeletal hyphae, and hyaline, thick-walled, ellipsoid, non-truncate, and cyanophilous basidiospores.Vanderbyliella tianmuensis (B.K. Cui & C.L. Zhao) B.K. Cui & Xing JiType species: Vanderbylia.Etymology: Vanderbyliella (Lat.) refers to the morphological similarity to Basidiocarps are annual to perennial, pileate, and hard corky to woody hard when dry. Pileal surface is clay-buff, orange-brown to yellowish brown, glabrous, and concentrically sulcate. Pore surface is buff to pale brown when dry; pores are round to angular; dissepiments thin, entire. Context cream to pale brown, corky to hard corky. Tubes are buff to pale brown, hard corky to woody hard. Hyphal system is dimitic; generative hyphae with clamp connections; skeletal hyphae are strongly dextrinoid and CB+; tissues are unchanged in KOH. Cystidia are absent; cystidioles are present. Basidiospores are ellipsoid, non-truncate, hyaline, thick-walled, smooth, dextrinoid or not, and CB+.Perenniporia tianmuensis B.K. Cui & C.L. Zhao and an unknown species, grouped together and formed a strongly supported clade  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847394Perenniporia tianmuensis B.K. Cui & C.L. Zhao, Mycoscience 54: 236 (2013).Basionym: Perenniporia tianmuensis, see Zhao and Cui [For a detailed description of  and Cui .Vanderbyliella tianmuensis was first described in Perenniporia from China by Zhao and Cui [Notes:  and Cui ; it is cSpecimens examined: CHINA. Zhejiang, Lin\u2019an County, Tianmushan Nature Reserve, on the basis of dead angiosperm trees, 10 October 2005, Cui 2648 .Xanthoperenniporia B.K. Cui & Xing Ji, gen. nov.MycoBank: MB 847361Differs from other genera by its resupinate basidiocarps with yellow pore surface, weakly dextrinoid, and cyanophilous skeletal hyphae, hyaline, thick-walled, ellipsoid, truncate, and cyanophilous basidiospores.Xanthoperenniporia tenuis (Schwein.) B.K. Cui & Xing JiType species: Xanthoperenniporia (Lat.) refers to resembling Tropicoporia but with a yellow pore surface.Etymology: Basidiocarps are annual to perennial, resupinate to reflexed-effused, and corky when dry. Pore surface is cream to yellow when fresh, buff to yellow; pores are round to angular. Subiculum is thin, cream, buff to pale yellowish brown. Tubes are concolorous with pore surface. Hyphal system is dimitic to trimitic, generative hyphae with clamp connections; skeletal hyphae are dextrinoid or weakly dextrinoid and CB+; tissues are unchanged in KOH. Cystidia are absent, cystidioles are usually present. Basidiospores are ellipsoid, truncate, hyaline, thick-walled, smooth, dextrinoid or not, and CB+.Perenniporia maackiae (Bondartsev & Ljub.) Parmasto, P. punctata Hai J. Li & Jing Si, P. subcorticola Chao G. Wang & F. Wu and P. tenuis (Schwein.) Ryvarden clustered together and formed a single clade with good support  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847395Fomitopsis maackiae Bondartsev & Ljub., Botanicheskie Materialy 15: 103 (1962).Basionym: Perenniporia maackiae (Bondartsev & Ljub.) Parmasto, Ann. Bot. fenn. 32(4): 223 (1995).= Perenniporia maackiae, see Cui et al. [For a detailed description of i et al. .Xanthoperenniporia maackiae grows mainly on Maackia. Xanthoperenniporia subcorticola is similar to X. maackiae by sharing resupinate basidiocarps with yellow pore surfaces, similar sized pores, and the presence of cystidioles, but X. subcorticola has smaller basidiospores (4.2\u20135 \u00d7 3.5\u20134.2 \u00b5m) [X. maackiae has larger basidiospores (5.4\u20136.3 \u00d7 3.8\u20135.0 \u00b5m).Notes: \u20134.2 \u00b5m) , and X. Maackia, 26 August 2014, Cui 11531 (BJFC). Jilin, Antu County, Changbaishan Nature Reserve, on dead tree of Maackia, 11 September 2014, Dai 14780 (BJFC).Specimens examined: CHINA. Heilongjiang, Yichun, Dailing, Liangshui Nature Reserve, on fallen branch of Xanthoperenniporia punctata (Hai J. Li & Jing Si) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847396Perenniporia punctata Hai J. Li & Jing Si, Phytotaxa 360(1): 56 (2018).Basionym: Perenniporia punctata, see Li et al. [For a detailed description of i et al. .Xanthoperenniporia punctata was recently described in Perenniporia from China by Li et al. [Notes: i et al.  and is cQuercus, 14 August 2017, Dai 17916 .Specimens examined: CHINA. Hubei, Yichang, Wufeng County, Chaibuxi National Forestry Park, on angiosperm stump, 15 August 2017, Dai 17923 , on rotten wood of Xanthoperenniporia subcorticola (Chao G. Wang & F. Wu) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847397Perenniporia subcorticola Chao G. Wang & F. Wu, MycoKeys 69: 62 (2020).Basionym: Perenniporia subcorticola, see Wang et al. [For a detailed description of g et al. .Xanthoperenniporia subcorticola was recently described in Perenniporia as P. subcorticola Chao G. Wang & F. Wu by Wang et al. [Perenniporia corticola by having a yellow pore surface, dimitic hyphal system, and truncate and dextrinoid basidiospores of almost the same size, but P. corticola differs from P. subcorticola by having arboriform skeletal hyphae and dendrohyphidia [Notes: g et al. . It is shyphidia .Pinus, 21 October 2005, Dai 7330 .Specimens examined: CHINA. Fujian, Wuyishan County, Wuyishan Nature Reserve, on rotten wood of Xanthoperenniporia tenuis (Schwein.) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847398Polyporus tenuis Schwein., Trans. Am. phil. Soc., New Series 4: 159 (1832).Basionym: Perenniporia tenuis (Schwein.) Ryvarden, Norw. J Bot. 20: 9 (1973).= Perenniporia tenuis, see Cui et al. [For a detailed description of i et al. .Xanthoperenniporia punctata is similar to X. tenuis in having resupinate basidiocarps, pale yellow to buff-yellow pore surface, and similar sized basidiospores, but the former differs in its smaller pores (6\u20139 per mm), and the absence of cystidioles and non-dextrinoid basidiospores [Notes: iospores , the latVitex, 25 August 2016, Dai 17026 (BJFC).Specimens examined: CHINA. Hubei, Yichang, Wufeng County, Chaibuxi National Forestry Park, on rotten angiosperm stump, 15 August 2017, Dai 17935 (BJFC). Shanxi, Yangcheng County, Manghe Nature Reserve, on fallen trunk of XanthoperenniporiaKey to species of X. punctata1. Basidiospores non-dextrinoid\u00a0.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a01. Basidiospores dextrinoid\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262Maackia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 X. maackiae2. Basidiocarps resupinate to reflexed-effused, growing on 2. Basidiocarps resupinate, growing on other trees\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263X. tenuis3. Basidiocarps annual, pores 4\u20136 per mm\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0X. subcorticola3. Basidiocarps perennial, pores 7\u20138 per mm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Yuchengia B.K. Cui & K.T. Steffen, Nordic J. Bot. 31(3): 333 (2013).MycoBank: MB 563490Yuchengia narymica (Pil\u00e1t) B.K. Cui, C.L. Zhao & K.T. Steffen, Nordic J. Bot. 31(3): 333 (2013).Type species: Basidiocarps are annual to perennial, resupinate, corky when fresh, and hard corky when dry. Pore surface is cream, yellowish buff to tan; pores are round to angular, dissepiments thin, entire. Subiculum is cream to pale ochraceous. Tubes are concolorous with pore surface. Hyphal system is dimitic; generative hyphae with clamp connections; skeletal hyphae amyloid or not, acyanophilous or weakly cyanophilous, dissolving in KOH. Cystidia are absent; cystidioles are present. Basidiospores are ellipsoid, truncate or not, hyaline, thick-walled, smooth, IKI\u2212, and CB+.Yuchengia narymica was first described as Trametes narymica Pil\u00e1t [Perenniporia by Pouzar [Yuchengia to accommodate Perenniporia narymica based on molecular data and morphological characteristics.Notes: The type species ca Pil\u00e1t  and latey Pouzar . Zhao ety Pouzar  proposedYuchengia kilemariensis (Spirin & Shirokov) B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847399Perenniporia kilemariensis Spirin & Shirokov, Folia Cryptogamica Petropolitana (Sankt-Peterburg) 6: 38 (2005).Basionym: Perenniporia kilemariensis, see Spirin et al. [For a detailed description of n et al. .Yuchengia kilemariensis was originally described from Russia as Perenniporia kilemariensis by Spirin et al. [Yuchengia. However, it has resupinate basidiocarps, a dimitic hyphal system with skeletal hyphae dissolving in KOH, and ellipsoid and non-dextrinoid basidiospores. These characters fit Yuchengia well. Moreover, in ITS + nLSU and five-gene phylogenetic analysis, the sequence of Y. kilemariensis from the type specimen fell into Yuchengia  B.K. Cui & Xing Ji, comb. nov.MycoBank: MB 847400Wrightoporia subadusta Z.S. Bi & G.Y. Zheng, Bull. Bot. Res., Harbin 7(4): 76 (1987).Basionym: Perenniporia subadusta (Z.S. Bi & G.Y. Zheng) Y.C. Dai, Ann. Bot. fenn. 39(3): 180 (2002).= Murinicarpus subadustus (Z.S. Bi & G.Y. Zheng) B.K. Cui & Y.C. Dai, Fungal Di-versity 97: 255 (2019).= Perenniporia cystidiata Y.C. Dai, W.N. Chou & Sheng H. Wu, Mycotaxon 83: 209 (2002).= Wrightoporia subadusta and Perennipori cystidiata, Cui et al. [Perenniporia and proposed Murinicarpus. Murinicarpus has the same characters with Microporellus Murrill, stipitate basidiocarps, dextrinoid skeletal hyphae, non-truncate and non-dextrinoid basidiospores [Microporellus in the current study.Notes: After studying type materials of i et al.  found thiospores , and it Perenniporia and related generaKey to species of Microporellus1. Basidiocarps stipitate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20261. Basidiocarps resupinate, effused-reflexed to pileate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262Abundisporus2. Basidiospores pale yellowish\u00a0\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a02. Basidiospores hyaline\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a033. Skeletal hyphae amyloid\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202643. Skeletal hyphae inamyloid\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20266Minoporus4. Basidiocarps pileate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20264. Basidiocarps resupinate to effused-reflexed\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a05Amylosporia5. Basidiospores amyloid\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Yuchengia5. Basidiospores inamyloid\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20266. Basidiocarps with rhizomorphs\u00a0.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a076. Basidiocarps without rhizomorphs\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a08Aurantioporia7. Pore surface orange to orange-brown \u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Rhizoperenniporia7. Pore surface grayish white to pale buff\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a08. Cystidia present\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a098. Cystidia absent\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a010Hornodermoporus9. Basidiocarps pileate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Cystidioporia9. Basidiocarps resupinate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202610. Basidiospores non-truncate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a01110. Basidiospores truncate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a01511. Basidiocarps pileate\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a01211. Basidiocarps resupinate\u00a0.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a013Vanderbylia12. Basidiospores obovoid \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Vanderbyliella12. Basidiospores ellipsoid\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Neoporia13. Basidiocarps annual\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202613. Basidiocarps annual to perennial\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202614Luteoperenniporia14. Hyphal system dimitic\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Poriella14. Hyphal system dimitic to trimitic\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Perenniporiopsis15. Basidiocarps osseous\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a015. Basidiocarps corky to woody hard\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202616Niveoporia16. Pore surface white when fresh, usually with reddish brown sterile margin\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202616. Pore surface cream, buff, yellowish to cinnamon, without reddish brown sterile margin\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a01717. Basidiocarps resupinate to pileate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a01817. Basidiocarps resupinate\u00a0\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a019Truncospora18. Basidiospores > 9 \u00b5m in length\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Truncatoporia18. Basidiospores < 9 \u00b5m in length\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202619. Tissues brown to black in KOH\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a02019. Tissues unchanged in KOH\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202621Dendroporia20. Dendrohyphidia present at dissepiment edges, basidiospores non-dextrinoid\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Citrinoporia20. Dendrohyphidia usually absent at dissepiment edges, basidiospores dextrinoid\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Xanthoperenniporia21. Pore surface yellow\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a021. Pore surface usually cream to cinnamon\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a022Macrosporia22. Basidiospores \u2265 9 \u00b5m in length\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a022. Basidiospores \u2264 9 \u00b5m in length\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a023Tropicoporia23. Skeletal hyphae usually non-dextrinoid\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a023. Skeletal hyphae usually dextrinoid\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a024Perenniporia24. Basidiospores < 6 \u00b5m in length\u00a0\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Macroporia24. Basidiospores > 6 \u00b5m in length\u00a0.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u00a0Perenniporia in the core polyporoid clade, with species of Perenniporia forming more than one lineage. Our phylogenetic analyses confirmed that Perenniporia s. l. is polyphyletic and is nested in the core polyporoid clade.The phylogenetic analyses carried out by Binder et al.  placed PAbundisporus Ryvarden is closely related to some Perenniporia species. Ryvarden [Abundisporus to accommodate the species with pale yellowish and non-dextrinoid basidiospores, previously accepted in Loweporus J.E. Wright. Our phylogenetic analyses showed that Abundisporus is closely related to Cystidioporia, Macrosporia, and Niveoporia, but distant from the Perenniporia s. s. as previous studies [Abundisporus is distinguished from these genera by its more or less pinkish basidiocarps, non-truncate, and pale yellowish basidiospores [Ryvarden  establis studies ,19,31. Miospores .Truncospora, typified by T. ochroleuca (Berk.) Pil\u00e1t, was established by Pil\u00e1t [Perenniporia s. l. This genus has often been treated as a synonym of Perenniporia since its establishment [Perenniporia detrita (Berk.) Ryvarden, P. ochroleuca (Berk.) Pil\u00e1t and P. ohiensis (Berk.) Ryvarden formed a morphologically homogeneous group that could be recognized at the genus level. Later phylogeny showed that these three species formed a monophyletic clade that was distinct from Perenniporia s. s. [Truncospora mainly differs from Perenniporia s. s. by having button-shaped to ungulate basidiocarps, variably dextrinoid skeletal hyphae, and longly ovoid basidiospores [by Pil\u00e1t . The genby Pil\u00e1t ,69. Theslishment ,33,34,70lishment  consideria s. s. ,19,31. Miospores .Vanderbylia was established by Reid [Perenniporia [Vanderbylia is an independent genus that is distant from Perenniporia s. s. [Vanderbylia is closely related to Vanderbyliella, but the latter has ellipsoid and dextrinoid or non-dextrinoid basidiospores. Vanderbylia differs from Perenniporia s. s. by its pileate basidiocarps, dextrinoid skeletal hyphae, and non-truncate and subglobose to amygdaliform basidiospores. by Reid ; species by Reid ,17. It wnniporia ,72. Receia s. s. ,17,18,19Hornodermoporus was established by Teixeira [H. martius (Berk.) Teixeira. The genus is characterized by its pileate basidiocarps with a black crust at the pileal surface, a dimitic hyphal system with strongly dextrinoid and cyanophilous skeletal hyphae, the presence of cystidia, truncate, oblong-ellipsoid, and strongly dextrinoid basidiospores. Hornodermoporus was treated as a synonym of Perenniporia [Perenniporia s. s. in having pileate basidiocarps with a black crusted pileal surface, dextrinoid skeletal hyphae, the presence of cystidia, and oblong-ellipsoid basidiospores. In our phylogeny, the separation of Hornodermoporus from Perenniporia is supported, as previously reported [Teixeira  and typinniporia ,58, but reported ,20,36,74Amylosporia, Murinicarpus, Perenniporiopsis, Poriella, and Yuchengia have recently been segregated from Perenniporia s. l. These genera were phylogenetically distant from Perenniporia s. s. in our study. Amylosporia differs from Perenniporia s. s. by its amyloid skeletal hyphae and basidiospores [Amylosporia hattorii and Perenniporia amylodextrinoidea Gilb. & Ryvarden both have amyloid skeletal hyphae, but the former has amyloid and larger basidiospores (10\u201312 \u00d7 5.5\u20137.5 \u03bcm) [Murinicarpus, newly proposed by Cui et al. [Wrightoporia subadusta = Perenniporia cystidiata, has same characters with Microporellus and is a synonym of Microporellus. It is different from Perenniporia s. s. by its stipitate basidiocarps, dextrinoid skeletal hyphae, presence of thick-walled cystidia, non-truncate and non-dextrinoid basidiospores [Perenniporiopsis is distinguished from Perenniporia s. s. by its basidiocarps waxy when fresh and rigidly osseous when dry, dextrinoid skeletal hyphae, and large basidiospores [Poriella as having non-dextrinoid basidiospores. Perenniporia africana and P. ellipsospora are congeneric, although they have dextrinoid basidiospores; they are transferred to Poriella. Poriella is distinguished from Perenniporia s. s. by its unbranched, dextrinoid skeletal hyphae, and non-truncate basidiospores. Perenniporia kilemariensis, which has inamyloid skeletal hyphae and truncate basidiospores is transferred to Yuchengia, so the definition of Yuchengia is revised also in this study. Yuchengia differs from Perenniporia s. s. by its non-dextrinoid basidiospores, unbranched \u201camyloid\u201d skeletal hyphae.iospores . Amylosp\u20137.5 \u03bcm) , and the\u20137.5 \u03bcm) . Murinici et al.  based oniospores . Perenniiospores . Chen etiospores  describeAurantioporia, Citrinoporia, Cystidioporia, Dendroporia, Luteoperenniporia, Macroporia, Macrosporia, Minoporus, Neoporia, Niveoporia, Rhizoperenniporia, Tropicoporia, Truncatoporia, Vanderbyliella, and Xanthoperenniporia from Perenniporia s. l. was well supported by phylogenetic analyses of ITS + nLSU and ITS + nLSU + mtSSU + TEF1 + TBB1. In the ITS + nLSU gene phylogenetic tree, Yuchengia is closely related to Poriella; in the combined five-gene phylogenetic analysis, Yuchengia was related to Minoporus, Neoporia, Poriella, and Vanderbyliella. However, Yuchengia has resupinate basidiocarps, amyloid or not, acyanophilous or weakly cyanophilous skeletal hyphae, and non-dextrinoid basidiospores; Minoporus produces pileate basidiocarps, cyanophilous skeletal hyphae, and truncate and dextrinoid basidiospores; Neoporia has dextrinoid and cyanophilous skeletal hyphae and dextrinoid basidiospores; Poriella has dextrinoid and cyanophilous skeletal hyphae [Vanderbyliella has pileate basidiocarps and dextrinoid and cyanophilous skeletal hyphae.The segregation of l hyphae ; VanderbPerenniporia s. l. have a new generic placement. For four species in the phylogenetic tree, including Perenniporia tephropora (Mont.) Ryvarden, P. subtephropora B.K. Cui & C.L. Zhao, P. eugeissonae P. Du & Chao G. Wang and P. straminea (Bres.) Ryvarden, due to unclear phylogenetic relationships and insufficient morphological difference from Perenniporia s. s. species, their taxonomic positions were remained in Perenniporia s. l.According to phylogenetic analyses and morphological characters, some species of Perenniporia s. l., 44 species of Perenniporia s. l. with available sequences. According to phylogenetic evidence and morphological characteristics, 15 new genera were set up, 2 new species were described, and 37 new combinations were proposed. Species lacking reliable molecular sequences were not included in this study.In summary, we carried out a comprehensive study on"}
{"text": "In this paper, a class of fractional deterministic and stochastic susceptible-infected-removed- susceptible (SIRS) epidemic models with vaccination is proposed. For the fractional deterministic SIRS epidemic model, the existence of solution and the stability of equilibrium points are analyzed by using dynamic method. Then, the appropriate controls are established to effectively control the disease and eliminate it. On this basis, the fractional stochastic SIRS epidemic model with vaccination is further considered, and a numerical approximation method is proposed. The correctness of the conclusion is verified by numerical simulation. S), infected (I), and recovered (R). The SIR infectious disease model was established by using kinetic method. The spreading law and prevalence trend were also studied. In the past 20 years, the research on the dynamics of infectious diseases has developed rapidly in the world, and a large number of mathematical models have been used to analyze various infectious diseases. Most of these mathematical models are applicable to the study of general laws of various infectious diseases  and CDt\u03b1f(t) \u2208 C, 0 < \u03b1 \u2264 1. If CDt\u03b1f(t) \u2265 0, \u2200t \u2208 , then for \u2200t \u2208 , f(t) is a nondecreasing function; if CDt\u03b1f(t) \u2264 0, \u2200t \u2208 , then for \u2200t \u2208 , f(t) is a nonincreasing function.Suppose x(t) \u2208 \u211d be a continuous and derivable function. Then, for any t \u2265 0, one hasLet x(t) \u2208 \u211d+ be a continuous and derivable function. Then, for any t \u2265 0, one hasLet f \u2208 C, F \u2208 C, there exist \u03c7, \u03c70 and T such that for \u2200t \u2265 T, the following inequalitySuppose X(t) = T \u2208 \u211d+3, t \u2265 0 under the initial conditions -(3) of f satisfies the condition (4) of x1(t) = S(t), x2(t) = I(t), x3(t) = R(t), x1(0) = S(0) = S0, x2(0) = I(0) = I0, x3(0) = R(0) = R0,Obviously, the vector function he model  can be rf) = A0 + A1X(t) + x1(t)A2X(t) + x2(t)A3X(t). Then,Set . Then,1ft,Xt=A0+N(t) = S(t) + I(t) + R(t), we haveSecondly, it is proved that all the solutions of fractional model  are unifCDt\u03b1NR(t) + \u03bcNR(t) = \u039b \u2212 \u03b5I \u2264 \u039b. According to Lemma 9 in . Because I(0) \u2265 0 and E\u03b1,1 > 0, so I(t) \u2265 0. From the third equation of model  \u2265 R(0)E\u03b1,1[\u2212a4t\u03b1]. Because R(0) \u2265 0 and E\u03b1,1 > 0, so R(t) \u2265 0. Hence, all the solutions of fractional model  > 0. To calculate the basic reproduction number of model  and rewrite in the matrix formIn order to study the equilibrium points of model , let CDten model  has a diof model , we reorof model  by settiE0 = . Then, the basic reproduction number R0 is the spectral radius of \u2131\ud835\udcb1\u22121, which takes the form R0 = a1S0/a3.The Jacobian of Equation  around ER0 < 1, then the disease-free equilibrium point E0 of model , and \u03bb3 = a3(R0 \u2212 1). Obviously, \u03bb1 < 0, \u03bb2 < 0, and the sign of \u03bb3 depends on R0. If and only if the R0 < 1, all of eigenvalues of J|E0 are negative. Hence, model  \u2208 \ud835\udc9f|CDt\u03b1V = 0}. When t\u27f6+\u221e, M\u27f6{E0}. So, E0 is the unique greatest positive invariant of M. According to generalized Lyapunov-Lasalle's invariance principle \u03c5, where t0 is the initial time, \u03bb \u2265 0, \u03c5 > 0, m(0) = 0, m(X) \u2265 0, and m(X) is locally Lipschitz on X \u2208 \u211d3 with the Lipschitz constant m0.The trivial solution of the following initial value problemX = . If there exists a continuously differentiable function V: . Combining the above formula with ) = V)/k1, \u03bb = k3/k2, and\u2009\u03c5 = 1/k0, by E0 of model , whereThe endemic equilibrium is equilibrium of model  in whichR0 > 1, then there exists the endemic equilibrium point E\u2217; otherwise, it does not exist.If a2a4 \u2212 ep = \u03bc(\u03bc + p + e) > 0 and ce + a4(\u03b8 \u2212 a3) = \u2212(\u03bc + e)(\u03bc + \u03b5) \u2212 \u03bcc < 0.After simple calculation, R0 > 1, then 1 \u2212 R0 < 0. Thus, I\u2217 > 0, thereby R\u2217 > 0. Consequently, the endemic equilibrium point exists, if R0 > 1.If E\u2217 isThe Jacobian matrix of model  around EJ|E\u2217 can be obtained as follows:A1 = a1I\u2217 + a2 + a4 > 0, A2 = a1I\u2217(2\u03bc + \u03b5 + c + e) + \u03bc(\u03bc + p + e) > 0, Through simple calculation, the characteristic equation of J|E\u2217 satisfy |arg\u03bb| > \u03b1\u03c0/2, then the endemic equilibrium point E\u2217 of model  = 18A1A2A3 + (A1A2)2 \u2212 4A3A13 \u2212 4A23 \u2212 27A32. According to  \u00d7 [(S \u2212 S\u2217) + (I \u2212 I\u2217)] + \u03c41a1(I \u2212 I\u2217)(S \u2212 S\u2217) + \u03c42(R \u2212 R\u2217)[c(I \u2212 I\u2217) + p(S \u2212 S\u2217) \u2212 a4(R\u2212R\u2217)] = \u2212a2(S \u2212 S\u2217)2 \u2212 (a3 \u2212 \u03b8)(I \u2212 I\u2217)2 \u2212 \u03c42a4(R \u2212 R\u2217)2 + (S \u2212 S\u2217)(I \u2212 I\u2217)(\u03b8 \u2212 a3 \u2212 a2 + \u03c41a1) + (e + \u03c42p)(S \u2212 S\u2217)(R \u2212 R\u2217) + (e + \u03c42c)(I \u2212 I\u2217)(R \u2212 R\u2217). Because \u03c41 = (a2 + a3 \u2212 \u03b8)/a1, so \u03b8 \u2212 a3 \u2212 a2 + \u03c41a1 = 0. Thus,a3 \u2212 \u03b8 = \u03bc + \u03b5 + c > 0. Let \u03c42 > 0, such that \u03c42a4 \u2212 ((e + \u03c42p)2/3a2) \u2212 ((e + \u03c42c)2/3(a3 \u2212 \u03b8)) \u2265 0. Thus, CDt\u03b1V \u2264 0. CDt\u03b1V = 0 if and only if S = S\u2217, I = I\u2217, and R = R\u2217. Set M = { \u2208 \ud835\udc9f|CDt\u03b1V = 0}. When t\u27f6+\u221e, M\u27f6{E\u2217}. So, E\u2217 is the unique greatest positive invariant of M. Hence, the disease-free equilibrium point E\u2217 of model  = 1, I(0) = 0.4, and R(0) = 0. In this case, after the calculation, we have E0 =  and R0 = 0.68 < 1. Then, according to E0 is locally asymptotically stable.In model , let \u039b =a1 = 0.8 are the same as those in E\u2217 = , R0 = 1.36 > 1, A1 = 0.70912, A2 \u2248 0.1662, A3 = 0.0144, A1A2 \u2248 0.1178557 > A3, and D(P) \u2248 0.000839 > 0. By E\u2217 is locally asymptotically stable.The values of the parameters except To support our results, we present computer simulations shown in Figures 1E0 = . Figures 44E\u2217 = .Figures 11R0 < 1, is globally asymptotically stable. That is, infectious diseases will be eliminated from the area. When R0 > 1, E0 is unstable, and E\u2217 is globally asymptotically stable and becomes endemic. In order to avoid the formation of E\u2217, we take measures to effectively control the disease, so that when R0 > 1, E0 can also be globally asymptotically stable, to eliminate the disease. To this end, we apply controller U so that model  represent isolation rate and cure rate, respectively, and consider the controller of the following form:In medicine, two methods are commonly used to exert control over an infected person: one is effective medication, and the other is to isolate those who have the disease from those who are susceptible. Let U = \u2212k1a1SI \u2212 k2I and 0 < k1 < 1, k2 > 0, then when R0 > 1, E0 is globally asymptotically stable in \ud835\udc9f.If V = I, thenConsider the Lyapunov function CDt\u03b1V = 0 if and only if I = 0. And E0 is the largest invariant set on I = 0, so E0 is globally asymptotically stable.\u039b = 0.2, a1 = 0.8, a2 = 0.26, a3 = 0.5, a4 = 0.34, \u03b8 = 0.05, e = 0.14, c = 0.15, p = 0.06, \u03b5 = 0.1, \u03bc = 0.2, S(0) = 1, I(0) = 0.4, and\u2009R(0) = 0. In this case, after the calculation, we have E0 =  and R0 = 1.36 > 1. Let k1 and k2 take different values that satisfy the conditions in 7k1 and k2 are, the faster the rate of disease I tends to 0; thus, the disease is eliminated. That is, the controller selected in this part is effective.To support our results, we present numerical simulations shown in Figures 7\u03c3i(0) = 0, i = 1, 2, 3.\u03c3i > 0 represents the intensity of white noise, which is an independent standard Brownian motion. The disease-free equilibrium obtained when I(t) \u2261 0 is the same point as deterministic model  = (1/2)S2(t) + (1/2)I2(t) + (1/2)R2(t). By We extend SIRS model  to stochic model . The staCDt\u03b1S=\u039b\u2212aCDt\u03b1S=\u039b\u2212ab + ((\u03b8 + e + p)/2) \u2212 a1I \u2212 a2 + \u03c31(d\u03c31/dt) < 0, a1S \u2212 a3 + ((b + \u03b8 + c)/2) + \u03c32(d\u03c32/dt) < 0, \u2212a4 + ((b + e + p + c)/2) + \u03c33(d\u03c33/dt) < 0. Then, CDt\u03b1V \u2264 0. Hence, the stochastic SIRS model T, G(X) = , d\u03c3/dt = T and the mapping F(X) = (F1(X), F2(X), F3(X))T with F1(X) = \u039b \u2212 a1SI + \u03b8I + eR \u2212 a2S, F2(X) = a1SI \u2212 a3I, F3(X) = cI + pS \u2212 a4R. By Fractional stochastic SIRS model  can be wGLDt\u03b1) is defined as follows:t = nh. For details, see [tn = nh, wj\u03b1 satisfies recursive relation as follows:From the perspective of the numerical implementation, the fractional derivative definition of the Grunwald-Letnikov is the most direct. So, it is more appropriate to solve the fractional stochastic epidemic model. The Grunwald-Letnikov fractional derivative  is a continuous integrable function, then the relation between Grunwald-Letnikov and Caputo fractional derivative is as follows:If \u03c3i are real constants, \u03bei(n) represent the three-dimensional Gaussian white noise processes, i = 1, 2, 3, and\u03b4ij represents Kronecker delta and \u03b4(t1 \u2212 tj) represents the Dirac function.For details, see \u201335. Mode\u039b = 0.2, a1 = 0.4, a2 = 0.26, a3 = 0.5, a4 = 0.34, \u03b8 = 0.05, e = 0.14, c = 0.15, p = 0.06, S(0) = 1, I(0) = 0.4, and R(0) = 0. In this case, after the calculation, we have E0 =  and R0 = 0.68 < 1. Numerical simulations are performed for the different fractional order values. Figures 13\u03c31 = 0.4, \u03c32 = 0.75, \u03c33 = 0.35. Figures 16S(t) for different fractional derivative values 1, 0.8, and 0.6. Figures 19I(t) for different fractional derivative values 1, 0.8, and 0.6. Figures 22R(t) for different fractional derivative values 1, 0.8, and 0.6. Observing the images, we found that different random perturbations and different fractional derivative values affect the speed of the system to the equilibrium point.In this part, we present numerical simulations of stochastic fractional SIRS epidemic model . Let \u039b =R0 > 1, so that the disease can be eliminated. On the other hand, by introducing noise into the disease transmission term, a fractional stochastic SIRS epidemic model with vaccination is further considered. The stability result of the stochastic fractional SIRS model at the equilibrium point is given. A numerical approximation method for fractional stochastic SIRS epidemic model is proposed. The correctness of the conclusion is verified by numerical simulation in each section. Our study shows that the fractional stochastic epidemic models based on virus dynamics are more realistic. This theory can provide a solid foundation for the study of similar diseases and has a wide range of applications in the biomedical field. For example, a stochastic delayed infectious disease model can be considered to study the effect of incubation periods on disease dynamics. In addition, our proposed theory can also be used to study other infectious diseases, such as HIV, COVID-19, and tuberculosis. We leave these problems to future work.This paper investigates a class of deterministic and stochastic fractional SIRS epidemic models with vaccination. The models studied in this paper are more general. Specifically, this paper extends the deterministic model of  to the S"}
{"text": "In the above abstract, the author \u201cMich\u00e8le Pr\u00e9vost\u201d was originally credited as \u201cMichele Provost\u201d; this error has since been corrected."}
{"text": "G be a graph, and the number of components of G is denoted by c(G). Let t be a positive real number. A connected graph G is t-tough if tc(G \u2212 S) \u2264 |S| for every vertex cut S of V(G). The toughness of G is the largest value of t for which G is t-tough, denoted by \u03c4(G). We call a graph G Hamiltonian if it has a cycle that contains all vertices of G. Chv\u00e1tal and other scholars investigate the relationship between toughness conditions and the existence of cyclic structures. In this paper, we establish some sufficient conditions that a graph with toughness is Hamiltonian based on the number of edges, spectral radius, and signless Laplacian spectral radius of the graph.Let MR subject classifications: 05C50, 15A18. G =  be a finite simple undirected graph with vertex set V(G) = {v1, v2, \u2026, vn} and edge set E(G). Write by m = |E(G)| the number of edges and n = |V(G)| the number of vertices of the graph G, respectively. The set of neighbors of a vertex v in graph G is denoted by NG(v). Let vi \u2208 V(G), we denote by di = dvi = dG(vi) = |NG(vi)| the degree of vi. Denote by \u03b4(G) [\u0394(G)] or simply \u03b4 (\u0394) the minimum (maximum) degree of G. Let  be a nondecreasing degree sequence of G, that is, d1 \u2264 d2 \u2264 \u22ef \u2264 dn. For convenience, we use G, where xk is the number of vertices of degree k in the graph G. We denote a bipartite graph with bipartition  by using G. We denote the cycle and the complete graph on n vertices by using Cn and Kn, respectively. We use Km, n to denote a complete bipartite graph with two parts having m, n vertices, respectively. Let G and H be two disjoint graphs. We denote by G+H the disjoint union of G and H, which is a graph with vertex set V(G) \u222a V(H) and edge set E(G) \u222a E(H). If G1 = G2 = \u2026 = Gk, we denote G1 + G2 + \u22ef + Gk by kG1. We denote by G \u2228 H the join of G and H, which is a graph obtained from the disjoint union of G and H by adding edges joining every vertex of G to every vertex of H. Let Kn \u2212 1 + v denote the complete graph Kn \u2212 1 together with an isolated vertex v. Other undefined symbols reference can be seen in Bondy and Murty (Let nd Murty  and Bauend Murty .adjacency matrix of G is A(G) = (aij), where aij = 1 if vi and vj are adjacent in G and aij = 0 otherwise. Let D(G) be the degree diagonal matrix of G, i.e., D(G) = diag{dG(v1), dG(v2), \u2026, dG(vn)}. The matrix Q(G) = D(G) + A(G) is called the signless Laplacian matrix of G. The largest eigenvalue of A(G), denoted by \u03bc(G), is called to be the spectral radius of G. The largest eigenvalue of Q(G), denoted by q(G), is called to be the signless Laplacian spectral radius of G.The G is called a Hamilton graph if it has a Hamilton cycle, and then we also call G Hamiltonian. The number of components of G is denoted by c(G). Let t be a positive real number. A connected graph G is t-tough if tc(G \u2212 S) \u2264 |S| for every vertex cut S of V(G). The toughness of G is the largest value of t for which G is t-tough, denoted by \u03c4(G). If G is a complete graph, take \u03c4(Kn) = \u221e for all n \u2265 1. If G is not a complete graph, G. Obviously, a t-tough graph is s-tough for all s < t.A cycle (path) containing every vertex of a graph is called a Hamilton cycle (path) of the graph. Graph NP-complete problem. In recent years, the study of Hamiltonian problem using spectrum graph theory has received extensive attention, and some meaningful results are obtained, such as Fiedler and Nikiforov  of the graph G, if there is a one-to-one mapping \u03c6 from vertex set V(G) of the graph to the components of the vector X; simply written Xu = \u03c6(u).At the beginning of this section, we first give some definitions. Let A(G) corresponding to the eigenvector X if and only if X \u2260 0,When \u03bc is an eigenvalue of the adjacency matrix characteristic equation of G.The Equation (2.1) is called the q is an eigenvalue of signless Laplacian matrix Q(G) corresponding to the eigenvector X if and only if X \u2260 0,When signless Laplacian characteristic equation of G.The Equation (2.2) is called the Lemma 2.1. Ho\u00e0ng , and G is t-tough, then \u03b4(G) \u2265 2t.Proof Let \u03b4(G) = dv, S = NG(v), then |S| = |NG(v)| = \u03b4(G). We can get c(G \u2212 S) \u2265 2. Because G is not a complete graph, bythenG) \u2265 2t.thus, we can get \u03b4(The proof is completed.\u25a0Lemma 2.6. Jung (G be a graph without a Hamiltonian cycle and at least 11 vertices. Then.6. Jung  let G bei) there exist two non-adjacent vertices x, y such that d(x) + d(y) \u2264 |V(G)| \u2212 5 or(ii) there exist for some t \u2265 1 vertices x1, x2, \u2026, xt such that G \u2212 x1 \u2212 \u22ef \u2212 xt has at least t + 1 components. + d(y) \u2264 |V(G)| \u2212 5.Proof Because G has no Hamiltonian cycle, G is not a complete graph. If there exist for some s \u2265 1 vertices x1, x2, \u2026, xs such that G \u2212 x1 \u2212 \u22ef \u2212xs has at least s + 1 components, bythena contradiction.The proof is completed.\u25a0G, denoting by G by recursively joining pairs of nonadjacent vertices whose degree sum is at least n until no such pair remains, refer to Bondy and Chv\u00e1tal  and simple connected graph with n(\u2265 8t) vertices and m edges. IfthenG is Hamiltonian when t \u2208 {1, 2} and n \u2265 8t.(i) G is Hamiltonian when t = 3 and n > 9t.(ii) Proof Suppose that G is not a Hamilton graph. By Lemma 2.1, there exists a positive integer k for dk \u2264 k, such that dn \u2212 k + t \u2264 n \u2212 k \u2212 1. Then we havethusk \u2212 2t)(2n \u2212 3k \u2212 5t \u2212 1) \u2264 0. Next, we discuss three cases.Since Case 1t = 1.n \u2265 8t = 8, (k \u2212 2)(2n \u2212 3k \u2212 6) \u2264 0. By Lemma 2.5, \u03b4 (G)\u2265 2t = 2. Since \u03b4(G) \u2264 dk \u2264 k, then k \u2265 2.In this case, Case 1.1 (k \u2212 2)(2n \u2212 3k \u2212 6) = 0, i.e., k = 2; or k \u2260 2 and 2n \u2212 3k \u2212 6 = 0.Case 1.1.1k = 2.G is a graph with d2 \u2264 2, dn \u2212 1 \u2264 n\u22123, dn \u2264 n \u2212 1, and we have G must be with degree sequence .In this case, G are non-adjacent, G must be the graph K1 \u2228 (Kn \u2212 3 \u2212 uv + wu + zv), where u, v \u2208 V(Kn \u2212 3), w, z \u2209 V(Kn \u2212 3), is Hamiltonian, a contraction.If two 2-degree vertices of G are adjacent, G must be the graph K1 \u2228 (K2 + Kn \u2212 3), but If two 2-degree vertices of Case 1.1.2k \u2260 2 and 2n \u2212 3k \u2212 6 = 0.n \u2264 11 because n = 9, k = 4. Then d4 \u2264 4, d6 \u2264 4, d9 \u2264 8, and we have G must be with degree sequence , and G = K3 \u2228 (3K2) is Hamiltonian, a contradiction.In this case, we can get Case 1.2 (k \u2212 2)(2n \u2212 3k \u2212 6) < 0.k \u2265 3 and 2n \u2212 3k \u2212 6 < 0. Since n \u2265 2k + 1. From these results, we have 4k + 2 \u2264 2n \u2264 3k + 5, that is k \u2264 3. Thus, we have k = 3, n = 7. A contradiction with known condition n \u2265 8.In this case, Case 2t = 2.k \u2212 4)(2n \u2212 3k \u2212 11) \u2264 0. By Lemma 2.5, \u03b4(G)\u2265 2t = 4. Since \u03b4(G) \u2264 dk \u2264 k, then k \u2265 4.In this case, (Case 2.1 (k \u2212 4)(2n \u2212 3k \u2212 11) = 0, i.e., k = 4; or k \u2260 4 and 2n \u2212 3k \u2212 11 = 0.Case 2.1.1k = 4.G is a graph with d4 \u2264 4, dn \u2212 2 \u2264 n \u2212 5, dn \u2264 n \u2212 1, and we have G is the graph with degree sequence n \u2212 6, (n \u2212 1)2), and n \u2265 8t = 16 for t = 2. Let S is the set containing four 4 \u2212 degree vertices of G, and there exist two non-adjacent vertices in S by Corollary 2.7.In this case, Lemma 2.8, S form a complete graph Kn \u2212 4 in Kn \u2212 4 is adjacent to one vertex of S, it must be adjacent to all vertices of S. Moreover, there are at least 2 vertices of the Kn \u2212 4 adjacent to all vertices of S because there exist two non-adjacent vertices in S by Corollary 2.7.By Kn \u2212 4 adjacent to all vertices of S, then K2 \u2228  is Hamiltonian, then If there are 2 vertices of the Kn \u2212 4 adjacent to all vertices of S, then K3 \u2228 (Kn \u2212 7 + 2K2) is Hamiltonian, then If there are 3 vertices of the Kn \u2212 4 adjacent to all vertices of S, then K4 \u2228 (Kn \u2212 8 + 4K1) is Hamiltonian, then If there are 4 vertices of the Kn \u2212 4 adjacent to all vertices of S, Ki \u2228 (Kn \u2212 4 \u2212 i + 4K1) is Hamiltonian, then If there are more than 4 vertices of the Case 2.1.2k \u2260 4 and 2n \u2212 3k \u2212 11 = 0.n \u2264 21 because t \u2264 n, hence n = 16, k = 7. Then d7 \u2264 7, d11 \u2264 8, d16 \u2264 15, and we have 7, 84, 155). Then there are no two vertices x, y that are not adjacent such that d(x) + d(y) \u2264\u2223 V(G) \u2223 \u2212 5. By Corollary 2.7, we can get G is Hamiltonian, a contradiction.In this case, we can get 16 \u2264 Case 2.2 (k \u2212 4)(2n \u2212 3k \u2212 11) < 0.k \u2265 5 and 2n \u2212 3k \u2212 11 < 0. Since n \u2265 2k + 1. From these results, we have 4k + 2 \u2264 2n \u2264 3k + 10, that is k \u2264 8. Thus, we have 16 = 8t \u2264 n \u2264 17.In this case, n = 16, we have k = 7 from 4k + 2 \u2264 2n \u2264 3k + 10. At this time 2n \u2212 3k \u2212 11 = 0, which contradicts to 2n \u2212 3k \u2212 11 < 0.When n = 17, we have k = 8 from 4k + 2 \u2264 2n \u2264 3k + 10. In this case, d8 \u2264 8, d11 \u2264 8, d17 \u2264 16. We have When Corollary 2.7, we can get G is Hamiltonian, a contradiction.When When From 10,166) and . They are not graphic, a contradiction.(1) for degree sequence . If the corresponding graphs are not Hamiltonian, there must exist two non-adjacent 6-degree vertices by Corollary 2.7, and the corresponding graphs are isomorphic to K6 \u2228 (C9 + 2K1) or K6 \u2228 (C4 + C5 + 2K1) or K6 \u2228 (C6 + K3 + 2K1). We can find these graphs are Hamiltonian, a contraction.(2) for degree sequence , , and  in x, y that are not adjacent such that d(x) + d(y) \u2264 \u2223V(G)\u2223 \u2212 5. By Corollary 2.7, we can get that G is Hamiltonian, a contradiction.(3) the other degree sequence except (2n \u2212 3k \u2212 16) \u2264 0. By Lemma 2.5, \u03b4(G) \u2265 2t = 6. Since \u03b4 \u2264 dk \u2264 k, then k \u2265 6.In this case, (Case 3.1 (k \u2212 6)(2n \u2212 3k \u2212 16) = 0, i.e., k \u2212 6 = 0 or k \u2260 6 and 2n \u2212 3k \u2212 16 = 0.Case 3.1.1k = 6.G is a graph with d6 \u2264 6, dn \u2212 3 \u2264 n \u2212 7, dn \u2264 n \u2212 1. We have G is n \u2212 9, (n \u2212 1)3), and we have n > 9t = 27 because t = 3. Let S is the set containing six 6 \u2212 degree vertices of G.In this case, Lemma 2.8, S form a complete graph Kn \u2212 6 in Kn \u2212 6 is adjacent to one vertex of S, it must be adjacent to all vertices of S. Moreover, there are at least 2 vertices of the Kn \u2212 6 adjacent to all vertices of S because there exist two non-adjacent vertices in S by Corollary 2.7.By Kn \u2212 6 adjacent to all vertices of S, then K2 \u2228 (Kn \u2212 8 + C6) is Hamiltonian, then If there are 2 vertices of the Kn \u2212 6 adjacent to all vertices of S, then K3 \u2228 (Kn \u2212 9 + C6) is Hamiltonian, then If there are 3 vertices of the Kn \u2212 6 adjacent to all vertices of S, then K4 \u2228 (Kn \u2212 10 + C6) and K4 \u2228 (Kn \u2212 10 + 2C3) are Hamiltonian, then If there are 4 vertices of the Kn \u2212 6 adjacent to all vertices of S, then K5 \u2228 (Kn \u2212 11 + 3K2) is Hamiltonian, then If there are 5 vertices of the Kn \u2212 6 adjacent to all vertices of S, then Cn(G) is Hamiltonian, a contradiction.If there are 6 vertices of the Kn \u2212 6 adjacent to all vertices of S, Ki \u2228 (Kn \u2212 6 \u2212 i + 6K1)(i > 6) is Hamiltonian, then If there are more than 6 vertices of the Case 3.1.2k \u2260 6 and 2n \u2212 3k \u2212 16 = 0.n \u2264 31 because n > 9t = 27. Since 2n \u2212 3k \u2212 16 = 0, then n = 29, k = 14. So d14 \u2264 14, d18 \u2264 14, d29 \u2264 28. We have 18, 2811). For this degree sequence, we can get that there are no two vertices x, y that are not adjacent such that d(x) + d(y) \u2264 \u2223V(G)\u2223 \u2212 5. By Corollary 2.7, we can get G is Hamiltonian, a contradiction.In this case, we can get 28 \u2264 Case 3.2 (k \u2212 6)(2n \u2212 3k \u2212 16) < 0.k \u2265 7 and 2n \u2212 3k \u2212 16 < 0. Since n \u2265 2k + 1. From these results, we have 4k + 2 \u2264 2n \u2264 3k + 15, that is k \u2264 13. Thus, we have n \u2264 27. A contradiction with known condition n > 9t = 27. \u25a0In this case, Theorem 3.2 Let G be a t \u2212 tough and simple connected graph with n vertices and m edges. IfthenG is Hamiltonian when t \u2208 {1, 2} and n \u2265 8t.(i) G is Hamiltonian when t = 3 and n > 9t.(ii) Proof Suppose that G is not Hamiltonian. Since Kn is Hamiltonian, then G \u2260 Kn. By theorem conditions, we can get n \u2265 8 when t \u2208 {1, 2, 3}, thus G \u2260 Kn \u2212 11, .Lemma 2.2,By thenTheorem 3.1, we get G is Hamiltonian, a contradiction. \u25a0By Theorem 3.3 Let G be a t \u2212 tough and simple connected graph with n vertices and m edges. IfthenG is Hamiltonian when t \u2208 {1, 2} and n \u2265 8t.(i) G is Hamiltonian when t = 3 and n > 9t.(ii) Proof Suppose that G is not a Hamilton graph. Since Kn is Hamiltonian, then G \u2260 Kn and G \u2260 Kn \u2212 1 + v. By theorem conditions, we can get n \u2265 8 when t \u2208 {1, 2, 3}, thus G \u2260 Kn, G \u2260 Kn \u2212 11, .Lemma 2.3,By thenTheorem 3.1, we get G is Hamiltonian, a contradiction.By \u25a0We suggest the following general problems.Problem 1 Let G be a t \u2212 tough and simple connected graph with n vertices and m edges. Ifthent \u2208 {1, 2} and n \u2265 8t, G is pancyclic.(i) when t = 3 and n > 9t, G is pancyclic.(ii) when Problem 2 Let G be a t \u2212 tough and simple connected graph with n vertices and m edges. Ifthent \u2208 {1, 2} and n \u2265 8t, G is pancyclic.(i) when t = 3 and n > 9t, G is pancyclic.(ii) when Problem 3 Let G be a t \u2212 tough and simple connected graph with n vertices and m edges. Ifthent \u2208 {1, 2} and n \u2265 8t, G is pancyclic.(i) when t = 3 and n > 9t, G is pancyclic.(ii) when The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.GY provide writing ideas and master the thesis as a whole. GC contributed significantly to analysis and manuscript preparation. TY performed the data analyses and wrote the manuscript. HX helped perform the analysis with constructive discussions. All authors contributed to the article and approved the submitted version.This study was supported by the National Natural Science Foundation of China (No. 11871077), the NSF of Anhui Province (1808085MA04 and 1908085MC62), the NSF of Anhui Provincial Department of Education (KJ2020A0894 and KJ2021A0650), Graduate Scientific Research Project of Anhui Provincial Department of Education (YJS20210515), Research and Innovation Team of Hefei Preschool Education College (KCTD202001), and Graduate offline course graph theory (2021aqnuxskc02).The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher."}
{"text": "R-submodule and \u03b1-intuitionistic fuzzy (\u03b1-IF) submodule. The proof that the level set of LIF-3 is contained in the level set of \u03b1-IF is given, and it is exclusively employed to define linear codes over \u03b1-IF submodule. Further, \u03b1-IF cyclic codes are presented along with their fundamental properties. Finally, an application based on genetic code is presented, and it is found that the technique of defining codes over \u03b1-IF submodule is entirely applicable in this scenario. More specifically, a mapping from the \u212464 module to a lattice L (comprising 64 codons) is considered, and \u03b1-IF codes are defined along with the respective degrees.In the last few decades, the algebraic coding theory found widespread applications in various disciplines due to its rich fascinating mathematical structure. Linear codes, the basic codes in coding theory, are significant in data transmission. In this article, the authors' aim is to enlighten the reader about the role of linear codes in a fuzzy environment. Thus, the reader will be aware of linear codes over lattice valued intuitionistic fuzzy type-3 (LIF-3)  \ud835\udd3d2, among which the binary codes are the simplest. Say, in a binary field \ud835\udd3d2, two codewords 110 and 111 are considered; then, both codewords comprise three bits. Now, when a message carrying 111 is sent in binary coding, the received message should contain 111 theoretically. Often, the transmission is corrupted, and the received message could be carrying 110. This error can be corrected by understanding the coding theory, and data transmission can be made more reliable and accurate. Linear codes are the most basic error-correcting codes which are the subspaces of a vector space. These codes are crucial to data transmission and data storage . The choice of linearization creates a problem; to overcome this drawback, it is replaced by Lattice homomorphism \u03b1 : L\u27f6, and it is said to be LIFS-3.Now, when dealing with the data transmission, one cannot always get crisp zeros and ones with data transmission ambiguity being an often occurring scenario. There may be plenty of reasons behind the emergence of uncertainty. These may be lack of knowledge, chance, imprecision, lack of information, and complexity. The imprecision inherent in the process can be dealt with by using fuzzy logic. Fuzzy sets (IF) are opposite to ordinary sets (crisp sets). In crisp set theory, things are categorized by the values 0 and 1, but the full membership and nonmembership are determined in a fuzzy set. This technique copes mathematically with the vagueness of determining boundaries by assigning grades of membership to the elements. Fuzzy set theory as a generalization of the classical set theory was established by Zadeh  in whichtanassov  proposedd Stoeva  proposed\u2323 cev i\u02d9  extendedn-dimensional vector space and defined fuzzy codes which are fuzzy subsets of n-tuples over the field. Hamming distance is also defined between two fuzzy codewords. Hall and Dial  is a linearization function satisfying \u2113(f(x))+\u2113(g(x)) \u2264 1.Let S be a non-empty set and L be a complete lattice with top element T and bottom element B. Consider f : S\u27f6L, g : S\u27f6L which are membership and nonmembership functions; then, a lattice valued intuitionistic fuzzy set of type-3 (LIFS-3) is the set , where \u03b1 : L\u27f6 is a lattice homomorphism with \u03b1(T)=1, \u03b1(B)=0 satisfyingLet \u03b1(f(s))+\u03b1(g(s)) \u2264 1 for all s \u2208 S.and R is a set equipped with two binary operations, namely, addition + and multiplication \u00b7 and satisfying the following axioms:R, +) is an Abelian group.\u27f6 is a semigroup.(a \u00b7 (b+c)=a \u00b7 b+a \u00b7 c and (b+c) \u00b7 a=b \u00b7 a+c \u00b7 a for all a, b, c \u2208 R.Multiplication is distributive over addition; that is, Ring theory is one of the extensions of group theory that encompasses a broad set of present study topics in mathematics, computer science, and mathematical/theoretical physics. They have a wide range of applications in the studies of geometric objects and topology, and their connections to other fields of algebra are quite well understood in several contexts. A ring R is called commutative if a \u00b7 b=b \u00b7 a for all a, b \u2208 R. Consider a commutative ring R. An R-module is a set M together with a binary operation addition \u2018\u2295' and scalar multiplication \u2018\u2217', where \u2217 : R \u00d7 M\u27f6M; then, for all r, s \u2208 R, a, \u2009b \u2208 M, we have the following:r(a \u2295 b)=ra \u2295 rb.r \u2295 s)a=ra \u2295 sa.(rs)\u2217a=r(s\u2217a). is a lattice valued intuitionistic fuzzy subset of type-3, where f : M\u27f6L and g : M\u27f6L, then A is called an LIF-3 submodule of M if for r \u2208 R and a, b \u2208 M we have the following:f(a+b) \u2265 f(a)\u2227f(b).f(r.a) \u2265 f(a).g(a+b) \u2264 g(a)\u2228g(b).g(r.a) \u2264 g(a).If f\u03b1, g\u03b1 : M\u27f6 be two composition functions, where f\u03b1=\u03b1of and g\u03b1=\u03b1og. Then, \u03b1-intuitionistic fuzzy (\u03b1-IF) submodule if for r \u2208 R and x, y \u2208 M we have the following:f\u03b1(x+y) \u2265 f\u03b1(x)\u2227f\u03b1(y).f\u03b1(r.y) \u2265 f\u03b1(x).g\u03b1(x+y) \u2264 g\u03b1(x)\u2228g\u03b1(y).g\u03b1(r.y) \u2264 g\u03b1(x).Let IfA=is an LIF-3 submodule, then\u03b1-IF submodule, thenMoreover, if LetA=be an LIF-3 submodule. Then, the necessary and sufficient conditions forAto be anR-module are as follows:f(k1x+k2y) \u2265 f(x)\u2227f(y),g(k1x+k2y) \u2264 g(x)\u2228g(y),x, y \u2208 M and k1, k2 \u2208 R.\u2200\u2009An LIF-3 submoduleA=of a ringRis called an LIF-3 ideal if for eachx, y \u2208 R,f(x \u2212 y) \u2265 f(x)\u2227f(y).f(x.y) \u2265 f(x)\u2228f(y).g(x \u2212 y) \u2264 g(x)\u2228g(y).g(x.y) \u2265 g(x)\u2227g(y).LetA=be an LIF-3 submodule; then, the level sets are defined as\u03b1 -IF submodule In a similar fashion, we can define level sets for an We can also write thatM=\u2124pkn which is a \u2124pk-module, then an LIF-3 submodule A of M is termed as an LIF-3 linear code having length n over \u2124pk.If we consider a module LetA=be an LIF-3 submodule andbe an\u03b1-IF submodule. LetAl1l2andbe the level sets forAand, where\u03b1(l1)=t1and\u03b1(l2)=t2. Then,Al1l2 and A and m \u2208 Al1l2; then, f(m) \u2265 l1 and g(m) \u2264 l2 imply that f\u03b1(m) \u2265 \u03b1(l1) and g\u03b1(m) \u2264 \u03b1(l2), where \u03b1(l1), \u03b1(l2) \u2208 . From this, we get that m \u2208 A\u03b1(l1)\u03b1(l2). Thus, Al1l2\u2286A\u03b1(l1)\u03b1(l2). Let \u03b1(l1)=t1 and \u03b1(l2)=t2; then, Al1l2\u2286At1t2.Let \u03b1-IF submodule, so we will use \u03b1-IF submodule for further discussion of codes. Let us consider a module M=\u2124pkn which is a \u2124pk module, and let \u03b1-IF submodule; then, \u03b1-IF linear code of length n over the module \u2124pk.From the above proposition, we have concluded that the level set of LIF-3 submodule is contained in Consider\u21244-module and latticeL={0,1, a, b, c, d, }having 0, 1 as bottom and top elements witha \u2264 c, a \u2264 d, andb \u2264 c. Letf, g : \u21244\u27f6Land\u03b1 : L\u27f6be defined as\u03b1(0)=\u03b1(b)=0, \u03b1(a)=0.2=\u03b1(c), and \u03b1(d)=\u03b1(1)=1. Then, \u03b1 -IF \u21244 module; therefore, \u03b1 -IF linear code.Let \u03b1-IF submodule t \u2208 , is said to be the degree of that element t and it is denoted by pt. In example 1, p0=2, p1=2, p0.2=2.Consider an Consider an\u03b1-IF submoduleis an\u03b1-IF linear code over\u2124pkiffare linear codes.\u03b1-IF submodule and k be non-empty subset of \u03b1-IF characteristic function of k is denoted by \u03c7k=\u2329f\u03b1\u03c7k, g\u03b1\u03c7k\u232a, where f\u03b1\u03c7k, g\u03b1\u03c7k : M\u27f6; then,Let Let\ud835\udc9ebe a subset of an\u03b1-IF submodule\u2124pkn; then,\ud835\udc9eis an\u03b1-IF linear code over\u2124pkiff\u03c7Cfor\ud835\udc9eis an\u03b1-IF linear code over the same ring\u2124pk.\ud835\udc9e be a linear code implying that \ud835\udc9e is an \u03b1-IF submodule. Now, if x, y \u2208 \ud835\udc9e, then by \u03c7C, we haveLet \ud835\udc9e is an \u03b1-IF submodule, then x+y, \u2009r.x \u2208 \ud835\udc9e and for all r \u2208 R imply that f\u03b1\u03c7C(x+y)=1=1\u22271=f\u03b1\u03c7C(x)\u2227f\u03b1\u03c7C(y) and f\u03b1\u03c7C(r.x)=1=f\u03b1\u03c7C(x). Similarly, g\u03b1\u03c7C(x+y) \u2264 g\u03b1\u03c7C(x)\u2228g\u03b1\u03c7C(y) and g\u03b1\u03c7C(r.x) \u2264 g\u03b1\u03c7C(x). Thus, the \u03c7C of \ud835\udc9e is an \u03b1-IF submodule; hence, it is an \u03b1-IF linear code.As \u03c7C be a linear code; hence, it is an \u03b1-IF submodule. Let x, y \u2208 \ud835\udc9e; then, by Conversely, suppose that the f\u03b1\u03c7C(x+y) \u2265 f\u03b1\u03c7C(x)\u2227f\u03b1\u03c7C(y)=1\u22271=1 and f\u03b1\u03c7C(r.x) \u2265 f\u03b1\u03c7C(x)=1. Similarly, g\u03b1\u03c7C(x+y) \u2264 g\u03b1\u03c7C(x)\u2228g\u03b1\u03c7C(y)=0\u22280=0 and g\u03b1\u03c7C(r.x) \u2265 g\u03b1\u03c7C(x)=0; hence, x+y, \u2009r.x \u2208 \ud835\udc9e; thus, \ud835\udc9e is an \u03b1-IF submodule, so \ud835\udc9e is an \u03b1-IF linear code.imply that Letbe an\u03b1-IFS of\u2124pkn. is an\u03b1-IF linear code over\u2124pkiffis an\u03b1-IF linear code.Letbe an\u03b1-IF fuzzy submodule; then,is also a linear code such that fort1, t2 \u2208 , f\u03b1(x)=t1andg\u03b1(x)=t2; then,Ais said to be a trivial\u03b1-IF linear code.LetM=\u2124pknbe a module over the ring\u2124pk. Then, two\u03b1-IF submodulesandare said to be orthogonal ift1, t2 \u2208 , we haveM.Furthermore, for all Letbe two\u03b1-IF submodules which are defined asandAsf\u03b1t1={a \u2208 M : f\u03b1(a) \u2265 t1}, we can compute the valuesf\u03b1t1fort1 \u2208 .Similarly, the remaining values given in Hence, LetMbe\u2124pkmodule andbe two\u03b1-IF submodules. If for allw \u2208 M,f\u03b1(w)=rorg\u03b1(w)=r, wherer \u2208 , then setis not orthogonal to setLetbe an\u03b1-IF submodule, sois a linear code over\u2124pkhaving lengthn. Consider setsAs the set\u2124pkn is finite, im(f\u03b1) and im(g\u03b1) are also finite. Consider an order t1 \u2265 t2 \u2265 \u22ef\u2265tr; then, the set im(f\u03b1) satisfies this order. Similarly, suppose t1\u2032 \u2264 t2\u2032 \u2264 \u2026\u2264tr\u2032 and this order is satisfied by im(g\u03b1). Gtjtj\u2032 . Hence, A can be obtained from the matrices Gt1t1\u2032, Gt2t2\u2032,\u2026, Gtrtr\u2032.LetM=\u2124pknbe a finite module andbe an\u03b1-IF submodule ofM. Then, there is an\u03b1-IF submodulesuch thatis orthogonal toif and only if|Im(f\u03b1)|, |Im(g\u03b1)| > 1, and for anyr \u2208 Im(f\u03b1), there exists an element\u03b7 \u2208 Im(f\u03b1)withf\u03b1r=(f\u03b1\u03b7)T. Similarly, for any elementp \u2208 Im(g\u03b1), there exist\u00f0 \u2208 Im(g\u03b1)such thatg\u03b1p=(g\u03b1\u00f0)T.M be a module and \u03b1-IF submodule of M. Suppose that |Im(f\u03b1)|=s > 1 and |Im(g\u03b1)|=k > 1 and for any element r \u2208 Im(f\u03b1) there is an element \u03b7 \u2208 Im(f\u03b1) such that f\u03b1r=(f\u03b1\u03b7)T. Moreover, for any element p \u2208 Im(g\u03b1), there exists an element \u00f0 \u2208 Im(g\u03b1) such that g\u03b1p=(g\u03b1\u00f0)T.Now, consider an order for Im(f\u03b1) and Im(g\u03b1) asLet M asj=1,\u2026, m. Now, if we define an \u03b1-IF set f\u03b1\u2032, g\u03b1\u2032 : M\u27f6, then for a \u2208 M, we have f\u03b1\u2032(a)=1 \u2212 tm\u2212j+1 and g\u03b1\u2032(a)=1 \u2212 tm\u2212j+1\u2032.By using these compositions, we can also define sets which form partition of module f\u03b1)={t1 \u2265 t2 \u2265 \u22ef\u2265tm} imply that f\u03b1t1\u2286f\u03b1t2\u2286\u22ef\u2286f\u03b1tm. Similarly, for Im(g\u03b1)={t1\u2032 \u2264 t2\u2032 \u2264 \u22ef\u2208\u2264tm\u2032}, we have g\u03b1tm\u2286\u22ef\u2286g\u03b1t2\u2286g\u03b1tm. As for any element r \u2208 Im(f\u03b1), there is an element \u03b7 \u2208 Im(\u03b1\u00b0f) such that f\u03b1r=(\u03b1\u00b0f\u03b7)T. Similarly, for any p \u2208 Im(g\u03b1), there is an element \u00f0 \u2208 Im(g\u03b1) such that g\u03b1p=(g\u03b1\u00f0)T, as in finite module there is a property related to orthogonality by which f\u03b1tj=(f\u03b1tm\u2212j+1)T. Accordingly, we have f\u03b11\u2212tm\u2212j+1\u2032={a \u2208 M : f\u03b1\u2032(a) \u2265 1 \u2212 tm\u2212j+1}=MjUMj\u22121U \u22ef UM1=f\u03b1tj=(f\u03b1tm\u2212j+1)T.As we know, Im(g\u03b1tj\u2032=(g\u03b1tm\u2212j+1\u2032)T and g\u03b11\u2212tm\u2212j+1\u2032\u2032={a \u2208 M : g\u03b1\u2032(a) \u2265 1 \u2212 tm\u2212j+1\u2032}=M1U \u2026 UMj\u22121UMj=g\u03b1tj=(g\u03b1tm\u2212j+1\u2032)T. Thus, \u03b1-IF submodule.Furthermore, \u03b1-IF submodules such that these two sets are orthogonal to each other; then, we have |Im(f\u03b1)| > 1 and |Im(g\u03b1)| > 1. For all t, t\u2032 \u2208 , we also have f\u03b11\u2212t1\u2032=(f\u03b1t)T and g\u03b11\u2212t1\u2032\u2032=(g\u03b1t\u2032)T; then, for any element r \u2208 Im(f\u03b1), there exist \u03b7 \u2208 Im(f\u03b1) such that f\u03b1r=(f\u03b1\u03b7)T, and for any element p \u2208 Im(g\u03b1), there exist \u00f0 \u2208 Im(g\u03b1) such that g\u03b1p=(g\u03b1\u00f0)T; this is due to the reason that im(f\u03b1)\u2032={1 \u2212 t : t \u2208 f\u03b1} and im(g\u03b1\u2032)={1 \u2212 t\u2032 : t\u2032 \u2208 g\u03b1}.Conversely, suppose that Suppose that, andbe three\u03b1-IF fuzzy submodules of a moduleMsuch thatis orthogonal toandis orthogonal to; then,\u03b1-IF fuzzy submodules of a module M such that b \u2208 f\u03b1t1\u2212\u2032, for t \u2208 . Then, for a \u2208 f\u03b1t, \u2329a, b\u232a=0. b \u2208 f\u03b1t1\u2212\u2033. Thus, f\u03b1t1\u2212\u2032\u2286f\u03b1t1\u2212\u2033. Therefore, f\u03b1t\u2033\u2286f\u03b1t\u2032. By following the same strategy, we get f\u03b1t\u2032\u2286f\u03b1t\u2033. Thus, f\u03b1\u2032=f\u03b1\u2033. In a similar manner, we can show that g\u03b1\u2032 =\u2009 g\u03b1\u2033. Hence, Let LetMbe a finite\u2124pkmodule andbe an\u03b1-IF submodule ofM; if there exists\u03b1-IFSsuch thatis orthogonal to the submodule, thenis an\u03b1-IF submodule of moduleM.Consider two\u03b1-IF linear codesand; then, these two\u03b1-IF codes are said to be equivalent if the level setsand forandare equivalent.\u03b1-IF cyclic codes and some important results related to these codes.Cyclic codes have long since been one of the most interesting families of codes because of their rich algebraic structure, and these codes play an important role in coding theory. In this section, we will discuss Consider an\u03b1-IF submoduleof module\u2124pkn; then,is called an\u03b1-IF cyclic code having lengthnover\u2124pkif for each \u2208 \u2124pknwe haveThe\u03b1-IF submoduleis an\u03b1-IF cyclic code over\u2124pkif and only ifare cyclic codes over\u2124pk.\u03b1-IF submodule and i=0,1,\u2026, n \u2212 1 we have Let As which imply that w0, w1,\u2026, wn\u22121) \u2208 \u2124pkn such that f\u03b1 < f\u03b1. Suppose t1\u2032=f\u03b1. Similarly, t2\u2032=g\u03b1 imply that Conversely, suppose that Let\u2124pknbe a module andbe an\u03b1-IF submodule.is an\u03b1-IF cyclic code on module\u2124pknif and only if the characteristic function of a level setis an\u03b1-IF cyclic code on\u2124pkn.Consider a module\u2124pkn; then,is an\u03b1-IF cyclic code on module\u2124pknif and only if for each \u2208 \u2124pknwe have\u2124pkn be a module and \u03b1-IF cyclic code on module \u2124pkn; then, we haveLet Thus,f\u03b1 and g\u03b1; then, by A is an \u03b1-IF cyclic code.Conversely, suppose that the above equality holds for Consider two\u03b1-IF cyclic codesand; then, we have the following:is an\u03b1-IF cyclic code.is an\u03b1-IF cyclic code.is an\u03b1-IF cyclic code.(1)\u03b1-IF modules of module \u2124pkn such that \u03b1-IF cyclic codes corresponding to \u03b1-IF modules. Then, for  \u2208 \u2124pkn,Let In a similar manner, we have\u03b1-IF modules, we get again an \u03b1-IF module; thus, \u03b1-IF cyclic code.(2)w0, w1,\u2026, wn\u22121) \u2208 \u2124pkn,Now, for (g\u03b1+g\u03b1\u2032) \u2264 (g\u03b1+g\u03b1\u2032). Thus, \u03b1-IF cyclic code.Similarly, we can show ((3)Proof follows from (2).\u2009By taking intersection of two If\u2124pknis a module, thenis said to be an\u03b1-IF cyclic code iff the non-empty level setsare\u03b1-IF ideals of the factor ring(\u2124pk[X]/(Xn \u2212 1)).\u2124pkn and a factor ring (\u2124pk[X]/(Xn \u2212 1)). Define a mapping as \u03d5 : \u2124pkn\u27f6(Zpk[X]/(Xn \u2212 1)) which is also an isomorphism. Let b= \u2208 \u2124pkn such that \u03d5(b)=\u2211j=0m\u22121bjXj.Consider a module \u03b1-IF cyclic code; this implies that \u03b1-IF cyclic code over \u2124pk. Cyclic codes are ideals in factor ring, which implies that Suppose that t1, t2 \u2208 , the set \u03d5, then the level set \u03b1-IF cyclic code; thus, \u03b1-IF cyclic code over ring \u2124pk.Conversely, suppose that for \u2124pk is a finite ring and if f\u03b1) and im(g\u03b1) are also finite; then, we have gjk(X) \u2208 \u2124pk[X] is the generator polynomial for gj+1k(X)/gjk(X), j=1,\u2026, r \u2212 1.As LetS={g1k(X), g2k(X),\u2026, grk(X)}be a set of polynomials such that the polynomial gj(X)\u2009divides\u2009Xn \u2212 1 for each j=1,\u2026, r. Ifgj+1k(X)|gjk(X)forj=1,2,\u2026, r \u2212 1and\u2124pkn=gj+1k(X), then the set of polynomials determines an\u03b1-IF cyclic code, wheregjk(X)is the collection of level cut cyclic subcodes ofProof follows from \u03b1-IF gray code by using the compositions f\u03b1 and g\u03b1. Consider a map \u03b7 : \u212422\u27f6\u212422; then, \u03b7 is called a gray map defined as \u03b7(0)=00, \u03b7(1)=01, \u03b7(2)=11, \u03b7(3)=10.The gray code, which is also called reflected binary code, is an ordering of the binary numeral system such that two successive values vary in a single bit. We will define Consider a mapping\u03b7 : \u212422\u27f6\u212422which is a gray map. Suppose thatS(\u212422)andS(\u212422)be two\u03b1-IF fuzzy subsets of\u212422and\u212422. Forf\u03b1, g\u03b1 \u2208 S(\u212422), an\u03b1-IF gray map\u03b7\u2217 : S(\u212422)\u27f6S(\u212422)is defined asConsider two mappingsf\u03b1, \u2009g\u03b1 : \u21244\u27f6defined as follows:By definition, we havea=0,Now, for \u03b1-IF gray map corresponding to these mappings are given in The values for \u2124pk to \u2124ppk\u22121. Using this map, an \u03b1-IF generalized gray map can be defined.Now, we consider generalizing fuzzy gray map which is a mapping from A mapping\u03b7\u2217 : S(\u2124pk)\u27f6S(\u2124ppk\u22121)is said to be an\u03b1-IF generalized gray map if for anyf\u03b1, \u2009g\u03b1 \u2208 S(\u2124pk)we haveIfis an\u03b1-IF code over\u2124pk, then it is said to be an\u03b1-IF\u2124pk-linear code if it is an image under the\u03b1-IF generalized gray map of a linear code over\u2124pk.Consider an\u03b1-IF codeC; then, it is an\u03b1-IF\u2124pkcyclic code ifCis an\u03b1-IF\u2124pk-linear code and also if it is the image under the\u03b1-IF generalized gray map of a cyclic code over\u2124pk.Consider mappingsf\u03b1, g\u03b1 : \u212426\u27f6whereandThen,C= is an \u03b1 -IF linear code having length six over \u21242 . Now, if we consider another code where f\u03b1, \u2009g\u03b1 : \u212443\u27f6 defined asC= is an \u03b1 -IF linear code having length 3 over \u21244.An\u03b1-IF gray map\u03b7\u2217is a bijective map.\u03b7\u2217 is a one-to-one function.This is due to the fact that The coding theory deals with encoding and decoding a message. During transmission of data, errors may occur; they can be detected and corrected by using different procedures such as parity check, syndrome decoding, and redundancy check which are for ordinary codes. Similarly, a method can be adopted in fuzzy codes for the confirmation about the codeword, that is, whether it belongs to the transmitted code or not. This can be done by considering the level set from which one can determine the confirmation degree as to whether a received codeword belongs to the original code. This method can be explained through the following example.Consider a module\u212424, where\u03b1-IF submodule where f\u03b1, g\u03b1 : \u212424\u27f6 andLet which is a subset of\u212424and also a linear code. Now, by transmitting this code, the received codeword has errors; assume that the received codewords are{000; .01, 010; 0.001, 011; 0.01, 100; 0.1, 101; 0.01, 110; \\\\0.1, 001; 0.01, 111; 0.999}.f\u03b1, g\u03b1 corresponding to the codewords of code C.t1 > 0.9 and t2 \u2264 0.01 that the received codeword is in C.The level set is defined as \u2124pkm be a module and \u03b1-IF submodule. If \ud835\udc9e is an \u03b1-IF linear code, then the level set \ud835\udc9et1t2 of \ud835\udc9e is also an \u03b1-IF linear code. As the elements in \u2124pkm are the words of length m over the alphabet {0,1,2,\u2026, pk \u2212 1} and the level set \ud835\udc9et1t2 also consists of the elements of \u2124pkm, then for a member a \u2208 \ud835\udc9et1t2 and for any integer r \u2265 0, the sphere of radius r and center a is defined as follows.Suppose Consider a moduleM=\u2124pkm; then, the sphere having radiusr(where\u20090 \u2264 r \u2264 m))containswords.\u2124pkm. Let x be a fixed word in \u2124pkm. Then, the number of the words which differ from x in m position is Consider a module LetM=\u212422, where\u212422={00,01,10,11}. Here,k=1, p=2, m=2, and0 \u2264 r \u2264 2. If we take a wordx=10, then its distances from the remaining codewords ared=1, d=2, d=1, and the sum of these distances is equal to 4. Now,, which is also equal to 4.Amino acid sequence of a protein can be determined from the sequence or order of DNA and RNA molecules . Althoug\u21244. Bennenni et al. . Suppose f, g : \u212464\u27f6L where L is a lattice consisting of 64 codons; the values for these functions and their composition are shown in \u03b1-IF submodule so it is an \u03b1-IF linear code. We will use the following methodology.In the presented study, codon structure is investigated by involving the lattice structure of codons along with module over a ring. As there are 64 possible codons, we take L comprises 64 codons and a module Z64={0,1,\u2026, 63} which also consists of 64 elements. Let a=0 \u2208 Z64; we know that f, g : Z64\u27f6L. Then, f(0)=UUU, g(0)=AAA, which are top and bottom elements of the lattice L as shown in , assign the values to these functions as f\u03b1(0)=1 \u2208 , g\u03b1(0)=0 \u2208 , where f\u03b1(0)+g\u03b1(0) \u2264 1. Similarly, values can be assigned to the remaining codons, which is shown in \u03b1-IF submodule with module Z64 is referred to as a linear code. By using t \u2208 ; for example, for t=1, the degree is 1 and for t=0.3 the degree is 10. The remaining values can be computed in a similar way. Degrees of respective codons are also given in Consider the lattice shown in . As f\u03b1, U show hydrophobic amino acids, and those with second base A show hydrophilic amino acids. There are total 16 codons that specify hydrophobic and hydrophilic amino acids, so we can define an \u03b1-IF linear code by considering the \u212416 module. Let f, g : \u212416\u27f6L and f\u03b1, g\u03b1 : \u212416\u27f6, where lattice L consists of 16 elements. A comparison can be made between the two properties of amino acid by employing the concept of degree given in The most important properties of amino acids are their hydrophobic and hydrophilic properties. As in lattice diagram, codons with second base \u03b1-IF submodule are defined. These two structures are related by means of their level sets; that is, the level set of LIF-3 submodule is contained in the level set of \u03b1-IF submodule. After obtaining this result, further codes are discussed over \u03b1-IF submodule. Linear codes and cyclic codes are defined over \u03b1-IF submodule. Some important properties and results related to these codes are investigated. It is also concluded that this concept of \u03b1-IF linear codes is entirely applicable in genetic code. There are 64 codons that specify different amino acids, so Z64 module is considered here and \u03b1-IF linear codes are defined over Z64 module.Communication systems are designed for data transmission, working on the principle of encoding and decoding information. In classical coding theory, different procedures can be adopted to detect and correct errors that may arise during communication. The communication process is highly influenced by vagueness, inaccuracy, imprecision, and uncertainty. The classical coding methodologies are sometimes not efficient in handling such situations. Therefore, fuzzy logic is a viable option to handle such type of information. Lattice valued intuitionistic fuzzy sets are the generalization of basic fuzzy sets that incorporated the degree of both membership and nonmembership. Thus, they constitute a more efficient framework to model uncertain data. In this article, LIF-3 submodule and \u03b1-IF submodule by involving \u2124pkn module over a ring \u2124pk instead of the field \ud835\udd3d. Moreover, there are several generalizations of fuzzy sets like picture fuzzy sets [\u03b1-IF codes are defined by considering the modules Z64 and Z16. This methodology seems to be attractive for further investigation of genetic codes.In literature, conventional codes such as hamming codes and Hadamard codes are defined, which are constructed by considering vector spaces and fields. Modules are the extension of vector spaces. Thus, this work can be extended further by employing the present methodology and by using modules instead of vector spaces and fields. This can be done by investigating these codes over zzy sets , Pythagozzy sets , hesitanzzy sets , and neuzzy sets  where th"}
{"text": "Electromechanical phonon-cavity systems are man-made micro-structures, in which vibrational energy can be coherently transferred between different degrees of freedom. In such devices, the energy transfer direction and coupling strength can be parametrically controlled, offering great opportunities for both fundamental studies and practical applications such as phonon manipulation and sensing. However, to date the investigation of such systems has largely been limited to linear vibrations, while their responses in the nonlinear regime remain yet to be explored. Here, we demonstrate nonlinear operation of electromechanical phonon-cavity systems, and show that the resonant response differs drastically from that in the linear regime. We further demonstrate that by controlling the parametric pump, one can achieve nonlinearity-mediated digitization and amplification in the frequency domain, which can be exploited to build high-performance MEMS sensing devices based on phonon-cavity systems. Our findings offer intriguing opportunities for creating frequency-shift-based sensors and transducers. Electromechanical phonon cavity systems offer advantages for solid state implementation, but thus far investigation has been limited to the linear regime. Here, Miao et al demonstrate non-linear operation of an electromechanical phonon cavity system and the drastic difference to the linear regime. As the degree of freedom (DOF) in the system increases , nonlinearity gives rise to unique phenomena such as synchronization8, chaos9, internal resonance10, and generation of frequency comb13, by enabling and modulating coherent energy transfer between the different DOFs14.Nonlinearity is ubiquitous in real-world physical systems. Among the many man-made structures, resonant microelectromechanical and nanoelectromechanical systems (MEMS/NEMS) offer great opportunities for designing, tuning, and exploiting nonlinearities and have enabled the exploration of nonlinear processes and nonlinearity-dictated device properties, such as energy dissipation15 offer the unique capability of phonon manipulation by parametrically coupling a mechanical resonance to a phonon cavity17, in many aspects analogous to a photon-cavity22, which has enabled a plethora of exotic physical phenomena such as self-cooling24, induced transparency26, parametric amplification27, and quantum squeezing29. These exquisite functions are realized through a pump signal in the cavity\u2019s sideband, which parametrically controls the dynamical coupling and backaction between the resonant mode and the cavity32.Among multi-DOF systems, electromechanical phonon-cavity systems35, such as sensing.One key feature that differentiates the phonon-cavity systems is that they operate entirely in the electromechanical domain, and thus are much more advantageous for implementation using monolithic solid-state devices. Further, the characteristic frequency of the phonon cavity is typically in the same frequency band as the resonant mode , which greatly simplifies the signal transduction, making such systems promising for frequency-shift-based applications14, research to date has been largely confined to linear operations, i.e., with limited vibration amplitude, which has plagued the exploration of nonlinear processes in these systems. Here we study nonlinear operation in a microelectromechanical phonon-cavity system and show that the response clearly differs from that in the linear regime. More importantly, leveraging the nonlinearity-mediated bi-stability, we demonstrate digitization and amplification modes of signal sensing enhancement in the frequency domain, which can be further tuned by the degree of nonlinearity in the vibration response.However, in such parametrically-coupled systems, despite the nonlinear nature of the coupling between the different DOFs\u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096969.1\u2009Hz) and a flexural mode resonance (at frequency \u03c92\u2009=\u20092\u03c0\u2009\u00d7\u200916649.6\u2009Hz). In this work, we specifically design our device to facilitate the observation of dynamical coupling in the nonlinear regime. This is achieved by the small capacitance gap, large capacitance area, and high-quality factor of the resonator (see \u201cMethods\u201d). By activating different combinations of the 12 electrodes , with clear nonlinearity-mediated bi-stability . We denote the critical frequency where such bi-stability occurs as \u03c9b, which depends on the driving amplitude and the sweep direction  and downward (high to low) frequency sweeps Fig.\u00a0. We now eps Fig.\u00a0, the eff\u03c9p has very strong and distinct effects on the resonant response: when pumping in the vicinity of the red side band  as \u03c9p approaches \u03c9red, producing a dip-like feature in the 2D color plot; when pumping around the blue side band  with a sudden jump at \u03c9p\u2009\u2248\u2009\u03c9blue, resulting in a spike-like feature in the 2D color plot  in the frequency down-sweeps, regardless the position of \u03c9b. This is clearly visible in the 3D plots  are the displacement and natural frequency of the resonant modes, respectively. Specifically, i\u2009=\u20091 corresponds to the mode of interest  being the harmonic driving force), and i\u2009=\u20092 corresponds to the phonon cavity . The energy dissipation rates are given by \u03b3i, and cij  give the intra-modal (i\u2009=\u2009j) and intermodal (i\u2009\u2260\u2009j) parametric coupling coefficients, with \u03c9p being the frequency of parametric pump. All these terms are normalized using the effective mass of the oscillator. \u03b5 is introduced as a scaling parameter, based on the assumption that all terms multiplied by \u03b5 are small compared with the terms on the left side of the equations. This assumption is true for steady-state vibration of high Q resonators, where the energy stored in the resonator  is much greater than dissipation or transfer between modes. This mathematical treatment allows us to apply multiple-scale approximation using the standard multidimensional Newton-Raphson algorithm40 (see SI 2.1 and 2.3\u20132.4 for details), from which we are able to numerically solve the above equations in the vicinity of the cavity sidebands (\u03c9p\u2009\u2248\u2009\u03c9red and \u03c9p\u2009\u2248\u2009\u03c9blue), with the results shown in Fig.\u00a0To understand the unique behavior of the nonlinear response in our phonon-cavity system, we examine its equations of motion by introducing Duffing nonlinearity into this 2-DOF system\u03c9p near \u03c9red, and the spike-like feature for \u03c9p near \u03c9blue. Such unique behavior can be qualitatively understood, in a simplified picture, by considering the nonlinearity-induced amplitude bi-stability in the resonance mode, together with its parametric-pump-controlled coupling to the phonon-cavity: in downward frequency sweeps, the resonant response initially follows the upper branch among the two stable solutions from the softening Duffing equation, before jumping to the lower branch at \u03c9b. When \u03c9p approaches \u03c9red, the parametric pump causes phonon to be removed from the resonant mode . In contrast, when \u03c9p\u2009\u2248\u2009\u03c9blue, phonons are created in\u00a0the resonant mode .The numerical results show very good agreement with the experimental data  and frequency (f) of a resonator, leading to an A-f relationship. Second, in phonon-cavity systems, the amplitude of a given mode is affected by the frequency difference between some other modes (such as the pump and the cavity modes), which gives rise to an f-A effect. By operating a phonon cavity device in a nonlinear regime, one can excite both effects, resulting in an f-A-f transduction, enabling signal amplification entirely within the frequency domain. While quantitative explanations for specific details (such as the asymmetry in \u03c9p for the dip-like and spike-like features) require more in-depth analyses and evaluation of multiple expressions derived from Eq.\u00a0It is also interesting to analyze the observed phenomena by considering two different effects. First, with the sudden jump between its two stable branches, Duffing bi-stability produces a distinct relationship between the amplitude (\u03c92) can function as the front-end transducer, which converts the input signal to a shift in \u03c92 .We now show that such tuning of nonlinearity-mediated bi-stability, via parametric coupling to the phonon cavity, can be exploited to realize distinct functions in the frequency domain, offering unique capabilities for sensing and signal transduction. In resonant electromechanical transducers, typically, a change in an input physical quantity  engenders a frequency shift, which is then measured or processed. In our devices, the phonon-cavity , as the input change in \u03c92 is equivalent to a shift of the operation point , as clearly demonstrated in both experiment  can be continuously and smoothly controlled by adjusting the pump strength Vp and the degree of nonlinearity (varying Vd), as shown in Fig.\u00a0In the first scenario, we utilize the horizontal red-blue boundary in Fig.\u00a0\u03c9p slightly below \u03c9blue), outlined by the small circles in Fig.\u00a0\u03c92 from the input engenders a continuous and amplified output (change in \u03c9b), for which we define the gain A as:Vp and the degree of nonlinearity (adjusting Vd). We find that deeper nonlinear vibration or stronger parametric coupling can produce larger amplification , even exceeding one order of magnitude (red data series). Second, there is a trade-off between gain and linear dynamic range: a larger gain is associated with a smaller dynamic range. By carefully adjusting Vd and Vp, one can choose the optimal operating condition.In the other scenario, we operate along the sloped red-blue boundary .Vf is applied (blue lines) on the electrodes to shift the frequency of mode 2 (\u03c92). In practice, the shift of \u03c92 can be caused by external factors such as temperature, pressure, acceleration, and angular velocity. Here, in order to demonstrate the feasibility of the proposed functions, we use the negative stiffness due to electrostatic force to simulate the frequency shift of mode 2 due to such external factors. To minimize the effect from environment variables, the MEMS resonator is measured in a vacuum with a constant temperature of 30\u2009\u00b0C. The frequency output of mode 1 (\u03c9b) is monitored (purple lines) using PLL in addition to the frequency sweep measurements ).The PLL set-up is shown in Fig.\u00a0Vf to control \u03c92, and monitor the frequency output \u03c9b using PLL. The PLL data  used in simulation and experiments, the results may not exactly agree quantitatively, but qualitatively all key features and behaviors are successfully reproduced experimentally.To verify the signal sensing enchancement functions, we operate the phonon cavity system in corresponding parameter spaces. Figure\u00a0Similarly, the experimental results for the red pump case Fig.\u00a0 agree we42. Our results show that, by harnessing nonlinearity in phonon-cavity system, intriguing opportunities emerge for high-performance frequency-shift-based sensors and transducers50. Specifically, compared with MEMS sensors and transducers using single-DOF nonlinear resonators51 or multiple coupled resonators53, the nonlinear phonon-cavity system in this work could potentially offer greater responsivity, additional control, smaller device footprint, and simplified system design, thus resulting in improved performance in MEMS-based sensing and signal transduction applications.These unique nonlinearity-mediated frequency-shift functions can be exploited for constructing phonon-cavity-based sensors and transducers. It is worth noting that operating devices in the frequency domain have a number of advantages, as \u201ctime and frequency are the most accurately measurable of all physical quantities\u201dThe electromechanical structure is designed to facilitate coherent energy transfer between the two resonant modes via parametric pump-controlled intermodal coupling. A three-segment beam is used to couple the\u00a0two masses together, facilitating intermodal coupling structurally. A large mass area and the small capacitive gap in the device design make it easier for the resonator to exhibit nonlinearity, without requiring excessive amplitude. The design of multiple groups of electrodes under the resonator allows us to efficiently adjust the frequency of the different modes and implement dynamic coupling at the same time. The quality factor of the resonator can be effectively controlled by the vacuum packaging process.2. The device is operated under room temperature. The device structure, including its cross-section, and the fabrication process are detailed in Fig.\u00a0The die embedding the vacuum-sealed MEMS resonators is electrically packaged in a ceramic leadless chip carrier with a pressure of 0.1\u2009Pa. The structure layer is 40-\u03bcm-Si and the electrode layer is 6-\u03bcm-Si. The gap between the structure layer and the electrode layer is 2\u2009\u03bcm. The lateral size of the resonator is about 3\u2009mm\u2009\u00d7\u20093\u2009mm. The insulating layer underneath the electrode layer is 2-\u03bcm-SiOA number of instruments are used in the measurement. The bias voltages are generated by a low noise voltage source (ITECH IT6233). The drive and pump signals are provided by a two-channel lock-in amplifier (Zurich Instruments HF2LI). The response motion of the resonator is detected by a capacitance\u2013voltage (C/V) converting scheme based on a charge amplifier and measured by the lock-in amplifier. Detailed measurement setup and wiring diagram are available in SI 1.2.Vd\u2009=\u20090.2, 0.4, 0.6, 0.8, 1.0\u2009mVpk and Vd\u2009=\u20091, 2, 3, 4, 5\u2009mVpk, respectively. The results in (c) and (d) are both measured with a frequency downward sweep.The linear and nonlinear responses of both resonant modes are first characterized in the absence of the parametric pump. The measurement condition for the dataset shown in Fig.\u00a0The coherent energy transfer in the phonon cavity is characterized by scanning the parametric pump frequency near either of the cavity sidebands. The measurement and plotting details for Fig.\u00a0\u03c9p\u2009\u2248\u2009\u03c9red , the device is measured with Vd\u2009=\u20090.05\u2009mVpk, Vp\u2009=\u20095.0\u2009Vpk, and plotted using \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096886.0\u2009Hz, \u03c9red\u2009=\u20092\u03c0\u2009\u00d7\u20099592.0\u2009Hz. In the linear regime when \u03c9p\u2009\u2248\u2009\u03c9blue , the device is measured with Vd\u2009=\u20090.05\u2009mVpk, Vp\u2009=\u20092.0\u2009Vpk, and plotted using \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096958.0\u2009Hz, \u03c9blue\u2009=\u20092\u03c0\u2009\u00d7\u200923,582.0\u2009Hz.In the linear regime when \u03c9p\u2009\u2248\u2009\u03c9red , the device is measured with Vd\u2009=\u20091.5\u2009mVpk, Vp\u2009=\u20095.0 Vpk. and plotted using \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096886.0\u2009Hz, \u03c9red\u2009=\u20092\u03c0\u2009\u00d7\u20099592.5\u2009Hz. When \u03c9p\u2009\u2248\u2009\u03c9blue , the device is measured with Vd\u2009=\u20091.0\u2009mVpk, Vp\u2009=\u20091.5\u2009Vpk. and plotted using \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096958.0\u2009Hz, \u03c9blue\u2009=\u20092\u03c0\u2009\u00d7\u200923,575.0\u2009Hz.In the nonlinear regime upward sweep, when \u03c9p\u2009\u2248\u2009\u03c9red , the device is measured with Vd\u2009=\u20091.5 mVpk, Vp\u2009=\u20095.0\u2009Vpk. and plotted using \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096886.0\u2009Hz, \u03c9red\u2009=\u20092\u03c0\u2009\u00d7\u20099592.5\u2009Hz. When \u03c9p\u2009\u2248\u2009\u03c9blue , the device is measured with Vd\u2009=\u20091.0\u2009mVpk, Vp\u2009=\u20091.5\u2009Vpk. and plotted using \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096958.0\u2009Hz, \u03c9blue\u2009=\u20092\u03c0\u2009\u00d7\u200923,575.0\u2009Hz.In the nonlinear regime downward sweep, when \u03c9d\u2009=\u20092\u03c0\u2009\u00d7\u20090.05\u2009Hz, \u2206\u03c9p\u2009=\u20092\u03c0\u2009\u00d7\u20090.2\u2009Hz. Note that the plot center frequencies are slightly adjusted for each panel for easy comparison across the different measurement conditions, in order to best illustrate the key findings. Similarly, the driving and pumping strength for the data presented in each panel are also chosen to best illustrate the key findings.In all measurements in Fig.\u00a0The numerical codes are constructed using C language. The numerical settings for producing the data in Fig.\u00a0\u03c9p\u2009\u2248\u2009\u03c9red (a), the device response is calculated with Vd\u2009=\u20090.05\u2009mVpk, Vp\u2009=\u20095.0\u2009Vpk, and plotted using \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096886.0\u2009Hz, \u03c9red\u2009=\u20092\u03c0\u2009\u00d7\u20099592.0\u2009Hz. In the linear regime when \u03c9p\u2009\u2248\u2009\u03c9blue (b), the device response is calculated with Vd\u2009=\u20090.05\u2009mVpk, Vp\u2009=\u20092.0\u2009Vpk, and plotted using \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096958.0\u2009Hz, \u03c9blue\u2009=\u20092\u03c0\u2009\u00d7\u200923,582.0\u2009Hz.In the linear regime when \u03c9p\u2009\u2248\u2009\u03c9red (c), the device response is calculated with Vd\u2009=\u20091.5\u2009mVpk, Vp\u2009=\u20095.0\u2009Vpk, and plotted using \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096886.0\u2009Hz, \u03c9red\u2009=\u20092\u03c0\u2009\u00d7\u20099592.5\u2009Hz. When \u03c9p\u2009\u2248\u2009\u03c9blue (d), the device response is calculated with Vd\u2009=\u20091.0\u2009mVpk, Vp\u2009=\u20092.0\u2009Vpk, and plotted using \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096958.0\u2009Hz, \u03c9blue\u2009=\u20092\u03c0\u2009\u00d7\u200923,575.0\u2009Hz.In the nonlinear regime upward sweep, when \u03c9p\u2009\u2248\u2009\u03c9red (e), the device response is calculated with Vd\u2009=\u20091.5\u2009mVpk, Vp\u2009=\u20095.0\u2009Vpk. and plotted using \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096886.0\u2009Hz, \u03c9red\u2009=\u20092\u03c0\u2009\u00d7\u20099592.5\u2009Hz. When \u03c9p\u2009\u2248\u2009\u03c9blue (f), the device response is calculated with Vd\u2009=\u20091.0\u2009mVpk, Vp\u2009=\u20092.0\u2009Vpk, and plotted using \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096958.0\u2009Hz, \u03c9blue\u2009=\u20092\u03c0\u2009\u00d7\u200923,575.0\u2009Hz.In the nonlinear regime downward sweep, when \u03c9d\u2009=\u20092\u03c0\u2009\u00d7\u20090.05\u2009Hz, \u2206\u03c9p\u2009=\u20092\u03c0\u2009\u00d7\u20090.2\u2009Hz. Note that the plot center frequencies are slightly adjusted for each panel for easy comparison across the different measurement conditions, in order to best illustrate the key findings. Similarly, the driving and pumping strength for the data presented in each panel are also chosen to best illustrate the key findings.In all calculations presented in Fig.\u00a0The numerical settings for producing the data in Fig.\u00a0Vd\u2009=\u20090.05\u2009mVpk, Vp\u2009=\u20092.0\u2009Vpk, \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096958.0\u2009Hz, and \u03c9p\u2009=\u20092\u03c0\u2009\u00d7\u200923,582.0\u2009Hz. The frequency steps are \u2206\u03c9d\u2009=\u20092\u03c0\u2009\u00d7\u20090.05\u2009Hz, and \u2206\u03c92\u2009=\u20092\u03c0\u2009\u00d7\u20090.2\u2009Hz. The line plots  are taken from data in (b) with \u2206\u03c92\u2009=\u20090 (c) and \u2206\u03c92\u2009=\u20092\u03c0\u2009\u00d7\u20091.0\u2009Hz (d).In the linear operation (b), Vd\u2009=\u20091.0\u2009mVpk, Vp\u2009=\u20092.0\u2009Vpk, \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096958.0\u2009Hz, and \u03c9p\u2009=\u20092\u03c0\u2009\u00d7\u200923,575.0\u2009Hz. The frequency steps are \u2206\u03c9d\u2009=\u20092\u03c0\u2009\u00d7\u20090.05\u2009Hz, and \u2206\u03c92\u2009=\u20092\u03c0\u2009\u00d7\u20090.2\u2009Hz. The line plots  taken from data in (d) when \u2206\u03c92\u2009=\u20090 (g) and \u2206\u03c92\u2009=\u20092\u03c0\u2009\u00d7\u20091.0\u2009Hz (h).In the nonlinear operation with downward sweep (f), Vd\u2009=\u20093.0\u2009mVpk, Vp\u2009=\u20090.6\u2009Vpk, \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096929.52\u2009Hz, and \u03c9p\u2009=\u20092\u03c0\u2009\u00d7\u200923,552.51\u2009Hz. The frequency steps are \u2206\u03c9d\u2009=\u20092\u03c0\u2009\u00d7\u20090.01\u2009Hz, \u2206\u03c92\u2009=\u20092\u03c0\u2009\u00d7\u20090.01\u2009Hz.In the zoom-in plot (i), Vp\u2009=\u20091.0, 1.5, 2.0\u2009Vpk when Vd is scanned from 2.0\u2009mVpk to 3.0\u2009mVpk. The step of Vd is 0.1\u2009mVpk.The data of digitization in (j) are calculated for Vd is 3.0\u2009mVpk and Vp\u2009=\u20090.35, 0.45, 0.50\u2009Vpk. The step of \u2206\u03c92 is 2\u03c0\u2009\u00d7\u20090.01\u2009Hz.The data of amplification in (k) are calculated for Vp is 0.50\u2009Vpk and Vd\u2009=\u20092.7, 2.9, 3.0\u2009mVpk. The step of \u2206\u03c92 is 2\u03c0\u2009\u00d7\u20090.01\u2009Hz.The data of amplification in (l) are calculated for The measurement settings for producing Fig.\u00a0Vd\u2009=\u20091.25\u2009mVpk, Vp\u2009=\u20091.5\u2009Vpk, \u03c91\u2009=\u20092\u03c0\u2009\u00d7\u20096538.0\u2009Hz, and \u03c9p\u2009=\u20092\u03c0\u2009\u00d7\u200923,048.0\u2009Hz. The steps are \u2206\u03c9d\u2009=\u20092\u03c0\u2009\u00d7\u20090.02\u2009Hz, and \u2206Vf\u2009=\u20090.05\u2009Vpk.In the nonlinear operation at blue pump with downward sweep (a), Vd\u2009=\u20091.25\u2009mVpk, Vp\u2009=\u20091.5\u2009Vpk, and \u03c9p\u2009=\u20092\u03c0\u2009\u00d7\u200923,048.0\u2009Hz. Vf is scanned from 0 to 1\u2009Vpk and the step of Vf is 0.05\u2009Vpk.When using PLL to test the phonon-cavity system at blue pump (b), Vp\u2009=\u20091.5\u2009Vpk when Vd is scanned from 0.75\u2009mVpk to 1.25\u2009mVpk. The step of Vd is 0.05\u2009mVpk.The experiment of digitization at blue pump in (c), Vd\u2009=\u20091.25\u2009mVpk, Vp\u2009=\u20091.5\u2009Vpk, and \u03c9p\u2009=\u20092\u03c0\u2009\u00d7\u200923,048.0\u2009Hz. Vf is scanned from 0.445 to 0.495\u2009pk and the step of Vf is 5\u2009mVpk.The experiment of amplification at blue pump in (d), \u03c92 and output signal \u03c9b under different operation conditions: when there is no parametric pump when there is a blue pump and when there is a red pump applied to the phonon-cavity system.The experiment result of frequency noise is shown in Fig.\u00a0\u03c3 is shown in Fig.\u00a0The time-domain data is shown in Fig.\u00a0Supplementary InformationPeer Review File"}
{"text": "G = (V(G), E(G)) be a graph with no loops, numerous edges, and only one component, which is made up of the vertex set V(G) and the edge set E(G). The distance d between two vertices u, v that belong to the vertex set of H is the shortest path between them. A k-ordered partition of vertices is defined as \u03b2 = {\u03b21, \u03b22, \u2026, \u03b2k}. If all distances d are finite for all vertices v \u2208 V, then the k-tuple , d, \u2026, d) represents vertex v in terms of \u03b2, and is represented by r(v|\u03b2). If every vertex has a different presentation, the k-partition \u03b2 is a resolving partition. The partition dimension of G, indicated by pd(G), is the minimal k for which there is a resolving k-partition of V(G). The partition dimension of Toeplitz graphs formed by two and three generators is constant, as shown in the following paper. The resolving set allows obtaining a unique representation for computer structures. In particular, they are used in pharmaceutical research for discovering patterns common to a variety of drugs. The above definitions are based on the hypothesis of chemical graph theory and it is a customary depiction of chemical compounds in form of graph structures, where the node and edge represent the atom and bond types, respectively.Let  L = {l1, l2, \u2026, lk} is a graph's ordered set of vertices and v \u2208 G, then the k-tuple r(v|L) = , r, \u2026, r). The notation r is the representation of v with regard to L, and the symbol L is said to be a resolving set if the different vertices of G have different representations regard to L. H's metric dimension, indicated by dim(H), is the minimal number of vertices in the resolving set. The task of computing a graph's locating set is a Non-deterministic Polynomial time problem or NP-hard  = {d, d, \u2026, d} are named as k-tuple representations. If each v in V(G) has a unique representation with regard to \u03b2, then the resolving partition of the vertex set is termed \u03b2, and the least value of the resolving partition set of V(G) is called the partition dimension of G and is indicated as pd(G)  = {1, 2, 3, \u2026, n} has E(H) = {, 1 \u2264 x \u2264 y \u2264 n}, iffy \u2212 1 = tq for some q, 1 \u2264 q \u2264 p  = 2 if and only if G = Pn\u201d.Theorem 2 Chartrand et al. (\u03d5 be a resolving partition of \u03b5(\u22ce) and \u03f51, \u03f52 \u2208 \u03b5(\u22ce). If d = d for all vertices w \u2208 \u03b5(\u22ce)\\, then \u03f51, \u03f52 belong to different classes of \u03d5.\u201dd et al.  \u201cLet \u03d5 bpd of Toeplitz graph with two generators 1 and t in Section 2 and Toeplitz graph partition dimension with three generators 1, 2, and t in Section 3.This study's findings the pd of the Toeplitz graph Tn\u23291, t\u232a, for t \u2265 2 the pd of the graph is three.The coming section is containing the discussion on the Theorem 2.1. A Toeplitz graph withn \u2265 4 isTn\u23291, 2\u232a. After that, pd = 3.Proof. Let the Toeplitz graph with n \u2265 4 is Tn\u23291, 2\u232a Then we will show that the Toeplitz graph with generators 1 and 2 consist a resolving partition set, \u03b2 = {\u03b21, \u03b22, \u03b23} with three elements, where \u03b21 = {v1}, \u03b22 = {vk}k\u22610(mod 2), \u03b23 = {vk}k\u22611(mod 2). Let \u03b2 = {\u03b21, \u03b22, \u03b23} resolve the vertices of graph G with V(G) = \u03b21 \u222a \u03b22 \u222a \u03b23.k = 1, 2, \u2026, n. In terms of resolving partition set \u03b2, we have the following representations of vk.When Because all of the representations of different vertices are distinctpd \u2265 3. Suppose on contrary that pd = 2. We know that pd(G) = 2, iff G is a path graph by Theorem 1, it is not possible for Tn\u23291, 2\u232a. Thus,Conversely: Now, we will show that Hence, from Inequalities (2) and (3), we haveTheorem 2.2. Let a Toeplitz graphTn\u23291, 3\u232a withn \u2265 5. Thenpd = 3.Proof. Let a Toeplitz graph Tn\u23291, 3\u232a with n \u2265 5. We will show that the Toeplitz graph with generators \u23291, 3\u232a consist of a resolving partition set, \u03b2 = {\u03b21, \u03b22, \u03b23} with three elements, where \u03b21 = {v1}, \u03b22 = {v2, \u2026, vt}, \u03b23 = {vt+1, \u2026, vn}. There are two cases for \u03b2:k \u2264 3, then we can write the representation of vk with respect to \u03b2 asCase 1: If 1 \u2264 Tn\u23291, 3\u232a.where k \u2264 n, then we can write the representation of vk with respect to \u03b2 aCase 2: If 4 \u2264 k \u2212 1 \u2261 j(mod 3), this shows that all the representations are different, thus,where pd \u2265 3. On contrary suppose that pd = 2. Theorem 1 demonstrates that pd(G) = 2, iff G is a path graph, then it is not possible for Tn\u23291, 3\u232a. Thus,Conversely: We will prove that Hence, from Inequalities (4) and (5), we haveTheorem 2.3. Let a Toeplitz graph with notationTn\u23291, t\u232a with even generatort \u2265 4, n \u2265 t+2. Thenpd = 3.Proof. Let a Toeplitz graph with notation Tn\u23291, t\u232a with even generator t \u2265 4, n \u2265 t+2. The Toeplitz graph with generators \u23291, t\u232a consisting of a resolving partition set will be demonstrated. \u03b2 = {\u03b21, \u03b22, \u03b23}, where \u03b21 = {v1}, 3 = {\u2200vk|vk \u2209 \u03b21, \u03b22}. There are three cases with respect to vk, which are the following;vk with regard to resolving partition set \u03b2;Case 1: When where vk with respect to resolving partition set \u03b2;Case 2: When z = 1 when z = 0.where vk with respect to resolving partition set \u03b2;Case 3: When z = 1 when k = 1 and otherwise z = 0.where It is clear that no two vertices have the same representation, implying that there are not any two vertices with the same representation.pd \u2265 3. Suppose on the contrary that pd = 2. We know that by Theorem 1, it is not possible for even t of graph Tn\u23291, t\u232a. Thus,On contrary, we shall now demonstrate that Hence, from Inequalities (6) and (7), we haveTheorem 2.4. Let a Toeplitz graphTn\u23291, t\u232a with oddt \u2265 5, n \u2265 t+2. Thenpd = 3.Proof. Let a Toeplitz graph Tn\u23291, t\u232a with odd t \u2265 5, n \u2265 t+2. We will show that the Toeplitz graph with generators 1 and t, consists of a resolving partition set, \u03b2 = {\u03b21, \u03b22, \u03b23} with three elements, where \u03b21 = {v1, v2, vt+1}, 3 = {\u2200vk|vk \u2209 {\u03b21, \u03b22}}.1There are two cases for \u03b2where 2There are two cases for \u03b2where 3, we have the following valuesFor \u03b21, \u03b22, and \u03b23From all these cases of \u03b2We conclude that all representations are unique, and no two vertices have identical representations.pd \u2265 3. Suppose on the contrary that pd = 2. We know that by Theorem 1, it is not possible for odd t of graph Tn\u23291, t\u232a. SoIn contrary, we shall now demonstrate that Hence, from Inequalities (8) and (9), we can say thatTn\u23291, 2, t\u232a. If t = 3, 4, 5, and t = 2i, i \u2265 3, n \u2265 t + 2 then partition dimension is 4.In this section, we are going to discuss the partition dimension of Theorem 3.1. LetTn\u23291, 2, t\u232a be a Toeplitz graph. Thenpd = 4.Proof. We split our theorem into three cases.t = 3, 4, 5.Case A: When Tn\u23291, 2, t\u232a be a Toeplitz graph with t = 3, 4, 5, n \u2265 t+2, then we will show that vertices of the Toeplitz graph with three generators consist of a resolving partition set, \u03b2 = {\u03b21, \u03b22, \u03b23, \u03b24} where \u03b21 = {v1}, \u03b22 = {v2}, \u03b23 = {v3, \u2026, vt}, and \u03b24 = {vt+1, \u2026, vn}. Then there are the three cases that follow:Let k\u22611(mod t), then we can write the unique position of vk regarding \u03b2 as;Case 1: If Tn\u23291, 2, t\u232awhere k\u22612, 3(mod t), we can write the representations of vk regarding \u03b2 as;Case 2: If Tn\u23291, 2, t\u232a.where k \u2261 4, 5(mod t), we can write the representations of vk with respect to \u03b2 asCase 3: If where t = 6, 8.Case B: When 1, \u03b22, \u03b23, \u03b24} be a resolving partition set. Where \u03b21 = {v1}, \u03b22 = {v2, vt}, \u03b23 = {v3, \u2026, vt\u22122}, and \u03b24 = {vt\u22121, vt+1, \u2026, vn}. We have different cases on vk, which are following;Let \u03b2 = {\u03b21;There are two cases for \u03b2z1 = 1 when k = even, z = 1 when k \u2261 0(mod 8) and otherwise both are 0.where 2;There are three cases for \u03b2z = 0 when 5 \u2264 k \u2264 t\u22121 and otherwise z = 2, and z1 has defined in \u03b21.where 3;There are three cases for \u03b24;There is the only case for \u03b2It is clear that no two vertices have the same representation, implying that there are no two vertices with the same representation.t = 2i, i \u2265 5.Case C: When 1, \u03b22, \u03b23, \u03b24} be a resolving partition set. Where \u03b21 = {v2}, \u03b22 = {v6}, \u03b23 = {va}, and \u03b24 = {\u2200vk|vk \u2209 \u03b21, \u03b22, \u03b23}. where Let \u03b2 = {\u03b2vk with regard to the resolving partition set \u03b2.The following is a representation of all vertices 1;There are two cases for \u03b22;There are four cases for \u03b23;There are four cases for \u03b2z1 = 0 when k = even, otherwise 1 and z2 = 1 when k = odd, otherwise 0.where 4;There is the only case for \u03b2It is clear that no two vertices have the same representation, implying that there does not exist two vertices with the same representation.Converse A, B, and C:pd \u2265 4. On contrary, suppose that pd = 3.We will show that pd is 3. If n consists of sets of different resolving partition set that are following:Different cases on behalf of our assumption that 1 = {v1, v2}, \u03b22 = {v3, v4}, Case 1: \u03b21 = {v1, v3}, \u03b22 = {v2, v4}, Case 2: \u03b21 = {v1, v2, v3}, \u03b22 = {v4}, Case 3: \u03b21 = {v1, v2}, \u03b22 = {v3}, Case 4: \u03b21 = {v1}, \u03b22 = {v2, v3}, Case 5: \u03b21 = {v1, v2, v4}, \u03b22 = {v3}, Case 6: \u03b21 = {v1, v3, v4}, \u03b22 = {v2}, Case 7: \u03b21 = {v1}, \u03b22 = {v2, v4}, Case 8: \u03b21 = {v1}, \u03b22 = {v2, v3, v4}, Case 9: \u03b21 = {v1}, \u03b22 = {v2, v3, v5}, Case 10: \u03b21 = {v1, v5}, \u03b22 = {v2, v4}, Case 11: \u03b21 = {v1, v4}, \u03b22 = {v3, v5}, Case 12: \u03b21 = {v1, v4}, \u03b22 = {v2, v5}, Case 13: \u03b21 \u2261 1(mod 3), \u03b22 \u2261 2(mod 3), \u03b23 \u2261 0(mod 3), then we have the following different vertices with same representation; Case 14: \u03b2Tn\u23291, 2, t\u232a into three resolving partition sets. Thus,According to the above cases, we can easily conclude that our supposition is wrong, and we can not resolve the vertices of Hence, from Inequalities (1-3) and (13), we can say thatpd=3, where t \u2265 2 and if Toeplitz graph consists of three generators, pd) = 4, where t = 3, 5 and t = 2i and i \u2265 2.In this study, we looked at different families of Toeplitz graphs and established that the partition dimension of each family is the constant, if the Toeplitz graph consists of two generators, then In this paper, inequality (1) also satisfied the metric dimension results  = 1, is constant, bounded or unbounded?The partition dimension of the Toeplitz graph with two generators Open Problem 2.k \u2265 2, s \u2265 3, t \u2265 4 and gcd = 1, is constant, bounded or unbounded?The partition dimension of the Toeplitz graph with three generators Open Problem 3.If the generators of the Toeplitz graph are increasing then the partition dimension either increasing or decreasing ?The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.RL conceived of the presented idea. AK developed the theory and performed the computations. AA and MA verified the analytical methods. GI and MN investigated and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript.This study was supported by the National Science Foundation of China (11961021 and 11561019), Guangxi Natural Science Foundation (2020GXNSFAA159084), and Hechi University Research Fund for Advanced Talents (2019GCC005).The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher."}
{"text": "In this DFT\u2010mechanistic work, the origin of the observed ring\u2010size effect is examined in detail using 1\u2010hexene, CH2=CH2 and H2 as substrates. It is shown that, at room temperature, both the N,N\u2019,N\u2019\u2019,N\u2019\u2019\u2019,N\u2019\u2019\u2019\u2019\u2010stabilized dimer and the monomer are not coordinated by THF in solution, while the corresponding N,N\u2019,N\u2019\u2019,N\u2019\u2019\u2019\u2010stabilized structures are coordinated by one THF molecule mimicking the fifth N\u2010coordination. Catalytic 1\u2010alkene hydrogenation may occur via anti\u2010Markovnikov addition over the terminal Ca\u2212H bonds of transient monomers, followed by faster Ca\u2212C bond hydrogenolysis. The higher catalytic activity of the larger N,N\u2019,N\u2019\u2019,N\u2019\u2019\u2019,N\u2019\u2019\u2019\u2019\u2010stabilized dimer is due to not only easier formation of but also due to the higher reactivity of the catalytic monomeric species. In contrast, despite unfavorable THF\u2010coordination in solution, the smaller N,N\u2019,N\u2019\u2019,N\u2019\u2019\u2019\u2010stabilized dimer shows a 3.2\u2005kcal\u2009mol\u22121 lower barrier via a dinuclear cooperative Ca\u2212H\u2212Ca bridge for H2 isotope exchange than the large N,N\u2019,N\u2019\u2019,N\u2019\u2019\u2019,N\u2019\u2019\u2019\u2019\u2010stabilized dimer, mainly due to less steric hindrance. The observed ring\u2010size effect can be understood mainly by a subtle interplay of solvent, steric and cooperative effects that can be resolved in detail by state\u2010of\u2010the\u2010art quantum chemistry calculations.Recently, it was shown that the double Ca\u2212H\u2212Ca\u2010bridged calcium hydride cation dimer [LCaH The catalytic activity of double Ca\u2212H\u2212Ca\u2010bridged calcium hydride cation dimer complexes is influenced by using macrocyclic tetra\u2010 or pentadentate ligands. Extensive dispersion\u2010corrected DFT calculations disclose that this influence is due to a subtle balance of solvent, steric and dinuclear cooperative effects that also depend on the substrate. Further hydrogenolysis of the Ca\u2212C bond of +mA is kinetically 4.1\u2005kcal\u2009mol\u22121 more favorable (via +mTS2) than the preceding alkene addition step and is \u221221.7\u2005kcal\u2009mol\u22121 exergonic to release the hexane product CH3CH2Bu along with +1\u2009m\u2009\u22c5\u2009THF, followed by exergonic dimerization of two +1\u2009m\u2009\u22c5\u2009THF complexes to regenerate the catalyst 2+1\u2009\u22c5\u2009THF. The catalytic hydrogenation of CH2=CHBu via this monomeric mechanism is \u221224.4\u2005kcal\u2009mol\u22121 exergonic over a sizeable barrier of 24.6\u2005kcal\u2009mol\u22121, consistent with the moderate heating at 60\u2009\u00b0C required experimentally.2=CHCye (Cye=cyclohexenyl) substrate.2=CH2) is used as substrate instead, the overall catalytic hydrogenation becomes \u221227.6\u2005kcal\u2009mol\u22121 over a 5.4\u2005kcal\u2009mol\u22121 lower barrier of 19.2\u2005kcal\u2009mol\u22121, suggesting a much faster reaction even at room temperature. Due to reduced steric hindrance, direct CH2=CH2 addition to a Ca\u2212H\u2212Ca bridge of cation dimer 2+1 becomes kinetically only 0.5\u2005kcal\u2009mol\u22121 less favorable . For comparison, similar addition of CH2=CHBu and CH2=CH2 to a Ca\u2212H\u2212Ca bridge of dimer cation +3 encounters relatively higher barriers of 26.3 and 21.8\u2005kcal\u2009mol\u22121 , respectively, consistent with the lower catalytic activity of +3 compared to 2+1\u2009\u22c5\u2009THF as observed experimentally.As shown in Figure\u20052+2 stabilized by larger L5 ligands, the anti\u2010Markovnikov\u2010selective addition of CH2=CHBu to the terminal Ca\u2212H bond of THF\u2010free+2\u2009m is still 1.5\u2005kcal\u2009mol\u22121 endergonic over a sizeable barrier of 23.1\u2005kcal\u2009mol\u22121 (via +mTS3) to form the calcium alkyl complex +mB. Further hydrogenolysis of the Ca\u2212C bond of +mB is kinetically 4.9\u2005kcal\u2009mol\u22121 more favorable (via +mTS4) than the preceding alkene addition step and is \u221219.0\u2005kcal\u2009mol\u22121 exergonic to release the hexane product CH3CH2Bu along with +2\u2009m, followed by exergonic +2\u2009m dimerization to regenerate the catalyst 2+2. Compared with the above 2+1\u2009\u22c5\u2009THF\u2010catalyzed hydrogenation of CH2=CHBu, the 2+2\u2010catalyzed one is indeed kinetically 1.5\u2005kcal\u2009mol\u22121 more favorable and insensitive to THF coordination, consistent with the higher catalytic activity observed for the 2+2 catalyst stabilized by larger L5 ligands.2=CH2 is used as substrate instead, the overall barrier for catalytic hydrogenation is reduced to only 17.5\u2005kcal\u2009mol\u22121 via the cation monomer +2\u2009m, consistent with the rapid reaction observed even at 0\u2009\u00b0C.2=CH2 addition to a Ca\u2212H\u2212Ca bridge of L5\u2010stabilized 2+2 encounters a high free energy barrier of 36.6\u2005kcal\u2009mol\u22121 and thus is kinetically highly disfavored. Moreover, compared with 2+1\u2009\u22c5\u2009THF\u2010catalyzed hydrogenation of CH2=CHBu, the 1.5\u2005kcal\u2009mol\u22121 lower barrier computed for the catalyst 2+2 is due to not only the 0.9\u2005kcal\u2009mol\u22121 lower free energy for the monomer +2\u2009m formation, but also to the 0.6\u2005kcal\u2009mol\u22121 lower alkene addition barrier to THF\u2010free+2\u2009m than to THF\u2010coordinated +1\u2009\u22c5\u2009THF. In other words, the observed \u201cligand ring\u2010size\u201d effectAs shown in Figure\u20052 isotope exchange, the dimeric mechanism becomes more important due to evidently reduced steric hindrance and stronger dinuclear cooperative effects. As seen in Figure\u20052 isotope exchange may occur via a cooperative Ca\u2212H\u2212Ca bridge of THF\u2010free cation dimer 2+1 over a low barrier of 16.9\u2005kcal\u2009mol\u22121 (via 2+TS5) after THF elimination,2 isotope exchange via a Ca\u2212H\u2212Ca bridge of THF\u2010coordinated 2+1\u2009\u22c5\u2009THF becomes kinetically 5.0\u2005kcal\u2009mol\u22121 less favorable (via 2+TS5a), mainly due increased steric hindrance. On the other hand, the cation monomer +1\u2009m\u2009\u22c5\u2009THF is 7.8\u2005kcal\u2009mol\u22121 higher in free energy but intrinsically 8.9\u2005kcal\u2009mol\u22121 more reactive than 2+1\u2009\u22c5\u2009THF, and thus is kinetically 1.1\u2005kcal\u2009mol\u22121 more favorable for catalytic H2 isotope exchange (via +mTS5). However, 2+1\u2009\u22c5\u2009THF remains kinetically 3.9\u2005kcal\u2009mol\u22121 less active than 2+1 for the catalytic H2 isotope exchange. Considering the overall reaction of 2+1\u2009\u22c5\u2009THF+H2\u21922+TS5+THF, it is clear that such H2 isotope exchange is disfavored by the THF coordination by about 2.8\u2005kcal\u2009mol\u22121. For comparison, a similar dimeric mechanism for cation +3\u2010catalyzed H2 isotope exchange  encounters a moderate but higher free energy barrier of 21.7\u2005kcal\u2009mol\u22121, thus is kinetically less efficient than 2+1.When very small dihydrogen is used as substrate for catalytic H2+2, a similar cooperative Ca\u2212H\u2212Ca\u2010bridge\u2010mediated H2 isotope exchange may also occur but over a 3.2\u2005kcal\u2009mol\u22121 higher barrier of 20.1\u2005kcal\u2009mol\u22121 (via 2+TS6). Interestingly, the terminal Ca\u2212H mediated H2 isotope exchange via the THF\u2010free cation monomer +2\u2009m becomes kinetically competitive over nearly the same barrier of 20.0\u2005kcal\u2009mol\u22121 (via +mTS6). In contrast to the 2+1\u2010catalyzed H2 isotope exchange that is disfavored by THF coordination within stable 2+1\u2009\u22c5\u2009THF, those reactions catalyzed by both the cation dimer 2+2 and the cation monomer +2\u2009m stabilized by the larger L5 ligand are not disfavored by the THF solvent but remain kinetically 3.2\u2005kcal\u2009mol\u22121 less favorable, in sharp contrast to the cases of catalytic hydrogenation of unactivated 1\u2010alkenes. In other words, no simple \u201cligand ring\u2010size\u201d effects can be expected for calcium hydride catalysts, with the catalytic activity actually being due to a subtle balance between solvent, steric and dinuclear cooperative effects.As seen in Figure\u20052+1\u2009\u22c5\u2009THF and 2+2 as catalysts stabilized by macrocyclic L4 and larger L5 ligands and using CH2=CHBu, CH2=CH2 and H2 as substrates of different size, the origin of the experimentally observed ligand ring\u2010size effect is explored by extensive dispersion\u2010corrected DFT calculations. It is disclosed that the catalytic activity of calcium hydride cation dimer catalysts can be influenced by a subtle interplay of solvent, steric and substrate\u2010dependent dinuclear cooperative effects that may change the underlying mechanism in 1\u2010alkene hydrogenation and H2 isotope exchange reactions.By using cationic dimers solv=3.18\u2005\u00c5). The density\u2010fitting RI\u2212J approachAll DFT calculations are performed with the TURBOMOLE 7.4 suite of programs.\u22121 to account for the 1\u2005mol\u2009L\u22121 reference concentration in solution. To check the effects of the chosen DFT functional on the reaction energies and barriers, single\u2010point calculations at both TPSS\u2010D3\u22121 (mean\u00b1standard deviation) though as expected 0.5\u00b11.5\u2005kcal\u2009mol\u22121 higher barriers are found at the PW6B95\u2010D3 level. In our discussion, the more reliable PW6B95\u2010D3+COSMO\u2010RS free energies  are used unless specified otherwise. The applied DFT methods in combination with the large AO basis set provide usually accurate electronic energies leading to errors for chemical energies (including barriers) on the order of typically 1\u20132\u2005kcal\u2009mol\u22121. This has been tested thoroughly for the huge data base GMTKN55More accurate solvation free energies in THF solution are computed with the COSMO\u2010RS modelThe authors declare no conflict of interest.1As a service to our authors and readers, this journal provides supporting information supplied by the authors. Such materials are peer reviewed and may be re\u2010organized for online delivery, but are not copy\u2010edited or typeset. Technical support issues arising from supporting information (other than missing files) should be addressed to the authors.Supporting InformationClick here for additional data file."}
{"text": "We prove that every countable left\u2010ordered group embeds into a finitely generated left\u2010ordered simple group. Moreover, if the first group has a computable left\u2010order, then the simple group also has a computable left\u2010order. We also obtain a Boone\u2013Higman\u2013Thompson type theorem for left\u2010orderable groups with recursively enumerable positive cones. These embeddings are Frattini embeddings, and isometric whenever the initial group is finitely\u00a0generated. Finally, we reprove Thompson's theorem on word problem preserving embeddings into finitely generated simple groups and observe that the embedding is isometric. A landmark result in combinatorial group theory and computability on groups is the Boone\u2013Higman theorem. It states that a finitely generated group has decidable word problem if and only if it embeds into a simple subgroup of a finitely presented group  of piecewise homeomorphisms of flows of the suspension, see Definition\u00a0T. In particular, it is finitely generated if H is so. Just as in .The \u03a6t that sends  to  is a homeomorphism and defines a flow \u03a6 on \u03a3, the suspension flow, so that  is a dynamical system as well. The orbits of the suspension flow are homeomorphic to the real\u00a0line.The map H(\u03c6) the group of homeomorphisms of \u03a3 that preserves the orbits of the suspension flow, and by H0(\u03c6) the subgroup of H(\u03c6) that, in addition, preserves the orientation on each\u00a0orbit.We denote by 3.2C be a clopen subset of X and let J\u2282R be of diameter <1. The embedding of C\u00d7J into X\u00d7R descends to an embedding into \u03a3 that we denote by \u03c0C,J.Let C\u2282X and subset J of diameter <1 in R, the map \u03c0C,J is a chart for the suspension, whose image is denoted by UC,J. If z is in the interior of UC,J, then \u03c0C,J is a chart at\u00a0z.Definition 3.1(Dyadic chart) Let C be a clopen subset of X, and let J be a dyadic interval of length <1 in R. Then \u03c0C,J:C\u00d7J\u21aa\u03a3 is called dyadic chart.For every clopen Definition 3.2(Dyadic map) A dyadic map is a map f of real numbers such that f(x)=\u03bbx+c, where \u03bb is a power of 2 and c is a dyadic\u00a0rational.Definition 3.3(\u21a6.FJ denotes the group of piecewise dyadic homeomorphisms of J with finitely many breakpoints.Definition 3.4\u03c0C,J be a dyadic chart and let f\u2208FJ. Then fC,J is the map in T(\u03c6) whose restriction to UC,J is given byFC,J be the subgroup of T(\u03c6) generated by the elements fC,J for all f\u2208FJ.Let We recall that T(\u03c6) is infinite, simple, left\u2010ordered and finitely generated  and I1:=.Lemma 3.7(. Let C(J)\u2282bij(J) denote the subgroup of all piecewise homeomorphisms with dyadic breakpoints on J. The subgroup of C(J) of orientation preserving bijections is denoted by C+(J).Example 3.9C(J).Every countable group embeds into Let us fix a half\u2010open interval Example 3.10J, and therefore into C+(J).Every countable left\u2010orderable group embeds into the group of orientation preserving homeomorphisms of J is dense in J, the next lemma is a basic property of the (piecewise) continuity.Lemma 3.11C(J) is uniquely determined by its values on non\u2010dyadic rational points on J. Moreover, every function from C(J) is continuous at non\u2010dyadic rational\u00a0points.Every function in Since the set of non\u2010dyadic rational points of To construct respective embeddings into finitely generated simple groups, we propose the following extension of the construction in .3.4G of C(J).Definition 3.12\u03c4\u03a3,J:G\u2192bij(\u03a3) as follows: for each g\u2208G, let \u03c4\u03a3,J(g):=g\u03a3,J, where g\u03a3,J is defined byg\u03a3,J is the identity map\u00a0elsewhere.Define Let us fix a subgroup Definition 3.13T)(The group  Let G be a subgroup of C(J). We define T as the subgroup of bij(\u03a3) generated by \u03c4\u03a3,J(G) and T(\u03c6).We extend Definition\u00a0Lemma 3.14G embeds into T by g\u21a6g\u03a3,J. Moreover, if G is finitely generated, then T is finitely generated as\u00a0well.The group T and the fact that T(\u03c6) is finitely generated. For the first statement, it is enough to notice that, by definition, g\u03a3,J is an identity map if and only if g=1.\u25a1The second statement follows from the definition of Definition 3.15((Non\u2010dyadic) rational points) A point \u2208\u03a3 is called a rational point on \u03a3 if t\u2208Q. If, in addition, t is not dyadic, we say that  is a non\u2010dyadic rational point.Lemma 3.16\u03a3.There exists a dense and recursive set of non\u2010dyadic rational points in X:={x1,x2,\u2026}\u2282X that is dense in X, for example, the set of proper ternary fractions. Moreover, for all i\u2208N, let Ri\u2282\u03a3 be defined asLet us choose a recursive countable subset R=\u222ai=1\u221eRi. Note that each of Ri is a recursive set. Therefore, since X is also recursive by our choice, then we get that R is recursive as well.\u25a1We denote Lemma 3.17G is a subgroup of C(J), then the elements of T are uniquely defined by their values on any (countable) dense set of non\u2010dyadic rational points of \u03a3. Moreover, the elements of T are continuous at non\u2010dyadic rational points of \u03a3.If \u03a3. Let R\u2282\u03a3 be such a set. Let us define X\u2282X such that for each x\u2208X there exists t\u2208Q such that \u2208R. Since R is dense, X is dense as\u00a0well.By Lemma\u00a0X\u2282X is dense in X, by Lemma\u00a0{\u2223x\u2208X,t\u2208R} is dense in \u03a3. Therefore, the elements of T are uniquely defined by their restrictions to the \u03a6\u2010orbits of the elements  for x\u2208X. Now, the lemma follows from the combination of this observation with Lemma\u00a0\u25a1Note that since 3.5To prove simplicity results, we use the following standard\u00a0tool.Y be a set, and H a group acting faithfully on Y. Then the rigid stabilizer of a subset U\u2282Y is the subgroup of H whose elements move only points from U. We denote the rigid stabilizer of U by RiSt(U).Let T(\u03c6) in  be an element of Thompson's group such that J\u2229f(J)=\u2205. Then fX\u00d7 is in T(\u03c6) and separates UX\u00d7J from UX,f(J). By the previous claim, fX,J\u2208N. Therefore, the first derived subgroup of the rigid stabilizer of the interior of UX\u00d7J is in N by Lemma\u00a0\u03c4\u03a3,J(G) is in the rigid stabilizer of the interior of UX,J. Thus \u03c4\u03a3,J(G)\u2032 is in N. As G is assumed to be perfect, this yields the claim.\u25a1Let T=\u27e8\u03c4\u03a3,J(G),T(\u03c6)\u27e9.\u25a1Now, to conclude the proof of Lemma\u00a03.6Lemma 3.20G\u2a7dC+(J), then the group T is left\u2010orderable. Moreover, if G is finitely generated and consists of computable functions, then there exists a left\u2010order on T with recursively enumerable positive\u00a0cone.If G\u2a7dC+(J), the action of T on \u03a6\u2010orbits of elements of X\u2282\u03a3 is orientation\u00a0preserving.First of all, note that since R={,,\u2026} be a fixed, recursively enumerated and dense subset of non\u2010dyadic rationals in \u03a3. The existence of such sets is by Lemma\u00a0Let f\u2208T, define f>1 if for the smallest index k\u2208N such that f\u2260 we have f= such that qk>sk. Therefore, by Lemma\u00a0f\u22601 either f>1 or f\u22121>1 and for f1,f2>1, f1f2>1. By Lemma\u00a0T.Now for R is recursive. Therefore, to check whether f>1, we can consecutively compute the values f,f,\u2026. We stop at the first k such that f\u2260. By Lemma\u00a0f\u22601, and since G consists of computable maps, this procedure recursively enumerates the positive cone of the above defined left\u2010order.\u25a1Recall that the set Corollary 3.21T(\u03c6) has a left\u2010order with recursively enumerable positive\u00a0cone.The group 4J is a fixed interval that is strictly contained in \u21a6. is called a chart. The maps hi are local representations of\u00a0h.Each of the triples Remark 4.2 tables of Thompson in )+1,t\u21a6f|(t\u22121)+1), }.Definition 4.6G\u2010dyadic maps) Let h\u2208T, and let {}1\u2a7di\u2a7dn be a chart representation of h such that for every hi, 1\u2a7di\u2a7dn, one of the following takes place:(I)hi is a dyadic map on Ii;(II)hi=f\u039b is the composition a dyadic map f and a G\u2010dyadic map \u039b. Then, the representation {}1\u2a7di\u2a7dn is called canonical. Also, charts for which hi corresponds to (I) or (II) are called charts of type (I) or (II),\u00a0respectively.4.2Definition 4.9(Inverse) Let {}1\u2a7di\u2a7dn be a chart representation of h\u2208T. Then {}1\u2a7di\u2a7dn is a chart representation of h\u22121 and is called the inverse of the initial\u00a0one.The following operations can be applied to go from one chart representation to another.Definition 4.10(Refinements) The following operations on charts are called refinement.(1)I=I0\u222aI1, then  can be replaced by ,f|I0) and ,f|I1).If (2)J=J0\u222aJ1, then  can be replaced by (C\u00d7f\u22121(J0),C\u00d7J0,f|f\u22121(J0)) and (C\u00d7f\u22121(J1),C\u00d7J1,f|f\u22121(J1)).If (3)C=C0\u222aC1, then  can be replaced by  and .If Definition 4.11(Reunions) A reunion is the inverse operation of a\u00a0refinement.Definition 4.12(Shifts) A shift (of order m\u2208Z) is replacing a triple  by (\u03c6m(C)\u00d7(I\u2212m),\u03c6m(C)\u00d7(J\u2212m),t\u21a6f(t+m)\u2212m).Remark 4.13T. In particular, chart representations of elements from T are not\u00a0unique.A chart representation obtained by a refinements, reunion, or a shift on its charts corresponds to the same element from Remark 4.14G are computable, the operations Since the functions in Lemma 4.15T is finitely generated and each of the generators can be represented by a canonical chart representation, which can be algorithmically\u00a0determined.T(\u03c6) given by Lemma\u00a0G can be represented by a canonical chart representation by definition.\u25a1Indeed, the generators of Lemma 4.16The inverse, refinements and shifts preserve the canonicity of chart\u00a0representations.We will prove only that shift operations on charts of type (II) preserve the canonicity of chart representations, as the rest of statements of the lemma are\u00a0straightforward.m is applied, is . Then a shift of order m would transform it into the chart (\u03c6m(Ci)\u00d7(Ii\u2212m),\u03c6m(Ci)\u00d7(Ji\u2212m),\u039b\u223c), where \u039b\u223c:Ii\u2212m\u2192Ji\u2212m is defined as \u039b\u223c(x)=\u039b(x+m)\u2212m.Suppose that the initial chart of type (II), on which a shift operation of order \u039b=fg1f1\u2026gnfn is decomposed as in Definition\u00a0\u039b\u223c=f\u223cg1f1\u2026gnf\u223cn, where f\u223cn(x)=fn(x+m) and f\u223c(x)=f(x)\u2212m. The chart \u039b\u223c is also a G\u2010dyadic map. Therefore, (\u03c6m(Ci)\u00d7(Ii\u2212m),\u03c6m(Ci)\u00d7(Ji\u2212m),\u039b\u223c) satisfies the definition of charts of type (II) from Definition\u00a0\u25a1Suppose that Definition 4.17(Composition) Let {}1\u2a7di\u2a7dn and {}1\u2a7di\u2a7dm be chart representations such that \u22c3Ii=\u22c3Ji\u2032=. Then we say that the chart representationRemark 4.18f,f\u2032\u2208T, respectively, then the composition chart representation corresponds to ff\u2032.Note that if, in Definition\u00a0Remark 4.19Note that if the two chart representations in Definition\u00a0Lemma 4.20h\u2208T be given by a canonical chart representation {}1\u2a7di\u2a7dn. Then there is an algorithm to determine a canonical chart representation {}1\u2a7dj\u2a7dn\u2032 of h such that \u22c3Jj\u2032=.Let 1\u2a7di\u2a7dn:We describe the algorithm. For Ji\u2282 do nothing, go to i+1;if Ji\u2229 and Ji\u2216 is non\u2010empty, let Ji1=Ji\u2229, Ji2=Ji\u2216 and apply a refinement  from Definition\u00a0Lemma 4.21h\u2208T be given by a canonical chart representation {}1\u2a7di\u2a7dn. Then there is an algorithm to determine a canonical chart representation {}1\u2a7dj\u2a7dn\u2032 of h such that \u22c3Ij\u2032=.Let \u25a1The proof is analogous to the proof of Lemma\u00a0Lemma 4.22f,f\u2032\u2208T, given by their canonical chart representations, computes a canonical representation of ff\u2032.There is an algorithm that for any two elements {}1\u2a7di\u2a7dn and {}1\u2a7di\u2a7dm of, respectively, f and f\u2032 such that \u22c3Ii=\u22c3Ji\u2032=. Then their composition will be a canonical chart representation of ff\u2032 , given as a word in finite set of generators, outputs a canonical chart representation of f. In particular, every element from T has a canonical chart\u00a0representation.There exists an algorithm that for any input From the previous two lemmas we get:T have canonical chart representations, see Lemma\u00a0\u25a1It follows from Remark\u00a04.3T.Lemma 4.24h\u2208T and let {}1\u2a7di\u2a7dn be a chart representation of h. Then h=1 in T if and only if, for all 1\u2a7di\u2a7dn, we have hi=id (and Ji=Ii).Let The following observations are useful for studying the groups 1\u2a7di\u2a7dn. Recall that h maps  to  for \u2208Ci\u00d7Ii. As h=1, =. Thus, for any x\u2208Ci and t\u2208Ii, there is m\u2208Z such that (\u03c6m(x),t\u2212m)=). We conclude that hi(t)=t\u2212m and \u03c6m(x)=x. But \u03c6 is a minimal subshift, that is, every orbit of \u03c6 is dense. In particular, m=0. Therefore, hi=id. This yields one side of the assertion. The inverse assertion is trivial.\u25a1Let Lemma 4.25G\u2010dyadic map is equal to the identity, then the word problem in T is\u00a0decidable.If there is an algorithm to decide whether a h\u2208T one can algorithmically find a canonical chart representation for h. By Lemma\u00a0h=1 if and only if for any canonical chart representation of h the corresponding charts of types (I), and (II) are identity charts. If hi a local representation in a chart of type (I), hi is a piecewise dyadic map with finitely many breakpoints, so that we can algorithmically check whether hi=id.By Lemma\u00a0h=f\u039b:I\u2192I is a local representation in a chart of type (II), where f:I0\u2192I is dyadic and \u039b:In\u2192I0 a G\u2010dyadic map. We note that h=id on I if, and only if, \u039bf=id on I0. In fact, \u039bf is a G\u2010dyadic map, so that, by assumption, we can algorithmically check whether h=id.\u25a1Now suppose that Corollary 4.26T(\u03c6) is computably left\u2010orderable. In particular, the word problem in T(\u03c6) is\u00a0decidable.T(\u03c6) has a recursively enumerable positive cone. Thus, by Lemma\u00a0T(\u03c6) is computably left\u2010orderable.\u25a1By Lemma\u00a05Theorem 5.1G embeds into a finitely generated perfect group H. In addition:(1)G is computable, then H has decidable word problem;if (2)G is left\u2010ordered, then H is left\u2010ordered;if (3)G is computably left\u2010ordered, then the left order on H is computable;if (4)the embedding is a Frattini embedding. Moreover, in case (2) and (3), the order on H continues the order on G.Every countable group Our next goal is to prove the following.G is assumed to be finitely generated, assertion (1) is proved in \u2208\u03a3. As the action of T(\u03c6) on \u03a3 preserves the \u03a6\u2013orbits, T(\u03c6) acts on the \u03a6\u2010orbit of z0, the action is orientation\u2010preserving, and its orbits are dense. Finally, recall that the \u03a6\u2010orbit of z0 is homoemorphic to R. We fix such a homeomorphism. This induces an action of T(\u03c6) on R. We fix this action of T(\u03c6).Let us fix an action of C0 denote the group of functions from R to G of bounded support. The action of T(\u03c6) on R induces an action \u03c3 of T(\u03c6) on C0 such that for every h\u2208C0 and f\u2208T(\u03c6),Let permutational wreath productG\u2240RT(\u03c6) is defined as the semi\u2010direct product C0\u22ca\u03c3T(\u03c6), where, for  and \u2208C0\u22ca\u03c3T(\u03c6), :=(h1\u03c3(f1)(h2),f1f2).The g\u2208G, we define the following function g\u00af in C0:G\u00af:={g\u00af\u2223g\u2208G}.Definition 5.2(Splinter groups) The splinter group is the subgroup of the permutational wreath product G\u2240RT(\u03c6) generated by G\u00af and T(\u03c6). We denote it by Sp.For every T(\u03c6) and G are finitely generated. We note the following.Lemma 5.3G embeds into Sp with image G\u00af. Moreover, Sp is finitely\u00a0generated.The group Recall that Lemma 5.4Sp is\u00a0perfect.The splinter group Lemma 5.5G\u00af is in the first derived subgroup Sp\u2032 of Sp.The group To prove Lemma\u00a0f1 and f2\u2208T(\u03c6) be such that f1 maps \u2208(G\u2240\u27e8z\u27e9)\u27e8s\u27e9. Moreover, the element , regarded as a map \u27e8s\u27e9\u2192G\u2240\u27e8z\u27e9, has support \u2286{1}. In addition, (1) is a map \u27e8z\u27e9\u2192G such that )(1)=gi.As it is shown in Section\u00a0g,h\u2208G, their images in \u27e8c,s\u27e9 are conjugate. Let g\u00af and h\u00af be the images of g and h in \u27e8c,s\u27e9, respectively. In particular, g\u00af and h\u00af are elements of the form . Let \u2208\u27e8c,s\u27e9\u2a7d(G\u2240\u27e8z\u27e9)\u2240\u27e8s\u27e9 be such that g\u00af\u22121=h\u00af. Then we getsupp(fg\u00afsnf\u22121)=supp(g\u00af)={1}. On the other hand,n=0, hence g\u00af(1) is conjugate to h\u00af(1) in G\u2240\u27e8z\u27e9. Repeating this argument one more time with respect to the pair g\u00af(1),h\u00af(1)\u2208G\u2240\u27e8z\u27e9 and using the fact that (g\u00af(1))(1)=g and (h\u00af(1))(1)=h, we get that g is conjugate to h in G. Since g,h\u2208G are arbitrarily chosen elements from G, we get that the embedding from . Since every left\u2010ordered group embeds as a subgroup into Homeo+(J), we have the following.Proposition 6.1G embeds into a finitely generated left\u2010ordered group H. In addition, the order on H continues the order on G.Every countable left\u2010ordered group Let G be countable left\u2010orderable group. Then, by Theorem\u00a0G embeds into a finitely generated perfect left\u2010orderable group G1. On its own turn, since G1 is left\u2010orderable, it embeds into Homeo+(J). Let G2\u2a7dHomeo+(J) such that G2 is isomorphic to G1. Let H=T, see Definition\u00a0H has the required properties.\u25a1Let G is finitely generated), as required by Remark\u00a0Homeo+(J).We now construct an embedding as in the previous proposition, that, in addition, is Frattini and isometric m\u2208Z, there exists at most one x\u2208QI such that {x}d=m and {g(x)}d\u2a7d0.For each (2)x1\u2260x2\u2208QI and {x1}d={x2}d, then {g(x1)}d\u2260{g(x2)}d.If Let g is a bijection from I to J, then we say that g is strongly permuting the dyadic parts if it maps rational points to rational points and its restriction g\u2223QI:QI\u2192QJ satisfies Definition\u00a0Remark 6.5g:QI\u2192QJ is strongly permuting the dyadic parts, then, for each m\u2208Z, the set {{g(x)}d\u2223x\u2208QI,{x}d=m} is unbounded from\u00a0above.If If 0<i\u2a7dn:fi:Ii\u2192Ji such that fi(x)=\u03bbix+ci, where \u03bbi is a power of 2 and, for all i\u2209{0,n}, ci\u22600; andbijective dyadic maps gi:Ji\u2192Ii\u22121, whose restrictions to QJi strongly permute the dyadic parts.bijective maps Lemma 6.6\u039b=g1f1g2f2\u2026gnfn, then, for large enough N\u2208N, the set\u039b\u2260id.If Let us consider, for n.We will prove the lemma by induction on n=1. Then \u039b(x)=g1(\u03bb1x+c1). If c1=0 or N>{c1}d and {x1}d={x2}d=N, then, by Lemma\u00a0{\u03bb1x1+c1}d={\u03bb1x2+c1}d. The statement now follows as g1 strongly permutes the dyadic parts (see Remark\u00a0Let n>1. Then \u039b(x)=g1(\u03bb1\u039b2(x)+c1), where \u039b2=g2f2\u2026gnfn and c1\u22600. By inductive assumption, for any large enough N, there exists a sequence {xi}i=1\u221e such that {xi}d=N and limi\u2192\u221e{\u039b2(xi)}d=\u221e. By Lemma\u00a0i, {\u03bb1\u039b2(xi)+c1}d={c1}d, hence, the lemma follows as g1 strongly permutes the dyadic parts.\u25a1Next let 6.2J be a fixed closed interval in R with non\u2010empty interior. We prove:Proposition 6.7G be a countable\u00a0group.Let G is left\u2010orderable, then there is an embedding \u03a8:G\u21aaHomeo+(J) such that, for all g\u2208G\u2216{1}, the map \u03a8(g):J\u2192J is strongly permuting the dyadic parts and does not fix any rational interior point of J.If G is computably left\u2010orderable, then, in addition, all the maps \u03a8(g) can be taken to be\u00a0computable.If Let QJ={q0,q1,\u2026} such that the natural order on QJ is computable with respect to this\u00a0enumeration.As in the proof of Proposition\u00a0\u03a6:G\u2192QJ.Lemma 6.8G is enumerated and densely left\u2010ordered, then there is an enumeration G={gi1,gi2,\u2026} and an order preserving bijection \u0398:G\u2192QJ such that:(1)j, {\u0398(gij)}d\u2208N and {\u0398(gij)}d\u2209{{\u0398(gik)}d\u22231\u2a7dk<j};for odd (2)j, gij\u2209{gikgil\u22121gim\u22231\u2a7dk,l,m<j};for even (3)G is computably left\u2010ordered, then the enumeration G={gi1,gi2,\u2026} and the map j\u21a6\u0398(gij) are computable.if If We first strengthen Lemma\u00a0G={1=g0,g1,g2,\u2026} and QJ={0=r0,r1,r2,\u2026} be fixed (recursive) enumerations. We define \u0398:g0\u21a6r0 and \u0398:gij\u21a6rij, where  and  are permutations of  and , respectively, defined recursively as\u00a0follows.Let Step2n+1. Let G2n={gi1,\u2026,gi2n} and Q2n={ri1,\u2026,ri2n} be already defined. Let us define gi2n+1 as the element of the smallest index that is not in G2n. Suppose that gis<gi2n+1<git and that no element from G2n is in between gis and git. Then define ri2n+1\u2208QJ to be of the smallest index such that:(O1)ri2n+1\u2209Q2n and ris<ri2n+1<rit;(O2){ri2n+1}d\u2208N and {ri2n+1}d\u2209{{rij}d\u22231\u2a7dj\u2a7d2n}.Step2n+2. Let G2n+1:={gi1,\u2026,gi2n+1} and Q2n+1={ri1,\u2026,ri2n+1} be already defined. Let us define ri2n+2 as the rational of the smallest index that is not in Q2n+1. Suppose that ris<ri2n+2<rit and that no element from Q2n+1 is in between ris and rit. Then let us define gi2n+2\u2208G as the element of the smallest index such that:(E1)gi2n+2\u2209G2n+1 and gis<gi2n+2<git; and(E2)gi2n+2\u2209{gikgil\u22121gim\u22231\u2a7dk,l,m\u2a7d2n+1}.\u0398 defined this way is order preserving by (O1) and (E1). Condition (O2) yields assertion (1), and (E2) yields assertion (2). Finally, as the procedure is algorithmic, we also obtain assertion (3).\u25a1The bijection Proof of Proposition 6.7G is dense. Let the enumeration G={g0,g1,\u2026} and \u0398:G\u2192QJ satisfy the assertions of Lemma\u00a0\u0398:G\u2192QJ induces the embedding \u03c1G\u0398:G\u21aaHomeo+(J) according to \u03c1G\u0398(g)(\u0398(h))=\u0398(gh) for g,h\u2208G. Denote \u03a8=\u03c1G\u0398. Let h\u2208G\u2216{1}. By Lemmas\u00a0\u03a8(h) is strongly permuting the dyadic\u00a0parts.By Remark\u00a0QJ such that \u0398(gi)=ri and let ri\u2260rj\u2208QJ. We define rk=\u03a8(h)(ri)=\u0398(hgi) and rl=\u03a8(h)(rj)=\u0398(hgj), so that gk=hgi and gl=hgj.To this end, we enumerate i\u2260j such that {ri}d={rj}d and {rk}d,{rl}d\u2209N. Since {rk}d,{rl}d\u2209N, the indices k and l are even. Then, since gk=hgi=(hgj)(gj\u22121)(gi) and gl=hgj=(hgi)(gi\u22121)(gj), by (2) of Lemma\u00a0i or j. Let j>i. Then, since {ri}d={rj}d, we get the index j is even. Since gj=gi(hgi)\u22121(hgj), again by (2) of Lemma\u00a0We first show property (1) of Definition\u00a0ri\u2260rj\u2208QJ such that {ri}d={rj}d and suppose that {rl}d={rk}d. Without loss of generality, l>i,j,k . Then, since {rk}d={rl}d, by (1) of Lemma\u00a0l has to be even. Therefore, since \u0398(hgj)=rl and l is even, by (2) of Lemma\u00a0hgj\u2209{gmgn\u22121gp\u22231\u2a7dm,n,p<j}. On the other hand, since l>i,j,k, we get hgj=(hgi)(gi\u22121)(gj), a\u00a0contradiction.Next, we prove property (2) of Definition\u00a0\u25a1This completes the proof of Proposition\u00a06.3G be countable left\u2010orderable group. Then, by Theorem\u00a0G embeds into a finitely generated perfect left\u2010orderable group G1. Moreover, this embedding is a Frattini\u00a0embedding.Let J:=. Since G1 is left\u2010orderable, there is an embedding \u03a8:G1\u21aaHomeo+(J). Let G2=\u03a8(G1). By Proposition\u00a0G2\u2286Homeo+(J) strongly permute the dyadic parts and do not fix any rational interior point of J.Let G2\u2013dyadic maps, see Definition\u00a0Lemma 6.9\u039b be a G2\u2013dyadic map. If G2 has decidable word problem, then there is an algorithm to decide whether \u039b=id.Let For the definition of n>0 and, for all 0\u2a7di\u2a7dn, let Ji\u2282J and let gi:Ji\u2192Ii\u22121 be the restriction of an element of G2 such that gi\u2260id. Moreover, let fi:Ii\u2192Ji, given by fi(x)=\u03bbix+ci, be dyadic maps such that fi\u2260id.Let 0<i<n, Ji\u2282 and, by definition, \u03bbi is a power of 2, we get that ci\u22600 for 0<i<n. Then, by Lemma\u00a0\u039b:=g1f1g2f2\u2026gnfn\u2260id.Since, for all n=1 and f1=id, then \u039b=g1\u2208G. Then we decide using the algorithm for the word problem in G.\u25a1If Lemma 6.10G1\u21aaT is\u00a0isometric.The embedding Combining Lemma\u00a0Lemma 6.11G1\u21aaT is a Frattini\u00a0embedding.The embedding h,g\u2208G2 and t\u2208T. We assume that ht\u22121gt=1.Let t by a canonical chart representation  such that \u22c3iJi=; and represent t\u22121 by . We recall that ti(Ii)=Ji.We represent k be a index such that 1/2 is in the closure of Jk and such that (after applying a chart refinement if necessary) Jk\u2286J. As g is fixing 1/2, there is Jk\u2032\u2286Jk such that g(Jk\u2032)\u2286Jk and such that 1/2 is in the closure of Jk\u2032. We let Ik\u2032=tk\u22121(Jk\u2032) and Ik\u2032\u2032=tk\u22121gtk(Ik\u2032).Let  is in a chart representation of t\u22121gt. Up to applying the algorithm of Lemma\u00a0Ik\u2032\u2032 is in . Moreover, up to applying a chart refinement if necessary, we may assume that either Ik\u2032\u2032\u2229J is empty or consists of one point (1/4 or 1/2), or Ik\u2032\u2032\u2286J.Then the triple Ik\u2032\u2032\u2229J is empty or consists of one point, then  is in a chart representation of ht\u22121gt. Thus tk\u22121gtk=id on Ik\u2032 by Lemma\u00a0g acts as the identity on Jk\u2032. Since non\u2010trivial elements of G2 do not fix any rational interior points of J, the element g=1.If ,htk\u22121gtk) is in a chart representation of ht\u22121gt. Thus htk\u22121gtk=id on Ik\u2032 by Lemma\u00a0h(Ik\u2032\u2032)=Ik\u2032. This is only possible if Ik\u2032\u2286J. But then Proposition\u00a0tk acts as an element of G2. Since non\u2010trivial elements of G2 do not fix any rational interior points of J, this implies that g and h are conjugate in G1.\u25a1Otherwise, the triple We can now conclude Theorems\u00a0Proof of Theorems 1, 2 and Remark 1.2H=T.We let H is finitely generated left\u2010orderable and simple by Lemmas\u00a0G embeds into H, and the order on H extends the order on G. Moreover, by Lemma\u00a0G is a Frattini\u00a0embedding.The group G is computably left\u2010ordered, we may in addition assume that G1 is computably left\u2010ordered, see Theorem\u00a0g\u2208G1, \u03a8(g) is computable. Therefore, by Lemma\u00a0H is recursively enumerable. Moreover, by Lemma\u00a0H has decidable word problem. By Lemma\u00a0H is computable.\u25a1If Proof of Theorem 4G be finitely generated left\u2010orderable group with a recursively enumerated positive cone. If G has decidable word problem, then the left\u2010order on G is computable by Lemma\u00a0G embeds into a finitely generated computably left\u2010ordered simple group H. In particular, the word problem in H is decidable. Thus H can be defined by a recursively enumerable set of relations. By  and J= and that is identity outside of \u222ai=1\u221eDi. In particular, the map \u03bel:Dk\u2192Dm is dyadic.Remark 7.3\u03bel:J\u2192J, l\u2208N, are computable and continuous at non\u2010dyadic points. Also, they have finitely many (dyadic) breakpoints outside of any open neighborhood of 1/3.The maps For every \u03bb:G1\u2192C(J) by \u03bb(g(l))=\u03bel, for all l\u2208N.Remark 7.4\u03bb is an embedding of G1 into computable maps in C(J).The map Let us define \u039b=g1f1g2f2\u2026gnfn:In\u2192I0, where fi:Ii\u2192Ji and gi:Ji\u2192Ii\u22121, be a G2\u2010dyadic map as in Definition\u00a0Ji\u2286J=. Correspondingly, we say that a chart of type (II), see Definition\u00a0special if the local representation in this chart is of the form f0\u039b, where \u039b is a special G2\u2013dyadic\u00a0map.Let fi, gi, 1\u2a7di\u2a7dn, are computable.Lemma 7.5K1,K2,\u2026,Ks\u2286 fixes 1/3, then it acts as an element of G2.Special local representations are of the form By Lemma\u00a0G2 is decidable, this implies that there exists an algorithm that decides whether a special G2\u2010dyadic map represents the identity map. This, combined with Lemmas\u00a0Corollary 7.8T is\u00a0decidable.The word problem in Since the word problem for 7.2G\u21aaT is Frattini.Lemma 7.9G1\u21aaT is a Frattini\u00a0embedding.The embedding The embedding We adapt the proof of Lemma\u00a0h,g\u2208G2 and t\u2208T. We assume that ht\u22121gt=1.Let t by a canonical chart representation {} such that \u22c3iJi=; and represent t\u22121 by {}. We recall that ti(Ii)=Ji.We represent k be a index such that 1/3 is in the closure of Jk and such that (after applying a chart refinement if necessary) Jk\u2286J. As g is fixing 1/3, there is Jk\u2032\u2286Jk such that g(Jk\u2032)\u2286Jk and such that 1/3 is in the closure of Jk\u2032. We let Ik\u2032=tk\u22121(Jk\u2032) and Ik\u2032\u2032=tk\u22121gtk(Ik\u2032).Let  is in a chart representation of t\u22121gt. Up to applying the algorithm of Lemma\u00a0Ik\u2032\u2032 is in . Moreover, up to applying a chart refinement if necessary, we may assume that either Ik\u2032\u2032\u2229J is empty or consists of one point 1/3, or Ik\u2032\u2032\u2286J.Then the triple Ik\u2032\u2032\u2229J is empty or consists of one point 1/3, then  is in a chart representation of ht\u22121gt. Thus tk\u22121gtk=id on Ik\u2032 by Lemma\u00a0g acts as the identity on Jk\u2032. As 1/3 is in the closure of Jk\u2032, this implies that g=1.If ,htk\u22121gtk) is in a chart representation of ht\u22121gt. Thus htk\u22121gtk=id on Ik\u2032 by Lemma\u00a0tkhtk\u22121g:Jk\u2032\u2192Jk\u2032 has to be the identity as well. As g is fixing 1/3, tkhtk\u22121 has to fix 1/3.Otherwise, the triple tk was dyadic ), it would have to fix 1/3, so that tk=id by Lemma\u00a0tk is of type (II), we may assume that tk is special, see Lemma\u00a0tk acts as an element of G2.If g and h are conjugated by elements of G2. This implies that g and h are conjugated in\u00a0G1.\u25a1Thus Corollary 7.10G\u21aaT is\u00a0Frattini.The embedding Combining Lemma\u00a0Lemma 7.11G1\u21aaT is an isometric\u00a0embedding.The embedding X for G1, and denote the generating set of T(\u03c6) given by Lemma\u00a0Y. We denote the union of the bijective images of X and Y in T by Z and recall that Z generates T. We assume that all generating sets are symmetric. We denote by |.|A the word metric with respect to the generating set A.We fix a finite generating set g\u2208G2 and let t=z1\u22efzm be a reduced word in the alphabet Z that represents g\u22121\u2208T, so that tg=1. In addition, we assume that m=|g|Z. We represent every generator zi by a canonical chart representation, see Lemma\u00a0 for t. Recall that the maps ti are compositions ti=h1\u22efhmi, where each map hj is a local representation in the canonical chart representation of a generator in Z and mi\u2a7dm. In addition, up to applying the algorithm of Lemma\u00a0t, we may assume that \u22c3Ii=.Let Ik be an interval such that 1/3 is in the closure of Ik and such that (after applying a chart refinement if necessary) Ik\u2286J.Let (Ci\u00d7g\u22121(Ik),Ci\u00d7Ji,tig) is in a canonical chart representation of the identity. By Lemma\u00a0tig is the identity mapping. In particular, g\u22121(Ik)=Ji, so that Ji\u2286J and 1/3 is in the closure of Ji.Then ti is not dyadic ) by Lemma\u00a0ti=fg1f1\u22efgnfn is a chart of type (II), then, by Lemma\u00a0ti is special. Thus ti=g1\u22efgn\u2208G2  is\u00a0isometric.If Combining Lemma\u00a0Proof of Theorem 7.1T follows from Lemma\u00a0T is decidable provided that it is decidable in G2. By Corollary\u00a0G\u21aaT is Frattini. By Corollary\u00a0G is finitely generated. Therefore, the embedding G\u21aaT satisfies Theorem\u00a0\u25a1The simplicity of Journal of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not\u2010for\u2010profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of\u00a0mathematics.The"}
{"text": "Historic records suggest that a virulent form of treponematosis, sexually transmitted syphilis was introduced to Asia from Europe by the da Gama crew, who landed in India in 1498. Our objective is to assess the gross pathology of human skeletal remains from the Tang dynasty of China to test the presence of treponemal infection in East Asia before 1498. We interpret this paleopathological evidence in the context of site ecology and sociocultural changes during the Tang dynasty.We examined the gross pathology of 1598 human skeletons from Xingfulindai (AD 618 to AD 1279) archeological site located on the Central Plain of China. Using the modified diagnostic criteria defined by Hackett's classical work, we classify the pathology as consistent, strongly suggestive, or pathognomonic for treponemal infection.Twelve adult individuals from Xingfulindai had bone lesions suggestive of systemic pathology. Two of these individuals displayed a combination of lesion patterns pathognomonic of treponemal disease and one had lesions consistent with treponematosis. The radiocarbon dates for the bone samples from these skeletons place them before AD 1200.The location of Xingfulindai in a continental climatic zone is not typical for yaws and bejel ecology, because these strains occur in the tropics, or in hot, dry environments, respectively. The urban setting, where there is documented evidence for increased interaction between multiple ethnic groups and a developed institution of courtesans during the Tang dynasty, favors sexually transmitted syphilis as the more likely diagnosis. This study supports an earlier spread of syphilis to China than 1498. Syphilis in China before Vasco da Gama? Together with the tropical skin disease pinta, linked to T. carateum, these infectious diseases are collectively referred to as treponematoses , yaws, and endemic syphilis (bejel) are caused by three subspecies of the spirochete bacteria Historically, three alternative models for the introduction of treponemal infection to continental China can be found in the literature: the Vasco da Gama model Bi,\u00a0, the TraT. p. ssp pertenue (yaws) in Vietnam about 4000\u2009years ago. They consider the migration of farmers from Southern China into the region as a route of transmission for this disease. Finally, it is conceivable that different forms of treponemal diseases have been present for a long time but evaded historic documentation in some regions by being confused with other diseases  tomb area  is located in the suburb of Xi'an City, Shaanxi Province , rhinopharyngitis mutilans (gangosa) and saber shin, as well as generalized subperiosteal proliferative lesions, substantial sclerosis and osteitis, and involvement of the medullary cavity of the long bones have all been recorded in patients suffering from yaws, bejel, and syphilis  suis Grin,\u00a0. Therefois Grin,\u00a0.Treponema cases. These categories were assigned based on Hackett\u00a0 on the occipital and parietal bones  had cranial hyperostosis or widespread periostosis or osteitis on multiple long bones suggestive of systemic pathology , which produces bright red bumpy fruits reminiscent of syphilis sores. He noted that the disease started spreading from the Lingbiao area of Guangzhou. His text also described a treatment for the disease, including the use of mercury ointment:Consistent with the Vasco da Gama model, the earliest direct description of ST syphilis can be found in \u201cCompendium of Materia Medica\u201d \u672c\u8349\u7eb2\u76ee (c1578\u20101596) by Li Shizhen \u674e\u6642\u73cd Li,\u00a0, which i\u201c\u6768\u6885\u75ae\uff0c\u53e4\u65b9\u4e0d\u8f7d\uff0c\u4ea6\u65e0\u75c5\u8005\uff0c\u8fd1\u65f6\u8d77\u4e8e\u5cad\u8868\uff0c\u4f20\u53ca\u56db\u65b9\uff0c\u76d6\u5cad\u8868\u98ce\u571f\u5351\u708e\uff0c\u5c9a\u7634\u718f\u84b8\uff0c\u996e\u5556\u8f9b\u70ed\uff0c\u7537\u5973\u6deb\u7325\u3002\u6e7f\u70ed\u4e4b\u90aa\u79ef\u84c4\u65e2\u6df1\uff0c\u53d1\u4e3a\u6bd2\u75ae\u3002\u9042\u81f4\u4e92\u76f8\u4f20\u67d3\uff0c\u7136\u7686\u6deb\u90aa\u4e4b\u4eba\u75c5\u4e4b\u3002\u201cIn the past, no books mentioned syphilis (myrica sores) nor how to cure it. Syphilis probably started in the Guangxi and Guangdong area and began to spread. The fields there are low\u2010lying and the weather is hot, humid, and foggy. Thus, people there like to eat spicy food. Because of this, it makes symptoms worse, causing toxic sores which begins to infect others, but only infects those who had sexual behavior.\u201d\u533b\u7528\u8f7b\u7c89\uff0c\u94f6\u6731\u52ab\u5242\uff0c\u4e94\u3001\u4e03\u65e5\u5373\u6108\u3002\u5176\u6027\u71e5\u70c8\uff0c\u5584\u9010\u75f0\u6d8e\u3002\u6d8e\u4e43\u813e\u4e4b\u6db2\uff0c\u6b64\u7269\u5165\u80c3\uff0c\u6c14\u5f52\u9633\u660e\uff0c\u6545\u6d8e\u88ab\u52ab\uff0c\u968f\u706b\u4e0a\u5347\uff0c\u4ece\u5589\u988a\u9f7f\u7f1d\u800c\u51fa\uff0c\u6545\u75ae\u5373\u5e72\u75ff\u800c\u6108\u3002\u82e5\u670d\u4e4b\u8fc7\u5242\uff0c\u53ca\u7528\u4e0d\u5f97\u6cd5\uff0c\u5219\u6bd2\u6c14\u7a9c\u5165\u7ecf\u7edc\u7b4b\u9aa8\u4e4b\u95f4\uff0c\u83ab\u4e4b\u80fd\u51fa\u3002\u75f0\u6d8e\u65e2\u53bb\uff0c\u8840\u6db2\u8017\u6db8\uff0c\u7b4b\u5931\u6240\u517b\uff0c\u8425\u536b\u4e0d\u4ece\uff0c\u53d8\u4e3a\u7b4b\u9aa8\u631b\u75db\uff0c\u53d1\u4e3a\u75c8\u6bd2\u75b3\u6f0f\u3002\u4e45\u5219\u751f\u866b\u4e3a\u7663\uff0c\u624b\u8db3\u76b1\u88c2\uff0c\u9042\u6210\u5e9f\u75fc\u3002\u201cUse medical calomel (mercury chloride) or cinnabar (mercury sulfide) to cure it. It usually takes five to seven days to cure. These types of drugs are so strong that they will easily make the spleen produce saliva. When the drugs go into the stomach, it will produce good energy and send it to yangming. Then, the energy will go up with the saliva that is produced from the spleen via the throat to the mouth, and it will gradually make the sores smaller, eventually curing them. However, if the dose is too large, then it can worsen the symptoms because the overdose can be toxic and produce bad energy to go into the meridian system. What's even worse, the bad energy has no way to get out, which will result in having less saliva, low blood pressure, weak bones, malnutrition, and then causes bone pain. This ultimately results in ulcer and gum disease. If the patient ignores this for a while, it will cause skin ulcers, and their hands and feet will begin cracking and develop a chronic illness.\u201d\u5176\u7c7b\u6709\u6570\u79cd\uff0c\u6cbb\u5219\u4e00\u4e5f\u3002\u5176\u8bc1\u591a\u5c5e\u53a5\u9634\uff0c\u9633\u660e\u4e8c\u7ecf\uff0c\u800c\u517c\u4e4e\u4ed6\u7ecf\u3002\u90aa\u4e4b\u6240\u5728\uff0c\u5219\u5148\u53d1\u51fa\uff0c\u5982\u517c\u5c11\u9634\u592a\u9634\uff0c\u5219\u53d1\u4e8e\u54bd\u5589\u3002\u517c\u592a\u9633\u3001\u5c11\u9633\uff0c\u5219\u53d1\u4e8e\u5934\u8033\u4e4b\u7c7b\u3002 \u201cAlthough there are many types of syphilis, the principles are the same. Syphilis will send bad energy to jueyin and yangming. People will feel discomfort in different parts of their body, depending on where the bad energy moves to. For example, if the energy goes to shaoyin and taiyin, the patient will feel discomfort in their throat. If it moves to taiyang and shaoyang, then the patient will feel discomfort around the head and ears.\u201d In this passage, \u201cjueyin,\u201d \u201cyangming,\u201d \u201cshaoyin,\u201d \u201cshaoyang,\u201d \u201ctaiyang,\u201d and \u201ctaiyin\u201d are the circulatory tracks of traditional Chinese medicine, which can be thought as the major flows or meridians of the qi energy, with distinct locations in the body Needham,\u00a0:48. For Smilax glabra) is a good remedy to cure yangming disease, because it tastes sweet and light, which helps smooth the \u201cfire\u201d\u2014bad energy. [\u2026] Presently, the soufeng jiedu (\u641c\u98ce\u89e3\u6bd2) soup is used to treat syphilis. Curing patients with serious symptoms by drinking soufeng jiedu will take a month; those with light symptoms are cured within half a month. To make soufeng jiedu soup use one liang  of \u571f\u832f\u82d3 tufuling (Smilax glabra) and 1.5\u00a0g of \u858f\u82e1\u4ec1 yiyiren (Job's tears: Coix lacryma\u2010jobi), \u91d1\u94f6\u82b1 jinyinhua (Japanese honeysuckle Lonicera japonica), \u9632\u98ce fangfeng (Saposhnikovia divaricate), \u6728\u74dc m\u00f9gu\u0101 , \u6728\u901a mutong (chocolate vine Akebia quinata), and \u767d\u9c9c\u76ae b\u00e1ixi\u01cenp\u00ed  each respectively, as well as 1.2\u00a0g of \u7682\u835a\u5b50 zaojiazi (Acacia). For people who do not have enough good energy, one can add \u4eba\u53c2 rencan  2.1\u00a0g; people who have low blood pressure can add \u5f53\u5f52 danggui  2.1\u00a0g). Cook all of the herbs with two bowls of water and make soup. Drink the soup three times a day and avoid drinking tea, eating beef, lamb, and chicken; maintain abstinence. This is the secret formula.\u201d\u60df\u571f\u832f\u82d3\u6c14\u5e73\uff0c\u5473\u7518\u800c\u6de1\uff0c\u4e3a\u9633\u660e\u672c\u836f\u3002[\u2026] \u4eca\u533b\u5bb6\u6709\u641c\u98ce\u89e3\u6bd2\u6c64\u6cbb\u6768\u6885\u75ae\uff0c\u4e0d\u72af\u8f7b\u7c89\u3002\u75c5\u6df1\u8005\u6708\u4f59\uff0c\u6d45\u8005\u534a\u6708\u800c\u6108\u3002\u670d\u8f7b\u7c89\u800c\u7b4b\u9aa8\u631b\u75db\uff0c\u762b\u75ea\u4e0d\u80fd\u52a8\u5c65\u8005\uff0c\u670d\u4e4b\u4ea6\u6548\u3002\u5176\u65b9\u7528\u571f\u832f\u82d3\u4e00\u4e24\uff0c\u858f\u82e1\u4ec1\u3001\u91d1\u94f6\u82b1\u3001\u9632\u98ce\u3001\u6728\u74dc\u3001\u6728\u901a\u3001\u767d\u9c9c\u76ae\u5404\u4e94\u5206\uff0c\u7682\u835a\u5b50\u56db\u5206\u3002\u6c14\u865a\uff0c\u52a0\u4eba\u53c2\u4e03\u5206\uff1b\u8840\u865a\uff0c\u52a0\u5f53\u5f52\u4e03\u5206\u3002\u60df\u5fcc\u8336\u3001\u8089\u3001\u6cd5\u9762\uff0c\u623f\u52b3\u3002\u76d6\u79d8\u65b9\u4e5f\u3002 \u201cPowdered tu fuling . The text described two types of sores,  Morgan,\u00a0. However4.2caries sicca pathognomonic or strongly suggestive of treponematosis has been described on a skull from Fujian Province dated to the Song dynasty (AD 960\u20131279)  Zhang,\u00a0  Zhang,\u00a0. However) Zhang,\u00a0. Two pos) Zhang,\u00a0  Zhang,\u00a0. In this) Zhang,\u00a0 proposed) Zhang,\u00a0. Because) Zhang,\u00a0 findings) Zhang,\u00a0 dating tThe Island Southeast Asia and Pacific model developed by Buckley and Oxenham\u00a0, reviews4.3T. pallidum it is important to examine the environmental and sociocultural conditions that favored the emergence and spread of the new forms of treponemal disease. The Xingfulindai cases of treponemal infection, with direct radiocarbon dates that place skeletons prior to 1400s, and their association with the large urban population of Chang'an, present us with such an opportunity. According to the New Book of Tang (chapter 41.27:1669), the capital city of Chang'an had 362,921 households with a total population of 960,188 people within its limits and ruled over 20 provinces. The population of Chang'an was multiethnic, with a notable influx of people from the northwestern frontier area of imperial China as well as Persia , was extremely active, promoting an increased level of interaction, bringing into contact previously isolated areas Waugh,\u00a0. As suchIf these pathological cases from Xingfulindai represent individuals with sexually transmitted syphilis, it is important to investigate how this would have affected the community by considering the known transmission rate of this disease, and the functional effects on people. Unfortunately, unbiased research on sexual transmission of syphilis is rare ; data curation (lead); formal analysis ; funding acquisition ; investigation (lead); project administration ; resources ; supervision (lead); visualization . Guoshuai Gao: Data curation ; investigation . Xiangyu Zhang: Investigation (supporting). Bo Gao: Data curation (supporting); investigation (supporting); resources (supporting). Chenggang Duan: Data curation (supporting); investigation (supporting). Hong Zhu: Formal analysis (supporting); supervision (supporting). Aida R. Barbera: Formal analysis (supporting); investigation (supporting); writing \u2013 original draft (supporting). Si\u00e2n Halcrow: Conceptualization ; funding acquisition ; methodology (supporting); writing \u2013 original draft (supporting). Kate Pechenkina: Conceptualization (lead); formal analysis (lead); investigation ; methodology (lead); project administration ; resources (supporting); supervision ; visualization ; writing \u2013 original draft .Figure S1 Supplementary figuresClick here for additional data file.Table S1 Supplementary tablesClick here for additional data file."}
{"text": "Monobius Alatawi and Kamran, where all the leg tarsi in females have one midventral seta. Moreover, the genus Tillandsobius Bolland is synonymized with the genus Tycherobius Bolland and the genus Neophyllobius Berlese is categorized in two new subgeneric divisions. For the first time, a key to all known species of the genus Neophyllobius is provided. The ambiguities in the ventral idiosoma setal notation are highlighted and discussed.The present study erects a new genus, Monobius Alatawi and Kamran, is hereby proposed for the two already described species, viz; M. electrus (\u017bmudzi\u0144ski) and M. meyerae (Bolland). In addition, the monospecific genus Tillandsobius Bolland is synonymized with the genus Tycherobius Bolland due to variations in the setae number of tibiae I\u2013IV. Further, the genus Neophyllobius Berlese is categorized in two new subgeneric divisions as Neophyllobius Berlese and Monophyllobius Mirza. The number and position of the midventral setae on tarsi I\u2013IV are considered as strong diagnostic generic and subgeneric diagnostic characters. The present study also includes the key to all known species of the genus Neophyllobius. The morphological characters of ten poorly described Neophyllobius species were studied in detail through published literature. The ambiguities in the ventral idiosoma setal notation are highlighted and discussed. It is concluded that two intercoxal setae 3a\u20134a are always present on small platelets, paired aggenital setae (ag) are present anteriorly and paired genital setae (g) present posteriorly on genital shield. In addition, five records of new species for Saudi Arabia are reported along with re\u2013descriptions of three species.A new genus,  Neophyllobius Berlese ..NeophyllNeophyllobius has one or two midventral setae on tarsi I\u2013IV, if two setae are present, they are in a longitudinal line. While the genera Tillandsobius and Tycherobius always have two midventral setae on tarsi I\u2013II, consistently not in a longitudinal line and variously spaced (Tillandsobius (one species) and Tycherobius (25 species) are closely related.These three genera were also differentiated based on the position of midventral setae on tarsi I\u2013II. The genus y spaced . In thisNeophyllobius, Tillandsobius and Tycherobius. However, the number and position of the mid\u2013ventral setae on tarsi I\u2013II, remain a constant and persistent generic diagnostic character. Therefore, the monospecific genus Tillandsobius is hereby synonymized with the genus Tycherobius. In addition, we propose a new genus, Monobius Alatawi and Kamran gen. nov., for the two species  namely, M. meyerae (Bolland) and M. electrus (\u017bmudzi\u0144ski), described originally in the genus Neophyllobius. The diagnoses of the genera Neophyllobius and Tycherobius are modified and provided below. The morphological characters of these three genera, including the new genus, are summarized in Based on the evidence provided above, the number of setae on tibiae I\u2013IV does not represent a strong morphological character to differentiate the three genera urn:lsid:zoobank.org:act:3B429EFF-A148-46A5-8BE4-32127A03E707Type species: Neophyllobius electrus \u017bmudzi\u0144ski, 2020:3 [, 2020:3 .Diagnosis: Leg tarsi I\u2013II with one mid-ventral seta present on distal half and a proximal solenidion, tarsi I\u2013IV with 9\u20139\u20137\u20137 tactile setae.Remarks: The new genus Monobius, is morphologically closer to the genera Neophyllobius and Tycherobius  and distinct from Acamerobia, Camerobia, Bisetolobius and Decaphyllobius . It can be further distinguished from Neophyllobius and Tycherobius due to the presence of one midventral seta distally on leg tarsi I\u2013II vs. two midventral setae on leg tarsi I\u2013II in later two genera.Etymology: The generic epithet is derived from the diagnostic character of one midventral seta on all leg tarsi (mono = one)Monobius includes two species M. meyerae [M. electrus [Neophyllobius.The new genus  meyerae  and M. eelectrus . Both spNeophyllobius meyerae Bolland, 1991:63 [ 1991:63 .Remarks: The species M. meyerae (Bolland) is referred to the new genus Monobius due to presence of one midventral seta on all leg tarsi. The species can be differentiated from the second species of the genus M. electrus (\u017bmudzi\u0144ski) based on number of setae on femur III (2 vs. 3), number of dorsal body setae (15 vs. 14) and state of pdx setae (present vs. absent).Distribution: South AfricaNeophyllobius electrus \u017bmudzi\u0144ski, 2020:3 [, 2020:3 .Remarks: The species M. electrus (\u017bmudzi\u0144ski) is referred to the new genus Monobius due to presence of one midventral seta on all leg tarsi.Distribution: Fossil preserved in Baltic Amber, PolandTycherobius Bolland, 1986: 205 [Tillandsobius Bolland, 1986: 205; synonym nov.Diagnosis: Two midventral setae on tarsi I\u2013II, not present in a longitudinal line,Remarks: The monospecific genus Tillandsobius was erected by Bolland [Tycherobius only by difference in number of setae on tibiae I\u2013IV. As mentioned earlier, tibial setal counts are variable among the species of the genus Tycherobius and cannot be considered as a generic diagnostic character , tarsi I\u2013IV (10\u201310\u20137\u20137 vs. 7\u20137\u20138\u20138), femur II\u2013III (3\u20132 vs. 4\u20133) and differences in length of dorsal body setae .Distribution: Florida, USANeophyllobius Berlese, 1886:19 [Type species:\u00a0Neophyllobius elegans Berlese, 1886:19 [Diagnosis: Two midventral setae on tarsi I\u2013II, present in a longitudinal line.Neophyllobius, 114 species have two midventral setae on all leg tarsi. In contrast, 14 species have one midventral seta on leg IV and seven species with no such information available [Neophyllobius is categorized in two new subgenera; Neophyllobius Berlese and Monophyllobius Mirza, based on the number of midventral setae on leg tarsi III\u2013IV.Among the species of the genus vailable ,8,9,16. urn:lsid:zoobank.org:act:D34B5F98-1F09-45C8-8C64-62CA683C154EType species:\u00a0Neophyllobius elegans Berlese, 1886:19 [Diagnosis: Leg tarsi III\u2013IV always with two midventral setaeNumber of species included: 114urn:lsid:zoobank.org:act:8648D2D3-7F38-442A-8C15-680FB8D560A7Type species:\u00a0Neophyllobius texanus McGregor, 1950:66 [ 1950:66 .Diagnosis: Leg tarsus III often and tarsus IV always with one midventral setaEtymology: The subgeneric epithet refers to the presence of one midventral seta on leg tarsi III and IV (mono = one)Number of species included: 14N. combreticola, N. lorestanicus, N. denizliensis, T. emadi and C. southcotii. The species N. combreticola along with two previously reported species, viz; N. muscantribii and N. fissus are redescribed in detail. In addition, the species N. lorestanicus and N. denizliensis were previously misidentified as N. communis and N. hispanicus, respectively.The present research reported five new records of camerobiid mites from Saudi Arabia, viz; Neophyllobius combreticola Bolland, 1991:196 [1991:196 ; Beyzavi1991:196 .Redescription n = 3)Female  and one sword-like seta, tarsus with two eupathidia (acm\u03b6 and sul\u03b6), two simple setae (ba and va) and one small solenidion (\u03c9)  .Dorsum  present, genito-anal valves with a pair of genital setae (g) and three pairs of anal setae ps1, ps2 and ps3 \u20131(\u03ba)\u20131\u20131, tibiae 9(\u03c6)\u20138(\u03c6)\u20138(\u03c6)\u20137(\u03c6), tarsi 10(\u03c9)\u201310(\u03c9)\u20138\u20138. All tarsi with ambulacrum bearing a pair of claws and an empodium with two rows of tenent hairs, all genu setae long and whip like, dorsal seta on genu I 290\u2013300 Female  and one sword\u2013like seta, tarsus with two eupathidia (acm\u03b6 and sul\u03b6), two simple setae (ba and va) and one small solenidion (\u03c9).athosoma : 51\u201355 lDorsum , 4c 43\u201344, intercoxal setae length: 1a 32\u201335, 3a 34\u201346, 4a 25\u201327 (ag) 14\u201315, genito-anal valves with a pair of genital setae (g) 10\u201313 and three pairs of anal setae ps1, ps2 and ps3 \u20131(\u03ba)\u20131\u20131, tibiae 9(\u03c6)\u20138(\u03c6)\u20138(\u03c6)\u20137(\u03c6), tarsi 10(\u03c9)\u201310(\u03c9)\u20138\u20138. All tarsi with ambulacrum bearing a pair of claws and an empodium with two rows of tenent hairs, all genu setae long and whiplike, dorsal seta on genu I 218\u2013220 , Al\u2013Qatif, SA, 26\u00b034.36.7\u2032 N, 49\u00b059.07.1\u2032 E, 25 January 2014 and 11 December 2013, coll. Kamal Alsahwan; one female, date palm (Arecaceae), Al\u2013Imam University, Riyadh, SA, 24\u00b048.785\u2032 N, 46\u00b042.142\u2032 E, 19 March 2011, coll. Kamal Alsahwan.Previous Distribution: TrinidadNeophyllobius muscantribii Bolland, 1991:73 [ 1991:73 .Neophyllobius eragrostidis Bolland, 1991:73 [Eragrostis (plant), syn. by du Toit, Theron and Ueckermann, 1998 [ 1991:73 ; Eragrosnn, 1998 .Redescription n = 3)Female  and one sword-like seta, tarsus with two eupathidia (acm\u03b6 and sul\u03b6), two simple setae (ba and va) and one small solenidion (\u03c9).athosoma : 54\u201360 lDorsum  14\u201315 present, genito\u2013anal valves with a pair of genital setae (g) 7\u20139 and three pairs of anal setae   .Leg (\u03ba)\u20131(\u03ba)\u20131\u20131, tibiae 9(\u03c6)\u20138(\u03c6)\u20138(\u03c6)\u20137(\u03c6), tarsi 10(\u03c9)\u201310(\u03c9)\u20138\u20138. All tarsi with ambulacrum bearing a pair of claws and an empodium with two rows of tenent hairs, dorsal seta on genu I (38\u201341) twice as long as the segment , Asir, SA, 19\u00b015.884\u2032 N, 42\u00b005.261\u2032 E, 29 October 2019, coll. M. Kamran and H. M. S. Mushtaq, 2 females, Kavi plant and Sider like, Al\u2013Baha, SA, 19\u00b059.807\u2032 N, 41\u00b025.715\u2032 E, 25 April 2013, coll. Kamal Alsahwan.Previous Distribution: South AfricaNeophyllobius lorestanicus Khanjani, Hoseini, Yazdanpanah and Masoudian, 2014: 441 [014: 441 .Remarks: The species N. lorestanicus is a new record for the camerobiid mite fauna of Saudi Arabia and no distinct morphological differences were found between Saudi Arabian specimens of N. lorestanicus and the original description. It belongs to the Neophyllobius subgenus nov. based on presence of two in-line midventral setae on tarsi I\u2013IV.Material Examined: Two females, date palm (Arecaceae), Irqa, Riyadh, SA, 24\u00b041.130\u2032 N, 46\u00b034.550\u2032 E, 11 December 2013, coll. Kamal Alsahwan; one female, Acacia spp, (Fabaceae). Al\u2013Ahsa, SA, 25\u00b055.371\u2032 N, 48\u00b059.037\u2032 E, 22 October 2020, coll. J. H. Mirza, N. A. Elgoni, H. M. Sajid and M. W. Khan.Previous distribution: IranNeophyllobius denizliensis Akyol, 2020:88 [Remarks: The species N. denizliensis is a new record for the camerobiid mite fauna of Saudi Arabia. It was previously misidentified as N. hispanicus [Neophyllobius subgenus nov. based on presence of two in-line midventral setae on tarsi I\u2013IV. No morphological differences were found between Saudi Arabian specimens and original description of the N. denizliensis.spanicus . It beloMaterial Examined: One female, Tamarix spp., (Tamaricaceae), Tabuk, SA, 28\u00b030.653\u2032 N, 36\u00b028.168\u2032 E, September 29, 2020, coll. J. H. Mirza, H. M. S. Mushtaq and E. M. Khan; one female, Phoenix dactylifera (Arecaceae), Irqa, Riyadh, SA, 13 December 2020, coll. Kamal Alsahwan; one female, grasses (Poaceae). Al\u2013Imam University, Riyadh, SA, 24\u00b048.785\u2032 N, 46\u00b042.142\u2032 E, 7 April 2016, coll. M. Kamran.Previous distribution: TurkeyTycherobius Bolland, 1986:205 [1986:205 .Type species: Neophyllobius lombardinii Summers and Schlinger, 1955 [er, 1955 .Tycherobiusemadi Khanjani, Hajizadeh, Ostovan and Asali FayazTycherobius emadi Khanjani, Hajizadeh, Ostovan and Asali Fayaz, 2013:134 [2013:134 ; Hoseini2013:134 .Remarks: The species Tycherobius emadi is first time reported from Saudi Arabia and is morphologically similar to the original description [cription .Material Examined: One female, soil, and leaf debris, Tabuk, SA, 26\u00b058.271\u2032 N, 49\u00b040.220\u2032 E, 19 October 2019, coll. M. Kamran and H. M. S. Mushtaq.Previous distribution: IranCamerobia Southcott, 1957: 306 [957: 306 .Type species: Camerobia australis Southcott, 1957 [tt, 1957 Camerobia southcotti GersonCamerobia southcotti Gerson, 1972: 502 [972: 502 .Remarks: The species Camerobia southcotti is a new record for the camerobiid mite fauna in Saudi Arabia and is morphologically similar to the original description [cription .Material Examined: Three females, unidentified wild host plant, Jeddah, SA, 26\u00b058.271\u2032 N, 49\u00b040.220\u2032 E, 25 April 2016, coll. M. Kamran and J. H. Mirza.Previous distribution: IsraelThe following 10 species are not included in the key either due to lack of available literature or incomplete descriptions/illustrations. The available but scarce information from different literatures about these species are summarized in the Neophyllobius elegansBerleseNeophyllobius elegans Berlese, 1886:19 [ 1886:19 ; Canestr 1886:19 ; Bolland 1886:19 .Bolland  reportedTable 2Neophyllobius guajavaeChatterjee and GuptaNeophyllobius guajavae Chatterjee and Gupta, in Gupta, 2002:38 [ 2002:38 .N. natalensis.The description and illustrations provided are very poor and didn\u2019t help to place the species in the diagnostic key provided in present work. Although, the original authors have compared the species with N. guajavae belongs to the new subgenus Monophyllobius based on tarsi III\u2013IV, each, with one midventral seta (as only illustrated) and closely resembling N. variegata. The dorso-central setae in both the species are short where c1 is 1/2 and 1/3 the length of c1\u2013d1 in N. guajavae and N. variegata, respectively.Based on the present work, the species Neophyllobius hyderabadensisIndra, Rao and ThakurNeophyllobius hyderabadensis Indra, Rao and Thakur, 1980:121 [1980:121 .N. hyderabadensis . Hence, it was not possible to provide any conclusive remarks on this species.The original published description and illustration were not found. The first author contacted Mr. Mahran Zeity who published a new camerobiid species from India, but was not able to get any information about Neophyllobius mexicanusMcGregorNeophyllobius mexicanus McGregor, 1950:49 [ 1950:49 ; de Leon 1950:49 ; Bolland 1950:49 .The original description and illustration provided by McGregor  were notNeophyllobius ornatusWomersleyNeophyllobius ornatus Womersley, 1940:248 [1940:248 ; Bolland1940:248 ; Fan and1940:248 .N. fissus, N. aegyptium, N. niloticus due to the femur IV with one seta.The status of this species is unresolved. The type specimen is lost as reported by Bolland  and Fan N. ornatus could belong to group of eight species  where all leg tarsi have two mid-ventral setae and femur I\u2013IV with 4\u20133\u20132\u20131 setae.Based on available information and classification of species in the present work, Neophyllobius saxatilisHalbertNeophyllobius saxatilis Halbert, 1923:384 [1923:384 ; van Eyn1923:384 ; Bolland1923:384 .van Eyndhoven  reportedN. saxatlilis could be placed among 48 species of the new subgenus Neophyllobius having all leg tarsi with two in-line midventral setae, femur I\u2013IV with 4\u20133\u20132\u20132 setae, palp tarsus having two setae and two eupathidia, one solenidion on leg tarsi I\u2013IV, dorsal setae c1 reaching the bases of setae d1, setae d1 not reaching the bases of setae f1.In the present study, the Neophyllobius summersiMcGregorNeophyllobius summersi McGregor, 1950:67 [ 1950:67 ; Bolland 1950:67 .Neophyllobius species, to date, do not contain such palp chaetotaxy. Bolland [N. mangiferus to be close to N. summersi and differed from the latter on the basis of length of dorso-central setae and genu I\u2013IV setae. However, it was not possible to include N. summersi, in the present diagnostic key, even with its unique character of palpfemur.McGregor  first ti Bolland  added so Bolland  considerNeophyllobius vanderwieliOudemansNeophyllobius vanderwieli Oudemans, 1926:121 [1926:121 ; Bolland1926:121 .N. kamalii, N. karabagiensis and N. sycomorus) which share the same character.Bolland  mentioneNeophyllobius sp.This unidentified species was reported from Yemen by Ueckermann et al. . The autNeophyllobius sp.N. dichantii, but no morphological data was providedThis unidentified species was mentioned by Ripka et al. , which wNeophyllobius, with 50 new species and 35 species redescriptions. A detailed genus diagnosis was given including; the ventral idiosoma with three pairs of ventral setae, two pairs of genital setae and three pairs of anal setae. In the remarks, the author stated that pregenital setae were never found in any other species of Neophyllobius, described until that time, in contrast to what was illustrated by Kuznetsov and Livshits [Bolland ,16 proviLivshits ; presenc1a is found on the coxa I in all species of Neophyllobius and it was counted along with coxal setal count. The similar concept is evident in the genus diagnosis provided by Bolland [Neophyllobius described to date. If the three pairs of ventral setae include the coxal seta 1a, then it supports the absence of aggenital seta (ag) in adult females as reported by Bolland [1a, then the total number of setae on the ventral idiosoma according to Bolland [It is difficult to discern the ventral idiosomal chaetotaxy from the genus diagnosis provided by Bolland  as that  Bolland  and all  Bolland . However Bolland , add up Neophyllobius, one each of Tycherobius (T. dazkiriensis), Bisetolobius (B. varius) species are available till date. For the ease of understanding, we here consider them as four different ventral idiosoma chaetotaxy (VIC) descriptions. All these cases have a total of four pairs of setae excluding coxal and anal setae. The species representing these cases are also presented in the 3a, 4a, ag and g setae are present. The VIC #2 is represented by 10 species where paired 3a is absent. The VIC #3 is represented by 14 species of Neophyllobius and a species, T. dazkiriensis, where paired setae 4a is absent. The VIC #4 includes 13 species of Neophyllobius and a species B. varius, where paired aggenital setae (ag) is absent. It is noteworthy that, in the cases 2\u20134, where either one of the paired setae was absent, two pairs of genital setae were described and illustrated , N. lorioi , N. saberi  and N. zolfigolii .Adding to this confusion, four different ventral idiosoma descriptions and illustrations represented by 55 ustrated . In addiNeophyllobius, few descriptions are available for immature stages, of which most are poor, which makes it further difficult to understand the setal ontology. Paredes\u2013Le\u00f3n et al. [N. cibyci. It was reported that on the ventral idiosoma, the intercoxal seta 3a is present in larval stages where the setae 4a appear in protonymphs and aggenital setae (ag) is only present in adult females with one pair of genital (g) setae. The redefinition of the family Camerobiidae provided by Fan and Walter [4a, ag and g present in females.In the genus n et al.  studied Neophyllobius, the Tycherobius (other than the exception mentioned above) have similar situation as that presented in VIC#1. The females of the other two genera, Camerobia  and Acamerobia (one species) also follow the similar ventral idiosoma setal notation.In its support, the species of closely related genus of Camerobia, Decaphyllobius, Neophyllobius and Tycherobius have been reported. As a case study, the ventral idiosoma of all these species are presented ; which were two pairs of intercoxal setae 3a\u20134a always present on small platelets, a pair of aggenital seta (ag), present on ventral integument and on or off the anterior margin of genital shield and a pair of genital setae (g) present on sides of genital opening. We support this notation based on the evidence discussed above and recommend future works to follow it.Based on the understanding from all the published literature on the family Camerobiidae and after carefully observing the relative position of ventral setae in Saudi Arabian camerobiid species, we reach the conclusion that four pairs of setae were present on the ventral idiosoma, excluding setae 1a and three pairs of anal setae  setae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026N. yunusi Bolland *8 Tarsus II with 10(\u03c9) setae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026N. fani8\u2018 Tarsus II with 9 \u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 9291 Some of the dorso\u2013central setae reach or cross the bases of setae next in line  setae, genu IV setae not reaching base of corresponding tarsi \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 N. ayvalikensis Akyol93\u2018 Tarsi II with 9(+\u03c9) setae, genu IV setae reaching base of corresponding tarsi \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.............1c twice the length of 1b, genus I seta passes tibiae and genu II and IV seta passing tarsi ends \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026. N. lorioi Smiley and Moser94 Femur setae very short, femur I setae 3 and 4 nearly at one level, coxal setae 1c nearly as long as 1b, none of the genu setae reaching tibia end \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. N. acaciae Bolland94\u2018 Femur setae longer, coxal seta N. lobatus De Leon95 Femur I\u2013IV strongly serrulate at their proximal posterior margins \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202695\u2018 Femur I\u2013IV not strongly serrulate at their bases \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 9696 Femur I setae 3 and 4 not based on one or nearly one level \u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 9796\u2018 Femur I setae 3 and 4 based on one or nearly one level \u2026\u2026...........................11397 Genu I\u2013III and femur I\u2013IV setae strongly lanceolate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 9897\u2018 Genu I\u2013III setae not lanceolate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026... 991c longer than 1b\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. N. ceratoniae Bolland98 Distal setae on femur IV not passing genu border, coxal seta 1c shorter than 1b\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026N. lanceolatus Bolland98\u2018 Distal setae on femur IV passing genu border, coxal seta 99 Genu I setae not reaching second row of tibia setae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026... 10099\u2018 Genu I setae reaching or longer than second row of tibia setae \u2026\u2026\u2026\u2026\u2026.. 104100 Genu II setae not reaching second row of tibia setae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 101N. binisetosus Bolland100\u2018 Genu II setae reaching second row of tibia setae \u2026\u2026\u2026.. N. atriplicis Bolland101 Genu III setae not whip like and plumed at the top \u2026\u2026\u2026.. 101\u2018 Genu III setae whip like \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 102d1, e1 and f1 much longer than other dorsal setae, coxal setae 1c much shorter than 1b \u2026\u2026 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.N. ambulans Meyer102 Genu II setae not reaching the first row of tibiae setae, dorso\u2013central setae 102\u2018 Genu II setae reaching at least the first row of tibiae setae \u2026......................... 103N. cavumarboris Meyer and Ryke103 Dorso\u2013central setae d1 and e1 much longer than other dorsal setae \u2026\u2026\u2026 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. N. camelli Khanjani and Ueckermann103\u2018 Dorso\u2013central setae c1 longer than other dorsal setae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 104 Genu II setae not whip like \u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 105104\u2018 Genu II setae whip like \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 107105 Proximal setae on femur IV reaching the base of distal setae ......................... 106N. armeniaca Bolland105\u2018 Proximal setae on femur IV not reaching the base of distal setae\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026N. abiegnus Khaustov and Abramov106 Genu II seta long whiplike, crossing first row of setae on respective tibiae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 N. ayyildizi Ko\u00e7 and Madanlar106\u2018 Genu II seta short, not reaching first row of setae on respective tibiae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026 107 Genu I and II setae reach till second row of tibia setae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 108107\u2018 Genu I and II setae are passing tibial border \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 1091c:1b = 6:18 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 N. iramus De Leon108 Dorsal setae small, not pointed, body small, coxa 1 seta 1c:1b = 6:17 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 N. equalis De Leon108\u2018 Dorsal setae more broad and occupied with very strong stings, coxa I seta N. setosus Bolland109 Dorsal setae specially spinosed \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 109\u2018 Dorsal setae not specially spinosed \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 110110 First seta on femur II not reaching the third, proximal femur I seta not the longest \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026................................................. 111N. pruni Bolland110\u2018 First seta on the femur II the longest femur seta which passes easily the third, proximal femur I seta the longest \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 e1 longest among the dorso\u2013central setae\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026111 Setae N. askalensis Do\u011fan and Ayyild\u0131z\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026e1 not the longest among the dorso\u2013central setae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 112111\u2018 Setae N. viticola Bolland112 Proximal seta on femur I equal in length with the second, distal seta on femur IV long, passes genu border, first seta on femur I reaches the bases of the fourth \u2026\u2026................................................................................................. N. combreticola112\u2018 Proximal seta on femur I shorter than the second, distal seta on femur IV short, not passing genu border, first seta on femur I does not reach the fourth seta at all \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 113 Dorsal setae broad, setae d1 strong, genu II seta as long as genu, genu I and II setae not reaching first row of tibiae setae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 114113\u2018 Dorsal setae smaller, genu I and II setae at least reaching first row of tibiae setae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026... 115N. curtipilis De Leon114 Genual setae distinctly spinose \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. N. spatulus De Leon114\u2018 Genual setae linear and faintly spinose \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. N. sierrae McGregor115 Genu I setae reach till first row of tibiae setae, palp femur swollen, palp genu short \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 115\u2018 Genu I setae longer, palp femur longer \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 116N. americanus Banks116 Two eupathidia on the palp tarsus based very different in level, dorsal setae small with many spicules \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 116\u2018 Two eupathidia on the palp tarsus on similar level, dorsal setae broader with less spicules \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026... 117117 Genu III\u2013IV setae are passing tarsus end \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 118117\u2018 Genu III\u2013IV setae not reaching beyond end of respective legs \u2026\u2026\u2026\u2026\u2026\u2026 120pdx and c1 grouped on a small finely striated platelet \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. N. tescalicola Parades\u2013Leon et al.118 Dorso\u2013central setae pdx and c1 not grouped on a platelet \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 119118\u2018 Dorso\u2013central setae c1 and d1 are the same as the distance between setae c1\u2013d1 and d1\u2013e1 respectively \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. N. farrieri De Leon119 Lengths of dorsal setae c1 and d1 are distinctly longer than the distance between setae c1\u2013d1 and d1\u2013e1 respectively \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 N. cibyci Parades\u2013Leon et al.119\u2018 Dorsal setae sce shorter sci \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 121120 Setae sce equal in length to sci \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2126N. consobrinus De Leon120\u2018 Setae vi, ve and sci subequal in length \u2026\u2026\u2026\u2026\u2026\u2126N. inequalis De Leon121 Lateral setae vi and ve \u2026\u2026\u2026\u2026 \u2126N. piniphilus Bolland121\u2018 Lateral setae sci distinctly longer than e1 passes the bases of h1 \u2026\u2026\u2026\u2026.\u2026\u2026. N. aegyptium Soliman and Zaher122 Setae e1 do not reach the bases of h1 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 123122\u2018 Setae 123 Genu I seta short not reaching second row of setae on corresponding tibiae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 124123\u2018 Genu I seta long, extend beyond corresponding tarsus base \u2026\u2026\u2026\u2026\u2026\u2026 126N. ferrugineus FanTarsi I\u2013II with 8 setae each \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026124\u2018 Tarsi I\u2013II with 10 setae each \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 1251c twice as long as coxal seta 1b \u2026. N. lalbaghensis Zeity and Gowda125 Coxal seta 1c 1.5 times as long as coxal seta 1b \u2026 N. womersleyi Fan and Walter125\u2018 Coxal seta e1\u2013f1 two times longer as distance d1\u2013e1, setae f1 the longest \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 N. niloticus Bolland126 Distance e1\u2013f1 sub\u2013equal to d1\u2013e1 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 127126\u2018 Distance N. bamiensis Khanjani et al.127 Palp femur with both setae short, not reaching end of palp genu; long setae of genu I\u2013III not reaching end of respected legs \u2026\u2026\u2026\u2026\u2026\u2026\u2026. 127\u2018 Palp femur with one of two setae reaches end of tarsus \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 128d1 reaching base of e1, coxal setae 1c longer than 2c in length \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026... N. fissus128 Dorsal setae d1 extend beyond the bases of e1, coxal setae 1c almost equal to 2c in length \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026N. punctulatus Fan128\u2018 Dorsal setae * = species described based on male holotype.N. bisetalis and N. spatulus with two setae on coxae II as mentioned by Bolland and Swift [+ = two species, viz; nd Swift  and Bollnd Swift , respect\u03b1 = The character of genu I\u2013IV without solenidion is mentioned in the original description .N. sycomorus. However, Bolland [N. sycomorus with the illustrations of male and female.\u03b2 = The male specimens were not reported at the time of original description for  Bolland  providedN. consobrinus and N. inequalis suggesting later could be a deutonymph of former. However, these species are placed in the diagnostic key based on available information. but types of each species require re\u2013examination.\u2126 = These three species have minute differences among them. Bolland and Swift  have als"}
{"text": "The authors regret that there were two errors in the original article. In the \u201cExperimental details\u201d section on page 115205, \u201c1 M sodium sulfide at 70\u201380 \u00b0C for 24 h\u201d should have read \u201c0.5 M sodium sulfide at 70\u201380 \u00b0C for 24 h\u201d. Additionally, The Royal Society of Chemistry apologises for these errors and any consequent inconvenience to authors and readers."}
{"text": "Scientific Reportshttps://doi.org/10.1038/s41598-021-93324-3, published online 07 July 2021Correction to: The original version of this Article contained errors.In Table 2 legend, the symbol of \u201cpicomol\u201d was incorrectly given as \u201cnanomol\u201d.\u22121), salinity (S in yr\u22121), Total Alkalinity (TA in \u00b5mol\u00a0kg\u22121 yr\u22121), salinity-normalized alkalinity (nTA in \u00b5mol\u00a0kg\u22121 yr\u22121), total dissolved inorganic carbon (DIC in \u00b5mol\u00a0kg\u22121 yr\u22121), salinity-normalized dissolved inorganic carbon (nDIC in \u00b5mol\u00a0kg\u22121 yr\u22121), in\u00a0situ pH in total scale (pHT yr\u22121), total hydrogen ion concentrations ([H+]T in nanomol\u00a0kg\u22121 yr\u22121), ion carbonate concentration excess over aragonite saturation (exCO3 = in \u00b5mol\u00a0kg\u22121 yr\u22121), and anthropogenic CO2.\u201d\u201cAverage trends obtained with the seasonally detrended data the in\u00a0situ temperature (T in \u00b0C\u00a0yrnow reads:\u22121), salinity (S in yr\u22121), Total Alkalinity (TA in \u00b5mol\u00a0kg\u22121 yr\u22121), salinity-normalized alkalinity (nTA in \u00b5mol\u00a0kg\u22121 yr\u22121), total dissolved inorganic carbon (DIC in \u00b5mol\u00a0kg\u22121 yr\u22121), salinity-normalized dissolved inorganic carbon (nDIC in \u00b5mol\u00a0kg\u22121 yr\u22121), in\u00a0situ pH in total scale (pHT yr\u22121), total hydrogen ion concentrations ([H+]T in picomol\u00a0kg\u22121 yr\u22121), ion carbonate concentration excess over aragonite saturation (exCO3 = in \u00b5mol\u00a0kg\u22121 yr\u22121), and anthropogenic CO2.\u201d\u201cAverage trends obtained with the seasonally detrended data the in\u00a0situ temperature (T in \u00b0C\u00a0yrAdditionally, the article contains a repeated error where the symbol for \u201cpmol\u201d was incorrectly given as \u201cnmol\u201d in the Results section, under the subheading \u2018Acidifcation drivers\u2019, in Figure\u00a0Furthermore, in Figure\u00a0The original Article and accompanying Supplementary Information file have been corrected.Supplementary Information."}
{"text": "Rn into hypercubes of two differnet side lengths p or q. This generalizes the Pythagorean tiling in R2. We also show that this tiling is unique up to symmetries, which proves a variation of a conjecture by B\u00f6lcskei from 2001. For positive integers p and q, this tiling also provides a tiling of (Z/(pn+qn)Z)n.We construct a unilateral lattice tiling of  Keller  for {a,a+1,a+2,\u2026,b\u22121,b}\u2286Z and ei for the ith standard unit vector in Rn.We write 2Theorem 2p,q\u2208R+ with q<p and n\u2208N\u2a7e2. There exists a lattice tiling of Rn into hypercubes of side length p or q.Let We split Theorem\u00a0Remark 2After a preprint of this article appeared on arXiv, the author was informed by Mihalis Kolountzakis that Theorem\u00a0A be the following n\u00d7n matrix:Let Lemma 1n\u222an\u22121\u00d7 is a system of representatives of Rn/AZn up to null\u00a0sets.Theorem\u00a0We split the proof of Lemma\u00a0Claim 1Rn has a representative in .Every element Proof of Claim 2A1:Consider the following matrix a1,..,an be the columns of A and b1,\u2026,bn the columns of A1. We obtain A1 in the following way.b1=an.bk=bk\u22121\u2212ak\u22121, for k\u2208.Let l\u2208 such that xi<p for all i<l and xl\u2a7ep, then x\u2212bl is a representative of x in .\u25a1Now let Claim 3x=T\u2208 be such that xi\u2a7eq\u2212p for all i<l and xl<q\u2212p. Then x+bl is a representative of x in  such that xi<p for all i<l and xl\u2a7ep. From the definition of x, it follows that l<n and x\u2212bl is a representative of x in n\u222an\u22121\u00d7=C=\u22c3i\u2208NQi up to null sets, which concludes the proof.\u25a1We now prove Lemma\u00a0Remark 3n\u222an\u22121\u00d7 cannot be representatives of the same element in Rn/AZn as this would imply that there were neighborhoods of the two points corresponding to the same set in Rn/AZn of non\u2010zero Lebesgue measure.Two different points in the interior of Lemma 2i\u2208 and let ei be the ith standard unit vector, then pei\u2209AZn and qei\u2209AZn.Let To show that the tiling is unilateral, the followimg Lemma is sufficient.n with difference pei and therefore they are both different representatives of Rn/AZn, so pei\u2209AZn. Analogously, there are two points in the interior of \u2208Bn\u2032 is in the stabilizer of the tiling \u21d4 S is of the following form.si,j=si+1,j+1 for i<n,j<n.si,n=\u2212si+1,1 and sn,j=\u2212s1,j+1 for i<n,j<n.sn,n=s1,1.The group of symmetries of the tiling has to keep the cube invariant, so it is a supgroup of the hyperoctahedral group S be as above and i such that si,1=\u00b11. Without loss of generality, si,1=1. Now\u2018\u21d0\u2019: Let v:=qei\u00b1pej such that v\u2260ai and i\u2260j. Thenai\u2212v is not in AZn because it is a vector connecting two inner points of  there exists i\u2208 such that Saj=\u00b1ai. Now suppose si,j=\u00b11, then Saj=\u00b1ai and therefore\u25a1\u2018\u21d2\u2019: Let Theorem 4p,q\u2208N with gcd=1. Then pn+qn=min{l\u2208N|lei\u2208AZn} for all i\u2208. In particular, there exists a unilateral lattice tiling of (Z/(pn+qn)Z)n into hypercubes of side length p or q.Let On the periodicity of the tiling along the coordinate axis, we have the following result:i\u2208. From Cramer's rule, we know that the solution of Ax=ei in Qn is of the form x=k/det(A)=k/(pn+qn) for a k=T\u2208Zn and therefore (pn+qn)ei\u2208AZn. More specifically, ki=det(Ai)/(pn+qn), where Ai=. Calculating the determinant with the Laplace expansion along the last column, we get det(Ai)=qn\u22121 and since gcd=1 the statement follows.Let Remark 4q=2 and p=1, the tiling gives a lower bound on the maximal number of hypercubes of side length two that can be packed in (Z/(2n+1)Z)n, which coincides with the optimal solution .21.q=2 a4Rn into hypercubes of two sizes is equivalent to the tiling described in the previous sections. It is known that such a tiling does not exist if we only use cubes of one size  and so x:=T\u2208BZn with |xi|\u2a7dp for all i\u2208. Now x connects two inner points of the big cube, a contradiction.\u25a1Assume the contrary. Then there exists Lemma 4A small and a big cube that properly touch share a\u00a0corner.T be a big cube and S a small cube that properly touch. Without loss of generality, let T=n and S properly touches T in {q}\u00d7Rn\u22121. We write S=\u00d7\u2a02j=2n where xi\u2208 for all i\u2208.Let S touches T in a full side but S and T do not share a corner, then xi\u2208 for all i\u2208 and without loss of generality xn\u2209{0,p\u2212q}. There have to be big cubes T1,T2 that touch T in {q}\u00d7\u2a02j=2n\u22121\u00d7 and {q}\u00d7\u2a02j=2n\u22121\u00d7, respectively, for a \u03b5>0. Therefore, T1=\u00d7\u2a02j=2n\u22121\u00d7 and T2=\u00d7\u2a02j=2n\u22121\u00d7 where yi,zi\u2208 for all i\u2208. Now E:=\u00d7\u2a02j=2n\u22121\u00d7 cannot be filled by any cubes, because it touches a small cube and it touches T1 and T2 in opposite full sides of distance p  for all i\u2208. Let T3 be the big cube properly touching S in {q}\u00d7\u2a02j=2n\u22121\u00d7 and let T4 be the big cube properly touching S in {q+p}\u00d7\u2a02j=2n for a \u03b5>0. We write T4=\u00d7\u2a02j=2n with wi\u2208 for all i\u2208. If wn<xn, then E1:=\u00d7\u2a02j=2n\u22121\u00d7 cannot be filled by any cubes, because it touches a small cube and it touches T and T4 in opposite full sides of distance p. Otherwise E2:=\u00d7\u2a02j=2n\u22121\u00d7 cannot be filled by any cubes, because it touches a small cube and it touches T3 and T4 in opposite full sides of distance p \u00d7\u00d7\u2a02j=i+1n\u22121Ii(j)\u00d7 with Ii(j)\u2208{,}. Since Ti and Tj do not properly intersect, we have Ii(j)= or Ij(i)=. From Lemma\u00a0Ti\u2032:=\u2a02j=1i\u22121Ji(j)\u00d7\u00d7\u2a02j=i+1n\u22121Ji(j)\u00d7 withTi\u2032 and Tj cannot properly intersect, Ji(j)= or Ij(i)= and consequently {Ii(j),Ij(i)}={,}.Lemma 6i\u2208 such that Ti=\u2a02j=1i\u22121\u00d7\u00d7\u2a02j=i+1n\u22121\u00d7.There exists exactly one Now assume Tn:=n\u22121\u00d7 is also a cube in the tiling. Let T\u2032 be such that T\u2032 touches T in n\u22121\u00d7{q} then T\u2032=\u2a02j=1n\u22121\u00d7 with xj\u2208\u00d7. Since S is the unique small cube properly touching T in Rn\u22121\u00d7{q}, F:=\u00d7 is filled by big cubes properly touching T. Now consider E:=)\u00d7. E is touching F and T\u2032\u2032 in full opposite sides of distance p so it can only by filled by small cubes and since small cubes cannot touch, by at most one small cube T and x2:=T are inner points of E but their midpoint xm:=(x1+x2)/2=T is an inner point of Tn so E cannot be filled by a single convex set.\u25a1Assume without loss of generality Claim 2x1=p.It holds that Proof of Claim 2x1>p. Then E\u2032:=\u00d7n\u22122\u00d7 is touching T\u2032 and S in full opposite sides of distance at most q\u2212p so E\u2032 cannot be filled.\u25a1Assume T\u2032=\u00d7n\u22122\u00d7 and since two different cubes of the form \u2a02j=1i\u22121\u00d7\u00d7\u2a02j=i+1n\u22121\u00d7 would intersect, T\u2032 is the unique cube of this form.\u25a1Therefore S be a small cube in the tiling and without loss of generality let S:=n. Let T:={T1\u2212,\u2026,Tn\u2212,T1+,\u2026,Tn+} be the set of cubes properly touching S such that S touches Ti\u2212 in Ri\u22121\u00d7{\u2212p/2}\u00d7Rn\u2212i and Ti+ in Ri\u22121\u00d7{p/2}\u00d7Rn\u2212i for all i\u2208 and C:={c1\u2212,\u2026,cn\u2212,c1+,\u2026,cn+} their corresponding center point. We can now define a function \u03bd:T\u2192T that assigns every cube T\u2208T the cube T\u2032 such that  are in the same relation as  from Lemma\u00a0i. We also use \u03bd as a function \u03bd:C\u2192C with the obvious meaning. Consequently, for all i\u2208 there exists j\u2208\u2216{i} such that \u03bd(vi\u2212)\u2212vi\u2212=qei\u00b1pej and for symmetry reasons \u03bd(vi+)\u2212vi+=\u2212qei\u2213pej. Now let G:= be the directed graph with vertex set V:=T and edge set E:={)|T\u2208T} =Ti\u2213.Now again let G has out\u2010degree one so G contains a cycle. Let C:= be a cycle in G with center points , T\u00af1=T\u00afk and 3\u2a7dk\u2a7d2n+1. For i\u2208, let ji\u2208 such that T\u00afi\u2208{Tji\u2212,Tji+}. Then c\u00afi can be written as{ji,ji+1}\u228eSi\u2212\u228eSi+= and xi,xi+1\u2208{\u22121,1} andvi:=c\u00afi+1\u2212c\u00afi=\u2212xiqeji+xi+1peji+1, then \u2211i=1k\u22121vi=0 because ck=c1.Every vertex in Tj\u2212\u2208C\u21d4Tj+\u2208C for all j\u2208 and C is an even cycle. Without loss of generality let T\u00af1=T1\u2212, then for symmetry reasons T\u00afk/2+1=T1+ andTherefore ej1=ejk/2+1=e1 and {ej1,\u2026,ejk/2} are pairwise different. Consequently, k/2=n which concludes the proof.\u25a1where {b1,b2,\u2026,bn}\u2286BZn whereC:=, then SC=A where S=i,j\u2208\u2208Bn\u2032 andNow consider the sets of vectors det(C)=det(A)=det(B)=pn+qn and therefore CZn=BZn and A and B describe the same tiling up to\u00a0symmetries.Now Mathematika is owned by University College London and published by the London Mathematical Society. All surplus income from the publication of Mathematika is returned to mathematicians and mathematics research via the Society's research grants, conference grants, prizes, initiatives for early career researchers and the promotion of\u00a0mathematics."}
{"text": "The most frequently used model for simulating multireader multicase (MRMC) data that emulates confidence-of-disease ratings from diagnostic imaging studies has been the Roe and Metz (RM) model, proposed by Roe and Metz in 1997 and later generalized by Hillis (2012), Abbey et\u00a0al. (2013), and Gallas and Hillis (2014). A problem with these models is that it has been difficult to set model parameters such that the simulated data are similar to MRMC data encountered in practice. To remedy this situation, Hillis (2018) mapped parameters from the RM model to Obuchowski\u2013Rockette (OR) model parameters that describe the distribution of the empirical AUC outcomes computed from the RM model simulated data. We continue that work by providing the reverse mapping, i.e., by deriving an algorithm that expresses RM parameters as functions of the OR empirical AUC distribution parameters.We solve for the corresponding RM parameters in terms of the OR parameters using numerical methods.An algorithm is developed that results in, at most, one solution of RM parameter values that correspond to inputted OR parameter values. The algorithm can be implemented using an R software function. Examples are provided that illustrate the use of the algorithm. A simulation study validates the algorithm.The resulting algorithm makes it possible to easily determine RM model parameter values such that simulated data emulate a specific real-data study. Thus, MRMC analysis methods can be empirically tested using simulated data similar to that encountered in practice. The resulting data are called multireader multicase (MRMC) data. These studies are typically used to compare different imaging modalities with respect to reader performance. Often measures of reader performance are functions of the estimated receiver-operating-characteristic (ROC) curve, such as the area under the ROC curve (AUC). The Obuchowski and Rockette method (OR)The most frequently used model for simulating MRMC data that emulate confidence-of-disease ratings from such studies has been the model first proposed by Roe and MetzBecause RM model parameters are expressed in terms of the latent rating data distribution, in contrast to MRMC analysis results that are almost always expressed in terms of parameters that describe the distribution of the reader performance outcomes, it has been difficult to set RM model parameter values such that the simulated data exhibit characteristics that are similar to MRMC data encountered in practice. To remedy this situation, Gallas and HillisAn outline of this paper is as follows. In Sec.\u00a022.12.1.1Let Using the RM notation, the model is given as Roe and Metz constrain the sum of the error variance and variance components involving case to be equal to one: Without loss of generality, Roe and Metz impose the constraints 2.1.2Although Roe and Metz only consider simulations for equal test DV distributions for each reader, the model can be easily modified to allow for test DV distributions that differ in their median AUC values by not setting Note that the RM model that allows for test-dependent AUCs is completely defined by seven parameters: 2.1.3In practice, estimated binormal-model nondiseased and diseased distribution variances for a reader-test combination are often different, with diseased subjects typically having more variable test results. Thus to better emulate real data, HillisSimilar to the original RM model,The algorithm discussed in this paper will be for the RMH model, which includes the original RM model2.2Obuchowski and Rockette\u2013,,These error variance\u2013covariance parameters are typically estimated by averaging corresponding conditional-on-readers estimates computed using the jackknife,The Defining 33.1The RMH-to-OR mapping, previously derived by Hillis,In this section, we discuss the main points of the OR-to-RMH algorithm when the goal is to emulate data from a real study with the RMH model; i.e., to determine RMH parameter values such that the expected values of the OR parameter estimates from the simulated MRMC samples are described by the unspecified, which we assume throughout this section. Two other options for The 3.1.1The OR-to-RMH algorithm requires inputted values for It is possible that there are several unspecified, that maps Let However, it is possible for the OR-to-RMH algorithm to return a solution such that Eq.\u00a0(18) holds only approximately, i.e., The lower and upper limits for For example, 0.01,0.5  and b=2,5  #creates data frame with 5 rows, each = RM_values> RM_Table34 <- c> RM_Table34 <- c> RM_Table34 <- c> RM_Table34 <- c> print(RM_Table3)n0\u2003n1\u2003delta1\u2003delta2\u2003var_R\u2003var_TR\u2003var_C1\u200369\u200345\u20032.392224\u20032.957029\u20030.1223413\u20030.005180485\u20030.47169641.1\u2003138\u200390\u20032.392224\u20032.957029\u20030.1223413\u20030.005180485\u20030.47169641.2\u2003\u200245\u200369\u20032.392224\u20032.957029\u20030.1223413\u20030.005180485\u20030.47169641.3\u2003\u200269\u200345\u20032.645200\u20032.645200\u20030.1223413\u20030.005180485\u20030.47169641.4\u2003\u200269\u200345\u20031.275900\u20031.275900\u20030.1223413\u20030.005180485\u20030.4716964var_TC\u2003var_RC\u2003var_error\u2003b\u2003b_method\u2003mean_to_sig11\u20030.1222262\u20030.1091448\u20030.2969327\u20030.656081\u2003unspecified\u20034.5635531.1\u20030.1222262\u20030.1091448\u20030.2969327\u20030.656081\u2003unspecified\u20034.5635531.2\u20030.1222262\u20030.1091448\u20030.2969327\u20030.656081\u2003unspecified\u20034.5635531.3\u20030.1222262\u20030.1091448\u20030.2969327\u20030.656081\u2003unspecified\u20034.5635531.4\u20030.1222262\u20030.1091448\u20030.2969327\u20030.656081\u2003unspecified\u20034.563553mean_to_sig2\u2003Pr1_improper\u2003Pr2_improper1\u20035.64101\u20030.003896242\u20037.862956e-051.1\u20035.64101\u20030.003896242\u20037.862956e-051.2\u20035.64101\u20030.003896242\u20037.862956e-051.3\u20035.64101\u20030.003896242\u20037.862956e-051.4\u20035.64101\u20030.003896242\u20037.862956e-05> OR_values_Table3 <- RMH_to_OR(RM_Table3)> printn0\u2003n1\u2003AUC1\u2003AUC2\u2003var_R\u2003var_TR\u2003error_var1\u200369\u200345\u20030.8970000\u20030.9410000\u20030.001540000\u20032.080000e-04\u20030.00078800021.1\u2003138\u200390\u20030.8970000\u20030.9410000\u20030.001540000\u20032.080000e-04\u20030.00039125761.2\u200345\u200369\u20030.8970000\u20030.9410000\u20030.001540000\u20032.080000e-04\u20030.00063444271.3\u200369\u200345\u20030.9190000\u20030.9190000\u20030.001644069\u20037.426773e-05\u20030.00078900631.4\u200369\u200345\u20030.7500034\u20030.7500034\u20030.007014410\u20033.019443e-04\u20030.0023458109cov1\u2003cov2\u2003cov3\u2003corr1\u2003corr21\u20030.0003412041\u20030.0003388401\u20030.0002356121\u20030.4330000\u20030.43000001.1\u20030.0001703301\u20030.0001691406\u20030.0001176498\u20030.4353401\u20030.43229971.2\u20030.0002800701\u20030.0002778178\u20030.0001940871\u20030.4414426\u20030.43789271.3\u20030.0003644012\u20030.0003363961\u20030.0002513892\u20030.4618483\u20030.42635421.4\u20030.0012240655\u20030.0012083227\u20030.0009406161\u20030.5218091\u20030.5150981corr3\u2003b\u2003mean_to_sig1\u2003mean_to_sig2\u2003Pr1_improper1\u20030.2990000\u20030.656081\u20034.563553\u20035.641010\u20030.0038962421.1\u20030.3006966\u20030.656081\u20034.563553\u20035.641010\u20030.0038962421.2\u20030.3059174\u20030.656081\u20034.563553\u20035.641010\u20030.0038962421.3\u20030.3186150\u20030.656081\u20035.046146\u20035.046146\u20030.0007838341.4\u20030.4009769\u20030.656081\u20032.433985\u20032.433985\u20030.326185605Pr2_improper1\u20037.862956e-051.1\u20037.862956e-051.2\u20037.862956e-051.3\u20037.838340e-041.4\u20033.261856e-019.3> VanDyke_OR_orig_values <- data.frame> Table4_OR1 <- VanDyke_OR_orig_values #creates data frame with 3 rows,> # each the same as VanDyke_OR_orig_values> Table4_OR2 <- data.frame,+ b_input = c, mean_sig_input = c)> Table4_OR <- cbind> print[1] \"Original OR parameter values\"> print(Table4_OR)n0\u2003n1\u2003AUC1\u2003AUC2\u2003var_R\u2003var_TR\u2003var_error\u2003corr1\u2003corr2\u2003corr31\u200369\u200345\u20030.897\u20030.941\u20030.00154\u20030.000208\u20030.000788\u20030.433\u20030.43\u20030.2991.1\u200369\u200345\u20030.897\u20030.941\u20030.00154\u20030.000208\u20030.000788*\u20030.433\u20030.43\u20030.2991.2\u200369\u200345\u20030.897\u20030.941\u20030.00154\u20030.000208\u20030.000788*\u20030.433\u20030.43\u20030.299b_method\u2003b_input\u2003mean_sig_input1\u2003unspecified\u2003NA\u2003NA1.1\u2003mean_to_sigma\u2003NA\u20035.21.2\u2003specified\u20031\u2003NAmean_to_sigma = mean_to_sigma or specified it is not necessary to specify a value for var_error, or the value can be NA*Note that with > Table4_RMH <- OR_to_RMH(Table4_OR)> print[1] \"Table 4 RMH parameter values\"> print(Table4_RM)n0\u2003n1\u2003delta1\u2003delta2\u2003var_R\u2003var_TR\u2003var_C1\u200369\u200345\u20032.392224\u20032.957029\u20030.12234134\u20030.005180485\u20030.47169641.1\u200369\u200345\u20032.303940\u20032.847902\u20030.11347812\u20030.004805176\u20030.46746761.2\u200369\u200345\u20031.855834\u20032.293997\u20030.07362882\u20030.003117776\u20030.4498198var_TC\u2003var_RC\u2003var_error\u2003b\u2003b_method1\u20030.1222262\u20030.1091448\u20030.2969327\u20030.6560810\u2003unspecified1.1\u20030.1220955\u20030.1089342\u20030.3015027\u20030.6929693\u2003mean_to_sigma1.2\u20030.1215947\u20030.1080172\u20030.3205683\u20031.0000000\u2003specifiedmean_to_sig1\u2003mean_to_sig2\u2003Pr1_improper\u2003Pr2_improper1\u20034.563553\u20035.641010\u20030.003896242\u20037.862956e-051.1\u20035.200000\u20036.427723\u20030.001778344\u20032.748745e-051.2\u2003Inf\u2003Inf\u20030.000000000\u20030.000000e+00> Table5_true_values <- RM_to_OR(Table4_RM)> print[1] \"Table 4 True OR values\"> printn0\u2003n1\u2003AUC1\u2003AUC2\u2003var_R\u2003var_TR\u2003var_error\u2003cov11\u200369\u200345\u20030.897\u20030.941\u20030.00154\u20030.000208\u20030.0007880002\u20030.00034120411.1\u200369\u200345\u20030.897\u20030.941\u20030.00154\u20030.000208\u20030.0007664249\u20030.00033186201.2\u200369\u200345\u20030.897\u20030.941\u20030.00154\u20030.000208\u20030.0006584975\u20030.0002851294cov2\u2003cov3\u2003corr1\u2003corr2\u2003corr3\u2003b1\u20030.0003388401\u20030.0002356121\u20030.433\u20030.43\u20030.299\u20030.65608101.1\u20030.0003295627\u20030.0002291610\u20030.433\u20030.43\u20030.299\u20030.69296931.2\u20030.0002831539\u20030.0001968908\u20030.433\u20030.43\u20030.299\u20031.0000000mean_to_sig1\u2003mean_to_sig2\u2003Pr1_improper\u2003Pr2_improper1\u20034.563553\u20035.641010\u20030.003896242\u20037.862956e-051.1\u20035.200000\u20036.427723\u20030.001778344\u20032.748745e-051.2\u2003Inf\u2003Inf\u20030.000000000\u20030.000000e+009.49.4.1> VanDyke_OR_altered_values_a <- data.frame> RM_values = OR_to_RMWarning message: In OR_to_RM.defaultn0\u2003n1\u2003delta1\u2003delta2\u2003var_R\u2003var_TR\u2003var_C\u2003var_TC\u2003var_RC\u2003var_error\u2003b1\u200369\u200345\u2003NA\u2003NA\u2003NA\u2003NA\u2003NA\u2003NA\u2003NA\u2003NA\u2003NAb_method\u2003mean_to_sig1\u2003mean_to_sig2\u2003Pr1_improper\u2003Pr2_improper1\u2003unspecified\u2003NA\u2003NA\u2003NA\u2003NAx1\u2003x2\u2003x3\u2003x4\u2003x5\u2003x6\u2003x71\u20031.264641\u20031.563224\u2003NA\u2003NA\u2003NA\u2003NA\u2003NA9.4.2> VanDyke_OR_altered_values_b <- data.frame> RM_values <- OR_to_RMWarning message: In OR_to_RM.defaultn0\u2003n1\u2003delta1\u2003delta2\u2003var_R\u2003var_TR\u2003var_C\u2003var_TC\u2003var_RC\u2003var_error\u2003b1\u200369\u200345\u2003NA\u2003NA\u2003NA\u2003NA\u2003NA\u2003NA\u2003NA\u2003NA\u2003NAb_method\u2003mean_to_sig1\u2003mean_to_sig2\u2003Pr1_improper\u2003Pr2_improper1\u2003unspecified\u2003NA\u2003NA\u2003NA\u2003NAx1\u2003x2\u2003x3\u2003x4\u2003x5\u2003x6\u2003x71\u20031.264641\u20031.563224\u20030.06838082\u2003NA\u2003NA\u2003NA\u2003NA9.4.3> VanDyke_OR_altered_values_c <- data.frame> RM_values <- OR_to_RMWarning message: In OR_to_RM.default  Try changing (reduce or increase) the value of var_error.( b) Try using one of the other two b_method options, which should always work.> printn0\u2003n1\u2003delta1\u2003delta2\u2003var_R\u2003var_TR\u2003var_C\u2003var_TC\u2003var_RC\u2003var_error\u2003b1\u200369\u200345\u2003NA\u2003NA\u2003NA\u2003NA\u2003NA\u2003NA\u2003NA\u2003NA\u2003NAb_method\u2003mean_to_sig1\u2003mean_to_sig2\u2003Pr1_improper\u2003Pr2_improper1\u2003unspecified\u2003NA\u2003NA\u2003NA\u2003NAx1\u2003x2\u2003x3\u2003x4\u2003x5\u2003x6\u2003x71\u20031.264641\u20031.563224\u20030.06838082\u20030.07127637\u2003NA\u2003NA\u2003NA"}
{"text": "Huan Yuan, Yan Xu, Yi Luo, Jia\u2010Rong Zhang, Xin\u2010Xin Zhu, Jian\u2010Hui Xiao, Aging Cell, 2022, 21(9):e13686. https://doi.org/10.1111/acel.13686In the published version of Yuan et al.\u00a0, the autIn the Discussion, \u201cFigure S4\u201d should have been \u201cFigure S5\u201d and the revised sentence should read as follows:Figure S5), indicating that the accelerated aging model was successfully established.\u201d\u201cConsistent with the previous findings , long\u2010term injection of d\u2010gal reduced the activity of T\u2010AOC, GSH\u2010Px, and SOD and increased the content of MDA, AGEs, and RAGEs in the blood, kidney, liver, and heart  and (b\u2013k). The revised caption is given below.In Figure 3 legend, p16FIGURE 3 CaM/CaMKII and Nrf2/HO\u20101/NQO1 signaling pathways were attenuated after YWHAE knockdown. (a) Changes in 14\u20103\u20103\u03b5, t\u2010Nrf2, n\u2010Nrf2, HO\u20101, NQO1, CaM, p\u2010CaMKII, and CaMKII expression in hAMSCs upon different treatments. (b\u2013k) Relative expression levels of 14\u20103\u20103\u03b5, n\u2010Nrf2, t\u2010Nrf2, CaM, CaMKII, p\u2010CaMKII, HO\u20101, and NQO1. n\u00a0=\u00a03. n\u2010Nrf2, nuclear Nrf2; t\u2010Nrf2, total Nrf2; p\u2010CaMKII, phosphorylated CaMKII; Control, control group; H2O2, senescent group; GA\u2010D, GA\u2010D treatment group; Mock\u2010vehicle, GA\u2010D treatment group plus empty carrier; h\u201014\u20103\u20103\u03b5, GA\u2010D treatment group plus YWHAE overexpression; sh\u201014\u20103\u20103\u03b5, GA\u2010D treatment group plus YWHAE knockdown; x\u00af\u00a0\u00b1\u2009SD, mean\u2009\u00b1\u2009standard deviation. *p\u00a0<\u2009.05, **p\u00a0<\u2009.01.In Figure 5 legend, there is an error in citing part figure labels. \u201c(c)\u201d should be \u201c(i)\u201d; \u201c(d\u2013f)\u201d should be \u201c(j\u2013l)\u201d; \u201c(g\u2013l)\u201d should be \u201c(c\u2013h)\u201d and the revised caption is provided below.FIGURE 5 GA\u2010D enhanced the defense against oxidative stress on the sera in D\u2010gal\u2010caused aging mice. (a) The experimental design. (b) The swimming test of mice in the different groups. (c\u2013h) Activity of T\u2010AOC, SOD, MDA, GSH\u2010Px, AGEs, and RAGE in the sera of mice exposed to different treatments. (i) Histopathological organization of the liver, kidney, and heart tissues in different mice groups, analyzed via hematoxylin\u2013eosin (HE) staining. Scale bar: 200\u2009\u03bcm. (j\u2013l) The histological scores of liver, kidney, and heart tissues in different mouse groups. n\u00a0=\u00a06. AGEs, advanced glycation end products; GSH\u2010Px, glutathione peroxidase; igGG\u2010H, in vivo high\u2010dose GA\u2010D treatment group; igGG\u2010L, in vivo low\u2010dose GA\u2010D treatment group; igGG\u2010M, in vivo medium\u2010dose GA\u2010D treatment group; MDA, malondialdehyde; MG, model group ; NG, normal group; RAGEs, receptor for advanced glycation end products; SG, solvent group ; SOD, superoxide dismutase; T\u2010AOC, total antioxidant capacity; x\u00af\u00a0\u00b1\u2009SD, mean\u2009\u00b1\u2009standard deviation. *p\u00a0<\u2009.05, **p\u00a0<\u2009.01.In addition, the published version of Supplementary MaterialClick here for additional data file."}
{"text": "Therefore, we replace the wordings for all special cases considered in [\u201cexact lossless\u201d in the Abstract and Sections 1.2, 1.3 and 4.1 with \u201csimplified lossless\u201d.\u201cexact rate region\u201d in Sections 1.2, 1.3, 3.2, 4, 4.1, 4.2 and 6 with \u201csimplified rate region\u201d.\u201cThe \u2026 rate region \u2026 when \u2026 is the set of \u2026\u201d in Lemmas 1\u20134 with \u201cThe \u2026 rate region \u2026 when \u2026 includes the set of \u2026\u201d.\u201cthe lossless rate regions\u201d in Section 4.3 with \u201cthe further simplified lossless rate regions\u201d.\u201cthe exact lossless rate region\u201d in Section 4.3.1 with \u201cthe further simplified lossless rate region\u201d.\u201cthe lossless rate region\u201d in Sections 4.3.2 and 4.4 with \u201can inner bound for the lossless rate region\u201d.\u201cone exact region\u201d in the Abstract with \u201cone achievable region\u201d.\u201cevaluate the exact rate region\u201d in Section 1.2 with \u201cevaluate a simplified rate region\u201d.\u201cthe rate region for\u201d in Section 1.3 with \u201ca rate region for\u201d.\u201cthe exact lossless\u201d in Section 4.3.2 with \u201ca simplified lossless\u201d.Special case results given in Lemmas 1\u20134 and evaluated in Section 4.4 as an example are unfortunately not necessarily the exact rate regions for all channel models, although they are valid inner bounds. In more detail, the inner bounds given in Theorems 1 and 2 can be simplified for functions that are partially invertible with respect to Furthermore, below Remark 1, delete the sentence \u201ci.e., rate regions for which inner and outer bounds match\u201d. Since [The author states that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated."}
{"text": "We consider the convergence of the solution of a discrete\u2010time utility maximization problem for a sequence of binomial models to the Black\u2010Scholes\u2010Merton model for general utility functions. In previous work by D.\u00a0Kreps and the second named author a counter\u2010example for positively skewed non\u2010symmetric binomial models has been constructed, while the symmetric case was left as an open problem. In the present article we show that convergence holds for the symmetric case and for negatively skewed binomial models. The proof depends on some rather fine estimates of the tail behaviors of binomial random variables. We also review some general results on the convergence of discrete models to Black\u2010Scholes\u2010Merton as developed in a recent monograph by D.\u00a0Kreps. He shaped this field by applying masterfully the tools from stochastic analysis which he dominated so\u00a0well.Mark Davis has dedicated a large portion of his impressive scientific work to Mathematical Finance initiated by Black, Scholes, and Merton  Brownian motion as for the binomial model. But if you try to apply the discrete time reasoning from the binomial case as in Cox et\u00a0al.  to the tBut is this really the last word? From an economic point of view this sharp distinction between two similar approximations of the same object seems to be artificial. Can one find a more satisfactory answer? This question recently triggered the attention of David Kreps and led him to take up the theme of option pricing again, where he had made fundamental contributions some 40 years ago. This renewed interest resulted in the monograph Kreps  which ap\u03a9=C0, the space of all continuous functions \u03c9 from  to R whose value at 0 is 0. We let \u03c9 denote a typical element of \u03a9, with \u03c9(t) the value of \u03c9 at date t. Let P be Wiener measure on \u03a9.To analyze this phenomenon in proper generality let us fix some notation as in Kreps' monograph Kreps . We workS=e\u03c9(t) for the stock, taking the bond as numeraire. We know that there is a unique probability measure on \u03a9, denoted P\u2217, that is equivalent to P and, under which, S(t) is a martingale (Harrison and Kreps (We consider a Black\u2010Scholes\u2010Merton model of the form nd Kreps ).x:\u03a9\u2192R. We let X denote the space of bounded and contingent claims which are continuous with respect to the norm topology on \u03a9=C0. The well\u2010known \u201ccomplete markets\u201d result for the Black\u2010Scholes model says that, for every x\u2208X, x can uniquely be writtenS\u2010integrable integrand \u03b1.Contingent claims are Borel\u2010measurable functions n=1,2,\u2026, we have different probability measures Pn defined on \u03a9, with the following structure: For each n, the support of Pn consists of piecewise linear functions that, in particular, are linear on all intervals of the form , for k=0,\u2026,n\u22121. The interpretation is that Pn represents a probability distribution on paths of the log of the stock price in an n\u2010th discrete\u2010time economy, in which trading between the stock and bond is possible only at times t=k/n for k=0,\u2026,n\u22121. At time 1, the bond and stock liquidate in state \u03c9 at prices 1 and e\u03c9(1).Now suppose that for n\u2010th discrete\u2010time economy can implement (state\u2010dependent) self\u2010financing trading strategies (V(0),{\u03b1n(k/n),k=0,\u2026,n\u22121}), where the interpretation is that V(0) is the value of the consumer's initial portfolio, \u03b1n is the number of shares of stock held by the consumer after she has traded at time k/n, and, after time 0, bond holdings are adjusted that any adjustments in stock holdings at times k/n are financed with bond purchases/sales. In the n\u2010th economy, the consumer only knows at time k/n the evolution of the stock price up to and including that date. In the usual fashion, if V is the value of the portfolio formed by this trading strategy at time k/n in state \u03c9, then for all k=1,\u2026,n,Consumers in the n, Pn specifies an arbitrage\u2010free model of a financial market in the usual sense: It is impossible to find in the n\u2010th discrete\u2010time model a trading strategy (V(0),\u03b1n) with V(0)=0, V(1)\u22650Pn\u2010a.s., and V(1)>0 with Pn\u2010positive probability. This is true if and only if there exists a probability measure Pn\u2217 that is equivalent to Pn, under which {(e\u03c9(k/n),Fk/n);k=0,\u2026,n} is a martingale  for the n\u2010th discrete\u2010time model. Of course, in general there will be more than one emm Pn\u2217. However, with respect to any emm Pn\u2217, (V(k/n),Fk/n) is a martingale with respect to Pn\u2217. In particular, the expectation of V(1) under every emm Pn\u2217 is V(0).We maintain throughout the assumption that, for each g et\u00a0al. ). Such aXn:={x\u2208X:x(\u03c9)=V for some trading strategy (V(0),\u03b1n) for the nth discrete\u2010time economy}. We refer to Xn as the space of synthesizable claims in the n\u2010th discrete\u2010time economy.Let X be approximated by elements of Xn? During a visit of the second named author to Stanford University in the spring term 2019 we jointly took up this scheme in the paper Kreps and Schachermayer \u2264x\u00af})=1 for all n, where x_=inf\u03c9x(\u03c9) and x\u00af=sup\u03c9x(\u03c9).The claim A basic question treated in detail in Kreps' mongraph Kreps  is the fhermayer  and founx\u2010controlled risk can be attained:Theorem 1.2Pn on \u03a9=C0 weakly tend to Wiener measure P, and that, for some sequence {\u03b4n;n=1,\u2026,} of positive numbers tending to zero, Suppose that the probability measures x\u2208X can be asymptotically synthesized with x\u2010controlled risk. Moreover, fixing the claim x, the sequence of claims {xn} that asymptotically synthesize x can be chosen where, for (Vn(0),\u03b1n) the trading strategy that gives xn, Vn(0)\u2261EP\u2217[x], the Black\u2010Scholes\u2010Merton price of the claim x.Then every (continuous and bounded) The main result result of Kreps and Schachermayer  states tAs a particular example, the theorem applies, for example, to the trinomial model and, more generally, to a wide range of incomplete approximations of the Black\u2010Scholes model. The message is: replacing the notions of sub\u2010 and super\u2010replications by Definition\u00a0Theorem\u00a0Let us recapitulate the setting which is slightly more structured than the assumption of Theorem\u00a0\u03b6 with mean zero, variance one, and bounded support. For an i.i.d. sequence {\u03b6j;j=1,2,\u2026}, where each \u03b6k has the distribution of \u03b6, the law for the price of the stock at time k/n isFix a random variable \u03a9=C0. For each n, let Pn be the probability measure on \u03a9 such that the joint distribution of (\u03c9(0),\u03c9(1/n),\u2026,\u03c9(1)) matches the distribution of (\u03be(0),\u03be(1/n),\u2026,\u03be(1)), and such that \u03c9(t) for k/n<t<(k+1)/n is the linear interpolate of \u03c9(k/n) and \u03c9((k+1)/n). And let S:\u03a9\u2192R+ be defined by S=e\u03c9(t). Donsker's Theorem tells us that Pn weakly tends to P, where P is Wiener measure on C0.Again we embed this model into the standard state space x, trading as above either in the discrete market or in the continuous\u00a0limit.We imagine an expected\u2010utility\u2010maximizing agent who is endowed with initial wealth nth discrete\u2010time economy (where the stock and bond trade (only) at times 0,1/n,2/n,\u2026,(n\u22121)/n), does the optimal expected utility she can attain approach, as n\u2192\u221e, what she can optimally attain in the continuous\u2010time Black\u2010Scholes\u2010Merton\u00a0economy?The question addressed in Kreps  is: If wun(x) be the supremal expected utility she can attain in the nth discrete\u2010time economy if her initial wealth is x, and let u(x) be her supremal expected utility in the Black\u2010Scholes\u2010Merton economy. Kreps (liminfnun(x)\u2265u(x). And he proved limnun(x)=u(x) in the very special cases of U having either constant absolute or relative risk aversion. But he only conjectures that the second \u201chalf\u201d, or limsupnun(x)\u2264u(x) is true for general (sufficiently regular) U.Let y. Kreps  obtainedDefinition 1.3U is a strictly increasing, strictly concave, and continuously differentiable function U:R+\u2192R, which satisfies the Inada conditions that limx\u21920U\u2032(x)=\u221e and limx\u2192\u221eU\u2032(x)=0.A utility function To tackle this issue in proper generality we first need precise definitionsun(x) and u(x) as the maximal expected utility an agent can achieve from initial wealth x by admissibly trading in the markets defined by the measures Pn and P.As usual, we define the corresponding value functions U, its asymptotic elasticity Kramkov and Schachermayer ((U), is defined byU(x)=x\u03b1/\u03b1 for \u03b1\u2208, then AE(U)=\u03b1.For the utility function hermayer , writtenU implies that AE(U)\u22641 in all cases; if U is bounded above and if U(\u221e)>0, then AE(U)=0. But if U(\u221e)=\u221e, AE(U) can equal 1; an example is where U(x)=x/ln(x) for sufficiently large x.The concavity of Theorem 1.4U satisfies AE(U)<1. Then, for all x>0, the value function x\u21a6u(x) is finite\u2010valued and[KS 20 Theorem 1] Suppose that the utility function The following theorem gives an affirmative answer to Kreps' conjecture under the asymptotic elasticity condition.admitting the conditionAE(U)<1, this theorem settles the issue of convergence of the optimal expected utility in the discrete approximations of the Black\u2010Scholes\u2010Merton model in an economically satisfactory way. Note that we did not suppose the completeness of the discrete markets modeled by the measures Pn. In other words: the convergence of expected utility behaves well, independently of whether we are in the binomial or in the trinomial approximation. Also note that the assertion of finiteness of both terms in =1? For this case Kreps and the second named author found to their surprise that the answer to Kreps' conjecture turns out to be negative. More surprisingly: this pathology already happens in the framework of the binomial model!But, of course, at this stage the next question pops up. What happens for the\u2014admittedly somewhat pathological\u2014case of utility functions with AETo address this issue let us fix the notation for the special case of the binomial model in .p\u2208 we consider and i.i.d. sequence (\u03b1n)n=1\u221e of Bernoulli variables withp\u2208. Denote by \u03b6n the corresponding standardized variablesE[\u03b6n]=0 and Var[\u03b6n]=1. Again we denote by\u00a0\u03ben,k the scaled partial sumsfn,k=P.For arbitrary S(n)(k/n)=e\u03ben,k and extend S(n) by interpolation to continuous\u2010time processes as above, then (S(n)(t))0\u2264t\u22641 approximates the Black\u2010Scholes\u2010Merton model (S(t))0\u2264t\u22641 with S(t)=exp(\u03c9(t)) for 0\u2264t\u22641.If we again set \u03c9(1) under Pn equals the binomial distribution of \u03ben,n=\u03b61+\u22ef+\u03b6nn12.The distribution of the random variable u(x) and un(x) asP\u2217 and Pn\u2217 now are the unique equivalent martingale measures pertaining to the Black\u2010Scholes\u2010Merton model and its n\u2010th approximation,\u00a0respectively.Again we define the value functions p\u2208 we have E[\u03b6n3]>0. This is the case where things go astray, as demonstrated by the counterexample in Section\u00a09 of Kreps and Schachermayer , there is a utility function U satisfying the conditions of Definition\u00a0AE(U)=1) such that u(x) is a perfectly well\u2010behaved finite function while limn\u2192\u221eun(x)=\u221e, for all x>0. This phenomenon happens if E[\u03b6n3]>0 which means that the up\u2010tick of the log\u2010price is larger than the\u00a0down\u2010tick.When hermayer . If p\u2208, we haveIf The good news is that in this case everything works out as it should as stated in the subsequent theorem which is the main novel contribution of the present paper.Pn\u2217 and P\u2217 denote the unique equivalent martingale measures of the binomial and the Black\u2010Scholes\u2010Merton model,\u00a0respectively.The theorem will follow from the subsequent more technical version of\u00a0. As abovV:R+\u2192R the conjugate function of U, that is, V(y)=supx>0{U(x)\u2212xy}, and the corresponding dual value functions byProposition 1.6Under the assumptions of Theorem\u00a0As usual we denote by Proof of Theorem 1.5(admitting Proposition 1.6) We deduce from  we obtahermayer  from\u00a0=12\u03c0e\u2212x22. SetProposition 1.7p\u2208, andgn(w) is bounded from above uniformly in n\u2208N and w\u2208.For the remaining cases, ithat is, ow thatlognfn,kow thatlognfn,kow thatlognfn,ktor of\u00a0\u2264n+112nlog(k!)\u2265logk\u2212klog=\u03b1\u2032(12)=\u03b1\u2032\u2032(12)=\u03b1\u2032\u2032\u2032(12)=0 and for the forth derivative we have \u03b1iv(w)=\u22122/(1\u2212w)3\u22122/w3<0 for w\u2208, and thus each of the functions \u03b1\u2032\u2032\u2032(w), \u03b1\u2032\u2032(w), \u03b1\u2032(w), and \u03b1(w) is strictly negative and decreasing for w\u2208.We have \u03b2n(1/2)=25/(12n) and \u03b2n\u2032(w)=4\u22121/(2w)+1/(2(1\u2212w))>0 for w\u2208, thus \u03b2n(w) is strictly positive and strictly increasing for w\u2208.We have w\u2208 we have gn(w)\u2264bn(3/4). As limn\u2192\u221e\u03b2n(3/4)=1+log(2/3)<\u221e it follows that gn(w) is bounded from above for the interval under\u00a0consideration.For w\u2208 we have gn(w)\u2264\u03b1(3/4)n+\u03b2n(1\u22121/n). Now \u03b1(3/4)<0 and \u03b2n(1\u22121/n)\u223c12logn as n\u2192\u221e. Here the second term on the right hand side of\u00a0(\u03b1(3/4)n grows quicker than 12logn to conclude that gn(w) is negative for w\u2208 and sufficiently large n.For  side of\u00a0 is the lgn(w) for w\u2208 replaced by gn(1\u2212w) for w\u2208.The proof of\u00a0 is compl\u25a1Having proved\u00a0 and\u00a017)17) we meC>0 is sufficiently strong. But we we can do better than that. We may adapt the above argument to yield a constant C=1+\u03b5 for n sufficiently large. Indeed, analyzing the above proof of Proposition\u00a0 not at w=3/4, but at a point \u03b8\u2208, which is close to 1/2 to obtain a better constant C, for large enough n. The detailed argument is given in the proof of the following proposition, which sharpens Proposition\u00a0Proposition 2.1C>1 there is n0(C)>0 such that equations\u00a0(n\u2265n0(C).For any quations\u00a0\u201317) and andC>1 tquations\u00a0\u201319) hol holC>1 tFor the above proof of Proposition\u00a0\u03d1\u2208 and proceed as in the proof of the Proposition\u00a0w\u2208 and w\u2208. In the first case, when w\u2208, we have gn(w)\u2264\u03b2n(\u03d1) and\u03d1 and equals zero when \u03d1=1/2. In the second case, for w\u2208 we have gn(w)\u2264\u03b1(\u03d1)n+\u03b2n(1\u22121/n). Again \u03b1(\u03d1)<0 and \u03b2n(1\u22121/n)\u223c12logn as n\u2192\u221e, so that gn(w) is negative for w\u2208 and sufficiently large n.\u25a1We consider 3Proof of Proposition 1.6((asymmetric case) admitting Proposition 1.6) We fix p\u2208 and follow the steps from the symmetric case, but now we get instead of  and z1,1=(1\u2212p)/p. Straightforward asymptotic expansions for n\u2192\u221e yield\u03b4>0. For p\u2208 it follows from the asymptotics thatn sufficiently large. In fact, these inequalities are also true for p=1/2 as can be seen from (stead of  the follstead of :(48)azn,k\u22650 then \u2264Hy and the uniform integrability follows just as in the symmetric\u00a0case.Instead of  we now ck\u22650 then  yields Hzn,k\u22640 then \u2265H\u223cy, where H\u223cy(x)=V(y\u223ce\u2212x/2\u22121/8) with y\u223c=ye\u03b4. Due to the convexity of V we have v(y\u223c)>\u2212\u221e and the uniform integrability follows just as in the symmetric case.\u25a1If we are in the left tail and k\u22640 then  yields HProof of Proposition 1.7(asymmetric case) Following the steps from the symmetric case we now getgn(w)=\u03b1(w)n+\u03b2n(w), where w\u2208, and\u03b1(p)=\u03b1\u2032(p)=\u03b1\u2032\u2032(p)=0. As regards the third derivative we find \u03b1\u2032\u2032\u2032(w)=1\u22122p(p\u22121)2p2<0 for w\u2208, and thus \u03b1\u2032\u2032\u2032(w), \u03b1\u2032\u2032(w), \u03b1\u2032(w), and \u03b1(w) again are strictly negative and decreasing for w\u2208.equality\u00a0 becomes\u03b2n(p)=112(6np\u2212np2+1n\u22126log(\u22124(p\u22121)p)) and \u03b2n\u2032(w)=1p\u2212p2+12\u22122w\u221212w>0 for w\u2208, thus \u03b2n(w) is strictly positive and strictly increasing for w\u2208.We have \u03d1\u2208. For w\u2208 we have gn(w)\u2264\u03b2n(\u03d1). As limn\u2192\u221e\u03b2n(\u03d1)=\u22121/2log(\u03d1(1\u2212\u03d1))+hp(1\u2212p)\u221211\u2212p\u2212log2 it follows that gn(w) is bounded from above for the interval under consideration. For w\u2208 we have gn(w)\u2264\u03b1(\u03d1)n+\u03b2n(1\u22121/n). Now \u03b1(\u03d1)<0 and \u03b2n(1\u22121/n)\u223c1/2logn as n\u2192\u221e and thus gn(w) is negative for w\u2208 and sufficiently large n.\u25a1Similarly as in the proof of Proposition"}
{"text": "The study analyses the linguistic situation in the three Ukrainian oblasts on the Black Sea coast using survey data collected from 1,200 respondents before the Russian attack on Ukraine. At the end of the 18th century, this region was the core of a \u201cnew Russian\u201d governate during Tsarist times. Previously, the region had been ruled by Tatars and there were neither Russian nor Ukrainian settlements. From the 19th century onwards, the Ukrainian and Russian population dominated. Since the annexation of the Crimea, these oblasts represent a crucial part of the Kremlin\u2019s plan to establish an \u201cexpanded New Russia (Novorossiya)\u201d under Moscow\u2019s control \u2013 extending along the Ukrainian-Russian border and the northern Black Sea coast, reaching from Xarkiv to Odesa. This area is clearly at the forefront of Russia\u2019s current war goals since controlling it would allow them to establish the strategically important land bridge to Crimea.Linguistically, the area undoubtedly belongs to those regions of Ukraine where Russian was prominent, although apart from the Crimea at no time was there an ethnic Russian majority on the Black Sea coast \u2013 neither during Soviet times nor since Ukraine\u2019s independence. This means that the population with Ukrainian \u201cnationality\u201d also made strong use of Russian. This situation is being instrumentalized by Moscow as an argument for its military intervention to protect the Russian or Russian-speaking population.The study firstly describes the linguistic situation in the region, differentiating between the so-called mother tongue, the first language acquired and the principally-used language. It can be shown that the traditionally assumed dominance of Russian is actually far weaker when the population\u2019s \u201cmulticodality\u201d, including the mixed variety Sur\u017eyk, is included in the analysis. A differentiation is made between respondents with Ukrainian and Russian nationality throughout the analysis. Using statistical procedures such as principal component analysis and cluster analysis, the interdependencies between stated mother tongue, first language and multicodality are presented. Different motives for claiming a certain mother tongue can be identified among subgroups of respondents. The analysis focuses particularly on the questions of the extent to which central government measures to strengthen the position of Ukrainian since Ukraine\u2019s independence have changed respondents\u2019 preferences when choosing a code, and whether respondents have perceived social pressure for any form of shift. On the whole, it can be established that speakers with Ukrainian nationality who were primarily socialized in Russian have considerably increased their usage of Ukrainian, but without abandoning Russian. At best, this can also be established to a minimal extent for respondents with Russian nationality. Furthermore, since there is only extremely scant evidence that respondents encounter disapproval or censure from their environment for their choice of code , Moscow\u2019s claim of persecution, if not genocide of the Russian-speaking population is exposed as a blatant lie. Apart from the Crimean Peninsula, annexed by Russia in 2014, the Ukrainian oblasts on the Northern coast of the Black Sea (including the Sea of Azov), hold a prominent position in the predatory-prey pattern of Russia\u2019s neo-imperial aggression under Putin\u2019s government, of course together with the Donbas area and Xarkiv. The Black Sea oblasts would guarantee continental access to the Crimean naval port of Sevastopol, which is of major importance for Russian imperial interests \u2013 not only in the Black Sea, including the so-called Balkan area with the Russia-oriented Transnistrian pseudo-state, but also in the Mediterranean area as well as Africa and all of the Middle East. Historically, the territory of the contemporary Ukrainian Black Sea oblasts formed the core of so-called \u201cNovorossiya\u201d, literally New Russia, or, to be precise, the Novorossiya Governate of the Russian Empire established during the second half of the 18th century. In the first half of the last millennium, these territories did not have an autochthonous Slavic population, neither Ukrainian (or \u201cLittle Russian\u201d in the terminology of the Russian Empire) nor Russian (\u201cGreat Russian\u201d in the same terminology). Serfs fleeing from more central parts of the Polish-Lithuanian Commonwealth started to settle in these less controlled areas called \u201cWild Fields\u201d in the 15th century. There, over the course of time, the Zaporizhian Sich and the Cossack Hetmanate developed, a semi-autonomous polity, which came to an ultimate end by 1775. The following colonization of these territories in Tsarist times was multi-ethnic, with Ukrainians and Russians being the largest groups in the long run, with differences between the share of the two groups over time, regions, cities / towns and villages. By the end of the 19th century, it suffices here to state that the All-Russian Empire Census  providesThe historical background,2014Apart from the Crimea, in all oblasts mentioned, ethnic Russians were in the minority, Ukrainians in the majority. This was the case in the first and until today only census taken in independent Ukraine in 2001 as well as in the last census taken by the Soviet Union in 1989  in a society like Ukraine. Several conceptions of native language will be highlighted and related to the three codes. Second, information on respondents\u2019 \u201cprimary code\u201d will be provided, accompanied by a discussion of the methodological problems confronting quantitative analyses based on this concept in multilingual societies. Third, the contemporary linguistic practice or current usage of the three codes will be discussed and, subsequently, the code of linguistic socialization during childhood will be investigated. Last but not least, the question of shifts between the codes over people\u2019s lifetimes will be posed.Language usage in the sense of acknowledged languages or other codes, e.g. vernaculars, has an impact on the statement of one\u2019s native language or \u201cmother tongue\u201d. But this statement has at least partially a different character because it can be only loosely, sometimes more, sometimes less connected with respondents\u2019 actual linguistic practice,15The immediate question for the situation in Ukraine, not only in the three oblasts at issue here, is that of the ethnic groups of Ukrainians and Russians. We will start our analysis with this aspect. The results are presented in Fig.\u00a0One difference between Ukrainian and Russian respondents is obvious. The latter declare Russian, their \u201ctitular language\u201d, to be their native language with only very few exceptions, although every tenth respondent only reservedly agreed with the statement. Ukrainians do so with Ukrainian to an extent that is only a little lower, but every tenth respondent rejects Ukrainian as their mother tongue. The attitudes to the respective other language, i.e. Ukrainian or Russian, as well as to Sur\u017eyk are similar for both ethnic Ukrainians and Russians, although the acknowledgement of the other two codes among Ukrainians is a little higher. The acknowledgement of Russian as a mother tongue can be observed for roughly 550 respondents, only one fifth of whom are Russians, the others Ukrainians. Among the latter, three in four exhibit a reserved acknowledgement of Russian as their native language.The number of Russians among the respondents is thus comparatively small (but at least approximately corresponds to the share of Russian in the population of the region)After 2013 / 2014, i.e. after the Russian annexation of the Crimean Peninsula and in the light of the military conflict between the Russian Federation and independent Ukraine in the Ukrainian oblasts of Luhans\u2019k and Donec\u2019k, and in still other words, in times of severe pressure from the Russian Federation on Ukraine, it is far from surprising that many Ukrainians feel more Ukrainian than previously. Quite a number of Ukrainians  who mostly used Russian as their principal medium of communication before the cited conflicts or even in Soviet times, state that Putin has finally made them convinced Ukrainians. Accordingly, given the background of such a social atmosphere, it is not surprising that even on the Black Sea coast, which most earlier investigations described as overwhelmingly Russian speaking  for all three codes (missing responses are hard to interpret in this respect), then we can state the following: Only roughly 40 percent stated that they have just one native language: 30 percent Ukrainian, 10 percent Russian and less than 1 percent Sur\u017eyk. Another roughly 40 percent stated two native languages: 20 percent Ukrainian and Sur\u017eyk, 17 percent Ukrainian and Russian, 2 percent Russian and Sur\u017eyk. About 17 percent stated that all three codes are their native languages.A brief comment on reservedly rejecting one or the other codes: If these reserved rejections were to be accepted as some, although clearly restricted form of \u201cfelt mothertongueness\u201d, the percentage of Sur\u017eyk and Russian would increase considerably, the former reaching about 50 percent the latter about 66 percent. Ukrainian  would gain much less: close to 90 percent rather than a little more than 80 percent.Declaring a language, a variety, or a code \u2013 to use the most neutral expression \u2013 to be one\u2019s native language or \u201cmother tongue\u201d has, as has been mentioned above, at least to some degree a symbolic, attitudinal character, expressing identity, loyalty and the like. On the basis of a multifactorial approach it should thus be possible to grade the attachment to single codes. This, of course, holds especially when there is a \u201cchoice\u201d between two or more codes in multilingual or \u201cmulticodal\u201d societies. In this investigation, the respondents were presented with three statements, one for each code: \u201cUkrainian / Russian / Sur\u017eykStatement in Table\u00a0The notion of \u201cnative language\u201d or \u201cmother tongue\u201d is rather a layman\u2019s category and as such it is multi-layered. Trying to grasp some and, hopefully, the most important layers or aspects of the notion, the respondents were asked to evaluate nine statements of the type \u201cFor me, the native language is a language, which \u2026\u201d. Here, there were only two options for the evaluations, yes or no. The statements that were evaluated are shown in the column There is no need to present descriptive quantitative data on the statements in detail. All but one statement did not reach a majority of positive reactions. This exception was that two thirds of the respondents agreed on point (A), that the native language is a code they acquired during childhood. It should be underlined that this agreement is by no means exclusive. Almost all respondents who agreed with this evaluation also agreed with at least one if not more of the other nine as well, however with large differences. The other statements reached agreement rates of between a minimum of 27 percent (B \u2013 the language I speak best as well as H \u2013 the language I love and appreciate) and a maximum of 40 percent (E \u2013 the language of my parents). The latter, E, and A as the only point agreed upon by the majority are, of course, in a way connected with each other. Both typically refer to the language acquired during childhood, a process in which parents are involved, at least in most cases. Such interdependencies can be stated for other points as well.i.Positive reactions to the statements \u201cBy \u2018mother tongue\u2019 I understand \u2026\u201d F \u2013 \u201c\u2026 the language of my country\u201d; G \u2013 \u201c\u2026 of the nationality, I belong to\u201d; H \u2013 \u201c\u2026 I love and hold in a regard\u201d; I \u201c\u2026 of my fatherland\u201d hint at an emotional, rather patriotic approach (PATRIO) to the notion of \u201cnative language\u201d \u2013 the first component.ii.The second component with high values for B \u2013 \u201c\u2026 the language I speak best\u201d; C \u2013 \u201c\u2026 I use in everyday life\u201d; D \u2013 \u201c\u2026 I think in\u201d rather signal a practical approach (PRACT). based on the language(s) or code(s) in their current real linguistic situation.iii.The third component with high values for A \u2013 \u201c\u2026 I acquired as a child\u201d and E \u2013 \u201c\u2026the language of my parents\u201d is practical or realistic in a similar way to ii. However, it does not refer to the current linguistic practice, but rather to that during childhood. One could speak of an acquisition approach (AQUI) to the notion \u201cnative language\u201d. It should be noted that only the patriotic approach is a symbolic one, mirroring attitudes to a language or code. The practical and the acquisition approach are realistic approaches in the sense that has just been mentioned.In order to base this assumption not only on intuition, a factor component analysis has been calculated.Similar to Hentschel and Zeller\u2019s  investigk-means clustering). For a k-means cluster analysis the number of clusters has to be fixed in advance. One may either predetermine the clusters and thus their number to be tested by reasonable non-quantitative hypotheses or fix the ideal number of clusters by another variant of cluster analysis: hierarchical cluster analysis. This method can only be run on a smaller number of elements to be clustered, which have to be drawn by random selection from the larger set of elements. As there are no hypotheses or suggestions at hand on how many (and which) groups of respondents should be distinguished on the basis of the three components, the latter approach was taken. The hierarchical cluster analysis was executed on 50 randomly selected respondents and yielded an ideal number of four clusters for the analysis of centroid-clustering that was conducted on the data of all 1,200 respondents. The latter yielded the results illustrated in Table\u00a0The statistical method to achieve this aim is cluster analysis. For a high number of elements to be clustered  the adequate variant of cluster analysis is a centroid-based clustering symbolic patriotism (PATRIO)acquisition during childhood (ACQUI)an inclusive bundle of all components (INCL)On the basis of their approach to the concept of \u201cnative language\u201d, the fixed groups of respondents or speakers will from here on be called speakers with an orientation towards For the first three, the same abbreviation will be used as for the components, for the last one the new abbreviation \u201cINCL\u201d.As can be seen in the last two lines of Table\u00a0Regarding the quantitative data presented in the last lines of Table\u00a0\u0421\u0443\u0440\u0436\u0438\u043a \u044f \u0441\u0447\u0456\u0442\u0430\u044e \u0441\u0432\u043e\u0457\u043c \u0440\u0456\u0434\u043d\u0438\u043c \u044f\u0437\u0438\u043a\u043e\u043c. (...) \u0432\u0441\u0456 \u0430\u0431\u0449\u0430\u044e\u0442\u044c\u0441\u044f \u0441\u043e \u043c\u043d\u043e\u0439, \u0445\u0442\u043e \u0445\u043e\u0447\u0435, \u043f\u0438\u0442\u0430\u0454\u0442\u044c\u0441\u044f \u043d\u0430\u043b\u0430\u0434\u0438\u0442\u044c \u0441\u0432\u2019\u044f\u0437\u044c, \u043e\u0431\u0449\u0430\u0454\u0442\u044c\u0441\u044f \u043d\u0430 \u0441\u0443\u0440\u0436\u0438\u043a\u0454. (...) \u0420\u0456\u0434\u043d\u0430 \u043c\u043e\u0432\u0430\u2026 \u043d\u0443 \u044f\u043a\u043e\u0439 \u0442\u0438 \u043e\u0431\u0449\u0430\u0454\u0448\u0441\u044f \u043f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u043e, \u043d\u0443 \u0456\u043c\u0454\u043d\u043d\u0430 \u0442\u0432\u043e\u044f \u0443\u0434\u043e\u0431\u043d\u0430 \u0434\u043b\u044f \u0442\u0435\u0431\u0435.(1230I consider Sur\u017eyk my native language. (...) everyone communicates with me, who wants, tries to establish a connection, communicates in Sur\u017eyk. (...) Native language \u2026 well, in which you communicate constantly, well, the one that\u2019s convenient for you.current linguistic practice (PRACT) [\u0440\u043e\u0434\u043d\u043e\u0439 \u044f\u0437\u044b\u043a] \u0443\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0438\u0439. \u041d\u0435\u0437\u0430\u0432\u0456\u0441\u0456\u043c\u043e \u043e\u0442 \u0442\u043e\u0433\u043e \u0448\u043e \u044f \u043d\u0438\u043c \u043d\u0435 \u0440\u0430\u0437\u0433\u0430\u0432\u0430\u0440\u0456\u0432\u0430\u044e \u043f\u0440\u0430\u043a\u0442\u0456\u0447\u0435\u0441\u043a\u0456 (...). \u0423\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0438\u0439 \u044f\u0437\u0438\u043a \u0441\u0447\u0456\u0442\u0430\u044e \u0440\u043e\u0434\u043d\u0438\u043c, \u043f\u043e\u0441\u043a\u043e\u043a\u0443, \u044f\u043a \u044f \u0441\u043a\u0430\u0437\u0430\u043b\u0430, \u044f \u0440\u043e\u0434\u0456\u043b\u0430\u0441\u044c \u0432 \u0423\u043a\u0440\u0430\u0457\u043d\u0456, \u043f\u043e\u0441\u043a\u043e\u043a\u0443 \u044f\u0432\u043b\u044f\u0454\u0442\u044c\u0441\u044f \u0434\u0435\u0440\u0436\u0430\u0432\u043d\u043e\u044e \u043c\u043e\u0432\u043e\u044e \u0456 \u0454\u0441\u0442\u0454\u0441\u0442\u0432\u0454\u043d\u043d\u043e \u044f\u0432\u043b\u044f\u0454\u0442\u044c\u0441\u044f \u0456 \u0440\u043e\u0434\u043d\u043e\u044e \u043c\u043e\u0432\u043e\u044e \u043c\u043e\u0454\u0439. [my native language is] Ukrainian. Regardless of the fact that I do not speak it practically (...). I consider Ukrainian to be my native language, because, as I said, I was born in Ukraine, because it is the state language and, of course, it is also my native language.symbolic patriotism (PATRIO) \u0420\u0456\u0434\u043d\u0430 \u043c\u043e\u0432\u0430 \u2013 \u0446\u0435 \u043c\u043e\u0432\u0430, \u043d\u0430 \u044f\u043a\u0456\u0439 \u0442\u0438 \u0440\u043e\u0437\u043c\u043e\u0432\u043b\u044f\u0454\u0448 \u0437 \u0434\u0438\u0442\u0438\u043d\u0441\u0442\u0432\u0430. \u0417\u0430\u0440\u0430\u0437 \u0441\u043a\u0430\u0436\u0443 \u044f\u043a \u043f\u043e \u043f\u0456\u0434\u0440\u0443\u0447\u043d\u0438\u043a\u0443, \u0430\u043b\u0435 \u0446\u0435 \u043c\u043e\u0432\u0430, \u044f\u043a\u043e\u0457 \u0442\u0435\u0431\u0435 \u043d\u0430\u0432\u0447\u0438\u043b\u0430 \u043c\u0430\u0442\u0456\u0440, \u0430 \u043c\u043e\u044f \u043c\u0430\u043c\u0430 \u0440\u043e\u0437\u043c\u043e\u0432\u043b\u044f\u0454 \u043d\u0430 \u0440\u043e\u0441\u0456\u0439\u0441\u044c\u043a\u0456\u0439, \u0442\u043e\u043c\u0443 \u043c\u043e\u044f \u0440\u0456\u0434\u043d\u0430 \u043c\u043e\u0432\u0430 \u0432\u0441\u0435 \u0436 \u0442\u0430\u043a\u0438 \u0440\u043e\u0441\u0456\u0439\u0441\u044c\u043a\u0430.Native language is the language you have spoken since childhood. Now I\u2019m going to put it as in a textbook, but this is the language that your mother taught you, and my mother speaks Russian, so my native language is still Russian.acquisition during childhood (ACQUI) \u0440\u043e\u0434\u043d\u043e\u0439 [\u044f\u0437\u044b\u043a]... \u043d\u0443 \u043e\u043d \u0431\u043e\u043b\u044c\u0448\u0435 \u0441 \u0441\u0442\u043e\u0440\u043e\u043d\u044b \u0438\u043c\u0435\u043d\u043d\u043e, \u043c\u043d\u0435 \u043a\u0430\u0436\u0435\u0442\u0441\u044f, \u0433\u043e\u0441\u0443\u0434\u0430\u0440\u0441\u0442\u0432\u0430, \u0438 \u0441 \u0434\u0440\u0443\u0433\u043e\u0439 \u0441\u0442\u043e\u0440\u043e\u043d\u044b, \u0442\u043e\u0442, \u043d\u0430 \u043a\u043e\u0442\u043e\u0440\u043e\u043c \u0442\u0435\u0431\u0435 \u043f\u0440\u043e\u0449\u0435 \u0440\u0430\u0437\u0433\u043e\u0432\u0430\u0440\u0438\u0432\u0430\u0442\u044c. (...) \u0431\u043e\u043b\u044c\u0448\u0443\u044e \u0447\u0430\u0441\u0442\u044c \u0436\u0438\u0437\u043d\u0438 \u044f \u0433\u043e\u0432\u043e\u0440\u044e \u0438\u043c\u0435\u043d\u043d\u043e \u043d\u0430 \u043d\u0435\u043c [\u0443\u043a\u0440\u0430\u0438\u043d\u0441\u043a\u043e\u043c], \u0438 \u0432\u0441\u0435, \u0447\u0442\u043e \u0432\u043e\u043a\u0440\u0443\u0433 \u043c\u0435\u043d\u044f, \u044d\u0442\u043e \u0443\u043a\u0440\u0430\u0438\u043d\u0441\u043a\u043e\u0435, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u044f \u0441\u0447\u0438\u0442\u0430\u044e, \u0447\u0442\u043e \u0440\u043e\u0434\u043d\u043e\u0439 \u044f\u0437\u044b\u043a \u2013 \u044d\u0442\u043e \u0443\u043a\u0440\u0430\u0438\u043d\u0441\u043a\u0438\u0439. \u0410 \u0440\u0443\u0441\u0441\u043a\u0438\u0439 \u2013 \u044d\u0442\u043e \u043f\u0440\u043e\u0441\u0442\u043e, \u043e\u043f\u044f\u0442\u044c \u0436\u0435, \u043f\u0440\u043e\u0449\u0435 \u0432\u0441\u0435\u043c \u0433\u043e\u0432\u043e\u0440\u0438\u0442\u044c \u043d\u0430 \u043d\u0435\u043c, \u0432\u043e\u0442 \u043f\u043e\u0447\u0435\u043c\u0443-\u0442\u043e \u0442\u0430\u043a \u0432\u044b\u0445\u043e\u0434\u0438\u0442. native [language] ... well, it is more on the part of, I think, the state, and on the other hand, the one that is easier for you to speak. (...) most of my life I speak it [Ukrainian], and everything around me is Ukrainian, so I think that my native language is Ukrainian. And Russian is just, again, it\u2019s easier for everyone to speak it, that\u2019s why it turns out that way.an inclusive bundle of all components (INCL) The survey we conducted was accompanied by open, in-depth interviews from which we can present utterances made by interviewees which illustrate the abovementioned classification from (a) to (d):20Having determined the four groups of respondents according to their conceptions of native language, the first question to be asked is whether there are different inclinations in these four groups to declare one or the other of the three codes to be their native language. Figures\u00a0Here, only tables for Ukrainian and Russian are presented, because there are no significant differences between the four groups in their readiness to declare Sur\u017eyk as their native language. (We will return to the question of which aspects make respondent choose Sur\u017eyk as their native language later in the text.)Comparing Ukrainian and Russian, the first but not most important point to be made is that Russian is much less frequently declared as the native language in all four groups. This of course relates to the fact that the number of ethnic Ukrainians among the respondents is ten times higher than that of ethnic Russians, since both groups have been considered here. Of more interest are, however, the quantitative differences between the groups for each of the two codes. Most obvious in this respect is that a patriotic (PATRIO) approach to \u201cmother tongue\u201d as well as an inclusive (INCL) approach almost automatically make respondents declare Ukrainian to be their native language. As has been shown in Table\u00a0For those respondents for whom a rather realistic conception of native language is characteristic, be it oriented on current usage (PRACT) or on acquisition during childhood (AQUI), the share of unanimous acceptance of Ukrainian as their native language is clearly reduced to approximately 50 percent in both groups. When reserved approval is additionally taken into account, then three out of four respondents in the groups PRACT and ACQUI consider Ukrainian their native language. For Russian, it must first be noted that the readiness to declare it one\u2019s native language is much lower. Apart from that, the differences between the four groups with different conceptions of native language are diametrically opposite to the constellation for Ukrainian. Only about 10 percent with a patriotic or inclusive conception of native language unreservedly declare Russian to be their native language. Within respondents for whom current usage or acquisition in childhood is most important \u2013 among them almost all of the Russians involved in the study, we observe a very clear increase in the tendency to name Russian without hesitation as one\u2019s native language, up to 35 or 40 percent, respectively.Briefly recapitulating this point, the first thing to note is that Russian tends to be declared the native language for symbolic or patriotic reasons on a very low level , Ukrainian on a very high one.We now turn to the question of to what extent the respondents\u2019 estimation of their own linguistic practice, in terms of the significance of the three codes in current usage or usage during childhood, has an impact on their statement of specific \u201cmother tongues\u201d.The analysis of the primarily used code by KIIS , as illuTwo methodological points need to be clarified before we turn to our interpretation: First, in contrast to the question about the native language, only one answer was allowed for the primary code. Multiple options were not necessary because the respondents were asked to quantify their usage of each of the three codes, which allows for a detailed comparison. And these results formed the basis for Map\u00a0As can be seen from Fig.\u00a0In the following analyses it is not the primary code that is used, because it is inferior for mirroring the presence of the three codes in the tricodal society. The problem with the primary code is that for non-monocodal speakers we do not know on which grounds they select the primary code from the two or three codes they speak.With regard to the extent of current usage of the three codes in the three Black Sea oblasts, Hentschel and Taranenko , p. 284 These seven types correspond to the seven logically possible mono-, bi- and tricodal combinations of the three codes. As the types are based on a statistical cluster analysis of frequency estimations, a correspondence between the number of logically possible combinations and quantitatively based clusters would not necessarily be the case. A respondent from the monocodal type, say \u201cU\u201d for Ukrainian, usually does use the other two codes as well, but only to a quantitatively medium extent. This means that on quantitative grounds other types would be possible, e.g. \u201cstrong U and strong R\u201d vs. \u201cstrong U and R of medium strength\u201d etc. The names of the types are thus based on the interpretation of the results of the cluster analysis  among the monocodal speakers of Ukrainian. Second, two thirds of them are monocodal speakers of Russian, but, third, these respondents make up just one quarter of all monocodal speakers of Russian.For the seven types of speakers (respondents), based on their estimations of linguistic practice in everyday life, now the statements on their native languages will be compared and, in a second step, their statements about the first language or code they grew up with. Figures\u00a0i.Speaker type U (218 respondents): The vast majority, eight out of ten of speakers of the speaker type U , names only Ukrainian as their native language, one in ten names Ukrainian together with Sur\u017eyk. As has been mentioned above, there are almost no respondents with Russian \u201cnationality\u201d in this speaker type,ii.R (210): Monocodal users of Russian reveal a heterogeneous picture of \u201cmothertongueness\u201d. Only four out of ten report Russian to be their only native language. For the ethnic Russians among them, the relation is however eight out of ten. Four in ten from both ethnic groups in speaker type R name Russian together with Ukrainian, one in ten name only Ukrainian. This all implies that monocodal speakers of Russian with Ukrainian nationality  preferably declare Ukrainian as their native language, although they rarely  practise it. Only a quarter of them unreservedly names only Russian as their native language, almost half of them Russian and Ukrainian, one in six only Ukrainian. With the latter group there is thus a complete mismatch between native language and current usage of codes.26iii.S-U-R (216): Of the speakers with a declared \u201ctricodalism\u201d, indeed more than one third name all three codes as their mother tongue. The share of those naming Ukrainian and Sur\u017eyk (but not Russian) as their native languages is a little smaller. The remainder name Ukrainian and Russian or only Ukrainian with comparable frequency. With regard to other speaker types, a quantification is less meaningful due to small numbers. It suffices to say that speakers of the S-R type (96) mostly name all three codes as their native language, (similarly to S-U-R speakers), to a lesser degree Sur\u017eyk and Ukrainian (but not the combination of Sur\u017eyk and Russian). S-U speakers indeed preferably name Sur\u017eyk and Ukrainian or Ukrainian alone.We start with the three largest of the seven groups of speaker type, which make up  almost three quarters of all of the 890 respondents who provided self-quantifications on the use and the status of native language for all three codes: i.Native language only Ukrainian (280 respondents): Six in ten declaring Ukrainian to be their sole native language belong to the monocodal Ukrainian speaker type (U). The other speaker types are represented to a much less extent: by either one in ten or by even fewer.ii.Only Russian (105): Almost nine out of ten are indeed monocodal speakers of Russian (R), one in ten uses Russian and Sur\u017eyk frequently (S-R).iii.Ukrainian and Russian (146): One half belongs to the monocodal Russian speaker type (R), about two in ten are each either bilingual speakers of Ukrainian and Russian or tricodal speakers.iv.Ukrainian and Sur\u017eyk (180): This group is very heterogenous in terms of speaker types: Four out of ten are tricodal speakers, two out of ten come from the S-U type, one from each of the monocodal types U, R, S.v.Ukrainian, Russian and Sur\u017eyk (150): One half consists indeed of tricodal speakers, a good quarter of S-R speakers and one out of ten is a monocodal S speaker. The number of speakers of other combinations are too small to be commented upon.From the point of view of respondents\u2019 statements about their mother tongue one can make the following statements about native language: The general interpretation of these observations is as follows: Among those people in whose daily linguistic practice Russian, Sur\u017eyk or both codes play a major role, there is a considerable reluctance to name Russian as their native language , an even stronger one to name Sur\u017eyk  acquisition cf. Fig.\u00a0c. This iii.Ukrainian and Russian are named as the only first language to a more or less equal extent of roughly one third of the respondents. Sur\u017eyk is named by one fifth. It is presumably safe to assume that at least some of the respondents were reluctant to admit that they grew up with only Sur\u017eyk in their family. This vernacular is stigmatized \u2013 as has been mentioned above \u2013 as the code of the uneducated, especially when there is no \u201cliterary\u201d variety used as well. One might feel uneasy to reveal that this was the case in one\u2019s family. This means that all three codes might be on a similar level as the first linguistic code acquired, although Sur\u017eyk is clearly named less frequently than Ukrainian and Russian. Regarding the ethnic groups of the respondents, we observe that among Russians more than two thirds declare only Russian as their first language (the rest is distributed rather evenly over other constellations of first languages), among Ukrainians more than a quarter do so. This means, on the other hand, that among those who in the first years of their lives were socialized only in Russian there are four Ukrainians and only one Russian.This brings us back to the question of the extent to which the three codes are currently used \u201cin reality\u201d. Up to this point we have only discussed data based on respondents\u2019 self-estimations of their own linguistic practice. To a certain, but of course variable degree people are interested in presenting themselves in a good light, which is especially to be reckoned with for the symbolic aspects of the \u201cmother tongue\u201d, which for many, as has been shown, has a symbolic-patriotic significance. A similar approach cannot be completely ruled out even for statements on the codes frequently practised or on the ones acquired during childhood. We therefore now turn to the question of how respondents estimate the presence of the three codes in their surroundings. First, the situation in families will be looked at, which is of course a closer or more intimate social context for the respondents and, second, the situation outside the family. The corresponding questions were \u201cHow often is X used or to be heard?\u201d The presentation of the quantitative relations will be differentiated for respondents with Ukrainian and with Russian nationality, because they differ clearly between the two groups. Figure\u00a0The first thing to notice in this analysis is that there are very few refusals to answer this question. This underlines that personal statements in surveys can be influenced by a feeling of discomfort for some phenomena. The second point to be made is of course the differences between ethnic Ukrainians and ethnic Russians. It looks as if the two groups are living in two separate linguistic worlds. Nearly one half of the respondents of Ukrainian nationality Fig.\u00a0a report The most striking difference between the two ethnic groups is of course connected with the presence of Russian in their respective micro-worlds. More than 90 percent of the ethnic Russians state that it is frequent, mostly very frequent in their surroundings, both in their families and in other social contexts. In view of the widespread monocodalism of ethnic Russians already described above, this is of course less astonishing for family contexts than for other ones. The corresponding figures for ethnic Ukrainians are much lower: a little more than 50 percent use Russian frequently or very frequently.This leads us to the question of competence in Ukrainian and Russian and to the question of shifts from one language or code during the respondents\u2019 lifetimes.Since there are relatively large groups that neither use Ukrainian nor Russian frequently, the question of competence in the two languages arises.The judgements about the level of competence in the two languages without distinguishing between active and passive competence in oral or written form are more informative, as the next figures illustrate:As a matter of fact, even these quantitative data in the general overview of Fig.\u00a0Clear differences in the self-judgements on competence in Ukrainian and Russian become obvious when speaker types are discriminated; cf. Fig.\u00a0It is obvious that many monocodal speakers of Ukrainian report a rather reduced competence in Russian Fig.\u00a0a and vicSo far in this section, the two ethnic groups have been analysed together. What has to be added is a word on competence in Ukrainian among respondents of Russian nationality, almost all of whom, as has been outlined above, belong to the monocodal Russian speaker type. For this reason, a differentiation based on speaker types would not be meaningful. It is, however, possible to make a more precise comment on the results presented in Fig.\u00a0These data hint at a situation in which Sur\u017eyk is to a certain degree some sort of communicative bridge in everyday life between speakers oriented towards Ukrainian or Russian, at least for ethnic Ukrainians. This is mirrored in the answers to the aspect of tolerance towards the three codes. There were two questions in our survey on this topic: \u201cHave you ever felt embarrassed because of using X?\u201d and \u201cHave you ever been jeered because of using X?\u201d Both questions were asked for all three codes. There are extremely few positive answers on either question. Three out of 100 respondents have experienced occasions of embarrassment for using Sur\u017eyk, one of 100 for using Ukrainian and even fewer for using Russian. Still fewer experienced objections to their choice of codes, interestingly more for Ukrainian  than for Russian or Sur\u017eyk .(5)\u0446\u0435 [\u0441\u0443\u0440\u0436\u0438\u043a] \u0434\u0435\u043a\u043e\u043b\u0438 \u0432\u0438\u043a\u043b\u0438\u043a\u0430\u0454 \u0441\u043c\u0456\u0445, a\u043b\u0435 \u0442\u0430\u043a\u0438\u0439 \u043f\u043e\u0437\u0438\u0442\u0438\u0432\u043d\u0438\u0439, \u0442\u0438 \u0437 \u0446\u044c\u043e\u0433\u043e \u043f\u043e\u0441\u043c\u0456\u044f\u0432\u0441\u044f \u0456 \u0442\u043e\u0431\u0456 \u043d\u0443..., \u043d\u0435 \u0442\u0430\u043a, \u0449\u043e \u201c\u0433\u0430-\u0433\u0430-\u0433\u0430\u201d \u043d\u0430\u0441\u043c\u0456\u0445\u0430\u0454\u0448\u0441\u044f. \u0410 \u0446\u0435 \u043f\u0440\u043e\u0441\u0442\u043e \u043f\u0440\u0438\u043a\u043e\u043b\u044c\u043d\u043e, \u043d\u0430\u0432\u0456\u0442\u044c \u0434\u0435\u044f\u043a\u0456 \u0441\u043b\u043e\u0432\u0430 \u043c\u043e\u0436\u0435\u0448 \u043f\u0435\u0440\u0435\u0439\u043d\u044f\u0442\u0438 \u0456 \u043f\u043e\u0442\u0456\u043c \u0442\u0430\u043a \u0441\u0430\u043c\u043e \u0457\u0445 \u0432\u0438\u043a\u043e\u0440\u0438\u0441\u0442\u043e\u0432\u0443\u0432\u0430\u0442\u0438.It [Sur\u017eyk] is sometimes laughable, but rather positive, you laughed at it and you, well ..., not that you would go laughing \u201cha-ha-ha\u201d. And it\u2019s just cool, you can even adopt some words and then use them in the same way. In most cases, according to the respondents, it is Sur\u017eyk that is stigmatized, but in some cases the use of the Ukrainian (6) or Russian (7) languages also causes ridicule or objection. (6)\u0423 \u043d\u0430\u0441 \u043f\u043e\u043b\u0443\u0447\u0430\u0454\u0442\u044c\u0441\u044f \u0432 \u043c\u0435\u043d\u0435 \u0432 \u0441\u0435\u0441\u0442\u0440\u0438 \u0447\u043e\u043b\u043e\u0432\u0456\u043a, \u0432\u0456\u043d \u2013 \u0440\u0443\u0441\u043a\u0438\u0439, \u0441\u0430\u043c \u0437 \u041a\u0435\u043c\u0435\u0440\u043e\u0432\u0441\u043a\u043e\u0439 \u043e\u0431\u043b\u0430\u0441\u0442\u0456, \u043d\u043e \u0432\u0456\u043d \u043f\u043e\u043d\u0430\u0447\u0430\u043b\u0443 \u043d\u0430\u0441 \u043d\u0435 \u043f\u043e\u043d\u0456\u043c\u0430\u0432 \u0442\u0440\u043e\u0448\u043a\u0438, \u0430 \u043f\u043e\u0442\u043e\u043c \u0432\u0456\u043d \u0443\u0436\u0435 \u043f\u0440\u0438\u0432\u0438\u043a, \u0445\u043e\u0447 \u0432\u0456\u043d \u0441\u0430\u043c \u0440\u0443\u0441\u043a\u0438\u0439, \u0456 \u0432\u0436\u0435 \u0439\u043e\u043c\u0443 \u0456 \u0437\u0430 \u0441\u0454\u043c\u0434\u0435\u0441\u044f\u0442\u044c, \u0456 \u044f\u043a \u0432\u0456\u043d \u043d\u0435 \u0445\u043e\u0442\u0456\u0432 \u043f\u0435\u0440\u0435\u0432\u0435\u0441\u0442\u0438, \u044f\u043a \u0432\u0456\u043d \u043d\u0435 \u0445\u043e\u0442\u0456\u0432 \u043f\u0440\u043e\u043c\u043e\u0432\u0438\u0442\u044c \u0446\u0456 \u0441\u043b\u043e\u0432\u0430 \u0443\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0456, \u0442\u043e \u0432 \u043d\u044c\u043e\u0433\u043e \u0432\u0441\u044c\u043e \u0440\u0430\u0432\u043d\u043e \u043d\u0456\u044f\u043a. \u0422\u043e \u043c\u0438 \u0437 \u043d\u044c\u043e\u0433\u043e \u0433\u043b\u0443\u0437\u0443\u0432\u0430\u043b\u0438, \u0430 \u043d\u0435 \u0432\u0456\u043d \u0437 \u043d\u0430\u0441. We have, therefore, my sister\u2019s husband, he is Russian, himself from the Kemerovo region, but at first, he did not always understand us, and then he got used to it, although he is Russian, and he is over seventy, and as he wanted to translate, as he wanted to say these words in Ukrainian, he anyhow did not succeed. Then we made fun of him, not he of us.(7)\u042f \u0440\u0430\u0431\u043e\u0442\u0430\u043b\u0430 \u0432 \u0441\u0444\u0454\u0440\u0456 \u043e\u0431\u0441\u043b\u0443\u0436\u0438\u0432\u0430\u043d\u0456\u044f, \u0456 \u044f \u043d\u0430\u0447\u0430\u043b\u0430 \u0431\u0430\u043b\u0430\u043a\u0430\u0442\u0438, \u044f\u043a \u044f \u0431\u0430\u043b\u0430\u043a\u0430\u044e, \u0442\u043e \u0434\u0454\u0432\u043e\u0447\u043a\u0430 \u043e\u0434\u043d\u0430 \u043c\u0435\u043d\u0456 \u0441\u043a\u0430\u0437\u0430\u043b\u0430 \u0442\u0430\u043a: \u201c\u043d\u0454 \u0440\u0430\u0437\u0433\u0430\u0432\u0430\u0440\u0456\u0432\u0430\u0439\u0442\u0454 \u0441\u0430 \u043c\u043d\u043e\u0439 \u043d\u0430 \u0443\u043a\u0440\u0430\u0456\u043d\u0441\u043a\u0430\u0439 \u043c\u043e\u0432\u0454, \u0440\u0430\u0437\u0433\u0430\u0432\u0430\u0440\u0456\u0432\u0430\u0439\u0442\u0454 \u0441\u0430 \u043c\u043d\u043e\u0439 \u043d\u0430 \u0440\u0443\u0441\u0441\u043a\u043e\u043c \u044f\u0437\u0438\u043a\u0443\u201d. I worked in the service sector, and I started talking as I talk, and one girl told me: \u201cdo not talk to me in Ukrainian, talk to me in Russian.\u201d Respondents often even show self-confidence, despite ridicule of their language. (8)\u0412 \u0443\u043d\u0456\u0432\u0454\u0440\u0441\u0456\u0442\u0454\u0442\u0456, \u0442\u043e \u0434\u0430, \u043c\u0438 \u043f\u0440\u0438\u0457\u0445\u0430\u043b\u0438 \u0437 \u0441\u0435\u043b\u0430, \u0442\u043e \u0432 \u043d\u0430\u0441 \u0442\u0430\u043a\u0438\u0439 \u0442\u0443\u0442\u0430 \u043c\u0454\u0441\u043d\u0438\u0439 \u0434\u0456\u0430\u043b\u0454\u043a\u0442, \u043c\u043e\u0436\u043d\u0430 \u0441\u043a\u0430\u0437\u0430\u0442\u044c. \u0423\u0436\u0435 \u041a\u0440\u0438\u0432\u0438\u0439 \u0420\u043e\u0433, \u0433\u043e\u0440\u043e\u0434, \u0432\u0441\u0456 \u043f\u043e-\u0440\u0443\u0441\u0441\u043a\u0438 \u0442\u0430\u043c, \u0442\u043e \u0437 \u043d\u0430\u0441, \u043a\u0430\u043d\u0454\u0448\u043d\u043e, \u0432\u0441\u0456 \u043f\u0456\u0434\u0441\u043c\u0456\u044e\u0432\u0430\u043b\u0438\u0441\u044f, \u0456 \u043c\u0438 \u044f\u043a \u043f\u043e\u0447\u0438\u043d\u0430\u043b\u0438 \u043f\u043e-\u0443\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0438, \u043d\u0430\u043c \u0442\u0440\u043e\u0448\u043a\u0438 \u0431\u0443\u043b\u043e \u043d\u0435\u0443\u0434\u043e\u0431\u043d\u043e, \u0456 \u043c\u0438 \u043d\u0430\u0447\u0438\u043d\u0430\u043b\u0438, \u0435\u0442\u043e \u0436 \u043f\u0435\u0440\u0432\u0456 \u043a\u0443\u0440\u0441\u0438, \u043f\u0456\u0434 \u043d\u0438\u0445 \u043f\u0456\u0434\u0441\u0442\u0440\u0430\u044e\u0432\u0430\u0442\u044c\u0441\u044f, \u043d\u0443, \u0431\u0430\u043b\u0430\u043a\u0430\u043b\u0438 \u043f\u043e-\u0440\u043e\u0441\u0456\u0439\u0441\u044c\u043a\u0438. \u041d\u0443 \u0430 \u043f\u043e\u0442\u0456\u043c \u0443\u0436\u0435 \u044f\u043a \u0442\u0440\u043e\u0448\u043a\u0438 \u0441\u0442\u0430\u043b\u0438 \u0441\u0442\u0430\u0440\u0448\u0456, \u0442\u043e \u043c\u0438 \u0440\u0456\u0448\u0438\u043b\u0438, \u0447\u043e \u043c\u0438 \u0434\u043e\u043b\u0436\u043d\u0456 \u043f\u0456\u0434 \u043d\u0438\u0445 \u043f\u0456\u0434\u0441\u0442\u0440\u0430\u044e\u0432\u0430\u0442\u044c\u0441\u044f, \u043d\u0435\u0445\u0430\u0439 \u043b\u0443\u0447\u0448\u0435 \u0432\u043e\u043d\u0438 \u043f\u0456\u0434 \u043d\u0430\u0441. \u0406 \u043c\u0438 \u043d\u0430\u0447\u0430\u043b\u0438 \u0440\u043e\u0437\u043c\u043e\u0432\u043b\u044f\u0442\u044c \u043f\u043e-\u0441\u0432\u043e\u0454\u043c\u0443 \u0456, \u043d\u0443, \u0456 \u043f\u0435\u0440\u0435\u0441\u0442\u0430\u043b\u0438, \u043a\u043e\u043b\u0438 \u0432\u043e\u043d\u0438 \u0437 \u043d\u0430\u0441 \u0441\u043c\u0456\u044f\u043b\u0438\u0441\u044c, \u043c\u0438 \u043f\u0440\u043e\u0441\u0442\u043e \u043d\u0430 \u0446\u0435 \u0432\u0456\u0434\u043f\u043e\u0432\u0456\u0434\u0430\u043b\u0438: \u0445\u043b\u043e\u043f\u0446\u0456, \u0432\u0438 \u043b\u0443\u0447\u0448\u0435 \u043f\u0456\u0434\u0441\u0442\u0440\u0430\u044e\u0439\u0442\u0435\u0441\u044c \u043f\u0456\u0434 \u043d\u0430\u0441, \u0443\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0430 \u0423\u043a\u0440\u0430\u0457\u043d\u0430, \u0443\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0430 \u043c\u043e\u0432\u0430, \u043d\u0443, \u0442\u043e\u0431\u0442\u043e \u0432 \u0442\u0430\u043a\u043e\u043c \u043f\u043b\u0430\u043d\u0456. At the university, yes, we came from the village, we have such a local dialect here, as one can say. Kryvyi Rih, on the other hand, the city, all in Russian there, then of course, all of us laughed, and we started in Ukrainian, we felt a little uncomfortable, and we started \u2013 those were the first years of studying \u2013 to adapt to them, well, chatted in Russian. Well, then as we got a little older, we decided that we shouldn\u2019t adapt to them, but rather they to us. And we started talking in our own way and, well, we stopped when they laughed at us, we just answered: guys, you better adapt to us, Ukrainian Ukraine, Ukrainian language, well, that is, something like that. However, the number of instances of ridicule or objection described above is far too small to interpret their differences. What can be interpreted is the low level of embarrassment and objections regarding the practice of the codes, even for Sur\u017eyk. Of course, there are conversational situations in which one of the codes is more appropriate or inappropriate. And discomfort may result if expectations for the use or non-use of a code are not fulfilled. This seems however to be bound to specific situations rather than general attitudes in society  languages. Here again, several utterances from the in-depth interviews are offered as illustration:32(9)\u041a\u043e\u0433\u0434\u0430 \u044f \u0431\u044b\u043b\u0430 \u043c\u0430\u043b\u0435\u043d\u044c\u043a\u0430\u044f, \u0432 \u043d\u0430\u0448\u0435\u0439 \u0441\u0435\u043c\u044c\u0435 \u0432\u0441\u0435 \u043e\u0431\u0449\u0430\u043b\u0438\u0441\u044c \u043d\u0430 \u0441\u0443\u0440\u0436\u0438\u043a\u0435 \u2013 \u043c\u0430\u043c\u0430, \u043f\u0430\u043f\u0430, \u0431\u0430\u0431\u0443\u0448\u043a\u0430, \u0434\u0435\u0434\u0443\u0448\u043a\u0430. \u0422\u043e\u043c\u0443 \u0456 \u044f, \u0432 \u0434\u0435\u0442\u0441\u0442\u0432\u0435 \u0456 \u0437\u0430\u0440\u0430\u0437, \u0440\u0430\u0437\u0433\u043e\u0432\u0430\u0440\u0438\u0432\u0430\u044e \u043d\u0430 \u0441\u0443\u0440\u0436\u0438\u043a\u0443. (\u2026) \u044f \u0456 \u0437\u0430\u0440\u0430\u0437 \u043e\u0431\u0449\u0430\u044e\u0441\u044c \u043d\u0430 \u0441\u0443\u0440\u0436\u0438\u043a\u0443, \u0456 \u0437 \u0432\u0430\u043c\u0438. (\u2026) \u044f \u0440\u0430\u0437\u0433\u043e\u0432\u0430\u0440\u0438\u0432\u0430\u044e \u0442\u0430\u043a \u044f\u043a \u043c\u0435\u043d\u0456 \u0443\u0434\u043e\u0431\u043d\u043e, \u044f\u043a\u0449\u043e \u0432\u0436\u0435 \u043c\u0435\u043d\u0435 \u043d\u0435 \u043f\u043e\u043d\u0456\u043c\u0430\u044e\u0442, \u0442\u043e \u044f \u043d\u0430\u043c\u0430\u0433\u0430\u044e\u0441\u044c \u043f\u0435\u0440\u0435\u0439\u0442\u0438 \u043d\u0430 \u0442\u043e\u0439 \u044f\u0437\u0438\u043a, \u043d\u0430 \u044f\u043a\u0438\u0439 \u0442\u0440\u0435\u0431\u0430 \u2013 \u0440\u0443\u0441\u0441\u043a\u0438\u0439 \u0447\u0438 \u0443\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0438\u0439. \u0425\u043e\u0442\u044c \u043c\u0435\u043d\u0456 \u0446\u0435 \u0456\u043d\u043e\u0433\u0434\u0430 \u0431\u0443\u0432\u0430\u0454 \u0441\u043b\u043e\u0436\u043d\u043e..When I was a kid, everyone in our family spoke Sur\u017eyk \u2013 mom, dad, grandma, grandpa. That\u2019s why I, as a child and today, I speak Sur\u017eyk. (...) I still communicate in Sur\u017eyk, even with you. (\u2026) I speak as I feel comfortable, if somebody doesn\u2019t understand me, I try to switch to the language I need \u2013 Russian or Ukrainian. Although it is sometimes difficult for me.(c)\u041c\u0456\u0439 \u043e\u0441\u043d\u043e\u0432\u043d\u0438\u0439 \u044f\u0437\u0438\u043a \u2013 \u0446\u0435 \u0437\u043c\u0456\u0448\u0430\u043d\u0430, \u043d\u0443 \u0456\u043b\u0456 \u0441\u0443\u0440\u0436\u0438\u043a. \u0417 \u0434\u0454\u0442\u0441\u0442\u0432\u0430 \u043c\u0435\u043d\u0435 \u0443\u0447\u0438\u043b\u0438 \u0440\u0430\u0437\u0433\u043e\u0432\u0430\u0440\u044e\u0432\u0430\u0442\u044c \u0437\u0440\u0430\u0437\u0443 \u0441\u0438\u043d\u0445\u0440\u043e\u043d\u043e \u043d\u0430 \u0434\u0432\u043e\u0445 \u044f\u0437\u0438\u043a\u0430\u0445. \u042f \u0441\u0447\u0438\u0442\u0430\u044e, \u0449\u043e \u0446\u0435 \u043f\u0440\u0435\u043a\u0440\u0430\u0441\u043d\u043e, \u043f\u043e\u0442\u043e\u043c\u0443 \u0448\u043e \u0449\u0430\u0441 \u0446\u0435 \u043c\u0435\u043d\u0456 \u0434\u0443\u0436\u0435 \u0441\u0438\u043b\u044c\u043d\u043e \u043f\u043e\u043c\u0430\u0433\u0430\u0454, \u044f \u043c\u043e\u0436\u0443 \u043f\u0435\u0440\u0435\u0445\u043e\u0434\u0438\u0442\u0438 \u0437 \u0440\u0443\u0441\u043a\u043e\u0433\u043e \u043d\u0430 \u0443\u043a\u0440\u0430\u0457\u043d\u0441\u043a\u0438\u0439, \u0437 \u0443\u043a\u0440\u0430\u0457\u043d\u0441\u043a\u043e\u0433\u043e \u043d\u0430 \u0440\u0443\u0441\u043a\u0438\u0439 (...) \u041c\u0435\u043d\u0456 \u0432\u043e\u043e\u0431\u0449\u0435 \u0446\u0435 \u043a\u043e\u043c\u0444\u043e\u0440\u0442\u043d\u043e.My main language is mixed, well, or Sur\u017eyk. Since childhood, I was taught to speak synchronously in two languages. I think it\u2019s great, because right now it helps me a lot, I can switch from Russian to Ukrainian, from Ukrainian to Russian (...) for me in general it\u2019s comfortable.The above quotation provides an explanation for the enormous shift among speakers with exclusively Sur\u017eyk as the first code towards all three codes in everyday linguistic practice (S-U-R). This is simply a social necessity for communicating with the large number of speakers who practise only one of the two \u201creal\u201d languages. (d)\u041d\u0443, \u044f \u0437 \u0441\u0430\u043c\u043e\u0433\u043e \u0434\u0454\u0442\u0441\u0442\u0432\u0430 \u0431\u0430\u043b\u0430\u043a\u0430\u044e \u043d\u0430 \u0441\u0443\u0440\u0436\u0438\u043a\u0443, \u043c\u0435\u043d\u0456 \u0437\u0430\u0440\u0430\u0437 \u0432\u043e\u0441\u0454\u043c\u043d\u0430\u0434\u0446\u044f\u0442\u044c \u0433\u043e\u0434, \u0456 \u0442\u0443\u0442\u0430 \u0448\u043e\u0431 \u0441\u043a\u0430\u0437\u0430\u0442\u044c, \u0448\u043e \u043d\u0430 \u044f\u043a\u0438\u0445\u043e\u0441\u044c \u043e\u043f\u0440\u0435\u0434\u0454\u043b\u044c\u043e\u043d\u043d\u0438\u0445 \u2013 \u043d\u0454, \u0443 \u043d\u0430\u0441 \u0456\u043d\u043e\u0433\u0434\u0430 \u043f\u0440\u043e\u0441\u043a\u0430\u043a\u0443\u0454 \u0456 \u0443\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u0430 \u043c\u043e\u0432\u0430, \u0456 \u0440\u043e\u0441\u0456\u0439\u0441\u044c\u043a\u0430, \u0456 \u0442\u043e \u0432\u0441\u0435 \u044f\u043a\u043e\u0441\u044c, \u044f\u043a\u043e\u0441\u044c \u0432\u043f\u0435\u0440\u0435\u043c\u0454\u0448\u043a\u0443.Well, I\u2019ve talked in Sur\u017eyk since very early childhood, I\u2019m eighteen years old now, and here one can say that in certain [languages] \u2013 no, sometimes people hop, between Ukrainian and Russian, and all this is somehow, somehow mixed up.Speakers with exclusively Sur\u017eyk as their first code almost never give up practising Sur\u017eyk frequently. The number of respondents reporting only one code, Ukrainian or Russian, in their everyday lives tends to zero, the number of respondents using Ukrainian and Russian (but not Sur\u017eyk) is zero. Quite a few of them report one of the combinations of Sur\u017eyk with Ukrainian or Russian  as their major codes: roughly 15 percent each. (e)\u041f\u0456\u0434\u043f\u0440\u0438\u0454\u043c\u0446\u0456 \u0443 \u043d\u0430\u0441 \u0432\u0441\u0456 \u0441\u0443\u0440\u0436\u0438\u043a\u0438. \u041d\u0443 \u044f\u043a \u0447\u0430\u0441\u0442\u0438\u043d\u0430 \u0454\u0441\u0442\u044c \u0456 \u0440\u043e\u0441\u0456\u0439\u0441\u044c\u043a\u043e\u043c\u043e\u0432\u043d\u0456, \u0454\u0441\u0442\u044c \u0456 \u0443\u043a\u0440\u0430\u0457\u043d\u043e\u043c\u043e\u0432\u043d\u0456. \u041e\u0441\u044c, \u0434\u043e \u0440\u0435\u0447\u0456, \u043e\u0446\u0456 \u043b\u044e\u0434\u0438, \u0432\u043e\u043d\u0438 \u0443\u043a\u0440\u0430\u0457\u043d\u043e\u043c\u043e\u0432\u043d\u0456. \u0410\u043b\u0435 \u044f \u0432\u0430\u043c \u0441\u043a\u0430\u0436\u0443, \u0430 \u043d\u0430 \u0441\u0443\u0440\u0436\u0438\u043a \u043f\u0435\u0440\u0435\u0439\u0448\u043e\u0432, \u043e\u0441\u044c \u043f\u0456\u0434\u043f\u0440\u0438\u0454\u043c\u0435\u0446\u044c, \u0445\u043b\u043e\u043f\u0447\u0438\u043d\u0430 \u043e\u0446\u0435\u0439. \u0412\u0456\u043d \u0443\u043a\u0440\u0430\u0457\u043d\u043e\u043c\u043e\u0432\u043d\u0438\u0439, \u0432\u0456\u043d \u0437 \u0422\u0435\u0440\u043d\u043e\u043f\u043e\u043b\u044f, \u0430\u043b\u0435 \u043d\u0430 \u0441\u0443\u0440\u0436\u0438\u043a \u043f\u0435\u0440\u0435\u0439\u0448\u043e\u0432.Entrepreneurs are all \u201cSur\u017eyk-folk\u201d. Well, partially there are Russian-speaking ones, there are Ukrainian-speaking ones, too. By the way, these people are Ukrainian-speaking. But I\u2019ll tell you, there\u2019s one who has shifted to Sur\u017eyk, an entrepreneur, this guy. He is Ukrainian-speaking, he\u2019s from Ternopil, but he shifted to Sur\u017eyk.The S-U-R-type of frequent current usage makes up the second largest group in the case of monocodal linguistic socialization with Ukrainian or Russian (15 or 18 percent), after the majority groups of monocodal-oriented speakers of Ukrainian and Russian. For both groups of the previously monocodal Ukrainian- or Russian-speaking children, according to (e) and (d) Sur\u017eyk becomes a frequently used code in further stages of their lives. It is well known that even people with a linguistic socialization in one of the literary languages during childhood and an academic education under certain conditions have to turn to a (mixed) vernacular, for example in professions such as engineers, architects or veterinarians to be accepted by workmen or farmers. (f)\u042f \u0434\u0443\u043c\u0430\u044e, \u0449\u043e \u0442\u0456, \u044f\u043a\u0456 \u043f\u0440\u0438\u0457\u0436\u0434\u0436\u0430\u044e\u0442\u044c \u0432\u043e\u043d\u0438 \u043d\u0435 \u0441\u0438\u043b\u044c\u043d\u043e \u043c\u043e\u0436\u0443\u0442\u044c \u043f\u0440\u0438\u0441\u0442\u0440\u043e\u0457\u0442\u0438\u0441\u044c, \u0431\u043e \u0432\u043e\u043d\u0438 \u0432\u0438\u0440\u043e\u0441\u043b\u0438 \u043d\u0430 \u0442\u043e\u043c\u0443 \u044f\u0437\u0438\u043a\u0443. \u0410 \u0454\u0441\u043b\u0456 \u043b\u044e\u0434\u0438\u043d\u0430 \u0432\u0438\u0440\u043e\u0441\u043b\u0430 \u043d\u0430 \u0447\u0438\u0441\u0442\u043e \u0443\u043a\u0440\u0430\u0457\u043d\u0441\u044c\u043a\u043e\u043c \u044f\u0437\u0438\u043a\u0443 \u0447\u0438 \u043d\u0430 \u0447\u0438\u0441\u0442\u043e \u0440\u0443\u0441\u043a\u043e\u043c, \u0442\u043e \u0457\u0439 \u0442\u043e\u0436\u0435 \u0442\u044f\u0436\u043a\u043e \u043f\u0435\u0440\u0435\u0439\u0442\u0438 \u043d\u0430 \u0441\u0443\u0440\u0436\u0438\u043a. (\u2026) \u042f \u0437\u043d\u0430\u044e \u0442\u0430\u043a\u0438\u0445 \u043b\u044e\u0434\u0435\u0439, \u044f\u043a\u0456 \u0442\u0443\u0442 \u0436\u0438\u0432\u0443\u0442\u044c \u0433\u043e\u0434\u0430\u043c\u0438 \u0456 \u0432\u043e\u043d\u0438 \u043d\u0435 \u043f\u0435\u0440\u0435\u0439\u0448\u043b\u0438 \u043d\u0430 \u0441\u0443\u0440\u0436\u0438\u043a \u043f\u043e\u043b\u043d\u043e\u0441\u0442\u044e. \u0422\u0430\u043a \u043d\u0430 \u0440\u0443\u0441\u043a\u043e\u043c \u0432\u043e\u043d\u0438 \u0439 \u0433\u043e\u0432\u043e\u0440\u044f\u0442\u044c, \u043d\u0435 \u043c\u043e\u0436\u0443\u0442\u044c \u043f\u0435\u0440\u0435\u0439\u0442\u0438.I think that those who come here, they cannot adapt that much, because they grew up in that language. And if people grew up in pure Ukrainian or in pure Russian, it is also difficult for them to switch to Sur\u017eyk. (\u2026) I know people who have lived here for years and they have not completely shifted to Sur\u017eyk. They still speak Russian, they are not able to switch.Apart from the group of current S-U-R speakers, a switch from Russian to Ukrainian is almost absent for speakers with only Russian in their first linguistic socialization \u2013 compare the very low share of U-R speakers. By the way, this is a clear indication of the fact that there is no pressure on the Russian-speaking population of Ukraine to abandon their code of first choice. It is even the case that there is a slight tendency for monocodal Ukrainian children to later turn to Russian as well. However, this might be a reflexion of the former predominance of Russian in public life. (g)Last but not least, it should be realized that people with a monocodal Ukrainian background have more readily turned to only Russian in current usage than vice versa. On the other hand, those with a monocodal Russian background more readily tend to add Ukrainian to their repertoire of frequently used codes than those with monocodal Ukrainian background add Russian to it. Thus, generalizing these observations, we can state the following: If there has been a shift among people to Ukrainian, which is more than probable but cannot be determined here precisely, then people with Russian or Sur\u017eyk as their code of first linguistic socialization add Ukrainian to their repertoire, rather than abandoning their first code completely. A further detail to be added in connection with this generalization is that there are almost no differences between respondents of Ukrainian or Russian nationality in this respect: If their first language was Russian, which was almost exclusively the case with ethnic Russians, they maximally add Ukrainian or (more often) Ukrainian and Sur\u017eyk to their repertoire, more or less to the same extent. Whatever shifts have been accomplished by people during their lifetimes, these were most obviously not caused by social pressure from people with a different linguistic orientation.The major observations are as follows: The region this study concentrated on belongs to the primary territorial aims of the war being waged by Russia against Ukraine. In the article, we tried to present basic insights on the linguistic situation in 2020 / 2021 in the three oblasts of Odesa, Mykola\u00efv and Xerson, which are traditionally seen as a region of Russian language predominance, even by Ukrainian colleagues from social sciences. As has already been pointed out by Hentschel and Taranenko , but dueThe sections above contain a wealth of quantitative data and the main points will be summarized and interpreted here comprehensively.The balanced tricodalism that Hentschel and Taranenko  have empIt is clear that with speakers of Russian nationality, who made up one tenth of all our respondents, there can be no question of a balanced tricodalism. Their language is predominantly Russian: As their native language, as their most frequently used language and even as the language being used and heard in their daily surroundings \u2013 in each respect with a more or less clear tendency to exclusiveness. This group of respondents seems to live to a large degree in a linguistically different social world, as do some of the monocodal Russian-speaking Ukrainians.In regard to the question of shifting from one code to two or three codes, this study mirrors rather precisely that there is a certain shift to Ukrainian in current linguistic practice. For the majority of those with a monocodal Russian background during childhood, Russian definitely remains the only frequently used language. This is most extreme for respondents of Russian nationality: only a few have added Ukrainian or Ukrainian and Sur\u017eyk to their linguistic repertoire. Among the ethnic Ukrainians with a monocodal Russian background during childhood, almost one quarter reports a frequent use of Ukrainian as well, often accompanied with a frequent practice of Sur\u017eyk. The self-estimated competence in Ukrainian in this group is considerable. It can (for the moment) not be discerned to what extent this reflects a shift to Ukrainian, as it may be the case that younger respondents could take advantage of better teaching of Ukrainian in the education system. Nevertheless, it is more than obvious that hardly anyone with Russian as the only code used during childhood abandoned it, not even ethnic Ukrainians.Apart from the \u201ccompetition\u201d between Ukrainian and Russian in Ukraine society, it can be stated that Sur\u017eyk holds onto its position in the \u201carchitecture\u201d of linguistic codes pretty well, even on the Black Sea coast. Our figures suggest that it was (at least) on a comparable level with Ukrainian and Russian as the code of linguistic socialization during childhood. For approximately one half of our respondents it belonged to the frequently used codes even in later years, though only rarely as the only one in frequent use. Last but not least, people estimate the presence of Sur\u017eyk as much stronger outside their families.The most significant shift to be observed was that in the statement of their native language by ethnic Ukrainians. In the Soviet census of 1989 and in the only Ukrainian census of 2001, ethnic Ukrainians in the three oblasts at issue here clearly named Ukrainian less frequently as their native language than in our survey. In the former census, only little more than one half of the population did so, here it was up to nine out of ten speakers. The language policy in independent Ukraine has obviously at least tightened the symbolic-patriotic link between titular ethnic group and the \u201ctitular\u201d language. That linguistic practice changes much less rapidly, is, of course, not astonishing.This study, of course, prompts many further questions. Almost all results presented here must be related, for example to sociodemographic criteria, such as age, sex, education, profession, diverse political and linguistic attitudes etc. This will be undertaken in the project underpinning this paper in the near future by employing multifactorial statistical methods to a much larger extent and by deeper qualitative analyses of the material gathered. Only then will further interpretations of the empirical results be possible. Nevertheless, we hope to have given a rather differentiated description of the linguistic landscape.However, none of the figures reported above, especially the reports about the general lack of objections and ridicule of their usage of any code (apart from rare exceptions), support the Kremlin assertion of prosecution of parts of the population for linguistic reasons. Almost no one claimed that their usage of Russian was objected to by others; not more or even fewer claimed to have experienced objections to their speaking Ukrainian or Sur\u017eyk. Of course, there are different opinions about the enforcement of Ukrainian by the state government and the legal status of Russian in the Ukraine (cf. Zeller, The linguistic situation in the three Black Sea oblasts in independent Ukraine can by no means serve as a legitimation for the Kremlin war in the sense of saving Russians or Russian-speaking people from prosecution. The assertion of such prosecution, if not worse, is one of the masses of lies that the Kremlin regime has spread over the last decades. Unfortunately, it is obviously not only a vast majority of the Russian people that is ready to believe these lies, but even a large group of \u201cPutin-versteher\u201d in the western world that has shown such inclinations, at least until recently.The ongoing war will change many things, including linguistic practice and attitudes as well. The situation which we have just described for 2020 / 2021 will most probably change radically."}
{"text": "In Section 3.2, \u201cmale\u201d has been corrected to \u201chypertension\u201dIn Table 3, \u201cmale\u201d has been corrected to \u201chypertension\u201dIn Section 3.3, \u201cmale sex\u201d has been corrected to \u201chypertension\u201dIn the first paragraph of Discussion, \u201cmale\u201d has been corrected to \u201chypertension\u201dIn Conclusions, \u201cmale\u201d has been corrected to \u201chypertension\u201dIn the article titled \u201cNovel Insights into the Predictors of Obstructive Sleep Apnea Syndrome in Patients with Chronic Coronary Syndrome: Development of a Predicting Model\u201d , several"}
{"text": "Listeriosis outbreak caused by contaminated stuffed pork, Andalusia, Spain, July to October 2019\u2019 by Fern\u00e1ndez-Mart\u00ednez et al. published on 27 October 2022. The level of significance was noted as \u2018p values\u202f\u2009\u2265\u2009\u202f0.05\u2019 in the originally published version. This was corrected to \u2018p values\u202f\u2009\u2264\u2009\u202f0.05\u2019 on 28 October 2022. We apologise for any inconvenience this may have caused.A correction was made to the article \u2018"}
{"text": "We include some examples that illustrate these applications. The obtained results of our proposed definition can provide a suitable modeling guide to study many problems in mathematical physics, soliton theory, nonlinear science, and engineering.A newly proposed generalized formulation of the fractional derivative, known as Abu-Shady\u2013Kaabar fractional derivative, is investigated for solving fractional differential equations in a simple way. Novel results on this generalized definition is proposed and verified, which complete the theory introduced so far. In particular, the chain rule, some important properties derived from the mean value theorem, and the derivation of the inverse function are established in this context. Finally, we apply the results obtained to the derivation of the implicitly defined and parametrically defined functions. Likewise, we study a version of the fixed point theorem for  Da\u03b1(1) = 0, if \u03b1 is not a natural numberThe Riemann-Liouville derivative does not satisfy Fractional derivative statements do not possess some of the fundamental properties of classical derivatives, such as the product rule, the quotient rule, or the chain ruleD\u03b1D\u03b2f = D\u03b1+\u03b2fThese derived proposals, in general, do not satisfy f must be differentiable in the ordinary senseThe definition of the Caputo derivative implies that the function Fractional calculus is theoretically considered as a natural extension of classical differential calculus, which has attracted many researchers, both from a more theoretical point of view and for its diverse applications in sciences and engineering. Thus, from a more theoretical perspective, various definitions of fractional derivatives have been initiated. Fractional definitions try to satisfy the usual properties of the classical derivative; however, the only property inherent in these definitions is the property of linearity. On the contrary, some of the drawbacks that these derivatives present can be located in the following:More information on this definition of fractional derivative can be found in , 2.The locally formulated fractional derivative is established through certain quotients of increments. In this sense, Khalil et al.  introducRecently, Abu-Shady and Kaabar  introducThe GFFD definition is very important in studying various phenomena in science and engineering due to the powerful applicability of this definition in investigating many fractional differential equations in a very simple direction of obtaining analytical solutions without the need for approximate numerical methods or complicated algorithms like other classical fractional definitions. This definition is a modified version of the conformable definition to overcome all issues and advantages associated with the conformable one.Regarding the geometric behavior of GFFD, by following the previous research study concerning the fractional cords orthogonal trajectories in the sense of conformable definition , GFFD caOne of the limitations of GFFD is that GFFD is locally defined derivative, and some future works are needed to proposed nonlocal formulation of GFFD in order to preserve the nonlocality property of fractional calculus. However, nonlocal definitions come with many associated challenges while working on solving fractional differential equations. Therefore, the future studies will work on overcoming all these challenges.\u03b1-differential functions are proposed in \u03b1-differentiable functions are presented in \u03b1-differentiable functions are included. Some conclusions are drawn in The work is constructed as follows: The GFFD and its main properties are presented in f :  = a\u2009DGFFD[f] + b\u2009DGFFD[g], \u2200a, b \u2208 RDGFFD[tp] = (p\u0393(\u03b2)/\u0393(\u03b2 \u2212 \u03b1 + 1))tp\u2212\u03b1, \u2200p \u2208 RDGFFD[\u03c8] = 0, \u2200constant functions f(t) = \u03c8DGFFD[fg] = fDGFFD[g] + gDGFFD[f]DGFFD[(f/g)] = (gDGFFD[f] \u2212 fDGFFD[g]/g2)f is a differentiable function, then DGFFDf(t) = (\u0393(\u03b2)/\u0393(\u03b2 \u2212 \u03b1 + 1))t\u03b11\u2212(df/dt)(t).If, additionally, Let 0 < \u03b1-derivative of certain functions using GFFD is expressed as:DGFFD[1] = 0DGFFD[sin(kt)] = (k\u0393(\u03b2)/\u0393(\u03b2 \u2212 \u03b1 + 1))t\u03b11\u2212cos(kt)DGFFD[cos(kt)] = \u2212(k\u0393(\u03b2)/\u0393(\u03b2 \u2212 \u03b1 + 1))t\u03b11\u2212sin(kt)DGFFD[ekt] = (k\u0393(\u03b2)/\u0393(\u03b2 \u2212 \u03b1 + 1))t\u03b11\u2212ektThe generalized \u03b1-derivative of the following functions:DGFFD[(\u0393(\u03b2 \u2212 \u03b1 + 1)/\u03b1\u0393(\u03b2))t\u03b1] = 1DGFFD[sin((\u0393(\u03b2 \u2212 \u03b1 + 1)/\u03b1\u0393(\u03b2))t\u03b1)] = cos((\u0393(\u03b2 \u2212 \u03b1 + 1)/\u03b1\u0393(\u03b2))t\u03b1)DGFFD[cos((\u0393(\u03b2 \u2212 \u03b1 + 1)/\u03b1\u0393(\u03b2))t\u03b1)] = \u2212sin((\u0393(\u03b2 \u2212 \u03b1 + 1)/\u03b1\u0393(\u03b2))t\u03b1)DGFFD[e\u03b2 \u2212 \u03b1 + 1)/\u03b1\u0393(\u03b2))t\u03b1)((\u0393(] = e\u03b2 \u2212 \u03b1 + 1)/\u03b1\u0393(\u03b2))t\u03b1)f(a) = f(b)Let c \u2208 , such that DGFFDf(c) = 0.Then, \u2203a > 0, \u03b1 \u2208 Let c \u2208 , \u220bh = (\u0393(\u03b2 \u2212 \u03b1 + 1)/\u03b1\u0393(\u03b2)).Then, \u2203\u03b1-differentiable functions, introduced in (t) = 0. If g not is constant in a neighborhood of a > 0, we can find a t0 > 0 such that g(t1) \u2260 g(t2) for any t1, t2 \u2208 . Now, since g is continuous at a, for \u03b5 sufficiently small, we haveWe prove the result following a standard limit approach. First, if the function Making\u03b1-differentiability and assuming g(t) > 0, Equation DGFFDg(t) \u2260 0\u2200t \u2208 g(b) \u2260 g(a)DGFFDf(t) and DGFFDg(t) not annulled simultaneously on Let c \u2208 , \u220bThen, \u2203Consider the functionF is continuous on , generalized \u03b1 \u2212 DF on , and F(a) = F(b) = 0, then by c \u2208  such that DGFFDF(c) = 0. Using the linearity of DGFFD and the fact that the generalized \u03b1-derivative of a constant is zero, our result follows.Since g(t) = (\u0393(\u03b2 \u2212 \u03b1 + 1)/\u03b1\u0393(\u03b2))t\u03b1Observe that a > 0, \u03b1 \u2208 Let DGFFDf(t) = 0, for all t \u2208 , then, f is a constant on If DGFFDf(t) = 0 for all t \u2208 . Let t1, t2 \u2208  with t1 < t2. So, the closed interval  is contained in , and the open interval  is contained in .Suppose f is continuous on  and \u03b1 \u2212 DF on . So, by c \u2208  withHence, f(t2) \u2212 f(t1) = 0 and f(t2) = f(t1)Therefore, t1 and t2 are arbitrary numbers in  with t1 < t2, then f is a constant on .Since a > 0, \u03b1 \u2208  = DGFFDG(t)\u2200t \u2208 . Then, \u2203 a constant C such thatLet H(t) = F(t) \u2212 G(t), it can easily be proven.By simply applying the above theorem to \u03b1 \u2208 f is generalized Let a > 0, DGFFDf(t) > 0\u2200t \u2208 , then f is strictly increasing on If DGFFDf(t) < 0\u2200t \u2208 , then f is strictly decreasing on If Then, we havec between t1 and t2 withDGFFDf(c) > 0, then f(t2) > f(t1) for t1 < t2. Therefore, f is strictly increasing on , since t1 and t2 are arbitrary number of If DGFFDf(c) < 0, then f(t2) < f(t1) for t1 < t2. Therefore, f is strictly decreasing on , since t1 and t2 are arbitrary number of .If Following similar line of argument as given in the \u03b1 \u2208  if the f is generalized \u03b1 \u2212 DF\u2009 on I and generalized \u03b1-derivative is continuous on I.Let I \u2282  an open interval, I \u2282  an open interval, \u03b1 \u2208  = b for some a \u2208 I, and DGFFDf(a) \u2260 0. Then, there is an open neighborhood U of a in which f admits an inverse function f\u22121 of class C\u03b1 on the open neighborhood V = f(U) of b, and its generalized \u03b1-derivative isLet DGFFDf(t) is continuous in the open interval I, it is a known fact that there exists an open neighborhood U of a in which DGFFDf(t) has a constant sign (the sign of DGFDf(a)). From f that is strictly monotonic on U (increasing if DGFFDf(a) > 0 and decreasing if DGFFDf(a) < 0). Therefore, f is continuous and strictly monotonic on U, so there is the inverse function of the one-to-one function f : U\u27f6V, with V = f(U). This inverse f\u22121 : V\u27f6U is of class C\u03b1 and strictly monotonic (in the same sense that f is) on V. Equation (f(f\u22121(y)) = y for all y \u2208 V, in which the \u03b1-derivative (with respect to y) is calculated, applying the chain ruleSince Equation  can be o\u03b1 \u2212 DF functions are presented in this section.Some interesting applications of the results obtained on generalized F = 0 implicitly defines a function y = g(t) in a certain open interval I, if F) = 0\u2200t \u2208 I. Suppose that g(t) and F) are generalized \u03b1 \u2212 DF functions in an open interval I \u2282 (o.\u221e), then the derivative DGFFDg(t) can be found by calculating the generalized \u03b1-derivative of F), as a compound function, and canceling this derivative calculated.It is a known fact that an equation y = g(t) function at the point t = 8, such that g(8) = 1, and it is implicitly defined by the equationNow, we are going to calculate the derivative of the generalized 1/3-differentiable Calculating the 1/3\u2009-derivative in this equation, we obtaint = 8 and g(8) = 1 in the equation above, we haveTaking Finally, the generalized 1/3-derivative is given byt = t(\u03bb), y = y(\u03bb) be generalized \u03b1 \u2212 DF functions on an open interval I \u2282 (o.\u221e), with DGFFDt(\u03bb) \u2260 0\u2200\u03bb \u2208 I. If t = t(\u03bb) and y = y(\u03bb), define the function y = y(\u03bb(t)) = y(t) (where \u03bb(t) is the inverse function of t(\u03bb)), and then, the generalized \u03b1-derivative of this function is given byLet Note that the above expression is obtained by applying the chain rule and the derivation formula of the inverse function, both in the generalized sense.y = y(t) defined parametrically byt)Thus, for example, consider the function \u03b1-derivative and its respective proof. In addition, we will establish some important results about the iteration of the fixed point in the generalized \u03b1-derivative sense. However, we first present some basic concepts and necessary results for the developments that we are going to carry out.We present the fixed point theorem for generalized t = f(t). Here,\u2009f\u2009is a mapping from X\u27f6X. We assume that X is endowed with the metric d. A point t \u2208 X which satisfies t = f(t) is called a fixed point of f.Let the following fixed-point equation a > 0, \u03b1 \u2208 Let c \u2208 , \u220bh = (\u0393(\u03b2)/\u0393(\u03b2 \u2212 \u03b1 + 1)).Then, \u2203Consider the functiong satisfies the conditions of c \u2208 , \u220b0CTt\u03b1[g](c) = 0. ThereforeThen, the function Hence,\u03b1-derivative.Now, we establish the fixed point theorem for generalized \u2009a > 0 and \u03b1 \u2208 \u2009 and f(t) \u2208 , \u2200t \u2208 , then g has a fixed point at . If also, f is generalized \u03b1 \u2212 DF on  andf has a unique fixed point p at .Let f(a) = a or f(b) = b, the existence of the fixed point is obvious. Suppose that f(a) \u2260 a and f(b) \u2260 b, therefore f(a) > a and f(b) < b. Let g(t) = f(t) \u2212 t, clearly continuous on , we haveIf p \u2208 \u220bg(p) = 0; that isBy the classical intermediate value theorem, then \u2203f has a fixed point at p.Therefore, p and q are fixed points on  with p \u2260 q. By \u03be between p and q, and therefore, in , such thatp \u2212 q| < |p \u2212 q| which is contradiction, therefore p = q.Suppose also that Equation  is satisf, we choose an initial approximate value p0, and we obtain the succession {pn}n=0\u221e by taking pn = f(pn\u22121) for each n \u2265 1. If the succession {pn}n=0\u221e of converges p and f is a continuous function, thent = f(t) is obtained. This technique is called iterative technique of the fixed point or functional iteration.To approximate the fixed point of a function f converges a solution of the equation t = f(t) and that also chooses f correctly in such a way that it makes the convergence as quickly as possible.The following result provides a first step to determine a procedure that guarantees that the function a > 0 and \u03b1 \u2208  and f(t) \u2208 , for all t \u2208 . Also, suppose that f is generalized \u03b1 \u2212 DF on  withLet p0 is any number in , then, the succession defined byp in .If a, b]. On the other hand, since f applies to  itself, the sequence {pn}n=0\u221e is defined \u2200n \u2265 0 and pn \u2208 \u2200n. Using Equation . Applying the above equation inductively resultsFirst, by Equation  and the k < 1, it easily follows thatpn}n=0\u221e converges to p.Since f satisfies the hypotheses of If n \u2265 1, the procedure used in the proof of For m > n \u2265 1Hence, for By kn/1 \u2212 k, and the smaller k can be made, the faster the convergence will be. Convergence can be very slow if k is close to 1.It is clear that the speed of convergence depends on the factor Finally, we present an example that illustrates these last established results.\u03b1 = \u03b2 = 1/2. We can observe that f =  \u2282 . Also, f is continuous andf satisfies the hypotheses of \u22124. Taking p0 = 1, to obtain this precision, 54 iterations are required. Also, note that since the generalized 1/2-derivative DGFFDf(t) is negative, the successive approximations oscillate around the fixed point.Consider the function: Novel results regarding the Abu-Shady\u2013Kaabar fractional derivative have been investigated in this study which are extensions of the previous research study's results in . In part"}
{"text": "Women's death due to complications of pregnancy and childbirth is still high. Maternity waiting homes are one of the strategies to reduce it. However, there is limited evidence on the effect of using maternity waiting homes on birth outcomes, particularly in this study area. Therefore, this study was aimed to estimate the effect of staying in maternity waiting homes use on maternal and perinatal birth outcomes and its challenges in the Amhara region, Northwest Ethiopia 2018.Institutional-based comparative cross-sectional study using both quantitative and qualitative approaches was conducted. Data were collected using structured questionnaire interviews, in-depth interview and chart reviews. Propensity score matching analysis was used to estimate the effect of maternity waiting homes use on birth outcomes. Propensity score matching analysis was used to match potential differences in background characteristics that affect pregnancy outcomes between comparison groups. We used thematic analysis for qualitative data.A total of 548 pregnant mothers (274 stayed in maternity waiting homes 274 did not stay) took part in this study. The proportion of adverse birth outcomes of mothers who stayed in maternity waiting homes were 15(5.5%) which is lower than those who didn't stay 35 (12.8%). After matching with baseline covariates, mean difference of adverse maternal birth outcomes, the difference between didn't use maternity waiting home and used was 10.4%, at (t\u2009=\u20093.78) at 5% level of significance. Similarly, the mean adverse perinatal birth outcomes difference between mothers who didn't use MWHs and used was 11% (t\u2009=\u20094.33).Maternity waiting home showed a significant positive effect on birth outcomes. Mothers who stayed in the maternity waiting homes had low adverse maternal and perinatal birth outcomes compared to non-users. Accommodations and quality health care services were the challenges mothers faced during their stay in the maternity waiting homes. Therefore, all concerned bodies should give attention accordingly to maternity waiting home services to reduce adverse birth outcomes through the strengthening of the quality of health care provided. In order to decrease geographic obstacles to get medical care as soon as problems or labor start, the World Health Organization (WHO) has supported the use of Maternity Waiting Homes (MWHs) . MaternaCurrently, regardless of their risk level, maternal waiting rooms are hosting women traveling from rural areas and outside the service delivery region on the last trimester of pregnancy . MWHs arWorldwide, around 6.3 million live births resulted in death before the age of five . The neoThe usage of maternal health care services is poor, and maternal mortality in underdeveloped regions is still 15 times greater than in industrialized regions . Every yMost of these deaths can be avoided with prompt access to emergency obstetrical care, but the location of the women's residences in relation to the closest medical facility may also have an impact . The avaLike other developing nations, Ethiopia launched maternity waiting homes in 1985. There is, however, a dearth of research, especially in our subject area. Additionally, past research did not make an effort to make the group comparable to other factors that influence pregnancy outcomes and instead used straightforward cross-sectional studies. This study evaluated the impact of maternity waiting homes on maternal and perinatal birth outcomes as well as its problems in the Amhara Region, Northwest Ethiopia.Between September and December 2018, a comparative cross-sectional study situated in an institution and involving 548 rural mothers who gave birth in the East Gojjam Administrative Zone was done. One of the eleven administrative zones in Ethiopia's Amhara National Regional State (ANRS) is East Gojjam Zone.Utilizing two formulas for population proportions, the sample size was calculated. Using EPI-INFO software version 7.2.4, the total sample size was calculated by taking into account the percentage of stillbirths among mothers who were using the maternity waiting homes setting (P1\u2009=\u20091.2%), the percentage of stillbirths among non-users of the maternity waiting homes setting (P2\u2009=\u200910%), the level of significance at the 5% level, and the power at the 80% level . MaterniThen, we used a multistage sampling procedure to choose the study participants. First, seven districts\u2014Basoliben, Dejen, Hulete Eju Enese, Debre Elias, Enemay, and Sinan\u2014out of the 20 in the East Gojjam Zone were chosen by simple random sampling to make up 30% of the research area. Second, two public health facilities with maternity waiting homes were chosen, one from each area. Finally, the study subjects were chosen using a simple random sample.We used the exit interview method to gather information for both mothers who use the maternity waiting home and non-user mothers who give birth in the public health facility.The treatment variable was maternity waiting home use, whereas the outcome variables were maternal and perinatal delivery outcomes (good/poor). Any residence that is close to or is a part of a health facility and is designed for a pregnant woman to stay in before giving birth is known as a maternity waiting home. Any mother death, fistula, uterine rupture, antepartum hemorrhage (APH), postpartum hemorrhage (PPH), and eclampsia are examples of adverse maternal birth outcomes. Any stillbirth, sudden neonatal death, and birth asphyxia are examples of adverse perinatal birth outcomes. Obstacles for mothers: -any social and economic issues mothers' encountered while residing at the maternal waiting home.p- value at 0.05. The qualitative results were divided into 6 primary themes and were analyzed thematically.After reading over a number of pertinent pieces of literature, we created a structured questionnaire. Data were gathered by interviewers using in-depth interview approaches, chart reviews, and administered questions. The data collection includes 14 nurses who served as data collectors and seven nurses who served as supervisors. The data collectors and supervisors received a two-days training on the study's objectives and data gathering methods in order to ensure the accuracy of the data. Investigators oversaw the entire data collection process. To match baseline obstetric and medical characteristics that influence pregnancy outcomes, we used propensity score matching. Finally, using STATA software, we calculated the average treatment effect of treated (ATT)  of them were Amhara by ethnicity, had a mean age of 27.5 years (SD\u2009+\u20095.6 years), and 152 (20.7%) of the mothers were Para-I (one) .In this study, 55(10.04%) mothers experienced unfavorable maternal delivery outcomes in total. Of the negative birth outcomes recorded, mothers who did not use MWH accounted for 35 (63.6%) of the negative maternal birth outcomes. Compared to the poor maternal birth outcomes seen in mothers who used MWH (20), this was worse (7.31%) See .First the propensity score was predicted. To predict the propensity score values for the independent variables Chronic Hypertension, Diabetes Mellitus, HIV status, Cardiac Disease, Anemia, Previous C/S, History of APH, History of PPH . Logit model was employed after the propensity score value predication of the different matching methods were applied . Second In addition to balancing test, another important step in investigating the validity or performance of the propensity score matching estimation is verifying the common support or overlap condition. To demonstrate the common support estimated results and test propensity scores for the two groups of this study, the researcher employed balanced score (PS) graph. The following output shows that the identified region of common support is  .After accounting for other variables that influence pregnancy and birth outcomes, mothers who stayed in the waiting homes before giving birth significantly improved the outcomes of birth. Mean good maternal unfavorable birth outcomes differed by 10.4% between users of Waiting Home and non-users (t\u2009=\u20093.78) nearest neighbor matching, at a significance level of 95% see .As shown below  mothers`The qualitative results were divided into 6 primary themes and were analyzed thematically as shown below;Theme 1: Regarding rooms' space insufficiency and improper preparation:The in-depth interviewees revealed that the rooms of maternity waiting homes are not properly prepared and insufficient.\u12ad\u134d\u1209 \u12a0\u122b\u1275 \u12a0\u120d\u130b\u12ce\u127d \u12eb\u1209\u1275 \u1232\u1206\u1295 \u12ad\u134d\u1209 \u1260\u1242 \u1235\u120b\u120d\u1206\u1290 \u12a5\u1293 \u12dd\u1245\u1270\u129b \u1260\u1218\u1206\u1291 \u1265\u12d9 \u1290\u134d\u1230 \u1321\u122d \u12a5\u1293\u1276\u127d \u1232\u1218\u1321 \u1218\u1328\u1293\u1290\u1245 \u12eb\u130b\u1325\u121b\u120d\u1362 \u1208\u1206\u1235\u1352\u1273\u120d \u12f0\u1295\u1260\u129e\u127b\u127d\u1295 \u12e8\u1270\u12d8\u130b\u1301\u1275\u1295 \u1266\u1273\u12ce\u127d \u12a5\u12e8\u1270\u1320\u1240\u121d\u1295 \u1290\u12cd\u1362 \u12a5\u1293\u1276\u127d \u12e8\u1270\u1208\u12e8 \u1218\u1273\u1320\u1262\u12eb \u1264\u1275 \u12c8\u12ed\u121d \u123b\u12c8\u122d \u1218\u1320\u1240\u121d \u12a0\u12ed\u127d\u1209\u121d\u1362 \u12a5\u1293\u1276\u127d \u12e8\u1270\u1208\u12e8 \u1218\u1273\u1320\u1262\u12eb \u1264\u1275 \u12c8\u12ed\u121d \u123b\u12c8\u122d \u1218\u1320\u1240\u121d \u12a0\u12ed\u127d\u1209\u121d\u1362 \u12a5\u1293\u1276\u127d \u1208\u1201\u1209\u121d \u1230\u12cd \u12e8\u1270\u12d8\u130b\u1301 \u12ad\u134d\u120e\u127d\u1295 \u12ed\u130b\u122b\u1209\u1362 \u12a5\u1293\u1276\u127d \u12a5\u12da\u12eb \u12a5\u12eb\u1209 \u121d\u127e\u1275 \u12a0\u12ed\u1230\u121b\u1278\u12cd\u121d\u1362 \u1260\u12da\u1205 \u121d\u12ad\u1295\u12eb\u1275 \u12a5\u1293\u1276\u127d \u12a5\u1235\u12aa\u12c8\u1208\u12f1 \u12f5\u1228\u1235 \u12a5\u12da\u1205 \u1218\u1246\u12e8\u1275 \u12a5\u1295\u12f0\u121b\u12ed\u1348\u120d\u1309 \u12a0\u121d\u1293\u1208\u1201\u1362\u201d.\u201cTheme 2: Regarding availability of adequate accommodations in the maternity waiting homes:Likewise, in-depth interviewees elaborated that in some institutions, even if the maternity waiting homes' rooms have enough beds and space but it is not well constructed and in addition the quality of accommodation services is poor..\u2018\u2018 \u12ad\u134d\u1209 \u1260\u1242 \u1266\u1273 \u12a5\u1293 \u12a0\u120d\u130b \u1262\u1296\u1228\u12cd\u121d \u1325\u122b\u1271 \u12e8\u1270\u1313\u12f0\u1208 \u12a5\u1293 \u1260\u12f0\u1295\u1265 \u12eb\u120d\u1270\u1230\u122b \u1290\u12cd\u1362 \u130d\u12f5\u130d\u12f3\u12cd \u12a5\u1293 \u1323\u122a\u12eb\u12cd \u1260\u1246\u122d\u1246\u122e\u12ce\u127d \u12e8\u1270\u1308\u1290\u1263 \u1235\u1208\u1206\u1290 \u12ad\u134d\u1209 \u121d\u1279 \u12e8\u1206\u1290 \u12e8\u1219\u1240\u1275 \u1218\u1320\u1295 \u12e8\u1208\u12cd\u121d\u1361\u1361 \u1260\u120c\u120b \u12a0\u1290\u130b\u1308\u122d \u1260\u1240\u1295 \u12cd\u1235\u1325 \u12ed\u121e\u1243\u120d\u1364 \u1260\u120c\u120a\u1275 \u12f0\u130d\u121e \u1260\u1323\u121d \u12ed\u1240\u12d8\u1245\u12db\u120d. \u1260\u1270\u1328\u121b\u122a\u121d \u12e8\u130d\u12f5\u130d\u12f3\u12cd \u12cd\u1235\u1320\u129b \u12ad\u134d\u120d \u1260 \u201c\u127a\u1351\u12f5\u201d \u12e8\u1270\u1238\u1348\u1290 \u1260\u1218\u1206\u1291 \u1275\u128b\u1295\u1293 \u1241\u1295\u132b \u12a0\u120d\u134e \u12a0\u120d\u134e \u12ed\u122b\u1263\u1260\u1273\u120d\u1361\u1361 \u12ed\u1205 \u1260\u12a5\u1293\u1276\u127d \u120b\u12ed \u1260\u121a\u1296\u1229\u1260\u1275 \u130a\u12dc \u1270\u1328\u121b\u122a \u1225\u1243\u12ed \u12eb\u1235\u12a8\u1275\u120b\u120d\u1361\u1361 Theme 3: Regarding access to water in the maternity waiting homes:In some health care facilities, it can be exceedingly difficult to get pregnant women access to water..\u201c\u1260\u1324\u1293 \u1323\u1262\u12eb\u127d\u1295 \u12cd\u1235\u1325 \u12a8\u121a\u12a8\u1230\u1271\u1275 \u12cb\u1293 \u12cb\u1293 \u1309\u12f3\u12ee\u127d \u12a0\u1295\u12f1 \u12e8\u12cd\u1203 \u12a5\u1325\u1228\u1275 \u1290\u12cd::\u1290\u134d\u1230 \u1321\u122d \u12a5\u1293\u1276\u127d \u1260\u1242 \u12cd\u1203 \u121b\u130d\u1298\u1275 \u1263\u1208\u1218\u127b\u120b\u1278\u12cd \u1295\u1345\u1205\u1293\u1278\u12cd\u1295 \u1208\u1218\u1320\u1260\u1245 \u12a0\u12f3\u130b\u127d \u1206\u1296\u1263\u1278\u12cb\u120d\u1362 \u1308\u1293 \u12e8\u12c8\u1208\u12f1 \u12a5\u1293\u1276\u127d \u12a5\u1295\u12b3\u1295 \u123b\u12c8\u122d \u12a0\u12ed\u12c8\u1235\u12f1\u121d\u1362 \u12a5\u1293\u1276\u127d \u1260\u12a5\u1290\u12da\u1205 \u12ad\u134d\u120e\u127d \u12cd\u1235\u1325 \u1218\u1246\u12e8\u1275 \u12a0\u12ed\u1348\u120d\u1309\u121d \u121d\u12ad\u1295\u12eb\u1271\u121d \u12ed\u1205 \u1260\u121a\u12eb\u1235\u12a8\u1275\u1208\u12cd \u12a8\u134d\u1270\u129b \u121d\u127e\u1275 \u121b\u1323\u1275 \u121d\u12ad\u1295\u12eb\u1275\u1362 \u1260\u12a5\u122d\u130d\u1325 \u12a8\u12a8\u1270\u121b\u12cd \u201c\u1304\u122a\u12ab\u1295\u201d \u1260\u1218\u12cd\u1230\u12f5 \u12cd\u1203 \u120b\u121b\u1245\u1228\u1265 \u12ed\u121e\u12a8\u122b\u120d, \u130d\u1295 \u1260\u1242 \u12a0\u12ed\u12f0\u1208\u121d::\u201d Theme 4: Regarding availability of entertainment options in the maternity waiting homes:As per the in-depth interviewees in most maternity waiting homes there are no entertainment options..\u12ad\u134d\u1209 \u12a5\u1295\u12f0 \u1274\u120c\u126a\u12e5\u1295 \u12c8\u12ed\u121d \u120c\u120b \u12e8\u1218\u12dd\u1293\u129b \u12a0\u121b\u122b\u132e\u127d \u12e8\u1209\u1275\u121d\u1362 \u1208\u1218\u1270\u129b\u1275, \u12a8\u1313\u12f0\u129e\u127b\u1278\u12cd \u130b\u122d \u1208\u1218\u12c8\u12eb\u12e8\u1275 \u12c8\u12ed\u121d \u1260\u1261\u1293 \u1225\u1290 \u1225\u122d\u12d3\u1275 \u120b\u12ed \u1208\u1218\u1308\u1298\u1275 \u1265\u127b \u1208\u121a\u121e\u12ad\u1229 \u12a5\u1293\u1276\u127d, \u12ed\u1205 \u127d\u130d\u122d \u12ed\u1348\u1325\u122b\u120d\u201d \u201cTheme 5: Regarding availability of affordable food in the maternity waiting homes:According to the in-depth interviewees providing a sufficient and balanced diet for pregnant women who are staying at maternal waiting homes up to delivery and for a few days after delivery is full of difficulties.\u1260\u1270\u1208\u12ed \u1260\u12a5\u122d\u130d\u12dd\u1293 \u12a5\u1293 \u1321\u1275 \u1260\u121b\u1325\u1263\u1275 \u130a\u12dc \u1260\u1242 \u12a5\u1293 \u12e8\u1270\u1218\u1323\u1320\u1290 \u121d\u130d\u1265 \u1218\u1218\u1308\u1265 \u12eb\u1208\u12cd\u1295 \u1320\u1240\u121c\u1273 \u1208\u1290\u134d\u1230 \u1321\u122d \u12a5\u1293\u1276\u127d \u12a5\u1293\u1235\u1270\u121d\u122b\u1208\u1295\u1362 \u1208\u121d\u1233\u1363 \u12a5\u122b\u1275 \u12a5\u1293 \u1208\u1241\u122d\u1235 \u1230\u12d3\u1275 \u201c\u123a\u122e\u12c8\u1325\u201d \u1260\u12a5\u1295\u1300\u122b \u1265\u127b \u12a5\u12e8\u1240\u1228\u1260\u120b\u1278\u12cd \u12ed\u1218\u1308\u1263\u1209\u1362\u12ed\u1205 \u1260\u1240\u1325\u1273 \u12e8\u1235\u1290 \u121d\u130d\u1265 \u1275\u121d\u1205\u122d\u1275 \u12a8\u121d\u1295\u1230\u1320\u12cd \u130b\u122d \u12ed\u1243\u1228\u1293\u120d \u121d\u1295\u121d \u12a5\u1295\u12b3\u1295 \u1260\u1242 \u1260\u1300\u1275 \u1262\u1296\u122d\u121d \u12e8\u1205\u1265\u1228\u1270\u1230\u1261 \u12e8\u1308\u1295\u12d8\u1265 \u12f5\u130b\u134d \u1262\u1296\u122d\u121d \u12e8\u1290\u134d\u1230 \u1321\u122d \u12a5\u1293\u1276\u127d\u1295 \u12a5\u1293 \u12a0\u12f2\u1235 \u12e8\u121a\u12c8\u1208\u12f1 \u1205\u133b\u1293\u1275\u1295 \u1205\u12ed\u12c8\u1275 \u121b\u12f3\u1295 \u12a0\u120d\u127b\u120d\u1295\u121d\u1362 \u201c \u201c\u12a5\u1293\u1276\u127d \u1263\u120e\u127b\u1278\u12cd\u1295 \u12c8\u12ed\u121d \u120c\u120e\u127d \u12f0\u130b\u134a \u12e8\u1206\u1291 \u12e8\u1264\u1270\u1230\u1265 \u12a0\u1263\u120b\u1275\u1295 \u12c8\u12f0 \u12a0\u122b\u1270\u129b\u12cd \u12e8\u1245\u12f5\u1218 \u12c8\u120a\u12f5 \u1240\u1320\u122e\u1278\u12cd \u12a5\u1295\u12f2\u12eb\u1218\u1321 \u12ed\u1218\u12a8\u122b\u1209 \u1235\u1208\u12da\u1205\u121d \u12a5\u1235\u12a8 \u12c8\u120a\u12f5 \u12f5\u1228\u1235 \u12a5\u12da\u12eb \u1260\u1218\u1246\u12e8\u1275 \u1235\u120b\u1208\u12cd \u1325\u1245\u121d \u1218\u12c8\u12eb\u12e8\u1275 \u12ed\u127d\u120b\u1209\u1362\u201d .However, they eventually wish to go back to their house when the mother's health improves. There is a chance that they will give birth at home, especially if they go against the advice of medical professionals and return home. This is because they think the doctors won't be delighted to see them again. Therefore, we will inform the health extension workers about mothers who go home again to avoid having a baby at home.Reducing maternal and perinatal mortality can be aided by maternity waiting homes . AccordiMothers of non-MWH users were more likely to experience labor obstruction (2.9% vs. 1 percent). There is no discernible difference between MWH users and non-users in the frequency of uterine rupture, nevertheless. Postpartum hemorrhage (PPH) was more common in mothers who did not stay at the maternity waiting home compared to mothers who did. Early infant death rates were 2.2% (22 per 1000) for mothers who did not stay in MWH against 0.7 percent (7 per 1000) for those who did. Additionally, mothers who did not remain in the MWH had a higher percentage of stillbirths than mothers who did .This is consistent with a study conducted in a systemic review in low-income countries and rural Ethiopia .The results of this study demonstrated that staying in maternity waiting homes significantly decreased the likelihood of poor maternal birth outcomes. This is in line with research done in rural Ethiopia , 7. ThisSimilar to how the results of this study indicated that using maternity waiting at home before delivery significantly reduced poor perinatal birth outcomes. Compared to mothers who did not stay in the maternity waiting home, mothers who did tended to experience 10% less unfavorable perinatal outcomes. This is consistent with research from a systemic review done in low- and middle-income nations . This coThe waiting room was insufficient and not prepared to the required standard, according to the qualitative findings. Mothers find it uncomfortable when it gets crowded when they arrive. Mothers were not provided with a separate bathroom or shower. They distributed food from rooms set aside for everyone in the institution. In keeping with a study done in rural Zambia, there was no television in the mothers' waiting area . It was Even in the waiting area, mothers desired frequent visits from their medical professionals. However, the participant's experiences revealed that until mothers complain, it's possible that no one will check on their health. This research is consistent with a study carried out in southern Loa .Maternity waiting homes had a significant positive contribution to improving maternal and perinatal birth outcomes. Even though maternal waiting homes have significant contributions to the health of mothers and their neonates, mothers faced several challenges during their stay."}
{"text": "Analysis and results, paragraph 1 - the phrase \u201c$2415 billion\u201d should read \u201c$2.415 billion\u201d. The phrase \u201c$50,671 billion\u201d should read \u201c$50.671 billion\u201d. The phrase \u201c$91,256 billion\u201d should read \u201c$91.256 billion\u201d. Discussion Section, 1st paragraph, second to last sentence states: \u201cConversely, when public funding takes its minimum sample value and private funding takes its maximum sample value our model predicts the probability of FDA approval to be 97.3%\u201d The value 97.3% should be 99.3%. Discussion section, 1st paragraph, 1st sentence begins: \u201cOut study\u2019s findings show no statistically significant relationship \u2026\u201d. \u201cOut\u201d should be \u201cOur\u201d. The dataset titles in Fig. 3 were reversed. The corrected Fig."}
{"text": "The aim of this research article is to derive a new relation between rough sets and soft sets with an algebraic structure quantale by using soft binary relations. The aftersets and foresets are utilized to define lower approximation and upper approximation of soft subsets of quantales. As a consequence of this new relation, different characterization of rough soft substructures of quantales is obtained. To emphasize and make a clear understanding, soft compatible and soft complete relations are focused, and these are interpreted by aftersets and foresets. Particularly, in our work, soft compatible and soft complete relations play an important role. Moreover, this concept generalizes the concept of rough soft substructures of other structures. Furthermore, the algebraic relations between the upper (lower) approximation of soft substructures of quantales and the upper (lower) approximation of their homomorphic images with the help of soft quantales homomorphism are examined. In comparison with the different type of approximations in different type of algebraic structures, it is concluded that this new study is much better. Quantale theory was proposed by Mulvey. It is bIn 1982, Pawlak developed the famous rough set theory , which iThere are many problems that arise in different fields such as engineering, economics, and social sciences in which data have some sort of uncertainty. Well-known mathematical tools have so many limitations because these tools are introduced for particular circumstances. There are many theories to overcome uncertainty such as fuzzy set theory, probability theory, rough sets, and vague sets, but these are limited due to its design.In 1998, Molodtsov present the idea of soft set theory, which is a mathematical tool to overcome the adversities affecting the above theories . Many auBy using aftersets and foresets notions associated with SBR, a new approximation space is widely utilized these days. By using generalized approximation space based on SBR, different soft substructures in semigroups were approximated by Kanawal and Shabir . MotivatQ- module [There are several authors who introduced rough sets theory in algebraic structures and soft algebraic structures. Iwinski analyzes algebraic properties of rough sets . Qurashi- module . Idea of- module . Rough p- module . Fuzzy i- module . General- module . In 27]Q- module- module , 29. Rou- module .The following scheme is designed for the rest of the paper. Some essential explanations related to quantales, its substructures, soft substructures, and their corresponding sequels are connected in q]\u03a8 denotes the equivalence class of the relation containing q. Any definable set in \u0398 would be written as finite union of equivalence classes of \u0398. Let R\u2286\u0398 in general R is not a definable set in \u0398. However, the set R can be approximated by two definable sets in \u0398. The first one is called \u03a8-lower approximation (\u03a8 \u2212 Lappr) of R, and the second is called \u03a8-upper approximation (\u03a8 \u2212 Uappr). They are defined as follows:Let \u0398 be a nonempty finite set called the universe set and \u03a8 be an E.R  over \u0398. Let [Lappr of R in \u0398 is the greatest definable in \u0398 contained in R. The \u03a8 \u2212 Uappr of R in \u0398 is the least definable set in \u0398 containing R. For any nonempty subset R in \u0398, P(\u0398) \u00d7 P(\u0398) if P(\u0398) denotes the set of all subsets of \u0398.The \u03a8 \u2212 l, \u2009w, \u2009li, \u2009wi \u2208 \u0398. Then,  is called quantale.Let \u0398 be a complete lattice. Define an associative binary relation \u2218 on \u0398 satisfyingT1, \u2009T2, \u2009TI \u2282 \u0398, i \u2208 I. We define some notions as follows:Let 1 and \u03982.Throughout the paper, quantales are denoted by \u0398W\u2286\u0398. Then, W is called a subquantale of \u0398 if the following holds:w1\u00b0w2 \u2208 W, \u2200w1, w2 \u2208 W.i\u2208Iwi, \u2208W, \u2200wi, \u2208W.\u2228Let \u2205\u2260That is, \u0398 closed under \u2218 and arbitrary supremum.I\u2286\u0398 is called left (right) ideal if the following satisfied:u, \u2009v \u2208 I implies u\u2228v \u2208 1p \u2208 \u0398, u \u2208 I such that p \u2264 u implies p \u2208 Iq \u2208 \u0398 and u \u2208 I implies q\u00b0u \u2208 I(u\u00b0q \u2208 I)Let \u0398 be a quantale, \u2205\u2260I\u2286\u0398 is called ideal of \u0398 if it is left as well as right ideal.A nonempty subset ={0, \u2009p, \u2009q, \u2009r, \u20091} complete lattices are shown in Let \u0398p}, {0, q}, {0, \u2009p,\u2009q, \u2009r}, and \u0398 are all I of quantale \u0398.Then, \u0398 is a quantale. Then, {0}, {0, I\u2286\u0398 be an ideal. I is called prime ideal if, \u2200u, \u2009v \u2208 \u0398, u\u00b0v \u2208 I\u21d2u \u2208 I or v \u2208 I. I is called semiprime I if, \u2200\u2009u \u2208 \u0398, u\u00b0u \u2208 I\u21d2u \u2208 II is called primary I if, \u2200u, \u2009v \u2208 \u0398, u\u00b0v \u2208 I and u \u2209 I implies vn \u2208 I for some n \u2208 \u039d.Let \u2205\u2260C) is called a soft set over \u0398 if \u03a8 : C\u27f6P(\u0398) where C is a subset of E (the set of parameters).A pair  and  be two soft sets over \u0398. Then,  soft subset  if the following conditions are fulfilled:C1\u2286C2F(c)\u2286H(c), \u2200\u2009c \u2208 C1Let (C) be a soft set over \u0398 \u00d7 \u0398, that is, \u03a8 : C\u27f6P(\u0398 \u00d7 \u0398). Then,  is called a soft binary relation (SBR) over \u0398 \u00d7 \u0398. A SBR over \u03981 \u00d7 \u03982 is a soft set  over \u03981 \u00d7 \u03982. That is, \u03a8 : C\u27f6P(\u03981 \u00d7 \u03982).Let  be a soft set over quantale \u0398. Then,C) is called soft subquantale over \u0398 iff \u03a8(c) is a subquantale of \u0398, \u2200\u2009c \u2208 C is called soft ideal over \u0398 iff \u03a8(c) is an ideal of \u0398, \u2200\u2009c \u2208 C is called soft prime ideal over \u0398 iff \u03a8(c) is a prime ideal of \u0398, \u2200\u2009c \u2208 C is called soft semiprime ideal over \u0398 iff \u03a8(c) is a semiprime ideal of \u0398, \u2200\u2009c \u2208 C is called soft primary ideal over \u0398 iff \u03a8(c) is a primary ideal of \u0398, \u2200\u2009c \u2208 C be a SBR over \u03981 \u00d7 \u03982, where C\u2286E (parametric set). Then, \u03a8 : C\u27f6P(\u03981 \u00d7 \u03982). For a soft set  over \u03982, the LapprUapprF, C) w.r.t the afterset are essentially two soft sets over \u03981, which is defined asLet  over \u03981, the LapprUapprH, C) w.r.t the foreset are actually two soft sets over \u03982, which is defined asAnd for a soft set ={q2 \u2208 \u03982 :  \u2208 \u03a8(c)} is called the afterset of q1 and \u03a8(c)q2={q1 \u2208 \u03981 :  \u2208 \u03a8(c)} is called the foreset of q2.For all c \u2208 F, C) over \u03982, For each soft set  over \u03981, For each soft set (C) be a SBR over \u03981 \u00d7 \u03982, that is, \u03a8 : C\u27f6P(\u03981 \u00d7 \u03982). Then,  is called soft compatible relation (SCPR) if for all p, \u2009r, \u2009ji \u2208 \u03981 and q, \u2009s, \u2009ki \u2208 \u03982(i \u2208 I), we havep, q),  \u2208 \u03a8(c)\u21d2\u2208\u03a8(c) \u2208 \u03a8(c)\u21d2\u2208\u03a8(c) over \u03981 \u00d7 \u03982 is called soft complete relation (SCTR) with respect to the afterset if, for all p, \u2009r, \u2009 \u2208 \u03981, we havep\u03a8(c)\u2228r\u03a8(c)=(p\u2228r)\u03a8(c)p\u03a8(c)\u22182r\u03a8(c)=(p\u22181r)\u03a8(c)for all c \u2208 C.A SCPR  is called \u2228-complete w.r.t the aftersets if it satisfies only condition (1). A SCPR  is called \u00b0-complete w.r.t the aftersets if it satisfies only condition (2).A SCPR  over \u03981 \u00d7 \u03982 is called soft complete relation (SCTR) with respect to the foreset if for all q, \u2009s \u2208 \u03982, and we havec)q\u2228\u039b(c)s=\u039b(c)(q\u2228s)\u039b(c)q\u22181\u039b(c)s=\u039b(c)(q\u22182s)\u039b is called \u2228-complete w.r.t the foresets if it satisfies only condition (1).A SCPR  is called \u00b0-complete w.r.t the foresets if it satisfies only condition (2).A SCPR  be a SCPR with respect to the afterset over \u03981 \u00d7 \u03982. Then, for any two soft sets  and  over \u03982, we haveLet \u2229F1(c) \u2260 \u2205 and y2\u03a8(c)\u2229F2(c) \u2260 \u2205, so there exist elements l, \u2009m \u2208 \u03982 such that l \u2208 y1\u03a8(c)\u2229F1(c) and m \u2208 y2\u03a8(c)\u2229F2(c). Thus, l \u2208 y1\u03a8(c), \u2009m \u2208 y2\u03a8(c), \u2009l \u2208 F1(c) and m \u2208 F2(c). So  \u2208 \u03a8(c) and  \u2208 \u03a8(c) imply  \u2208 \u03a8(c); that is, (l\u22182m) \u2208 (y1\u22181y2)\u03a8(c). Also, l\u22182m\u2208F1(c)\u22182F2(c); therefore, l\u22182m \u2208 y1\u22181y2\u03a8(c)\u2229F1(c)\u22182F2(c). This shows that For arbitrary c \u2208 C, let x=y1\u2228y2 for some y1\u03a8(c)\u2229F1(c) \u2260 \u2205 and y2\u03a8(c)\u2229F2(c) \u2260 \u2205, so there exist elements l, \u2009m \u2208 \u03982 such that l \u2208 y1\u03a8(c)\u2229F1(c) and m \u2208 y2\u03a8(c)\u2229F2(c). Thus, l \u2208 y1\u03a8(c), \u2009m \u2208 y2\u03a8(c), \u2009l \u2208 F1(c), and m \u2208 F2(c). So  \u2208 \u03a8(c) and  \u2208 \u03a8(c) imply  \u2208 \u03a8(c); that is, (l\u2228m) \u2208 (y1\u2228y2)\u03a8(c). Also, l\u2228m \u2208 F1(c)\u2228F2(c); therefore, l\u2228m \u2208 y1\u2228y2\u03a8(c)\u2229F1(c)\u2228F2(c). This shows that Now, for arbitrary C) be a SCPR with respect to the foreset over \u03981 \u00d7 \u03982. Then, for any two soft sets  and  over \u03981, we haveLet  be a SCTR w.r.t the afterset over \u03981 \u00d7 \u03982. Then, for any two soft sets  and  over \u03982, we haveLet  and \u03a8F2(c) is empty, then (1) is obvious. Now, for arbitrary c \u2208 C, consider that \u03a8F1(c) \u2260 \u2205 and \u03a8F2(c) \u2260 \u2205. Then, \u03a8F1(c)\u00b01\u03a8F2(c) \u2260 \u2205. So, let x \u2208 \u03a8F1(c)\u00b01\u03a8F2(c). Then, x=y1\u00b01y2 for some y1 \u2208 \u03a8F1(c) and y2 \u2208 \u03a8F2(c). This implies that \u2205\u2260y1\u03a8(c)\u2286F1(c) and \u2205\u2260y2\u03a8(c)\u2286F2(c). As (y1\u00b01y2)\u03a8(c)=y1\u03a8(c)\u00b02\u03a8(c)\u2286F1(c)\u00b02F2(c). This shows that x=y1\u00b01y2 \u2208 \u03a8F1F2(c). Hence, (1) is proved.For arbitrary c \u2208 C, if at least one of \u03a8F1(c) and \u03a8F2(c) is empty, then (2) is obvious. Now, for arbitrary c \u2208 C, consider that \u03a8F1(c) \u2260 \u2205 and \u03a8F2(c) \u2260 \u2205. Then, \u03a8F1(c)\u2228\u03a8F2(c) \u2260 \u2205. So, let x \u2208 \u03a8F1(c)\u2228\u03a8F2(c). Then, x=y1\u2228y2 for some y1 \u2208 \u03a8F1(c) and y2 \u2208 \u03a8F2(c). This implies that \u2205\u2260y1\u03a8(c)\u2286F1(c) and \u2205\u2260y2\u03a8(c)\u2286F2(c). As (y1\u2228y2)\u03a8(c)=y1\u03a8(c)\u2228y2\u03a8(c)\u2286F1(c)\u2228F2(c). This shows that x=y1\u2228y2 \u2208 \u03a8FF21\u2228(c). Hence, (2) is proved.For arbitrary C) be a SCTR with respect to the foreset over \u03981 \u00d7 \u03982. Then, for any two soft sets  and  over \u03981, we haveLet  to be the SBR over \u03981 \u00d7 \u03982 and x\u03a8(c) \u2260 \u2205 for all x \u2208 \u03981, c \u2208 C, and \u03a8(c)y \u2260 \u2205 for all y \u2208 \u03982, c \u2208 C unless otherwise specified.Throughout this section, we consider  be a SBR over \u03981 \u00d7 \u03982 and  be a soft set over \u03982. If Uappr. 1, then  is called generalized upper soft (GUpS) subquantale of \u03981 w.r.t the aftersets. If Uappr1, then  is called GUpS ideal  of \u03981 w.r.t the aftersets.Let  be a SBR over \u03981 \u00d7 \u03982 and  be a soft set over \u03981. If Uappr2, then  is called generalized upper soft (GUpS) subquantale of \u03982 w.r.t the foresets. If Uappr2, then  is called GUpS ideal  of \u03982 w.r.t the foresets.Let  be a SCPR over \u03981 \u00d7 \u03982. If  is a soft subquantale of \u03982, then  is a GUpS subquantale of \u03981 w.r.t the aftersets.Let  is a soft subquantale, then c \u2208 C. Let i \u2208 I. Then, pi\u03a8(c)\u2229F(c) \u2260 \u2205. So, there exists qi \u2208 pi\u03a8(c)\u2229F(c). Thus, qi \u2208 pi\u03a8(c) and qi \u2208 F(c) since  is a SCPR. Therefore,  \u2208 \u03a8(c), i \u2208 I implies  \u2208 \u03a8(c). This implies that \u2228i\u2208Iqi \u2208 \u2228i\u2208Ipi\u03a8(c). Also, \u2228i\u2208Iqi \u2208 F(c)  is a soft subquantale). So, \u2228i\u2208Iqi \u2208 \u2228i\u2208Ipi \u2208 \u03a8(c)\u2229F(c). Hence, Suppose that (p1\u03a8(c)\u2229F(c) \u2260 \u2205 and q1 \u2208 p1\u03a8(c)\u2229F(c) and q2 \u2208 p2\u03a8(c)\u2229F(c). Thus, q1 \u2208 p1\u03a8(c), q1 \u2208 F(c), q2 \u2208 p2\u03a8(c), and q2 \u2208 F(c) since  is a SCPR. Therefore, ,  \u2208 \u03a8(c) implies (p1\u00b01q1), (p2\u00b02q2) \u2208 \u03a8(c). This implies that q1\u00b02q2 \u2208 p1\u00b0p2\u03a8(c). Also, q1\u00b02q2 \u2208 (c)  is a soft subquantale). So, q1\u00b02q2 \u2208 p1\u00b0p2\u03a8(c)\u2229F(c). Hence, Let With the same arguments, the next C) be a SCPR over \u03981 \u00d7 \u03982. If  is a soft subquantale of \u03981, then  is a GUpS subquantale of \u03982 w.r.t the foresets.Let  be a soft \u2228-complete relation over \u03981 \u00d7 \u03982 w.r.t the aftersets. If  is a soft left (right) ideal of \u03982, then  is a GUpS left (right) ideal of \u03981 w.r.t the aftersets.Let  is a soft left ideal of \u03982, then c \u2208 C. Let u1\u03a8(c)\u2229F(c) \u2260 \u2205 and u2\u03a8(c)\u2229F(c) \u2260 \u2205. So, there exists v1 \u2208 u1\u03a8(c)\u2229F(c) and v2 \u2208 u2\u03a8(c)\u2229F(c). Thus, v1 \u2208 u1\u03a8(c), v1 \u2208 F(c), v2 \u2208 u2\u03a8(c), and v2 \u2208 F(c) since  is a SCPR. Therefore,  \u2208 \u03a8(c); that is, v1\u2228v2 \u2208 (u1\u2228u2)\u03a8(c). Also, v1\u2228v2 \u2208 F(c)  is a soft left ideal). So, v1\u2228v2 \u2208 (u1\u2228u2)\u03a8(c)\u2229F(c). Hence, Suppose that \u2229F(c). Thus, v2 \u2208 u2\u03a8(c) and v2 \u2208 F(c). Since  is a soft \u2228-complete relation, therefore, v2 \u2208 u2\u03a8(c)=u1\u2228u2\u03a8(c)=u1\u03a8(c)\u2228u2\u03a8(c). This implies that v2=s\u2228t, for some s \u2208 u1\u03a8(c) and t \u2208 u2\u03a8(c). Thus, s \u2264 v2 and v2 \u2208 F(c) imply s \u2208 F(c) (as F(c) is ideal). So, s \u2208 u1\u03a8(c)\u2229F(c). Hence, Now, let p, \u2009x \u2208 \u03981 and x\u03a8(c)\u2229F(c) \u2260 \u2205. So, there exist q \u2208 x\u03a8(c)\u2229F(c). Thus, q \u2208 x\u03a8(c) and q \u2208 F(c). Since  is a soft left ideal so, y\u22182q \u2208 F(c) for any y \u2208 p\u03a8(c)\u2286\u03982. This implies that  \u2208 \u03a8(c). So,  \u2208 \u03a8(c); that is, y\u22182q \u2208 p\u22181x\u03a8(c). So, y\u22182q \u2208 p\u22181x\u03a8(c)\u2229F(c). Hence, Let C) be a SCTR over \u03981 \u00d7 \u03982 w.r.t the aftersets. If  is a soft prime ideal of \u03982, then  is a GUpS prime ideal of \u03981 w.r.t the aftersets.Let  is a soft prime ideal of \u03982, then c \u2208 C. Then, by F, C) is generalized upper soft ideal of \u03981. Let p1, \u2009p2 \u2208 \u03981 such that p1\u22181p2)\u03a8(c)\u2229F(c) \u2260 \u2205. So, there exist q \u2208 (p1\u22181p2)\u03a8(c)\u2229F(c). This implies that q \u2208 (p1\u22181p2)\u03a8(c) and q \u2208 F(c). Since  is a SCTR, q \u2208 (p1\u22181p2)\u03a8(c)=p1\u03a8(c)\u22182p2\u03a8(c). Thus, q=c\u22182d for some c \u2208 p1\u03a8(c) and d \u2208 p2\u03a8(c). Thus, c\u22182d \u2208 F(c) and  is a soft prime ideal of \u03982 so, c \u2208 F(c) or d \u2208 F(c). Thus, c \u2208 p1\u03a8(c)\u2229F(c) or d \u2208 p2\u03a8(c)\u2229F(c). Hence, Assume that (C) be a SCTR over \u03981 \u00d7 \u03982 w.r.t the aftersets. If  is a soft semiprime ideal of \u03982, then  is a GUpS semiprime ideal of \u03981 w.r.t the aftersets.Let  is a soft semiprime ideal of \u03982, then c \u2208 C. Then, by F, C) is generalized upper soft ideal of \u03981. Let p1 \u2208 \u03981 such that p1\u22181p1)\u03a8(c)\u2229F(c) \u2260 \u2205. So, there exist q \u2208 (p1\u22181p1)\u03a8(c)\u2229F(c). This implies that q \u2208 (p1\u22181p1)\u03a8(c) and q \u2208 F(c). Since  is a SCTR, q \u2208 (p1\u22181p1)\u03a8(c)=p1\u03a8(c)\u22182\u2009p1\u03a8(c). Thus, q=c\u2009\u22182\u2009c for some c \u2208 p1\u03a8(c). Thus, c\u22182c \u2208 F(c) and  is a soft semiprime ideal of \u03982 so, c \u2208 F(c). Thus, c \u2208 p1\u03a8(c)\u2229F(c). Hence, Assume that (C) be a SCTR over \u03981 \u00d7 \u03982 w.r.t the aftersets. If  is a soft primary ideal of \u03982, then  is a GUpS primary ideal of \u03981 w.r.t the aftersets.Let  is a soft primary ideal of \u03982, then C. Then, by F, C) is generalized upper soft ideal of \u03981. Let p1, \u2009p2 \u2208 \u03981 such that p1\u22181p2\u2208p1\u22181p2)\u03a8(c)\u2229F(c) \u2260 \u2205. So, there exist q \u2208 (p1\u22181p2)\u03a8(c)\u2229F(c). This implies that q \u2208 (p1\u22181p2)\u03a8(c) and q \u2208 F(c). Since  is a SCTR, q \u2208 (p1\u22181p2)\u03a8(c)=p1\u03a8(c)\u22182p2\u03a8(c). Thus, q=c\u22182d for some c \u2208 p1\u03a8(c) and d \u2208 p2\u03a8(c). Thus, c\u22182d \u2208 F(c) and  is a soft primary ideal of \u03982 so dn \u2208 F(c) for some n \u2208 \u2115. Also, dn \u2208 p2n\u03a8(c) for n \u2208 \u2115. Thus, dn \u2208 p2n\u03a8(c)\u2229F(c). Hence, Assume that C={c1, c2} and define SBR  over \u03981 \u00d7 \u03982 by the ruleLet C) is SCPR. The aftersets with respect to \u03a8(c1) and \u03a8(c2) are given as follows:Then,  over \u03982 by the ruleDefine soft set  is not a soft subquantale of \u03982. But 1. So  is a GUpSS\u0398 of \u03981 w.r.t the aftersets.Then, (c1) and \u03a8(c2) are given as follows:Foresets with respect to \u03a8 over \u03981 by the ruleDefine soft set  is not a soft subquantale of \u03981. But 2. So,  is a GUpS subquantale of \u03982 w.r.t the foresets.Then, ((2)C={c1, c2} and define SBR  over \u03981 \u00d7 \u03982 by the ruleNow, let We define \u2218c1) and \u03a8(c2) are given as follows:Aftersets with respect to \u03a8(C) is \u2228-complete relation over \u03981 \u00d7 \u03982 w.r.t the aftersets. Define soft set  over \u03982 by the ruleThen,  is not a soft ideal of \u03982. But 1. So,  is a GUpS ideal of \u03981 w.r.t the aftersets.Then, (C) over \u03981 \u00d7 \u03982 by the ruleNow, define SBR  and \u03a8(c2) are given as follows:Foresets with respect to \u03a8(C) is soft \u2228-complete relation over \u03981 \u00d7 \u03982 w.r.t the foresets. Define soft set  over \u03981 by the ruleThen,  is not a soft ideal of \u03981. But 2. So,  is a GUpS ideal of \u03982 w.r.t the foresets.Then, (Similar examples can be presented to justify that converse of Theorems 11C) be a SBR over \u03981 \u00d7 \u03982. Consider the soft set  over \u03982, if Lappr1, then  is called generalized lower soft (GLWS) subquantale of \u03981 w.r.t the aftersets. If Lappr1, then  is called GUpS ideal  of \u03981 w.r.t the aftersets.Let  be a SBR over \u03981 \u00d7 \u03982. Consider the soft set  over \u03981, if Lappr2, then  is called GLWS subquantale of \u03982 w.r.t the foresets. If Lappr2, then  is called GUpS ideal  of \u03982 w.r.t the foresets.Let  be a SCTR over \u03981 \u00d7 \u03982 w.r.t the aftersets. If  is a soft subquantale of \u03982, then  is a GLWS subquantale of \u03981 w.r.t the aftersets.Let  is a soft subquantale of \u03982 and C. Let ui\u03a8(c)\u2286M(c). Since  is a SCTR, therefore, \u2228i\u2208I(ui\u03a8(c))=(\u2228i\u2208Iui)\u03a8(c)\u2286M(c). Hence, Suppose that (u1\u03a8(c)\u2286M(c) and u2\u03a8(c)\u2286M(c). Since  is a SCTR and  is a soft subquantale, therefore, u1\u03a8(c)\u22182u2\u03a8(c)\u2286M(c)\u22182M(c) implies (u1\u22181u2)\u03a8(c)\u2286M(c). Hence, Now, let With the same arguments, next C) be a SCTR over \u03981 \u00d7 \u03982 w.r.t the foresets. If  is a soft subquantale of \u03981, then  is a GLWS subquantale of \u03982 w.r.t the foresets.Let  be a SCTR over \u03981 \u00d7 \u03982 w.r.t the aftersets. If  is a soft ideal of \u03982, then  is a GLWS ideal of \u03981 w.r.t the aftersets.Let  is a soft ideal of \u03982 and C. Let u1\u03a8(c)\u2286M(c) and u2\u03a8(c)\u2286M(c). Since  is a SCTR and  is a soft ideal of \u03982 so u1\u03a8(c)\u2228u2\u03a8(c)=(u1\u2228u2)\u03a8(c)\u2286M(c)\u2228M(c); that is, (u1\u2228u2)\u03a8(c)\u2286M(c) Hence, Suppose that  and v2 \u2208 u2\u03a8(c)\u2286M(c). So, v1\u2228v2 \u2208 (u1\u2228u2)\u03a8(c), that is, v1\u2228v2 \u2208 u2\u03a8(c)\u2286M(c). Since M(c) is ideal so v1 \u2264 v1\u2228v2 \u2208 M(c) implies v1 \u2208 M(c). Thus, u1\u03a8(c)\u2286M(c). Hence, Now, let u, \u2009y \u2208 \u03981 and y\u03a8(c)\u2286M(c). Consider v1\u2208(u\u22181y)\u03a8(c) since  is a SCTR so v1 \u2208 u\u2009\u03a8(c)\u22182y\u03a8(c). Thus, v1=c\u22182d for some c \u2208 u\u03a8(c) and d \u2208 y\u03a8(c). But y\u03a8(c)\u2286M(c) so d \u2208 M(c) and  is a soft ideal of \u03982; therefore, c\u22182d \u2208 M(c), that is, v1 \u2208 M(c). Thus, (u\u22181y)\u03a8(c)\u2286M(c). Hence, Now, let C) be a SCTR over \u03981 \u00d7 \u03982 w.r.t the aftersets. If  is a soft prime ideal of \u03982, then  is a GLWS prime ideal of \u03981 w.r.t the aftersets.Let  is a soft prime ideal of \u03982 and C. Then, by Theorem 4.19,  is GLWS ideal of \u03981. Let u1, \u2009u2 \u2208 \u03981 such that u1\u22181u2)\u03a8(c)\u2286M(c). Consider v \u2208 (u1\u22181u2)\u03a8(c)\u2286M(c). Since  is a SCTR, v \u2208 (u1\u22181u2)\u03a8(c)=u1\u03a8(c)\u22182u2\u03a8(c). Thus, v=c\u22182d for some c \u2208 u1\u03a8(c) and d \u2208 u2\u03a8(c). This implies that v=c\u22182d \u2208 M(c). As  is a soft prime ideal so, c \u2208 M(c) or d \u2208 M(c). Thus, c \u2208 u1\u03a8(c)\u2286M(c) or d \u2208 u2\u03a8(c)\u2286M(c). Hence, Assume that (C) be a SCTR over \u03981 \u00d7 \u03982 w.r.t the aftersets. If  is a soft semiprime ideal of \u03982, then  is a GLWS semiprime ideal of \u03981 w.r.t the aftersets.Let  is a soft semiprime ideal of \u03982 and C. Then, by M, C) is GLWS ideal of \u03981. Let u \u2208 \u03981 such that u\u22181u)\u03a8(c)\u2286M(c). Let v \u2208 u\u03a8(c). As  is a SCTR so v\u22182v \u2208 (u\u22181u)\u03a8(c)\u2286M(c). Since  is a soft semiprime ideal, v\u22182v \u2208 M(c) implies v \u2208 M(c). Thus, u\u03a8(c)\u2286M(c). Hence, Assume that (C) be a SCTR over \u03981 \u00d7 \u03982 w.r.t the aftersets. If  is a soft primary ideal of \u03982, then  is a GLWS primary ideal of \u03981 w.r.t the aftersets.Let  is a soft primary ideal of \u03982 and C. Then, by Theorem 4.19.,  is a GLWS ideal of \u03981. Let u1, \u2009u2 \u2208 \u03981 such that u1\u22181u2)\u03a8(c)\u2286M(c). Let v \u2208 (u1\u22181u2)\u03a8(c). Since  is a SCTR, v \u2208 u1\u03a8(c)\u22182u2\u03a8(c). Thus, v=c\u22182d for some c \u2208 u1\u03a8(c) and d \u2208 u2\u03a8(c). Thus, dn \u2208 u2n\u03a8(c) for some n \u2208 \u2115. Also, c\u22182d \u2208 M(c). As  is a soft primary ideal, c \u2209 M(c) and dn \u2208 M(c). Thus, u2n\u03a8(c)\u2286M(c). Hence, n \u2208 \u2115.Suppose that , and then, we established the relationship between homomorphic images and their approximation by SBR.\u03b7 : \u03981\u27f6\u03982 is called weak quantale homomorphism (WQH) if \u03b7(p\u22181q)=\u03b7(p)\u22182\u03b7(q) and \u03b7(p\u2228q)=\u03b7(p)\u2228\u03b7(q), where  and  are quantales. If \u03b7 is one-one, then \u03b7 is monomorphism. If \u03b7 is onto, then \u03b7 is called epimorphism, and if \u03b7 is bijective, then \u03b7 is called isomorphism between  and .A function H, C1) be a soft quantale over \u03981 and  be a soft quantale over \u03982. Then,  is said to soft weak homomorphic to  if there exist ordered pair of functions  satisfies the following\u03b7 : \u03981\u27f6\u03982 is onto WQH, that is, \u03b7(p\u22181q)=\u03b7(p)\u22182\u03b7(q) and \u03b7(p\u2228q)=\u03b7(p)\u2228\u03b7(q)\u03b6 : C1\u27f6C2 is surjective\u03b7(H(c1))=F(\u03b6(c1)), \u2200c1 \u2208 C1Let  of functions is SWQH. If in ordered pair  both \u03b7 and \u03b6 are one-to-one functions, then  is said to soft weak isomorphic to  and  is called SWQI.The ordered pair  be soft weak homomorphic to  with SWQH . Let  be a SBR over \u03982 and \u2286. Define \u03a81(c3)={}\u2208\u03981 \u00d7 \u03981:(\u03b7(x), \u03b7(y)) \u2208 \u03a82(c3){} be a SBR over \u03981. Then, the following holds:1, C3) is SCPR if  is SCPR is SWQI and  is SCPR w.r.t the aftersets (w.r.t the foresets), then  is SCPR w.r.t the aftersets (w.r.t the foresets)If  is SWQI, then \u03b7, \u03b6) be a SWQI. Then, Let (Let (and (2) are obviousH1, C1\u2032)\u2286 and for any c3 \u2208 C3, z \u2208 \u03982. Then, there exist a \u2208 \u03981 such that \u03b7(a)=z. Thus, x \u2208 a\u03a81(c3)\u2229H1(c1\u2032). So,  \u2208 \u03a81(c3) and x \u2208 H1(c1\u2032). Thus, (\u03b7(a), \u03b7(x)) \u2208 \u03a82(c3), that is, \u03b7(x) \u2208 \u03b7(a)\u03a82(c3). Also, \u03b7(x) \u2208 \u03b7(H1(c1\u2032)). So, \u03b7(a)\u03a82(c3)\u2229\u03b7(H1(c1\u2032))\u2260\u2205. This implies that Suppose (w\u03a82(c3)\u2229\u03b7(H1(c1\u2032)) \u2260 \u2205. This implies that y \u2208 w\u03a82(c3)\u2229\u03b7(H1(c1\u2032)). Thus, y \u2208 w\u03a82(c3) and y \u2208 \u03b7(H1(c1\u2032)). This implies that there exists x \u2208 H1(c1\u2032)\u2286\u03981 and x1 \u2208 \u03981 such that \u03b7(x)=y and \u03b7(x1)=w. So, =(\u03b7(x1), \u03b7(x))\u2208\u03a82(c3). This implies that  \u2208 \u03a81(c3). So, x \u2208 x1\u03a81(c3)\u2229H1(c1\u2032). Thus, Now, let H1, C1\u2032)\u2286 and for any c3 \u2208 C3, z \u2208 \u03982. Then, there exist a \u2208 \u03981 such that \u03b7(a)=z. Thus, a\u03a81(c3)\u2286H1(c1\u2032). Let x \u2208 z\u03a82(c3). Then, there exist y \u2208 \u03981 such that \u03b7(y)=x. So, \u03b7(y) \u2208 \u03b7(a)\u03a82(c3), that is, (\u03b7(a), \u2009\u03b7(y)) \u2208 \u03a82(c3). So,  \u2208 \u03a81(c3), that is, y \u2208 a\u03a81(c3)\u2286H1(c1\u2032). Thus, \u03b7(y)\u2208\u03b7(H1(c1\u2032)). So, \u03b7(a)\u03a82(c3)\u2286\u03b7(H1(c1\u2032)). Thus, Suppose (a \u2208 \u03981 such that \u03b7(a)=z and \u03b7(a)\u03a82(c3)\u2286\u03b7(H1(c1\u2032)). Let x \u2208 a\u03a81(c3), that is,  \u2208 \u03a81(c3). Then, (\u03b7(a), \u03b7(x)) \u2208 \u03a82(c3). Then, \u03b7(x) \u2208 \u03b7(a)\u03a82(c3)\u2286\u03b7(H1(c1\u2032)). So, \u03b7(x) \u2208 \u03b7(H1(c1\u2032)). This implies that x \u2208 H1(c1\u2032). So, a\u03a81(c3)\u2286H1(c1\u2032). Then, Now, let c3 \u2208 C3. Then, \u03b7 is bijection so Let With a similar technique, H, C1) be soft weak isomorphic to  with SWQI . Let  be a SCPR over \u03982 and \u2286. Define \u03a81(c3)={}\u2208\u03981 \u00d7 \u03981:(\u03b7(x), \u03b7(y))\u2208\u03a82(c3){} for any c3 \u2208 C3. Then, the following holds:1 iff 2 for all c3 \u2208 C31 iff 2 for all c3 \u2208 C31 iff 2 for all c3 \u2208 C31 iff 2 for all c3 \u2208 C31 iff 2 for all c3 \u2208 C3Let (1 for any c3 \u2208 C3. We will show that 2. By Let \u03b7(u)=p and \u03b7(v)=q. Since \u03b7, \u03b6) is SWQI so p\u2228q=\u03b7(u)\u2228\u03b7(v)=\u03b7(u\u2228v)\u2208Let p, q \u2208 \u03982 such that p \u2264 q and u \u2208 \u03981 and \u03b7(u)=p and \u03b7(v)=q. So, \u03b7(u) \u2264 \u03b7(v) implies \u03b7(u\u2228v)=\u03b7(u)u \u2264 v and Now, let p \u2208 \u03982 and u \u2208 \u03981 and \u03b7(u)=p and \u03b7(v)=q. Since \u03b7(u\u22181v)2.Finally, let 2 for any c3 \u2208 C3. We will show that 1.Conversely, suppose that \u03b7(u), \u03b7(v)\u2208Let u, v \u2208 \u03981 such that u \u2264 v and \u03b7(u\u2228v)=\u03b7(u)\u2228\u03b7(v)=\u03b7(v)\u2208\u03b7(u) \u2264 \u03b7(v). Since u \u2208 \u03981 and \u03b7(u) \u2208 \u03982 and u\u22181v\u2208Now, let The proof of (2)\u2013(5) is similar to the proof of (1).With the same arguments, the next H, C1) be soft weak isomorphic to  with SWQI . Let  be a SCTR over \u03982 and \u2286. Define \u03a81(c3)={}\u2208\u03981 \u00d7 \u03981:\u03b7(x), \u03b7y\u2208\u03a82(c3){} for any c3 \u2208 C3. Then, the following holds:1 iff 2 for all c3 \u2208 C31 iff 2 for all c3 \u2208 C31 iff 2 for all c3 \u2208 C31 iff 2 for all c3 \u2208 C31 iff 2 for all c3 \u2208 C3Let (Yang and Xu  introduc1 \u00d7 \u03982 is employed. This new relation over \u03981 \u00d7 \u03982 enables us to use the concept of aftersets and foresets to express the lower and upper approximation. Important results regarding to the approximation of soft substructures of quantales under SBR with some essential algebraic conditions such as compatible and complete relations are introduced. To emphasize and make a clear understanding, soft compatible and soft complete relations are focused, and these are interpreted by aftersets and foresets. Particularly, in our work, soft compatible and soft complete relations play an important role. Crux of these results is that whenever we approximate a soft algebraic structure of quantale, corresponding upper and lower approximations, are again the same kind of soft algebraic structure. Furthermore, we presented the soft quantale homomorphism and established the relationship of soft homomorphic images with their approximation under SBR.The new combined effect of an algebraic structure quantale with rough and soft sets is presented by using soft binary relation, in this paper. The soft substructures of quantales like soft subquantale and soft ideal are discussed. The approximation w.r.t aftersets and foresets of these substructures by SBR which is an extended notion of Pawlak's rough approximation space are presented. The more generalized version of approximation space implied from SBR over \u0398In future, one can use this work and generalize it to different soft algebraic structures such as soft quantale modules, soft hypergroups, soft hyperquantales, and soft hyperrings. One can take motivation from our generalized approximation space and define new approximation spaces."}
{"text": "We study steady axisymmetric water waves with general vorticity and swirl, subject to the influence of surface tension. This can be formulated as an elliptic free boundary problem in terms of Stokes' stream function. A change of variables allows us to overcome the generic coordinate\u2010induced singularities and to cast the problem in the form \u201cidentity plus compact,\u201d which is amenable to Rabinowitz's global bifurcation theorem, whereas no restrictions regarding the absence of stagnation points in the flow have to be made. Within the scope of this new formulation, local curves and global families of solutions, bifurcating from laminar flows with a flat surface, are\u00a0constructed. In the irrotational and swirl\u2010free setting, such waves were studied numerically by Vanden\u2010Broeck et\u00a0al.In this paper, we study axisymmetric water waves with surface tension, modeled by assuming that the domain is bounded by a free surface on which capillary forces are acting, and that in cylindrical coordinates z\u2010direction and ignore viscosity and gravity. In the irrotational swirl\u2010free case, the problem can be formulated in terms of a harmonic velocity potential. In contrast, we formulate the problem in terms of Stokes' stream function, which satisfies a second\u2010order semilinear elliptic equation\u00a0known alternatively in the literature as the Hicks equation, the Bragg\u2013Hawthorne equation, or the Squire\u2013Long equation  and p=p(x\u20d7) denote the velocity and the pressure, respectively, and \u03a9 is the fluid domain. In cylindrical coordinates , that is, x=rcos\u03d1, y=rsin\u03d1, and z=z, the velocity field u\u20d7 is expressed asu\u03d1\u22600. From the incompressibility and the axisymmetry of the flow, it follows that we can introduce Stokes' stream function \u03a8, such thatru\u03d1 is constant along streamlines, which we express as ru\u03d1=F(\u03a8) where F is an arbitrary function. The steady Euler equations\u00a0are then equivalent to the Bragg\u2013Hawthorne equationWe consider periodic axisymmetric capillary waves traveling at constant speed along the \u03a9={\u2208R2:0<r<d+\u03b7(z)} and its boundaries by \u2202\u03a9S={\u2208R2:r=d+\u03b7(z)} (free surface) and \u2202\u03a9C={\u2208R2:r=0} (center line). Although the latter could be considered as part of the domain, it is sometimes convenient to consider it as a boundary due to the appearance of inverse powers of r in the equations. On the free surface r=d+\u03b7(z), we have the kinematic boundary condition u\u20d7\u00b7n\u20d7=0, where n\u20d7=e\u20d7r\u2212\u03b7\u2032(z)e\u20d7z denotes a normal vector. Expressed in terms of \u03a8, this takes the form \u03a8z+\u03b7z\u03a8r=0 on \u2202\u03a9S. In addition, we have the dynamic boundary condition p=\u2212\u03c3\u03ba on \u2202\u03a9S, where\u2202\u03a9S and \u03c3>0 is the coefficient of surface tension. Using Bernoulli's law, we can eliminate the pressure and express this as\u2202\u03a9S, where Q is the Bernoulli constant. At the center line \u2202\u03a9C, the identity \u03a8z=rur shows that \u03a8z=0. Summarizing, we have following boundary value problem:F and \u03b3 are arbitrary functions of \u03a8.We next consider the boundary conditions. Assume that the fluid domain is given by 33.1\u2202\u03a9S and \u2202\u03a9C. We normalize \u03a8 such that it vanishes on \u2202\u03a9C and assign the name m to its value on \u2202\u03a9S. Thus, we deal with the equationsQ and m are constants. The fluid velocity is given byr=0, provided that \u03c8 is continuous at r=0, since then . Written in terms of \u03c8, Equation\u00a0/r is of class C1 and r\u03c8 is of class C2, both viewed as functions on {\u2208 the inverse map. Then, with \u03c8\u00af=\u03c8, that is,As for Bernoulli's equation\u00a0, we do nI\u03c8\u00af=\u03c8\u223c, is trans3.5\u03d5=0 on s=1 instead of functions \u03c8\u00af with variable boundary condition at s=1. Thus, we introduce, for any \u03bb\u2208R, the functionm=m(\u03bb):=d2\u03c8\u03bb(1), read of \u03d5=0 on s=ms of \u03d5,  and m=m, readaL\u03b7I\u03d5=\u2212\u03b31, and\u03d5s+d2=M with M compact. Meanwhile, we also clarify what Q exactly is, namely, we define it as an expression in . An advantage of rewriting the problem in this form is that it facilitates the use of Rabinowitz' global bifurcation theorem, which was originally formulated for such problems. Although it is probably possible to apply some later version adapted to problems that are not of the form \u201cidentity plus compact\u201d (\u03a90), Q=Q, and m=m(\u03bb); andthe tuple1solves  with \u03a9=H(iii)\u03c8\u2208Cper,e2,\u03b1(\u03a9\u00af) and satisfies  and I\u03d5\u2208C2,\u03b1(\u03a90I\u00af), provided F=0. Indeed, we have I\u03c8\u2208C2,\u03b1(\u03a9I\u00af) in this case since I\u03c8\u03bb\u2208C2,1(\u03a90I\u00af); in particular, \u03c8 satisfies  are solutions of \u2014they makticular,  has the M being smooth, the property that M is of class C2 follows from the property that A is of class C2; this, in turn, is guaranteed by assumption \u2208R\u00d7U\u03b5 be arbitrary. In the following, the quantities C can change from line to line, but are always shorthand for a certain expression in its arguments that remains bounded for bounded arguments. Moreover, let R>0 and suppose \u2225\u2225R\u00d7X\u2264R. Since \u03c8\u03bb is of class C1 with respect to \u03bb and L\u03b7 is elliptic uniformly in \u03b7 due to \u03b7+d\u2265\u03b5, we see thatM2 is compact on R\u00d7U\u03b5 because of the compact embedding of Cper2,\u03b1(\u03a90I\u00af) in Hper1(\u03a90I) and in Cper0,\u03b1(\u03a90I\u00af) combined withM1, we immediately find, in view of the obtained estimates for A,M1 is compact on R\u00d7U\u03b5 because C3,\u03b1 is compactly embedded in C2,\u03b1.\u25aaThe other operations in the definition of sumption . Now let44.1F and, in particular, its evaluation at a trivial solution. For simplicity, we always write A\u03b7 for A\u03b7\u03b4\u03b7, that is, the partial derivative of A with respect to \u03b7 evaluated at  and applied to a direction \u03b4\u03b7. The same applies similarly to expressions such as A\u03d5, L\u03b7\u03b7, and so\u00a0forth.We now want to calculate the partial derivative L\u03b7, which only depends on \u03b7 and not on \u03d5, leads toL\u03c6=f gives L\u03b4\u03c6+\u03b4L\u03c6=\u03b4f, we see that IA\u03b7 is the unique solution ofIA\u03d5 is the unique solution ofA=0 here, we haveLinearizing the operator M1. After a lengthy computation, we get the following results for the partial derivatives of M1 evaluated at a trivial solution , noticing that SA\u03b7=SA\u03d5=0 at such points:P is the projection onto the space of functions with zero\u00a0average.Next, we turn to \u2212c(\u03bb) is the z\u2010component of the velocity at the surface of the trivial laminar flow corresponding to \u03bb in view of \u03b4\u03a6. Hence, the \u201cgood unknown\u201d for the linearized problem in the new variables is actually \u03b4w\u2212(v\u2032\u2218\u03a6)\u03b4\u03a6. In particular, (\u03b4w\u2212(v\u2032\u2218\u03a6)\u03b4\u03a6)\u2218\u03a6\u22121 satisfies the linearized problem in the original variables. In our setting, the latter linearization is formal due to the free boundary, but it has a simpler structure than the linearization in the flattened variables.Lemma 3c(\u03bb)\u22600. ThenLetBefore we proceed with the investigation of local bifurcation, we first introduce an isomorphism, which facilitates the computations later and is sometimes called T(\u03bb) and [T(\u03bb)]\u22121 are well defined, and a simple computation shows that they are inverse to each other.\u25aaBoth T\u2010isomorphism, we introducec(\u03bb)\u22600. Now recall thatV=V[\u03b7] the unique solution off:=\u22122\u03c8\u03bb+s\u03c8s\u03bbc(\u03bb)S\u03b8+dc(\u03bb)A\u03b7S\u03b8+A\u03d5(2\u03c8\u03bb+s\u03c8s\u03bbc(\u03bb)S\u03b8)+V[S\u03b8] satisfiesIf=0 at |y|=1. Thus, recalling , L2(\u03bb)\u03b8 is the unique solution ofL1(\u03bb)\u03b8 is (in the set of L\u2010periodic functions with zero average) uniquely determined byLet us now consider a trivial solution . In view of the ecalling , 55), , w, wT\u2010isom4.3F. Clearly, in view of the T\u2010isomorphism, it suffices to study the kernel of L; here and in the following, we will suppress the dependency of L on \u03bb. From , provided L\u03b8=0. Thus, combining =\u2211k=0\u221e\u03b8k(s)cos(k\u03bdz) as a Fourier series. Then we easily see that\u03b80(1)=0 is already included in the definition of Y, and\u03bc\u2208R, let us now introduce the function \u03b2\u223c=\u03b2\u223c\u03bc,\u03bb, which is defined to be the unique solution of the singular Cauchy problem\u03b2\u223c\u2208C2,\u03b1 by the same argument as in Section\u00a0We now turn to the investigation of the kernel of  \u03bb. From , we infeombining  and 74)F, and therefore, \u2202s\u03b80(0)=0. Hence, \u03b80 is a multiple of \u03b2\u223c0,\u03bb. However, since necessarily \u03b80(1)=0, \u03b80 has to vanish identically in view of k\u22651 and notice as above that \u2202s\u03b8k(0)=0 provided (\u03b8k is a multiple of \u03b2\u223c\u2212(k\u03bd)2,\u03bb if and only if (\u03b2\u223c\u2212(k\u03bd)2,\u03bb(1)=0 and that (\u03b8k(1)=0. Since therefore also \u2202s\u03b8k(1)=0 by virtue of 2,\u03bb(1)\u22600 and define \u03b2\u2212(k\u03bd)2,\u03bb:=\u03b2\u223c\u2212(k\u03bd)2,\u03bb/\u03b2\u223c\u2212(k\u03bd)2,\u03bb(1). Hence, Equation\u00a0(\u03b8k if and only if the dispersion relation\u03b8k is a multiple of \u03b2\u2212(k\u03bd)2,\u03bb. We summarize our results concerning the kernel.Lemma 4\u03bb\u2208R with c(\u03bb)\u22600 and under the assumption =\u2211k=0\u221e\u03b8k(s)cos(k\u03bdz), is in the kernel of L(\u03bb) if and only if \u03b80=0 and for each k\u22651(a)\u03b8k=0, or(b)\u03b2\u223c\u2212(k\u03bd)2,\u03bb(1)\u22600, \u03b8k is a multiple of \u03b2\u223c\u2212(k\u03bd)2,\u03bb, and the dispersion relationD given in  is at first only defined if \u03b2\u223c\u03bc,\u03bb(1)\u22600. If this property fails to hold, we set D:=\u221e in the\u00a0following.Clearly, 4.4L can be written as an orthogonal complement with respect to a suitable inner product. This will be helpful later. To this end, we introduce the inner productf1,f2\u2208H0,per1(R), g1,g2:\u03a9\u223c0\u2192R with Ig1,Ig2\u2208Hper1(\u03a9\u223c0I), where \u03a9\u223c0:=xK is even self\u2010adjoint, we first note that obviously, H admits the fundamental decomposition H=\u00d7{0})+\u02d9({0}\u00d7C) into a positive and negative subspace. Associated with this decomposition is the fundamental symmetry\u27e8u1,u2\u27e9J==\u27e8d2\u03c61,\u03c62\u27e9Ls32+g(\u03bb)\u22121b1b2\u00af. The operator JK is self\u2010adjoint with respect to \u27e8\u00b7,\u00b7\u27e9J, since now the assumptions of Ref. J\u2010adjoint by an upper index \u27e8\u2217\u27e9, we haveJ\u27e8\u2217\u27e9=J and JK\u27e8\u2217\u27e9J=K\u2217 >0 if u\u22600. Thus, there exists exactly one negative eigenvector of K0; cf. again Ref. K0 has exactly one negative eigenvalue \u03bc0 and its other eigenvalues are positive. With the same proof as in Ref. C>0, the estimateF are arbitrary, we define the perturbation A via D(A)={u\u2208H:\u03c6\u2208D(T),b=\u03c6(1)} andA is densely defined and bounded, and we have K=K0+A. Now consider a real \u03bc<\u03bc0\u2212C\u2225A\u2225J. Because ofK\u2212\u03bcI is invertible in view of the Neumann series. This completes the proof.\u25aaThe first assertion is clear because K, as we see in what follows.Proposition 3q\u2212\u03bb denotes the negative part of q\u03bb. Then the operator K has only real eigenvalues, K has exactly one eigenvalue \u03bc<\u2212\u2225q\u2212\u03bb\u2225\u221e, and all its other eigenvalues satisfy \u03bc>\u2212\u2225q\u2212\u03bb\u2225\u221e. Moreover, all eigenvalues are algebraically\u00a0simple.Assume thatUnder a certain condition, we can infer even more properties of the spectrum of K and u=) an associated eigenvector. Due to Lemma\u00a0\u03c6(1)\u22600 =0 and thus \u03c6\u22610), it follows thatu cannot be neutral and \u03bc has to be real. Since, additionally, by Ref. H is a \u03c01\u2010space\u2014there exists exactly one nonpositive eigenvector of K, the first assertion follows immediately. The second statement is a direct consequence of the fact that all eigenvalues are real and no eigenvectors are neutral.\u25aaLet \u03bc be an eigenvalue of Remark 2c\u00b1 are not real). In particular, if \u03b3 and FF\u2032 are bounded, this condition is satisfied if \u201cc(\u03bb) is sufficiently large\u201d or, provided that additionally, F is bounded, if simply \u201c|c(\u03bb)| is sufficiently large.\u201dIf Equation\u00a0 holds, t5.2F.We now turn to a more detailed investigation of the conditions for local bifurcation for specific examples of \u03b3 and 5.2.1\u03b3=F=0. By (c(\u03bb)\u22600 if and only if \u03bb\u22600. Moreover, \u03b2\u223c=\u03b2\u223c\u2212(k\u03bd)2,\u03bb solvesI1 and K1 are modified Bessel functions of the first and second kinds. Since K1(x)\u2192\u221e as x\u21920, we necessarily have c2=0. Determining the remaining constant c1 yieldsdI1/dx=I0\u2212I1/x (cf. Ref. (k\u03bd)2d2\u22121>0 if D(\u2212(k\u03bd)2,\u03bb)=0, the dispersion relation D(\u2212(k\u03bd)2,\u03bb)=0 can hence be written as\u03bd>0, k\u2208N with k\u03bd>1/d and then \u03bb such that \u22600, Equation\u00a0 is one\u2010dimensional if this relation is satisfied for some k\u2208N and is trivial if it fails to hold for all k. Indeed, Equation\u00a0; see Ref. As a first example, we consider the case without vorticity and swirl, that is, =F=0. By  and 24)\u03b3=F=0. Bycf. Ref. ,(136)tten as\u03c3c(\u03bb)2=ktten as\u03c3c(\u03bb)2=ktten as\u03c3c(\u03bb)2=ktten as\u03c3c(\u03bb)2=kten as1\u03c3c(\u03bb)2=kD\u03bb(\u2212(k\u03bd)2,\u03bb)\u22600 always holds in view of c\u03bb(\u03bb)\u22600.Furthermore, it is therefore clear that the transversality condition q\u03bb=0 and h(\u03bb)=d\u22122+\u03c3\u22121d\u22121c(\u03bb)2>0.Moreover, it is easy to see that  is alway5.2.2\u03b3\u22600 is a constant and F=0. By Equations\u00a0(c(\u03bb)\u22600 if and only if \u03bb\u2260\u03b3d24. Noticing that \u03b2\u2212(k\u03bd)2,\u03bb is the same as in the previous example without vorticity, we, moreover, haveD(\u2212(k\u03bd)2,\u03bb)=0 for 1/c(\u03bb), we see that necessarily1/(\u03bdd)\u2208N. We now want to reformulate the second case. Clearly, Equation\u00a0 in view of 36+116<224. Therefore and because of I0(0)=1, I1(0)=0, and I1\u2032(0)=1/2, the function \u03c7:\u2192 is strictly monotonically increasing and onto. Hence,  such that \u03c7(x1)=4\u03c3d\u22125\u03b3\u22122 and x0:=x1/d. Thus, we have the equivalenceD(\u2212(k\u03bd)2,\u03bb)=0 for c(\u03bb) yieldsD(\u2212(k\u03bd)2,\u03bb) cannot vanish. Next, we search for solutions of (k and c(\u03bb) (and not k and \u03bb) since R\u220b\u03bb\u21a6c(\u03bb)\u2208R is bijective. Both for the first case (for which 1/(\u03bdd)\u2208N is necessary) and for the second case, we can easily first choose an appropriate k and then c(\u03bb) via =b\u00b1(x) withb\u2212(1) and possibly b\u00b1(0) are to be interpreted as the limit of the above expression as x tends to 1 or 0; the limit for x\u21921 exists because x=1 is a simple root of both nominator and denominator, and the limit for x\u21920 also as I1(0)=0 and I1\u2032(0)=1/2. Having clarified this, we see that b\u00b1 is smooth on  and continuous on .Let us now turn to the case  x=0. By , 161), , \u03be<1/2 azero. By , we ther\u03be=16/81. Differentiating (bx\u2212(0)=bxx\u2212(0)=bxxx\u2212(0)=0 yieldbxxxx\u2212(0)<0; hence, b\u2212 is strictly monotonically\u00a0decreasing.Let us now consider ntiating  twice mo1/(\u03bdd)\u2209N; in Figure\u00a0\u03be\u226516/81:If D(\u2212(k\u03bd)2,\u03bb)=0 can have at most one root k\u2208N.The dispersion relation 16/81\u2264\u03be<1/2 andIf \u03be<16/81:If IfIfIfTo summarize, for fixed \u03bb, we have therefore proved the following, provided 1/(\u03bdd)\u2208N andk=1/(\u03bdd).If \u03b3<0, these statements remain true after reversing all inequalities in the conditions for c(\u03bb) and changing maxb\u2212 to minb\u2212.If, however, D(\u2212(k\u03bd)2,\u03bb)=0 has exactly one solution k\u2208N. Since c\u03bb(\u03bb)\u22600, it holds that D\u03bb(\u2212(k\u03bd)2,\u03bb)\u22600 if and only if \u03be\u22641/2 or k\u03bdd>x1\u00a0otherwise.Next, let us turn to the transversality condition, fix \u03bb, and assume that q\u03bb=0 and h(\u03bb)=d\u22122+2\u03c3\u22121d\u22121\u03bb(4\u03bb\u2212\u03b3d2) by (h(\u03bb)>0 for all \u03bb\u2208R if \u03b32<8\u03c3d\u22125\u21d4\u03be>12, and in the case \u03be\u226412, h(\u03bb)>0 if and only if \u03bb\u2209 whereFinally, we have a look at . Here, q\u2212\u03b3d2) by . Therefo5.2.3\u03b3=0 and F(x)=ax for some a\u2208R =0). Having a look at, for example, \u22600 has to hold to allow for c(\u03bb)\u22600. Moreover, the dispersion relation can be written as\u03b6:=\u03bc/a2, b:=ad>0. Due to this relation and under the assumption J0(b)\u22600, which yields c\u03bb(\u03bb)\u22600, it is clear that the transversality condition is always satisfied, provided that the corresponding kernel is one\u2010dimensional. Since A depends in a rather complicated way on \u03b6, we do not further study the dispersion relation and the possibility of multidimensional kernels rigorously; it is, however, interesting to notice that, due to  for which A>0 and J0(b)\u22600, that is, exactly the points that allow for solutions of the dispersion relation (c(\u03bb)\u22600.The easiest case to include nonzero swirl is to take example, , we findtten asd\u03c3\u22121c(\u03bb)tten asd\u03c3\u22121c(\u03bb)6Theorem 3X be a Banach space, U\u2282R\u00d7X open, and F\u2208C. Assume that F admits the form F=x+f with f compact, and that Fx\u2208C). Moreover, suppose that F=0 and that Fx has an odd crossing number at \u03bb=\u03bb0. Let S denote the closure of the set of nontrivial solutions of F=0 in R\u00d7X and C denote the connected component of S to which  belongs. Then one of the following alternatives occurs:(i)C is unbounded;(ii)C contains a point  with \u03bb1\u2260\u03bb0;(iii)C contains a point on the boundary of U.Let The theory for local bifurcation having setup, we now turn to global bifurcation, which is, of course, the main motivation of our formulation \u201cidentity plus compact.\u201d To this end, we first state the global bifurcation theorem by Rabinowitz.U=X can be found in Ref. U.The proof of this theorem in the case Theorem 4\u03bb0\u22600 such that the dispersion relationD given by =0 in R\u00d7X and C denote the connected component of S to which  belongs. Then one of the following alternatives occurs:(i)C is unbounded in the sense that there exists a sequence \u2208C such\u00a0that(a)|\u03bbn|\u2192\u221e, or(b)\u2225\u03b7n\u2225C2,\u03b1\u2192\u221e, or(c)\u2225r2/p\u03b3(\u03a8n)+r2/p\u22122F(\u03a8n)F\u2032(\u03a8n)\u2225Lp(Z\u03b7n)\u2192\u221e with p:=52\u2212\u03b1, where Z\u03b7n denotes a L\u2010periodic instance of the axially symmetric fluid domain in R3 corresponding to \u03b7n and \u03a8n=r2((\u03d5n+d2(d+\u03b7n)2\u03c8\u03bbn)\u2218H[\u03b7n]\u22121) is the corresponding original Stokes stream function,n\u2192\u221e; as (ii)C contains a point  with \u03bb1\u2260\u03bb0;(iii)C contains a sequence  such that \u03b7n converges to some \u03b7 in C0,per,e2,\u03b2(R) for any \u03b2\u2208 and such that there exists z\u2208 withAssume  and thatlation1D(\u2212(k\u03bd)2Now we can prove the following result.F is of class C2 and admits the form \u201cidentity plus compact\u201d on each R\u00d7U\u03b5, \u03b5>0. Moreover, it is well known that F has an odd crossing number at , provided that F is a Fredholm operator with index zero\u2010 and one\u2010dimensional kernel, and the transversality condition holds. These properties, in turn, are consequences of the hypotheses of the theorem in view of Lemmas\u00a0F coincides with L(\u03bb0) up to an isomorphism. For each \u03b5>0, we can thus apply Theorem\u00a0U chosen to be the interior of R\u00d7U\u03b5. Thus, on each R\u00d7U\u03b5, C coincides with its counterpart obtained from Theorem\u00a0\u03b5>0 is arbitrary and R\u00d7U=\u22c3\u03b5>0(R\u00d7U\u03b5), it is evident that necessarilyC is bounded in R\u00d7X and (ii) fails to\u00a0hold.As was already observed in Lemma\u00a0C is bounded in R\u00d7X if (i)(a)\u2013(c) and  and, since  can be left out because of unique solvability of the Dirichlet problem associated with L\u03b7), and changes of variables via H[\u03b7] and via cylindrical coordinates in R5 and R3. In the above, \u03a9\u223c\u03b7 denotes a periodic instance of \u03a9\u03b7=H[\u03b7](\u03a90) and \u03a8, Z\u03b7 are analogously defined as in the statement of (c). Moreover, the constant C>0 can change in each\u00a0step.Let us investigate alternative (i) further. To show that it can be as stated above, we show that, in view of alternative (i) of Theorem\u00a0\u2013(c) and  fail to d, since  does notC0,per,e2,\u03b1(R) in C0,per,e2,\u03b2(R).\u25aaFinally, we turn to alternative (iii). If Equation\u00a0 holds, bRemark 3\u03c9\u03d1=\u03c9\u20d7\u00b7e\u20d7\u03d1=\u2212r\u03b3(\u03a8)\u2212(FF\u2032)(\u03a8)/r, satisfies \u2225r2/p\u22121\u03c9n\u03d1\u2225Lp(Z\u03b7n)\u2192\u221e as n\u2192\u221e.Alternative (i)(c) says that the angular component of the vorticity, in general, given by Remark 4In a two\u2010dimensional situation without surface tension (and with gravity), sometimes, an alternative such as (ii) above can be eliminated. The strategy to this end typically relies on maximum principle arguments, which, however, appear to be unavailable when capillary effects are taken into account. Therefore, it is unclear whether and, if so, how (ii) can be eliminated in the present\u00a0situation.Proposition 4(i)(b')(\u03b1)\u2225\u03b7n\u2225C1,\u03b1\u2192\u221e, or(\u03b2)\u2225|u\u20d7n|2\u2225C0,\u03b1(Sn)\u2192\u221e , or(\u03b3)|Q|\u2192\u221e (the Bernoulli constant is unbounded).In Theorem\u00a0We also have the following.\u25aaThis follows easily from the Bernoulli equation"}
{"text": "Hs(\u211d). If D > 0, the local well-posedness is given for s > 1/2 and for s > 3/2 if D=0.In this paper, we show the local well-posedness for the Cauchy problem for the equation of the Nagumo type in this equation (1) in the Sobolev spaces  D > 0 is a constant diffusion coefficient, \u03b1 \u2208  and \u03f5 > 0 is a small positive quantity. In . Thus, using the dominated convergence theorem, we have as follows:However, side of  is a intc on the interval  such thath\u27f60+(1/h)\u222btt+h\u2016V(t+h \u2212 \u03c4)F(u(\u03c4)) \u2212 F(u(t))\u2016s\u22122d\u03c4=0.Now, from the mean value theorem for integrals, there exists a value \u2202t+\u03b7(t)=Q\u03b7(t) \u2212 F(u(t)) in Hs\u22122(\u211d). where \u2202t+ is the right derivative. In similar way, we can conclude that the left derivative is \u2202t\u2212\u03b7(t)=Q\u03b7(t) \u2212 F(u(t)) in Hs\u22122(\u211d). So, \u03b7 \u2208 C1) and \u2202t\u03b7(t)=Q\u03b7(t) \u2212 F(u(t)). As V(t)\u03c8(x) is the solution of the linear problem (u(t)=V(t)\u03c8+\u03b7(t) \u2208 C1) and satisfies  is locally integrable on  for all n. It follows that u \u2208 \ud835\udcb3 for all n and satisfies \u2016un(t)\u2016s \u2264 \u2016\u03c8n\u2016s+M \u2264 \u03b3+M where \u03b3=supn\u2016\u03c8n\u2016s. Therefore, supt\u2208\u2016un(t)\u2016s \u2264 \u03b3+M for all n and supt\u2208\u2016u(t)\u2016s \u2264 \u03b3+M. Now, similar to the proof of the previous proposition, we have as follows:As E1/2,1((b\u0393(1/2))2T) is finite .Let D=0 using a priori estimate and the parabolic regularization method, the so-called vanishing viscosity method .\u03b7(t), a(t) and b(t) be real valued positive continuous functions defined on \u2286 : \u03a9(A(\u03c4))+\u03c4B\u03c4\u03a9(limr\u27f6\u221eG(r))}.Let This is a particular case of the theorem given in  [pp. 78]s > 3/2 be fixed. Then, F(u) satisfies the estimateu, w \u2208 Hs(\u211d), where L0=\u03f5x\u2211k=0n\u22121xkyn\u22121\u2212k+(\u03f5n/2)yn+x2+xy+y2+(\u03b1+1)(x+y).Let q=\u2211k=0n\u22121ukwn\u22121\u2212k. As s > 3/2 thus Hs(\u211d) and Hs\u22121(\u211d) are Banach algebras. Moreover, we have that Hs(\u211d)\u27f6Hs\u22121(\u211d) and Hs(\u211d)\u27f6L\u221e(\u211d). Thus, using the Cauchy\u2013Schwartz inequality, we have as follows:We define r \u2265 1 and s > 3/2 be fixed and h, v are real valued functions. Then, there exists a constant C=C such that(T. Kato). Let v, h\u2202xv)s| \u2264 C\u2016\u2202xh\u2016s\u22121\u2016v\u2016s2.In particular, |) be the corresponding solution of  > 0, depending on \u2016\u03c8\u2016s, such that uD can be extended to the interval , and there is a function \u03c1 \u2208 C]; .Let  problem  with iniution of  for someHs(\u211d) and Using the inner product in uD(t)\u2016s2 \u2264 \u03c1(t) for all t \u2208 , and therefore we have thatt \u2208 .From n=1, from (\u03c1(t) \u2264 (1/(\u2016\u03c8\u2016s2+2Cs(\u03b1+\u03f5+2)t)\u22121 \u2212 8Cst), for 0 \u2264 t \u2264 T\u2217 where Ts(\u03c8) such that 0 < Ts(\u03c8) < T\u2217, and therefore, we conclude thatt \u2208 .For the case =1, from  we have uD(t)\u2016s2 \u2264 \u03c1(t) for all t \u2208 .Let u0\u2016s2 \u2264 \u03c1(t) for all t \u2208 , where \u03c1(t) is as in Moreover, \u2016Ts=Ts(\u03c8) be as in Let uD(t))D>0 is a net which converges to a function u0 \u2208 C) in the L2\u2212 norm, uniformly over .First we will show that . Then,Let \u03c1 is the function defined in the proof of Let In order to bound the first term, we have as follows:We can bound the second term by the following:uD1 \u2212 uD2|uD12 \u2212 uD22|)0| \u2264 2M\u2016uD1 \u2212 uD2\u201602.The third term is bounded by | \u2212 uD2(t)\u201602 \u2264 C|D1 \u2212 D2| for all t \u2208 , and since L2(\u211d) is complete there exists the limit u0(t)=limD\u27f60uD(t) in L2(\u211d) uniformly with respect to t \u2208 , i.e.u0 \u2208 C).Applying Gronwall's inequality to the last relation, we show that there is a constant u0 \u2208 Hs(\u211d). Let t \u2208 . Since uD\u27f6u0 in L2(\u211d), as D\u27f60+, then there exists a subsequence {Dnj)(} such thatNow we show that We obtain by Fatou's Lemma as follows:uD\u21c0u0 in Hs(\u211d) for all t \u2208  as D\u27f60+.We must show that uD(t))D>0 is a weak Cauchy net in Hs(\u211d), uniformly with respect to t \u2208 . In fact, given \u03c6 \u2208 Hs(\u211d) and \u03f5 > 0, choosing \u03c6\u03f5 \u2208 Hs(\u211d) such that \u2016\u03c6 \u2212 \u03c6\u03f5\u2016s \u2264 \u03f5, thenD1,D2\u27f60+supt\u2208(uD1(t) \u2212 uD2(t)|\u03c6|)s=0.First of all, we will show that . Moreover, since the convergence is uniform for all \u03c6 \u2208 Hs(\u211d), we can conclude that u0 \u2208 Cw).Thus, we have that u0 \u2208 Cw1).Finally, we show that \u03c6 \u2208 Hs\u22122(\u211d). Then,t \u2208 . Since uD\u27f6u0 in L2(\u211d) and uD\u21c0u0 in Hs(\u211d), we have \u2202xuD\u21c0\u2202xu0 in Hs\u22121(\u211d) and \u2202x2uD\u21c0\u2202x2u0 in Hs\u22122(\u211d) uniformly on . Observe that if r > 1/2, fn\u21c0f in Hr(\u211d) and gn\u21c0g in Hr(\u211d) then fngn\u21c0fg in Hr(\u211d). After, we haveTs]. Thereby, taking the limit as D\u27f60+ in (Let \u211d). Then,uDt\u03c6s\u22122=\u03c8u0 be as in the preceding theorem, then u0 \u2208 AC).Let t \u2208  \u21a6 u(u \u2212 \u03b1)(u \u2212 1)+\u03f5unux is weakly continuous in Hs\u22122(\u211d) and the Sobolev space is separable, then applying the Bochner\u2013Pettis theorem, it is a strongly measurable function in Hs\u22122(\u211d). Therefore,u0 \u2208 AC).Since So, from  we concls > 3/2 and T > 0 be fixed, \u03c8j \u2208 Hs(\u211d), j=1,2, and vj \u2208 C)\u2229Cw)\u2229AC) two weak sense solutions to (D=0 such that vj(0)=\u03c8j, j=1,2. Then,L0 is as in the Let w(t)=v1(t) \u2212 v2(t). Since s > 3/2, we have s \u2212 2 > \u22121/2 > 1 \u2212 s and 1 \u2212 s > \u2212s, and also Hs\u22122(\u211d)\u27f6Hs\u2212(\u211d). Using the fact that w(t) is real valued we havet \u2208  is fixed, h is such that t+h \u2208 , and \u2329\u00b7|\u00b7|\u232as is the Hs duality bracket. As t \u2208  \u21a6 w(t) \u2208 Hs(\u211d) is bounded andHs\u22122(\u211d)\u27f6Hs\u2212(\u211d), from (Let d we havewt+hwt+h0unded and\u2202twt=limhR is given by (From given by . Applyin\u2202twt02\u2264L0\u03c8 \u2208 Hs(\u211d) with s > 3/2. Then, there exists a Ts=Ts(\u03c8) > 0 and a unique u0 \u2208 C) such thatLet u0 \u2208 C).From the previous results, there exists a unique solution of  in the c\u03c6 \u2208 Hs(\u211d) be such that \u2016\u03c6\u2016s=1. Then, we have |(u0|\u03c6|)s| \u2264 \u2016u0(t)\u2016s \u2264 \u03c1(t)1/2, for all \u03c6 \u2208 Hs(\u211d) and for all t \u2208 . Additionally,\u03c6 \u2208 Hs(\u211d). As we have \u2016\u03c8\u2016s=sup\u03c6\u2016s\u2016=1|(\u03c8|\u03c6|)s|, then taking supremum over \u2016\u03c6\u2016s=1 in (t\u27f60+\u2016u0(t)\u2016s=limsupt\u27f60+\u2016u0(t)\u2016s=\u2016\u03c8\u2016s, i.e., the limit of \u2016u0(t)\u2016s exists as t\u27f60+ and limt\u27f60+\u2016u0(t)\u2016s=\u2016\u03c8\u2016s. Since u(t)\u27f6\u03c8 weakly in Hs(\u211d) as t\u27f60+, it follows that limt\u27f60+u0(t)=\u03c8 in the norm of Hs(\u211d). Let t\u2032 \u2208 ; Hs+1(\u211d))\u2229C1]; Hs\u22121(\u211d)) be the corresponding solution of the problem ] which is independent of D. Then, uD can be extended, if necessary, to the interval , with \u03c8 viewed as an element of Hs(\u211d).Let r=s+1 to obtainApplying  with r=sNow, from inequality (3.12) and Theorem 3.2 in , we obtat, we have as follows:Therefore, using , 67) in in67) int \u2208  and therefore we can extend (if necessary) u=u(t) to  as a solution in Hs+1(\u211d). Thus, we conclude that Ts(\u03c8) \u2264 Ts+1(\u03c8). So, uD \u2208 C]; Hs+1(\u211d)) for D > 0. From , then we have u0 \u2208 C]; Hs+1(\u211d)).Observe that the last inequality is independent of Following Lemma 5 in  we have s > 3/2. For \u03c8 \u2208 Hs(\u211d) and \u03c4 > 0, we defineLet \u03c4\u27f60+\u2016\u03c8\u03c4 \u2212 \u03c8\u2016s=0, and there exists a constant C=C(s) such thatThen, lim\u03c4\u27f60+\u2016\u03c8\u03c4 \u2212 \u03c8\u2016s=0 uniformly on compact subsets of Hs(\u211d).Moreover, lim\u03c4\u27f60+\u2016\u03c8\u03c4 \u2212 \u03c8\u2016s=0. Now, to prove the uniformity on compact subsets, it is enough to show that \u03c8n\u27f6\u03c8 in Hs(\u211d) implies lim\u03c4\u27f60+\u2016\u03c8n\u03c4 \u2212 \u03c8n\u2016s=0 uniformly for n=1,2,\u2026, since sequential compactness is equivalent to compactness in metric spaces. Thus, observe thatNotice that \u03f5 > 0 be given and choose N such that if n \u2265 N, then \u2016\u03c8n \u2212 \u03c8\u2016s < (1/3)\u03f5. Thus, for \u03c40 > 0 small enough that 0 < \u03c4 < \u03c40 we haven \u2264 N. Now, if n \u2265 N then we haveLet n.Hence  holds foOn the other hand, we havee\u03c4(1+\u03be2)s/2\u2212 \u2212 e\u03c4(1+\u03be2)s/2\u2212| \u2264 |\u03c4 \u2212 \u03b8|(1+\u03be2)s/2, and then we haveFinally, using the mean value theorem, |The proof is complete.s > 3/2, \u03c8 \u2208 Hs(\u211d), \u03c8\u03c4 (for \u03c4 > 0) be as in the preceding lemma. If u0\u03c4 is solution of the problem (u0\u03c4(0)=\u03c8\u03c4, for all \u03c4 > 0, then there are constants C=C > 0 and \u03b7=\u03b7(s) \u2208  such that\u03c4 sufficiently small and 0 \u2264 \u03b8 \u2264 2\u03c4.Let  problem  with u0\u03c4\u03c40 > 0 be such that u0\u03c4(t) is well-define in  for all 0 < \u03c4 \u2264 \u03c40. Then,Let Now, the right-hand side of the inequality  will be First, we will estimate . ApplyinNow, we will estimate . Observes > 3/2, there is s0 such that 3/2 < s0+1 < s. From the Cauchy\u2013Schwartz inequality, we obtainFinally, we estimate . As s > q is defined in the proof of Now, we will estimate each term on the right-hand side of the last inequality. First, observe thatu0\u03c4)n \u2212 (u0\u03b8)n\u2016s0\u2016u0\u03c4\u2016s+1. From \u03c4 \u2264 \u03c40.We also estimate \u2016(s0/s). To estimate the term \u2016u0\u03c4 \u2212 u0\u03b8\u20160, observe thatFrom \u03b8 \u2264 \u03c4. Therefore, we haveFrom the term  is boundOf the bounds that were found for , 80b), , 80b),  we concle obtain .The following corollary follows immediately from F be a compact subset in Hs(\u211d). Suppose that \u03c8 \u2208 F, \u03c8\u03c4 and u0\u03c4 are defined as in the preceding result. Then, u0\u03c4 converges uniformly to u0, for all t \u2208 , as \u03c4\u27f60+.Let \u03c8 \u21a6 u0 is continuous in the following sense: let \u03c8j \u2208 Hs(\u211d), j=1,2,3,\u2026 such that \u03c8j\u27f6\u03c8 in Hs(\u211d) and uj0, \u2208 C; Hs(\u211d))\u2229C1) are the corresponding solutions of the problem =\u03c8j. Let T \u2208 . Then, there exists a positive integer N0=N0 such that Ts,j \u2265 T for all j \u2265 N0 andThe map  problem  with ini\u03c8 \u2208 Hs(\u211d) and let {\u03c8j}j\u2208\u2115 be a sequence in Hs(\u211d) such that converges to \u03c8. Suppose that u0, uj0,, u0\u03c4, uj0,\u03c4 are the corresponding solutions of .Let  problem  is local"}
{"text": "Page 3, Table 1, column 4: \u201c\u2212\u00a3441\u201d should read \u201c\u2212\u00a3681,\u201d and \u201c\u00a3680\u201d should read \u201c\u00a3227.\u201dVolume 10, no. 3, e00425-22, 2022,"}
{"text": "In particular, we show that such mappings are homeomorphisms when n=3 or when the branch set is empty. This gives a positive answer to the corresponding cases of a question of\u00a0Vuorinen.We study global injectivity of proper branched coverings from the open Euclidean  Notice that throughout the paper by domain, we mean a connected open subset of Rn. The structure of this set is tied to the topology and geometry of the mapping itself, but in general the structure of the branch set is not well understood. Even for the important special class of continuous, open, and discrete maps called quasiregular mappings, many properties of the branch set remain largely unknown, but the topic garners great interest. In his ICM address \u2192f(\u03a9) and any x\u2208\u03a9\u2229f\u22121(\u03b2(0)) there exists a path \u03b1:\u2192\u03a9 for which f\u2218\u03b1=\u03b2 and \u03b1(0)=x. Such a path is called a lift of \u03b2(under f).Let Finally, a fundamental technique in the study of branched covers is the path\u2010lifting. For the terminology and basic theory of this technique, we refer to  there exists a maximal lift \u03b3:I\u2192\u03a9 of \u03b2 such that \u03b3(0)=x, where I is a subinterval of  of type  or  since by the continuity of f we have f(\u03b3(b))=\u03b2(b). This is again a contradiction with the maximality of \u03b3, and so the original claim holds true; I= and we may choose \u03b1=\u03b3.\u25a1Note first that since 3f:Bn\u2192f(Bn)\u2282Rn is in fact a covering map defined on a contractible n\u2010manifold. This observation can be used to show that the image f(Bn) is actually an Eilenberg\u2013MacLane spaceK, that is, a path\u2010connected space whose fundamental group is isomorphic to a group G and which has contractible universal covering space, see, for instance, \u2192X from x1 to xi. With the help of these paths define a function[f\u2218\u03b3i] denotes the homotopy class of the loop f\u2218\u03b3i in Y. Since X is simply connected, any two paths joining x1 and xi are homotopic and thus their images under f are also homotopic. This shows that the function \u03a8 is well defined. The function \u03a8 is also surjective. To see this, notice that for any loop \u03b3 in Y that starts and ends at y lifts to a path \u03b3\u223ci that joins x1 to some xi\u2208f\u22121(y). Thus it follows that\u03a8. Therefore\u25a1Take a point Proof of Theorem 1.2f:Bn\u2192f(Bn)\u2282Rn is a proper branched covering such that Bf=\u2205. Denotef is a covering map, and since Bn is simply connected, it is the universal cover of Y. This means, by definition, that Y is an Eilenberg\u2013MacLane space K for G=\u03c01(Y), see  of the path \u03b3=\u03b30 that keeps the endpoints fixed at x1 and x2. On the other hand \u03b31 is a constant path as a lift of a constant path. Therefore we have x1=x2 and thus f is injective. Especially, f:Bn\u2192Y is then a global homeomorphism as a continuous and open bijection.\u25a1In order to see that 4E^ of a set E\u2282\u03a9 is called a boundary component ofE if its closure in \u03a9 is not compact. Note that a set E can have several boundary components.Lemma 4.1K\u2282f(\u03a9) be a non\u2010empty compact set. DenoteV\u2282\u03a9 is a domain such that C\u2282V\u2282V\u00af\u2282\u03a9. Then for the set U\u2254f\u22121(f(V)) we have the following:(a)U is a path\u2010connected open set such that f(U)=f(V),(b)U is a normal domain for f,(c)f|U:U\u2192f(U) is a proper branched cover,(d)f|\u03a9\u2216U\u00af:\u03a9\u2216U\u00af\u2192f(\u03a9)\u2216f(U\u00af) is a proper branched cover and \u03a9\u2216U\u00af=f\u22121(f(\u03a9\u2216U\u00af)), and(e)E\u2282\u03a9 is a boundary component of \u03a9\u2216U\u00af, then f(E) is a boundary component of f(\u03a9)\u2216f(U\u00af). Moreover, if a point y is contained in a boundary component of f(\u03a9)\u2216f(U\u00af), then all its preimages are contained in boundary components of \u03a9\u2216U\u00af.if a set LetWe start by proving a lemma which provides a useful collection of large normal domains in the setting of the Vuorinen question. In what follows a connected component V is a domain and f an open map, f(V) is open and so is its preimage U under the continuous map f.(a)U we first note that f(V) is a domain containing the compact set K. Therefore, for any point x\u2208U we may connect f(x) and K with a path \u03b1:\u2192f(V). By Lemma \u03b1 has a lift \u03b1\u223c:\u2192\u03a9 with \u03b1\u223c(0)=x and by the definition of U we have |\u03b1\u223c|\u2282U. On the other hand \u03b1\u223c(1)\u2208C\u2282V, and so each point x\u2208U can be connected with a path to an interior point of the connected set V\u2282U. This implies that U is path\u2010connected.For the second claim of (a) we simply note that(b)\u2202f(U)=f(\u2202U). Openness of f gives the inclusiony\u2208f(\u2202U). If U\u2229f\u22121(y)\u2260\u2205, then U is a neighborhood of one of the preimages of y. Then by the openness of f we see that f(U)=f(V) is a neighborhood of y. This implies that U=f\u22121(f(V)) is a neighborhood of all the points in the preimage of y. Therefore, we havey\u2208f(\u2202U). Thus, we have proved that U\u2229f\u22121(y)=\u2205 and so y\u2208\u2202f(U) since y\u2208f(U)\u00af. This gives usBy (a) it is enough to show that (c)U is a domain by (a) and the restriction of a branched covering to a domain is a branched covering. To show that f|U:U\u2192f(U) is proper, we fix a compact set A\u2282f(U) and note that f\u22121(A)\u2282\u03a9 is compact since f is proper. Now as U=f\u22121(f(U)), we have that f\u22121(A)\u2282U, and so we see that (f|U)\u22121(A) is compact. Thus f|U is proper.The set (d)\u2202f(U)=f(\u2202U), we see that alsof(\u03a9). As in part (c), the properness will follow after we show thatx\u2208f\u22121(f(\u03a9\u2216U\u00af)). Suppose, toward a contradiction, that x\u2209\u03a9\u2216U\u00af. Then either x\u2208\u2202U or x\u2208U. In the first case we have by applying (f(x)\u2208\u2202f(\u03a9\u2216U\u00af), which is not possible because by the choice of x we have f(x)\u2208f(\u03a9\u2216U\u00af). In the second case we get by the definition of U that f\u22121(f(x))\u2282U. Thus, we have x\u2209\u03a9\u2216U\u00af which again goes against our assumptions. Therefore, we conclude that x\u2208\u03a9\u2216U\u00af and soFirst we note that the restriction of a branched covering to an open set is a branched covering. Since by part (b) we have at alsof(\u2202(\u03a9\u2216U\u00af(e)E be first some (not necessarily a boundary) component of \u03a9\u2216U\u00af. First we show that E is mapped onto some component of f(\u03a9)\u2216f(U\u00af). It is clear that f(E) is contained in some component C1 of f(\u03a9)\u2216f(U\u00af). If C1\u2260f(E), we find a point y\u2208C1\u2216f(E) and a sequence {yj}j\u2208N in f(E) converging to y. Notice that by the definition of U we have f\u22121(y)\u2282\u03a9\u2216U\u00af. Choose a sequence {xj}j\u2208N in E such that f(xj)=yj. Since f\u22121 is compact, we see that {xj} has a convergent subsequence , which converges to a point x. By continuity we see that x\u2208f\u22121(y) and thus x\u2209E but since y\u2209f(U\u00af) we have x\u2208\u03a9\u2216U\u00af. Thus x is in a different component than all xj. This is a contradiction and therefore every component of \u03a9\u2216U\u00af is mapped onto some component of f(\u03a9)\u2216f(U\u00af) by the mapping f.Let E is a boundary component of \u03a9\u2216U\u00af and its image is not a boundary component of f(\u03a9)\u2216f(U\u00af), then since f is proper, f\u22121(f(E)) would have a compact closure which is not possible as E is a boundary component. This proves the first claim. Continuity now implies that other components are not mapped to boundary components, which implies the second claim.\u25a1If Note that since \u03c01(X) of a space X.Definition 4.2\u03a9\u2282Rn has torsion\u2010free fundamental group at infinity if for any compact set K\u2282\u03a9 there exists a domain V\u2283K with V\u00af being compact in \u03a9 and such that \u03c01(\u03a9\u2216V\u00af) is torsion\u2010free; recall that a group is torsion\u2010free if no element g\u2260e has the property that gj=e for some j\u2208N, where e is the neutral element of the\u00a0group.We say that a domain The proof of Theorem\u00a0simply connected at infinity, see, for example, \u2192C with \u03b1(0)=x0 and \u03b1(1)=x1. The image of this path, \u03b2\u2254f\u2218\u03b1:\u2192f(ER), is a loop based at y0\u2254f(x0). If \u03b2 was zero\u2010homotopic in f(ER), we could lift the homotopy with the covering map f|ER:ER\u2192f(ER) into a homotopy in ER contracting the path \u03b1 to a point without changing the endpoints of the path, see [1.30]. This is not possible when k\u2a7e2, and so we must have [\u03b2]\u22600 in \u03c01(f(ER),y0). Likewise, since f is a proper map, its restriction to B\u2216f\u22121(f(Bf)) is also a covering map and so \u03b2 is not zero\u2010homotopic also in f(B)\u2216K\u2282f(B)\u2216f(Bf).Let \u03b3:\u2192ER, see again Figure\u00a0\u03b31=\u03b1. Then, when \u03b3k:\u2192ER has been defined and if \u03b3k(0)\u2260\u03b3k(1), we define \u03b3k+1 by lifting the path \u03b2 from the point \u03b3k(1) with Lemma \u03b3k. Since the covering map is a local homeomorphism, this procedure is well defined and since f\u22121(f(x0)) is finite, it terminates after at most k\u00a0steps.Next we construct a loop n\u2a7e3 the loop \u03b3 can be contracted to a point in the spherical shell B\u2216B\u00afR and thus in Es. This contracting homotopy can then be pushed with the covering map f|Es into f(Es), and so we see that[\u03b2] is a non\u2010trivial torsion element in \u03c01. Since, as noted before, [\u03b2]\u22600 also in \u03c01(f(B)\u2216K) and clearly [\u03b2]m=0 in \u03c01(f(B)\u2216K), we see that [\u03b2] is also a non\u2010trivial torsion element in f(B)\u2216K. This is a contradiction and so the original claim holds.\u25a1But now we note thatProposition 4.4\u03a9\u2282R3 be a domain. Then \u03c01(\u03a9) is\u00a0torsion\u2010free.Let Our proof in dimension three relies on the following result of Papakyriakopoulos, see [R3 has torsion\u2010free fundamental group, in particular, it has torsion\u2010free fundamental group at infinity. This yields the proof of Theorem\u00a0By Proposition\u00a0Proof of Theorem 1.1f:B3\u2192f(B3)\u2282R3 be a proper branched covering and denote Y\u2254f(B3). By Proposition K\u2282Y, the fundamental group of Y\u2216K is torsion\u2010free. Thus Y has torsion\u2010free fundamental group at infinity, and the claim follows from Proposition \u25a1Let Remark 4.5f restricted to the open ball B3 is a proper branched covering onto a space which is an open 3\u2010manifold outside one singular point. Furthermore Bf={0}, so in particular the branch is non\u2010empty but compact. Similar examples appear from universal covers of homology spheres. Thus we must assume that the image of f is a manifold. We do remark that we do not know if the Vuorinen question holds for mappings f:B3\u2192N where N is a 3\u2010manifold not necessarily embeddable into R3.As mentioned in the introduction e, e.g.,  for the Remark 4.6essentially properf:B3\u2192R3 with Bf=\u2205 that are not homeomorphisms. In the same paper it is claimed that one can obtain a proper branched covering by restricting the above mapping f to a ball Br with radius r<1 arbitrarily close to 1. However, no detailed argument is provided and the claim is not true for essentially proper branched coverings in general. It turns out that every open continuous map g:Bn\u2192R3 is essentially proper. This can be seen as follows: Let K\u2282g(Bn) be an arbitrary compact set. Since g is open, the sets {g(Br)}r\u2208 form an open covering of K. Thus, by compactness we find a ball Br0 such that B\u00afr0\u2229g\u22121(K) is compact and K=g(B\u00afr0\u2229g\u22121(K)). Especially, the mapping mentioned in (In  it was pioned in  is essenBulletin of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not\u2010for\u2010profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of\u00a0mathematics.The"}
{"text": "Experimental & Molecular Medicine 10.1038/s12276-023-00925-1, published online 13 January 2023Correction to: After online publication of this article, the authors noticed few errors in the Fig. 1b and Fig. 1c that were inadvertently introduced owing to a technical error. The unit for melatonin\u2019s concentration is (pg/mg brain) rather than (ng/mg brain). The concentrations of all groups is divided by 100 to report the exact values as 16.52\u2009\u00b1\u20090.60  rather than 1652\u2009\u00b1\u200960 ; 20.31\u2009\u00b1\u20090.87  rather than 2031\u2009\u00b1\u200987 ; 13.41\u2009\u00b1\u20090.88  rather than 1341\u2009\u00b1\u200988 ; 33.44\u2009\u00b1\u20090.32  rather than 3344\u2009\u00b1\u200932 ; 18.79\u2009\u00b1\u20091.12  rather than 1879\u2009\u00b1\u2009112 ; 24.70\u2009\u00b1\u20090.99  rather than 2470\u2009\u00b1\u200999 . Figure 1C, the value of all the groups is multiplied by 0.05, a missed factor in the calculation, to report the result as 0.36\u2009\u00b1\u20090.01 rather than 7.2\u2009\u00b1\u20090.3 ; 0.38\u2009\u00b1\u20090.02 rather than 7.6\u2009\u00b1\u20090.4 ; 0.69\u2009\u00b1\u20090.08 rather than 13.8\u2009\u00b1\u20091.7 ; 0.18\u2009\u00b1\u20090.01 rather than 3.6\u2009\u00b1\u20090.2 ; 0.39\u2009\u00b1\u20090.02 rather than 7.9\u2009\u00b1\u20090.4; 0.26\u2009\u00b1\u20090.02 rather than 5.4\u2009\u00b1\u20090.5; . The Fig. 1b and 1c are replaced with correct versions.p\u2009<\u20090.01) reduced by constant darkness (29\u2009\u00b1\u20098) but not by melatonin (112\u2009\u00b1\u200927) in comparison to that in the vehicle group .\u201d Is replaced by \u201cImmature OPCs were significantly (**p\u2009<\u20090.01) reduced by constant darkness (29\u2009\u00b1\u20098) but not by melatonin (112\u2009\u00b1\u200927) in comparison to that in the vehicle group (264\u2009\u00b1\u200948).\u201dAuthors also corrected the following typo in section of results: \u201cImmature OPCs were significantly (**The authors apologize for any inconvenience caused.The original article has been corrected."}
{"text": "The bivariate or multivariate distribution can be used to account for the dependence structure between different failure modes. This paper considers two dependent competing failure modes from Gompertz distribution, and the dependence structure of these two failure modes is handled by the Marshall\u2013Olkin bivariate distribution. We obtain the maximum likelihood estimates (MLEs) based on classical likelihood theory and the associated bootstrap confidence intervals (CIs). The posterior density function based on the conjugate prior and noninformative (Jeffreys and Reference) priors are studied; we obtain the Bayesian estimates in explicit forms and construct the associated highest posterior density (HPD) CIs. The performance of the proposed methods is assessed by numerical illustration. It is extremely common that the failure of a product or a system contains several competing failure modes in reliability engineering; any failure mode will lead to the failure result. Competing risks' data contain the failure time and the corresponding failure mode, which can be modeled by the competing risks' model and has been commonly performed in many research fields, such as engineering and medical statistics. Previous studies have mostly assumed the competing failure modes to be independent; Wang et al. , Ren andIn addition to using copula function to handle the relationship between different competing failure modes, the bivariate or multivariate distribution also can be used to account for the correlation between different failure modes. The Marshall\u2013Olkin distribution , which hThe Gompertz distribution is a widely used growth model which has been studied extensively; Ismail  studied In the rest of this paper, we will present the model description and some properties. f is a Gompertz distribution; the density function and reliability function of it are\u03bb is shape parameter and \u03b8 is scale parameter.Suppose that U0, \u2009U1, \u2009and\u2009\u2009U2 are three independent Gompertz variables with different scale parameters, that is, U0 ~ GP, U1 ~ GP, and U2 ~ GP. Let T1=min and T2=min; we obtain T1 ~ GP and T2 ~ GP. Then, the pair of variables  follows the MOGP distribution denoted by  ~ MOGP. When \u03b80=0, the two variables T1\u2009and\u2009T2 are independent and T1\u2009and\u2009T2 will be dependent when \u03b80 > 0; hence, \u03b80 can be regarded as a correlation coefficient between T1\u2009and\u2009T2.Suppose T1, \u2009T2) can be written asThe joint PDF of  are presented in fT1,\u2009T2 is a unimodal function.The surface plots of n identical products into test, and each product has two dependent failure modes with lifetimes T1, \u2009T2,  ~ MOGP. Then, the system lifetime is X=min ~ MOGP. Let \u03b4l0=I(Tl1=Tl2), \u03b4l1=I(Tl1 < Tl2), and \u03b4l2=I(Tl1 > Tl2), for l=1, \u22ef, n, where I(\u00b7) is an indicator function. Then, we can compute n0=\u2211l\u03b4l0, n1=\u2211l\u03b4l1, \u2009n2=\u2211l\u03b4l2, and n=n0+n1+n2.Put l=1, \u22ef, n, \u03b4l0=I(Tl1=Tl2), \u03b4l1=I(Tl1 < Tl2), and \u03b4l2=I(Tl1 > Tl2), We haveFor l=1, \u22ef, n, we have \u03b4l0+\u03b4l1+\u03b4l2=1,For \u03b4l0, \u03b4l1, \u03b4l2) ~ Multinomial, \u03b81/(\u03b80+\u03b81+\u03b82), \u03b82/(\u03b80+\u03b81+\u03b82)).Therefore, . Set the first partial derivation of log\u2009\u2009L about \u03b80, \u2009\u03b81, \u2009\u03b82, \u2009\u03bb to 0, i.e.,The MLEs of \u03b80, \u2009\u03b81, \u2009and\u2009\u03b82 asFrom , we get L, we obtain\u03bb.Substituting \u22022h(\u03bb)/\u2202\u03bb2 < 0, which implies that h(\u03bb) is concave. Some iterative schemes can be used to find the MLE for \u03bb, such as Newton\u2013Raphson algorithm.We can show that \u03b80, \u2009\u03b81, \u2009\u03b82, \u2009and\u2009\u03bb. The Bootstrap method is a resampling method to estimate some statistical characteristics for the unknown parameters by taking samples from the original samples repeatedly; the obtained samples are called Bootstrap samples. This method has a great practical value since it does not need to assume the overall distribution or construct the pivot quantity. We generate the Bootstrap sample by the following three steps:n and observed data , we get the estimates Step 1: for the fixed value of n, x1\u2217, x2\u2217, \u22ef, xn\u2217). Then, get the MLEs Step 2: for the values of M times to obtain M sets of the values Step 3: repeat Step 2 Since it is hard to construct the exact CIs for the unknown parameters, we consider the Bootstrap method to construct CIs for parameters \u03b80, \u2009\u03b81, \u2009\u03b82, \u2009\u03bb at 1 \u2212 \u03b3 confidence level asBased on the Bootstrap sample and by percentile Bootstrap (Boot-P) method, we construct the Boot-P CIs for \u03bb is known. Denote \u03b8=\u03b80+\u03b81+\u03b82, which has a Gamma prior with hyperparameters a and b asIn this section, we suppose the shape parameter \u03b80/\u03b8+\u03b81/\u03b8+\u03b82/\u03b8=1, so given \u03b8,  follows a Dirichlet prior with hyper parameters c0, \u2009c1, and c2, that is,Due to \u03b80,\u2009\u03b81, \u2009and\u2009\u03b82 becomesc=c0+c1+c2.Therefore, the joint prior of \u03b80,\u2009\u03b81, \u2009\u03b82) asAccording to Jeffreys , Jeffreyni=n \u00b7 \u03b8i/(\u03b80+\u03b81+\u03b82), \u2009i=0, \u20091, \u20092, so I can be written asFrom Thus, the Jeffreys prior is given by\u03c02, the joint posterior distribution of  is proper.Based on the Jeffreys prior \u03b80,\u2009\u03b81, \u2009\u03b82) based on \u03c02 asFrom  and 7),,7), we o\u03c02 with respect to \u03b80,\u2009\u03b81, \u2009and\u2009\u03b82, we obtainA=\u2211i=1n(e\u03bbxi \u2212 1) and B is a beta function.Integrating \u03b80,\u2009\u03b81, \u2009\u03b82) based on \u03c02 is proper.Thus, the joint posterior distribution of  to  is one-to-one with the inverse transformation \u03b80=\u03bc0\u03bc1, \u03b81=\u03bc0\u03bc2, and \u03b82=\u03bc0(1 \u2212 \u03bc1 \u2212 \u03bc2). The Jacobian matrix of the transformation has the formBernardo  and BergThe likelihood function  becomes can be written asThe Fisher information matrix of } and {, \u03bc0}, the reference priors are the same, which is given by \u03b80,\u2009\u03b81, \u2009\u03b82) is Under the ordering groups {\u03bc0, \u2009\u03bc1, \u2009\u03bc2}, {\u03bc0, \u2009\u03bc2, \u03bc1}, {\u03bc1, \u2009\u03bc0, \u2009\u03bc2}, and {\u03bc1, \u2009\u03bc2, \u2009\u03bc0}, the reference priors are the same, which is given by \u03b80,\u2009\u03b81, \u2009\u03b82) is Under the ordering groups {\u03bc2, \u2009\u03bc0, \u2009\u03bc1} and {\u03bc2, \u2009\u03bc1, \u03bc0}, the reference priors are the same, which is given by \u03b80,\u2009\u03b81, \u2009\u03b82) is Under the ordering groups {(i)\u03bc0, \u2009\u03bc1, \u2009\u03bc2) isThe Fisher information matrix of } and {, \u03bc0} is the same as in [The reference prior for the ordering groups {me as in , which i(ii)I1 isThe inverse of (iii)h1=1/\u03bc02, h2=1/\u03bc1(1 \u2212 \u03bc1), and h3=(1 \u2212 \u03bc1)/(\u03bc2(1 \u2212 \u03bc1 \u2212 \u03bc2)).According the notations in , we obtak={|ak0 < \u03bc0 < bk0, \u2009ak1 < \u03bc1, \u2009ak2 < \u03bc2, \u2009\u03bc1+\u03bc2 < dk}, such that ak0, \u2009ak1, \u2009ak2\u27f60, bk0\u27f6\u221e, and dk\u27f61, as k\u27f6\u221e. Then, we haveChoose the compact sets \u03a9where Then, we get the reference prior as\u03bc0\u2217, \u2009\u03bc1\u2217, \u2009\u03bc2\u2217) is an inner point of \u03a9k.where .Similarly, under the ordering group {\u03bc1, \u2009\u03bc0, \u2009\u03bc2} isThe Fisher information matrix of {I2 isThe inverse of h1=1/\u03bc1(1 \u2212 \u03bc1), h2=1/\u03bc02, and h3=(1 \u2212 \u03bc1)/(\u03bc2(1 \u2212 \u03bc1 \u2212 \u03bc2)).Similarly, we obtain k={|ak0 < \u03bc1, \u2009ak1 < \u03bc0 < bk1, \u2009ak2 < \u03bc2, \u2009\u03bc1+\u03bc2 < dk}, such that ak0, \u2009ak1, \u2009ak2\u27f60, bk1\u27f6\u221e, and dk\u27f61, as k\u27f6\u221e. Then, we haveChoose the compact sets \u03a9where \u03bc1\u2217, \u2009\u03bc0\u2217, \u2009\u03bc2\u2217) be an inner point of \u03a9k; we get the reference prior asLet .Similarly, under the ordering group {(v)\u03bc2, \u2009\u03bc1, \u2009\u03bc0} isThe Fisher information matrix of {I3 isThe inverse of h1=1/\u03bc2(1 \u2212 \u03bc2), h2=(1 \u2212 \u03bc2)/(\u03bc1(1 \u2212 \u03bc1 \u2212 \u03bc2)), and h3=1/\u03bc02.Then, we obtain k={|ak0 < \u03bc2, \u2009ak1 < \u03bc1, \u2009\u03bc2+\u03bc1 < dk, ak2 < \u03bc0 < bk2}, such that ak0, \u2009ak1, \u2009ak2\u27f60, bk2\u27f6\u221e, and dk\u27f61, as k\u27f6\u221e. Then, we haveChoose the compact sets \u03a9\u03bc2\u2217, \u2009\u03bc1\u2217, \u2009\u03bc0\u2217) be an inner point of \u03a9k, we obtain the reference prior asLet . According to the one-to-one transformation from  to , we can obtain the reference priors \u03c02, \u03c03, \u03c04 from \u03c9R1, \u03c9R2, and\u2009\u03c9R3, respectively.Similarly, under the ordering group {\u03c03 and \u03c04, the posterior distributions of  are proper.Based on the reference priors \u03b80,\u2009\u03b81, \u2009\u03b82) based on reference prior \u03c03 and \u03c04 are, respectively, asThe joint posterior distributions of  and \u03c04 with respect to \u03b80,\u2009\u03b81, \u2009and\u2009\u03b82, respectively, we obtainIntegrating \u03b80,\u2009\u03b81, \u2009\u03b82) based on \u03c03 and \u03c04 are proper.Thus, the posterior distributions of  based on \u03c01, \u2009\u03c02, \u2009\u03c03, \u2009and\u2009\u03c04 are, respectively, asw1=\u0393(\u2211i=02ci)ba\u2212c0\u2212c1\u2212c2/\u0393(a) and w2=\u220fi=02bci/\u0393(ci).The joint posterior distributions of , we, we12), \u03b80, \u2009\u03b81, \u2009\u03b82, \u2009and\u2009\u03b8 can be constructed by the Monte Carlo method studied by Chen and Shao [The HPD credible intervals of parameters and Shao .n and the observed data , compute the Bayesian estimates of \u03c01, \u2009\u03c02, \u2009\u03c03, \u2009and\u2009\u03c04, respectively.Step 1: given the value of M times; we obtain M sets of the values \u03c01, \u2009\u03c02, \u2009\u03c03, \u2009and\u2009\u03c04, respectively. Arrange them in the ascending order, we obtainStep 2: repeat Step 1 \u03b3 confidence level asStep 3: compute the CIs at 1 \u2212 \u2009 Step 4\u03b8v, v=0,1,2, and \u03b8 are the shortest intervals among w=1,\u2009\u20092, \u22ef, M \u2212 (1 \u2212 \u03b3)M, respectively.: the HPD CIs for \u03bb is known. The initial values for parameters  are . The initial values for the hyperparameters a, \u2009b, \u2009c0, \u2009c1, \u2009and\u2009c2 are all 0.001. Take the sample size n\u2009=\u200910, 20, 30, and 50. Generate the random samples  from MOGP by the following steps:n, generate n samples u01, u02, \u22ef, un0 from GP, u11, u12, \u22ef, un1 from GP, and u21, u22, \u22ef, un2 from GP. Then, we obtain Tl1=min and Tl2=min, \u2009l=1,2, \u22ef, n.\u2009 Step 1: for a fixed value xl, \u03b4l0, \u03b4l1, \u03b4l2), \u2009l=1,2, \u22ef, n, where xl=min, \u03b4l0=I(Tl1=Tl2), \u03b4l1=I(Tl1 < Tl2), and \u03b4l2=I(Tl1 > Tl2).\u2009 Step 2: compute  of the MLEs, the average lengths (ALs), and coverage probabilities (CPs) of the 95% Boot-P CIs, and the MSEs of the Bayesian estimates, the ALs, and CPs of the 95% HPD CIs, which are shown in Table 2Table 2n, the Bayesian estimates based on \u03c01, \u03c02, \u2009and\u2009\u03c04 are smaller than the MSEs of MLEs. The MSEs of Bayesian estimates of \u03b80\u2009and\u2009\u03b82 based on \u03c04 are smaller than that based on \u03c01, \u03c02, \u2009and\u2009\u03c03. The MSEs of Bayesian estimates of \u03b81 based on \u03c03 are smaller than that based on \u03c01, \u03c02, \u2009and\u2009\u03c04. The MSEs of Bayesian estimates of \u03b8 based on \u03c01 are smaller than that based on \u03c02, \u03c03, \u2009and\u2009\u03c04.The MSEs of MLEs and Bayesian estimates decrease as the sample size increases. For given sample size The CPs of Boot-P and HPD CIs are all close to 0.95. The ALs of Boot-P and HPD CIs decrease; the associated CPs increase when the sample size increases. The CPs of HPD CIs based on Bayesian estimates are larger than the CPs of Boot-P CIs based on MLEs.\u03bb, \u03b80, \u03b81, \u03b82) as , we use the procedures mentioned above to generate U0, \u2009U1, \u2009and\u2009U2 from GP, GP, and GP, respectively. We then get T1=min and T2=min; the latent lifetime of the system is min. The simulated data are listed in \u03b80, \u2009\u03b81, \u2009\u03b82, \u2009and\u2009\u03b8 are shown in \u03b80, \u2009\u03b81, \u2009\u03b82, \u03b8) are close to the true value.For illustrative purposes, with initial value for parameters .\u03b80, \u2009\u03b81, \u2009\u03b82, \u2009and\u2009\u03b8 are shown in The MLEs, Bayesian estimates, and associated 95% CIs for parameters This paper discussed the point estimates and CIs for the parameters of the dependent competing risks' model from MOGP distribution. We studied the appropriateness of the posteriors based on conjugate prior and Jeffreys and Reference priors, obtained the Bayesian estimates in closed forms, and constructed the associated HPD CIs. From the simulations results, the use of the Bayesian method can be recommended if the priors are available. The results of the illustrative analysis show that the proposed methods work well; from the lengths of CIs, we can conclude the Bayesian estimates are better than MLEs in general."}
{"text": "In the digital era, the health information presented on virtual platforms plays a pivotal role in supporting people\u2019s active and healthy life. The ageing, especially ageing women, are more likely to seek and accept health information through online media platforms. The study shows that short health videos on social media platforms are extremely popular among ageing women in China for the accessing of virtual coaching. Adopting the qualitative methodology of in-depth interview and discourse analysis, the study investigates virtual coaching with short health video practice among 39 Chinese ageing women in different fields, who are all over sixty years old. Specifically, with the analytical tools of transitivity and generic structure analysis, the study explores the impacts of short health videos on Chinese ageing women\u2019s cognition, behavior and interpersonal relationships. The result shows that virtual practice and coaching via short health videos can build health awareness and a dynamic new lifestyle, and motivate women to positively practice physical activity and maintain positive interpersonal relationships. Factors affecting the effectiveness of short health videos are discussed for future research in the field of modeling and intervention. As a convenient and effective means of conveying health and medical information to the public, short health videos on social media platforms have been popular among the ageing in China, penetrating into their daily life and behavioral interactions. Exposed to digital platforms, ageing women, who are reported as more relationship-oriented or more vulnerable in some specific health areas in the existing research ,2,3, areRegarding the topics of short health videos and ageing groups\u2019 virtual health behavior, scholars at home and abroad have studied the topic from different perspectives.The research on short health videos has been conducted from multiple perspectives. From the perspective of current and developmental situations, scholars ,5 have fWith the penetration of information technology, the ageing are unconsciously involved in the digital wave. Especially, the sense of familiarity and presence created by short video platforms further strengthens their participation. This positive and convenient network interaction is beneficial not only to their individual health but also to their emotional connection with others. Specifically, the viewing and posting of short videos can create opportunities to alleviate social isolation and loneliness among the ageing. Especially when participating in a common activity with family members, friends and other network members, the ageing can meet their needs for information, interpersonal relationships, and social life, which reduces their loneliness and has a positive impact on their individual behavioral tendencies ,14,15,16In terms of acceptance behaviors, a survey finds that the ageing are mainly affected by four factors when accepting information technology: needs satisfaction, perceived usability, support availability, and public acceptance, among which needs satisfaction and support availability are relatively more important . ManaginThe social and emotional support from family members and friends, as well as the extension of provided network communication, further affects the social relations of the ageing group . In termIn the existing literature, research on short video users mainly focuses on the youth group, and those on the ageing mainly concern the digital gap, intergenerational compensation, life improvement, etc., among which health videos are briefly overviewed as a tool element. In addition, current research on the health of ageing women is mainly concentrated on the fields of sports, medicine and others, focusing on the impact of physiological functions on their body, psychology and behavior, focusing on the specific topics of cancer, sex, suicide and depression ,25. In aTherefore, based on the existing literature, the study, taking Chinese ageing women as an important group in the video-based information audience, investigates the impacts of virtual practice and coaching using short videos from the perspective of cognition, behavior and interpersonal relationships.In order to gain a firsthand understanding of Chinese ageing women\u2019s cognition, behavior and interpersonal relationship with virtual practice and coaching via short videos, 47 participants were recruited by 40 postgraduates at the School of Journalism and Communication, Shanxi University during the winter vacation of 2022. These postgraduates have professional knowledge and skills in qualitative methods in social studies and were informed of the research objectives before the interview. Most of the participants are the postgraduates\u2019 family members or relatives or neighbors, who are mostly from the urban areas or the different counties of Shanxi Province in China. The inclusion criteria for selection were: (1) female over sixty years; and (2) possessing a smart phone and having had the experience of viewing the short health videos for at least half a year.The exploration of social media interaction can be approached by various methods, among which qualitative research has been widely and continuously chosen by researchers . Two typIn-depth interview is a widely employed and valued qualitative research methodology which involves comprehensive conversations between the researcher and interviewee, with a task-based exploration of the interviewee\u2019s subjective experiences, meaning-making, accounting processes, and unspoken assumptions about life and the social world in general . An in-dIn-depth interviews provide access to the context of people\u2019s behavior, opening a way for the researcher to understand the meaning of that behavior. A basic assumption in in-depth interviewing research is that the meaning people make of their experience affects the way they live that experience .The information from the interviewees is very useful for understanding the current situation regarding Chinese ageing women\u2019s engagement with short health videos. The questions specifically asked during the interviews are: (1) What are the reasons for watching short videos? (2) Do you think short videos are beneficial? Has there been any change in your health awareness? (3) Have you made any adjustment to your living or eating habits after watching the short videos? (4) Will you follow the content or advice of the short videos? If not, what factors affect this? (5) Would you like to share or recommend the short videos with/to your family members or friends? Who do you usually share with? By forwarding the links or by recounting the information? (6) Has watching short videos influenced or improved your family life? Has there been any change or improvement in building your family relationships?Discourse analysis is the discipline devoted to the investigation of the relationship between form and function in verbal communication , in whicRegarding the theoretical and methodological framework for analyzing the interviewees\u2019 narratives, the study takes a discourse-oriented perspective based on Systemic Functional Linguistics ,32 and CThe central idea in Systemic Functional Linguistics is that language is a semiotic system of choice making. Transitivity analysis carries out the description of the representation of social events and social actors and construes the experiential meaning through clauses that are constituted of participants, processes and circumstances as is displayed in Genre is defined as \u201ca socially ratified way of using language in connection with a particular type of social activity\u201d  (p. 14).Starting by examining the transitivity patterns and generic structures of the interviewees\u2019 utterances based on the discourse analytic process, an interpretation and explanation of the cognitive, behavioral and interpersonal impacts of the virtual practice and coaching of short health videos on Chinese ageing women will be categorized and summarized. The detailed procedures for the study are displayed in During the data collection stage, the data of eight participants were excluded due to missing crucial information for the analysis. Thus, the final sample comprises 39 Chinese women, whose age ranges from 60 to 80 years (Mean = 64.5). The basic information on the participants is listed in Educational level and job experience. The participants have varied educational degrees: 18 of them have high school education, six have a college degree, 10 have middle school educational level and five only have the experience of primary education. Most of them have job experience, as is shown in Health situation. Living conditions. Most of the participants live with family, except that No. 15 lives alone and No. 37 lives with three room mates in a dormitory of a hotel where she works as a cook.Practice with short health videos. In general, the participants have an intensive and varied use of smart phones, including viewing short videos for at least between half a year to five or six years. The daily frequency for practicing with short videos is relatively high, from half an hour to three or four hours a day. The social media platforms the participants usually rely on include Kwai, Tik Tok, WeChat Short Video, Toutiao, etc., among which Kwai and Tik Tok are much more popular. At the beginning of their practice with social media platforms, most of them report that their children or grandchildren offered coaching or help with downloading the APPs. The contents of the short videos they mainly focus on include senile diseases, such as high blood pressure and diabetes, aerobics or dancing, and healthy eating. The providers of the videos include two rough categories: official medical institutions, and individuals and relevant groups. The former is mainly composed of health agencies and hospital accounts, and the latter includes both professional and non-professional health practitioners.a little bit)\u201d, No. 39 responds \u201cm\u00e9iy\u01d2u du\u014dd\u00e0de bi\u00e0nhu\u00e0 BA (not much has been changed BA)\u201d.The analysis of the sample to assess the cognitive change of the interviewees shows that 32 give the definitely positive answer that virtual practice and coaching via short videos are beneficial, and their cognition regarding health problems has changed after watching the videos. One interviewee (No. 12), a doctor in a village, gives the negative answer that there is no change in her cognition. The rest of the interviewees respond positively but in a partial degree or hesitation; for example, No. 27 answers \u201cdu\u014dsh\u01ceo y\u01d2u y\u00ecdi\u01cen (As is exemplified in think) or \u201cr\u00e8nw\u00e9i\u201d (think) and the attributive relational process \u201csh\u00ec\u201d (be) are the most recurring and important meaning resources that enable the interviewees to depict their cognitive state. The circumstances are mainly the positive comments on the effect of watching short videos, such as \u201ct\u01d0ngh\u01ceo\u201d (very good), \u201cy\u01d2u d\u00e0ol\u01d0\u201d (reasonable), \u201dy\u01d2uy\u00ec\u201d, \u201cy\u01d2uh\u01ceoch\u00f9\u201d, \u201csh\u00f2uy\u00ecf\u011biqi\u01cen\u201d , \u201cy\u01d2ub\u0101ngzh\u00f9\u201d (helpful), \u201cy\u01d2usu\u01d2g\u01ceibi\u00e0n\u201d (changed). More specifically, the interviewees construe their cognitive change by using the mental process of \u201czh\u00f9y\u00ec\u201d (pay attention to), \u201cgu\u0101nzh\u00f9\u201d (pay attention to), \u201czh\u01cengw\u00f2le\u201d (acquire), \u201czh\u012bd\u00e0o\u201d (know), \u201cb\u0101ngzh\u00f9 w\u01d2 li\u01ceoji\u011b le\u201d (help me understand). The behavioral processes, such as \u201cc\u01cein\u00e0\u201d (adopt) and \u201cf\u01cengzh\u00e0o\u201d (imitate) are graded ones in the interviewees\u2019 cognitive change, i.e., they begin to accept useful information and take actions after virtual coaching on the short video health platforms.The cognitive mental process \u201cju\u00e9de\u201d , eating habits and other aspects in the interviewees\u2019 narration. In this part of the statement, as is best exemplified in the following two transcriptions of No. 20 and No. 30, the main type of generic structure is exemplification, in which the contrasts between \u201cy\u01d0qi\u00e1n\u201d or \u201czh\u012bqi\u00e1n\u201d (before) and \u201cxi\u00e0nz\u00e0i\u201d (now) or after watching a short video are made.Y\u01d0qi\u00e1n BA, b\u00fa\u00e0i ch\u016bq\u00f9, \u00e0i z\u00e0iji\u0101l\u01d0 d\u0101izhe. Xi\u00e0nz\u00e0ile k\u00e0nle y\u01d0h\u00f2u, r\u00e8nw\u00e9i r\u00e9nji\u0101 shu\u014dde h\u00e1i t\u01d0ngdu\u00ec de, y\u01d2uxi\u0113 f\u00e1nx\u012bnsh\u00ec, ch\u016bq\u00f9 h\u00e9 bi\u00e9r\u00e9n du\u014d ji\u0101oli\u00fa ji\u0101oli\u00fa, du\u014d w\u00e1nwan, du\u014d du\u00e0nli\u00e0n du\u00e0nli\u00e0n, hu\u00edji\u0101h\u00f2u zh\u00e8ge f\u00e1nx\u012bnsh\u00ec ji\u00f9 w\u00e0ngle, t\u01d0ngh\u01ceode. Xi\u00e0nz\u00e0i w\u01d2 m\u011biti\u0101n z\u01ceosh\u00e0ng q\u01d0l\u00e1i sh\u01d2uxi\u0101n h\u0113 y\u012bb\u0113i w\u0113nk\u0101ishu\u01d0, r\u00e1nh\u00f2u chu\u0101nd\u00e0ih\u01ceo ji\u00f9 h\u00e9 l\u00ednj\u016bmen q\u012bdi\u01cenl\u00e1izh\u014dng q\u00f9 c\u016bnw\u00e0i m\u00e0nz\u01d2u. W\u01d2 y\u01d0qi\u00e1n ch\u012bf\u00e0n b\u01d0ji\u00e0o k\u01d2uw\u00e8izh\u00f2ng, xi\u00e0nz\u00e0i b\u00f9g\u01cen k\u01d2uw\u00e8i zh\u00f2ngle, g\u00e0o ji\u0101l\u01d0der\u00e9n y\u011bsh\u00ec, ch\u012bf\u00e0n y\u00e0o sh\u01ceoch\u012by\u00e1n, k\u01d2uw\u00e8i ch\u012bde q\u012bng d\u00e0n y\u012bdi\u01cen, p\u00edngsh\u00ed z\u00e0i y\u01d0nsh\u00ed sh\u00e0ng sh\u01ceoch\u012b y\u00f3un\u00ecde. P\u00edngsh\u00ed ch\u012bx\u00ecli\u00e1ng, ch\u012bxi\u0113 sh\u00e0nsh\u00ed xi\u0101nw\u00e9i, b\u00fay\u00e0o l\u01ceoch\u012b b\u00e1imi\u00e0n, d\u00e0m\u01d0; du\u014dch\u012b qi\u00e1om\u00e0imi\u00e0n, y\u00f9m\u01d0mi\u00e0n, f\u01cenzh\u00e8ngsh\u00ec du\u014dch\u012b c\u016bli\u00e1ng BA. (No. 20)BeforeAfter watching it, I think what they say is quite right. When there are some annoying things, go out to communicate more with others, play more exercises, I will forget the annoying things after returning home, which is good. Now after I wake up every morning, first I will drink a glass of warm water, then get dressed and go out for a slow walk outside the village at seven o\u2019clock with my neighbors. I used to have a heavy taste in eating, but now I don\u2019t dare to have a heavy taste. I also tell my family that we should eat less salt and less greasy food. Usually we eat refined grains, we should eat some dietary fiber, such as more buckwheat noodles, cornmeal, anyway, eat more coarse grains, instead of white noodles or rice all the time., I didn\u2019t like going out, I liked staying at home. zh\u012bqi\u00e1n k\u01d2uw\u00e8i pi\u0101nxi\u00e1n, ch\u01ceoc\u00e0i zu\u00f2 f\u00e0n sh\u00edy\u00e1n hu\u00ec f\u00e0ngdedu\u014d, xi\u00e0nz\u00e0i y\u01d0j\u012bng sh\u01ceoch\u012b sh\u00edy\u00e1n, h\u00e1iy\u01d2u x\u012bnl\u00e0 c\u00ecj\u00ec de sh\u00edw\u00f9, y\u00f3un\u00ec de r\u00f2ul\u00e8i ch\u012bde y\u011bsh\u01ceole, ch\u012bde sh\u016bc\u00e0i, h\u00falu\u00f3bo, d\u00f2uzh\u00ecp\u01d0n hu\u00ec du\u014dy\u012bxi\u0113. (No. 30)B\u01d0r\u00fa z\u00e0i sh\u0113nghu\u00f3 x\u00edgu\u00e0n f\u0101ngmi\u00e0n, shu\u00ecji\u00e0o zh\u012bqi\u00e1n y\u00f2ng r\u00e8shu\u01d0 p\u00e0oji\u01ceo, y\u012bb\u0101n hu\u00ec p\u00e0o b\u00e0ng\u00e8du\u014d xi\u01ceosh\u00ed, zu\u00f2yizu\u00f2 sh\u01d2ub\u00f9 y\u01d0j\u00ed t\u00f3ub\u00f9 de \u00e0nm\u00f3 du\u00e0nli\u00e0n, y\u01d2uzh\u00f9y\u00fa b\u0101ngzh\u00f9 shu\u00ecmi\u00e1n; z\u00e0i y\u01d0nsh\u00ed x\u00edgu\u00e0n f\u0101ngmi\u00e0n, before I prefer the salty taste and like putting more salt in cooking. Now I eat less salty, spicy food and less greasy meat, and eat more vegetables, carrots and soy products.For example, in terms of living habits, I wash my feet in hot water before going to bed, usually for more than half an hour, and do a massage on hands and head to help sleep; in terms of eating habits, The changes in living habits narrated by the ageing women are displayed as different aspects regarding their health, such as time rescheduling, as in No. 14\u2032s narration \u201cw\u01censh\u00e0ng y\u00e0o z\u01ceoshu\u00ec, b\u00f9y\u00e0o ch\u0101ogu\u00f2 sh\u00edy\u012bdi\u01cen\u201d , \u201cg\u0113nzhe sh\u00ecp\u00edn zu\u00f2 j\u01d0ngzhu\u012bc\u0101o\u201d . The reasons for virtual coaching are mainly because they have diseases  or they are not strong (as No. 39). Most of them can follow virtual coaching to exercise for a long term . In the interviewees\u2019 narration, the effects of exercise through virtual coaching are obvious, such as \u201cy\u01d2u xi\u00e0ogu\u01d2\u201d (effective), \u201ct\u01d0ng sh\u00edj\u00ec\u201d , \u201ct\u01d0ng gu\u01ceny\u00f2ng\u201d (very helpful), \u201ct\u01d0ng y\u01d2u j\u012bngsh\u00e9n\u201d (very energetic), \u201ch\u01ceole\u201d (improved), \u201cq\u00edngku\u00e0ng h\u01ceodu\u014dle\u201d (much better), \u201dhu\u01cenji\u011b\u201d (relieved). Expectations are optional in their statement, but generally they expect that virtual coaching will be effective and they can be healthier.By observing the sample, as to the question whether they can follow the virtual coaching in order to exercise, the study finds that most of the interviewees can do so, as is indicated in I should make a change, but I can\u2019t control myself. I can\u2019t change it. It seems that I am still too young); No. 16, who is 72, says \u201cL\u01cende zu\u00f2, yu\u00e1ny\u012bn ji\u00f9sh\u00ec sh\u0113nt\u01d0 h\u00e1ik\u011by\u01d0, y\u00e0osh\u00ec j\u012bngch\u00e1ng sh\u0113ngb\u00ecng, k\u011bnd\u00ecng ji\u00f9 g\u00e8ng zh\u00f9zh\u00f2ng y\u012bxi\u0113\u201d .As for the interviewees who don\u2019t follow virtual coaching; the reasons can be attributed to their relatively young age and strong and healthy body, for example, No. 15, who is 63, states \u201c\u00c0nl\u01d0 y\u012bngg\u0101i g\u01ceibi\u00e0n, d\u00e0nsh\u00ec z\u00ecj\u01d0 k\u00f2ngzh\u00ec b\u00f9zh\u00f9, g\u01ceibi\u00e0n b\u00f9li\u01ceo, k\u00e0nl\u00e1i h\u00e1ish\u00ec ni\u00e1nl\u00edng xi\u01ceo degu\u00f2\u201d  share by forwarding the link, 8 (22%) use oral narration, 16 (44%) use both of the above, and 32 interviewees believe that viewing or following these videos has a positive impact on improving family life.The observation of transitivity finds the processes adopted by participants are mainly material processes, that is, the action of sharing, or verbal processes, the action of telling. The analysis of process types, Goal or Receiver (the one to which/whom the process is extended), and adverbial elements can best display the interviewees\u2019 sharing activities.ji\u01cenji\u00e9 y\u00ecdi\u01cen ji\u00f9sh\u00ec, y\u01d2uy\u00f2ng de, zh\u00f2ngy\u00e0o de, f\u0113nxi\u01ceng g\u011bi q\u00edt\u0101 r\u00e9n\u201d , \u201cg\u0113n bi\u00e9r\u00e9n sh\u00ecp\u00edn, t\u00e0nt\u01ceo du\u01cen sh\u00ecp\u00edn l\u01d0 de y\u01cengsh\u0113ng zh\u012bshi\u201d , \u201cg\u011bi t\u0101men ji\u01cengsh\u00f9 y\u00edbi\u00e0n, hu\u00f2zh\u011b g\u011bi t\u0101men sh\u00ecf\u00e0n y\u00edxi\u00e0 zh\u00e8ge d\u00f2ngzu\u00f2\u201d , t\u014dngch\u00e1ng hu\u00ec zhu\u01cenf\u0101 d\u00e0o q\u00fanl\u01d0, t\u00f3ngsh\u00ed y\u011bhu\u00ec shu\u014dj\u01d0j\u00f9 .36 of the interviewees would like to share useful videos to others. As to the processes, besides unilaterally recommending, sharing or forwarding the information, 53% would take part in more communication, for example, \u201ct\u00f3ngxu\u00e9q\u00fan, ch\u016bzh\u014dng t\u00f3ngxu\u00e9, sh\u012bf\u00e0n t\u00f3ngxu\u00e9, t\u00f3ngsh\u00ec\u201d , \u201cy\u00edg\u00e8 w\u00e9ich\u00edle s\u00ecni\u00e1n de d\u00e0yu\u0113, 50 r\u00e9n de f\u011bns\u012bq\u00fan\u201d , \u201cy\u012bku\u00e0i d\u01cep\u00e1i de p\u00e9ngy\u01d2umen\u201d . Among the Goals or Receivers, the participants also target their information to certain groups, for example, \u201cy\u00f3uq\u00ed sh\u00ec b\u00fa sh\u00e0ngb\u0101n de r\u00e9n\u201d ,\u201d \u201cW\u01d2 m\u00e8imei ma, t\u0101 b\u00fash\u00ec t\u00e1ngni\u00e0ob\u00ecng\u2026. Du\u00ec t\u0101 y\u01d2uy\u00ec, w\u01d2 ji\u00f9 g\u011bi t\u0101 f\u0101gu\u00f2q\u00f9 le.\u201d .The Goals or Receivers are not limited to their family members or relatives. The interviewees would share or communicate with a wide range of people, for example, \u201cf\u0101g\u011bi d\u014duy\u00e0o f\u0101ch\u016bq\u00f9 ne\u201d \u201cY\u00f3uq\u00ed\u201d used by No. 4 (a teacher) and \u201ck\u011b\u00e0i zhu\u01cenf\u0101 YA\u201d \u201ct\u00e8bi\u00e9 \u00e0i f\u0101\u201d, \u201cm\u011biti\u0101n d\u014duy\u00e0o f\u0101, m\u00e9iy\u01d2u y\u00ecti\u0101n b\u00f9 f\u0101 de\u201d used by No. 10  indicate their great desire and willingness to share the videos which they think useful and helpful to their family members, relatives and friends.3.d\u014duy\u00e0o f\u0101ch\u016bq\u00f9 NE. Y\u00f3uq\u00ed sh\u00ec b\u00fa sh\u00e0ngb\u0101n de r\u00e9n w\u01d2 ji\u00f9 g\u011bi du\u014d f\u0101xi\u0113 li\u00e0nji\u0113, y\u011br\u00e0ng t\u0101men z\u00e0iji\u0101 bi\u0101n k\u0101nh\u00e1izi bi\u0101n du\u00e0nli\u00e0n sh\u0113nt\u01d0. (No. 4)F\u0101z\u00e0i p\u00e9ngy\u01d2uqu\u0101n le h\u00e9 q\u012bny\u01d2umen de q\u00fanl\u01d0, h\u00e1iy\u01d2u hu\u00ec f\u0101g\u011bi t\u00f3ngxu\u00e9q\u00fan, ch\u016bzh\u014dng t\u00f3ngxu\u00e9, sh\u012bf\u00e0n t\u00f3ngxu\u00e9, t\u00f3ngsh\u00ec, I send  to the moments (on Wechat) and all the groups, including the groups of relatives and friends, junior middle school classmates, normal college classmates, and colleagues. Especially for those who are not working, I send more links, and let them exercise at home while taking care of their grandchildren.4.\u00e0i f\u0101b\u00f9 YA. Ji\u00f9sh\u00ec n\u00e0 sh\u00edh\u00f2u g\u0101ngg\u0101ng n\u00e1zhe sh\u01d2uj\u012b w\u00e1n MA HA, AIYA k\u011b xi\u01ceng zhu\u01cenf\u0101 NE, h\u01ceoxi\u00e0ng bi\u00e9r\u00e9n d\u014du k\u00e0n b\u00f9ji\u00e0n, w\u01d2 g\u01cenj\u01d0n b\u01ce zh\u00e8ge zhu\u01cenf\u0101 ch\u016bq\u00f9 HA\u2026. h\u01ceoxi\u00e0ng yu\u00e1nl\u00e1i f\u01cenzh\u00e8ng t\u00e8bi\u00e9 \u00e0i f\u0101, m\u011biti\u0101n d\u014duy\u00e0o f\u0101, m\u00e9iy\u01d2u y\u00ecti\u0101n b\u00f9 f\u0101 de, y\u01d2ush\u00edh\u00f2u h\u00e1i f\u0101 h\u01ceoj\u01d0ti\u00e1o NE. En f\u0113nxi\u01ceng d\u00e0o p\u00e9ngy\u01d2u qu\u0101n HA, f\u0101ch\u016bq\u00f9. \u2026 ji\u00f9sh\u00ec g\u0101ng n\u00e1sh\u00e0ng sh\u01d2uj\u012b w\u00e1nde sh\u00edh\u00f2u k\u011b xi\u01ceng f\u0101 NE. AIYA, k\u011b y\u00e0o f\u0101 NE, f\u0101 g\u011bi zh\u00e8ge f\u0101 g\u011bi n\u00e0ge. (No. 10)\u2026 w\u01d2 yu\u00e1nl\u00e1i y\u011bsh\u00ec, \u2026 I liked forwarding so much, and loved posting. When I was first playing with my mobile phone, I loved forwarding so much. It seemed that no one else could see it, so I quickly forwarded. \u2026It seemed that I really loved posting, every day. Sometimes I posted several pieces in one day. Well, I shared them on moments, sending all of them out. \u2026When I first got my phone, I just wanted to post. I wanted to forward links, to this one or to that one.The adverbs of degree or the mood particles used by the interviewees can clearly display their attitudes towards interpersonal behavior. Referring to their sharing activities, in the following transcriptions, the expressions \u201cEveryone feels very happy. No. 23), \u201cM\u00e0nman p\u00e9ngy\u01d2u y\u011b bi\u00e0ndu\u014dle\u201d , \u201cG\u0113n t\u0101de ji\u0101oli\u00fa du\u014dle y\u01d0h\u00f2u, xi\u01ceox\u00edfuer zh\u012bd\u00e0ole w\u01d2de y\u00f2ngy\u00ec, du\u00ec w\u01d2 y\u011by\u01d2ule g\u01ceigu\u0101n.\u201d .The analysis of the sample also shows that sharing short health videos improve the interviewees\u2019 family life, and 32 of the interviewees give an affirmation and positive comments, for example, \u201cD\u00e0ji\u0101 d\u014du ju\u00e9de t\u01d0ng k\u0101ix\u012bn de.\u201d  use the expressions \u201csh\u0101ow\u0113i y\u01d2uxi\u0113\u201d and \u201csh\u0101ow\u0113i g\u01ceish\u00e0nle y\u00ecdi\u01cen BA\u201d to indicate very few or slight improvements in their family relationship. Four indicate that the sharing of the videos doesn\u2019t have any or much impact on family life or relationship: \u201cm\u00e9iy\u01d2u sh\u00e9nme bi\u00e0nhu\u00e0\u201d , \u201cbi\u00e0nhu\u00e0 b\u00fad\u00e0\u201d , \u201cdu\u00ec ji\u0101t\u00edng sh\u0113nghu\u00f3 y\u01d0ngxi\u01ceng b\u00fad\u00e0 BA\u201d , and \u201cm\u00e9iy\u01d2u t\u00e0id\u00e0de y\u01d0ngxi\u01ceng\u201d . However, No. 26 adds that the change mainly lies in diet (y\u01d0nsh\u00ed f\u0101ngmi\u00e0n hu\u00ec y\u01d2u y\u01d0ngxi\u01ceng) and No. 33 admits that the change is mainly reflected in the individual (zh\u01d4y\u00e0o sh\u00ec du\u00ec g\u00e8r\u00e9n de y\u01d0ngxi\u01ceng).Sometimes I get addicted to watching videos and spend a long time, which affects my family members. No. 21), \u201cg\u0113n ji\u0101r\u00e9n de ji\u0101oli\u00fa bi\u00e0n sh\u01ceole, xi\u00e1nxi\u00e1 sh\u00edji\u0101n j\u012bb\u011bn hu\u0101z\u00e0ile k\u00e0nsh\u00ecp\u00edn sh\u00e0ng\u201d .While reporting the positive impacts, six of the interviewees (15%) refer to negative impacts, for example, \u201cy\u01d2u sh\u00edh\u00f2u k\u00e0n sh\u00ecp\u00edn k\u00e0n sh\u00e0ngy\u01d0n, k\u00e0nde sh\u00edji\u0101n t\u00e0i ch\u00e1ng, y\u01d0ngxi\u01cengle ji\u0101r\u00e9n de xi\u016bx\u00ed\u201d  or \u201cr\u00e8nw\u00e9i\u201d (think) and the attributive relational process \u201csh\u00ec\u201d (be) with positive comments, such as \u201dy\u01d2uy\u00ec\u201d , \u201cy\u01d2ub\u0101ngzh\u00f9\u201d (helpful) about their cognitive change after adopting the virtual health information by expressing \u201czh\u01cengw\u00f2le\u201d (acquire), \u201cb\u0101ngzh\u00f9 w\u01d2 li\u01ceoji\u011b le\u201d (help me understand), and even graded behavioral processes, such as \u201cc\u01cein\u00e0\u201d (adopt), \u201cf\u01cengzh\u00e0o\u201d (imitate) or \u201cg\u0113nzhe sh\u00ecp\u00edn zu\u00f2...\u201d (follow the video to do...). The main generic arrangements, i.e., the contrast between their cognition before and after their watching the short videos with optional examples, the sequence of cognitive changes and actions to be taken, and the affirmation that these videos are definitely beneficial, along with some optional comments, uncover the representation of modification by contrasting, sequence and affirmation patterns.At the individual level, according to the knowledge, attitude and practice (KAP) theory ,39, peopforward), or verbal processes, the action of telling, such as \u201ct\u00e0nt\u01ceo\u201d (discuss), \u201cji\u01cengsh\u00f9\u201d (retell), shu\u014dj\u01d0j\u00f9 (add a few words). The Goals or Receivers with whom the participants share or communicate are also not limited to their family members or relatives, but to a wide range of people, especially to those who need the specific health information. The adverbs of degree or the mood particles used by the participants vividly indicate their desire and willingness to share the short videos with their family members, relatives and friends. As to whether sharing short health videos improves the interviewees\u2019 family life, most of them display an affirmative attitude by unveiling their representations with positive comments and evaluations.At the interpersonal level, in the Chinese context of collectivist culture, the ageing need more information technology especially regarding health information to connect with others, due to their closer interpersonal relationships . They maAs is indicated in many studies ,42, percI now cook with less oil and less salt, which is good for blood vessels... get more sun exposure, because some experts say that calcium in food is just a part of it and cannot be fully absorbed. It is good to go out and get some sun exposure.\u201d In contrast, for those without any disease, their willingness and motivation towards behavioral change are not very strong. For example, No. 16, who is 72 years old, says that she is still healthy and too lazy, but she would pay more attention if she often got sick. Therefore, ageing women with diseases are more inclined to accept health information in short videos, and make adjustments to their lifestyle habits and eating habits or physical exercises.The analysis in According to the interviews, there is also a strong correlation between age and changes in cognition and behavior. Some ageing women in their early 60s make no major changes in acquired behaviors after watching health videos; on the contrary, elder women are more willing to receive virtual health coaching and adjust their health behaviors after viewing the short videos. For example, the 61-year-old No. 19 responds that there has not much change in her cognition and exercise behavior; and the 63-year-old No. 15 also mentions in the interview that she can\u2019t control herself and make changes because she thinks she is still young.Interpersonal communication plays an important role in changing attitudes and plays a key role in persuading individuals to adopt healthy behaviors ,44. At tmy husband has high blood lipids. At the beginning, I imitated the short videos to make some light meals... Gradually, he began to accept it. And now my husband often comes home for dinner, and the family relationship is more harmonious\u201d; and No. 30 often shares health information on eating habits to family members because \u201cthe granddaughter has bad eating habits, likes fried foods such as hamburgers, and rarely eats vegetables, which leads to her weak physical condition and often being ill\u201d. Besides family members and relatives, the interviewees also target their information to a certain group, who have a great need for specific health information, for example, those who are not working (No. 4) and those who are in their eighties (No. 11).The interpersonal relationship structure is determined by the needs of human beings . Some inChildren are young and they have their own opinions. The views and living habits between the ageing and young are also different. My sharing health knowledge with them will be opposed,... so later on I will not communicate with them about this. No longer share!\u201d.In addition, the study also finds that interactive communication between the interviewee and the communicator has an important impact on the willingness to share. According to Altman & Taylor , communiAlthough the rapid development of transportation and information makes co-presence face-to-face interaction between people increasingly unnecessary , to the Maybe it has something to do with my previous job. If I think it\u2019s good, I want to promote it to everyone so that they can enjoy it\u201d; No. 6 accounts the partial reason why she doesn\u2019t like sharing health information, \u201cI don\u2019t know how to forward it, I just watch it myself\u201d.Besides the above mentioned essential factors, others such as occupation and technology skill have less impact on interpersonal improvement, but are also mentioned by the interviewees. For example, No. 13, a retired doctor, says \u201cThe study provides clear evidence that short health videos on social media platforms can build ageing women\u2019s health awareness, motivate behavioral modification and improve interpersonal relationships. Factors affecting these changes can be taken into consideration for future research in the field of short health videos in terms of modeling and intervention. At the same time, the study also attempts to apply and extend the notion of transitivity analysis and generic structure analysis to health communication research. It shows the great potential of discourse analysis, as a complementary investigation tool in analyzing qualitative material, especially on the effectiveness of short health videos in modifying cognition, behavior and interpersonal relationship."}
{"text": "Arthrobotrys is the most complex genus of Orbiliaceae nematode-trapping fungi. Its members are widely distributed in various habitats worldwide due to their unique nematode-trapping survival strategies. During a survey of nematophagous fungi in Yunnan Province, China, twelve taxa were isolated from terrestrial soil and freshwater sediment habitats and were identified as six new species in Arthrobotrys based on evidence from morphological and multigene  phylogenetic analyses. These new species i.e., Arthrobotrys eryuanensis, A. jinpingensis, A. lanpingensis, A. luquanensis, A. shuifuensis, and A. zhaoyangensis are named in recognition of their places of origin. Morphological descriptions, illustrations, taxonomic notes, and a multilocus phylogenetic analysis are provided for all new taxa. In addition, a key to known species in Arthrobotrys is provided, and the inadequacies in the taxonomic study of nematode-trapping fungi are also discussed. Arthrobotrys superba Corda)   and bigger than those of A. sphaeroides .Notes: The phylogenetic analyses revealed that ) and 8% 7/589 bp  conidia ,39,40, wArthrobotrys shuifuensis F. Zhang & X.Y. Yang sp. nov.  long, 3\u20135 \u00b5m  wide at base, gradually tapering upwards to apex, 1.5\u20133.5 \u00b5m  wide at apex, erect, septate, unbranched or rarely branched, hyaline, producing several separate nodes by repeated elongation of conidiophores, with each node consisting of 2\u20138 papilliform bulges and bearing polyblastic conidia. Conidia 17\u201336 \u00d7 5\u201312.5 \u00b5m , oblong or capsule-shaped, narrower towards the lower and pointed base, 1-septate, median septum, hyaline, rough-walled. Chlamydospores 6\u201318 \u00d7 3\u20137.5\u00b5m , cylindrical, in chains, hyaline, rough-walled. Capturing nematodes with adhesive networks.Additional specimen examined: CHINA, Yunnan Province, Zhaotong City, Shuifu County, 28\u00b032\u203231.80\u2033 N, 104\u00b019\u20329.50\u2033 E, from terrestrial soil, 16 June 2017, F. Zhang. YXY48.Arthrobotrys shuifuensis is the closest species to A. arthrobotryoides, there are 9.6% (57/596 bp) differences in ITS sequence between them. Morphologically, this species is similar to A. arthrobotryoides in their capsule-shaped, 1-septate conidia, whereas the conidia of A. shuifuensis are significantly longer than those of A. arthrobotryoides . In addition, the conidiophores of A. arthrobotryoides are unbranched and produces a continuous irregularly swollen node at apex, while the conidiophores of A. shuifuensis are branched, producing several separate nodes with the repeated elongation of the conidiophores . In addition, these three species differ slightly in the number of septation on conidia; A. zhaoyangensis produces 1\u20133-septate conidia (mostly 3-septate), while the conidia of A. sinensis are 2-septate; A. sphaeroides sometimes produces aseptate conidia [Notes: Phylogenetic analysis revealed that the systematic position of  conidia ,79.1.Conidia 0\u20131-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202621.Conidia multi-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026302.Conidia mostly aseptate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.Conidia mostly 1-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202663.A. botryosporaConidiophores branched near apex, producing a node at each branch, or producing several separate nodes by repeated elongation; conidia ovate, with a papilliform bulge at the base\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263.Conidiophores unbranched\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202644.A. amerosporaConidiophores with a cluster short denticles at apex; conidia obovoid, 15\u201331 (23.5) \u00d7 10\u201320 (15.9) \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20264.Conidiophores producing several clusters of short denticles by repeated elongation.\u202655.A. yunnanensisConidia elongated, ellipsoid\u2013cylindrical, 0\u20131-septate, mostly non-septate, 17.5\u201332.5 (22.6) \u00d7 2.75\u20137.5 (5.5) \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265.A. nonseptataConidia elongated, ellipsoidal, non-septate, constricted at the base, 11\u201316.8 \u00d7 5\u20136.6 \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20266.Conidia develop on short denticles\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202676.Conidia develop on nodes.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026137.Conidia curved\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202687.Conidia straight\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026108.A. musiformisConidiophores unbranched, conidia in loose capitate arrangement at apex; conidia ellipsoid, mostly curved, 20\u201347.5 (30.9) \u00d7 7\u201312.5 (10.3) \u03bcm\u2026\u2026\u2026\u2026\u2026\u2026\u20268.Conidiophores branched, producing several clusters short denticles by repeated elongation\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202699.A. shahriariConidiophores simple or occasionally branched; conidia elongate-obovoid or elongate-ellipsoidal, 1-septate, straight or curved, 33.5\u201357 \u00d7 11\u201315.5 \u00b5m\u2026\u2026\u20269.A. eryuanensisConidiophores branched; macroconidia 1-septate, straight or slightly curved, 18\u201344.5 (28.4) \u00d7 5\u201311.5 (8.7) \u00b5m, microconidia aseptate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202610.A. perpastaConidiophores producing short denticles by repeated elongation; conidia 1-septate near the base, obpyriform, sometimes constricted at the septum, 24\u201332.5 \u00d7 12.5\u201320 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202610.Conidiophores with clustered short denticles at apex; conidia in loose capitate arrangement at apex\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20261111.A. javanicaConidia clavate, 1-septate at median or submedian, slightly constricted at the septum, 20\u201337.5 (27.9) \u00d7 7.5\u201310 (8.8) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202611.Conidia obovoid or obpyriform\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20261212.A. obovataConidia obovoid, 1-septate near the base, apical cell much larger, smaller at basal cell, 28.5\u201332 (30) \u00d7 18\u201320.5 (20) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202612.A. koreensisConidia obpyriform, 1-septate at submedian, slightly constricted at the septum, 21.4\u201326.9 \u00d7 11.6\u201315.6 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202613.Conidia develop on short denticles or obscure nodes of conidiophores\u2026\u2026\u2026\u20261413.Conidia develop in clusters on swollen nodes of conidiophores\u2026\u2026\u2026\u2026\u2026\u2026\u20261714.A. chazaricaConidiophores branched, producing short denticles by repeated elongation; conidia obovate, elongate\u2013obovate, 22.5\u201332 \u00d7 11\u201322.5 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202614.Conidiophores unbranched; conidia clavate or pyriform\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20261515.A. pseudoclavataConidia develop on apical conidiophores, conidia clavate, 0 or 1-septate, constricted at the base, 30\u201345 \u00d7 8\u201311 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 15.Conidia pyriform, 1-septate near the basal, apical cell much larger, smaller at basal cell; conidiophores producing several short denticles by repeated elongation\u2026\u2026\u20261616.A. paucisporaConidia perceptibly constricted at the septum, 25\u201333.8 \u00d7 12.5\u201316.3 \u00b5m\u202616.A. cystosporiaConidia non-constricted, 25\u201335 \u00d7 18\u201324 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202617.Conidiophores branched\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20261817.Conidiophores unbranched\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262418.Conidia 1-septate at median\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20261918.Conidia 1-septate at submedian\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262119.A. superbaConidia elongate\u2013elliptical or cylindrical, 7.5\u201327.5 (15.8) \u00d7 5\u201310.5 (6.6) \u00b5m\u202619.Conidia short elliptical to oblong or capsule-shaped\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262020.A. arthrobotryoidesConidiophores occasionally branched, with distinct continuous swollen apical nodes; conidia ellipsoidal, 20\u201322 \u00d7 9\u201310 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202620.A. shuifuensisConidiophores usually branched, bearing conidia on slightly swollen nodes; conidia capsule-shaped, 17\u201336 (27.2) \u00d7 5\u201312.5 (8.2) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202621.A. robustaConidiophores bearing conidia on apical nodes; conidia oblong\u2013pyriform, 20\u201327.5 (24.4) \u00d7 7.5\u201312.5 (10.8) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202621.Conidiophores producing several separate nodes by repeated elongation\u2026\u2026\u2026\u20262222.A. cladodesConidia ellipsoid, elongate\u2013obovate, 10\u201320 (17.5) \u00d7 5\u20138 (6.2) \u00b5m\u2026\u2026\u2026\u2026\u202622.Conidia obovoid, obpyriform, or ovoid\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262323.A. latisporaConidia subglobose or elliptical, 14.8\u201321.5 (18.3) \u00d7 10.1\u201316.3 (13.5) \u00b5m\u2026\u2026 23.Orbilia jesu-lauraeConidia obovoid or obpyriform, 1 septum at submedian, slightly constricted at the septum, 14\u201326 \u00d7 7.5\u201313 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202624.Condia develop on apical node of conidiophores\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262524.Conidiophores producing several separate nodes by repeated elongation\u2026\u2026\u2026\u20262725.A. flagransConidia non-constricted at the septum, obconical or ellipsoidal, 25\u201350 \u00d7 10\u201315 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202625.Conidia obconical or pyriform, constricted at the septum\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262626.A. apscheronicaConidia larger size, constricted at septum, 21\u201342 (30.5) \u00d7 8\u201315 (12.7) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202626.A. conoidesConidia small size, perceptibly constricted at the septum, 15\u201337.5 (28.4) \u00d7 7.5\u201314.5 (11.8) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202627.Conidia 1-septate at median\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262827.Conidia 1-septate at submedian\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.2928.A. anomalaConidiophores producing continuously expanded node or several separate nodes by repeated elongation; conidia cylindric, long ellipsoid, larger size, 13\u201322 \u00d7 3\u20137 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202628.A. dendroidesConidiophores producing several slightly swollen nodes by repeated elongation; conidia ovate, oblong, cylindric, smaller size, 10\u201320 (14.6) \u00d7 2.5\u20135 (4) \u00b5m\u202629.A. jinpingensisConidia obpyriform or drop-shaped, some with a bud-like projection at the base, smaller size, 11.2\u201326.4(18.6) \u00d7 6.6\u201314.4(10.8) \u00b5m\u2026\u2026\u202629.A. oligosporaConidia pyriform or obovoid, slightly constricted at the septum, larger size, 17\u201335 (23) \u00d7 8.5\u201316 (12) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202630.Conidia without largest cell, with several septa, uniformly distributed among conidial cells \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263130.Conidia with largest cell\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263731.A. iridisY-shaped conidia develop on conidiophores\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202631.Conidia other type, never Y-shaped\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263232.Conidiophores branched\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.3332.Conidiophores unbranched\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263433.A. dianchiensisMacroconidia spindle-shape or clavate, with 1\u20137-septate, mostly 2\u20135, 37.5\u2013100 (70) \u00d7 10\u201317.5 (14.3) \u00b5m, microconidia spindle-shape, 0 or 1-septate\u2026\u2026\u2026\u202633.A. tabrizicaConidia elongate\u2013ovate to elongate\u2013doliform or ellipsoidal, with 1\u20133-septate, 28.5\u201356 \u00d7 11.5\u201322.5 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u202634.Conidia bearing on apical conidiophores\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263534.Conidiophores producing several cluster conidia by repeated elongation\u2026\u2026\u2026\u20263635.A. multiformisSeveral conidia develop on apical conidiophores, macroconidia elongate-fusiform, clavate, 4\u201312-septate; microconidia clavate, cylindric\u2013clavate, 0 or 1-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202635.A. shizishannaConidiophores bearing single conidium; conidia clavate, sometimes slightly curved, 2\u20139-septate, 22.5\u201373.8 (50.6) \u00d7 5\u201310 (6.6) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202636.A. polycephalaConidiophores with inconspicuous short denticles; macroconidia fusoid-shaped, curved, 2\u20134-septate, mostly 3\u20134, 30\u201350 (45.1) \u00d7 8\u201316.5 (12.2) \u00b5m, microconidia ellispsoid, slightly curved, 1 or 2-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202636.A. pyriformiConidiophores producing several short denticles by repeated elongation; conidia elongate\u2013pyriform, 1\u20133-septate, mostly 2 or 3, 17\u201338 \u00d7 6.5\u201311.5 \u00b5m\u2026\u2026\u202637.Conidiophores branched\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263837.Conidiophores unbranched\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026.\u2026\u20264338.Conidiophores bear a single conidium\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263938.Conidiophores bear several conidia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20264039.A. globosporaConidia globose or obpyriform, 1\u20132-septate, 25\u201337.5 \u00d7 15\u201322.5 \u00b5m\u2026\u202639.A. sinensisConidia subspherical or obovoid or subfusiform, 1\u20133-septate, 23.5\u201330 (27.6) \u00d7 17\u201325 (20) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202640.Conidia in capitate arrangement at apex of conidiophores\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20264140.Conidia in non-capitate arrangement on conidiophores\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20264241.A. azerbaijanicaConidia obovoid or ellipsoidal, 1\u20134-septate, mostly 1, 18\u201336 (28.1) \u00d7 12\u201320 (15.3) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202641.A. oviformisConidia pyriform, 1\u20132-septate, mostly 1, 7.5\u201322.5 (15.8) \u00d7 5\u201310 (6.6) \u00b5m\u202642.A. indicaConidiophores bearing 1 conidium, sometimes 2 conidia; conidia elliptic, top-shaped, 0\u20132-septate, 17.5\u201330 (23.2) \u00d7 12.5\u201320 (14.8) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202642.A. oudemansiiConidiophores bearing several conidia; macroconidia subfusiform, 2\u20134-septate, 40\u201365 (52) \u00d7 17\u201323 (20) \u00b5m, microconidia obovoid, aseptate \u2026\u2026\u202643.Conidiophores bear several conidia \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20264443.Conidiophores bear a single conidium \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265244.Conidiophores bear several conidia near apex by repeated elongation\u2026\u2026\u2026\u20264544.Conidiophores bear several conidia at apex\u2026\u2026\u2026\u2026\u2026\u2026\u20264745.A. vermicolaConidia elongate\u2013ellipsoidal to broadly fusiform, 1\u20133-septate, mostly1 or 2, 25\u201350 \u00d7 17.5\u201325 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202645.Conidiophores producing denticles by repeated elongation, conidia fusiform, elongate\u2013fusoid or clavate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20264646.A. mangrovisporaConidia variable in shape, broadly turbinate to elongate\u2013fusoid, ellipsoidal, fusiform, clavate, 1\u20133-septate, mostly 2, 25\u201350 (38.9) \u00d7 12\u201324 (17.3) \u00b5m\u2026\u2026\u2026\u202646.A. scaphoidesConidia fusiform, sometimes slightly curved, 1\u20136-septate, mostly 2\u20133, 36.6\u201379.3 (57) \u00d7 11\u201317.5 (14) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202647.A. gampsosporaConidia spindle-shaped, curved,1\u20134-septate, 25\u201376 \u00d7 7\u201316 \u00b5m\u2026\u2026\u202647.Conidia straight\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20264848.Conidia 0\u20133-septate, mostly 1 or 2\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20264948.Conidia 1\u20134-septate, mostly 3 or 4\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265049.A. microscaphoidesConidia cymbiform or fusiform, mostly 2-septate, 22.5\u201345 (27.2) \u00d7 10\u201320 (13.9) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202649.A. clavisporaConidia pyriform, clavate, mostly 1 or 2 septate, 25\u201340 (17.5) \u00d7 7.5\u201319 (15.4) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202650.A. thaumasiaConidia small, subfusiform, 1\u20134-septate, mostly 3, 30\u201360 (36.2) \u00d7 15\u201330(20.2) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202650.Conidia larger, ellipsoidal, fusoid\u2013ellipsoidal\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265151.A. psychrophilaConidia ellipsoidal, fusoid\u2013ellipsoidal, 2\u20134-septate, mostly 4, 46\u201370 (62.3) \u00d7 21\u201329 (24.7) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202651.A. megalosporaConidia fusiform, elongate\u2013ellipsoidal or obovoid, 2\u20134-septate, mostly 3 or 4, 40\u201375 \u00d7 18\u201335 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202652.A. eudermataConidia spindle-shaped, globose, 1\u20133-septate, mostly 2 or 3, 37\u201355 (49) \u00d7 17.5\u201335 (28) \u00b5m, microconidia ellipsoid, aseptate\u2026\u2026\u2026\u2026\u202652.Without microconidia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265353.A. janusConidia turbinate or napiform, 1\u20132-septate, mostly 1 near the base, the largest cell at the apex of conidia, 15\u201326 (22.5) \u00d7 17.5\u201337.5 (28.5) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202653.Conidia with more than 2 septa; the largest cell is located in the apex or center of conidia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265454.The septum of the conidia is not more than 3\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265554.Conidia 1\u20135-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265855.Conidia clavate, obovoid, or subspherical, 0\u20133-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265655.Conidia fusiform, 2\u20133-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265756.A. cookedickinsonConidia clavate or obovoid, 1\u20133-septate, mostly 2\u20133, 30\u201352.5 (42) \u00d7 15\u201322.5 (17.6) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202656.A. sphaeroidesConidia subspherical or obovoid, 0\u20133-septate, mostly 2\u20133, 20\u201340 (32) \u00d7 17\u201325 (20.4) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202657.A. rutgerienseConidia globose or subfusiform, 2\u20133-septate, 27\u201347.5 (32.2) \u00d7 17.5\u201327.5 (22) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202657.A. fusiformisConidia spindle-shaped, fusiform or ellipsoidal, 2\u20133-septate, 32.5\u201347.5 (41) \u00d7 12.5\u201317.5 (15.5) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202658.Conidia 1\u20135-septate, mostly 3 or 4\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.5958.Conidia 1\u20134-septate, mostly 2 or 3\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20266159.A. xiangyunensisConidia variable in shape, obpyriform, broadly turbinate, subfusiform, elongate-fusoid or clavate, 1\u20135-septate, 27\u201372 (55.8) \u00d7 14.5\u201328.5 (21.9) \u00b5m\u202659.Conidia ellipsoid, obpyriform or subfusiform, 2\u20135-septate, mostly 3 or 4\u2026\u2026\u2026\u2026\u20266060.A. reticulataConidia ellipsoid, fusiform, 50\u201365 \u00d7 20\u201325 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202660.A. longiphoraConidia obpyriform or subfusiform, 40\u201390 (54) \u00d7 15\u201327.5 (18) \u00b5m\u2026\u2026\u2026\u2026 61.Conidia mostly 2-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20266261.Conidia mostly 3-septate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20266362.A. luquanensisConidia obovate, obpyriform or drop-shaped; the distal cell is much smaller, the largest cell usually at the apex, 28\u201353.5 (40.9) \u00d7 17\u201332.5 (26.3) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026 62.A. guizhouensisConidia obpyriform or subfusiform, the largest cell usually at the centre, 30.5\u201371.5 (52.7) \u00d7 18.5\u201328.5 (23.9) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202663.A. zhaoyangensisConidia, subglobose, obovoid to obpyriform, 25.5\u201352 (35.4) \u00d7 14\u201332 (22.9) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202663.Conidia fusiform or ellipsoid\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20266464.A. salinaConidia fusiform to ellipsoid, 32.5\u201352.5 \u00d7 12.5\u201317.5 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 64.A. lanpingensisConidia mostly subfusiform, 31.1\u201355.2 (45.4) \u00d7 13.5\u201324.3 (19.7) \u00b5m\u2026\u2026\u2026Arthrobotrys, which contains the greatest number of species. Morphologically, 61 species of Arthrobotrys can be divided into different groups according to the morphologies of their conidiophores and conidia [Arthrobotrys species are unclear. The reason for this dilemma is the lack of molecular data for many species, and the existing data cannot provide a stable phylogenetic placement. Therefore, to thoroughly analyse the taxonomy of nematode-trapping fungi, we should use more comprehensive molecular data in future studies.In this phylogenetic analysis, 65 species of nematode-trapping fungi used in this study were clustered into two large clades according to their mechanisms of catching nematodes. Clade I contained species that catch nematodes with adhesive trapping devices (adhesive nets and knobs). Clade II contained species that catch nematodes with active traps (constricting rings). Within clade I, species were clustered into two clades according to their trap types: one clade contained all species that produce adhesive nets, and the other contained those species that produce adhesive knobs. The results were consistent with previous studies ,15,57,80 conidia ; howeverA. eryuanensis and A. shuifuensis could be easily distinguished from known species based on their distinct morphological characteristics. The remaining four species required more detailed characteristics (such as the size of conidia) to be identified from known species. When mycologists measure the size of conidia, they are accustomed to uniformly calculating the size data of conidia with different shapes and septate numbers, and the sizes of these conidia usually show significant differences. This causes the size range of conidia to be too extensive for effective comparisons of different species [The emergence of molecular phylogenetic methods has led to unprecedented breakthroughs in the study of fungal taxonomy. Phylogenetic studies based on only a few molecular barcodes cannot provide sufficient and reliable information for the definition of fungal species; therefore, morphological descriptions of each species are still extremely important ,82. Howe species ,19. (2)  species ,19. As ahttp://www.speciesfungorum.org (accessed on 6 March 2022)). These data indicated that the excavation of nematode-trapping fungi seems to have reached a plateau, and over time, it is unlikely that many new species will be discovered. However, in recent years, we have investigated nematode-trapping fungi in Yunnan Province and collected 10 new species (four previously published and six reported in this study) [After the first nematode-trapping fungus was established in 1839 , the hiss study) ,84,85. H"}
{"text": "Scientific Reports 10.1038/s41598-022-23991-3, published online 15 November 2022Correction to: The original version of this\u00a0Article contained an error in the y-axis\u00a0labels of\u00a0Figure\u00a0\u201c100000\u201dnow reads:\u201c100\u201d\u201c10000\u201dnow reads:\u201c10\u201d\u201c1000\u201dnow reads:\u201c1\u201dThe original Figure\u00a0The original Article has been corrected."}
{"text": "Scientific Reportshttps://doi.org/10.1038/s41598-021-02511-9, published online 29 July 2022Correction to: The original version of this Article contained errors in Table 1.The values for \u201cAbove secondary school education\u201d in section \u201cEducational status\u201d in column \u201cFrequency\u201d were incorrectly given as \u201c56\u201d instead of \u201c57\u201d, and in column \u201cPercentage\u201d the values were incorrectly given as \u201c23.4\u201d instead of \u201c23.8\u201d. Moreover, in column \u201cPercentage\u201d, the value for \u201cNo education\u201d was rounded down from \u201c31.4\u201d to \u201c31.2\u201d, and the value for \u201cPrimary school education\u201d was rounded down from \u201c21.3\u201d to \u201c21.2\u201d. The correct and incorrect values appear below.\u2020\u201d was rounded down from \u201c1.3\u201d to \u201c1.2\u201d. The correct and incorrect values appear below.Additionally, the data for \u201cProtestant\u201d in section \u201cReligion\u201d was omitted. The value \u201c50\u201d was added in column \u201cFrequency\u201d and the value \u201c20.8\u201d was added in column \u201cPercentage\u201d. Moreover, in column \u201cPercentage\u201d, the value for \u201cOtherIncorrect:Correct:The original Article has been corrected."}
{"text": "We employ some results about continued fraction expansions of Herglotz\u2013Nevanlinna functions to characterize the spectral data of generalized indefinite strings of Stieltjes type. In particular, this solves the corresponding inverse spectral problem through explicit\u00a0formulas. Stieltjes continued fractions played a decisive role in the solution of the inverse spectral problem for Krein strings , 27, 28.1m be a Herglotz\u2013Nevanlinna function, that is, the function m is defined and analytic on C\u2216R, maps the upper complex half\u2010plane into the closure of the upper complex half\u2010plane and satisfies the symmetry relationa is a real constant, b is a non\u2010negative constant and \u03c1 is a non\u2010negative Borel measure on R which is subject to the growth restrictiona and b, as well as the measure \u03c1 in this integral representation are uniquely determined by the function m and may be recovered\u00a0explicitly.Let s0,s1,\u2026 the moments of the measure \u03c1, that is, we setK\u2208N0, the function m allows the asymptotic expansion|z|\u2192\u221e along the imaginary axis, provided the moments of the measure \u03c1 exist up to order 2K. Here, the real constants s\u22122 and s\u22121 are defined byTheorem 1.1K\u2208N0, the moments of the measure \u03c1 exist up to order 2K if and only if the function m allows the asymptotic expansion|z|\u2192\u221e along the imaginary axis for some real constants s\u22122,\u2026,s2K.For any fixed As long as they exist, we will denote with entation\u00a0 that form. To this end, we first introduce the Hankel determinantsc\u2208Ck one computes thatk is zero. For future reference, we also state the useful relations\u03c1 exist up to order 2K, then the relations\u00a0 is the smallest integer k\u2208N0 for which the sequence \u03941,0,\u2026,\u03941,k has precisely n non\u2010zero elements. One observes that the increasing function \u03ba is defined in such a way that \u03941,\u03ba(1),\u2026,\u03941,\u03ba(N+1) enumerates all non\u2010zero members of the sequence \u03941,0,\u2026,\u03941,D. As it follows from relation\u00a0=0, this determines \u03ba recursively.Proposition 1.2m is a rational Herglotz\u2013Nevanlinna function, then it admits the continued fraction expansion\u03c50,\u2026,\u03c5N and the real constants \u03c90,\u2026,\u03c9N are given byl1,\u2026,lN and the non\u2010negative constant r are given byIf Let us suppose for now that ows from\u00a0 that therelation\u00a0 that them is a rational Herglotz\u2013Nevanlinna function. For each t\u2208R, we define the rational Herglotz\u2013Nevanlinna function mt bymt will be denoted in a natural way with an additional subscript t. In particular, note that m0 coincides with our initial function m and hence so do the associated quantities . As each of the Hankel determinants depends analytically on t, we may conclude that the determinants \u0394t,1,0,\u2026,\u0394t,1,\u03ba(N+1) are all non\u2010zero as long as t\u22600 is small enough. Indeed, even if \u03941,k does vanish for some k\u2208{1,\u2026,\u03ba(N+1)\u22121}  is certainly non\u2010zero), then this holds because the derivative\u03941,k=0.Suppose that t\u22600 is small enough, the function mt admits a Stieltjes continued fraction expansion [K=\u03ba(N+1), the non\u2010zero real constants \u03c9t,1,\u2026,\u03c9t,K are given bylt,1,\u2026,lt,K as well as the non\u2010negative constant rt are given byK is zero and that the alternative expression in\u00a0=qt,K clearly converge pointwise to the rational Herglotz\u2013Nevanlinna function q\u03ba(N+1) defined byr is given by\u00a0 converge pointwise to some rational Herglotz\u2013Nevanlinna function q\u03ba(n+1). If \u03941,\u03ba(n)+1 does not vanish, then \u03ba(n+1)=\u03ba(n)+1 and we infer from\u00a0 converge pointwise to the Herglotz\u2013Nevanlinna function q\u03ba(n) defined via\u03c5n, \u03c9n and ln are given by\u00a0+1 vanishes, one has \u03ba(n+1)=\u03ba(n)+2 and using the second expression in\u00a0(\u03c9n is given by\u00a0(k=\u03ba(n)+1 as well as k=\u03ba(n)+2, in which the \u03941,\u03ba(n)+1 terms vanish, we infer that the last limit is equal tok=\u03ba(n)+2, this becomes\u03c5n is given by\u00a0(\u03c5n is positive as relation\u00a0(k=\u03ba(n)+1 shows that the numerator on the left\u2010hand side is not zero. Now observe that in the current case, the functions qt,\u03ba(n) satisfyz\u2208C\u2216R. With the help of the limits above, a computation then shows that the functions qt,\u03ba(n) converge pointwise to the rational Herglotz\u2013Nevanlinna function q\u03ba(n) defined by\u00a0, then the expansion in Proposition\u00a0\u03ba(n)=n\u22121 for all n\u2208{1,\u2026,N+1} and thus it follows from the formulas in\u00a0+1 is not zero. On the other side, if \u03941,\u03ba(n)+1 is zero, then relation\u00a0+1 vanishes and also that \u03c5n+|\u03c9n|>0 for all n\u2208{1,\u2026,N}.The constants in the continued fraction in Proposition\u00a0relation\u00a0 shows threlation\u00a0 allows uIt is not difficult to see that any continued fraction of the form in Proposition\u00a0m1 and m2 be Herglotz\u2013Nevanlinna functions and denote with \u03c11 and \u03c12 the corresponding measures in the respective integral representations.Lemma 1.4K\u2208N0 and suppose(1)m2(z)/z\u2192\u03b2 for some positive \u03b2 as |z|\u2192\u221e along the imaginary axis. Then the moments of the measure \u03c12 exist up to order 2K if and only if the moments of the measure \u03c11 exist up to order 2K+2.Assume that (2)m1(z)/z\u21920 and m2(z)/z\u21920 as |z|\u2192\u221e along the imaginary axis. Then the moments of the measure \u03c12 exist up to order 2K if and only if the moments of the measure \u03c11 exist up to order 2K.Assume that Let Before we proceed to non\u2010rational Herglotz\u2013Nevanlinna functions, let us first provide two auxiliary results. In order to state them, let m2(z)/z\u2192\u03b2 for some positive \u03b2 as |z|\u2192\u221e along the imaginary axis. If the moments of the measure \u03c12 exist up to order 2K, then from Theorem\u00a0|z|\u2192\u221e along the imaginary axis for some real constants s2,\u22121,\u2026,s2,2K. This implies|z|\u2192\u221e along the imaginary axis. Since the first term on the right\u2010hand side is a rational function, we conclude from Theorem\u00a0\u03c11 exist up to order 2K+2. Conversely, if the moments of the measure \u03c11 exist up to order 2K+2, then we have|z|\u2192\u221e along the imaginary axis for some real constants s1,1,\u2026,s1,2K+2 and thus|z|\u2192\u221e along the imaginary axis. Like before, we may conclude again that the moments of the measure \u03c12 exist up to order 2K by invoking Theorem\u00a0We begin with the case when m1(z)/z\u21920 and m2(z)/z\u21920 as |z|\u2192\u221e along the imaginary axis. If the moments of the measure \u03c12 exist up to order 2K, then from Theorem\u00a0|z|\u2192\u221e along the imaginary axis for some real constants s2,\u22121,\u2026,s2,2K. Since m1(z)/z\u21920 as |z|\u2192\u221e along the imaginary axis, we conclude that s2,\u22121 is not zero and thus we have|z|\u2192\u221e along the imaginary axis. As the first term on the right\u2010hand side is a rational function, we infer from Theorem\u00a0\u03c11 exist up to order 2K. The converse direction follows by symmetry.\u25a1Now let us assume that Lemma 1.5m\u223c1 and m\u223c2 withmi for i\u2208{1,2} admit the continued fraction expansionN\u2208N, non\u2010negative constants \u03c50,\u2026,\u03c5N\u22121, real constants \u03c90,\u2026,\u03c9N\u22121 and positive constants l1,\u2026,lN with \u03c5n+|\u03c9n|>0 for all n\u2208{1,\u2026,N\u22121}, then|z|\u2192\u221e along the imaginary axis, where the integer K\u2208N0 is given byj is equal to one whenIf there are Herglotz\u2013Nevanlinna functions i\u2208{1,2}, let us define Herglotz\u2013Nevanlinna functions qi,1,\u2026,qi,N byn\u2208{1,\u2026,N\u22121} recursively viami=\u03c50z+\u03c90+qi,1 by assumption. A computation then shows that|z|\u2192\u221e along the imaginary axis for all n\u2208{1,\u2026,N\u22121}, which yields|z|\u2192\u221e along the imaginary axis. Since this entails that|z|\u2192\u221e along the imaginary axis, it remains to note that|z|\u2192\u221e along the imaginary axis.\u25a1For m.Theorem 1.6m is a non\u2010rational Herglotz\u2013Nevanlinna function and let K\u2208N0. Then the moments of the measure \u03c1 exist up to order 2K if and only if there is a Herglotz\u2013Nevanlinna function m\u223c, an integer N\u2208N, non\u2010negative constants \u03c50,\u2026,\u03c5N, real constants \u03c90,\u2026,\u03c9N\u22121 and positive constants l1,\u2026,lN with \u03c5n+|\u03c9n|>0 for all n\u2208{1,\u2026,N\u22121} andm admits the continued fraction expansionSuppose that With the help of these two lemmas, we are now ready to add another item to the equivalence in Theorem\u00a0m admits such a continued fraction expansion. We define the rational Herglotz\u2013Nevanlinna function m1 by replacing m\u223c in the continued fraction on the right\u2010hand side of\u00a0/z\u21920 as |z|\u2192\u221e along the imaginary axis. This allows us to write\u03c51 and a Herglotz\u2013Nevanlinna function m\u223c, which is the claimed expansion. Now let K\u2208N and suppose that the assertion holds for all lesser integers. As before, we may write m like in\u00a0(\u03c51 and a Herglotz\u2013Nevanlinna function m\u223c with m\u223c(z)/z\u21920 as |z|\u2192\u221e along the imaginary axis. In the case when K=1 and \u03c51\u22600, this is the required expansion. Otherwise, we conclude from Lemma\u00a0\u03c1\u223c corresponding to the function m\u223c exist up to orderm\u223c then admits a continued fraction expansion of the claimed form, we readily see that so does m (it only remains to note that \u03c51+|\u03c91|>0 in this expansion since q(z)/z\u21920 as |z|\u2192\u221e along the imaginary axis).\u25a1For the converse direction, we will use induction. To this end, let us first consider the case when o writem(z)=\u2212s\u2212o write1m(z)=\u2212sm is a non\u2010rational Herglotz\u2013Nevanlinna function such that the moments of the measure \u03c1 exist up to order 2K for some K\u2208N0. Under these assumptions, the determinants \u03940,k are well defined for all k\u2208{0,\u2026,K+1} and positive in view of\u00a0 is the smallest integer k\u2208{0,\u2026,K} for which the sequence \u03941,0,\u2026,\u03941,k has precisely n non\u2010zero elements. One observes that the increasing function \u03ba is defined in such a way that \u03941,\u03ba(1),\u2026,\u03941,\u03ba(N) enumerates all non\u2010zero members of the sequence \u03941,0,\u2026,\u03941,K. As it follows from relation\u00a0=0, this determines \u03ba recursively.Corollary 1.7m is a non\u2010rational Herglotz\u2013Nevanlinna function such that the moments of the measure \u03c1 exist up to order 2K for some K\u2208N0, then there is a Herglotz\u2013Nevanlinna function m\u223c withm admits the continued fraction expansion\u03c50,\u2026,\u03c5N\u22121 and the real constants \u03c90,\u2026,\u03c9N\u22121 are given by\u00a0=K\u22121 and the function m\u223c satisfiesIf given by\u00a0 and the given by\u00a0. FurtherAs in the rational case, we are able to provide explicit formulas for the constants in the continued fraction expansion in terms of the Hankel determinants. To this end, let us suppose that  view of\u00a0, as the relation\u00a0 that them is a non\u2010rational Herglotz\u2013Nevanlinna function such that the moments of the measure \u03c1 exist up to order 2K for some K\u2208N0. According to Theorem\u00a0m admits a continued fraction expansion of the form\u00a0 the corresponding function\u00a0, we may obtain expressions for the constants in this expansion in terms of the Hankel determinants by comparison with Proposition\u00a0N ,\u03941,\u03ba(2),\u2026 enumerates all non\u2010zero elements of the sequence \u03941,0,\u03941,1,\u2026 (of which there are infinitely many as the sequence does not contain any consecutive zeros).Corollary 1.8m is a non\u2010rational Herglotz\u2013Nevanlinna function. Then all moments of the measure \u03c1 exist if and only if there are Herglotz\u2013Nevanlinna functions m\u223c1,m\u223c2,\u2026, non\u2010negative constants \u03c50,\u03c51,\u2026, real constants \u03c90,\u03c91,\u2026 and positive constants l1,l2,\u2026 with \u03c5n+|\u03c9n|>0 for all n\u2208N such that for every N\u2208N the function m admits the continued fraction expansion\u03c50,\u03c51,\u2026 and the real constants \u03c90,\u03c91,\u2026 are given by\u00a0 be a generalized indefinite string so that L\u2208 and \u03c5 is a non\u2010negative Borel measure on [0,L). We consider the corresponding spectral problem of the formz is a complex spectral parameter. Of course, this differential equation\u00a0has to be understood in a distributional sense  such thatdf. In this case, the constant df is uniquely determined and will henceforth always be denoted with f\u2032(0\u2212) for apparent\u00a0reasons.A solution of\u00a0 is a funLet refer to  for more is the corresponding Weyl\u2013Titchmarsh function m defined on C\u2216R by\u03c8 is a non\u2010trivial solution of the differential equation\u00a0\u21a6m establishes a one\u2010to\u2010one correspondence between generalized indefinite strings and Herglotz\u2013Nevanlinna functions. In the following, we are going to use our findings from the last section\u00a0to characterize those Herglotz\u2013Nevanlinna functions that correspond in this way to generalized indefinite strings that begin with a discrete part. To be more precise, we consider generalized indefinite strings  of the formN\u2208N0, increasing points x1,\u2026,xN+1 in , real weights \u03c90,\u2026,\u03c9N and non\u2010negative weights \u03c50,\u2026,\u03c5N, where \u03b4x denotes the unit Dirac measure centered at a point x\u2208[0,L). For such coefficients, the solutions \u03c8 of the differential equation\u00a0 has the formL, the increasing points x1,\u2026,xN in , the real weights \u03c90,\u2026,\u03c9N and the non\u2010negative weights \u03c50,\u2026,\u03c5N are given byIf the Weyl\u2013Titchmarsh function It was shown in [equation\u00a0 are piecelation\u03c8\u2032 such thatx1,\u2026,xN are given bym\u223c corresponding to  admits the same continued fraction expansion as m in view of\u00a0 has the claimed form and the explicit formulas follow readily from the ones in Proposition\u00a0\u25a1Suppose that the Weyl\u2013Titchmarsh function  view of\u00a0, we concRemark 2.3w\u2208Lloc2[0,L) is the normalized anti\u2010derivative of the distribution \u03c9 so thatw is piecewise constant. After plugging in the expressions from\u00a0+1 is zero or\u00a0not.For generalized indefinite strings as in Proposition\u00a0\u2211j=0n\u22121\u03c9identity\u00a0. For the,\u03ba(n),\u222b0xnw(x)ons from\u00a0, it remaelations\u00a0 and\u00a0, real weights \u03c90,\u2026,\u03c9N\u22121 and non\u2010negative weights \u03c50,\u2026,\u03c5N with \u03c5n+|\u03c9n|>0 for all n\u2208{1,\u2026,N\u22121} and has the formSuppose that the Weyl\u2013Titchmarsh function We now proceed to add another item to the equivalences in Theorem\u00a0\u03c1 exist up to order 2K so that the function m admits the continued fraction expansion\u00a0 that satisfyx1,\u2026,xN are given by\u00a0, which guarantees that  has the claimed\u00a0form.Assume first that the moments of the measure xpansion\u00a0 by Theorgiven by\u00a0 and it irding to\u00a0, the Weyrding to\u00a0. It thenl1,\u2026,lN given bym\u223c  has the formx1,\u2026,xN in , the real weights \u03c90,\u2026,\u03c9N\u22121 and the non\u2010negative weights \u03c50,\u2026,\u03c5N\u22121 are given by\u00a0=K\u22121 and one hasIf the Weyl\u2013Titchmarsh function given by\u00a0. FurtherIt is again possible to find explicit expressions for the weights and their positions in Theorem\u00a0\u25a1In view of Corollary\u00a0Remark 2.6|z|\u2192\u221e along the imaginary axis, of the Weyl\u2013Titchmarsh function m uniquely determines the generalized indefinite string  near the left endpoint. This can be viewed as a variant of local inverse uniqueness results as in [Theorem\u00a0ts as in , 33, 38 ts as in , 2439.One can infer from Lemma\u00a0Corollary 2.7m is not rational. Then all moments of the spectral measure \u03c1 exist if and only if there are increasing points x1,x2,\u2026 in , real weights \u03c90,\u03c91,\u2026 and non\u2010negative weights \u03c50,\u03c51,\u2026 with \u03c5n+|\u03c9n|>0 for all n\u2208N such that the generalized indefinite string  has the formx1,x2,\u2026 in , the real weights \u03c90,\u03c91,\u2026 and the non\u2010negative weights \u03c50,\u03c51,\u2026 are given by\u00a0 whose coefficients are supported on discrete sets, we just need to introduce one additional condition on the spectral data.Corollary 2.8m is not rational. Then all moments of the spectral measure \u03c1 exist andx1,x2,\u2026 in  with xn\u2192L, real weights \u03c90,\u03c91,\u2026 and non\u2010negative weights \u03c50,\u03c51,\u2026 with \u03c5n+|\u03c9n|>0 for all n\u2208N such that the generalized indefinite string  has the formx1,x2,\u2026 in , the real weights \u03c90,\u03c91,\u2026 and the non\u2010negative weights \u03c50,\u03c51,\u2026 are given by\u00a0(Suppose that the Weyl\u2013Titchmarsh function given by\u00a0.In order to make sure that the Weyl\u2013Titchmarsh function \u03c1({0})=L\u22121, which holds for arbitrary generalized indefinite strings; see [\u25a1This is a consequence of Corollary\u00a0Remark 2.9\u03c9 is supported on a discrete set and the measure \u03c5 vanishes identically. More precisely, these indefinite strings are determined by the additional conditions that\u03941,1,\u03941,2,\u2026 is\u00a0zero.In conjunction with the solution of the inverse spectral problem for generalized indefinite strings in\u00a0[As already indicated before, this last result can be used to prove Corollary\u00a0Bulletin of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not\u2010for\u2010profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of\u00a0mathematics.The"}
{"text": "Counterfeiting has become a prevalent business worldwide, resulting in high losses for many businesses. Considerable attention has been paid to research an individual attitude toward purchasing luxury counterfeit products in the offline context. However, there is currently lesser-known literature on the given phenomenon in the context of social commerce. Moreover, researchers observed that counterfeiting consumption is associated with consumer ethical values or beliefs. Practitioners and researchers are keen to find those factors that affect consumers\u2019 ethical consumption behavior to reduce pirated products\u2019 demand. However, the role of religion in shaping ethical behavior is less documented in the counterfeiting context. Therefore, this study investigated the effect of religiosity on the counterfeiting of luxury products in Pakistan. A five-dimensional Islamic religiosity model was adopted to understand the consumption phenomena. For quantitative research, cross-sectional data were collected from the generation M of Pakistan through self-administrative questionnaires. A total of 394 valid responses from active online users were collected to empirically examine the conceptual model by employing the partial least square structural equation model (PLS-SEM). The results reveal that all five dimensions of religiosity negatively affect the attitude of generation M. Moreover, it is found that knowledge has the highest negative effect on attitude, followed by orthopraxis, experience, central duties, and basic duties. The study also explains the theoretical and practical implications of the research. Finally, limitations and future research were also discussed. The virulence of the COVID-19 coronavirus pandemic has shaken the world . This paReligion plays an extremely important role in guiding human behavior. Although, the association between religion and buying behavior was already established . DespiteSecond, past research has found various significant predictors accountable for luxury counterfeit buying intention in the offline context but few in the online context . The intThird, the research mainly focused on generation M\u2019s attitude toward counterfeiting products. Generation M refers to millennial Muslims of the new generation . This MuReligion is an essential cultural value because it is unique, extensive, and dictates social institutes that profoundly affect the individual and society\u2019s perception, belief, and behavior . TheoretNevertheless, there is still limited research addressing the role of religiousness in counterfeiting and digital piracy phenomena . PreviouAccording to the Quran verses, counterfeiting is prohibitive in Islam because Islam said any activity leading to such action is considered a fraud. Islam forbids all kinds of cheating and all deceiving acts, whether in buying and selling fraud or between people in any other matter. All Muslims are urged to be honest and trustworthy in everything they do in all situations.\u064a\u0670\u06e4\u0640\u0627\u064e\u064a\u0651\u064f\u0647\u064e\u0627 \u0627\u0644\u0651\u064e\u0630\u0650\u064a\u06e1\u0646\u064e \u0627\u0670\u0645\u064e\u0646\u064f\u0648\u06e1\u0627 \u0644\u064e\u0627 \u062a\u064e\u062e\u064f\u0648\u06e1\u0646\u064f\u0648\u0627 \u0627\u0644\u0644\u0651\u0670\u0647\u064e \u0648\u064e\u0627\u0644\u0631\u0651\u064e\u0633\u064f\u0648\u06e1\u0644\u064e \u0648\u064e\u062a\u064e\u062e\u064f\u0648\u06e1\u0646\u064f\u0648\u06e1\u06e4\u0627 \u0627\u064e\u0645\u0670\u0646\u0670\u062a\u0650\u0643\u064f\u0645\u06e1 \u0648\u064e\u0627\u064e\u0646\u06e1\u0640\u062a\u064f\u0645\u06e1 \u062a\u064e\u0639\u06e1\u0644\u064e\u0645\u064f\u0648\u06e1\u0646\u064e\u200f\u201cO you, who have believed, do not deceive Allah and the Messenger or deceive your trusts while you know (the consequences)\u201d (Quran 8:27).\u0648\u064e\u0644\u064e\u0627 \u062a\u064f\u062c\u064e\u0627\u062f\u0650\u0644\u0652 \u0639\u064e\u0646\u0650 \u0627\u0644\u0651\u064e\u0630\u0650\u064a\u0652\u0646\u064e \u064a\u064e\u062e\u0652\u062a\u064e\u0627\u0646\u064f\u0648\u0652\u0646\u064e \u0627\u064e\u0646\u0652\u0641\u064f\u0633\u064e\u0647\u064f\u0645\u0652 \u06d7 \u0627\u0650\u0646\u0651\u064e \u0627\u0644\u0644\u0651\u0670\u0647\u064e \u0644\u064e\u0627 \u064a\u064f\u062d\u0650\u0628\u0651\u064f \u0645\u064e\u0646\u0652 \u0643\u064e\u0627\u0646\u064e \u062e\u064e\u0648\u0651\u064e\u0627\u0646\u064b\u0627 \u0627\u064e\u062b\u0650\u064a\u0652\u0645\u064b\u0627\u06d9\u201cAnd do not argue on behalf of those who deceive themselves. Indeed, Allah loves not one who is a habitually sinful deceiver\u201d (Quran 4:107).\u0625\u0650\u0646\u0651\u064e \u0671 \u06cc\u0651\u064e\u0623\u0652\u0645\u064f\u0631\u064f\u0643\u064f\u0645\u0652 \u0623\u064e\u0646 \u062a\u064f\u0624\u064e\u062f\u0651\u064f\u0648\u0627\u06df \u0671\u0644\u0652\u0623\u064e\u0645\u064e\u0646\u064e\u062a\u0650 \u0625\u0650\u0644\u064e\u0649\u0670\u0653 \u0623\u064e\u06be\u0652\u0644\u0650\u06be\u064e\u0627\u0648\u064e\u0625\u0650\u0630\u064e\u0627 \u062d\u064e\u0643\u064e\u0645\u0652\u062a\u064f\u0645 \u0628\u064e\u06cc\u0652\u0646\u064e \u0671\u0644\u0646\u0651\u064e\u0627\u0633\u0650 \u0623\u064e\u0646 \u062a\u064e\u062d\u0652\u0643\u064f\u0645\u064f\u0648\u0627\u06df \u0628\u0650\u0671\u0644\u0652\u0639\u064e\u062f\u0652\u0644\u0650 \u0625\u06da\u0650\u0646\u0651\u064e \u0671 \u0646\u0651\u064e\u0650\u0639\u0650\u0645\u0651\u064e\u0627 \u06cc\u064e\u0639\u0650\u0638\u064f\u0643\u064f\u0645 \u0628\u0650\u06be\u0650 \u0625\u06e6\u0653\u06d7\u0650\u0646\u0651\u064e \u0671 \u0643\u0651\u064e\u0627\u0646\u064e \u0633\u064e\u0645\u0650\u06cc\u0639\u06e2\u064b\u0627 \u0628\u064e\u0635\u0650\u06cc\u0631\u06ed\u064b\u0627\u201cIndeed, Allah orders you to render the trust to their owners, and when you judge between people to judge with justice. Excellent is that which Allah instructs you. Indeed, Allah is ever Hearing and seeing\u201d (Quran 4:58)\u0625\u0650\u0646\u0651\u064e \u0671 \u06cc\u0651\u064e\u064f\u062f\u064e\u0641\u0650\u0639\u064f \u0639\u064e\u0646\u0650 \u0671\u0644\u0651\u064e\u0630\u0650\u06cc\u0646\u064e \u0621\u064e\u0627\u0645\u064e\u0646\u064f\u0648\u0653\u0627\u06df \u0625\u06d7\u0650\u0646\u0651\u064e \u0671 \u0644\u0651\u064e\u0627 \u06cc\u064f\u062d\u0650\u0628\u0651\u064f \u0643\u064f\u0644\u0651\u064e \u062e\u064e\u0648\u0651\u064e\u0627\u0646\u06e2\u064d \u0643\u064e\u0641\u064f\u0648\u0631\u064d\u201cLo! Allah defended those who are true. Lo! Allah loved not each deceitful ingrate\u201d (Quran 22:38).\u0648\u064e\u0623\u064e\u0648\u0652\u0641\u064f\u0648\u0627\u06df \u0671\u0644\u0652\u0643\u064e\u06cc\u0652\u0644\u064e \u0625\u0650\u0630\u064e\u0627 \u0643\u0650\u0644\u0652\u062a\u064f\u0645\u0652 \u0648\u064e\u0632\u0650\u0646\u064f\u0648\u0627\u06df \u0628\u0650\u0671\u0644\u0652\u0642\u0650\u0633\u0652\u0637\u064e\u0627\u0633\u0650 \u0671\u0644\u0652\u0645\u064f\u0633\u0652\u062a\u064e\u0642\u0650\u06cc\u0645\u0650 \u0630\u06da\u064e\u0644\u0650\u0643\u064e \u062e\u064e\u06cc\u0652\u0631\u06ed\u064c \u0648\u064e\u0623\u064e\u062d\u0652\u0633\u064e\u0646\u064f \u062a\u064e\u0623\u0652\u0648\u0650\u06cc\u0644\u06ed\u064b\u201cAnd measure full when you measure. And weigh with an even balance. This is better, and its end is good\u201d (Quran 17:35).\u0648\u064e\u06cc\u0652\u0644\u064c \u0644\u0650\u0644\u0652\u0645\u064f\u0637\u064e\u0641\u0651\u0650\u0641\u0650\u06cc\u0646\u064e \u0627\u0644\u0651\u064e\u0630\u0650\u06cc\u0646\u064e \u0625\u0650\u0630\u064e\u0627 \u0627\u0643\u0652\u062a\u064e\u0627\u0644\u064f\u0648\u0627 \u0639\u064e\u0644\u064e\u0649 \u0627\u0644\u0646\u0651\u064e\u0627\u0633\u0650 \u06cc\u064e\u0633\u0652\u062a\u064e\u0648\u0652\u0641\u064f\u0648\u0646\u064e \u0648\u064e\u0625\u0650\u0630\u064e\u0627 \u0643\u064e\u0627\u0644\u064f\u0648\u06be\u064f\u0645\u0652 \u0623\u064e\u0648\u0652 \u0648\u064e\u0632\u064e\u0646\u064f\u0648\u06be\u064f\u0645\u0652 \u06cc\u064f\u062e\u0652\u0633\u0650\u0631\u201cWoe to the defrauders who use short measures, who, when they measure [a commodity bought] from the people, take the full Measure, but diminish when they measure or weigh for them\u201d.\u06cc\u064e\u0670\u0653\u0623\u064e\u06cc\u0651\u064f\u06be\u064e\u0627 \u0671\u0644\u0651\u064e\u0630\u0650\u06cc\u0646\u064e \u0621\u064e\u0627\u0645\u064e\u0646\u064f\u0648\u0627\u06df \u0644\u064e\u0627 \u062a\u064e\u0623\u0652\u0643\u064f\u0644\u064f\u0648\u0653\u0627\u06df \u0623\u064e\u0645\u0652\u0648\u0670\u064e\u0644\u064e\u0643\u064f\u0645 \u0628\u064e\u06cc\u0652\u0646\u064e\u0643\u064f\u0645 \u0628\u0650\u0671\u0644\u0652\u0628\u064e\u0670\u0637\u0650\u0644\u0650 \u0625\u0650\u0644\u0651\u064e\u0622 \u0623\u064e\u0646 \u062a\u064e\u0643\u064f\u0648\u0646\u064e \u062a\u0650\u062c\u064e\u0670\u0631\u064e\u0629\u064b \u0639\u064e\u0646\u062a\u064e\u0631\u064e\u0627\u0636\u064d\u06e2 \u0645\u0651\u0650\u0646\u0643\u064f\u0645\u0652 \u0648\u06da\u064e\u0644\u064e\u0627 \u062a\u064e\u0642\u0652\u062a\u064f\u0644\u064f\u0648\u0653\u0627\u06df \u0623\u064e\u0646\u0641\u064f\u0633\u064e\u0643\u064f\u0645\u0652 \u0625\u06da\u0650\u0646\u0651\u064e \u0671 \u0643\u0651\u064e\u0627\u0646\u064e \u0628\u0650\u0643\u064f\u0645\u0652 \u0631\u064e\u062d\u0650\u06cc\u0645\u06ed\u064b\u0627\u201cO you who have believed, do not consume one another\u2019s wealth unjustly but only (lawful) business by mutual consent. And do not kill yourselves (or one another). Indeed, Allah is to you ever merciful\u201d (Quran 4:29).\u0648\u064e\u0644\u064e\u0627 \u062a\u064e\u0623\u0652\u0643\u064f\u0644\u064f\u0648\u0653\u0627\u06df \u0623\u064e\u0645\u0652\u0648\u064e\u0644\u064e\u0643\u064f\u0645\u0628\u064e\u06cc\u0652\u0646\u064e\u0643\u064f\u0645 \u0628\u0650\u0671\u0644\u0652\u0628\u064e\u0637\u0650\u0644\u0650 \u0648\u064e\u062a\u064f\u062f\u0652\u0644\u064f\u0648\u0627\u06df \u0628\u0650\u06be\u064e\u0622 \u0625\u0650\u0644\u064e\u0649 \u0671\u0644\u0652\u062d\u064f\u0643\u0651\u064e\u0627\u0645\u0650 \u0644\u0650\u062a\u064e\u0623\u0652\u0643\u064f\u0644\u064f\u0648\u0627\u06df \u0641\u064e\u0631\u0650\u06cc\u0642\u06ed\u064b\u0627 \u0645\u0651\u0650\u0646\u0652 \u0623\u064e\u0645\u0652\u0648\u064e\u0644\u0650 \u0671\u0644\u0646\u0651\u064e\u0627\u0633\u0650 \u0628\u0650\u0671\u0644\u0652\u0625\u0650\u062b\u0652\u0645\u0650 \u0648\u064e\u0623\u064e\u0646\u062a\u064f\u0645\u0652 \u062a\u064e\u0639\u0652\u0644\u064e\u0645\u064f\u0648\u0646\u064e\u201cAnd do not consume one another\u2019s wealth unjustly or send it [in bribery] to the rulers in order that [they might aid] you [to] consume a portion of the wealth of the people in sin, while you know [it is unlawful]\u201d (Quran 2:188).Hadith is the saying of the Prophet Muhammad (PBUH). Hadith and Sunnah are both critical aspects of Islam. Hadith and Sunnah are two of Islam\u2019s key aspects. Hadith plays an essential role in everyone\u2019s life because it forms the person you are and who you have become along this life\u2019s journey. Following the Prophet Muhammad\u2019s hadith or Sunnah (PBUH), Allah\u2019s ways and the messages he sent down from the heavens will be followed to help us achieve our goals.Prophet Muhammad\u2019s (PBUH) saying!\u201cThe one who deceives is not one of us\u201d \u201cDon\u2019t deceive someone who trusts you so you can\u2019t cheat him\u201d (Wasa\u2019 il ul-Shia Vol. 12 pages 364).He (PBUH) also said: \u201cAn honest and truthful Muslim trader shall be held with the martyrs on the Day of Resurrection.\u201d .\u201cIf both parties were to speak the truth and explain the faults and attributes (of the goods), then they would be blessed in their transaction, and if they said lies and concealed anything, then their transaction\u2019s blessings would be lost\u201d (Bukhari and Muslim Resolution No. (101).The Prophet has been recorded to state that it is not permitted to sell goods without making clear about everything, nor is it allowed for anybody who knows about the goods\u2019 defects, not to mention them. An act of dishonesty is the same as cheating. One of the worst forms of fraud is dishonesty. A dishonest person is always likely to defraud others as often as possible such as misuse, false claim, intimate good, and so on.The primary duties of Muslims contain all religious beliefs and attitudes. The Muslims believe that there is only one God (Allah), \u201cIndeed, your God is one\u201d (37:4), and Muhammad is the Prophet, \u201co Muhammad! They say \u2018we testify that you are the last Messenger of Allah\u201d (63:1). Religious teachings are based on three things, the \u201cQuran\u201d and the \u201cHadith\u201d (Muhammad\u2019s documented saying and acts), and the shari\u2019a (Religious law) offers answers to all ethical questions . The Mus\u201cRather, to Allah belong the hereafter and the first life\u201d (Quran 53:25).The second dimension is the fulfillment of the core religious duties. These consist of following more or less the \u201cfive pillars of Islam\u201d duties which consist of religious beliefs and practices : (1) \u201cShThe experience dimension includes people\u2019s perceptions and practices of their faith . ExperieThe knowledge aspect involves knowledge of the person about religion. The contents of the Quran and Sunnah are usually the primary source of Islamic knowledge. Believers are expected to know a minimum of these contents. Muslims call for the in-depth understanding of equality for all human beings, a strong sense of brotherhood, good or bad deeds, including morality, modesty, humility, trustworthiness, duty, justice, patience, fairness, and tolerance, ethics in dealing, care, empathy, and compassion .\u201cHe also says, we have revealed to you the book which clarifies every matter\u201d (Quran 16:89).\u201cIndeed this Quran guides to the path which is clearer and straighter than any other\u201d .The Orthopraxis dimension of religiosity plays an important and distinct role in Muslim religiosity . OrthoprH1(a): Basic Duties dimension of religiosity negatively affect consumers\u2019 attitude toward purchasing luxury counterfeit products.H1(b): Central Duties dimension of religiosity negatively affects consumers\u2019 attitude toward purchasing luxury counterfeit products.H1(c): The experience dimension of religiosity negatively affects consumers\u2019 attitudes toward purchasing luxury counterfeit products.H1(d): The knowledge dimension of religiosity negatively affects consumers\u2019 attitudes toward purchasing luxury counterfeit products.H1(e): Orthopraxis dimension of religiosity negatively affect consumers\u2019 attitude toward purchasing luxury counterfeit products.A survey method was employed to validate the conceptual model because the quantitative research method predicts individual responses and examines the relationship between constructs . Many reA 7-point Likert scale was used to measure all these constructs that ranged from strongly disagree = 1 to strongly agree = 7. A quantitative approach using the survey questionnaire method was used to examine the influence of religiosity on Pakistani Muslim youth\u2019s attitude toward counterfeit products. The questionnaire comprises two parts. Section A will be the attitude (5 items) , basic dIn the current study, the non-probability sampling technique of purposive sampling was employed. It is suitable and more appropriate when the sampling frame is not available and population is unknown . MoreoveStructural Equation Modeling (SEM) has an advantage for statistical analysis in terms of efficiency, accuracy, and convenience over traditional multivariate statistical techniques . MoreoveThe measurement model is used to observe the relationship between observed data and latent variables, and it also explains the calculation of variables. The advantage of this model was to assess the valuation of validity and reliability test. Construct reliability is measured using outer loading, and internal consistency of reliability was measured through composite reliability. Moreover, convergent reliability is measured through the average variance extracted (AVE) . To asceNext discriminating validity, Heterotrait-Monotrait correlation ratio (HTMT) TEST, was used to measure discriminating validity because it is more potent than other methods. According to t-values, coefficient of determination (R2), effect size (f2), and predictive relevance (Q2). Using the bootstrapping method , path coefficient significance was measured. The results indicate that all hypotheses are accepted and significant toward individuals\u2019 attitudes to purchasing counterfeit products. However, findings suggested that the knowledge dimension has the most significant negative impact on attitude to use luxury counter fitting products (47%), followed by orthopraxis dimension (19%), experience dimension (13%), central duties (8%), and basic religiosity (7%)  see . The R2 (7%) see .Counterfeiting is a centuries-old crime, the proliferation of counterfeit products has become a global phenomenon, and indeed is a serious business issue all over the nation . It\u2019s fap > 0.05). The highest impact of the knowledge dimension is in line with the expectations that more knowledge and study of basic tenets and scriptures of Islam influence the followers to act accordingly. This finding also collaborates with the previous studies . The results are also in agreement with previous research . One plausible explanation for the significant results is that when the individual is more afraid of the creator, they abide by what is considered good  and avoid what is deemed bad (haram), which can be seen in their practices. Muslims follow the ALLAH instructions because they know that their excellent behavior rewards them and their bad behavior punished them in the hereafter  show a negative association of religious duties with attitude toward counterfeit products with a weak effect. The results are consistent with the earlier study. Previous studies also suggest that a person with strong religious commitment is less likely to engage in unethical behavior such as illegal use of the drug  have a weak negative relationship with attitude toward the purchase of counterfeit products. Some previous researches also validate the finding of the study. Basic duties are interconnected with the core or fundamental values of Islam, which differentiate Muslims from non-Muslims. The main reasons for the results are the previous author characterized these beliefs into three types. The first type of belief explains the exitance of the divine, the second type of belief explains the purpose of the divine, and the third describes the ethical structure of the whole region  on specific counterfeiting products like cosmetics, digital piracy, and online books. Third, many other factors or variables like culture, income, and education should be added to conduct further research. Fourth, this study only focuses on young Muslim consumers. Further studies can consider other religions like Christians, Hindus, Jews, and Buddhism.The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.Ethical review and approval was not required for the study on human participants in accordance with the local legislation and institutional requirements. Written informed consent from the patients/participants or patients/participants legal guardian/next of kin was not required to participate in this study in accordance with the national legislation and the institutional requirements.SA and HZ: conceptualization and writing of original manuscript. MA and NK: review and improve. PP: data collection and analysis. All authors contributed to the article and approved the submitted version.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher."}
{"text": "The industry-based internet of things (IIoT) describes how IIoT devices enhance and extend their capabilities for production amenities, security, and efficacy. IIoT establishes an enterprise-to-enterprise setup that means industries have several factories and manufacturing units that are dependent on other sectors for their services and products. In this context, individual industries need to share their information with other external sectors in a shared environment which may not be secure. The capability to examine and inspect such large-scale information and perform analytical protection over the large volumes of personal and organizational information demands authentication and confidentiality so that the total data are not endangered after illegal access by hackers and other unauthorized persons. In parallel, these large volumes of confidential industrial data need to be processed within reasonable time for effective deliverables. Currently, there are many mathematical-based symmetric and asymmetric key cryptographic approaches and identity- and attribute-based public key cryptographic approaches that exist to address the abovementioned concerns and limitations such as computational overheads and taking more time for crucial generation as part of the encipherment and decipherment process for large-scale data privacy and security. In addition, the required key for the encipherment and decipherment process may be generated by a third party which may be compromised and lead to man-in-the-middle attacks, brute force attacks, etc. In parallel, there are some other quantum key distribution approaches available to produce keys for the encipherment and decipherment process without the need for a third party. However, there are still some attacks such as photon number splitting attacks and faked state attacks that may be possible with these existing QKD approaches. The primary motivation of our work is to address and avoid such abovementioned existing problems with better and optimal computational overhead for key generation, encipherment, and the decipherment process compared to the existing conventional models. To overcome the existing problems, we proposed a novel dynamic quantum key distribution (QKD) algorithm for critical public infrastructure, which will secure all cyber\u2013physical systems as part of IIoT. In this paper, we used novel multi-state qubit representation to support enhanced dynamic, chaotic quantum key generation with high efficiency and low computational overhead. Our proposed QKD algorithm can create a chaotic set of qubits that act as a part of session-wise dynamic keys used to encipher the IIoT-based large scales of information for secure communication and distribution of sensitive information. We applied the class \u201cRandom\u201d in Java for the chaotic production of qubit variant states. We used octa-qubit bases to capture the matched patterns at the side of the receiver. The octal states are represented as {, , , , , , , } to signify the states of the qubits over the randomly directed transmission qubits as shown in The originator produces the range of values from 0 to 1, which is mapped with the full direction of the qubit; at the receiver side, we need to select the matched base according to our octal state representation. The considerable equivalent random numbers generated as per the movement of the qubit are {0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1} as represented in public String createRandomBase{double base_num = qubit.randomGenerator;\u2003\u2003\u2003\u00a0if (base_num <0 && base_num <0.25)\u2003\u2003\u2003\u00a0\u2003\u2003\u2003\u00a0base = \u201c\u00b1\u201d;else if (base_num >0.25 && base_num<0.5)) \u2003\u2003\u2003\u00a0\u2003\u2003\u2003\u00a0base = \u201c\u253c\u201d;else if (base_num >0.5 && base_num<0.75) \u2003\u2003\u2003\u00a0\u2003\u2003\u2003\u00a0base = \u201c\u04fe\u201d;else if (base_num >0.75 && base_num<1))\u2003\u2003\u2003\u00a0\u2003\u2003\u2003\u00a0base = \u201cX\u201d;return base;\u2003\u2003\u2003\u00a0}According to the selected base, the SOP sequence is chaotic. The base values and the SOPs of the above snippet are represented as the polarizer angle base \u201c\u253c\u201d means to capture only the precise vertical and the horizontally directed matched qubits {, }. The polarizer angle base \u201cX\u201d means to capture only the precise left and right orthogonal-directed matched qubits {, }. Here, we added two more variants to also capture other directions of the qubit. The polarizer angle base \u201c\u00b1\u201d means to capture different directions of the qubit in between the straight rectilinear and straight orthogonal positions that are to capture the directed matched qubits { } of the top half circle. The polarizer angle base \u201c\u04fe\u201d means to capture other directions of the qubit in between the straight rectilinear and straight orthogonal positions that are to capture the directed matched qubits { } of the bottom half-circle. The entire process is given as:public String CreateSop(String base)\u2003{RL1 = \u201c\u253c\u201d;\u2003SRL1 = \u201c\u00b1\u201d;RL2 = \u201c\u253c\u201d;\u2003SRL2 = \u201c\u00b1\u201d;LOG1 = \u201cX\u201d;\u2003SLOG1 = \u201c\u04fe\u201d;ROG2 = \u201cX\u201d;\u2003SROG2 = \u201c\u04fe\u201d;if || base.equals(RL2))\u2003\u2003\u2003\u2003{double base_num = qubit.randomGenerator;\u2003\u2003\u2003\u2003\u00a0if (base_num >0.25 && base_num <0.5)\u2003\u2003\u2003\u2003\u00a0\u2003\u2003\u2003\u2003\u00a0sop = \u201c|\u201d;else\u2003\u2003\u2003\u2003\u00a0\u2003\u2003\u2003\u2003\u00a0sop = \u201c\u2014\u201d;\u2003\u2003}else if || base.equals(ROG2))\u2003\u2003\u2003\u2003{double base_num = qubit.randomGenerator;\u2003\u2003\u2003\u2003\u00a0if (base_num > 0.75 && base_num <1)\u2003\u2003\u2003\u2003\u00a0\u2003\u2003\u2003\u2003\u00a0sop = \u201c/\u201d;else\u2003\u2003\u2003\u2003\u00a0\u2003\u2003\u2003\u2003\u00a0sop = \u201c\\\\\u201d;\u2003\u2003}else if || base.equals(SRL2))\u2003\u2003\u2003\u2003{double base_num = qubit.randomGenerator;\u2003\u2003\u2003\u2003\u00a0if (base_num >0 && base_num < 0.25)\u2003\u2003\u2003\u2003\u00a0\u00b1_|\u201d;\u2003\u2003\u2003\u2003\u00a0sop = \u201celse\u2003\u2003\u2003\u2003\u00a0\u00b1_\u2014\u201d;\u2003\u2003\u2003\u2003\u00a0sop = \u201c\u2003\u2003}else if || base.equals(SROG2))\u2003\u2003\u2003\u2003{double base_num = qubit.randomGenerator;\u2003\u2003\u2003\u2003\u00a0if (base_num > 0.5 && base_num < 0.75)\u2003\u2003\u2003\u2003\u00a0\u2003\u2003\u2003\u2003\u00a0sop = \u201c \u04fe_/\u201d;else\u2003\u2003\u2003\u2003\u00a0\u2003\u2003\u2003\u2003\u00a0sop = \u201c \u04fe_\\\\\u201d;\u2003\u2003}return sop;\u2003\u2003\u2003}The central portion of secure end-to-end transmission by using the proposed QKD technique is the measurement of the multi-qubits of Singh by King. This means, we need to address how King can produce his set of bits; these bits may be a part of the private key. The calculation of the qubits of Singh correspondingly abolishes the generated bits, letting Singh transmit his base set to King only, and access or capture by third-party users or intruders is impossible. The states of the multi-qubits calculation process function through predicting the base of Singh first alongside the base of King. If the chaotic grounds of Singh and King are accorded, the matched pattern bit sets will be used as part of the secret key for the encipherment and decipherment process. In a counter case, the chaotic bases of Singh and King are not accorded. Afterward, the restrained bit sets are discarded. In both cases, the multi-qubit set should be destroyed after matched bits are generated. The finally produced bits approach yields the bit 1 if the SOP is \u201c|\u201d or \u201c\u2044\u201d or \u201c\u00b1_|\u201d or \u201c\u04fe_/\u201d;\\ and 0 if the sop is \u201c\u2014\u201c or \u201c\\\u201d or \u201c\u00b1_\u2014\u201d or \u201c\u04fe_\\\\\u201d. The qubit measurement and bitrate production functions are specified as:public int bitCreate (String sop){if )\u2003\u2003\u2003\u2003\u00a0\u00a0\u00a0 return 1;\u2003\u2003\u2003\u00a0\u00a0else if )\u2003\u2003\u00a0\u00a0\u00a0\u00a0 return 1;\u2003\u2003\u2003\u00a0\u00a0else if )\u2003\u00a0\u00a0\u00a0 return 1;\u2003\u2003\u2003\u00a0\u00a0else if )\u2003\u00a0\u00a0\u00a0 return 1;\u2003\u2003\u2003\u00a0\u00a0else if )\u2003\u00a0\u00a0\u00a0\u00a0\u00a0 return 0;\u2003\u2003\u2003\u00a0\u00a0else if )\u2003\u00a0\u00a0\u00a0\u00a0\u00a0return 0;\u2003\u2003\u2003\u00a0\u00a0else if )\u2003\u00a0\u00a0return 0;\u2003\u2003\u2003\u00a0\u00a0else if )\u2003return 0;\u2003\u2003\u2003\u00a0\u00a0else return null;\u2003\u2003\u2003\u00a0\u00a0}public int qubitMeasure {int bitset;\u2003\u2003\u2003\u00a0\u00a0if )\u2003\u2003\u2003\u00a0\u00a0\u2003\u2003\u2003\u00a0\u00a0{null;\u2003\u2003\u2003\u00a0\u00a0bitset = bitCreate(myqubit.sop); singqubit = \u2003\u2003\u2003\u00a0\u00a0}else\u2003\u2003\u2003\u00a0\u00a0\u2003\u2003\u2003\u00a0\u00a0{\u2003\u2003\u2003\u00a0\u00a0singqubit.base= base_;\u2003\u2003\u2003\u00a0\u00a0singqubit.sop = singqubit.createSop(base_);\u2003\u2003\u2003\u00a0\u00a0bitset = bitCreate(singqubit.sop);null;\u2003\u2003\u2003\u00a0\u00a0singqubit = \u2003\u2003\u2003\u00a0\u00a0}return bitset;\u2003\u2003\u2003\u00a0\u00a0}To provide more security to encrypt the sensitive information of users, we need to produce a secret quantum key with a reasonable count of qubits. Here, we show the list of methods used to generate a good number of qubits; from these, we extract the final secret key for the encipherment and decipherment process. The first step of the algorithm is that Singh needs to pass a random integer, RI. Here, RI is the count of qubits he plans to produce for the transmission. Recall that we should have a base and SOP for the construction of qubits. Singh initially makes a list of commands by invoking the random base method, and as per that list, he produces his list of SOPs mapped with the list of qubits. Afterward, Singh utilizes both lists to make the final list of mapped qubits with their respective bases, which he will send to King. Later, King will obtain the qubit list of Singh; he utilizes the RI supplied by the count to produce his base list as Singh already has done. King also creates his bit set list with the help of using the qubit measurement process. Now that both Singh and King have their base and bit set lists, both shall interchange their base lists to construe their lists of bit sets. If their occurred bases are matched, then the resultant bit list is used to generate a secret key. Otherwise, it is discarded. The result is a shared secret key, whose size is approximately half of RI. To evaluate this, we maintain the approximate size of the secret key related to RI. For every running instance, the proportion varies and progressively touches just about half of the actual length. In existing models, due to the production of shorter key length and after filtration, the final key is much shorter and is vulnerable to attacks specifically in Social Internet of Things (SIoT) and related applications, blockchain based mobile applications, wireless body area network based applications, computer vision based applications , fuzzy oThe proposed model is highly chaotic and dynamic. Despite this, it works fine to exhibit the notion and how it performs compared to previous QKD models. In this paper, we only demonstrated how keys are exchanged between IIoT devices in the absence of attacks and theoretically proved that this model provides more security from all types of quantum attacks compared to traditional models. In addition, this model uses less qubit transmission rate and key exchange rate along with minimal computational overhead and shows more efficacy with regards to error-rate measurement to detect whether snooping has happened or not. In future work, we can add an unauthorized person or intruder and demonstrate how to address different advanced quantum attacks at one time while sensitive data are transmitted among multiple IIoT devices. It could be possible to examine various privacy and security parameters of the proposed standard, such as the minimum bit size needed to detect intruders. At the same time, with the communication of qubits among IoT and IIoT devices, we can enhance their security by strengthening the projected model with a multi-photon multi-state entanglement approach using Qiskit for working with quantum computers at the level of pulses, circuits, and IoT and IIoT application modules."}
{"text": "This article has been withdrawn by the authors. The authors and thejournal conclude that the images between and within Figures\u00a02D and 3D were reused.Specifically, the image \u201cpcDNA\u201d in Figure\u00a03D was a duplication of the image\u201cGW9662\u00a0+ Vehicle\u201d in Figure\u00a02D. Similarly, the image \u201cp300WT\u00a0+ 15(S)-HETE\u201d inFigure\u00a03D was captured from the well \u201cPFS1YF\u00a0+ 15(S)-HETE\u201d in Figure\u00a02D. The image\u201cp300WT\u201d in Figure\u00a03D was captured from the well \u201cpcDNA\u00a0+ 15(S)-HETE\u201d in Figure\u00a03Ditself. Additional potential internal reuse is in Figure\u00a02D, STAT1(WT)\u00a0+ 15(S)-HETE\u201dand Figure\u00a02D, \u201cPFS1YF\u00a0+ 15(S)-HETE\u201d. The authors state that these imageduplications do not affect the conclusions of the paper. The authors apologize forthis oversight."}
{"text": "This study gives a critical examination of the performances of three top key opinion leaders (KOLs) on social media in China to explore whether gender serves as an important social factor in their interaction with their followers. A critical analysis of their Weibo posts has revealed that they construct different gender identities through their preferential choices of personality traits, speech acts, and addressing terms. This can be explained in terms of their target consumers and the products they promote. It concludes that although gender can serve as an important factor in top KOLs\u2019 performances on social media, it can be appropriated and exploited in varied ways to serve different communicative purposes. China has the world\u2019s largest live-streaming market in terms of the scale of participation and revenue generation . The COVSince KOLs play important roles in diffusing information, setting agenda, and influencing others\u2019 decision-making or behaviour , how KOLPrevious studies on gendered interaction have identified different personality traits associated with males and females . For exaNevertheless, disputes remain over whether gender differences in language use will dissolve on social media. While the anonymity of social media might possibly lead to the hiding or invisibility of gender disparity, some previous studies have demonstrated that gender differences are still evident even when communication moves from offline to online . Some otBesides, studies on gendered interaction \u201creached a deadlock partly because of the research question itself (looking for differences between women and men) and its presumptions (taking \u2018gender\u2019 for granted)\u201d , p. 411.A KOL is defined as \u201can individual who has a great amount of influence on the decision making, attitudes and behaviors of other people\u201d , p. 511 This study focuses on three top KOLs, namely Li Jiaqi (Li), Viya, and Luo Yonghao (Luo). They were selected because of their influences as well as their distinctive gender performances. Li was once a makeup seller at a L\u2019Or\u00e9al counter in Nanchang (a second-tier city in China). It is quite rare in China for a man selling cosmetics to women . HoweverViya was the first female KOL of Taobao Live , and sheLuo was the \u201cfirst generation internet influencer\u201d in China , p. 88, st 2021 to 30 September 2021 (6 months). They are all \u201cV-users\u201d (verified users on Weibo) with millions of followers. As a hybridized form of Twitter and Facebook, Sina Weibo is one of the most popular social media platforms in China. As a social networking platform, Weibo serves as a crucial channel for interpersonal communication . It is aThis study gives a critical discourse analysis (CDA) of gender construction in the Weibo posts of three top KOLs in Chinese mainland. CDA views language use as a social practice and underlines the significance of examining language use in its social contexts . It viewThe constitutive effects of language can be identified in three aspects: (1) social knowledge; (2) social identities; and (3) social relations . While sHow do the three KOLs vary in their choice of linguistic features?How do they construct relations between themselves and their followers?What gender images do they construct for themselves on their social media?How do they contribute to their respective communicative purposes?In order to reveal the dynamic relations between gender and interaction, this study conducts a top-down analysis of these posts at three levels: (1) personality traits; (2) speech acts; and (3) addressing terms. Previous studies have demonstrated that men and women tend to display different personality traits. Men tend to be \u201cdetermined,\u201d \u201ceffective,\u201d \u201cfighting,\u201d \u201cforceful,\u201d \u201cconfident,\u201d \u201cchampion,\u201d and \u201crational\u201d (masculine traits), whereas women tend to be \u201ccaring,\u201d \u201cwarm,\u201d \u201ccompassionate,\u201d \u201cunderstanding,\u201d \u201ccongenial,\u201d \u201chumble,\u201d and \u201cempathetic\u201d (feminine traits) . This stSpeech acts refer to the linguistic acts associated with the speaker\u2019s utterance . Zhang aThey are followed by a close analysis of the self- and other-addressing terms used by the three KOLs on their social media. All the pronouns and nominal forms used to address themselves and others are examined . They arThe coding of personality traits and speech acts was performed by the first and second authors. The two authors coded the data independently after the initial training and discussion of the coding criteria and procedure. The discrepancies in some results were resolved through further discussion. The inter-rater reliability was measured with Cohen\u2019s Kappa. The average percent agreement of all coded categories was greater than 90%, and the Cohen\u2019s Kappa was 0.91, indicating a strong agreement between the two coders .zhinan\u201d  image.They also show distinct differences in their preferences for specific personality traits. Luo is noted for his highest preferences for the masculine traits of \u201cforceful\u201d (50%) and \u201crational\u201d (16%). Li shows the highest preferences for the feminine traits of \u201ccaring\u201d (28%) and \u201ccongenial\u201d (20%) and the masculine trait of \u201cforceful\u201d (23%). Viya shows the highest preferences for the feminine trait of \u201cunderstanding\u201d (20%) and the masculine traits of \u201cforceful\u201d (26%) and \u201cconfident\u201d (14%). It shows that the personality trait \u201cforceful\u201d is valued by all three KOLs, which can be attributed to the primary function of their Weibo posts, i.e., to promote their products and engage the followers to buy their products (see Example 1). However, it is much more highlighted in Luo\u2019s Weibo posts than in the other two KOLs\u2019, suggesting that Luo prefers to deliver information about his products in a more straightforward way than the other two KOLs. This contributes to the construction of his \u201cBesides, Luo shows a \u201crational\u201d trait by questioning, challenging or criticizing some social phenomenon or popular ideas (see Example 2). He disaligns himself with the public and the popular ideas, thus further consolidating his image of being the \u201cspiritual leader\u201d of some youngsters or middle-aged men. In other words, Luo does two things on his Weibo posts: promoting the products and constructing a critical image for himself. More often than not, they are achieved on separate posts. Examples are as follows:(1) Forceful:\u5168\u573a\u5e73\u57475\u6298!\u5168\u573a\u5e73\u57475\u6298!\u5168\u573a\u5e73\u57475\u6298! .Average 50% off for all items! Average 50% off for all items! Average 50% off for all items!(2) Rational:\u867d\u7136 Chord \u662f\u4eba\u7c7b\u505a\u8fc7\u7684\u6700\u4e11\u7684\u89e3\u7801\u5668\uff0c\u4f46\u97f3\u8d28\u786e\u5b9e\u597d\u2026\u4f60\u770b\uff0c\u4ec5\u4ece\u5546\u4e1a\u89d2\u5ea6\u6765\u8bf4\uff0c\u7535\u5b50\u4ea7\u54c1\u7684\u8bbe\u8ba1\u6ca1\u90a3\u4e48\u91cd\u8981\u3002.Although Chord is the ugliest decoder made by humans, its sound quality is indeed good\u2026You see, just considering business, the design of electronic products is not that important.guimi\u201d  of his overwhelming female followers. Usually, they are achieved on the same Weibo post. That means Li promotes his products through constructing himself as a congenial and understanding friend of his followers. As Example 5 shows, Li shows a \u201cforceful\u201d trait by announcing the news on the one hand and a \u201ccaring\u201d trait by asking the followers whether they are ready or not on the other hand. In example 6, Li shows a \u201cforceful\u201d trait by announcing the news and a \u201ccongenial\u201d trait by showing he and his followers share the same hobby at the same time.By contrast, although Li also promotes his products by underlining the \u201cforceful\u201d trait, he shows equal, if not more, preferences for the \u201ccongenial\u201d and \u201ccaring\u201d traits (see Examples 3 and 4). In other words, Li shows alignment with the followers by constructing himself as a \u201c(3) Caring:\u590f\u5929\u4e86\uff0c\u6709\u6ca1\u6709\u60f3\u5403\u7684\u96f6\u98df\u5440\u7279\u522b\u662f\u90a3\u79cd\u79c1\u85cf\u96f6\u98df\u6216\u8005\u5bb6\u4e61\u7f8e\u98df\uff0c\u7ed9\u6211\u6284\u6284\u4f5c\u4e1a\u5440 .Summer is coming, have any snacks wanna try? Especially those personal treasures or hometown specialties, let me copy your homework.(4) Congenial:\u6211\u7684\u5bb5\u591c\u548c\u6700\u8fd1\u56f0\u56f0\u7684Never\u80d6\u6b7b\u6211\u7b97\u4e86 .My night meal and the recent sleepy Never will become too fat(5) Forceful\u2009+\u2009Caring:\u6b63\u5f0f\u5b98\u5ba3!#\u6240\u6709\u5973\u751f\u7684offer#\u5012\u8ba1\u65f62\u5929\u54af\u3002\u51c6\u5907\u597d\u4e86\u5417\uff0c9\u670827\u53f712:00\u89c1!.th!Official announcement! # All Girls\u2019 Offer # Countdown to 2\u2009days. Ready? See you on 12:00 Sep. 27(6) Forceful\u2009+\u2009Congenial:\u4eca\u665a10\u70b9\uff0c\u4ed6\u6765\u54af!\u6211\u5c31\u5dee\u62a5\u8eab\u4efd\u8bc1\u53f7\u5566!.Tonight 10 O\u2019clock, he\u2019s coming! Only the ID number has not been disclosed!Viya is different in that she promotes her products by constructing herself as an expert in the field. On the one hand, she highlights that she understands what the followers need; on the other hand, she underlines her confidence in the quality and prices of her products. This is often achieved on the same posts (see Examples 7 and 8). In other words, she promotes her products because she believes that it is the best choice for the followers. The image she constructs for herself is a professional image rather than a gendered image. Examples are as follows:(7) Forceful\u2009+\u2009understanding:99\u7206\u54c1\u65e5!vivo\u3001\u96c5\u8bd7\u5170\u9edb\u65b0\u54c1\u9996\u53d1!\u7389\u6cfd\u3001\u96c5\u840cACE\u3001OLAY\u8d85\u7ea2\u74f6\u3001\u7b14\u8bb0\u672c\u7535\u8111\u3001\u767e\u5a01\u5564\u9152\u2026\u5168\u5bb6\u4eba\u90fd\u9700\u8981\u7684\u597d\u7269\u5c31\u5728\u4eca\u665a! .\u4eca\u665a#\u8587\u5a05\u76f4\u64ad\u95f4# 99 Top-sellers\u2019 Day! Vivo, Est\u00e9e Lauder debut! Dr. Yu, Yaman ACE, Olay Regenerist, laptop, Budweiser\u2026Handy products for the whole family are here tonight!Tonight #Viya\u2019s Live Room# (8) Forceful\u2009+\u2009Confident:\u96f6\u98df\u8282\u5f00\u5403\u5566!\u6587\u548c\u53cb\u3001\u597d\u5229\u6765\u3001\u661f\u5df4\u514b\u3001\u81ea\u55e8\u9505\u3001\u9a6c\u8fed\u5c14\u51b0\u68cd\u3001\u597d\u5229\u6765\u751c\u54c1\u3001\u5317\u6d77\u7267\u573a\u65b0\u54c1\u9178\u5976..0.60\u2009+\u2009\u65e0\u9650\u56de\u8d2d\u7684\u7206\u6b3e\u96f6\u98df\uff0c\u4e0d\u6765\u4e8f\u70b8! .\u4eca\u665a #\u8587\u5a05\u76f4\u64ad\u95f4# Snack\u2019s Day Start eating! Wenheyou, Hollyland, Starbucks, self-heating hotpot, Modern ice cream, Hollyland dessert, Beihai Ranch new arrival yogurt\u202660\u2009+\u2009repurchase top-selling snacks, you\u2019ll suffer great losses if not come!Tonight #Viya\u2019s Live Room# They highlight different personality traits to construct different gender images for themselves on their Weibo posts. Luo prefers to construct a \u201cstraight male\u201d image for himself, Li a \u201cladybro\u201d image while Viya a \u201cprofessional\u201d image. This can be attributed to the types of products they promote and the target followers of their Weibo posts.Besides, the three KOLs also show distinct differences in the choice of specific speech acts. Among all speech acts, \u201cdirective\u201d and \u201cpromoting self and others\u201d are most frequently used by three KOLs. \u201cPromoting self and others\u201d is related to the primary function of their Weibo posts, i.e., to promote their products (see Example 10). \u201cDirective\u201d is used to ask the followers to buy their products (see Example 9).(9) Directive:\u73b0\u5728\u5c31\u6765\u6211\u76f4\u64ad\u95f4\u62a2\u8d2d!.\u4f60\u6ca1\u770b\u9519\uff0c\u4eca\u5929\u5468\u4e94\uff0c\u5168\u573a\u5e73\u5747\u4e94\u6298\uff0cnow come and snap up in my live room!You did not misread it, today\u2019s Friday, average 50% off for all items, (10) Promoting self and others:\u4eca\u665a#\u8587\u5a05\u96f6\u98df\u8282# \u6765\u5566!\u6587\u548c\u53cb\u3001\u80af\u5fb7\u57fa\u3001\u5fc5\u80dc\u5ba2\u3001\u661f\u5df4\u514b..\u8d85\u591a\u7f8e\u98df\u8ba9\u7231\u7231\u7231\u4e0d\u5b8c!.Tonight #Viya\u2019s Snack Day# Comes! Wenheyou, KFC, Pizza Hut, Starbucks\u2026too many snacks to enjoy them all!As Among the three KOLs, Li shows the lowest preference for \u201cpromoting self and others,\u201d which suggests that he values soft sell over hard sell. This can be further seen from Li\u2019s highest preferences for \u201cself-reporting moment and information\u201d and \u201celiciting response.\u201d They suggest that Li prefers to interact with the followers by sharing his private life and engaging the public. Hereby he constructs himself as a congenial and caring ladybro of his followers. Examples are as follows:(11) Self-reporting moment and information:\u5230\u4e0a\u6d77\u4e86\uff0c\u5f00\u59cb\u5de5\u4f5c\u7528\u7740\u8d35\u5987\u9762\u819c\u56de\u56de\u8840\u6211\u56f0\uff0c\u6211\u611f\u89c9\u4eca\u665a\u76f4\u64ad\u4f1a\u80e1\u8a00\u4e71\u8bed .Arrive in Shanghai, start working, refresh myself with a lady\u2019s facemask, I\u2019m sleepy, feeling that I\u2019ll wander in my talk tonight.(12) Eliciting Response:\u4eca\u665a\u5237\u9178\u5c0f\u8bfe\u5802\u54e6 \u4f60\u4eec\u6709\u4ec0\u4e48\u95ee\u9898\u8981\u95ee\u5417#\u674e\u4f73\u7426\u76f4\u64ad##\u674e\u4f73\u7426\u5c0f\u8bfe\u5802# .Tonight peeling class Do you have any questions # Li Jiaqi\u2019s Live Room# #Li Jiaqi\u2019s small class#.By contrast, Luo shows his highest preference for \u201cshowing judgement and appreciation.\u201d That means that Luo is never afraid of expressing his own opinions towards some social phenomena or some things, thus constructing himself a key opinion leader or spiritual leader of youngsters and middle-aged men, as in the following:(13) Showing judgement and appreciation:\u5356\u4e86\u4e00\u5e74\u591a\u7684\u8d27\uff0c\u6709\u4e9b\u540c\u4e8b\u5df2\u7ecf\u6e10\u6e10\u4e0d\u8bf4\u4eba\u8bdd\u4e86\uff0c\u6bd4\u5982\u67d0\u4e2a\u9009\u54c1\u73b0\u8d27\u5c11\uff0c\u4e0d\u591f\u5356\uff0c\u8bf4\u6210\u201c\u5e93\u5b58\u6df1\u5ea6\u4e0d\u8db3\u201d \u4e0d\u8bf4\u4eba\u8bdd\u7684\u5438\u5f15\u529b\u7a76\u7adf\u5728\u54ea\u91cc\u5462?.Have been selling for over 1 year, some colleagues have gradually not been able to speak like a normal man, for example \u201csome product is in short supply, insufficient for sale,\u201d they\u2019ll say \u201cstock in deep short supply\u201d what\u2019s the appeal in saying unlike human?Old Luo, 6), \u201c\u7f57\u8001\u5e08\u201d , and \u201c\u8fd9\u4e2a\u661f\u7403\u4e0a\u6700\u4f18\u79c0\u7684\u8f6f\u4ef6\u4ea7\u54c1\u7ecf\u7406\u201d . Viya prefers to highlight her authority in her profession, such as \u201c\u7ea2\u8272\u63a8\u8350\u5b98\u201d , \u201c\u54c1\u9274\u56e2\u56e2\u957f\u201d , \u201c\u54c1\u724c\u661f\u9274\u5b98\u201d , \u201c\u5929\u732b\u5c0f\u9ed1\u76d2\u2018\u5f00\u65b0\u5b98\u2019\u201d , \u201c\u676d\u5dde\u5e02\u53cd\u8bc8\u9a97\u5ba3\u4f20\u5927\u4f7f\u201d , and \u201c\u5fae\u535a\u7535\u5546\u53f7\u9996\u5e2d\u597d\u7269\u4f18\u9009\u5b98\u201d . She prefers to highlight her title as an official to construct her social esteem in her profession. In other words, she aligns with the established social recognition. Therefore, both Luo and Viya construct themselves as having a higher social status than the followers. However, Luo prefers to use self-designated labels, while Viya prefers to highlight social recognition. By contrast, Li prefers to locate himself at a lower power position to his followers by addressing himself as \u201c\u4f73\u7426\u201d , \u201c\u5c0f\u674e\u201d , \u201c\u5c0f\u674e\u8001\u5e08\u201d , \u201c\u674e\u8001\u5934\u201d , \u201c\u674e\u65f6\u9ae6\u201d , \u201c\u53e4\u98ce\u5c0f\u674e\u201d , \u201c\u53e4\u88c5\u5c0f\u674e\u201d , \u201c\u674e\u59d3\u7ecf\u7eaa\u4eba\u201d , \u201c\u56fd\u98ce\u5c0f\u674e\u201d . Through using self-depreciating terms, nicknames, and first names, Li seeks to construct solidarity with his followers.In their choice of nominal addressing terms, Luo prefers to highlight his expertise and achievements in his field. In other words, he prefers to brag about his prestige and expertise, as can be seen from such expressions as \u201c\u8001\u7f57\u201d , \u201c\u795e\u7ecf\u75c5\u201d , \u201c\u771f\u8001\u8d56\u201d , \u201c\u6d41\u6c13\u201d , \u201c\u571f\u8c6a\u5927\u54e5\u6d0b\u8d22\u4e3b\u5927\u59d0\u201d , \u201c\u5404\u8def\u5584\u957f\u4ec1\u7fc1\u201d . In other words, he prefers to attack or satirize some people, thus constructing an aggressive or cynical image for himself. However, Viya prefers to use some nicknames or internet buzzwords to address others, such as \u201c\u5c0f\u4f19\u4f34\u4eec\u201d , \u201c\u8587\u5a05\u7684\u5973\u4eba\u9a91\u58eb\u4eec\u201d , \u201c\u76c6\u53cb\u4eec\u201d , \u201c\u65f6\u5c1a\u8fbe\u4eba\u4eec\u201d , \u201c\u4f53\u9a8c\u5b98\u4eec\u201d . These terms help to either construct intimacy between herself and her followers or show respect to her followers.Their different preferences can also be seen from the nominal terms they used to address others. Luo uses not only some ironic terms but also some abusive terms to address others, including \u201cXX\u201d (7), \u201c\u8d31\u4eba\u4eec\u201d , \u201c(\u6240\u6709)\u5973\u751f\u7537\u751f\u201d , \u201cMM\u4eec\u201d, \u201c\u7f8e\u7709\u4eec\u201d , \u201c\u5b9d\u5988\u5b9d\u7238\u4eec\u201d . This further demonstrates that Li prefers to use gender as a specific strategy in his promotion.Although Li also uses some internet buzzwords to address his followers, he also uses some intimate addressing terms to address his followers by their genders, such as \u201c(\u6240\u6709)\u5973\u751f\u201d  image. He prefers to highlight the masculine traits of \u201cforceful\u201d and \u201crational,\u201d the speech acts of \u201cdirective\u201d and \u201cshowing judgement and appreciation,\u201d and the use of positive self-addressing terms and negative other-addressing terms. Li tends to construct a \u201cguimi\u201d (ladybro) image. He prefers to underline the feminine personality traits of \u201ccaring\u201d and \u201ccongenial,\u201d the speech acts of \u201cself-reporting moment and information\u201d and \u201celiciting response,\u201d and the self-denigration self-addressing terms and respectful other-addressing terms. Viya tends to construct a \u201cprofessional\u201d image. She prefers to underscore both the masculine traits of \u201cforceful\u201d and \u201cconfident\u201d and the feminine trait of \u201cunderstanding,\u201d the speech acts of \u201cpromoting self and others\u201d and \u201cself-reporting mood,\u201d and the respectful self- and other-addressing terms.An analysis of the posts of the three top KOLs at three different levels finds that they tend to construct different images. Luo tends to construct a \u201czhinan\u201d who is rational, critical, and brave enough to show his judgment towards some social phenomena. In other words, he attracts his followers by distinguishing himself from his followers. By contrast, since the products Li promotes are primarily female products and most of his followers are female. He has to cater to the interest of his followers by constructing himself as a congenial and caring \u201cguimi\u201d (ladybro) of his followers. In order to establish solidarity with female followers, he has to resort to \u201cself-reporting moment and information\u201d and \u201celiciting response\u201d to attract his followers. In between is Viya. Although she is a female, she promotes products to both males and females. She attracts her followers by her expertise in selecting high-quality products with the lowest prices. Therefore, gender does not serve as a prominent promoting strategy because she sells products to both males and females.Their different ways of identity constructions can be attributed to their target consumers and the products they promote. Since the majority of Luo\u2019s followers are males, they support him because of his sharp words and entrepreneurial spirits. Therefore, Luo prefers to construct himself as a \u201cTherefore, different KOLs tend to construct different images on their social media. Although previous studies show that feminine traits are gaining advantage over time  as it caSocial media provides a platform for people to construct distinguishing personalities . AlthougThis study reveals not only the different strategies by different top KOLs to interact with their followers on social media but, more importantly, the dynamic relations between gender and interaction. It demonstrates that gender serves as not only a social constraint but also an important social resource which can be appropriated and explored for branding on social media . TherefoThe raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.ML contributed to the research design, data description and interpretation, paper writing and revising, and funding support. RZ analyzed the data, and contributed to the literature review and paper drafting. JF was responsible for the quality control and proofreading. All authors contributed to the article and approved the submitted version.This work was funded by the Hong Kong Polytechnic University .The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher."}
{"text": "Current methods of the conversion between a rotation quaternion and Euler angles are either a complicated set of multiple sequence-specific implementations, or a complicated method relying on multiple matrix multiplications. In this paper a general formula is presented for extracting the Euler angles in any desired sequence from a unit quaternion. This is a direct method, in that no intermediate conversion step is required  and it is general because it works with all 12 possible sequences of rotations. A closed formula was first developed for extracting angles in any of the 12 possible sequences, both \u201cProper Euler angles\u201d and \u201cTait-Bryan angles\u201d. The resulting algorithm was compared with a popular implementation of the matrix-to-Euler angle algorithm, which involves a quaternion-to-matrix conversion in the first computational step. Lastly, a single-page pseudo-code implementation of this algorithm is presented, illustrating its conciseness and straightforward implementation. With an execution speed 30 times faster than the classical method, our algorithm can be of great interest in every aspect. R \u2208 SO(3), where SO(3) is the group of invertible 3 \u00d7 3 matrices such that det(R) = 1 and When dealing with 3D orientation problems, many different formalisms can be used to describe a given rotation , each ofR = R2R1 is the rotation matrix corresponding to a rotation by R1 followed by a rotation by R2. 3D rotation matrices have some numerical shortcomings, however. For example, as many as 9 numbers (and 6 constraints) are required to represent a 3 degree of freedom rotation, and it can be difficult and computationally costly to orthogonalize a rotation matrix numerically  or  = [323] is equivalent to the sequence ZYZ). A Python implementation can be found on  \u2212 q[j]\u2003b \u2190 q[i] + q[k] \u00d7 \u03b5\u2003c \u2190 q[j] + q[0]\u2003d \u2190 q[k] \u00d7 \u03b5 \u2212 q[i]\u2003elsea \u2190 q[0]\u2003b \u2190 q[i]\u2003c \u2190 q[j]\u2003d \u2190 q[k] \u00d7 \u03b5\u2003end\u03b8+ \u2190 atan2\u03b8\u2212 \u2190 atan2switchvalue of \u03b82docase 0 do\u2003\u03b81 \u2190 0 // For simplicity, we are setting \u2003\u2003\u03b83 \u2190 2 \u00d7 \u03b8+ \u2212 \u03b81\u2003\u2003case\u03c0/2 do\u2003\u03b81 \u2190 0\u2003\u2003\u03b83 \u2190 2 \u00d7 \u03b8\u2212 + \u03b81\u2003\u2003otherwise do\u2003\u03b81 \u2190 \u03b8+ \u2212 \u03b8\u2212\u2003\u2003\u03b83 \u2190 \u03b8+ + \u03b8\u2212\u2003\u2003end\u2003endifnot_properthen\u03b83 \u2190 \u03b5 \u00d7 \u03b83\u2003\u03b82 \u2190 \u03b82 \u2212 \u03c0/2\u2003end\u03b81, \u03b83 \u2190  // \u201cwrap\u201d assures \u03b81, \u03b83 \u2208 2 = 16 floating point multiplications , 4 \u00d7 3 = 12 multiplications by 2 and 3 \u00d7 3 + 6 = 15 additions/subtractions. This conversion step alone is more than enough to make an algorithm based on [Many operations are required to convert a quaternion into a rotation matrix. Using the homogeneous formula from \u221212. The execution times required in our tests for each sequence (and their ratios) are presented in the In this section, a performance comparison between our method and the Shuster method is presented. We adapted the SciPy library in order to compile the algorithm as described in Section 4. A real data set comprising the orientation of a spinning object with 3284 data points was used to compare the efficiency of the two algorithms. The full implementation and data set can be downloaded from [The Euler angles are still a useful intuitive 3D orientation parametrization. A fast method of conversion to/from any other set of parameters can therefore be of great interest for displaying or analyzing data, for instance. In this study, we therefore developed a general formula for this conversion which is concise, easy to implement and easy to debug. In addition, the fact that our method is about 30 times faster than the method proposed by , which r"}
{"text": "Following publication of the original article , we haveFigure 1: \u201c111 patients received\u2009\u2265\u20091 risdiplam dose:\u201d"}
{"text": "The article was published without the keywords \u201cDrug prescription, aged 80 and over, primary care\u201d.In \u201cAbstract\u201d section, \u201cConclusions\u201d is corrected as \u201cConclusion\u201d.The heading sizes for \u201cAim\u201d and \u201cEthics approval\u201d are corrected.The sentence \u201cIn summary, this study examined medication\u2026\u201d till \u201c\u2026 prescription might be helpful in the very old.\u201d should be placed under the \u201cConclusion\u201d section.The section \u201cConflict of interest\u201d is corrected as \u201cConflicts of interest\u201d.The original article has been corrected."}
{"text": "Key Words to Understand China: The Fight Against COVID-19 for daily precautions, policy publicity, and information dissemination. Based on the translations and relevant data from Chinese and English media, this study makes a tentative analysis of the linguistic, ideological, and cultural features of English used in the translations. The findings are that China English is outstanding with respect to translating some Chinese-specific words and expressions in the translations, which generally results from the method of literal rendering. It is suggested that literal translation and transliteration can be used more frequently in English texts related to big events in order to strengthen the Chinese characteristics of China English as a major variety of world Englishes.During the COVID-19 outbreak, the China Academy of Translation translated and officially released five groups of  Although the origin of COVID-19 has not yet been confirmed, the virus caused by SARS-CoV-2 instead of viral escape from a laboratory \u201cmost likely arose in bats, and then spread to humans via an as-yet unidentified intermediary animal,\u201d according to the World Health Organization (WHO).ril 2020 . In respanguages . The latanguages , 2020c. Some scholars have so far studied the characteristics of English as a first, second or foreign language used to report seminal global events, such as wars, PHEICs, terrorist attacks, and sports games. For example, It is of great importance to study English used in reports on big events, such as COVID-19, especially in countries and regions where English is used as a second or foreign language because such English has a great influence on English varieties, such as China English. As a major variety of world Englishes, China English, formerly labeled \u201cChinglish,\u201d refers to \u201cEnglish with Chinese characteristics\u201d , p. 1, aA Tale of Two Cities: \u201cIt was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity.\u2004.\u2004..\u201d  at its WeChat public platform by the middle of April 2020, and the first three groups in more than a dozen foreign languages were officially approved and released by the CFLPA via various media and channels, such as the website of the Translators Association of China  Xinhuanet-English version; (4) The New York Times, The Guardian, US Today, The Australian, and New Scientist. There are 52,417 words/characters in the corpus of this study, the inclusion of the linguistic innovations into the sample is purely thematic, and the new verbal signs which occur once in the corpus are included into the sample. A few abbreviations are used in the case study for the convenience of discussion, including PY, WT, LT, and CT which, respectively, refer to Chinese Pinyin version, word-for-word translation, literal translation, and the CAT\u2019s translation of Chinese examples. In the following sections, we analyze the linguistic, ideological, and cultural features of English used in the CAT\u2019s translations from bilingual, translation, and bicultural perspectives.The KWUC-FAC is a Chinese-English bilingual version which consists of six sections, including \u201cDecisions by the Central Leadership,\u201d \u201cAnti-Epidemic Guidelines and Arrangements,\u201d \u201cEffective Measures,\u201d \u201cAbout COVID-19,\u201d \u201cInternational Aid,\u201d and \u201cBrave Fighters\u201d . In each section, the key words, phrases or sentences in the Chinese government documents are used as titles and followed by a detailed account of them. There are a total of 615 key words in the five groups in which 181 words in the first three groups had been authorized by the CFLPA. A comparative approach is taken in this study to compare Chinese originals with their English translations, and these translations are also compared with similar English expressions in Chinese and Western media in order to summarize the features of epidemic-related China English. Data for this project is taken from the websites or WeChat public platforms of the following sources: (1) China Academy of Translation; (2) 1\u2003\u2003\u2003\u2003\u65b0\u589e\u2003\u2003\u786e\u8bca\u2003\u2003\u75c5\u4f8bx\u012bnz\u0113ng\u2003\u2003qu\u00e8zh\u011bn\u2003\u2003b\u00ecngl\u00ecPY:\u2003\u2003CT:\u2003\u2003newly confirmed cases2\u2003\u2003\u5bf9\u2003\u2003\u6297\u75ab\u2003\u2003\u533b\u52a1\u2003\u2003\u4eba\u5458\u2003\u2003\u4fdd\u62a4\u3001\u2003\u2003\u5173\u5fc3\u3001\u2003\u2003\u7231\u62a4du\u2003\u2003k\u00e0ngy\u00ec\u2003\u2003y\u012bw\u00f9\u2003\u2003r\u00e9nyu\u00e1n\u2003\u2003b\u0103oh\u00f9\u2003\u2003\u2003\u2003gu\u0101nx\u012bn\u2003\u2003\u00e0ih\u00f9PY:\u2003\u2003CT:\u2003\u2003to provide full protection and care to medical workers fighting against the epidemic3\u2003\u2003\u6253\u8d62\u2003\u2003\u75ab\u60c5\u2003\u2003\u9632\u63a7\u2003\u2003\u2003\u72d9\u51fb\u6218d\u0103y\u00edng\u2003\u2003y\u00ecq\u00edng\u2003\u2003f\u00e1ngk\u00f2ng\u2003\u2003\u2003z\u016dj\u012bzh\u00e0nPY:\u2003\u2003CT:\u2003\u2003to fight and win the battle against the epidemicTranslation can be defined as \u201cthe replacement of textual material in one language (SL) by equivalent textual material in another language (TL)\u201d , p. 20. d\u0103, \u201cfight\u201d) in the set phrase \u201c\u6253\u8d62\u201d  seems unnecessary and thus can be omitted because \u201cwin the battle\u201d presupposes \u201cfight the battle.\u201d In Chinese, omission of some sentence part gives rise to obscurity in meaning or semantic implicitness which is an important feature of Chinese 5\u2003\u2003\u5916\u9632\u2003\u2003\u8f93\u5165\u3001\u2003\u2003\u5185\u9632\u2003\u2003\u2003\u6269\u6563w\u00e0if\u00e1ng\u2003\u2003sh\u016br\u00f9,\u2003\u2003n\u00e8if\u00e1ng\u2003\u2003ku\u00f2s\u00e0n\u2003\u2003WT\u2003\u2003outside prevent entering, inside prevent spreadingPY:\u2003\u2003CT:\u2003\u2003preventing the coronavirus from entering and spreading within a region6\u2003\u2003\u5e94\u6536\u2003\u2003\u5c3d\u6536\u3001\u2003\u2003\u5e94\u6cbb\u2003\u2003\u5c3d\u6cbby\u012bngsh\u014du\u2003\u2003j\u00ecnsh\u014du,\u2003\u2003y\u012bngzh\u00ec\u2003\u2003j\u00ecnzh\u00ec\u2003\u2003WT:\u2003\u2003should admit all admitted, should treat all treatedPY:\u2003\u2003CT:\u2003\u2003admitting all suspected and confirmed cases for treatmentIn examples 1 and 2, the expressions \u201cnewly confirmed cases\u201d and \u201cfighting against the epidemic\u201d are acceptable, but they are not as concise as \u201cnew infections\u201d and \u201cfight the coronavirus\u201d used by Western media . In exam Chinese , p. 101,w\u016dh\u00e0nsh\u00ec) and the object is \u201cfour groups of people\u201d  which include confirmed cases, suspected cases, febrile patients who cannot rule out the possibility of infection, and close contacts. Likewise, in examples 5 and 6, it is not clear what should be prevented from entering from outside and spreading inside, and who should be admitted and treated. The CAT\u2019s translations illustrate that they are \u201ccoronavirus\u201d and \u201csuspected and confirmed cases.\u201d Specifically, the context indicates that as for \u201c\u5916\u9632\u8f93\u5165\u3001\u5185\u9632\u6269\u6563,\u201d the subject is \u201cthe community\u201d  and the object is \u201cimported cases\u201d  and \u201cthe outbreak\u201d ; as for \u201c\u5e94\u6536\u5c3d\u6536\u3001\u5e94\u6cbb\u5c3d\u6cbb,\u201d the subject is \u201cthe hospitals\u201d  and the object is \u201cconfirmed and suspected cases\u201d . Therefore, the implied context-bound meaning should be clarified by adding proper words in translating in order to help the international community to understand China\u2019s official guidelines on the epidemic control completely. Parataxis is another feature of Chinese in which phrases and clauses are often juxtaposed without the use of connectors, while English is a hypotactic language in which phrases and clauses are joined with conjunctions or prepositions In example 4, the Chinese original does not tell us who should carry out the screening work and what should be screened, and according to the context, the subject is \u201cWuhan City\u201d  is used four times in the original. This lexical repetition is a striking feature of Chinese which can be viewed as a special form of hypotaxis . But the word \u201c\u96c6\u4e2d\u201d is omitted in the translation, and its meaning is implied by \u201cdedicated\u201d and \u201call.\u201d Let us look at one more example:8\u2003\u2003\u65e9\u2003\u2003\u53d1\u73b0\u3001\u2003\u2003\u65e9\u2003\u2003\u62a5\u544a\u3001\u2003\u2003\u65e9\u2003\u2003\u9694\u79bb\u3001\u2003\u2003\u65e9\u2003\u2003\u6cbb\u7597z\u0103o\u2003\u2003f\u0101xi\u00e0n,\u2003\u2003z\u0103o\u2003\u2003b\u00e0og\u00e0o,\u2003\u2003z\u0103o g\u00e9l\u00ed,\u2003\u2003\u2003\u2003z\u0103o\u2003\u2003zh\u00ecli\u00e1oPY:\u2003\u2003WT:\u2003\u2003early discovery, early report, early isolation, early treatmentCT:\u2003\u2003early detection, reporting, isolation and treatmentIn example 7, the Chinese original is a typical paratactic sentence in which four clauses are put together without using a connector in between. However, in the CAT\u2019s translation, the Chinese paratactic sentence is transformed into a hypotactic one in the English version via the use of such connectors as \u201cin,\u201d \u201cby,\u201d \u201cfrom,\u201d \u201cand,\u201d and \u201cwith.\u201d Moreover, the word \u201c\u96c6\u4e2d\u201d (z\u0103o) is used four times in the original, while its literal rendering \u201cearly\u201d is used only once in the English version because repetition should, in many cases, be avoided in English via ellipsis or substitution 10\u2003\u2003\u5bf9\u53e3\u2003\u2003\u652f\u63f4\u2003\u2003du\u00eck\u014fu\u2003\u2003zh\u012byu\u00e1nPY: CT:\u2003\u2003pairing assistance In example 8, the word \u201c\u65e9\u201d , \u201c\u594b\u6218\u201d , \u201c\u4f5c\u6218\u201d , \u201c\u6218\u6597\u201d , \u201c\u5907\u6218\u201d , \u201c\u6218\u80dc\u201d , \u201c\u6253\u8d62\u201d , \u201c\u963b\u51fb\u6218\u201d , \u201c\u6218\u6597\u529b\u201d , \u201c\u653b\u575a\u6218\u201d , \u201c\u603b\u4f53\u6218\u201d , \u201c\u4eba\u6c11\u6218\u4e89\u201d , \u201c\u6218\u7ebf\u201d , \u201c\u524d\u7ebf\u201d , \u201c\u9632\u7ebf\u201d , \u201c\u519b\u4ee4\u72b6\u201d , \u201c\u5821\u5792\u201d , and \u201c\u727a\u7272\u201d . The use of war metaphors in the Chinese discourse of battling the coronavirus reflects the militarism of the Chinese people 12\u2003\u2003\u8fd9\u2003\u2003\u662f\u2003\u2003\u52a8\u5458\u4ee4,\u2003\u2003\u4e5f\u662f\u2003\u2003\u5ba3\u8a00\u4e66,\u2003\u2003\u66f4\u662f\u2003zh\u00e8\u2003\u2003sh\u00ec\u2003\u2003d\u00f2ngyu\u00e1nling,\u2003\u2003y\u011bsh\u00ec\u2003\u2003xu\u0101ny\u00e1nsh\u016b,\u2003\u2003g\u00e8ngsh\u00ec\u2003PY: \u2003\u2003\u519b\u4ee4\u72b6\u3002j\u016bnl\u00ecngzhu\u00e0ng.WT: \u2003\u2003This is mobilization order, also is manifesto, moreover is military order.CT: \u2003\u2003This is a mobilization and an order.In containing the coronavirus spread, great achievements had been achieved in China under the leadership of the CPC and the Central Government whose efforts were acknowledged by the international community. Just as the WHO pointed out, \u201cChina has rolled out perhaps the most ambitious, agile and aggressive disease containment effort in history\u201d . The scie people , p. 127,r\u00e9nm\u00edn zh\u00e0nzh\u0113ng, \u201cpeople\u2019s war\u201d), \u201c\u603b\u4f53\u6218\u201d , and \u201c\u963b\u51fb\u6218\u201d , but none of them is retained in the translation. Instead, the general expression \u201cthe nation\u2019s war\u201d is used to cover and represent their meanings. Similarly, the powerful expressions of \u201c\u52a8\u5458\u4ee4\u201d , \u201c\u5ba3\u8a00\u4e66\u201d , and \u201c\u519b\u4ee4\u72b6\u201d  are not faithfully represented in the English version in example 12 because \u201c\u4ee4\u201d  in \u201c\u52a8\u5458\u4ee4,\u201d \u201c\u5ba3\u8a00\u4e66,\u201d and \u201c\u519b\u4ee4\u201d  in \u201c\u519b\u4ee4\u72b6\u201d are not translated, resulting in the weakening of the expected effectiveness of publicity. Other types of metaphor are also seen in Xi\u2019s talks. For example:13\u2003\u2003\u751f\u547d\u2003\u2003\u2003\u2003\u91cd\u4e8e\u2003\u2003\u6cf0\u5c71\u3002\u2003\u2003sh\u0113ngm\u00ecng\u2003\u2003zh\u00f2ngy\u00fa\u2003\u2003t\u00e0ish\u0101n.PY: LT: \u2003\u2003Life is weightier than Mount Tai.CT: \u2003\u2003Saving lives is of paramount importance.14\u2003\u2003\u75ab\u60c5\u2003\u2003\u5c31\u662f\u2003\u2003\u547d\u4ee4\uff0c\u2003\u2003\u9632\u63a7\u2003\u2003\u5c31\u662f\u2003\u2003\u8d23\u4efb\u3002y\u00ecq\u00edng\u2003\u2003ji\u00f9sh\u00ec\u2003\u2003m\u00ecngling,\u2003\u2003f\u00e1ngk\u00f2ng\u2003\u2003ji\u00f9sh\u00ec\u2003\u2003z\u00e9r\u00e8n.PY:\u2003\u2003 LT: \u2003\u2003COVID-19 is the order; its prevention and control is the responsibility.CT: \u2003\u2003Go where there is epidemic, fight it till it perishes.Three war-related expressions are used in the original in example 11, including \u201c\u4eba\u6c11\u6218\u4e89\u201d In the Chinese sentences in examples 13 and 14, people\u2019s life is compared to Mount Tai, and the epidemic to a military order, which emphasized on the significance of safeguarding the people\u2019s lives and controlling the virus spread. The belief contained in example 13 that the life of all people is of equal importance constitutes a sharp contrast to some Western leaders\u2019 philosophy of herd immunity and natural elimination which put elderly people at great risk . In the sh\u00e8ng, \u201cwin\u201d), President Xi clearly pointed out the key to battling the virus by saying that China would eventually win the war of battling the epidemic if Wuhan and Hubei could successfully control the virus, and he called on the whole nation to provide assistance to them. This example shows that there are also good repetitions of words in English, just like the opening sentence in Charles Dickens\u2019 A Tale of Two Cities. It should be noted that unique political language was often seen in the CPC\u2019s guidelines, and some expressions excerpted from Xi\u2019s talks had become slogans for all Party members and cadres in combating the coronavirus. For example:16\u2003\u2003\u8ba9\u2003\u2003\u515a\u65d7\u2003\u2003\u5728\u2003\u2003\u9632\u63a7\u2003\u2003\u75ab\u60c5\u2003\u2003\u6597\u4e89\u2003\u2003\u2003\u2003\u7b2c\u4e00\u7ebfr\u00e0ng\u2003\u2003d\u0103ngq\u00ed\u2003\u2003z\u00e0i\u2003\u2003f\u00e1ngk\u00f2ng\u2003\u2003y\u00ecq\u00edng\u2003\u2003d\u00f2uzh\u0113ng\u2003\u2003d\u00ecy\u012bxi\u00e0nPY: \u2003\u2003\u9ad8\u9ad8\u2003\u2003\u98d8\u626c\u3002g\u0101og\u0101o pi\u0101oy\u00e1ng.CT: \u2003\u2003Let the Party flag fly high on the front line of the anti-epidemic war.17\u2003\u2003\u2003\u4e00\u4e2a\u2003\u2003\u515a\u5458\u2003\u2003\u2003\u5c31\u662f\u2003\u2003\u4e00\u9762\u2003\u2003\u65d7\u5e1c,\u2003\u2003\u4e00\u4e2a\u2003\u2003\u652f\u90e8\u2003\u2003\u5c31\u662f\u2003\u2003y\u012bg\u00e8\u2003\u2003d\u0103ngyu\u00e1n\u2003\u2003\u2003ji\u00f9sh\u00ec\u2003\u2003y\u012bmi\u00e0n\u2003\u2003q\u00edzh\u00ec,\u2003\u2003y\u012bg\u00e8\u2003\u2003zh\u012bb\u00f9\u2003\u2003ji\u00f9sh\u00ecPY: \u2003\u2003\u4e00\u5ea7\u2003\u2003\u5821\u5792\u3002y\u012bzu\u00f2\u2003\u2003b\u0103ol\u011bi.CAT:\u2003\u2003A Party member is like a flag and a Party branch is like a fortress.COVID-19 first broke out in Wuhan, Hubei\u2019s capital city in January 2020, and Hubei was the hardest-hit province in China. Therefore, the central task of the whole country focused on the epidemic control in Wuhan and Hubei. The English version in example 15 is a literal rendering, and by repeating the word \u201c\u80dc\u201d (d\u0103ngyu\u00e1n g\u00e0nb\u00f9 xi\u00e0ch\u00e9n) during the outbreak, which means the deep community engagement of all Party members and cadres in the epidemic prevention and control . Finally, the shift in emotional coloring is sometimes seen in translating commendatory terms. For example, as a commendatory word, \u201c\u6b89\u804c\u201d  is rendered as the neutral word \u201cdie\u201d; \u201c\u5d07\u9ad8\u201d  in \u201c\u5d07\u9ad8\u7684\u656c\u610f\u201d  is also a commendatory word but omitted, resulting in the weakening of the original commendatory coloring .The political slogans in examples 16 and 17 have special connotations in the context of the CPC leadership. Confronted with the epidemic crisis, President Xi asked all Party members and Party organizations at all levels to take the lead to bring into full play their abilities by going deep into grass-roots communities and villages to fight the virus. This was clearly illustrated in the catchphrase \u201c\u515a\u5458\u5e72\u90e8\u4e0b\u6c89\u201d 19\u2003\u2003\u96f7\u795e\u5c71\u2003\u2003\u2003\u2003\u533b\u9662l\u00e9ish\u00e9nsh\u0101n\u2003\u2003y\u012byu\u00e0nPY: \u2003\u2003CT: \u2003\u2003Leishenshan Hospital20\u2003\u2003\u4e2d\u897f\u533b\u2003\u2003\u2003\u2003\u5e76\u7528zh\u014dngx\u012by\u012b\u2003\u2003b\u00ecngy\u00f2ngPY:\u2003\u2003 CT: \u2003\u2003combined use of TCM and Western medicineDue to the lockdown and self-quarantine, COVID-19 had a great impact on China\u2019s politics, economy, culture, healthcare, education, travel, and daily life. This impact was reflected in the CAT\u2019s translations. For example:\u65b9\u8231\u533b\u9662,\u201d f\u0101ngc\u0101ng y\u012byu\u00e0n) were built to deal with suspected or mild cases , and it can be argued that the Chinese original \u201c\u65b9\u8231\u201d  in the name of hospitals of this category might allude to Noah\u2019s ark which is translated into Chinese as \u201c\u8bfa\u4e9a\u65b9\u821f\u201d (nu\u00f2y\u00e0 f\u0101ngzh\u014du). As the saying goes, God help those who help themselves. The self-reliance of the Chinese nation played a decisive role in controlling COVID-19. As seen in example 20, in the treatment of infected people, traditional Chinese medicine (TCM) and Western medicine were combined because they have their own advantages, and their combined use \u201chad achieved good effects during China\u2019s fight against SARS in 2003\u201d . During the period of lockdown, hundreds of millions of people had been placed in some form of isolation in order to contain the virus spread 22\u2003\u2003\u2003\u2003\u89e3\u51b3\u2003\u2003\u597d\u2003\u2003\u751f\u6d3b\u2003\u2003\u2003\u5fc5\u9700\u54c1\u4f9b\u5e94\u2003\u2003\u2003\u2003\u7684\u2003\u2003PY: \u2003\u2003ji\u011bju\u00e9\u2003\u2003h\u0103o\u2003\u2003sh\u0113nghu\u00f3\u2003\u2003\u2003b\u00ecx\u016bp\u012dn\u2003\u2003g\u00f2ngy\u012bng\u2003\u2003de\u201c\u6700\u540e\u2003\u2003\u4e00\u2003\u2003\u516c\u91cc\u201d\u2003\u2003\u95ee\u9898\u201czu\u00ech\u00f2u\u2003\u2003y\u012b\u2003\u2003g\u014dngl\u012d\u201d\u2003\u2003 w\u00e8nt\u00edCT: \u2003\u2003to ensure the \u201clast kilometer\u201d delivery of daily necessities23\u2003\u2003\u2003\u2003\u6b66\u6c49\u2003\u2003\u5feb\u9012\u2003\u2003\u5c0f\u54e5\u2003\u2003\u6c6a\u52c7\uff1a\u2003\u2003\u7ec4\u7ec7\u2003\u2003\u5fd7\u613f\u8005PY: \u2003\u2003w\u01d4h\u00e0n\u2003\u2003ku\u00e0id\u00ec\u2003\u2003xi\u0103og\u0113\u2003\u2003w\u0101ng\u2003\u2003y\u014fng:\u2003\u2003z\u016dzh\u012b\u2003\u2003zh\u00ecyu\u00e0nzh\u011b\u63a5\u9001\u2003\u2003\u533b\u62a4\u2003\u2003\u4eba\u5458ji\u0113s\u00f2ng y\u012bh\u00f9 r\u00e9nyu\u00e1nCT: \u2003\u2003Wang Yong: a courier volunteerWuhan was the epicenter of the epidemic in China and the first city in lockdown, with more than 50,000 infections. As the number of infected people skyrocketed, the hospitals available in Wuhan were unable to tackle confirmed cases. As a result, two big hospitals were built within 20 days, a \u201cChinese speed\u201d in times of urgency, and they were named \u201cHuoshenshan Hospital\u201d and \u201cLeishenshan Hospital.\u201d As illustrated in examples 18 and 19, \u201cHuoshenshan\u201d and \u201cLeishenshan\u201d are the Chinese Pinyin versions or transliterations of \u201c\u706b\u795e\u5c71\u201d and \u201c\u96f7\u795e\u5c71\u201d whose literal meanings are \u201cFire God Mountain\u201d and \u201cThunder God Mountain.\u201d It was believed that the Fire God and the Thunder God could help the Chinese to stop the epidemic because they take control of fire and lightning which can kill the virus . Moreoves spread . The supku\u00e0id\u00ec xi\u01ceog\u0113) who were praised by Western media because they took the risk of losing their lives to do so On behalf of the CPC Central Committee, Vice Premier Sun Chunlan, together with the Central Guiding Team, arrived in Wuhan on January 27, 2020, and found that daily necessities could not be sent to residents fast and smoothly . Therefore, it was proposed that the governors and city mayors should take full responsibility for food supplies, such as Chinese cabbage, pakchoi seedlings, and musang king durians , and ensure the \u201clast kilometer\u201d smooth delivery of food and other daily necessities. With the help of tens of thousands of community cadres and courier volunteers, such as Wang Yong, ample and timely daily supplies were guaranteed, as indicated in examples 21, 22, and 23. As for quarantined city dwellers who were unable to do the cooking by themselves, the meals were brought to them by thousands of \u201cdeliverymen\u201d  The outbreak forced China\u2019s Ministry of Education (MOE) to postpone the commencement of the spring semester, and launch the \u201cHome Study Initiative\u201d  . Based oChina Daily, 27 February 2020). As a form of gratitude, there appeared many English versions of the verse line by the Chinese, such as \u201cMiles apart, but close at heart\u201d in example 25 and \u201cRivers low, mountains high; the same moon in the sky\u201d by Professor Zhao Yanchun, the translator of the Chinese classic Three Character Classic.The Tokyo-based Japan Youth Development Association donated 20,000 masks to Hubei, on the boxes of which was written the Chinese verse line \u201c\u5c71\u5ddd\u5f02\u57df, \u98ce\u6708\u540c\u5929\u201d , and this powerful poetic message had a very positive response on Chinese social media  as \u201cChina will do everything possible to combat it\u201d  is not as good as the literal rendering of \u201cBetter be in vain in nine out of ten prevention efforts than lose the defense line in one out of ten thousand efforts.\u201d Although a number of translation methods are employed in the CAT\u2019s translations of the 25 examples above with a varying use frequency, such as annotation (2 times), free translation (2 times), addition (3 times), omission (8 times), and literal translation/transliteration (10 times), it can be argued that the methods of literal translation and transliteration can best keep Chinese characteristics in rendering Chinese-specific words and expressions, such as \u201cfight and win\u201d for \u201c\u6253\u8d62,\u201d \u201cactively win\u201d for \u201c\u79ef\u6781\u4e89\u53d6,\u201d and \u201cHuoshenshan Hospital\u201d for \u201c\u706b\u795e\u5c71\u533b\u9662.\u201d It is not advisable to achieve clarity in meaning and brevity in expression at the cost of losing Chinese uniqueness in foreign publicity.Lexical repetition can be retained in translating in most cases in order to achieve semantic emphasis and represent Chinese characteristics. For instance, the CAT\u2019s translation of \u201cearly detection, reporting, isolation and treatment\u201d for \u201c\u65e9\u53d1\u73b0, \u65e9\u62a5\u544a, \u65e9\u9694\u79bb, \u65e9\u6cbb\u7597\u201d can be modified as \u201cearly detection, early reporting, early isolation, early treatment,\u201d which can better show China\u2019s fast responses to the coronavirus. War metaphors as an important feature in the epidemic-related Chinese discourse can also be maintained, as illustrated in \u201cpeople\u2019s war, total warfare, tough fighting\u201d for \u201c\u4eba\u6c11\u6218\u4e89, \u603b\u4f53\u6218, \u963b\u51fb\u6218\u201d which is better than the CAT\u2019s generalization of \u201cthe nation\u2019s war.\u201d Similarly, the CAT\u2019s generalization of \u201c\u5b81\u53ef\u5341\u9632\u4e5d\u7a7a, \u4e0d\u53ef\u5931\u9632\u4e07\u4e00\u201d , a world peace and security-oriented concept proposed by China . The pandemic has proved that all of us live in one and the same world, and only when the whole world should be united as one, could the virus be wiped out. However, the ideological confrontation seems to continue to exist between China and the West. The fact that COVID-19 first broke out in China made some Westerners call it \u201cChinese virus\u201d or \u201cChina virus,\u201d as used in Western media\u2019s headlines, such as \u201cSchools isolate students in danger of China virus\u201d . The WHO named the new virus \u201cCOVID-19\u201d for the very purpose of preventing the use of such prejudiced terms which are related to \u201ca geographical location, an animal, an individual or a group of people\u201d , decided to go on a voyage to Japan; after six attempts and the loss of his eyesight, Jianzhen finally arrived, and made a significant contribution to the spread of Buddhism in Japan\u201d . The story behind the verse line invoked both shared history and mutual appreciation for ancient poetry in Chinese. Many Chinese people were touched by the poetic expression of support from the Japanese during the COVID-19 outbreak.During the extended lockdown, millions of families were forced to live in confined spaces in Wuhan and other cities in Hubei. Many problems, such as boredom and physical abuse, took place in these families . MoreoveEnglish was used widely and frequently in Chinese media during the COVID-19 outbreak for daily precautions, policy publicity, and information dissemination, and the CAT\u2019s translations provided the media with importance reference for the use of English. China English was outstanding with respect to translating some Chinese-specific words and expressions in the translations, and it generally resulted from the method of literal rendering. With this method, some of the CAT\u2019s translations can be improved with more Chinese characteristics if we take linguistic acceptability into consideration at the same time. In some sense, translation, especially literal translation, is an important way to develop and enrich China English. As for ideological publicity, by virtue of lexical repetitions and use of war metaphors, the greatest importance was attached to people\u2019s lives in President Xi\u2019s talks which highlighted high efficiency and strategic rationality in combating the virus. Although there were some problems during the period of lockdown, such as boredom and physical abuse, poetic message and musical power brought comfort and warmth to quarantined people in cold China in the late winter and early spring of 2020."}
{"text": "The original version of this article unfortunately contained a mistake. The wrong Table 2 was published and in Table 5, document measures in the column \u201cRange\u201d were mistakenly listed as dates.The corrected Tables 3D\u22122D\u201dIn the section \u201cSingle linear regression analysis of the angle \u03c1 and the \u0394Both equations should have a \u201cminus\u201d sign in the beginning :p\u2009<\u20090.0001, R2\u2009=\u20090.0446, Fig.\u00a05c). On the left, angle \u03c1 showed a linear regression relationship with the difference of AV angles \u03943D\u22122D .. On the left, angle \u03c1 showed a linear regression relationship with the difference of AV angles \u03943D\u22122D .\u201d\u201cwhich means that angle \u03c1It should be:\u03943D\u22122D on the right \u201d.\u201cwhich means that angle \u03c1 has a significant negative influence on"}
{"text": "Experimental design applications for discriminating between models have been hampered by the assumption to know beforehand which model is the true one, which is counter to the very aim of the experiment. Previous approaches to alleviate this requirement were either symmetrizations of asymmetric techniques, or Bayesian, minimax, and sequential approaches. Here we present a genuinely symmetric criterion based on a linearized distance between mean\u2010value surfaces and the newly introduced tool of flexible nominal sets. We demonstrate the computational efficiency of the approach using the proposed criterion and provide a Monte\u2010Carlo evaluation of its discrimination performance on the basis of the likelihood ratio. An application for a pair of competing models in enzyme kinetics is\u00a0given. The crucial problem is that one typically cannot construct an optimal model\u2010discrimination design without already knowing which model is the true one, and what are the true values of its parameters. In this respect, the situation is analogous to the problem of optimal experimental design for parameter estimation in nonlinear statistical models  linear, T\u2010optimality, albeit being feasible, are not a natural choice. We further suppose that we do not use the full prior distribution of the unknown parameters of the models, which rules out Bayesian approaches such as Felsenstein , int(\u03981), and \u03b51,\u2026,\u03b5n are unobservable random errors. For both k=0,1 and any x\u2208X, we will assume that the functions \u03b7k are differentiable on int(\u0398k); the gradient of \u03b7k in \u03b8k\u2208int(\u0398k) will be denoted by \u2207\u03b7k. Our principal assumption is that one of the models is true but we do not know which, that is, for k=0 or for k=1 there exists \u03b8\u00afk\u2208\u0398k such that yi=\u03b7k+\u03b5i.Let N, where \u03c32\u2208. The assumption of the same variances of the errors for both models is plausible if, for instance, the errors are due to the measurement device and hence do not significantly depend on the value being measured. The situation with different error variances requires a more elaborate approach, compared with Fedorov and P\u00e1zman \u2212logL(\u03b8^1)<>0. Under the normality, homoskedasticity, and independence assumptions, this decision is equivalent to a decision based on the proximity of the vector (yi)i=1n of observations to the vectors of estimated mean values )i=1n and )i=1n.The choice of the best discrimination rule based on the observations is generally a nontrivial problem. However, it is natural to compute the maximum likelihood estimates m0\u2260m1 to counterbalance favoring models with greater number of parameters, Cox (L(\u03b8^0)/L(\u03b8^1)(em1/em0)n/n\u223c, which corresponds to the Bayesian information criterion; see Schwarz  be the design used for the collection of data prior to the decision, and assume that model \u03b70 is true, with the corresponding parameter value \u03b8\u00af0. Note that this comes without loss of generality and symmetry as we can equivalently assume model \u03b71 to be true. Then, the probability of the correct decision based on the likelihood ratio is equal to(yi)i=1n follows the normal distribution with mean )i=1n and covariance \u03c32In.Let d, where Ej is the set of all possible mean values of the observations under the model j, j=0,1, and d denotes the infimum distance between all pairs of elements of two sets, is developed as\u00a0follows.Clearly, probability , and denote y:=(yi)i=1n, \u03b7j(\u03b8j):=)i=1n for j=0,1. Note that we can express P[d\u2264d]. Now, let R=\u2225\u03b5\u2225, where \u03b5=y\u2212\u03b70(\u03b8\u00af0), be the norm of the vector of errors. Assuming R\u2264d/2 we obtaind/2\u2264d) and consequently[R\u2264d/2] implies the event [d\u2264d], that is, d, which depends on the underlying experimental design. Although this maximization is much simpler than maximizing d through linearization, as will be explained in the following\u00a0section.Consider a fixed experimental design 1.1\u03b70=\u03b80x and \u03b71=e\u03b81x. Furthermore, for the moment we assume just two observations y1,y2 at fixed design points x1=\u22121 and x2=1, respectively. In this case evidently \u03b8^0=y2\u2212y12 and \u03b8^1 is the solution of 2e\u2212\u03b8(y1\u2212e\u2212\u03b8)\u22122e\u03b8(y2\u2212e\u03b8)=0, which for \u22122\u2264y1\u22642 is the log root of the polynomial \u03b34\u2212\u03b33y2+\u03b3y1\u22121. Figure\u00a0Let 2\u03b8\u223c0\u2208int(\u03980) and \u03b8\u223c1\u2208int(\u03981) be nominal parameter values, which satisfy the basic discriminability condition\u03b70\u2260\u03b71 for some x\u2208X. Let us introduce regions \u0398\u223c0\u2286int(\u03980)\u2286Rm and \u0398\u223c1\u2286int(\u03981)\u2286Rm containing \u03b8\u223c0 and \u03b8\u223c1; we will consequently call \u0398\u223c0 and \u0398\u223c1flexible nominal sets. It is evident that optimal designs depend on the parameter spaces in the same way as on our flexible nominal sets  be a design. Let us perform the following particular linearization of Model \u03b7k=0,1 in \u03b8\u223ck:Fk(D) is the n\u00d7m matrix given byak(D) is the n\u2010dimensional vector\u03b5=T is a vector of independent N\u00a0errors.Let ak(D) plays an important role and, although it is known, we cannot subtract it from the vector of observations, as is usual when we linearize a single nonlinear regression model. However, if \u03b7k corresponds to the standard linear model then ak(D)=0n for any D.Note that for the proposed method the vector 2.1\u03b80\u2208\u0398\u223c0,\u03b81\u2208\u0398\u223c1. The criterion \u03b4 can be viewed as an approximation of the nearest distance d of the mean\u2010value surfaces of the models, in the neighborhoods of the vectors )i=1n and )i=1n; see the illustrative Figure\u00a0Consider the design criterionD= represented by the counting measure \u03be on X defined as# means the size of a set. Let \u03b8\u223c=T. For all x\u2208X let\u03b80\u2208\u0398\u223c0, \u03b81\u2208\u0398\u223c1 and \u03b8=T we haveM in Equations\u00a0i\u00b7 is the ith row of the matrix , with parameter \u03b8 and independent, homoskedastic errors \u03b51,\u2026,\u03b5n with mean 0; we will call response difference model.We will now express the \u03b4\u2010criterion as a function of the design 2.2D, expression \u03b42(D|\u03b8) is a quadratic function of \u03b8=T. Moreover, both \u03b4(D|\u03b8) and \u03b42(D|\u03b8) are convex because they are compositions of an affine function of \u03b8 and convex functions \u2225.\u2225 and \u2225.\u22252, respectively. Clearly, if the flexible nominal sets are compact, convex, and polyhedral, optimization For a fixed design \u03b4(D|\u03b8) as follows. As\u0398\u223c:=\u0398\u223c0\u00d7\u0398\u223c1 in the response difference model with artificial observationsAlternatively, we can view the computation of \u0398\u223c0=\u0398\u223c1=Rm, the infimum in R package bvls; see Stark and Parker  be defined via formula \u03b4app2 is linear on \u039e. Moreover, let\u03b4app2 is positive homogeneous and concave on \u039e.For The following simple proposition collects the analytic properties of a natural analogue of \u03b4 defined on the linear vector space \u039e of all finite signed measures on \u03b4app2 implies that an s\u2010fold replication of an exact design leads to an s\u2010fold increase of its \u03b42 value. Consequently, a natural and statistically interpretable definition of relative \u03b4\u2010efficiency of two designs D1 and D2 is given by \u03b42(D1)/\u03b42(D2), provided that \u03b42(D2)>0.Positive homogeneity of D be the set of all n\u2010point designs. A design D\u2217\u2208D will be called \u03b4\u2010optimal, if\u0398\u223c0={\u03b8\u223c0} and \u0398\u223c1={\u03b8\u223c1}, then \u03b4(D\u2217) is strictly positive. However, for larger flexible nominal sets it can happen that \u03b4(D\u2217)=0.Let As the evaluation of the \u03b4\u2010criterion is generally very rapid, the calculation of a \u03b4\u2010optimal, or nearly \u03b4\u2010optimal design is similar to that for standard design criteria. For instance, in small problems we can use complete\u2010enumeration and in larger problems we can employ an exchange heuristic, such as the KL exchange algorithm :=Rm, \u0398\u223c1(\u221e):=Rm, such that r can be considered a tuning (set) parameter governing the size of the flexible nominal sets. In \u0398\u223c0(1) and \u0398\u223c1(1) are \u201cunit\u201d nondegenerate compact cuboids centered on respective nominal parameters. For any design D and r\u2208, we define\u03b4r\u2010optimal values of the problem will be denoted byProposition 2.2D be a design. Functions \u03b4r2(D), \u03b4r(D), o2(r), o(r) are nonincreasing and convex in r on the entire interval . (b) There exists r\u2217<\u221e, such that for all r\u2265r\u2217: (i) o(r)=o(\u221e); (ii) Any \u03b4\u221e\u2010optimal design is also a \u03b4r\u2010optimal\u00a0design.(a) Let For simplicity, we will focus on cuboid flexible nominal sets centered at the nominal parameter values. This choice can be justified by the results of Sidak , in partD be an n\u2010point design and let 0\u2264r1\u2264r2\u2208.(a) Let \u03b4r12(D)\u2265\u03b4r22(D) follows from definitions o2(r1)\u2265o2(r2) follows from the fact that a maximum of nonincreasing functions is a nonincreasing function. Monotonicity of \u03b4r(D) and o(r) in r can be shown\u00a0analogously.Inequality \u03b4r2(D) in r, let \u03b1\u2208 and let r\u03b1=\u03b1r1+(1\u2212\u03b1)r2. For all r\u2208, let \u03b8^r denote a minimizer of \u03b4r2(D|\u00b7) on \u0398\u223c(r):=\u0398\u223c0(r)\u00d7\u0398\u223c1(r). Convexity of \u03b42(D|\u03b8) in \u03b8 and a simple fact \u03b1\u03b8^r1+(1\u2212\u03b1)\u03b8^r2\u2208\u0398\u223c(r\u03b1) yield\u03b4r2(D) is convex in r. The convexity of \u03b4r(D) in r can be shown analogously. The functions o2 and o, as pointwise maxima of a system of convex functions, are also\u00a0convex.To prove the convexity of D of size n, the function \u03b4\u221e2(D|\u00b7) is nonnegative and quadratic on R2m, therefore its minimum is attained in some \u03b8D\u2208R2m. There is only a finite number of exact designs of size n, and \u0398\u223c(r)\u2191rR2m, which means that there exists r\u2217<\u221e such that \u03b8D\u2208\u0398\u223c(r\u2217) for all designs D of size n. Let r\u2265r\u2217. We haveD(\u221e) be any \u03b4\u221e\u2010optimal n\u2010trial design. The equality (i) and the fact that \u03b4r(D(\u221e)) and o(r) are nonincreasing with respect to r gives\u25a1(b) For any design  of relevant set parameters; increasing the set parameter beyond r\u2217 leaves the optimal designs as well as the optimal value of the \u03b4\u2010criterion unchanged. We will call any such r\u2217 a set upper bound.The second part of Proposition\u00a0r\u2217. Our experience shows that it usually requires only a small number of recomputations of the \u03b4r\u2010optimal design, even if rini is small and q is close to 1, resulting in a good set upper bound r\u2217 , as well as to obtain \u03b4r\u2010optimal designs in steps 2 and 9 of the algorithm itself.Algorithm\u00a0Algorithm 12.4X={1.00,1.01,\u2026,2.00}, \u03b8\u223c0=e, and \u03b8\u223c1=1. Note that these nominal values satisfy \u03b70=\u03b71. Moreover, let us set \u0398\u223c0(1)= and \u0398\u223c1(1)=, and let the required size of the experiment be n=6. First, we computed the value o2(\u221e)\u22480.02614. Next, we used Algorithm\u00a0rini=0.3 and q=1+10\u22126, which returned a set upper bound r\u2217\u22480.6787 after as few as seven computations of \u03b4r\u2010optimal designs. Informed by r\u2217, we computed \u03b4r\u2010optimal designs for r=0.01,0.1,0.2,\u2026,0.7. The resulting \u03b4r\u2010optimal designs are displayed in Figure\u00a0\u0398\u223c(r)s are very narrow, the \u03b4r\u2010optimal design is concentrated in the design point x=2, effectively maximizing the difference between \u03b70 and \u03b71. For larger values of r, the \u03b4r\u2010optimal design has a 2\u2010point and ultimately a 3\u2010point\u00a0support.Consider the models from the motivating example. Let r\u2217, beyond which the values of \u03b4r are constantly 0 for all designs. These cases can be identified by solving a linear programming (LP) problem, as we show next.Proposition 2.3D\u00af be the design that performs exactly one trial in each point of X. Consider the following LP problem with variables r\u2208R, \u03b80\u2208Rm, \u03b81\u2208Rm:T. Then, r\u2217 is a finite set upper bound. Moreover, o(r)=0 for all r\u2208.Let For some pairs of competing models there exists a set upper bound D and its nonreplication version Dnr we have \u03b4r(Dnr)=0 implies \u03b4r(D)=0. Moreover, if D2\u2ab0D1 in the sense that D2 is an augmentation of D1 then \u03b4r(D2)=0 implies \u03b4r(D1)=0. Now let T be a solution of r\u2265r\u2217 and let D be any design. Definition of \u03b4r and the form of \u03b4r(D\u00af)=0. From D\u00af\u2ab0Dnr we see that then \u03b4r(Dnr)=0, hence \u03b4r(D)=0. The proposition follows.\u25a1From the expression r\u2217 obtained using Proposition\u00a0n, that is, it is a set upper bound simultaneously valid for all design sizes. The basic discriminability condition implies that r\u2217\u22600.Note that a0(D\u00af) and a1(D\u00af) are zero. Therefore, T for any r\u22650 such that both \u0398\u223c0(r) and \u0398\u223c1(r) cover 0m. That is, for the case of linear models, there is a finite set upper bound r\u2217 beyond which the \u03b4r\u2010values of all designs vanish. However, the same holds for specific nonlinear models, including the ones from Section\u00a0Proposition 2.4r\u2217 such that o(r)=0 for all r\u2208.Assume that both competing regression models are linear provided that we consider a proper subset of their parameters as known constants. Then  has a fiIf the competing models are linear, vectors erefore,  has a fek0<m components of \u03b80 converts Model 0 to a linear model. More precisely, let \u03b801,\u2026,\u03b80m denote the components of \u03b80 and assume that\u03b3j(0), j=k0+1,\u2026,m. Choose \u03b8^0 such that \u03b8^0j=\u03b8\u223c0j for j=1,\u2026,k0, and \u03b8^0j=0 for j=k0+1,\u2026,m. Make an analogous assumption for Model 1 and also define \u03b8^1 analogously. It is then straightforward to verify that for the design D\u00af from Proposition\u00a0Fk(D\u00af)\u03b8^k+ak(D\u00af)=0d, where d=#X, for both k=0,1. Therefore, any T such that \u03b8^0\u2208\u0398\u223c0(r) and \u03b8^1\u2208\u0398\u223c1(r) is a solution of \u25a1Without loss of generality, assume that fixing the first In the following, we numerically demonstrate that the \u03b4 design criterion leads to designs which yield a high probability of correct\u00a0discrimination.3y is alternatively modeled asx1 denotes the concentration of the substrate and x2 the concentration of an inhibitor. The data used in Bogacka et\u00a0al. , \u03b80,2,\u03b81,2\u2208 designs for \u03bb and employ them for model discrimination. Note that also this method is not fully symmetric as it requires a nominal value for \u03bb for linearization of In Atkinson , the two\u03b8\u223c01=\u03b8\u223c11=\u03b8\u223c21=10, \u03b8\u223c02=\u03b8\u223c12=\u03b8\u223c22=4.36, \u03b8\u223c03=2.58, \u03b8\u223c13=5.16, and \u03b8\u223c23=3.096. However, note that particularly for model X=X1\u00d7X2=\u00d7.The nominal values used in Atkinson  obviouslT\u2212optimal designs assuming \u03bb=0 (A1) and \u03bb=1 (A4), a compound T\u2010optimal design (A3), and a Ds\u2010optimum (A2) for the encompassing model . We will compare our \u03b4\u2010optimal designs against exact versions of these designs, properly rounded by the method of Pukelsheim and Rieder . We then simulated the data\u2010generating process under each model for N=10000 times and calculated the total percentages of correct discrimination (hit rates) when using the likelihood ratio as decision\u00a0rule.As we are assuming equal variances for the two models we are using the estimate for the error standard deviation \u03b41,\u03b42, and \u03b43, which represent a range of different nominal intervals. Specifically we chose \u0398\u223ck=[\u03b8\u223ck1\u00b1r\u03c3\u223ck1]\u00d7[\u03b8\u223ck2\u00b1r\u03c3\u223ck2]\u00d7[\u03b8\u223ck3\u00b1r\u03c3\u223ck3]k=0,1, where we chose \u03b8\u223ckj=\u03b8^kj and \u03c3\u223ckj=\u03c3^kj for k=0,1 and j=1,2,3. The tuning parameter r was set to three levels: r=1 (which is close to the lower bound of still providing a regular design), r=5 and r=15 , respectively. To make the latter more precise: the models in considerations are such that if we fix the last two out of the three parameters, then they become one\u2010parameter linear models. Therefore, using Proposition\u00a0r\u2217. Solving r\u2217\u224864.02. Note that the same bound is valid for all design sizes n. Designs A1\u2013A4 and \u03b41 all contain four support points, while \u03b42 has six and \u03b43 has five, respectively. A graphical depiction of the designs is given in Figure\u00a0We are comparing the designs A1\u2013A4 to three specific \u03b4 designs Robustness study: As we would like to avoid comparing designs only when the data are generated from the nominal values , we perturbed the data\u2010generating process by drawing parameters from uniform distributions drawn at \u03b8\u223c\u00b1c\u00d7\u03c3\u223c\u03b8, where c then acts as a perturbation parameter. Under these settings all these designs fare pretty well as can be seen from Table\u00a0A4 and \u03b42 seem to outperform the other competing designs by usually narrow margins except perhaps for A1, which is consistently doing worst. Note that in a real situation the true competitors of \u03b4\u2010optimal designs are just A2 and A3 as it is unknown beforehand which model is\u00a0true.3.2n=60 is nearly perfect, we are required to inflate the error variance. However, using additive normal errors in the data\u2010generating process and inflating the variance by a large enough factor, would generate a large number of negative observations, which renders likelihood estimation invalid. So, the data\u2010generating process was adapted to use multiplicative log\u2010normal errors. The observations were then rescaled to match the means from the original process. This way we are at liberty to inflate the error variance by any factor without producing faulty observations. Note that now the data\u2010generating process does not fully match the assumptions under which the designs were generated, but this can just be considered an extended robustness study as it holds for all compared designs equally. We could of course also have calculated the designs under the same data\u2010generating process, but as the fit of the model to the original data is not greatly improved and models As the discriminatory power of all the designs for 5\u00d7\u03c3^ (and c=0). The respective designs \u03b41\u20133 were qualitatively similar to those given in Figure\u00a0n=60 observations from these designs a thousand\u00a0times.Perturbation of the parameters here did not exhibit a discernible effect, while the error inflation still does. For brevity, we here report only again the results for using 1 (remember it being the T\u2010optimum design assuming \u03b70 true), while \u03b41 and \u03b42 come close (and beat the true competitors A2 and A3) with A1 again being clearly the\u00a0worst.The corresponding boxplots of the correct classification rates are given in Figure\u00a04T\u2010optimality, can be undertaken with efficient routines of quadratic optimization that in general enhance the speed of computations by an order of magnitude. An optimal exact design problem is a problem of discrete optimization, and the efficiency of its solution critically depends on the speed of evaluation of the design criterion. By a series of approximations, we substituted the theoretically ideal but numerically infeasible computation of the probability of correct discrimination with a simple convex optimization, which can be solved rapidly and reliably. Combined with the proposed methodology of flexible nominal sets, we can construct an entire sequence of exact experimental designs efficient for discrimination between models. Also it was shown in an example that resulting designs are competitive in their actual discriminatory\u00a0abilities.We have presented a novel design criterion for symmetric model discrimination. Its main advantage is that design computations, unlike for The notion of flexible nominal sets may have independent merit. Note again the distinction between parametric spaces and flexible nominal sets . Parametric spaces usually encompass all theoretically possible values of the parameters, while flexible nominal sets can contain the unknown parameters with very high likelihood, and still be significantly smaller than the original parameter spaces. In this paper, we do not completely specify the process of constructing the flexible nominal sets, but if we perform a two stage experiment, with a second, discriminatory phase, the potential specification through confidence intervals is an important\u00a0problem.As the approach suggested offers a fundamentally new way of constructing discriminatory designs, many properties are yet unexplored. A nonexhaustive list of questions\u00a0follows.Sequential procedure. The proposed method lends itself naturally to a two\u2010stage procedure, where parameter estimates and confidence intervals are employed as nominal values in the second stage. Even sequential generation of design points can be straightforwardly\u00a0implemented.Approximate designs. Proposition\u00a0\u03b4app2 is concave on the set of all approximate designs. Therefore, it is possible to work out a minimax\u2010type equivalence theorem for \u03b4\u2010optimal approximate designs, and use specific convex optimization methods to find a \u03b4\u2010optimal approximate designs numerically. For instance, it would be possible to employ methods analogous to Burclov\u00e1 and P\u00e1zman , a natural criterion closely related to \u03b4r\u2010optimality can be defined as\u03b4\u223cr requires a multivariate nonconvex optimization for the evaluation in each design D, which entails possible numerical difficulties and a long time to compute an optimal design. However, the \u03b4r\u2010optimal design, which can be computed rapidly and reliably, can serve as efficient initial design for the optimization of \u03b4\u223cr. Note that if \u0398\u223c0 is a singleton containing only the nominal parameter value for Model 0, the \u03b4r\u2010optimal designs could potentially be used as efficient initial designs for computing the exact version of the criterion of T\u2010optimality.Selection of the best design from a finite set of possible candidates. As most proposals for the construction of optimal experimental designs, the method depends on the choice of some tuning parameters or even on entire prior distributions (in the Bayesian approach), which always results in a set of possible designs. It would be interesting to develop a comprehensive Monte\u2010Carlo methodology for the choice of the best design out of this pre\u2010selected small set of candidate designs. A useful generalization of the rule would take into account possibly unequal losses for the wrong\u00a0classification.Noncuboid sets. The methodology could certainly be extended to other types of flexible nominal sets, particularly when we are interested in functional relations among the parameters. However, then the particularly efficient box constrained quadratic programming algorithm could not be\u00a0utilized.Higher order approximations. As a referee remarked it is possible to employ tighter approximations of the sets of mean values of responses than the one that we suggest. For instance, it would be possible to use the local curvature of the mean\u2010value function. However, this may also lead to the loss of numerical efficiency of the\u00a0method.More than two rival models. Another referee remark leads us to point out the natural extension to investigate a weighted sum or the minimum \u03b4 over all paired comparisons. The implications of these suggestions, however, requires deeper\u00a0investigations.Different error variances. Yet another referee requested a clarification for how to proceed in case of unequal error variances for the two models. In case the functional form of these variances are known simple standardizations of the models will suffice. All other cases, including dependencies of the errors, will require more elaborate\u00a0strategies.Combination with other criteria. The proposed method can produce poor or even singular designs for estimating model parameters. Because of this problem, already mentioned in Atkinson and Fedorov (DT\u2010optimality. The same approach is possible for \u03b4\u2010optimality. However, our numerical experience suggests that for a large enough size of the flexible nominal set, the \u03b4\u2010optimal designs tend to be supported on a set that is large enough for estimability of the parameters, without any combination with an auxiliary criterion. A detailed analysis goes beyond the scope of this\u00a0paper. Fedorov , Atkinso Fedorov  used a cThe authors declare that there is no conflict of\u00a0interest.This article has earned an Open Data badge for making publicly available the digitally\u2010shareable data necessary to reproduce the reported results. The data is available in the Reproducible Research\u201d for making publicly available the code necessary to reproduce the reported results. The results reported in this article could fully be reproduced.This article has earned an open data badge \u201cSUPPORTING INFORMATIONClick here for additional data file."}
{"text": "This article has been withdrawn by the authors. The Journal statesthat the immunoblots corresponding to \u2018Rac1\u2019 in figure\u00a04B and 4H were duplicated.The immunoblots corresponding to \u2018beta-Tubulin\u2019 in figure\u00a04B and \u2018Pak2\u2019 in figure\u00a05Ewere duplicated. Possible micrograph reuse for \u2018RhoA ASO3\u00a0+ Thrombin\u2019 panel infigure\u00a04C and \u2018Control ASO\u2019 panel in figure\u00a010D. Additional potential reuse in \u2018Pak2ASO2\u00a0+ Thrombin\u2019 panel in figure\u00a05C and \u2018G-alpha-qASO2\u2019 panel in figure\u00a010D.Possible reuse in immunoblot panels for \u2018GTP-RhoA\u2019 in figure\u00a04E and \u2018GTP-Rac1\u2019 infigure\u00a010J. Additional potential reuse between micrograph panels \u2018SCH79797+Thrombin\u2019in figure\u00a08A and \u2018Control ASO\u2019 in figure\u00a02C. There was possible image reuse betweenimmunoblots \u2018beta-Tubulin\u2019 panel in figure\u00a07A and \u2018Pak2\u2019 panel in figure\u00a07F.Finally, there was possible image reuse for \u2018PY20\u2019 immunoblot panel in figures\u00a07Dand 9E, which represent different experimental conditions. Due to these immunoblotand image duplications, the authors wish to withdraw this article."}
{"text": "These maps can be realized as attractors of disc homeomorphisms in such a way that the attractors vary continuously (in Hausdorff distance on the disc) with the change of bonding map as a parameter. Furthermore, for generic Lebesgue measure\u2010preserving maps f the background Oxtoby\u2013Ulam measures induced by Lebesgue measure for f on the interval are physical on the disc and in addition there is a dense set of maps f defining a unique physical measure. Moreover, the family of physical measures on the attractors varies continuously in the weak* topology; that is, the parametrized family is statistically stable. We also find an arc in the generic Lebesgue measure\u2010preserving set of maps and construct a family of disk homeomorphisms parametrized by this arc which induces a continuously varying family of pseudo\u2010arc attractors with prime ends rotation numbers varying continuously in . It follows that there are uncountably many dynamically non\u2010equivalent embeddings of the pseudo\u2010arc in this family of\u00a0attractors.The main goal of this paper is to study topological and measure\u2010theoretic properties of an intriguing family of strange planar attractors. Building toward these results, we first show that any generic Lebesgue measure\u2010preserving map  We provide a parametrized family of strange attractors where all the attracting sets are homeomorphic but nevertheless, as we shall see later, they exhibit a variety of rich dynamical behavior and have good measure\u2010theoretic and statistical properties. The attracting sets of this family are all homeomorphic to the one\u2010dimensional space of much interest in Continuum Theory and beyond, called the pseudo\u2010arc. A continuum is a non\u2010empty compact connected metric space. The pseudo\u2010arc may be regarded as the most intriguing planar continuum not separating the plane. On the one hand its structure is quite complicated, since it does not contain any arc. On the other hand it reflects much regularity in its shape, since it is homeomorphic to any of its proper subcontinua. For the history of the pseudo\u2010arc and numerous results connecting it to other mathematical fields we refer the reader to the introduction in residual iated in  that theTheorem 1.1G\u03b4 set T\u2282C\u03bb(I) such that if f\u2208T then for every \u03b4>0 there exists a positive integer n so that fn is \u03b4\u2010crooked.There is a dense Here we prove another topological property of Lebesgue measure\u2010preserving maps, which might be the most surprising of the properties yet; namely we prove:\u03b4\u2010crookedness is not an easy\u2010to\u2010state property (see Definition\u00a0\u03b4\u2010crookedness in the sense of Theorem\u00a0Corollary 1.2C\u03bb(I)\u2010generic map as a single bonding map is the\u00a0pseudo\u2010arc.The inverse limit with any The One should note that all interesting \u201cglobal\u201d dynamics in interval maps can be reflected in Lebesgue measure\u2010preserving maps  and let f^:I^\u2192I^ be the natural extension off (or the shift homeomorphism). A natural projection\u03c00:I^\u2192I defined by \u03c00(x)=x0semi\u2010conjugatesf^ to f.At this point, let us mention that one can view inverse limits with single bonding maps as the simplest invertible dynamical extensions of the dynamics given by the bonding map. Let us state this fact more precisely. Denote by Y let g:Y\u2192Y be an invertible dynamical system and let p:Y\u2192I factor g to f. Then p factors through \u03c00: that is, f^ is the minimal invertible system which extends f \u2282C(I) the class of interval maps with the dense set of periodic points and by CDP(I)\u00af its closure. Building on the well\u2010known properties of interval maps mentioned earlier (see Remark\u00a0Corollary 1.3CDP(I)\u00af\u2010generic map as a single bonding map is the\u00a0pseudo\u2010arc.The inverse limit with any Denote by C(I) (where C(I) denotes the class of all continuous interval maps). On the other hand, Bing 27]). ThiC ) the set of all continuous mappings from a metric space X to a metric space Y . We equip the space C with the metric of uniform convergence \u03c1. Let D\u2282R2 denote a closed topological disk. We say that a compact set K\u2282D\u2282R2 is the  attractor of h:D\u2192D in D if for every x\u2208D\u2216\u2202D, the omega limit set\u03c9h(x)\u2282K and for some z\u2208K we have that \u03c9h(z)=K.Denote by K\u2282D\u2216\u2202D\u2282R2 we can associate the circle of prime ends as the compactification of D\u2216K. If h\u223c:R2\u2192R2 preserves orientation and h\u223c(K)=K, h\u223c(D)=D then h\u223c induces an orientation\u2010preserving homeomorphism of the prime ends circle, and therefore it gives a natural prime ends rotation number. The prime ends rotation number allows one to study boundary dynamics of underlying global attractors and distinguish their embeddings from dynamical point of view, see Definition\u00a0Theorem 1.5T be a dense G\u03b4 subset of C\u03bb(I) from Theorem\u00a0{ft}t\u2208\u2282T\u2282C\u03bb(I) and a parametrized family of homeomorphisms {\u03a6t}t\u2208\u2282H varying continuously with t having \u03a6t\u2010invariant pseudo\u2010arc attractors \u039bt\u2282D for every t\u2208 so that(a)\u03a6t|\u039bt is topologically conjugate to f^t:I^f\u2192I^f.(b){\u039bt}t\u2208 vary continuously in the Hausdorff metric.The attractors (c)\u03a6t vary continuously with t in the interval .Prime ends rotation numbers of homeomorphisms (d){\u039bt}t\u2208.There are uncountably many dynamically non\u2010equivalent planar embeddings of the pseudo\u2010arc in the family Let To a non\u2010degenerate and non\u2010separating continuum C\u03bb(I) and it would be interesting to know if this set itself is pathwise connected. Third, let us mention that the homeomorphism group of the pseudo\u2010arc contains no non\u2010degenerate continua  should be continuous. There is also no immediate argument why homeomorphisms ht\u2218\u03a6t|\u039bt\u2218ht\u22121 are different. Since in this work we are interested only in embeddings from dynamical perspective we leave the following question about the topological nature of embeddings open.Question 1t\u2260t\u2032\u2208 the attractors \u039bt and \u039bt\u2032  non\u2010equivalently\u00a0embedded?Are for every This result is interesting also from several other aspects. First, it is the first example in the literature (to our knowledge) of a parametrized family of strange attractors where the attractors are proven to be homeomorphic, yet the boundary dynamics on the attractor is very rich. This result underlines the fact that pseudo\u2010arc is among one\u2010dimensional continua a special object with respect to its flexibility to permit a variety of different dynamical behavior. Furthermore, let us also note that Theorem\u00a0continua , so Theosults of  in state equivalent if there is a homeomorphism h:R2\u2192R2 such that h(\u039bt)=\u039bt\u2032. Note that h does not have to intertwine the dynamics, so Theorem\u00a0Recall that two planar embeddings are called C\u03bb(I), and we take Theorem\u00a0M is called Oxtoby\u2013Ulam (OU) or good if it is non\u2010atomic, positive on open sets, and assigns zero measure to the boundary of manifold M (if it exists) \u2282R denote the unit interval. Let diam(A) denote the diameter of A\u2282I. Let \u03bb denote the Lebesgue measure on the underlying Euclidean space. By C(I) we denote the family of all continuous interval maps. Furthermore, let C\u03bb(I)\u2282C(I) denote the family of all continuous Lebesgue measure\u2010preserving functions of I. We equip both C(I) and C\u03bb(I) with the metric of uniform convergence\u03c1: we shall use B for the open ball of radius \u03be centered at x\u2208X and for a set U\u2282X we shall denoted to denote the Euclidean distance on the underlying Euclidean space. We say that a map f is piecewise linear (or piecewise affine) if it has finitely many critical points  and is linear on every interval of monotonicity . We say that an interval map f is leo if for every open interval J\u2282I there exists a non\u2010negative integer n so that fn(J)=I. This property is also sometimes referred in the literature as topological exactness.Let 2.2Definition 2.1f\u2208C(I), let a,b\u2208I and let \u03b4>0. We say that f is \u03b4\u2010crooked betweena and b if for every two points c,d\u2208I such that f(c)=a and f(d)=b, there is a point c\u2032 between c and d and there is a point d\u2032 between c\u2032 and d such that |b\u2212f(c\u2032)|<\u03b4 and |a\u2212f(d\u2032)|<\u03b4. We will say that f is \u03b4\u2010crooked if it is \u03b4\u2010crooked between every pair of\u00a0points.Let For an easier visualization of the concept defined in the following definition we refer the reader to Figure\u00a0Definition 2.2(cr[n])n=1\u221e\u2282N be the sequence defined in the following way: cr[1]:=1, cr[2]:=2 and cr[n]:=2cr[n\u22121]+cr[n\u22122] for each n\u2a7e3.Let The following two definitions and the first part of the third definition were introduced in . We willDefinition 2.3g1 and g2 be two maps of I into itself such that g1(0)=g2(0)=0 and g1(1)=g2(1)=1. Suppose m\u2a7e3 is an integer and s is a real number such that 0<s<1/2. Then \u03d5 is the function of I into itself defined by the formula:Let Definition 2.4n\u2a7e3 denoten\u2208N, let simplen\u2010crooked map\u03c3n:I\u2192I be defined in the following way :=1\u2212\u03c3n(t) for each t\u2208I. Let \u03c3nL:=\u03c3n| where \u03c3n|:\u2192, \u03c3nR:=\u03c3n| where \u03c3n|:\u2192. Similarly as above let \u03c3\u2212nL and \u03c3\u2212nR denote the reflections of \u03c3nL and \u03c3nR,\u00a0respectively.For each integer Definition 2.5n\u2a7e3 the maximal number of intervals of monotonicity of \u03c3n of the same length is denoted by #\u03c3n.For every integer Definition 2.6f:\u2192 where ,\u2282R and a<b,a\u2032<b\u2032 is an odd function around the point\u2208R2 if the graph of f| equals the rotation for angle \u03c0 of the graph of f| around .We say that a function Observation 2.7n\u2a7e3(1)\u03c3n is a piecewise linear continuous function.(2)#\u03c3n=cr[n]. In particular, for even n, #\u03c3n is even and it is odd for odd n.it holds that (3)\u03c3n has uniform slope being \u00b1cr[n]n.(4)\u03c3n is an odd function around the point , that is, it holds that \u03c3\u2212nL(t)=\u03c3nR(t+1/2) and \u03c3\u2212nR(t+1/2)=\u03c3nL(t) for all t\u2208.For every integer \u03b5=3/n, however let us note that this estimate is far from optimal for n small.Remark 2.8\u03b5>0 and n is sufficiently large to ensure 2/n<\u03b5, the map \u03c3n is \u03b5\u2010crooked.By Proposition\u00a03.5 in , if \u03b5>0 We will often use the following remark with Remark 2.9\u03b5\u2010crooked with small \u03b5>0 it cannot be a small perturbation of the identity map. To work with small perturbations of identity cr[n]+cr[n\u22121]n.a continuous and piecewise linear function with the uniform slope (2).an odd function around the point For each odd integer For each odd integer formula\u03bb^n,k(t)Lemma 2.11n\u2a7e7 and every integer k\u2a7e1 it holds that if t,s\u2208 are such that |\u03bb^n,k(t)\u2212\u03bb^n,k(s)|<n\u22121n then \u03bb^n,k is 3n\u2010crooked between \u03bb^n,k(t) and \u03bb^n,k(s).For every odd integer \u03bb^n,k is generated using rescaled \u03c3\u2212(n\u22121)R, \u03c3n, and \u03c3\u2212(n\u22121)L that are properly shifted vertically; to simplify the notation in this proof we will refer to the three parts of the definition of \u03bb^n,k simply by \u03c3\u2212(n\u22121)R, \u03c3n, and \u03c3\u2212(n\u22121)L while remembering about the rescaling and shift. Therefore, \u03bb^n,k consists of blocks of the formSi for i\u2208{1,\u2026,2(n+k)} and which gives a well\u2010defined order on\u00a0them.First note that the function a,b\u2208, so that |a\u2212b|<n\u22121n. If points c,d\u2208, where f(c)=a and f(d)=b, are contained in some Si for i\u2208{1,\u2026,2(n+k)}, the claim follows directly from Remark\u00a0S1 or S2(n+k) note that the images are restrictions of \u03c3\u2212(n\u22121) and thus we can again use Remark\u00a0Fix c,d\u2208 are contained in two adjacent blocks Sj<Sj+1, \u03bb^n,k|Sj=\u03c3\u2212(n\u22121) and \u03bb^n,k|Sj+1=\u03c3n for some j\u2208{1,\u2026,2(n+k)\u22121}. Assume that c\u2208Sj and d\u2208Sj+1 . Then, since \u03bb^n,k(Sj)\u2282\u03bb^n,k(Sj+1) it follows that there always exist c\u2032\u2208Sj+1 such that c<c\u2032<d and so that f(c\u2032)=a. Thus we obtain the claim using Remark\u00a0Now assume that points c,d\u2208 are contained in two non\u2010adjacent blocks, c\u2208Sj and d\u2208Sj\u2032 for |j\u2212j\u2032|\u2a7e2. If a\u2208\u03bb^n,k(Sj\u2032) or b\u2208\u03bb^n,k(Sj) then we can find two adjacent blocks between Sj and Sj\u2032 and use the arguments from the preceding paragraph as we assumed that |a\u2212b|<n\u22121n. Note that a\u2208\u03bb^n,k(Sj\u2032) or b\u2208\u03bb^n,k(Sj) holds always except if \u03bb^n,k|Sj=\u03bb^n,k|Sj\u2032=\u03c3n, |j\u2212j\u2032|=2, and n\u22122n\u2a7d|a\u2212b|\u2a7dn\u22121n. But in this case observe that 3n\u2010crookedness of \u03c3n assures that we can find a point between c and d in either Sj or Sj\u2032 with the required image value.\u25a1Now let us assume that Definition 2.12n,k\u2208N define the flip mapFor all Definition 2.13n\u2a7e7 and every k\u2208N define the map \u03bbn,k:I\u2192I byFor every odd integer Now we have all the ingredients to define the final map with which we will work in this section.t\u2208I.for all \u03bbn,k for n=7 and note that the properly scaled \u201cmiddle\u201d building blocks of \u03bbn,k are as in Figure\u00a0Observation 2.14n\u2a7e7 and each integer k\u2a7e1 the map \u03bbn,k is(1)\u00b1(cr[n]+cr[n\u22121]),a continuous piecewise linear function with the uniform slope being (2),an odd function around the point (3)\u03bbn,k(jn+k\u22121)=jn+k\u22121 for all j\u2208{0,\u2026,n+k\u22121}.such that For each odd integer See Figure\u00a0\u03bbn,k preserves Lebesgue measure for all odd n\u2a7e7 and k\u2a7e1.Definition 2.15j\u2208{1,\u2026,n+k\u22121} and some fixed integer k\u2a7e1 and odd integer n\u2a7e7 denote by Ij:=1n+k\u22121\u2282I and letjth vertical strip andjth horizontal strip. Let p1:I\u00d7I\u2192I (p2:I\u00d7I\u2192I) be the natural projection onto the first (second) coordinate. Furthermore, define the maximal number of intervals of monotonicity of \u03bbn,k of the diameter exactly 1n+k\u22121 in Vj  byFor every Now we will turn to the proof that the function Observation 2.16n\u2a7e7 and each integer k\u2a7e1 it holds that(1)\u03bbn,k|Ij is an odd function around the point (2j\u221212(n+k\u22121),2j\u221212(n+k\u22121)) for any j\u2208{n+12,\u2026,k+n\u221212}.the function (2)#(Vj)=cr[n]+cr[n\u22121] for any j\u2208{1,\u2026,n+k\u22121} (since function Fl does not change #(Vj)).(3)diam)=nn+k\u22121 if j\u2208{n+12,\u2026,k+n\u221212}.(4)diam)=n+2j\u221212(n+k\u22121) if j\u2208{1,\u2026,n\u221212}.For each odd integer \u03bbn,k preserves Lebesgue measure we will also implicitly use the following simple observation.Observation 2.17E differentiable interval map f preserves Lebesgue measure if and only iff\u2208C\u03bb(I) and any non\u2010degenerate interval J\u2282I,Comp(f\u22121(J)) denotes the set of all connected components of f\u22121(J).A piecewise monotone and up to a finite points To check whether Proposition 2.18n\u2a7e7 and each integer k\u2a7e1 the map \u03bbn,k preserves Lebesgue measure on I.For each odd integer #(Vj)=cr[n]+cr[n\u22121] for all j\u2208{1,\u2026,n+k\u22121} and since by Observation\u00a0\u03bbn,k has uniform slope  we only need to show that #(Hj)=cr[n]+cr[n\u22121] since diam(p1(Vj))=diam(p2(Hj\u2032)) for any j,j\u2032\u2208{1,\u2026,n+k\u22121}. We will consider boxesBj,l:=Vj\u2229Hl and the number of injective branches of \u03bbn,k of Bj,l denoted by # for all j,l\u2208{1,\u2026,n+k\u22121}, see Figure\u00a0From Observation\u00a0j\u2208{n+12,\u2026,k+n\u221212}. Then #(Vj)=#+\u22ef+#+\u22ef+#=#+2#+2#+\u22ef+2#, due to Observation\u00a0n is odd. But note that =# for all m\u2208{j\u2212n\u221212,\u2026,j+n\u221212} and thus #(Hj)=#+\u22ef+#+\u22ef+#=#+\u22ef+#+\u22ef+#=#(Vj) which finishes this part of the\u00a0proof.First assume that j\u2208{1,\u2026,n\u221212}\u222a{k+n+12,\u2026,n+k\u22121}. By Observation\u00a0j\u2208{1,\u2026,n+12}. By Definition\u00a0#(Hj)=#+#\u22ef+#+\u22ef+#=#+2#+2#+\u22ef+2#=#(Vj) .\u25a1Now assume that Equation\u00a0 and ther\u03bbn,k fits in the context of Proposition 5 of  defined for every odd integer n\u2a7e7 and every integer k\u2a7e1 by\u03bbn,k from \u03bb\u223cn,k applying the flip function from Definition\u00a0a,b\u2208 so that |a\u2212b|<\u03b5 we have that the map \u03bb\u223cn,k is 3\u03b3\u2010crooked between a and b. However, since one obtains \u03bbn,k from \u03bb\u223cn,k and the flip function by the definition at most decreases the distances between the function values, the claim follows immediately.Now let us prove that for every (iii)A\u2282I is a subinterval so that diam(A)<2(cr[n]+cr[n\u22121])(n+k\u22121), then by Observation\u00a0diam)>(cr[n]+cr[n\u22121])diam(A)2>diam(A). If a subinterval A\u2282I is such that \u03b3>diam(A)\u2a7e2(cr[n]+cr[n\u22121])(n+k\u22121), then it follows that diam)\u2a7e\u03b3>diam(A) because A contains at least one full interval of monotonicity of the diameter of image being \u03b3. Now assume that \u03b3\u2a7ddiam(A)\u2a7d2\u03b3. Then there are x,y\u2208A such that \u03bbn,k(y)=max\u03bbn,k(Ij) and \u03bbn,k(x)=min\u03bbn,k(Ij\u2032) where |j\u2212j\u2032|\u2a7d1 and A\u2229Ij\u2260\u2205, A\u2229Ij\u2032\u2260\u2205. Note that A is contained in at most three different intervals Ii and \u03bbn,k(Ii) covers itself and two neighboring intervals on each side, provided it is not the interval containing endpoints 0 or 1 ,\u03bbn,k(y)]\u2282\u03bbn,k(A), in particular diam)\u2a7ediam(A). If A contains 0 or 1 it follows from the construction of \u03bbn,k that A\u2282\u03bbn,k(A) and thus subsequently diam)\u2a7ediam(A); even more, diam)\u2a7en+12(n+k\u22121). When diam(A)>2\u03b3, then there are j<j\u2032 such that Ij\u222a\u2026\u222aIj\u2032\u2282A and diam(Ij\u222a\u2026\u222aIj\u2032) is maximal possible for the interval Ij\u222a\u2026\u222aIj\u2032 under inclusion . But then clearly A\u2282 \u03b5/2=n\u221212(n+k\u22121). Namely, |\u03bb\u223cn,k(x)\u2212\u03bb\u223cn,k(y)|\u2a7en+1n+k\u22121 where x and y are as in the previous paragraph. After applying the flip function we obtain |\u03bbn,k(x)\u2212\u03bbn,k(y)|\u2a7en+12(n+k\u22121), see Figure\u00a0This part follows from the arguments in the last paragraph if one notes that (c)Ij the first maximal value of \u03bbn,k lies \u03b3 apart for all j\u2208{1,\u2026,k+n\u221212} and it is exactly \u03b3 greater from the maximal value of \u03bbn,k(Ij\u22121) for all j\u2208{2,\u2026,k+n\u221212} and we will use this fact heavily. Furthermore, if we denote \u03b1:=\u03b3cr[n]+cr[n\u22121] that is \u03b1 is the length of the smallest interval of monotonicity, then \u03bbn,k(xi)=max\u03bbn,k for each i\u2208{1,\u2026,k+n\u221212\u22121}, where xi are as in Observation\u00a0\u03b3\u2a7ddiam(A)<2\u03b3 the conclusion follows from Observation\u00a0For this part Observation\u00a0A\u2282I be an interval such that diam(A)\u2a7e2\u03b3, say A:=\u2282I. Then there is maximal j such that Ij\u2282A. If j\u2a7ek+n\u221212 then max\u03bbn,k(A)=1 and so \u03bbn,k=\u03bbn,k where b+r\u2a7d1 for some non\u2010negative real number r. So assume that j<k+n\u221212. Next, let j\u2032 be the maximal number such that xj\u2032\u2208, where xj\u2032 is as in Observation\u00a0j\u2032=k+n\u221212 then \u03bbn,k(xj\u2032)=1 and clearly xj\u2032\u2212xj<r+\u03b3 so \u03bbn,k\u2282B,r+\u03b3). For the last case, fix any x\u2208. Then, by the repetative structure of building blocks of the graph \u03bbn,k \u2a7d\u03bbn,k(y)+s\u03b3. But this shows that \u03bbn,k\u2282B,r+\u03b3), which concludes the proof.\u25a1Let piecewise linear maps that preserve Lebesgue measure\u03bb by PL\u03bb and piecewise linear maps that are leo and preserve\u03bb by PL\u03bb(leo). If additionally they satisfy Markov property, which means there is a partition 0=a0<a1<\u2026<an=1 such that for each i the map f is monotone and there are s<t such that f=, then we call them Markov piecewise linear leo maps that preserve\u03bb. The set of all such maps is denoted PLM\u03bb(leo).Definition 2.21f\u2208C(I) is called admissible, if |f\u2032(t)|\u2a7e4 for every t\u2208I for which f\u2032(t) exists and f is\u00a0leo.A piecewise linear map We denote the set of \u03bbn,k from Lemma\u00a0Lemma 2.22f:I\u2192I be an admissible map. Let \u03b7 and \u03b4 be two positive real numbers. Then there is an admissible map F:I\u2192I and there is a positive integer n such that Fn is \u03b4\u2010crooked and \u03c1<\u03b7. Moreover, if f\u2208C\u03bb(I), such F can be also chosen to be in C\u03bb(I).Let Having the appropriate Lebesgue measure\u2010preserving perturbations F=f\u2218\u03bbn,k and proceed with the proof as in \u2282I\u2192I we say that they are \u03bb\u2010equivalent if for each Borel set A\u2208B(I) (where B(I) from now on denotes Borel\u03c3\u2010algebra on I) it holds that,f\u2208C\u03bb(I) and \u2282I we denote by C the set of all continuous maps \u03bb\u2010equivalent to f|. We defineFor maps In what follows we will also need perturbations of maps that preserve Lebesgue measure, similarly as in .DefinitDefinition 2.26f\u2208C\u03bb(I) and \u2282I. For any fixed m\u2208N, let us define the map h=h\u27e8f;,m\u27e9:\u2192I by :h\u27e8f;,m\u27e9\u2208C for each m and h\u27e8f;,m\u27e9\u2208C\u2217 for each m odd. In particular, if h=h\u27e8f;,m\u27e9, m odd, we will speak of regularm\u2010fold window perturbationh of f .Let The following definition is illustrated by Figure\u00a0Lemma 2.27\u03b5>0 and every leo map f\u2208C\u03bb(I) there exists F\u2208C\u03bb(I) such that F is admissible and \u03c1<\u03b5.For every For more details on the perturbations from the previous definition we refer the reader to .Lemma 2C\u03bb(I), so let us start with such map g with \u03c1<\u03b5/2. Let us choose a Markov partition 0=a0<a1<\u2026<an=1 for g. Periodic points are dense for a leo map, so including points from periodic orbits as points in the partition, we may also require that |ai+1\u2212ai|<\u03b4 for any fixed \u03b4>0. In particular, we have that g is monotone on each interval , diam)<\u03b5/2 and g= for some indices k<k\u2032. Now, repeating construction in Lemma\u00a05 of  by its regular m\u2010fold window perturbation, with odd and sufficiently large m. This way F is admissible with \u03c1\u2a7ddiam)<\u03b5/2. Window perturbations are invariant for C\u03bb(I), hence F\u2208C\u03bb(I). Clearly also g=F=. Therefore, for each i there is n\u2208N such that Fn=I. But then, if we fix any open set U\u2282I, then since slope on intervals of monotonicity of F is at least 4, there is M\u2208N such that FM(U) contains three consecutive intervals of monotonicity, and therefore FM+1(U)\u2283.\u25a1By Lemma\u00a0mma\u00a05 of  we constLemma 2.28 be two maps so that \u03c1<\u03b5. If f is \u03b4\u2010crooked, then F is (\u03b4+2\u03b5)\u2010crooked.The following lemma is an essential ingredient in the construction of pseudo\u2010arc using inverse limits in .Lemma 2Now we are ready to prove the main theorem of this\u00a0section.Proof of Theorem 1.1k\u2a7e1 let the set Ak\u2282C\u03bb(I) be contained in the set of maps f such that fn is (1/k\u2212\u03b4)\u2010crooked for some n and some sufficiently small \u03b4>0. First observe that Ak is dense. Namely, by Lemma\u00a0C\u03bb(I). If we start with such a map g then first applying Lemma\u00a0g to a map f\u2208Ak by an arbitrarily small perturbation. But if f\u2208Ak and n,\u03b4 are constants from the definition of Ak, then by Lemma\u00a0B\u2282Ak. This shows that Ak contains an open dense set. But then the setG\u03b4 and clearly each element f\u2208T satisfies the conclusion of the theorem.\u25a1For any 2.3CDP(I)\u2282C(I) the family of interval maps with a dense set of periodic points. First note that CDP(I) is not a closed space. However, since CDP(I)\u00af is closed in C(I) it is thus a complete space as well. Now we state a useful remark that is given and explained in the introduction of  let f\u223ct be defined by f\u223ct(27)=f\u223ct(47)=f\u223ct(1721)=f\u223ct(1)=0 and f\u223ct(37)=f\u223ct(57)=f\u223ct(1921)=1 and piecewise linear between these points on the interval . Furthermore on the interval x\u2208 let:Proposition 4.1t\u2208, the map f\u223ct\u2208C\u03bb(I).For every For what follows we refer the reader to Figure\u00a0d by f\u223ct2=f\u223ct(47)=f\u223c0,f\u223c1\u2208C\u03bb(I). For x\u2208 it holds that s0:=|f\u223c0\u2032(x)|=7 and s1:=|f\u223c1\u2032(x)|=212; thus s1/s0=3/2. Note that for any t\u2208 it holds that for x\u2208 and y\u2208I either there exist 3 points of f\u223ct\u22121 in  where f\u223ct has slope 212 or 2 points where ft\u223c has slope 7. Therefore, invoking Observation\u00a0f\u223ct\u2208C\u03bb(I) for all t\u2208.\u25a1Applying Observation\u00a0Observation 4.2{f\u223ct}t\u2208 is a family of continuous piecewise linear maps varying continuously with t\u2208. Furthermore, f\u223c0 is an 8\u2010fold map with f\u223c0(0)=0 and f\u223c1 is a 9\u2010fold map so that f\u223c1(0)=1 and f\u223c1(1)=0. Moreover, since it holds that \u03bbn,k(0)=0 and \u03bbn,k(1)=1 for any odd n and k\u2208N it holds for g\u223ct:=f\u223ct\u2218\u03bbn1,k1\u2218\u2026\u2218\u03bbnm,km for any k1,\u2026,km\u2a7e1 and odd n1,\u2026,nm\u2a7e1 that g\u223c0(0)=0, g\u223c1(0)=1, and g\u223c1(1)=0.Observation 4.3t\u2208 and for all points x\u2208I where f\u223ct\u2032 is defined it holds that 212\u2a7e|f\u223ct\u2032(x)|\u2a7e7.For every Observation 4.4t\u2208 and any subinterval A\u2282I which does not contain two critical points it holds that diam(f\u223ct(A))>3diam(A).For every Due to the previous observation we obtain the following.Proposition 4.5t\u2208, the map f\u223ct is\u00a0leo.For every \u2282I and any t\u2208. By Observation N\u2208N such that f\u223ctN contains two critical points. But then, by the definition of f\u223ct, we obtain that f\u223ctN+1=I. Indeed f\u223ct is leo for all t\u2208.\u25a1Fix any non\u2010degenerate interval Corollary 4.6t\u2208, the map f\u223ct is\u00a0admissible.For every Combining Observation\u00a0{f\u223ct}t\u2208 with the help of Lemma\u00a0{ft}t\u2208\u2282C\u03bb(I) of continuous maps varying continuously with t as well.Lemma 4.7\u03b2>0. Let g\u223ct:=f\u223ct\u2218\u03bbn1,k1\u2218\u2026\u2218\u03bbnm,km for some k1,\u2026,km\u2a7e1 and odd n1,\u2026,nm\u2a7e1. There is an integer N\u2a7e0 so that for every 0\u2a7da\u2a7db\u2a7d1 with b\u2212a>\u03b2 and all t\u2208 it holds that g\u223ctN=I.Let Now we will perturb the maps N so that 3N\u03b2>1. By Observation\u00a0\u03bbn,k does not shrink intervals  contains two critical points of f\u223ct. By the definition of f\u223ct we have g\u223ctj+1=I.\u25a1Take any Lemma 4.8\u03b7 and \u03b4 be two positive real numbers fixed for the whole family {g\u223ct}t\u2208 where g\u223ct:=f\u223ct\u2218\u03bbn1,k1\u2218\u2026\u2218\u03bbnm,km for some k1,\u2026,km\u2a7e1 and odd n1,\u2026,nm\u2a7e1. Then there is a positive integer N such that for every t\u2208 there exists an admissible map G\u223ct:I\u2192I so that G\u223ctN is \u03b4\u2010crooked and \u03c1<\u03b7. Moreover, G\u223ct\u2208C\u03bb(I) and G\u223ct=g\u223ct\u2218\u03bbnm+1,km+1 for some km+1\u2a7e1 and odd nm+1\u2a7e1.Let f in the Lemma from  and . All the maps g\u223ct have slopes bounded from the above by the same constant, call it s, since slopes of all f\u223ct are uniformly bounded from the above and all the maps in the composition are piecewise linear. Also, Lemma\u00a0N for all g\u223ct which plays the role of Proposition\u00a06 in  we use Lemma\u00a0{F\u223ct}t\u2208\u2208A1. But if {F\u223ct}t\u2208\u2208A1 and m,\u03b4 are constants from the definition of A1, then by Lemma\u00a0B\u2282A1. For the second step we take the family {F\u223ct}t\u2208. Proceeding as in the rest of the proof of Theorem\u00a0Ak the family {ft}t\u2208\u2282T of continuous maps varying with t.\u25a1The procedure we take is the same as in the proof of Theorem\u00a0{ft}t\u2208, see \u2192D which decomposes D into a continuously varying family of arcs {\u03b1}x\u2208\u2202D\u2282C, so that \u03b1(I) are pairwise disjoint except perhaps at the endpoints \u03b1, where one requires that \u03b1\u2208I. We can then associate a retractionr:D\u2192I defined by r)=\u03b1 for every x\u2208\u2202D corresponding to the given decomposition. The map is boundary retract, but we need to maintain the disc, so we will collapse only the \u201cinner half\u201d of it (see definition of R below). Recall also, that a continuous map between two compact metric spaces is called a near\u2010homeomorphism, if it is a uniform limit of\u00a0homeomorphisms.Now we will briefly describe standard parametrized BBM construction for the family ,1], see  for moresmashR:D\u2192D as a near\u2010homeomorphism so that:Having the above decomposition in arcs we define unwrapping of {ft}t\u2208\u2282T as a continuously varying family f\u00aft:D\u2192D of orientation\u2010preserving homeomorphisms so that for all t:(i)suppf\u00aft\u2282{\u03b1;s\u2a7e1/2},(ii)R\u2218f\u00aft|I=ft, For the purpose of easier discussion afterward let us fix the unwrappings to be the \u201crotated graphs\u201d of the corresponding functions, following . Moreover, it follows from the choice of unwrapping that every point from the interior of D^t is attracted to I^t, therefore, I^t is a global attractor for H^t and thus \u039bt is a global attractor for \u03a6t as well. By Theorem 3.1 from  vary continuously with t and the attractors {\u039bt}t\u2208 vary continuously in Hausdorff\u00a0metric.As a consequence of (i) we obtain that for all  theorem , D^t:=liows from  that \u03a6t|3.1 from  {\u03a6t}t\u2208 is a family of orientation\u2010preserving homeomorphisms on a topological disk D\u2282R2 continuously varying with t. For every t\u2208 let Kt\u2282intD be a non\u2010degenerate sphere non\u2010separating continuum, invariant under \u03a8t, and assume that {Kt}t\u2208 vary continuously with t in the Hausdorff metric. Then the prime ends rotation numbers vary continuously with t\u2208.To a non\u2010degenerate and non\u2010separating continuum by Barge\u00a0.Lemma 4Definition 4.11X and Y be metric spaces. Suppose that F:X\u2192X and G:Y\u2192Y are homeomorphisms and E:X\u2192Y is an embedding. If E\u2218F=G\u2218E we say that the embedding E is a dynamical embedding of  into . If E, respectively, E\u2032, are dynamical embeddings of  resp.  into , respectively, , and there is a homeomorphism H:Y\u2192Y\u2032 so that H(E(X))=E\u2032(X\u2032) which conjugates G|E(X) with G\u2032|E\u2032(X\u2032) we say that the embeddings E and E\u2032 are dynamically equivalent.Let Finally, let us define how we distinguish the embeddings from the dynamical perspective. In what follows we generalize the definition from  of equivRemark 4.12Y=Y\u2032=R2 and X,X\u2032 are pseudo\u2010arcs (in particular plane non\u2010separating continua). Thus, the dynamical equivalence from Definition\u00a0H conjugates G with G\u2032 on all R2.In our case Definition 4.13x\u2208K\u2282R2 is accessible if there exists an arc A\u2282R2 such that A\u2229K={x} and A\u2216{x}\u2282R2\u2216K.We say that a point We will also use the following definition.Now let us prove the main theorem of this\u00a0section.Proof of Theorem 1.5Items (a) and (b) follow directly from Theorem 3.1 of .\u039b0=h(I^0) has an accessible point h0) fixed under \u03a60. We choose a horizontal radial arc Q0\u2282D that has an endpoint in 0\u2208I\u2282D. Note that by the definition of H0 it holds that H0(Q0)=Q0 and H0|Q0 is a near\u2010homeomorphism. Thus, J0:=lim\u2190 is an arc by the result of Brown  looks like precisely , however we do not delve in that aspect of research in this\u00a0work.While the embeddings from Theorem\u00a05C\u03bb(I) is a complete space in C(I) with the supremum metric. However, the space C\u03bb(I) is not equicontinuous and thus by Arzel\u00e1\u2013Ascoli theorem C\u03bb(I) is not compact. Therefore, we cannot apply the parametrized BBM construction from [C\u03bb(I) but a priori only from the topological perspective). Thus this section\u00a0can be viewed as a generalization of the preceding section\u00a0with the additional measure\u2010theoretic\u00a0ingredients.Note that the set ion from  directlyion from  for some5.1X is a compact metric space and that f:X\u2192X is continuous and onto and recall that we denote by X^:=lim\u2190 and by \u03c0n:X^\u2192I the coordinate projections maps. Recall also that B(X) denotes the \u03c3\u2010algebra of Borel sets in X. First we will need the following standard result.Theorem 5.1 Suppose ) is a separable Borel space and that f:X\u2192X is onto and B(X)\u2010measurable. Let B(X^) be the smallest \u03c3\u2010algebra on X^ such that all the projection maps \u03c0i are measurable. If {\u03bcn}n\u2208N0 is a sequence of probability measures on B(X) such that \u03bcn(A)=\u03bcn+1(f\u22121(A)) for all A\u2208B(X), then there exists a unique probability measure \u03bc^ on B(X^) such that \u03bc^(\u03c0n\u22121(A))=\u03bcn(A) for all A\u2208B(X) and each n\u2208N0.In this subsection we give some measure\u2010theoretic results that are required later in the construction. Suppose M(X) denote the set of all invariant probability measures on the Borel\u03c3\u2010algebra B(X). For any \u03bc\u2208M(X) a continuous function f:X\u2192X induces a map f\u2217:M(X)\u2192M(X) given by\u2208lim\u2190(M(X),f\u2217) can be uniquely extended to a probability measure on X^, that is we have a function:Another result that we use is from . Let M Suppose f:X\u2192X is a continuous function on a compact metric space. Let B(X^) be the smallest \u03c3\u2010algebra such that all the projection maps \u03c0i are measurable and let \u03bc^=G). Then \u03bc^ is f^\u2010invariant and \u03c3\u2010invariant.Theorem\u00a06 in  shows thDefinition 5.3\u03bc be an f\u2010invariant measure on X. Set B\u03bc is a basin of\u03bc for f if for all g\u2208C(X) and x\u2208B\u03bc:\u03bcphysical forf if there exists a basin B\u03bc of \u03bc for f and a measurable set B so that B\u2282B\u03bc and \u03bb(B)>0.Let \u03bd^ for the natural extension f^:X^\u2192X^ is called inverse limit physical measure if \u03bd^ has a basin B^\u03bd^ so that \u03bb(\u03c00(B^\u03bd^))>0.An invariant measure Theorem 5.4(Theorems 11 and 12 from ) If \u03bc is a physical measure for f:X\u2192X where X is an Euclidean space, then the induced measure \u03bc^ on X^ is an inverse limit physical measure for the natural extension f^. In particular, there is a basin B^\u03bc^:=\u03c00\u22121(B) of \u03bc^ for f^ with \u03bb(B)>0.M(I) be the space of Borel probability measures on I equipped with the Prokhorov metricD defined by\u03bc,\u03bd\u2208M(I). The following (asymmetric) formulaD coincides with the weak\u2217topology for measures, in particular M(I) equipped with the metric D is a compact metric space (for more details on Prokhorov metric and weak* topology the reader is referred to [Let erred to ).5.2M is called Oxtoby\u2013Ulam (OU) or good if it is non\u2010atomic, positive on open sets, and assigns zero measure to the boundary of manifold M (if it exists) [\u03bb^f on the inverse limit lim\u2190 using Lebesgue measure \u03bb on D, where the map f is a near\u2010homeomorphism of D and identity on \u2202D. Then we will find a homeomorphism \u0398f:lim\u2190\u2192D and define a push\u2010forward measure \u03bbf=(\u0398f)\u2217\u03bb^f. By this construction it is clear that \u03bbf is an\u00a0OU\u2010measure.In what follows, we will adjust Oxtoby\u2013Ulam technique of full Lebesgue measure transformation  to the c exists) , 46. In \u0398f, we need the following definitions.Definition 5.5lim\u2190 be an inverse limit where {Xi}i\u2a7e0 are continua and {fi:Xi+1\u2192Xi}i\u2a7e0 a collection of continuous maps. A sequence (ai)i\u2a7e0 of positive real numbers is a Lebesgue sequence for lim\u2190 if there is a sequence (bi)i\u2a7e0 of positive real numbers such that:(1)\u2211i=0\u221ebi<\u221e,(2)xi,yi\u2208Xj and any i<j, if |xi\u2212yi|<aj, then |fi+1\u2218\u2026\u2218fj(xi)\u2212fi+1\u2218\u2026\u2218fj(yi)|<bj.for any  A sequence (ci)i\u2a7e0 of positive real numbers is a measure sequence for lim\u2190 if:(1)\u2211i=n+1\u221eci<cn/2 for any n\u2a7e0,(2)x^\u2260y^\u2208lim\u2190 there exists a non\u2010negative integer N so that |xN+1\u2212yN+1|>cN.for any two points Let To provide a parametrized version of Brown's theorem and in particular to construct a continuously varying family of homeomorphisms Now we are ready to prove the main theorem of this\u00a0section.Proof of Theorem 1.6C\u03bb(I) be the family of Lebesgue measure\u2010preserving maps and let Q=\u2229n=0\u221eQn\u2282C\u03bb(I) be the intersection of open dense sets Qn satisfying Theorem\u00a0\u03bb), and Theorem\u00a03 from [{fi}i=0\u221e\u2282Q that are dense in Q. By assumptions we know that each of these maps is leo, Lebesgue measure is ergodic measure for each fi (it is even weakly mixing), has the shadowing property and by Theorem\u00a0\u03b4>0 there exists N\u2208N so that fiN is \u03b4\u2010crooked. Let S\u2282I be a set of full Lebesgue measure such that any x\u2208S is generic point of all fi.Let m\u00a03 from   and \u03c1<\u03bein/4<bn/8 where the sequence \u03bein will be specified later (we will need its faster convergence). Note that \u03c1<\u03bein/4+\u03bejn+1/4<\u03bein/2 provided that \u03bein>\u03bejn+1. All the lower indices will be specified\u00a0later.We also fix a sequence n there exist indices {jin}i\u2208N and \u03b4in<2\u2212n such that if we denote Ain:=B then An:=\u222aiAin is dense (and open by the definition) in Q, {fi}i=0\u221e\u2282An and Ain\u2229Ajn=\u2205 for i\u2260j. Simply, for any i the set {\u03c1:j\u2260i} is countable, and so we can choose \u03b4in outside this set, making construction of consecutive Ain possible by induction . We can also make each \u03b4in arbitrarily small, in particular for m>n and any i,j we may require that if Aim\u2229Ajn\u2260\u2205, then Aim\u2282Ajn. We can also require that each Ain\u2282Qn, since each Qn is open and contains all functions fi.Note that for each \u03b4in, because we will need them to be sufficiently small as will be specified\u00a0later.In our construction we will implement additional requirements on values of D\u2282R2, let I\u2282int(D) be the unit interval on which the BBM construction will take place, let I\u2282int(D1)\u2282D1\u2282int(D2)\u2282D2\u2282int(D) where D1 and D2 are two closed discs and let R:D\u2192D be a near\u2010homeomorphism, such that R(D2)=I, R|D\u2216D2 is one\u2010to\u2010one and R is identity on the boundary of D. We also require that the smash R is done along radial lines. It is not hard to provide an analytic formula defining R. These maps and discs are fixed throughout the whole\u00a0construction.For a closed disk f\u2208C\u03bb(I) we construct an unwrapping f\u00af:D\u2192D in the following way:(1)f\u00af(I)\u2282intD1 and as usually in BBMs f\u00af|I is a rotated graph of f,(2)f\u00af is identity on D\u2216D1,(3)R\u2218f\u00af|I=f and every point in intD is attracted to I under iteration of R\u2218f\u00af where I is identified with I in a standard way. We also denote f\u223c=f\u00af|I. One of the main features of the construction will be to ensure that unwrappings within the family that we construct vary continuously with f.Now let us briefly recall how BBM construction is performed in general, for more details see Section\u00a0U\u2282D2\u2216D1 of positive Lebesgue measure in D is transformed onto set R\u2218f\u00af(U)\u2282I of positive Lebesgue measure on I. It is a consequence of Fubini's theorem, since the smash R is performed along radial lines, and so the base of integration needs to have positive Lebesgue\u00a0measure.One important property to notice here is that any set {Ci}i=1\u221e\u2282intD1 be a collection of Cantor sets such that Ci\u2229I=\u2205, Ci\u2229Cj=\u2205 and \u03bb(D1)=\u2211i=1\u221e\u03bb(Ci). In other words, these Cantor sets fill densely intD1\u2216I and carry full Lebesgue measure. Such family of Cantor sets can be chosen using standard arguments. We may require that \u222aiR(Ci)\u2282S, because the union of radial lines over S has the full Lebesgue\u00a0measure.Let \u03b4i0. When fji0 is fixed, we also have f\u00afji0 and so the images f\u00afji0(Ck) are explicitly determined as well. Therefore, we may require thatNow it is a good moment to set the first restriction on ai0>0 be such that if d<ai0 for x,y\u2208D then d(hi0(x),hi0(y))<b0/2. We also require that 16\u03b4i0<b0. This implies that if map H:D\u2192D satisfies \u03c1<4\u03b4i0 and d<ai0 thenLet k,l such that Ak1\u2282Al0. Then fjk1\u2208B. Construct a homeomorphism G:D\u2192D such that G(x)=x for all x\u2209B(hl0(I),2\u03b4l0) and for x\u2208I we have\u03c1<2\u03b4l0, because we move the graph along horizontal lines. Similarly, we construct a map H:D\u2192D such that H(x)=x for x\u2209B(f\u00afjl0(C1),\u03b4l0),x\u2208B(f\u00afjl0(C1),\u03b4l0).Now let us explain how the first step of the construction will be made, the reader is also referred to Figure\u00a0\u22c3i=1\u221eCi and sufficiently well approximate small portions covering Cantor set H(f\u00afjl0(C1)), then define maps between these small portions and selected Cantor sets, and then extend the map to a homeomorphism on sufficiently small neighborhoods where these translations of small portions take place. This is possible, because our Cantor sets are in the plane, so are tamely embedded\u03c1<4\u03b4l0, hk1|I=f\u223cjk1 and hk1(C1)\u2282\u22c3i=1\u221eCi. Similarly as above, we decrease \u03b4k1 so thathk1(I) and Cantor set hk1(C2) so we need the neighborhoods of sets disjoint \u03c1<2\u03c1<\u03beln/2,(2)hkn+1|I=f\u223cjkn+1,(3)hln|Cr=hkn+1|Cr for r=1,\u2026,n,(4)hkn+1(Cr)\u2282\u222ai=1\u221eCi for r=1,\u2026,n+1,(5)\u03b4kn+1<\u03b4ln/2 and \u03bekn+1<\u03beln/2.This construction can be extended by induction in the following way. If \u03b4in and \u03bein so that we are able to repeat arguments from [Now we will perform an additional adjustment of the constants nts from . That isAjnn is already constructed for some index jn\u2208N and Aj11\u2283Aj22\u2283\u2026\u2283Ajnn. Let hjii be a homeomorphism corresponding to Ajii. There exists a positive real number ajnn such that if d<ajnn then\u03b4jn+1n+1 for Ajn+1n+1\u2282Ajnn is adjusted with the correspondence to the condition \u03bejn+1n+1<ajnn/8 . This will ensure that if uniform limit Rjnn\u2218hjnn\u2192F exists, then(ajnn)n=1\u221e is a Lebesgue sequence for {Rjnn\u2218hjnn}n=1\u221e and (bn)n=1\u221e.Assume that the set T\u2208C is given a priori and F was obtained as its perturbation. Fix any i,n>0 and let \u03b3>0 be such that \u03c1<\u03b3. Then for any x\u2208D we haveT is fixed, we have \u03c1<1/n for each i=0,1,\u2026,n, provided that \u03b3 was sufficiently small . Therefore, taking \u03b4jnn, \u03bejnn sufficiently small, we can require that if uniform limit Rjnn\u2218hjnn\u2192F exists, theni=0,1,\u2026,n. For each jn\u2208N we pick a real number cjnn>0 in such a way that cjnn<18ckn\u22121 where Akn\u22121\u2283Ajnn and additionally, if d<cjnn for some x,y\u2208D, thenx^,y^\u2208lim\u2190 then there is M\u2208N0 such that d>\u03b3 for some \u03b3>0. Take n>M such that 1/n<\u03b3. Thend((Rjnn\u2218hjnn)n\u2212M(xn+1),(Rjnn\u2218hjnn)n\u2212M(yn+1))<1/4n then we have a contradiction with the choice of \u03b3, therefore d>cjnn, meaning that {cjnn}n=1\u221e is a measure sequence for F. We additionally require that for each n\u2208N0 we haveAssume that a map g\u2208A=\u2229n=1\u221eAn. Then there are indices i=in such that g\u2208Ain=B where jn:=jinn. Consider the associated sequence of homeomorphisms hjnn:D\u2192D. For any n<m we havehjnn form a Cauchy sequence in C. Thus there exists a map Fg obtained as the uniform limit of the maps hjnn. But then Fg|I=g\u223c as g\u223c is a uniform limit of maps f\u223cjn=hjnn|I. Furthermore hjnn|Cr=hjrr|Cr for all n\u2a7er and therefore Fg(Cr)\u2282\u222ai=1\u221eCi.Now we will turn our attention to the implications of the construction. Assume that the above inductive construction has been performed and fix any F:D\u00d7A\u2192D\u00d7A byf,g\u2208A and \u03b5>0 there is \u03b4>0 such that if \u03c1<\u03b4 then there are n and jn such that 2\u2212n+4<\u03b5 and additionally \u03c1<4\u03b4jnn<2\u2212n+2 and \u03c1<4\u03b4jnn<2\u2212n+2. Namely, for sufficiently small \u03b4 we have f,g\u2208B\u2282Ajnn. This shows that F is\u00a0continuous.Let us define a map F=(R(Fg(x)),g)=(g\u223c(x),g) for each x\u2208I. If we fix any set of positive Lebesgue measure U\u2282D2\u2216D1 then R\u2218Fg(U)=R(U) and R(U) has positive one\u2010dimensional Lebesgue measure on I. But then by Fubini's theorem there is a set Sg\u2282U such that \u03bb(Sg)>0 and R(Sg) is contained in the set of generic points of g, in particular any point x\u2208Sg under iteration of Fgrecovers the Lebesgue measure onI, that is, the measure 1n\u2211i=0n\u03b4(Fgi(x)) converges in weak* topology to the Lebesgue measure on I.Now we will deduce properties (c) and (d). By the definition it holds that Fi:=R\u2218Ffi and take any set U\u2282intD of positive Lebesgue measure. We can write U=\u222aj=0\u221eUj as a disjoint union of sets Uj such that j is the minimal index such that Fij(Uj)\u2282D2. Note that for any j>0 we have Fij(Uj)=Rj(Uj)\u2282D2\u2216I. In particular, if Y\u2282Fij(Uj) is such that \u03bb(Y)=0 then also \u03bb(R\u2212j(Y))=0, where the latter formula makes sense, because R\u22121 is well defined on D2\u2216I. But then Ffi(\u22c3j=0\u221eRj(U\u223cj))\u2282\u222ai=1\u221eCi for some sets U\u223cj\u2282Uj satisfying \u03bb(Uj)=\u03bb(U\u223cj) and thereforej. But then there is a set S\u223ci of full Lebesgue measure in D such that for each x\u2208S\u223ci there is N\u2208N0 such that FiN(x)\u2208S. This means that every point in S\u223ci is eventually transferred into a generic point of fi, which means that the orbit of x under Fi recovers the one\u2010dimensional Lebesgue measure on I. This shows that the Lebesgue measure on I is a physical measure for each F and it is unique physical measure for a dense set of functions g\u2208C\u03bb(I) (this dense set corresponds with the maps {fi}i=0\u221e from the start of the construction). In fact it is unique each time when generic points of g contain the set S and may have (but not necessarily has) other physical measures in remaining\u00a0cases.But now consider the special case of map D^:=lim\u2190. Now we are going to define a map \u0398:D^\u2192D\u00d7A byDenote g\u2208\u2229n=1\u221eAinn. We can write g as the second coordinate in lim\u2190, since it is a constant sequence of gs; thus we can also write \u0398g:=\u0398:D^g\u2192D, where D^g:=lim\u2190. Since we have satisfied its assumptions, by Theorem\u00a01 from [\u0398 is well defined. Furthermore, by Theorem\u00a02 from [\u0398 is a homeomorphism for each g\u2208A, because it is a composition of a homeomorphism with projection onto the first coordinate in the inverse limit defined by homeomorphisms lim\u2190.where m\u00a01 from  \u0398 is welm\u00a02 from , it hold\u2208D^ thenNote that if \u03b5>0 and n sufficiently large 1 denotes the natural projection to the first coordinate)g\u2208\u2229i=1\u221eAjil and every x^ such that \u2208D^. Thus as a consequence, for \u03b4 sufficiently small, all f,g\u2208\u2229i=1\u221eAjil and d<\u03b4 we have1),\u03981)<3\u03b5 which proves that \u0398 is\u00a0continuous.But by  we have+1so by  we obtaif we define a homeomorphism \u03a6f:=\u0398f\u2218F^\u2218\u0398f\u22121:D\u2192D. Abusing the notation, for the following inverse limit spaces we will identify f with the interval map f|I. Denote \u039bf:=\u03a6f(I^f), where I^f:=lim\u2190 and note that by Corollary\u00a0\u039bf is the pseudo\u2010arc for every f\u2208A. We can write \u03a6:=(\u0398\u00d7id)\u2218F^\u2218(\u0398\u00d7id)\u22121 and put \u03a6f=\u03a6 showing that the family \u03a6f is varying continuously. This also shows that the family {\u039bf}f\u2208A varies continuously in Hausdorff\u00a0metric.For each \u03bc^f be an invariant measure induced on the inverse limit I^f using Lebesgue measure \u03bb on I and define a push\u2010forward measure \u03bcf=(\u0398f)\u2217\u03bc^f. Formally, the measure \u03bc^f is defined on the space I^f\u2282lim\u2190, however we can also view it as a measure on the space D^. Let us show that measures \u03bc^f vary continuously in the weak* topology in D^. By the definition \u03bc^f(\u03c0n\u22121(B))=\u03bb(B\u2229I) for any Borel set B\u2282D and every n\u2208N0  such that \u03c0n(P^g)=P. Denote \u03bd:=1|P|\u2211q\u2208P\u03b4q and \u03bd^g:=1|P^g|\u2211q^\u2208P^g\u03b4q^. Fix any \u03b5>0 and let us assume that 2\u2212n<\u03b5/2. There is \u03b3>0 such that if \u03c1<\u03b3, x^,y^\u2208lim\u2190 and d(\u03c0n(x^),\u03c0n(y^))<2\u03b3 then d<\u03b5. We may also assume that for any two consecutive points p,q\u2208P with respect to the ordering in I we have d<\u03b3/2 and that \u03b3>0 is sufficiently small so that if x^\u2208lim\u2190, y^\u2208lim\u2190 satisfy \u03c0n(x^)=\u03c0n(y^), then d<\u03b5.By Theorem\u00a0ails see ). Take a\u03c1<\u03b3 we haveq^\u2208P^ satisfies q^\u2208\u03c0n\u22121(B(\u03c0n(B^),2\u03b3) then there is z^\u2208B^ such that d)<2\u03b3 and therefore d<\u03b5. This givesD<\u03b5  denotes the Prokhorov metric on M(I) defined in the end of Subsection\u00a0q^\u2208P^g there is p^\u2208P^f such that \u03c0n(q^)=\u03c0n(p^) and therefore D<\u03b5. This gives that D<3\u03b5 provided that \u03c1<\u03b3. Indeed, the function f\u21a6\u03bc^f is\u00a0continuous.Note that since \u03b1\u2208C, then by identifying \u0398 to the projection on the first coordinate \u03b1\u2218\u0398\u2208C and we have already proven that for any fi\u2192f from A,If f\u21a6\u03bcf where each \u03bcf is the push\u2010forward measure on D defined by \u03bcf:=\u0398\u2217\u03bc^f. It is clear from the definition that the support of \u03bcf is \u039bf\u2282D.\u25a1We therefore have thatProceedings of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not\u2010for\u2010profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of\u00a0mathematics.The"}
{"text": "L-fuzzy set by incorporating membership functions, nonmembership functions from a nonempty set X to any lattice L, and lattice homomorphism from L to the interval . In this article, lattice-valued intuitionistic fuzzy subgroup type-3 (LIFSG-3) is introduced. Lattice-valued intuitionistic fuzzy type-3 normal subgroups, cosets, and quotient groups are defined, and their group theocratic properties are compared with the concepts in classical group theory. LIFSG-3 homomorphism is established and examined in relation to group homomorphism. The research findings are supported by provided examples in each section.The lattice-valued intuitionistic fuzzy set was introduced by Gerstenkorn and Tepavcevi as a generalization of both the fuzzy set and the  U as described by Zadeh is based on the formulation of a function \u03bc from U to the closed interval . The function is called a membership function, whereas the images of elements of U under this function are called membership grades. For instance, let U be a collection of finite groups, p(x) be the total number of subgroups in x \u2208 U. If q(x) is the total number of normal subgroups computed by a student in x, then \u03bc(x)=q(x)/p(x) defines a fuzzy membership grade to the normal subgroups in x. But there is a chance that if the group order is large and the student is unable to compute all the normal subgroups, then \u03bc(x) will be greater than the one reported by the student. This leads us toward the concept of nonmembership grades first introduced by Atanassov  is replaced by a partially ordered set L. Atanassov ; and termed their finding lattice-valued intuitionistic fuzzy set of a second type (LIFS-2). Different properties, such as the decomposition theorem and synthesis, were established for these fuzzy sets. However, the choice of a linearization map makes LIFS-2 less capable of dealing with basic set operations. For instance, the union of two LIFS-2s need not be a LIFS-2. Thus, the map was replaced with lattice homomorphism \u03b1 : L\u27f6, and the refinement is called a lattice-valued intuitionistic fuzzy set type-3 and abbreviated as LIFS-3.Over the years, several other generalizations of fuzzy sets have been introduced depending upon various parameters of uncertainty, vagueness, and imprecision by employing membership, nonmembership, hesitancy, and indeterminacy grades. In all these generalizations, the grades are real numbers ranging between 0 and 1. The interval  inherits the natural partial order from the set of real numbers and constitutes a lattice. Partial ordering and fuzzy uncertainties are key features of real-life problems with infinite solutions or no solution at all. So it is quite obvious to think about the replacement of  by any suitable lattice. Goguen  introductanassov  presente\u02c7 cev i\u00b4  refined The term group was first used by \u00c9variste Galois in the 1830s for the set of roots of polynomial equations. However, the modern-day definition of the group was established in 1870. Since then, significant research has been carried out in this area, and now the group is one of the most important algebraic structures providing a basic structure for several mathematical branches including analysis, game theory, coding theory, and algebraic geometry. Groups have strong applications in different scientific fields, especially symmetric groups, which play a vital role in theocratical physics and quantum mechanics. In genetics, the four-codon basis constitutes a group isomorphic to the Klein four-group. Gene mutation can be identified by establishing group homomorphism on copies of the sixty-four codon system. The coset diagram depicting group action has a close link with the crystal structure in chemistry. After Zadeh's invention, many researchers attempted to use and replace the ordinary set with the fuzzy set in various theocratical and experimental areas.G is a fuzzy subgroup if and only if all the level sets are subgroups of G  and refinement is abbreviated as (LIFS-3).Gerstenkorn and Tepa \u02c7 cev i\u00b4  refined \ud835\udcae, L, \u03b6\u2111L, \u03be\u2111L, \u2118), with \ud835\udcae a nonempty set, L is a complete lattice with top and bottom elements T and B, respectively, \u03b6\u2111L : X\u27f6L and \u03be\u2111L : \ud835\udcae\u27f6L are membership and nonmemberships functions. The map \u2118 : L\u27f6 is a lattice homomorphism with \u2118(T)=1, \u2118(B)=0)+\u2118(\u03be\u2111L(s)) \u2264 1.Such that for every S={a, b, c, d, e, f} and the lattice L={B, r, s, t, \u03c4} w.r.t partial orderConsider \u2118 : L\u27f6, \u03b6\u2111L : S\u27f6L and \u03be\u2111L : \ud835\udcae\u27f6LDefine s \u2208 S, \u2118(\u03b6\u2111L(s))+\u2118(\u03be\u2111L(s)) \u2264 1 imply that  is a LIFS-3 of \ud835\udcae.Then, for every G, membership \u03b6IL\u2009 : \u2009G\u2009\u27f6\u2009L, nonmembership \u03beIL\u2009 : \u2009G\u2009\u27f6\u2009L, and a lattice homomorphism \u2118\u2009 : \u2009L\u2009\u27f6\u2009L. The combination of LIFS \u2212 3 and group will provide us a new refined algebraic structure which we can use effectively in real world problems.We will define lattice-valued intuitionistic fuzzy subgroups (LIFSG) by using a group G, lattice L with top and bottom elements T and B, respectively, and lattice homomorphism \u2118 : L\u27f6. A LIFS-3 of G, that is, IL\u2009=\u2009 formulates a LIFSG-3 of G provided that for every x, y \u2208 G,\u03b6IL(xy) \u2265 \u03b6IL(x)\u2227\u03b6IL(y)\u2009and\u2009\u03beIL(xy)\u2009 \u2264 \u2009\u03beIL(x)\u2228\u03beIL(y)\u03b6IL(x)=\u03b6IL(x\u22121)\u2009and\u2009\u03beIL(x)=\u03beIL(x\u22121)For a group L be a lattice  and G be a group. Let IL\u2009=\u2009 be a LIFSG-3 of G. For t1\u2009 \u2208 \u2009Im\u03b6IL and t2\u2009 \u2208 \u2009Im\u03beIL, the -cut set (or level set) is a subgroup of G, called -cut subgroup (or level subgroup).Let Recall the definition of cut sets in IFS where these sets are defined as follows:e be the identity element in G. Then, for any u \u2208 G\u2009e \u2208 uu\u22121Let \u03b6IL(e) is an upper bound for Im\u03b6IL.By the property of LIFSG, we have\u03beIL(e) \u2264 \u03beIL(u)\u2200 u \u2208 G, this implies that \u03beIL(e) is an lower bound for Im\u2009\u03beIL. We get that e \u2208 ILt1, t2) \u2264 t2, this implies that uv\u22121 \u2208 ILt1, t2) be a LIFS-3 such that for each t1, t2\u2009 \u2208 \u2009L, the -cut set ILt1, t2)( andCase 1: Let \u03b6IL(u)=\u21131 and \u03b6IL(v)=\u21132 and \u21133=\u21131\u2227\u21132. Then \u21133 \u2264 \u21131\u2009and\u2009\u21132 and uv \u2208 IL\u21133, t2)(. We get that\u2009 Suppose \u03beIL(v)=\u21135 and \u21136=\u21134\u2228\u21135. Then, \u21136 \u2265 \u21134\u2009and\u2009\u21135 and uv \u2208 ILt1, \u21136)(. Now using the same argument as above it is easy to show that\u2009 If (ii)u \u2208 ILt1, t2) \u2265 t3 \u2265 t1, \u2009\u03beIL(u) \u2264 t2 \u2264 t5, \u2009\u03beIL(v) \u2264 t4 \u2264 t5Now, (3)t1\u2009 \u2265 \u2009t3, \u2009t4\u2009 \u2264 \u2009t2. As L is a lattice so t1\u2009\u2227\u2009t3 exist. Suppose t6\u2009=\u2009t1\u2009\u2227\u2009t3 then t1, t3\u2009 \u2265 \u2009t6,Suppose for all \u03b6IL(v) \u2265 t3 \u2265 t6, \u2009\u03b6IL(u) \u2265 t1 \u2265 t6, \u2009\u03beIL(v) \u2264 t4 \u2264 t2Now G sustain the axioms of LIFSG-3. Hence, IL is a LIFSG-3 of G.On the basis of previous discussion we conclude that the elements of G and lattice L with T and B as the top and bottom element, the LIFS-3 IL= is a LIFSG-3 of G if and only ifFor every group G and lattice L with T and B as the top and bottom element, let H\u2009 \u2264 \u2009G. For t \u2208 L, define \u03b6Ht and \u03beHt from G to L asFor a group \u2118 : L\u27f6 be a lattice homomorphism. Then, \u2118(T)=1 and \u2118(B)=0. Suppose \u2118(t)=0.2, then G.Let \u2118 : L\u27f6 be a lattice homomorphism defined as \u2118(T)=1 and \u2118(B)=0 and \u2118(t)=0.2. For g\u2009 \u2208 \u2009G we have two cases:g\u2009 \u2208 \u2009H, \u03b6Ht(g)=t, \u2009\u03beHt(g)=tCase 1: Let g \u2208 G/H, \u03b6Ht(g)=B, \u2009\u03beHt(g)=T(2) Case 2: Let Let g1, g2\u2009 \u2208 \u2009G, we have three cases:g1, g2\u2009 \u2208 \u2009H, this implies that g1g2\u2009 \u2208 \u2009Hg1 \u2208 H, g2 \u2208 G/H, this implies that g1g2\u2009 \u2209 \u2009H(2)\u03b6Ht(g1)=t, \u03b6Ht(g2)=B, \u03b6Ht(g1g2)=B\u2009 Thus, \u03beHt(g1g2)=T=\u03beHt(g1)\u2228\u03beHt(g2).Similarly, (3)g1, g2 \u2208 G/H, then two cases arises:g1g2 \u2208 G/H(b)g1g2 \u2208 HNow, for From previous discussion we get thatG.s1 \u2208 Im\u2009\u03b6Ht, s2 \u2208 Im\u2009\u03beHtNow, if t, t)\u2212 cut subgroupThen, . Then, using this definition, we proved some useful results in this section.G, lattice L with top element T and bottom element B and lattice homomorphism \u2118 : L\u27f6. Let IL= be a LIFSG-3. Then, IL is called LIFNSG-3 G ifFor a group G, lattice L with top element T and bottom element B and lattice homomorphism \u2118 : L\u27f6. If IL= is a LIFNSG-3 of G, then for t1 \u2208 Im\u2009\u03b6IL and t2 \u2208 Im\u2009\u03beILFor a group IL is a LIFNSG-3 of G. Let y\u2009 \u2208 \u2009ILt1,t2 and x\u2009 \u2208 \u2009G. Then, \u03b6IL(y) \u2265 t1\u2009and\u2009\u03beIL(y) \u2264 t2 this implies thatSuppose xyx\u22121 \u2208 ILt1, t2) is a LIFNSG-3 of G if and only ifFor a group IL is a LIFNSG-3 of G. Then, \u03b6IL(xyx\u22121)=\u03b6IL(y)\u2200 x, y \u2208 G, implies that \u03b6IL(xy)=\u03b6IL(yx). Similarly we get that \u03beIL(xy)=\u03beIL(yx). Conversely, assume that \u03b6IL(xy)=\u03b6IL(yx)\u2009and\u2009\u03beIL(xy)=\u03beIL(yx)\u2200 x, \u2009y \u2208 G. Then, \u03b6IL((xy)x\u22121)=\u03b6IL(x\u22121(xy))=\u03b6IL(y) and \u03beIL((xy)x\u22121)=\u03beIL(x\u22121(xy))=\u03beIL(y).Suppose IL= be a LIFSG-3 of G. Then IL is LIFNSG-3 if and only if for all x, y\u2009 \u2208 \u2009G,Let IL is LIFNSG-3 of G. Then, \u03b6IL=\u03b6IL(xyx\u22121y\u22121) implies that \u03b6IL \u2265 \u03b6IL(xyx\u22121)\u2227\u03b6IL(y\u22121). We get that \u03b6IL \u2265 \u03b6IL(y). Similarly, \u03beIL \u2264 \u03beIL(y). Conversely, assume that \u03b6IL \u2265 \u03b6IL(y)\u2009and\u2009\u03beIL \u2264 \u03beIL(y). Then \u03b6IL(xyx\u22121)=\u03b6IL(xyx\u22121y\u22121y). We get that \u03b6IL(xyx\u22121) \u2265 \u03b6IL\u2227\u03b6IL(y).Suppose \u03b6IL(y)=\u03b6IL(x\u22121xyx\u22121x) implies \u03b6IL(y) \u2265 \u03b6IL(x)\u2227{\u03b6IL(xyx\u22121)\u2227\u03b6IL(x)}. If \u03b6IL(x)\u2227\u03b6IL(xyx\u22121)=\u03b6IL(x), then \u03b6IL(y) \u2265 \u03b6IL(x)\u2200 x, y \u2208 G. \u21d2\u2009\u03b6IL is constant. In this case,As \u03b6IL(x)\u2227\u03b6IL(xyx\u22121)=\u03b6IL(xyx\u22121)If \u03beIL(x)\u2228\u03beIL(xyx\u22121)=\u03beIL(x), then this implies that \u03beIL(y) \u2264 \u03beIL(x)\u2200 x, y \u2208 G. Then, \u03beIL is constant. In this case \u03beIL(y)=\u03beIL(xyx\u22121). If \u03beIL(x)\u2228\u03beIL(xyx\u22121)=\u03beIL(xyx\u22121)If \u03b6IL(xyx\u22121)=\u03b6IL(y). Similarly, from (\u03beIL(xyx\u22121)=\u03beIL(y). Thus, IL is LIFNSG-3 of G. Hence proved.From  and 38)38), we gly, from  and 39)\u03b6IL be a LIFSG-3 of G, such that all -cut subgroups of IL are normal in G. Then, IL is a LIFNSG-3.Let G is finite this implies that Im\u03b6IL and Im\u2009\u03beIL are finite sets. Suppose Im\u2009\u03b6IL={t1, t2,\u2026, tk} with t1 < t2,\u2026, <tk and Im\u2009\u03beIL={s1, s2,\u2026, sm} with s1 > s2,\u2026, >sm\u22121 > sm. Consider \ud835\udcb3={ILti, sj)-cut subgroup of IL. As each As G, it can be expressed as union of \ud835\udc9e={xyx\u22121 : x \u2208 G} and y \u2208 ILti, sj)=\u03b6IL(y) and \u03beIL(xyx\u22121)=\u03beIL(y) for all x\u2009 \u2208 \u2009G and y \u2208 ILti, sj)=\u03b6IL(y) and \u03beIL(xyx\u22121)=\u03beIL(y)\u2200 x, y \u2208 G. This implies that IL is a LIFNSG-3 of G.It is normal in In the first section, we introduced the fundamental concepts of the factor group and group homomorphism. In this section, we will discuss these two important features of classical group theory for LIFSG-3.G, lattice L with top and bottom element T and B. Consider the lattice homomorphism \u2118 : L\u27f6. Let IL= be a LIFSG-3 of G. For x\u2009 \u2208 \u2009G define two mapsFor a group Then,ILx= is a LIFS-3 of G. The LIFS-3This implies that G is called the lattice-valued intuitionistic fuzzy coset type-3 (LIFC-3) of G induced by x and IL.where IL is a LIFNSG-3 of G, then for any x\u2009 \u2208 \u2009G,If IL is a LIFNSG-3 of G. Then, for x\u2009in\u2009GSuppose \u03beIL. Hence proved.Similarly, we proved for IL= be a LIFNSG-3 of G. Let G/IL={ILx : x \u2208 G} be the collection of all LIFC-3 of G induced by x\u2009 \u2208 \u2009G and IL. Then, G/IL is a group under the binary operation ILxoILy= and G/IL, whereLet ILx1, ILx2, ILy1, ILy2 \u2208 G/IL such that ILx1=ILy1\u2009and\u2009ILx2=ILy2Let \u03b6ILx1x2(g)=\u03b6IL(g(x1x2)\u22121) implies that \u03b6ILx1x2(g) \u2265 \u03b6IL(gy2\u22121y1\u22121)\u2227\u03b6IL(y1y2x2\u22121x1\u22121). As \u03b6ILx1\u2009=\u2009\u03b6ILy1 and \u03b6ILx2\u2009=\u2009\u03b6ILy2Consider g\u2009=\u2009y1y2x2\u22121, then \u03b6IL(y1y2x2\u22121x1\u22121)=\u03b6IL(y2x2\u22121), because IL is a lattice-valued intuitionistic fuzzy normal subgroup type-3. As \u03b6IL(gx2\u22121)=\u03b6IL(gy2\u22121), implies that \u03b6IL(y1y2x2\u22121x1\u22121)=\u03b6IL(e). Thus, we get thatIf Similarly, we get\u2118(\u03b6ILx1x2(g))+\u2118(\u03beILx1x2(g))=\u2118(\u03b6IL(g(x1x2)\u22121))+\u2118(\u03beIL(g(x1x2)\u22121))Now we have, \u03b6ILe(g)=\u03b6IL(ge\u22121)=\u03b6IL(g)\u2009and\u2009\u03beILe(g)=\u03beIL(ge\u22121)=\u03beIL(g)\u2200 g \u2208 G, this implies that ILe\u2009=\u2009IL and for any ILx \u2208 G/IL.Thus, the binary operation is well defined. The associativity of composition of functions implies that the given binary operation is associative. ConsiderILx\u2009o\u2009ILe\u2009=\u2009ILx this implies that ILe is the identity element in G/IL. Let ILx \u2208 G/IL, where x\u2009 \u2208 \u2009G. As G is a group so x\u22121\u2009 \u2208 \u2009G, implies that ILx\u22121 \u2208 G/IL andG/IL is a group.Similarly, G/IL. Hence proved.Consider IL be a LIFNSG-3 of G. Then, G denoted by LIFQSG \u2212 3.Let G be a group and IL be a LIFNSG-3 of G. The map \u03c6 : G\u27f6G/IL defined by \u03c6(x)=ILx\u2200 xnG is epimorphism withLet x, y\u2009 \u2208 \u2009G, then \u03c6(xy)=ILxy=ILxoILy=\u03c6(x)o\u03c6(y) this implies that \u03c6 is a group homomorphism. Clearly, \u03c6 is onto,Let st isomorphism theorem we get that G/G\u03b6IL\u03beIL\u2245G/IL. Hence proved.From 1G be a group and IL= be a lattice-valued intuitionistic fuzzy subgroup type-3 of G. Suppose Let is a LIFSG-3 of Let Similarly, \u03d1o\u03beIL)(a\u22121)=\u03d1o\u03beIL(a). Now,JL is a LIFSG-3 of Similarly,  be a LIFNSG-3 of G. If Let IL is a LIFNSG-3 of G. Let Suppose \u03d1o\u03beIL(uvu\u22121)=\u03beIL(\u03d1\u22121(v))=\u03d1o\u03beIL(v)\u21d2JL is a LIFNSG-3 of Similarly, G from the LIFS-3 of In the following proposition we will induce LIFSG-3 of IL= is a LIFSG-3 of G. Where \u2200\u2009g \u2208 GLet g1, g2\u2009 \u2208 \u2009G. Then,Let \u03beJLo\u2009\u03c6(g1g2) \u2264 \u03beJLo\u2009\u03c6(g2)\u2228\u03beJLo\u2009\u03c6(g2). Now,Similarly, \u03beJLo\u2009\u03c6(g\u22121)=\u03beJLo\u2009\u03c6(g).Similarly, G, L, \u03b6JLo\u03c6, \u03beJLo\u03c6, \u2118) is a LIFSG-3 of G. Hence proved.\u21d2=\u03beJLo\u03c6(v). \u21d2\u2009IL is a LIFNSG-3 of G. Hence proved.Similarly, G and L be a lattice with top element T and bottom element B and \u2118 : L\u27f6 be a lattice homomorphism. Let IGLIFSG\u22123 and G and Let Defined byis bijective.From previous propositions we know that ifThen,Now, if\u03d1 is bijectiveAs Similarly,\u03d1 is bijective, this implies \u03beIL1\u2009=\u2009\u03beIL2. Thus, we get that, ifAs then G, L, \u03b6JLo\u03d1, \u03beJLo\u03d1, \u2118) \u2208 IGLIFSG\u22123 and then (\u03d1o(\u03beJLo\u2009\u03d1)=\u03beJL. This implies that Similarly, G be a group, L and T and B and \u2118 : L\u27f6 be lattice homomorphisms such that \u2118(T)=1, \u2009\u2118(B)=0. If IL= be a LIFSG-3 of G, then G. WhereLet are defined asandg1, g2\u2009 \u2208 \u2009G. Then,Let fo\u03beIL(g1g2) \u2264 fo\u03beIL(g1)\u2228fo\u03beIL(g2). Let g \u2208 G andSimilarly, M1 and M2 are of the formThen, Thus,Hence, we get the required result.G and L and T and B and \u2118 : L\u27f6 be lattice homomorphisms such that \u2118(T)=1, \u2118(B)=0. Let IL= be a LIFSG-3 of G. Then,Let is a LIFSG-3 of IL= is a LIFSG-3 of G, then from Theorem 4If G. From Proposition 9is a LIFSG-3 of is a LIFSG-3 of G, lattice L with top and bottom elements T and B, respectively, lattice homomorphism \u2118 : L\u27f6. Let IL= be a LIFNSG-3 of G. Then, the setFor a group G andG/G\u03b6IL\u03beIL.is normal in N\u2009\u22b2\u2009G and ILN= is a LIFNSG-3 of G/N, then INL= is a LIFNSG-3 of G, whereConversely, if are defined asIL be a LIFNSG-3 of G. Then, \u03b6IL(x) \u2264 \u03b6IL(e) and \u03beIL(x) \u2265 \u03beIL(e)\u2200 x \u2208 G, implies that G\u03b6IL\u03beIL is a cut subgroup so it is normal in G and allow to construct factor group G/G\u03b6IL\u03beIL. Now, for the map Let Similarly, for the map AlsoG/G\u03b6IL\u03beIL. Conversely, supposeG/N. Define \u03b6INL : G\u27f6L asbe a LIFNSG-3 of Then,\u03beINL : G\u27f6L asDefine Then,INL is a LIFNSG-3 of G.\u21d2G are exactly the subgroups of G, and conversely, any LIFS-3 of G whose level sets are subgroups of G is LIFSG-3. Lattice-valued intuitionistic fuzzy normal subgroups type-3 and lattice-valued intuitionistic fuzzy factor subgroups type-3 of G are governed by normal subgroups and factor groups of G. Structure preserving LIFSG-3 maps are also discussed, and it is observed that they can be derived by extending group homomorphism. In the future, the notion foundations laid in this article can be used to find the Abelian subgroups of finite p-groups [L. The research findings can be utilized for application in algebra [The article is about the study of group theocratic concepts in a lattice-valued intuitionistic fuzzy type-3 environment. The structure is established by introducing lattice homomorphism, membership and nonmembership grades obeying certain laws for the binary operation defined on the group and inverses of elements under that operation. It is concluded that the level sets of LIFSG-3 of a group p-groups , verify p-groups , 36, comp-groups , aggregap-groups , and funp-groups  for lattp-groups \u201342 where algebra , 44 and  algebra  to handl"}
{"text": "We study the boundary continuity of solutions to fully nonlinear elliptic equations. We first define a capacity for operators in non-divergence form and derive several capacitary estimates. Secondly, we formulate the Wiener criterion, which characterizes a regular boundary point via potential theory. Our approach utilizes the asymptotic behavior of homogeneous solutions, together with Harnack inequality and the comparison principle. For the existence of a solution u (in a suitable sense) to the Dirichlet problemu = f on \u2202\u2126 is satisfied by the upper Perron solution, in general. Instead, we are forced to discover an additional condition for the boundary \u2202\u2126, which enables us to capture the boundary behavior of Let \u2126 be an open and bounded subset in x0 \u2208 \u2202\u2126 is regular with respect to \u2126, iff \u2208 C(\u2202\u2126). One simple characterization of a regular boundary point is to find a barrier function; see \u2202\u2126 such as an exterior sphere condition or an exterior cone condition have been invoked to guarantee the boundary continuity at x0 \u2208 \u2202\u2126 for a variety of elliptic operators.To be precise, we say a boundary point x0 \u2208 \u2202\u2126 is regular if and only if the Wiener integral diverges, i.e.2 is defined by the variational capacity of the Laplacian operator. Surprisingly, the Wiener criterion becomes both a sufficient and necessary condition for the regularity of a boundary point. Here the notion of capacity is used to measure the \u2018size\u2019 of sets in view of given differential equations. Roughly speaking, x0 \u2208 \u2202\u2126 is regular if and only if \u2126c is \u2018thick\u2019 enough at x0 in the potential theoretic sense.On the other hand, Wiener  developeDj(aijDi), where aij is bounded and measurable, and for the Laplacian operator. For the p-Laplacian operator , Maz\u2019ya  function u in \u2126 is a (viscosity) F-supersolution [resp. (viscosity) F-subsolution] in \u2126, when the following condition holds:x0 \u2208 \u2126, \u03c6 \u2208 C2(\u2126) and u \u2212 \u03c6 has a local minimum at x0, thenu \u2212 \u03c6 has a local maximum at x0, then F(D2\u03c6(x0)) \u2265 0.]if u \u2208 C(\u2126) a (viscosity) F-solution if u is both an F-subsolution and an F-supersolution.We say that Lemma 2.2. Suppose that a lower semi-continuous function u is an F-supersolution in \u2126. ThenProof. We argue by contradiction: suppose that\u03c6 \u2208 C2(\u2126), it follows that u \u2212 \u03c6 has a local minimum at x0 and so we can test this function. Therefore,Then for any Theorem 2.3.(Stability) Let {uk}k\u22651 \u2282 C(\u2126) be a sequence of F-solutions in \u2126. Assume that ukconverges uniformly in every compact set of \u2126 to u. Then u is an F-solution in \u2126.(Compactness) Suppose that {uk}k\u22651 \u2282 C(\u2126) is a locally uniformly bounded sequence of F-solutions in \u2126. Then it has a subsequence that converges locally uniformly in \u2126 to an F-solution.Theorem 2.4 (Harnack convergence theorem). Let {uk}k\u22651 \u2282 C(\u2126) be an increasing sequence of F-solutions in \u2126. Then the function u = limk\u2192\u221eukis either an F-solution or identically +\u221e in \u2126.Proof. If u(x) < \u221e for some x \u2208 \u2126, it follows from Harnack inequality that u is locally bounded in \u2126. The interior C\u03b1-estimate yields that the sequence uk is equicontinuous in every compact subset of \u2126. Thus, applying Arzela-Ascoli theorem and We demonstrate two essential tools for Perron\u2019s method, namely, the comparison principle and the solvability of the Dirichlet problem in a ball.Theorem 2.5 . Lef, there exists an increasing sequence of continuous functions {fn} such that fn \u2192 f pointwise as n \u2192 \u221e.In the previous theorem, Theorem 2.6 (The solvability of the Dirichlet problem). Let \u2126 satisfy a uniform exterior cone condition and f \u2208 C(\u2202\u2126). Then there exists a unique F-solutionof the Dirichlet problemProof. The existence depends on the construction of global barriers achieving given boundary data and the standard Perron\u2019s method; see  = \u2212\u2133[u]; for example, any linear operator L and p-Laplcian operators \u0394p possess this property. Then we havex0 \u2208 \u2202\u2126 is regular iff \u2208 C(\u2202\u2126). Nevertheless, for the general fully nonlinear operator F, we do not have this property. Therefore, it seems that we have to require both conditions simultaneously, when we define a regular point for F. To the best of our knowledge, it is unknown whether the two conditions in the definition are redundant. One possible approach to show that only one condition is essential is to prove that f is resolutive whenever f is continuous on \u2202\u2126; see Before we define a barrier function, which characterizes a regular boundary point, we shortly deal with the resolutivity of boundary data:Definition 2.17 (Resolutivity). We say that a bounded function f on \u2202\u2126 is (F-)resolutive if the upper and the lower Perron solutions f is resolutive, we write Lemma 2.18. Let \u2126 be a bounded open set of, let f and g be bounded functions on \u2202\u2126, and let c be any real number.If f = c on \u2202\u2126, then f is resolutive and Hf = c in \u2126.andIf f is resolutive, then f + c is resolutive and Hf+c = Hf + c.If c > 0, then cf = cHffor c \u2265 0.If f \u2264 g, thenandf does not implyx \u2208 \u2202\u2126. However, the converse is true in some sense:Note that the resolutivity of Lemma 2.19. Let \u2126 be an open and bounded subset of \u2126. Suppose that there exists F-harmonic h in \u2126 such thatfor any x \u2208 \u2202\u2126. Then. In particular, f is resolutive.Proof. Since h \u2208 \ud835\udcb0f \u2229 \u2112f, we have Lemma 2.20. If u is a bounded F-superharmonic (or F-subharmonic) function on the bounded open set \u2126 such that f(x) = lim\u2126\u220by\u2192xu(y) exists for all x \u2208 \u2202\u2126, then f is a resolutive boundary function.Proof. Obviously, we have u \u2208 \ud835\udcb0f and so F-harmonic in \u2126 andf is resolutive. An analogous argument works for the F-subharmonic case. \u25a12.3.Definition 2.21 (Barrier). Let x0 \u2208 \u2202\u2126. A function upper barrier [resp. lower barrier] in \u2126 at the point x0 ifw+ [resp. w\u2212] is F-superharmonic [resp. F-subharmonic] in \u2126;y\u2192x\u2126\u220bw+(y) > 0 [resp. lim supy\u2192x\u2126\u220bw\u2212(y) < 0] for each x \u2208 \u2202\u2126 \\ {x0};lim infw+ is positive in \u2126 and a lower barrier w\u2212 is negative in \u2126. Moreover, under the condition (F2), cw+ is still an upper barrier for any constant c > 0 and an upper barrier w+. See also  byWe define the Lemma 3.1. Suppose that \u2131 is a family of F-superharmonic functions in \u2126, locally uniformly bounded below. Then the lower semi-continuous regularization s of inf \u2131,is F-superharmonic in \u2126.Proof. Since \u2131 is locally uniformly bounded below, s is lower semi-continuous. Fix an open D \u2282\u2282 \u2126 and let F-harmonic function satisfying h \u2264 s on \u2202D. Then h \u2264 u in D whenever u \u2208 \u2131. It follows from the continuity of h that h \u2264 s in D. \u25a1Definition 3.2 .\u03c8 : \u2126 \u2192 capacity potential of E in \u2126.In particular, we call the function Remark 3.3. For an operator in divergence form, there exists an alternative method to define the capacity potential. For simplicity, suppose that the operator is given by the p-Laplacian. Let \u2126 be bounded and K \u2282 \u2126 be a compact set. For \u03c8 \u2261 1 on K, the p-harmonic function u in \u2126 \\ K with capacity potential of K in \u2126 and denoted by \u211b. Here note that \u211b is independent of the particular choice of \u03c8 and the existence of the capacity potential is guaranteed by the variational method. Indeed, both definitions of capacity potentials coincide; see  part in F-superharmonic function Br2 such that U of u across K, which is defined byU is the lower semi-continuous regularization of the function v, whereU = u in Br2 \\ K and so U is F-superharmonic in Br2 \\ K.Now we consider a canonical lower semi-continuous extension \u00a0K.See  for detaF-superharmonic in Br2, for any \u03b5 > 0 and z0 \u2208 Br2\\K. Indeed, the convexity of F immediately guarantees that F-superharmonic in Br2 \\ K. On the other hand, since K. In other words, for any \u03c6 \u2208 C2(\u2126), x0 \u2208 K. Thus, recalling the equivalence of F-supersolution and F-superharmonic function , consider the upper Perron solution f(x0) = 0 and max\u2202\u2126 |f| \u2264 1. For \u03b5 > 0, we can choose a ball B with center x0 such that \u2202(2B) \u2229 \u2126 \u2260 \u2205 and |f| < \u03b5 in 2B \u2229 \u2202\u2126. Then we defineB, F-harmonic in \u2126 \u2229 2B. On the other hand, by u is continuous in \u2126 and by the pasting lemma, u is F-superharmonic in \u2126. Moreover, it can be easily checked thatu \u2208 \ud835\udcb0f and so v \u2208 \u2112f and so,\u03b5 > 0 is arbitrary, we conclude thatx0 is regular. \u25a1Next, we exhibit a converse direction of the above lemma: i.e. a characterization of an irregular boundary point. We expect that this lemma may be employed to prove the necessity of the Wiener criterion for the general case.Lemma 3.22 (Characterization of an irregular boundary point). If there exists a constant \u03c1 > 0 such that the capacity potential u = u\u03c1ofwith respect to B2\u03c1(x0) satisfies the inequalitythen the boundary point x0 \u2208 \u2202\u2126 is irregular.Proof. Since the capacity potential u is the lower semi-continuous regularization, we haveu\u03c1\u2032 \u2264 u\u03c1 when 0 < \u03c1\u2032 < \u03c1. Thus, we can choose a sufficiently small \u03c1 > 0 such that (\u2202B\u03c12(x0) \u2260 \u2205.we haveu(x0)=lif on \u2202(\u2126 \u2229 B\u03c12(x0)) such that f(x) = 3/2 if x \u2208 \u2202\u2126 \u2229 B\u03c1/2(x0), 0 \u2264 f(x) \u2264 3/2 if x \u2208 \u2202\u2126 \u2229 (B\u03c1(x0) \\ B\u03c1/2(x0)) and f(x) = 0 on the remaining part of \u2202(\u2126 \u2229 B\u03c12(x0)). Then we consider the lower Perron solution B\u03c12(x0). For this purpose, let v \u2208 \u2112f(\u2126\u2229B\u03c12(x0)) and g is given by ) First, for x \u2208 \u2202\u2126 \u2229 B\u03c1(x0), we have(on x \u2208 \u2202\u2126 \u2229 (B\u03c12(x0) \\ B\u03c1(x0)), we haveNext, for \u2202B\u03c12(x0)) Similarly, we obtain. Recalling x0 is irregular with respect to \u2126. \u25a1Therefore, 4.In this section, we prove the sufficiency of the Wiener criterion and its sequential corollaries, via the potential estimates. More precisely, we first develop quantitative estimates for the capacity potential Definition 4.1. We say that a set E is F-thick at z if the Wiener integral diverges, i.e.Remark 4.2. Recalling c > 0 which is independent of t > 0 such that form of :\u222b01caparing in , 40.Now we can state an equivalent form of our main theorem, c is both F-thick and x0 \u2208 \u2202\u2126, then x0 is regular. To prove this statement, we need several auxiliary lemmas regarding the capacity potential.If \u2126Lemma 4.3. Fix a ball B. Suppose that K \u2282 B\u2032 is compact andIf 0 < \u03b3 < 1 and K\u03b3 \u2254 {x \u2208 B : v(x) \u2265 \u03b3} \u2282 B\u2032, thenProof. We write F-superharmonic in B and so is v/\u03b3 due to (F2). Since v \u2265 \u03b3 in K\u03b3, we have v/\u03b3 \u2265 1 in K\u03b3. Thus, Clearly, B\\K\u03b3 whereRecalling x \u2208 \u2202K\u03b3. Since u is F-superharmonic and v/\u03b3 is F-harmonic in B \\ K\u03b3, the comparison principle leads to u \u2265 v/\u03b3 in B \\ K\u03b3 and soThen for Consequently, we conclude that\u25a1Lemma 4.4. Fix a ball B = B2r(x0). Let K \u2282 Br = Br(x0) be a compact set andThen there exists a constant c > 0 which is independent of K and r such thatfor any x \u2208 Br.Proof. Denotev is a non-negative F-solution in B\\K, Harnack inequality yields that there exists a constant c1 > 0 independent of r > 0 such thatSince B \\ Br/56 implies thatMorevoer, the strong maximum principle in K and r.Here we applied KM \u2282 B\u2032, we can apply Now since Finally, combining , 4.3) a a4.3) anWe may rewrite the previous lemma asLemma 4.5. Let x0 \u2208 \u2202\u2126, \u03c1 > 0 andThen for all 0 < r \u2264 \u03c1, there exists a constant c > 0 such thatfor any x \u2208 Br(x0).Proof. Denote r \u2264 \u03c1 and let k be the integer with 2k\u2212\u03c1 < r \u2264 2k1\u2212\u03c1.i = 0, 1, 2, \u2026Then write for et \u2265 1 + t, estimate  is given byFurthermore, we claim that Thus, for u \u2265 ui+1 in Di+1 and so Bi+1.Therefore, by the comparison principle, v0 \u2265 u1 \u2265 \u22ef \u2265 uk in Bk, which implies thatRepeating the argument above, we conclude that t \u2264 s \u2264 2t, thenn, \u03bb, \u039b and these results also hold for Finally, the result follows fromNow we are ready to prove the sufficiency of the Wiener criterion, Proof of Theorem 1.2. Let x0 \u2208 \u2202\u2126, \u03c1 > 0 and definer \u2264 \u03c1, there exist a constant c1, c2 > 0 such thatx \u2208 Br(x0). Letting r \u2192 0+, we conclude thatThen applying \u03c1 > 0 can be arbitrarily chosen, an application of x0 \u2208 \u2202\u2126 is a regular boundary point.  at x0 \u2208 \u2202\u2126, then we can deduce the continuity of the Perron solution at x0 under a relaxed condition:On the other hand, if additional information is imposed on the boundary data Corollary 4.6. Suppose that f \u2208 C(\u2202\u2126) attains its maximum [resp. minimum] at x0 \u2208 \u2202\u2126. If \u2126cis F-thick [resp. thick] at x0 \u2208 \u2202 \u2126, thenProof. Similarly as in the proof of the previous theorem, this corollary is the consequence of f \u2208 C(\u2202\u2126) is resolutive, then we are able to obtain a quantitative estimate for the modulus of continuity.Furthermore, if the given boundary data Lemma 4.7 (The modulus of continuity). Suppose that \u2126 is an open and bounded subset ofLet f \u2208 C(\u2202\u2126).If x0 \u2208 \u2202\u2126 with f(x0) = 0, then for 0 < r \u2264 \u03c1, we haveandwhere \u2126r \u2254 \u2126 \u2229 Br(x0) and \u2202\u2126\u03c12 \u2254 \u2202\u2126 \u2229 B\u03c12(x0).Furthermore, if f is resolutive, then we have the quantitative estimate:whereProof. Let B\u03c12.w \u2254 1 \u2212 v and writeThen let f(x0) = 0, we have max\u2202\u2126f \u2265 0 and u is F-subharmonic and s is F-harmonic in \u2126\u03c12. Moreover,Note that since we assumed s \u2265 u in \u2126\u03c12 and so \u03c12.Thus, the comparison principle yields that On the other hand, let\u2202\u2126\u03c12.By the same argument, we derive w  such that x0 \u2208 \u2202\u2126, then \u2126 satisfies an exterior corkscrew condition at x0. Thus, the following corollary obtained from the  Wiener criterion is a generalized result of Note that if \u2126 satisfies an exterior cone condition at Corollary 4.9 (Exterior corkscrew condition). Suppose that \u2126 satisfies an exterior corkscrew condition at x0 \u2208 \u2202\u2126. Then x0is a regular boundary point. Moreover, if f is H\u00f6lder continuous at x0and is resolutive, then Hfis H\u00f6lder continuous at x0.Proof. A small modification of n, \u03bb, \u039b and \u03b4. Thus, if x0 satisfies an exterior corkscrew condition, then we havex0 is a regular boundary point by the Wiener criterion.f(x0) = 0 by adding a constant for f, if necessary. Since f is resolutive, we can apply the quantitative estimate obtained in Next, for the second statement, we may assume Heref is H\u00f6lder continuous at x0: there exists a constant C > 0 such that |f(x)| = |f(x) \u2212 f(x0)| \u2264 C|x \u2212 x0|\u03b3 \u2264 C\u03c1\u03b3 for x \u2208 \u2202\u2126\u03c12.x0:\u2126 satisfies an exterior corkscrew condition at \u03c1 = r/21, we conclude that the Perron solution Hf is H\u00f6lder continuous at x0. \u25a1Thus, choosing Remark 4.10 (Example). In this example, we suppose n = 2, \u03bb < \u039b. Then it immediately follows that\u2202\u2126.We consider a domain \u03b1*(F) < 0, we know that a single point has non-zero capacity. More precisely, recalling the homogeneous solution for F is given byc = c > 0 such thatSince Therefore, we havec is F-thick at 0.In other words, \u2126On the other hand, since c is not In other words, \u2126f1 \u2208 C(\u2202\u2126) is a boundarye data given byLet f1 is resolutive) andThen clearly the function f1 attains its maximum at 0 and \u2126c is F-thick at 0.Alternatively, one can apply f2 \u2208 C(\u2202\u2126) is a boundary data given byLet \u03b5 > 0, we have \u03b5 \u2192 0, we conclude Then since the zero function belongs to Therefore, we deduce that 5.F. Indeed, our strategy is to employ the argument made in [p-Wiener criterion for p-Laplacian operator with p > n \u2212 1. Since the assumption p > n \u2212 1 was essentially imposed to ensure the capacity of a line segment is non-zero in [In this section, we provide the necessity of the Wiener criterion, under additional structure on the operator  made in  which pr-zero in , we begiLemma 5.1. Suppose that F is convex and \u03b1*(F) > s for some s > 0. Let K be a compact subset in  \u210bs(K) < \u221e, where \u210bsis the s-dimensional Hausdorff measure. ThenProof. For any \u03b4 > 0, defineK by balls Bi with diameter ri not exceeding \u03b4. Then since K is compact, for each \u03b4 \u2208 , there exist finitely many open balls ri < \u03b4, F. Here we may assume minx|=1|V (x) = 1 by normalizing V. If we let Wi is non-negative and F-superharmonic in Wi(x) \u2265 1 on Bi.Now we consider the homogeneous solution F is convex, W is F-superharmonic in W \u2265 1 on W \u2265 1 on K. Therefore, \u03b1* > s. Letting \u03b4 \u2192 0, we finish the proof. \u25a1Finally, we let  we used  and \u03b1* >K is given by a line segment L, whose Hausdorff dimension is exactly 1.Now we prove the partial converse statement of Lemma 5.2. Suppose that F is concave and \u03b1*(F) < 1. Let L = {x0 +se : ar \u2264 s \u2264 br} be a line segment in Br(x0), where e is an unit vector in  0 < a < b < 1 are constants satisfyingThenProof. Note that since L is a line segment, for any \u03b4 > 0, one can cover L by open balls Bi = B\u03b43(xi), 1 \u2264 i \u2264 N(\u03b4) where xi \u2208 L, |xi \u2212 xj| \u2265 2\u03b4 whenever i \u2260 j, and \u03b5 > 0, there exist a sufficiently small \u03b4 > 0 and corresponding cover K\u03b4 such thatIf we denote V is given by\u03b1*(F) \u2208 . Note that if \u03b1* < 0, then a single point has a positive capacity , writeOn the other hand, for simplicity, we suppose that the homogeneous solution capacity  and the F is concave, W is F-subharmonic in Since y \u2208 \u2202Bi for some i. Then for j \u2260 i, we have(On \u03b1* < 1.Here we used the condition \u2202Br2) For z \u2208 \u2202Br2,. Suppose that F is concave and \u03b1*(F) < 1. Let K be a compact subset in Br(x0) such that K meets  \u2208 , where 0 < a < b < 1 are constants satisfyingThen there exists a constant c = c such thatProof. The proof is similar to the one of xt)( \u2208 K\u2229S(t) for all t \u2208 . In particular, for small \u03b4 > 0, we define xi \u2254 xar+2\u03b4i) so that\u03b4 > 0, we define a set K\u03b4 by\u03b5 > 0, there exists a sufficiently small \u03b4 > 0 such thatNote that V is given by\u03b1*(F) \u2208 . For each i = 1, 2, \u22ef , N(\u03b4), writeOn the other hand, for simplicity, we suppose that the homogeneous solution F is concave, W is F-subharmonic in Since \u2202K\u03b4) For y \u2208 \u2202K\u03b4, let y \u2208 \u2202Bi for some i. Then for j \u2260 i, we have(On \u03b1* < 1.Here we used the condition \u2202Br2) For z \u2208 \u2202Br2,, letLet Lemma 5.4. Suppose that F is concave and \u03b1*(F) < 1. Then, there exists a constant c1 > 0 depending only on n, \u03bb, \u039b such that: ifthen the set A\u03b3contains a sphere  \u2208 .Proof. For 0 < \u03b3 < 1, let E\u03b3 \u2254 {x \u2208 Br2 : u(x) \u2265 \u03b3}. We argue by contradiction: suppose that A\u03b3 does not contain any S(t) for t \u2208 . Then the set E\u03b3 meets S(t) for all t \u2208  and we havea = 1/10 and b = 1/5.On the other hand, by Combining two estimates above, we obtainTherefore, by choosing Now we are ready to prove the necessity of the Wiener criterion, Proof of Theorem 1.3. For simplicity, we write Br = Br(x0). Suppose that \u2126c is not F-thick at x0 \u2208 \u2202\u2126, i.e.\u03b5 > 0 to be determined, choose r1 > 0 small enough so thatFor ri+1 = ri/2 andSet Applying i, choose a regular domain Ei such that Next, by Lemma 2.27 and bi \u2264 (c0 +1)\u03b5 for i = 2, 3, \u22ef . Moreover, let \u03b3i = c1 \u00b7 bi, the setS(ti) for some ti \u2208 . Now by selecting \u03b3i < 1. In particular, since u2 = 1 on E2 and S(t2) \u2282 A2, we conclude that S(t2) \u2282 \u2126.Then we havef \u2208 C(\u2202\u2126) be the boundary function defined byNext, let Then we have the following results for the lower Perron solution r > 0 large enough so that \u2126 \u2282 Br. Moreover, set a domain f0 \u2208 C(\u2202\u21260) byBr is regular, we have v \u2208 \u2112f(\u2126) and v \u2264 w in \u2126 using the comparison principle.Then since Therefore, we conclude that F-harmonic in \u2126 and u \u2264 0 in S(t2), we claim thatFor E3 is a regular domain, we haveIndeed, since u \u2264 u3 in S(t3) \u2282 A3, we observe thatThus, the comparison principle yields that u \u2212 \u03b33 instead of u), we conclude thatIterating this argument (for example, consider leads to .u, the estimate (x0 \u2208 \u2202\u2126 is an irregular boundary point. \u25a1Finally, recalling the definition of estimate  is equiv"}
{"text": "Wireless sensor networks (WSNs) have been developed recently to support several applications, including environmental monitoring, traffic control, smart battlefield, home automation, etc. WSNs include numerous sensors that can be dispersed around a specific node to achieve the computing process. In WSNs, routing becomes a very significant task that should be managed prudently. The main purpose of a routing algorithm is to send data between sensor nodes (SNs) and base stations (BS) to accomplish communication. A good routing protocol should be adaptive and scalable to the variations in network topologies. Therefore, a scalable protocol has to execute well when the workload increases or the network grows larger. Many complexities in routing involve security, energy consumption, scalability, connectivity, node deployment, and coverage. This article introduces a wavelet mutation with Aquila optimization-based routing (WMAO-EAR) protocol for wireless communication. The presented WMAO-EAR technique aims to accomplish an energy-aware routing process in WSNs. To do this, the WMAO-EAR technique initially derives the WMAO algorithm for the integration of wavelet mutation with the Aquila optimization (AO) algorithm. A fitness function is derived using distinct constraints, such as delay, energy, distance, and security. By setting a mutation probability P, every individual next to the exploitation and exploration phase process has the probability of mutation using the wavelet mutation process. For demonstrating the enhanced performance of the WMAO-EAR technique, a comprehensive simulation analysis is made. The experimental outcomes establish the betterment of the WMAO-EAR method over other recent approaches. With the rapid advancement in Future Internet technologies, the mobile Internet, the Internet of Things (IoT), the physical world, and the sensor cloud are regularly getting more connected and moving faster toward the always-connected model . In thisIn WSNs, numerous types of research have involved routing strategy, corporeal design, procedure for sensing capability, power management, and security issues of SNs. The lifetime of SNs is the major problem for WSNs, since SNs have constrained energy resources. The routing algorithm has played a crucial role in the SN\u2019s lifetime . RoutingThis article introduces a wavelet mutation with Aquila optimization-based routing (WMAO-EAR) protocol for wireless communication. The presented technique aims to accomplish energy-aware routing in WSNs. To do this, it initially derives the WMAO algorithm for the integration of wavelet mutation with the Aquila optimization (AO) algorithm. A fitness function (FF) is derived using distinct constraints, such as delay, energy, distance, and security. Once the mutation probability P has been set, every individual next to the exploitation and exploration process has the probability of mutation using the wavelet mutation process. For establishing the enhanced performance of the WMAO-EAR technique, a comprehensive simulation analysis is made.Jagadeesh and Muthulakshmi  examinedAl-Otaibi et al.  developeSrikanth et al.  establisIn this study, a new WMAO-EAR algorithm is proposed for effectual and energy-aware wireless communication. The presented WMAO-EAR technique aims to accomplish energy-aware routing in WSNs.Approaches utilized to carry out the presented routing protocol were the network model, energy model, and energy-dissipation model . In the \u03b1 times more than that of the intermediate and normal node energy is \u03b2 times more than normal node energy. Now, the \u03b1 via WSNs involve link heterogeneity, energy heterogeneity, and computational heterogeneity. Energy heterogeneity is regarded as the most important to ensure optimal network performance. In the study, we discussed three levels of energy heterogeneity: advanced, normal, and intermediate nodes. Intermediate node initial energy is between advanced and normal node initial energy, given that m and b represent the percentage of advanced nodes and intermediate nodes correspondingly. Advanced node energy is d is formulated as follows.Reception and data transmission are two fundamental functions in WSNs. Usually, the data-communication method expends lots of energy when compared to data reception. Now, energy cost during In Equation (4), Furthermore, energy costs during d indicates a predetermined range Consider that the n sensor is positioned in the region of M size and the sensor is static. Every sensor is regarded as being aware of the identification and location of other SNs. Also, consider that the advanced node\u2019s location is predetermined, while the normal and intermediate nodes are placed randomly. All the nodes transmit information to neighboring CH for data aggregation. The distance between the BS and nodes is Abualigah et al.  introducThe next strategy used to perform the Levy flight Levy, and the quality function rand\u00a0rand shows a random number. 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Correspondingly, \u03c3 represents the wavelet mutating coefficient. Its formula is [\u03c6 signifies an arbitrary number within \u22122.5\u03b1 and 2.5\u03b1. The a is the scaling parameter and its expression is [s indicates a given constant. The mutant individual In the WMAO algorithm, the mutation was considered a significant technique to support the technique\u2019s jump from local optimum . In thisrmula is :(19)\u03c3=1\u03b1ssion is :(20)\u03b1=s\u22c5This procedure assures that an individual with superior fitness will be entering the next iteration, thereby enhancing the convergence speed and optimization capability of the algorithm.In the presented WMAO-EAR technique, an FF is derived using distinct constraints, such as delay, energy, distance, and security. The main concept of the proposed algorithm is to decrease the communication distance between the selected nodes and CH . It alsoEquation (25) illustrates the FF for distance, where Equation (28) illustrates the FF of energy. The value Equation (29) shows the FF of delay This section inspects the routing performance of the WMAO-EAR method on WSNs. The presented WMAO-EAR model is tested under two node counts (NC) of 100 and 300. A comparative NODN assessment of the WMAO-EAR model with other models is provided in Eventually, with 2500 rounds, the WMAO-EAR method attained a lower NODN of 2, whereas the CE-EC, SEED, NEH-CP, and HCEHUC approaches attained higher NODN of 18, 44, 97, and 158, respectively. These experimental values show the enhancements of the WMAO-EAR technique over other existing models.In this study, a new WMAO-EAR approach was devised for effective and energy-aware wireless communication. The presented WMAO-EAR technique aims to accomplish an energy-aware routing process in WSNs. To do this, the WMAO-EAR technique initially derives the WMAO algorithm for the integration of wavelet mutation with the AO algorithm. An FF is derived using distinct constraints, such as delay, energy, distance, and security. Once the mutation probability P is set, every individual next to the exploitation and exploration stage process has a probability of mutation using the wavelet mutation process. To demonstrate the enhanced performance of the WMAO-EAR technique, a comprehensive simulation analysis has been presented. The experimental results establish the superiority of the WMAO-EAR method over other recent approaches. In the future, lightweight cryptographic solutions can be applied to boost secure communication."}
{"text": "Here we demonstrate the piRNAi assay by quantifying acute and inherited silencing in the ribonucleotidyltransferase rde-3 (ne3370) mutant.In the absence ofrde-3,acute silencing was reduced but still detectable, whereas inherited silencing was abolished.We recently developed a piRNA-based silencing assay (piRNAi) to study small-RNA mediated epigenetic silencing: acute gene silencing is induced by synthetic piRNAs expressed from extra-chromosomal array and transgenerational inheritance can be quantified after array loss. The assay allows inheritance assays by injecting piRNAs directly into mutant animals and targeting endogenous genes ( C. elegans is a convenient model for studying small RNA-mediated inherited silencing due to the animal's short generation time (three days) and the ability to identify molecular pathways in genetic screens . Epigenetic silencing of an endogenous gene is often done by targeting a temperature-sensitive gain-of-function allele ofoma-1(zu405)with dsRNA, and silencing persists for up to three generations . For a visual read-out, single-copy transgenes with GFP expression in the germline  have been engineered to contain endogenous piRNA binding sites in the 3' UTR . For transgenes, piRNA-induced silencing persists longer and sometimes indefinitely. Genetic factors required for small RNA-mediated inherited silencing have primarily been identified by crossing silenced piRNAgfpsensor strains into mutant genetic backgrounds . However, introducing mutations by genetic crosses raises several concerns. First, there are several examples of mating causing changes in epigenetic inheritance. For example, the lack of transgene pairing during meiosis after a cross can lead to permanent transgene silencing via PRG-1-dependent mechanisms , and mating can induce multigenerational silencing inherited for over 300 generations . Moreover, Dodson and Kennedy (2019) characterized a transgenerational disconnect between the genotype and phenotype (sensitivity to exogenous RNAi) ofmeg-3/4mutants for more than seven generations after a genetic cross. Second, crosses frequently require molecular genotyping, which makes it cumbersome to perform many biological replicates. Third, there are some concerns about using transgenes as a proxy for endogenous gene silencing. For example, most piRNA sensor strains include synthetic piRNA binding sites in the 3' UTR , but endogenous genes are resistant to piRNA silencing when targeting their 3' UTRs . Moreover, transgene insertion site , non-coding DNA structures , coding sequence , and transgene structure  can influence epigenetic silencing. These observations suggest that transgenes may not fully recapitulate the balance between silencing foreign DNA and protecting endogenous gene expression . Finally, distinguishing between silencing initiation and maintenance phases is complicated using genetic crosses. Experiments require crossing mutant alleles to sensor strains, de-repress silencing, and outcrossing mutations to monitorde novoestablishment of silencing .et al., 2022; Gajicet al., 2022). Using piRNAi, we identified two endogenous targets,him-5 andhim-8, that inherit silencing for three and six generations, respectively .him-5 andhim-8 mutants are generally healthy but have a similar loss-of-function phenotype that is easy to score . We reasoned that piRNA-mediated silencing ofhim-5orhim-8might be useful as a tool to directly test the role of gene mutations in initiating and maintaining inherited silencing. Here, we show that piRNAi can be used to test acute and inherited silencing inrde-3, a gene also known asmut-2.We recently developed a method called piRNA interference (piRNAi) that can efficiently silence both transgenes and endogenous genes by expressing synthetic piRNAs from arrays generated by injection  and RNA interference (RNAi) .In vitro, RDE-3 has ribonucleotidyltransferase activity  and, in vivo, rde-3is required for the addition of non-templated poly (UG) tails to the 3\u2019 end of mRNAs targeted by RNAi and repressed transposons . pUGylated mRNAs are templates for RNA-dependent RNA polymerases (RdRPs), resulting in small RNA amplification and inherited silencing . RDE-3 is required to maintain the silencing of piRNA transgene sensors . However, the role ofrde-3in initiating silencing is unclear; re-introducing RDE-3 led to rapid re-silencing of agfp::cdk-1transgene, but variable and incomplete re-silencing of agfp::csr-1transgene . Also,rde-3mutants are insensitive to the injection of dsRNA targetingunc-22but are sensitive to dsRNA expressed from transgenes . These conflicting results could be caused by differences between transgenes, the effects of mating, or the levels of the primary silencing dsRNA. We, therefore, decided to use piRNAi to test the role of RDE-3 in the initiation and maintenance of silencing of an endogenous gene. We targetedhim-5 with six synthetic guide piRNAs (sg-piRNAs)  in wild-type (N2) animals andrde-3(ne3370) mutants.rde-3 is a mutator strain and is relatively unhealthy, with a small brood size and infrequently produces males. To account for an elevated male frequency in the mutant population, we generated transgenicrde-3animals with non-targeting sg-piRNAs as a control. Targetinghim-5with piRNAi resulted in an increased frequency of males in N2 animals but a significantly lower male frequency inrde-3animals  . However, male frequency inrde-3animals was significantly increased compared to negative controls . We tested the role of RDE-3 in maintaining silencing by losing the piRNAi trigger (the piRNAi arrays with a Pmyo-2::mCherryfluorescent marker) and scoring male frequency in the following generations. In wild-type animals, the male frequency remains elevated for at least three generations after the primary piRNAs targetinghim-5are lost , consistent with prior observations . In contrast, we could not detect an inherited elevation of male frequency inrde-3mutants . The initial frequency of males was relatively low inrde-3animals, which limits our ability to make strong conclusions. However, our results support a model where primary piRNAs can post-transcriptionally silence a target transcript (him-5mRNA) at reduced efficiency, butrde-3is required for small RNA amplification and transcriptional silencing. These results support the observations by Chenet al. (2005) that persistently high somatic expression of dsRNA targetingunc-22from a plasmid causes a phenotype. In contrast, a single transient injection ofin vitrotranscribed dsRNA is inefficient. Presumably, RDE-3 amplifies the primary trigger by generatinghim-5pUG RNA templates for RdRP-mediated 22G amplification; these secondary RNAs are subsequently used to set up transcriptional silencing via repressive chromatin marks deposited by thehrde-1dependent nuclear RNAi pathway.him-5orhim-8) is easy to score in various genetic backgrounds and allows distinguishing between silencing initiation and maintenance of endogenous genes.More generally, we demonstrate that piRNAi can be used as a tool to directly test genetic factors required for acute and inherited silencing of endogenous genes. Elevated male frequency . The injection mix for all experiments consisted of ~15-20 ng/\u00b5l of synthetic dsDNA piRNA fragments (Twist Bioscience), 12.5 ng/\u00b5l of a plasmid encoding hygromycin resistance (pCFJ782), and 2 ng/\u00b5l of a fluorescent co-injection marker Pmyo-2::mCherry (pCFJ90). The total concentration of the injection mix was adjusted to 100 ng/\u00b5l with a 1kb DNA ladder . This mix was injected into young adult hermaphrodite animals and allowed to recover on standard NGM plates seed with OP50 bacteria. 36-48 hours post-injection, 500 \u00b5l of 4 mg/ml stock of Hygromycin solution  was topically added to the bacterial lawn of injection plates to select for transgenic (F1) progeny. A single healthy transgenic F2 adult was picked from each plate to generate a clonal strain, and pharyngeal mCherry fluorescence was visually confirmed.Quantification of male frequency.Quantification of male frequency was performed as previously reported by . Briefly, six virgin L4 hermaphrodites were picked to freshly seeded NGM plates with hygromycin selection to select for the piRNAi array. The frequency of males was determined using a dissection microscope and by visual inspection of 100 adult animals on plates incubated on ice for 30 minutes to immobilize animals.Inherited silencing assay.Six virgin L4 animals were picked to non-selective NGM plates to obtain a mixed progeny population with and without the piRNAi array. Males were not quantified in this mixed population; however, L4 animals carrying the sg-piRNAs were propagated in parallel, and their progeny were scored for males (G0). In the following generation, non-transgenic L4 animals were carefully picked from the mixed population based on the absence of pharyngeal mCherry expression (a marker for the piRNAi array). The progeny of these animals was quantified for male frequency (G1). Male frequency was quantified in all following generations by picking L4s until the male frequency was below 1%.Data quantification and statistics. Independently generated transgenic animals were treated as biological replicates. piRNA-mediated silencing is stochastic , and the data do not follow a normal distribution. We performed statistical tests using one-side parametric Mann-Whitney tests to account for this.Software. Statistical analysis was performed with GraphPad Prism (v 9.4.1), figures were generated with Adobe Illustrator (v 26.4.1), and the manuscript was written with Microsoft Word (v 16.63.1).List of strains, plasmids, and piRNAi fragments used in this study.StrainsN2 Standard wildtype strain (Brenner 1974)rde-3(ne3370) IWM286Plasmidsmyo-2::mCherry::unc-54 UTR pCFJ90 Prps-0::HygroRpCFJ782 PpiRNAi fragments him-5(six targeting piRNAs in upper-case):T288cgcgcttgacgcgctagtcaactaacataaaaaaggtgaaacattgcgaggatacatagaaaaaacaatacttcgaattcatttttcaattacaaatcctgaaatgtttcactgtgttcctataagaaaacattgaaacaaaatattaagTGAGTTAGCTTTCCGGAGCTTctaattttgattttgattttgaaatcgaatttgcaaatccaattaaaaatcattttctgataattagacagttccttatcgttaattttattatatctatcgagttagaaattgcaacgaagataatgtcttccaaatactgaaaatttgaaaatatgttTCCTCACGAAAAACCTGCCTAttGccagaactcaaaatatgaaatttttatagttttgttgaaacagtaagaaaatcttgtaattactgtaaactgtttgctttttttaaagtcaacctacttcaaatctacttcaaaaattataatgtttcaaattacataactgtgtATGCAGAGAGATCAGTAGGTActgtagagcttcaatgttgataagatttattaacacagtgaaacaggtaatagttgtttgttgcaaaatcggaaatctctacatttcatatggtttttaattacaggtttgttttataaaataattgtgtgatggatattattttcagacctcatactaatctgcaaaccttcaaacaatatgtgaagtctactctgtttcactcaaccattcatttcaatttggaaaaaaatcaaagaaatgttgaaaaattttcctgtttcaacattatgacaaaaatgttatgattttaataaaaacaaTCGATCACTGTTGACAATCACttctgtttttcttagaagtgttttccggaaacgcgtaattggttttatcacaaatcgaaaacaaacaaaaatttttttaattatttctttgctagttttgtagttgaaaattcactataatcatgaataagtgagctgcccaagtaaacaaagaaaatttggcagcggccgacaactaccgggttgcccgatttatcagtggaggaTAATCCGGCACGTAGAATGTAtctaatgtgatgtacacggttttcatttaaaaacaaattgaaacagaaatgactacattttcaaattgtctatttttgctgtgtttattttgccaccaacaaTCGATGCGACCAACTGTTTTTtcaatctagtaaactcacttaatgcaattcctccagccacatatgtaaacgttgtatacatgcagaaaacggttttttggttttaatgggaacttttgacaaattgttcgaaaatcttaagctgtcccatttcagttgggtgatcgatttT119 Control (non-targeting piRNAs):ctcggtcaattaaagaaagaCATTTTTCATCGGATTTGCTActaaaaaataattttaaAAACGATCATATGCAAATCCAgtgaaactttattcaaaccaaaacgtttaatcagctaattgaaacattaaaaattttatgattttgttagtttttctagcaatgtcaatgcaatcaaataattttcaagtaagatgtttaatgagttatagacttttttattaaatttttgaaaaaaaaaccgatttcagatttaagtaaaattatctctgcttctgctgcattgctgcgaaacaaaaattcctttctgtgcaaagtatagtATAAACGAGGAGCACAAATGAgtgacaattagaaatctcaccgggttttctagatcatctgaaacatataattttaaaaaattgacaccttgttcaacTGTTGCACATATCACTTTTGAtcgaaacattaaatgtctcatgatttttaaagctcttttagaacagtcgCCAATCCCCTTATCCAATTTAttgaaaacaattttctagcgagatgttaaatgagtttgttgaaacagtagattttcgtgtaaacttttgaaaacaaaaattacgttttaaataaaattatatccacttcagcagtgtgcccttgaaacaaaaaagctcgatcaaaaaatttattttttgtgaatggccaccaacttttcaggcaaaattacaaaaaaacataaaatttactgtttcaaaaagttaatataattttggcagcgcatatacctacacTGAATTTTGGCAGAGGCAATTacctctttttgaaaataaag"}
{"text": "In the originally published version of this manuscript, there were errors within Table 4. In the line for variable \u201cAlfieri stitch\u201d the value should read: \u201c3 (2.9)\u201d instead of \u201c3 (13.3)\u201d. For variable \u201cPosterior semi-annulus suture plasty\u201d should read: \u201c8 (7.8)\u201d instead of \u201c8 (6.7)\u201d. These errors are now corrected in online."}
{"text": "Nature Neurology 10.1038/s41593-022-01107-4, published online 11 July 2022.y axis of Fig. 5d should have been \u201c\u221220\u201d and \u201c\u221240\u201d. Further, the scale values near the spike waveforms in Fig. 2a, now reading \u201c100 \u03bcs\u201d appeared originally as \u201c10 \u03bcs.\u201d The errors have been corrected in the HTML and PDF versions of the article.In the version of this article initially published, \u201c\u22122\u201d and \u201c\u22124\u201d on the"}
{"text": "The authors wish to amend the sections of the manuscript detailed below. These corrections are due to inaccuracies in the adsorption activity, as well as in the surface area of the adsorbent.The corrections are as follows:2 g\u22121\u201d should instead read as \u201c4.55 nm and the surface area is 120.7 m2 g\u22121\u201d.(1) On page 1, the sentence including \u201c4.13 nm and the surface area is 112.9 m\u22121\u201d should instead read as \u201c483 to 2190 mg g\u22121\u201d.(2) On page 1, in the sentence including \u201c490 to 2210 mg g2 g\u22121\u201d should instead read as \u201c120.7 m2 g\u22121\u201d, and \u201c4.13 nm\u201d should instead read as \u201c4.55 nm\u201d.(3) On page 3 (left column), the sentence including \u201c16.00 m\u22121\u201d should instead read as \u201c790 mg g\u22121\u201d.(4) On page 4 (left column), the sentence including \u201c808 mg g\u22121\u201d should instead read as \u201c849 to 1000 mg g\u22121\u201d.(5) Page 5 (right column), the sentence including \u201c868 to 1000 mg g\u22121\u201d should instead read as \u201c417, 425, 476 and 483 mg g\u22121\u201d.(6) Page 5 (right column), the sentence including \u201c422, 431, 481 and 490 mg g\u22121\u201d should instead read as \u201c1800, 2043, 2110 and 2190 mg g\u22121\u201d.(7) Page 5 (right column), the sentence including \u201c1812, 2061, 2164 and 2210 mg g\u22121\u201d should instead read as \u201c849 to 1000 mg g\u22121\u201d.(8) Page 5 (left column), the sentence including \u201c868 to 1000 mg gThe Royal Society of Chemistry apologises for these errors and any consequent inconvenience to authors and readers."}
{"text": "In the aggregation of uncertain information, it is very important to consider the interrelationship of the input information. Hamy mean (HM) is one of the fine tools to deal with such scenarios. This paper aims to extend the idea of the HM operator and dual HM (DHM) operator in the framework of complex intuitionistic fuzzy sets (CIFSs). The main benefit of using the frame of complex intuitionistic fuzzy CIF information is that it handles two possibilities of the truth degree (TD) and falsity degree (FD) of the uncertain information. We proposed four types of HM operators: CIF Hamy mean (CIFHM), CIF weighted Hamy mean (CIFWHM), CIF dual Hamy mean (CIFDHM), and CIF weighted dual Hamy mean (CIFWDHM) operators. The validity of the proposed HM operators is numerically established. The proposed HM operators are utilized to assess a multiattribute decision-making (MADM) problem where the case study of tourism destination places is discussed. For this purpose, a MADM algorithm involving the proposed HM operators is proposed and applied to the numerical example. The effectiveness and flexibility of the proposed method are also discussed, and the sensitivity of the involved parameters is studied. The conclusive remarks, after a comparative study, show that the results obtained in the frame of CIFSs improve the accuracy of the results by using the proposed HM operators. MADM is an essential process of decision-making (DM) science whose objective is to see the best options from the arrangement of likely ones. In DM, a singular requirement is to evaluate the given choices by different classes, such as single, range, and so for appraisal purposes. Nevertheless, in various fanciful conditions, it is, for the most part, pursuing for the person to convey their choices as a new number. For this, the frame of the fuzzy set (FS)  was dever\u03bce\u03c0i\u03b8\u03bc2 where r\u03bc, \u03b8\u03bc \u2208  and such kind of framework describes two different aspects of an uncertain phenomena. Moreover, the idea of CFS recently got a large number of attractions, and some useful work may be found in  and \u03bd\u00c7(x) : Y\u27f6 provided that 0 \u2264 \u03bc\u00c7(x)+\u03bd\u00c7(x) \u2264 1 and hesitancy degree represent vc(x)=1 \u2212 (\u03bc\u00c7(x)+\u03bd\u00c7(x)), (x) \u2208 . Furthermore, \u00c7=) denotes an intuitionistic fuzzy number CIFN.An IFS is of the form of \u00c7 = {, \u03bd\u00c7(x))|x \u2208 y} such that \u03bc\u00c7(x) = r\u03bc\u00c7(x)e\u03c0i\u03b8\u03bc\u00c7(x)2 and \u03bd\u00c7(x) = s\u03bd\u00c7(x)e\u03c0i\u03a6\u03bd\u00c7(x)2 where r\u03bc\u00c7(x), \u03a6s\u03bd\u00c7(x) denote the amplitude terms and \u03b8\u03bc\u00c7(x)\u03a6,\u03bd\u00c7(x) denote the phase terms of \u03bc\u00c7(x) and \u03bd\u00c7(x), respectively, from  provided that 0 \u2264 r\u03bc\u00c7(x) + s\u03bd\u00c7(x) \u2264 1 and 0 \u2264 \u03b8\u03bc\u00c7(x) + \u03a6s\u03bd\u00c7(x) \u2264 1 and hesitancy degree represent \u210f(x) = 1 \u2212 (r\u03bc\u00c7(x)e\u03c0i\u03b8\u03bc\u00c7(x)2 + s\u03bd\u00c7(x)e\u03c0i\u03bd\u00c7(x)2), \u210f(x) \u2208 . Furthermore, \u00c7 = (r\u03bc\u00c7(x)e\u03c0i\u03b8\u03bc\u00c7(x)2, s\u03bd\u00c7(x)e\u03c0i\u03a6\u03bd\u00c7(x)2) represents a complex intuitionistic fuzzy number CIFN.A CIFS is of the form of \u00c7 = (r\u03bc\u00c7(x)e\u03c0i\u03b8\u03bc\u00c7(x)2, s\u03bd\u00c7(x)e\u03c0i\u03bd\u00c7(x)2) be a CIFN. The score function S is as follows:Let\u00c7 = (r\u03bc\u00c7(x)e\u03c0i\u03b8\u03bc\u00c7(x)2, r\u03bd\u00c7(x)e\u03c0i\u03a6\u03bd\u00c7(x)2) be a CIFN. The accuracy function \u2032H is as follows:LetWe gave \u00c71=(0.3ei2\u03c0(0.45), 0.5ei2\u03a0(0.25)) and \u00c72=(0.6ei2\u03a0(0.70), 0.2ei2\u03a0(.25)) be two CIFNs. By using Definitions \u00c71)=1/2((0.3 \u2212 0.5)+(0.45 \u2212 0.25))=1/2(\u22120.20+0.20)=0 \u2208 \u2009\u0161(\u00c72)=1/2((0.6 \u2212 0.2)+(0.70 \u2212 0.25))=1/2(0.40+0.45)=0.48 \u2208 \u2009\u0161(H(\u00c71)=1/2((0.3+0.5)+(0.45+0.25))=1/2(0.80+0.70)=0.75 \u2208 \u2009H(\u00c72)=1/2((0.6+0.2)+(0.70+0.25))=1/2(0.8+0.95)=0.88 \u2208 \u2009Let 0.3ei2\u03c00.5, 0.5ei2\u00c71 = (r\u03bc1(x)e\u03c0i\u03b8\u03bc1(x)2, r\u03bd1(x)e\u03c0i\u03a6\u03bd1(x)2) and \u00c72 = (r\u03bc2(x)e\u03c0i\u03b8\u03bc2(x)2, r\u03bd2(x)e\u03c0i\u03a6\u03bd2(x)2) be two CIFNs. ThenS(\u00c71) < S(\u00c72), then \u00c71 < \u00c72If S(\u00c71) > S(\u00c72), then \u00c71 > \u00c72If S(\u00c71) = S(\u00c72), then:If H(\u00c71) > H(\u00c72), then\u00c71 > \u00c72H(\u00c71) < H(\u00c72), then\u00c71 < \u00c72H(\u00c71) = H(\u00c72), then\u00c71 \u2248 \u00c72.Let \u00c71 = (r\u03bc1(x)e\u03c0i\u03b8\u03bc1(x)2, r\u03bd1(x)e\u03c0i\u03a6\u03bd1(x)2) and \u00c72 = (r\u03bc2(x)e\u03c0i\u03b8\u03bc2(x)2, r\u03bd2(x)e\u03c0i\u03a6\u03bd2(x)2) be two CIFNs. Then some fundamental operations are defined as follows:\u00c71\u2286\u00c72\u27far\u03bc1(x) \u2264 r\u03bc2(x), \u2009\u03b8\u03bc1(x) \u2264 \u03b8\u03bc2(x) and r\u03bd1(x) \u2265 r\u03bd2(x), \u2009\u03a6\u03bd1(x) \u2265 \u03a6\u03bd2(x)\u00c71 = \u00c72\u27far\u03bc1(x) = r\u03bc2(x), \u2009\u03b8\u03bc1(x) = \u03b8\u03bc2(x)r\u00c71(x) = r\u00c72(x), \u03b8\u00c71(x) = \u0394\u00c72(x) and r\u03bd1(x) = r\u03bd2(x), \u2009\u03a6\u03bd1(x) = \u03a6\u03bd2(x)\u00c71\u2032 = {(r\u03bd1(x)ei\u03a6\u03bd1(x), r\u03bc1(x)ei\u03b8\u03bc1(x))}Let\u00c71 = (r\u03bc1(x)e\u03c0i\u03b8\u03bc1(x)2, r\u03bd1(x)e\u03c0i\u03a6\u03bd1(x)2) and \u00c72 = (r\u03bc2(x)e\u03c0i\u03b8\u03bc2(x)2, r\u03bd2(x)e\u03c0i\u03a6\u03bd2(x)2) be two CIFNs and \u03bb > 0 be a real number. Then\u03bb\u00c71 = (1 \u2212 (1 \u2212 r\u03bc1(x))\u03bbe\u03c0i(1 \u2212 (1 \u2212 \u03b8\u03bc1(x))\u03bb)2, (r\u03bd1\u03bb(x) \u00b7 e\u03c0i(\u03a6\u03bd1(x))\u03bb2))\u00c71\u03bb = ((r\u03bc1\u03bb(x)e\u03c0i(\u03b8\u03bc1(x))\u03bb2), 1 \u2212 (1 \u2212 r\u03bd1(x))\u03bbe\u03c0i(1 \u2212 (1 \u2212 \u03a6\u03bd1(x))\u03bb)2)LetIn this section, we recall the basic definition of HM operator. We also discuss the HM operators that are previously defined. Furthermore, we point out towards the limitations of such existing HM operators that lead us to propose some new HM operators.Consider the HM operators defined for real numbers.x is such that 1 \u2264 x \u2264 n and Ckx represent the binomial coefficient, that is, Cnx = x!/n!(x \u2212 n)!The HM operator is as follows:x)=\u00c7 if \u00c7i=\u00c7, HMx) \u2264 HMx) if \u00c7i \u2264 \u03c9i, HM\u00c7i) \u2264 HMx) \u2264 max\u00c7imin(x)=(1/k)\u2211i=1k\u00c7iFor arithmetic mean operator HMx)=(\u220fi=1k\u00c7i)x1/For geometric mean operator HMThe HM operator is likely to satisfy the following properties:We recall the definition of the DHM operator for real numbers.The DHM operator is defined as follows:Ij = (r\u03bcj(x), sj(x)), j = 1,2,\u2026, k be the collection of IFNs. Then intuitionistic fuzzy Hamy mean (IFHM) operator is defined as follows:Let Ij = (, ), j = 1,2,\u2026, k be the collection of interval-valued IFNs (IVIFNs). Then interval-valued intuitionistic fuzzy Hamy mean operator is defined as follows:LetAll the above-discussed HM operators deal with two real values TD and FD. Consider a scenario with TD and FD having further two aspects, then the operators discussed above become unable to deal with such information. Therefore, we aim to propose the concept of HM operator in the framework of CIFS because such an operator can deal with two aspects of TD and FD at a time.In this section, we introduced the idea of HM operators in the framework of CIFSs. We also proved that the CIFHM operator satisfied the basic properties of AO. We give example to support the proposed operator. First, consider the HM operators based on CIFNs as follows.\u00c7j=(r\u03bcj(x)ei\u03b8\u03bcj(x), svj(x)ei\u03a6\u03bdj(x)), j=1,2,\u2026, k, be the collection of CIFNs. Then, the CIFHM operator is defined as follows:Let \u00c7j=(r\u03bcj(x)e\u03c0i\u03b8\u03bcj(x)2, s\u03bdje\u03c0i\u03a6\u03bdj(x)2), j=1,2,\u2026, k be the collection of CIFNs. Then the aggregated value of the CIFHM operator is also a CIFN such thatLet This theorem has two parts: first, we derive the formula given in equation  as follor\u03bc\u2009(x)+s\u03bd(x) \u2264 1 and 0 \u2264 \u03b8\u03bc(x)+\u03a6\u03bd(x) \u2264 10 \u2264 Now, we prove that equation  represenr\u03bc(x) \u2264 1 and 0 \u2264 \u03b8\u03bc(x) \u2264 1 We haveWe know that 0 \u2264 Similarly,r\u03bc(x)e\u03c0i\u03b8\u03bc(x)2 \u2264 1 and 0 \u2264 s\u03bd(x)e\u03c0i\u03a6\u03bd(x)2 \u2264 1 so we haveSince 0 \u2264 Now, we prove that the CIFHM operator satisfies the properties of the aggregation function in Theorems \u00c7j=(r\u03bcj(x)ei\u03b8(\u03bc/j)(x), s\u03bdj(x)ei\u0394\u03bdj(x)), j=1,2,\u2026, k be the collection of all identical values of CIFNs. ThenLet\u00c7j=(r\u03bcj(x)ei\u03b8(\u03bc/j)(x), s\u03bdj(x)ei\u03a6\u03bdj(x))=(r(x)ei\u03b8(x), s(x)ei\u03a6(x))=\u00c7, j=1,2,\u2026, k. ThenWe know that \u00c7j=(r\u03bcj(x)ei\u03b8\u03bc/j(x), s\u03bdj(x)ei\u03a6\u03bdj(x)), j=1,2,\u2026, k and Dj(x)=(g\u03bcj(x)ei\u03b1\u03bc/j(x), h\u03bd\u2009j(x)ei\u03b2\u03bd/j(x)), j=1,2,\u2026, k be two sets of CIFNs. If \u00c7j(x) \u2264 Dj(x), that is, r\u03bcj(x) \u2264 g\u03bcj(x), \u03b8\u03bcj(x) \u2264 \u03b1\u03bcj(x) and s\u03bdj(x) \u2264 h\u03bdj(x), \u03a6\u03bdj(x) \u2264 \u03b2\u03bdj(x) thenLet \u00c7j \u2264 Dj, that is, r\u03bcj(x) \u2264 g\u03bcj(x), \u03b8\u03bcj(x) \u2264 \u03b1j(x) and sj(x) \u2264 hj(x), \u03a6j(x) \u2264 \u03b2j(x). ThenWe know that r\u03bcj(x)e\u03c0i\u03b8\u03bc(x)2 \u2264 ge\u03c0i\u03b1\u03bc(x)2. In a similar way, we can investigate the value of s\u03bdj(x)e\u03c0i\u03a6\u03bd(x)2 \u2265 h\u03bd(x)e\u03c0i\u03b2\u03bd(x)2.(1)r\u03bcj(x)e\u03c0i\u03b8\u03bc(x)2 < g\u03bcj(x)e\u03c0i\u03b1\u03bc(x)2 and s\u03bd(x)e\u03c0i\u03a6\u03bd(x)2 > h\u03bd(x)e\u03c0i\u03b2\u03bd(x)2, thenIf (2)r\u03bcj(x)e\u03c0i\u03b8\u03bc(x)2=g\u03bcj(x)e\u03c0i\u03b1\u03bc(x)2 and s\u03bd(x)e\u03c0i\u03a6\u03bd(x)2=h\u03bd(x)e\u03c0i\u03b2\u03bd(x)2, thenIf According to equation (19), \u00c7j=(r\u03bcj(x)ei\u03b8\u03bc\u2009j(x), s\u03bd\u2009j(x)ei\u03a6\u03bd\u2009j(x)), j=1,2,\u2026, k be the collection of CIFNs. If \u00c7j\u2212=min and \u00c7j+=max, thenLet From boundedness property:CIFHMxHMx) = \u00c7+.From monotonicity property:In a decision-making problem, the weights of all attributes and the experts sometimes matter. So we discuss the influence of weights on the HM operator in this section and develop the weighted HM operator as follows.\u00c7j=(r\u03bcj(x)ei\u03b8\u03bcj(x), s\u03bdj(x)ei\u03a6\u03bdj(x)), j=1,2,\u2026, k be the collection of CIFNs with weight vector wi=T, \u2009wi \u2208  and \u2211i=1nwi=1. Then the CIFWHM operator is defined as follows:Let \u00c7j=(r\u03bcj(x)(x)e\u03c0i\u03b8\u03bcj(x)2, s\u03bdj(x)e\u03c0i\u03a6\u03bdj(x)2), j=1,2,\u2026, k be the collection of CIFNs. Then the aggregated value of the CIFWHM operator is also a CIFN such thatLet r\u03bc(x)+s\u03bd(x) \u2264 1 and 0 \u2264 \u03b8\u03bc(x)+\u0394\u03bd(x) \u2264 10 \u2264 Now, we have to show that is a CIFN.r\u03bc(x) \u2264 1 and 0 \u2264 \u03b8\u03bc(x) \u2264 1 We haveWe know that 0 \u2264 Similarly,r\u03bc(x)e\u03c0i\u03b8\u03bc(x)2 \u2264 1 and 0 \u2264 s\u03bd(x)e\u03c0i\u03a6\u03bd(x)2 \u2264 1 so we haveSince 0 \u2264 We gave \u00c71=0.45e\u03c0i(0.3)2, 0.62e\u03c0i(0.41)2, \u00c72=0.2e\u03c0i(0.7)2, 0.52e\u03c0i(0.6)2, \u00c73=0.5e\u03c0i(0.6)2, 0.2e\u03c0i(0.91)2, \u00c74=0.7e\u03c0i(0.8)2, 0.42e\u03c0i(0.15)2 be the CIFNs with weight vector w=. Then we use the proposed CIFWHM operator to aggregate the given CIFNs. Suppose that x=2.Let \u00c7j=(r\u03bcj(x)ei\u03b8\u03bcj(x), s\u03bdj(x)ei\u03a6\u03bdjj(x)), \u2009j=1,2,\u2026, k be the collection of all identical values of CIFNs. ThenLet Similar to \u00c7j=(r\u03bcj(x)ei\u03b8\u03bcj(x), s\u03bdj(x)ei\u03a6\u03bdj(x)), \u2009j=1,2,\u2026, k and Dj=(g\u03bcj(x)ei\u03b1\u03bcj(x), h\u03bdj(x)ei\u03b2\u03bdj(x)), j=1,2,\u2026, k be two sets of CIFNs. Then r\u03bcj(x) < g\u03bcj(x), \u03b8\u03bc\u2009j(x) < \u03b1\u03bc\u2009j(x) and s\u03bdj(x) > h\u03bdj(x), \u2009\u03a6\u03bdj(x) > \u03b2\u03bdj(x). ThenLet Similar to \u00c7j=(r\u03bcj(x)ei\u03b8\u03bcj(x), s\u03bdj(x)ei\u03a6\u03bdj(x)), \u2009j=1,2,\u2026, k be the collection of CIFNs. IfLet thenFrom boundedness property:x = \u00c7+.CIFWHMFrom monotonicity property:From From property 4, we haveIn this section, we use the idea of the DHM operator in the framework of CIFSs. We prove the validity of the proposed AO. We also give a numerical example to support the proposed CIFDHM operator.\u00c7j=(r\u03bcj(x)ei\u03b8\u03bcj, s\u03bdj(x)ei\u03a6\u03bdj), \u2009j=1,2,\u2026, k be the collection of CIFNs. Then CIFDHM operator is defined as follows:Let \u00c7j=(r\u03bcj(x)ei\u03b8\u03bcj, s\u03bdj(x)ei\u03a6\u03bdj), \u2009j=1,2,\u2026, k be the collection of CIFNs. Then CIFDHM operator is defined as follows:Let The proof is analogous to the proof of The CIFDHM operator also satisfies the basic properties of aggregation as discussed in Theorems Now we will elaborate on the concepts of DHM operator in the framework of CIFSs by keeping the weight under observation.\u00c7j=(r\u03bcj(x)ei\u03b8\u03bcj(x), s\u03bdj(x)ei\u03a6\u03bdj(x)), j=1,2,\u2026, k be the collection of CIFNs with weight vector wi=T, \u2009wi \u2208 , and \u2211i=1nwi=1. Then the CIFWDHM operator is defined as follows:Let \u00c7j=(r\u03bcj(x)ei\u03b8\u03bcj(x), s\u03bdj(x)ei\u03a6\u03bdj(x)), j=1,2,\u2026, k be the collection of CIFNs. Then the aggregated value of the CIFWDHM operator is also a CIFN and is given byLet The proof is similar to the proof of We gave \u00c71=0.55e\u03c0i(0.45)2, 0.70e\u03c0i(0.85)2, \u00c72=0.40e\u03c0i(0.70)2, 0.82e\u03c0i(0.77)2, \u00c73=0.60e\u03c0i(0.40)2, \u20090.72e\u03c0i(0.60)2, \u00c74=0.50e\u03c0i(0.75)2, 0.52e\u03c0i(0.35)2 be the CIFNs with the weight vector of the attributes be w=. Then we use the proposed CIFWDHM operator to investigate the CIFN. Suppose that x=2.Let The CIFWDHM operator also satisfies the basic properties of aggregation as discussed in Theorems A CIFS is an extension of an IFSs in which the sum of TD and FD lies on interval ; each TD and FD has two aspects: amplitude terms and phase terms. In this section, we developed a MADM method based on CIFWHM and CIFWDHM operators to solve a problem involving the development of the tourism industry. First, we proposed the MADM algorithm, and then we present a comprehensive example to utilize the proposed algorithm.\u2009 Step 1: Information are collected from the decision-maker about the tourism industry . All information are in the form of CIFNSs.\u2009 Step 2: This step involves the utilization of the proposed CIFWHM and CIFWDHM operator to aggregate the CIF information depicted in the decision matrix given in step 1.\u2009 Step 3: This step is about the analysis of the aggregated information based on score values of CIFNs by using \u2009 Step 4: In this step, we investigate the score values of all aggregated information for ordering and ranking the attributes.In this subsection, we describe the steps of the algorithm as follows:The average increase in the economy induces ever-greater competitive surroundings for tourism enterprises. The essential competition of the tourism industry depends on the tour destination. In the tourism environment, to assure the regular development of a visitor vacation spot, it is essential to take measures to increase tourism vacation spots to enhance competitiveness. The vital role of the tourism industry is to analyze and evaluate the vacation spot competitiveness, which can display the attraction strength of a vacation spot and imply the direction for the efficient allocation of assets. Therefore, it is a crucial manner for determining the vacation spot development mode and route to analyze the important elements of tourism destination and extract qualitative and quantitative assessment index, which can uphold long-term competitive benefit and the nonstop improvement of vacation spot. This is a problem of interest, and we try to investigate this problem using our proposed AOs and adopt the tourism destination problem from .Ai that have to be evaluated. These five destinations are to be examined based on four attributes where G1: is the attractiveness of tourism sources, G2 is the infrastructure and development of the tourism industry, G3 is supporting force of the tourism environment, and G4 is the tourist demand. The five possible vacationer locations are to be assessed with IVIFNs that are weighting vectors w= as shown in \u2009 Step 1: The decision matrix containing the information about the five alternatives by anonymous decision-makers is provided in x=3. All aggregated results are shown in \u2009 Step 2: Aggregate CIFNs are shown in the decision matrix in \u2009 Step 3: By utilizing Consider five possible tourist destinations A3 by using the CIFWHM operator and A1 by using the CIFWDHM operator at the parameter of x=3. Consider In Another view of the results of Tables x on the results. We see the effect on ordering and ranking of the results by the variation of the parameter x=1,2,3,4 obtained by using CIFWHM and CIFWDHM operators. We listed the analysis of the results after varying the parameter x in Tables In this subsection, we see the impact of parameter A3 as the best tourist destination when we take x=1,2,3,4 in the case of the CIFWDHM operator also. Variation of x is studied in both cases, and we reached the following results:x does not create any impact on ranking results.Variation of x may have an impact in some other cases depending on the data we dealt with must be analyzed each time in each case.Variation of We analyzed In this section, we aim to analyze the aggregated results obtained using the CIFHM operators with the aggregated results obtained using the AOs of Akram et al. , Wang etA3 is the suitable destination. Moreover, CIFHWG operator by Akram et al. [A5 and A1 as suitable destinations, respectively. The only reason for having different results is that the operators discussed in [From m et al.  and CIFMm et al.  gave A5 ussed in , 39, 43 CIFHM and CIFDHM operators are the generalizations of the results discussed in , 40 and CIFWHM and CIFWDHM operators deal with such information that has two aspects of the TD and FD denoted by amplitude term and phase termThe advantages of the proposed work are as follows:We proposed CIFHM and CIFDHM operators based on CIFSsWe also investigated the basic properties of the proposed work in the form of idempotency, monotonicity, and boundednessWe elaborated the concepts of the DHM operator in the framework of CIFSs by keeping the weight under observationWe analyzed the consequences of the proposed work through application and numerical examples to illustrate the CIFSs. In this paper, we utilized the idea of HM operators to elaborate the CIFSs in the framework of CIFHM operators to find the reliability of CIFSs. A CIFS has two aspects: TD and FD; TD has also two aspects amplitude terms and phase terms; and similarly, FD has two parts: amplitude terms and phase terms of FD that carried the more flexible information. The main contribution and key factors in this paper are as follows:"}
{"text": "The concept of resolving set and metric basis has been very successful because of multi-purpose applications both in computer and mathematical sciences. A system in which failure of any single unit, another chain of units not containing the faulty unit can replace the originally used chain is called a fault-tolerant self-stable system. Recent research studies reveal that the problem of finding metric dimension is NP-hard for general graphs and the problem of computing the exact values of fault-tolerant metric dimension seems to be even harder although some bounds can be computed rather easily. In this article, we compute closed formulas for the fault-tolerant metric dimension of lattices of two types of boron nanotubes, namely triangular and alpha boron. These lattices are formed by cutting the tubes vertically. We conclude that both tubes have constant fault tolerance metric dimension 4. Computer networks are graphs with vertices representing hosts, servers, or hubs, and edges as connecting mediums between them. Vertex is actually a possible location to find faults or some damaged devices in a computer network. This idea somehow created an urge in Slater and independently in Harary and Melter  to uniquA moving point in a graph may be located by finding the distance from the point to the collection of sonar stations that have been properly positioned in the graph. Thus finding a minimal but sufficiently large set of labeled vertices such that a robot can find its position, is a well-established problem known as robot navigation. This sufficiently large set of labeled vertices is a resolving set of the graph space and the corresponding cardinality is the metric dimension. Similarly, on another node, a real-world problem is the study of networks whose structure has not been imposed by a central authority but has arisen from local and distributed processes. It is very difficult and expensive to obtain a map of all nodes and the links between them. A commonly used technique is to obtain a local view of the network from various locations and combine them to obtain a good approximation for the real network. Metric dimension also has some applications in this aspect as well.G, and metric dG : V(G) \u00d7 V(G) \u2192 \u2115 \u222a {0}, where \u2115 is the set of positive integers and dG is the minimum number of edges in any path between x and y. Let W = {w1, w2, ..., wk} be an ordered set of vertices of G and let v be a vertex of G. The representation r(v|W) of v with respect to W is the k\u2212tuple , dG, ..., dG). If distinct vertices of G have a distinct representation with respect to W, then W is called a resolving set of G  is its cardinality.Consider a simple, connected graph Wn to be n \u2265 7. Caceres et al. (n \u2265 7. Tomescu and Javaid (Jn2 to be n \u2265 4. A particular metric-feature of the family of graphs is independent of metric dimension on the particular element of the family. A connected graph has a constant metric dimension if \u03b2(G) = k, where k \u2208 \u2115 is fixed. This feature has been presented in Imran et al. , is the minimum cardinality of such \u015a. A family Recent development in this context has paved way for a new related concept known as fault-tolerance in the metric dimension. Suppose that, in a network, o Slater , 2002 ano Slater . A resols Hayes, . Slater s Hayes,  introducm\u00d7n and triangular boron nanotubes Tm\u00d7n. These lattices are formed by cutting both tubes vertically. Kwun et al.  = , if 1 \u2264 q \u2264 lFor k is even, a is odd, and If a is odd, k is even, and If k is even, a is even, and If k is even, a is even, and If a = 2l, r = , if 1 \u2264 b \u2264 lFor a = 2l \u2212 1, r = , if 1 \u2264 q \u2264 lFor k is even, a is odd, and If k is even, a is odd, and If k is even, a is even, and If k is even, a is even, and If a = 2l, r = , if 1 \u2264 b \u2264 lFor Case II: Let k > 2l.y = rl + p \u2212 1. If r \u2265 2, rl < k \u2264 (r + 1)l, and k is odd.Let a = rl + p, where 1 \u2264 p \u2264 l and a is oddFor a is evenIf k is even, r \u2265 2, and rl < k \u2264 (r + 1)l thenIf y = rl + p \u2212 1. For a = rl + p where 1 \u2264 p \u2264 l and a is oddLet a is evenIf F is an FTRS for Tk,l. Therefore, These vectors are distinct in at least two coordinates. So Theorem 2.2. Let Tk,l denote the graph of k \u00d7 l, 2D-lattice of triangular boron nano tubes. Then Proof. For the lower bound on Case I:k \u2264 l.Tk,l. Let F = {vi,j, va,b, vr,s} where i < a < r and j < b < s be an FTRS for Tk,l. Then F1 = F\\{v} is a R.S. for each v \u2208 F. We discuss the following possibilities:We claim that any set of cardinality 3 is not an FTRS for Possibility 1: When all vertices in F lie on the same row.F lie in the first row, then i = a = r = 1. Let F1 = {vj1,, vb1,} then r = r, a contradiction.(i) If all vertices in F lie in p \u2212 th row and 1 < p. Let F1 = {vp,j, vp,b} then either r = r or r = r, a contradiction.(ii) If all vertices in Possibility 2: When two vertices in F lie on the same row.i = a = l and r \u2260 l. Let F1 = {vl,j, vl,q} then either r = r or r = r, a contradiction.WLOG we may suppose that Possibility 3: When the vertices in F lie on three different rows.F lie on same column, let j = b. If j = b = 1 and F1 = {v1,1, va,1} then r = r, a contradiction.(i) Two vertices in F1 = {vi,1, vj,1}, 1 \u2264 i < j < k then either r = r or r = r, a contradiction.(ii) If F1 = {vi,b, vj,b}, 1 \u2264 i < j < k, and 1 < q < l then either r = r or r = r, a contradiction.(iii) If S lie on different columns. Let S1 = {Vi,j, Va,b}, i < a then(iv) If all vertices in j < b then either r = r or r = r, a contradiction.(a) If j > b then either r = r or r = r, a contradiction.(b) If F is not an FTRS for Tm,n. So From the above discussion, we conclude that Case II:k > lTk,l having cardinality 3, it is enough to prove that any set of vertices having two elements is not an R.S. for Tk,l. Let F1 = {vi,j, va,b} be an R.S. for Tk,l. There are three possibilitiesIn order to prove that there is no FTRS for Possibility 1: If vi,j, va,b are taken from same row, i.e., i = a thenF1 = {v1,1, vl1,} then l is even and l is odd, a contradiction.(i) If F1 = {vi,j, vi,b}, 1 \u2264 i < k, and 1 \u2264 j < b < l then r = r, a contradiction.(ii) If F1 = {vi,j, vi,b}, 1 \u2264 i < k, and 1 < j < b = l then r = r, a contradiction.(iii) If F1 = {vk,j, vk,b} and 1 \u2264 j < b < l then r = r or r = r, a contradiction.(iv) If F1 = {vk,j, vk,b} and 1 < j < b = l then r = r, a contradiction.(v) If F1 = {vi,1, vi,l} and 1 < i < k then r = r, a contradiction.(vi) If F1 = {vk,1, vk,l} then r = r, a contradiction.(vii) If Possibility 2: If vi,j, va,b are taken from same column, i.e., j = b thenF1 = {v1,1, vk,1} then r = r, a contradiction.(i) If F1 = {vi,1, vj,1} where 1 \u2264 i < j < m then r = r or r = r, a contradiction.(ii) If F1 = {vi,1, vj,1} and 1 < i < j = k then r = r or r = r, a contradiction.(iii) If F1 = {vi,b, vj,b}, 1 \u2264 i < j < k, and 1 < b < l then r = r or r = r, a contradiction.(iv) If F1 = {vi,b, vj,b}, 1 < i < j \u2264 k, and 1 < b < l then r = r or r = r, a contradiction.(v) If F1 = {vi,b, vj,b}, i = 1, j = k, and 1 < q < l then r = r, a contradiction.(vi) If F1 = {vi,l\u22121, vj,l\u22121} and i < j < k then r = r or r = r, a contradiction.(vii) If F1 = {vl1,, vk,l} then r = r or r = r, a contradiction.(viii) If Possibility 3: If vi,j, va,b are taken from different rows and different columns, i.e., i \u2260 p,j \u2260 q. Let F1 = {vi,j, va,b} and i < aj < b then r = r or r = r, a contradiction.(i) If j > b then r = r or r = r, a contradiction.(ii) If a = k and i = 1 then r = r or r = r or r = r, a contradiction.(iii) If Tk,l. So Thus no set having two vertices is an R.S. for The next theorem gives Theorem 2.3. Let \u03b1k,l denote the graph of k \u00d7 l 2D-lattice of \u03b1 boron nano tubes. Then Proof. Consider k \u00d7 l 2D-lattice of \u03b1-boron Nanotubes. We denote this graph by \u03b1k,l. In general, the tube is \u201ccut\u201d vertically and the upper left triangle is pointing up as shown in k,l is partitioned as {v1,1, v1,2, v1,3, ...., vl1,, v2,1, v2,2, ...., vl2,, v4,1, v4,2, ......, vl4,, v5,1, v5,2, ......, vl5,, v7,1, v7,2, ......, vl7,, ......, vk,1, vk,2, vk,3, ...., vk,l} \u222a {v3,1, v3,2, v3,4, v3,5, v3,7, v3,8, ...., vl3,, v6,1, v6,3, v6,4, v6,6, v6,7, v6,9, ...., vl6,, v9,1, v9,2, v94,, v9,5, v9,7, v9,8, ....,l9,, ......}.Case I: Whenk < l.F = {v1,1, vl1,, vk,1, vk,l}. We show that F is an FTRS for \u03b1k,l. We give the distance vectors of all vertices va,b of \u03b1k,l relative to F.Let k be odd and k \u2260 6q + 1, q \u2208 Z+.Let a = 1For a = 2For a is odd, a \u2260 3p, p \u2208 Z+, and In general, if a is odd, a \u2260 3p, p \u2208 Z+, and If a is even, a \u2260 3p, p \u2208 Z+, and If a is even, a \u2260 3p, p \u2208 Z+, and If k = 6p + 1.Let a is odd, a \u2260 3p, k \u2208 Z+, and If a is odd, a \u2260 3p, p \u2208 Z+, and If a is even, a \u2260 3p, p \u2208 Z+, and If a is even, a \u2260 3p, p \u2208 Z+, and If a is odd, a = 3p, p \u2208 Z+, b \u2260 3q thenIf a is odd, a = 3p, p \u2208 Z+, b \u2260 3q thenIf a is even, a = 3p, p \u2208 Z+, b \u2260 3q \u2212 1 thenIf a is even, a = 3p, p \u2208 Z+, b \u2260 3q \u2212 1 thenIf k be even and k \u2260 6p+2, p \u2208 Z+.Let a is odd, a \u2260 3p, p \u2208 Z+, and If a is odd, a \u2260 3p, p \u2208 Z+, and If a is even, a \u2260 3p, p \u2208 Z+, and If a is even, a \u2260 3p, p \u2208 Z+, and If m = 6k+2.Let a is odd, a \u2260 3p, p \u2208 Z+, and If a is odd, a \u2260 3p, p \u2208 Z+, and If a is even, a \u2260 3p, p \u2208 Z+, and If a is even, a \u2260 3p, p \u2208 Z+, and If a is odd, a = 3p, p \u2208 Z+, b \u2260 3q, and q \u2208 Z+ thenIf a is odd, a = 3p, p \u2208 Z+, b \u2260 3q, and q \u2208 Z+ thenIf a is even, a = 3p, p \u2208 Z+, b \u2260 3q \u2212 1, and q \u2208 Z+ thenIf a is even, a = 3p, k \u2208 Z+, b \u2260 3q \u2212 1, and q \u2208 Z+ thenIf Case II:k \u2265 2l.r \u2265 2, rl \u2264 k \u2264 (r + 1)l, and k is odd thenIf a = rn + p where 0 \u2264 p \u2264 l \u2212 1, a \u2260 3p, and a is oddFor a is evenIf a = 3p, a is odd, b \u2260 3q, and q \u2208 Z+If a = 3k, a is even, b \u2260 3q \u2212 1, and q \u2208 Z+If k is even, r \u2265 2, and rl \u2264 k \u2264 (r + 1)l thenIf a = rl + p where 0 \u2264 p \u2264 l \u2212 1For a is odd, a \u2260 3pIf a is even, a \u2260 3pIf a = 3p, a is odd, b \u2260 3q, and q \u2208 Z+If a = 3p, a is even, b \u2260 3q \u2212 1, and q \u2208 Z+If F is an FTRS for \u03b1k,l. Therefore These representations are distinct in at least two coordinates. So Case I: k \u2265 l.k,l) = 3 [19], so Since \u03b2 = r, a contradiction.(i) If all vertices in F lie in p \u2212 th row and 1 < p \u2260 3l, l \u2208 Z+. Let F1 = {vp,j, vp,b} then either r = r or r = r, a contradiction.(ii) If all vertices in i = a = r = 3q, l \u2208 Z+. Let F1 = {vq, j3, vq, b3} then either r = r or r = r, a contradiction.(iii) If Possibility 2: When two of vertices in F lie on the same row.i = a = p and r \u2260 l. Let F1 = {vp,j, vp,b} then either r = r or r = r, a contradiction.WLOG let Possibility 3: If all the three vertices in F lie on three different rowsF lie on same column, let j = b. If j = b = 1 and F1 = {v1,1, va,1} then r = r, a contradiction.(i) Two vertices in F1 = {vi,1, vj,1}, 1 \u2264 i < j < k then either r = r or r = r, a contradiction.(ii) If F1 = {vi,b, vj,b}, 1 \u2264 i < j < k, and 1 < b < l then either r = r or r = r, a contradiction.(iii) If F lie on different columns. Let F1 = {vi,j, va,b}, i < a then(iv) If all vertices in j < b then either r = r or r = r, a contradiction.(I) If j > b then either r = r or r = r, a contradiction.(II) If F is not an FTRS for \u03b1k,l. So Thus In this article, we computed the fault-tolerant metric dimension of triangular and alpha boron nanotubes. In both cases, we proved that this dimension is 4. Hence, these tubes are families of a constant fault-tolerant metric dimension. These facts can be used in the networking of nano-devices using these tubes and nano-engineering.Formal analysis was done by MM. Investigation and draft are done by ZH. Both authors contributed to the article and approved the submitted version."}
{"text": "The author wishes to make the following correction to this paper :In the original article, there were following mistakes:k\u2032+5/K\u20325K\u20326\u201d should be replaced with \u201ck\u2032+5K\u20325K\u20326\u201d. Parameters were incorrectly assigned in the legend for Figure 5. The incorrect parameters \u201cK\u20325K\u20326\u201d should be replaced with \u201c1/(K\u20325K\u20326)\u201d. Parameters were incorrectly assigned in the legend for Figure 6. The incorrect parameters \u201cK\u2032DPAk+4\u201d should be replaced with \u201ck+4K\u2032DPA\u201d. A common term was incorrectly used in the legend for Table 2. The incorrect parameters \u201cSteady-state ATPase\u201d should be replaced with \u201cSteady-state ATPase Activity\u201d. A parameter was incorrectly assigned in the legend for Table 4. The incorrect parameters \u201cIn the original article, there was a mistake in Table 3 as published. Rate and equilibrium constants, and corresponding values were incorrectly assigned. The corrected In the original article, there were mistakes in Tables 4 and 5 as published. Rate and equilibrium constants and a unit were incorrectly assigned. The corrected There were following errors in the original article. K\u20325K\u20326\u201d should be replaced with \u201c1/(K\u20325K\u20326)\u201d.Paragraph 8. The incorrect constants \u201cK\u2032DPA\u201d should be replaced with \u201c1/K\u2032DPA\u201d. The incorrect constants \u201cK\u2032DPAk\u2032+4\u201d should be replaced with \u201ck\u2032+4K\u2032DPA\u201d.Paragraph 10. The incorrect constants \u201cK\u2032DPA\u201d should be replaced with \u201c1/K\u2032DPA\u201d. Paragraph 11. The incorrect constants \u201cA correction has been made to 2. Results and Discussion: K\u20325K\u20326\u201d should be replaced with \u201c1/K\u20325K\u20326\u201d.A correction has been made to 3. Conclusions, Paragraph 1. The incorrect constants \u201cThe authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated."}
{"text": "Bolbelasmusunicornis  is critically reviewed throughout its range with emphasis on the Czech Republic and Slovakia. The species has been reliably recorded from 377 localities in 19 countries. New records are given from 152 localities of Bulgaria, Czech Republic, Germany, Hungary, Italy, Moldova, Poland, Romania, Serbia, Slovakia, Turkey, and Ukraine. For Germany, the species is recorded for the first time in 54 years. The occurrence of the species in Switzerland is confirmed by two historical specimens from Z\u00fcrich. The only known historical specimen labelled \u201cKaukasus\u201d is given, which could originate from Russia, where this species has not been recorded before . All published faunistic data from across the range are presented here in full, in several cases supplemented by details subsequently obtained by the author. Distribution maps are compiled separately for the Czech Republic and Slovakia, and for the entire range. A separate map is also available for Hungary, where approximately one-third of the known localities are located. Statistical data concerning the flight activity of adults, seasonal dynamics for part of the distribution area, details of records and notes on the bionomy and ethology of the species are provided. Possible feeding strategies for adults and larvae of B.unicornis are discussed, as well as current knowledge of the natural history of various representatives of the subfamily Bolboceratinae. A monitoring method for the species is proposed.The distribution of  Bolbelasmusunicornis  is a European species of earth-borer beetle extending into the western Asian part of Turkey with the centre of distribution in the Pannonian Basin . For this reason, it has been listed as a species of special conservation in many European countries. At the instigation of Slovakia, it has been included in Annexes II and IV of the Habitat Directive of the European Union (species in need of strict protection). As very few faunistic records are known from most countries, each new record is critically important to increase our knowledge to implement appropriate conservation strategies for the species. For more than 50 years the species has not been recorded in France, Slovenia, Bosnia and Herzegovina, Albania, and Moldova. It is probably extinct in France, Switzerland, Poland, and the Czech Republic.Schrank, 789 is a Scarabaeusunicornu by S.aeneas by th century, the species was often confused with Scarabaeusquadridens Fabricius, 1781 from India and later synonymised with it . Nothing is known about the immature stages and the diet of adults and larvae. However, some authors assumed that both adults and larvae feed on hypogeous fungi . In the congeneric species B.gallicus  and B.brancoi Hillert & Kr\u00e1l, 2016, this ability has also been recorded in larvae ; Given its secretive lifestyle and lack of knowledge of effective collecting methods, the distribution and bionomy of gi e.g., . Adults,Bolbocerodema Nikolajev, 1973 is considered here to be a subgenus of the genus Bolbocerosoma Schaeffer, 1906, in accordance with Bolboceratinae as a subfamily of Geotrupidae is consistent with The nomenclature used in this research follows PP \u2013 P\u0159\u00edrodn\u00ed pam\u00e1tka , PR \u2013 P\u0159\u00edrodn\u00ed rezervace (= Nature Reserve), and NPR \u2013 N\u00e1rodn\u00ed p\u0159\u00edrodn\u00ed rezervace . The abbreviation FSLG means flying slowly low above the ground. The following acronyms are used for time zones: CEST \u2013 Central European Summer Time, and EEST \u2013 Eastern European Summer Time. The abbreviation representing a collector/observer (see list below) with no further details mentioned means the collector and depository are identical (leg. and coll.). All details regarding observations of adults of B.unicornis (in particular their flight activities) were provided by the listed participants of these observations. The material has been identified by the author, the curators of the collections, or the observers and collectors listed.Faunistic records from the Czech Republic and Slovakia are divided into paragraphs beginning with a number representing the code of the faunistic square that refers to the Central European grid for mapping fauna and flora Fig. . For othThe following systems are used to transliterate cited literature and geographical or personal names in the Cyrillic and Armenian scripts: BGN/PCGN 2013 Agreement for Bulgarian, BGN/PCGN 1947 System for Russian, BGN/PCGN 2005 Agreement for Serbian, BGN/PCGN 2019 Agreement for Ukrainian, and BGN/PCGN 1981 System for Armenian.For the distribution map of the Czech Republic and Slovakia, the records are divided into three time periods: the records before 1960, records between 1960\u20131999, and records after 1999 Fig. , 18. TheStatistics on flights of adults were compiled for eight localities (seven Slovak and one Serbian), for which detailed data were available Tables \u20138. A tabThe graph of seasonal dynamics was generated with data obtained from countries of the Pannonian Basin for which data on a minimum of 30 specimens were available Fig. .The dates of Panzer\u2019s works are adopted from ABC Attila Bal\u00e1zs, \u010camovce, SlovakiaABZ Andrii Ivanovych Bachynskyi (\u0410\u043d\u0434\u0440\u0456\u0439 \u0406\u0432\u0430\u043d\u043e\u0432\u0438\u0447 \u0411\u0430\u0447\u0438\u043d\u0441\u044c\u043a\u0438\u0439), Zalishchyky, UkraineADW Alexander Dostal, Vienna, AustriaAGB Andr\u00e1s G\u00f3r, Biatorb\u00e1gy, HungaryAHB Adam Hergovits, Bratislava, SlovakiaAKB Attila Kot\u00e1n, Budapest, HungaryAMK Andr\u00e1s M\u00e1t\u00e9, Kecskem\u00e9t, HungaryAPC Alexandru-Mihai Pintilioaie, Com\u0103ne\u0219ti, RomaniaAPE Attila P\u00e1l, \u00c9rd, HungaryAPO Anton\u00edn Peutlschmid, Olomouc, Czech RepublicARC Adrian Ruic\u0103nescu, Cluj-Napoca, RomaniaASH Ale\u0161 Sedl\u00e1\u010dek, Hranice, Czech RepublicASK Artur Anatoliiovych Shekhovtsov (\u0410\u0440\u0442\u0443\u0440 \u0410\u043d\u0430\u0442\u043e\u043b\u0456\u0439\u043e\u0432\u0438\u0447 \u0428\u0435\u0445\u043e\u0432\u0446\u043e\u0432), Kharkiv, UkraineAUP \u00c1kos Uherkovich, P\u00e9cs, HungaryBBO Boris Buben\u00edk Sr., Ostrava, Czech RepublicBCK Csaba B\u00e1n, Kecskem\u00e9t, HungaryBJN Ji\u0159\u00ed Brestovansk\u00fd Jr., Neratovice, Czech RepublicBJO Boris Buben\u00edk Jr., Ostrava, Czech RepublicBKL Bence Krajcsovszky, L\u00e1batlan, HungaryBMP Marek Bunalski, Pozna\u0144, PolandBSP \u2020 Svatopluk B\u00edl\u00fd, Prague, Czech RepublicBVK Bohdan Mykolaiovych Vasko (\u0411\u043e\u0433\u0434\u0430\u043d \u041c\u0438\u043a\u043e\u043b\u0430\u0439\u043e\u0432\u0438\u0447 \u0412\u0430\u0441\u044c\u043a\u043e), Kyiv, UkraineCBE Csaba Bartha, Eger, HungaryCBK Csaba B\u00edr\u00f3, Kecskem\u00e9t, HungaryCKZ Csaba Kutasi, Zirc, HungaryCMI Cosmin-Ovidiu Manci, Ia\u0219i, RomaniaCSB Csaba Szab\u00f3ky, Budapest, HungaryCSS Csaba Szinet\u00e1r, Szombathely, HungaryCVK Csaba Vad\u00e1sz, Kecskem\u00e9t, HungaryCWP Christian Wieser, Pischeldorf, AustriaDCO Dan \u010cag\u00e1nek, Otrokovice, Czech RepublicDHH David Hrebe\u0148, Hav\u00ed\u0159ov, Czech RepublicDHP David Hor\u00e1k, Prost\u011bjov, Czech RepublicDJP Daniel Ju\u0159ena, Prost\u011bjov, Czech RepublicDKC Denis Keith, Chartres, FranceDKP David Kr\u00e1l, Prague, Czech RepublicDPB Dragan Pavi\u0107evi\u0107 (\u0414\u0440\u0430\u0433\u0430\u043d \u041f\u0430\u0432\u0438\u045b\u0435\u0432\u0438\u045b), Belgrade, SerbiaDPS Dmytro Protopopov (\u0414\u043c\u0438\u0442\u0440\u043e \u041f\u0440\u043e\u0442\u043e\u043f\u043e\u043f\u043e\u0432), Semyhiria, UkraineDRW Dominik Rabl, Vienna, AustriaDVB Dalibor V\u0161iansk\u00fd, Brno, Czech RepublicDVH Du\u0161an Vac\u00edk, Hranice, Czech RepublicDVZ Daniel V\u00edt, Zl\u00edn, Czech RepublicEJB Eduard Jendek, Bratislava, SlovakiaFKD Ferenc Klecska, Dunaharaszti, HungaryFPT Filip Pavel, T\u00fdni\u0161t\u011b nad Orlic\u00ed, Czech RepublicFSB Filip \u0160trba, Bratislava, SlovakiaFSP Franti\u0161ek \u0160t\u011bp\u00e1nek, P\u0159erov, Czech RepublicFTK Florian Theves, Weingarten (Baden), GermanyFTR Filip Trnka, Rychnov nad Kn\u011b\u017enou, Czech RepublicFTV Filip Trojan, Velk\u00e9 N\u011bm\u010dice, Czech RepublicGAB \u00c1d\u00e1m G\u00f3r, Biatorb\u00e1gy, HungaryGPB Gergely Petr\u00e1nyi, Budapest, HungaryGML Geoffrey Miessen, Li\u00e8ge, BelgiumGSB Gy\u0151z\u0151 Sz\u00e9l, Budapest, HungaryHDO Hryhorii Mykolaiovych Demydov (\u0413\u0440\u0438\u0433\u043e\u0440\u0456\u0439 \u041c\u0438\u043a\u043e\u043b\u0430\u0439\u043e\u0432\u0438\u0447 \u0414\u0435\u043c\u0438\u0434\u043e\u0432), Odessa, UkraineHMS Heinz Mitter, Steyr, AustriaHTR Hennadii Tarasenko (\u0413\u0435\u043d\u043d\u0430\u0434\u0456\u0439 \u0422\u0430\u0440\u0430\u0441\u0435\u043d\u043a\u043e), Rzhyshchiv, UkraineIIB Ionu\u021b \u0218tefan Iorgu, Bucharest, RomaniaIJN Ivo Jeni\u0161, N\u00e1klo, Czech RepublicIMO Ivo Martin\u016f, Olomouc, Czech RepublicIMP Ivan Marvan, Pardubice, Czech RepublicIPO Ivan Palou\u0161ek, Olomouc, Czech RepublicIRB Imre Retez\u00e1r, Budapest, HungaryITV Ilja Trojan, Velk\u00e9 N\u011bm\u010dice, Czech RepublicJAH \u2020 Josef Ad\u00e1mek, Hradec Kr\u00e1lov\u00e9, Czech RepublicJBB Jan Bezd\u011bk, Brno, Czech RepublicJCM Josef Chyb\u00edk, Mod\u0159ice, Czech RepublicJDB J\u00e1nos Dobos, Budapest, HungaryJDC Ji\u0159\u00ed Dvo\u0159\u00e1k, \u010cel\u010dice, Czech RepublicJHH Ji\u0159\u00ed H\u00e1jek, Hrobce, Czech RepublicJHL Jan Hele\u0161ic, Lu\u017eice (near Hodon\u00edn), Czech RepublicJHM Jarom\u00edr Hanu\u0161, Moravsk\u00e9 Bud\u011bjovice, Czech RepublicJHP Ji\u0159\u00ed Hanzl\u00edk, P\u0159erov\u2013Popovice, Czech RepublicJJP \u2020 Josef Jur\u010d\u00ed\u010dek, Prague, Czech RepublicJKB J\u00e1n Kodada, Bratislava, SlovakiaJKJ Jaroslav Kal\u00e1b, Jina\u010dovice, Czech RepublicJKO Jind\u0159ich Kuja, Olomouc, Czech RepublicJKP Josef Kro\u0161l\u00e1k, Plze\u0148, Czech RepublicJKV Josef Kadlec, Varnsdorf, Czech RepublicJMB J\u00f3zsef Muskovits, Budapest, HungaryJMD \u2020 Jaroslav M\u00e1slo, Doln\u00ed Radechov\u00e1, Czech RepublicJMH Jan Mat\u011bj\u00ed\u010dek, Hradec Kr\u00e1lov\u00e9, Czech RepublicJPH Jan Pelik\u00e1n, Hradec Kr\u00e1lov\u00e9, Czech RepublicJPP Ji\u0159\u00ed Plech\u00e1\u010d, Pecka, Czech RepublicJRC Jind\u0159ich Ry\u0161av\u00fd Jun., \u010cesk\u00e9 Bud\u011bjovice, Czech RepublicJRS Jaroslav Resl, Sn\u011b\u017en\u00e9 (near Nov\u00e9 M\u011bsto nad Metuj\u00ed), Czech RepublicJSP Jan Schneider, Prague, Czech RepublicJST J\u00f3zsef S\u00e1r, Teklafalu, HungaryJSU Ji\u0159\u00ed Spru\u017eina, \u00dast\u00ed nad Labem, Czech RepublicJTK Jaroslav Turna, Kostelec na Han\u00e9, Czech RepublicJVP \u2020 Jan Vitner, Prague, Czech RepublicJZJ Jaroslav \u017d\u00e1k, Jezernice, Czech RepublicKBB Karol Bucsek, Bratislava, SlovakiaKDO Karel Dole\u017eel, Olomouc, Czech RepublicKFW Katrin Fuchs, Vienna, AustriaKHE Kriszti\u00e1n Harmos, Eger, HungaryKLP Ji\u0159\u00ed Kl\u00edcha, Prague, Czech RepublicKPH Karel Pils, Hlohovec, SlovakiaKPL Krzysztof Pa\u0142ka, Lublin, PolandKPV \u2020 Karel Pol\u00e1\u010dek, Vysok\u00e9 M\u00fdto, Czech RepublicKRU Kv\u011btoslav Resl, Uhersk\u00fd Brod, Czech RepublicKVB Kl\u00e1ra Varga B\u00e1nn\u00e9, Kecskem\u00e9t, HungaryKVM Kirill Vladimirovich Makarov (\u041a\u0438\u0440\u0438\u043b\u043b \u0412\u043b\u0430\u0434\u0438\u043c\u0438\u0440\u043e\u0432\u0438\u0447 \u041c\u0430\u043a\u0430\u0440\u043e\u0432), Moscow, RussiaKVS V\u00e1clav K\u0159ivan, \u0160t\u011bm\u011bchy, Czech RepublicLAB L\u00e1szl\u00f3 \u00c1d\u00e1m, Budapest, HungaryLEN Ladislav Ernest, Nymburk, Czech RepublicLFI Lucian Fusu, Ia\u0219i, RomaniaLHI Lucian H\u0103nceanu, Ia\u0219i, RomaniaLKK Ladislav Kandrn\u00e1l, Kunovice (near Uhersk\u00e9 Hradi\u0161t\u011b), Czech RepublicLKM \u013dudev\u00edt Ka\u0161ovsk\u00fd, Martin, SlovakiaLMB Levente Erik Moharos, Budakeszi, HungaryLMN Ladislav Mi\u0161ko, Nov\u00e9 Z\u00e1mky, SlovakiaLMO Lubo\u0161 Mazal, Olomouc, Czech RepublicLMT Ladislav Mencl, T\u00fdnec nad Labem, Czech RepublicLNB L\u00e1szl\u00f3 N\u00e1dai, Budapest, HungaryLRL Luciano Ragozzino Lerma, ItalyMBO Michal Bedna\u0159\u00edk, Olomouc, Czech RepublicMBF Marco Bastianini, Follonica, ItalyMBP Milan Brabec, Prague, Czech RepublicMHP Marcel H\u00e1jek, Plze\u0148, Czech RepublicMJR Martin Jagelka, Roho\u017en\u00edk, SlovakiaMKJ Martin Kejzlar, Jev\u00ed\u010dko, Czech RepublicMKY Mark Yuriyevich\u2019 K\u2019alashyan (\u0544\u0561\u0580\u056f \u0545\u0578\u0582\u0580\u056b\u0587\u056b\u0579 \u0554\u0561\u056c\u0561\u0577\u0575\u0561\u0576), Yerevan, ArmeniaMLB M\u00e1rk Luk\u00e1tsi, Budapest, HungaryMLS Milo\u0161 Lo\u0161\u0165\u00e1k, \u0160umperk, Czech RepublicMNB Martin N\u011bmec, Brno, Czech RepublicMNR Milan Nikod\u00fdm, Roztoky (near Prague), Czech RepublicMOB Oto Majzlan, Bratislava, SlovakiaMPK Michal Pikner, Kn\u011b\u017epole (near Uhersk\u00e9 Hradi\u0161t\u011b), Czech RepublicMPN Milo\u0161 Popovi\u0107 (\u041c\u0438\u043b\u043e\u0448 \u041f\u043e\u043f\u043e\u0432\u0438\u045b), Ni\u0161, SerbiaMPP Milan Sl\u00e1ma, Prague, Czech RepublicMPV Milan Pilar\u010d\u00edk, Velk\u00e9 Pavlovice, Czech RepublicMRM Mikl\u00f3s Ringler, Munich, GermanyMRV Milan Ry\u0161av\u00fd, Vlko\u0161 (near P\u0159erov), Czech RepublicMSB Milan \u0160trba, Bratislava, SlovakiaMSN Marko \u0160\u0107iban (\u041c\u0430\u0440\u043a\u043e \u0428\u045b\u0438\u0431\u0430\u043d), Novi Sad, SerbiaMSZ Miloslav \u0160anda, \u017datec, Czech RepublicMTM Maximilian Teodorescu, M\u0103gurele, RomaniaMTS Milan Truba\u010d\u00edk, Star\u00e9 M\u011bsto (near Uhersk\u00e9 Hradi\u0161t\u011b), Czech RepublicMVP Martin Volf, Prague, Czech RepublicMYP Vladislav Mal\u00fd, Prague, Czech RepublicMZB Miroslav Z\u00faber, Bradlec, Czech RepublicMZK Miroslav Ihorovych Zaika (\u041c\u0438\u0440\u043e\u0441\u043b\u0430\u0432 \u0406\u0433\u043e\u0440\u043e\u0432\u0438\u0447 \u0417\u0430\u0457\u043a\u0430), Kyiv, UkraineMZP Zdena Martinov\u00e1 & Zden\u011bk Znamen\u00e1\u010dek, Prague, Czech RepublicNKB Na\u010fa Kazdov\u00e1, Brno, Czech RepublicNPB Norbert Pometk\u00f3, Budapest, HungaryOHS Oliver Hillert, Sch\u00f6neiche bei Berlin, GermanyOKO Oleksandr Oleksandrovych Kolomeichuk (\u041e\u043b\u0435\u043a\u0441\u0430\u043d\u0434\u0440 \u041e\u043b\u0435\u043a\u0441\u0430\u043d\u0434\u0440\u043e\u0432\u0438\u0447 \u041a\u043e\u043b\u043e\u043c\u0435\u0439\u0447\u0443\u043a), Odessa, UkraineOMB \u2020 Ott\u00f3 Merkl, Budapest, HungaryOSD Oleksandr Oleksiiovych Sukhenko (\u041e\u043b\u0435\u043a\u0441\u0430\u043d\u0434\u0440 \u041e\u043b\u0435\u043a\u0441\u0456\u0439\u043e\u0432\u0438\u0447 \u0421\u0443\u0445\u0435\u043d\u043a\u043e), Dnipro, UkraineOSO Ond\u0159ej Sabol, Ostrava\u2013Nov\u00e1 B\u011bl\u00e1, Czech RepublicOVK Oleksii Vasyliuk (\u041e\u043b\u0435\u043a\u0441\u0456\u0439 \u0412\u0430\u0441\u0438\u043b\u044e\u043a), Kyiv, UkrainePCB Petr \u010cechovsk\u00fd, Brno, Czech RepublicPFS P\u00e9ter Farkas, S\u00e1gv\u00e1r, HungaryPIL Pavel Imr\u00ed\u0161ek, Louny, Czech RepublicPJH Josef Pavlas, Hav\u00ed\u0159ov, Czech RepublicPJL Pavel J\u00e1chymek, Luha\u010dovice, Czech RepublicPKB P\u00e9ter Kov\u00e1cs, Budapest, HungaryPKG Peter Kurina, Gajary, SlovakiaPMB Peter Mih\u00e1lik, Bratislava, SlovakiaPPB Petr Pachol\u00e1tko, Brno, Czech RepublicPSZ Petr St\u0159edul\u00ednsk\u00fd, Zl\u00edn, Czech RepublicPVP Petr V\u010deli\u010dka, Prague, Czech RepublicPZW Petr Z\u00e1bransk\u00fd, Vienna, AustriaRCP \u2020 Radek \u010cervenka, Prague, Czech RepublicRCR Roman Cs\u00e9falvay, Rohovce\u2013Kyselica, SlovakiaREE R\u00f3bert Enyedi, Eger, HungaryRFO \u2020 Rostislav Forn\u016fsek, Olomouc, Czech RepublicRGM Rudolf Gabzdil, Michalovce, SlovakiaRHB Roman Hergovits, Bratislava, SlovakiaRHK Rostyslav Pavlovych Herasymov (\u0420\u043e\u0441\u0442\u0438\u0441\u043b\u0430\u0432 \u041f\u0430\u0432\u043b\u043e\u0432\u0438\u0447 \u0413\u0435\u0440\u0430\u0441\u0438\u043c\u043e\u0432), Kyiv, UkraineRKP Radim Kl\u00ed\u010d, Prost\u011bjov, Czech RepublicRMU Robert Mach\u00e1lek, Uhersk\u00e9 Hradi\u0161t\u011b, Czech RepublicRPM Riccardo Pittino, Milan, ItalyRSG Roland \u0160tefanovi\u010d, Galanta, SlovakiaRSP Radim \u0160igut, Paskov, Czech RepublicRVO \u2020 Radovan Veigler, Olomouc, Czech RepublicRZJ Renata \u017d\u00e1kov\u00e1, Jezernice, Czech RepublicSBP S\u00e1ndor B\u00e9rces, Pom\u00e1z, HungarySBS Stoyan Beshkov, Sofia, BulgariaSDP Stanislav Dole\u017eal, Plze\u0148\u2013Bo\u017ekov, Czech RepublicSIB S\u00e1ndor Ilniczky, Budapest, HungarySKP Ji\u0159\u00ed Sk\u00fdpala, Prague, Czech RepublicSPP Serge Peslier, Perpignan, FranceSRB Richard Schn\u00fcrmacher, Bratislava, SlovakiaSRL Stefan Rabl, Lengenfeld, AustriaSTK Serhii Vasylovych Tsykal (\u0421\u0435\u0440\u0433\u0456\u0439 \u0412\u0430\u0441\u0438\u043b\u044c\u043e\u0432\u0438\u0447 \u0426\u0438\u043a\u0430\u043b), Kyiv, UkraineSZM Stefano Ziani, Meldola, ItalyTBK Torsten Bittner, Wei\u00dfenberg, GermanyTDB Tam\u00e1s Deli, B\u00e9k\u00e9scsaba, HungaryTDS Tibor Danyik, Szarvas, HungaryTGK Tom\u00e1\u0161 Grulich, Doln\u00ed Stud\u00e9nky\u2013Kr\u00e1lec, Czech RepublicTKB \u2020 Tibor Kov\u00e1cs Sr., B\u00e1tonyterenye\u2013Kisterenye, HungaryTKG Tibor Kov\u00e1cs Jr., Gy\u00f6ngy\u00f6s, HungaryTKH Tom\u00e1\u0161 Kopeck\u00fd, Hradec Kr\u00e1lov\u00e9, Czech RepublicTKK Tam\u00e1s Kiss, Kecskem\u00e9t, HungaryTNB Tam\u00e1s N\u00e9meth, G\u00f6d\u00f6ll\u0151, HungaryTSH Tibor Spev\u00e1r, Hlohovec, SlovakiaTVP Tom\u00e1\u0161 Vendl, Prague, Czech RepublicVDP V\u00e1clav Dongres, Plze\u0148, Czech RepublicVGG \u2020 Vadim Gennad\u2019yevich Grach\u00ebv (\u0412\u0430\u0434\u0438\u043c \u0413\u0435\u043d\u043d\u0430\u0434\u044c\u0435\u0432\u0438\u0447 \u0413\u0440\u0430\u0447\u0451\u0432), Moscow, RussiaVJP Vojt\u011bch Ji\u0159\u00ed\u010dek, Prost\u011bjov, Czech RepublicVKK Vitalii Volodymyrovych Kavurka (\u0412\u0456\u0442\u0430\u043b\u0456\u0439 \u0412\u043e\u043b\u043e\u0434\u0438\u043c\u0438\u0440\u043e\u0432\u0438\u0447 \u041a\u0430\u0432\u0443\u0440\u043a\u0430), Kyiv, UkraineVKS V\u00edt\u011bzslav Kub\u00e1\u0148, \u0160lapanice, Czech RepublicVLP Jarom\u00edr Vali\u0161, P\u0159erov, Czech RepublicVMP Vlastimil Mihal, P\u0159erov, Czech RepublicVPB Vilmos Polonyi, Budapest, HungaryVRH Vladislav \u0158eb\u00ed\u010dek, Hradi\u0161tko, Czech RepublicVSI Valentin Sz\u00e9n\u00e1si, Isaszeg, HungaryVSK Vasyl Musiiovych Serhiienko (\u0412\u0430\u0441\u0438\u043b\u044c \u041c\u0443\u0441\u0456\u0439\u043e\u0432\u0438\u0447 \u0421\u0435\u0440\u0433\u0456\u0454\u043d\u043a\u043e), Kyiv, UkraineVSM Viktor Viktorovych Strenada (\u0412\u0456\u043a\u0442\u043e\u0440 \u0412\u0456\u043a\u0442\u043e\u0440\u043e\u0432\u0438\u0447 \u0421\u0442\u0440\u0435\u043d\u0430\u0434\u0430), Mykolaiv, UkraineVTH Vladim\u00edr Thomka, Humenn\u00e9, SlovakiaVTO Viacheslav Anatoliiovych Trach (\u0412\u044f\u0447\u0435\u0441\u043b\u0430\u0432 \u0410\u043d\u0430\u0442\u043e\u043b\u0456\u0439\u043e\u0432\u0438\u0447 \u0422\u0440\u0430\u0447), Odessa, UkraineVUB Viorel Ungureanu, Buz\u0103u, RomaniaVVO Vladim\u00edr Vyhn\u00e1lek, Olomouc, Czech RepublicVZO Vladim\u00edr Zeman, Olomouc, Czech RepublicWBW Wolfgang Barries, Vienna, AustriaWHS Walter Heinz, Schwanfeld, GermanyYHS Yurii Mykolaiovych Geriak (\u042e\u0440\u0456\u0439 \u041c\u0438\u043a\u043e\u043b\u0430\u0439\u043e\u0432\u0438\u0447 \u0490\u0435\u0440\u044f\u043a), Sambir, UkraineYKL Yurii Vasylovych Kanarskyi (\u042e\u0440\u0456\u0439 \u0412\u0430\u0441\u0438\u043b\u044c\u043e\u0432\u0438\u0447 \u041a\u0430\u043d\u0430\u0440\u0441\u044c\u043a\u0438\u0439), Lviv, UkraineYKO Yevhenii Vitaliiovych Khalaim (\u0404\u0432\u0433\u0435\u043d\u0456\u0439 \u0412\u0456\u0442\u0430\u043b\u0456\u0439\u043e\u0432\u0438\u0447 \u0425\u0430\u043b\u0430\u0457\u043c), Odessa, UkraineYSK Yurii Yevhenovych Skrylnyk (\u042e\u0440\u0456\u0439 \u0404\u0432\u0433\u0435\u043d\u043e\u0432\u0438\u0447 \u0421\u043a\u0440\u0438\u043b\u044c\u043d\u0438\u043a), Kharkiv, UkraineZBB Zoran Bo\u017eovi\u0107 (\u0417\u043e\u0440\u0430\u043d \u0411\u043e\u0436\u043e\u0432\u0438\u045b), Batajnica, SerbiaZBP Zs\u00f3fia Mocskonyi B\u00e9rcesn\u00e9, Pom\u00e1z, HungaryZCP Zden\u011bk \u010cerm\u00e1k, Prost\u011bjov, Czech RepublicZDP \u2020 Zden\u011bk Dole\u017eal, Plze\u0148, Czech RepublicZKB Zolt\u00e1n K\u00f6rmendy, Budapest, HungaryZKM Zden\u011bk Kraus, Mikulovice (near Znojmo), Czech RepublicZLB Zden\u011bk La\u0161t\u016fvka, Brno, Czech RepublicZVP Zden\u011bk Vancl, Police nad Metuj\u00ed, Czech RepublicBNMS Brukenthal National Museum, Sibiu, RomaniaBZLA Biologiezentrum Linz, AustriaCMZCCroatian Natural History Museum, Zagreb, CroatiaCUIR Alexandru Ioan Cuza University, Ia\u0219i, RomaniaETHZ Entomological collection of the Swiss Federal Institute of Technology, Z\u00fcrich, SwitzerlandFGBI Franziskaner Gymnasium Bozen, Bolzano, ItalyFMNHFinnish Museum of Natural History LUOMUS, University of Helsinki, Helsinki, FinlandGANM \u201cGrigore Antipa\u201d National Museum of Natural History, Bucharest, RomaniaGUNU Nizhyn Gogol State University, Nizhyn, UkraineHNHMHungarian Natural History Museum, Budapest, HungaryIECAInstitute of Entomology, Biology Centre of the Czech Academy of Sciences, \u010cesk\u00e9 Bud\u011bjovice, Czech RepublicIZCMInstitute of Zoology of the Academy of Sciences of Moldova, Chi\u0219in\u0103u, Republic of MoldovaJHIS Jovan Had\u017ei Institute of Biology of the Research Centre of the Slovenian Academy of Sciences and Arts, Ig, SloveniaLKKA Landesmuseums f\u00fcr K\u00e4rnten, Klagenfurt am W\u00f6rthersee, AustriaMCASMuseo Civico Archeologico e di Scienze Naturali \u201cFederico Eusebio\", Alba, ItalyMCZRMuseo Civico di Zoologia, Rome, ItalyMFSNMuseo Friulano di Storia Naturale, Udine, ItalyMHKC Museum of Eastern Bohemia in Hradec Kr\u00e1lov\u00e9, Hradec Kr\u00e1lov\u00e9, Czech RepublicMHNGMus\u00e9um d\u2019histoire naturelle de Gen\u00e8ve, Geneva, SwitzerlandMIZPMuseum and Institute of Zoology of the Polish Academy of Sciences, Warsaw, PolandMJMC Muzeum jihov\u00fdchodn\u00ed Moravy ve Zl\u00edn\u011b, Zl\u00edn, Czech RepublicMKPC Muzeum Komensk\u00e9ho v P\u0159erov\u011b, P\u0159erov, Czech RepublicMMBCMoravian Museum, Brno, Czech RepublicMNBG Leibniz-Institut f\u00fcr Evolutions- und Biodiversit\u00e4tsforschung, Museum f\u00fcr Naturkunde, Berlin, GermanyMMGH M\u00e1tra Museum of the Hungarian Natural History Museum, Gy\u00f6ngy\u00f6s, HungaryMMSHM\u00f3ra Ferenc Museum, Szeged, HungaryMNFI Natural History Museum \u201cLa Specola\u201d, Florence, ItalyMNHNMus\u00e9um national d\u2019Histoire naturelle, Paris, FranceMNSA Museum Nieder\u00f6sterreich, Sankt P\u00f6lten, AustriaMPGU Moscow Pedagogical State University, Moscow, RussiaMSNB Museo di Scienze Naturali dell\u2019Alto Adige, Bolzano, ItalyMSNGMuseo Civico di Storia Naturale \u201cGiacomo Doria\u201d, Genoa, ItalyMSNMMuseo Civico di Storia Naturale, Milan, ItalyMTDG Senckenberg Naturhistorische Sammlungen, Museum f\u00fcr Tierkunde, Dresden, GermanyMUSEMuseo delle Scienze, Trento, ItalyMZLUBiological Museum, Lund University, Lund, SwedenMZSFMus\u00e9e zoologique de l\u2019universit\u00e9 et de la ville de Strasbourg, Strasbourg, FranceNHMBNaturhistorisches Museum Basel, SwitzerlandNHMDNatural History Museum of Denmark, University of Copenhagen, Copenhagen, DenmarkNHMK State Natural History Museum of V. N. Karazin Kharkiv National University, Kharkiv, UkraineNHMLNatural History Museum, London, United KingdomNHMU National Science and Natural History Museum of the National Academy of Sciences of Ukraine, Kyiv, UkraineNHMWNaturhistorisches Museum Wien, Vienna, AustriaNMAGNaturmuseum Augsburg, GermanyNMBENaturhistorisches Museum Bern, SwitzerlandNMCM National Museum of Ethnography and Natural History, Chi\u0219in\u0103u, Republic of MoldovaNMEGNaturkundemuseum Erfurt, GermanyNMPCNational Museum, Prague, Czech RepublicNMSBNational Museum of Natural History, Sofia, BulgariaPMSLSlovenian Museum of Natural History, Ljubljana, SloveniaRBINRoyal Belgian Institute of Natural Sciences, Brussels, BelgiumRMNH Naturalis Biodiversity Centre (formerly Rijksmuseum van Natuurlijke Historie), Leiden, NetherlandsSDEISenckenberg Deutsches Entomologisches Institut, M\u00fcncheberg, GermanySIZKI. I. Schmalhausen Institute of Zoology of National Academy of Sciences of Ukraine, Kyiv, UkraineSMLU State Museum of Natural History, Lviv, UkraineSMNKStaatliches Museum f\u00fcr Naturkunde Karlsruhe, GermanySMNSStaatliches Museum f\u00fcr Naturkunde Stuttgart, GermanySMOCSilesian Museum, Opava, Czech RepublicSNMS Slovak National Museum\u2013Natural History Museum, Bratislava, SlovakiaTLMFTiroler Landesmuseum Ferdinandeum, Innsbruck, AustriaTMLS Tekovsk\u00e9 m\u00fazeum v Leviciach, Levice, SlovakiaUMJGUniversalmuseum Joanneum, Graz, AustriaVMHS Vihorlatsk\u00e9 m\u00fazeum Humenn\u00e9, SlovakiaZFMKZoologishes Forschungsmuseum Alexander Koenig, Bonn, GermanyZMNU Zoological Museum of the Taras Shevchenko National University, Kyiv, UkraineZMPC Z\u00e1pado\u010desk\u00e9 muzeum v Plzni, Plze\u0148, Czech RepublicZINR Zoological Institute of Russian Academy of Sciences, Saint Petersburg, RussiaZSMGStaatliche Naturwissenschaftliche Sammlungen Bayerns, Zoologische Staatssammlung, Munich, GermanyZUDH Department of Nature Conservation, Zoology and Game Management, University of Debrecen, Debrecen, HungaryGEOTRUPIDAE Latreille, 1802Family: Bolboceratinae Mulsant, 1842Subfamily: BOLBELASMINI Iablokoff-Khnzorian, 1977Tribe: Bolbelasmus Boucomont, 1911Genus: Bolbelasmus Boucomont, 1911Subgenus: Bolbelasmus (Bolbelasmus) unicornis Species: Published data5354: \u201cKummer\u201d [= Hrad\u010dany near Mimo\u0148], 1 \u2642 flying in the evening, no other data .her data . This sp7067: Hovorany, 6.v.1941, Jan Roubal leg.  labelled as Bolbelasmusunicornis, with black specimens of the same species, correctly labelled as Odonteusarmiger, have been found in the Hude\u010dek\u2019s collection in MKPC; no specimens of B.unicornis were discovered in this collection .? her data , 1930; Bher data . These t6870: \u201cUngarisch Hradisch\u201d [= Uhersk\u00e9 Hradi\u0161t\u011b] env., Morava River valley, no other data ; \u010cej\u010d env., \u201cMansonova step\u201d [= so-called \u201cManson\u2019s steppe\u201d], 48\u00b055'32.1\"N, 16\u00b058'46.6\"E, ca. 210 m a.s.l., 20.vi.1986, 1 \u2640 FSLG after sunset, PCB; 17.vi.1988, 1 \u2640 FSLG after sunset, PCB; 21.vi.1988, 2 \u2640\u2640 FSLG after sunset, VKS; 27.vi.1988, 8 spec. FSLG after sunset , VKS; 29.vi.1988, 2 spec. FSLG after sunset, VKS; 19.vi.1989, 1 \u2640 FSLG after sunset, JTK; 16.vi.1995, 1 \u2642 FSLG after sunset, VKS  , 24.v.1985, 2 spec., RFO ; 17.vii.1990, 1 \u2642 and 1 \u2640, 5.v.1992, 1 \u2642, MTS , 103 m a.s.l., 5.vii.1975, plant materials alluviated by flooded Danube and Ipe\u013e rivers, 1 \u2640, VKS leg. et coll., 1 \u2642, PPB leg., coll. VKS .7781: \u201cPlachti[n]ce\u201d , 5.vi.1938, [Rudolf] Schwarz leg. , SKP leg., coll. SPP , LMT (er), LMT .7494: Slansk\u00e1 Huta env., 48\u00b034'54.8\"N, 21\u00b028'31.7\"E, 600 m a.s.l., 24.vii.1972, 1 \u2640 crawling on the ground after sunset, ZLB obs. + photo \u2013 see Fig. ZLB pers. comm., 2022).7596: Ladmovce, 9.viii.1982, 2 \u2642\u2642 excavated from their burrows from a depth of 8 cm, and 1 \u2640 from a depth of ca. 20 cm, LMT , A. Mackov\u00e1 leg. , 132\u2013133 m a.s.l., 31.vii.2009, 1 \u2640, at UV light, KBB leg., coll. DVH; 18.viii.2014, 1 \u2640, at light, KBB obs.; 20.viii.2014, 1 \u2642 and 1 \u2640 FSLG after sunset, AHB and RHB obs.; 27.viii.2014, 2 \u2640\u2640 FSLG after sunset, AHB obs.; 5.ix.2014, 3 \u2642\u2642 and 2 \u2640\u2640 FSLG after sunset, PKG and RHB obs.; 3.vi.2015, 11 \u2642\u2642 and 4 \u2640\u2640 flying slowly 10\u201320 cm above the ground after sunset, AHB and RHB obs.; 5.vi.2015, 5 \u2642\u2642 and 1 \u2640 flying slowly 10\u201320 cm above the ground after sunset, PKG and RHB obs.; 7.vi.2015, 4 \u2642\u2642 FSLG after sunset, AHB obs.; 26.viii.2015, 2 \u2640\u2640 flying slowly ca. 0.5 m above the ground at 20.20 and 20.30 CEST, DJP obs.; 28.viii.2015, 1 \u2640 flying ca. 10\u201320 cm above the ground at 20.27 CEST, AHB obs.; 29.v.2016, 7 spec. FSLG after sunset, together with ca. 15 spec. of Od.armiger, EJB, MSB and RHB obs.; 7.vi.2016, 11 spec.  FSLG at 21.25\u201321.45 CEST, AHB and RHB obs. , DJP, FSB, MSB and PKG obs. ; Bratislava \u2013 Podunajsk\u00e9 Biskupice, Kop\u00e1\u010d Island, PR Kop\u00e1\u010dsky ostrov, ca. 48\u00b05'41.97\"N, 17\u00b09'43.14\"E, 132 m a.s.l., 13.vi.2006, 1 \u2640, Malaise trap, MOB leg., coll. VKS; 30.v.2016, 2 \u2642\u2642, and 3 \u2640\u2640 FSLG after sunset, together with ca. 15 spec. of Od.armiger, EJB and RHB obs.; 7.vi.2016, 1 spec. FSLG after sunset, MSB obs.; 23.vi.2016, 2 \u2642\u2642 FSLG at 21.32 and 21.38 CEST, MSB obs.; 1.vii.2016, 2 \u2642\u2642 FSLG at 21.27 and 21.43 CEST, MSB obs.; 19.vii.2016, 4 spec. FSLG after sunset, EJB and JKB obs.; 20.vii.2016, 7 spec. FSLG after sunset, EJB and JKB obs.; ca. 48\u00b05'39.5\"N, 17\u00b09'42.3\"E, and 48\u00b05'43.8\"N, 17\u00b09'30.4\"E, 14.viii.2016, 7 spec. FSLG after sunset, DJP, ITV and VKB obs.; ca. 48\u00b05'45.8\"N, 17\u00b09'41.9\"E, 19.v.2018, 1 \u2642 and 1 \u2640 FSLG after sunset, MRV and VMP obs.; 9.vi.2018, 7 \u2642\u2642 and 9 \u2640\u2640 FSLG after sunset, together with ca. 20 spec. of Od.armiger, JHP, MRV and VMP obs.; 14.vii.2018, 1 \u2642 and 4 \u2640\u2640 FSLG after sunset, JRC and MRV obs.7869\u20137969: \u201c\u0160tef\u00e1nikovce\u201d [= Rovinka near Dunajsk\u00e1 Lu\u017en\u00e1], ca. 130 m a.s.l., May 1949, tens of spec. observed during the day sitting on the tops of the grass blades above the water on a flooded steppe meadow (after the flood), Josef Marvan obs., 2 spec. (\u2642 and \u2640) leg., coll. IMP.7969: Bratislava \u2013 \u010cunovo, PR Ostrovn\u00e9 l\u00fa\u010dky , TSH obs.7772: \u0160opor\u0148a, [ca. 122 m a.s.l.], July 1952, 1 \u2642, Kotek leg., coll. MHKC.7373: Modrovka, [ca. 170 m a.s.l.], 15.vii.1979, 1 \u2640, Mrklovsk\u00fd leg., coll. Ladislav Boj\u010duk deposited in MHKC.7074: Tren\u010d\u00edn \u2013 Zlatovce [env.], June 1926, [Rudolf] \u010cepel\u00e1k leg., 1 \u2642 and 1 \u2640 in coll. Josef Gottwald deposited in NHMB, 2 \u2640\u2640 in coll. Paolo Luigioni deposited in MCZR, 1 \u2642 in coll. Zden\u011bk Tesa\u0159 deposited in SNMS, 1 \u2642 in coll. MIZP; Tren\u010d\u00edn \u2013 Zlatovce [env.], June 1926, 1 \u2640, collector not specified, coll. TLMF; [Tren\u010d\u00edn \u2013] Zlatovce [env.], no other data, 2 \u2640\u2640 in coll. Zden\u011bk Tesa\u0159 deposited in SNMS; June 1931, L[adislav] Korbel leg., 1 \u2642 in coll. JJP, 1 \u2640 in coll. Ladislav Dan\u011bk deposited in MHKC; June 1935, 1 \u2642 and 1 \u2640 (ex coll. Johann Peter Wolf), \u201ccol. Kardasch\u201d [= Gregor Kardasch leg.], coll. ETHZ.7174: \u201cTrencsen Ungarn\u201d , undated, 1 \u2642 and 1 \u2640 (ex coll. Engelbert Pawlik) in coll. NMPC, 1 \u2642 and 1 \u2640 in coll. FMNH, 2 spec. in coll. ZSMG, 1 spec. in coll. MTDG, 2 \u2640\u2640 (ex coll. P. Franck) in coll. MIZP, 1 \u2640  in coll. SMNS; Tren\u010d\u00edn, no other data, 3 spec. in coll. NHMW, 1 \u2642 in coll. MNBG, 1 spec in coll. SZM; \u201cTrencin Slow.\u201d , no other data, 1 \u2640 in coll. Leopold Mader deposited in MNSA; \u201cTren\u010d\u00edn, Tch\u00e9coslovaquie\u201d , undated, 1 \u2642 and 1 \u2640,\u201ccoll. J[oseph] Clermont\u201d, coll. Jacques Baraud deposited in MNHN; \u201cSlovakia Tren\u010d\u00edn\u201d, 2.vii.[year not specified], no other data, 1 \u2640 in coll. MHKC; Tren\u010d\u00edn, undated, [Rudolf] \u010cepel\u00e1k [leg.], 6 \u2642\u2642 and 5 \u2640\u2640 in coll. Leopold Mader deposited in MNSA, 5 \u2642\u2642 and 4 \u2640\u2640 in coll. SNMS, 3 \u2642\u2642 and 5 \u2640\u2640 in coll. MHNG, 2 \u2642\u2642 and 3 \u2640\u2640 in coll. MNBG, 3 \u2642\u2642 and 1 \u2640  in coll. SDEI, 2 \u2642\u2642 and 2 \u2640\u2640 in coll. Henri Coiffait deposited in MNHN, 3 \u2642\u2642 and 1 \u2640 (ex coll. Johann Peter Wolf) in coll. ETHZ, 3 \u2642\u2642 and 1 \u2640 in coll. MHKC, 2 \u2642\u2642 (ex coll. Sten Stockmann) in coll. FMNH, 1 \u2642 and 1 \u2640 in coll. MIZP, 1 \u2642 and 1 \u2640 in coll. SMNS, 1 \u2642 in coll. RBIN, 1 \u2642 in coll. Ladislav Dan\u011bk deposited in MHKC, 1 \u2640 in coll. Jacques Baraud deposited in MNHN, 1 \u2642 in coll. Georg Frey deposited in NHMB, 1 \u2642 and 1 \u2640 in coll. Vladim\u00edr Zoufal deposited in MMBC, 1 \u2640 in coll. Emil Jagemann deposited in MMBC, 2 spec. in coll. ZSMG, 1 \u2642 (ex coll. Antonio Porta) in coll. MSNM, 1 spec. (head and pronotum missing) in coll. RMNH, 2 \u2642\u2642 in coll. LEN, 1 \u2642 in coll. DKC, 1 \u2640 in coll. VKS; Tren\u010d\u00edn, undated [most likely late 1920s/early 1930s], Z. Zeman leg., 1 \u2640 in coll. SMNS, 1 \u2642 in coll. VKS; Tren\u010d\u00edn, undated, 1 \u2642, V[il\u00e9m] Steidl [leg.], coll. MIZP; Tren\u010d\u00edn, undated, [Ladislav] Krejc\u00e1rek [leg.], 2 \u2642\u2642 and 1 \u2640 in coll. TMLS  in coll. NHMB, 1 \u2642 in coll. FMNH; Tren\u010d\u00edn, 1931, 3 \u2642\u2642 and 3 \u2640\u2640, [Rudolf] \u010cepel\u00e1k [leg.], coll. Paolo Luigioni deposited in MCZR; Tren\u010d\u00edn, May 1931, Dr A[lois] Richter leg., 1 \u2640 in coll. NMPC, 1 \u2640 in coll. MJMC; \u201cTrencsin\u201d [= Tren\u010d\u00edn], undated, 1 spec., S. Kardasch [leg.], coll. SMNK; Tren\u010d\u00edn, June 1935, 1 spec., G[regor] Kardasch [leg.], coll. SMNK; Tren\u010d\u00edn, 1936, 2 spec., [Rudolf] \u010cepel\u00e1k [leg.], coll. SMNK; Tren\u010d\u00edn, June 1936, 1 \u2642 and 1 \u2640, [Rudolf] \u010cepel\u00e1k [leg.], coll. VKS; Tren\u010d\u00edn, July [19]36, 1 \u2642 in coll. Jan Vol\u00e1k deposited in MHKC.TMLS see , 1 \u2640 in 7274: [Pova\u017esk\u00fd Inovec Mts], Inovec [hill env.], 1 \u2642, [Ladislav] Krejc\u00e1rek [leg.], coll. Josef Gottwald deposited in NHMB.7674: Nitra [env.], 1950, no other data, 1 \u2642 in coll. MHKC.8177: \u0160t\u00farovo env., Beliansk\u00e9 kopce hills, \u201cHegyfarok\u201d [= Modr\u00fd vrch], 47\u00b049'8.09\"N, 18\u00b039'32.4\"E, ca. 150 m a.s.l., 20.viii.2005, 1 spec., at light after midnight, 1.ix.2005, 1 spec., at light after midnight, 14.vi.2006, 2 spec., at light, 15.vi.2006, 1 spec., at light, 15.vi.2007, 1 spec., at light, 29.vii.2008, 2 spec., at light, 30.vii.2008, 1 spec., VVO obs.; Modr\u00fd vrch, PR V\u0155\u0161ok, 47\u00b049'6\"N, 18\u00b039'33\"E, ca. 150 m a.s.l., 22.v.2014, 1 \u2642, at light, OSO obs.; 47\u00b049'13.5\"N, 18\u00b039'21.5\"E, ca. 195 m a.s.l., 4.vi.2015, 1 \u2640 at UV light at 21.30\u20130.30 CEST and 1 \u2640 at UV light (the same trap) at 1.15 CEST (5.vi.2015), OSO obs. ; 6.vi.2015, 1 \u2642 FSLG after sunset, anonymous observer from the Czech Republic obs. (moreover 1 spec. of Och.integriceps in the light trap was observed); 47\u00b049'9.54\"N, 18\u00b039'26.4\"E, ca. 170 m a.s.l., 27.v.2015, 1 spec. FSLG after sunset, and 1 spec. at light, two anonymous observers from the Czech Republic obs.8078: Zalaba, 47\u00b058'8.8\"N, 18\u00b042'29.2\"E, ca. 150 m a.s.l., June 1975, 1 spec. crawling on the ground on a sandy slope sparsely covered with black locust trees (Robiniapseudoacacia) at ca. 19.00 CEST, JAH.8178: Bajtava, 16.vi.2006, 1 spec., at light, VVO obs.; Kamenica nad Hronom, \u010cierna hora hill  on a path, ONV obs., 5 \u2642\u2642 and 3 \u2640\u2640 flying ca. 30\u2013100 cm above the ground at 19.52\u201320.07 CEST, together with 1 \u2642 of Od.armiger, DJP and ONV obs.  on a path, DJP, JCM and IMO obs., 1 \u2642 flying relatively quickly ca. 1 m above the ground at 19.43 CEST and 1 \u2642 flying very slowly ca. 10 cm above the ground at 19.47 CEST, DJP and IMO obs.  and at light (3 spec.), OSO obs.; Kamenica nad Hronom env., ca. 530 m NNE of the hilltop of \u010cierna hora hill, 47\u00b050'26\"N, 18\u00b043'50.7\"E, 180 m a.s.l., 30.v.2011, 1 \u2640 FSLG after sunset, small steppe hillside near an oak forest, FSP obs.; Kov\u00e1\u010dov, [110 m a.s.l.], 8.viii.1965, 1 \u2642, K[arel] Pol\u00e1\u010dek leg., coll. MHKC; Ch\u013eaba, 11.vi.1985, 1 \u2642 and 1 \u2640 J. Hladn\u00fd leg., coll. JZJ.ill Fig. , ca. 47\u00b0bs. Fig. , and 15 bs. Fig. , 1 spec 8178\u20138278: \u201cParka\u0148\u201d [= \u0160t\u00farovo], [ca. 110 m a.s.l.], 1934, no other data, 1 \u2640 in coll. MHKC; 1940, no other data, 1 \u2642 in coll. MHKC; \u0160t\u00farovo, July 1967, 1 \u2640, collector unknown, coll. ASH.8179: Ch\u013eaba env., Mo\u010diar (the site near the confluence of the Danube and Ipe\u013e rivers), 47\u00b049'14.53\"N, 18\u00b050'52.72\"E, 110 m a.s.l., 12.vi.2014, 1 \u2642 FSLG at 21.40 CEST, together with 2 \u2640\u2640 of Od.armiger, OSO obs.7785: Cerov\u00e1 vrchovina Mts, Hajn\u00e1\u010dka \u2013 Bukov\u00e1 env., ca. 48\u00b013'51.39\"N, 19\u00b058'26.32\"E, 1.vi.1978, 1 \u2642 crawling on the ground in the afternoon in sunlight, IJN leg. [storage of the specimen unknown]; Cerov\u00e1 vrchovina Mts, Hajn\u00e1\u010dka \u2013 Bukov\u00e1 env., \u201ccircular pasture under v\u00e1ggon\u201d , 23.vi.1990, 1 \u2640 flying at 21.28 CEST, JVP leg., ex original coll. JVP, currently in coll. NMPC; Hajn\u00e1\u010dka \u2013 western edge of the village, 48\u00b012'48.2\"N, 19\u00b056'52.1\"E, ca. 275 m a.s.l., 7.vi.2010, 1 \u2640 FSLG after sunset, together with more spec. of Od.armiger and Och.chrysomeloides, PVP obs.; 8.vi.2010, 1 spec. FSLG after sunset, together with more spec. of Od.armiger and Och.chrysomeloides, PVP obs.; Hajn\u00e1\u010dka \u2013 Bukov\u00e1 env., steppe hillside , 48\u00b013'37.24\"N, 19\u00b058'23.73\"E, 340\u2013390 m a.s.l., 27.v.2008, 5 \u2640\u2640 FSLG at 21.10\u201321.35 CEST, 22 \u00b0C, no wind, together with 20 spec. of Od.armiger and 19 spec. of Och.chrysomeloides, DJP and FSP obs.  \u2642 crawling on the T-shirt spread out on the ground near the edge of the forest, under an oak tree (Quercuscerris) at 19.55 CEST, 1 \u2642 flying relatively quickly and zigzag ca. 1 m above the ground and 3 \u2640\u2640 flying slowly ca. 0.5 m above the ground at 21.10\u20131.40 CEST, 21 \u00b0C, no wind to light air, together with 20 spec. of Od.armiger and 14 spec. of Och.chrysomeloides, DJP, KDO and PJL obs. , RCR obs.; 8.vii.2015, 9 \u2642\u2642 and 6 \u2640\u2640 FSLG after sunset (ca. 21.11 CEST), RCR obs.; 29.vi.2017, 4 \u2642\u2642 and 8 \u2640\u2640 FSLG after sunset (ca. 21.07 CEST), RCR obs.; Hajn\u00e1\u010dka, Lapos, 48\u00b013'32.37\"N, 19\u00b057'50.58\"E, ca. 350 m a.s.l., 24.vii.2020, 12 \u2642\u2642 and 9 \u2640\u2640 FSLG at 21.05\u201321.30 CEST, 15 \u00b0C, RCR obs.7785\u20137885: Cerov\u00e1 vrchovina Mts, Gemersk\u00fd Jablonec, 48\u00b012'0.44\"N, 19\u00b059'24.31\"E, 250\u2013265 m a.s.l., steppe hillside with shrubbery of Prunusspinosa and Rosacanina on the hilltop, 4.vii.2009, 1 \u2642 a 3 \u2640\u2640 FSLG at 21.30\u201321.50 CEST, FPT and JPH obs. , DJP obs. ; Jestice \u2013 K\u00f6k\u00e9nyes, 48\u00b012'45.84\"N, 20\u00b02'50.77\"E, 250 m a.s.l., 23.vi.2020, 1 \u2642 flying after sunset, ABC obs.; 6.vii.2020, 1 \u2642 and 1 \u2640 flying after sunset, ABC obs.; Jestice \u2013 Iv\u00e1nk\u00fata env., 48\u00b012'30.9\"N, 20\u00b05'5.37\"E, 254 m a.s.l., 7.vi.2015, 2 \u2642\u2642 FSLG at 20.25 CEST, edge of an oak forest, RCR obs.7489: Slovak Karst, \u201cRaka\u0165a\u201d [= Rakyta Cottage] env., 48\u00b035'29.7\"N, 20\u00b034'01.45\"E, ca. 540 m a.s.l., 5.vii.1988, 1 \u2640 excavated from its burrow, DKP . In addition to the already known localities, Published data\u201cGallia\u201d, no other data .\u201cAlsace\u201d, no other data .Grand Est, Bas-Rhin, Strasbourg, ca. 140 m a.s.l., \u201cin coll. Dr Puton \u2013 Jules Bourgeois pers. comm.\u201d, no other data , which, according to Denis Keith (pers. comm.), is dubious and probably based on mislabelled material. The site (a mountainous area) does not meet the known requirements of the species and its occurrence here seems to be highly improbable. The same applies to the Mont Cenis specimen from the Abeille de Perrin\u2019s collection in the MNHN.For France, Published dataBaden-W\u00fcrttemberg, Markgr\u00e4flerland, Neuenburg am Rhein \u2013 Gri\u00dfheim, \u201cGri\u00dfheimer Trockenaue\u201d, , 2.vi.1967, 1 \u2642, at light, Hans Messmer leg., photo + coll. Richard Disch (rd Disch .Bavaria (Bayern), \u201cBavaria\u201d, no other data  \u2013 these records will be published with additional details at a later date .Bavaria (Bayern), Upper Bavaria (Oberbayern), Ingolstadt, [ca. 370 m a.s.l.], 9.ix.[18]92, 1 spec. Dr K[arl] Daniel [leg.], \u201cFundortverwechslung\u201d [= locality mistaken], coll. ZSMG . The new records presented from Baden represent the first known data on the species\u2019 occurrence in Germany after 54 years , Basel, undated, 1 spec., Ed. Bernoulli leg. (lli leg. .Republic and Canton of Ticino (Repubblica e Cantone Ticino), no other data, Villa [leg.] , \u201cTigurini\u201d [= Z\u00fcrich], undated, 2 \u2640\u2640, collector unknown, \u201cMus. Drews.\u201d , coll. NHMD , no other data , 1 \u2642, [Flaminio] Baudi [di Selve] [leg.], and 1 \u2642, L. Carrara [leg.], no other data, coll. MNFI , no other data,  \u2013 this specimen, labelled \u201c92\", is still in the Bertolini\u2019s collection deposited in MUSE; ? Provincia autonoma di Trento, Torcegno env., \u201cin the mountains above Torcegno\u201d, undated, 3 spec., together with Od.armiger, [Giovanni] Costesso leg., coll. Stefano de Bertolini ; Venezia Tridentina, no other data , 1 \u2640 at actinic light at 21.15 CEST , 10.ix.2018, 1 \u2642 in flight at 20.15 CEST , 12.x.2018, 1 \u2640 in flight at 20.00 CEST, 15.v.2019, 1 \u2642 and 1 \u2640 in flight at 21.15 CEST , 16.v.2019, 2 \u2642\u2642 and 1 \u2640 in flight at 21.00\u201321.15 CEST (air temperature 17 \u00b0C), 1 \u2640 crawling on the ground at 21.20 CEST, 24.v.2019, 5 \u2642\u2642 in flight at 21.20\u201321.35 CEST , 26.v.2019, 1 \u2642 in flight at 21.20\u201321.35 CEST , 1.vi.2019, 1 \u2642 and 2 \u2640\u2640, in flight at 21.40 CEST , 6.vi.2019, 2 \u2642\u2642 and 1 \u2640, in flight at 21.00\u201321.20 CEST , 7.vi.2019, 2 \u2642\u2642 in flight at 21.35 CEST , Paolo Glerean and Gabriele Stefani obs. (ani obs. .Tuscany (Toscana), no other data, coll. Dr L[ucas] von Heyden , coll. SDEI.\u201cItalia, Sella [it is not clear whether it is a geographical name or the name of a person]\u201d, no other data, 1 \u2642 in coll. Piedmont (Piemonte), \u201cPedem.\u201d, , no other data, 1 \u2642 in coll. RBIN; \u201cPedemt.\u201d [= Piedmont], no other data, 3 \u2642\u2642 and 1 \u2640 in coll. Maurice Pic deposited in MNHN; \u201cPedemont.\u201d [= Piedmont], no other data, 1 \u2642 and 1 \u2640 (ex coll. Christian Drewsen) in coll. NHMD, 1 \u2642 (ex coll. Carl Gustaf Thomson) in coll. MZLU, 1 spec. in coll. NHMW; \u201cPedemont.\u201d [= Piedmont], undated, L[\u00e9on Marc Herminie] Fairm[aire] [leg.], 1 \u2642 and 1 \u2640 (ex coll. Fredrik Wilhelm M\u00e4klin) in coll. FMNH; \u201cAlp. Pedemont.\u201d [= Alpes Pedemontium], undated, 2 \u2642\u2642 and 1 \u2640, [Vittore] Ghiliani [leg.], coll. NHMD; \u201cPi\u00e9mont\u201d [= Piedmont], undated, 1 \u2642 in coll. Elz\u00e9ar Abeille de Perrin deposited in MNHN, 1 \u2640 in coll. Antoine Boucomont deposited in MNHN, 1 \u2640 in coll. Jacques Baraud deposited in MNHN, 1 \u2642 in coll. NMPC, 1 \u2640, in coll. Alfonz Gspan deposited in PMSL, 1 \u2642  in coll. NHML; \u201cPiemont\u201d [= Piedmont], no other data, 1 \u2642 and 1 \u2640 in coll. MNBG, 1 \u2642 and 1 \u2640 in coll. RBIN; \u201cPiemont\u201d [= Piedmont], \u201ccoll. Rottenberg\u201d, 1 \u2640 in coll. SDEI; \u201cPiemont\u201d [= Piedmont], \u201ccoll. [Carl] Felsche\u201d, 1 spec. in coll. MTDG; \u201cPiemonte\u201d [= Piedmont], \u201ccolezz. Alzona\u201d [= coll. Alzona], 1 \u2640 in coll. MSNM; Citt\u00e0 metropolitana di Torino, Rivarossa, [ca. 285 m a.s.l.], no other data, 1 \u2640 in coll. Leopold Mader deposited in MNSA; \u201cTurin\u201d [= Torino], no other data, 1 \u2642 and 2 \u2640\u2640 in coll. Sylvain Augustin de Marseul deposited in MNHN, 1 \u2642 and 1 \u2640 in coll. NHMD, 1 spec. in coll. ZSMG; Torino, 25.vii.[year not specified], no other data, 1 \u2642 in coll. Georg Frey deposited in NHMB; Torino, \u201calluvioni Po\u201d [= alluvial materials of the Po river], ca. 230 m a.s.l., 1871, 1 \u2640, L. Fea leg., coll. MSNG; Torino, no other data, 1 \u2642 in coll. FMNH, 1 spec. in coll. NHMW; Borgofranco d\u2019Ivrea, [ca. 250 m a.s.l.], undated, 1 \u2642, L. Demarchi leg., coll. MSNG; Provincia di Alessandria, Lerma, ca. 300 m a.s.l., May 1995, 1 \u2642, in the morning accidentally dug up from the soil in the orchard; 8.iv.2014, 1 \u2642, in the morning accidentally dug up from the soil in the orchard; 4.v.2014, 1 \u2642 in the morning on the ground and 1 \u2640 at UV light at 21.30 CEST, after several days of rain; 11.v.2014, 1 \u2640, accidentally dug up from the soil in the garden at 16.00 CEST; 3.viii.2014, 1 \u2640 flying around the light at 21.30 CEST; 16.v.2015, 1 \u2642, at UV light at 21.30 CEST, rain in the morning and the day before, very wet, 17 \u00b0C; 17.v.2015, 1 \u2642, at UV light at 21.15 CEST, wet, 17 \u00b0C; 20.vi.2015, 1 \u2640, at UV light at 22.00 CEST, heavy rainfall in previous days, vegetation and soil heavily saturated with water, 17 \u00b0C; 21.vi.2015, 1 \u2640, at UV light at 22.00, wet, 17 \u00b0C; 29.vi.2015, 1 \u2640, at UV light at 21.50 CEST, 23 \u00b0C, LRL obs. (see obs. see .Lombardy (Lombardia), Provincia di Varese, Casorate Sempione, ca. 280 m a.s.l., October 1958, 1 \u2642, at light, A. Bilardo leg., ex original coll. Giovanni Mariani, currently deposited in coll. RPM . Records from Sicily  , which is evident both from the drawing of the specimen and from its description; in addition, this specimen was allegedly lost . This study presents new records from the third known locality with a recent occurrence of the species in Italy (Lerma).In the collection of Zden\u011bk Tesa\u0159 deposited in m Sicily  refer to898) see . Benasso898) see  is apparPublished dataMazovian Voivodeship (Wojew\u00f3dztwo mazowieckie), Warsaw \u2013 Saska K\u0119pa, 80\u201385 m a.s.l., undated, 2 spec., Antoni Waga leg. , Opole County, Z\u0142otniki, ca. 155 m a.s.l., undated, 1 \u2640, Ludwik Fryderyk Hildt leg. (ldt leg. .\u015awi\u0119tokrzyskie Voivodeship (Wojew\u00f3dztwo \u015bwi\u0119tokrzyskie), Kielce County, Ch\u0119ciny, 1 spec., no other data, , Lublin env., no other data, Baumgarten leg. , Wadowice County (Po\u00adwiat wadowicki), \u201cWadowice, Hal.\u201d , 1 \u2640, undated, Smolik [leg.], DJP det., coll. NMBE.\u015awi\u0119tokrzyskie Voivodeship (Wojew\u00f3dztwo \u015bwi\u0119tokrzyskie), Kielce County, \u201cG\u00f3ry Stokrzyskie [env.], Ga\u0142\u0119zice, [Mt.] G\u00f3ra Ostr\u00f3wka\u201d , July 1921, 1 \u2642, J. Czarnocki [leg.], \u201cPolonia, [coll.] Sz[ymon] Tenenbaum\u201d, coll. MIZP , Linz, Scharlinz, ca. 250 m a.s.l., 25.v.1936, 1 \u2642, [Johann] Wirthumer leg., coll. BZLA , no other data ; Gro\u00df-Enzersdorf \u2013 M\u00fchlleiten env., 48\u00b010'34\"N, 16\u00b033'6.6\"E, 159 m a.s.l., 24.vi.2019, 1 \u2642 flying up to 0.5 m above the ground at 21.40 CEST, meadow adjacent to the forest, 22 \u00b0C, gentle breeze, ADW obs. + photo , no other data, coll. Dr Lucas von Heyden , with no further details , Villach, Teufelsgraben, 1 spec. with no other data , Grazer Bergland, H\u00f6rgas [near Gratwein-Stra\u00dfengel], undated, 1 \u2642 [10.6 mm], G[ustav] Wallaberger Sr. leg., coll. UMJG , J. Gunczy obs., photo Gernot Kunz , \u201cNied. Oesterr.\u201d [= Nieder\u00f6sterreich], no other data, 1 \u2642 in coll. NMBE; Melk, undated, 23 spec. in coll. NHMW, 1 \u2642 and 1 \u2640  in coll. Georg Frey deposited in NHMB, 1 \u2642  in coll. MHNG  in coll. Georg Frey deposited in NHMB  in coll. MHNG;  \u201cDonau-Auen\u201d, undated, 1 \u2642 and 1 \u2640, F[ranz] Bl\u00fchweiss leg., 1 \u2642 and 1 \u2640, Fr. Reiss leg., ex original coll. Rudolf Petrovitz, currently in coll. MHNG;  Donauauen, no other data, 1 \u2640 in coll. TLMF, 10 spec. in coll. NHMW;  Donauauen, undated, 1 \u2640, F[ranz] Bl\u00fchweiss [leg.], coll. MNBG; \u201cMarchfeld, Oberweiden\u201d, no other data, 1 \u2640 in coll. MNBG; Oberweiden, Steppe [= steppe], 7.viii.1959, 1 \u2640, J[osef] Gusenleitner leg., coll. BZLA.MHNG cf. ; Wachau,NHMB cf. ; \u201cUmg. WVienna (Wien), \u201cWien\u201d [= Vienna], no other data, 2 \u2640\u2640 in coll. Vladim\u00edr Zoufal deposited in MMBC, 1 spec. in coll. MTDG, 1 \u2640 in coll. BZLA; \u201cWien\u201d [= Vienna], undated, 1 \u2642, J[osef] Moser leg., coll. BZLA; \u201cVienne\u201d [= Vienna], no other data, 1 \u2642 and 1 \u2640 in coll. Albert Sicard deposited in MNHN; \u201cWien Umg.,\u201d [= Vienna env.], no other data, 1 \u2642 in coll. Leopold Mader deposited in MNSA; \u201cWien, Umgebg.\u201d [= Vienna env.], undated, 1 \u2640, F. Schade [leg.], coll. Jaroslav Matou\u0161ek deposited in MMBC; \u201cWien Umgebgebung\u201d, undated, 2 \u2642\u2642, A[dolf] Hoffmann leg., coll. TLMF; \u201cUmg. Wien\u201d [= Vienna env.], undated, Ad[olf] Hoffmann [leg.], 1 \u2640 (ex coll. P. Franck) in coll. MIZP, 1 \u2642 in coll. SMNS, 1 \u2640 in coll. Alfonz Gspan deposited in PMSL; \u201cHochwasser bei Wien\u201d [= flood near Vienna], no other data, 1 \u2640 (ex coll. Adolf Hoffmann) in coll. Jan Roubal deposited in SNMS; Vienna, Donau [= Danube river], Hochwasser [= flood], undated, 1 \u2642  in coll. MHNG; Vienna, \u201cDonau\u00fcberschwemmung\u201d [= flooded Danube river], September 1920, 1 spec., R. F. Lang [leg.], coll. NHMW; Vienna env., undated, 1 \u2640, Carl Mandl [leg.], coll. Georg Frey deposited in NHMB; Vienna env., undated, 1 \u2642, Matuschka [leg.], ex original coll. Josef Breit (Vienna), currently in coll. Georg Frey deposited in NHMB; Vienna, \u201cInundationsgebiet\u201d [= inundation area of the Danube river], undated, 3 spec.  in coll. NHMW, 1 \u2642  in coll. Georg Frey deposited in NHMB; Vienna, Prater, no other data, 1 spec. in coll. NHMW.CommentIn Austria, this species is known from six of the nine Austrian states. A recent attempt to rediscover the species at suitable sites along the Traun River in Upper Austria  was unsuPublished dataWestern Transdanubia (Nyugat-Dun\u00e1nt\u00fal), Vas County, \u201cMolna-Szecs\u0151d\u201d [= Molnaszecs\u0151d], 10.vi. [turn of the 19th and 20th century], ca. 180 m a.s.l., 1 spec. inside the digestive system of Cuculuscanorus, Ern\u0151 Csiki obs. , Kom\u00e1rom-Esztergom County, \u201cSz\u0151ny\u201d [a part of the current Kom\u00e1rom city], 6.viii.1901, 2 spec. inside the digestive system of Upupaepops, Ern\u0151 Csiki obs. ; Kom\u00e1rom-Esztergom County, Csolnok, no other data ; Oroszl\u00e1ny env., Majkpuszta, Majki-hegy, 14.vi.1997, 1 \u2640, at light. CKZ ; Fej\u00e9r County, Adony, no other data .iki obs. ; Kom\u00e1romiki obs. ; Kom\u00e1romher data , 28.v.18ght. CKZ ; Fej\u00e9r Cher data , 1 \u2642, unSouthern Transdanubia (D\u00e9l-Dun\u00e1nt\u00fal), Somogy County, Fony\u00f3d, ca. 140 m a.s.l., undated, 1 \u2642, Viktor Stiller leg., coll. HNHM ; Somogy County, Ordacsehi, Csehi-berek, 21.vii.2004, Gy\u00f6rgy Rozner leg. ; Somogy County, Balatonf\u00f6ldv\u00e1r, no other data ; Somogy County, Balatonvil\u00e1gos \u2013 Balatonaliga, 1.viii.1980, collector unknown, 1 spec. in coll. HNHM ; Somogy County, Szenna, 9.vi.1998, Gy\u00f6rgy Rozner leg. ; Tolna County, H\u0151gy\u00e9sz, 46\u00b030'38\"N, 18\u00b025'55\"E, 24.vii.1994, 1 spec. at light, collector not specified ; Tolna County, B\u00e1taap\u00e1ti env., Nagy-m\u00f3r\u00e1gyi-v\u00f6lgy [valley], Quercetum, 15.vii.2004, 1 \u2640, OMB leg., coll. HNHM ; Baranya County, Sellye, finding in truffle (Tuber sp.), no other data ; Baranya County, \u201cSzabolcs\u201d [= P\u00e9cs \u2013 Szabolcs or Mecsekszabolcs] env., \u201cSzarvasn\u00f3ta\u201d, ca. 46\u00b08'8\"N, 18\u00b015'46\"E, beginning of June 1880, 1 \u2642 and 1 \u2640, the female was digging a hole into the ground at the edge of a fo\u00adrest footpath like Copris, and it seemed that the male was helping her with this work, Dr Ern\u0151 Kaufmann leg. , Veszpr\u00e9m County, P\u00e1pa env., no other data , June 1895, 1 \u2640, Fe\u00adrenc Wachsmann leg., coll. HNHM ; Veszpr\u00e9m County, Balatonalm\u00e1di, 5.ix.1940, 1 \u2642, Ern\u0151 Csiki leg., coll. HNHM ; Veszpr\u00e9m County, V\u00e1szoly env., \u00d6reg-hegy, 250\u2013290 m a.s.l., 3.vii.1999, 1 spec., IRB leg. ; Veszpr\u00e9m County, Paloznak 17.viii.1961, Frigyes Nov\u00e1k leg., coll. HNHM ; Pest County, Buda hills (Budai-hegys\u00e9g), no other data ; Pest County, \u201cKis-Szent-Mikl\u00f3s\u201d or \u201c\u0150rszentmikl\u00f3s\u201d [= \u0150rbotty\u00e1n \u2013 \u0150rszentmikl\u00f3s], 1876, dry oak forest on the hill, 1 spec. on the ground in the grass in the evening , Karoly Saj\u00f3 leg. ; Pest County, Pilis hegys\u00e9g, no other data ; Pest County, Szigetszentmikl\u00f3s, 6.vi.1954, 1 \u2640, Mikl\u00f3s Natt\u00e1n leg., coll. HNHM ; Pest County, Dabas, no other data ; Pest County, T\u00e1borfalva env., shooting and training area, , 11.vii.2012, 1 spec., at light, SIB obs. ; Pest County, G\u00f6d\u00f6ll\u0151 env., no other data ; Pest County, G\u00f6d\u00f6ll\u0151, 55 Erd\u0151sz\u00e9l Street, , 2005, no other data, VSI leg., ; 47\u00b029'11.48\"N, 19\u00b023'19.52\"E, 3.vi.1972, 1 spec., IRB leg. ; Pest County, Isaszeg, no other data ; 1908, 1 \u2640, 1909, 1 \u2640, June 1917, 1 \u2642, Hug\u00f3 Diener leg., coll. HNHM ; June 1929, 1 spec., Hug\u00f3 Diener leg., coll. HNHM , 17.vi.2002, 1 spec., at light, SIB obs., 19.vi.2004, 1 spec., at light, SIB obs., 29.v.2005, 1 spec., at light, SIB obs. ; Pest County, Biatorb\u00e1gy, 27.vi.1999, 1 \u2642, at light, AGB leg., coll. HNHM ; Pest County, Nagykov\u00e1csi env., Julianna-major, 10.vi.1985, 1 spec., at light, 18.vii.1985, 1 spec., at light, Dezs\u0151 Szal\u00f3ki leg. , 8.vi.1991, 2 \u2640\u2640, on Glomusmacrocarpum, Cynodonto-Festucetum, LAB leg., coll. HNHM ; Buda [currently western part of Budapest], \u201cGraberl\u201d [a historical excursion destination in the Buda surroundings], 13.v.1798 (!), 1 spec., T\u00f3bi\u00e1s Koy leg. ; Budapest env., Kamaraerd\u0151, 25.iv.1920, 1 \u2642 and 30.v.1922, 1 \u2640, Hug\u00f3 Diener leg., coll. HNHM ; [Budapest \u2013] R\u00e1kos, no other data ; Budapest \u2013 Cinkota, no other data ; Budapest \u2013 M\u00e1rtonhegy, 17.iii.1949, 1 \u2642, J\u00f3zsef Sz\u0151cs leg., coll. HNHM ; Budapest \u2013 Nagyt\u00e9t\u00e9ny, undated, 1 \u2642, Seb\u0151 Endr\u0151di leg., coll. HNHM ; Budapest \u2013 R\u00e1kosszentmih\u00e1ly, 15.viii.1930, 1 \u2642, at light, Jen\u0151 Gy\u0151rffy leg., coll. HNHM ; Budapest \u2013 Sz\u00e9pv\u00f6lgy, 23.vi.1975, OMB leg. , Csongr\u00e1d-Csan\u00e1d County, Szeged \u2013 Kiskundorozsma env., Nagysz\u00e9k, 16.\u201323.vi.1989, 1 spec., pitfall trap with ethylene glycol, B\u00e9la Gask\u00f3 leg., coll. MMSH ; B\u00e9k\u00e9s County, Dombegyh\u00e1z env., Trianon border mound, 46\u00b018'17.54\"N, 21\u00b08'43.38\"E, 99 m a.s.l., 9.vi.2013, 1 \u2642, pitfall trap on a narrow strip of grass with loess soil, TDS and TDB leg., coll. HNHM ; B\u00e1cs-Kiskun County, \u201cPesz\u00e9r\u201d [= Kunpesz\u00e9r] env., no other data , Heves County, M\u00e1tra Mts, Galyatet\u0151, 10.vii.1959, S\u00e1ndor Szab\u00f3 leg. .ab\u00f3 leg. ; Borsod-Northern Great Plain (\u00c9szak-Alf\u00f6ld), Hajd\u00fa-Bihar County, \u201cDebreczen\u201d [= Debrecen] env., ca. 1860\u20131880, 1 spec., J\u00f3zsef T\u00f6r\u00f6k leg. , Gy\u0151r-Moson-Sopron County, Gy\u0151r \u2013 Lik\u00f3cs env., ca. 47\u00b042'52.5\"N, 17\u00b041'45\"E, 2019, 115 m a.s.l., pitfall traps, no other data , Fej\u00e9r County, Cs\u00f3r, ca. 150 m a.s.l., 21.v.2014, 1 \u2640, at light on a steppe, MPK leg., coll. DCO; 28.v.2016, 1 \u2642, at light, DVZ obs.; Fej\u00e9r County, Cs\u00e1kber\u00e9ny, Bucka hill, 47\u00b020'51.65\"N, 18\u00b021'35.32\"E, 230 m a.s.l., 11.vi.1987, 1 spec., at light, CSB obs.; Fej\u00e9r County, G\u00e1nt env., K\u00f6ves-v\u00f6lgy [valley], 47\u00b024'19.94\"N, 18\u00b022'47.67\"E, 280 m a.s.l., 14.vi.2019, 1 \u2640 flying after sunset, VSI obs.; Fej\u00e9r County, Nagykar\u00e1csony, 46\u00b052'49.4\"N, 18\u00b043'27.1\"E, 150 m a.s.l., 2.vi.2021, 2 \u2642\u2642 and 1 \u2640 FSLG after sunset, 17\u201318 \u00b0C, light breeze, TDS obs.; Fej\u00e9r County, Adony env., 47\u00b05'17.2\"N, 18\u00b049'10.3\"E, 120 m a.s.l., 1.vi.2021, 10 spec. FSLG after sunset, 11\u201315 \u00b0C, no wind, TDB and TDS obs.; Kom\u00e1rom-Esztergom County, K\u00f6rnye, no other data, 1 \u2642 in coll. RBIN; Kom\u00e1rom-Esztergom County, Esztergom env., Kis-Str\u00e1zsa-hegy hill, 47\u00b044'59.210\"N, 18\u00b044'35.07\"E, 210 m a.s.l., 23.iv.2006, 1 spec., at light (mercury-vapor lamp), VPB; Kom\u00e1rom-Esztergom County, Keszt\u00f6lc env., 47\u00b043'13.4\"N, 18\u00b047'43.3\"E, 17.x.2014, 260 m a.s.l., 1 \u2640 excavated from its burrow from a depth of 60 cm, loess steppe with abundant occurrence of Lethrusapterus , TVP ; Kom\u00e1rom Esztergom County, M\u00e1riahalom env., 47\u00b037'28.3\"N, 18\u00b041'21.68\"E, 190 m a.s.l., 31.vii.2020, 1 \u2640, at light, BKL obs. + photo (DJP det.); Veszpr\u00e9m County, Nagyv\u00e1zsony env., 47\u00b01'40.73\"N, 17\u00b042'38.62\"E, 315 m a.s.l., 12.vi.2009, 1 \u2640 flying ca. 10 cm above the ground after sunset, KLP; 16.vi.2016, 3 \u2642\u2642 FSLG after sunset, JHH, JPP, JSUMSZ, MPV and PIL obs.; Veszpr\u00e9m County, V\u00e1szoly env., \u00d6reg-hegy, 250\u2013290 m a.s.l., 3.vii.1999, 1 spec., IRB leg., coll. SZM; Veszpr\u00e9m County, \u00d6rv\u00e9nyes, 46\u00b055'8.3\"N, 17\u00b048'26.07\"E, 150 m a.s.l., 16.vi.2019, 1 \u2640 flying after sunset, forest pasture, VSI obs.; Veszpr\u00e9m County, Fels\u0151\u00f6rs, \u00d6reg-hegy, 47\u00b00'57.59\"N, 17\u00b058'52.72\"E, 214 m a.s.l., 7.viii.2018, 1 \u2642, dead near the light in a garden, FKD obs. + photo (DJP det.); Veszpr\u00e9m County, Bakony Mts, Lit\u00e9r, [ca. 200 m a.s.l.], 14.vii.2014, 1 \u2640, IRB leg., coll. GML.Southern Transdanubia (D\u00e9l-Dun\u00e1nt\u00fal), Somogy County, Balatonendr\u00e9d, 46\u00b050'52\"N, 17\u00b059'18\"E, 174 m a.s.l., 11.v.1989, 1 \u2640 excavated from its burrow together with 1 \u2642 of Od.armiger, VRH; Somogy County, S\u00e1gv\u00e1r, Jaba-v\u00f6lgy [valley], 46\u00b049'28.29\"N, 18\u00b02'32.93\"E, 180 m a.s.l., 25.ix.2017, 1 \u2640 crawling on the ground, PFS obs. + photo (DJP det.); Somogy County, Balatonvil\u00e1gos \u2013 Balatonaliga, 10.vi.1983, 1 spec., at light, SIB obs.; Baranya County, Zselic Mts, Mozsg\u00f3, ca. 150 m a.s.l., 27.vii.2017, 1 \u2642, at light, MRM; Baranya County, Dr\u00e1vaszabolcs, 4/c K\u00f6zt\u00e1rsas\u00e1g t\u00e9r Street, 45\u00b048'20.95\"N, 18\u00b012'43.74\"E, 91 m a.s.l., 28.vi.2020, 1 \u2640 dead under the lamp, JST; Baranya County, Vill\u00e1nyi-hegys\u00e9g Mts, Nagyhars\u00e1ny env., Sz\u00e1rsomly\u00f3 hill, ca. 145 m a.s.l., 22.v.1977, 1 \u2642, at light., AUP; Baranya County, Erd\u0151smecske, ca. 240 m a.s.l., 18.viii.2012, 1 spec., 31.vii.2016, 1 spec., 27.v.2017, 1 spec., REE obs.Central Hungary (K\u00f6z\u00e9p-Magyarorsz\u00e1g), Pest County, Zs\u00e1mb\u00e9k, June 2016, 1 \u2640, students of Department of Zoology, Charles University, Prague leg., coll. DKP deposited in NMPC; Pest County, Biatorb\u00e1gy, 47\u00b027'54.501\"N, 18\u00b051'0.515\"E, ca. 190 m a.s.l., 24.vii.2021, 1 \u2642, at light, GAB obs. + photo (DJP det.); Pest County, Nagymaros env., Rig\u00f3-hegy hill, 47\u00b046'31.63\"N, 18\u00b056'11.65\"E, ca. 300 m a.s.l., 21.iv.2019, 1 \u2642, night sweeping, TNB leg., coll. HNHM; Pest County, Szentendre \u2013 Izb\u00e9g env., 47\u00b041'47.61\"N, 19\u00b01'40.06\"E, 195 m a.s.l., 9.vi.2014, 1 spec., at light (mercury-vapor lamp), GBP and APE obs.; Pest County, P\u00f3csmegyer env., 47\u00b043'44.5\"N, 19\u00b06'25.7\"E, 110 m a.s.l., 11.viii.2006, 1 spec., 20.vi.2008, 1 spec., 18.vi.2010, 1 spec., pitfall traps without attractant, SBP and ZBP leg., 16.ix.2014, 1 \u2640, pitfall trap, SBP leg. [storage of the specimens unspecified]; Pest County, Pom\u00e1z env., Szam\u00e1r-hegy hill, 47\u00b039'28.7\"N, 18\u00b058'43.06\"E, ca. 185 m a.s.l., 2.vii.2019, 1 \u2642 flying after sunset, VSI obs.; Pest County, Pom\u00e1z, Majd\u00e1npola, 47\u00b038'27.7\"N, 19\u00b00'18.61\"E, 190 m a.s.l., 1.viii.2019, 1 \u2642 at light (mercury-vapor lamp), SIB obs., 1 m FSLG after sunset, VSI leg., coll. HNHM; Pest County, Budakeszi env., Hossz\u00fa-d\u0171l\u0151, 200 m a.s.l., 5.vi.1991, 2 \u2642\u2642 and 1 \u2640, Cynodonto-Festucetum, on Glomusmacrocarpum, LAB leg., coll. GML (pair) and JMB (1 \u2642); Pest County, Budakeszi, 5.vi.2013, 1 spec., 6.vii.2014, 1 spec., 19.vii.2014, 1 spec., 26.vii.2014, 1 spec., 3.vi.2015, 1 spec., 28.v.2016, 1 spec., 4.vi.2018, 1 spec., all at light, SIB obs.; Pest County, Budakeszi, gliding airport, ca. 200 m a.s.l., 5.vii.1991, 1 spec., LNB leg., coll. SZM; Pest County, Budakeszi, Farkas-hegy env., gliding airport, 47\u00b028'39.7\"N, 18\u00b054'50\"E, ca. 200 m a.s.l., 6.v.2018, 2 \u2642\u2642 flying after sunset, TNB obs. (1 \u2642 in coll. HNHM); 23.v.2019, 1 m flying after sunset, 12.vi.2019, 1 \u2640, night sweeping, 17.vi.2019, 1 \u2642, night sweeping, 27.vi.2019, 1 \u2642, night sweeping, 2.vii.2019, 1 \u2640 flying after sunset, TNB obs., 22.ix.2019, 1 spec., TNB obs., 47\u00b028'55.2\"N, 18\u00b055'6.25\"E, 18.vi.2018, 2 \u2642\u2642 and 1 \u2640 flying after sunset, 20.vi.2018, 4 \u2640\u2640 flying after sunset, 5.vii.2018, 2 \u2642\u2642 and 3 \u2640\u2640 flying after sunset, 10.vii.2018, 2 \u2642\u2642 flying after sunset, 12.ix.2018, 1 \u2642 excavated from its burrow, TNB leg., coll. HNHM; Pest County, Budakeszi \u2013 Nagysz\u00e9n\u00e1szug, ca. 47\u00b029'11.6\"N, 18\u00b055'26.3\"E, ca. 230 m a.s.l., 18.vi.2018, 1 spec., 9.vi.2019, 3 spec. in a private garden, LMB obs.; Pest County, Buda\u00f6rs env., Farkas-hegy, 47\u00b028'27.29\"N, 18\u00b056'40.42\"E, ca. 335 m a.s.l., 8.vi.2019, 1 spec., OMB obs.; 22.vi.2021, 1 \u2642 FSLG after sunset, VSI obs.; Pest County, T\u00f6r\u00f6kb\u00e1lint, Nagy-Mez\u0151, 47\u00b025'31.01\"N, 18\u00b057'31.04\"E, 216 m a.s.l., 18.vi.2019, 1 \u2640 flying after sunset, VSI obs.; Budapest, T\u00e9t\u00e9nyi-fenns\u00edk env., 47\u00b025'2.309\"N 18\u00b058'59.332\"E, 180 m a.s.l., 6.viii.2021, 1 \u2642, at light, MLB obs. + photo, DJP det.; Budapest, \u201cPest\u201d [currently eastern part of Budapest], no other data, 2 spec in coll. MNHN; \u00dajpest [currently part of Budapest], undated, 1 \u2642, Robert Meusel [leg.], coll. Jo\u017ee Staudacher deposited in PMSL; Budapest, no other data, 7 spec. in coll. NHMW, 2 \u2642\u2642 and 1 \u2640  in coll. Georg Frey deposited in NHMB, 1 \u2642  in coll. Jacques Baraud deposited in MNHN, 1 \u2642 in coll. MNBG, 1 \u2642 in coll. DKC; Budapest, undated, [Hug\u00f3] Diener [leg.], 2 \u2642\u2642 and 1 \u2640  in coll. Georg Frey depo\u00adsited in NHMB, 3 spec. in coll. ZSMG, 1 spec. in coll. SMNK, 1 \u2642 in coll. DKC; Budapest, Ofen [= Buda], undated, [E.] Merkl [leg.], 2 \u2642\u2642 and 2 \u2640\u2640 (ex coll. St\u00f6cklein) in coll. Georg Frey deposited in NHMB; \u201cBuda-Pesth\u201d [= Budapest], undated, 1 \u2642 and 1 \u2640, E. Merkl leg., coll. NMPC; Budapest, 1890, \u201ccoll. O. Leonhard\u201d, no other data, 2 \u2642\u2642 in coll. SDEI; Budapest, 1895, 1 \u2642 and 1 \u2640, [Hug\u00f3] Diener [leg.], coll. SDEI; Budapest, 1899, 2 \u2640\u2640, [Hug\u00f3] Diener [leg.], coll. MSNG; Budapest, H\u00e1rmashat\u00e1rhegy Airfield, 47\u00b033'11.133\"N, 18\u00b058'29.279\"E, 276 m a.s.l., 7.vi.2019, 1 spec., NPB obs.; Pest County, Dunakeszi, gliding airport, 47\u00b036'51.79\"N, 19\u00b08'55.91\"E, 125 m a.s.l., 10.vi.2019, 1 \u2640 flying after sunset, VSI obs.; Pest County, Bugyi env., Nemes-\u00fcrb\u0151, ca. 47\u00b010'55.9\"N, 19\u00b011'24.7\"E, 92 m a.s.l., 7.vii.2018, 3 spec., Hunor Gy\u0151rfy obs.; Pest County, Bugyi, \u00dcrb\u0151puszta, 47\u00b09'52.47\"N, 19\u00b010'21.22\"E, 91 m a.s.l., 10.vi.2019, 1 \u2642 flying after sunset, VSI obs.; Pest County, Tat\u00e1rszentgy\u00f6rgy env., Ord\u00edt\u00f3, ca. 47\u00b02'13.8\"N, 19\u00b017'40.2\"E, ca. 95 m a.s.l., 5.vii.1999, 4 spec., AMK obs.; Pest County, Tat\u00e1rszentgy\u00f6rgy env., Rohanka-d\u0171l\u0151, 47\u00b03'48.05\"N, 19\u00b020'26.47\"E, 98 m a.s.l., date not available [end of 20th or beginning of 21st century], 1 spec. flying after sunset, AMK obs.; Pest County, Tat\u00e1rszentgy\u00f6rgy env., Szabad-r\u00e9t, ca. 47\u00b03'14.07\"N, 19\u00b018'1.37\"E, 94 m a.s.l., 29.vi.2018, 1 spec., CVK obs.; Pest County, Tat\u00e1rszentgy\u00f6rgy env., Sz\u00e9na-d\u0171l\u0151, ca. 47\u00b01'42.25\"N, 19\u00b017'26.7\"E, ca. 100 m a.s.l., 21.vi.1998, 3 \u2640\u2640, AMK obs.; Pest County, Nagytarcsa env., K\u00fcd\u0151i-hegy hill, 47\u00b032'21.43\"N, 19\u00b019'11.77\"E, 230 m a.s.l., 8.vi.2003, 1 \u2642 and 2 \u2640\u2640, at light (mercury-vapor lamp), VSI obs. (1 spec. in coll. HNHM), 47\u00b032'13.92\"N, 19\u00b019'10.72\"E, 21.iv.2006, 1 \u2640, at light (mercury-vapor lamp), VSI obs., 47\u00b031'59.96\"N, 19\u00b019'22.96\"E, ca. 250 m a.s.l., 19.vi.2018, 1 \u2642, night sweeping, VSI leg., coll. HNHM, 1 \u2640, at UV light, SIB obs., 47\u00b032'17.59\"N, 19\u00b019'16.03\"E, 18.vi.2013, 1 \u2642 and 2 \u2640\u2640, at light (mercury-vapor lamp), 25.vii.2019, 1 \u2640, at UV light, VSI obs.; Pest County, Csom\u00e1d, \u00d6reg-hegy, 47\u00b039'29.88\"N, 19\u00b012'38.05\"E, 15.vi.2002, 1 \u2642 and 1 \u2640, at light (mercury-vapor lamp), VSI obs. (\u2642 in coll. HNHM); Pest County, G\u00f6d\u00f6ll\u0151, 55 Erd\u0151sz\u00e9l Street, 47\u00b036'11.3\"N, 19\u00b023'23.6\"E, 250 m a.s.l., 15.vi.2004, 1 \u2642, at light (mercury-vapor lamp), VSI leg., coll. HNHM, 5.viii.2004, 1 \u2640, 7.viii.2004, 1 \u2640, at light (mercury-vapor lamp), VSI obs.; Pest County, G\u00f6d\u00f6ll\u0151 env., Fah\u00e1z-tet\u0151 hill, 47\u00b037'10.12\"N, 19\u00b025'8.94\"E, 255 m a.s.l., 19.v.2004, 1 m and 2 \u2640\u2640, at light (mercury-vapor lamp), VSI obs. (1 f in coll. HNHM), 47\u00b037'5.81\"N, 19\u00b025'9.68\"E, 26.vi.2017, 1 \u2640, at light (mercury-vapor lamp), VSI, TNB and AKB obs.; Pest County, G\u00f6d\u00f6ll\u0151 env., Per\u0151c-oldal, 47\u00b034'5.754\"N, 19\u00b020'8.424\"E, ca. 250 m a.s.l., 30.vi.2019, 1 \u2640, Csan\u00e1d Sz\u00e9n\u00e1si leg., coll. HNHM; Pest County, V\u00e1c\u00adkis\u00fajfalu, Sz\u00e9lesek, 47\u00b042'40.24\"N, 19\u00b019'36.32\"E, 180 m a.s.l., 24.vii.2018, 1 \u2640 flying after sunset, VSI obs.; Pest County, Pest County, Galgam\u00e1csa env., Ecskendi Forest, \u00d6rd\u00f6g-\u00e1rok area, 47\u00b044'20.32\"N, 19\u00b025'17.34\"E, 235 m a.s.l., 5.vi.2015, 1 \u2640, at light (marcury-vapor lamp), VSI obs.; Pest County, Domonyv\u00f6lgy, B\u00e1r\u00e1nyj\u00e1r\u00e1s, 47\u00b037'23.8\"N, 19\u00b024'1.94\"E, 220 m a.s.l., 21.v.2004, 1 \u2642 and 1 \u2640, at light (mercury-vapor lamp), VSI obs. (1 spec. in coll. HNHM); Pest County, G\u00f6d\u00f6ll\u0151 - M\u00e1riabesny\u0151 env., 47\u00b035'38.59\"N, 19\u00b024'4.82\"E, ca. 190 m a.s.l., 13.vi.2013, 1 \u2642, ZKB obs. + photo (DJP det.); Pest County, Isaszeg, 29 Erd\u0151 Street, 47\u00b031'23.412\"N, 19\u00b023'33.87\"E, 19.vi.2003, 1 \u2642, at light, VSI obs.; Pest County, Isaszeg env., Szarkaberki-v\u00f6lgy [valley] 47\u00b032'14.86\"N, 19\u00b022'11.26\"E, ca. 210 m a.s.l., 27.vi.2019, 2 \u2642\u2642 and 1 \u2640 flying after sunset, VSI obs., 1 \u2642, at UV light, SIB leg., coll. HNHM; 23.vi.2020, 10 spec. SIB obs.; 1.vii.2020, 1 spec., at light, SIB obs.; Pest County, Isaszeg env., K\u0151malmi t\u00f6lgyes, 47\u00b033'51.65\"N, 19\u00b025'48.93\"E, ca. 250 m a.s.l., 9.v.2004, 1 \u2640, at light (mercury-vapor lamp), VSI leg., coll. HNHM; Pest County, Dabas, 20.v.2012, 1 spec., at light, SIB obs.; Pest County, P\u00e9cel, 5.vi.2018, 1 spec., at light, SIB obs.; Pest County, P\u00e9cel env., 47\u00b029'49.85\"N, 19\u00b022'56.56\"E, ca. 200 m a.s.l., 12.vi.2010, 1 \u2642, at light (mercury-vapor lamp), JDB obs.; Pest County, P\u00e9cel env., Trianoni-eml\u00e9km\u0171, 47\u00b028'28.86\"N, 19\u00b022'10.78\"E, ca. 255 m a.s.l., 15.iv.2015, 1 spec., LNB obs.; Pest County, Cs\u00e9vharaszt, 24.vi.2004, 1 spec., at light., SIB obs.; Pest County, Albertirsa env., Goly\u00f3fog\u00f3-v\u00f6lgy [valley], 47\u00b015'52.86\"N, 19\u00b037'59.73\"E, 150 m a.s.l., 1.vii.2019, 2 \u2642\u2642 flying after sunset, SIB and VSI obs., 2 \u2640\u2640, at UV light, SIB obs. (1 \u2640 in coll. HNHM); Pest County, T\u00f3alm\u00e1s, Boldogk\u00e1ta-puszta, 47\u00b030'22.77\"N, 19\u00b042'2.44\"E, 110 m a.s.l., 28.vi.2019, 1 \u2642 and 1 \u2640 flying after sunset, VSI obs. (\u2640 in coll. HNHM), 1 spec., at light, SIB obs.; Pest County, T\u00e1pi\u00f3bicske, Gombai-patak [stream] bank, 47\u00b022'12.5\"N, 19\u00b038'43.6\"E, 120 m a.s.l., 3.vii.2019, 1 \u2642 flying after sunset, VSI obs.; Pest County, T\u00e1pi\u00f3bicske, Fels\u0151-T\u00e1pi\u00f3 [stream] bank, 47\u00b023'58.92\"N, 19\u00b041'25.29\"E, 111 m a.s.l., 20.vii.2020, 1 \u2642 flying after sunset, VSI obs.Southern Great Plain (D\u00e9l-Alf\u00f6ld), B\u00e1cs-Kiskun County, Kunpesz\u00e9r env., Als\u00f3-Pesz\u00e9ri-r\u00e9tek, ca. 47\u00b03'50.129\"N, 19\u00b017'57.59\"E, 93 m a.s.l., 8.vi.1996, 2 \u2642\u2642, 23.vi.1998, 1 spec., 10.vi.2002, 2 \u2642\u2642 and 1 \u2640, at light, AMK obs.; B\u00e1cs-Kiskun County, Kunpesz\u00e9r env., Pesz\u00e9ri-erd\u0151 forest, ca. 100 m a.s.l., 6.vi.1998, 6 spec., 21.vi.1999, 3 spec., 30.vi.1999, 2 \u2642\u2642, 11.vii.1999, 1 spec., AMK obs.; 26.vi.2018, 2 spec., at light, REE obs.; 28.vi.2018, 3 spec., at light, REE and CVK obs.; 4.vii.2018, 4 spec., 5.vii.2018, 7 spec., at light, REE obs.; 6.vii.2018, 1 spec., CVK obs.; 9.vii.2020, 1 spec., 13.vii.2020, 1 spec., at light, REE obs.; 22.vii.2020, 1 spec., 29.vii.2020, 1 spec., 30.vii.2020, 5 spec., 31.vii.2020, 2 spec., 8.viii.2020, 1 spec., 16.viii.2020, 10 spec., 18.viii.2020, 7 spec., 19.viii.2020, 6 spec., 20.viii.2020, 8 spec., Botond Kozma obs.; B\u00e1cs-Kiskun County, Kunadacs env., Hunga\u00adrian meadow viper Conservation Centre, ca. 47\u00b01'27.807\"N, 19\u00b017'21.286\"E, ca. 100 m a.s.l., 29.vi.2018, 1 spec., Vad\u00e1sz Csaba obs.; B\u00e1cs-Kiskun County, Kunadacs env., Hetvenholdas, ca. 47\u00b00'56.34\"N, 19\u00b016'54.34\"E, ca. 97 m a.s.l., 27.ix.2016, 1 \u2640 FSLG after sunset, AMK obs.; B\u00e1cs-Kiskun County, Kunadacs, Nagy-erd\u0151 forest, date not available [21st century], 1 spec. caught after sunset, AMK obs.; B\u00e1cs-Kiskun County, Kunadacs, Peregi-d\u0171l\u0151, ca. 46\u00b057'0.4\"N, 19\u00b017'29.4\"E, 95 m a.s.l., 6.vi.2006, 1 spec., AMK obs.; B\u00e1cs-Kiskun County, Kunadacs env., Szabadsz\u00e1ll\u00e1si-legel\u0151, ca. 46\u00b056'6\"N, 19\u00b018'15\"E, 94 m a.s.l., 9.vi.2006, 1 spec., AMK obs.; B\u00e1cs-Kiskun County, P\u00e1hi, P\u00e1hi-r\u00e9tek, ca. 100 m a.s.l., 10.vii.2020, 3 spec., CBK and REE obs.; B\u00e1cs-Kiskun County, Kiskunhalas env., pasture, 46\u00b024'10.97\"N, 19\u00b030'33.08\"E, 122 m a.s.l., 7.vi.2021, 2 \u2642\u2642 and 1 \u2640, at light just after sunset, together with 1 \u2640 of Od.armiger, TKK obs. + photo (DJP det.); B\u00e1cs-Kiskun County, Kecskem\u00e9t \u2013 Hunyadiv\u00e1ros, 46\u00b055'6.114\"N, 19\u00b042'43.794\"E, 115 m a.s.l., 29.vi.2021, 1 \u2640, at light, BCK and KVB obs. + photo (DJP det.).Northern Hungary (\u00c9szak-Magyarorsz\u00e1g), N\u00f3gr\u00e1d County, Koz\u00e1rd, village area, 47\u00b054'53.31\"N, 19\u00b037'7.07\"E, 180 m a.s.l., 28.vii.2020, 1 \u2640, at light , KHE obs.; N\u00f3gr\u00e1d County, Koz\u00e1rd env., Majors\u00e1gi-hegy hill, 47\u00b054'59.87\"N, 19\u00b036'37.52\"E, ca. 240 m a.s.l., 1.viii.2020, 1 \u2640 flying after sunset, KHE obs.; N\u00f3gr\u00e1d County, Koz\u00e1rd env., Poh\u00e1nka hill, 47\u00b054'56.21\"N, 19\u00b037'29.88\"E, 225 m a.s.l., 29.vii.2020, 1 \u2642 and 1 \u2640 flying after sunset, KHE obs.; N\u00f3gr\u00e1d County, B\u00e1tonyterenye \u2013 Kisterenye, V\u00e1ci Mih\u00e1ly Street, 48\u00b00'32.22\"N, 19\u00b049'46.48\"E, 190 m a.s.l., 28.v.1978, 1 \u2642 FSLG after sunset, TKB and TKG obs.; Heves County, Tarna\u00adlelesz env., Pataji-far, 48\u00b07'32.66\"N, 20\u00b09'32.12\"E, 475 m a.s.l., 9.vi.2016, 1 \u2642, in the grass during the day, shrubby edge of an oak forest (Quercuscerris), CBE obs.CommentBolbelasmusunicornis has been recorded several times as food for some birds  in several localities of the Austro-Hungarian Empire, including two Hungarian, two Slovak, and one Romanian locality , \u201cCarniolia, Bolbocerasquadridens Fabr.\u201d, undated, 1 \u2642, Ferdinand Joseph Schmidt leg., coll. F. J. Schmidt deposited in PMSL, Savo Brelih revid. , \u201cLeonhard\u201d [= Lenart v Slovenskih goricah], no other data, 1 spec. J. N. Spitzy leg. et coll. , Moslavina , no other data [19th century] , Osijek-Baranja County, Osijek env., no other data [19th century], Vukas [leg.] , \u201cDalmatia\u201d, no other data, 1 \u2640 in coll NMPC; \u201cDalmat.\u201d [= Dalmatia], no other data, 1 \u2640 in coll. NMPC.Commentth century. The only recent record (Gradi\u0161te) is given by MMBC and NMPC are presented in this study.In Croatia, the species is known only from four old records from the 19Published dataZFMK , Zavidovi\u0107i env., Gostovi\u0107 river valley, no other data, K\u00e1roly Kendi leg.  or Republika Srpska (\u0420\u0435\u043f\u0443\u0431\u043b\u0438\u043a\u0430 \u0421\u0440\u043f\u0441\u043a\u0430), Babin potok [river], no other data, 1 spec. in coll. Ren\u00e9 Mik\u0161i\u0107 [currently deposited in CMZC] (in CMZC] .CommentOnly four old records from Bosnia and Herzegovina have been published. No recent findings are known.Published dataVojvodina (\u0412\u043e\u0458\u0432\u043e\u0434\u0438\u043d\u0430), Srem District (\u0421\u0440\u0435\u043c\u0441\u043a\u0438 \u043e\u043a\u0440\u0443\u0433), Mt. Fru\u0161ka gora (\u0424\u0440\u0443\u0448\u043a\u0430 \u0433\u043e\u0440\u0430), village of Vrdnik (\u0412\u0440\u0434\u043d\u0438\u043a), June 2016, 1 \u2640, at light, collector unknown, coll. DKP deposited in NMPC , Mala Ivan\u010da (\u041c\u0430\u043b\u0430 \u0418\u0432\u0430\u043d\u0447\u0430) env., Grkovo (\u0413\u0440\u043a\u043e\u0432\u043e), Tre\u0161nja Forest (\u0428\u0443\u043c\u0430 \u0422\u0440\u0435\u0448\u045a\u0430), 14.v.1986, 1 \u2640 dug up beneath a hazel shrub together with Tuber fungi DPB leg. et coll. , Bor District (\u0411\u043e\u0440\u0441\u043a\u0438 \u043e\u043a\u0440\u0443\u0433), \u0110erdap National Park (\u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0438 \u043f\u0430\u0440\u043a \u0402\u0435\u0440\u0434\u0430\u043f), 6 km WSW of Tekija (\u0422\u0435\u043a\u0438\u0458\u0430), 27.\u201328.v.2014, 2 \u2642\u2642 and 1 \u2640, collector not specified, coll. DKP [deposited in NMPC] , Srem District (\u0421\u0440\u0435\u043c\u0441\u043a\u0438 \u043e\u043a\u0440\u0443\u0433), In\u0111ija (\u0418\u043d\u0452\u0438\u0458\u0430) env., Kr\u010dedin (\u041a\u0440\u0447\u0435\u0434\u0438\u043d), 19.viii.2006, 2 \u2642\u2642, at light, steppe meadow near the Danube river, LMN leg., coll. RSG and VVO; 45\u00b010'04.5\"N, 20\u00b008'15.4\"E, 98 m a.s.l., 1.vii.2013, 1 \u2642, at light at 21.25 CEST, ZBB obs. + photo , MSN obs. + photo , Bor District (\u0411\u043e\u0440\u0441\u043a\u0438 \u043e\u043a\u0440\u0443\u0433), Leskovo (\u041b\u0435\u0441\u043a\u043e\u0432\u043e) env., 44\u00b018'17.28\"N, 21\u00b056'54.96\"E, ca. 400 m a.s.l., 20.vi.2020, 1 \u2642 crawling on the ground near the road at 18.42 CEST, MPN obs. + photo  FSLG at 20.35\u201321.00 CEST, steppe hillside  near an oak-beech forest, DHH (22 spec.), RKP (16 spec.), ZCP (11 spec.), TGK (1 spec.), and PSZ (1 spec.) leg., coll. OSD, DHH, DJP, DKP, GML, LMO, MBF, PSZ, RKP, TGK, VJP and ZCP , IMO (7 spec.), MKJ (2 spec.) leg. et coll., 4 \u2642\u2642 and 3 \u2640\u2640 in coll. GML , Sauk, 10.vi.1958, 1 spec., 10.\u201320.vi.1961, 1 spec., Xhelo Murraj leg. , S\u0103laj County, Zal\u0103u env., 3.viii.1973, 1 \u2640, forest, collector unknown, coll. OHS , Suceava County, \u201cMihoweny\u201d [= Mihoveni], 1 \u2642 with no other data .Muntenia, Giurgiu County, Comana, no other data, Arnold Lucien Montandon leg., Jules Bourgeois det. , Tulcea County, Babadag [env.], [100\u2013200 m a.s.l.], 1989\u20132000, no other data , Sibiu County, \u201cTranssylv. Alpen\u201d [= Transsilvanische Alpen ], \u201cR.Turm Pa\u017fs\u201d [= Roter-Turm-Pass (Pasul Turnu Ro\u0219u)], 350\u2013450 m a.s.l., 1917, 1 \u2642 and 1 \u2640, Dr Maertens [leg.], coll. MNBG; Cluj County, Suatu, ca. 46\u00b046'39\"N, 23\u00b058'24\"E, ca. 365 m a.s.l., August 1997, 1 \u2640, at light, steppe hillside with sparsely scattered oak trees, ARC , Bac\u0103u County, Com\u0103ne\u0219ti , July 2004, 1 \u2642, dead inside the collector\u2019s house (pro\u00adbably attracted by the light), APC; 31.vii.2010, 1 \u2640, at light, APC; 8.viii.2011, 1 \u2642, Barber pitfall trap, APC leg., coll. CMI; Ia\u0219i County, H\u00e2rl\u0103u env., P\u00eercovaci env., 47\u00b028'28.29\"N, 26\u00b047'22.17\"E, 240 m a.s.l., 24.vi.2021, 1 \u2640, LHI obs. + photo (DJP det.); Ia\u0219i County, Ia\u0219i \u2013 Rediu, Iazul T\u0103ute\u0219ti, 47\u00b013'33.4\"N, 27\u00b028'06.7\"E, 120 m a.s.l., 28.vii.2021, 3 \u2640\u2640, at light, MJR leg., coll. PKG; Ia\u0219i County, Ia\u0219i \u2013 Miroslava, Valea lui David, 47\u00b011'38\"N, 27\u00b028'2.114\"E, ca. 90 m a.s.l., 9.vii.2021, 2 \u2640\u2640, together with 1 \u2642 of Od.armiger, LHI obs. + photo (DJP det.); Ia\u0219i County, B\u00e2rnova Forest , 4.vii.2005, 1 \u2642, found accidentally on the ground during the day, LFI leg., coll. CUIR  env., Kulhav\u00e1 sk\u00e1la hill env., 44\u00b042'11.47\"N, 21\u00b043'41.49\"E, 357 m a.s.l., 1.vi.2012, 1 elytron on a path going through a pasture, BJN; Cara\u0219-Severin County, Svat\u00e1 Helena (Sf\u00e2nta Elena), 44\u00b040'29.8\"N, 21\u00b042'35\"E, 325 m a.s.l., 18.vi.2017, 1 \u2640 FSLG after sunset, ZCP, 44\u00b040'57.73\"N, 21\u00b042'19\"E, 350 m a.s.l., 23.vi.2017, 4 spec. FSLG after sunset, ZCP obs. (1 \u2642 leg. et coll.); Cara\u0219-Severin County, Mehadia, undated [19th century], 1 \u2640, \u201cex. coll. [Otto] Staudinger\u201d, coll. MTDG; Mehedin\u021bi County, Tisov\u00e9 \u00dadol\u00ed , ca. 44\u00b032'36.7\"N, 22\u00b010'20.4\"E, ca. 420 m a.s.l., 28.v.2008, 1 \u2642 flying slowly up to 0.5 m above a path crossing a forest-steppe meadow at 21.45 EEST (= 40 min after sunset), JKV.Muntenia, \u201cBukarest\u201d [= Bucharest], undated, 1 \u2642, V[ladim\u00edr] Zoufal leg., coll. Vladim\u00edr Zoufal deposited in MMBC; Teleorman County, Poroschia, [ca. 40 m a.s.l.], no other data, 1 \u2640 in coll. GANM; Buz\u0103u County, M\u0103gura, M\u0103n\u0103stirea Ciolanu [= Ciolanu Monastery], 5.vii.2014, 1 \u2642, at light (160 W mercury-vapor lamp), beech forest, VUB.Dobruja (Dobrogea), Tulcea County, Agighiol, 12.vi.1993, 1 \u2640, Ioana Matache leg., coll. GANM; Tulcea County, Babadag [env.], [100\u2013200 m a.s.l.], 20.vi.1958, 1 \u2642, 20.vi.1968, 1 \u2642, Nicolae S\u0103vulescu leg., coll. GANM; 11.vii.1985, 1 \u2642, at light (mercury-vapor lamp), foot of a forest-steppe loess hill, JHM leg., coll. VKS , 44\u00b048'55.47\"N, 28\u00b041'23.15\"E, 110 m a.s.l., 6.vi.2016, 1 spec., IIB, 44\u00b049'04.0\"N, 28\u00b040'57.9\"E, 140 m a.s.l., 10.vi.2016, ca. 30 spec. FSLG after sunset, MVP obs. ; Constan\u021ba County, B\u0103neasa \u2013 Canaraua Fetei, ca. 44\u00b03'13.28\"N, 27\u00b040'15.07\"E, ca. 115 m a.s.l., 17.vii.1965, 1 \u2642, Nicolae S\u0103vulescu leg., coll. GANM; Constan\u021ba County, Albe\u0219ti env., Hagieni Forest, ca. 50 m a.s.l., 20.vi.1964, 1 \u2640, collector unknown, coll. GANM; Constan\u021ba County, Hagieni, ca. 50 m a.s.l., 18.vi.1995, 1 \u2640, at light, CWP leg., coll. LKKA.CommentFor Romania, which can be considered one of the countries at the centre of the species\u2019 distribution, surprisingly small amounts of data have been published. New records from 22 Romanian localities are presented here.Published dataC\u0103l\u0103ra\u0219i District (Raionul C\u0103l\u0103ra\u0219i), Bularda near Dereneu, ca. 165 m a.s.l., 16.vi.1931, 3 \u2642\u2642 and 4 \u2640\u2640, Nicolai Zubowsky leg., coll. N. Zubowsky deposited in NMCM , D\u0103nceni, ca. 170 m a.s.l., 31.v.1929, 1 \u2642, Nicolai Zubowsky leg., coll. N. Zubowsky deposited in NMCM (City of Chi\u0219in\u0103u (Municipiul Chi\u0219in\u0103u), Chi\u0219in\u0103u, [ca. 100 m a.s.l.], 20.v. and 10.vii.[between 1900\u20131915], no other data , Chi\u0219in\u0103u, 20.iv.1912, 1 \u2640, Nicolai Zubowsky leg., Valeriu Derjanschi det., coll. Rodion Stepanov (box No. 10) deposited in NMCM.Anenii Noi District (Raionul Anenii Noi), H\u00eerbov\u0103\u021b env., H\u00eerbov\u0103\u021b Forest, ca. 285 m a.s.l., June 1970, 1 \u2642, Rodion Stepanov leg., Valeriu Derjanschi det., coll. R. Stepanov (box No. 28) deposited in IZCM (for incomplete data on this record see CommentThe first known record from Moldova (Chi\u0219in\u0103u) is mentioned by Published data\u201cGubernia podolska\u201d  .\u201cVolhynien\u201d , undated, 2 spec., prof. Bresser leg.  \u2013 this rIvano-Frankivsk Oblast (\u0406\u0432\u0430\u043d\u043e-\u0424\u0440\u0430\u043d\u043a\u0456\u0432\u0441\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Chornohora (\u0427\u043e\u0440\u043d\u043e\u0433\u043e\u0440\u0430) [mountain range], 9.viii.1939, 1 \u2642, collector unknown, coll. SIZK (? ll. SIZK  \u2013 the naTernopil Oblast (\u0422\u0435\u0440\u043d\u043e\u043f\u0456\u043b\u044c\u0441\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Ternopil Raion (\u0422\u0435\u0440\u043d\u043e\u043f\u0456\u043b\u044c\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Zboriv (\u0417\u0431\u043e\u0440\u0456\u0432), 19.viii.1937, 1 \u2640; collector unknown, coll. SIZK , Bukovina (\u0411\u0443\u043a\u043e\u0432\u0438\u043d\u0430), Chernivtsi Raion (\u0427\u0435\u0440\u043d\u0456\u0432\u0435\u0446\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Chernivtsi (\u0427\u0435\u0440\u043d\u0456\u0432\u0446\u0456), 4 spec. with no other data , Vinnytsia Raion (\u0412\u0456\u043d\u043d\u0438\u0446\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Vinnytsia (\u0412\u0456\u043d\u043d\u0438\u0446\u044f) env., August 1928, 1 \u2642, caught in flight in the evening, collector unknown , Odessa Raion (\u041e\u0434\u0435\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Odessa (\u041e\u0434\u0435\u0441\u0430), 1827\u20131831, no other data ; Rozdilna Raion (\u0420\u043e\u0437\u0434\u0456\u043b\u044c\u043d\u044f\u043d\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), 4 km NW of Butsynivka (\u0411\u0443\u0446\u0438\u043d\u0456\u0432\u043a\u0430) village, 4.vi.2011, 1 \u2640, at UV light, YKO leg., coll. VTO .her data , Odessa her data , 2001, \u201cher data ; Odessa Kyiv Oblast (\u041a\u0438\u0457\u0432\u0441\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Kiyv (\u041a\u0438\u0457\u0432), old town, May 1839, 4 spec. under a dead dog, June 1870, 1 spec. on a grassy path, Johann Heinrich Hochhuth leg. ; Obukhiv Raion (\u041e\u0431\u0443\u0445\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Hryhorivka (\u0413\u0440\u0438\u0433\u043e\u0440\u0456\u0432\u043a\u0430), 6.vi.1928, 1 \u2642, collector unknown, coll. Ye. M. Savchenko deposited in NHMU , Zvenyhorodka Raion (\u0417\u0432\u0435\u043d\u0438\u0433\u043e\u0440\u043e\u0434\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Talne (\u0422\u0430\u043b\u044c\u043d\u0435), 1 \u2642 with no other data ; Cherkasy Raion (\u0427\u0435\u0440\u043a\u0430\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Kaniv (\u041a\u0430\u043d\u0456\u0432) env., Kaniv Nature Reserve (\u041a\u0430\u043d\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u043f\u0440\u0438\u0440\u043e\u0434\u043d\u0438\u0439 \u0437\u0430\u043f\u043e\u0432\u0456\u0434\u043d\u0438\u043a), no other data , Novhorod-Siverskyi Raion (\u041d\u043e\u0432\u0433\u043e\u0440\u043e\u0434-\u0421\u0456\u0432\u0435\u0440\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Novhorod-Siverskyi (\u041d\u043e\u0432\u0433\u043e\u0440\u043e\u0434-\u0421\u0456\u0432\u0435\u0440\u0441\u044c\u043a\u0438\u0439) env., 51\u00b059'N, 33\u00b016'E, 18.vii.2003, 1 spec., I. V. Porokhniach leg., coll. GUNU , Shostka Raion (\u0428\u043e\u0441\u0442\u043a\u0438\u043d\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Matskove (\u041c\u0430\u0446\u043a\u043e\u0432\u0435) env., ca. 51\u00b028'48\"N, 33\u00b053'24\"E, ca. 150 m a.s.l., 28.vii.2018, 1 spec., MZK , Lubny Raion (\u041b\u0443\u0431\u0435\u043d\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Lubny (\u041b\u0443\u0431\u043d\u0438), [ca. 160 m a.s.l.], July (year and nummber of specimens not specified), Kruhlik [leg.], coll. Provincial Museum of Poltava ( Poltava .Dnipropetrovsk Oblast (\u0414\u043d\u0456\u043f\u0440\u043e\u043f\u0435\u0442\u0440\u043e\u0432\u0441\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Synelnykove Raion (\u0421\u0438\u043d\u0435\u043b\u044c\u043d\u0438\u043a\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Raivka (\u0420\u0430\u0457\u0432\u043a\u0430), 1.viii.2000, 1 spec., A. M. Sumarokov leg. , Mukachevo Raion (\u041c\u0443\u043a\u0430\u0447\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), \u201cSch\u00f6nb Ungarn\u201d , [ca. 190 m a.s.l.], \u201ccoll. Kirsch\u201d, undated, 1 \u2642, coll. MTDG.Ivano-Frankivsk Oblast (\u0406\u0432\u0430\u043d\u043e-\u0424\u0440\u0430\u043d\u043a\u0456\u0432\u0441\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Kosiv Raion (\u041a\u043e\u0441\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Pistyn (\u041f\u0456\u0441\u0442\u0438\u043d\u044c), [ca. 400 m a.s.l.], undated, 1 \u2642,\u201cA. St?kl\u201d [the third letter is illegible] leg., coll. SMLU.Ternopil Oblast (\u0422\u0435\u0440\u043d\u043e\u043f\u0456\u043b\u044c\u0441\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Chortkiv Raion (\u0427\u043e\u0440\u0442\u043a\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), \u201cTorskie, pow[iat] Zaleszcz[yki]\u201d , [ca. 250 m a.s.l.], 27.vi.[19]33, 1 \u2642, collector unknown, coll. MIZP; Dniester Canyon National Nature Park (\u041d\u0430\u0446\u0456\u043e\u043d\u0430\u043b\u044c\u043d\u0438\u0439 \u043f\u0440\u0438\u0440\u043e\u0434\u043d\u0438\u0439 \u043f\u0430\u0440\u043a \u00ab\u0414\u043d\u0456\u0441\u0442\u0440\u043e\u0432\u0441\u044c\u043a\u0438\u0439 \u043a\u0430\u043d\u044c\u0439\u043e\u043d\u00bb), Chortkiv Raion (\u0427\u043e\u0440\u0442\u043a\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Horodok (\u0413\u043e\u0440\u043e\u0434\u043e\u043a), 48\u00b038'18.96\"N, 25\u00b050'11.04\"E, ca. 140 m a.s.l., 6.vii.2018, 6 \u2642\u2642 and 2 \u2640\u2640 FSLG after sunset, steppe meadow on the terrace of the Dniester (\u0414\u043d\u0456\u0441\u0442\u0435\u0440) river, YKL, YHS and ABZ leg. coll. YKL and YHS , Bukovina (\u0411\u0443\u043a\u043e\u0432\u0438\u043d\u0430), Chernivtsi Raion (\u0427\u0435\u0440\u043d\u0456\u0432\u0435\u0446\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), \u201cCzernowitz\u201d [= Chernivtsi (\u0427\u0435\u0440\u043d\u0456\u0432\u0446\u0456)], no other data, 1 \u2642 in coll. Georg Frey deposited in NHMB, 1 \u2642 and 2 \u2640\u2640  in coll. Georg Frey deposited in NHMB, 2 \u2642\u2642 and 3 \u2640\u2640 in coll. ZMNU, 1 \u2642 and 2 \u2640\u2640 in coll. UMJG.Vinnytsia Oblast (\u0412\u0456\u043d\u043d\u0438\u0446\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), \u201c\u041a\u0438\u0435\u0432\u0441\u043a\u0430\u044f \u0433[\u0443\u0431\u0435\u0440\u043d\u0438\u044f], \u0421\u043a\u0432\u0438\u0440\u0441\u043a\u0438\u0439 \u0443[\u0435\u0437\u0434]\u201d , \u201c\u0418\u043b\u044c\u0438\u043d\u0446\u044b\u201d [= Illintsi (\u0406\u043b\u043b\u0456\u043d\u0446\u0456)], 14.vi.[year not specified], [ca. 215 m a.s.l.], 1 \u2640, collector not specified, coll. ZINR (probably one of the two specimens mentioned by Odessa Oblast (\u041e\u0434\u0435\u0441\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Bilhorod-Dnistrovskyi Raion (\u0411\u0456\u043b\u0433\u043e\u0440\u043e\u0434-\u0414\u043d\u0456\u0441\u0442\u0440\u043e\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Karolino-Buhaz (\u041a\u0430\u0440\u043e\u043b\u0456\u043d\u043e-\u0411\u0443\u0433\u0430\u0437), Studentska (\u0421\u0442\u0443\u0434\u0435\u043d\u0442\u0441\u044c\u043a\u0430) railway station, ca. 46\u00b09'56.34\"N, 30\u00b033'24.58\"E, 22 m a.s.l., 15.vi.2017, 1 \u2642 crawling on the ground, OKO.Kyiv Oblast (\u041a\u0438\u0457\u0432\u0441\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Bila Tserkva Raion (\u0411\u0456\u043b\u043e\u0446\u0435\u0440\u043a\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), \u201cHa\u0142ajki, Kijow[ska] g[ubernia]\u201d , [ca. 190 m a.s.l.], [probably 19th Century], no other data, 1 \u2642 in coll. MIZP; Bucha Raion (\u0411\u0443\u0447\u0430\u043d\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Muzychi (\u041c\u0443\u0437\u0438\u0447\u0456), ca. 160 m a.s.l., 18.vii.2006, 1 \u2642, at light, M. Nesterov leg., coll. SIZK; \u201cKiew\u201d [= Kyiv (\u041a\u0438\u0457\u0432)], undated, 1 \u2640 in Hartmann [leg.], coll. NMPC; \u201cKieff\u201d [= Kyiv (\u041a\u0438\u0457\u0432)], May [19]05, 1 \u2642, Shelushko [leg.], coll. ZINR; Fastiv Raion (\u0424\u0430\u0441\u0442\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Novosilky (\u041d\u043e\u0432\u043e\u0441\u0456\u043b\u043a\u0438), ca. 180 m a.s.l., 21.vii.2012, 1 \u2640, M. Nesterov leg., coll. SIZK; Obukhiv Raion (\u041e\u0431\u0443\u0445\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Mali Dmytrovychi (\u041c\u0430\u043b\u0456 \u0414\u043c\u0438\u0442\u0440\u043e\u0432\u0438\u0447\u0456), 50\u00b012'59\"N, 30\u00b032'29\"E, ca. 160 m a.s.l., 17.vii.2010, 1 \u2642, at light, together with 1 \u2642 of Od.armiger, VSK leg., coll. KLP; 29.v.2014, 1 \u2642 and 1 \u2640, at light, RHK; 25.v.2016, 1 \u2642, 28.v.2016, 1 \u2640, 13.vi.2020, 1 \u2642 and 2 \u2640\u2640, at light, STK; Obukhiv Raion (\u041e\u0431\u0443\u0445\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Rzhyshchiv (\u0420\u0436\u0438\u0449\u0456\u0432), Taras Shevchenko Park (\u041f\u0430\u0440\u043a \u0456\u043c\u0435\u043d\u0456 \u0422\u0430\u0440\u0430\u0441\u0430 \u0428\u0435\u0432\u0447\u0435\u043d\u043a\u0430), 49\u00b057'58.1\"N, 31\u00b002'39.5\"E, 112 m a.s.l., 18.ix.2021, 1 \u2640 crawling on the ground at 16:22 EEST, HTR obs. + photo (DJP det.); Obukhiv Raion (\u041e\u0431\u0443\u0445\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Rzhyshchiv (\u0420\u0436\u0438\u0449\u0456\u0432) env., area of the Ecological Research Centre \u201cHlyboki Balyky (\u0413\u043b\u0438\u0431\u043e\u043a\u0456 \u0431\u0430\u043b\u0438\u043a\u0438)\u201d, 49\u00b057'44.082\"N, 31\u00b07'8.094\"E, ca. 150 m a.s.l., 18.vi.2021, 1 \u2640, at light OVK obs. + photo + recorded an audio track of its stridulation (DJP det.); 49\u00b057'43.729\"N, 31\u00b07'9.782\"E, 19.vi.2021, 1 \u2642, together with 1 \u2642 and 2 \u2640\u2640 of Od.armiger, OVK obs. + photo (DJP det.); Myronivka Raion (\u041c\u0438\u0440\u043e\u043d\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Tulyntsi (\u0422\u0443\u043b\u0438\u043d\u0446\u0456), ca. 150 m a.s.l., 9.vi.2020, 1 \u2640, at light, STK; Myronivka Raion (\u041c\u0438\u0440\u043e\u043d\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Velykyi Bukryn (\u0412\u0435\u043b\u0438\u043a\u0438\u0439 \u0411\u0443\u043a\u0440\u0438\u043d) env., 49\u00b057'13\"N, 31\u00b018'8\"E, 155 m a.s.l., 27.vi.2009, 1 \u2640, at light, VSK.Cherkasy Oblast (\u0427\u0435\u0440\u043a\u0430\u0441\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Cherkasy Raion (\u0427\u0435\u0440\u043a\u0430\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Kaniv (\u041a\u0430\u043d\u0456\u0432) env., Kaniv Nature Reserve (\u041a\u0430\u043d\u0456\u0432\u0441\u044c\u043a\u0438\u0439 \u043f\u0440\u0438\u0440\u043e\u0434\u043d\u0438\u0439 \u0437\u0430\u043f\u043e\u0432\u0456\u0434\u043d\u0438\u043a), 49\u00b043'12\"N, 31\u00b031'19\"E, ca. 200 m a.s.l., 20.vi.1984, 6 spec. excavated from their burrows, steppe slope in a hornbeam forest, KVM and VGG leg., coll. MKY and MPGU.Kirovohrad Oblast (\u041a\u0456\u0440\u043e\u0432\u043e\u0433\u0440\u0430\u0434\u0441\u044c\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442\u044c), Oleksandriia Raion (\u041e\u043b\u0435\u043a\u0441\u0430\u043d\u0434\u0440\u0456\u0439\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Semyhiria (\u0421\u0435\u043c\u0438\u0433\u0456\u0440\u2019\u044f) env., 49\u00b00'29.52\"N, 32\u00b054'21.24\"E, 135 m a.s.l., 2.vii.2020, 1 \u2642, at light, DPS obs. + photo , Dnipro Raion (\u0414\u043d\u0456\u043f\u0440\u043e\u0432\u0441\u044c\u043a\u0438\u0439 \u0440\u0430\u0439\u043e\u043d), Dnipro (\u0414\u043d\u0456\u043f\u0440\u043e) [Dnipropetrovsk until 19 May 2016], Tunelna Balka tract (\u0422\u0443\u043d\u0435\u043b\u044c\u043d\u0430 \u0431\u0430\u043b\u043a\u0430) , research area of the Institute of Grain Crops of NAAS of Ukraine (\u0406\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u0437\u0435\u0440\u043d\u043e\u0432\u0438\u0445 \u043a\u0443\u043b\u044c\u0442\u0443\u0440 \u041d\u0410\u0410\u041d \u0423\u043a\u0440\u0430\u0457\u043d\u0438), 48\u00b022'58.2\"N, 35\u00b002'01.7\"E, 143 m a.s.l., June 1978, remains of a dead specimen (elytra) on the ground near the greenhouse, OSD leg. et coll. , Oryahovo (\u041e\u0440\u044f\u0445\u043e\u0432\u043e), no other data (her data .Ruse Province (\u041e\u0431\u043b\u0430\u0441\u0442 \u0420\u0443\u0441\u0435), Vetovo (\u0412\u0435\u0442\u043e\u0432\u043e), no other data (her data .Razgrad Province (\u041e\u0431\u043b\u0430\u0441\u0442 \u0420\u0430\u0437\u0433\u0440\u0430\u0434), Razgrad (\u0420\u0430\u0437\u0433\u0440\u0430\u0434) env., 16.v.1905, in the barracks area, number of specimens not specified, Andrey Markovich leg, 13.vi.1907, in the vineyards, number of specimens not specified, Andrey Markovich leg. (ich leg. .Pleven Province (\u041e\u0431\u043b\u0430\u0441\u0442 \u041f\u043b\u0435\u0432\u0435\u043d), Pleven (\u041f\u043b\u0435\u0432\u0435\u043d), April [year and collector not specified] (ecified] .Sofia Province (\u0421\u043e\u0444\u0438\u0439\u0441\u043a\u0430 \u043e\u0431\u043b\u0430\u0441\u0442), Gorna Malina (\u0413\u043e\u0440\u043d\u0430 \u041c\u0430\u043b\u0438\u043d\u0430) \u2013 \u201c\u0414\u0417\u0421\u201d , ca. 650 m a.s.l., 7.vii.1969, 2 \u2642\u2642 and 1 \u2640 excavated from the soil from a depth of ca. 10 cm on a pasture (northern slope), collector not specified , Shumen (\u0428\u0443\u043c\u0435\u043d), ca. 200 m a.s.l., 1914, 2 \u2642\u2642, Hanu\u0161 leg., coll. MYP (oll. MYP .Burgas Province (\u041e\u0431\u043b\u0430\u0441\u0442 \u0411\u0443\u0440\u0433\u0430\u0441), \u201cMichurin (\u041c\u0438\u0447\u0443\u0440\u0438\u043d)\u201d [= Tsarevo (\u0426\u0430\u0440\u0435\u0432\u043e)], 29.\u201330.vi.1982, 1 \u2642, at light., BSP , Dulovo (\u0414\u0443\u043b\u043e\u0432\u043e) env., Karakuz forest (\u0433\u043e\u0440\u0430 \u041a\u0430\u0440\u0430\u043a\u0443\u0437) , 14.vi.1952, 1 \u2640, P[encho Stefanov] Drenski leg., coll. NMSB , Dimovo (\u0414\u0438\u043c\u043e\u0432\u043e) env., steppe meadow near the Archar (\u0410\u0440\u0447\u0430\u0440) river, 43\u00b045'28.7\"N, 22\u00b044'51.1\"E, 110 m a.s.l., 26.vi.2010, 3 \u2642\u2642 and 2 \u2640\u2640 FSLG after sunset just before a storm, together with several spec. of Och.chrysomeloides, no wind, 26 \u00b0C, ASH , Oreshak (\u041e\u0440\u0435\u0448\u0430\u043a) env., 43\u00b017'50.67\"N, 27\u00b053'47.29\"E, 300 m a.s.l., 6.vii.2020, 1 \u2640 flying up to 0.5 m above the grass at ca. 22.00 EEST, forest-steppe clearing in an oak forest, MTM obs. + photo , Edirne Province, ca. 15 km E of Edirne , 27.iii.[19]88, 1 \u2640, WHS leg., coll. DKP deposited in NMPC ( in NMPC .Published dataAegean Region (Ege B\u00f6lgesi), Denizli Province, Denizli env., [\u00c7\u00fcr\u00fcksu River valley], \u201cGoundely\u201d [= Goncal\u0131] [railway station env.], ca. 200 m a.s.l., May [19]26, 1 \u2642, [Hans] Kulzer leg., coll. ZSMG , Ayd\u0131n Province, [B\u00fcy\u00fck Menderes River valley], \u201cBereketli (Denizli)\u201d [= Bereketli near Nazilli], ca. 80 m a.s.l., 5.vii.1965, 1 \u2642, [Helio] Pierotti and [Antonello] Perissinotto leg., DJP det. (2021), coll. Helio Pierotti deposited in MSNG \u2013 this record was published under a misidentification as Bolbelasmustauricus Petrovitz, 1973  (see Only the three records mentioned above are known for Turkey. A record from Osmaniye Province (Kadirli) reported by 895) see .Great BritainPublished dataEast of England, Cambridgeshire, marshes between Peterborough and Wisbech, beginning of summer 1807, 1 \u2642 and 1 \u2640, plant materials alluviated by flooded River Nene, together with 2 \u2642\u2642 and 3 \u2640\u2640 of Od.armiger, William Skrimshire leg. . Bolbelasmusunicornis was included in checklists, catalogues, and monographs dealing with the scarabaeoid fauna of several countries as follows: France .Figure B.unicornis is Warsaw (Poland), while the northernmost locality with a recent record is Novhorod Siverskyi (northern Ukraine). The southernmost historical locality is Denizli (southwestern Turkey), while the southernmost recent localities are Babin Kal (Serbia) and Oreshak (Bulgaria). The westernmost historical locality is Mulhouse , while the westernmost recent localities are Bruchsal  and Lerma . The eas\u00adternmost locality with a recent record of the species is Kocherezhky (Ukraine), which is also the easternmost known point of occurrence of the species.The northernmost known historical locality of B.unicornis spend most of their time underground. Above-ground activity is limited to short flight periods after sunset. Exceptionally, adults have been observed crawling on the ground during daylight hours (see Faunistic records). Flight statistics from each site are shown in Tables B.gallicus, only 5% of the 830 individuals found were females . The average duration of flights was of 25 minutes, with the minimum and maximum limits of 4 and 63 minutes, respectively. The ave\u00adrage air temperature during flights was 21 \u00b0C with limits of 14 and 26 \u00b0C. However, it is likely that the beetles are able to fly at lower temperatures, as has been observed, for example, in the Australian bolboceratine Blackburniuminsigne , adults of which have been found flying to lights at 4\u20136 \u00b0C . Similarly, flights of adults of Odonteusarmiger were observed during rain (Ivo Jeni\u0161 and Ilja Trojan pers. comm.). Adults of B.unicornis usually fly very slowly at a height of 20\u201350 cm above the ground, sometimes literally hovering in the same spot. However, in windy conditions they have been observed to fly faster and also at greater heights, ca. 1\u20132 m above the ground. Individuals flying quickly around a pile of logs at the edge of a forest were observed by the author near the village of Hajn\u00e1\u010dka in southern Slovakia (see Faunistic records). This phenomenon was also observed in Od.armiger (Ilja Trojan and Ivo Jeni\u0161 pers. comm.): adults of this species were flying around the fallen oak trunk and piles of wet logs after sunset. When disturbed, the flying specimens of B.unicornis either immediately fell into the grass and buried themselves or accelerated their flight, increasing the height from the ground and flying away. This also applies to disturbances caused by too strong light source, e.g., from a headlamp. During flights, most beetles show light-aversion and avoid light sources; the individuals that were attracted to light were single cases only. These were mostly long-distance flights that occurred later in the night. Very rarely a few individuals did fly to the illuminated canvas just after sunset when it was not yet completely dark . In most Czech and Slovak localities, adults of B.unicornis were flying together with Od.armiger and Och.chrysomeloides or O.integriceps . When excavating adults, sometimes two to three individuals were found in a single burrow, even of the same sex  in North Carolina, but he never found specimens of the same sex in the same burrow. On several occasions, individuals of B.unicornis, Od.armiger, and Och.chrysomeloides (or O.integriceps) have been found together in a single burrow  together with sporocarps of Tuber sp. in the Belgrade District of Serbia. These findings support earlier hypotheses about the mycetophagy of the species  stored a mass of sporocarps of ectomycorrhizal basidiomycete Rhizopogonpachyphloes in their burrows, and Rh.nigrescens. In the European species Od.armiger, Rh.luteolus, partially decayed, together with two individuals of Anoplotrupesstercorosus , and another individual feeding on a sporocarp of Glomusmicrocarpum. Furthermore, these authors reported that near burrows dug by adults of Od.armiger kept in captivity, sporocarps of Endogonelactiflua were found. In contrast, adults of the genus Eucanthus, for example, probably do not ingest any food at all  feed on sporocarps of arbuscular mycorrhizal fungi  were repeatedly found under dry horse and cow dung in the Far East of Russia, in cavities covered with white mould, thus suggesting that the adults are mycetophagous; no burrows were observed under the dung. In Australia, dissections of bolboceratines and analysis of their faeces were carried out mainly by Blackbolbus, Blackburnium, Bolboleaus and Bolborhachium species contained large quantities of spores of various species of hypogeous fungi , as well as immature unidentified sporocarp tissue, unidentified ascomycetes, or glomeralean hyphae and spores with varying quantities of soil. These authors also reported that only six of 120 specimens of bolboceratines collected while in flight , and only 34 of 114 bolboceratines collected from burrows had food in their intestines. It is likely that the beetles feed only intermittently and possibly spend protracted periods without a meal. In many cases, their burrows may serve to provide them only with shelter until their next foray. In several genera, B.unicornis, it was not possible to dissect the individuals to determine the intestinal contents due to its strict protection in all EU countries. In the burrows of some Australian species of the genera Blackbolbus, Blackburnium, Bolborhachium, and Elephastomus, pieces of sporocarps of Scleroderma sp., Hysterangium sp., and unspecified hypogeous fungi of the families Hymenogasteraceae and Clathraceae have been found, but with no eggs or larvae present in the vicinity  have been found on the sporocarps of hypogeous fungi . Scleroderma sporocarps found in burrows of Blackbolbusfrontalis  were inhabited by numerous nitidulid beetles identified as Thalycrodesmixtum Kirejtshuk & Lawrence, 1992, and two sporocarps identified as Hysterangium sp. found in soil close to a burrow of Blackbolbusfrontalis were infested with nematodes. Mycetophagy of adults of the genus Ochodaeus, representatives of which were collected together with B.unicornis, was also recently confirmed  measured 7.2 \u00d7 5.9 mm and 8.1 \u00d7 6.5 mm. The largest female of B.anneae measured 15.1 mm in length, while the largest female of B.recticorne measured 18.8 mm in length. According to Blackburniumreichei  weighed 45\u201356% as much as the females that laid them and measured 9.5\u201310.5 \u00d7 7.5\u20139.0 mm. On the other hand, the eggs of the North American bolboceratine Odonteusdarlingtoni are not so large compared to the adults: they measure ca. 2.4 \u00d7 1.5 mm, whereas the adults are ca. 10 mm in length , nevertheless, according to B.gallicus fixes its giant egg to the ceiling of a small egg-shaped brood cell using soil mixed with its own excreta. All the cells found by Rahola Fabra were empty, which means that they did not contain anything that could provide food for the future larvae. Similarly, Od.armiger contained no provision, but in a few cases he found pieces of unspecified fungi or humus in the burrows. Also, Od.armiger. The fact that the brood cells did not contain any material collected by females differs from what Bolboceratinae, where females lined their brood cells with material brought in from outside  that could be a food for the larva. In two species of the genus Bolborhachium the brood cells were filled with fine black humus, perhaps mixed with fungi  is unlikely to account for all (if any) of their weight gain. As the contents of the larval intestine were hygroscopic, perhaps larvae ingest salts and/or humic and fulvic acids that enable them to absorb water  where the head and mandibles are strongly sclerotised and the legs well developed with strong tarsal claws, the larvae of all known bolboceratines are degenerate. According to Blackbolbushoplocephalus  provides the most extreme example of degeneration known to date. Its immobility and vestigial appen\u00addages  suggested it was a non-feeding, resting stage. Importantly, though, the mandibles of the second instar (judging from its exuvia) were equally feeble and consistent with a no-feeding hypothesis . Odonteusdarlingtoni, O.liebecki , and Bolbocerosomafarctum  feed on humus, carefully sifting it from a provision of humus-rich sand filling the lower ends of burrows. This is consistent with earlier observations by Robert J. Sim, who assumed that females of Odonteussimi  lay their eggs in humus formed into an elongated mass at the lower ends of the burrows , while the food of larvae could be fine soil humus and/or soil bacteria. The previous hypothesis that the larvae of B.unicornis feed on sporocarps of hypogeous fungi , the period between discovery of an egg and hatching of the larva was 15\u201335 days. According to Houston, duration of the larval stage in Bolborhachiumrecticorne ranged from 63 to 95 days (n = 6) and for one Blackburniumreichei larva, it was 44 days. One larva of Bolboleaushiaticollis pupated 81 days after being found while another (hatched from an egg) survived for at least 13 months before dying. Based on the Houston\u2019s data, development from egg to adult in Bolborhachiumrecticorne could require 129\u2013159 days or more. As newly emerged adults remained in their natal cells for at least 30 days while their integuments harden and darken, total development time (egg to active adult) might require 6 months or more, according to Houston. Bolboleaushiaticollis, mature larvae enter a dormant stage, thereby extending the development time even further. Blackbolbushoplocephalus that remained dormant for 105 days before pupating.Life cycles have been documented for only a few bolboceratine species , 2016. HB.unicornis is similar, it is very likely that only adults, both old and newly emerged, overwinter. This assumption is supported by numerous records where both old, dark-coloured individuals with heavily abraded teeth of fore tibiae and fresh, pale-coloured individuals with sharp protibial teeth have been recorded at the beginning of the season (pers. obs.). Some bolboceratines have overlapping generations. For example, in the genus Odonteus, eggs, larvae, pupae, and adults have been observed together in a single branching burrow , sandy-loess substrate , gravelly-sandy-loess substrate , loess substrate  or limestone substrate . Characteristic habitats are steppe or forest-steppe pastures at the edges of oak forests , of oak-beech or beech forests , oak-hornbeam or hornbeam forests , and of shrub zones . In the Kiskuns\u00e1g National Park it occurs in the Pannonic sand dune thicket  , while further south the species occurs in areas of more extensive forest cover . In eastern Ukraine (Dnipro City) the species occurs in sparse oak forest  . Occurreest Fig. . Also, test Fig. . Other tRobiniapseudoacacia or Ailanthusaltissima in steppe and forest-steppe habitats. Furthermore, the extensive removal of shrubs such as Crataegus sp., Rosacanina, Corylusavellana, and trees  and taller herbaceous plants, as well as too intensive sheep grazing seem to have negative effects on the presence of B.unicornis , Odonteusarmiger, Ochodaeuschrysomeloides, O.integriceps, Gymnopleurus spp., Carabusmontivagus Palliardi, 1825, C.scabriusculus Olivier, 1795, Capnodistenebrionis , Perotislugubris , Ptosimaundecimmaculata , Sphenoptera spp., and Agrilusalbogularis Gory, 1841 (observations by many collectors including the author).The B.unicornis has been recorded varies between 20 and 800 m a.s.l. The average altitude of all known localities for which it could be at least approximately determined (n = 351) is 220 m a.s.l. It is therefore a species of lowland and lower hills.The elevation of the sites where Vespacrabro Linnaeus, 1758), as reported by The most effective method for monitoring this species is to capture adults during their flights after sunset with a net using a flashlight, preferably a headlamp, in suitable microhabitat. This collecting method was employed as early as the 1920s by Rudolf \u010cepel\u00e1k see . The effAlternatively, during the day, one may find the beetles in their burrows, which are indicated by small piles of excavated soil at the entrances . These push-ups are similar to those of some large ground-nesting bees but the push-ups of the bees are conical, composed of uniformly loose soil about a central entrance, whereas the soil pushed up by the earth-borer beetles tends to form an irregular pile of lumps Figs , 15E, F.Light trapping appears to be ineffective to capture this species cf. also , as the Bolbelasmusunicornis is also difficult to find due to the fact that observable activity of adults (flights and digging underground tunnels with push-ups) occurs only after heavy rains, when the soil is damp and loose enough for the beetles to burrow easily. During the dry periods, when the soil is hard, and also in winter, the beetles are buried deeper in the ground and show no above-ground activity, making it very difficult to find them.B.unicornis from the diary of the excellent Czech coleopterist Rudolf \u010cepel\u00e1k , \u013dutov (P\u00e1lenice hill), and certainly from Tren\u010d\u00edn southwards everywhere on the south-eastern slopes.The area of Mal\u00e1 hora is sparsely covered with grass, which reaches 40\u201350 cm in places. If it is a quiet evening (no wind), preferably without moonlight, at 9 pm they start flying about 20\u201330 cm above the ground. In my right hand I have a net with a handle about 10 cm long, not white but dark, and in my left hand a torch. I bend down and suddenly hear \u2018zzzzz...\u2019. If I feel it\u2019s very close, I shine the torch and immediately catch it with the net. If I miss, it falls into the grass, where I would look for it vainly. It flies like a bee collecting pollen. If the grass is cut, they fly quite fast, likeGeotrupes. The flights continue until ca. 9.30 pm. Sometimes I hear it on the ground, stridulating like a longhorn beetle. And then we see: the female digging a hole and the male removing away the excavated soil. I dug a hole to see what they were looking for, but all I found were healthyOrnithogalumorGageabulbs.\u201d"}
{"text": "\u211br, \u211bc} > 1, the effects of parameters on the basic reproduction numbers, and the effect of parameter on the infectious groups. Finally, the stakeholders must focus on minimizing the transmission rates and increasing the recovery (removed) rate for both racism and corruption action which can be considered prevention and controlling strategies.Racism and corruption are mind infections which affect almost all public and governmental sectors. However, we cannot find enough published literatures on mathematical model analyses of racism and corruption coexistence. In this study, we have contemplated the dynamics of racism and corruption coexistence in communities, using deterministic compartmental model to analyze and suggest proper control strategies to stakeholders. We used qualitative and comprehensive mathematical methods and analyzed both the racism model in the absence of corruption and the corruption model in the absence of racism. We have computed basic reproduction numbers by applying the next generation matrix method. The developed model has a disease-free equilibrium point that is locally asymptotically stable whenever the reproduction number is less than one. Additionally, we have done sensitivity analysis to observe the effect of the parameters on the incidence and transmission of the mind infections that deduce the transmission rates of both the racism and corruption are highly sensitive. The numerical simulation we have simulated showed that the endemic equilibrium point of racism and corruption coexistence model is locally asymptotically stable when max{\u2009 Three decades ago, David Mason, a sociologist, who defined the term \u201cinstitutional racism\u201d would remain a political catchphrase devoid of analytical rigor . Marx anCorruption is an unpleasant act for communities in general; however, it does not trouble and upset everyone on an equal level. Although corruption is harmful to society as a whole, it frequently has a greater negative impact on existing marginalized groups and is primarily practiced in developing nations \u201317. The Today in our globe, ethnic and racial diversity is increasing rapidly , 33. ThoOne of the phenomena that can be presented by the mathematical model is the impact and expansion of racism-corruption coexistence among the communities. In this study, we have reviewed a literature done by other scholars to examine the spread and transmission of racism and corruption single existence as infectious diseases with a mathematical modelling approach as stated in , 39\u201345, N into seven distinct classes. These classes are those individuals who are susceptible to corruption or racism \u2009S\u2009(t), those who are corrupted \u2009C(t), those who stopped corruption R1(t), those who are racist \u2009R(t), those who stopped racism \u2009R2(t), and those who are both corrupted and racist C1(t) and those who stopped both corruption and racism at the same time\u2009R3(t).We have assumed that all the parameters used in this mode are nonnegative. The recruitment rate entering into the susceptible class is from birth and immigration. Moreover, we have considered that the susceptible individuals are equally likely to be corrupt and/or racialist and the corrupt and/or racialist individual compels susceptible individuals into corruption and/or racism practice(s) as they effectively interact. Upon being recovered, individuals become either susceptible or honest from the act of corruption and/or racism. Using the above basic assumptions and descriptions, we have divided the total population \u03c91 \u2265 1\u2009 is the modification parameter that increases infectivity and \u2009\u03b2 is the corruption transmission rate. Moreover, we have used the racism mass action incidence rate given by\u03c92 \u2265 1 is the modification parameters that increase infectivity and \u2009\u03b1 is the racism transmission rate.The susceptible individuals become corrupted with standard incidence rate given byUsing the model assumptions and descriptions stated above, the flow chart of the racism and corruption dynamics is given byUsing Before we analyze the racism-corruption coexistence model , we needC = C1 = , R1 = R3 = 0, so that we do have the dynamical systemWe have derived the mathematical model of racism in the absence of corruption from the full racism and corruption coexistence model by making \u2009S(t), \u2009R(t), and \u2009R2(t) of the racism dynamical system (t > 0.The solutions \u2009l system  are nonn\u03c4 = sup{t > 0 : S\u2009(t) > 0, R(t) > 0\u2009and\u2009R2(t) > 0}.Let us define \u2009S(t), \u2009R(t), and R2(t) are continuous so that we can say \u03c4 > 0. If \u03c4 = +\u221e, then positivity holds. Nevertheless, if \u20090 < \u03c4 < +\u221e, then\u2009S(t) = 0 or (t) = 0\u2009R2(t) = 0.Since all, dS/dt = \u039b + \u03b82R2 \u2212 (\u03bbR + \u03bc)S. Then, after applying the integrating factor method with some mathematical calculations, we have obtained S(\u03c4) = gS(0) + g\u222b0\u03c4exp\u03bbR + \u03bc)dt\u222b((\u03c0\u03bb + \u03b82R2)dt > 0, where \u2009g = exp\u03bc\u03c4 + \u222b0\u03c4(\u03bbR + \u03bc))\u2212 > 0, R2(t) > 0. Moreover using the definition of \u2009\u03c4, the solution \u2009S(\u03c4) > 0 so that S(\u03c4) \u2260 0. Using the same procedure all, the solutions of the dynamical system are nonnegative.From the first equation of the racism model, we do have R = 0 which gives following result. Er0 =  = .Racism-free equilibrium point of the racism model in the absences of corruption is obtained by making the right-hand side of equation is equal to zero providing that the racist class is equal to zero as The reproduction number is the average number of people that become racist because of the entry of one racial person into a completely susceptible population in the absence of intervention. Moreover, reproduction number utilizes to determining the effect of the control measures and to understand the capability of the corruption to disseminate in the entire community when the control strategies are applied .\u211br which is manipulated by Van den Driesch, Pauline, and James Warmouth next-generation matrix approach [FV\u22121 = [\u2202\u2131i(\u2009ErO)/\u2202xj][\u2202\u03bdi(\u2009ErO)/\u2202xj]\u22121, where \u2009\u2131i\u2009 is the rate of appearance of new infection in compartment \u2009i\u2009, \u03bdi\u2009 is the transfer of infections from one compartment i\u2009 to another, and Er0 is the disease-free equilibrium point \u2009ErO =  = The reproduction number of racism in the absence of corruption model denoted by \u2009approach  is the l\u2131i(x) and the transition matrix \ud835\udcb1i(x) are given byThe general transmission matrix \u2009Then, we have obtainedFV\u22121 are {0, \u03b1\u039b/((\u03b32 + \u03bc)\u03bc)}.Thus the eigenvalues of The reproduction number of racism in the absence of corruption model is given byEr0 =  =  of the system (4) is locally asymptotically stable if the reproduction number \u2009\u211br < 1, and it is unstable if\u2009\u211br > 1.The racism-free equilibrium point J(\u2009Er0) of the model \u2009(4) given byThe Jacobean matrix at racism-free equilibrium point is \u03bb1 = \u2212\u03bc, \u03bb2 = \u03b1(\u039b/\u03bc) \u2212 (\u03b32 + \u03bc), and \u03bb3 = \u2212(\u03b82 + \u03bc). But \u03bb2 = \u03b1(\u039b/\u03bc) \u2212 (\u03b32 + \u03bc) can have the form \u03bb2 = (\u03b1\u039b/\u03bc)((\u03b1\u039b/(\u03bc(\u03b32 + \u03bc))) \u2212 1) = (\u03b1\u039b/\u03bc)(\u211br \u2212 1). Moreover, \u03bb2 = (\u03b1\u039b/\u03bc)(\u211br \u2212 1) < 0 if and only if \u211br < 1; hence, all the eigenvalues are negative which implies the racism-free equilibrium point is locally asymptotically stable if and only if \u211br < 1; otherwise, it is unstable.By using Software Wolfram Mathematica, we have the eigenvalues \u2009\u211br < 1.The racism-free equilibrium is globally asymptotically stable if\u2009To prove the global asymptotic stability (GAS) of the racism-free equilibrium point, we have used the method of Lyapunov functions.l1 such that; l1 = aR, where\u2009a = \u03bc/\u03b1\u039b. We defined a Lyapunov function \u03bbR = \u03b1R\u2009 and N = S + R + R1 = \u03bb/\u03bc. But we do have dl1/dt = 0 if R = 0\u2009 or R = 1. From this fact, we do have  is the only singleton set in { \u2208 \u03a91 : dl1/dt = 0}. Therefore, by the principle of LaSalle (1976), the racism-free equilibrium point is globally asymptotically stable if\u2009\u211br < 1.Moreover, R \u2260 0. Suppose the endemic equilibrium point of the model is denoted by\u2009Er\u2217 = .It is mandatory to be sure about number of endemic equilibrium of the model before investigating the global asymptotic stability of the disease-free equilibrium point (DFE). The endemic equilibrium point of the dynamical system of  is solve\u03bbR(t) = \u03b1(R(t)), and we have derived the following:The corresponding force of infection is \u03bbR = 0 or (\u03bbR + \u03bc)(\u03b82 + \u03bc)(\u03b32 + \u03bc) \u2212 \u03b82\u03b32\u03bbR = \u03b1\u039b(\u03b82 + \u03bc) after simplification and rearrangement of the terms; we have \u03bbR = 0 or \u03bbR = (\u03b32 + \u03bc)(\u03b82 + \u03bc)/(\u03b32 + \u03b82 + \u03bc)((\u03b1\u039b/(\u03bc(\u03b32 + \u03bc))) \u2212 1). \u21d2Er\u2217 =  exist when \u2009\u211br > 1 whereTherefore, there is unique endemic equilibrium point for the racism model in the absence of corruption given by \u211br > 1.The racism model in the absence of corruption has a unique endemic equilibrium point whenever \u2009Er\u2217 = \u2009 is locally asymptotically stable if the \u2009\u211br > 1, otherwise unstable. To deduce the local stability of the endemic equilibrium point, we use the method of Jacobian matrix and Routh Hurwitz stability criteria. The corresponding Jacobian matrix of the dynamical system at the endemic equilibrium point Er\u2217 = \u2009isThe endemic equilibrium point \u2009A = \u2013(((\u03b1\u039b\u03bbR(\u03b82 + \u03bc))/((\u03bbR + \u03bc)(\u03b82 + \u03bc)(\u03b32 + \u03bc) \u2212 \u03b82\u03b32\u03bbR)) + \u03bc), B = \u2212((\u03b1\u039b(\u03b82 + \u03bc)(\u03b32 + \u03bc))/((\u03bbR + \u03bc)(\u03b82 + \u03bc)(\u03b32 + \u03bc) \u2212 \u03b82\u03b32\u03bbR)), C = (\u03b1\u039b\u03bbR(\u03b82 + \u03bc))/((\u03bbR + \u03bc)(\u03b82 + \u03bc)(\u03b32 + \u03bc) \u2212 \u03b82\u03b32\u03bbR), D = ((\u03b1\u039b(\u03b82 + \u03bc)(\u03b32 + \u03bc))/((\u03bbR + \u03bc)(\u03b82 + \u03bc)(\u03b32 + \u03bc) \u2212 \u03b82\u03b32\u03bbR)) \u2212 (\u03b32 + \u03bc), E = \u03b32, and F = \u2212(\u03b82 + \u03bc). a0 = 1,\u2009a1 = \u2212(A + D + F), a2 = (AF + DF \u2212 BC + AD), and a3 = (ADF + EC\u03b82 \u2212 BCF).Then, the characteristic equation of the above Jacobian matrix is given bya0 = 1 > 0 and a1 = (\u2212\u03b82\u03b32\u03bbR/((\u03bbR + \u03bc)(\u03b82 + \u03bc)(\u03b32 + \u03bc) \u2212 \u03b82\u03b32\u03bbR))(1 \u2212 (\u03b1\u039b/((\u03b32 + \u03bc)\u03bc))). But \u2009\u211br > 1. Now, we can determine the local stability of endemic equilibrium point by applying the Routh-Hurwitz criteria on\u2009a0\u03bb3 + a1\u03bb2 + a2\u03bb + a3 = 0. Following the same algebraic manipulation, all the coefficients of the characteristic's polynomial are positives whenever \u2009b1 > 0 if \u2009\u211br > 1.\u21d2In the same procedure,c1 > 0 if\u2009\u2009\u211br > 1.\u21d2\u211br > 1.We have observed that the first column of the Routh-Hurwitz array has no sign change; thus, the endemic equilibrium point of the dynamical system is locally asymptotically stable for\u2009R = R2 = C1 = 0 so that we do have the dynamical system. The mathematical model of corruption in the absence of racism is obtained from the full racism and corruption coexistence model  by makinS(t), \u2009C(t), and \u2009R1(t) of the dynamical system of corruption model (t > 0.The solutions \u2009on model  are nonn\u03c4 = sup{t > 0 : S\u2009(t) > 0, C(t) > 0, and\u2009R1(t) > 0}.Let us define\u2009S(t), \u2009C(t), and R1(t) are continuous so that we can say \u03c4 > 0. If \u03c4 = +\u221e, then positivity holds. Nevertheless, if \u20090 < \u03c4 < +\u221e, then \u2009S(t) = 0 or (t) = 0 and\u2009R1(t) = 0.Since all, dS/dt = \u039b + \u03b81R1 \u2212 (\u03bbC + \u03bc)S.From the first equation of the racism model, we do have S(\u03c4) = fS(0) + f\u222b0\u03c4exp\u03bbc + \u03bc)dt\u222b((\u039b + \u03b81R1)dt > 0, where \u2009f = exp\u03bc\u03c4 + \u222b0\u03c4(\u03bbc + \u03bc))\u2212 > 0, R1(t) > 0. Moreover, using the definition of \u2009\u03c4, the solution \u2009S(\u03c4) > 0 so that S(\u03c4) \u2260 0. Using the same procedure, all the solutions of the dynamical system are nonnegative.Then, applying the integrating factor method with some mathematical calculations, we have obtained C = 0 which gives result\u2009\u2009EcO =  = .Corruption-free equilibrium point of the corruption model in the absences of racism is obtained by making the right-hand side of equation equal to zero providing that the corrupted class is equal to zero as \u211bc which is manipulated by next-generation matrix approach [FV\u22121 = [(\u2202\u2131i(\u2009EcO))/\u2202xj][(\u2202\u03bdi(\u2009EcO))/\u2202xj]\u22121, where \u2009\u2131i\u2009 is the rate of appearance of new infection in compartment \u2009i\u2009, \u03bdi\u2009 is the transfer of infections from one compartment i\u2009 to another where 1 \u2264 i, j \u2264 m, m is the number of infected compartments, and Ec0 is the disease-free equilibrium point EO =  = .The reproduction number of corruption in the absence of racism model denoted by \u2009approach  is the l\u2131i(x) and the transmission matrix \u2009\ud835\udcb1i(x) are given byThe general transition matrix \u2009Then, we have obtainedFV\u22121\u2009 are {0, \u03b2/(\u03b31 + \u03bc)}.Thus, the eigenvalues of \u211bC = \u03b2/(\u03b31 + \u03bc).Therefore, the reproduction number of the corruption model in the absence of racism is given by \u2009ECO =  =  of the system  of the model , and \u03bb3 = (\u03b81 + \u03bc).Using Software Wolfram Mathematica, we have obtained the eigenvalues \u2009\u03bb2 = \u03b2 \u2212 (\u03b31 + \u03bc) can have the form \u2009\u03bb2 = (\u03b32 + \u03bc)((\u03b2/(\u03b32 + \u03bc)) \u2212 1) = (\u03b32 + \u03bc)(\u211bc \u2212 1). Moreover, \u03bb2 = (\u03b32 + \u03bc)(\u211bc \u2212 1) < 0 if and only if \u211bc < 1; hence, all the eigenvalues are negative which implies the disease-free equilibrium point is locally asymptotically stable if and only if \u211bc < 1; otherwise, it is unstable.But \u211bc < 1.The corruption-free equilibrium is globally asymptotically stable if\u2009To prove the global asymptotic stability (GAS) of the corruption-free equilibrium point, we have used the method of Lyapunov functions.l3 such that l3 = bC\u2009 where\u2009b = \u03bc/\u039b, \u21d2dl3/dt = (\u03bbC\u03bcS/\u039b) \u2212 C.We defined a Lyapunov function \u03bbC = \u03b2C/N and N = S + C + R2 = \u03bb/\u03bc\u2009, \u21d2dl2/dt \u2264 [\u2009\u211br \u2212 1]C, and thus (dl2/dt) < 0\u2009 if \u2009\u211bc < 1.But we do have dl2/dt = 0 if C = 0\u2009 or \u2009\u211bc = 1. From this fact, we do have  is the only singleton set in { \u2208 \u03a9 : dl2/dt = 0}. Therefore, by the principle of LaSalle (1976), racism-free equilibrium point is globally asymptotically stable if\u2009\u211bC < 1.Moreover, C \u2260 0. Suppose the endemic equilibrium point of the model is denoted by EC\u2217 = . The corresponding force of infection is \u03bbC(t) = (\u03b2/N)(C(t)), and we have derived the following. It is crucial to be sure about the number of endemic equilibrium of the model before investigating the global asymptotic stability of the DFE. The endemic equilibrium point of the dynamical system of  is solveAfter some algebraic simplification and rearrangement of the terms, we haveEc\u2217 = , whereTherefore, there is unique endemic equilibrium point for corruption model in the absence of racism given by Ec\u2217 = \u2009 is locally asymptotically stable if the \u2009\u211bC > 1, otherwise unstable.The endemic equilibrium point \u2009See the Appendix.\u03a9 = { \u2208 \u211d+7, N \u2264 (\u039b/\u03bc)}.The mathematical modeling is the representation of real world phenomena that can be demonstrated by dealing with different quantitative and qualitative attributes. In this newly extended model, we have represented human populations, which cannot be negative. Therefore, we need to show that all the state variables in our model are always nonnegative as well as the solutions of the dynamical system remains positive with positive initial conditions in the bounded region given by S(t), \u2009C(t), R(t), C1(t), R1(t), R2(t),\u2009and R3(t) of the racism and corruption coexistence model (t > 0.The solutions \u2009ce model  are nonn\u03c4 = sup{t > 0 : S\u2009(t) > 0, C(t) > 0, R(t) > 0, C1(t) > 0, R1(t) > 0, R2(t) > 0, and\u2009R3(t) > 0}.By defining,\u2009S(t), \u2009C(t), R(t), C1(t), R1(t), R2(t),\u2009and R3(t) are continuous so that we can deduce that \u2009\u03c4 > 0. If \u03c4 = +\u221e, then positivity holds. However, if \u20090 < \u03c4 < +\u221e, then \u2009S(t) = 0 or \u2009C(t) = 0 or R(t) = 0 or C1(t) = 0\u2009 or \u2009R1(t) = 0 or \u2009R2(t) = 0 or\u2009R3(t) = 0.All dS/dt = \u039b + \u03b82R2 + \u03b83R3 + \u03b81R1 \u2212 (\u03bbR + \u03bbC + \u03bc)S, and applying the integrating factor method and after some calculations, we have obtained S(\u03c4) = hS(0) + h\u222b0\u03c4exp\u03bbR + \u03bbC + \u03bc))dt\u222b((\u039b + \u03b82R2 + \u03b83R3 + \u03b81R1)dt > 0, where \u2009h = exp\u03bc\u03c4 + \u222b0\u03c4(\u03bbR + \u03bbC + \u03bc))\u2212 > 0, \u2009R2(t) > 0, \u2009R3(t) > 0\u2009 and \u2009R1 > 0. Finally, using the definition of \u2009\u03c4, the solution \u2009S(\u03c4) > 0 so that S(\u03c4) \u2260 0.From the first equation of the racism and corruption coexistence model, we do have Following the same procedure all the solutions of the dynamical system are nonnegative.+7.The region dN/dt) \u2264 \u039b \u2212 \u03bcN. By incorporating standard comparison theorem, we have obtained \u222b(dN/(\u039b \u2212 \u03bcN)) \u2264 \u222bdt and integrating both sides gives \u22121/\u03bcln(\u039b \u2212 \u03bcN) \u2264 t + c, where c is some constant, and after some mathematical calculations and simplifications, we have obtained 0 \u2264 N\u2009(t) \u2264 (\u039b/\u03bc). This result implies all the possible solutions of the given dynamical system with positive initial conditions are bounded.Since all the state variables are nonnegative in the absence of infections, we have obtained \u2009 = .The racism and corruption-free equilibrium point of the model is obtained by making the right-hand side of the system  is equal\u211bcr the Van den Driesch, Pauline, and James Warmouth next-generation matrix approach [FV\u22121 = [\u2202\u2131i(EO)/\u2202xj][\u2202\u03bdi(EO)/\u2202xj]\u22121, where \u2009\u2131i\u2009 is the rate of appearance of new infection in compartment \u2009i\u2009, \u03bdi\u2009 is the transfer of infections from one compartment i\u2009 to another where 1 \u2264 i, j \u2264 m, m is the number of infected compartments, and Ech0 is the disease-free equilibrium point EO =  = .The reproduction number of racism and corruption coexistence model denoted by \u2009approach  is the l\u2131i(x) and the transition matrix \u2009\ud835\udcb1i(x) are given byThe general transmission matrix \u2009Then, after some calculations, we have obtainedA = (\u03b31 + \u03bc), B = \u2212\u03c31, C = (\u03b32 + \u03bc), D = \u2212\u03c32, and E = (\u03c32 + \u03b33 + \u03c31 + \u03bc).where \u2009FV\u22121\u2009 are {0, \u03b1\u039b/(\u03b32 + \u03bc)\u03bc, \u03b2/(\u03b31 + \u03bc)}. Thus, the reproduction number of racism and corruption coexistence denoted by \u2009\u211brc and is\u2009max\u2009{\u039b\u03b1/(\u03b32 + \u03bc)\u03bc, \u03b2/(\u03b31 + \u03bc)}.The eigenvalues of \u211brc = max{\u2009\u211br, \u211bc} = max{\u039b\u03b1/(\u03b32 + \u03bc)\u03bc, \u03b2/(\u03b31 + \u03bc)}.That means \u2009EO =  =  of the model is locally asymptotically stable if the reproduction number \u2009\u211brc < 1, and it is unstable if\u2009\u211brc > 1.The racism and corruption free equilibrium point J(X\u2009) of the model (The Jacobean matrix \u2009he model  is givenJ(EO) as \u03bb1 = \u2212\u03bc, \u03bb2 = \u03b2\u2013(\u03b31 + \u03bc), \u03bb3 = \u03b1(\u03c0\u03bb/\u03bc) \u2212 (\u03b32 + \u03bc), \u03bb4 = \u2212(\u03c32 + \u03b33 + \u03c31 + \u03bc), \u03bb5 = \u2212(\u03b81 + \u03bc), \u2009\u03bb6 = \u2212(\u03b82 + \u03bc),\u2009\u03bb7 = \u2212(\u03b83 + \u03bc).Using Wolfram Mathematica, we have obtained the eigenvalues of \u03bb2 and \u03bb3\u2009 can be rewritten as follows: \u03bb2 = \u03b2\u2013(\u03b31 + \u03bc) = (\u03b31 + \u03bc)((\u03b2/(\u03b31 + \u03bc)) \u2212 1) and \u03bb3 = \u03b1(\u039b/\u03bc) \u2212 (\u03b32 + \u03bc) = (\u03b32 + \u03bc)((\u039b\u03b1/(\u03b32 + \u03bc)\u03bc) \u2212 1).But \u211brc < 1. Therefore, the racism and corruption coexistence free equilibrium point is locally asymptotically stable if and only if \u211brc < 1; otherwise, it is unstable.Hence, all the eigenvalues are negative if \u2009E\u2217 =  which occurs when the mind infection persist in the community, and we computed by making the right hand side of the model as zero and obtained asThe racism-corruption coexistence endemic equilibrium point of the full model  is denot\u211bc < 1\u2009 and \u211br < 1, respectively, implying that there is no endemic equilibrium point if \u211brc < 1 for the coexistence model , if \u2009\u211br > 1 is the corruption-free (racism persistence) equilibrium point. The analysis of the equilibrium E1\u2217 is similar to the endemic equilibrium Er\u2217 in the model , if \u211bc > 1 is the racism-free (corruption persistence) equilibrium point. The analysis of the equilibrium E2\u2217 is similar to the endemic equilibrium Ec\u2217 in equations  is the racism-corruption coexistence persistence equilibrium pointThe summary of the racism-corruption mind infection persistence equilibrium points: The explicit computation of the mind infection persistence equilibrium point of the coinfection model  in termsS = x1, C = x2, R = x3, C1 = x4, and R1 = x5, R2 = x6, and R3 = x7 such that \u2009N = x1 + x2 + x3 + x4 + x5 + x6 + x7. Furthermore, by using vector notation \u2009X = T, the Racism-Corruption coexistence model (dX/dt = F(X) with F = T, as follows:In this section, we apply the center manifold theory given by \u03bbR = \u03b1(x3 + \u03c92x4) and \u03bbC = (\u03b2/N)(x2 + \u03c91x4).with ECR0, denoted by J(ECR0), and this gives usThen, the method entails evaluating the Jacobian of the system  at the D\u211bCR = 1, and suppose that \u03b2 = \u03b2\u2217\u2009is chosen as a bifurcation parameter.Consider \u211bCR = 1 as \u2009\u211bCR = \u03b2/(\u03b31 + \u03bc) = 1 and solving for \u03b2, we have obtained\u2009\u03b2 = \u03b2\u2217 = \u03b31 + \u03bc. From J\u03b2\u2217 as \u03bb1 = \u2212\u03bc, \u03bb2 = 0 or \u03bb3 = \u03b1x10 \u2212 (\u03b32 + \u03bc) = \u03b1\u039b/\u03bc \u2212 (\u03b32 + \u03bc) = (\u03b32 + \u03bc)[(\u03b1\u039b/\u03bc(\u03b32 + \u03bc)) \u2212 1] = (\u03b32 + \u03bc)[\u2009\u211bR \u2212 1] < 0\u2009 if \u2009\u211bR < 1\u2009 or \u03bb4 = \u2212(\u03c32 + \u03b33 + \u03c31 + \u03bc) or \u03bb5 = \u2212(\u03b81 + \u03bc) or \u03bb6 = \u2212(\u03b82 + \u03bc), and \u03bb7 = \u2212(\u03b83 + \u03bc).After some steps of the calculation, we have obtained the eigenvalues of J(ECR0) of equation  has right eigenvectors associated with the zero eigenvalue given by u = T asM1 = \u2212\u03b2\u2217\u03c91 \u2212 \u03b1\u03c92x10, M2 = \u03b1x10 \u2212 (\u03b32 + \u03bc), and M3 = \u2212(\u03c32 + \u03b33 + \u03c31 + \u03bc).In eigenvectors of \u2009e system  at \u03b2 = \u03b2Then solving equation  the righ\u03b2 = \u03b2\u2217 given by v = T asSimilarly, the left eigenvector associated with the zero eigenvalues at Then, solving equation  the lefta and b are obtained asAfter some steps of calculations, the bifurcation coefficients a is negative.Thus, the coefficient \u2009Moreover,u2 > 0 and v2 > 0 then a < 0 and b > 0, the Racism-Corruption coexistence model  = (\u2202\u211brc/\u2202\u03b5)\u2217(\u03b5/\u211brc)\u2009 as stated in literatures [The normalized forward sensitivity index of a variable racism and corruption coexistence model  basic reeratures \u201350.\u211br\u2009 and \u211bc\u2009 since\u2009\u211brc = max{\u211br, \u211bc}.The sensitivity indices enable us to examine the relative importance of different parameters in racism and corruption incidence and prevalence. The most sensitive parameter has the magnitude of the sensitivity index larger than that of all other parameters. We can calculate the sensitivity index in terms of Using the parameter values given in \u211cr manipulated with parameters values from \u211br = 3.51\u2009 at the racism transmission rate \u2009\u03b1 = 1.37, which imply that racism spreads throughout the community. Moreover, sensitivity analysis given in \u039b and racism transmission rate \u03b1 are highly affecting the racism reproduction number\u2009\u211cr.\u211bc = 6.03 at the corruption expansion rate \u2009\u03b2 = 1.51, which imply that corruption spreads throughout the community. Moreover, sensitivity analysis given in \u039b and corruption transmission rate \u2009\u03b2 are highly affecting the racism reproduction number\u2009\u211bc.Here, with the given parameter values in In this subsection, sensitivity analysis of the racism and corruption coexistence transmission dynamics is performed to identify the most influential parameters for the spread as well as for the control of coexistence mind infection transmission in the community. The results of sensitivity analysis based on the set of parameters values given in In this section, numerical simulation has been performed with MATAB ode45 code to analyze the effect of some parameters that causes for conducting this illegal activity. Most specifically, we investigated the stability of the endemic equilibrium point of the main coexistence model , paramet\u211brc = max{\u211br, \u211bc} = max{3.51, 6.03} = 6.03 > 1. The numerical simulation justified that the physical phenomenon that can be stated as the expansion and spreading of racism-corruption coexistence activity is consistently occurred throughout the considered community in the study.\u03b32 on the racism reproduction number \u2009\u211br. The plot demonstrates that when the value of \u2009\u03b32 increases, the racism reproduction number is going down, and whenever the value of \u03b32 > 0.059 implies that \u2009\u211br < 1. In other words, it mean as the power of treatment rate increases the number of racial individual will decrease. Therefore, the stakeholder shall concentrate on maximizing the value of racism recovery rate \u03b32\u2009 by applying possible interventions strategies to prevent and control the problem of racism.\u03b31 and corruption reproduction \u2009\u211bc. The plot shows that that when the value of corruption recovery rate \u2009\u03b31 increases, the corruption reproduction number decreases, and whenever the value of \u03b31 > 0.508 implies that \u2009\u211bc < 1. This means that as the power treatment rate increases, the number of corrupted individual decreases. Moreover, the result tells us the stakeholder shall concentrate on maximizing the values of \u03b31\u2009 to prevent and control the expansion of racism in the community under consideration.\u03b2 on the corruption reproduction number \u2009\u211bc. The displayed simulation states that when the value of \u2009\u03b2\u2009 increases, the corruption reproduction number is going up and the value of \u03b2 < 0.17\u2009 implies that \u211bC < 1. Therefore, the stakeholders are expected to minimizing the values of corruption transmission rate \u03b2 to control corruption expansion in the community.\u211br. The plot demonstrates that whenever the value of \u2009\u03b1\u2009 increases, the racism reproduction number increases, and the value of \u03b1 < 0.322\u2009 implies that \u211br < 1. Therefore, the stakeholders are expected to minimizing the values of racism transmission rate \u2009\u03b1\u2009 to control the racism expansion in the community.In \u03b33\u2009 increases from 0.54 to 0.73 the number of racism-corruption co-occurrence in the community decreases. The figure deduces that when the values of \u03b33 increases, the number of individual conduction both racism and corruption among population is going down. This means as the power of treatment increases the racial and corrupted coexistence class in the model become decreases. Therefore, the stakeholders shall expect to maximize the values of parameter \u2009\u03b33 to control the expansion of the racism-corruption coexistence in the community.In In \u211br = 3.51 at \u03b1 = 1.37 and \u211bc = 6.03 at \u2009\u03b2 = 1.51, i.e., \u2009\u211bcr = max{\u2009\u211br, \u211bc} = max{3.51,6.03} = 6.03. Using numerical simulation, we have verified the qualitative result that the endemic equilibrium point of the racism and corruption coexistence model is locally asymptotically stable when \u211bcr = max{\u2009\u211br, \u211bc} = max{3.51,6.03} = 6.03 > 1. Also numerical simulation results showed that whenever the racism transmission rate increases the racism mind infection transmission increases, the corruption transmission rate increases, the corruption mind infection transmission increases, the racism recovery rate increases, the racism mind infection transmission decreases, the corruption recovery rate increases, and the corruption mind infection transmission decreases.Nowadays, racism and corruption coexistence is a major problem affecting nations throughout the world, but literatures on prevention and controlling its expansion through the community were rare. In this work, we have developed a first new nonlinear compartmental deterministic mathematical model on the transmission dynamics of racism and corruption coexistence expansion. The developed model has disease-free equilibrium points that are both locally asymptotically and globally asymptotically stable whenever their corresponding basic reproduction number is less than one. All the model mind infection endemic equilibrium points were both locally asymptotically and globally-asymptotically stable whenever their corresponding reproduction number is less than unity. The model did not have the phenomenon of backward bifurcation. The sensitivity analysis of the model showed us the racism transmission and corruption transmission rates are the most sensitive parameters which have a direct effect on the racism and corruption coexistence mind infection transmission in the community. Also the racism recovery rate and corruption recovery rate have high indirect impact on the basic reproduction numbers of the racism model and corruption model, respectively. Using the parameter values given in \u211br and \u2009\u211bc, we shall give future directions for the stakeholders in the community. The results we have obtained have a crucial role for stakeholders, as it governs the eradication and/or persistence of racism, corruption, and racism-corruption coexistence which are illegal activities in a community. Stakeholders shall concentrate on decreasing the racism transmission rate, the corruption transmission rate, and increasing or maximizing the values of racism and corruption recovery rates that are used to minimize and possibly to eradicate the problem from the community.Moreover, based on the impact of some changes of parameters on the corresponding reproduction number \u2009Finally, we recommend the governments of nations to introduce, apply and ensure anticorruption and antidiscriminatory laws, and take the bold measures to beak the interconnection of corruption and racism. We want to remark the whole community stay unite to identify common problems and committed to research and advocacy from societies. The international institutions shall be collaborated for better understanding of these two interlinked problems and set up monitoring and investigation bodies. In the limitations of this study, the next potential researchers can incorporate them and extend this study: optimal control approach, stochastic approach, fractional order derivative approach, environmental impacts, age, and spatial structure, whenever possible validating the model by applying appropriate real data."}
{"text": "G. efibulatus belonging to sect. Androsacei, G. iodes and G. sinopolyphyllus belonging to sect. Impudicae and G. strigosipes belonging to sect. Levipedes are proposed as new to science. The detailed descriptions, colour photos of basidiomata and line-drawings of microscopic structures are provided. The comparisons with closely related species and a key to known species of Gymnopus s. str. reported with morpho-molecular evidence in China is also given.Nine collections of gymnopoid fungi were studied based on morpho-molecular characteristics. The macromorphology was made according to the photograph of fresh basidiomata and field notes, while the micromorphology was examined via an optical microscope. Simultaneously, the phylogenetic analyses were performed by maximum likelihood and Bayesian inference methods based on a combined dataset of nrITS1-nr5.8S-nrITS2-nrLSU sequences. Integrated analysis of these results was therefore,  Gymnopus (Pers.) Roussel is a group of white-spored macrofungi with collybioid, rarely tricholomatoid, marasmioid or omphalinoid habit and distributed worldwide  7\u20139 \u00d7 (3.5\u2013)4\u20134.5(5.5\u2013) \u00b5m 1.63\u20132.05(\u20132.27), Q = 1.85), ellipsoid to oblong, hyaline, thin-walled. Basidia [n = 20] 19.5\u201332 \u00d7 6\u20138 \u00b5m, clavate, 4-spored. Basidioles: [n = 20] 20.5\u201328 \u00d7 4.5\u20138 \u00b5m, clavate. Cheilocystidia [n = 22] 12\u201340.5 \u00d7 5\u201315 \u00b5m, narrowly clavate to broadly clavate, often with more or less finger-like apical projections, sometimes lobed or forming Siccus-type broom cells, thin-walled, hyaline. Pleurocystidia absent. Pileipellis: an entangled, repent cutis of cylindrical, thin-walled, sometimes coarsely incrusted, otherwise smooth hyphae, terminal cells, diverticulate, lobed to irregularly branched, almost coralloid, mixed with some subglobose cells, slightly brownish in KOH. Stipitipellis: a cutis composed of cylindrical hyphae, parallelly arranged, often smooth, sometimes with scattered diverticula, sometimes dextrinoid, otherwise inamyloid, slightly thick-walled, hyaline. Caulocystidia: absent. Clamp connections: absent.Additional specimen examined: China, Chongqing City, Gold Buddha Mountain National Scenic Area, 29\u00b01\u203243\u2033 N, 107\u00b011\u20321\u2033 E, elev. 2098 m, on dead broadleaves, 11 August 2020, J.P. Li, Z.Z. Cen, Q.Y. Lin, M. Wang, HGASMF01-11995.G. efibulatus is strongly reminiscent of G. pallipes J.P. Li and Chun Y. Deng and G. cremeostipitatus Anton\u00edn, Ryoo and Ka in the field. However, both G. pallipes from China and G. cremeostipitatus from South Korea, with smaller basidiospores  and the latter one with a pubescent stipe and scattered-to-frequent caulocystidia helps distinguish them from the new species  and the clamped structures  5.5\u20137.5(\u20138.5) \u00d7 3\u20134(\u20134.5) \u00b5m 1.73\u20132.30(\u20132.36), Q = 1.99), oblong, hyaline, thin-walled. Basidia [n = 20] 18.5\u201329 \u00d7 4.5\u20136 \u00b5m, clavate, 4-spored, hyaline, thin-walled. Basidioles [n = 20] 21\u201328 \u00d7 5\u20137.5 \u00b5m, hyaline, thin-walled. Cheilocystidia [n = 30] 13.5\u201335.5 \u00d7 3\u20136.5 \u00b5m, cylindrical or narrowly clavate, irregular, sometimes with forked, rostrate or one irregular filiform apical projection(s), thin-walled. Pleurocystidia absent. Pileipellis a cutis consisting of interwoven arranged, cylindrical, sometimes slightly incrusted, otherwise smooth hyphae, terminal cells cylindrical to subclavate, irregular to lobate, sometimes with lateral diverticula or projections, turn green in KOH. Stipitipellis a cutis composed of cylindrical hyphae, parallelly arranged, hyaline, thin-walled. Caulocystidia [n = 20] 25.5\u201375 \u00d7 5\u20139 \u00b5m, cylindrical, sometimes with scattered irregular lobate, obtuse apical projections. Clamp connections present.Additional specimens examined: China, Guizhou Province, Qiandongnan Miao and Dong Autonomous Prefecture, Liping Country, Deshun village, 26\u00b013\u203236\u2033 N, 109\u00b019\u203244\u2033 E, elev. 842.4 m, on dead broadleaves, 28 August 2020, J.P. Li, D.F. Wei, M. Wang, HGASMF01-10069; Hunan Province, Xiangxi Tujia and Miao Autonomous Prefecture, Yongshun County, Xiaoxi Town, Xiaoxi Village, 28\u00b048\u203220\u2033 N, 110\u00b015\u203227\u2033 E, elev. 512 m, on dead broadleaves, 22 July 2021, L.N. Liu, HMJAU 60388.G. iodes resembles G. iocephalus (Berk. and M.A. Curtis) Halling, G. similis Anton\u00edn, R. Ryoo and K.H. Ka and G. variicolor Anton\u00edn, Ryoo, Ka and Tom\u0161ovsk\u00fd by having a striate pileus, an unpleasant odour and similar-sized basidiospores, which agree to their phylogenetic relationship. However, G. iocephalus from the USA, differs by the pileipellis hyphae turning blue in KOH and the hymenium lacking cheilocystidia  (4.5\u2013)5\u20137 \u00d7 (2.5\u2013)3\u20134 \u00b5m 1.58\u20132.13(\u20132.3), Q = 1.85), ellipsoid to oblong, hyaline, thin-walled. Basidia: [n = 30] 20\u201329.5 \u00d7 5\u20137.5 \u00b5m, clavate, 4-spored. Basidioles: [n = 20] 18.5\u201328 \u00d7 5.5\u20137 \u00b5m, clavate. Cheilocystidia: [n = 30] 13\u201343 \u00d7 2.5\u20138.5 \u00b5m, narrowly to irregular clavate, smooth, or with one or more projections or irregular and branched outgrowth at the apex, hyaline, thin-walled. Pleurocystidia: absent. Pileipellis: a cutis consisting of interwoven, cylindrical hyphae, branched, smooth, often with oily contents, sometimes with scattered diverticula, terminal cells cylindrical, sometimes irregularly branched, coralloid at the apex. Stipitipellis: a cutis consisting of interwoven, cylindrical hyphae, smooth or with scattered diverticula, thin-walled. Caulocystidia: [n = 21] 24\u201362.5 \u00d7 3\u20135.5 \u00b5m, cylindrical, often tapering towards the apex. Clamp connections: present.Additional specimens examined: China, Jilin Province, Baishan City, Linjiang City, near Tieshigou Ravine, 41\u00b056\u203249\u2033 N, 126\u00b044\u203240\u2033 E, elev. 966 m, on dead broadleaves, 20 July 2021, J.P. Li, N.G. Pan, X. Wang, HMJAU 60387.G. atlanticus V. Coimbra and Wartchow, G. densilamellatus Anton\u00edn, Ryoo and Ka, G. hariolorum (Bull.) Anton\u00edn, Halling and Noordel., G. polyphyllus (Peck) Halling, and G. virescens A.W. Wilson, Desjardin and E. Horak are similar to the new species. However, G. atlanticus from Brazil, differs by having a sulcate pileus margin, less lamellulae tiers (l = 3) and smaller basidiospores (7.5 \u00d7 3.6 \u00b5m)  (5.5\u2013)6\u20137(\u20137.5) \u00d7 3\u20133.5(\u20134) \u00b5m 1.72\u20132.16(\u20132.2), Q = 1.94), oblong, hyaline, thin-walled. Basidia [n = 20] 18\u201331 \u00d7 4.5\u20136.5 \u00b5m, clavate, 4-spored. Basidioles: [n = 20] 20\u201332 \u00d7 4.5\u20136.5 \u00b5m, clavate. Cheilocystidia: [n = 50] 13\u201338.5 \u00d7 3.5\u201313 \u00b5m, clavate to narrowly clavate, subfusoid, irregular, lobed, sometimes with filiform apical projection, hyaline, thin-walled. Pleurocystidia: absent. Pileipellis: a cutis consisting of interwoven, cylindrical hyphae, smooth, terminal cells lobed, irregular branched, coralloid, forming a Dryophila-structure. Stipitipellis: a cutis composed of cylindrical hyphae, parallelly arranged, hyaline, slightly thick- to thick-walled. Caulocystidia: [n = 20] 14.5\u201341 \u00d7 3\u20136 \u00b5m, cylindrical, hyaline, thin-walled. Clamp connections present.Additional specimen examined: China, Guizhou Province, Tongren City, Yanhe County, Huangtu town, Huaxi village, Shengjiling Ridge, 28\u00b042\u203245\u2033 N, 108\u00b016\u203227\u2033 E, elev. 765 m, on dead broadleaves, 27 October 2020, A. Xu, HMAS 295797.G. sect. Levipedes (Qu\u00e9l.) Halling with brownish-coloured pileus and similar lamellae spacing, G. agricola Murrill, G. hybridus (K\u00fchner and Romagn.) Anton\u00edn and Noordel., G. loiseleurietorum  Anton\u00edn and Noordel., G. sepiiconicus (Corner) A.W. Wilson, Desjardin and E. Horak, G. spongiosus (Berk. and M.A. Curtis) Halling and G. vitellinipes A.W. Wilson, Desjardin and E. Horak are close to the new species. However, G. agricola, from the USA, can be distinguished by its estriate pileus margin, a cartilaginous and non-strigose stipe [G. hybridus, from France, and G. sepiiconicus, from South Solomons, are characterized by the non-strigose stipe and the absence of caulocystidia [G. vitellinipes, from Indonesia, has larger basidiospores (8.3\u20139.3 \u00d7 4\u20134.4 \u00b5m) and a poorly developed Dryophila-structure in the pileipellis [G. loiseleurietorum, from Austria, differs by the absence of true cheilocystidia and hyphae turn green in KOH [G. spongiosus, from the USA, has smaller basidiospores (8.4 \u00d7 3.6 \u00b5m) that often turn olive green in alkali [Notes: Amongst the known species within se stipe ; G. hybrcystidia ,14; G. veipellis ; G. loisn in KOH ; G. sponn alkali .G. spongiosus.Phylogenetic analyses agree with the morphological study, which showed the new species is close to Gymnopus s. str. with morpho-molecular evidence in ChinaKey to species within Dryophila-type structures \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 2.Terminal cells of pileipellis broad, mostly inflated, mixed with irregularly branched elements and some resembling \u2212Dryophila-type structures \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 3.Terminal cells of pileipellis coralloid, more or less diverticulate, lobed to irregularly branched, or with 2.G. omphalinoidesPileus generally deeply umbilicate; lamellae ventricose \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.... \u2212G. schizophyllusPileus more or less depressed; lamellae linear to arcuate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.... 3.Siccus-type broom cells, stipitipellis with dextrinoid hyphae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 4.Rhizomorphs present, cheilocystidia consist of \u2212Siccus-type broom cell, stipitipellis without dextrinoid hyphae\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5.Rhizomorphs absent, cheilocystidia never a 4.G. pallipesClamp connections present \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. \u2212G. efibulatusClamp connections absent \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.5.Basidiomata with unpleasant odour \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026... 6.\u2212Basidiomata with negligible odour \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 10.6.Lamellae not close or crowded \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 7.\u2212Lamellae close or crowded \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..... 9.7.G. alliifoetidissimusPileus general white overall \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026... \u2212Pileus not white \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 8.8.G. similisPileus light brown, orange white to greyish orange when old \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. \u2212G. iodesPileus almost reddish lilac overall when drying \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 9.G. densilamellatusPileipellis consist of incrusted hyphae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. \u2212G. sinopolyphyllusPileipellis without incrusted hyphae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026....... 10.G. strigosipesCaulocystidia present \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026... \u2212Caulocystidia not recorded \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..... 11.11.Stipe smooth or tomentose \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026... 12.\u2212Stipe with hairs \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.......... 15.12.Pileipellis made up of smooth hyphae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 13.\u2212Pileipellis with incrusted hyphae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.... 14.13.G. macrosporusBasidia sterigmata extremely long, up to 32 \u00b5m \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 \u2212G. tiliicolaBasidia sterigmata normally long \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026........... 14.G. longusBasidia sterigmata extremely long, up to 33 \u00b5m \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 \u2212G. globulosusBasidia sterigmata normally long \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 15.Pileus tomentose or pileipellis with incrusted hyphae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 16.\u2212Pileus without tomenta and pileipellis made up of smooth hyphae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 17.16.G. tomentosusPileus tomentose, pileipellis made up of smooth hyphae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 \u2212G. longisterigmaticusPileus without tomenta, pileipellis with incrusted hyphae \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 17.G. erythropusPileus estriate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.... \u2212Pileus striate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026....... 18.18.G. striatusStipe longitudinally striate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026....... \u2212G. changbaiensisStipe smooth \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. Gymnopus s. str. have been reported in China as yet, of which 15 taxa were reported based on morpho-molecular evidence [Gymnopus species and DNA barcodes. The newly proposed species, except G. efibulatus, a member of G. sect. Androsacei, embrace the current sectional concept well. Four odorous fungi, namely, G. iocephalus, G. iodes, G. similis and G. variicolor formed an independent clade implying their close affinities phylogenetically. When revisiting these four gymnopoid fungi in morphology, it is not hard to find that they share the striate pileus and the distant to subdistant lamellae [G. densilamellatus, G. polyphyllus and G. sinopolyphyllus, forming an independent clade, share the very close to crowded lamellae [G. sect. Androsacei is still an unsolved clade thus far, phylogenetically. Formally, Li et al. discussed the sectional circumscription and noted the broom cells were absent or weakly present in the pileipellis of several taxa [G. efibulatus also lack the bloom cells but subglobose cells were observed. Accordingly, this section is worthy of further exploration in the future based on morphology inferred from more materials and multilocus phylogenetic analyses.A total of 28 known species of evidence ,17,18,33lamellae ,12. Similamellae ,12. Besi"}
{"text": "Scientific Reports 10.1038/srep43191, published online 27 February 2017Correction to: This Article contains errors in Figure\u00a05 and Figure\u00a07.\u2212/\u2212\u2009+\u2009PDCD4\u201d group shows partial overlap with Figure 5e, the \u201cM, WT\u201d group. The corrected Figure\u00a0In Figure\u00a05i in the \u201cM, miR-21Additionally, in Figure\u00a07d, the \u201cWT, aged\u201d group overlaps with Figure\u00a06d, the \u201cWT, OVX\u201d group. The corrected Figure"}
{"text": "Andrena Fabricius, 1775, described by Ferdinand Morawitz from the collection of Aleksey Fedtschenko deposited in the Zoological Museum of the Moscow State University and in the Zoological Institute, Russian Academy of Sciences, St. Petersburg (Russia), are critically reviewed. Precise information with illustrations of types for 52 taxa is provided; of these 39 species are valid and thirteen are invalid (ten synonyms and three homonyms). Lectotypes are here designated for the following 24 nominal taxa: Andrenaacutilabris Morawitz, 1876, A.aulica Morawitz, 1876, A.bicarinata Morawitz, 1876, A.combusta Morawitz, 1876, A.comparata Morawitz, 1876, A.corallina Morawitz, 1876, A.discophora Morawitz, 1876, A.fedtschenkoi Morawitz, 1876, A.initialis Morawitz, 1876, A.laeviventris Morawitz, 1876, A.leucorhina Morawitz, 1876, A.mucorea Morawitz, 1876, A.nitidicollis Morawitz, 1876, A.oralis Morawitz, 1876, A.planirostris Morawitz, 1876, A.ravicollis Morawitz, 1876, A.rufilabris Morawitz, 1876, A.sarta Morawitz, 1876, A.smaragdina Morawitz, 1876, A.sogdiana Morawitz, 1876, A.subaenescens Morawitz, 1876, A.tuberculiventris Morawitz, 1876, A.turkestanica Morawitz, 1876, and A.virescens Morawitz, 1876.The type specimens of the genus  Zoological Museum of the Moscow State University, Moscow (ZMMU) and in the Zoological Institute of the Russian Academy of Sciences, St. Petersburg (ZISP). In the first part, the family Halictidae was considered , the second part of Ferdinand Morawitz\u2019s critical study on the bees collected by Aleksey Fedtschenko 1869\u20131871 Expeditions in \u201cTurkestan\u201d was published. In the first volume, \u201cenuinae\u201d , Morawitrenidae\u201d , the rem species . The spe species  and the Andrena Fabricius, 1775 is the one of the largest bee genera, numbering about 1550 species worldwide, of which about 950 are known from the Palaearctic region , who visited the ZMMU from 26.03.1975 to 01.04.1975  received a red label with the inscription \u201cLectotypus Warncke 1975.\u201d All other type specimens were left without nomenclatural status labels. However, unlike other groups of bees  and Zoological Institute of the Russian Academy of Sciences, St. Petersburg (ZISP). The data from each label are separated by two slashes (//). Square brackets are used for English translations and when information is added to specimen label information  or published data .All of the material listed below was examined for this study. In the following list, the taxa are treated in alphabetical order of the names used in the original descriptions. Each entry includes the name of the taxon in its original combination, the complete reference to the original description of the species  and a list of type specimens present in the collections of the Zoological Museum of the Moscow State University, Moscow . Illustrations were obtained by montaging from an image series that covers different focal planes into a single in-focus image with the program Helicon Focus 7. The final illustrations were post-processed for contrast and brightness using Adobe Photoshop software.The classification and current species status mostly follow Andrena Fabricius, 1775Genus Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff1.Morawitz, 18769B7E5D93-5C81-551C-BEC4-7012E1FA0BC7Andrenaacutilabris Morawitz, 1876: 165 (key), 175, \u2640, \u2642.Tashkent (Uzbekistan).Uzbekistan: Tashkent, Urmitan.41\u00b018'N, 69\u00b016'E] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // acutilabris [handwritten by F. Morawitz] // Paralectotypus Andr.acutilabris Mor., design. Osychnjuk, 1980 <red label> // Lectotypus Andrenaacutilabris Morawitz, 1876, design. Astafurova et al., 2022 <red label> // Zoological Institute St. Petersburg INS_HYM_0000300 [ZISP].\u2640, \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a  // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // acutilabris Mor. [handwritten by F. Morawitz] // Paralectotypus Andrenaacutilabris Mor., design. Osychnjuk, 1980 <red label> [ZISP]; 6 \u2642, 2., 5.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a; 1 \u2640, 11.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // Lectotypus Warncke 1975 // Paralectotypus Andrenaacutilabris Morawitz, 1876, design. Astafurova et al., 2022 <identical red labels on each paralectotype specimen> [ZMMU].Andrena (Nobandrena) acutilabris Morawitz, 1876.Turkmenistan, Uzbekistan, Tajikistan, Kazakhstan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff2.Morawitz, 18764EB0DE5A-2A36-5D91-BFE3-B0E72FF30969Andrenaahenea Morawitz, 1876: 164, 166 (key), 210, \u2640, \u2642.Samarkand (Uzbekistan).Uzbekistan: Samarkand.39\u00b039'N, 66\u00b057'E] // Andrenaahenea Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus Andrenaahenea Morawitz, 1876, design. ZMMU].\u2640, designated by Osychnjuk et al. 2008: 51, 16.[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a .1 \u2642, 16.[III.1876] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // Paralectotypus Andrena (Euandrena) ahenea Morawitz, 1876.Uzbekistan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff3.Morawitz, 1876C47DE2A4-D7F6-5EA9-AF77-05732ADA4358Andrenaamoena Morawitz, 1876: 164 (key), 211, \u2640.Chardara (Kazakhstan).Kazakhstan: Syr Darja River near Chardara.41\u00b018'N, 67\u00b057'E] // Andrenaamoena Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus Andrenaamoena Morawitz, 1876, design. ZMMU].\u2640, designated by (3 \u2640). 3 \u2640, 25.[IV.1871] // \u0427\u0430\u0440\u0434\u0430\u0440\u0430 [Chardara] // Paralectotypus Andrenaamoena Morawitz, 1876, design. ZMMU].Andrena (Aciandrena) amoena Morawitz, 1876  lateralis Morawitz, 1876 , Caucasus, Turkey, Israel, Iran, Afghanistan, Central Asia.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff5.Morawitz, 1876FBBADC3D-7CF1-5500-B7F1-E508B1E2BC1BAndrenaaulica Morawitz, 1876: 162, 166 (key), 187, \u2640, \u2642.Panjakent (Tajikistan).Uzbekistan: Tashkent, Samarkand, Oalyk Gorge; Tajikistan: Panjakent, Fan [River], Varzaminor [= Ayni], Iori Gorge.39\u00b030'N, 67\u00b036'E] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andrenaaulica Morawitz, \u2640 [handwritten by F. Morawitz] // Paralectotypus Andr.aulica F. Mor., design. Osychnjuk, 1980 <red label> // Lectotypus Andrenaaulica Morawitz, 1876, design. Astafurova et al., 2022 // Zoological Institute St. Petersburg INS_HYM_0000299 [ZISP].\u2640, \u041f\u044f\u043d\u0434\u0436\u0438\u043a\u0435\u043d\u0442\u044a  // Andr.ferghanica F. Mor, \u2642, Popov det, 1936; 1 \u2640, \u0417\u0430\u0440\u0430\u0432\u0448. \u0434\u043e\u043b. // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // aulica Mor. [handwritten by F. Morawitz]; Paralectotypus Andrenaaulica Mor., design. Osychnjuk, 1980 <identical red labels on each paralectotype specimen> [ZISP]; 4 \u2640, 8.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a; 9 \u2640, 20., 21., 23., 26., 30.[III], 9.[V.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a; 3 \u2640, 28., 29.[IV.1869] // \u041f\u044f\u043d\u0434\u0436\u0438\u043a\u0435\u043d\u0442\u044a; 1 \u2640, 12.[VI.1870] // \u0424\u0430\u043d\u044a; 2 \u2640, 7.[VI.1870] // \u0412\u0430\u0440\u0437\u0430\u043c\u0438\u043d\u043e\u0440\u044a; 1 \u2640, 2 \u2642, 25.[III], 2., 7.[V.1869] // \u0417\u0430\u0440\u0430\u0432\u0448. \u0434\u043e\u043b. // Paralectotypus Andrenaaulica Morawitz, 1876, design. Astafurova et. al., 2022 <identical red labels on each paralectotype specimen> [ZMMU].Andrenaaulica Mor. Typ. [handwritten by F. Morawitz] // Lectotypus Warncke 1975; ZMMU]. However, the label date (31.[II.1869]) does not correspond any date mentioned for type series by There is one female specimen labelled by Warcke as \u201cLectotype\u201d .42\u00b005'N, 68\u00b010'E] // Andrenabairacumensis Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Holotypus Andrenabairacumensis Mor., 1876 <red label, labelled by Yu. Astafurova> [ZMMU].\u2640, 4.[V.1871] // \u0411\u0430\u0439\u0440\u0430\u043a\u0443\u043c\u044a ; Tajikistan: Panjakent.41\u00b018'N, 69\u00b016'E] // Andrenabicarinata Mor. [handwritten by F. Morawitz] // Paralectotypus Andrenabicarinata Mor., design. Osychnjuk, 1980 <red label> // Andrenatuberculiventris Mor. A. Osytshnjuk det. // Lectotypus Andrenabicarinata Morawitz, 1876, design. Astafurova et al., 2022 <red label> // Zoological Institute St. Petersburg INS_HYM_0000298 [ZISP].\u2640, 11.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a  // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andrenabicarinata Mor. [handwritten by F. Morawitz] // Paralectotypus Andrenabicarinata Mor., design. Osychnjuk, 1980 < red label > // Andrenatuberculiventris, D.A. Sidorov det.; 1 \u2640, 5.[IV.1871] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andrenabicarinata F.Mor, Cotype!, F. Morawitz det. // Andrenatuberculiventris, D.A. Sidorov det. [ZISP]; 1 \u2640, 5.[V.1871] // \u0423\u0440\u043c\u0438\u0442\u0430\u043d\u044a [Urmitan] // Lectotypus Warncke 1975 <red label> // A.tuberculiventris [det. A.Osytshnjuk]; 1 \u2642, 5.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // Andrenatuberculiventris, D.A. Sidorov det.; 1 \u2640, 29.[VI.1869] // Katty-Kurgan // Andrenatuberculiventris, D.A. Sidorov det.; 1 \u2640, 2.[V.1871] // \u0423\u0440\u043c\u0438\u0442\u0430\u043d\u044a [Urmitan] // Andrenatuberculiventris, D.A. Sidorov det. // Paralectotypus Andrenabicarinata Morawitz, 1876, design. Astafurova et al., 2022 <identical red labels on each paralectotype specimen> [ZMMU].Andrena (Parandrenella) tuberculiventris Morawitz, 1876 , 205, \u2640.Samarkand (Uzbekistan).Uzbekistan: Samarkand.39\u00b039'N, 66\u00b057'E] // Andrenacapillosa Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 // Lectotypus Andrenacapillosa Morawitz, 1876, design. ZMMU].\u2640, designated by (5 \u2640). 1 \u2640, <golden circle>, 27.[II.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // capillosa Mor. Typ. [handwritten by F. Morawitz]; 4 \u2640, 27.[II.1869], 3., 4.[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Andrenacapillosa Mor., design. Osychnjuk, 1980 <identical red labels on each paralectotype specimen> [ZISP].Andrena (Euandrena) capillosa Morawitz, 1876.Description of male: Uzbekistan, Tajikistan, Kazakhstan. The record from Russia (Western Siberia) by Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff9.Morawitz, 187671B337F1-1FD1-5B2B-A154-0B57EB5FE4C0Andrenacarinifrons Morawitz, 1876: 164 (key), 198, \u2640.Turkistan Province (Kazakhstan).Kazakhstan: between the Karak Mts and Bairkum.42\u00b046'N, 67\u00b024'E] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andrenacarinifrons Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus Andrenacarinifrons Morawitz, 1876 design. by Osytchjuk et al., 2005 <red label, labelled by Yu. Astafurova> // Zoological Institute St. Petersburg INS_HYM_0000200.\u2640, designated by Andrena (Carinandrena) carinifrons Morawitz, 1876.Description of male: Turkmenistan, Uzbekistan, Kazakhstan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff10.Morawitz, 18766B61927A-F633-5629-AF93-B179FBF73B45Andrenacombusta Morawitz, 1876: 163, 165 (key), 189, \u2640, \u2642.Kattakurgan (Uzbekistan).Uzbekistan: Tashkent and near Kattakurgan.39\u00b053'N, 66\u00b015'E] // Andrenacombusta Mor. [handwritten by F. Morawitz] // Lectotypus Warncke, 1975 <red label> // Lectotypus Andrenacombusta Morawitz, 1876, design. Astafurova et al., 2022 <red label> [ZMMU].\u2640, 29. [IV.1869] // \u0412\u0435\u0440\u0445\u043d.[\u0438\u0439] \u0417\u0430\u0440\u0430\u0432\u0448.[\u0430\u043d]  // \u0412\u0435\u0440\u0445\u043d. \u0417\u0430\u0440\u0430\u0432\u0448. [Upper Zaravshan]; 1 \u2642, 11.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // Paralectotypus Andrenacombusta Morawitz, 1876, design. Astafurova et al., 2022 <identical red labels on each paralectotype specimen> [ZMMU].Andrena (Truncandrena) combusta Morawitz, 1876.Azerbaijan, Turkey, Syria, Iran, Afghanistan, Uzbekistan, Tajikistan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff11.Morawitz, 1876172B32CF-0E9E-56B0-BEA5-69B9EF64D435Andrenacomparata Morawitz, 1876: 166 (key), 188, \u2642.Tashkent (Uzbekistan).Uzbekistan: Tashkent and Samarkand.41\u00b018'N, 69\u00b016'E] // Andrenacomparata Mor. [handwritten by F. Morawitz] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andr.aulica F. Mor, \u2642, Popov 1936 det. // Paralectotypus Andr.comparata Mor., design. Osychnjuk, 1980 <red label> // Lectotypus Andrenacomparata Morawitz, 1876, design. Astafurova et al., 2022 <red label> // Zoological Institute St. Petersburg INS_HYM_0000297 [ZISP].\u2642, 9.[III.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a  // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // Andrenacomparata Mor. Typ. [handwritten by F. Morawitz] // Andr.aulica F.Mor, \u2642, Popov det. 1936 // Paralectotypus Andrenacomparata Mor., design. Osychnjuk, 1980 <red label> [ZISP]; 1 \u2642, 21.[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // Andrenacomparata Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Andrenaaulica, D.A. Sidorov det., 2022; 7 \u2642, 21., 26., 30.[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // Paralectotypus Andrenacomparata Morawitz, 1876, design. Astafurova et al., 2022 <identical red labels on each paralectotype specimen> [ZMMU].Andrena (Plastandrena) aulica Morawitz, 1876 (synonymised by Andrena (Plastandrena) bimaculata  by Andrenaaulica, above).Listed as Russia (European part), Caucasus, Iran, Central Asia, India.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff12.Morawitz, 187650D305FD-02E6-5309-B6E0-03759C1DE0F6Andrenacorallina Morawitz, 1876: 162 (key), 203, \u2640.Urmetan (Tajikistan).Tajikistan: Urmitan.39\u00b026'N, 68\u00b015'E] // Andrenacorallina Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 // Lectotypus Andrenacorallina Morawitz, 1876, design. Astafurova et al., 2022 <red label> [ZMMU].\u2640, 2.[V.1869] // \u0423\u0440\u043c\u0438\u0442\u0430\u043d\u044a  // Andrenadiscophora Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus Andrenadiscophora Morawitz, 1876, design. Astafurova et al., 2022 <red label> [ZMMU].\u2640, 8.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a  // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // discophora Mor. \u2642, Typ. [handwritten by F. Morawitz]; 1 \u2640, 4 \u2642, \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andrenadiscophora Mor. [handwritten by F. Morawitz]; 1 \u2640, <golden circle> // 11.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // discophora Mor. \u2640, Typ. [handwritten by F. Morawitz]; 2 \u2640, 8.[IV.1871] and 27.[III.1987] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]; 1 \u2640, 11.[IV.1981] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andr.discophora F. Mor., Popov det. // Paralectotypus Andrena Mor., design. Osychnjuk, 1980 <identical red labels on each paralectotype specimen> [ZISP]; 16 \u2640, 4 \u2642, [28.II\u201311.IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // Paralectotypus Andrenadiscophora Morawitz, 1876, design. Astafurova et al., 2022 <identical red labels on each paralectotype specimen> [ZMMU].Andrena (Trachandrena) discophora Morawitz, 1876.Uzbekistan, Tajikistan, Kazakhstan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff14.Morawitz, 1876507C9637-6F8D-5654-8D5F-0E328CC6C9BCAndrenaFedtschenkoi Morawitz, 1876: 162, 165 (key), 184, \u2640, \u2642.Chardara (Kazakhstan).Kazakhstan: Kysyl-Kum [desert] near draw-well Chakany, Karak steppe, Chardara; Uzbekistan: Zeravshan River valley.41\u00b018'N, 67\u00b057'E] // AndrenaFedtschenkoi Mor. [handwritten by F. Morawitz] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Andrenafedtschenkoi Mor., design. Osychnjuk, 1980 <red label> // Lectotypus Andrenafedtschenkoi Morawitz, 1876, design. Astafurova et al., 2022 <red label> // Zoological Institute St. Petersburg INS_HYM_0000296.\u2640, \u0427\u0430\u0440\u0434\u0430\u0440\u0430  // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // fedtschenkoi F. Morawitz, \u2640 [handwritten by F. Morawitz]; 3 \u2642, \u0427\u0430\u0440\u0434\u0430\u0440\u0430 // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // AndrenaFedtschenkoi Mor. [handwritten by F. Morawitz]; 1 \u2642, 25.[IV.1869] // \u0417\u0430\u0440\u0430\u0432\u0448\u0430\u043d.[\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430] [Zeravshan River valley] // fedtschenkoi F. Morawitz, \u2642 Typ. [handwritten by F. Morawitz] // Paralectotypus Andrenafedtschenkoi Mor., design. Osychnjuk, 1980 <red label> [ZISP]; 1 \u2640, 28.[IV.1871] // \u041a\u0438\u0437\u0438\u043b\u044a\u043a\u0443\u043c // AndrenaFedtschenkoi Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label>; 4 \u2640, 4 \u2642, 25.[IV.1869], 12., 13., 18., 23.[V.1869] // \u0417\u0430\u0440\u0430\u0432\u0448. \u0434\u043e\u043b.; 3 \u2642, 25.[IV.1871] // \u0427\u0430\u0440\u0434\u0430\u0440\u0430; 1 \u2640, 5.[V.1871] // \u041a\u0430\u0440\u0430\u043a.[\u0441\u043a\u0430\u044f] \u0441\u0442\u0435\u043f\u044c [Karak steppe] // Paralectotypus Andrenafedtschenkoi Morawitz, 1876, design. Astafurova et al., 2022 <identical red labels on each paralectotype specimen> [ZMMU].Andrena (Ulandrena) fedtschenkoi Morawitz, 1876.Turkmenistan, Uzbekistan, Tajikistan, Kazakhstan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff15.Morawitz, 187609C1F838-36F7-50E1-B769-D69BF110A5D9Andrenaferghanica Morawitz, 1876: 163 (key), 189, \u2640.Alai Mts, Kavuk Pass (Kyrgyzstan).Kyrgyzstan: Kavuk Pass.39\u00b040'N, 72\u00b015'E] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andrenaferghanica Morawitz [handwritten by F. Morawitz] // Lectotypus Andrenaferghanica Mor., design. Osychnjuk, 1980 <red label> // Holotypus Andrenaferghanica Mor., 1876 <red label, labelled by Yu. Astafurova> // Zoological Institute St. Petersburg INS_HYM_0000261 [ZISP].\u2640, 24.[VI.1871] // \u0410\u043b\u0430\u0439  // Andrenaflavitarsis Mor. \u2640 [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus Andrenaflavitarsis Morawitz, 1876, design. ZMMU].\u2640, designated by . 1 \u2640, 5.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andrenaflavitarsis Mor. [handwritten by F. Morawitz] // Andrenaflavitarsis Mor., design. Osychnjuk, 1980 < red label > [ZISP]; 1 \u2642, 16.[III.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // Andrenaflavitarsis Mor. [handwritten by F. Morawitz] // Paralectotypus Andrenaflavitarsis Morawitz, 1876, design. ZMMU].Andrena (Euandrena) flavitarsis Morawitz, 1876.Uzbekistan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff17.Morawitz, 18767919437C-157C-5B68-91C2-DB45AB668844Andrenafuscicollis Morawitz, 1876: 164, 165 (key), 208, \u2640, \u2642.Tashkent (Uzbekistan).Uzbekistan: Tashkent.41\u00b018'N, 69\u00b016'E] // Andrenafuscicollis Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus Andrenafuscicollis Morawitz, 1876, design. ZMMU].\u2642, designated by . 1 \u2642, <golden circle>, 1.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // fuscicollis Mor. \u2642 [handwritten by F. Morawitz]; 4 \u2642, 5., 10.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]; 1 \u2640, <golden circle>, 10.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // fuscicollis Mor. Typ., \u2640 [handwritten by F. Morawitz] // Paralectotypus Andrenafuscicollis Mor., design. Osychnjuk, 1980 <identical red labels on each paralectotype specimen> [ZISP]; 4 \u2640, 1.,10., 11.[IV.1871] \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // Paralectotypus Andrenafuscicollis Morawitz, 1876, design. ZMMU].Andrena (Fuscandrena) fuscicollis Morawitz, 1876.The collection date in the original publication is \u201cfrom 11 February to 11 March\u201d. However, the lectotype and paralectotype specimens designated by A. Osytshnjuk are labelled as collected in April [green label]. Probably \u201cMarch\u201d was mistakenly mentioned in the Morawitz\u2019 publication instead of April.Turkmenistan, Uzbekistan, Tajikistan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff18.Morawitz, 1876C32A70DA-D344-5CB0-8FDB-BF59457419E3Andrenahieroglyphica Morawitz, 1876: 163 (key), 192, \u2640.Sokh Enclave (Uzbekistan).Uzbekistan: near Sokh.39\u00b057'N, 71\u00b007'E] // Andrenahieroglyphica Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Holotypus Andrenahieroglyphica Mor., 1876 <red label, labelled by Yu. Astafurova> [ZMMU].\u2640, 28.[VI.1871] // \u0421\u043e\u0445\u044a  // Andrenainfirma Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus Andrenainfirma Morawitz, 1876, design. ZMMU].\u2642, designated by by . 1 \u2642, 8.[VI.1870] // \u0412\u0430\u0440\u0437\u0430\u043c\u0438\u043d\u043e\u0440\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]; 1 \u2642, <golden circle> // 21.[VI.1870] // \u0418\u0441\u043a\u0430\u043d\u0434\u0435\u0440\u044a [Iskander] // infirma Mor. Typ. [handwritten by F. Morawitz]; 1 \u2642, 21.[VI.1870] // \u0418\u0441\u043a\u0430\u043d\u0434\u0435\u0440\u044a // Andrenainfirma Mor. [handwritten by F. Morawitz] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Andrenainfirma Mor., design. Osytshnjuk, 1980 <identical red labels on each paralectotype specimen> [ZISP]; 1 \u2640 [without head], 9.[VI.1869] // \u0412\u0430\u0440\u0437\u0430\u043c\u0438\u043d\u043e\u0440\u044a // \u0424\u0435\u0434\u0447\u0435\u043d\u043a\u043e [Fedtschenko leg.]; 1 \u2642, 26.[VI.1871] // \u0427\u0438\u0431\u0443\u0440\u0433\u0430\u043d\u044a [Khodzha-Chiburgan River] // Paralectotypus Andrenainfirma Morawitz, 1876., design. ZMMU].Andrena (Melandrena) infirma Morawitz, 1876.Afghanistan, Central Asia.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff20.Morawitz, 1876FC99A7E8-DD11-5754-AFDC-B0A7621D887EAndrenainitialis Morawitz, 1876: 164, 166 (key), 199, \u2640, \u2642.Shardara District (Kazakhstan).Kazakhstan: \u201cbetween Keles [River] and Kosaral [Lake]\u201d.41\u00b010'N, 68\u00b006'E] // Andrenainitialis Mor. [handwritten by F. Morawitz] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Andrenainitialis Mor., design. Osychnjuk, 1980 // Lectotypus Andrenainitialis Morawitz, 1876, design. Astafurova et al., 2022 <red label> // Zoological Institute St. Petersburg INS_HYM_0000295.\u2640, 24.[IV.1871] // \u041a\u043e\u0441\u0430\u0440\u0430\u043b\u044a  \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Andrenainitialis Mor., design. Osychnjuk, 1980 [ZISP]; 1 \u2640, 24.[IV.1871] // \u041a\u043e\u0441\u0430\u0440\u0430\u043b\u044a // Andrenainitialis Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975; 1 \u2640, 24.[IV.1871] // \u041a\u043e\u0441\u0430\u0440\u0430\u043b\u044a; 1 \u2642, 24.[IV.1871] // \u041a\u043e\u0441\u0430\u0440\u0430\u043b\u044a // Andrenalateralis, D.A. Sidorov det., 2022 // Paralectotypus Andrenainitialis Morawitz, 1876, design. Astafurova et al., 2022 <identical red labels on each paralectotype specimen> [ZMMU].Andrena (incertae sedis) initialis Morawitz, 1876.Andrena (incertae sedis) lateralis Morawitz, 1876, as was mentioned by Male unknown. The single male from the type series belongs to Uzbekistan, Kazakhstan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff21.Morawitz, 1876D63503E2-A06A-541E-8961-220548D83CFAAndrenalaeviventris Morawitz, 1876: 163 (key), 182, \u2640.Obburdon (Tajikistan).Uzbekistan: Gus [near Urgut]; Tajikistan: Pyandzhikent, Obburden [Obburdon].40\u00b025'N, 69\u00b018'E] // Andrenalaeviventris Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus Andrenalaeviventris Morawitz, 1876, design. Astafurova et al., 2022 <red label> [ZMMU].\u2640, 4.[VI.1870] // \u041e\u0431\u0431\u0443\u0440\u0434\u0435\u043d\u044a .1 \u2640, 24.[V.1869] // \u0417\u0430\u0440\u0430\u0432\u0448.[\u0430\u043d\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430] [Zaravshan River Valley] // Paralectotypus Andrena (Aciandrena) laeviventris Morawitz, 1876.Male unknown , 200, \u2640, \u2642.Shardara District (Kazakhstan).Kazakhstan: \u201cbetween Keles [River] and Kosaral [Lake]\u201d, Kyzyl-Kum [desert]; Uzbekistan: Urmitan [near Katty-Kurgan]; Tajikistan: Varzaminor [Ayni], near Obburden [Obburdon].41\u00b010'N, 68\u00b006'E] // Andrenalateralis Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus Andrenalateralis Morawitz, 1876., design. ZMMU].\u2640, designated by by . 1 \u2640, 24.[VI.1871] // \u041a\u043e\u0441\u0430\u0440\u0430\u043b\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andrenalateralis F.Mor. Popov 1935 det.; 1 \u2640, 1 \u2642, <golden circle>, 24.[VI.1871] // \u041a\u043e\u0441\u0430\u0440\u0430\u043b\u044a // lateralis Mor. Typ. [handwritten by F. Morawitz]; 2 \u2642, 25.[IV.1871] // \u0427\u0430\u0440\u0434\u0430\u0440\u0430 [Shardara] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andrenalateralis Mor. [handwritten by F. Morawitz] // Paralectotypus Andrenalateralis Mor., design. Osychnjuk, 1980 <identical red labels on each paralectotype specimen> [ZISP]; 4 \u2640, 24.[IV.1871] // \u041a\u043e\u0441\u0430\u0440\u0430\u043b\u044c; 1 \u2640, 30.[IV.1871] // \u041a\u0438\u0437\u0438\u043b\u043a\u0443\u043c\u044a [Kyzyl-Kum desert]; 9 \u2640, \u041e\u0431\u0431\u0443\u0440\u0434\u0435\u043d\u044a [Obburden]; 2 \u2640, \u0412\u0430\u0440\u0437\u0430\u043c\u0438\u043d\u043e\u0440\u044a [Varzaminor]; 9 \u2642, 25.[IV.1871] // \u0427\u0430\u0440\u0434\u0430\u0440\u0430 // Lectotypus Andrenalateralis Morawitz, 1876, design. ZMMU].Andrena (incertae sedis) lateralis Morawitz, 1876.Europe, Russia (to East Siberia), Caucasus, Turkey, Israel, Iran, Afghanistan, Central Asia.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff23.Morawitz, 18766D8B14C6-E3AC-5BD3-A0AC-A334A76E8EEEAndrenaleucorhina Morawitz, 1876: 165 (key), 169, \u2642.Shardara (Kazakhstan).Kazakhstan: Syr-Darja River, near Chardara.41\u00b018'N, 67\u00b057'E] // Andrenaleucorhina Mor. [handwritten by F. Morawitz] // Lectotypus Warmcke 1975 <red label> // Lectotypus Andrenaleucorhina Morawitz, 1876 design. Astafurova et al., 2022 <red label> [ZMMU].\u2642, 25.[IV.1871] // \u0427\u0430\u0440\u0434\u0430\u0440\u0430  // \u0427\u0430\u0440\u0434\u0430\u0440\u0430 [Chardara] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andrenaleucorhina Morawitz [handwritten by F. Morawitz] // Paralectotypus Andrenaleucorhina Morawitz, 1876 design. Astafurova et al., 2022 <red label> .Andrena (Ulandrena) abbreviata Dours, 1873 , 181, \u2642.Tashkent (Uzbekistan).Uzbekistan: Tashkent.41\u00b018'N, 69\u00b016'E] // Andrenalucidicollis Mor. [handwritten by F. Morawitz] // Lectotypus Warncke, 1975 <red label> // Syntypus Andrenalucidicollis Mor., 1876 <red label> [ZMMU].\u2642, 5.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a  // maculipes Mor. Typ. [handwritten by F. Morawitz] // Lectotypus Andrenamaculipes Mor., design. Osychnjuk, 1980 <red label> // Lectotypus Andrenamaculipes Morawitz, 1876, design. ZISP].\u2642, designated by . 2 \u2640, 2 \u2642, \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andrenamaculipes F. Moraw. [handwritten by F. Morawitz]; 1 \u2640, <golden circle>, 16.[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // maculipes Mor. Typ. [handwritten by F. Morawitz] // Paralectotypus Andr.maculipes Mor., design. Osychnjuk, 1980 <identical red labels on each paralectotype specimen> [ZISP]; 7 \u2640, 7, 16.[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a; 2 \u2640, 4.[III.1869] // \u0417\u0430\u0440\u0430\u0432\u0448.[\u0430\u043d\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430] [Zaravshan River Valley] // Paralectotypus Andrenamaculipes Morawitz, 1876, design. ZMMU].Andrena (Chrysandrena) maculipes Morawitz, 1876.ZMMU instead of the ZISP.Central Asia.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff26.Morawitz, 1876AFACB171-0C0A-54F1-AC08-46299ECAF5A7Andrenamajalis Morawitz, 1876: 163 (key), 182, \u2640.Bairkum (Kazakhstan).Kazakhstan: Bairakum, Karak steppe; Uzbekistan: Zeravshan River valley, Dzham Gorge, Oalyk Gorge between Oalyk and Aksay.42\u00b005'N, 68\u00b010'E] // Andrenamajalis Mor. [handwritten by F. Morawitz] // Lectotypus Warmcke, 1975 <red label> // Lectotypus Andrenamajalis Morawitz, 1876., design. ZMMU].\u2640, designated by (14 \u2640). 3 \u2640, the same labels as in the lectotype; 3 \u2640, 5.[V.1871] // \u041a\u0430\u0440\u0430\u043a\u0441\u043a.[\u0430\u044f] \u0441\u0442\u0435\u043f\u044c [Karak steppe] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]// Paralectotypus Andr.majalis Mor., design. Osychnjuk, 1980 <identical red labels on each paralectotype specimen> [ZISP]; 4 \u2640, 4.[V.1871] // \u0411\u0430\u0439\u0440\u0430\u043a\u0443\u043c\u044a; 1 \u2640, 5.[V.1871] // \u041a\u0430\u0440\u0430\u043a\u0441\u043a.[\u0430\u044f] \u0441\u0442\u0435\u043f\u044c; 3 \u2640, 16.,17.[V.1869] // \u0417\u0430\u0440\u0430\u0432\u0448.[\u0430\u043d\u0441\u043a\u0430\u044f] \u0434\u043e\u043b.[\u0438\u043d\u0430] [Zaravshan River Valley] // Paralectotypus Andrenamajalis Morawitz, 1876 design. Osychnjuk et al., 2008 <identical red labels on each paralectotype specimen, labelled by Yu. Astafurova> [ZMMU].Andrena (Euandrena) majalis Morawitz, 1876.Description of male: Central Asia.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff27.Morawitz, 1876055E3059-8E4C-54CB-BF56-36294DB2D4EEAndrenamordax Morawitz, 1876: 165 (key), 196, \u2642.Ayni (Tajikistan).Tajikistan: Varzaminor.39\u00b023'N, 68\u00b032'E] // Andrenamordax Mor. [handwritten by F. Morawitz] // Lectotypus Warmcke 1975 <red label> // Lectotypus Andrenamordax Morawitz, 1876, design. ZMMU].\u2642, designated by (3 \u2642). 2 \u2642, 8.[VI.1870] // \u0412\u0430\u0440\u0437\u0430\u043c\u0438\u043d\u043e\u0440\u044a // Andrenamordax Mor. [handwritten by F. Morawitz] // Paralectotypus Andrenamordax Mor., design. Osychnjuk, 1980 <red label> [ZISP]; 1 \u2642, 9.[VI.1870] // \u0412\u0430\u0440\u0437\u0430\u043c\u0438\u043d\u043e\u0440\u044a // Paralectotypus Andrenamordax Morawitz, 1876, design. ZMMU].Andrena (Hoplandrena) mordax Morawitz, 1876.Description of female: Uzbekistan, Tajikistan, Kyrgyzstan, Kazakhstan, China.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff28.Morawitz, 18765A85590B-42AF-594E-9EFF-0A4FAA607D96Andrenamucorea Morawitz, 1876: 164, 165 (key), 212, \u2640, \u2642.Between Tashkent and Keless River (Uzbekistan).Uzbekistan: between Tashkent and Keless River, Urmitan [near Katty-Kurgan]; Kazakhstan: \u201cbetween Keles [River] and Kosaral [Lake]\u201d, Kyzyl Kum [desert].41\u00b015'N, 69\u00b010'E] // Andrenamucorea Mor. [handwritten by F. Morawitz] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Andr.mucorea Mor., design. Osychnjuk, 1980 <red label> // Lectotypus Andrenamucorea Morawitz, 1876 design. Astafurova et al., 2022 <red label> // Zoological Institute St. Petersburg INS_HYM_0000294 [ZMMU].\u2640, 23.[IV.1871] // \u041a\u0435\u043b\u0435\u0441\u044a [River]  // \u041a\u0435\u043b\u0435\u0441\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andrenamucorea Mor. [handwritten by F. Morawitz]; 1 \u2640, 24.IV.1871 // \u041a\u043e\u0441\u0430\u0440\u0430\u043b\u044a [Kosaral] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]; 1 \u2640, \u041a\u043e\u0441\u0430\u0440\u0430\u043b\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // mucorea F. Mor., \u2640 [handwritten by F. Morawitz] // Paralectotypus Andr.mucorea Mor., design. Osychnjuk, 1980 <identical red labels on each paralectotype specimen > [ZISP]; 1 \u2640, 10.[V.1869] // \u0423\u0440\u043c\u0438\u0442\u0430\u043d\u044a [Urmitan] // Andrenamucorea Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label>; 7 \u2640, 22., 23.[IV.1871] // \u041a\u0435\u043b\u0435\u0441\u044a; 2\u2640, 9., 10.[V.1869] // \u0423\u0440\u043c\u0438\u0442\u0430\u043d\u044a // Paralectotypus Andrenamucorea Morawitz, 1876, design. Astafurova et al., 2022 <identical red labels on each paralectotype specimen> [ZMMU].Andrena (Poecilandrena) mucorea Morawitz, 1876.Uzbekistan, Tajikistan, Kazakhstan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff29.Morawitz, 187691DEE2B8-6FED-5FB1-9E67-2823415B3F1FAndrenanigrita Morawitz, 1876: 166 (key), 196, \u2642.Iskanderkul Lake (Tajikistan).Tajikistan: Iskanderkul Lake.39\u00b004'N, 68\u00b022'E] // Andrenanigrita Mor. [handwritten by F. Morawitz] // Lectotypus Warmcke 1975 <red label> // Holotypus Andrenanigrita Mor., 1876 <red label, labelled by Yu. Astafurova> [ZMMU].\u2642, 18.[VI.1870] // \u0418\u0441\u043a\u0430\u043d\u0434\u0435\u0440\u044a .Description of female: The lectotype designation by Uzbekistan, Tajikistan, Kyrgyzstan, Kazakhstan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff30.Morawitz, 18766C0500DB-E21A-586A-B461-FBC4277C16E2Andrenanitidicollis Morawitz, 1876: 162, 166 (key), 180, \u2640, \u2642.Kyzylkum Desert (Kazakhstan).Kazakhstan: Kyzyl-kum Desert; Karak mountains .41\u00b058'N, 67\u00b003'E] // Andrenanitidicollis Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus Andrenanitidicollis Morawitz, 1876, design. Astafurova et al., 2022 <red label> [ZMMU].\u2640, 29.[IV.1871] // \u041a\u0438\u0437\u0438\u043b\u044a\u043a\u0443\u043c\u044a .1 \u2642, 6.[V.1871] // \u041a\u0438\u0437\u0438\u043b\u044a\u043a\u0443\u043c\u044a // Paralectotypus Andrena (incertae sedis) nitidicollis Morawitz, 1876.Kazakhstan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff31.Morawitz, 18761D4DCA7C-1F1C-5485-AD49-3D7F45537F8AAndrenanupta Morawitz, 1876: 163, 166 (key), 191, \u2640, \u2642.Tashkent (Uzbekistan).Uzbekistan: Tashkent and Samarkand.41\u00b018'N, 69\u00b016'E] // Andrenanupta Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus Andrenanupta Morawitz, 1876, design. ZMMU].\u2640, designated by . 1 \u2640, the same label as in the lectotype; 2 \u2640, 2 \u2642, \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // nupta F. Mor. [handwritten by F. Morawitz] // Paralectotypus Andr.nupta Mor., design. Osychnjuk, 1980 <identical red labels on each paralectotype specimen> [ZISP]; 2 \u2640, 1 \u2642, 27.[III.1871], 3., 8.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a; 1 \u2642, 5.[IV.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a [Samarkand] // Paralectotypus Andrenanupta Morawitz, 1876, design. ZMMU].Andrena (Euandrena) nupta Morawitz, 1876.Uzbekistan, Kazakhstan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff32.Morawitz, 18769B29B2C9-3AE3-56A6-B52E-07F70DB32B87Andrenaoralis Morawitz, 1876: 162 (key), 177, \u2640.Tashkent (Uzbekistan).Uzbekistan: near Tashkent.41\u00b018'N, 69\u00b016'E] // oralis Mor. Typ. [handwritten by F. Morawitz] // Lectotypus Andrenaoralis Mor., design. Osychnjuk, 1980 <red label> // Lectotypus, Andrenaoralis Morawitz, 1876, design. Astafurova et al., 2022 <red label> // Zoological Institute St. Petersburg INS_HYM_0000201 [ZISP].\u2640, <golden circle> // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a  // Lectotypus Warncke 1975 <red label> // Paralectotypus Andrenaoralis Morawitz, 1876, design. Astafurova et al., 2022 <red label> [ZMMU].1 \u2640, 5.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442 // Andrena (Orandrena) oralis Morawitz, 1876.Andrenasogdiana Morawitz, 1876 , Turkey, Turkmenistan, Uzbekistan, Kazakhstan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff33.Morawitz, 18769532633F-6932-54FE-9D00-E3F908AAA849Andrenapannosa Morawitz, 1876: 162 (key), 197, \u2640.Khodzha-Chiburgan Gorge (Kyrgyzstan).Kyrgyzstan: Khodzha-Chiburgan Gorge.39\u00b048'N, 70\u00b041'E] // pannosa Mor. Typ. [handwritten by F. Morawitz] // Lectotypus Andrenapannosa Mor., design. Osychnjuk, 1980 <red label> // Lectotypus, Andrenapannosa Morawitz, 1876 design. Astafurova et al., 2022 <red label> // Zoological Institute St. Petersburg INS_HYM_0000287 [ZISP].\u2640, <golden circle>, 21.[VI.1871] // \u0427\u0438\u0431\u0443\u0440\u0433\u0430\u043d\u044a  // \u0427\u0438\u0431\u0443\u0440\u0433\u0430\u043d\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andrenapannosa Mor. [handwritten by F. Morawitz] // Paralectotypus Andr.pannosa Mor., design. Osychnjuk, 1980 <red label> [ZISP]; 1 \u2640, 21.[VI.1871] // \u0427\u0438\u0431\u0443\u0440\u0433\u0430\u043d\u044a // Paralectotypus Andrenapannosa Morawitz, 1876, design. Astafurova et al., 2022 <identical red labels on each paralectotype specimen> [ZMMU].Andrena (Euandrena) pannosa Morawitz, 1876.ZMMU. This specimen is a female labelled \u201c21.[06.1971] // Soch [Uzbek enclave] Andrenapannosa Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975.\u201d However, according to the original description , the type location is \u201cKhodzha-Chiburgan Gorge\u201d ; thus, the lectotype published by ZISP from \u201cChiburgan\u201d, which correspond to the original description of Morawitz. One of these females labelled by A. Osytshnjuk as \u201clectotype\u201d is designated here as a lectotype of Andrenapannosa.Description of male: Uzbekistan, Kyrgyzstan, Tajikistan, Kazakhstan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff34.Morawitz, 1876B23B5039-ED84-5D62-A6C3-AB2075AB4F9BAndrenaplanirostris Morawitz, 1876: 163, 165 (key), 174, \u2640, \u2642.Tashkent (Uzbekistan).Uzbekistan: near Tashkent.41\u00b018'N, 69\u00b016'E] // Andrenaplanirostris Mor. [handwritten by F. Morawitz] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Andr.planirostris Mor., design. Osychnjuk, 1980 <red label> // Lectotypus Andrenaplanirostris Morawitz, 1876, design. Astafurova et al., 2022 <red label> // Zoological Institute St. Petersburg INS_HYM_0000256 [ZISP].\u2640, 23.[III.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a  // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // planirostris Mor. Typ. \u2640 [handwritten by F. Morawitz]; 9 \u2640, 23., 25.[III.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz]; 1 \u2642, \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // planirostris Mor. [handwritten by F. Morawitz] // Paralectotypus Andr.planirostris Mor., design. Osychnjuk, 1980 <red label> [ZISP]; 10 \u2640, 1 \u2642, 11., 22., 23., 24., 25., 26. \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // Paralectotypus Andrenaplanirostris Morawitz, 1876, design. Astafurova et al., 2022 <identical red labels on each paralectotype specimen> [ZMMU].Andrenaplanirostris Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975; ZMMU]. However, the label date does not correspond to any date [11\u201326.III.1871) mentioned for the type series by There is a female specimen labelled by Warcke as a \u201cLectotypus\u201d [<golden circle> // 5.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442 // Andrena (Planiandrena) planirostris Morawitz, 1876.Uzbekistan, Kazakhstan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff35.Morawitz, 187642502708-0CC4-593B-85A3-AEC547D2C9ADAndrenapunctifrons Morawitz, 1876: 164 (key), 202, \u2640.Khodzha-Chiburgan River (Kyrgyzstan).Kyrgyzstan: \u201cKhodzha-Chiburgan River\u201d.39\u00b048'N, 70\u00b041'E] // Andrenapunctifrons Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Holotypus Andrenapunctifrons Mor., 1876 <red label> [ZMMU].\u2640, \u0427\u0438\u0431\u0443\u0440\u0433\u0430\u043d\u044a  // Andrenapunctiventris Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Holotypus Andrenapunctiventris Mor., 1876 <red label> [ZMMU].\u2640, 15.[VI.1870] // \u0418\u0441\u043a\u0430\u043d\u0434\u0435\u0440\u044a  // Andrenaquadrifasciata Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Holotypus Andrenaquadrifasciata Mor., 1876 <red label> [ZMMU].\u2640, 7.[VII.1871] // \u0428\u0430\u0433\u0438\u043c\u0430\u0440\u0434\u0430\u043d\u044a  // ravicollis Mor. Typ. [handwritten by F. Morawitz] // Paralectotypus Andrenaravicollis Mor., design. Osychnjuk, 1980 <red label> // Lectotypus Andrenaravicollis Morawitz, 1876, design. Astafurova et al., 2022 <red label> // Andrenaleucorhina, D.A. Sidorov det., 2022 // Zoological Institute St. Petersburg INS_HYM_0000286 [ZISP].\u2640, <golden circle> // 25.[IV.1871] // \u0427\u0430\u0440\u0434\u0430\u0440\u0430 .1 \u2640, 25.[IV.1871] // \u0427\u0430\u0440\u0434\u0430\u0440\u0430 // Andrenaravicollis Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975; ZMMU]. However, the label date does not correspond the date [25.III.1871] mentioned for the type series by There is a female specimen (in bad condition) labelled by Warcke as a \u201cLectotypus\u201d [27.[IV.1871] // \u0427\u0430\u0440\u0434\u0430\u0440\u0430 // Andrena (Ulandrena) abbreviata Dours, 1873 , 180, \u2640.NW of Bairkum (Kazakhstan).Uzbekistan: Katty-Kurgan; Kazakhstan: Karak steppe.42\u00b046'N, 67\u00b024'E] // Andrenarufilabris Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus Andrenarufilabris Morawitz, 1876, design. Astafurova et al., 2022 <red label> [ZMMU].\u2640, 5.[V.1871] // \u041a\u0430\u0440\u0430\u043a\u0441\u043a.[\u0430\u044f] \u0441\u0442\u0435\u043f\u044c  and 6.[V.1869] // \u041a\u0430\u0442\u0442\u044b \u041a\u0443\u0440\u0433\u0430\u043d\u044a // Andrenasatra, D.A. Sidorov det., 2022 // Paralectotypus, Andrenarufilabris Morawitz, 1876, design. Astafurova et al., 2022 <red label> [ZMMU].Andrena (Simandrena) sarta Morawitz, 1876 , 167, \u2640, \u2642.Anzob (Tajikistan).Tajikistan: Anzob.39\u00b004'N, 68\u00b052'E] // Andrenarufina Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 // Lectotypus Andrenarufina Morawitz, 1876, design. ZMMU].\u2642, designated by Andrenarufina Morawitz, 1876, design. ZMMU].\u2640, 75. // \u0422\u0443\u0440\u043a\u0435\u0441\u0442.[\u0430\u043d\u0441\u043a\u0438\u0439] \u043a\u0440.[\u0430\u0439] [Turkestan] // Paralectotypus, Andrena (Cnemidandrena) rufina Morawitz, 1876.Uzbekistan, Tajikistan, Kazakhstan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff41.Morawitz, 18764801D484-8040-534D-9125-A0496776FEA9Andrenasarta Morawitz, 1876: 164 (key), 171, \u2640.30 km SE of Kozhatogai, Turkistan Province (Kazakhstan).Kazakhstan: steppe between Tashkent and Syrdarya River.41\u00b047'N, 68\u00b023'E] // satra Mor. [handwritten by Osytshnjuk] // Lectotypus Warncke 1975 // Lectotypus Andrenasarta Morawitz, 1876, design. Astafurova et al., 2022 <red label> [ZMMU].\u2640, 20.[V.1871] // \u0421\u0442\u0435\u043f\u044c \u043c.[\u0435\u0436\u0434\u0443] \u0421.[\u044b\u0440] [-] \u0434.[\u0430\u0440\u044c\u0435\u0439] \u0438 \u0422.[\u0430\u0448\u043a\u0435\u043d\u0442\u043e\u043c]  // Lectotypus Warncke, 1975 // Lectotypus Andrenasemiaenea Morawitz, 1876, ZMMU].\u2640, designated by (4 \u2640). 3 \u2640, 25.IV.1871 // \u0427\u0430\u0440\u0434\u0430\u0440\u0430 [Chardara] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Andr.semiaenea Mor., design. Osychnjuk, 1980 <red label> [ZISP]; 1 \u2640, 24.[IV.1871] // \u041a\u043e\u0441\u0430\u0440\u0430\u043b\u044c [Kosaral] // Paralectotypus Andenasemiaenea Mor., design. ZMMU].Andrena (Poecilandrena) semiaenea Morawitz, 1876.Lepidandrena by Listed as Afghanistan, Uzbekistan, Kazakhstan, Mongolia.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff43.Morawitz, 18761DE20C98-571A-51BA-9C30-4D65FD27CD34Andrenasmaragdina Morawitz, 1876: 164, 165 (key), 211, \u2640, \u2642.Samarkand (Uzbekistan).Uzbekistan: Samarkand.39\u00b039'N, 66\u00b057'E] // Andrenasmaragdina Mor. [handwritten by F. Morawitz] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Andrenasmaragdina Mor., design. Osychnjuk, 1980 <red label> // Lectotypus Andrenasmaragdina Morawitz, 1876, design. Astafurova et al., 2022 <red label> // Zoological Institute St. Petersburg INS_HYM_0000293 [ZISP].\u2640, \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a  // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // Andrenasmaragdina Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label>; 1 \u2640, 1 \u2642, 10., 28.[II.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // Paralectotypus Andrenasmaragdina Morawitz, 1876, design. Astafurova et al., 2022 <red label> [ZMMU].Andrena (Notandrena) smaragdina Morawitz, 1876.Poecilandrena by Listed as Uzbekistan, Tajikistan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff44.Morawitz, 18763BEFB11C-F0B3-50CF-BB78-B4EBFD4EF6E0Andrenasogdiana Morawitz, 1876: 165 (key), 177, \u2642.Samarkand (Uzbekistan).Uzbekistan: Samarkand.39\u00b039'N, 66\u00b057'E] // Andrenasogdiana Mor. [handwritten by F. Morawitz] // Paralectotypus Andrenasogdiana Mor., design. Osychnjuk, 1980 <red label> // Andrenaoralis, \u2642, D.A. Sidorov det. 2022 // Lectotypus Andrenasogdiana Morawitz, 1876, design. Astafurova et al., 2022 <red label> // Zoological Institute St. Petersburg INS_HYM_0000292 [ZISP].\u2642, 11.[IV.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a  mentioned for the type series by There is a male specimen labelled by Warncke as a \u201cLectotypus\u201d , Turkey, Turkmenistan, Uzbekistan, Kazakhstan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff45.Morawitz, 18761462616D-C19B-5A54-94EC-7FD24029765FAndrenasordida Morawitz, 1876: 163, 166 (key), 173, \u2640, \u2642.Tashkent (Uzbekistan).Uzbekistan: near Tashkent.41\u00b018'N, 69\u00b016'E] // Andrenasordida Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus Andrenasordida Morawitz, 1876, design. ZMMU].\u2640, designated by . 1 \u2640, 1 \u2642, the same label as in the lectotype; 4 \u2640, \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // sordida [handwritten by F. Morawitz] // Paralectotypus Andr.sordida Mor., design. Osychnjuk, 1980 <identical red labels on each paralectotype specimen> [ZISP]; 78 \u2640, 1 \u2642, 1., 3., 5., 8.[IV.1871], 28.[III.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // Paralectotypus Andrenasordida Mor. design. ZMMU].Andrena (Leucandrena) sordidella Viereck, 1918, replacement name for A.sordida Morawitz, 1876 .Uzbekistan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff46.Morawitz, 1876D0F5D5E8-812B-5682-9441-5276A1375D7DAndrenasubaenescens Morawitz, 1876: 164, 166 (key) 207, \u2640, \u2642.Samarkand (Uzbekistan).Uzbekistan: Samarkand.39\u00b039'N, 66\u00b057'E] // subaenescens Mor. Typ. [handwritten by F. Morawitz] // Paralectotypus Andrenasubaenescens Mor., design. Osychnjuk, 1980 <red label> // Lectotypus Andrenasubaenescens Morawitz, 1876, design. Astafurova et al., 2022 <red label> // Zoological Institute St. Petersburg INS_HYM_0000255 [ZISP].\u2640, <golden circle> // 27.[II.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a  // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Andr.sordida Mor., design. Osychnjuk, 1980 <red label> [ZISP]; 1 \u2640, 27.[II.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // Andrenasubaenescens Mor. [handwritten by F. Morawitz] // Lectotypus Warncke, 1975 <red label>; 4 \u2640, 1 \u2642. 27., 28.[II\u201318.III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // Paralectotypus Andrenasubaenescens Morawitz, 1876, design. Astafurova et. al., 2022 <identical red labels on each paralectotype specimen> [ZMMU].Andrena (Poecilandrena) subaenescens Morawitz, 1876.Uzbekistan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff47.Morawitz, 18766C6D8114-958B-53F9-A23C-DC08B526346FAndrenatemporalis Morawitz, 1876: 165 (key), 204, \u2642.Samarkand (Uzbekistan).Uzbekistan: Samarkand.39\u00b039'N, 66\u00b057'E] // Andrenatemporalis Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus, Andrenatemporalis Morawitz, 1876, design. ZMMU].\u2642, designated by (7 \u2642). 1 \u2642, <golden circle> // 20.[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // temporalis Mor. Typ. [handwritten by F. Morawitz]; 1 \u2642, \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Andrenatemporalis F. Morawitz, \u2642 [handwritten by F. Morawitz] // Paralectotypus Andr.temporalis Mor., design. Osychnjuk, 1980 <red label> [ZISP]; 5 \u2642, 16., 20.[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // Paralectotypus, Andrenatemporalis Morawitz, 1876, design. ZMMU].Andrena (Notandrena) hieroglyphica Morawitz, 1876 , 184, \u2642.Tashkent (Uzbekistan).Uzbekistan: Tashkent.41\u00b018'N, 69\u00b016'E] // Andrenatuberculiventris Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus Andrenatuberculiventris Morawitz, 1876, design. Astafurova et al., 2022 <red label> [ZMMU].\u2642, 10.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a  // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // tuberculiventris Mor. Typ. [handwritten by F. Morawitz] // Paralectotypus Andr.tuberculiventris Mor., design. Osychnjuk, 1980 <red label> [ZISP]; 2 \u2642, 10.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // Paralectotypus Andrenatuberculiventris Morawitz, 1876, design. Astafurova et al., 2022 <red label> [ZMMU].Andrena (Parandrenella) tuberculiventris Morawitz, 1876 (according to Andrena (Parandrenella) bicarinata Morawitz, 1876 by Listed as Uzbekistan, Tajikistan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff49.Morawitz, 1876D1059395-B44F-5680-B205-0F4EA578CEE0Andrenaturkestanica Morawitz, 1876: 162, 166 (key), 192, \u2640, \u2642.Urgut, Zeravshan River valley (Uzbekistan).Uzbekistan: Zeravshan River valley; steppe between Ulus and Dzham.39\u00b024'N, 67\u00b013'E] // turkestanica F. Mor. \u2640, [handwritten by F. Morawitz] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Andr.turkestanica Mor., design. Osychnjuk, 1980 <red label> // Lectotypus Andrenaturkestanica Morawitz, 1876, design. Astafurova et al., 2022 <red label> // Zoological Institute St. Petersburg INS_HYM_0000285 [ZISP].\u2640, \u0412\u0435\u0440\u0445\u043d.[\u0438\u0439] \u0417\u0430\u0440\u0430\u0432\u0448.[\u0430\u043d]  // Paralectotypus Andr.turkestanica Mor., design. Osychnjuk, 1980 <red label> [ZISP]; 1 \u2640, 12.[V.1869] // \u0412\u0435\u0440\u0445\u043d. \u0417\u0430\u0440\u0430\u0432\u0448. // Andrenaturkestanica Mor. // Lectotypus Warncke 1975 <red label> // Paralectotypus Andrenaturkestanica Morawitz, 1876, design. Astafurova et al., 2022 <identical red labels on each paralectotype specimen> [ZMMU].Andrena (Melanapis) fuscosa Erichson, 1835 , Caucasus, Turkey, Syria, Lebanon, Israel, Jordan, Iran, Afghanistan, Pakistan, Central Asia, India.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff50.Morawitz, 1876EFE42609-8123-5324-8A1B-58479B7E0AD7Andrenaurmitana Morawitz, 1876: 165 (key), 175, \u2642.Urmetan (Tajikistan).Tajikistan: near Urmitan [Urmetan].39\u00b026'N, 68\u00b015'E] // Andrenaurmitana Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Holotypus Andrenaurmitana Mor., 1876 <red label> [ZMMU].\u2642, 2.[V.1869] // \u0423\u0440\u043c\u0438\u0442\u0430\u043d\u044a  // virescens Mor., \u2640 [handwritten by F. Morawitz] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Andr.virescens Mor., design. Osychnjuk, 1980 <red label> // Lectotypus Andrenavirescens Morawitz, 1876, design. Astafurova et al., 2022 <red label> // Zoological Institute St. Petersburg INS_HYM_0000254 [ZISP].\u2640, \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a  // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // Paralectotypus Andr.virescens Mor., design. Osychnjuk, 1980 <red label> [ZISP]; 1 \u2642, 27.[II.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // Andrenavirescens Mor. [handwritten by F. Morawitz] // Lectotypus Warncke, 1975 <red label>; 5 \u2640, 1 \u2642, 10., 23., 24.[III.1871], 8.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a; 1 \u2640, 2 \u2642, 27.[II.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a ; 1 \u2640, 22.[IV.1871] // \u041a\u0435\u043b\u0435\u0441\u044a // // Paralectotypus, Andrenavirescens Morawitz, 1876, design. Astafurova et al., 2022 <identical red labels on each paralectotype specimen> [ZMMU].Andrena (Poecilandrena) virescens Morawitz, 1876.Uzbekistan, Tajikistan.Taxon classificationAnimaliaHymenopteraAndrenidae\ufeff52.Morawitz, 1876ACA99AF2-42E2-566A-AEAA-5EDD94580840Andrenaviridigastra Morawitz, 1876: 164, 165 (key), 206, \u2640, \u2642.Tashkent (Uzbekistan).Uzbekistan: Tashkent, Samarkand.41\u00b018'N, 69\u00b016'E] // Andrenaviridigastra Mor. [handwritten by F. Morawitz] // Lectotypus Warncke 1975 <red label> // Lectotypus, Andrenaviridigastra Morawitz, 1876, design. ZMMU].\u2640, designated by . 1 \u2640, <golden circle>, 16.[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // viridigastra Mor., \u2640, Typ. [handwritten by F. Morawitz]; 2 \u2640, 27.[II.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a [Tashkent] // \u043a.[\u043e\u043b\u043b\u0435\u043a\u0446\u0438\u044f] \u0424. \u041c\u043e\u0440\u0430\u0432\u0438\u0446\u0430 [Collection of F. Morawitz] // viridigastra Mor. [handwritten by F. Morawitz]; 1 \u2642, <golden circle> // 16.[III.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a // viridigastra Mor. \u2642, Typ. [handwritten by F. Morawitz] // Paralectotypus Andr.viridigastra Mor., design. Osychnjuk, 1980 <identical red labels on each paralectotype specimen> [ZISP]; 20 \u2640, 14 \u2642, 28.[II.1871], 16., 23., 24., 25., 27.[III.1871], 5., 8., 10., 11.[IV.1871] // \u0422\u0430\u0448\u043a\u0435\u043d\u0442\u044a; 6 \u2640, 17 \u2642, 16., 18., 21., 25[III.1869] // \u0421\u0430\u043c\u0430\u0440\u043a\u0430\u043d\u0434\u044a // Paralectotypus, Andrenaviridigastra Morawitz, 1876, design. ZMMU].Andrena (Melandrena) viridigastra Morawitz, 1876.Turkmenistan, Uzbekistan, Tajikistan, Kazakhstan."}
{"text": "In this paper, we proposed a stochastic SVEI brucellosis model with stage structure by introducing the effect of environmental white noise on transmission dynamics of brucellosis. By Has'minskii theory and constructing suitable Lyapunov functions, we established sufficient conditions on the existence of ergodic stationary distribution for the considered model. Moreover, we also established sufficient condition for extinction of the disease. Finally, two examples with numerical simulations are given to illustrate the main results of this paper. Brucella, of which there are six species: B. abortus, B. melitensis, B. suis, B. ovis, B. canis, and B. neotomae  such that they are continuously twice differentiable in x and once in t. The differential operator L of equation , thenVt = \u2202V/\u2202t, Vx = ((\u2202V/\u2202x1), \u22ef, (\u2202V/\u2202xd)), Vxx = (\u22022V/\u2202xi\u2202xj)d\u00d7d. By It\u00f4's formula, if x(t) \u2208 \u211dd, thenIf Next, we present a result about the existence of stationary distribution (see Has'minskii ).X(t) be a homogeneous Markov process in Ed  and be described by the following stochastic differential equation:Let The diffusion matrix is defined as follows:X(t) has a unique ergodic stationary distribution \u03a0(\u00b7) if there exists a bounded domain D \u2282 Ed with regular boundary \u0393 andThe Markov process A1: there is a positive number M such that \u2211i,j=1daij(x)\u03bei\u03bej \u2265 M|\u03be|2, x \u2208 D, \u03be \u2208 \u211dd.A2: there exists a nonnegative C2-function V such that LV is negative for any Ed\\D. Thenx \u2208 Ed, where f(\u00b7) is a function integrable with respect to the measure \u03c0.In studying the dynamical behavior of an epidemic model, the first importance is whether the solution is global and positive. Hence, in the following theorem, we will study the existence and uniqueness of the global positive solution, which is a prerequisite for researching the long-term behavior of model .X0 = (S1(0), S2(0), V(0), E(0), I(0)) \u2208 \u211d+5, there is a unique solution X(t) = (S1(t), S2(t), V(t), E(t), I(t)) of system , S2(0), V(0), E(0), I(0)) \u2208 \u211d+5, there is a unique local solution (S1(t), S2(t), V(t), E(t), I(t)) on . For each integer n > n0, define the stopping time as follows:Since the coefficients of system  satisfy ion time . To show\u03c4n is increasing as n\u27f6\u221e. Let \u03c4\u221e = limn\u27f6\u221e\u03c4n, then \u03c4\u221e \u2264 \u03c4e a.s. In what follows, we need to verify \u03c4\u221e = \u221e a.s. If this assertion is violated, there is a constant T > 0 and an \u03b5 \u2208  such that \u2119{\u03c4\u221e \u2264 T} > \u03b5. As a result, there exists an integer n1 \u2265 n0 such thatThroughout this paper, we set inf\u2205 = \u221e . It is easy to see that C2-function V: \u211d+5\u27f6\u211d+ byDefine a Using ItBy applying the following invariant set of model  which is\u03b6 is positive constant which is independent of S1, S2, V, E, I, and t, we can getSince T\u2227\u03c4\u03b5 and taking expectations, then we can obtainIntegrating both sides  from 0 t\u03a9\u03b5 = {\u03c4\u03b5 \u2264 t} for n \u2265 n1 by (P(\u03a9n) \u2265 \u03b5. Notice that for every \u03c9 \u2208 \u03a9\u03b5, there is at least one of S1, S2, V, E, and I that equal either n or 1/n. Hence, S1, S2, V, E, and I are no less thanSet  \u2265 n1 by , P(\u03a9n) \u2265a\u2227b donates the minimum of a and b. In view of (\u03a9(\u03c9) is the indicator function of \u03a9n. Let n\u27f6\u221e leads to the contradictionConsequently, view of  and 23)(23)VS1\u03c4\u03b5\u03c4\u221e = \u221e a.s.Therefore, we must have The difference between model  and the Define a parameterR0s > 1, then system (\u03a0(\u00b7) and it has the ergodic property.Assume that n system  has a unS1(0), S2(0), V(0), E(0), I(0)) \u2208 \u211d+5), there is a unique global solution  \u2208 \u211d+5..In view of The diffusion matrix of system  is givenChoose A1 in Then the condition C2-function Q : \u211d+5\u27f6\u211d in the following from\u03c7 is a constant satisfying 0 < \u03c7 < 2\u03bc/\u03c312\u2228\u03c322\u2228\u03c332\u2228\u03c342\u2228\u03c352,M > 0 satisfies the following conditionConstruct a Uk = ((1/k), k) \u00d7 ((1/k), k) \u00d7 ((1/k), k) \u00d7 ((1/k), k) \u00d7 ((1/k), k). Furthermore, Q is a continuous function. Hence, Q must have a minimum point +5. Then we define a nonnegative C2-function V : \u211d+5\u27f6\u211d+ as follows:It is easy to check thatMaking use of It\u03bb is defined in , we, we45), D such that the condition A2 in \u03b5i > 0 are sufficiently small constants satisfying the following conditions:F, G, H, J, K, and L are positive constants which can be seen from , ,  \u2264 \u22121 on \u211d+5\\D, which is equivalent to proving it on the above ten domains.Next, we will show that S1, S2, V, E, I) \u2208 D1, one can get thatIf  \u2208 D2, we haveC is defined in , LV \u2264 \u22121 for any  \u2208 D2.In view of , we can S1, S2, V, E, I) \u2208 D3, one can see thatIf  \u2208 D4, one can see thatIf ,\u2009LV \u2264 \u22121 for any  \u2208 D4.In view of , we can S1, S2, V, E, I) \u2208 D5, one can see thatIf , LV \u2264 \u22121 for any  \u2208 D5.We can obtain that for sufficiently small S1, S2, V, E, I) \u2208 D6, one can see thatIf  \u2208 D7, one can see thatIf  \u2208 D8, one can see thatLV \u2264 \u22121\u2009on\u2009D8.If  \u2208 D9, we obtainIf  \u2208 D10, it follows thatIf  if R0s = \u03b71\u03b72\u03b2\u03b5\u03b8/((\u03bc1 + d + \u03b71 + (\u03c312/2))(\u03b8 + \u03bc2 + (\u03c322/2))(\u03bc2 + \u03b72 + (\u03c342/2))(\u03bc2 + c + (\u03c352/2))) > 1. Note that the expression of R0s coincide with the threshold R0 of the deterministic system , S2(t), V(t), E(t), I(t) be the solution of system (S1(0), S2(0), V(0), E(0), I(0)) \u2208 \u211d+5. If (\u03b2(\u03b5 + 1)(A\u03c4 + b)(\u03bc2 + \u03b72))/\u03bc\u03c4\u03b72 < ((\u03c352/2) + \u03bc2 + c)\u2227(\u03c342/2), then the disease I(t) will extinct exponentially with probability one, i.e., moreoverLet f system  with anyE + (\u03bc2 + \u03b72/\u03b72)I], we haveApplying Itt, and the fact that limt\u27f6\u221eBi(t)/t = 0, i = 4, 5 [Integrating the above inequality from 0 to i = 4, 5 , yields\u03c9 \u2208 \u03a9, \u2203T = T(\u03c9) such thatFor any \u03b2(\u03b5 + 1)(A\u03c4 + b)(\u03bc2 + \u03b72))/\u03bc\u03c4\u03b72 < ((\u03c352/2) + \u03bc2 + c)\u2227(\u03c342/2).\u03bek\u2009 are the Gaussian random variables which follow standard normal distribution N, and \u03c3i, 1 \u2264 i \u2264 5, are intensities of white noises.In this section, we will give two numerical examples to illustrate the main theoretical results obtained in this paper. The numerical simulation method can be found in , 22, 23.A = 45, \u03b2 = 0.05, \u03bc1 = 0.05, \u03bc2 = 0.06, \u03b5 = 3.2, \u03c4 = 0.01, b = 0.3, \u03b71 = 0.25, \u03b72 = 0.5, and \u03b8 = 1.1, \u03b4 = 0.01, c = 0.02, \u03c31 = 0.2, \u03c32 = 0.05, \u03c33 = 0.31, \u03c34 = 0.03, and \u03c35 = 0.02. It is clear that conditions of R0s = \u03b71\u03b72\u03b2\u03b5\u03b8/((\u03bc1 + d + \u03b71 + (\u03c312/2))(\u03b8 + \u03bc2 + (\u03c322/2))(\u03bc2 + \u03b72 + (\u03c342/2))(\u03bc2 + c + (\u03c352/2))) = 1.17080128We take parameters as S1(t), S2(t), V(t), E(t), I(t) are given in The histogram and the smoothing curves of the probability density functions of A = 1000, \u03b2 = 0.0001, \u03bc1 = 0.1, \u03bc2 = 0.25\u03b5 = 0.18, \u03c4 = 0.002, b = 1.5, \u03b71 = 1.06, \u03b72 = 3.4, and \u03b8 = 0.1, \u03b4 = 0.4, c = 0.05, \u03c31 = 0.2, \u03c32 = 0.5, \u03c33 = 0.31, \u03c34 = 1.65, and \u03c35 = 1.5. It is clear that conditions of We take parameters as S1(t), S2(t), V(t), E(t) and extinction of I(t) for stochastic model (S1(0), S2(0), V(0), E(0), I(0)) = .The curves on the persistence of ic model  are giveIn this paper, we consider the stochastic perturbations for deterministic model  and deriR0s > 1 and applying the theory of stochastic differential equations, Has'minskii theory, Ito's formula, and Lyapunov function method, we obtained some sufficient conditions on the existence of ergodic stationary distribution of model (In this paper, firstly, we have considered the stochastic perturbations for deterministic system  and estaof model . We alsoof model . Especia"}
{"text": "Adolescence can manifest different in norm and in illness. It\u2019s important to find common characteristics of adaptation with different types of ontogenesis, or leading manifestations of diseaseThree adolescence (boys&girls) sample: normal \u2013 22, middle age 16, cardio pathology \u2013 7, middle age 16, psychopathology \u2013 12, middle age 15Direct self-esteem by Dembo-Rubinstein (DR) test and indirect self-esteem by color attitude test by Etkind (CAT), Structure of Temperament Questionnaire (STQ-77), Buss-Perry Aggression Questionnaire (BPAQ).Significant differences  were obtained on scales BRAQ \u201cHostility\u201d , \u201cCommon aggression\u201d , STQ-77 \u201cPhysical Endurance\u201d , \u201cPhysical Tempo\u201d , \u201cSocial Endurance\u201d , \u201cSocial Tempo\u201d , \u201cPlasticity\u201d , \u201cSelf-confidence\u201d , \u201cNeuroticism\u201d ; gaps DR-CAT for scales \u201cHealth\u201d , \u201cHappiness\u201d . Pearson correlation coefficient between STQ-77, BRAQ and Gaps DR-CAT found in normal group: Gap DR-CAT \u201cHealth\u201d \u2013 STQ-77 \u201cPhysical Endurance\u201d , Gap DR-CAT \u201cSmart\u201d - STQ-77 \u201cIntellectual Endurance\u201d , Gap DR-CAT \u201cHappiness\u201d \u2013 BRAQ \u201cHostility\u201d , Gap DR-CAT \u201cHappiness\u201d - STQ-77 \u201cImpulsivity\u201d , \u201cNeuroticism\u201d . Correlation was founded in cardio pathology group: Gap DR-CAT \u201cSmart\u201d \u2013 BRAQ \u201cPhysical aggression\u201d , \u201cAnger\u201d , \u201cCommon aggression\u201d , Gap DR-CAT \u201cHappiness\u201d \u2013 BRAQ \u201cPhysical aggression\u201d , \u201cAnger\u201d , \u201cCommon Aggression\u201d . For psychopathology wasn\u2019t found correlations.Comparative study of personality traits of adolescents with different types of ontogenesis  is important for evaluating their adaptation and determining targets of psychotherapeutic work."}
{"text": "Covariance estimation is essential yet underdeveloped for analysing multivariate functional data. We propose a fast covariance estimation method for multivariate sparse functional data using bivariate penalized splines. The tensor\u2010product B\u2010spline formulation of the proposed method enables a simple spectral decomposition of the associated covariance operator and explicit expressions of the resulting eigenfunctions as linear combinations of B\u2010spline bases, thereby dramatically facilitating subsequent principal component analysis. We derive a fast algorithm for selecting the smoothing parameters in covariance smoothing using leave\u2010one\u2010subject\u2010out cross\u2010validation. The method is evaluated with extensive numerical studies and applied to an Alzheimer's disease study with multiple longitudinal outcomes. Let x(k)k=1,\u2026,p be a set of p random functions with each function in L2(T). Assume that the p\u2010dimensional vector x(t)=x(1),\u2026,x(p)\u22a4\u2208Rp has a p\u2010dimensional smooth mean function,\u03bc(t)=E{x(t)}=Ex(1)(t),\u2026,Ex(p)(t)\u22a4=\u03bc(1)(t),\u2026,\u03bc(p)(t)\u22a4. Define the covariance function asC=E(x(s)\u2212\u03bc(s))(x(t)\u2212\u03bc(t))\u22a4=Ckk\u20321\u2264k,k\u2032\u2264p and Ckk'=Covx(k)(s),x(k')(t). Then, the covariance operator \u0393:H\u2192H associated with the kernel C can be defined such that for any f\u2208H, the kth element of \u0393f is given by Ck=,\u2026,Ckp)\u22a4. Note that \u0393 is a linear, self\u2010adjoint, compact, and non\u2010negative integral operator. By the Hilbert\u2013Schmidt theorem, there exists a set of orthonormal bases {\u03a8\u2113}\u2113\u22651\u2208H, \u03a8\u2113=\u03a8\u2113(1),\u2026,\u03a8\u2113(p)\u22a4, and <\u03a8\u2113,\u03a8\u2113\u2032>H=\u2211k=1p\u222b\u03a8\u2113(k)(t)\u03a8\u2113\u2032(k)(t)dt=1\u2113=\u2113\u2032, such that d\u2113 is the \u2113th largest eigenvalue corresponding to \u03a8\u2113. Then, the multivariate Mercer's theorem gives Ckk\u2032=\u2211\u2113=1\u221ed\u2113\u03a8\u2113(k)(s)\u03a8\u2113(k\u2032)(t). As shown in Saporta (x(t) has the multivariate Karhunen\u2013Lo\u00e8ve representation,x(t)=\u03bc(t)+\u2211\u2113=1\u221e\u03be\u2113\u03a8\u2113(t), where \u03be\u2113=<x\u2212\u03bc,\u03a8\u2113>H are the scores with E(\u03be\u2113)=0 and E(\u03be\u2113\u03be\u2113\u2032)=d\u21131\u2113=\u2113\u2032. The covariance operator \u0393 has the positive semidefiniteness property; that is, for any a=\u22a4\u2208Rp, the covariance function of a\u22a4x, denoted by Ca, satisfies that for any sets of time points \u2282T with an arbitrary positive integer q, the square matrix [Ca]{1\u2264i,j\u2264q}\u2208Rq\u00d7q is positive semidefinite.Let  Saporta , x(t) ha2.2yij(k),tij(k):i=1,\u2026,n;k=1,\u2026,p;j=1,\u2026,mik, where tij(k)\u2208T is the observed time point, yij(k) is the observed kth response, n is the number of subjects, and mik is the number of observations for subject i's kth response. The model is xi(t)=xi(1)(t),\u2026,xi(p)(t)\u22a4\u2208H, \u03f5ij(k) are random noises with zero means and variances \u03c3k2 and are independent across i,j, and k.Suppose that the observed data take the form Ckk\u2032. We adopt a three\u2010step procedure. In the first step, empirical estimates of the covariance functions are constructed. Let rij(k)=yij(k)\u2212\u03bc(k)tij(k) be the residuals and Cij1j2(kk\u2032)=rij1(k)rij2(k\u2032) be the auxiliary variables. Note thatECij1j2(kk\u2032)=Ckk\u2032tij1(k),tij2(k\u2032)+\u03c3k21{k=k\u2032,j1=j2} for 1\u2009\u2264\u2009j1\u2009\u2264\u2009mik,1\u2009\u2264\u2009j2\u2009\u2264\u2009mik\u2032. Thus, Cij1j2(kk\u2032) is an unbiased estimate of Ckk\u2032tij1(k),tij2(k\u2032) whenever k\u2260k\u2032 or j1\u2260j2. In the second step, the noisy auxiliary variables are smoothed to obtain smooth estimates of the covariance functions. For smoothing, we use bivariate P\u2010splines . Let r^ij(k)=yij(k)\u2212\u03bc^(k)tij(k) and \u0108ij1j2(kk\u2032)=r^ij1(k)r^ij2(k\u2032), the actual auxiliary variables.The goal is to estimate the covariance functions P\u2010splines model Ckk\u2032 uses tensor\u2010product splines Gkk\u2032 for 1\u2009\u2264\u2009k,k\u2032\u2009\u2264\u2009p. Specifically, Gkk\u2032=\u22111\u2264\u03b31,\u03b32\u2264c\u03b8\u03b31\u03b32(kk\u2032)B\u03b31(s)B\u03b32(t), where \u0398kk\u2032=\u03b8\u03b31\u03b32(kk\u2032)1\u2264\u03b31,\u03b32\u2264c\u2208Rc\u00d7c is a coefficient matrix, {B1(\u00b7),\u2026,Bc(\u00b7)} is the collection of B\u2010spline basis functions in T, and c is the number of equally spaced interior knots plus the order (degree plus 1) of the B\u2010splines. Because Ckk\u2032=Ck\u2032k=Covx(k)(s),x(k\u2032)(t), it is reasonable to impose the assumption that Gkk\u2032=Gk\u2032k. Therefore, in the rest of the section, we consider only k\u2009\u2264\u2009k\u2032.The bivariate D\u2208R(c\u22122)\u00d7c denote a second\u2010order differencing matrix such that for a vector a=\u22a4\u2208Rc,Da=\u22a4\u2208Rc\u22122. Also let \u2016\u00b7\u2016F be the Frobenius norm. For the cross\u2010covariance function Ckk\u2032 with k<k\u2032, the bivariate P\u2010splines estimate the coefficient matrix \u0398kk\u2032 by \u0398^kk\u2032, which minimizes the penalized least squares \u03bbkk\u20321 and \u03bbkk\u20322 are two non\u2010negative smoothing parameters that balance the model fit and smoothness of the estimate and will be determined later. Indeed, the column penalty \u2016D\u0398kk\u2032\u2016F2 penalizes the second\u2010order consecutive differences of the columns of \u0398kk\u2032 and similarly, the row penalty \u2016D\u0398kk\u2032\u22a4\u2016F2 penalizes the second\u2010order consecutive differences of the rows of \u0398kk\u2032. The two penalty terms are essentially penalizing the second\u2010order partial derivatives of Gkk\u2032 along the s and t directions. The two smoothing parameters are allowed to differ to accommodate different levels of smoothing along the two directions.Let Ckk with k=1,\u2026,p, we conduct bivariate covariance smoothing by enforcing the following constraint on the coefficient matrix \u0398kk  is a symmetric function. Then, the coefficient matrix \u0398kk and the error variance \u03c3k2 are jointly estimated by \u0398^kk and \u03c3^k2, which minimize the penalized least squares \u0398kk and \u03bbk is a smoothing parameter. Note that the two penalty terms in Equation\u00a0\u0398kk is symmetric, and thus, only one smoothing parameter is needed for auto\u2010covariance estimation.It follows that 2.2.1k and k\u2032 with k\u2009\u2264\u2009k\u2032. Let \u03b8kk\u2032=vec(\u0398kk\u2032)\u2208Rc2 be a vector of the coefficients and b(t)={B1(t),\u2026,Bc(t)}\u22a4\u2208Rc denotes the B\u2010spline base. Then, \u0108ij1j2(kk\u2032) for each pair of k and k\u2032. Let ri(k)=ri1(k),\u2026,rimik(k)\u22a4\u2208Rmik, C^i(kk\u2032)=ri(k)\u2297ri(k\u2032)\u2208Rmikmik\u2032 and C^(kk\u2032)=C^1(kk\u2032),\u22a4,\u2026,C^n(kk\u2032),\u22a4\u22a4\u2208RNkk\u2032, where Nkk\u2032=\u2211i=1nmikmik\u2032 is the total number of auxiliary responses for the pair of k and k\u2032. As for the B\u2010splines, let bi(k)=bti1(k),\u2026,btimik(k)\u2208Rc\u00d7mik, Bi(kk\u2032)=bi(k\u2032)\u2297bi(k)\u22a4\u2208R(mikmik\u2032)\u00d7c2, and B(kk\u2032)=B1(kk\u2032),\u22a4,\u2026,Bn(kk\u2032),\u22a4\u22a4\u2208RNkk\u2032\u00d7c2.We first introduce the notation. Let vec(\u00b7) be an operator that stacks the columns of a matrix into a column vector and denote by \u2297 the Kronecker product. Fix Ckk\u2032 with k<k\u2032, the penalized least squares in Equation\u00a0P1=Ic\u2297D\u22a4D and P2=D\u22a4D\u2297Ic. The expression in Equation\u00a0\u03b8kk\u2032. Therefore, we derive that Ckk\u2032 is \u0108kk\u2032={b(t)\u2297b(s)}\u22a4\u03b8^kk\u2032.For estimation of the cross\u2010covariance functions \u03c7k\u2208Rc(c+1)/2 be a vector obtained by stacking the columns of the lower triangle of \u0398kk, and let Gc\u2208Rc2\u00d7c(c+1)/2 be a duplication matrix such that \u03b8kk=Gc\u03b7k =vec(Imik)\u2208Rmik2 and Z(k)=Z1(k),\u22a4,\u2026,Zn(k),\u22a4\u22a4\u2208RNkk. Finally, let \u03b2k=\u03c7k\u22a4,\u03c3k2\u22a4\u2208Rc\u02dc with c\u02dc=c(c+1)/2+1. It follows that the penalized least squares in Equation\u00a0X(k)=B(kk),Z(k)\u2208RNkk\u00d7c\u02dc and Q=blockdiagGc\u22a4(Ic\u2297DD\u22a4)Gc\u22a4,0\u2208Rc\u02dc\u00d7c\u02dc. Therefore, we obtain \u03b8^kk=Gc\u03c7^k, and the estimate of the auto\u2010covariance function Ckk is \u0108kk={b(t)\u2297b(s)}\u22a4\u03b8^kk.Let The above estimates of covariance functions may not lead to a positive semidefinite covariance operator and thus have to be refined. We pool all estimates together, and we shall use the following proposition.Proposition 1Ckk\u2032=b(s)\u22a4\u0398kk\u2032b(t). Let G=\u222bb(t)b(t)\u22a4dt\u2208Rc\u00d7c and assume that G is positive definite ,\u22a4,\u2026,u\u2113(p),\u22a4\u22a4\u2208Rpc is the associated eigenvector with u\u2113(k)\u2208Rc and such that \u03a8\u2113(k)(t)=b(t)\u22a4G\u221212u\u2113(k).Assume that \u03a8\u2113(k)(t) are linear combinations of the B\u2010spline basis functions, which means that they can be straightforwardly evaluated, an advantage of spline\u2010based methods compared with other smoothing methods for which eigenfunctions are approximated by spectral decompositions of the covariance functions evaluated at a grid of time points.The proof is provided in Appendix A. Proposition \u0398^kk\u2032, the estimate of the coefficient matrix \u0398kk\u2032, the spectral decomposition of [G12\u0398^kk\u2032G12]kk\u2032 gives us estimates d^\u2113 and u^\u2113=u^\u2113(1),\u22a4,\u2026,u^\u2113(p),\u22a4\u22a4. We discard negative d^\u2113 to ensure that the multivariate covariance operator is positive semidefinite, and this leads to a refined estimate of the coefficient matrix \u0398kk\u2032, \u0398\u02dckk\u2032=G\u221212\u2211\u2113:d^\u2113>0d^\u2113u^\u2113(k)u^\u2113(k\u2032),\u22a4G\u221212. Then, the refined estimate of the covariance functions is C\u02dckk\u2032=b(s)\u22a4\u0398\u02dckk\u2032b(t). Proposition \u03a8\u02dc\u2113(k)(t)=b(t)\u22a4G\u221212u^\u2113(k).Once we have ion of G2\u0398^kk\u2032G12For principal component analysis or curve prediction in practice, one may select further the number of principal components by either the proportion of variance explained (PVE) \u22121B\u22a4\u2208RN\u00d7N, Si=Bi(B\u22a4B+\u03bb1P1+\u03bb2P2)\u22121B\u22a4\u2208Rmi2\u00d7N, and Sii=Bi(B\u22a4B+\u03bb1P1+\u03bb2P2)\u22121Bi\u22a4\u2208Rmi2\u00d7mi2.We shall also now suppress the subscript (Imi2\u2212Sii)\u22122=Imi2+2Sii, which results in the generalized cross\u2010validation, denoted by iGCV, Then, a shortcut formula for Equation\u00a0nd Huang  and Xiaond Huang , the iCVS is of dimension 2,500\u00d72,500 if n=100 and mi=m=5 for all i. Thus, we need to further simplify the formula.Although iGCV is much easier to compute than iCV, the formula in Equation\u00a0Gn=B\u22a4B, B\u02dc=BGn\u22121/2\u2208RN\u00d7c2, B\u02dci=BiGn\u22121/2\u2208Rmi2\u00d7c2, f=B\u02dc\u22a4C^\u2208Rc2, fi=B\u02dci\u22a4C^i\u2208Rc2, and Li=B\u02dci\u22a4B\u02dci\u2208Rc2\u00d7c2. Also let P\u02dc1=Gn\u22121/2P1Gn\u22121/2\u2208Rc2\u00d7c2, P\u02dc2=Gn\u22121/2P2Gn\u22121/2\u2208Rc2\u00d7c2, and \u2211=Ic2+\u03bb1P\u02dc1+\u03bb2P\u02dc2. Then, Equation\u00a0Let \u03a3 has two smoothing parameters. Following Wood (\u2211=I+\u03c1{wP\u02dc1+(1\u2212w)P\u02dc2}, where \u03c1=\u03bb1+\u03bb2 represents the overall smoothing level and w=\u03bb1\u03c1\u22121\u2208 is the relative weight of \u03bb1. We conduct a two\u2010dimensional grid search of , as follows. For a given w, let Udiag(s)U\u22a4 be the eigendecompsition of wP\u02dc1+(1\u2212w)P\u02dc2, where U\u2208Rc2\u00d7c2 is an orthonormal matrix and s=\u2208Rc2 is the vector of eigenvalues. Then, \u2211\u22121=Udiag(d\u02dc)U\u22a4 with d\u02dc=1/(1+\u03c1s)\u2208Rc2.Note that ing Wood , we use Proposition 2f\u02dci=U\u22a4fi\u2208Rc2, f\u02dc=U\u22a4f\u2208Rc2, g=f\u02dc\u2299f\u02dc\u2212\u2211i=1nf\u02dci\u2299f\u02dci\u2208Rc2, L\u02dci=U\u22a4LiU\u2208Rc2\u00d7c2, and F=\u2211i=1n(f\u02dcif\u02dc\u22a4)\u2299L\u02dci\u2208Rc2\u00d7c2.Let \u2299 stand for the point\u2010wise multiplication. Then, w, note that only d\u02dc depends on \u03c1 and needs to be calculated repeatedly, and all other terms need to be calculated only once. The entire algorithm is presented in Algorithm 1. We give an evaluation of the complexity of the proposed algorithm. Assume that mi=m for all i. The first initialization (Step 1) requires O(nm2c2+nc4+c6) computations. For each w, the second initialization (Step 2) also requires O{nc4min+c6} computations. For each \u03c1, Steps 3\u20138 require O(nc4) computations. Therefore, the formula in Proposition mis are small.The proof is provided in Appendix A. For each 2.3xi(t) is generated from a multivariate Gaussian process. Suppose that we want to predict the ith multivariate response xi(t) at {si1,\u2026,sim} for m\u2009\u2265\u20091. Let yi(k)=yi1(k),\u2026,yimik(k)\u22a4 be the vector of observations at ti1(k),\u2026,timik(k) for the kth response. Let \u03bci(k),o=\u03bc(k)ti1(k),\u2026,\u03bc(k)timik(k)\u22a4 be the vector of the kth mean function at the observed time points. Let yi=yi(1),\u22a4,\u2026,yi(p),\u22a4\u22a4 and \u03bcio=\u03bci(1),o,\u22a4,\u2026,\u03bci(p),o,\u22a4\u22a4. Let \u03bcin=\u03bc(1)(si1),\u2026,\u03bc(1)(sim),\u2026,\u03bc(p)(si1),\u2026,\u03bc(p)(sim)\u22a4 be the vector of mean functions at the time points for prediction.For prediction, assume that the smooth curve bi(k),o=bti1(k),\u2026,btimik(k)\u22a4 and bin=bsi1,\u2026,bsim\u22a4. Next, let Bio=blockdiagbi(1),o,\u2026,bi(p),o and Bin=Ip\u2297bin. Then, Cov(yi) given by Bio\u0398Bio,\u22a4+blockdiag, and Cov(xi) and Cov are given by Bin\u0398Bin,\u22a4 and Bio\u0398Bin,\u22a4, respectively. Let \u0398\u02dc=\u0398\u02dckk\u20321\u2264k,k\u2032\u2264p\u2208Rpc\u00d7pc. Plugging in the estimates, we predict xi by \u03bc^io=\u03bc^(1)ti1(1),\u2026,\u03bc^(1)timi1(1),\u2026,\u03bc^(p)ti1(p),\u2026,\u03bc^(p)timip(p)\u22a4 is the estimate of \u03bcio, \u03bc^in=\u03bc^(1)(si1),\u2026,\u03bc^(1)(sim),\u2026,\u03bc^(p)(si1),\u2026,\u03bc^(p)(sim)\u22a4 is the estimate of \u03bcin, V^i=Bio\u0398\u02dcBio,\u22a4+blockdiag\u03c3^12Imi1,\u2026,\u03c3^p2Imip. An approximate covariance matrix for x^i is % point\u2010wise confidence interval for the kth response is given by Var^xi(k)(sij)|yi can be extracted from the diagonal of Cov^(xi|yi).It follows that L\u2009\u2265\u20091 scores \u03bei=\u22a4 for the ith subject. Note that \u03bei\u2113=\u222b\u03a8\u2113(t)\u22a4{xi(t)\u2212\u03bc(t)}dt. With a similar derivation as above, xi(t)\u2212\u03bc(t) can be predicted by Ip\u2297b(t)\u22a4\u0398\u02dcBio,\u22a4V^i\u22121(yi\u2212\u03bc^io). By Proposition \u03a8\u2113(k)(t) are estimated by b(t)\u22a4G\u221212u^\u2113(k), and thus, \u03a8\u2113(t)\u22a4=u^\u2113\u22a4{Ip\u2297G\u221212b(t)}. It follows that Finally, we predict the first 3We evaluate the finite sample performance of the proposed method (denoted by mFACEs) against mFPCA via a synthetic simulation study and a simulation study mimicking the Alzheimer's Disease Neuroimaging Initiative (ADNI) data in the real data example. Here, we report the details and results of the former as the conclusions remain the same for the latter, and details are provided in the 3.1p=3 responses. The mean functions are \u03bc(t)=\u22a4. We first specify the auto\u2010covariance functions. Let \u03a61(t)=2sin(2\u03c0t),2cos(4\u03c0t),2sin(4\u03c0t)\u22a4, \u03a62(t)=2cos(\u03c0t),2cos(2\u03c0t),2cos(3\u03c0t)\u22a4, and\u03a63(t)=2sin(\u03c0t),2sin(2\u03c0t),2sin(3\u03c0t)\u22a4. Also let Ckk=\u03a6k(s)\u22a4\u039bkk\u03a6k(t),k=1,2,3. For the cross\u2010covariance functions, letCkk\u2032=\u03c1\u03a6k(s)\u22a4\u039bkk12\u039bk\u2032k\u203212\u03a6k\u2032(t) for k\u2260k\u2032, where \u03c1\u2208 is a parameter to be specified. The induced covariance operator from the above specifications is proper; see Lemma \u03c1kk\u2032=Ckk\u2032/CkkCk\u2032k\u2032 is bounded by \u03c1. Hence, \u03c1 controls the overall level of correlation between responses: if \u03c1=0, then the responses are uncorrelated from each other.We generate data by model\u00a0 (2\u03c0t),5t\u22122]\u22a4. We f\u2113=1,\u2026,9, we simulate the scores \u03bei\u2113 from N, where d\u2113 are the induced eigenvalues. Next, we simulate the white noises \u03f5ij(k) fromN, where \u03c3\u03f52 is determined according to the signal\u2010to\u2010noise ratio SNR=\u2211\u2113d\u2113/(p\u03c3\u03f52). Here, we let SNR=2. For each response, the sampling time points are drawn from a uniform distribution in the unit interval, and the number of observations for each subject, mik, is generated from a uniform discrete distribution on {3,4,5,6,7}. Thus, the sampling points vary not only from subject to subject but also across responses within each subject.The eigendecomposition of the multivariate covariance function gives nine nonzero eigenvalues with associated multivariate eigenfunctions; hence, for n and the correlation parameter \u03c1. We let n=100,200, or 400. We let \u03c1=0.5, which corresponds to a weak correlation between responses as the average absolute correlation between responses is only 0.36. Another value of \u03c1 is 0.9, which corresponds to a moderate correlation between responses as the average absolute correlation between responses is about 0.50.We use a factorial design with two factors: the number of subjects In total, we have six model conditions, and for each model condition, we generate 200 datasets. To evaluate the prediction accuracy of the various methods, we draw 200 additional subjects as testing data. The true correlation functions and a sample of the simulated data are shown in the \u0108kk\u2032 be an estimate of Ckk\u2032, and then RISE are given by \u2113th eigenfunction, we use the integrated square errors (ISE), which are defined as d^\u2113/d\u2113.We compare mFACEs and mFPCA in terms of estimation accuracy of the covariance functions, the eigenfunctions and eigenvalues, and prediction of new subjects. For covariance function estimation, we use the relative integrated square errors (RISE). Let For predicting new curves, we use the mean integrated square errors (MISE), which are given by 3.2\u03c1=0.9), the advantage of mFACEs is substantial even for the small sample size n=100.Figure % of the total variation in the functional data for \u03c1=0.5, and it is 80% for \u03c1=0.9. Figure Figures R package face . We define the relative efficiencies of different methods as the ratios of MISEs with respect to those of univariate FPCA; see Figure In summary, mFACEs shows competing performance against alternative methods.4http://ida.loni.ucla.edu/.The ADNI is a two\u2010stage longitudinal observational study launched in year 2003 with the primary goal of investigating whether serial neuroimages, biological markers, clinical and neuropsychological assessments can be combined to measure the progression of AD =Ckk\u2032/CkkCk\u2032k\u2032. The plot indicates two groups of biomarkers, ADAS\u2010Cog 13 and FAQ, in one group and RAVLT.imme, RAVLT.learn, and MMSE in another group. The biomarkers within the groups are positively correlated and negatively correlated between groups, which makes sense as high values of ADAS\u2010Cog 13 and FAQ and low values for the other biomarkers suggest of AD. Next, we display in Figure % and 11% of the total variance in the functional part of the data. The eigenfunctions reveal how the five biomarkers covary and how a subject's trajectories of biomarkers deviate from the population mean. Indeed, we see from Figure % point\u2010wise confidence bands for three subjects. We focus on predicting the trajectories over the first 4\u00a0years as there are more observations. We can see that the confidence bands are getting wider at the later time points because of fewer observations.We analyse the five longitudinal biomarkers using mFACEs. For better visualization, we plot in Figure 4.2\u0177ij(k) is the predicted value of the kth biomarker for the ith subject at time tij(k). We conduct two types of validation: an internal validation and an external validation. For the internal validation, we perform a 10\u2010fold cross\u2010validation to the combined data of ADNI\u20101 and ADNI\u20102. For the external validation, we fit the model using only the ADNI\u20101 data and then predict ADNI\u20102 data. Figure We compare the proposed mFACEs with mFPCA and mFPCA(CE) for predicting the five longitudinal biomarkers. The prediction performance is evaluated by the average squared prediction errors (APE), 5The prevalence of multivariate functional data has sparked much research interests in recent years. However, covariance estimation for multivariate sparse functional data remains underdeveloped. We proposed a new method, mFACEs, and its features include the following: (a) A covariance smoothing framework is proposed to tackle multivariate sparse functional data; (b) an automatic and fast fitting algorithm is adopted to ensure the scalability of the method; (c) eigenfunctions and eigenvalues can be obtained through a one\u2010time spectral decomposition, and eigenfunctions can be easily evaluated at any sampling points; and (d) a multivariate extension of the conditional expectation approach \u22121/2 as in Chiou et\u00a0al. dt\u22121/2 as in Happ and Greven . As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this paper. A complete listing of ADNI investigators can be found at: http://adni.loni.usc.edu/wp\u2010content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf.Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database (mfaces (https://github.com/cli9/mfaces) contains an open\u2010source implementation of the proposed method described in the article. Demo zip file contains codes to demonstrate the proposed method with a simulated dataset in Section 3.The Supporting info itemClick here for additional data file."}
{"text": "Scientific Reports 10.1038/s41598-018-24153-0, published online 12 April 2018Correction to: The original version of this Article contained errors in Equation\u00a07, which was incorrectly given as:The correct Equation\u00a07 appears below.As a result of the changes to Equation\u00a07, the Abstract,\u22125\u00a0RI unit (RIU) obtained in Zone II for a 1-cm sensor length.\u201d\u201cThe sensors can be employed over a very wide dynamic RI range from 1.316 to over 1.608 at a wavelength of 1550\u2009nm, with the best resolution of 2.2447\u2009\u00d7\u200910now reads:\u22125\u00a0RI unit (RIU) obtained in Zone II for a 1-cm sensor length.\u201d\u201cThe sensors can be employed over a very wide dynamic RI range from 1.316 to over 1.608 at a wavelength of 1550\u2009nm, with the best resolution of 2.2406\u2009\u00d7\u200910Additionally, in Figure\u00a05a, the unit of fiber diameter \u201c\u03bcm\u201d was incorrectly given as \u201cmm\u201d.The original Figure\u00a0In the final paragraph of the Discussion,\u22125\u00a0RIU and 3.2634\u2009\u00d7\u200910\u22125\u00a0RIU, respectively, compared to the other two Zones. For Zone I, the best resolution is achieved by the 4-cm long sensor with a minimum detection level of 1.5438\u2009\u00d7\u200910\u22123\u00a0RIU while the 1-cm and 2.5-cm sensors are capable of resolutions of 5.1952\u2009\u00d7\u200910\u22123\u00a0RIU and 1.7462\u2009\u00d7\u200910\u22123\u00a0RIU, respectively.\u201d\u201cIt is also in this Zone that the 2.5-cm and 4-cm sensors have the best relative resolutions of 2.9919\u2009\u00d7\u200910now reads:\u22125\u00a0RIU and 3.2517\u2009\u00d7\u200910\u22125\u00a0RIU, respectively, compared to the other two Zones. For Zone I, the best resolution is achieved by the 4-cm long sensor with a minimum detection level of 1.6116\u2009\u00d7\u200910\u22123\u00a0RIU while the 1-cm and 2.5-cm sensors are capable of resolutions of 5.5905\u2009\u00d7\u200910\u22123\u00a0RIU and 1.7528\u2009\u00d7\u200910\u22123\u00a0RIU, respectively.\u201cIt is also in this Zone that the 2.5-cm and 4-cm sensors have the best relative resolutions of 2.9847\u2009\u00d7\u200910In the second paragraph of the Conclusions,\u22125\u00a0RIU is achieved for the 1-cm sensor.\u201d\u201cFor Zone II, the best sensor resolution of 2.2447\u2009\u00d7\u200910now reads:\u22125\u00a0RIU is achieved for the 1-cm sensor.\u201d\u201cFor Zone II, the best sensor resolution of 2.2406\u2009\u00d7\u200910Lastly, as a result of the changes to Equation\u00a07, the data in Table S1, S2, S3, S5, S6 and S7 in the Supplementary Information was incorrect.The original The original Article and accompanying Supplementary Information file has now been corrected.Supplementary Information."}
{"text": "Scientific Reportshttps://doi.org/10.1038/s41598-019-54197-9, published online 28 November 2019Correction to: \u03c3 is conductivity, R is electrical resistance measured, A is cross-sectional area of the cylinder, and l is length of the cylinder. As a result, the electric conductivity values in the original version of the Article were about 100 times smaller than the correct values.The original version of this Article contained a calculation error in electric conductivity values. As illustrated in the Methods section, the inner diameter of the glass tube for the measurement was 1.0\u00a0mm. However, the authors incorrectly used 10.0\u00a0mm for the calculations. In addition, an incorrect equation Consequently, in the Abstract,2 carbon components, and exhibits a conductivity above 10\u22122 S cm\u22121 and a specific surface area of 650\u2009m2\u00a0g\u22121.\u201d\u201cThe BDND comprises BDD and spnow reads:2 carbon components, and exhibits a conductivity above 1 S cm\u22121 and a specific surface area of 650\u2009m2\u00a0g\u22121.\u201d\u201cThe BDND comprises BDD and spIn addition, in the Results and Discussion section, under the subheading \u2018Preparation of the boron-doped nanodiamond\u2019,\u22129 S cm\u22121 to 3.5\u2009\u00d7\u200910\u22124 S cm\u22121 with a deposition time of 1\u00a0h. The BDND prepared with 8\u00a0h deposition time exhibited a conductivity sufficient for an electrochemical electrode material (2.0\u2009\u00d7\u200910\u22122 S cm\u22121).\u201d\u201cThe electrical conductivity of the as-deposited BDND was increased rapidly from 1.8\u2009\u00d7\u200910now reads:\u22127 S cm\u22121 to 1.4\u2009\u00d7\u200910\u22122 S cm\u22121 with a deposition time of 1\u00a0h. The BDND prepared with 8\u00a0h deposition time exhibited a conductivity sufficient for an electrochemical electrode material (4.0 S cm\u22121).\u201d\u201cThe electrical conductivity of the as-deposited BDND was increased rapidly from 7.2\u2009\u00d7\u200910Finally, the y-axis scale and conductivity value in Figure 1 was incorrect. The original Figure\u00a0The original Article has been corrected."}
{"text": "Nature Communications 10.1038/s41467-021-22353-3, published online 23 April 2021.Correction to: The original version of this Article contained errors in inline Equations in the last paragraph of the \u201cSteering of GHZ states\u201d subsection, which incorrectly read:\u03c1\u2009=\u2009p|GHZ\u03d5N+1\u232a\u2329GHZ\u03d5N+1|\u2009+\u2009(1\u2212p)N+1, using the same measurements we obtain FQB|A \u2009\u2265\u2009p2N2/[p\u2009+\u2009 2(1\u2212p)/2N], 4Var QB|A \u2009\u2264\u2009(1\u2212p)N/2N\u2009+\u2009p(1\u2212p)N2. Whenever p \u226b\u20092N\u2212, the criterion witnesses steering.For a mixture The correct form should read:\u03c1\u2009=\u2009p|GHZ\u03d5N+1\u232a\u2329GHZ\u03d5N+1|\u2009+\u2009(1\u2212p)N+1, using the same measurements we obtain FQB|A \u2009\u2265\u2009p2N2/[p\u2009+\u2009 2(1\u2212p)/2N], 4Var QB|A \u2009\u2264\u2009(1\u2212p)N\u2009+ p(1\u2212p)N2. For large N, whenever p \u2a86\u20091/\u221aN, the criterion witnesses steering.For a mixture This has been corrected in the PDF and HTML versions of the Article.The original version of the\u00a0\u03c3z by Alice, Bob\u2019s conditional states are easily found to gived = 2N . With a \u03c3x measurement, (3) results inFor a measurement of p \u226b\u20091/d = 2N\u2212, we can neglect the terms involving d. Then FQB|A  \u2a86 pN2, VarQB|A  \u2a85 p(1\u2212p)N2< pN2 for p < 1.When The correct version reads instead\u03c3z by Alice, Bob\u2019s conditional states are easily found to giveFor a measurement of \u03c3x measurement, (3) results ind = 2N . When p \u226b\u20091/d = 2N\u2212, we can neglect the term involving d. Then FQB|A  \u2a86 pN2, and the differenceN > (1\u2212p)/p2. For large N, this condition approximates to p\u2009\u2a86\u20091/\u221aN.With a The HTML has been updated to include a corrected version of the\u00a0Updated\u00a0Supplementary\u00a0Information"}
{"text": "Similarity measures (SM) and correlation coefficients (CC) are used to solve many problems. These problems include vague and imprecise information, excluding the inability to deal with general vagueness and numerous information problems. The main purpose of this research is to propose an m-polar interval-valued neutrosophic soft set (mPIVNSS) by merging the m-polar fuzzy set and interval-valued neutrosophic soft set and then study various operations based on the proposed notion, such as AND operator, OR operator, truth-favorite, and false-favorite operators with their properties. This research also puts forward the concept of the necessity and possibility operations of mPIVNSS and also the m-polar interval-valued neutrosophic soft weighted average operator (mPIVNSWA) with its desirable properties. Cosine and set-theoretic similarity measures have been proposed for mPIVNSS using Bhattacharya distance and discussed their fundamental properties. Furthermore, we extend the concept of CC and weighted correlation coefficient (WCC) for mPIVNSS and presented their necessary characteristics. Moreover, utilizing the mPIVNSWA operator, CC, and SM developed three novel algorithms for mPIVNSS to solve the multicriteria decision-making problem. Finally, the advantages, effectiveness, flexibility, and comparative analysis of the developed algorithms are given with the prevailing techniques. Multicriteria decision-making (MCDM) is an essential condition for decision scientific discipline. The decision-maker should judge the choices stated by the diverse forms of distinguishing perspectives. Though, in quite a lot of situations, it is tough for someone to undertake it because of numerous uncertainties in the data. One is due to lack of expertise or ravishment of policies. Thus, to measure the given disadvantages and thinking tools, a succession of philosophies had been projected. Zadeh introduced the notion of the fuzzy set (FS)  to resolMaji et al.  expandedCorrelation plays a significant part in statistics as well as engineering science. The joint association of two variable quantities can be utilized to appraise the interdependency of the correlation qualitative analysis. In addition to using probabilistic strategies for noticeably pragmatic engineering science complications, you also can locate various boundaries to probabilistic methods. However, the bodily structure has numerous exceptions, the improvement is challenging, and it is difficult to obtain exact consequences. Thus, due to the wide variety of incomprehensible info, the consequences of probability theory are not able to provide professionals with suitable information. In addition, in natural world concerns, there is not any priggish reason out to deal along with distinguished statistical information. Due to the preliminary limitations, the outcomes of probability theory are not conducive to specialists. So, probability theory is not very adequate to resolve the insecurity explicit in the data. Several assessors around the world have prearranged and suggested different strategies to solve anxiety-related complications. Wang et al.  developeGerstenkorn and Ma\u0144ko  proposedIn the past few years, quite a lot of mathematicians have progressed numerous methodologies such as similarity measures, CC, and aggregation operators (AOs) along with their applications in DM. Harish  offered In this epoch, specialists believe that real life is touching the track of multipolarization. Therefore, there is no distrust that the multipolarization of information has played a significant part in the prosperity of many arenas of science and technology. In neurobiology, multipolar neurons in the brain accumulate a lot of info from other neurons. In the whole manuscript, the motivation for expanding and mixing this research work is gradually given. We prove that under any appropriate circumstances, different hybrid structures comprehending FS will be transformed into distinct privileges of mPIVNSS. This study will be the utmost versatile form that can be used to merge data in daily life complications. The organization of the current research is such as follows: some basic concepts are presented in In the following section, we recalled some fundamental concepts which help us to construct the structure of the following study.(See ).\ud835\udcb0 be a universe, and \ud835\udc9c be an NS on \ud835\udcb0 defined as \ud835\udc9c={<u, u\ud835\udc9c(u), v\ud835\udc9c(u), w\ud835\udc9c(u) > :u \u2208 \ud835\udcb0}, where u, v, w: \ud835\udcb0\u27f6]0\u2212, 1+ and 0 \u2264 u\u03b1(u)+v\u03b1(u)+w\u03b1(u) \u2264 3, for all \u03b1=1,2,\u2026, m; and u \u2208 \ud835\udcb0.Let (See ).\ud835\udcb0 be the universal set and \u2130 be the set of attributes concerning \ud835\udcb0. Let \ud835\udcab(\ud835\udcb0) be the power set of \ud835\udcb0 and \ud835\udc9c\u2286\u2130. A pair  is called an SS over \ud835\udcb0 and its mapping is given asLet It is also defined as(See ).\ud835\udcb0 be the universal set and \u2130 be the set of attributes concerning \ud835\udcb0. Let \ud835\udcab(\ud835\udcb0) be the set of neutrosophic sets over \ud835\udcb0 and \u2009\ud835\udc9c\u2286\u2130. A pair  is called an NSS over \ud835\udcb0, and its mapping is given asLet (See ).\ud835\udcb0 be a universal set; then, the interval-valued neutrosophic set can be expressed by the set \ud835\udc9c={u, (u\ud835\udc9c(u), v\ud835\udc9c(u), w\ud835\udc9c(u)) : u \u2208 \u2009\ud835\udcb0}, where u\ud835\udc9c, v\ud835\udc9c, and w\ud835\udc9c are truth, indeterminacy, and falsity membership functions for \ud835\udc9c, respectively, u\ud835\udc9c, v\ud835\udc9c, and w\ud835\udc9c\u2286 for each u \u2208 \ud835\udcb0, whereLet u \u2208 \ud835\udcb0, 0 \u2264 u\ud835\udc9c(u)+v\ud835\udc9c(u)+w\ud835\udc9c(u) \u2264 3, and IVN (\ud835\udcb0) represent the family of all interval-valued neutrosophic sets on \ud835\udcb0.For each point, (See ).\ud835\udcb0 be a universe of discourse and \u2130 be a set of attributes, and an m-polar neutrosophic soft set (mPNSS) \u2118\u211c over \ud835\udcb0 defined asu\u03b1(u), v\u03b1(u), and w\u03b1(u) represent the truthiness, indeterminacy, and falsity, respectively, u\u03b1(u), v\u03b1(u), w\u03b1(u)\u2286 and 0 0 \u2264 u\u03b1(u)+v\u03b1(u)+w\u03b1(u) \u2264 3, for all \u03b1=1,2,3,\u2026, m; e \u2208 \u2130 and u \u2208 \ud835\udcb0. Simply an m-polar neutrosophic number (mPNSN) can be expressed as \u2118={\u2329u\u03b1, v\u03b1, \u2009w\u03b1\u232a}, where 0 \u2264 u\u03b1+v\u03b1+w\u03b1 \u2264 3 and \u03b1=1,2,3,\u2026, m.Let (See ).\ud835\udcb0 be a universe of discourse and \u2130 be a set of attributes, and IVNSS \u2118\u211c over \ud835\udcb0 defined asu\u211c(u)=, v\u211c(u)=, and w\u211c(u)= represents the interval truthiness, indeterminacy, and falsity, respectively, u\u211c(u), v\u211c(u), w\u211c(u)\u2286 and 0 \u2264 u\u211c\ud835\udd32(u)+v\u211c\ud835\udd32(u)+w\u211c\ud835\udd32(u) \u2264 3, for each e \u2208 \u2130 and u \u2208 \ud835\udcb0.Let m, representing m criteria of the object, but mPFS cannot handle indeterminacy and falsity objects. NS bargains with truth, falsity, one any choice specifications containing indeterminacy, but are not able to deal with multicriteria, multiple sources, and multiple polarities information fusion of possible choices. To conquer this question, Deli et al. , v\u03b1(u)=, and w\u03b1(u)= represent the interval truthiness, indeterminacy, and falsity, respectively; u\u03b1(u), v\u03b1(u), w\u03b1(u)\u2286 and 0 \u2264 u\u03b1\ud835\udd32(u)+v\u03b1\ud835\udd32(u)+w\u03b1\ud835\udd32(u) \u2264 3 for all \u03b1=1,2,3,\u2026, m; e \u2208 \u2130 and u \u2208 \ud835\udcb0. Simply, an m-polar interval-valued neutrosophic soft number (mPIVNSN) can be expressed as u\u03b1\ud835\udd32(u)+v\u03b1\ud835\udd32(u)+w\u03b1\ud835\udd32(u) \u2264 3 and \u03b1=1,2,3,\u2026, m.Let \u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, \u2118\u211c is called an m-polar interval-valued neutrosophic soft subset of \u2118\u2112, if\u03b1=1,2,3,\u2026, m; e \u2208 \u2130 and u \u2208 \ud835\udcb0.Let \u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, \u2118\u211c=\u2118\u2112, if\u03b1=1,2,3,\u2026, m; e \u2208 \u2130 and u \u2208 \ud835\udcb0.Let \u2118\u211c be an mPIVNSS over \ud835\udcb0. Then, empty mPIVNSS can be represented as Let \u2118\u211c be an mPIVNSS over \ud835\udcb0. Then, universal mPIVNSS can be represented as Let \u2118\u211c be an mPIVNSS over \ud835\udcb0. Then, the complement of mPIVNSS is defined as follows:Let \u2118\u211c be an mPIVNSS over \ud835\udcb0, thenIf LetThen, by using Similarly, we can prove 2 and 3.\u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then,Let \u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then,Let \u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then,Let As we know,By using Now, by using Now,By using Hence,Similar to assertion 1.\u2118\u211c, \u2118\u2112, and \u2118\u210b be three mPIVNSSs over \ud835\udcb0. Then,Let As we know,Hence,Similarly, we can prove other results.\u2118R and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, their extended union is defined asLet \ud835\udcb0={u1, u2} be a universe of discourse and E={e1, e2, e3, e4} be a set of attributes, and \u211c={e1, e2} and \u2112=Unsupported{e3, e4}\u2286E. Consider 3-PIVNSSs \u2118\u211c and \u2118\u2112 over \ud835\udcb0 can be represented as follows:Assume Then,\u2118R and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, their extended intersection is defined asLet Generally, if \u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, their difference is defined as follows:Let \u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, their addition is defined as follows:Let \u2118\u211c be an mPIVNSS over \ud835\udcb0. Then, its scalar multiplication is represented as \u2118\u211c. Let \u2118\u211c be the mPIVNSS over \ud835\udcb0. Then, its scalar division is represented as Let \u2118\u211c be an mPIVNSS over \ud835\udcb0. Then, the truth-favorite operator on \u2118\u211c is denoted by Let \u2118\u211c be an mPIVNSS over \ud835\udcb0. Then, the false-favorite operator on \u2118\u211c is denoted by Let \u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, their AND operator is represented by \u2118\u211c\u2227\u2118\u2112 and defined as follows:Let \u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, their OR operator is represented by \u2118\u211c\u2228\u2118\u2112 and defined as follows:Let Reconsider \u2118\u211c be an mPIVNSS. Then, necessity operation on mIVPNSS is represented by \u2295\u2118\u211c and defined as follows:Let \u2118\u211c be the mPIVNSS over \ud835\udcb0. Then, possibility operation on mIVPNSS is represented by \u2297\u2118\u211c and defined as follows:Let \u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then,Let \ud835\udcb0.As we know,\u2118\u211c\u222a\u03f5\u2118\u2112=\u2118\u210b:Let By using \u2118\u2112\u222a\u03f5 \u2295 \u2118\u211c=\u2135, where \u2295\u2118\u211c and \u2295\u2118\u2112 are given as follows by using the definition of necessity operation.Assume \u2295By using \u2118\u211c\u2009\u222a\u03f5\u2009\u2118\u2112)=\u2295\u2118\u2112\u222a\u03f5 \u2295 \u2118\u211c.Therefore,  represents the correlation among them. Then, the subsequent estates hold.\ud835\udc9emPIVNSS=\u03c2mPIVNSS(\u2118\u211c)\ud835\udc9emPIVNSS=\u03c2mPIVNSS(\u2118\u2112)Let The proof is trivial.\u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, the CC between them is given as \u03b4mPIVNSS and can be stated as follows:Let \u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, CC between them satisfies the following properties.\u03b4mPIVNSS \u2264 10 \u2264 \u03b4mPIVNSS=\u03b4mPIVNSS\u2118\u211c=\u2118\u2112, that is, \u2200j, \u03b1, u\u03b1\u2113\u211c(uj)=u\u03b1\u2113\u2112(uj), u\u03b1\ud835\udd32\u211c(uj)=u\u03b1\ud835\udd32\u2112(uj), v\u03b1\u2113\u211c(uj)=v\u03b1\u2113\u2112(uj), v\u03b1\ud835\udd32\u211c(uj)=v\u03b1\ud835\udd32\u2112(uj), w\u03b1\u2113\u211c(uj)=w\u03b1\u2113\u2112(uj), w\u03b1\ud835\udd32\u211c(uj)=w\u03b1\ud835\udd32\u2112(uj), then \u03b4mPIVNSS=1.If Let The proof is obvious.\u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, their CC has also been given as \u03b4mPIVNSS1 and is expressed as follows:Let \u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, CC between them satisfies the following properties.\u03b4IVIFSS1 \u2264 10 0 \u2264 \u03b4IVIFSS1=\u03b4IVIFSS1\u2118\u211c=\u2118\u2112, that is, \u2200j, \u03b1, u\u03b1\u2113\u211c(uj)=u\u03b1\u2113\u2112(uj), u\u03b1\ud835\udd32\u211c(uj)=u\u03b1\ud835\udd32\u2112(uj), v\u03b1\u2113\u211c(uj)=v\u03b1\u2113\u2112(uj), v\u03b1\ud835\udd32\u211c(uj)=v\u03b1\ud835\udd32\u2112(uj), w\u03b1\u2113\u211c(uj)=w\u03b1\u2113\u2112(uj), w\u03b1\ud835\udd32\u211c(uj)=w\u03b1\ud835\udd32\u2112(uj), then \u03b4IVIFSS1=1.If Let The proof is obvious.\u03b3={\u03b31, \u03b32, \u03b33,\u2026,\u03b3n}T, where \u03b3i > 0, \u2211i=1n\u03b3i=1.These days, it is important to discuss the weight of mPNSS for practical life. When professionals set different weights for each alternative, the decision may be different. Therefore, it is perfectly correct for experts to weigh the recent decision. Suppose the weight of professionals can be stated as \u2118\u211c and \u2118\u2112 are two mPIVNSS over \ud835\udcb0. Then, their WCC is given as \u03b4WmPIVNSS and stated as follows:Let \u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, their WCC also given as \u03b4WmPIVNSS1 is defined as follows:Let \u03b3={(1/n), (1/n),\u2026, (1/n)}, then \u03b4WmPIVNSS and \u03b4WmPIVNSS1 are reduced to \u03b4mPIVNSS and \u03b4mPIVNSS1, respectively, defined in Definitions If we consider \u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, WCC between them satisfies the following properties.\u03b4WmPIVNSS \u2264 10 \u2264 \u03b4WmPIVNSS=\u03b4WmPIVNSS\u2118\u211c=\u2118\u2112, that is, \u2200j, \u03b1, u\u03b1\u2113\u211c(uj)=u\u03b1\u2113\u2112(uj), u\u03b1\ud835\udd32\u211c(uj)=u\u03b1\ud835\udd32\u2112(uj), v\u03b1\u2113\u211c(uj)=v\u03b1\u2113\u2112(uj), v\u03b1\ud835\udd32\u211c(uj)=v\u03b1\ud835\udd32\u2112(uj), w\u03b1\u2113\u211c(uj)=w\u03b1\u2113\u2112(uj), w\u03b1\ud835\udd32\u211c(uj)=w\u03b1\ud835\udd32\u2112(uj), then \u03b4WmPIVNSS=1.If Let The proof is obvious.\u03b2={\u03b21, \u03b22, \u03b23,\u2026, \u03b2s} for assessment under the team of experts such as \ud835\udcb0={u1, u2, u3,\u2026, un} with weights \u03a9=T, such that \u03a9i > 0, \u2211i=1n\u03a9i=1. Let \u2130={e1, e2,\u2026, em} be a set of attributes with weights, and \u03b3=T be a weight vector for parameters such as \u03b3i > 0, \u2211j=1m\u03b3j=1. The team of experts {ui: i=1,2,\u2026, n} evaluate the alternatives {\u03b2z)(=(u\u03b1ijz), where u\u03b1ij(z)=, v\u03b1ijz)(=, and w\u03b1ijz)(=, where 0 \u2264 u\u03b1\u2113(u), u\u03b1\ud835\udd32(u), v\u03b1\u2113(u), v\u03b1\ud835\udd32(u), w\u03b1\u2113(u), w\u03b1\ud835\udd32(u) \u2264 1 and 0 \u2264 u\u03b1ij\ud835\udd32(u)+v\u03b1ij\ud835\udd32(u)+w\u03b1ij\ud835\udd32(u) \u2264 3. So, \u2112ij(z)=(, ) for all i, j.Assume a set of \u201cs\u201d alternatives such as The flowchart of the offered algorithm is shown in In the past few years, many mathematicians developed various methodologies to solve MCDM problems, such as aggregation operators for different hybrid structures, CC, similarity measures, and decision-making applications. Some operational laws and mPIVNSWA with its decision-making approach have been established for mPIVNSS which is an extension of the interval-valued neutrosophic weighted aggregation operator . The ide\u2118\u211c=, , \u2009, \u2118\u211c1=, , \u2009, and \u2118\u211c2=, , \u2009 be three mPIVNSNs, and the basic operators for mPIVNSNs are defined as when \u03b4 > 0.\u2118\u211c1 \u2295 \u2118\u211c2=\u2329, , \u2009\u232a\u2118\u211c1 \u2297 \u2118\u211c2=\u2329, , \u232a\u03b4\u2118\u211c=\u2329, , \u2009\u232a\u2118\u211c)\u03b4=\u2329, , \u2009\u232a=\u03b4\u2118\u211c2 \u2295 \u03b4\u2118\u211c1\u2118\u211c1 \u2297 \u2118\u211c2)\u03b4=(\u2118\u211c1)\u03b4 \u2297 (\u2118\u211c2)\u03b4(\u03b41\u2118\u211c1 \u2295 \u03b42\u2118\u211c1=(\u03b41 \u2295 \u03b42)\u2118\u211c1\u2118\u211c1)\u03b41 \u2297 (\u2118\u211c1)\u03b42=(\u2118\u211c1)\u03b41+\u03b42(\u2118\u211c \u2295 \u2118\u211c1) \u2295 \u2118\u211c2=\u2118\u211c \u2295 (\u2118\u211c1 \u2295 \u2118\u211c2)(\u2118\u211c \u2297 \u2118\u211c1) \u2297 \u2118\u211c2=\u2118\u211c \u2297 (\u2118\u211c1 \u2297 \u2118\u211c2)(Let The proof of the above laws is straightforward by using \u2118\u211ceij=, ,\u2009 be a collection of mPIVNSNs. \u03a9i and \u03b3j are the weight vectors for expert's and parameters, respectively, with given conditions \u03a9i > 0, \u2211i=1n\u03a9i=1, \u03b3j > 0, \u2211j=1m\u03b3j \u2009=\u20091, where . Then, the mPIVNSWA operator defined as mPIVNSWA: \u0394n\u27f6\u0394 is defined as follows:Let \u2118\u211ceij=, ,\u2009 be a collection of mPIVNSNs, where , and the aggregated value is also an interval-valued neutrosophic soft number, such asLet We can prove this easily by using IFSWA .\u2118\u211c=, , \u2009 be an mPIVNSN; then the score, accuracy, and certainty functions for an mPIVNSN, respectively, defined as follows:\ud835\udd4a(\u2118\u211c)=(1/6m)(u\u03b1\u2113(u)+u\u03b1\ud835\udd32(u)+1 \u2212 v\u03b1\u2113(u)+1 \u2212 v\u03b1\ud835\udd32(u)+1 \u2212 w\u03b1\u2113(u)+1 \u2212 w\u03b1\ud835\udd32(u))(\u2118\u211c)\ud835\udd38(\u2118\u211c)=(1/4m)(4+u\u03b1\u2113(u)+u\u03b1\ud835\udd32(u) \u2212 w\u03b1\u2113(u) \u2212 w\u03b1\ud835\udd32(u))(\u2118\u211c)(2)\u2102(\u2118\u211c)=(1/2m)(2+u\u03b1\u2113(u)+u\u03b1\ud835\udd32(u))(\u2118\u211c), where \u03b1=1,2,\u2026, m(3)Let \u2118\u211c and \u2118\u211c1 be two mPIVNSSs. Then, the comparison approach is presented as follows:\u2118\u211c) > (\u2118\u211c1), then \u2118\u211c is superior to \u2118\u211c1(1)If (\u2118\u211c)=(\u2118\u211c1) and (\u2118\u211c) > \ud835\udd38(\u2118\u211c1), then \u2118\u211c is superior to \u2118\u211c1(2)If (\ud835\udd4a(\u2118\u211c)=\ud835\udd4a(\u2118\u211c1), \ud835\udd38(\u2118\u211c)=\ud835\udd38(\u2118\u211c1), and \u2102(\u2118\u211c) > \u2102(\u2118\u211c1), then \u2118\u211c is superior to \u2118\u211c1(3)If \ud835\udd4a(\u2118\u211c)=\ud835\udd4a(\u2118\u211c1), \ud835\udd38(\u2118\u211c) > \ud835\udd38(\u2118\u211c1), and \u2102(\u2118\u211c)=\u2102(\u2118\u211c1), then \u2118\u211c is indifferent to \u2118\u211c1 and can be denoted as \u2118\u211c ~ \u2118\u211c1If Let \u03b2={\u03b21, \u03b22, \u03b23,\u2026, \u03b2s} for assessment under the team of experts such as \ud835\udcb0={u1, u2, u3,\u2026, un} with weights \u03a9=T, such that \u03a9i > 0, \u2211i=1n\u03a9i=1. Let \u2130={e1, e2,\u2026, em} be a set of attributes with weights \u03b3=T be a weight vector for parameters such as \u03b3i > 0, \u2211j=1m\u03b3j=1. The team of experts {ui: i=1,2,\u2026, n} evaluate the alternatives {\u03b2z)(=(u\u03b1ijz), where u\u03b1ij(z)=, v\u03b1ijz)(=, and w\u03b1ijz)(=, where 0 \u2264 u\u03b1\u2113(u), u\u03b1\ud835\udd32(u), v\u03b1\u2113(u), v\u03b1\ud835\udd32(u), w\u03b1\u2113(u), w\u03b1\ud835\udd32(u) \u2264 1 and 0 \u2264 u\u03b1ij\ud835\udd32(u)+v\u03b1ij\ud835\udd32(u)+w\u03b1ij\ud835\udd32(u) \u2264 3. So, \u0394k=(, , ) for all i, j. Experts give their preferences for each alternative in term of mPIVNSNs by using the mPIVNSWA operator in the form of\u0394k=(, , ). Compute the score values for each alternative and analyze the ranking of the alternatives.Assume a set of \u201cs\u201d alternatives such as The flowchart of the offered algorithm is shown in \u2118R1 and \u2118R2 be two mPIVNSSs over the universe of discourse \ud835\udcb0={u1, \u2009u2,\u2026, \u2009uj}. Then, a cosine similarity measure between \u2118R1 and \u2118R2 is defined asLet \u2118\u211c, \u2118\u2112, and \u2118Q be three mPIVNSSs. Then, the following properties hold.\ud835\udcaemPIVNSS1 \u2264 10 \u2264 \ud835\udcaemPIVNSS1=\ud835\udcaemPIVNSS1\u2118\u211c\u2286\u2118\u2112\u2286\u2118Q, then \ud835\udcaemPIVNSS1 \u2264 \ud835\udcaemPIVNSS1 and \ud835\udcaemPIVNSS1 \u2264 \ud835\udcaemPIVNSS1If Let By using the above definition, the proof of these properties can be done easily.\u2118R1 and \u2118R2 be two mPIVNSSs over the universe of discourse \ud835\udcb0={u1, \u2009u2,\u2026, \u2009uj}. Then, the set-theoretic similarity measure between \u2118R1 and \u2118R2 is defined asLet \u2118\u211c and \u2118\u2112 be two mPIVNSSs over \ud835\udcb0. Then, the following properties hold.\ud835\udcaemPIVNSS2 \u2264 10 0 \u2264 \ud835\udcaemPIVNSS2=\ud835\udcaemPIVNSS2\u2118\u211c\u2286\u2118\u2112\u2286\u2118Q, then \ud835\udcaemPIVNSS2 \u2264 \ud835\udcaemPIVNSS2 and \ud835\udcaemPIVNSS2 \u2264 \ud835\udcaemPIVNSS2If Let By using the above definition, the proof of these properties can be done easily.In this section, we utilized the developed approaches based on correlation coefficient, mPIVNSWA operator, and similarity measures for decision making.\u03b2(1), \u03b2(2), \u03b2(3), \u03b2(4)}. A team of three experts has been hired by the president of the institution {u1, u2, u3} with weights T for final scrutiny and provide the selection criteria, as given in e1\u2009=\u2009experience, e2\u2009=\u2009publications, and e3\u2009=\u2009research quality with weights T. Each expert gives his preferences for each alternative in the form of mPIVNSNs under the considered parameters given in Tables A university calls for the appointment of a vacant position of associate professor. For further assessment, four candidates  choose after preliminary review such as {\u03b2(1), \u03b2, \u03b2(3), \u03b2\u03b2(1), \u03b2(2), \u03b2(3), \u03b2(4)} be a set of alternatives who are shortlisted for interview and \u2130={e1=experience,e2=publications,e3=research\u2009quality} be a set of parameters for the selection of associate professor. Let \u211c and \u2112\u2286\u2130; then, we construct the 3-PIVNSS \u2118\u211c(e) according to the requirement of university management such as follows:Assume {\u2118\u2112t)((e) for each alternative according to experts, where t\u2009=\u20091, 2, 3, 4.Construct 3-PIVNSS 6.2.1.\u03b4mPIVNSS(\u2118\u211c(e), \u2118\u2112(1)(e)), \u03b4mPIVNSS(\u2118\u211c(e), \u2118\u2112(2)(e)), \u03b4mPIVNSS(\u2118\u211c(e), \u2118\u2112(3)(e)), and \u03b4mPIVNSS(\u2118\u211c(e), \u2118\u2112(4)(e)) by using equation , \u2118\u2112(2)(e)), \u03b4mPIVNSS(\u2118\u211c(e), \u2118\u2112(3)(e)), and \u03b4mPIVNSS(\u2118\u211c(e), \u2118\u2112(4)(e)) given as. \u03b4mPIVNSS(\u2118\u211c(e), \u2118\u2112(2)(e))=(25.04/28.6727)=0.87330, \u03b4mPIVNSS(\u2118\u211c(e), \u2118\u2112(3)(e))=(23.73/29.4968)=0.80449, and \u03b4mPIVNSS(\u2118\u211c(e), \u2118\u2112(4)(e))=(24.58/28.7433)=0.85516. This shows that \u03b4mPIVNSS(\u2118\u211c(e), \u2118\u2112(1)(e)) > \u2009\u03b4mPIVNSS(\u2118\u211c(e), \u2118\u2112(2)(e)) > \u03b4mPIVNSS(\u2118\u211c(e), \u2118\u2112(4)(e))\u2009 > \u03b4mPIVNSS(\u2118\u211c(e), \u2118\u2112(3)(e)). The above-obtained ranking shows that \u03b2(1) is the best alternative. So, the ranking of other alternatives is given as \u03b2(1) > \u03b2(2) > \u03b2(4) > \u03b2(3). Graphical results are shown in Similarly, we can find the CC between 6.2.2.\u2009 Step 1: experts evaluate the scores for each alternative in the form of mPIVNSNs given in Tables 21=, , , \u03942=, , , \u03943=, , , and \u03944=, , .\u2009 \u2009Step 2: utilizing equation , the opi\ud835\udd4a(\u03941)=0.2045, \ud835\udd4a(\u03942)=0.2004, \ud835\udd4a(\u03943)=0.1709, and \ud835\udd4a(\u03944)=0.1828.\u2009 \u2009Step 3: utilizing equation , compute\ud835\udd4a(\u03941) > \ud835\udd4a(\u03942) > \ud835\udd4a(\u03944) > \ud835\udd4a(\u03943). So, \u03b2(1) > \u03b2(2)\u2009 > \u03b2(4) > \u03b2(3); hence, the alternative \u03b2(1) is the most suitable alternative for the position of associate professor. Graphical representation of the obtained results is shown in \u2009 \u2009Step 4: so, alternatives' ranking is as follows: 6.2.3.\u03b4mPIVNSS1(\u2118\u211c(e), \u2118\u2112(1)(e)), \u03b4mPIVNSS1(\u2118\u211c(e), \u2118\u2112(2)(e)), \u03b4mPIVNSS1(\u2118\u211c(e), \u2118\u2112(3)(e)), and \u03b4mPIVNSS1(\u2118\u211c(e), \u2118\u2112(4)(e)) by using equation , \u2118\u2112(2)(e)), \u03b4mPIVNSS1(\u2118\u211c(e), \u2118\u2112(3)(e)), and \u03b4mPIVNSS1(\u2118\u211c(e), \u2118\u2112(4)(e)) given as \u03b4mPIVNSS1(\u2118\u211c(e), \u2118\u2112(2)(e))=(1/9)(46.77/28.6727)=0.18124, \u03b4mPIVNSS1(\u2118\u211c(e), \u2118\u2112(3)(e))=(1/9)(45.11/29.4968)=\u20090.16992, and \u03b4mPIVNSS1(\u2118\u211c(e), \u2118\u2112(4)(e))=(1/9)(46.45/28.7433)=0.17956. This shows that \u03b4mPIVNSS1(\u2118\u211c(e), \u2118\u2112(1)(e)) > \u2009\u03b4mPIVNSS1(\u2118\u211c(e), \u2118\u2112(2)(e)) > \u03b4mPIVNSS1(\u2118\u211c(e), \u2118\u2112(4)(e)) > \u03b4mPIVNSS1(\u2118\u211c(e), \u2118\u2112(3)(e)), which shows that alternative \u03b2(1) is the most appropriate and similar to \u2118\u211c(e). So, alternatives ranking is given as \u03b2(1) > \u03b2(2)\u2009 > \u03b2(4) > \u03b2(3).Similarly, we can find the cosine similarity measure between \u03b4mPIVNSS2(\u2118\u211c(e), \u2118\u2112(1)(e)), \u03b4mPIVNSS2(\u2118\u211c(e), \u2118\u2112(2)(e)), \u03b4mPIVNSS2(\u2118\u211c(e), \u2118\u2112(3)(e)), and \u03b4mPIVNSS2(\u2118\u211c(e), \u2118\u2112(4)(e)). From Tables \u03b4mPIVNSS2(\u2118\u211c(e), \u2118\u2112(1)(e))=0.17889, \u03b4mPIVNSS2(\u2118\u211c(e), \u2118\u2112(2)(e))=0.17548, \u03b4mPIVNSS2(\u2118\u211c(e), \u2118\u2112(3)(e))=0.16735, and \u03b4mPIVNSS2(\u2118\u211c(e), \u2118\u2112(4)(e))=0.17766. This shows that \u03b4mPIVNSS2(\u2118\u211c(e), \u2118\u2112(1)(e))\u2009 > \u03b4mPIVNSS2(\u2118\u211c(e), \u2118\u2112(4)(e)) > \u03b4mPIVNSS2(\u2118\u211c(e), \u2118\u2112(2)(e)) > \u03b4mPIVNSS2(\u2118\u211c(e), \u2118\u2112(3)(e)). So, \u03b2(1) is the best alternative using the set-theoretic similarity measure, and the ranking of other alternatives is given as \u03b2(1) > \u03b2(4)\u2009 > \u03b2(2) > \u03b2(3). Graphical representation of results is shown in Now, we compute the set-theoretic similarity measure by using \u2118\u211c(e), \u2118\u2112(e)), \u03b4mPIn the next section, we are going to talk about utility, ease, and management with the help of a planned method. We also made a tentative assessment of the following with planned techniques and some existing methods.Through this study and comparison, it can be determined that the results obtained from the proposed approach are either more general than the methods available. Although, on the whole, the DM method associated with the usual DM methods adjusts the additional information to overcome the hesitation. Also, the various hybrid structures of FS are becoming a special feature of mPIVNS, with some suitable conditions being added. General information related to the object can be described accurately and analytically, as given in It turns out to be a contemporary problem. Why do we have to particularize novel algorithms according to the present novel structure? There are several indications that the recommended methodology can be exceptional compared to other existing methods. We remember the fact that IFS, picture fuzzy set, FS, hesitant fuzzy set, NS, and other fuzzy sets have been restricted by the mixed structure and cannot provide complete information regarding the situation. But, the proposed model in this study be the utmost appropriate for MCDM because it can handle three types of information such as truth, falsity, and indeterminacy. Comparative analysis with some common methods is given in Chen et al.'  multipol\u03b2(1) is the finest alternative for the position of associate professor. The proposed approach can be compared to other available methods and observed that our proposed methodologies deliver the most reliable results comparative to available techniques. We observe one most interesting fact in our obtained results that our proposed methodologies deliver the same optimal and worst choices. The comparison of our proposed methodologies with some existing approaches is given in We recommend some novel algorithms under mPIVNSS by utilizing the developed mPIVNSS such as the mPIVNSWA operator, correlation coefficient, and similarity measures in the following section. Subsequently, we utilize the suggested algorithms to a realistic problem, namely, for the selection of an appropriate associate professor. It can be observed that The research concludes that the results obtained from the planned point of view exceed the results of the prevailing theories. Therefore, compared to existing techniques, established AOs, similarity measures, and CC handled uncertain and confusing information efficiently. However, under the current DM strategy, the main advantage of the planned method is that it can accommodate additional information in the data compared to existing techniques. This is a useful tool for resolving misinformation and vagueness in the DM method. The advantage of a planned approach with measures related to the current approach is avoiding the consequences based on negative reasons.In this study, a novel hybrid structure has been established by merging two independent structures m-polar fuzzy set and interval-valued neutrosophic soft set which is known as mPIVNSS. Some fundamental operations with their properties have been introduced for mPIVNSS. We have developed the CC and WCC with their properties in the content of mPIVNSS and also defined some operational laws for mPIVNSS and established a novel operator such as m-polar interval-valued neutrosophic weighted aggregation operator based on developed operational laws. To compute the similarity measure between two mPIVNSS, the idea of cosine and set-theoretic similarity measures have been established. Three novel algorithms based on mPIVNSS have been constructed to solve MCDM problems, correlation coefficient, mPIVNSWA operator, and similarity measures. A comparative analysis was also performed to demonstrate the proposed method. Finally, the projected ideas presented high constancy and functionality for decision-makers in the decision-making process. Based on the results acquired, it has been terminated and the above approach is extremely appropriate for finding the problem of MCDM in today's life. In the future, anyone can be introduced to the multipolar interval-valued neutrosophic weighted geometric operator with its decision-making approach. Furthermore, the concept of mPIVNSS will be extended to a multipolar interval-valued neutrosophic hypersoft set with their basic operators. The proposed impression can be functional to moderately a lot of problems in real life, including the therapeutic career, computing, artificial intelligence, pattern recognition, and finances."}
{"text": "Pythagorean fuzzy soft set (PFSS) is the most powerful and effective extension of Pythagorean fuzzy sets (PFS) which deals with the parametrized values of the alternatives. It is also a generalization of intuitionistic fuzzy soft set (IFSS) which provides us better and precise information in the decision-making process comparative to IFSS. The core objective of this work is to construct some algebraic operations for PFSS such as OR-operation, AND-operation, and necessity and possibility operations. Furthermore, some fundamental properties have been established for PFSS utilizing the developed operations. Moreover, a decision-making technique has been offered for PFSS based on a score matrix. To demonstrate the validity of the proposed approach, a numerical example has been presented. Finally, to ensure the practicality of the established approach, a comprehensive comparative analysis has been presented. The obtained results show that our developed approach is most effective and delivers better information comparative to prevailing techniques. Zadeh  introduc2\u2009+\u2009NMG2 \u2264 1. As a generalized set of IFS, PFS has a close relationship with IFS. The PFS accurately access the uncertain facts than IFS. Muhammad Zulqarnain et al.  are of membership grade and nonmembership functions respectively with 0 \u2264 \u2032\u03d2A(P)+\u03d1A(P) \u2264 1 and A \u2282 \u2115.Let X be a collection of objects, then a PFS A over X is defined asA(P), \u03d1A(P) : X\u27f6 are membership and nonmembership grade functions, respectively. Furthermore, 0 \u2264 \u2032\u03d2A(P)2+\u03d1A(P)2 \u2264 1 and I=1 \u2212 \u2032\u03d2(P)2 \u2212 \u03d1A(P)2 is called degree of indeterminacy.Let A(P)+\u03d1A(P) \u2264 1 and I=1 \u2212 \u2032\u03d2A(P) \u2212 \u03d1A(P), whereas in PFS, we have condition 0 \u2264 \u2032\u03d2A(P)2+\u03d1A(P)2 \u2264 1 and I=1 \u2212 \u2032\u03d2A(P)2 \u2212 \u03d1A(P)2. We can say that a PFS is the general case of IFS.We can see from the above definitions that the only difference is in the conditions, i.e., in IFS, we deal with the condition 0 \u2264 \u03d2X be a universal set and \u2115 be set of attributes, then a pair  is called a PFSS over X where \u03a9 : \u2115\u27f6\u2118KX is a mapping and \u2118KX is known as the collection of all PFS subsets of universal set X.A(P), \u03d1A(P) : A\u27f6 are of membership grade and nonmembership functions respectively with 0 \u2264 \u2032\u03d2A(P)2+\u03d1A(P)2 \u2264 1, degree of indeterminacy A \u2282 \u2115.Let ij=\u2329\u2032\u03d2ij, \u03d1ij\u232a. For calculating the ranking of alternatives, Zulqarnain et al. . It is notified that the score function is unable to differentiate the PFSNs in some cases. For example, let \u210b11\u2009=\u20090.3162, \u20090.4472 and \u210b12\u2009=\u20090.5477, \u20090.6324, then according to the definition of score function, we have S(\u210b11)\u2009=\u2009\u22120.1 and S(\u210b12)\u2009=\u2009\u22120.1. So, in this case, it is impossible to find the finest alternative utilizing the score function. To handle this drawback, an accuracy function has been developed in which the sum of squares of membership and nonmembership such asA(\u210bij) \u2208 .For the sake of readers' convenience, we express the PFSN as \u210bn et al.  introducij and \u211bij, the following comparison laws are defined:(1)S(\u210bij) > S(\u211bij), then \u210bij > \u211bijIf (2)S(\u210bij)=S(\u211bij), thenA(\u210bij) > A(\u211bij), then \u210bij > \u211bijIf A(\u210bij)=A(\u211bij), then \u210bij=\u211bijIf If Thus, to compare two PFSNs \u210bMatrices play a vital role in numerous areas of life such as calculation strategies, managing the magnitude of several engineering complications, medical science, and social science.FA, E) becomes a soft Pythagorean set softer than X, then subset X \u2208 E is defined differently by RA={, e \u2208 a, P \u2208 FA}. Let RA be identified by its MG and NMG functions such as \u2032\u03d2RA : X \u00d7 E\u27f6 and \u03d1RA : X \u00d7 E\u27f6.\u03b1 \u00d7 \u03b2 order.If \u2009=\u2009{P, (\u2032\u03d2A(P), \u03d1A(P))|P \u2208 A} and \u2009=\u2009{P, (\u2032\u03d2B(P), \u03d1B(P))|P \u2208 B} be two PFSS, where \u2032\u03d2A(P), \u2032\u03d2B(P), \u03d1A(P), \u03d1B(P) \u2208 . Then, OR-operation between them is written as follows:Let \u2009=\u2009{P, (\u2032\u03d2A(P), \u03d1A(P))|P \u2208 A} and \u2009=\u2009{P, (\u2032\u03d2B(P), \u03d1B(P))|P \u2208 B} be two PFSS, where \u2032\u03d2A(P), \u2032\u03d2B(P), \u03d1A(P), \u03d1B(P) \u2208 . Then, AND-operation between them is written as follows:Let \u2009=\u2009{P, (\u2032\u03d2A(P), \u03d1A(P))|P \u2208 A}, \u2009=\u2009{P, (\u2032\u03d2B(P), \u03d1B(P))|P \u2208 B}, and  = {P, (\u2032\u03d2C(P), \u03d1C(P))|P \u2208 C} be three PFSS \u2032\u03d2A(P), \u2032\u03d2B(P), \u2032\u03d2C(P), \u03d1A(P), \u03d1B(P), \u03d1C(P) \u2208 .K, A)\u2228\u2009=\u2009\u2228\u2228\u2228)\u2009=\u2009\u2228)\u2228\u2009=\u2009{P, (\u2032\u03d2A(P), \u03d1A(P))|P \u2208 A} and \u2009=\u2009{P, (\u2032\u03d2B(P), \u03d1B(P))|P \u2208 B} be two PFSS, where \u2032\u03d2A(P), \u2032\u03d2B(P), \u03d1A(P), \u03d1B(P) \u2208 . Then, utilizing (1) Let \u2009=\u2009{P, (\u2032\u03d2A(P), \u03d1A(P))|P \u2208 A}, \u2009=\u2009 {P, (\u2032\u03d2B(P), \u03d1B(P))|P \u2208 B}, and \u2009=\u2009 {P, (\u2032\u03d2C(P), \u03d1C(P))|P \u2208 C} be three PFSS \u2032\u03d2A(P), \u2032\u03d2B(P), \u03d1A(P), \u03d1B(P) \u2208 .(2) Let \u2009=\u2009{P, (\u2032\u03d2A(P), \u03d1A(P))|P \u2208 A}, \u2009=\u2009{P, (\u2032\u03d2B(P), \u03d1B(P))|P \u2208 B}, and \u2009= {P, (\u2032\u03d2C(P), \u03d1C(P))|P \u2208 A} be three PFSS \u03bcA(P), \u03bcB(P), \u03d1A(P), \u03d1B(P) \u2208 .K, A)\u2227\u2009=\u2009\u2227\u2227\u2227)\u2009=\u2009\u2227)\u2227\u2009=\u2009{P, (\u2032\u03d2A(P), \u03d1A(P))|P \u2208 A} and \u2009=\u2009{P, (\u2032\u03d2B(P), \u03d1B(P))|P \u2208 B} be two PFSS, where \u2032\u03d2A(P), \u2032\u03d2B(P), \u03d1A(P), \u03d1B(P) \u2208 . Then, using (1) Let \u2009=\u2009{P, (\u2032\u03d2A(P), \u03d1A(P))|P \u2208 A}, \u2009=\u2009{P, (\u2032\u03d2B(P), \u03d1B(P))|P \u2208 B}, and \u2009= {P, (\u2032\u03d2C(P), \u03d1C(P))|P \u2208 C} be three PFSS \u2032\u03d2A(P), \u2032\u03d2B(P), \u2032\u03d2C(P), \u03d1A(P), \u03d1B(P), \u03d1c(P) \u2208 . Then, same as the above utilizing (2) Let \u2009=\u2009{P, (\u2032\u03d2A(P), \u03d1A(P))|P \u2208 A} and \u2009=\u2009{P, (\u2032\u03d2B(P), \u03d1B(P))|P \u2208 B} be two PFSS. Then, De Morgan Laws are given as follows:K, A)\u2228)0=O\u2227O\u2227)0=O\u2228O\u2009=\u2009{P, (\u2032\u03d2A(P), \u03d1A(P))|P \u2208 A} and \u2009=\u2009{P, (\u2032\u03d2B(P), \u03d1B(P))|P \u2208 B} be two PFSS. Then,(1) Let \u2009=\u2009{P, (\u2032\u03d2A(P), \u03d1A(P))|P \u2208 A} and \u2009=\u2009{P, (\u2032\u03d2B(P), \u03d1B(P))|P \u2208 B} be two PFSS. Then,(2) Let \u2009=\u2009{P, (\u2032\u03d2A(P), \u03d1A(P))|P \u2208 A} be a PFSS. Then, the necessity operation on PFSS is denoted by \u2009 \u2295 \u2009 and defined as follows:Let \u2009=\u2009{P, (\u2032\u03d2A(P), \u03d1A(P))|P \u2208 A} be a PFSS, then the possibility operation on PFSS is denoted by \u2297 and written as follows:Let \u2009=\u2009{P, (\u2032\u03d2A(P), \u03d1A(P))|P \u2208 A} be a PFSS. Then, the following properties are satisfied.K, A)O]O\u2009=\u2009\u2297[\u2009 \u2295 \u2009O]O\u2009= \u2295 \u2009[\u2297]\u2009= \u2295 \u2009\u2009 \u2295 \u2009[\u2009 \u2295 \u2009]\u2009=\u2009\u2297\u2009 \u2295 \u2009[\u2297]\u2009= \u2295 \u2009\u2297[\u2009 \u2295 \u2009]\u2009=\u2009\u2297\u2297\u2009= \u2295 \u2009\u2227\u2009 \u2295 \u2009\u2009 \u2295 \u2009[\u2228]=\u2009 \u2295 \u2009\u2228\u2009 \u2295 \u2009\u2009 \u2295 \u2009=\u2297\u2227\u2297\u2297[\u2228]=\u2297\u2228\u2297\u2297 to choose the best bike available in the market. Then, PFSM can be written by using parameters as follows:Let These four PFSS are represented by the following PFSM, respectively:By using score matrix definition,By using the definition of utility matrix,Now,x1, the 1st brand having maximum value, in this fashion, we conclude from the judgment of four experts that Honda is the best brand for business.From the above results g1 > g2 > g5 > g6 > g3 > g4.Ranking of alternative For comparative analysis, we made the comparison of our proposed method with different methods by using their algorithm. Through this process, we can easily show the credibility of our proposed method.Here, we use the method of Rathika and Subramanian , in whic\u2009 Step 1: input IFSSs and obtain intuitionistic fuzzy soft matrices (IFSM)\u2009 Step 2: take complement of IFSS and obtain complement IFSM from these setsA \u2212 B \u2212 C \u2212 D), (Ac \u2212 Bc \u2212 Cc \u2212 Dc), and value matrix of these IFSMs\u2009 Step 3: find , (Ac+Bc+Cc+Dc), and value matrix of these IFSMs\u2009 Step 3: find  from these sets\u2009 Step 3: evaluate the cross-product of FSM\u2009 Step 4: enumerate the optimum subscript matrix\u2009 Step 5: compute the best alternative that is having max valueWe haveg1 > g3 > g2 > g5 > g4 > g6.Ranking of alternative Furthermore, we use the algorithm of Rathika et al. , in whic\u2009 Step 1: input IFSSs and obtain IFSMs\u2009 Step 2: take complement of IFSS and obtain complement IFSM from these setsA \u2212 B \u2212 C \u2212 D), (Ac \u2212 Bc \u2212 Cc \u2212 Dc), and value matrix of these IFSMs\u2009 Step 3: find (\u2009 Step 4: compute the score of IFSMs and complement IFSMsSi for xi in-universe\u2009 Step 5: calculate the total score xi of the total score matrix Si has maximum value which is the best alternative\u2009 Step 6: we conclude that element Si has more values besides one value, then rerun the development by repeating the parameter.If max g1 > g3 > g4 > g2 > g5 > g6.Ranking of alternative Also, we use the technique of Chetia and Das , in whicStep 1: select the FSS set of parametersStep 2: obtain FSMs from these setsStep 3: find the union of these FSMsStep 4: enumerate the weight along with the individual item by taking rowwise sum membership valuesStep 5: compute the best alternative that is having max valueNow, weights matrix of the bikes of different brands,g1 > g4 > g3=g5 > g6 > g2.Ranking of alternative The results obtained through existing methodologies with the proposed technique with their score values are given in With the ongoing research and comparison, the results obtained by our proposed method and existing procedures are in In this paper, we developed some logical operators such as OR-operation and AND-operation for PFSS with their fundamental characteristics. Also, some novel operations such as necessity and possibility operations have been presented with their fundamental properties. A decision-making approach has been constructed for PFSM based on a score matrix and utility matrix. To confirm the validity of our established approach, a comprehensive numerical example has been developed. To express the legitimacy, effectiveness, and efficiency of our proposed method, a logical comparison between the current work and the proposed approach is also provided. The obtained consequences show that the developed technique is more reliable compared to existing techniques. Future research will focus on offering several other operators under the PFSS environment to address decision-making issues. Many other structures such as topological, algebra, and orderly structures can be developed and discussed in the environment under consideration. The proposed idea can be applied to many issues in real life, including the medical profession, pattern recognition, and economics."}
{"text": "The purpose of this paper is to define the concept of -fuzzy sets and discuss their relationship with other kinds of fuzzy sets. We describe some of the basic set operations on -fuzzy sets. -Fuzzy sets can deal with more uncertain situations than Pythagorean and intuitionistic fuzzy sets because of their larger range of describing the membership grades. Furthermore, we familiarize the notion of -fuzzy topological space and discuss the master properties of -fuzzy continuous maps. Then, we introduce the concept of -fuzzy points and study some types of separation axioms in -fuzzy topological space. Moreover, we establish the idea of relation in -fuzzy set and present some properties. Ultimately, on the basis of academic performance, the decision-making approach of student placement is presented via the proposed -fuzzy relation to ascertain the suitability of colleges to applicants. The concept of fuzzy sets was proposed by Zadeh . The theThe idea of intuitionistic fuzzy sets suggested by Atanassov  is one oThe concept of fuzzy topological spaces was introduced by Chang . He studThe main purpose of this paper is to introduce the concept of -fuzzy sets and compare them with the other types of fuzzy sets. We introduce the set of operations for the -fuzzy sets and explore their main features. Following the idea of Chang, we define a topological structure via -fuzzy sets as an extension of fuzzy topological space, intuitionistic fuzzy topological space, and Pythagorean fuzzy topological space. We discuss the main topological concepts in -fuzzy topological spaces such as continuity and compactness. In addition, the concept of relation to -fuzzy sets is investigated. Finally, an improved version of max-min-max composite relation for -fuzzy sets is proposed.In this section, we initiate the notion of -fuzzy sets and study their relationship with other kinds of fuzzy sets. Then, we furnish some operations to -fuzzy sets.X be a universal set. Then, the -fuzzy set -FS) D is defined by the following:\u03b1D(r) : X\u27f6 is the degree of membership and \u03b2D(r) : X\u27f6 is the degree of non-membership of r \u2208 X to D, with the conditionLet r \u2208 X to D is defined byThe degree of indeterminacy of \u03b1D(r))3+(\u03b2D(r))2+(\u03c0D(r))5=1, and \u03c0D(r)=0 whenever (\u03b1D(r))3+(\u03b2D(r))2=1. In the interest of simplicity, we shall mention the symbol D= for the -FS D={\u2329r, \u03b1D(r), \u03b2D(r)\u232a : \u2009r \u2208 X}.It is clear that   is the degree of membership and \u03b2K(r) : X\u27f6 is the degree of non-membership of every r \u2208 X to K.Let et (IFS)  (resp. Pet (IFS)  and Fermet (IFS) ) is defi\u03b1D(r)=0.9 and \u03b2D(r)=0.5 for X={r}. We obtain 0.9+0.5=1.40 > 1 and (0.9)2+(0.5)2=1.06 > 1 which means that D= neither follows the condition of IFS nor follows the condition of PFS. On the other hand, (0.9)3+(0.5)2=0.979 < 1 which means we can apply the -FS to control it. That is, D= is a -FS.To illustrate the importance of -FS to extend the grades of membership and non-membership degrees, assume that The set of -fuzzy membership grades is larger than the set of intuitionistic membership grades and Pythagorean membership grades.r1, r2 \u2208 , we haveIt is well known that for any two numbers Then, we getHence, the space of -fuzzy membership grades is larger than the space of intuitionistic membership grades and Pythagorean membership grades. This development can be evidently recognized in X={rj : \u2009j=1,\u2026, k} be a universal set and D be -FS. If \u03c0D(rj)=0, then Let D is -FS and \u03c0D(rj)=0 for rj \u2208 X; then,Presume that D be -FS and r \u2208 X such that \u03b2D(r)=0.82 and \u03c0D(r)=0. Then, Let \u03b4 be a positive real number (\u03b4 > 0). If D1= and D2= are two -FSs, then their operations are defined as follows:D1\u2229D2=.D1 \u222a D2=.D1c=.Let We will use supremum \u201csup\u201d instead of maximum \u201cmax\u201d and infimum \u201cinf\u201d instead of minimum \u201cmin\u201d if the union and the intersection are infinite.D1= and D2= are both -FSs. Then,D1\u2229D2===.D1 \u222a D2===.D1c=.\u03b4=4.\u03b4=4.Assume that L1= and L2= be two -FSs; then, the following properties hold:L1\u2229L2=L2\u2229L1.L1 \u222a L2=L2 \u222a L1.L1\u2229L2) \u222a L2=L2.(L1 \u222a L2)\u2229L2=L2.==L2\u2229L1.The proof is similar to (1).L1\u2229L2) \u222a L2= \u222a ===L2.(The proof is similar to (3).From L1=, L2= and L3= be three -FSs and \u03b4 > 0; then,L1\u2229(L2\u2229L3)=(L1\u2229L2)\u2229L3.L1 \u222a (L2 \u222a L3)=(L1 \u222a L2) \u222a L3.\u03b4(L1 \u222a L2)=\u03b4L1 \u222a \u03b4L2.L1 \u222a L2)\u03b4=L1\u03b4 \u222a L2\u03b4.(2)The proof is similar to (1).(3)(4)The proof is similar to (3).For the three -FSs Lc is -FS for any -FS L.In the following result, we claim that L1= and L2= be two -FSs such that L1c and L2c are -FSs. Then,L1\u2229L2)c=L1c \u222a L2c.(L1 \u222a L2)c=L1c\u2229L2c.(2)The proof is similar to (1).For the two -FSs D1= and D2= be two -FSs; then,D1=D2 if and only if \u03b1D1=\u03b1D2 and \u03b2D1=\u03b2D2.D1 \u2265 D2 if and only if \u03b1D1 \u2265 \u03b1D2 and \u03b2D1 \u2264 \u03b2D2.D2 \u2282 D1 or D1\u2283D2 if D1 \u2265 D2.Let D1= and D2= for X={x}, then D1=D2.If D1= and D2= for X={x}, then D2 \u2264 D1 and D2 \u2282 D1.If In this section, we formulate the concept of -fuzzy topology on the family of -fuzzy sets whose complements are -fuzzy sets and scrutinize main properties. Then, we define -fuzzy continuous maps and give some characterizations. Finally, we establish two types of -fuzzy separation axioms and reveal the relationships between them.\u03c4 be a family of -fuzzy subsets of a non-empty set X. IfX, 0X \u2208 \u03c4 where 1X= and 0X=,1D1\u2229D2 \u2208 \u03c4, for any D1, D2 \u2208 \u03c4,i\u2208IDi \u2208 \u03c4, for any {Di}i\u2208I \u2282 \u03c4,\u222athen \u03c4 is called a -fuzzy topology on X and  is a -fuzzy topological space. We call D an open -FS if it is a member of \u03c4 and call its complement a closed -FS.Let \u03c4={1X, 0X} the indiscreet -fuzzy topology on X. If \u03c4 contains all -fuzzy subsets, then we call \u03c4 the discrete -fuzzy topology on X.We call \u03c4={1X, 0X, D1, D2, D3, D4, D5} be the family of -fuzzy subsets of X={x1, x2}, whereLet \u03c4 is -fuzzy topology on X.Hence, D on a set X is a -fuzzy set having the form D={\u2329r, \u03b1D(r), 1 \u2212 \u03b1D(r)\u232a : \u2009r \u2208 X}. Then, every fuzzy topological space  in the sense of Chang is obviously a -fuzzy topological space in the form \u03c4={D : \u2009\u03b1D \u2208 \u03c41} whenever we identify a fuzzy set in X whose membership function is \u03b1D with its counterpart D={\u2329r, \u03b1D(r), 1 \u2212 \u03b1D(r)\u232a : \u2009r \u2208 X}. Similarly, one can note that every intuitionistic fuzzy topology (Pythagorean fuzzy topology) is -fuzzy topology. The following examples explain this note.We showed that every fuzzy set \u03c4={1X, 0X, D1, D2} as family of fuzzy subsets of X={x}, whereConsider \u03c4 is fuzzy topology on X, and hence it is -fuzzy topology.Then, \u03c4={1X, 0X, D1, D2} be the family of -fuzzy subsets on X={x1, x2} whereLet \u03c4 is -fuzzy topology. On the other hand, \u03c4 is neither intuitionistic fuzzy topology nor Pythagorean fuzzy topology.Hence, X, \u03c4) be a -fuzzy topological space and D={\u2329x, \u03b1D(x), \u03b2D(x)\u232a : \u2009x \u2208 X} be a -FS in X. Then, the -fuzzy interior and -fuzzy closure of D are, respectively, defined byD)=\u2229{H : H is a closed -FS in X and D \u2282 H}.cl(D)=\u222a{G : G is an open -FS in X and G \u2282 D}.int be a -fuzzy topological space and D be any -FS in X. Then,D) is an open -FS.int(D) is a closed -FS.cl(X)=cl(1X)=1X and int(0X)=cl(0X)=0X.int in D={\u2329c1, 0.67, 0.81\u232a, \u2329c2, 0.75, 0.74\u232a}, then int(D)=0X and cl(D)=1X.Consider the -fuzzy topological space  be a -fuzzy topological space and D1, D2 be -FSs in X. Then, the following properties hold:D1) \u2282 D1 and D1 \u2282 cl(D1).int \u2282 int(D2) and cl(D1) \u2282 cl(D2).If D1 is an open -FS if and only if D1=int(D1).D1 is a closed -FS if and only if D1=cl(D1).Let ((1) and (2) are obvious.(3) and (4) follow from X, \u03c4) be a -fuzzy topological space and D1, D2 be -FSs in X. Then, the following properties hold:D1) \u222a int(D2) \u2282 int(D1 \u222a D2).int(D1\u2229D2) \u2282 cl(D1)\u2229cl(D2).cl(D1\u2229D2)=int(D1)\u2229int(D2).int(D1) \u222a cl(D2)=cl(D1 \u222a D2).cl(Let ((1) and (2) follows from (1) of the above theorem.D1\u2229D2) \u2282 int(D1) and int(D1\u2229D2) \u2282 int(D2), we obtain int(D1\u2229D2) \u2282 int(D1)\u2229int(D2). On the other hand, from the facts int(D1) \u2282 D1 and int(D2) \u2282 D2, we have int(D1)\u2229int(D2) \u2282 D1\u2229D2 and int(D1)\u2229int(D2) \u2208 \u03c4; we see that int(D1)\u2229int(D2) \u2282 int(D1\u2229D2), and hence int(D1\u2229D2)=int(D1)\u2229int(D2).(3): since int((4) can be proved similar to (3).X, \u03c4) be a -fuzzy topological space and D be -FS in X. Then, the following properties hold:Dc)=int(D)c.cl(Dc)=cl(D)c.int(Dc)c=int(D).cl(Dc)c=cl(D).int(Let (We only prove (1); the other parts can be proved similarly.D={\u2329x, \u03b1D(x), \u03b2D(x)\u232a : \u2009x \u2208 X} and suppose that the family of open -fuzzy sets contained in D is indexed by the family {\u2329x, \u03b1Ui(x), \u03b2Ui(x)\u232a : \u2009i \u2208 J}. Then, int(D)={\u2329x, \u2228\u03b1Ui(x), \u2227\u03b2Ui(x)\u232a}. Therefore, int(D)c={\u2329x, \u2227\u03b2Ui(x), \u2228\u03b1Ui(x)\u232a}. Now, Dc={\u2329x, \u03b2D(x), \u03b1D(x)\u232a} such that \u03b1Ui \u2264 \u03b1D, \u03b2Ui \u2265 \u03b2D for each i \u2208 J. This implies that {\u2329x, \u03b2Ui(x), \u03b1Ui(x)\u232a : \u2009i \u2208 J} is the family of all closed -fuzzy sets containing Dc. That is, cl(Dc)={\u2329x, \u2227\u03b2Ui(x), \u2228\u03b1Ui(x)\u232a}. Hence, cl(Dc)=int(D)c.Let f : X\u27f6Y be a map and A and B be -fuzzy subsets of X and Y, respectively. The functions of membership and non-membership of the image of A, denoted by f[A], are, respectively, calculated byLet B, denoted by f\u22121[B], are, respectively, calculated byThe functions of membership and non-membership of preimage of f[A] and f\u22121[B] are -fuzzy subsets, consider \u03b3A(z))5=\u03b1A(z))3+(\u03b2A(z))2. If f\u22121(y) is non-empty, then we obtainTo show that f\u22121(y)=\u03d5 leads to the fact that (\u03b1f[A](y))3+(\u03b2f[A](y))2=1.In contrast, f\u22121[B].It is easy to prove the case of f : X\u27f6Y be a map s.t. A and B are -fuzzy subsets of X and Y, respectively. Then, we havef\u22121[Bc]=f\u22121[B]c.f[A]c\u2286f[Ac].B1\u2286B2, then f\u22121[B1]\u2286f\u22121[B2] where B1 and B2 are -fuzzy subsets of Y.If A1\u2286A2, then f[A1]\u2286f[A2] where A1 and A2 are -fuzzy subsets of X.If f[f\u22121[B]]\u2286B.A\u2286f\u22121[f[A]].Let (1)v \u2208 X and let B be a -fuzzy subset of Y. Then,Consider \u03b2f\u22121[Bc](v)=\u03b2f\u22121[B]c(v). Therefore, f\u22121[Bc]=f\u22121[B]c, as required.Similarly, one can have (2)w \u2208 Y such that f\u22121(w) \u2260 \u03d5 and for any -fuzzy subset A of X, we can writeFor any Now from , we havef\u22121(w)=\u03d5. Following a similar technique, we obtain \u03b2f[Ac](w) \u2264 \u03b2f[A]c(w), which means that f[A]c\u2286f[Ac].The proof is easy when (3)B1\u2286B2. Then, for each v \u2208 X, \u03b1f\u22121[B1](v)=\u03b1B1(f(v)) \u2264 \u03b1B2(f(v))=\u03b1f\u22121[B2](v). Also, \u03b2f\u22121[B1](v) \u2265 \u03b2f\u22121[B2](v). Hence, we obtain the desired result.Assume that (4)A1\u2286A2 and w \u2208 Y. The proof is easy when f(w)=\u03d5. So, presume that f(w) \u2260 \u03d5. Then,Assume that \u03b1f[A1] \u2264 \u03b1f[A2] follows. Similarly, we have \u03b2f[A1] \u2265 \u03b2f[A2].Thus, (5)w \u2208 Y s.t. f(w) \u2260 \u03d5, we find thatFor any \u03b1f[f\u22121[B]](w)=0 \u2264 \u03b1B(w) when f(w)=\u03d5. Similarly, we have \u03b2f[f\u22121[B]](w)=0 \u2265 \u03b2B(w).On the other hand, we have (6)v \u2208 X, we haveFor any \u03b2f\u22121[f[A]] \u2264 \u03b2A.Similarly, we have The proof of the following result is easy, and hence it is omitted.X and Y be two non-empty sets and f : X\u27f6Y be a map. Then, the following statements are true:f[\u222ai\u2208IAi]=\u222ai\u2208If[Ai] for any -fuzzy subset Ai of X.f\u22121[\u222ai\u2208IBi]=\u222ai\u2208If\u22121[Bi] for any -fuzzy subset Bi of Y.f[A1\u2229A2] \u2282 f[A1]\u2229f[A2] for any two -fuzzy subsets A1 and A2 of X.f\u22121[\u2229i\u2208IBi]=\u2229i\u2208If\u22121(Bi) for any -fuzzy subset Bi of Y.Let A and U are two -fuzzy subsets. We call U a neighborhood of A, briefly nbd, if there exists an open -fuzzy subset E such that A\u2286E\u2286U.In a -fuzzy topological space, consider that A is open iff it contains a nbd of its each subset.A -fuzzy subset The proof is easy.f : \u27f6 is said to be -fuzzy continuous if for any -fuzzy subset A of X and for any nbd V of f[A] there is a nbd U of A s.t. f[U]\u2286V.A map f : \u27f6:f is -fuzzy continuous.A of X and each nbd V of f[A], there is a nbd U of A s.t. for each B\u2286U, we obtain f[B]\u2286V.For each -FS A of X and for each nbd V of f[A], there is a nbd U of A s.t. U\u2286f\u22121[V].For each -FS A of X and for each nbd V of f[A], f\u22121[V] is a nbd of A.For each -FS The following statements are equivalent for a map f be a -fuzzy continuous map. Consider A as a -FS of X and V as a nbd of f[A]. Then, there is a nbd U of A s.t. f[U]\u2286V. If B\u2286U, we obtain f[B]\u2286f[U]\u2286V.\u2009(1)\u21d2(2): let A as a -FS of X and V as a nbd of f[A]. According to (2), there is a nbd U of A s.t. for each B\u2286U, we find f[B]\u2286V. Therefore, B\u2286f\u22121[f[B]]\u2286f\u22121[V]. Since B is chosen arbitrarily, U\u2286f\u22121[V].\u2009(2)\u21d2(3): assume A is a -FS of X and V is a nbd of f[A]. According to (3), there is a nbd U of A s.t. U\u2286f\u22121[V]. Since U is a nbd of A, there is an open -FS K of X s.t. A\u2286K\u2286U. On the other hand, we obtain A\u2286K\u2286f\u22121[V] because U\u2286f\u22121[V]. This means that f\u22121[V] is a nbd of A.\u2009(3)\u21d2(4): presume A is a -FS of X and V is a nbd of f[A]. By hypothesis, f\u22121[V] is a nbd of A. So, there is an open -FS K of X s.t. A\u2286K\u2286f\u22121[V] which means f[K]\u2286f[f\u22121[V]]\u2286V. Moreover, K is an open -FS, so it is a nbd of A. Hence, we obtain the proof that f is -fuzzy continuous.\u2009(4)\u21d2(1): suppose that f : \u27f6 is -fuzzy continuous iff f\u22121[B] is an open -FS of X for each open -FS B of Y.A map f as a -fuzzy continuous map. Consider an open -FS B of Y s.t. A\u2286f\u22121[B]. This directly gives that f[A]\u2286B. It follows from V of f[A] satisfying V\u2286B. Now, f is -fuzzy continuous, so by (4) of f\u22121[V] is a nbd of A. Also, it follows from (3) of f\u22121[V]\u2286f\u22121[B]. So, f\u22121[B] is a nbd of A. Since A is an arbitrary subset of f\u22121[B], then by f\u22121[B] is open.Necessity: presume A is a -FS of X and V is a nbd of f[A]. Then, \u03c42 contains a -FS L of s.t. f[A]\u2286L\u2286V. By hypothesis, f\u22121[L] is an open -FS. Also, we have A\u2286f\u22121[f[A]]\u2286f\u22121[L]\u2286f\u22121[V]. Thus, f\u22121[V] is a nbd of A which demonstrates that f is -fuzzy continuous.Presume We build the following two examples such that the first one provides a -fuzzy continuous map, whereas the second one presents a fuzzy map that is not -fuzzy continuous.X={a1, a2} with the -fuzzy topology \u03c41={1X, 0X, A1} and Y={b1, b2} with the -fuzzy topology \u03c42={1Y, 0Y, B1}, whereConsider f : X\u27f6Y be defined as follows:Let Y, 0Y, and B1 are open -fuzzy subsets of Y, thenX. Thus, f is -fuzzy continuous.Since 1X={a1, a2} with the -fuzzy topology \u03c41={1X, 0X} and Y={b1, b2} with the -fuzzy topology \u03c42={1Y, 0Y, B1}, where B1={\u2329b1, 0.82, 0.62\u232a, \u2329b2, 0.52, 0.90\u232a}.Consider f : X\u27f6Y be defined as follows:Let B1 is an open -fuzzy subset of Y, but f\u22121[B1]={\u2329a1, 0.82, 0.62\u232a, \u2329a2, 0.52, 0.90\u232a} is not an open -fuzzy subset of X, f is not -fuzzy continuous.Since f : \u27f6 is -fuzzy continuous.B of Y we have that f\u22121[B] is a closed -fuzzy subset of X.For each closed -fuzzy subset f\u22121[B])\u2286f\u22121[cl(B)] for each -fuzzy set in Y.cl(f\u22121[int(B)]\u2286int(f\u22121[B]) for each -fuzzy set in Y.The following are equivalent to each other:They can be easily proved using Theorems Y, \u03c4) be a -fuzzy topological space and f : X\u27f6Y be a map. Then, there is a coarsest -fuzzy topology \u03c41 over X such that f is -fuzzy continuous.Let -fuzzy subsets \u03c41 is the coarsest -fuzzy topology over X such that f is -fuzzy continuous.(1)x \u2208 X thatWe can write for any \u03b2f\u22121[0Y](x)=\u03b2X0(x) for any x \u2208 X which implies f\u22121[0Y]=0X. Now, as 0Y \u2208 \u03c4, we have 0X=f\u22121[0Y] \u2208 \u03c41. In a similar manner, it is easy to see that 1X=f\u22121[1Y] \u2208 \u03c41.Similarly, we immediately have (2)D1, D2 \u2208 \u03c41. Then, for i=1,2, there exists Bi \u2208 \u03c4 such that f\u22121[Bi]=Di which implies \u03b1f\u22121[Bi]=\u03b1Di and \u03b2f\u22121[Bi]=\u03b2Di. Thus, we obtain for any x \u2208 X thatAssume that \u03b2D1\u2229D2=\u03b2f\u22121[B1\u2229B2]. Hence, we get D1\u2229D2 \u2208 \u03c41.Similarly, it is not difficult to see that (3)Di}i\u2208I is an arbitrary subfamily of \u03c41. Then, for any i \u2208 I, there exists Bi \u2208 \u03c41 such that f\u22121[Bi]=Di which implies \u03b1f\u22121[Bi]=\u03b1Di and \u03b2f\u22121[Bi]=\u03b2Di. Therefore, one can get for any x \u2208 X thatAssume that {We prove that \u03b2i\u2208IDi\u222a=\u03b2f\u22121[\u222ai\u2208IBi]. Thus, we have \u222ai\u2208IDi \u2208 \u03c41.On the other hand, it is easy to see that f is trivial. Now, we prove that \u03c41 is the coarsest -fuzzy topology over X such that f is -fuzzy continuous. Let \u03c42\u2286\u03c41 be a -fuzzy topology over X such that f is -fuzzy continuous. If B \u2208 \u03c41, then there is V \u2208 \u03c4 such that f\u22121[V]=B. Since f is -fuzzy continuous with respect to \u03c42, we have B=f\u22121[V] \u2208 \u03c42. Hence, \u03c42=\u03c41, as required.From Separation axioms are one of the most important and popular notions in topological studies. They have been studied and applied to model some real-life issues in soft setting as explained in , 17.X \u2260 \u2205 and x \u2208 X be a fixed element in X. Suppose that r1 \u2208  are two fixed real numbers such that r13+r22 \u2264 1. Then, a -fuzzy point pr1, r2), \u03b2p(x)\u232a} is defined to be a -fuzzy set of X as follows.y \u2208 X. In this case, x is called the support of pr1, r2)-fuzzy point pr1, r2)-fuzzy set D={\u2329x, \u03b1D(x), \u03b2D(x)\u232a} of X denoted by pr1, r2)(x \u2208 D if r1 \u2264 \u03b1D(x) and r2 \u2265 \u03b2D(x). Two -fuzzy points are said to be distinct if their supports are distinct.Let D1={\u2329x, \u03b1D1(x), \u03b2D1(x)\u232a} and D2={\u2329x, \u03b1D2(x), \u03b2D2(x)\u232a} be two -fuzzy sets of X. Then, D1\u2286D2 if and only if pr1, r2)-fuzzy point pr1, r2), and x, y \u2208 X. A -fuzzy topological space  is said to be(1)T0 if for each pair of distinct -fuzzy points pr1, r2)-fuzzy sets L and K such that(2)T1 if for each pair of distinct -fuzzy points pr1, r2)-fuzzy sets L and K such thatLet X, \u03c4) be a -fuzzy topological space. If  is T1, then  is T0.Let -fuzzy topology \u03c4={1X, 0X, D}, where D={\u2329c1, 1,0\u232a, \u2329c2, 0,1\u232a}. Then,  is T0 but not T1 because there does not exist an open -fuzzy set K such that K={\u2329x, 0,1\u232a, \u2329y, 1,0\u232a}.Consider X, \u03c4) be a -fuzzy topological space, r1, r3 \u2208 . If  is T0, then for each pair of distinct -fuzzy points pr1, r2)(x) \u2260 cl(y).Let  be T0 and pr1, r2)-fuzzy points of X. Then, there exist two open -fuzzy sets L and K such thatLet -fuzzy set which does not contain pr1, r2)(y) is the smallest closed -fuzzy set containing pr3, r4)(y)\u2286Lc, and therefore pr1, r2)(y). Consequently, cl(x) \u2260 cl(y).Let X, \u03c4) be a -fuzzy topological space. If px is closed -fuzzy set for every x \u2208 X, then,  is T1.Let x is a closed -fuzzy set for every x \u2208 X. Let pr1, r2)-fuzzy points of X; then, x \u2260 y implies that pxc and pyc are two open -fuzzy sets such thatSuppose X, \u03c4) is T1.Thus, -FSs. Moreover, we provide a numerical example to elaborate on how we can apply the composite relations to obtain the optimal choices.X and Y be two (crisp) sets. The -fuzzy relation R -FR) from X to Y is a -FS of X \u00d7 Y characterized by the degree of membership function \u03b1R and degree of non-membership function \u03b2R. The -FR R from X to Y will be denoted by R(X\u27f6Y). If D is a -FS of X, then(1)R(X\u27f6Y) with D is a -FS C of Y denoted by C\u2009=\u2009R o D and is defined byThe max-min-max composition of the -FR (2)R(X\u27f6Y) with D is a -FS C of Y denoted by C\u2009=\u2009R o D, such thatThe improved composite relation of Let Q(X\u27f6Y) and R(Y\u27f6Z) be two -FRs. Then, for all  \u2208 X \u00d7 Z and n \u2208 Y,(1)R o Q is the -fuzzy relation from X to Z defined byThe max-min-max composition (2)R o Q is the -fuzzy relation from X to Z such thatThe improved composite relation Let The improved composite and max-min-max composite relations for -fuzzy sets are calculated by the following:D1 and D2 be two -fuzzy sets for X={x1, x2, x3, x4}. Assume thatLet D1 and D2 as follows:By using Definitions x1, x2, x3, x4) in D1 and D2, respectively, is 0.7, 0.5, 0.6, and 0.8. Also, the maximum value of the non-membership values of the elements  in D1 and D2, respectively, is 0.79, 0.87, 0.85, and 0.69. From It is obvious that the minimum value of the membership values of the elements 45), we oWe localize the idea of -FR as follows.S={r1,\u2026, rl} be a finite set of subjects related to the colleges, C={b1,\u2026, bm} be a finite set of colleges, and A={t1,\u2026, tn} be a finite set of students. Suppose that we have two -FRs, U(A\u27f6S) and R(S\u27f6C), such that\u03b1U denotes the degree to which the student (t) passes the related subject requirement (r).\u2009\u03b2U denotes the degree to which the student (t) does not pass the related subject requirement (r).\u2009\u03b1R denotes the degree to which the related subject requirement (r) determines the college (b).\u2009\u03b2R denotes the degree to which the related subject requirement (r) does not determine the college (b).\u2009Let T=RoU is the composition of R and U. This describes the state in which the students, ti, with respect to the related subject requirement, rj, fit the colleges, bk. Thus,ti \u2208 A and bk \u2208 C, where i, j, and k take values from 1,\u2026, n.\u03b1RoU and \u03b2RoU of the composition T\u2009=\u2009R o U are as follows , College of Medicine (M), College of Agricultural Engineering Sciences (AE), College of Sport Sciences (Sp), College of Science (S)} be the set of colleges the students are vying for -FS. We illustrated that this type produces membership grades larger than intuitionistic and Pythagorean fuzzy sets which are already defined in the literature. However, Fermatean fuzzy sets give a larger space of membership grades than -FS. We summarize the relationships in terms of the space of membership and non-membership grades in the following figure.Regarding topological structure, we illustrated that every fuzzy topology in the sense of Chang (intuitionistic fuzzy topology and Pythagorean fuzzy topology) is a -fuzzy topology. In contrast, every -fuzzy topological space is a Fermatean fuzzy topological space because every -fuzzy subset of a set can be considered as a Fermatean fuzzy subset. The next example elaborates that Fermatean fuzzy topological space need not be a -fuzzy topological space.X={x1, x2}. Consider the following family of Fermatean fuzzy subsets \u03c4={1X, 0X, D1, D2}, whereLet X, \u03c4) is a Fermatean fuzzy topological space, but  is not a -fuzzy topological space.Observe that -fuzzy sets and studied their relationship with intuitionistic fuzzy, Pythagorean fuzzy, and Fermatean fuzzy sets. In addition, some operators on -fuzzy sets are defined and their relationships have been proved. The notions of -fuzzy topology, -fuzzy neighborhood, and -fuzzy continuous mapping were studied. Furthermore, we introduced the concept of -fuzzy points and studied separation axioms in -fuzzy topological space. We also introduced the concept of relation to -fuzzy sets, called -FR. Moreover, based on academic performance, the application of -FSs was explored on student placement using the proposed composition relation.In future work, more applications of -fuzzy sets may be studied; also, -fuzzy soft sets may be studied. In addition, we will try to introduce the compactness and connectedness in -fuzzy topological spaces. The motivation and objectives of this extended work are given step by step in this paper."}
{"text": "Wymioty s\u0105 cz\u0119sto spotykanym objawem w populacji pediatrycznej. Ich etiologia jest zr\u00f3\u017cnicowana, pocz\u0105wszy od \u0142agodnych zaburze\u0144 czynno\u015bciowych, a\u017c do ci\u0119\u017ckich zagra\u017caj\u0105cych \u017cyciu chor\u00f3b uk\u0142adowych. Najcz\u0119\u015bciej wyst\u0119puj\u0105 w przebiegu chor\u00f3b przewodu pokarmowego, ale mog\u0105 dotyczy\u0107 tak\u017ce wielu innych schorze\u0144 zlokalizowanych poza nim. Z uwagi na zr\u00f3\u017cnicowan\u0105 etiologi\u0119 obejmuj\u0105c\u0105 r\u00f3\u017cne uk\u0142ady i narz\u0105dy niekiedy mog\u0105 sprawia\u0107 trudno\u015bci diagnostyczne. W niniejszym artykule przedstawiono przypadek zespo\u0142u Panayiotopou/osa \u2013 dzieci\u0119cej padaczki potylicznej o wczesnym pocz\u0105tku . Charakterystyczn\u0105 cech\u0105 tego zespo\u0142u s\u0105 napady z objawami ze strony uk\u0142adu autonomicznego lub wyst\u0119powanie autonomicznych stan\u00f3w padaczkowych. Dominuj\u0105cymi objawami s\u0105 wymioty i nudno\u015bci, kt\u00f3re w pierwszej kolejno\u015bci najcz\u0119\u015bciej sugeruj\u0105 zapalenie \u017co\u0142\u0105dka czy jelit, b\u0105d\u017a migren\u0119 lub proces rozrostowy w o\u015brodkowym uk\u0142adzie nerwowym. Rzadko w diagnostyce r\u00f3\u017cnicowej bierze si\u0119 pod uwag\u0119 mo\u017cliwo\u015b\u0107 wyst\u0119powania wymiot\u00f3w jako elementu napadu padaczkowego. Celem naszej pracy jest zwr\u00f3cenie uwagi na trudno\u015bci w zdefiniowaniu w\u0142a\u015bciwej przyczyny nawracaj\u0105cych wymiot\u00f3w. Niejednokrotnie, pomimo szczeg\u00f3\u0142owo zebranego wywiadu oraz szerokiej diagnostyki, dopiero bezpo\u015brednie stwierdzenie charakterystycznych objaw\u00f3w dla danego zespo\u0142u pozwala na ustalenie ostatecznego rozpoznania. Helicobacter pylori, zapalenie w\u0105troby, zapalenie trzustki, ostre schorzenia chirurgiczne  czy zesp\u00f3\u0142 cyklicznych wymiot\u00f3w. W ustaleniu ich etiologii podstawowe znaczenie ma dok\u0142adnie zebrany wywiad zawieraj\u0105cy pytania o pocz\u0105tek, cz\u0119stotliwo\u015b\u0107, czas trwania wymiot\u00f3w oraz objawy towarzysz\u0105ce [Wymioty s\u0105 nieswoistym, cz\u0119sto wyst\u0119puj\u0105cym u dzieci objawem, o zr\u00f3\u017cnicowanej etiologii, pocz\u0105wszy od \u0142agodnych zaburze\u0144 czynno\u015bciowych, a\u017c do ci\u0119\u017ckich zagra\u017caj\u0105cych \u017cyciu chor\u00f3b uk\u0142adowych. W wi\u0119kszo\u015bci przypadk\u00f3w problem ust\u0119puje szybko i samoistnie. Wymioty najcz\u0119\u015bciej wyst\u0119puj\u0105 w przebiegu chor\u00f3b przewodu pokarmowego, ale mog\u0105 dotyczy\u0107 tak\u017ce wielu innych schorze\u0144 zlokalizowanych poza nim . Etiologrzysz\u0105ce ,2.Celem pracy jest przedstawienie trudno\u015bci diagnostycznych u pacjentki, u kt\u00f3rej przyczyn\u0105 uporczywych wymiot\u00f3w ostatecznie okaza\u0142a si\u0119 by\u0107 padaczka. Po prawie 2 latach wyst\u0119powania nawracaj\u0105cych wymiot\u00f3w, kilkukrotnych hospitalizacjach, przeprowadzeniu wielokierunkowej diagnostyki, dopiero po zaobserwowaniu charakterystycznych objaw\u00f3w w trakcie pobytu w Klinice rozpoznano zesp\u00f3\u0142 Panayiotopoulosa, nale\u017c\u0105cy do padaczek o pocz\u0105tku w dzieci\u0144stwie .Dziewczynka 11-letnia zosta\u0142a przyj\u0119ta do Kliniki z powodu nawracaj\u0105cych wymiot\u00f3w i b\u00f3l\u00f3w brzucha celem diagnostyki i leczenia. Z wywiadu wiadomo, \u017ce sporadyczne wymioty pojawi\u0142y si\u0119 u dziewczynki w 9 r\u017c. W pierwszym roku ich wyst\u0119powania nie obserwowano cykliczno\u015bci ani towarzysz\u0105cych dolegliwo\u015bci, natomiast w 10 r\u017c. ich cz\u0119stotliwo\u015b\u0107 uleg\u0142a nasileniu, a rodzice zaobserwowali, \u017ce wymioty poprzedza\u0142y: z\u0142e samopoczucie, l\u0119k , zawroty g\u0142owy, senno\u015b\u0107 oraz silny b\u00f3l g\u0142owy. Najcz\u0119\u015bciej epizody trwa\u0142y 1 dzie\u0144, ust\u0119powa\u0142y samoistnie. W ci\u0105gu miesi\u0105ca poprzedzaj\u0105cego hospitalizacj\u0119 w Klinice dolegliwo\u015bci wyst\u0119powa\u0142y \u015brednio co tydzie\u0144. Wymiociny nigdy nie zawiera\u0142y niepokoj\u0105cych domieszek, nie by\u0142o konieczno\u015bci wyr\u00f3wnywania zaburze\u0144 jonowych oraz poda\u017cy p\u0142yn\u00f3w infuzyjnych.Dziewczynka z tego powodu by\u0142a wielokrotnie hospitalizowana w r\u00f3\u017cnych o\u015brodkach w Polsce. W wyniku przeprowadzonej diagnostyki wykluczono: proces rozrostowy OUN w oddziale neurologii , zaburzenia gospodarki w\u0119glowodanowej w oddziale endokrynologii, borelioz\u0119 oraz padaczk\u0119 w kolejnym oddziale neurologii (na podstawie prawid\u0142owego 24-godzinnego zapisu EEG nie potwierdzono padaczki). Dziewczynk\u0119 dwukrotnie konsultowa\u0142 psycholog, nie stwierdzaj\u0105c istotnych nieprawid\u0142owo\u015bci. Z wywiadu wynika r\u00f3wnie\u017c, \u017ce rodzicom kilkukrotnie sugerowano konieczno\u015b\u0107 konsultacji psychiatrycznej. Poza tym rodzice zg\u0142osili pojawianie si\u0119 grudkowych, sw\u0119dz\u0105cych zmian sk\u00f3rnych w zgi\u0119ciach \u0142okciowych i na twarzy.Przy przyj\u0119ciu do Kliniki stan dziecka by\u0142 dobry. Z odchyle\u0144 od normy w badaniu przedmiotowym stwierdzono: pojedyncze zmiany grudkowe w zgi\u0119ciach \u0142okciowych. W badaniach laboratoryjnych bez nieprawid\u0142owo\u015bci. Ze wzgl\u0119du na zg\u0142aszane w trakcie epizod\u00f3w zawroty g\u0142owy dziewczynka by\u0142a konsultowana przez laryngologa, kt\u00f3ry nie potwierdzi\u0142 cech ostrego uszkodzenia cz\u0119\u015bci obwodowej zmys\u0142u r\u00f3wnowagi. Konsultuj\u0105cy psychiatra stwierdzi\u0142 mo\u017cliwo\u015b\u0107 wyst\u0119powania zachowa\u0144 hipochondrycznych.W kolejnym dniu hospitalizacji dziewczynka od rana zg\u0142asza\u0142a pogorszenie samopoczucia i zawroty g\u0142owy. W godzinach popo\u0142udniowych, w pozycji le\u017c\u0105cej, wyst\u0105pi\u0142y jako\u015bciowe zaburzenia \u015bwiadomo\u015bci pod postaci\u0105 pogorszenia kontaktu logicznego, z towarzysz\u0105cymi automatyzmami w formie g\u0142o\u015bnego prze\u0142ykania \u015bliny utrzymuj\u0105ce si\u0119 oko\u0142o 10 minut oraz wymioty. Nast\u0119pnie dosz\u0142o do utraty przytomno\u015bci i wyst\u0105pi\u0142 uog\u00f3lniony napad padaczkowy toniczno-kloniczny trwaj\u0105cy oko\u0142o 2 minut. Po odzyskaniu przytomno\u015bci obserwowano stopniowe ust\u0119powanie ilo\u015bciowych zaburze\u0144 \u015bwiadomo\u015bci (senno\u015b\u0107 ponapadow\u0105).Na podstawie przebiegu klinicznego i otrzymanych wynik\u00f3w bada\u0144 wysuni\u0119to podejrzenie zespo\u0142u Panayiotopoulosa. Dziewczynk\u0119 przekazano do Oddzia\u0142u Neurologii Dzieci\u0119cej celem dalszej diagnostyki i leczenia. Ponownie wykonano EEG, w kt\u00f3rym w stanie czuwania w odprowadzeniach potylicznych stwierdzono czynno\u015b\u0107 napadow\u0105 o charakterze iglic, w pozosta\u0142ych odprowadzeniach, bez nieprawid\u0142owo\u015bci. Do leczenia w\u0142\u0105czono na sta\u0142e preparat kwasu walproinowego. Od tej pory, tj. przez kolejne 8 miesi\u0119cy, do dzi\u015b nie wyst\u0105pi\u0142 \u017caden epizod drgawkowy, nie obserwowano zaburze\u0144 \u015bwiadomo\u015bci ani wymiot\u00f3w.Zesp\u00f3\u0142 Panayiotopoulosa to dzieci\u0119ca padaczka potyliczna o wczesnym pocz\u0105tku . Po raz Zesp\u00f3\u0142 Panayiotopoulosa zosta\u0142 stosunkowo niedawno wyodr\u0119bniony, dlatego pocz\u0105tkowo cz\u0119sto\u015b\u0107 wyst\u0119powania nie by\u0142a jednoznacznie ustalona , 10. ObePocz\u0105tek objaw\u00f3w wyst\u0119puje mi\u0119dzy 1-14 rokiem \u017cycia, ze szczytem miedzy 3-6 r\u017c. (76% przypadk\u00f3w) , 14, 15.Do napad\u00f3w autonomicznych koniecznych do rozpoznania zespo\u0142u Panayiotopoulosa nale\u017c\u0105: nudno\u015bci,wymioty, hipersaliwacja, sinica, blado\u015b\u0107 pow\u0142ok sk\u00f3rnych, zaburzenia termoregulacji, b\u00f3le g\u0142owy, zaburzenia rytmu serca i oddechu ,16. WymiNudno\u015bci i wymioty wyst\u0119puj\u0105ce u pacjent\u00f3w z zespo\u0142em Panayiotopoulosa cz\u0119sto pocz\u0105tkowo nie s\u0105 traktowane, jako element napadu padaczkowego, ale sugeruj\u0105 podejrzenie zapalenia \u017co\u0142\u0105dka i jelit, choroby lokomocyjnej b\u0105d\u017a migreny, alergii pokarmowej, natomiast do\u0142\u0105czenie si\u0119 napad\u00f3w po\u0142owiczych i uog\u00f3lnionych nasuwa podejrzenie zapalenia m\u00f3zgu lub procesu rozrostowego w o\u015brodkowym uk\u0142adzie nerwowym , 12.W diagnostyce r\u00f3\u017cnicowej u przedstawionej pacjentki wzi\u0119to pod uwag\u0119 wszystkie powy\u017csze przyczyny, jak r\u00f3wnie\u017c uwzgl\u0119dniono dzieci\u0119ce zespo\u0142y epizodyczne, takie jak: zesp\u00f3\u0142 cyklicznych wymiot\u00f3w, migren\u0119 brzuszn\u0105 i przedsionkow\u0105, \u0142agodne napadowe zawroty g\u0142owy, \u0142agodny napadowy kr\u0119cz szyi . PacjentWynik badania neurologicznego pacjent\u00f3w z zespo\u0142em Panayiotopoulosa jest prawid\u0142owy. Choroba najcz\u0119\u015bciej nie powoduje zaburze\u0144 rozwoju psychoruchowego oraz intelektualnego dziecka. Czasami w okresie nasilenia napad\u00f3w u cz\u0119\u015bci dzieci mog\u0105 wyst\u0105pi\u0107 niewielkie zaburzenia funkcji wykonawczych i j\u0119zykowych .W zapisach EEG mi\u0119dzynapadowych wyst\u0119puj\u0105 zmiany napadowe wieloogniskowe pod postaci\u0105 iglic i zespo\u0142\u00f3w iglicy z fal\u0105 woln\u0105 o zmiennej lokalizacji w kolejnych badaniach. U 75% pacjent\u00f3w z zespo\u0142em Panayiotopoulosa w badaniu rejestrowano zmiany w okolicy potylicznej, u 25% zmiany w okolicy innej ni\u017c potyliczna, tylko u 2% wyst\u0119puj\u0105 wy\u0142adowania uog\u00f3lnione, natomiast u 9% nie stwierdzono zmian napadowych , 16, 19.EEG stwierdzono uog\u00f3lnione wy\u0142adowania zespo\u0142\u00f3w iglicy z fal\u0105 woln\u0105, zar\u00f3wno w zapisach wykonanych w stanie czuwania, jak i we \u015bnie, kt\u00f3re by\u0142y na pocz\u0105tku jedyn\u0105 manifestacj\u0105 zespo\u0142u . W zdecyZesp\u00f3\u0142 Panayiotopoulosa uwa\u017ca si\u0119 za padaczk\u0119 zwykle niewymagaj\u0105c\u0105 leczenia p/padaczkowego, napady maj\u0105 tendencj\u0119 do ust\u0119powania , ale opiZa rozpoznaniem zespo\u0142u Panayiotopoulosa u omawianej pacjentki przemawia: typowy wiek wyst\u0119powania objaw\u00f3w, typowe objawy autonomiczne pod postaci\u0105: wymiot\u00f3w, b\u00f3lu g\u0142owy, zwrotu ga\u0142ek ocznych, zaburze\u0144 \u015bwiadomo\u015bci (w naszym przypadku dziewczynka by\u0142a \u201ejakby nieobecna\u201d), l\u0119ku oraz g\u0142o\u015bnego prze\u0142ykania \u015bliny, zmiany w EEG (w odprowadzeniach potylicznych czynno\u015b\u0107 napadowa o charakterze iglic), prawid\u0142owy rozw\u00f3j dziecka oraz efekt kliniczny zastosowanego leczenia p/padaczkowego. Zesp\u00f3\u0142 Panayiotopoulosa jest chorob\u0105 o \u0142agodnym przebiegu, zwykle niezaburzaj\u0105c\u0105 prawid\u0142owego rozwoju psychoruchowego, dobrze znan\u0105 neurologom, ale jeszcze s\u0142abo pediatrom.Nawracaj\u0105ce wymioty u dzieci mog\u0105 by\u0107 objawem r\u00f3\u017cnych schorze\u0144, w tym padaczki.Wyst\u0105pienie zaburze\u0144 orofaryngealnych i uog\u00f3lnionych drgawek w trakcie hospitalizacji w Klinice pozwoli\u0142o na ustalenie w\u0142a\u015bciwego rozpoznania."}
{"text": "Deficyt aktywno\u015bci transaldolazy nale\u017cy do wrodzonych b\u0142\u0119d\u00f3w metabolizmu na szlaku przemiany pentoz, kt\u00f3ry do tej pory rozpoznano i opisano u 33 pacjent\u00f3w, w tym 4 z Polski.W artykule przedstawiono obraz kliniczny, patogenez\u0119 i diagnostyk\u0119 choroby. Autorzy przedstawili ponadto w\u0142asn\u0105 propozycj\u0119 algorytmu diagnostyki deficytu transaldolazy. Szlak pentozofosforanowy obejmuje ci\u0105g reakcji biochemicznych, zachodz\u0105cych w cytozolu, kt\u00f3rych produktami ko\u0144cowymi s\u0105 rybozo-5-fosforan i NADPH. Sumaryczne r\u00f3wnanie reakcji cyklu pentozofosforanowego jest nast\u0119puj\u0105ce:Deficyt aktywno\u015bci transaldolazy (ang. + + H2O \u2192 rybozo-5-fosforan + 2NADPH + 2H+ + CO2.glukozo-6-fosforan + 2NADPTa droga przemian biochemicznych spe\u0142nia dwie role:wytwarzanie NADPH (zredukowany dinukleotyd nikotynoamidoadeninowy), kt\u00f3ry spe\u0142nia rol\u0119 donora proton\u00f3w i elektron\u00f3w w redukcyjnych procesach biosyntezy.dostarczanie reszt rybozy do biosystezy nukleotydowi kwas\u00f3w nukleinowych.Szczeg\u00f3ln\u0105 intensywno\u015b\u0107 wykazuje on w okresie wzrastania p\u0142odu oraz w pierwszych latach \u017cycia , 2, 3.Przemiany na szlaku pentozofosforanowym dzielimy na dwie fazy: nieodwracaln\u0105 faz\u0119 oksydacyjn\u0105 i odwracaln\u0105 faz\u0119 nieoksydacyjn\u0105. Transaldolaza (EC 2.2.1.2), obok transketolazy, jest enzymem odwracalnej fazy cyklu pentozofosforanowego i katalizuje nast\u0119puj\u0105c\u0105 reakcj\u0119:sedoheptulozo-7-fosforan + gliceraldehydo-3-fosforan \u2192 erytrozo-4-fosforan + fruktozo-6-fosforanW wyniku deficytu transaldolazy dochodzi do nagromadzenia polioli \u2013 erytritolu, arabitolu, rybitolu, sedoheptitolu, perseitolu oraz siedmiow\u0119glowych cukr\u00f3w \u2212 sedoheptulozy, mannoheptulozy i fosfosedoheptulozy.Celem pracy jest analiza obrazu klinicznego, patogenezy i diagnostyki deficytu aktywno\u015bci transaldolazy na podstawie danych z pi\u015bmiennictwa oraz do\u015bwiadcze\u0144 w\u0142asnych. Jak do tej pory, w dost\u0119pnych publikacjach nie zosta\u0142 przedstawiony algorytm diagnostyczny, st\u0105d autorzy przedstawili propozycj\u0119 w\u0142asn\u0105 algorytmu diagnostyki TALDO.2Deficyt aktywno\u015bci transaldolazy zosta\u0142 po raz pierwszy opisany przez Verhoeven i wsp., w 2001 r., u kilkumiesi\u0119cznego niemowl\u0119cia z wewn\u0105trzmacicznym op\u00f3\u017anieniem wzrastania, hepatosplenomegali\u0105 i zaburzeniami krzepni\u0119cia . Biopsja), w tym 4 z Polski [4-18]. Wi\u0119kszo\u015b\u0107 (79%) z nich pochodzi\u0142a od spokrewnionych rodzic\u00f3w, w tym 12 z jednego arabskiego rodu [Dotychczas, TALDO opisano u 33 pacjent\u00f3w , w tym 4ego rodu .early-onset TALDO) \u2013 wyst\u0105pienie objaw\u00f3w w okresie prenatalnym lub w ci\u0105gu pierwszych 3 miesi\u0119cy \u017cycia, stanowi\u0105c\u0105 chorob\u0119 o ci\u0119\u017ckim przebiegu i niepomy\u015blnym rokowaniu \u2013 oraz posta\u0107 objawiaj\u0105c\u0105 si\u0119 p\u00f3\u017aniej (ang. late-onset TALDO) \u2013 powy\u017cej 3. miesi\u0105ca \u017cycia, stanowi\u0105c\u0105 powoli post\u0119puj\u0105c\u0105 chorob\u0119 z mo\u017cliwym d\u0142ugoletnim prze\u017cyciem [Obraz kliniczny TALDO jest zr\u00f3\u017cnicowany, jednak zawsze obejmuje post\u0119puj\u0105ce uszkodzenie w\u0105troby oraz nerek (dysfunkcja cewek nerkowych) . Wyr\u00f3\u017cnize\u017cyciem .Wi\u0119kszo\u015b\u0107 przypadk\u00f3w opisanych w literaturze stanowi\u0142o TALDO o wczesnym pocz\u0105tku. U noworodk\u00f3w i niemowl\u0105t stwierdzano ma\u0142\u0105 mas\u0119 urodzeniow\u0105 \u2013 odnotowywano ju\u017c wewn\u0105trzmaciczne op\u00f3\u017anienie wzrastania, powi\u0119kszenie w\u0105troby i \u015bledziony, niedokrwisto\u015b\u0107, ma\u0142op\u0142ytkowo\u015b\u0107, zaburzenia krzepni\u0119cia, a w kilku przypadkach obrz\u0119k p\u0142odowy. W niekt\u00f3rych przypadkach przebieg ci\u0105\u017cy charakteryzowa\u0142 si\u0119 masywnym przyrostem masy cia\u0142a u matek oraz du\u017cym i nieprawid\u0142owym \u0142o\u017cyskiem. Blisko po\u0142owa tych pacjent\u00f3w zmar\u0142a w okresie niemowl\u0119cym, najcz\u0119\u015bciej w wyniku krwawienia, jako efektu koagulopatii w przebiegu uszkodzenia w\u0105troby , 17, 18.Nieco odmiennie przedstawia si\u0119 przebieg deficytu transaldolazy o wczesnym pocz\u0105tku u polskich pacjent\u00f3w, kt\u00f3rzy mimo pocz\u0105tkowo z\u0142ego rokowania , wkroczyli w drug\u0105 dekad\u0119 \u017cycia (3 z 4 pacjent\u00f3w) z objawami skompensowanej marsko\u015bci w\u0105troby i kontrolowanej tubulopatii (poprzez leczenie substytucyjne) .TALDO o p\u00f3\u017anym pocz\u0105tku wyst\u0105pienia objaw\u00f3w, mimo potencjalnie lepszego rokowania, prowadzi powoli do post\u0119puj\u0105cego uszkodzenia w\u0105troby i nerek. LeDuc i wsp. opisali 8-letniego ch\u0142opca, u kt\u00f3rego TALDO zosta\u0142o wykryte przypadkowo (skrining rodzinny) . Al-ShamW przebiegu TALDO w w\u0105trobie stwierdza si\u0119 charakterystyczne drobnoguzkowe w\u0142\u00f3knienie, a ostatecznie marsko\u015b\u0107 w\u0105troby. Z czasem u chorych z d\u0142u\u017cszym okresem prze\u017cycia pojawiaj\u0105 si\u0119 objawy tubulopatii \u2013 hiperkalciuria, bia\u0142komocz cewkowy (k\u0142\u0119buszkowo-cewkowy) \u2013 zwykle jako pierwsze objawy, aminoaciduria, glukozuria, kwasica kanalikowa \u2013 oraz stopniowo rozwija si\u0119 przewlek\u0142a choroba nerek . W obrazhaemangioma cavernosum). W literaturze znajduj\u0105 si\u0119 ponadto opisy pacjent\u00f3w z cutis laxa oraz hipertrychoz\u0105 [Charakterystyczne dla TALDO s\u0105 zmiany sk\u00f3rne \u2013 poszerzona siatka naczy\u0144 i teleangiektazje na sk\u00f3rze tu\u0142owia (zw\u0142aszcza plec\u00f3w) oraz tendencja do tworzenia naczyniak\u00f3w jamistych , a badania hormonalne wykaza\u0142y u cz\u0119\u015bci z nich hipogonadyzm hipergonadotropowy.W niekt\u00f3rych przypadkach opisywane s\u0105 wrodzone wady serca, do najcz\u0119stszych nale\u017c\u0105 dro\u017cny przew\u00f3d t\u0119tniczy oraz ubytek przegrody mi\u0119dzyprzedsionkowej , 17, 18.Rozw\u00f3j psychoruchowy i umys\u0142owy pacjent\u00f3w s\u0105 prawid\u0142owe , 17, 18.3Deficyt aktywno\u015bci transaldolazy prowadzi do kumulacji w p\u0142ynach ustrojowych polioli \u2212 erytritolu, arabitolu, rybitolu, sedoheptitolu, perseitolu i siedmiow\u0119glowych cukr\u00f3w \u2212 sedoheptulozy, mannoheptulozy i fosfosedoheptulozy , 21, 22.Diagnostyka opiera si\u0119 na stwierdzeniu wydalania z moczem, jak i kumulacji w surowicy (metod\u0105 spektrometrii mas) rybitolu, arabitolu, erytritolu, oraz o najwy\u017cszej czu\u0142o\u015bci sedoheptulozy i sedoheptulozo-7-fosforanu , 21, 22.TALDO1)  .W patogenezie schorzenia podkre\u015bla si\u0119 toksyczny efekt dzia\u0142ania gromadzonych metabolit\u00f3w (ufosforylowanych cukr\u00f3w) na hepatocyty i kanaliki nerkowe. Inna hipoteza podkre\u015bla znaczenie stresu oksydacyjnego, kt\u00f3ry jest efektem braku NADPH koniecznego dla odtwarzania zredukowanej formy glutationu \u2013 kofaktora peroksydazy glutationowej przeciwdzia\u0142aj\u0105cej stresowi oksydacyjnemu , 4. Redu4Leczenie pacjent\u00f3w z TALDO jest wy\u0142\u0105cznie objawowe. Upo\u015bledzenie czynno\u015bci syntetycznej w\u0105troby mo\u017ce wi\u0105za\u0107 si\u0119 z konieczno\u015bci\u0105 okresowej poda\u017cy albumin czy osoczowych czynnik\u00f3w krzepni\u0119cia. Podobnie, upo\u015bledzenie czynno\u015bci cewek nerkowych mo\u017ce wymaga\u0107 suplementacji nieorganicznych fosforan\u00f3w, poda\u017cy alfa-kalcydiolu czy wyr\u00f3wnywania zaburze\u0144 elektrolitowych.Dyskusyjne s\u0105 wskazania do ewentualnego przeszczepienia w\u0105troby. Niekt\u00f3rzy autorzy sugeruj\u0105 mo\u017cliwo\u015b\u0107 uszkodzenia przeszczepionej w\u0105troby , 15, 16.Przeszczepienie w\u0105troby u polskich pacjent\u00f3w wobec wzgl\u0119dnie stabilnego ich stanu i r\u00f3wnoczesnego uszkodzenia nerek nie jest obecnie brane pod uwag\u0119 . Ze wzgl5U pacjent\u00f3w z niewyja\u015bnionym post\u0119puj\u0105cym uszkodzeniem w\u0105troby i nerek wskazane jest wykonanie badania kumulacji polioli. Potwierdzenia/wykluczenia choroby wymaga r\u00f3wnie\u017c ka\u017cdy przypadek noworodka/niemowl\u0119cia z hepatosplenomegali\u0105, niedokrwisto\u015bci\u0105, ma\u0142op\u0142ytkowo\u015bci\u0105 i zaburzeniami krzepni\u0119cia, zw\u0142aszcza urodzonego z ma\u0142\u0105 urodzeniow\u0105 mas\u0105 cia\u0142a (oraz du\u017cym przyrostem masy cia\u0142a matki w trakcie trwania ci\u0105\u017cy).W\u0142\u00f3knienie, marsko\u015b\u0107 w\u0105troby o nieustalonej etiologii zawsze wymaga badania st\u0119\u017cenia polioli."}
{"text": "W ostatnich latach obserwuje si\u0119 sta\u0142e zwi\u0119kszanie cz\u0119sto\u015bci nowych zachorowa\u0144 na nieswoiste zapalenia jelit u dzieci, st\u0105d na ca\u0142ym \u015bwiecie trwaj\u0105 badania maj\u0105ce na celu okre\u015blenie prawdopodobnych czynnik\u00f3w predysponuj\u0105cych do wyst\u0105pienia choroby.Ocena wyst\u0119powania wczesnych czynnik\u00f3w ryzyka u dzieci z nieswoistym zapaleniem jelit.Do grupy badanej zakwalifikowano 60 dzieci z nieswoistym zapaleniem jelit, w wieku 2-19 lat, b\u0119d\u0105cych pod opiek\u0105 Kliniki Pediatrii, Alergologii i Gastroenterologii. Rozpoznanie choroby ustalono na podstawie obowi\u0105zuj\u0105cych kryteri\u00f3w [ESPGHAN]. W\u015br\u00f3d metod badanych zastosowano zwalidowany kwestionariusz ankiety w\u0142asnej konstrukcji. Grup\u0119 kontroln\u0105 stanowi\u0142o 60 dzieci bez nieswoistego zapalenia jelit.Analiza drogi porodu w grupie badanej wykaza\u0142a, \u017ce 14 (23%) dzieci zosta\u0142o urodzonych drog\u0105 ci\u0119cia cesarskiego, a 46 (77%) si\u0142ami natury. U dzieci urodzonych drog\u0105 ci\u0119cia cesarskiego ryzyko wyst\u0105pienia choroby Le\u015bniowskiego-Crohna by\u0142o nieistotnie wi\u0119ksze, natomiast u dzieci z wrzodziej\u0105cym zapaleniem jelita grubego - nieistotnie mniejsze w por\u00f3wnaniu z grup\u0105 kontroln\u0105. \u015aredni czas karmienia piersi\u0105 w grupie badanej by\u0142 nieistotnie kr\u00f3tszy w por\u00f3wnaniu z grup\u0105 kontroln\u0105. Wcze\u015bniactwo i ekspozycja na dym nikotynowy w 1 roku \u017cycia zwi\u0119ksza\u0142y nieistotnie ryzyko wyst\u0105pienia wrzodziej\u0105cego zapalenia jelita grubego. Ci\u0119\u017cki przebieg choroby dotyczy\u0142 prawie po\u0142owy dzieci nara\u017conych na antybiotykoterapi\u0119 w 1r\u017c., ale nie by\u0142a to r\u00f3\u017cnica istotna w por\u00f3wnaniu do dzieci o \u0142agodniejszym przebiegu nieswoistego zapalenia jelit.Nie wykazano istotnych r\u00f3\u017cnic w zakresie wyst\u0119powania wczesnych czynnik\u00f3w ryzyka u dzieci z nieswoistym zapaleniem jelit w por\u00f3wnaniu do zdrowych dzieci. Wy\u0142onienie czynnik\u00f3w mog\u0105cych mie\u0107 wp\u0142yw na rozw\u00f3j choroby wymaga dalszych bada\u0144. Nieswoiste zapalenia jelit (NZJ) stanowi\u0105 grup\u0119 schorze\u0144 charakteryzuj\u0105cych si\u0119 przewlek\u0142ym procesem zapalnym o z\u0142o\u017conej i prawdopodobnie wieloczynnikowej etiologii. Dwiema najcz\u0119\u015bciej wyst\u0119puj\u0105cymi jednostkami chorobowymi w tej grupie s\u0105 choroba Le\u015bniowskiego - Crohna (ch. L-C) oraz wrzodziej\u0105ce zapalenie jelita grubego (WZJG). Na NZJ wskazywa\u0107 mog\u0105 objawy takie jak: nawracaj\u0105ce b\u00f3le brzucha, biegunka (czasami z domieszk\u0105 krwi lub \u015bluzu), niedokrwisto\u015b\u0107, utrata masy cia\u0142a lub te\u017c niezwi\u0105zane z infekcj\u0105 podwy\u017cszenie wyk\u0142adnik\u00f3w stanu zapalnego.W ci\u0105gu ostatnich lat obserwuje si\u0119 sta\u0142e zwi\u0119kszanie cz\u0119sto\u015bci nowych zachorowa\u0144 na NZJ . Por\u00f3wnuWyniki przeprowadzonych dotychczas bada\u0144 wskazuj\u0105 na udzia\u0142 w patogenezie NZJ nie tylko czynnik\u00f3w genetycznych, ale r\u00f3wnie\u017c \u015brodowiskowych . Uznaje Ludzkie jelito stanowi przyk\u0142ad z\u0142o\u017conego ekosystemu, w kt\u00f3rym znajduje si\u0119 bogata i zr\u00f3\u017cnicowana mikrobiota, kt\u00f3ra posiada kluczowy wp\u0142yw na prawid\u0142owe funkcjonowanie przewodu pokarmowego cz\u0142owieka. Na rozw\u00f3j mikrobioty ju\u017c od najwcze\u015bniejszego okresu \u017cycia ma wp\u0142yw wiele czynnik\u00f3w, np. spos\u00f3b karmienia dziecka, przebyte infekcje czy te\u017c antybiotykoterapia w pierwszych latach \u017cycia oraz droga porodu. Kolejnym wa\u017cnym elementem wp\u0142ywaj\u0105cym na kszta\u0142towanie si\u0119 mikrobioty przewodu pokarmowego jest rodzaj karmienia w pierwszym roku \u017cycia . Istotn\u0105 rol\u0119 mo\u017ce odgrywa\u0107 r\u00f3wnie\u017c fakt przebycia w pierwszych miesi\u0105cach \u017cycia antybiotykoterapii, co ma wp\u0142yw zar\u00f3wno na bakterie chorobotw\u00f3rcze, jak r\u00f3wnie\u017c mikrobiot\u0119 fizjologiczn\u0105.Bior\u0105c pod uwag\u0119 znaczenie wczesnego okresu \u017cycia dziecka na jego zdrowie w przysz\u0142o\u015bci, mo\u017cliwy jest zwi\u0105zek stanu noworodka po urodzeniu, sposobu od\u017cywiania oraz innych czynnik\u00f3w oko\u0142oporodowych z rozwojem NZJ. Kolejnym czynnikiem, o udokumentowanym znaczeniu, w rozwoju chor\u00f3b cywilizacyjnych, w tym NZJ, jest palenie papieros\u00f3w . Na znacCelem badania jest ocena cz\u0119sto\u015bci wyst\u0119powania i zwi\u0105zku wybranych czynnik\u00f3w \u015brodowiskowych w okresie pre- i postnatalnym z rozwojem NZJ w grupie dzieci pozostaj\u0105cych pod opiek\u0105 Kliniki Pediatrii, Alergologii i Gastroenterologii w Bydgoszczy.Pocz\u0105tkowo badaniem obj\u0119to 105 pacjent\u00f3w z NZJ pozostaj\u0105cych pod sta\u0142\u0105 opiek\u0105 Kliniki Pediatrii, Alergologii i Gastroenterologii CM SU nr 1 w Bydgoszczy w okresie od 1 stycznia 2015 do 30 czerwca 2016 r. Grup\u0119 badan\u0105 stanowi\u0142o 66 dzieci z NZJ, z czego u 36  rozpoznano ch. L-C., u 24  WZJG, a u 6 9,09%) NZJ nieokre\u015blone. Z uwagi na nisk\u0105 liczebno\u015b\u0107 grupy dzieci choruj\u0105cych na posta\u0107 nieokre\u015blon\u0105 NZJ, pacjent\u00f3w tych pomini\u0119to w dalszej analizie. Zatem ostatecznie do badanej grupy zaliczono 60 dzieci. Schemat przebiegu badania zamieszczono na Rycinie 1. NZJ rozpoznawano na podstawie obowi\u0105zuj\u0105cych kryteri\u00f3w klinicznych, endoskopowych, histopatologicznych i radiologicznych zgodnie z zaleceniami ESPGHAN [,09% NZJ Zgodnie z klasyfikacj\u0105 parysk\u0105 badane dzieci podzielono na trzy grupy wiekowe: < 10 lat, 10-17 lat oraz \u2265 17 lat [\u015aredni wiek rozpoznania ch. L.-C. wynosi\u0142 12,04 \u00b1 3,42 lat (mediana 12 lat), a WZJG 10,60 \u00b1 4,27 lat (mediana 11 lat) . U 15  dzieci z NZJ, chorob\u0119 rozpoznano w wieku poni\u017cej 10 lat, u 5  w wieku 17 lat i powy\u017cej. Wiek rozpoznania NZJ u pozosta\u0142ych dzieci, tj. u 40  mie\u015bci\u0142 si\u0119 mi\u0119dzy 10-17 lat. W badanej grupie u 22  dzieci stosowano leczenie biologiczne. Charakterystyk\u0119 grupy badanej i kontrolnej przedstawiono w U Manna-Whitneya, a dla rozk\u0142ad\u00f3w zmiennych kategoryzowanych \u2013 test chi-kwadrat. W celu wyodr\u0119bnienia czynnik\u00f3w maj\u0105cych istotne znaczenie dla wyst\u0105pienia WZJG i ch. L-C zastosowano model regresji logistycznej  oraz iloraz szans wraz z podaniem przedzia\u0142\u00f3w ufno\u015bci. Iloraz szans dla zmiany jednostkowej zastosowanych parametr\u00f3w oraz iloraz szans dla zmiany r\u00f3wnej zakresowi analizowanych zmiennych zosta\u0142 obliczony z 95% przedzia\u0142ami ufno\u015bci. We wszystkich por\u00f3wnaniach przyj\u0119to poziom istotno\u015bci statystycznej p<0.05. Analiz\u0119 wynik\u00f3w bada\u0144 przeprowadzono przy u\u017cyciu programu komputerowego STATISTICA 13.1 .Metoda statystyczna: W przypadku danych o charakterze ilo\u015bciowym wyniki opisano przy u\u017cyciu nast\u0119puj\u0105cych parametr\u00f3w: liczba przypadk\u00f3w (N), warto\u015b\u0107 \u015brednia (M), odchylenie standardowe (SD), minimum (Min), maksimum (Max) oraz mediana (Me). Dane o charakterze zmiennych jako\u015bciowych (kategoryzowanych) zosta\u0142y opisane przez zestawienie wzgl\u0119dnej liczby przypadk\u00f3w (N) i ich procentowego udzia\u0142u w badanej grupie. W celu wykrycia istnienia pomi\u0119dzy badanymi grupami ewentualnych r\u00f3\u017cnic w warto\u015bciach poszczeg\u00f3lnych czynnik\u00f3w ilo\u015bciowych zastosowano test Analiza drogi porodu w grupie badanej wykaza\u0142a, \u017ce 14  dzieci zosta\u0142o urodzonych drog\u0105 ci\u0119cia cesarskiego, a 46  si\u0142ami natury. W\u015br\u00f3d dzieci z rozpoznan\u0105 ch. L-C jedena\u015bcioro  urodzi\u0142o si\u0119 drog\u0105 ci\u0119cia cesarskiego, a 25  si\u0142ami natury, natomiast w\u015br\u00f3d dzieci z WZJG troje  urodzi\u0142o si\u0119 drog\u0105 ci\u0119cia cesarskiego, a 21  si\u0142ami natury. W grupie kontrolnej si\u0142ami natury urodzi\u0142o si\u0119 46  dzieci, a drog\u0105 ci\u0119cia cesarskiego \u2013 14  badanych. Nie wykazano r\u00f3\u017cnic w zakresie drogi porodu mi\u0119dzy dzie\u0107mi z NZJ a dzie\u0107mi z grupy kontrolnej , oraz mi\u0119dzy dzie\u0107mi z ch. L-C a dzie\u0107mi z WZJG . \u015aredni czas trwania ci\u0105\u017cy u matek dzieci z grupy badanej wynosi\u0142 39,27\u00b11,73 i by\u0142 por\u00f3wnywalny z grup\u0105 kontroln\u0105 . Z pierwszej ci\u0105\u017cy urodzi\u0142o si\u0119 20  dzieci z ch. L-C, 9  z WZJG oraz 33  dzieci z grupy kontrolnej. Por\u00f3wnuj\u0105c te trzy grupy wykazano, \u017ce dzieci z WZJG nieistotnie rzadziej pochodzi\u0142y z pierwszej ci\u0105\u017cy ni\u017c dzieci z ch. L-C oraz dzieci z grupy kontrolnej .U dziewi\u0119ciu (15%) matek dzieci z NZJ wyst\u0119powa\u0142y poronienia co najmniej 1 raz w \u017cyciu przed urodzeniem badanego dziecka, z czego 6  stanowi\u0142y matki dzieci z ch. L-C, a 3  matki dzieci z WZJG. Analogiczne dane w grupie kontrolnej dotyczy\u0142y matek 7  dzieci .Troje (5%) dzieci z NZJ urodzi\u0142o si\u0119 z mas\u0105 < 2500 g, z czego jedno choruje na ch. L-C , a 2 dzieci na WZJG . W grupie kontrolnej 5  dzieci urodzi\u0142o si\u0119 z nisk\u0105 urodzeniow\u0105 mas\u0105 cia\u0142a. Nie wykazano istotnych r\u00f3\u017cnic w zakresie \u015bredniej masy urodzeniowej mi\u0119dzy dzie\u0107mi z NZJ a grup\u0105 kontroln\u0105, jak r\u00f3wnie\u017c mi\u0119dzy dzie\u0107mi z ch. L-C a WZJG \u015arednia ocena w skali Apgar dzieci z grupy badanej wynosi\u0142a 9,15 \u00b11,16 i by\u0142a por\u00f3wnywalna z ocen\u0105 dzieci z grupy kontrolnej. Wi\u0119kszo\u015b\u0107 pacjent\u00f3w z grupy badanej, jak i kontrolnej \u2013 odpowiednio 55  oraz 56  - urodzi\u0142o si\u0119 w stanie dobrym, z punktacj\u0105 8-10 pkt. w skali Apgar. Punktacja ta dotyczy\u0142a 29 dzieci z ch. L-C oraz 21  dzieci w WZJG.\u015aredni czas karmienia piersi\u0105 (wy\u0142\u0105cznie lub w spos\u00f3b mieszany) w grupie badanej by\u0142 nieistotnie kr\u00f3tszy w por\u00f3wnaniu z grup\u0105 kontroln\u0105 . Wy\u0142\u0105cznie karmionych piersi\u0105 przez 1-6 miesi\u0119cy by\u0142o 24 (40%) dzieci, z czego u 13  rozpoznano ch. L-C , natomiast u 11  dzieci - WZJG . W grupie kontrolnej wy\u0142\u0105czne karmienie piersi\u0105 dotyczy\u0142o 25  pacjent\u00f3w.Antybiotyk w 1. roku \u017cycia stosowano u 22  dzieci z NZJ oraz u 24 (40%) dzieci w grupie kontrolnej . Nieistotnie cz\u0119\u015bciej antybiotykoterapia dotyczy\u0142a dzieci z ch. L-C, tj. 15 , w por\u00f3wnaniu z pacjentami z WZJG, tj. 7 , .W pierwszym roku \u017cycia 16  dzieci z NZJ by\u0142o cho\u0107 raz hospitalizowanych. W grupie kontrolnej dotyczy\u0142o to 20  dzieci, . W\u015br\u00f3d dzieci, kt\u00f3re by\u0142y hospitalizowane w 1 roku \u017cycia 10  stanowi\u0142y dzieci z ch. L-C  oraz 6  z WZJG (co stanowi 25% badanych z tym rozpoznaniem).Matki 6 (10%) dzieci z NZJ pali\u0142y papierosy w czasie trwania ci\u0105\u017cy, z czego u 2 dzieci rozpoznano ch. L-C, a u 4 WZJG, natomiast w grupie kontrolnej dotyczy\u0142o to 4  matek .W badanej grupie dzieci, na ekspozycj\u0119 na dym tytoniowy  w 1 roku \u017cycia nara\u017cono 26  dzieci z NZJ, natomiast w grupie kontrolnej - 24 (40%) dzieci, .W grupie badanej eksponowanych na dzia\u0142anie dymu tytoniowego w okresie niemowl\u0119cym by\u0142o 16  dzieci z ch. L-C, oraz 10  dzieci z WZJG.Powy\u017csze dane przedstawiono w Celem oceny znaczenia wybranych czynnik\u00f3w \u015brodowiskowych w rozwoju NZJ przeprowadzono wieloczynnikow\u0105 analiz\u0119 regresji logistycznej. Nie wykazano wyst\u0119powania istotnego zwi\u0105zku NZJ z wcze\u015bniactwem, kolejno\u015bci\u0105 urodzenia, rodzajem porodu i antybiotykoterapi\u0105, hospitalizacj\u0105 w 1 roku \u017cycia oraz ekspozycj\u0105 na dym tytoniowy . Ryzyko Cho\u0107 wykazano r\u00f3\u017cnice w cz\u0119sto\u015bci antybiotykoterapii, hospitalizacji w 1r\u017c, okresie karmienia piersi\u0105 w zale\u017cno\u015bci od p\u0142ci dzieci z NZJ, oraz stwierdzono, \u017ce NZJ o wczesnym pocz\u0105tku cz\u0119\u015bciej dotyczy\u0142 dzieci poddanych ekspozycji na dym tytoniowy w ci\u0105\u017cy ni\u017c NZJ, kt\u00f3ry rozwin\u0105\u0142 si\u0119 u dzieci po 10r\u017c, to r\u00f3\u017cnice te by\u0142y nieistotne statystycznie. Ci\u0119\u017cki przebieg choroby dotyczy\u0142 prawie po\u0142owy dzieci nara\u017conych na antybiotykoterapi\u0119 w 1r\u017c., ale nie by\u0142a to r\u00f3\u017cnica istotna w por\u00f3wnaniu do dzieci o \u0142agodniejszym przebiegu NZJ .Mimo bada\u0144 prowadzonych od lat patogeneza NZJ nadal nie jest znana. W\u015br\u00f3d potencjalnych czynnik\u00f3w sprawczych bierze si\u0119 pod uwag\u0119 predyspozycj\u0119 gene-tyczn\u0105, zmiany mikrobioty jelitowej, defekty odporno\u015bci nabytej i wrodzonej oraz inne czynniki \u015brodowiskowe . Wy\u0142onieW badaniu w\u0142asnym nie wykazano istotnych r\u00f3\u017cnic w zakresie wyst\u0119powania wybranych czynnik\u00f3w \u015brodowiskowych, takich jak: czynniki oko\u0142oporodowe, antybiotykoterapia i hospitalizacje w 1 roku \u017cycia oraz ekspozycja na dym tytoniowy u dzieci z NZJ w por\u00f3wnaniu do zdrowych dzieci.Dysbioza w zakresie mikrobioty przewodu pokarmowego bywa zwi\u0105zana z wieloma schorzeniami, w tym r\u00f3wnie\u017c z NZJ. Charakteryzuje j\u0105 zmniejszona r\u00f3\u017cnorodno\u015b\u0107 tych bakterii przewodu pokarmowego, kt\u00f3re wykazuj\u0105 dzia\u0142anie przeciwzapalne \u2013 np. Faecalibacterium oraz zwi\u0119kszona ilo\u015b\u0107 Enterobacteriaceae (np. E. coli) . Znacz\u0105cZaburzenia w sk\u0142adzie mikrobioty przewodu pokarmowego mog\u0105 pojawi\u0107 si\u0119 ju\u017c w najwcze\u015bniejszych latach \u017cycia. Wp\u0142yw na ni\u0105 ma m.in. droga porodu, a co za tym idzie naturalna kolonizacja dziecka mikrobiot\u0105 matki podczas przej\u015bcia przez kana\u0142 rodny. St\u0105d te\u017c powsta\u0142a hipoteza, \u017ce urodzenie metod\u0105 ci\u0119cia cesarskiego mo\u017ce prowadzi\u0107 do dysbiozy, zwi\u0119kszaj\u0105c ryzyko zachorowania na NZJ. Sevelsted i wsp. stwierdzili istotn\u0105 statystycznie zale\u017cno\u015b\u0107 pomi\u0119dzy urodzeniem dziecka metod\u0105 ci\u0119cia cesarskiego a wzrostem ryzyka p\u00f3\u017aniejszego zachorowania na NZJ, astm\u0119 oraz og\u00f3lnoustrojowe choroby tkanki \u0142\u0105cznej . R\u00f3wnie\u017cWed\u0142ug najnowszych danych pochodz\u0105cych z 150 kraj\u00f3w na ca\u0142ym \u015bwiecie, obecnie 18,6% wszystkich porod\u00f3w odbywa si\u0119 drog\u0105 ci\u0119cia cesarskiego, a w Europie wska\u017anik ten si\u0119ga 25% . W badanKolejnym istotnym czynnikiem ryzyka NZJ i innych chor\u00f3b cywilizacyjnych jest stosowanie nieprawid\u0142owej diety. Amre i wsp. wykazali zale\u017cno\u015b\u0107 pomi\u0119dzy stosowaniem diety niezbilansowanej pod wzgl\u0119dem zawarto\u015bci kwas\u00f3w t\u0142uszczowych, warzyw oraz owoc\u00f3w, a zwi\u0119kszonym ryzykiem zachorowania na ch. L-C . Nie wiaZmiany r\u00f3wnowagi mikrobioty jelitowej mog\u0105 by\u0107 powodowane r\u00f3wnie\u017c poprzez stosowanie antybiotyk\u00f3w. Ponadto wiadomo, \u017ce leki te moduluj\u0105 odpowied\u017a immunologiczn\u0105, a tym samym mog\u0105 wp\u0142ywa\u0107 na rozw\u00f3j NZJ. Wykazano zwi\u0105zek pomi\u0119dzy antybiotykoterapi\u0105 w wieku niemowl\u0119cym i wczesnym dzieci\u0144stwie, a p\u00f3\u017aniejsz\u0105 zwi\u0119kszon\u0105 predyspozycj\u0105 do zachorowania na NZJ, zw\u0142aszcza ch. L-C ,27,28. NKolejnym wa\u017cnym czynnikiem rozwoju chor\u00f3b cywilizacyjnych, w tym NZJ, jest nara\u017cenie na dzia\u0142anie dymu tytoniowego. Udowodniono, i\u017c palacze z ch. L-C maj\u0105 ci\u0119\u017cszy przebieg choroby ni\u017c osoby niepal\u0105ce, a zaprzestanie palenia mo\u017ce z\u0142agodzi\u0107 ten proces . W badanJednym z cel\u00f3w naszego badania by\u0142a pr\u00f3ba oceny zwi\u0105zku czynnik\u00f3w oko\u0142oporodowych takich jak \u2013 czas trwania ci\u0105\u017cy, urodzeniowa masa cia\u0142a oraz punktacja w skali Apgar z rozwojem NZJ w badanej populacji dzieci. Na podstawie przeprowadzonych bada\u0144 nie wykazali\u015bmy jednak istotnych r\u00f3\u017cnic w tym zakresie mi\u0119dzy dzie\u0107mi z NZJ a dzie\u0107mi zdrowymi.Jak nam wiadomo, badanie w\u0142asne jest pierwszym badaniem podejmuj\u0105cym pr\u00f3b\u0119 oceny zwi\u0105zku wybranych czynnik\u00f3w \u015brodowiskowych z rozwojem NZJ w populacji polskich dzieci. S\u0142abym punktem badania jest ma\u0142a liczebno\u015b\u0107 grupy badanej, z czego mo\u017ce wynika\u0107 brak istotnych zale\u017cno\u015bci statystycznych. Przeprowadzone badanie traktujemy jako badanie wst\u0119pne wymagaj\u0105ce kontynuacji.Nie wykazano istotnych r\u00f3\u017cnic w zakresie wyst\u0119powania wczesnych czynnik\u00f3w ryzyka u badanych dzieci z NZJ w por\u00f3wnaniu do zdrowych dzieci. Wy\u0142onienie czynnik\u00f3w mog\u0105cych mie\u0107 wp\u0142yw na rozw\u00f3j NZJ wymaga dalszych bada\u0144."}
{"text": "Zapalenie jelita cienkiego i okr\u0119\u017cnicy wywo\u0142ane bia\u0142kami pokarmowymi (food protein-induced enterocolitis syndrome \u2013 FPIES) jest rodzajem lgE-nieza/e\u017cnej alergii pokarmowej, o r\u00f3\u017cnym stopniu ci\u0119\u017cko\u015bci. Ostra posta\u0107 choroby manifestuje si\u0119 wymiotami, nadmiern\u0105 senno\u015bci\u0105 i blado\u015bci\u0105 sk\u00f3ry, kt\u00f3re pojawiaj\u0105 si\u0119 zwykle w ci\u0105gu 7-4 godzin od spo\u017cycia pokarmu wyzwalaj\u0105cego, i mo\u017ce powadzi\u0107 do wstrz\u0105su. Pokarmy indukuj\u0105ce objawy to najcz\u0119\u015bciej: mleko krowie, soja, ry\u017c i owies. Przewlek\u0142a posta\u0107 FP/ES jest typowa dla niemowl\u0105t karmionych mlekiem modyfikowanym lub mieszank\u0105 sojow\u0105 i objawia si\u0119 nawracaj\u0105cymi wymiotami, biegunk\u0105 oraz s\u0142abymi przyrostami masy cia\u0142a. U wi\u0119kszo\u015bci pacjent\u00f3w z FP/ES, do ustalenia rozpoznania i zidentyfikowania pokarm\u00f3w wyzwalaj\u0105cych objawy wystarcza szczeg\u00f3\u0142owa analiza wywiadu chorobowego. W przypadkach w\u0105tpliwych przydatna jest doustna pr\u00f3ba prowokacji pokarmowej. Leczenie FP/ES polega na eliminacji pokarm\u00f3w wywo\u0142uj\u0105cych objawy, monitorowaniu post\u0119pu choroby oraz edukacji opiekun\u00f3w. Wi\u0119kszo\u015b\u0107 dzieci nabywa tolerancj\u0119 pokarmow\u0105 w wieku 3-5 lat Niekiedy mog\u0105 by\u0107 oznak\u0105 zapalenia jelita cienkiego i okr\u0119\u017cnicy wywo\u0142anego bia\u0142kami pokarmowymi (food protein-induced enterocolitis syndrome \u2013 FPIES), armowymi , 6, 7, 8armowymi , 4, 8. Warmowymi . W ninieCho\u0107 pierwszy opis FPIES pochodzi z 1940 r., to schorzenie po raz pierwszy zosta\u0142o formalnie zdefiniowane w po\u0142owie lat siedemdziesi\u0105tych ubieg\u0142ego wieku . Wg Mi\u0119dChoroba ujawnia si\u0119 zwykle mi\u0119dzy 2 a 7 m\u017c., przy czym FPIES indukowany bia\u0142kami mleka krowiego (BMK) czy soj\u0105 dotyczy zwykle niemowl\u0105t <6 m\u017c., a indukowany pokarmami sta\u0142ymi \u2013 niemowl\u0105t mi\u0119dzy 6 a 12 m\u017c., zgodnie z czasem wprowadzania pokarm\u00f3w do diety [do diety . FPIES wwcze\u015bnie, tj. poni\u017cej 9 m\u017c. lub p\u00f3\u017ano, tj. powy\u017cej 9 m\u017c. [\u0142agodnej postaci choroby typowe s\u0105: nawracaj\u0105ce wymioty z/lub bez biegunki, blado\u015b\u0107 i niewielka senno\u015b\u0107. W ci\u0119\u017ckiej postaci wyst\u0119puj\u0105: nawracaj\u0105ce, nasilone wymioty z/lub bez biegunki, blado\u015b\u0107, senno\u015b\u0107, odwodnienie, hipotensja, wstrz\u0105s i zaburzenia metaboliczne. W zale\u017cno\u015bci od przebiegu wyr\u00f3\u017cnia si\u0119 ostr\u0105 i przewlek\u0142\u0105 posta\u0107 choroby [FPIES nale\u017cy \u201etraktowa\u0107, jako potencjalny stan nag\u0142y, stanowi\u0105cy zagro\u017cenie \u017cycia, objawiaj\u0105cy si\u0119 wymiotami o op\u00f3\u017anionym pocz\u0105tku i/lub wodnist\u0105/krwist\u0105 biegunk\u0105, kt\u00f3re w ostateczno\u015bci mog\u0105 spowodowa\u0107 niestabilno\u015b\u0107 hemodynamiczn\u0105 oraz spadek ci\u015bnienia t\u0119tniczego krwi, u co najmniej 15% pacjent\u00f3w\u201d [ej 9 m\u017c. . Przebie choroby , 8, 12.FPIES - posta\u0107 ostra wyst\u0119puje, gdy pokarm wyzwalaj\u0105cy jest spo\u017cywany w pewnych odst\u0119pach czasu lub po okresie jego wcze\u015bniejszej eliminacji. Charakterystyczne dla tej postaci s\u0105 nawracaj\u0105ce wymioty, rozpoczynaj\u0105ce si\u0119 w ci\u0105gu 1-4 godzin od przyj\u0119cia pokarmu, z towarzysz\u0105c\u0105 nadmiern\u0105 senno\u015bci\u0105 i blado\u015bci\u0105 sk\u00f3ry. Niekiedy wyst\u0119puje wodnista biegunka (czasem z krwi\u0105 lub \u015bluzem), kt\u00f3ra pojawia si\u0119 zwykle w ci\u0105gu 5-10 godzin od spo\u017cycia pokarmu, najcz\u0119\u015bciej w ci\u0105gu doby. Objawy zwykle ust\u0119puj\u0105 w ci\u0105gu 24 godzin. Rozw\u00f3j dziecka jest zazwyczaj prawid\u0142owy, a pomi\u0119dzy epizodami ostrego FPIES nie obserwuje si\u0119 \u017cadnych niepokoj\u0105cych objaw\u00f3w. Op\u00f3\u017aniony pocz\u0105tek oraz brak objaw\u00f3w sk\u00f3rnych i oddechowych sugeruj\u0105 reakcj\u0119 og\u00f3lnoustrojow\u0105, ale odmienn\u0105 od anafilaksji. Ci\u0119\u017ckie reakcje mog\u0105 prowadzi\u0107 do hipotermii, methemoglobinemii, rozwoju kwasicy metabolicznej, niedoci\u015bnienia i wstrz\u0105su, co mo\u017ce pocz\u0105tkowo sugerowa\u0107 posocznic\u0119 , 12. NalKryteria rozpoznania postaci ostrej FPIES: pacjent powinien spe\u0142ni\u0107 kryterium wi\u0119ksze i co najmniej 3 kryteria mniejsze [mniejsze .Kryterium wi\u0119ksze: wymioty pojawiaj\u0105ce si\u0119 w ci\u0105gu 1-4 godzin po spo\u017cyciu podejrzanego pokarmu i brak typowych IgE-zale\u017cnych alergicznych objaw\u00f3w sk\u00f3rnych lub oddechowych.Kryteria mniejsze:Drugi (lub kolejny) epizod nawracaj\u0105cych wymiot\u00f3w po spo\u017cyciu podejrzanego pokarmu.Nawracaj\u0105ce wymioty w ci\u0105gu 1-4 h po spo\u017cyciu pokarmu.Nadmierna senno\u015b\u0107.Nadmierna blado\u015b\u0107 sk\u00f3ry.Konieczno\u015b\u0107 wizyty w oddziale ratunkowym.Konieczno\u015b\u0107 zastosowania p\u0142ynoterapii do\u017cylnej.Biegunka w ci\u0105gu 24 godzin (zwykle 5-10 godzin od spo\u017cycia podejrzanego pokarmu).Hipotensja.Hipotermia.FPIES \u2013 posta\u0107 przewlek\u0142a rozwija si\u0119 u niemowl\u0105t karmionych regularnie mieszank\u0105 mleczn\u0105 lub sojow\u0105. Obj awia si\u0119 okresowymi wymiotami, przewlek\u0142\u0105 biegunk\u0105, zaburzeniami przyrostu masy cia\u0142a i rozwoju fizycznego. Ci\u0119\u017ckie postaci przewlek\u0142ego FPIES mog\u0105 tak\u017ce prowadzi\u0107 do odwodnienia i wstrz\u0105su. Po wyeliminowaniu szkodliwego pokarmu objawy ust\u0119puj\u0105 w ci\u0105gu 3-10 dni [3-10 dni . Ponowne3-10 dni , 11, 14.Kryteria rozpoznania postaci przewlek\u0142ej FPIES: brak jednoznacznych kryteri\u00f3w wi\u0119kszych i mniejszych, ale okre\u015blono objawy typowe dla \u0142agodnej i ci\u0119\u017ckiej postaci przewlek\u0142ego FPIES [go FPIES ,8.Posta\u0107 \u0142agodna \u2013 pokarm indukuj\u0105cy objawy spo\u017cywany jest w ni\u017cszych dawkach . Objawy: okresowe wymioty i/lub biegunka, zwykle s\u0142abe przyrosty masy cia\u0142a (< 10 g/dzie\u0144 u m\u0142odych niemowl\u0105t*), ale bez cech odwodnienia i kwasicy metabolicznej [olicznej , 12.Posta\u0107 ci\u0119\u017cka \u2013 pokarm indukuj\u0105cy objawy wyst\u0119puje regularnie w diecie chorego (np. mleko modyfikowane). Objawy: okresowo wyst\u0119puj\u0105ce, intensywne wymioty, biegunka (r\u00f3wnie\u017c z krwi\u0105), czasami odwodnienie i kwasica metaboliczna [boliczna , 12.Najwa\u017cniejszym kryterium rozpoznania przewlek\u0142ego FPIES jest ust\u0105pienie objaw\u00f3w w ci\u0105gu kilku dni po wyeliminowaniu szkodliwego pokarmu(\u00f3w) z diety i nag\u0142y nawr\u00f3t objaw\u00f3w po ich ponownym spo\u017cyciu. Bez potwierdzenia dodatnim wynikiem OFC, rozpoznanie przewlek\u0142ego FPIES pozostaje domniemane.wiek pacjenta, miejsce zamieszkania czy wsp\u00f3\u0142-wyst\u0119powanie IgE-zale\u017cnej alergii na pokarmy .Na obraz kliniczny FPIES maj\u0105 wp\u0142yw takie czynniki jak: Niemowl\u0119ta z FPIES wywo\u0142anym przez bia\u0142ka mleka krowiego (BMK) lub soj\u0119 w wieku poni\u017cej 2 miesi\u0119cy, wykazuj\u0105 istotnie wi\u0119ksze ryzyko wyst\u0105pienia biegunki, krwi w kale i op\u00f3\u017anienia rozwoju fizycznego, w por\u00f3wnaniu ze starszymi niemowl\u0119tami, kt\u00f3re cz\u0119\u015bciej prezentuj\u0105 same wymioty, bez biegunek .Istnienie r\u00f3\u017cnic narodowo\u015bciowych w obrazie klinicznym FPIES sugeruj\u0105 wyniki bada\u0144 dzieci o r\u00f3\u017cnym miejscu zamieszkania. Nomura i wsp. zaobserwowali wyst\u0119powanie gor\u0105czki u 13% oraz krwawych stolc\u00f3w u 47% badanych niemowl\u0105t z Japonii, a a\u017c 47% dzieci mia\u0142o dodatnie wyniki specyficznych przeciwcia\u0142 IgE (Sie) . Ten fennast\u0119puj\u0105ca: wymioty \u2013 100%, nadmierna senno\u015b\u0107 \u2013 77%, biegunka \u2013 25% i biegunka z krwi\u0105 \u2013 4,5%, a dodatni wynik PTS odnotowano jedynie u 5% chorych . PrzewlePatofizjologia FPIES nie jest do ko\u0144ca poznana. Uwa\u017ca si\u0119, \u017ce g\u0142\u00f3wn\u0105 rol\u0119 w rozwoju procesu zapalnego w jelicie pe\u0142ni\u0105 antygenowo swoiste limfocyty T, kt\u00f3re w wyniku aktywacji przez alergeny pokarmowe uwalniaj\u0105 cytokiny prozapalne . Wzrost Dla FPIES nie ma charakterystycznych bada\u0144, a rozpoznanie opiera si\u0119 przede wszystkim na wywiadzie, w kt\u00f3rym stwierdza si\u0119 wyst\u0119powanie powtarzalnych, charakterystycznych objaw\u00f3w po spo\u017cyciu okre\u015blonego pokarmu, z popraw\u0105 po jego eliminacji ,6. Je\u015bliWykonanie OFC w celach diagnostycznych powinno by\u0107 zarezerwowane w przypadku, gdy , 5, 24:wywiad chorobowy jest niejasny,pokarm wyzwalaj\u0105cy jest niezidentyfikowany,przebieg objaw\u00f3w jest nietypowy, np. objawy pojawiaj\u0105 si\u0119 po kilku minutach od spo\u017cycia przy braku slgE lub dolegliwo\u015bci utrzymuj\u0105 si\u0119 pomimo eliminacji podejrzanego pokarmu z diety,podejrzewa si\u0119 przewlek\u0142y FPIES .Protok\u00f3\u0142 wykonania OFC:Dawk\u0119 pokarmu prowokuj\u0105cego okre\u015blono jako 0,06 do 0,6 g bia\u0142ka/kg masy cia\u0142a  [Ka\u017cda OFC wymaga \u015bcis\u0142ego nadzoru lekarza. Przed pr\u00f3b\u0105 zaleca si\u0119 zabezpieczenie obwodowego dost\u0119-pu do\u017cylnego (50% dzieci z dodatnim wynikiem OFC mo\u017ce wymaga\u0107 leczenia p\u0142ynami do\u017cylnymi) , 2, 6. Dy cia\u0142a) , 6, 24. y cia\u0142a) ,6, 24. Wy cia\u0142a) , 6, 24. y cia\u0142a) ,8.Kryterium wi\u0119ksze dodatniej OFC: wymioty pojawiaj\u0105ce si\u0119 w ci\u0105gu 1-4 godzin po spo\u017cyciu podejrzanego pokarmu i brak klasycznych IgE-zale\u017cnych alergicznych objaw\u00f3w sk\u00f3rnych lub oddechowych.Kryteria mniejsze dodatniej OFC:Nadmierna senno\u015b\u0107Blado\u015b\u0107Biegunka w ci\u0105gu 5-10 godzin po spo\u017cyciu pokarmuHipotoniaHipotermiaZwi\u0119kszona liczba neutrofil\u00f3w o ponad 1500 kom\u00f3rek/ ml krwi w stosunku do liczby wyj\u015bciowej (wzrost ten osi\u0105ga maksimum po 6 godzinach od spo\u017cycia pokarmu spustowego).Przyjmuje si\u0119, \u017ce lekarz prowadz\u0105cy mo\u017ce uzna\u0107 wynik OFC za dodatni, nawet, je\u015bli zosta\u0142o spe\u0142nione tylko kryterium wi\u0119ksze, bez \u017cadnych kryteri\u00f3w mniejszych, poniewa\u017c nie zawsze istnieje mo\u017cliwo\u015b\u0107 oceny liczby neutrofil\u00f3w w odpowiednim czasie. W przypadku ujemnego wyniku OFC zaleca si\u0119 regularne spo\u017cywanie pokarmu .Rozpoznanie FPIES nie jest \u0142atwe i cz\u0119sto bywa op\u00f3\u017anione , 14, 17.mleko krowie i soja. Alergia na te pokarmy cz\u0119sto wsp\u00f3\u0142istnieje ze sob\u0105. W badaniach Ruffner i wsp., u oko\u0142o po\u0142owy pacjent\u00f3w z FPIES indukowanym mlekiem krowim, obserwowano reakcj\u0119 na soj\u0119 [U 83% pacjent\u00f3w, FPIES jest wywo\u0142ywany przez 1 alergen, u 17% przez 2 alergeny a u 5-10% przez 3 alergeny , 25. Do  na soj\u0119 , 27, 28. na soj\u0119 , 30.ziarna zb\u00f3\u017c , kurczak, indyk, wo\u0142owina, ryby (szczeg\u00f3lnie w Hiszpanii i W\u0142oszech), orzechy, warzywa , ro\u015bliny str\u0105czkowe: orzeszki ziemne, zielony groszek, soczewica; owoce  i owoce morza [FPIES mo\u017ce rozwin\u0105\u0107 si\u0119 wskutek reakcji na pokarmy uzupe\u0142niaj\u0105ce takie jak: ce morza , 4, 17. ce morza . DzieckoDane na temat wsp\u00f3\u0142wyst\u0119powania FPIES na kilka r\u00f3\u017cnych pokarm\u00f3w s\u0105 zr\u00f3\u017cnicowane. W USA odnotowywano najwi\u0119cej przypadk\u00f3w reakcji na 2 lub 3 pokarmy, natomiast w Australii, a przede wszystkim we W\u0142oszech, wyra\u017anie dominuje \u201e1-pokarmowe\u201d FPIES [Pomimo tego, i\u017c FPIES jest postaci\u0105 IgE-niezale\u017cnej alergii pokarmowej (AP), u wielu pacjent\u00f3w obserwuje si\u0119 wsp\u00f3\u0142wyst\u0119powanie chor\u00f3b atopowych. Rodzinne obci\u0105\u017cenie atopi\u0105 dotyczy 54% pacjent\u00f3w z FPIES . Oko\u0142o 3U niemowl\u0105t z przewlek\u0142ym FPIES w badaniach laboratoryjnych obserwuje si\u0119 r\u00f3\u017cny stopie\u0144 niedokrwisto\u015bci, hipoalbuminemii i leukocytoz\u0119 z eozynofili\u0105. Badanie stolca mo\u017ce ujawni\u0107 obecno\u015b\u0107 krwi utajonej, neutrofil\u00f3w, eozynofil\u00f3w, kryszta\u0142\u00f3w Charcota-Leydena i/lub substancji redukuj\u0105cych .Nie zaleca si\u0119 rutynowego wykonywania bada\u0144 endoskopowych z biopsj\u0105 . Mog\u0105 on90% z nich wyniki punktowych test\u00f3w sk\u00f3rnych (PTS), jak r\u00f3wnie\u017c slgE s\u0105 ujemne w momencie wst\u0119pnej diagnozy [Nie nale\u017cy rutynowo ocenia\u0107 slgE dla alergen\u00f3w pokarmowych u pacjent\u00f3w z FPIES. U diagnozy , 27, 30.diagnozy , 18, 19.Zwi\u0105zek mi\u0119dzy FPIES a obecno\u015bci\u0105 Sie mo\u017cna t\u0142umaczy\u0107 w dwojaki spos\u00f3b. Zwi\u0119kszona przepuszczalno\u015b\u0107 jelitowa u chorych z FPIES, mo\u017ce skutkowa\u0107 zwi\u0119kszon\u0105 przenikalno\u015bci\u0105 alergen\u00f3w pokarmowych, co prowadzi do wytworzenia Sie. Odwrotnie, wyst\u0119puj\u0105ce miejscowo w b\u0142onie \u015bluzowej jelit przeciwcia\u0142a IgE, mog\u0105 u\u0142atwia\u0107 wychwyt alergen\u00f3w, a w konsekwencji prowadzi\u0107 do indukcji \u201eog\u00f3lnego\u201d procesu zapalnego jelit.Bior\u0105c pod uwag\u0119 potencjalny udzia\u0142 alergenowo swoistych limfocyt\u00f3w T, w patofizjologii FPIES oraz w mechanizmie powstawania odczyn\u00f3w podczas wykonywania ATP, metoda ta wydaje si\u0119 by\u0107 obiecuj\u0105cym narz\u0119dziem diagnostycznym. Istniej\u0105 pojedyncze badania, kt\u00f3re oceniaj\u0105 przydatno\u015b\u0107 test\u00f3w w diagnostyce FPIES , 31. OkrOstre epizody FPIES mog\u0105 przebiega\u0107 w spos\u00f3b \u0142agodny (1-2 epizody wymiot\u00f3w), umiarkowany (>3 epizod\u00f3w wymiot\u00f3w i nadmierna senno\u015b\u0107) lub ci\u0119\u017cki . Ten ostatni mo\u017ce prowadzi\u0107 do wyst\u0105pienia wstrz\u0105su hipowolemicznego. W\u00f3wczas nale\u017cy niezw\u0142ocznie za\u0142o\u017cy\u0107 obwodowy dost\u0119p do\u017cylny i rozpocz\u0105\u0107 intensywn\u0105 resuscytacj\u0119 p\u0142ynem izotonicznym  oraz roztw\u00f3r dekstrozy w postaci ci\u0105g\u0142ego wlewu do\u017cylnego (terapia podtrzymuj\u0105ca). Pojedyncza dawka do\u017cylnego metyloprednizolonu , przypuszczalnie mo\u017ce ograniczy\u0107 rozw\u00f3j procesu zapalnego, jednak \u017cadne badania nie rekomenduj\u0105 ich bezwzgl\u0119dnego zastosowania. Nale\u017cy r\u00f3wnie\u017c monitorowa\u0107 i w razie potrzeby korygowa\u0107 kwasic\u0119 metaboliczn\u0105 i zaburzenia elektrolitowe. W ci\u0119\u017ckich reakcjach pacjenci mog\u0105 wymaga\u0107 tak\u017ce tlenoterapii, wentylacji mechanicznej, zastosowania wazopresor\u00f3w, wodorow\u0119glan\u00f3w lub b\u0142\u0119kitu metylenowego w methemoglobinemii. Auto-strzykawki z adrenalin\u0105 nie s\u0105 rutynowo zalecane dla pacjent\u00f3w z FPIES. Wyj\u0105tek stanowi\u0105 chorzy, z wsp\u00f3\u0142istniej\u0105c\u0105 alergi\u0105 IgE-zale\u017cn\u0105, kt\u00f3rzy w opinii lekarza prowadz\u0105cego, s\u0105 potencjalnie nara\u017ceni na wyst\u0105pienie anafilaksji .Niezale\u017cnie od ci\u0119\u017cko\u015bci objaw\u00f3w ostrego epizodu FPIES, u dzieci >6 miesi\u0119cy, mo\u017cna rozwa\u017cy\u0107 zastosowanie ondansetronu  jako leczenia wspomagaj\u0105cego wymioty , 23, 32.Na podstawie danych wskazuj\u0105cych, \u017ce wi\u0119kszo\u015b\u0107 ostrych epizod\u00f3w zdarza si\u0119 w domu, ust\u0119puje bez leczenia, i nie ko\u0144czy si\u0119 zgonem, Sopo i wsp. sugeruj\u0105, \u017ce w trakcie ostrego, \u0142agodnego epizodu FPIES u dzieci powy\u017cej 1 r.\u017c, mo\u017cna przyj\u0105\u0107 postaw\u0119 wyczekuj\u0105c\u0105, ale tylko pod warunkiem mo\u017cliwego dost\u0119pu do natychmiastowego za\u0142o\u017cenia drogi do\u017cylnej i podania steryd\u00f3w. U dzieci m\u0142odszych zawsze wskazane jest natychmiastowe za\u0142o\u017cenie dost\u0119pu do\u017cylnego. Wa\u017cnym elementem post\u0119powania jest poinformowanie opiekun\u00f3w o konieczno\u015bci doustnego nawadniania. Ponadto wskazane jest zaopatrzenie pacjenta w pisemny plan post\u0119powania , 32.Do podstawowych zasad post\u0119powania z dzieckiem z FPIES nale\u017cy:ustalenie odpowiedniej diety eliminacyjnejedukacja opiekun\u00f3w dzieckaleczenie objaw\u00f3w po ekspozycji na alergizuj\u0105ce pokarmy monitorowanie post\u0119pu choroby, w tym nabywania tolerancji pokarmowejprzeprowadzenie konsultacji dietetycznej, celem ustalenia zasad przestrzegania diety eliminacyjnej i zastosowania suplementacji zapobiegaj\u0105cej wyst\u0105pieniu niedobor\u00f3w \u017cywieniowych , 33.Niemowl\u0119tom z podejrzeniem FPIES wywo\u0142anym przez BMK lub soj\u0119 zaleca si\u0119 unikanie wszelkich form tych pokarm\u00f3w, w tym produkt\u00f3w pieczonych, chyba\u017ce s\u0105 one ju\u017c zawarte w diecie i dobrze tolerowane , 5, 32. Z uwagi na ryzyko wsp\u00f3\u0142wyst\u0119powania FPIES na mleko i soj\u0119, nale\u017cy rozwa\u017cy\u0107 nadz\u00f3r lekarza podczas wprowadzania soi do diety i odwrotnie , 26, 27.Wi\u0119kszo\u015b\u0107 niemowl\u0105t nie reaguje na alergeny pokarmowe obecne w mleku matki, co wynika z faktu trawienia bia\u0142ek pokarmowych przez matk\u0119 oraz obecno\u015bci w mleku matczynym IgA i TGF\u00df. Wg Nowak-W\u0119grzyn i wsp. u \u017cadnego dziecka karmionego piersi\u0105 nie obserwowano objaw\u00f3w FPIES na pokarm spo\u017cywany przez matk\u0119 . Objawy Czas ponownego wprowadzenia wcze\u015bniej eliminowanego pokarmu do diety dziecka jest zmienny. Zale\u017cy od dotychczasowego przebiegu choroby. Spos\u00f3b wprowadzenia  powinien by\u0107 zawsze wsp\u00f3lnie ustalony mi\u0119dzy lekarzem i opiekunem dziecka, uwzgl\u0119dniaj\u0105c wiek pacjenta, liczb\u0119 pokarm\u00f3w wyzwalaj\u0105cych, wyniki slgE, ci\u0119\u017cko\u015b\u0107 poprzednich reakcji FPIES, a tak\u017ce komfort opiekuna oraz dost\u0119p i odleg\u0142o\u015b\u0107 do lokalnych oddzia\u0142\u00f3w ratunkowych .nie zaleca si\u0119 op\u00f3\u017anionego wprowadzania pokarm\u00f3w uzupe\u0142niaj\u0105cych do diety u tych dzieci [S\u0142odkie ziemniaki i zielony groszek zakwalifikowano do grupy wysokiego ryzyka, w kt\u00f3rej znalaz\u0142y si\u0119 tak\u017ce: mleko krowie, soja, p\u0142atki owsiane, ry\u017c, dr\u00f3b, jajo kurze, ryby i banany. Za bezpieczne dla pacjent\u00f3w z FPIES uznano owoce, takie jak: truskawki, jagody, \u015bliwki, brzoskwinie, arbuz i awokado, w\u015br\u00f3d pokarm\u00f3w mi\u0119snych - jagni\u0119cin\u0119 i wo\u0142owin\u0119, a w\u015br\u00f3d ziaren - komos\u0119 ry\u017cow\u0105 i proso. Autorzy podkre\u015blaj\u0105 jednak, i\u017c zawsze nale\u017cy bra\u0107 pod uwag\u0119 preferencje \u017cywieniowe danego dziecka. W\u00f3wczas pomocne mo\u017ce by\u0107 u\u017cycie produkt\u00f3w z grupy umiarkowanego ryzyka, kt\u00f3re powszechnie wyst\u0119puj\u0105 w diecie polskich niemowl\u0105t, jak np. marchewka, bia\u0142y ziemniak, zielona fasolka, jab\u0142ko, gruszka, kaszka pszenna czy chrupki/p\u0142atki kukurydziane lub j\u0119czmienne. Wed\u0142ug Venter i Groetch wprowadzanie zb\u00f3\u017c mo\u017cna rozpocz\u0105\u0107 od kukurydzy, nast\u0119pnie j\u0119czmienia, owsa i na ko\u0144cu ry\u017cu [Dzieci z FPIES indukowanym BMK lub soj\u0105 maj\u0105 zwi\u0119kszone ryzyko reakcji na pokarmy sta\u0142e, najcz\u0119\u015bciej ry\u017c lub owies, jednak h dzieci . Bazuj\u0105ch dzieci , 33. Zal\u0144cu ry\u017cu , 38.U niemowl\u0105t z ci\u0119\u017ckim FPIES na BMK i/lub soj\u0119 zaleca si\u0119 nadzorowane przez lekarza wprowadzanie pokarm\u00f3w sta\u0142ych. Nale\u017cy je wprowadza\u0107 pojedynczo, z nast\u0119pow\u0105 4-dniow\u0105 obserwacj\u0105 , 38. Z ustarszy wiek pacjenta w czasie wyst\u0105pienia pierwszego epizodu FPIES czy ustalenia diagnozy oraz dodatni wynik PTS, wi\u0105\u017ce si\u0119 z wolniejszym nabywaniem tolerancji pokarmowej. Caubet i wsp. r\u00f3wnie\u017c obserwowali wp\u0142yw wsp\u00f3\u0142wyst\u0119powania IgE-zale\u017cnej AP na wyd\u0142u\u017cenie czasu nabywania tolerancji u pacjent\u00f3w z FPIES [Rozw\u00f3j tolerancji na BMK i soj\u0119 nast\u0119puje wcze\u015bniej, ni\u017c na pokarmy sta\u0142e. W badaniu Lee i wsp. z Australii, w wieku 3 lat 88% dzieci tolerowa\u0142o mleko krowie, a 87% ry\u017c, podczas gdy procent pacjent\u00f3w toleruj\u0105cych jaja i ryby wynosi\u0142 odpowiednio: 12,5% i25% . Autorzy z FPIES . P\u0142e\u0107, w z FPIES . Wyniki  z FPIES . Wg Katz z FPIES . \u015aredni  z FPIES . W bryty z FPIES .Nie ma zalece\u0144 co do konkretnego czasu wykonania OFC celem oceny nabycia tolerancji pokarmowej u pacjent\u00f3w z FPIES, jednak zazwyczaj s\u0105 one podejmowane wci\u0105gu 12 do 18 miesi\u0119cy od ostatniej reakcji . FPIES jFPIES jest postaci\u0105 IgE-niezale\u017cnej alergii pokarmowej o r\u00f3\u017cnym stopniu ci\u0119\u017cko\u015bci, wyst\u0119puj\u0105c\u0105 u niemowl\u0105t i ma\u0142ych dzieci.Ostra posta\u0107 choroby manifestuje si\u0119 wymiotami, nadmiern\u0105 senno\u015bci\u0105 i blado\u015bci\u0105 sk\u00f3ry, kt\u00f3re pojawiaj\u0105 si\u0119 zwykle w ci\u0105gu 1-4 godzin od spo\u017cycia pokarmu, i/lub biegunk\u0105 5-10 godzin po ekspozycji.Przewlek\u0142a posta\u0107 FPIES jest typowa dla niemowl\u0105t karmionych mlekiem modyfikowanym lub mieszank\u0105 sojow\u0105 i objawia si\u0119 nawracaj\u0105cymi wymiotami, biegunk\u0105 oraz zaburzeniami przyrostu masy cia\u0142a.Pokarmami, kt\u00f3re najcz\u0119\u015bciej indukuj\u0105 FPIES s\u0105: mleko krowie, soja, ry\u017c i owies.Rozpoznanie opiera si\u0119 na wywiadzie; w w\u0105tpliwych przypadkach przydatna jest doustna pr\u00f3ba prowokacji pokarmem.Leczenie polega na zastosowaniu diety eliminacyjnej.Wi\u0119kszo\u015b\u0107 dzieci nabywa tolerancj\u0119 pokarmow\u0105 w wieku 3-5 lat."}
{"text": "The original version of this article unfortunately contained a mistake. The keywords \u201cNarcolepsy\u201d and \u201cPitolisant\u201d are missing.Keywords should be Craniopharyngioma \u00b7 Sleep \u00b7 Hypothalamic dysfunction \u00b7 Sleep-related breathing disorders \u00b7 Hypersomnolence \u00b7 Narcolepsy \u00b7 Pitolisant"}
{"text": "Post\u0119puj\u0105ca rodzinna cholestaza wewn\u0105trzw\u0105trobowa typu 3 (PFIC-3) nale\u017cy do grupy rodzinnych cholestaz wewn\u0105trzw\u0105trobowych, dziedziczonych w spos\u00f3b autosomalny recesywny. Patogeneza choroby wi\u0105\u017ce si\u0119 z obecno\u015bci\u0105 patogennych wariant\u00f3w molekularnych w genie ABCB4. Dotychczas, w literaturze opisano ok. 200 pacjent\u00f3w z r\u00f3\u017cnymi schorzeniami w\u0105troby i dr\u00f3g \u017c\u00f3\u0142ciowych, stanowi\u0105cych ekspresj\u0119 kliniczn\u0105 PFIC-3.Celem pracy jest charakterystyka patogenezy, obrazu klinicznego, diagnostyki oraz leczenia PFIC-3 na podstawie aktualnego przegl\u0105du pi\u015bmiennictwa. Progressive Familial Intrahepatic Cholestasis type 3, PFIC-3) [OMIM 602347], obok post\u0119puj\u0105cej rodzinnej cholestazy wewn\u0105trzw\u0105trobowej typu 1 i 2  oraz deficytu bia\u0142ka TJP-2 (ang. Tight Junction Protein 2) nale\u017cy do grupy rodzinnych cholestaz wewn\u0105trzw\u0105trobowych, dziedziczonych w spos\u00f3b autosomalny recesywny, b\u0119d\u0105cych przyczyn\u0105 post\u0119puj\u0105cego uszkodzenia w\u0105troby , po\u0142o\u017conego na chromosomie 7q21 i koduj\u0105cego bia\u0142ko MDR3 (ang. Multi-Drug Resistance class III), nale\u017c\u0105ce do rodziny bia\u0142ek ABC (ang. ATP-binding cassette) i podrodziny B (ABCB) [Patogeneza choroby wi\u0105\u017ce si\u0119 z obecno\u015bci\u0105 patogennych wariant\u00f3w molekularnych genu B (ABCB) , 6, 7.ABCB4 zidentyfikowano u pacjent\u00f3w z r\u00f3\u017cnymi schorzeniami w\u0105troby i dr\u00f3g \u017c\u00f3\u0142ciowych:Obecno\u015b\u0107 patogennych wariant\u00f3w molekularnych w co najmniej jednym allelu genu przej\u015bciowa cholestaza niemowl\u0119ca,idiopatyczna przewlek\u0142a (\u22656 miesi\u0119cy) cholestaza u starszych dzieci i doros\u0142ych,Low Phospholipid-Associated Cholelithiasis),kamica \u017c\u00f3\u0142ciowa zwi\u0105zana z niskim st\u0119\u017ceniem fosfolipid\u00f3w w \u017c\u00f3\u0142ci ,wewn\u0105trzw\u0105trobowa cholestaza ci\u0119\u017carnych (ang. cholestaza polekowa (indukowana np. stosowaniem \u015brodk\u00f3w antykoncepcyjnych) ,10,11,12Wszystkie powy\u017csze zaburzenia stanowi\u0105 ekspresj\u0119 kliniczn\u0105 post\u0119puj\u0105cej rodzinnej cholestazy wewn\u0105trzw\u0105trobowej typu 3.Szacuje si\u0119, \u017ce zaledwie 1/3 wszystkich przypadk\u00f3w PFIC-3 ma sw\u00f3j pocz\u0105tek w okresie niemowl\u0119cym (bardzo rzadko w okresie noworodkowym). Zwykle pierwsze objawy choroby pojawiaj\u0105 si\u0119 w p\u00f3\u017anym dzieci\u0144stwie, a tak\u017ce mog\u0105 zosta\u0107 zaobserwowane dopiero u doros\u0142ych.W zale\u017cno\u015bci od wieku wyst\u0105pienia pierwszych objaw\u00f3w, obraz kliniczny PFIC-3 jest r\u00f3\u017cny. U najm\u0142odszych dzieci choroba przebiega zwykle w postaci cholestazy z towarzysz\u0105cym \u015bwi\u0105dem sk\u00f3ry (czasami \u015bwi\u0105d sk\u00f3ry jest jedynym objawem); w badaniu przedmiotowym z odchyle\u0144 mo\u017cna stwierdzi\u0107 hepatomegali\u0119, rzadziej splenomegali\u0119, natomiast w wynikach bada\u0144 laboratoryjnych podwy\u017cszon\u0105 aktywno\u015b\u0107 aminotransferaz i gamma-glutamylotranspeptydazy oraz podwy\u017cszone st\u0119\u017cenie kwas\u00f3w \u017c\u00f3\u0142ciowych w surowicy.W naturalnym przebiegu PFIC-3 dochodzi do rozwoju nadci\u015bnienia wrotnego, st\u0105d objawy kliniczne u starszych dzieci czy doros\u0142ych mog\u0105 wynika\u0107 z powik\u0142a\u0144 nadci\u015bnienia wrotnego, np. krwawienia z \u017cylak\u00f3w prze\u0142yku. W badaniu przedmiotowym typowo stwierdza si\u0119 w\u00f3wczas splenomegali\u0119, a w wynikach bada\u0144 laboratoryjnych cechy hipersplenizmu , 4, 8-9.hepatocarcinoma czy cholangiocarcinoma, a tak\u017ce zwi\u0119kszone ryzyko wyst\u0105pienia cholestazy polekowej czy cholestazy ci\u0119\u017carnych [Pacjenci z PFIC-3 maj\u0105 zwi\u0119kszone ryzyko rozwoju kamicy p\u0119cherzyka \u017c\u00f3\u0142ciowego czy wewn\u0105trzw\u0105trobowych dr\u00f3g \u017c\u00f3\u0142ciowych (st\u0105d wielokrotnie mylnie objawy \u0142\u0105czone s\u0105 ze stwierdzan\u0105 kamic\u0105), zwi\u0119kszone ryzyko rozwoju \u0119\u017carnych , 12, 13.Danych literaturowych dotycz\u0105cych obrazu klinicznego PFIC-3 w populacji pediatrycznej jest stosunkowo niewiele. Do tej pory ukaza\u0142y si\u0119 opisy pojedynczych przypadk\u00f3w lub serie kilku przypadk\u00f3w. Jedn\u0105 z pierwszych publikacji by\u0142 opis 2 przypadk\u00f3w autorstwa de Vree i wsp. z 1998 roku. Pierwszy opis dotyczy\u0142 3-miesi\u0119cznego ch\u0142opca z nawracaj\u0105cymi epizodami \u017c\u00f3\u0142taczki cholestatycznej i \u015bwi\u0105du sk\u00f3ry, kt\u00f3ry trafi\u0142 do o\u015brodka referencyjnego w wieku 3 lat. Biopsja w\u0105troby wykonana w tym wieku wykaza\u0142a obecno\u015b\u0107 dokonanej marsko\u015bci narz\u0105du. Z uwagi na brak odpowiedzi na leczenie UDCA, kilka miesi\u0119cy po rozpoznaniu choroby wykonano zabieg przeszczepienia w\u0105troby. Drugi opis dotyczy\u0142 8-miesi\u0119cznego ch\u0142opca, u kt\u00f3rego jedynym objawem choroby by\u0142 \u015bwi\u0105d sk\u00f3ry. W wieku 3 lat wykonano biopsj\u0119 w\u0105troby, kt\u00f3ra wykaza\u0142a znaczne w\u0142\u00f3knienie narz\u0105du, bez dokonanej marsko\u015bci narz\u0105du. W leczeniu stosowano UDCA, jednak\u017ce z uwagi na cz\u0119\u015bciow\u0105 odpowied\u017a i progresj\u0119 choroby, w wieku 9 lat wykonano zabieg przeszczepienia w\u0105troby.Gastroenterology, gdzie dokonali charakterystyki fenotypu choroby w grupie 17 pacjent\u00f3w [Wi\u0119ksz\u0105 kohort\u0119 pacjent\u00f3w z PFIC-3 przedstawili Jacquemin i wsp. w 2001 roku na \u0142amach acjent\u00f3w . \u015aredni Journal of Paediatric Gastroenterology and Nutrition opublikowali wyniki wieloo\u015brodkowego badania przeprowadzonego we W\u0142oszech, w kt\u00f3rym opisali grup\u0119 28 dzieci z PFIC-3 [Z kolei, Colombo i wsp. w 2011 roku na \u0142amach z PFIC-3 . MedianaABCB4; mediana wieku pierwszych objaw\u00f3w wynosi\u0142a 2 lata.Podobny odsetek pacjent\u00f3w z PFIC-3 w grupie pacjent\u00f3w z cholestatyczn\u0105 chorob\u0105 w\u0105troby o nieustalonej etiologii, przedstawili Gordo-Gilart i wsp. w 2016 roku. W grupie 48 dzieci z przewlek\u0142\u0105 cholestatyczn\u0105 chorob\u0105 w\u0105troby o nieznanej przyczynie, u 9 pacjent\u00f3w zidentyfikowano patogenne mutacje w 1 allelu genu ATP8B1, ABCB11, ABCB4, TJP2. W\u015br\u00f3d badanej grupy zidentyfikowano 6 doros\u0142ych pacjent\u00f3w z PFIC-3, w tym jednego pacjenta z hepatocarcinoma.Warto podkre\u015bli\u0107, \u017ce o diagnostyce w kierunku PFIC-3 nale\u017cy tak\u017ce pami\u0119ta\u0107 u starszych dzieci oraz w\u015br\u00f3d doros\u0142ych. Vitale i wsp. w 2017 roku opublikowali wyniki badania, do kt\u00f3rego w\u0142\u0105czono 48 pacjent\u00f3w, w wieku powy\u017cej 6 lat z kryptogenn\u0105 cholestaz\u0105 (definiowan\u0105 jak\u0105 wzrost aktywno\u015bci gamma-glutamylotranspeptydazy lub obecno\u015b\u0107 \u015bwi\u0105du sk\u00f3ry z towarzysz\u0105cym wzrostem st\u0119\u017cenia kwas\u00f3w \u017c\u00f3\u0142ciowych w surowicy utrzymuj\u0105ce si\u0119 co najmniej 6 miesi\u0119cy), u kt\u00f3rych przeprowadzono badania molekularne metod\u0105 NGS (sekwencjonowanie nowej generacji) w oparciu o panel 4 gen\u00f3w: Journal of Hepatology ukaza\u0142y si\u0119 bardzo ciekawe wyniki pracy Doge i wsp. [ABCB4. Co ciekawe, a\u017c u 13 pacjent\u00f3w pierwsze objawy zaobserwowano dopiero w wieku doros\u0142ym. W\u015br\u00f3d kobiet wyst\u0105pienie cholestazy obserwowano w zwi\u0105zku ze stosowaniem doustnych \u015brodk\u00f3w antykoncepcyjnych lub w trakcie trwania ci\u0105\u017cy. Do chwili obecnej ukaza\u0142o si\u0119 wiele artyku\u0142\u00f3w wi\u0105\u017c\u0105cych obecno\u015b\u0107 patogennych wariant\u00f3w molekularnych genu ABCB4 z wyst\u0105pieniem wewn\u0105trzw\u0105trobowej cholestazy ci\u0119\u017carnych  [W 2017 roku na \u0142amach e i wsp. . W\u015br\u00f3d 4 ATP8B1) .Rozpoznanie PFIC-3 jest ustalane na podstawie:wywiad\u00f3w \u2013 rodzinne wyst\u0119powanie choroby, ale tak\u017ce poszczeg\u00f3lnych fenotyp\u00f3w: cholestazy, w tym cholestazy ci\u0119\u017carnych czy polekowej, kamicy p\u0119cherzyka \u017c\u00f3\u0142ciowego czy wewn\u0105trzw\u0105trobowych dr\u00f3g \u017c\u00f3\u0142ciowych,obrazu klinicznego \u2013 \u017c\u00f3\u0142taczka (mo\u017cliwe postaci bez\u017c\u00f3\u0142taczkowe), hepatomegalia, rzadziej hepatosplenomegalia, \u015bwi\u0105d sk\u00f3ry \u2013 czasami \u015bwi\u0105d sk\u00f3ry jest jedynym objawem klinicznym,bada\u0144 laboratoryjnych \u2013 hiperbilirubinemia z przewag\u0105 st\u0119\u017cenia bilirubiny bezpo\u015bredniej (cholestaza), podwy\u017cszona aktywno\u015b\u0107 gamma-glutamylotranspeptydazy (co odr\u00f3\u017cnia PFIC-3 od PFIC-1 czy PFIC-2), podwy\u017cszone st\u0119\u017cenie kwas\u00f3w \u017c\u00f3\u0142ciowych, podwy\u017cszona aktywno\u015b\u0107 aminotransferaz,bada\u0144 obrazowych \u2013 ultrasonografia jamy brzusznej mo\u017ce wykaza\u0107 powi\u0119kszenie w\u0105troby czy \u015bledziony, a tak\u017ce obecno\u015b\u0107 z\u0142og\u00f3w w p\u0119cherzyku \u017c\u00f3\u0142ciowym czy drogach \u017c\u00f3\u0142ciowych,bada\u0144 histopatologicznych i immunohistochemicznych \u2013 cholestazy/zast\u00f3j \u017c\u00f3\u0142ci, proliferacja przewodzik\u00f3w \u017c\u00f3\u0142ciowych, tworzenie rozetek hepatocyt\u00f3w, cechy zapalenia i/lub w\u0142\u00f3knienia, badanie immunohistochemiczne z przeciwcia\u0142ami p/MDR3 \u2013 wynik negatywny , 8-9, 18wynik\u00f3w bada\u0144 dodatkowych:Cech\u0105 charakterystyczn\u0105 cholestazy w przebiegu PFIC-3 jest podwy\u017cszona aktywno\u015b\u0107 GGTP, kt\u00f3ra wymaga r\u00f3\u017cnicowania z innymi przyczynami cholestazy zewn\u0105trzw\u0105trobowej \u2013 atrezja dr\u00f3g \u017c\u00f3\u0142ciowych, torbiele i inne wady dr\u00f3g \u017c\u00f3\u0142ciowych, kamica p\u0119cherzyka i dr\u00f3g \u017c\u00f3\u0142ciowych, czy wewn\u0105trzw\u0105trobowej \u2013 niedob\u00f3r alfa- 1-antyproteazy, zesp\u00f3\u0142 Alagille\u2019a, mukowiscydoza, toksyczne i polekowe uszkodzenie w\u0105troby. Dla cholestazy zewn\u0105trzw\u0105trobowej typowe jest stwierdzenie odbarwionych stolc\u00f3w , brak lub ma\u0142y p\u0119cherzyk \u017c\u00f3\u0142ciowy w badaniu USG jamy brzusznej. Cholangiografia metod\u0105 rezonansu magnetycznego pozwala wykluczy\u0107 stwardniaj\u0105ce zapalenie dr\u00f3g \u017c\u00f3\u0142ciowych lub inne patologie dr\u00f3g \u017c\u00f3\u0142ciowych. Prawid\u0142owe st\u0119\u017cenie alfa-1-antyproteazy w surowicy oraz fenotyp alfa-1-antyproteazy pozwalaj\u0105 wykluczy\u0107 jej deficyt. Prawid\u0142owy wynik badania przesiewowego noworodk\u00f3w w kierunku mukowiscydozy (dost\u0119pny we wszystkich wojew\u00f3dztwach w Polsce od czerwca 2009 roku) oraz prawid\u0142owe st\u0119\u017cenie chlork\u00f3w w pocie (w sytuacji braku wyniku badania przesiewowego lub w sytuacjach w\u0105tpliwych) pozwalaj\u0105 wykluczy\u0107 mukowiscydoz\u0119. Nieobecno\u015b\u0107 wady serca, kr\u0119g\u00f3w motylich, prawid\u0142owy wynik badania oftalmologicznego w lampie szczelinowej oraz nieobecno\u015b\u0107 cech dysmorfii twarzy pozwalaj\u0105 oddali\u0107 podejrzenie od zespo\u0142u Alagille\u2019a ,20,21.ATP7B, jednak\u017ce powy\u017cszy przypadek pokazuje, \u017ce PFIC3, obok pierwotnego stwardniaj\u0105cego zapalenia dr\u00f3g \u017c\u00f3\u0142ciowych i pierwotnego \u017c\u00f3\u0142ciowego zapalenia w\u0105troby (dawniej pierwotnej marsko\u015bci \u017c\u00f3\u0142ciowej w\u0105troby) nale\u017cy doda\u0107 do grupy jednostek chorobowych przebiegaj\u0105cych ze zwi\u0119kszonym odk\u0142adaniem miedzi w w\u0105trobie, sugeruj\u0105cym chorob\u0119 Wilsona [Boga i wsp. przedstawili opis przypadku 15-letniego ch\u0142opca ze splenomegali\u0105, u kt\u00f3rego w toku diagnostyki wykonano biopsj\u0119 w\u0105troby, kt\u00f3ra wykaza\u0142a obecno\u015b\u0107 st\u0142uszczenia drobnokropelkowego oraz zwi\u0119kszon\u0105 zawarto\u015b\u0107 miedzi (>250 \u03bcg/g suchej tkanki) w bioptacie w\u0105troby sugeruj\u0105c podejrzenie choroby Wilsona. Analiza molekularna nie wykaza\u0142a obecno\u015bci patogennych wariant\u00f3w molekularnych w genie ext generation sequencing; NGS) to najbardziej zaawansowana wysokoprzepustowa metoda badania genomu cz\u0142owieka, kt\u00f3ra pozwala na jednoczesne genotypowanie miliard\u00f3w par zasad w znacznie kr\u00f3tszym czasie i za ni\u017csz\u0105 cen\u0119, w stosunku do dotychczas powszechnie stosowanego w diagnostyce genetycznej klasycznego sekwencjonowania wed\u0142ug Sangera. Mo\u017ce ona obejmowa\u0107 analiz\u0119 ca\u0142ego genomu , dotyczy\u0107 mapowania zmian w sekwencji koduj\u0105cej wszystkich gen\u00f3w, stanowi\u0105cej 2% genomu,  lub obejmowa\u0107 analiz\u0119 wybranego panelu gen\u00f3w (od kilkudziesi\u0119ciu do kilku tysi\u0119cy). Dzi\u0119ki zastosowaniu metody sekwencjonowania nowej generacji w wielu o\u015brodkach na \u015bwiecie, ale tak\u017ce i w Polsce (m.in. w IP-CZD) mo\u017cliwe s\u0105 przesiewowe testy molekularne wykorzystuj\u0105ce panel kilkudziesi\u0119ciu-kilkuset gen\u00f3w. Takie badanie mo\u017ce stanowi\u0107 mniej kosztown\u0105 procedur\u0119 i pozwoli\u0107 na postawienie rozpoznania w znacznie kr\u00f3tszym czasie ni\u017c opisany wy\u017cej tok diagnostyczny [Ostateczne potwierdzenie rozpoznania PFIC-3 stanowi zwykle wynik badania molekularnego. Sekwencjonowanie nowej generacji  stosowany zwykle w dawce 10-20 mg/kg m.c./dob\u0119, jednak\u017ce w ok. po\u0142owie przypadk\u00f3w PFIC-3 terapia UDCA jest nieskuteczna. W przypadkach z cholestaz\u0105 wskazana jest suplementacja witamin rozpuszczalnych w t\u0142uszczach . Z kolei, w przypadku rozwoju nadci\u015bnienia wrotnego i zwi\u0105zanych z nim powik\u0142a\u0144 stosowane jest leczenie endoskopowe (opaskowanie \u017cylak\u00f3w prze\u0142yku), a w przypadku rozwoju niewydolno\u015bci w\u0105troby pacjenci wymagaj\u0105 przeszczepienia narz\u0105du , 8-9. Co\u0119\u017carnych , 12, 13.Partial External Biliary Diversion, PEBD) w grupie pacjent\u00f3w z PFIC-3. Lemoine i wsp. w 2017 roku na \u0142amach Journal of Pediatric Surgery przedstawili d\u0142ugofalow\u0105 obserwacj\u0119 skuteczno\u015bci zabiegu PEBD w\u015br\u00f3d pacjent\u00f3w z PFIC, w tym u jednego pacjenta z PFIC3 [Istniej\u0105 tak\u017ce pojedyncze doniesienia na temat pr\u00f3b zastosowania zabiegu cz\u0119\u015bciowego zewn\u0119trznego odprowadzenia \u017c\u00f3\u0142ci (ang.  z PFIC3 . Diao i  z PFIC3 .ABCB4, powoduj\u0105ca ca\u0142kowity brak ekspresji bia\u0142ka MDR3, wi\u0105\u017ce si\u0119 z ci\u0119\u017ckim przebiegiem PFIC-3 w postaci szybszej progresji do marsko\u015bci w\u0105troby i s\u0142abszej odpowiedzi na leczenie UDCA [ABCB4, maj\u0105 wi\u0119ksze szanse na pozytywne rezultaty leczenia UDCA.Wykazano, \u017ce obecno\u015b\u0107 mutacji w obu allelach genu nie UDCA , 16, 17."}
{"text": "Scientific Reports 10.1038/s41598-021-81696-5, published online 26 January 2021Correction to: The original version of this Article contained errors, that affected the descriptive statistics. Inferential statistics\u00a0remained unaffected.In Table 1, the Cohen\u2019s d values were incorrectly given. The correct and incorrect values appear below.Incorrect: Correct: As a result, in the Method section, under the subheading \u2018Statistical procedures\u2019,t(398)\u2009= \u2212\u20092.055,\u00a0p\u2009<\u20090.05,\u00a0d\u2009= \u2212\u20090.212), driven by higher scores in VIQ (t(398)\u2009= \u2212\u20092.942,\u00a0p\u2009<\u20090.05,\u00a0d\u2009= \u2212\u20090.314).\u201d\u201cThe samples differed significantly in IQ scores, with higher scores in the ASD+\u2009sample (now reads:t(398)\u2009= \u2212\u20092.055,\u00a0p\u2009<\u20090.05,\u00a0d\u2009= \u2212\u20090.230), driven by higher scores in VIQ (t(398)\u2009= \u2212\u20092.942,\u00a0p\u2009<\u20090.05,\u00a0d\u2009= \u2212\u20090.326).\u201d\u201cThe samples differed significantly in IQ scores, with higher scores in the ASD+\u2009sample \u2009=\u20091.698,\u00a0p\u2009=\u20090.090,\u00a0d\u2009=\u20090.186), DDF (t(398)\u2009= \u2212\u20090.555,\u00a0p\u2009=\u20090.579,\u00a0d\u2009= \u2212\u20090.061), and EOT (t(398)\u2009=\u20090.488,\u00a0p\u2009=\u20090.626,\u00a0d\u2009=\u20090.053). The groups did not differ significantly in their levels of AQ (t(398)\u2009= \u2212\u20091.693,\u00a0p\u2009=\u20090.091,\u00a0d\u2009= \u2212\u20090.185) and BDI (t(398)\u2009=\u20091.579,\u00a0p\u2009=\u20090.115,\u00a0d\u2009=\u20090.173).\u201d\u201cThe ASD+\u2009sample did not significantly differ from the ASD\u2212 sample in any TAS-20 subdomains, DIF (now reads:t(398)\u2009=\u20091.698,\u00a0p\u2009=\u20090.090,\u00a0d\u2009=\u20090.183), DDF (t(398)\u2009= \u2212\u20090.555,\u00a0p\u2009=\u20090.579,\u00a0d\u2009= \u2212\u20090.060), and EOT (t(398)\u2009=\u20090.488,\u00a0p\u2009=\u20090.626,\u00a0d\u2009=\u20090.054). The groups did not differ significantly in their levels of AQ (t(398)\u2009= \u2212\u20091.693,\u00a0p\u2009=\u20090.091,\u00a0d\u2009= \u2212\u20090.187) and BDI (t(398)\u2009=\u20091.579,\u00a0p\u2009=\u20090.115,\u00a0d\u2009=\u20090.172).\u201d\u201cThe ASD+\u2009sample did not significantly differ from the ASD\u2212 sample in any TAS-20 subdomains, DIF , DIF , and DDF , but not with EOT . In the ASD\u2212 sample, BDI significantly increased with AQ , and with DIF , but not with DDF  or EOT . Considering correlations of autism and alexithymia traits in the ASD\u2009+\u2009sample, DIF significantly increased with AQ  and DDF . Similarly, AQ significantly increased with DIF  and DDF  in the ASD\u2212 sample. EOT was not correlated with AQ in either sample .\u201d\u201cRegarding correlations with depressive symptoms in the ASD\u2009+\u2009sample, BDI scores significantly increased with AQ , DIF , and DDF , but not with EOT . In the ASD\u2212 sample, BDI significantly increased with DIF , but not with AQ , DDF  or EOT . Considering correlations of autism and alexithymia traits in the ASD\u2009+\u2009sample, DIF significantly increased with AQ  and DDF . Similarly, AQ significantly increased with DIF  and DDF  in the ASD\u2212 sample. EOT was not correlated with AQ in either sample .\u201d\u201cRegarding correlations with depressive symptoms in the ASD\u2009+\u2009sample, BDI scores significantly increased with AQ (The original Article has been corrected."}
{"text": "Impatiens longshanensis (The LSID for the name Impatiens longshanensis is: 77219154-1) sp. nov. and I. lihengiana (The LSID for the name I. lihengiana is: 77219153-1) sp. nov., from Hunan, China, are described and illustrated here. The molecular phylogenetic study suggests that I. longshanensis and I. lihengiana should be placed in the I. sect. Impatiens. A detailed description, diagnostic characters between the two new species and allied species, pollen and seed morphology, and color photographs are provided. In addition, based on wide sampling, we found that the longifilamenta group, an endemic group to China, whose members have basal lobes of lateral united petals with long filamentous hairs, shows significant morphological variability. In this paper, we discuss the taxonomic significance of morphological characteristics within this group. Based on a literature review and observation of living materials in the field, an updated identification key for this group is also proposed. Hydrocera Blume ex Wight & Arnott [Impatiens Linnaeus [Impatiens is one of the most species-rich genera of angiosperms, having more than 1000 species. These are mainly distributed in the mountainous regions of the tropics and subtropics, and have tropical Africa, Madagascar, South India and Sri Lanka, Sino-Himalaya, and South-East Asia as their centers of diversity [Impatiens species occur in diverse habitats, such as in forest understories, roadside ditches, valleys, abandoned fields, along streams and in seepages, usually in mesic or wet conditions, although some species can tolerate drier habitats [Impatiens are grown all over the world as ornamental plants [Impatiens have been discovered and described in recent years [Impatiens distributed in China [Balsaminaceae comprises two genera, the monotypic & Arnott  and ImpaLinnaeus . Impatieiversity ,4. The giversity ,6. Impatl plants . Many nent years ,13,14,15in China ,17. MostImpatiens is notoriously difficult to classify [Impatiens are usually fleshy plants, with fine and fragile flowers, usually folded and coalesced in dried specimens, hence losing their original shape and therefore difficult to reconstruct. The botanist Hooker called them \u201ca terror to botanists\u201d and \u201cdeceitful above all plants, and desperately wicked\u201d. Yu et al. [Impatiens into two subgenera, I. subgen. Clavicarpa and subgen. Impatiens [Semeiocardium, Racemosae, Fasciculatae, Impatiens, Tuberosae, Scorpioidae, and Uniflorae.As is well known, classify ,18. Morpu et al.  divided mpatiens , based oImpatiens was once treated as I. sect. longifilamenta [I. sect. Impatiens [The longifilamenta group in ilamenta , then mempatiens . This grImpatiens growing on wet shady habitats under an evergreen forest. Specimens were collected carefully, and the flowers and fruits were preserved in formalin-acetic-alcohol (FAA) solution for further identification. After careful examination of the relevant specimens and literature [During the botanical explorations from 2012 to 2020 in Longshan County, Hunan province, the authors encountered two interesting species of terature ,16,17, tImpatiens longshanensis Y. Y. Cong & Y. X. Song, sp. nov. Type: China, Hunan, Longshan County, Bamian Mountain, Zisheng Bridge, under moist and shady places, 109\u00b015\u203205.92\u2033 E, 28\u00b052\u203230.24\u2033 N, altitude 1194 m, 7 October 2020, Yi-Yan Cong 35,443 .Impatiens longshanensis is morphologically similar to I. dicentra Franch. ex Hook. f., but differs due to its suborbicular lamina base (vs. lamina base cuneate); green, equilateral lateral sepals, 5.5\u20137 mm wide, coarsely dentate on both sides, and inconspicuously thickened abaxial midvein ; lower sepal 1.85\u20132.3 cm deep (vs. 3\u20135 cm deep); reniform dorsal petal (vs. orbicular); oblong basal lobe (vs. lanceolate); and dolabriform distal lobes (vs. lanceolate).China, Hunan, Longshan County, Bamian Mountain, 109\u00b026\u203237\u2033 E, 28\u00b096\u203258\u2033 N, altitude 1336 m, 27 July 2013, Yan Xiao LS-2238 (CSH).Annual herb, 45\u201380 cm tall. Stem erect, slender, slightly ridged base 0.5\u20130.7 cm in diam, well branched, nodes swollen in lower part. Leaves alternate, petiole 1\u20133 cm. Lamina 4.5\u20138 \u00d7 1.2\u20133.9 cm, ovate or narrowly elliptic, membranaceous, glabrous on both surfaces, apex cuspidate, base suborbicular, with 3\u20134 pairs stipitate glands at basal margin, margin crenate-serrate, setose between marginal teeth, lateral veins 5\u20138 pairs. Inflorescences in upper leaf axils, 1-flowered; peduncles short, 0.6\u20131.1 cm long. Pedicels 2-bracteate, lower bracts linear, upper bracts ovate, persistent. Flowers pale yellow, large, 3\u20134.5 cm long. Sepals: lateral sepals 2, broadly ovate-orbicular, equilateral, 6.5\u20139 mm long, 5.5\u20137 mm wide, green, acuminate at apex, coarsely dentate on both sides, abaxial midvein inconspicuously thickened; lower sepal red striate, saccate, 1.85\u20132.3 cm deep excluding spur, mouth 1.2\u20131.6 cm wide, anterior gradually narrowed downward into a long beak, base gradually narrowed into a spur, ca 1 cm, incurved, 2-lobed. Petals: dorsal petal reniform, 11\u201313.5 mm long, 16\u201322 mm wide, base suborbicular, apex emarginate, shortly rostellate, abaxial midvein cristate, green, lateral united petals not clawed, 1.55\u20132.1 cm long, 2-lobed, basal lobe 8\u201312 \u00d7 6\u20138.5 mm, oblong, apex with a filamentous long hair, distal lobes 18\u201325 \u00d7 8.5\u201313 mm, dolabriform, margin entire, apex obtuse, abruptly narrowed into a short filamentous hair, auricle inflexed. Stamens 5, 5\u20137.5 mm long, filaments linear, free for about 1/2 of their length. Anthers ovoid, joined into a ring surrounding the ovary apex, apex obtuse, 2\u20132.5 mm long; ovary superior, 4\u20135.5 mm long, 5-carpellate, erect, fusiform, placentation axile. Capsule linear, 1.6\u20132.5 cm long, fleshy, 5-valved. Seeds many, subellipsoid.The specific epithet \u201clongshanensis\u201d refers to the locality of the type specimen, Longshan County, Hunan, China.Phenology: Flowering and fruiting were observed in the field from September to November.Impatiens longshanensis and I. dicentra are oblong in polar view, 4-colpate, exine with irregularly reticulate ornamentation, dense granules in lumina, the former average size of E1 \u00d7 E2 = 26.12 (23.83\u201327.71) \u00d7 15.05 (14.37\u201315.56) \u03bcm (Pollen grains: 5.56) \u03bcm A\u2013C, the 5.56) \u03bcm D\u2013F. The I. longshanensis subellipsoid, 2.47 \u00d7 2.22 mm, ratio of L (length)/W (width) = 1.11, under SEM /W (width) = 1.14, under SEM .Impatiens davidii Franch, but differing in having narrowly elliptic or narrowly ovate-elliptic leaf blades (vs. ovate-oblong or ovate-lanceolate); petiole 1.5\u20132.5 cm (vs. 4\u20138 cm); lateral sepals yellow-green, purple spotted, 1-veined ; lower sepal funnelform (vs. saccate); dorsal petal apex long rostellate (vs. short rostellate); lateral united petals not clawed, 2.5\u20132.8 cm long ; and basal lobe ovate-lanceolate (vs. oblong).The new species is morphologically similar to Annual herb, 60\u201375 cm tall. Stem erect, slender, base 0.5\u20130.7 cm in diam, rarely branched. Leaves alternate, petiole 1.5\u20132.5 cm. Lamina 6\u201312.5 \u00d7 3\u20134.2 cm, narrowly elliptic or narrowly ovate-elliptic, membranaceous, glabrous on both surfaces, apex cuspidate, base cuneate, margin crenate-serrate, setose between marginal teeth, lateral veins 6\u20138 pairs. Inflorescences in upper leaf axils, 1-flowered; peduncles 0.6\u20131.1 cm long. Pedicels 2-bracteate, lower bracts linear, upper bracts ovate ca. 0.5 cm long, ca. 0.3 cm wide, apex long acuminate. Flowers yellow, large, 4\u20134.5 cm long. Sepals: lateral sepals 2, yellow-green, purple spotted, suborbicular, 8\u201310 mm long, ca. 8 mm wide, 1-veined, acuminate at apex, lower sepal red striate, funnelform, 2\u20132.2 cm deep excluding spur, mouth 18\u201322 mm wide, base gradually narrowed into a spur, 10\u201312 mm long, incurved, 2-lobed. Petals: dorsal petal suborbicular, 15\u201317 mm long, 12\u201314 mm wide, apex long rostellate, abaxial midvein cristate, green, lateral united petals not clawed, 2.5\u20132.8 cm long, 2-lobed, basal lobe 10\u201312.5 \u00d7 2\u20133.5 mm, ovate-lanceolate, apex with a filamentous long hair, ca. 1 mm long, distal lobes 19\u201322 \u00d7 10\u201312 mm, dolabriform, apex obtuse, constricted into a filamentous hair, auricle inflexed. Stamens 5, ca. 5 mm long, filaments linear, free for about 1/2 of their length. Anthers ovoid, joined into a ring surrounding the ovary apex, apex obtuse, ca. 4.5 mm long; ovary fusiform, superior, ca. 5 mm, 5-carpellate, erect, placentation axile. Capsule linear, 3.3\u20133.5 cm long, fleshy, 5-valved.The specific epithet \u201clihengiana\u201d is given in honor of Prof. Heng Li, a taxonomist at Kunming Institute of Botany, Chinese Academy of Sciences, who has made significant contributions to plant taxonomy.Phenology: Flowering and fruiting were observed in the field from September to November.Impatiens lihengiana is currently known only from the type locality in Longshan County, Hunan Province, China  \u00d7 15.28 (13.15\u201316.92) \u03bcm (Pollen grains: 6.92) \u03bcm G\u2013I, the 6.92) \u03bcm J\u2013L. The Impatiens can be divided into I. subgen. Clavicarpa and I. subgen. Impatiens. Within I. subgen. Impatiens, several sections can be recognized. The molecular phylogenetic analysis of Impatiens based on ITS and atpB-rbcL supported ten species, namely, I. soulieana, I. lecomtei, I. platychlaena, I. bullatisepala, I. davidii, I. dicentra, I. fissicornis, I. tayemonii, and our two proposed new species, to cluster into a clade which belongs to I. sect. Impatiens , whereas the apex of distal lobes in other species in this clade is entire, and the apex is constricted into filamentous hair. Thus, the phylogenetic hypothesis is congruent with the observed morphological distinctiveness. Interestingly, the apex of distal lobes in Impatiens oblongipetala is also retuse, but more phylogenetic evidence is needed to clearly understand its taxonomic status. The molecular data supported that subclade . MorpholBased on morphological characteristics, there are significant variations in the structure of the flower within this group A\u2013F. The Based on many years of field observations, in addition to literature consultation, the following identification key is proposed, and hence the description herein.-1a Basal lobes of lateral united petals with a filamentous long hair, distal lobes of lateral united petals apex retuse with aseta \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2-1b Basal lobes of lateral united petals with a filamentous long hair, distal lobes of lateral united petals apex entire with a filamentous long hair or a seta \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 3-I. oblongipetala2a Flowers pale rose pink; lateral sepals abaxial midvein narrowly thickened; distal lobes of lateral united petals oblong or suboblong \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -I. soulieana2b Flowers yellow; lateral sepals abaxial midvein narrowly carinate; distal lobes of lateral united petals dolabriform \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -3a Lower sepal funnelform or navicular \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4-3b Lower sepal saccate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 13-I. shennongensis4a Lower sepal navicular, spur absent \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -4b Lower sepal funnelform, spur 2-lobed \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 5-I. weihsiensis5a Lateral sepals margin 4-or 5-denticulate on one side \u2026\u2026\u2026\u2026\u2026\u2026\u2026 -5b Lateral sepals margin entire \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6-6a Lateral united petals clawed \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 7-6b Lateral united petals not clawed \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 8-I. toxophora7a Lateral sepals broadly ovate-cordate, abaxial midvein cristate \u2026\u2026\u2026 -I. tayemonii7b Lateral sepals ovate-orbicular, abaxial midvein without cristate \u2026\u2026 -8a Lateral sepals abaxial midvein carinate narrowly cristate or with a spinelike appendage \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 9-8b Lateral sepals abaxial midvein without carinate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 12-I. lecomtei9a Flowers pink \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -9b Flowers yellow \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 10-I. cornutisepala10a lateral sepals ovate, abaxial midvein a spinelike appendage; dorsal petal reniform \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -10b lateral sepals orbicular, abaxial midvein carinate or narrowly cristate; dorsal petal orbicular \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202611-I. brevipes11a Bracts subulate; Lateral sepals 10\u201320 mm in diam, abaxial midvein acutely carinate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -I. mussotii11b Bracts ovate; Lateral sepals ca. 8 mm in diam, abaxial midvein narrowly cristate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -I. lihengiana12a Flowers yellow; lateral sepals suborbicular, 1-veined \u2026\u2026\u2026\u2026\u2026\u2026 -I. gongchengensis12b Flowers pale purple or purple-red; Lateral sepals ovate-orbicular,7-veined \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -13a Lateral sepals margin dentate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 14-13b Lateral sepals margin entire \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 18-I. dicentra14a Lateral sepals inequilateral, coarsely dentate on one side \u2026\u2026\u2026\u2026\u2026\u2026 -14b Lateral sepals equilateral, dentate on both side \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 15-15a Lateral sepals margin irregularly fimbriate-lacerate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 16-15b Lateral sepals margin coarsely dentate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 17-I. lacinulifera16a Flowers yellow, to 4.5 cm deep; lateral sepals margin and abaxial midvein irregularly fimbriate-lacerate; dorsal petal orbicular \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -I. platyceras16b Flowers pale purple, 2\u20133 cm deep; lateral sepals margin irregularly lacerate; dorsal petal broadly reniform \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -I. fissicornis17a Lateral united petals clawed; lower sepal with a hooked spur; dorsal petal suborbicular \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -I. longshanensis17b Lateral united petals not clawed; lower sepal with a incurved spur; dorsal petal reniform \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -18a Lateral united petals clawed \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 19-18b Lateral united petals not clawed \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 22-19a Dorsal petal orbicular or suborbicular \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 20-19b Dorsal petal broadly ovate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 21-I. davidii20a Lateral sepals yellow, broadly ovate, 9-veined \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -I. vittata20b Lateral sepals pale green, orbicular, abaxial midvein with a small sac at base \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -I. plicatisepala21a Lateral sepals abaxially plicated; basal lobes of lateral united petals oblong \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -I. bullatisepala21b Lateral sepals lateral veins reticulate and sunk on abaxial surface with bullate projections among veins; basal lobes of lateral united petals ovate to elliptic \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -I. platychlaena22a Lateral sepals abaxial midvein not thickened, many veined \u2026\u2026\u2026 -22b Lateral sepals abaxial midvein fine or slightly thickened, carinate \u2026\u2026\u2026\u2026\u2026 23-I. waldheimiana23a Lateral sepals apex aristate-acuminate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -23b Lateral sepals apex without aristate-acuminate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 24-I. robusta24a Flowers ca. 4 cm deep; lateral sepals abaxial midvein fine, turgid; basal lobes of lateral united petals oblate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 -I. conaensis24b Flowers 2.5\u20133 cm deep; lateral sepals abaxial midvein slightly thickened, carinate; basal lobes of lateral united petals ovate-lanceolate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 Impatiens sect. Impatiens. I. longshanensis is similar to I. dicentra in its single-flowered inflorescence, yellow flowers, and saccate lower sepal. However, I. longshanensis can easily be distinguished from similar species by its green equilateral lateral sepals, reniform dorsal petal, oblong basal lobe, and dolabriform distal lobes. A more detailed morphological comparison between I. longshanensis and I. dicentra is provided in I. lihengiana is superficially similar to I. davidii in having single-flowered inflorescence, dolabriform distal lobes, and 4-colpate pollen grains, but differs by its yellow-green lateral sepals, purple spotted, 1-veined, funnelform lower sepal, not clawed lateral united petals, and ovate-lanceolate basal lobe. A detailed comparison of similar species is given in Based on morphological evidence and molecular phylogenetic study, the two new species should be placed in Impatiens. In addition, this group is endemic to China, and mainly distributed in western Sichuan, Hubei, southern Henan, Guizhou, northwest Guangxi, and Jiangxi, Taiwan, and it favors montane elevations (1000\u20133000 m). Overall, the evidence from the combinations of morphology and phylogeny shows that the longifilamenta group is a cohesive group. For this group, an apex with a filamentous appendage in basal lobes of lateral united petals is the most noticeable feature distinguishing it from other species of Impatiens taxa. For example, in the previous literature [Impatiens platychlaena are purple, but in this study, the color changes gradually, ranging from purple, yellow with purple spots to pure yellow, based on years of field observation . Dried pollen grains and seeds were carefully mounted on circular metal stubs using double-sided adhesive tape, sputter coated with gold using the JEC-3200 Auto Fine Coater, and then examined and photographed using the JSM-IT500 SEM. Pollen characters were described according to the literature ,24, and Impatiens were chosen to construct a phylogenetic tree with Hydrocera trifloral as outgroups. DNA sequences of these 34 species were downloaded from GenBank, with the exception of the two new species, I. dicentra, I. platychlaena, and I. longialata. Species names and GenBank accession numbers are provided in Genomic DNA of the two new species was extracted from silica gel-dried leaves  using Mag-MK Plant Genomic DNA extraction kits . PCR product sequencing was carried out using TSINGKE Biological Technology. Thirty-three representative species from atpB-rbcL [For phylogenetic analysis, two molecular markers were used: ITS and tpB-rbcL ,29. PhyltpB-rbcL . SequenctpB-rbcL . SequenctpB-rbcL . The bestpB-rbcL . The BaytpB-rbcL  under thtpB-rbcL  under thtpB-rbcL  bootstratpB-rbcL ."}
{"text": "The unique\u2010continuation property from sets of positive measure is here proven for the many\u2010body magnetic Schr\u00f6dinger equation. This property guarantees that if a solution of the Schr\u00f6dinger equation vanishes on a set of positive measure, then it is identically zero. We explicitly consider potentials written as sums of either one\u2010body or two\u2010body functions, typical for Hamiltonians in many\u2010body quantum mechanics. As a special case, we are able to treat atomic and molecular Hamiltonians. The unique\u2010continuation property plays an important role in density\u2010functional theories, which underpins its relevance in quantum chemistry. The unique\u2010continuation property from sets of positive measure for the Schr\u00f6dinger equation states that a solution cannot vanish unless it is equal to zero everywhere. This property is an important ingredient in the proof of the Hohenberg\u2010Kohn theorem, which is a fundamental result in quantum chemistry stating that the one\u2010body particle density determines the external potential. This work addresses this property and its relevance in current density functional theory that extends density functional theory to include systems with magnetic fields. Hohenberg and Kohn showed for systems without magnetic fields that the one\u2010body ground\u2010state particle density determines the electric  potential up to a constant.In the presence of magnetic fields though, the approach of Hohenberg and Kohn to set up a universal density functional requires more than just the particle density due to the fact that an additional vector potential enters the system's Hamiltonian. Both the paramagnetic current density and the total  current density have been suggested as basic variables alongside the particle densityregularized energy functional was taken in References For more detailed accounts on the existence of generalized Hohenberg\u2010Kohn theorems within CDFT see References N), which is a difficult task. In the present work we obtain results that are adapted to the many\u2010body Schr\u00f6dinger equation and that furthermore include vector potentials, building on the results of KurataKlocn and its generalization Klocn,\u03b4, with n = 3 and n = 6  CDFT. It addresses the property that a solution of the magnetic Schr\u00f6dinger equation cannot vanish on a set of positive measure, a property called unique continuation, see Definition 1. Unique continuation is also a fundamental property for solutions of the magnetic Schr\u00f6dinger equation in its own right and has been well\u2010studied.2\u03c8 in \u03c11/2 = \u03c8\u2009>\u20090 and the following relation to the scalar potential v must hold e is the ground\u2010state energy. Conversely, given a particle density \u03c1 we can ask if a potential v exists such that the given \u03c1 is the ground\u2010state density of that potential. For the one\u2010particle case, this problem has been studied by Englisch and EnglischN\u2010representable densities). Corollary 3 in Reference \u03c11/2\u2009\u2264\u2009C\u03c11/2 and \u03c1\u22121\u2208Lloc1 besides N\u2010representability) provides sufficient conditions for one\u2010particle v\u2010representability, that is, v can be computed from \u03c1 as given in Equation \u03c1 is the ground\u2010state density of that v.In the most simple setting of only one particle and without vector potential, it is known that the (unique) ground state N\u2010electron case without magnetic field, we first recall the Hohenberg\u2010Kohn theorem: Given two systems, if \u03c11 = \u03c12 then v1 = v2\u2009+\u2009constant, where \u03c1k, k = 1,2, is the ground\u2010state particle density of the corresponding system defined by the potential vk. The proof of this result relies on the fact that if \u03c8 is a ground state of both systems, then \u2211k=1Nv1xk\u2212v2xk\u03c8=constant\u00d7\u03c8. If \u03c8 does not vanish on a set of positive (Lebesgue) measure we have v1 = v2+ constant almost everywhere. The proof can then be completed by means of the variational principle, using the Hohenberg\u2010Kohn argument by reductio ad absurdum.Returning to the general N interacting, nonrelativistic (spinless) particles subjected to both a scalar and a vector potential. The fundamental question then is, whether any eigenfunction of the corresponding Hamiltonianunique continuation.Definition 1N\u03c8 = e\u03c8 has the unique\u2010continuation property (UCP) from sets of positive (Lebesgue) measure if a solution that satisfies \u03c8 = 0 on a set of positive measure is identically zero. Furthermore, the Schr\u00f6dinger equation is said to have the strong UCP if whenever \u03c8 vanishes to infinite order at some point x0, that is, for all m\u2009>\u20090We say that the Schr\u00f6dinger equation HRemark 1N.The strong UCP implies the weak UCP. The UCP from sets of positive measure allows us to conclude \u03c8\u2009\u2260\u20090 almost everywhere for any eigenfunction of HIn this article we address the more general case of \u03c8|\u2009\u2264\u2009|\u03be1||\u2207\u03c8|\u2009+\u2009|\u03be2||\u03c8|.\u03be1\u2208Llocn\u211dn and \u03be2\u2208Llocn/2\u211dn, the corresponding differential equation \u0394\u03c8 = \u03be1 \u00b7\u2207\u03c8\u2009+\u2009\u03be2\u03c8 has the UCP from sets of positive measure.Llocp constraints become more restrictive with increasing particle number, since the dimension of the configuration space n enters in the conditions. Directly applied to HN\u03c8 = e\u03c8 this means that if a solution \u03c8 in the Sobolev space Hloc2N/N+2\u211d3N vanishes on a set of positive measure,A belongs to Lloc3N\u211d3N, then \u03c8 is identically zero.There exists a considerable amount of literature that treats the UCP for differential inequality |\u0394v\u2208Llocp\u211d3, p\u2009>\u20092.Such results are used by Lammert,3HN be as in Equation HN\u03c8 = e\u03c8. We write HN = TA\u2009+\u2009V\u2009+\u2009U, where TA=\u2211j=1Ni\u2207j+Axj2,\u2207j=\u2202\u2202xj1\u2202\u2202xj2\u2202\u2202xj3 and xj=xj1xj2xj3\u2208\u211d3 are the coordinates of the jth electron. Here we use TA instead of \u210f = 2me = 1 and qe = \u22121, such that the Laplace operator appears without a factor 1/2.Let the Hamiltonian V is a one\u2010body potential given by Vx=\u2211j=1Nvxj with U between the electrons is modeled by u on W = V\u2009+\u2009U. Furthermore, B = \u2207\u2009\u00d7\u2009A. With the notation A\u00afx=Axjj=1N, the Schr\u00f6dinger equation is rewritten asWA=W+A\u00af2+i\u2207\u22c5A\u00af.The electric potential f\u2208Lloc2\u211dn belongs to the Sobolev space Hlock\u211dn if f has weak derivatives up to order k that belong to Lloc2\u211dn. Let the set of infinitely differentiable functions with compact support on C0\u221e\u211d3N. We say that \u03c8\u2208Hloc1\u211d3N is a solution of Equation \u03c6\u2208C0\u221e\u211d3NA function Theorem 2Let N\u2009\u2265\u20091. Assume thatWA\u2208Lloc3N/2\u211d3Nand each component of A is an element ofLloc3N\u211d3N. Then the Schr\u00f6dinger equation has the UCP from sets of positive measure, that is, if a solution\u03c8\u2208Hloc2N/N+2\u211d3Nvanishes on a set of positive measure then it is identically zero.\u2009\u2264\u20092 for all N. Following KurataLlocp constraints, with p proportional to N, can be avoided.Definition 3f\u2208Lloc1\u211dn belongs to the Kato class Klocn, n\u2009\u2260\u20092, if for every R\u2009>\u20090, limr\u21920+\u03b7Kr;f=0, whereA function Furthermore, f\u2208Klocn,\u03b4\u2282Klocn, \u03b4\u2009>\u20090, if for every R\u2009>\u20090If one employs Theorem 2 with f = f+\u2009\u2212\u2009f\u2212, where f\u2212 (f+) is the negative (positive) part of f given by We write Assumption 1 1r0\u2009>\u20090SupposeRemark 3by,Qy\u2208Kloc3N,\u03b4, for some \u03b4\u2009>\u20090, as a sufficient condition for Equation Remark 1.2 in KurataRemark 4AssumptionKlocn, n = 3 N being the dimensionality of the underlying configuration space. This condition is optimal in the sense that the class cannot be enlarged to smaller orders than n = 3 N, or the UCP will be lost. This follows from the inclusion Llocp\u2282Klocn for all p\u2009>\u2009n/2 and a sharp counterexample provided in Reference p, p\u2009<\u2009n/2. So if the order of the Kato class would be any m\u2009<\u2009n then it also includes Llocp with m/2\u2009<\u2009p\u2009<\u2009n/2 and that is ruled out by the given counterexample.The main condition in With the notation above, we formulate.V and U. In the sequel we use the notation |F| for the Frobenius norm  of a matrix j,l.Theorem 4Suppose Assumption. If\u03c8\u2208Hloc2\u211d3Nis a solution of (3) and vanishes to infinite order atthen \u03c8 is identically zero. Thus the Schr\u00f6dinger equation has the strong UCP.Lemma 1Let n\u2009\u2265\u20093, be fixed, x=x1\u2026xn\u2208\u211dn,A\u02dc=A\u02dc1\u2026A\u02dcn:\u211dn\u2192\u211dn,W\u02dc:\u211dn\u2192\u211d, F=Fj,lj,l=1nwithFj,l=\u2202A\u02dcj/\u2202xl\u2212\u2202A\u02dcl/\u2202xj, and suppose that is then fulfilled. The choice \u00c3 = A impliesTA=i\u2207+A\u02dc2and Equation (3) can be written as Equation (9).Furthermore assume, for some Each component ofA\u2208Lloc4\u211d3yieldsA\u02dcj\u2208Lloc4\u211d3Nfor j = 1, \u2026, 3 N. From\u2207\u22c5A\u02dc=\u2211k=1N\u2207k\u22c5Axkand\u2207\u22c5A\u2208Lloc2\u211d3, we obtain\u2207\u22c5A\u02dc\u2208Lloc2\u211d3N. Since a = |\u00c3|2, it holds that|A\u02dc|2\u2208Kloc3N. Moreover, the matrix F satisfiessince F contains N repeated blocks of sub matrices of the formThis establishes Equation W\u223c = W\u2009\u2212\u2009e and Equation Remark 5AssumptionKloc3N is replaced by Kloc3N+Flocp\u211d3N, 1\u2009<\u2009p\u2009\u2264\u20093 N/2. Here Flocp\u211d3N is the Fefferman\u2010Phong class and in this case a solution must be an element of Hloc2\u211d3N\u2229Lloc\u221e\u211d3N, and there is an additional condition on V\u2212.As stated in Remark 1.1 in Kurata,From Equation 4v is locally Theorem 6Suppose Assumption 1. If in additionv\u2212\u2208Lloc3/2\u211d3and\u03c8\u2208Hloc2\u211d3Nsolvesand vanishes on a set of positive measure, then \u03c8 is identically zero. Consequently, the Schr\u00f6dinger equation has the UCP from sets of positive measure.Remark 6x1, x2) = u\u2032(x1\u2009\u2212\u2009x2) \u00b7\u2207W, we have with the choice x\u00af0=x0\u2026x0\u2208\u211d3N, for fixed q1;x0\u2009and q2;x0. (See below the proof of Corollary W corresponding to the molecular case.) Furthermore,Because NH modeling atoms and molecules, and where the exponents in the integrability constraints are independent of the particle number N.Corollary 8For N\u2009\u2265\u20092 andfixed, supposeFurther, letv\u2212\u2208Lloc3/2\u211d3,v\u2208Kloc3, andu\u2208Kloc6, as well asQx0\u2009satisfyingwithq1;x0\u2208Kloc3,\u03b4andq2;x0\u2208Kloc6,\u03b4. Then the Schr\u00f6dinger equationhas the UCP from sets of positive measure.We can now formulate our main result that includes NH modeling atoms and molecules in magnetic fields if just Equation Corollary 9For N\u2009\u2265\u20092, suppose the magnetic field is such that Equation holds. Then withwhereand Zj\u2009>\u20090 are the positions and charges of thenuclei, respectively, the UCP from sets of positive measure holds for the Schr\u00f6dinger equation.In particular, the magnetic Schr\u00f6dinger equation has the UCP from sets of positive measure for 5\u03c8, define the particle density and the paramagnetic current density according toIn the presence of a magnetic field, no equivalence of a  Hohenberg\u2010Kohn result exists at present.A we may compute the total current density by the sum j=j\u03c8p+\u03c1\u03c8A. Now, fix the particle number N as well as the two\u2010particle interaction u . If \u03c8 is a ground state for some v and A, that is, H\u03c8 = e\u03c8, where e is the ground\u2010state energy, then \u03c1\u03c8, j\u03c8p, and j=j\u03c8p+\u03c1\u03c8A are called ground\u2010state densities of H. Whether the ground\u2010state particle density \u03c1 and the total current density j determine v and A (up to a gauge transformation) is still an open question in the general case.\u03c1,jp) it is well\u2010known that this density pair does not determine the potentials v and A.For a vector potential H and H have the same ground\u2010state particle density  and \u2207\u2009\u00d7\u2009A1 = \u2207\u2009\u00d7\u2009A2 = B. Suppose that vk, Ak for k = 1,2, and B fulfill Assumption 1 f such that A1 = A2 \u2212\u2009\u2207f, the variational principle yields\u03c82 is the ground state of H. Switching the indices, we find thatNow, assume that two systems with Hamiltonians Hv,A1 and \u03c82 is a ground state of both H and H, which leads toConsequently, \u03c82\u2009\u2260\u20090 and it follows 1v = v2\u2009+\u2009constant.Theorem 10Assume \u2207\u2009\u00d7\u2009A1 =\u2009\u2207\u2009\u00d7\u2009A2 =\u2009B and that vk, Akfor k = 1,2, and B fulfill Assumption 1 and take the requirements of Theoremfor H and H to hold. If the ground\u2010state particle densities satisfy \u03c11 =\u2009\u03c12, then v1 =\u2009v2 +\u2009C almost everywhere for some constant C.Remark 7Theorem 10 is the Hohenberg\u2010Kohn theorem for BDFT, first established by Grayce and HarrisHowever, Theorem 6 allows us to conclude Corollary 11Assume Assumption 1 and the requirements of Theorem 6 for H and H and that the ground\u2010state densities fulfill \u03c1 = \u03c11 =\u2009\u03c12, j = j1 =\u2009j2. For systems with jp =\u20090, it follows B1 =\u2009B2(even A1 =\u2009A2holds) and v1 =\u2009v2 +\u2009C for some constant C.Theorem 10 can be used to obtainProof. For systems with pj = 0, j1 = j2 implies\u03c11 = \u03c12 = \u03c1. Theorem \u03c1\u2009>\u20090 almost everywhere and we may conclude A1 = A2. Theorem v1 = v2\u2009+\u2009C for some constant C. \u25a16Proof of Theorem 6In the sequel let D = 3 N. By Assumptionthe strong UCP holds for Equationby TheoremNext, we follow the proof of Theorem 1.2 given after Lemma 3.3 in Regbaouiand Lemma A.2 in Lammert. = u\u2032(x1\u2009\u2212\u2009x2).) We start by showing the following inverse Poincar\u00e9 inequality for solutions of the Schr\u00f6dinger equation:x0=x0;jj=1N\u2208\u211dD and r\u2009\u2264\u2009r0For an arbitrary point C is a positive constant that depends on r0\u2009>\u20090, v, and A (but is independent of u\u2009\u2265\u20090).Here h\u2208C0\u221eB2rx0 that satisfies h(x) = 1 if |x\u2009\u2212\u2009x0|\u2009\u2264\u2009r, h\u2009\u2264\u20091 for |x\u2009\u2212\u2009x0|\u2009\u2264\u20092r, and |\u2207h(x)|\u2009\u2264\u20092r\u22121. In the Schr\u00f6dinger equation Choose We now bound each of the terms of the right hand side in Equation h\u2207\u03c8\u22252\u2225\u03c8\u2207h\u22252. Using the inequality 2ab\u2009\u2264\u2009a2/6\u2009+\u20096b2, we obtain an upper boundIt is immediate that the first term is less or equal to 2\u2225I1=\u22122i\u222b\u211dDA\u00af\u22c5\u2207\u03c8h2\u03c8\u00afdx. The Cauchy\u2010Schwarz inequality together with 2ab\u2009\u2264\u2009a2/6\u2009+\u20096b2 yieldTo continue, let WA and write e\u2212WA=e\u2212W\u2212A\u00af2\u2212i\u2207\u22c5A\u00af. ThusW = V+\u2009+\u2009U+\u2009\u2212\u2009V\u2212\u2009\u2265\u2009\u2212V\u2212 thatFor the last term of the right hand side in Equation \u0398=V\u2212+\u2223e\u2223+6A\u00af2+\u2223\u2207\u22c5A\u00af\u2223, from Equations Define \u0398x\u2264\u2211j=1N\u03981xj holds. Furthermore, we haveI2.With the notation v\u2212\u2208Lloc3/2\u211d3, A2\u2208Lloc2\u211d3, and \u2207\u22c5A\u2208Lloc2\u211d3, and it follows that \u03981\u2208Lloc3/2\u211d3. To bound the term I2 from above, we closely follow Lammert\u03c1\u02dcx1=N\u222b\u211d3N\u22121h\u03c82dx2\u22efdxN. For M\u2009>\u20090 we let M\u2032=\u2016\u03981\u03c7B2r\u03c7{\u03981\u2265M}\u20163/2, where the characteristic function of a set X is denoted X\u03c7. H\u00f6lder's inequality gives\u03c1\u02dc3\u2264C\u2207\u03c1\u02dc1/222. A direct computation of \u2207(\u03c1\u02dc1/2), using the definition of \u03c1\u02dc, shows that \u2207\u03c1\u02dc1/222\u2264\u222b\u211dD\u2207h\u03c82dx . Given \u03b5\u2009>\u20090 there is an r0 = r0(\u03b5) so that (cf. (3.11) in RegbaouiSuppose C. Applying the Cauchy\u2010Schwarz inequality to the right hand side of Equation C. Since |E|\u2009\u2265\u2009|E \u2229 Br|, Equation Lemma 3.3 in Regbaouifr=\u222bBr\u03c82dx, fix an integer n and choose \u03b5\u2009>\u20090 so that C\u2032\u03b5D2//(1\u2009\u2212\u2009\u03b5)2 = 2n\u2212. Then Equation f(r)\u2009\u2264\u20092n\u2212f(2r). By iterationr and k chosen such that r0 depends on n. Consequently f vanishes to infinite order, that is, for all m there is an r0(m) such thatIntroduce the function \u03c8 = 0 follows now by the strong UCP given by Theorem 4. \u25a1Proof of Corollary 11, by assumption, is an element ofThis is a consequence of the proof of Theorem 6, since \u0398Lloc3/2\u211d3. \u25a1Proof of Corollary 2We first demonstrate that the conditions of Corollaryfulfills Assumption Due to the particular form of the potentials, we make use of the following: Letf1\u2208Kloc3,\u03b4andf2\u2208Kloc6,\u03b4. Then both\u2211k=1Nf1xkandare elements ofKloc3N,\u03b4. Similar statements for Kncan be found in Simon(Example F) and Aizenman\u2010Simon(Theorem 1.4). We prove our claim by direct computations. DefineI1\u03b4andI2\u03b4according toThat We next demonstrate thatq =  and note thatTo show Equation J1\u03b4 is finite sinceq = , thenNow, Proof of Corollary 3We first reduce the molecular case to the atomic one. Since the UCP from sets of positive measure is local, it can be applied to any open set in the domain individually. So instead of one singularity (the y of Assumption), we can treat an arbitrary (yet countable) number of singularities if they do not have an accumulation point. For this just choose an open cover {Uj} ofwhere each Ujcontains not more than one nucleusIt remains to show that allqxnuc;j,bxnuc;jbelong to the respective local Kato classes and we are done if we prove the results for atoms.Corollary Z\u2009>\u20090, and v\u2212\u2208Lloc3/2\u211d3 and with the choice x\u00afnuc=xnuc\u2026xnuc\u2208\u211dD, we have with Qxnucx=Qx\u00afnucx the equalityIn the sequel we let q1,xnuc=v\u2212 and q2,xnuc=0.Thus, in this case we can choose \u03b4\u2009<\u20091, we claim that V,U\u2208KlocD,\u03b4. By the first part it suffices to show v\u2208Kloc3,\u03b4 and u\u2208Kloc6,\u03b4. For v\u2208Kloc3,\u03b4, we introduce polar coordinates with radius s and polar angle t. Then it holds that y \u00b7 x = \u2212s|x|cost. For Furthermore, for 0\u2009<\u2009t, use |s\u2009+\u2009|x|\u2009\u2212\u2009|s \u2212\u2009|x|||\u22642|x|, and the conclusion is obtained for v. In a similar fashion, for u we establish that with u\u2208Kloc6,\u03b4, sinceWe integrate over The atomic case is now a consequence of Corollary 7In this work we were able to show the unique\u2010continuation property from sets of positive measures for the important case of the many\u2010body magnetic Schr\u00f6dinger equation for classes of potentials that are independent of the particle number. This is crucial in order to not artificially restrict the permitted potentials in large systems. We further specifically addressed molecular Hamiltonians, thus covering most cases that usually arise in physics."}
{"text": "Second, we quantified the strain field in the aortic tissue in reference to photo-bleached markers. It was found in the radial-circumferential plane that the largest strain direction was\u2009\u2212\u200921.3\u00b0\u2009\u00b1\u200911.1\u00b0, and the zero normal strain direction was 28.1\u00b0\u2009\u00b1\u200910.2\u00b0. Thus, the SFs in aortic SMCs were not in line with neither the largest strain direction nor the zero strain direction, although their orientation was relatively close to the zero strain direction. These results suggest that SFs in aortic SMCs undergo stretch, but not maximal and transmit the force to nuclei under intraluminal pressure.Stress fibers (SFs) in cells transmit external forces to cell nuclei, altering the DNA structure, gene expression, and cell activity. To determine whether SFs are involved in mechanosignal transduction upon intraluminal pressure, this study investigated the SF direction in smooth muscle cells (SMCs) in aortic tissue and strain in the SF direction. Aortic tissues were fixed under physiological pressure of 120\u00a0mmHg. First, we observed fluorescently labeled SFs using two-photon microscopy. It was revealed that SFs in the same smooth muscle layers were aligned in almost the same direction, and the absolute value of the alignment angle from the circumferential direction was 16.8\u00b0\u2009\u00b1\u20095.2\u00b0 . SFs work as a force-transmitting and force-focusing molecular \u201cdevice\u201d and transmit external forces to distant places in cells  production rate, while those cyclically stretched perpendicular to their major axis do not  from Chubu Kagaku Shizai, Nagoya, Japan. Descending thoracic aortas were obtained, as described previously  plane, we embedded fixed aortic samples in 3% agar solution  for 15\u00a0min at 4\u00a0\u00b0C and then sliced them perpendicular to the longitudinal (z) axis into 150-\u03bcm-thick section using a DTK-1000 microslicer .To observe SFs in the radial-circumferential (3), was filled with 200\u2009\u00d7\u2009diluted Alexa Fluor 488 phalloidin  in PBS(\u2013) including 0.2% bovine serum albumin , incubated for 4\u00a0h at room temperature, and washed thrice with PBS(\u2013).Next, we placed the sections inside a flow cell made with a 25\u2009\u00d7\u200960 C025601 coverslip  and a 18\u2009\u00d7\u200918 C218181 coverslip (Matsunami) glued together with 86-\u03bcm-thick NW-10 double-sided tape . The flow cell, box shape  equipped with a mode-locked Ti:sapphire laser . We observed samples using optical filters  and a LUMPLFLN 60XW objective lens .z direction with the z interval of 1\u00a0\u03bcm. For tubular aortic samples pressurized at 15\u00a0mmHg to obtain the reference state of strain, the longitudinal-circumferential planes were imaged from the outside of the adventitia with the r interval of 1\u00a0\u03bcm. To compare the SF direction between local positions, we imaged the ventral, dorsal, and lateral sides of the aorta.For aortic samples fixed at 120\u00a0mmHg and sectioned, the radial-circumferential planes at 15\u201330\u00a0\u03bcm depth from the surface were imaged from z direction and implemented the maximum intensity projection in the z direction. Next, we defined the circumferential direction on the maximum intensity projection image as the overall direction of ELs, which we determined using the Directionality function . First, we stacked radial-circumferential plane images of EL and SFs in the We made strain markers by photobleaching ELs to the samples to simultaneously measure the SF direction and local strain. The detailed process is given . The normal strain in the SF direction was calculated as follows:We determined the strain in reference to strain marker positions at 15\u00a0mmHg using isoparametric mapping with a first-order shape function, as described previously . Data were shown as the mean\u2009\u00b1\u2009standard deviation. Absolute values of SF angles biased from zero were evaluated using the Figure\u00a0\u03b1SF\u2009=\u2009\u2009\u2212\u20092.6\u00b0\u2009\u00b1\u200917.6\u00b0 (n\u2009=\u200996). We found no SFs in the range\u2009\u2212\u20091\u00b0\u2009\u2264\u2009\u03b1SF\u2009<\u20097\u00b0, indicating that SFs are not aligned in the circumferential direction. In addition, no SFs were aligned in the range \u03b1SF\u2009<\u2009\u2009\u2212\u200933\u00b0 or 31\u00b0\u2009\u2264\u2009\u03b1SF. The frequency distribution showed two distinct peaks, indicating that SFs are aligned in a specific direction. Dividing the distribution into two by the border of 0\u00b0, that is, positive and negative data, and fitting the Gaussian distribution to each gave us the mean of \u03b1SF as\u2009\u2212\u200917.7\u00b0 (n\u2009=\u200957) and 16.0\u00b0 (n\u2009=\u200939) for negative and positive distributions, respectively. Taking the absolute value of \u03b1SF, the mean was |\u03b1SF|=\u200916.8\u00b0\u2009\u00b1\u20095.2\u00b0, which is significantly different from the circumferential direction (0\u00b0).Figure\u00a0\u03b1SF| was 18.7\u00b0\u2009\u00b1\u20095.2\u00b0 (n\u2009=\u200926) on the ventral, 15.5\u00b0\u2009\u00b1\u20094.9\u00b0 (n\u2009=\u200931) in the lateral, and 17.0\u00b0\u2009\u00b1\u20095.3\u00b0 (n\u2009=\u200939) in the dorsal side. There was no significant difference between the three groups.Figure\u00a0We tested 9 mice and 87 sliced samples and identified markers in only 9 regions of 4 samples (2 mice). Typical image of ELs and SFs, observed simultaneously, is presented in Fig.\u00a0\u03b5\u03b8\u03b8, \u03b5rr, and \u03b5\u03b8r in nine regions where we clearly observed SFs. \u03b5\u03b8\u03b8 was positive (0.13\u2009\u00b1\u20090.04), and \u03b5rr was either positive or negative (\u2212\u20090.09\u2009\u00b1\u20090.07). We also obtained both positive and negative values for \u03b5\u03b8r (\u2212\u20090.07\u2009\u00b1\u20090.12). Regions 3\u20135 in Fig.\u00a0Figure\u00a0\u03b11, \u03b12, \u03b1min, and \u03b1SF in each region from the circumferential direction. In all regions, the signs of \u03b11 and \u03b1SF were different. The signs of the angles in region 4 were completely opposite to those in other regions. Unifying the direction of shear strains by changing the sign of data in region 4 gave us \u03b1SF\u2009=\u200915.5\u00b0\u2009\u00b1\u20092.7\u00b0, \u03b11\u2009=\u2009\u2009\u2212\u200921.3\u00b0\u2009\u00b1\u200911.1\u00b0, \u03b12\u2009=\u200968.7\u00b0\u2009\u00b1\u200911.1\u00b0, and \u03b1min\u2009=\u200928.1\u00b0\u2009\u00b1\u200910.2\u00b0. These results showed that SFs do not align in the first principal strain direction but that the SF angle is closer to the zero normal strain angle .Figure\u00a0\u03b5\u03b8\u03b8 and \u03b51 with \u03b5SF. \u03b5SF was 0.06\u2009\u00b1\u20090.04, which was significantly smaller than \u03b5\u03b8\u03b8 (0.13\u2009\u00b1\u20090.04) and \u03b51 (0.19\u2009\u00b1\u20090.07). However, \u03b5SF was significantly larger than 0.Figure\u00a0\u03b11 , SFs avoid stretch by aligning perpendicular to the direction of the cyclic stretch  was smaller compared to our previous study (\u03b5\u03b8\u03b8\u2009=\u20090.38)  obtained in the present study is almost equivalent to the one at 40\u00a0mmHg in the previous study, we consider that \u03b11 obtained in the present study would not vary so much from that at 120\u00a0mmHg. Therefore, it could be certain that \u03b11 is far different from \u03b1SF.The strain magnitude and direction might be affected by the sample preparation in this study, compared with those in unfixed samples subjected to the same loading conditions. The normal circumferential strain at 120\u00a0mmHg in this study . Also, if circumferential-radial shear strain is released, SFs rotates circumferentially, resulting in a higher \u03b1SF and lower \u03b5SF . Thus, the effect of strain release on \u03b1SF and \u03b5SF differs depending on what strain was released. Further investigations are required to determine more precise value of \u03b1SF and evaluate whether \u03b1SF equals to \u03b1min or not.Although SFs are not in the largest strain direction, they are also not in line with \u03b1SF is slightly different from \u03b1min probably because of the necessity of tension in the actomyosin network to assemble and maintain the SF structure"}
{"text": "\u211b1 < 1. Using center manifold theory, we have verified that both the pneumonia only submodel and the HIV/AIDS-pneumonia coepidemic model undergo backward bifurcations whenever \u211b2 < 1\u2009 and \u211b3 = max{\u211b1, \u211b2} < 1, respectively. Thus, for pneumonia infection and HIV/AIDS-pneumonia coinfection, the requirement of the basic reproduction numbers to be less than one, even though necessary, may not be sufficient to completely eliminate the disease. Our sensitivity analysis results demonstrate that the pneumonia disease transmission rate \u2009\u03b22 and the HIV/AIDS transmission rate \u2009\u03b21 play an important role to change the qualitative dynamics of HIV/AIDS and pneumonia coinfection. The pneumonia infection transmission rate \u03b22 gives rises to the possibility of backward bifurcation for HIV/AIDS and pneumonia coinfection if \u211b3 = max{\u211b1, \u211b2} < 1, and hence, the existence of multiple endemic equilibria some of which are stable and others are unstable. Using standard data from different literatures, our results show that the complete HIV/AIDS and pneumonia coinfection model reproduction number is \u211b3 = max{\u211b1, \u211b2} = max{1.386, 9.69\u2009} = 9.69\u2009 at \u03b21 = 2 and \u03b22 = 0.2\u2009 which shows that the disease spreads throughout the community. Finally, our numerical simulations show that pneumonia vaccination and treatment against disease have the effect of decreasing pneumonia and coepidemic disease expansion and reducing the progression rate of HIV infection to the AIDS stage.In this paper, we proposed and analyzed a realistic compartmental mathematical model on the spread and control of HIV/AIDS-pneumonia coepidemic incorporating pneumonia vaccination and treatment for both infections at each infection stage in a population. The model exhibits six equilibriums: HIV/AIDS only disease-free, pneumonia only disease-free, HIV/AIDS-pneumonia coepidemic disease-free, HIV/AIDS only endemic, pneumonia only endemic, and HIV/AIDS-pneumonia coepidemic endemic equilibriums. The HIV/AIDS only submodel has a globally asymptotically stable disease-free equilibrium if  HIV/AIDS remains a major global health problem affecting approximately 70 million people worldwide causing significant morbidity and mortality WHO, 2018) [8 [1]. OvPneumonia is one of the leading airborne infectious diseases caused by microorganisms such as bacteria, viruses, or fungi. It has been the common cause of morbidity and mortality in adults, children under five years of age, and HIV-mediated immunosuppression worldwide, and it is a treatable respiratory lung infectious disease , 10\u201314. A coepidemic is the coexistence of two or more infections on a single individual at the population level . HIV/AIDMathematical and statistical models of infectious diseases have, historically, provided useful insight into the transmission dynamics and control of infectious diseases . MathemaBabaei et al.  developeLimited mathematical modeling research analysis has been conducted on HIV/AIDS-pneumonia coepidemics, for prevention and controlling of the disease transmission with controlling and prevention mechanisms; however, theoretical sources such as , 20, 21 We are motivated by the above studies especially the HIV/AIDS-pneumonia coexistence in the community; therefore, in this study, we considered the three center for disease control and prevention (CDC) stages of the HIV infection which are acute HIV infection, chronic HIV infection, and AIDS stage; we presented and analyzed a mathematical model describing the transmission dynamics of HIV/AIDS and pneumonia coinfection in a population where treatment for HIV/AIDS and both vaccination and treatment for pneumonia are available, respectively, in the community. Our model will be used to evaluate the effect of treatment at every infection stage of the HIV/AIDS only model, pneumonia only model, HIV/AIDS-pneumonia coinfection model, and effect of vaccination for pneumonia only model as control strategies for minimizing incidences of coinfections in the target population. The paper is organized as follows. The model is formulated in N(t) into twelve distinct classes as susceptible class to both HIV and pneumonia infections Y1(t), pneumonia vaccine class Y2\u2009(t)\u2009, pneumonia-infected class Y3(t), acute HIV-infected class Y4(t), chronic HIV-infected class Y5(t), AIDS patient class Y6(t), acute HIV-pneumonia coepidemic class Y7(t), chronic HIV-pneumonia coepidemic class Y8(t), AIDS-pneumonia coepidemic class Y9(t), pneumonia treatment class Y10(t), HIV/AIDS treatment class Y11(t) entered from the three infection stages Y4(t), Y5(t), and Y6(t), and HIV/AIDS-pneumonia coepidemic treatment class Y12(t) entered from Y7(t), Y8(t), and Y9(t) cases such thatAccording to CDC the three HIV/AIDS infection stages, we divide the human population \u03c13 \u2265 \u03c12 \u2265 \u03c11 \u2265 1 is the modification rate that increases infectivity and \u03b21 is the HIV/AIDS contagion rate.The susceptible class acquires HIV at the standard incidence rate given by\u03c93 > \u03c92 > \u03c91 is the modification rate that increases infectivity and \u03b22 is the pneumonia contagion rate.The susceptible class acquires pneumonia at the mass action incidence rate\u03c0 and (1 \u2212 \u03c0) fraction of population entered to the vulnerable classA fraction of the population has been vaccinated before the disease outbreak at the portion of \u03c4 and from pneumonia-treated class in which those individuals who lose their temporary immunity by the rate\u2009\u03b8The susceptible class is increased from the vaccinated class in which those individuals who are vaccinated but did not respond to vaccination with the waning rate of \u03f5\u2009 of the serotype not covered by the vaccine where 0 \u2264 \u03f5 < 1Assume vaccination is not 100% effective, so vaccinated individuals also have a chance of being infected with proportion Individuals in a given compartment are homogeneousY6(t) and Y9(t) classes due to their reduced daily activitiesAssume no HIV transmission from dIndividuals in each class are subject to natural mortality rate The human population is variableWe assumed there is no dual-infection transmission simultaneouslyAssume HIV has no vertical transmission and pneumonia is not naturally recoveredNo permanent immunity for pneumonia-infected individuals and become susceptible again after treatmentTo construct the complete coepidemic dynamical system, we have assumed the following:In this subsection using parameters in From With initial conditions,The sum of all the differential equations in  is(6)dNThe model is mathematically analyzed by proving various theorems and algebraic computation dealing with different quantitative and qualitative attributes. Since the system deals with human populations which cannot be negative, we need to show that all the state variables are always nonnegative well as the solutions of system  remain pt > 0 in the bounded region given in , Y2(t), Y3(t), Y4(t), Y5(t), \u2009Y6(t), Y7(t), Y8(t), Y9(t), Y10(t), Y11(t), and Y12(t) of system  > 0, Y2(0) > 0, Y3(0) > 0, Y4(0) > 0, Y5(0) > 0, Y6(0) > 0, Y7(0) > 0, Y8(0) > 0, Y9(0) > 0, Y10(0) > 0, Y11(0), and Y12(0) > 0; then, for all t > 0, we have to prove that Y1\u2009(t) > 0, Y2(t) > 0, Y3(t) > 0,\u2009Y4(t) > 0, Y5(t) > 0, Y6(t) > 0, Y7(t) > 0, Y8(t) > 0, Y9(t) > 0, \u2009Y10(t) > 0, Y11(t) > 0, and Y12(t) > 0.Assume \u03c4 = sup{t > 0 : Y1\u2009(t) > 0, Y2(t) > 0, Y3(t) > 0, \u2009Y4(t) > 0, Y5(t) > 0, Y6(t) > 0, Y7(t) > 0, Y8(t) > 0, Y9(t) > 0, Y10(t) > 0, Y11(t) > 0, and\u2009Y12(t) > 0}.Define: Y1(t), Y2(t), Y3(t), Y4(t), \u2009Y5(t), Y6(t), Y7(t), Y8(t), Y9(t), Y10(t), Y11(t), and Y12(t)(t), we deduce that \u03c4 > 0. If \u03c4 = +\u221e, then positivity holds. But, if 0 < \u03c4 < +\u221e, Y1(\u03c4) = 0 or Y2(\u03c4) = 0 or Y3(\u03c4) = 0 or Y4(\u03c4) = 0 or Y5(\u03c4) = 0 or Y6(\u03c4) = 0 or Y7(\u03c4) = 0 or Y8(\u03c4) = 0 or Y9(\u03c4) = 0 or Y10(\u03c4) = 0 or Y11(\u03c4) = 0 or Y12(0) = 0.From the continuity of\u2009dY1/dt = (1 \u2212 \u03c0)\u039b + \u03b8Y10 + \u03c4Y2 \u2212 (d + \u03bbHC + \u03bbPC)Y1.Here, from the first equation of the model , we haveY1(\u03c4) = M1Y1(0) + M1\u222b0\u03c4expd + \u03bbHc(t) + \u03bbPc(t))dt\u222b(((1 \u2212 \u03c0)\u039b + \u03b8Y10(t) + \u03c4Y2(t))dt > 0\u2009where\u2009M1 = expd\u03c4 + \u222b0\u03c4(\u03bbHC(w) + \u03bbPC(w)dw)\u2212 > 0, and from the definition of\u2009\u03c4, we see that Y2(t) > 0, \u2009Y10(t) > 0, and also the exponential function is always positive; then, the solution Y1(\u03c4) > 0; hence, Y1(\u03c4) \u2260 0. From the second equation of the model (dY2/dt = \u03c0\u039b \u2212 (d + \u03c41 + \u03f5\u03bbPc + \u03bbHc)Y2 and we have got Y2(\u03c4) = M1Y2(0) + M1\u222b0\u03c4expd + \u03c41 + \u03f5\u03bbPc(t) + \u03bbHc(t))dt\u222b((\u03c0\u039b)dt > 0, where M1 = expd\u03c4 + \u03c41\u03c4 + \u222b0\u03c4(\u03bbHc(w) + \u03f5\u03bbPc(w)dw)\u2212 > 0, and also, the exponential function always is positive; then, the solution\u2009Y2(\u03c4) > 0; hence, Y2(\u03c4) \u2260 0. Similarly, all the remaining state variables Y3(\u03c4) > 0; hence,\u2009Y3(\u03c4) \u2260 0 and Y4(\u03c4) > 0; hence, Y4(\u03c4) \u2260 0 and Y5(\u03c4) > 0; hence, Y5(\u03c4) \u2260 0 and Y6(\u03c4) > 0; hence, Y6(\u03c4) \u2260 0 and Y7(\u03c4) > 0; hence, Y7(\u03c4) \u2260 0 and Y8(\u03c4) > 0; hence, Y8(\u03c4) \u2260 0 and Y9(\u03c4) > 0; hence, Y9(\u03c4) \u2260 0 and Y10(\u03c4) > 0; hence, Y10(\u03c4) \u2260 0 and Y11(\u03c4) > 0; hence, Y11(\u03c4) \u2260 0 and Y12(\u03c4) > 0; hence Y12(\u03c4) \u2260 0. Thus, based on the definition of\u2009\u03c4, it is not finite which means \u03c4 = +\u221e, and hence, all the solutions of system ) \u2264 \u222bdt and integrating both sides gives \u2212(1/d)ln(\u039b \u2212 dN) \u2264 t + c where c is some constant, and after some steps of calculations, we have got 0 \u2264 N\u2009(t) \u2264 \u039b/d which means all possible solutions of system  = Y1(t) + Y4(t) + Y5(t) + Y6(t) + Y11(t) and the HIV/AIDS single infection force of infection is given by \u03bbH = (\u03b21/N1)(Y4 + \u03c11Y5) with initial conditions Y1(0) > 0, Y4(0) \u2265 0, Y5(0) \u2265 0, Y6(0) \u2265 0, and\u2009Y11(0) \u2265 0.where the total population is Here, the detailed HIV/AIDS submodel model analysis is given in .Y4 = Y5 = Y6 = Y7=\u2009Y8=Y9 = Y11 = Y12 = 0, which is given byFrom model , we haveY1(0) > 0, Y2(0) \u2265 0, Y3(0) \u2265 0, Y10(0) \u2265 0, total population \u2009N2(t) = Y1(t) + Y2(t) + Y3(t) + Y10(t), and pneumonia force of infection \u2009\u03bbP = \u03b22Y3(t).With initial conditions, \u03a92 is positively invariant and a global attractor of all positive solutions of submodel  as zero and setting the infectious class and treatment class to zero as Y10 = \u039b(d + \u03c41) \u2212 \u039b\u03c0d/d(d + \u03c41) and Y20 = \u039b\u03c0/(d + \u03c41) such that E20 =  = ((\u039b(d + \u03c41) \u2212 \u039b\u03c0d/d(d + \u03c41)), (\u039b\u03c0/d + \u03c41), 0, 0).\u211b2 using the van den Driesch and Warmouth next-generation matrix approach [FV\u22121 = [\u2202\u2131i(E20)/\u2202xj][\u2202\u03bdi(E20)/\u2202xj]\u22121 where \u2131i is the rate of appearance of new infection in compartment i, \u03bdi is the transfer of infections from one compartment i to another, and E20 is the disease-free equilibrium point. Then, after a long calculation, we have gotThe effective reproduction number measures the average number of new infections generated by a typically infectious individual in a community when some strategies are in place, like vaccination or treatment. We calculate the effective reproduction number \u2009approach . The EffThen, using Mathematica, we have gotFV\u22121\u2009isThe characteristic equation of the matrix \u211b2) of FV\u22121 of the pneumonia submodel (\u211b2 = (\u03b22\u03f5\u039b\u03c0d + \u03b22\u039b(d + \u03c41) \u2212 \u03b22\u039b\u03c0d)/(d(d + \u03c41)(d + \u03ba + dP)). Here, \u211b2 is the effective reproduction number for pneumonia infection.Then, the spectral radius  submodel  is localE20 = ((\u039b(d + \u03c41) \u2212 \u039b\u03c0d)/(d(d + \u03c41)), \u039b\u03c0/(d + \u03c41\u2009), 0, 0) and Routh Hurwitz stability criteria. The Jacobian matrix of a dynamical system  \u2212 \u03b22\u039b\u03c0d)/(d(d + \u03c41))) \u2212 (d + \u03ba + dP).Then, the characteristic equation of the above Jacobian matrix is given by\u03bb1 = \u2212d < 0 or\u2009\u03bb2 = \u2212(d + \u03c41) < 0 or\u2009\u03bb3 = (d + \u03ba + dP)[\u211b2 \u2212 1] < 0 if\u2009\u211b2 < 1 or \u03bb4 = \u2212(d + \u03b8) < 0. Therefore, since all the eigenvalues of the characteristics polynomial of the system . Moreover, let \u03bbP\u2217 = \u03b22Y3\u2217 be the associated pneumonia mass action incidence rate (\u201cforce of infection\u201d) at an equilibrium point. To find conditions for the existence of an arbitrary equilibrium point(s) for which pneumonia infection is endemic in the population, the equations of model (\u03bbP\u2217 = \u03b22Y3\u2217 at an endemic equilibrium point. Setting the right-hand sides of the equations of model (Y2\u2217 = \u03c0\u039b/(\u03f5\u03bbP\u2217 + d + \u03c41), Y10\u2217 = \u03baY3\u2217/(d + \u03b8) and substitute Y2\u2217 and Y10\u2217\u2009 in to Y1\u2217, we obtain Y1\u2217 = ((1 \u2212 \u03c0)\u039b + \u03c41Y2\u2217 + \u03b8TP\u2217)/(d + \u03bbP\u2217) = ((1 \u2212 \u03c0)\u039b/(d + \u03bbP\u2217)d + \u03bbP\u2217) + (\u03c0\u039b\u03c41/(\u03f5\u03bbP\u2217 + d + \u03c41)(d + \u03bbP\u2217)) + (\u03b8\u03b3Y3\u2217/(d + \u03b8)(d + \u03bbP\u2217)) and substitute Y2\u2217 and Y1\u2217 in\u2009Y3\u2217, we obtainLet an arbitrary endemic equilibrium point of pneumonia-only dynamical system  be denotof model  are solvof model  to zero Y3\u2217 in to pneumonia submodel  = a2\u03bbP\u22172 + a1\u03bbP\u2217 + a0 = 0 so that the quadratic equation can be analyzed for the possibility of multiple equilibriums. It is worth noting that the coefficient\u2009a2\u2009is always positive and a0 is positive (negative) if \u211bP is less than (greater than) unity, respectively. Hence, we have established the following result.Here, the nonzero equilibrium(s) of the model  satisfiea0 < 0 Exactly one unique endemic equilibrium if a1<0, and a0 = 0 or a12 \u2212 4a2a0 = 0Exactly one unique endemic equilibrium if\u2009a0 > 0\u2009, a1 < 0, and\u2009a12 \u2212 4a2a0 > 0Exactly two endemic equilibriums if No endemic equilibrium otherwiseThe pneumonia submodel  has the \u211b2 < 1; examples of the existence of backward bifurcation phenomenon in mathematical epidemiological models, and the causes, can be seen in [\u2202\u03bdi(E30)/\u2202xj]\u22121 where\u2009\u2131i\u2009is the rate of appearance of new infection in compartment\u2009i\u2009, \u03bdi\u2009is the transfer of infections from one compartment i\u2009to another, and E30 is the disease-free equilibrium point E30 = (\u039b(d + \u03c41) \u2212 \u039b\u03c0d/d(d + \u03c41), \u039b\u03c0/d + \u03c41, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0).Here, we calculated the HIV/AIDS-pneumonia coinfection effective reproduction number\u2009of model  using thF is given byV is given byAfter long detailed calculations, the transition matrix D1 = d + \u03ba + dP,\u2009D2 = d + \u03ba1 + \u03b41,\u2009D3 = d + \u03ba2 + \u03b42,\u2009D4 = d + \u03ba3 + dA,\u2009D5 = d + dP + \u03c31 + \u03b43,\u2009D6 = d + dP + \u03c32 + \u03b44,\u2009and\u2009D7 = d + dAP + \u03b53.where\u2009Then, by using Mathematica, we have gotFV\u22121 is given byThe characteristic equation of the matrix\u2009A1 = (\u03c0\u03f5\u039b\u03b22/(d + \u03c41)) + \u03b22[(\u2212\u03c0\u039bd + \u039b[d + \u03c41]/d[d + \u03c41])]/D1, B = (\u03b21/D2) + (\u03b41\u03b21\u03c11/D2D3); \u2009then, the eigenvalues of FV\u22121 are \u03bb1 = \u03b22\u03f5\u03c0d\u039b + \u03b22\u039b[d + \u03c41] \u2212 \u03b22\u03c0\u039bd/D1(d + \u03c41)\u2009or\u2009\u03bb2 = (\u03b21/D2) + \u03b41\u03b21\u03c11/D2D3\u2009 or \u03bb3 = \u03bb4 = \u03bb5 = \u03bb6 = \u03bb7 = \u03bb8 = \u03bb9 = \u03bb10 = 0.where FV\u22121 which is given byThus, the effective reproduction number of the HIV/AIDS-pneumonia coinfection dynamical system  is the d\u211b3 = max{\u03bb1, \u03bb2} = max{\u03b22\u03f5\u03c0d\u039b + \u03b22\u039b[d + \u03c41] \u2212 \u03b22\u03c0\u039bd/D1(d + \u03c41), (\u03b21/D2) + (\u03b41\u03b21\u03c11/D2D3)}. Here, \u2009\u211b2 = \u03b22\u03f5\u03c0d\u039b + \u03b22\u039b[d + \u03c41] \u2212 \u03b22\u03c0\u039bd/(d + \u03ba + dP)(d + \u03c41) is the effective reproduction number for pneumonia-only infected individual and \u211b1 = (\u03b21/(d + \u03ba1 + \u03b41)) + (\u03b21\u03c11\u03b41/(d + \u03ba1 + \u03b41)(d + \u03ba2 + \u03ba2)) is the basic reproduction for HIV/AIDS-only infected individual.\u211b1 represent the basic reproduction number for HIV/AIDS submodel, \u2009\u211b2 and \u211b3 are the effective reproduction numbers for the pneumonia submodel and HIV/AIDS-pneumonia coinfection model, respectively. The following three outcomes are possible: (i) for \u211b1 < 1, the HIV/AIDS submodel disease-free steady state E1 is globally stable in the region \u03a91, and HIV is not spreading in the community; (ii) for \u211b2 < 1, then E2 is not globally stable in the region \u03a92, and pneumonia may spread through the community; (iii) for \u211b3 < 1, the steady state E3\u2009is not globally stable in the region \u03a9, and HIV/AIDS-pneumonia coinfection may spread through the community.Here, \u03c31\u2009or\u2009\u03c32\u2009or\u2009\u03c33) are present in the expression for\u2009\u211b3, indicating no impact of treating coinfected population on\u2009\u211b3.Note that none of the parameters corresponding to coinfection treatment  of model (E30 is given byThe Jacobian matrix of model  at E30 iZ1 = \u03b22\u03f5Y20 + \u03b22Y10 \u2212 (d + \u03ba + dP),\u2009Z2 = (\u03b21/N0)Y10 \u2212 (d + \u03ba1 + \u03b41), \u2009Z3 = \u2212(d + \u03ba2 + \u03b42), \u2009Z4 = \u2212(d + \u03ba3 + dA),\u2009Z5 = \u2212(d + dP + \u03c31 + \u03b43),\u2009Z6 = \u2212(d + dP + \u03c32 + \u03b44),\u2009Z7 = \u2212(d + dAP + \u03c33), and\u2009Z8 = \u2212(d + \u03b8).where\u2009J\u2009(E30) is given byThen, the characteristic equation of the Jacobian matrix After detailed calculations, we have got that\u03bb1 = \u03bb2 = \u03bb3 = \u2212d < 0 or \u2009\u03bb4 = (d + \u03ba + dP)[\u211b2 \u2212 1] < 0 if \u211b2 < 1 or \u03bb5 = \u2212(d + \u03c41) < 0 or \u03bb6 = \u2212(d + \u03ba3 + dA) < 0 or \u03bb7 = \u2212(d + dP + \u03c31 + \u03b43) < 0 or \u03bb8 = \u2212(d + dP + \u03c32 + \u03b44) < 0 or \u03bb9 = \u2212(d + dAP + \u03c33) < 0 or \u03bb10 = \u2212(d + \u03b8) < 0\u2009or a2\u03bb2 + a1\u03bb + a0 = 0 for a2 = 1,\u2009a1 = (d + \u03ba2 + \u03b42) + (d + \u03ba1 + \u03b41)[1 \u2212 (Y10/N0)\u211bY4] > 0 if \u211bY4 < 1 and\u2009a0 = (d + \u03ba2 + \u03b42)(d + \u03ba1 + \u03b41)\u2009where\u2009\u03c13 \u2265 \u03c12 \u2265 \u03c11 \u2265 1 and\u2009\u03bbPC = \u03b22[x3 + \u03c91x7 + \u03c92x8 + \u03c93x9]\u2009 where\u2009\u03c93 \u2265 \u03c92 \u2265 \u03c91 \u2265 1; then, the method entails evaluating the Jacobian of system , and this gives usF1 = \u2212\u03b22Y10,\u2009F2 = \u2212\u03b21(Y10/(Y10 + Y20)),\u2009F3 = \u2212\u03b21\u03c11(Y10/(Y10 + Y20)), F4 = \u2212\u03b21\u03c12(Y10/(Y10 + Y20)) \u2212 \u03b22\u03c91Y10,\u2009F5 = \u2212\u03b21\u03c13(Y10/(Y10 + Y20)) \u2212 \u03b22\u03c92Y10,\u2009F6 = \u2212\u03b22\u03c93Y10, \u2009F7 = \u2212(d + \u03c41),\u2009F8 = \u2212\u03f5\u03b22Y20,\u2009F9 = \u2212\u03b21(Y20/(Y10 + Y20)),\u2009F10 = \u2212\u03b21\u03c11(Y20/(Y10 + Y20)),Further, by using vector notation \u2009follows:dx1dt=f1=F11 = \u2212\u03f5\u03b22\u03c91VP0 \u2212 \u03b21\u03c12(YP0/(Y10 + YP0)), F12 = \u2212\u03f5\u03b22\u03c92VP0 \u2212 \u03b21\u03c13(Y20/(Y10 + Y20)),\u2009F13 = \u2212\u03f5\u03b22\u03c93Y20,\u2009F14 = \u03f5\u03b22Y20 + \u03b22Y10 \u2212 (d + \u03ba + dP),\u2009F15 = \u03f5\u03b22\u03c91Y20 + \u03b22\u03c91Y10,\u2009F16 = \u03f5\u03b22\u03c92Y20 + \u03b22\u03c92Y10,\u2009F17 = \u03f5\u03b22\u03c93Y20 + \u03b22\u03c93Y10, F18 = \u03b21(Y10/(Y10 + Y20)) \u2212 (d + \u03ba1 + \u03b41),\u2009F19 = \u03b21\u03c11(Y10/(Y10 + Y20)),\u2009F20 = \u03b21\u03c12(Y10/(Y10 + Y20)),\u2009F21 = \u03b21\u03c13(Y10/(Y10 + Y20)),\u2009F22 = \u2212(d + \u03ba2 + \u03b42),\u2009F23 = \u2212(d + \u03ba3 + dA),\u2009F24 = \u2212(d + dP + \u03c31 + \u03b43),\u2009F25 = \u2212(d + dP + \u03c32 + \u03b44),\u2009F26 = \u2212(d + dAP + \u03c33), and \u2009F27 = \u2212(d + \u03b8).\u211b2 > \u211b1, and \u211b3 = 1, so that \u211b2 = 1. Furthermore, let \u03b22 = \u03b2\u2217 is chosen as a bifurcation parameter. Solving for \u03b22 from \u211b2 = 1 as \u211b2 = \u03b22\u03f5\u039b\u03c0d + \u03b22\u039b(d + \u03c41) \u2212 \u03b22\u039b\u03c0d/d(d + \u03c41)(d + \u03ba + dP) = 1, we have got the value \u03b2\u2217 = \u03b22 = d(d + \u03c41)(d + \u03ba + dP)/\u03f5\u039b\u03c0d + \u039b(d + \u03c41) \u2212 \u039b\u03c0d.Without loss of generality, consider the case when J(E30) of the system  < 0 or \u03bb3 = 0\u2009 or \u03bb4 = \u2212d < 0 or\u2009\u03bb5 = \u2212d < 0 or\u2009\u03bb6 = \u2212(d + \u03b8) < 0 or \u03bb7 = \u2212(d + dAP + \u03ba3) < 0 or\u2009\u03bb8 = \u2212(d + \u03ba3 + dA) < 0 or \u03bb9 = \u2212(d + dP + \u03c31 + \u03b43) < 0 or \u03bb10 = \u2212(d + dP + \u03c31 + \u03b44) < 0\u2009ora2 = 1 > 0, a1 = (d + \u03ba1 + \u03b41)[1 \u2212 (Y10/(Y10 + Y40))\u211bY4] + (d + \u03ba2 + \u03b42) > 0 if \u211bY4 < 1, and a0 = a0 = (d + \u03ba1 + \u03b41)(d + \u03ba2 + \u03b42)[1 \u2212 (Y10/(Y10 + Y20))\u211b1] > 0 if \u211b1 < 1.After solving the Jacobian e system  at the D\u211b1 < 1, and hence, both eigenvalues are negative. It follows that the Jacobian J(E30) of . Hence, the center manifold theory  where G1=\u03b8\u03ba/d(d + \u03b8) and G2 = \u03f5\u03b2\u2217\u03c41VP0 + \u03b2\u2217(d + \u03c41)Y10 + \u03f5\u03b2\u2217dY20/d(d + \u03c41). Thus, the bifurcation coefficient a is positive whenever\u2009G1 > G2. Furthermore, b = v3u3/\u2202x3\u2202\u03b22) = v3u3(\u03f5Y20 + Y10) > 0.\u27f9a = 2\u211b3 = \u211b2 = 1 whenever\u2009G1 > G2.Hence, it follows from in Castillo-Chavez and Song  that moda = G1 > G2 > 0Model  will unda = G1 > G2 < 0Model  will und\u211b3\u2009that depends differentiably on a parameter p is defined as SI(p) = (\u2202\u211b3/\u2202p)\u2217(p/\u211b3) [Definition. The normalized forward sensitivity index of a variable\u2009)\u2217(p/\u211b3) .\u211b1\u2009and \u211b2\u2009since\u2009\u211b3 = max{\u211b1, \u211b2}. Sensitivity analysis results and the numerical simulation are given in this section with parameters values given in N0 is the total number of the initial population of complete model .In this paper, with parameter values in \u211b2 = 9.69 at \u03b22 = 0.2 imply that pneumonia spreads throughout the community and also we have got the indices as shown in \u039b and the pneumonia spreading rate \u03b22. The foremost sensitive negative parameter is treatment rate of pneumonia (\u03ba) which is inversely related to the effective reproduction number \u211b2, i.e., a smaller amount of increase in this parameter value will lead to a greater amount of reduction in the effective reproduction number while a smaller amount of decrement will cause a big increment in the basic reproduction number. Epidemiologically, the most sensitive parameters to \u211b1 and \u211b2 which can be controlled through interventions and preventions are found to be\u2009\u03b21 and\u2009\u03b22, respectively.Similarly, with parameter values in In this section, numerical simulation is performed for complete HIV/AIDS-pneumonia coepidemic model . With od\u03c0 on the pneumonia effective reproduction number \u2009\u211b2. The figure reflects that when the value of\u2009\u03c0 increases, the pneumonia effective reproduction number is going down, and whenever the value of \u03c0 > 0.64 imply \u211b2 < 1. Therefore, public policymakers must concentrate on maximizing the values of pneumonia vaccination portion \u03c0 to prevent and control pneumonia spreading.In this subsection, as we see in \u03b22 on the pneumonia effective reproduction number \u211b2 by keeping the other rates as in \u03b22 increases, the pneumonia effective reproduction number \u211b2 increases, and whenever the value of \u03b22 < 0.022 implies \u211b2 < 1. Therefore, public policymakers must concentrate on minimizing the values of pneumonia spreading rate \u03b22 to minimize pneumonia effective reproduction number\u2009\u211b2.In this subsection, as we see in \u03ba in decreasing the number of pneumonia-only infectious populations. The figure reflects that when the values of \u03ba increase, the number of pneumonia-only infectious population is going down. Therefore, public policymakers must concentrate on maximizing the values of treatment rate \u03ba to pneumonia disease.In this subsection, as we see in \u03ba1,\u2009\u03ba2, and \u03ba3 in decreasing the number of acute HIV only, chronic HIV only, and AIDS-infected population, respectively. The figures reflect that when the values of \u03ba1,\u2009\u03ba2, and \u03ba3 increase, the number of acute HIV only, chronic HIV only, and AIDS-infected population is going down, respectively. Therefore, public policymakers must concentrate on maximizing the values of treatment rate of individuals to HIV/AIDS infection.In this subsection, as we see in Figures 6\u03b21 on the acute HIV-pneumonia coepidemic population\u2009Y7. The figure reflects that as the value of the transmission rate (\u03b21) of HIV/AIDS is increased, the coepidemic population increases, which means the expansion of coepidemic of HIV/AIDS-pneumonia will increase. To control coepidemic of HIV/AIDS-pneumonia, decreasing the spreading rate of HIV/AIDS is important. Therefore, stakeholders must concentrate on decreasing the spreading rate of HIV/AIDS by using the treatment and appropriate method of prevention mechanism to bring down the expansion of coepidemic in the community.In this section, we see in \u03c31,\u2009\u03c32, and \u03c33 in decreasing the number of acute HIV and pneumonia, chronic HIV and pneumonia, and AIDS and pneumonia coinfectious population, respectively. The figures reflect that when the values of\u2009\u03c31,\u2009\u03c32, and \u03c33 increase, the number of acute HIV-pneumonia, chronic HIV-pneumonia, and AIDS-pneumonia coepidemic population is going down, respectively. Therefore, public policymakers must concentrate on maximizing the values of treatment rates of HIV/AIDS-pneumonia coepidemic population.In this subsection, as we see in Figures 10In \u211b3 = max{\u211b1, \u211b2} = max{1.386, 9.69\u2009} = 9.69\u2009 at \u03b21 = 2 and \u03b22 = 0.2. From our numerical result, we recommend that public policymakers must concentrate on maximizing the values of pneumonia vaccination portion and treatment rate of individuals to pneumonia disease. Finally, some of the main epidemiological findings of this study include pneumonia vaccination and treatment against disease has the effect of decreasing the pneumonia and coepidemic disease expansion and prevalence and reducing the progression rate of HIV infection to the AIDS stage and the HIV/AIDS prevalence.A realistic compartmental mathematical model on the spread and control of HIV/AIDS-pneumonia coepidemic incorporating pneumonia vaccination and treatment for both infections are available at each stage of the infection in a population constructed and analyzed. We have shown the positivity and boundedness of the complete HIV/AIDS-pneumonia coepidemic model. Using center manifold theory, we have shown that the pneumonia-only infection and the complete HIV/AIDS-pneumonia coepidemic models undergo the phenomenon of backward bifurcation whenever their corresponding effective reproduction numbers are less than one. The complete model has a disease-free equilibrium that is locally asymptotically stable whenever the maximum of the reproduction numbers of the two submodels described above is less than one. Numerical simulation shows that the complete HIV/AIDS-pneumonia coepidemic model endemic equilibrium point is locally asymptotically stable when its effective reproduction number is greater than one. These results have important public health implications, as they govern the elimination and/or persistence of the two diseases in a community. By analyzing the various associated reproduction numbers, we have shown that the impact of some parameters changes on the associated reproduction numbers to give future recommendations for the stakeholders in the community. From the numerical result, we have got the complete model reproduction number is Due to conflict in our country Ethiopia, it is difficult to incorporate experimental data in the study."}
{"text": "Eozynofilowe zapalenie prze\u0142yku (EoE) jest obok choroby refluksowej najcz\u0119\u015bciej rozpoznawan\u0105 przewlek\u0142\u0105 chorob\u0105 zapaln\u0105 prze\u0142yku, wyst\u0119puj\u0105c\u0105 zar\u00f3wno u dzieci jak i u pacjent\u00f3w doros\u0142ych. Objawy kliniczne EoE zale\u017c\u0105 od wieku pacjenta i czasu trwania choroby. U niemowl\u0105t i m\u0142odszych dzieci wyst\u0119puj\u0105 zaburzenia karmienia, upo\u015bledzenie rozwoju fizycznego, wymioty, b\u00f3le brzucha. U dzieci starszych i doros\u0142ych dominuj\u0105 zaburzenia po\u0142ykania pokarm\u00f3w sta\u0142ych, zatrzymanie k\u0119sa pokarmowego w prze\u0142yku, b\u00f3le w klatce piersiowej. W obrazie endoskopowym prze\u0142yku stwierdza si\u0119 zmiany zapalne b\u0142ony \u015bluzowej, bruzdy, pier\u015bcienie i b\u0142ony oraz nast\u0119powe zw\u0119\u017cenie \u015bwiat\u0142a prze\u0142yku. W badaniu histologicznym stwierdza si\u0119 zapalenie prze\u0142yku z naciekiem kom\u00f3rek kwasoch\u0142onnych. W 2017 roku mi\u0119dzynarodowa robocza grupa ekspert\u00f3w pod kierunkiem A.J. Lucendo i auspicjami towarzystw naukowych: UEG \u2013 United European Gastroenterology, ESPGHAN \u2013 European Society of Pediatric Gastroenterology, Hepatology and Nutrition, EAACI \u2013 European Academy of Allergy and Clinical Immunology, EUREOS \u2013 Eupropean Society of Eosinophilic Oesophagitis, opracowa\u0142a zalecenia diagnostyczne i lecznicze EoE, stosuj\u0105c metod\u0119 GRADE . Mi\u0119dzynarodowa grupa ekspert\u00f3w sk\u0142ada\u0142a si\u0119 z gastroenterolog\u00f3w, alergolog\u00f3w, pediatr\u00f3w, laryngolog\u00f3w, patolog\u00f3w i epidemiolog\u00f3w. Szeroki udzia\u0142 specjalist\u00f3w z r\u00f3\u017cnych dziedzin medycyny pozwoli\u0142 w opracowanych zaleceniach uwzgl\u0119dni\u0107 r\u00f3\u017cne aspekty tej choroby. Herpes simplex virus, mycoplasma, H. pylori), stosowaniu antybiotyk\u00f3w, inhibitor\u00f3w pompy protonowej i in. W immunopatogenezie EoE alergeny pokarmowe i wziewne powoduj\u0105 odpowied\u017a limfocyt\u00f3w Th2, cytokin, pobudzenie mastocyt\u00f3w i aktywacje czynnik\u00f3w maj\u0105cych potencjalny udzia\u0142 w remodelingu prze\u0142yku [Eozynofilowe zapalenie prze\u0142yku (EoE) jest przewlek\u0142\u0105 chorob\u0105 zapaln\u0105 charakteryzuj\u0105c\u0105 si\u0119 dysfunkcj\u0105 prze\u0142yku i naciekiem zapalnym b\u0142ony \u015bluzowej prze\u0142yku z dominacj\u0105 granulocyt\u00f3w kwasoch\u0142onnych. Eozynofilowe zapalenie prze\u0142yku zosta\u0142o po raz pierwszy opisane w 1978 roku przez Landresa i wsp. , 3. Od tprze\u0142yku .W kilku wcze\u015bniejszych zaleceniach opracowanych przez ekspert\u00f3w towarzystw naukowych zosta\u0142y przedstawione kryteria diagnostyczne i zalecenia lecznicze tej choroby , 14, 15.Najnowsze rekomendacje dotycz\u0105ce diagnostyki i leczenia EoE u dzieci i doros\u0142ych zosta\u0142y opublikowane w 2017 roku . WymieniEozynofilowe zapalenie prze\u0142yku jest przewlek\u0142\u0105 chorob\u0105 zwi\u0105zan\u0105 z odpowiedzi\u0105 immunologiczn\u0105 prze\u0142yku, kt\u00f3rej towarzysz\u0105 zmiany histologiczne \u015bciany prze\u0142yku z miejscowym naciekiem zapalnym, z dominacj\u0105 eozynofilii oraz r\u00f3\u017cnorodne, zale\u017cne od wieku i czasu trwania zapalenia objawy kliniczne spowodowane dysfunkcj\u0105 prze\u0142yku. W diagnostyce tej choroby nale\u017cy wykluczy\u0107 inne przyczyny eozynofilii miejscowej i systemowej, a objawy kliniczne i zmiany histopatologiczne nie powinny by\u0107 interpretowane oddzielnie. Wycofane zosta\u0142y z poprzedniej definicji okre\u015blenia, \u017ce EoE jest chorob\u0105 prze\u0142yku \u201ezale\u017cn\u0105 od antygenu\u201d oraz poj\u0119cie \u201eeozynofilii prze\u0142yku odpowiadaj\u0105cej na leczenie inhibitorami pompy protonowej\u201d , stosowanego od 2011 roku jako kryterium diagnostyczne, gdy\u017c jedynie pacjenci nieodpowiadaj\u0105cy na terapi\u0119 PPI albo alternatywnie z prawid\u0142ow\u0105 pH-metri\u0105 prze\u0142yku mogli mie\u0107 rozpoznane EoE. Wed\u0142ug tych za\u0142o\u017ce\u0144 tylko GERD i choroby zale\u017cne od kwasu solnego odpowiadaj\u0105 na leczenie PPI. Obecnie uwa\u017ca si\u0119, \u017ce remisja kliniczna i histologiczna na terapii PPI jest raczej cz\u0119\u015bci\u0105 przebiegu EoE ni\u017c oddzieln\u0105 jednostk\u0105 chorobow\u0105, a pacjenci odpowiadaj\u0105cy i nieodpowiadaj\u0105cy na terapi\u0119 PPI maj\u0105 nak\u0142adaj\u0105ce si\u0119 cechy fenotypowe i genetyczne , 15.Eozynofilowe zapalenie prze\u0142yku jest rozpoznawane w ka\u017cdym wieku. U dzieci cz\u0119sto\u015b\u0107 rozpozna\u0144 wzrasta wraz z wiekiem, u pacjent\u00f3w doros\u0142ych szczyt zachorowa\u0144 wyst\u0119puje pomi\u0119dzy 30-50 rokiem \u017cycia. W zale\u017cno\u015bci od szeroko\u015bci geograficznej zachorowalno\u015b\u0107 na EoE wynosi od 3 do 13 nowych zachorowa\u0144 na 100 000 mieszka\u0144c\u00f3w/rok w Europie, USA i Kanadzie . Natomia2), tzn. powi\u0119kszeniu mikroskopowym 400-krotnym. Ponadto w badaniu histologicznym bioptatu b\u0142ony \u015bluzowej prze\u0142yku mog\u0105 wyst\u0119powa\u0107 mikroropnie z\u0142o\u017cone z eozynofilii (skupiska co najmniej 4 eozynofilii), przerost i/lub w\u0142\u00f3knienie warstwy podstawnej, poszerzenie przestrzeni mi\u0119dzykom\u00f3rkowej, wyd\u0142u\u017cenie warstwy brodawkowej. Nale\u017cy zaznaczy\u0107, \u017ce objawy kliniczne nie koreluj\u0105 dobrze z histologiczn\u0105 aktywno\u015bci\u0105 EoE. Dlatego niezb\u0119dne jest badanie histopatologiczne do monitorowania choroby.Obraz kliniczny eozynofilowego zapalenia prze\u0142yku zale\u017cy od wieku pacjenta i fenotypu choroby. U niemowl\u0105t i m\u0142odszych dzieci najcz\u0119stszymi objawami s\u0105: niepok\u00f3j, trudno\u015bci w karmieniu i odmowa przyjmowania pokarmu, ulewanie, wymioty oraz b\u00f3l w nadbrzuszu, prowadz\u0105ce do zahamowania rozwoju dziecka. U pacjent\u00f3w starszych podobnie jak u doros\u0142ych wyst\u0119puj\u0105 zaburzenia po\u0142ykania pokarm\u00f3w sta\u0142ych z epizodami zaklinowania k\u0119sa pokarmu (food impaction) w prze\u0142yku, zgaga, b\u00f3l w klatce piersiowej. Pacjenci cz\u0119sto staraj\u0105 si\u0119 pokona\u0107 istniej\u0105ce zaburzenia w po\u0142ykaniu pokarm\u00f3w sta\u0142ych popijaniem posi\u0142k\u00f3w wod\u0105, unikaj\u0105 pokarm\u00f3w sta\u0142ych, co przyczynia si\u0119 do zmniejszenia dolegliwo\u015bci a tak\u017ce do op\u00f3\u017anienia wykonania badania endoskopowego i rozpoznania choroby. U os\u00f3b doros\u0142ych w\u015br\u00f3d objaw\u00f3w choroby dysfagia wyst\u0119puje w 70-80%, zatrzymanie pokarmu w 33-54% . W badanEozynofilowe zapalenie prze\u0142yku nale\u017cy r\u00f3\u017cnicowa\u0107 z innymi chorobami przebiegaj\u0105cymi z eozynofili\u0105 prze\u0142yku jak: choroba refluksowa prze\u0142yku, choroby infekcyjne prze\u0142yku, eozynofilowe zapalenie \u017co\u0142\u0105dka i jelita cienkiego, choroba trzewna, achalazja, choroby tkanki \u0142\u0105cznej, HES (Hypereosinophilic Syndrome), nadwra\u017cliwo\u015b\u0107 na leki i inne. W chorobie reffluksowej prze\u0142yku mo\u017cemy obserwowa\u0107 niewielk\u0105 eozynofili\u0119 prze\u0142yku, nie przekraczaj\u0105c\u0105 5/eozynofilii/hpt. Nale\u017cy podkre\u015bli\u0107, \u017ce EoE i GERD stanowi\u0105 odr\u0119bne jednostki chorobowe, mimo podobnych objaw\u00f3w, kt\u00f3re mog\u0105 wyst\u0119powa\u0107 niezale\u017cnie od siebie, wzgl\u0119dnie mog\u0105 si\u0119 na siebie nak\u0142ada\u0107.remodeling), zw\u0142\u00f3knienie, zw\u0119\u017cenie i zaburzenie po\u0142ykania. Leczenie przeciwzapalne eozynofilowego zapalenia prze\u0142yku mo\u017ce ograniczy\u0107 progresj\u0119 choroby. Eozynofilowe zapalenie prze\u0142yku ma wp\u0142yw na jako\u015b\u0107 \u017cycia pacjent\u00f3w poprzez negatywne oddzia\u0142ywania psychologiczne na aktywno\u015b\u0107 fizyczn\u0105 i spo\u0142eczn\u0105. Brak jest dowod\u00f3w, \u017ce EoE jest stanem chorobowym, kt\u00f3ry mo\u017ce prowadzi\u0107 do rozwoju nowotworu prze\u0142yku.Nieleczone eozynofilowe zapalenie prze\u0142yku najcz\u0119\u015bciej prowadzi do wyst\u0119powania przewlek\u0142ych objaw\u00f3w chorobowych zwi\u0105zanych z dysfunkcj\u0105 prze\u0142yku spowodowan\u0105 zapaleniem, kt\u00f3re mo\u017ce powodowa\u0107 przebudow\u0119 \u015bciany prze\u0142yku  indukuje remisj\u0119 histologiczn\u0105 u oko\u0142o \u00be chorych dzieci i doros\u0142ych. U pacjent\u00f3w doros\u0142ych eliminacja 4 alergen\u00f3w pokarmowych indukowa\u0142a remisj\u0119 u oko\u0142o 50% chorych, natomiast eliminacja 2 alergen\u00f3w (mleko i gluten) u 40% pacjent\u00f3w. Bardzo cz\u0119sto odpowiednia eliminacja alergen\u00f3w pokarmowych pozwala zatem na utrzymanie remisji wolnej od leczenia farmakologicznego, co jest du\u017c\u0105 zalet\u0105. Nale\u017cy zaznaczy\u0107, \u017ce przydatno\u015b\u0107 test\u00f3w alergicznych w identyfikacji alergen\u00f3w jest do\u015b\u0107 niska. Dobre efekty obserwowano tak\u017ce po zastosowaniu diety elementarnej . Dieta eflutykazon, kt\u00f3ry w indukcji remisji u dzieci stosuje si\u0119 w dawce 880-1760 \u03bcg/dob\u0119, a u pacjent\u00f3w doros\u0142ych w dawce 1760 \u03bcg/dob\u0119. Dawki podtrzymuj\u0105ce wynosz\u0105: u dzieci 440-880 \u03bcg/dob\u0119, u doros\u0142ych 880-1760 \u03bcg/dob\u0119. Kortykosteroidy stosowane miejscowo s\u0105 bardziej bezpieczne i maj\u0105 mniej dzia\u0142a\u0144 niepo\u017c\u0105danych w por\u00f3wnaniu ze stosowanymi w szczeg\u00f3lnie ci\u0119\u017ckich przypadkach glikokortykosterydami systemowo. Z tych glikokortykosteroid\u00f3w - budesonid w indukcji remisji u dzieci jest stosowany w dawce 1-2 mg/dob\u0119, natomiast 2-4 mg/dob\u0119 u doros\u0142ych. W podtrzymaniu remisji dawka wynosi:1mg/dob\u0119 u dzieci i 2mg/dob\u0119 u pacjent\u00f3w doros\u0142ych. Jednak glikokortykosteroidy stosowane og\u00f3lnie nie s\u0105 rekomendowane w EoE [Zar\u00f3wno u dzieci jak i u pacjent\u00f3w doros\u0142ych skuteczn\u0105 indukcj\u0119 histologiczn\u0105 EoE uzyskuje si\u0119 kortykosteroidami podawanymi miejscowo do prze\u0142yku. Terapia jest najbardziej skuteczna gdy lek jest stosowany w postaci po\u0142ykanego lepkiego syropu, w\u00f3wczas kontakt leku z dystaln\u0105 cz\u0119\u015bci\u0105 prze\u0142yku jest d\u0142u\u017cszy. Zalecany jest ne w EoE .omeprazolu: u dzieci 1-2 mg/kg/dob\u0119 podzielone w dw\u00f3ch dawkach; doro\u015bli 20-40 mg/2 x dob\u0119. Wykazano wi\u0119ksz\u0105 skuteczno\u015b\u0107 PPI, u pacjent\u00f3w z nieprawid\u0142owym zapisem pH-metrii oraz gdy PPI by\u0142 stosowany 2 x dob\u0119 w por\u00f3wnaniu ze stosowaniem 1 x dob\u0119. U pacjent\u00f3w, kt\u00f3rzy odpowiedzieli na leczenie PPI d\u0142ugotrwa\u0142a terapia podtrzymuj\u0105ca PPI by\u0142a skuteczna w podtrzymaniu remisji [Liczne badania wykaza\u0142y, \u017ce terapia inhibitorami pompy protonowej (PPI-Proton Pump Inhibitor) jest skuteczna w indukcji remisji EoE zar\u00f3wno u dzieci jak i pacjent\u00f3w doros\u0142ych. U pacjent\u00f3w obserwowano zar\u00f3wno ust\u0105pienie objaw\u00f3w klinicznych (ponad 60%) jak i eozynofilii (ponad 50%). Rekomendowane dawki  remisji . Jednak  remisji .omalizumab), przeciwko TNF-alfa (infliximab), jak r\u00f3wnie\u017c kromoglikan sodu i leki przeciwhistaminowe nie mia\u0142y wp\u0142ywu na objawy i eozynofili\u0119 prze\u0142yku, podobnie jak brak jest wystarczaj\u0105cych dowod\u00f3w aby rekomendowa\u0107 antagonist\u0119 receptora leukotrienowego (montelukast).Wymienione leczenie diet\u0105 eliminacyjn\u0105, inhibitorami pompy protonowej czy miejscowo kortykosteroidami jest proponowane jako leczenie przeciwzapalne pierwszego rzutu. U pacjent\u00f3w z dysfagi\u0105, utkni\u0119ciem k\u0119sa pokarmowego w prze\u0142yku, nieodpowiadaj\u0105cych na leczenie przeciwzapalne powinno by\u0107 wykonane endoskopowe rozszerzanie prze\u0142yku. Endoskopowe rozszerzanie prze\u0142yku zmniejsza zaburzenia po\u0142ykania u ponad \u00be doros\u0142ych pacjent\u00f3w ze zw\u0119\u017conym \u015bwiat\u0142em prze\u0142yku, natomiast pozostaje bez wp\u0142ywu na zapalenie prze\u0142yku. W wybranych ci\u0119\u017ckich przypadkach EoE nie reaguj\u0105cych na wymienione metody leczenia przeciwzapalnego mog\u0105 by\u0107 przydatne zar\u00f3wno w indukcji jak i utrzymaniu d\u0142ugotrwa\u0142ej remisji: azatiopryna i 6-merkaptopuryna. Stosowanie przeciwcia\u0142 przeciwko interleukinie-5, przeciwcia\u0142 przeciwko Il-13 nie mia\u0142o wp\u0142ywu na objawy EoE i w niewielkim stopniu zmniejsza\u0142o eozynofili\u0119 prze\u0142yku. Stosowane przeciwcia\u0142a przeciwko IgE (Na rycinie 1 przedstawiono algorytm leczenia w eozynofilowym zapaleniu prze\u0142yku wg Lucendo i wsp. ."}
{"text": "Deficyt kortyzolu stanowi rzadk\u0105 przyczyn\u0119 cholestazy niemowl\u0119cej.Celem pracy by\u0142o przedstawienie patogenezy cholestazy w przebiegu deficytu kortyzolu orazcharakterystyka wybranych zaburze\u0144 towarzysz\u0105cych deficytowi. Zgodnie z najnowszymi 2017 rok) rekomendacjami Europejskiego i P\u00f3\u0142nocnoameryka\u0144skiego Towarzystwa Gastroenterologii, Hepatologii i \u017bywienia Dzieci (ESPGHAN i NASPGHAN), cholestaz\u0119 rozpoznaje si\u0119, gdy st\u0119\u017cenie bilirubiny bezpo\u015bredniej w surowicy przekracza warto\u015b\u0107 1 mg/dl, niezale\u017cnie od st\u0119\u017cenia bilirubiny ca\u0142kowitej [7 rok rekDiagnostyka r\u00f3\u017cnicowa cholestazy jest trudna, z uwagi na mnogo\u015b\u0107 przyczyn. Rokowanie uzale\u017cnione jest od postawionego rozpoznania i dotyczy nie tylko post\u0119pu uszkodzenia w\u0105troby, ale r\u00f3wnie\u017c obj\u0119cia procesem chorobowym innych narz\u0105d\u00f3w i uk\u0142ad\u00f3w, w tym o\u015brodkowego uk\u0142adu nerwowego. Prawid\u0142owo postawione rozpoznanie pozwala na szybkie wdro\u017cenie w\u0142a\u015bciwego leczenia.Pierwszy opis przypadku cholestazy zwi\u0105zanej z deficytem kortyzolu pojawi\u0142 si\u0119 ju\u017c w 1956 roku i dotyczy\u0142 niemowl\u0119cia z wrodzon\u0105 wielohormonaln\u0105 niedoczynno\u015bci\u0105 przysadki (WNP). Dotychczas, w literaturze opisano ok. kilkadziesi\u0105t przypadk\u00f3w niemowl\u0105t z cholestaz\u0105 i WNP , 8, 9.Patomechanizm cholestazy w przebiegu WNP jest z\u0142o\u017cony, jednak najprawdopodobniej g\u0142\u00f3wn\u0105 rol\u0119 odgrywa deficyt kortyzolu na skutek niedostatecznego wydzielania przysadkowej adrenokortykotropiny (ACTH), czyli wt\u00f3rna niedoczynno\u015b\u0107 kory nadnerczy .Kortyzol jest hormonem bior\u0105cym udzia\u0142 w regulacji syntezy i transportu kwas\u00f3w \u017c\u00f3\u0142ciowych, m.in. poprzez udzia\u0142 w regulacji transkrypcji gen\u00f3w koduj\u0105cych bia\u0142ka BSEP, MDR3 oraz MRP2, pe\u0142ni\u0105cych funkcje transporter\u00f3w kwas\u00f3w \u017c\u00f3\u0142ciowych (ryc. 1) . Grammat-/Cl- za po\u015brednictwem receptora glikokortykoidowego, wywiera dodatkowy efekt \u017c\u00f3\u0142ciop\u0119dny [Kortyzol oddzia\u0142uj\u0105c na niedojrza\u0142e hepatocyty stymuluje ich r\u00f3\u017cnicowanie w kom\u00f3rki wewn\u0105trzw\u0105trobowych dr\u00f3g \u017c\u00f3\u0142ciowych. Ponadto, pobudzaj\u0105c kana\u0142y HCO3ciop\u0119dny .W patogenezie cholestazy w przebiegu WNP bierze si\u0119 r\u00f3wnie\u017c pod uwag\u0119 znaczenie niedoboru hormonu wzrostu (GH) oraz hormonu tyreotropowego (TSH) , 7, 8, 9Niedob\u00f3r kortyzolu w okresie niemowl\u0119cym zdarza si\u0119 rzadko, a przyczyny deficytu kortyzolu mog\u0105 by\u0107 wrodzone lub nabyte i mog\u0105 prowadzi\u0107 do uszkodzenia nadnerczy \u2013 pierwotna niedoczynno\u015b\u0107 nadnerczy, lub dotyczy\u0107 przysadki i podwzg\u00f3rza \u2013 wt\u00f3rna i trzeciorz\u0119-dowa niedoczynno\u015b\u0107 nadnerczy.Niedob\u00f3r kortyzolu stwierdzamy mi\u0119dzy innymi we wrodzonej wielohormonalnej niedoczynno\u015bci przysadki, izolowanym niedoborze ACTH, urazie przysadki, po glikokortykoterapii, we wrodzonym przero\u015bcie kory nadnerczy, wrodzonej hipoplazji nadnerczy (zesp\u00f3\u0142 DAX-1), adrenoleukodystrofii (X-ALD), wylewach do nadnerczy, kiedy zniszczeniu ulega ponad 90% mi\u0105\u017cszu nadnerczy oraz w rodzinnym niedoborze glikokortykoid\u00f3w spowodowanym niewra\u017cliwo\u015bci\u0105 nadnerczy na ACTH.W etiologii wielohormonalnej niedoczynno\u015bci przysadki rozpatruje si\u0119 wp\u0142yw czynnik\u00f3w \u015brodowiskowych i genetycznych. W przypadku wrodzonego charakteru schorzenia, ujawnienie objaw\u00f3w choroby mo\u017ce nast\u0105pi\u0107 w okresie niemowl\u0119cym, a nawet wczesnodzieci\u0119cym. Cz\u0119sto\u015b\u0107 wyst\u0119powania ocenia si\u0119 na oko\u0142o 1 na 50 000 urodze\u0144, z czego w 10 do 35% obserwuje si\u0119 wsp\u00f3\u0142wyst\u0119powanie cholestazy . WrodzonPoza cholestaz\u0105, objawami kt\u00f3re mog\u0105 nasuwa\u0107 podejrzenie WNP, s\u0105 epizody nawracaj\u0105cej hipoglikemii. W zale\u017cno\u015bci od stopnia niedoboru hormon\u00f3w przysadkowych  oraz kortyzolu, hipoglikemia mo\u017ce manifestowa\u0107 si\u0119 niepokojem, zaburzeniami oddychania czy drgawkami ju\u017c w 1. dobie \u017cycia dziecka , 7, 8, 9O wielohormonalnej nieodczynno\u015bci przysadki warto tak\u017ce pomy\u015ble\u0107 w przypadku obecno\u015bci zaburze\u0144 rozwojowych oczu i linii po\u015brodkowej cia\u0142a  oraz nieprawid\u0142owo\u015bci dotycz\u0105cych rozwoju narz\u0105d\u00f3w p\u0142ciowych (u ch\u0142opc\u00f3w mikropenis czy wn\u0119trostwo) , 9, 16.pituitary stalk interruption syndrome, PSIS), w kt\u00f3rym obrazowanie metod\u0105 rezonansu magnetycznego pozwala uwidoczni\u0107 przerwan\u0105 ci\u0105g\u0142o\u015b\u0107 szypu\u0142y przysadki, kt\u00f3rej zwykle to-warzyszy brak lub ektopia tylnego p\u0142ata przysadki oraz hipoplazja przedniego p\u0142ata przysadki. W pracy Mauvaix i wsp., opublikowanej w 2016 roku, b\u0119d\u0105cej retrospektywn\u0105 analiz\u0105 pacjent\u00f3w z PSIS rozpoznanym przed uko\u0144czeniem 1. roku \u017cycia, \u017c\u00f3\u0142taczka cholestatyczna by\u0142a obecna u 31% pacjent\u00f3w, i obok epizod\u00f3w hipoglikemii stanowi\u0142a jeden z najwcze\u015bniejszych objaw\u00f3w klinicznych sugeruj\u0105cych niedoczynno\u015b\u0107 przysadki. Co wi\u0119cej, st\u0119\u017cenie kortyzolu w surowicy w grupie pacjent\u00f3w z cholestaz\u0105 by\u0142o znacznie bardziej obni\u017cone ni\u017c w grupie bez cholestazy [Podstaw\u0119 rozpoznania WNP stanowi\u0105 badania hormonalne. Obecno\u015b\u0107 wady strukturalnej w okolicy podwzg\u00f3rzowo-przysadkowej zwi\u0119ksza prawdopodobie\u0144-stwo wyst\u0105pienia WNP . Na uwagTGIF, SHH, CDON, GPR161, PROKR2, HESX1, OTX2, LHX3/LHX4, SIX6, PTX2; w izolowanym niedoborze ACTH \u2013 TBX19. Powi\u0119kszenie przysadki mo\u017ce wyst\u0119powa\u0107 u pacjent\u00f3w z mutacj\u0105 w genie PROP1 [W diagnostyce wykorzystuje si\u0119 r\u00f3wnie\u017c badania molekularne. Genezy wad strukturalnych OUN z to-warzysz\u0105cym zespo\u0142em PSIS upatruje si\u0119 mutacjach gen\u00f3w: ie PROP1 , 18. Badie PROP1 , 8, 9. Wfamilial glucocorticoid deficiency, FGD) [W diagnostyce r\u00f3\u017cnicowej cholestazy niemowl\u0119cej przebiegaj\u0105cej z deficytem kortyzolu i prawid\u0142owym/pod-wy\u017cszonym st\u0119\u017ceniem ACTH (pierwotna niedoczynno\u015b\u0107 kory nadnerczy) opr\u00f3cz zaburze\u0144 peroksysomalnych, lizosomalnych i mitochondrialnych nale\u017cy bra\u0107 pod uwag\u0119 rodzinny niedob\u00f3r glikokortykosteroid\u00f3w  , 23, 24.MC2R, MRAP, MCM4, NNT, AAAS [Rodzinny niedob\u00f3r GKS jest heterogenn\u0105, bardzo rzadk\u0105 chorob\u0105 zwi\u0105zan\u0105 z deficytem glikokortyko-steroid\u00f3w (zw\u0142aszcza kortyzolu) oraz oporno\u015bci\u0105 kory nadnerczy na ACTH. Dziedziczy si\u0119 autosomalnie rece-sywnie i w wi\u0119kszo\u015bci przypadk\u00f3w zwi\u0105zany jest z mutacj\u0105 genu koduj\u0105cego receptor ACTH, u pozosta\u0142ych mutacje mog\u0105 dotyczy\u0107 rejonu regulatorowego genu koduj\u0105cego receptor ACTH lub innych czynnik\u00f3w odpowiedzialnych za r\u00f3\u017cnicowanie kory nadnerczy  w opisywanych wy\u017cej patologiach. W WNP mamy do czynienia z wt\u00f3rn\u0105 niedoczynno\u015bci\u0105 kory nadnerczy, a wi\u0119c niskiemu st\u0119\u017ceniu kortyzolu w surowicy towarzyszy niskie st\u0119\u017cenie ACTH w osoczu. Z kolei, w FGD mamy do czynienia z pierwotn\u0105 niedoczynno\u015bci\u0105 kory nadnerczy, a wi\u0119c niskiemu st\u0119\u017ceniu kortyzolu w surowicy towarzyszy prawid\u0142owe st\u0119\u017cenie ACTH w osoczu.Normy st\u0119\u017ce\u0144 kortyzolu w surowicy w godzinach porannych wynosz\u0105 ok. 5-25 \u03bcg/dl. W zaburzeniach endokrynologicznych przebiegaj\u0105cych z faktycznym defi-cytem kortyzolu zwykle obecne s\u0105 objawy kliniczne hipokortyzolemii, w tym m.in. epizody hipoglikemii. Nale\u017cy pami\u0119ta\u0107, \u017ce u niemowl\u0105t dolna granica normy st\u0119\u017cenia kortyzolu w surowicy, jest ni\u017csza , 27. NajU niemowl\u0105t z cholestaz\u0105 o nieustalonej etiologii nale\u017cy wzi\u0105\u0107 pod uwag\u0119 deficyt kortyzolu i w zwi\u0105zku z tym uwzgl\u0119dni\u0107 badanie st\u0119\u017cenia kortyzolu  w panelu wykonywanych bada\u0144. Co wi\u0119cej, takie post\u0119powanie jest zalecane w najnowszych rekomendacjach Europejskiego i P\u00f3\u0142nocnoameryka\u0144skiego Towarzystwa Gastroenterologii, Hepatologii i \u017bywienia Dzieci (ESPGHAN i NASPGHAN) dotycz\u0105cych cholestazy niemowl\u0119cej."}
{"text": "Drobne drogi oddechowe s\u0105 miejscem powstawania zmian patologicznych w przebiegu wielu chor\u00f3b jak np. astma lub mukowiscydoza, cz\u0119sto ju\u017c we wczesnym ich stadium. Ta cz\u0119\u015b\u0107 dr\u00f3g oddechowych jest jednak pomijana w konwencjonalnych badaniach czynno\u015bciowych uk\u0142adu oddechowego i z tego powodu cz\u0119sto nazywana jest \u201ecich\u0105 stref\u0105 p\u0142uc\u201d. W niniejszej pracy przedstawiono podstawy teoretyczne testu wyp\u0142ukiwania azotu metod\u0105 wielokrotnych oddech\u00f3w (MBNW \u2212 ang. multi-breath nitrogen washout) w diagnostyce chor\u00f3b drobnych dr\u00f3g oddechowych. Om\u00f3wiono zagadnienia techniczne zwi\u0105zane z przygotowaniem pacjent\u00f3w pediatrycznych do przeprowadzenia badania oraz przebieg wykonywania testu. Mo\u017cliwo\u015bci zastosowania klinicznego opisanej metody stanowi\u0105 nadal przedmiot wielu bada\u0144 oraz budz\u0105 nadzieje na wype\u0142nienie luki w testach czynno\u015bciowych drobnych dr\u00f3g oddechowych. Ze wzgl\u0119du na zaanga\u017cowanie autor\u00f3w w diagnostyk\u0119 i leczenie chorych na mukowiscydoz\u0119 w pracy opisano r\u00f3wnie\u017c do\u015bwiadczenia w\u0142asne dotycz\u0105ce wykorzystania tego badania w tej grupie pacjent\u00f3w. Obecnie metoda znajduje si\u0119 w fazie intensywnie prowadzonych analiz zwi\u0105zanych z wykryciem wczesnych stadi\u00f3w choroby oskrzelowo-p\u0142ucnej w przebiegu mukowiscydozy, kiedy jeszcze wyniki innych bada\u0144 czynno\u015bciowych s\u0105 prawid\u0142owe lub niemo\u017cliwe do wykonanie z uwagi na wiek pacjenta. Korelacja z metodami obrazowymi (tomografia komputerowa klatki piersiowej) i nasileniem zmian strukturalnych mo\u017ce w przysz\u0142o\u015bci ograniczy\u0107 liczb\u0119 wykonywanych bada\u0144 radiologicznych, a tym samym zmniejszy\u0107 nara\u017canie pacjenta na promieniowanie jonizuj\u0105ce. Wprowadzenie test\u00f3w oceniaj\u0105cych funkcj\u0119 p\u0142uc u niemowl\u0105t i dzieci przedszkolnych z mukowiscydoz\u0105 i innymi chorobami drobnych dr\u00f3g oddechowych mo\u017ce zmodyfikowa\u0107 post\u0119powanie kliniczne i poprawi\u0107 rokowanie. Swoj\u0105 nazw\u0119 zawdzi\u0119czaj\u0105 brakiem mo\u017cliwo\u015bci ich pe\u0142nej oceny na podstawie dotychczas wykonywanych bada\u0144 czynno\u015bciowych uk\u0142adu oddechowego, w tym przede wszystkim spirometrii. Wiadomo jednak, \u017ce w niekt\u00f3rych jednostkach chorobowych, takich jak mukowiscydoza (CF \u2212 ang. cystic fibrosis), astma, przewlek\u0142a obturacyjna choroba p\u0142uc (POCHP) wcze\u015bnie dochodzi do zaburze\u0144 w drobnych drogach oddechowych. Badania histopatologiczne tkanki p\u0142ucnej pobranej po\u015bmiertnie oraz podczas biopsji przezoskrzelowych in vivo, potwierdzaj\u0105 udzia\u0142 w patogenezie astmy zar\u00f3wno centralnych jak i obwodowych dr\u00f3g oddechowych. Podobnie badania autopsyjne tkanki p\u0142ucnej chorych na mukowiscydoz\u0119 opisuj\u0105 zmiany patologiczne w drobnych oskrzelach. Ju\u017c we wczesnym stadium choroby oskrzelowo-p\u0142ucnej w badaniach obrazowych uwidaczniane s\u0105: pogrubienie \u015bcian oskrzeli, \u201eobjaw pu\u0142apki powietrznej\u201d, korki \u015bluzowe i rozstrzenie oskrzeli. Ze wzgl\u0119du na niewielki wp\u0142yw oporu obwodowych dr\u00f3g oddechowych na ich op\u00f3r ca\u0142kowity, wyniki bada\u0144 spirometrycznych cz\u0119sto mieszcz\u0105 si\u0119 jeszcze w granicach normy lub ze wzgl\u0119du na wiek i brak wsp\u00f3\u0142pracy z dzieckiem, testy te nie mog\u0105 by\u0107 wykonane. Do niedawna przebieg choroby dr\u00f3g oddechowych np. w mukowiscydozie lub astmie by\u0142 monitorowany jedynie na podstawie wynik\u00f3w spirometrii \u2212 metody odzwierciedlaj\u0105cej nieprawid\u0142ow\u0105 funkcj\u0119 dr\u00f3g oddechowych w zaawansowanym stadium choroby .Wykrycie zmian w obwodowych drogach oddechowych umo\u017cliwia wdro\u017cenie odpowiedniego leczenia. Z tego powodu istnieje du\u017ca potrzeba stosowania w diagnostyce chor\u00f3b p\u0142uc metod nieinwazyjnych pozwalaj\u0105cych na jak najwcze\u015bniejsze wykrywanie zaburze\u0144 w zakresie drobnych dr\u00f3g oddechowych przy jednoczesnym nieskomplikowanym sposobie ich wykonania, umo\u017cliwiaj\u0105cym badanie dzieci oraz s\u0142abo wsp\u00f3\u0142pracuj\u0105cych pacjent\u00f3w doros\u0142ych (np. os\u00f3b starszych).Wprowadzenie testu wyp\u0142ukiwania gazu metod\u0105 wielokrotnych oddech\u00f3w (MBW \u2212 ang. multi-breath washout) pozwala na wykrycie nieprawid\u0142owej funkcji dr\u00f3g oddechowych charakteryzuj\u0105cej si\u0119 zaburzon\u0105 dystrybucj\u0105 wentylacji oraz patologiczn\u0105 pu\u0142apk\u0105 powietrzn\u0105 m.in. u dzieci z astm\u0105 lub mukowiscydoz\u0105 ju\u017c we wczesnym stadium choroby , 2, 3, 4Drogi oddechowe pod wzgl\u0119dem czynno\u015bciowym mo\u017cna podzieli\u0107 na stref\u0119 przewodz\u0105c\u0105 ( nie oddechow\u0105), czyli oskrzela i oskrzeliki doprowadzaj\u0105ce powietrze do p\u0119cherzyk\u00f3w p\u0142ucnych oraz stref\u0119 oddechow\u0105, w kt\u00f3rej odbywa si\u0119 wymiana gazowa.2 powierzchni wymiany gazowej pomi\u0119dzy wdychanym powietrzem a krwi\u0105 naczy\u0144 w\u0142osowatych. Z\u0142o\u017cona struktura p\u0142uc zapewnia efektywne mieszanie gaz\u00f3w i prawid\u0142ow\u0105 dystrybucj\u0119 wentylacji. Obwodowe drogi oddechowe tworz\u0105 oskrzela poni\u017cej si\u00f3dmej generacji, o wewn\u0119trznej \u015brednicy poni\u017cej 2 mm (u os\u00f3b doros\u0142ych), nie posiadaj\u0105ce \u015bciany chrz\u0119stnej (ryc.1). Stanowi\u0105 one 95% ca\u0142kowitej pojemno\u015bci p\u0142uc ale tylko 10-20% ca\u0142kowitego oporu dr\u00f3g oddechowych. Konwencjonalne badania czynno\u015bciowe p\u0142uc dostarczaj\u0105 informacji o du\u017cych oskrzelach pomijaj\u0105c drobne drogi oddechowe.W wyniku dychotomicznych ale niesymetrycznych podzia\u0142\u00f3w tworz\u0105 one ok. 23 generacje (pokolenia) . JednostW strefie przewodz\u0105cej dominuje transport gazu za pomoc\u0105 konwekcji, a liniowa szybko\u015b\u0107 przep\u0142ywu gazu jest wzgl\u0119dnie wysoka. W przypadku obwodowych dr\u00f3g oddechowych transport oraz wymiana gaz\u00f3w za pomoc\u0105 konwekcji i dyfuzji molekularnej maj\u0105 r\u00f3wnowa\u017cny udzia\u0142 tworz\u0105c tzw. \u201efront dyfuzyjno-konwekcyjny\u201d.Pr\u0119dko\u015b\u0107 liniowa gazu w obwodowych drogach oddechowych jest stosunkowo ma\u0142a, a udzia\u0142 oporu dr\u00f3g oddechowych i zwi\u0105zane z tym faktem ograniczenie przep\u0142ywu pozostaj\u0105 niewielkie. Z tego powodu zmiany patologiczne w obwodowych drogach oddechowych nie zawsze s\u0105 wykrywane podczas spirometrii .Pierwsze opisy technik wyp\u0142ukiwania gaz\u00f3w znacznikowych pojawi\u0142y si\u0119 ponad 60 lat temu.6) metod\u0105 pojedynczego oddechu (SBW \u2212 ang. single-breath washout) lub wielokrotnych oddech\u00f3w (MBW). Wspomniane metody SBW oraz MBW zosta\u0142y opracowane i po raz pierwszy opisane przez Becklake w 1952 roku. Dzi\u0119ki rozwojowi techniki jaki nast\u0105pi\u0142 w XX i XXI mo\u017cliwe jest wykonywanie analiz w czasie rzeczywistym z jednoczesnym wykorzystaniem zaawansowanych metod obliczeniowych, co umo\u017cliwia ocen\u0119 nie tylko stopnia homogenno\u015bci wymiany gazowej, ale r\u00f3wnie\u017c zlokalizowanie tocz\u0105cych si\u0119 proces\u00f3w chorobowych. Test wyp\u0142ukiwania azotu metod\u0105 wielkokrotnych oddech\u00f3w jako wymagaj\u0105cy minimalnej wsp\u00f3\u0142pracy z pacjentem znalaz\u0142 swoje zastosowanie zw\u0142aszcza w\u015br\u00f3d pacjent\u00f3w pediatrycznych. Po wielu latach test\u00f3w w laboratoriach badawczych metoda ta zosta\u0142a zaakceptowana przez grupy ekspert\u00f3w ERS/ATS i ECFS-CTN [IGW-wyp\u0142ukiwanie gazu oboj\u0119tnego jako metoda pionierska opisana w 1940 roku, umo\u017cliwia pomiar heterogenno\u015bci wentylacji (VI \u2212 ang. ventilation inhomogenity) i wnikliw\u0105 ocen\u0119 funkcji drobnych dr\u00f3g oddechowych . Jednak ECFS-CTN , 9, 10.Testy SBW i MBW odzwierciedlaj\u0105 przede wszystkim funkcj\u0119 drobnych dr\u00f3g oddechowych - g\u0142\u00f3wne miejsce mieszania si\u0119 powietrza wdechowego z wydechowym. Na niejednorodno\u015b\u0107 wentylacji p\u0142uc mo\u017ce wp\u0142ywa\u0107 wiele czynnik\u00f3w. Najwa\u017cniejsze z nich to:\u2212 r\u00f3\u017cnice w wymianie gazowej pomi\u0119dzy r\u00f3\u017cnymi obszarami p\u0142uc,\u2212 sekwencyjne nape\u0142nianie i opr\u00f3\u017cnianie jednostek p\u0142uc,\u2212 asymetria w budowie dystalnej cz\u0119\u015bci p\u0142uc.Ze wzgl\u0119du na wp\u0142yw grawitacji i niesymetryczne podzia\u0142y dr\u00f3g oddechowych nawet u os\u00f3b zdrowych wyst\u0119puje niewielkiego stopnia niejednorodno\u015b\u0107 wentylacji p\u0142uc. Zaburzenia wymiany gazowej zwi\u0105zane z chorob\u0105 dodatkowo j\u0105 nasilaj\u0105 .W przebiegu mukowiscydozy, a tak\u017ce innych chor\u00f3b drobnych dr\u00f3g oddechowych dochodzi do nieodwracalnych zmian w wyniku proces\u00f3w w\u0142\u00f3knienia i post\u0119puj\u0105cej destrukcji tkanki p\u0142ucnej, a tak\u017ce zaburze\u0144 potencjalnie odwracalnych zwi\u0105zanych z miejscowym stanem zapalnym i zaleganiem \u015bluzu . AntybioW chorobach uk\u0142adu oddechowego zmiany zachodz\u0105ce w obwodowych drogach oddechowych powoduj\u0105 niejednorodno\u015b\u0107 wentylacji. Test MBNW z powodzeniem stosowany jest w celu oceny homogeniczno\u015bci wentylacji stanowi cenne \u017ar\u00f3d\u0142o informacji o stopniu oraz lokalizacji niejednorodno\u015bci wentylacji p\u0142uc, a tak\u017ce jest dobrym uzupe\u0142nieniem spirometrii, gdy\u017c analizie podlegaj\u0105 w g\u0142\u00f3wnej mierze procesy zachodz\u0105ce na poziomie 8-23generacji oskrzelik\u00f3w o \u015brednicy poni\u017cej 2mm. Wytyczne dotycz\u0105ce procedur oraz protok\u00f3\u0142 badania zosta\u0142y przedstawione w najnowszym dokumencie konsensusu ERS/ATS , 10.2 kaniu (ryc. 3). W celu okre\u015blenia aktualnego st\u0119\u017cenia N2 stosuje si\u0119 technik\u0119 po\u015bredni\u0105 polegaj\u0105c\u0105 na analizie w czasie rzeczywistym st\u0119\u017ce\u0144 dwutlenku w\u0119gla (CO2) oraz tlenu O2. Kolejnym krokiem jest wyliczanie st\u0119\u017cenia N2 wg wzoru:W Pracowni Bada\u0144 Czynno\u015bciowych P\u0142uc Centrum Leczenia Mukowiscydozy w Dziekanowie Le\u015bnym test MBNW wykonywany jest przy u\u017cyciu urz\u0105dzenia EXHALYZER\u00ae D ECO MEDICS AG. W tym przypadku wykorzystuje si\u0119 Njako gaz oboj\u0119tny ulegaj\u0105cy wyp\u0142u2 podczas p\u0142ukania *F oznacza u\u0142amkowe st\u0119\u017cenie gazu; FAr (Argon) traktuje si\u0119 jako sta\u0142\u0105 cz\u0119\u015b\u0107 FN2 zawartym w powietrzu atmosferycznym, przez 100% O2 medyczny. W przypadku test\u00f3w z zastosowaniem gaz\u00f3w znacznikowych np. SF6 nale\u017cy pami\u0119ta\u0107 o konieczno\u015bci przeprowadzenia fazy inhalacji tym gazem (ryc. 4).Test MBNW z zastosowaniem azotu jako wyp\u0142ukiwanego gazu oboj\u0119tnego polega na stopniowym zast\u0119powaniu N2, natomiast uruchomione zostaje podawanie medycznego 100% O2. Stopniowe wyp\u0142ukiwanie azotu z uk\u0142adu oddechowego, jest prowadzone do momentu osi\u0105gni\u0119cia w powietrzu wydychanym 1/40 warto\u015bci pocz\u0105tkowej st\u0119\u017cenia N2. Poszczeg\u00f3lne etapy badania zosta\u0142y przedstawione w Po ustabilizowaniu oddech\u00f3w pacjenta, nast\u0119puje zatrzymanie podawania powietrza a tym samym poda\u017cy NBadanie mo\u017cna uzna\u0107 za wykonane poprawnie w momencie, gdy w co najmniej dw\u00f3ch prawid\u0142owo przeprowadzonych pr\u00f3bach uzyskano wsp\u00f3\u0142czynniki LCI 2,5% nier\u00f3\u017cni\u0105ce si\u0119 wi\u0119cej ni\u017c 5% . KorelacWyniki pochodz\u0105ce z badania MBW:FRC \u2013 ang. Functional Residual Capacity \u2212 czynno\u015bciowa pojemno\u015b\u0107 zalegaj\u0105ca. Parametr ten okre\u015bla ilo\u015b\u0107 powietrza, kt\u00f3ra pozostaje w p\u0142ucach po wykonaniu spokojnego wydechu.LCI \u2013 ang. Lung Clearance Index \u2013 wska\u017anik oczyszczania p\u0142uc \u2212 jest najcz\u0119\u015bciej stosowanym wska\u017anikiem w te\u015bcie MBW. Wsp\u00f3\u0142czynnik LCI wyliczany jest ze wzoru:LCI= CEV/FRC, gdzieCEV \u2013 Cumulative Expired Volume- skumulowana obj\u0119to\u015b\u0107 wydechowa, b\u0119d\u0105ca sum\u0105 wydychanych obj\u0119to\u015bci oddechowych.FRC \u2013 czynno\u015bciowa pojemno\u015b\u0107 zalegaj\u0105ca.LCI dostarcza informacji o tym, ile razy obj\u0119to\u015b\u0107 gazu w p\u0142ucach na starcie wyp\u0142ukiwania musi by\u0107 wymieniona (ang. TO \u2013 turnover), aby doprowadzi\u0107 do wyeliminowania azotu z p\u0142uc do 1/40  st\u0119\u017cenia wyj\u015bciowego. Dotychczasowe badania wykaza\u0142y, \u017ce LCI jest przydatnym wska\u017anikiem czynno\u015bci p\u0142uc u chorych na CF, a tak\u017ce ma wi\u0119ksz\u0105 czu\u0142o\u015b\u0107 w r\u00f3\u017cnych przedzia\u0142ach wiekowych w por\u00f3wnaniu z spirometri\u0105 , 15, 16.W pe\u0142nej analizie wynik\u00f3w badania MBW nale\u017cy uwzgl\u0119dni\u0107 r\u00f3wnie\u017c:Scond (phase III slope index of conductive ventilation inhomogeneity) \u2013 wska\u017anik zaburze\u0144 w strefie przewodz\u0105cej proksymalnej do oskrzelik\u00f3w ko\u0144cowych.Sacin (phase III slope index of acinar ventilation inhomogeneity ) \u2013 wska\u017anik zaburze\u0144 wentylacji w p\u0119cherzykach p\u0142ucnychSnIII  \u2013 wska\u017aniki dysfunkcji wentylacji w najbardziej obwodowych cz\u0119\u015bciach p\u0142uc.Niezwykle istotna jest r\u00f3wnie\u017c ocena kszta\u0142tu oraz d\u0142ugo\u015b\u0107 krzywej wyp\u0142ukiwania azotu, a tak\u017ce kszta\u0142tu krzywej fazy p\u0119cherzykowej (SnIII) .Przyk\u0142adowe wyniki badania MBNW u pacjent\u00f3w w\u0142asnych przedstawiono na rycinie 5.Z naszego do\u015bwiadczenia wynika, \u017ce test wyp\u0142ukiwania azotu metod\u0105 wielokrotnych oddech\u00f3w (MBNW) jest przyjazn\u0105 dla pacjenta, nieinwazyjn\u0105 metod\u0105 oceniaj\u0105c\u0105 jednorodno\u015b\u0107 dystrybucji wentylacji w drobnych drogach oddechowych poprzez rejestrowanie wyp\u0142ukiwania oboj\u0119tnego gazu znacznikowego podczas spokojnego oddychania. W Pracowni Bada\u0144 Czynno\u015bciowych P\u0142uc Centrum Leczenia Mukowiscydozy w Dziekanowie Le\u015bnym badanie to jest wykonywane rutynowo w celu obserwacji zmian wsp\u00f3\u0142czynnika oczyszczania p\u0142uc u pacjent\u00f3w w wieku od 2 do 28 lat. Nale\u017cy jednak pami\u0119ta\u0107, \u017ce LCI jest szeroko stosowanym parametrem oceniaj\u0105cym ca\u0142kowit\u0105 niejednorodno\u015b\u0107 wentylacji w r\u00f3\u017cnych grupach wiekowych od niemowl\u0105t do os\u00f3b starszych. W jednostkach chorobowych, w kt\u00f3rych dochodzi do zaj\u0119cia drobnych dr\u00f3g oddechowych okaza\u0142 si\u0119 by\u0107 bardziej czu\u0142ym parametrem ni\u017c konwencjonalne badania czynno\u015bciowe i lepiej koreluj\u0105cym z uszkodzeniami strukturalnymi p\u0142uc. Nowe podej\u015bcie do analizy krzywych wyp\u0142ukiwania np. znormalizowana analiza nachylenia fazy III, dostarcza dalszych danych na temat lokalizacji zmian w obr\u0119bie drobnych dr\u00f3g oddechowych i post\u0119pu choroby. MBW mo\u017ce by\u0107 stosowana w diagnostyce i monitorowaniu leczenia wielu chor\u00f3b, jak mukowiscydoza, astma, zarostowe zapalenie oskrzelik\u00f3w , 4, 5. PNa prze\u0142omie ostatnich dekad oczekiwana d\u0142ugo\u015b\u0107 i jako\u015b\u0107 \u017cycia chorych na mukowiscydoz\u0119 znacz\u0105co wzros\u0142a, nadal jednak powik\u0142ania oskrzelowo-p\u0142ucne pozostaj\u0105 g\u0142\u00f3wn\u0105 przyczyn\u0105 \u015bmierci. Wraz z coraz lepsz\u0105 opiek\u0105 i leczeniem post\u0119p choroby uleg\u0142 znacznemu spowolnieniu. Nale\u017cy podkre\u015bli\u0107, \u017ce choroba p\u0142uc rozpoczyna si\u0119 ju\u017c w pierwszych miesi\u0105cach \u017cycia, cz\u0119sto jeszcze przed wyst\u0105pieniem objaw\u00f3w klinicznych i pogorszeniem wynik\u00f3w spirometrycznych. Jednak ze wzgl\u0119du na brak wsp\u00f3\u0142pracy z pacjentem i metod diagnostycznych dostosowanych do m\u0142odego wieku, dopiero od 4-5 r\u017c. rozpoczyna si\u0119 wykonywanie rutynowych bada\u0144 czynno\u015bciowych p\u0142uc oceniaj\u0105cych stopie\u0144 zaawansowania i progresj\u0119 choroby oskrzelowo-p\u0142ucnej. Obecnie wiadomo, \u017ce FEV 1 nie jest dobrym wska\u017anikiem w ocenie wczesnej choroby p\u0142uc w CF , 4, 17. Metod\u0105 referencyjn\u0105 do rozpoznawania wczesnych i zaawansowanych zmian strukturalnych w p\u0142ucach jest tomografia komputerowa o wysokiej rozdzielczo\u015bci (HRCT ang. high resolution computer tomography). Jednak nara\u017cenie na promieniowanie jonizuj\u0105ce ogranicza jego wykorzystanie jako narz\u0119dzia do monitorowania przebiegu choroby. W ostatnich latach wzros\u0142o zainteresowanie HRCT jako metody oceny wczesnego stadium choroby oskrzelowo-p\u0142ucnej, kt\u00f3ra nie jest wykrywalna w konwencjonalnej spirometrii. R\u00f3wnie\u017c pr\u00f3by stosowania HRCT jako punktu ko\u0144cowego w badaniach klinicznych u ma\u0142ych dzieci z CF wydaj\u0105 si\u0119 obiecuj\u0105ce. W celu zminimalizowania dawki kumulacyjnej opracowano protoko\u0142y o zmniejszonej dawce promieniowania jonizuj\u0105cego.Wykonywanie HRCT u m\u0142odszych pacjent\u00f3w jest jednak czasoch\u0142onne i obci\u0105\u017caj\u0105ce. To procedura wymagaj\u0105ca sedacji lub znieczulenia og\u00f3lnego, powoduj\u0105ca wzrost niepokoju i l\u0119ku u pacjent\u00f3w i ich rodzin. Natomiast MBW jest nieinwazyjn\u0105, bezpieczn\u0105 i czu\u0142\u0105 metod\u0105 oceniaj\u0105c\u0105 wczesny okres choroby p\u0142uc w CF , 15, 18..18. KilkOstatnio stwierdzono, \u017ce parametry niehomogeniczno\u015bci wentylacji mierzone podczas MBW mog\u0105 by\u0107 por\u00f3wnywalne do HRCT w diagnostyce zmian p\u0142ucnych u pacjent\u00f3w z CF . LCI ma Metoda ta pozwala na wczesne rozpoznanie, monitorowanie choroby p\u0142uc oraz ocen\u0119 wynik\u00f3w stosowania nowych strategii leczenia w celu zatrzymania lub spowolnienia progresji zmian p\u0142ucnych. MBW jest r\u00f3wnie czu\u0142\u0105 metod\u0105 co niskodawkowa HRCT w wykrywaniu wczesnych zmian p\u0142ucnych u m\u0142odych pacjent\u00f3w z CF i mo\u017ce by\u0107 stosowana jako nieinwazyjna metoda zar\u00f3wno w rutynowej opiece, jak i w badaniach klinicznych.Badanie MBW wesz\u0142o ju\u017c w schemat corocznych bada\u0144 bilansowych w niekt\u00f3rych o\u015brodkach mukowiscydozy. Zgodnie z zaleceniami LCI powinno stanowi\u0107 rutynow\u0105 cz\u0119\u015b\u0107 corocznej oceny i by\u0107 wykonywane u wszystkich dzieci w wieku \u22655 lat .Mo\u017cliwo\u015b\u0107 rozpoznawania wczesnych zaburze\u0144 dr\u00f3g oddechowych w tych \u201ecichych latach,\u201d kiedy FEV1 mie\u015bci si\u0119 w granicach normy, jest szczeg\u00f3lnie przydatne w badaniu nowych terapii u niemowl\u0105t i ma\u0142ych dzieci z \u0142agodn\u0105 chorob\u0105 oskrzelowo-p\u0142ucn\u0105 . LCI zacW o\u015brodku autor\u00f3w od stycznia 2017 r. prowadzone s\u0105 badania oceniaj\u0105ce przydatno\u015b\u0107 testu MBNW w monitorowaniu przebiegu choroby oskrzelowo-p\u0142ucnej u pacjent\u00f3w z mukowiscydoz\u0105. Pod opiek\u0105 Centrum Leczenia Mukowiscydozy w Dziekanowie Le\u015bnym pozostaje 400 pacjent\u00f3w od wieku niemowl\u0119cego do 18 r\u017c. Chorzy maj\u0105 wykonywane badania czynno\u015bciowe uk\u0142adu oddechowego podczas bada\u0144 bilansowych oraz przed i po leczeniu zaostrze\u0144 zmian oskrzelowo-p\u0142ucnych. W fazie bada\u0144 wst\u0119pnych pozostaje korelacja zaburze\u0144 homogenno\u015bci wentylacji (LCI) z innymi badaniami jak spirometria, bodypletyzmografia, oscylometria oraz wp\u0142yw leczenia zaostrze\u0144 zmian oskrzelowo-p\u0142ucnych na jego zmienno\u015b\u0107. Niemniej interesuj\u0105cym zagadnieniem jest zale\u017cno\u015b\u0107 przewlek\u0142ego zaka\u017cenia dr\u00f3g oddechowych flor\u0105 patogenn\u0105 w tym przede wszystkim pa\u0142eczk\u0105 ropy b\u0142\u0119kitnej (Pseudomonas aeruginosa) oraz gronkowcem z\u0142ocistym (Staphylococcus aureus) i nasilenia zaburze\u0144 wentylacji p\u0142uc u chorych na mukowiscydoz\u0119. Po podsumowaniu wynik\u00f3w, badania te b\u0119d\u0105 one przedmiotem kolejnej pracy.Astma oskrzelowa jest przewlek\u0142\u0105 chorob\u0105 zapaln\u0105 ca\u0142ego drzewa oskrzelowego, nie tylko du\u017cych oskrzeli ale i drobnych dr\u00f3g oddechowych. Potwierdzaj\u0105 to badania histopatologiczne, w kt\u00f3rych stwierdza si\u0119 m.in. zwi\u0119kszon\u0105 liczb\u0119 kom\u00f3rek zapalnych w obr\u0119bie drobnych dr\u00f3g oddechowych i przestrzeni p\u0119cherzykowych. Dane te wskazuj\u0105 na du\u017cy potencja\u0142 nieinwazyjnych bada\u0144 oceniaj\u0105cych czynno\u015b\u0107 drobnych dr\u00f3g oddechowych, nie tylko u\u0142atwiaj\u0105cych rozpoznanie astmy szczeg\u00f3lnie u m\u0142odszych dzieci, lecz tak\u017ce ocen\u0119 i monitorowanie jej przebiegu , 4, 5.Udzia\u0142 drobnych dr\u00f3g oddechowych w astmie zosta\u0142 potwierdzony w wielu badaniach. Ca\u0142kowity wzrost niehomogenno\u015bci wentylacji (VI) wyra\u017cony wzrostem LCI stwierdzono u chorych na astm\u0119 w por\u00f3wnaniu z grup\u0105 kontroln\u0105. Badania IGW ( zar\u00f3wno MBW jak i SBW) potwierdzi\u0142y przewlek\u0142e zmiany patologiczne cz\u0119\u015bci przewodz\u0105cej uk\u0142adu oddechowego . U dziecNiehomogenno\u015b\u0107 wentylacji (VI) wydaje si\u0119 by\u0107 wa\u017cnym czynnikiem predykcyjnym nadreaktywno\u015bci dr\u00f3g oddechowych w astmie niezale\u017cnie od stanu zapalnego . Jest onIGW mo\u017ce by\u0107 bardziej czu\u0142ym narz\u0119dziem diagnostycznym w rozpoznawaniu astmy ni\u017c spirometria. LCI i Scond r\u00f3\u017cnicowa\u0142y nawracaj\u0105ce \u015bwisty wieku przedszkolnego od zdrowej grupy kontrolnej  w przeciMBW mo\u017ce by\u0107 r\u00f3wnie\u017c u\u017cytecznym nieinwazyjnym markerem przebudowy dr\u00f3g oddechowych (remodelingu). W grupie doros\u0142ych z \u0142agodn\u0105 astm\u0105 udokumentowano wzrost Scond, nie w pe\u0142ni odwracalny po zastosowaniu leku rozszerzaj\u0105cego oskrzela. W grupie chorych na astm\u0119 \u0142agodn\u0105 do umiarkowanej udowodniono wzrost zaburze\u0144 wentylacji w gronkach z nieodwracalnym wzrostem Sacin. Dane te s\u0105 zgodne z ewolucj\u0105 proces\u00f3w naprawczych drobnych dr\u00f3g oddechowych i mog\u0105 wskazywa\u0107 na remodeling dr\u00f3g oddechowych. Badania histologiczne du\u017cych dr\u00f3g oddechowych sugeruj\u0105, \u017ce proces ten rozpoczyna si\u0119 w m\u0142odym wieku i zosta\u0142 udokumentowany u dzieci w wieku szkolnym. Wzrost LCI by\u0142 tylko cz\u0119\u015bciowo odwracalny w stabilnej, \u0142agodnej i przetrwa\u0142ej astmie dzieci\u0119cej  i mo\u017ce sPopraw\u0119 w Sacin bez zmian w Scond wykazano u doros\u0142ych chorych na astm\u0119 po zmianie preparatu glikokortykosteroidu wziewnego na zawieraj\u0105cy bardzo ma\u0142e cz\u0105steczki . PacjencJak przedstawiono, zastosowanie MBW u chorych na astm\u0119 mo\u017ce poprawi\u0107 zrozumienie proces\u00f3w patofizjologicznych, szczeg\u00f3lnie dotycz\u0105cych nadreaktywno\u015bci oskrzeli i dostarczy\u0107 dodatkowego narz\u0119dzia do oceny odpowiedzi klinicznej na stosowane leczenie. Wysoka czu\u0142o\u015b\u0107 badania stwarza mo\u017cliwo\u015b\u0107 wczesnego rozpoznania i leczenia, a w przysz\u0142o\u015bci poprawy rokowania , 4, 5.Post\u0119p w leczeniu BPD (ang. bronchopulmonary dysplasia), wprowadzenie surfaktantu oraz r\u00f3\u017cnych metod wentylacji, spowodowa\u0142 wyodr\u0119bnienie dw\u00f3ch populacji chorych na BPD: star\u0105\u201d \u2212 przed stosowaniem surfaktantu i \u201enow\u0105\u201d \u2212 po jego wprowadzeniu surfaktantu. Obecnie dzieci urodzone przedwcze\u015bnie z przewlek\u0142\u0105 chorob\u0105 p\u0142uc prezentuj\u0105 odmienny obraz kliniczny i patomorfologiczny ni\u017c te sprzed \u201eery surfaktantu\u201d. Mimo, \u017ce w obu typach dochodzi do przerwania wewn\u0105trz\u0142onowego rozwoju p\u0142uc, to \u201estara\u201d BPD charakteryzuje si\u0119 m.in. intensywnym, rozlanym w\u0142\u00f3knieniem przegr\u00f3d p\u0119cherzykowych, utrzymuj\u0105cym si\u0119 a\u017c do wczesnego wieku przedszkolnego. Nowe \u201eBPD\u201d charakteryzuje si\u0119 \u0142agodniejszym, bardziej rozproszonym procesem w\u0142\u00f3knienia. Sugeruje to, \u017ce w przebiegu BPD dochodzi do mieszanych zaburze\u0144 restrykcyjno-obturacyjnych. W zale\u017cno\u015bci od sposobu post\u0119powania w okresie noworodkowym r\u00f3\u017cny jest stopie\u0144 obturacji dr\u00f3g oddechowych, a tym samym zaburze\u0144 homogenno\u015bci wentylacji .Wst\u0119pne badania niemowl\u0105t z BPD opisuj\u0105 wzrost niejednorodno\u015bci wentylacji i obni\u017cenie FRC . StwierdPrzydatno\u015b\u0107 MBW w BPD pozostaje niejasna. Bior\u0105c pod uwag\u0119 przewa\u017caj\u0105ce zaburzenia restrykcyjne oraz rozproszony charakter zmian w \u201enowym\u201d BPD, przydatno\u015b\u0107 MBW mo\u017ce by\u0107 niewielka w stosunku do tradycyjnych bada\u0144 , 4, 5.Zarostowe zapalenie oskrzelik\u00f3w (BO \u2212 ang. bronchiolitis obliterans) jest chorob\u0105 zapaln\u0105 drobnych dr\u00f3g oddechowych, przebiegaj\u0105c\u0105 w spos\u00f3b niejednorodny z zw\u0142\u00f3knieniem i zanikiem dystalnych dr\u00f3g oddechowych. Mo\u017ce by\u0107 spowodowana infekcj\u0105 wirusow\u0105, uszkodzeniem chemicznym lub zaburzeniami odporno\u015bci. Badania sugeruj\u0105 przydatno\u015b\u0107 MBW we wczesnym wykrywaniu BO zar\u00f3wno po transplantacji p\u0142uc  jak i poPierwotna dyskineza rz\u0119sek i mukowiscydoza to choroby o autosomalnie recesywnym typie dziedziczenia, z dominuj\u0105cymi objawami klinicznymi ze strony uk\u0142adu oddechowego. Obie choroby charakteryzuj\u0105 si\u0119 zaburzeniem klirensu \u015bluzowo-rz\u0119skowego, przewlek\u0142ym bakteryjnym zaka\u017ceniem dr\u00f3g oddechowych, zapaleniem neutrofilowym i stopniow\u0105 progresj\u0105 zmian oskrzelowo-p\u0142ucnych. Etiologia tych zaburze\u0144 jest jednak odmienna. Do zaburze\u0144 klirensu \u015bluzowo-rz\u0119skowego w PCD dochodzi w wyniku nieprawid\u0142owej budowy lub /i funkcji aparatu rz\u0119skowego. Przebieg choroby jest znacznie \u0142agodniejszy ni\u017c CF a czas prze\u017cycia znacznie d\u0142u\u017cszy.Analogicznie do wynik\u00f3w bada\u0144 u chorych na mukowiscydoz\u0119, u kt\u00f3rych stwierdza si\u0119 korelacj\u0119 wska\u017anika LCI z nieprawid\u0142owo\u015bciami w HRCT klatki piersiowej, zak\u0142adano, \u017ce badanie MBW b\u0119dzie r\u00f3wnie\u017c przydatne do oceny choroby p\u0142uc w pierwotnej dyskinezie rz\u0119sek. Zbadano zale\u017cno\u015b\u0107 pomi\u0119dzy LCI, wynikami spirometrii i HRCT u chorych w zaawansowanym stadium PCD. W przeciwie\u0144stwie do pacjent\u00f3w z CF w badanej grupie chorych nie stwierdzono korelacji pomi\u0119dzy FEV1 a LCI ani pomi\u0119dzy HRCT, LCI i FEV1. Wysuni\u0119to przypuszczenie, ze r\u00f3\u017cnica w korelacji wynika by\u0107 mo\u017ce z odmienno\u015bci zaj\u0119cia du\u017cych i drobnych dr\u00f3g oddechowych w tych dw\u00f3ch jednostkach chorobowych .Z kolei w innym badaniu prospektywnym por\u00f3wnywano LCI z FEV1 z wynikami bada\u0144 obrazowych- HRCT klatki piersiowej u chorych z \u0142agodn\u0105 do umiarkowanej PCD. Warto\u015b\u0107 LCI korelowa\u0142a z wynikiem HRCT i ze zmianami takimi jak pogrubienie \u015bcian oskrzeli, korki \u015bluzowe i rozstrzenie oskrzeli. W por\u00f3wnaniu z FEV 1 wska\u017ank LCI by\u0142 bardziej czu\u0142ym parametrem w wykrywaniu nieprawid\u0142owo\u015bci strukturalnych. Udowodniono, \u017ce pomiar LCI u chorych na PCD ma znaczenie kliniczne i jest czulszym wska\u017anikiem koreluj\u0105cym z nieprawid\u0142owo\u015bciami strukturalnymi w HRCT ni\u017c FEV1.Wrodzona przepuklina przeponowa  powoduj\u0105c zmniejszenie si\u0119 przestrzeni wewn\u0105trz klatki piersiowej w okresie alweolaryzacji, prowadzi do rozwoju hipoplazji p\u0142uc. Badania przeprowadzone u dzieci w wieku szkolnym wskazywa\u0142y na zaburzenia o typie obturacji o r\u00f3\u017cnym stopniu ci\u0119\u017cko\u015bci . W badanNiemowl\u0119ta z przetrwa\u0142ym tachypnoe mog\u0105 prezentowa\u0107 r\u00f3\u017cne zaburzenia oddechowo-kr\u0105\u017ceniowe. MBW mo\u017ce u\u0142atwi\u0107 r\u00f3\u017cnicowanie zaburze\u0144 obturacjnych drobnych dr\u00f3g oddechowych jak grudkowe zapalenie oskrzelik\u00f3w (ang Follicular bronchitis \u2212 FB) od chor\u00f3b restrykcyjnych z prawid\u0142owym LCI i obni\u017conym FRC . PotrzebRozw\u00f3j zmian patologicznych w przebiegu wielu chor\u00f3b drobnych dr\u00f3g oddechowych obserwowany jest ju\u017c u najm\u0142odszych dzieci. W celu ograniczenia lub zapobiegania uszkodzeniu p\u0142uc ogromne znaczenie maj\u0105 wczesne rozpoznanie i rozpocz\u0119cie w\u0142a\u015bciwego leczenia. Wzrost zainteresowania MBW jako badania zapewniaj\u0105cego wgl\u0105d w procesy patologiczne drobnych dr\u00f3g oddechowych przyczyni\u0142 si\u0119 do coraz cz\u0119stszego jego stosowania. Zosta\u0142o ono uznane za czu\u0142e, bezpieczne i przydatne narz\u0119dzie do badania funkcji p\u0142uc i ich odpowiedzi na r\u00f3\u017cne czynniki uszkadzaj\u0105ce. Z naszego do\u015bwiadczenia wynika, \u017ce poprzez ocen\u0119 niejednorodno\u015bci wentylacji p\u0142uc daje mo\u017cliwo\u015b\u0107 rozpoznania zaburze\u0144 ju\u017c w pocz\u0105tkowym stadium wielu chor\u00f3b, cz\u0119sto kiedy jeszcze wyniki konwencjonalnych bada\u0144 spirometrycznych s\u0105 prawid\u0142owe. To proste badanie wymagaj\u0105ce od pacjenta spokojnego oddychania, bez konieczno\u015bci wykonywania forsownych oddech\u00f3w, pozwala na bezinwazyjn\u0105 ocen\u0119 drobnych dr\u00f3g oddechowych w r\u00f3\u017cnych grupach wiekowych : od populacji dzieci\u0119cej po osoby doros\u0142e \u017ale wsp\u00f3\u0142pracuj\u0105ce. Obecnie d\u0105\u017cy si\u0119 do standaryzacji wynik\u00f3w i okre\u015blenia norm przydatnych w diagnostyce, monitorowaniu przebiegu i leczenia chor\u00f3b drobnych dr\u00f3g oddechowych. Zastosowanie test\u00f3w oceniaj\u0105cych funkcj\u0119 p\u0142uc u niemowl\u0105t i dzieci przedszkolnych z mukowiscydoz\u0105 i innymi chorobami drobnych dr\u00f3g oddechowych mo\u017ce zmodyfikowa\u0107 post\u0119powanie kliniczne i poprawi\u0107 rokowanie , 4, 5. W"}
{"text": "After publication of the original article , the autThese errors have been corrected in the original article. Please find below a summary of the errors that have been corrected:\u201c6.7 million\u201d had been written instead of \u201c8.8 million\u201d\u201c90.5%\u201d had been written instead of \u201c61%\u201dIn the Results of the Abstract:\u201c50.92%\u201d had been written instead of \u201c38.82%\u201dIn Table 2, in the row \u2018AOM Myringotomy\u2019:\u201823.8%\u201d had been given for the effectiveness estimate of PCV13 against all-cause pneumonia\u2019, while the value should read \u201823.7%\u2019.In the subsection \u2018All-cause pneumonia effectiveness\u2019:\u201c17\u201d and \u201c17\u201d, respectively, had been detailed instead of \u201c19\u201d and \u201c19\u201d, respectivelyIn Table 6, in the row \u2018Death due to IPD/pneumonia\u2019, under columns \u2018PCV13 (A)\u2019 and \u2018PHiD-CV (B)\u2019, respectively:\u201cNTD 557 957\u201d had been written instead of \u201cNTD 557 887\u201dIn Table 6, in the row \u2018All-cause pneumonia\u2019, under the column \u2018Difference (B\u2013A)\u2019:\u2212\u20090.6 had been written instead of \u2212\u00a05.9\u2212\u20096.6 had been written instead of \u2212\u20098.8\u2212\u20093.9 had been written instead of \u2212\u20094.8\u2212\u20090.8 had been written instead of \u2212\u20091.2\u2212\u20097.1 had been written instead of \u2212\u20099.3\u2212\u200914.4 had been written instead of \u2212\u200919.60.15 had been written instead of 5.5\u2212\u20096.7 had been written instead of \u2212\u20098.8\u2212\u200988.2 had been written instead of \u2212\u00a090.4In Table 8, column \u2018Cost (millions)\u2019 (in descending order):8 had been written instead of 146 had been written instead of 521 had been written instead of 2238 had been written instead of 3938 had been written instead of \u2212\u200912In Table 8, column \u2018QALY\u2019 (in descending order):\u201c\u2212\u00a06.7 million\u201d had been written instead of \u201c\u2212\u00a08.8 million\u2019In the Footnote of Table 8:\u201cThe supplementary Fig. 2 stresses the general context and observations that were made in the present study.\u201dAt the end of the \u2018Conclusion\u2019, the following sentence was missing:The authors apologize for any inconvenience caused."}
{"text": "Atopowe zapalenie sk\u00f3ry jest przewlek\u0142\u0105 chorob\u0105 zapaln\u0105, przebiegaj\u0105c\u0105 z nawrotowymi zaostrzeniami, uporczywym \u015bwi\u0105dem, rumieniem, sucho\u015bci\u0105 sk\u00f3ry wskutek uszkodzenia bariery nask\u00f3rkowej i zaka\u017ceniami gronkowcowymi. Czynnikami wywo\u0142uj\u0105cymi s\u0105 mutacje w genie koduj\u0105cym filagryn\u0119, rozregulowanie uk\u0142adu immunologicznego, zmiany dotycz\u0105ce mikrobiomu sk\u00f3ry i lipid\u00f3w w stratum corneum, niedob\u00f3r peptyd\u00f3w antymikrobiologicznych AMPs. Choroba dotyczy g\u0142\u00f3wnie dzieci, powoduj\u0105c znaczne pogorszenie jako\u015bci \u017cycia, a jej pierwsze objawy wyst\u0119puj\u0105 w oko\u0142o 90% przypadk\u00f3w przed uko\u0144czeniem 5. r.\u017c. Przez lata termin \u201eemolienty\u201d odnoszono do substancji nat\u0142uszczaj\u0105cych stosowanych w celu uelastycznienia i zmi\u0119kczenia sk\u00f3ry w chorobach przebiegaj\u0105cych z szorstko\u015bci\u0105, z\u0142uszczaniem i sucho\u015bci\u0105 sk\u00f3ry. Obecnie wiadomo, \u017ce emolienty mog\u0105 tak\u017ce dzia\u0142a\u0107 nawil\u017caj\u0105co poprzez zatrzymywanie wody w sk\u00f3rze, st\u0105d cz\u0119sto terminy \u201eemolient\u201d i \u201esubstancja nawil\u017caj\u0105ca\u201d s\u0105 u\u017cywane zamiennie. Zgodnie z najnowszymi rekomendacjami towarzystw dermatologicznych podstawow\u0105 terapi\u0105 w atopowym zapaleniu sk\u00f3ry jest d\u0142ugotrwa\u0142e stosowanie preparat\u00f3w emoliencyjnych aplikowanych regularnie na sk\u00f3r\u0119 oraz dodawanych do k\u0105pieli. Emolienty mog\u0105 by\u0107 stosowane w monoterapii lub \u2013w okresach zaostrze\u0144 \u2013w skojarzeniu z miejscowo stosowanymi kortykosteroidami lub inhibitorami kalcyneuryny. Badania kliniczne wykaza\u0142y, \u017ce systematyczne stosowanie preparat\u00f3w emoliencyjnych pomaga zar\u00f3wno przywr\u00f3ci\u0107 funkcje ochronne bariery sk\u00f3rnej, jak i wp\u0142ywa na zmniejszenie zu\u017cycia miejscowo stosowanych kortykosteroid\u00f3w u niemowl\u0105t, dzieci i doros\u0142ych pacjent\u00f3w z atopowym zapaleniem skory. Wyniki bada\u0144 oraz wieloletnie do\u015bwiadczenie kliniczne dowodz\u0105 skuteczno\u015bci i bezpiecze\u0144stwa stosowania preparat\u00f3w emoliencyjnych. W pracy przedstawiono aktualny stan wiedzy na temat emolient\u00f3w, tj. przegl\u0105d sk\u0142adnik\u00f3w preparat\u00f3w emoliencyjnych, w\u0142a\u015bciwo\u015bci i mechanizmy dzia\u0142ania emolient\u00f3w, a tak\u017ce om\u00f3wiono ich znaczenie w atopowym zapaleniu sk\u00f3ry oraz wyniki bada\u0144 klinicznych przeprowadzonych z udzia\u0142em dzieci z atopowym zapaleniem sk\u00f3ry. Przez dziesi\u0105tki lat okre\u015blenie \u201eemolient\u201d dotyczy\u0142o wy\u0142\u0105cznie substancji b\u0119d\u0105cej sk\u0142adnikiem produktu kosmetycznego, kt\u00f3rej zadaniem by\u0142o nat\u0142uszczenie, zmi\u0119kczenie, wyg\u0142adzenie i po\u015brednie nawil\u017cenie sk\u00f3ry, i takie nazewnictwo zastosowano w niniejszej pracy. Obecnie w terminologii kosmetologicznej i dermatologicznej spotykamy si\u0119 coraz cz\u0119\u015bciej z tym okre\u015bleniem w odniesieniu do gotowego produktu kosmetycznego o w\u0142a\u015bciwo\u015bciach nawil\u017caj\u0105cych i nat\u0142uszczaj\u0105cych, w sk\u0142ad kt\u00f3rego wchodzi jeden lub kilka emolient\u00f3w. Poniewa\u017c jednak taki kosmetyk zawiera nie tylko emolienty, ale te\u017c inne sk\u0142adniki recepturowe, w niniejszej pracy nosi on nazw\u0119 produktu emoliencyjnego. Z punktu widzenia idealnego dzia\u0142ania takiego kosmetyku najlepiej jest, je\u015bli zawiera on emolienty w po\u0142\u0105czeniu z substancjami wi\u0105\u017c\u0105cymi i zatrzymuj\u0105cymi wod\u0119 w sk\u00f3rze, czyli tzw. humektantami. Dzi\u0119ki temu sk\u0142adniki lipofilowe (emolienty) i hydrofilowe (humektanty) uzupe\u0142niaj\u0105 swoje dzia\u0142anie, zapobiegaj\u0105c sucho\u015bci sk\u00f3ry.emolliere\u201d, tzn. zmi\u0119kcza\u0107. S\u0105 jednymi z najstarszych sk\u0142adnik\u00f3w kosmetyk\u00f3w piel\u0119gnacyjnych. Oleje ro\u015blinne o w\u0142a\u015bciwo\u015bciach emoliencyjnych stosowano ju\u017c w staro\u017cytnym Egipcie i Chinach, lanolin\u0119 \u2013 w staro\u017cytnej Grecji, a historia wazeliny si\u0119ga drugiej po\u0142owy XIX w. Obecnie emolienty prze\u017cywaj\u0105 \u201edrug\u0105 m\u0142odo\u015b\u0107\u201d, a ich stosowanie nie jest ju\u017c tylko intuicyjne, lecz tak\u017ce poparte wynikami bada\u0144. Substancje te znalaz\u0142y zastosowanie zar\u00f3wno w leczeniu, jak i profilaktyce wielu chor\u00f3b sk\u00f3ry, zw\u0142aszcza w atopowym zapaleniu sk\u00f3ry (AZS), \u0142uszczycy, wyprysku kontaktowym. Stosowanie preparat\u00f3w emoliencyjnych ogranicza cz\u0119sto konieczno\u015b\u0107 leczenia glikokortykosteroidami i inhibitorami kalcyneuryny, wyd\u0142u\u017ca okres remisji objaw\u00f3w, a niekiedy nawet zapobiega nawrotom choroby [Emolienty s\u0105 cenionymi sk\u0142adnikami kosmetyk\u00f3w dla niemowl\u0105t, dzieci i os\u00f3b doros\u0142ych, zw\u0142aszcza ze sk\u00f3r\u0105 atopow\u0105. S\u0105 stosowane zar\u00f3wno w kosmetykach pozostaj\u0105cych na sk\u00f3rze , jak i w preparatach myj\u0105cych . Ich nazwa pochodzi od \u0142aci\u0144skiego \u201e choroby .Transepidermal Water Loss), okre\u015blaj\u0105ca przeznask\u00f3rkow\u0105 utrat\u0119 wody. W wyniku utworzenia okluzji emolienty zatrzymuj\u0105 wod\u0119 w nask\u00f3rku, powoduj\u0105c zmniejszenie TEWL [Natural Moisturizing Factor). W sk\u0142ad NMF wchodz\u0105 substancje o silnym dzia\u0142aniu nawil\u017caj\u0105cym, g\u0142\u00f3wnie aminokwasy i ich pochodne, kwas piroglutaminowy (PCA), kwas urokainowy, sole i jony nieorganiczne, kwas mlekowy, mocznik. Naturalny czynnik nawil\u017caj\u0105cy, kt\u00f3ry stanowi 15-30% masy warstwy rogowej nask\u00f3rka, wp\u0142ywa reguluj\u0105co na wi\u0105zanie wody w sk\u00f3rze, a jego zmniejszenie o ok. 10% powoduje znacz\u0105c\u0105 sucho\u015b\u0107 sk\u00f3ry. Stosowanie emolient\u00f3w prowadzi do wyra\u017anej poprawy uwodnienia warstwy rogowej nask\u00f3rka oraz zmi\u0119kczenia i wyg\u0142adzenia sk\u00f3ry. Ponadto, dzi\u0119ki zdolno\u015bci odtwarzania lipid\u00f3w nask\u00f3rka wymytych przez substancje powierzchniowo czynne zawarte w preparatach myj\u0105cych, emolienty wp\u0142ywaj\u0105 na regeneracj\u0119 nask\u00f3rka i przywracaj\u0105 jego prawid\u0142ow\u0105 funkcj\u0119. Poprawa stanu nawil\u017cenia i nat\u0142uszczenia sk\u00f3ry wp\u0142ywa na jej elastyczno\u015b\u0107, a zdolno\u015b\u0107 do regeneracji nask\u00f3rka zabezpiecza przed uszkodzeniem, p\u0119kni\u0119ciami i nadmiernym \u0142uszczeniem oraz wnikaniem niepo\u017c\u0105danych substancji egzogennych. D\u0142ugotrwa\u0142e i systematyczne stosowanie kosmetyk\u00f3w emoliencyjnych zmniejsza odczucie \u015bwi\u0105du sk\u00f3ry, kt\u00f3ry jest jednym z najbardziej dokuczliwych objaw\u00f3w AZS, zmniejsza tak\u017ce stan zapalny sk\u00f3ry.Wi\u0119kszo\u015b\u0107 emolient\u00f3w wykazuje dzia\u0142anie epidermalne, gdy\u017c nie przenikaj\u0105 one przez nask\u00f3rek, lecz tworz\u0105 na jego powierzchni cienki film \u2212 barier\u0119 okluzyjn\u0105 \u2212 zabezpieczaj\u0105c\u0105 przed nadmiernym odparowywaniem wody z g\u0142\u0119bszych warstw sk\u00f3ry. Inne, jak np. lanolina i ceramidy, penetruj\u0105 do struktur cementu mi\u0119dzykom\u00f3rkowego warstwy rogowej nask\u00f3rka, trwale odbudowuj\u0105c barier\u0119 nask\u00f3rkow\u0105 . Stopie\u0144nie TEWL , dzi\u0119ki Ze wzgl\u0119du na r\u00f3\u017cnorodno\u015b\u0107 emolient\u00f3w, odmienne mechanizmy ich dzia\u0142ania nawil\u017caj\u0105cego, regeneracyjnego, przeciw\u015bwi\u0105dowego oraz zdolno\u015bci poprawy zaburzonej charakterystyki mikrobiomu sk\u00f3ry u chorych na AZS, istotny jest odpowiedni, indywidualny dob\u00f3r preparatu emoliencyjnego w zale\u017cno\u015bci od stanu klinicznego pacjenta i oczekiwanego efektu terapeutycznego.Grupa emolient\u00f3w obejmuje substancje naturalne oraz otrzymywane syntetycznie. W pi\u015bmiennictwie brak jest jednoznacznej klasyfikacji emolient\u00f3w. Wyst\u0119puje du\u017ca r\u00f3\u017cnorodno\u015b\u0107 podzia\u0142\u00f3w zar\u00f3wno pod wzgl\u0119dem budowy chemicznej, w\u0142a\u015bciwo\u015bci, jak i \u017ar\u00f3d\u0142a otrzymywania tych substancji. Nie u\u0142atwia te\u017c sprawy fakt, \u017ce w niekt\u00f3rych opracowaniach angloj\u0119zycznych termin \u201emoisturisers\u201d (preparaty o dzia\u0142aniu nawil\u017caj\u0105cym) jest stosowany zamiennie z terminem \u201eemollients\u201d (preparaty emoliencyjne) , 5.Przedstawiony poni\u017cej podzia\u0142 emolient\u00f3w stanowi pr\u00f3b\u0119 po\u0142\u0105czenia i uproszczenia r\u00f3\u017cnych system\u00f3w klasyfikacji tych zwi\u0105zk\u00f3w.Caprylic/Capric Trigliceryde). W\u0142a\u015bciwo\u015bci emoliencyjne wykazuj\u0105 tak\u017ce inne zwi\u0105zki z tej grupy, m.in. alkohol cetylowy i kwas stearynowy.W tej grupie wa\u017cne miejsce zajmuj\u0105 syntetyczne estry kwas\u00f3w t\u0142uszczowych, zw\u0142aszcza mirystynian izopropylu i mirystynian mirycylu, oraz otrzymywany z oleju palmowego palmitynian izopropylu. Nie tylko nat\u0142uszczaj\u0105 i zmi\u0119kczaj\u0105 nask\u00f3rek, ale tak\u017ce umo\u017cliwiaj\u0105 wymian\u0119 tlenow\u0105 i wodn\u0105, nie pozostawiaj\u0105c uczucia lepko\u015bci. S\u0105 cz\u0119sto stosowane jako dodatek do olej\u00f3w mineralnych, gdy\u017c powoduj\u0105 cz\u0119\u015bciowe otwarcie warstwy okluzyjnej powsta\u0142ej po zastosowaniu parafin, dzi\u0119ki czemu umo\u017cliwiaj\u0105 wymian\u0119 mi\u0119dzy sk\u00f3r\u0105 i otoczeniem. Cennym sk\u0142adnikiem pochodzenia naturalnego jest trigliceryd kaprylowo-kaprynowy , znanej z du\u017cej skuteczno\u015bci u pacjent\u00f3w z AZS, a tak\u017ce jest popularnym pod\u0142o\u017cem wielu ma\u015bci.Zawarto\u015b\u0107 ceramid\u00f3w w sk\u00f3rze zmniejsza si\u0119 z wiekiem, s\u0105 te\u017c wyp\u0142ukiwane przez \u015brodki myj\u0105ce, co zaburza funkcje ochronne sk\u00f3ry i powoduje zwi\u0119kszenie TEWL. Ceramidy s\u0105 cz\u0119sto stosowane w po\u0142\u0105czeniu z innymi sk\u0142adnikami naturalnego p\u0142aszcza lipidowego: cholesterolem, kwasami t\u0142uszczowymi, skwalenem, fosfolipidami, co przyczynia si\u0119 do odbudowy bariery sk\u00f3rno-nask\u00f3rkowej i nawil\u017cenia sk\u00f3ry. Ich skuteczno\u015b\u0107 w postaci zwi\u0119kszenia uwodnienia sk\u00f3ry i zmniejszenia TEWL oraz odnowy bariery nask\u00f3rkowej zosta\u0142a potwierdzona w badaniach . Dzi\u0119ki Prunus dulcis) i s\u0142onecznika (Helianthus annuus) oraz zawieraj\u0105ce cenny kwas gamma-linolenowy oleje wiesio\u0142kowy (Oenothera biennis i O. paradoxa) i og\u00f3recznikowy . Bardzo korzystne w\u0142a\u015bciwo\u015bci maj\u0105 te\u017c: olej z oliwek, czyli bogata w kwas oleinowy oliwa, olej z pestek winogron (Vitis vinifera), kie\u0142k\u00f3w pszenicy (Triticum vulgare Germ Oil), mas\u0142osza , orzech\u00f3w makadamii (Macadamia ternifolia), awokado (Persea gratissima), jojoby (Simmondsia chinensis), nasion lnu (Linum usitatissimum). Dzi\u0119ki korzystnemu dzia\u0142aniu na sk\u00f3r\u0119 oleje te s\u0105 stosowane tak\u017ce w kosmetykach dla dzieci ze sk\u00f3r\u0105 atopow\u0105. Ponadto cennym surowcem ro\u015blinnym jest owies (Avena sativa); uzyskany z niego ekstrakt \u2212 mleczko owsiane \u2212 ma dzia\u0142anie \u0142agodz\u0105ce i przeciwzapalne [Oleje ro\u015blinne ze wzgl\u0119du na swe w\u0142a\u015bciwo\u015bci piel\u0119gnacyjne s\u0105 cz\u0119sto stosowane w kosmetykach dla niemowl\u0105t i dzieci. S\u0105 bardzo cenionymi surowcami ze wzgl\u0119du na zawarto\u015b\u0107 nienasyconych kwas\u00f3w t\u0142uszczowych i innych sk\u0142adnik\u00f3w wp\u0142ywaj\u0105cych na odbudow\u0119 lipid\u00f3w nask\u00f3rka, m.in. poprzez udzia\u0142 w tworzeniu cementu mi\u0119dzykom\u00f3rkowego . Najcz\u0119\u015bwzapalne . Nale\u017cy Now\u0105, cenn\u0105 grup\u0105 emolient\u00f3w s\u0105 polihydroksylowane oleje ro\u015blinne, kt\u00f3re charakteryzuj\u0105 si\u0119 korzystniejszymi w\u0142a\u015bciwo\u015bciami fizyko-chemicznymi, tj. lepsz\u0105 rozprowadzalno\u015bci\u0105, mniejsz\u0105 kleisto\u015bci\u0105 itp., przy zachowanym bardzo dobrym dzia\u0142aniu nawil\u017caj\u0105cym sk\u00f3r\u0119.vernix caseosa), co t\u0142umaczy bardzo korzystne dzia\u0142anie lanoliny na barier\u0119 nask\u00f3rkow\u0105 noworodk\u00f3w [Lanolina (wosk owczy) jest emolientem otrzymywanym z we\u0142ny owczej, reguluj\u0105cym gospodark\u0119 wodn\u0105 sk\u00f3ry i zmi\u0119kczaj\u0105cym nask\u00f3rek. Jest szeroko stosowana w kosmetyce i farmacji ze wzgl\u0119du na zdolno\u015b\u0107 znacznego wi\u0105zania wody. Nawil\u017caj\u0105co dzia\u0142aj\u0105 jej sk\u0142adniki, g\u0142\u00f3wnie sterole i ich estry oraz wolne i zestryfikowane kwasy t\u0142uszczowe, kt\u00f3re penetruj\u0105 do struktur cementu mi\u0119dzykom\u00f3rkowego. Lanolina i jej pochodne nale\u017c\u0105 do emolient\u00f3w trwale odbudowuj\u0105cych barier\u0119 nask\u00f3rkow\u0105, gdy\u017c efekty ich dzia\u0142ania utrzymuj\u0105 si\u0119 d\u0142u\u017cej ni\u017c w przypadku emolient\u00f3w okluzyjnych . Struktuworodk\u00f3w . W przesCera alba, Cera flava, Beeswax). Cennym surowcem kosmetycznym jest skwalen  - biogenny emolient o du\u017cym powinowactwie do sk\u00f3ry, sk\u0142adnik p\u0142aszcza lipidowego. Tworzy na powierzchni sk\u00f3ry film okluzyjny, pe\u0142ni\u0105cy rol\u0119 bariery ochronnej.W\u0142a\u015bciwo\u015bci emoliencyjne wykazuje te\u017c wosk pszczeli (Petrolatum), olej parafinowy , cerezyn\u0119, ozokeryt, wosk mikrokrystaliczny . Ich w\u0142a\u015bciwo\u015bci fizyczne s\u0105 zbli\u017cone do w\u0142a\u015bciwo\u015bci t\u0142uszcz\u00f3w ro\u015blinnych i zwierz\u0119cych, jednak w odr\u00f3\u017cnieniu od nich charakteryzuj\u0105 si\u0119 doskona\u0142\u0105 stabilno\u015bci\u0105, m.in. nie ulegaj\u0105 rozk\u0142adowi pod wp\u0142ywem UV i tlenu. Wywieraj\u0105 silne dzia\u0142anie okluzyjne, tworz\u0105c na powierzchni sk\u00f3ry warstw\u0119 lipidow\u0105, dzi\u0119ki czemu dzia\u0142aj\u0105 zmi\u0119kczaj\u0105co, nat\u0142uszczaj\u0105co i po\u015brednio nawil\u017caj\u0105co, gdy\u017c zatrzymuj\u0105 wod\u0119 w sk\u00f3rze. Jednym z najcz\u0119\u015bciej stosowanych emolient\u00f3w jest wazelina, kt\u00f3ra nie tylko nawil\u017ca sk\u00f3r\u0119, ale tak\u017ce wp\u0142ywa na odbudow\u0119 kwa\u015bnego p\u0142aszcza lipidowego pe\u0142ni\u0105cego rol\u0119 bariery ochronnej sk\u00f3ry [antimicrobial peptides), a tym samym przyczynia si\u0119 do zwalczania patogen\u00f3w bakteryjnych, w tym Staphylococcus aureus [Parafiny s\u0105 grup\u0105 surowc\u00f3w obejmuj\u0105c\u0105 m.in. stosowan\u0105 od 1872 r. wazelin\u0119  . Wbrew iIch inna nazwa to polimery filmotw\u00f3rcze, gdy\u017c tworz\u0105 na powierzchni sk\u00f3ry niewidoczny i odporny na zmywanie film okluzyjny, kt\u00f3ry jednak nie zaburza jej funkcji. Najpopularniejszymi substancjami z tej grupy s\u0105 polidimetylosiloksan (dimetykon), cyklopolidimetylosiloksan (cyklometykon), cyklopentasiloksan, cykloheksasiloksan. Podobnie jak parafiny, silikony s\u0105 oboj\u0119tne chemicznie i wyj\u0105tkowo stabilne, tj. odporne na dzia\u0142anie UV, tlenu i wody. Ich niewielka ilo\u015b\u0107 dodana do kosmetyku emoliencyjnego zawieraj\u0105cego olej parafinowy znacznie poprawia jego w\u0142a\u015bciwo\u015bci sensoryczne.Jak ju\u017c wspomniano, humektanty s\u0105 substancjami o w\u0142a\u015bciwo\u015bciach higroskopijnych, kt\u00f3re wykazuj\u0105 zdolno\u015b\u0107 wi\u0105zania i zatrzymania wody w warstwie rogowej nask\u00f3rka. Do grupy tej nale\u017c\u0105 m.in. gliceryna, mocznik, kwas hialuronowy, glikole, aminokwasy, kwas mlekowy i mleczan sodu, alantoina, pantenol. Niekt\u00f3re z nich wchodz\u0105 w sk\u0142ad naturalnego czynnika nawil\u017caj\u0105cego NMF. Odpowiednie nawodnienie warstwy rogowej zapewnia w\u0142a\u015bciw\u0105 elastyczno\u015b\u0107 sk\u00f3ry, chroni j\u0105 przed uszkodzeniami oraz zapewnia utrzymanie homeostazy sk\u00f3ry.Gliceryna jest popularnym sk\u0142adnikiem preparat\u00f3w emoliencyjnych, dzia\u0142aj\u0105cym nie tylko nawil\u017caj\u0105co, ale tak\u017ce u\u0142atwiaj\u0105cym wch\u0142anianie innych substancji.Mocznik w st\u0119\u017ceniu do 10% wywiera dzia\u0142anie nawil\u017caj\u0105ce, co jest szczeg\u00f3lnie korzystne u dzieci z atopowym zapaleniem sk\u00f3ry. W pi\u015bmiennictwie znajduj\u0105 si\u0119 jednak doniesienia o zwi\u0119kszonym, toksycznym st\u0119\u017ceniu mocznika we krwi u noworodk\u00f3w i niemowl\u0105t po zastosowaniu preparat\u00f3w zawieraj\u0105cych mocznik . Zwi\u0105zekstratum corneum z jednoczesnym zmniejszeniem TEWL, dzi\u0119ki czemu zmniejsza si\u0119 sucho\u015b\u0107 i \u015bwi\u0105d sk\u00f3ry. Utworzenie ochronnego filmu na powierzchni sk\u00f3rnych zmian atopowych i korzystny wp\u0142yw w postaci przyspieszenia gojenia zadrapa\u0144 i zmian wypryskowych oraz regeneracji uszkodzonej sk\u00f3ry, wp\u0142ywaj\u0105 na odtworzenie prawid\u0142owej bariery nask\u00f3rkowej [xerosis), z\u0142uszczaniem, \u015bwi\u0105dem, uszkodzeniem i (lub) podra\u017cnieniem sk\u00f3ry.Ze wzgl\u0119du na silne dzia\u0142anie nawil\u017caj\u0105ce kwasu hialuronowego jego stosowanie wp\u0142ywa na popraw\u0119 uwodnienia k\u00f3rkowej . Podkre\u015bAminokwasy, kwas mlekowy i mleczan sodu s\u0105 sk\u0142adnikami NMF i zapewniaj\u0105 skuteczne nawil\u017cenie warstwy rogowej nask\u00f3rka. Pantenol silnie nawil\u017ca i zmi\u0119kcza sk\u00f3r\u0119, wp\u0142ywa tak\u017ce aktywizuj\u0105co na podzia\u0142y kom\u00f3rkowe, pobudzaj\u0105c wzrost oraz odnow\u0119 kom\u00f3rek nask\u00f3rka i sk\u00f3ry w\u0142a\u015bciwej. Dzia\u0142a regeneruj\u0105co, \u0142agodzi podra\u017cnienia (w tym powsta\u0142e w wyniku oparze\u0144 s\u0142onecznych) i przyspiesza gojenie uszkodze\u0144 sk\u00f3ry . PodobneAtopowe zapalenie sk\u00f3ry jest chorob\u0105 o z\u0142o\u017conej etiopatogenezie, charakteryzuj\u0105c\u0105 si\u0119 nieprawid\u0142ow\u0105 budow\u0105 i dzia\u0142aniem warstwy rogowej nask\u00f3rka. Jest najcz\u0119\u015bciej wyst\u0119puj\u0105c\u0105 zapaln\u0105 chorob\u0105 sk\u00f3ry u dzieci. Jej pierwsze objawy w oko\u0142o 60% przypadk\u00f3w manifestuj\u0105 si\u0119 w okresie niemowl\u0119cym, a w oko\u0142o 90% przypadk\u00f3w \u2212 przed uko\u0144czeniem pi\u0105tego roku \u017cycia . Zachorostratum corneum [stratum corneum, niedob\u00f3r peptyd\u00f3w antymikrobiologicznych (AMPs) [G\u0142\u00f3wnym czynnikiem wywo\u0142uj\u0105cym AZS s\u0105 mutacje w genie koduj\u0105cym filagryn\u0119, kt\u00f3ra jest bia\u0142kiem odpowiadaj\u0105cym za prawid\u0142ow\u0105 funkcj\u0119 bariery nask\u00f3rkowej . Mutacje corneum , 20, a t corneum . W\u015br\u00f3d ch (AMPs) .S. aureus, bakteria kolonizuj\u0105ca od 60 do 100% pacjent\u00f3w [S. epidermidis [S. aureus nad gronkowcami komensalnymi, co sugeruje, \u017ce bakterie komensalne mog\u0105 chroni\u0107 przed rozwojem AZS [S. aureus [S. aureus nad innymi szczepami, co zaburza prawid\u0142owe funkcjonowanie bariery sk\u00f3rnej. Ostatnio wykazano, \u017ce u niemowl\u0105t kolonizacja sk\u00f3ry przez S. aureus koreluje dodatnio z wyst\u0105pieniem AZS w p\u00f3\u017aniejszym okresie niemowl\u0119cym oraz \u017ce nasilenie kolonizacji poprzedza o 2 miesi\u0105ce wyst\u0105pienie pierwszych objaw\u00f3w AZS [Zmiany w mikrobiomie, czyli zaburzenia r\u00f3wnowagi mikroflory bakteryjnej zasiedlaj\u0105cej sk\u00f3r\u0119, odgrywaj\u0105 znacz\u0105c\u0105 rol\u0119 w etiopatogenezie AZS. Na sk\u00f3rze pacjent\u00f3w z ci\u0119\u017ckim AZS dominuje acjent\u00f3w  z t\u0105 chodermidis . W mikroojem AZS . Bakteriojem AZS . Istotn\u0105. aureus , 23. Wytaw\u00f3w AZS .Przed 2000 r. przeprowadzono pi\u0119\u0107 , a po tyWyniki wi\u0119kszo\u015bci bada\u0144 wykaza\u0142y, \u017ce stosowanie u doros\u0142ych pacjent\u00f3w z AZS preparat\u00f3w emoliencyjnych wzmacnia barier\u0119 nask\u00f3rkow\u0105 i znacznie poprawia stan sk\u00f3ry, wp\u0142ywaj\u0105c na zmniejszenie sucho\u015bci, szorstko\u015bci i z\u0142uszczania sk\u00f3ry oraz odczucia \u015bwi\u0105du, a nawet prowadzi do istotnego wyd\u0142u\u017cenia fazy regresji objaw\u00f3w ,30,31,32vs 7,5%; p>0,05). Simpson i wsp. (2011) w badaniu z udzia\u0142em 127 pacjent\u00f3w w wieku \u2265 3 lat odnotowali szybsze ust\u0119powanie objaw\u00f3w chorobowych w 7., 14. i 21. dniu stosowania 2 razy dziennie preparatu emoliencyjnego  jednocze\u015bnie z preparatem steroidowym, ni\u017c po stosowaniu samego steroidu  [Podobne wyniki uzyskano w badaniach przeprowadzonych u niemowl\u0105t i dzieci , 37, 38.S. aureus, z jednoczesnym obni\u017ceniem pH sk\u00f3ry i zwi\u0119kszeniem liczby szczep\u00f3w Streptococcus salivarius, bakterii wykazuj\u0105cych w\u0142a\u015bciwo\u015bci immunomodulacyjne [Stosowanie preparat\u00f3w emoliencyjnych u niemowl\u0105t i os\u00f3b doros\u0142ych z AZS zmniejsza o 50-78% kolonizacj\u0119 ulacyjne . Najnowsulacyjne , 40. Nieulacyjne . Wykazanulacyjne .prescription emollient devices), jakim jest Atopiclar (MAS063DP), przeprowadzone zar\u00f3wno u doros\u0142ych, jak i u dzieci. Preparaty te opr\u00f3cz sk\u0142adnik\u00f3w hydrofilowych i lipofilowych zawieraj\u0105 tak\u017ce zwi\u0105zki dzia\u0142aj\u0105ce przeciwzapalnie. Atopiclar jest hydrolipidowym kremem emoliencyjnym, kt\u00f3ry opr\u00f3cz substancji nawil\u017caj\u0105cych i odbudowuj\u0105cych barier\u0119 nask\u00f3rkow\u0105  zawiera zwi\u0105zki o dzia\u0142aniu przeciwzapalnym i przeciw\u015bwi\u0105dowym . Wyniki 4 wieloo\u015brodkowych bada\u0144 randomizowanych [Na uwag\u0119 zas\u0142uguj\u0105 wyniki randomizowanych bada\u0144 skuteczno\u015bci emoliencyjnego wyrobu medycznego z aptecznej grupy tzw. PEDs  b\u0105d\u017a inhibitorami kalcyneuryny (terapia proaktywna)."}
{"text": "Scientific Reports 10.1038/s41598-021-03968-4, published online 19 January 2022Correction to: The original version of this Article contained a repeated error where the \u2018As a result,\u02d9o2max\u201d.\u201cnow reads:o2max\u201d.\u201c\u201c\u02d9E\u201d.now reads:\u201c\u201co2\u201d.now reads:o2\u201d.\u201c31P magnetic resonance spectroscopy data processing,\u2019Furthermore, in the Methods section, under the subheading \u2018m is the [ADP] at which oxidative ATP synthesis is taken to be half maximal (25 \u00b5mol/L) and n (2.2) is a Hill coefficient that describes the relationship between 35,72,73.\u201d\u201cwhere now reads:m is the [ADP] at which oxidative ATP synthesis is taken to be half maximal (25 \u00b5mol/L) and n (2.2) is a Hill coefficient that describes the relationship between V and [ADP]35,72,73.\u201d\u201cwhere V is the initial rate of PCr resynthesis, Kst column \u2018Visit\u2019,In Table 4, in the 1\u201c\u02d9E (L/min)\u201d.now reads:\u201cThe original Article has been corrected."}
{"text": "Pani Profesor Krystyna Bo\u017ckowa by\u0142a na tyle bogat\u0105 postaci\u0105, \u017ce nie spos\u00f3b jest zakre\u015bli\u0107 ramy tej charakterystyki. Powiedzie\u0107, \u017ce jest lekarzem, nauczycielem, naukowcem i spo\u0142ecznikiem to zdecydowanie za ma\u0142o.W Instytucie Matki i Dziecka pracowa\u0142a od 1960 r, kolejno na stanowiskach kierownika Kliniki Pediatrii, z-cy Dyrektora IMiD ds. naukowo-badawczych (9 lat) i wreszcie Dyrektora Instytutu (20 lat). Jest tw\u00f3rc\u0105 oryginalnej polskiej koncepcji medycyny wieku rozwojowego, integruj\u0105cej specjalno\u015bci pediatryczne. Jej zas\u0142ug\u0105 jest konsekwentne wcielanie tej koncepcji w \u017cycie. Konsekwencja \u2013 to cecha, kt\u00f3r\u0105 z pewno\u015bci\u0105 Pani Profesor posiada\u0142a i kt\u00f3rej uczy\u0142a swoich asystent\u00f3w, czego i ja do\u015bwiadczy\u0142am. \u201eJe\u015bli powiedzia\u0142e\u015b A, musisz powiedzie\u0107 Bi Ci tak dalej. Je\u015bli obroni\u0142e\u015b doktorat, nast\u0119pnego dnia zacznij planowa\u0107 habilitacj\u0119, nie zatrzymuj si\u0119, szkoda czasu, id\u017a dalej\u201d.W bogatej dzia\u0142alno\u015bci naukowej Pani Profesor Bo\u017ckowej mo\u017cna wyodr\u0119bni\u0107 4 g\u0142\u00f3wne kierunki bada\u0144: rozw\u00f3j funkcji biochemicznych organizmu dziecka, reaktywno\u015b\u0107 ustroju rosn\u0105cego, zw\u0142aszcza na leki, wrodzone choroby metaboliczne i \u017cywienie. Dzi\u015b mo\u017cemy powiedzie\u0107, \u017ceta wizja sprawdzi\u0142a si\u0119 w pe\u0142ni. Od lat 70-80. zesz\u0142ego stulecia stale poszerzany jest w IMiD panel bada\u0144 przesiewowych wykonywanych u wszystkich noworodk\u00f3w w Polsce, opracowywany pod wzgl\u0119dem merytorycznym, metodycznym i organizacyjnym. W kolejnych pokoleniach dziesi\u0105tki dzieci zdiagnozowanych w okresie bezobjawowym i uchronionych przed ci\u0119\u017ckimi uszkodzeniami kt\u00f3rych, je\u015bli ju\u017c zaistniej\u0105, nie da si\u0119 odwr\u00f3ci\u0107. To wszystko zacz\u0119\u0142o si\u0119 w czasach prza\u015bnych, niepor\u00f3wnywalnych do tego, co jest naszym dzisiejszym do\u015bwiadczeniem. Nie istnia\u0142y granty badawcze ani stypendia naukowe, ale Pani Profesor potra+\u0142a uzyska\u0107 dla Instytutu wsp\u00f3\u0142prac\u0119 ameryka\u0144sk\u0105, kt\u00f3ra umo\u017cliwi\u0142a zacz\u0105tki pediatrii metabolicznej w Polsce.Pani Profesor Bo\u017ckowa potra+\u0142a wydoby\u0107 z doniesie\u0144 \u015bwiatowych to, co budowa\u0142o nowe trendy w ramach medycyny wieku rozwojowego i zainteresowa\u0107 nimi swoich wsp\u00f3\u0142pracownik\u00f3w. Szczeg\u00f3lnie interesowa\u0142a si\u0119 tym, co w r\u00f3\u017cnych specjalno\u015bciach medycznych dotyczy\u0142o mukowiscydozy. Cieszy\u0142a si\u0119 ogromnie tym, \u017ce wreszcie w Polsce, w Instytucie Matki i Dziecka powsta\u0142 o\u015brodek leczenia tej choroby na miar\u0119 XXI wieku.Wielki dorobek dydaktyczny Pani Profesor wyra\u017ca si\u0119 szkoleniem specjalistycznym ordynator\u00f3w oddzia\u0142\u00f3w pediatrycznych w kraju oraz lekarzy z zagranicy na mi\u0119dzynarodowych kursach organizowanych w IMiD we wsp\u00f3\u0142pracy ze \u015awiatow\u0105 Organizacj\u0105 Zdrowia. Oczami m\u0142odego w\u00f3wczas lekarza patrzy\u0142am na goszcz\u0105cych w Instytucie pracownik\u00f3w ochrony zdrowia z r\u00f3\u017cnych stron \u015bwiata, tak\u017ce z Afryki, w tradycyjnie kolorowych sukniach i zawojach na g\u0142owach. Na tle naszej szarej w\u00f3wczas rzeczywisto\u015bci wygl\u0105dali jak rajskie ptaki. Profesor Bo\u017ckowa stworzy\u0142a szko\u0142\u0119 pediatryczn\u0105, specjalizuj\u0105c wielu lekarzy, kieruj\u0105c licznymi przewodami doktorskimi i habilitacyjnymi. Cech\u0105 tej szko\u0142y by\u0142 wysoki poziom fachowy (zawsze by\u0142a bardzo wymagaj\u0105cym szefem i nie uznawa\u0142a taryfy ulgowej bez wzgl\u0119du na okoliczno\u015bci), umiej\u0119tno\u015b\u0107 pracy zespo\u0142owej  oraz humanistyczny typ lekarza spo-\u0142ecznika . Nieustannie dopingowa\u0142a nas lekarzy w trakcie specjalizacji i m\u0142odych doktorant\u00f3w do wyt\u0119\u017conej pracy: \u201eprzecie\u017c poprzestanie na badaniu pacjent\u00f3w i wydawaniu zalece\u0144 to zbyt ma\u0142o, to wkr\u00f3tce ka\u017cdego znudzi. Trzeba analizowa\u0107 to, z czego buduje si\u0119 w\u0142asne do\u015bwiadczenie i pisa\u0107, pisa\u0107, pisa\u0107. Dop\u00f3ki nie napiszesz, my\u015blisz, \u017ce wszystko wiesz, a dopiero jak zaczniesz pisa\u0107, oka\u017ce si\u0119, ile tu jeszcze znak\u00f3w zapytania pozosta\u0142o do rozstrzygni\u0119cia. A jak czasem uda ci si\u0119 wymy\u015ble\u0107 lub odnale\u017a\u0107 co\u015b naprawd\u0119 oryginalnego, poczujesz rado\u015b\u0107 tw\u00f3rcy i prawdziwego badacza\u201d. I jeszcze jedno wspomnienie: kiedy kto\u015b narzeka\u0142, \u017ce z tym lub z tamtym profesorem trudno si\u0119 porozumie\u0107, Pani Profesor mawia\u0142a: \u201ewszyscy profesorowie s\u0105 trudnymi lud\u017ami, ka\u017cdy na sw\u00f3j spos\u00f3b, bo \u017ceby dobrn\u0105\u0107 do nominacji profesorskiej wr\u0119czanej przez Prezydenta RP trzeba pokona\u0107 tyle trudno\u015bci, wykaza\u0107 taki hart ducha, \u017ce to nie mo\u017ce pozosta\u0107 bez \u015bladu\u201d.Pani Profesor Bo\u017ckowa by\u0142a koordynatorem Centralnego Programu Badawczo-Rozwojowego \u201eOchrona zdrowia matki, dziecka i rodziny\u201d i koordynatorem mi\u0119dzynarodowym w zakresie pediatrii kraj\u00f3w socjalistycznych. Jako specjalista krajowy w zakresie ochrony zdrowia dzieci i m\u0142odzie\u017cy, a tak\u017ce jako dyrektor Instytutu by\u0142a Profesor Bo\u017ckowa niestrudzonym adwokatem dzieci w zabezpieczaniu ich potrzeb zdrowotnych i spo\u0142ecznych. Pe\u0142ni\u0142a wiele funkcji w licznych towarzystwach naukowych. By\u0142a ekspertem \u015awiatowej Organizacji Zdrowia i cz\u0142onkiem honorowym wielu towarzystw naukowych krajowych i zagranicznych, cz\u0142onkiem Komitet\u00f3w i Komisji PAN, wielu Rad Naukowych. By\u0142a aktywnym dzia\u0142aczem spo\u0142ecznym \u2013 m.in. w Rz\u0105dowej Komisji Ludno\u015bciowej, w Krajowej Radzie Kobiet.Developmental Period Medicine/Medycyny Wieku Rozwojowego, czasopisma, kt\u00f3re jest kontynuacj\u0105 poprzednich czasopism Instytutu Matki i Dziecka.Od 1996 roku Pani Profesor Bo\u017ckowa by\u0142a Redaktorem Naczelnym Dzi\u0119ki doskona\u0142ej znajomo\u015bci warsztatu naukowego Pani Profesor inspirowa\u0142a autor\u00f3w do zaj\u0119cia si\u0119 najnowszymi tematami z zakresu medycyny wieku rozwojowego. Szczeg\u00f3lnie ceni\u0142a badania interdyscyplinarne oraz podkre\u015bla\u0142a znaczenie publikowania ich rezultat\u00f3w.Pochyla\u0142a si\u0119 nad ka\u017cd\u0105 prac\u0105, wk\u0142adaj\u0105c wiele wysi\u0142ku by by\u0142a doskona\u0142a zar\u00f3wno merytorycznie jak i j\u0119zykowo. Wszystko to przyczyni\u0142o si\u0119 do tego, \u017ce czasopismo jest jednym z bardziej licz\u0105cych si\u0119 w Polsce.Developmental Period Medicine/Medycyna Wieku Rozwojowego. Ci\u0105gle zadziwia\u0142a nas bystro\u015bci\u0105 swego umys\u0142u i gotowo\u015bci\u0105 dyskusji o nauce, jej sukcesach i zagro\u017ceniach. Potrafi\u0142a zara\u017ca\u0107 nas swoim entuzjazmem i ch\u0119ci\u0105 odkrywania tego, co jeszcze przed nami. Tym wszystkim sprawi\u0142a, \u017ce dzi\u015b czujemy si\u0119 osieroceni.P\u00f3\u0142tora roku temu, z okazji 65. rocznicy istnienia Instytutu Matki i Dziecka jednym z punkt\u00f3w programu by\u0142a laudacja, podzi\u0119kowanie Pani Profesor Bo\u017ckowej za lata Jej owocnej pracy na rzecz polskiej pediatrii. Dzi\u015b my\u015bl\u0119, jak dobrze si\u0119 sta\u0142o, \u017ce zdo\u0142ali\u015bmy to zrobi\u0107 wtedy, gdy Pani Profesor by\u0142a jeszcze z nami. W ostatnich latach spotykali\u015bmy Pani\u0105 Profesor na posiedzeniach Rady Naukowej i w pokoju redakcyjnym czasopisma"}
{"text": "In our published article, there was an error in the unit of measurement on pages 4, 5, and 15, which was mainly about the dosage of agonist peptides and which should be \u201c\u03bcM\u201d rather than \u201cmM.\u201d The specific changes are as follows:.1In line 6 of paragraph 4 on page 4 of the text, 10\u2009mM needs to be changed to 10\u2009\u03bcM..2The dosage of agonist peptides in .3The 10\u2009mM in lines 4 and 8 of the legend in .4On page 15, the 10\u2009mM in line 8 of the description of the method for enzyme activity analysis needs to be changed to 10\u2009\u03bcM, and \u201cagonist peptide concentration gradients of 0.3125\u2009mM, 0.625\u2009mM, 1.25\u2009mM, 2.5\u2009mM, 5\u2009mM, 10\u2009mM, 20\u2009mM, and 40\u2009mM\u201d in line 20 needs to be changed to \u201c0.3125\u2009\u03bcM, 0.625\u2009\u03bcM, 1.25\u2009\u03bcM, 2.5\u2009\u03bcM, 5\u2009\u03bcM, 10\u2009\u03bcM, 20\u2009\u03bcM, and 40\u2009\u03bcM.\u201dVolume 12, no. 1, e03299-20, 2021,"}
{"text": "Pod\u0142o\u017cem fenyloketonurii (PKU) jest dziedziczony autosomalnie recesywnie niedob\u00f3r hydroksylazy fenyloalaniny, kt\u00f3ry powoduje nadmierne gromadzenie si\u0119 fenyloalaniny (Phe) we krwi i m\u00f3zgowiu. Ograniczenie poda\u017cy Phe w diecie ma na celu umo\u017cliwienie utrzymywania si\u0119 st\u0119\u017cenia tego aminokwasu we krwi w rekomendowanym przedziale 120-360 \u03bcmol/L. W praktyce klinicznej ocena tolerancji Phe opiera si\u0119 na cz\u0119stych oznaczeniach st\u0119\u017cenia aminokwasu w powi\u0105zaniu z analiz\u0105 jad\u0142ospis\u00f3w. Nieleczona PKU u kobiety w wieku prokreacyjnym mo\u017ce prowadzi\u0107 do rozwoju zespo\u0142u fenyloketonurii matczynej u potomstwa.by\u0142a analiza wp\u0142ywu ci\u0105\u017cy wielop\u0142odowej na tolerancj\u0119 fenyloalaniny, wykorzystuj\u0105ca obserwacje ci\u0105\u017c pojedynczych i bli\u017aniaczych u kobiet chorych na PKU stosuj\u0105cych diet\u0119 niskofenyloalaninow\u0105.Retrospektywnie analizowano ci\u0105\u017c\u0119 pojedyncz\u0105 i bli\u017aniacz\u0105 u 3 chorych na PKU, stosuj\u0105cych diet\u0119 niskofenyloalaninow\u0105. Wszystkie pacjentki by\u0142y obj\u0119te opiek\u0105 wyspecjalizowanego dietetyka oceniaj\u0105cego tolerancj\u0119 Phe. Analizowano wiek ci\u0105\u017cowy wprowadzenia diety, przyrost masy cia\u0142a w czasie ci\u0105\u017cy, odsetek oznacze\u0144 Phe poza zakresem referencyjnym oraz pomiary noworodk\u00f3w, z kt\u00f3rych \u017caden nie chorowa\u0142 na PKU.Ca\u0142kowity wzrost tolerancji Phe i jego wzorzec w ci\u0105\u017cach pojedynczych i mnogich r\u00f3\u017cni\u0142 si\u0119 istotnie u ka\u017cdej z pacjentek. W ci\u0105\u017cach pojedynczych i mnogich tolerancja Phe wzros\u0142a o 579%/468%, 674%/261% i 427%/236% u chorych z genotypem Q383X/R408W, EX3DEL/EX3DEL, R281L/R408W. W ostatnich 10 tygodniach ci\u0105\u017cy wzrost tolerancji Phe si\u0119ga\u0142 odpowiednio 62%/149%, 33%/64% i 37%/40%. Stopie\u0144 zwi\u0119kszenia masy cia\u0142a ci\u0119\u017carnej oraz p\u0142odu nie umo\u017cliwia\u0142 przewidywania zmian tolerancji Phe.Poznanie tolerancji Phe w ci\u0105\u017cy pojedynczej u chorej na PKU nie by\u0142o pomocne w prognozowaniu dopuszczalnej poda\u017cy tego aminokwasu w ci\u0105\u017cy bli\u017aniaczej. Niezb\u0119dne s\u0105 dalsze badania nad metabolizmem Phe w ci\u0105\u017cy w celu opracowania zalece\u0144 u\u0142atwiaj\u0105cych kompleksow\u0105 opiek\u0119 nad chorymi na PKU w wieku rozrodczym. PAH, powoduj\u0105cych zmniejszenie aktywno\u015bci hydroksylazy fenyloalaninowej (PAH), kt\u00f3ra, g\u0142\u00f3wnie w w\u0105trobie, przekszta\u0142ca aminokwas fenyloalanin\u0119 (Phe) w tyrozyn\u0119. Trwa\u0142emu uszkodzeniu o\u015brodkowego uk\u0142adu nerwowego i innym objawom PKU mo\u017cna skutecznie zapobiega\u0107 poprzez wczesne zastosowanie diety niskofenyloalaninowej [Fenyloketonuria klasyczna OMIM 261600; PKU) jest najcz\u0119\u015bciej wyst\u0119puj\u0105cym wrodzonym zaburzeniem metabolizmu aminokwasowo-bia\u0142kowego [600; PKU aninowej , 4 jako aninowej , co zbioCi\u0105\u017ce wielop\u0142odowe, w praktyce w wi\u0119kszo\u015bci bli\u017aniacze 97-98%), wyst\u0119puj\u0105 coraz cz\u0119\u015bciej, co przede wszystkim wi\u0105\u017ce si\u0119 ze stosowaniem inwazyjnych technik wspomaganego rozrodu oraz wzrastaj\u0105cym wiekiem kobiet zachodz\u0105cych w ci\u0105\u017c\u0119. W Europie na prze\u0142omie XIX i XX wieku cz\u0119sto\u015b\u0107 rodzenia si\u0119 bli\u017ani\u0105t wynosi\u0142a 14-15/1000 porod\u00f3w i zmala\u0142a do ok. 9-10/1000 porod\u00f3w w latach 70-tych XX wieku, koreluj\u0105c z urbanizacj\u0105 i obni\u017caniem si\u0119 \u015bredniego wieku matek w coraz mniej dzietnych rodzinach. Ostatnie lata charakteryzuje szybki wzrost cz\u0119sto\u015bci ci\u0105\u017c bli\u017aniaczych do ponad 15/1000 porod\u00f3w (>3% noworodk\u00f3w stanowi\u0105 bli\u017ani\u0119ta). Ci\u0105\u017ca mnoga stanowi czynnik ryzyka hipotro' i p\u0142odu i powik\u0142a\u0144 oko\u0142oporodowych [-98%, wysPodobnie, jak w populacji og\u00f3lnej ci\u0105\u017ce mnogie stanowi\u0105 \u017ar\u00f3d\u0142o problem\u00f3w medycznych, a w cz\u0119\u015bci i etycznych, tak\u017ce w rodzinach os\u00f3b chorych na PKU. Znane s\u0105 przypadki selektywnej aborcji jednego bli\u017ani\u0119cia po ustaleniu rozpoznania PKU . BadaniaPAH por\u00f3wnuj\u0105c rodze\u0144stwa, w tym zar\u00f3wno bli\u017ani\u0119ta dwujajowe [Badacze zajmuj\u0105cy si\u0119 PKU wielokrotnie wykorzystywali model bli\u017ani\u0105t , 18, 19.wujajowe , jak i jwujajowe , 21. W 1wujajowe  przedstawujajowe . Niekiedwujajowe .Przedstawiano r\u00f3\u017cny przebieg kliniczny ci\u0105\u017c i laktacji u jednojajowych bli\u017aniaczek chorych na PKU , 26. W 1PAH, jak i z dwoma zmutowanymi allelami. U kobiet chorych na PKU, leczonych diet\u0105 niskofenyloalaninow\u0105, obserwuje si\u0119 zazwyczaj istotny wzrost tolerancji Phe w ci\u0105\u017cy [PAH a jeden p\u0142\u00f3d by\u0142 homozygot\u0105 PAH R408W. W ci\u0105\u017cach z heterozygotycznym p\u0142odem udokumentowano systematyczny du\u017cy wzrost tolerancji Phe, natomiast ci\u0105\u017c\u0119 z homozygot\u0105 mutacji PAH charakteryzowa\u0142y trudno\u015bci w dopasowaniu diety do zapotrzebowania bia\u0142kowo-kalorycznego, co skutkowa\u0142o m.in. du\u017cym przyrostem masy cia\u0142a ci\u0119\u017carnej. Pr\u00f3by zwi\u0119kszenia poda\u017cy newralgicznego aminokwasu bezwarunkowo skutkowa\u0142y przekroczeniami g\u00f3rnej granicy dopuszczalnego st\u0119\u017cenia Phe we krwi [W 1984 r. Levy i wsp.  postulow w ci\u0105\u017cy . W 2008r w ci\u0105\u017cy  przedsta we krwi . Na tej Celem pracy by\u0142a analiza wp\u0142ywu ci\u0105\u017cy wielop\u0142odowej na tolerancj\u0119 fenyloalaniny, wykorzystuj\u0105ca obserwacje ci\u0105\u017c pojedynczych i bli\u017aniaczych u kobiet chorych na PKU stosuj\u0105cych diet\u0119 niskofenyloalaninow\u0105.Przeprowadzono retrospektywn\u0105 analiz\u0119 dokumentacji medycznej kobiet chorych na PKU z wywiadem ci\u0105\u017cy mnogiej i pojedynczej, podczas kt\u00f3rych leczono je diet\u0105 niskofenyloalaninow\u0105 w Poradni Fenyloketonurii przy Klinice Pediatrii IMiD. Zidenty' kowano 3 pacjentki, posiadaj\u0105ce potomstwo z jednej ci\u0105\u017cy bli\u017aniaczej i jednej pojedynczej. W momencie zaj\u015bcia w ci\u0105\u017c\u0119 wiek kobiet mie\u015bci\u0142 si\u0119 w przedziale 24-32 lata.Analizowano tolerancj\u0119 Phew okresie prekoncepcyjnym oraz w ci\u0105\u017cy w zale\u017cno\u015bci od liczby p\u0142od\u00f3w, masy cia\u0142a i jej przyrostu u ci\u0119\u017carnej, wska\u017anika masy cia\u0142a (BMI) ci\u0119\u017carnej, masy urodzeniowej potomstwa i estymowanych jej zmian w okresie \u017cycia p\u0142odowego z wykorzystaniem siatek centylowych dla dzieci z ci\u0105\u017c pojedynczych i bli\u017aniaczych , 37. ZmiZgodnie z pi\u015bmiennictwem rozbie\u017cno\u015b\u0107 co najmniej 15% uznano za istotn\u0105 klinicznie , 39.U wszystkich trzech chorych na PKU pierwsza ci\u0105\u017ca by\u0142a pojedyncza  do maksymalnie 1338 mg , ryc. 1, 2 i 3. Zaawansowaniu ci\u0105\u017cy towarzyszy\u0142 wzrost tolerancji Phe w stosunku do masy cia\u0142a chorych na PKU  i IV. U U ka\u017cdej chorej stwierdzono wi\u0119kszy wzrost tolerancji Phe w ci\u0105\u017cy pojedynczej ni\u017c bli\u017aniaczej zar\u00f3wno przez pierwsze 34 tygodnie (cezura przyj\u0119ta w zwi\u0105zku z zako\u0144czeniem si\u0119 ci\u0105\u017cy bli\u017aniaczej pacjentki 1 w 35 tygodniu i dost\u0119pno\u015bci\u0105 ostatnich oblicze\u0144 tolerancji Phe z przedostatniego tygodnia) jak i przez ca\u0142y czas jej trwania .Wielko\u015b\u0107 wzrostu Phe przez ca\u0142\u0105 ci\u0105\u017c\u0119 pojedyncz\u0105 a tak\u017ce jej zmiany w okresie bezpo\u015brednio poprzedzaj\u0105cym rozwi\u0105zanie, czy ich powi\u0105zanie z mas\u0105 cia\u0142a ci\u0119\u017carnej oraz estymowan\u0105 mas\u0105 p\u0142odu, nie umo\u017cliwia\u0142y przewidywania ilo\u015bciowej tolerancji Phe w ci\u0105\u017cy bli\u017aniaczej. Postulowanego, przez badaczy niemieckich, proporcjonalnego udzia\u0142u w\u0105troby p\u0142odu, bez defektu metabolizmu, w zwi\u0119kszaniu tolerancji Phe  nie potwMniejszy w por\u00f3wnaniu z ci\u0105\u017c\u0105 pojedyncz\u0105 wzrost tolerancji Phe w ci\u0105\u017cy bli\u017aniaczej u kobiet chorych na PKU jest zaskakuj\u0105cy. Dla wyja\u015bnienia tego zjawiska niezb\u0119dne b\u0119dzie odst\u0105pienie od praktyki wy\u0142\u0105czania pacjentek z ci\u0105\u017c\u0105 mnog\u0105 z prac obserwacyjnych po\u015bwi\u0119conych zmianom tolerancji Phe u chorych w wieku prokreacyjnym ,33, co pPAH od ojca, b\u0119d\u0105cego bezobjawowym nosicielem, wynosi tylko ok. 1:120 [W 1977 r. na podstawie przegl\u0105du pi\u015bmiennictwa Pueschel i wsp.  sugerowak. 1:120  i ich pok. 1:120 , jednak\u017ck. 1:120 , 55. W nk. 1:120 . Urodzenk. 1:120 , 53.PAH. U dw\u00f3ch z 3 pacjentek wyst\u0119powa\u0142a ponad 15% procentowa r\u00f3\u017cnica masy cia\u0142a przed ci\u0105\u017c\u0105 pojedyncz\u0105 i bli\u017aniacz\u0105 a przypuszczalnie oddzia\u0142ywanie na p\u0142\u00f3d przyrostu masy cia\u0142a kobiety w ci\u0105\u017cy zale\u017cy od jej przedci\u0105\u017cowego BMI [Istotn\u0105 niedoskona\u0142o\u015bci\u0105 naszej pracy by\u0142o obj\u0119cie analiz\u0105 ci\u0105\u017c, w kt\u00f3rych diety niskofenyloalaninowej nie wprowadzono prekoncepcyjnie, co stanowi aktualnie og\u00f3lnie przyj\u0119t\u0105 rekomendacj\u0119 , 6. Popuwego BMI , 58. Pozwego BMI , 43.Jako zalety pracy wymagaj\u0105 podkre\u015blenia homogenno\u015b\u0107 etniczna pacjentek, niestosowanie technik wspomaganego rozrodu oraz niewyst\u0119powanie ekspozycji na dym tytoniowy, jak i nadzorowanie diety niskofenyloalaninowej przez tego samego dietetyka we wszystkich ci\u0105\u017cach.Poznanie tolerancji Phe w ci\u0105\u017cy pojedynczej u chorej na PKU i jej zale\u017cno\u015bci od wzrostu masy cia\u0142a matki i p\u0142odu nie umo\u017cliwia satysfakcjonuj\u0105cego prognozowania dopuszczalnej poda\u017cy tego aminokwasu w ci\u0105\u017cy bli\u017aniaczej. Opracowanie zalece\u0144 u\u0142atwiaj\u0105cych kompleksow\u0105 opiek\u0119 nad chorymi na PKU w wieku rozrodczym wymaga dalszych bada\u0144.Wszystkie oznaczenia st\u0119\u017cenia fenyloalaniny wykonano w Zak\u0142adzie Bada\u0144 Przesiewowych i Diagnostyki Metabolicznej Instytutu Matki i Dziecka, Kierownik: dr Mariusz O\u0142tarzewski. W dotarciu do archiwalnych pozycji pi\u015bmiennictwa nieocenionej pomocy udzielili bibliotekarze mgr Ewa Plewko (Biblioteka Naukowa Instytutu Matki i Dziecka) i mgr Grzegorz \u015awi\u0119\u0107kowski (Biblioteka Naukowa Narodowego Instytutu Zdrowia Publicznego \u2013 Pa\u0144stwowego Zak\u0142adu Higieny)."}
{"text": "Much less investigated has been a potential function for the distribution of F-actin plus and minus ends. In syncytial Drosophila embryos, Rho1 signaling is high between actin caps, i.e. the cortical intercap region. Capping protein binds to free plus ends of F-actin to prevent elongation of the filament. Capping protein has served as a marker to visualize the distribution of F-actin plus ends in cells and in vitro. In the present study, we probed the distribution of plus ends with capping protein in syncytial Drosophila embryos. We found that capping proteins are specifically enriched in the intercap region similar to Dia and MyoII but distinct from overall F-actin. The intercap enrichment of Capping protein was impaired in dia mutants and embryos, in which Rok and MyoII activation was inhibited. Our observations reveal that Dia and Rok-MyoII control Capping protein enrichment and support a model that Dia and Rok-MyoII control the organization of cortical actin cytoskeleton downstream of Rho1 signaling.Rho signaling with its major targets the formin Dia, Rho kinase (Rok) and non-muscle myosin II (MyoII, encoded by This article has an associated First Person interview with the first authors of the paper. Summary: Plus ends of actin filaments are enriched at cortical regions rich in Rho signaling in syncytial Drosophila embryos depending on the actin regulator Dia and Rho kinase. Underlying the plasma membrane of cells, a thin layer of cortical F-actin fulfills manifold functions such as playing a central role in cell polarity, providing a scaffold for membrane associated proteins, and ensuring mechanical stiffness, among others. Cortical F-actin contains F-actin regulators, Myosin motor proteins and other cortical proteins. Linker proteins belonging to the ERM protein family mediate attachment of cortical F-actin to the plasma membrane. The F-actin cortex contributes to segregation of proteins and establishment of cortical domains, including polarity proteins that are crucial for formation and maintenance of cell polarity .Drosophila development. Following a uniform cortex in preblastoderm stages, the cortex becomes patterned with the arrival of the nuclei at the cortex (blastoderm stages). During interphases, two cortical domains mark the cap region above the nuclei and intercap region in between caps  slides towards the plus ends of actin filaments  specifically binds as a dimer to plus ends, where it catalyzes the incorporation of G-actin to promote filament elongation . Similarilaments  and induin vitro . The Troin vitro .Dia and MyoII  are both controlled by Rho signaling. In early embryos, Dia is involved in the formation of pole cells, the metaphase furrow and the cellularization furrow, and together with Cip4 in the exclusion of lateral markers from the furrow canal .Drosophila embryos with the distribution of the heterodimeric capping protein and test the function of two targets of Rho signaling, Dia and MyoII, in distribution of capping proteins.Unlike actomyosin in muscle cells, the F-actin cortex is generally assumed to consist of a random meshwork of filaments with uniformly distributed plus ends and plus end binding proteins. In contrast, it is also expected that spatially patterned Rho1 signaling will impinge on the structure, dynamics and organization of the F-actin cortex. These contrasting views have not been much tested in a physiological context. In the present study, we compare the spatial pattern of Rho1 signaling in early Drosophila embryos is structured into two regions during interphase: actin caps and the region between actin caps, namely the intercap region . This finding is consistent with the intercap enrichment of MyoII as previously reported . We labeled Dia by inserting a GFP tag before the stop codon by CRISPR-mediated integration at the endogenous dia locus . The Dia-GFP construct is fully functional, since we employed a homozygous stock, in which Dia-GFP completely substitutes Dia. It is worth noting that the pattern is rather uniform after Dia immunostaining in fixed embryos . The staining pattern of Dia-GFP depends on formin activity, since the intercap restriction was strongly reduced after treatment with the formin inhibitor SMIFH2, whereas overall F-actin caps were not affected by drug treatment .Rho signaling controls F-actin in two ways, at least \u2013 nucleation and elongation via Dia and contractility via Rok and MyoII A. The cop region B. Both rp region . For exap region . Using ap region , we analreported  and of DFig.\u00a0S3)  . FurtherFig.\u00a0S3) . In summHaving identified a restriction of Rho signaling and two of its targets to the intercap region, we analyzed markers for the organization of cortical F-actin. We found that the plus-end binding proteins Cpa and Cpb as well as the minus-end binding protein Tmod were enriched at intercaps.Fig.\u00a0S4C). The distinct distribution pattern of Cpa and overall F-actin became also obvious in axial sectioning of frontal views .Cpa immunostaining revealed a strong enrichment outside of actin caps in intercaps during syncytial blastoderm interphases distinct from overall F-actin distribution A. A simial views B. StainiFigs\u00a0S4G and S5). In frontal views, we observed an enrichment of the CpaGFP puncta in regions of basal sections outside of actin caps beyond a diffuse staining of caps and intercaps overlapping with the F-actin label Moesin-GFP. The puncta appeared more prominent than in fixed embryos . The GFP or Cherry tagging did not affect subcellular distribution, since the staining pattern of fixed wild-type and CpaGFP embryos was comparable . The Cpa puncta were mobile, with a distribution of their velocities in the range of a few \u00b5m/min . We assayed Tmod distribution in living embryos with a functional GFP fusion protein, expressed from the endogenous locus . Strikingly, Cpa was almost uniformly distributed at the cortex in dia mutants, including actin caps as compared to the sharp lines representing the intercap regions  . The overall MyoII levels and restriction to the intercap region appeared comparable between wild type and dia mutants. Yet, the staining in the submicron scale appeared grainier, which could be caused by a partially disrupted and less continuous actomyosin network in the intercap regions . We conclude that Cpa depends on both targets of Rho signaling: Rok-MyoII activity and Dia.As Rho signaling controls both Dia and MyoII in parallel, we hypothesized that MyoII may also control Cpa enrichment at intercaps. We indirectly inhibited MyoII by injecting the Rho kinase inhibitor Y-27632 (ig.\u00a0S4A) . We cannig.\u00a0S4A) A, reminihibition B,C. Dia hibition . We test regions D,E  with Utrophin-GFP, which stably binds to F-actin allowing assay of the exchange kinetics , CpaGFP, Cpa-mCherry and Dia-GFP (the present study), sy5dia , MyoII3xGFP (Drosophila Stock Center), and utrophin-GFP (Table S1).Fly stocks used were , diasy5 , Histone, diasy5 , Moesin-oII3xGFP , Rho1 seoII3xGFP , Tmod-GFphin-GFP  . The coding sequence of eGFP or mCherry with a 5\u2032-terminal linker sequence was inserted into the 3\u2032 region of the target gene.LINKER, egfp; UTR): 5\u2032-CCAGTTGCCCATCACCAGGACCAAGATCGACTGGAGCAAAATCGTCTCGTACAGCATTGGCAAGGAACTGAAGACGCAAGGCGTGGGCatggtgagcaagggcgaggagctgttcaccggggtggtgcccatcctggtcgagctggacggcgacgtaaacggccacaagttcagcgtgtccggcgagggcgagggcgatgccacctacggcaagctgaccctgaagttcatctgcaccaccggcaagctgcccgtgccctggcccaccctcgtgaccaccctgacctacggcgtgcagtgcttcagccgctaccccgaccacatgaagcagcacgacttcttcaagtccgccatgcccgaaggctacgtccaggagcgcaccatcttcttcaaggacgacggcaactacaagacccgcgccgaggtgaagttcgagggcgacaccctggtgaaccgcatcgagctgaagggcatcgacttcaaggaggacggcaacatcctggggcacaagctggagtacaactacaacagccacaacgtctatatcatggccgacaagcagaagaacggcatcaaggtgaacttcaagatccgccacaacatcgaggacggcagcgtgcagctcgccgaccactaccagcagaacacccccatcggcgacggccccgtgctgctgcccgacaaccactacctgagcacccagtccgccctgagcaaagaccccaacgagaagcgcgatcacatggtcctgctggagttcgtgaccgccgccgggatcactctcggcatggacgagctgtataagTAAGACCAGGAGGTGCAAGCGGAAGAGGAGCGGACACAGTCGACAAAGCGGCCCAATAGTCTTCCACTTGCCATGGCCAAGCGAGCATAGGAACCGATCAC-3\u2032.The gene region of CpaGFP is as follows : 5\u2032-CGACATTCAAGGCAATGCGTCGCCAGTTGCCCATCACCAGGACCAAGATCGACTGGAGCAAAATCGTCTCGTACAGCATTGGCAAAGAACTGAAGACGCAATCTAGAatggtgagcaagggcgaggaggataacatggccatcatcaaggagttcatgcgcttcaaggtgcacatggagggctccgtgaacggccacgagttcgagatcgagggcgagggcgagggccgcccctacgagggcacccagaccgccaagctgaaggtgaccaagggtggccccctgcccttcgcctgggacatcctgtcccctcagttcatgtacggctccaaggcctacgtgaagcaccccgccgacatccccgactacttgaagctgtccttccccgagggcttcaagtgggagcgcgtgatgaacttcgaggacggcggcgtggtgaccgtgacccaggactcctccctgcaggacggcgagttcatctacaaggtgaagctgcgcggcaccaacttcccctccgacggccccgtaatgcagaagaagaccatgggctgggaggcctcctccgagcggatgtaccccgaggacggcgccctgaagggcgagatcaagcagaggctgaagctgaaggacggcggccactacgacgctgaggtcaagaccacctacaaggccaagaagcccgtgcagctgcccggcgcctacaacgtcaacatcaagttggacatcacctcccacaacgaggactacaccatcgtggaacagtacgaacgcgccgagggccgccactccaccggcggcatggacgagctgtacaagtagCTGCAGATAACTTCGTATAATGTATGCTATACGAAGTTATGCTAGC-3\u2032.Cpa-mCherry : 5\u2032-CGGACGCGTGTCACCAACGGACAACTAATGACCCGCGAAATGATCCTCAACGAGGTTCTAGGCTCCGCGTCTAGAatggtgagcaagggcgaggagctgttcaccggggtggtgcccatcctggtcgagctggacggcgacgtaaacggccacaagttcagcgtgtccggcgagggcgagggcgatgccacctacggcaagctgaccctgaagttcatctgcaccaccggcaagctgcccgtgccctggcccaccctcgtgaccaccctgacctacggcgtgcagtgcttcagccgctaccccgaccacatgaagcagcacgacttcttcaagtccgccatgcccgaaggctacgtccaggagcgcaccatcttcttcaaggacgacggcaactacaagacccgcgccgaggtgaagttcgagggcgacaccctggtgaaccgcatcgagctgaagggcatcgacttcaaggaggacggcaacagcctggggcacaagctggagtacaactacaacagccacaacgtctatatcatggccgacaagcagaagaacggcatcaaggtgaacttcaagatccgccacaacatcgaggacggcagcgtgcagctcgccgaccactaccagcagaacacccccatcggcgacggccccgtgctgctgcccgacaaccactacctgagcacccagtccgccctgagcaaagaccccaacgagaagcgcgatcacatggtcctgctggagttcgtgaccgccgccgggatcactctcggcatggacgagctgtacaagtaaCTGCAGATAACTTCGTATAATGTATGCTATACGAAGTTATGCTAGC-3\u2032.Dia-GFP ; rabbit anti-Cpa .The full length Cpa coding sequence was cloned into pGEX-His, leading to a GST domain at the N-terminus and a 6\u00d7 His tag at the C-terminus. The Cpa antibody was raised in rabbit against recombinant GST-Cpa-His6 protein expressed in For histological staining, embryos were fixed by formaldehyde or heat fixation using standard methods described previously . After fThe ROCK inhibitor Y-27632 (Sigma) was injected into embryos expressing CpaGFP during preblastoderm at a concentration of 10\u2005mM. Embryos were fixed after reaching syncytial blastoderm and stained as described above after hand-removal of the vitelline membrane.Table S2.Details on antibodies, stains and inhibitors are given in Embryo extracts were produced by freezing dechorionated embryos in liquid nitrogen. The frozen embryos were homogenized with a pestle in 2\u00d7 Laemmli buffer to generate an extract with 1.5\u2005embryos/\u00b5l. The extract was heated to 95\u00b0C for 10\u2005min and loaded onto a 10% SDS gel. Blotting onto a nitrocellulose membrane was performed for 1.5\u2005h and the membrane was then blocked for 1\u2005h in PBS+5% milk powder. Incubation with primary antibodies in PBT was performed overnight at 4\u00b0C followed by washing four times with PBT and incubation with the secondary antibodies for 2\u2005h at room temperature in PBT. The membrane was washed four times with PBT and then imaged using an Odyssey CLx infrared imaging system (LI-COR Biosciences) with 16-bit depth. Images were processed using Adobe Photoshop and Illustrator.dia germline clone embryos expressing CpaGFP were live imaged with a frame size of 488\u00d7488 pixels (32.16\u00d732.16\u2005\u00b5m), a z-stack step size of 0.5\u2005\u00b5m and a frame rate of 120\u2005s.Imaging was performed using a Zeiss LSM 780 confocal microscope equipped with Airyscan or LSM980 Airyscan 2. Fixed samples were imaged using an LCI Plan Neofluar 63\u00d7/water NA 1.3 objective and live imaging was performed with a Plan Neofluar 63\u00d7/oil NA 1.4 objective. Embryos for live imaging were handled as described previously . Embryosdia embryos (dia embryos. Three embryos per genotype were used for quantification.F-actin in fixed wild-type and  embryos  was quandia germline clone embryos expressing utrophin-GFP (t\u2212Imin)/(Imax\u2212Imin). Where It is the intensity at time t, Imin is the intensity immediately post-bleach and Imax is the intensity pre-bleach.FRAP experiments were conducted with wild-type and phin-GFP C,D. The Fig.\u00a0S4A). The fluorescence intensity of Dia-GFP in living embryos during interphase 13 with or without SMIFH2 treatment was measured with frames of 28.0\u00d71.67\u2005\u00b5m at the intercap region. The fluorescence intensities of Dia-GFP were normalized.The formin inhibitor SMIFH2 (Abcam) was dissolved in DMSO with a concentration of 2\u2005mM and injected into syncytial embryos expressing Dia-GFP  were collected and injected with the ROCK inhibitor Y-27632. Imaging was performed using a spinning disk microscope 20\u2005min after injection.Embryos expressing Dia-GFP or MyoII-GFP ("}
{"text": "In our lives, we cannot avoid the uncertainty. Randomness, rough knowledge, and vagueness lead us to uncertainty. In mathematics, the fuzzy set (FS) theory and logics are used to model uncertain events. This article defines a new concept of complex picture fuzzy relation (CPFR) in the field of FS theory. In addition, the types of CPFRs are also discussed to make the paper more fruitful. Today's complex network architecture faces the ever-changing threats. The cyber-attackers are always trying to discover, catch, and exploit the weaknesses in the networks. So, the security measures are essential to avoid and dismantle such threats. The CPFR has a vast structure composed of levels of membership, abstinence, and nonmembership which models uncertainty better than any other structures in the theory. Moreover, a CPFR has the ability to cope with multivariable problems. Therefore, this article proposes modeling techniques based on the complex picture fuzzy information which are used to study the effectiveness and ineffectiveness of different network securities against several threats and cyber-attack practices. Moreover, the strength and preeminence of the proposed methods are verified by studying their comparison with the existing methods. Uncertainty is an inevitable part of human life which has many causes ranging from just falling short of conviction to almost complete absence of awareness or belief. Probability measures the uncertainty due to randomness. Mostly mathematics deals with precise and accurate information. Modeling uncertainty has long been a crux for mathematicians. In 1965, fuzzy sets (FSs) and logics were presented by Zadeh  symbolizes the level of membership of D.Let Y be a universe and D be a collection of elements in Y. Then, D is called a complex fuzzy set (CFS) ifM(d) : Y\u27f6{z : \u2009z \u2208 \u2102, |z| \u2264 1} symbolizes the level of membership of D and z is a complex number. Another form of a CFS is\u03b1(d) : Y\u27f6 is called the amplitude value of level of membership, and \u03c1(d) : Y\u27f6 is called the phase value of level of membership.Let D={d, \u03b1(d)e\u03c1(d)\u03c0j2 : \u2009d \u2208 Y} and F={f, \u03b1(f)e\u03c1(f)\u03c0j2 : \u2009f \u2208 Y} are two CFSs in universe Y, then the Cartesian product between D and F is given as follows:\u03b1 : Y\u27f6 and \u03c1 : Y\u27f6 symbolize the amplitude value and phase value of the level of membership of the Cartesian product D \u00d7 F defined as follows:If R, i.e., R\u2286D \u00d7 F.Any subcollection of a Cartesian product between two CFSs is said to be a complex fuzzy relation (CFR) which is denoted by D={e\u03c0j2(1/2)), e\u03c0j2(2/3)), e\u03c0j2(3/5))} is a CFS, then the Cartesian product on D isIf R isThe CFR Y be a universe and D be a collection of elements in Y. Then, D is called an intuitionistic fuzzy set (IFS) if it is of the form as follows:m(d) : Y\u27f6 symbolizes the level of membership of D and n(d) : Y\u27f6 symbolizes the level of nonmembership of D such that 0 \u2264 m(d)+n(d) \u2264 1.Let Y be a universe and D be a collection of elements in Y. Then, D is called a complex intuitionistic fuzzy set (CIFS) ifM(d) : Y\u27f6{z1 : \u2009z1 \u2208 \u2102, |z1| \u2264 1} symbolizes the level of membership of D, N(d) : Y\u27f6{z2 : \u2009z2 \u2208 \u2102, |z2| \u2264 1} symbolizes the level of nonmembership of D, and z1\u2009and\u2009z2 are complex numbers such that 0 \u2264 |z1|+|z2| \u2264 1. Another form of a CIFS is\u03b1M(d) : Y\u27f6 is called the amplitude value of level of membership, \u03c1M(d) : Y\u27f6 is called the phase value of level of membership, \u03b1N(d) : Y\u27f6 is called the amplitude value of level of nonmembership, and \u03c1N(d) : Y\u27f6 is called the phase value of level of nonmembership such that 0 \u2264 \u03b1M(d)+\u03b1N(d) \u2264 1 and 0 \u2264 \u03c1M(d)+\u03c1N(d) \u2264 1.Let Y be a universe and D be a collection of elements in Y. Then, D is called a picture fuzzy set (PFS) if it is of the formm(d) : Y\u27f6 symbolizes the level of membership of D, a(d) : Y\u27f6 symbolizes the level of abstinence of D, and n(d) : Y\u27f6 symbolizes the level of nonmembership of D such that 0 \u2264 m(d)+a(d)+n(d) \u2264 1.Let Y be a universe and D be a collection of elements in Y. Then, D is called a complex picture fuzzy set (CPFS) ifM(d) : Y\u27f6{z1 : \u2009z1 \u2208 \u2102, |z1| \u2264 1} symbolizes the level of membership of D, A(d) : Y\u27f6{z2 : z2 \u2208 \u2102, |z2| \u2264 1} symbolizes the level of abstinence of D, N(d) : Y\u27f6{z3 : \u2009z3 \u2208 \u2102, |z3| \u2264 1} symbolizes the level of nonmembership of D, and z1, z2, \u2009and\u2009z3 are complex numbers such that for any nonnegative integer q, 0 \u2264 |z1|+|z2|+|z3| \u2264 1. Another form of a CPFS is\u03b1M(d) : Y\u27f6 is called the amplitude value of level of membership, \u03c1M(d) : Y\u27f6 is called the phase value of level of membership, \u03b1A(d) : Y\u27f6 is called the amplitude value of level of abstinence, \u03c1A(d) : Y\u27f6 is called the phase value of level of abstinence, \u03b1N(d) : Y\u27f6 is called the amplitude value of level of nonmembership, and \u03c1N(d) : Y\u27f6 is called the phase value of level of nonmembership such that 0 \u2264 \u03b1M(d)+\u03b1A(d)+\u03b1N(d) \u2264 1 and 0 \u2264 \u03c1M(d)+\u03c1A(d)+\u03c1N(d) \u2264 1.Let In this section, the novel concepts of complex picture fuzzy relation (CPFR), Cartesian products between two CPFSs, and different types of CPFRs are introduced. Every definition is supported by an example.D={d, \u03b1M(d)e\u03c1M(d)\u03c0j2, \u03b1A(d)e\u03c1A(d)\u03c0j2, \u2009\u03b1N(d)e\u03c1N(d)\u03c0j2 : \u2009d \u2208 Y} and F={d, \u03b1M(f)e\u03c1M(f)\u03c0j2, \u03b1A(f)e\u03c1A(f)\u03c0j2, \u2009\u03b1N(f)e\u03c1N(f)\u03c0j2 : \u2009d \u2208 Y} are two CPFSs in universe Y, then the Cartesian product between D and F is given as follows:\u03b1M : Y\u27f6, \u03b1N : Y\u27f6, and \u2009\u03b1N : Y\u27f6 symbolize the amplitude values of levels of membership, abstinence, and nonmembership of the Cartesian product D \u00d7 F, respectively. \u03c1M : Y\u27f6, \u03c1N : Y\u27f6, and \u2009\u03c1N : Y\u27f6 symbolize the phase values of levels of membership, abstinence, and nonmembership of the Cartesian product D \u00d7 F, respectively. These values for D \u00d7 F are defined asIf R, i.e., R\u2286D \u00d7 F.Any subcollection of a Cartesian product of two CPFSs is said to be a complex fuzzy relation (CPFR) which is denoted by D \u00d7 D{, 1/3e\u03c0j2(1/20), 1/5e\u03c0j2(1/10), 1/3e\u03c0j2(1/4)), , 1/3e\u03c0j2(1/20), 1/5e\u03c0j2(1/10), 1/3e\u03c0j2(1/4)), , 2/7e\u03c0j2(1/20), 1/5e\u03c0j2(1/10), 1/3e\u03c0j2(1/4)), , 1/3e\u03c0j2(1/20), 1/5e\u03c0j2(1/10), 1/3e\u03c0j2(1/4)), , 2/5e\u03c0j2(1/5), 2/5e\u03c0j2(3/10), 2/10e\u03c0j2(1/5)), , 2/7e\u03c0j2(1/10), 3/10e\u03c0j2(3/10), 1/4e\u03c0j2(1/5)), , 2/7e\u03c0j2(1/10), 1/5e\u03c0j2(1/10), 1/3e\u03c0j2(1/4)), , 2/7e\u03c0j2(1/10), 3/10e\u03c0j2(3/10), 1/4e\u03c0j2(1/5)), , 2/7e\u03c0j2(1/10), 3/10e\u03c0j2(7/10), 1/4e\u03c0j2(1/6))}IfR isThe CPFR The converse of a CPFRR={, \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2 : \u2009 \u2208 R} is defined as R\u22121={, \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2 : \u2009 \u2208 R}.R given as follows:The converse of CPFR R is a complex picture reflexive fuzzy relation (CP-reflexive-FR) if \u2200e\u03c1M(d)\u03c0j2, \u03b1A(d)e\u03c1A(d)\u03c0j2, \u2009\u03b1N(d)e\u03c1N(d)\u03c0j2) \u2208 D implies , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R.A CPFR R is a CP-reflexive-FR on a CPFS D as follows:The relation R is a complex picture irreflexive fuzzy relation (CP-irreflexive-FR) if \u2200e\u03c1M(d)\u03c0j2, \u03b1A(d)e\u03c1A(d)\u03c0j2, \u2009\u03b1N(d)e\u03c1N(d)\u03c0j2) \u2208 D implies , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2209 R.A CPFR R is a CP-irreflexive-FR on a CPFS D given in equation (The following relation equation :(21)R=d,R is a complex picture symmetric fuzzy relation (CP-symmetric-FR) if , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R implies , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R.A CPFR R is a CP-symmetric-FR on a CPFS D given in equation (The following relation equation :(22)R=d,R is a complex picture asymmetric fuzzy relation (CP-asymmetric-FR) if , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R implies , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2209 R.A CPFR R is a CP-asymmetric-FR on a CPFS D given in equation (The following relation equation :(23)R=d,R is a complex picture antisymmetric fuzzy relation (CP-antisymmetric-FR) if , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R and , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R implyA CPFR R is a CP-antisymmetric-FR on a CPFS D given in equation (The following relation equation :(25)R=d,R is a complex picture transitive fuzzy relation (CP-transitive-FR) if , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R and , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R imply , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R.A CPFR R is a CP-transitive-FR on a CPFS D given in equation (The following relation equation :(26)R=d,R is a complex picture complete fuzzy relation (CP-complete-FR) if \u2200e\u03c1M(d)\u03c0j2, \u03b1A(d)e\u03c1A(d)\u03c0j2, \u2009\u03b1N(d)e\u03c1N(d)\u03c0j2) \u2208 D and e\u03c1M(f)\u03c0j2, \u03b1A(f)e\u03c1A(f)\u03c0j2, \u2009\u03b1N(f)e\u03c1N(f)\u03c0j2) \u2208 D imply , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R or , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R.A CPFR R is a CP-complete-FR on a CPFS D given in equation (The following relation equation :(27)R=d,R is a complex picture equivalence fuzzy relation  if R isCP-reflexive-FRCP-symmetric-FRCP-transitive-FRA CPFR R is a CP-equivalence-FR on a CPFS D given in equation (The following relation equation :(28)R=d,R is a complex picture preorder fuzzy relation (CP-preorder-FR) if R isCP-reflexive-FRCP-transitive-FRA CPFR R is a CP-preorder-FR on a CPFS D given in equation (The following relation equation :(29)R=d,R is a complex picture strict order fuzzy relation (CP-strict order-FR) if R isCP-irreflexive-FRCP-transitive-FRA CPFR R is a CP-strict order-FR on a CPFS D given in equation (The following relation equation :(30)R=d,R is a complex picture partial order fuzzy relation  if R isCP-reflexive-FRCP-antisymmetric-FRCP-transitive-FRA CPFR R is a CP-partial order-FR on a CPFS D given in equation (The following relation equation :(31)R=d,R is a complex picture linear order fuzzy relation (CP-linear order-FR) if R isCP-reflexive-FRCP-antisymmetric-FRCP-transitive-FRCP-complete-FRA CPFR R is a CP-antisymmetric-FR on a CPFS D given in equation (The following relation equation :(32)R=d,R1 and R2 is a complex picture composite fuzzy relation (CP-composite-FR) which is defined as follows:The composition of two CPFRs R is a CP-composite-FR between CPFRs R1 and R2:The following relation d modulo R is defined asd, \u03b1M(d)e\u03c1M(d)\u03c0j2, \u03b1A(d)e\u03c1A(d)\u03c0j2, \u2009\u03b1N(d)e\u03c1N(d)\u03c0j2) and a CP-equivalence-FR R.The equivalence class of R is CP-equivalence-FR on a CPFS D given in equation (The following relation equation :(37)R=d,D modulo R areThe equivalence classes of each element in This section presents some results and properties of CP-symmetric-FRs, CP-transitive-FRs, CP-composite-FRs, and CP-equivalence-FRs.R is a CP-symmetric-FR on a CPFS Diff\u2009R=R\u22121.A CPFR R is a CP-symmetric-FR on a CPFS D, thenAssume that f, d), \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R\u22121.However, , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R, and we have , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R\u22121 which implies that , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R.\u2200, \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R\u2218R, then there exists an element f \u2208 Y such that , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R and , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R which imply that Assume that R\u2218R\u2286R.Hence, R\u2218R\u2286R, then the composition of CPFRs implies thatConversely, assume that d, f), \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R and , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R, we haveFor , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R\u2218R, then there exists an element f in Y such thatConversely, assume that , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R and , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R. R is a CP-equivalence-FR. Therefore, the CP-transitive-FR R implies that49) proveR is a CP-equivalence-FR on a CPFS D, then , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 Riff\u2009R[d]=R[f].If R is a CP-equivalence-FR on a CPFS D. So, we can define the equivalence classes for R. Let R[d]=R[f], then for some g in Y,g, \u03b1M(g)e\u03c1M(g)\u03c0j2, \u03b1A(g)e\u03c1A(g)\u03c0j2, \u2009\u03b1N(g)e\u03c1N(g)\u03c0j2) \u2208 R[d] implies , \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R.52), we hConversely, assume thatg, \u03b1M(g)e\u03c1M(g)\u03c0j2, \u03b1A(g)e\u03c1A(g)\u03c0j2, \u2009\u03b1N(g)e\u03c1N(g)\u03c0j2) \u2208 R[f].Using the definition of CP-transitive-FR on equations  and 56)56), we hHence,Likewise, assume thatUsing the definition of CP-transitive-FR on equations  and 61)61), we hg, d), \u03b1Me\u03c1M\u03c0j2, \u03b1Ae\u03c1A\u03c0j2, \u2009\u03b1Ne\u03c1N\u03c0j2) \u2208 R, which implies that e\u03c1M(g)\u03c0j2, \u03b1A(g)e\u03c1A(g)\u03c0j2, \u2009\u03b1N(g)e\u03c1N(g)\u03c0j2) \u2208 R[d].21), the This section contains the applications of CPFSs, CPFRs, and their types. Network security is an expansive word that covers a group of technologies, devices, techniques, and procedures. Network security can be defined in Layman's terms as a collection of rules and configurations that are designed to defend and safeguard the integrity, confidentiality, and accessibility of data and computer networks via software and hardware technologies. All industries, organizations, and enterprises require a degree of network security solutions to protect them from the expanding cyber threats in today's wild world.\u2009 List the securities and the threats\u2009 Make a set of securities and a set of threats that are to be studied\u2009 Convert the two sets to CPFSs by carefully assigning each element the level of membership, level of abstinence, and level of nonmembership\u2009 Find the Cartesian product between the two CPFSs\u2009 Read and interpret the numerical resultsIn the following subsections, some common threats that are faced by networks, the network security techniques, and the relationships among them are discussed. The algorithm for the used method is portrayed through (1)Network Access Control (NAC). NAC prevents the potential attackers from infiltrating the network. It is set at granular levels, e.g., granting administrators full access to the network and refusing access to particular private folders or stopping their personal devices from connecting to the network:(2)Antivirus (AV). Antivirus software keeps an organization protected from viruses, ransomware, worms, and Trojans: (3)Firewall (FW). FW acts as a blockade between the trusted internal network and the untrusted external networks. The rules of blockade and authorization of traffic to a network are configured by administrators:(4)Virtual Private Network (VPN). VPNs build a connection to the network from a different endpoint or location.Different methods of securing a network are discussed as follows. The level of membership, level of abstinence, and level of nonmembership are also assigned:The most common network security threats one may encounter are explored as follows:The most common network threats in cybersecurity for a daily Internet user are computer viruses. Generally, computer viruses are pieces of software and codes that are written to be spread from one computer to another. They disable security settings, send spam, corrupt, and steal data and information from a computer, and even they can delete everything on a hard drive. They enter through e-mail attachments or downloaded from specific websites to infect the computers on a network:Adware is any software which tracks data through browsing habits. Based on those habits, the advertisements and pop-ups are shown. The adware can slow down computer's processor and Internet connection speed. Adware downloaded without consent is considered malicious.Spyware is similar to adware. It is secretly installed on a computer which contains key loggers for recording personal information such as e-mail addresses, passwords, and credit card information:Many servers use SQL for storing website data. With the progression of technology, the network security threats have also advanced which lead the threat of SQL injection attacks.SQL injection attacks exploit security vulnerabilities in the application's software to target data-bases. They use wicked code to achieve secretive data and change and even destroy that data. It is one of the most dangerous privacy issues for data confidentiality:A Trojan horse or Trojan is a malicious bit of attacking code or software which is hidden behind a genuine program. It tricks people into running it willingly. They spread often by e-mail attachments and clicking on a false advertisement.It records passwords by logging keystrokes, hijack webcams, and steal sensitive data on a computer.Man-in-the-middle attacks are cybersecurity attacks that allow the attacker to eavesdrop on communication between two targets. It can listen to a private communication.DNS spoofing, IP spoofing, ARP spoofing, HTTPS spoofing, Wi-Fi hacking, and SSL hijacking are some of the types of MITM attacks:Now that, the effectiveness and ineffectiveness of each network security and threat are analyzed, and we carry out the following mathematics.S and T that represent the sets of securities and threats, respectively, are as follows:Since we have the following two CPFSs, S and T is deliberated by The conception of Cartesian product is used to find out the ability of certain securities against specific threat. Thus, finding the Cartesian product between the CPFSs S \u00d7 T describes the association between that pair, i.e., the influences and impacts of a security on a threat. The levels of membership tell the efficacy of a network security to grab a specific threat with respect to some time. The levels of abstinence indicate the no effects or neutral effect of a security against a certain threat. The levels of nonmembership designate the inefficiency or ineptness of a security against a certain threat. For example, the ordered pair , (3/5)e\u03c0j(4/9), (1/5)e\u03c0j(1/5), (2/11)e\u03c0j(2/15)) emphasizes that the antivirus software can positively tackle the threats and risks to the network by viruses. Further, the numbers explain that the levels of uselessness and inefficiency are low, i.e., the levels of abstinence and nonmembership, respectively. More precisely, the complex picture fuzzy values are translated as the level of security that antivirus software provides against the vulnerabilities of a virus is 60% with respect to (4/9) time units, the level of neutral effects is 20% with respect to 1/5 time units, and the chances of risks via viruses evading the antivirus software are 18% with respect to 2/15 time units. In case of the securities, the longer durations of time in the level of membership is thought to be better, while the smaller time frame in the levels of nonmembership is better. Obviously, the levels of abstinence describe the neutral effects.Each pair of elements in the Cartesian product In this section, the reliability of the proposed structures of CPFRs is verified by carrying out the comparative study among the proposed and preexisting structures such as FRs, CFRs, IFRs, CIFRs, and PFRs.The structures of FR, IFR, and PFR have one similarity, that is, the real-valued level of memberships, level of abstinence, and level of nonmemberships. Therefore, they are limited to only one-dimensional problems. These structures cannot model periodicity and problems with multivariable.FRs only discuss the level of membership. IFRs discuss the level of membership and level of nonmembership. So, these structures are completely swept out of the competition.On the other hand, PFRs discuss all the three levels, i.e., level of membership, level of abstinence, and level of nonmembership. A detailed comparison is given as follows.S and T representing the set of securities and the set of threats, respectively:Let us consider the problem deliberated in R between S and T is given in The PFR R that it gives the real-valued information of level of membership, level of abstinence, and level of nonmembership. So, it does not indicate the time frame for the relation.It is clear from the above PFR Hence, these structures have certain limitations, and thus they give limited information.The structures of CFR and CIFR consist of complex valued functions.The CFRs only discuss the level of membership. Therefore, they cannot provide sufficient solution to the problem in application.The CIFR talks about the level of membership and level of nonmembership. Thus, CIFRs are used to solve the problem in S and T representing the set of securities and the set of threats, respectively:Let us consider the problem deliberated in The details of abbreviations used in the above sets are described in R between S and T is given in The CIFR It is clear from This article introduced the novel concepts of complex picture fuzzy relations (CPFRs) using the idea of Cartesian products between two complex picture fuzzy sets (CPFSs). Moreover, the types of CPFRs such as CP-equivalence-FR, CP-partial order-FR, CP-total order-FR, and CP-composite-FR were also studied with the help of definitions, suitable examples, properties, and results. Furthermore, the ground-breaking modeling techniques, based on the proposed picture fuzzy information, were introduced. These modeling methods were then used to model the problems of network and communication securities. The application problem was modeled and solved to achieve the required results, i.e., the level of effectiveness, ineffectiveness, and no effects of the network security methods against the threats that are faced by the network and communication systems. Finally, a comparative study had been carried out that verified the preeminence of the proposed methods over the existing methods. In future, these innovative concepts can be further extended to other generalizations of fuzzy set theory and fuzzy logic which will produce robust modeling techniques."}
{"text": "G. Dixon, MD\u201d and \u201cCaitlyn Lutfy, MPH.\u201d On page 970, in Table 3, in the row for \u201cSocial participation ability,\u201d in the columns for \u201cPost\u2013COVID-19 patients,\u201d \u201cControl patients,\u201d and \u201cmean difference\u201d the summary scale T-score mean standard deviations and mean differences should have read, \u201c46.6 (44.7 to 48.6),\u201d \u201c50.5 (50.0 to 51.1),\u201d and \u201c \u22124.2 (\u22126.4 to \u22122.0),\u201d respectively. In the report \u201cOutcomes Among Patients Referred to Outpatient Rehabilitation Clinics After COVID-19 diagnosis \u2014 United States, January 2020\u2013March 2021,\u201d on page 967, the following authors\u2019 names should have read, \u201cMeredith"}
{"text": "The aim of the study was the analysis of trends of infant and neonatal mortality in Poland in 1995-2015, overall and by gestational age, main groups of causes and age at death.Data from birth and death certificates from 1995, 2000, 2005, 2010, 2014 and 2015 were used. Infant, neonatal, postneonatal, perinatal and gestational age \u2013 specific mortality rates were presented. Main groups of causes of deaths were determined according to the International Classification of Diseases ICD-10.In Poland, in 1995-2015 infant mortality decreased more than three-fold, similarly to neonatal and postneonatal mortality. Early neonatal mortality decreased almost four-fold, stillbirths rate - twofold and perinatal mortality - almost three-fold. The progress, to the greatest extend was related to preterm births. Infant mortality in this group decreased from 128.5 per 1000 preterm live births in 1995 to 36.8 in 2015. The main causes of all infant deaths are perinatal conditions and congenital anomalies.The decrease of infant and neonatal mortality has been continued in the last twenty years and affected mainly preterm births born from the shorter and shorter gestations, what implicates growing demand for long lasting health care and rehabilitation. Deaths of infants and neonates born at term and not related to congenital anomalies are relatively rare and need individual assessment. Umieralno\u015b\u0107 niemowl\u0105t jest prawdopodobnie jednym z najdawniej u\u017cywanych (negatywnych) wska\u017anik\u00f3w stanu zdrowia \u2013 obrazuj\u0105cym stopie\u0144, w jakim spo\u0142ecze\u0144stwa s\u0105 w stanie uchroni\u0107 dziecko od ryzyka utraty \u017cycia. Jest to jednocze\u015bnie wska\u017anik od bardzo dawna i powszechnie dost\u0119pny analizom, z kt\u00f3rych na przyk\u0142ad wiadomo, \u017ce na prze\u0142omie XIX i XX wieku na ziemiach polskich umiera\u0142o co 5 niemowl\u0119, w latach 20-tych XX wieku wsp\u00f3\u0142czynnik umieralno\u015bci niemowl\u0105t wynosi\u0142 oko\u0142o 150 na 1000 urodze\u0144 \u017cywych, w 1950 roku \u2013 111, w 1960 roku \u2013 55, w 1990 roku \u2013 19 a dzi\u015b \u2013 4. Podobne do zr\u00f3\u017cnicowania w czasie, widoczne jest zr\u00f3\u017cnicowanie umieralno\u015bci niemowl\u0105t mi\u0119dzy krajami. Przeci\u0119tnie, w najbiedniejszych krajach \u015bwiata wsp\u00f3\u0142czynnik umieralno\u015bci niemowl\u0105t kszta\u0142tuje si\u0119 obecnie na poziomie oko\u0142o 50 zgon\u00f3w na 1000 urodze\u0144 \u017cywych, podczas gdy w najbogatszych \u2013 oko\u0142o 5 [W krajach rozwini\u0119tych, w efekcie przemian cywilizacyjnych, w tym technologicznych jakie zachodzi\u0142y w ostatnich dziesi\u0119cioleciach, ryzyko zgonu dziecka w pierwszym roku \u017cycia jest niewielkie a zgon dziecka urodzonego o czasie staje si\u0119 powoli zdarzeniem rzadko wyst\u0119puj\u0105cym. Dlatego mo\u017ce, od pewnego czasu uznano, \u017ce w krajach rozwini\u0119tych kwestie zdrowia dziecka i matki zosta\u0142y rozwi\u0105zane i wobec wci\u0105\u017c nie w pe\u0142ni opanowanych problem\u00f3w zdrowotnych doros\u0142ych , nie zas\u0142uguj\u0105 na zwi\u0119kszon\u0105 uwag\u0119.Wsp\u00f3\u0142czynniki umieralno\u015bci niemowl\u0105t, wsp\u00f3\u0142czynniki umieralno\u015bci noworodk\u00f3w czy wsp\u00f3\u0142czynniki umieralno\u015bci oko\u0142oporodowej nadal odzwierciedlaj\u0105 jednak zmiany, jakie zachodz\u0105 w sferze opieki zdrowotnej a ich zr\u00f3\u017cnicowanie sugeruje kierunki dzia\u0142a\u0144, kt\u00f3re powinny by\u0107 podj\u0119te w celu zapobiegania dramatycznym zdarzeniom prowadz\u0105cym do \u015bmierci dziecka.Oczywi\u015bcie, jest po\u017c\u0105dane, aby wska\u017aniki stanu zdrowia najm\u0142odszej populacji opiera\u0107 nie tylko na zgonach, ale tak\u017ce na zachorowaniach a nawet rozwija\u0107 wska\u017aniki pozytywne, ale zanim to osi\u0105gniemy, musimy dokonywa\u0107 jak najg\u0142\u0119bszych analiz danych, kt\u00f3re s\u0105 dost\u0119pne.per se \u015bwiadczy przede wszystkim o dojrza\u0142o\u015bci noworodka i determinuje jego szanse prze\u017cycia poza organizmem matki.Niniejsza publikacja nawi\u0105zuje do wcze\u015bniejszych og\u00f3lnokrajowych analiz umieralno\u015bci niemowl\u0105t i noworodk\u00f3w, zdrowia matek oraz opieki oko\u0142oporodowej prowadzonych przez Instytut Matki i Dziecka od ponad 50 lat a nast\u0119pnie artyku\u0142\u00f3w pierwszej autorki na ten temat i prac badawczych wykonywanych przez ni\u0105 w ramach projektu Unii Europejskiej Peristat , 3, 4. OW niniejszym opracowaniu przyczyny zgon\u00f3w niemowl\u0105t i noworodk\u00f3w przedstawiono tylko wed\u0142ug g\u0142\u00f3wnych rozdzia\u0142\u00f3w Mi\u0119dzynarodowej Statystycznej Klasyfikacji Chor\u00f3b i Problem\u00f3w Zdrowotnych. Stosowanie tej klasyfikacji do bezpo\u015bredniej analizy szczeg\u00f3\u0142owych pojedynczych przyczyn zgon\u00f3w noworodk\u00f3w i niemowl\u0105t od lat nie by\u0142o uznawane za uzasadnione, poniewa\u017c niekt\u00f3re pojedyncze przyczyny znajduj\u0105ce si\u0119 w r\u00f3\u017cnych rozdzia\u0142ach klasyfikacji, faktycznie reprezentuj\u0105 wsp\u00f3ln\u0105 etiologi\u0119, kt\u00f3ra nie jest widoczna, gdy przyczyny te analizowane s\u0105 w rozproszeniu. Na przyk\u0142ad, niekt\u00f3re choroby uwarunkowane genetycznie s\u0105 kodowane nie tylko w rozdziale dotycz\u0105cym wad wrodzonych, ale i w rozdziale dotycz\u0105cym chor\u00f3b uk\u0142adu nerwowego oraz w rozdziale dotycz\u0105cym zaburze\u0144 wydzielania wewn\u0119trznego, stanu od\u017cywienia i przemian metabolicznych. W ostatnich latach powsta\u0142y nowe propozycje grupowania pojedynczych przyczyn zgon\u00f3w noworodk\u00f3w , 6 i w pCelem opracowania jest analiza dynamiki umieralno\u015bci niemowl\u0105t i noworodk\u00f3w w Polsce w okresie dwudziestolecia 1995-2015, z uwzgl\u0119dnieniem czasu trwania ci\u0105\u017cy, z kt\u00f3rej urodzi\u0142 si\u0119 noworodek, g\u0142\u00f3wnych grup przyczyn oraz wieku w momencie zgonu.W pracy wykorzystano zg\u0142oszenia urodzenia noworodka i karty zgon\u00f3w niemowl\u0105t z lat 1995, 2000, 2005, 2010, 2014 i 2015. Dokumenty te, po wype\u0142nieniu przez pracownik\u00f3w s\u0142u\u017cby zdrowia i pracownik\u00f3w Urz\u0119d\u00f3w Stanu Cywilnego, s\u0105 przekazywane do G\u0142\u00f3wnego Urz\u0119du Statystycznego, gdzie s\u0105 przetwarzane g\u0142\u00f3wnie dla cel\u00f3w statystyki ludno\u015bci. Instytut Matki i Dziecka od ponad 30 lat wykorzystuje zanonimizowane indywidualne dane ze zg\u0142osze\u0144 urodzenia noworodka jako narz\u0119dzie s\u0142u\u017c\u0105ce monitorowaniu stanu zdrowia niemowl\u0105t, noworodk\u00f3w i matek na poziomie og\u00f3lnokrajowym. W Polsce istnieje jeszcze jedno \u017ar\u00f3d\u0142o danych dotycz\u0105ce zgon\u00f3w oko\u0142oporodowych \u2013 sprawozdawczo\u015b\u0107 w formie zagregowanej ze szpitali (formularz MZ-29), jednak liczba zgon\u00f3w w wieku 0-6 dni w tym \u017ar\u00f3dle jest niedoszacowana, poniewa\u017c sprawozdawczo\u015b\u0107 nie uwzgl\u0119dnia ewentualnych zgon\u00f3w po przekazaniu noworodka do innego szpitala.Obliczono wsp\u00f3\u0142czynniki umieralno\u015bci niemowl\u0105t (liczba zgon\u00f3w w pierwszym roku \u017cycia na 1000 urodze\u0144 \u017cywych), wsp\u00f3\u0142czynniki umieralno\u015bci noworodk\u00f3w , wsp\u00f3\u0142czynniki umieralno\u015bci oko\u0142oporodowej (suma liczby zgon\u00f3w w wieku 0-6 dni i martwych urodze\u0144 na 1000 urodze\u0144 \u017cywych i martwych), cz\u0105stkowe wsp\u00f3\u0142czynniki umieralno\u015bci wed\u0142ug czasu trwania ci\u0105\u017cy (liczba zgon\u00f3w w danej grupie czasu trwania ci\u0105\u017cy w stosunku do liczby urodze\u0144 \u017cywych w tej grupie czasu trwania ci\u0105\u017cy) oraz wsp\u00f3\u0142czynniki i struktur\u0119 umieralno\u015bci w podziale na g\u0142\u00f3wne grupy przyczyn wed\u0142ug Mi\u0119dzynarodowej Statystycznej klasyfikacji Chor\u00f3b i Problem\u00f3w Zdrowotnych . Wyodr\u0119bniono grup\u0119 \u201eWybranych stan\u00f3w rozpoczynaj\u0105cych si\u0119 w okresie oko\u0142oporodowym\u201d (P00-P96) oraz grup\u0119 \u201eWad rozwojowych wrodzonych, zniekszta\u0142ce\u0144 i aberracji chromosomowych\u201d (Q00-Q99). Przyczyn zgon\u00f3w nie uwzgl\u0119dniono dla roku 1995, poniewa\u017c w\u00f3wczas stosowana by\u0142a inna \u2013 dziewi\u0105ta rewizja klasyfikacji.Zgodnie z przyj\u0119t\u0105 na \u015bwiecie a tak\u017ce w Polsce praktyk\u0105 wsp\u00f3\u0142czynniki obliczono dziel\u0105c liczb\u0119 zgon\u00f3w zarejestrowanych w danym roku kalendarzowym przez liczb\u0119 urodze\u0144 \u017cywych zarejestrowanych w tym samym roku kalendarzowym.Braki danych dotycz\u0105cych czasu trwania ci\u0105\u017cy dla urodze\u0144 \u017cywych i zgon\u00f3w uzupe\u0142niono proporcjonalnie wed\u0142ug znanego rozk\u0142adu czasu trwania ci\u0105\u017cy. Dla urodze\u0144 \u017cywych uzupe\u0142nienia te dotyczy\u0142y 3005 przypadk\u00f3w w 1995 roku, 95 \u2013w 2000 roku, 33 \u2013w 2005 roku, 2 \u2013w 2010 roku, 5 \u2013w 2014 roku i 1589 \u2013w 2015 roku. Dla zgon\u00f3w uzupe\u0142nienia te dotyczy\u0142y 298 przypadk\u00f3w w 1995 roku, 59 \u2013w 2000 roku, 37 \u2013w 2005 roku, 2 \u2013 w 2010 roku, 1 \u2013w 2014 roku i 8 \u2013w 2015 roku.Wyniki podano zar\u00f3wno dla 2014 roku, jak i 2015 roku, poniewa\u017c w 2015 roku, na skutek wprowadzenia b\u0142\u0119dnego rozporz\u0105dzenia w sprawie wzor\u00f3w karty urodzenia i karty martwego urodzenia  przesta\u0142y by\u0107 dost\u0119pne dane dotycz\u0105ce martwych urodze\u0144. Podany tu niepor\u00f3wnywalny wsp\u00f3\u0142czynnik umieralno\u015bci oko\u0142oporodowej i martwych urodze\u0144 w 2015 roku pochodzi ze wspomnianej wcze\u015bniej rutynowej zagregowanej sprawozdawczo\u015bci Ministerstwa Zdrowia [W okresie ostatnich 20 lat umieralno\u015b\u0107 niemowl\u0105t w Polsce zmniejszy\u0142a si\u0119 ponad trzykrotnie , podobnie jak umieralno\u015b\u0107 noworodk\u00f3w  i umieralno\u015b\u0107 w okresie ponoworodkowym  jest ponad dwudziestokrotnie wy\u017csze, ni\u017c w przypadku dziecka urodzonego o czasie (\u226537 tygodnia ci\u0105\u017cy). Im kr\u00f3tszy jest czas trwania ci\u0105\u017cy, tym mniejsze s\u0105 szanse prze\u017cycia. Jeszcze 20 lat temu (w 1995 roku) w okresie niemowl\u0119cym umiera\u0142o oko\u0142o 80% dzieci urodzonych przed 28 tygodniem ci\u0105\u017cy, 30% dzieci urodzonych w 28-31 tygodniu, oko\u0142o 8% dzieci urodzonych w 32-34 tygodniu i oko\u0142o 3% dzieci urodzonych w 35,36 tygodniu , notuje si\u0119 tak\u017ce post\u0119p wyra\u017cony spadkiem umieralno\u015bci niemowl\u0105t z 5,3 na 1000 urodze\u0144 \u017cywych \u226537 tygodnia ci\u0105\u017cy w 1995 roku do 1,4 w 2015 roku.W zwi\u0105zku z radykalnym zmniejszaniem si\u0119 ryzyka zgonu w grupie dzieci urodzonych przedwcze\u015bnie, co dotyczy zw\u0142aszcza okresu noworodkowego, mo\u017cna zastanawia\u0107 si\u0119 nad tym, czy zmniejszenie ryzyka zgonu bezpo\u015brednio po urodzeniu nie skutkuje zwi\u0119kszeniem ryzyka zgonu w dalszym okresie \u017cycia niemowl\u0119cia.Wydaje si\u0119, \u017ce tak nie jest, poniewa\u017c nie obserwuje si\u0119 jednoczesnego zwi\u0119kszenia umieralno\u015bci w okresie ponoworodkowym tych niedojrza\u0142ych i nierzadko chorych dzieci . Uwzgl\u0119dG\u0142\u00f3wnymi przyczynami zgon\u00f3w noworodk\u00f3w i niemowl\u0105t s\u0105 choroby okresu oko\u0142oporodowego, zw\u0142aszcza zwi\u0105zane z wcze\u015bniactwem (oko\u0142o 50% zgon\u00f3w niemowl\u0105t i 60% zgon\u00f3w noworodk\u00f3w) oraz wady wrodzone (oko\u0142o 30% zgon\u00f3w niemowl\u0105t i noworodk\u00f3w) . Noworodki urodzone przedwcze\u015bnie, o ile umieraj\u0105, to najcz\u0119\u015bciej w nast\u0119pstwie wcze\u015bniactwa . To w\u0142a\u015bnie w zakresie obni\u017cenia si\u0119 umieralno\u015bci z powodu chor\u00f3b okresu oko\u0142oporodowego u niemowl\u0105t przedwcze\u015bnie urodzonych \u2013 z 51,4 zgon\u00f3w na 1000 urodze\u0144 \u017cywych przed 37 tygodniem ci\u0105\u017cy w 2000 roku do 26,8 w 2015 roku \u2013 dokonano w ostatnim 20-leciu najwi\u0119kszy post\u0119p.Ponadto, ponad 20% zgon\u00f3w noworodk\u00f3w urodzonych przedwcze\u015bnie powodowanych jest wadami rozwojowymi wrodzonymi, zniekszta\u0142ceniami i aberracjami chromosomowymi (2015 rok), z czego prawie 30% wrodzonymi wadami rozwojowymi uk\u0142adu kr\u0105\u017cenia a zw\u0142aszcza serca (kody Q20-Q28 klasyfikacji ICD-10).Zgon noworodka urodzonego o czasie \u2013 o ile nast\u0119puje, to g\u0142\u00f3wnie z powodu wady wrodzonej , ale tak\u017ce z powodu wszystkich pozosta\u0142ych przyczyn, w tym oko\u0142oporodowych, na przyk\u0142ad niedotlenienia wewn\u0105trzmacicznego i zamartwicy urodzeniowej (ponad 30% tzw. przyczyn oko\u0142oporodowych w grupie noworodk\u00f3w urodzonych o czasie w 2015 roku) czy zespo\u0142u nag\u0142ej \u015bmierci niemowl\u0105t (ponad 20% pozosta\u0142ych przyczyn zgon\u00f3w w tej w grupie noworodk\u00f3w). Og\u00f3lnie, zgony niemowl\u0105t urodzonych o czasie zdarzaj\u0105 si\u0119 rzadko a przyczyny tych zgon\u00f3w s\u0105 bardzo zr\u00f3\u017cnicowane .Umieralno\u015b\u0107 niemowl\u0105t w Polsce w ci\u0105gu ostatnich 20 lat uleg\u0142a znacznemu obni\u017ceniu i trendy spadkowe dotyczy\u0142y zar\u00f3wno umieralno\u015bci noworodkowej jak i w okresie ponoworodkowym. Najwi\u0119kszy spadek obserwowany by\u0142 w latach 1995-2005: umieralno\u015bci noworodkowej o 4,5 na 1000 urodze\u0144 \u017cywych a umieralno\u015bci postneonatalnej o 1,0 na 1000 urodze\u0144 \u017cywych. W latach nast\u0119pnych obni\u017canie obu umieralno\u015bci mia\u0142o wolniejsze tempo, bardziej zbli\u017cone do liniowego: odpowiednio 0,9 i 0,5 w okresie 5 lat.Polska zbli\u017ca si\u0119 stopniowo do poziomu osi\u0105ganego przez kraje Europy Zachodniej (kraje Unii Europejskiej przed 2004 rokiem). W roku 1995 wsp\u00f3\u0142czynniki umieralno\u015bci neonatalnej wynosi\u0142y 10,1 na 1000 urodze\u0144 \u017cywych dla Polski i 3,7 dla kraj\u00f3w Europy Zachodniej, natomiast w roku 2000 - odpowiednio 5,6 oraz 3,1 na 1000 urodze\u0144 \u017cywych . PodobniW Polsce na przestrzeni ostatnich 20 lat obserwowane jest obni\u017cenie umieralno\u015bci zar\u00f3wno w\u015br\u00f3d niemowl\u0105t donoszonych, jak i przedwcze\u015bnie urodzonych. Najwi\u0119ksze zmiany \u2013 ponad 4-krotne zmniejszenie ryzyka zgonu \u2013 dotyczy\u0142y niemowl\u0105t urodzonych w 28-31, 32-34 i 35-36 tygodniu ci\u0105\u017cy. Prawie 4-krotne zmniejszenie \u2013 do\u015b\u0107 niskiego ju\u017c \u2013 ryzyka zgonu mia\u0142o te\u017c miejsce w grupie niemowl\u0105t urodzonych po 36 tygodniu ci\u0105\u017cy. Stosunkowo najmniej, cho\u0107 r\u00f3wnie\u017c znacz\u0105co bo prawie 2-krotnie obni\u017cy\u0142o si\u0119 ryzyko zgonu niemowl\u0105t urodzonych najwcze\u015bniej \u2013 pomi\u0119dzy 22 i 27 tygodniem ci\u0105\u017cy. Jednak\u017ce MacDorman i wsp., kt\u00f3ry por\u00f3wna\u0142 umieralno\u015b\u0107 niemowl\u0105t z 2010 roku wed\u0142ug czasu trwania ci\u0105\u017cy w wybranych krajach europejskich i Stanach Zjednoczonych stwierdzi\u0142, \u017ce wsp\u00f3\u0142czynniki w Polsce s\u0105 2-3-krotnie wy\u017csze ni\u017c w innych krajach rozwini\u0119tych [Jednak z drugiej strony nale\u017cy tak\u017ce zauwa\u017cy\u0107, \u017ce poniewa\u017c obni\u017cenie umieralno\u015bci dzieci urodzonych przedwcze\u015bnie nie jest odzwierciedleniem zmniejszenia cz\u0119sto\u015bci wyst\u0119powania wcze\u015bniactwa w Polsce (odsetek urodze\u0144 \u017cywych z porod\u00f3w przed 37 tygodniem trwania ci\u0105\u017cy jest do\u015b\u0107 stabilny w ci\u0105gu ostatnich 20 lat i wynosi\u0142 6-7% w tym okresie), to skutkuje wi\u0119ksz\u0105 prze\u017cywalno\u015bci\u0105 tej grupy dzieci, cz\u0119sto wymagaj\u0105cych d\u0142ugotrwa\u0142ej opieki i rehabilitacji. Fakt ten niesie za sob\u0105 implikacje dla organizacji opieki zdrowotnej i systemu +nansowania ochrony zdrowia w kraju.G\u0142\u00f3wne przyczyny umieralno\u015bci niemowl\u0105t w Polsce to choroby okresu oko\u0142oporodowego i wady wrodzone. Odpowiadaj\u0105 one odpowiednio za ok. 50% i 30% zgon\u00f3w w trakcie pierwszego roku \u017cycia i proporcje te nie zmienia\u0142y si\u0119 znacz\u0105co w trakcie ostatnich 15 lat.Oko\u0142o 30% wynosi tak\u017ce odsetek zgon\u00f3w noworodkowych z powodu wad wrodzonych w Polsce prezentowany w raporcie EURO-PERISTAT z roku 2010, co stawia j\u0105 na 9 miejscu spo\u015br\u00f3d 27 kraj\u00f3w (region\u00f3w) pod wzgl\u0119dem udzia\u0142u wad jako przyczyny zgon\u00f3w w tym okresie [Dane z Rejestru Wielkopolskiego \u2013 pe\u0142noprawnego cz\u0142onka sieci rejestr\u00f3w wad wrodzonych EUROCAT pokazuj\u0105, \u017ce Polska nale\u017cy do kraj\u00f3w o przeci\u0119tnej cz\u0119sto\u015bci wyst\u0119powania wad wrodzonych  . PowodemAnaliza zmian umieralno\u015bci noworodk\u00f3w i niemowl\u0105t w Polsce w czasie oraz por\u00f3wnania mi\u0119dzynarodowe pokazuj\u0105 z jednej strony sukcesy \u2013 zdecydowan\u0105 popraw\u0119 wska\u017anik\u00f3w umieralno\u015bci dotycz\u0105c\u0105 wszystkich analizowanych subpopulacji w tej grupie wieku w ci\u0105gu ostatnich 20 lat. Z drugiej jednak strony odst\u0119p, jaki wci\u0105\u017c dzieli Polsk\u0119 od kraj\u00f3w Europy Zachodniej wymaga dzia\u0142a\u0144 w zakresie dw\u00f3ch g\u0142\u00f3wnych przyczyn odpowiedzialnych za zdrowie niemowl\u0105t: chor\u00f3b okresu oko\u0142oporodowego i wad wrodzonych. Konieczna jest dalsza poprawa systemu opieki zdrowotnej w zakresie wczesnej diagnostyki i skutecznej terapii, tak\u017ce w okresie ci\u0105\u017cy. Jednocze\u015bnie, wyst\u0119puje pilna potrzeba zintensyfikowania prac badawczych nad mo\u017cliwo\u015bciami zapobiegania wcze\u015bniactwu i wadom wrodzonym. Istotna jest te\u017c pog\u0142\u0119biona analiza zgon\u00f3w dzieci wypisywanych po urodzeniu ze szpitala jako zdrowe, zw\u0142aszcza w aspekcie potencjalnych uwarunkowa\u0144 spo\u0142eczno-ekonomicznych.W ostatnim dwudziestoleciu, post\u0119p w zakresie zmniejszania umieralno\u015bci niemowl\u0105t jest kontynuowany. Wyra\u017ca si\u0119 on przede wszystkim zwi\u0119kszaniem szans prze\u017cycia dzieci urodzonych z ci\u0105\u017c o coraz kr\u00f3tszym czasie trwania.G\u0142\u00f3wnymi przyczynami zgon\u00f3w niemowl\u0105t i noworodk\u00f3w jest wcze\u015bniactwo i wady wrodzone. Nat\u0119\u017cenie zgon\u00f3w z powodu wcze\u015bniactwa maleje, przy braku zmian w zakresie jego wyst\u0119powania. Oznacza to, \u017ce zwi\u0119ksza si\u0119 populacja dzieci wymagaj\u0105cych nierzadko d\u0142ugotrwa\u0142ej opieki medycznej i rehabilitacji. Jednocze\u015bnie, wyst\u0119puje pilna potrzeba zintensyfikowania prac badawczych nad mo\u017cliwo\u015bciami zapobiegania wcze\u015bniactwu i wadom wrodzonym.Zgony niemowl\u0105t i noworodk\u00f3w urodzonych o czasie i z przyczyn organicznych innych ni\u017c wady wrodzone nie wyst\u0119puj\u0105 cz\u0119sto. Konieczne jest monitorowanie indywidualnych przypadk\u00f3w takich zgon\u00f3w."}
{"text": "Przewlek\u0142e b\u00f3le brzucha s\u0105 bardzo cz\u0119sto wyst\u0119puj\u0105c\u0105 dolegliwo\u015bci\u0105 w populacji dzieci i m\u0142odzie\u017cy. W wi\u0119kszo\u015bci przypadk\u00f3w pod\u0142o\u017cem s\u0105 czynno\u015bciowe zaburzenia przewodu pokarmowego. Jednak u kilku procent dzieci powodem utrzymuj\u0105cego si\u0119 przewlekle b\u00f3lu brzucha s\u0105 schorzenia organiczne zlokalizowane w obr\u0119bie przewodu pokarmowego i poza przewodem pokarmowym, w tym r\u00f3wnie\u017c procesy nowotworowe. W\u015br\u00f3d przyczyn organicznych, opr\u00f3cz tych cz\u0119sto spotykanych, takich jak: nietolerancje i alergie pokarmowe, choroba refluksowa prze\u0142yku, przewlekle zapalenie \u017co\u0142\u0105dka i dwunastnicy czy zaka\u017cenia uk\u0142adu moczowego, w diagnostyce nale\u017cy uwzgl\u0119dni\u0107 r\u00f3wnie\u017c przyczyny bardzo rzadkie, na przyk\u0142ad choroby nowotworowe, a w\u015br\u00f3d nich guzy w obr\u0119bie jamy brzusznej. W opisywanym przez nas przypadku, u 6-letniej dziewczynki z przewlek\u0142ymi b\u00f3lami brzucha, objawami o charakterze refluksu \u017co\u0142\u0105dkowo-prze\u0142ykowego i zaparciami, z rozpoznan\u0105 wcze\u015bniej alergi\u0105 pokarmow\u0105 i nietolerancj\u0105 laktozy, z uwagi na wyst\u0105pienie objaw\u00f3w alarmowych, ustalono wskazania do poszerzenia diagnostyki. Uwidoczniony w badaniu radiologicznym klatki piersiowej cie\u0144 kr\u0105g\u0142y na wysoko\u015bci przepony, okaza\u0142 si\u0119 by\u0107 w badaniu komputerowym, guzem okolicy przykr\u0119gos\u0142upowej z cechami charakterystycznymi dla neuroblastoma. Po makroskopowo ca\u0142kowitej resekcji guza, na podstawie wyniku badania histopatologicznego, ustalono rozpoznanie: ganglineuroblastoma. Obecno\u015b\u0107 objaw\u00f3w alarmowych w badaniu podmiotowym i przedmiotowym u dzieci z b\u00f3lami brzucha sugeruje wy\u017csze prawdopodobie\u0144stwo organicznej przyczyny dolegliwo\u015bci i zawsze powinno sk\u0142ania\u0107 do przeprowadzenia poszerzonej diagnostyki. Nerwiak zarodkowy zwojowy (ganglioneuroblastoma) nale\u017cy do chor\u00f3b bardzo rzadkich, w wi\u0119kszo\u015bci przypadk\u00f3w zlokalizowany jest pierwotnie w obr\u0119bie jamy brzusznej, a najcz\u0119stszym zwi\u0105zanym z nim objawem jest b\u00f3l brzucha. Przewlek\u0142y b\u00f3l brzucha, rozumiany jako d\u0142ugo utrzymuj\u0105cy si\u0119, ci\u0105g\u0142y b\u0105d\u017a nawracaj\u0105cy, dotyczy znacznej cz\u0119\u015bci populacji dzieci i m\u0142odzie\u017cy. Cz\u0119sto\u015b\u0107 wyst\u0119powania przewlek\u0142ych b\u00f3l\u00f3w brzucha u dzieci nie jest dok\u0142adnie poznana. Szacuje si\u0119, \u017ce problem ten jest powodem od 2 do 4% wszystkich wizyt w gabinetach pediatrycznych . W badanNiespe\u0142na 6-letnia dziewczynka zosta\u0142a przyj\u0119ta do kliniki z powodu przewlek\u0142ych b\u00f3l\u00f3w brzucha, kt\u00f3re uleg\u0142y nasileniu w ci\u0105gu ostatnich kilku dni przed przyj\u0119ciem. We wst\u0119pnej ocenie, uwag\u0119 zwraca\u0142a r\u00f3wnie\u017c tendencja do zapar\u0107 utrzymuj\u0105ca si\u0119 od okresu niemowl\u0119cego oraz objawy refluksu \u017co\u0142\u0105dkowo-prze\u0142ykowego. Wywiad rodzinny by\u0142 nieobci\u0105\u017cony. Dziewczynka urodzi\u0142a si\u0119 z ci\u0105\u017cy pierwszej, porodu o czasie, drog\u0105 ci\u0119cia cesarskiego z powodu po\u0142o\u017cenia miednicowego p\u0142odu, z prawid\u0142ow\u0105 mas\u0105 cia\u0142a, w stanie dobrym (9 punkt\u00f3w Apgar). Karmiona by\u0142a piersi\u0105 do 2 m.\u017c., nast\u0119pnie z powodu nasilonych kolek, wprowadzona zosta\u0142a mieszanka o znacznym stopniu hydrolizy, z dobrym efektem. Dieta by\u0142a rozszerzana, zgodnie z zaleceniami, od 6 m.\u017c. Od tego czasu wyst\u0119powa\u0142y zaparcia. Dziewczynka oddawa\u0142a stolce co kilka dni z wysi\u0142kiem, o twardej konsystencji, bez patologicznych domieszek, przyjmowa\u0142a postaw\u0119 retencyjn\u0105. Od oko\u0142o 4 r.\u017c. zacz\u0119\u0142a zg\u0142asza\u0107 b\u00f3le brzucha. Po 3 miesi\u0105cach utrzymywania si\u0119 dolegliwo\u015bci, zosta\u0142a skierowana do oddzia\u0142u pediatrycznego szpitala rejonowego celem diagnostyki. Wykonane w\u00f3wczas badania laboratoryjne, poza nieznaczn\u0105 nadp\u0142ytkowo\u015bci\u0105, pozostawa\u0142y w granicach normy, w tym badanie serologiczne w kierunku choroby trzewnej (przeciwcia\u0142a przeciw transglutaminazie tkankowej w klasach IgA i IgG). W badaniu ultrasonograficznym jamy brzusznej opisany zosta\u0142 prawid\u0142owy obraz narz\u0105d\u00f3w wewn\u0119trznych i przestrzeni zaotrzewnowej. Po kilku dniach pobytu w domu, z powodu nasilenia dolegliwo\u015bci b\u00f3lowych brzucha z podejrzeniem niedro\u017cno\u015bci jelit, przyj\u0119ta zosta\u0142a ponownie do szpitala. Po wykluczeniu potrzeby interwencji chirurgicznej (na podstawie prawid\u0142owego wyniku badania USG jamy brzusznej i obserwacji klinicznej), dziewczynka zosta\u0142a wypisana do domu z zaleceniami stosowania preparatu makrogolu i wlewek doodbytniczych. Leczenie by\u0142o kontynuowane w domu z przej\u015bciow\u0105 popraw\u0105. Po kolejnych czterech miesi\u0105cach, z powodu nawracaj\u0105cych b\u00f3l\u00f3w brzucha i zapar\u0107, ponownie hospitalizowana by\u0142a w oddziale pediatrycznym. W toku przeprowadzonej diagnostyki, u pacjentki zosta\u0142y ustalone rozpoznania: nietolerancja laktozy, alergia pokarmowa oraz zaparcia. W leczeniu zastosowana zosta\u0142a dieta eliminacyjna, kt\u00f3ra od tego czasu by\u0142a \u015bci\u015ble przestrzegana. Po 3 miesi\u0105cach ponowne nasili\u0142y si\u0119 dolegliwo\u015bci b\u00f3lowe zlokalizowane w nadbrzuszu oraz dodatkowo dziewczynka uskar\u017ca\u0142a si\u0119 na b\u00f3le o charakterze k\u0142ucia w okolicy lewego pod\u017cebrza, wybudzaj\u0105ce kilkakrotnie w nocy. Ponadto wyst\u0119powa\u0142y nasilone odbijania, nawracaj\u0105ce czkawki, uczucie cofania pokarmu, poranna chrypka, chrz\u0105kanie, uczucie cia\u0142a obcego w gardle, od d\u0142u\u017cszego czasu pokas\u0142ywanie. Stolce oddawa\u0142a co 3 dni, z wysi\u0142kiem, o zbitej konsystencji, bez patologicznych domieszek. Dziewczynka wykazywa\u0142a niezaburzon\u0105 aktywno\u015b\u0107 fizyczn\u0105 oraz prawid\u0142owe \u0142aknienie. W wywiadzie zwraca\u0142 uwag\u0119 ubytek masy cia\u0142a oko\u0142o 1 kg w ci\u0105gu ostatnich 4 tygodni.Przy przyj\u0119ciu do kliniki dziewczynka by\u0142a w stanie og\u00f3lnym dobrym, eutroficzna . W badaniu fizykalnym, poza niewielk\u0105 skolioz\u0105 w odcinku piersiowym i ubytkami pr\u00f3chniczymi, nie stwierdzono odchyle\u0144 od normy. Wykonane badania laboratoryjne z nieprawid\u0142owo\u015bci ujawni\u0142y jedynie nieznacznie przy\u015bpieszone OB (16 mm), przy prawid\u0142owym CRP  oraz liczb\u0119 p\u0142ytek krwi powy\u017cej g\u00f3rnej granicy normy (PLT 504 10^3/uL). W badaniu ultrasonograficznym jamy brzusznej zosta\u0142 opisywany prawid\u0142owy obraz narz\u0105d\u00f3w wewn\u0119trznych i przestrzeni zaotrzewnowej.W trakcie pobytu w klinice dziewczynka codziennie zg\u0142asza\u0142a b\u00f3l o charakterze k\u0142ucia zlokalizowany w lewym pod\u017cebrzu, r\u00f3wnie\u017c wybudzaj\u0105cy ze snu. Ponadto utrzymywa\u0142 si\u0119 suchy kaszel o niewielkim nasileniu. W wykonanym zdj\u0119ciu RTG klatki piersiowej uwidoczniony zosta\u0142 w rzucie kr\u0119gos\u0142upa, na wysoko\u015bci przepony, s\u0142abo odgraniczony kr\u0105g\u0142y cie\u0144 wielko\u015bci oko\u0142o 4x4cm, mog\u0105cy sugerowa\u0107 przepuklin\u0119 przeponow\u0105. Zdj\u0119cie RTG klatki piersiowej z widoczn\u0105 zmian\u0105 przedstawiono na rycinie 1.Wynik wykonanego badania kontrastowego g\u00f3rnego odcinka przewodu pokarmowego nie potwierdzi\u0142 obecno\u015bci przepukliny przeponowej. Dziewczynka zosta\u0142a zakwalifikowana do badania TK klatki piersiowej z kontrastem, kt\u00f3re uwidoczni\u0142o przykr\u0119gos\u0142upowo po stronie prawej (na wysoko\u015bci Th9-L2) hipodensyjn\u0105, mi\u0119kkotkankow\u0105 mas\u0119 o wymiarach 1,4x3,5x5,1 cm, z drobnymi zwapnieniami, przylegaj\u0105c\u0105 do trzon\u00f3w kr\u0119g\u00f3w i tylnych odcink\u00f3w dolnych prawych \u017ceber, do op\u0142ucnej i do prawej odnogi przepony, ponadto obni\u017cenie wysoko\u015bci trzon\u00f3w kr\u0119g\u00f3w Th7 i 8 opisane jako stan po z\u0142amaniu kompresyjnym oraz s\u0142abo r\u00f3\u017cnicuj\u0105ce si\u0119, rozsiane ogniska rozrzedzenia struktury kostnej kr\u0119g\u00f3w uwidocznionych w badaniu, mog\u0105ce odpowiada\u0107 zmianom o charakterze przerzutowym. Obraz guza w TK klatki piersiowej przedstawiono na rycinie 2.Dziewczynka przekazana zosta\u0142a do Kliniki Transplantacji Szpiku, Onkologii i Hematologii Dzieci\u0119cej celem dalszej diagnostyki i leczenia. Po makroskopowo ca\u0142kowitej resekcji guza, na podstawie wyniku badania histopatologicznego, zosta\u0142o ustalone rozpoznanie: ganglineuroblastoma nodular wariant, stroma rich type, bez amplifikacji onkogenu MYCN. Po zabiegu dziewczynka czu\u0142a si\u0119 dobrze, zg\u0142aszane wcze\u015bniej dolegliwo\u015bci b\u00f3lowe brzucha ust\u0105pi\u0142y. Po oko\u0142o 3 miesi\u0105cach od resekcji, dziewczynka zacz\u0119\u0142a uskar\u017ca\u0107 si\u0119 na b\u00f3le plec\u00f3w zlokalizowane na pograniczu kr\u0119gos\u0142upa piersiowego i l\u0119d\u017awiowego. Wykonana kilkakrotnie scyntygrafia ko\u015bci (MIBG) oraz badanie MRI kr\u0119gos\u0142upa nie potwierdzi\u0142y wznowy miejscowej ani obecno\u015bci ognisk przerzutowych. Obecnie dziewczynka pozostaje pod opiek\u0105 tutejszej kliniki z powodu przewlek\u0142ych zapar\u0107 i nawracaj\u0105cych objaw\u00f3w o charakterze refluksu \u017co\u0142\u0105dkowo-prze\u0142ykowego bez cech zapalenia prze\u0142yku (w endoskopii prawid\u0142owy obraz b\u0142ony \u015bluzowej g\u00f3rnego odcinka przewodu pokarmowego). Obj\u0119ta jest r\u00f3wnie\u017c sta\u0142\u0105 opiek\u0105 Kliniki Transplantacji Szpiku, Onkologii i Hematologii Dzieci\u0119cej bez cech wznowy procesu nowotworowego.H. pylori, choroba trzewna oraz zaka\u017cenia w obr\u0119bie uk\u0142adu moczowego. W\u015br\u00f3d rzadszych przyczyn nale\u017cy uwzgl\u0119dni\u0107: bezoar, zapalenie w\u0105troby i trzustki, kamic\u0119 \u017c\u00f3\u0142ciow\u0105, kamic\u0119 uk\u0142adu moczowego. Natomiast do przyczyn bardzo rzadkich zaliczamy eozynofilowe zapalenie prze\u0142yku, nieswoiste zapalenia jelit, nawracaj\u0105ce wg\u0142obienia, uchy\u0142ek Meckela, torbiele dr\u00f3g \u017c\u00f3\u0142ciowych, schorzenia ginekologiczne (torbiele jajnik\u00f3w), ch\u0142oniaki, guzy zlokalizowane w obr\u0119bie jamy brzusznej [Wed\u0142ug klasycznej definicji Apleya i Naisha, nawracaj\u0105cy b\u00f3l brzuch charakteryzuje wyst\u0105pienie co najmniej 3 epizod\u00f3w b\u00f3lu brzucha w ci\u0105gu 3 miesi\u0119cy, o nasileniu ograniczaj\u0105cym normaln\u0105 aktywno\u015b\u0107 . Jednak rzusznej , 3.Zadaniem lekarza, a jednocze\u015bnie wyzwaniem, jest ustalenie w kt\u00f3rym przypadku przyczyna mo\u017ce by\u0107 organiczna i wymaga poszerzenia diagnostyki. Cz\u0119sto\u015b\u0107, nasilenie, lokalizacja b\u00f3lu oraz jego wp\u0142yw na ograniczenie aktywno\u015bci chorego nie wykazuj\u0105 warto\u015bci w r\u00f3\u017cnicowaniu przyczyny dolegliwo\u015bci. R\u00f3wnie\u017c cz\u0119sto zg\u0142aszane objawy towarzysz\u0105ce w postaci niech\u0119ci do jedzenia, nudno\u015bci, wymiot\u00f3w, wzd\u0119\u0107, czy wsp\u00f3\u0142istniej\u0105ce zaburzenia o charakterze psychologicznym, nie pozwalaj\u0105 jednoznacznie r\u00f3\u017cnicowa\u0107 przyczyn organicznych i czynno\u015bciowych . Jednak Objawy alarmowe w wywiadzie oraz badaniu przedmiotowym wzbudzaj\u0105ce podejrzenie choroby organicznej przedstawiono w Wed\u0142ug obowi\u0105zuj\u0105cych IV Kryteri\u00f3w Rzymskich do jednostek przebiegaj\u0105cych z czynno\u015bciowym b\u00f3lem brzucha nale\u017c\u0105: dyspepsja czynno\u015bciowa, zesp\u00f3\u0142 jelita dra\u017cliwego, migrena brzuszna oraz nieokre\u015blony czynno\u015bciowy b\u00f3l brzucha. R\u00f3wnie\u017c zaparciom czynno\u015bciowym mog\u0105 towarzyszy\u0107 przewlek\u0142e b\u00f3le brzucha . B\u00f3l brzW omawianym przez nas przypadku, u dziewczynki b\u00f3le brzucha o zmiennym nasileniu wyst\u0119powa\u0142y z cz\u0119stotliwo\u015bci\u0105 od kilku razy dziennie do kilku razy w miesi\u0105cu, od oko\u0142o 2 lat. Od okresu niemowl\u0119cego obserwowano zaparcia, kt\u00f3re pocz\u0105tkowo spe\u0142nia\u0142y kryteria dla rozpoznania schorzenia czynno\u015bciowego: typowy pocz\u0105tek zwi\u0105zany ze zmian\u0105 diety, brak objaw\u00f3w alarmowych, bolesne wypr\u00f3\u017cnienia i twarde stolce oddawane 2x w tygodniu, postawa retencyjna . W toku We wst\u0119pnej diagnostyce przewlek\u0142ych b\u00f3l\u00f3w brzucha opr\u00f3cz dok\u0142adnego badania podmiotowego i przedmiotowego z uwzgl\u0119dnieniem objaw\u00f3w alarmowych, nale\u017cy wykona\u0107 podstawowe badania laboratoryjne, do kt\u00f3rych nale\u017c\u0105: morfologia krwi obwodowej z rozmazem, wyk\u0142adniki stanu zapalnego , enzymy trzustkowe i w\u0105trobowe, badanie przesiewowe w kierunku choroby trzewnej, badanie ka\u0142u na obecno\u015b\u0107 krwi utajonej i badanie biochemiczne moczu. Diagnostyka obrazowa w pocz\u0105tkowym etapie powinna obejmowa\u0107 badanie ultrasonograficzne jamy brzusznej , 14. Na International Neuroblastoma Pathology Classification wyr\u00f3\u017cnia si\u0119 dwa podtypy histopatologiczne: GNB intermixed (Schwannian stroma-rich) oraz GNB nodular (composite Schwannian stroma-rich/stroma-dominant and stroma-poor), o zdecydowanie lepszym rokowaniu w przypadku GNB intermixed [Neuroblastoma (NB), ganglioneuroblastoma (GNB) i ganglioneuroma (GN) nale\u017c\u0105 do guz\u00f3w wsp\u00f3\u0142czulnego uk\u0142adu nerwowego wywodz\u0105cych si\u0119 z prymitywnych kom\u00f3rek wsp\u00f3\u0142czulnych zwoj\u00f3w nerwowych . Stanowitermixed , 17. Gantermixed . Mo\u017ce wytermixed . Obraz ktermixed , 19. PodInternational Neuroblastoma Risk Group Staging System (INRGSS) [Rozpoznanie guz\u00f3w neuroblastycznych opiera si\u0119 przede wszystkim na badaniach obrazowych. W badaniu ultrasonograficznym guz mo\u017ce by\u0107 opisany jako niejednorodna echogenicznie masa ze zwapnieniami i cechami rozpadu. Badaniem z wyboru pozostaje tomografia komputerowa, kt\u00f3ra umo\u017cliwia dok\u0142adniejsz\u0105 ocen\u0119 pochodzenia, rozmiar\u00f3w i lokalizacji guza, wyst\u0119powania zwapnie\u0144, zaj\u0119cia regionalnych w\u0119z\u0142\u00f3w ch\u0142onnych, stopnia naciekania s\u0105siaduj\u0105cych tkanek. Rezonans magnetyczny (MRI) jest wykorzystywany do oceny naciekania pierwotnego guza na struktury kana\u0142u kr\u0119gowego oraz do wykrywania przerzut\u00f3w do w\u0105troby u niemowl\u0105t i ma\u0142ych dzieci . Guzy po(INRGSS) . W przyp(INRGSS) .W przypadku ganglioneuroblastoma rokowanie co do odpowiedzi na leczenie i d\u0142ugo\u015bci prze\u017cycia jest zdecydowanie lepsze ni\u017c dla neuroblastoma. Guz mo\u017ce ulega\u0107 spontanicznej regresji (1-2%) lub r\u00f3\u017cnicowaniu do ganglioneuroma. \u0141agodniejszy przebieg kliniczny i lepsze rokowanie obserwowane jest r\u00f3wnie\u017c w przypadku pacjent\u00f3w z GNB intermixed w por\u00f3wnaniu do podtypu nodular , 21.Dok\u0142adny wywiad i badanie przedmiotowe wraz z uwzgl\u0119dnieniem objaw\u00f3w alarmowych oraz odpowiednio dobrane badania dodatkowe, stanowi\u0105 kluczowe narz\u0119dzia w identyfikacji pacjent\u00f3w z chorob\u0105 organiczn\u0105. W diagnostyce r\u00f3\u017cnicowej nale\u017cy r\u00f3wnie\u017c uwzgl\u0119dni\u0107 rzadkie przyczyny przewlek\u0142ych b\u00f3l\u00f3w brzucha, w tym mo\u017cliwo\u015b\u0107 wyst\u0105pienia choroby onkologicznej. Obraz kliniczny guz\u00f3w neuroblastyczych jest niezwykle r\u00f3\u017cnorodny, a lekarze powinni by\u0107 \u015bwiadomi nie tylko klasycznych objaw\u00f3w choroby, ale r\u00f3wnie\u017c tych nietypowych. Ustalenie wskaza\u0144 do poszerzenia diagnostyki pozwala na szybkie postawienie rozpoznania i wdro\u017cenie odpowiedniego post\u0119powania terapeutycznego, co przek\u0142ada si\u0119 na popraw\u0119 wska\u017anik\u00f3w prze\u017cycia i zminimalizowanie nieodwracalnych uszkodze\u0144."}
{"text": "In the original article, there was a mistake in In the original article, there was also a mistake in the Appendix table \u201cThe Authenticity Scale: Original, Russian, and Corresponding English Versions\u201d as published. The text of Item 1 (both English and Russian Wording) was incorrectly highlighted in bold. The text should be in normal (regular) style. The corrected Appendix table \u201cThe Authenticity Scale: Original, Russian, and Corresponding English Versions\u201d appears below.The authors apologize for these errors and state that they do not change the scientific conclusions of the article in any way. The original article has been updated.AppendixAuthenticity Scale: Original, Russian, and corresponding English versions.The \u0418\u043d\u0441\u0442\u0440\u0443\u043a\u0446\u0438\u044f:\u041f\u043e\u0436\u0430\u043b\u0443\u0439\u0441\u0442\u0430, \u043f\u0440\u043e\u0447\u0442\u0438\u0442\u0435 \u0441\u043f\u0438\u0441\u043e\u043a \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u043d\u044b\u0445 \u0443\u0442\u0432\u0435\u0440\u0436\u0434\u0435\u043d\u0438\u0439 \u0438 \u043e\u0446\u0435\u043d\u0438\u0442\u0435 \u0438\u0445 \u0441 \u0442\u043e\u0447\u043a\u0438 \u0437\u0440\u0435\u043d\u0438\u044f \u0442\u043e\u0433\u043e, \u043d\u0430\u0441\u043a\u043e\u043b\u044c\u043a\u043e \u043e\u043d\u0438 \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440\u0438\u0437\u0443\u044e\u0442 \u0412\u0430\u0448\u0438 \u043f\u0440\u0438\u0432\u044b\u0447\u043a\u0438 \u0438 \u043f\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u0435. \u041f\u043e\u0441\u0442\u0430\u0432\u044c\u0442\u0435 \u0433\u0430\u043b\u043e\u0447\u043a\u0443 \u0432 \u044f\u0447\u0435\u0439\u043a\u0435 \u043f\u043e\u0434 \u0442\u0435\u043c \u043e\u0442\u0432\u0435\u0442\u043e\u043c, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u043f\u043e\u0434\u0445\u043e\u0434\u0438\u0442 \u0412\u0430\u043c.Instruction:Please read the list of statements provided and rate them in terms of how they characterize your habits and behavior. Please tick the answer that best describes you."}
{"text": "In this paper, the phrase should be corrected as follows:For \u201cwith 50 \u03bcM ionomycin/1 \u03bcM phorbol\u201dread \u201cwith 0.5 \u03bcM ionomycin/10 nM phorbol\u201d."}
{"text": "This article appoints a novel model of rough set approximations (RSA), namely, rough set approximation models build on containment neighborhoods RSA (CRSA), that generalize the traditional notions of RSA and obtain valuable consequences by minifying the boundary areas. To justify this extension, it is integrated with the binary version of the honey badger optimization (HBO) algorithm as a feature selection (FS) approach. The main target of using this extension is to assess the quality of selected features. To evaluate the performance of BHBO based on CRSA, a set of ten datasets is used. In addition, the results of BHOB are compared with other well-known FS approaches. The results show the superiority of CRSA over the traditional RS approximations. In addition, they illustrate the high ability of BHBO to improve the classification accuracy overall the compared methods in terms of performance metrics. In recent days, the high dimensionality  became aRecently, the rough set theory (RS) , 23 has z\u211b, \u211bz neighborhoods c\u2009= {z \u2208 \u039b : \u2229\u2113=1n(zC\u2113z)\u2286M}\u2009=\u2009n)(E\u2297L(M).(z\u2208n)(E\u2297L(M), then \u2229\u2113=1n(zC\u2113z)\u2286M. Since Suppose z\u2208n)(E\u2297L(M), then \u2229\u2113=1n(zC\u2113z)\u2286M. In view of (z \u2208 \u2229\u2113=1n(zC\u2113z). Hence, z \u2208 M. So, n)(E\u2297L(M)\u2286M.Let n)(E\u2297L(n)(E\u2297L(M))\u2286n)(E\u2297L(M). The other side, let z\u2208n)(E\u2297L(M), then \u2229\u2113=1n(zC\u2113z)\u2286M. Let y\u2208\u2229\u2113=1n(zC\u2113z). Hence, in view of \u2113=1n(yC\u2113y)\u2286\u2229\u2113=1n(zC\u2113z) for all y\u2208\u2229\u2113=1n(zC\u2113z). Consequently, \u2229\u2113=1n(yC\u2113y)\u2286M for all y\u2208\u2229\u2113=1n(zC\u2113z) and so y\u2208n)(E\u2297L(M) for all y\u2208\u2229\u2113=1n(zC\u2113z). Hence, \u2229\u2113=1n(zC\u2113z)\u2286\u2009n)(E\u2297L(M). So, z\u2208n)(E\u2297L(n)(E\u2297L(M)), which implies that n)(E\u2297L(M)\u2286n)(E\u2297L(n)(E\u2297L(M)) and so n)(E\u2297L(M)\u2009=\u2009n)(E\u2297L(n)(E\u2297L(M)).According to properties 5 and 7, we get Statements 2, 4, and 6 are obvious.Mostly, the opposite of  of Propo1, \u211b2 are two binary relations on \u039b\u2009=\u2009{\u03b21, \u03b22, \u03b23, \u03b24} such that1\u2009=\u2009{, , , , , }\u2009\u211b2\u2009=\u2009{, , , , , , }\u2009\u211bConsider \u211bM\u2009=\u2009{\u03b22}, \u03b24}, then (2)E\u2297L(M)\u2009=\u2009\u2205, \u03b24}, \u03b22, \u03b24}. So, If M={\u03b22}, In view of \u2113, \u2113 \u2208 {1,2,\u2026, n} is a dominance relation on \u039b and M\u2286\u039b, then the coming statements hold:M\u2286n)(E\u2297L(n)(E\u2297U((M))M\u2287n)(E\u2297U(n)(E\u2297L(M))n)(E\u2297U(M)\u2009=\u2009n)(E\u2297L(n)(E\u2297U(M))n)(E\u2297L(M)\u2009=\u2009n)(E\u2297U(n)(E\u2297L(M))If each \u211bx \u2208 M. We shall prove that \u2229\u2113=1n(xC\u2113x)\u2286n)(E\u2297U(M), i.e., \u2229\u2113=1n(yC\u2113y)\u2229M \u2260 \u2205, \u2200y\u2208\u2229\u2113=1n(xC\u2113x). In view of 7\u2022 of Proposition 3, x\u2208n)(E\u2297U(M). Let y \u2208 \u2229\u2113=1n(xC\u2113x). Since each \u211b\u2113, \u2113 \u2208 {1,2,\u2026, n} is a dominance relation on \u039b, then by Corollary 3, \u2229\u2113=1n(yC\u2113y)\u2229M=\u2229\u2113=1n(xC\u2113x)\u2229M \u2260 \u2205. So the required has proven.Let Similar to 1.n)(E\u2297L(n)(E\u2297U(M))\u2286n)(E\u2297U(M). On the other hand, in view of 8\u2022 of n)(E\u2297U(M)\u2286n)(E\u2297L(n)(E\u2297U(n)(E\u2297U(M)))\u2009=\u2009n)(E\u2297L(n)(E\u2297U(M)). So, n)(E\u2297U(M)\u2009= n)(E\u2297L(n)(E\u2297U(M)).Obviously, from 7 of Similar to 3.\u2113, \u2113 \u2208 {1,2,\u2026, n} on \u039b, except the dominance  relation.The next example shows that the statements 1\u2009and\u20093 of \u03b21, \u03b22, \u03b23, \u03b24} and \u25b3 is an identity relation on \u039b, then we discuss the following cases:1, \u211b2 are two binary relations on \u039b defined as \u211b1\u2009=\u2009{, , , , , , , , , , , }\u211b2\u2009=\u2009{, , , , , , , }Then we get \u2229\u2113=12(\u03b21C\u2113\u03b21)\u2009=\u2009{\u03b21}, \u2229\u2113=12(\u03b22C\u2113\u03b22)\u2009=\u2009{\u03b22}, \u2229\u2113=12(\u03b23C\u2113\u03b23)\u2009=\u2009{\u03b23}, \u2229\u2113=12(\u03b24C\u2113\u03b24)\u2009=\u2009{\u03b21, \u03b24}.Suppose \u211bM\u2009=\u2009{\u03b24}, then (2)E\u2297U(M)\u2009=\u2009{\u03b24}, (2)E\u2297L((2)E\u2297U(M))\u2009=\u2009\u2205.If 1, \u211b2 are two reflexive relations on \u039b defined as \u211b1\u2009=\u2009\u25b3\u222a{, , , } and \u211b2\u2009=\u2009\u25b3\u222a{, , , , , }.Suppose \u211b\u2113=12(\u03b21C\u2113\u03b21)\u2009=\u2009{\u03b21}, \u2229\u2113=12(\u03b22C\u2113\u03b22)\u2009=\u2009{\u03b22}, \u2229\u2113=12(\u03b23C\u2113\u03b23)\u2009=\u2009{\u03b23}, \u2229\u2113=12(\u03b24C\u2113\u03b24)\u2009=\u2009{\u03b23, \u03b24}.\u2229M\u2009=\u2009{\u03b24}, then (2)E\u2297U(M)\u2009=\u2009{\u03b24}, (2)E\u2297L((2)E\u2297U(M))\u2009=\u2009\u2205.If 1, \u211b2 are two tolerance relations on \u039b defined as \u211b1\u2009=\u2009\u25b3\u222a{, , , , , , , } and \u211b2\u2009=\u2009\u25b3\u222a{, , , , , , , , , }.Suppose \u211b\u2113=12(\u03b21C\u2113\u03b21)\u2009=\u2009{\u03b21}, \u2229\u2113=12(\u03b22C\u2113\u03b22)\u2009=\u2009{\u03b22, \u03b24}, \u2229\u2113=12(\u03b23C\u2113\u03b23)\u2009=\u2009\u039b, \u2229\u2113=12(\u03b24C\u2113\u03b24)\u2009=\u2009{\u03b24}.\u2229M\u2009=\u2009{\u03b21, \u03b22}, we find that (2)E\u2297U(M)\u2009=\u2009{\u03b21, \u03b22, \u03b23}, (2)E\u2297L((2)E\u2297U(M))\u2009=\u2009{\u03b21}.If 1, \u211b2 are two symmetric and transitive relations on \u039b defined as \u211b1\u2009=\u2009{, , , , , , }\u211b2\u2009=\u2009{, , , , , , , , }Suppose \u211b\u2113=12(\u03b21C\u2113\u03b21)\u2009=\u2009{\u03b21, \u03b24}, \u2229\u2113=12(\u03b22C\u2113\u03b22)\u2009=\u2009\u039b, \u2229\u2113=12(\u03b23C\u2113\u03b23)\u2009=\u2009{\u03b23, \u03b24}, \u2229\u2113=12(\u03b24C\u2113\u03b24)\u2009=\u2009{\u03b24}\u2229M\u2009=\u2009{\u03b21, \u03b23}, we find that (2)E\u2297U(M)\u2009=\u2009{\u03b21, \u03b22, \u03b23}, (2)E\u2297L((2)E\u2297U(M))\u2009=\u2009\u2205.If If \u039b\u2009=\u2009{n)(E\u2297L(.) satisfies Kuratowski's interior axioms and so one can deduce a topology n)(\u22a4 on \u039b that is given by n)(\u22a4\u2009=\u2009{M\u2286\u039b : \u2009n)(E\u2297L(M)=M}.According to the statements 2, 3, 7, and 8 of Next, we have the following remark using \u2113, \u2113 \u2208 {1,2,\u2026, n} is a dominance relation on \u039b, then \u2229\u2113=1nzC\u2113z is n)(\u22a4C\u2113 produced from zC\u2113z is finer than the topology n)(\u22a4\u2113\u211b produced from z\u211b\u2113z is finer than the topology n)(\u22a4\u2113\u211b produced from z\u211b\u2113z and the topology n)(\u22a4C\u2113\u232a\u2329\u2009=\u2009n), , , , , , }\u2009\u211b2\u2009=\u2009{, , , , , , , , }\u2009\u211b\u2113=12(\u03b21\u211b\u2113\u03b21)\u2009=\u2009\u2205, \u2229\u2113=12(\u03b22\u211b\u2113\u03b22)\u2009=\u2009{\u03b22}, \u2229\u2113=12(\u03b23\u211b\u2113\u03b23)\u2009=\u2009\u2205, \u2229\u2113=12(\u03b24\u211b\u2113\u03b24)\u2009=\u2009\u2205\u2009\u2229\u2113=12(\u03b21C\u2113\u03b21)\u2009=\u2009{\u03b21}, \u2229\u2113=12(\u03b22C\u2113\u03b22)\u2009=\u2009{\u03b22, \u03b23}, \u2229\u2113=12(\u03b23C\u2113\u03b23)\u2009=\u2009{\u03b23}, \u2229\u2113=12(\u03b24C\u2113\u03b24)\u2009=\u2009{\u03b24}\u2009\u2229\u2113=12(\u211b\u2113\u2329\u03b21\u232a\u211b\u2113)\u2009=\u2009\u2229\u2113=12(C\u2113\u2329\u03b21\u232aC\u2113)\u2009=\u2009{\u03b21}, \u2229\u2113=12(\u211b\u2113\u2329\u03b22\u232a\u211b\u2113)\u2009=\u2009\u2229\u2113=12(C\u2113\u2329\u03b22\u232aC\u2113)\u2009=\u2009{\u03b22}\u2009\u2229\u2113=12(\u211b\u2113\u2329\u03b23\u232a\u211b\u2113)\u2009=\u2009\u2229\u2113=12(C\u2113\u2329\u03b23\u232aC\u2113)\u2009=\u2009{\u03b22, \u03b23}, \u2229\u2113=12(\u211b\u2113\u2329\u03b24\u232a\u211b\u2113)\u2009=\u2009\u2229\u2113=12(C\u2113\u2329\u03b24\u232aC\u2113)\u2009=\u2009{\u03b24}\u2009\u2229n)(\u22a4\u2113\u211b\u2009=\u2009P(\u039b)\u2009n)(\u22a4C\u2113\u232a\u2329\u2009=\u2009n)\u2113, \u2113 \u2208 {1,2,\u2026, n} is reflexive, n)(\u22a4C\u2113\u2288n)(\u22a4\u2113\u211b\u2288n)(\u22a4C\u2113If each \u211bThe converse of 1 and 2 of (2)n)(\u22a4C\u2113 are not comparable(3)n)(\u22a4C\u2113 are not comparable(4)\u2113, \u2113 \u2208 {1,2,\u2026, n} is reflexive, then n)(\u22a4\u2113\u211b\u2286n)(\u22a4\u2113\u232a\u2329\u211bIf each \u211b(5)\u2113, \u2113 \u2208 {1,2,\u2026, n} is transitive, then n)(\u22a4\u2113\u232a\u2329\u211b\u2286n) that is defined asC > 1 stands for a constant value, T represents the total number of iterations, and t indicates the current iteration.The steps of HBO begin by setting the initial of value of agents using the following:lowed by , the expThe next step is to update the solutions using the operators of Digging stage. This is performed based on the cardioid movements formulated asZnew stands for the new value of Zi, Zb represents the best solution found so far, and r3, r4, r5, and r6 are random numbers. (B) is a constant number. (F) is a parameter used to control the search direction and it is the value determined using the following equation:In equation , Znew stI stands for the smell intensity of the prey (xb) and it is used to represent the distance between the xb and xi. It is formulated asr7 is a random number. The steps of HBO are given in Meanwhile, the solutions can be updated using operators of Honey stage. This process is achieved using the following formula:We present the steps of the developed feature selection based on modified HBO and combined it with CRSA approximations in this section. fk, \u2003k=1,2,\u2026, d) of the dataset corresponding to each solution dimension. Then, such dataset is split into training and testing sets. As well as, through the interval , the generation of the initial value for N agents Z is performed. Furthermore, the computation of the value of the objective function of Zi, \u2003i=1,2,\u2026, N with determining the finest among them is performed. Through utilizing the HBO operators, modernize the agents. The modernization process is repeated till the stopping condition is met. Each phase is discussed in detail as follows.The initialization of HBOCRSA is represented in identifying the features number  given as in equation  given byThe second step in the developed method contains choosing the features that are more relevant regarding ones in equation . This isquation (FitZi=\u03b7\u00d7ZPOSC(D) represents the positive region defined as\u03b3C(D) aims for computing the features approximating power.In equation , POSC(D)efined as6POSCD=\u222aBZb, after that the agents are modernized by utilizing the HBO operators given in Finally, the third step aims for identifying the finest agent Zb and perform classification for the reduced set by utilizing the KNN classifier for the features quality assessing.Meeting the stopping condition for the proposed technique can be tested and returning the finest solution as the solutions accepted with repeating the strides existed in the second stage. Moreover, the testing set is used to assess the relevant features existed in the finest solution The performance of the developed method can be justified through the modified CRSA which is considered as RS extension with concluding some of the experiments. Through our work and experiments, we depended on using ten datasets, in addition to comparing the results with other FS approaches. In this experiment, the proposed BHBA is used to determine the relevant feature from the given data. To conduct this, the CRSA is used as a part of the fitness function that is defined in equation .The quality of HBOCRSA is validated by utilizing ten datasets with varying dimensionality. The datasets, gathered from various fields, are taken online from UCI , and thoThe efficiency of HBOCRSA can be validated and justified, where the dataset is split into 80% and 20% training and testing sets, respectively, where these percentages form the whole number. Through 30 independent times, each algorithm has been conducted for guaranteeing the comparison quality. The comparisons are performed through using the algorithms of the salp swarm algorithm (SSA), self-adaptive differential evolution (SaDE), grey-wolf optimization (GWO), genetic algorithm (GA), and teaching-learning-based optimization (TLBO) as competing ones for feature selection. With taking the note that each algorithm parameter is set regarding its implementation, the iteration number and the population size are set to 15 and 20 which are considered the common parameters. Thereafter, each solution in the population has dimension which equals to the features number for each dataset.(i)Acc) exemplifies the accuracies average; overall, the runs number (Nr=30) is given asAverage accuracy (AVG(ii)|BZBest|) calculates the features average chosen by each algorithm through all runs and it is given asAverage number of the selected features , namely, the rough set approximation models depending on containment neighborhoods (CRSA), that generalize the classical notions of (RSA) and derive a number of distinguished results. To evaluate the appropriateness of this model, it is applied to enhance the classification of contrasting dataset by utilizing it an objective function for distinct feature selection approach. This has been carried out by applying the binary version of honey badger optimization algorithm (BHBO) as feature selection method. The effects of HBOCRSA are compared with different MH techniques, which involve GWO, LSHADE, SSA, GA, TLBO, and SaDE. A class of ten datasets is employed to assess the performance of the developed method. The experiential results clarified the high performance of the developed method as FS approach which owns accuracy better than other methods. Furthermore, the number of the selected features acquired utilizing the HBOCRSA is smaller than the other methods. Additionally, the accomplishment of the model of (CRSA) is better than classical (RS) approach that is based on the factors of performance measures. Based on the favourable consequences gained from the developed method, it can be applied in diverse scopes, for instance, cloud computing, image processing, and IoT applications. In addition, it can be reconstructed as multiobjective technique and used to several sets of reality multiobjective problems involving feature, selection, and engineering issues and others."}
{"text": "RM) T cells in mouse and human melanoma-associated vitiligo skin form large lymphoid aggregates with CXCL16-expressing dendritic cells. CD11c depletion or disruption of the CXCR6-CXCL16 axis results in loss of skin TRM cells and tumor immunity.Tissue-resident memory (T RM) T cells are emerging as critical components of the immune response to cancer; yet, requirements for their ongoing function and maintenance remain unclear. APCs promote TRM cell differentiation and re-activation but have not been implicated in sustaining TRM cell responses. Here, we identified a novel role for dendritic cells in supporting TRM to melanoma. We showed that CD8 TRM cells remain in close proximity to dendritic cells in the skin. Depletion of CD11c+ cells results in rapid disaggregation and eventual loss of melanoma-specific TRM cells. In addition, we determined that TRM migration and/or persistence requires chemotaxis and adhesion mediated by the CXCR6/CXCL16 axis. The interaction between CXCR6-expressing TRM cells and CXCL16-expressing APCs was found to be critical for sustaining TRM cell\u2013mediated tumor protection. These findings substantially expand our knowledge of APC functions in TRM T-cell homeostasis and longevity.Tissue-resident memory (T RM) cells are a unique subset of memory cells persisting at initial sites of challenge   . CXCR6 i (Bonzo) , 19, 20. (Bonzo) , 22, CXC (Bonzo) , 24. CXCnfection . Howeverrstitium . Recentlcination . Thus, t and DCs . CXCR6 i and DCs . HoweverRM cells in maintaining their long-term skin residence through direct aggregation with transmembrane CXCL16-expressing DCs. Short-term depletion of CD11c+ cells led to a reduction in skin CXCL16 expression and CD8 TRM cell dispersal, whereas longer depletion led to TRM cell loss. While CXCR6-deficient CD8 effector T cells were capable of infiltrating primary tumors, CD8 TRM cell positioning within MAV-affected skin was disrupted, resulting in defective anti-tumor memory responses. Together, these data define a new role for antigen presenting cells in capturing and holding CD8 TRM cells within peripheral tissues through CXCL16/CXCR6-mediated migration and/or adhesion.In this study, we report an important requirement for CXCR6 expression on CD8 TRM cells are present in abundant numbers in the skin of depigmented (MAV affected) but not unaffected mice , 3130, 3ted mice . MoreoveRM cells . Using ited skin . In addited skin . The appted skin , 35, 36. T cells  indicatiRM cells co-clustered with APCs, MAV-affected and unaffected skin sections were stained with antibodies specific for CD11c, a marker expressed on a number of APCs including DCs and macrophages  receptor controlled by the CD11c promoter (CD11c.DTR) for which administration of DT selectively depletes CD11c+ cells. MAV was established in CD11c.DTR+ and CD11c.DTRneg littermate mice followed by three doses of DT over the course of 7 d to determine the effects of short-term CD11c depletion on CD8 TRM cell clusters   . Luc+Pmeesection . Beginniesection . Luc+Pmeof CXCR6 . Total Cof CXCR6 . The perof CXCR6 . Howeverof CXCR6 . CXCR6 eperiment . To deteperiment . TogetheRM cells and that its ligand CXCL16 is expressed on the surface of closely associated CD11c+ cells. To determine whether CXCR6 and CXCL16 interaction is required for leukocyte clustering in skin, MAV was established in wild-type and CXCR6-deficient (CXCR6\u2212/\u2212) mice. CD8 effector T cells develop in response to regulatory T cell depletion, infiltrate B16F10 tumors and mediate melanocyte destruction in the skin. Effector T cell dependent melanocyte killing becomes evident in wild-type mice 20\u201330 d post tumor resection with the regrowth of white hair at the surgical site in 60\u201370% of treated mice  and cDC2 dermal DCs , indicating this is likely a mixed population of cells comprised of Langerhans cells and cCD2 cells. Future studies that combine spatial and transcriptional data will be informative in identifying the specific myeloid cells that co-aggregate with and are required for skin CD8 TRM cell maintenance.Whereas CD11cve tract . Howeverve tract , 39, 40.ve tract , 40. Whec+ cells , 49. Howrmal DCs , 50. TraRM cell clustering with CD11c+ cell also remains to be determined. Similar to other memory T-cell subsets, CD8 TRM cells require homeostatic cytokines for their long-term survival. In the skin, hair follicle derived IL-15 can support TRM cell survival  (no. 00029821) and performed in accordance with ethical guidelines and regulation. De-identified skin biopsy samples from patients with progressing MAV were obtained from the Department of Surgical Oncology at DHMC. Melanoma patients developed vitiligo before or during immunotherapy with nivolumab and/or ipilimumab. Healthy, control, skin was obtained from the DHMC Pathology core.CXCR6\u2212/\u2212 (stock #:005693), CD11c.DTR (stock #:004509), and Thy1.1+Pmel (stock #:005023) mice were purchased from The Jackson Laboratory and bred in-house. To generate CXCR6\u2212/\u2212Thy1.1+Pmel mice, Thy1.1+Pmel were crossed with mice CXCR6\u2212/\u2212 mice. All studies were performed in accordance with the Institutional Animal Care and Use Committee (IACUC) Guidelines at Dartmouth College. Animals were housed within the specific pathogen-free section of the Center for Comparative Medicine and Research at Dartmouth College.C57BL/6 and CD45.1 mice were purchased from Charles River Breeding Laboratories. All mouse studies were performed in accordance with the IACUC Guidelines at Dartmouth College. Animals were housed within the specific pathogen-free section of the Center for Comparative Medicine and Research at Dartmouth College in standard cages containing a maximum of five mice per cage. Mice had ad libitum access to water and food. In vivo studies used a minimum of five mice per group. Sample size was altered after experiments were initiated only if tumors re-grew or if the surgical site failed to heal. As indicated, the final data were pooled from identical experiments. All experiments were conducted with fixed end points and were performed in either duplicate or triplicate.131Cs irradiator on three consecutive days before i.v. injection of CD11c.DTR BM cells (one donor:one recipient). Chimeric mice were used 6 wk post reconstitution.7-wk-old CD45.1 mice received 350 rad per day of \u03b3 irradiation from a i.d.) in C57BL/6 mice. Before using the cell line for experiments, it was tested by the Infectious Microbe PCR Amplification Test and authenticated by the Research Animal Diagnostic Laboratory (RADIL) at the University of Missouri. Tumor cells were cultured in RPMI-1640 containing 10% of heat-inactivated FBS in a humidified incubator at 37\u00b0C with 7% CO2.To ensure reproducible growth, the B16F10 (B16) mouse melanoma cell line was passaged intradermally (i.d. on the right flank with 2 \u00d7 105 B16 cells on day 0. On days 4 and 10 mice were treated with anti-CD4 mAb (clone GK1.5) 250 \u03bcg i.p. and tumors were surgically excised on day 12 . Purified na\u00efve T cells were adoptively transferred into recipients or activated before retroviral transduction. CD8 T cells were activated in a 24-well plate at a density of 1 \u00d7 106 per well in complete T-cell medium  with Dynabeads Mouse T-Activator CD3/CD28  and 50 ng/ml recombinant mouse IL-2 . T cell cultures were passaged with fresh complete T-cell medium and cytokines as needed.CD8 T cells were isolated from pooled LNs and spleen of na\u00efve C57BL/6, EF154513 and EF154514) separated by P2A self-cleaving sequence were synthesized as a genomic block (gblock) from IDT and cloned into pCMV2.1 MSCV-based retrovirus upstream of an internal ribosomal entry sequence and GFP. Pmel gblock sequence : (5\u2032- gccggaattcagatctaccatgaaatccttgagtgtttcactagtggtcctgtggctccagtttaattgggtgagaagccagcagaaggtgcagcagagcccagaatccctcactgtctcagaggga gccatggcctctctcaactgcactttcagtgatcgttcttctgacaacttcaggtggtacagac agcattctgggaaaggccttgaggtgctggtgtccatcttctctgatggtgaaaaggaagaa ggcagttttacagctcacctcaatagagccagcctgcatgttttcctacacatcagagagccgcaacccagtgactctgctctctacctctgtgcagtgaacacaggaaactacaaatacgtctttggagcaggtaccagactgaaggttatagcacacatccagaacccagaacctgctgtgtaccagttaaaagatcctcggtctcaggacagcaccctctgcctgttcaccgactttgactcccaaatcaatgtgccgaaaaccatggaatctggaacgttcatcactgacaaaactgtgctgga catgaaagctatggattccaagagcaatggggccattgcctggagcaaccagacaagc ttcacctgccaagatatcttcaaagagaccaacgccacctaccccagttcagacgttccc tgtgatgccacgttgactgagaaaagctttgaaacagatatgaacctaaactttcaaaacc tgtcagttatgggactccgaatcctcctgctgaaagtagccggatttaacctgctcatgacgctgcggctgtggtccagcggctccggagccacgaacttctctctgttaaagcaagcaggagacgtggaagaaaaccccggtcccatgggcaccaggcttcttggctgggcagtgggcccgcggcttcttggctgggcagtgttctgtctccttgacacagtactgtctgaagctggagtcacccagtctcccagatatgcagtcctacaggaagggcaagctgtttccttttggtgtgaccctatttctggacatgataccctttactggtatcagcagcccagagaccaggggccccagcttctagtttactttcgggatgaggctgttatagataattcacagttgccctcggatcgattttctgctgtgaggcctaaaggaactaactccactctcaagatccagtctgcaaagcagggcgacacagccacctatctctgtgccagcagtttccacagggactataattcgcccctctactttgcggcaggcacccggctcactgtgacagaggatctgagaaatgtgactccacccaaggtctccttgtttgagccatcaaaagcagagattgcaaacaaacaaaaggctaccctcgtgtgcttggccaggggcttcttccctgaccacgtggagctgagctggtgggtgaatggcaaggaggtccacagtggggtcagcacggaccctcaggcctacaaggagagcaattatagctactgcctgagcagccgcctgagggtctctgctaccttctggcacaatcctcgcaaccacttccgctgccaagtgcagttccatgggctttcagaggaggacaagtggccagagggctcacccaaacctgtcacacagaacatcagtgcagaggcctggggccgagcagactgtgggattacctcagcatcctatcaacaaggggtcttgtctgccaccatcctctatgagatcctgctagggaaagccaccctgtatgctgtgcttgtcagtacactggtggtgatggctatggtcaaaagaaagaattcatgactcgagtgtttaaacgtcgacggtatcgataagcttcgggatc-3\u2032).The Pmel TCR \u03b1 and \u03b2 chain sequences  with a luciferase reporter gene using an MSCV-based retrovirus or TCR\u03b1/TCR\u03b2 specific for gp100 (described above). At 6 h post-transduction, the medium was replaced with 6 ml of complete T-cell medium. At a total of 48 h post transfection, the medium containing virus was collected and passed through a 0.45 \u03bcm filter.293T cells were plated at a density of 2.5 \u00d7 10\u2212/\u2212Thy1.1+Pmel and Thy1.1+Pmel CD8 T cells were transduced with a PpyRE9 retroviral supernatant to generate Luc+Pmel CD8 T cells. CD8 T cells from WT mice were first transduced with Pmel retroviral supernatant containing 8 \u03bcg/ml polybrene and 50 ng/ml recombinant mouse IL-2 then centrifuged at 1,500g for 1.5 h, followed by an additional transduction with PpyRE9 retroviral supernatant. In all cases, CD8 T cells were rested at 37\u00b0C for 5 h before replacing the retroviral supernatant with fresh complete T-cell medium containing 50 ng/ml recombinant mouse IL-2. Luc+Pmel CD8 T cells were selected with 400 \u03bcg/ml G418 for 48 h before adoptive transfer. WT mice transduced with Pmel and PpyRE9 were FACs sorted based on expression of GFP and anti-mouse Vb13-PE  and cultured for an additional 24 h in complete T-cell medium containing 50 ng/ml recombinant mouse IL-2 before adoptive transfer into recipient mice at a dose of 2.5\u20135 \u00d7 104 cells/mouse.24 h after activation, CXCR62 of specified regions of interest.Mice were first shaved then injected i.p. with 200 \u03bcl of 15 mg/ml of the luciferase substrate, d-luciferin  in PBS and imaged after 8 min with a Xenogen IVIS-200 system (PerkinElmer). Photon emission was detected with acquisition times ranging from 5 s to 3 min. Analysis of the images was performed using Living Image software (PerkinElmer) by obtaining average radiance per second per cmi.d. injections of DT  at the site of depigmentation over the course of 1 wk. In experiments where CD11c+ cell depletion was continuously maintained, CD11c.DTR bone marrow chimeric mice received 0.25 \u03bcg of DT i.p. every 3 d beginning on day 30 post-surgery and lasting until termination of the experiment.For short-term depletion of CD11c expressing cells, CD11c.DTR mice received 3\u201350 ng 2 patch of skin overlapping the surgery site as well as distal skin were harvested. Lymphoid tissues were mechanically dissociated. Skin was minced and incubated in 2 mg/ml Collagenase Type IV , 0.2 mg/ml DNase I (Sigma-Aldrich), and 2% FBS in HBSS at 37\u00b0C for 25 min with a magnetic stir bar. Remaining skin fragments were mechanically dissociated in RPMI-1640 containing 10% FBS and 2 mM EDTA. Cell suspensions were first stained for live cells with Zombie Aqua Fixable Viability dye  then Fc receptors were blocked using anti-CD16 and anti-CD32 antibodies . Samples were stained for 30 min on ice with various antibody combinations. Antibodies from BioLegend: anti-mouse CD45-APC/Fire 750 , anti-mouse CD8\u03b1-PerCP/Cynine5.5, -PE/Cy7 , anti-mouse Thy1.1-PerCP/Cynine5.5 , anti-mouse CD103-Alexa Fluor 647, -FITC , anti-mouse CD44-APC/Fire 750 , anti-mouse CD62L-Brilliant Violet 510 , anti-mouse CXCR6-Brilliant Violet 421 , anti-mouse CD11c Alex Fluor 647 , anti-mouse CD11b-Alexa Fluor 488 ; Bioss: anti-mouse CXCL16 Alexa Fluor 488 . For intracellular staining of CXCR6, cells were fixed/permeabilized using reagents from the Foxp3 Staining Kit  following the manufacturer\u2019s protocol. Flow cytometry was performed on a MACSQuant 10 Analyzer (Miltenyi), and data were analyzed using FlowJo software (Tree Star).On indicated days after tumor excision, mice were euthanized and inguinal (tumor-draining) lymph nodes, spleen, and a 2-cmFor histological examination, tissues were fixed with 10% formalin in phosphate-buffered saline, embedded in paraffin, and cut into 4 \u00b5m sections. The following primary antibodies were used to stain the paraffin embedded sections: anti-human CD8  anti-human CD11c , anti-human CXCL16 , anti-human CXCR6 , and Fontana Masson to detect melanin. IHC stains were performed using Leica Bond Max and RX Automated stainer (Leica Microsystems), Bond Epitope Retrieval 2 (Leica Microsystem), and ChromoPlex 1 Dual Detection for BOND kit (Leica Microsystem) all according to the manufacturers\u2019 instructions. Immunohistochemical stains of human tissues were conducted by the Dartmouth Pathology Shared Resources.Excised skins were incubated in 4% PFA for 15 min at 4\u00b0C followed by a 1-h incubation in 30% sucrose all in PBS. Skins were embedded in optimum cutting temperature . 10 \u03bcm sections were cut using a cryostat, air-dried, fixed in cold methanol, and then rehydrated in PBS. Sections were blocked with 5% BSA, 1% goat serum, 1% rat serum and 1% donkey serum in PBS for 1 h at RT and then stained overnight at 4\u00b0C with combinations of the following directly conjugated antibodies from BioLegend: anti-mouse CD8\u03b2-Alexa Fluor 555 (clone 53-6.7), anti-mouse CD11c Alexa Fluor 647 (clone N418), and anti-mouse CD11b Alexa Fluor 488 ; Bioss: anti-mouse CXCL16 Alexa Fluor 488 ; Hoechst 33324 . Slides were mounted using ProLong Diamond Antifade Reagent  and left overnight at room temperature to set. To detect GFP, skin was rehydrated in PBS, permeabilized with 0.25% Triton X-100 in PBST (PBS and 1% Tween 20) for 10 min at RT, then blocked with 5% BSA and 0.1% Triton X-100 in PBS at RT for 1 h. Anti-mouse GFP-Alexa Fluor 555  was diluted in 1% BSA with 0.1% Triton X-100 and incubated ON at 4\u00b0C.Purified anti-mouse CD8\u03b2  was conjugated to Alexa Fluor 555 using the Alexa Fluor 555 Antibody Labeling Kit  following the manufacturer\u2019s protocol.Images were acquired with a Zeiss LSM 800 microscope fitted with GaAsP detectors, using a 40\u00d7 Plan-Apochromat 1.4NA objective. Patient skin and whole murine skin scans were acquired with the PerkinElmer Vectra three automated Olympus upright BX51 fluorescence microscope using the 10\u00d7 UPlan SApo NA 0.40 WD and the 20\u00d7 UPlan SApo NA 0.75 WD objectives.2 and distances between cells in patient skin was manually quantified/measured using Fiji (ImageJ). inForm image analysis software (PerkinElmer) was used for colocalization measurements. Signal crosstalk was eliminated by first creating a spectral library to define the spectral curve for each chromogen . The spectral library was then used to unmix the signals on the multicolored slides by recognizing the unique spectral curves. After spectral unmixing, the pixel-based colocalization image analysis of the inForm software package was used to determine the percentage of cells co-expressing target markers.Number of cells/cm+ cell. The epidermis was first masked, CD8 T cells were identified using the spots tool, whereas CD11c+ cells were identified using the surface tool. After identification of both cell types Imaris\u2019 shortest distance tool was used to calculate the distances between the two.Images of murine skin were analyzed and processed with Fiji (ImageJ) and Imaris 9.5 software (Bitplane). To determine the shortest distance between two CD8 T cells, ND plugin for ImageJ was used according to the developer\u2019s instructions . Images +CD11bneg and CD11c+CD11b+ monocytes were sorted from MAV skin 50 d after surgical removal of B16F10 tumor. Each tissue was digested as described above, two mice were pooled before staining, for a total of three experimental samples. Single-cell suspensions were prepared for each tissue and stained with anti-mouse CD8a-PerCP/Cynine5.5 , anti-mouse CD11c Alexa Fluor 647 , anti-mouse CD11b Alexa Fluor 488 , and anti-mouse CD45.2 Alexa Fluor 488 . The various cell populations were sorted directly into 200 \u03bcl QIAGEN RLT buffer using an ARIA-II cell sorter (BD Biosciences). Total RNA was purified using RNeasy Mini Kit  following the manufacturer\u2019s protocol. From the extracted RNA, cDNA was made using the SMART-Seq v4 Ultra Low Input Kit (Takara) and 10 cycles of cDNA amplification. Libraries were generated from 10 ng cDNA using the Nextera DNA Flex library prep kit (Illumina). Libraries underwent quality control by Fragment Analyzer (Agilent) and Qubit (Thermo Fisher Scientific) to determine the size distribution and the quantity of the libraries. Libraries were sequenced on a NextSeq 500 (Illumina). Fastq files were aligned to the mm10 genome using bowtie2 .Differentially expressed genes were determined using R package DESeq2 , P-valuet test was used, when comparing three distinct groups one-way ANOVA was used. All statistical analyses were performed in Prism 8 software (GraphPad) and data were considered significant if P \u2264 0.05.When comparing statistical differences between two groups an unpaired two-tailed GSE180647.Bulk RNA-seq data generated can be found at Gene Expression Omnibus under accession code"}
{"text": "Padaczka jest cz\u0119st\u0105 chorob\u0105 neurologiczn\u0105, diagnozowan\u0105 u 0,8-1% populacji \u015bwiatowej. Jest to choroba g\u0142\u00f3wnie wieku rozwojowego, gdy\u017c 80% zachorowa\u0144 dotyczy os\u00f3b poni\u017cej 20 r.\u017c. Przyczyny padaczki s\u0105 zr\u00f3\u017cnicowane, jednak zawsze uwa\u017cana by\u0142a za chorob\u0119 o pod\u0142o\u017cu genetycznym, do czego w chwili obecnej nie mamy ju\u017c w\u0105tpliwo\u015bci. Genetyka padaczek przesz\u0142a rewolucj\u0119 od momentu identyfikacji pierwszego \u201egenu padaczkowego\u201d. Sta\u0142o si\u0119 tak za spraw\u0105 wprowadzenia sekwencjonowania nast\u0119pnej generacji jako podstawowej metody badawczej i diagnostycznej, oraz zmiany podej\u015bcia badawczego, z analizy padaczek rodzinnych do bada\u0144 pod\u0142o\u017ca, g\u0142\u00f3wnie sporadycznych, encefalopatii padaczkowych. W kr\u00f3tkim czasie w oparciu o identyfikacje gen\u00f3w sprawczych wyodr\u0119bniono ponad 50 zespo\u0142\u00f3w wczesnodzieci\u0119cych encefalopatii padaczkowych. Obecnie sekwencjonowanie nast\u0119pnej generacji, eksomowe lub panelowe wykorzystywane jako powszechne narz\u0119dzie w diagnostyce encefalopatii padaczkowych o skuteczno\u015bci diagnostycznej na poziomie 30-40%. Uzyskanie \u201ediagnozy genetycznej\u201d w coraz wi\u0119kszej liczbie przypadk\u00f3w pozwala na zastosowanie odpowiedniej dla danego przypadku farmakoterapii. Jako \u017ce encefalopatie padaczkowe uznawane s\u0105 za choroby modelowe dla padaczek, opracowywane dla nich strategie terapeutyczne mog\u0105 przyczyni\u0107 si\u0119 tak\u017ce do opracowania leczenia cz\u0119stych postaci tej choroby. World Health Organisation, WHO), na \u015bwiecie na padaczk\u0119 choruje ponad 50 milion\u00f3w os\u00f3b, a liczb\u0119 nowych zachorowa\u0144 okre\u015bla si\u0119 na 2,4 miliona/rok. W Europie liczba chorych szacowana jest na oko\u0142o 3,6 miliona, w Polsce to oko\u0142o 350 tysi\u0119cy os\u00f3b. Padaczka jest przede wszystkim chorob\u0105 wieku rozwojowego, bo a\u017c 80% zachorowa\u0144 stwierdza si\u0119 poni\u017cej dwudziestego roku \u017cycia. W pierwszym roku \u017cycia odnotowuje si\u0119 najwy\u017csz\u0105 cz\u0119sto\u015b\u0107 zachorowania na ci\u0119\u017ckie, lekooporne zespo\u0142y padaczkowe o z\u0142ym rokowaniu dla pacjent\u00f3w \u2013 encefalopatie padaczkowe . W tej te\u017c grupie wiekowej najwa\u017cniejsz\u0105 rol\u0119 w etiologii tych chor\u00f3b odgrywaj\u0105 czynniki genetyczne.Padaczka to choroba, kt\u00f3ra towarzyszy cz\u0142owiekowi od zawsze, a pierwsze o niej wzmianki znajdziemy ju\u017c w dokumentach babilo\u0144skich. To jedna z najcz\u0119stszych chor\u00f3b neurologicznych dotykaj\u0105ca 0,8-1% wszystkich populacji. Wed\u0142ug danych \u015awiatowej Organizacji Zdrowia  dotycz\u0105ce klasyfikacji padaczek, ostateczna diagnoza uwzgl\u0119dnia\u0107 musi nie tylko rodzaj napad\u00f3w czy typ padaczki, ale tak\u017ce, w r\u00f3wnym stopniu jej etiologi\u0119 - w tym pod\u0142o\u017ce genetyczne choroby [Pomimo, \u017ce padaczka towarzyszy cz\u0142owiekowi od tysi\u0119cy lat to tak naprawd\u0119 zacz\u0119to j\u0105 dok\u0142adniej poznawa\u0107 i rozumie\u0107 dopiero w ostatnich wiekach . Najpier choroby .Epilepsy: Its Symptoms, Treatment and Relation to Other Chronic Convulsive Diseases\u201d pisa\u0142 \u201ePoczynaj\u0105c od prac staro\u017cytnych, autorzy byli i s\u0105 nadal zgodni, co do teorii, \u017ce padaczka jest przede wszystkim chorob\u0105 dziedziczn\u0105\u2026\u201d \u2013 nie potrafiono jednak tego udowodni\u0107. Dzisiaj pod\u0142o\u017ce genetyczne padaczek/zespo\u0142\u00f3w padaczkowych nie podlega ju\u017c dyskusji. Badania genetyczne sta\u0142y si\u0119 r\u00f3wnie\u017c elementem r\u00f3\u017cnicowej diagnostyki zespo\u0142\u00f3w padaczkowych. Szczeg\u00f3lnie dotyczy to zespo\u0142\u00f3w z grupy encefalopatii padaczkowych. identyfikacja pod\u0142o\u017ca molekularnego pozwala tu na jednoznaczn\u0105 weryfikacj\u0119 rozpoznania, a co najwa\u017cniejsze, w coraz wi\u0119kszej liczbie przypadk\u00f3w mo\u017cliwo\u015b\u0107 zastosowania odpowiedniego, dla zidentyfikowanego defektu molekularnego, leczenia czy post\u0119powania terapeutycznego. Tak wi\u0119c odnosz\u0105c si\u0119 do dyskusji rozpocz\u0119tej identyfikacj\u0105 pierwszych \u201egen\u00f3w padaczkowych\u201d, a kt\u00f3rej wyrazem by\u0142a debata na \u0142amach czasopisma Epilepsia \u201eDoes genetic information in humans help us treat patients?\u201d [ang. Next Generation Sequencing, NGS).W XIX w. John Russell Reynolds w swej pracy \u201etients?\u201d  zdecydowang. Whole Exom Sequencing, WES), albo wybrane geny/obszary genomu \u2013 sekwencjonowanie panelowe (celowane). Ka\u017cda z wymienionych powy\u017cej metod identyfikuje szereg wariant\u00f3w DNA, w\u015br\u00f3d kt\u00f3rych trzeba zidentyfikowa\u0107 te, kt\u00f3re konkretnie odpowiedzialne s\u0105 za fenotyp badanego. Nie zawsze jest to mo\u017cliwe, nie zawsze r\u00f3wnie\u017c potrafimy jednoznacznie okre\u015bli\u0107 patogenno\u015b\u0107 wytypowanych wariant\u00f3w, nawet je\u017celi zlokalizowane s\u0105 w znanych i wi\u0105zanych z dan\u0105 jednostk\u0105 chorobow\u0105 genach.NGS to bardzo wydajne narz\u0119dzie pozwalaj\u0105ce na r\u00f3wnoczesn\u0105 analiz\u0119 fragment\u00f3w DNA pokrywaj\u0105cych ca\u0142y genom \u2013 sekwencjonowanie ca\u0142ogenomowe , tylko fragmenty koduj\u0105ce genomu (eksom) \u2013 sekwencjonowanie eksomowe , zar\u00f3wno w procesie diagnostycznym, co do identyfikacji przyczyny choroby , jak i w zakresie poradnictwa genetycznego . Medycyna genomowa, poprzez liczb\u0119 uzyskiwanych w badaniach danych, nie zawsze mo\u017cliwych do interpretacji, zdecydowanie zwi\u0119ksza ten zakres niepewno\u015bci. Dlatego ju\u017c na etapie kierowania pacjenta na badanie i w procesie poradnictwa genetycznego problem ten powinien by\u0107 wyra\u017anie akcentowany. Szczeg\u00f3lnie \u017ce w ostatnich latach popularny sta\u0142 si\u0119 przekaz, \u017ce poznanie sekwencji genomu lub wszystkich gen\u00f3w pacjenta/badanego na pewno pozwoli na zidentyfikowanie przyczyn jego choroby . Nale\u017cy Pierwszym etapem rozwoju genetyki padaczek by\u0142o wykazanie udzia\u0142u czynnik\u00f3w genetycznych w rozwoju tej choroby, a pierwszym krokiem badania rodzinne i badania bli\u017ani\u0105t prowadzone na pocz\u0105tku XX w. Wykaza\u0142y one istotno\u015b\u0107 udzia\u0142u czynnik\u00f3w dziedzicznych w rozwoju padaczki, lecz by\u0142y niedopracowane metodologicznie, mi\u0119dzy innymi pod k\u0105tem kryteri\u00f3w w\u0142\u0105czenia w odniesieniu do etiologii choroby. Istotn\u0105 rol\u0119 odegra\u0142y dopiero badania prowadzone w latach 50-tych przez Lennox\u2019a, w kt\u00f3rych udowodni\u0142 rodzinne predyspozycje wyst\u0119powania padaczki. W badaniach bli\u017ani\u0105t wykaza\u0142 ponadto, \u017ce ryzyko wyst\u0105pienia padaczki \u201eidiopatycznej\u201d u bliskich krewnych pacjenta jest wy\u017csze ni\u017c padaczki symptomatycznej . Co ciekang. common) lub rzadkie, monogenowe . Okre\u015blenie padaczki cz\u0119ste jest poj\u0119ciem og\u00f3lnym i obejmuje jednostki, kt\u00f3re wyst\u0119puj\u0105 u ~1 na 200 os\u00f3b z nawracaj\u0105cymi napadani uog\u00f3lnionymi lub cz\u0119\u015bciowymi [ang. Genetic Generalized Epilepsies, GGE) w tym m\u0142odzie\u0144cz\u0105 padaczk\u0119 miokloniczn\u0105  i idiopatyczn\u0105 padaczk\u0119 z napadami nie\u015bwiadomo\u015bci , a tak\u017ce padaczki ogniskowe  obejmuj\u0105ce padaczk\u0119 skroniow\u0105  i Idiopatyczn\u0105 padaczk\u0119 ogniskow\u0105 . W przypadku tej grupy chor\u00f3b nie mo\u017cna jednoznacznie okre\u015bli\u0107 \u201eczynnika genetycznego/genu\u201d odpowiedzialnego za wyst\u0105pienie choroby. W genomowych badaniach asocjacyjnych , przeprowadzonych dla odpowiednio licznych kohort pacjent\u00f3w zidentyfikowano szereg loci i polimorfizm\u00f3w powi\u0105zanych z t\u0105 grup\u0105 padaczek oraz wykazano, \u017ce powszechne/cz\u0119ste polimorfizmy jednego nukleotydu  w naszym genomie odpowiadaj\u0105 za oko\u0142o 26% zmienno\u015bci fenotypowej padaczek. Szacunki z tych bada\u0144 wskazuj\u0105 na zaanga\u017cowanie co najmniej 400 loci zwi\u0105zanych z podatno\u015bci\u0105 do wyst\u0105pienia padaczek, aczkolwiek mo\u017ce ich by\u0107 nawet kilka tysi\u0119cy \u2212 ka\u017cde o ma\u0142ym czynniku ryzyka [Padaczki, podobnie jak i wiele innych chor\u00f3b o pod\u0142o\u017cu genetycznym mo\u017cemy podzieli\u0107 na tzw. cz\u0119ste (\u015bciowymi . Cz\u0119ste u ryzyka . Chorym Zespo\u0142y padaczkowe rzadkie \u2212 monogenowe (o dziedziczeniu mendlowskim) to choroby, w przypadku kt\u00f3rych mutacja pojedynczego genu jest niezb\u0119dna i wystarczaj\u0105ca dla wyst\u0105pienia okre\u015blonego fenotypu, oraz zwi\u0105zana z jego segregacj\u0105 w rodzinie. Pomimo, \u017ce zespo\u0142y te stanowi\u0105 niewielki odsetek w grupie padaczek to identyfikacja ich gen\u00f3w sprawczych pozwala nast\u0119pnie na identyfikacj\u0119 zar\u00f3wno szlak\u00f3w molekularnych jak i kom\u00f3rkowych zaanga\u017cowanych w ich etiopatogenez\u0119.- CHRNA4, kt\u00f3rego mutacje stanowi\u0142y pod\u0142o\u017ce nocnej padaczki czo\u0142owej o dziedziczeniu autosomalnym dominuj\u0105cym . Przez nast\u0119pne kilka lat badaj\u0105c rodzinne przypadki r\u00f3\u017cnych rodzaj\u00f3w epilepsji/zespo\u0142\u00f3w padaczkowych z zastosowaniem analizy sprz\u0119\u017ce\u0144 zidentyfikowano kilkana\u015bcie gen\u00f3w, kt\u00f3rych mutacje stanowi\u0142y pod\u0142o\u017ce molekularne choroby. Kodowane przez nie bia\u0142ka w wi\u0119kszo\u015bci stanowi\u0142y kana\u0142y/podjednostki kana\u0142\u00f3w jonowych zale\u017cnych od napi\u0119cia lub ligandu. Wtedy te\u017c powsta\u0142o poj\u0119cie padaczek jako \u201ekana\u0142opatii\u201d (ang. channelopathies).Pocz\u0105tek genetyki padaczek monogenowych to rok 1994, gdy zidentyfikowano pierwszy gen SCN1A. Wykazano, \u017ce jest zwi\u0105zany z wyst\u0119powaniem zespo\u0142u uog\u00f3lnionej padaczki z drgawkami gor\u0105czkowymi plus  o dziedziczeniu dominuj\u0105cym i zmiennym obrazie fenotypowym [ang. Dravet Syndrome, DS). W tym wypadku by\u0142y to jednak g\u0142\u00f3wnie mutacje o charakterze de novo [Jednym ze zidentyfikowanych w tym okresie gen\u00f3w, by\u0142 koduj\u0105cy bia\u0142ko Nav1.1 - podjednostk\u0119 \u03b1 napi\u0119ciowozale\u017cnego kana\u0142u sodowego, gen otypowym . W 2001  de novo . Wykazanera kana\u0142opatii\u201d (ang. \u201echannelopathy era\u201d) okres, do 2001 roku, identyfikacji niewielkiej liczby gen\u00f3w koduj\u0105cych g\u0142\u00f3wnie kana\u0142y jonowe , 2. \u201emroczne czasy\u201d (ang. \u201edark ages\u201d) genetyki padaczek, kiedy prowadzono szereg bada\u0144 asocjacyjnych, bez konkluzywnych i powtarzalnych wynik\u00f3w, oraz 3. \u201eera genomiki\u201d rozpocz\u0119ta okresem bada\u0144 z zastosowaniem por\u00f3wnawczej hybrydyzacji genomowej do mikromacierzy  a nast\u0119pnie zdominowana przez metod\u0119 NGS \u2013 okres \u201eba\u0144ki NGS\u201d (ang. \u201eNGS buble\u201d), kt\u00f3ry trwa do dzisiaj [er\u0105 encefalopatii padaczkowych\u201d.Prze\u0142omem w badaniach nad pod\u0142o\u017cem molekularnym padaczek, zwi\u0105zanym z rozwojem molekularnych technik badawczych, by\u0142 rozw\u00f3j genomiki i przej\u015bcie z analizy poszczeg\u00f3lnych gen\u00f3w do analizy na poziomie genomowym. Patrz\u0105c z perspektywy roku 2017 mo\u017cna wyr\u00f3\u017cni\u0107 kilka etap\u00f3w rozwoju bada\u0144 na padaczkami. S\u0105 to, jak okre\u015blaj\u0105 autorzy koncepcji: 1. \u201e dzisiaj , 14. W bang. Copy Number Variations, CNVs) odgrywaj\u0105 istotn\u0105 rol\u0119 u os\u00f3b z padaczk\u0105. Obecnie szacuje si\u0119, \u017ce oko\u0142o 4% chorych z encefalopatiami padaczkowymi ma patogenne CNVs, a u kolejnych 4% identyfikowane s\u0105 warianty potencjalnie patogenne [STXBP1, u pacjenta z zespo\u0142em Ohtahara [GRIN2A u pacjent\u00f3w z EEs [STXBP1 jest jednym z najcz\u0119\u015bciej identyfikowanych gen\u00f3w u os\u00f3b z EEs o szerokim spektrum fenotypowym, a GRIN2A u chorych z zespo\u0142em padaczki z afazj\u0105. Warto wspomnie\u0107 tu r\u00f3wnie\u017c o genie CHD2, kt\u00f3ry zosta\u0142 zidentyfikowany przez analiz\u0119 mikrodelecji u pacjenta z wczesnodzieci\u0119c\u0105 encefalopatia padaczkow\u0105 , ale tak\u017ce w ramach pierwszych kompleksowych, wieloo\u015brodkowych badaniach eksomowych z zastosowaniem NGS [Analiza mikrorearan\u017cacji z zastosowaniem aCGH pozwoli\u0142a na zidentyfikowanie nawracaj\u0105cych mikrodelecji 15q13.3, 15q11.2, and 16p13.11, identyfikowanych u pacjent\u00f3w z GGEs z cz\u0119sto\u015bci\u0105 oko\u0142o 1% ka\u017cda. P\u00f3\u017aniejsze badania wykaza\u0142, \u017ce zmiany liczby kopii  73 \u201egeny neurorozwojowe\u201d zwi\u0105zane z du\u017cymi wadami rozwojowymi m\u00f3zgu i padaczk\u0105, (3) 536 \u201egeny zwi\u0105zane z padaczk\u0105\u201d czyli powoduj\u0105ce jednostki chorobowe, w kt\u00f3rych padaczka stanowi jeden z objaw\u00f3w, (4) 284 \u201egeny potencjalnie zwi\u0105zane z padaczk\u0105\u201d, kt\u00f3rych udzia\u0142 w patogenezie choroby wymaga weryfikacji [Na pocz\u0105tku 2017 r. Wang i wsp. [yfikacji .CDKL5, gdzie u wi\u0119kszo\u015bci pacjent\u00f3w najpierw wyst\u0119puje op\u00f3\u017anienie rozwojowe [SCN 1A do zahamowania rozwoju cz\u0119sto dochodzi w drugim roku \u017cycia i wydaje si\u0119, \u017ce bez bezpo\u015bredniego wp\u0142ywu napad\u00f3w padaczkowych [prawdziwe\u201d EEs w przypadku, kt\u00f3rych u pocz\u0105tkowo prawid\u0142owo rozwijaj\u0105cych si\u0119 dzieci aktywno\u015b\u0107 padaczkowa prowadzi do zaburze\u0144 neurorozwojowych; przyk\u0142adem s\u0105 tu zespo\u0142y ci\u0105g\u0142ych wy\u0142adowa\u0144 iglica-fala podczas snu wolnofalowego  czy zesp\u00f3l Landau-Kle+nera , 2. grupa obejmuj\u0105ca choroby, w przypadku kt\u00f3rych zaburzenia kognitywne s\u0105 obserwowane ju\u017c w momencie wyst\u0105pienia napad\u00f3w i s\u0105 wynikiem zaburze\u0144 formowania sieci neuronalnej. Bioelektryczna aktywno\u015b\u0107 napadowa m\u00f3zgu w tym mo\u017ce stanowi\u0107 tylko czynnik pogarszaj\u0105cy. Ta grupa obejmuje zespo\u0142y o charakterystycznym wieku zachorowania i charakterystyce kliniczno-elektro-encefalograficznej; przyk\u0142adem s\u0105 tu zesp\u00f3\u0142 Ohtahara (ang. Ohtahara Syndrome), wczesna miokloniczna encefalopatia  oraz niekt\u00f3re przypadki zespo\u0142\u00f3w Westa, Dravet czy Lennox\u2019a-Gastaut  [SCN8A zale\u017cna encefalopatia, EIEE13 wg. OMIM). Na pewno jest to propozycja warta rozwa\u017cenia, szczeg\u00f3lnie je\u017celi we\u017amiemy pod uwag\u0119 heterogenno\u015b\u0107 genetyczn\u0105 i fenotypow\u0105 tych zespo\u0142\u00f3w (ryc. 2).Encefalopatie padaczkowe traktowane s\u0105 jako zespo\u0142y, w kt\u00f3rych objawem wiod\u0105cym jest padaczka, stanowi\u0105ca pierwszy cel terapeutyczny, a g\u0142\u00f3wnym objawem wsp\u00f3\u0142wyst\u0119puj\u0105cym jest zahamowanie lub regres rozwoju. Stanowi\u0105 one bardzo specyficzn\u0105, cho\u0107 heterogenn\u0105, grup\u0119 rzadkich zespo\u0142\u00f3w padaczkowych ujawniaj\u0105cych si\u0119 we wczesnym okresie rozwojowym. W przypadku EEs objawy zazwyczaj wyst\u0119puj\u0105 ju\u017c w niemowl\u0119ctwie, a 40% padaczek wyst\u0119puj\u0105cych przed 3 r.\u017c. nale\u017cy do tej grupy. EIEEs cechuje wsp\u00f3\u0142wyst\u0119powanie okre\u015blonego obrazu klinicznego, charakterystycznego zapisu EEG oraz nieprawid\u0142owy rozw\u00f3j psychoruchowy pacjent\u00f3w. S\u0105 to ci\u0119\u017ckie, lekooporne zespo\u0142y padaczkowe o z\u0142ym rokowaniu, w kt\u00f3rych, wyst\u0119powanie napad\u00f3w i/lub podklinicznych wy\u0142adowa\u0144 epileptogennych, jak si\u0119 przyjmuje, stanowi pod\u0142o\u017ce post\u0119puj\u0105cych zaburze\u0144 funkcjonalnych m\u00f3zgu prowadz\u0105c do zahamowania (niekiedy regresu) rozwoju pacjenta . Obecnieozwojowe . Natomiaczkowych . Mo\u017cliweS., LGS) . Bior\u0105c de novo w genomie cz\u0142owieka [de novo. Udowodni\u0142y to pierwsze badania z zastosowaniem eksomowego sekwencjonowania nast\u0119pnej generacji \u2013 WES prowadzone dla pacjent\u00f3w z NI [analiza tr\u00f3jek\u201d , tzn. por\u00f3wnanie sekwencji eksom\u00f3w probanta i jego rodzic\u00f3w. Przy za\u0142o\u017ceniu, \u017ce wariant patogenny jest dominuj\u0105cy i de novo, brano pod uwag\u0119 tylko te zmiany, kt\u00f3re zidentyfikowane u probanta nie mia\u0142y charakteru dziedzicznego. Dzi\u0119ki takiemu podej\u015bciu badawczemu, w stosunkowo kr\u00f3tkim czasie zidentyfikowano szereg gen\u00f3w, kt\u00f3rych mutacje stanowi\u0105 pod\u0142o\u017ce molekularne encefalopatii padaczkowych. W chwili obecnej w bazie OMIM wyodr\u0119bnionych jest 55 typ\u00f3w EIEEs przypisanych konkretnym genom, a lista na pewno nie jest jeszcze ci\u0105gle zamkni\u0119ta [Niepe\u0142nosprawno\u015b\u0107 intelektualna (NI), to ci\u0119\u017ckie zaburzenie neurorozwojowymi zdecydowanie redukuj\u0105ce p\u0142odno\u015b\u0107 os\u00f3b chorych, a mimo to wyst\u0119puj\u0105ce stosunkowo cz\u0119sto w populacji ludzkiej (1-2%). Pr\u00f3b\u0105 wyt\u0142umaczenia tego zjawiska by\u0142a hipoteza kompensacji utraty alleli w populacji poprzez wysok\u0105 cz\u0119sto\u015b\u0107 mutacji z\u0142owieka . EIEEs pt\u00f3w z NI , 27, kt\u00f3t\u00f3w z NI . W obu pamkni\u0119ta .Du\u017cym problemem w badaniach genetycznych zespo\u0142\u00f3w z grupy EIEEs pozostaje ich heterogenno\u015b\u0107 genetyczna i kliniczna, ale tak\u017ce fakt ewolucji objaw\u00f3w klinicznych w czasie, co niejednokrotnie powoduje, \u017ce jednoznaczn\u0105 diagnoz\u0119 mo\u017cna postawi\u0107 oceniaj\u0105c rozw\u00f3j choroby retrospektywnie.SCN 1A (obecnie szacowana cz\u0119sto\u015b\u0107 populacyjna to 1:22 000). Mutacje w genie SCN1A s\u0105 r\u00f3wnie\u017c przyczyn\u0105 \u0142agodniejszego, dziedzicznego zespo\u0142u GEFS+2 (Ryc. 1), gdzie obserwujemy cz\u0119sto wewn\u0105trzrodzinn\u0105 zmienno\u015b\u0107 obrazu klinicznego z wyst\u0105pieniem DS w\u0142\u0105cznie [Jednym z najlepiej poznanych zespo\u0142\u00f3w z tej grupy, zar\u00f3wno pod wzgl\u0119dem obrazu klinicznego, jak i pod\u0142o\u017ca molekularnego, jest zesp\u00f3\u0142 Dravet  najcz\u0119\u015bciej powodowany mutacjami w genie w\u0142\u0105cznie , 30.SCN1A. Mutacje w genie SCN1A identfikowane s\u0105 ponad 90% pacjent\u00f3w z diagnoz\u0105 DS, u pozosta\u0142ych mog\u0105 to by\u0107 mutacje de novo w genach SCN2A, SCN1B, GABRA1, GABRG2, KCNA2, HCN1, PCDH19, CHD2, STXBP1 (Ryc.2) [SCN1A-zale\u017cnych mo\u017ce by\u0107 szersze ni\u017c przypuszczano, prezentuj\u0105c przypadki chorych z heterozygotyczn\u0105 mutacj\u0105 missens p.> r226Met o fenotypie wczesnej encefalopatii padaczkowej, ale o zdecydowanie ci\u0119\u017cszym przebiegu ni\u017c DS [Rozpoznanie kliniczne, czy podejrzenie DS jest wskazaniem do wykonania analizy genu  (Ryc.2) . Najnowsu ni\u017c DS .PCDH19, SCN8A i CHD2. Zespo\u0142y te zosta\u0142 wy\u0142\u0105czone z grupy zespo\u0142\u00f3w podobnych do DS  i okre\u015blone jako odr\u0119bne jednostki, odpowiednio jako EIEE9 i EIEE13 oraz dzieci\u0119ca encefalopatia padaczkowa zwi\u0105zana z genem CHD2  [Dok\u0142adna analiza obrazu klinicznego dla wi\u0119kszej grupy pacjent\u00f3w ze zidentyfikowanymi mutacjami patogennymi w okre\u015blonych genach pozwala na wyodr\u0119bnienie nowych jednostek chorobowych na podstawie cech r\u00f3\u017cnicuj\u0105cych je od innych o podobnym fenotypie. Tak by\u0142o r\u00f3wnie\u017c w przypadku zespo\u0142\u00f3w zwi\u0105zanych z mutacjami w genach CHD2 EE) , 34, 19.okres krytyczny\u201d formowania r\u00f3wnowagi pomi\u0119dzy procesami pobudzenia i hamowania neuronalnego OUN. Jest to okres, w kt\u00f3rym charakterystyczne jest obni\u017cenie progu pobudliwo\u015bci zwi\u0105zane z nier\u00f3wnowag\u0105 ekspresji np. kana\u0142\u00f3w jonowych pobudzaj\u0105cych w stosunku do hamuj\u0105cych, skutkuj\u0105c\u0105 podwy\u017cszon\u0105 podatno\u015bci\u0105 na wyst\u0105pienie napad\u00f3w, kt\u00f3re mog\u0105 prowadzi\u0107 do zaburzenia proces\u00f3w synaptogenezy i prawid\u0142owego rozwoju sieci neuronalnych [Choroby monogenowe stanowi\u0105 niewielki odsetek chor\u00f3b genetycznie uwarunkowanych, ale to analiza ich pod\u0142o\u017ca \u2013 identyfikacja genu sprawczego, pozwala na poznanie mechanizm\u00f3w i szlak\u00f3w kom\u00f3rkowych zaanga\u017cowanych w ich etiopatogenez\u0119. Badania identyfikuj\u0105ce geny, kt\u00f3rych mutacje zwi\u0105zane s\u0105 z ekspresj\u0105 danego fenotypu, stanowi\u0105 jednak dopiero pierwszy etap prowadz\u0105cy do poznania patomechanizm\u00f3w poszczeg\u00f3lnych chor\u00f3b. EIEEs to bardzo heterogenna grupa jednostek, kt\u00f3rych obraz kliniczny jest wynikiem okre\u015blonych dysfunkcji uk\u0142adu nerwowego. Okres rozwojowy m\u00f3zgu, a w szczeg\u00f3lno\u015bci ostatni okres rozwoju prenatalnego i pierwsze dwa lata rozwoju postnatalnego to tzw. \u201eonalnych . W\u015br\u00f3d gonalnych , 24, 20.SCN8A czy kontrolowanie temperatury w przypadku EIEE6\u2013 zespo\u0142u Dravet, gen SCN1A).Diagnostyka zespo\u0142\u00f3w padaczkowych, a w szczeg\u00f3lno\u015bci encefalopatii padaczkowych zawsze stanowi\u0142a du\u017ce wyzwanie ze wzgl\u0119du na heterogenno\u015b\u0107 genetyczn\u0105, ale tak\u017ce du\u017c\u0105 zmienno\u015b\u0107 fenotypow\u0105 nawet przy tym samym pod\u0142o\u017cu genetycznym. Dla niekt\u00f3rych zespo\u0142\u00f3w mo\u017cliwe jest okre\u015blenie korelacji genotyp-fenotyp, jednak w przypadku EIEEs jest to niezwykle trudne. Problemem jest tu zmienno\u015b\u0107 fenotypowa zespo\u0142\u00f3w, w tym ewolucja objaw\u00f3w klinicznych w czasie  i konieczno\u015b\u0107 postawienia diagnozy jak najwcze\u015bniej. Zazwyczaj obraz kliniczny jest wtedy jeszcze niejednoznaczny, diagnoza mo\u017ce mie\u0107 wp\u0142yw na rodzaj stosowanej farmakoterapii . Wykrywalno\u015b\u0107 mutacji patogennych dla tej metody wynosi oko\u0142o 4-5% [SCN1A) czy zesp\u00f3\u0142 Retta (gen MECP2) w pierwszej kolejno\u015bci konieczna jest dok\u0142adna analiza tych gen\u00f3w \u2013 sekwencjonowanie z pe\u0142nym pokryciem wszystkich ekson\u00f3w oraz identyfikacja rearan\u017cacji wewn\u0105trz genowych (delecji/duplikacji ekson\u00f3w), co wymaga zastosowania metod o wy\u017cszej rozdzielczo\u015bci ni\u017c CMA (np. MLPA). Alternatywn\u0105 metod\u0105 jest zastosowanie sekwencjonowania panelowego, czyli r\u00f3wnoleg\u0142a analiza szeregu gen\u00f3w metod\u0105 NGS. Wykrywalno\u015b\u0107 mutacji przy zastosowaniu sekwencjonowania panelowego szacuje \u015brednio si\u0119 na 20%, w przypadku pacjent\u00f3w o wczesnym wieku zachorowania warto\u015b\u0107 ta jest wy\u017csza i wynosi do 40% [SCN1A, MECP2, SLC2A czy STXBP1. Ograniczeniem bada\u0144 celowanych jest fakt, \u017ce panele obejmuj\u0105 tylko te geny, kt\u00f3re w danym momencie s\u0105 znane. [Wyb\u00f3r schematu diagnostyki pacjenta z zastosowaniem analizy DNA musi uwzgl\u0119dnia\u0107 dane kliniczne, adekwatno\u015b\u0107 proponowanego testu zar\u00f3wno pod wzgl\u0119dem wykrywalno\u015bci mutacji w kontek\u015bcie badanego fenotypu jak i ogranicze\u0144 technicznych proponowanej metody. W przypadku padaczek/encefalopatii padaczkowych o wyborze post\u0119powania diagnostycznego decyduj\u0105 wiek zachorowania i przebieg kliniczny choroby pacjenta. Najliczniejsz\u0105 grup\u0119 kierowan\u0105 na badania stanowi\u0105 chorzy z tzw. o\u0142o 4-5% , 16. Je\u017ci do 40% . Panele \u0105 znane. . LogicznPost\u0119p technologiczny sekwencjonowania materia\u0142u genetycznego, kt\u00f3ry nast\u0105pi\u0142 w ostatnim czasie spowodowa\u0142 nag\u0142e zmiany w genetyce padaczek, g\u0142\u00f3wnie poprzez gwa\u0142towny wzrost zidentyfikowanych liczby gen\u00f3w, kt\u00f3rych mutacje stanowi\u0105 pod\u0142o\u017ce chor\u00f3b z tej grupy, a szczeg\u00f3lnie encefalopatii padaczkowych. Mo\u017cliwe sta\u0142o si\u0119 wprowadzenie test\u00f3w diagnostycznych z wykorzystaniem sekwencjonowania eksomowego i panelowego. Skuteczno\u015b\u0107 diagnostyczna tych test\u00f3w oceniana jest na oko\u0142o 30% (dla EIEEs do 40%). Pomimo \u017ce testy genetyczne pozwalaj\u0105 w chwili obecnej na postawienie diagnozy molekularnej w szeregu przypadkach, to ich wyniki ci\u0105gle sk\u0142aniaj\u0105 do stawiania wi\u0119kszej liczby pyta\u0144 ni\u017c daj\u0105 odpowiedzi. Systematycznie jednak poszerzaj\u0105 nasz\u0105 wiedz\u0119 na temat etiopatofizjologii zespo\u0142\u00f3w padaczkowych.katastroficzne\u201d zespo\u0142\u00f3w [Czy postawienie diagnozy molekularnej u pacjenta, czyli poznanie pod\u0142o\u017ca molekularnego choroby, b\u0119dzie mia\u0142o dla tego pacjenta znaczenie, je\u017celi chodzi o terapi\u0119, pytali w 2008 Delgado-Escueta i Bourgeois . Miejmy zespo\u0142\u00f3w . Jako \u017ce"}
{"text": "Zesp\u00f3\u0142 \u0142amliwego chromosomu X  jest, po zespole Downa, najcz\u0119stsz\u0105 dziedziczn\u0105 przyczyn\u0105 niepe\u0142nosprawno\u015bci intelektualnej (NI). Stopie\u0144 niepe\u0142nosprawno\u015bci intelektualnej pacjent\u00f3w z FXS jest r\u00f3\u017cny i zale\u017cy g\u0142\u00f3wnie od p\u0142ci. Zaburzeniom intelektualnym towarzysz\u0105 dodatkowe objawy takie jak op\u00f3\u017anienie rozwoju psychoruchowego, zaburzenia zachowania czy emocji.W ponad 99% przypadk\u00f3w, choroba spowodowana jest wyst\u0119powaniem mutacji dynamicznej w genie FMR1 zlokalizowanym na chromosomie X. W wyniku ekspansji tr\u00f3jki nukleotyd\u00f3w CGG (>200 powt\u00f3rze\u0144) dochodzi do zahamowania ekspresji genu, a tym samym znacznego obni\u017cenia poziomu bia\u0142ka FMRP kodowanego przez gen FMR1. Ekspansja sekwencji CGG do zakresu premutacji (55-200 powt\u00f3rze\u0144 CGG) warunkuje wyst\u0105pienie nosicielstwa FXS i chor\u00f3b FMR1-zale\u017cnych takich jak: przedwczesne wygasanie funkcji jajnik\u00f3w zwi\u0105zane z zespo\u0142em \u0142amliwego chromosomu X  oraz zespo\u0142u dr\u017cenia i ataksji zwi\u0105zanego z zespo\u0142em \u0142amliwego chromosomu X . W przypadku obu tych chor\u00f3b objawy kliniczne wyst\u0119puj\u0105 dopiero u ludzi doros\u0142ych.Celem pracy jest przybli\u017cenie aktualnej wiedzy dotycz\u0105cej pod\u0142o\u017ca molekularnego i epidemiologii zespo\u0142u \u0142amliwego chromosomu X oraz innych chor\u00f3b FMR1-zale\u017cnych. Niepe\u0142nosprawno\u015b\u0107 intelektualna (NI) jest jednym z cz\u0119stszych objaw\u00f3w obserwowanych w zespo\u0142ach genetycznych. Definiuje si\u0119 j\u0105 jako stan, ujawniaj\u0105cy si\u0119 przed uko\u0144czeniem 18 roku \u017cycia, w kt\u00f3rym sprawno\u015b\u0107 intelektualna jest istotnie ni\u017csza od przeci\u0119tnej . TowarzyFragile X Syndrome, FXS). Jest on tak\u017ce wymieniany jako jedna z cz\u0119stszych genetycznych przyczyn autyzmu. Po raz pierwszy zesp\u00f3\u0142 \u0142amliwego chromosomu X zosta\u0142 opisany w 1943 roku przez Jamesa Purdona Martina i Juli\u0119 Bell, dlatego czasem nazywany jest r\u00f3wnie\u017c zespo\u0142em Martina-Bell. Badacze w swojej pracy opisali rodzin\u0119, w kt\u00f3rej niepe\u0142nosprawno\u015b\u0107 intelektualna w stopniu znacznym zosta\u0142a stwierdzona u 11 m\u0119\u017cczyzn, podczas gdy kobiety nie wykazywa\u0142y objaw\u00f3w choroby. Rodow\u00f3d rodziny wskazywa\u0142 na dziedziczenie choroby sprz\u0119\u017cone z p\u0142ci\u0105. Potwierdzi\u0142y to badania Herberta Lubsa, kt\u00f3ry w 1969 r. opublikowa\u0142 wyniki swoich spostrze\u017ce\u0144, w kt\u00f3rych stwierdzi\u0142, \u017ce u niekt\u00f3rych os\u00f3b z NI, analiza kariotypu wskazuje na obecno\u015b\u0107 charakterystycznego ,,przew\u0119\u017cenia\u2019\u2019, zlokalizowanego blisko ko\u0144ca d\u0142ugiego ramienia chromosomu X, w pozycji Xq27.3 [FMR1 , kt\u00f3rego mutacje s\u0105 przyczyn\u0105 zespo\u0142u \u0142amliwego chromosomu X [Jedn\u0105 z cz\u0119stszych chor\u00f3b genetycznie uwarunkowanych, w kt\u00f3rych przebiegu obserwowana jest niepe\u0142nosprawno\u015b\u0107 intelektualna jest zesp\u00f3\u0142 \u0142amliwego chromosomu X .Gen  choroby . PonadtoFMR1 jest wykonywane rutynowo, zaraz po analizie kariotypu. Umo\u017cliwia ono nie tylko potwierdzenie rozpoznania klinicznego, lecz r\u00f3wnie\u017c jest wskazaniem do badania nosicielstwa zespo\u0142u w rodzinie, co pozwala na identyfikacj\u0119 potencjalnych pacjent\u00f3w z FXPOI i FXTAS. Ponadto ostateczne potwierdzenie badaniem molekularnym rozpoznania klinicznego zespo\u0142\u00f3w FXS, FXPOI i FXTAS u\u0142atwia przyj\u0119cie odpowiednich strategii w rehabilitacji i leczeniu. Umo\u017cliwia tak\u017ce uzyskanie odpowiedniej porady genetycznej dla ca\u0142ej rodziny pacjenta, dotycz\u0105cej ryzyka wyst\u0105pienia choroby u kolejnych os\u00f3b z rodziny.W przypadku pacjent\u00f3w z niepe\u0142nosprawno\u015bci\u0105 intelektualn\u0105, badanie molekularne w kierunku zespo\u0142u \u0142amliwego chromosomu X i identyfikacji mutacji dynamicznej w genie FMR1-zale\u017cnych.Celem pracy jest przedstawienie objaw\u00f3w klinicznych, epidemiologii i pod\u0142o\u017ca molekularnego zespo\u0142u \u0142amliwego chromosomu X i chor\u00f3b 1FMR1, kt\u00f3rego mutacje odpowiedzialne s\u0105 za wyst\u0105pienie objaw\u00f3w choroby jest genem silnie konserwowanym w toku ewolucji. Podobie\u0144stwo sekwencji aminokwasowej w grupie ssak\u00f3w jest wysokie, przyk\u0142adowo mi\u0119dzy cz\u0142owiekiem a mysz\u0105 wynosi 97%. Ponadto wz\u00f3r ekspresji bia\u0142ka w poszczeg\u00f3lnych tkankach jest podobny u wszystkich gatunk\u00f3w [FMR1 sk\u0142ada si\u0119 z 17 ekson\u00f3w i koduje bia\u0142ko FMRP  o wielko\u015bci 71 kDa [Gen gatunk\u00f3w . Ludzki i 71 kDa .Ssacze bia\u0142ko FMRP oraz jego paralogi autosomalne \u2212 FXR1P i FXR2P stanowi\u0105 ma\u0142\u0105 rodzin\u0119 bia\u0142ek wi\u0105\u017c\u0105cych cz\u0105steczki RNA. Bia\u0142ka te wykazuj\u0105 wysokie podobie\u0144stwo sekwencji aminokwasowej >60%), za\u015b geny je koduj\u0105ce prawdopodobnie powsta\u0142y wskutek duplikacji jednego genu . Bia\u0142ka z rodziny FMRP zawieraj\u0105 trzy charakterystyczne, homologiczne domeny KH odpowiedzialne za wi\u0105zanie i regulacj\u0119 cz\u0105steczek RNA (m.in. mRNA i rRNA). O tym jak istotn\u0105 rol\u0119 pe\u0142ni\u0105 te domeny w aktywno\u015bci bia\u0142ka \u015bwiadczy efekt mutacji punktowej p.Ile304Asn, kt\u00f3ra powoduje zmiany w obr\u0119bie drugiej domeny KH. Analiza struktury krystalicznej bia\u0142ka z mutacj\u0105 wykaza\u0142a, \u017ce jej obecno\u015b\u0107 zak\u0142\u00f3ca prawid\u0142owe fa\u0142dowanie domeny KH, co w konsekwencji zaburza wi\u0105zanie cz\u0105steczek RNA przez bia\u0142ko i powoduje utrat\u0119 jego funkcji [0%, za\u015b gKolejn\u0105 charakterystyczn\u0105 domen\u0105 jest kaseta RGG \u2013 domena bogata w reszty argininy i glicyny, zlokalizowana w C-ko\u0144cowej cz\u0119\u015bci bia\u0142ka, kt\u00f3ra wi\u0105\u017ce cz\u0105steczki RNA posiadaj\u0105ce w swojej budowie charakterystyczn\u0105 struktur\u0119 bogat\u0105 w nukleotydy guaninowe . Bia\u0142ko Bia\u0142ko FMRP ulega ekspresji g\u0142\u00f3wnie w kom\u00f3rkach nerwowych, zw\u0142aszcza tych maj\u0105cych du\u017c\u0105 liczb\u0119 synaps np. zlokalizowanych w obr\u0119bie hipokampu czy m\u00f3\u017cd\u017cku. Bia\u0142ko stanowi cz\u0119\u015b\u0107 polirybosomu i jest zaanga\u017cowane w proces translacji bia\u0142ek odpowiedzialnych m.in. za plastyczno\u015b\u0107 synaptyczn\u0105 , 10. Bra2untranslated region) na ko\u0144cu 5\u2019 genu FMR1 (Rycina 1). Termin ,,mutacja dynamiczna\u2019\u2019 zosta\u0142 wprowadzony w celu wyr\u00f3\u017cnienia unikatowej cechy jak\u0105 maj\u0105 powt\u00f3rzone sekwencje DNA, czyli mo\u017cliwo\u015bci spontanicznego wyd\u0142u\u017cania si\u0119 (tzw. ekspansji). Udowodniono, \u017ce zwielokrotnienie liczby powt\u00f3rze\u0144 nukleotydowych mo\u017ce mie\u0107 istotne implikacje kliniczne \u2013 obecno\u015b\u0107 mutacji dynamicznej w r\u00f3\u017cnych genach stanowi przyczyn\u0119 wyst\u0105pienia oko\u0142o 30 r\u00f3\u017cnych chor\u00f3b neurologicznych, neurodegeneracyjnych czy nerwowo-mi\u0119\u015bniowych. Opr\u00f3cz FXS, do grupy chor\u00f3b wywo\u0142anych tym typem mutacji zaliczamy m.in. chorob\u0119 Huntingtona, dystro/ \u0119 miotoniczn\u0105 typu 1 i 2, wybrane ataksje rdzeniowo-m\u00f3\u017cd\u017ckowe, niekt\u00f3re rodzaje stwardnienia zanikowego bocznego czy ot\u0119pienia czo\u0142owo-skroniowego. Wszystkie z wymienionych chor\u00f3b, mimo \u017ce dotycz\u0105 r\u00f3\u017cnych gen\u00f3w i r\u00f3\u017cnych sekwencji powt\u00f3rzonych, wykazuj\u0105 te same w\u0142a\u015bciwo\u015bci, np. dziedziczenie choroby i jej progresja uzale\u017cnione s\u0105 od wyj\u015bciowej liczby powt\u00f3rze\u0144 nukleotydowych. Niestabilne powt\u00f3rzenia wykazuj\u0105 r\u00f3wnie\u017c specyficzne w\u0142a\u015bciwo\u015bci strukturalne \u2212 udowodniono, \u017ce mog\u0105 one przybiera\u0107 charakterystyczne formy przestrzenne np. kszta\u0142t spinki do w\u0142os\u00f3w. Struktury te sprzyjaj\u0105 \u201epo\u015blizgowi\u201d polimerazy DNA, co zaburza procesy replikacji, rekombinacji, naprawy DNA i jednocze\u015bnie prowadzi do dalszego zwielokrotnienia powt\u00f3rze\u0144 sekwencji nukleotydowych [Jak wspomniano wcze\u015bniej, za znacz\u0105c\u0105 wi\u0119kszo\u015b\u0107 (>99%) przypadk\u00f3w zespo\u0142u \u0142amliwego chromosomu X odpowiedzialna jest mutacja dynamiczna, zwi\u0105zana ze zwielokrotnieniem liczby powt\u00f3rze\u0144 tr\u00f3jnukleotydowych CGG w regionie niekoduj\u0105cym . Liczba powt\u00f3rze\u0144 tej tr\u00f3jki nukleotydowej mo\u017ce znajdowa\u0107 si\u0119 w zakresie od 5 do ponad 750. Zgodnie z wytycznymi European Molecular Genetics Quality Network (EMQN) i American College of Medical Genetics (ACMG), w zale\u017cno\u015bci od liczby powt\u00f3rze\u0144 CGG wyr\u00f3\u017cnia si\u0119 trzy podstawowe typy alleli (Sekwencja nukleotydowa (CGG)n w genie y alleli . PierwszFMR1, wyodr\u0119bnia si\u0119 niestabilne allele z zakresu premutacji (55-200 powt\u00f3rze\u0144) i allele z zakresu pe\u0142nej mutacji, kt\u00f3r\u0105 stwierdza si\u0119 przy liczbie powt\u00f3rze\u0144 powy\u017cej 200. Dodatkowo, w przypadku obecno\u015bci pe\u0142nej mutacji dochodzi do hypermetylacji regionu promotorowego genu, co prowadzi do wyciszenia jego ekspresji, a w konsekwencji do braku bia\u0142ka FMRP, co skutkuje zaburzeniami w postaci objaw\u00f3w zespo\u0142u \u0142amliwego chromosomu X [Ponadto w przypadku genu mosomu X .FMR1 mamy do czynienia z tzw. antycypacj\u0105 matczyn\u0105 \u2013 do ekspansji powt\u00f3rze\u0144 od premutacji do zakresu pe\u0142nej mutacji dochodzi w kom\u00f3rkach jajowych, w zwi\u0105zku, z czym choroba przekazywana jest dzieciom przez matki nie wykazuj\u0105ce objaw\u00f3w choroby. Ojciec \u2013 nosiciel mo\u017ce przekaza\u0107 allel z zakresu premutacji swoim c\u00f3rkom, kt\u00f3re stan\u0105 si\u0119 w ten spos\u00f3b nosicielkami choroby i b\u0119d\u0105 mia\u0142y blisko 50% ryzyko posiadania chorego potomstwa. Nie odnotowano przypadk\u00f3w ekspansji powt\u00f3rze\u0144 CGG do zakresu pe\u0142nej mutacji w linii ojcowskiej [U os\u00f3b b\u0119d\u0105cych nosicielami allelu z zakresu premutacji znacznie cz\u0119\u015bciej dochodzi do ekspansji liczby powt\u00f3rze\u0144 sekwencji CGG, r\u00f3wnie\u017c do zakresu pe\u0142nej mutacji. Do zwi\u0119kszenia liczby powt\u00f3rze\u0144 dochodzi najprawdopodobniej w kom\u00f3rkach rozrodczych, w trakcie procesu replikacji materia\u0142u genetycznego lub znacznie rzadziej, na pocz\u0105tkowych etapach rozwoju zarodka. W przypadku genu cowskiej .FMR1, a tym samym nie jest zahamowana jego transkrypcja i synteza bia\u0142ka FMRP. Jak wspomniano wcze\u015bniej allele z zakresu premutacji s\u0105 niestabilne i istnieje ryzyko ich ekspansji do pe\u0142nej mutacji w nast\u0119pnych pokoleniach. Ryzyko to jest zale\u017cne od liczby powt\u00f3rze\u0144 CGG i szacuje si\u0119, \u017ce w przypadku alleli >99 powt\u00f3rze\u0144 wzrasta ono do 100%. Sugeruje si\u0119 tak\u017ce, \u017ce stabilno\u015b\u0107 alleli mo\u017ce zale\u017ce\u0107 od obecno\u015bci przerw w sekwencji (CGG)n w postaci sekwencji AGG. Zazwyczaj w prawid\u0142owych, stabilnych allelach obecne s\u0105 dwie sekwencje AGG (co oko\u0142o 10 powt\u00f3rze\u0144 CGG). W przypadku alleli niestabilnych (premutacji) przerw AGG jest znacznie mniej lub nie wyst\u0119puj\u0105 wcale.Allele z zakresu premutacji do\u015b\u0107 powszechnie wyst\u0119puj\u0105 w populacji (1/250-450 ch\u0142opc\u00f3w i 1/130-200 dziewczynek). W rodzinach obci\u0105\u017conych zespo\u0142em \u0142amliwego chromosomu X cz\u0119sto\u015b\u0107 wyst\u0119powania premutacji jest wy\u017csza ni\u017c pe\u0142nej mutacji. U nosicieli premutacji nie stwierdza si\u0119 hypermetylacji genu FMR1, na skutek metylacji cytozyn zawartych w d\u0142ugim ci\u0105gu powt\u00f3rzonej sekwencji. Metylacja tego regionu genu prowadzi do zahamowania jego ekspresji, a tym samym do zmniejszenia lub braku syntezy bia\u0142ka FMRP, co bezpo\u015brednio przek\u0142ada si\u0119 na ujawnienie objaw\u00f3w choroby. Przy wyst\u0105pieniu pe\u0142nej mutacji proces metylacji zachodzi zar\u00f3wno u m\u0119\u017cczyzn jak i u kobiet. Warto jednak podkre\u015bli\u0107 fakt, \u017ce w sytuacji, je\u017celi u kobiet zmutowany gen b\u0119dzie znajdowa\u0142 si\u0119 na inaktywowanym chromosomie X, to pomimo stwierdzenia w badaniu molekularnym pe\u0142nej mutacji osoba taka, mo\u017ce nie wykazywa\u0107 \u017cadnych objaw\u00f3w choroby [Jak wspomniano wy\u017cej pe\u0142n\u0105 mutacj\u0119 w przypadku FXS stwierdza si\u0119 przy wyst\u0105pieniu ponad 200 powt\u00f3rze\u0144 CGG. Sytuacja ta powoduje hypermetylacj\u0119 promotora oraz regionu 5\u2019UTR genu FMR1, takie jak delecja ca\u0142ego lub fragmentu genu czy mutacje punktowe, w tym mutacje typu missens (zmiana aminokwasu). Efektem tych mutacji b\u0119dzie, podobnie jak w przypadku pe\u0142nej mutacji, brak (m\u0119\u017cczy\u017ani) lub znacz\u0105co obni\u017cony poziom (kobiety) bia\u0142ka FMRP, a w przypadku zmian aminokwasowych, powstanie bia\u0142ka o zmienionej aktywno\u015bci lub specyficzno\u015bci wzgl\u0119dem cz\u0105steczek, z kt\u00f3rymi si\u0119 \u0142\u0105czy [Ekspansja powt\u00f3rze\u0144 tr\u00f3jnukleotydowych CGG jest odpowiedzialna za wyst\u0105pienie >99% przypadk\u00f3w zespo\u0142u \u0142amliwego chromosomu X. U 1% pacjent\u00f3w opisywane s\u0105 jednak inne zmiany w obr\u0119bie genu i\u0119 \u0142\u0105czy .3FMR1. Podstawowym objawem zespo\u0142u \u0142amliwego chromosomu X jest niepe\u0142nosprawno\u015b\u0107 intelektualna, kt\u00f3ra stwierdzana jest u wszystkich ch\u0142opc\u00f3w z pe\u0142n\u0105 mutacj\u0105 i ma zazwyczaj stopie\u0144 umiarkowany lub g\u0142\u0119boki. U oko\u0142o 50% kobiet stwierdza si\u0119 NI w stopniu lekkim. U wi\u0119kszo\u015bci pacjent\u00f3w FXS niepe\u0142nosprawno\u015b\u0107 intelektualna ujawnia si\u0119 ju\u017c w okresie wczesnodzieci\u0119cym [Osoby z zespo\u0142em \u0142amliwego chromosomu X wykazuj\u0105 szerokie spektrum objaw\u00f3w klinicznych, kt\u00f3re w znacznej mierze uzale\u017cnione s\u0105 od p\u0142ci pacjenta oraz obecno\u015bci zmian zwi\u0105zanych z mutacj\u0105 dynamiczn\u0105 w genie ieci\u0119cym .Niepe\u0142nosprawno\u015bci intelektualnej obserwowanej u pacjent\u00f3w z FXS mog\u0105 towarzyszy\u0107 r\u00f3wnie\u017c takie objawy jak: op\u00f3\u017anienie rozwoju psychoruchowego, zw\u0142aszcza w zakresie rozwoju mowy oraz zaburzenia zachowania i problemy emocjonalne . U ok. 2Poza niepe\u0142nosprawno\u015bci\u0105 intelektualn\u0105 i op\u00f3\u017anieniem rozwoju psychoruchowego, u os\u00f3b z FXS wyst\u0119puj\u0105 tak\u017ce specyficzne cechy dysmorficzne \u2013 charakterystyczna jest d\u0142uga, w\u0105ska twarz i odstaj\u0105ce uszy. Objawy te wyst\u0119puj\u0105 u oko\u0142o 70% pacjent\u00f3w i s\u0105 silniej wyra\u017cone u os\u00f3b starszych. Za dodatkowe objawy choroby uznaje si\u0119 tak\u017ce: nadmiern\u0105 ruchliwo\u015b\u0107 staw\u00f3w (67% chorych), p\u0142askostopie (71% chorych), wypadanie zastawki mitralnej, a tak\u017ce makroorchidyzm (70% chorych) pojawiaj\u0105cy si\u0119 u ch\u0142opc\u00f3w po okresie dojrzewania .4FMR1, co ma toksyczny efekt wzgl\u0119dem kom\u00f3rek nerwowych. Wykazano, \u017ce cz\u0105steczki mRNA zawieraj\u0105ce premutacj\u0119 gromadz\u0105 si\u0119 w du\u017cej ilo\u015bci w astrocytach i neuronach powoduj\u0105c ich przedwczesne obumieranie, czego efektem jest wyst\u0105pienie objaw\u00f3w klinicznych choroby [FMR1 w m\u00f3zgach  [FMR1. Aktywno\u015b\u0107 ta by\u0142a 2 i 10-krotnie wy\u017csza u m\u0119\u017cczyzn maj\u0105cych odpowiednio powy\u017cej i poni\u017cej 100 powt\u00f3rze\u0144 sekwencji CGG [Zesp\u00f3\u0142 dr\u017cenia i ataksji zwi\u0105zany z zespo\u0142em FXS zosta\u0142 opisany po raz pierwszy w 2001 r. u pi\u0119ciu dziadk\u00f3w, kt\u00f3rzy mieli wnuki z potwierdzonymi badaniami molekularnymi zespo\u0142em \u0142amliwego chromosomu X . W przec choroby . Badaniakrotnie) . Ponadtoncji CGG .Jak wspomniano wy\u017cej zesp\u00f3\u0142 dr\u017cenia i ataksji zwi\u0105zany z zespo\u0142em \u0142amliwego chromosomu X zosta\u0142 po raz pierwszy opisany w 2001 przez doktor Randi Hagerman. Jest to choroba neurodegeneracyjna wieku doros\u0142ego (objawy w wieku 50-60 lat). U pacjent\u00f3w obserwuje si\u0119 g\u0142\u00f3wnie dr\u017cenie , problemy z poruszaniem si\u0119 nasilaj\u0105ce si\u0119 z wiekiem (57%) oraz post\u0119puj\u0105cy deficyt poznawczy i neuropsychologiczny . Ponadto mog\u0105 wyst\u0119powa\u0107 u nich dodatkowe objawy, takie jak neuropatia obwodowa (60%), impotencja (80%), zaburzenia czynno\u015bci p\u0119cherza moczowego i jelit (30-55%). Jak podaj\u0105 dane literaturowe, ryzyko rozwoju choroby uzale\u017cnione jest od wielko\u015bci premutacji -u os\u00f3b posiadaj\u0105cych allele zawieraj\u0105ce <70 powt\u00f3rze\u0144 CGG jest ono ni\u017csze ni\u017c u os\u00f3b o wi\u0119kszej liczbie powt\u00f3rze\u0144 .FMR1 jest r\u00f3wnie\u017c istotnym czynnikiem wyst\u0105pienia przedwczesnego wygasania funkcji jajnik\u00f3w zwi\u0105zanego z zespo\u0142em \u0142amliwego chromosomu X, definiowanego jako spontaniczna menopauza wyst\u0119puj\u0105ca u kobiet przed uko\u0144czeniem 40 roku \u017cycia, kt\u00f3ra obejmuje szereg dysfunkcji jajnik\u00f3w. FXPOI wyst\u0119puje u ok. 16-20% kobiet, b\u0119d\u0105cych nosicielkami premutacji w genie FMR1 i po raz pierwszy informacja o tej zale\u017cno\u015bci zosta\u0142a opisana w 1994 roku przez Schwartza i wsp\u00f3\u0142pracownik\u00f3w. Ich badania wykaza\u0142y, \u017ce kobiety z premutacj\u0105 mia\u0142y wy\u017csze ryzyko wyst\u0105pienia nieregularnych miesi\u0105czek ni\u017c kobiety z allelami z zakresu prawid\u0142owego i z pe\u0142n\u0105 mutacj\u0105 [Nosicielstwo premutacji w genie  mutacj\u0105 .Poza tym wykazano, \u017ce u kobiet b\u0119d\u0105cych nosicielkami premutacji dysfunkcja jajnik\u00f3w zale\u017cna jest od liczby powt\u00f3rze\u0144 CGG, jednak zwi\u0105zek ten nie jest liniowy. U kobiet o \u015bredniej liczbie powt\u00f3rze\u0144 CGG (80-100 CGG) FXPOI pojawia si\u0119 cz\u0119\u015bciej (32%) i wcze\u015bniej w stosunku do grup nosicielek o mniejszej ni\u017c 80 i wi\u0119kszej ni\u017c 100 liczbie powt\u00f3rze\u0144 CGG. Przypuszcza si\u0119, \u017ce u zdrowych kobiet FMRP mo\u017ce odgrywa\u0107 rol\u0119 w aktywacji p\u0119cherzyk\u00f3w i starzeniu si\u0119 jajnik\u00f3w. U nosicielek premutacji gromadz\u0105cy si\u0119 mRNA mo\u017ce wywiera\u0107 toksyczny efekt na funkcjonowanie kom\u00f3rek jajowych. Teoria ta sprawdzana jest w badaniach prowadzonych na p\u0119cherzykach pobranych z jajnik\u00f3w samic myszy z premutacj\u0105. Wst\u0119pne obserwacje wskazuj\u0105 na wysokie st\u0119\u017cenie FMRP w kom\u00f3rkach jajowych podczas folikulogenezy. By\u0107 mo\u017ce allele z zakresu \u015bredniej premutacji warunkuj\u0105 powstawanie najwi\u0119kszej ilo\u015bci mRNA bia\u0142ka FMRP w jajnikach, a tym samym generuj\u0105 najwi\u0119ksz\u0105 toksyczno\u015b\u0107, kt\u00f3ra zaburza ten proces. Sugeruje si\u0119 r\u00f3wnie\u017c, \u017ce w zale\u017cno\u015bci od liczby powt\u00f3rze\u0144 CGG mRNA mog\u0105 przybiera\u0107 r\u00f3\u017cne konformacj\u0119 np. spinki do w\u0142os\u00f3w, co mo\u017ce wp\u0142ywa\u0107 na oddzia\u0142ywanie z r\u00f3\u017cnymi bia\u0142kami i cz\u0105steczkami mRNA .5FMR1- zale\u017cnych (FXTAS i FXPOI), w wielu krajach podejmuje si\u0119 pr\u00f3by jej oceny poprzez badania przesiewowe noworodk\u00f3w [FMR1 DNA methylation\u2019\u2019, w ramach kt\u00f3rego, w okresie od kwietnia 2006 do wrze\u015bnia 2008 roku przebadano ok. 36 tysi\u0119cy noworodk\u00f3w p\u0142ci m\u0119skiej. Stwierdzono, \u017ce cz\u0119sto\u015b\u0107 wyst\u0119powania FXS w badanej populacji wynosi 1 na 5161 m\u0119\u017cczyzn [Jak wspomniano wcze\u015bniej, cz\u0119sto\u015b\u0107 wyst\u0119powania zespo\u0142u \u0142amliwego chromosomu X szacowana jest na 1/4000 u ch\u0142opc\u00f3w i 1/5000-8000 u dziewczynek, co jednak nie jest warto\u015bci\u0105 dok\u0142adn\u0105. St\u0105d te\u017c w celu jednoznacznej oceny cz\u0119sto\u015bci wyst\u0119powania zespo\u0142u \u0142amliwego chromosomu X oraz innych chor\u00f3b worodk\u00f3w . Przyk\u0142am\u0119\u017cczyzn .Podobne badanie przesiewowe prowadzone by\u0142o w Kanadzie, jednak badano nie tylko noworodki, lecz r\u00f3wnie\u017c ich matki, co umo\u017cliwi\u0142o, poza ocen\u0105 cz\u0119sto\u015bci wyst\u0119powania choroby, analiz\u0119 stabilno\u015bci przekazywania alleli o r\u00f3\u017cnej d\u0142ugo\u015bci. W trakcie badania przeanalizowano pr\u00f3bki DNA od 24449 anonimowych par matka - dziecko. Mutacja dynamiczna u noworodk\u00f3w badana by\u0142a wtedy, gdy u matki stwierdzono obecno\u015b\u0107 allelu zawieraj\u0105cego powy\u017cej 45 powt\u00f3rze\u0144 CGG. W ten spos\u00f3b zidentyfikowano 2 ch\u0142opc\u00f3w z pe\u0142n\u0105 mutacj\u0105, kt\u00f3rych matki by\u0142y nosicielkami alleli zawieraj\u0105cych 80 i 100 powt\u00f3rze\u0144 CGG. Na tej podstawie cz\u0119sto\u015b\u0107 wyst\u0119powania mutacji u ch\u0142opc\u00f3w oszacowano na 1/6078 urodze\u0144 . W EuropFMR1 nie wi\u0105\u017ce si\u0119 z wyst\u0119powaniem objaw\u00f3w klinicznych typowych dla zespo\u0142u \u0142amliwego chromosomu X, w tym niepe\u0142nosprawno\u015bci intelektualnej. Nie zmienia to jednak faktu, \u017ce u cz\u0119\u015bci nosicieli choroby obserwuje si\u0119 zaburzenia psychologiczne, takie jak: problemy z koncentracj\u0105 i uczeniem si\u0119, nadpobudliwo\u015b\u0107, op\u00f3\u017anienie rozwoju czy problemy emocjonalne. Ponadto stwierdzono, \u017ce u ok. 40% m\u0119\u017cczyzn i 16% kobiet w starszym wieku, b\u0119d\u0105cych nosicielami premutacji, wyst\u0119puje zesp\u00f3\u0142 dr\u017cenia i ataksji zwi\u0105zany z FXS. U oko\u0142o 16-20% kobiet z premutacj\u0105 wyst\u0119puje przedwczesne wygasanie funkcji jajnik\u00f3w zwi\u0105zane z zespo\u0142em \u0142amliwego chromosomu X [Dzi\u0119ki badaniom epidemiologicznym uda\u0142o si\u0119 tak\u017ce okre\u015bli\u0107 cz\u0119sto\u015b\u0107 wyst\u0119powania premutacji  w populacji og\u00f3lnej, kt\u00f3ra wynosi 1/130-200 u kobiet oraz 1/250-450 u m\u0119\u017cczyzn. Obecno\u015b\u0107 premutacji w genie mosomu X , 35, 36.6FMR1 wi\u0105\u017ce si\u0119 ze zwi\u0119kszonym ryzykiem dziecka chorego na FXS.Wczesne rozpoznanie choroby jest kluczowe dla podj\u0119cia odpowiedniego leczenia objawowego i rehabilitacji. Umo\u017cliwia tak\u017ce obj\u0119cie rodziny chorego dziecka w\u0142a\u015bciw\u0105 opieka medyczn\u0105, w tym poradnictwem genetycznym. Coraz cz\u0119\u015bciej podejmowane s\u0105 pr\u00f3by opracowania skutecznej terapii, kt\u00f3ra mia\u0142aby zminimalizowa\u0107 skutki niedoboru bia\u0142ka FMRP w kom\u00f3rkach nerwowych, a tym samym z\u0142agodzi\u0107 objawy kliniczne u pacjent\u00f3w. W profilaktyce problem\u00f3w z p\u0142odno\u015bci\u0105 wa\u017cna jest identyfikacja pacjentek z zespo\u0142em przedwczesnego wygasania funkcji jajnik\u00f3w zwi\u0105zanego z zespo\u0142em \u0142amliwego chromosomu X, gdy\u017c obecno\u015b\u0107 premutacji w genie Warto wspomnie\u0107, \u017ce diagnostyka w kierunku zespo\u0142u \u0142amliwego chromosomu X jest jednym z cz\u0119\u015bciej zlecanych bada\u0144 diagnostycznych nie tylko u dzieci z NI, lecz r\u00f3wnie\u017c u pacjent\u00f3w z op\u00f3\u017anieniem rozwoju psychoruchowego, op\u00f3\u017anionym rozwojem mowy czy zaburzeniami zachowania, w tym zaburzeniami ze spektrum autyzmu. Zdarzaj\u0105 si\u0119 pacjenci, u kt\u00f3rych zostaje potwierdzone rozpoznanie tego zespo\u0142u, jeszcze przed rozwini\u0119ciem pe\u0142nego obrazu klinicznego choroby. Szczeg\u00f3lnie istotne jest przeprowadzenie badania w rodzinach, w kt\u00f3rych niepe\u0142nosprawno\u015b\u0107 intelektualna wyst\u0119puje g\u0142\u00f3wnie u m\u0119\u017cczyzn, w kilku pokoleniach, co wskazuje na defekt genetyczny dziedzicz\u0105cy si\u0119 w spos\u00f3b sprz\u0119\u017cony z p\u0142ci\u0105 (z chromosomem X). Wykluczenie zespo\u0142u FXS powinno by\u0107 wst\u0119pem do poszukiwania defekt\u00f3w w innych genach zlokalizowanych na chromosomie X, kt\u00f3rych mutacje punktowe lub delecje/duplikacje mog\u0105 by\u0107 przyczyn\u0105 wyst\u0105pienia NI. Tego typu badania, wykorzystuj\u0105ce analiz\u0119 sprz\u0119\u017ce\u0144, sekwencjonowanie metod\u0105 Sangera, technik\u0119 MLPA czy wreszcie techniki wysokoprzepustowe takie jak por\u00f3wnawcza hybrydyzacja genomowa do mikromacierzy czy sekwencjonowanie nast\u0119pne generacji s\u0105 r\u00f3wnie\u017c prowadzone w Zak\u0142adzie Genetyki Medycznej Instytutu Matki i Dziecka. Badania wykonywane s\u0105 m.in. w ramach projekt\u00f3w badawczych finansowanych ze \u017ar\u00f3de\u0142 zewn\u0119trznych i umo\u017cliwi\u0142y identyfikacj\u0119 defektu molekularnego oraz okre\u015blenie ryzyka ponownego wyst\u0105pienia choroby w kilkudziesi\u0119ciu rodzinach , 38, 39.Pragniemy serdecznie podzi\u0119kowa\u0107 Pracownikom Zak\u0142adu Genetyki Medycznej IMiD, kt\u00f3rzy uczestniczyli w diagnostyce i badaniach naukowych nad zespo\u0142em \u0142amliwego chromosomu X oraz innych zaburze\u0144 intelektualnych o pod\u0142o\u017cu genetycznym, kt\u00f3rzy nie s\u0105 wsp\u00f3\u0142autorami tej pracy."}
{"text": "Hadron spectroscopy provides a way to understand the dynamics of the strong interaction. For light hadron systems, only phenomenological models or lattice quantum chromodynamics (QCD) are applicable, because of the failure of perturbation expansions for QCD at low energy. Experimental data on light hadron spectroscopy are therefore crucial to provide necessary constraints on various theoretical models. Light meson spectroscopy has been studied using charmonium decays with the Beijing Spectrometer Experiment (BES) at the Beijing Electron-Positron Collider, operating at 2.0\u20134.6 GeV center-of-mass energy, for nearly three decades. Charmonium data with unprecedented statistics and well-defined initial and final states provide BESIII with unique opportunities to search for glueballs, hybrids and multi-quark states, as well as perform systematic studies of the properties of conventional light mesons. In this article, we review BESIII results that address these issues. We review BESIII results on the study of glueballs and multi-quark states, which shed light on the understanding of the strong interaction. JPC are arranged in representations of the SU(3) group, and this led to the quark model by Gell-Man and Zweig\u00a0,,\u2212p   \u2212p [\\doc[B\u00a0[D\u00a0[K\u00a0 mesons, [B\u00a0[D\u00a0[K\u00a0. An attrf0(1500) and f0(1710) are main competitors for the lightest 0++ glueball candidates, since they are copiously produced in gluon-rich processes and both have masses that are near the LQCD predicted values. The inclusion of data from radiative J/\u03c8 decays provides a source that is complementary to hadronic production experiments.The scalar resonances J/\u03c8 decays to \u03c0+\u03c0\u2212 and \u03c00\u03c00 have been studied by the MARKIII [9\u00a0J/\u03c8 events accumulated with the BESIII detector [J/\u03c8 \u2192 \u03b3\u03c00\u03c00 decays [f0(1500) and f0(1710). The \u03c00\u03c00 invariant mass spectrum for the selected J/\u03c8 \u2192 \u03b3\u03c00\u03c00 events is shown in Fig.\u00a0f2(1270) signal, a shoulder on the high mass side of f2(1270), an enhancement at \u223c1.7\u00a0GeV/c2 and a peak at \u223c 2.1\u00a0GeV/c2 are evident. A mass-independent PWA was performed, where the amplitudes for radiative J/\u03c8 decays to \u03c00\u03c00 are constructed in the radiative multipole basis, as described in detail in Appendix A of\u00a0[++ amplitudes as a function of ++ structures just below 1.5\u00a0GeV/c2 and near 1.7\u00a0GeV/c2. In the mass-dependent PWA, the s dependence of the \u03c0\u03c0 interaction (where s is the invariant mass squared of the two pions) is parameterized as a coherent sum of resonances, each described by a Breit\u2013Wigner line shape with resonance properties, e.g. the mass, width and branching fraction, that are extracted from the fit. The preceding BESII experiment [J/\u03c8 \u2192 \u03b3\u03c0+\u03c0\u2212 and \u03b3\u03c00\u03c00, using relativistic covariant tensor amplitudes constructed from Lorentz-invariant combinations of the polarization and four-momentum vectors of the initial- and final-state particles, with helicity \u00b11\u2009J/\u03c8 initial states [f0(1500) and f0(1710) decaying to \u03c0\u03c0 are listed in Table Radiative  MARKIII , DM2 [45 MARKIII  and BES  MARKIII ,59 experdetector , J/\u03c8 \u2192 \u03b30 decays  were usedix A of\u00a0. The comperiment  performel states . The PWAl states . The meaecays [f0500 and fJ/\u03c8 radiative decays to \u03b7\u03b7 and 8\u2009J/\u03c8 events collected with the BESIII detector, the decays of J/\u03c8 \u2192 \u03b3\u03b7\u03b7 were investigated [2 are apparent. A mass-dependent PWA was carried out, and the results indicate that the peak at around 1.5 GeV/c2 is mainly from the well-established tensor state f0(1500). The statistical significance of the f0(1500) signal is 8\u03c3. The peaks around 1.7 and 2.1 GeV/c2 are dominated by f0(1710) and f0(2100), respectively, and the significance for the presence of a tensor f2(2340) state is 7.6 \u03c3. The red histogram in Fig. ++ and 2++ components, respectively.Scalar and tensor glueball candidates were also studied with stigated . The blaensor f2240 state ensor f2240 state KSKS system produced in radiative J/\u03c8 decays was performed [9\u2009J/\u03c8 decays collected by the BESIII detector. The black dots with error bars in Fig. KSKS for the selected \u03b3KSKS events. Three significant peaks in the KSKS mass spectrum around 1.5, 1.7 and 2.2\u00a0GeV/c2 are observed. A mass-dependent amplitude analysis was applied to extract the parameters and product branching fractions of the resonances that parameterized the KSKS invariant mass spectrum as a sum of Breit\u2013Wigner line shapes. In addition, a mass-independent analysis was performed to obtain the function that describes the dynamics of the KSKS system while making minimal assumptions about the properties and number of poles in the amplitudes. The two approaches give consistent results. The red histogram in Fig. ++ and 2++ components, respectively. The dominant scalar contributions come from f0(1500), f0(1710), and f0(2200). The tensor spectrum in J/\u03c8 \u2192 \u03b3KSKS is dominated by the well-known f2(2340) is needed in the fit.A study of the erformed  using 1.f0(1500) and f0(1710) scalars and the f2(2340) tensor in J/\u03c8 \u2192 \u03b3\u03b7\u03b7 and \u03b3KSKS are listed in Table f0(1710) are about an order of magnitude larger than that for f0(1500). A contribution from f2(2340) is needed in both the J/\u03c8 \u2192 \u03b3\u03b7\u03b7 and J/\u03c8 \u2192 \u03b3KSKS channels. The mass of the tensor state f2(2340) is consistent with the LQCD prediction for a pure tensor glueball.The measured product branching fractions for the f0500 and fJ/\u03c8 \u2192 \u03b3\u03c6\u03c6 decay events [9\u2009J/\u03c8 data sample, is shown as black dots with error bars in Fig.\u00a0c signal and clear structures at lower \u03c6\u03c6 invariant masses are observed. Both mass-dependent and mass-independent PWA were performed for the M(\u03c6\u03c6) < 2.7\u00a0GeV/c2 region with results that are consistent. In addition to three dominant 0\u2212+ pseudoscalar states \u03b7(2225), \u03b7(2100) and X(2500), three tensors, f2(2010), f2(2300) and f2(2340), and one scalar f0(2100) contribute significantly in the PWA fit. The green short-dashed, the red dash\u2013dot and the blue long-dashed histograms in Fig. JPC = 0\u2212+, 0++ and 2++, respectively from the model-dependent PWA fit, and the red solid histogram shows the total contribution from all components, which is in good agreement with data. The statistical significance of f2(2340) \u2192 \u03c6\u03c6 is 11\u03c3.The \u03c6\u03c6 invariant mass distribution for selected radiative y events , from th and f2230, and onf0(1500) and f0(1710) in different decay modes in Fig. We show a comparison between the product branching fractions for the scalar glueball candidates f0100 and f0f0(1500) and f0(1710) production rates in J/\u03c8 radiative decays. A comparison of the measured production rates with those obtained from LQCD calculations for a scalar glueball is given in Table By taking entclass1pt{minimaf0(1710) in gluon-rich J/\u03c8 radiative decays is close to LQCD calculations for a scalar glueball and is about an order of magnitude larger than that for f0(1500). This might suggest that f0(1710) has a larger gluonic component than f0(1500). Studies of f0(1500) and f0(1710) production in other gluon-favored and gluon-disfavored processes will be crucial to conclusively establish the scalar glueball. For the f2(2340) tensor state, the LQCD prediction for the production rate of a pure-gauge tensor glueball in radiative J/\u03c8 decays \u00a0[TensorGlueball/\u0393total = 1.1(2) \u00d7 10\u22122. The presence of f2(2340) in the \u03b7\u03b7\u00a0[KSKS [f2(2340) might be a candidate for the tensor glueball. However, the current measured production rate for f2(2340), based on the observed \u03b7\u03b7, f2(2340) are needed.The production rate for  decays \u00a0 is \u0393Tensn the \u03b7\u03b7\u00a0, KSKS [6\u03b7\u03b7\u00a0[KSKS  and \u03c6\u03c6\u00a0[\u03b7\u03b7\u00a0[KSKS  final stI = 0,\u2009JPC = 0\u2212+ pseudoscalars are the \u03b7 and \u03b7\u2032. The small number of expected radial excitations for 0\u2212+ states in the quark model provides a clean and promising environment for the search of pseudoscalar glueballs.The ground states of the 2, \u03b7(1440), was first observed in +\u03c0\u2212 with \u03b7(1440) \u2192 \u03b7\u03c0+\u03c0\u2212 and \u2212p process\u00a0[J/\u03c8 radiative decays\u00a0[2\u00a0[a0(980)\u03c0 or direct et\u00a0al. [A pseudoscalar state around 1440 MeV/cocument}\u00a0, and fur process\u00a0 and J/\u03c8 e decays\u00a0,72. Conse decays\u00a0,74, due ecays\u00a0[2\u00a0. SubsequV region  or in\u00a0[7V region . However [et\u00a0al.  claimed 8\u2009J/\u03c8 events collected with the BESIII detector, the decays of J/\u03c8 \u2192 \u03b3\u03c0+\u03c0\u2212\u03c00 and \u03b33\u03c00 were studied [f0(980)\u03c00 was observed for the first time with a statistical significance larger than 10\u03c3 in both the charged (f0(980) \u2192 \u03c0+\u03c0\u2212, Fig. f0(980) \u2192 \u03c00\u03c00, Fig. With 2.25 \u00d7 10 studied . The isoocument}\u00a0,77,78, wy BESIII .J/\u03c8 \u2192 \u03b3\u03b7(1405/1475) \u2192 \u03b3\u03c00f0(980) \u2192 \u03b33\u03c0 stimulated many theoretical efforts to understand the nature of \u03b7(1405/1475). With the assumption that only one 0\u2212+ exists around 1.4 GeV/c2, the triangle singularity mechanism was found to play a more dominant role than a0(980) \u2212 f0(980) mixing, and it can produce the anomalously large isospin violations in \u03b7(1405) \u2192 \u03c0+\u03c0\u2212\u03c00, according to [The anomalous large isospin violations in  \u2192 \u03b3\u03c00f09 \u2192 \u03b33\u03c0 strding to .J/\u03c8 decays to \u03b3\u03b7(1405/1475) and \u03b7(1405/1475) \u2192 \u03b3\u03c6, with 1.3 \u00d7 109\u2009J/\u03c8 events at BESIII \u00a0[The \u03b7(1405/1475) state was also observed in  BESIII \u00a0. The obs2 from LQCD calculations, while the existence of any pseudoscalar states above 2.0 GeV/c2 is not well established experimentally. In J/\u03c8 \u2192 \u03b3\u03b7\u2032\u03c0+\u03c0\u2212 decays at BESIII [X(1835) by BESII [\u2032\u03c0+\u03c0\u2212 invariant mass distribution. In addition, two additional states, X(2120) and X(2370), are observed with statistical significances larger than 7.2\u03c3 and 6.4\u03c3, respectively. The mass of the X(2370) state is measured to be c2 from a one-dimensional fit. The X(2370) state has been further confirmed in the J/\u03c8 radiative decays  in the two decay modes agree with each other, and coincide with the mass of the lightest pseudoscalar glueball from LQCD calculations, which makes X(2370) a candidate for the lightest pseudoscalar glueball. However, it is crucial to determine its spin parity and observe it in more decay modes before this conclusion can be firmly established.The mass for the lightest pseudoscalar glueball is expected to be higher than 2.3 GeV/ct BESIII , the obsby BESII  was conf) and X230, are obJPC quantum numbers, in which case they would not mix with conventional Hybrid states are color-singlet combinations of constituent quarks and gluons, such as a \u2212+ exotic hybrid candidates, i.e. \u03c01(1400) and \u03c01(1600), which decay into different final states, such as \u03b7\u03c0, \u03b7\u2032\u03c0, f1(1285)\u03c0, b1(1235)\u03c0 and \u03c1\u03c0, were reported in different reactions. The evidence for \u03c01(2015) has also been reported. Reviews of the experimental status on these isovector 1\u2212+ exotic states can be found in\u00a0[8 \u03c8(3686) events collected with BESIII, an amplitude analysis is applied to \u03c8(3686) \u2192 \u03b3\u03c7c1, \u03c7c1 \u2192 \u03b7\u03c0+\u03c0\u2212 to search for \u03c01(1400), \u03c01(1600) and \u03c01(2015) [\u2212+ state in the \u03b7\u03c0 invariant mass spectrum, and upper limits for the branching fractions \u03c7c1 \u2192 \u03c01(1400)\u00b1\u03c0\u2213, \u03c7c1 \u2192 \u03c01(1600)\u00b1\u03c0\u2213 and \u03c7c1 \u2192 \u03c01(2015)\u00b1\u03c0\u2213, with subsequent \u03c01(X)\u00b1 \u2192 \u03b7\u03c0\u00b1 decay, are established. BESIII searches for isovector exotic states in \u03b7\u2032\u03c0 invariant mass spectra are ongoing.The observation of isovector 10) and \u03c0100, whichfound in\u00a0,86\u201388. W\u03c01(2015) . Figure\u00a0\u2212+ states. The theoretical predictions for their main decay modes are f1(1285)\u03b7, a1\u03c0 and \u03b7\u03b7\u2032, etc. [J/\u03c8 events that were recently accumulated by BESIII provide an ideal laboratory for the search for such states.There is no evidence for the existence of isosinglet 1\u03b7\u2032, etc. . The 10 JP = 0\u2212 by BESIII\u00a0[2 and width of 2 at the 90An anomalously proton-antiproton \\document} decays\u00a0 and CLEO} decays\u00a0. This sty BESIII\u00a0, with a tructure\u00a0; howeveresonance\u00a0.X(1835) state was first observed by the BESII experiment as a peak in the \u03b7\u2032\u03c0+\u03c0\u2212 invariant mass distribution in J/\u03c8 \u2192 \u03b3\u03b7\u2032\u03c0+\u03c0\u2212 decays\u00a0[2 and 2; the X(1835) state was also observed in the JP = 0\u2212 by a model-dependent PWA\u00a0[X(1835) decaying into \u03b3\u03c6 was recently observed in J/\u03c8 \u2192 \u03b3\u03b3\u03c6\u00a0[The \u2212 decays\u00a0\u00a0 and X(1835) is really a \u2212X(1835) at the \u2032\u03c0+\u03c0\u2212 line shape of X(1835) with high statistical precision therefore provides valuable information that helps clarify the natures of X(1835) and One of the theoretical interpretations of the natures of ocument}\u00a0 suggestsnd state\u00a0. If X, and another is the coherent sum of two resonant amplitudes.With 1.09 \u00d7 10X(1835) couples to \u2032\u03c0+\u03c0\u2212 above the f0(980) \u2192 \u03c0+\u03c0\u2212 line shape at the X(1835) line shape:T is the decay amplitude, \u03c1out is the phase space for J/\u03c8 \u2192 \u03b3\u03b7\u2032\u03c0+\u03c0\u2212, s is the square of the \u03b7\u2032\u03c0+\u03c0\u2212 system mass, \u03c1k is the phase space for decay mode k and s:g2 of all decay modes other than 0 is the maximum two-body decay phase space volume\u00a0[In the first model, we assume that state  formula\u00a0, definede volume\u00a0 and \\docX(1835) and 2 and 2.The fit results for this model are shown in Fig.\u00a0given in\u00a0, the polmentclass2pt{minimX(1835) and another resonance with mass close to the \u2032\u03c0+\u03c0\u2212 mass spectrum around 2 is performed. This fit yields a narrow resonance below the 2 and 2, with a statistical significance larger than 7\u03c3. The fit results for the second model are shown in Fig.\u00a0In the second model, we assume that the distortion comes from the interference between mentclasspt{minimaBased on current data samples, two models fit the data with similar fit qualities. Both fits suggest the existence of either a broad state with strong couplings to Continuous experimental efforts are being made to search for and study glueballs, hybrids and multi-quark states from charmonium decays, supported by the huge statistics data samples accumulated at BESIII.f0(1710) in gluon-rich J/\u03c8 radiative decays is about an order of magnitude higher than that for f0(1500) and is close to LQCD calculations for the production rate of a scalar glueball, under current circumstance. This suggests that f0(1710) can have a larger gluonic component than f0(1500). Studies of f0(1500) and f0(1710) in other gluon-favored and gluon-disfavored processes with improved analysis techniques will be crucial to further refine this conclusion. The mass of the f2(2340) tensor state matches the LQCD expectation for a pure tensor glueball. This, and its copious production in J/\u03c8 radiative decays to \u03b7\u03b7, KSKS and \u03c6\u03c6, might suggest that f2(2340) is a candidate of the tensor glueball. However, the current measured production rates for f2(2340) appear to be substantially lower than LQCD expectations. Since no dominant glueball decay mode can be expected, due to the flavor blindness of glueball decays, searches for additional f2(2340) decay modes are necessary. In light of the observation of X(2370) in J/\u03c8 \u2192 \u03b3\u03b7\u2032\u03c0+\u03c0\u2212 and J/\u03c8 event sample and the clean environment in J/\u03c8 \u2192 \u03b3\u03b7\u2032KSKS decays will make an amplitude analysis and the determination of the spin parity of X(2370) possible.We have found that the production rate for \u2212+ exotic hybrid candidates \u03c01(1400), \u03c01(1600) and \u03c01(2015) were seen in the \u03c8(3686) \u2192 \u03b3\u03c7c1, \u03c7c1 \u2192 \u03b7\u03c0+\u03c0\u2212 decay process with 4.48 \u00d7 108 \u03c8(3686) events collected with BESIII. As of yet, no evidence for an isoscalar 1\u2212+ exotic hybrid has been found. The 10 billion J/\u03c8 event sample will provide an ideal laboratory for the search of isoscalar 1\u2212+ exotic hybrids in f1(1285)\u03b7, a1\u03c0 and \u03b7\u03b7\u2032, etc. decay channels.In searching for hybrid states with exotic quantum numbers, no significant signals for the isovector 12, more data are needed to further study the J/\u03c8 \u2192 \u03b3\u03b7\u2032\u03c0+\u03c0\u2212 process. Also, line shapes for other radiative decay channels should be studied near the In order to elucidate further the nature of the states around"}
{"text": "Zespo\u0142y preekscytacji to coraz cz\u0119\u015bciej wykrywane jednostki chorobowe w populacji dzieci i m\u0142odzie\u017cy. Ich istot\u0105 jest obecno\u015b\u0107 dodatkowej drogi/dr\u00f3g przewodzenia w sercu, przez kt\u00f3r\u0105 impuls przewodzony jest szybciej ni\u017c fizjologicznie. Prowadzi to do szybszego pobudzenia kom\u00f3r i mo\u017ce by\u0107 pod\u0142o\u017cem powstawania gro\u017anych arytmii. Najcz\u0119stszym zespo\u0142em preekscytacji jest zesp\u00f3\u0142 Wolffa-Parkinsona--White\u2019a, kt\u00f3ry dotyczy do 2/1000 os\u00f3b. Obecno\u015b\u0107 dodatkowej drogi przewodzenia mo\u017ce skutkowa\u0107 wyst\u0105pieniem powa\u017cnych konsekwencji, poczynaj\u0105c od cz\u0119stoskurczu nadkomorowego, a ko\u0144cz\u0105c na nag\u0142ym zgonie sercowym. Istniej\u0105 zar\u00f3wno inwazyjne, jak i nieinwazyjne metody diagnostyczne zespo\u0142\u00f3w preekscytacji. Post\u0119powanie terapeutyczne obejmuje farmakoterapi\u0119 oraz zabieg przerwania ci\u0105g\u0142o\u015bci dodatkowej drogi przewodzenia zwany ablacj\u0105, co pozwala trwale usun\u0105\u0107 przyczyn\u0119 arytmii. Zespo\u0142y preekscytacji definiowane s\u0105, jako jednostki chorobowe, w kt\u00f3rych dochodzi do przedwczesnej aktywacji miokardium kom\u00f3r przez impuls, kt\u00f3ry jest przewodzony przez nieprawid\u0142ow\u0105 drog\u0119 i omija fizjologiczne op\u00f3\u017anienie w w\u0119\u017ale przedsionkowo-komorowym. Zosta\u0142o opisanych kilka typ\u00f3w zespo\u0142\u00f3w preekscytacji, a r\u00f3\u017cni je od siebie anatomia nieprawid\u0142owej dodatkowej drogi przewodzenia . W zespoCz\u0119sto\u015b\u0107 wyst\u0119powania zespo\u0142\u00f3w preekscytacji u dzieci wynosi od 1:250 do 1:1000 . Wyr\u00f3\u017cniNajcz\u0119\u015bciej spotykan\u0105 postaci\u0105 preekscytacji wyst\u0119puj\u0105c\u0105 u dzieci jest typ Wolffa-Parkinsona-White\u2019a . Ten typIstniej\u0105 trzy g\u0142\u00f3wne kryteria, kt\u00f3re musz\u0105 zosta\u0107 spe\u0142nione, aby potwierdzi\u0107 preekscytacj\u0119 o typie WPW w 12-odprowadzeniowym zapisie EKG . S\u0105 to:krotki odstep PR (ponizej) dolnej granicy normy odpowiednio do wieku pacjenta \u2013 0,08 s poni\u017cej 3 r.\u017c., 0,1 s do 16 r.\u017c. i 0,12s powy\u017cej 16 r.\u017c.) manifestuj\u0105cy wcze\u015bniejsz\u0105 depolaryzacj\u0119 kom\u00f3r [obecna fala delta na ramieniu wstepuj\u0105cym za\u0142amkaR reprezentuj\u0105ca stosunkowo chaotyczn\u0105 i powoln\u0105 depolaryzacj\u0119 kom\u00f3r poprzez dodatkow\u0105 drog\u0119 przedsionkowo-komorow\u0105 poszerzony zespo\u0142 QRS  , 9.Dodatkowo mog\u0105 by\u0107 obecne zaburzenia repolaryzacji manifestuj\u0105ce si\u0119 zmian\u0105 odchylenia fali ST-T w przeciwn\u0105 stron\u0119 ni\u017c wektor zespo\u0142u QRS , 8.Badanie elektrokardiograftczne pozwala na zlokalizowanie dodatkowej drogi przewodzenia w 1 z 5 g\u0142\u00f3wnych po\u0142o\u017ce\u0144: po lewej stronie serca u oko\u0142o 50% pacjent\u00f3w, po prawej stronie serca u 20%, w regionie przegrody u oko\u0142o 30% pacjent\u00f3w, a ok. 7% pacjent\u00f3w ma wiele po\u0142\u0105cze\u0144 dodatkowych , 7.U pacjent\u00f3w z cechami preekscytacji w EKG tylko ok. 50% wykazuje jakiekolwiek objawy, z czego najcz\u0119stszym jest ko\u0142atanie serca, kt\u00f3re jest manifestacj\u0105 nawrotnego cz\u0119stoskurczu przedsionkowo-komorowego (AVRT). M\u0142odsze dzieci opisuj\u0105 ten objaw, jako b\u00f3l w klatce piersiowej, pulsowanie w gardle, a nawet b\u00f3l brzucha. Mo\u017ce wyst\u0105pi\u0107 r\u00f3wnie\u017c zas\u0142abni\u0119cie czy omdlenie. Nast\u0119pstwem utrwalonego AVRT mo\u017ce by\u0107 migotanie przedsionk\u00f3w, a tak\u017ce zatrzymanie kr\u0105\u017cenia w mechanizmie migotania kom\u00f3r. Niemowl\u0119ta z AVRT mog\u0105 prezentowa\u0107 niespecyficzne objawy, takie jak dra\u017cliwo\u015b\u0107, niech\u0119\u0107 do jedzenia, objawy ze strony przewodu pokarmowego, dr\u00f3g oddechowych, niepok\u00f3j, a nawet wstrz\u0105s. Objawy podczas cz\u0119stoskurczu r\u00f3\u017cni\u0105 si\u0119 w zale\u017cno\u015bci od wieku, szybko\u015bci cz\u0119stoskurczu oraz wsp\u00f3\u0142istniej\u0105cych chor\u00f3b serca .Wyr\u00f3\u017cniamy podzia\u0142 na dwie grupy tachyarytmii wyst\u0119puj\u0105ce w przebiegu zespo\u0142u WPW ze wzgl\u0119du na to czy dodatkowa droga przewodzenia jest integraln\u0105 cz\u0119\u015bci\u0105 p\u0119tli reentry czy te\u017c nie. W pierwszym przypadku mamy do czynienia z cz\u0119stoskurczem ortodromowym lub antydromowym, natomiast w drugim jest to najcz\u0119\u015bciej tachyarytmia przedsionkowa oraz cz\u0119stoskurcz komorowy. Podczas cz\u0119stoskurczu ortodromowego p\u0119tl\u0119 reentry tworzy: mi\u0119sie\u0144 przedsionk\u00f3w, uk\u0142ad bod\u017acoprzewodz\u0105cy, dodatkowa droga przewodzenia (pobudzana wstecznie) oraz mi\u0119sie\u0144 kom\u00f3r. W tym przypadku zespo\u0142y QRS s\u0105 najcz\u0119\u015bciej w\u0105skie. Natomiast w cz\u0119stoskurczu antydromowym jest przewodzenie przez dodatkow\u0105 drog\u0119 w kierunku zst\u0119puj\u0105cym i wsteczna aktywacja uk\u0142adu bod\u017aco-przewodz\u0105cego z pobudzeniem przedsionk\u00f3w. W tym przypadku uwidacznia si\u0119 maksymalna preekscytacja kom\u00f3r, co w obrazie EKG daje szerokie zespo\u0142y QRS  rycina .Preekscytacja jest relatywnie cz\u0119st\u0105 przyczyn\u0105 cz\u0119stoskurczu nadkomorowego. Oko\u0142o 60% dzieci z AVRT wykazuje objawy arytmii w pierwszym roku \u017cycia, najcz\u0119\u015bciej w 3-4 miesi\u0105cu \u017cycia. U ponad 20% nowo zdiagnozowanych dzieci z AVRT stwierdza si\u0119 w EKG zesp\u00f3\u0142 WPW po przywr\u00f3ceniu rytmu zatokowego. Wiele z tych dzieci wraz z wiekiem traci predyspozycje do cz\u0119stoskurczu w wyniku zmiany w\u0142a\u015bciwo\u015bci elektrycznych dodatkowych dr\u00f3g przewodzenia . UjawnieCechy zapisu EKG, kt\u00f3re mog\u0105 sugerowa\u0107 wyst\u0105pienie zespo\u0142u WPW, czyli najcz\u0119stszej formy zespo\u0142u preekscytacji, w analizie 12 odprowadzeniowego EKG mo\u017cna znale\u017a\u0107 u 1,5-3,1 na 1000 os\u00f3b. Rzeczywista cz\u0119sto\u015b\u0107 wyst\u0119powania zespo\u0142u WPW mo\u017ce by\u0107 wi\u0119ksza, poniewa\u017c nie we wszystkich przypadkach cechy preekscytacji widoczne s\u0105 na standardowym 12-odprowadzeniowym zapisie EKG, w momencie, gdy nie wyst\u0119puje napadowy cz\u0119stoskurcz [Do metod diagnostycznych u pacjent\u00f3w z zespo\u0142ami preekscytacji zalicza si\u0119:EKG: ocena rodzaju zespo\u0142u preekscytacji.Holter EKG: ocenia nasilenie zaburzen rytmu orazumo\u017cliwia wykrycie utajonej/intermituj\u0105cej postaci zespo\u0142u WPW, kt\u00f3ra nie zawsze jest uchwytna w momencie wykonywania standardowego EKG.Badanie echokardiograficzne: ocenia anotomie serca (wsp\u00f3\u0142istnienie towarzysz\u0105cych wad serca) oraz funkcj\u0119 lewej komory.refractory period of accessory pathway -APERP). Wad\u0105 tego testu jest fakt, i\u017c potrafi on ca\u0142kowicie wykluczy\u0107 ryzyko nag\u0142ego zgonu tylko u 8% pacjent\u00f3w zar\u00f3wno dzieci, jak i doros\u0142ych [Proba wysi\u0142kowa: nieinwazyjny i prosty w wykonaniu test. Nag\u0142y zanik fali delta podczas pr\u00f3by wysi\u0142kowej koreluje z nast\u0119powym efektywnym okresem refrakcji dodatkowej drogi  propafenon w bolusie co 10 minut 0,2-1 mg/ kg m.c/dawk\u0119, nast\u0119pnie wlew iv. 4-7 mcg/kg m.c./min., 2) amiodaron w bolusie iv. w 5% glukozie przez 5-10 min. 5 mg/kg, nast\u0119pnie 1-2 h wlewy iv. lub 3) sotalol 0,2-1,5 mg/ kg m.c./dawk\u0119 we wlewie iv. przez 30-60 min. Adenozyna mo\u017ce by\u0107 tak\u017ce przydatnym narz\u0119dziem diagnostycznym, poniewa\u017c u pacjent\u00f3w z niewielkim stopniem preekscytacji oraz ukryt\u0105 dodatkow\u0105 drog\u0105 przewodzenia mo\u017ce ona pom\u00f3c uwidoczni\u0107 preekscytacj\u0119 kom\u00f3r . W przypTerapia AVRT u pacjent\u00f3w posiadaj\u0105cych dodatkow\u0105 drog\u0119 przewodzenia zale\u017cna jest od wieku pacjenta, klinicznych objaw\u00f3w preekscytacji oraz towarzysz\u0105cych strukturalnych chor\u00f3b serca. Celem terapii jest zapobieganie nawrotom AVRT, a tak\u017ce zmniejszenie ryzyka powa\u017cnych zaburze\u0144 rytmu, w tym migotania kom\u00f3r lub nag\u0142ego zatrzymana kr\u0105\u017cenia. Istniej\u0105 dwie podstawowe opcje terapeutyczne: zachowawcza terapia farmakologiczna oraz inwazyjna - ablacja.W leczeniu przewlek\u0142ym, zabezpieczaj\u0105cym pacjenta przed napadami cz\u0119stoskurczu leki antyarytmiczne, g\u0142\u00f3wnie doustne preparaty beta- bloker\u00f3w (II klasa lek\u00f3w antyarytmicznych), s\u0105 zwykle lekami pierwszego rzutu. Nie wp\u0142ywaj\u0105 one bezpo\u015brednio na przewodnictwo przez dodatkow\u0105 drog\u0119, ale ograniczaj\u0105 epizody AVRT, kt\u00f3re mog\u0105 wywo\u0142a\u0107 migotanie przedsionk\u00f3w, przez co po\u015brednio mog\u0105 one zmniejszy\u0107 ryzyko nag\u0142ego zgonu . Beta \u2013 Za najbardziej efektywn\u0105 metod\u0119 leczenia zespo\u0142\u00f3w preekscytacji uznana jest ablacja dodatkowej drogi przewodzenia. Jest ona skutecznym zabiegiem dla objawowych pacjent\u00f3w zagro\u017conych nag\u0142ym zgonem sercowym, kt\u00f3rzy s\u0105 w odpowiednim wieku . Polega Powa\u017cne dzia\u0142ania niepo\u017c\u0105dane przypisywane ablacji to blok przedsionkowo-komorowy, perforacja serca, perforacja t\u0119tnicy wie\u0144cowej oraz powik\u0142ania zakrzepowo-zatorowe. Zgony odnotowywane, jako powik\u0142anie pediatrycznych ablacji nast\u0105pi\u0142y wskutek perforacji serca, urazu mi\u0119\u015bnia sercowego, zakrzep\u00f3w wie\u0144cowych lub m\u00f3zgowych i komorowych zaburze\u0144 rytmu serca. U pediatrycznych pacjent\u00f3w odsetek zgon\u00f3w po ablacji wynosi 0.22% .W badaniu opisuj\u0105cym przyczyny nag\u0142ych zgon\u00f3w sercowych u sportowc\u00f3w, zesp\u00f3\u0142 WPW odpowiedzialny by\u0142 za oko\u0142o 1% NZK. W Polsce zgodnie z Rozporz\u0105dzeniem Ministra Zdrowia z dnia 22 lipca 2016, ka\u017cde dziecko przed otrzymaniem orzeczenia o zdolno\u015bci do uprawiania okre\u015blonej dyscypliny sportowej musi mie\u0107 wykonany m.in. zapis EKG. W ostatnich latach zwi\u0119kszenie liczby wykonywanych spoczynkowych zapis\u00f3w EKG, jako skrining przed dopuszczeniem do zaj\u0119\u0107 sportowych, doprowadzi\u0142o do zwi\u0119kszonej rozpoznawalno\u015bci asymptomatycznych pacjent\u00f3w z preekscytacj\u0105 o typie WPW. Optymalne leczenie tej grupy pacjent\u00f3w ci\u0105gle pozostaje problematyczne. Podczas gdy udany zabieg ablacji eliminuje ryzyko nag\u0142ego zgonu sercowego u bezobjawowych pacjent\u00f3w z cechami preekscytacji w zapisie EKG, to kierowanie ka\u017cdego pacjenta na ten zabieg mo\u017ce nie\u015b\u0107 za sob\u0105 r\u00f3wnie\u017c wi\u0119ksze ryzyko powik\u0142a\u0144. Wst\u0119pna ocena uczestnictwa bezobjawowych pacjent\u00f3w pediatrycznych w zaj\u0119ciach sportowych, w\u0142\u0105czaj\u0105c zaj\u0119cia WF, powinna zawiera\u0107 wykonanie zapisu spoczynkowego EKG . U ka\u017cdePodsumowuj\u0105c, zespo\u0142y preekscytacji, w szczeg\u00f3lno\u015bci zesp\u00f3\u0142 WPW, jest istotnym problemem zw\u0142aszcza w populacji dzieci i m\u0142odzie\u017cy, kt\u00f3re uprawiaj\u0105 sport czy te\u017c wybieraj\u0105 przysz\u0142y zaw\u00f3d. Wa\u017cne jest wczesne wykrywanie zespo\u0142u WPW, kt\u00f3re umo\u017cliwia kontrolowanie cz\u0119stoskurczu oraz zabezpiecza przed gro\u017anymi dla \u017cycia arytmiami. W dzisiejszych czasach jest to szczeg\u00f3lnie wa\u017cne ze wzgl\u0119du na powi\u0119kszaj\u0105c\u0105 si\u0119 liczb\u0119 o\u015brodk\u00f3w pediatrycznych wykonuj\u0105cych zabieg ablacji, kt\u00f3ry daje szans\u0119 na normalne \u017cycie dzieci z tym istotnym problemem zdrowotnym."}
{"text": "Scientific Reports 10.1038/s41598-021-85641-4, published online 18 March 2021Correction to: The original version of this Article contained an error in Table 3 where a decimal point was missing in column \u201c500\u201d for crude protein parameter,bc\u201d\u201c5848\u2009\u00b1\u20094.1now reads:bc\u201d\u201c584.8\u2009\u00b1\u20094.1The original Article has been corrected."}
{"text": "Feeling of meaning in life is extremely crucial factor of mental health. The lack of it can result in various disorders. Many authors, especially those connected with current of humanistic psychology underline the teenagers\u2019 life sense.The aim of the paper was to examine the level of satisfaction with life, the frequency of psychosomatic complaints by junior high school students as well as the estimation of economical status of family and the analysis of meaning in life with above mentioned factors.The research was carried out in 2015 at 70 schools from all over the country, in group of 3695 lower secondary school students of I-III classes at the age of 13-17 . The analysis connected with meaning in life using the shorten version of Purpose in Life Test (PIL) as well as analysis of life satisfaction using Cantril scale were taken up. What is more, the subjective physical complaints using single-factor shorten scale and economic status of family with the usage of material resources FAS scale  were examined. The statistical analysis included a one-way analysis of variance (ANOVA), t-student test post-hoc test as well as multivariate logistic regression model.The average level of meaning in life among the examined students was 24,7 points , the boys achieved higher score than girls. The students satisfied with life , with rare physical complaints  and from affluent families  were significantly characterized by higher average level of meaning in life than students who were dissatisfied with their life, often or fairly suffer from health complaints and live in families of at most average level of affluence.The meaning in life is positively connected with satisfaction with life, lack of subjective complaints and family affluence. Because there is a lack of analysis linked with school teenagers\u2019 meaning in life in Polish literature, another research involved not only shorten but also full version of this tool should be conducted. Viktor Emil Frankl (1992) w ksi\u0105\u017cce \u201eCz\u0142owiek w poszukiwaniu sensu\u201d, stwierdza, i\u017c nie nale\u017cy poszukiwa\u0107 jakiego\u015b abstrakcyjnego sensu \u017cycia, gdy\u017c ka\u017cdy cz\u0142owiek ma swoj\u0105 misj\u0119, kt\u00f3r\u0105 powinien wype\u0142ni\u0107 i w kt\u00f3rej ma szans\u0119 odnale\u017a\u0107 sens swego istnienia. Powy\u017csza koncepcja sk\u0142ania ku my\u015bleniu, i\u017c cz\u0142owiek w ka\u017cdym momencie mo\u017ce czu\u0107 potrzeb\u0119 podj\u0119cia trudu, kt\u00f3rym jest odnalezienie oraz wype\u0142nienie swojego \u017cycia sensem. Poszukiwanie sensu \u017cycia mo\u017cna wi\u0119c uzna\u0107 za podstawow\u0105 motywacj\u0119 w \u017cyciu cz\u0142owieka [W istniej\u0105cej literaturze mo\u017cna znale\u017a\u0107 r\u00f3\u017cny spos\u00f3b definiowania poj\u0119cia sensu \u017cycia. Na przyk\u0142ad, l Frankl 992 w ksiPoczucie sensu \u017cycia to niezwykle istotny wska\u017anik zdrowia psychicznego, kt\u00f3rego deficyt mo\u017ce prowadzi\u0107 do wielu zaburze\u0144 oraz braku motywacji do podejmowania trud\u00f3w dnia codziennego. W konsekwencji mo\u017ce spowodowa\u0107 du\u017c\u0105 oboj\u0119tno\u015b\u0107, prowadz\u0105c\u0105 do rezygnacji z \u017cycia, zar\u00f3wno u osoby doros\u0142ej, jak i m\u0142odzie\u017cy , 4. WielW okresie dojrzewania i dorastania pojawia si\u0119 wskazywanie sobie odleg\u0142ych cel\u00f3w i podejmowanie intensywnych dzia\u0142a\u0144 zmierzaj\u0105cych do ich realizacji. R\u00f3\u017cni\u0105 si\u0119 one w zale\u017cno\u015bci od warunk\u00f3w, w jakich m\u0142odzie\u017c \u017cyje. Formowane s\u0105 nowe rodzaje motywacji, zmianie ulegaj\u0105 tre\u015bci i organizacja prze\u017cy\u0107 uczuciowych. Wyst\u0119puj\u0105ce w tym czasie trudno\u015bci dotycz\u0105 m.in. nadania sensu w\u0142asnemu \u017cyciu.Poszukiwanie w\u0142asnej to\u017csamo\u015bci zwi\u0105zanej z poczuciem sensu \u017cycia sk\u0142ada si\u0119, wed\u0142ug K. Obuchowskiego  z trzech\u2013 Faza identyfikacji \u2013 uto\u017csamianie si\u0119 z zewn\u0119trznymi wzorami.\u2013 Faza kosmiczna \u2013 oderwanie od rzeczywisto\u015bci, roz/mach, chaos w poszukiwaniu celu i sensu \u017cycia.\u2013 Faza dojrza\u0142ego sensu \u017cycia  \u2013 umiej\u0119tno\u015b\u0107 okre\u015blenia siebie i sensu swojego \u017cycia. Faza ta kszta\u0142tuje si\u0119 stopniowo, w okresie dojrzewania p\u0142ciowego i dorastania.Zadowolenie z \u017cycia, jako poznawczy komponent subiektywnego samopoczucia, zajmuje wiele uwagi naukowc\u00f3w w dziedzinie psychologii pozytywnej , 9, 10. well-being), czyli dobrego samopoczucia, psychicznego i spo\u0142ecznego, kt\u00f3re mo\u017ce by\u0107 najlepiej okre\u015blone przez dan\u0105 osob\u0119. Obszar zdrowia subiektywnego mo\u017ce zawiera\u0107 szereg \u015bwiadomych i nie\u015bwiadomych element\u00f3w, np. poczucia \u201erezerwy\u201d zdrowia czy poczucia niedoboru zdrowia lub dysfunkcji [Z poj\u0119ciem zdrowia subiektywnego, zaproponowanym przez C. Ry= , \u0142\u0105czy ssfunkcji . ZnamienNier\u00f3wno\u015bci spo\u0142eczne w zdrowiu s\u0105 podstawowym elementem spo\u0142ecznych bada\u0144 epidemiologicznych, a status socjoekonomiczny (SES) jest \u015bci\u015ble zwi\u0105zany ze zdrowiem . ZnaczenBior\u0105c pod uwag\u0119, \u017ce zadowolenie z \u017cycia, zdrowie subiektywne oraz status materialny rodziny s\u0105 niezwykle wa\u017cnymi zmiennymi w badaniach dzieci i m\u0142odzie\u017cy szkolnej, zdecydowano si\u0119 sprawdzi\u0107 ich zwi\u0105zek r\u00f3wnie\u017c z poczuciem sensu \u017cycia m\u0142odzie\u017cy w wieku 13-17 lat.Celem pracy by\u0142o zbadanie poziomu sensu i zadowolenia z \u017cycia oraz cz\u0119sto\u015bci odczuwania dolegliwo\u015bci subiektywnych przez uczni\u00f3w gimnazjum, a tak\u017ce ocena statusu ekonomicznego rodziny. Podj\u0119to r\u00f3wnie\u017c analiz\u0119 zwi\u0105zku mi\u0119dzy poczuciem sensu \u017cycia a pozosta\u0142ymi zmiennymi kontekstowymi.Postawiono dwa pytania badawcze:1. Czy wymienione powy\u017cej zmienne s\u0105 r\u00f3\u017cnicowane przez p\u0142e\u0107 i wiek uczni\u00f3w?2. Czy zadowolenie z \u017cycia, dolegliwo\u015bci somatyczne i status ekonomiczny rodziny s\u0105 zwi\u0105zane z poczuciem sensu \u017cycia?\u201e\u015arodowisko fizyczne i spo\u0142eczne oraz jako\u015b\u0107 funkcjonowania szko\u0142y a zdrowie subiektywne i zachowania zdrowotne\u201d (Nr 2013/09/B/HS6/03438). Na przeprowadzenie badania uzyskano zgod\u0119 Komisji Bioetycznej przy Instytucie Matki i Dziecka.Praca przedstawia wyniki analiz prowadzonych z wykorzystaniem danych uzyskanych z og\u00f3lnopolskiego badania ankietowego w szko\u0142ach w 2015 roku, w ramach projektu NCN pt.: Badaniem obj\u0119to grup\u0119 3695 uczni\u00f3w klas I-III gimnazjum w wieku 13-17 lat (1754 ch\u0142opc\u00f3w i 1941 dziewcz\u0105t), w 70 szko\u0142ach z terenu ca\u0142ego kraju, po uprzednim uzyskaniu zgody dyrektora szko\u0142y oraz uczni\u00f3w i ich rodzic\u00f3w. \u0179r\u00f3d\u0142o danych stanowi anonimowa ankieta audytoryjna pt. \u201eZdrowie i szko\u0142a\u201d, wype\u0142niana przez uczni\u00f3w w tradycyjnej formie papierowej lub on line.Podj\u0119te w pracy analizy dotyczy\u0142y nast\u0119puj\u0105cych zagadnie\u0144:Poczucie sensu \u017cycia, badano przy u\u017cyciu Testu Sensu \u017bycia \u2013 Purpose in Life Test (PIL) autorstwa J. C. Crumbaugha i L. T. Maholicka, w polskiej adaptacji Z. P\u0142u\u017cek [1. . P\u0142u\u017cek . W badan. P\u0142u\u017cek  Sk\u0142ada sW \u017cyciu: nie mam \u017cadnych cel\u00f3w ani do niczego nie d\u0105\u017c\u0119 (1) \u2212 mam bardzo wyra\u017ane cele i d\u0105\u017cenia (7);Moje istnienie jest: zupe\u0142nie bezcelowe (1) \u2013 celowe i sensowne (7);Ka\u017cdy dzie\u0144: niesie ze sob\u0105 co\u015b nowego (7) \u2013 jest zawsze taki sam (1);W d\u0105\u017ceniu do cel\u00f3w \u017cyciowych: nigdy nie mia\u0142em powodzenia (1) \u2013 uda\u0142o mi si\u0119 zaspokaja\u0107 moje potrzeby (7);Uwa\u017cam, \u017ce moje szanse na znalezienie sensu \u017cycia, celu i roli w \u017cyciu: s\u0105 bardzo du\u017ce (7) \u2212 s\u0105 praktycznie \u017cadne (1);Doszed\u0142em do wniosku, \u017ce: brak mi celu (1) \u2212 mam wyra\u017ane cele daj\u0105ce pe\u0142ne zadowolenie (7).Dok\u0142adniejsze informacje dotycz\u0105ce zastosowanej skali zosta\u0142y przedstawione w odr\u0119bnym opracowaniu .Zadowolenie z \u017cycia, badano z zastosowaniem skali wizualnej, b\u0119d\u0105cej adaptacj\u0105 tzw. skali Cantrila i stosowanej w badaniach HBSC  od 2002 roku [2. 002 roku . Skala tObok jest rysunek drabiny. Na g\u00f3rze drabiny jest liczba 10 \u2013 umownie oznaczaj\u0105ca \u017cycie, kt\u00f3re wydaje Ci si\u0119 najlepsze. Na dole drabiny jest liczba 0 \u2013 oznaczaj\u0105ca \u017cycie, kt\u00f3re wydaje Ci si\u0119 najgorsze. Pomy\u015bl, jakie jest teraz Twoje \u017cycie i w kt\u00f3rym miejscu drabiny Ty stan\u0105\u0142by\u015b. Wstaw X w jedn\u0105 kratk\u0119 w drabinie cyfry, kt\u00f3ra znajduje si\u0119 w tym miejscu.Przyj\u0119to podzia\u0142 dychotomiczny i za\u0142o\u017cono \u017ce osoby, kt\u00f3re wybra\u0142y poziom 0-5 punkt\u00f3w s\u0105 niezadowolone ze swojego \u017cycia, natomiast te, kt\u00f3re uzyska\u0142y 6-10 punkt\u00f3w to osoby zadowolone z \u017cycia.Dolegliwo\u015bci subiektywne, badano za pomoc\u0105 skr\u00f3conej, jednoczynnikowej skali b\u0119d\u0105cej cz\u0119\u015bci\u0105 skali cz\u0105stkowej Problemy zdrowia fizycznego. Skala ta wraz z dwoma innymi skalami cz\u0105stkowymi  pochodzi z kwestionariusza CHIP-AE  [W ci\u0105gu ostatnich 4 tygodni, przez ile dni naprawd\u0119 \u017ale si\u0119 czu\u0142e\u015b?; budzi\u0142e\u015b si\u0119 z uczuciem zm\u0119czenia?; mia\u0142e\u015b b\u00f3le g\u0142owy? Pytanie3. Edition) . Do zastzaznacz znakiem X jedn\u0105 kratk\u0119 w ka\u017cdym wierszu, a ka\u017cde stwierdzenie mia\u0142o kategorie odpowiedzi: wcale; od 1 do 3 dni; od 4 do 6 dni; od 7 do 14 dni; od 15 do 28 dni. Dla potrzeb analiz kategoriom tym przyporz\u0105dkowano odpowiednio od 0 do 4 punkt\u00f3w. Odpowiedzi zosta\u0142y wykorzystane do opracowania indeksu sumarycznego przyjmuj\u0105cego zakres od 0 do 12 punkt\u00f3w. Im wy\u017cszy wynik uzyskany na skali, tym cz\u0119\u015bciej badana m\u0142odzie\u017c odczuwa\u0142a dolegliwo\u015bci somatyczne w ci\u0105gu ostatnich 4 tygodni. Dokonano podzia\u0142u na trzy kategorie w zale\u017cno\u015bci od cz\u0119sto\u015bci odczuwania dolegliwo\u015bci somatycznych: rzadko (0-1 punkt); przeci\u0119tnie (2-6 punkt\u00f3w); cz\u0119sto (7-12 punkt\u00f3w).poprzedzone by\u0142o instrukcj\u0105: Status materialny rodziny oceniano z zastosowaniem skali zasob\u00f3w materialnych - FAS . Stopniowo modyfikowana skala FAS wykorzystywana jest w badaniach HBSC jako wiarygodny miernik statusu spo\u0142ecznoekonomicznego rodziny [biedne (0-4 punkty), raczej biedne (5-6 punkt\u00f3w), przeci\u0119tne (7-9 punkt\u00f3w) i zamo\u017cne (10-13 punkt\u00f3w). Kategoryzacji dokonano zgodnie z krajowymi wytycznymi, uwzgl\u0119dniaj\u0105cymi du\u017cy odsetek rodzin z wynikiem w skali FAS poni\u017cej 7 punkt\u00f3w [4.  rodziny , 28. W owysoki poziom poczucia sensu \u017cycia (powy\u017cej 26 punkt\u00f3w) oraz brak wysokiego poczucia sensu \u017cycia (poni\u017cej 26 punkt\u00f3w). Do modelu regresji w\u0142\u0105czono r\u00f3wnie\u017c p\u0142e\u0107 oraz klas\u0119 (od I do III) gimnazjum jako zmienne kontrolowane. We wszystkich analizach za poziom istotno\u015bci przyj\u0119to p<0,05. Zastosowano program SPSS v. 17.W pierwszej cz\u0119\u015bci pracy przedstawiono wyniki analiz jednowymiarowych, dotycz\u0105ce \u015bredniego poziomu sensu \u017cycia, poziomu zadowolenia z \u017cycia oraz cz\u0119sto\u015bci odczuwania dolegliwo\u015bci subiektywnych. Dalsze analizy prowadzono na zmiennych skategoryzowanych. Do zbadania r\u00f3\u017cnic w poziomie poczucia sensu \u017cycia u uczni\u00f3w w zale\u017cno\u015bci od ich poziomu zadowolenia z \u017cycia oraz cz\u0119sto\u015bci odczuwania dolegliwo\u015bci subiektywnych i statusu ekonomicznego rodziny zastosowano jednoczynnikow\u0105 analiz\u0119 wariancji (ANOVA), test t-Studenta oraz tzw. test posthoc czyli test por\u00f3wna\u0144 wielokrotnych. Analizy te s\u0105 stosowane gdy analiza wariancji informuje, \u017ce s\u0105 istotne statystycznie r\u00f3\u017cnice jednak nie wiadomo, kt\u00f3re z por\u00f3wnywanych grup r\u00f3\u017cni\u0105 si\u0119 miedzy sob\u0105 W celu okre\u015blenia prawdopodobie\u0144stwa wysokiego poziomu sensu \u017cycia w kontek\u015bcie wybranych zmiennych, oszacowano wielowymiarowy model regresji logistycznej, dziel\u0105c umownie zmienn\u0105 dotycz\u0105c\u0105 poczucia sensu \u017cycia na 2 kategorie: 11.11.2W toku analizy zebranego materia\u0142u stwierdzono, \u017ce ch\u0142opcy charakteryzowali si\u0119 istotnie wy\u017cszym poziomem sensu \u017cycia ni\u017c dziewcz\u0119ta . Ustalono r\u00f3wnie\u017c, \u017ce klasa gimnazjum istotnie r\u00f3\u017cnicuje \u015bredni poziom sensu \u017cycia uczni\u00f3w . Analizuj\u0105c powy\u017csze zmienne po skategoryzowaniu, odnotowano r\u00f3\u017cnice w poziomie zadowolenia z \u017cycia w zale\u017cno\u015bci od p\u0142ci i poziomu edukacji (klasy gimnazjum). R\u00f3\u017cnice te by\u0142y istotne statystycznie tylko w zale\u017cno\u015bci od p\u0142ci, przy czym ch\u0142opcy istotnie cz\u0119\u015bciej ni\u017c dziewcz\u0119ta byli zadowoleni z \u017cycia . Cz\u0119sto\u015b\u0107 odczuwania przez m\u0142odzie\u017c dolegliwo\u015bci subiektywnych by\u0142a r\u00f3\u017cnicowana istotnie statystycznie zar\u00f3wno przez p\u0142e\u0107, jak i klas\u0119 gimnazjum . Dziewcz\u0119ta oraz uczniowie III klasy gimnazjum istotnie cz\u0119\u015bciej ni\u017c ch\u0142opcy i uczniowie m\u0142odsi deklarowali nasilone odczuwanie dolegliwo\u015bci subiektywnych .1.3Analiza wariancji ANOVA oraz por\u00f3wnania wielokrotne wskaza\u0142y na istotne statystycznie r\u00f3\u017cnice w \u015brednim poziomie sensu \u017cycia mi\u0119dzy uczniami w zale\u017cno\u015bci od ich zadowolenia z \u017cycia i cz\u0119sto\u015bci odczuwania dolegliwo\u015bci fizycznych oraz statusu ekonomicznego rodziny . Uczniow2Do modelu regresji logistycznej w\u0142\u0105czono wszystkie analizowane zmienne, a tak\u017ce p\u0142e\u0107 i klas\u0119 gimnazjum. Kategoriami odniesienia by\u0142y: brak zadowolenia z \u017cycia; cz\u0119sto odczuwane dolegliwo\u015bci fizyczne; rodzina biedna; III klasa gimnazjum i p\u0142e\u0107 \u2212 dziewcz\u0119ta. Analizy wykaza\u0142y, \u017ce zadowolenie z \u017cycia, odczuwanie dolegliwo\u015bci fizycznych rzadko i z przeci\u0119tn\u0105 cz\u0119stotliwo\u015bci\u0105 oraz co najmniej przeci\u0119tny status ekonomiczny rodziny s\u0105 zwi\u0105zane z wysokim poziomem sensu \u017cycia . StwierdPersonal Meaning Profile (PMP). W przytoczonym badaniu wzi\u0119\u0142a tak\u017ce udzia\u0142 niepor\u00f3wnywalnie mniejsza pr\u00f3ba nastolatk\u00f3w \u2013 104 osoby. Nie bez znaczenia mog\u0105 tu tak\u017ce pozostawa\u0107 r\u00f3\u017cnice kulturowe. Ch\u0142opcy te\u017c istotnie bardziej ni\u017c dziewcz\u0119ta byli zadowoleni z \u017cycia. Podobne wyniki uzyskano w badaniach HBSC 2014 [W niniejszej analizie pos\u0142u\u017cono si\u0119 wynikami badania ankietowego przeprowadzonego na pr\u00f3bie 3695 uczni\u00f3w w wieku 13-17 lat (I-III klasy gimnazjum). W pracy postawiono dwa pytania badawcze. Odpowiadaj\u0105c na pierwsze z nich, dotycz\u0105ce r\u00f3\u017cnicowania analizowanych zmiennych przez p\u0142e\u0107 i wiek, wykazano, \u017ce ch\u0142opcy istotnie cz\u0119\u015bciej ni\u017c dziewcz\u0119ta wykazywali si\u0119 wy\u017cszym poziomem poczucia sensu \u017cycia. Rathi i Rastogi (2007) uzyskali w swych analizach inne wyniki . W przepBSC 2014  Badania BSC 2014  OdwrotneBSC 2014 . W ich aBSC 2014 . Drugie BSC 2014  wykaza\u0142yBSC 2014 . Uwa\u017cno\u015bBSC 2014 . Mi\u0119dzykBSC 2014 . Wyniki BSC 2014 . PodobneBSC 2014 .Bior\u0105c pod uwag\u0119 zwi\u0105zek poczucia sensu \u017cycia z zadowoleniem z \u017cycia, dolegliwo\u015bciami subiektywnymi oraz statusem ekonomicznym rodziny, warto uwzgl\u0119dni\u0107 ten spos\u00f3b my\u015blenia w dalszych badaniach i dzia\u0142aniach z zakresu polityki zdrowotnej i edukacyjnej. Przede wszystkim nale\u017ca\u0142oby wykorzysta\u0107 potencjalny ochronny wp\u0142yw poczucia sensu \u017cycia na zdrowie, w programach promocji zdrowia, kt\u00f3re powinny zawiera\u0107 wi\u0119kszy \u0142adunek tre\u015bci dotycz\u0105cych zdrowia psychicznego, odporno\u015bci psychicznej czy radzenia sobie ze stresem. W niekt\u00f3rych prywatnych lub licz\u0105cych ma\u0142\u0105 liczb\u0119 uczni\u00f3w szko\u0142ach publicznych w Polsce prowadzone s\u0105 spotkania mentoringowe a tak\u017ce tutoriale maj\u0105ce na celu popraw\u0119 dba\u0142o\u015bci m\u0142odzie\u017cy o zdrowie. W wi\u0119kszych plac\u00f3wkach o\u015bwiatowych warto wdro\u017cy\u0107 serie spotka\u0144 socjoterapeutycznych w kilkunastoosobowych grupach lub klasach maj\u0105cych na celu nauk\u0119 korzystania ze wsparcia spo\u0142ecznego. W Polsce brakuje aktualnych bada\u0144 dotycz\u0105cych sensu \u017cycia m\u0142odzie\u017cy szkolnej, dlatego celowe by\u0142oby przeprowadzenie kolejnych analiz z uwzgl\u0119dnieniem innych zmiennych, takich jak: przemoc r\u00f3wie\u015bnicza czy subiektywna witalno\u015b\u0107 a tak\u017ce por\u00f3wna\u0144 poziomu poczucia sensu \u017cycia u nastolatk\u00f3w zdrowych oraz z chorobami przewlek\u0142ymi w powi\u0105zaniu z innymi zmiennymi kontekstowymi.Niew\u0105tpliwymi ograniczeniami przeprowadzonego badania by\u0142o u\u017cycie w nim skr\u00f3conych skal lub pojedynczych pyta\u0144 dotycz\u0105cych analizowanych zagadnie\u0144. Jest to charakterystyczne ograniczenie dotycz\u0105ce du\u017cych bada\u0144 populacyjnych, kt\u00f3re prowadzone s\u0105 na kilkutysi\u0119cznych pr\u00f3bach, za pomoc\u0105 wielow\u0105tkowych ankiet. Analizy te maj\u0105 na celu, przede wszystkim informowa\u0107 o obecno\u015bci pewnych zale\u017cno\u015bciach i wyznacza\u0107 kierunki dalszych, pog\u0142\u0119bionych eksploracji. Nale\u017cy jednak uwzgl\u0119dni\u0107 fakt, \u017ce s\u0105 to pytania walidowane w badaniach mi\u0119dzynarodowych, dlatego rzetelno\u015b\u0107 uzyskanych przy ich zastosowaniu wynik\u00f3w jest bardzo wysoka.W niniejszym badaniu zastosowano r\u00f3wnie\u017c skr\u00f3con\u0105, 6-itemow\u0105 wersj\u0119 kwestionariusza PIL, opracowan\u0105 przez J. \u017byci\u0144sk\u0105 i J. Januszka . ZdecydoPomimo tego, niniejsza praca jest jednym z nielicznych opracowa\u0144 obejmuj\u0105cych poczucie sensu \u017cycia m\u0142odzie\u017cy szkolnej. Zar\u00f3wno ta, jak i wcze\u015bniejsze analizy, dotycz\u0105ce tej problematyki  , stanowiWed\u0142ug przeprowadzonych analiz, predyktorem wysokiego poziomu poczucia sensu \u017cycia wydaje si\u0119 by\u0107 zadowolenie z \u017cycia, dobre samopoczucie fizyczne oraz wy\u017csza zamo\u017cno\u015b\u0107 rodziny.Skala PIL w wersji 6-punktowej jest godnym polecenia narz\u0119dziem w badaniach populacyjnych, przyst\u0119pnym poznawczo r\u00f3wnie\u017c dla m\u0142odzie\u017cy szkolnej.Poniewa\u017c w literaturze polskiej brakuje analiz dotycz\u0105cych poczucia sensu \u017cycia m\u0142odzie\u017cy szkolnej, warto kontynuowa\u0107 i pog\u0142\u0119bia\u0107 badania, z uwzgl\u0119dnieniem zar\u00f3wno skr\u00f3conej, jak i pe\u0142nej wersji tego narz\u0119dzia."}
{"text": "Arabic language is a challenging language for automatic processing. This is due to several intrinsic reasons such as Arabic multi-dialects, ambiguous syntax, syntactical flexibility and diacritics. Machine learning and deep learning frameworks require big datasets for training to ensure accurate predictions. This leads to another challenge faced by researches using Arabic text; as Arabic textual datasets of high quality are still scarce. In this paper, an intelligent framework for expanding or augmenting Arabic sentences is presented. The sentences were initially labelled by human annotators for sentiment analysis. The novel approach presented in this work relies on the rich morphology of Arabic, synonymy lists, syntactical or grammatical rules, and negation rules to generate new sentences from the seed sentences with their proper labels. Most augmentation techniques target image or video data. This study is the first work to target text augmentation for Arabic language. Using this framework, we were able to increase the size of the initial seed datasets by 10 folds. Experiments that assess the impact of this augmentation on sentiment analysis showed a 42% average increase in accuracy, due to the reliability and the high quality of the rules used to build this framework. Arabic language is considered the most widely spoken language among the Semitic languages . It is aRecently, many efforts have investigated the Arabic language whether to analyze the text , parse sSentiment analysis is the task of processing data, mainly textual, in order to determine its polarity, i.e., positive, negative, or neutral . This taDeep learning has received unprecedented attention in recent years and provided state-of-the-art results in many fields including sentiment analysis . HoweverThe syntax of Arabic language is complex \u2014as seversame labels as the input sentences. By comparison, applying the set of rules described in (3) as aforementioned, generates new sentences with opposite labels to the input sentences. Experiments proved the viability and effectiveness of the augmentation framework by running three experiments using three datasets. The size of the original datasets substantially increased and the generated sentences were of high quality.For building the framework, initially the Stanford Arabic Parser was used to generate the parse trees of Arabic sentences. Afterwards, the augmentation rules generated were used on these trees, to generate several equivalent parse trees for the original sentences utilizing Arabic morphology, syntax, synonyms and negation particles. These augmentation rules can be broadly divided into: (1) rules which alter or swap branches of the parse trees as per Arabic syntax and thus generate new sentences with the same labels; (2) rules which generate new parse trees by utilizing the synonyms of words in these sentences, and also generate new sentences with the same original labels; (3) rules which insert negation particles into the sentences and thus generate new sentences with opposite labels. It is worth mentioning here that the work in this paper addresses text augmentation for sentiment analysis. This means that the labels of the investigated sentences are either neutral, positive or negative. Applying the sets of rules described in (1) or (2) above will generate new sentences with the The rest of this article is organized as follows: \u201cRelated Work\u201d briefly describes the related literature works. \u201cArabic Language Properties\u201d explains the properties of Arabic language. \u201cDescription of Framework\u201d explains the design of the transformation rules which are the core of the augmentation framework. \u201cNegation\u201d describes the implementation of the framework. \u201cEvaluation\u201d demonstrates the experiments which were carried out to assess the effectiveness of the proposed work. And finally \u201cConclusion\u201d summarizes the conclusion of this work.This section describes related studies which have utilized Arabic WordNet as a component of frameworks. It also describes the related work which addresses data augmentation.WordNet  is a larWordNet has been very useful as it was used to build many Natural Language Processing applications, Information Retrieval, term expansion and document representations . For exaHowever, many efforts have been reported to adapt WordNet for other languages, such as WordNets for European languages  and FrenSeveral researchers have targeted extending Arabic WordNet. For example, in the work reported in Data augmentation is a technique that is used to increase the size of datasets and preserve the labels at the same time. It became popular with deep learning networks as they require training on huge datasets to secure high accuracies . ExtendiOn the other hand, data augmentation is limited when dealing with textual data. This is due to the very difficult definition and standardization of specific rules or transformations that preserve the meaning of the produced textual data . BasicalThe works reported in The work reported in k score words as a substitute. In a similar study, As it can be seen from the above literature, most of the existing augmentation techniques address image or audio data and less work addresses text augmentation. In this regard, it should be mentioned that no work addresses Arabic text augmentation. The current proposed framework is substantially different from text augmentation which relies on the replacement of words by their synonyms. On the other hand, it utilizes the rich syntax and grammar of the Arabic language in order to generate transformation rules, that are subsequently used to generate new sentences based on seed sentences.Arabic language is one of the Semitic languages. It consists of 28 basic letters. Several Arabic letters change their shapes based on their location in the word. For example, the letter (\u0633) has the shape (\u0633\u0640\u0640) when it is located at the beginning of the word, the shape (\u0640\u0633\u0640) when it is located at the middle of the word, (\u0640\u0633) when it is located at the end of the word but connected to the previous letter, and (\u0633) when it comes at the end of the word but disconnected from the previous letter. Arabic is an inflectional language that is written from right to left. The following three subsections provide background about Arabic language.Study \u201cdarasa \u062f\u064e\u0631\u064e\u0633\u064e \u201dScholar \u201cdAris \u062f\u0627\u0631\u0650\u0633 \u201dLesson \u201cdaros \u062f\u064e\u0631\u0652\u0633 \u201dTeacher \u201cmudaris \u0645\u064f\u062f\u064e\u0631\u0650\u0633 \u201dSchools \u201cmdAris \u0645\u062f\u0627\u0631\u0650\u0633 \u201dSchool \u201cmadrasop \u0645\u064e\u062f\u0631\u064e\u0633\u0652\u0629\u201dStudy \u201cmudArasop \u0645\u064f\u062f\u0627\u0631\u064e\u0633\u0652\u0629\u201dMorphology is the structure of words. The morphology of Arabic language is complex but systematic\u2014where there are two ways to build a word in Arabic: derivation and agglutination. The derivation is a way of generating stems from a list of roots; based on three basic letters (\u0641\u060c \u0639\u060c \u0644) for trilateral roots. For example, by using the root word \u201c\u062f\u0631\u0633\u201d that rhymes with \u201c\u0641\u0639\u0644\u201d one can generate the following stems:The second way to build words in Arabic language is agglutination. In this way, the words are built by adding affixes to the word. These affixes could be prefixes at the beginning of words such as (\u0627\u0633\u062a\u060c \u062a\u0645 \u060c \u062a\u060c \u0627\u0646), infixes in the middle of the word (such as \u0627), or suffixes at the end of the word such as (\u0629 \u060c \u0627\u0621 \u060c \u0627\u0646).In Arabic scripts, the sentence has two types or categories . Each type has its own grammar and rules. The nominal sentence, in Arabic, consists of a subject (Almubtada) and predicate (Alkhabar). The normal order is that the subject is followed by the predicate but in certain cases, it is allowed to swap them (e.g. the sentence \u201c\u0623\u0646\u0652\u062a \u064e \u0645\u062c\u062a\u0647\u062f\u064c \u201d which means \u201cYou are diligent\u201d could be \u201c\u0645\u062c\u062a\u0647\u062f\u064c \u0623\u0646\u0652\u062a \u064e\u200f\u201d). The subject in the nominal sentence can be Noun, Pronoun or Number while the predicate can be Singular Noun, Adverb, Preposition, Nominal sentence, or Verbal Sentence.The verbal sentence in Arabic, like in many other languages, consists of Verb (V), Subject (S) and Object (O) without a specific order, which means that the order of verbal sentences could be: VSO, VOS, SVO or VOS. Additionally, in Arabic language diacritics, prefixes and suffixes are used to represent gender. Therefore, the absence of diacritics can create ambiguity and might change the meaning.Fatha: symbolized as an italic score on the top of the letter.Dma: symbolized as a small (\u0648) letter on the top of the letter.Ksra: symbolized as an italic underscore on the bottom of the letter.Sokon: symbolized as a small circle on the top of the letter.One of the Arabic language features is the diacritics that are written above or underneath its letters. Diacritics are small vowel marks that represent three short vowels . They are used to regulate and control the letters and pronunciation. Therefore, diacritics have a huge effect on the text and its meaning, removing them may lead to morphological-lexical and morphological-syntactical ambiguities. For example, the word (nEm) (\u0646\u0639\u0645) has the meaning \u2018Yes\u2019 if it was written (naEom \u0646\u064e\u0639\u0652\u0645), while it means \u2018graces\u2019 if it was written (niEm \u0646\u0650\u0639\u0645). The basic diacritics of Arabic language are:As a first step, clear definitions of Arabic grammar rules were specified. These rules include specifications for nominal sentences, verbal sentences, questions, verbs, adjectives, pronouns, prepositions, conjunctions and numbers. These defined grammar-based rules were represented using the Stanford Arabic parser tagset. RULE 1: DTNN+ADJ \u2192 ADJ+DTNNExample: Alrjl mHbwb (\u0627\u0644\u0631\u062c\u0644 \u0645\u062d\u0628\u0648\u0628) \u2192 mHbwb Alrjl (\u0645\u062d\u0628\u0648\u0628 \u0627\u0644\u0631\u062c\u0644)Parse: (ROOT (S (NP (DTNN \u0627\u0644\u0631\u062c\u0644)) (ADJP (JJ \u0645\u062d\u0628\u0648\u0628)))) \u2192 (ROOT (ADJP (JJ \u0645\u062d\u0628\u0648\u0628) (NP (DTNN \u0627\u0644\u0631\u062c\u0644))))RULE 2: NN+ADJ \u2192 ADJ+NNExample: mAlk rA}E (\u0645\u0627\u0644\u0643 \u0631\u0627\u0626\u0639) \u2192 rA}E mAlk (\u0631\u0627\u0626\u0639 \u0645\u0627\u0644\u0643)Parse: (ROOT (S (NP (NNP \u0645\u0627\u0644\u0643)) (ADJP (JJ \u0631\u0627\u0626\u0639)))) \u2192 (ROOT (FRAG (NP (JJ \u0631\u0627\u0626\u0639)) (NP (NNP \u0645\u0627\u0644\u0643))))RULE 3: DTNN+NN \u2192 NN+DTNNExample: Alrjl $jAE (\u0627\u0644\u0631\u062c\u0644 \u0634\u062c\u0627\u0639) \u2192 $jAE Alrjl (\u0634\u062c\u0627\u0639 \u0627\u0644\u0631\u062c\u0644)Parse: (ROOT (S (NP (DTNN \u0627\u0644\u0631\u062c\u0644)) (NP (NNP \u0634\u062c\u0627\u0639)))) \u2192 (ROOT (ADJP (JJ \u0634\u062c\u0627\u0639) (NP (DTNN \u0627\u0644\u0631\u062c\u0644))))RULE 4: NN+NN \u2192 NN+NN(swap)Example: AHmd Swth rA}E (\u0627\u062d\u0645\u062f \u0635\u0648\u062a\u0647 \u0631\u0627\u0626\u0639) \u2192 rA}E AHmd Swth (\u0631\u0627\u0626\u0639 \u0627\u062d\u0645\u062f \u0635\u0648\u062a\u0647)Parse: (ROOT (S (NP (NNP \u0627\u062d\u0645\u062f)) (NP (NN \u0635\u0648\u062a\u0647) (JJ \u0631\u0627\u0626\u0639)))) \u2192 (ROOT (FRAG (NP (JJ \u0631\u0627\u0626\u0639)) (NP (NNP \u0627\u062d\u0645\u062f) (NNP \u0635\u0648\u062a\u0647))))RULE 5: NN+DTNN \u2192 NN+DTNNExample: Ebd AlrHmn xlwq (\u0639\u0628\u062f \u0627\u0644\u0631\u062d\u0645\u0646 \u062e\u0644\u0648\u0642)\u2192xlwq Ebd AlrHmn (\u062e\u0644\u0648\u0642 \u0639\u0628\u062f \u0627\u0644\u0631\u062d\u0645\u0646)Parse: (ROOT (FRAG (NP (NNP \u0639\u0628\u062f)) (NP (DTNNP \u0627\u0644\u0631\u062d\u0645\u0646)) (NP (NNP \u062e\u0644\u0648\u0642)))) \u2192 (ROOT (NP (NNP \u062e\u0644\u0648\u0642) (NNP \u0639\u0628\u062f) (DTNNP \u0627\u0644\u0631\u062d\u0645\u0646)))RULE 6: DTNN+DTNN \u2192 DTNN+DTNN(swap)Example: Ebd AlrHmn AlrHym (\u0639\u0628\u062f \u0627\u0644\u0631\u062d\u0645\u0646 \u0627\u0644\u0631\u062d\u064a\u0645) \u2192 AlrHym Ebd AlrHmn (\u0627\u0644\u0631\u062d\u064a\u0645 \u0639\u0628\u062f \u0627\u0644\u0631\u062d\u0645\u0646)Parse: (ROOT (FRAG (NP (NNP \u0639\u0628\u062f)) (NP (DTNNP \u0627\u0644\u0631\u062d\u0645\u0646)) (NP (DTNNP \u0627\u0644\u0631\u062d\u064a\u0645)))) \u2192 (ROOT (S (NP (DTNNP \u0627\u0644\u0631\u062d\u064a\u0645)) (NP (NNP \u0639\u0628\u062f) (DTNNP \u0627\u0644\u0631\u062d\u0645\u0646))))RULE 7: ADJ+ADJ \u2192 DTNN+DTNN (sawap)Example: AlftAp Aljmylp mjthdp (\u0627\u0644\u0641\u062a\u0627\u0629 \u0627\u0644\u062c\u0645\u064a\u0644\u0629 \u0645\u062c\u062a\u0647\u062f\u0629) \u2192 AlftAp mjthdp Aljmylp (\u0627\u0644\u0641\u062a\u0627\u0629 \u0645\u062c\u062a\u0647\u062f\u0629 \u0627\u0644\u062c\u0645\u064a\u0644\u0629)Parse: (ROOT (NP (DTNN \u0627\u0644\u0641\u062a\u0627\u0629) (DTJJ \u0627\u0644\u062c\u0645\u064a\u0644\u0629) (DTJJ \u0645\u062c\u062a\u0647\u062f\u0629))) \u2192 (ROOT (NP (DTNN \u0627\u0644\u0641\u062a\u0627\u0629) (DTJJ \u0645\u062c\u062a\u0647\u062f\u0629) (DTJJ \u0627\u0644\u062c\u0645\u064a\u0644\u0629)))RULE 8: PP+(NN+DTNN) \u2192 place them in the beginning and reverse the sentenceExample: Ebr Alm$rf En $kr AlfSl (\u0639\u0628\u0631 \u0627\u0644\u0645\u0634\u0631\u0641 \u0639\u0646 \u0634\u0643\u0631 \u0627\u0644\u0641\u0635\u0644) \u2192 En $kr AlfSl Ebr Alm$rf (\u0639\u0646 \u0634\u0643\u0631 \u0627\u0644\u0641\u0635\u0644 \u0639\u0628\u0631 \u0627\u0644\u0645\u0634\u0631\u0641)Parse: (ROOT (S (VP (VBD \u0639\u0628\u0631) (NP (DTNN \u0627\u0644\u0645\u0634\u0631\u0641)) (PP (IN \u0639\u0646) (NP (NN \u0634\u0643\u0631) (NP (DTNN \u0627\u0644\u0641\u0635\u0644))))))) \u2192 (ROOT (S (PP (IN \u0639\u0646) (NP (NN \u0634\u0643\u0631) (NP (DTNN \u0627\u0644\u0641\u0635\u0644)))) (NP (NN \u0639\u0628\u0631) (NP (DTNN \u0627\u0644\u0645\u0634\u0631\u0641)))))RULE 9: PP+DTNN \u2192 place them in the beginning and revese the sentencesExample: bAsm yqdm $y}A mn AlfkAhAt (\u0628\u0627\u0633\u0645 \u064a\u0642\u062f\u0645 \u0634\u064a\u0626\u0627\u064b \u0645\u0646 \u0627\u0644\u0641\u0643\u0627\u0647\u0627\u062a) \u2192 mn AlfkAhAt bAsm yqdm $y}A (\u0645\u0646 \u0627\u0644\u0641\u0643\u0627\u0647\u0627\u062a \u0628\u0627\u0633\u0645 \u064a\u0642\u062f\u0645 \u0634\u064a\u0626\u0627\u064b)Parse: ROOT (S (NP (NNP \u0628\u0627\u0633\u0645)) (VP (VBP \u064a\u0642\u062f\u0645) (NP (NP (NN \u0634\u064a\u0626\u0627)) (PP (IN \u0645\u0646) (NP (DTNNS \u0627\u0644\u0641\u0643\u0627\u0647\u0627\u062a))))))) \u2192 (ROOT (S (PP (IN \u0645\u0646) (NP (NN \u0627\u0644\u0641\u0643\u0627\u0647\u0627\u062a) (NP (NNP \u0628\u0627\u0633\u0645)))) (VP (VBP \u064a\u0642\u062f\u0645) (NP (NN \u0634\u064a\u0626\u0627)))))RULE 10: PP+\u2192 place them in the beginnig and reverse the sentenceExample: tSAdq mE Al*}Ab ElY >n ykwn f>sk mstEdA (\u062a\u0635\u0627\u062f\u0642 \u0645\u0639 \u0627\u0644\u0630\u0626\u0627\u0628 \u0639\u0644\u0649 \u0623\u0646 \u064a\u0643\u0648\u0646 \u0641\u0623\u0633\u0643 \u0645\u0633\u062a\u0639\u062f ) \u2192 Al*}Ab ElY >n ykwn tSAdq mE Al*}Ab f>sk mstEdA (\u0639\u0644\u0649 \u0623\u0646 \u064a\u0643\u0648\u0646 \u062a\u0635\u0627\u062f\u0642 \u0645\u0639 \u0627\u0644\u0630\u0626\u0627\u0628 \u0641\u0623\u0633\u0643 \u0645\u0633\u062a\u0639\u062f\u0627\u064b)Parse: (ROOT (S (VP (VBP \u062a\u0635\u0627\u062f\u0642) (NP (NN \u0645\u0639) (NP (DTNN \u0627\u0644\u0630\u0626\u0627\u0628))) (PP (IN \u0639\u0644\u0649) (NP (DTNN \u0623\u0646))) (S (VP (VBP \u064a\u0643\u0648\u0646) (NP (NNP \u0641\u0623\u0633\u0643)) (ADJP (JJ \u0645\u0633\u062a\u0639\u062f\u0627))))))) \u2192 (ROOT (S (PP (IN \u0639\u0644\u0649) (NP (DTNN \u0623\u0646))) (VP (VBP \u064a\u0643\u0648\u0646) (S (VP (VBP \u062a\u0635\u0627\u062f\u0642) (NP (NN \u0645\u0639) (NP (NNP \u0627\u0644\u0630\u0626\u0627\u0628) (NNP \u0641\u0623\u0633\u0643))) (ADJP (JJ \u0645\u0633\u062a\u0639\u062f\u0627)))))))RULE 11: Wh-prounoun at the end of the sentences \u2192 Move it to the beginningExample: njH Al*y *hb AlY Almdrsp (\u0646\u062c\u062d \u0627\u0644\u0630\u064a \u0630\u0647\u0628 \u0627\u0644\u0649 \u0627\u0644\u0645\u062f\u0631\u0633\u0629) \u2192 Al*y *hb AlY Almdrsp njH (\u0627\u0644\u0630\u064a \u0630\u0647\u0628 \u0627\u0644\u0649 \u0627\u0644\u0645\u062f\u0631\u0633\u0629 \u0646\u062c\u062d)Parse: (ROOT (S (VP (VBD \u0646\u062c\u062d) (SBAR (WHNP (WP \u0627\u0644\u0630\u064a)) (S (VP (VBD \u0630\u0647\u0628) (PP (IN \u0627\u0644\u0649) (NP (DTNN \u0627\u0644\u0645\u062f\u0631\u0633\u0629))))))))) \u2192 (ROOT (S (SBAR (WHNP (WP \u0627\u0644\u0630\u064a)) (S (VP (VBD \u0630\u0647\u0628) (PP (IN \u0627\u0644\u0649) (NP (DTNN \u0627\u0644\u0645\u062f\u0631\u0633\u0629)))))) (VP (VBD \u0646\u062c\u062d))))RULE 12: Special adverb+) \u2192 )+Special adverbExample: AlEfw End Almqdrp (\u0627\u0644\u0639\u0641\u0648 \u0639\u0646\u062f \u0627\u0644\u0645\u0642\u062f\u0631\u0629)\u2192End Almqdrp AlEfw (\u0639\u0646\u062f \u0627\u0644\u0645\u0642\u062f\u0631\u0629 \u0627\u0644\u0639\u0641\u0648)Parse: (ROOT (NP (NP (DTNN \u0627\u0644\u0639\u0641\u0648)) (NP (NN \u0639\u0646\u062f) (NP (DTNN \u0627\u0644\u0645\u0642\u062f\u0631\u0629))))) \u2192 (ROOT (NP (NN \u0639\u0646\u062f) (NP (NP (DTNN \u0627\u0644\u0645\u0642\u062f\u0631\u0629)) (NP (DTNN \u0627\u0644\u0639\u0641\u0648)))))Example: frH Alwld bxbr AlrHlp qbl >n y*hb (\u0641\u0631\u062d \u0627\u0644\u0648\u0644\u062f \u0628\u062e\u0628\u0631 \u0627\u0644\u0631\u062d\u0644\u0629 \u0642\u0628\u0644 \u0623\u0646 \u064a\u0630\u0647\u0628) \u2192 qbl >n y*hb frH Alwld bxbr AlrHlp (\u0642\u0628\u0644 \u0623\u0646 \u064a\u0630\u0647\u0628 \u0641\u0631\u062d \u0627\u0644\u0648\u0644\u062f \u0628\u062e\u0628\u0631 \u0627\u0644\u0631\u062d\u0644\u0629)Parse: (ROOT (S (NP (NP (NN \u0641\u0631\u062d) (NP (DTNN \u0627\u0644\u0648\u0644\u062f))) (NP (NP (NN \u0628\u062e\u0628\u0631) (NP (DTNN \u0627\u0644\u0631\u062d\u0644\u0629))) (NP (NN \u0642\u0628\u0644) (NP (DTNN \u0623\u0646))))) (VP (VBP \u064a\u0630\u0647\u0628)))) \u2192 (ROOT (S (NP (NN \u0642\u0628\u0644) (NP (DTNN \u0623\u0646))) (VP (VBP \u064a\u0630\u0647\u0628) (NP (NN \u0641\u0631\u062d) (NP (DTNN \u0627\u0644\u0648\u0644\u062f))) (NP (NN \u0628\u062e\u0628\u0631) (NP (DTNN \u0627\u0644\u0631\u062d\u0644\u0629))))))RULE 13: Pronoun+(NN |VB | ADJ) \u2192 (NN |VB | ADJ)+PronounExample: hy tjyd AlxyATp (\u0647\u064a \u062a\u062c\u064a\u062f \u0627\u0644\u062e\u064a\u0627\u0637\u0629) \u2192 tjyd hy AlxyATp (\u062a\u062c\u064a\u062f \u0647\u064a \u0627\u0644\u062e\u064a\u0627\u0637\u0629)Parse: (ROOT (S (NP (PRP \u0647\u064a)) (VP (VBP \u062a\u062c\u064a\u062f) (NP (DTNN \u0627\u0644\u062e\u064a\u0627\u0637\u0629))))) \u2192 (ROOT (S (VP (VBP \u062a\u062c\u064a\u062f) (NP (PRP \u0647\u064a)) (NP (DTNN \u0627\u0644\u062e\u064a\u0627\u0637\u0629)))))Example: Ant rjl krym (\u0623\u0646\u062a \u0631\u062c\u0644 \u0643\u0631\u064a\u0645) \u2192 rjl krym Ant (\u0631\u062c\u0644 \u0627\u0646\u062a \u0643\u0631\u064a\u0645)Parse: (ROOT (S (NP (PRP \u0627\u0646\u062a)) (NP (NP (NN \u0631\u062c\u0644)) (NP (NNP \u0643\u0631\u064a\u0645))))) \u2192 (ROOT (NP (NP (NN \u0631\u062c\u0644) (NP (PRP \u0627\u0646\u062a))) (NP (NNP \u0643\u0631\u064a\u0645))))Example: hy Al>jml (\u0647\u064a \u0627\u0644\u0623\u062c\u0645\u0644) \u2192 hy Al>jml (\u0627\u0644\u0623\u062c\u0645\u0644 \u0647\u064a)Parse: (ROOT (S (NP (PRP \u0647\u064a)) (NP (NNP \u0627\u0644\u0627\u062c\u0645\u0644)))) \u2192 (ROOT (S (VP (VBP \u0627\u0644\u0627\u062c\u0645\u0644) (NP (PRP \u0647\u064a)))))RULE 14: (NN|DTNN)+VB \u2192 VB+(NN|DTNN)Example: Alwld y>kl qlylA (\u0627\u0644\u0648\u0644\u062f \u064a\u0623\u0643\u0644 \u0642\u0644\u064a\u0644\u0627\u064b) \u2192 y>kl Alwld qlylA (\u064a\u0623\u0643\u0644 \u0627\u0644\u0648\u0644\u062f \u0642\u0644\u064a\u0644\u0627\u064b)Parse: (ROOT (S (NP (DTNN \u0627\u0644\u0648\u0644\u062f)) (VP (VBP \u064a\u0623\u0643\u0644) (NP (NN \u0642\u0644\u064a\u0644\u0627))))) \u2192 (ROOT (S (VP (VBP \u064a\u0623\u0643\u0644) (NP (DTNN \u0627\u0644\u0648\u0644\u062f)) (NP (NN \u0642\u0644\u064a\u0644\u0627)))))Example: bAsm yqdm $y}A mn AlfkAhAt (\u0628\u0627\u0633\u0645 \u064a\u0642\u062f\u0645 \u0634\u064a\u0626\u0627 \u0645\u0646 \u0627\u0644\u0641\u0643\u0627\u0647\u0627\u062a) \u2192 yqdm bAsm $y}A mn AlfkAhAt (\u064a\u0642\u062f\u0645 \u0628\u0627\u0633\u0645 \u0634\u064a\u0626\u0627 \u0645\u0646 \u0627\u0644\u0641\u0643\u0627\u0647\u0627\u062a)Parse: (ROOT (S (NP (NNP \u0628\u0627\u0633\u0645)) (VP (VBP \u064a\u0642\u062f\u0645) (NP (NP (NN \u0634\u064a\u0626\u0627)) (PP (IN \u0645\u0646) (NP (DTNNS \u0627\u0644\u0641\u0643\u0627\u0647\u0627\u062a))))))) \u2192 (ROOT (S (VP (VBP \u064a\u0642\u062f\u0645) (NP (NNP \u0628\u0627\u0633\u0645)) (NP (NP (NN \u0634\u064a\u0626\u0627)) (PP (IN \u0645\u0646) (NP (DTNNS \u0627\u0644\u0641\u0643\u0627\u0647\u0627\u062a)))))))RULE 15: NN+ \u2192 +NNExample: Alwld <n ydrs ynjH (\u0627\u0644\u0648\u0644\u062f \u0625\u0646 \u064a\u062f\u0631\u0633 \u064a\u0646\u062c\u062d) \u2192 <n ydrs Alwld ynjH (\u0625\u0646 \u064a\u062f\u0631\u0633 \u0627\u0644\u0648\u0644\u062f \u064a\u0646\u062c\u062d)Parse: (ROOT (S (NP (DTNN \u0627\u0644\u0648\u0644\u062f) (DTJJ \u0625\u0646)) (VP (VBP \u064a\u062f\u0631\u0633) (S (VP (VBP \u064a\u0646\u062c\u062d)))))) \u2192 (ROOT (S (VP (VBD \u0625\u0646) (S (VP (VBP \u064a\u062f\u0631\u0633) (NP (DTNN \u0627\u0644\u0648\u0644\u062f)) (S (VP (VBP \u064a\u0646\u062c\u062d))))))))Example: bAsm lw ynAm mbkrA lA ytEb (\u0628\u0627\u0633\u0645 \u0644\u0648 \u064a\u0646\u0627\u0645 \u0645\u0628\u0643\u0631\u0627 \u0644\u0627 \u064a\u062a\u0639\u0628) \u2192 lw ynAm bAsm mbkrA lA ytEb 0(\u0644\u0648\u064a\u0646\u0627\u0645 \u0628\u0627\u0633\u0645 \u0645\u0628\u0643\u0631\u0627 \u0644\u0627 \u064a\u062a\u0639\u0628 )Parse: (ROOT (S (NP (NNP \u0628\u0627\u0633\u0645)) (SBAR (IN \u0644\u0648) (S (VP (VBP \u064a\u0646\u0627\u0645) (ADJP (JJ \u0645\u0628\u0643\u0631\u0627))))) (PRT (RP \u0644\u0627)) (VP (VBP \u064a\u062a\u0639\u0628)))) \u2192 (ROOT (S (SBAR (IN \u0644\u0648) (S (VP (VBP \u064a\u0646\u0627\u0645) (NP (NNP \u0628\u0627\u0633\u0645)) (ADJP (JJ \u0645\u0628\u0643\u0631\u0627))))) (VP (PRT (RP \u0644\u0627)) (VBP \u064a\u062a\u0639\u0628))))RULE 16: VB+(NN|DTNN) \u2192 (NN+DTNN)+VBExample: wqE Alwld ElY Al>rD (\u0648\u0642\u0639 \u0627\u0644\u0648\u0644\u062f \u0639\u0644\u0649 \u0627\u0644\u0623\u0631\u0636) \u2192 Alwld wqE ElY Al>rD (\u0627\u0644\u0648\u0644\u062f \u0648\u0642\u0639 \u0639\u0644\u0649 \u0627\u0644\u0623\u0631\u0636)Parse: (ROOT (S (VP (VBD \u0648\u0642\u0639) (NP (DTNN \u0627\u0644\u0648\u0644\u062f)) (PP (IN \u0639\u0644\u0649) (NP (DTNN \u0627\u0644\u0627\u0631\u0636)))))) \u2192 (ROOT (S (NP (DTNN \u0627\u0644\u0648\u0644\u062f)) (VP (VBD \u0648\u0642\u0639) (PP (IN \u0639\u0644\u0649) (NP (DTNN \u0627\u0644\u0623\u0631\u0636))))))RULE 17: VB+) \u2192 Special-character+(NN|DTNN))+VBExample: Elmt >n AlwfA' Sfp EZymp (\u0639\u0644\u0645\u062a \u0623\u0646 \u0627\u0644\u0648\u0641\u0627\u0621 \u0635\u0641\u0629 \u0639\u0638\u064a\u0645\u0629) \u2192 >n AlwfA' Elmt Sfp EZymp (\u0623\u0646 \u0627\u0644\u0648\u0641\u0627\u0621 \u0639\u0644\u0645\u062a \u0635\u0641\u0629 \u0639\u0638\u064a\u0645\u0629)Parse: (ROOT (S (VP (VBD \u0639\u0644\u0645\u062a) (NP (NN \u0623\u0646) (NP (DTNN \u0627\u0644\u0648\u0641\u0627\u0621))) (NP (NN \u0635\u0641\u0629) (JJ \u0639\u0638\u064a\u0645\u0629))))) \u2192 (ROOT (S (NP (NN \u0623\u0646) (NP (DTNN \u0627\u0644\u0648\u0641\u0627\u0621))) (VP (VBD \u0639\u0644\u0645\u062a) (NP (NN \u0635\u0641\u0629) (JJ \u0639\u0638\u064a\u0645\u0629)))))RULE 18: +) \u2192 )+Example: ln >Elm >n AlwfA' Sfp EZymp (\u0644\u0646 \u0623\u0639\u0644\u0645 \u0623\u0646 \u0627\u0644\u0648\u0641\u0627\u0621 \u0635\u0641\u0629 \u0639\u0638\u064a\u0645\u0629) \u2192>n AlwfA' ln >Elm Sfp EZymp (\u0623\u0646 \u0627\u0644\u0648\u0641\u0627\u0621 \u0644\u0646 \u0623\u0639\u0644\u0645 \u0635\u0641\u0629 \u0639\u0638\u064a\u0645\u0629)Parse: (ROOT (S (VP (PRT (RP \u0644\u0646)) (VBP \u0623\u0639\u0644\u0645) (NP (NN \u0623\u0646) (NP (DTNN \u0627\u0644\u0648\u0641\u0627\u0621))) (NP (NN \u0635\u0641\u0629) (JJ \u0639\u0638\u064a\u0645\u0629))))) \u2192 (ROOT (S (NP (NN \u0623\u0646) (NP (DTNN \u0627\u0644\u0648\u0641\u0627\u0621))) (VP (PRT (RP \u0644\u0646)) (VBP \u0623\u0639\u0644\u0645) (NP (NN \u0635\u0641\u0629) (JJ \u0639\u0638\u064a\u0645\u0629)))))RULE 19: +(NN|DTNN) \u2192 (NN|DTNN)+Example: ln >Elm Alwld n$yT (\u0644\u0646 \u0623\u0639\u0644\u0645 \u0627\u0644\u0648\u0644\u062f \u0646\u0634\u064a\u0637) \u2192 Alwld ln >Elm n$yT (\u0627\u0644\u0648\u0644\u062f \u0644\u0646 \u0623\u0639\u0644\u0645 \u0646\u0634\u064a\u0637)Parse: (ROOT (S (VP (PRT (RP \u0644\u0646)) (VBP \u0627\u0639\u0644\u0645) (NP (DTNN \u0627\u0644\u0648\u0644\u062f)) (ADJP (JJ \u0646\u0634\u064a\u0637))))) \u2192 (ROOT (S (NP (DTNN \u0627\u0644\u0648\u0644\u062f)) (VP (PRT (RP \u0644\u0646)) (VBP \u0627\u0639\u0644\u0645) (ADJP (JJ \u0646\u0634\u064a\u0637)))))RULE 20: )+VB \u2192 VB+)Example: >n AlwfA' yEml mEk (\u0627\u0646 \u0627\u0644\u0648\u0641\u0627\u0621 \u064a\u0639\u0645\u0644 \u0645\u0639\u0643) \u2192 yEml >n AlwfA' mEk (\u064a\u0639\u0645\u0644 \u0623\u0646 \u0627\u0644\u0648\u0641\u0627\u0621 \u0645\u0639\u0643)Parse: (ROOT (S (NP (NN \u0623\u0646) (NP (DTNN \u0627\u0644\u0648\u0641\u0627\u0621))) (VP (VBP \u064a\u0639\u0645\u0644) (NP (NN \u0645\u0639\u0643)))) \u2192 (ROOT (S (VP (VBP \u064a\u0639\u0645\u0644) (NP (NN \u0623\u0646) (NP (DTNN \u0627\u0644\u0648\u0641\u0627\u0621) (DTJJ \u0645\u0639\u0643))))))RULE 21: )+ \u2192 +)Example: >n AlwfA' lA yHlw mEk (\u0627\u0646 \u0627\u0644\u0648\u0641\u0627\u0621 \u0644\u0627 \u064a\u062d\u0644\u0648 \u0645\u0639\u0643) \u2192 lA yHlw >n AlwfA' mEk (\u0644\u0627 \u064a\u062d\u0644\u0648 \u0623\u0646 \u0627\u0644\u0648\u0641\u0627\u0621 \u0645\u0639\u0643)Parse: (ROOT (S (NP (NN \u0623\u0646) (NP (DTNN \u0627\u0644\u0648\u0641\u0627\u0621))) (VP (PRT (RP \u0644\u0627)) (VBP \u064a\u062d\u0644\u0648) (NP (NN \u0645\u0639\u0643))))) \u2192 (ROOT (S (VP (PRT (RP \u0644\u0627)) (VBP \u064a\u062d\u0644\u0648) (NP (NN \u0623\u0646) (NP (DTNN \u0627\u0644\u0648\u0641\u0627\u0621) (DTJJ \u0645\u0639\u0643))))))RULE 22: CD+(NN|DTNN|VB) \u2192 CD+(NN|DTNN|VB)Example: njH TAlb fy Altwjyhy (\u0646\u062c\u062d 15 \u0637\u0627\u0644\u0628 \u0641\u064a \u0627\u0644\u062a\u0648\u062c\u064a\u0647\u064a) \u2192 fy Altwjyhy njH TAlb (\u0641\u064a \u0627\u0644\u062a\u0648\u062c\u064a\u0647\u064a \u0646\u062c\u062d 15 \u0637\u0627\u0644\u0628)Parse: (ROOT (S (VP (VBD \u0646\u062c\u062d) (NP (CD 15) (NP (NN \u0637\u0627\u0644\u0628))) (PP (IN \u0641\u064a) (NP (ADJP (DTJJ \u0627\u0644\u062a\u0648\u062c\u064a\u0647\u064a))))))) \u2192 (ROOT (S (PP (IN \u0641\u064a) (NP (ADJP (DTJJ \u0627\u0644\u062a\u0648\u062c\u064a\u0647\u064a)))) (VP (VBD \u0646\u062c\u062d) (NP (CD 15) (NP (NN \u0637\u0627\u0644\u0628))))))RULE 23: WH-Adverb+) | ) \u2192 ) | )+WH-AdverbExample: kyf kl AlnAs y>klwn (\u0643\u064a\u0641 \u0643\u0644 \u0627\u0644\u0646\u0627\u0633 \u064a\u0623\u0643\u0644\u0648\u0646) \u2192 kl AlnAs kyf y>klwn (\u0643\u0644 \u0627\u0644\u0646\u0627\u0633 \u0643\u064a\u0641 \u064a\u0623\u0643\u0644\u0648\u0646)Parse: (ROOT (SBARQ (WHADVP (WRB \u0643\u064a\u0641)) (S (NP (NOUN_QUANT \u0643\u0644) (NP (DTNN \u0627\u0644\u0646\u0627\u0633))) (S (VP (VBP \u064a\u0623\u0643\u0644\u0648\u0646)))))) \u2192 (ROOT (S (NP (NOUN_QUANT \u0643\u0644)) (VP (NP (DTNN \u0627\u0644\u0646\u0627\u0633)) (SBAR (WHADVP (WRB \u0643\u064a\u0641)) (S (VP (VBP \u064a\u0623\u0643\u0644\u0648\u0646))))))Extensive experiments, showed that the Arabic Stanford parsing is not very accurate especially for the adverbs and negation words. This will adversely affect the system by generating wrong synonyms for the sentences. Therefore, there existed the need for declaring our own list of some adverbs, negation and special words because Stanford Parser does not assign the proper labels, as expected. These lists are presented in The general framework is illustrated in The Arabic WordNet browser is free and publicly available. It uses a locally-stored database of Arabic data in XML format\u2014where words of the same meaning are linked through pre-defined lexical relations. Furthermore, the interface is modeled on the European language WordNet interface; hence, it contains the basic components with additional Arabic components. However, the performance of the Arabic WordNet is not satisfactory when compared with other WordNets. For example, the Arabic WordNet contains only 9.7% of the Arabic lexicon, while the English WordNet covers 67.5% of the English lexicon. Also, the Arabic WordNet synsets are linked only through hyponymy, synonymy and equivalence; correspondingly seven semantic relations are used in the English WordNet. However, since the main goal is generating the synonyms of the words, the limitation of the Arabic WordNet did not substantially affect the work. Also, to avoid the noise caused by diacritics, only the first top five synsets in each synonyms list were considered. Employing the synonyms and the transformation rules, enables us to generate a huge number of sentences that are equivalent, in meaning and label, to the original input sentence. Every extracted synonym, using Arabic WordNet, creates a new sentence from the input sentence. Subsequently, these sentences are processed by the transformation module which selectively applies the proper transformation rules and generates even more sentences with the same meaning and label to the original sentence. Meaning, here, is defined in the loose sense of being suitable for sentiment analysis and is not from a linguistics perspective. From a linguistic perspective, synonymous sentences represent close meanings but not exactly the same. As an example, one can generate 47 sentences from the simple verbal sentence (\u201c\u0623\u0643\u0644 \u0627\u0644\u0648\u0644\u062f \u0627\u0644\u062a\u0641\u0627\u062d\u0629\u201d) (The boy ate the apple) using only the synonyms and transformation rules (i.e. without using the negation module which would generate even more sentences). This module is responsible for generating parse trees for the original input sentence; and the generated sentences using Stanford Arabic parser tagset. With parse trees, it becomes easier to apply the suitable transformation rules to a given sentence, and it also facilitates the infusion of negation particles into sentences as described in the next section. Negating a sentence in Arabic means inserting one of the negation particles used in Arabic into an affirmative sentence. Every negation particle, in Arabic, has its own rules in terms of the type of verbs or nouns it affects and in terms of the position in the sentence in which it is inserted. Negating a sentence will result in a new sentence that has an opposite meaning to the original input sentence. The label of the input sentence is also flipped from positive to negative. In addressing the negation problem, we adopted the Negation-aware Framework presented by After defining the negation rules, the system is able to negate a set of sentences and generate all possible variations of these sentences as a result of inserting negation particles regardless if the sentences are nominal or verbal sentences. The following subsections describe, thoroughly, three experiments that were designed to test the accuracy of the proposed augmentation framework. Firstly, an assessment for the impact of the proposed framework on sentiment analysis was made. Secondly, we tested the correctness of each transformation rule. Finally, the accuracy of the Negation module was tested and formulated.souq.com. The data was annotated with three labels . In this experiment, and before performing any changes on the original data, the data was tested using several supervised classifiers . The data was divided into 70% for training and 30% for testing. All the classifiers used word embedding that is generated using AraVec with a dimension equals to 300 (The aim of this experiment is to classify product reviews into positive, negative or neutral reviews. The focus of this experiment is not the classifier, but to assess the resulting accuracy of using the proposed framework when enlarging the size of the dataset. To perform the first experiment, we used a subset of a public dataset of product reviews  which cos to 300 . After tThe aim of this experiment was to test the accuracy of each transformation rule independently. To achieve this goal, it was preferable to design a small artificial dataset, which consists of 40 statements with positive sentiment, 32 statements with negative sentiment and 27 neutral statements. A total of 99 sentences were carefully designed to align with the 23 transformation rules. Each sentence was processed by the augmentation tool, and thus several sentences were generated for each input sentence. The generated sentences were manually inspected to test their validity. Rule accuracy is a measure that evaluates the ability of a given rule to generate correct and meaningful sentences. Rule accuracy is calculated by dividing the number of correct sentences generated by a given rule by the number of all sentences generated by that rule. \u201cA correct sentence\u201d means a grammatically correct and meaningful sentence. The goal of the third experiment is to assess the capability of the Negation module in order to generate correct sentences. A small artificial dataset which consists of 26 positive sentences and 24 negative sentences was created for this purpose. It should be mentioned here that the Negation module is responsible for inserting proper negation particles into the input sentences. Negation flips the polarity of the input sentence. This means that positive sentences will become negative and vice versa. All the resulted sentences from the Negation module are correct with their respective labels properly flipped.In this study, a novel data augmentation framework for Arabic textual datasets for sentiment analysis was presented. In total, 23 transformation rules were designed to generate new sentences from the input ones. These rules were designed after carefully inspecting Arabic morphology and syntax. To increase the number of generated sentences for every rule, Arabic WordNet was used to swap the words with their respective synonyms. These rules preserve the labels of the input sentences. This means that if the input sentence has a positive label then the generated sentences also have positive labels. By the same token, if the label of the input sentence is negative, the labels of the generated sentences are also negative. The same is true for the neutral label. A Negation module was also designed to insert negation particles into Arabic sentences. This module inverts or flips the labels of the generated sentences, as this is the effect of negation particles on the polarity of statements. Experimentally, we tested the proposed framework by conducting three experiments. The first experiment has demonstrated the effect of increasing the dataset size, using the augmentation tool, on classification. As expected, the accuracy improved in all the classifiers. This indicates that the quality of the generated sentences was high. The second experiment was designed to test the accuracy of each transformation rule. An artificial dataset was designed for this purpose. All rules scored extremely high accuracies. The third and last experiment used an artificial dataset to assess the quality of the generated sentences from the Negation module. The experiment reveals that all generated sentences were correct with proper associated labels.10.7717/peerj-cs.469/supp-1Supplemental Information 1Click here for additional data file.10.7717/peerj-cs.469/supp-2Supplemental Information 2Click here for additional data file.10.7717/peerj-cs.469/supp-3Supplemental Information 3Click here for additional data file.10.7717/peerj-cs.469/supp-4Supplemental Information 4Click here for additional data file."}
{"text": "Nadwaga i oty\u0142o\u015b\u0107 z ich konsekwencjami pod postaci\u0105 cukrzycy typu 2 i chor\u00f3b uk\u0142adu kr\u0105\u017cenia, stanowi\u0105 du\u017cy problem zdrowia publicznego. Wed\u0142ug ostatnich doniesie\u0144 wa\u017cn\u0105 rol\u0119 w \u201eepidemii\u201d oty\u0142o\u015bci mo\u017ce odgrywa\u0107 mikrobiota jelitowa. Z uwagi na to, \u017ce mikrobiota jelit mo\u017ce wp\u0142ywa\u0107 na mas\u0119 cia\u0142a, wra\u017cliwo\u015b\u0107 na insulin\u0119 czy metabolizm glukozy i lipid\u00f3w pozwala wysun\u0105\u0107 hipotez\u0119, \u017ce zmiany w obr\u0119bie mikrobioty mog\u0105 mie\u0107 znaczenie w patogenezie oty\u0142o\u015bci i zespo\u0142u metabolicznego. Nadwaga i oty\u0142o\u015b\u0107 oraz ich konsekwencje pod postaci\u0105 m.in. chor\u00f3b uk\u0142adu kr\u0105\u017cenia, zaburze\u0144 lipidowych czy cukrzycy typu II, stanowi\u0105 olbrzymi problem zdrowia publicznego od kilku lat. Rozw\u00f3j oty\u0142o\u015bci ma zwi\u0105zek z wieloma czynnikami, zar\u00f3wno genetycznymi, jak i \u015brodowiskowymi. Od kilku lat podnosi si\u0119 rol\u0119 jelita i organizm\u00f3w go zasiedlaj\u0105cych w patomechanizmie oty\u0142o\u015bci.Przew\u00f3d pokarmowy cz\u0142owieka jest skolonizowany przez kompleks 10 mld drobnoustroj\u00f3w nazywany kiedy\u015b mikroflor\u0105 jelitow\u0105. Poj\u0119cie mikrobiomu zosta\u0142o wprowadzone w 2001 r. przez Joshua Lederberg\u2019a i okre\u015bla\u0142o ca\u0142o\u015b\u0107 ekologicznego \u015brodowiska z\u0142o\u017conego z drobnoustroj\u00f3w komensalicznych, symbiotycznych i chorobotw\u00f3rczych. Obecnie termin ten okre\u015bla zesp\u00f3\u0142 wszystkich gen\u00f3w mikroorganizm\u00f3w \u017cyj\u0105cych w i na ciele cz\u0142owieka, natomiast zesp\u00f3\u0142 tych mikrob\u00f3w okre\u015bla si\u0119 poj\u0119ciem mikrobioty jelitowej .Do badania mikrobioty cz\u0142owieka wykorzystuje si\u0119 analiz\u0119 16S rRNA oraz bada si\u0119 z\u0142o\u017cono\u015b\u0107 pr\u00f3bek na drodze sekwencjonowania materia\u0142u genetycznego uzyskanego bezpo\u015brednio ze \u015brodowiska. To podej\u015bcie jest nazywane \u201emetagenomik\u0105\u201d. Termin metagenomika zosta\u0142 zaproponowany przez prof. J.Handelsma\u2019a w roku 1998 .Ogromn\u0105 rol\u0119 w badaniach, opr\u00f3cz metagenomiki, odgrywaj\u0105 r\u00f3wnie\u017c badania mRNA (meta-transkryptomika), bia\u0142ek (metaproteomika) i sieci metabolicznych (metainteraktomika), z uwagi na to, \u017ce sama metagenomika nie zapewnia bezpo\u015brednich informacji o tym, kt\u00f3re geny s\u0105 w danych warunkach funkcjonalne .Metagenomowe analizy ludzkiego mikrobiomu wykaza\u0142y, \u017ce w jelicie znajduje si\u0119 3,3 miliona unikatowych gen\u00f3w, czyli 150 razy wi\u0119cej ni\u017c w naszym w\u0142asnym genomie, a r\u00f3\u017cnorodno\u015b\u0107 bakterii jelitowych jest szacowana na ponad 1000 gatunk\u00f3w .Badania, dzi\u0119ki kt\u00f3rym, mo\u017cliwe jest poznanie wp\u0142ywu mikrobioty na organizm ludzki s\u0105 mo\u017cliwe dzi\u0119ki badaniom na myszach germ-free  oraz analizie genu 16SrRNA u wielu mikroorganizm\u00f3w. Gen ten, wielko\u015bci 1,5 kb, zawiera w swojej strukturze silnie konserwowane sekwencje, utrwalane w toku ewolucji. S\u0142u\u017cy on do klasyfikacji mikroorganizm\u00f3w i pozwala udokumentowa\u0107 histori\u0119 ich ewolucji oraz tworzenie drzew filogenetycznych , 8. GrupKompleksowe badania pozwoli\u0142y sklasyfikowa\u0107 4 typy mikroorganizm\u00f3w jelitowych, kt\u00f3re stanowi\u0105 94-98% wszystkich izolowanych drobnoustroj\u00f3w, s\u0105 to: Firmicutes (64%), Bacteroides(23%), Proteobacteria(8%), Actinobacteria (3%). Dominuj\u0105 Bacteroides i Firmicutes. Bakterie te s\u0105 obecne w ca\u0142ym przewodzie pokarmowym, ale najwi\u0119ksza ich liczba jest w jelicie grubym \u2212 mi\u0119dzy 10 do 10 a 10 do 12 kom\u00f3rek/gram tre\u015bci pokarmowej [Do niedawna uwa\u017cano, \u017ce noworodek, a szczeg\u00f3lnie jego jelito jest ja\u0142owe. W 2014 r. pojawi\u0142y si\u0119 doniesienia o obecno\u015bci bakterii w \u0142o\u017cysku i p\u0142ynie owodniowym w okresie prenatalnym. Uwa\u017ca si\u0119, \u017ce pochodz\u0105 one z przewodu pokarmowego matki , 12, 13.Mleko kobiece mo\u017cna zaliczy\u0107 do naturalnych synbiotykow, bowiem zawiera naturalne oligosacharydy \u2212 HMO (Human milk oligosacharydes) i bakterie probiotyczne, wp\u0142ywaj\u0105ce na sk\u0142ad mikroflory jelitowej dziecka. U noworodk\u00f3w urodzonych silami natury w 3 dobie \u017cycia zmniejsza si\u0119 liczebno\u015b\u0107 bakterii z rodzaju Clostridium na korzy\u015b\u0107 Bifidobacterium , 16.Noworodek urodzony przez ci\u0119cie cesarskie jest pozbawiony kontaktu z mikrobiot\u0105 jelitow\u0105 matki, co ma wp\u0142yw na sk\u0142ad jego mikrobioty jelitowej. Po porodzie w jelicie noworodk\u00f3w cz\u0119\u015bciej hoduje si\u0119 Clostridium dificile oraz bakterie ze \u015brodowiska szpitalnego . ZmienioWa\u017cnym czynnikiem wp\u0142ywaj\u0105cym na sk\u0142ad mikrobioty niemowl\u0119cia jest antybiotykoterapia matki w czasie ci\u0105\u017cy oraz w pierwszych tygodniach \u017cycia dziecka. Przyjmowanie antybiotyk\u00f3w w czasie 2 i 3 trymestru ci\u0105\u017cy przez kobiety zwi\u0119ksza ryzyko rozwoju oty\u0142o\u015bci o 84% u potomstwa w por\u00f3wnaniu z grup\u0105 dzieci, kt\u00f3rych matki nie przyjmowa\u0142y antybiotyk\u00f3w. Wed\u0142ug bada\u0144, antybiotykoterapia mo\u017ce mie\u0107 wp\u0142yw na ryzyko wzrostu masy cia\u0142a u dzieci w wieku 2 lat, zw\u0142aszcza dotyczy to antybiotyk\u00f3w makrolidowych podawanych w pierwszych 6 miesi\u0105cach \u017cycia i cz\u0119\u015bciej ni\u017c 1 raz .Badania kohortowe KOALA z 2002r. (n=2834) wykaza\u0142y, \u017ce sk\u0142ad mikrobioty jelitowej we wczesnym dzieci\u0144stwie wp\u0142ywa na mas\u0119 cia\u0142a dzieci w p\u00f3\u017aniejszych latach, a dok\u0142adniej \u2212 kolonizacja Bacteroides fragilis u dzieci w 1 miesi\u0105cu \u017cycia wi\u0105\u017ce si\u0119 z wy\u017cszym BMI dzieci w wieku 10 lat. Dotyczy\u0142o to dzieci, kt\u00f3rych matki mia\u0142y ma\u0142\u0105 ilo\u015b\u0107 b\u0142onnika w diecie w okresie ci\u0105\u017cy . Oko\u0142o 2Jednym z najistotniejszych czynnik\u00f3w wp\u0142ywaj\u0105cych na sk\u0142ad mikroflory jelitowej jest dieta. Wiele przeprowadzonych bada\u0144 wykazuje, \u017ce zmiana diety z nisko- na wysokot\u0142uszczow\u0105 powodowa\u0142a znacz\u0105ce r\u00f3\u017cnice ilo\u015bciowe w sk\u0142adzie mikrobioty jelitowej. Obserwowano spadek liczebno\u015bci bakterii nale\u017c\u0105cych do typu Bacteroidetes przy jednoczesnym, znacz\u0105cym wzro\u015bcie liczebno\u015bci Firmicutes i Proteobacteria. Zmiany te by\u0142y niezale\u017cne od wyst\u0119powania, b\u0105d\u017a braku objaw\u00f3w oty\u0142o\u015bci u badanych os\u00f3b . PodobneW ostatnich latach mikrobiota jelitowa zosta\u0142a zidentyfikowana jako determinanta rozwoju oty\u0142o\u015bci, zar\u00f3wno w badaniach na zwierz\u0119tach, jak i na ludziach. Badania Ley\u2019a i wsp., opublikowane w 2005 r. pokaza\u0142y po raz pierwszy r\u00f3\u017cnice w mikrobiocie jelit myszy szczup\u0142ych i oty\u0142ych . MikrobiProwadzono r\u00f3wnie\u017c badania przeniesienia ludzkiej mikrobioty jelitowej do jelit myszy germ-free. Skolonizowanie ich ludzk\u0105 mikrobiot\u0105 powodowa\u0142o, \u017ce przekazywa\u0142y cech\u0119 potomstwu i dodatkowo zwi\u0119ksza\u0142y r\u00f3\u017cnorodno\u015b\u0107 mikrobioty. Por\u00f3wnania sekwencji 16S rRNA bakterii z pr\u00f3bek ka\u0142u ludzi doros\u0142ych o r\u00f3\u017cnym stopniu pokrewie\u0144stwa, pokaza\u0142y m.in. wyra\u017anie wy\u017csze podobie\u0144stwo mikroflory jelit pomi\u0119dzy monozygotycznymi bli\u017aniakami ni\u017c w przypadku niespokrewnionych os\u00f3b \u017cyj\u0105cych w takich samych warunkach \u015brodowiskowych np. ma\u0142\u017ce\u0144stw . NatomiaKolejne badania polega\u0142y na badaniu sk\u0142adu mikrobioty jelit myszy po wprowadzeniu diety wysokot\u0142uszczowej. Okaza\u0142o si\u0119, \u017ce dieta taka powoduje zmiany sk\u0142adu mikrobioty jelita ju\u017c po up\u0142ywie jednego dnia. Badania wykaza\u0142y r\u00f3wnie\u017c mo\u017cliwo\u015b\u0107 wywo\u0142ania oty\u0142o\u015bci u zwierz\u0105t na skutek przeniesienia mikrobioty ludzi oty\u0142ych do jelit zdrowych zwierz\u0105t .Turnbaugh i wsp. udowodnili, \u017ce sk\u0142ad mikrobioty jelitowej wp\u0142ywa na mas\u0119 cia\u0142a. Przeprowadzili oni transfer mikroorganizm\u00f3w z jelit homozygotycznych myszy oty\u0142ych ob/ob (myszy z genetycznie uwarunkowanym brakiem leptyny wynikaj\u0105cym z mutacji typu nonsens w 105 kodonie genu ob.) i myszy o prawid\u0142owej masie cia\u0142a do jelit myszy germ free  o prawid\u0142owej masie cia\u0142a. Po dw\u00f3ch tygodniach zaobserwowano, \u017ce myszy, kt\u00f3rym przeszczepiono mikroflor\u0119 od myszy oty\u0142ych, pozyskiwa\u0142y wi\u0119cej kalorii z po\u017cywienia i wykazywa\u0142y szybsze odk\u0142adanie tkanki t\u0142uszczowej. Myszy oty\u0142e w por\u00f3wnaniu z myszami szczup\u0142ymi mia\u0142y o 50% mniejsz\u0105 zawarto\u015b\u0107 Bacteroides i proporcjonalny wzrost Firmicutes, zaobserwowano r\u00f3wnie\u017c wzrost gen\u00f3w zwi\u0105zanych z wykorzystaniem energii z po\u017cywienia, co mo\u017ce by\u0107 przyczyn\u0105 rozwoju oty\u0142o\u015bci , 27.W innym eksperymencie wykazano, \u017ce w przeciwie\u0144stwie do myszy posiadaj\u0105cych mikroflor\u0119 jelit, zwierz\u0119ta GF nie maj\u0105 tendencji do tycia, pomimo spo\u017cywanej wysokot\u0142uszczowej i wysokocukrowej diety . KolejneMikrobiota jelitowa wp\u0142ywa na homeostaz\u0119 energetyczn\u0105 organizmu gospodarza, co jest nazywane \u201ehipotez\u0105 magazynowania\u201d (the storage hypothesis) . Jest kiWykazano, \u017ce popularne diety maj\u0105ce na celu zmniejszenie masy cia\u0142a, oparte na spo\u017cywaniu przede wszystkim bia\u0142ek i ma\u0142ej ilo\u015bci w\u0119glowodan\u00f3w, mog\u0105 powodowa\u0107 zmian\u0119 sk\u0142adu populacji bakterii w jelicie grubym cz\u0142owieka i ich mikrobiologicznej aktywno\u015bci, a co za tym idzie, ich wp\u0142ywu na zdrowie gospodarza. Wprowadzaj\u0105c okre\u015blone diety i pobieraj\u0105c pr\u00f3bki ka\u0142u od badanych os\u00f3b do analizy 16S rRNA przy pomocy ( uorescencyjnej hybrydyzacji in-situ (FISH), wykazano, \u017ce redukcja ilo\u015bci w\u0119glowodan\u00f3w w spo\u017cywanych posi\u0142kach prowadzi do spadku zawarto\u015bci w jelitach bakterii produkuj\u0105cych ma\u015blan: Roseburia spp., Eubacterium recitale oraz Bifidobacterium spp., ale nie stymuluje zmian liczby Bacteroides spp., za\u015b spo\u017cywanie wi\u0119kszej ilo\u015bci w\u0119glowodan\u00f3w skutkuje zwi\u0119kszeniem og\u00f3lnej liczby kom\u00f3rek bakterii , 40.Potwierdza to hipotez\u0119, \u017ce bakteryjna mikroflora nie tylko umo\u017cliwia wydajniejsze wykorzystywanie w\u0119glowodan\u00f3w zawartych w po\u017cywieniu, ale te\u017c ma zdolno\u015b\u0107 do modulowania przetwarzania pokarmu i magazynowania t\u0142uszczu przez gospodarza , 28, 36.Mikrobiota przewodu pokarmowego, nazywana przez badaczy \u201enowym organem w obr\u0119bie ludzkiego organizmu\u201d, ma znacz\u0105cy wp\u0142yw na funkcjonowanie organizmu od okresu prenatalnego. Nowoczesne badania umo\u017cliwiaj\u0105 lepsze zrozumienie funkcji mikrobioty jelitowej, w szczeg\u00f3lno\u015bci w \u015bwietle narastaj\u0105cej w \u015bwiecie oty\u0142o\u015bci i zwi\u0105zanych z ni\u0105 powik\u0142a\u0144."}
{"text": "In this paper, we study the problem of exponential stability for the Hopfield neural network with time-varying delays. Different from the existing results, we establish new stability criteria by employing the method of variation of constants and Gronwall's integral inequality. Finally, we give several examples to show the effectiveness and applicability of the obtained criterion. Since Hopfield  proposedu\u2217 of system (x(t)=u(t) \u2212 u\u2217 and obtainThey shift the equilibrium point f system  to the o1 and \u03942 and positive scalars \u03c4m, \u03c4M, \u03c1m, \u03c1M, \u03b2k, \u03b3j, n1, and n2, there exist symmetric definite matrices P > 0, Ws > 0, Ss > 0, Qk > 0, and Rj > 0, positive definite diagonal matrices V > 0, U > 0, \u039b1 > 0, and \u039b2 > 0, and matrices Gr and J such that the following inequalities hold, s=1,2,3:They constructed a new Lyapunov\u2013Krasovskii functional and then used Jensen's integral inequality to obtain the following criterion for system . The oriThese symbols are defined in .P, Ws, Ss, Qk, and Rj exist?Do the matrices For such complex matrix inequalities, how does one ensure the existence of the unknown matrices?If they exist, how are they represented?There are some problems with this result:If one does not solve these problems, the stability of the original equation remains unsolved. In fact, the stability depends only on the coefficient matrices of the system, not on the existence of those unknown matrices.We have also paid attention to some recent research results \u201330. TheiTo solve this problem, in this paper, we will use the technique of integral inequality to construct a new stability criterion, which is only related to the coefficient matrix and independent of those unknown matrices.x(t)=(x1(t),x2(t),\u2026,xn(t))T denotes the neuron state vector, g(x(t))=(g1(x1(t)), g2(x2(t)),\u2026,gn(xn(t)))Tis the activation function, and g(x(t \u2212 \u03c4(t))) is the time-delay term. A={diag\u2009a1, a2,\u2026, an}, B=(bij)n\u00d7n, and C=(cij)n\u00d7n are the interconnected matrices with appropriate dimensions. The initial state \u03ba(t) is a continuously differentiable vector function. U is the bias value, and \u03c4(t) denotes transmission delay and satisfies 0 \u2264 \u03c4(t) \u2264 \u03c4, \u03c4\u2032(t) \u2264 \u03c4\u2217 \u2264 1, where \u03c4 and \u03c4\u2217 are constants.Gronwall's integral inequality plays an important role in the qualitative theory of differential equations. Many researchers extended it and used it to solve numerous problems \u201336. Howen \u00d7 n matrix M=(aij)n\u00d7n and the n-dimensional vector as follows:In this paper, we define the norms of the gi satisfy the following conditions:gi: R\u27f6R is continuous and differentiable, and gi(0)=0gi is bounded on R, that is, \u2016gi(t)\u2016 \u2264 Gi for t \u2208 R, and Gi is a constantgi(y) \u2212 gi(x)\u2016\u2016\u2264Li\u2016\u2016y \u2212 x\u2016 for all y, x \u2208 R, where Li is a constant, and let L=max{L1, L2,\u2026, Ln}\u2016We assume that all activation functions A\u22121(B+C)\u2016L < 1 and the activation function g satisfies conditions (i)\u2013(iii), then the equilibrium point of system T is the equilibrium point of system A, the inverse matrix A\u22121 of A exists; therefore, =(wij)n\u00d7n and I=A\u22121U, then  is a continuous mapping of Rn\u27f6Rn; then, H(u) is also a continuous mapping of Rn\u27f6Rn. According to the definition of the norm of the n-dimensional vector and assumption (ii), we have thatG=max{G1, G2,\u2026, Gn}.From conditions (i)\u2013(iii), x|\u2016x\u2016 \u2264 \u03c1} is a bounded convex set and H(u) is a continuous mapping of \u03a9\u27f6\u03a9 . According to Brouwer's fixed-point theorem, there must exist u\u2217 \u2208 \u03a9 such that H(u\u2217)=u\u2217. As formula T is another equilibrium point of system \u2016L > 0, we have \u2211j=1n|uj\u2217 \u2212 vj\u2217|=0 and v\u2217=u\u2217. This equation shows that the equilibrium point is unique.According to the condition 1 \u2212 \u2016x\u2217 and y(t)=x(t) \u2212 x\u2217. In this situation, system (f(y)=g(y+x\u2217) \u2212 g(x\u2217), f(y(t))=(f1(y1(t)), f2(y2(t)),\u2026,fn(yn(t))T, and the initial state is \u03b7(t)=x(t) \u2212 x\u2217, t \u2208 . The meaning of the other symbols is the same as that of system (fi(z) be a continuous function that satisfies a Lipschitz condition for all z \u2208 R. That is, assume thatLi > 0 and for all z1, z2 \u2208 R.Let the equilibrium point of system  be x\u2217 an, system  can be rf system . Let actM \u2265 1 and \u03b1 > 0, such thatSystem  is said K be a nonnegative constant and v(t) and p(t) are nonnegative and continuous functions on the interval \u03b1 \u2264 t \u2264 \u03b2 and satisfy the inequalityLet In this section, we discuss the global exponential stability condition for the trivial solution of system .The linear term in system  can be eThe fundamental solution matrix of  is(20)ext=0 and the corresponding initial value be \u03b7(0)=T, then the solution of system  satisfies conditions (i)\u2013(iii) with the Lipschitz constant L; ifSuppose that the activation function f system  is globat=0, the initial value is \u03b7=T; by using the method of constant variation, we obtain that the solution of system L(1/1 \u2212 \u03c4\u2217)exp(\u03c9\u03c4)) \u2212 \u03c9 < 0, then the delay system L\u2009\u2009exp(\u03c9\u03c4)) \u2212 \u03c9 < 0, the trivial solution of system =tanh(u) satisfies the Lipschitz condition with the Lipschitz constant L=1.We consider the following two-dimensional neural network model without delay:\u03c9=2, \u2016B\u2016=1.4\u2009and\u2009L\u2016B\u2016 \u2212 \u03c9=\u22120.6 < 0. When t=0, the initial value is (x(0), y(0))=. According to We take\u03c9=1, \u2016B\u2016=30, and L\u2016B\u2016 \u2212 \u03c9=28 > 0. When t=0, the initial value is (x(0), y(0))=. The state rail diagram of the system is shown in If we take fi(u)=tanh(u), i=1,2, satisfies the Lipschitz condition and L=1.We consider the following two-dimensional neural network model with constant delay:\u03c4=0.1, \u03c9=2, \u2016B\u2016=0.8, and \u2016C\u2016=1, then ((\u2016B\u2016+\u2016C\u2016)L\u2009\u2009exp(\u03c9\u03c4)) \u2212 \u03c9 \u2248 \u22120.0107 < 0. According to t=0, the initial value is (x(0), y(0))=. The state rail diagram of the system is shown in If we takefi(u)=tanh(u), i=1,2, satisfies the Lipschitz condition andL=1.We consider the following two-dimensional neural network model with variable delay:\u03c4(t)=0.1\u2009\u2009sin(t)+0.1, then \u03c9=2, \u2016B\u2016=0.5, \u2016C\u2016=0.6, \u03c4=0.2, \u2009\u03c4\u2217=0.1, and ((\u2016B\u2016+\u2016C\u2016)L(1/1 \u2212 \u03c4\u2217)exp(\u03c9\u03c4)) \u2212 \u03c9 \u2248 \u22120.1767 < 0. According to t=0, the initial value is(x(0), y(0))=, and the state rail diagram of the system is shown in If we take\u03c4(t)=0.1\u2009\u2009sin(t)+0.1, then \u03c9=2, \u2016B\u2016=11, \u2016C\u2016=10.3, \u03c4=0.2, \u03c4\u2217=0.1, \u2009and\u2009((\u2016B\u2016+\u2016C\u2016)L(1/1 \u2212 \u03c4\u2217)exp(\u03c9\u03c4)) \u2212 \u03c9 \u2248 33.31 > 0. When t=0, the initial value is (x(0), y(0))=. The state rail diagram of the system is shown in If we takeIn , the autIn this work, we have studied the exponential stability for the Hopfield neural network with a time-varying delay. We use the method of variation of constants of ordinary differential equations to obtain an equation satisfied by the state variable of the neural network. Then, we used Gronwall's inequality to analyze this system and obtained new criteria for the exponential stability of the neural networks with time-varying delay. Our result is related only to the coefficient matrix of the system and not to the existence of the other unknown matrices. It is easy to test the exponential stability for specific systems by using these criteria."}
{"text": "Tipula (L.) eleniyasp. nov. is described as new to science, and variations in the male terminalia in two populations are noted. Two subspecies (quadridentataquadridentata and quadridentatapaupera) are elevated to species rank. Detailed photo\u2019s complement the descriptions of all five species , and data on ecology and distribution patterns are included as well as identification keys to males and females. Tipulacaucasica is recorded from the West Caucasus and Tipulaquadridentata is recorded from Dagestan (Russia) for the first time. Parallel evolution is traced in the male terminalia of the new species and in several non caucasica species group of Palaearctic Lunatipula.The  Lunatipula Edwards in the world fauna  caucasica Riedel, Tipula (Lunatipula) armata Riedel , and Tipula (Lunatipula) aurita Riedel. caucasica was first proposed by Lunatipula from the Caucasus  at altitudes of 1200\u20132000 m. The new species is characterized by a projection on the inner gonostylus of the male, the structure of this shape is not known in any of the 502 known species, including the 360 Palaearctic species.There are 502 species in the subgenus ld fauna . This isld fauna , 1983, cCaucasus , 2018. Ihttp://www.heliconsoft.com/heliconsoft-products/helicon-focus). Photographs of lateral and dorsal view and details of the structure of head, pronotum and scutum were made with a Canon 5D Mark IV digital camera equipped with a Canon MP-E 65mm f/2.8 1\u20135 \u00d7 macro lens and Canon Macro Twin-Lite MT-26EX-RT flash. Adobe Photoshop CC 2019 software was used to edit the pictures. Measurement were made with an MBS\u201310 microscope with a scale installed in the focal plane of the 8\u00d7 eyepiece .All crane flies were collected by sweep-net and then pinned. The genitalia were macerated in warm 10% KOH for ca. one hour to remove soft tissue, and then rinsed in distilled water. Cleared genitalia are preserved in glycerol in micro vials pinned with their respective specimens. Specimens were studied with an Olympus SZ61 stereo microscope. A Nikon d7000 digital camera equipped with combined Tamron 70\u2013300 / 4\u20135.6 and EL-Nikkor 50/2.8 lenses or Mitutoyo M Plan Apo 10X Microscope objective lens was used to capture partially focused images of each specimen or structure. These were stacked using the Helicon Focus (version 7.6.4) software (Tipula (Lunatipula) caucasica were unavailable for this study.For citing label data on type specimens, a slash / separates each label. Square brackets  are used to indicate additional information not on the original label. Original spelling is retained, including punctuation. In some cases, holotypes were marked by a red label without text and some paratypes were not initially marked. In such cases, specimens with locality labels corresponding to those in the publications of We generally follow the terminology of All measurements of the new species were made on pinned material.Species distributions are given according to The original descriptions and measurements by IEMT Tembotov Institute of Ecology of Mountain Territories of Russian Academy of Sciences, Nalchik, Russia; VPMC private collection of Valentin E. Pilipenko, Moscow, Russia; ZISPZoological Institute Russian Academy of Sciences, St. Petersburg, Russia. All specimens from the caucasica species group deposited in ZISP and collected before 1964 have been identified by E.N.Savchenko unless specifically noted.Abbreviations for institutional and private collections used herein: Taxon classificationAnimaliaDipteraTipulidaeEdwards, 193152FEDC5D-FC63-5689-B524-1DC67EEADBF9Diagnosis (after T.quadridentata). Caucasian endemics. Savchenko included the following species in the caucasica group: caucasica, quadridentataquadridentata, quadridentatapaupera, talyshensis.Taxon classificationAnimaliaDipteraTipulidaeRiedel, 1920219A9352-B860-5688-9DBE-B0947CC84DEATipula (Lunatipula) caucasicaZISP \u2022 1 male; \u201c\u0426\u0445\u0440\u0430-\u0426\u0445\u0430\u0440\u043e \u0431\u043b\u0438\u0437 \u0411\u0430\u043a\u0443\u0440\u0438\u0430\u043d\u0438\u201d [Tskhra-Tskharo near Bakuriani]; 29 Jun. 1958; Kurnakov leg.; ZISP \u20221 male; \u201c\u041b\u0435\u0431\u0435\u0440\u0434\u0435, \u041c\u0435\u043d\u0433\u0440\u0435\u043b\u0438\u044f, \u0413\u0440\u0443\u0437\u0438\u044f\u201d ; 17 Jul. 1959; Savenko leg.; ZISP \u2022 1 male; \u201c\u0410\u0434\u0436\u0430\u0440\u043e-\u0418\u043c\u0435\u0440\u0435\u0442\u0438\u043d\u0441\u043a. \u0445\u0440. \u0413\u043e\u0440\u044b \u0443 \u0421\u0430\u0438\u0440\u043c\u0435, \u0413\u0440\u0443\u0437\u0438\u044f\u201d ; 23 May 1958; Kurnakov; ZISP \u2022 2 males, 1 female; \u201d\u0443\u0449. \u0426\u0432\u0430\u043d\u0438\u0430\u0442-\u0425\u0435\u0432\u0438 \u0410\u0434\u0436\u0430\u0440\u043e-\u0418\u043c\u0435\u0440\u0435\u0442\u0438\u043d\u0441\u043a. \u0445\u0440. \u0413\u0440\u0443\u0437\u0438\u044f\u201d ; 23 Jun. 1958; Kurnakov leg; / \u201c\u0412\u044b\u0441\u043e\u043a\u043e\u0442\u0440\u0430\u0432\u044c\u0435 \u0443 \u0432\u0435\u0440\u0445\u043d. \u0433\u0440\u0430\u043d\u0438\u0446\u044b \u043b\u0435\u0441\u0430 \u0443 \u0440\u0443\u0447\u044c\u044f\u201d [High grass at the edge of the border of the forest by the stream]; ZISP \u2022 2 males; \u201c\u0441. \u041e\u0442\u0445\u0430\u0440\u0430, \u0413\u0443\u0434\u0430\u0443\u0442\u0441\u043a. \u0440-\u043d, \u0410\u0431\u0445\u0430\u0437\u0438\u044f, \u0431\u0435\u0440. \u0440\u0435\u0447\u043a\u0438, \u043a\u0443\u0441\u0442\u044b.\u201d ; 30 Jun. 1958; Kurnakov; ZISP \u20222 males; \u201c\u0410\u0431\u0445\u0430\u0437\u0438\u044f, \u0441. \u041e\u0442\u0445\u0430\u0440\u0430 \u043f\u0440\u0435\u0434\u0433\u043e\u0440\u044c\u044f \u0411\u0437\u044b\u0431\u0441\u043a\u043e\u0433\u043e \u0445\u0440.\u201d ; 20 May 1958; Kurnakov; / \u201c\u0414\u0443\u0431\u043e\u0432\u043e-\u0433\u0440\u0430\u0431\u043e\u0432\u044b\u0439 \u043b\u0435\u0441, \u043a\u0443\u0441\u0442\u044b Asalia\u201d ; ZISP; Azerbaijan \u2022 1 male; \u201c\u0411\u0435\u043b\u043e\u043a\u0430\u043d\u044b, \u0445\u0440. \u0410\u0445\u043a\u0435\u043c\u0430\u043b. \u0410\u0437\u0435\u0440\u0431. 200 \u043c\u201d  alt. 200 m]; 15 Jun. 1964; Pastukhov leg.; ZISP; Russia \u2013 Krasnodarskiy kray \u2022 1 male; Sotshi prov. Pontica; 6 May 1932; B.Rohdendorf leg.; / \u201cT.caucasica Ried det. Lacksch.\u201d [terminalia in Canada balsam pinned with specimen]; ZISP \u2022 1 female, Khosta, Caucasian Reserve, Tiso-samshitovaya rosha [Yew-and-Boxwood Tree Grove]; 43\u00b031'656\"N, 39\u00b052'467\"E; alt. 54 m; collected at light, 8 May 2018; V. Lantsov leg.; IEMT \u2022 2 males, 2 females; same collection data as for preceding, 43\u00b031'688\"N, 39\u00b052'561\"E; alt. 35 m; 8 May 2018; V. Lantsov leg. / Lime-beech  forest with yew (Taxusbaccata) and hornbeam (Carpinusbetulus) in tree layer and butcher\u2019s-broom (Ruscuscolchicus \u2013 dominant and Ruscusaculeatus) in ground layer [one of the females was collected in the same community in a damp place near a yew log]; IEMT \u2022 2 males, 2 females, same collection data as for preceding, 43\u00b031'780\"N, 39\u00b052'518\"E; alt. 98 m; 9 May 2018; V. Lantsov leg.; / Hornbeam \u2013 ash  forest with addition of yew (Taxusbaccata) and lime (Tiliabegoniifolia) with blackberry (Rubussanctus) and fig (Ficuscarica) in shrub layer and with butcher\u2019s-broom (Ruscuscolchicus) in ground layer; ZISP \u2022 4 males, 4 females, same collection data as for preceding, 43\u00b031'775\"N, 39\u00b052'423\"E; alt. 101 m; 9 May 2018; V. Lantsov leg.; / Hornbeam  forest with butcher\u2019s-broom (Ruscuscolchicus) and fern (Aspleniumscolopendrium) in ground layer; ZISP \u2022 1 male and 1 female (in copula), 1 female; same collection data as for preceding, 43\u00b031'913\"N, 39\u00b052'463\"E; alt. 153 m; 9 May 2018, V. Lantsov leg. / Ash-lime (Fraxinusexcelsior + Tiliabegoniifolia) forest with hornbeam (Carpinusbetulus) and with black berry (Rubusanatolicus) and miscellaneous herbs in ground layer; IEMT \u2022 2 males, 1 female, same collection data as for preceding, 43\u00b031'825\"N, 39\u00b052'653\"E; alt. 99 m; 10 May 2018, V. Lantsov leg.; IEMT \u2022 1 male, same collection data as for preceding, 43\u00b031'656\"N, 39\u00b052'467\"E; alt. 54 m; V. Lantsov leg.; ZISP \u2022 2 males, 1 female, same collection data as for preceding,43\u00b032'257\"N, 39\u00b052'682\"E; alt. 94 m; 10 May 2018, V. Lantsov leg.; / beech  forest with addition of yew (Taxusbaccata) and hornbeam (Carpinusbetulus) with bunch grass  in ground layer; IEMT \u2022 6 males, 3 females; Krasnodar Territory, environs Shakhe River higher than Salokh-Aul; 5\u20138 Jun. 2011; V. Pilipenko leg.; VPMC; Russia \u2013 Dagestan \u2022 1 female, \u201c\u0425\u043e\u0447\u0430\u043b-\u0414\u0430\u0433, \u0430\u043b\u044c\u043f. \u043e\u0431\u043b. \u0414\u0430\u0433\u0441\u0442. [\u0414\u0430\u0433\u0435\u0441\u0442\u0430\u043d]\u201d ]; 29 Jun. 1909 Mlokosevich leg.; ZISP \u2022 1 female; Tlyaratinsky district, near village Salda, stream on the right bank of the river Dzhurmut; 41\u00b058'388\"N, 46\u00b030'446\"E; alt. 1792 m; 2 Jul. 2016, V. Lantsov leg.; / \u201cOak (Quercusiberica) sparse forest with Populustremula and Acerplatanoides in tree layer, Rosa canina, Spiraeacrenata and Cotoneasterintegerrimus in shrub, and clover (Trifoliumcanescens) and miscellaneous herbs in ground layer; ZISP \u2022 2 males, 1 female; near village Betelda, stream on the left bank of the river Dzhurmut; 41\u00b056'853\"N, 46\u00b032'424\"E; alt. 1850 m; 3 Jul. 2016, V. Lantsov leg.; / Willow  shrub with herbaceous layer (Epilobiummontanum) (dominant), Cardamineseidlitziana, Menthacaucasica, Veronicaanagallis-aquatica and Myosotonaquaticum community near spring above the flood plain of the left bank of the river; ZISP \u2022 2 males; near village Genekolob, on the high left bank of the river Dzhurmut; 41\u00b057'392\"N, 46\u00b031'276\"E; alt. 2011 m; 4 Jul. 2016, V. Lantsov leg.; / Oak (Quercusiberica) forest with mountain-ash (Sorbusaucuparia), wayfaring tree (Viburnumlantana) in shrub layer [the lower border of a forest and a forb-cereal meadow]; ZISP \u2022 2 males; same collection data as for preceding, 41\u00b057'449\"N, 46\u00b031'228\"E; alt. 1977 m; 4 Jul. 2016, V. Lantsov leg.; ZISP \u2022 2 males, between the villages of Gerel and Genekolob, the right bank of the river Dzhurmut, 41\u00b055'619\"N, 46\u00b034'468\"E; alt. 1925 m; 5 Jul. 2016, V. Lantsov leg.; / Above the floodplain terrace with a rare herbaceous birch (Betularaddeana) forest with willow  (coppice forest); ZISP.Georgia \u2022 1 male, 3 females; \u201c\u0443\u0449. \u0411\u0430\u043d\u0438-\u0425\u0435\u0432\u0438 \u0431\u043b. \u0411\u043e\u0440\u0436\u043e\u043c\u0438, \u0410\u0434\u0436\u0430\u0440\u043e-\u0418\u043c\u0435\u0440\u0435\u0442\u0438\u043d\u0441\u043a. \u0445\u0440. ; 26 Jun. 1958; Kurnakov leg.; Dark stripes medially on lighter background extending entire length of scutum. Inner gonostylus posteriorly with wedge-shaped projection directed anteriorly. Tergite 9 at apex with three rounded notches, middle one usually deepest and widest.Adult male Fig. . GeneralHead. Light grey, sometimes bluish  of wings barely reaching base of discal cell. Female terminalia  to 2011 m (in Dagestan).From end of April to the beginning of July.Fagusorientalis, Taxusbaccata, Carpinusorientalis, Quercusiberica, Populustremula, Acerplatanoides, etc., sometimes in communities near springs with Salixcaprea.Specimens were captured in diverse forest habitats (see above) predominantly in mesophytic moderately moist deciduous or mixed communities with Endemic to the Caucasus. It was recently noted for Russia (Dagestan) for the first time . CurrentAccording to Taxon classificationAnimaliaDipteraTipulidae35B73B80-118E-5247-AD3A-67AD3A54266Chttp://zoobank.org/2847C4F8-9727-4FC3-BECD-C552B75382C8Holotype: Russia \u2022 1 male; Krasnodarskiy Kray, Sochi env. Psekhako Mt., 43\u00b041'28\"N, 40\u00b022'E; alt. ~ 2000 m; 14\u201318 Jun. 2008; K. Tomkovich leg.; ZISP. Holotype in good condition; however, left antenna, front right and left hind legs missing. Paratypes: Russia \u2022 1 male; same data as for holotype \u2022 2 males; Krasnodarskiy Kray, Kamyshanova Polyana env. [Biological station of Krasnodar State University], 44\u00b016'91\"N, 40\u00b004'46\"E; alt. 1200 m; 26 Jul. 2018; V..Pilipenko leg.; ZISP.Male. General coloration grey with silver pruinescence. Tergite 9 with deep rounded notches either side of slightly grooved cone-shaped projection. Sternite 8 with a pair of small appendages, each bearing a long medially curved spine, broad base of appendages with fringe of long whitish hairs. Outer gonostylus small, triangular, slightly thickened distally, covered with long setae. Inner gonostylus with small rod-like outgrowth in middle of outer edge.st flagellomere 0.5 mm. Length (mm) of leg segments, fore (1), mid (2), and hind (3); successively femur, tibia, 1st and 2nd tarsomeres: 1 , 2  and 3 . Length (mm) of 3rd, 4th, and 5th tarsomeres of all three pairs of legs, approximately the same: 1.0; 0.6; 0.5 mm, respectively.Adult Fig. . Male boHead ; with short but clearly visible arolium between ones.Abdomen  Fig. , variantThe new species is readily separable from all other species of the subgenus by the presence of an outgrowth medially, on the outer edge of the inner gonostylus Figs ; 14G.Adults were collected at altitudes ranging from 1200\u20132000 m.Adults are active from middle of June through the end of July.Carpinusbetulus), oriental beech (F\u0101gus orient\u0101lis), Nordman fir (Abiesnordmanniana), ash vulgaris (Fraxinusexcelsior), field maple (Acercampestre), colchis holly (Ilexcolchica), etc.Mixed moderately moist mesophytic plants, shady communities that include common hornbeam (Endemic to the Caucasus: currently known from the West Caucasus.Tipula (Lunatipula) eleniya sp. nov. is named after the mother of the first author, Elena Nikolaevna Lantsova.Taxon classificationAnimaliaDipteraTipulidaeSavchenko, 19640273951C-175D-51F0-98B0-12BAF0F9EB34Tipula (Lunatipula) quadridentatapauperaHolotype: Georgia \u20221 male; \u201c\u0417\u0430\u0433\u043e\u0440 [\u0417\u0430\u0433\u0430\u0440 \u0421\u0432\u0430\u043d\u0435\u0442\u0438\u044f], \u0413\u0440\u0443\u0437. \u0421\u0421\u0420\u201d ; alt. 2623 m; 19 Jul. 1957; R. Savenko leg.; ZISP. / \u201cTipula 4-dentata paupera ssp. n. det. Savchenko\u201d / \u201cHolotypus\u201d  / \u201cTipula (Lunatipula) paupera Sav., stat. nov. Lantsov, Pilipenko, 2020\u201d [white label]. Thorax  smeared, obscuring coloration. Preservation of legs: forelegs both with only trochanters; midlegs: left with part of femur, right only with femur; hind legs: right missing. Wings slightly crumpled. Paratypes. Georgia \u20221 male; \u201c\u0443\u0440. \u041b\u0430\u0435\u0434\u0438\u043b\u044c (\u041a\u043e\u0440\u0443\u043b\u0434\u0430\u0448 [\u041a\u043e\u0440\u0443\u043b\u0434\u0430\u0448\u0438] \u2013 \u0417\u0430\u0433\u0430\u0440)\u201d [Laedil [territory] (Koruldash [Koruldashi] \u2013 Zagar)]; 5 Aug. 1957; Savenko leg.; ZISP. / \u201cParatypus\u201d  / \u201cTipula (Lunatipula) paupera Sav., stat. nov. Lantsov, Pilipenko, 2020\u201d [white label] \u2022 2 males; \u201c\u0421\u043f\u0443\u0441\u043a \u0441 \u043f\u0435\u0440\u0435\u0432\u0430\u043b\u0430 \u0411\u0430\u0441\u0441\u0430 \u0432 \u041d\u0430\u043a\u0440\u0443 [\u0432 \u0434\u043e\u043b\u0438\u043d\u0443 \u0440. \u041d\u0430\u043a\u0440\u0430], \u0421\u0442\u0430\u0432\u0440\u043e\u043f\u043e\u043b\u044c\u0441\u043a\u0438\u0439 \u043a\u0440.\u201d [Descent from the Bassa Pass to Nakra [to the valley of the river Nakra], Stavropol kr. ; 5 Aug. 1956; L. Arens leg.; ZISP. / \u201cParatypus\u201d  / \u201cTipula (Lunatipula) paupera Sav., stat. nov. Lantsov, Pilipenko, 2020\u201d [white label].Male. Gonocoxite with two elongate, pointed teeth, one dorsally and one ventrally. Tergite 9 with two projections posteriorly, separated by deep wide notch. Paired appendages of sternite 8 widely spaced, base shorter than wide, gap between not masked by setae. Apical appendages of sternite 9 elongate, narrow distally, with dense bundle of relatively short golden yellow setae at tip.Adult male Fig. . GeneralHead .The species was described and treated as a subspecies of caucasica group by the number and arrangement of setae dorsally on the head  and one of the paratypes was collected \u201con the descent from the Bassa Pass\u201d, the height of which is 3057 m. It can be assumed that this species occurs in high-mountainous habitats.Adults were collected from the last third of July to early August.Data absent.Endemic to the Caucasus-currently known from the southern slopes of the Greater Caucasus (Georgia).Taxon classificationAnimaliaDipteraTipulidaeSavchenko, 1964E104034B-F2F2-51DB-8C14-5337C9D914B4Tipula (Lunatipula) quadridentataquadridentataHolotype. Russia \u2022 1 male; \u201d\u043e\u043a\u043e\u043b\u0438\u0446\u044b \u0421\u0442\u0430\u0432\u0440\u043e\u043f\u043e\u043b\u044f, \u0431\u0430\u0439\u0440\u0430\u0447\u043d.[\u044b\u0439] \u043b\u0435\u0441\u201d  [small forest in steppe ravines]; 25 May 1954; [S.] Medvedev leg.; ZISP. / \u201c\u044e\u0436\u043d. \u0441\u043a\u043b. \u043f\u043e\u0434 \u043f\u043e\u043b\u043e\u0433\u043e\u043c [\u043b\u0435\u0441\u0430]\u201d [southern slopes under the canopy forest] / \u201cT. (Lunatipula) quadridenta sp. n., \u043e\u043f\u0440. \u0415.\u0421\u0430\u0432\u0447\u0435\u043d\u043a\u043e\u201d [det. E. Savchenko]. The specimen is very badly damaged and glued together. The head is glued to the thorax; the abdomen is broken in half and glued together; two legs of uncertain position are glued to the specimen, one without coxa and trochanter, and the other missing the last four tarsal segments. Paratypes. Russia \u20221 male; \u201c\u0421\u0442\u0430\u0440\u044b\u0439 \u043b\u0435\u0441 \u043a \u044e\u0433\u0443 \u043e\u0442 \u0421\u0442\u0430\u0432\u0440\u043e\u043f\u043e\u043b\u044f\u201d [Old forest south of Stavropol]; 25 May 1954;[S.] Medvedev leg.; ZISP. / \u201c\u043d\u0438\u0436. \u0447\u0430\u0441\u0442\u044c \u0441\u0435\u0432. \u0441\u043a\u043b\u043e\u043d\u0430\u201d [lower part of northern slope] \u2022 1 female, \u201c\u0413\u0435\u043e\u0440\u0433\u0438\u0435\u0432\u0441\u043a\u043e\u0435 \u043b\u0435\u0441\u043d\u0438\u0447\u0435\u0441\u0442\u0432\u043e \u0422\u0443\u0430\u043f\u0441\u0438\u043d\u0441\u043a. \u0440-\u043d\u201d [Georgievskoe forestry Tuapse District]; 21 May 1954; K. Arnoldi leg.; ZISP \u20221 male, same collection data as for preceding; 22 May 1954 \u2022 3 males, 2 females; \u201c\u0433. \u041b\u044b\u0441\u0430\u044f, 800\u2013900 \u043c, \u0422\u0443\u0430\u043f\u0441\u0438\u043d\u0441\u043a. \u0440-\u043d\u201d ; 26 May 1954; K. Arnoldi leg.; ZISP; / \u201c\u0432\u0435\u0440\u0448\u0438\u043d\u043d\u0430\u044f \u043b\u0443\u0433\u043e\u0432\u0438\u043d\u0430 \u0438 \u043e\u043f\u0443\u0448\u043a\u0430 \u043b\u0435\u0441\u0430 800\u2013900 m\u201d .ZISP / \u201cT. (Lunatipula) quadridentata sp. n. \u043e\u043f\u0440. \u0415.\u041d.\u0421\u0430\u0432\u0447\u0435\u043d\u043a\u043e\u201d \u2022 1 female, \u201c\u043c. [\u043c\u044b\u0441] \u041f\u0435\u043d\u0430\u0439, \u043a \u044e\u0433\u0443 \u043e\u0442 \u041d\u043e\u0432\u043e\u0440\u043e\u0441\u0441\u0438\u0439\u0441\u043a\u0430\u201d  24. V. [1]956; Gilyarov leg.; ZISP \u2022 1 male, 1 female; \u201c\u043e\u043a\u0440. \u0441\u0442. \u0421\u043c\u043e\u043b\u0435\u043d\u0441\u043a\u043e\u0439, \u0421\u0438\u0432\u0435\u0440\u0441\u043a\u043e\u0433\u043e \u0440-\u043d\u0430, \u041a\u0440\u0430\u0441\u043d\u043e\u0434\u0430\u0440. \u043a\u0440.\u201d  20 May 1963; Savchenko leg.; / \u201c\u043e\u043f\u0443\u0448.[\u043a\u0430] \u0441\u043c\u0435\u0448.[\u0430\u043d\u043d\u043e\u0433\u043e] \u043f\u0440\u0435\u0434\u0433\u043e\u0440\u043d.[\u043e\u0433\u043e] \u043b\u0435\u0441\u0430\u201d [edge of mixed foothill forest]; ZISP \u2022 1 male; \u201c\u043e\u043a\u0440. \u0441\u0442. \u041a\u0440\u0438\u0432\u0435\u043d\u043a\u043e\u0432\u0441\u043a\u043e\u0439, \u0422\u0443\u0430\u043f\u0441.[\u0438\u043d\u0441\u043a\u043e\u0433\u043e] \u0440-\u043d\u0430 \u041a\u0440\u0430\u0441\u043d\u043e\u0434\u0430\u0440.[\u0441\u043a\u043e\u0433\u043e] \u043a\u0440\u0430\u044f\u201d ; 25 May 1963; Savchenko leg.; / \u201c\u043e\u043f\u0443\u0448\u043a\u0430 \u043b\u0438\u0441\u0442\u0432.[\u0435\u043d\u043d\u043e\u0433\u043e] \u043b\u0435\u0441\u0430 \u0443 \u0440\u0435\u043a\u0438, \u0432\u0434\u043e\u043b\u044c \u0433\u043e\u0440\u043d.[\u043e\u0433\u043e] \u043f\u043e\u0442\u043e\u043a\u0430\u201d ; ZISP \u2022 1 male; Khosta, Krasnodarskiy Kray, Caucasian Reserve, Tiso-samshitovaya rosha [Yew-and-Boxwood Tree Grove]; 43\u00b032'014\"N, 39\u00b052'621\"E; alt. 135 m; 10 May 2018; V. Lantsov leg.; / Fern-butcher community on the rocky slopes of the right side of the gorge of the Khosta River; IEMT \u2022 1 male; Khosta, Krasnodarskiy Kray, Caucasian Reserve, Tiso-samshitovaya rosha [Yew-and-Boxwood Tree Grove]; 43\u00b031'656\"N, 39\u00b052'467\u2019\u2019 E; alt. 54 m; 12 May 2018, V. Lantsov leg.; / collected on light; IEMT \u2022 2 males ; North Caucasus, Krasnodarskiy Kray, near village Mezmay; 44\u00b011'291\"N, 39\u00b058'090\"E; alt. 808 m; 15 May 2018; V. Lantsov leg.; / Beech , fruit tree  forest with Fraxinusexcelsior in under growth, Sambucusnigra, Cornusmas, Rosa canina, Crataegus sp., in shrub layer and sedge Carexpendula, cereals and herbs in ground layer ; IEMT \u2022 1 male; Krasnodar Territory, Apsheron District, environs of village Mezmay, Guam Gorge, left slope, 44\u00b012'409\"N, 39\u00b055'056\"E, alt. 547 m; 24 May 2019; V. Lantsov leg.; / Landslide foot, community with butterbur [Petasitesalbus] as dominant along the banks of the stream, open places at the edge of the forest; ZISP \u2022 8 males ; Krasnodar Territory, Seversky District, in vicinity of village Thamaha, 3 May 2016; S. Kustov leg.; ZISP \u2022 7 males, 1 female  ; Krasnodar Territory, Seversky District, environs of village Plancheskaya, 3 May 2016. S. Kustov leg.; ZISP \u2022 5 males ; Krasnodar Territory, environs of village Bolshoy Utrish; 1 May 2008; E. Hachikov. leg.; ZISP \u2022 3 males ; Krasnodar Territory, municipality Anapa, environs of village Sukko, Kvashin\u2019s Gorge, Dolgaya Niva territory; 44\u00b047'20\"N, 37\u00b028'33\"E; alt. 67 m; 6\u20138 May 2016; S. Kustov, V. Gladun. leg.; ZISP \u2022 5 males  ; Krasnodar Territory, river Shakhe Gorge; 43\u00b052'46\"N, 39\u00b050'00\"E; 2 May 2012; V. Pilipenko leg.; VPMC \u2022 3 males; ; Khosta, Krasnodarskiy Kray, Caucasian Reserve, Tiso-samshitovaya rosha [Yew-and-Boxwood Tree Grove]; 43\u00b032'014\"N, 39\u00b052'621\"E; 8 May 2012; V. Pilipenko leg.; VPMC \u2022 4 males ; Krasnodar Territory, 13 km to the N from Sochi, Sukhoy Canyon; 43\u00b032\u2019N, 39\u00b056\u2019E; 5 May 2014; V. Pilipenko leg.; VPMC; RUSSIA \u2013 Dagestan \u2022 5 males; Makhachkala, Tarki Distr.; 42\u00b056'57\"N, 47\u00b029'41\"E; alt. 220 m; 1 May 2019., V. Pilipenko leg.; cemetery on the hillside; VPMC \u2022 1 male, 1 female; same locality; 10 May 2019., V. Pilipenko leg.; VPMC \u2022 1 male, Tarki-Tau Mt.; 42\u00b056'28\"N, 47\u00b028'08\"E; alt. 450 m; 2 May 2019; V. Pilipenko leg.; oak forest; VPMC.Russia \u2022 9 males, 8 females; \u201c\u043a\u0443\u0440\u043e\u0440\u0442 \u201c\u0413\u043e\u0440\u044f\u0447\u0438\u0439 \u041a\u043b\u044e\u0447\u201d \u0445\u0440. \u041a\u043e\u0442\u0445.[\u0441\u043a\u0438\u0439] \u041a\u0440\u0430\u0441\u043d\u043e\u0434\u0430\u0440.[\u0441\u043a\u0438\u0439] \u043a\u0440. \u0434\u0443\u0431.[\u043e\u0432\u044b\u0439] \u043b\u0435\u0441\u201d [Goryachiy Klyuch resort Koth[skiy] Ridge, Krasnodar.[sky] District oak. Forest]; 18 May 1956; Gilyarov leg.; Tergite 9 with four widely spaced teeth and with three rounded notches distally; caudal margin of gonocoxite with two dentate projections. Paired appendages of sternite 8 with wide base bearing thick yellow setae distally covering gap between. Cercus long and straight; hypogynial valve only slightly longer than width at base.Adult male Fig. . GeneralHead  by tergite 9 with four widely spaced teeth and with three small notches at the apex and a small spine beneath  to 800\u2013900 m .Adults were collected from throughout the month of May.Specimens are found in moderately humid woody deciduous communities.Endemic to the Caucasus; currently known from the West Caucasus  and from the East Caucasus .Taxon classificationAnimaliaDipteraTipulidaeSavchenko, 19641D18358B-8391-55CF-B3A1-33D56252432ATipula (Lunatipula) talyshensisHolotype: Azerbaijan \u20221 male, \u201c\u0440-\u043d \u041b\u0435\u0440\u0438\u043a, \u0410\u0437\u0435\u0440\u0431\u0430\u0439\u0434\u0436. \u0421\u0421\u0420 26. VI. [1]954 \u0414\u0436\u0430\u0444\u0430\u0440\u043e\u0432\u201d  ; 26 Jun. 1954; Jafarov; / \u201cTipulatalyshensis det. Savchenko sp. n.\u201d [white label] / [Original red label without text] / \u201cHolotypus\u201d [red label]; ZISP. Holotype in good condition .Malescaucasica group only)Females  eleniya was compared with those of the 502 known species including the 360 Palaearctic species of the subgenus Lunatipula. No direct matches were found, but outgrowths of various shapes on the middle of the inner gonostylus were found in some Palaearctic species: two species from Turkey (Tipula (Lunatipula) auriculata Mannheims, 1963 and Tipula (Lunatipula) horsti Theischinger, 1982), one from China (Tipula (Lunatipula) oreada Alexander, 1933), and two from Kyrgystan (Tipula (Lunatipula) milkoi Pilipenko, 2005 and Tipula (Lunatipula) zarnigor Savchenko, 1954). The latter species has also been recorded in Tajikistan and northeastern Afghanistan (The inner gonostylus of hanistan . The spe"}
{"text": "Metaphtonymy is identified as a special rhetoric figure that specifies the interaction between metaphor and metonymy and which is pervasive in literary works. How and why do trainee translators translate metaphtonymy? Using task analysis, semi-structured discourse-based interviews, and a questionnaire survey among 30 master of translation and interpreting (MTI) trainee translators, this study investigates their translation approaches adopted when translating the metaphtonymies in Chinese extracted prose and explores the effects of their choices. It is found that they mainly employed three approaches: omission, modification, and retainment, with omission being the most, and retainment the least frequent. The main factors attributing to each approach range from the prominence degrees and cross-cultural adaptation abilities of the metaphtonymies, rhetorical awareness of translators, and transference competence to their translation knowledge sub-competence. This study suggests that trainee translators should be instructed to systematically construct rhetoric knowledge, and the teaching design should emphasize the competence of trainees of identifying rhetorical devices and their competence of shifting rhetoric between languages. Since the 1980s, metonymy and metaphor have been widely accepted as different ways of thinking rather than as mere stylistic devices , from the School of Foreign Languages, Northeast Normal University, each of whom was enrolled in the second year. All the participants were native speakers of Chinese and had learned English for at least 15 years. They all passed the Test for English majors\u2013Band 8 organized annually by the National Foreign Languages Teaching Advisory Board under the Ministry of Education of the People's Republic of China (hereinafter abbreviated as PRC), which proves that they have acquired high-level competence in the English language and culture. Each participant demonstrated translation competence, obtaining a Level-2 certificate of the China Accreditation Test for Translators and Interpreters, a state-level vocational qualification examination entrusted by the Ministry of Human Resources and Social Security of the PRC. In addition, they have completed two courses in translation theory and four practical courses in translation strategy, literary translation, English and Chinese rhetoric and Chinese grammar, and rhetoric and writing. The participants provided their written informed consent forms prior to the translation task and were rewarded with course credits. All participants were informed that their translations were only for academic use.The participants involved in the study were 30 MTI students , with ages ranging from 23 to 25 years  were submitted in word documents via E-mail.All the participants attended the translation workshop, an optional course designed and taught in the last term of the second year for students of MTI. They were from Class One opened by the first author. At the end of the term, they were given the task of translating the selected Chinese prose into English. The PDF version of the material (the ST) was sent to each participant The translations of participants were to be used to assess their achievements in the course. Each participant had access to the internet and the library for information related to the prose and the writer. The participants could refer back to translation dictionaries when needed. It was assumed that the participants would be familiar with the genre of prose, as literary works including prose had been used as STs in their translation classes.Cultural Sojourn written by Yu Qiuyu, one of the most influential contemporary Chinese writers. As important prose of his volume Cultural Sojourn published in 1992, the full text of the prose was longer than their previous translation assignments, thus, the ST was abridged from Section One and the first five paragraphs of Section Two, consisting of 1,403 words. Second, the ST should be the one that has never been translated into English. The selected prose meets this criterion, and no translations would be used as a reference by the participants. The last criterion is related to the number of metaphtonymies in the ST. Ten metaphtonymies were identified based on the below-mentioned identification procedure.Consulting the criteria for evaluating interpreting materials proposed by Zheng and Xiang , we madevia similarity, the linguistic unit is labeled as metaphor; if via contiguity, it is labeled as metonymy, (4) determine if the linguistic metaphor conveys metonymic meaning via conceptual mapping, or if the linguistic metonymy embodies metaphorical meaning via conceptual mapping. If it is one of the cases in (4), label the linguistic unit as a metaphtonymy. The third and fourth steps were also used to examine if the translations of metaphtonymies identified in the ST were metaphors, metonymies, or metaphtonymies. When deciding the basic meaning and contextual meaning of the linguistic unit, we referred to Xin Hua Ci Dian, a Chinese authoritative dictionary, and some journals analyzing the prose. With few exceptions, all the explanations of the metaphtonymic expressions rely on that source, and the exceptions will be identified as they appear. As per the procedure, 10 metaphtonymic lexical entries presented in 1-TT30) were analyzed by the authors in terms of the way that metaphtonymies in the ST were translated by each participant, thus, translation approaches were categorized.As for the identification procedures, we modified MIPVU established by Steen et al.  and FiveA questionnaire was given to all the participants before they handed in their translations, which was returned to the researchers with their translations. The questionnaire consists of 14 questions focusing on their knowledge of metaphor, metonymy, and metaphtonymy (3 questions), their identification of metaphor, metonymy, and metaphtonymy in the translation task (3 questions), their attitudes toward handling the metaphtonymies in translation (4 questions), and the challenges they met in translating those figurative linguistic units (4 questions). They could refer back to their translations when they filled in the questionnaire.1-P10) was given back their translations and the ST for reference. Metaphtonymies and their translations (if translated by the participant) have been highlighted with a red line by the authors in the ST and their translations. Subsequently, each of them was asked to explain what metaphor and metonymy are, and then the interviewer elaborated what metaphtonymy meant to them. Finally, the participants were encouraged to reflect on their translating processes, especially, the processes of translating the highlighted linguistic units, and above all the reasons why those metaphtonymies were handled that way. The interviews were organized in Chinese and finally were recorded and transcribed. In this study, each question of the interviewer and the response of the interviewee has been translated into English by the authors.During the next day, 10 participants (one-third of the group) were randomly selected to attend a retrospective discourse-based semi-structured interview. In the interview, each participant , is a metaphtonymic expression, in which \u201c\u6597\u58eb\u201d (its basic meaning is fighters or warriors), metonymically stands for the Chinese patriotic demonstration initiated on May 4, 1919, and \u201c\u4e94\u56db\u201d (its basic meaning is May Fourth) metaphorically refers to the demonstration camp mainly constituted by college students, intellectuals and citizens, etc. \u201c\u6597\u58eb\u201d and \u201c\u4e94\u56db\u201d interact conceptually with each other and combine into a phrase. The three translations keep the conceptual interaction relation by literally translating \u201c\u4e94\u56db\u201d into May Fourth and \u201c\u6597\u58eb\u201d into fighters, soldiers, or warriors. In the above ST, \u201c\u4e94\u56db\u6597\u58eb\u201d  and \u201c\u72d7\u201d (its basic meaning is dog) comprise the source domain and map onto the target domain of greedy and duplicitous people, and Han Yu metonymically used the phrase to name his colleagues who served the royal families. Example 4 kept one of the metaphors in the ST using \u201cflies-like cowards,\u201d while Example 5 creatively replaced the metaphtonymic relations in the ST adopting \u201cpiggy Shylock\u201d in which a new interaction occurs between a new pair of metaphor and metonymy. The word \u201cpiggy\u201d metaphorically prompts the source domain \u201cpig\u201d in which some bad features of pigs, such as laziness and greed, are mapped onto the target domain of avaricious people. Shylock, a classical figure in Shakespeare's Omission occurs through ellipsis, where the translation of the ST offers no metaphorical and metonymic equivalents that are employed in the TT, thus leading to the deletion of the interaction between metaphor and metonymy in the TT.ST3: \u6587\u4eba\u4eec\u7684\u8863\u886b\u6b65\u5c65\u3001 \u8c08\u5410\u884c\u6b62 \u3001 \u5c45\u5ba4\u5e03\u7f6e\u3001\u4ea4\u9645\u5f80\u6765 \uff0c\u90fd\u4e0e\u4e66\u6cd5\u6784\u6210\u548c\u8c10 \u3002Wen ren men de yi shan bu lv, tan tu xing zhi, ju shi bu zhi, jiao ji wang lai, dou yu shu fa gou cheng he xie.TT: Example 6. The literati's dressing codes and behavioral manners, living conditions and communication habits were all in harmony with calligraphy.Example 7. The literati's living styles and social interactions were all in harmony with calligraphy.In ST3, the metaphoricity of \u201c\u8c08\u5410\u884c\u6b62\u201d resides in the compound \u201c\u8c08\u5410,\u201d in which the word \u201c\u5410\u201d means throwing up something out of the mouth of an individual. The basic meaning of \u201c\u8c08\u201d is talking, and combined with \u201c\u5410,\u201d \u201c\u8c08\u5410\u201d means making utterances. The abstract content of these utterances is metaphorically contrasted with some specific things thrown out of the mouth. In \u201c\u884c\u6b62,\u201d \u201c\u6b62,\u201d an interchangeable word with \u201c\u8dbe,\u201d whose meaning is toes, first word stands for foot, then refers to walking manners and is finally combined with \u201c\u884c\u201d to metonymically mean behavior patterns, thereby forming a metonymic chain. \u201c\u8c08\u5410\u201d and \u201c\u884c\u6b62\u201d form a phrase emphasizing the interactive relationship between metaphor and metonymy. Example 6 employs \u201cbehavioral manner,\u201d the metonymic meaning of \u201c\u8c08\u5410\u884c\u6b62.\u201d While Example 7 uses \u201cliving styles,\u201d an umbrella term, to refer to \u201cdressing code,\u201d \u201cbehavior manner,\u201d and \u201croom layout,\u201d which are denoted by \u201c\u8863\u886b\u6b65\u5c65,\u201d \u201c\u8c08 \u5410\u884c\u6b62,\u201d \u201c\u5c45\u5ba4\u5e03\u7f6e,\u201d respectively. The two examples adopt the omission approach to translate the metaphtonymic phrase \u201c\u8c08\u5410\u884c\u6b62\u201d in the TT.Besides the different rates of translation approaches to each metaphtonymy, every trainee translator shows considerable variations in the three approaches employed in their TTs, which are listed in 2 and TT23, retainment was not adopted.In all the 30 TTs, the omission approach was more frequently employed to translate the metaphtonymic expressions including \u201c\u5b98\u5c60\u5bb0\u8f85,\u201d \u201c\u4fa0\u9aa8\u8d64\u80c6,\u201d \u201c\u8102\u817b\u7c89\u6e0d,\u201d \u201c\u5f97\u5fc3\u5e94\u624b,\u201d and \u201c\u8c08\u5410\u884c\u6b62.\u201d These four-character structures constitute idioms originating from some Chinese historic allusions, which meant challenges to the thirty translators because they are culture-specific, and find no equivalents in target language culture. However, analysis of the translations reveals that most translators had a good command of the contextual meanings of these metaphtonymic expressions but chose to omit their metaphtonymic provenance.The modification approach accounts for nearly one-third of the 30 TTs. It is assumed that most translators attempted to sustain the rhetoric force in the ST using the equivalent metaphor or metonymy or modified metaphtonymic expressions in the TT.2 and TT23 discussed. These two examples presented the metonymic meaning of \u201c\u94a2\u7b14\u6587\u5316\u201d and translated them explicitly as \u201cforeign advanced culture\u201d and \u201cthe great culture of western countries.\u201dRetainment was used by 28 trainee translators to render \u201c\u94a2\u7b14\u6587\u5316,\u201d except the two of TT\u94a2\u7b14\u6587\u5316 \u3002ST4: \u4e94\u56db\u6597\u58eb\u4eec\u81ea\u5df1\u4e5f\u4f7f\u7528\u6bdb\u7b14\uff0c\u4f46\u4ed6\u4eec\u662f\u7528\u6bdb\u7b14\u5728\u547c\u5524\u7740Wu si dou shi men zi ji ye shi yong mao bi, dan ta men shi yong mao bi zai hu huan zhe gang bi wen hua.foreign advanced culture.TT: Example 8. The fighters of May Fourth Movement used brushes, but they were using them to call upon the great culture of western countries.Example 9. The warriors of the Movement also used brushes, however, they called for The questionnaires and interviews centered on the translation process of metaphtonymic expressions of the participants, with the intention to probe how they identified metaphtonymies of the ST, reveal their attitudes toward translating these rhetorical figures of the ST and understands the difficulties they encountered when translating these figurative languages.In the questionnaire, questions 1\u20136 were related to their understandings of metaphor, metonymy, metaphtonymy and whether they could identify them in the ST. Nearly 93% of the participants reported that they had sufficient knowledge of metaphor and metonymy, the rest reported that they had a basic knowledge of metaphor and metonymy, while 80% of the answers of the participants revealed that they knew metaphtonymy but seldom encountered, analyzed, or relied on them in practice. Almost 70% of the participants admitted that they knew what metaphtonymy and its working mechanism were because they translated some cognitive linguistics papers in which they saw this term and referred back to common dictionaries and journals. The rest explained that they found some lexical units entailed both metaphorical meaning and metonymic meaning after referring to dictionaries and analyzing the context in which these lexical units emerged, but they did not know the term used to name these phenomena.All the participants reported that the ST was well-written in which metaphors and metonymies were widely used. When asked to list some specific examples, they could give at least five for each group. However, as to metaphtonymies, most of them could list \u201c\u5b98\u5c60\u5bb0\u8f85,\u201d \u201c\u4fa0\u9aa8\u8d64\u80c6,\u201d \u201c\u8747\u8425\u72d7\u82df,\u201d \u201c\u8102\u817b\u7c89\u6e0d,\u201d or \u2018\u4e94\u56db\u6597\u58eb,\u201d while a few participants categorized \u201c\u8747\u8425\u72d7\u82df\u201d into metaphor, but \u201c\u8102\u817b\u7c89\u6e0d\u201d and \u201c\u5b98\u5c60\u5bb0\u8f85\u201d into metonymies. Three of the participants in the interview explained that the metaphoricity of \u201c\u8747\u8425\u72d7\u82df\u201d was highly prominent, for instance:Interviewer: \u2018\u4e3a\u4ec0\u4e48\u4f60\u8ba4\u4e3a\u2018 \u8747\u8425\u72d7\u82df '\u662f\u9690\u55bb?\u201d [Why did you identify \u201c\u8747\u8425\u72d7\u82df\u201d as a metaphor?]4: \u201c\u5f88\u660e\u663e, \u8fd9\u662f\u4e2a\u9690\u55bb\u3002\u5728\u6587\u4e2d\uff0c\u4f5c\u8005\u628a\u90e8\u5206\u4f20\u7edf\u6587\u4eba\u6bd4\u55bb\u6210 \u2018\u82cd\u8747 '\u6216 \u2018\u72d7'\u3002\u67e5 \u4e86\u76f8\u5173\u8d44\u6599 \u540e \uff0c\u624d \u4e86\u89e3\u5230\u8fd9 \u77ed\u8bed \u662f\u6bd4\u55bb\u4e0d\u62e9\u624b\u6bb5\u8ffd\u6c42\u540d\u5229\u7684\u4eba. PAll the interviewees reported that they first searched for in their mental lexicon the meaning of the 10 metaphtonymies highlighted in their TTs by the teacher, and then judged the meaning based on linguistic context. If they were uncertain, they continued to look it up into dictionaries or the internet to find out its basic and extended meaning and then reanalyzed the sentence in the ST. After the meaning of the lexical unit was decided, they would search for its equivalent in the TT. Taking the response of Participant 9 as an example:Interviewer: \u201c\u80fd \u8bf7 \u4f60 \u56de\u5fc6\u4e0b\u9690\u55bb\u8f6c\u55bb\u4e92\u52a8\u8868\u8fbe\u201c\u4e94\u56db\u6597\u58eb\u201d\u7684\u7ffb\u8bd1\u8fc7\u7a0b \u5417? [Could you please retrospect your translating process of the metaphtonymic expression \u201c\u4e94\u56db\u6597\u58eb\u201d?]9: \u201c\u8bfb\u5230\u8fd9\u4e2a\u53e5\u5b50, \u5c31\u60f3\u5230 \u2018\u4e94\u56db '\u662f\u6307 \u2018\u4e94\u56db\u8fd0\u52a8 ', \u6597\u58eb\u662f\u6bd4\u55bb\u90a3\u4e9b \u8fd0\u52a8 \u9886\u5bfc \u8005\u548c \u53c2\u4e0e\u8005\u3002\u867d\u7136\u4e94\u56db\u8fd0\u52a8\u7684\u5386\u53f2 \u65e9 \u5c31\u5b66\u8fc7, \u4f46 \u8bd1\u524d \u8fd8\u662f \u4e0a\u7f51 \u67e5\u9605\u4e86\u76f8\u5173\u4ecb\u7ecd \u3002 \u5728\u82f1\u8bed\u91cc\u6597\u58eb\u5bf9\u5e94\u7684\u8bcd\u6709 soldier, warrior \u7b49, \u67e5\u9605 \u5b57\u5178 \u540e, \u6211\u9009\u62e9\u4e86 soldier, \u56e0\u4e3a \u5b83\u7684\u8bed\u4e49\u8272\u5f69\u66f4\u79ef\u6781\u3002\u4e94\u56db\u8fd0\u52a8\u4e0a\u6587\u5df2\u7ecf\u63d0\u8fc7, \u6240\u4ee5\u8bd1\u4e3a The soldiers of May Fourth \u4e0d\u4f1a\u8ba9\u82f1\u8bed\u8bfb\u8005\u56f0\u60d1\u3002\u201d PAs to the rhetorical devices of the ST, the questionnaires show that nearly 33.3% of the participants thought it was extremely important to sustain or modify the figurative languages in the TT, those who thought it was important or unimportant accounted for, respectively, 53.3% or 13.4%. Overall, most participants attached importance to the metaphtonymies in the TT in terms of their equivalents of the ST. Their responses show that they were more concerned with the equivalence of stylistic features between the ST and the TT.They thought the ST was prose, characterized by highly poetic language, for which it was the duty of translator to preserve. The response of participant 1 is instructive in this respect:Interviewer: \u201c\u4f60\u8ba4\u4e3a\u5728\u8bd1\u6587\u4e2d\u4fdd\u7559\u539f\u6587\u4e2d\u7684\u8f6c\u55bb\u3001\u9690\u55bb\u7b49\u4fee\u8f9e\u91cd\u8981\u5417\uff1f\u4e3a\u4ec0\u4e48\uff1f\u201d 1: \u201c\u6211\u8ba4\u4e3a\u4fdd\u7559\u539f\u6587\u4e2d\u7684\u4fee\u8f9e\u5f88\u91cd\u8981\u3002 \u6211\u4e5f\u662f\u8fd9\u4e48\u505a\u7684\u3002 \u4f8b\u5982\uff0c \u5728\u82f1\u8bed\u91cc\u627e \u4e0d\u5230 \u201c\u8747\u8425\u72d7\u82df \u201d\u7684\u5bf9\u5e94\u9879\uff0c\u6211\u5c31\u7528\u4e86 \u201cpiggy Shylock\u201d \u8fd9\u4e2a\u8f6c\u55bb\u6765\u4ee3\u66ff, \u8fd9\u4e5f\u7b97\u521b\u9020\u6027\u7ffb\u8bd1\u5427 \u3002 \u539f\u6587\u662f\u6563\u6587\uff0c\u8bed\u8a00\u7f8e\uff0c\u4fee\u8f9e\u591a\uff0c\u610f\u5883 \u8fd8\u5f88 \u6df1\u3002\u5982\u679c\u7ffb\u8bd1\u7684\u65f6\u5019\uff0c\u4e0d\u4fdd\u7559\u8fd9\u4e9b\u7279\u5f81\uff0c\u53ea\u7528 \u7b80\u5355 \u8bed\u8a00\uff0c\u8bd1\u6587\u5c31\u4e0d\u7f8e\u4e86\uff0c\u4e0d\u80fd\u8868\u8fbe\u4f59\u79cb\u96e8\u7684\u5199\u4f5c\u610f\u56fe\u4e86\u3002 \u201d PAbout 86.6% of the participants thought it was necessary to translate figurative languages into their equivalents in the TT. However, Interviewer: \u6e90\u8bed\u548c\u76ee\u7684\u8bed\u5b58\u5728\u5dee\u522b, \u8fd9\u5bf9\u4f60\u7ffb\u8bd1\u5305\u62ec\u9690\u55bb\u8f6c\u55bb\u4e92\u52a8\u73b0\u8c61\u5728\u5185\u7684\u4fee\u8f9e \u4ea7\u751f\u5f71\u54cd\u4e86\u5417\uff1f [Were there any influences from the differences between the source language and target language on your handling of the figurative languages including metaphtonymy?]7: \u662f\u554a, \u6709\u5f71\u54cd\u3002\u6c49\u8bed\u56db\u5b57\u683c\u7ed3\u6784, \u5bf9\u4ed7\u4f18\u7f8e, \u6bd4\u5982\u8bf4\u8fd9\u7bc7\u6563\u6587\u4e2d\u7684 \u201c\u5f97\u5fc3\u5e94\u624b\u201d \u7ffb\u8bd1\u7684\u65f6\u5019\u5982\u679c\u628a \u201c\u5fc3 \u201d\u548c \u201c\u624b \u201d\u90fd\u7ffb\u8bd1\u51fa\u6765, \u5c31\u4f1a\u4f7f\u82f1\u8bed\u53e5\u5b50\u5197\u957f, \u4e0d\u7b26\u5408\u82f1\u8bed\u7684\u4e60\u60ef\u3002\u6240\u4ee5, \u8981\u4e48\u4fdd\u7559\u4e00\u4e2a, \u8981\u4e48\u610f\u8bd1. PThe analysis of the TT translated by the trainee translators revealed that three main approaches were employed, namely, retainment, modification, and omission, although there were some variations among the individual participants to some extent. Combined with the analysis of the translations of participants, the results of the questionnaires and interviews show that there were at least three underlying factors affecting how the participants handled these metaphtonymic expressions in the TT.The first factor contributing to the three translation approaches was metaphtonymic expressions, specifically, their prominence degrees in the ST and cross-cultural adaptation abilities in the TT. Figures of rhetoric occur when an expression deviates from expectation of the receiver, but the expression is still interpreted as an appropriate one (McQuarrie and Mick, The second factor was metaphtonymy awareness of the trainee translators and their transference competence of bilingual rhetoric. The competence of translators in recognizing the rhetorical devices of the ST when translating figurative language requires them to adopt appropriate strategies Smith, , which sThe bilingual sub-competence, a very basic componential element of translation competence, is made up of pragmatic, socio-linguistic, textual, and lexical-grammatical knowledge in each language (PACTE GROUP, The last factor seemed to be the translation knowledge sub-competence of participants, namely, the knowledge of the principles of translators that guide their translation processes (PACTE GROUP, This study examined the performance of Chinese students of MTI in translating metaphtonymic devices in an extracted Chinese prose, based on the analysis of the TT, combined with questionnaires and interviews after translation. It was found that the participants most frequently deleted the interactive relations between metaphor and metonymy of the ST by explicitly presenting in the TT the extending meaning of the metaphtonymic expressions. Moreover, the participants modified the ST metaphtonymies as metaphors or metonymies in the TT or creatively employed a new English metaphtonymy. However, not all the participants attempted to retain the ST metaphtonymies in the TT. In other words, retainment was employed less frequently, a subtle violation of the rhetorical equivalence principle.The analysis of the questionnaires and interviews shed light on the factors contributing to the three approaches employed by the participants to translate metaphtonymies. It seems that the three approaches were influenced by the prominent cultural features of the ST metaphtonymies and whether they had lexical equivalents in the target language system. In addition, the rhetoric awareness and rhetoric shifting intercultural competence of translators comprised the underlying reasons for the choices of translators in handling metaphtonymies. The principles and standards to which the translators adhered also exerted influence on the way they dealt with the ST metaphtonymies. It can be concluded that metaphorical and metonymic thinking of translators emerges when they comprehend and render the metaphtonymies.ex-post-facto task analysis and semi-structured interviews should be developed, such as think-aloud protocols and also eye-tracking protocols for capturing attention fixation points in the ST.The findings raise several implications for training translators. In training practice mainly related to rhetoric and translation, trainees should be instructed to systematically construct knowledge of the rhetorical structures, including analyzing metaphor, metonymy, and the interactions between them. In addition, teaching design should center on competence of trainees in identifying multiple rhetorical devices, and their competence in shifting rhetoric between languages. Instruction of trainees should emphasize the features of different genres of texts and the importance of mastering appropriate translating principles and strategies. The findings also provide potentially important issues for future studies. The translation material was narrowed down to Chinese prose, one type of literary work, but metaphtonymy is not the sole province of literature, it exists in all types of prose and is distributed over many different genres and occasions. Additional research should focus on scientific and political texts, to enhance the reliability and validity of the present findings. Moreover, the sample of trainee translators in the present study was in relatively small quantity and limited to Chinese students of MTI. Therefore, it is suggested that the follow-up research should consider comparative studies of novice and expert translators, with the hope of teasing out other factors contributing to the translation of metaphtonymies. Finally, methods extending beyond The original contributions presented in the study are included in the article/Ethical review and approval was not required for the study on human participants in accordance with the local legislation and institutional requirements. Written informed consent for participation was not required for this study in accordance with the national legislation and the institutional requirements.SJ designed the study and the manuscript, analyzed the data, and wrote all sections. ZL collected, annotated, and analyzed the data. TO reviewed and edited the manuscript. All authors contributed to the conception of the study and analyzed and discussed the data.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest."}
{"text": "Furthermore, by utilizing the developed concept, a list of q-ROFR Einstein weighted averaging and geometric aggregation operators are presented which are based on algebraic and Einstein norms. Similarly, some interesting characteristics of these operators are initiated. Moreover, the concept of the entropy and distance measures is presented to utilize the decision makers' unknown weights as well as attributes' weight information. The EDAS  methodology plays a crucial role in decision-making challenges, especially when the problems of multicriteria group decision-making (MCGDM) include more competing criteria. The core of this study is to develop a decision-making algorithm based on the entropy measure, aggregation information, and EDAS methodology to handle the uncertainty in real-word decision-making problems (DMPs) under q-rung orthopair fuzzy rough information. To show the superiority and applicability of the developed technique, a numerical case study of a real-life DMP in agriculture farming is considered. Findings indicate that the suggested decision-making model is much more efficient and reliable to tackle uncertain information based on q-ROFR information.The main purpose of this manuscript is to present a novel idea on the  In the history of agriculture, the domestication of plants and animals, as well as the manufacturing and dissemination techniques for cultivating them productively, is documented. Agriculture began independently in several places of the world and included a broad range of taxa. Farming was well known on the Nile's banks by 8000 BC. Around this time, agriculture evolved independently in the Far East, most likely in China, with rice as the primary crop rather than wheat. Overstretched water supplies, high levels of deforestation, and decreased soil fertility have all resulted from modern farming practices. Since there is insufficient water to continue farming as is, how vital water, ground, and environment resources are used to increase crop yields must be reevaluated. Giving ecosystems importance, understanding environmental and livelihood tradeoffs, and balancing the rights of a range of users and interests may be a solution. Inequities that occur as a result of such steps, such as water reallocation from poor to wealthy and land clearing to make room for more profitable farmland, need to be tackled. Technological advances aid in the provision of tools and services to farmers in order to help them become more prosperous. Conservation tillage, a farming technique that helps avoid land loss due to deforestation, reduces water pollution, and improves carbon sequestration, is one example of a technology-enabled innovation.To meet the growing demand for food, farming, which was never an easy job to begin with, now needs more analytics and technology. In one case, mathematicians, hydrologists, and farmers met in California to formulate a strategy that would reduce the amount of water used for crops while still making a profit for the farmers and satisfying market demand. The mathematical model used data including plant growth properties and water requirements to determine which crops to plant, when to plant them, and which areas should be left unplanted. Farmers were satisfied to wisely use their own and community tools, while mathematicians were happy to collaborate with business experts.Pawlak  initiate\u03bc, \u03bd), where \u03bc represents the positive grade and \u03bd represents the negative grade function, with condition that \u03bc2+\u03bd2 \u2264 1. Many authors contribute to Pythagorean FSs: Ding and Liu  and \u03bdz(\u03b4) \u2208  are known as positive and negative membership grades of \u03b4 and (\u03bcz(\u03b4))2+(\u03bdz(\u03b4))2 \u2264 1, \u2200\u2009\u03b4 \u2208 M.Suppose a nonempty set M be a nonempty set. A q-ROFS Z in the universe M is a set having the form\u03bcz(\u03b4) \u2208  and \u03bdz(\u03b4) \u2208  represent positive and negative membership grades of \u03b4 and (\u03bcz(\u03b4))q+(\u03bdz(\u03b4))q \u2264 1 with q > 2, \u2200\u2009\u03b4 \u2208 M.Let Z=\u2329\u03b4, \u03bcz(\u03b4), \u03bdz(\u03b4)\u232a is represented as Z= and is called q-rung orthopair number (q-ROFN).For simplicity, M and \u03b6 \u2208 M \u00d7 M is a crisp relation. Then,\u03b6 is reflexive if \u2009 \u2208 \u03b6, \u2200\u2009\u2118 \u2208 M\u03b6 is symmetric if \u2118, \u2202\u2208M and \u2009 \u2208 \u03b6, then \u2009 \u2208 \u03b6\u03b6 is transitive if \u2118, \u2202, d \u2208 M, \u2009 \u2208 \u03b6, and  \u2208 \u03b6, then \u2009 \u2208 \u03b6Suppose a universal set M and any arbitrary relation over a set M is \u03b6 \u2208 M \u00d7 M. Now, define \u03b6\u2217 : M\u27f6P(M) as a mapping:\u03b6\u2217(\u2118) is an object's successor neighborhood \u2118 w.r.t \u03b6. Crisp approximation space (AS) is defined as the pair . The lower and upper approximation (Lo and Up A) of \u00a3 w.r.t AS  for each \u00a3\u2286M are now designated and defined asSuppose a nonempty set As a result, M to be a universe set and \u03b6 \u2208 q \u2212 ROFS(M \u00d7 M) to be any q-ROF relation on a set M. Then,\u03b6 is reflexive if \u03bc\u03b6=1 and \u03bd\u03b6=0, \u2200\u2009\u2118 \u2208 M\u03b6 is symmetric if  \u2208 M \u00d7 M, \u03bc\u03b6\u2009=\u2009\u03bc\u03b6, and \u03bd\u03b6\u2009=\u2009\u03bd\u03b6\u03b6 is transitive if  \u2208 M \u00d7 M, \u03bc\u03b6 \u2265 \u2228M\u2202\u2208[\u03bc\u03b6\u2228\u03bc\u03b6], and \u03bd\u03b6=\u2227M\u2202\u2208[\u03bd\u03b6\u2227\u03bd\u03b6]Consider q-ROFS will be developed here to acquire the notion of the q-ROF rough set (q-ROFRS) and describe its fundamental operational laws.The hybrid notion of the rough set and M to be a universe set and for any subset \u03b6 \u2208 q \u2212 ROFS(M \u00d7 M) to be any nonempty q-ROF relation on a set M. The pair  is thus referred to as q-ROF AS. The lower and upper approximation (Lo and Up A) of \u2112 w.r.t AS  are two q-ROFSs for any \u2112\u2286q \u2212 ROFS(M), which are defined asq > 2. As q-ROFSs, q-ROFRS.Consider q-ROF rough value (q \u2212 ROFRV), and its collection is known as q \u2212 ROFRS(M).For simplicity, q-ROFRS concept.We now set an example for better clarifying the M={\u21181, \u21182, \u21183, \u21184} and  be the q-ROF AS with \u03b6 \u2208 q \u2212 ROFS(M \u00d7 M) being any nonempty q-ROF relation on a set M  q \u2212 ROFRS.Therefore,Suppose e \u2208 \u2115). The operational laws can be defined as follows.(1)(2)(3)(4)Suppose t(e1)=log((2 \u2212 e1)/e1) and s(e1)=log((1+e1)/(1 \u2212 e1)) to t and s operators,(1)(2)(3)(4)Through assigning Einstein norm generator q-ROFRVs, we use score function for their comparison.To compare two or more So) and accuracy (Ao) functions:Suppose So(\u03b6(\u21121)) > So(\u03b6(\u21122)), \u21d2\u2009\u03b6(\u21121) > \u03b6(\u21122)If So(\u03b6(\u21121))=So(\u03b6(\u21122)), \u21d2If Ao(\u03b6(\u21121)) > Ao(\u03b6(\u21122)), \u21d2\u2009\u03b6(\u21121) > \u03b6(\u21122)If Ao(\u03b6(\u21121))=Ao(\u03b6(\u21122)), \u21d2\u2009\u03b6(\u21121)=\u03b6(\u21122)If Suppose M, \u03b6)\u2208q-ROF approximation space. Consider \u03b6(\u21121) \u222a \u03b6(\u21122)=\u03b6(\u21122) \u222a \u03b6(\u21121)\u03b6(\u21121)\u2229\u03b6(\u21122)=\u03b6(\u21122)\u2229\u03b6(\u21121)\u03b6(\u21121))c)c=\u03b6(\u21121), where (\u03b6(\u21121))c is the complement of \u03b6(\u21121)((\u03b6(\u21121) \u222a \u03b6(\u21122))c=(\u03b6(\u21121))c\u2229(\u03b6(\u21122))c(\u03b6(\u21121)\u2229\u03b6(\u21122))c=(\u03b6(\u21121))c \u222a (\u03b6(\u21122))c\u2208q-ROF approximation space. Consider \u03b6(\u21121) \u2295 \u03b6(\u21122)=\u03b6(\u21122) \u2295 \u03b6(\u21121)\u03b6(\u21121) \u2297 \u03b6(\u21122)=\u03b6(\u21122) \u2297 \u03b6(\u21121)\u03b2\u00b7(\u03b6(\u21121) \u2295 \u03b6(\u21122))=(\u03b2\u00b7\u03b6(\u21121) \u2295 \u03b2\u00b7\u03b6(\u21122))\u03b6(\u21121) \u2297 \u03b6(\u21122))\u03b2=(\u03b6(\u21121))\u03b2 \u2297 (\u03b6(\u21122))\u03b2(Consider (q-ROFRVs.Aggregation information (AInf) plays a vital role in integrating data into a single format and solving decision-making problems (DMPs). Throughout this portion, we present a list of innovative aggregation information based on various standard-based operating regulations for M, \u03b6)\u2208q-ROF AS. Let \u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)) are T, i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1.Consider \u2208q-ROF approximation space. Consider \u03b21, \u03b22,\u2026\u03b2n)T is the weight information of (\u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)), i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1. Then, WA AInf is a mapping Dn\u27f6D, i.e.,Consider  Step 1: for By the induction method:n=\u03b4; the result is true.\u2009Step 2: consider for n=\u03b4+1; the result is true.\u2009Step 3: consider for Hence, \u2200 positive integers, the given result is valid.q-ROFRV. So, by q-ROFRVs. Therefore, WA(\u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)) is also a q-ROFRV under q-ROF AS .From the above analysis, q-ROF rough weighted averaging operator are initiated in Some important properties of the M, \u03b6)\u2208q-ROF AS. Let \u03b21, \u03b22,\u2026\u03b2n)T be the weight information of (\u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)), i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1. Then, some important properties of the q-ROF rough weighted averaging operator are described as follows:(1)e \u2208 \u2115, thenIdempotency: if (2)Boundedness: let (3)Monotonicity: let Consider \u2208q-ROF AS. Let \u03be(1), \u03be(2), \u03be(3),\u2026, \u03be(n)), \u03be(e) is represented as the order, and the weight of (\u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)) is T, i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1.Consider \u2208q-ROF AS. Let \u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)) be T, i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1. Then, OWA AInf is a transformation Dn\u27f6D, i.e.,Consider  to be a q-ROF AS. Consider \u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)) be T, i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1. Then, some important properties of the q-ROF rough ordered weighted averaging operator are described as follows:(1)e \u2208 \u2115, thenIdempotency: if (2)Boundedness: let (3)Monotonicity: let Consider \u2208q-ROF AS. Consider \u03be(1), \u03be(2), \u03be(3),\u2026, \u03be(n)) is represented by \u03be(e) such that \u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)) is T, i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1. Also, T represent the corresponding weight of (\u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)), i.e., \u03b7e \u2265 0; \u2211e=1n\u03b7e=1.Consider \u2208q-ROF AS. Consider \u03b21, \u03b22,\u2026\u03b2n)T to be the weight of (\u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)), i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1. Then, HWA AInf is a mapping Dn\u27f6D with associated weight T, i.e., \u03b7e \u2265 0, \u2211e=1n\u03b7e=1, such thatConsider \u2208q-ROF approximation space. Let \u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)) be T, i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1. Then, some important properties of the q-ROF rough hybrid weighted averaging operator are described as follows:(1)e \u2208 \u2115, thenIdempotency: if (2)Boundedness: let (3)Monotonicity: let Consider that \u2208q-ROF approximation space. Let \u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)) is T, i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1.Consider that \u2208q-ROF AS. Let \u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)) be T, i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1. Then, WG AInf is a transformation Dn\u27f6D, i.e.,Consider , \u03b6(\u21122),\u2026, \u03b6(\u2112n)) is also a q-ROFRV under q-ROF AS .From the above analysis, q-ROF rough weighted geometric operator are initiated in Some important properties of the M, \u03b6)\u2208q-ROF AS. Consider \u03b21, \u03b22,\u2026\u03b2n)T to be the weight of (\u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)), i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1. Then, some important properties of the q-ROF rough weighted geometric operator are described as follows:(1)e \u2208 \u2115, thenIdempotency: if (2)Boundedness: let (3)Monotonicity: let Consider \u2208q-ROF approximation space. Let \u03be(e) is denoted as the order according to (\u03be(1), \u03be(2), \u03be(3),\u2026, \u03be(n)) and T and the weight of (\u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)) is T, i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1.Consider \u2208q-ROF approximation space. Consider e \u2208 \u2115) and T to be the weight information of (\u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)) such that \u03b2e \u2265 0; \u2211e=1n\u03b2e=1. Then, the mapping of OWG AInf is Dn\u27f6D, i.e.,Consider that \u2208q-ROF approximation space. Let \u03b21, \u03b22,\u2026\u03b2n)T and the weight of (\u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)) be T, i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1.(1)e \u2208 \u2115, thenIdempotency: if (2)Boundedness: let (3)Monotonicity: let Consider \u2208q-ROF AS. Let \u03be(e) is represented as the order according to (\u03be(1), \u03be(2), \u03be(3),\u2026, \u03be(n)) such that \u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)) is T, i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1. Also, T represent the associated weight of (\u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)), i.e., \u03b7e \u2265 0; \u2211e=1n\u03b7e=1.Consider \u2208q-ROF AS. Let \u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)) be T, i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1. Then, HWG AInf is a transformation Dn\u27f6D with associated weight T, i.e., \u03b7e \u2265 0; \u2211e=1n\u03b7e=1, i.e.,Consider \u2208q-ROF approximation space. Let \u03b21, \u03b22,\u2026\u03b2n)T be the weight of (\u03b6(\u21121), \u03b6(\u21122),\u2026, \u03b6(\u2112n)), i.e., \u03b2e \u2265 0; \u2211e=1n\u03b2e=1. Then, some important properties of the q-ROF rough hybrid weighted geometric operator are described as follows:(1)e \u2208 \u2115, thenIdempotency: if (2)Boundedness: let (3)Monotonicity: let Consider T being the weights, i.e., \u03b2t \u2208 , \u2211t=1h\u03b2t=1. Allow a group of decision makers (DMs) kth alternative \u2137k under the tth attribute \u2112t, and \u03b7s \u2208 , q-ROF rough values.\u2009Step 1: construct the experts' evaluation matrices \u2009where the number of experts is represented by \u2009Step 2: evaluate normalized experts' matrices q-ROF Einstein rough weighted averaging operator.\u2009Step 3: evaluate the collective expert information based on the \u2009Step 4: evaluate the value of the AVS by using suggested aggregation operators for considering the alternative w.r.t the attribute.\u2009Step 5: utilizing the value of the AVS, evaluate the values of the PDAS and NDAS as follows:Spi and negative Sni weighted distance as follows:\u2009Step 6: evaluate the positive Spi and negative Sni weighted distances as\u2009Step 7: now, evaluate normalized positive NSpi and NSni, appraisal score ASoi can be calculated as follows:\u2009Step 8: utilizing the value of ASoi value.\u2009Step 9: rate the alternatives and select the higher We present a methodology for dealing with uncertainties in decision-making (DM) while dealing with Throughout this part, a practical MAGDM problem involving determining an acceptable mode of farming among various types of agrifarming is used to ensure that the established approach is applicable and feasible.1): develop productive, self-sufficient, and cost-effective production systems that earn well. Another advantage is that farming gives us a decent income as well as jobs, food, and services to the majority of people who are currently poor. Furthermore, it enables the development of rural areas and the establishment of social connections between the rural and urban worlds.Good crop production (\u21372): manage the quality of air, water, and soil while preserving and protecting biodiversity and territories. Agriculture's first benefit is environmental protection as it reduces deforestation and natural resource depletion, increases biodiversity, and reduces carbon emissions.Environmental protection (\u21373): improve the quality in which natural resources are used. Another key difficulty that green agriculture faces is the rapid degradation and loss of natural resources. The availability of natural resources improves farming and benefits us.Natural resources' availability (\u21374): increase food production and distribution energy efficiency. With the world's population increasing and persistently high levels of hunger and poverty, sustainable agriculture yields must solve the problem of food security by generating more in less time.Food security and productivity : certain farming factors, such as soil and time of ripeness, have an effect on the quality of the products. On wheat, barley, rice, and other cereals, maturity level and degree of dryness matter.High quality production (\u21122): since the labor cost in agriculture, i.e., farming, is so high, qualified employees and manual labor are in high demand.Limiting the need for manual labor (\u21123): in the field of agriculture, there is an interesting idea for decreasing production costs which includes the use of robots. We must deal with certain uncontrollable variables such as environmental conditions, buying various brands of seeds, and using a huge number of chemicals.Decreasing the cost of production (\u21124): the use of automation, according to scientists, technologists, scholars, and farmers, will solve the complicated project in a quick and easy manner.Completion of a time-consuming project (\u21125): to maintain a consistent location, the farm must be managed using artificial intelligence from seeding to harvesting.Consistent role to complete a project  is listed in \u2009Step 9: alternatives' ranking \u2137The invited decision makers are composed of three experts.2 is the best among the alternatives based on the above computational process, and thus, it is strongly suggested.We obtained that alternative \u2137q-ROFRS. This concept will provide a more versatile and efficient basis for fuzzy system modeling and decision-making under uncertainties due to the implementation of the concept of the rough set (RS) theory. Based on the developed concept, a list of aggregation operators such as q-ROFR weighted averaging and geometric operators are established based on algebraic and Einstein norms. Furthermore, the basic desirable characteristics of developed operators are discussed in detail. Moreover, the concept of the entropy and distance measures is presented to determine the decision makers' unknown weights as well as attributes' weight information. Furthermore, we have successfully applied the proposed approach to a MADMP involving the selection of the best agrifarming robots in agriculture. In contrast to some current methods, numerical results indicate that the q-ROFRS-based method is more realistic and versatile in real-world applications. To demonstrate the feasibility and superiority of the proposed methods, a comparative review of the final ranking and optimal decision in robotic agrifarming calculated by the suggested methods with some previous methods is given. We will expand this work in the future to Frank aggregation operators and Hamacher operators to solve a variety of real-world problems and make decisions under uncertainties in different fields such as computational intelligence and medical diagnosis.In this manuscript, we have presented a new FS extension called"}
{"text": "Due to its prevalence and its health-related, economic and social consequences, childhood and adult obesity is a complex, medical and civilizational problem, which has been on the increase in the last decade. The results of multi-center investigations reveal that genetic factors play an essential role in the etiopathogenesis of obesity, particularly in the case of extreme cases with very early onset. The Body Mass Index (BMI) is one of the most frequently used indicators of obesity and shows a strong genetic component with a 40-70% degree of heritability. The three types of genetically conditioned obesity are: (1) isolated (nonsyndromic) monogenic obesity, (2) syndromic monogenic obesity associated with dysmorphic features and/or congenital defects, caused by mutations in specific gene(s), (3) chromosomal aberrations, including submicroscopic changes. The most prevalent common (complex) obesity is linked to the presence of various changes in different genomic loci, which are subject to interactions and modifications by environmental , as well as epigenetic  and epistatic (gene-gene interaction) factors. Recent investigations using the modern methods of genome-wide association studies (GWAS), bioinformatics and proteomics, have made it possible to elucidate 8 key genes among the 97 genes most likely to play significant roles in the metabolic effects of obesity. The results of investigations on the pathogenesis of complex obesity do not as yet clarify the potential pathogenic significance of these genomic changes in humans. This article discusses the neuro-endocrinological regulation of the sensation of hunger and thirst, the clinical consequences of mutations in genes associated with the melanocortin pathway, and the features of the most common obesity syndromes, including syndromes conditioned by genomic imprinting. A diagnostic algorithm for cases of suspected syndromic obesity is proposed. Oty\u0142o\u015b\u0107 i nadmierna masa cia\u0142a stanowi\u0105 powa\u017cny problem zdrowotny w skali ca\u0142ej populacji i zwykle jest wynikiem nadmiernego gromadzenia tkanki t\u0142uszczowej w organizmie. Powszechnie stosowan\u0105 miar\u0105 stosowan\u0105 dla os\u00f3b doros\u0142ych jest wska\u017anik masy cia\u0142a . Nadwaga jest definiowana jako przekroczenie warto\u015bci BMI 25, a oty\u0142o\u015b\u0107 30. W zwi\u0105zku ze znaczn\u0105 zmienno\u015bci\u0105 \u015brednich warto\u015bci BMI w okresie rozwojowym definicj\u0119 nadwagi i oty\u0142o\u015bci u dzieci do 18 roku \u017cycia opiera si\u0119 obecnie na warto\u015bciach centylowych masy cia\u0142a  [Od ko\u0144ca lat 80 XX wieku obserwuje si\u0119 sta\u0142\u0105 tendencj\u0119 do wzrostu warto\u015bci BMI u os\u00f3b doros\u0142ych o 0,4-0,5 w ci\u0105gu dekady . Ro\u015bnie Podstawowych przyczyn tego stanu upatruje si\u0119 w zmianach stylu \u017cycia, zw\u0142aszcza coraz \u0142atwiejszej dost\u0119pno\u015bci do \u017cywno\u015bci wysokoenergetycznej oraz zmniejszaj\u0105cej si\u0119 aktywno\u015bci fizycznej w ostatnim p\u00f3\u0142wieczu, a tak\u017ce czynnik\u00f3w hormonalnych zwi\u0105zanych z przemian\u0105 i wydatkowaniem energii podlegaj\u0105cej o\u015brodkowej regulacji osi przysadka-podwzg\u00f3rze oraz wzajemnego oddzia\u0142ywania r\u00f3\u017cnych czynnik\u00f3w, w tym czynnik\u00f3w genetycznych. W latach 90 tych zwr\u00f3ci\u0142o uwag\u0119 badaczy rodzinne wyst\u0119powanie oty\u0142o\u015bci. W rodzinach, w kt\u00f3rych oboje rodzice s\u0105 otyli, ryzyko oty\u0142o\u015bci u dziecka jest 10 krotnie wi\u0119ksze od populacyjnego . ZastosoZgodnie ze stanem wsp\u00f3\u0142czesnej wiedzy oty\u0142o\u015b\u0107 jest chorob\u0105 z\u0142o\u017con\u0105, charakteryzuj\u0105c\u0105 si\u0119 zr\u00f3\u017cnicowan\u0105 ekspresj\u0105 kliniczn\u0105 oraz heterogenn\u0105, etiologi\u0105.Obecnie najcz\u0119\u015bciej w praktyce klinicznej wyr\u00f3\u017cnia si\u0119 trzy typy oty\u0142o\u015bci:oty\u0142o\u015b\u0107 jednogenowa (monogenic obesity) \u2013 tzw. oty\u0142o\u015b\u0107 izolowana uwarunkowana mutacjami pojedynczych gen\u00f3w.oty\u0142o\u015b\u0107 syndromiczna, b\u0119d\u0105ca jednym z objaw\u00f3w zespo\u0142u genetycznie uwarunkowanego\u2212zespo\u0142y dysmorficzne uwarunkowane monogenowo , w tym choroby uwarunkowane rodzicielskim pi\u0119tnowaniem genomowym.3oty\u0142o\u015b\u0107 powszechna (common obesity) wyst\u0119puj\u0105c\u0105 najcz\u0119\u015bciej, uwarunkowan\u0105 poligenowo, wieloczynnikowo.Znacz\u0105cy post\u0119p w badaniach nad genetyk\u0105 oty\u0142o\u015bci przynios\u0142a ostatnia dekada XX wieku, dzi\u0119ki wcze\u015bniejszym obserwacjom zebranym w badaniach do\u015bwiadczalnych na modelu zwierz\u0119cym, maj\u0105cych na celu poznanie z\u0142o\u017conego mechanizmu o\u015brodkowej regulacji wydzielania hormon\u00f3w, bia\u0142ek i neuroprzeka\u017anik\u00f3w zapewniaj\u0105cych utrzymanie homeostazy zwi\u0105zanej z poborem i wydatkowaniem energii oraz ich wp\u0142ywu na uczucie g\u0142odu i syto\u015bci. Pierwsze odkrycia dotyczy\u0142y lokalizacji podwzg\u00f3rzowego uk\u0142adu regulacji g\u0142odu i syto\u015bci, co by\u0142o mo\u017cliwe dzi\u0119ki eksperymentalnym uszkodzeniom podwzg\u00f3rza u szczur\u00f3w. Badania do\u015bwiadczalne na modelach mysich doprowadzi\u0142y do odkrycia u skrajnie oty\u0142ych myszy ob/ ob i db/db mutacji genu leptyny i jej receptora, co mia\u0142o fundamentalne znaczenie dla lepszego poznania neurohormonalnych mechanizm\u00f3w moduluj\u0105cych nadmierne odk\u0142adanie tkanki t\u0142uszczowej, a w szczeg\u00f3lno\u015bci roli podwzg\u00f3rza w kontroli syto\u015bci i g\u0142odu .Podstawowym o\u015brodkiem regulacji apetytu, kt\u00f3ry nosi nazw\u0119 szlaku melanokortynowego, jest j\u0105dro \u0142ukowate w podwzg\u00f3rzu. Jego aktywacja przez leptyn\u0119 (\u201ehormon syto\u015bci\u201d wydzielany przez kom\u00f3rki t\u0142uszczowe), a w mniejszym stopniu tak\u017ce przez insulin\u0119, za po\u015brednictwem odpowiednich receptor\u00f3w aktywuje wydzielanie pro-opiomelanokortyny. Dalsza przemiana w \u03b1- i \u03b2- melenokorytyn\u0119 umo\u017cliwia aktywacj\u0119 receptora melanokortyny 4 (MS4R) w j\u0105drze przykomorowym podwzg\u00f3rza, co uruchamia sygna\u0142 syto\u015bci. Z kolei grelina (hormon wydzielany w przez kom\u00f3rki ok\u0142adzinowe \u017co\u0142\u0105dka), pobudza \u0142aknienie poprzez stymulacj\u0119 kom\u00f3rek wydzielaj\u0105cych neuropeptyd Y (NPY) oraz bia\u0142ko z rodziny Agouti ( AGgRP) w j\u0105drze \u0142ukowatym podwzg\u00f3rza, kt\u00f3re hamuj\u0105 dzia\u0142anie receptora MS4R i wyzwalaj\u0105 uczucie g\u0142odu. Aktywacja cz\u0119\u015bci sygnalizuj\u0105cej syto\u015b\u0107 jednocze\u015bnie hamuje dzia\u0142anie anoreksygenne melanokortyny.Dotychczas opisano zaledwie kilkana\u015bcie rodzin z mutacjami genu leptyny LEP powoduj\u0105cymi utrat\u0119 funkcji genu. Gen koduj\u0105cy leptyn\u0119 u cz\u0142owieka, homologiczny do mysiego genu ob, zlokalizowany jest na d\u0142ugim ramieniu chromosomu 7q31.3 , 12, 13.W kom\u00f3rkach podwzg\u00f3rza znajduje si\u0119 receptor leptyny, kt\u00f3ry pod wp\u0142ywem leptyny inicjuje proces hamowania apetytu. W 1995 r. Tartaglia i wsp.  zidentyfWyniki licznych bada\u0144 nad patogenez\u0105 oty\u0142o\u015bci, szczeg\u00f3lnie zapocz\u0105tkowanej w wieku dzieci\u0119cym doprowadzi\u0142y do wykrycia wielu innych gen\u00f3w szlaku leptyna-melanokortyna w podwzg\u00f3rzu. W\u015br\u00f3d nich wymienia si\u0119 takie geny jak: POMC, PSK1, MC4R, BDNF. Z ci\u0119\u017ckimi postaciami dzieci\u0119cej oty\u0142o\u015bci koreluje si\u0119 geny: NTRK2B oraz SIM1.POMC, skutkuje hiperfagi\u0105 i wczesn\u0105 oty\u0142o\u015bci\u0105 w wyniku zaburzenia funkcji szlaku sygna\u0142owego melanokortyny. W\u015br\u00f3d najbli\u017cszych krewnych dzieci b\u0119d\u0105cych homozygotami mutacji POMC (z brakiem proopiomelanokortyny) zaobserwowano cz\u0119stsze wyst\u0119powanie oty\u0142o\u015bci u heterozygot pod wzgl\u0119dem mutacji POMC. Sugeruje to, \u017ce nawet heterozygotyczne mutacje tego genu mog\u0105 r\u00f3wnie\u017c skutkowa\u0107 rozwojem oty\u0142o\u015bci o mniejszym nasileniu [W 1998 roku H. Krude i wsp. opisali obecno\u015b\u0107 homozygotycznych mutacji genu pro-opiomelanokortyny (POMC), stanowi\u0105cego prekursor dla wytwarzania hormon\u00f3w, w tym adrenokortykotropiny (ACTH) i hormonu stymuluj\u0105cego melanocyty (MSH) u kilkorga dzieci z objawami wczesnej oty\u0142o\u015bci z nadmiernym apetytem oraz kryz\u0105 nadnerczow\u0105 z hipoglikemi\u0105 i hiponatremi\u0105 w okresie noworodkowym. U os\u00f3b pochodzenia europejskiego dodatkowo obserwowano objawy niedoboru melaniny pod postaci\u0105 jasnej karnacji sk\u00f3ry i rudych w\u0142os\u00f3w oraz niekiedy cholestaz\u0119 i \u017c\u00f3\u0142taczk\u0119 . W surowasileniu .MC4R) zmapowanego w regionie 8q21 s\u0105 najcz\u0119stsz\u0105 przyczyn\u0105 izolowanej oty\u0142o\u015bci z pocz\u0105tkiem we wczesnym dzieci\u0144stwie i wyst\u0119puj\u0105 w 1-6% przypadk\u00f3w ekstremalnej oty\u0142o\u015bci u dzieci i m\u0142odocianych. W odr\u00f3\u017cnieniu od wcze\u015bniej opisanych, rozw\u00f3j oty\u0142o\u015bci wyst\u0119puje u os\u00f3b b\u0119d\u0105cych nosicielami heterozygotycznych mutacji tego genu  [MC4R) dodatkowo stwierdza sie hyperinsulinemi\u0119 oraz przy\u015bpieszenie wzrastania. Nadmierny apetyt ujawnia si\u0119 w pierwszym roku \u017cycia, natomiast z wiekiem staje si\u0119 mniej nasilony [PCSK1, SIM1, BDNF, NTRK2 oraz GSHR.Mutacje genu receptora melanoktyny 4 (inuj\u0105ce) . U nosicnasilony . TerapiaPCSK1  ) zmapowanego w regionie 5q15 zaburzaj\u0105 funkcj\u0119 PC1/3 (prohormon konwertazy1), enzymu zaanga\u017cowanego w proces odszczepienia wielu prekursor\u00f3w hormon\u00f3w peptydowych, zwi\u0105zanych z regulacj\u0105 przyjmowania pokarm\u00f3w, homeostaz\u0105 glukozy i homeostaz\u0105 energii, na przyk\u0142ad proopiomelanokortyn\u0105, proinsulin\u0105, proglukagonem i proergrelin\u0105. By\u0142 on jednym z pierwszych gen\u00f3w korelowanych z monogeniczn\u0105 oty\u0142o\u015bci\u0105 wczesn\u0105. W sporadycznych przypadkach nosicieli homozygotycznych mutacji skutkuj\u0105cych utrat\u0105 funkcji genu PCSK1 obserwowano hiperfagi\u0119, wczesny rozw\u00f3j oty\u0142o\u015bci, hipogonadyzm hipogonadotropowy, hipokortyzolemi\u0119, hipoglikemi\u0119 poposi\u0142kow\u0105, podwy\u017cszony poziom POMC, a we wczesnym wieku dzieci\u0119cym zaburzenia wch\u0142aniania i przewlek\u0142e ci\u0119\u017ckie biegunki. Badania ostatnich lat dotycz\u0105cych genomu w r\u00f3\u017cnych populacjach wykaza\u0142y ponadto silne sprz\u0119\u017cenie pomi\u0119dzy polimorfizmami PCSK1 i zwi\u0119kszonym ryzykiem oty\u0142o\u015bci [Mutacje genu oty\u0142o\u015bci .SIM1 (zmapowanego w regionie 6q16.3), wykazuj\u0105cego swoj\u0105 aktywno\u015b\u0107 transkrypcyjn\u0105 w j\u0105drze przykomorowym podwzg\u00f3rza opisano u dziewczynki z powsta\u0142\u0105 de novo zr\u00f3wnowa\u017con\u0105 translokacj\u0105 mi\u0119dzy chromosomami 1p22.1 i 6q16.2. Hiperfagia, wczesny rozw\u00f3j oty\u0142o\u015bci oraz podobie\u0144stwo fenotypowe do zespo\u0142u Pradera i Williego sugerowa\u0142o rol\u0119 genu SIM1 w patogenezie ci\u0119\u017ckiej oty\u0142o\u015bci o wczesnym pocz\u0105tku. [SIM1 i objawami skrajnej oty\u0142o\u015bci przypominaj\u0105cymi zesp\u00f3\u0142 Pradera i Williego. Badania funkcjonalne r\u00f3\u017cnych zmian w genie SIM1 potwierdzaj\u0105 istnienie zwi\u0105zku przyczynowego utraty funkcji genu SIM1 a objawami ci\u0119\u017ckiej oty\u0142o\u015bci\u0105 zar\u00f3wno z obecno\u015bci\u0105 jak i bez fenotypu podobnego do zespo\u0142u Pradera i Wiliego [Uszkodzenie funkcji genu ocz\u0105tku. . By\u0142o to Wiliego .BDNF (zmapowanego w regionie 11p14.1), zaanga\u017cowanego w synaptyczn\u0105 funkcj\u0119 przekazywania sygna\u0142u MC4R opisano u dziecka ze skrajn\u0105 oty\u0142o\u015bci\u0105 i nadmiernym apetytem, nadpobudliwo\u015bci\u0105, upo\u015bledzonymi funkcjami poznawczymi oraz os\u0142abion\u0105 pami\u0119ci\u0105 [Mutacje genu pami\u0119ci\u0105 .NTRK2, genu koduj\u0105cego receptor TrkB wykryto u dziecka z ci\u0119\u017ck\u0105 oty\u0142o\u015bci\u0105 i nadmiernym apetytem oraz op\u00f3\u017anionym rozwojem, trudno\u015bciami w nauce, zaburzon\u0105 pami\u0119ci\u0105 kr\u00f3tkotrwa\u0142\u0105 oraz obni\u017conym progiem b\u00f3lowym [GSHR, u os\u00f3b z niskim wzrostem oraz stosunkowo p\u00f3\u017anym, w okresie dojrzewania, rozwojem oty\u0142o\u015bci [Mutacje genu  b\u00f3lowym . Opisanooty\u0142o\u015bci .Opisane powy\u017cej przyk\u0142ady gen\u00f3w o potwierdzonej roli w patogenezie oty\u0142o\u015bci o wczesnym pocz\u0105tku nie wyczerpuj\u0105 zagadnienia zwi\u0105zanego z przyczynami oty\u0142o\u015bci monogenowej \u2013 niesyndromicznej, a z uwagi na niezwykle rzadkie wyst\u0119powanie mutacji w tych genach nie mog\u0105 t\u0142umaczy\u0107 problemu oty\u0142o\u015bci w skali populacji. Liczne badania populacyjne prowadzone w ci\u0105gu ostatniej dekady z zastosowaniem wysokoporzepustowego sekwencjonowania nast\u0119pnej generacji jak r\u00f3wnie\u017c badania funkcjonalne wykazuj\u0105 obecno\u015b\u0107 wielu polimorficznych zmian w obr\u0119bie wymienionych powy\u017cej gen\u00f3w jak r\u00f3wnie\u017c w nowych genach \u015bci\u015ble skorelowanych z ryzykiem oty\u0142o\u015bci zar\u00f3wno o wczesnym pocz\u0105tku jak i w wieku dojrza\u0142ym. Zestawienie skutk\u00f3w klinicznych omawianych gen\u00f3w oty\u0142o\u015bci monogenowej - niesyndromicznej przedstawiono w Oty\u0142o\u015b\u0107 zespo\u0142owa (syndromiczna) rozpatrywana jest w kontek\u015bcie r\u00f3\u017cnego rodzaju cech klinicznych powi\u0105zanych z okre\u015blonymi cechami fenotypu, wadami wrodzonymi, niepe\u0142nosprawno\u015bci\u0105 intelektualn\u0105 oraz specyficznymi zaburzeniami zachowania. Z analiz Kaur i wsp. opublikowanych w 2017 roku przeprowadzonych na podstawie danych uzyskanych w r\u00f3\u017cnych bazach, takich jak: MEDLINE, EMBASE, CINAHL, Pubmed, Orphanet, Web of Science oraz Cochrane Library databases wynika, \u017ce do 2016 roku, ukaza\u0142o si\u0119 a\u017c 13.719 doniesie\u0144 naukowych dotycz\u0105cych oty\u0142o\u015bci zespo\u0142owej u ludzi. Na podstawie tego przegl\u0105du wy\u0142oniono 79 zespo\u0142\u00f3w monogenowych z oty\u0142o\u015bci\u0105 w obrazie klinicznym, z czego w 19  zespo\u0142\u00f3w patologia molekularna warunkuj\u0105ca chorob\u0119 jest znana, w 11  cz\u0119\u015bciowo poznana. Z pozosta\u0142ej grupy 49 (62%) chor\u00f3b, w 27 (34%) zmapowano potencjalne loci chromosomowe, za\u015b w pozosta\u0142ych 22(27%) chorobach nie s\u0105 dotychczas poznane, ani potencjalny gen, ani loci chromosomowe warunkuj\u0105ce wyst\u0105pienia okre\u015blonych objaw\u00f3w. Trudno\u015bci w ustaleniu etiologii tych zespo\u0142\u00f3w w du\u017cej mierze wynikaj\u0105 z niezwykle rzadkiego ich wyst\u0119powania w populacji oraz w wielu przypadkach z\u0142o\u017conego mechanizmu dziedziczenia, podlegaj\u0105cego regulacji poprzez czynniki epigenetyczne, mozaikowo\u015b\u0107 czy te\u017c wsp\u00f3\u0142dzia\u0142anie gen\u00f3w modyfikuj\u0105cych. Oszacowanie cz\u0119sto\u015bci wyst\u0119powania by\u0142o mo\u017cliwe tylko dla 12 spo\u015br\u00f3d 79 zespo\u0142\u00f3w i wynosi ona od 1:565 do <1:1,000,000 w\u015br\u00f3d \u017cywo urodzonych. Rozw\u00f3j nowoczesnych technik diagnostycznych, technik molekularnych, a tak\u017ce technik obrazowych  przyczynia si\u0119 do lepszego poznania, kt\u00f3re cz\u0119\u015bci m\u00f3zgu i potencjalne geny wykazuj\u0105ce w nim swoj\u0105 ekspresj\u0119 odpowiadaj\u0105 za zaburzenie homeostazy energetycznej, hiperfagi\u0119, wzrost BMI i rozw\u00f3j oty\u0142o\u015bci. W przysz\u0142o\u015bci mo\u017ce to przyczyni\u0107 si\u0119 do ustalenia korelacji fenotypowo - genotypowej r\u00f3\u017cnych postaci oty\u0142o\u015bci i opracowania celowanych lek\u00f3w do jej skutecznego leczenia .Oty\u0142o\u015b\u0107 zespo\u0142owa uwarunkowana jest obecno\u015bci\u0105 mutacji okre\u015blonego genu (\u00f3w) dziedziczonymi autosomalnie dominuj\u0105co lub sprz\u0119\u017conych z chromosomem X, aberracjami chromosomowymi, zmianami liczby kopii fragment\u00f3w DNA (Copy Number Variations \u2212 CNVs) w zespo\u0142ach mikrodelecji/mikroduplikacji okre\u015blanych mianem tzw. chor\u00f3b genomowych. Co ciekawe, w niekt\u00f3rych spo\u015br\u00f3d tych zespo\u0142\u00f3w obserwuje si\u0119 du\u017c\u0105 heterogenno\u015b\u0107 genetyczn\u0105 czyli wyst\u0119powanie bardzo podobnego fenotypu klinicznego mimo zr\u00f3\u017cnicowanego, wielogenowego uwarunkowania. Plejotropowy charakter skutk\u00f3w klinicznych okre\u015blonej zmiany w genomie powoduje, \u017ce diagnostyka tych zespo\u0142\u00f3w na poziomie klinicznym nierzadko jest bardzo trudna i wymaga \u015bcis\u0142ej wsp\u00f3\u0142pracy wielu specjalist\u00f3w, w szczeg\u00f3lno\u015bci w zakresie biologii molekularnej.Do najlepiej poznanych zespo\u0142\u00f3w oty\u0142o\u015bci syndromicznej nale\u017c\u0105: zesp\u00f3\u0142 Pradera i Williego, zesp\u00f3\u0142 Bardeta i Biedla, zesp\u00f3\u0142 Cohena, zesp\u00f3\u0142 B\u00f6rjeson i Lehmana, zesp\u00f3\u0142 Alstr\u00f6ma, zesp\u00f3\u0142 Simpsona i Golabiego, zesp\u00f3\u0142 Carpentera, zesp\u00f3\u0142 Wilsona i Turnera, zesp\u00f3\u0142 Smitha i Magenis, zesp\u00f3\u0142 disomii chromosomu 14 (z. Temple), osteodystrofia Albrighta typu 1 oraz niekt\u00f3re zespo\u0142y mikrodelecji i mikroduplikacji przedstawione w Zesp\u00f3\u0142 Pradera i Williego (PWS) (OMIM \u2212 1762700) jest najcz\u0119stsz\u0105 przyczyn\u0105 genetycznie uwarunkowanej oty\u0142o\u015bci oraz modelowym przyk\u0142adem choroby uwarunkowanej rodzicielskim pi\u0119tnowaniem genomowym (\u201egenomic imprinting\u201d). W ostatnich latach sta\u0142 si\u0119 r\u00f3wnie\u017c chorob\u0105 modelow\u0105 do badania mechanizm\u00f3w regulacji \u0142aknienia i rozwoju oty\u0142o\u015bci u cz\u0142owieka.W\u015br\u00f3d g\u0142\u00f3wnych objaw\u00f3w PWS wyst\u0119puje: znacznego stopnia hipotonia mi\u0119\u015bniowa w okresie noworodkowym i niemowl\u0119cym, brak/s\u0142aby odruch ssania, trudno\u015bci w karmieniu, brak/s\u0142aby przyrost masy cia\u0142a w okresie niemowl\u0119cym, hiperfagia, oty\u0142o\u015b\u0107, niedob\u00f3r wysoko\u015bci cia\u0142a, ma\u0142e d\u0142onie i stopy, g\u0119sta lepka \u015blina zasychaj\u0105ca w k\u0105cikach ust, cechy dysmorfii twarzy, hipogonadyzm hipogonadotropowy, op\u00f3\u017anienie rozwoju psychoruchowego oraz specyficzny typ zaburze\u0144 zachowania zwi\u0105zanych z hiperfagi\u0105 . Do niedawna opisywano 2 fazy zaburze\u0144 od\u017cywiania w PWS: okres niemowl\u0119cy z trudno\u015bciami w karmieniu i okres hiperfagii z rozwojem oty\u0142o\u015bci powy\u017cej 3 roku \u017cycia .Z ostatnich bada\u0144 wynika, \u017ce przej\u015bcie z fazy s\u0142abego \u0142aknienia i niedoboru masy cia\u0142a do fazy hiperfagii jest procesem z\u0142o\u017conym i sk\u0142ada si\u0119 z 7 faz z pocz\u0105tkiem ju\u017c w okresie p\u0142odowym. Zwykle do drugiego roku \u017cycia rozw\u00f3j fizyczny jest odpowiedni do wieku, natomiast pomi\u0119dzy 2,1\u20134,5 rokiem \u017cycia nast\u0119puje nadmierny przyrost masy cia\u0142a bez wzrostu \u0142aknienia ani zwi\u0119kszonej poda\u017cy kalorii. Wzmo\u017cone \u0142aknienie (hiperfagia) i nadmierny przyrost masy cia\u0142a rozwija si\u0119 zwykle pomi\u0119dzy 4-8 rokiem \u017cycia, natomiast faza niepohamowanego \u0142aknienia nast\u0119puje powy\u017cej 8 lat. Fakty te powinny by\u0107 uwzgl\u0119dnione w leczeniu dietetycznym i profilaktyce oty\u0142o\u015bci pocz\u0105wszy od okresu niemowl\u0119cego . HiperfaBadania nad patogenez\u0105 oty\u0142o\u015bci w PWS prowadzone od wielu lat lecz jak dot\u0105d nie doprowadzi\u0142y do ostatecznego wyja\u015bnienia przyczyny hyperfagii. Wiadomo, \u017ce jest to proces bardzo z\u0142o\u017cony, zwi\u0105zany z zaburzeniem o\u015brodkowej regulacji osi podwzg\u00f3rze \u2013 przysadka, w tym funkcji neuroprzeka\u017anik\u00f3w zwi\u0105zanych z regulacj\u0105 o\u015brodk\u00f3w g\u0142odu i syto\u015bci w podwzg\u00f3rzu.Potwierdzaj\u0105 to wyniki bada\u0144 z zastosowaniem funkcjonalnego MRI i PET w trakcie ogl\u0105dania wysokokalorycznego po\u017cywienia oraz po posi\u0142ku, kt\u00f3re wykaza\u0142y brak aktywacji region\u00f3w odpowiedzialnych za uczucie syto\u015bci po spo\u017cyciu posi\u0142ku oraz zwi\u0119kszon\u0105 aktywacj\u0119 region\u00f3w odpowiedzialnych za uczucie g\u0142odu i motywacji (hipokamp oraz kora oczodo\u0142owo\u2013czo\u0142owa) .St\u0119\u017cenie greliny u chorych z PWS, tzw. \u201ehormonu g\u0142odu\u201d wytwarzanego przez kom\u00f3rki ok\u0142adzinowe \u017co\u0142\u0105dka jest stale na bardzo wysokim poziomie, 3- krotnie wy\u017cszym ni\u017c u zdrowych os\u00f3b. Obni\u017cenie st\u0119\u017cenia greliny po posi\u0142ku nie skutkuje jednak zmniejszeniem uczucia g\u0142odu. W j\u0105drach przykomorowych podwzg\u00f3rza u chorych z PWS stwierdza si\u0119 zmniejszon\u0105 liczb\u0119 i obj\u0119to\u015b\u0107 neuron\u00f3w produkuj\u0105cych oksytocyn\u0119, kt\u00f3ra jest neuropeptydem hamuj\u0105cym \u0142aknienie. Wyniki bada\u0144 do\u015bwiadczalnych zar\u00f3wno na modelach mysich jak i u chorych z PWS sugeruj\u0105, \u017ce ma\u0142a dawka donosowo podawanej oksytocyny jest bezpieczna i mo\u017ce wp\u0142ywa\u0107 na redukcj\u0119 apetytu oraz popraw\u0119 w zakresie zale\u017cnych od hiperfagii zaburze\u0144 zachowania .W patogenezie PWS odgrywaj\u0105 rol\u0119 3 g\u0142\u00f3wne defekty molekularne: delecje w regionie 15q11.2-13 (~75% przypadk\u00f3w), matczyna disomia 15 (mUPD15) w oko\u0142o ~24% przypadk\u00f3w oraz defekty imprintingowe (~1-3% przypadk\u00f3w). Mimo wielu lat bada\u0144 nadal nie uda\u0142o si\u0119 ustali\u0107 gen\u00f3w odpowiedzialnych za ekspresj\u0119 kluczowych dla zespo\u0142u objaw\u00f3w. W ostatnich latach sugeruje si\u0119 znaczenie delecji (podlegaj\u0105cego imprintingowi) skupiska gen\u00f3w SNORD-116, koduj\u0105cych ma\u0142e j\u0105derkowe RNA  jako kluczowego genu odpowiedzialnego za hiperfagi\u0119 i oty\u0142o\u015b\u0107 w PWS .U chorych z zespo\u0142em Pradera i Williego od 2006 roku w Polsce z powodzeniem stosuje si\u0119 terapi\u0119 rekombinowanym hormonem wzrostu (rGH) (od 2 roku \u017cycia), co pozwala na popraw\u0119 bilansu energetycznego, obni\u017cenie ca\u0142kowitej masy cia\u0142a przy jednoczesnym zwi\u0119kszeniu bezt\u0142uszczowej masy cia\u0142a (zwi\u0119kszenie si\u0142y mi\u0119\u015bniowej), a w konsekwencji przyspieszenie rozwoju psychoruchowego uzyskanie wzrostu zbli\u017conego do r\u00f3wie\u015bnik\u00f3w, spektakularn\u0105 normalizacj\u0119 fenotypu oraz popraw\u0119 jako\u015bci \u017cycia. , 37, 38.Zesp\u00f3\u0142 Bardeta i Biedla (BBS) (OMIM \u2013 209900) jest rzadkim zespo\u0142em, dziedziczonym autosomalnie recesywnie, w kt\u00f3rym oty\u0142o\u015bci centralnej rozwijaj\u0105cej si\u0119 na prze\u0142omie 1 i 2 roku \u017cycia, towarzysz\u0105: niepe\u0142nosprawno\u015b\u0107 intelektualna i zmienne problemy z zachowaniem, polidaktylia pozaosiowa d\u0142oni i st\u00f3p (68-80% chorych) oraz hipogonadyzm i hipogenitalizm u pacjent\u00f3w p\u0142ci m\u0119skiej. \u015arednie BMI szacuje si\u0119 na 31,5\u201336,6 kg/m2. W p\u00f3\u017aniejszym okresie rozwija si\u0119 post\u0119puj\u0105ca dystrofia siatk\u00f3wki lub zwyrodnienie barwnikowe (90% chorych), prowadz\u0105ce cz\u0119sto do \u015blepoty nocnej u m\u0142odych doros\u0142ych. W obrazie choroby stwierdza si\u0119 wady uk\u0142adu kielichowo-miedniczkowego oraz torbielowato\u015b\u0107 nerek, oraz zwykle \u0142agodn\u0105 ich niewydolno\u015bci\u0105, a tak\u017ce wrodzone wady serca, kardiomiopati\u0119, nadci\u015bnienie t\u0119tnicze oraz cukrzyc\u0119 z insulinooporno\u015bci\u0105 i nietolerancj\u0105 glukozy.Zesp\u00f3\u0142 Bardeta i Biedla nale\u017cy do tzw. grupy ciliopatii i charakteryzuje si\u0119 heterogenno\u015bci\u0105 genetyczn\u0105 \u2013 jest uwarunkowany zmianami w co najmniej 20 r\u00f3\u017cnych genach: BBS1, BBS2, ARL6 (BBS3), BBS4, BBS5, MKKS (BBS6), BBS7, TTC8 (BBS8), BBS9, BBS10, TRIM 32 (BBS11), BBS12, MKS1 (BBS13), CEP290 (BBS14), WDPCP (BBS15), SDCCAG8 (BBS16), LZTFL1 (BBS17), BBIP1 (BBS18). Najcz\u0119\u015bciej mutacje stwierdza si\u0119 w genach: BBS1 \u2013 23,1%, BBS10 \u2013 20%, BBS2 \u2013 8,1%, BBS9 \u2013 6% oraz MKKS (BBS6) \u2013 5,8%. W oko\u0142o 20% przypadk\u00f3w nie identyfikuje si\u0119 mutacji w \u017cadnym z wymienionych gen\u00f3w . ZazwyczZesp\u00f3\u0142 Alstr\u00f6ma (OMIM \u2013 203800) jest rzadkim zespo\u0142em zaliczanym do grupy ciliopatii uwarunkowanym mutacjami w genie ALMS1 zmapowanym w regionie 2p13.1. Dziedziczy si\u0119 autosomalnie recesywnie. Chorych charakteryzuje hiperfagia pojawiaj\u0105ca si\u0119 ju\u017c w wieku niemowl\u0119cym, kt\u00f3rej towarzyszy hiperinsulinomia o nasileniu nieproporcjonalnym w stosunku do oty\u0142o\u015bci, przewlek\u0142a hiperglikemia, a nast\u0119pnie cukrzyca typu 2 oraz umiarkowana oty\u0142o\u015b\u0107 brzuszna. Objawami przepowiadaj\u0105cymi wyst\u0105pienie prowadz\u0105cej od \u015blepoty neuropatii z degeneracj\u0105 czopk\u00f3w i pr\u0119cik\u00f3w mog\u0105 by\u0107 oczopl\u0105s i \u015bwiat\u0142owstr\u0119t obecne ju\u017c w pierwszym roku \u017cycia. Powoli post\u0119puj\u0105cy niedos\u0142uch zmys\u0142owo-nerwowy wykrywany jest z regu\u0142y oko\u0142o 5 roku \u017cycia. U cz\u0119\u015bci chorych ju\u017c w pierwszych latach \u017cycia jest diagnozowana kardiomiopatia, za\u015b w drugiej lub trzeciej dekadzie post\u0119puj\u0105ca nefropatia [fropatia .Zesp\u00f3\u0142 Cohena (OMIM \u2013 216550) jest dziedziczony autosomalnie recesywnie i uwarunkowany mutacjami w genie VPS13B zmapowanym w regionie 8q22.2. Stosunkowo cz\u0119ste wyst\u0119powanie choroby stwierdza si\u0119 w populacji \u017byd\u00f3w Ashkenazyjskich oraz Finlandii. Osoby dotkni\u0119te tym zespo\u0142em zwykle cechuje op\u00f3\u017anienie rozwoju psychoruchowego z hipotoni\u0105 i nadmiern\u0105 wiotko\u015bci\u0105 staw\u00f3w w okresie niemowl\u0119cym i wczesnodzieci\u0119cym, niepe\u0142nosprawno\u015b\u0107 intelektualna w stopniu umiarkowanym b\u0105d\u017a znacznym, cz\u0119sto brak/znaczne op\u00f3\u017anienie rozwoju mowy, padaczka (wyst\u0119puj\u0105ca prawie u wszystkich chorych), ma\u0142og\u0142owie, ataksja, charaktery-styczny fenotyp behawioralny (mi\u0142a i pogodna osobowo\u015b\u0107), niski wzrost, smuk\u0142e, d\u0142ugie palce. d\u0142oni, w\u0105skie stopy. W\u015br\u00f3d charakterystycznych cech dysmorfii, wymienia si\u0119 takie jak: twarz hipotoniczna, wysoki grzbiet nosa, brwi o falistym kszta\u0142cie, cienka warga g\u00f3rna, tendencja do otwartych ust, wysuni\u0119cie do przodu g\u00f3rnych siekaczy. Po pi\u0105tym roku \u017cycia stwierdza si\u0119 post\u0119puj\u0105c\u0105 kr\u00f3tkowzroczno\u015b\u0107 cz\u0119sto skojarzon\u0105 z dystrofi\u0105 siatk\u00f3wki i naczyni\u00f3wki. W badaniach obrazowych stwierdza si\u0119 hipoplazje m\u00f3\u017cd\u017cku oraz du\u017ce cia\u0142o modzelowate. Wa\u017cnym diagnostycznie objawem jest okresowa leukopenia/neutropenia o \u0142agodnym nasileniu, cz\u0119sto ju\u017c w okresie niemowl\u0119cym. Oty\u0142o\u015b\u0107 brzuszna \u015bredniego stopnia, pojawia si\u0119 ko\u0142o 5 roku \u017cycia [ku \u017cycia .Zesp\u00f3\u0142 B\u00f6rjesona, Forssmana i Lehmanna (OMIM \u2013 301900) jest bardzo rzadkim zespo\u0142em dziedzicz\u0105cym si\u0119 recesywnie w spos\u00f3b sprz\u0119\u017conym z chromosomem X. Przyczyn\u0105 choroby s\u0105 mutacje w genie PHF 6 zmapowanym w regionie Xq26.2. Niemowl\u0119ta p\u0142ci m\u0119skiej mog\u0105 wykazywa\u0107 objawy fenotypowe przypominaj\u0105ce zesp\u00f3\u0142 Pradera i Williego . W\u015br\u00f3d cech dysmorfii wymienia si\u0119: pogrubienie rys\u00f3w twarzy, du\u017ce odstaj\u0105ce ma\u0142\u017cowiny uszne, g\u0142\u0119boko osadzone ga\u0142ki oczne, w\u0105skie szpary powiekowe, szerokie rozstawienie palc\u00f3w st\u00f3p. Zwracaj\u0105 uwag\u0119 mi\u0119siste d\u0142onie, hipoplazja dystalnych i \u015brodkowych paliczk\u00f3w d\u0142oni . Niepe\u0142nosprawno\u015b\u0107 intelektualna zwykle umiarkowanego lub znacznego stopnia, ma\u0142og\u0142owie, padaczka, niskoros\u0142o\u015b\u0107, skolioza, kifoza, kr\u00f3tkowzroczno\u015b\u0107 oraz u cz\u0119\u015bci pacjent\u00f3w cechy polineuropatii czuciowo-nerwowej dope\u0142niaj\u0105 obraz kliniczny choroby. Znacznego stopnia oty\u0142o\u015b\u0107 z towarzysz\u0105c\u0105 nasilon\u0105 ginekomasti\u0105 rozwija si\u0119 dopiero w p\u00f3\u017anym dzieci\u0144stwie, a w wieku dojrza\u0142ym jest zwykle znacznego stopnia. Matki chorych m\u0119\u017cczyzn b\u0119d\u0105ce nosicielkami mutacji genu PHF6 mog\u0105 wykazywa\u0107 dyskretne cechy dysmorfii z pogrubieniem rys\u00f3w twarzy [w twarzy .Zesp\u00f3\u0142 Albrighta (OMIM \u2013 103580), inne nazwy zespo\u0142u: Albright hereditary osteodystrophy (AHO), rzekoma niedoczynno\u015b\u0107 przytarczyc typu 1a, dziedziczna osteodystrofia Albrighta, jest grup\u0105 chor\u00f3b metabolicznych, charakteryzuj\u0105cych si\u0119 oporno\u015bci\u0105 tkanek docelowych na parathormon. Wyr\u00f3\u017cnia si\u0119 cztery typy choroby: Ia, Ib, Ic i II. Najbardziej znana jest posta\u0107 1A, choroba dziedziczona w spos\u00f3b autosomalny dominuj\u0105cy spowodowana mutacjami w genie GNAS1, zmapowanym w regionie 20q13.32, kt\u00f3ry koduje podjednostk\u0119 alfa bia\u0142ka Gs. Ekspresja objaw\u00f3w zwi\u0105zana jest z rodzicielskim pi\u0119tnowaniem genomowym (matczyny imprinting genomowy). Chorych cechuje zauwa\u017calna oko\u0142o 3-5 roku \u017cycia niskoros\u0142o\u015b\u0107, charakterystyczna, okr\u0105g\u0142a twarz, pe\u0142ne policzki, op\u00f3\u017anione wyrzynanie z\u0119b\u00f3w, hipoplazja szkliwa i wady zgryzu, wady kostne z kr\u00f3tk\u0105 szyj\u0105, kr\u00f3tkimi ko\u015b\u0107mi \u015br\u00f3dr\u0119cza i \u015br\u00f3dstopia oraz palcami (zw\u0142aszcza 4 i 5), osteopenia i deformacje ko\u015bci d\u0142ugich, niekiedy kraniosynostoza i zgrubienie pokrywy czaszki, ektopiczne kostnienie tkanek mi\u0119kkich oraz skr\u00f3canie dystalnego paliczka kciuka (od trzeciej do pi\u0105tej ko\u015bci \u015br\u00f3dr\u0119cza), napady t\u0119\u017cyczki i parestezji, po\u0142\u0105czone z oporno\u015bci\u0105 na dzia\u0142anie hormonu wzrostu, tyreotropiny, gonadotropin. W cz\u0119\u015bci przypadk\u00f3w wyst\u0119puje niepe\u0142nosprawno\u015b\u0107 intelektualna [ektualna .Zesp\u00f3\u0142 Carpentera  (OMIM \u2013 201000) uwarunkowany jest mutacjami w genie RAB23 (zmapowanyw regionie 6p12.1) lub genie MEGF8 (zmapowany w regionie 19q13.2), dziedziczy si\u0119 autosomalnie recesywnie. W obrazie choroby wyst\u0119puj\u0105 takie cechy jak: makrocefalia (wie\u017cowaty kszta\u0142t czaszki spowodowany nieprawid\u0142owym zarastaniem szw\u00f3w czaszkowych), syndaktylia sk\u00f3rna palc\u00f3w d\u0142oni i st\u00f3p, polidaktylia przedosiowa , niedos\u0142uch przewodzeniowy, wrodzone wady serca i niepe\u0142nosprawno\u015b\u0107 intelektualna. Oty\u0142o\u015b\u0107 wyst\u0119puje u wi\u0119kszo\u015bci chorych ju\u017c w pierwszych latach \u017cycia [ch \u017cycia .Zesp\u00f3\u0142 \u0142amliwego chromosomu X, FraX (OMIM \u2013 300624) jest spowodowany nadmiern\u0105 (>200) ekspansj\u0105 niestabilnych powt\u00f3rze\u0144 tr\u00f3jnukleotydowych CGG w promotorze genu FMR1, zmapowanym w regionie Xq27.3. Nale\u017cy do najcz\u0119stszych po zespole Downa przyczyn niepe\u0142nosprawno\u015bci intelektualnej u p\u0142ci m\u0119skiej. Choroba dziedziczy si\u0119 dominuj\u0105co w spos\u00f3b sprz\u0119\u017cony z chromosomem X. Charakterystyczn\u0105 pe\u0142n\u0105 ekspresj\u0119 objaw\u00f3w zespo\u0142u FraX obserwuje si\u0119 u os\u00f3b p\u0142ci m\u0119skiej . Kobiety nosicielki mutacji genu FMR1 wykazuj\u0105 \u0142agodn\u0105 ekspresj\u0119 objaw\u00f3w pod postaci\u0105 trudno\u015bci szkolnych (w oko\u0142o 50% przypadk\u00f3w). U niekt\u00f3rych chorych wyst\u0119puje ponadto nadmierny apetyt, oty\u0142o\u015b\u0107 i niskoros\u0142o\u015b\u0107. Co ciekawe, oty\u0142o\u015b\u0107 wyst\u0119puje tak\u017ce u os\u00f3b z delecjami obejmuj\u0105cymi gen FMR1 [gen FMR1 .Zesp\u00f3\u0142 MEHMO , Epileptic seizures \u2013 padaczka, Hypogenitalism \u2013 hipogenitalizm, Microcephaly \u2013 ma\u0142og\u0142owie, Obesity \u2013 oty\u0142o\u015b\u0107) (OMIM \u2013 300148) nale\u017cy do bardzo rzadko wyst\u0119puj\u0105cych zespo\u0142\u00f3w oty\u0142o\u015bci sprz\u0119\u017conej z chromosomem X, a jego nazwa jest akronimem wiod\u0105cych objaw\u00f3w zespo\u0142u. Locus genu choroby znajduje si\u0119 najprawdopodobniej w regionie Xp21.1-p22.13, natomiast gen warunkuj\u0105cy objawy nie jest dotychczas poznany. Znacznego stopnia oty\u0142o\u015b\u0107 rozwija si\u0119 w niemowl\u0119ctwie. Niekiedy opisywane s\u0105 znaczne op\u00f3\u017anienie rozwoju, niski wzrost, wzmo\u017cone napi\u0119cie mi\u0119\u015bniowe, wyg\u00f3rowane odruchy \u015bci\u0119gniste, oczopl\u0105s, nadmierna pobudliwo\u015b\u0107. Zwykle chorzy ch\u0142opcy umieraj\u0105 po 2 roku \u017cycia. Badania laboratoryjne wskazuj\u0105 na zaburzone dzia\u0142anie kompleks\u00f3w 1, 3 i 4 \u0142a\u0144cucha oddechowego w mitochondriach. Nieprawid\u0142owe, powi\u0119kszone mitochondria widoczne w obrazach z mikroskopii elektronowej kwalifikuj\u0105 zesp\u00f3\u0142 jako chorob\u0119 mitochondrialn\u0105 [ndrialn\u0105 .Badanie chromosom\u00f3w cz\u0142owieka umo\u017cliwi\u0142o poznanie przyczyn znanych wcze\u015bniej chor\u00f3b genetycznych. Stopniowy rozw\u00f3j technik badania chromosom\u00f3w, w tym FISH (@uorescencyjna hybrydyzacja in situ) i mikromacierzy (aCGH), umo\u017cliwi\u0142 wykrywanie submikroskopowych rearan\u017cacji chromosomowych, takich jak mikrodelecje i mikroduplikacje chromosomowe. Skutkiem mikrodelecji/mikroduplikacji chromosomowych jest brak funkcji/nadekspresja wielu przyleg\u0142ych gen\u00f3w znajduj\u0105cych si\u0119 w regionie utraconego lub zduplikowanego odcinka chromosomu. Skutki kliniczne zale\u017c\u0105 od wielko\u015bci zmiany, i jej zawarto\u015bci genetycznej, a w niekt\u00f3rych mikrodelecjach od ich rodzicielskiego pochodzenia (choroby imprintingowe) \u2212 np. zesp\u00f3\u0142 delecji 15q11-13. W wi\u0119kszo\u015bci przypadk\u00f3w zmiany tego typu powstaj\u0105 de novo, natomiast w niekt\u00f3rych s\u0105 one produktem zr\u00f3wnowa\u017conych rearan\u017cacji chromosomowych, odziedziczonych od jednego z rodzic\u00f3w.Trisomia chromosomu 21 pary w zespole Downa \u2013 typowym cechom zespo\u0142u  towarzyszy oty\u0142o\u015b\u0107, w okresie dzieci\u0119cym u oko\u0142o 15-50%, a w wieku doros\u0142ym u 45-90% chorych. Czynnikiem sprzyjaj\u0105cym oty\u0142o\u015bci jest zmniejszona aktywno\u015b\u0107 fizyczna wynikaj\u0105ca z hipotonii mi\u0119\u015bniowej, jak i zmniejszone spoczynkowe zu\u017cycie energii. U dzieci z zespo\u0142em Downa obserwowano podwy\u017cszone st\u0119\u017cenie leptyny, wydaje si\u0119 te\u017c, \u017ce do oty\u0142o\u015bci mo\u017ce istotnie przyczynia\u0107 si\u0119 niedoczynno\u015b\u0107 tarczycy [tarczycy .Zesp\u00f3\u0142 mikrodelecji 1p36 (OMIM \u2212 607862) \u2013 w obrazie klinicznym stwierdza si\u0119 cechy niepe\u0142nosprawno\u015bci intelektualnej zwykle w stopniu g\u0142\u0119bokim, brak ekspresji mowy, zaburzenia zachowania (agresja), mnogie wady strukturalne, g\u0142\u00f3wnie m\u00f3zgu (ma\u0142og\u0142owie u 38% chorych) i serca , du\u017ce p\u00f3\u017ano zarastaj\u0105ce ciemi\u0119 przednie cechy dysmorfii oraz oty\u0142o\u015b\u0107 w wieku dojrza\u0142ym [ojrza\u0142ym .Zesp\u00f3\u0142 mikrodelecji 16p11.2 (OMIM \u2212 611913) \u2013 jedna z pierwszych wykrytych mikrodelecji identy-fikowana jest u oko\u0142o 0,3-0,7% pacjent\u00f3w z cechami niepe\u0142nosprawno\u015bci intelektualnej i zaburze\u0144 zachowania. W jej obrazie klinicznym dominuj\u0105: op\u00f3\u017anienie rozwoju intelektualnego, zaburzenia poznawcze i behawioralne oraz zaburzenia zachowania , zaburzenia ze spektrum autyzmu, choroby psychiatryczne . W przypadku obj\u0119cia delecj\u0105 genu SH2B1, zaanga\u017cowanego w szlak sygna\u0142owy leptyny, dodatkowo pojawia si\u0119 nadmierny apetyt, wcze\u015bnie rozwijaj\u0105ca si\u0119 oty\u0142o\u015b\u0107 i ci\u0119\u017cka insulinooporno\u015b\u0107 [oporno\u015b\u0107 .Zesp\u00f3\u0142 mikrodelecji 6q16.2, obejmuje gen SIM1. Obraz kliniczny przypomina zesp\u00f3\u0142 Pradera i Williego (Prader-Willi-like phenotype), z hipotoni\u0105 i trudno\u015bciami w karmieniu w okresie noworodkowym z nast\u0119pow\u0105 oty\u0142o\u015bci\u0105 i g\u0142\u0119bokiego stopnia niepe\u0142nosprawno\u015bci\u0105 intelektualn\u0105 oraz kr\u00f3tkimi ko\u0144czynami, kt\u00f3rym niestale towarzysz\u0105: nadmierny apetyt, wady serca, nieznaczna dysmorfia twarzowa [twarzowa .Inne nowo opisane zespo\u0142y mikrodelecyjne s\u0105 zwi\u0105zane z regionem 13q34, z op\u00f3\u017anionym wzrastaniem i rozwojem psychoruchowym, \u0142agodn\u0105 dysmorfi\u0105 twarzow\u0105, oty\u0142o\u015bci\u0105 znacznego stopnia i niepe\u0142nosprawno\u015bci\u0105 intelektualn\u0105  oraz z rZesp\u00f3\u0142 mikroduplikacji 7q11.23 (OMIM \u2212 609757), obejmuj\u0105cej region krytyczny dla zespo\u0142u Williamsa. Nadekspresja genu YWHAG zlokalizowanego w regionie duplikacji jest zwi\u0105zana z oty\u0142o\u015bci\u0105, op\u00f3\u017anieniem rozwoju mowy, \u0142agodn\u0105 niepe\u0142nosprawno\u015bci\u0105 intelektualn\u0105, niekiedy wadami serca lub OUN, a cz\u0119sto problemami behawioralnymi, w tym ADHD, niepokojem zwi\u0105zanym z kontaktami spo\u0142ecznymi, zaburzeniami ze spektrum autyzmu [ autyzmu . W 2016  autyzmu .Oty\u0142o\u015b\u0107 jest te\u017c opisywana jako cecha w innych zespo\u0142ach mikrodelecyjnych, w tym 2p25, 2q37, 3pter, 7q22.1, 9q34, 10p15.3, 14q11.2, 14q12, 10q22, Xq27, a tak\u017ce z zespo\u0142ach mikroduplikacji 6q21-22, 6q27, 14q11.2 i Xq21 . W tabelNa rycinie 1 przedstawiono schemat post\u0119powania diagnostycznego w przypadku podejrzenia oty\u0142o\u015bci syndromicznej.Schemat post\u0119powania diagnostycznego w przypadku podejrzenia oty\u0142o\u015bci syndromicznej powinien uwzgl\u0119dnia\u0107 szczeg\u00f3\u0142ow\u0105 analiz\u0119 rodowodu i ocen\u0119 kliniczn\u0105  oraz analiz\u0119 wynik\u00f3w bada\u0144 dodatkowych: , ocen\u0119 okulistyczn\u0105 celem ukierunkowania procesu diagnostycznego. W\u015br\u00f3d bada\u0144 genetycznych w diagnostyce oty\u0142o\u015bci nale\u017cy uwzgl\u0119dni\u0107 kariotyp, FISH, testy metylacyjne, aCGH oraz sekwencjonowanie okre\u015blonych gen\u00f3w odpowiedzialnych za ekspresj\u0119 objaw\u00f3w podejrzewanego zespo\u0142u monogenowego. W przypadku, kiedy na podstawie oceny cech klinicznych, rodowodowych, wynik\u00f3w bada\u0144 biochemicznych, hormonalnych i radiologicznych istnieje podejrzenie wyst\u0105pienia choroby o wielogenowym uwarunkowaniu (np. zesp\u00f3\u0142 Bardeta i Biedla) mo\u017cna rozwa\u017cy\u0107 wykonanie bada\u0144 metod\u0105 sekwencjonowania nast\u0119pnej generacji (NGS) z zastosowaniem WES/zestaw TruSight One. W niekt\u00f3rych przypadkach nale\u017cy zastosowa\u0107 kilka uzupe\u0142niaj\u0105cych si\u0119 metod diagnostycznych  w celu ustalenia typu defektu molekularnego odpowiedzialnego za wyst\u0105pienie cech klinicznych choroby. Wyniki przeprowadzonych bada\u0144 stan\u0105 si\u0119 podstaw\u0105 do weryfikacji molekularnej rozpoznania klinicznego oraz oszacowania ryzyka genetycznego w rodzinie, co jest nieod\u0142\u0105cznym elementem poradnictwa genetycznego.W dalszych badania nad genetycznym uwarunkowaniem oty\u0142o\u015bci stosowano trzy r\u00f3\u017cne metody poszukiwa\u0144. Pierwsz\u0105 by\u0142o typowanie gen\u00f3w kandydackich zwi\u0105zanych z regulacj\u0105 metabolizmu w oparciu o badania na zwierz\u0119tach lub in vitro. Weryfikacja wytypowanych gen\u00f3w polega na por\u00f3wnaniu cz\u0119sto\u015bci wyst\u0119powania mutacji danego genu w grupie os\u00f3b z dan\u0105 cech\u0105 (w tym przypadku oty\u0142o\u015bci\u0105) a wyst\u0119powania mutacji danego genu w grupie os\u00f3b zdrowych. Ten spos\u00f3b by\u0142 z powodzeniem stosowany w identyfikacji gen\u00f3w odpowiadaj\u0105cych za powszechne choroby i cechy z\u0142o\u017cone. Jednak w przypadku oty\u0142o\u015bci nie uzyskano zadawalaj\u0105cych rezultat\u00f3w, mimo przebadania ponad 200 gen\u00f3w zwi\u0105zanych z regulacj\u0105 apetytu, metabolizmem glukozy i t\u0142uszcz\u00f3w oraz rozwojem tkanki t\u0142uszczowej .Drug\u0105 metod\u0105 by\u0142a analiza sprz\u0119\u017ce\u0144 polimorfizm\u00f3w mikrosatelitarnych rozsianych regularnie w ca\u0142ym genomie z regionami wi\u0105\u017c\u0105cymi si\u0119 ze zwi\u0119kszonym ryzykiem choroby. Ten spos\u00f3b umo\u017cliwi\u0142 identyfikacj\u0119 licznych gen\u00f3w odpowiedzialnych za choroby jednogenowe, jednak zestawienie wynik\u00f3w z 37 bada\u0144 na ponad 31 tysi\u0105cach os\u00f3b nie wskaza\u0142o na istnienie g\u0142\u00f3wnego genu odpowiedzialnego za oty\u0142o\u015b\u0107 u ludzi . Na og\u00f3\u0142Bardziej obiecuj\u0105ce wyniki przynios\u0142o zastosowanie asocjacji polimorfizm\u00f3w pojedynczych nukleotyd\u00f3w . W tej metodzie wykorzystuje si\u0119 dziesi\u0105tki tysi\u0119cy polimorficznych wariant\u00f3w DNA ograniczonych do pojedynczych nukleotyd\u00f3w (SNP) w celu wykrycia r\u00f3\u017cnic w cz\u0119sto\u015bci wyst\u0119powania u os\u00f3b chorych i zdrowych. Kolejne etapy pozwoli\u0142y wyr\u00f3\u017cni\u0107 kilkadziesi\u0105t gen\u00f3w lub region\u00f3w wyra\u017anie zwi\u0105zanych z oty\u0142o\u015bci\u0105. Pierwszy z nich ujawni\u0142 zwi\u0105zek wariantu genu FTO (Fat mass- and Obesity-associated), w drugim etapie wykryto zwi\u0105zek SNP w pobli\u017cu genu MC4R. W kolejnej turze bada\u0144 powi\u0105zano oty\u0142o\u015b\u0107 z polimorfizmem nukleotyd\u00f3w w obr\u0119bie lub w pobli\u017cu gen\u00f3w: SH2B1, KCTD15, TMEM18 oraz NEGR1. Kilka innych polimorfizm\u00f3w by\u0142o bliskich osi\u0105gni\u0119cia znamienno\u015bci statystycznej lub nie potwierdzono ich znaczenia przez r\u00f3\u017cne grupy badawcze . OstatecWyniki tych bada\u0144 nie doprowadzi\u0142y do wykrycia pojedynczego genu bezpo\u015brednio wywo\u0142uj\u0105cego oty\u0142o\u015b\u0107 u cz\u0142owieka, lecz raczej licznych zmian o niewielkim sumuj\u0105cym si\u0119 dzia\u0142aniu. Wiele z nich wykazuje dzia\u0142anie plejotropowe, zwi\u0119kszaj\u0105c ryzyko innych chor\u00f3b, w tym cukrzycy, zaburze\u0144 lipidowych. Szacuje si\u0119, \u017ce wp\u0142yw odkrytych licznych loci na BMI mo\u017ce odpowiada\u0107 ponad 20% zmienno\u015bci . Dalsze Cytowane powy\u017cej badania odzwierciedlaj\u0105 zakres z\u0142o\u017conej i bardzo trudnej problematyki zwi\u0105zanej z poszukiwaniem zwi\u0105zku przyczynowego pomi\u0119dzy identyfikowanymi w badaniach GWAS tysi\u0105cami wariant\u00f3w w r\u00f3\u017cnych loci w cz\u0119\u015bci koduj\u0105cej genomu, kt\u00f3rych efekt fenotypowy w odniesieniu do ryzyka oty\u0142o\u015bci pozostaje cz\u0119sto niepewny. Nie mo\u017cna wykluczy\u0107, \u017ce tzw. odziedziczalno\u015b\u0107 oty\u0142o\u015bci jest wyolbrzymiona, poniewa\u017c nie mo\u017cna \u015bci\u015ble oddzieli\u0107 znaczenia licznych czynnik\u00f3w modyfikuj\u0105cych, w tym czynnik\u00f3w \u015brodowiskowych dzia\u0142aj\u0105cych na organizm ju\u017c w trakcie \u017cycia p\u0142odowego (np. dieta wysokot\u0142uszczowa u matki w trakcie trwania ci\u0105\u017cy mo\u017ce implikowa\u0107 wyst\u0105pienie cech zespo\u0142u metabolicznego u potomstwa) jak r\u00f3wnie\u017c wp\u0142yw czynnik\u00f3w epigenetycznych, kt\u00f3re mog\u0105 by\u0107 badanie z zastosowaniem nowych technik diagnostycznych jakimi s\u0105 specyficzne mikromacierze genomowe umo\u017cliwiaj\u0105ce okre\u015blenie metylacji DNA okre\u015blonych region\u00f3w w genomie. Potwierdzaj\u0105 to wieloletnie badania epidemiologiczne, kliniczne i na zwierz\u0119tach do\u015bwiadczalnych wskazuj\u0105ce na rol\u0119 stresu oksydacyjnego w patogenezie oty\u0142o\u015bci poprzez zwi\u0119kszenie proliferacji preadipocyt\u00f3w. Oty\u0142o\u015b\u0107 sama w sobie mo\u017ce powodowa\u0107 og\u00f3lnoustrojowy stres oksydacyjny poprzez r\u00f3\u017cne mechanizmy biochemiczne zwi\u0105zane z fosforylacj\u0105 oksydacyjn\u0105. W\u015br\u00f3d czynnik\u00f3w kt\u00f3re przyczyniaj\u0105 si\u0119 do stresu oksydacyjnego w oty\u0142o\u015bci wymienia si\u0119 takie jak: hyperglikemia, podwy\u017cszone st\u0119\u017cenia lipid\u00f3w, niedobory minera\u0142\u00f3w i witamin, przewlek\u0142e infekcje, hyperleptynemia, zwi\u0119kszona aktywno\u015b\u0107 mi\u0119\u015bni dla zapobiegania wzrostu masy cia\u0142a. .Przedstawione w pracy informacje nie wyczerpuj\u0105 szerokiego zagadnienia jakim jest oty\u0142o\u015b\u0107 i jej genetyczne aspekty. Rozw\u00f3j nowoczesnych technologii w biologii molekularnej bioinformatyki, transkryptomiki, proteomiki, biochemii sprzyja w ostatniej dekadzie post\u0119powi w wiedzy na temat r\u00f3\u017cnych aspekt\u00f3w zwi\u0105zanych z oty\u0142o\u015bci\u0105 i jej powa\u017cnymi w skutkach nast\u0119pstwami zdrowotnymi. Wyniki dotychczasowych wielokierunkowych bada\u0144 jednocze\u015bnie wskazuj\u0105 na potrzeb\u0119 dalszego pog\u0142\u0119biania wiedzy w tej dziedzinie i ukierunkowuj\u0105 poszukiwanie skutecznej terapii farmakologicznej w przysz\u0142o\u015bci."}
{"text": "Wickerhamomyces  are isolated from various habitats and distributed throughout the world. Prior to this study, 35 species had been validly published and accepted into this genus. Beneficially, Wickerhamomyces species have been used in a number of biotechnologically applications of environment, food, beverage industries, biofuel, medicine and agriculture. However, in some studies, Wickerhamomyces species have been identified as an opportunistic human pathogen. Through an overview of diversity, taxonomy and recently published literature, we have updated a brief review of Wickerhamomyces. Moreover, two new Wickerhamomyces species were isolated from the soil samples of Assam tea (Camellia sinensis var. assamica) that were collected from plantations in northern Thailand. Herein, we have identified these species as W. lannaensis and W. nanensis. The identification of these species was based on phenotypic  and molecular analyses. Phylogenetic analyses of a combination of the internal transcribed spacer (ITS) region and the D1/D2 domains of the large subunit (LSU) of ribosomal DNA genes support that W. lannaensis and W. nanensis are distinct from other species within the genus Wickerhamomyces. A full description, illustrations and a phylogenetic tree showing the position of both new species have been provided. Accordingly, a new combination species, W. myanmarensis has been proposed based on the phylogenetic results. A new key for species identification is provided.Ascomycetous yeast species in the genus  Wickerhamomyces was first proposed by Kurtzman et al. . Moreover, W. anomalus and W. rabaulenis have been discovered in Antarctica (King George Island) [W. psychrolipolyticus has been discovered from Japan [Wickerhamomyces species have been successfully isolated from various habitats, as has been summarized in Most species of the genus analyses ,2,4,5,6.analyses . A phyloanalyses  has suggmarensis  should bspecies) . An incrhe world . It has y Europe  species,species) ,80,81,82 Island)  and Ocea Island) , respectom Japan . ConsequWickerhamomyces have been used in a variety of industries including the medicinal, agricultural, biofuel, food and beverage industries, and a number of others [W. anomalus for biotechnological applications. For example, the W. anomalus strains CBS261, HN006 and HN010 are capable of excessively producing ethyl acetate. As a result, this species has been used in the brewing of Baijiu (Chinese liquor) and in winemaking to improve the aroma and quality of the finished product [Many species of f others . Most pr product ,84,85.Wickerhamomyces anomalus strains BS91 and DMKU-RP04 could effectively inhibit plant pathogenic fungi and have been used as a biological control agent in agriculture [W. anomalus strains SDBR-CMU-S1-06 and Wa-32 have exhibited plant growth promotion potential by solubilizing insoluble minerals, producing indole-3-acitic acid (IAA) and siderophores, and by secreting various extracellular enzymes [W. anomalus are known to produce killer toxins that possess antimicrobial and larvicidal activities [Wickerhamomyces bovis and W. silvicola have been observed to produce mycocin, which exhibited fungicidal activity [W. lynferdii and W. sydowiorum have been recognized as relevant yeast species for the improvement of the fermentation processes for coffee cherries and cocoa, respectively [W. subpelliculosus has been used as an alternative to baker\u2019s yeast [W. chambardii could produce amylase and cellulase enzymes that could be used to produce bioethanol from corn straw [W. anomalus and W. edaphicus [W. siamensis [W. rabaulensis [W. psychrolipolyticus [W. mucosus [iculture ,87,88. N enzymes ,89. Moretivities ,91. Wickactivity ,93. In aectively ,94. Furt\u2019s yeast , while Wrn straw ,97. Prevdaphicus ,98,99, tiamensis , xylitolaulensis , cellulaolyticus  and the  mucosus  could beWickerhamomyces species  have been responsible for the spoilage of beer and bakery products [W. anomalus, W. myanmarensis and W. onychis have also been reported, but only with patients with serious illness [W. anomalus has been labeled a biosafety level 1 organism by the European Food Safety Authority [On the other hand, some products ,105,106. illness ,110,111.uthority  and is cWickerhamomyces species, namely W. anomalus, W. ciferrii, W. edaphicus, W. rabaulensis, W. siamensis, W. sydowiorum, W. tratensis and W. xylosicus, have been reported in Thailand [Camellia sinensis var. assamica) plantations in northern Thailand [Wickerhamomyces that represent potentially new species. In our present study, we have described them into two novel species. These two novel species are introduced based on their phenotypic  and molecular characteristics. To confirm their taxonomic status, phylogenetic relationship was determined by analysis of the combined sequence dataset of the D1/D2 domains of LSU and ITS sequences.Currently, only eight Thailand ,46,70,78Thailand ,114. In Thailand , we obtaC. sinensis var. assamica) plantations in Thep Sadej, Doi Saket District, Chiang Mai Province and Sri Na Pan, Muang District, Nan Province, northern Thailand [Five yeasts strains  isolated from soils of Assam tea . Final PCR products were sent to 1st Base Company Co., Ltd.,  for sequencing. The obtained sequences were used to query GenBank via BLAST .Each yeast strain was grown in 5 mL of yeast extract-malt extract broth in 18 \u00d7 180 mm test tubes with shaking at 150 rpm on an orbital shaker in the dark for two days. Yeast cells were harvested by centrifugation at 11,000 rpm and washed three times with sterile distilled water. Genomic DNA was extracted from yeast cells using DNA Extraction Mini Kit  following the manufacturer\u2019s protocol. The ITS region and D1/D2 domains of LSU gene were amplified by polymerase chain reactions (PCR) using ITS1/ITS4 primers  and NL1/Phylogenetic analysis was carried out based on the combined dataset of ITS and D1/D2 domains of LSU sequences. Sequences from this study along with those obtained from previous studies and the GenBank database were selected and provided in Wickerhamomyces strains (including 37 type strains obtained from either previous studies or the present study) and two sequences  of the outgroup  belonged to the genus Wickerhamomyces according to the phylogenetic results of Arastehfar et al. [W. nanensis), were clearly distinguished from the previously known species of Wickerhamomyces. Moreover, three yeast strains in this study, SDBR-CMU-S2-02, SDBR-CMU-S2-15, and CMU-S3-06 (described here as W. lannaensis) formed a sister clade to W. ochangensis with high support (BS = 100% and PP = 1.0).The sequences of five yeast strains were deposited in the GenBank database . The alioutgroup . Our phyr et al.  and Shimr et al. . MoreoveMycobank No.: 841356lannaensis\u201d refers to Lanna kingdom the historic name of northern Thailand, the collection locality of the type strain of the species.Etymology: \u201cC. sinensis var. assamica) plantation, May 2017, J. Kumla and N. Suwannarach, Holotype: Thailand, Chiang Mai Province, Thep Sadej, Doi Saket District, in soil from Assam tea (Description: The streak culture on YMA after two days at 30 \u00b0C is circular from (1\u20132 mm in diameter), white to cream color, smooth surface, dull-shining, entire margin, and raised elevation. After growth on YMA at 30 \u00b0C for two days, the cells are spheroidal to short ovoidal (3.6\u20133.8 \u00d7 2.4\u20132.6 \u00b5m), occur singly or in budding pairs. Pseudohyphae and true hyphae were absent. Ascospores were not obtained for individual strains and strain pairs on YMA, 5% MEA, PDA and V8 agar after incubation at 30 \u00b0C for one month. Urea hydrolysis and diazonium blue B reactions are negative. Fermentation tests, glucose is delayed positive, but galactose, maltose, sucrose, trehalose, melibiose, lactose, raffinose, and xylose are negative. D-glucose, D-xylose, rhamnose, cellobiose, salicin, inulin (weak), glycerol, D-glucitol, D-mannitol, D-glucono-1,5-lactone, D-gluconate, DL-lactate (weak), succinate, and ethanol are assimilated. No growth was observed in L-sorbose, N-acetyl glucosamine, D-ribose, L-arabinose, D-arabinose, sucrose, maltose, \u03b1,\u03b1-trehalose, \u03b1-methyl-D-glucoside, melibiose, lactose, raffinose, melezitose, soluble starch, erythritol, ribitol, galactitol, myo-inositol, 2-ketogluconic acid, 5-ketogluconic acid, D-glucuronate, D-galacturonic acid, citrate, methanol, and xylitol. For the assimilation of nitrogen compounds, growth on ammonium sulfate, potassium nitrate, sodium nitrite, ethylamine HCl, cadaverine, and creatine (weak) are positive and on L-lysine is latent positive.Growth in the vitamin-free medium is weak positive. Growth was observed at 15 \u00b0C and 30 \u00b0C, but not at 35, 37, 40, 42 and 45 \u00b0C. Growth in the presence of 50% glucose is positive, but growth in the presence of 0.01% cycloheximide, 0.1% cycloheximide, 60% glucose, 10% NaCl with 5% glucose and 15% NaCl with 5% glucose are negative. Acid formation is negative.C. sinensis var. assamica) plantation, September 2016, J. Kumla and N. Suwannarach, SDBR-CMU-S2-02, SDBR-CMU-S2-06.Additional strains examined: Thailand, Nan Province, Muang District, Sri Na Pan, in soil from Assam tea ; additional strains SDBR-CMU-S2-02  and SDBR-CMU-S2-06 .W. lannaensis formed a monophyletic clade in a well-supported clade and was found to be closely related to W. ochangensis  plantation, September 2016, J. Kumla, N. Suwannarach and S. Khuna, Holotype: Thailand, Nan Province, Muang District, Sri Na Pan, in soil from Assam tea (Description: The streak culture on YMA after two days at 30 \u00b0C is circular from (1\u20132 mm in diameter), white to cream color, smooth surface, dull-shining, entire margin, and raised elevation. After growth on YMA at 30 \u00b0C for two days, the cells are spheroidal to short ovoidal (3.8\u20134.0 \u00d7 2.4\u20132.5 \u00b5m), occur singly or in budding pairs. Pseudohyphae (4.8\u20136.9 \u00d7 2.2\u20132.9 \u00b5m) were produced in Dalmau plate culture on 5% MEA and PDA after 7 days at 25 \u00b0C, but true hyphae are not obtained. Ascospores were not observed for individual strains and strain pairs on YMA, 5% MEA, PDA and V8 agar after incubation at 30 \u00b0C for one month. Urea hydrolysis and diazonium blue B reactions are negative. Fermentation test, glucose is delayed positive, but galactose, maltose, sucrose, trehalose, melibiose, lactose, raffinose, and xylose are not positive. D-glucose, D-galactose, cellobiose, salicin, glycerol, D-mannitol, D-glucono-1,5-lactone, DL-lactate (weak), succinate, citrate, and ethanol are assimilated. No growth was observed in L-sorbose, N-acetyl glucosamine, D-ribose, D-xylose, L-arabinose, D-arabinose, rhamnose, sucrose, maltose, \u03b1,\u03b1-trehalose, \u03b1-methyl-D-glucoside, melibiose, lactose, raffinose, melezitose, inulin, soluble starch, erythritol, ribitol, D-glucitol, galactitol, myo-inositol, 2-ketogluconic acid, 5-ketogluconic acid, D-gluconate, D-glucuronate, D-galacturonic acid, methanol, and xylitol. For the assimilation of nitrogen compounds, growth on ammonium sulfate, potassium nitrate (weak), sodium nitrite (weak), ethylamine HCl, l-lysine, and creatine (slow) are positive, but cadaverine is not. Growth in the vitamin-free medium is weak. Growth was observed at 15 \u00b0C and 30 \u00b0C, but not at 35, 37, 40, 42 and 45 \u00b0C. Growth in the presence of 50% glucose and acid formation are positive, but growth in the presence of 0.01% cycloheximide, 0.1% cycloheximide, 60% glucose, 10% NaCl with 5% glucose, and 15% NaCl with 5% glucose are negative.C. sinensis var. assamica) plantation, September 2016, J. Kumla and N. Suwannarach, SDBR-CMU-S2-14.Additional strain examined: Thailand, Nan Province, Muang District, Sri Na Pan, in soil from Assam tea ; additional strain SDBR-CMU-S2-14 .W. nanensis were similar to W. chambardii. However, W. chambardii differed from W. nanensis by its ascospore formation and could not assimilate D-mannitol [W. nanensis and W. chambardii as different species. Moreover, W. nanensis formed a monophyletic clade in a well-supported clade and was separated from other Wickerhamomyces species  J. Kumla, N. Suwannarach and S. Lumyong, comb. nov.Mycobank No.: 841356Pichia myanmarensis Nagats., H. Kawas. and T. Seki, Int. J. Syst. Evol. Microbiol. 55: 1381, 2005.Basionym: P. myanmarensis, belongs to the genus Wickerhamomyces and has a close phylogenetic relationship with W. anomalus  a. Raffinose is assimilated\u2026\u2026\u2026\u2026.\u2026\u2026.\u2026..\u2026\u2026.\u2026.\u2026.\u2026.\u2026.\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u20263\u00a0\u00a0\u00a0\u00a0\u00a0\u2003W. kurtzmaniib. Raffinose is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026.3.(2) a. Citrate is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u20264\u00a0\u00a0\u00a0\u00a0\u00a0\u2003W. orientalisb. Citrate is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026.\u2026\u2026\u2026\u2026\u2026......\u2026.4.(3) a. Ribitol is assimilated\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265\u00a0\u00a0\u00a0\u00a0\u00a0\u2003W. spegazziniib. Ribitol is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u20265.W. edaphicus(4) a. Growth at 37 \u00b0C\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026.\u2026\u2026.\u00a0\u00a0\u00a0\u00a0\u00a0\u2003b. Growth is absent at 37 \u00b0C\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026..\u2026\u2026.\u2026\u2026\u2026\u2026\u202666.W. sydowiorum(5) a. Ascospores observed on 5% MEA\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026....\u2026\u00a0\u00a0\u00a0\u00a0\u00a0\u2003W. arborariusb. Ascospores not observed on 5% MEA\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026..\u2026\u2026.\u20267.(1) a. Raffinose is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20268\u00a0\u00a0\u00a0\u00a0\u00a0\u2003b. Raffinose is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026.198.(7) a. Nitrate is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026...\u2026\u2026\u2026\u2026.9\u00a0\u00a0\u00a0\u00a0\u00a0\u2003b. Nitrate is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026159.(8) a. L-Rhamnose is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026.10\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0b. L-Rhamnose is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..1210.W. ciferrii(9) a. L-Arabinose is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026...\u2026.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0b. L-Arabinose is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u20261111.W. psychrolipolyticus(10) a. Sucrose is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u00a0\u00a0W. xylosivorusb. Sucrose is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026...\u2026...12.(9) a. Growth in vitamin-free medium\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202613\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2003W. subpelliculosusb. Growth is absent in vitamin-free medium\u2026\u2026\u2026.\u2026..\u2026\u2026\u2026..13.(12) a. Soluble starch is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.14\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003W. lynferdiib. Soluble starch is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026....14.W. myanmarensis(13) a. D-Arabinose is assimilated\u2026\u2026\u2026\u2026\u2026..\u2026\u2026..\u2026.\u2026\u2026\u2026\u2026\u2026\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003W. anomalusb. D-Arabinose is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u202615.(8) a. Ribitol is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.16\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003b. Ribitol is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026.\u2026\u20261716.W. strasburgensis(15) a. Galactose is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.....\u2026\u2026\u2026\u2026\u2026.\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003W. rabaulensisb. Galactose is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026..\u2026\u2026\u202617.W. patagonicus(15) a. Growth in vitamin-free medium\u2026\u2026\u2026.\u2026\u2026\u2026....\u2026\u2026\u2026\u2026..\u2026\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003b. Growth is absent in vitamin-free medium\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026...1818.W. onychis(17) a. Citrate is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026...\u2026.\u2026.\u2026\u2026\u2026\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003W. siamensisb. Citrate is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026..\u2026\u2026\u202619.(7) a. 2-Keto-D-gluconate is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026.20\u00a0\u00a0\u00a0\u2003\u2003b. 2-Keto-D-gluconate is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u20262120.W. mucosus(19) a. D-Glucitol is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026.\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003W. xylosicusb. D-Glucitol is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026.21.(19) a. D-Arabinose is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202622\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003b. D-Arabinose is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.2322.W. sylviae(21) a. Growth at 37 \u00b0C\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026.\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003W. morib. Growth is absent at 37 \u00b0C\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026.23.(21) a. Galactose is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026...\u2026\u2026..24\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003b. Galactose is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026\u2026\u2026....\u2026\u2026\u2026\u2026\u2026.2724.W. silvicola(23) a. L-Arabinose is assimilated\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026.\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003b. L-Arabinose is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026\u2026\u2026....\u2026...\u2026\u2026.2525.W. scolytoplatypi(24) a. Sucrose is assimilated\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026.\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003b. Sucrose is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026.\u2026\u2026\u20262626.W. nanensis(24) a. D-Mannitol is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026.\u2026\u2026.\u2026\u2026\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003W. chambardiib. D-Mannitol is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026..\u2026.\u2026\u2026\u2026.\u2026....\u2026.27.W. pijperi(23) a. L-Sorbose is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026...\u2026..\u2026\u2026\u2026\u2026..\u2026\u2026...\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003b. L- Sorbose is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026.\u2026.\u2026\u2026\u2026\u2026\u20262828.(27) a. D-Xylose is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026.29\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003W. tratensisb. D-Xylose is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026..\u2026.\u2026\u2026\u2026\u2026.29.(28) a. Sucrose is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202630\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003b. Sucrose is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u20263630.(29) a. Cellobiose is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026....31\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003W. queroliaeb. Cellobiose is not assimilated\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.31.(30) a. D-Glucitol is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026.32\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003W. chaumierensisb. D-Glucitol is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026.32.(31) a. Growth at 37 \u00b0C\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026....33\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003b. Growth is absent at 37 \u00b0C\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u20263433.W. bovis(32) a. L-Arabinose is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.....\u2026\u2026\u2026..\u2026\u2026\u2026\u2026.\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003W. canadensisb. L-Arabinose is not assimilated\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026.34.(32) a. Nitrate is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026.35\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003W. hampshirensisb. Nitrate is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026...\u2026...35.W. bisporus(34) a. True hyphae are formed\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003W. alnib. True hyphae are not formed\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u202636.W. menglaensis(29) a. Citrate is assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026..\u2026\u2026\u2026..\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003b. Citrate is not assimilated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263737.W. ochangensis(36) a. Growth at 37 \u00b0C\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026..\u2026\u2026..\u2026..\u2026\u2026\u2026\u2026\u00a0\u00a0\u00a0\u00a0\u00a0\u2003\u2003W. lannaensisb. Growth is absent at 37 \u00b0C\u2026\u2026\u2026\u2026\u2026\u2026..\u2026\u2026\u2026...\u2026..\u2026\u2026.\u2026\u2026.Wickerhamomyces species are based primarily on phenotypical characteristics. These are further recognized as relevant morphological, biochemical, and physiological characteristics [Wickerhamomyces were originally classified into various yeast genera [Wickerhamomyces was proposed by Kurtzman et al. [Candida, Hansenula, Pichia, and Williopsis [Wickerhamomyces species. Species of the genus Wickerhamomyces are known to be widely distributed throughout the world and have been isolated from various habitats as shown in Wickerhamomyces consisted of 35 accepted and published species according to molecular phylogenetic analysis. Our phylogenetic results were similar to those of Arastehfar et al. [P. myanmarensis should be placed in the genus Wickerhamomyces. Consequently, we have proposed that this yeast species be named W. myanmarensis.Traditional methods of identification and characterization for the eristics ,128. Howt genera ,50,70,75lliopsis ,70. Therr et al.  who indiWickerhamomyces anomalus was first species reported in Thailand in 2002 [W. edaphicus has been discovered in Thailand [Wickerhamomyces species have been found [W. siamensis, W. tratensis, and W. xylosicus were only known to be from Thailand [Wickerhamomyces species, namely W. lannaensis and W. nanensis, that were isolated from soil collected from Assam tea plantations in northern Thailand were proposed based on identification through molecular phylogenetic and phenotypic  analyses. Therefore, effective identification of the Wickerhamomyces species has increased the number of species found in Thailand to 10 species and has led to 38 global species. This present discovery has increased the number of species of yeast known to be from Thailand and is considered important in terms of stimulating deeper investigations of yeast varieties in Thailand. Ultimately, these findings will help researchers gain a better understanding of the distribution and ecology of Wickerhamomyces.Yeast diversity has been investigated in various habitats throughout different regions of Thailand ,46,70,78Thailand . Until nen found ,70,78,82Thailand ,70,82. IWickerhamomyces have been investigated, and some strains have been used in a variety of biotechnology, food, and beverage industries, as well as in medical and agricultural fields [Wickerhamomyces species can survive in a variety of environments, climate change has had an impact on both the terrestrial biome and the aquatic environment. These environments are known to serve as habitats for a number of microorganisms [Wickerhamomyces. Therefore, in addition to studying the diversity and distribution of newly identified species, future research should focus on the effects of climate change on Wickerhamomyces.Many species of the genus l fields ,101,102.rganisms ,133,134"}
{"text": "Correction to: EJNMMI Radiopharm Chem 6, 11 (2021)https://doi.org/10.1186/s41181-021-00126-zFollowing publication of the original article ,\u201d, it should read \u201cCatalysts.\u201d-In page 8, 6-[18F]fluoro-3,4-dimethoxybenzyl bromide\u201d and \u201cSep-PakC18-Plus reactor.\u201d, it should read \u201c6-[18F]fluoro-3,4-dimethoxybenzaldehyde\u201d and \u201cSep-PakC18-Plus\u201d, respectively.-Pages 8\u20139, where it reads \u201cFig.\u00a08, where it reads \u201c22 or 23\u201d, it should read \u201c23 or 24\u201d and, number 24 under the structure, should be disregarded.-Page 9, \u201cKrasikova also prepared nitropiperonal12, using a combination of cPTC,22.\u201d it should read \u201cKrasikova also prepared 6-[18F]FDOPA , using a combination of22.\u201d-Page 9, where it reads \u201cusing cPTC as alkylation agent.\u201d it should read \u201cusing different catalysts.\u201d-Page 10, caption of Table\u00a02, where it reads -Page 10, Table\u00a02, line 1, where it reads \u201cPTC\u201d, should read \u201ccatalyst\u201d.18F]FDOPA\u201d.-Page 12, Table\u00a03, entry 3, where it reads \u201c31\u2009\u00b1\u20093\u201d, it should read \u201c31\u2009\u00b1\u20093*\u201d, and where it reads \u201cn.d. not described\u201d, it should read \u201cn.d. not described. *RCY for protected 6-[18F]FDOPA with 31 \u00b1 3% RCY\u201d it should read \u201cthe protected 6-[18F]FDOPA with 31 \u00b1 3% RCY\u201d.-Page 13, where it reads, \u201c6-.-There are multiple occurrences of [18F]. They should read fluoro-l-DOPA by Cu-mediated fluorination of a BPin precursor. Org Biomol Chem. 2019;17(38):8701\u20135.The original article (Neves et al."}
{"text": "Backusella chlamydospora sp. nov., B. koreana sp. nov., and B. thermophila sp. nov., as well as two new records, B. oblongielliptica and B. oblongispora, were found in Cheongyang, Korea, during an investigation of fungal species from invertebrates and toads. All species are described here using morphological characters and sequence data from internal transcribed spacer sequences of ribosomal DNA and large subunit of the ribosomal DNA. Backusella chlamydospora is different from other Backusella species by producing chlamydospores. Backusella koreana can be distinguished from other Backusella species by producing abundant yeast-like cells. Backusella\u00a0thermophila is characterized by a variable  columellae and grows well at 37 \u00b0C. Multigene phylogenetic analyses of the combined ITS and LSU rDNA sequences data generated from maximum likelihood and MrBayes analyses indicate that B. chlamydospora, B. koreana, and B. thermophila form distinct lineages in the family Backusellaceae. Detailed descriptions, illustrations, phylogenetic tree, and taxonomic key to the Backusella species present in Korea are provided.Three novel fungal species,  Backusella  was established by Ellis and Hesseltine in 1969 . . B. korerangiola . TherefoBackusella are saprobic and commonly isolated from soil and leaf litter [Backusella has not been involved in human infections [B. chlamydospora, B. koreana, and B. thermophila), and two newly recorded species (B. oblongielliptica and B. oblongispora) of Backusella were first discovered from invertebrate and toad samples. The presence of Backsuella species in different substrates reflects their ecological importance in ecosystems. Further studies on the fungal diversity of the niches are needed to understand the relationships between these organisms in ecosystems.Most species of f litter ,9. Some f litter ,8,13. Hofections . InteresBackusella, have not yet been described and that invertebrate samples may be a rich source of novel species of fungi.The identification of these novel species expands the range of potential habitats of the members of this genus. This suggests that unknown micro-fungi, especially those belonging to the genus Backusella species in KoreaKey to B. chlamydospora1. Chlamydospores present\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 1. Chlamydospores absent\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 2B. koreana2. Yeast-like cells present \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2. Yeast-like cells not observed\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 33. Sporangia commonly bigger than 150 \u00b5m diam., sporangiola not formed\u2026\u2026\u2026 53. Sporangia up to 130 \u00b5m diam., sporangiola formed\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 4B. circina4. Unispored sporangiola abundant \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20264. Unispored sporangiola rarely present, or not observed \u2026\u2026\u2026\u2026\u2026\u2026................... 6B. oblongielliptica5. Columellae subglobose to ellipsoidal; sporangiospores oblong to ellipsoidal, (28\u2013)30\u201340 \u00d7 12\u201315(\u201318) \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026B. oblongispora5. Columellae ellipsoidal to pyriform; sporangiospores ellipsoidal, 8.5\u201312 \u00d7 14\u201318 \u00b5m\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026............ B. locustae6. Sporangiospores globose to subglobose; no growth at 37 \u00b0C \u2026\u2026\u2026\u2026\u2026. B. thermophila6. Sporangiospores ellipsoidal; good growth and sporulation at 37 \u00b0C \u2026"}
{"text": "Both IGFBP\u20103\u2010 and TGF\u2010\u03b2\u2010stimulated PPase activities in cell lysates are abolished when cells are co\u2010treated with TGF\u2010\u03b2/IGFBP\u20103 antagonist or RAP (LRP\u20101/T\u03b2R\u2010V antagonist). However, the IGFBP\u20103\u2010stimulated PPase activity, but not TGF\u2010\u03b2\u2010stimulated PPase activity, is sensitive to inhibition by okadaic acid (OA). In addition, OA or PP2Ac siRNA reverses IGFBP\u20103 growth inhibition, but not TGF\u2010\u03b2 growth inhibition, in Mv1Lu and 32D cells. These suggest that IGFBP\u20103\u2010 and TGF\u2010\u03b2\u2010stimulated PPases are identical to PP2A and PP1, respectively. By Western blot/phosphorimager/immunofluorescence\u2010microscopy analyses, IGFBP\u20103 and TGF\u2010\u03b2 stimulate T\u03b2R\u2010V\u2010mediated IRS\u20102\u2010dependent activation and cytoplasm\u2010to\u2010nucleus translocation of PP2Ac and PP1c, resulting in dephosphorylation of p130/p107 and pRb, respectively, and growth arrest. Small molecule TGF\u2010\u03b2 enhancers, which potentiate TGF\u2010\u03b2 growth inhibition by enhancing T\u03b2R\u2010I\u2013T\u03b2R\u2010II\u2010mediated canonical signaling and thus activating T\u03b2R\u2010V\u2010mediated tumor suppressor signaling cascade (T\u03b2R\u2010V/IRS\u20102/PP1/pRb), could be used to prevent and treat carcinoma.The TGF\u2010\u03b2 type V receptor (T\u03b2R\u2010V) mediates growth inhibition by IGFBP\u20103 and TGF\u2010\u03b2 in epithelial cells and loss of T\u03b2R\u2010V expression in these cells leads to development of carcinoma. The mechanisms by which T\u03b2R\u2010V mediates growth inhibition (tumor suppressor) signaling remain elusive. Previous studies revealed that IGFBP\u20103 and TGF\u2010\u03b2 inhibit growth in epithelial cells by stimulating T\u03b2R\u2010V\u2010mediated IRS\u20101/2\u2010dependent activation and cytoplasm\u2010to\u2010nucleus translocation of IGFBP\u20103\u2010 or TGF\u2010\u03b2\u2010stimulated protein phosphatase (PPase), resulting in dephosphorylation of pRb\u2010related proteins  or pRb, and growth arrest. To define the signaling, we characterized/identified the IGFBP\u20103\u2010 and TGF\u2010\u03b2\u2010stimulated PPases in cell lysates and nucleus fractions in Mv1Lu cells treated with IGFBP\u20103 and TGF\u2010\u03b2, using a cell\u2010free assay with  A549 cellshuman Caucasian lung carcinoma cellsCDKcyclin\u2010dependent kinaseCHO cellsChinese hamster ovary epithelial cellsD32 cellsmurine 32D myeloid cellsEMTepithelial mesenchymal transitionIGF\u20101insulin\u2010like growth factor\u20101IGF\u20102insulin\u2010like growth factor\u20101IGFBP\u20103insulin\u2010like growth factor\u2010binding protein\u20103IRinsulin receptorIRS\u20101/2insulin receptor substrate\u20101/2LRP\u20101low density lipoprotein receptor\u2010related protein\u20101Mv1Lu cellsmink lung epithelial cellsOAokadaic acidp107p130, pRb\u2010related proteinsPAI\u20101plasminogen activator inhibitor\u20101PP1protein phosphatase 1cPP136\u2010kDa PP1 catalytic subunitPP2Aprotein phosphatase 2APP2A\u2010B56a 56\u2010kDa substrate\u2010recognition B subunit of PP2AcPP2A37\u2010kDa PP2A catalytic subunitPPaseprotein phosphatasepRbretinoblastoma protein (p105)RAPreceptor\u2010associated proteinsiRNAsmall interfering RNATGF\u2010\u03b2transforming growth factor\u2010\u03b2T\u03b2R\u2010Itype I TGF\u2010\u03b2 receptorT\u03b2R\u2010IItype II TGF\u2010\u03b2 receptorT\u03b2R\u2010IIItype III TGF\u2010\u03b2 receptorT\u03b2R\u2010Vtype V TGF\u2010\u03b2 receptor125\u03b2st to 65th of human TGF\u2010\u03b21TGF\u2010\u03b2 peptide antagonist containing amino acid residues 411d (0.3\u00a0\u00b5g/ml or 10\u00a0nM) for binding to T\u03b2R\u2010V,Insulin\u2010like growth factor\u2010binding protein\u20103 (IGFBP\u20103) is a growth regulator which exhibits IGF\u2010dependent and \u2010independent growth inhibitory activities in target cells.\u2212/\u2212 cells) and H1299 human non\u2010small cell lung carcinoma cells.\u2212/\u2212 cells express both T\u03b2R\u2010I and T\u03b2R\u2010II, and respond to TGF\u2010\u03b2\u2010stimulated T\u03b2R\u2010I/T\u03b2R\u2010II\u2010mediated transcriptional activation of extracellular matrix (ECM)\u2010related genes, such as PAI\u20101.\u2212/\u2212 cells exhibit a spindle\u2010shaped fibroblastoid morphology, frequently observed in invasive carcinoma cells.\u2212/\u2212 cells with T\u03b2R\u2010V/LRP\u20101 cDNA confers sensitivity to either TGF\u2010\u03b2 or IGFBP\u20103 growth inhibition and restores normal squamous epithelial morphology.IGFBP\u20103 and TGF\u2010\u03b2 do not inhibit growth in cells lacking T\u03b2R\u2010V, such as homozygous LRP\u20101\u2010deficient mouse embryonic fibroblasts (PEA\u201013 cells), CHO cells deficient in LRP\u20101  and pRb (p105) in the nucleus, respectively, in epithelial cells and growth arrest.We previously demonstrated that IGFBP\u20103 and TGF\u2010\u03b2 inhibit growth in epithelial cells by stimulating T\u03b2R\u2010V\u2010mediated tumor suppressor signaling which involves IRS\u20101/2\u2010dependent activation and cytoplasm\u2010to\u2010nucleus translocation of IGFBP\u20103\u2010 or TGF\u2010\u03b2\u2010stimulated protein phosphatase (PPase), and dephosphorylation of retinoblastoma family proteins in the nucleus, resulting in cell growth arrest.22.132P]ATP, [32P]\u2010orthophosphate and [methyl\u20103H] thymidine (67\u00a0Ci/mmol) were purchased from ICN Biochemicals . Okadaic acid (OA) was purchased from Tocris. IGFBP\u20103 and TGF\u2010\u03b21 (TGF\u2010\u03b2) were purchased from Peprotech. Insulin (A11382II) was purchased from Gibco. Primary antibodies against IRS\u20101 (sc\u2010398), IRS\u20102 (sc\u2010390761), PP1c  (sc\u20107482), pRb (p105) (sc\u201065230), p107 (sc\u2010250), p130 (sc\u2010374521), phosphorylated Smad2 (P\u2010Smad2) (sc\u2010135644), Ser 270\u2010phosphorylated IRS\u20101/2 (P\u2010IRS\u20101/2) (sc\u201017192), \u03b2\u2010actin (sc\u201047778), and lamin B (sc\u20106216) were purchased from Santa Cruz Biotechnology. Rabbit antibodies against N\u2010 and C\u2010terminal of human LRP\u20101 (T\u03b2R\u2010V) were purchased from Sigma Chemical Co. and Abcam, respectively. Rabbit polyclonal antibodies against hyperphosphoryrated Rb (P\u2010Rb) (#8516) and PP2Ac  (#2038) were purchased from Cell Signaling Technology. Alexa Fluor 488\u2010 and 594\u2010conjugated secondary antibodies were purchased from Thermo Fisher. Secondary antibodies conjugated with horseradish peroxidase  and enhanced chemiluminescence (ECL) kit (Perkin\u2010Elmer Life Sciences) were used to develop immunoblots. TGF\u2010\u03b2 peptide antagonist [\u03b2125], a dual TGF\u2010\u03b2/IGFBP\u20103 antagonist, was synthesized as previously described.All chemicals used in the experiments were prepared as a 10\u00a0mM stock solution in DMSO. The final concentration of DMSO in all experiments was 0.1% or lower, which had no effect on IGFBP\u20103 and TGF\u2010\u03b2 activity. Human receptor\u2010associated protein (RAP) was provided by Dr. Dudley K. Strickland .  ATP (200\u00a0cpm/pmol), 10\u00a0mM MgCl2, and 1.5\u00a0Unit/ml of the catalytic subunit of protein kinase A in a final volume of 3\u00a0ml. After overnight incubation at room temperature, the solution was filtered on a column (1.5\u00a0\u00d7\u00a020\u00a0cm) of Sephadex 50G equilibriated in 50\u00a0mM Tris\u2013HCl containing 10% glycerol and 1\u00a0mM benzamidine. Before stimulation with IGFBP\u20103 or TGF\u2010\u03b2, cells were treated with or without 25\u00a0\u00b5g/ml of RAP (receptor\u2010associated protein) and 30\u00a0\u00b5g/ml of TGF\u2010\u03b2 peptide antagonist (\u03b2125) in serum\u2010free DMEM or DMEM/Ham's F\u201012 medium for 10\u00a0min. The cells were stimulated with IGFBP\u20103 (0.6\u00a0\u00b5g/ml) or TGF\u2010\u03b2 (40\u00a0pM) for 3\u00a0hr. The cells were washed with cold phosphate\u2010buffered saline (PBS), detached with 50\u00a0mM Tris\u2013HCl pH 7.0 containing 0.25\u00a0M sucrose, 5\u00a0mM EDTA, and pelleted at 1,500\u00a0rpm for 5\u00a0min at 4\u00b0C. The cells were then lysed in 50\u00a0\u00b5l of homogenization buffer .32P\u2010labeled substrate in a final volume of 0.05\u00a0ml. Reactions in triplicates were initiated with the 32P\u2010labeled casein at 30\u00b0C, and after a 10\u00a0min reaction period, 0.1\u00a0ml of 10% trichloroacetic acid (TCA) was added. The mixture was centrifuged at 12,000\u00a0g for 2\u00a0min in a microcentrifuge. About 0.1\u00a0ml of the supernatant was then added to 1\u00a0ml scintillation counting liquid, and radioactivity was determined.The PPase activity assay mixtures were composed of 50\u00a0mM Tris\u2013HCl, pH 7.0 containing 10% glycerol, 1\u00a0mM benzamidine, 0.1\u00a0mM PMSF, 14\u00a0mM mercaptoethanol, 0.1\u00a0mg of bovine serum albumin (BSA), PPase\u2010containing sample (cell lysates or nucleus extracts containing 5\u00a0\u00b5g protein), and 2\u2013103\u00a0cpm; 200\u00a0cpm/pmol phosphate). This non\u2010specific PPase activity was subtracted from the total PPase activity in the cell lysates from cells treated with IGFBP\u20103 or TGF\u2010\u03b2 in order to estimate IGFBP\u20103\u2010stimulated or TGF\u2010\u03b2\u2010stimulated PPase activity. For this reason, the mean (\u00b1SD) of the non\u2010specific PPase activity from triplicates was taken as 0\u00a0cpm in cells treated with vehicle only.The lysates from cells treated with vehicle only exhibited non\u2010specific PPase activity (IGFBP\u20103\u2010 or TGF\u2010\u03b2\u2010independent PPase activity with certain ~102.4c and hyperphosphorylated pRb (P\u2010Rb), the images were analyzed in three dimensions using an AxioObserver Z1 Apotome microscope (Zeiss). Colocalization was evaluated in single optical planes taken through the entire z\u2010axis of each cell. All images were acquired using identical intensity and photodetector gain to allow quantitative comparisons of relative levels of immunoreactivity between samples. All images were cropped and sized using ImageJ.One milliliter of culture media containing approximately 5,000\u201310,000 Mv1Lu cells was added to a 35\u00a0mm culture dish containing a square coverslip. Mv1Lu cells grown on coverslips were treated with IGFBP\u20103 or TGF\u2010\u03b2. Cells were then fixed in 4% paraformaldehyde for 15\u00a0min followed by permeabilization. Fixed cells were blocked with 5% BSA in PBS for 20\u00a0min at room temperature (RT) and then incubated with an appropriate primary antibody solution overnight at 4\u00b0C. Fixed cells were incubated with Alexa Fluor\u2010conjugated secondary antibodies for 1\u00a0hr at RT. Samples were observed with a Zeiss AxioObserver Z1 microscope (Zeiss), and images were captured using AxioVision Rev 4.6 software. To determine the nuclear localization and the colocalization of PP12.52, 0.1\u00a0mM EDTA, 1\u00a0mM dithiothreitol (DTT), and 0.5\u00a0mM phenylmethylsulfonyl fluoride . The homogenates were centrifuged for 30\u00a0s at 500\u00a0g at 4\u00b0C to eliminate any unbroken tissue. The supernatants were incubated on ice for 20\u00a0min, vortexed for 30\u00a0s after the addition of 50\u00a0\u03bcL of 10% Nonidet P\u201040 , and then centrifuged for 1\u00a0min at 5,000\u00a0g at 4\u00b0C. The crude nucleus pellet was suspended in 200\u00a0\u03bcL of ice\u2010cold extraction buffer  and incubated on ice for 30\u00a0min, mixed frequently, and centrifuged at 12,000\u00a0g at 4\u00b0C for 15\u00a0min. The supernatants were collected as nucleus extracts for Western blot and PPase activity assay. Protein concentration was determined using a bicinchoninic acid assay kit with BSA as the standard .Nuclear extracts of the cells were prepared by hypotonic lysis followed by high salt extraction. Briefly, cell pellets were homogenized in 0.5\u00a0mL of ice\u2010cold lysis buffer, composed of 10\u00a0mM HEPES pH 7.9, 10\u00a0mM KCl, 2\u00a0mM MgCl2.6c siRNA oligonucleotide corresponding to nucleotide sequence 5\u2019\u2010xxx\u20103\u2019 (ON\u2010TARGETplus SMARTpool Cat #: L\u2010040657\u201300) and negative control siRNA were obtained from Dharmacon. PP2Ac siRNA and negative control siRNA were resuspended in in RNase\u2010free water and stored at \u221280\u00b0C. Transfection of siRNA was carried out using electroporation . Three million cells in 600\u00a0\u00b5l of RPMI 1640 were incubated with siRNA in a 0.4\u00a0cm cuvette for 5\u00a0min on ice before electroporation . After additional 5\u2010min incubation on ice, cells were re\u2010suspended in 12\u00a0ml of RPMI 1640 supplemented with glutamine and 10% FCS  without antibiotic. Antibiotics (1% penicillin/streptomycin) were added at 6\u00a0hr after electroporation. All measurements were performed at 24 or 72 hr after transfection.Murine PP2A2.7c siRNA knocked\u2010down 32D cells were determined by the measurement of [methy\u20103H] thymidine incorporation into cellular DNA as described previously.3H] thymidine incorporation into cellular DNA was determined by incubation of cells with [methy\u20103H] thymidine for 6\u00a0hr.Growth of OA\u2010treated Mv1Lu cells and PP2A2.86 cells) were lysed with 100\u00a0\u00b5l, 50\u00a0mM Tris\u2013HCl, pH 7.0 containing 1% Triton X\u2010100, 150\u00a0mM NaCl, 5\u00a0mM EDTA, and 0.1\u00a0mM PMSF. Cell lysates were subjected to 7.5% SDS\u2010PAGE and Western blotting using specific antibodies (Santa Cruz Biotechnology) as described previously.Seventy\u2010two hours after siRNA transfection, 32D and Mv1Lu cells (3\u00a0\u00d7\u00a0102.96 cells) grown in 6\u2010well plate were washed and incubated in phosphate\u2010free DMEM for 1\u00a0hr to deplete intracellular phosphate. After 2\u00a0hr of incubation with [32P] orthophosphate at 37\u00b0C in a CO2 incubator, cells were treated with 1\u00a0\u00b5g/ml of IGFBP\u20103 and/or OA (and RAP) for 16\u00a0hr. Cell lysates were prepared by suspending cells in 600\u00a0\u00b5l of lysis buffer and p130 or p107 was immunoprecipitated with a rabbit polyclonal antibody against the N\u2010terminal domain of p130 or p107. The p130 or p107 antibody complex was captured with a protein G\u2010coated agarose beads. The immunoprecipitated proteins were resolved using 7.5% SDS\u2010PAGE. The gel was dried and autoradiographed by a phosphorimager.Mv1Lu and 32D cells (3\u00a0\u00d7\u00a0102.10t\u2010test was used for determining the significance of a difference between two (vehicle only and sample) means. It was mainly used to compare the means between two groups (vehicle only and one specific concentration of IGFBP\u20103 or TGF\u2010\u03b2). The values were presented as mean\u00a0\u00b1\u00a0SD. p\u00a0<\u00a00.05 was considered significant.Two\u2010tailed unpaired Student's 33.132P\u2010phosphorylated casein, which was generated by 32P\u2010phosphate\u2010labeling (32P\u2010labeling) of casein (dephosphorylated) with protein kinase A in the presence of \u03b3\u201032P\u2010ATP, was incubated with cell lysates of Mv1Lu cells treated with or without IGFBP\u20103 or TGF\u2010\u03b21 (TGF\u2010\u03b2). After incubation, 32P\u2010phosphate released from 32P\u2010casein via the action of stimulated PPase in cell lysates and nucleus extracts were separated from remaining 32P\u2010casein by 10% trichloroacetic acid (TCA) precipitation in the presence of a carrier protein (BSA). The 32P\u2010phosphate released was recovered in the supernate of the 10% TCA solution. The IGFBP\u20103\u2010 and TGF\u2010\u03b2\u2010stimulated PPase activities were estimated by subtracting the radioactivity of 32P\u2010phosphate released by cell lysates or nucleus extracts of cells treated without IGFBP\u20103 or TGF\u2010\u03b2 from that released by cell lysates or nucleus extracts of cells treated with IGFBP\u20103 or TGF\u2010\u03b2. Using this assay, we characterized the kinetics, IGFBP\u20103 or TGF\u2010\u03b2 concentration dependence and OA sensitivity of the IGFBP\u20103\u2010 or TGF\u2010\u03b2\u2010stimulated PPase activity in Mv1Lu cells. As shown in Figure\u00a0d (50\u00a0pM) of TGF\u2010\u03b2 binding to T\u03b2R\u2010V3\u00a0cpm; 200\u00a0cpm/pmol phosphate)  Figure\u00a0. However) Figure\u00a0. OA at 1) Figure\u00a0. These r3.2125)\u2212/\u2212 cells, respectively). As shown in Figure\u00a0125 alone stimulated non\u2010specific (IGFBP\u20103\u2010 or TGF\u2010\u03b2\u2010independent) PPase activity  PPase activity which was statistically indifferent from that treated with RAP or \u03b2125 alone, suggesting that RAP or \u03b2125 completely abolished the IGFBP\u20103\u2010 and TGF\u2010\u03b2\u2010stimulated PPase activities  and did not exhibit IGFBP\u20103\u2010stimulated PPase activity. IGFBP\u20103 and RAP appeared to exert additive effects on stimulating non\u2010specific (IGFBP\u20103\u2010independent) PPase activity in CHO\u2010LRP\u20101\u2212/\u2212 cells  seen in wild\u2010type CHO\u2010K1 cells  which exhibited a spindle\u2010shaped fibroblastoid morphology, frequently observed in invasive carcinoma cells.\u2212/\u2212 cells (carcinoma cells) appeared to greatly upregulate non\u2010specific (IGFBP\u20103\u2010independent) PPase activity in these CHO\u2010LRP\u20101\u2212/\u2212 cells.The T\u03b2R\u2010V has been identified as the IGFBP\u20103 receptor which mediates its IGF\u2010independent growth inhibitory activity.y Figure\u00a0. Co\u2010treas Figure\u00a0. In Figus Figure\u00a0. Howevers Figure\u00a0. LRP\u20101  induced by IGFBP\u20103 and TGF\u2010\u03b2. As shown in Figure\u00a0c), a 65\u2010kDa scaffolding A subunit, and a 56\u2010kDa substrate\u2010recognizing B subunit (PP2A\u2010B56).c siRNA transfection on IGFBP\u20103\u2010induced growth inhibition in these murine 32D cells. This PP2Ac siRNA was developed based on the murine sequence. 32D cells were transfected with control siRNA, 2 and 4\u00a0nM PP2Ac siRNA by electroporation and treated with IGFBP\u20103. As shown in Figure\u00a0c siRNA (2 and 4\u00a0nM) reversed the growth inhibition induced by IGFBP\u20103   . This is consistent with the inability of murine PP2Ac siRNA to significantly downregulate mink PP2Ac  Figure\u00a0 as deter3.5c siRNA in murine 32D cells. These suggest that the IGFBP\u20103\u2010stimulated PPase is identical to PP2A. We hypothesized that IGFBP\u20103 induces growth inhibition by stimulating IRS\u20102\u2010dependent activation and cytoplasm\u2010to\u2010nucleus translocation of PP2Ac in Mv1Lu cells. To test this, Mv1Lu cells were treated with 0, 2, and 10\u00a0nM , IGFBP\u20103 for 2\u00a0hr. The cytoplasm and nucleus fractions in treated cells were then isolated and subjected to 7.5% SDS\u2010PAGE followed by Western blot analysis. As shown in Figure\u00a0c in the nucleus fraction by 1.5\u2010 to 1.7\u2010fold (n = 3) as compared to that in cells treated with vehicle only (0\u00a0nM IGFBP\u20103). These results suggest that IGFBP\u20103 promotes cytoplasm\u2010to\u2010nucleus translocation of PP2Ac (likely as the IRS\u20102 complex) in Mv1Lu cells.As described above, IGFBP\u20103\u2010stimulated PPase activity and IGFBP\u20103\u2010induced growth inhibition are blocked or reversed by co\u2010treatment with very low concentrations of OA in Mv1Lu and 32D cells and by transfection with PP2A3.632P\u2010orthophosphate at 37\u00b0C for 1\u00a0hr, washed, and incubated with 0.3\u00a0\u00b5g/ml (10\u00a0nM) IGFBP\u20103 in the presence of excess unlabeled orthophosphate in the medium. After 2\u00a0hr at 37\u00b0C, 32P\u2010labeled cell lysates were immunoprecipitated with specific antibodies to p130 and p107 and analyzed by 7.5% SDS\u2010PAGE and quantified by a phosphorimager . As shown in Figure\u00a0PP2A plays a critical multi\u2010faceted role in the regulation of the cell cycle. It has been implicated in dephosphorylation of two retinoblastoma protein (pRb)\u2010related proteins, p130 and p107, which interact primarily with E2F4 and E2F5 and are most active in G0\u2010the quiescent phase of the cell cycle.3.7c) and at least one regulatory subunit which directs PP1c to different substrates or sites. To test this, we performed immunofluorescence microscopy in Mv1Lu cells treated with 40\u00a0pM TGF\u2010\u03b2 at 37\u02daC for 0 and 1\u00a0hr using antibodies to T\u03b2R\u2010V (LRP\u20101) and PP1c , which is the PP1c\u2010dephosphorylated product of pRb in the nucleus. To demonstrate the subsequent cytoplasm\u2010to\u2010nucleus translocation of PP1c and its effect on dephosphorylation of pRb in the nucleus, we performed immunofluorescence analysis in Mv1Lu cells treated with 40\u00a0pM TGF\u2010\u03b2 at 37\u02daC for 0, 1, and 2\u00a0hr using specific antibodies to PP1c and hyperphosphorylated pRb (P\u2010Rb). We reasoned that TGF\u2010\u03b2 promotes nucleus accumulation of PP1c and should accordingly decrease the amount of P\u2010Rb, its target substrate, in the nucleus. PP1 specifically dephosphorylates pRb in the nucleus of target cells.c and P\u2010Rb in the nucleus  present in the cytoplasm and nucleus fractions are identified as hyperphosphorylated (as a slow\u2010migrating form) and hypophosphorylated (as a fast\u2010migrating form) forms of pRb, respectively, based on its mobility on 7.5% SDS\u2010PAGE.3.8c, PP2Ac, pRb (Rb), phosphorylated Smad2 (P\u2010Smad2), phosphorylated IRS\u20101/2 , lamin B, and \u03b2\u2010actin using 7.5% SDS\u2010PAGE and quantitative Western blot analysis with specific antibodies to PP1c, PP2Ac, and others after subcellular cytoplasm/nucleus fractionation of Mv1Lu and A549 cells treated with 40\u00a0pM TGF\u2010\u03b2 for 0, 1, and 2\u00a0hr. As shown in Figure\u00a0c , P\u2010IRS\u20102 and P\u2010Smad2 by 1.5\u2010 to 2\u2010fold (n\u00a0=\u00a03) in the nucleus fraction in Mv1Lu and A549 cells. In contrast, TGF\u2010\u03b2 did not significantly increase the amount of PP2Ac in the nucleus fraction in these cells. Interestingly, Rb present in the cytoplasm and nucleus fractions were identified as phosphorylated (as a slow\u2010migrating form) and dephosphorylated (as a fast\u2010migrating form) forms of Rb, respectively, based on its mobility on 7.5% SDS\u2010PAGE.c and P\u2010IRS\u20102, and correspondingly increases the amount of dephosphorylated Rb (as a fast\u2010migrating form of Rb), which is the PP1\u2010dephosphorylated product of Rb, in the nucleus.To further support the hypothesis that TGF\u2010\u03b2 stimulates cytoplasm\u2010to\u2010nucleus translocation of PP1, we determined the subcellular localization of PP1c Figure\u00a0, dephospc Figure\u00a0, P\u2010IRS\u20102c Figure\u00a0, and P\u2010Sc Figure\u00a0 in the nc Figure\u00a0e,h. Afte4c docking motifs (FxxR/KxR/K),Here, we have provided evidence revealing that IGFBP\u20103 inhibits growth in epithelial cells by stimulating the T\u03b2R\u2010V\u2010mediated growth inhibition signaling cascade which involves IRS\u20102, PP2A, and pRb\u2010related proteins, p130 or p107. We propose a revised version of our previously published model\u2212/\u2212 cells with LRP\u20101 (T\u03b2R\u2010V) cDNA restores the growth inhibitory response to IGFBP\u20103 and TGF\u2010\u03b2, and normal epithelial morphologySeveral lines of evidence suggest that IGFBP\u20103 acts as a potential tumor suppressor gene. First, aberrant promoter methylation of IGFBP\u20103 gene, which silences its expression, is detected in human gastric cancer, colorectal cancer, breast cancer, and malignant mesothelioma cancer.125\u2010labeled IGFBP\u20103 affinity labeling (binding/cross\u2010linking) followed by immunoprecipitation using antiserum to T\u03b2R\u2010V in epithelial cells and other cell types.125\u2010labeled IGFBPs affinity labeling.d of 10\u00a0nM.125, which contains a minimal active site motif of WS/CXD in TGF\u2010\u03b2 and IGFBP\u20103 molecules, blocks TGF\u2010\u03b2 and IGFBP\u20103 binding to TGF\u2010\u03b2 receptors in epithelial cells and reverses growth inhibition induced by either TGF\u2010\u03b2 or IGFBP\u20103 in these cells.d of IGFBP\u20103 binding to TMEM219/IGFBP\u20103R has been estimated to be 125\u00a0nM.T\u03b2R\u2010V is the only cell surface IGFBP\u20103 receptor identified by IThe transcriptional activation and growth inhibition activities of TGF\u2010\u03b2 have generally been thought to be mediated by the canonical T\u03b2R\u2010I/T\u03b2R\u2010II/Smad2/3/4 signaling cascade.c, resulting in dephosphorylation of pRb (p105) in the nucleus, as demonstrated by Western blot analysis following subcellular fractionation (to yield cytoplasm and nucleus fractions) and immunofluorescence analysis. (8) PP1 as well as PP2A are the master regulators of the eukaryotic cell cycle.We previously proposed a model for the mechanism by which TGF\u2010\u03b2 inhibits growth in epithelial cells by binding to a site between cell surface subdomains I and II of T\u03b2R\u2010V in target cells. In this model, TGF\u2010\u03b2 stimulates sequential association of IRS\u20101 or IRS\u20102 and a Ser/Thr\u2010specific PPase with the cytoplasmic tail of T\u03b2R\u2010V by inducing T\u03b2R\u2010V dimerization via its covalently linked homodimeric structure. In the ternary complexes, the Ser/Thr\u2010specific PPase becomes activated and dephosphorylate IRS\u20101/2. Dephosphorylated IRS\u20101\u2010PPase or IRS\u20102\u2010PPase binary complexes dissociate from the cytoplasmic tail of T\u03b2R\u2010V and undergo IRS\u20101/2\u2010dependent translocation from cytoplasm to the nucleus where the PPase dephosphorylates pRb (retinoblastoma protein) or pRb\u2010related proteins, resulting in growth arrest. This model lacked the identity of PPase, IRS\u20101/2, and retinoblastoma family proteins which are involved in TGF\u2010\u03b2\u2010stimulated T\u03b2R\u2010V\u2010mediated tumor suppressor (growth inhibition) signaling.c docking motifs of FrssfR (residues 438\u2013443) and FefRpR (residues 298\u2013303), respectively (PP1c docking affinity: IRS\u20102\u00a0>\u00a0IRS\u20101). IRS\u20102 appears to comprise of overlapping high\u2010affinity PP1 and PP2A docking motifs of FefRpR (residues 298\u2013303) and LkeLfE (residues 294\u2013299), respectively, suggesting that PP1 and PP2A docking to IRS\u20102 are mutually exclusive. After dephosphorylation of IRS\u20102 by activated PP1 in the ternary complex, dephosphorylated IRS\u20102\u2010PP1 binary complexes dissociate from the cytoplasmic tail of T\u03b2R\u2010V and enter the nucleus via the nucleus\u2010targeting function of IRS\u20102. In the nucleus, PP1 dephosphorylates pRb (p105), resulting in cell growth arrest. In normal epithelial cells, TGF\u2010\u03b2 potently inhibits cell growth (~100% growth inhibition at 1\u20135\u00a0pM) by stimulating the T\u03b2R\u2010V\u2010mediated signaling cascade (T\u03b2R\u2010V/IRS\u20102/PP1/pRb) to dephosphorylate (activate) pRb by PP1 in concert with canonical TGF\u2010\u03b2 signaling (T\u03b2R\u2010I/T\u03b2R\u2010II/Smad2/3/4).c itself possesses a high\u2010affinity PP2A\u2010B56 docking motif of LlrLfE (residues 82\u201387), on the way to bind to T\u03b2R\u2010I (Figure\u00a0c also recruits PP2A to form the high\u2010affinity\u2010bound T\u03b2R\u20101\u2010PPc\u2010PP2A complex. PP2A recruited by T\u03b2R\u20101 as the high\u2010affinity\u2010bound PP1\u2010PP2A complexc\u2010PP2A by T\u03b2R\u2010I to suppress non\u2010Smad signaling.In this model Figure\u00a0, TGF\u2010\u03b2, I Figure\u00a0, PP1c alTGF\u2010\u03b2 is known to act as a tumor suppressor and a tumor promoter during tumorigenesis. The mechanism of switching TGF\u2010\u03b2 from a tumor suppressor to a tumor promoter in the process of tumorigenesis remains unclear.Activation of tumor suppressors or their signaling for the treatment of human cancers has been a long sought, yet elusive, strategy for an effective therapy.The authors declare that there are no competing financial interest associated with this work.C.L.C. performed most of the experiments in the Department of Biochemistry and Molecular Biology, Saint Louis University School of Medicine. F.W.H. was involved in identifying the negative regulation of non\u2010Smad pathways by T\u03b2R\u2010V. S.S.H and J.S.H. were involved in research designing and finalizing the manuscript."}
{"text": "Poephila cincta) at fledgling and adult stages. Our results revealthat in both the fledgling and the adult, cranial musculature shows clear and complexpartitioning in the Musculus adductor mandibulae externus that isconsistent with other families within Passeriformes. We quantified jaw-muscle sizes andfound that the adult showed a decrease in muscle mass in comparison to the fledglingindividual. We propose that this could be the result of low sample size or a physiologicaleffect of parental care in Passeriformes. Our study shows that high-resolutionvisualization techniques are informative at revealing morphological discrepancies forstudies that involve small specimens such as Passeriformes especially with carefulspecimen selection criteria.Synopsis Dietary requirements and acquisition strategies change throughout ontogenyacross various clades of tetrapods, including birds. For example, birds hatch withcombinations of various behavioral, physiological, and morphological factors that placethem on an altricial\u2013precocial spectrum. Passeriformes (=songbirds) in particular, afamily constituting approximately more than half of known bird species, displays the mostdrastic difference between hatchling and adults in each of these aspects of their feedingbiology. How the shift in dietary resource acquisition is managed during ontogenyalongside its relationship to the morphology of the feeding apparatus has been largelyunderstudied within birds. Such efforts have been hampered partly due to the small size ofmany birds and the diminutive jaw musculature they employ. In this study, we used standardand diffusible iodine-based contrast-enhanced computed tomography in conjunction withdigital dissection to quantify and describe the cranial musculature of the Black-throatedFinch  v\u00e0 v\u1edbi ch\u1ee5p c\u1eaft l\u1edbp t\u0103ng c\u01b0\u1eddng \u0111\u1ed9 t\u01b0\u01a1ng ph\u1ea3n d\u1ef1a tr\u00ean i-\u1ed1t khu\u1ebfch t\u00e1n , b\u00ean c\u1ea1nh v\u1edbi vi\u1ec7c gi\u1ea3i ph\u1eabuk\u1ef9 thu\u1eadt s\u1ed1, \u0111\u1ec3 \u0111\u1ecbnh l\u01b0\u1ee3ng v\u00e0 m\u00f4 t\u1ea3 h\u1ec7 c\u01a1 \u0111\u1ea7u c\u1ee7a lo\u00e0i Di c\u1ed5 \u0111en  \u1edf giai \u0111o\u1ea1n r\u1eddi t\u1ed5 v\u00e0 khi tr\u01b0\u1edfng th\u00e0nh. K\u1ebft qu\u1ea3 ph\u00e2nt\u00edch cho th\u1ea5y, \u1edf c\u1ea3 chim non m\u1edbi r\u1eddi t\u1ed5 v\u00e0 chim tr\u01b0\u1edfng th\u00e0nh, h\u1ec7 c\u01a1 \u0111\u1ea7u c\u00f3 s\u1ef1 ph\u00e2n chiaph\u1ee9c t\u1ea1c v\u00e0 r\u00f5 r\u1ec7t t\u1ea1i b\u00f3 c\u01a1 c\u1eafn ngo\u00e0i (Musculus adductor mandibulaeexternus), t\u01b0\u01a1ng t\u1ef1 c\u00e1c h\u1ecd chim kh\u00e1c trong b\u1ed9 S\u1ebb. Khi \u0111\u1ecbnh l\u01b0\u1ee3ng k\u00edch th\u01b0\u1edbc c\u1ee7ab\u00f3 c\u01a1 n\u00e0y, ch\u00fang t\u00f4i ph\u00e1t hi\u1ec7n kh\u1ed1i l\u01b0\u1ee3ng c\u01a1 c\u1ee7a c\u00e1 th\u1ec3 tr\u01b0\u1edfng th\u00e0nh th\u1ea5p h\u01a1n c\u1ee7a c\u00e1 th\u1ec3m\u1edbi r\u1eddi t\u1ed5. \u0110\u00e2y c\u00f3 th\u1ec3 l\u00e0 h\u1ec7 qu\u1ea3 c\u1ee7a c\u1ee1 m\u1eabu nh\u1ecf ho\u1eb7c l\u00e0 m\u1ed9t t\u00e1c \u0111\u1ed9ng sinh l\u00fd trong qu\u00e1tr\u00ecnh ch\u0103m con c\u1ee7a th\u00e0nh vi\u00ean b\u1ed9 S\u1ebb. Nghi\u00ean c\u1ee9u c\u1ee7a ch\u00fang t\u00f4i cho th\u1ea5y c\u00e1c k\u1ef9 thu\u1eadt tr\u1ef1cquan h\u00f3a v\u1edbi \u0111\u1ed9 ph\u00e2n gi\u1ea3i cao r\u1ea5t h\u1eefu d\u1ee5ng trong vi\u1ec7c ph\u00e2n bi\u1ec7t h\u00ecnh th\u00e1i \u1edf nh\u1eefng nghi\u00eanc\u1ee9u d\u00f9ng m\u1eabu v\u1eadt c\u00f3 k\u00edch th\u01b0\u1edbc nh\u1ecf nh\u01b0 chim thu\u1ed9c b\u1ed9 S\u1ebb, nh\u1ea5t l\u00e0 khi c\u00f3 ph\u01b0\u01a1ng ph\u00e1p l\u1ef1ach\u1ecdn m\u1eabu v\u1eadt k\u0129 l\u01b0\u1ee1ng.Nhi\u1ec1u nh\u00e1nh \u0111\u1ed9ng v\u1eadt c\u00f3 t\u1ee9 chi, bao g\u1ed3m c\u1ea3 chim, c\u00f3 nhu c\u1ea7u dinh d\u01b0\u1ee1ng v\u00e0 chi\u1ebfn thu\u1eadtki\u1ebfm \u0103n li\u00ean t\u1ee5c thay \u0111\u1ed5i xuy\u00ean su\u1ed1t qu\u00e1 tr\u00ecnh sinh tr\u01b0\u1edfng c\u1ee7a ch\u00fang. V\u00ed d\u1ee5, chim non n\u1edfd\u01b0\u1edbi s\u1ef1 t\u00e1c \u0111\u1ed9ng c\u1ee7a c\u00e1c y\u1ebfu t\u1ed1 h\u00e0nh vi, sinh l\u00fd, v\u00e0 h\u00ecnh th\u00e1i \u0111\u1ec3 x\u00e1c \u0111\u1ecbnh th\u1eddi \u0111i\u1ec3m ch\u00fangc\u00f3 th\u1ec3 t\u1ef1 ki\u1ebfm \u0103n. B\u1ed9 S\u1ebb  Throughout ontogeny, shifts in food resource acquisition are common across vertebrates. Such chThe avian feeding apparatus is part of a kinetic skull held together by ligaments and withmotions powered by a series of muscles  that enhPoephila cincta), a seed-eating songbird in theOld-World tropics and Australasia with a short, thick, and conical bill. Juvenile orsub-adult birds rapidly achieve average adult size or larger as a combined result of thesteady diet from their parents while not having to perform many energy-consuming tasks, suchas flying and active foraging  were each acquired as deceased individuals from the Tulsa Zoo, Inc.. The fledgling had not molted to adult plumage and was\u223c50\u2009days from hatching at time of death based on records kept by the Tulsa Zoo, whereas theadult black-throated finch had mature plumage and was >180\u2009days old at the time of death.The finch specimens were initially stored frozen. All specimens were chemically fixed in 10%neutral buffered formalin for \u223c2\u2009weeks. Specimens were then pre-stain CT-scanned to captureskull morphologies, using grayscale thresholding in Avizo\u00a9 version9.0, 9.3, and 9.5  to generate skeletal models.Computed tomographic data were collected on a GE phoenixv|tome|x s240 high-resolution microfocusCT system  at the American Museum of Natural HistoryMicroscopy and Imaging Facility  and on a Nikon XTH 225 ST high-resolutionmicrofocus CT system  at DENTSPLY\u2019s Research and Design Facility .All unstained specimens were scanned at resolutions of <70-\u03bc isometric voxel sizes toobtain the degree of detail necessary to identify bony landmarks. All scanning parametersare listed in One fledgling (unknown sex) and one adult female black-throated finch (2KI) for 10 or 14\u2009days  of Lugol\u2019s iodine  to calculate the mass of each jaw muscle. The left-sidemusculature was reconstructed in both specimens for consistency. The following muscles were3D rendered for the black-throated finch based on work by Musculus adductor mandibulae externus caudalis(MAMEC); M. adductor mandibulae externus rostralis lateralis (MAMERL);M. adductor mandibulae externus rostralis medialis (MAMERM); M.adductor mandibulae externus rostralis temporalis (MAMERT); M. adductormandibulae externus ventralis (MAMEV); M. adductor mandibulaecaudalis (MAMC); M. depressor mandibulae (MDM); M.protractor pterygoideus (MPP); M. protractor quadratus (MPQ);M. pseudotemporalis profundus (MPsP); M. pseudotemporalissuperficialis (MPsS); M. pterygoideus dorsalis (MPtD); andM. pterygoideus ventralis (MPtV) , using the \u201cMeasurements\u201d tool. During image-stack processing, weutilized Fiji  to crop, rotate, and re-slicethe global axes of the image stack so that they were orthogonal in the standard anatomicalplanes. Following segmentation of the pre-injection skeletal scans, the diceCT image stackswere processed secondarily. The anatomy of the skull and left-side jaw musculature wasmanually reconstructed in Avizo based on grayscale value differences. Each muscle was firstdelineated in the plane that was easiest to discern and evaluate, which in this case was thetransverse plane, to differentiate it from adjacent muscle bellies before more thoroughsegmentation was performed. The initial step generated a tubular schematic of muscles andtheir attachment points. A more thorough segmentation was then performed on each muscleusing a combination of the \u201cBrush\u201d tool and the \u201cInterpolation\u201d tool. When the thoroughsegmentation in one plane of view was performed, at least one other view was simultaneouslymonitored in order to identify, corroborate, and confirm the muscle boundaries seen in theprimary plane view. Muscle boundaries were determined based on sharp differentiation betweengrayscale values that usually denotes muscles and dense, unstained connective tissues ormuscles and bones/cartilage  and os opticus  fiber morphologies  in comparison to thefledgling specimen of the same species. Several lines of reasoning supported that the CTimage stack of the adult finch did not have any apparent distortion that could be due tomuscle shrinkage from the iodine staining. In the scan, the muscle fibers did not appearstraightened and \u201crigid,\u201d and no prominent gaps were apparent between adjacent muscle fiberbundles. The space between the jaw muscles of the fledgling were more prominent than in theadult, but the fledgling\u2019s brain tissues are still flushed against the cranial cavity see.LikewisOther factors that could have contributed to this mass difference might be intraspecificdifferences such that the adult female we sampled was a particularly small individual orthat the fledgling was a particularly large individual. Because this species is not reportedas sexually dimorphic, we interpret that the sex of the fledgling should not have been afactor regarding overall mass differences. However, breeding conditions might be animportant consideration for the adult. Other female songbirds, such as house wrens, havebeen reported to lose mass during breeding season either for contributing body tissues tooffspring production  or due tBetter criteria for specimen collection, including information about sex and in what seasonthe specimen was collected, is stressed for collecting small songbirds because slightdifferences can contribute to apparently significant changes. Further research shouldincorporate additional specimens \u2014including hatchlings, additionalfledglings, and male and female adults\u2014to fully explore these findings. These factors arealso important considering that digital dissection is a time-consuming process, and the lownumber of individuals in this study limits some interpretations. Digital dissection speedcan be improved by utilizing interpolation tools  along wiobab007_Supplementary_DataClick here for additional data file."}
{"text": "Motivation: SNP-based kinship analysis with genome-wide relationship estimation and IBD segment analysis methods produces results that often require further downstream process- ing and manipulation. A dedicated software package that consistently and intuitively imple- ments this analysis functionality is needed.Results: Here we present the skater R package for SNP-based kinship analysis, testing, and evaluation with R. The skater package contains a suite of well-documented tools for importing, parsing, and analyzing pedigree data, performing relationship degree inference, benchmarking relationship degree classification, and summarizing IBD segment data.Availability: The skater package is implemented as an R package and is released under the MIT license at https://github.com/signaturescience/skater. Documentation is available at https://signaturescience.github.io/skater. R version: R version 4.0.4 (2021-02-15)skater package version: 0.1.0 or KING, and methods that rely on identity by descent (IBD) segment detection, such as GERMLINE, hap-IBD, and IBIS. Recent efforts focusing on benchmarking these methods have been aided by tools for simulating pedigrees and genome-wide SNP data. Analyzing results from genome-wide SNP-based kinship analysis or comparing analyses to simulated data for benchmarking have to this point required writing one-off analysis functions or utility scripts that are seldom distributed with robust documentation, test suites, or narrative examples of usage. There is a need in the field for a well-documented software package with a consistent design and API that contains functions to assist with downstream manipulation, benchmarking, and analysis of SNP-based kinship assessment methods. Here we present the skater package forSNP-basedkinshipanalysis,testing, andevaluation withR.Inferring familial relationships between individuals using genetic data is a common practice in population genetics, medical genetics, and forensics. There are multiple approaches to estimating relatedness between samples, including genome-wide measures, such as those implemented in PlinkUse Cases section below. The package adheres to \u201ctidy\u201d data analysis principles, and builds upon the tools released under the tidyverse R ecosystem.The skater package provides an intuitive collection of analysis and utility functions for SNP-based kinship analysis. Functions in the package include tools for importing, parsing, and analyzing pedigree data, performing relationship degree inference, benchmarking relationship degree classification, and summarizing IBD segment data, described in full in thehttp://CRAN.R-project.org/package=skater. Users can install skater in R by executing the following code:install.packages(\"skater\")The skater package is hosted in the Comprehensive R Archive Network (CRAN) which is the main repository for R packages:https://github.com/signaturescience/skater. The development version may contain new features which are not yet available in the version hosted on CRAN. This version can be installed using theinstall_github function in the devtools package:install.packages(\"devtools\")devtools::install_githubAlternatively, the development version of skater is available on GitHub atWhen installing skater, other packages which skater depends on are automatically installed, including magritr, tibble, dplyr, tidyr, readr, purrr, kinship2, corrr, rlang, and others. that many R users will already have installed. Use cases are demonstrated in detail below. In summary, the skater package has functions for:\u2022Reading in various output files produced by commonly used tools in SNP-based kinship analysis\u2022Pedigree parsing, manpulation, and analysis\u2022Relationship degree inference\u2022Benchmarking and assessing relationship classification accuracy\u2022IBD segment analysis post-processingMinimal system requirements for installing and using skater include R (version 3.0.0 or higher) and several tidyverse packageshttps://signaturescience.github.io/skater/.A comprehensive reference for all the functions in the skater package is available atskater package provides a collection of analysis and utility functions forSNP-basedkinshipanalysis,testing, andevaluation as anR package. Functions in the package include tools for working with pedigree data, performing relationship degree inference, assessing classification accuracy, and summarizing IBD segment data.library(skater)The and other genetic analysis tools is the.fam file. A.fam file is a tabular format with one row per individual and columns for unique IDs of the mother, father, and the family unit. The package includesread_fam to read files in this format:famfile <- system.filefam <- read_fam(famfile)fam##\u2003#\u2003A tibble:\u200364 x 6##\u2003\u2003\u2003fid\u2003\u2003\u2003 id\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 dadid\u2003\u2003\u2003\u2003\u2003\u2003\u2003 momid\u2003\u2003\u2003\u2003\u2003\u2003 sex\u2003affected##\u2003\u2003\u2003<chr>\u2003\u2003 <chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <chr>\u2003\u2003\u2003\u2003\u2003 <int>\u2003\u2003 <int>## 1\u2002\u2003testped1 testped1_g1-b1-s1\u20030\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 0\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 1\u2003\u2003\u2003\u2003 1## 2\u2002\u2003testped1 testped1_g1-b1-i1\u20030\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 0\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 2\u2003\u2003\u2003\u2003 1## 3\u2002\u2003testped1 testped1_g2-b1-s1\u20030\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 0\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 1\u2003\u2003\u2003\u2003 1## 4\u2002\u2003testped1 testped1_g2-b1-i1 testped1_g1-b1-s1 testped1_g1-b1-i1\u00a02\u2003\u2003\u2003\u2003 1## 5\u2002\u2003testped1 testped1_g2-b2-s1\u20030\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 0\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 1\u2003\u2003\u2003\u2003 1## 6\u2002\u2003testped1 testped1_g2-b2-i1 testped1_g1-b1-s1 testped1_g1-b1-i1\u2002 2\u2003\u2003\u2003\u2003 1## 7\u2002\u2003testped1 testped1_g3-b1-i1 testped1_g2-b1-s1 testped1_g2-b1-i1\u2002 2\u2003\u2003\u2003\u2003 1## 8\u2002\u2003testped1 testped1_g3-b2-i1 testped1_g2-b2-s1 testped1_g2-b2-i1\u2002 1\u2003\u2003\u2003\u2003 1## 9\u2002\u2003testped2 testped2_g1-b1-s1\u20030\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 0\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 2\u2003\u2003\u2003\u2003 1## 10\u2003testped2 testped2_g1-b1-i1\u20030\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 0\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 1\u2003\u2003\u2003\u2003 1##\u2003#\u2026 with 54 more rowsPedigrees define familial relationships in a hierarchical structure. One of the common formats used by PLINK.fam formated files can then be translated to thepedigree structure used by thekinship2 package. The \u201cfam\u201d format may include multiple families, and thefam2ped function will collapse them all into atibble with one row per family:peds <- fam2ped(fam)peds##\u2003#\u2003A tibble:\u20038 x 3##\u2003\u2003 fid\u2003\u2003\u2003 data\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 ped##\u2003\u2003 <chr>\u2003\u2003 <list>\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <list>##\u20031\u2003testped1\u2003<tibble\u2003[8 x 5]>\u2003<pedigree>##\u20032\u2003testped2\u2003<tibble\u2003[8 x 5]>\u2003<pedigree>##\u20033\u2003testped3\u2003<tibble\u2003[8 x 5]>\u2003<pedigree>##\u20034\u2003testped4\u2003<tibble\u2003[8 x 5]>\u2003<pedigree>##\u20035\u2003testped5\u2003<tibble\u2003[8 x 5]>\u2003<pedigree>##\u20036\u2003testped6\u2003<tibble\u2003[8 x 5]>\u2003<pedigree>##\u20037\u2003testped7\u2003<tibble\u2003[8 x 5]>\u2003<pedigree>##\u20038\u2003testped8\u2003<tibble\u2003[8 x 5]>\u2003<pedigree>Family structures imported fromtibble is nested by family ID. Thedata column contains the individual family information, while theped column contains the pedigree object for that family. Using standard tidyverse operations, the resulting tibble can be unnested for any particular family:peds %>%\u2003dplyr::filter(fid==\"testped1\") %>%\u2003tidyr::unnest(cols=data)## # A tibble:\u20038 x 7##\u2003\u2002 fid\u2003\u2003\u2003 id\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2002 dadid\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002momid\u2003\u2003\u2003sex\u2003affected\u2002\u2003\u2003ped##\u2003\u2002 <chr>\u2003\u2003 <chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002<chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002<chr>\u2003\u2003<int>\u2003\u2003 <dbl>\u2003<list>## 1\u2003testped1 testped1_g1-b1-s1 <NA>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002 <NA>\u2003\u2003\u2003\u2002\u2002 1\u2003\u2003\u2002\u2002\u2002\u2003\u20021\u2003<pedig~## 2\u2003testped1 testped1_g1-b1-i1 <NA>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002 <NA>\u2003\u2003\u2003\u2002\u2002 2\u2003\u2003\u2002\u2002\u2002\u2003\u20021\u2003<pedig~## 3\u2003testped1 testped1_g2-b1-s1 <NA>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002 <NA>\u2003\u2003\u2003\u2002\u2002 1\u2003\u2003\u2002\u2002\u2002\u2003\u20021\u2003<pedig~## 4\u2003testped1 testped1_g2-b1-i1 testped1_g1-b1-s1 testped1_~\u2002\u20022\u2003\u2003\u2002\u2002\u2002\u2003\u20021\u2003<pedig~## 5\u2003testped1 testped1_g2-b2-s1 <NA>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002 <NA>\u2003\u2003\u2003\u2002\u2002 1\u2003\u2003\u2002\u2002\u2002\u2003\u20021\u2003<pedig~## 6\u2003testped1 testped1_g2-b2-i1 testped1_g1-b1-s1 testped1_~\u2002 2\u2003\u2003\u2002\u2002\u2002\u2003\u20021\u2003<pedig~## 7\u2003testped1 testped1_g3-b1-i1 testped1_g2-b1-s1 testped1_~\u2002 2\u2003\u2003\u2002\u2002\u2002\u2003\u20021\u2003<pedig~## 8\u2003testped1 testped1_g3-b2-i1 testped1_g2-b2-s1 testped1_~\u2002 1\u2003\u2003\u2002\u2002\u2002\u2003\u20021\u2003<pedig~In the example above, the resultingmar orcex can be used to adjust aesthetics):peds$ped[[1]]## Pedigree object with 8 subjects## Bit size= 4plot, cex=.7)A single pedigree can also be inspected or visualized Theped2kinpair function takes a pedigree object and produces a pairwise list of relationships between all individuals in the data with the expected kinship coefficients for each pair.Theped2kinpair(peds$ped[[1]])##\u2003#\u2003A tibble:\u200336 x 3##\u2003\u2003\u2003id1\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002id2\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002k##\u2003\u2003\u2003<chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003<chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<dbl>##\u2003 1\u2003testped1_g1-b1-s1\u2003testped1_g1-b1-s1\u20030.5##\u2003 2\u2003testped1_g1-b1-i1\u2003testped1_g1-b1-s1\u20030##\u2003 3\u2003testped1_g1-b1-s1\u2003testped1_g2-b1-s1\u20030##\u2003 4\u2003testped1_g1-b1-s1\u2003testped1_g2-b1-i1\u20030.25##\u2003 5\u2003testped1_g1-b1-s1\u2003testped1_g2-b2-s1\u20030##\u2003 6\u2003testped1_g1-b1-s1\u2003testped1_g2-b2-i1\u20030.25##\u2003 7\u2003testped1_g1-b1-s1\u2003testped1_g3-b1-i1\u20030.125##\u2003 8\u2003testped1_g1-b1-s1\u2003testped1_g3-b2-i1\u20030.125##\u2003 9\u2003testped1_g1-b1-i1\u2003testped1_g1-b1-i1\u20030.5##\u200310\u2003testped1_g1-b1-i1\u2003testped1_g2-b1-s1\u20030##\u2003# \u2026 with 26 more rowsThe function can be run on a single family:kinpairs <-\u2003peds %>%\u2003dplyr::mutate) %>%\u2003dplyr::select %>%\u2003tidyr::unnest(cols=pairs)kinpairs##\u2003#\u2003A\u2003tibble: 288 x 4##\u2003\u2003\u2003fid\u2003\u2003\u2003\u2002 id1\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003id2\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003k##\u2003\u2003\u2003<chr>\u2003\u2002\u2003 <chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<dbl>##\u2003 1\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g1-b1-s1\u20030.5##\u2003 2\u2003testped1\u2003testped1_g1-b1-i1\u2003testped1_g1-b1-s1\u20030##\u2003 3\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g2-b1-s1\u20030##\u2003 4\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g2-b1-i1\u20030.25##\u2003 5\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g2-b2-s1\u20030##\u2003 6\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g2-b2-i1\u20030.25##\u2003 7\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g3-b1-i1\u20030.125##\u2003 8\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g3-b2-i1\u20030.125##\u2003 9\u2003testped1\u2003testped1_g1-b1-i1\u2003testped1_g1-b1-i1\u20030.5##\u200310\u2003testped1\u2003testped1_g1-b1-i1\u2003testped1_g2-b1-s1\u20030##\u2003# \u2026 with 278 more rowsThis function can also be mapped over all families in the pedigree:ped2kinpair over allped objects in the inputtibble, and that relationships are not shown for between-family relationships.Note that this mapsped2kinpair or other kinship estimation software.The skater package includes functions to translate kinship coefficients to relationship degrees. The kinship coefficients could come fromdibble function creates adegreeinferencetibble, with degrees up to the specifiedmax_degree (default=3), expected kinship coefficient, and lower (l) and upper (u) inference ranges as defined in Manichaikul et al. Degree 0 corresponds to self/identity/monozygotic twins, with an expected kinship coefficient of 0.5, with inference range >=0.354. Anything beyond the maximum degree resolution is considered unrelated (degreeNA). Note also that while the theoretical upper boundary for the kinship coefficient is 0.5, the inference range for 0-degree  extends to 1 to allow for floating point arithmetic and stochastic effects resulting in kinship coefficients above 0.5.dibble## # A tibble: 5 x 4##\u2003 degree\u2003\u2003\u2003k\u2003\u2003\u2003\u2003 l\u2003\u2003\u2003\u2003 u##\u2003 <int>\u2003 <dbl>\u2003\u2003 <dbl>\u2003 <dbl>## 1\u2003\u2003 0\u20030.5\u2003\u2003\u2003 0.354\u2003 1## 2\u2003\u2003 1\u20030.25\u2003\u2003\u20030.177\u2003 0.354## 3\u2003\u2003 2\u20030.125\u2003\u2003\u20020.0884\u2002 0.177## 4\u2003\u2003 3\u20030.0625\u2003 0.0442\u20030.0884## 5\u2003\u2003NA\u20030\u2003\u2003\u2003\u2002 -1\u2003\u2003\u2002\u2003 0.0442Themax_degree default is 3. Change this argument to allow more granular degree inference ranges:dibble (max_degree = 5)## # A tibble: 7 x 4##\u2003 degree\u2003\u2003\u2003k\u2003\u2003\u2003 l\u2003\u2003\u2003 u##\u2003 <int>\u2003 <dbl>\u2003 <dbl>\u2003 <dbl>## 1\u2003\u2003 0\u20030.5\u2003\u2003 0.354\u2003 1## 2\u2003\u2003 1\u20030.25\u2003\u20030.177\u2003 0.354## 3\u2003\u2003 2\u20030.125\u2003 0.0884\u20030.177## 4\u2003\u2003 3\u20030.0625\u20030.0442\u20030.0884## 5\u2003\u2003 4\u20030.0312\u20030.0221\u20030.0442## 6\u2003\u2003 5\u20030.0156\u20030.0110\u20030.0221## 7\u2003\u2003NA\u20030\u2003\u2002\u2003\u2003-1\u2003\u2003\u2002\u2003 0.0110The degree inferencedibble function will emit a warning withmax_degree >=10, and will stop with an error at >=12.Note that the distance between relationship degrees becomes smaller as the relationship degree becomes more distant. Thekin2degree function infers the relationship degree given a kinship coefficient and amax_degree up to which anything more distant is treated as unrelated. Example first degree relative:kin2degree## [1] 1Thekin2degree## [1] NAExample 4th degree relative, but using the default max_degree resolution of 3:kin2degree## [1] 4Example 4th degree relative, but increasing the degree resolution:kin2degree function is vectorized over values ofk, so it can be used inside of amutate on atibble of kinship coefficients:# Get two pairs from each type of relationship we have in kinpairs:kinpairs_subset <-\u2003kinpairs %>%\u2003dplyr::group_by(k) %>%\u2003dplyr::slice(1:2)kinpairs_subset##\u2003#\u2003A tibble:\u200310 x 4##\u2003#\u2003Groups:\u2003\u2003k [5]##\u2003\u2003\u2003fid\u2003\u2003\u2003\u2002 id1\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002id2\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 k##\u2003\u2003\u2003<chr>\u2003\u2003\u2002 <chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002 <dbl>##\u2003 1\u2003testped1\u2003testped1_g1-b1-i1\u2003testped1_g1-b1-s1\u20030##\u2003 2\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g2-b1-s1\u20030##\u2003 3\u2003testped1\u2003testped1_g3-b1-i1\u2003testped1_g3-b2-i1\u20030.0625##\u2003 4\u2003testped2\u2003testped2_g3-b1-i1\u2003testped2_g3-b2-i1\u20030.0625##\u2003 5\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g3-b1-i1\u20030.125##\u2003 6\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g3-b2-i1\u20030.125##\u2003 7\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g2-b1-i1\u20030.25##\u2003 8\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g2-b2-i1\u20030.25##\u2003 9\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g1-b1-s1\u20030.5##\u200310\u2003testped1\u2003testped1_g1-b1-i1\u2003testped1_g1-b1-i1\u20030.5# Infer degree out to third degree relatives:kinpairs_subset %>%\u2003dplyr::mutate )##\u2003#\u2003A tibble:\u200310 x 5##\u2003#\u2003Groups:\u2003\u2003k [5]##\u2003\u2003\u2003fid\u2003\u2003\u2003\u2002 id1\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u2003id2\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 k\u2003\u2002degree##\u2003\u2003\u2003<chr>\u2003\u2003\u2002 <chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <dbl>\u2003 <int>##\u2003 1\u2003testped1\u2003testped1_g1-b1-i1\u2003testped1_g1-b1-s1\u20030\u2003\u2003\u2003\u2003\u2003 NA##\u2003 2\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g2-b1-s1\u20030\u2003\u2003\u2003\u2003\u2003 NA##\u2003 3\u2003testped1\u2003testped1_g3-b1-i1\u2003testped1_g3-b2-i1\u20030.0625\u2003\u2002\u2003 3##\u2003 4\u2003testped2\u2003testped2_g3-b1-i1\u2003testped2_g3-b2-i1\u20030.0625\u2003\u2003\u2002 3##\u2003 5\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g3-b1-i1\u20030.125\u2003\u2003\u2003\u20022##\u2003 6\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g3-b2-i1\u20030.125\u2003\u2003\u2003\u20022##\u2003 7\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g2-b1-i1\u20030.25\u2003\u2003\u2003\u2002 1##\u2003 8\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g2-b2-i1\u20030.25\u2003\u2003\u2003\u2002 1##\u2003 9\u2003testped1\u2003testped1_g1-b1-s1\u2003testped1_g1-b1-s1\u20030.5\u2003\u2003\u2003\u2003\u20030##\u200310\u2003testped1\u2003testped1_g1-b1-i1\u2003testped1_g1-b1-i1\u20030.5\u2003\u2003\u2003\u2003\u20030TheOnce estimated kinship is converted to degree, it may be of interest to compare the inferred degree to truth. When aggregated over many relationships and inferences, this approach can help benchmark performance of a particular kinship analysis method.confusion_matrix function from Clark to provide standard contingency table metrics  with a new reciprocal RMSE (R-RMSE) metric. Theconfusion_matrix function on its own outputs a list with four objects:1.tibble with calculated accuracy, lower and upper bounds, the guessing rate and p-value of the accuracy vs. the guessing rate.A2.tibble with contingency table statistics calculated for each class. Details on the statistics calculated for each class can be reviewed on the help page for?confusion_matrix.A3.matrix with the contingency table object itself.A4.vector with the reciprocal RMSE (R-RMSE). The R-RMSE represents an alternative to classification accuracy when benchmarking relationship degree estimation and is calculated using the formula in (1). Taking the reciprocal of the target and predicted degree results in larger penalties for more egregious misclassifications  than misclassifications at more distant relationships . The +0.5 adjustment prevents division-by-zero when a 0th-degree  relative pair is introduced.AThe skater package adapts akinpairs data from above and randomly flip ~20% of the true relationship degrees:# Function to randomly flip levels of a factor randomflip <- function ifelse(runif(length(x))<p, sample(unique(x)), x)# Infer degree (truth/target) using kin2degree, then randomly flip 20% of themset.seed(42)kinpairs_inferred <- kinpairs %>%\u2003dplyr::mutate) %>%\u2003dplyr::mutate) %>%\u2003dplyr::mutate(degree_inferred=randomflip (degree_truth))kinpairs_inferred##\u2003#\u2003A tibble:\u2003288 x 6##\u2003\u2002\u2003fid\u2003\u2003\u2003 id1\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2002\u2003\u2003id2\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003k\u2002\u2002\u2003\u2002degree_truth\u2002degree_inferred##\u2003\u2002\u2003<chr>\u2003\u2003 <chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002<chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003<dbl> <chr>\u2003\u2003\u2003\u2003\u2002<chr>## 1 testped1 testped1_g1-b1-s1 testped1_g1-b1-s1 0.5\u2003\u20030\u2003\u2003\u2003\u2003\u2003\u2003 0## 2 testped1 testped1_g1-b1-i1 testped1_g1-b1-s1 0\u2003\u2003\u2003unrelated\u2003\u2003unrelated## 3 testped1 testped1_g1-b1-s1 testped1_g2-b1-s1 0\u2003\u2003\u2003unrelated\u2003\u2003unrelated## 4 testped1 testped1_g1-b1-s1 testped1_g2-b1-i1 0.25\u20031\u2003\u2003\u2003\u2003\u2003\u2003 1## 5 testped1 testped1_g1-b1-s1 testped1_g2-b2-s1 0\u2003\u2003\u2003unrelated\u2003\u2003unrelated## 6 testped1 testped1_g1-b1-s1 testped1_g2-b2-i1 0.25\u20031\u2003\u2003\u2003\u2003\u2003\u2003 1## 7 testped1 testped1_g1-b1-s1 testped1_g3-b1-i1 0.125\u20022\u2003\u2003\u2003\u2003\u2003\u2003 2## 8 testped1 testped1_g1-b1-s1 testped1_g3-b2-i1 0.125\u20022\u2003\u2003\u2003\u2003\u2003\u2003 1## 9 testped1 testped1_g1-b1-i1 testped1_g1-b1-i1 0.5\u2003\u20020\u2003\u2003\u2003\u2003\u2003\u2003 0## 10 testped1 testped1_g1-b1-i1 testped1_g2-b1-s1 0\u2003\u2003\u2002unrelated\u2003\u2003unrelated## #\u2026 with 278 more rowsTo illustrate the usage, this example will start with theconfusion_matrix function will return all four objects noted above:confusion_matrix##\u2003$Accuracy##\u2003#\u2003A tibble:\u20031 x 5##\u2003\u2002Accuracy\u2002\u2018Accuracy\u2003LL\u2018\u2003\u2018Accuracy\u2003UL\u2018\u2003\u2018Accuracy\u2003Guessing\u2018\u2003\u2018Accuracy\u2003P-value\u2018##\u2003\u2003\u2003\u2002<dbl>\u2003\u2003\u2003\u2003\u2003 <dbl>\u2003\u2003\u2003\u2003\u2003 <dbl>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <dbl>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003<dbl>##\u20031\u2003\u2002 0.812\u2003\u2003\u2003\u2003\u2003 0.763\u2003\u2003\u2003\u2003\u2003 0.856\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 0.333\u2003\u2003\u2003\u2003\u2003\u2003\u2003 1.09e-62####\u2003$Other##\u2003#\u2003A tibble:\u20036 x 15##\u2003\u2003 Class\u2002\u2003\u2003\u2003\u2003 N\u2002\u2018Sensitivity/Re~\u2002\u2018Specificity/TN~\u2002\u2018PPV/Precision\u2018\u2002NPV\u2003\u2002\u2018F1/Dice\u2018##\u2003\u2003 <chr>\u2003 <dbl>\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <dbl>\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <dbl>\u2003\u2003\u2003\u2003\u2003\u2003<dbl>\u2003<dbl>\u2003\u2003\u2003<dbl>##\u20031\u20020\u2003\u2003\u2003\u2003\u200364\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 0.75\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u20020.964\u2003\u2003\u2003\u2003\u2003\u20030.857\u20030.931\u2003\u2003\u20030.8##\u20032\u20021\u2003\u2003\u2003\u2003\u200372\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 0.806\u2003\u2003\u2003\u2003\u2003\u2002\u2003 0.944\u2003\u2003\u2003\u2003\u2003\u20030.829\u20030.936\u2003\u2003\u20030.817##\u20033\u20022\u2003\u2003\u2003\u2003\u200348\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 0.833\u2003\u2003\u2003\u2003\u2003\u2003\u2002 0.967\u2003\u2003\u2003\u2003\u2003\u20030.833\u20030.967\u2003\u2003\u20030.833##\u20034\u20023\u2003\u2003\u2003\u2003\u2003 8\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u20020.75\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u20030.936\u2003\u2003\u2003\u2003\u2003\u2002\u20030.25\u2003 0.992\u2003\u2003\u20030.375##\u20035\u2002unrelated\u200296\u2003\u2003\u2003\u2003\u2003\u2003 \u2003\u20030.854\u2003\u2003\u2003\u2003\u2003\u2003\u2002 0.958\u2003\u2003\u2003\u2003\u2003\u20030.911\u20030.929\u2003\u2003\u20030.882##\u20036\u2002Average\u2002\u200357.6\u2003\u2003\u2003\u2003\u2003\u2003\u2003 0.799\u2003\u2003\u2003\u2003\u2003\u2002\u2003 0.954\u2003\u2003\u2003\u2003\u2003\u20030.736\u20030.951\u2003\u2003\u20030.741##\u2003# \u2026 with 8 more variables: Prevalence <dbl>, Detection Rate <dbl>,##\u2003#\u2003 Detection Prevalence <dbl>, Balanced Accuracy <dbl>, FDR <dbl>, FOR <dbl>,##\u2003#\u2003 FPR/Fallout <dbl>, FNR <dbl>####\u2003$Table##\u2003\u2003\u2003\u2003\u2003\u2003 Target##\u2003Predicted\u2003\u20030\u2003 1\u2003 2\u20033\u2003unrelated##\u2003\u20030\u2003\u2003\u2003\u2003\u2002 48\u2003 4\u2003 2\u20031\u2003\u2003\u2003\u2003\u2002\u20031##\u2003\u20031\u2003\u2003\u2003\u2003\u2002\u20035\u200358\u2003 4\u20030\u2003\u2003\u2003\u2003\u2002\u20033##\u2003\u20032\u2003\u2003\u2003\u2003\u2003\u20020\u2003 3\u200340\u20031\u2003\u2003\u2003\u2003\u2002\u20034##\u2003\u20033\u2003\u2003\u2003\u2003\u2003\u20028\u2003 4\u2003 0\u20036\u2003\u2003\u2003\u2003\u2002\u20036##\u2003\u2003unrelated\u20033\u2003 3\u2003 2\u20030\u2003\u2003\u2003\u2002\u200382####\u2003$recip_rmse##\u2003[1] 0.4665971Next, running thepurrr::pluck can be used to isolate just the contingency table:confusion_matrix %>%\u2003purrr::pluck(\"Table\")##\u2003\u2003\u2003\u2003\u2003\u2003 Target##\u2003Predicted\u2003\u20030\u2003 1\u2003\u20022\u20033\u2003unrelated##\u2003\u20030\u2003\u2003\u2003\u2003\u2002 48\u2003 4\u2002\u20032\u20031\u2003\u2003\u2003\u2003\u2003\u20021##\u2003\u20031\u2003\u2003\u2003\u2003\u2002\u20035\u200358\u2002\u20034\u20030\u2003\u2003\u2003\u2003\u2003\u20023##\u2003\u20032\u2003\u2003\u2003\u2003\u2003\u20020\u2003 3\u200340\u20031\u2003\u2003\u2003\u2003\u2002\u2002 4##\u2003\u20033\u2003\u2003\u2003\u2003\u2002\u20038\u2003 4\u2003\u20020\u20036\u2003\u2003\u2003\u2003\u2002\u2002\u20036##\u2003\u2003unrelated\u20033\u2003 3\u2002\u20032\u20030\u2003\u2003\u2003\u2003\u2002 82Standard tidyverse functions such asconfusion_matrix function includes an argument to output in a tidy (longer=TRUE) format, and the example below illustrates how to spread contingency table statistics by class:confusion_matrix %>%\u2003purrr::pluck(\"Other\") %>%\u2003tidyr::spread  %>%\u2003dplyr::relocate  %>%\u2003dplyr::mutate_if##\u2003#\u2003A tibble:\u200314 x 7##\u2003\u2003\u2003Statistic\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 \u2003\u2002\u20180\u2018\u2003\u2003 \u20181\u2018\u2003\u2003 \u20182\u2018\u2003\u2003\u2003\u20183\u2018\u2003 unrelated\u2003 Average##\u2003\u2003\u2003<chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u2003\u2003\u2002 <dbl>\u2003 <dbl>\u2003 <dbl>\u2003\u2003<dbl>\u2003\u2003\u2003 <dbl>\u2003\u2003 <dbl>##\u2003 1\u2003Balanced Accuracy\u2003\u2003\u2003\u2002\u2002 0.86\u2003\u20030.88\u2003\u20030.9\u2003\u2002 0.84\u2003\u2003\u2003\u2003 0.91\u2003\u2003\u20030.88##\u2003 2\u2003Detection Prevalence\u2002\u2002\u2003\u20030.19\u2003\u20030.24\u2003\u20030.17\u2003\u20020.083\u2003\u2003\u2003\u20030.31\u2003\u2003\u20030.2##\u2003 3\u2003Detection Rate\u2003\u2003\u2003\u2002\u2003\u2003\u20030.17\u2003\u20030.2\u2003\u2003 0.14\u2002\u20030.021\u2003\u2003\u2003\u20030.28\u2003\u2003\u20030.16##\u2003 4\u2003F1/Dice\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u2002\u2003\u2003 0.8\u2003\u2003 0.82\u2003\u20030.83\u2003\u20020.38\u2003\u2003\u2003\u2003 0.88\u2003\u2003\u20030.74##\u2003 5\u2003FDR\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u2003 0.14\u2003\u20030.17\u2003\u20030.17\u2003\u20020.75\u2003\u2003\u2003\u2003 0.089\u2003\u2002 0.26##\u2003 6\u2003FNR\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u2003\u2003 0.25\u2003\u20030.19\u2003\u20030.17\u2003\u20020.25\u2003\u2003\u2003\u2003 0.15\u2003\u2003\u20030.2##\u2003 7\u2003FOR\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u2003\u2003\u2003\u2003 0.069\u2003 0.064\u2003 0.033\u2002 0.0076\u2003\u2003\u2002 0.071\u2003\u2003 0.049##\u2003 8\u2003FPR/Fallout\u2003\u2003\u2003\u2003\u2002\u2003\u2003\u2003 0.036\u2002 0.056\u2003 0.033\u2002 0.064\u2003\u2003\u2002\u2003\u20020.042\u2003\u2003 0.046##\u2003 9\u2003N\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u200364\u2003\u2003\u200372\u2003\u2003\u200348\u2003\u2003\u2003 8\u2003\u2003\u2003\u2003\u2003 96\u2003\u2003\u2003\u200358##\u200310\u2003NPV\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u2003\u2002\u2003 0.93\u2003\u20030.94\u2003\u20030.97\u2003\u20030.99\u2003\u2003\u2003\u2003 0.93\u2003\u2003\u20030.95##\u200311\u2003PPV/Precision\u2002\u2003\u2003\u2003\u2002\u2003\u2003 0.86\u2003\u20030.83\u2003\u20030.83\u2003\u20030.25\u2003\u2003\u2003\u2003 0.91\u2003\u2003\u20030.74##\u200312\u2003Prevalence\u2002\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u20030.22\u2003\u20030.25\u2003\u20030.17\u2003\u20030.028\u2003\u2003\u2003\u20030.33\u2003\u2003\u20030.2##\u200313\u2003Sensitivity/Recall/TPR\u2003\u20030.75\u2003\u20030.81\u2003\u20030.83\u2003\u20030.75\u2003\u2003\u2003\u2003 0.85\u2003\u2003\u20030.8##\u200314\u2003Specificity/TNR\u2003\u2003\u2003\u2003\u2002\u2003 0.96\u2003\u20030.94\u2003\u20030.97\u2003\u20030.94\u2003\u2003\u2003\u2003 0.96\u2003\u2003\u20030.95The and IBIS detect shared IBD segments between individuals. The skater package includes functionality to take those IBD segments, compute shared genomic centimorgan (cM) length, and converts that shared cM to a kinship coefficient. In addition to inferred segments, these functions can estimate \u201ctruth\u201d kinship from simulated IBD segments. Theread_ibd function reads pairwise IBD segments from IBD inference tools and from simulated IBD segments. Theread_map function reads in genetic map in a standard format which is required to translate the total centimorgans shared IBD to a kinship coefficient using theibd2kin function. See?read_ibd and?read_map for additional details on expected format.Tools such as hap-IBD,read_ibd function reads in the pairwise IBD segment format. Input to this function can either be inferred IBD segments from hap-IBD (source=\"hapibd\") or simulated segments (source=\"pedsim\"). The first example below uses data in thehap-ibd output format:hapibd_filepath <- system.filehapibd_seg <- read_ibdhapibd_seg##\u2003#\u2003A tibble:\u20033,954 x 6##\u2003\u2003\u2003id1\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u2003id2\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u2003chr\u2003\u2003\u2003start\u2003\u2003\u2003\u2003end\u2003\u2002length##\u2003\u2003\u2003<chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003<chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<dbl>\u2003\u2003\u2003<dbl>\u2003\u2003\u2003<dbl>\u2003 <dbl>##\u2003 1\u2003testped1_g1-b1-s1\u2003testped1_g3-b1-i1\u2003\u2003\u20031\u2003197661576\u2003234863602\u2003\u200347.1##\u2003 2\u2003testped1_g2-b2-i1\u2003testped1_g3-b1-i1\u2003\u2003\u20031\u2003197661576\u2003231017545\u2003\u200339.8##\u2003 3\u2003testped1_g3-b1-i1\u2003testped1_g3-b2-i1\u2003\u2003\u20031\u2003197661576\u2003212799139\u2003\u200320.3##\u2003 4\u2003testped3_g1-b1-s1\u2003testped3_g3-b2-i1\u2003\u2003\u20031\u2003\u20032352146\u2003 10862397\u2003\u200317.7##\u2003 5\u2003testped3_g2-b2-i1\u2003testped3_g3-b2-i1\u2003\u2003\u20031\u2003\u20032352146\u2003 10862397\u2003\u200317.7##\u2003 6\u2003testped1_g1-b1-s1\u2003testped1_g2-b1-i1\u2003\u2003\u20031\u2003\u20033328659\u2003 64123868\u2003\u200386.4##\u2003 7\u2003testped1_g1-b1-s1\u2003testped1_g3-b1-i1\u2003\u2003\u20031\u2003\u20033328659\u2003 33476811\u2003\u200351.2##\u2003 8\u2003testped1_g2-b2-s1\u2003testped1_g3-b2-i1\u2003\u2003\u20031\u2003\u20035003504\u2003 32315147\u2003\u200345.9##\u2003 9\u2003testped2_g1-b1-i1\u2003testped2_g3-b1-i1\u2003\u2003\u20031\u2003240810528\u2003248578622\u2003\u200315.9##\u200310\u2003testped2_g1-b1-i1\u2003testped2_g2-b2-i1\u2003\u2003\u20031\u2003241186056\u2003249170711\u2003\u200315.5##\u2003#\u2003\u2026 with 3,944 more rowsTheread_map. Software for IBD segment inference and simulation requires a genetic map. The map loaded for kinship estimation should be the same one used for creating the shared IBD segment output. The example below uses a minimal genetic map that ships withskater:gmap_filepath <- system.filegmap <- read_map(gmap_filepath)gmap##\u2003#\u2003A tibble:\u200328,726 x 3##\u2003\u2003\u2003 chr value\u2003\u2003\u2003\u2002\u2002 bp##\u2003\u2003 <dbl> <dbl>\u2003\u2002\u2002\u2003<dbl>##\u2003 1\u2003\u2003 1\u20030\u2003\u2003\u2003\u2002\u2003752721##\u2003 2\u2003\u2003 1\u20030.0292\u20031066029##\u2003 3\u2003\u2003 1\u20030.0829\u20031099342##\u2003 4\u2003\u2003 1\u20030.157\u2003 1106473##\u2003 5\u2003\u2003 1\u20030.246\u2003 1152631##\u2003 6\u2003\u2003 1\u20030.294\u2003 1314015##\u2003 7\u2003\u2003 1\u20030.469\u2003 1510801##\u2003 8\u2003\u2003 1\u20030.991\u2003 1612540##\u2003 9\u2003\u2003 1\u20031.12\u2003\u20031892325##\u200310\u2003\u2003 1\u20031.41\u2003\u20031916587##\u2003# \u2026 with 28,716 more rowsIn order to translate the shared genomic cM length to a kinship coefficient, a genetic map must first be read in withibd2kin function takes the segments and map file and outputs atibble with one row per pair of individuals and columns for individual 1 ID, individual 2 ID, and the kinship coefficient for the pair:ibd_dat <- ibd2kinibd_dat##\u2003#\u2003A tibble:\u2003196 x 3##\u2003\u2003\u2003id1\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u2003id2\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002 kinship##\u2003\u2003\u2003<chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003<chr>\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <dbl>##\u2003 1\u2003testped1_g1-b1-i1\u2003testped1_g1-b1-s1\u20030.000316##\u2003 2\u2003testped1_g1-b1-i1\u2003testped1_g2-b1-i1\u20030.261##\u2003 3\u2003testped1_g1-b1-i1\u2003testped1_g2-b2-i1\u20030.263##\u2003 4\u2003testped1_g1-b1-i1\u2003testped1_g2-b2-s1\u20030.000150##\u2003 5\u2003testped1_g1-b1-i1\u2003testped1_g3-b1-i1\u20030.145##\u2003 6\u2003testped1_g1-b1-i1\u2003testped1_g3-b2-i1\u20030.133##\u2003 7\u2003testped1_g1-b1-i1\u2003testped2_g1-b1-i1\u20030.000165##\u2003 8\u2003testped1_g1-b1-i1\u2003testped2_g1-b1-s1\u20030.000323##\u2003 9\u2003testped1_g1-b1-i1\u2003testped2_g2-b1-i1\u20030.000499##\u200310\u2003testped1_g1-b1-i1\u2003testped2_g2-b1-s1\u20030.000318##\u2003#\u2003\u2026 with 186 more rowsTheThe skater R package provides a robust software package for data import, manipulation, and analysis tasks typically encountered when working with SNP-based kinship analysis tools. All package functions are internally documented with examples, and the package contains a vignette demonstrating usage, inputs, outputs, and interpretation of all key functions. The package contains internal tests that are automatically run with continuous integration via GitHub Actions whenever the package code is updated. The skater package is permissively licensed (MIT) and is easily extensible to accommodate outputs from new genome-wide relatedness and IBD segment methods as they become available.1.http://CRAN.R-project.org/package=skater.Software available from:2.https://github.com/signaturescience/skater.Source code available from:3.https://doi.org/10.5281/zenodo.5761996.Archived source code at time of publication:4.Software license: MIT License.SDT, VPN, and MBS developed the R package.All authors contributed to method development.SDT wrote the first draft of the manuscript.All authors assisted with manuscript revision.All authors read and approved the final manuscript.No competing interests were disclosed.This work was supported in part by award 2019-DU-BX-0046 (Dense DNA Data for Enhanced Missing Persons Identification) to B.B., awarded by the National Institute of Justice, Office of Justice Programs, U.S. Department of Justice and by internal funds from the Center for Human Identification. The opinions, findings, and conclusions or recommendations expressed are those of the authors and do not necessarily reflect those of the U.S. Department of Justice. This article describes an R package that provides a suite of tools for performing kinship and IBD segment analysis. Key features include methods for reading and analyzing pedigree data, inferring relationships, benchmarking relationship degree classification, and IBD segment analyses. The use cases in this article and the R reference manual on CRAN together provide exemplary documentation for skater. Users unfamiliar with pedigree-style analyses and/or who desire a clean, well documented interface for performing these analyses can find a great deal of utility in this R package.Are the conclusions about the tool and its performance adequately supported by the findings presented in the article?YesIs the rationale for developing the new software tool clearly explained?YesIs the description of the software tool technically sound?YesAre sufficient details of the code, methods and analysis (if applicable) provided to allow replication of the software development and its use by others?YesIs sufficient information provided to allow interpretation of the expected output datasets and any results generated using the tool?YesReviewer Expertise:Relatedness in large datasets.I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. ad hoc scripts for parsing, cleaning, and manipulating various result files. Arguably, the best solution to this problem is to collect related scripts into an R package, which can be properly documented, version-controlled, and potentially shared with others. Theskater package presented in this paper is precisely such a package, aimed at the users of several popular software for relatedness estimation, including KING, PLINK, hap-IBD, and others. A specific goal of the package is to provide tools that work consistently across these programs.For many researchers, myself included, R is the language-of-choice when it comes to downstream analysis and tidying-up the output of other programs. This inevitably leads to a disorganized body of The paper is written in clear language, well structured, and easy to follow. Parts of the paper resemble a user manual, including detailed code examples. I enjoyed playing around with the package, which is well organized and documented. In particular, I appreciate the thorough input checks and helpful error messages when things go wrong. The package may not offer a ton of novel methods yet, but I can certainly see myself using it the next time I run some of these relatedness programs.It would be helpful to clarify what sort of applications skater is intended for. The introduction mentions \"relationship inference in population genetics, medical genetics, and forensics\", but that seems overly broad. The focus is on simple measures like the kinship coefficient and relatedness degree  suggesting large-scale applications rather than small-scale pedigree analysis as e.g. in forensic case work. Indeed, all the programs referenced in the paper are intended for large-scale population studies. Furthermore, it seems to be an underlying assumption that all individuals are noninbred , and also that they are human (by the hardcoded 3560 cM genome length). To be clear, I think these limitations and assumptions are perfectly fine, but it would be better to state (or discuss) them more clearly.A connection to \"SNP-based methods\" is mentioned repeatedly, but never really explained. Is there one? I note at least one of the cited programs (hap-IBD) is designed to work with sequencing data. Does it matter for skater how the kinship estimates were obtained?The function `confusion_matrix` produces an impressive array of summary stats when comparing inferred kinship degrees to true ones. This is great, but I wonder about the origin of this function. In my understanding, it is a modified version of a function from another package, confusionMatrix, written by Michael Clark (not an author of the paper). Several related functions appear to be copied verbatim into skater. All of this is clearly marked in the code, with Clark rightfully listed as author, and Clark's package does come with a permissive license (MIT), but I still find it a bit odd. Why not simply import it? That way, bug fixes and improvements in the original version would automatically propagate to skater, avoiding the confusion of multiple versions. If the problem is that confusionMatrix is not on CRAN, perhaps one could reach out to make this happen?In the example illustrating `confusion_matrix`, the authors construct a dataset by modifying a previous one, by \"randomly flipping ~20% of the true degrees\". The term \"flip\" sounds misplaced to me here, since the variable isn't dichotomous. Also, the procedure doesn't change 20%, only 0.2 * 4/5 = 16% on average, since the new values are generated from the complete set and stay unchanged with probability 1/5. Regardless of these minor issues, I must admit I found the example rather artificial. It would be much more interesting to see confusion_matrix applied to a real dataset! That could also motivate some comments on how its output should be interpreted. Finally, I cannot resist offering a couple of suggestions for the skater package itself:The `read_fam` for reading .fam files is very strict, insisting on space-separated columns and disallowing any format deviations. I note that e.g. PLINK, and several other R packages that read pedigree files, allow variations like tab-separated columns and missing phenotype column; perhaps this could be useful in the skater package too.Separating parent-offspring pairs from full sibs is a crucial step in many relatedness studies, and often possible by a simple analysis of the output of e.g. KING. Does skater offer such differentiation? If not, it might make a nice addition to the package in the future. Below are a few issues I have with the manuscript.Are the conclusions about the tool and its performance adequately supported by the findings presented in the article?PartlyIs the rationale for developing the new software tool clearly explained?YesIs the description of the software tool technically sound?YesAre sufficient details of the code, methods and analysis (if applicable) provided to allow replication of the software development and its use by others?YesIs sufficient information provided to allow interpretation of the expected output datasets and any results generated using the tool?YesReviewer Expertise:Statistical genetics, pedigree analysis, RI confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. The authors developed an R package to evaluate the performance of different relatedness inference tools. Both the R package and the manuscript are user-friendly and easy to follow. This tool should be timely and valuable to geneticists  who have a need to compare different relatedness inference tools and choose a tool that performs the best for their own genetic datasets. The manuscript is very well written.How about only one parent available, with Father ID 1234 and mother ID 0? This is allowed and considered as PLINK format. How does the proposed tool handle it, e.g., ask the user to reformat it?How about both parents only with IDs available. E.g., Father ID is 1234 and mother ID is 5678, and none of them have appeared again in Individual IDs?\u00a0 Again, the comment above is quite minor and may only require some clarification. The overall quality of this manuscript is great. One very minor comment is about the format requirement for the developed tool. Is the required format basically PLINK with expectations that some commonly used PLINK formats are still not allowed, or all PLINK format is acceptable? E.g.,Are the conclusions about the tool and its performance adequately supported by the findings presented in the article?YesIs the rationale for developing the new software tool clearly explained?YesIs the description of the software tool technically sound?YesAre sufficient details of the code, methods and analysis (if applicable) provided to allow replication of the software development and its use by others?YesIs sufficient information provided to allow interpretation of the expected output datasets and any results generated using the tool?YesReviewer Expertise:My area of research is statistical genetics, specifically developing methods and tools for relatedness inference.I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard."}
{"text": "Scientific Reports 10.1038/srep20968, published online 10 February 2016Correction to: This Article contains errors in Figure\u00a04 and 6.In the assembly of Figure\u00a04B, images from Figure\u00a02C were inadvertently incorporated for the \u201cshRNA-sfrp2\u201d, \u201cshRNA-sfrp2\u2009+\u2009miR218\u201d, and \u201cmiR218 inhibitor\u201d panels.For Figure\u00a04C, the flow cytometry data for the \u201cshRNA-sfrp2\u2009+\u2009miRNA217 inhibitor\u201d and \u201cmiR218 inhibitor\u201d panels were incorrect.In Figure\u00a06A, incorrect images were used for the \u201cmimic con\u201d condition.The correct Figures"}
{"text": "MicroRNAs (miRNAs) participate in the reactivation of \u03b3\u2010globin expression in \u03b2\u2010thalassemia. However, the miRNA transcriptional profiles of pediatric \u03b2\u2010thalassemia remain unclear. Accordingly, in this study, we assessed miRNA expression in pediatric patients with \u03b2\u2010thalassemia.Differentially expressed miRNAs in pediatric patients with \u03b2\u2010thalassemia were determined using microRNA sequencing.Hsa\u2010miR\u2010483\u20103p, hsa\u2010let\u20107f\u20101\u20103p, hsa\u2010let\u20107a\u20103p, hsa\u2010miR\u2010543, hsa\u2010miR\u2010433\u20103p, hsa\u2010miR\u20104435, hsa\u2010miR\u2010329\u20103p, hsa\u2010miR\u201092b\u20105p, hsa\u2010miR\u20106747\u20103p and hsa\u2010miR\u2010495\u20103p were significantly upregulated, whereas hsa\u2010miR\u20104508, hsa\u2010miR\u201020a\u20105p, hsa\u2010let\u20107b\u20105p, hsa\u2010miR\u201093\u20105p, hsa\u2010let\u20107i\u20105p, hsa\u2010miR\u20106501\u20105p, hsa\u2010miR\u2010221\u20103p, hsa\u2010let\u20107g\u20105p, hsa\u2010miR\u2010106a\u20105p, and hsa\u2010miR\u201017\u20105p were significantly downregulated in pediatric patients with \u03b2\u2010thalassemia. After integrating our data with a previously published dataset, we found that hsa\u2010let\u20107b\u20105p and hsa\u2010let\u20107i\u20105p expression levels were also lower in adolescent or adult patients with \u03b2\u2010thalassemia. The predicted target genes of hsa\u2010let\u20107b\u20105p and hsa\u2010let\u20107i\u20105p were associated with the transforming growth factor \u03b2 receptor, phosphatidylinositol 3\u2010kinase/AKT, FoxO, Hippo, and mitogen\u2010activated protein kinase signaling pathways. We also identified 12 target genes of hsa\u2010let\u20107a\u20103p and hsa\u2010let\u20107f\u20101\u20103p and 21 target genes of hsa\u2010let\u20107a\u20103p and hsa\u2010let\u20107f\u20101\u20103p, which were differentially expressed in patients with \u03b2\u2010thalassemia. Finally, we found that hsa\u2010miR\u2010190\u20105p and hsa\u2010miR\u20101278\u20105p may regulate hemoglobin switching by modulation of the B\u2010cell lymphoma/leukemia 11A gene.The results of the study show that several microRNAs are dysregulated in pediatric \u03b2\u2010thalassemia. Further, the results also indicate toward a critical role of let7\u00a0miRNAs in the pathogenesis of pediatric \u03b2\u2010thalassemia, which needs to be investigated further. p\u2010value <0.05, 111\u00a0microRNAs were upregulated in \u03b2\u2010thalassemia patients, while 85\u00a0microRNAs were downregulated in \u03b2\u2010thalassemia patients. Those microRNAs could clearly distinguish the normal cohorts from the \u03b2\u2010thalassemia patients. Hsa\u2010miR\u20102100\u20103p, hsa\u2010microRNA\u201015a\u20105p, hsa\u2010microRNA\u201016\u20105p, and hsa\u2010miR\u2010503\u20105p were all downregulated in pediatric \u03b2\u2010thalassemia patients. We also found five let7\u00a0microRNAs hsa\u2010let\u20107b\u20105p, hsa\u2010let\u20107i\u20105p, hsa\u2010let\u20107f\u20105p, hsa\u2010let\u20107e\u20105p, and hsa\u2010let\u20107d\u20105p and were downregulated in pediatric thalassemia patients.We performed microRNA sequencing to identify the microRNA expression profiling of pediatric \u03b2\u2010thalassemia. Totally, 530\u00a0microRNAs were identified. Based on criteria of fold change >1.5 and  The red cells were lysed using PAXgene Blood RNA Kit. The remaining mononuclear cells were used for further RNA isolation. The information and clinical conditions of the participants were also collected.2.2Total RNA from mononuclear cells was isolated using a miRNeasy Mini Kit (Qiagen) according to the manufacturer's protocol. Briefly, mononuclear cells were lysed using lysis reagent, and 140\u00a0\u03bcl chloroform was added. The upper aqueous phase was then mixed with 100% ethanol, and the mixture was transferred to the column, washed, and eluted with RNase\u2010free water.2.3Total RNA was used to prepare the miRNA sequencing library. After linker ligation, cDNA synthesis, and polymerase chain reaction (PCR) amplification, 135\u2013155\u2010bp PCR amplification fragments were selected. The library was denatured into single\u2010stranded DNA, captured on an Illumina flow cell, amplified into clusters, and sequenced for 51 cycles using an Illumina NextSeq 500\u00a0sequencer (Illumina).2.4http://bioconductor.org/packages/release/bioc/html/edgeR.html) in R statistics software,p value less than 0.05, and CPM greater than or equal to 1.After sequencing, Solexa Chastity software was used for quality control. The linkers were removed using Cutadapt,2.5https://github.com/CJ\u2010Chen/TBtools), respectively.Volcano plots and Venn diagrams were generated using Fancy Volcano Plot and Wonderful Venn in TBtools software  in R statistics software.Unsupervised heatmaps were generated using \u201cpheatmap\u201d package http://www.targetscan.org/vert_72/)The targets of miRNAs were predicted using miRDB (2.8https://david.ncifcrf.gov).p\u00a0<\u00a00.05.The enriched biological processes and KEGG signaling pathways were determined using the Database for Annotation, Visualization, and Integrated Discovery Web site . Microarray expression was analyzed using R software .The gene expression matrix from patients with \u03b2\u2010thalassemia and normal controls was deposited in the GSE56088 dataset from the GEO Web site (2.10let7\u00a0miRNAs and predicted target genes was created using Cytoscape (http://www.cytoscape.org/).A network of 2.11t tests. Statistical significance was set at p\u00a0<\u00a00.05.Box plots were generated using GraphPad Prism software. Statistical analysis was performed using Student's 33.1Peripheral blood samples from five children diagnosed with \u03b2\u2010thalassemia in our hospital and five healthy children were collected to identify differentially expressed miRNAs. The clinical characteristics of \u03b2\u2010thalassemia and normal individuals are shown in Table\u00a0Next, we performed miRNA sequencing to identify the miRNA expression profile of pediatric \u03b2\u2010thalassemia. In total, 530\u00a0miRNAs were identified; among these, 111\u00a0miRNAs were upregulated, whereas 85\u00a0miRNAs were downregulated in patients with \u03b2\u2010thalassemia. Three hundred and 330\u00a0miRNAs showed no differential expression in pediatric \u03b2\u2010thalassemia and normal cohorts, respectively Figure\u00a0. All difhsa\u2010miR\u2010210 regulates \u03b3\u2010globin expression through the transcription factor BCL11A,miR\u201015a and miR\u201016\u20101 elevate \u03b3\u2010globin expression through the transcription factor MYB.hsa\u2010miR\u2010503\u00a0has been shown to be downregulated in patients with \u03b2\u2010thalassemia.hsa\u2010miR\u20102100\u20103p, hsa\u2010miR\u201015a\u20105p, hsa\u2010miR\u201016\u20105p, and hsa\u2010miR\u2010503\u20105p were all downregulated in pediatric patients with \u03b2\u2010thalassemia compared with that in normal individuals  \u03b2 receptor signaling pathway, and somatic stem cell population maintenance /AKT, FoxO, Hippo, and mitogen\u2010activated protein kinase (MAPK) signaling pathways  of BCL11A (Figure\u00a0hsa\u2010miR\u20101278\u20105p targeted the 3\u2032 UTRs of BCL11A (Figure\u00a0hsa\u2010miR\u2010190\u20105p and hsa\u2010miR\u20101278\u20105p were both downregulated in patients with \u03b2\u2010thalassemia compared with those in normal controls (Figure\u00a0BLC11A is a critical transcription factor that regulates hemoglobin switching.A Figure\u00a0. In addiA Figure\u00a0. Moreoves Figure\u00a0.hsa\u2010miR\u2010190\u20105p and hsa\u2010miR\u20101278\u20105p. In addition to BCL11A, hsa\u2010miR\u2010190\u20105p also targeted ZNF99, FNDC3A, ORC4, PFDN4, ZNF382, EPC2, PHF20L1, ASAP2, WDR44, ZNF529, NHLRC2, ZFC3H1, CHMP7, YTHDF3, and TAPBP genes (Figure\u00a0hsa\u2010miR\u2010190\u20105p also targeted NRBF2, ZRANB2, PAX8, DNAJB14, TIPARP, LHX6, IGF2BP2, and STK40 (Figure\u00a0IGF2BP2 and STK40 were also target genes of hsa\u2010let\u20107a\u20103p and hsa\u2010let\u20107f\u20101\u20103p (Figure\u00a0hsa\u2010miR\u2010190\u20105p and hsa\u2010miR\u20101278\u20105p in hemoglobin switching and \u03b2\u2010thalassemia need to be studied in greater detail.We further identified the target genes of s Figure\u00a0. Further0 Figure\u00a0. Interesp Figure\u00a0. However4let7\u00a0miRNAs, that is, hsa\u2010let\u20107b\u20105p, hsa\u2010let\u20107i\u20105p, hsa\u2010let\u20107f\u20105p, hsa\u2010let\u20107e\u20105p, and hsa\u2010let\u20107d\u20105p, may be involved in the reactivation of \u03b3\u2010globin expression and HbF synthesis in pediatric and adult patients with \u03b2\u2010thalassemia.\u03b2\u2010Thalassemia is a heterogeneous disease, and the clinical manifestations of \u03b2\u2010thalassemia in pediatric and adult patients may be different.LIN28B/let7 axis directly regulates BCL11A expression to promote hemoglobin switching.hsa\u2010let\u20107a and hsa\u2010let\u20107b reactivated the expression of HbF in erythroid cells.hsa\u2010let\u20107a was upregulated in pediatric \u03b2\u2010thalassemia, whereas hsa\u2010let\u20107b was downregulated in pediatric \u03b2\u2010thalassemia. The functions of hsa\u2010let\u20107i\u20105p, hsa\u2010let\u20107f\u20105p, hsa\u2010let\u20107e\u20105p, and hsa\u2010let\u20107d\u20105p in the regulation of \u03b3\u2010globin expression or HbF synthesis in erythroid cells have not been reported. Furthermore, we identified 21 target genes of hsa\u2010let\u20107b\u20105p and hsa\u2010let\u20107i\u20105p, which were differentially expressed in patients with \u03b2\u2010thalassemia. We also found that the target genes of hsa\u2010let\u20107b\u20105p and hsa\u2010let\u20107i\u20105p were associated with the PI3K/AKT, FoxO, Hippo, and MAPK signaling pathways. How those pathways involved in the pathology of \u03b2\u2010thalassemia should be further studied.Consistent with these observations, reports have shown that the Hsa\u2010miR\u2010210 and hsa\u2010let\u20107b\u20105p regulate \u03b3\u2010globin expression through BCL11A.miR\u201015a and miR\u201016\u20101 elevate \u03b3\u2010globin expression through the transcription factor MYB.hsa\u2010miR\u2010210, hsa\u2010let\u20107b\u20105p, miR\u201015a, and miR\u201016\u20101 were all downregulated in \u03b2\u2010thalassemia. We believe that our data could help to identify more miRNAs associated with the BCL11A transcription factor. Indeed, our findings showed that two miRNAs, that is, hsa\u2010miR\u2010190\u20105p and hsa\u2010miR\u20101278\u20105p, may regulate hemoglobin switching by targeting BCL11A. However, the functions of these miRNAs should be studied further.BCL11A,let7\u00a0miRNAs and their target genes are abnormally dysregulated in pediatric \u03b2\u2010thalassemia. However, there were some limitations to the integrated analysis of the different datasets. Because of differences in cohorts and approaches, our analysis could not fully reveal the miRNA profiles associated with pediatric and adult \u03b2\u2010thalassemia. In addition, identification of differentially expressed miRNA target genes in patients with \u03b2\u2010thalassemia using the GSE56088 dataset may also have some bias. In our subsequent studies, we will collect a large cohort of \u03b2\u2010thalassemia cases comprising patients of different ages and perform miRNA sequencing and mRNA sequencing simultaneously. Additionally, the functions of hsa\u2010let\u20107i\u20105p, hsa\u2010let\u20107f\u20105p, hsa\u2010let\u20107e\u20105p, and hsa\u2010let\u20107d\u20105p in the regulation of \u03b3\u2010globin expression will be studied in greater detail.To the best of our knowledge, this is the first study to identify differentially expressed miRNAs, particularly in pediatric \u03b2\u2010thalassemia. Our results suggest that The authors declare that they have no conflicts of interest.HW performed the data analysis and wrote the manuscript. MH collected the blood samples. SY, YL, and YH helped with the collection of blood samples. HL and LP designed the study and supervised the work."}
{"text": "With the course of this research, it will prove that subsemigroup, the set of all right h-bi-ideals, and set of all left h-bi-ideals are bands for h-regular semiring. Moreover, it will be demonstrated that if semigroup of all h-bi-ideals (B(H), \u2217) is semilattice, then H is h-Clifford. This research will also explore the classification of minimal h-bi-ideal.Semigroups are generalizations of groups and rings. In the semigroup theory, there are certain kinds of band decompositions which are useful in the study of the structure of semigroups. This research will open up new horizons in the field of mathematics by aiming to use semigroup of  Primary idea of semigroup and monoid is given by Wallis . Wallis h-ideals, a new class of ideals proposed by Iizuka  ={{x1 + x2 + x3 \u22ef +xn} \u2208 {u} \u222a {u2} \u222a uHu}. Then, B[u] be a subsemiring of semiring H. Further, \u2200h \u2208 H and x, y \u2208 {u} \u222a {u2} \u222a uHu. We get xhy \u2208 Hu which refers that B[u]HB[u]\u2286B[u] and thus B[u] be a bi-ideal of H.Let H be a semiring and let take u from H. Then, the principal h-bi-ideal of H generated by u is given by:Let x, y\u2208Bh(u). Then, \u2203m, n \u2208 H and u \u2208 H such that x + u + u2 + umu + w = u + u2 + umu + w and y + u + u2 + unu + w\u2032 = u + u2 + unu + w\u2032 for w \u2208 H. Then, we have:H, +) is semillatice, wherexy \u2208 Bh(u). Also, x + y \u2208 Bh(u). Thus, Bh(u) is a subsemiring of H. Similarly, xmy \u2208 Bh(u) for all x, y \u2208 Bh(u) and m \u2208 H. Thus, Bh(u) is a bi-ideal of H. Indeed h\u2212closure of B[u]. Hence, Bh(u) is a h-bi-ideal of H. Let u \u2208 B and let B be a h-bi-ideal of semiring H. Let x \u2208 Bh(u), then there exists m \u2208 H such that:Let u \u2208 B implies that u + u2 + umu + w \u2208 B and so, x \u2208 B. Hence, Bh(u)\u2286B. Thus, Bh(u) is the least h-bi-ideal of H, which contains u.Now, h-bi-ideals, the set of all left h-ideals, and the set of all right h-ideals shall be each semigroup with respect to the product of subsets of H specified in the usual way: \u2217:P(H) \u00d7 P(H)\u27f6P(H) by The set of all H. Acknowledge receipt by P(H) of all subsets of H. We represent set of all h-bi-ideal by B(H), set of all left h-ideal by \u2112(H), and set of right h-ideal by \u211b(H). We define a binary operation \u2217:P(H) \u00d7 P(H)\u27f6P(H) by Instead of defining binary operation on these three sets individually, we consider them to be semigroups of all subsets of MN = {mn | m \u2208 M, n \u2208 N}; so,As P(H) is a semigroup under the usual operation \u2217. Also, since L1 and L2 are left h-ideal, then h-ideal. So, \u2217:P(H) \u00d7 P(H)\u27f6P(H) induced a binary operation on \u2112(H); in the interests of comfort, we signify with the same symbol \u201c\u2217.\u201d Thus, \u2112(H) is a semigroup under the defined operation \u201c\u2217.\u201d Analogously, (\u211b(H) is a semigroup under the same binary operation \u201c\u2217.\u201d Now, we can show that the same holds for B(H).We will check that M and N be the h-bi-ideals of semiring H. Let m \u2208 M and n \u2208 N such thatj + j\u2032 = j\u2032\u2032and mn + mn = mn. Since  is semillatice, soj1 = mnj\u2032 + jmn + jj\u2032. \u21d2u + v & uv both belong to h \u2208 H, we have uhv + mnhmn + j1h = mnhmn + j1h implies that h-closure of MN & h-subset of H. Thus, h-bi-ideal of H. This shows that B(H) is a semigroup.Let S = {0, d, e, f} defined by the following tables. Now, consider the semiring h-ideal is S itself. Obviously, this h-ideal is h-bi-ideal which forms a semigroup.The only H be a semigroup, then H is said to be a regular semigroup if for each t \u2208 H, there exists u\u2032 \u2208 H such that t = tu\u2032t. Bourne [S to be regular if u\u2032, v\u2032 \u2208 H exists for each a \u2208 H that t + tu\u2032t = tv\u2032t. Adhikari [k-regularity; let H be a semiring, if  is a semillatice, then t + tu\u2032t = tv\u2032t. Adding tu\u2032t + tv\u2032t to both sides, that t + t(u\u2032 + v\u2032)t = t(u\u2032 + v\u2032)t. We can say that a semiring H \u2208 HL+ is called k-regular if and only if for any t \u2208 H, there is u\u2032 \u2208 H such that t + tu\u2032t = tu\u2032t.Suppose . Bourne  has descAdhikari  defined H is called h-regular if \u2200t\u2032 \u2208 H, then \u2203u\u2032, v\u2032 \u2208 H such that t\u2032 + t\u2032u\u2032t\u2032 + w = t\u2032v\u2032t\u2032 + w, for w \u2208 H.A semiring H be a semiring, if  is a semillatice, then t + tu\u2032t+w = tv\u2032t + w. Adding tu\u2032t + tv\u2032t to both sides, then t + t(u\u2032 + v\u2032)t + w = t(u\u2032 + v\u2032)t + w. We can say that a semiring H \u2208 HL+ is called h-regular if and only if for any t \u2208 H, there is u\u2032 \u2208 H such that t + tu\u2032t + w = tu\u2032t + w.Let H. H is h-regularh-bi-ideal B of H, For every h-bi-ideal M of H, For every generalized The following conditions are identical for the semiring iii)\u21d2(ii) We know that every h-bi-ideal is a generalized h-bi-ideal from definition, so it is obvious.(i)\u21d2(iii) Assume that H is h-regular and let M be the generalized h-bi-ideal of H. Then, MHM\u2286M refers to u \u2208 M, since H is h-regular, then there exists m\u2032 \u2208 H such that u + um\u2032u + w = um\u2032u + w for w \u2208 H. Now, um\u2032u \u2208 MHM implies that (ii)\u21d2(i) Let u\u2032 \u2208 H, consider the h-bi-ideal Bh(u\u2032). Then, n1, n2, n3, n4 \u2208 Bh(u\u2032), and h1, h2 \u2208 H such thatn = n1 + n2 + n3 + n4 \u2208 Bh(u\u2032) and h = h1 + h2 \u2208 H. Hence, \u2203h \u2208 H such thatt = u\u2032 + u\u20322 + u\u2032h2 + u\u20322h2 + h3u\u20322 + h2w\u2032 + w\u2032h2, and w\u2032w\u2032 = w. Thus, H is h-regular.(\u211b(H) and (\u2112(H)) are bands for h-regular semiring H.Subsemigroup \u211b(H) and \u2112(H) are two subsemigroups. Let R\u2032 \u2208 \u211b(H) and u \u2208 R\u2032, then \u2203m \u2208 H such that u + umu + w = umu + w for w \u2208 H. Now,Let um \u2208 R\u2032H\u2286R\u2032 implies that R\u2032\u2286R\u20322. Also, it is obvious that R\u20322\u2286R\u2032. Thus, it can be written as R\u2032 = R\u20322. Hence, \u211b(H) is a band.\u2112(H) is a band that can prove dually.Similarly, h-bi-ideals B(H) is a product of \u211b(H) and \u2112(H) semigroups. Bear in mind that \u211b(H)\u2112(H) = {RL | R\u2032 \u2208 \u211b(H), L\u2032 \u2208 \u2112(H)}.In the following theorem, we will prove that semigroup of all B(H) = \u211b(H)\u2112(H) for h-regular semiring H.Show that B(H) = \u211b(H)\u2112(H), we have to show that B(H)\u2286\u211b(H)\u2112(H) and \u211b(H)\u2112(H)\u2286B(H). Suppose R\u2032 \u2208 \u211b(H) and L\u2032 \u2208 \u2112(H) and we represent B is h-subsemiring of H. Now, B \u2208 B(H). Hence, \u211b(H)\u2112(H)\u2286B(H).To prove that B \u2208 B(H). We represent L\u2032 is left h-ideal of H. Now, R\u2032 is right h-ideal of H. Now, H is h-regular, and we know that H is h-regular iffh-bi-ideal B of H. So, B(H)\u2286\u211b(H)\u2112(H). Thus, B(H) = \u211b(H)\u2112(H). Hence, proved.Now, suppose h-bi-ideals of the h-Clifford semiring and the left h-Clifford semiring.In this segment, we are characterizing the semigroup of all H be a semiring, then H is called a h-Clifford semiring ifH be a h-regular semiring, i.e., for every t \u2208 H, then \u2203u, v \u2208 H such that t + tut + w = tvt + w; for h \u2208 Ht \u2208 HLet e of semiring H is h-idempotent of H if e + e2 + w = e2 + w for w \u2208 H. Eh(H) is representation of set of all h-idempotents.An element h-regular semiring , the given following statements are identicalH be h-Clifford semiringu \u2208 H(ii)h-ideals as well as all right h-ideals are two sided and (iii) All left h-ideal and for all right h-ideals R(iv)h-ideals L1, L2 of H and h-ideals R1, R2 of H(v)In i)\u21d2(ii) Suppose H is h-Clifford and we want to show that u, v \u2208 H, then \u2203t \u2208 H such thatH is h-regular, then there is h1, h2 \u2208 H such that u + uh1u + w = uh1u + w and v + vh2v + w\u2032 = vh2v + w\u2032. Now, u + uh1u + uh1tv + w + w\u2032 = uh1u + uh1tv + w + w\u2032; this implies that u + uh1u + uh1tv + w = uh1u + uh1tv + w, where w + w\u2032 = w. Hence, by  is a semillatice, these implies thatx = h1 + h2 + h1t + h2t. Thus, ux and vx are h-idempotents. Since H is a h-Clifford semiring, there are s1, s2 in H such that u + vs1v + w = vs1v + w and vxu + us2u + w\u2032\u2032 = us2u + w\u2032\u2032for w\u2032\u2032 \u2208 H. Hence, \u21d2(iii) Suppose that u \u2208 H. For this, let t1 \u2208 H such that v + ut1 + w\u2032 = ut1 + w\u2032 for w\u2032 \u2208 H. Since H is h-regular, there is t2 \u2208 H for which u + ut2u + w = ut2u + w. Thus, there is t = t1 + t2 \u2208 H such that v + ut + w\u2032 = ut + w\u2032 and u + utu + w = utu + w. This implies that x \u2208 H such that ut + xu + w = xu + w. Now, v + ut + xu + w\u2032 = ut + xu + w\u2032 implies that v + xu + w\u2032 = xu + w\u2032, that is, (iii)\u21d2(iv) Lets take u from left h-ideal L, and t \u2208 H, then L is two-sided h-ideal. Similarly, this will be hold for right h-ideal R. Let M and N are two h-ideals of H. Then, MN contain in M\u2229N. Let u \u2208 M\u2229N, since H is h-regular, there is h \u2208 H such that u + utu + w = utu + w for w \u2208 H. So, u will be in (iv)\u21d2(v) This is trivial.(v)\u21d2(i) Let u \u2208 H and e be h-idempotent in H. Then, from (v), we can write H is h-regular, there is t \u2208 H such that ue + uetue + w1 = uetue + w1 for w1 \u2208 H and so,(ue + ((uet)u)(e(tue)) + w1 = ((uet)u)(e(tue)) + w1 implies that p \u2208 H such that ue + pu + w2 = pu + w2. Now, \u21d2(vi) This is trivial.(vi)\u21d2(i) Let u \u2208 H and e \u2208 Eh(H). Since H is h-regular semiring, there is t \u2208 H such that ue + uetue + w5 = uetue + w5 for w5 \u2208 H, and so,(ue + (uet)u(etu)e + w = (uet)u(etu)e + w implies that p \u2208 H such that ue + pu + w6 = pu + w6. Hence, , \u2217) is semilattice, then prove that H is h-Clifford.Let B(H) is semillatice, then H is h-regular semiring. Let u, v \u2208 H, then uv \u2208 Bh(u)Bh(v) = Bh(v)Bh(u) implies that uv + rs + w = rs + w for some r \u2208 Bh(v) and s \u2208 Bh(u) and for w \u2208 H; since H is h-regular, there are x, y \u2208 H such that r + vxv + w\u2032 = vxv + w\u2032 and s + uyu + w\u2032\u2032 = uyu + w\u2032\u2032 for w\u2032, w\u2032\u2032 \u2208 H. Then, we have uv + rs + w = rs + wz = xvuy and w\u2032\u2032\u2032 = vxvw\u2032 + w\u2032uyu + w\u2032w\u2032\u2032 \u2208 H.Assume that H is a h-Clifford semiring.Hence, h-bi-ideals B(H) is a left normal band if and only if semiring is a left h-Clifford.The set of all H be h-Clifford semiring. Let L, M, N \u2208 B(H) and l \u2208 L, m \u2208 M, and n \u2208 N, we have t + lmn + w = lmn + w for w \u2208 H. Since H is h-regular; there is s \u2208 H such that lmn + lmnslmn + w1 = lmnslmn + w1 for w1 \u2208 H, which implies that t + lmnslmn + w = lmnslmn + w, from which we get t + lxlmnxnsls1n + w = lxlmnxnsls1n + w as l + lxl + w\u2032 = lxl + w\u2032, n + nxn + w\u2032\u2032 = nxn + w\u2032\u2032, and mn + s1m + w\u2032\u2032\u2032 = s1m + w\u2032\u2032\u2032 for some x, s1 \u2208 H and w\u2032, w\u2032\u2032, w\u2032\u2032\u2032 \u2208 H. Again, since lm + s2l + w1 = s2l + w1 and nsls1 + s3n + w2 = s3n + w2; for some s2, s3 \u2208 H and w1, w2 \u2208 H, after rearranging the above expression, we get t + (lxs2l)(nxs3n)m + w = (lxs2l)(nxs3n)m + w. Thus, L\u2217M\u2217N\u2286L\u2217N\u2217M; similarly, it can be proved that L\u2217N\u2217M is subset of L\u2217M\u2217N. Hence, L\u2217M\u2217N = L\u2217N\u2217M, and so, B(H) is a left normal band.Let h-bi-ideals B(H) be a left normal band and we are to prove that H is a left h-Clifford. Thus, H is h-regular, we know that semigroup B(H) is regular under \u2217 iff H is h-regular. Using this result, let u, v \u2208 H, then there is t \u2208 H such that uv + uvtuv + w = uvtuv + w which implies that B(H) is a left normal band, then uv + opq + w\u2032 = opq + w\u2032, where o \u2208 Bh(uvt), p \u2208 Bh(v), q \u2208 Bh(u), and w\u2032 \u2208 H. Since H is h-regular, there is h \u2208 H such that q + uhu + w\u2032\u2032 = uhu + w\u2032\u2032. Now, uv + opq + w\u2032 = opq + w\u2032 implies that uv + (opuh)u + w\u2032\u2032\u2032 = (opuh)u + w\u2032\u2032\u2032, for w\u2032\u2032\u2032 \u2208 H. Thus, H is left h-Clifford.Conversely, suppose that set of all h-ideals in this section and will work on semiring of minimal left h-ideals and minimal right h-ideals. Also, we will look at those semirings H \u2208 HL+ that have a minimal left(right)h-ideal, and a minimal h-bi-ideal. Bm(H) stands for the class of all minimal h-bi-ideals. Likewise, we will denote minimal left and minimal right h-ideals with \u2112m(H) and \u211bm(H).We will define semiring in terms of minimal H be a semiring and N be the h-bi-ideal of H, then N be a minimal h-bi-ideal if for every nontrivial h-bi-ideal U of H, U\u2286N follows that U = N.Let N\u2286H is a minimal h-bi-ideal of H if and only if h-ideal R1, L1 of H, respectively.Let \u2205\u2260N\u2286H, suppose that N is a minimal h-bi-ideal of H, then h-ideal and left h-ideal, respectively. Let us take R2\u2286R1 is a right h-ideal of H, then h-bi-ideal of H, this follows that N; R2 = R1. Thus, h-ideal of H. Likewise, h-ideal of H, since h-bi-ideal and N is a minimal h-b-ideal of Let \u2205\u2260Bm(H) is set of all minimal h-bi-ideals, then Bm(H) is a subsemigroup of B(H).If h-bi-ideal M1and M2 in Bm(H). Let G \u2208 B(H) be such that M1, we have Bm(H) would be subsemigroup of B(H).Suppose two minimal \u2112m(H) and \u211bm(H) are subsemigroup of B(H) which can justify easily.As the same pattern, the set h-ideals is \u2112m(H) and L1\u2208\u2112(H), then L1\u2217Lm\u2032 = Lm\u2032, where Lm\u2032 is in \u2112m(H). Hence, \u2112m(H) be a left ideal of semigroup, i.e., (\u2112(H), \u2217), in particular a subsemigroup. Further, (\u2112m(H), \u2217) be a right zero band.(a) If a nonempty class of all minimal left h-ideals is \u211bm(H) and R1\u2208\u211b(H), then R1\u2217Rm\u2032 = Rm\u2032, where Rm\u2032 is in \u211bm(H). Hence, \u211bm(H) be a right ideal of semigroup, i.e., (\u211b(H), \u2217), in particular, a subsemigroup. Further, (\u211bm(H), \u2217) be a left zero band.(b) If a nonempty class of all minimal right L1 \u2208 \u2112(H) and Lm\u2032\u2208\u2112m(H), take the left h-ideal L1\u2217Lm\u2032. Then, L1\u2217Lm\u2032 implies Lm\u2032 and by minimality of Lm\u2032 shows L1\u2217Lm\u2032 = Lm\u2032.(a) For some (b) This proceeds dually.h-bi-ideal Bm(H) for a semiring H is nonempty, then \u2200Mm, Gm \u2208 Bm(H), and B \u2208 B(H); there are Mm\u2217B\u2217Gm \u2208 Bm(H), Mm\u2217Mm = Mm, and Mm\u2217B\u2217Mm = Mm. As a consequence, semigroup's bi-ideal is the set of all minimal h-bi-ideals (Bm(H), \u2217), in particular, a subsemigroup. Further, (Bm(H), \u2217) is rectangular band.If set of all minimal Bm(H) is set of all minimal h-bi-ideal and B(H) denotes the set of h-bi-ideal, then suppose Mm, Gm \u2208 Bm(H) and B \u2208 B(H). Denote the bi-ideal Mm\u2217B\u2217Gm by G. Let F \u2208 B(H) such that F\u2286G, then Mm shows that Mm = F\u2217Mm. Similarly, Gm = Gm\u2217F; therefore, G = F. Thus, the h-b-ideal Mm\u2217B\u2217Gm \u2208 Bm(H). Furthermore, Bm(H) be a subsemigroup of B(H). As a result, Bm(H) is bi-ideal of B(H). For some Mm \u2208 Bm(H), let us assume the h-bi-ideal Mm\u2217Mm. Then, Mm shows that Mm\u2217Mm = Mm. Since, also, Mm\u2217Mm\u2217Mm is h-bi-ideal, from Mm it means that Mm\u2217B\u2217Mm = Mm.Let h-regular semiring along their h-ideals. The different subclasses of the h-regular semiring are described by their h-bi-ideal semigroup. We have shown that B(H) = \u211b(H)\u2112(H) for h-regular semiring H. We have characterized a new class of semigroup of h-bi-ideals in h-Clifford semiring. Lastly, we worked on minimal h-ideal along their h-bi-ideals and proved that Bm(H) is a rectangular band. This research further can be extended with h-quasi ideals and k-ideals. The other idea of its extension is in the field of biology by using in growth-fragmentation-coagulation equation.This research represents"}
{"text": "Oty\u0142o\u015b\u0107 u dzieci prowadzi do zaburze\u0144 metabolicznych i zmian strukturalnych w naczyniach, kt\u00f3re s\u0105 przyczyn\u0105 rozwoju chor\u00f3b uk\u0142adu kr\u0105\u017cenia i cukrzycy typu 2.badania by\u0142a ocena czynnik\u00f3w ryzyka mia\u017cd\u017cycy u dzieci oty\u0142ych badanych w Oddziale Hospitalizacji Jednego Dnia Instytutu Matki i Dziecka.Zbadano 75 dzieci w wieku 6-12 lat  z BMI >97 centyla, stanowi\u0105c\u0105 grup\u0119 badan\u0105. Grup\u0119 kontroln\u0105 stanowi\u0142o:36 dzieci w wieku 5-10 lat  z BMI 75-90.Analizowano dane z wywiadu dotycz\u0105ce oty\u0142o\u015bci, chor\u00f3b uk\u0142adu kr\u0105\u017cenia i zaburze\u0144 lipidowych w rodzinie, u badanych dzieci analizowano BMI, obw\u00f3d pasa, lipidogram, st\u0119\u017cenie insuliny, grubo\u015b\u0107 kompleksu b\u0142ony wewn\u0119trznej i \u015brodkowej t\u0119tnicy szyjnej (IMT) w obu grupach.U 82,6% oty\u0142ych dzieci z grupy badanej stwierdzono w rodzinie oty\u0142o\u015b\u0107, a u 72% choroby uk\u0142adu kr\u0105\u017cenia i zaburzenia przemiany lipidowej, u dzieci z grupy kontrolnej: u 34,2% by\u0142a oty\u0142o\u015b\u0107 w rodzinie a u 36,8% choroby uk\u0142adu kr\u0105\u017cenia i zaburzenia lipidowe. \u015aredni obw\u00f3d pasa u oty\u0142ych dzieci wynosi\u0142 72,7 cm, w grupie kontrolnej 59,9 cm . \u015arednie st\u0119\u017cenie cholesterolu, LDL, tr\u00f3jgliceryd\u00f3w by\u0142o wy\u017csze w grupie badanej . Poziom insuliny u oty\u0142ych by\u0142 prawie dwukrotnie wy\u017cszy ni\u017c w grupie kontrolnej . \u015arednia grubo\u015b\u0107 IMT u oty\u0142ych dzieci wynosi\u0142a 0,36 mm ; \u00b10,059 SD (strona prawa) i 0,37 mm; \u00b10,033 SD (strona lewa). W grupie kontrolnej: 0,323 mm; \u00b10,087 SD (strona prawa), 0,32mm; \u00b10,082 SD (strona lewa), r\u00f3\u017cnice mi\u0119dzy grupami by\u0142y istotne statystycznie. Znaleziono dodatni\u0105 korelacj\u0119 pomi\u0119dzy obwodem pasa i poziomem insuliny. Wnioski: Oty\u0142o\u015b\u0107 i zaburzenia lipidowe cz\u0119\u015bciej dotycz\u0105 dzieci obci\u0105\u017conych rodzinnie oty\u0142o\u015bci\u0105 i chorobami uk\u0142adu kr\u0105\u017cenia. Zaburzenia lipidowe statystycznie cz\u0119\u015bciej wyst\u0119puj\u0105 u dzieci oty\u0142ych. IMT jest znamiennie wi\u0119ksza u dzieci oty\u0142ych w por\u00f3wnaniu z grup\u0105 kontroln\u0105 co sugeruje, \u017ce zmiany mia\u017cd\u017cycowe struktury naczy\u0144 szyjnych mog\u0105 wyst\u0105pi\u0107 u dzieci oty\u0142ych ju\u017c we wczesnym dzieci\u0144stwie. Pozwala je wykry\u0107 nieinwazyjna ultrasonograficzna metoda oceny IMT. Dzieci z oty\u0142o\u015bci\u0105, szczeg\u00f3lnie trzewn\u0105 maj\u0105 wy\u017cszy poziom insuliny, co mo\u017ce przyczyni\u0107 si\u0119 do rozwoju insulinooporno\u015bci.U rodzic\u00f3w brak jest dostatecznej wiedzy o skutkach oty\u0142o\u015bci i jego wp\u0142ywu na wyst\u0119powanie mia\u017cd\u017cycy i chor\u00f3b uk\u0142adu kr\u0105\u017cenia. Nadwaga i oty\u0142o\u015b\u0107 stanowi\u0105 narastaj\u0105cy problem w medycynie rozwojowej na ca\u0142ym \u015bwiecie. Wed\u0142ug raportu opracowanego przez The International Obesity Task Force co pi\u0105te dziecko w Europie ma nadwag\u0119 lub jest oty\u0142e . R\u00f3wnie\u017cOty\u0142o\u015b\u0107 jest jednym z najwa\u017cniejszych czynnik\u00f3w rozwoju mia\u017cd\u017cycy, kt\u00f3ra jest procesem chorobowym, a jego istot\u0105 s\u0105 zmiany zapalno-proliferacyjne prowadz\u0105ce do uszkodzenia \u015bciany t\u0119tnicy. Zmiany mia\u017cd\u017cycowe mog\u0105 rozwija\u0107 si\u0119 jednocze\u015bnie w r\u00f3\u017cnych naczyniach. Podstaw\u0105 ich rozwoju jest dysfunkcja \u015br\u00f3db\u0142onka naczy\u0144 , 7, 8. CMia\u017cd\u017cyca, prowadz\u0105ca do uszkodzenia i zw\u0119\u017cenia naczy\u0144 krwiono\u015bnych nale\u017cy do g\u0142\u00f3wnych zdrowotnych zagro\u017ce\u0144 cywilizacyjnych wsp\u00f3\u0142czesnego \u015bwiata. Procesy te zaczynaj\u0105 si\u0119 ju\u017c we wczesnym dzieci\u0144stwie i mog\u0105 prowadzi\u0107 do rozwoju chor\u00f3b uk\u0142adu kr\u0105\u017cenia w wieku \u015brednim i starszym , 10, 11.Post\u0119py w dziedzinie diagnostyki obrazowej pozwalaj\u0105 na mierzenie i ocen\u0119 grubo\u015bci \u015br\u00f3db\u0142onka naczy\u0144, kt\u00f3ry pierwszy ulega procesom patologicznym w du\u017cych naczyniach obwodowych oraz w nerkach.tunica intima), b\u0142on\u0119 \u015brodkow\u0105 (tunica media) oraz b\u0142on\u0119 zewn\u0119trzn\u0105 (tunica adventiva). Kompleksem intima-media nazywamy struktury \u015bciany naczyniowej mierzone od linii granicznej przydanka\u2013b\u0142ona \u015brodkowa do linii granicznej b\u0142ona wewn\u0119trzna\u2013 \u015bwiat\u0142o naczynia. Pomiar grubo\u015bci kompleksu b\u0142ony wewn\u0119trznej i \u015brodkowej t\u0119tnicy szyjnej wsp\u00f3lnej (IMT \u2212 intima-media thickness) pozwala na ocen\u0119 wczesnych zmian mia\u017cd\u017cycowych stosuj\u0105c nieinwazyjn\u0105 metod\u0119 ultrasonograficzn\u0105 [W t\u0119tnicach mo\u017cemy wyr\u00f3\u017cni\u0107 b\u0142on\u0119 wewn\u0119trzn\u0105 , kt\u00f3re stanowi\u0142y grup\u0119 badan\u0105 i 36 dzieci z prawid\u0142ow\u0105 mas\u0105 cia\u0142a (BMI 75-90 centyl), kt\u00f3re stanowi\u0142y grup\u0119 kontroln\u0105. W\u015br\u00f3d dzieci oty\u0142ych by\u0142o 36 ch\u0142opc\u00f3w i 39 dziewczynek, w grupie kontrolnej 18 ch\u0142opc\u00f3w i 18 dziewczynek.Do wykonania bada\u0144 zastosowano nast\u0119puj\u0105ce metody:wywiad dotycz\u0105cy wyst\u0119powania rodzinnych czynnik\u00f3w ryzyka mia\u017cd\u017cycy ,ocena wska\u017anika BMI,pomiar obwodu pasa,badania biochemiczne: lipidogram, TSH , fT3, fT4, st\u0119\u017cenie glukozy i insuliny we krwi na czczo,badanie usg szyi \u2013 pomiar grubo\u015bci kompleksu b\u0142ony wewn\u0119trznej i \u015brodkowej \u015bciany t\u0119tnicy szyjnej wsp\u00f3lnej (IMT).Badania krwi by\u0142y wykonywane przy pomocy aparatu Cobas-Integra, usg szyi za pomoc\u0105 aparatu usg firmy Philips U22. Pomiaru kompleksu b\u0142ony wewn\u0119trznej i \u015brodkowej t\u0119tnicy szyjnej dokonywano na \u015bcianie dolnej t\u0119tnicy szyjnej, w pozycji typowej, w projekcji pod\u0142u\u017cnej, przy u\u017cyciu g\u0142owicy 7,5 MHz. Wykonywano dwa pomiary ze strony prawej i lewej, oznaczano \u015bredni\u0105 z pomiar\u00f3w. Poziomy oznaczanych wska\u017anik\u00f3w lipidogramu por\u00f3wnywano do norm podanych przez NCEP  zale\u017cnych od wieku dzieci (za punkt odniesienia przyj\u0119to warto\u015bci prawid\u0142owe) .Wszystkie oty\u0142e dzieci zosta\u0142y skierowane na konsultacje dietetyczne.2, do badania korelacji \u2013 wsp\u00f3\u0142czynnik korelacji rho-Spearmana.Do analizy statystycznej zastosowano: test Ko\u0142mogorowa-Smirnowa, test U Manna-Whitney\u2019a, \u03c72(1)=26,53; p<0,001.Z danych z wywiadu wynika, \u017ce w grupie dzieci z oty\u0142o\u015bci\u0105 (grupa badana) dodatni wywiad rodzinny w kierunku oty\u0142o\u015bci  dotyczy\u0142 82,6% dzieci. W grupie kontrolnej oty\u0142o\u015b\u0107 w rodzinie dotyczy\u0142a tylko 34,2% dzieci. R\u00f3\u017cnica jest istotna statystycznie: \u03c7W grupie badanej 32% dzieci mia\u0142o oty\u0142ych obydwoje rodzic\u00f3w; 17,3% dzieci mia\u0142o tylko oty\u0142\u0105 matk\u0119, natomiast 16% dzieci mia\u0142o oty\u0142ego tylko ojca.2(1)=13,00; p<0,001.W tabeli I przedstawiono sumarycznie dane z wywiadu.W grupie badanej 72% mia\u0142o obci\u0105\u017cony wywiad rodzinny dotycz\u0105cy r\u00f3wnie\u017c chor\u00f3b uk\u0142adu kr\u0105\u017cenia i zaburze\u0144 lipidowych, podczas gdy w grupie kontrolnej dotyczy\u0142o to 36,8% dzieci. R\u00f3\u017cnica mi\u0119dzy grupami jest istotna statystycznie: \u03c7Dzieci z grupy badanej mia\u0142y \u015brednio 7,97 lat, za\u015b z grupy kontrolnej \u2013 8,02 roku. Grupy nie r\u00f3\u017cni\u0142y si\u0119 istotnie pod wzgl\u0119dem wieku: t(110)=-0,15; p>0,05.\u015arednie BMI w grupie dzieci oty\u0142ych wynosi\u0142o \u2212 24,3 , a w grupie kontrolnej z prawid\u0142ow\u0105 mas\u0105 cia\u0142a \u2212 19,00 .\u015aredni obw\u00f3d pasa w grupie badanej wynosi\u0142 72,7 cm , w grupie kontrolnej 59,9 cm , r\u00f3\u017cnica by\u0142a istotna statystycznie, t(103)=8,171;p<0,001.Prawid\u0142owe warto\u015bci sk\u0142adnik\u00f3w lipidogramu wg NCEP dla dzieci z grupy kontrolnej, do kt\u00f3rych por\u00f3wnywano wyniki bada\u0144 uzyskanych badanych przez nas dzieci by\u0142y nast\u0119puj\u0105ce: cholesterol ca\u0142kowity <170 mg/dl, LDL<110 mg/dl, HDL>45 mg/dl, tr\u00f3jglicerydy (TG): do 9 r.\u017c. <75 mg/dl, 10-18 r.\u017c.<90 mg/dl.Dzieci z oty\u0142o\u015bci\u0105 mia\u0142y znacz\u0105co wy\u017csze poziomy cholesterolu ca\u0142kowitego i jego frakcji LDL oraz tr\u00f3jgliceryd\u00f3w. \u015arednie st\u0119\u017cenie cholesterolu w tej grupie wynosi\u0142o 167 mg/dl , a w grupie kontrolnej 124 mg/dl . R\u00f3\u017cnica by\u0142a znacz\u0105ca statystycznie, t(111)=9,12; p<0,001. Warto\u015bci cholesterolu ca\u0142kowitego powy\u017cej 170 mg/dl dotyczy\u0142y 47% dzieci z oty\u0142o\u015bci\u0105 (z grupy badanej).\u015arednie st\u0119\u017cenie frakcji LDL cholesterolu w grupie badanej wynosi\u0142o 101 mg/ dl , a w grupie kontrolnej 72 mg/ dl , r\u00f3\u017cnica by\u0142a istotna statystycznie, t(110)=6,870;p<0.001. Warto\u015bci LDL>110 mg/ dl obserwowano u 33% dzieci oty\u0142ych.\u015arednie st\u0119\u017cenie tr\u00f3jgliceryd\u00f3w w grupie badanej by\u0142o wy\u017csze i wynosi\u0142o 86 mg/ dl  w por\u00f3wnaniu z grup\u0105 kontroln\u0105, u kt\u00f3rej by\u0142o na poziomie 63 mg/ dl , r\u00f3\u017cnica by\u0142a istotna statystycznie, t(100)-3,028; p=0,003. St\u0119\u017cenie tr\u00f3jgliceryd\u00f3w >75mg/dl dotyczy\u0142o 61% dzieci z oty\u0142o\u015bci\u0105.W grupie badanej (z oty\u0142o\u015bci\u0105) ni\u017csze st\u0119\u017cenie frakcji HDL cholesterolu (<45 mg/ dl) obserwowano u 21% dzieci oty\u0142ych.St\u0119\u017cenie insuliny na czczo u dzieci z grupy badanej by\u0142o prawie dwukrotnie wy\u017csze i wynosi\u0142o 10,27 mU/ l, w grupie kontrolnej by\u0142o r\u00f3wne 5,88 mU/l, t(109)=3,054; p<0,01. Wszystkie badane dzieci mia\u0142y we krwi prawid\u0142owy poziom hormon\u00f3w tarczycy oraz poziom glukozy na czczo. Na podstawie bada\u0144 obrazowych stwierdzono, \u017ce u dzieci oty\u0142ych \u015brednia grubo\u015b\u0107 kompleksu \u015bciany t\u0119tnicy szyjnej wewn\u0119trznej i \u015brodkowej po stronie prawej wynosi\u0142a 0,36 mm; \u00b10,059 , a po stronie lewej 0,37 mm; \u00b10,033 . Nie by\u0142a to r\u00f3\u017cnica istotna statystycznie.W grupie kontrolnej warto\u015bci te by\u0142y nast\u0119puj\u0105ce: po stronie prawej 0,32 mm; \u00b10,087 , po stronie lewej 0,32mm ; \u00b10,082 . Nie by\u0142a to r\u00f3\u017cnica istotna statystycznie. U dzieci oty\u0142ych, stanowi\u0105cych grup\u0119 badan\u0105, \u015brednia grubo\u015b\u0107 kompleksu \u015bciany wewn\u0119trznej i \u015brodkowej t\u0119tnicy szyjnej wsp\u00f3lnej by\u0142a wi\u0119ksza ni\u017c w grupie kontrolnej, szczeg\u00f3lnie dotyczy\u0142o to t\u0119tnicy lewej, r\u00f3\u017cnica by\u0142a istotna statystycznie, t(111)=2,394,p <0,01 po stronie lewej, natomiast po stronie prawej - t(111)=2,609, p<0,05. Dane zebrano w tabeli II.Analizowano r\u00f3wnie\u017c korelacje pomi\u0119dzy badanymi parametrami w grupie dzieci oty\u0142ych. Nie znaleziono istotnej korelacji pomi\u0119dzy BMI, st\u0119\u017ceniem cholesterolu i jego frakcjami oraz poziomem tr\u00f3jgliceryd\u00f3w a grubo\u015bci\u0105 kompleksu \u015bciany t\u0119tnicy szyjnej wewn\u0119trznej i \u015brodkowej. Znaleziono dodatni\u0105 korelacj\u0119 pomi\u0119dzy poziomem insuliny na czczo a obwodem pasa u dzieci oty\u0142ych, im wi\u0119kszy by\u0142 obw\u00f3d pasa tym wy\u017cszy by\u0142 poziom insuliny, r=0,313, p<0,05.Wszystkie badane dzieci oty\u0142e zosta\u0142y skierowane na konsultacj\u0119 dietetyczn\u0105, ale tylko 35/75 dzieci zg\u0142osi\u0142o si\u0119 na wizyt\u0119: pod sta\u0142\u0105 opiek\u0105 poradni dietetycznej pozosta\u0142o 14/35 dzieci, 11/35 nie zg\u0142osi\u0142o si\u0119 na kolejny wyznaczony termin wizyty, a 10/35 dzieci by\u0142o tylko na jednej wizycie i nie zg\u0142osi\u0142y si\u0119 na kolejne. 29/75 dzieci z grupy badanej zg\u0142osi\u0142o si\u0119 na badanie kontrolne do Oddzia\u0142u Hospitalizacji Jednego Dnia, 20/29 z nich zg\u0142osi\u0142o si\u0119 po roku od pierwszej wizyty, a jedynie 9/29 po 2 latach. Siedmioro z 29 dzieci by\u0142o dwukrotnie na wizycie kontrolnej, po roku i po 2 latach.Nale\u017cy podkre\u015bli\u0107, \u017ce tylko po\u0142owa oty\u0142ych dzieci skorzysta\u0142a z zaplanowanych konsultacji dietetycznych i tylko 18% pozosta\u0142o pod sta\u0142\u0105 opiek\u0105 dietetyka. Na wizyty kontrolne do Oddzia\u0142u zg\u0142asza\u0142a si\u0119 ma\u0142a liczba pacjent\u00f3w pomimo zaprosze\u0144 listownych lub telefonicznych.Oty\u0142o\u015b\u0107 jest narastaj\u0105cym problemem u dzieci polskich. Najcz\u0119\u015bciej mamy do czynienia z oty\u0142o\u015bci\u0105 prost\u0105 zwi\u0105zan\u0105 z nadmiern\u0105 poda\u017c\u0105 energii w diecie. Jedynie w wyj\u0105tkowych przypadkach jest to oty\u0142o\u015b\u0107 wt\u00f3rna do innych chor\u00f3b lub uwarunkowana genetycznie. Na rozw\u00f3j oty\u0142o\u015bci pierwotnej (prostej) ma wp\u0142yw wiele czynnik\u00f3w, ale przede wszystkim tryb \u017cycia, nawyki \u017cywieniowe, ma\u0142a aktywno\u015b\u0107 <zyczna lub jej brak. Oty\u0142o\u015b\u0107 cz\u0119\u015bciej dotyczy dzieci, kt\u00f3rych rodzice s\u0105 otyli, co wykazano w badaniu w\u0142asnym, jak i innych autor\u00f3w , 17.Dzieci oty\u0142e cz\u0119\u015bciej maj\u0105 zaburzenia gospodarki lipidowej, szczeg\u00f3lnie dotyczy to st\u0119\u017cenia tr\u00f3jgliceryd\u00f3w, cholesterolu ca\u0142kowitego i jego frakcji LDL. Wed\u0142ug ostatnich doniesie\u0144 Polskiego Towarzystwa Lipidologicznego, to w\u0142a\u015bnie frakcja LDL cholesterolu ca\u0142kowitego ma najwi\u0119ksze znaczenie dla rozwoju blaszki mia\u017cd\u017cycowej i nag\u0142ych zdarze\u0144 sercowo-naczyniowych u doros\u0142ych.W badaniu w\u0142asnym \u015brednie st\u0119\u017cenie cholesterolu i jego frakcji LDL oraz tr\u00f3jgliceryd\u00f3w by\u0142o znamiennie wy\u017csze u dzieci z oty\u0142o\u015bci\u0105, co jest zgodne z doniesieniami wielu innych autor\u00f3w , 18, 19.Wiele danych wskazuje, \u017ce markery i czynniki ryzyka sercowo-naczyniowego wykrywane w dzieci\u0144stwie, cz\u0119sto stwierdzane s\u0105 r\u00f3wnie\u017c u tych samych os\u00f3b w wieku doros\u0142ym. Najcz\u0119\u015bciej mamy do czynienia z nadmiern\u0105 mas\u0105 cia\u0142a, zaburzeniami w zakresie gospodarki lipidowej czy w\u0119glowodanowej , 20, 21.Bogalusa Heart Study wykaza\u0142y istotn\u0105 korelacj\u0119 mi\u0119dzy czynnikami ryzyka rozwoju mia\u017cd\u017cycy, takimi jak, nadci\u015bnienie t\u0119tnicze i oty\u0142o\u015b\u0107 we wczesnym okresie \u017cycia, a zmianami mia\u017cd\u017cycowymi w aorcie i w t\u0119tnicach wie\u0144cowych w wieku doros\u0142ym [Wi\u0119kszo\u015b\u0107 dzieci ze zdiagnozowanymi, ale nieleczonymi zaburzeniami lipidowymi b\u0119dzie charakteryzowa\u0142a si\u0119 nimi tak\u017ce w p\u00f3\u017aniejszym okresie \u017cycia. Obserwacje z doros\u0142ym .Prze\u0142omowe znaczenie dla zrozumienia patogenezy schorze\u0144 uk\u0142adu kr\u0105\u017cenia mia\u0142o dokonane ponad 20 lat temu odkrycie, \u017ce \u015br\u00f3db\u0142onek naczyniowy jest organem endo- i parakrynnym. Jego czynno\u015b\u0107 odgrywa wa\u017cn\u0105 rol\u0119 w regulacji napi\u0119cia \u015bciany naczynia, proliferacji kom\u00f3rek mi\u0119\u015bni g\u0142adkich, adhezji monocyt\u00f3w i leukocyt\u00f3w oraz proces\u00f3w krzepni\u0119cia, poprzez uwalnianie szeregu mediator\u00f3w rozszerzaj\u0105cych naczynia utrzymuje w\u0142a\u015bciwe napi\u0119cie \u015bciany t\u0119tnicy. Nieprawid\u0142owa funkcja \u015br\u00f3db\u0142onka traktowana jest jako wst\u0119pny etap rozwoju blaszki mia\u017cd\u017cycowej , 22. CelW badanej grupie dzieci w niniejszej pracy \u015brednia grubo\u015b\u0107 IMT u dzieci oty\u0142ych wynosi\u0142a: w lewej t\u0119tnicy 0,37 mm, w prawej 0,36 mm i by\u0142a znamiennie statystycznie wy\u017csza ni\u017c w grupie kontrolnej . Podobne wyniki uzyskali inni badacze. W pracy Leite i wsp., oceniaj\u0105cej grubo\u015b\u0107 IMT w grupie 150 portugalskich oty\u0142ych dzieci , \u015brednia grubo\u015b\u0107 kompleksu u oty\u0142ych dzieci wynosi\u0142a \u2013 0,472 mm, w grupie kontrolnej \u2212 0,418 mm , w badanAnalizowano r\u00f3wnie\u017c korelacje pomi\u0119dzy badanymi parametrami w grupie dzieci oty\u0142ych i nie znaleziono istotnej korelacji pomi\u0119dzy BMI, st\u0119\u017ceniem cholesterolu i jego frakcjami oraz poziomem tr\u00f3jgliceryd\u00f3w a grubo\u015bci\u0105 kompleksu \u015bciany t\u0119tnicy szyjnej wewn\u0119trznej i \u015brodkowej (IMT). Wed\u0142ug wielu autor\u00f3w taka korelacja istnieje. W cytowanej wcze\u015bniej pracy Litwina i wsp. autorzy dowiedli, \u017ce w kszta\u0142towaniu warto\u015bci grubo\u015bci IMT, opr\u00f3cz biochemicznych czynnik\u00f3w ryzyka rozwoju chor\u00f3b sercowonaczyniowych rol\u0119 predyktora uszkodzenia \u015bciany naczy\u0144 pe\u0142ni tak\u017ce odpowiednio wysoka warto\u015b\u0107 BMI . Do podoW badaniach w\u0142asnych zwr\u00f3cono uwag\u0119 na grup\u0119 dzieci, kt\u00f3re zg\u0142osi\u0142y si\u0119 na badania kontrolne. Obserwowano nast\u0119puj\u0105c\u0105 zale\u017cno\u015b\u0107: gdy BMI tych dzieci wzrasta\u0142o to r\u00f3wnie\u017c wzrasta\u0142a grubo\u015b\u0107 IMT, natomiast w przypadku obni\u017cenia si\u0119 warto\u015bci BMI zmniejsza\u0142a si\u0119 grubo\u015b\u0107 kompleksu. Z uwagi jednak na ma\u0142\u0105 grup\u0119 dzieci, u kt\u00f3rych takie zale\u017cno\u015bci obserwowano zjawisko to wymaga dalszych badaniach. Jest wysoce prawdopodobne, \u017ce obni\u017cenie masy cia\u0142a u oty\u0142ych dzieci mo\u017ce zahamowa\u0107, a nawet by\u0107 mo\u017ce cofn\u0105\u0107 rozpoczynaj\u0105cy si\u0119 proces mia\u017cd\u017cycowy w naczyniach.Insulina jest hormonem bezpo\u015brednio dzia\u0142aj\u0105cym na kom\u00f3rki naczy\u0144 poprzez: zmniejszenie produkcji tlenku azotu, zwi\u0119kszenie apoptozy kom\u00f3rek \u015br\u00f3db\u0142onka, obni\u017cenie wychwytu glukozy, dodatkowo ma wp\u0142yw na ekspresj\u0119 czynnik\u00f3w wzrostowych i adhezyjnych oraz wzrost i migracj\u0119 kom\u00f3rek mi\u0119\u015bni g\u0142adkich naczy\u0144. U os\u00f3b z oty\u0142o\u015bci\u0105 brzuszn\u0105 dochodzi do rozwoju insulinooporno\u015bci, kt\u00f3ra jest jednym z czynnik\u00f3w rozwoju choroby niedokrwiennej serca. Rola hiperinsulinizmu w patogenezie mia\u017cd\u017cycy jest z\u0142o\u017cona i nie do ko\u0144ca poznana , 26, 30.W badanej grupie dzieci st\u0119\u017cenie insuliny na czczo by\u0142o prawie dwukrotnie wy\u017csze u dzieci oty\u0142ych ni\u017c w grupie kontrolnej . Znaleziono dodatni\u0105 korelacj\u0119 pomi\u0119dzy poziomem insuliny na czczo a obwodem pasa u dzieci oty\u0142ych, im wi\u0119kszy by\u0142 obw\u00f3d pasa tym wy\u017cszy by\u0142 poziom insuliny . Podobne obserwacje przedstawiaj\u0105 inni autorzy , 33. OtyBrak dostatecznej wiedzy w\u015br\u00f3d rodzic\u00f3w oty\u0142ych dzieci dotycz\u0105cej konsekwencji zdrowotnych zwi\u0105zanych z nadmiern\u0105 mas\u0105 cia\u0142a wymaga podejmowania bardziej skutecznych dzia\u0142a\u0144 u\u015bwiadamiaj\u0105cych spo\u0142ecze\u0144stwo o skutkach nieprawid\u0142owego stylu \u017cycia.Oty\u0142o\u015b\u0107 i zaburzenia lipidowe cz\u0119\u015bciej dotycz\u0105 dzieci obci\u0105\u017conych rodzinnie oty\u0142o\u015bci\u0105 i chorobami uk\u0142adu kr\u0105\u017cenia.Zaburzenia lipidowe statystycznie cz\u0119\u015bciej wyst\u0119puj\u0105 u dzieci oty\u0142ych ni\u017c u dzieci z prawid\u0142ow\u0105 mas\u0105 cia\u0142a.Grubo\u015b\u0107 kompleksu \u015bciany t\u0119tnicy szyjnej wewn\u0119trznej i \u015brodkowej jest znamiennie wi\u0119ksza u dzieci oty\u0142ych w por\u00f3wnaniu z grup\u0105 kontrolna co sugeruje, \u017ce zmiany mia\u017cd\u017cycowe struktury naczy\u0144 szyjnych mog\u0105 wyst\u0105pi\u0107 u dzieci oty\u0142ych ju\u017c we wczesnym dzieci\u0144stwie.Wczesne zmiany mia\u017cd\u017cycowe u dzieci pozwala wykry\u0107 nieinwazyjna ultrasonograficzna metoda oceny grubo\u015bci kompleksu b\u0142on wewn\u0119trznej i \u015brodkowej t\u0119tnicy szyjnej.Dzieci z oty\u0142o\u015bci\u0105, szczeg\u00f3lnie trzewn\u0105 maj\u0105 wy\u017cszy poziom insuliny co mo\u017ce przyczyni\u0107 si\u0119 do rozwoju insulinooporno\u015bci.Wczesne leczenie dzieci oty\u0142ych mo\u017ce zahamowa\u0107 proces rozwoju mia\u017cd\u017cycy i zapobiec rozwojowi chor\u00f3b uk\u0142adu kr\u0105\u017cenia u tych dzieci w przysz\u0142o\u015bci.Brak dostatecznej wiedzy w\u015br\u00f3d rodzic\u00f3w oty\u0142ych dzieci dotycz\u0105cej konsekwencji zdrowotnych zwi\u0105zanych z nadmiern\u0105 mas\u0105 cia\u0142a wymaga podejmowania bardziej skutecznych dzia\u0142a\u0144 u\u015bwiadamiaj\u0105cych spo\u0142ecze\u0144stwo o skutkach nieprawid\u0142owego stylu \u017cycia prowadz\u0105cego do oty\u0142o\u015bci i jej powik\u0142a\u0144."}
{"text": "Agudelo et al. discover expanded clones of similar antibodies from the memory B cells of tick-borne encephalitis virus\u2013infected and vaccinated donors; however, the most potent and broadly neutralizing antibodies derive from donors recovering from natural infection. 50s below 1 ng/ml, were found only in individuals who recovered from natural infection. These antibodies also neutralized other tick-borne flaviviruses, including Langat, louping ill, Omsk hemorrhagic fever, Kyasanur forest disease, and Powassan viruses. Structural analysis revealed a conserved epitope near the lateral ridge of EDIII adjoining the EDI\u2013EDIII hinge region. Prophylactic or early therapeutic antibody administration was effective at low doses in mice that were lethally infected with TBEV.Tick-borne encephalitis virus (TBEV) is an emerging human pathogen that causes potentially fatal disease with no specific treatment. Mouse monoclonal antibodies are protective against TBEV, but little is known about the human antibody response to infection. Here, we report on the human neutralizing antibody response to TBEV in a cohort of infected and vaccinated individuals. Expanded clones of memory B cells expressed closely related anti-envelope domain III (EDIII) antibodies in both groups of volunteers. However, the most potent neutralizing antibodies, with IC Tick-borne flaviviruses are responsible for a series of emerging infectious diseases including fatal encephalitis. Like other flaviviruses, the tick-borne encephalitis virus (TBEV) envelope protein (E) is composed of three structural domains . There is no specific therapy for TBE, and treatment is limited to supportive care. For those individuals who survive, long-term sequelae are common .Although TBEV vaccines are available, immunity requires regular boosting, and vaccination is less effective in the young and elderly. Vaccination requires administration of three separate doses spaced over up to 2 yr, with booster doses recommended at intervals of 3\u20135 yr . BreakthRecovered individuals can produce potent serologic neutralizing activity against TBEV, and the serum can be cross-reactive against other tick-borne flaviviruses, but the nature of the antibodies that mediate these effects is not known . PostexpHere, we report on the molecular properties of human anti-TBEV antibodies with exceptional neutralizing potency and breadth that are effective in protection and early therapy in mice.We analyzed sera from 141 individuals hospitalized with TBE during the 2011 and 2018 outbreaks in the Czech Republic . Samples were obtained at the time of hospitalization during the encephalitic phase of disease . In agre5 dilution for neutralization using luciferase-expressing TBEV reporter virus particles  for the top 28 infected individuals varied from 0.37 to 6.7 \u00d7 106 . In total, 776 IgG antibody heavy and light chain gene pairs were amplified by RT-PCR and sequenced , Far Eastern (TBEVFE), and Siberian  ranging from 0.2 to 12 ng/ml  as low as 0.02 ng/ml (50 of 8.3 ng/ml). Seven antibodies, all isolated from infected donors, were potent neutralizers of TBEV RVPs with IC50s below 1 ng/ml  for binding to the EDIIIs of Langat virus (LGTV), LIV, OHFV, KFDV, and Powassan lineage I and II viruses  and light chain (LC) contacts to the EDI\u2013EDIII hinge and the BC loop and light chain contacts to the DE loop of the EDIII (2 of surface area on the EDIII (333 \u00c52 by the VH and 265 \u00c52 by the VL). T025 inserts Asp100HC and Trp94LC into a cleft in the EDIII, making a salt bridge (Asp100HC\u2013Lys311EDIII) and hydrogen bonds to the EDIII , consistent with 100% sequence conservation between these three strains of virus in the EDIII epitope residues  from the broad and potent antibody T025 (IGVH3\u201330/IGVK3\u201315) in complex with the EDIII domains of all three subtypes of TBEV . The strhe EDIII . The anthe EDIII . Crystalresidues .WE structure to a 3.9 \u00c5 cryo-EM structure of a mouse monoclonal antibody (19/1786) bound to the TBEV virion . All mice treated with the isotype control antibody died by day 10 (n = 6). In contrast, even the lowest dose of T025 was protective; all but 1 of the 24 mice receiving the antibody survived  24 h before challenge with 10 d later . All 12 Tick-borne flaviviruses can cause a fulminant encephalitis for which there is no effective therapy. This group of viruses are a growing public health concern in Europe, Asia, and North America. Among disease-causing tick-borne flaviviruses, TBEV is prevalent in Central Europe and Russia. Although there is a great deal of information on the polyclonal humoral immune response to TBEV , there iHuman neutralizing antibody responses to pathogens frequently converge on the same IGV genes. Examples include neutralizing antibodies to HIV-1, influenza, Zika, hepatitis B, and SARS-CoV-2 viruses . Antibod50s ranging from 12\u20131,180 ng/ml . VH1\u201369 antibodies are highly represented in the human repertoire and are also common among broadly neutralizing antibodies to influenza, hepatitis C, and HIV-1 viruses , and they were only found in convalescent individuals. The absence of this class of potent antibodies in the vaccinees examined is consistent with the lower levels of serum neutralizing potency in this group. Finally, VH3\u201348 antibodies are also potent neutralizers of several related tick-borne flaviviruses including KFDV, LGTV, LIV, and OHFV, with IC50s of 1\u201336 ng/ml.Among the neutralizing antibodies to TBEV, VH1\u201369 and VH3\u201348 were recurrent in multiple donors. VH1\u201369 was paired with a variety of light chain genes to produce neutralizing antibodies that were found among vaccinees and recovered individuals. This group of antibodies varied broadly in neutralizing activity with IC viruses . Anti-TBAntibodies to a number of different flaviviruses, including dengue and Zika virus, can be protective if administered before and even after infection . In RussAntibodies against EDIII are often among the most potent raised during flavivirus infection . The broWhile most antibodies to flavivirus EDIII are thought to be virus specific , T025 isAvailable TBEV vaccines were developed over 30 yr ago and consist of inactivated virus grown on chick embryo cells. Vaccination is TBEV specific, requires priming and two boosts, and results in 90\u2013100% seroconversion depending on the vaccine used . AdditioSamples of peripheral blood were obtained upon consent from individuals previously hospitalized with confirmed TBEV infection or individuals previously vaccinated against TBEV in \u010cesk\u00e9 Bud\u011bjovice, Czech Republic, under protocols approved by the ethical committees of the Hospital in \u010cesk\u00e9 Bud\u011bjovice , the Biology Center of the Czech Academy of Sciences , and The Rockefeller University (IRB DRO-0984). Clinical data were obtained at the treating hospital, and severity of disease was evaluated according to the following scale: mild, flu-like symptoms with meningeal irritation defined as meningitis, characterized by fever, fatigue, nausea, headache, back pain, arthralgia/myalgia and neck or back stiffness; moderate, previous symptoms together with tremor, vertigo, somnolence, and photophobia defined as meningoencephalitis; severe, prolonged neurological consequences including ataxia, titubation, altered mental status, memory loss, quantitative disturbance of consciousness, and palsy revealed as encephalitis, encephalomyelitis, or encephalomyeloradiculitis .Peripheral blood mononuclear cells (PBMCs) were obtained by gradient centrifugation using Ficoll and stored in liquid nitrogen in freezing media (90% FCS and 10% DMSO). Prior to experiments, aliquots of sera  were heat-inactivated at 56\u00b0C for 1 h and then stored at 4\u00b0C.Escherichia coli and purified from inclusion bodies as previously reported  or 301\u2013397 for strains Sofjin  and Vasilchenko  were used to produce untagged EDIII proteins or EDIII proteins containing a C-terminal 6XHis-Avitag. Constructs encoding untagged EDIIIs of other tick-borne flaviviruses were constructed similarly . Expression plasmids were transformed into BL21(DE3) E. coli and induced with 1 mM isopropyl \u03b2-D-1-thiogalactopyranoside at 37\u00b0C for 4 h. The cells were lysed and the insoluble fraction containing inclusion bodies was solubilized and refolded in 400 mM L-arginine, 100 mM Tris-base, pH 8.0, 2 mM EDTA, 0.2 mM phenyl-methylsulfonyl fluoride, 5 mM reduced and 0.5 mM oxidized glutathione, and 10% glycerol at 4\u00b0C. Refolded protein was purified by size exclusion chromatography  in 20 mM Tris, pH 8.0, 150 mM NaCl, and 0.02% NaN3. EDIIIs were concentrated to 10\u201320 mg/ml.EDIII antigens were expressed in reported . Express3. Fabs were concentrated to \u223c15 mg/ml.T025 Fabs for structural studies were produced and purified as described in previous studies . Brieflyhttps://github.com/stratust/igpipeline). Nucleotide somatic hypermutation and CDR3 length were also analyzed using in-house R and Perl scripts, as described previously , based on 776 IGH CDR3 sequences from this study and 22,654,256 IGH CDR3 sequences from public databases of memory B cell receptor sequences  and conjugated to streptavidin-PE  and streptavidin-Alexa Fluor 647 . Ovalbumin  was biotinylated using the EZ Sulfo-NHS-LC-Biotinylation kit according to the manufacturer\u2019s instructions  and conjugated to streptavidin BV711 . Biotinylation was confirmed by ELISA before use in flow cytometry.Avi-tagged TBEV\u2212CD8\u2212CD14\u2212CD16\u2212ZombieNIR\u2212CD20+Ova\u2212EDIII-PE+EDIII-AF647+ B cells were sorted using a FACS Aria III (Becton Dickinson) into individual wells of 96-well plates. Each well contained 4 \u00b5l of a lysis buffer comprising 0.5\u00d7 PBS, 10 mM DTT, and 3,000 U/ml RNasin Ribonuclease Inhibitors . Sorted cells were snap-frozen on dry ice and then stored at \u221280\u00b0C. Antibody sequences are derived from memory B cells because they originate from small CD20+ cells, and the antibody genes were PCR amplified using IgG-specific primers.PBMCs from sample 111 were enriched for B cells via positive selection using CD19 microbeads . PBMCs from all other donors were enriched for B cells by negative selection . All selection protocols were performed according to the manufacturer\u2019s instructions. Enriched B cells were incubated for 30 min on ice in FACS buffer  with fluorophore-labeled EDIII and ovalbumin, and in the presence of anti-human antibodies anti-CD3-APC-eFluro 780 , anti-CD8-APC-eFluro 780 , anti-CD14-APC-eFluro 780 , anti-CD16-APC-eFluro 780 , anti-CD20-PECy7 , and Zombie NIR . Single CD3RNA from single cells was reverse transcribed using SuperScript III Reverse transcription . The resulting cDNA was stored at \u221220\u00b0C until amplification of the variable Ig heavy (IGH), Ig light (IGL), and Ig kappa (IGK) genes by nested PCR followed by Sanger sequencing. Amplicons from the first PCR reaction were used as template for nested PCR-amplification followed by sequence- and ligation-independent cloning into antibody expression vectors as previously described . RecombiA West Nile virus subgenomic replicon-expressing plasmid encoding Renilla luciferase (pWNVII-Rep-REN-IB) and a ZIKV CprME expression plasmid had previously been obtained from Ted Pierson , was amplified with primers DFRp1532 (5\u2032-GGA\u200bATT\u200bCGC\u200bGGC\u200bCGC\u200bCTC\u200bAGG-3\u2032) and DFRp1533 (5\u2032-GCG\u200bGCT\u200bCGA\u200bGTT\u200bAAT\u200bTAA-3\u2032) before cloning at the NotI and PacI sites of plasmid pPOWV-LB-CprME (see below), resulting in pTBEV-WE-CprME.Synthetic DNA with CprME coding sequence  corresponding to TBEV, Western European subtype strain Neudoerfl (GenBank accession no. L06436.1 with six synonymous changes (in lowercase and bold) to reduce complexity; 5\u2032-CTA\u200bCTT\u200bGGC\u200bAGT\u200bACA\u200bTCT\u200bACG\u200bTAT\u200bTAG\u200bTCA\u200bTCG\u200bCTA\u200bTTA\u200bCCA\u200bTGG\u200bTGA\u200bTGC\u200bGGT\u200bTTT\u200bGGC\u200bAGT\u200bACA\u200bTCA\u200bATG\u200bGGC\u200bGTG\u200bGAT\u200bAGC\u200bGGT\u200bTTG\u200bACT\u200bCAC\u200bGGG\u200bGAT\u200bTTC\u200bCAA\u200bGTC\u200bTCC\u200bACC\u200bCCA\u200bTTG\u200bACG\u200bTCA\u200bATG\u200bGGA\u200bGTT\u200bTGT\u200bTTT\u200bGGC\u200bACC\u200bAAA\u200bATC\u200bAAC\u200bGGG\u200bACT\u200bTTC\u200bCAA\u200bAAT\u200bGTC\u200bGTA\u200bACA\u200bACT\u200bCCG\u200bCCC\u200bCAT\u200bTGA\u200bCGC\u200bAAA\u200bTGG\u200bGCG\u200bGTA\u200bGGC\u200bGTG\u200bTAC\u200bGGT\u200bGGG\u200bAGG\u200bTCT\u200bATA\u200bTAA\u200bGCA\u200bGAG\u200bCTC\u200bTCT\u200bGGC\u200bTAA\u200bCTA\u200bGAG\u200bAAC\u200bCCACT\u200bGCT\u200bTAC\u200bTGG\u200bCTT\u200bATC\u200bGAA\u200bATT\u200bAAT\u200bACG\u200bACT\u200bCAC\u200bTAT\u200bAGG\u200bGAG\u200bACC\u200bCAA\u200bGCT\u200bGGC\u200bTAG\u200bTTA\u200bAGC\u200bTAT\u200bCAA\u200bCAA\u200bGGA\u200bATT\u200bCGC\u200bGGC\u200bCGC\u200bCAG\u200bGCTATG\u200bATG\u200bACC\u200bACT\u200bTCT\u200bAAA\u200bGGA\u200bAAG\u200bGGG\u200bGGC\u200bGGT\u200bCCC\u200bCCT\u200bAGG\u200bCGC\u200bAAG\u200bCTT\u200bAAA\u200bGTG\u200bACC\u200bGCA\u200bAAT\u200bAAG\u200bTCG\u200bCGA\u200bCCA\u200bGCA\u200bACG\u200bAGC\u200bCCA\u200bATG\u200bCCA\u200bAAG\u200bGGC\u200bTTC\u200bGTG\u200bCTG\u200bTCG\u200bCGC\u200bATG\u200bCTG\u200bGGG\u200bATT\u200bCTT\u200bTGG\u200bCAC\u200bGCC\u200bGTG\u200bACA\u200bGGC\u200bACG\u200bGCC\u200bAGA\u200bCCC\u200bCCA\u200bGTG\u200bCTG\u200bAAA\u200bATG\u200bTTC\u200bTGG\u200bAAA\u200bACG\u200bGTA\u200bCCA\u200bCTG\u200bCGC\u200bCAG\u200bGCG\u200bGAG\u200bGCT\u200bGTT\u200bCTG\u200bAAG\u200bAAG\u200bATA\u200bAAG\u200bAGA\u200bGTT\u200bATC\u200bGGG\u200bAAC\u200bTTG\u200bATG\u200bCAG\u200bAGC\u200bCTT\u200bCAC\u200bATG\u200bAGA\u200bGGG\u200bCGT\u200bCGC\u200bAGG\u200bTCA\u200bGGT\u200bGTG\u200bGAC\u200bTGG\u200bACT\u200bTGG\u200bATT\u200bTTT\u200bTTG\u200bACG\u200bATG\u200bGCG\u200bTTG\u200bATG\u200bACC\u200bATG\u200bGCC\u200bATG\u200bGCA\u200bACC\u200bACC\u200bATC\u200bCAC\u200bCGG\u200bGAC\u200bAGG\u200bGAA\u200bGGA\u200bTAC\u200bATG\u200bGTT\u200bATG\u200bCGG\u200bGCC\u200bAGT\u200bGGA\u200bAGG\u200bGAC\u200bGCT\u200bGCA\u200bAGC\u200bCAG\u200bGTC\u200bAGG\u200bGTA\u200bCAA\u200bAAC\u200bGGA\u200bACG\u200bTGC\u200bGTC\u200bATC\u200bCTG\u200bGCA\u200bACA\u200bGAC\u200bATG\u200bGGA\u200bGAG\u200bTGG\u200bTGT\u200bGAA\u200bGAT\u200bTCA\u200bATC\u200bACC\u200bTAC\u200bTCT\u200bTGC\u200bGTC\u200bACG\u200bATT\u200bGAC\u200bCAG\u200bGAG\u200bGAA\u200bGAA\u200bCCC\u200bGTT\u200bGAC\u200bGTG\u200bGAC\u200bTGC\u200bTTC\u200bTGC\u200bCGA\u200bGGT\u200bGTT\u200bGAT\u200bAGG\u200bGTT\u200bAAG\u200bTTA\u200bGAG\u200bTAT\u200bGGA\u200bCGC\u200bTGT\u200bGGA\u200bAGG\u200bCAA\u200bGCT\u200bGGA\u200bTCT\u200bAGG\u200bGGG\u200bAAA\u200bAGG\u200bTCT\u200bGTG\u200bGTC\u200bATT\u200bCCA\u200bACA\u200bCAT\u200bGCA\u200bCAA\u200bAAA\u200bGAC\u200bATG\u200bGTC\u200bGGG\u200bCGA\u200bGGT\u200bCAT\u200bGCA\u200bTGG\u200bCTT\u200bAAA\u200bGGT\u200bGAC\u200bAAT\u200bATT\u200bCGA\u200bGAT\u200bCAT\u200bGTC\u200bACC\u200bCGA\u200bGTC\u200bGAG\u200bGGC\u200bTGG\u200bATG\u200bTGG\u200bAAG\u200bAAC\u200bAAG\u200bCTT\u200bCTA\u200bACT\u200bGCC\u200bGCC\u200bATT\u200bGTG\u200bGCC\u200bTTG\u200bGCT\u200bTGG\u200bCTC\u200bATG\u200bGTT\u200bGAT\u200bAGT\u200bTGG\u200bATG\u200bGCC\u200bAGA\u200bGTG\u200bACT\u200bGTC\u200bATC\u200bCTC\u200bTTG\u200bGCG\u200bTTG\u200bAGT\u200bCTA\u200bGGG\u200bCCA\u200bGTG\u200bTAC\u200bGCC\u200bACG\u200bAGG\u200bTGC\u200bACG\u200bCAT\u200bCTT\u200bGAG\u200bAAC\u200bAGA\u200bGAT\u200bTTT\u200bGTG\u200bACA\u200bGGA\u200bACT\u200bCAA\u200bGGG\u200bACC\u200bACC\u200bAGA\u200bGTG\u200bTCC\u200bCTA\u200bGTT\u200bTTG\u200bGAA\u200bCTT\u200bGGA\u200bGGC\u200bTGC\u200bGTG\u200bACC\u200bATC\u200bACA\u200bGCT\u200bGAG\u200bGGC\u200bAAG\u200bCCA\u200bTCC\u200bATT\u200bGAT\u200bGTA\u200bTGG\u200bCTC\u200bGAA\u200bGAC\u200bATT\u200bTTT\u200bCAG\u200bGAA\u200bAGC\u200bCCG\u200bGCT\u200bGAA\u200bACC\u200bAGA\u200bGAA\u200bTAC\u200bTGC\u200bCTG\u200bCAC\u200bGCC\u200bAAA\u200bTTG\u200bACC\u200bAAC\u200bACA\u200bAAA\u200bGTG\u200bGAG\u200bGCT\u200bCGC\u200bTGT\u200bCCA\u200bACC\u200bACT\u200bGGA\u200bCCG\u200bGCG\u200bACA\u200bCTT\u200bCCG\u200bGAG\u200bGAG\u200bCAT\u200bCAG\u200bGCT\u200bAAT\u200bATG\u200bGTG\u200bTGC\u200bAAG\u200bAGA\u200bGAC\u200bCAA\u200bAGC\u200bGAC\u200bCGT\u200bGGA\u200bTGG\u200bGGA\u200bAAC\u200bCAC\u200bTGtGGaTTcTTcGGG\u200bAAG\u200bGGC\u200bAGT\u200bATA\u200bGTG\u200bGCT\u200bTGT\u200bGCA\u200bAAG\u200bTTT\u200bGAA\u200bTGC\u200bGAG\u200bGAA\u200bGCA\u200bAAA\u200bAAA\u200bGCT\u200bGTG\u200bGGC\u200bCAC\u200bGTC\u200bTAT\u200bGAC\u200bTCC\u200bACA\u200bAAG\u200bATC\u200bACG\u200bTAT\u200bGTT\u200bGTC\u200bAAG\u200bGTT\u200bGAG\u200bCCC\u200bCAC\u200bACA\u200bGGG\u200bGAT\u200bTAC\u200bTTG\u200bGCT\u200bGCA\u200bAAT\u200bGAG\u200bACC\u200bAAT\u200bTCA\u200bAAC\u200bAGG\u200bAAA\u200bTCA\u200bGCA\u200bCAG\u200bTTT\u200bACG\u200bGTG\u200bGCA\u200bTCC\u200bGAG\u200bAAA\u200bGTG\u200bATC\u200bCTG\u200bCGG\u200bCTC\u200bGGC\u200bGAC\u200bTAT\u200bGGA\u200bGAT\u200bGTG\u200bTCG\u200bCTG\u200bACG\u200bTGT\u200bAAA\u200bGTG\u200bGCA\u200bAGT\u200bGGG\u200bATT\u200bGAT\u200bGTC\u200bGCC\u200bCAA\u200bACT\u200bGTG\u200bGTG\u200bATG\u200bTCA\u200bCTC\u200bGAC\u200bAGC\u200bAGC\u200bAAG\u200bGAC\u200bCAC\u200bCTG\u200bCCT\u200bTCT\u200bGCA\u200bTGG\u200bCAA\u200bGTG\u200bCAC\u200bCGT\u200bGAC\u200bTGG\u200bTTT\u200bGAG\u200bGAC\u200bTTG\u200bGCG\u200bCTG\u200bCCC\u200bTGG\u200bAAA\u200bCAC\u200bAAG\u200bGAC\u200bAAC\u200bCAA\u200bGAT\u200bTGG\u200bAAC\u200bAGT\u200bGTG\u200bGAG\u200bAAA\u200bCTT\u200bGTG\u200bGAA\u200bTTT\u200bGGA\u200bCCA\u200bCCA\u200bCAT\u200bGCT\u200bGTG\u200bAAA\u200bATG\u200bGAT\u200bGTT\u200bTTC\u200bAAT\u200bCTG\u200bGGG\u200bGAC\u200bCAG\u200bACG\u200bGCT\u200bGTG\u200bCTG\u200bCTC\u200bAAA\u200bTCA\u200bCTG\u200bGCA\u200bGGA\u200bGTT\u200bCCG\u200bCTG\u200bGCC\u200bAGT\u200bGTG\u200bGAG\u200bGGC\u200bCAG\u200bAAA\u200bTAC\u200bCAC\u200bCTG\u200bAAA\u200bAGC\u200bGGC\u200bCAT\u200bGTT\u200bACT\u200bTGT\u200bGAT\u200bGTG\u200bGGA\u200bCTG\u200bGAA\u200bAAG\u200bCTG\u200bAAA\u200bCTG\u200bAAA\u200bGGC\u200bACA\u200bACC\u200bTAC\u200bTCC\u200bATG\u200bTGT\u200bGAC\u200bAAA\u200bGCA\u200bAAG\u200bTTC\u200bAAA\u200bTGG\u200bAAG\u200bAGA\u200bGTT\u200bCCT\u200bGTG\u200bGAC\u200bAGC\u200bGGC\u200bCAT\u200bGAC\u200bACA\u200bGTA\u200bGTC\u200bATG\u200bGAG\u200bGTA\u200bTCA\u200bTAC\u200bACA\u200bGGA\u200bAGC\u200bGAC\u200bAAG\u200bCCA\u200bTGT\u200bCGG\u200bATC\u200bCCG\u200bGTG\u200bCGG\u200bGCT\u200bGTG\u200bGCA\u200bCAT\u200bGGT\u200bGTC\u200bCCA\u200bGCG\u200bGTT\u200bAAT\u200bGTA\u200bGCC\u200bATG\u200bCTC\u200bATA\u200bACC\u200bCCC\u200bAAT\u200bCCA\u200bACC\u200bATT\u200bGAA\u200bACA\u200bAAT\u200bGGT\u200bGGC\u200bGGA\u200bTTC\u200bATA\u200bGAA\u200bATG\u200bCAG\u200bCTG\u200bCCA\u200bCCA\u200bGGG\u200bGAT\u200bAAC\u200bATC\u200bATC\u200bTAT\u200bGTG\u200bGGA\u200bGAC\u200bCTT\u200bAGC\u200bCAG\u200bCAG\u200bTGG\u200bTTT\u200bCAG\u200bAAA\u200bGGC\u200bAGT\u200bACC\u200bATT\u200bGGT\u200bAGA\u200bATG\u200bTTT\u200bGAA\u200bAAA\u200bACC\u200bCGC\u200bAGG\u200bGGA\u200bTTG\u200bGAA\u200bAGG\u200bCTC\u200bTCT\u200bGTG\u200bGTT\u200bGGA\u200bGAA\u200bCAT\u200bGCA\u200bTGG\u200bGAC\u200bTTT\u200bGGC\u200bTCA\u200bGTA\u200bGGC\u200bGGG\u200bGTA\u200bCTG\u200bTCT\u200bTCT\u200bGTG\u200bGGG\u200bAAG\u200bGCA\u200bATC\u200bCAC\u200bACG\u200bGTG\u200bCTG\u200bGGG\u200bGGA\u200bGCT\u200bTTC\u200bAAC\u200bACC\u200bCTT\u200bTTT\u200bGGtGGtGTT\u200bGGA\u200bTTC\u200bATC\u200bCCT\u200bAAG\u200bATG\u200bCTG\u200bCTG\u200bGGG\u200bGTT\u200bGCT\u200bCTG\u200bGTC\u200bTGG\u200bTTG\u200bGGA\u200bCTA\u200bAAT\u200bGCC\u200bAGG\u200bAAT\u200bCCA\u200bACG\u200bATG\u200bTCC\u200bATG\u200bACG\u200bTTT\u200bCTT\u200bGCT\u200bGTG\u200bGGG\u200bGCT\u200bTTG\u200bACA\u200bCTG\u200bATG\u200bATG\u200bACA\u200bATG\u200bGGA\u200bGTT\u200bGGG\u200bGCATAA\u200bTAG\u200bTTA\u200bATT\u200bAAC\u200bTCG\u200bAGC\u200bCGC\u200bGGT\u200bTCG\u200bAAG\u200bGTA\u200bAGC\u200bCT-3\u2032) was PCR amplified with primers DFRp1511 (5\u2032-ATC\u200bTAC\u200bGTA\u200bTTA\u200bGTC\u200bATC\u200bGCT\u200bATT\u200bA-3\u2032) and DFRp1514 (5\u2032-ACC\u200bGCG\u200bGCT\u200bCGA\u200bGTT\u200bAAT\u200bTAA-3\u2032) and cloned at the Eco105I and SacII sites of plasmid pZIKV-HPF-CprME (Synthetic DNA containing the CprME sequence (underlined) of POWV-LB strain , resulting in a fragment fusing the CMV promoter with POWV-DTV C-encoding sequences (bolded in primer RU-O-26690). Using as template DTVp1 (ATG\u200bGTG\u200bACT\u200bACT\u200bTCT\u200bAAA\u200bGGA\u200bAAG-3\u2032) and RU-O-26711 (5\u2032-GTT\u200bTCC\u200bCCA\u200bTCCTCTATCG\u200bCTC\u200bTG-3\u2032), with bolded nucleotides indicating synonymous mutations introduced to ablate the SacII site. DNA was amplified using DTVp1 as template and oligos RU-O-26710  and RU-O-26688 (5\u2032-TTC\u200bGAA\u200bCCG\u200bCGG\u200bCTG\u200bGGT\u200bCCT\u200bATT\u200bATGC\u200bTCC\u200bGAC\u200bTCC\u200bCAT\u200bTGT\u200bCAT\u200bCAT\u200bC-3\u2032) to generate a fragment overlapping the killed SacII site to the end of the E protein coding region followed by a SacII site. The three DNA fragments were annealed, extended and then PCR amplified using primers RU-O-24611 and RU-O-26688. The resulting DNA fragment was digested with SnaBI and SacII and cloned into similarly digested pZIKV-HPF-CprME to generate pPOWV-DTV-CprME.A three-piece assembly PCR strategy was used. DNA upstream of the CMV promoter in pZIKV-HFP-CprME to just downstream of the beginning of the C-encoding region was PCR amplified with primers RU-O-24611 (5\u2032-CTT\u200bGAC\u200bCGA\u200bCAA\u200bTTG\u200bCAT\u200bGAA\u200bG-3\u2032) and RU-O-26690 , was amplified with primers DFRp1532 and DFRp1533 before cloning at the NotI and PacI sites of plasmid pPOWV-LB-CprME (see above), resulting in pKFDV-W-377-CprME.Synthetic DNA with the CprME-flanked sequence of KFDV, strain W-377 (GenBank accession no. NC_003690 (A590G and A1893C).The CprME of LGTV, isolate TP21-636, was amplified from a plasmid kindly provided by Dr. Sonja Best  with primers DFRp1563 (5\u2032-GGA\u200bATT\u200bCGC\u200bGGC\u200bCGC\u200bCTC\u200bAGG\u200bATG\u200bGCC\u200bGGG\u200bAAG\u200bGCC\u200bGTT\u200bCTA-3\u2032) and DFRp1566 (5\u2032-CCG\u200bCGG\u200bCTC\u200bGAG\u200bTTA\u200bATT\u200bAAC\u200bTAT\u200bTAG\u200bGCT\u200bCCA\u200bACC\u200bCCC\u200bAGA\u200bGTC\u200bAT-3\u2032) before cloning at the NotI and PacI sites of plasmid pPOWV-LB-CprME, resulting in pLGTV-TP21-636-CprME. There are two nucleotide mutations from GenBank accession no. KP144331), was amplified with primers DFRp1532 and DFRp1533 before cloning at the NotI and PacI sites of plasmid pPOWV-LB-CprME, resulting in pLIV-LI3/1-CprME.Synthetic DNA with the CprME-flanked sequence of LIV, isolate LI3/1 (GenBank accession no. NC_005062), was amplified with primers DFRp1532 and DFRp1533 before cloning at the NotI and PacI sites of plasmid pPOWV-LB-CprME, resulting in pOHFV-CprME.Synthetic DNA with the CprME-flanked sequence of OHFV, strain Bogoluvovska  containing 20 mM Hepes and 10% FBS. For the next 72 h, in 24-h intervals, RVP-containing supernatants were harvested and filtered through a 0.45-\u03bcm filter and frozen at \u221280\u00b0C, and media were replaced with DMEM containing 20 mM Hepes and 10% FBS. Frozen RVPs were later thawed and titrated on Huh-7.5 cells to determine the dilution of RVPs at which cells express 106 relative light units in the absence of sera or antibody.RVPs were produced by cotransfecting 1 \u00b5g pWNVII-Rep-REN-IB plasmid with 3 \u00b5g of the flavivirus CprME plasmid of choice into the permissive cell line Lenti-X 293T using Lipofectamine 2000  according to the manufacturer\u2019s instructions. Cells were seeded 24 h previously at 1050 and IC50 were determined by nonlinear regression analysis using Prism software (GraphPad). In the cross-neutralization screening against the panel of flavivirus RVPs, recombinant antibodies were assayed at 1 \u00b5g/ml final concentration using the protocol described above, and the results were compared with no antibody control.96-well plates were seeded with 7,500 Huh-7.5 cells/well in 50 \u00b5l DMEM (Gibco) supplemented with 10% FBS and 1% nonessential amino acids. After 24 h, 100 \u00b5l diluted RVPs were combined with 100 \u00b5l diluted sera or antibody, incubated for 1 h at 37\u00b0C, and then 50 \u00b5l of the mix were added in triplicate to the plated cells. RVPs are diluted appropriately in BA-1 diluent (Medium 199 [Lonza] supplemented with 1% BSA and 100 units/ml penicillin/streptomycin) to achieve the desired relative light unit expression. After an additional 24 h of incubation at 37\u00b0C, media was aspirated off the cells, replaced with 35 \u00b5l lysis buffer , and the plates were frozen at \u221280\u00b0C. 15 \u00b5l of the subsequently thawed lysis buffer was used for Renilla luciferase expression measurement using the Renilla Luciferase Assay System . Sera were either diluted to 1:600,000 final concentration for TBEV RVP neutralization screening or serially diluted to generate curves. Recombinant monoclonal antibodies were used at 10 \u00b5g/ml final concentration and serially diluted 1:3 for neutralization assays. NT2 fragments conjugated to HRP  were diluted 1:5,000 in PBS-T, added to the plates, and incubated again for 1 h at room temperature. Between each step, the plates were washed with PBS-T four times. Plates were finally developed using TMB substrate ; the reaction was stopped using 1 M sulfuric acid and the plates read at 450 nm. Sera were screened for binding at 1:500 dilution. Recombinant monoclonal antibodies were diluted to 10 \u00b5g/ml and serially diluted 1:3; the half effective concentration (EC50) was determined by nonlinear regression analysis using Prism 8 (GraphPad). For cross-binding assays, recombinant antibodies were assayed at 1 \u00b5g/ml according to the protocol described above using the panel of flavivirus EDIII proteins. The anti-HIV monoclonal antibody 10\u20131074 was used as isotype control  were coated overnight with 250 ng of the EDIII protein in PBS per well at room temperature; plates were then blocked with 0.1 mM EDTA, 0.05% Tween, and 2% BSA in PBS for 2 h at room temperature. Samples were diluted in PBS-T, added to plates, and incubated for an additional 1 h at room temperature. Secondary goat anti-human-IgG Fab\u2032 control . Antibod control  were conhttp://www.arboviruscollection.cz/index.php?lang=en). The virus was originally isolated from the blood of a diseased 10-yr-old child in Brno, Czech Republic (formerly Czechoslovakia), in 1953. Prior to the use in in vitro and in vivo experiments, the virus was propagated in suckling mouse brains and/or BHK-21 cells.The low-passage TBEV strain Hypr was provided by the Collection of Arboviruses, Institute of Parasitology, Biology Centre of the Czech Academy of Sciences, Ceske Budejovice, Czech Republic (PS (porcine kidney stable) cells  were cul5 cells per well) were added to 24-well tissue culture plates. After 4 h of incubation at 37\u00b0C with 0.5% CO2, each well was overlaid with carboxymethylcellulose (1.5% in L-15 medium). After a 5-d incubation at 37\u00b0C and 0.5% CO2, the cell monolayers were visualized using naphthalene black. Viral titers were expressed as PFU per milliliter.To determine virus titer, plaque assays were performed as previously described , with sl4 PS cells were added per well. After 4-d incubation (37\u00b0C), the cytopathic effect was monitored microscopically, and cell viability was measured using the Cell Counting Kit-8 (Dojindo Molecular Technologies) according to the manufacturer\u2019s instructions. IC50 was calculated from two independent experiments done in octuplicates using GraphPad Prism .The virus neutralization test was performed as described previously , with seVirus neutralization was also assayed using a fluorescence-based virus neutralization test. Monoclonal antibodies  were incubated with TBEV, and PS cells were infected as described above. The cells were incubated for 48 h at 37\u00b0C. The cell monolayers were fixed with cold acetone/methanol (1:1), blocked with 10% FBS, and incubated with mouse anti-flavivirus antibody . After washing, the cells were labeled with secondary goat anti-mouse antibody conjugated to FITC  and counterstained with DAPI (diluted to 1 \u00b5g/ml) to visualize cell nuclei. The fluorescence signal was recorded with an Olympus IX71 epifluorescence microscope and processed by ImageJ software.Data were analyzed using Mann\u2013Whitney tests or ANOVA and Tukey\u2019s multiple comparison tests as specified and comparison of survival curves was analyzed by log-rank (Mantel\u2013Cox) test, calculated in GraphPad Prism (version 8.4.3). P values < 0.05 were considered significant.WE EDIII-His-Avitag complex  were obtained by combining 0.2 \u00b5l crystallization complex with 0.2 \u00b5l of 0.1 M sodium citrate tribasic dihydrate, pH 5.0, 10% PEG 6000 in sitting drops at 22\u00b0C. Crystals of T025 Fab\u2013TBEVFE EDIII-His-Avitag complex  were obtained by combining 0.2 \u00b5l crystallization complex with 0.2 \u00b5l of 0.1 M sodium citrate tribasic dihydrate, pH 5.0, 10% PEG 6000 in sitting drops at 22\u00b0C. Crystals of T025 Fab\u2013TBEVSi EDIII complex  were obtained by combining 0.2 \u00b5l crystallization complex with 0.2 \u00b5l of 5% (\u00b1)-2-methyl-2,4-pentanediol, 0.1 M Hepes, pH 7.5, 10% PEG 10,000 in sitting drops at 22\u00b0C. Crystals were cryoprotected with 25% glycerol before being cryopreserved in liquid nitrogen.Complexes for crystallization were produced by mixing Fab and antigen at a 1:1 molar ratio and incubating at room temperature for 1\u20132 h. Crystals of T025 Fab\u2013TBEVWE EDIII complex structure was solved by molecular replacement using the VHVL domains from PDB accession no. 2GHW, the CHCL domains from PDB accession no. 4OGX, and TBEV EDIII from PDB accession no. 6J5F as search models in PHASER . The protocol was approved by the Committee on the Ethics of Animal Experimentation of the Institute of Parasitology and the Departmental Expert Committee for the Approval of Projects of Experiments on Animals of the Czech Academy of Sciences (permit no. 4253/2019).Specific pathogen\u2013free BALB/c mice were obtained from ENVIGO RMS. Sterilized pellet diet and water were supplied ad libitum. In all experiments, female mice aged 6\u20138 wk were used. Mice were housed in individually ventilated plastic cages (Techniplast) with wood-chip bedding, with a constant temperature of 22\u00b0C, a relative humidity of 65%, and under a 12-h light/dark cycle. Three mice per group were used in experiments. Mice were inoculated i.p. 1 d before or 1, 3, or 5 d after infection with monoclonal antibodies T025 or 10\u20131074 in 200 \u03bcl PBS and infected subcutaneously with 100 PFU TBEV-Hypr (propagated eight times in suckling mouse brains). Mice were monitored for symptoms and survival over time and euthanized when reaching a humane endpoint.FE and TBEVSi by ELISA, screening results for antibody binding and neutralization of related tick-borne flaviviruses, and neutralization curves for selected antibodies against related tick-borne flaviviruses. 50 and IC50 values for all antibodies tested. Table S1lists clinical data from TBEV-infected and vaccinated donors.Click here for additional data file.Table S2lists sequences of anti-TBEV-EDIII IgG antibodies.Click here for additional data file.Table S3lists highly similar antibody sequences shared across individuals.Click here for additional data file.Table S4lists sequences of cloned recombinant monoclonal antibodies.Click here for additional data file.Table S5lists effective and inhibitory concentrations of recombinantly expressed monoclonal antibodies.Click here for additional data file.Table S6lists data collection and refinement statistics for the crystal structures.Click here for additional data file."}
{"text": "In the column headed \u201cGSH cohort: Controls (n\u2009=\u200971)\u201d and in the row \u201cPUT, more affected\u201d, the value was incorrectly stated as \u201c9.07\u2009\u00b1\u20091.27\u201d, where the correct value is \u201c8.07\u2009\u00b1\u20091.27\u201d. The original article has been corrected.In Table 1 of this article, the value in the column headed \u201cGSH cohort: Controls ("}
{"text": "Page 11: In Table 1, columns headed \u201cResistance phenotype\u201d and \u201cAntibiotic resistance gene(s),\u201d all instances of \u201cAmp\u201d and \u201cblaTEM-1B\u201d, respectively, should not be boldface.Volume 6, no. 4, e00729-21, 2021,"}
{"text": "We look at fractional Langevin equations (FLEs) with generalized proportional Hadamard\u2013Caputo derivative of different orders. Moreover, nonlocal integrals and nonperiodic boundary conditions are considered in this paper. For the proposed equations, the Hyres\u2013Ulam (HU) stability, existence, and uniqueness (EU) of the solution are defined and investigated. In implementing our results, we rely on two important theories that are Krasnoselskii fixed point theorem and Banach contraction principle. Also, an application example is given to bolster the accuracy of the acquired results. In recent years, fractional calculus has gained great importance by numerous renowned mathematicians. The most essential feature of this topic is that it allows us to execute integrations and differentiations in any order, not necessarily integer ones Such a benefit has been encouraged by the applications in various areas, conceivably including fractal phenomena which appear in many sciences such as physics and engineering. Also, discovering the fractional derivatives and their generalizations was done by well-known mathematicians such as Riemann, Caputo, Hadamard, Euler, Liouville, Laplace, Laurent, and Fourier and was constantly the major path of research in the fractional calculus area. Moreover, these derivatives will provide us new chances to get generalized solutions of fractional differential equations \u20136. It isAccording to varied literature, like \u201317, it hOur paper arranged as follows. In \ud835\udd38\u2102r to refer to all absolutely continuous functions \u03be such that its derivative of order (r \u2212 1) is absolutely continuous on , where r \u2208 \u2115 and e is the renowned Euler's number.We use the set \u03be : +1 and q > 0.The qth-order fractional HCD is given byThe q.Rahman et al.  recentlyqth-order generalized proportional Hadamard fractional integral is given byq > 1 and \u03f1 \u2208 , and \u03c7 \u2208 C.The qth-order of the one side of a broader range of fractional proportional derivative is given bym=[q]+1, \u03f1 \u2208 , and q > 0.The qth-order of the one side of Caputo-broader range of fractional proportional derivative is given bym=[q]+1, \u03f1 \u2208 , and q > 0.The \u03c7(t)=ln(t) in Definitions Obviously, if we put By using Proposition 3.1, Proposition 4.1, and Remark 1.1 in , we obtaq, p \u2208 \u2102 such that Re(p) > 0 and Re(q) > 0. Then, for any \u03f1 \u2208  and m=[Re(q)]+1, we haveq,\u03f1et((\u03f1 \u2212 1)/\u03f1)ln\u2009(ln\u2009t)p\u22121)(\u03d6)=(\u0393(p)/\u03f1q\u0393(p+q))e\u03d6((\u03f1 \u2212 1)/\u03f1)ln\u2009(ln\u2009\u03d6)q+p\u22121/\u03f1)ln\u2009(ln\u2009t)p\u22121)(\u03d6)=(\u03f1q\u0393(p)/\u0393(p \u2212 q))e\u03d6((\u03f1 \u2212 1)/\u03f1)ln\u2009(ln\u2009\u03d6)p\u2212q\u22121 > 0, \u03f1 \u2208 , m=[Re(q)]+1, \u03be \u2208 L1, and (\u03d6) \u2208 \ud835\udd38\u2102m. Then,Let Ml=/\u0393(l+1)\u03f1l).Also, the fractional differential equationAlso,m \u2212 r by l has been modified. Hence, the solution of the FDE (By applying equation  in 12),,12), we  the FDE  is givenThe solution of the following fractional Langevin equation in the integral formq,\u03f1 to equation =0\u21d2M4=0, z\u2032(1)=0\u21d2M3=0, and z\u2033(1)=0\u21d2M2=0.From Therefore,From the conditions,t=1 in t=e in  is a Bananch space and its norm is defined by \u2016\ud835\udd23\u2016=supt\u2208\ud835\udd23(\ud835\udd31) such that, for each \ud835\udd23 \u2208 \ud835\udc9e, it implies that \ud835\udd23 : \u27f6\u211d is continuous.Suppose that the space )| \u2264 k(t)\u2200 ) \u2208  \u00d7 \u211d\u2009and\u2009k(t) \u2208 , with\u2009supt|k(t)|=\u2016k\u2016|\u03be) \u2212 \u03be)| \u2264 \u03c7|z(t) \u2212 z1(t)|\u2200 t \u2208 , z, z1 \u2208 \ud835\udc9e\u2009and\u2009\u03c7 > 0|Let us imagine that the continuous function e] ifThen, the FLEs  has a mi\ud835\udd2f=(\u03d21\u2016k\u2016/(1 \u2212 \u03d22)), whereChoose Define the closed ball By combining the operators e].Keep this in mind First, we will show that et((\u03f1 \u2212 1)/\u03f1)ln\u2009 \u2264 et \u2212 ln\u2009u)((\u03f1 \u2212 1)/\u03f1) such thatSecond, we will demonstrate that et((\u03f1 \u2212 1)/\u03f1)ln\u2009 \u2264 et \u2212 ln\u2009u)((\u03f1 \u2212 1)/\u03f1)((\u03f1 \u2212 1)/\u03f1)ln( < et1/u))((\u03f1 \u2212 1)/\u03f1))((\u03f1 \u2212 1)/\u03f1) \u2212 \ud835\udd0d1z(t2)|\u27f60.Taking the limit as \ud835\udd0d1 is an equicontinuous as a result of this. We may conclude that \ud835\udd0d1 compacts on z so that z=\ud835\udd0d1z. As a consequence, equation , (II) and\u2135=supt\u2208|\u03be| < \u221e\u2009 (I3) Let us imagine that the continuous function e].Ifthe FLEs  have a u\ud835\udd2f1 > (\u03d21\u2135/(1 \u2212 \u03d21 \u2212 \u03d22)), to define the closed ball Choose \ud835\udd0d is contractive.u \u2208  and For each Therefore,et2/u)((\u03f1 \u2212 1)/\u03f1)ln( < et1/u))((\u03f1 \u2212 1)/\u03f1))((\u03f1 \u2212 1)/\u03f1) which satisfiesIt is said that the integral equation  is HU st\u03be :  \u00d7 \u211d\u27f6\u211d satisfies inferences (I) and (II), then equation , (I2), and (I3) are fulfilled.\u03c7=(1/4), we get \u03d23=0.372486 < 1, \u03d22=0.0565441, and \u03d21=1.98731. Hence, all assumptions of By putting \u03c7\u03d21+\u03d22=0.553373 < 1, this implies that On the other hand, Fractional Langevin equations have a considerable role in modeling varied physical phenomena. For instance, they have been employed for describing single-file prevalence  and the"}
{"text": "Helicobacter pylori (H.\u00a0pylori) uses several outer membrane proteins for adhering to its host's gastric mucosa, an important step in establishing and preserving colonization. Several adhesins  have been characterized in terms of their three-dimensional structure. A recent addition to the growing list of outer membrane porins is LabA (LacdiNAc-binding adhesin), which is thought to bind specifically to GalNAc\u03b21-4GlcNAc, occurring in the gastric mucosa. LabA47-496 protein expressed as His-tagged protein in the periplasm of E.\u00a0coli and purified via subtractive IMAC after TEV cleavage and subsequent size exclusion chromatography, resulted in bipyramidal crystals with good diffraction properties. Here, we describe the 2.06\u00a0\u200b\u00c5 resolution structure of the exodomain of LabA from H.\u00a0pylori strain J99 (PDB ID: 6GMM). Strikingly, despite the relatively low levels of sequence identity with the other three structurally characterized adhesins (20\u201349%), LabA shares an L-shaped fold with SabA and BabA. The \u2018head\u2019 region contains a 4\u00a0\u200b+\u00a0\u200b3 \u03b1-helix bundle, with a small insertion domain consisting of a short antiparallel beta sheet and an unstructured region, not resolved in the crystal structure. Sequence alignment of LabA from different strains shows a high level of conservation in the N- and C-termini, and identifies two main types based on the length of the insertion domain (\u2018crown\u2019 region), the \u2018J99-type\u2019 (insertion ~31\u00a0\u200bamino acids), and the H.\u00a0pylori \u201826695 type\u2019 (insertion ~46\u00a0\u200bamino acids). Analysis of ligand binding using Native Electrospray Ionization Mass Spectrometry (ESI-MS) together with solid phase-bound, ELISA-type assays could not confirm the originally described binding of GalNAc\u03b21-4GlcNAc-containing oligosaccharides, in line with other recent reports, which also failed to confirm LacdiNAc binding.  \u20222.06 \u00c5 resolution X-ray structure of LabA shares L-type fold with SabA and BabA, despite low sequence homology.\u2022Avoidance of multiple N-terminal truncations of LabA expressed in E. coli periplasm after redesign of expression construct.\u2022ESI-MS suggests very low binding affinity for lacDiNac or related carbohydrates, calling into question current ligand.\u2022Sequence analysis across different strains indicates 2 types of LabA (\u2018J99- vs 26695-type\u2019) with high overall conservation. H.\u00a0pylori establishes chronic infection in the stomach despite the hostile environment with its acidic conditions and the abundance of proteolytic enzymes. As a result of this adaptation, approximately half the world's population is infected with the bacterium  is a carbohydrate structure presented on the gastric mucous cells, carried by the mucin MUC5AC, which has been suggested as a receptor for H.\u00a0pylori adherence -2-acetamido-2-deoxy-\u03b2-D-glucopyranoside, 530.52\u00a0\u200bDa) was purchased from Carbosynth . Chitotriose (\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc, MW 627.59\u00a0\u200bDa), chitotetraose (\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc, MW 830.79\u00a0\u200bDa) and chitohexaose (\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc, MW 1237.17\u00a0\u200bDa) were purchased from Dextra . HMO1 (\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b2)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 488.17\u00a0\u200bDa), HMO2 -[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-Glc, MW 488.17\u00a0\u200bDa), HMO11 -\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-Glc, MW 853.31\u00a0\u200bDa), HMO12 -\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-Glc, MW 853.31\u00a0\u200bDa), HMO18 -[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-Glc, MW 999.36\u00a0\u200bDa), HMO19 -\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 1072.38\u00a0\u200bDa), HMO21 (\u03b1-D-Neu5Ac-(2\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b3)-[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b4)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 1144.40\u00a0\u200bDa), HMO22 (\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b2)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b3)-[\u03b1-D-Neu5Ac-(2\u00a0\u200b\u2192\u00a0\u200b6)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 11440.40\u00a0\u200bDa), HMO25 -[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b4)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 1364.50\u00a0\u200bDa), HMO26 -[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b2)]-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 691.25\u00a0\u200bDa), and HMO27 -[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b2)]-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-GlcNAc(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 1056.39\u00a0\u200bDa) were purchased from Elicityl SA ; HMO3 (\u03b1-D-Neu5Ac-(2\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 633.21\u00a0\u200bDa), HMO4 (\u03b1-D-Neu5Ac-(2\u00a0\u200b\u2192\u00a0\u200b6)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 633.21\u00a0\u200bDa), HMO5 (\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b2)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-Glc, MW 634.23\u00a0\u200bDa), HMO6 -\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 707.25\u00a0\u200bDa), HMO7 -\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 707.25\u00a0\u200bDa), HMO8 (\u03b1- L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b2)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 853.31\u00a0\u200bDa), HMO9 -[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b4)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 853.31\u00a0\u200bDa), HMO10 -[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 853.31\u00a0\u200bDa), HMO13 (\u03b1-D-Neu5Ac-(2\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 998.34\u00a0\u200bDa), HMO14 (\u03b1-D-Neu5Ac-(2\u00a0\u200b\u2192\u00a0\u200b6)-[\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)- \u03b2-D-Glc, MW 998.34\u00a0\u200bDa), HMO15 (\u03b1-D-Neu5Ac-(2\u00a0\u200b\u2192\u00a0\u200b6)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 998.34\u00a0\u200bDa), HMO16 (\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b2)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b3)-[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b4)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 999.36\u00a0\u200bDa), HMO24 -[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b6)-[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b2)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 1144.40\u00a0\u200bDa), HMO32 -[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b6)-[\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b3)-[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b4)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 1364.50\u00a0\u200bDa), and HMO35 --[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b6)-[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b2)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b3)-[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b4)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 1510.55\u00a0\u200bDa) were purchased from IsoSep ; HMO17 -[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b4)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-Glc, MW 999.36\u00a0\u200bDa), HMO20 -\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b6)-[\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 1072.38\u00a0\u200bDa), HMO28 (\u03b1-D-Neu5Ac-(2\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc, MW 674.24\u00a0\u200bDa), HMO29 (\u03b1-D-Neu5Ac-(2\u00a0\u200b\u2192\u00a0\u200b6)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc, MW 674.24\u00a0\u200bDa), HMO30 -[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b2)]-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b3)-[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b4)]-\u03b2-D-GlcNAc(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 674.24\u00a0\u200bDa), HMO31 -[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b6)-[\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 1218.44\u00a0\u200bDa), and HMO33 -[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-[\u03b1-L-Fuc-(1\u00a0\u200b\u2192\u00a0\u200b3)]-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 1364.50\u00a0\u200bDa) were purchased from Dextra ; HMO23 (\u03b1-D-Neu5Ac-(2\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b3)-[\u03b1-D-Neu5Ac-(2\u00a0\u200b\u2192\u00a0\u200b6)]-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 1289.44\u00a0\u200bDa), and HMO34 -\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-GlcNAc-(1\u00a0\u200b\u2192\u00a0\u200b3)-\u03b2-D-Gal-(1\u00a0\u200b\u2192\u00a0\u200b4)-\u03b2-D-Glc, MW 1072.38\u00a0\u200bDa) were purchased from CarboSynth . A 1.0\u00a0\u200bmM stock solution of each glycan was prepared in deionized water and stored at \u221220\u00a0\u200b\u00b0C until used.2.2E.\u00a0coli XL10-Gold which had previously been developed and optimized for BabA J99 (AAD05605.1), as described in (AE000511.1) (constructs see 21\u2013517K 26695 into pOPE101: FOR PvuII: 5\u2032-CAGTAGCAGCTGGAAGACAACGGCTTTTTTGTG-3\u2032 and REV (BamHI) 5\u2032-GCTGCTGGATCCCTTCTTCTTCTTCTTCTTGAGTTCTTGACTCCTAGATTG-3\u2019 (restriction sites underlined). The reverse primer introduces a hexalysine tag at the C-terminus of the recombinant protein. All oligonucleotide primers were from Merck, UK. The sequences of all recombinant plasmids were confirmed by Sanger sequencing . The only modification applied was the use of 0.2\u00a0\u200bmM Isopropyl-\u03b2-D-thiogalactopyranoside (IPTG) for induction of protein expression, instead of 0.1\u00a0\u200bmM IPTG used previously. The proteins were expressed in 6\u00a0\u200bL of bacterial culture prior to harvesting and reconstituted with 600\u00a0\u200bmL of each of the two different cell lysis buffers as described in (47-496 J99 was obtained by gene synthesis by ThermoScientific Fisher (GeneArt) (sequence available in supplementary data), and subcloned into pOPE101.The pOPE101 plasmid  construct and protocol for periplasmic expression in BabA J99  was alsoribed in  and LabAribed in . In totaribed in B with a 2.32.3.1The combined periplasmic extracts were incubated for a maximum of 2\u00a0\u200bh with 5\u00a0\u200bmL of Ni Sepharose 6 Fast Flow resin  at 4\u00a0\u200b\u00b0C and then loaded on a gravity Econo-Column\u00ae chromatography column. The flowthrough was collected and the column was washed with ten column volumes (CV) of washing buffer, consisting of 20\u00a0\u200bmM Tris-Cl, pH 7.4 and 300\u00a0\u200bmM NaCl. Finally, the protein was consecutively eluted from the column with 3\u20135x CV of 20, 40, 100 and 200\u00a0\u200bmM and 10x CV of 500\u00a0\u200bmM imidazole in washing buffer. The protein content of the different fractions was analyzed with electrophoresis; the proteins were separated on NuPAGE 4\u201312% Bis-Tris protein gels  and the gels were stained with InstantBlue\u2122 .2.3.2Approximately 30\u00a0\u200bmg of IMAC purified protein, at a concentration 1\u00a0\u200bmg/mL, were mixed with 100\u00a0\u200b\u03bcL of 3\u00a0\u200bmg/mL Tobacco Etch Virus (TEV) protease (in-house AstraZeneca product). The cleavage reaction was left to happen overnight at 4\u00a0\u200b\u00b0C in a dialysis setup. The reaction mixture was inserted in Spectrum\u2122 Spectra/Por\u2122 1 RC dialysis membrane tubing made of regenerated cellulose dialysis with a MWCO of 6000\u20138000\u00a0\u200bDa , and was dialysed against 5\u00a0\u200bL of 20\u00a0\u200bmM Tris-Cl pH 7.4, 300\u00a0\u200bmM NaCl, in order to remove any residual imidazole from the IMAC elution fractions.2.3.3The overnight reaction mixture was incubated for 1\u00a0\u200bh at 4\u00a0\u200b\u00b0C with 1\u00a0\u200bmL of Ni Sepharose 6 Fast Flow resin and then loaded on a gravity column. The flow through was collected and the column was washed again with 10 CV. The contents of the column were then eluted with increasing imidazole step gradient, at 20, 40, 250 and 500\u00a0\u200bmM imidazole. Each eluted fraction was 5 CV. The collected fractions were analyzed with electrophoresis followed by InstantBlue staining, as previously.2.3.4The purest IMAC fractions were concentrated to 5\u00a0\u200bmL, using a Vivaspin sample concentrator with a molecular weight cut-off of 30,000\u00a0\u200bDa . The concentrated protein samples were loaded onto a HiLoad 16/60 Superdex 75 (120\u00a0\u200bmL) gel filtration column , previously equilibrated with buffer containing 25\u00a0\u200bmM Bicine, pH 8.4, and 150\u00a0\u200bmM NaCl, connected to an \u00c4KTA purifier system . The buffer was chosen based on buffer optimization results (data not shown). The flow rate was set at 1 mL/min. The fractions containing protein were analyzed with electrophoresis and InstantBlue staining, as previously.2.4The molecular weight of purified proteins was determined by liquid chromatography (LC) \u2013 time-of-flight (ToF) mass spectrometry (MS). Approximately 5\u00a0\u200b\u03bcg of purified protein sample was loaded onto an Agilent 1100 Series LC  which was coupled to a time-of-flight Q-ToF Premier mass spectrometer , equipped with an electron spray ionizer for acquisition in a positive ionization mode. The software MassLynx  was used to analyse the data.2.547-496 J99 grew in 28% poly(ethylene glycol) methyl ether 2000 and 0.1\u00a0\u200bM Bis-Tris (pH 6.5). Crystals appeared within one week of incubation at 20\u00a0\u200b\u00b0C.Protein samples were concentrated to 20\u00a0\u200bmg/mL and centrifuged for the removal of aggregated protein, before dispensing the crystal plates. Crystallization was performed using the sitting drop vapour diffusion method in 96-well MRC crystallization plates  and dispensed with the assistance of the Mosquito\u00ae Robot . Crystallization trials used commercial and proprietary sparse-matrix screens. Each droplet contained protein sample in 25\u00a0\u200bmM Bicine, pH\u00a0\u200b=\u00a0\u200b8.4, and 150\u00a0\u200bmM NaCl mixed with a precipitant solution, at a volume ratio 1:1, and was equilibrated against 50\u00a0\u200b\u03bcL of the precipitant solution at 4 and 20\u00a0\u200b\u00b0C. Bipyramidal crystals for LabA2.6AIMLESS (4ZH7) as a search model in the program phaser  and were indexed and integrated using the XDS package . AnisotrAIMLESS . Phases m phaser . The modm phaser  and recim phaser  and Bustm phaser . RMSD va2.72.7.1a, HSA-Leb, HSA-Ley, HSA-sialyl-Lex  and HSA-LacdiNAc , were immobilized at a concentration of 5.0\u00a0\u200b\u03bcg/mL on Maxisorp plates  and incubated with LabA21-496 and LabA21-496-6\u00a0\u200bK at a concentration range from 5 to 20\u00a0\u200b\u03bcg/mL. After washing off unbound glycans and proteins, the wells were sequentially incubated, with three washes between the steps, with a mouse-anti-c-Myc biotinylated antibody  and the conjugate streptavidin-HRP , both at a 1:2000 dilution, in order to complex with bound recombinant LabA and enable tetramethylbenzidine chromogenic detection. Absorbance was measured at 450\u00a0\u200bnm on a Spark\u00ae 10\u00a0\u200bM multimode microplate reader .For enzyme-linked immunosorbent assay (ELISA), HSA , HSA-Le2.7.2ref) were each buffer-exchanged into 200\u00a0\u200bmM aqueous ammonium acetate (pH 6.8) using a 10\u00a0\u200bkDa\u00a0\u200bMW cutoff Amicon Ultra-4 centrifugal filter . Protein concentrations were estimated by UV absorption (280\u00a0\u200bnm). These stock solutions were stored at \u221220\u00a0\u200b\u00b0C until used.LabA 26695 and a single chain antibody (which served as Pref) was loaded into the nanoESI tip. To perform ESI, a voltage of ~1\u00a0\u200bkV was applied to a platinum wire in contact with the solution. Mass spectra were acquired using a sampling cone voltage of 30\u00a0\u200bV and an extraction cone voltage of 2\u00a0\u200bV. The source pressure was 3.2\u00a0\u200bmbar and the temperature was 60\u00a0\u200b\u00b0C. The source wave velocity and wave height were 200\u00a0\u200bm\u00a0\u200bs\u22121 and 0.2\u00a0\u200bV, respectively. Gas flow rates were 2\u00a0\u200bmL\u00a0\u200bmin\u22121 in Trap, 180\u00a0\u200bmL\u00a0\u200bmin\u22121 in helium cell and 90\u00a0\u200bmL\u00a0\u200bmin\u22121 in ion mobility cell. Ions were transmitted through the Trap and Transfer ion guides using voltages of 5\u00a0\u200bV and 2\u00a0\u200bV, respectively. At least 150 scans were collected for every acquisition. Data acquisition and processing were carried out using MassLynx (v 4.1).All ESI-MS measurements were performed in positive ion mode using a Synapt G2 ESI-quadrupole-ion mobility separation-time-of-flight mass spectrometer , equipped with a nanoflow ESI source. The nanoESI tips were produced from borosilicate capillaries  pulled to ~5\u00a0\u200b\u03bcm outer-diameter using a P-1000 micropipette puller . For each measurement, approximately 10\u00a0\u200b\u03bcL of sample solution  for the interactions between LabA 26695 and the glycan ligands were quantified using the direct ESI-MS assay  of the ligand-bound (PL) to free protein (P) ions, after correction for nonspecific ligand binding (0) and ligand ([L]0), eq Association constants  with the following codes: 6GMM. All remaining data are contained within the article or the supporting information.33.121-496 from H.\u00a0pylori strain J99 was produced by periplasmic expression in E.\u00a0coli as recently described (21-496 J99 (47-496) after TEV cleavage.LabAescribed , with C-escribed  and sizeescribed . Althoug-496 J99 . The mai-496 J99  E and F.-496 J99  A and B,47-496  added by the cloning and left after restriction with TEV . However, the ratio of the main  product to the secondary product  was increased and the purity of the protein sample used for protein crystallization screening was enhanced.Following recombinant expression, the protein was cleaved with TEV-protease. The two-step purification was repeated for the removal of the enzyme and the protein was analyzed C\u2013E. TEV 47-496 F reveale3.247-496 construct were performed in 384 different sets of conditions in sparse-matrix screens. Within seven days, protein crystals were obtained in a crystallization drop containing 28% poly-(ethylene glycol) methyl ether 2000 and 100\u00a0\u200bmM Bis-Tris at a pH 6.5 and in the absence of salt, on a plate stored at 20\u00a0\u200b\u00b0C. The crystals displayed a bipyramidal morphology with maximum dimensions of 60\u00a0\u200b\u00d7\u00a0\u200b150\u00a0\u200b\u03bcm . The crystal parameters, as well as data processing and structure refinement statistics are shown in The crystals belonged to the space group Coot  formed an antiparallel coil bundle, similar to a tetratricopeptide repeat motif, at a near perpendicular angle to the \u2018handle\u2019 region, creating a kinked (or L-shaped) tertiary structure. The connecting features between the \u03b1-helices were: (i) A 48-amino acid segment between the \u03b1-2 and \u03b1-3 helices; this connecting segment, which extended out of the core of the head region, contained two small antiparallel \u03b2-sheet strands (\u03b2-1 and \u03b2-2); (ii) A 42-amino acid segment between the \u03b1-3 and \u03b1-4 helices; this segment contained the crown region, consisting of a two-strand antiparallel \u03b2-sheet (\u03b2-3 and \u03b2-4) between T204 and N230, and a disordered loop within the crown, between N211 and E224; (iii) A 28-amino acid loop between the \u03b1-4 and \u03b1-5 helices, with two short \u03b1-helices; (iv) A 21-amino acid loop between the \u03b1-5 and \u03b1-6 helices; (v) A 28-amino acid segment between the \u03b1-7 and \u03b1-8 helices, containing a short \u03b1-helix and a disordered loop between G380 and D388. LabA has three disulphide bonds , all three located in the head region Coot  is 0.921\u00a0\u200b\u00c5; without pruning, the RMSD across all 402 pairs rises to 5.493\u00a0\u200b\u00c5. Using the central (\u2018head\u2019) part of the molecule (ranging from S57 to V462 in BabA) results in a low RMSD of 0.874\u00a0\u200b\u00c5 for 208 pruned atom pairs (without pruning: 4.959\u00a0\u200b\u00c5). The pruned atoms in the \u2018head\u2019 region correspond to the inserted \u2018crown\u2019 region , as the crown is much smaller and contains an unstructured region in LabA. Without pruning, the RMSD for AA 57\u2013462 rises to 4.780\u00a0\u200b\u00c5. Matching the handle regions D27-P56 and V461\u2013K528 gives RMSD values of 0.465 and 1.147\u00a0\u200b\u00c5, respectively. Overall, these results show that the handle and head regions are very similar between LabA and BabA, but are connected at slightly different angles, with the largest discrepancies between the two structures found in the crown region.Despite the relatively low sequence identity, all these proteins share high conformational similarity in their head regions, where a 4\u00a0\u200b+\u00a0\u200b3 \u03b1-helix bundle was present; however, significant differences were observed in the crown region, which may be related to each protein's ligand binding specificity. The \u2018crown\u2019 regions of LabA and HopQ  compris3.3H.\u00a0pylori to human gastric mucins. To verify this binding ability, a sandwich-type ELISA was carried out in order to test the binding of the recombinantly expressed LabA21-496 and LabA21-496-6\u00a0\u200bK  in the absence and presence of LacdiNAc. A reference protein  was added to the latter solution to correct the mass spectrum for the occurrence of nonspecific ligand-protein binding during the ESI process  was initially performed on aqueous ammonium acetate  solutions containing LabA process . However47-496 26695 with LacdiNAc was measured. 47-496 26695 (2.5\u00a0\u200b\u03bcM), Pref (3.2\u00a0\u200b\u03bcM) and LacdiNAc (15\u00a0\u200b\u03bcM). Only one isoform of LabA was detected, with a MW of 51,277\u00a0\u200bDa. After correction of nonspecific binding using the reference protein method (\u22121 (mean\u00a0\u200b\u00b1\u00a0\u200bs.d.). We next decided to investigate binding to chito-oligosaccharides due to the structural similarities to LacdiNAc. The affinities of chitotriose, chitotetraose and chitohexaose (structures see ref (3.2\u00a0\u200b\u03bcM) and three chito-glycans (each 48\u00a0\u200b\u03bcM). The affinities for the three oligosaccharides ranged from 1000\u00a0\u200bM\u22121 to 2000 M\u22121.Binding of this newly produced LabAn method , the aff47-496 26695 and no binding at all of LabA21-496 J99 in ELISA-like format (\u22121).Having established only weak binding affinity of LabAe format  for Lacde format , all of e format . A repre4While our initial attempts to crystallize LabA protein were unsuccessful, removal of the C-terminal tags and addition of an N-terminal TEV cleavage site resulted in suitable crystals. It has been reported that excessive solubility can play a role in making protein crystallization more challenging. In particular, highly soluble protein versions bearing a C-terminal pentalysine tag have been found to yield small needle-shaped crystals, similar to those obtained by us here S inapproAt the same time, limiting the proportion of unstructured regions in the protein was also considered during the design of the new expression construct. It was known from the crystal structure of BabA J99 that the three C-terminal tags, used for the enhancement of solubility, detection by Western Blotting and purification with IMAC, were not visible in the crystal structure. This confirmed the conformational flexibility of these sequences. We removed the C-terminal hexalysine and c-Myc-tags and moved the hexahistidine tag needed for IMAC to the N-terminus, upstream of the TEV cleavage site. This approach resulted in the new recombinant LabA variant which yielded protein crystals sufficient for X-ray crystallography during initial screening already, without the requirement for further optimization.H.\u00a0pylori.Overall, the protein adopts a similar structure to the previously described adhesins of H.\u00a0pylori adhesins  adopt a three-dimensional L-shape, suggesting that this structure may fulfil some yet to be discovered functional role. The highest similarity of the LabA amino acid sequence to that of BabA was corroborated by the superimposition of their crystal structures and the obtained low RMSD values. The similarity among all adhesins was most pronounced in the handle and head and particularly low in the crown region, where the glycan binding site of BabA is found. While this explains the lack of binding affinity of LabA for Leb, it does not provide any information about the actual binding site of this protein. In the absence of a ligand, the crown appears partially disordered in most adhesin structures.Despite low amino acid sequence similarity, all four crystallized H.\u00a0pylori \u2018J99-type\u2019, with an overall total length of approx. 686\u2013691 amino acids and a shorter insertion domain of approx. 31\u00a0\u200bamino acids, and the H.\u00a0pylori \u201826695 type\u2019, with an approximate total length of 702\u2013711 amino acids and a longer insertion domain of approx. 46\u00a0\u200bamino acids. In the absence of a ligand co-crystal structure, the exact significance of these structural variations between strains remains unclear. While BabA also shows strain variation in the crown region/insertion domain, the variations in this region are stronger in BabA, which has led to the identification of two hypervariable/diversity loops   in whichloops (N  (Moonensloops (N ). In conloops  and SabA , and in line with the obtained structure for LabA (6GMM), the N-terminal region is located close to the membrane insertion domain, which is buried deep in the outer membrane of H.\u00a0pylori, and is a very unlikely binding site, as it would not be able to access its putative ligand on epithelial cells.However, our ESI-MS screening indicated that LabA only weakly binds to Lac/LacNAc-containing oligosaccharides, without any clear specificity. Furthermore, we were not able to detect any binding to LacdiNAc-HSA in an ELISA-type assay. We do not believe that the omission of the first 46\u00a0\u200bN-terminal amino acids in the improved construct can be responsible for the lack of LacdiNAc binding for two reasons. First, the longer LabAa values summarized in Because of our inability to detect any binding to LacdiNAc, we extended our study to the binding of 35 HMOs with similar structures and to three chito-oligosaccharides of different lengths . The latH.\u00a0pylori J99 or 26695 strain  using a variety of methods, including whole bacteria binding assays in microtiter plates. This was not due to a lack of expression or translation of LabA protein, as this could be identified by LC-MS/MS. The complete lack of binding of H.\u00a0pylori bacteria to LacdiNAc, together with our inability to detect specific and high affinity binding of the recombinant LabA protein, suggests that LacdiNAc might not be the physiological ligand for this adhesin, and that the glycan preference of LabA remains to be elucidated. Alternatively, the function of LabA in promoting binding may be accessory, rather than direct, supporting the structure and/or binding activity of another adhesin with specificity for LacdiNAc as determined by This lack of binding to LacdiNAc is in good agreement with the recent observation by Mthembu and co-workers . The aut10.13039/501100000266EPSRC (Grant EP/L01646X), 10.13039/100004325AstraZeneca R&D and the 10.13039/501100000837University of Nottingham. Y.C., L.N. and J.S.K. acknowledge funding from the 10.13039/501100003178Alberta Glycomics Centre. Ross Overman is currently employed by Leaf Expression Systems. FHF is currently fully funded by a grant from the LOEWE DRUID .This research has been supported through joint funding from Vasiliki Paraskevopoulou: Investigation, Methodology, Validation, Writing - original draft, Writing - review & editing. Marianne Schimpl: Investigation, Formal analysis, Methodology, Validation, Writing - review & editing. Ross C. Overman: Methodology, Conceptualization, Writing - review & editing. Snow Stolnik: Funding acquisition, Supervision, Writing - review & editing. Yajie Chen: Investigation, Methodology, Writing - review & editing. Linh Nguyen: Investigation, Methodology, Writing - review & editing. G. Sebastiaan Winkler: Funding acquisition, Supervision, Writing - review & editing. Paul Gellert: Methodology, Conceptualization, Writing - review & editing, Funding acquisition, Supervision. John S. Klassen: Conceptualization, Methodology, Writing - review & editing. Franco H. Falcone: Funding acquisition, Project administration, Supervision, Writing - original draft, Writing - review & editing.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper."}
{"text": "Nature Communications 10.1038/S41467-020-20875-W, published online 10 February 2021.Correction to: t test were reported with the incorrect degrees of freedom, incorrectly written as \u201ct18858\u2009=\u2009\u221220.915, p\u2009<\u20090.001.\u201d The correct version replaced \u201ct18858\u201d with \u201ct18558.\u201d This has been corrected in both the PDF and HTML versions of the Article.The original version of this Article contained an error in the \u201cMethods\u201d section, under the \u201cDecision-making task\u201d heading. The results of a The original version of this Article also contained an error in the \u201cMethods\u201d section titled \u201cBehavioural modelling of value-guided decisions\u201d in which Eqs. 5 and 6 inadvertently swapped during production. The incorrect version statedu(m) and w(p), for reward magnitudes and probabilities, Eqs. 5 and 6, respectively)56.Since it is known that humans do not weigh magnitudes and probabilities in a statistically optimal way, we considered systematic distortions in the weighting of reward information in our models (pO are the objective reward probabilities and \u03b3 is a free parameter used to fit subjective reward probabilities. Subjective magnitudes were estimated by:where The correct version statesw(pO) and u(mO) for reward probabilities and magnitudes, Eqs. 5 and 6, respectively)56.Since it is known that humans do not weigh magnitudes and probabilities in a statistically optimal way, we considered systematic distortions in the weighting of reward information in our models (pO are the objective reward probabilities and \u03b3 is a free parameter used to fit subjective reward probabilities. Subjective magnitudes were estimated by:where The original version of this Article also contained an error in the Methods section under the heading \u201cMRS data acquisition\u201d, in which the terms \u201crM1 and lM1\u201d were incorrectly ordered. The incorrect version statedx\u2009=\u2009\u221228.97\u2009\u00b1\u20090.82, y\u2009=\u2009\u221218.48\u2009\u00b1\u20090.92, z\u2009=\u200951.86\u2009\u00b1\u20090.59 and MNI x\u2009=\u200931.90\u2009\u00b1\u20090.71, y\u2009=\u2009\u221214.76\u2009\u00b1\u20091.06, z\u2009=\u200949.76\u2009\u00b1\u20090.88 for rM1 and lM1, respectively.Average M1 voxel centroids in standard space were estimated at MNI The correct version statesx\u2009=\u2009\u221228.97\u2009\u00b1\u20090.82, y\u2009=\u2009\u221218.48\u2009\u00b1\u20090.92, z\u2009=\u200951.86\u2009\u00b1\u20090.59 and MNI x\u2009=\u200931.90\u2009\u00b1\u20090.71, y\u2009=\u2009\u221214.76\u2009\u00b1\u20091.06, z\u2009=\u200949.76\u2009\u00b1\u20090.88 for lM1 and rM1, respectively.Average M1 voxel centroids in standard space were estimated at MNI"}
{"text": "Analyses of children and young people mortality continue to be an important component of health monitoring of this population. Such analyses provide the basis to assess the overall trends, the structure of the causes of death over longer periods, and the differences between Poland and other countries.The purpose of the current study is to present the current status and the direction of changes since 2000 with regard to the level and underlying causes of mortality in children and adolescents aged 1-19 years in Poland on the background of statistics for leading European countries.Interactive databases available online: the National Demographic Database provided by the Central Statistical Office and the International WHO-MDB Database were used. Poland, constantly belonging to Eur-B category, was compared with the combined group of 27 leading countries, classified as a very low total mortality group (Eur-A) according to WHO. Linear trends of overall and cause-specific mortality in 2000\u20132013 were estimated. The causes of death have been presented according to the main classes of the 10th Revision of the International Statistical Classification of Diseases and Related Health Problems (ICD-10). External and other causes were adopted as the two principal categories.In 2015, 1471 deaths of persons aged 1-19 were recorded in Poland . Changes in children and adolescents mortality by age have a nonlinear nature (U-shaped), and the lowest level is recorded at the age of 5-9 years.According to 2014 data, 50.2% of deaths of children and adolescents aged 1-19 years occurred due to external causes, including non-intentional and intentional ones. This percentage increased from 18.4% in the 1-4 age group to 68.6% at the age of 15-19 years. Apart from external causes, the dominating causes of death are malignant neoplasms, congenital defects, or nervous system and respiratory system diseases. The ranking of those causes of death changes in successive age groups and over time. When age is considered, a higher proportion of congenital defects and respiratory system diseases was found in mortality younger children and a higher proportion of circulatory system diseases and undefined cases in mortality of adolescents. When trends were studied, a continuing elimination of infectious diseases was observed together with growing impact of rare diseases in all age groups.The excess mortality of Polish population at age 1-19 by comparison to Eur-A countries increased from 21% in 2000 to 56% in 2013, mainly due to unfavourable trends in adolescents. The rate of decline in the mortality of young children (1-4 years) was greater than in Eur-A countries, both in case of external and other causes. In the age group 5-14 years the higher rate of change was sustained only with regard to external causes. Among adolescents and young adults, the distance between Poland and Eur-A countries increased during the studied period. The shape of trend in the 15-24 age group was unfavourable for Poland, mainly with respect to external causes. This observation could be in part explained by increasing suicide trend in Poland since 2008, coexisting with rather constant level in Eur-A countries.The mortality rate among the population aged 1-19 years in Poland is systematically decreasing, but it still exceeds the average level recorded in leading European countries, particularly in relation to adolescents. When assessing the ability to reduce mortality in Poland to the level of Eur-A countries, attention must be paid to the causes considered as avoidable. Further studies ought to focus on the trends and international comparisons only foreshadowed in this study with regard to individual diagnoses, discussing possible preventive measures. Introduction of an ICD-11 classification will enable more accurate coding of causes of death, including a more precise analysis of the burden of rare diseases, which are an increasing challenge to public health in the population at the developmental age. Analizy umieralno\u015bci stanowi\u0105 wa\u017cny element oceny sytuacji zdrowotnej spo\u0142ecze\u0144stwa. Umieralno\u015b\u0107 niemowl\u0105t i przeci\u0119tne trwanie \u017cycia s\u0105 ci\u0105gle traktowane jako kluczowe wska\u017aniki rozwoju ekonomicznego . Poziom Wed\u0142ug raport\u00f3w \u015awiatowej Organizacji Zdrowia (WHO), poziom umieralno\u015bci dzieci i m\u0142odzie\u017cy stale si\u0119 obni\u017ca, jednak nadal wyst\u0119puj\u0105 znaczne r\u00f3\u017cnice mi\u0119dzy krajami . Kraje pLancet opublikowa\u0142o seri\u0119 artyku\u0142\u00f3w [Okresowo powtarzane analizy umieralno\u015bci dzieci i m\u0142odzie\u017cy , 9, jak rtyku\u0142\u00f3w , 16, 17 rtyku\u0142\u00f3w .Dane dotycz\u0105ce zdrowia populacji, w tym statystyki zgon\u00f3w, s\u0105 wykorzystywane dla usprawnienia dzia\u0142ania systemu ochrony zdrowia na poziomie og\u00f3lnokrajowym i lokalnym. Wspomaga to polityk\u0119 zdrowotn\u0105 i programy zdrowotne na r\u00f3\u017cnych etapach ich tworzenia, od ustalania grup ryzyka i priorytet\u00f3w do oceny stopnia realizacji za\u0142o\u017conych cel\u00f3w . Od ponaW grupie dzieci i m\u0142odzie\u017cy wyst\u0119puje specyficzna struktura przyczyn zgon\u00f3w, zwi\u0105zana z du\u017co wi\u0119kszym ni\u017c u os\u00f3b doros\u0142ych udzia\u0142em tzw. przyczyn zewn\u0119trznych, kt\u00f3re mog\u0105 wynika\u0107 ze zdarze\u0144 przypadkowych lub dzia\u0142a\u0144 zamierzonych.Wymagaj\u0105 one innego podej\u015bcia do prewencji ni\u017c pozosta\u0142e choroby spowodowane przyczynami wewn\u0119trznymi, obejmuj\u0105ce narz\u0105dy i uk\u0142ady organizmu. Stanowi\u0105 one najwi\u0119ksze obci\u0105\u017cenie populacji w zwi\u0105zku z potencjalnie utraconymi latami \u017cycia i latami \u017cycia z niepe\u0142nosprawno\u015bci\u0105. Obejmuj\u0105 one zar\u00f3wno zgony z powodu wypadk\u00f3w komunikacyjnych, jak r\u00f3wnie\u017c w wyniku przemocy, w tym samob\u00f3jstwa i zab\u00f3jstwa. Powodzenie prewencji wypadkowej w du\u017cej mierze zale\u017cy od czynnik\u00f3w zlokalizowanych poza sektorem ochrony zdrowia, szczeg\u00f3lnie w zakresie prewencji pierwotnej. Teoretycznie, wypadki (i ich skutki) w wi\u0119kszo\u015bci uznaje si\u0119 za mo\u017cliwe do unikni\u0119cia, przy wdro\u017ceniu skutecznych dzia\u0142a\u0144, szczeg\u00f3lnie w obszarze legislacji i zmian \u015brodowiska ) zycznego. Zmiany te ukierunkowane s\u0105 na popraw\u0119 bezpiecze\u0144stwa dzieci w \u015brodowisku wychowania, nauki, sportu i rekreacji, w tym popraw\u0119 bezpiecze\u0144stwa uczestnik\u00f3w ruchu drogowego , 22.W ostatnich latach coraz cz\u0119\u015bciej podejmowany jest te\u017c temat chor\u00f3b rzadkich lub ultrarzadkich w kontek\u015bcie umieralno\u015bci populacji w wieku rozwojowym, jak r\u00f3wnie\u017c w planowaniu kompleksowej opieki zdrowotnej i jej koszt\u00f3w w odniesieniu do tej grupy pacjent\u00f3w. Podnoszona jest r\u00f3wnie\u017c zasadno\u015b\u0107 osobnej kodyfikacji tych chor\u00f3b. Wiele chor\u00f3b przewlek\u0142ych, w tym te rzadkie, zaczyna by\u0107 bardziej eksponowana w statystykach stanu zdrowia starszych dzieci i m\u0142odzie\u017cy, z powodu ograniczenia \u015bmiertelno\u015bci w wieku niemowl\u0119cym oraz wyeliminowania innych, dominuj\u0105cych wcze\u015bniej przyczyn. Zgodnie z definicj\u0105, choroby rzadkie to choroby zagra\u017caj\u0105ce \u017cyciu lub powoduj\u0105ce przewlek\u0142\u0105 niepe\u0142nosprawno\u015b\u0107, o maksymalnym regionalnym zasi\u0119gu od 1 na 10 000 do 1 na 1000 os\u00f3b. W definicji Unii Europejskiej, zawartej w regulacji Parlamentu Europejskiego i Rady Europy z dnia 16 grudnia 1999 roku  podane jgarbage codes \u2013 termin przyj\u0119ty przez WHO), b\u0119d\u0105cych skutkiem nieprawid\u0142owych zapis\u00f3w w karcie zgonu, uniemo\u017cliwiaj\u0105cych nadanie odpowiedniego kodu przyczyny. Wydaje si\u0119 tak\u017ce, \u017ce wprowadzenie przez Narodowy Fundusz Zdrowia w 2008 roku tzw. jednorodnych grup pacjent\u00f3w (JGP) [Wydaje si\u0119, \u017ce aby analizy umieralno\u015bci skutecznie wspomaga\u0142y ocen\u0119 systemu opieki medycznej i stopnia wdra\u017cania dzia\u0142a\u0144 prewencyjnych, nale\u017cy te\u017c odnie\u015b\u0107 si\u0119 krytycznie do jako\u015bci uzyskiwanych danych. W \u015bwietle dost\u0119pnej wiedzy, ma\u0142o jest tego typu analiz i cz\u0119\u015bciej dotycz\u0105 one doros\u0142ych ni\u017c dzieci i m\u0142odzie\u017cy . Specjal\u00f3w (JGP) , mog\u0142o sCelem opracowania jest przedstawienie aktualnych danych na temat umieralno\u015bci dzieci i m\u0142odzie\u017cy w wieku 1-19 lat w Polsce oraz pr\u00f3ba oceny jako\u015bci dost\u0119pnych informacji.Sformu\u0142owano nast\u0119puj\u0105ce pytania badawcze:Jaki jest obecny poziom umieralno\u015bci dzieci i m\u0142odzie\u017cy w Polsce wed\u0142ug p\u0142ci i wieku?Jak kszta\u0142tuje si\u0119 umieralno\u015b\u0107 dzieci i m\u0142odzie\u017cy w Polsce na tle wzorcowych kraj\u00f3w europejskich w \u015bwietle najnowszych danych?Czym r\u00f3\u017cni\u0105 si\u0119 obserwowane w Polsce trendy umieralno\u015bci w grupach wieku oraz z powodu g\u0142\u00f3wnych kategorii przyczyn od trend\u00f3w w innych krajach?Czy system kodowania przyczyn zgon\u00f3w jest wystarczaj\u0105co dok\u0142adny, aby u\u0142atwi\u0107 definiowanie obszar\u00f3w priorytetowych w systemie opieki zdrowotnej nad dzie\u0107mi i m\u0142odzie\u017c\u0105.http://demografia.stat.gov.pl/bazademografia/Tables.aspx).Wykorzystano og\u00f3lnie dost\u0119pne bazy danych demograficznych zamieszczone na stronie internetowej G\u0142\u00f3wnego Urz\u0119du Statystycznego (GUS). Szczeg\u00f3\u0142owe dane dotycz\u0105ce stanu ludno\u015bci i zgon\u00f3w wed\u0142ug wieku znajduj\u0105 si\u0119 w zak\u0142adce \u201ewyniki bada\u0144 bie\u017c\u0105cych\u201d  dost\u0119pn\u0105 na stronie internetowej \u015awiatowej Organizacji Zdrowia, wersja z lipca 2016 roku. Pozwala ona na interaktywne tworzenie zestawie\u0144 w r\u00f3\u017cnych przekrojach  Ograniczeniem tej bazy jest pos\u0142ugiwanie si\u0119 z g\u00f3ry zdefiniowan\u0105 list\u0105 przyczyn zgon\u00f3w oraz niejednolite zasady agregacji grup wieku w odniesieniu do r\u00f3\u017cnych przyczyn. W zwi\u0105zku z tym trudno by\u0142o uzyska\u0107 wszystkie zaplanowane zestawienia, dok\u0142adnie dla przedzia\u0142u wieku 1-19 lat.W zestawieniach mi\u0119dzynarodowych wykorzystano baz\u0119 WHO-MDB  [Przedstawiono dane z ostatniego dost\u0119pnego roku, co w przypadku krajowych zestawie\u0144 opieraj\u0105cych si\u0119 na danych GUS oznacza 2015 rok lub 2014 rok (wg przyczyn), za\u015b 2013 rok w przypadku zestawie\u0144 mi\u0119dzynarodowych uzyskanych z bazy WHO-MDB.Wsp\u00f3\u0142czynniki umieralno\u015bci liczono na 100 000 os\u00f3b danej grupy wieku, przyjmuj\u0105c za mianownik stan ludno\u015bci na 30 czerwca danego roku. Niewielkie r\u00f3\u017cnice w warto\u015bciach wsp\u00f3\u0142czynnik\u00f3w obliczanych na podstawie danych GUS wzgl\u0119dem zestawie\u0144 WHO mog\u0105 wynika\u0107 ze standaryzacji oraz zdefiniowanej populacji bazowej. W por\u00f3wnaniach mi\u0119dzynarodowych zestawianych wed\u0142ug wieku uwzgl\u0119dniono trzy dost\u0119pne grupy , w\u0142\u0105czaj\u0105c te\u017c w niekt\u00f3rych analizach m\u0142odych doros\u0142ych do 29 lat lub sporadycznie niemowl\u0119ta (jako \u0142\u0105czna grupa 0-14 lat).Przyczyny zgonu przedstawiono wed\u0142ug g\u0142\u00f3wnych klas odpowiadaj\u0105cych X Rewizji Mi\u0119dzynarodowej Statystycznej klasyfikacji Chor\u00f3b i Problem\u00f3w Zdrowotnych, klasyfikacja trzycyfrowa (z rozszerzeniem). Za nadrz\u0119dne kategorie przyj\u0119to przyczyny zewn\u0119trzne (klasy od V do Y) oraz pozosta\u0142e przyczyny (klasy od A do R). Lista klas przyczyn znajduje si\u0119 dalej w \u0179r\u00f3d\u0142em danych o zgonach ludno\u015bci Polski jest, dokument ,,Karta zgonu\u201d, kt\u00f3ry zawiera informacj\u0119 o przyczynie zgonu bezpo\u015bredniej, wt\u00f3rnej i wyj\u015bciowej. Przy opracowywaniu danych o zgonach wed\u0142ug przyczyn przyjmuje si\u0119 wyj\u015bciow\u0105 przyczyn\u0119, czyli chorob\u0119 stanowi\u0105c\u0105 pocz\u0105tek procesu chorobowego, kt\u00f3ry doprowadzi\u0142 do zgonu albo uraz czy zatrucie, w wyniku kt\u00f3rego nast\u0105pi\u0142 zgon .Analizuj\u0105c tendencj\u0119 zmian, oszacowano trend liniowy w latach 2000-2013. Przyj\u0119to umownie, \u017ce wsp\u00f3\u0142czynnik regresji liniowej opisuje \u015brednie tempo zmian umieralno\u015bci w badanym okresie, a wy\u017csza warto\u015b\u0107 wsp\u00f3\u0142czynnika determinacji R-kwadrat (jako\u015bci dopasowania modelu) \u015bwiadczy o lepszym dopasowaniu funkcji trendu do danych rzeczywistych.Wed\u0142ug danych z 2015 roku, populacja dzieci i m\u0142odzie\u017cy w wieku 1-19 lat liczy\u0142a w Polsce 7,4 miliona os\u00f3b. W por\u00f3wnaniu z pocz\u0105tkiem okresu obj\u0119tego analiz\u0105 zmniejszy\u0142a si\u0119 ona o oko\u0142o 2,3 mln os\u00f3b, a w strukturze wed\u0142ug wieku obni\u017cy\u0142 si\u0119 udzia\u0142 starszej m\u0142odzie\u017cy, z chwil\u0105 gdy wiek adolescencji osi\u0105gn\u0119\u0142y roczniki \u201eni\u017cu demogra) cznego\u201d. Za wa\u017cn\u0105, ale trudn\u0105 do wymiernego wykazania zmian\u0119 nale\u017cy uzna\u0107 wzrost udzia\u0142u dzieci i m\u0142odzie\u017cy z przewlek\u0142ymi problemami zdrowotnymi, wynikaj\u0105cy z coraz lepszej prze\u017cywalno\u015bci noworodk\u00f3w z ma\u0142a mas\u0105 urodzeniow\u0105 i coraz lepszych mo\u017cliwo\u015bci leczenia chor\u00f3b uznawanych wcze\u015bniej za letalne.W 2015 roku zarejestrowano w Polsce 1474 zgony os\u00f3b w wieku 1-19 lat, w tym 963 zgony ch\u0142opc\u00f3w (65%) i 511 dziewcz\u0105t (35%). Aktualne wska\u017aniki umieralno\u015bci wed\u0142ug wieku, p\u0142ci i miejsca zamieszkania przedstawiono w Analizuj\u0105c umieralno\u015b\u0107 dzieci i m\u0142odzie\u017cy w tradycyjnych grupach wieku, stwierdzono najni\u017cszy jej poziom w wieku 5-9 lat, a najwy\u017cszy w wieku 15-19 lat. Znacz\u0105cy spadek umieralno\u015bci mo\u017cna zauwa\u017cy\u0107 w kolejnych rocznikach najm\u0142odszej grupy wieku (1-4 lat). W drugim roku \u017cycia (12-24 miesi\u0105ce) zanotowano 30,7 zgon\u00f3w na 100 000 populacji, podczas gdy w pi\u0105tym roku \u017cycia (48-60 miesi\u0119cy) ju\u017c trzy razy mniej .Na rycinie 1 przedstawiono wsp\u00f3\u0142czynniki umieralno\u015bci na 100 000 populacji wed\u0142ug p\u0142ci i dok\u0142adnych rocznik\u00f3w od pierwszego do dziewi\u0119tnastego uko\u0144czonego roku \u017cycia. R\u00f3\u017cnice mi\u0119dzy ch\u0142opcami i dziewcz\u0119tami zaczynaj\u0105 narasta\u0107 w drugiej dekadzie \u017cycia, a po 14 roku \u017cycia s\u0105 coraz wi\u0119ksze. U dziewcz\u0105t zagro\u017cenie zgonem jest zbli\u017cone w skrajnych rocznikach (krzywa ma dok\u0142adny kszta\u0142t litery U), podczas gdy u ch\u0142opc\u00f3w r\u00f3\u017cnica mi\u0119dzy skrajnymi rocznikami jest trzykrotna, na niekorzy\u015b\u0107 m\u0142odzie\u017cy.Na podobne prawid\u0142owo\u015bci wskazywa\u0142y poprzednie opracowania danych polskich, jak r\u00f3wnie\u017c zestawienia zagraniczne , 9, 14. W przyczyn zewn\u0119trznych \u2013 niezamierzonych, zamierzonych lub o nieustalonej intencji (klasa V-Y). W 2014 roku zmar\u0142o z tej przyczyny 769 os\u00f3b w wieku 1-19 lat. Udzia\u0142 przyczyn zewn\u0119trznych w statystyce zgon\u00f3w systematycznie zwi\u0119ksza si\u0119 w kolejnych grupach wieku \u2212 od 18,4% w wieku 1-4 lat, do 68,6% w wieku 15-19 lat.Po\u0142owa zgon\u00f3w nast\u0105pi\u0142a z tzw. nowotwory. Kategoria ta obejmuje g\u0142\u00f3wnie nowotwory z\u0142o\u015bliwe (klasa C). W 2014 roku zmar\u0142o z tej przyczyny 214 os\u00f3b w wieku 1-19 lat, co stanowi 14,0% wszystkich zgon\u00f3w. Wsp\u00f3\u0142czynniki na 100 000 populacji s\u0105 nieznacznie podwy\u017cszone w skrajnych grupach wieku w por\u00f3wnaniu z dzie\u0107mi w wieku 5-14 lat.Drug\u0105 klas\u0119 przyczyn zgon\u00f3w stanowi\u0105 wrodzone wady rozwojowe (klasa Q). W 2014 roku zmar\u0142y z tej przyczyny 154 osoby w wieku 1-19 lat, co stanowi 10,1% wszystkich zgon\u00f3w. Wsp\u00f3\u0142czynniki zgon\u00f3w na 100 000 populacji wyra\u017anie obni\u017caj\u0105 si\u0119 w kolejnych grupach wieku. W drugim i trzecim roku \u017cycia jest to jeszcze dominuj\u0105ca przyczyna, przewy\u017cszaj\u0105ca udzia\u0142 wypadk\u00f3w i nowotwor\u00f3w z\u0142o\u015bliwych. W opracowaniach dotycz\u0105cych ca\u0142ej populacji (\u0142\u0105cznie z doros\u0142ymi) wady wrodzone s\u0105 niezauwa\u017can\u0105 przyczyn\u0105 zgon\u00f3w. Wed\u0142ug danych z 2014 roku, 77% zgon\u00f3w z powodu wrodzonych wad rozwojowych nast\u0105pi\u0142o w pierwszym roku \u017cycia, 8% w wieku 1-19 lat i jedynie 15% u os\u00f3b doros\u0142ych, od 20 roku \u017cycia.Trzeci\u0105 z kolei klas\u0119 przyczyn zgon\u00f3w dzieci i m\u0142odzie\u017cy stanowi\u0105 choroby uk\u0142adu nerwowego (klasa G). W 2014 roku zmar\u0142y z tej przyczyny 104 osoby w wieku 1-19 lat, co stanowi 6,8% wszystkich zgon\u00f3w. Udzia\u0142 procentowy tej klasy chor\u00f3b w rankingu przyczyn umieralno\u015bci zmienia si\u0119 w kolejnych grupach wieku, cho\u0107 warto\u015bci wsp\u00f3\u0142czynnik\u00f3w utrzymuj\u0105 si\u0119 na stabilnym poziomie. Wobec og\u00f3lnie niskiej umieralno\u015bci, w wieku 10-14 lat, choroby uk\u0142adu nerwowego zajmuj\u0105 obecnie trzeci\u0105 pozycj\u0119 w rankingu przyczyn zgon\u00f3w.Kolejn\u0105 wa\u017cn\u0105 klas\u0105 przyczyn zgon\u00f3w dzieci i m\u0142odzie\u017cy s\u0105 choroby uk\u0142adu oddechowego (klasa J \u2212 92 przypadki w 2014 roku), choroby uk\u0142adu kr\u0105\u017cenia (klasa I \u2212 66 przypadk\u00f3w) oraz objawy i stany niezaklasyfikowane wcze\u015bniej (klasa R \u2212 57 przypadk\u00f3w). Choroby uk\u0142adu oddechowego stanowi\u0105 wi\u0119ksze zagro\u017cenie dla m\u0142odszych dzieci, podczas gdy wsp\u00f3\u0142czynniki zgon\u00f3w z powodu chor\u00f3b uk\u0142adu kr\u0105\u017cenia oraz stan\u00f3w niedok\u0142adnie zaklasyfikowanych zwi\u0119kszaj\u0105 si\u0119 u m\u0142odzie\u017cy w wieku 15-19 lat.Spo\u015br\u00f3d pozosta\u0142ych klas przyczyn zgon\u00f3w dzieci i m\u0142odzie\u017cy, warto zwr\u00f3ci\u0107 uwag\u0119 na chor\u00f3b zaka\u017anych i paso\u017cytniczych (po\u0142\u0105czona klasa A i B) jako jednej z dziewi\u0119ciu najwa\u017cniejszych klas przyczyn zgon\u00f3w zaczyna mie\u0107 w \u015bwietle najnowszych statystyk znaczenie historyczne. W 2014 roku zanotowano w wieku 1-19 lat tylko 14 przypadk\u00f3w zgon\u00f3w z tej przyczyny. Stosunkowo cz\u0119\u015bciej notowane s\u0105 one do 36 miesi\u0105ca \u017cycia. Mo\u017cna wi\u0119c by\u0142oby zacz\u0105\u0107 w\u0142\u0105cza\u0107 t\u0119 klas\u0119 do \u201einnych przyczyn\u201d.Wyr\u00f3\u017cnienie w Analizuj\u0105c przyczyny zgon\u00f3w wed\u0142ug szczeg\u00f3\u0142owych kod\u00f3w ICD-10, mo\u017cna zauwa\u017cy\u0107 znaczn\u0105 ich r\u00f3\u017cnorodno\u015b\u0107 oraz powa\u017cne problemy klasyfikacyjne. W 2014 roku 1531 zgon\u00f3w dzieci i m\u0142odzie\u017cy w wieku 1-19 lat zosta\u0142o zakodowanych wed\u0142ug 415 r\u00f3\u017cnych przyczyn (kod trzycyfrowy z rozszerzeniem), z czego w 247 przypadkach by\u0142 to pojedynczy przypadek. Zidentyfikowano 521 zgon\u00f3w zakodowanych z rozszerzeniem \u201e9\u201d, co oznacza przypadek bli\u017cej nieokre\u015blony (BNO). Tego typu tendencja do wybierania rozpozna\u0144 BNO wyst\u0119powa\u0142a ju\u017c wcze\u015bniej. Na przyk\u0142ad, dziesi\u0119\u0107 lat wcze\u015bniej, w 2005 roku udzia\u0142 tego typu rozpozna\u0144 wynosi\u0142 w tej samej grupie wieku nawet 43%.W\u015br\u00f3d chor\u00f3b zaka\u017anych i paso\u017cytniczych, dominuj\u0105 zgony z powodu posocznicy. Nale\u017cy pami\u0119ta\u0107, \u017ce s\u0105 to pojedyncze przypadki (A39-A41 \u2212 9 zgon\u00f3w w wieku 1-19 lat).W klasie nowotwor\u00f3w najwi\u0119ksza liczba zgon\u00f3w dotyczy\u0142a nowotwor\u00f3w z\u0142o\u015bliwych m\u00f3zgu (C71 \u2212 87 przypadk\u00f3w) i ostrej bia\u0142aczki limfoblastycznej (C91.0 \u2013 24 przypadki). W odniesieniu do pozosta\u0142ych rozpozna\u0144 z klasy C zanotowano w 2014 roku pojedyncze przypadki zgon\u00f3w dzieci i m\u0142odzie\u017cy wieku 1-19 lat, w tym najwi\u0119cej z powodu: ziarnicy i ch\u0142oniak\u00f3w (12 zgon\u00f3w) nowotwor\u00f3w z\u0142o\u015bliwych ko\u015bci (10 zgon\u00f3w), nowotwor\u00f3w z\u0142o\u015bliwych tkanek mi\u0119kkich (9 zgon\u00f3w) oraz nowotwor\u00f3w z\u0142o\u015bliwych nerki (6 zgon\u00f3w). Bez retrospektywnego wywiadu trudno jest oceni\u0107, jak cz\u0119sto u tej samej osoby pojawi\u0142y si\u0119 tzw. drugie pierwotne nowotwory.W klasie \u201eE\u201d zaburze\u0144 wydzielania wewn\u0119trznego, stanu od\u017cywienia i przemian metabolicznych, w wieku 1-19 lat dominowa\u0142y jako przyczyna zgon\u00f3w choroby metaboliczne, np. zw\u0142\u00f3knienie wielotorbielowate (r\u00f3\u017cne postacie mukowiscydozy E84 \u2013 10 przypadk\u00f3w), rzadkie choroby pod postaci\u0105 zaburze\u0144 spichrzania lipid\u00f3w (E75 \u2013 9 przypadk\u00f3w) lub mukopolisacharydozy (E76.0 \u2013 6 przypadk\u00f3w). Z powodu r\u00f3\u017cnych powik\u0142a\u0144 cukrzycy insulinozale\u017cnej (E10.7 i E10.8) zmar\u0142y 3 osoby. Pozosta\u0142e zgony w tej klasie dotyczy\u0142y pojedynczych przypadk\u00f3w spowodowanych innymi rzadkimi chorobami metabolicznymi. Warto podkre\u015bli\u0107, \u017ce na tle og\u00f3lnych zestawie\u0144, klasa \u201eE\u201d kojarzona jest g\u0142\u00f3wnie z cukrzyc\u0105 i jej powik\u0142aniami (E10-E14), co potwierdza obserwacja, \u017ce w 2014 roku 94% zgon\u00f3w ca\u0142ej ludno\u015bci kodowanych w klasie \u201eE\u201d nast\u0105pi\u0142o z tej przyczyny.W klasie chor\u00f3b uk\u0142adu nerwowego najwi\u0119cej przypadk\u00f3w zgon\u00f3w spowodowanych by\u0142o dzieci\u0119cym pora\u017ceniem m\u00f3zgowym , padaczk\u0105 . Tak dominuj\u0105ca cz\u0119\u015b\u0107 BNO w tej grupie spowodowana by\u0142a by\u0107 mo\u017ce podaniem kodu choroby wed\u0142ug g\u0142\u00f3wnej manifestacji klinicznej, a zmar\u0142e dzieci mog\u0142y mie\u0107 inne zaburzenia, najcz\u0119\u015bciej genetyczne, b\u0119d\u0105ce przyczyn\u0105 mi\u0119dzy innymi padaczki. Kolejnymi przyczynami zgon\u00f3w by\u0142y pierwotne zaburzenia mi\u0119\u015bniowe z subklasy chor\u00f3b po\u0142\u0105cze\u0144 nerwowo-mi\u0119\u015bniowych (G71 \u2013 15 przypadk\u00f3w) oraz zapalenie m\u00f3zgu i/lub rdzenia kr\u0119gowego .W klasie chor\u00f3b uk\u0142adu kr\u0105\u017cenia, zgony najcz\u0119\u015bciej nast\u0119powa\u0142y z powodu zatrzymania akcji serca (I46 \u2013 22 przypadki), rzadziej z powodu niewydolno\u015bci serca (I50 - 8 przypadk\u00f3w). Dominowa\u0142y zgony m\u0142odzie\u017cy w wieku 15-19, a po\u0142owa przypadk\u00f3w by\u0142a r\u00f3wnie\u017c niedok\u0142adnie okre\u015blona (I46.9 i I50.9).W klasie chor\u00f3b uk\u0142adu oddechowego w 2014 roku zanotowano najwi\u0119cej zgon\u00f3w z powodu zapalenia p\u0142uc (J12 do J18 \u2212 \u0142\u0105cznie 76 przypadk\u00f3w), w tym szczeg\u00f3lnie zapalenia p\u0142uc wywo\u0142anego przez niezidentyfikowany czynnik zaka\u017any (J18 \u2013 58 przypadk\u00f3w).W statystyce zgon\u00f3w w wieku 1-19 z powodu wrodzonych wad rozwojowych dominowa\u0142y wady uk\u0142adu kr\u0105\u017cenia (Q20-Q28: \u0142\u0105cznie 84 przypadki), w tym najcz\u0119\u015bciej \u201einne wrodzone wady rozwojowe serca\u201d . Na dalszym miejscu wed\u0142ug cz\u0119sto\u015bci zgon\u00f3w by\u0142y wrodzone wady rozwojowe uk\u0142adu nerwowego  oraz aberracje chromosomowe niesklasyfikowane gdzie indziej .W klasie \u201eR\u201d objaw\u00f3w, cech chorobowych i nieprawid\u0142owych wynik\u00f3w bada\u0144 niesklasyfikowanych gdzie indziej, przyczyn\u0105 zgonu prawie w stu procentach by\u0142y \u201eniedok\u0142adnie okre\u015blone lub inne przyczyny zgonu\u201d, kodowane jako R96 do R99. Na 56 takich przypadk\u00f3w zarejestrowanych w 2014 roku, 41 dotyczy\u0142o m\u0142odzie\u017cy w wieku15-19 lat. Opracowania zagraniczne wskazuj\u0105 na znaczne zr\u00f3\u017cnicowanie mi\u0119dzy krajami, je\u015bli chodzi o dopuszczanie klasyfikacji przyczyn zgon\u00f3w do klasy R. Wykazano na przyk\u0142ad powszechne stosowanie tego typu praktyki w Danii, wobec 13-krotnie ni\u017cszych odpowiednich wsp\u00f3\u0142czynnik\u00f3w w Finlandii .W\u015br\u00f3d nieintencjonalnych zewn\u0119trznych przyczyn zgon\u00f3w dzieci i m\u0142odzie\u017cy w wieku 1-19 lat zarejestrowano w 2014 roku 314 zgony w wypadkach komunikacyjnych (V1-V99). Przewa\u017cnie by\u0142y to wypadki u\u017cytkownika samochodu (124 zgony w klasie V40-V49) lub osoby pieszej (68 zgon\u00f3w w klasie V3-V9). Poza wypadkami komunikacyjnymi, wyst\u0119puje kategoria tzw. uraz\u00f3w wypadkowych (W00-X59), gdzie zanotowano przede wszystkim \u015bmiertelne utoni\u0119cia  oraz wypadkowe zatrucia (22 przypadki \u2212 X41-X49).Wypadki spowodowane dzia\u0142aniami zamierzonymi obejmuj\u0105 zamierzone samouszkodzenie oraz napa\u015b\u0107, co w przypadku zdarze\u0144 ze skutkiem \u015bmiertelnym oznacza odpowiednio samob\u00f3jstwo lub zab\u00f3jstwo. W 2014 roku w populacji dzieci i m\u0142odzie\u017cy zarejestrowano 208 przypadk\u00f3w samob\u00f3jstw (X60-X84), w tym 13 w wieku 10-14 lat i 195 w wieku 15-19 lat. Osoby w wieku 1-19 lat rzadko by\u0142y o) arami zab\u00f3jstwa (15 przypadk\u00f3w w 2014 r. wg kod\u00f3w X85-Y09). Liczba zgon\u00f3w spowodowanych zdarzeniami o nieokre\u015blonym zamiarze by\u0142a wi\u0119ksza (Y11-Y34 \u2013 53 przypadki).Podsumowuj\u0105c analiz\u0119 szczeg\u00f3\u0142owych przyczyn zgon\u00f3w, warto zwr\u00f3ci\u0107 uwag\u0119 na choroby, kt\u00f3re w \u015bwietle standard\u00f3w mi\u0119dzynarodowych nie powinny stanowi\u0107 przyczyny zgonu (najcz\u0119\u015bciej uwzgl\u0119dnia si\u0119 przypadki zgonu przed 75 rokiem \u017cycia) przy odpowiedniej opiece medycznej. S\u0105 to g\u0142\u00f3wnie przyczyny pozawypadkowe. Pos\u0142uguj\u0105c si\u0119 list\u0105 kod\u00f3w ICD-10 i aktualnie obowi\u0105zuj\u0105c\u0105 na \u015bwiecie list\u0105 chor\u00f3b uleczalnych i do unikni\u0119cia , stwierdW W odniesieniu do przyczyn poza wypadkowych, \u015bredni roczny spadek wsp\u00f3\u0142czynnika zgon\u00f3w oszacowano na 0,248 w Polsce i 0,298 w krajach Eur-A. W odniesieniu do przyczyn zewn\u0119trznych tempo zmian by\u0142o wyra\u017anie wolniejsze w Polsce ni\u017c w krajach Eur-A . Je\u015bli ograniczymy analizy do tzw. zdarze\u0144 niezamierzonych , to umieralno\u015b\u0107 obni\u017ca\u0142a si\u0119 \u015brednio rocznie o 0,324 w Polsce oraz o 0,450 zgon\u00f3w na 100 000 os\u00f3b w wieku 1-19 lat w krajach Eur-A.Powy\u017csze tendencje zmian prze\u0142o\u017cy\u0142y si\u0119 na zmian\u0119 wska\u017anika nadumieralno\u015bci polskiej populacji dzieci i m\u0142odzie\u017cy. Podczas gdy w przypadku przyczyn zewn\u0119trznych nadumieralno\u015b\u0107 wzros\u0142a w latach 2000-2013 z 27,6% a\u017c do 109,3%, w odniesieniu do pozosta\u0142ych przyczyn zgon\u00f3w by\u0142 do wzrost z 14,2% do 25,4%.W U m\u0142odzie\u017cy i m\u0142odych doros\u0142ych r\u00f3\u017cnice w tempie zmian na niekorzy\u015b\u0107 Polski dotyczy\u0142y obu g\u0142\u00f3wnych kategorii przyczyn, ale by\u0142y wi\u0119ksze w odniesieniu do przyczyn zewn\u0119trznych . Cz\u0119\u015bcioObszerniejsza analiza tendencji zmian umieralno\u015bci z powodu szczeg\u00f3\u0142owych przyczyn, w tym przyczyn uznanych za mo\u017cliwe do unikni\u0119cia wymaga odr\u0119bnego opracowania. Mo\u017cna jedynie wskaza\u0107 kilka przyk\u0142ad\u00f3w, kt\u00f3re \u015bwiadcz\u0105 czasem na korzy\u015b\u0107, a czasem na niekorzy\u015b\u0107 Polski. O jako\u015bci systemu kodowania przyczyn i po\u015brednio te\u017c o jako\u015bci opieki medycznej \u015bwiadczy udzia\u0142 przyczyn nieokre\u015blonych w statystyce zgon\u00f3w (klasa R). Wed\u0142ug zalece\u0144 WHO, wpisanie kodu z klasy R w karcie zgonu jest ostateczno\u015bci\u0105, w przypadku gdy wyczerpano mo\u017cliwo\u015bci uzyskania precyzyjnej diagnozy. W por\u00f3wnaniu z krajami Eur-A, w Polsce notowana jest ni\u017csza umieralno\u015bci z tych przyczyn u m\u0142odszych dzieci, ale wy\u017csza u starszych dzieci i u m\u0142odzie\u017cy .U m\u0142odzie\u017cy i m\u0142odych doros\u0142ych zarysowa\u0142 si\u0119 do 2011 roku silny trend spadkowy, owocuj\u0105cy zr\u00f3wnaniem wsp\u00f3\u0142czynnik\u00f3w z krajami Eur-A, jednak tej korzystnej tendencji nie uda\u0142o si\u0119 utrzyma\u0107 w latach 2012-14. Alarmuj\u0105ce z kolei wydaj\u0105 si\u0119 zestawienia dotycz\u0105ce zgon\u00f3w z powodu zapale\u0144 p\u0142uc, uznanych za typow\u0105 przyczyn\u0119 mo\u017cliw\u0105 do unikni\u0119cia . Wsp\u00f3\u0142czynniki zgon\u00f3w dzieci i m\u0142odzie\u017cy s\u0105 w Polsce nadal wy\u017csze ni\u017c w przoduj\u0105cych krajach regionu europejskiego, ale dystans wzgl\u0119dem kraj\u00f3w wzorcowych stopniowo si\u0119 redukuje. Za punkt odniesienia przyj\u0119to 27 kraj\u00f3w Eur-A, o ni\u017cszej umieralno\u015bci doros\u0142ych i dzieci. Grupa ta obejmuje wi\u0119kszo\u015b\u0107 kraj\u00f3w pierwotnej Unii Europejskiej (UE-15), Islandi\u0119, Izrael, Norwegi\u0119, Szwajcari\u0119 oraz wybrane tzw. ma\u0142e pa\u0144stwa . Pod tym wzgl\u0119dem opracowanie r\u00f3\u017cni si\u0119 od innych, uznaj\u0105cych za punkt odniesienia ca\u0142\u0105 Uni\u0119 Europejsk\u0105 lub tzw. kraje UE-15 . Polska W analizowanym okresie 2000-2014, zanotowano w Polsce wyra\u017an\u0105 tendencj\u0119 spadkow\u0105 umieralno\u015bci dzieci i m\u0142odzie\u017cy. Tempo spadku zmienia si\u0119 jednak w grupach wieku i wed\u0142ug przyczyn zgon\u00f3w. Niew\u0105tpliwym sukcesem ostatnich lat jest bardzo szybki spadek umieralno\u015bci m\u0142odszych dzieci (1-4) z powodu przyczyn zewn\u0119trznych i pozosta\u0142ych oraz starszych dzieci z powodu przyczyn zewn\u0119trznych. Stosunkowo niski udzia\u0142 przyczyn nieokre\u015blonych w statystyce zgon\u00f3w ma\u0142ych dzieci te\u017c przemawia na korzy\u015b\u0107 systemu opieki zdrowotnej nad t\u0105 grup\u0105 wieku. Niepokoi\u0107 powinno natomiast zbyt wolne tempo spadku umieralno\u015bci m\u0142odzie\u017cy i m\u0142odych doros\u0142ych oraz wysoki udzia\u0142 przyczyn nieokre\u015blonych w statystyce zgon\u00f3w. W ca\u0142ej populacji wieku rozwojowego nale\u017cy zwraca\u0107 te\u017c uwag\u0119 na okresy wzrostu cz\u0119sto\u015bci zgon\u00f3w z wybranych przyczyn.Health for the World\u2019s Adolescents: a second chance in the second decade) [Prace wskazuj\u0105ce na m\u0142odzie\u017c jako grup\u0119 wieku zaniedbywan\u0105 w opiece medycznej pojawiaj\u0105 si\u0119 od 30 lat. Zaowocowa\u0142o to mi\u0119dzy innymi opracowaniem przez WHO globalnej strategii H4WA . Dost\u0119pne klasyfikacje chor\u00f3b i przyczyn zgon\u00f3w nie daj\u0105 podstaw do wnioskowania o rzeczywistych okoliczno\u015bciach zgonu, w tym o czasie i rodzaju podj\u0119tej interwencji oraz wsp\u00f3\u0142wyst\u0119puj\u0105cych czynnikach \u015brodowiskowych. Wed\u0142ug Sidebothama i wsp., kluczowe znaczenie ma opracowanie krajowych procedur identyfikacji stopnia ci\u0119\u017cko\u015bci przypadku [W wielu cytowanych opracowaniach podkre\u015blano, \u017ce ka\u017cdy zgon dziecka lub nastolatka powinien by\u0107 traktowany jako osobista tragedia jego rodziny, jednak wnikliwa analiza poszczeg\u00f3lnych przypadk\u00f3w, mo\u017ce s\u0142u\u017cy\u0107 opracowaniu strategii poprawy opieki zdrowotnej nad dzie\u0107mi. Jak podkre\u015blaj\u0105 Roberts i Jackson, opr\u00f3cz analiz tzw. obci\u0105\u017cenia chorobami, polegaj\u0105cych na \u201eilo\u015bciowym\u201d monitorowaniu chorobowo\u015bci i umieralno\u015bci, nale\u017ca\u0142oby poddawa\u0107 ocenie wdra\u017cane rozwi\u0105zania i metody post\u0119powania medycznego w przypadku konkretnych chor\u00f3b przewlek\u0142ych i o ostrym przebiegu . Autorzyrzypadku .Przyj\u0119ty w pracy okres obserwacji pozwoli\u0142 na nakre\u015blenie \u015bredniookresowych zmian w strukturze przyczyn zgon\u00f3w. Za sukces nale\u017cy uzna\u0107 zmniejszenie udzia\u0142u chor\u00f3b zaka\u017anych i paso\u017cytniczych w statystykach zgon\u00f3w dzieci i m\u0142odzie\u017cy. Du\u017co trudniejsze wydaje si\u0119 ograniczenie cz\u0119sto\u015bci wyst\u0119powania i skutk\u00f3w chor\u00f3b, kt\u00f3re maj\u0105 bardziej z\u0142o\u017cone uwarunkowania i wymagaj\u0105 dzia\u0142a\u0144 wielosektorowych. Ci\u0105gle trudno jest te\u017c na podstawie dost\u0119pnych \u017ar\u00f3de\u0142 danych zebra\u0107 informacje o chorobach rzadkich, kt\u00f3re nie stanowi\u0105 odr\u0119bnej klasy w istniej\u0105cych systemach kodowania. Mo\u017cna mie\u0107 nadziej\u0119, \u017ce wprowadzenie klasyfikacji ICD-11, cz\u0119\u015bciowo rozwi\u0105\u017ce zwi\u0105zane z tym problemy. Nale\u017cy te\u017c s\u0105dzi\u0107, \u017ce poza doniesieniami kazuistycznymi wiedz\u0119 na temat chor\u00f3b rzadkich, mo\u017cliwo\u015bci ich diagnozowania i leczenia b\u0119dzie pog\u0142\u0119bia\u0107 wsp\u00f3\u0142praca mi\u0119dzynarodowa."}
{"text": "R0S on the SDE solution. On the one hand, when R0S < 1, the SDE has an illness-free solution set under gentle additional conditions. This implies that the epidemic can be eliminated with a likelihood of 1. On the other hand, when R0S > 1, the SDE has an endemic stationary circulation under mild additional conditions. This prompts the stochastic regeneration of the epidemic. Also, we show that arbitrary fluctuations can reduce the infection outbreak. Hence, valuable procedures can be created to manage and control epidemics.The spread of epidemics has been extensively investigated using susceptible-exposed infectious-recovered-susceptible (SEIRS) models. In this work, we propose a SEIRS pandemic model with infection forces and intervention strategies. The proposed model is characterized by a stochastic differential equation (SDE) framework with arbitrary parameter settings. Based on a Markov semigroup hypothesis, we demonstrate the effect of the proliferation number  Many biological and human populations have been facing the threat of viral epidemics. The spread of such epidemics typically leads to large death tolls and significant economic and healthcare costs. The Ebola outbreak in early 2014 led to the loss of thousands of lives in Africa . ThousanRecently, perturbations have been incorporated into deterministic models of pandemics under reasonable conditions. Subsequent models have been proposed under stochastic assumptions. Gray et al.  proposedIn this study, the main contributions are introducing a susceptible-exposed-infectious-recovered-susceptible (SEIRS) epidemic model with infection forces and investigating how changes in conditions, hatching time, and other parameter settings affect the epidemic dynamics. In particular, we extend the SDE formulation of Cai et al.  and fineS, E, I, and R, respectively. The total population N is given by N=S+E+I+R. The SEIR model accepts that the recovered people might lose their immunity and reemerge in the susceptible state. The SEIR model is applicable to numerous infectious epidemics such as H7N9, bacterial loose bowels, typhoid fever, measles, dengue fever, and AIDS +(\u03bc\u03b3/\u03bc+\u03b3 \u2212 \u03b2\u03bc)I\u2217=0.The endemic equilibrium terms, namely, F(I)=\u039b \u2212 \u03b7\u03bc2f(I\u2217)/\u03b2[\u03b7+\u03b1(\u03bc+\u03bd+\u03b4)]+(\u03bc\u03b3/\u03bc+\u03b3 \u2212 \u03b2\u03bc)I\u2217. Based on the assumption (H1), the function F(I) is decreasing. Since Define F(I)=0 possesses a unique positive solution I\u2217 if R0 > 1. Therefore, a unique endemic equilibrium The equation t > 0.The next lemma demonstrates that the solutions for model  are limi\ud835\udd4f. Also, every direction of model (Model  is decidof model  will in N(t)=S(t)+E(t)+I(t)+R(t), we have the following:Joining all conditions in  and consHereafter, by integrating , we obtaThis concludes the proof of the lemma.Here, the global asymptotic stability of the epidemic-free equilibrium E0 is investigated. In particular, we prove S \u2212 \u039b/\u03bc)2+\u03b81E+\u03b82I, Construct the following Lyapunov function: V\u2009\u2009=1/2\u2009=\u2009f(0)+\u2009f\u2032(0)I+o(I), we get the following:If Moreover, since we have the following:\u03bc+\u03b2(I+\u03b1E)/f(I)](S \u2212 \u039b/\u03bc)2 \u2212 \u03b7f(0)\u03b5/f(I)I \u2212 \u03b82(\u03bc+v+\u03b4)I \u2212 R0\u03b81(\u03bc+\u03b7)+\u03b5\u03b7/f(I)f\u2032(0)I2. Note the nonnegativity of the functions S, E, I, \u2009and\u2009R. Also, note that the relationships in the right side of the last inequality are nonpositive; i.e., dV/dt \u2264 0, if and only if dV/dt=0 Consequently, the best invariant S=\u039b/\u03bc, \u2009E=0, \u2009I=0, \u2009and\u2009R=0 set in { : dV/dt=0} is a singleton {E0}.Hence, dV/dt \u2264 \u2212), t \u2208 , where \u03c7 is a characteristic function. SinceDX\u03c6h0;=\u03b5V \u2212 1/2\u03b52\u03c8(T)v+1/6\u03b53\u03c82(T)v+o(\u03b53), where v, \u03c8(T)v, and\u03c82(T)v are straightly autonomous and the subsidiary DX0;\u03d5 has a rank of 3.Thus, X0 \u2208 \u03a9\u2009\u2009and\u2009X \u2208 \u03a9, we demonstrate the existence of a control work \u03d5 and T\u2009>\u20090 for which X\u03c6(0)=X0\u2009and\u2009X\u03c6(T)=X. Set \u03c9\u03d5=x\u03d5+y\u03d5+z\u03d5. Model  \u2208 :0 < x, z < \u039b/\u03bc, \u039b/\u039b+\u03bc < w < \u039b/\u03bcand\u2009x, z, <w}. For any  \u2208 \u2009\u03a00\u2009an\u2009\u2009d\u2009.\u2009 \u2208 \u2009\u03a00, it can be claimed that there exists a control function \u03d5\u2009and\u2009T > 0 for whichLet \u03a0x\u03d5(0), w\u03d5(0), z\u03d5(0))=and(x\u03d5(T), w\u03d5\u2009(T), z\u03d5(T))=. We create the function \u03d5 in the next steps. First of all, we determine a positive constant T and a differentiable function w\u03d5 : \u27f6, for which+m, \u039b/\u03bc \u2212 m), then we have the following equation:If C2 function \u03c9\u03d5 : \u27f6(\u039b/(\u03bc+\u03b3)+m, \u039b/\u03bc \u2212 m) can be obtained for which\u03c9\u03d5 satisfies +m, \u039b/\u03bc \u2212 m) is constructed so that\u03c9\u03d5 satisfies +m, \u039b/\u03bc \u2212 m) to a C2 function \u03c9\u03d5 on the whole segment  for which \u039b \u2212 (\u03bc+\u03b3)\u00b7\u03c9\u03d5(t) < \u2212(\u03bc+\u03b3)m < \u03c9\u03d5\u2032(t) < \u03bcm < \u039b \u2212 \u03bc\u03c9\u03d5(t) for . So, the function \u03c9\u03d5 satisfies , \u2009g2,\u2009and\u2009g3 are introduced in  is a positive solution of model  for whichNow, for almost every (a)Z0 \u2208 : the conclusion is obvious from (ous from .(b)Z0 \u2208 : assume on the contrary that our claim is not true. Then, there would be \u03a9\u2032 \u2208 \u03a9 with Prob (\u03a9\u2032)\u2009>\u20090 for which Z0 \u2208 . From (Zt(w) is carefully expanding on [0,\u221e) for any w \u2208 \u03a9\u2032 . Consequently,\u03b3). From , Zt(w) iFrom , we get Hence,t\u222b0t\u03c32\u03b12Ss2/f2(Is)ds \u2264 \u03c32\u03b12\u039b2/\u03bc2f2(0) < +\u221e, and using the strong law of large numbers for martingales [Since 1/tingales , we get St, Et, It, and f(It), we obtain the following:Therefore, taking into consideration the continuity of the functions This contradicts the limit (c)Z0 \u2208 : we use again a proof by contradiction with arguments similar to those in (b) to deduce that there is \u03a9\u2032 \u2208 \u03a9 with Prob (\u03a9\u2032)\u2009>\u20090 for whichActually, three cases exist.w \u2208 \u03a9\u2032, we get the following:Using  and for w \u2208 \u03a9\u2032. Hence,Hence, t\u27f6\u221eIt=0 a.s. and the claim follows. Remark 7: from Lemmas U\u2217, then sup U\u2217\u2009=\u2009\u03a0.This is contradictory to the assumption that limR0s\u2009>\u20091, the semigroup {P (t)} t\u2009\u2265\u20090 is either sweeping with respect to minimal sets or asymptotically stable.Assume that P (t)} t\u2009\u2265\u20090 is a fundamental Markov semigroup with a constant kernel k for t\u2009>\u20090. Then, the appropriation of  possesses a density U, which fulfills (t)} t\u2009\u2265\u20090 can be restricted to the space L0 (\u03a0). As indicated by f\u2009\u2208\u2009D, we have the following:By fulfills . From LeP(t)}, t\u2009\u2265\u20090 is asymptotically stable or is sweeping with respect to minimal sets.Thus, from f(I)=1+aI2. For the convenience of display, the simulation is set as 100 times 100 in the space-time range, the abscissa represents the time, and the ordinate represents the number of patients. The simulations can help us to investigate how the ecological perturbations and the harmfully idle periods influence the spread of epidemics. In particular, we consider the global characteristics of a general SDE model with infection forces for both the deterministic case (without infection forces) and the stochastic case (with infection forces). In the first set of simulations, the parameters of the stochastic model are set as follows: \u03bb\u2009=\u20090.23, \u03bc\u2009=\u20090.01, \u03b1\u2009=\u20090.36, \u03b2\u2009=\u20090.52, \u03b3\u2009=\u20090.45, \u03c3\u2009=\u20090.6, \u03b4\u2009=\u20090.31, v\u2009=\u20090.13, \u03b7\u2009=\u20090.25, and a\u2009=\u20090.1 \u2009=\u2009. It is easy to see that the system is oscillating. Next, we study how environmental oscillations affect the spread of epidemics by reviewing the global dynamics of the general SEIRS model.The results in \u03bb\u2009=\u20090.001, \u00b5\u2009=\u20090.01, \u03b1\u2009=\u20090.35, \u03b2\u2009=\u20090.6, \u03b3\u2009=\u20090.05, \u03c3\u2009=\u20090.15, a\u2009=\u20095, \u03b4\u2009=\u20090.05, \u03bd\u2009=\u20090.6, \u03b7\u2009=\u20090.33, R0s\u2009=\u20091.5329\u2009>\u20091, and R0\u2009=\u20090.9877\u2009<\u20091. The simulation results for the deterministic model are shown in \u03c3\u2009=\u20090.15, 0.35, 0.55, and 0.75, while keeping the other parameters unchanged.We take the parameter values as follows: The results are shown in Figures 3\u03b1, the system has stronger disturbance and worse control ability. Therefore, the incubation period is an important variable in disease control. The existence of the incubation period will lead to the difficulty of disease control.Figures 7\u03bb\u2009=\u20090.001, \u03bc\u2009=\u20090.01, \u03b1\u2009=\u20090.75, \u03b2\u2009=\u20090.1, \u03b3\u2009=\u20090.25, \u03c3\u2009=\u20090.35, a\u2009=\u20090.001, \u03b4\u2009=\u20090.05, \u03bd\u2009=\u20090.1, and \u03b7\u2009=\u20090.33.We computed the time series and confidence intervals of each variable, as shown in Figures There are many variables in the system. We only discussed several representative variables in detail. In the actual disease control, we can discuss the influence of each variable on the system, so as to better control the spread of disease.Several groups of simulation results show that the conclusion of this study is correct. In the actual disease model control, we should pay attention to the types of diseases and fully consider the interference of random factors. The establishment of control variables in this study can provide basic theoretical basis and model reference for the simulation of subsequent infectious disease models.Worldwide populations have been largely and negatively impacted by infectious disease outbreaks, which had detrimental effects socially and economically . IndividNatural infection forces affect the spread of epidemics. In this study, we investigated the components of a stochastic SEIRS model with a general contamination force. The stochastic effects were considered by incorporating a multiplicative background noise in the development conditions of both the susceptible and exposed populations.Ros can be used to control the stochastic elements of a SDE model based on the Markov semigroup assumptions. If Ros\u2009<\u20091, and with gentle additional conditions, the SDE framework has a disease-free solution set, which implies the eradication of the epidemic with a likelihood of 1. When Ros\u2009>\u20091, and again under mild additional conditions, the SDE framework has an endemic equilibrium. This prompts the stochastic persistence of the disease.Our investigations uncover two important perspectives. Firstly, the generation number R0S is the main control variable of random infectious disease model control, which should be considered in practice. In addition, the change in initial value may also lead to uncontrollable results of the system, which brings greater challenges to infectious disease control.The number"}
{"text": "The HA and the influence of distance on the HA both significantly decreased gradually over the last decades (p\u202f<\u20090.01). For the first and only time, the HA reversed into an away advantage (AA) for the season 2019/2020 (p\u202f<\u20090.01). The influence of distance on HA has been significant (p\u202f<\u20090.01) in the past (before about 1990) and contributed roughly by about half, compared to a\u00a0situation without HA or AA. It increases with distance and saturates at around 100\u202fkm. Such saturation behaviour is in line with results from higher divisions of other countries with similar travelling distances such as Italy, Turkey and England. However, the distance-dependent contribution to HA has been approximately halved and reduced to an insignificant amount today. Furthermore, the temporal HA reduction is significantly larger for large distances compared to short distances (p\u202f<\u20090.01). Reporting and quantifying a\u00a0reduction (p\u202f<\u20090.01) of the distance-dependent contribution to HA over a\u00a0time span of 57\u00a0years is novel.A\u00a0statistical analysis is presented that investigates the dependence of team cities\u2019 geographical distances on the effect of home advantage (HA) for 57\u00a0years of the men\u2019s German first soccer division (\u201cBundesliga\u201d), including 17,376 matches (seasons starting from 1964 to 2020). The data shows that the HA can clearly be evidenced in the past and present (statistical  The home advantage (HA) is a\u00a0well-known phenomenon in soccer worldwide , ri\u202f=\u20090.5 (draw), ri\u202f=\u20091 (win)) for the respective team and indicating its performance/ability. This way, artificial mathematical alterations are avoided , which is a\u00a0common measure in the literature. A\u00a0mathematical analysis reveals that here\u00a0g is usually slightly larger than\u00a0r on average by an offset of \u03943P\u202f:=\u2009<g>\u202f\u2212\u2009<r>\u202f\u2248\u202f+\u20090.01\u202f=\u20091% when changing to a\u00a03-point counting system (see Discussion for details). The average team performance or result for a\u00a0specific ensemble of N\u00a0matches is marked as\u00a0r(t) or r(d) (g(t) or g(d) alike), as a\u00a0function of time\u00a0t or distance\u00a0d, respectively. Averaging of r\u00a0runs over all N\u00a0match results of all home teams, neglecting the respective result for the away team. Therefore, an averaged value of r\u202f=\u20090.5 would indicate that there is no HA and a\u00a0value of r\u202f=\u20091 would mean that the home team would always win . Thus, the HA is larger with increasing\u00a0r. Consequently, the HA will be regarded with respect to 0.5 and the term r\u202f\u22120.5 is used for analysis.Match results of the men\u2019s German first soccer division from years 1964 to 2020 (denoting the year in which the respective season started) were taken from online databases  is calculated by the mathematical distance on a\u00a0great circle viaRE\u202f=\u20096371\u202fkm is the average earth\u2019s radius   and as a\u00a0function of the distance\u00a0d as r(d). The applied discretisations for evaluation are \u0394t\u202f=\u20091 a\u00a0(1\u00a0year) in time and \u0394d\u202f=\u200920\u202fkm in distance. This means that a\u00a0data point\u00a0r(d) contains match results from the semi-open interval . All data points are provided with error bars, which denote the standard error confidence interval s\u202f=\u2009\u03c3\u202f/\u2009\u221aN, where\u00a0\u03c3 is the standard deviation of\u00a0r and\u00a0N is the number of considered matches for the respective data point. The discretisation is necessary to ensure satisfying large enough match counts N(d) or N(t) per datum.The average result\u00a0p\u2011values\u202f<\u20090.05) via t-test to check whether defective parameters are distinguishable. Bivariate Pearson correlation coefficients\u00a0c in the closed interval  are calculated and Gaussian error propagation is applied.The software Gnuplot is used for diagrams and fits. It applies the Levenberg\u2013Marquardt algorithm  for all home games of the m\u202f=\u200918 active teams in the respective season  is shown in Fig.\u00a0r>\u202f=\u20090.631\u202f\u00b1\u20090.003 (<g>\u202f=\u200963.4%) is clearly distinct from 0.5; therefore an HA is present  is decreasing with a\u00a0high anti-correlation of c\u202f=\u2009\u22120.82): In the 1970s roughly r(t)\u202f\u2248\u20090.7  and from the 1990s onwards r(t)\u202f\u2248\u20090.6. In order to investigate the two regimes separately, two time ensembles are defined: From 1964 to 1989 (dataset\u00a0A with <rA>\u202f=\u20090.679\u202f\u00b1\u20090.004 and 7890 matches with c\u202f=\u2009\u22120.10 and <rA>\u202f=\u2009<gA>\u202f=\u200967.9%) and thereafter from 1990 to 2020 (dataset\u00a0B with <rB>\u202f=\u20090.591\u202f\u00b1\u20090.004 and 9486 matches with c\u202f=\u2009\u22120.59 and <rB>\u202f\u2248\u2009<gB>\u202f=\u200959.8%). They are isolated by a\u00a0vertical black line in Fig.\u00a0r(t) drops to 0.65 and below. This way, the majority of data points r(t) of A\u00a0lie above B\u00a0and both datasets can be analysed separately.The per-year averaged performance r(d) as a\u00a0function of the cities\u2019 distance\u00a0d is shown in Fig.\u00a0dA>\u202f=\u2009(282\u202f\u00b1\u2009163) km and <dB>\u202f=\u2009(325\u202f\u00b1 169) km . One observes that the HA is larger for dataset\u00a0A (higher r(d)) compared to dataset\u00a0B for the majority of distances\u00a0d and r(d) decreases slightly for small\u00a0d. A\u00a0saturation of r(d) can be spotted around 100\u202fkm (for dataset\u00a0A), but generally, r(d) is comparatively constant. A\u00a0linear fit (not shown) for data points d\u202f\u2265\u2009100\u202fkm to each dataset separately reveals that the slopes cannot be unambiguously classified as positive or negative according to error (both slopes << 0.01 per 100\u202fkm). This finding is also in line with calculated correlation coefficients\u00a0c (of distance\u00a0d vs. result\u00a0r) to be around zero for both datasets (0\u202f\u2248\u2009| cA/B (d\u202f\u2265\u2009100\u202fkm)\u00a0| << 0.01). Still, for both datasets\u00a0r(0\u202fkm)\u202f>\u20090.5 is clearly valid, which indicates that a\u00a0negligible distance\u00a0d is still connected with a\u00a0detectable HA. Nevertheless, as mentioned above, a\u00a0slight decrease in the HA of the home team can be observed for short distances\u00a0d, especially for dataset\u00a0A. Accordingly, for d\u202f\u2264\u2009100\u202fkm small positive correlations of\u00a0(A) 0.070 (p\u202f<\u20090.01) and\u00a0(B) 0.021 (insignificant) as well as positive linear slopes of\u00a0(A) 0.071 and\u00a0(B) 0.020 (each per 100\u202fkm) can be found in this region. In order to investigate the influence of distance more closely, an exponential function according to Eq.\u00a0r0 represents the maximal result value\u00a0r(d) and thus maximal HA and r\u221e (<\u202f0) and d0 (>\u202f0\u202fkm) represent the distance-dependent contribution and influence on HA. The motivation for an exponential saturating function is that the average performance r(d) first increases, but then seems to stabilize for large distances\u00a0d, as pointed out before. The results of the least square fit procedure are shown in Table\u00a0Discretised plots of the averaged performances \u2009282\u202f\u00b1\u200916 km and <r(0\u202fkm)\u202f=\u2009r0\u202f+\u2009r\u221e\u202f>0.5 (p\u202f<\u20090.01), which means that the HA cannot be solely explained by the geographical/travelling distance\u00a0d alone, as reported before  play each other  decreased over time , an influence of the distance\u00a0d on the HA is present (see Eq.\u00a0p\u202f<\u20090.01) decreased . On the one hand, shorter distances\u00a0d reduce the HA of the home team and on the other hand, the HA saturates for larger distances\u00a0d. The total influence of distance is smaller compared to the distance-independent influences on HA, since |r\u221e|\u202f<\u2009|r*| is significant (p\u202f<\u20090.01) in the past\u00a0(A) and the present\u00a0(B). The results suggest that up to half (p\u202f<\u20090.01) of the total HA has been explicable by distance-related effects in the past (see \u03b2A in Table\u00a0\u03b2B).Since\u00a0 see Eq.\u00a0. Howeverr\u221e\u202f=\u2009\u22120.082 (r0\u202f=\u20090.643) from the dataset he used, coinciding with the distance-dependent influence r\u221e=\u2009\u22120.076\u202f\u00b1\u20090.026 found here of dataset\u00a0A (1964\u20131989) in Table\u00a0r\u221e\u202f=\u2009\u22120.040 and r0\u202f=\u20090.617 derived from the dataset they used) comparable with the value of r\u221e\u202f=\u2009\u22120.018\u202f\u00b1\u20090.025 found here for dataset\u00a0B (1990\u20132020). The same trend has been described for the Italian Seria\u00a0A . However, the number of same city derbies is only about 10% (40 of 389 matches with d\u202f<\u200920\u202fkm), thus playing a\u00a0minor role here.These findings will be discussed in the context of selected other countries. Pollard  analysedataset\u00a0A 964\u20131989 g with 3\u00a0(win), 1\u00a0(draw) and 0\u00a0(loss) points per match (3-point counting system), which is a\u00a0common measure for many soccer divisions worldwide , including the German Bundesliga since 1995. As noted before, for the 2\u2011point counting system (1994 and earlier) the identity r\u202f=\u2009g holds. For the years from 1995 to 2020 (defined as dataset\u00a0C as a\u00a0subset of dataset\u00a0B), we find very high correlations of c\u202f=\u20090.9997 between the two types of analysis for rC(t) with gC(t) as well as for rC(d) with gC(d). However, g is usually (except for the year 2019 with AA) slightly larger than the result value\u00a0r by an offset of \u03943P\u202f=\u202f+\u20090.0083\u202f\u00b1\u20090.0026\u202f\u2248\u2009(0.8\u202f\u00b1\u20090.3) %\u202f\u2248\u20091% on average (<gC>\u202f=\u20090.594\u202f\u00b1\u20090.003 =\u200959.4% and <rC>\u202f=\u20090.586\u202f\u00b1\u20090.003\u202f= 58.6%), which has to be taken into account. All according fit parameters for g(d)\u202f=\u2009g0\u202f+\u2009g\u221e\u00a0\u22c5 exp(\u2212d\u202f/\u2009d0g) equivalent to Eq.\u00a0g0\u202f=\u20090.597\u202f\u00b1\u20090.011\u202f=\u200959.7%, g\u221e\u202f=\u2009\u22120.021\u202f\u00b1\u20090.026\u202f=\u2009\u22122.1%, d0g\u202f=\u2009(122\u202f\u00b1 313) km, g*\u202f=\u20090.076\u202f\u00b1\u20090.037\u202f=\u20097.6%, gHA\u202f= 0.097\u202f\u00b1\u20090.011\u202f=\u20099.7%, \u03b1g\u202f=\u2009(3.6\u202f\u00b1\u20094.4) %, \u03b2g\u202f=\u2009(22\u202f\u00b1\u200929) %) coincide well with fit parameters of r(d)  km, r*\u202f=\u20090.069\u202f\u00b1\u20090.034, rHA\u202f=\u20090.089\u202f\u00b1\u20090.009, \u03b1\u202f=\u2009(3.4\u202f\u00b1\u20094.3) %, \u03b2\u202f=\u2009(22\u202f\u00b1\u200930) %) for dataset\u00a0C and even with parameters of dataset\u00a0B (1990\u20132020) within the margin of error (see Table\u00a0\u03943P\u202f\u2248\u20091%). Thus, teams with comparatively higher HA were marginally advantaged in gaining points at home  from past\u00a0(A) to present\u00a0(B), indicating a\u00a0reduction of the distance-dependent influence on HA. It is argued here that Oberhofer et\u00a0al. (r\u221e has decreased and lost significance over time (from p\u202f<\u20090.01 for\u00a0A to insignificant for\u00a0B).Oberhofer et\u00a0al.  also anar et\u00a0al.  did not r et\u00a0al. . Corresprteam,home> of home matches divided by the average result of all matches <rteam> of the team to account for the different playing strengths of teams) has been compared with the average distance <dteam> travelled (not shown). Indeed, there are no outliers for the largest distances <dteam> and also slope (\u22120.006 per 100\u202fkm with s\u202f=\u2009\u00b1260%) as well as correlation (c\u202f=\u2009\u22120.05) are even slightly negative and insignificant according to margin of error .For Brazil  , while a\u00a0decreasing trend of HA had already set in during the years before  is clearly visible in the year 1990, which is the year of German reunification. Never again after this year did the HA reach result values r(t) of 0.65 or above  in the whole history of the Bundesliga happened in 2019, reversing the HA into an AA for the first and only time (r\u202f=\u20090.430\u202f<\u20090.5). In this year, the global pandemic of COVID-19 spread, also imposing social disruptions. During this time, many soccer matches worldwide  were held without spectators. From the analysis of tens of thousands of those matches in European major and minor leagues, Wunderlich et\u00a0al. (r\u202f=\u20090.551\u202f>\u20090.5), even when spectators were reduced to less than 0.2\u00a0million. For the total dataset (1964\u20132020), we even find a\u00a0high anti-correlation (c\u202f=\u2009\u22120.572\u202f<\u20090 with p\u202f<\u20090.001) between r(t) and the number of spectators (not shown). Surprisingly, this is a\u00a0hint that more spectators could even reduce HA, however superimposed by others covariates  by Jacklin  due to lsee Fig.\u00a0 and mighsee Fig.\u00a0 or accelh et\u00a0al.  found a\u00a0h et\u00a0al.  claimed h et\u00a0al.  found a\u00a0h et\u00a0al. . Due to h et\u00a0al. . In contp\u202f<\u20090.01), but its distance-dependent and distance-independent contribution both decreased over the decades (p\u202f<\u20090.01). This is the first time that a\u00a0reduction in the distance-dependent HA is reported for Germany  and its dependence on geographical (travelling) distance has been investigated. The HA is clearly present (Factors that might explain the reductions of the distance-dependent and distance-independent influence on HA have been discussed. These include improved travel conditions (and strategies) nowadays reducing travel fatigue (Pollard & Pollard, d on a\u00a0perfect sphere is slightly different from the real geographical distance or travelling distance or travelling time. These quantities as well as direct measurements of the away team\u2019s travel fatigue (Pollard & Pollard, Following earlier reports (Nevill et\u00a0al.,"}
{"text": "Binaural Squelch With the SONNET, the head shadow effect was \u2212 1.2\u00a0dB (range\u2009\u2212\u20095.9\u20120.6) in natural mode, 0.9\u00a0dB (range\u2009\u2212\u20094.3\u20126.1), and in adaptive mode, and\u2009\u2212\u20090.4\u00a0dB (range\u2009\u2212\u20094.4\u20122.5) in the omnidirectional mode\u201d was published incorrectly.In the original publication of the article, under the section Binaural effects, the following sentence \u201cBinaural Squelch With the SONNET, the binaural squelch effect was \u2212 1.2\u00a0dB (range\u2009\u2212\u20095.9\u20120.6) in natural mode, 0.9\u00a0dB (range\u2009\u2212\u20094.3\u20126.1), and in adaptive mode, and \u2212 0.4\u00a0dB (range\u2009\u2212\u20094.4\u20122.5) in the omnidirectional mode.\u201dThe correct sentence should read as \u201cBinaural Summation\u201d. The correct paragraph should read as below,In addition, there are some mistakes in the paragraph \u201cBinaural Summation With the SONNET, the binaural summation effect was \u2212 0.6\u00a0dB (range\u2009\u2212\u20091.1\u20122.7) in natural mode, 0.0\u00a0dB (range\u2009\u2212\u20091.3\u20122.2), and in adaptive mode, and 0.2\u00a0dB (range\u2009\u2212\u20091.2\u20122.1) in the omnidirectional mode. No significant differences were found between SONNET modes. With the OPUS 2/RONDO, the binaural summation effect was \u2212 0.4\u00a0dB (range\u2009\u2212\u20091.1\u20121.6). No significant differences were found between SONNET modes and the OPUS 2/RONDO. .The original article has been updated."}
{"text": "In this study, the predefined time synchronization problem of a class of uncertain chaotic systems with unknown control gain function is considered. Based on the fuzzy logic system and varying-time terminal sliding mode control technology, the predefined time synchronization between the master system and the slave system can be realized by the proposed control method in this study. The simulation results confirm the theoretical analysis. In recent decades, chaotic synchronization has been a research hotspot. The main reason is its wide application, such as in the fields of secure communication, biological systems, and so on \u20136. Up toIn this study, the predefined time synchronization of the main system and the slave system is considered. The main highlights are as follows: the synchronization of two uncertain chaotic systems is realized by the varying-time sliding mode control method, and the case where the controller gain is unknown is considered. The rest of this study is organized as follows. Some preliminaries are given in \u03be1, \u03be2 \u2208 \u211d are the states of system  \u2208 \u211d is a nonlinear function.The master system is described as\u03be\u02d91=\u03be2,\u03be\u02d9\u03b71, \u03b72 \u2208 \u211d are the states of system  \u2208 \u211d is a nonlinear function, u \u2208 \u211d is the control input, and g is a control gain function.The slave system is described as\u03b7\u02d91=\u03b72,\u03b7\u02d9e1=\u03b71 \u2212 \u03be1, e2=\u03b72 \u2212 \u03be2. The aim of this study is to design a varying-time terminal sliding mode control method, so that the synchronization error e1 reaches a small neighborhood of zero in the predefined time. According to =\u03b72(0).States f1 and f2 are unknown but bounded.g is unknown strictly positive and there exists a positive constant \u03c7, such that g > \u03c7.\u03be2(0)=\u03b72(0) in T is a preset time, 0 < q/p < 1, q and p are the odds, \u03b1, \u03b2 are the design positive constant, and \u03bb1, \u03bb2, and \u03bb3 satisfy the following conditions:(1)In order to ensure that the initial value of system  belongs (2)t=T, i.e.,The sliding mode  is conti(3)In order to ensure that sliding mode  can quicIn order to achieve the aim of this study, the time-varying terminal sliding mode is considered:Letq/p with respect to time t may appear singular problem, and we modify The derivation of \u0394\u03be1, \u03be2, \u03b71, \u03b72 are measurable, unknown functions f1, f2, and g can be estimated by fuzzy logic systems [f1, f2, and g, there exist \u03b8f1\u2217T\u03c6f1, \u03b8f2\u2217T\u03c6f2, and \u03b8g\u2217T\u03c6g, such that\u03b5f1, \u03b5f2\u03b5g, and \u03b5g are the bounded fuzzy estimation errors, \u03b8f1\u2217T, \u03b8f2\u2217T, and \u03b8g\u2217T are the ideal weight vectors, and \u03c6f1, \u03c6f2, and \u03c6g are the Gaussian functions.Since  systems , 25. Forz with respect to t can be obtained asFrom  and 4),,4), the k1 is a design positive constant, and \u03b8f1\u2217, \u03b8f2\u2217, and \u03b8g\u2217. The parameter adaptation laws of \u03b3f1, \u03b3f2, \u03b3g, \u03b4f1, \u03b4f2, and \u03b4g are the design positive constants. Let \u03b5(t)=\u03b5f1+\u03b5f2+\u03b5gue. Obviously, \u03b5(t) is bounded, i.e., there exists a positive constant \u03b5\u2217, such that |\u03b5(t)| \u2264 \u03b5\u2217.Now, design the controller asUnder Assumptions V1 with respect to t yieldsConsider the following Lyapunov function:^g. From , derivatSubstituting  and 13)13) to 1 yields. Obviously, if V2 will monotonically decrease only to enter \u03a9e. Therefore, we obtain the convergence range of the tracking error e1.Let f1=\u2212102(1 \u2212 cos\u2009\u2009\u03be1)2/sin3\u03be1+sin\u2009\u2009\u03be1 \u2212 0.5\u03be2 \u2212 0.05\u03be23+35.7\u2009\u2009sin(2t)sin\u2009\u2009\u03be1. For the slave system =\u2212102(1 \u2212 cos\u2009\u2009\u03b71)2/sin3\u03b71+sin\u2009\u2009\u03be1 \u2212 0.5\u03b72 \u2212 0.05\u03b723+35.5\u2009\u2009sin(2t)sin\u2009\u2009\u03b71, g=5+sin\u2009\u2009\u03b72. Obviously, g > \u03c7\u225c3. The initial values \u03be1(0)=\u22121, \u03be2(0)=1, \u03b71(0)=2, and \u03b72(0)=1. The fuzzy membership functions are selected as\u03c1=\u03be1, \u03be2, \u03b71, \u03b72; j=1,2,3,4,5. First, select a group of parameters as T=2, k1=3, q=3, p=5, \u03b2=3, \u03b1=\u22125, \u03bb1=\u22125/4, \u03bb2=5, \u03bb3=14/23/5, and the simulation results are shown in Figures 1\u03be1 of master system , u fluctuates at T=2s, and then, the controller has a small chattering phenomenon.In this section, the chaotic gyroscope system  is taken1t, \u03be1, \u03be=\u2212102.Obviously, the proposed control method  in this In this study, the predefined time synchronization problem of uncertain chaotic systems was investigated. The fuzzy logic system was used to estimate the unknown function. A time-varying sliding mode was constructed. The proposed varying-time terminal sliding mode control method in this study made all signals bounded and the synchronization error entered a small neighborhood of zero after the predefined time. Simulation results show the effectiveness of the method."}
{"text": "Analysis of genomes evolving via block-interchange events leads to a combinatorial problem of sorting by block-interchanges, which has been studied recently to evaluate the evolutionary relationship in distance between two biological species since block-interchange can be considered as a generalization of transposition. However, for genomes consisting of multiple chromosomes, their evolutionary history should also include events of chromosome fusions and fissions, where fusion merges two chromosomes into one and fission splits a chromosome into two.n2) time algorithm to efficiently compute and obtain a minimum series of fusions, fissions and block-interchanges required to transform one circular multi-chromosomal genome into another, where n is the number of genes shared by the two studied genomes. In addition, we have implemented this algorithm as a web server, called FFBI, and have also applied it to analyzing by gene orders the whole genomes of three human Vibrio pathogens, each with multiple and circular chromosomes, to infer their evolutionary relationships. Consequently, our experimental results coincide well with our previous results obtained using the chromosome-by-chromosome comparisons by landmark orders between any two Vibrio chromosomal sequences as well as using the traditional comparative analysis of 16S rRNA sequences.In this paper, we study the problem of genome rearrangement between two genomes of circular and multiple chromosomes by considering fusion, fission and block-interchange events altogether. By use of permutation groups in algebra, we propose an FFBI is a useful tool for the bioinformatics analysis of circular and multiple genome rearrangement by fusions, fissions and block-interchanges. For the past two decades, genome rearrangements have been studied and can be modelled to learn more about the evolution of mitochondrial, chloroplast, viral, bacterial and mammalian genomes . To evaln2) time algorithm, where n is the number of genes, to solve the so-called block-interchange distance problem that is to find a minimum series of block-interchanges for transforming one linear chromosome into another. Later, we  and  of \u03b1p. That is, \u03c12\u03c11 is a block-interchange event affecting on \u03b1 by swapping  and , two non-intersecting blocks in \u03b1.Recently, Lin et al.  further \u03b1 into another I can be expressed by a product of 2-cycles, say \u03c1k\u03c1k-1...\u03c11, such that \u03c1k\u03c1k-1...\u03c11\u03b1 = I . This property implies that I\u03b1-1 contains all information that can be utilized to derive \u03c11, \u03c12,...,\u03c1kfor transforming \u03b1 into I.As discussed above, any series of fusions, fissions and block-interchanges required to transform one circular multi-chromosomal genome \u03b1 as a product of 2-cycles [\u03b1, let f(\u03b1) denote the number of the disjoint cycles in the cycle decomposition of \u03b1. Notice that f(\u03b1) counts also the non-expressed cycles of length one. For example, if \u03b1 =   is a permutation of E = {1, 2,..., 5}, then f(\u03b1) = 3, instead of f(\u03b1) = 2, since \u03b1 =   (3). Then the following lemma shows the lower bound of the number of 2-cycles in any product of 2-cycles of a permutation.It is well known that every permutation can be written as a product of 2-cycles. For example,  =   . However, there are many ways of expressing a permutation 2-cycles . Given aLemma 1 Let \u03b1 be an arbitrary permutation of E = {1, 2,..., n}. If \u03b1 can be expressed as a product of m 2-cycles, say \u03b1 = \u03b11\u03b12...\u03b1m with each \u03b1i being 2-cycle, then m \u2265 n - f(\u03b1).Proof. We prove this lemma by induction on m. The lemma is true if m = 0, since \u03b1 = 1 then, meaning that f(\u03b1) = n, and hence m = n - f(\u03b1) = 0. Suppose now that the lemma holds for any permutation that can expressed as a product of less than m 2-cycles, where m > 0. Let \u03b1' = \u03b12\u03b13, ...,\u03b1m. Then by the induction hypothesis, we have m - 1 \u2265 n - f(\u03b1'). Since \u03b1 = \u03b11\u03b1' and \u03b11 is a 2-cycle, \u03b11 operates on \u03b1' either as a fusion by joining two cycles of \u03b1' into one cycle  = f(\u03b1') - 1) or as a fission by splitting one cycle of \u03b1' into two cycles  = f(\u03b1') + 1). Whichever \u03b11 operates on \u03b1', we have f(\u03b1) \u2265 f(\u03b1') - 1. As a result, m = (m - 1) + 1 \u2265 n - f(\u03b1') + 1 = n - (f(\u03b1') - 1) \u2265 n - f(\u03b1). \u00a0\u00a0\u00a0 \u25a1n genes can be expressed by a permutation of E = {1, 2,..., n}. Given two such genomes G1 and G2 over the same gene set E, the genome rearrangement distance between G1 and G2, denoted by d, is defined to be the minimum number of events needed to transform G1 into G2, where the events allowed to take place are fusions, fissions and block-interchanges. In this section, we shall show that there is an optimal series of events required to transform G1 into G2 such that all fusions come prior to all block-interchanges, which come before all fissions. Here, such an optimal scenario of genome rearrangements is referred as in canonical order.As mentioned previously, each circular multi-chromosomal genome with Lemma 2 d = d.Proof. Let \u03a6 = <\u03c31, \u03c32,...,\u03c3\u03b4> be an optimal series of events required to transform G1 into G2. Clearly, \u03a6' = <\u03c3\u03b4, \u03c3\u03b4-1,...,\u03c31> is an optimal series of events for transforming G2 into G1 by reversing the role of every event \u03c3i, where 1 \u2264 i \u2264 \u03b4, such that \u03c3i is a fission  in \u03a6' if \u03c3i is a fusion  in \u03a6.Lemma 3 There is an optimal series of events required to transform G1 into G2 such that every fission occurs after every fusion and block-interchange.Proof. Let \u03a6 = <\u03c31, \u03c32,...,\u03c3\u03b4> be an optimal series of events needed to transform G1 into G2. Of course, if every fission occurs after every fusion and block-interchange in \u03a6, then the proof is done. Now, suppose that not every fission occurs after every fusion or block-interchange in \u03a6. Then let i be the largest index in \u03a6 such that \u03c3i is a fission preceding \u03c3i+1 that is either a fusion or a block-interchange. We can then obtain a new optimal series \u03a6' = <\u03c31,...,\u03c3i-1, \u03c3i+2,...,\u03c3\u03b4> to transform G1 into G2 such that \u03c3i splits a chromosome \u03b1 into \u03b11 and \u03b12. If \u03c3i+1 is a fusion, then we assume that it joins two chromosomes \u03b21 and \u03b22into \u03b2; otherwise, if \u03c3i+1 is a block-interchange, then assume that it affects \u03b21 such that \u03b21 becomes \u03b2 through a block-interchange. Clearly, if neither \u03b21 nor \u03b22 is created by \u03c3i, then the desired series \u03a6' is obtained by swapping \u03c3i and \u03c3i+1 in \u03a6 . If both \u03b21 and \u03b22 are created by \u03c3i, then the net rearrangement of \u03c3i (a split operation) followed by \u03c3i+1 (a joint operation) either has no effect on \u03b1 or becomes a block-interchange affecting \u03b1. By removing \u03c3i and \u03c3i+1 from \u03a6 or replacing them with an extra block-interchange, we thus obtain a new optimal series of the events transforming G1 into G2 with strictly less than \u03b4 events, a contradiction. Hence, we assume that only one of \u03b21 and \u03b22 is created by \u03c3i and without loss of generality, let \u03b21 = \u03b11. Now, we consider the following two cases.\u03c3i+1 is a fusion. For simplicity of discussion, we let \u03b1 = , \u03b22 = , \u03c3i =  and \u03c3i+1 = , where 1 <x <y - 1 and y <z. Then the net rearrangement caused by \u03c3i and \u03c3i+1 is to transform \u03b1 and \u03b2 into \u03b12 =  and \u03b2 = . In fact, this rearrangement can also be done by first joining \u03b1 and \u03b22 into  via \u03c3i+1 and then splitting  into \u03b12 and \u03b2 via . In other words, we can obtain \u03a6' by letting \u03c3i+1 and x, y).Case 1: \u03c3i+1 is a block-interchange. Clearly, the net rearrangement caused by \u03c3i and \u03c3i+1 is to transform \u03b1 into \u03b2 and \u03b12, which is equivalent to the rearrangement by first applying \u03c3i+1to \u03b1 and then further splitting it into \u03b2 and \u03b12 via \u03c3i. Then \u03a6' is obtained by swapping \u03c3i and \u03c3i+1 in \u03a6.Case 2: G1 into G2 such that all fissions come after all fusions and block-interchanges.In other words, we can always obtain \u03a6' from \u03a6 according to the method described above. Repeating this process on the resulting \u03a6', we can finally obtain an optimal series of events that are required to transform Lemma 4 There is an optimal series of events required to transform G1 into G2 in a canonical order such that all fusions come before all block-interchanges, which come before all fissions.Proof. Let \u03a6 = <\u03c31, \u03c32,..., \u03c3\u03b4> be an optimal series of events required to transform G1 into G2. If there are no fusions or block-interchanges, then the proof is completed. If not, according to Lemma 3, we may assume that all fusions and block-interchanges occur earlier than all fissions. Let i be the index of the last non-fission in \u03a6 and also let G' be the resulting genome after all \u03c31, \u03c32,...\u03c3i have affected G1. Since \u03a6 is optimal, it is straightforward to see that \u03a6i = <\u03c31, \u03c32,...,\u03c3i> is an optimal series of fusions and block-interchanges needed to transform G1 into G'. As discussed in the proof of Lemma 2, \u03c3i, \u03c3i-1,...,\u03c31> is an optimal series of fissions and block-interchanges for transforming G' into G1. Moreover, by Lemma 3, we can obtain G' into G1 such that all block- interchanges in G1 into G', and all its fusions occur before all its block-interchanges. Therefore, there is an optimal series of events needed to transform G1 into G2 such that all fusions come earlier than all block-interchanges, which come before all fissions. \u00a0\u00a0\u00a0 \u25a1G1 and G2 are two given linear multi-chromosomal genomes, where G1 =   and G2 =  . Then the optimal scenario between them is a fission, splitting  into (1) , followed by a fusion, joining (1) and  to . However, this optimal scenario can not be transformed into another in the canonical order according to the steps as described in Lemmas 3 and 4. Actually, there is no an optimal scenario between such two linear genomes using any two rearrangement events that begin with a fusion.It is worth mentioning that an optimal scenario in a canonical order does not necessarily exist for linear multi-chromosomal genomes. For example, suppose that \u03b1 and I be two given circular multi-chromosomal genomes over the same gene set E = {1, 2,...,n}. Here, we assume that the genes in I are sorted in the order of increasing and consecutive numbers, and that gene i + 1 is on the right side of gene i within the same chromosome, where 1 \u2264 i \u2264 n - 1. For example, I =    if I has three circular chromosomes with two, three and four genes, respectively. In this case, the computation of d and its corresponding optimal scenario can be considered as a problem of sorting \u03b1 using the minimum set of operations, including fusions, fissions and block-interchanges.Let \u03c1\u03bb\u03c1\u03bb - 1 ... \u03c11 is a product of 2-cycles that corresponds to an optimal series \u03a6 of fusions, fissions and block-interchanges for transforming \u03b1 into I. Then \u03a6 = \u03c1\u03bb\u03c1\u03bb - 1...\u03c11 = I\u03b1-1 and by Lemma 1, \u03bb \u2265 n - f(I\u03b1-1). If \u03a6' = \u03bb 2-cycles with \u03a6' = I\u03b1-1, then the number of 2-cycles in \u03a6' that function as fusions or fissions must be greater than or equal to that in \u03a6; otherwise, the total number of fusions, fissions and block-interchanges in \u03a6' for transforming \u03b1 into I must be less than that in \u03a6, a contradiction. The reason is that a fusion or fission requires only one 2-cycle for rearrangement, whereas a block-interchange requires two 2-cycles. In other words, the number of 2-cycles serving as the fusions and fissions is minimum in any optimal series of events.Suppose that d and its optimal scenario of rearrangement events in a canonical order. Let \u03c7(\u03b1) and \u03c7(I) denote the numbers of chromosomes in \u03b1 and I, respectively, and let \u03b1 = \u03b11\u03b12...\u03b1\u03c7(\u03b1) and I = I1I2...I\u03c7(I). Then, an undirected graph \u03b1,I) = }.\u2022 I = {I1, I2,...,I\u03c7(I)}.\u2022 \u03b1i, Ij) | 1 \u2264 i \u2264 \u03c7 (\u03b1), 1 \u2264 j \u2264 \u03c7 (I), and \u03b1i and Ij have at least a common gene}.\u2022 \u03b1 = \u03b11\u03b12...\u03b15 =      and I = I1I2 ... I6 =      . Then the induced graph \u03b1,I) is shown in Figure \u03b1 and I are independent sets in \u03b1, I) . A connected component of \u03b1,I) is defined to be a maximal subgraph of \u03b1,I) such that there exists a path between any pair of vertices in this subgraph. For example, the induced \u03b1,I) as shown in Figure Ik of I there are two genes that appear in two different chromosomes \u03b1i and \u03b1j of \u03b1, then  \u2208 \u03b1j, Ik) \u2208 \u03b1i and \u03b1j belong to the same connected component in \u03b1,I).For instance, suppose that 1, 2,...,\u03c9} denote the collection of all connected components in \u03b1,I). For each 1 \u2264 i \u2264 \u03c9, let \u03b2i and Ji denote the chromosomes in \u03b1 and I, respectively, whose corresponding vertices belong to i in \u03b1,I). Let gene(\u03b2i) and gene(Ji) be the collections of the genes in all chromosomes of \u03b2i and Ji, respectively. Then gene(\u03b2i) = gene(Ji) and gene(\u03b2i) \u2229 gene(\u03b2j) = \u2205 for any 1 \u2264 j \u2260 i \u2264 \u03c9. Let ni be the number of genes in gene(\u03b2i). Clearly, n = n1 + n2 + ... + n\u03c9. In addition, it can be verified that I\u03b1-1 = (J1J2J\u03c9f(I\u03b1-1) = f.According to the properties above, we then find a product \u03a6 of 2-cycles so that \u03a6\u03b1, I) of a given instance \u03b1 and I has exactly one connected component. Let \u03a6 = <\u03c31, \u03c32, ...., \u03c3\u03b4> be an optimal series of events for transforming \u03b1 into I in which all fusions precede all block-interchanges that further precede all fissions. Let nfu, nbi and nfi denote the numbers of fusions, block-interchanges and fissions, respectively, in \u03a6. Then \u03b4 = nfu + nbi + nfi. In the following, we shall show that \u03a6 can be expressed by a product of n - f(I\u03b1-1) 2-cycles in which the number of 2-cycles functioning as the fusions and fissions is minimum.To simplify our discussion, throughout the rest of this section we assume that the induced \u03b1i and \u03b1j in \u03b1 with (\u03b1iIk) \u2208 \u03b1j, Ik) \u2208 \u03b1i and \u03b1j to one chromosome; otherwise, Ik can not be formed from \u03b1 by a fission later. Since all needed fusions come together in the beginning of \u03a6, nfu = \u03c7(\u03b1) - 1, which is the lower bound of the number of fusions required in any optimal series of events for transforming \u03b1 into I. After these nfu fusions, the resulting \u03b1 becomes only one chromosome. Since the next nbi block-interchanges are intra-chromosomal mutations, we have nfi = \u03c7(I) - 1. Actually, \u03c7(I) - 1 is the minimum number of the required fissions in any optimal series of events for transforming \u03b1 into I, since it is the minimum number of the fusions used in the corresponding optimal series of events to transform I into \u03b1.It should be noticed that the chromosomes considered here are disjoint . Hence, for any two chromosomes \u03c1, we use x \u2208 \u03c1 to denote that x is a number in \u03c1. For any two x \u2208 \u03c1 and y \u2208 \u03c1, they are said to be adjacent in \u03c1 if \u03c1(x) = y or \u03c1(y) = x. Next, we show a way to derive nfu 2-cycles from I\u03b1-1 such that these 2-cycles function as the fusions that join all chromosomes of \u03b1 to a single one, if \u03b1 has multiple chromosomes, where nfu = \u03c7(\u03b1) - 1. For simplicity, later in the text we use \"cycle in I\u03b1-1\" to represent \" cycle in the cycle decomposition of I\u03b1-1\" in meaning, unless a possible confusion may arise.Given any cycle Lemma 5 Let \u03b1i and \u03b1j be any two disjoint cycles in \u03b1. Then there must exist a cycle in I\u03b1-1 that contains two numbers x and y such that x \u2208 \u03b1i and y \u2208 \u03b1j.Proof. Since we assume that the induced \u03b1, I) contains exactly and only one connected component, and \u03b1i and \u03b1j contain some numbers u and v, respectively, such that both u and v are in a cycle Ik of I. Notice that u \u2209 \u03b1j and v \u2209 \u03b1i. Suppose that there is no cycle in I\u03b1-1 that contains two numbers x and y such that they are in these two different cycles of \u03b1, say x \u2208 \u03b1i and y \u2208 \u03b1j. Then all numbers in any cycle of I\u03b1-1 are contained in some cycle of \u03b1. Without loss of generality, let Ik =  and let p <q for simplifying the discussion. For each 1 \u2264 x \u2264 p, let \u03b1(ax) = bx. Then we have I\u03b1-1(bx) = ax+1 (since I\u03b1-1\u03b1 = I), which means that both bx and ax+1 are in the same cycle of I\u03b1-1 and hence they are also in the same cycle of \u03b1. If ax is in \u03b1i, then bx is also in \u03b1i, which further leads to ax+1 \u2208 \u03b1i. Since u = a1 is in \u03b1i, all a2, a3, ..., ap are in \u03b1i. As a result, both of u and v are in \u03b1i, a contradiction. Hence, there exists a cycle in I\u03b1-1 that contains x and y such that x \u2208 \u03b1i and y \u2208 \u03b1j.The following lemma can be easily verified.Lemma 6  =   , where 1 \u2264 i <j.\u03b1i and \u03b1j of \u03b1, we can find two numbers x and y in a cycle of I\u03b1-1, say \u03b2, such that x \u2208 \u03b1i and y \u2208 \u03b1j. Let \u03b2 = , where q \u2265 2. Then we consider the following two cases. Case 1: x and y are adjacent in \u03b2. For simplicity, let x = aq-1 and y = aq. Then by Lemma 6, \u03b2 =  (aq) . Case 2: x and y are not adjacent in \u03b2. Let x = ap and y = aq, where 1 \u2264 p <q - 1. Then \u03b2 =    according to Lemma 6. In other words, we can derive a 2-cycle  from \u03b2 such that it can join \u03b1i and \u03b1j to one cycle. After \u03b1i, and \u03b1j are joined together via , the number of the cycles (including 1-cycles) in the resulting I\u03b1-1 increases by one. Repeatedly based on the procedure above, we can derive consecutive nfu 2-cycles from I\u03b1-1, say \u03c61, \u03c62,...,\u03c7 (\u03b1) cycles in \u03b1 to a single one, where nfu = \u03c7(\u03b1) - 1. In other words, \u03c61, \u03c62,...,\u03c7(\u03b1) - 1 fusions that transform genome \u03b1 with \u03c7(\u03b1) chromosomes into a genome, denoted by \u03b1', with a single chromosome. Clearly, we have \u03b1' = \u03c61\u03b1, I\u03b1'-1 = I\u03b1-1 \u03c61\u03c62 ... f(I\u03b1'-1) = f(I\u03b1-1) + nfu. Hence, we can immediately claim the following.According to Lemma 5, for any two cycles Claim 1 \u03b1' = \u03c61\u03b1, I\u03b1'-1 = I\u03b1-1 \u03c61\u03c62 ... f(I\u03b1'-1) = f(I\u03b1-1) + nfu, where nfu = \u03c7(\u03b1) - 1.\u03c7(I) > 1. Similarly as the discussion above, we can derive consecutive nfi 2-cycles from \u03b1'I-1, say \u03c81, \u03c82, ..., I with \u03c7(I) chromosomes into a genome, denoted by I', with only one chromosome, where nfi = \u03c7(I) - 1 and \u03b1'I-1 is the inverse of I\u03b1'-1 -1). Then we have I' = \u03c81I (hence \u03c81\u03c82 ... I' = I), \u03b1'I'-1 = \u03b1'I-1\u03c81\u03c82 ... f(\u03b1'I'-1) = f(\u03b1'I-1) + nfi. Conversely, we can use \u03c81 as fissions to split I' with one chromosome into I with nfi chromosomes. Since \u03b1'I-1 is the inverse of I\u03b1'-1, it can be easily obtained from I\u03b1'-1 by just reversing the order of the numbers in each cycle of I\u03b1'-1 and hence f(\u03b1'I-1) = f(I\u03b1'-1), which leads to f(\u03b1'I'-1) = f(I\u03b1'-1) + nfi. As a result, we have f(I'\u03b1'-1) = f(\u03b1'I-1) = f(I\u03b1'-1) + nfi since I'\u03b1'-1 = (\u03b1'I-1)-1. Therefore, the following claim can be obtained.Without loss of generality, we now suppose that Claim 2 \u03c81\u03c82 ... I' = I, \u03b1'I'-1 = \u03b1'I-1\u03c81\u03c82 ... and f(I'\u03b1'-1) = f(I\u03b1'-1) + nfi, where nfi = \u03c7(I) - 1.\u03b1' and I' now are the genomes with only one chromosome. Then based on the algorithm proposed by Lin et al. [I'\u03b1'-1 to transform \u03b1' into I'. Certainly, these nbi block-interchanges can be further expressed by a product of 2nbi 2-cycles, say \u03b1' into I', where I'\u03b1'-1 = Notice that both n et al. , we can Claim 3 I' = \u03b1'.\u03c81\u03c82...\u03b1 = I (hence \u03a6 = I\u03b1-1) can be easily verified by Claims 1, 2 and 3 as follows.Now we let \u03a6 = nfu + (n - f(I'\u03b1'-1)) + nfi) 2-cycles that can transform \u03b1 into I. More clearly, \u03a6 first uses \u03c61, \u03c62,...,nfu fusions) to transform \u03b1 into \u03b1', then uses nbi block-interchanges) to transform \u03b1' into I', and finally uses \u03c81 (acting as nfi fissions) to transform I' into I. By Claims 1 and 2, we can show that (nfu + (n - f(I'\u03b1'-1)) + nfi = n - f(I\u03b1-1) as follows.In other words, \u03a6 is a product of (nfu + (n - f(I'\u03b1'-1)) + nfinfu + (n - f(I\u03b1'-1) + nfi) + nfi \u00a0\u00a0\u00a0 (by Claim 2)= nfu + (n - f(I\u03b1-1) + nfu - nfi) + nfi \u00a0\u00a0\u00a0 (by Claim 1)= n - f(I\u03b1-1)= \u03c7(\u03b1) - 1 and \u03c7(I) - 1 are the lower bounds of the numbers of fusions and fissions, respectively, required in any optimal series of rearrangement events for transforming \u03b1 into I. Hence, the number of 2-cycles in \u03a6 that function as the fusions and fissions is minimum. Along with that \u03a6 = I\u03b1-1 can be expressed as a product of n - f(I\u03b1-1) 2-cycles, we thus conclude that \u03a6 is an optimal series of the events that transform \u03b1 into I with first nfu fusions, then nbi block-interchanges and finally nfi fissions, where nfu = \u03c7(\u03b1) -1, nfi = \u03c7(I) - 1.As mentioned before, Lemma 7 fu fusions come before all nbi block-interchanges that come before all nfi fissions, where There is an optimal series of the events needed to transform \u03b1 into I in a canonical order such that all nnfu = \u03c7(\u03b1) - 1, and nfi = \u03c7(I) - 1.\u03b1 =    and I =     for an example. It should be straightforward to see that \u03b1,I) is a connected bipartite graph with \u03c7(\u03b1) = 3 and \u03c7(I) = 4, and I\u03b1-1 =    and hence f(-2I\u03b1) = 5, since two 1-cycles  and (5)) are not explicitly shown. First, we are to find two 2-cycles \u03c61 and \u03c62 (since nfu = \u03c7(\u03b1) -1 = 2) from I\u03b1-1 to transform genome \u03b1 with three chromosomes into genome \u03b1' with exactly one chromosome. To this purpose, we let \u03c61 =  and \u03c62 = , since I\u03b1-1 =    . Then by Claim 1, \u03b1' = \u03c62\u03c61\u03b1 =  and I\u03b1'-1 = I\u03b1-1\u03c61\u03c62 =  . Next, we need to find three 2-cycles \u03c81, \u03c82 and \u03c83 (since nfi = \u03c7(I) - 1 = 3) from \u03b1'I-1, which is equal to (I\u03b1'-1)-1 =   =    , to transform I into I' with only one chromosome. By letting \u03c81 = , \u03c82 =  and \u03c83 = , we have I' = \u03c83\u03c82\u03c81 =  and \u03b1'I'-1 = \u03b1'I-1\u03c81\u03c82\u03c83 =  according to Claim 2. Finally, we will find two 2-cycles n - f(I\u03b1-1) - nfu - nfi = 12 - 5 - 2 - 3 = 2) from I'\u03b1'-1 that act as a block-interchange to transform \u03b1' into I', where I'\u03b1'-1 = (\u03b1'I'-1)-1 =  =  . By letting \u03b1' =    = , which indeed equals I'. Consequently, we find an optimal series of events \u03a6= \u03c81\u03c82\u03c83\u03c62\u03c61 that transform \u03b1 into I .Let us take d between two given circular multi-chromosomal genomes \u03b1 and I, and also to generate an optimal scenario of the required rearrangement events in a canonical order. In Algorithm Sorting-by-ffbi, the purpose of Step 2.3.3  is to find two numbers x and y that are both in some cycle of \u03b3 = Ji\u03b3 = \u03b2i. By Lemma 5, such x and y exist. In fact, they can be found using the following simple approach. For simplicity, let \u03b3k =  to compute the genome rearrangement distance Input: Two circular multi-chromosomal genomes \u03b1 and I.Output: d and a minimum series \u03a6 of events required to transform \u03b1 into I.1:\u00a0\u00a0\u00a0Find all connected components 1, 2,...,\u03c9 in graph \u03b1,I);ni the number of genes in i. */\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0/* Denote by 2:\u00a0\u00a0\u00a0for each i, 1 \u2264 i \u2264 \u03c9,do\u03b2i (resp. Ji) the collection of chromosomes in \u03b1 (resp. I) whose corresponding vertices are in i. */\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0/* Denote by 2.1:\u00a0\u00a0\u00a0Compute Ji\u03b3 = Ji2.2:\u00a0\u00a0\u00a0nfu = \u03c7(\u03b2i) - 1, nfi = \u03c7(Ji) -1, \u03b4i = nfu + nbi + nfi;2.3:\u00a0\u00a0\u00a0if \u03c7(\u03b2i) > 1 then\u00a0\u00a0\u00a0/* To compute \u03c61, \u03c62,...,2.3.1:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0for each cycle of \u03b2i do\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Create a set to contain all the numbers in this cycle;endfor\u00a0\u00a0\u00a02.3.2:\u00a0\u00a0\u00a0/* Let \u03b31\u03b32 ... \u03b3p be the cycle decomposition of the current \u03b3 and\u03b3q =  and union;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0endfor\u00a0\u00a0\u00a02.3.4:\u00a0\u00a0\u00a0\u03c61\u03b2i and \u03b3 = \u03b3\u03c61\u03c62...\u03b3 is Jiendif\u00a0\u00a0\u00a02.4:\u00a0\u00a0\u00a0if \u03c7(Ji) > 1 then/* To compute \u03c81, \u03c82,...,2.4.1:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u03b3 = \u03b3-1/* New \u03b3 becomes 2.4.2:\u00a0\u00a0\u00a0for each cycle of Ji do\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Create a set to contain all the numbers in this cycle;endfor\u00a0\u00a0\u00a0\u00a0\u00a0\u00a02.4.3:\u00a0\u00a0\u00a0/* Let \u03b31\u03b32...\u03b3p be the cycle decomposition of the current \u03b3 and\u03b3q =  and union;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0endfor\u00a0\u00a0\u00a02.4.5:\u00a0\u00a0\u00a0 = \u03c81Ji and \u03b3 = \u03b3\u03c81\u03c82...\u03b3 is endif\u00a0\u00a0\u00a02.5:\u00a0\u00a0\u00a0/* To compute 2.5.1:\u00a0\u00a0\u00a0\u03b3 = \u03b3-1;/* New \u03b3 becomes 2.5.2:\u00a0\u00a0\u00a02.5.3:\u00a0\u00a0\u00a0for j = 1 to nbi dox and y in \u03b3;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Arbitrarily choose two adjacent elements a1,a2...,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0/* Let a1, a2,...,a1 = x and assume y = ak;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Circularly shift ;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0for h = 1 to ni do\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0ah) = h;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0index  such that\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Find two adjacent elements u) \u2264 k - 1 and index(v) \u2265 k;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0index;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u03b3 = \u03b3\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0endfor\u00a0\u00a0\u00a03:\u00a0\u00a0\u00a0Let i \u2264 \u03c9;.4:\u00a0\u00a0\u00a0Output d = 1\u03a62...\u03a6\u03c9;Theorem 1 Given two circular multi-chromosomal genomes \u03b1 and I over the same gene set E = {1,2,..., n}, the problem of computing the genome rearrangement distance between \u03b1 and I using fusions, fissions and block-interchanges can be solved and an optimal series of such events in a canonical order can be obtained in n2) time.Proof. As discussed above, Algorithm Sorting-by-ffbi transforms \u03b1 into I using the minimum number of fusions, fissions and block-interchanges. Next, we follow to analyze its time-complexity. Notice that given an undirected graph with p vertices and q edges, all the connected components in this graph can be found in p + q) time using depth-first search or breadth-first search [n2) time for computing the connected components in the induced bipartite graph \u03b1, I), since in the worst case, the number of edges in \u03b1,I) is \u03c7(\u03b1) \u00d7 \u03c7(I) and \u03c7(\u03b1) = n) and \u03c7(I) = n). In Step 2, there are \u03c9 outer iterations, each computing the minimum series of events needed to transform \u03b2i into Ji, where 1 \u2264 i \u2264 \u03c9. Clearly, Steps 2.1 costs ni) time for computing Jinfu inner iterations in Step 2.3, each with the purpose of finding two numbers x and y that are both in the same cycle in \u03b3 and, however, in the different cycles in \u03b2i. In the worst case, Step 2.3 needs ni find-set operations and nfu union operations to finish its overall process. Note that Step 2.3.1 can be implemented by initially creating a set for each number in gene(\u03b2i) and then performing ni - \u03c7(\u03b2i) union operations to generate \u03c7(\u03b2i) sets with each corresponding to a cycle in \u03b2i, where \u03c7(\u03b2i) = nfu + 1. Hence, the total number of union operations is ni - 1 in Step 2.3. In fact, these find-set and union operations can be implemented in ni) time using the so-called \"static disjoint set union and find\" algorithm proposed by Gabow and Tarjan [ni) time. By the same principle, it can be verified that the time cost of Step 2.4 is ni). As for Step 2.5, adopted from our previous work [nbini) time, where  = ni). As a result, the time cost of Step 2 is \u03b6n), where \u03b6 is the maximum nbi among all iterations in Step 2 and \u03b6 <n. Clearly, Steps 3 and 4 cost constant time. Therefore, the total time-complexity of Algorithm Sorting-by-ffbi is n2).t search . As a red Tarjan . In otheous work , it takeVibrio genomes, we identified and constructed a table of orthologous genes that are putatively not involved in horizontal gene transfer (HGT) events by adopting the so-called symmetrical best hits (SymBets for short). In principle, two genes match and give SymBets if they are more similar to each other than they are to any other genes from the compared genomes [Vibrio genomes. First, the GenePlot [Vibrio genomes. Next, after removing all one-to-many or many-to-many SymBets, the three resulting tables of SymBet genes were joined to give a new one of one-to-one orthologous genes for all the three Vibrio genomes by using a rule as follows. If genes a (from genome A) and b (from genome B),b and c (from genome C), and c and a are all one-to-one SymBet pairs, then a, b and c are considered as one-to-one orthologous genes for the genomes A,B and C. In other words, the SymBet relationships among a, b and c result in a triangle. Finally, those genes that were involved in the putative HGT events detected and available in the Horizontal Gene Transfer Database [To analyze the rearrangement of three  genomes ,34. Dete genomes ,34. Part genomes ,34. For GenePlot  program Database  were theCLL and HTC contributed equally to this work. CLL conceived of the study, participated in the design and analysis of algorithm and drafted the manuscript. YLH participated in the algorithm design and software development, and carried out the bioinformatics experiments. TCW participated in the software development. HTC participated in the design and coordination of this study as well as in drafting the manuscript.1The Institute for Genome Research (TIGR) offers a website of Comprehensive Microbial Resource (CMR) that provides information on all of the publicly available and complete bacterial genomes."}
{"text": "We describe Support Vector Machine (SVM) applications to classification and clustering of channel current data. SVMs are variational-calculus based methods that are constrained to have structural risk minimization (SRM), i.e., they provide noise tolerant solutions for pattern recognition. The SVM approach encapsulates a significant amount of model-fitting information in the choice of its kernel. In work thus far, novel, information-theoretic, kernels have been successfully employed for notably better performance over standard kernels. Currently there are two approaches for implementing multiclass SVMs. One is called external multi-class that arranges several binary classifiers as a decision tree such that they perform a single-class decision making function, with each leaf corresponding to a unique class. The second approach, namely internal-multiclass, involves solving a single optimization problem corresponding to the entire data set (with multiple hyperplanes).Each SVM approach encapsulates a significant amount of model-fitting information in its choice of kernel. In work thus far, novel, information-theoretic, kernels were successfully employed for notably better performance over standard kernels. Two SVM approaches to multiclass discrimination are described: (1) internal multiclass (with a single optimization), and (2) external multiclass (using an optimized decision tree). We describe benefits of the internal-SVM approach, along with further refinements to the internal-multiclass SVM algorithms that offer significant improvement in training time without sacrificing accuracy. In situations where the data isn't clearly separable, making for poor discrimination, signal clustering is used to provide robust and useful information \u2013 to this end, novel, SVM-based clustering methods are also described. As with the classification, there are Internal and External SVM Clustering algorithms, both of which are briefly described. SVMs are fast, easily trained, discriminators ,2, for wSVMs use variational methods in their construction and encapsulate a significant amount of discriminatory information in their choice of kernel. In reference  informatThe SVM discriminators are trained by solving their KKT relations using the Sequential Minimal Optimization (SMO) procedure . A chunkBinary Support Vector Machines (SVMs) are based on a decision-hyperplane heuristic that incorporates structural risk management by attempting to impose a training-instance void, or \"margin,\" around the decision hyperplane .ik, where index i labels the M feature vectors (1 \u2264 i \u2264 M) and index k labels the N feature vector components (1 \u2264 i \u2264 N). For the binary SVM, labeling of training data is done using label variable yi = \u00b1 1 (with sign according to whether the training instance was from the positive or negative class). For hyperplane separability, elements of the training set must satisfy the following conditions: w\u03b2xi\u03b2-b \u2265 +1 for i such that yi = +1, and w\u03b2xi\u03b2-b \u2264 -1 for yi = -1, for some values of the coefficients w1, ..., wN, and b (using the convention of implied sum on repeated Greek indices). This can be written more concisely as: yi(w\u03b2xi\u03b2-b) - 1 \u2265 0. Data points that satisfy the equality in the above are known as \"support vectors\" (or \"active constraints\").Feature vectors are denoted by x\u03b2xi\u03b2 - b = 0. The boundary hyperplanes on the two classes of data are separated by a distance 2/w, known as the \"margin,\" where w2 = w\u03b2w\u03b2. By increasing the margin between the separated data as much as possible the optimal separating hyperplane is obtained. In the usual SVM formulation, the goal to maximize w-1 is restated as the goal to minimize w2. The Lagrangian variational formulation then selects an optimum defined at a saddle point of L = (w\u03b2w\u03b2)/2 - \u03b1\u03b3y\u03b3(w\u03b2x\u03b3\u03b2-b) - \u03b10, where \u03b10 = \u03a3\u03b3\u03b1\u03b3, \u03b1\u03b3 \u2265 0 (1 \u2264 \u03b3 \u2264 M). The saddle point is obtained by minimizing with respect to {w1, ...,wN,b} and maximizing with respect to {\u03b11, ..., \u03b1M}. If yi(w\u03b2xi\u03b2-b) - 1 \u2265 0, then maximization on \u03b1i is achieved for \u03b1i = 0. If yi(w\u03b2xi\u03b2-b) - 1 = 0, then there is no constraint on \u03b1i. If yi(w\u03b2xi\u03b2-b) - 1 < 0, there is a constraint violation, and \u03b1i \u2192 \u221e. If absolute separability is possible the last case will eventually be eliminated for all \u03b1i, otherwise it's natural to limit the size of \u03b1i by some constant upper bound, i.e., max(\u03b1i) = C, for all i. This is equivalent to another set of inequality constraints with \u03b1i \u2264 C. Introducing sets of Lagrange multipliers, \u03be\u03b3 and \u03bc\u03b3(1 \u2264 \u03b3 \u2264 M), to achieve this, the Lagrangian becomes:Once training is complete, discrimination is based solely on position relative to the discriminating hyperplane: w\u03b2w\u03b2)/2 - \u03b1\u03b3[y\u03b3(w\u03b2x\u03b3\u03b2-b)+\u03be\u03b3] + \u03b10 + \u03be0C - \u03bc\u03b3\u03be\u03b3, where \u03be0 = \u03a3\u03b3\u03be\u03b3, \u03b10 = \u03a3\u03b3\u03b1\u03b3, and \u03b1\u03b3 \u2265 0 and \u03be\u03be \u2265 0 (1 \u2264 \u03b3 \u2264 M).L =  = \u03b10 -  and \u03b1\u03b3y\u03b3 = 0, where only the variations that maximize in terms of the \u03b1\u03b3 remain (known as the Wolfe Transformation). In this form the computational task can be greatly simplified. By introducing an expression for the discriminating hyperplane: fi = w\u03b2xi\u03b2 - b = \u03b1\u03b3y\u03b3x\u03b3\u03b2xi\u03b2 - b, the variational solution for L(\u03b1) reduces to the following set of relations : (i) \u03b1i = 0 \u21d4 yifi \u2265 1, (ii) 0 < \u03b1i < C \u21d4 yifi = 1, and (iii) \u03b1i = C \u21d4 yifi \u2264 1. When the KKT relations are satisfied for all of the \u03b1\u03b3 (with \u03b1\u03b3y\u03b3 = 0 maintained) the solution is achieved.  and the \u03b1's associated with the negatives to 1/N(-), where N(+) is the number of positives and N(-) is the number of negatives.) Once the Wolfe transformation is performed it is apparent that the training data  above) enter into the Lagrangian solely via the inner product xi\u03b2xj\u03b2. Likewise, the discriminator fi, and KKT relations, are also dependent on the data solely via the xi\u03b2xj\u03b2 inner product.At the variational minimum on the {wi\u03b2xj\u03b2 are possible and are usually formulated in terms of the family of symmetric positive definite functions (reproducing kernels) satisfying Mercer's conditions  Substituting the above equations into the Lagrangian and after simplification reduces into the dual formalism:Where all \u03b1i,j\u2211m(\u03b4im - \u03b1im)(\u03b4jm - \u03b1jm)(Kij + \u03b2) - \u03b2\u2211i,m\u03b4im\u03b1imMaximize: -1/2\u2211im, \u2211m\u03b1im = 1, i = 1...1; m = 1...kConstraint: 0 \u2264 \u03b1ij = xi.xj is the Kernel generalization. In vector notation:Where Ki,j(\u0394yi - Ai)(\u0394yj - Aj)(Kij + \u03b2) - \u03b2\u2211i\u0394yiAiMaximize: -1/2\u2211i, Ai. 1 = 1, i = 1 ...1Constraint: 0 \u2264 Ai = \u0394yi - Ai. Hence after ignoring the constant: -1/2\u2211i,j\u03c4i.\u03c4j(Kij + \u03b2) + \u03b2\u2211i\u0394;yi\u03c4i, subject to: \u03c4i \u2264 \u0394yi, \u03c4i.1 = 0, i = 1 ...l. The dual is solved  using the decomposition method.Let \u03c4i,j\u03c4im.\u03c4jm(Kij + \u03b2) - \u03b2\u2211i,m\u03b4im\u03c4imMinimize: 1/2\u2211i \u2264 \u0394yi, \u03c4i.1 = 0, i = 1 ...lConstraint: \u03c4The Lagrangian of the dual is:i,j,m\u03c4im.\u03c4jm(Kij + \u03b2) - \u03c6\u2211i,m\u03b4im\u03c4im - \u2211i,muim(\u03b4im - \u03c4im) - \u2211ivi\u2211m\u03c4imL = 1/2\u2211im \u2265 0Subject to uim:We take the gradient of the Lagrangian with respect to \u03c4\u03c4m[L] = \u2211i\u03c4jm(Kij + \u03b2) - \u03b2\u03b4im + uim - vi = 0\u25bci\u03c4jm(Kij + \u03b2) - \u03b2\u03b4im + uim - vi = 0 and fim = \u2211i\u03c4jm(Kij + \u03b2) - \u03b2\u03b4im, then f(\u03c4) = fim + uim - vi = 0. By KKT conditions we get two more equations:Introducing f(\u03c4) = \u2211im(\u03b4im - \u03c4im) = 0 and uim \u2265 0uim = \u03c4im, then uim \u2265 0, hence fim \u2264 vi. Case II: if \u03c4im < \u03b4im, then uim = 0, hence fim = vi. Note: There is atleast one 'm' for all i such that \u03c4im < \u03b4im is satisfied.Case I: if \u03b4Therefore combining Case I & II, we get:m{fim} \u2264 vi \u2264 minm: \u03c4im < \u03b4im{fim}maxm{fim} \u2264 minm: \u03c4im < \u03b4im{fim}Or maxm{fim} - minm: \u03c4im < \u03b4im{fim} \u2264 \u03b5Or maxim < \u03b4im implies that \u03b1im > 0. Since \u2211m\u03b1im = 1, for any i each \u03b1im is treated as the probability that the data point belongs to class m. Hence we define KKT violators as:Note: \u03c4m{fim} - minm: \u03c4im < \u03b4im{fim} > \u03b5 for all i.maxUsing the method in  to solvei,j\u03c4i.\u03c4j(Kij + \u03b2) + \u03b2\u2211i\u0394yi\u03c4iQ(\u03c4) = -1/2\u2211i \u2264 \u0394yi, \u03c4i.1 = 0, i = 1 ...lSubject to: \u03c4Expanding in terms of a single '\u03c4' vector:p(\u03c4p) = -1/2Ap(\u03c4p. \u03c4p) - Bp.\u03c4p + CpQWhere:p = Kpp + \u03b2Ap = -\u03b2\u0394yp + \u2211i\u2260p\u03c4i(Kip + \u03b2)Bp = -1/2\u2211i,j\u2260p\u03c4i.\u03c4j(Kij + \u03b2) + \u03b2\u2211i\u2260p\u03c4i\u0394yiCp', we have to minimize:Therefore ignoring the constant term 'Cp(\u03c4p) = 1/2Ap(\u03c4p. \u03c4p) + Bp.\u03c4pQp \u2264 \u0394yp and \u03c4p.1 = 0Subject to: \u03c4The above equation can also be written as:p(\u03c4p) = 1/2Ap(\u03c4p + Bp/Ap).(\u03c4p + Bp/Ap) - Bp.Bp/2ApQp + Bp/Ap) & D = (\u0394yp + Bp/Ap) in the above equation. Hence, after ignoring the constant term Bp.Bp/2Ap and the multiplicative factor 'Ap' we have to minimize:Substitute v = (\u03c42Q(v) = 1/2v.v = 1/2||v||Subject to: v \u2264 D and v.1 = D.1 - 1The Lagrangian is given by:2 - \u2211m\u03c1m(Dm - vm) - \u03c3[\u2211m(vm - Dm) + 1]L(v) = 1/2||v||m \u2264 0Subject to: \u03c1m = vm + \u03c1m - \u03c3 = 0. By KKT conditions we have: \u03c1m(Dm - vm) = 0 & \u03c1m \u2265 0, also vm \u2264 Dm. Hence by combining the above in-equalities, we have: vm = Min{Dm, \u03c3}, or \u2211mvm = \u2211mMin{Dm, \u03c3} = \u2211mDm - 1. The above equation uniquely defines the '\u03c3' that satisfies the above equation AND that '\u03c3' is the optimal solution of the quadratic optimization problem.  that satisfies the equation: |(\u03c3l - \u03c3l+1)/\u03c3l| \u2264 tolerance. The initial value for '\u03c3' is set to \u03c31 = 1/K[\u2211mDm - 1].Update rule for '\u03c4': Once we have '\u03c3', \u03c4newm = vm - Bpm/(Kpp + \u03b2), or:newm = vm - fpm/(Kpp + \u03b2) + \u03c4oldm\u03c4i} be a data set of 'N' points in Rd. Using a non-linear transformation \u03c6, we transform 'x' to some high-dimensional space called Kernel space and look for the smallest enclosing sphere of radius 'R'. Hence we have: ||\u03c6(xj) - a ||2 \u2264 R2 for all j = 1,...,N; where 'a' is the center of the sphere. Soft constraints are incorporated by adding slack variables '\u03b6j':Let {xj) - a ||2 \u2264 R2 + \u03b6j for all j = 1,...,N||\u03c6(xj \u2265 0Subject to: \u03b6We introduce the Lagrangian as:2 - \u2211j\u03b2j(R2 + \u03b6j - ||\u03c6(xj) - a ||2) - \u2211j\u03b6j\u03bcj + C\u2211j\u03b6jL = Rj \u2265 0, \u03bcj \u2265 0,Subject to: \u03b2j\u03b6j is a penalty term. Setting to zero the derivative of 'L' w.r.t. R, a and \u03b6 we have: \u2211j\u03b2j = 1; a = \u2211j\u03b2j\u03c6(xj); and \u03b2j = C - \u03bcj.where C is the cost for outliers and hence C\u2211Substituting the above equations into the Lagrangian, we have the dual formalism as:i,j\u03b2i\u03b2jKij where 0 \u2264 \u03b2i \u2264 C; Kij = exp(-||xi - xj||2/2\u03c32)W = 1 - \u2211i\u03b2i = 1Subject to: \u2211j\u03bcj = 0 and \u03b2j(R2 + \u03b6j - ||\u03c6(xj) - a ||2) = 0.By KKT conditions we have: \u03b6j' if \u03b6j > 0, then \u03b2j = C and hence it lies outside of the sphere i.e. R2 < ||\u03c6(xj) - a ||2. This point becomes a bounded support vector or BSV. Similarly if \u03b6j = 0, and 0 < \u03b2j < C, then it lies on the surface of the sphere i.e. R2 = ||\u03c6(xj) - a ||2. This point becomes a support vector or SV. If \u03b6j = 0, and \u03b2j = 0, then R2 > ||\u03c6(xj) - a ||2 and hence this point is enclosed with-in the sphere.In the kernel space of a data point 'xStaphylococcus aureus, which has a molecule-sized channel opening for partial capture, if not translocation, of biomolecules drawn in by electrophoretic forces (such as DNA)  for all mWhere fCase I:im = 0 for m S.T fm = fmmaxIf \u03b1iyi > 0 and hence \u03b6i = 0Implies \u03b1yi - fmmax - 1 \u2265 0Hence fCase II:im > 0 for m S.T fm = fmmax and \u03b1iyi > \u03b1imIf 1 > \u03b1i = 0Implies \u03b6yi - fmmax - 1 = 0Hence fCase III:im > 0 for m S.T fm = fmmax and \u03b1iyi \u2264 \u03b1imIf 1 \u2265 \u03b1i > 0Implies \u03b6yi - fmmax - 1 + \u03b6i = 0Hence fyi - fmmax - 1 < 0Or fThe SVM Decision Tree shown in Fig. simultaneous curves . The problem is that tuning and optimizing a single decision tree is already a large task, even for five species } = fyi, and Let fm = maxm{fm(xi)} for all m \u2260 yi, then we define the margin as: (fyi - fm), hence data point xi is dropped if (fyi - fm) = Confidence Parameter. *C was required to achieve 100% accuracy.) The results are shown in Table all species).Suppose we define the criteria for dropping weak data as the margin: For any data point xst 0.0000*C was reThe SVM-Internal approach to clustering was originally defined by . Data poThe minimal enclosing sphere, when mapped back into the data space, can separate into several components; each enclosing a separate cluster of points. The width of the kernel (say Gaussian) controls the scale at which the data is probed while the soft margin constant helps to handle outliers and over-lapping clusters. The structure of a dataset is explored by varying these two parameters, maintaining a minimal number of support vectors to assure smooth cluster boundaries.We have used the algorithm defined in  to identIn each comparison we sub-divide the line segment connecting the two data points into 20 parts; hence we obtain 19 different points on this line segment. The two data points belong to the same cluster only if all the 19 points lie inside the cluster. Given the cost of evaluating utmost 19 points for every comparison, the need to eliminate comparisons that do not have an impact on the cluster connectivity becomes even more important. Finally we have used Depth First Search (DFS) algorithm for the cluster harvest. Results are shown in Tables a priori knowledge of each vector's class. The algorithm works by first running a Binary SVM against a data set, with each vector in the set randomly labeled, until the SVM converges , to a multiclass  SVM, an FSA-based nanopore spike detector, and an HMM-based channel current feature extraction. New, web-accessible, channel current analysis tools, have also been developed for kinetic feature extraction (via channel current sub-level lifetimes), and clustering. The website is designed using HTML and CGI scripts that are executed to process the data sent when a form filled in by the user is received at the web server \u2013 results are then e-mailed to the address indicated by the user. The interface to this and all other software described is available via the group Home Page: see Fig. . The SVMAdaptive feature extraction and discrimination, in the context of SVMs, can be accomplished by small batch reprocessing using the learned support vectors together with the new information to be learned. The benefit is that the easily deployed properties of SVMs can be retained while at the same time co-opting some of the on-line adaptive characteristics familiar from on-line learning with neural nets. This is also compatible with the chunking processing that is already implemented. A situation where such adaptation might prove necessary in nanopore signal analysis is if the instrumentation was found to have measurable, but steady, drift (at a new level of sensitivity for example). At the forefront of online adaptation, where the discrimination and feature extraction optimizations are inextricably mixed, further progress may derive benefit from the Information-Geometrical methods of S. Amari -25.In a parallel datarun to that indicated in Fig. If SVM performance on the full HMM parameter set  offers equivalent performance after rejecting weak data, then the possibility for significant improvement with selection on good parameters. An AdaBoost method is being used to select HMM parameters by representing each feature vector component as an independent Na\u00efve Bayes classifier (trained on the data given), that then comprise the pool of experts in the AdaBoost algorithm -34. The \u2022 External Multi-class SVM gave best results with Sentropic Kernel while Internal Multi-class SVM gave best results with AbsDiff kernel.\u2022 Internal Multi-class approach overcomes the need to search for the best performing tree out of many possibilities. This is a huge advantage especially when the number of classes is large.\u2022 Using a margin to define the drop zone for the internal multi-class approach produced far better results i.e. fewer data were dropped to achieve 100% accuracy.\u2022 Additional benefit of using the margin is that the drop zone tuning to achieve 100% accuracy becomes trivial.Clustering Methods were also examined. The results show that our SVM-based clustering implementations can separate data into proper clusters without any prior knowledge of the elements' classification. this can be a powerful resource for insight into data linkages (topology).\u2022 External and Internal SVM The Nanopore Detector is operated such that a stream of 100 ms samplings are obtained (throughput was approximately one sampling per 300 ms in ).2 transition probabilities, and an a posteriori information from the Viterbi path solution which is, essentially, a de-noised histogram of the bloackade sub-level occupation probabilities . This ii,j\u03b2i\u03b2jKij where 0 \u2264 \u03b2i \u2264 C; Kij = exp(-||xi - xj||2/2\u03c32), also \u2211i\u03b2i = 1. For any data point 'xk', the distance of its image in kernel space from the center of the sphere is given by: R2(xk) = 1 - 2\u2211i\u03b2iKik + \u2211i,j\u03b2i\u03b2jKij. The radius of the sphere is R = {R(xk) | xk is a Support Vectors}, hence data points which are Support Vectors lie on cluster boundaries. Outliers are points that lie outside of the sphere and therefore they do not belong to any cluster i.e. they are Bounded Support Vectors. All other points are enclosed by the sphere and therefore they lie inside their respective cluster. KKT Violators are given as: (i) If 0 < \u03b2i < C and R(xi) \u2260 R; (ii) If \u03b2i = 0 and R(xi) > R; and (iii) If \u03b2i = C and R(xi) < R.The dual formalism is: 1 - \u2211\u03b2 {\u2211i,j\u03b2i\u03b2jKij - 1}. In the SMO decomposition, in each iteration we select \u03b2i & \u03b2j and change them such that f(\u03b2) reduces. All other \u03b2's are kept constant for that iteration. Let us denote \u03b21 & \u03b22 as being modified in the current iteration. Also \u03b21 + \u03b22 = (1 - \u2211i = 3\u03b2i) = s, a constant. Let \u2211i = 3\u03b2iKik = Ck, then we obtain the SMO form: f = \u03b221 + \u03b222 + \u2211i,j = 3\u03b2i\u03b2jKij + 2\u03b21\u03b22K12 + 2\u03b21C1 + 2\u03b22C2. Eliminating \u03b21: f(\u03b22) = (s - \u03b22)2 + \u03b222 + \u2211i,j = 3\u03b2i\u03b2jKij + 2(s - \u03b22)\u03b22K12 + 2(s - \u03b22)C1 + 2\u03b22C2. To minimize f(\u03b22), we take the first derivative w.r.t. \u03b22 and equate it to zero, thus f'(\u03b22) = 0 = 2\u03b22(1 - K12) - s(1 - K12) - (C1 - C2), and we get the update rule: \u03b22new = [(C1 - C2)/2(1 - K12)] + s/2. We also have an expression for \"C1 - C2\" from: R(x12) - R(x22) = 2(\u03b22 - \u03b21)(1 - K12) - 2(C1 - C2), thus C1 - C2 = [R(x22) - R(x12)]/2 + (\u03b22 - \u03b21)(1 - K12), substituting, we have:The Wolfe dual is: f(\u03b2) = Min 1new = \u03b21old - [R(x22) - R(x12)]/[4(1 - K12)]\u03b2Compute 'C': if percent outliers = n and number data points = N, then: C = 100/(N*n)Initialize \u03b2: Initialize m = int(1/C) - 1 number of randomly chosen indices to 'C'i\u03b2i = 1Initialize two different randomly chosen indices to values less than 'C' such that \u22112(xi) for all 'i' based on the current value of \u03b2.Compute Ri < C; Set II if \u03b2i = 0; and Set III if \u03b2i = C.Divide data into three sets: Set I if 0 < \u03b22_low = Max{ R2(xi) | 0 \u2264 \u03b2i < C} and R2_up = Min{ R2(xi) | 0 < \u03b2i \u2264 C}.Compute RIn every iteration execute the following two paths alternatively until there are no KKT violators:1. Loop through all examples Keep count of number of KKT Violators.2_low - R2_up < 2*tol.2. Loop through examples belonging only to Set I  until RExamine Example Subroutinea. Check for KKT Violation. An example is a KKT violator if:2(xi) > R2_up; choose R2_up for joint optimizationSet II and R2(xi) < R2_low; choose R2_low for joint optimizationSet III and R2(xi) > R2_up + 2*tol OR R2(xi) < R2_low - 2*tol; choose R2_low or R2_up for joint optimization depending on which gives a worse KKT violatorSet I and Rb. Call the Joint Optimization subroutineJoint Optimization Subroutine12) where K12 is the kernel evaluation of the pair chosen in Examine Examplea. Compute \u03b7 = 4(1 - K2(x2) - R2(x1)]/\u03b7b. Compute D = [R2), \u03b21} = L1c. Compute Min{(C - \u03b21), \u03b22} = L2d. Compute Min{ for all 'i' based on the changes in \u03b21 & \u03b22h. Re-compute R2_low & R2_up based on elements in Set I, R2(x1) & R2(x2)i. Re-compute RThe SVM-clustering software is written in Perl. It runs data on a separate Binary SVM also written in Perl. This SVM uses a C file for kernel calculations. The data run on the SVM is created by running raw data through a tFSA/HMM(written in C), which creates a data set that contains 151 feature vectors for each element. The following is a simple step-by-step description of the basic algorithm used for SVM-clustering on this data:1. Start with a set of data vectors .4. After initial convergence is obtained for the randomly labeled data set, relabel the misclassified data vectors, which have confidence factor values greater than some threshold.5. Rerun the SVM on the newly relabeled data set.6. Continue relabeling and rerunning SVM until no vectors in the data set are misclassified (or there is no improvement).The paper was written by SWH and AY. The external clustering work was contributed by CM. The channel current feature vector extraction used to create the data sets was performed by ML."}
{"text": "In particular, we have chosen log y1 = log y0 + (\u03b2L \u2212 \u03b2H)log d*, holding the continuity of y(d).\u201d"}
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{"text": "Moreover, we describe two pathways leading to the precipitation of the \u03b1-phase mediated by diffusion-based orthorhombic structures, \u03b1\u2033lean and \u03b1\u2033iso. Via coupling the lattice parameters to composition both phases evolve into \u03b1 through rejection of Nb. These findings have the potential to promote new microstructural design approaches for Ti\u2013Nb alloys and \u03b2-stabilized Ti-alloys in general.Ti-alloys represent the principal structural materials in both aerospace development and metallic biomaterials. Key to optimizing their mechanical and functional behaviour is in-depth know-how of their phases and the complex interplay of diffusive vs. displacive phase transformations to permit the tailoring of intricate microstructures across a wide spectrum of configurations. Here, we report on structural changes and phase transformations of Ti\u2013Nb alloys during heating by in situ synchrotron diffraction. These materials exhibit anisotropic thermal expansion yielding some of the largest linear expansion coefficients (+\u2009163.9\u00d710 Complex phase transformations in \u03b2-stabilised titanium alloys can dramatically change their \u03b1 and \u03b2 microstructures, providing tailorability for aerospace or biomaterial applications. Here the authors show that Ti-Nb alloys exhibit giant thermal expansions and identify two new pathways that lead to \u03b1 phase formation. Certain alloy compositions display low Young\u2019s moduli (E) below 80\u2009GPa after rapid cooling5, providing suitable starting points for the development of novel low-modulus alloys for load-bearing implant applications. Due to their high strength to density ratio \u03b2-stabilized Ti-alloys have also become increasingly used for various aerospace applications such as airframes and landing gears3.Ti-alloys are the workhorses in modern aerospace design and engineering of metallic biomaterials. In particular the class of \u03b2-stabilized Ti-alloys represents highly promising multifunctional materials with desirable structural and functional properties for various biomedical and engineering applications7. Up to the present day, these features have been stirring ever increasing interest.Due to the thermoelastic martensitic transformation of the body centred cubic (bcc) \u03b2-phase to orthorhombic martensite \u03b1\u2033 this alloy family also demonstrates shape memory (SM) behaviour and superelasticity (SE)iso precipitates10. The transformation and precipitation pathways occurring during aging decide the morphology, size and arrangement of the precipitating products12.The mechanical and functional properties of these alloys are optimized by controlled adjustment of the microstructural parameters via complex thermomechanical processing paths. For instance, the mechanical behaviour of \u03b2 and near-\u03b2 Ti alloys can be significantly improved by uniform dispersions of fine \u03b1 and/or \u03c9iso was described13. Recently, a unique nanolaminate structure consisting of \u03b1\u2033 martensite and planar complexions of a thermal \u03c9ath was reported14. Furthermore, for two commercial \u03b2-stabilized Ti\u2013Mo\u2013Fe\u2013Al and Ti\u2013Al\u2013Mo\u2013V\u2013Fe\u2013Cr alloys, Ivasishin et al. observed the formation of an intermediate orthorhombic phase exhibiting the same crystal structure as \u03b1\u2033 martensite during early stages of \u03b1-precipitation15. More recently, employing in situ diffraction methods this phase was also detected upon heat treatment of Ti\u2013Al\u2013Mo\u2013Cr\u2013Sn\u2013Zr, Ti\u2013Al\u2013Mo\u2013V\u2013Cr\u2013(Zr) and Ti\u2013V\u2013Fe\u2013Al alloys used for advanced structural aircraft components18. For some alloys a \u03b2-stabilizer lean structure, denoted \u03b1\u2033lean, was observed during the decomposition of \u03b1\u2033 martensite19. It was reported that decomposition of \u03b1\u2033 martensite into \u03b1 and \u03b2 phases by aging below the austenite start temperature As involves a spinodal mechanism21 implying continuous changes in chemical composition.The great diversity and complex interplay of diffusion-driven vs. displacive phase transformations in \u03b2-stabilized Ti-alloys stimulates their further exploration. For instance, several years ago a novel coupled diffusional-displacive transformation mechanism occurring during formation of isothermal \u03c9Ms below RT. Furthermore, \u03b2 needs to be stable enough against \u03c9ath formation and additional \u03b1 precipitation. Otherwise precipitation of \u03b1 and/or \u03c9iso as well as martensite formation takes place upon cooling after aging, thus modifying the constitution of phases at the temperature of interest22. These difficulties can be overcome by the use of in situ diffraction methods permitting direct observation of the microstructural evolution caused by heat treatments. In situ diffraction methods, employing, e.g. synchrotron radiation, are therefore the first choice to capture phase reactions and transitions taking place over narrow temperature and time intervals directly at the critical temperature23.Detailed knowledge about transformation and precipitation processes is the foundation for the advancement of tailored thermomechanical treatment routes. However, following these processes through diffraction patterns recorded ex situ at room temperature (RT) is challenging. First, due to the variations in cell parameters caused by thermal expansion and secondly, because of transformations potentially taking place during cooling from the aging temperature to RT. Only when the phases resulting from aging neither decompose nor transform during cooling the microstructure formed at the aging temperature will be observable at RT. For the \u03b2-phase this only applies, when it contains sufficient \u03b2-stabilizers to suppress the martensite start temperature 7 this system serves as a prototype to study SM and SE in Ni-free Ti-based alloys. Its importance is underlined by the fact that most of the recently developed low-modulus as well as SM and SE \u03b2-stabilized Ti-based alloys are modifications of the Ti\u2013Nb system23. Hence, a better understanding of the binary base alloy system will help explain, at least to some extent, the alloys derived from it.Since the discovery of SM in Ti\u2013NbcNb alloys  adapted from previous studies24. At the heating rate employed (10\u2009\u00b0C\u2009min\u22121), reversion of \u03b1\u2033 martensite followed by substantial \u03c9iso precipitation occurs for c\u2009\u2265\u200928.5  to track these transformations in situ for the same alloy formulations and heating rate as in Fig.\u00a0c\u2009=\u200916, 21 and 28.5 and of \u03b1\u2033 with some austenitic \u03b2-phase for c\u2009=\u200936, in agreement with the literature26. Figure\u00a023.The initial microstructure consisted of orthorhombic martensite \u03b1\u2033 for iso. In addition, Nb-depleted \u03b1\u2033, denoted \u03b1\u2033lean, and a thermally formed phase exhibiting the same crystal structure as \u03b1\u2033, denoted \u03b1\u2033iso, were observed. Using the diffractograms . The aspect ratios influence the shape of the orthorhombic \u03b1\u2033 unit cell and determine whether it resembles more closely \u03b1\u2032 or \u03b2. If b\u03b1\u2033/a\u03b1\u2033\u2009=\u2009\u03b1\u2033 are arranged in a hexagonal pattern identical to the atoms on (0001)\u03b1\u2032 for hexagonal close-packed (hcp) martensite \u03b1\u2032. On the other hand, for b\u03b1\u2033/a\u03b1\u2033\u2009=\u2009c\u03b1\u2033/a\u03b1\u2033\u2009=\u200926. Both limiting values are indicated in Fig.\u00a0\u03b1\u2033 through the fractional coordinate y in Wyckoff position 4c of space group Cmcm25.Figure\u00a0ams Fig.\u00a0 the lattb\u03b1\u2033/a\u03b1\u2033 and c\u03b1\u2033/a\u03b1\u2033 at 50\u2009\u00b0C in Fig.\u00a0c\u2009=\u200916 to c\u2009=\u200936, b\u03b1\u2033/a\u03b1\u2033 and c\u03b1\u2033/a\u03b1\u2033, respectively, drop from 1.686 to 1.514 and from 1.567 to 1.460, thus coming closer to the limit of 1.414 corresponding to \u03b2. Thus, Nb-lean \u03b1\u2033 is structurally closer to hcp \u03b1\u2032 whereas Nb-rich \u03b1\u2033 is more similar to bcc \u03b2.The lattice parameters of \u03b1\u2033 martensite are strongly affected by the Nb content. Likewise the shape of the orthorhombic unit cell of \u03b1\u2033 varies with Nb content, as seen from c\u2009=\u200916 to 210\u2009\u00b0C for c\u2009=\u200936  \u03b1L along b\u03b1\u2033 is negative, corresponding to a contraction of the b\u03b1\u2033 spacing upon heating. The magnitude of \u03b1L along a\u03b1\u2033 and along b\u03b1\u2033 grows with increasing Nb content. \u03b1L along c\u03b1\u2033 decreases up to c\u2009=\u200928.5, but is largest for c\u2009=\u200936.For \u03b1\u2033 martensite in the present alloys a\u03b1\u2033 and c\u03b1\u2033 spacings expand at a rate of 163.9\u00d710\u22126\u2009\u00b0C\u22121 and 24.4\u00d710\u22126\u2009\u00b0C\u22121, respectively, the b\u03b1\u2033 spacing contracts by \u221295.1\u00d710\u22126\u2009\u00b0C\u22121 between 50\u2009\u00b0C and 210\u2009\u00b0C. These rates are comparable to or even larger in magnitude than those of materials exhibiting colossal thermal expansion (defined as |\u03b1L|\u2009\u2265\u2009100\u00d710\u22126\u2009\u00b0C\u22121) such as Ag3[Co(CN)6]33. The present values for the expansion rates of the b\u03b1\u2033 and c\u03b1\u2033 spacings fully agree with recently published data for \u03b1\u2033 martensite in Ti\u201335.4Nb30 NiGa alloys31 exhibit similarly strong anisotropic thermal expansion. Still, in most cases their expansion rates are smaller in magnitude than for Ti\u201336Nb and Ti\u201328.5Nb 6]29, the fulleride Sm2.75C6028 and the extremely loosely bound solid Xe below 75\u2009K27  are among the largest (both positive and negative) ever reported for solid crystalline metallic systems, be they isotropic or anisotropic. Typical values for 36. In the present case of \u03b1\u2033 martensite in Ti\u2013Nb, the expansion and contraction along the unit cell edges partially compensate each other. Yet, the positive \u03b1L along a\u03b1\u2033 and c\u03b1\u2033 dominate over the negative \u03b1L along b\u03b1\u2033 leading to an expansion rate \u03b1V of the bulk between 24.7\u00d710\u22126\u2009\u00b0C\u22121 and 91.0\u00d710\u22126\u2009\u00b0C\u22121. The bulk expansion of \u03b2-phase in Ti\u201336Nb turns out slightly smaller (84.3\u00d710\u22126\u2009\u00b0C\u22121) than that of \u03b1\u2033 (91.0\u00d710\u22126\u2009\u00b0C\u22121).The variation of the unit cell volume due to anisotropic thermal expansion may be positive or negativec\u2009=\u2009 decomposition of \u03b1\u2033 into \u03b1 and \u03b2 phases takes place, which is accompanied by an exothermic event  with temperature is inverted relative to the thermal expansion behaviour discussed previously. While upon heating below the temperature interval of decomposition \u03b1\u2033 martensite becomes more similar to austenite \u03b2, it becomes more \u03b1-like during decomposition. This is reflected by the behaviour of b\u03b1\u2033/a\u03b1\u2033 and c\u03b1\u2033/a\u03b1\u2033 which first decrease towards \u03b1\u2033 martensite is preserved in Ti\u201316Nb up to 600\u2009\u00b0C and in Ti\u201321Nb up to 510\u2009\u00b0C. No precipitation products were detected below these temperatures suggesting that \u03b1\u2033 maintains its initial composition present at RT. \u03b1\u2033 fully decomposes into \u03b1 and \u03b2 by further heating of Ti\u201316Nb to 630\u2009\u00b0C and of Ti\u201321Nb to 590\u2009\u00b0C. During decomposition, the lattice parameters of \u03b1\u2033lean with those of \u03b1\u2033 martensite at RT suggests that \u03b1\u2033lean progressively rejects Nb thereby approaching equilibrium \u03b1. In this process a\u03b1, c\u03b1  and no contributions of \u03b1\u2033 are required to fully model the observed intensities. The initial evolution of the {101\u03050}\u03b1 reflection below 560\u2009\u00b0C \u03b1\u2033 of a second orthorhombic component \u03b1\u2033lean,II. Since for both phases the modeled profiles reached comparable agreement with the experimental data it is problematic to unambiguously assign one of these phases.A close look at the diffraction patterns for Ti\u201321Nb in Fig.\u00a0Q\u2009\u22121 below 560\u2009\u00b0C .5% Fig.\u00a0. They arc\u2009=\u200916 and 21, \u03b1\u2033 reverts martensitically to austenite \u03b20 causing an endothermic event for c\u2009=\u200928.5 and 36  under isothermal conditions37.The refined lattice parameters and unit cell aspect ratios of \u03b1\u2033iso with temperature is inverted relative to the structural change of \u03b1\u2033 martensite due to thermal expansion. This strongly suggest that compositional changes take place in \u03b1\u2033iso. For instance, for a Ti\u2013Mo-based alloy (TIMETAL-LCB) analytical scanning transmission electron microscopy clearly evidenced that the \u03b2-stabilizer content of \u03b1\u2033iso precipitates was progressively reduced with aging time10.The variation of the unit cell geometry of \u03b1\u2033iso unit cell towards a more hcp-like structure  are of particular interest, since they permit designing composites with zero net thermal expansion when combined with positive thermal expansion materials.Aside from reaching giant values in case of Nb-rich alloy formulations, the thermal expansion rates of the present alloys can be adjusted across a wide range by simply modifying their composition. This represents an advantage over many ceramic materials which often show rather limited controllability of their thermal expansion30. For each of the present alloys this process takes place across the entire temperature range in which \u03b1\u2033 martensite is present. Irrespective of composition, b\u03b1\u2033/a\u03b1\u2033 and c\u03b1\u2033/a\u03b1\u2033 continuously decrease with temperature towards 1.414 \u03b1\u2033 take positions which further increase the similarity of \u03b1\u2033 to \u03b2. This would manifest itself in an increase of the fractional coordinate y of Wyckoff position 4c of space group Cmcm when \u03b1\u2033 is heated towards Af25.The polycrystalline nature of the irradiated volume \u2009\u2192\u2009\u03b1\u2009+\u2009\u03b2\u03b1\u2033\u2009\u2192\u2009\u03b1\u2033lean\u2009+\u2009\u03b1\u2033lean,II (+\u03b1\u2033rich\u2009+\u2009\u03b2)\u2009\u2192\u2009\u03b1\u2009+\u2009\u03b2\u03b1\u2033\u2009\u2192\u2009\u03b1\u2033The in situ data clearly evidences that during martensite decomposition \u03b1\u2033lean as well as Nb-enriched \u03b1\u2033rich form. Nb-enriched \u03b1\u2033rich and/or \u03b2 is expected to take up the Nb expelled from \u03b1\u2033lean. \u03b1\u2033rich eventually evolves into \u03b2. Although \u03b1\u2033rich and \u03b2 were not detected for Ti\u201321Nb, the presence of at least one of these phases is necessary for mass conservation during Nb-expulsion from \u03b1\u2033lean. In addition, in the first pathway \u03b1 nucleates, whereas in the second pathway a second Nb-lean component \u03b1\u2033lean,II nucleates and develops into \u03b1.For both pathways Nb-depleted \u03b1\u2033c\u2009=\u200916 is very similar to that for c\u2009=\u200921. Again Nb-depleted \u03b1\u2033lean forms from the quenched-in \u03b1\u2033. However, no evidence for \u03b1 or \u03b1\u2033lean,II was found and the transformation sequence observed for Ti\u201316Nb during heating was thus:The decomposition process of \u03b1\u2033 for When heating continues after \u03b1\u2033 decomposition the reflections of \u03b1 gradually weaken, which corresponds to the endothermic conversion of \u03b1 to \u03b2 while approaching the \u03b1\u2013\u03b2 transus Fig.\u00a0.iso. Comparing the lattice parameters of \u03b1\u2033iso  by arc-melting followed by cold-crucible casting22. Analysis of the Nb and O contents of the as-cast alloys by wet chemical analysis and hot gas extraction showed that the deviation from the nominal Nb content and the maximal O content was <0.1 wt.% each. The cast rods were homogenized for 24\u2009h at 1000\u2009\u00b0C under Ar and quenched into water (HQ treatment). Prior to quenching the parent \u03b2-phase exhibited a grain size of (300\u2009\u00b1\u2009150)\u2009\u00b5m. Specimens for light microscopy were prepared by mechanical polishing followed by etching in an aqueous solution of 2 vol.% HF and 6.5 vol.% HNO3.Four binary Ti\u2013\u22121 was achieved with a resistively heated modified Linkam hot stage purged with Ar. Diffraction patterns were recorded by a 2-dimensional (2D) image-plate detector (FReLoN) with 2048\u2009\u00d7\u20092048 pixels centered on the transmitted beam. Example raw 2D-diffraction patterns are shown in Supplementary Fig.\u00a0\u03bb\u2009=\u20090.20664\u2009\u00c5 and a beam cross-section of 50\u2009\u00d7\u200950\u2009\u00b5m2 were used. The sample-to-detector distance and zero-point shift were calibrated at RT using standard CeO2 powder. Data collection was done every \u00b0C using an exposure time of 0.2\u2009s. Calibration, background subtraction and azimuthal integration of the 2D patterns into 1-dimensional patterns were performed with Fit2D40. Azimuthal integration was carried out over the entire 360\u00b0 in order to minimize potential effects of texture and graininess. Diffraction angles are reported in terms of the wavevector transfer Q\u2009=\u20094\u03c0\u2219sin(\u03b8)/\u03bb, where \u03b8 denotes the semi-angle between the incident and the diffracted beam.In situ SXRD was conducted at the ID11 beamline of the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. Thin rod-shaped samples (diameter\u2009\u2248\u2009800\u2009\u00b5m) were extracted from the HQ material by electrical discharge machining and subjected to a solution treatment at 1000\u2009\u00b0C for 4\u2009h in Ar followed by water quenching. In situ heating of the samples from RT to 760\u2009\u00b0C at a rate of 10\u2009\u00b0C\u2009minlean and \u03b1\u2033iso). For each phase detected the lattice parameters were determined depending on temperature by structureless (Le Bail) refinements utilizing the FULLPROF programme suite41. The reflection profiles were simulated with Thompson\u2013Cox\u2013Hastings pseudo-Voigt functions42 and the background was modelled by linear segments, which were included in the refinements.On the patterns obtained phase analysis was carried out. For each phase the space group and occupied Wyckoff position(s) are listed in Table\u00a0The datasets generated and analyzed during the current study are available from the corresponding author on request.Supplementary InformationPeer Review File"}
{"text": "In the line \u2018Elective content varies markedly across universities (1 to 122\u00a0h)\u2019, the number \u201c122\u201d should be \u201c222\u201d.In the line \u2018In Australia, medical school curricula contain a median of 2.55\u00a0h of compulsory intellectual disability content\u2019, there should be a word \u201cunits\u201d inserted between \u201ccurricula\u201d and \u201ccontain\u201d. The full corrected sentence should read: \u2018In Australia, medical school curricula units contain a median of 2.55\u00a0h of compulsory intellectual disability content\u2019.After publication of the original article , the aut"}
{"text": "Scientific Reports6: Article number: 3774010.1038/srep37740; published online 11232016; updated on 05022017The original version of this Article contained errors resulting from the incorrect calculation of formation energy in 2D\u2009+\u20092D\u2009+\u20092D\u2009\u2192\u200921H\u2009+\u20094He\u2009+\u2009212\u2009J/m3\u201d.\u201cNow reads:2D\u2009+\u20092D\u2009+\u20092D\u2009\u2192\u200921H\u2009+\u20094He\u2009+\u2009e-\u2009+\u200915\u2009J/m3\u201d.\u201cIn addition, there was an error in the \u2018Introduction section\u2019, where\u201cIncrease in paraeomagnetic magnitude between 2.7\u20132.1 billion years\u201dNow reads:\u201cIncrease in palaeomagnetic magnitude between 2.7\u20132.1 billion years\u201dThere were also errors in There were also errors in Under the sub-heading \u201cExistence of a D-rich inner core\u201d,21\u2009kg (=100\u2009Mkm3\u2009\u00d7\u20092\u2009Mg/m3\u2009\u00d7\u20094/20\u2009\u00d7\u20091/0.006), the volume of Fe-D crystals up through the present is 4.99\u2009\u00d7\u20091017\u2009m3 (6.67\u2009\u00d7\u20091021/13.36 [Mg/m3]), resulting in an unreacted (Fe-D) crystal volume of 4.37\u2009\u00d7\u20091017 m3 (=4.99\u2009\u00d7\u20091017\u2009\u2212\u20096.23\u2009\u00d7\u20091016)\u201d.\u201cOn the other hand, because the incorporated gross weight of D from the creation of Earth\u2019s core to present time is estimated as 6.67\u2009\u00d7\u200910Now reads:21\u2009kg (=100\u2009Mkm3\u2009\u00d7\u20092\u2009Mg/m3\u2009\u00d7\u20094/20\u2009\u00d7\u20091/0.006), the volume of Fe-D crystals up through the present is 4.99\u2009\u00d7\u20091017\u2009m3 (6.67\u2009\u00d7\u20091021/13.36 [Mg/m3]), resulting in an unreacted (Fe-D) crystal volume of 4.39\u2009\u00d7\u20091017 m3 (=4.99\u2009\u00d7\u20091017\u2009\u2212\u20096.00\u2009\u00d7\u20091016)\u201d.\u201cOn the other hand, because the incorporated gross weight of D from the creation of Earth\u2019s core to present time is estimated as 6.67\u2009\u00d7\u200910Finally, there was an error in the \u2018Conclusions\u2019 section where:Now reads:These errors have now been corrected in the HTML and PDF versions of this Article."}
{"text": "The Middle East respiratory syndrome (MERS) coronavirus, a newly identified pathogen, causes severe pneumonia in humans. MERS is caused by a coronavirus known as MERS-CoV, which attacks the respiratory system. The recently defined receptor for MERS-CoV, dipeptidyl peptidase 4 (DPP4), is generally expressed in endothelial and epithelial cells and has been shown to be present on cultured human nonciliated bronchiolar epithelium cells. In this paper, a class of novel four-dimensional dynamic model describing the infection of MERS-CoV is given, and then global stability of the equilibria of the model is discussed. Our results show that the spread of MERS-CoV can also be controlled by decreasing the expression rate of DPP4. The Middle East respiratory syndrome (MERS) coronavirus, a newly identified pathogen, causes severe pneumonia in humans, with a mortality of nearly 44%. Human-to-human spread has been demonstrated, raising the possibility that the infection could become pandemic . A colorT(t), I(t), and v(t) denote the concentration of uninfected cells, infected cells, and free viruses at time t, respectively. The constant \u03bb > 0 is the rate at which new uninfected cells are generated (from a pool of precursor cells). The constants d > 0 and \u03b2 \u2265 0 are the death rate of uninfected cells and the rate constant characterizing infection of the cells, respectively. The constant d1 > 0 is the death rate of the infected cells due to either viruses or immune responses. The infected cells produce new viruses at the rate d1N during their life, on average having the length 1/d1, where N > 0 is some integer number. The constant c > 0 is the rate at which the viruses are cleared, and the average lifetime of a free virus is 1/c.It is well-known that dynamic models are still playing important roles in describing the interactions among uninfected cells, free viruses, and immune responses . A thT\u02d9=\u03bb\u2212\u03b2vtTD(t) represents the concentration of DPP4 on the surface of uninfected cells, which can be recognized by surface spike (S) protein of MERS-CoV )v(t)T(t). It is assumed that DPP4 is produced from the surface of uninfected cells at the constant rate \u03bb1 > 0. DPP4 is destroyed, when free viruses try to infect uninfected cells, at the rate \u03b21(\u03b2D(t))v(t)T(t), and is hydrolyzed at the rate \u03b3D(t). Here, \u03b21 \u2265 0 and \u03b3 > 0 are constants. It is assumed that there is no undestroyed DPP4 on the surface of infected cells. All other parameters in model . InfectT\u02d9=\u03bb\u2212\u03b2DtvT(0) \u2265 0, I(0) \u2265 0, v(0) \u2265 0, and D(0) \u2265 0. It is not difficult to show that the solution (T(t), I(t), v(t), D(t)) with the initial condition is existent, unique, bounded, and nonnegative for all t \u2265 0  > 0 and D(t) > 0 for all t > 0). If T(0) > 0, I(0) > 0, v(0) > 0, and D(0) > 0, it is easily proven that the corresponding solution (T(t), I(t), v(t), D(t)) is positive for all t \u2265 0.The initial condition of model  is givena < 1, \u03bc = min\u2061{d, (1 \u2212 a)d1, c}.Furthermore, it can be easily shown that the set to model , where 0The purpose of the paper is to study local and global stability of the equilibria of model  by usingR0 = N\u03b2\u03bb\u03bb1/cd\u03b3. Model  = . If R0 > 1, model , where, for \u03b21 = 0, v = v\u2217 = d\u03b3(R0 \u2212 1)/\u03b2\u03bb1, for \u03b21 > 0, v = v\u2217 > 0 is the positive root of the equation \u03b2\u03b21c2v2 \u2212 Nc\u03b2(\u03bb1 + \u03bb\u03b21)v + Ncd\u03b3(R0 \u2212 1) = 0, and The basic reproductive ratio of the virus for model  is R0 = \u03b3. Model  always h1, model  also hasFirst, we have the following result.\u03a91 = {\u2223 \u2208 \u03a9, T > 0, D > 0}, the infection-free equilibrium E0 =  is globally asymptotically stable when R0 < 1 and globally attractive when R0 = 1.With respect to the set T, I, v, D), Jacobian matrix of model (E0 is f(\u03c1) = (\u03c1 + d)(\u03c1 + \u03b3)[\u03c12 + (c + d1)\u03c1 + cd1(1 \u2212 R0)] = 0. Clearly, if R0 < 1, all roots of f(\u03c1) = 0 have negative real parts. Hence, E0 is local asymptotic stability by Routh-Hurwitz criterion. If R0 = 1, f(\u03c1) = 0 has the zero root \u03c1 = 0 and three negative roots. Hence, E0 is linearly stable.At any equilibrium (of model  is(5)J=\u2212U is continuous on \u03a91 and positive definite with respect to E0 and satisfies condition (ii) of Definition 1.1 in [\u03a9 = \u03a9\u2216\u03a91. Calculating the derivative of U along the solutions of model  \u2208 \u03a9, U < +\u221e}. Let M be the largest subset in Q which is invariant with respect to the set model \u2223 \u2208 \u03a9, T = T0, D = D0}. From the invariance of M and model \u2223 \u2208 \u03a9, T > 0, I > 0, v > 0, D > 0}, the infected equilibrium E\u2217 is locally asymptotically stable when R0 > 1. In addition, if (2d\u03b3\u03bca)2 \u2265 \u03b21\u03b22\u03bb1\u03bb3N2, where 0 < a < 1, \u03bc = min\u2061{d, (1 \u2212 a)d1, c}, the infected equilibrium E\u2217 is globally asymptotically stable.With respect to the set E\u2217 is ai > 0\u2009\u2009. Furthermore, by using Matlab program, it can been shown that \u03943 = a1a2a3 \u2212 a12 \u2212 a32a0 has 200 items in which all items are positive. Hence, E\u2217 is local asymptotic stability by Routh-Hurwitz criterion.The characteristic equation at of model  at E\u2217 isW is continuous on \u03a92 and positive definite with respect to E\u2217 and satisfies condition (ii) of Definition 1.1 in [\u03a9 = \u03a9\u2216\u03a92. Calculating the derivative of W along the solutions of model 2/8\u03b3 \u2265 0 and \u03b3/2D \u2212 T(\u03b2\u03b21v)2/8\u03b21d \u2265 0 are equivalent to the inequality 4d\u03b3 \u2265 \u03b21\u03b22TDv2. Since T(t) \u2264 T0, D \u2264 D0, and v(t) \u2264 \u03bbN/\u03bca for all t \u2265 0, we have that the inequality 4d\u03b3 \u2265 \u03b21\u03b22TDv2 holds, if the condition (2d\u03b3\u03bca)2 \u2265 \u03b21\u03b22\u03bb1\u03bb3N2 in dW/dt \u2264 0 on \u03a92.On the other hand, notice the inequality in : (12)\u2212xzQ = {dW/dt = 0\u2223 \u2208 \u03a9, W < +\u221e}. Let M be the largest subset in Q which is invariant with respect to the set of model \u2223 \u2208 \u03a9, T = T\u2217, D = D\u2217}. From the invariance of M and model  and R0 = 0.90909 < 1. E0 is asymptotically stable.Let us first give some numerical simulations on the orbits of model . Take th\u03b3 = 0.05, and all the other parameters are the same as above. We can also compute the values of the infection-free equilibrium E0, the infected equilibrium E\u2217, and the basic reproductive ratio, E0 = , E\u2217 = , and R0 = 2 > 1. E\u2217 is asymptotically stable. We would like to point out here that, based on the numerical simulations, the condition (2d\u03b3\u03bca)2 \u2265 \u03b21\u03b22\u03bb1\u03bb3N2 may be further weakened or even removed.Let us take io, E0 = 0,0, 0,20R0 = N\u03b2\u03bb\u03bb1/cd\u03b3, let us give some simple discussions on the interactions between the protein DPP4 and the virus infection. Usually, in the absence of any drug treatment, all the parameters in model (R0 can be regarded as relatively fixed constants. If some drug treatment measures are taken, the effectiveness of the treatment can be reflected in the regulation of the parameter \u03b3. For example, by increasing the value of \u03b3, the value of the basic reproductive ratio of R0 can be changed from greater than 1 to less than 1. In the numerical simulations in this section, \u03b3 = 0.05 and R0 = 2 > 1. If increasing \u03b3 from \u03b3 = 0.05 to \u03b3 = 0.11, R0 = 0.9090 < 1.Finally, by using the basic reproductive ratio in model  and the"}
{"text": "Leptonetela Kratochv\u00edl, 1978 that is endemic to karst systems in Eurasia using DNA barcoding. We analyzed 624 specimens using one mitochondrial gene fragment (COI). The results show that DNA barcoding is an efficient and rapid species identification method in this genus. DNA barcoding gap and automatic barcode gap discovery (ABGD) analyses indicated the existence of 90 species, a result consistent with previous taxonomic hypotheses, and supported the existence of extreme male pedipalpal tibial spine and median apophysis polymorphism in Leptonetela species, with direct implications for the taxonomy of the group and its diversity. Based on the molecular and morphological evidence, we delimit and diagnose 90 Leptonetela species, including the type species Leptonetela kanellisi. Forty-six of them are previously undescribed. The female of Leptonetela zhaiLeptonetela tianxinensis  comb. nov. is transferred from the genus Leptoneta Simon, 1872;the genus Guineta Lin & Li, 2010 syn. nov. is a junior synonym of Leptonetela; Leptonetela gigachela comb. nov. is transferred from Guineta. The genus Sinoneta Lin & Li, 2010 syn. nov. is a junior synonym of Leptonetela; Leptonetela notabilis comb. nov. and Leptonetela sexdigiti comb. nov. are transferred from Sinoneta; Leptonetela sanchahe Wang & Li nom. nov. is proposed as a replacement name for Sinoneta palmata because Leptonetela palmata is preoccupied.Extreme environments, such as subterranean habitats, are suspected to be responsible for morphologically inseparable cryptic or sibling species and can bias biodiversity assessment. A DNA barcode is a short, standardized DNA sequence used for taxonomic purposes and has the potential to lessen the challenges presented by a biotic inventory. Here, we investigate the diversity of the genus  Subterranean ecosystems, such as caves and cracks, are evident mainly in karst areas, which represent nearly 4% of the rocky outcrops of the world. These environments are marked by permanent darkness, a lack of diurnal and annual rhythms, and extremely scarce food sources . Many stDNA barcoding relies on the use of a standardized DNA region as a tag for accurate and rapid species identification  and has Leptonetela is discontinuously distributed in the South China karst and the Balkan Peninsula, a karstic region in Europe. The genus has 54 catalogued species . Rapid and accurate identification within this genus is difficult due to congeneric species sharing similar morphological traits, a lack of obvious morphological differences between closely related species and some species only differ in one or a few quantitative differences, such as the location, length ratio or thickness of the male pedipalpal tibial spines and the number of teeth on the median apophysis.The South China karst, a United Nations Educational, Scientific and Cultural Organization (UNESCO) World Heritage Site since 2007, is noted for its karst features and landscapes as well as rich biodiversity. Numerous subterraean species have been reported in this region, especially invertebrate fauna . The spi species , and witLeptonetela and investigate the diversity of the genus. The standard molecular barcode, cytochrome c oxidase subunit \u2160 (COI) was used. A species discovery method, automatic barcode gap discovery (ABGD)  to test taxonomic value of morphological characters used in traditional methods of classification.In this study, we test the usefulness of DNA barcoding for species identification in the subterranean genus y (ABGD) , and a sy (ABGD)  were botLeptonetela individuals from 122 populations (caves) (Supplementary Table S1) in Eurasia  barcode region using the primer pairs LCOI490/HCO2198 , 2.0 \u03bcL of dNTP Mix (2.5 mmol/L), 1 \u03bcL of each forward and reverse 10 \u03bcmol/L primer, 1 \u03bcL of DNA template, and 0.25 \u03bcL Taq DNA polymerase . Double-stranded PCR products were visualized by agarose gel electrophoresis (1% agarose). PCR products were purified and sequenced by Sunny Biotechnology Co., Ltd  using the ABI 3730XL DNA analyser. Sequences were aligned using ClustalW in Mega 6.0  following the manufacturer's protocol. We amplified the cytochrome c oxidase subunit \u2160  in RAXML v. 7.0.3 with the GTRCAT model . One hunCOI barcode dataset (see Supplementary Table S1) using two species delineation methods. DNA barcoding gap analyses require an a priori species designation. Therefore, we divided the 624 Leptonetela individuals of 122 populations (caves) into 90 putative species based on morphological characters and geographic information. In our DNA barcoding gap analysis, we examined the overlap between the mean intraspecific and interspecific Kimura 2-parameter (K2P)   and uncodistance  for eachdistance .The automatic barcode gap discovery procedure (ABGD) , which does not require assigning samples to putative species, calculates all pairwise distances in the dataset, evaluates intraspecific divergences, and then sorts the samples into candidate species using the calculated distances. We performed ABGD analyses online (<a href=\"http://wwwabi.snv.jussieu.fr/public/abgd/\" target=\"_blank\">http://wwwabi.snv.jussieu.fr/public/abgd/</a>), using three different distance metrics: Jukes-Cantor (JC69) , Kimura 2-parameter (K2P) , and simple distance  . We analyzed the data using two different values for the parameters Pmin (0.0001 and 0.001), Pmax (0.1 and 0.2), and relative gap width (X=1 or 1.5), with all other parameters at default values. The terminology and the measurements in this paper generally follow This article conforms to the requirements of the amended International Code of Zoological Nomenclature. All nomenclatural acts contained within this published work have been registered in ZooBank. The ZooBank LSIDs (Life Science Identifiers) can be resolved and the associated information viewed by appending the LSID to the prefix \"<a href=\"http://zoobank.org/\" target=\"_blank\">http://zoobank.org/</a>\". The LSID for this publication is: urn:lsid:zoobank.org:pub:7ECB1BDC-8893-4D0F-8BEA-17ECE327FC47 L. kanellisi and L. robustispina, four or more DNA barcodes were generated. All nucleotides were translated into functional protein sequences in the correct reading frame, with no stop codons or indels observed. Similar to other arthropod studies, our data indicated a high AT-content for this mitochondrial gene fragment: the mean sequence compositions were A=20.5%, C=12.6%, G=24.4%, T=41.4%.In total, 624 DNA barcodes were analyzed. A full list of the analyzed specimens can be found in Supplementary Table S1. Fragment lengths of the analyzed DNA barcodes ranged from 107 (0.005%) to 617 bp (89%). For all populations, except Leptonetela is monophyletic, with the node highly supported  whereas interspecific distances were between 3.1/3% and 31.9/25% (K2P/uncorrected p-distance). Maximum intraspecific distances > 3% were found for two species, including L. reticulopecta (4.3/4.0%), and L. pentakis Lin & Li, 2010 (5.3/5%). The lowest interspecific distance were revealed for the two species pairs L.changtu Wang & Li sp. nov. with L. chuan Wang & Li sp. nov. and L. kangsa Wang & Li sp. nov. with L. shibingensis Guo, Yu & Chen, 2016 with a value of 3.1/3%. Minimum interspecific pairwise distances < 5%, and > 3% were found for two species pairs: L. shibingensis with L. shanji Wang & Li sp. nov. and L. dao Wang & Li sp. nov. with L. xiaoyan Wang & Li sp. nov. The mean interspecific distance between the 90 tentative species was 17.9/15.6% (K2P/uncorrected p-distance), and the mean intraspecific distance within each species was 0.2% (both K2P and uncorrected p-distance) in Leptonetela. A histogram of the gap and overlap between intra-and interspecies genetic distances are show in DNA barcoding gap analysis: Based on our COI dataset, using the originally specified parameter combinations and partitions resulted mostly in 90 distinct species that correspond to the 90 species observed in the previous taxonomic hypotheses based on morphology. The result was the same regardless of the model of evolution employed (Jukes-Cantor (JC), K2P, Simple Distance). The settings P min/P max=0.0001/0.2 yielded the most significant P values. However, at lower values of prior intraspecific distance (P), recursive partitioning of ABGD recognized more species , our data represent the first published DNA barcodes.DNA barcoding is widely recognized as a useful tool for species identification across the animal kingdom . Our resCOI showed high genetic structure between populations within species .Classically, geographic isolation is considered a primary feature of troglobitic taxa . Our DNACOI sequence divergence. The interspecific genetic divergences between L. chuan Wang & Li sp. nov. and L. changtu Wang & Li sp. nov., L. kangsa Wang & Li sp. nov. and L. shibingensis, as well as between L. shibingensis and L. shanji Wang & Li sp. nov. was 3.1/3.0% based on K2P/uncorrected and p-distance models. Compared with other species L. chuan Wang & Li sp. nov., and L. changtu Wang & Li sp. nov., L. kangsa Wang & Li sp. nov., L. shibingensis and L. shanji Wang & Li sp. nov. are more closely distributed. In morphology, L. chuan Wang & Li sp. nov. and L. changtu Wang & Li sp. nov. can be distinguished by the shape of the median apophysis and the conductor ; L. kangsa Wang & Li sp. nov., L. shibingensis and L. shanji Wang & Li sp. nov. can be distinguished by the location and pattern of male pedipalpal tibial spines . Nevertheless, we found two species with maximum pairwise distance > 3%, including L. reticulopecta (specimens from Tianshegnqiao Cave are clearly distant from the rest) with 4.3/4.0%, L. pentakis (specimens from Liaoya cave is clearly distant from the rest) with 5.3/5.0%. Then we achieved a threshold of 3.11/3.0% (K2P/uncorrected and p-distance), excluding taxa from Tianshegnqiao Cave and Liaoya Cave. This threshold was interestingly close to the 3% commonly used in barcoding literature L. strinatii  1\u00a0\u00a0Spermathecae thin and loosely twisted\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026-\u00a0\u00a0Not as above\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202622\u00a0\u00a0Male pedipalp with median apophysis\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263-\u00a0\u00a0Male pedipalp without median apophysis\u2026\u2026\u2026\u2026\u2026\u2026\u202693\u00a0\u00a0Median apophysis like pine needles, sclerotized\u2026\u2026\u20264-\u00a0\u00a0Not as above\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026334\u00a0\u00a0Median apophysis appears as 4 pine needle-like appendages\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265-\u00a0\u00a0Median apophysis divided into more or less than 4 pine needle-shaped appendages\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20266L. chakousp.nov.5\u00a0\u00a0Tibia \u2160 spine strong, conspicuous, with bifurcated tip\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. grandispina Lin & Li, 2010-\u00a0\u00a0Tibia \u2160 spine strong, located at the middle of tibia prolaterally\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20266\u00a0\u00a0Cymbium roughly double the length of bulb\u2026\u2026\u2026\u2026\u2026\u20267-\u00a0\u00a0Cymbium roughly the same length as bulb\u2026\u2026\u2026\u2026\u2026\u2026\u20268L. liuzhai Wang & Li sp. nov.7\u00a0\u00a0Median apophysis divided into 15 pine needle-like appendages\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. shuilian Wang & Li sp. nov.-\u00a0\u00a0Median apophysis divided into 2 pine needle-like appendages\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. pentakis Lin & Li, 20108\u00a0\u00a0Cymbium constricted medially, median apophysis divided into 5 pine needle-like appendages\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. dao Wang & Li sp. nov.-\u00a0\u00a0Cymbium not constricted medially, median apophysis divided into 2 pine needle-like appendages\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20269\u00a0\u00a0Male pedipalp with 5 tibial spines prolaterally\u2026\u2026\u2026\u202610-\u00a0\u00a0Male pedipalp with more than 5 tibial spines prolaterally\u20262910\u00a0\u00a0Cymbium constricted and wrinkled medially\u2026\u2026\u2026\u2026\u202611-\u00a0\u00a0Cymbium not constricted or wrinkled medially\u2026\u2026\u2026\u20262211\u00a0\u00a0Tibial spines slender and without bifurcated tip\u2026\u2026\u2026\u202612-\u00a0\u00a0Tibial spines strong or with bifurcated tip\u2026\u2026\u2026\u2026\u2026\u2026\u20261612\u00a0\u00a0Prolateral lobe tongue-shaped\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202613L. sanyan Wang & Li sp. nov.-\u00a0\u00a0Prolateral lobe absent\u2026\u2026\u202613\u00a0\u00a0Pedipalpal tibia with one spine significantly longer than other spines\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202614L. meitan Lin & Li, 2010-\u00a0\u00a0Pedipalp tibia \u2160, \u2161 spines nearly the same length\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202614\u00a0\u00a0Conductor bamboo leaf-shaped in ventral view\u2026\u2026\u2026\u202615L. liangfeng Wang & Li sp. nov.-\u00a0\u00a0Conductor C-shaped in ventral view\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. suae Lin & Li, 201015\u00a0\u00a0Embolus and conductor long, intersecting\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. tongzi Lin & Li, 2010-\u00a0\u00a0Embolus and conductor short, not intersecting\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202616\u00a0\u00a0Pedipalpal tibia \u2160 spine with bifurcated tip\u2026\u2026\u2026\u2026\u2026\u2026\u202617-\u00a0\u00a0Pedipalpal tibia \u2160 spine without bifurcated tip\u2026\u2026\u2026\u2026\u20261917\u00a0\u00a0Pedipalpal tibia \u2160 spine strong, asymmetrically bifurcate\u202618L. danxia Lin & Li, 2010-\u00a0\u00a0Pedipalpal tibia \u2160 spine slender, symmetrically bifurcate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. andreevi Deltshev, 198518\u00a0\u00a0Pedipalpal tibia \u2160 spine located proximally at tibia, thin spines \u2161, \u2164 and \u2165 arranged in a triangle, conductor bamboo leaf-shaped in ventral view\u2026\u2026\u2026L. furcaspina Lin & Li, 2010-\u00a0\u00a0Pedipalpal tibia \u2160 spine located at distal 1/3 of tibia, conductor C shaped in ventral view\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202619\u00a0\u00a0Pedipalpal tibia \u2160 spine longest\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202620-\u00a0\u00a0Pedipalpal tibia \u2161 spine longest\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202621L. langdong Wang & Li sp. nov.20\u00a0\u00a0Pedipalpal tibia \u2160 spine bent distally, conductor reduced\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. yaoiWang & Li, 2011-\u00a0\u00a0Pedipalpal tibia \u2160 spine not bent distally, conductor semicircular in ventral view\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. lineataWang & Li, 201121\u00a0\u00a0Eyes absent, pedipalpal tibia \u2162, \u2164 and \u2165 spines more slender than \u2160, \u2161 spines\u2026\u2026\u2026\u2026\u2026L. caucasica Dunin, 1990-\u00a0\u00a0Six eyes, pedipalpal tibial spines equally strong\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202622\u00a0\u00a0Conductor developed\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202623-\u00a0\u00a0Conductor reduced\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20262623\u00a0\u00a0Pedipalpal tibia \u2160 spine without bifurcated tip\u2026\u2026\u2026\u2026\u202624L. anshun Lin & Li, 2010-\u00a0\u00a0Pedipalpal tibia \u2160 spine with bifurcated tip, other spines concentrated distally, tip of conductor bifurcated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202624\u00a0\u00a0Conductor bamboo leaf-shaped in ventral view\u2026\u2026\u2026\u202625L. dashui Wang & Li sp. nov.-\u00a0\u00a0Conductor C-shaped in ventral view, pedipalpal tibia \u2160 spine longest\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. qiangdao Wang & Li sp. nov.25\u00a0\u00a0Pedipalpal tibia \u2160 spine strong, prolateral bulbal lobe reduced\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. nuda -\u00a0\u00a0Pedipalpal tibia \u2160 spine slender, prolateral bulbal lobe tongue-shaped\u2026\u2026\u2026\u2026\u2026L. curvispinosa Lin & Li, 201026\u00a0\u00a0Cymbium with a distal and proximal spine prolaterally, pedipalpal tibial spines equidistant\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026-\u00a0\u00a0Not as above\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202627L. wangjia Wang & Li sp. nov.27\u00a0\u00a0Pedipalpal tibia \u2160 spine slender, asymmetrically bifurcated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026-\u00a0\u00a0Not as above\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202628L. maxillacostata Lin & Li, 201028\u00a0\u00a0Pedipalpal tibia \u2160, \u2161, and \u2162 spines concentrated in the mid of tibia, 2 additional spines located distally, prolateral bulbal lobe reduced\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. chenjia Wang & Li sp. nov.-\u00a0\u00a0Pedipalpal tibia \u2160 spine longest, located far from others, prolateral lobe small, tongue shaped\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202629\u00a0\u00a0Male pedipalp with 6 tibial spines retrolaterally\u2026\u2026\u2026\u202630-\u00a0\u00a0Male pedipalp with 7 tibial spines retrolaterally\u2026\u2026\u2026\u202632L. gang Wang & Li sp. nov.30\u00a0\u00a0Pedipalpal tibia \u2160, \u2161 spines strong, equally length, \u2161 spine asymmetrically bifurcated, conductor reduced\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026-\u00a0\u00a0Pedipalpal tibial spines slender, not bifurcated, conductor developed\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202631L. gigachela 31\u00a0\u00a0Pedipalpal tibia with 2 large spines prolaterally, cymbium not constricted medially, earlobe-shaped process absent, and cymbium long, twice the length of bulb\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. wenzhu Wang & Li sp. nov.-\u00a0\u00a0Pedipalpal tibia without prolateral spines, cymbium constricted medially, retrolaterally attaching an earlobe-shaped process, cymbium less than twice the length of bulb\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. rudong Wang & Li sp. nov.32\u00a0\u00a0Cymbium with 1 horn-shaped spine on the earlobe-shaped process, conductor thin, triangular in ventral view\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. la Wang & Li sp. nov.-\u00a0\u00a0Earlobe-shaped process of cymbium without spine, conductor broad, C shaped in ventral view\u202633\u00a0\u00a0Median apophysis like a pointed process or lamelliform..34-\u00a0\u00a0Median apophysis finger-shaped or harrow-like\u2026\u2026\u20265034\u00a0\u00a0Cymbium not constricted medially, earlobe-shaped process reduced\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202635-\u00a0\u00a0Cymbium constricted medially, earlobe-shaped process developed\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20263835\u00a0\u00a0Male pedipalpal tibia with 6 spines retrolaterally\u2026\u2026\u2026\u2026\u202636-\u00a0\u00a0Male pedipalpal tibia only with 5 spines retrolaterally\u2026\u202637L. bama Lin & Li, 201036\u00a0\u00a0Pedipalpal tibia with 4 long spines prolaterally, the retrolateral \u2160 spine longest, \u2161 \u2162 spines short and strong, median apophysis pointed, conductor bamboo leaf-shaped\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. yangi Lin & Li, 2010-\u00a0\u00a0Pedipalpal tibia with 3 long spines prolaterally, the retrolateral \u2160 spine longest and strongest, median apophysis \"M\"-shaped, conductor reduced\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. liping Lin & Li, 201037\u00a0\u00a0Pedipalpal tibia with 1 long spine prolaterally, the retrolateral \u2160 spine longest and strongest, median apophysis pointed, conductor bamboo leaf-shaped\u2026\u2026\u2026L. mayang Wang & Li sp. nov.-\u00a0\u00a0Pedipalpal tibia with 3 long spines prolaterally, the retrolateral spines \u2160 slender, and longest, median apophysis obtuse triangle shaped, conductor narrow, triangular\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202638\u00a0\u00a0Cymbium with 1 strong spine on the earlobe-shaped process\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202639-\u00a0\u00a0No spine on the earlobe-shaped process\u2026\u2026\u2026\u2026\u2026\u2026\u20264339\u00a0\u00a0Male pedipalp tibia with 5 spines retrolaterally\u2026\u2026\u2026\u202640-\u00a0\u00a0Male pedipalp tibia with more than 5 spines retrolaterally\u202641L. jiahe Wang & Li sp. nov.40\u00a0\u00a0Cymbium with 1 curved spine retrolaterally, median apophysis pointed, with 3 sclerotized apices distally, conductor C shaped\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. panbao Wang & Li sp. nov.-\u00a0\u00a0Cymbium without curved spine retrolaterally, median apophysis punctate in ventral view, conductor vestigial\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. jiulong Lin & Li, 201041\u00a0\u00a0Pedipalpal tibia \u2160 spine strong, \u2161 spine asymmetrically bifurcated, median apophysis lamelliform, conductor triangular\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026-\u00a0\u00a0Pedipalpal tibia \u2160 spine slender, not bifurcated\u2026\u2026\u2026\u202642L. parlongaWang & Li, 201142\u00a0\u00a0Pedipalpal tibia with 3 long spines prolaterally, 6 spines retrolaterally, median apophysis semicircular\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. mitaWang & Li, 2011-\u00a0\u00a0Pedipalpal tibia with 5 long spines prolaterally, 7 spines retrolaterally, median apophysis mita-shaped, embolus with 1 tooth distally\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202643\u00a0\u00a0Pedipalpal tibia with 5 spines retrolaterally\u2026\u2026\u2026\u2026\u2026\u202644-\u00a0\u00a0Pedipalpal tibia with more than 5 spines retrolaterally\u2026\u20264944\u00a0\u00a0Pedipalpal tibia with 3 long spines prolaterally\u2026\u2026\u2026\u2026\u2026\u202645-\u00a0\u00a0Pedipalpal tibia with 1 or 2 long spines prolaterally\u2026\u2026\u20264745\u00a0\u00a0Conductor C shaped in ventral view\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202646L. xianren Wang & Li sp. nov.-\u00a0\u00a0Conductor bamboo leaf-shaped in ventral view, retrolateral spines \u2160 longest, median apophysis triangular\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. rudiculaWang & Li, 201146\u00a0\u00a0Pedipalpal tibia \u2160 spine longest, the rest concentrated at distal end of tibia, median apophysis spatula-shaped in ventral view\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. longli Wang & Li sp. nov.-\u00a0\u00a0Pedipalpal tibia \u2160 spine longest, \u2160, \u2161, and \u2162 spines equally strong, median apophysis single quote shaped, \" \u2032 \" in ventral view\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. pungitiaWang & Li, 201147\u00a0\u00a0Pedipalpal tibia with 1 long spine prolaterally, median apophysis tongue-shaped, conductor triangular\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026-\u00a0\u00a0Pedipalpal tibia with 2 long spines prolaterally\u2026\u2026\u2026\u2026\u2026\u202648L. chiosensisWang & Li, 201148\u00a0\u00a0Pedipalpal tibia \u2160 spine strongest, \u2162-\u2164 spines in a triangular arrangement, median apophysis punctate, conductor triangular\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. feilong Wang & Li sp. nov.-\u00a0\u00a0Pedipalpal tibia \u2160 spine longest, spine \u2160-\u2162 equally strong, \u2163-\u2164 situated distally median apophysis \"m\"-shaped, conductor triangular\u2026\u2026\u2026\u2026L. tiankeng Wang & Li sp. nov.49\u00a0\u00a0Pedipalpal tibia with 6 spines retrolaterally, tibia \u2160 spine close to others, median apophysis flake-like, sclerotized distally, conductor broad, undulate distally\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. lophacantha -\u00a0\u00a0Pedipalpal tibia with 7 spines retrolaterally, tibia \u2160 spine distant from others, median apophysis small worm-shaped, conductor thin, triangular\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202650\u00a0\u00a0Median apophysis index finger like\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202651-\u00a0\u00a0Median apophysis harrow-like\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202672L. xinhua Wang & Li sp. nov.51\u00a0\u00a0Embolus bifurcated\u2026\u2026\u2026\u2026\u2026-\u00a0\u00a0Embolus not bifurcated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265252\u00a0\u00a0Base of median apophysis swollen\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202653-\u00a0\u00a0Base of median apophysis not swollen\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20265653\u00a0\u00a0Male pedipalpal tibia with 5 spines retrolaterally\u2026\u2026\u202654L. quinquespinata -\u00a0\u00a0Male pedipalpal tibia with 6 slender spines retrolaterally, spines \u2160 longest, conductor smooth, semicircular\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. jinsha Lin & Li, 201054\u00a0\u00a0Pedipalpal tibia \u2160 spine much stronger than \u2161, asymmetrically bifurcated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026-\u00a0\u00a0Pedipalpal tibia \u2160 spine similarly strong as \u2161, not bifurcated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202655L. gubin Wang & Li sp. nov.55. Cymbium constricted medially, earlobe-shaped process with 2 long, curved spines retrolaterally, base of median apophysis distinctly swollen, conductor smooth, broad, semicircular\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. lujia Wang & Li sp. nov.-\u00a0\u00a0Cymbium not constricted medially, earlobe-shaped process small, base of median apophysis slightly swollen, conductor rugose, thin, triangular\u2026\u2026\u2026\u2026\u2026L. wuming Wang & Li sp. nov.56\u00a0\u00a0Median apophysis bifurcated distally\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026-\u00a0\u00a0Median apophysis not bifurcated distally\u2026\u2026\u2026\u2026\u2026\u2026\u20265757\u00a0\u00a0Pedipalpal tibia \u2160 spine located at the base of tibia\u2026\u202658-\u00a0\u00a0Pedipalpal tibia \u2160 spine located medially\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202659L. shibingensis Guo, Yu & Chen, 201658\u00a0\u00a0Pedipalpal tibia \u2160 spine asymmetrically bifurcated, tibia with 4 long spines prolaterally\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. kangsa Wang & Li sp. nov.-\u00a0\u00a0Pedipalpal tibia \u2160 spine not bifurcated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202659\u00a0\u00a0Male pedipalp tibia with 6 spines retrolaterally\u2026\u2026\u2026\u202660-\u00a0\u00a0Male pedipalp tibia with 5 spines retrolaterally\u2026\u2026\u2026\u202662L. xiaoyan Wang & Li sp. nov.60\u00a0\u00a0Pedipalpal tibia with 4 spines prolaterally, cymbium with 1 curved spine at the base of retrolateral surface, median apophysis weakly sclerotized\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026-\u00a0\u00a0Not as above\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202661L. oktocantha Lin & Li, 201061\u00a0\u00a0Male pedipalp tibia with 2 spines prolaterally, conductor short, broad and rugose\u2026\u2026\u2026L. hexacantha Lin & Li, 2010-\u00a0\u00a0Male Pedipalp tibia without spine prolaterally, conductor smooth, semicircular\u2026\u2026\u2026\u2026\u202662\u00a0\u00a0Median apophysis curved distally\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202663-\u00a0\u00a0Median apophysis not curved distally\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202665L. mengzongensisWang & Li, 201163\u00a0\u00a0Cymbium with 1 horn-shaped spine on the earlobe-shaped process retrolaterally, tibia spines gradually shorted, conductor smooth, C shaped\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026-\u00a0\u00a0Cymbium without spine on the earlobe-shaped process retrolaterally\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202664L. hamata Lin & Li, 201064\u00a0\u00a0Male pedipalp tibia with 2 long setae prolaterally, tibia \u2160 \u2161 and \u2162 spines equally in length, conductor broad, semicircular\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. tetracantha Lin & Li, 2010-\u00a0\u00a0Male pedipalp tibia with 4 long spines prolaterally, tibia \u2160 \u2161 spines equally in length, conductor long, curved distally\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202665\u00a0\u00a0Pedipalpal tibia \u2160 spur strong\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202666-\u00a0\u00a0Pedipalpal tibia \u2160 spine slender\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20267066\u00a0\u00a0Pedipalpal tibia \u2160 spine asymmetrically bifurcated\u2026\u2026\u202667L. reticulopecta Lin & Li, 2010-\u00a0\u00a0Pedipalpal tibia \u2160 spine not bifurcated, conductor broad, C shaped, median apophysis distinctly sclerotized\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202667\u00a0\u00a0Median apophysis tapering\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202668-\u00a0\u00a0Median apophysis blunt\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202669L. shanji Wang & Li sp. nov.68\u00a0\u00a0Pedipalpal tibia \u2160 spine located at the middle of tibia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. digitata Lin & Li, 2010-\u00a0\u00a0Pedipalpal tibia \u2160 spine located at the basal of tibia\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. tianxinensis 69\u00a0\u00a0Pedipalpal tibia \u2161-\u2164 spines slender flexible, \u2160 and \u2161 spines equally length, conductor shorter than median apophysis\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. nanmu Wang & Li sp. nov.-\u00a0\u00a0Pedipalpal tibia \u2161 spine slender, \u2162 spine strong, conductor longer than median apophysis\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. huoyan Wang & Li sp. nov.70\u00a0\u00a0Pedipalpal tibia \u2160 spine located at the base of tibia, other spines concentrated distally on tibia, conductor smooth, semicircular\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026-\u00a0\u00a0Not as above\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202671L. geminispina Lin & Li, 201071\u00a0\u00a0Pedipalpal tibia \u2160 \u2161 spines adjacent, the rest short, concentrated distally, outermost plumose, tibia with 2 spines prolaterally, conductor bifurcate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. tianxingensisWang & Li, 2011-\u00a0\u00a0Pedipalpal tibia \u2160-\u2163 spines spaced at regular intervals, \u2163 and \u2164 adjacent, tibia \u2160-\u2162 equal in length, conductor short, C shaped\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202672\u00a0\u00a0Median apophysis harrow-like, horrow pin reduced to sclerotized spots\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202673-\u00a0\u00a0Median apophysis harrow-like, horrow pin not reduced\u202675L. liuguan Wang & Li sp. nov.73\u00a0\u00a0Pedipalpal tibial spines slender, equally strong, median apophysis long, half the length of bulb\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026-\u00a0\u00a0Pedipalpal tibial spines not equally strong, median apophysis short, 1/5 the length of bulb\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202674L. penevi Wang & Li, 201674\u00a0\u00a0Pedipalpal tibia \u2160, \u2161 spines equally strong, stronger than other spines, \u2162-\u2164 in triangular arrangement, cymbium constricted medially, with one curved spine at the base of constriction retrolaterally\u2026\u2026\u2026\u2026L. changtu Wang & Li sp. nov.-\u00a0\u00a0Pedipalpal tibia \u2160 \u2161 \u2162 spines equally strong, stronger than other spines, \u2162-\u2164 not triangular arrangement, cymbium not constricted medially\u2026\u2026\u2026\u202675\u00a0\u00a0Median apophysisi harrow-like, the horrow pin not constant in size\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202676-\u00a0\u00a0Median apophysisi harrow-like, the horrow pin constant in size\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20268076\u00a0\u00a0Pedipalpal tibia \u2160 spine not bifurcated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202677L. lianhua Wang & Li sp. nov.-\u00a0\u00a0Pedipalpal tibia \u2160 spine strong, asymmetrically bifurcated, other 4 spines slender, median apophysis with 5 small teeth and 1 large, horn-shaped tooth\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202677\u00a0\u00a0Pedipalpal tibia \u2160 spine longest\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202678-\u00a0\u00a0Pedipalpal tibia \u2161 spine longest\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202679L. megaloda 78\u00a0\u00a0Median apophysis palmate, with six teeth distally\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. niubizi Wang & Li sp. nov.-\u00a0\u00a0Median apophysis antler-like, with 4 small teeth and 1 large tooth, which bears 2 small teeth\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. hangzhouensis 79\u00a0\u00a0Two large teeth on the periphery of median apophysis, 2 small teeth in the middle\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. microdonta -\u00a0\u00a0Two large teeth on the periphery of median apophysis, 5 small teeth in the middle.. 80\u00a0\u00a0Median apophysis short and broad\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202681-\u00a0\u00a0Median apophysis long and thin\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20268781\u00a0\u00a0Pedipalpal tibia \u2160 spine strongest\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202682-\u00a0\u00a0Pedipalpal tibia \u2160 spine not strongest\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202683L. identica 82\u00a0\u00a0Pedipalpal tibia \u2160, \u2161 spines equally strong, stronger than other 3 spines, median apophysis with 6 small teeth apically\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. meiwang sp.nov.-\u00a0\u00a0Pedipalpal tibia \u2161, \u2162 spines equally strong, spine \u2161 longest, median apophysis with 5 sharp teeth apically\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202683\u00a0\u00a0Pedipalpal tibia \u2160 spine bifurcated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202684-\u00a0\u00a0Pedipalpal tibia \u2160 spine not bifurcated\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20268984\u00a0\u00a0Distal edge of median apophysis with 6 teeth\u2026\u2026\u2026\u2026\u202685-\u00a0\u00a0Distal edge of median apophysis with 5 or 10 teeth\u202686L. zakou Wang & Li sp. nov.85\u00a0\u00a0Teeth of median apophysis needle-shaped, earlobe-shaped process of cymbium absent; in the female, anterior margin of atrium with one pointed process medially\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. sexdentataWang & Li, 2011-\u00a0\u00a0Teeth of median apophysis normal, cymbium with earlobe-shaped process, female anterior margin of atrium without pointed process\u2026\u2026\u2026\u2026\u2026\u2026L. longyu Wang & Li sp. nov.86\u00a0\u00a0Distal edge of median apophysis with 5 teeth, conductor C shaped, tip of conductor undulate in the female, anterior margin of atrium with one pointed process medially\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. shicheng Wang & Li sp. nov.-\u00a0\u00a0Distal edge of median apophysis with 10 teeth, conductor C shaped, distal edge of conductor smooth; in the female, anterior margin of atrium without pointed process\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202687\u00a0\u00a0Pedipalpal tibia \u2161 spine tapering\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202688L. flabellarisWang & Li, 2011-\u00a0\u00a0Pedipalpal tibia \u2160 spine blunt\u2026L. palmata Lin & Li, 201088\u00a0\u00a0Distal edge of median apophysis with 5 teeth, conductor short, C shaped\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. kanellisi -\u00a0\u00a0Distal edge of median apophysis with 7 teeth, conductor long, triangular shaped\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202689\u00a0\u00a0Pedipalpal tibia with clusters of short spines dorsally\u2026\u202690-\u00a0\u00a0Pedipalpal tibia without clusters of short spines dorsally\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202691L. encun Wang & Li sp. nov.90\u00a0\u00a0Distal edge of median apophysis linear, with 8 teeth\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. robustispina -\u00a0\u00a0Distal edge of median apophysis semicircular, with 12 teeth\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202691\u00a0\u00a0Base of pedipalpal tibia swollen\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202692-\u00a0\u00a0Base of pedipalpal tibia not swollen\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u20269692\u00a0\u00a0Pedipalpal tibia \u2160 spine bifurcate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202693L. notabilis -\u00a0\u00a0Pedipalpal tibia \u2160 spine trifurcate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202693\u00a0\u00a0Conductor triangular, longer than median apophysis, median apophysis with 7 teeth\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202694-\u00a0\u00a0Conductor C shaped, shorter than median apophysis, median apophysis with 6 teeth\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202695L. shuang Wang & Li sp. nov.94\u00a0\u00a0Spermathecae not twisted distally\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. sanchahe Wang & Li nom. nov.-\u00a0\u00a0Spermathecae twisted distally\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. sexdigiti 95\u00a0\u00a0Spermathecae weakly twisted\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. lihu Wang & Li sp. nov.-\u00a0\u00a0Spermathecae strongly twisted\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202696\u00a0\u00a0Pedipalpal tibia \u2160 spine strongset\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202697-\u00a0\u00a0Pedipalpal tibia \u2160, \u2161, \u2162 spines equally strong, stronger than other spines\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202610397\u00a0\u00a0Pedipalpal tibia \u2161 spine bifurcate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202698-\u00a0\u00a0Pedipalpal tibia \u2160 spine not bifurcate\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202610198\u00a0\u00a0Pedipalpal tibia \u2161-\u2164 spine slender, curved, equally strong\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202699-\u00a0\u00a0Pedipalpal tibia \u2161-\u2164 spine not equally strong\u2026\u2026\u2026\u2026100L. arvanitidisi Wang & Li, 201699\u00a0\u00a0Distal edge of median apophysis with 6 teeth\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. erlong Wang & Li sp. nov.-\u00a0\u00a0Distal edge of median apophysis with 5 teeth\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. tawo Wang & Li sp. nov.100\u00a0\u00a0Distal edge of median apophysis with 4 teeth, tibia \u2161, \u2162 spines equally strong, stronger than other 2 spines\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. paragamiani Wang & Li, 2016-\u00a0\u00a0Distal edge of median apophysis with 3 teeth, tibia \u2162-\u2164 spines equally strong, slender than spine \u2161\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026101\u00a0\u00a0Pedipalpal tibia \u2161-\u2164 spines equally strong\u2026\u2026\u2026\u2026\u2026\u2026102L. deltshevi -\u00a0\u00a0Pedipalpal tibia \u2162-\u2164 spines equally strong, slender than spine \u2161, distal edge of median apophysis with 4 teeth\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. gittenbergeriWang & Li, 2011102\u00a0\u00a0Distal edge of median apophysis with 5 teeth, conductor C shaped\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. zhaiWang & Li, 2011-\u00a0\u00a0Distal edge of median apophysis with 6 teeth, conductor semicircular\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026L. thracia Gasparo, 2005103\u00a0\u00a0Pedipalpal tibia \u2160 \u2161 spines equally strong\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026-\u00a0\u00a0Pedipalpal tibia \u2161 and \u2162 spines equally strong\u2026\u2026\u2026104L. dabian Wang & Li sp. nov.104\u00a0\u00a0Distal edge of median apophysis with 3 teeth, tibia with 3 large spines prolaterally\u2026\u2026\u2026L. chuan Wang & Li sp. nov.-\u00a0\u00a0Distal edge of median apophysis with 6 teeth, tibia with 6 long setae prolaterally\u2026\u2026\u2026\u2026Type species:Leptonetela kanellisi  from Greece.Diagnosis. The genus Leptonetela can be distinguished from other leptonetid genera by the following combination of male pedipalpal characters: femur lacking spines and tibia with a longitudinal row of spines on the retrolateral surface.Redescription. Carapace yellowish or white. Sternum shield-shaped. Opisthosoma gray, ovoid, covered with short hairs. Male pedipalpal patella with one short spine dorso-distally; tibia with trichobothria dorsally; cymbium with strong, thorny spine distally; bulb yellowish, ovoid, with two appendages inserted ventrally, median apophysis chitinous, conductor membranous, median apophysis and conductor absent in some species, embolus transparent, membranous. Female genital area covered with short hairs. Vulva with a pair of spermathecae and sperm ducts, spermathecae twisted and weakly sclerotized.Distribution. Greece, Turkey, Georgia, Azerbaijan, Vietnam and China.Type material. Holotype: male (IZCAS), Chakou Cave, N27.93\u00b0, E106.14\u00b0, Shalang, Shibao Town, Gulin County, Luzhou City, Sichuan Province, China, 20 April 2014, Y. Li, H. Zhao & Y. Lin leg. Paratypes: 1 male and 3 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. dao Wang & Li sp. nov., L. grandispina Lin & Li, 2010, L. liuzhai Wang & Li sp. nov., L. pentakis Lin & Li, 2010, and L. shuilian Wang & Li sp. nov., but can be distinguished by the male pedipalpal tibia with 5 spines retrolaterally, the basal spine strong, conspicuous and with a bifurcate tip ; the median apophysis divided into 4 pine needle like structures ; from L. dao Wang & Li sp. nov., L. grandispina, L. pentakis by the conductor reduced in this new species (L. liuzhai Wang & Li sp. nov. by the cymbium 1.3 times longer than bulb (L. liuzhai Wang & Li sp. nov. and L. shuilian Wang & Li sp. nov.).cate tip  . Total length 2.25 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.27 (gth 2.27 -B. Carapgth 2.27 : spermatDistribution. China (Sichuan).Type material. Holotype: male (IZCAS), Dao Cave, N27.19\u00b0, E105.06\u00b0, Shuanglong, Salaxi County, Bijie City, Guizhou Province, China, 18 November 2011, H. Chen & Z. Zha leg. Paratypes: 1 male and 20 females, same data as holotype; 5 males and 7 females, Shanlanqiao Cave, N26.28\u00b0, E106.04\u00b0, Shanlanqiao, Qianyanqiao Town, Anshun City, Guizhou Province, China, 4 November 2011, H. Chen & Z. Zha leg.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. chakou Wang & Li sp. nov., L. grandispina Lin & Li, 2010, L. liuzhai Wang & Li sp. nov.L. pentakis Lin & Li, 2010, and L. shuilian Wang & Li sp. nov., but can be separated from L. chakou Wang & Li sp. nov., L. grandispina, L. liuzhai Wang & Li sp. nov. and L. pentakis by median apophysis divided into 2 pine needlelike ; from L. chakou Wang & Li sp. nov., L. grandispina by the tibial spines slender (L. chakou Wang & Li sp. nov. and \u2161 spines in L. grandispina strong); from L. chakou Wang & Li sp. nov., L. pentakis by the cymbium not constricted medially (L. liuzhai Wang & Li sp. nov. and L. shuilian Wang & Li sp. nov. by the cymbium 1.2 times longer than bulb (L. liuzhai Wang & Li sp. nov. and L. shuilian Wang & Li sp. nov.).edlelike  . Total length 2.28 . Similar to male in color and general features, but larger and with longer legs. Total length 2.76 (gth 2.76 -B. Carapgth 2.76 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), nameless Cave, N25.27\u00b0, E107.43\u00b0, Longli, Liuzhai Town, Nandan County, Hechi City, Guangxi Zhuang Autonomous Region, China, 29 January 2015, Y. Li & Z. Chen leg. Paratypes: 2 males and 6 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. chakou Wang & Li sp. nov., L. dao Wang & Li sp. nov., L. grandispina Lin & Li, 2010, L. pentakis Lin & Li, 2010, and L. shuilian Wang & Li sp. nov. but can be separated from L. chakou Wang & Li sp. nov., L. dao Wang & Li sp. nov., L. grandispina, and L. pentakis by the male pedipalpal cymbium double the length of bulb, median apophysis divided into 15 pine needle-like structures ; from L. chakou Wang & Li sp. nov., and L. grandispina by the tibial spines slender (L. chakou Wang & Li sp. nov. and \u2161 spines in L. grandispina strong); from L. chakou Wang & Li sp. nov., and L. pentakis by the cymbium not constricted medially in this new species . Total length 2.25 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.50 (gth 2.50 -B. Carapgth 2.50 : spermatDistribution. China (Guangxi).Type material. Holotype: male (IZCAS), Shuilian Cave, N24.43\u00b0, E106.97\u00b0, Pingle, Fengshan County, Hechi City, Guangxi Zhuang Autonomous Region, China, 22 March 2015, Y. Li & Z. Chen leg. Paratypes: 6 males and 4 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. chakou Wang & Li sp. nov., L. dao Wang & Li sp. nov., L. grandispina Lin & Li, 2010, L. pentakis Lin & Li, 2010, and L. liuzhai Wang & Li sp. nov. but can be separated from L. chakou Wang & Li sp. nov., L. dao Wang & Li sp. nov., L. grandispina Lin & Li, 2010, L. pentakis Lin & Li, 2010 by the male pedipalpal cymbium double the length of bulb; from L. chakou Wang & Li sp. nov., L. grandispina, L. liuzhai Wang & Li sp. nov. and L. pentakis by the median apophysis divided into 2 pine needlelike structures in L. chakou Wang & Li sp. nov. ; from L. chakou Wang & Li sp. nov. and L. grandispina by the tibial spines slender (L. chakou Wang & Li sp. nov. and \u2161 spines in L. grandispina strong); from L. chakou Wang & Li sp. nov. and L. pentakis by the cymbium not constricted medially in this new species.sp. nov.  . Total length 2.25 . Similar to male in color and general features, but smaller and with shorter legs. Total length 2.10 (gth 2.10 -B. Carapgth 2.10 : spermatDistribution. China (Guangxi).Type material. Holotype: male (IZCAS), Chenjia Cave, N28.38\u00b0, E108.67\u00b0, Tianba, Songtao County, Tongren City, Guizhou Prvince, China, 9 March 2013, H. Zhao & J. Liu leg. Paratypes: 1 male and 2 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. anshun Lin & Li, 2010, L. suae Lin & Li, 2010, L. tongzi Lin & Li, 2010, L. meitan Lin & Li, 2010, L. liangfeng Wang & Li sp. nov., and L. sanyan Wang & Li sp. nov., but can be distinguished by the male pedipalal tibia \u2160 spine far apart from the other 4 spines ; is also similar to L. huoyan Wang & Li sp. nov., but can be distinguished by the absent median apophysis, reduced conductor (L. huoyan Wang & Li sp. nov.).4 spines , conduct4 spines  . Total length 2.50 . Similar to male in color and general features, but smaller and with shorter legs. Total length 2.25 (gth 2.25 -B. Carapgth 2.25 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Liangfeng Cave, N28.32\u00b0, E107.84\u00b0, Tian, Fengle Town, Wuchuan County, Zunyi City, Guizhou Province, China, 7 August 2012, H. Zhao leg. Paratypes : 1 male and 2 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. anshun Lin & Li, 2010, L. suae Lin & Li, 2010, L. tongzi Lin & Li, 2010, L. meitan Lin & Li, 2010, L. chenjia Wang & Li sp. nov., and L. sanyan Wang & Li sp. nov., but can be distinguished by the male pedipalpal bulb conductor C shaped ; from L. anshun by the tibia \u2160 spine slender not bifurcated (L. anshun).C shaped  . Total length 2.28 . Similar to male in color and general features, but smaller and with shorter legs. Total length 2.14 (gth 2.14 -B. Carapgth 2.14 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Sanyan Cave, N29.15\u00b0, E107.60\u00b0, Heyi, Yangxi Town, Daozhen County, Guizhou Province, China, 30 May 2011, Z. Zha leg. Paratypes: 1 male and 2 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. anshun Lin & Li, 2010, L. suae Lin & Li, 2010, L. tongzi Lin & Li, 2010, L. meitan Lin & Li, 2010, L. chenjia Wang & Li sp. nov., and L. liangfeng Wang & Li sp. nov., but can be separated from all above except L. tongzi by in the male conductor C shaped in this new species ; from L. tongzi by in the female atrium triangular, anterior margin of the atrium undulate (L. tongzi). species  . Total length 1.78 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.03 (gth 2.03 -B. Carapgth 2.03 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Wangjia Cave, N26.98\u00b0, E107.94\u00b0, Gaoqi, Nongchang Town, Huangpin County, Guizhou Province, China, 4 March 2012, H. Zhao & J. Liu leg. Paratype: 1 female, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. danxia Lin & Li, 2010, and L. yaoiL. danxia, bamboo leaf-shaped in L. yaoi), from L. yaoi by the tibia \u2160 spine slender, asymmetrically bifurcated (L. yaoi); from L. danxia by the unwrinkled cymbium (L. danxia).furcated  . Total length 2.13 .Type material. Holotype: male (IZCAS), Qiangdao Cave, N25.83\u00ba, E109.04\u00ba, Guandong Town, Congjiang County, Qiandongnan Prefecture, Guizhou, China, 16 March 2013, H. Zhao & J. Liu leg. Paratypes: 3 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. furcaspina Lin & Li, 2010, L. langdong Wang & Li sp. nov. and L. dashui Wang & Li sp. nov., but can be distinguished by the male pedipalpal tibia \u2160 spine strong .e strong , conducte strong  . Total length 1.80 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.25 (gth 2.25 -B. Carapgth 2.25 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Menglonggong Cave, N27.07\u00b0, E107.76\u00b0, Langdong Village, Huangping County, Qiandongnan Prefecture, Guizhou Province, China, 3 March 2013, H. Zhao & J. Liu leg. Paratypes: 1 male and 2 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. furcaspina Lin & Li, 2010, L. qiangdao Wang & Li sp. nov., and L. dashui Wang & Li sp. nov., but can be distinguished by the male pedipalpal tibia \u2160 spine strong, tip curved .p curved . conductp curved  : total length 2.25 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.25 (gth 2.25 -B. Carapgth 2.25 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Dashui Cave, N26.61\u00b0, E106.61\u00b0, Shijicheng, Jinyang New Urban Area, Guiyang City, Guizhou Province, China, 18 June 2011, Z. Zha leg. Paratypes: 1 male and 2 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. furcaspina Lin & Li, 2010, L. qiangdao Wang & Li sp. nov., and L. langdong Wang & Li sp. nov., but can be distinguished by the slender male pedipalpal tibia \u2160 spine, \u2161 \u2162 spines curved basally . basally , conduct basally , . Total length 1.88 . Similar to male in color and general features, but larger and with shorter legs. Total length 1.93 (gth 1.93 -B. Carapgth 1.93 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Gang Cave, N26.87\u00b0, E108.91\u00b0, Tunhou, Nanming Town, Jianhe County, Kaili City, Guizhou Province, China, 15 December 2011, Z. Zha leg. Paratypes: 15 males and 6 females, same data as holotype; 4 males and 5 females, Long Cave, N26.85\u00b0, E108.79\u00b0, Longtang, Liangshang Town, Sansui County, Kaili City, Guizhou Province, China, 18 December 2011, Z. Zha leg; 5 males and 5 females, Shenxian Cave, N26.87\u00b0, E108.89\u00b0, Shixing, Xiaolan Country, Nanming Town, Jianhe County, Kaili City, Guizhou Province, China, 16 December 2011, Z. Zha leg; 5 females, Niu Cave, N26.86\u00b0, E108.93\u00b0, Cenge, Nanming Town, Jianhe County, Kaili City, Guizhou Province, China, 14 December 2011, Z. Zha leg.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. jiulong Lin & Li, 2010, but can be distinguished by the male pedipalpal tibia with 6 spines retrolaterally, tibia \u2161 spine thickest, \u2160, \u2161 spines equally length, tibia \u2161 spine asymmetrically bifurcated (L. jiulong).furcated , median furcated  . Total length 2.63 . Similar to male in color and general features, but smaller and with shorter legs. Total length 2.38 (gth 2.38 -B. Carapgth 2.38 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), La Cave in Xiaoyakou, N25.80\u00b0, E104.95\u00b0, Puan County, Qianxinan Prefecture, Guizhou Province, China, 14 July 2012, H. Zhao leg. Paratypes: 3 males and 5 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. rudong Wang & Li sp. nov. and L. wenzhu Wang & Li sp. nov. but can be distinguished from L. wenzhu Wang & Li sp. nov. by the male pedipalpal tibia with 7 spines retrolaterally ; from L. rudong Wang & Li sp. nov. by the tibia with 4 long setae prolaterally (L. rudong Wang & Li sp. nov.); from L. rudong Wang & Li sp. nov., and L. wenzhu Wang & Li sp. nov. by the conductor broad, C shaped .aterally  . Total length 2.97 . Similar to male in color and general features, but smaller and with shorter legs. Total length 2.81 (gth 2.81 -B. Carapgth 2.81 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Rudong Cave, N25.57\u00b0, E110.62\u00b0, Longpan Mountain, Dongtian, Xing'an County, Guilin City, Guangxi Zhuang Autonomous Region, China, 11 July 2009, C. Wang & Z. Yao leg. Paratypes: 1 male and 3 females, same data as holotype; 3 females, Gouya Cave, N25.46\u00b0, E110.11\u00b0, Hufeng, Guanyang County, Guilin City, Guangxi Zhuang Autonomous Region, China, 30 August 2009, C. Wang & Z. Yao leg; 2 females, Jiulong Cave, N25.46\u00b0, E110.09\u00b0, Shifeng, Guanyang County, Guilin City, Guangxi Zhuang Autonomous Region, China, 30 August 2009, C. Wang & Z. Yao leg.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. la Wang & Li sp. nov., and L. wenzhu Wang & Li sp. nov. but can be distinguished by the male pedipalpal tibia with 2 long setae, 2 spines prolaterally, 1 long seta, and 6 spines retrolaterally, cymbium with 1 horn-shaped spine on the earlobe-shaped process . process -D, condu process  . Total length 2.12 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.15 , Wenzhu Cave, N25.44\u00b0, E105.13\u00b0, Longchang Town, Xingren City, Guizhou Province, China, 16 July 2012, H. Zhao leg. Paratypes: 1 male and 2 females, same data as holotype; 4 females, Xiaoya Cave, N25.44\u00b0, E105.13\u00b0, Yaqiao Town, Xingren City, Guizhou Province, China, 16 July 2012, H. Zhao leg.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. rudong Wang & Li sp. nov., and L. la Wang & Li sp. nov. but can be distinguished by the male pedipalpal tibia with 6 spines retrolaterally .aterally , conductaterally  . Total length 2.63 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.88 (gth 2.88 -B. Carapgth 2.88 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Underground River, N25.27\u00b0, E107.44\u00b0, Longli, Liuzhai Town, Nandan County, Hechi City, Guangxi Zhuang Autonomous Region, China, 29 January 2015, Y. Li & Z. Chen leg. Paratypes: 3 males and 4 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. chiosensisL. panbao Wang & Li sp. nov., but can be distinguished by the male pedipalpal tibia \u2160, \u2161 and \u2162 spines equally strong .y strong , conducty strong  . Total length 1.88 . Similar to male in color and general features, but larger and with shorter legs. Total length 1.95 (gth 1.95 -B. Carapgth 1.95 : spermatDistribution. China (Guangxi).Type material. Holotype: male (IZCAS), Panbao Cave, N28.38\u00b0, E108.67\u00b0, Panbao, Shichang Town, Songtao County, Tongren City, Guizhou Province, China, 8 March 2013, H. Zhao & J. Liu leg. Paratypes: 2 male and 4 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. chiosensisL. Longli Wang & Li sp. nov., but can be distinguished by the male pedipalpal tibial spines slender, equally strong, cymbium with 1 strong spine on the earlobe-shaped process . process , conduct process  . Total length 2.38 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.50 (gth 2.50 -B. Carapgth 2.50 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Feilong Cave, N26.44\u00b0, E107.02\u00b0, Longli Town, Qiannan Prefecture, Guizhou Province, China, 27 July 2012, H. Zhao leg. Paratypes: 9 females, same data as holotype; 1 female, Lianhua Cave, N26.43\u00b0, E106.95\u00b0, Lianhua Town, Qiannan Prefecture, Guizhou Province, China, 27 July 2012, H. Zhao leg.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. yangi Lin & Li, 2010 and L. jiahe Wang & Li sp. nov., but can be distinguished from L. yangi by the male pedipalpal cymbium constricted medially, attaching an earlobe-shaped process (L. yangi), from L. jiahe Wang & Li sp. nov. by the median apophysis \"m\"-shaped, conductor triangular . process , conduct process  . Total length 2.31 . Similar to male in color and general features, but smaller and with shorter legs. Total length 2.13 (gth 2.13 -B. Carapgth 2.13 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Jiahe Cave, N25.25\u00b0, E110.20\u00b0, Lingui Town, Lingui County, Guilin City, Guangxi Zhuang Autonomous Region, China, 20 December 2013, H. Zhao leg. Paratypes: 3 males and 5 females, same data as holotype; 6 males and 5 females, Flytiger Cave, N25.25\u00b0, E110.20\u00b0, Lingui Town, Lingui County, Guilin City, Guangxi Zhuang Autonomous Region, China, 20 December 2013, H. Zhao leg.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. yangi Lin & Li, 2010, and L. feilong Wang & Li sp. nov., but can be distinguished by the male pedipalpal cymbium with 1 short spine on the earlobe-shaped process, and 1 curved, long spine retrolaterally .aterally , median aterally  . Total length 2.43 : tibia with 5 slender spines retrolaterally, spines \u2160, \u2161 and \u2162 equally strong, stronger than other spines, spines \u2160 longest. Cymbium constricted medially, retrolaterally attaching to 1 curved spine and an earlobe-shaped process, with 1 short spine. Embolus triangular, and prolateral lobe absent. Median apophysis shaped like pointed process, with 3 sclerotized spots distally. Conductor C shaped . Similar to male in color and general features, but smaller and with shorter legs. Total length 2.30 (gth 2.30 -B. Carapgth 2.30 : spermatDistribution. China (Guangxi).Type material. Holotype: male (IZCAS), Xianren Cave, N29.73\u00b0, E110.31\u00b0, Yvpingxini, Zouma Town, Hefeng County, Enshi Tujia and Miao Autonomous Prefecture, Hubei Province, China, 27 January 2011, Y. Li & J. Liu leg. Paratypes: 2 males and 3 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. liping Lin & Li, 2010, and L. parlongaL. liping, 6 slender spines in L. parlonga); median apophysis triangular ; from L. parlonga by the cymbium retrolaterally with 1 horn-shaped spine on the earlobe-shaped process in L. parlonga.iangular  . Total length 2.23 . Similar to male in color and general features but larger and with shorter legs. Total length 2.38 (gth 2.38 -B. Carapgth 2.38 : spermatDistribution. China (Hubei).Type material. Holotype: male (IZCAS), Tiankeng Cave, N26.64\u00b0, E104.80\u00b0, Hegou, Dewu Town, Zhongshan County, Liupanshui City, Guizhou Province, China, 9 November 2011, H. Chen & Z. Zha leg. Paratypes: 4 males and 5 females, same data as holotype; 2 females, Luoshui Cave, N26.64\u00b0, E104.80\u00b0, Hegou, Dewu Town, Zhongshan County, Liupanshui City, Guizhou Province, China, 9 November 2011, H. Chen & Z. Zha leg.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. rudiculaL. rudicula).Description. Male (holotype). Total length 2.03 . Similar to male in color and general features, but smaller and with shorter legs. Total length 1.93 (gth 1.93 -B. Carapgth 1.93 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Mayang Cave, N28.55\u00b0, E108.06\u00b0, Quankou, Dejiang County, Tongren City, Guizhou Province, China, 10 August 2012, H. Zhao leg. Paratype: 1 female, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species can be distinguished from all other species of the genus by the male pedipalpal cymbium with one curved, short spine medially in retrolateral view, median apophysis triangular, spermathecae not tightly twisted, just spiraled in the female.Description. Male (holotype). Total length 2.13 .Type material. Holotype: male (IZCAS), Gubin River, N26.50\u00b0, E107.52\u00b0, Gubin, Xingshan Town, Majiang County, Shengkaili City, Guizhou Province, China, 28 November 2011, H. Chen & Z. Zha leg. Paratypes: 22 males and 14 females, same data as holotype; 4 males and 5 females, nameless Cave, N26.50\u00b0, E107.52\u00b0, Gubin, Xingshan Town, Majiang County, Shengkaili City, Guizhou Province, China, 28 November 2011, H. Chen & Z. Zha leg.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. jinsha Lin & Li, 2010, L. quinquespinata , L. xinhua Wang & Li sp. nov., L. lujia Wang & Li sp. nov. and L. xinhua Wang & Li sp. nov. but can be distinguished by the male pedipalpal tibia with 4 slender spines prolaterally, 5 slender spines retrolaterally, with spines \u2160, \u2161 equal length, cymbium with 2 long curved spines on earlobe-shaped process retrolaterally ; from L. jinsha, L. lujia Wang & Li sp. nov. and L. xinhua Wang & Li sp. nov. by the semicircular conductor, base of median apophysis distinctly swollen, 4 times wider than the tip .aterally  . Total length 1.88 . Similar to male in color and general features, but larger and with longer legs. Total length 2.30 (gth 2.30 -B. Carapgth 2.30 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Wuming Cave, N26.48\u00ba, E107.54\u00ba, Lujia Bridge, Gubin, Xingshan Town, Majiang County, Kaili City, Guizhou Province, China, 29 November 2011, H. Chen & Z. Zha leg. Paratypes: 1 male and 2 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. jinsha Lin et Li, 2010, L. quinquespinata , L. xinhua Wang & Li sp. nov. and L. gubin Wang & Li sp. nov. but can be distinguished by the male pedipalpal tibia with 4 long setae prolaterally, 5 slender spines retrolaterally, with spines \u2160 longest, spines \u2161 \u2162 equal length ; from L. gubin and L. quinquespinata by the slightly swollen base of the median apophysis .l length , conductl length , . Total length 1.72 . Similar to male in color and general features, but larger and with shorter legs. Total length 1.70 (gth 1.70 -B. Carapgth 1.70 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), nameless Cave, N27.85\u00ba, E111.31\u00ba, Caojia Town, Xinhua County, Loudi City, Hunan Province, China, 24 March 2016, Y. Li & Z. Chen leg. Paratypes: 3 males and 2 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. jinsha Lin & Li, 2010, L. quinquespinata , L. lujia Wang & Li sp. nov., and L. gubin Wang & Li sp. nov., but can be distinguished by the bifurcated embolus, male pedipalpal tibia with 5 slender spines prolaterally, 5 slender spines retrolaterally, conductor triangular ; from L. gubin and L. quinquespinata by the base of median apophysis slightly swollen .iangular , : total length 1.78 : similar to male in color and general features, but with a larger body size and shorter legs. Total length 1.95 (gth 1.95 -B. Prosogth 1.95 : spermatDistribution. China (Hunan).Type material. Holotype: male (IZCAS), Kangsagulie Cave, N26.79\u00ba, E108.21\u00ba, Datang, Geyi Town, Taijiang County, Kaili City, Guizhou Province, China, 5 December 2011, H. Chen & Z. Zha leg. Paratypes: 7 males and 6 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. shibingensis and L. wuming Wang & Li sp. nov., but can be distinguished by the median apophysis index finger-like in prolaterally view, tip bifurcated ; from L. wuming Wang & Li sp. nov., by the tibia \u2160 spine located at the middle of tibia, embolus with 1 basal tooth .furcated  . Total length 2.07 . Similar to male in color and general features, but smaller and with shorter legs. Total length 2.02 (gth 2.02 -B. Carapgth 2.02 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Wuming Cave, N25.43\u00b0, E105.62\u00b0, Dabei Town, Zhenfeng County, Guizhou Province, China, 18 July 2012, H. Zhao leg. Paratypes: 2 males and 5 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. kangsa Wang & Li sp. nov., and L. shibingensis but can be distinguished by the male pedipalpal bulb embolus without basal tooth (L. kangsa Wang & Li sp. nov. and L. shibingensis by the tibia spines \u2160 located at the base of the tibia (L. kangsa Wang & Li sp. nov. and L. shibingensis).al tooth , . Total length 1.50 . Similar to male in color and general features, but larger and with shorter legs. Total length 3.02 (gth 3.02 -B. Carapgth 3.02 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Shanji Cave, N27.28\u00b0, E107.82\u00b0, Xiaguihua, Xiaosai Town, Yuqing County, Zunyi City, Guizhou Province, China, 15 August 2012, H. Zhao leg. Paratypes: 3 males and 3 females, same data as holotype; 2 females, Guanyin Cave, N27.32\u00b0, E107.71\u00b0, Hongjun, Longxi Town, Yuqing County, Zunyi City, Guizhou Province, China, 15 August 2012, H. Zhao leg; 3 females, Liangfeng Cave, N27.27\u00b0, E107.76\u00b0, Xiaosai Town, Yuqing County, Zunyi City, Guizhou Province, China, 14 August 2012, H. Zhao leg.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. digitata Lin & Li, 2010, L. hamata Lin & Li, 2010 and L. tetracantha Lin & Li, 2010, but can be distinguished by the male pedipalpal tibia spines \u2160 strong, located medially ; from L. digitata by themedian apophysis not curved (median apophysis curved in L. digitata).medially  . Total length 2.08 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.40 (gth 2.40 -B. Carapgth 2.40 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Gejiaxiaoyan Cave, N27.11\u00b0, E105.24\u00b0, Shanjiao, Zhuchang Town, Bijie City, Guizhou Province, China, 27 January 2011, H. Chen & Z. Zha leg. Paratypes: 2 males and 6 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. curvispinosa Lin & Li, 2010, but can be distinguished by the male pedipalpal tibia with 4 large spines prolaterally, 6 large spines retrolaterally (L. curvispinosa).aterally , median aterally  . Total length 1.67 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.12 (gth 2.12 -B. Carapgth 2.12 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Heyuantou nameless Cave, N29.25\u00b0, E109.35\u00b0, Huoyan Street, Guitang Dam Town, Longshan County, Hubei Province, China, 15 January 2014, Y. Li & Y. Lin leg. Paratypes: 1 male and 2 females, same data as holotype; 1 male and 4 females, nameless Cave, N29.61\u00b0, E109.17\u00b0 Jieping, Xianfeng County, Enshi Tujia and Miao Autonomous Prefecture, Hubei Province, China, 17 January 2014, Y. Li & Y. Lin leg.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. anshun Lin & Li, 2010 and L. chenjia Wang & Li sp. nov., but can be distinguished by on the male pedipalpal bulb median apophysis slightly sclerotized, index finger like, conductor broad, semicircular ; from L. anshun by the tibia spines \u2160 slender (L. anshun).circular  . Total length 2.25 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.53 (gth 2.53 -B. Carapgth 2.53 : spermatDistribution. China (Hubei).Type material. Holotype: male (IZCAS), Liuguan Cave, N26.15\u00b0, E106.46\u00b0, Mengqiu, Baiyunshan Town, Changshun County, Guizhou Province, China, 23 December 2010, Z. Zha & Z. Chen leg. Paratypes: 2 female, same data as holotype; 1 male, Fenghuang Cave, N26.09\u00b0, E106.39\u00b0, Shenglian, Zhonghuo Town, Changshun County, Guizhou Province, China, 23 December 2010, Z. Zha & Z. Chen leg.Etymology. The specific name refers to the typelocality; noun.Diagnosis. This new species is similar to L. penevi Wang & Li, 2016 and L. changtu Wang & Li sp. nov. but can be distinguished by on the male pedipalpal bulb median apophysis long, and half the length of bulb ; male pedipalpal tibia spines slender, equally strong . of bulb  . Total length 1.88 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.08 (gth 2.08 -B. Carapgth 2.08 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Nanmu Cave, N28.10\u00b0, E110.08\u00b0, Pushi Town, Luxi County, Hunan Province, China, 5 April 2016, Y. Li & Z. Chen leg. Paratypes: 3 males and 2 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. tianxingensis, but can be distinguished by on the male pedipalpal bulb conductor longer than median apophysis (L. tianxingensis); male pedipalpal tibia \u2162 spine strong (L. tianxingensis).pophysis  : total length 1.70 : similar to male in color and general features, but with a larger body size and longer legs. Total length 1.98 (gth 1.98 -B. Prosogth 1.98 : spermatDistribution. China (Hunan).Type material. Holotype: male (IZCAS), Changtu Cave, N27.14\u00b0, E105.43\u00b0, Honglin, Qianxi Town, Bijie County, Guizhou Province, China, 23 November 2011, Z. Zha & Z. Zha leg. Paratypes: 1 male and 10 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. penevi Wang & Li, 2016 and L. liuguan Wang & Li sp. nov. but can be distinguished by the male pedipalpal tibia spines \u2160, \u2161, \u2162 equally strong, stronger than other two spines ; from L. liuguan Wang & Li sp. nov. by median apophysis short, 1/5 the length of bulb (L. liuguan Wang & Li sp. nov.); from L. penevi by the cymbium not constricted .o spines  . Total length 2.33 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.70 (gth 2.70 -B. Carapgth 2.70 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Lianhua Cave, N25.48\u00b0, E114.09\u00b0, Niedou Town, Chongyi County, Jiangxi Province, China, 24 April 2013, Y. Luo & J. Liu leg. Paratypes: 3 males and 10 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. niubizi Wang & Li sp. nov. but can be distinguished by the male pedipalpal tibia with 5 spines retrolaterally, with \u2160 spine strongest, tip bifurcated, the other 4 spines slender, 2 of them longer than \u2160 spine (L. niubizi Wang & Li sp. nov.). \u2160 spine ; tip of  \u2160 spine  . Total length 2.00 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.25 (gth 2.25 -B. Carapgth 2.25 : spermatDistribution. China (Jiangxi).Type material. Holotype: male (IZCAS), Niubizi Cave, N27.62\u00b0, E106.67\u00b0, Leshan Town, Zunyi County, Zunyi City, Guizhou Province, China, 1 August 2012, H. Zhao leg. Paratypes: 7 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. lianhua Wang & Li sp. nov. but can be distinguished by the male pedipalp tibia with 5 slender spines retrolaterally, spines \u2160 longest, not bifurcated (L. lianhua Wang & Li sp. nov.).furcated , median furcated  . Total length 2.53 . Similar to male in color and general features, but with a larger body size and shorter legs. Total length 2.60 (gth 2.60 -B. Carapgth 2.60 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Longyu Cave, N29.40\u00b0, E110.09\u00b0, Cili County, Hunan Province, China, 5 June 2011, Z. Zha leg. Paratypes: 4 males and 5 females, same data as holotype, 5 males and 6 females, Niuerduo Cave, N29.404\u00b0, E110.73\u00b0, Cili County, Hunan Province, China, 9 April 2016, Y. Li & Z. Chen leg.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. sexdentataL. shicheng Wang & Li sp. nov., L. zakou Wang & Li sp. nov. and L. meiwang Wang & Li sp. nov. but can be distinguished by median apophysis harrow-like, tip with 5 small teeth ; from L. shicheng Wang & Li sp. nov. by the tip of conductor undulate (L. shicheng Wang & Li sp. nov.); from L. zakou Wang & Li sp. nov. by the teeth of median apophysis needle-shaped in L. zakou Wang & Li sp. nov.; from L. meiwang Wang & Li sp. nov. by the tibia spines \u2160 strongest, tip asymmetrically bifurcated (tibia spines \u2161 strongest in L. meiwang Wang & Li sp. nov.).ll teeth  . Total length 1.63 . Similar to male in color and general features, but larger and with longer legs. Total length 2.05 (gth 2.05 -B. Carapgth 2.05 : spermatDistribution. China (Hunan).Type material. Holotype: male (IZCAS), Shicheng Cave, N27.31\u00b0, E109.07\u00b0, Jiangwu, Shanshi Town, Lianhua County, Pingxiang City, Jiangxi Province, China, 14 November 2015, Z. Chen & G. Zhou leg. Paratypes: 2 males and 5 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. sexdentataL. longyu Wang & Li sp. nov., L. zakou Wang & Li sp. nov. and L. meiwang Wang & Li sp. nov. but can be distinguished by the harrow-like median apophysis, with 10 small teeth distally ; conductor smooth ; from L. zakou Wang & Li sp. nov. by the teeth of median apophysis needle-shaped in L. zakou Wang & Li sp. nov.; from L. meiwang Wang & Li sp. nov. by the tibia spines \u2160 strongest, tip asymmetrically bifurcated (L. meiwang Wang & Li sp. nov.).distally  . Total length 2.40 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.60 (gth 2.60 -B. Carapgth 2.60 : spermatDistribution. China (Jiangxi).Type material. Holotype: male (IZCAS), Zakou Cave, N29.35\u00b0, E109.58\u00b0, Hongyanxi Town, longshan City, Hunan Province, China, 10 January 2016, Z. Chen & Z. Wang leg. Paratypes: 3 males and 5 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. sexdentataL. longyu Wang & Li sp. nov., L. shicheng Wang & Li sp. nov., and L. meiwang Wang & Li sp. nov. but can be distinguished by on the male pedipalpal bulb median apophysis with 6 teeth, needle-shaped ; from L. shicheng Wang & Li sp. nov. by the distally undulate conductor (L. shicheng Wang & Li sp. nov.); from L. meiwang Wang & Li sp. nov. by the tibia \u2160 spine strongest, tip asymmetrically bifurcated (L. meiwang Wang & Li sp. nov.).e-shaped  . Total length 1.75 . Similar to male in color and general features, but larger and with shorter legs. Total length 1.70 (gth 1.70 -B. Carapgth 1.70 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Meiwang Cave, N28.09\u00b0, E111.43\u00b0, Nanhua, Zhenshang Town, Lodi County, HuNan Province, China, 27 March 2016, Y. Li & Z. Chen leg. Paratypes: 1 male and 1 female, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. sexdentataL. longyu Wang & Li sp. nov., L. shicheng Wang & Li sp. nov. and L. zakou Wang & Li sp. nov. but can be distinguished by the harrow-like median apophysis, with 5 sharp teeth distally ; conductor short, reduced ; from L. zakou Wang & Li sp. nov. by the teeth of median apophysis needle-shaped in L. zakou Wang & Li sp. nov.distally , tibia \u2161distally  : total length 1.75 .Type material. Holotype: male (IZCAS), Xianren Cave, N29.18\u00b0, E109.95\u00b0, Xianren, Tawo Town, Yongshun County, Hunan Province, China, 14 January 2016, Z. Chen & Z. Wang leg. Paratypes: 2 males and 2 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. arvanitidisi Wang & Li, 2016, L. paragamiani Wang & Li, 2016 and L. erlong Wang & Li sp. nov. but can be distinguished by on the male pedipalpal bulb median apophysis with 4 teeth distally ; tibia spines \u2160 strongest, tip asymmetrically bifurcated, spines \u2161, \u2162 equally strong, stronger than other 2 ; from L. arvanitidisi by the conductor C tile-shaped (L. arvanitidisi).distally  . Total length 1.90 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.00 (gth 2.00 -B. Carapgth 2.00 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Erlong Cave, N27.82\u00b0, E110.23\u00b0, Siqian Town, Chenxi County, Huaihua City, Hunan Province, China, 19 March 2016, Y. Li & Z. Chen leg. Paratypes: 4 males and 2 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. arvanitidisi Wang & Li, 2016, L. paragamiani Wang & Li, 2016 and L. tawo Wang & Li sp. nov. but can be distinguished by on the male pedipalpal bulb median apophysis with 5 teeth distally ; from L. paragamiani and L. tawo Wang & Li sp. nov. by the tibia spines \u2161 -\u2164 slender, curved, and equally strong ; from L. arvanitidisi by the conductor C tile-shaped (L. arvanitidisi).distally  : total length 1.95 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.30 (gth 2.30 -B. Prosogth 2.30 : spermatDistribution. China (Hunan).Type material. Holotype: male (IZCAS), Wuming Cave, N25.75\u00b0, E107.92\u00b0, Dabian, Sandong Town, Sandu County, Qiannan Prefecture, Guizhou, China, 22 March 2013, H. Zhao & J. Liu leg. Paratypes: 2 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. thraciaL. chuan Wang & Li sp. nov., but can be distinguished by the male pedipalal tibia with 3 spines prolaterally, 5 slender spines, retrolaterally ; tip of median apophysis bent upwards, with 3 larger teeth distally ; conductor thin, tongue shaped (L. thracia and L. chuan Wang & Li sp. nov.).aterally -D . Total length 2.38 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.40 (gth 2.40 -B. Carapgth 2.40 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Chuan Cave, N27.08\u00b0, E105.67\u00b0, Yangchangba Town, Dafang County, Guizhou Province, China, 13 March 2011, H. Chen & Z. Zha leg. Paratype: 1 female, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. thraciaL. dabian Wang & Li sp. nov., but can be distinguished by the male pedipalpal tibia with 7 long setae prolaterally, 5 slender spines retrolaterally, with spines \u2160, \u2161, \u2162 equally strong, stronger than others ; tip of median apophysis bent downwards, with 5 larger teeth distally ; from L. dabian Wang & Li sp. nov. by the triangular conductor (L. dabian Wang & Li sp. nov.).n others  . Total length 2.10 .Type material. Holotype: male (IZCAS), nameless Cave, N25.10\u00b0, E107.65\u00b0, Lihu Town, Nandan County, Hechi City, Guangxi Zhuang Autonomous Region, China, 31 January 2015, Y. Li & Z. Chen leg. Paratypes: 2 males and 5 females, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. notabilis , L. sexdigiti ; and L. shuang Wang & Li sp. nov., but can be separated from L. notabilis by the male pedipalpal tibia spines \u2160 bifurcate (L. notabilis); from L. shuang Wang & Li sp. nov. by the conductor C tile-shaped, distal edge of median apophysis with 6 teeth (L. shuang Wang & Li sp. nov.); from L. sexdigiti by the strongly twisted spermathecae (spermathecae loosely twisted in L. sexdigiti).ifurcate  . Total length 2.13 . Similar to male in color and general features, but larger and with shorter legs. Total length 2.50 (gth 2.50 -B. Carapgth 2.50 : spermatDistribution. China (Guangxi).Type material. Holotype: male (IZCAS), Shuang Cave, N25.93\u00b0, E107.26\u00b0, Bailong Town, Pingtang County, Qiannan Prefecture, Guizhou Province, China, 24 July 2012, H. Zhao leg. Paratypes: 2 females, same data as holotype; 2 males and 6 females, Dongkou Cave, N25.93\u00b0, E107.25\u00b0, Longxiang, Bailong Town, Pingtang County, Qiannan Prefecture, Guizhou Province, China, 25 July 2012, H. Zhao leg.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. notabilis , L. sexdigiti , and L. lihu Wang & Li sp. nov., but can be separated from L. notabilis by the male pedipalp tibia spines \u2160 bifurcate (L. notabilis); from L. sexdigiti and L. lihu Wang & Li sp. nov. by the conductor triangular, distal edge of median apophysis with 7 teeth (L. sexdigiti and L. lihu Wang & Li sp. nov.); from L. sexdigiti by in the female spermathecae strongly twisted (L. sexdigiti).ifurcate  . Total length 2.00 . Similar to male in color and general features, but smaller and with shorter legs. Total length 1.98 (gth 1.98 -B. Carapgth 1.98 : spermatDistribution. China (Guizhou).Type material. Holotype: male (IZCAS), Encun Cave, N25.08\u00b0, E107.59\u00b0, En, Chengguan Town, Nandan County, Hechi City, Guangxi Zhuang Autonomous Region, China, 30 January 2015, Y. Li & Z. Chen leg. Paratypes: 1 male and 1 female, same data as holotype.Etymology. The specific name refers to the type locality; noun.Diagnosis. This new species is similar to L. robustispina  but can be distinguished by the male pedipalpal tibia with 5 spines retrolaterally, with spines \u2160 longest, spines \u2160, \u2161, \u2162 equally strong, stronger than others (L. robustispina).n others , distal n others  . Total length 2.00 .Leptonetela zhaiMaterial examined. 4 females (IZCAS), Rudong Cave, N25.57\u00b0, E110.62\u00b0, Longpan Mountain, Dongtian, Hucheng Town, Xing'an County, Guilin City, Guangxi Zhuang Autonomous Region, China, 08 November 2012, Z. Chen & Z. Zhao leg.Description. Male. See Female. Total length 2.12 (gth 2.12 -B. Carapgth 2.12 : spermatDistribution. China (Guangxi).Remarks. The female of the species is described for the first time. Females of Leptonetelazhai were collected from the same cave where the male holotype of L. zhaiLeptoneta tianxinensis Tong & Li, 2008: 382, Type material examined. Paratypes: 12 males, 6 females (IZCAS), Tianxin Cave, N33.35\u00b0, E111.88\u00b0, Sandaohe, Qilipo Town, Neixiang County, Henan Province, China, 24 June 2005, Q. Wang & Y. Tong leg.Remarks. Our research showed that this species should be transferred to the genus Leptonetela, based on the result of DNA barcoding and morphological characters such as the pedipalpal femur lacking spines and the tibia with one strong spine retrolaterally.Guineta gigachela Lin & Li, 2010: 6, Type material examined. Holotype: male (IZCAS), Qingzi Cave, N26.51\u00b0, E107.99\u00b0, Mianxi, Sankeshu Town, Kaili City, Guizhou Province, China, 26 May 2007, Y. Li & J. Liu leg. Paratypes: 2 males and 12 females, same data as holotype.Remarks. Our studies showed that that Guineta Lin & Li, 2010 syn. nov. should be a junior synonym of Leptonetela Kratochv\u00edl, 1978.Sinoneta notabilis Lin & Li, 2010: 83, Type material examined. Holotype: male (IZCAS), Hebiandong Cave, Kaikou Town, Duyun City, N26.00\u00b0, E107.20\u00b0, Guizhou Province, China, 8 May 2006, Y. Li leg. Paratypes: 1 male and 1 female, same data as holotype.Remarks. Our studies showed that Sinoneta Lin & Li, 2010 syn. nov. should be a junior synonym of Leptonetela Kratochv\u00edl, 1978.Sinoneta sexdigiti Lin & Li, 2010: 87, Type material examined. Holotype: male (IZCAS), Qiaotou Cave, Dashan, Shuangliu Town, Kaiyang County, N26.05\u00b0, E107.85\u00b0, Guizhou Province, China, 11 May 2006, Y. Li & Z. Yang leg. Paratypes: 5 males and 29 females, same data as holotype.Qianleptoneta palmata Chen, Jia & Wang, 2010: 2902, Sinoneta palmataQianleptoneta).Material examined. 1 male and 1 female (IZCAS), Sanchahe Cave, N26.53\u00b0, E107.70\u00b0, Sanchahe, Jialiang Town, Libo County, Guizhou Province, China, 16 May 2011, C. Wang & L. Lin leg.Etymology. The specific name refers to the type locality; noun.Remarks. Qianleptoneta palmata was collected from Sanchahe Cave in Guizhou, China and published by Chen et al. in December 2010. Qianleptoneta palmata Chen et al, 2010 to the genus Sinoneta Lin & Li, 2010. Nevertheless, in this study our results confirmed that Qianleptoneta palmata belonged to the genus Leptonetela.Leptonetela palmata is a preoccupied name (secondary homonym) for a species collected from Dixian Cave in Guizhou, China and published by Lin & Li, in August 2010. Subsequently, Leptonetela sanchahe Wang & Li nom. nov. is proposed for the taxon from Sanchahe Cave, in Guizhou, China.Yi Wu and Guo Zheng helped to prepare photos of the manuscript. Sarah C. Crews kindly checked the English of the manuscript. Peter J\u00e4ger, Yanfeng Tong and Yucheng Lin provided valuable comments on an early version of the manuscript."}
{"text": "AbstractCHIOC), comprising types deposited between 1979 and 2016, is presented to complement the first list of all types that was published in 1979. This part encompasses Acanthocephala, Nematoda and the other non-helminth phyla Cnidaria, Annelida, and Arthropoda. Platyhelminthes was covered in the first (Monogenoidea) and second (RhabditophoraTrematoda and Cestoda) parts of the catalogue published in September 2016 and March 2017, respectively. The present catalogue comprises type material for 116 species distributed across five phyla, nine classes, 50 families, and 80 genera. Specific names are listed systematically, followed by type host, infection site, type locality, and specimens with their collection numbers and references. Species classification and nomenclature are updated.The third part of the catalogue of type material in the Helminthological Collection of the Oswaldo Cruz Institute/FIOCRUZ ( CHIOC), Rio de Janeiro, Brazil, contains helminths that form part of the fauna of Brazil, and other countries, and are from a wide range of hosts captured in a variety of different biomes. The samples are holotypes, paratypes, and representative specimens of Platyhelminthes, Acanthocephala, Nematoda, and other non-helminth phyla. The CHIOC holds around 38,400 samples of helminth parasites from South America and other continents, and represents the largest such collection in Latin America and one of the largest collections worldwide : 408 nematodes, 216 digenetic trematodes, 11 monogenoids, 52 acanthocephalans, 28 cestodes, and four of pentastomids . Thus, the present catalogue is the third list of type species held in this collection, and encompasses those of Acanthocephala, Nematoda, and the non-helminth phyla Cnidaria, Annelida and Arthropoda that have been deposited in CHIOC since 1979. The purpose of this article is to inform the scientific community about the types deposited in CHIOC as of December 1, 2016, and follows the articles of the International Code of Zoological Nomenclature , 4\u201310% formaldehyde (with or without 2% acetic acid), 70% ethanol-formaldehyde-acetic acid (AFA) or as microscope slide preparations. All the material is available for consultation, but holotypes are not loaned. Unless otherwise stated, all type material is in good condition.PageBreakspecies are arranged alphabetically. The information on each entry is presented in the following format:The catalogue is arranged taxonomically as phyla, classes, orders, families, genera, and species, under the original spelling and combinations. Phyla and classes are arranged phylogenetically, starting with helminth phyla. Orders, families, genera, and 1. Original genus-species combination with author(s) and year of publication. An asterisk (*) denotes the type species of the genus.2. Type host: updated scientific name, author(s) and year, with original scientific name in square brackets (when changed), followed by principal taxonomic group in parentheses.3. Infection site in the host.4. Type locality: country, province or state, department, specific locality and coordinates (if available).CHIOC catalog number. Categories for types follow articles 73\u201375 of the Code (5. Primary type status: sex (if applicable/ possible), the Code .6. Remark sections are inserted when necessary and include additional information about host, locality, or status of the types.CHIOC.7. References include publications in which the species was described and those that mention type specimens in the Acanthocephala follows Nematoda follows Myxozoa follows Annelida follows Beesley et al. (2000) and Copepoda follows The valid names adopted for parasitized hosts follow specific bibliographies. Diplopod names are in accordance with P\u00e9rez-Asso (1996); beetles with BMNHBritish Museum of Natural History, Collection at the Department of Zoology, Natural History Museum, London, England;CHFC Helminthological Collection of the Science College, University of the Republic, Montevideo, Uruguay;CHIBB Reference Helminthological Collection of the Parasitology Department, Bioscience Institute, Paulista State University (UNESP), Botucatu, S\u00e3o Paulo, Brazil;CHIOCHelminthological Collection of the Oswaldo Cruz Institute, FIOCRUZ, Rio de Janeiro, Brazil;CHIP\u2013URG Helminthological Collection of the Laboratory of Ichthyoparasitology, University of Rio Grande, Rio Grande, Rio Grande do Sul, Brazil;CHMLP/MLPHelminthological Collection of the Museum of La Plata, La Plata, Buenos Aires, Argentina;CMNPACanadian Museum of Nature Invertebrate Collections-Parasites, Ottawa, Canada;CNHE National Collection of Helminths, Institute of Biology, National Autonomous University of Mexico, Mexico City, Mexico;CZACCHelminthological Collection of the Zoological Collections, Institute of Ecology and Systematic, Havana, Cuba;FCAV/UNESP Helminthological Collection of the Parasitic Diseases Laboratory, Department of Preventive Veterinary Medicine, Paulista State University, S\u00e3o Paulo, Brazil;HWMLHarold W. Manter Laboratory, University of Nebraska State Museum, Lincoln, Nebraska, USA;INPANational Institute for Amazon Research, Manaus, Amazonas, Brazil;IPCAS/IPCR/ASCR Institute of Parasitology, Academy of Sciences of the Czech Republic, \u010cesk\u00e9 Bud\u011bjovice, Czech Republic;MNHN/INVENational Museum of Natural History, Paris, France;MNRJCarcinological Collection of the National Museum of Rio de Janeiro, Rio de Janeiro, Brazil;RITRoyal Belgian Institute of Natural Sciences, Brussels, Belgium;RMCACollection of the Royal Museum of Central Africa, Tervuren, Belgium;USNMUnited States National Museum, Smithsonian Institution, Washington, D.C, USA;USNPCUnited States National Parasitological Collection, Beltsville, Maryland, USA.Acanthocephala, Nematoda and other non-helminth phyla, such as Cnidaria (Myxozoa), Annelida (Polychaeta and Hirudinea) and Arthropoda (Copepoda), in CHIOC, from Brazil and other countries of the world, for a period of more than 35 years of parasitological studies. This catalogue adds 38 primary types of Acanthocephala, represented by eight species distributed among eight families and eight genera; approximately 420 primary types of Nematoda, represented by 99 species, distributed across 36 families and 64 genera; four primary types of Myxozoa, represented by one family, two genera and two species; eight primary types of one species of Polychaeta and six primary types of one species of Hirudinea; and 22 primary types of Copepoda, represented PageBreakby five species, distributed in three families, and four genera. The most representative families were the nematode families Cucullanidae, with 18 species, followed by Pharyngodonidae, with seven.This database and bibliographic survey provides the diversity types of Monogenoidea, Trematoda, and Cestoda  was the host family that exhibited the greatest diversity of parasites, with ten species.One hundred and sixteen parasites of 101 species of vertebrate and eight species of invertebrate hosts were catalogued. The invertebrate hosts were beetles, diplopods, and crustaceans. Most of the type species recorded (47%) were parasites of bony fishes, such as those of  Cestoda , 2017. OAcanthocephala listed, five species were parasites of bony fishes, one species was a parasite of frogs (Bufonidae), one species was a parasite of birds , and one species was a parasite of crab-eating foxes (Canidae). Among the nematodes, five species were parasites of beetles  and two were parasites of diplopods (Rhinocricidae and Spirobolellidae). Species of nematode parasites of fishes included 41 of bony fishes and only one of cartilaginous fishes (Arhynchobatidae). Three nematode species were parasites of frogs (Bufonidae). The nematodes parasites of reptiles included five species parasitic on the suborder Autarchoglossa , five parasites of snakes , and three parasites of lizards . Nine nematode species were parasites of birds . Species of nematode parasites of mammals included 16 parasites of rodents , four parasites of the order Carnivora , two parasites of armadillos (Dasypodidae), two parasites of opossums (Didelphidae), and one parasite of bats (Molossidae). Myxozoans and copepods parasitized bony fishes. The single polychaeta species was a parasite of freshwater crayfish (Parastacidae) and the single Hirudinea species was a parasite of turtles (Podocnemididae).Among the CHIOC from 1979 to 2016 includes, in total, 423 species of parasites from 251 different hosts caught in almost all Brazilian states, and almost all continents, with the exception of the Middle East and Oceania , coming from all regions of the country. Other countries with cataloged material are from America , the Caribbean (Cuba) and Africa (Democratic Republic of Congo). This third part of the catalogue of type material housed in  Oceania , 2017. MPageBreakTaxon classificationAnimaliaOligacanthorhynchidaOligacanthorhynchidaeGomes, Olifiers, Souza, Barbosa, D\u2019Andrea, & Maldonado Jr., 2015Cerdocyonthous  (Carnivora: Canidae).Small intestine.18\u00b059'S, 56\u00b039'W).Brazil, Mato Grosso do Sul State, Nhumirim Ranch , 30812 c .Taxon classificationAnimaliaGyracanthocephalaQuadrigyridae*Noronha, 1992Gymnothoraxocellatus Agassiz, 1831 (Osteichthyes: Muraenidae).Intestine.Brazil, Bahia State, Arembepe.CHIOC 32742 a\u2013b (\u2642\u2642), c (\u2640).CHIOC 32742 a\u2013c referred as type material in the original description.Taxon classificationAnimaliaNeoechinorhynchidaNeoechinorhynchidaeBrasil-Sato & Pavanelli, 1998Pimelodusmaculatus Lac\u00e9p\u00e8de, 1803 (Osteichthyes: Pimelodidae).Anterior intestine.Brazil, Minas Gerais State, S\u00e3o Francisco River basin, Tr\u00eas Marias.CHIOC 33718 a.\u2642 CHIOC 33718 b , c, e (\u2642\u2642), d (\u2640).USNPC.Other paratypes deposited in Taxon classificationAnimaliaEchinorhynchidaArhythmacanthidaeVieira, Felizardo & Luque, 2009Pseudopercisnumida Miranda-Ribeiro, 1903 (Osteichthyes: Pinguipedidae).Intestine.22\u00b052'43.26\"S, 42\u00b01'11.55\"W).Brazil, Rio de Janeiro State, Cabo Frio (CHIOC 37103.\u2642 CHIOC 36672 (\u2642), 37104 , 37105 (\u2640).Taxon classificationAnimaliaEchinorhynchidaDiplosentidae*Salgado-Maldonado & Santos, 2000Sciadespassany  [= Ariuspassany] (Osteichthyes: Ariidae).Intestine.2\u00b00'N, 50\u00b024'W).Brazil, Amap\u00e1 State, Maraca Island , c\u2013j (\u2640\u2640).CNHE and BMNH.Other paratypes deposited in Taxon classificationAnimaliaEchinorhynchidaEchinorhynchidaeLent & Santos, 1989Atelopusoxyrhynchus Boulenger, 1903 (Amphibia: Bufonidae).Small intestine.Venezuela, M\u00e9rida.CHIOC 32173.\u2642 CHIOC 31667, 32174 , 32175 a\u2013b (\u2642\u2642).PageBreakTaxon classificationAnimaliaEchinorhynchidaGymnorhadinorhynchidae*Braicovich, Lanfranchi, Farber, Marvaldi, Luque & Timi, 2014Decapteruspunctatus  (Osteichthyes: Carangidae).Intestine.22\u00b053'S, 42\u00b000'W).Brazil, Rio de Janeiro State, Cabo Frio , 35941 c (\u2642), d\u2013e (\u2640\u2640).Two additional paratypes deposited in the IPCAS collection.Taxon classificationAnimaliaPolymorphidaPolymorphidaeMonteiro, Amato & Amato, 2006Phalacrocoraxbrasilianus  .Small and large intestine.30\u00b000'S, 51\u00b015'W).Brazil, Rio Grande do Sul State, Gua\u00edba Lake, Gua\u00edba , 36630 c (\u2640).Taxon classificationAnimaliaTrichinellidaCapillariidaeSantos, Moravec & Venturieri, 2008Arapaimagigas  (Osteichthyes: Arapaimidae).Anterior part of intestine and pyloric caeca.00\u00b005'30\"S, 49\u00b034'50\"W).Brazil, Par\u00e1 State, Mexiana Island (Amazon River delta), natural canals and breeding tanks of fish farm at Santo Ambr\u00f3sio Farm , 35559 c\u2013d .Taxon classificationAnimaliaTrichinellidaCapillariidaeRodrigues, 1992Liolaemuslutzae Mertens, 1938 (Iguania: Liolaemidae).Small intestine.Brazil, Rio de Janeiro State, Maric\u00e1.CHIOC 32796 a.\u2642 CHIOC 32796 b (2\u2642\u2642), c (3\u2642\u2642), d (4\u2640\u2640), e (5\u2640\u2640), f (6\u2640\u2640), g (7\u2640\u2640), h (8\u2640\u2640), i (9\u2640\u2640).Taxon classificationAnimaliaTrichinellidaTrichuridaeLopes Torres, Nascimento, Menezes, Garcia, Santos, Maldonado Jr., Miranda, Lanfredi & Souza, 2011Thrichomysapereoides Lund, 1839 (Rodentia: Echimyidae).Cecum.19\u00b002'14\"S, 41\u00b049'59\"W).Brazil, Minas Gerais State, Capit\u00e3o Andrade , 35710 c (\u2642), 37365 a\u2013b (\u2642\u2642), 37365 c (\u2640).Lopes Torres et al. (2011).Taxon classificationAnimaliaTrichinellidaTrichuridaeGomes, Lanfredi, Pinto & Souza, 1992Oligoryzomysnigripes  [= Oryzomysnigripes] (Rodentia: Cricetidae).Large intestine.28\u00b045'S, 52\u00b015'W).Brazil, Rio Grande do Sul State, Arvorezinha (CHIOC 32790 a.\u2642 CHIOC 32790 b (\u2640), 32791 a\u2013b, f\u2013g, m (\u2640\u2640), c\u2013e, h\u2013l (\u2642\u2642).PageBreakTaxon classificationAnimaliaRhabditidaAcuariidae*Vicente, Pinto & Noronha, 1980Pitangussulphuratus  (Aves: Tyrannidae).Stomach.Brazil, Rio de Janeiro State, Rio de Janeiro, Raimundo Island.CHIOC 31780 a.\u2642 CHIOC 31780 b\u2013c, g\u2013h (\u2640\u2640), d\u2013f (\u2642\u2642).There is no paratype \u201ci\u201d as indicated in the original description, which was a mistake.Taxon classificationAnimaliaRhabditidaAnisakidaeMoravec, Kohn & Fernandes, 1994Bryconhilarii  (Osteichthyes: Bryconidae).Stomach.Brazil, Paran\u00e1 State, Paran\u00e1 River, Foz do Igua\u00e7\u00fa.CHIOC 32961.\u2640 Holotype and allotype deposited in the IPCAS collection.Taxon classificationAnimaliaRhabditidaAnisakidaeMartins & Yoshitoshi, 2003Leporinusmacrocephalus Garavello & Britski, 1988 (Osteichthyes: Anostomidae).Stomach.Brazil, S\u00e3o Paulo State, Batatais.CHIOC 34675 a.\u2642 CHIOC 34675 b , 34675 c (\u2642\u2642), 34675 d (\u2640\u2640).Taxon classificationAnimaliaRhabditidaAnisakidaeSantos, Lent & Gibson, 2004Riorajaagassizii  (Chondrichthyes: Arhynchobatidae).Intestine.23\u00b030'S, 45\u00b000'W).Brazil, S\u00e3o Paulo State, off Ubatuba  (Arhynchobatidae). Other paratypes deposited in the BMHN collection.Taxon classificationAnimaliaRhabditidaAnisakidaeMelo, Santos, Giese, Santos & Santos, 2011Satanopercajurupari  (Osteichthyes: Cichlidae).Intestine.01\u00b027'21\"S, 48\u00b030'14\"W).Brazil, Par\u00e1 State, Bel\u00e9m, Guam\u00e1 River .CHIOC cited as \u201c35716\u201d in the original description due to a mistake.Paratype from Taxon classificationAnimaliaRhabditidaAnisakidaePereira, Tavares, Scholz & Luque, 2015Platydorasarmatulus  (Osteichthyes: Doradidae).Intestine.19\u00b034'S, 57\u00b000'W).Brazil, Mato Grosso do Sul State, Miranda River , c (hologenophore).Other paratypes deposited in the IPCAS collection.Taxon classificationAnimaliaRhabditidaAproctidaeRodrigues & Rodrigues, 1980Guiraguira  (Aves: Cuculidae).Ocular cavity.Brazil, S\u00e3o Paulo State, Ilha Seca.CHIOC 31808 a.\u2642 CHIOC 31808 b , 31808 c\u2013d (\u2640\u2640).Taxon classificationAnimaliaRhabditidaAscaridiidaePanizzutti, Santos, Vicente, Muniz-Pereira & Pinto, 2003Crotalusdurissus Linnaeus, 1758 (Serpentes: Viperidae).Stomach.Brazil, Paran\u00e1 State, Itaip\u00fa Binacional Reserve, Foz do Igua\u00e7\u00fa .CHIOC 34937 a.CHIOC 34937 b\u2013f.Panizzuti et al. (2003).Taxon classificationAnimaliaRhabditidaAscaridiidaeSiqueira, Panizzutti, Muniz-Pereira & Pinto, 2005Bothropsjararaca  (Serpentes: Viperidae).Stomach.22\u00b030'39\"S, 43\u00b011'4\"W), 857 m high.Brazil, Rio de Janeiro State, Petr\u00f3polis, Serra das Araras , 36232 c (\u2640), d (anterior extremity).Taxon classificationAnimaliaRhabditidaAspidoderidaeJim\u00e9nez-Ruiz, Gardner & Varela-Stokes, 2006Dasypusnovemcinctus Linnaeus, 1758 (Cingulata: Dasypodidae).Large intestine.31\u00b019'34\"N, 98\u00b009'33\"W), 311 m high.United States, Texas, El Pedregal, 18 miles north by road (U.S. 281) from Lampasas , 35430 , 35431 .CHIOC collected from Mexico, but there are some inconsistencies with regard to the collecting localities in the original description and the cataloging data. Holotype, allotype, and other paratypes are deposited in the HWML collection. Additional paratypes are deposited in CMNPA, CNHE, and USNM.There are two paratypes in Taxon classificationAnimaliaRhabditidaAspidoderidaePinto, Kohn, Fernandes & Mello, 1982Nectomyssquamipes  (Rodentia: Cricetidae).Small intestine.Brazil, Goi\u00e1s State, Formosa.CHIOC 31879 a.\u2642 CHIOC 31879 b\u2013f.Taxon classificationAnimaliaRhabditidaCamallanidaeFerraz & Thatcher, 1990Osteoglossumbicirrhosum  (Osteichthyes: Osteoglossidae).Intestine.Brazil, Amazonas State, Anavilhanas Archipelago, Negro River.CHIOC 32557 a\u2013b .INPA collection. CHIOC numbers were not included in the original description.Holotype and allotype deposited in the Taxon classificationAnimaliaRhabditidaCamallanidaeMartins, Garcia, Piazza & Ghiraldelli, 2007Xiphophorusmaculatus  (Osteichthyes: Poeciliidae).Intestine.Brazil, S\u00e3o Paulo State, Araraquara.allotype \u2640 and paratypes.CHIOC 35283.\u2642, PageBreakTaxon classificationAnimaliaRhabditidaCamallanidaeFerraz & Thatcher, 1992Hypophthalmusedentatus Spix & Agassiz, 1829 (Osteichthyes: Pimelodidae).Intestine.Brazil, Amazonas State, Marchantaria Island, Solim\u00f5es River.CHIOC 32756 (\u2642), 32757 (\u2640).INPA. CHIOC numbers were not included in the original description.Holotype, allotype and other paratypes deposited in the collection of Taxon classificationAnimaliaRhabditidaCamallanidaeKohn & Fernandes, 1988Hypostomusalbopunctatus  (Osteichthyes: Loricariidae).Intestine.Brazil, Paran\u00e1 State, Igua\u00e7\u00fa River, Hydroelectric Power station Salto Os\u00f3rio.CHIOC 32430 a.\u2642 CHIOC 32430 b .Taxon classificationAnimaliaRhabditidaCamallanidaeGiese, Santos & Lanfredi, 2009Ageneiosusucayalensis Castelnau, 1855 (Osteichthyes: Auchenipteridae).Intestine.Brazil, Par\u00e1 State, Bel\u00e9m, Guajar\u00e1 Bay .CHIOC 35604 a.\u2642 CHIOC 35604 b , 35604 c.Taxon classificationAnimaliaRhabditidaCamallanidaeKohn & Fernandes, 1988Corydoraspaleatus  .Intestine.Brazil, Paran\u00e1 State, Igua\u00e7\u00fa River, Hydroelectric Power station Salto Os\u00f3rio.CHIOC 32432 a.\u2642 CHIOC 32431 a , 32431 b (\u2642), 32432 b\u2013c (\u2640\u2640).Taxon classificationAnimaliaRhabditidaCarnoyidaeGarc\u00eda & Morffe, 2014Nesoboluspiedra P\u00e9rez-Asso, 1996 (Diplopoda: Rhinocricidae).Hind gut.20\u00b000'32.68\"N, 75\u00b037'18.8\"W).Cuba, Santiago de Cuba province, La Gran Piedra, La Isabelica, .CZACC.Holotype female and other paratypes males and females deposited in Taxon classificationAnimaliaRhabditidaCosmocercidae\u00c1vila, Str\u00fcssmann & Silva, 2010Iphisaelegans Gray, 1851 .Large intestine.15\u00b007'S, 58\u00b058'W).Brazil, Mato Grosso State, S\u00e3o Domingos Valley .CHIBB.Three paratypes deposited in Taxon classificationAnimaliaRhabditidaCrenosomatidaeVieira, Muniz-Pereira, Lima, Moraes Neto, Guimar\u00e3es & Luque, 2012Galictiscuja  (Carnivora: Mustelidae).Bronchi and bronchioles.21\u00b076'S, 43\u00b021'W).Brazil, Minas Gerais State, Juiz de Fora , 35813 c (\u2642), 35813 d (2\u2640\u2640).One paratype male and two females deposited in the IPCAS collection.Taxon classificationAnimaliaRhabditidaCucullanidaeGiese, Lanfredi & Santos, 2010AgeneiosusucayalensisSmall intestine.Brazil, Par\u00e1 State, Bel\u00e9m, Guajar\u00e1 Bay .CHIOC 35657 a.\u2642 CHIOC 35657 b .Taxon classificationAnimaliaRhabditidaCucullanidaePereira, Vieira & Luque, 2014Balistescapriscus Gmelin, 1789 .Intestine.Brazil, Rio de Janeiro State, Angra dos Reis Bay .CHIOC 35896 a.\u2642 CHIOC 35896 b , 35897 .Two paratypes males and two females deposited in the IPCAS collection.Taxon classificationAnimaliaRhabditidaCucullanidaeMoravec, Kohn & Fernandes, 1993Auchenipterusnuchalis  (Osteichthyes: Auchenipteridae).Intestine.Brazil, Paran\u00e1 State, Foz do Igua\u00e7\u00fa, Reservoir of the hydroelectric power station of Itaip\u00fa.CHIOC 32951.\u2642 One specimen deposited, no paratypes.Taxon classificationAnimaliaRhabditidaCucullanidaePereira Jr. & Costa, 1996Micropogoniasfurnieri  (Osteichthyes: Sciaenidae).Digestive tract.PageBreakBrazil, Rio Grande do Sul State, Rio Grande.CHIOC 33655.\u2640 CHIOC 33656 .Other paratypes deposited in CHIP-URG.Pereira Jr. and Costa (1996).Taxon classificationAnimaliaRhabditidaCucullanidaePereira, Vieira & Luque, 2015 in Vieira et al. (2015)Lophiusgastrophysus Miranda-Ribeiro, 1915 (Osteichthyes: Lophiidae).Intestine.22\u00b052' S, 42\u00b001' W).Brazil, Rio de Janeiro State, Cabo Frio , 36746 .One paratype male and one female deposited in the IPCAS collection.Taxon classificationAnimaliaRhabditidaCucullanidaePereira, Vieira & Luque, 2015 in Vieira et al. (2015)Pagruspagrus  (Osteichthyes: Sparidae).Intestine.22\u00b052'S, 42\u00b001'W).Brazil, Rio de Janeiro State, Cabo Frio , 36748 (\u2640).Taxon classificationAnimaliaRhabditidaCucullanidaePereira, Vieira & Luque, 2015 in Vieira et al. (2015)Pseudopercissemifasciata Intestine.22\u00b052' S, 42\u00b001' W).Brazil, Rio de Janeiro State, Cabo Frio , 36750 .One paratype male deposited in the IPCAS collection.Taxon classificationAnimaliaRhabditidaCucullanidaeOxydorasniger  (Osteichthyes: Doradidae).Intestine.Brazil, Amazonas State, Solim\u00f5es River near Manaus.PageBreakCHIOC 32456 a\u2013b .INPA collection. CHIOC numbers were not included in the original description.Holotype and allotype deposited in the Taxon classificationAnimaliaRhabditidaCucullanidaeMoreira, Rocha & Costa, 2000Trachelyopterusstriatulus  [= Parauchenipterusstriatulus] (Osteichthyes: Auchenipteridae).Intestine.19\u00b050'S, 42\u00b035'W).Brazil, Minas Gerais State, Central Lake of the State Park of Rio Doce .Taxon classificationAnimaliaRhabditidaCucullanidaeMoravec, Kohn & Fernandes, 1993Pimelodellalateristriga  (Osteichthyes: Heptapteridae).Intestine.Brazil, Paran\u00e1 State, Gua\u00edra.CHIOC 32826 (\u2642).Holotype deposited in the ASCR collection.Taxon classificationAnimaliaRhabditidaCucullanidaeMoravec, Kohn & Fernandes, 1997Pterodorasgranulosus  (Osteichthyes: Doradidae).Intestine.Brazil, Paran\u00e1 State, Gua\u00edra (Paran\u00e1 River basin), Reservoir of Itaip\u00fa.allotype \u2640 and paratype.CHIOC 33532.\u2642, CHIOC cited as \u201c33697\u201d in the original description due to a mistake. Other paratypes deposited in the ASCR collection.Paratype from Taxon classificationAnimaliaRhabditidaCucullanidaeMoravec, Kohn & Fernandes, 1993Pseudoplatystomacorruscans  (Osteichthyes: Pimelodidae).Intestine.PageBreakBrazil, Paran\u00e1 State, Paran\u00e1 River, Gua\u00edra.CHIOC 32829 a.\u2642 CHIOC 32829 b , 32829 c\u2013d, 32910, 32913, 32914.Other paratypes deposited in the ASCR collection.Taxon classificationAnimaliaRhabditidaCucullanidaeMoravec, Kohn & Fernandes, 1997Rhamphichthysrostratus  (Osteichthyes: Rhamphichthyidae).Intestine.Brazil, Paran\u00e1 State, Santa Helena, reservoir of the hydroelectric power station of Itaip\u00fa.CHIOC 33533.\u2640 CHIOC 33533 (2\u2640\u2640)CHIOC cited as \u201c33716\u201d in the original description due to a mistake. Other paratypes deposited in the ASCR collection.Type material from Taxon classificationAnimaliaRhabditidaCucullanidaeLacerda, Takemoto, Marchiori, Martins & Pavanelli, 2013Cichlapiquiti Kullander & Ferreira, 2006 (Osteichthyes: Cichlidae).Intestine.10\u00b066'55\"S, 48\u00b042'36\"W).Brazil, Tocantins State, Lajeado reservoir .CNHE.Other paratypes deposited in Taxon classificationAnimaliaRhabditidaCucullanidaeTimi, Lanfranchi, Tavares & Luque, 2009Umbrinacanosai Berg, 1895 (Osteichthyes: Sciaenidae).Posterior end of intestine.38\u00b008'S, 57\u00b032'W).Argentina, Buenos Aires Province, Mar del Plata (CHIOC 35615 (5\u2642\u2642), 35616 (2\u2640\u2640), 35617 (5\u2642\u2642), 35618 (5\u2640\u2640).CHIOC 35617 and 35618 collected in Pedra de Guaratiba , Rio de Janeiro  from the sciaenid fish Micropogoniasfurnieri. Holotype, allotype, and other paratypes deposited in CHMLP.PageBreakTaxon classificationAnimaliaRhabditidaCucullanidaePaschoal, Vieira, Cezar & Luque, 2014Orthopristisruber  (Osteichthyes: Haemulidae).Intestine.Brazil, coast of the Rio de Janeiro State .CHIOC 35894 a.\u2642 CHIOC 35894 b , 35895 .One paratype male and two females deposited in the IPCAS collection.Taxon classificationAnimaliaRhabditidaCucullanidaeMoravec, Kohn & Fernandes, 1997PimelodusmaculatusIntestine.Brazil, Paran\u00e1 State, Gua\u00edra (Paran\u00e1 River basin), Reservoir of Itaip\u00fa.CHIOC 33534.\u2642 Holotype cited as \u201c33717\u201d in the original description due to a mistake.Taxon classificationAnimaliaRhabditidaCucullanidaePereira Jr. & Costa, 1996MicropogoniasfurnieriDigestive tract.Brazil, Rio Grande do Sul State, Rio Grande.CHIOC 33649.\u2640 CHIOC 33650 , 33651 (\u2640), 33652 (\u2640), 33653 a\u2013b (\u2642\u2642), 33654 (\u2642).Other paratypes deposited in CHIP-URG.Pereira Jr. and Costa (1996).Taxon classificationAnimaliaRhabditidaCystidicolidaePinto, Vicente & Noronha, 1984Trachinotuscarolinus  (Osteichthyes: Carangidae).Stomach.PageBreak22\u00b052'23\"S, 42\u00b020'20\"W).Brazil, Rio de Janeiro State, Araruama , e, g (\u2642\u2642), 32033 a\u2013b .Taxon classificationAnimaliaRhabditidaCystidicolidaePereira, Pereira & Luque, 2014Holocentrusadscensionis  (Osteichthyes: Holocentridae).Anterior intestine and caecum.Brazil, Rio de Janeiro State, Angra dos Reis Bay.CHIOC 35879 a.\u2642 CHIOC 35879 b , 35880 (\u2640), 35881 (\u2642), 35882 (\u2642).Two paratypes males and one female deposited in the IPCAS collection.Taxon classificationAnimaliaRhabditidaCystidicolidaePereira, Timi, Vieira & Luque, 2012Mullusargentinae Hubbs & Marini, 1933 (Osteichthyes: Mullidae).Stomach and intestine.22\u00b055'S, 43\u00b012'W).Brazil, Rio de Janeiro State, Rio de Janeiro (CHIOC 35802 a.\u2642 CHIOC 35802 b (15\u2640\u2640), 35803 a , 35803 b (12\u2642\u2642).Other paratypes deposited in the IPCAS collection.Taxon classificationAnimaliaRhabditidaCystidicolidaePereira, Pereira, Timi & Luque, 2013Kyphosussectatrix  (Osteichthyes: Kyphosidae).Stomach.23\u00b000'S, 44\u00b010'W).Brazil, Rio de Janeiro State, Angra dos Reis , 35849 a\u2013b .PageBreakTaxon classificationAnimaliaRhabditidaDiplotriaenidaePinto, Vicente & Noronha, 1981Xiphocolaptesalbicollis  (Aves: Dendrocolaptidae).Body cavity.Brazil, Rio de Janeiro State, Angra dos Reis.CHIOC 31826 a.\u2642 CHIOC 31826 b\u2013d, g (\u2642\u2642), e\u2013f, h (\u2640\u2640), 31827 a (\u2642), b\u2013c (\u2640\u2640).Taxon classificationAnimaliaRhabditidaDracunculidaeMoravec & Santos, 2009Eunectesmurinus  (Serpentes: Boidae).Subcutaneous tissue (inside skin swelling) and body cavity (mesentery near lung).00\u00b005'30\"S, 49\u00b034'50\"W).Brazil, Par\u00e1 State, Mexiana Island (Amazon River delta), Santo Ambr\u00f3sio Farm .Gizzard.Brazil, Esp\u00edrito Santo State, Concei\u00e7\u00e3o da Barra.CHIOC 32783 a.\u2642 CHIOC 32783 b (\u2642), 32783 c, e\u2013f (\u2640\u2640), 32783 d .PageBreakTaxon classificationAnimaliaRhabditidaHeligmonellidae*Gon\u00e7alves, Pinto & Durette-Desset, 2007Dasyproctafuliginosa Wagler, 1832 (Rodentia: Dasyproctidae).Stomach.Brazil, Amazonas State, Barcelos, Jauari waterway, left margin of the Arac\u00e1 River.CHIOC 35044 a.\u2642 CHIOC 34844, 35044 b , c\u2013f (\u2640\u2640), g\u2013k (\u2642\u2642), l\u2013m (synlophe \u2642\u2642), n (synlophe).Dasyproctaleporina  was cited as the type host in the original description, but D.fuliginosa is the host of all type material of CHIOC as indicated in the cataloguing data. Other male and female paratypes deposited in the MNHN collection.Taxon classificationAnimaliaRhabditidaHeligmonellidaeSim\u00f5es, Santos & Maldonado Jr., 2012Oecomysmamorae  (Rodentia: Cricetidae).Small intestine.19\u00b034'54\"S, 56\u00b014'62\"W).Brazil, Mato Grosso do Sul State, Rio Negro Farm, Aquidauana , 35667 .Taxon classificationAnimaliaRhabditidaHeligmonellidaeWeirich, Catzeflis & Jim\u00e9nez, 2016Oecomysauyantepui Tate, 1939Small intestine.04\u00b033'708\"N, 52\u00b026'590\"W), 197 m high.French Guiana, Municipality of Roura, Cacao  (Rodentia: Cricetidae).Small intestine.23\u00b009'07.50\"S, 44\u00b013'44.20\"W).Brazil, Rio de Janeiro State, Angra dos Reis, Ilha Grande (CHIOC 35928 a.\u2642 CHIOC 35928 b\u2013c (7\u2642\u2642).Taxon classificationAnimaliaRhabditidaHeligmonellidaeSouza, Digiani, Sim\u00f5es, Luque, Rodrigues-Silva & Maldonado Jr., 2009OligoryzomysnigripesSmall intestine.22\u00b012'44\"S, 42\u00b048'40\"W).Brazil, Rio de Janeiro State, Serra dos \u00d3rg\u00e3os, Teres\u00f3polis , 36925 b .Taxon classificationAnimaliaRhabditidaHeterakidae\u00c1vila & Silva, 2009Hoplocercusspinosus Fitzinger, 1843 (Iguania: Hoplocercidae).Small and large intestine.10\u00b09'0\"S, 59\u00b027'0\"W).Brazil, Mato Grosso State, Aripuan\u00e3 .CHIOC 3561\u201d in the original description due to a mistake. Other paratypes deposited in CHIBB.Holotype and allotype cited as \u201cPageBreakTaxon classificationAnimaliaRhabditidaHeterakidaeVicente, Pinto & Noronha, 1993Crypturellusvariegatusvariegatus  (Aves: Tinamidae).Intestine.Brazil, Esp\u00edrito Santo State.CHIOC 32850 a.\u2642 CHIOC 32850 b\u2013e (\u2642\u2642), f\u2013j (\u2640\u2640), 32851 a\u2013d (\u2642\u2642).Taxon classificationAnimaliaRhabditidaHystrignathidaeMorffe & Garc\u00eda, 2010Passalusinterstitialis Eschischoltz, 1829 .Gut caeca.Cuba, La Habana province, Jaruco, Escaleras de Jaruco.CHIOC 37823 (2\u2640\u2640).CHIOC number was not included in the original description. Holotype female and additional paratypes females deposited in CZACC.Taxon classificationAnimaliaRhabditidaHystrignathidaeMorffe & Garc\u00eda, 2010Coleoptera: Passalidae).Unidentified, short, blackish passalid beetle .CHIOC number was not included in the original description. Holotype female and additional paratypes females deposited in CZACC.PageBreakTaxon classificationAnimaliaRhabditidaHystrignathidae*Morffe & Garc\u00eda, 2013Pentalobusbarbatus  .Gut caeca.Democratic Republic of Congo, Ituri province, Mogwalu.CHIOC 37903 (\u2640).CHIOC number was not included in the original description. Holotype and paratype females deposited in CZACC. Other paratypes deposited in the RMCA collection.Taxon classificationAnimaliaRhabditidaHystrignathidae*Morffe & Garc\u00eda, 2013Didimoidescf.parastictus  .Gut caeca.Democratic Republic of Congo, Ituri province, Mogwalu.CHIOC 37904 (\u2640).CHIOC number was not included in the original description. Holotype and paratype females deposited in CZACC. Other paratypes deposited in the RMCA collection.Taxon classificationAnimaliaRhabditidaHystrignathidae*Morffe & Garc\u00eda, 2013Didimus sp. .Gut caeca.01\u00b019'S, 29\u00b022'E).Democratic Republic of Congo, Kivu Region, Katale (CHIOC 37822 a\u2013d (\u2642\u2642).CHIOC number was not included in the original description. Female holotype and male and female paratypes deposited in CZACC. Other paratypes deposited in the RMCA collection.PageBreakTaxon classificationAnimaliaRhabditidaMetastrongylidaeVieira, Muniz-Pereira, Lima, Moraes Neto, Guimar\u00e3es & Luque, 2013Pumayagouaroundi  (Carnivora: Felidae).Pulmonary arteries.21\u00b041'20\"S, 43\u00b020'40\"W).Brazil, Minas Gerais State, Juiz de Fora , 35812 c\u2013d .Taxon classificationAnimaliaRhabditidaMolineidae*Hoppe & Nascimento, 2007DasypusnovemcinctusMucosa and lumen of the cecum and colon.Brazil, Mato Grosso do Sul State, Aquidauana.CHIOC 35448 .FCAV/UNESP helminthological collection of Jaboticabal, S\u00e3o Paulo State.Male holotype and female allotype deposited in the Taxon classificationAnimaliaRhabditidaMolineidaeSantos, Geise, Maldonado & Lanfredi, 2008Rhinellamarina  [= Chaunusmarinus] (Amphibia: Bufonidae).Small intestine.01\u00b028'03\"S, 48\u00b020'18\"W).Brazil, Par\u00e1 State, Bel\u00e9m , 36855 c, f (\u2640\u2640), d\u2013e (\u2642\u2642).PageBreakTaxon classificationAnimaliaRhabditidaMolineidaeLent & Santos, 1989AtelopusoxyrhynchusSmall intestine.Venezuela, M\u00e9rida.and allotype \u2640. CHIOC 31668.\u2642 CHIOC 32355.Taxon classificationAnimaliaRhabditidaOnchocercidaeAlmeida & Vicente, 1984Canisfamiliaris Linnaeus, 1758 (Carnivora: Canidae).Subcutaneous and intramuscular tissue.Brazil, Rio de Janeiro State, Rio de Janeiro.CHIOC 32176 a.\u2642 CHIOC 32176 b .Taxon classificationAnimaliaRhabditidaOnchocercidaeMoraes Neto, Lanfredi & Souza, 1997Akodoncursor  (Rodentia: Muridae).Abdominal cavity.22\u00b042'30\"S, 42\u00b037'34\"W).Brazil, Rio de Janeiro State, Rio Bonito, Catimbau Grande .Other paratypes deposited in the MNHN collection.Moraes Neto et al. (1997).Taxon classificationAnimaliaRhabditidaOnchocercidaeBain, Petit & Diagne, 1989Oecomystrinitatis Body cavity and lung.Brazil, Par\u00e1 State, Caraj\u00e1s.PageBreakCHIOC 32643 (2\u2640\u2640).CHIOC number was not included in the original description.Holotype and allotype deposited in the MNHN collection. Taxon classificationAnimaliaRhabditidaOnchocercidaeBain, Petit & Diagne, 1989NectomyssquamipesBody cavity and lung.Brazil, S\u00e3o Paulo State, S\u00e3o Paulo.CHIOC 31056 c.\u2640 CHIOC 31056 a (\u2642), 31056 b , 32642 .CHIOC 31056 d was dismounted and received the number 32642.The slide Taxon classificationAnimaliaRhabditidaOnchocercidaeVicente, Pinto & Noronha, 1980Megarynchuspitangua  (Aves: Tyrannidae).Infraorbital sinus.Brazil, Rio de Janeiro State, Angra dos Reis.CHIOC 31782 a.\u2642 CHIOC 31782 b\u2013e (\u2642\u2642), f\u2013j (\u2640\u2640).Taxon classificationAnimaliaRhabditidaOxyuridae*Jim\u00e9nez-Ruiz & Gardner, 2003Oxymycterusparamensis Thomas, 1902 (Rodentia: Cricetidae).Cecum and large intestine.19\u00b049'S, 63\u00b058'W), 1130 m high.Bolivia, Chuquisaca Department, 2 km SW of Monteagudo , 3152 m high. Holotype, allotype, and paratypes deposited in the HWML collection. Other two paratypes deposited in CNHE.Paratype from PageBreakTaxon classificationAnimaliaRhabditidaOxyuridae*Feij\u00f3, Torres, Maldonado Jr. & Lanfredi, 2008Gracilinanusagilis  (Didelphimorphia: Didelphidae).Cecum.18\u00b059'00\"S, 56\u00b039'00\"W).Brazil, Mato Grosso do Sul State, Pantanal, Nhumirim Farm , 35518 b.CHIOC 35517 was collected in the Rio Negro Farm, Nhecol\u00e2ndia Region , Pantanal, Mato Grosso do Sul State.Taxon classificationAnimaliaRhabditidaPharyngodonidae*Moravec, Kohn & Fernandes, 1992PimelodellalateristrigaIntestine.Brazil, Paran\u00e1 State, Paran\u00e1 River, Gua\u00edra.CHIOC 32718 a\u2013b, e (\u2642\u2642), c\u2013d, f\u2013i (\u2640\u2640), 32719 a\u2013d (\u2640\u2640).Holotype, allotype, and other paratypes deposited in the IPCAS collection. Additional paratypes deposited in MNHN.Taxon classificationAnimaliaRhabditidaPharyngodonidaeMoravec, Kohn & Fernandes, 1992Megalancistrusparananus  (Osteichthyes: Locariidae).Intestine.Brazil, Paran\u00e1 State, Paran\u00e1 River, Gua\u00edra.CHIOC 32649, 32721 a\u2013b .Holotype, allotype, and other paratypes deposited in the IPCAS collection. Additional paratypes deposited in MNHN.PageBreakTaxon classificationAnimaliaRhabditidaPharyngodonidaeMoravec, Kohn & Fernandes, 1992Trachydorasparaguayensis  (Osteichthyes: Doradidae).Intestine.24\u00b020'S, 52\u00b038'W).Brazil, Paran\u00e1 State, Foz do Igua\u00e7\u00fa, Reservoir of the hydroelectric power of Itaip\u00fa  .Intestine.Brazil, Minas Gerais State, Volta Grande Reservoir.CHIOC 33852 a.\u2642 CHIOC 33852 b , 33852 c (5\u2642\u2642), 33852 d (7\u2640\u2640).Taxon classificationAnimaliaRhabditidaPharyngodonidae*Moravec, Kohn & Fernandes, 1992Rhinelepisaspera Spix & Agassiz, 1829 (Osteichthyes: Locariidae).Intestine.Brazil, Paran\u00e1 State, Paran\u00e1 River, Gua\u00edra.CHIOC 32652, 32720 a (\u2642), b (\u2640).Holotype, allotype, and other paratypes deposited in the IPCAS collection. Additional paratypes deposited in MNHN.Taxon classificationAnimaliaRhabditidaPharyngodonidaeVicente, Vrcibradic, Muniz-Pereira & Pinto, 2000Notomabuyafrenata  [= Mabuyafrenata] (Autarchoglossa: Scincidae).PageBreakLarge intestine.Brazil, S\u00e3o Paulo State, Valinhos.CHIOC 33965 a.. \u2642 CHIOC 33965 b , CHIOC 33965 c\u2013e (\u2642\u2642), f\u2013h (\u2640\u2640).. Taxon classificationAnimaliaRhabditidaPharyngodonidaeVicente, Vrcibradic, Rocha & Pinto, 2002Aspronemadorsivittatum  [= Mabuyadorsivittata] (Autarchoglossa: Scincidae).Large and small intestine.Brazil, Rio de Janeiro State, National Park of Itatiaia, Prateleiras.CHIOC 34539 a.\u2640 CHIOC 34539 b\u2013c (\u2640\u2640), 34540 a\u2013b (\u2642\u2642), 34541 a\u2013b (\u2640\u2640), c (\u2642), d (2\u2640\u2640), e\u2013f (larvae).CHIOC 34541 a\u2013f collected in the Ecological Station of Itirapina, S\u00e3o Paulo State.Paratypes Taxon classificationAnimaliaRhabditidaPhilometridaeC\u00e1rdenas, Moravec, Fernandes & Morais, 2012Pygocentrusnattereri Kner, 1858 .Oculo-orbits and nasal mucosa.03\u00b050'32.8\"S, 62\u00b034'32.4\"W), Coari.Brazil, Amazonas State, Marac\u00e1 Lake (CHIOC 35777.\u2640 CHIOC 35778 (2\u2640\u2640), 35779 (\u2640), 35780 (\u2640), 35781 (\u2640), 35782.CHIOC 35779 collected in the Baixio Lake , Iranduba; CHIOC 35780 collected in the Iauara Lake , Manacapuru; CHIOC 35781 collected in the Ara\u00e7\u00e1 Lake , Codaj\u00e1s; CHIOC 35782 collected in the Anan\u00e1 Lake , Anori . Other paratypes deposited in the INPA collection.PageBreakTaxon classificationAnimaliaRhabditidaPhysalopteridaePereira, Alves, Rocha, Souza Lima & Luque, 2014Salvatormerianae  (Autarchoglossa: Teiidae).Stomach.21\u00b047'32\"S, 43\u00b022'6\"W).Brazil, Minas Gerais State, Juiz de Fora, Parque da Lajinha , 35885 c (\u2642), 35885 d (2\u2640\u2640).Taxon classificationAnimaliaRhabditidaPhysalopteridaeS\u00e3o Luiz, Sim\u00f5es, Lopes Torres, Barbosa, Santos, Giese, Rocha & Maldonado Jr., 2015Cerradomyssubflavus  (Rodentia: Cricetidae).Stomach.20\u00b013'28.30\"S, 46\u00b030'39.20\"W).Brazil, Minas Gerais State, S\u00e3o Roque de Minas, Serra da Canastra National Park .S\u00e3o Luiz et al. (2015).Taxon classificationAnimaliaRhabditidaPhysalopteridaeLopes Torres, Maldonado Jr. & Lanfredi, 2009GracilinanusagilisStomach.19\u00b015'01\"S, 57\u00b001'29\"W).Brazil, Mato Grosso do Sul State, Pantanal, Alegria Farm (CHIOC 35651 a.\u2642 CHIOC 35651 b (\u2642), 35652 a , 36652 b (\u2640).Lopes Torres et al. (2009).Taxon classificationAnimaliaRhabditidaPhysalopteridaePereira, Alves, Rocha, Lima & Luque, 2012Salvatormerianae [= Tupinambismerianae]Stomach.Brazil, Minas Gerais State, Juiz de Fora.CHIOC 35811 a.\u2642 PageBreakCHIOC 35811 b , 35811 c .Taxon classificationAnimaliaRhabditidaQuimperiidae*Moravec, Kohn & Fernandes, 1992PterodorasgranulosusIntestine.Brazil, Paran\u00e1 State, Paran\u00e1 River near Gua\u00edra.CHIOC 32722 a.\u2642 CHIOC 32722 b .CHIOC 32722 \u201cc\u201d as indicated in the original description, which was a mistake. Other paratype deposited in the IPCAS collection.There is no paratype Taxon classificationAnimaliaRhabditidaRaphidascarididaeKnoff, Felizardo, I\u00f1iguez, Maldonado Jr., Torres, Pinto & Gomes, 2012Paralichthysisosceles Jordan, 1891 .Abdominal cavity, abdominal musculature, stomach, stomach mucosa, mesentery, intestine, heart serosa, kidney serosa, liver serosa, ovary, ovary serosa, and spleen serosa.Brazil, Rio de Janeiro State, Angra dos Reis .CHIOC 37523 a (L3 larvae).CHIOC 37523 b\u2013e, 35771 (L3 larvae).Taxon classificationAnimaliaRhabditidaRhabdiasidaeBarrella, Santos & Silva, 2010Spilotespullatus Linnaeus, 1758 (Serpentes: Colubridae).Lungs.23\u00b06'S, 48\u00b055'W).Brazil, S\u00e3o Paulo State, Avar\u00e9 (CHIOC 35653 a.\u2640 PageBreakCHIOC 35653 b (5\u2640\u2640).CHIBB.Other paratypes deposited in Barella et al. (2010).Taxon classificationAnimaliaRhabditidaRhabdiasidaeSantos, Melo, Silva, Giese & Furtado, 2011RhinellamarinaLungs.01\u00b028'03\"S, 48\u00b020'18\"W).Brazil, Par\u00e1 State, Bel\u00e9m (CHIOC 35705 a.Hermaphrodite \u2640 CHIOC 35705 b (ten paratypes).Taxon classificationAnimaliaRhabditidaRhabdiasidaeMorais, Aguiar, M\u00fcller, Narciso, Silva & Silva, 2016Bothropsmoojeni Hoge, 1966Lungs.21\u00b003'04.9\"S, 51\u00b052'52.6\"W).Brazil, S\u00e3o Paulo State, Castilho, Private Reserve of Natural Heritage \u2018Foz do Rio Aguape\u00ed\u2019 .CHIOC 35628 .CHIOC cited as \"35525\" in the original description due to a mistake. Holotype and allotype deposited in the FCAV/UNESP of Jaboticabal, S\u00e3o Paulo State.Paratype from PageBreakTaxon classificationAnimaliaRhabditidaRictulariidaeCardia, Tebaldi, Fornazari, Menozzi, Langoni, Nascimento, Bresciani & Hoppe, 2015Eumopsglaucinus  (Chiroptera: Molossidae).Mucosa of the small intestine.22\u00b052'47\"S, 48\u00b026'42\"W).Brazil, S\u00e3o Paulo State, Botucat\u00fa .FCAV/UNESP of Ara\u00e7atuba, S\u00e3o Paulo State. Paratypes from CHIOC cited as \u201c35824\u201d in the original description due to a mistake. CHIOC 35826 collected in Ja\u00fa , S\u00e3o Paulo State.Holotype, allotype and additional paratypes deposited in Taxon classificationAnimaliaRhabditidaStrongyloididaeRodrigues, Vicente & Gomes, 1985Kerodonrupestris  (Rodentia: Caviidae).Small intestine.Brazil, Piau\u00ed State, Floriano Peixoto.CHIOC 32172 a.\u2640 CHIOC 32172 b\u2013e (\u2640\u2640).Taxon classificationAnimaliaRhabditidaSubuluridaeVicente, Van Sluys, Fontes & Kiefer, 2000Eurolophosaurusnanuzae  [= Tropidurusnanuzae] (Iguania: Tropiduridae).Large and small intestine.19\u00b020'S, 43\u00b044'W).Brazil, Minas Gerais State, Serra do Cip\u00f3 , 34196 b, c (\u2642\u2642), 34197 a, c, e, g (\u2640\u2640), 34197 b, d, f (\u2642\u2642), 33853, 33854 (\u2642), 33855 (\u2642), 33856 (\u2642).PageBreakTaxon classificationAnimaliaRhabditidaTetrameridaeVicente, Pinto e Noronha, 1996Nyctanassaviolaceacayennensis  (Aves: Ardeidae).Gizzard.Brazil, Rio de Janeiro State, Rio de Janeiro.CHIOC 33182 a.\u2642 CHIOC 33182 b, f\u2013h (\u2642\u2642), c\u2013e (\u2640\u2640).Taxon classificationAnimaliaRhabditidaTetrameridaePinto & Vicente, 1995Theristicuscaudatuscaudatus  (Aves: Threskiornithidae).Gizzard .Brazil, Mato Grosso do Sul State, Salobra.CHIOC 33173 a.\u2642 CHIOC 33173 b\u2013c (\u2642\u2642), 33186 c .Taxon classificationAnimaliaRhabditidaTrichostrongylidae*Pinto & Gomes, 1985Opisthocomushoazin  (Aves: Opisthocomidae).Proventriculus and gizzard.Brazil, Amazonas State, Paran\u00e1 de Cambixa, Careiro Island.CHIOC 32024 a.\u2642 CHIOC 32024 b\u2013e (\u2640\u2640), 32025 a\u2013b (\u2640\u2640), 32026 a\u2013b (\u2642\u2642), c\u2013d (\u2640\u2640).PageBreakTaxon classificationAnimaliaRhabditidaViannaiidaeDurette-Desset, Gon\u00e7alves & Pinto, 2006DasyproctafuliginosaSmall intestine.0\u00b058'29\"S, 62\u00b055'27\"W).Brazil, Amazonas State, Barcelos, Jauari waterway, left margin of the Arac\u00e1 River, Tr\u00eas Barracas settlement , 35418 (7\u2640\u2640).Taxon classificationAnimaliaRhabditidaViannaiidaeDurette-Desset, Gon\u00e7alves & Pinto, 2006DasyproctafuliginosaSmall intestine.0\u00b058'29\"S, 62\u00b055'27\"W).Brazil, Amazonas State, Barcelos, Jauari waterway, left margin of the Arac\u00e1 River, Tr\u00eas Barracas settlement , 35420 (7\u2642\u2642), 35421 (4\u2640\u2640), 35052 a\u2013k (\u2642\u2642).Taxon classificationAnimaliaRhabditidaXustrostomatidae*Garc\u00eda & Morffe, 2015Spirobollelus sp. (Diplopoda: Spirobolellidae).Hind gut.Cuba, Mayabeque Province, San Jos\u00e9 de las Lajas, La Jaula.CHIOC 38211 a (\u2640), 38211 b (\u2642).CHIOC number was not included in the original description. Male holotype and male and female paratypes deposited in CZACC. Other paratypes deposited in the RIT collection.PageBreakTaxon classificationAnimaliaBivalvulidaMyxobolidaeMartins & Onaka, 2006Cyphocharaxnagelii  (Osteichthyes: Curimatidae).Gill filaments.Brazil, S\u00e3o Paulo State, S\u00e3o Jos\u00e9 do Rio Pardo, Rio do Peixe Reservoir.CHIOC 34986.CHIOC 34818 (fixed gills).CHIOC samples cited as \u201cspecimens deposited\u201d in the original description. No additional type material is deposited in other collections.Taxon classificationAnimaliaBivalvulidaMyxobolidaeMartins & Onaka, 2006CyphocharaxnageliiGill filaments.Brazil, S\u00e3o Paulo State, S\u00e3o Jos\u00e9 do Rio Pardo, Rio do Peixe Reservoir.CHIOC 34987.CHIOC 34834 (fixed gills).CHIOC samples cited as \u201cspecimens deposited\u201d in the original description. No additional type material is deposited in other collections.Taxon classificationAnimaliaEunicidaHistriobdellidaeAmato, 2001Parastacusbrasiliensis  (Decapoda: Parastacidae).Branchial chamber.29\u00b037'S, 50\u00b047'W).Brazil, Rio Grande do Sul State, Taquara , 34335 a\u2013b, e (\u2640\u2640), c\u2013d, f (\u2642\u2642).USNM collection.Other paratypes deposited in the Taxon classificationAnimaliaRhynchobdellidaOzobranchidae*Peralta, Matos & Serra-Freire, 1998Podocnemisexpansa  (Testudines: Podocnemididae).Carapace, plastron, head, over the eyes, pelvic appendages, cloaca, mouth and nostrils.Brazil, Par\u00e1 State, Bel\u00e9m, Zoobotanical Park of the Em\u00edlio Goeldi Museum.CHIOC 37524 a.CHIOC 33598, 37524 b\u2013e.CHIOC 33598 by the authors after the publication. CHIOC 33598 was indicated as holotype and paratypes in the original description.The holotype and some paratypes received new numbers because part of the type material was mounted from Taxon classificationAnimaliaPoecilostomatoidaBomolochidaeCentropomusundecimalis  (Osteichthyes: Centropomidae).Gills.23\u00b001'S, 44\u00b019'W).Brazil, Rio de Janeiro State, Angra dos Reis (CHIOC 34819 (10\u2640\u2640).Acantholochus Cressey, 1984 by Hamaticolax by MNRJ carcinological collection.Species placed in PageBreakTaxon classificationAnimaliaPoecilostomatoidaChondracanthidae*Luque & Alves, 2003Genypterusbrasiliensis Regan, 1903 (Osteichthyes: Ophidiidae).Oral cavity.Brazil, coastal zone of the Rio de Janeiro State .CHIOC 34892 (5\u2640\u2640), 34893 (3\u2642\u2642).MNRJ carcinological collection.Female holotype, male allotype, and other male and female paratypes deposited in the Taxon classificationAnimaliaPoecilostomatoidaErgasilidaeSantos, Thatcher & Brasil-Sato, 2007Pygocentruspiraya Gill filaments and nasal fossae.18\u00b012'59\"S, 45\u00b017'34\"W).Brazil, Minas Gerais State, Upper S\u00e3o Francisco River, Tr\u00eas Marias Reservoir (CHIOC 36841.\u2640 CHIOC 35502 (\u2640), 35503 (\u2640), 35504 (\u2640), 36842 (\u2640), 36843 (\u2640), 36844 (\u2640).CHIOC 35504, 36843, and 36844 collected from the serrasalmid fish Serrasalmusbrandtii L\u00fctken, 1875.Taxon classificationAnimaliaPoecilostomatoidaErgasilidaeEngers, Boeger & Brand\u00e3o, 2000Rhamdiaquelen  (Osteichthyes: Heptapteridae).Gills filaments.Brazil, Rio Grande do Sul State, Barragem do Capan\u00e9, Cachoeira do Sul.CHIOC 34008 a.\u2640 CHIOC 33840 a\u2013c, 34008 b\u2013f (\u2640\u2640).HWML and USNM.Other paratypes deposited in the collections of PageBreakTaxon classificationAnimaliaPoecilostomatoidaErgasilidaeTavares & Luque, 2005Aspistorluniscutis  (Osteichthyes: Ariidae).Gills.23\u00b001'S, 44\u00b019'W).Brazil, Rio de Janeiro State, Angra dos Reis (CHIOC 35393.\u2640 CHIOC 35392 (5\u2640\u2640), 35394 (5\u2640\u2640)."}
{"text": "The authors have requested that the following changes be made to their paper .In Figure 1, the caption was changed to \u201cFigure 1. 10\u2013step method for estimating free sugars content \u201d \u201d \u201d \u201d \u201d \u201d \u201d \u201d \u201d \u201d ["}
{"text": "Background: Theophanes Chrysobalantes'De curatione is a little known but highly relevant therapeutic manual dating to the tenth century AD. The text has come down to us in \u00a0an unusually large number of manuscripts, most of which transmit a mainstream version of the text.Methods: In the present article, three versions deriving from the mainstream text are being examined. For this, these versions are being compared to the mainstream text, in order to understand the aim behind the alterations and additions they were subjected to. The overarching goal is to understand, why these changes were made, and how skilled the editors were. It is a rather unusual approach, as divergent versions are usually not examined in research literature, since they are secondary to the original text.Results and conclusions: The results clearly show that the text was redacted several times, but not by highly sophisticated editors. The general aims of the redactions were to make the text easier to understand. Here, the main aim of the editor was obviously to restore the text to its original form. This article focusses on a completely different type of revision, namely a rephrased, augmented or otherwise heavily altered text. These are not that common as far as standard medical manuals are concerned, and they are not usually the focus of scholarly analysis.ItDe Curatione, a little known but highly relevant therapeutic manual that was in its mainstream form composed in the tenth century AD3. This work contains a substantial collection of therapeutic instructions arranged in a head to foot order, along with some miscellaneous material at the end. During my research on the transmission of the work I found that four manuscripts transmit a significantly altered text compared to the mainstream text. These are in alphabetical order4:The samples that are going to be discussed below come from manuscripts transmitting Theophanes Chrysobalantes-th century, f. 1r-141v5.Escorial, Real Biblioteca, T III 1, 16-th century, f. 1r-54v6.Florence, Biblioteca Laurenziana Plut. 75, 6, 14-th century, f. 1r-86r7.London, Wellcome Library MSL 135, 16-th century, f. 121r-149r8.Palermo, Biblioteca Centrale della Regione Siciliana XIII C 3, 169. So in total, we are dealing with three versions ofDe Curatione, which differ significantly from the mainstream tradition. In the following, I am going to describe these three versions very briefly. The main research question to be addressed is, what purpose these alterations served.The manuscripts from Florence and the Wellcome Library present a near identical text, and are therefore closely related and probably siblingsDe Curatione, the first chapter showing significant discrepancies from the mainstream text is chapter 4 (Bernard). As it is common for Byzantine therapeutic texts, the chapter starts with a heading and a brief description of a disease, which is then followed by several alternative recipes for medical treatment. Generally, in this genre, the descriptive part of the chapter tends to have a stable transmission, whereas the transmission of the recipes is notoriously unstable10. In this case, however, the descriptive parts also show significant differences, and since this is unusual, these will form the basis for this article.Starting, or rather collating, from the beginning ofDe Curatione as far as the text itself is concerned. Rather, it is the selection and organisation of excerpts that is of interest.Below, I present a comparison of the mainstream text as edited in the Bernard edition, and two versions, the Florence/Wellcome group and the Escorial version. The Palermo manuscript will be discussed at a later point in the present article, as it does not show any significant variants against the mainstream version in this part of11 \u03c0\u03b1\u03c1\u1f70 \u03c4\u1f74\u03bd \u03ba\u03b5\u03c6\u03b1\u03bb\u1f74\u03bd \u1f22 \u03c6\u03bb\u03ad\u03b3\u03bc\u03b1\u03c4\u03bf\u03c2 \u1f01\u03bb\u03bc\u03c5\u03c1\u03bf\u1fe6 \u1f22 \u03c7\u03bf\u03bb\u03ce\u03b4\u03bf\u03c5\u03c2 \u03ba\u03b1\u1f76 \u03bc\u03b5\u03bb\u03b1\u03b3\u03c7\u03bf\u03bb\u03b9\u03ba\u03bf\u1fe6 \u03b1\u1f35\u03bc\u03b1\u03c4\u03bf\u03c2. \u03ba\u03b1\u1f76 \u03b4\u03b5\u1fd6 \u03ba\u03b5\u03bd\u03bf\u1fe6\u03bd \u03c0\u03c1\u1ff6\u03c4\u03bf\u03bd \u03c4\u1f78 \u1f45\u03bb\u03bf\u03bd \u03c3\u03ce\u03bc\u03b1 \u03b4\u03b9\u1f70 \u03c6\u03bb\u03b5\u03b2\u03bf\u03c4\u03cc\u03bc\u03bf\u03c5 \u03ba\u03b1\u1f76 \u03ba\u03b1\u03b8\u03ac\u03c1\u03c3\u03b5\u03c9\u03c2 \u03c4\u03bf\u1fe6 \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03ba\u03c4\u03bf\u1fe6\u03bd\u03c4\u03bf\u03c2 \u03c7\u03c5\u03bc\u03bf\u1fe6. \u1f14\u03c0\u03b5\u03b9\u03c4\u03b1 \u03ba\u03b9\u03bc\u03c9\u03bb\u03af\u03b1\u03bd \u1f00\u03c0\u03bf\u03b2\u03c1\u03ad\u03be\u03b1\u03c2 \u1f55\u03b4\u03b1\u03c4\u03b9, \u03bc\u03af\u03be\u03bf\u03bd \u03c3\u03b5\u03cd\u03c4\u03bb\u03bf\u03c5 \u03c7\u03c5\u03bb\u1f78\u03bd \u03ba\u03b1\u1f76 \u03ba\u03b1\u03c4\u03ac\u03c7\u03c1\u03b9\u03b5.4. \u03a0\u03b5\u03c1\u1f76 \u03c0\u03b9\u03c4\u03c5\u03c1\u03b9\u03ac\u03c3\u03b5\u03c9\u03c2. \u1f21 \u03c0\u03b9\u03c4\u03c5\u03c1\u03af\u03b1\u03c3\u03b9\u03c2 \u03bb\u03b5\u03c0\u03c4\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03c0\u03b9\u03c4\u03c5\u03c1\u03bf\u03b5\u03b9\u03b4\u1ff6\u03bd \u03c3\u03c9\u03bc\u03ac\u03c4\u03c9\u03bd \u1f10\u03ba \u03c4\u1fc6\u03c2 \u1f10\u03c0\u03b9\u03c6\u03b1\u03bd\u03b5\u03af\u03b1\u03c2 \u03c4\u1fc6\u03c2 \u03ba\u03b5\u03c6\u03b1\u03bb\u1fc6\u03c2 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fe6 \u1f04\u03bb\u03bb\u03bf\u03c5 \u03c3\u03ce\u03bc\u03b1\u03c4\u03cc\u03c2 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f00\u03c0\u03cc\u03c4\u03b7\u03be\u03b9\u03c2 \u03c7\u03c9\u03c1\u1f76\u03c2 \u1f11\u03bb\u03ba\u03ce\u03c3\u03b5\u03c9\u03c2. \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03ba\u03b1\u03ba\u03bf\u03c7\u03c5\u03bc\u03af\u03b1\u03c2 \u1f00\u03bd\u03b5\u03bd\u03b5\u03c7\u03b8\u03b5\u03af\u03c3\u03b7\u03c2Translation: On pityriasis. Pityriasis is a melting of smooth and bran like bodies from the surface of the head or of the rest of the body, without rupturing. It happens when bad humours are brought up around the head either consisting of salty phlegm or bilious and melancholic blood. And one first needs to empty the whole body through blood letting and evacuation of the superfluous humour. Then rinse earth from Cimolus with water, mix with beet root juice and apply.12) \u03c4\u1f70 \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u1f00\u03bb\u03cc\u03b7\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1fc6\u03c2 \u03c0\u03b9\u03ba\u03c1\u1fb6\u03c2...8. \u03a0\u03b5\u03c1\u1f76 \u1f00\u03bb\u03c9\u03c0\u03b5\u03ba\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u1f40\u03c6\u03b9\u03ac\u03c3\u03b5\u03c9\u03c2. \u1f08\u03bb\u03c9\u03c0\u03b5\u03ba\u03af\u03b1 \u1f10\u03c3\u03c4\u03b9\u03bd \u1f45\u03c4\u03b1\u03bd \u03b1\u1f31 \u03c4\u03c1\u03af\u03c7\u03b5\u03c2 \u1f00\u03c0\u03bf\u03c0\u03ad\u03c3\u03c9\u03c3\u03b9 \u03c4\u1fc6\u03c2 \u03ba\u03b5\u03c6\u03b1\u03bb\u1fc6\u03c2 \u03b4\u03b9\u1f70 \u03c7\u03c5\u03bc\u1f78\u03bd \u03b8\u03b5\u03c1\u03bc\u1f78\u03bd \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b1\u03b2\u03c1\u03c9\u03c4\u03b9\u03ba\u1f78\u03bd \u03ba\u03cc\u03c0\u03c4\u03bf\u03bd\u03c4\u03b1 \u03c4\u1f70\u03c2 \u1fe5\u03af\u03b6\u03b1\u03c2 \u03b1\u1f50\u03c4\u1ff6\u03bd. \u0394\u03b9\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 \u03b4\u1f72 \u1f21 \u1f40\u03c6\u03af\u03b1\u03c3\u03b9\u03c2 \u03c4\u1ff7 \u03c3\u03c7\u03ae\u03bc\u03b1\u03c4\u03b9. \u1f29 \u03b3\u1f70\u03c1 \u1f40\u03c6\u03af\u03b1\u03c3\u03b9\u03c2 \u03bb\u03b5\u03b9\u03bf\u03c4\u03ad\u03c1\u03b1 \u1f10\u03c3\u03c4\u1f76\u03bd \u1f65\u03c3\u03c0\u03b5\u03c1 \u03bf\u1f31 \u1f44\u03c6\u03b5\u03b9\u03c2. \u1f29 \u03b4\u1f72 \u1f00\u03bb\u03c9\u03c0\u03b5\u03ba\u03af\u03b1 \u03c4\u03c1\u03b1\u03c7\u03c5\u03c4\u03ad\u03c1\u03b1 \u03c0\u03bf\u03bb\u03bb\u1ff7. \u03a3\u03c4\u03bf\u03c7\u03ac\u03b6\u03bf\u03c5 \u03b4\u1f72 \u1f00\u03c0\u1f78 \u03c4\u1fc6\u03c2 \u03c7\u03c1\u03bf\u03b9\u1fb6\u03c2 \u03c4\u03bf\u1fe6 \u03b4\u03ad\u03c1\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u1f78\u03bd \u03c0\u03bb\u03b5\u03bf\u03bd\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1 \u03c7\u03c5\u03bc\u1f78\u03bd \u03ba\u03b1\u1f76 \u03c4\u03bf\u03cd\u03c4\u03bf\u03c5 \u03c4\u1f74\u03bd \u03ba\u03ac\u03b8\u03b1\u03c1\u03c3\u03b9\u03bd \u03c0\u03c1\u1ff6\u03c4\u03bf\u03bd \u03c0\u03bf\u03b9\u03bf\u1fe6. \u0395\u1f30 \u03bc\u1f72\u03bd \u1f10\u03c0\u1f76 \u03c4\u1f78 \u03bc\u03b5\u03bb\u03ac\u03bd\u03c4\u03b5\u03c1\u03bf\u03bd \u1f22 \u03bb\u03b5\u03c5\u03ba\u03cc\u03c4\u03b5\u03c1\u03cc\u03bd \u1f10\u03c3\u03c4\u03b9\u03bd \u1f21 \u03c7\u03c1\u03bf\u03b9\u1f70 \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u1f31\u03b5\u03c1\u1fb6\u03c2 \u1f00\u03bd\u03c4\u03b9\u03b4\u03cc\u03c4\u03bf\u03c5 \u03ba\u03b1\u03b8\u03b1\u03af\u03c1\u03bf\u03bc\u03b5\u03bd. \u0395\u1f30 \u03b4\u1f72 \u1f10\u03c0\u1f76 \u03c4\u1f78 \u1f60\u03c7\u03c1\u03cc\u03c4\u03b5\u03c1\u03bf\u03bd (\u03ba\u03af\u03c4\u03c1\u03b9\u03bd\u03bf\u03bd Ro113 (yellow Ro1), the aloe- and bitter [remedies]\u2026On alopecia and ophiasis. Alopecia is when the hair of the head falls out because of a hot and corrosive humour cutting off the roots. Ophiasis has a different form. For ophiasis is smoother like snakes. Alopecia is far more rough. Assess the superfluous humour from the colour of the skin, and first remove it. If the colour is towards more black or white, we purge with the holy (antidote). If it is paler14:The text given below has been transcribed from Plut. 75, 6 4. \u03a0\u03b5\u03c1\u1f76 \u03c0\u03b9\u03c4\u03c5\u03c1\u03b9\u03ac\u03c3\u03b5\u03c9\u03c2. \u1f29 \u03c0\u03b9\u03c4\u03c5\u03c1\u03af\u03b4\u03b1 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03bc\u1f72\u03bd \u1f10\u03c0\u1f76 \u03c0\u03bb\u03ad\u03bf\u03bd \u03b5\u1f30\u03c2 \u03c4\u1f74\u03bd \u03ba\u03b5\u03c6\u03b1\u03bb\u03ae\u03bd. \u0393\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03b5\u1f30\u03c2 \u03c4\u1f78 \u1f04\u03bb\u03bb\u03bf \u03b4\u03ad\u03c1\u03bc\u03b1 \u03c4\u1fc6\u03c2 \u03c3\u03b1\u03c1\u03ba\u1f78\u03c2 \u03c4\u1fc6\u03c2 \u03bb\u03b5\u03c5\u03ba\u1fc6\u03c2 \u03ba\u03b1\u1f76 \u03c0\u03b9\u03c4\u03c5\u03c1\u03bf\u03b5\u03b9\u03b4\u03bf\u1fe6\u03c2 \u03c7\u03c9\u03c1\u1f76\u03c2 \u1f11\u03bb\u03ba\u03ce\u03c3\u03b5\u03c9\u03c2. \u0393\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b4\u1f72 \u1f45\u03c4\u03b1\u03bd \u1f00\u03bd\u03ad\u03bb\u03b8\u03b7 \u1f55\u03bb\u03b7 \u03c4\u03b9\u03c2 \u03b5\u1f30\u03c2 \u03c4\u1f74\u03bd \u03ba\u03b5\u03c6\u03b1\u03bb\u1f74\u03bd \u1f22 \u1f00\u03c0\u1f78 \u03c6\u03bb\u03ad\u03b3\u03bc\u03b1\u03c4\u03bf\u03c2 \u1f01\u03bb\u03bc\u03c5\u03c1\u03bf\u1fe6 \u1f22 \u03c7\u03bf\u03bb\u03ce\u03b4\u03bf\u03c5\u03c2 \u03ba\u03b1\u1f76 \u03bc\u03b5\u03bb\u03b1\u03b3\u03c7\u03bf\u03bb\u03b9\u03ba\u03bf\u1fe6 \u03b1\u1f35\u03bc\u03b1\u03c4\u03bf\u03c2. \u039a\u03b1\u1f76 \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03c0\u03c1\u1ff6\u03c4\u03bf\u03bd \u1f35\u03bd\u03b1 \u03ba\u03b5\u03bd\u03ce\u03c3\u03b7\u03c2 \u1f00\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u1f45\u03bb\u03bf\u03c5 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u1f78\u03bd \u03c0\u03bb\u03b5\u03bf\u03bd\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1 \u03c7\u03c5\u03bc\u1f78\u03bd \u03ba\u03b1\u1f76 \u03b4\u03b9\u1f70 \u03c6\u03bb\u03b5\u03b2\u03bf\u03c4\u03bf\u03bc\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u1f11\u03c4\u03ad\u03c1\u03b1\u03c2 \u03ba\u03b1\u03b8\u03ac\u03c1\u03c3\u03b5\u03c9\u03c2. \u1f1c\u03c0\u03b5\u03b9\u03c4\u03b1 \u03ba\u03b9\u03bc\u03c9\u03bb\u03af\u03b1\u03bd \u03b2\u03c1\u03ad\u03be\u03b1\u03c2 \u1f55\u03b4\u03b1\u03c4\u03b9 \u03bc\u03af\u03be\u03bf\u03bd \u03ba\u03b1\u1f76 \u03c3\u03b5\u03cd\u03c4\u03bb\u03bf\u03c5 \u03b6\u03c9\u03bc\u1f78\u03bd \u03ba\u03b1\u1f76 \u1f04\u03bb\u03b5\u03b9\u03c6\u03b5.Translation: On pityriasis. Pityriasis happens mainly on the head. It also happens on the other white and bran like skin of the flesh without rupturing. It happens when some matter goes up into the head or from salty phlegm or bilious and melancholic blood. And first you should empty the superfluous humour from the whole body both through blood letting and other removal. Then rinse earth from Cimolus with water, mix also beet root juice and apply.8. \u03a0\u03b5\u03c1\u1f76 \u1f00\u03bb\u03c9\u03c0\u03b5\u03ba\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u1f40\u03c6\u03b9\u03ac\u03c3\u03b5\u03c9\u03c2. \u1f08\u03bb\u03c9\u03c0\u03b5\u03ba\u03af\u03b1 \u1f10\u03c3\u03c4\u03af\u03bd, \u1f45\u03c4\u03b1\u03bd \u03b1\u1f31 \u03c4\u03c1\u03af\u03c7\u03b5\u03c2 \u1f00\u03c0\u03bf\u03c0\u03ad\u03c3\u03c9\u03c3\u03b9 \u03c4\u1fc6\u03c2 \u03ba\u03b5\u03c6\u03b1\u03bb\u1fc6\u03c2 \u1f00\u03c0\u1f78 \u03b1\u1f30\u03c4\u03af\u03b1\u03c2 \u03b8\u03b5\u03c1\u03bc\u1ff6\u03bd \u03c7\u03c5\u03bc\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b1\u03b2\u03c1\u03c9\u03c4\u03b9\u03ba\u1ff6\u03bd \u03c4\u03c1\u03c9\u03b3\u03cc\u03bd\u03c4\u03c9\u03bd \u03ba\u03b1\u1f76 \u03ba\u03bf\u03c0\u03c4\u03cc\u03bd\u03c4\u03c9\u03bd \u03c4\u1f70\u03c2 \u1fe5\u03af\u03b6\u03b1\u03c2 \u03c4\u1ff6\u03bd \u03c4\u03c1\u03b9\u03c7\u1ff6\u03bd. \u0394\u03b9\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u1f14\u03c7\u03b5\u03b9 \u1f10\u03bd\u03b1\u03bb\u03bb\u03b1\u03b3\u1f74\u03bd \u1f21 \u1f40\u03c6\u03af\u03b1\u03c3\u03b9\u03c2 \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f78 \u03c3\u03c7\u1fc6\u03bc\u03b1. \u1f29 \u03b3\u1f70\u03c1 \u1f40\u03c6\u03af\u03b1\u03c3\u03b9\u03c2 \u03bb\u03b5\u03b9\u03bf\u03c4\u03ad\u03c1\u03b1 \u1f10\u03c3\u03c4\u03b9 \u03ba\u03b1\u1f76 \u1f41\u03bc\u03b1\u03bb\u03c9\u03c4\u03ad\u03c1\u03b1 \u1f65\u03c3\u03c0\u03b5\u03c1 \u03bf\u1f31 \u1f44\u03c6\u03b5\u03b9\u03c2. \u1f29 \u03b4\u1f72 \u1f00\u03bb\u03c9\u03c0\u03b5\u03ba\u03af\u03b1 \u1f10\u03c3\u03c4\u03b9 \u03c4\u03c1\u03b1\u03c7\u03b5\u1fd6\u03b1. \u03a3\u03c4\u03bf\u03c7\u03ac\u03b6\u03bf\u03c5 \u03b4\u1f72 \u1f00\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u03c7\u03c1\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u03bf\u1fe6 \u03b4\u03ad\u03c1\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u1f78\u03bd \u03c0\u03bb\u03b5\u03bf\u03bd\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1 \u03c7\u03c5\u03bc\u1f78\u03bd \u03ba\u03b1\u1f76 \u03c0\u03bf\u03af\u03b5\u03b9 \u03c0\u03c1\u1ff6\u03c4\u03bf\u03bd \u03c4\u03bf\u03cd\u03c4\u03bf\u03c5 \u03c4\u1f74\u03bd \u1f10\u03ba\u03b2\u03bf\u03bb\u1f74\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f74\u03bd \u03ba\u03ac\u03b8\u03b1\u03c1\u03c3\u03b9\u03bd. \u039a\u03b1\u1f76 \u03b5\u1f30 \u03bc\u1f72\u03bd \u1f14\u03c3\u03c4\u03b9 \u03c4\u1f78 \u03c7\u03c1\u1ff6\u03bc\u03b1 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78 \u03bc\u03b5\u03bb\u03b1\u03bd\u03ce\u03c4\u03b5\u03c1\u03bf\u03bd \u1f22 \u03c0\u03c1\u1f78\u03c2 \u03bb\u03b5\u03c5\u03ba\u03ce\u03c4\u03b5\u03c1\u03bf\u03bd \u03c0\u03bf\u03b9\u03bf\u1fe6 \u03bc\u1f72\u03bd \u03c4\u1f74\u03bd \u03ba\u03ac\u03b8\u03b1\u03c1\u03c3\u03b9\u03bd \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u1f31\u03b5\u03c1\u1fb6\u03c2 \u1f00\u03bd\u03c4\u03b9\u03b4\u03cc\u03c4\u03bf\u03c5. \u0395\u1f30 \u03b4\u1f72 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78 \u1f60\u03c7\u03c1\u03cc\u03c4\u03b5\u03c1\u03bf\u03bd \u03ba\u03b1\u1f76 \u03ba\u03b9\u03c4\u03c1\u03b9\u03bd\u03ce\u03c4\u03b5\u03c1\u03bf\u03bd \u03ba\u03b1\u03b8\u03b1\u03af\u03c1\u03bf\u03bc\u03b5\u03bd \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u1f00\u03bb\u03cc\u03b7\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1fc6\u03c2 \u03c0\u03b9\u03ba\u03c1\u1fb6\u03c2\u2026Translation: On alopecia and ophiasis. Alopecia is when the hair of the head falls off because of an aetiology of hot and corrosive eating away humours that cut off the roots of the hair. Ophiasis differs and diverges according to its form. For ophiasis is smoother and more even like snakes. Alopecia is rough. Assess the superfluous humour from the colour of the skin, and first expel and remove it. And if the colour is more towards black or towards white, remove it through the holy antidote. And if it is more towards paler and yellower, we remove it through the aloe- and the bitter [remedy]\u20264. \u03a0\u03b5\u03c1\u1f76 \u03c0\u03b9\u03c4\u03c5\u03c1\u03b9\u03ac\u03c3\u03b5\u03c9\u03c2. \u1f29 \u03c0\u03b9\u03c4\u03c5\u03c1\u03af\u03b1\u03c3\u03b9\u03c2 \u03bb\u03b5\u03c0\u03c4\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03c0\u03b9\u03c4\u03c5\u03c1\u03bf\u03b5\u03b9\u03b4\u1ff6\u03bd \u03bb\u03b5\u03c0\u03b9\u03b4\u03af\u03c9\u03bd \u1f10\u03ba \u03c4\u1fc6\u03c2 \u1f10\u03c0\u03b9\u03c6\u03b1\u03bd\u03b5\u03af\u03b1\u03c2 \u03c4\u1fc6\u03c2 \u03ba\u03b5\u03c6\u03b1\u03bb\u1fc6\u03c2 \u03ba\u03b1\u1f76 \u03c4\u03bf\u1fe6 \u03bb\u03bf\u03b9\u03c0\u03bf\u1fe6 \u03c3\u03ce\u03bc\u03b1\u03c4\u03bf\u03c2 \u1f00\u03c0\u03cc\u03c4\u03b5\u03be\u03b9\u03c2 \u1f10\u03c3\u03c4\u1f76\u03bd \u1f04\u03bd\u03b5\u03c5 \u1f11\u03bb\u03ba\u03ce\u03c3\u03b5\u03c9\u03c2. \u0393\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b4\u1f72 \u1f10\u03ba \u03ba\u03b1\u03ba\u03bf\u03c7\u03c5\u03bc\u03af\u03b1\u03c2 \u1f00\u03bd\u03b5\u03bd\u03b5\u03c7\u03b8\u03b5\u03af\u03c3\u03b7\u03c2 \u03c0\u03b5\u03c1\u1f76 \u03c4\u1f74\u03bd \u03ba\u03b5\u03c6\u03b1\u03bb\u1f74\u03bd \u1f22 \u03c6\u03bb\u03ad\u03b3\u03bc\u03b1\u03c4\u03bf\u03c2 \u1f01\u03bb\u03bc\u03c5\u03c1\u03bf\u1fe6 \u1f22 \u03c7\u03bf\u03bb\u03ce\u03b4\u03bf\u03c5\u03c2 \u03ba\u03b1\u1f76 \u03bc\u03b5\u03bb\u03b1\u03b3\u03c7\u03bf\u03bb\u03b9\u03ba\u03bf\u1fe6 \u03b1\u1f35\u03bc\u03b1\u03c4\u03bf\u03c2. \u039a\u03b1\u1f76 \u03c0\u03c1\u1ff6\u03c4\u03bf\u03bd \u03b4\u03b5\u1fd6 \u03c4\u1f78\u03bd \u1f30\u03b1\u03c4\u03c1\u1f78\u03bd \u03ba\u03b5\u03bd\u03bf\u1fe6\u03bd \u03c4\u1f78 \u03c3\u1ff6\u03bc\u03b1 \u1f10\u03ba \u03c4\u03bf\u1fe6 \u03c0\u03bb\u03b5\u03bf\u03bd\u03b5\u03ba\u03c4\u03bf\u03cd\u03bd\u03c4\u03bf\u03c2 \u03c7\u03c5\u03bc\u03bf\u1fe6 \u03b4\u03b9\u1f70 \u03c6\u03bb\u03b5\u03b2\u03bf\u03c4\u03bf\u03bc\u03af\u03b1\u03c2 \u1f22 \u03ba\u03b1\u03b8\u03ac\u03c1\u03c3\u03b5\u03c9\u03c2 \u03c0\u03c1\u03bf\u03c3\u03b7\u03ba\u03bf\u03cd\u03c3\u03b7\u03c2 \u03c4\u1ff7 \u03c7\u03c5\u03bc\u1ff7. \u0395\u1f36\u03c4\u03b1 \u03c7\u03c1\u1fc6\u03c3\u03b8\u03b1\u03b9 \u1f00\u03bb\u03b5\u03af\u03bc\u03bc\u03b1\u03c3\u03b9\u03bd. \u1f1c\u03c3\u03b5\u03c4\u03b1\u03b9 \u03b4\u1f72 \u1f04\u03bb\u03b5\u03b9\u03bc\u03bc\u03b1 \u1f21 \u03ba\u03b9\u03bc\u03c9\u03bb\u03af\u03b1 \u03c0\u03c1\u03bf\u03c3\u03b2\u03c1\u03b1\u03c7\u03b5\u1fd6\u03c3\u03b1 \u1f55\u03b4\u03b1\u03c4\u03b9 \u03ba\u03b1\u1f76 \u03c4\u03b5\u03cd\u03c4\u03bb\u1ff3 \u03c7\u03c5\u03bc\u1ff7 \u03bc\u03b9\u03c7\u03b8\u03b5\u1fd6\u03c3\u03b1 \u1f00\u03bb\u03b5\u03b9\u03c6\u03bf\u03bc\u03ad\u03bd\u03b7.Translation: On pityriasis. Pityriasis is a melting of smooth and bran like scales from the periphery of the head and the rest of the body without rupturing. It happens from bad humours that are brought up around the head or from salty phlegm or bilious and melancholic blood. And first the doctor has to empty the superfluous humour from the body through blood letting or a form of purging that is appropriate for the humour. Then to use salves. The salve will be earth from Cimolus rinsed with water and mixed with beet root juice applied.8. \u03a0\u03b5\u03c1\u1f76 \u1f00\u03bb\u03c9\u03c0\u03b5\u03ba\u03af\u03b1\u03c2 \u03ba\u03b1\u1f76 \u1f40\u03c6\u03b9\u03ac\u03c3\u03b5\u03c9\u03c2. \u1f08\u03bb\u03c9\u03c0\u03b5\u03ba\u03af\u03b1 \u1f10\u03c3\u03c4\u03b9 \u03bc\u03ac\u03b4\u03b7\u03c3\u03b9\u03c2 \u03c4\u03c1\u03b9\u03c7\u1ff6\u03bd \u03ba\u03b5\u03c6\u03b1\u03bb\u1fc6\u03c2, \u1f00\u03c0\u03bf\u03c0\u03af\u03c0\u03c4\u03bf\u03c5\u03c3\u03b9 \u03b4\u1f72 \u03b1\u1f31 \u03c4\u03c1\u03af\u03c7\u03b5\u03c2 \u03b4\u03b9\u1f70 \u03c7\u03c5\u03bc\u1f78\u03bd \u03b8\u03b5\u03c1\u03bc\u1f78\u03bd \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b1\u03b2\u03c1\u03c9\u03c4\u03b9\u03ba\u1f78\u03bd \u03c4\u1f70\u03c2 \u1fe5\u03af\u03b6\u03b1\u03c2 \u03b1\u1f50\u03c4\u1ff6\u03bd \u1f00\u03c0\u03bf\u03c3\u03c0\u1ff6\u03bd\u03c4\u03b1. \u0394\u03b9\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 \u03b4\u1f72 \u1f21 \u1f40\u03c6\u03af\u03b1\u03c3\u03b9\u03c2 \u03c4\u1ff7 \u03c3\u03c7\u03ae\u03bc\u03b1\u03c4\u03b9. \u1f29 \u03b3\u1f70\u03c1 \u1f40\u03c6\u03af\u03b1\u03c3\u03b9\u03c2 \u03bb\u03b5\u03b9\u03bf\u03c4\u03ad\u03c1\u03b1 \u1f10\u03c3\u03c4\u1f76\u03bd \u1f65\u03c3\u03c0\u03b5\u03c1 \u03bf\u1f31 \u1f44\u03c6\u03b5\u03b9\u03c2. \u1f29 \u03b4\u1f72 \u1f00\u03bb\u03c9\u03c0\u03b5\u03ba\u03af\u03b1 \u03c4\u03c1\u03b1\u03c7\u03c5\u03c4\u03ad\u03c1\u03b1 \u03c0\u03bf\u03bb\u03bb\u1ff7. \u03a3\u03c4\u03bf\u03c7\u03ac\u03b6\u03bf\u03c5 \u03b4\u1f72 \u1f00\u03c0\u1f78 \u03c4\u1fc6\u03c2 \u03c4\u03bf\u1fe6 \u03b4\u03ad\u03c1\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c7\u03c1\u03bf\u03b9\u1fb6\u03c2 \u03c4\u1f78\u03bd \u03c0\u03bb\u03b5\u03bf\u03bd\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1 \u03c7\u03c5\u03bc\u1f78\u03bd \u03ba\u03b1\u1f76 \u03c4\u03bf\u03cd\u03c4\u03bf\u03c5 \u03c4\u1f74\u03bd \u03ba\u03ac\u03b8\u03b1\u03c1\u03c3\u03b9\u03bd \u03c0\u03c1\u1ff6\u03c4\u03bf\u03bd \u03c0\u03bf\u03b9\u03bf\u1fe6. \u0395\u1f30 \u03bc\u1f72\u03bd \u03b3\u1f70\u03c1 \u1f10\u03c0\u1f76 \u03c4\u1f78 \u03bc\u03b5\u03bb\u03ac\u03bd\u03c4\u03b5\u03c1\u03bf\u03bd \u1f22 \u03bb\u03b5\u03c5\u03ba\u03cc\u03c4\u03b5\u03c1\u03bf\u03bd \u1f10\u03c3\u03c4\u1f76\u03bd \u1f21 \u03c7\u03c1\u03bf\u03b9\u1f70 \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u1f31\u03b5\u03c1\u1fb6\u03c2 \u03ba\u03b1\u03b8\u03b1\u03af\u03c1\u03bf\u03bc\u03b5\u03bd. \u0395\u1f30 \u03b4\u1f72 \u1f10\u03c0\u1f76 \u03c4\u1f78 \u1f60\u03c7\u03c1\u03cc\u03c4\u03b5\u03c1\u03bf\u03bd \u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u1f00\u03bb\u03cc\u03b7\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1fc6\u03c2 \u03c0\u03b9\u03ba\u03c1\u1fb6\u03c2\u2026Translation: On alopecia and ophiasis. Alopecia is baldness of the hair of the head. The hair falls off because of a hot and corrosive humour that detaches their roots. Ophiasis has a different form, for ophiasis is smoother like snakes. Alopecia is much rougher. Assess the superfluous humour from the colour of the skin, and first remove it. For if the colour is more towards blacker or whiter, we clean it through the Holy [remedy]. If it is towards the paler, through the aloe- and the bitter [remedy]\u2026In the samples presented above, the transmission of the mainstream text is fairly consistent, and also sufficiently consistent with the Bernard edition. As it is to be expected, a number of manuscripts show minor variants, but except one which will be discussed later on, these variants are irrelevant for the argument. This gives us an ideal backdrop for our analysis.15. This is not an uncommon occurrence in medieval manuscript transmission \u2013 in fact, the two main principles a textual critic has to work with arelectio brevior potior andlectio difficilior potior, the shorter variant is the more likely  one and the more difficult variant is the likely  one16. They also replaced a word that is slightly difficult to understand, namely \u1f00\u03bd\u03b5\u03bd\u03b5\u03c7\u03b8\u03b5\u03af\u03c3\u03b7\u03c2 \u201cbrought up\u201d, along with a preposition referring to it17. This word forms part of a sentence explaining that pityriasis is caused by bad humours being \u201cbrought up around the head\u201d. This sentence is not easily understood without any knowledge of ancient medical terminology, and it may at first sight give the impression that it was somehow corrupted in the course of the transmission. However, it was already extant in an important source of Theophanes, Paul of Aegina18. The word used in Paul in the majority of manuscripts is \u1f00\u03bd\u03b5\u03bd\u03b5\u03c7\u03b8\u03b5\u03af\u03c3\u03b7\u03c2, with one witness reading \u1f00\u03bd\u03b1\u03c7\u03b5\u03b8\u03b5\u03af\u03c3\u03b7\u03c2 \u201cpoured up\u201d.First, a comparison of the mainstream text versus the Florence/Wellcome group. In the chapter on pityriasis, the editor rephrased the text, making it easier to understand, and slightly more vernacular19. In the remainder of the passage discussed in this article, the editor just rephrases the text without any major changes to the content.In the mainstream version, and in Paul, the phrase essentially means that bad humours are being excreted through the scalp, which then manifests in bran-like deposits forming on the surface of the skin. The editor of the Florence/Wellcome version rephrased this, stating that pityriasis is caused by \u201cmatter\u201d going into the head. This makes the text easier to understand, but also fairly vague. The \u201cbad humours\u201d become neutral \u201cmatter\u201d, and it is not clear that the matter is ultimately excreted through the skin. Even more problematic is that the editor then abruptly reverts to readings close to the mainstream text a few words later, saying that alternatively pityriasis may also be caused by two different humours, which obviously does not make senseIn the second paragraph, the editor of the Florence/Wellcome version made a number of changes to the text, which do not have any bearing on the content. For instance, they added a synonym, e.g. in the phrase \u03b4\u03b9\u03b1\u03b2\u03c1\u03c9\u03c4\u03b9\u03ba\u1ff6\u03bd \u03c4\u03c1\u03c9\u03b3\u03cc\u03bd\u03c4\u03c9\u03bd, two words meaning \u201ceating away\u201d. Otherwise, the text is slightly simplified, but not remarkably so. On two occasions, however, the editor makes decisive changes. At the beginning, they use the word \u03b1\u1f30\u03c4\u03af\u03b1 \u201ccause\u201d, to clarify that they are moving on from a description of the symptoms to an analysis of the causes of the disease. This is certainly correct, but not strictly necessary, as this would have been evident to a medically trained reader anyway. The second intervention is of a more drastic nature.20. This word is not extant in the corresponding passage in Paul of Aegina21, which was the source for the mainstream version. It is, however, extant in one of the manuscripts transmitting the mainstream version, Ro1, which replaces the word \u201cpaler\u201d with \u201cyellow\u201d. But, to complicate matters, Ro1 comes from an entirely different end of the stemma of the Theophanes tradition than the Florence/Wellcome version, and none of the manuscripts within its branch share the reading \u03ba\u03af\u03c4\u03c1\u03b9\u03bd\u03bf\u03bd22. Therefore, Ro1 cannot be the source for the Florence/Wellcome version, unless one assumes some form of contamination23.After a brief outline of the aetiology of this type of hair loss, i.e. an abundance of three possible humours, the mainstream version advises to assess the colour of the skin to determine which humour it is, and then administer the correct treatment. A colour resembling black or white would necessitate one specific treatment, a more pale colour another. Here, the editor of the Florence/Wellcome version adds another option after \u201cmore pale\u201d, namely yellow. Here, they use a late, and somewhat vernacular, word to describe yellow, \u03ba\u03af\u03c4\u03c1\u03b9\u03bd\u03bf\u03bdDe Curatione: neither of the two versions of Ioannes archiatrus talk about yellow skin24. This text goes back to a source that was very close to Theophanes. Moreover, the chapter on alopecia in Leo medicus (another text somewhat related to Theophanes) does not discuss the colour of the skin at all25. Alexander of Tralles, whoseTherapeutics share some links with the wider transmission of Theophanes, discusses skin colour in his chapter on alopecia, but does not mention any yellow26. The source for all these authors, including Theophanes and Paul, is, directly or indirectly, Galen's vast and influential workMethod of Healing, who equally does not mention yellow27.Consequently, this small detail turns out to be quite significant, as it appears in two ends of the transmission independently. Curiously, it is absent from both the source of Theophanes and some other texts that are connected toNext, I am going to examine the relationship between the mainstream transmission and the Escorial manuscript. The first part of the chapter on pityriasis is almost identical, except for one synonym. However, the editor suddenly adds some important information further on. Rather than just saying that \u201cone\u201d should remove the offending humour from the body, the editor specifies that this should be done by a \u201cdoctor\u201d. The editor then adds a sentence to clarify the structure of the chapter, and the final sentence of the excerpt is rephrased, but without any major changes to the content covered.In the second chapter, the editor of the Escorial manuscript rephrased the first part of the text, again without making any major changes to the meaning. As a result of these changes, the Escorial version is easier to understand than the mainstream text. The remainder of the text is almost identical to the mainstream.iatrosophion. The title is followed by a pinax, or table of contents, and then by the beginning of Theophanes'De curatione. On f. 145v to 146r, respectively, it contains the chapters on pityriasis and alopecia discussed in this article without any major alterations. This is then followed by some more chapters from Theophanes, roughly up to chapter 18 (Bernard) and f. 148v respectively. From f. 149r the text has been taken from another source. So in essence, an editor excerpted some chapters from Theophanes and used them to form part of a new handbook28.The final manuscript to be discussed, Palermo XIII C 3, is of a different nature. It is not so much a rephrased text than a rearranged one. According to its title on f. 121r, the section of the manuscript contains anOverall, what does the analysis of these three versions contribute to our research question? What was the motivation behind these changes? As for the first two versions discussed, a general trend was quite clearly to simplify the text, which could manifest in a slightly more vernacular wording. It is not difficult to guess the motives behind a mere simplification: the editor tried to make the text more accessible. Adding redundant synonyms had the same purpose. If a user did not understand one way of phrasing something, then perhaps he would have been able to understand the other. All this could be done while remaining well within the syntax and lexicon of the learned elite of the time. The motive to adjust the text a bit more to the vernacular does, on the other hand, raise a number of important questions. First of all, one is left wondering who the intended audience of these versions was. And then, it also raises the question of the date at which these changes were made.29, in the first half of the tenth century, which we can confidently use as a terminuspost quem30. The terminusante quem is the fourteenth and sixteenth century respectively, as this is the date of the manuscripts. During this period, the Greek vernacular was already in existence; this can be determined from a few vernacular words transmitted in other sources, such as for instance Leo medicus on the medical side31, but we are not in a position to reconstruct the development of the vernacular fully because of a lack of sources32. Overall, though, it is safe to say that a slight adjustment towards the vernacular fits well into the general linguistic background of the time.As for the date, Theophanes lived, according to his own wordsA slightly simplified and slightly more vernacular version would be more accessible to users who are literate, and educated enough to understand the classicising learned Byzantine idiom that was used in writing, but who were more comfortable with an idiom that was closer to the Greek spoken in everyday life. It is, however, well worth bearing in mind that the idiom was indeed just very slightly adjusted towards the vernacular. The editor did not translate the text.aitia \u201ccause\u201d, and the latter \u201cdoctor\u201d. This shows at least a basic understanding of medicine and a familiarity with the structure of medical handbooks.Both editors had at the very least basic medical training along with some philological skills. Even though the process of redacting the text seems basic and straightforward to us, it does actually take some confidence and determination to prepare, in the eyes of the editor, an improved version of a text. As far as their medical training is concerned, both the editors of the Florence/Wellcome and the Escorial version add some new information \u2013 the former the word33. In any case, the addition of an identifiable colour such as yellow \u2013 the word \u03ba\u03af\u03c4\u03c1\u03b9\u03bd\u03bf\u03bd means \u201clemon coloured\u201d \u2013 makes the text more precise. In Classical Greek, names of colours are notoriously difficult to interpret, and in many cases we can only guess what they may have referred to. In Galen's description of the scalp colour in alopecia patients, the words black and white are used \u2013 this much is clear \u2013 and then the rather unspecified \u201cpale\u201d. This terminology was very much the state of the art in Galen's time. We cannot be entirely certain about the Greek Middle Ages, as more basic research needs to be carried out on the development and characteristics of this idiom.We cannot be certain where the word \u201cyellow\u201d originated, which was added in the Florence/Wellcome version. Perhaps it was already in existence in the manuscript that was used by the editor; alternatively, it may have come from a common, possibly even oral source such as an oral teaching tradition34. Alternatively, they may have made a conscious decision to select certain passages from one source and others from another source. In any case, the way it was done reveals that we are yet again dealing with an editor who had at least basic medical training \u2013 they knew how to structure a therapeutic handbook \u2013 and who also had some philological skills.The Palermo manuscript presents a somewhat different picture, as far as the use and modification of sources is concerned. Here, someone compiled a text using more than one manuscript, and we cannot be certain about the reasons behind this decision. It is equally possible that the editor only had fragmentary sources at their disposal and then decided to stitch it all together to form a manualSo altogether, the analysis has shown that three different people with similar skill sets worked on and with Theophanes' text. They were confident handling the material, yet they were not part of the top range of brilliant scholars of the time. What we can see here is a more intermediary layer of abilities. Their attitude towards an author such as Theophanes was very different from the attitude of ordinary scribes who aimed to preserve the works of earlier authorities as accurately as possible.All data is presented in the main text. I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. To all four referees:First of all many thanks for your input. I would like to address a few points from the reviews in a comment rather than in the main text body of my article.The question about the historical context of the transmission is a very good one, and MC already provided all there is to say in her review.The reason why I kept my discussion of the Palermo manuscript brief, and focused on the Theophanes passages in question is because a PhD student in Germany is working on this manuscript. We divided the topic between us so that there is not going to be any undue overlap.The \u201cvernacular\u201d:I use the word \u201cvernacular\u201d to describe the medieval, and to a certain extent also early modern, idiom (that was mostly) spoken in the Greek world. This term is generally understood in anglophone scholarship, but it is far from ideal for all the reasons outlined by the reviewers. However, until a more fitting term is found, I shall continue to use \u201cvernacular\u201d.Clavis Sanationis.We have only very few edited testimonies of texts that are solely written in the vernacular, but many more exist. Some more evidence can be extracted from other sources, such as for instance Leo medicus, as referenced in my article. The pronunciation of the vernacular is for instance described in Latin in some lemmata of Simon of Genoa'sThere are indeed a number of major studies in the history of the Greek language. However, these studies all use different terminology to describe what I call the vernacular. Here are some samples:Medieval and Modern Greek. Cambridge: CUP. On p. 55, he talks about informal, living speech and subliterary texts representing an uneasy balance between the purist ideal and the speech of the people.- R. Browning 1969.Greek. A History of the Language and its Speakers. London: Longman. On p. 153-155 he talks about vernacular Greek.- G. Horrocks 1997.A History of the Greek Language. From its Origins to the Present. Leiden: Brill. On p. 245 he talks about popular texts.- F. Rodriguez Adrados 2005.A Companion to the Ancient Greek Language, Chichester: Blackwell, p. 539-563. Many thanks to Petros Bouras-Vallianatos to pointing this out.A good analysis can be found in: D. Holton, I. Manolessou, \u201cMedieval and Early Modern Greek\u201d, in E. Bakker (2005).The general debate about the nature of modern Greek is also ongoing, which is certainly going to have an influence on the discussion on the medieval vernacular.Altogether, it is quite clear that the question of diglossia in the Middle Ages cannot be resolved here, but perhaps the matters raised in the reviews might help further the discussion.The \u201cmainstream\u201d version:I use the word \u201cmainstream\u201d to describe the most frequently transmitted version of a medical text.The odd \u03c0\u03b1\u03c1\u03ac and the syntax of the versions:I have thought about this a lot, when I wrote the article, and then again when I read the review, and I came to the conclusion that I shall leave it as it stands, and also in particular as we would need a better understanding of the medieval grammar and lexicon to resolve this.The syntax has indeed been simplified in the versions.kitrinon:The colour \u03ba\u03af\u03c4\u03c1\u03b9\u03bd\u03bf\u03bdth century manuscript (starts from line 3 of the second paragraph).http://gallica.bnf.fr/ark:/12148/btv1b10722308m/f189.item The language of the text could very well be consistent with an earlier, medieval date.It is notoriously difficult to identify colours in ancient or medieval texts. This is in particular the case as only very few sources have even been edited. Surprisingly, a TLG search did indeed yield one good source. It is transmitted in the Par. gr. 2329 f. 184v, a 17kitrina. Consider egg yolk and the dry bile of akitrinou bull.\u039a\u03b1\u1f76 \u03ba\u03c1\u03cc\u03ba\u03bf\u03bd \u03b2\u03ac\u03bb\u03b5 \u03ba\u03b1\u1f76 \u03ba\u03bf\u03c1\u03ba\u03bf\u03c5\u03bc\u1f70\u03bd, \u03ba\u03b1\u1f76 \u03bc\u03ad\u03bb\u03b9, \u03ba\u03b1\u1f76 \u1f04\u03bb\u03bb\u03b1 \u03ba\u03af\u03c4\u03c1\u03b9\u03bd\u03b1\u00b7 \u03bd\u03cc\u03b5\u03b9 \u03ba\u03c1\u03cc\u03ba\u03bf\u03c5\u03c2 \u1f60\u1ff6\u03bd \u03ba\u03b1\u1f76 \u03c7\u03bf\u03bb\u1f74\u03bd \u03b2\u03bf\u1f78\u03c2 \u03ba\u03b9\u03c4\u03c1\u03af\u03bd\u03bf\u03c5 \u03be\u03b7\u03c1\u03ac\u03bd. - And add saffron and turmeric and honey and other things that areThis recipe is mentioned in connection with a procedure to make a metal appear like gold, and it is fairly clear that the colour the author had in mind was yellow.Citron or lemon:These modern English words designate two distinct types of fruit. However, it would be problematic to assume that these two words consistently described two distinct types of fruit in Classical and Medieval Greek as well. Citrons and lemons look very similar indeed, and it is certainly possible that these names were not used consistently. This would also be supported by the fact that the word for lemon is derived from the Greek word for citron in at least two different European language that immediately come to my mind, German and Polish.We would need to gather more Greek textual evidence to address this question. What is currently available on the TLG does not suffice, and the situation may be complicated by regional dialects as well.But perhaps it is worth pointing out that both words are relatively rarely attested on the TLG, which in turn raises a lot of questions.Theological texts:catenae. For this reason, I adopted some editorial methods from the Nestle Aland edition of the New Testament for my edition of Ioannes archiatrus.For some reason, theological scholarship does not have a problem with texts that were augmented later on, for instance This is a valuable and well researched article of a proper scientific standard. Title, abstract and structure are fine. Conclusions are well drawn. It concerns the history of medicine in the Byzantine period, a rich but seriously under-researched period, apart from the history of manuscripts, which is the topic of the article. The article shows that the scribes in certain parts of the tradition were modifying the text and simplifying it for their own purposes, in contrast with the accurate reproduction of the text in the scholarly branch of the tradition. This process is to be expected in Byzantium and throughout the other cultures - Syrian, Arabic, Hebrew\u00a0and Western European - which adopted the Greek medical corpus. Minor suggestions I would have for the author are:kitrinon in texts of the period. The citron is the citrus plant most likely to be familiar in Byzantium, more so than in texts in the time of Theophrastus or Galen. Its taste and flavours are not lemony any more than an apple tastes like a pear, nor is the colour lemon-yellow. I think it is more green-yellow and quite different from standard yellow words in Greek. Check the use of In the first extract, the author might add, by way of explaining the simplification of the text, that a clause has replaced a genitive absolute. Is it a grammatical simplification as well as a muddled scientific modification? I very much like the conclusion that the scribes were medically literate and had a scientific interest in where or not a 'doctor' was needed, whether or not 'bad humour' was the medical term that was appropriate, and whether or not a Galenic 'cause' should be inserted into the text. These findings need to filter through to the medical and social history of Byzantium.I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. De curatione. It is an important piece of work not only for the useful insights it offers into the different versions of the mainstream text but mostly for its observations on the working methods of different editors, their intellectual background and intended audiences. It will hopefully (re)emphasise the need for similar research which must be conducted in this significant area of study focusing on the circulation and reception of Byzantine medical works.This is an interesting article, which has a persuasive argument revolving around the transmission of sections from Theophanes Chrysobalantes\u2019Page 2, paragraph 2: I think it might be worth clarifying how the term \u201cmainstream\u201d is used throughout. Does this refer to the most widely circulated version of the text?De curatione a therapeutic manual only, or important for its value as a diagnostic manual as well?Page 2, paragraph 2: isPage 2, paragraph 3: What are the exact folio numbers preserving Theophanes\u2019 text in the Palermo manuscript?1, who provides a lengthy discussion on the addressee of Theophanes\u2019 work. \u00a0Page 2, note 3: Constantine Porphyrogenitus\u2019 name needs to be referred to consistently (compared to its occurrence in notes 27 and 28). Also, it is worth consulting and citing the useful article by Joseph A. M. Sonderkamp (1984)Page 2, note 3: Is Bernard\u2019s edition the only available edition of the text? If not, it would be worthwhile to refer to earlier[?] editions.Page 2, \u201cTranscription of the mainstream text\u201d: Why \u201ctranscription\u201d? It is stated above that the mainstream text is printed in line with Bernard\u2019s edition. So why not just say \u201cedition of the mainstream text\u201d? Furthermore, it would be helpful to provide references to page number and lines in Bernard\u2019s edition for all the passages commented upon in the article.Page 2, passage 4 On pityriasis: Since the name of diseases (here and below) are transliterated, they should be given in italics for the reader\u2019s convenience.Page 2, passage 4 \u201cIt happens when bad humours are brought up around the head\u201d: \u03c0\u03b1\u03c1\u03ac + accusative denotes movement towards a person or a thing, so maybe \u201cto\u201d/\u201cinto\u201d the head .Page 2, passage 4, \u201c\u2026cleaning of the superfluous humour\u201d: Better: \u201cevacuation\u201dPage 2-3, passage 8: The translation for \u201c\u0395\u1f30 \u03bc\u1f72\u03bd ... \u03ba\u03b1\u03b8\u03b1\u03af\u03c1\u03bf\u03bc\u03b5\u03bd\u201d is missing.Page 3, passage 8, \u201cthe hair of the head falls out\u201d: maybe \u201cthe hair falls out from the head\u201dPage 2, passage 8, \u201cand first remove it\u201d: perhaps \u201cand the first thing you should do is to purge it\u201dPage 2, passage 8, \u201cthe aloe- and bitter\u201d: rather \u201cby means of aloe and bitter...\u201d in line with \u201c\u03b4\u03b9\u1f70 \u03c4\u1fc6\u03c2 \u1f31\u03b5\u03c1\u1fb6\u03c2 \u1f00\u03bd\u03c4\u03b9\u03b4\u03cc\u03c4\u03bf\u03c5\u201d of the previous sentence.Page 3, Transcription of the Florence/Wellcome group, passage 4 \u201con the other white and bran like skin of the flesh\u201d: \u201con the rest of the skin of the white and bran-like flesh\u201dPage 3, Transcription of the Florence/Wellcome group, passage 4 \u201c\u2026or from salty phlegm\u2026\u201d : \u201c\u2026consisting either of salty phlegm \u2026\u201dPage 3, note 7: it is worth mentioning that the Wellcome manuscript is not listed in Sonderkamp\u2019s sigla.2.Page 6: How do you define the term \u201cvernacular\u201d? Any relevant literature? See, for example, Robert\u00a0Browning (1983)De curatione? Please consider adding some further examples.Overall, do we find similar editorial \u201calterations\u201d in other sections of Theophanes\u2019 The article is suitable for publication in its current form. Some suggestions are provided below, which the author may want to consider and/or address:I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Epitome de curatione morborum.This article considers in a concrete way the thorny question of the revisions medical treatises of the Byzantine era may have undergone. It takes as an example a treatise composed in Byzantium in the 10th century by a certain Theophanes Chrysobalantes, thepityriasis andalopecia, and compares their text as given by the reference edition (published at the end of the 18th century) and by three different revised versions, which B.\u00a0Zipser identified. She shows that in these three versions, the text has been rewritten - on the one hand to adapt it to the linguistic evolutions of Greek language, and on the other hand to become easier to understand. The analysis is well conducted and well argued. The results are convincing. Its approach is relevant. It takes two concrete examples, namely two chapters devoted respectively to the diseasesConcerning the question of the evolution of Greek language in the Byzantine period and the birth of the so-called \"vernacular\" Greek, there must\u00a0exist\u00a0a bibliography in the domain of linguistics. I do not know it personally , but I presume it would be useful, in order to ensure the credibility of this argument, to mention some basic publications in this field. I can not conceive that no one has ever studied the evolution of Greek language in medieval time.Concerning the dating of the revised versions and the conclusions of the article, it seems to me that the three revisions discussed here should be clearly distinguished. Generally, moreover, it would be interesting to provide the readers with some\u00a0context on\u00a0the manuscripts that contain these versions. This would help to find some data to answer to the important question asked by B. Zipser: \"[...] one is left wondering who the intended audience of these versions was.\" Even without depth research, one can get information by taking into account the basic bibliography of the manuscripts. For example :Los manuscritos griegos de El Escorial copiados por Jacobo Diasorino: estudio paleogr\u00e1fico y codicol\u00f3gico, Madrid, Univ. Complutense, 2011 (which I have not read)1 - The ms. Escorial T.III.1 was copied towards the middle of the 16th century by Iakobos Diassorinos . Given the biography of the copyist, it is probable that this took place in Western Europe; there may be information about this in the Masters thesis of P. Garc\u00eda Bueno,\u00a0 Now, Diassorinos is well known as a forger of texts; also, he was quite interested in medicine. There is therefore\u00a0a good chance that the revised version transmitted by the Escorial manuscript would be from\u00a0Diassorinos himself. We can at least ask the question, especially as we do not find this version in any previous manuscript. - The Palermo manuscript comes from Crete, where it was realized in the first half of the 16th century, probably around 1525 . This manuscript is a huge medical compendium (more than 1000 folios!), gathering material from many different sources. No other manuscript resembles it. Could we make the hypothesis that it was its own copyists who did the editing work? We can at least ask the question, since these men already seem to have accomplished a sort of editorial work, by gathering scattered information.2, mentioned in the bibliography of the Biblioteca Laurenziana website). The London manuscript MSL 135, which contains a version very similar to that of the Florence manuscript, was copied in the East, in the middle of the 16th century . Except if it descends from the Florence manuscript (a fact which must be stated clearly), it must go back to a common model. This common model, which must have been\u00a0prior to the second half of the 14th century, was therefore still in the Orient towards the middle of the 16th century. The London manuscript therefore\u00a0confirms that the revision it preserves, together with the Florence manuscript, goes back to the middle of the 14th century at the latest, and that it circulated in Constantinople and its region, where it may have been made. One could make this hypothesis for the origin\u00a0of this revised version even if, actually, nothing proves it. - As for the Florence manuscript, it was copied towards the beginning of the second half of the 14th century in Constantinople . The article would benefit from distinguishing them. \u00a0 However, I have some remarks, which do not detract from the value of the conclusions but aim to make the article more precise: These remarks are only suggestions for further researches; they do not call into question the quality of the article.Real Bibioteca, \u03a4 III 1.The first mention of the Escorial ms.: one could add the Library\u2019s name\u00a0: Escorial,Cat\u00e1logo de los c\u00f3dices griegos de la Biblioteca de El Escorial, vol.\u00a01, Madrid : Imprenta Hel\u00e9nica, 1936, p.\u00a0506-507.3Note 5: There exists a more recent catalogue\u00a0: Revilla, Alejo (O.S.A.),-238v (unless this indication is erroneous).The first mention of the Palermo ms.: add the end folios of the text : f.121r\u201cThe manuscripts from Florence and the Wellcome Library present a near identical text, and are therefore siblings\u201d: one could imagine that the Wellcome ms. may be a copy of the Florence ms. Is this impossible?http://mss.bmlonline.it/s.aspx?Id=AVg6Otf3ADdoerJpI7oR&c=VIII#/book \u00a0.Note\u00a06: it could be useful to add the link to the digitized images:Note\u00a011: since Sonderkamp\u2019s book is not easily accessible, the reader would benefit from the indication of the exact shelfmarks of the manuscripts corresponding to Sonderkamp\u2019s sigla (for example: which ms. is \u201cRo1\u201d?).\u03b9\u03ba\u03bf\u1fe6\u00a0; \u03b1\u1f35\u03bc\u03b1\u03c4\u03bf\u03c2\u00a0; \u03ba\u03b5\u03bd\u1f7d\u03c3\u1fc3\u03c2 ; chap.\u00a08: \u03bb\u03b5\u03c5\u03ba\u1f79\u03c4\u03b5\u03c1\u03bf\u03bd.\"Transcription of the Florence/Wellcome group\u201d: chap.\u00a04: read (or say that you reproduce the ms.\u2019orthography) \u03bc\u03b5\u03bb\u03b1\u03b3\u03c7\u03bf\u03bb\u03b1\u03bd\u03b5\u1f77\u03b1\u03c2; \u1f00\u03c0\u1f79\u03c4\u03b7\u03be\u03b9\u03c2\u00a0; \u03bc\u03b5\u03bb\u03b1\u03b3\u03c7\u03bf\u03bb\u03b9\u03ba\u03bf\u1fe6\u00a0; chap.\u00a08: \u0395\u1f30 \u03b4\u1f72.\u201cTranscription of T III 1\u201d: chap.\u00a04: read (or say that you reproduce the ms.\u2019orthography) \u1f10\u03c0\u03b9\u03c6Paragraph beginning with \"After a brief outline [...]\": one could imagine that the replacement of the classical \u1f60\u03c7\u03c1\u1f79\u03c4\u03b5\u03c1\u03bf\u03bd by the more recent \u03ba\u1f77\u03c4\u03c1\u03b9\u03bd\u03bf\u03bd can have occurred independently in many different manuscripts. It could be just a sort of linguistic update. So B. Zipser is right in presuming no special relation between the Florence/Wellcome version and the Ro1 ms.Therapeutics the same book asDe curatione?Paragraph beginning with \u201cThe final manuscript [...]\u201d: Are Theophanes\u2019Note 29: the reader would appreciate a few more explanations (namely that Leo explicitly quotes a \u201cvernacular\u201d word?). One can not understand this without having a look at Leo\u2019s text.Last phrase: \u201cTheir attitude towards an author such as Theophanes was very different from the attitude of ordinary scribes who aimed to preserve the works of earlier authorities as accurately as possible\u201d: this is not entirely true for all texts. While sacred texts and, in general, texts from famous and most known authors were (mostly) transcribed with respect, it is obvious that scientific and technical texts, above all those attributed to little known authors (to tell nothing about anonymous texts), were much more subject to changes and rewordings. Alas, Theophanes is not the only medical author in this case. Remarks concerning details:I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. This is a solid and useful article, analyzing how and why revisions were introduced to a widely diffused medical text authored in the tenth century. Although the article's conclusions are drawn on the basis of a specific text and its versions in concrete manuscripts (as is appropriate), they can be applied more broadly to the manuscript transmission of other medical  texts.In the four manuscripts under examination, what other texts is Theophanes Chrysobalantos copied with? Do these other texts present characteristics similar to the revised versions of Theophanes Chrysobalantos? In other words, can these manuscripts as integral objects (and not as vehicles for single texts) be construed as professional manuals for practitioners with a particular profile, perhaps reflected in the register at which most of the texts they contain is pitched?In the Escorial manuscript, could the specification that a doctor has to empty the superfluous humor etc, help us understand anything at all about a possible division of labor between doctors and medical assistants? Of course, this one mention of a doctor is not enough. The manuscript could, perhaps, be scrutinized for more clues (which it may or may not contain).What can physical characteristics of the codices , tell us about the post-Byzantine reception of Byzantine medicine ? Do these sixteenth-century manuscripts appear to have been created within an Ottoman or early modern European context? What does this tell us about the role of Byzantine medicine in early Ottoman or early modern European practice? Inquiry in the above three directions may allow a more detailed sociology of the redactors, scribes, and users of Theophanes' text.\u00a0 The article is acceptable as is,\u00a0 but would gain in interest and depth if the author could add some discussion on the following: An issue that does not affect the argument and overall scholarly contribution of the article, but one that the author may wish to address if she chooses to introduce revisions is the following: the article's current discussion of what a vernacular register is and when it is introduced into writing is somewhat old fashioned . All languages with long written traditions have multiple written registers (closer or more remote from the high brow canon). When vernacularisms become manifest in written literature largely depends on how long the surviving written record is. Vernacularisms in written Greek are evident since the translations collectively known as the Septuagint, the compilation of the Christian gospels , the publication of the acts of the church councils, or the composition of best sellers of monastic literature, such as the Spiritual Meadow by John Moschos\u2014in other words, considerably earlier than the tenth century. Given that the earliest surviving Greek medical manuscripts are from the tenth century (e.g. Paul of Aegina now in Paris) and that, in their overwhelming majority, date from the twelfth century and later, our ability to discern vernacularisms in the medical texts earlier than these dates is limited. But this does not mean that a linguistic vernacularization of medical texts was not part of earlier medical practice and training.I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard."}
{"text": "Nowadays, chaos generators are an attractive field for research and the challenge is their realization for the development of engineering applications. From more than three decades ago, chaotic oscillators have been designed using discrete electronic devices, very few with integrated circuit technology, and in this work we propose the use of field-programmable gate arrays (FPGAs) for fast prototyping. FPGA-based applications require that one be expert on programming with very-high-speed integrated circuits hardware description language (VHDL). In this manner, we detail the VHDL descriptions of chaos generators for fast prototyping from high-level programming using Python. The cases of study are three kinds of chaos generators based on piecewise-linear (PWL) functions that can be systematically augmented to generate even and odd number of scrolls. We introduce new algorithms for the VHDL description of PWL functions like saturated functions series, negative slopes and sawtooth. The generated VHDL-code is portable, reusable and open source to be synthesized in an FPGA. Finally, we show experimental results for observing 2, 10 and 30-scroll attractors. Chaos theory deals with nonlinear and complex dynamic behavior that is associated to unpredictable phenomena. The main characteristic is that small changes in the initial conditions lead to drastic changes in the results. It is deterministic because one knows its model parameters and it is unpredictable because one does not know the evolution of the trajectories, and then one cannot predict its behavior.Three decades ago, the author in  introducn-double scrolls . It was a generalization of Chua\u2019s circuit where the 1-double scroll corresponds to the classical double scroll one. Generating more than 2-scrolls was the challenge after Chua\u2019s circuit. In 2004 , and one must decide how many places to use for the integer and fractional parts  is updated to x1 + h\u2217y1, where x1 and y1 are initial conditions and h is the step sizeState variable yn[j] is updated to y1 + h\u2217z1, where y1 and z1 are initial conditions and h is the step sizeState variable x1 is between the break points values B[q] and B[q+1]. If it is satisfied thenVerifies if variable d1PWL is equated to d1\u2217k[p], where d1 corresponds to coefficient d1 in k[p] to a saturation level in Eq (2)Variable n \u2212 1) times, where n is the number of scrolls being generatedLoop to iterate 2(x1 is between the break points B[q+1] and B[q+2]. If it is satisfied thenVerifies if d1PWL is equated to d1\u2217(((k[p+1]-k[p])/(B[q+2]-B[q+1]))\u2217(x1-((B[q+1]+B[q+2])/2))+((k[p+1]+k[p])/2))If step 9 is not satisfied thend1PWL is equated to d1\u2217k[p+1]q by 2Increases p by 1Increases zn[j] is updated to z1+h(-a\u2217x1 \u2212 b\u2217y1 \u2212 c\u2217z1 + d1PWL), where a, b, c are coefficients in Eq (1)State variable x1 is updated to the next iterationState variable y1 is updated to the next iterationState variable x1 is updated to the next iterationState variable j from step 1 is incremented to the next iteration, until st is accomplished.Index To generate even and odd number of scrolls from Algorithm 1. Generating 2-scrolls using saturated function serieswhile j<st:1 \u2003\u2003p = 02 \u2003\u2003\u2003q = 03 \u2003\u2003\u2003xn.insert4 \u2003\u2003\u2003yn.insert5 \u2003\u2003\u2003if x1 > B[q] and x1 < B[q+1]:6 \u2003\u2003\u2003d1PWL = d1\u2217k[p]7 \u2003\u2003\u2003\u2003while q < 2\u2217(n-1):8 \u2003\u2003\u2003if x1 >= B[q+1] and x1 <= B[q+2]:9 \u2003\u2003\u2003\u2003d1PWL = d1\u2217(((k[p+1]-k[p])/(B[q+2]-B[q+1]))\u2217(x1-((B[q+1]+B[q+2])/2))+((k[p+1]+k[p])/2))10 \u2003\u2003\u2003\u2003\u2003elif x1 > B[q+2] and x1 < B[q+3]:11 \u2003\u2003\u2003\u2003d1PWL = d1\u2217k[p+1]12 \u2003\u2003\u2003\u2003\u2003q = q+213 \u2003\u2003\u2003\u2003p = p+114 \u2003\u2003\u2003\u2003zn.insert)15 \u2003\u2003\u2003x1 = xn[j]16 \u2003\u2003\u2003y1 = yn[j]17 \u2003\u2003\u2003z1 = zn[j]18 \u2003\u2003\u2003j = j+119 \u2003\u2003\u2003f(x) must be described as shown in Eqs  is lower than the break point B[0]. If it is satisfied thenVerifies if PWL is equated to m\u2217(x-B[0])+k[0]), where m is the slope, B[0] the break point and k[0] the amplitudeVariable n \u2212 3) times, where n is the number of scrolls being generatedLoop to iterate (2x in f(x) is between the break points B[j] and B[j+1]. If it is satisfied thenVerifies if PWL is equated to m[j]\u2217(x-B[j+1-j1])+k[j+1-j1]j is equal to n-3. If it is satisfied thenVerifies if j is equated to j+2Index j1 is equated to 1Index j fails the above thenVerifies if j is equated to j+1Index x in f(x) is between the break points B[n-2] and B[n-1]. If it is satisfied thenVerifies if PWL is equated to m[n-2]\u2217x, where m[n-2] correponds to slope m1 or m2 in Eqs (2 in Eqs  and 6)PWL is eqx is higher than B[2\u2217n-3]. If it is satisfied thenVerifies if PWL is equated to m\u2217(x-B[2\u2217n-3])+k[2\u2217n-3]x[n] is updated to x+h\u2217), where alpha is coefficient \u03b1 in x and y are initial conditions, and h is the step sizeState variable y[n] is updated to y+h\u2217gamma\u2217(x-y+z), where gamma is coefficient \u03b3 in z is the initial conditionState variable z[n] is updated to z+h\u2217(-beta\u2217y), where beta is coefficient \u03b2 in Eq (4)State variable x is updated to the next iterationState variable y is updated to the next iterationState variable z is updated to the next iterationState variable i from step 1 is incremented to the next iteration, until st is accomplished.Index For Chua\u2019s circuit based on negative slopes, to generate even and odd number of scrolls from n in Eqs  and 6),,f(x) musAlgorithm 2. Generating 2-scrolls using negative slopeswhile i < st:1 \u2003\u2003j = 02 \u2003\u2003\u2003j1 = 03 \u2003\u2003\u2003if x < B[0]:4 \u2003\u2003\u2003PWL = m\u2217(x-B[0])+k[0])5 \u2003\u2003\u2003\u2003while j < 2\u2217n-3:6 \u2003\u2003\u2003if x >= B[j] and x < B[j+1]:7 \u2003\u2003\u2003\u2003PWL = m[j]\u2217(x-B[j+1-j1])+k[j+1-j1]8 \u2003\u2003\u2003\u2003\u2003if j == n-3:9 \u2003\u2003\u2003\u2003\u2003j = j + 210 \u2003\u2003\u2003\u2003\u2003\u2003j1 = 111 \u2003\u2003\u2003\u2003\u2003\u2003else:12 \u2003\u2003\u2003\u2003\u2003j = j + 113 \u2003\u2003\u2003\u2003\u2003\u2003if x >= B[n-2] and x < B[n-1]:14 \u2003\u2003\u2003\u2003PWL = m[n-2]\u2217x15 \u2003\u2003\u2003\u2003\u2003if x >= B[2\u2217n-3]:16 \u2003\u2003\u2003\u2003PWL = m\u2217(x-B[2\u2217n-3])+k[2\u2217n-3]17 \u2003\u2003\u2003\u2003\u2003xn.insert))18 \u2003\u2003\u2003yn.insert)19 \u2003\u2003\u2003zn.insert)20 \u2003\u2003\u2003x = xn[i]21 \u2003\u2003\u2003y = yn[i]22 \u2003\u2003\u2003z = zn[i]23 \u2003\u2003\u2003i = i + 124 \u2003\u2003\u2003f(x) must be described as shown in Eqs (st timesLoop to iterate x in f(x) is lower than the break point B[0]. If it is satisfied thenVerifies if PWL is equated to m(x \u2212 B[0]) + k[0], where m is the slope, B[0] the break point and k[0] the amplitudeVariable x in f(x) is higher than B[n-2], where n is the number of scrolls being generated. If it is satisfied thenVerifies if PWL is equated to m\u2217(x-B[n-2])+k[2\u2217n-3], where m is the slope, B[n-2] the break point and k[2\u2217n-3] the amplituden-2 timesLoop to iterate x in f(x) is between the break points B[j] and B[j+1]. If it is satisfied thenVerifies if PWL is equated to m[j]\u2217(x-(B[j+1]+B[j])/2)x[n] is updated to x+h\u2217alpha(y-PWL)), where alpha is coefficient \u03b1 in x and y are initial conditions, and h is the step sizeState variable y[n] is updated to y+h\u2217gamma(x-y+z), where gamma is coefficient \u03b3 in z the initial conditionState variable z[n] is updated to z+h(-beta\u2217y), where beta is coefficient \u03b2 in Eq (7)State variable x is updated to the next iterationState variable y is updated to the next iterationState variable z is updated to the next iterationState variable i from step 1 is incremented to the next iteration, until st is accomplished.Index For Chua\u2019s circuit based on sawtooth function, to generate even and odd number of scrolls from n in Eqs  and 9),,f(x) musAlgorithm 3. Generating 2-scrolls using sawtooth functionwhile i < st:1 \u2003\u2003if x <= B[0]:2 \u2003\u2003\u2003PWL = m\u2217(x-B[0])+k[0]3 \u2003\u2003\u2003\u2003elif x > B[n-2]:4 \u2003\u2003\u2003PWL = m\u2217(x-B[n-2])+k[2\u2217n-3]5 \u2003\u2003\u2003\u2003for j in range(n-2):6 \u2003\u2003\u2003if x > B[j] and x <= B[j+1]:7 \u2003\u2003\u2003\u2003PWL = m[j]\u2217(x-(B[j+1]+B[j])/2)8 \u2003\u2003\u2003\u2003\u2003xn.insert))9 \u2003\u2003\u2003yn.insert)10 \u2003\u2003\u2003zn.insert)11 \u2003\u2003\u2003x = xn[i]12 \u2003\u2003\u2003y = yn[i]13 \u2003\u2003\u2003z = zn[i]14 \u2003\u2003\u2003i = i+115 \u2003\u2003\u2003From the pseudocodes listed above, one can infer the kind of digital hardware for generating chaotic behavior. For instance, to generate more than 2-scrolls or odd scrolls, one just needs to extend the PWL descriptions from Eqs , 5), ,  and 9),,9), and k1 = -1, k2 = 1, B1 = -0.0165, B2 = 0.0165, a = 0.7, b = 0.7, c = 0.7, and d1 = 0.7. The double-scroll attractor is shown in For the chaotic oscillator based on saturated function series: m1 = -0.276, m2 = -3.3036, B1 = -0.1, B2 = 0.1, \u03b1 = 10, \u03b2 = 15, \u03b3 = 1, k1 = 0.3036, k2 = -0.3036. The double-scroll attractor is shown in For Chua\u2019s circuit using negative slopes: m = 0.25, B1 = 0, \u03b1 = 10, \u03b2 = 16, \u03b3 = 1 and k = 0.25. The double-scroll attractor is shown in For Chua\u2019s circuit using sawtooth function: To compute the fixed-point format being used for the VHDL descriptions, one needs to simulate the chaotic oscillator to know parameters like coefficient values and PWL characteristics, namely: break points, amplitudes, and slopes. Those parameter values for generating 2-scroll attractors for the three chaotic oscillators are the following:Other simulation results for generating 30-scrolls are shown in Figs From the simulation results, one can identify the ranges of the state variables and parameters that will serve to define the computer arithmetic to translate simulation parameters to VHDL-code. The binary digits will have integer and fractional parts and our algorithm converts real numbers to their 2\u2019s complement. Basically, the integer part is converted through successive divisions and the fractional part with multiplications. From the 2\u2019s complement numbers, one can generate VHDL code by interconnecting the required blocks that solve a discretized system of equations, e.g. Eqs , 11) an an11) anFor example: Using 32 bits, the ranges of the state variables for generating 2-scrolls in Again, by using 32 bits, the ranges of the state variables for generating 30-scrolls in As one can infer, an important issue when implementing a dynamical system using fixed-point arithmetic is related to the necessary number of bits for the fractional part. The required number of bits can be estimated by trial and error techniques until observing the number of scrolls in the phase-space portrait, as already shown in , ch. 8. e1, e2, e3] = I, and I is the identity matrix of size 3 \u00d7 3. Thus, ei, for i = 1, 2, 3, are each unitary column vector of the identity matrix I.To measure the Lyapunov exponents, the initial state of the chaotic oscillator is set tot > 0, then the states in the three new observational systems will be y = T. The observational system is integrated by several steps until an orthonormalization period TO is reached. After this, the state of the variational system is orthonormalized by using the standard Gram-Schmidt method. The next integration is carried out by using the new orthonormalized vectors as initial conditions T is the state after the matrix  is orthonormalized, then the Lyapunov exponent \u03bbi, for i = 1, 2, 3 is evaluated byThe Lyapunov exponents measure the long time sensitivity of the flow in TO was set to 5 seconds and the period T to 2000 seconds, respectively. The number of bits for the integer part was established by looking at the phase-space portraits of the expanded system, so that 8-bits are good enough for the integer part. For the fractional part, we test the behavior by using from 10 to 30 bits, therefore, In this work the simulation of the expanded system was carried on with fixed integer arithmetic, using Forward Euler method with a time-step of 0.001. The computation of the Lyapunov exponents using st listed in the pseudocodes in Sect. 3. It also processes the initial conditions from the first iteration loop.As detailed in , 21, thex and y are easy to implement as shown in z needs more hardware, as shown in From the discretized equations of the oscillator based on saturated functions in x and their corresponding PWL functions for negative slopes and sawtooth. As state variables y and z have the same description from Eqs x and theAgain, for both Chua\u2019s chaotic oscillators based on negative slopes and sawtooth function, the number of building blocks required in the Chaotic Oscillator Unit from Very High-Speed Integrated Circuit Hardware Description Language, it was developed around 1980 at the request of the U.S. Department of Defense. At the beginning, the main goal of VHDL was the electric circuit simulation; however, tools for synthesis and implementation in hardware based on VHDL behavior or structure description files were developed later. With the increasing use of VHDL, the need for standardized was generated. In 1986, the Institute of Electrical and Electronics Engineers (IEEE) standardized the first hardware description language through the 1076 and 1164 standards. Nowadays, VHDL is technology/vendor independent, then VHDL codes are portable and reusable.VHDL is the acronym of In Subsection 4.1 one can see the main parameters of a 2-scroll chaotic oscillator, like: coefficient values, break points, amplitudes, and slopes. They are used to perform a high-level simulation to identify the ranges of the state variables and iteration parameters that serve to define the computer arithmetic to translate simulation parameters to VHDL-code. Our approach can automatically generate VHDL code by interconnecting the required blocks that solve a discretized system of equations, e.g. Eqs \u201312), as, as12), Step 1: Select the number of scrolls being generated, coefficient values and characteristics of the PWL function of the desired multi-scroll chaotic oscillator. The break points are directly related to the number of scrolls and must be provided from the left (the most negative) to the right (the most positive), according to Figs Step 2: The high-level simulation is performed and our approach verifies the number of desired scrolls. From simulation data, the fixed-point format is established, as discussed in Subsection 4.1, where the ranges in Figs Step 3: Our approach creates a file containing the libraries and code for the blocks implementing The VHDL-codes for all the required blocks are shown in the following Algorithms. As one sees, they are described using 28 bits as the word length. This is not an issue, since our approach can generate the codes automatically. The signals are also described for each case.CLK: This is the master clock of the FPGA. All the operations are performed based on the clock frequency.RST: Reset of the system, puts the outputs to zero. Restarts the system.N1 and N2: Input data.OUTS: Output data.sena1: Internal signal with the double of bits from the word length to perform the multiplication.The blocks multiplier, adder and subtractor have the following signal descriptions:Algorithm 4. Multiplier in VDHLentity multiplier is1 \u2003\u2003port;5 \u2003\u2003\u2003N2: in std_logic_vector(27 downto 0);6 \u2003\u2003\u2003OUTS: out std_logic_vector(27 downto 0) := (others=>\u20180\u2019));7 \u2003\u2003\u2003end multiplier;8 \u2003\u2003architecture code of multiplier is9 \u2003\u2003signal sena1: signed(55 downto 0) := (others=>\u20180\u2019);10 \u2003\u2003begin11 \u2003\u2003process12 \u2003\u2003begin13 \u2003\u2003if RST = \u20181\u2019 then14 \u2003\u2003\u2003OUTS <= (others => \u20180\u2019);15 \u2003\u2003\u2003\u2003elsif rising_edge(CLK) then16 \u2003\u2003\u2003sena1 <= (signed(N1)\u2217signed(N2));17 \u2003\u2003\u2003\u2003OUTS <= std_logic_vector(sena1(51 downto 24));18 \u2003\u2003\u2003\u2003end if;19 \u2003\u2003\u2003end process;20 \u2003\u2003end code;21 \u2003\u2003Algorithm 5. Adder in VHDLentity adder is1 \u2003\u2003port;5 \u2003\u2003\u2003N2: in std_logic_vector(27 downto 0);6 \u2003\u2003\u2003OUTS: out std_logic_vector(27 downto 0) := (others => \u20180\u2019));7 \u2003\u2003\u2003end adder;8 \u2003\u2003architecture code of adder is9 \u2003\u2003begin10 \u2003\u2003process11 \u2003\u2003begin12 \u2003\u2003if RST = \u20181\u2019 then13 \u2003\u2003\u2003OUTS <= (others => \u20180\u2019);14 \u2003\u2003\u2003\u2003elsif rising_edge(CLK) then15 \u2003\u2003\u2003OUTS <= std_logic_vector(signed(N1) + signed(N2));16 \u2003\u2003\u2003\u2003end if;17 \u2003\u2003\u2003end process;18 \u2003\u2003end code;19 \u2003\u2003Algorithm 6. Subtractor in VHDLentity subtractor is1 \u2003\u2003port;5 \u2003\u2003\u2003N2: in std_logic_vector(27 downto 0);6 \u2003\u2003\u2003OUTS: out std_logic_vector(27 downto 0) := (others => \u20180\u2019));7 \u2003\u2003\u2003end subctractor;8 \u2003\u2003architecture code of subtractor is9 \u2003\u2003begin10 \u2003\u2003process11 \u2003\u2003begin12 \u2003\u2003if RST = \u20181\u2019 then13 \u2003\u2003\u2003OUTS <= (others => \u20180\u2019);14 \u2003\u2003\u2003\u2003elsif rising_edge(CLK) then15 \u2003\u2003\u2003OUTS <= std_logic_vector(signed(N1) \u2212 signed(N2));16 \u2003\u2003\u2003\u2003end if;17 \u2003\u2003\u2003end process;18 \u2003\u2003end code;19 \u2003\u2003Algorithm 7. Iterations control block in VHDLentity IterCtrl is1 \u2003\u2003port;5 \u2003\u2003\u2003Yini: in std_logic_vector(27 downto 0);6 \u2003\u2003\u2003Zini: in std_logic_vector(27 downto 0);7 \u2003\u2003\u2003Xout: out std_logic_vector(27 downto 0) := (others => \u20180\u2019);8 \u2003\u2003\u2003Yout: out std_logic_vector(27 downto 0) := (others => \u20180\u2019);9 \u2003\u2003\u2003Zout: out std_logic_vector(27 downto 0) := (others => \u20180\u2019));10 \u2003\u2003\u2003end IterCtrl;11 \u2003\u2003architecture code of IterCtrl is12 \u2003\u2003signal dz: std_logic_vector(27 downto 0) := \u201c0000000000000000000000000000\u201d;13 \u2003\u2003signal dy: std_logic_vector(27 downto 0) := \u201c0000000000000000000000000000\u201d;14 \u2003\u2003signal dx: std_logic_vector(27 downto 0) := \u201c0000000101100110011001100110\u201d;15 \u2003\u2003begin16 \u2003\u2003process17 \u2003\u2003variable count: integer := 0;18 \u2003\u2003begin19 \u2003\u2003if RST = \u20181\u2019 then20 \u2003\u2003\u2003dx <= (others => \u20180\u2019);21 \u2003\u2003\u2003\u2003dy <= (others => \u20180\u2019);22 \u2003\u2003\u2003\u2003dz <= (others => \u20180\u2019);23 \u2003\u2003\u2003\u2003elsif rising_edge(CLK) then24 \u2003\u2003\u2003if count = 8 then25 \u2003\u2003\u2003\u2003count := 0;26 \u2003\u2003\u2003\u2003\u2003dx <= Xini;27 \u2003\u2003\u2003\u2003\u2003dy <= Yini;28 \u2003\u2003\u2003\u2003\u2003dz <= Zini;29 \u2003\u2003\u2003\u2003\u2003else30 \u2003\u2003\u2003\u2003count := count +1;31 \u2003\u2003\u2003\u2003\u2003end if;32 \u2003\u2003\u2003\u2003end if;33 \u2003\u2003\u2003end process;34 \u2003\u2003Xout <= dx;35 \u2003\u2003Yout <= dy;36 \u2003\u2003Zout <= dz;37 \u2003\u2003end code;38 \u2003\u2003CLK and RST are the same for all blocks.Xini, Yini and Zini: Feeds input signals to the Chaotic Oscillator Unit, until accomplishing st iterations as sketched in Sect. 3.st iterations.Xout, Yout and Zout: Output data that will be processed by the Iterations control block to perform dx, dy and dz: Initial conditions for each iteration. These registers save the output data of the state variables and then feeds them as input signals Xini, Yini and Zini.count: Signal to update the initial values to the Chaotic Oscillator Unit at each iteration. This signal is enabled after 8 clock cycles (CLKs), which is the time required by the maximum number of series-connected blocks.Iterations control block (Algorithm 7) has the signals:Algorithm 8. Comparator for implementing saturated functions in VHDLentity comparator is1 \u2003\u2003port;5 \u2003\u2003\u2003dato_sat1: in std_logic_vector(27 downto 0);6 \u2003\u2003\u2003dato_pen0: in std_logic_vector(27 downto 0);7 \u2003\u2003\u2003dato_X: in std_logic_vector(27 downto 0);8 \u2003\u2003\u2003dato_S: out std_logic_vector(27 downto 0) := (others => \u20180\u2019));9 \u2003\u2003\u2003end comparator;10 \u2003\u2003architecture code of comparator is11 \u2003\u2003constant B0: std_logic_vector(27 downto 0) := \u201c1000000000000000000010101000\u201d;12 \u2003\u2003constant B1: std_logic_vector(27 downto 0) := \u201c1111111110111100011010101000\u201d;13 \u2003\u2003constant B2: std_logic_vector(27 downto 0) := \u201c0000000001000011100101011000\u201d;14 \u2003\u2003constant B3: std_logic_vector(27 downto 0) := \u201c0111111111111111111111101111\u201d;15 \u2003\u2003begin16 \u2003\u2003process17 \u2003\u2003begin18 \u2003\u2003if RST = \u20191\u2019 then19 \u2003\u2003\u2003dato_S<= (others => \u20180\u2019);20 \u2003\u2003\u2003\u2003elsif rising_edge(CLK) then21 \u2003\u2003\u2003if dato_X > B0 AND dato_X < B1 then22 \u2003\u2003\u2003\u2003dato_S <= dato_sat0;23 \u2003\u2003\u2003\u2003\u2003elsif dato_X > B2 AND dato_X < B3 then24 \u2003\u2003\u2003\u2003dato_S <= dato_sat1;25 \u2003\u2003\u2003\u2003\u2003else26 \u2003\u2003\u2003\u2003dato_S <= dato_pen0;27 \u2003\u2003\u2003\u2003\u2003end if;28 \u2003\u2003\u2003\u2003end if;29 \u2003\u2003\u2003end process;30 \u2003\u2003end code;31 \u2003\u2003The PWL functions are implemented using comparators. If the PWL segments increase, as detailed in Sect. 3, then also the number of comparisons do. The comparator blocks have CLK and RST signals as the previous blocks.dato_sat0: Input data for the first level of saturation.dato_sat1: Input data for the second level of saturation.dato_pen0: Input data for the slope.x to perform comparisons.dato_X: Input data of the state variable dato_S: Output data that takes the value from one input like dato_sat0, dato_sat1, or dato_pen0.B0, B1, B2 and B3: Constants to represent the break points from the left to the right of the PWL functions.The PWL function for the oscillator based on saturated functions is implemented by Algorithm 8, where:dato_pen0: Input data for the first slope located on the left of the PWL function.dato_pen1: Input data for the last slope, which is the same as the first.dato_penm: Input data for the central slope.x to perform comparisons.dato_X: Input data of the state variable dato_S: Output data that takes the value from one input like dato_pen0, dato_pen,1 or dato_penm.B0 and B1: Constants to represent the break points from the left to the right of the PWL functions.The PWL function for the oscillator based on negative slopes is implemented by Algorithm 9, where:Algorithm 9. Comparator for implementing negative slopes in VHDLentity comparador is1 \u2003\u2003port;5 \u2003\u2003\u2003dato_pen1: in std_logic_vector(27 downto 0);6 \u2003\u2003\u2003dato_X: in std_logic_vector(27 downto 0);7 \u2003\u2003\u2003dato_S: out std_logic_vector(27 downto 0) := (others => \u20180\u2019));8 \u2003\u2003\u2003end comparador;9 \u2003\u2003architecture complicado of comparador is10 \u2003\u2003constant B0: std_logic_vector(27 downto 0) := (others => \u20180\u2019)11 \u2003\u2003begin12 \u2003\u2003process13 \u2003\u2003begin14 \u2003\u2003if RST = \u20191\u2019 then15 \u2003\u2003\u2003dato_S<= (others => \u20180\u2019);16 \u2003\u2003\u2003\u2003elsif rising_edge(CLK) then17 \u2003\u2003\u2003if dato_X < B0 then18 \u2003\u2003\u2003\u2003dato_S <= dato_pen0;19 \u2003\u2003\u2003\u2003\u2003else20 \u2003\u2003\u2003\u2003dato_S <= dato_pen1;21 \u2003\u2003\u2003\u2003\u2003end if;22 \u2003\u2003\u2003\u2003end if;23 \u2003\u2003\u2003end process;24 \u2003\u2003end complicado;25 \u2003\u2003Algorithm 10. Comparator for implementing sawtooth function in VHDLentity comparator is1 \u2003\u2003port;5 \u2003\u2003\u2003dato_pen1: in std_logic_vector(27 downto 0);6 \u2003\u2003\u2003dato_penm: in std_logic_vector(27 downto 0);7 \u2003\u2003\u2003dato_X: in std_logic_vector(27 downto 0);8 \u2003\u2003\u2003dato_S: out std_logic_vector(27 downto 0) := (others => \u20180\u2019));9 \u2003\u2003\u2003end comparator;10 \u2003\u2003architecture complicado of comparador is11 \u2003\u2003constant B0: std_logic_vector(27 downto 0) := \u201c1111111100110011001100110100\u201d;12 \u2003\u2003constant B1: std_logic_vector(27 downto 0) := \u201c0000000011001100110011001100\u201d;13 \u2003\u2003begin14 \u2003\u2003process15 \u2003\u2003begin16 \u2003\u2003if RST = \u20191\u2019 then17 \u2003\u2003\u2003dato_S<= (others => \u20180\u2019);18 \u2003\u2003\u2003\u2003elsif rising_edge(CLK) then19 \u2003\u2003\u2003if dato_X < B0 then20 \u2003\u2003\u2003\u2003dato_S <= dato_pen0;21 \u2003\u2003\u2003\u2003\u2003elsif dato_X >= B1 then22 \u2003\u2003\u2003\u2003dato_S <= dato_pen1;23 \u2003\u2003\u2003\u2003\u2003else24 \u2003\u2003\u2003\u2003dato_S <= dato_penm;25 \u2003\u2003\u2003\u2003\u2003end if;26 \u2003\u2003\u2003\u2003end if;27 \u2003\u2003\u2003end process;28 \u2003\u2003end complicado;29 \u2003\u2003dato_pen0: Input data for the first slope.dato_pen1: Input data for the last slope, which is the same as the first.x to perform comparisons.dato_X: Input data of the state variable dato_S: Output data that takes the value from one input like dato_pen0, dato_pen1.B0: Constant to represent the break point.The PWL function for the oscillator based on sawtooth function is implemented by Algorithm 10, and it is similar as the last two comparators, where:The generated VHDL-code is ready to be synthesized into an FPGA, so that the following Section shows experimental results.x is shown on the top left-side, y on the bottom left-side, and the phase-space portrait x \u2212 y on the right-side. One can count the number of scrolls from the phase-space portraits, and from Lyapunov exponents evaluation, one can infer that the more scrolls are generated the more complex behavior. In this manner, and from the experimental results, it is clear that engineering applications like in [The generated VHDL-codes for the synthesis of multi-scroll chaotic oscillators based on PWL functions were implemented in the Altera\u2019s FPGA EP4CGX150DF31C7 Cyclone IV GX. The used resources are listed in We have introduced an approach programmed in Python for the generation of VHDL-code associated to multi-scroll chaotic oscillators that are based on PWL functions. The pseudocodes for simulating three kinds of chaotic oscillators were listed to infer the FPGA implementation of the PWL functions based on saturated functions series, negative slopes and sawtooth one. From the high-level simulation, our algorithm determines the fixed-point format to interconnect digital blocks associated to the discretized equations of the chaotic oscillators. The PWL functions are then implemented by using comparator blocks that can increase the number of comparisons according to the number of scrolls being generated. It was highlighted that these tasks are performed from the description of the dynamical equations and PWL functions, to the interconnection of the digital blocks and generation of the VHDL-code, which is portable, reusable and open source to be synthesized in an FPGA of any vendor.It is worthy mentioning that the FPGA resources depend on the discretization approach. For instance, Subsection 4.2 showed that Forward Euler and the fourth-order Runge-Kutta methods computed similar values of the Lyapunov exponents, and also when using floating point and fixed integer arithmetic. However, the number of bits for the fractional part matters. As shown in Finally, it can be concluded that our approach for generating VHDL descriptions can estimate the number of required blocks from the equations listed in"}
{"text": "There is an error in Table 3. The \u201cMean\u201d value for \u201cMaize (frequency per week)\u201d is missing. Please see the correct There are a number of errors in Table 4. The \u201cMedian\u201d values for \u201cPlasma iron (\u03bcg/L)\u201d, \u201cFerritin (\u03bcg/L)\u201d, and \u201cTransferrin saturation (%)\u201d are missing. The \u201cPercent\u201d values for \u201c< 8.3\u201d under \u201csTfR (mg/L)\u201d, \u201c\u2264 1.0\u201d under \u201cAGP (g/L)\u201d, and \u201c< 125\u201d under \u201cHemoglobin (g/L)\u201d are missing. Please see the correct The publisher apologizes for the errors."}
{"text": "Genome rearrangement describes gross changes of chromosomal regions, plays an important role in evolutionary biology and has profound impacts on phenotype in organisms ranging from microbes to humans. With more and more complete genomes accomplished, lots of genomic comparisons have been conducted in order to find genome rearrangements and the mechanisms which underlie the rearrangement events. In our opinion, genomic comparison of different individuals/strains within the same species (pan-genome) is more helpful to reveal the mechanisms for genome rearrangements since genomes of the same species are much closer to each other.Pseudomonas aeruginosa and Escherichia coli, and investigate the inversion events among different strains of the same species. We find an interesting phenomenon that long  inversion regions are flanked by a pair of Inverted Repeats (IRs). This mechanism can also explain why the breakpoint reuses for inversion events happen. We study the prevalence of the phenomenon and find that it is a major mechanism for inversions. The other observation is that for different rearrangement events such as transposition and inverted block interchange, the two ends of the swapped regions are also associated with repeats so that after the rearrangement operations the two ends of the swapped regions remain unchanged. To our knowledge, this is the first time such a phenomenon is reported for transposition event.We study the mechanism for inversion events via core-genome scaffold comparison of different strains within the same species. We focus on two kinds of bacteria, Pseudomonas aeruginosa and Escherichia coli strains, IRs were found at the two ends of long sequence inversions. The two ends of the inversion remained unchanged before and after the inversion event. The existence of IRs can explain the breakpoint reuse phenomenon. We also observed that other rearrangement operations such as transposition, inverted transposition, and inverted block interchange, had repeats (not necessarily inverted) at the ends of each segment, where the ends remained unchanged before and after the rearrangement operations. This suggests that the conservation of ends could possibly be a popular phenomenon in many types of chromosome rearrangement events.In both The online version of this article (doi:10.1186/s12864-017-3655-0) contains supplementary material, which is available to authorized users. Comparative genomics studies show that genome rearrangement events often occur between two genomes. Genome rearrangement events play important role in speciation. The rearrangement operations include deletions, insertions, inversion, transposition, block interchange, translocation, fission and fusion, etc. Here we study the mechanism for inversion events via core-genome scaffold comparison of different strains within the same species.Drosophila melanogaster and two closely related species,D. simulans and D. yakuba, and reconstructed the molecular events that underlie their origin [P-value <10\u22123) [P<10\u22124).By comparing two genomes, we can find candidate rearrangement operations. However, the set of rearrangement operations to transform one genome into the other is not unique in many cases. Computing the rearrangement operations between two genomes under different assumptions is an active area, where intensive research have been conducted . It is rr origin . Rajarame <10\u22123) . Darmon e <10\u22123) . The asse <10\u22123) , 9\u201312. Ae <10\u22123) . They alDrosophila genus [Saccharomyces pastorianus [Pevzner and Tesler found extensive breakpoint reuse for inversion events in mammalian evolution when comparing human and mouse genomic sequences \u201315. Statla genus , 19 as wtorianus .It is well known that recombination (crossing-over) of homologous or non-homologous DNAs can lead to various genetic variations including inversions, transpositions, insertions/deletions, and will leave some direct or inverted repeats on both ends, and existing repeats can further promote more variations.Pseudomonas aeruginosa has been done by Ozer et al. [To study the rearrangement operations, comparison of different individuals/strains within the same species (pan-genomes) can be more helpful since strains within the same species are conserved. A pan-genome, or supra-genome, describes the full complement of genes in a clade , which can have large variation in gene content among closely related strains. Pan-genomes were first studied by Tettelin more than a decade ago . Severalr et al. . For panPseudomonas aeruginosa and Escherichia coli, and investigate the inversion events among different strains of the same species. We find an interesting phenomenon that long  inversion regions are flanked by pairs of Inverted Repeats (IRs) which are often Insertion Sequences (ISs). This mechanism also explains why the breakpoint reuses for inversion events happen. We study the prevalence of the phenomenon and find that it is a major mechanism for inversions. The other observation is that for different rearrangement events such as transposition and inverted block interchange, the two ends of the swapped regions are also associated with repeats so that after the rearrangement operations the two ends of the swapped regions remain unchanged. To our knowledge, this is the first time such a phenomenon is reported for transposition event.We study two types of bacteria, We develop a pipeline to generate the core-genome blocks, dispensable blocks and strain-specific blocks based on the multiple sequence alignment produced by Mugsy .Pseudomonas aeruginosa, before merging, there are 185 blocks in the core genome of the 25 strains. After merging, the scaffolds contain 69 blocks.We then develop a computer program to generate the scaffolds of the strains from the core-genome blocks by repeatedly merging two consecutive blocks appearing in all the strains of the same species. In this way, the number of distinct blocks in the core-genome scaffold is reduced dramatically. For example, for After that, we compute the inversion distance between two scaffolds. Computing the inversion distance between two scaffolds is a very hard and complicated combinatorial problem. Several algorithms have been developed. Due to the difficulty of algorithm design, most of the algorithms only consider inversion events. However, a transposition/block-interchange event can be represented as 3 inversion events, and an inverted transposition/block-interchange event can be represented as 2 inversion events. Therefore, some of the computed inversion events may not be real. There are algorithms dealing with inversion and other rearrangement events such as block interchanges simultaneously. However, the weights for different events are different . Thus, those algorithms still suffer from the problem of outputting inversions that are not real.Our strategy here is to eliminate some obvious (independent) transposition, inverted transposition, block interchange, and inverted block interchange events before computing the inversion distance between two scaffolds.G1=+1+2\u2026+n is the first input scaffold and G2=\u03c01\u03c02\u2026\u03c0n is a sign permutation of the n blocks over the set N={1,2,\u2026,n} of n distinct blocks, where each integer i\u2208N appear once in G2 in the form of either +i or \u2212i. All the rearrangement operations are on G2.For simplicity, we always assume that transposition swaps the order of two consecutive blocks/regions without changing their signs. A transposition  on regions \u03c0i,\u2026,\u03c0j\u22121 and \u03c0j\u2026\u03c0k\u22121 transforms the sign permutation \u03c01\u2026\u03c0i\u22121\u03c0i\u2026\u03c0j\u22121\u03c0j\u2026\u03c0k\u22121\u03c0k\u2026\u03c0n into \u03c01\u2026\u03c0i\u22121\u03c0j\u2026\u03c0k\u22121\u03c0i\u2026\u03c0j\u22121\u03c0k\u2026\u03c0n.A independent if it transforms the sign permutation \u03c01\u2026\u03c0i\u22122\u03c0i\u22121\u03c0i+1\u03c0i\u03c0i+2\u03c0i+3\u2026\u03c0n into \u03c01\u2026\u03c0i\u22122\u03c0i\u22121\u03c0i\u03c0i+1\u03c0i+2\u03c0i+3\u2026\u03c0n, where \u03c0i\u22121\u03c0i\u03c0i+1\u03c0i+2 is either +(q\u22121)+q+(q+1)+(q+2) or \u2212(q+2)\u2212(q+1)\u2212q\u2212(q\u22121) for {q\u22121,q,q+1,q+2}\u2286N={1,2,\u2026,n}. Though an independent transposition swaps two consecutive blocks \u03c0i+1 and \u03c0i instead of two regions \u03c0i,\u2026,\u03c0j\u22121 and \u03c0j\u2026\u03c0k\u22121 as in the definition of a general transposition, a pre-process allows us to merge two consecutive blocks if they are consecutive in both input genomes. Thus, we can still handle some cases for swapping two consecutive regions. For example, the genome +1+2+6+7+3+4+5+8 becomes +1+2+4+3+5 after merging +6+7 (represented as +4)and +3+4+5 (represented as +3) and re-number +8 as +5 in the new representation. An independent transposition can change +1+2+4+3+5 into +1+2+3+4+5. In terms of breakpoint graph, the two blocks \u03c0i+1\u03c0i in an independent transposition is involved in a 6-edge cycle and after the transformation the 6-edge cycle becomes three 2-edge cycles. In other words, the three breakpoints involved in the 6-edge cycle disappear after the transformation. See Fig. A transposition is inverted transposition swaps the order of two consecutive blocks/regions with one of the block\u2019s sign changed. An inverted transposition  on regions \u03c0i,\u2026,\u03c0j\u22121 and \u03c0j\u2026\u03c0k\u22121 transforms the sign permutation \u03c01\u2026\u03c0i\u22121\u03c0i\u2026\u03c0j\u22121\u03c0j\u2026\u03c0k\u22121\u03c0k\u2026\u03c0n into \u03c01\u2026\u03c0i\u22121\u2212\u03c0k\u22121\u2026\u2212\u03c0j\u03c0i\u2026\u03c0j\u22121\u03c0k\u2026\u03c0n or \u03c01\u2026\u03c0i\u22121\u03c0j\u2026\u03c0k\u22121\u2212\u03c0j\u22121\u2026\u2212\u03c0i\u03c0k\u2026\u03c0n.An independent if it transforms the sign permutation \u03c01\u2026\u03c0i\u22122\u03c0i\u22121\u2212\u03c0i+1\u03c0i\u03c0i+2\u03c0i+3\u2026\u03c0n or \u03c01\u2026\u03c0i\u22122\u03c0i\u22121\u03c0i+1\u2212\u03c0i\u03c0i+2\u03c0i+3\u2026\u03c0n into \u03c01\u2026\u03c0i\u22122\u03c0i\u22121\u03c0i\u03c0i+1\u03c0i+2\u03c0i+3\u2026\u03c0n, where \u03c0i\u22121\u03c0i\u03c0i+1\u03c0i+2 is either +(q\u22121)+q+(q+1)+(q+2) or \u2212(q+2)\u2212(q+1)\u2212q\u2212(q\u22121) for {q1,q,q+1,q+2}\u2286N={1,2,\u2026,n}.An inverted transposition is block interchange swaps the locations of two separated blocks without changing their signs. A block interchange  on regions \u03c0i\u2026\u03c0j and \u03c0k\u2026\u03c0l transforms \u03c01\u2026\u03c0i\u22121\u03c0k\u2026\u03c0l\u03c0j+1\u2026\u03c0k\u22121\u03c0i\u2026\u03c0j\u03c0l+1 \u2026 \u03c0n into \u03c01\u2026\u03c0i\u22121\u03c0i\u2026\u03c0j\u03c0j+1\u2026\u03c0k\u22121\u03c0k\u2026\u03c0l\u03c0l+1\u2026\u03c0n.A independent if it transforms the sign permutation \u03c01\u2026\u03c0i\u22121\u03c0k\u03c0i+1\u2026\u03c0k\u22121\u03c0i\u03c0k+1\u2026\u03c0n into \u03c01\u2026\u03c0i\u22121\u03c0i\u03c0i+1\u2026\u03c0k\u22121\u03c0k\u03c0k+1\u2026\u03c0n, where \u03c0i\u22121\u03c0i\u03c0i+1 is either +q+(q+1)+(q+2) or \u2212(q+2)\u2212(q+1)\u2212q and \u03c0k\u22121\u03c0k\u03c0k+1 is either +p+(p+1)+(p+2) or \u2212(p+2)\u2212(p+1)\u2212p for {q,q+1,q+2}\u2286N and {p,p+1,p+2}\u2286N. Similarly, the two blocks \u03c0k and \u03c0i are involved in two (interleaving) 4-edge cycles in the breakpoint graph and after the transformation, they become four 2-edge cycles. In other words, there are four breakpoints at the two ends of the two blocks, after the transformation, the four breakpoints disappear. See Fig. A block interchange is +(q+1)+q+ or \u2212 on regions \u03c0i\u2026\u03c0j and \u03c0k\u2026\u03c0l transforms \u03c01\u2026\u03c0i\u22121\u2212\u03c0l\u2026\u2212\u03c0k\u03c0j+1\u2026\u03c0k\u22121\u2212\u03c0j\u2026\u2212\u03c0i\u03c0l+1\u2026\u03c0n into \u03c01\u2026\u03c0i\u22121\u03c0i\u2026\u03c0j\u03c0j+1\u2026\u03c0k\u22121\u03c0k\u2026\u03c0l\u03c0l+1\u2026\u03c0n.An independent if it transforms the sign permutation \u03c01\u2026\u03c0i\u22121\u2212\u03c0k\u03c0i+1\u2026\u03c0k\u22121\u2212\u03c0i\u03c0k+1\u2026\u03c0n into \u03c01\u2026\u03c0i\u22121\u03c0i\u03c0i+1\u2026\u03c0k\u22121\u03c0k\u03c0k+1\u2026\u03c0n, where \u03c0i\u22121\u03c0i\u03c0i+1 is either +q+(q+1)+(q+2) or \u2212(q+2)\u2212(q+1)\u2212q and \u03c0k\u22121\u03c0k\u03c0k+1 is either +p+(p+1)+(p+2) or \u2212(p+2)\u2212(p+1)\u2212p for {q,q+1,q+2}\u2286N and {p,p+1,p+2}\u2286N. Again, there are four breakpoints at the two ends of the two blocks \u2212\u03c0i and \u2212\u03c0k, after the transformation, the four breakpoints disappear.An inverted block interchange is After eliminating independent transposition, inverted transposition, block interchange and inverted block interchange events, we use GRIMM-Synteny , 28 to cFinally, we developed a pipeline to compare sequences at the two ends of each inversion region to see whether a pair of inverted repeats exists. Once the inverted repeats are found, the pipeline can also search all the strains and mark down its positions in different strains.Pseudomonas aeruginosa strains PACS2, F22031,NCGM1900, LES431, NCGM2.S1, Carb01_63, SCV20265, UCBPP-PA14, VRFPA04, DSM_50071, 19BR, 213BR, B136-33, PA7, PA1, YL84, LESB58, M18, RP73, DK2, MTB1, PAO1, PA1R, NCGM1984,and FRD1 were downloaded from NCBI GenBank. The details of these 25 Pseudomonas aeruginosa strains are listed in Additional file Pseudomonas aeruginosa strain is in Additional file Complete genome sequences of 25 Pseudomonas aeruginosa strains NCGM1984, B136-33, YL84, M18, LESB58, SCV20265, LES431, UCBPP-PA14, DK2, MTB-1, DSM_50071, Carb01_63, and F22031. Group 2 contains 6 strains, which are strains RP73, 213BR, PA1, PA1R, 19BR, and PAO1. Groups 3-8 contain 1 strain each and the respective strains are PACS2, FRD1, NCGM2.S1, VRFPA04, NCGM1900, and PA7.Group 1 contains 13 strains, which are We computed the pairwise inversion distance between scaffolds after eliminating other kinds of independent rearrangement events such as transpositions, inverted-transpositions, block-interchanges, and inverted-block-interchanges. For each of the 8 scaffolds, we chose a scaffold with the minimum inversion distance (after eliminating other independent rearrangement events) to compare. The purpose was to compare two scaffolds with a small number of inversions so that we can observed real inversions between them. From Table B that appears four times in Both Scaffold 1 and Scaffold 5, where B appear as \u2212B once and as +B three times in Scaffold 1. The four occurrences of B form a pair of IRs at the two ends of each of the 3 inversion regions , we can see that for the first inversion (-B-6+B to -B6+B), the real cutting points (breakpoints) are at the left end of -B and the right end of +B, while for the other two inversions  1 to 31) with complete sequences from 17 genome families at NCBI\u2019s GenBank. These 31 strains are SE15, IAI39, EC4115, CFT073, CE10, O103:H2 str. 12009, C227-11, 536, K-12 substr. MG1655, ST2747, NA114, 042, O111:H- str. 11128, O145:H28str.RM13514, O104:H4 str. 2011C-3493, SE11, SS52, APEC O78, SMS-3-5, DH1Ec095, 1303, O157:H7 str. Sakai, 55989, B str. REL606, O83:H1 str. NRG 857C, UMN026, PCN033, 789, O127:H6 str. E2348/69, P12b, and ED1a. The detailed information of these 31 strains is listed in Additional file Escherichia coli strain is given in Additional file We selected 31 Escherichia coli strains EC4115, CE10, C227-11, K-12 substr. MG1655, ST2747, 042 O104:H4 str. 2011C-3493, SE11, SS52, APEC O78, DH1Ec095, 1303, O157:H7 str. Sakai, 55989, B str. REL606, O83:H1 str. NRG 857C, UMN026, PCN033, 789, O127:H6 str. E2348/69, and ED1a. Group 2 contains 3 strains, SE15, CFT073 and 536. Groups 3-9 contain 1 strain each and the respective strains are O145:H28 str. RM13514, SMS-3-5, P12b, IAI39, O103:H2 str. 12009, NA114, and O111:H- str. 11128.Group 1 contains 21 strains which are After computing pairwise inversion distance among the 9 scaffolds, we selected a scaffold with minimum inversion distance for each of the 9 scaffolds as shown in Table In total, there are 17 inversion events among the 8 distinct pairs in Table For all the inversions listed in Table We find a total of 12 different types of pairs of inverted repeats and use letters from +D/-D to +M/-M, +S/-S and +Q/-Q to label and differentiate these 12 pairs of IRs. The locations of these IRs in the scaffolds are shown in Fig. The three inversion steps from Scaffolds 1 to 7 are illustrated in Fig. It is worth pointing out that the two +Ms in Scaffold 1 form a pair of directed repeats (DRs). After inversion , the pair of directed repeats (DRs) of M becomes a pair of inverted repeats. This means that a pair of DRs has the potential to mediate inversions.We find an inverted block interchange between Scaffold 6 and 1 and we use Fig. The other explanation is that an inverted block interchange can be replaced by two inversions. Figure Figure mentclass2pt{minimThe other explanation is that the inverted transposition can be replaced by two inversions: the first inversion is from Blocks 16 to -15 and the second inversion is from Blocks -19 to -16 see Fig. . Both ofPseudomonas aeruginosa and Escherichia coli strains, IRs were found at the two ends of long sequence inversions. The two ends of the inversion remained unchanged before and after the inversion event. We also observed that other rearrangement operations such as transposition, inverted transposition, and inverted block interchange, had repeats (not necessarily inverted) at the ends of each segment, where the ends remained unchanged before and after the rearrangement operations. This suggests that the conservation of ends could possibly be a popular phenomenon in many types of chromosome rearrangement events. Past studies reveal that insertions and deletions (indels) can be mediated by directed repeats (DRs) [For both ts (DRs) , 30. Seqts (DRs) , 30. Howts (DRs) .The mechanism for breakpoint reuse is also interesting. The fact that long inversions are flanked by a pair of inverted repetitive elements can clearly explain why breakpoint reuse happens for inversions. Our observations show that the breakpoint reuse is actually the repeated segment reuse. The breakpoints at the nucleotides level for the reused repeat differ depending on the repeat is at the left or right end of the inversion.Pseudomonas aeruginosa and Escherichia coli strains. We have found that repeats were at the ends of different kinds of rearrangement events including inversion, transposition, inverted transposition, and inverted block interchange. In many cases, these repeats keep the ends of rearrangement events unchanged. This suggests that the conservation of ends could possibly be a popular phenomenon in many types of chromosome rearrangement events.In this paper, we have studied the rearrangement events for both Additional file 1Table S3. Information on Repeats found in Pseudomonas aeruginosa strains. Table S4. Scaffolds of each of the 25 Pseudomonas aeruginosa strains. Table S5. Physical positions of repeats and breakpoints of rearrangements in Pseudomonas aeruginosa strains. Table S6. Scaffolds of each of the 31 Escherichia coli Strains. Table S7. Physical positions of repeats and breakpoints of rearrangements in Escherichia coli strains. Table S8. Information on repeats found in the Escherichia coli Strains. (XLSX 103 kb)Additional file 2Cases where multiple transposition or block interchange events happen between two groups. (DOCX 20 kb)"}
{"text": "We propose a comprehensive delayed HBV model, which not only considers the immune response to both infected cells and viruses and a time delay for the immune system to clear viruses but also incorporates an exposed state and the proliferation of hepatocytes. We prove the positivity and boundedness of solutions and analyze the global stability of two boundary equilibria and then study the local asymptotic stability and Hopf bifurcation of the positive (infection) equilibrium and also the stability of the bifurcating periodic solutions. Moreover, we illustrate how the factors such as the time delay, the immune response to infected cells and viruses, and the proliferation of hepatocytes affect the dynamics of the model by numerical simulation. Hepatitis B virus (HBV) has become one of the serious infectious diseases threatening global human health, which can cause chronic liver infection and further result in liver inflammation, fibrosis, cirrhosis, or even cancer . Each yeMathematical modeling and analysis of the dynamics of such infectious viruses as HBV play important roles in understanding the factors that govern the infectious disease progression and offering insights into developing treatment strategies and guiding antiviral drug therapies . So far,Among these works, the development of virus models with immune responses is gaining much attention , 11, 12.In this paper, we will propose a more comprehensive model than those existing ones, which not only considers the immune response to both infected cells and viruses and a time delay for the immune system to clear viruses but also incorporates an exposed state and the proliferation of hepatocytes. We first discuss the existence of two boundary equilibria and one positive (infection) equilibrium. We then analyze the global stability of the two boundary equilibria, the local asymptotic stability and Hopf bifurcation of the positive equilibrium and also the stability of the bifurcating periodic solutions. Moreover, we perform numerical simulations to illustrate some of the theoretical results we obtain and also illustrate how the factors such as the immune response to infected cells and viruses and the proliferation of hepatocytes affect the dynamics of the model under time delay.The paper is structured as follows. In  Wang et al.  proposedBased on this model, we propose a new and comprehensive HBV model, which not only considers the immune response to both infected cells and viruses and a time delay for the immune system to clear viruses but also incorporates an exposed state and the proliferation of hepatocytes. To better understand our model, we illustrate its mechanism in x,\u2009\u2009e,\u2009\u2009y,\u2009\u2009v,\u2009\u2009z, and w denote the number of uninfected cells, exposed cells, infected cells, free viruses, CTLs, and alanine aminotransferases (ALT), respectively. The parameter \u03bb represents the natural production rate of uninfected cells. rx is a new term which is introduced to represent the proliferation of hepatocytes, where r is the proliferation rate. Parameters d, (and the following) a1,\u2009\u2009\u03b5,\u2009\u2009k4, and k7 represent the natural death rate of uninfected cells, exposed cells, infected cells, free viruses, CTLs, and ALT, respectively. \u03b2 represents the infection rate from uninfected cells to exposed cells and a2 the transfer rate from exposed cells to infected cells. The production rate of free viruses from infected cells is denoted by k, and the production rate of CTLs by k3. k5 represents the production rate of ALT from the extrahepatic tissue and k1k6 the production rate of ALT when the infected hepatocytes are killed by CTL. The immunity-induced clearance for infected cells is modeled by a term k1yz, where k1 represents the clearance rate of infected cells. Similarly, the immunity-induced clearance for free viruses is modeled by k2vz, where k2 represents the clearance rate of free viruses. \u03c4 is time delay. All the parameters in this paper are positive and d > r. For convenience, we define new parameter \u03c1 = d \u2212 r.The model is then given as follows:In this subsection, we prove the positivity and the boundedness of solutions of system .x(0) \u2265 0,\u2009\u2009e(0) \u2265 0,\u2009\u2009y(0) \u2265 0,\u2009\u2009w(0) \u2265 0,\u2009\u2009v(t) \u2265 0,\u2009\u2009z(t) \u2265 0,\u2009\u2009t \u2208 . From the first equation of system (x(t) = e0t(\u03c1+\u03b2v(s))ds\u2212\u222bx(0) + \u03bb\u222b0test[\u03c1+\u03b2v(\u03be)]d\u03be\u2212\u222bds; therefore, x(t) \u2265 0 for \u2200t > 0 if x(0) \u2265 0. Next, we consider the second, third, and fourth equation in system (e(t), y(t), v(t):We denote f system , we haven system  as a none(t) \u2265 0,\u2009\u2009y(t) \u2265 0,\u2009\u2009v(t) \u2265 0 if e(0) \u2265 0,\u2009\u2009y(0) \u2265 0,\u2009\u2009z(0) \u2265 0.Based on Theorem 2.1 in , we havez(t) = ek4t\u2212y(0) + \u222b0tk3v(t \u2212 \u03b3)z(t \u2212 \u03b3)ek4(t\u2212\u03b3)\u2212d\u03b3, we have z(t) \u2265 0,\u2009\u2009\u2200t > 0 if z(t) \u2265 0,\u2009\u2009t \u2208 .w(t) = ek7t\u2212w(0) + ek7t\u2212\u222b0t[(k5 + k1k6y(s)z(s)]ek7t\u2212ds, because y(t), z(t) \u2265 0, so we have w(t) \u2265 0,\u2009\u2009\u2200t > 0 if w(0) \u2265 0.G(t) as a linear combination of x,\u2009\u2009e,\u2009\u2009y,\u2009\u2009v,\u2009\u2009z:Hence, the nonnegative is proved. In what follows, we will study the boundedness of solutions. We define t\u2192\u221e\u2061G(t) \u2264 \u03bb/\u03b4, namely, x(t) + e(t) + y(t) + (a1/2k)v(t) + (a1k2/2kk3)z(t + \u03c4) \u2264 \u03bb/\u03b4. So we have 0 \u2264 x(t), e(t), y(t), v(t), z(t) \u2264 \u03bb/\u03b4 Because the boundedness of x(t), e(t), y(t), v(t), z(t),\u2009\u2009limt\u2192\u221e\u2061w(t) \u2264 (k5 + k1k6)\u03bb2/k7\u03b42.Therefore, we obtain limThe boundedness is proved.In this subsection, we study the equilibria of system . The metE00 in which x \u2260 0,\u2009\u2009w \u2260 0,\u2009\u2009e = y = v = z = 0 and an equilibrium without immune response E11 in which x \u2260 0,\u2009\u2009e \u2260 0,\u2009\u2009y \u2260 0,\u2009\u2009v \u2260 0,\u2009\u2009w \u2260 0,\u2009\u2009z = 0) and a positive (infection) equilibrium E22 in which x \u2260 0,\u2009\u2009E \u2260 0,\u2009\u2009y \u2260 0,\u2009\u2009v \u2260 0,\u2009\u2009z \u2260 0,\u2009\u2009w \u2260 0.The system  has two E00 = , where x0 = \u03bb/\u03c1, and w0 = k5/k7, and the basic reproductive number is obtained by the following method.The infection-free equilibrium is R0 = \u03c1(FV\u22121), whereR0 = a2kx0\u03b2/a1\u03b5(a1 + a2).Based on integral operator spectral radius, the basic reproductive number is E11 = , where x1 = a1\u03b5(a1 + a2)/a2k\u03b2,\u2009\u2009e1 = (a1\u03b5/a2k)v1,\u2009\u2009y1 = (\u03b5/k)v1,\u2009\u2009v1 = a2k\u03bb/a1\u03b5(a1 + a2) \u2212 \u03c1/\u03b2, and w1 = k5/k7. Similarly, we have the basic reproductive number is R1 = k3v1/k4 + kk3y1k1/a1k2k4 at E11 = .The equilibrium without immune response is E22 = , whereThe infected positive equilibrium is In this section, we will employ the direct Lyapunov method and LaSalle's invariance principle to establish the global asymptotic stability of the two boundary equilibria.E00 is globally asymptotically stable if and only if R0 < 1.The infection-free equilibrium See E11 is globally asymptotically stable if and only if R1 < 1.The equilibrium without immune response See E22.In this section, we will discuss the local asymptotic stability and Hopf bifurcation of the positive equilibrium E22 is as follows:The characteristic equation of system  at E22 DefineH above becomes\u03c4 = 0,  lie to the left of the imaginary axis when \u03c4 = 0. However, with \u03c4 increasing from zero, some of its roots may cross the imaginary axis to the right. In this case, there are some roots having positive real parts, and therefore the equilibrium E22 becomes unstable. Next, we will discuss the stability of system (E22 when \u03c4 > 0.From f system  at E22 We first divide  into two\u03bb = i\u03c9\u2009\u2009(\u03c9 > 0). Substituting \u03bb = i\u03c9 into  = 0 has a positive real root \u03c92.Suppose  has a pu i\u03c9 into  yields = 0 has no positive real roots, then the positive equilibrium E22 is locally asymptotically stable for any \u03c4 > 0.If G(x) = 0 has no positive real roots, then obviously  = 0 has j = 0,1, 2,\u2026.Substituting  i\u03c9 into , we obtaH = 0 has a pair of purely imaginary roots j and \u03bbnj)((\u03c4) = \u03b1nj)((\u03c4) + i\u03c9nj)((\u03c4) as the root of (\u03c4nj)((\u03c4nj) = 0 and Therefore, the characteristic equation  root of  near \u03c4nG(x) = 0 has some positive real roots, then E22 is locally asymptotically stable for \u03c4 \u2208 . Therefore, system  = F1\u03b4(\u03b8)d\u03b8 + F2\u03b4(\u03b8 + \u03c4)d\u03b8.By the Riesz representation theorem , there e\u03d5 \u2208 C, we further define\u03c6 \u2208 C, define\u03d5, \u03c6\u03b7(\u03b8) = \u03b7 and \u03d5 \u2208 C. Then A(0) and A\u2217(0) are adjoint operators.For n system  can be wh(\u03b8) and h\u2217(s) be the eigenvectors of A(0) and A\u2217(0) corresponding to the eigenvalues i\u03c90 and \u2212i\u03c90, respectively. We choose h(\u03b8) and h\u2217(s) as h\u2217(s), h(\u03b8)\u232a = 1 is satisfied. We give the detailed computation of (Let \u2217(s) as h\u03b8=1,h2,hg20,\u2009\u2009g11,\u2009\u2009g02, and g21, using the method given in  = sign[(dG/dx)|x=xn].Then the following values can be computed:ectively . From (Cu2\u2217 = \u2212Re(c1(0))/G\u2032(\u03c902). We obtain the following theorem.Let Assume the hypotheses (1), (2), and (3) at the beginning of u2\u2217 > 0\u2009\u2009(u2\u2217 < 0), then the bifurcating periodic solutions exist for \u03c4 > \u03c40\u2009\u2009(\u03c4 < \u03c40) in a \u03c40-neighborhood.(1) If \u03b22 < 0\u2009\u2009(\u03b22 > 0), the bifurcating periodic solutions are orbitally asymptotically stable as t \u2192 +\u221e\u2009\u2009(t \u2192 \u2212\u221e).(2) If \u03c40 = \u03c4n0(0), where c1(0)).(1) If \u03b22 < 0, namely, Re(c1(0)) < 0, then there exist stable periodic solutions for \u03c4 > \u03c4n0(0) in a \u03c40-neighborhood. So the bifurcating periodic solutions are orbitally asymptotically stable as t \u2192 +\u221e.(2) If In this section, we will numerically illustrate the theoretical results obtained above and also discuss how the factors such as the immune response to infected cells and viruses and the proliferation of hepatocytes affect the dynamics of the model under time delay.For the following simulations, we choose the parameter values for system  as follox(t) = 1,\u2009\u2009e(t) = 1,\u2009\u2009y(t) = 1,\u2009\u2009v(t) = 1,\u2009\u2009z(t) = 1, and w(t) = 1, where t \u2208 .We set the initial values to E22 =  and the critical time value \u03c4n0(0) = 0.041.With the parameter values given in , we have\u03c4 > 0.041, we obtain stable bifurcating periodic solutions. For example, when \u03c4 = 0.05, the simulation result is shown Figures When \u03c4 < 0.041, the bifurcating periodic solutions are unstable. For example, when \u03c4 = 0.03, the simulation result is shown in Figures E22 is asymptotically stable and the system will converge to E22.When \u03c4), the radius of limit cycle will increase. The simulation result is shown With the increasing of time delay  equilibrium and also the stability of the bifurcating periodic solutions. We also numerically illustrate the Hopf bifurcation and the stability of the bifurcating periodic solutions.Moreover, we numerically illustrate how the factors such as the time delay, the immune response to infected cells and viruses, and the proliferation of hepatocytes affect the dynamics of the model, which shows that the former two factors have a big effect on the model dynamics, while the latter one does not have a big effect."}
{"text": "There is an incorrect figure referenced in the third paragraph of the \u201c3.1 Grouping Taxis by Incomes\u201d section of the Results. In the section titled \u201cWorking time\u201d, \u201cFig 5\u201d should be listed as \u201cFig 6\u201d.There is an incorrect figure referenced in the fourth paragraph of the \u201c3.1 Grouping Taxis by Incomes\u201d section of the Results. In the section titled \u201cPassenger load\u201d, \u201cFig 6\u201d should be listed as \u201cFig 5\u201d."}
{"text": "Scholars of Luke\u2013Acts have struggled to define the apostles\u2019 proclamations of judgment on those who threatened the early Christian community. Ananias and Sapphira (Acts 4.32\u20135.11), Simon magus (8.4-25) and Bar-Jesus (13.4-12) all fall victim to the apostles\u2019 words of power, yet scholars have typically shied away from categorizing their speeches as curses. Close analysis of the structure, style, phonaesthetic and dramatic aspects of the Greek texts suggests, however, that Luke indeed intends the apostles\u2019 speeches to be heard as curses whilst simultaneously presenting them as legitimate acts of power. A comparison with Greek and Coptic \u2018magical\u2019 texts helps to place the curses of Acts in the context of cursing traditions in the wider ancient Mediterranean world. They have generally regarded Paul\u2019s words to Bar-Jesus as a form of a curse,7 An analysis of each episode will, in the first place, attend to the thematic, grammatical and phonaesthetic properties of the apostle\u2019s speech to highlight its dramatic power in a performance setting, both within the narrative and in the narrative\u2019s recitation amongst Christian communities. Secondly, in each case I will employ a comparative perspective to evaluate the curses of Acts in relation to a selection of Greek and Coptic \u2018magical\u2019 texts. This approach aims to demonstrate how an ancient audience of Luke\u2013Acts would have heard the apostles engaging with curse traditions typical of ancient Mediterranean culture while also viewing them as divinely authorized acts of power superior to acts of \u2018magic\u2019.This article, in contrast, will employ literary analysis to demonstrate key links between these three episodes in Acts, arguing that these conflict narratives should be read in continuity with each other.8 In fact, curses play an important role in the Jewish and Christian scriptures.9 In this article I will follow the more anthropologically informed definition of a curse employed by David Frankfurter in his analysis of blessings and curses in Roman Imperial and Late Antique Egypt (or the subversive power that plagues one following such a performance (as in \u201cher curse still rests on me\u201d)\u2019. However, I would also add the qualification that this subversive power can be operated in a way deemed to be socially legitimate or illegitimate. Thus I use \u2018curse\u2019 as a neutral term that is open to construal as either \u2018miracle\u2019 or \u2018magic\u2019 according to its performer\u2019s perceived authority and status. Following the studies of Poupon, Reimer, Marguerat and others,10 this article will be sensitive to the strategies employed by the author of Acts to navigate a context in which the apostles\u2019 words of power are susceptible to accusations of magic. It will go further than previous studies by placing the strategies themselves within cursing traditions of Late Antiquity, showing greater overlap between them than previously acknowledged and the potential tensions in the claim to distinction found in Acts. Given that scholars widely accept Paul\u2019s words to Bar-Jesus to be a form of a curse, we will begin with that episode, identifying the key features of the curse itself before progressing to discuss the earlier episodes in the narrative.The hesitancy of scholars to categorize the apostles\u2019 judgments as curses may be linked to the continued association of the label \u2018curse\u2019 with uncritical conceptions of \u2018magic\u2019 that define magic as some impure or primitive form of religion.ue Egypt : 158. Fr11 in a similar manner to the OT prophets . Bar-Jesus is described as a \u03c8\u03b5\u03c5\u03b4\u03bf\u03c0\u03c1\u03bf\u03d5\u03ae\u03c4\u03b7\u03c2 (13.6), the prophets\u2019 natural antagonist.12 Paul is \u2018filled with the holy spirit\u2019 (13.9) whereas Bar-Jesus is said to be \u2018full of all deceit and villainy\u2019 (13.10). Such rivalry is reminiscent of the prophetic clashes attested throughout the Jewish scriptures, none more famous than Elijah\u2019s challenge to the prophets of Baal (1 Kgs 18.1-40). The text also describes Bar-Jesus as \u03bc\u03ac\u03b3\u03bf\u03c2, which is reminiscent of Moses and Aaron\u2019s encounter with the magicians of Pharaoh (Exod. 7\u20139). These intertextual frames prepare the audience to expect a power conflict between Paul and Bar-Jesus.The beginning of Acts 13.4-12 sets the scene in Gentile Salamis and introduces some of the key themes that are to follow in the narrative. It employs intertextual allusions in order to parallel Paul with Bar-Jesus and to posit the pair as natural opponents. Paul is \u2018sent out by the Holy Spirit\u2019 (13.4),Paul\u2019s speech can be divided into the following parts and contains three themes that are also found in Peter\u2019s pronouncements in Acts 4.32\u20135.11 and 8.9-25.Paul begins by re-labelling his opponent. He then asks a rhetorical question which provides an effective contrast to his previous statements. His speech comes to a climax as he vividly curses his opponent by \u2018the hand of the Lord\u2019.Paul\u2019s pronouncement is notable for its direct style, created by its repeated re-labelling of the adversary. In addressing Bar-Jesus (\u2018son of Jesus\u2019) as \u03c5\u1f31\u1f72 \u03b4\u03b9\u03b1\u03b2\u03cc\u03bb\u03bf\u03c5 (13.10), Paul employs irony to accuse him of being an impostor : 617. InThe first of these three themes is the presence of the demonic, introduced through the re-branding of Bar-Jesus as \u03c5\u1f31\u1f72 \u03b4\u03b9\u03b1\u03b2\u03cc\u03bb\u03bf\u03c5. This designation is a major element in creating Paul\u2019s direct style of address. Its position in the syntax of the sentence also emphasizes deceit and villainy and possibly implies that the label \u2018son of the devil\u2019 is dependent on or deducible from Bar-Jesus\u2019 actions.The second theme, implicit in this instance, is the adversary\u2019s loss of status in the community of God. Despite appearing to be a \u1fbd\u0399\u03bf\u03c5\u03b4\u03b1\u1fd6\u03bf\u03c2, the text reveals that Bar-Jesus is in fact a \u03c8\u03b5\u03c5\u03b4\u03bf\u03c0\u03c1\u03bf\u03d5\u03ae\u03c4\u03b7\u03c2 (13.6), effectively undermining his legitimacy as a member of the house of Israel. In fact, by describing him as a \u03bc\u03ac\u03b3\u03bf\u03c2 Luke depicts Bar-Jesus as someone more akin to an antagonistic Gentile. In addition to identifying him as a \u03bc\u03ac\u03b3\u03bf\u03c2, the opening verses (6-7) characterize him in a fashion strongly reminiscent of the court magicians in the Jewish scriptures who attend the palaces of Gentile kings . Such intertextual allusions contribute to the re-labelling of Bar-Jesus as an outsider in relation to the community, as impious and atheistic as a Gentile magician. Paul\u2019s curse completes the stripping of Bar-Jesus\u2019 status as a Jewish holy man when he challenges him (13.10): \u03bf\u1f50 \u03c0\u03b1\u03cd\u03c3\u1fc3 \u03b4\u03b9\u03b1\u03c3\u03c4\u03c1\u03ad\u03d5\u03c9\u03bd \u03c4\u1f70\u03c2 \u1f41\u03b4\u03bf\u1f7a\u03c2 [\u03c4\u03bf\u1fe6] \u03ba\u03c5\u03c1\u03af\u03bf\u03c5 \u03c4\u1f70\u03c2 \u03b5\u1f50\u03b8\u03b5\u03af\u03b1\u03c2; Paul effectively accuses his opponent of perverting the nation, an accusation traditionally made against those who promoted idolatry and impurity in the OT  and for which dramatic punishments were commanded : 110. Pafull of all deceit and villainy\u2019 (13.10). Paul\u2019s searching gaze (13.9) reveals that Paul has insight into his adversary\u2019s inner self, an ability commonly associated with God and his prophets in the OT, as well as with Jesus in the gospels.13The third theme is the impurity of the adversary\u2019s heart. Paul describes Bar-Jesus as \u2018Finally, the reaction of the witness to Paul\u2019s miracle is also significant: \u2018When the proconsul saw what had happened, he believed\u2019 (13.12). Paul\u2019s curse, like his more positive acts of power, leads to conversion or else a newly found respect for the Christian movement.defixiones, amulets, ostraca and inscriptions provide valuable insights into the practice of cursing in ancient Mediterranean societies.14 Whilst the dating of the Coptic Christian curses in Meyer and Smith\u2019s collection varies widely (first century ce to eleventh or twelfth century), the texts in Preisendanz\u2019s Papyri Graecae Magicae . Whereas binding curses from classical Athens are very formulaic, curses of the imperial era are far more elaborate and explicit. In addition to appeals\u2019 : 13-14. defixiones in the way that they often address the local gods of the institutional cults, in their lack of magical formulae and their use of quasi-legal language, in their supplicatory posture, and in their desire for recompense on named enemies rather than anonymous ones , constitute a type of curse distinct from that of public cursing , suggesting the immediate fulfilment of Paul\u2019s pronouncement.28At first glance, Paul\u2019s judgment on Bar-Jesus appears the most curse-like because of his direct style of address and use of particular motifs. His use of the second person imperative (13.11a: \u03ba\u03b1\u1f76 \u03bd\u1fe6\u03bd \u1f30\u03b4\u03bf\u1f7a) draws the comparison to the mode of Faraone\u2019s first category of \u2018direct curse\u2019, \u2018which is a performative utterance, that is, a form of incantation by which the tic way\u2019 : 10. AltPGM IV.467-68 and IV.831-32 quote Homer\u2019s Iliad 8.424 in the form of a question.Paul\u2019s utterance also shares a number of stylistic features and motifs with the curse texts. Paul\u2019s use of epiplexis in his curse on Bar-Jesus  mirrors the style of some Greek curse texts that employ epiplexis seemingly as part of their rhetorical strategy. \u03c4\u03bf\u03bb\u03bc\u03ae\u03c3\u03b5\u03b9\u03c2 \u0394\u03b9\u1f78\u03c2 \u1f04\u03bd\u03c4\u03b1 \u03c0\u03b5\u03bb\u03ce\u03c1\u03b9\u03bf\u03bd \u1f14\u03b3\u03c7\u03bf\u03c2 \u1f00\u03b5\u1fd6\u03c1\u03b1\u03b9;29Will you dare to raise your mighty spear against Zeus?Devices such as rhetorical questions serve to heighten the drama of the curse ritual and, if performed publicly, had the potential to intimidate one\u2019s opponent.PGM XIVc.16-27 uses \u03ba\u03b1\u03c4\u03b1\u03b2\u03ac\u03bb\u03bb\u03c9 to convey sudden, harmful action as well as a spatial dynamic of descent.The most obvious parallel between Paul\u2019s curse and Greek and Coptic curse spells is the desire for the deity to strike the spell-caster\u2019s adversary. \u1f27\u03ba\u03ad \u03bc\u03bf\u03b9 \u03ba[\u03b1\u1f76] \u03b2\u03ac\u03b4\u03b9\u03c3\u03bf\u03bd \u03ba\u03b1\u1f76 \u03ba\u03b1\u03c4\u03ac\u03b2\u03b1\u03bb\u03b5 \u03c4\u1f78\u03bd \u03b4\u03b5\u1fd6\u03bd\u03b1 (\u1f22 \u03c4\u1f74\u03bd \u03b4\u03b5\u1fd6\u03bd\u03b1) \u1fe5\u03af\u03b3\u03b5\u03b9 \u03ba\u03b1\u1f76 \u03c0\u03c5|\u03c1\u03b5\u03c4\u1ff7 \u03b1\u1f50\u03c4\u1f78\u03c2 \u1f20\u03b4[\u03af]\u03ba\u03b7\u03c3\u03ad\u03bd \u03bc\u03b5 \u03ba\u03b1\u1f76 \u03c4\u1f78 \u1f07\u03b9\u03bc\u03b1 \u03c4\u03bf\u1fe6 \u03a4\u03c5\u03d5\u1ff6\u03bd\u03bf\u03c2 \u1f10\u03be\u03ad\u03c7\u03c5\u03c3\u03b5\u03bd \u03c0\u03b1\u03c1\u1fbd \u1f11\u03b1\u03c5|\u03c4\u1ff7 (\u1f22 \u03b1\u1f51\u03c4\u1fc7). \u03b4\u03b9\u1f70 \u03c4\u03bf[\u1fe6]\u03c4\u03bf \u03c4\u03b1\u1fe6\u03c4\u03b1 \u03c0\u03bf\u03b9\u1ff6PGM XIVc.25-27 = PDM xiv.690-94, trans. \u2018Come to me and go and strike down him, NN  with chills and fever. That very person has wronged me and he (or she) has spilled the blood of Typhon in his own (or her own) house. For this reason I am doing this\u2019 . Another implores: \u2018You must strike Prestasia and Tnounte and Eboneh, quickly, deservedly\u2019.30Such requests are common in Coptic spell texts. One speaker orders the deity to \u2018strike Philadelphe and her children\u2019 . The vivid image of descent conjured for the audience reflects the descent of Bar-Jesus\u2019s social status as he submits to the apostle.Many curse texts also emphasize the spatial movement of descent. They speak of bringing their victims \u2018down\u2019 in some manner : 121-23,PGM III.494-611 appeals to a deity who \u2018lowers darkness\u2019.Another parallel between the two groups of texts is their choice of affliction. Multiple Greek and Coptic curse texts seek to inflict blindness upon their enemies. \u03b4\u03b5\u1fe6\u03c1\u03cc \u03bc\u03bf\u03b9,|| \u03ba\u03cd\u03c1\u03b9\u03b5, \u1f41 \u03c0\u03bf\u03c4\u1f72 \u03c4[\u1f78] \u03d5\u1ff6\u03c2 \u1f00\u03bd\u03ac[\u03b3]\u03c9\u03bd, \u03c0\u03bf\u03c4\u1f72 \u03c4\u1f78 \u03c3\u03ba\u03cc\u03c4\u03bf\u03c2 \u03ba\u03b1\u03c4\u03ac|\u03b3\u03c9\u03bd <\u03ba\u03b1\u03c4\u1f70> \u03c4\u1f74\u03bd \u03c3\u03b5\u03b1\u03c5\u03c4\u03bf\u1fe6 \u03b4\u1fe6\u03bd\u03b1\u03bc\u03b9\u03bd [\u2026]\u2018Come to me, / lord, you who sometimes raise the light, sometimes lower the darkness [with] your own power\u2019 .33 One Coptic curse asks the deity to \u2018cause their eyes to fog and come out\u2019 . Another requests blindness to fall on its victim\u2019s \u2018two eyes\u2019 .Again, this spell evokes a descending spatial dynamic through its use of \u03ba\u03b1\u03c4\u03ac in \u03ba\u03b1\u03c4\u03ac\u03b3\u03c9\u03bd. Similarly, the prayer in PGM VII.764-65 describes the deity\u2019s ability to \u2018wane from light into darkness\u2019 (trans. Grese).ACM 28 begins: \u2018avenge me on the one who opposes me and on the one who has driven me from my place\u2019 (trans. Meyer). Others present their speakers as vulnerable persons or victims of violence, gossip or slander (ACM 89-94). Paul\u2019s curse depicts Bar-Jesus as an aggressor; he is an \u1f10\u03c7\u03b8\u03c1\u03cc\u03c2, an adversarial term in curse texts.34 He is also \u2018full of deceit (\u03b4\u03cc\u03bb\u03bf\u03c2)\u2019, a common term for unknown causes of death which Versnel includes as part of an \u2018idiom of suspicion\u2019 that characterizes \u2018prayers for justice\u2019 . However, Luke suggests that it is God who is the victim of Bar-Jesus\u2019 aggression (13.10), which helps to portray Paul\u2019s curse as an act of divine judgment rather than one of personal vengeance.Both Paul\u2019s curse and the \u2018magical\u2019 curse texts also share a sense of victimization: many Greek and Coptic curse texts are characterized by their speaker\u2019s desire for justice after being wronged. ACM 88, 92). Although the devil is not exactly the same as a demon in the NT, Paul shares the concept that his opponent is filled by a malevolent force. In Acts it seems especially important for the apostles to identify the satan as the force of evil in direct opposition to God , which appears to concur with the thought of the Bar-Jesus narrative.36 The curse texts also identify the demonic as both present in their enemies and as instruments of punishment. One curse includes the command: \u2018Send to him an evil demon\u2019, and another: \u2018You must make a demon descend upon her\u2019 . Another text requests that the deity brings physical pain and \u2018demonic madness\u2019 on its victim .The curse texts and Acts also propound the belief that the spell-caster\u2019s opponent is possessed by a demon (n to God . Howeverlasting image of the scene simply because the curse is not relieved within the narrative of Acts (1989: 84). But whilst it is the final image, the words \u1f04\u03c7\u03c1\u03b9 \u03ba\u03b1\u03b9\u03c1\u03bf\u1fe6 (13.11) prevent it from being the lasting one. At first glance, Paul\u2019s seeming allowance for repentance contrasts with some of the Greek and Coptic curses that command enduring afflictions on their opponents . However, Faraone has highlighted how many of the Greek binding curses seek not to harm aggressively but to restrict defensively (1991: 8). ACM 106 includes an intriguing clause in its command: \u2018Let them come and bring them down upon the body of N. child of N. \u2026 until I myself, N. child of N., have mercy on it\u2019 (trans. Emmel). If Faraone\u2019s argument is correct, namely, that Greek binding curses should be primarily characterized by their desire to control their opponents rather than by their will to harm, then Paul\u2019s curse on Bar-Jesus fits on the more defensive end of a spectrum of curse traditions that span from temporarily incapacitating a person, to causing illness and even killing. Luke\u2019s theological perspective does not place the apostle\u2019s speech outside the field of cursing but rather complements the type of curse that is cast.The limited duration of Bar-Jesus\u2019 blindness (13.11) has interested scholars and is often understood to be in keeping with the theological programme in Acts that desires sinners to repent and submit themselves to God\u2019s mercy. Garrett takes an opposing view in interpreting Bar-Jesus\u2019 blindness as the final and ACM 90, 90, 95). Luke\u2013Acts also connects to OT miracle stories through allusions. Paul describes the \u03c7\u03b5\u1f76\u03c1 \u03ba\u03c5\u03c1\u03af\u03bf\u03c5 working against Bar-Jesus, a probable allusion to Moses bringing the plagues against the Egyptians (\u03b4\u03ac\u03ba\u03c4\u03c5\u03bb\u03bf\u03c2 \u03b8\u03b5\u03bf\u1fe6 in LXX Exod. 8.15), as well as to Jesus whose exorcisms are described as \u1f10\u03bd \u03b4\u03b1\u03ba\u03c4\u03cd\u03bb\u1ff3 \u03b8\u03b5\u03bf\u1fe6 (Lk. 11.20). Paul\u2019s curse does not employ historiola, but it does allude to OT intertexts that, as well as framing the scene within certain narrative expectations, function to suggest that both Paul and the text of Luke\u2013Acts carry a similar authority to that of the OT. Both Paul\u2019s curse and the spell texts appeal to holy scriptures to acquire some form of status or power. In summary, a comparison of Paul\u2019s curse with Greek and Coptic curse texts has shown that they share a number of characteristics such as demonic possession and punishment, the desire for immediacy, the use of spatial imagery, and connections to historical demonstrations of divine power.A final parallel between Acts and the curse texts can be found in their common appeal to well-known miracle stories. A number of the spell texts from late Antiquity employ historiola as a means of accessing divine powers described in stories of past events. The curse texts often appeal to OT stories of God\u2019s judgment against figures such as Cain, Pharaoh or the Assyrian army , in the same way that they \u2018listened eagerly\u2019 to Simon (8.11), suggesting that the apostle competes for their attention as Paul did with Sergius Paulus. But it also firmly distinguishes between them by clarifying their contrasting sources of power. Philip demands attention through spirit-empowered signs, whilst Simon amazes the people by his \u2018magic\u2019. As with Bar-Jesus, Simon\u2019s association with \u03bc\u03b1\u03b3\u03b5\u03cd\u03c9 may have reminded audiences of notorious adversaries in the Jewish scriptures.In this episode, the bite of the curse comes at the beginning of the speech in the form of a sharp rebuke, followed by a series of verdicts and accusations.Like Acts 13.10-11, Peter\u2019s curse maintains the drama through its direct style of address. The second person pronoun is again heavily repeated, making use of \u03c3\u03bf\u03c5, \u03c3\u03bf\u03af and \u03c3\u03b5 eight times in 8.20-23. This focuses the audience\u2019s attention firmly on Peter\u2019s opponent and indicates that Peter has likewise fixed his gaze on him. In contrast, the double use of the first person pronoun in Simon\u2019s response to Peter shows Simon submitting to the direction of Peter\u2019s attention and verdict.The phonaesthetic features of Peter\u2019s curse also function to heighten the drama of its performance. The text employs sibilance in \u03c3\u03bf\u03c5 \u03c3\u1f7a\u03bd \u03c3\u03bf\u1f76 and \u03b5\u1f30\u03c2. It also creates assonant effects with repeated \u03b5\u03b9 and \u03b7 sounds (\u03b5\u1f34\u03b7 \u03b5\u1f30\u03c2 \u1f00\u03c0\u03ce\u03bb\u03b5\u03b9\u03b1\u03bd). Such devices give a fluid, poetic quality to phrases like \u03c4\u1f78 \u1f00\u03c1\u03b3\u03cd\u03c1\u03b9\u03cc\u03bd \u03c3\u03bf\u03c5 \u03c3\u1f7a\u03bd \u03c3\u03bf\u1f76 \u03b5\u1f34\u03b7 \u03b5\u1f30\u03c2 \u1f00\u03c0\u03ce\u03bb\u03b5\u03b9\u03b1\u03bd (8.20), at the same time functioning to hold the whole speech together as a textual unit.The thematic content of the speech itself also indicates continuity with Paul\u2019s own speech. Verse 20 suggests that Simon\u2019s crime is one involving greed and deceit. The text implies that Simon desires the ability to bestow the Holy Spirit because of its profiteering potential. Whilst Strelan interprets Simon as a sincere insider who mistakenly thinks he can become more fully integrated in the new movement by means of his money (2004: 213), there is better support for Klauck\u2019s interpretation that Simon is slipping back into old ways (2000: 20). Peter\u2019s insight into the state of Simon\u2019s heart not only implies that Simon\u2019s understanding of the Spirit is misplaced, but that his motivation is wrong.As with Acts 13.4-12, v. 21 shows that Peter, like Paul, has the ability to see into a person\u2019s heart. The repetition of \u03ba\u03b1\u03c1\u03b4\u03af\u03b1 draws attention to both Peter\u2019s power and Simon\u2019s true nature. That Simon\u2019s heart is \u2018not right before God\u2019 and that his intent requires forgiveness suggest a deep-rooted opposition to the apostles, rather than an earnestly misplaced one (contra Strelan).The element of excommunication that underlies Acts 13.4-12 is explicit in Acts 8.21. Because Simon is a member of the growing community of believers in Samaria, his chastisement by the community\u2019s leaders has to be publicly demonstrated in order to protect the community\u2019s purity and reputation. Verses 18-19 describe Simon\u2019s attempt to purchase the Holy Spirit, a gift often understood by early Christians as a sign of the community\u2019s inheritance (cf. 2 Cor. 1.22). As a result, Peter declares: \u2018You have no part or share in this\u2019 (Acts 8.21). Despite having claimed to have joined the Christian community, Simon\u2019s greed shows him to be an impostor. Peter\u2019s curse functions to publicly reverse Simon\u2019s reputation before the eyes of those who had previously listened eagerly to him (8.10). For the audience of Acts, Simon\u2019s \u2018true\u2019 status as an impostor should come as no surprise in the wake of Luke\u2019s characterization of Simon as one who practised the arts of the \u03bc\u03ac\u03b3\u03bf\u03b9, a label that may have invited suspicion in connection to LXX Dan. 2.2.40 Like Bar-Jesus, Simon is perceived to be opposed to righteousness . Whilst the standard labels for demonic powers are not used in the story of Simon magus, lexical and thematic links with Acts 13.4-12 and other exorcism narratives in Luke\u2013Acts suggest that readers should understand Simon, like Bar-Jesus, to be under the influence of demonic forces and in opposition to God; this in turn invites the reader to recognize a theological battle between God and evil underlying the narrative.Like Acts 13.10, the demonic is also present in Peter\u2019s opponent despite the absence of labels like \u03b4\u03b1\u03b9\u03bc\u03cc\u03bd\u03b9\u03bf\u03bd, \u03b4\u03b9\u03ac\u03b2\u03bf\u03bb\u03bf\u03c2, or \u03c3\u03b1\u03c4\u03b1\u03bd\u1fb6\u03c2. The absence of such terms leads Peter\u2019s speech shares a number of motifs with Greek and Coptic curses and also with some crucial elements of Faraone\u2019s second category of binding curses: \u2018the prayer formula\u2019 (1991: 10). Although it does not match the category\u2019s most defining characteristic \u2013 it is not addressed to God \u2013 it does manifest a number of its other aspects. It employs the third person optative, \u2018may your silver perish with you\u2019. Its disclosure that it is God who is the true victim of Simon\u2019s crime and its desire for confession also evoke Versnel\u2019s related category of \u2018prayers for justice\u2019 or \u2018judicial prayers\u2019 : 192.In \u2018prayers for justice\u2019 the speaker typically seeks vindication for some harm, real or perceived, often relating to stolen goods and/or to potential gossip or slander. In these prayers, the petitioner appears to \u2018transfer\u2019 the crime onto the deity, employing quasi-legal vocabulary : 175. Th41I consecrate to the mother of the gods the gold pieces that I have lost, all of them, so that the goddess will track them down and bring everything to light and will punish the guilty in accordance with her power and in this way will not be made a laughing stock.The act of \u2018handing over\u2019 the culprit to the deity or to some lesser power is the natural extension of the transferral of victimization onto the deity. In the eyes of the spell-caster, divine punishment functions as either irrevocable punishment or as a \u2018conditional and temporary means of pressure\u2019 which might bring the culprit to confession or the perceived damage to restitution : 80.This \u2018transfer\u2019 terminology is clearly seen in other words attributed to the apostles; Paul \u2018hands over\u2019 (\u03c0\u03b1\u03c1\u03b1\u03b4\u03af\u03b4\u03c9\u03bc\u03b9) troublesome figures to Satan for limited periods . In the case of Peter\u2019s speech, the transfer is implicit, as the apostle simply states that Simon\u2019s crime is against God and the Holy Spirit, using quasi-legal terminology such as \u1f00\u03b4\u03b9\u03ba\u03af\u03b1. Simon\u2019s response (8.24) acknowledges that he has been moved into the sphere of divine influence.42 In many curse texts, a public confession of guilt, and the humiliation it would bring, is as desirable as the return of stolen possessions because of the honour it would restore to the deity and to the wronged person .d person : 153.PGM IV.1227-64.43In addition to these aspects of \u2018prayers for justice\u2019, Peter\u2019s pronouncement shares other motifs with the curse texts. A number of scholars have recognized the lexical overlap of Peter\u2019s curse (8.20: \u03c4\u1f78 \u1f00\u03c1\u03b3\u03cd\u03c1\u03b9\u03cc\u03bd \u03c3\u03bf\u03c5 \u03c3\u1f7a\u03bd \u03c3\u03bf\u1f76 \u03b5\u1f34\u03b7 \u03b5\u1f30\u03c2 \u1f00\u03c0\u03ce\u03bb\u03b5\u03b9\u03b1\u03bd) with \u1f14\u03be\u03b5\u03bb\u03b8\u03b5, \u03b4\u03b1\u1fd6\u03bc\u03bf\u03bd,|| \u1f10\u03c0\u03b5\u03af \u03c3\u03b5 \u03b4\u03b5\u03c3\u03bc\u03b5\u03cd\u03c9 \u03b4\u03b5\u03c3\u03bc\u03bf\u1fd6\u03c2 \u1f00\u03b4\u03b1\u03bc\u03b1\u03bd\u03c4\u03af\u03bd\u03bf\u03b9\u03c2 | \u1f00\u03bb\u03cd\u03c4\u03bf\u03b9\u03c2, \u03ba\u03b1\u1f76 \u03c0\u03b1\u03c1\u03b1\u03b4\u03af\u03b4\u03c9\u03bc\u03af \u03c3\u03b5 \u03b5\u1f30\u03c2 \u03c4\u1f78 \u03bc\u03ad|\u03bb\u03b1\u03bd \u03c7\u03ac\u03bf\u03c2 \u1f10\u03bd \u03c4\u03b1\u1fd6\u03c2 \u1f00\u03c0\u03c9\u03bb\u03b5\u03af\u03b1\u03b9\u03c2.Come out, daimon, since I bind you with unbreakable adamantine fetters, and I deliver you into the black chaos in perdition .PGM IV.1227-64 is an exorcism spell rather than a curse text. However, this should not deter one from drawing a comparison with Peter\u2019s curse since the exorcism simply involves the client aiming the curse towards the demon rather than towards an individual. Both texts use \u1f00\u03c0\u03ce\u03bb\u03b5\u03b9\u03b1 as an appropriate punishment for their adversary and favour sibilant phonetics in their speech.44 The sense of a demonic presence in Peter\u2019s reference to \u2018bonds\u2019 also parallels many of the spell texts that describe their opponents as being possessed by a demonic power. The belief that a person could be overpowered by evil is present in both groups of texts.It should be acknowledged that PGM XL.1-18 curses her daughter\u2019s father who has apparently robbed her daughter of her funeral gifts.The style of Peter\u2019s curse on Simon and his money also parallels Greek curses against an individual\u2019s property as well as against their person. The client of \u03c4\u1fc6\u03c2 \u03b4\u1f72 | \u03ba\u03b1\u03c4\u03b1\u03b2\u03bf\u03b9\u1fc6\u03c2 \u1f10\u03bd\u03b8\u1fe6\u03c4\u03b1 \u03ba\u03b5\u03b9\u03bc\u03ad\u03bd\u03b7\u03c2, \u03ba\u03b1\u03ba\u1ff6\u03c2 \u1f00\u03c0\u03bf\u03bb\u03bb\u03cd\u03bf\u03b9\u03c4\u03bf \u03ba\u1f10\u03b3 \u03b3\u1fc6\u03b9 \u03ba\u1f10\u03bd \u03b8\u03b1\u03bb\u03ac\u03c3\u03c3\u03b7\u03b9 \u03ba\u03b1\u1f50\u03c4\u1f78\u03c2 | \u03ba\u03b1\u1f76 \u03c4\u1f70 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f51\u03c0\u1f78 \u03c4\u03bf\u1fe6 \u1f48\u03c3\u03b5\u03c1[\u03ac]\u03c0\u03b9\u03bf\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1ff6\u03bd \u03b8\u03b5\u1ff6\u03bd \u03c4\u1ff6\u03bd \u1f00\u03bc\u03c0\u1fbd \u1f48\u03c3\u03b5\u03c1\u1fb6\u03c0\u03b9 \u03ba\u03b1\u03b8\u03b7\u03bc\u03ad\u03bd\u03c9\u03bd [\u2026]As long as my cry for help is deposited here, he and what belongs to him should be utterly destroyed badly, both on earth and on sea, by Oserapis and the gods who sit together with Oserapis [\u2026] .The speaker of this curse adapts the punishment to the theme of the crime; her adversary is a thief, therefore his possessions are involved in his retribution. A similar style is employed in the first line of Peter\u2019s curse.Like historiola in the curse spells, Acts 8.4-25 consciously associates itself with historic instances of divine power. Both Peter\u2019s instruction and Simon\u2019s subsequent petition are pregnant with allusions to the OT. Peter\u2019s instruction to repent and Simon\u2019s positive reaction parallel the interaction between Jonah and the city of Nineveh (LXX Jon. 3.1-9), another narrative in which a man of God is sent to a Gentile land to proclaim the news of God\u2019s judgment. Connections include \u1f00\u03c0\u03ce\u03bb\u03b5\u03b9\u03b1 , \u03bc\u03b5\u03c4\u03b1\u03bd\u03bf\u03ad\u03c9  and \u03ba\u03b1\u03ba\u03af\u03b1 . The second part of Simon\u2019s petition is also evocative of the words of the narrator in Jon. 3.10. For an audience familiar with the Septuagint, these parallels frame Peter as a prophet in the mould of Jonah, who is sent by God to a non-Jewish audience. These OT allusions function to associate the apostles with the authority of the Jewish scriptures, an important strategy for presenting the apostles\u2019 curses as legitimate acts of power. The text presents Peter in line with the prophets, authoritative agents commissioned to communicate God\u2019s will to his people. Peter\u2019s curses are therefore legitimate in the same way that the prophets were authorized to curse on behalf of the Lord . Such strategies for presenting the apostolic curses as carrying divine authorization are vital for defending the apostles\u2019 actions against charges of magic.45 However, by comparing Peter\u2019s words with Greek curse texts from the \u2018magical\u2019 papyri, another dimension to Simon\u2019s reaction can be offered. Peter\u2019s use of \u03c7\u03bf\u03bb\u03ae connects his speech to a number of texts in the PGM where \u03c7\u03bf\u03bb\u03ae is frequently employed in lists of \u2018magical\u2019 ingredients to denote the gall or bile of animals required for the performances of spells.46PGM XII.401-44 even acknowledges that such ingredients were inscribed on religious statutes by temple scribes so that common people would not use them to practise magic. The use of \u03c7\u03bf\u03bb\u03ae, when combined with \u03b4\u03b5\u03c3\u03bc\u03cc\u03c2, a term that also has \u2018magical\u2019 connotations (binding spells were known as \u03ba\u03b1\u03c4\u03ac\u03b4\u03b5\u03c3\u03bc\u03bf\u03b9), suggests that Peter\u2019s curse could involve a knowing slur on Simon\u2019s association with magic (LSJ: 380).An obvious problem with interpreting Peter\u2019s words to Simon as a curse is that its effects do not appear to materialize. Strelan, one of the few scholars to interpret Peter\u2019s words as a curse, argues that Simon\u2019s request for a prayer represents a request for a remedial blessing in a fashion reminiscent of Judg. 17.1-4 (2004: 215). Strelan\u2019s reading is convincing, strengthened by the thematic and lexical parallels between these two narratives.PGM IX.1-14, a text focused on defeating an opponent\u2019s anger.However, Peter\u2019s use of \u03c7\u03bf\u03bb\u03ae also brings it into comparison with \u03b8\u03c5\u03bc\u03bf\u1fe6 \u03c3\u03b5 \u03c0\u03b1<\u03cd>\u03c3\u03c9 \u03ba\u03b1\u03af \u03c3\u03b5 \u03c0\u03c1\u03b1\u03b0\u03bd\u03c9 \u03c7\u03bf\u03bb\u1fc6\u03c2. \u1f10\u03bb\u03b8\u1f72 \u03ba\u03b1\u1f76 \u03b4\u03b9\u03b1\u03ba\u03c1\u03ac\u03c4\u03b5\u03b9 \u03c3\u03b9\u03b3\u1fc7 \u03c3\u03b9\u03b3\u1f74\u03bd \u03d5\u03ad\u03c1\u03c9\u03bd| \u03c4\u03b5 \u03c0\u03b1<\u1fe6>\u03c3\u03b9\u03bd \u03ba\u03b1\u1f76 \u03b8\u03c5\u03bc\u03bf\u1f7a<\u03c2> \u03c3\u03c4\u1fc6\u03c3\u03bf\u03bd \u03c8\u03c5\u03c7\u1ff6\u03bd \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd \u1f40\u03c1\u03b3\u03ac\u03c2 \u03c4\u03b5 \u03c0\u03ac\u03c3\u03b1\u03c2 \u03c3\u03b2\u03ad\u03c3\u03bf\u03bd, \u03d5\u03c1\u03ad\u03bd\u03b1\u03c2 \u1f41\u03c1\u03ba\u03c1\u03af\u03c3\u03b1\u03c2 [\u2026]I\u2019ll give you rest from wrathAnd soothe your raging.Come silently and bringSilence and keep it.Stop ev\u2019ry wrath in soulsAnd melt all angerOf those with temper [\u2026] .Spells against anger in the \u2018magical\u2019 papyri combine positive sentiments that express the desire to cure an opponent with negative sentiments aimed towards their anger.\u03ba\u03b1\u03c4\u03c5\u03c0\u03cc\u03c4\u03b1\u03be\u03bf\u03bd, \u03ba\u03b1\u03c4\u03b1\u03b4\u03bf\u1fe6\u03bb\u03c9\u03c3\u03bf\u03bd, \u03d5\u03af\u03bc\u03c9\u03c3\u03bf\u03bd \u03c4\u1f74\u03bd \u03c8\u03c5\u03c7\u03ae\u03bd, \u03c4\u1f78\u03bd \u03b8\u03c5\u03bc\u1f78\u03bd <\u03c4\u03bf\u1fe6 \u03b4\u03b5\u1fd6\u03bd\u03b1>PGM IX.9, trans. Hock).47Bring into subjection, enslave, and put to silence the soul, the wrath  [\u2026]  establishes Peter in a dominant position over his adversary.Again, Peter\u2019s speech against his adversary is direct. The second person pronoun and the possessive adjective are employed five times in the forms of \u03c3\u03bf\u03c5, \u03c3\u03b5, \u03c3\u03bf\u03b9, \u03c3\u03bf\u03c5 and \u03c3\u1fc7, focusing attention firmly on his opponent. He addresses Ananias by name, which functions to isolate Peter and Ananias from amongst the onlookers. That it is Peter who actively names Ananias . The three rhetorical questions function to communicate Peter\u2019s judgment on Ananias; he is relabelled as demonic, his deviancy is identified, and his status as an impostor is revealed.The final, declarative sentence does not issue a punitive command, but it is an incrimination that serves to pass Ananias over for divine judgment. The assonance of repeated \u03c9 sounds in the words \u03bf\u1f50\u03ba \u1f10\u03c8\u03b5\u03cd\u03c3\u03c9 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03b9\u03c2 \u1f00\u03bb\u03bb\u1f70 \u03c4\u1ff7 \u03b8\u03b5\u1ff7 gives the statement its musical quality. Through the structure and style of the text, Acts establishes the potency of Peter\u2019s words; his narration in v. 5 demonstrates its effects.Like his speech to Simon, Peter\u2019s speech to Ananias employs consonance and sibilance to enhance its rhetorical force. Repeated sigmas across \u03c3\u03bf\u03c5, \u03c8\u03b5\u03cd\u03c3\u03b1\u03c3\u03b8\u03b1\u03af \u03c3\u03b5 and \u03bd\u03bf\u03c3\u03d5\u03af\u03c3\u03b1\u03c3\u03b8\u03b1\u03b9 \u1f00\u03c0\u1f78 \u03c4\u1fc6\u03c2 \u03c4\u03b9\u03bc\u1fc6\u03c2 (5.3), combined with the alliteration of \u03c4\u1fc6\u03c2 \u03c4\u03b9\u03bc\u1fc6\u03c2 \u03c4\u03bf\u1fe6 \u03c7\u03c9\u03c1\u03af\u03bf\u03c5, serve to hold the speech together as a distinct unit of text: the repeated phonetics increase its fluidity as it crescendos towards its climax.The three themes of the two other episodes examined in this article are also present in Peter\u2019s rhetorical questions. First, Ananias is filled by the satan (5.3: reminiscent of Judas in Lk. 22.3), which parallels Peter who is \u2018filled with the Holy Spirit\u2019 (4.8). The presence of the demonic also places Peter\u2019s speech in line with the Coptic and Greek curse spells that identify their enemies with demonic activity and wish demonic punishment upon them.filled \u2026 to lie\u2019 by the satan. The third aspect shared with the other speeches is its excommunicatory function.49 Ananias is described as laying \u2018only a part\u2019 (\u03bc\u03ad\u03c1\u03bf\u03c2) of his proceeds at the apostle\u2019s feet (5.2). Ironically, the text implies that Ananias dies at Peter\u2019s feet and makes it explicit that Sapphira ends up there too (5.10), the same place where the money was laid. In this way, the punishment \u2013 as in some curse texts \u2013 mirrors the crime. In bringing only a part of the proceeds to the apostles, Ananias and Sapphira lose their part in the community of believers.Secondly, impure motives are also present. \u03ba\u03b1\u03c1\u03b4\u03af\u03b1 is repeated three times in the pericope. Just as Bar-Jesus was said to be \u2018full of all deceit and villainy\u2019 (13.10), so Ananias\u2019s heart is \u201850 The text emphasizes this effect through the repetition of \u03c0\u03af\u03c0\u03c4\u03c9 in the description of Ananias and Sapphira immediately falling down .Whilst Peter\u2019s speech itself does not include any explicit demand for divine striking, the effect of his words is one that suggests that his adversaries have been struck by the invisible hand of God.The curse-like nature of Peter\u2019s words is more explicit in his speech to Sapphira. With regard to the narrative of the episode, Peter\u2019s utterance comes at a point when he has seen the effects of his words on Ananias and has waited three hours for Sapphira\u2019s return (5.7). Peter incriminates her for the same crime as Ananias, but his verdict carries an even stronger tone of judgment.Peter again expresses his incrimination of Sapphira as a rhetorical question (5.9b). He identifies God as the victim of his opponent\u2019s deceit. Beginning with an emphatic \u1f30\u03b4\u03bf\u1f7a\u0300, Peter\u2019s verdict employs notable sibilance in the words \u03ba\u03b1\u1f76 \u1f10\u03be\u03bf\u03af\u03c3\u03bf\u03c5\u03c3\u03af\u03bd \u03c3\u03b5 to end his curse with a sharp, staccato flourish. The speech is then immediately followed by its fulfilment, as Luke portrays the effect by employing the modifier \u1f14\u03c0\u03b5\u03c3\u03b5\u03bd \u03b4\u1f72 \u03c0\u03b1\u03c1\u03b1\u03c7\u03c1\u1fc6\u03bc\u03b1 (5.10). Such emphasis on immediacy characterizes Peter\u2019s speech as words of power that cause Sapphira\u2019s death.If the audience understands Peter\u2019s words to Sapphira as a curse, then it follows that Peter\u2019s words to Ananias should be understood in a similar light. The first incident mirrors the second; both husband and wife are accused of the same crime, both suffer the same punishment, and both are carried out by the young men. The effects of Peter\u2019s words are immediate in both encounters: in v. 5 the text\u2019s use of participles  draws the audience\u2019s attention to the aspects of cause and effect.defixiones are written, or the senselessness of an accompanying figurine.51 Often, the characteristics of the corpse by which a curse was buried were also invoked: \u2018Just as this corpse lies useless, so too may everything be useless for NN\u2019.52 Jordan explains: \u2018That the living should be affected by the conditions of the dead with whom curse tablets are deposited is a common idea in Greek magic\u2019 (1999: 118). In Acts 5.9c Peter declares, in dramatic fashion, that the death of Ananias will be the model for Sapphira\u2019s own fate. Grimly, she falls down just as he fell down, becoming as useless as he is. This parallelism could simply be attributed to Luke\u2019s use of irony or accomplished literary style . In this context, Peter\u2019s proclamation would represent a curse at the harsher end of the spectrum.Peter\u2019s \u2018verdict\u2019 on Sapphira is even more intriguing when it is compared with \u2018persuasive analogy\u2019, the third style of curse in Faraone\u2019s typology. These curses reflect a belief that a ritual action might \u2018urge\u2019 the victim to become like something to which it is dissimilar : 8. AttrThe image of Ananias and Sapphira falling at the feet of Peter , juxtaposed with the upwards movement of the young members of the community (5.6), also evokes motifs from the Greek curse texts.\u03ba\u03b1\u1f76 \u03d5\u03af\u03bc\u03c9\u03c3\u03bf\u03bd, | \u1f51\u03c0\u03cc\u03c4\u03b1\u03be\u03bf\u03bd, \u03ba\u03b1\u03c4\u03b1\u03b4\u03bf\u03cd\u03bb\u03c9\u03c3\u03bf\u03bd \u03c4\u1f78\u03bd \u03b4\u03b5\u1fd6\u03bd\u03b1 \u03c4\u1ff7 \u03b4\u03b5\u1fd6\u03bd\u03b1 \u03ba\u03b1\u1f76 \u03c0\u03bf\u03af\u03b7|\u03c3\u03bf\u03bd \u03b1\u1f50\u03c4\u03cc\u03bd, \u1f51\u03c0\u1f78 \u03c4\u03bf\u1f7a\u03c2 \u03c0\u03cc\u03b4\u03b1\u03c2 \u03bc\u03bf\u03b9 \u1f14\u03bb\u03b8\u1fc3. |PGM VII.966-68, trans. Hock).Put to silence, subordinate, enslave him, NN, to him, NN, and cause him to come under my feet .The common themes that can be identified in the apostles\u2019 speeches strongly suggest that the confrontations with Bar-Jesus, Simon magus and Ananias and Sapphira should be read in continuity with each other. As a group of texts, they reveal important information concerning the attitude of the early church towards deviant and threatening behaviour. In each of these three narratives, the opponents\u2019 crimes are associated with deceit and greed, a tendency that appears to complement Luke\u2019s concern for the poor and destitute. Taken together, the texts also shed light on the nature of accusations of magic: Simon and Bar-Jesus\u2019 involvement with magic is associated with deceit and greed, but it also functions as an ethno-classification that emphasizes the foreign territory in which the apostles find themselves. Literary analysis of the three narratives suggests that Luke crafts each text in such a way that audiences would have heard the apostles\u2019 speeches as dramatic words of power. In contrast to interpreters who downplay the apostles\u2019 responsibility in the drama, this article has argued that the text emphasizes the relation between the apostles\u2019 pronouncements and the subsequent afflictions.voces magicae,55 whereas they do employ common styles of performance, showing elements of \u2018direct curses\u2019, \u2018prayers for justice\u2019 and \u2018persuasive analogies\u2019. Luke also frames the curses in agonistic contexts \u2013 the usual context for binding curses \u2013 as Paul and Peter compete with other wonder-workers and benefactors. Their accusations against impure motives mirror the concern for \u2018evil thoughts\u2019 and Schadenfreude in the curse texts , portraying Paul and Peter as genuine men of God who are comparable to the prophets of the Jewish scriptures. Their Spirit-powered signs are superior to anything practised by magicians, and they have the authority to judge those inside the Christian community. Rather than explicitly petitioning the deity for action in the manner of the supplicants of the curse spells, the apostles speak for and about the deity in their capacity as his agents. However, for the reasons stated above, Peter\u2019s curse on Sapphira strains these strategies to their limits, which is reflected in the \u2018great fear\u2019 of those who hear about the encounter (5.11). In this episode, Luke must rely on the sheer authority of the charismatic leader within the Christian community (4.30-31) to legitimate his actions.Smith\u2019s topography of religions  offers a"}
{"text": "Edition of December 2017, vol. 109 (6), Supl. 2, p. 1-34In the \u201cAtualiza\u00e7\u00e3o das Diretrizes Brasileiras de Valvopatias: Abordagem das Les\u00f5esAnatomicamente Importantes\u201d, published as a supplement to the Arquivos Brasileiros deCardiologia , the following corrections shouldbe considered:In table 18, column 5, line 6: replacement of \u201cIIb\u201d with \u201cIIb C\u201d.In Table 24, line 4: the items \u201cIB, IB, IC\u201d should be aligned with the phrase\u201dFra\u00e7\u00e3o de eje\u00e7\u00e3o < 50%\u201d. In line 5, items \u201cIIa, IIa B, I C\u201d should be alignedwith the phrase \u201c Aus\u00eancia de reserva inotr\u00f3pica no teste ergom\u00e9trico e/ou baixacapacidade funcional\u201d. On line 6, remove item \u201cB\u201d.In table 24, section \u201cTratamento cir\u00fargico convencional\u201d, change from \u201cSemreserva contr\u00e1til\u201d to \u201cSem reserva contr\u00e1til + escore de c\u00e1lcio valvarelevado\u201d.In table 43, section \u201c TAVI ritmo sinusal \u201c, change of recommendations asfollows:Varfarina \u2013 Correct: ESC - IIb B*.AAS + clopidogrel  \u2013Correct: SBC - IIa C; AHA - IIb C; ESC - IIa C. NOACs \u2013 Correct: SBC \u2013III.Inclusion of the name of Dr. Samira Kaissar Nasr Ghorayeb in the document.Edition of April 2016, vol. 106 (4), Supl. 1, p. 1-23In the \u201cIII Diretrizes da Sociedade Brasileira de Cardiologia sobre An\u00e1lise e Emiss\u00e3o deLaudos Eletrocardiogr\u00e1ficos\u201d, published as a supplement to the Arquivos Brasileiros deCardiologia , the following corrections shouldbe considered:On page 1, section 2.1., text correction \u201cO eixo de P pode variar entre -30\u00ba e +90\u00ba\u201d to\u201cO eixo de P pode variar entre 0\u00ba e +90\u00ba\u201d."}
{"text": "Page\u00a01; line\u00a022: \u201cwas found within the pNPD4_2 plasmid\u201d should read \u201cwas found within the chromosome.\u201dVolume 5, no. 29, e00375-17, 2017,"}
{"text": "Property \u03b6s\u2009=\u2009f(Jvm) for Jvm\u2009>\u20090 and for Jvm\u2009\u2248\u20090 and property \u03b6s\u2009=\u2009f(\u0394C1) are calculated. Moreover, results of a simultaneous influence of \u0394P and \u0394\u03c0 on a value of coefficient \u03b6s when Jvm\u2009=\u20090 and Jvm\u2009\u2260\u20090 are investigated and a graphical representation of the dependences obtained in the research is provided. Also, mathematical relationships between the coefficient \u03b6s and a concentration Rayleigh number (RC) were studied providing a relevant graphical representation. In an experimental test, aqueous solutions of glucose and ethanol were used.In this paper, the authors investigate the membrane transport of aqueous non-electrolyte solutions in a single-membrane system with the membrane mounted horizontally. The purpose of the research is to analyze the influence of volume flows on the process of forming concentration boundary layers (CBLs). A mathematical model is provided to calculate dependences of a concentration polarization coefficient ( Lp), the reflection coefficient (\u03c3m) and the diffusive permeability coefficient (\u03c9m). Usefulness of the classical as well as a modified form of the Kedem\u2013Katchalsky equations has been confirmed repeatedly (\u03b4l\u03b4h)\u22121,\u03c61\u2009=\u2009[2Lp\u0394P(Dh\u03b4l \u2013 Dl\u03b4h) \u2013 4DlDh]\u03b4l\u22121\u03b4h\u22121,\u03c62\u2009=\u20094LpDlDh (\u0394P \u2013 \u03c3m\u0394\u03c0)\u03b4l\u22121\u03b4h\u22121,\u03bc0\u2009=\u20094Lp\u03c3m2RTDl\u03b4h\u2009+\u2009Dh\u03b4l)\u03b4l\u22121\u03b4h\u22121,\u03bc1\u2009=\u20094Lp\u03c3m\u0394\u03c0\u03c9mRT(Dl\u03b4h\u2009+\u2009Dh\u03b4l) \u03b4l\u22121\u03b4h\u22121.Including Eq.  and 13)13) in EqLp, \u03c3m and \u03c9m), solutions , volume flux (Jvm) and thicknesses of CBL  \u2009+\u20092(\u0394\u03c0)2 (Dh\u03b4l \u2013 Dl\u03b4h) \u2013 4RTDh\u03b4l +ZDl\u03b4h)\u03c3m\u0394\u03c0 ]\u03b4l\u22121\u03b4h\u221213\u2009=\u2009[Lp3\u03c3m\u03b4l\u03b4h\u0394\u03c0(\u0394P)2 \u2013 2(Dh\u03b4l \u2013 Dl\u03b4h)Lp2\u03c3m\u0394\u03c0\u0394P \u2013 2Lp3\u03c3m2\u03b4l\u03b4h\u0394P(\u0394\u03c0)2 \u2013ZDlDhLp\u03c3m\u0394\u03c0\u2009+\u20094Lp2\u03c3m2RTDh\u03b4l\u2009+\u2009Dl\u03b4h)\u0394P \u20134Lp\u03c3m\u03c9mRT(Dh\u03b4l\u2009+\u2009Dl\u03b4h)\u0394\u03c0]\u03b4l\u22121\u03b4h\u2212144\u2009=\u2009[4LpDlDh\u03c3m\u0394\u03c0\u2009+\u2009Lp3\u03b4l\u03b4h(\u0394P)2\u03c3m\u0394\u03c0 ]\u03b4l\u22121\u03b4h\u22121ZIncluding Eq.  in Eq. , we obtaLp\u2009=\u20095\u2009\u00d7\u200910\u221212 m3N\u22121s\u22121. The values of the reflection coefficient and diffusive permeability coefficient of the membrane for the glucose and ethanol are respectively \u03c3m1\u2009=\u20090.068, \u03c9m1\u2009=\u20098\u2009\u00d7\u200910\u221210 mol N\u22121s\u22121, \u03c3m2\u2009=\u20090.025 and \u03c9m2\u2009=\u200914.3\u2009\u00d7\u200910\u221210 mol N\u22121s\u22121. The diffusion of each individual component in the solution is characterized by the following coefficients: D1\u2009=\u20090.69\u2009\u00d7\u200910\u22129 m2s\u22121 and D2\u2009=\u20091.57\u2009\u00d7\u200910\u22129 m2s\u22121. The volumes of the compartments  were the same and equal to 200\u00a0cm3. For the low glucose and ethanol concentration, we have \u03c1h\u2009=\u2009\u03c1l(1\u2009+\u2009\u03b11Ch1\u2009+\u2009\u03b12Ch2), \u03bdh\u2009=\u2009\u03bdl(1\u2009+\u2009\u03b31Ch1\u2009+\u2009\u03b32Ch2) with the coefficients \u03b11\u2009=\u2009\u03c1l\u22121\u2202\u03c1/\u2202C1\u2009=\u20096.01\u2009\u00d7\u200910\u22125 m3 mol\u22121, \u03b31\u2009=\u2009\u03bdl\u22121\u2202\u03bd/\u2202C1\u2009=\u20093.95\u2009\u00d7\u200910\u22124 m3 mol\u22121, \u03b12\u2009=\u2009\u03c1l\u22121\u2202\u03c1/\u2202C2\u2009=\u2009\u22129.02\u2009\u00d7\u200910\u22126 m3 mol\u22121and \u03b32\u2009=\u2009\u03c1l\u22121\u2202\u03bd/\u2202C2\u2009=\u20091.82\u2009\u00d7\u200910\u22125 m3 mol\u22121  }\u03b12\u2009=\u2009\u22120,5JvmRClDl\u03bdl\u03c1l\u03b13\u2009=\u2009\u2212RClDl2\u03bdl\u03c1l\u03b21\u2009=\u2009g(\u2202\u03c1/\u2202C){\u03b6s\u03c9m\u0394\u03c0 \u2013 Jvm[0.5 (Ch \u2013 Cl)\u2009+\u2009\u03b6s\u03c3m(Ch\u2009+\u2009Cl)]}\u03b22\u2009=\u20090.5JvmRChDh\u03bdh\u03c1h\u03b23\u2009=\u2009\u2212RChDh2\u03bdh\u03c1h.Taking into account Eq.  and 13)13) in EqP\u2009=\u20090 is Jvm\u2009=\u2009\u2212Lp\u03c3m\u03b6sD\u0394\u03c0, consequently \u0394\u03c0\u2009=\u2009\u2212Jvm(Lp\u03c3m\u03b6sD)\u22121. Assuming that Cl\u2009=\u20090, we have Ch\u2013Cl\u2009=\u2009Ch\u2009+\u2009Cl\u2009=\u2009Ch\u2009=\u2009\u2212Jvm(Lp\u03c3m\u03b6sDRT)\u22121. Moreover, assuming that the CBL thickness is \u03b4l\u2009=\u2009Dl(2RT\u03c9m)\u22121(\u03b6s\u22121 \u2013 1) and \u03c1l\u2009=\u2009\u03c1h\u2009=\u2009\u03c10, then Eq. . The spatial graph of this function presents the dependence of concentration polarization coefficient \u03b6s on the Rayleigh number (RCl) and the volume flux (Jvm). It is not essential to present Eq.  was made for the Nephrophan membrane and aqueous ethanol solution  and the volume flux (Jvm). The graph also shows the significant influence of the two parameters mentioned earlier on the value of the concentration polarization coefficient.A graph of dependence ion Fig.\u00a0. The shaion Fig.\u00a0 proves t\u03b6s\u2009=\u2009f(Jvm), \u03b6s\u2009=\u2009f(\u0394C), \u03b6s\u2009=\u2009f and it is possible to evaluate the influence of osmotic flux (Jvm) and/or the simultaneous operation of osmotic forces (\u0394\u03c0) and hydrostatic forces ((\u0394P) on the value of the concentration polarization coefficient (\u03b6s). Equations , allowing the evaluation of the numerical relations between the concentration polarization coefficient (\u03b6s), the osmotic flux (Jvm) and the concentration Rayleigh number (RC). The results of the research carried out confirmed the significant role of concentration boundary layers in osmotic and diffusive transport, in particular their applicative aspect in technology and medicine, as mentioned in the Introduction [Equations  and 17)17), deriquations -22) and and\u03b6s\u2009=oduction , 32\u201334. oduction , 31, 33."}
{"text": "Scientific Reports7:10141; doi:10.1038/s41598-017-09415-7; Article published online 04 September 2017This Article contains errors in Figure 4, where \u201cEremis acutirostris\u201d should read \u201cEremias acutirostris\u201d, \u201cAcananthodactylus cantoris\u201d should read \u201cAcanthodactylus cantoris\u201d, and \u201cPedioplais burchelli\u201d should read \u201cPedioplanis burchelli\u201d. The correct Figure 4 appears below as Figure"}
{"text": "An operational method to construct UMEBs containing d(d\u2032 \u2212 1) maximally entangled vectors is established, and two UMEBs in d(d\u2032 \u2212 r) maximally entangled vectors in r = 1, 2, \u2026, d \u2212 1. Correspondingly, two UMEBs in We study unextendible maximally entangled bases (UMEBs) in  Quantum teleportation, which can be used for distributed quantum learning5 and even in organisms6, is a essential element in quantum information processing. Maximally entangled states attract much attention due to their importance in ensuring the highest fidelity and efficiency in quantum teleportation7. A pure state |\u03c8\u232a is said to be a d\u2009\u2297\u2009d\u2032 (d\u2009<\u2009d\u2032) maximally entangled state if and only if for an arbitrary given orthonormal basis {|iA\u232a} of subsystem A, there exists an orthonormal basis {|iB\u232a} of subsystem B such that |\u03c8\u232a can be written as 8.Quantum entanglement lies in the heart of the quantum information processing. It plays important roles in many fields such as quantum teleportation, quantum coding, quantum key distribution protocol, quantum non-locality13 and plays an important role in Van der Waals interaction in transformation optics14. It is tightly related to entanglement. While, it is proven that the unextendible product bases (UPBs) reveal some nolocality without entanglement16. The UPB is a set of incomplete orthogonal product states in bipartite quantum system dd\u2032 vectors which have no additional product states are orthogonal to each element of the set17.Nonlocality is a very useful concept in quantum mechanics17:A UPB in Obviously, they are all product states.\u03c8\u232a, which is orthogonal to |\u03d5i\u232a. If |\u03c8\u232a is a product state, it can be expressed as |\u03c8\u232a = (a|0\u232a + b|1\u232a + c|2\u232a)\u2009\u2297\u2009(a\u2032|0\u232a + b\u2032|1\u232a + c\u2032|2\u232a), where a2 + b2 + c2 = a\u20322 + b\u20322 + c\u20322 = 1. No loss of generalization, we assume that a, a\u2032 \u2260 0. From |\u03c8\u232a is orthogonal to |\u03d50\u232a, we have b\u2032 = a\u2032 \u2260 0. Due to |\u03c8\u232a is orthogonal to |\u03d54\u232a, we can conclude that c\u2032 \u2260 0. Because of that |\u03c8\u232a is orthogonal to |\u03d51\u232a, we have b = a \u2260 0. Owing to that |\u03c8\u232a is orthogonal to |\u03d53\u232a, we obtain that c = b \u2260 0. And |\u03c8\u232a is orthogonal to |\u03d52\u232a, we get that c\u2032 = b\u2032 \u2260 0. That is to say, a = b = c = a\u2032 = b\u2032 = c\u2032 = \u03c8\u232a is equal to |\u03d54\u232a, instead of being orthogonal to |\u03d54\u232a. Therefore, |\u03c8\u232a can not be a product state. That\u2019s why the set There exist a nonzero pure state |18 generalized the notion of UPB to unextendible maximally entangled bases (UMEB): a set of incomplete orthogonal maximally entangled states in bipartite quantum system dd\u2032 vectors which have no additional maximally entangled vectors orthogonal to all of them. They state that UMEBs can be used to construct examples of states for which 1-copy entanglement of assistance (EoA) is strictly smaller than the asymptotic EoA and find quantum channels that are unital but not convex mixtures of unitary operations18.Bravyi and Smolin19:Let {|0\u232a, |1\u232a} be a orthogonal base of Obviously, they are all maximally entangled states.\u03c8\u232a is orthogonal to |\u03d5i\u232a, it\u2019s sure that |\u03c8\u232a = (a|0\u232a + b|1\u232a)\u2009\u2297\u2009|2\u2032\u232a, where a2 + b2 = 1. In other words, |\u03c8\u232a must be a product state, rather than maximally entangled states. Hence If a nonzero pure state |n maximally entangled vectors is usually expressed as a n-number UMBE, when n is smaller than dd\u2032. Chen and Fei19 provided a way to construct d2-member UMEBs in et al.20 and Li et al.21 constructed two sets of UMEBs in d < d\u2032) independently. Wang et al.22 put forward a method of constructing UMEBs in d = p or 2p, where p is a prime and p = 3 mod 4. They also presented a 23-member UMEB in 23. Then Guo26 proposed a scenario of constructing UMEBs via the space decomposition, which improves the previous work about UMEBs.The number of the vectors in a UMEB is less than the dimension of the bipartite system space. Therefore a UMEB in d(d\u2032 \u2212 1)-number UMEB and then present explicit constructions of UMEBs in d(d\u2032 \u2212 r)-member UMEBs in r = 1, 2, \u2026, d \u2212 1, and give two examples in In this paper, we give two methods of constructing UMEBs in d < d\u2032. Let us first recall some basic notions and lemmas24. Let {|k\u232a} and Md\u00d7d\u2032 be the Hilbert space of all d \u00d7 d\u2032 complex matrices equipped with the inner product defined by \u2329A|B\u232a = Tr(A\u2020B) for any A, B\u2009\u2208\u2009Md\u00d7d\u2032. If Md\u00d7d\u2032, where \u2329Ai|Aj\u232a = d\u03b4ij, then there is a one-to-one correspondence between {|\u03d5i\u232a} and {Ai} as follows26:Sr(|\u03d5i\u232a) denotes the Schmidt number of |\u03d5i\u232a. Obviously, |\u03d5i\u232a is a maximally entangled pure state in Cd\u2009\u2297\u2009Cd\u2032 iff (d)1/2Ai is a d \u00d7 d\u2032 singular-value-1 matrix .Throughout this paper, we assume that Cd\u2009\u2297\u2009Cd\u2032 is called a maximally entangled basis (MEB) of Cd\u2009\u2297\u2009Cd\u2032. A set of pure states 19:(i)\u03d5i\u232a, i = 1, 2, 3 ... n are all maximally entangled states.|(ii)(iii)n < dd\u2032, and if a pure state |\u03c8\u232a satisfies that \u2329\u03d5i|\u03c8\u232a = 0, i = 1, 2, 3... n, then |\u03c8\u232a can not be maximally entangled.A basis Md\u00d7d\u2032 is called single-value-1 Hilbert-Schmidt basis (SV1B) of Md\u00d7d\u2032. A set of d \u00d7 d\u2032 matrices Md\u00d7d\u203224:(i)Ai, i = 1, 2, 3 ... n are all single-value-1 matrices.(ii)i, j = 1, 2, 3... n.(iii)n < dd\u2032, and if a matrix X satisfies that Tr(X\u2020Ai) = 0, i = 1, 2, 3... n, then X can not be a single-value-1 matrix.A Hilbert-Schmidt basis Md\u00d7d\u2032 iff Cd\u2009\u2297\u2009Cd\u2032, and Md \u00d7 d\u2032 iff Cd\u2009\u2297\u2009Cd\u2032. Therefore, for convenience, we may just call an SV1B Md \u00d7 d\u2032 an MEB Md\u00d7d\u2032 a UMEB It is obvious that 24.In deriving our main results, we need the following lemma in ref.\u03d5i\u232a} is a MEB in M1 and {|\u03c8i\u232a} is a UMEB in \u03d5i\u232a}\u222a{|\u03c8i\u232a} is a UMEB in M24. If {|\u03d5i\u232a} is a MEB in M1 and \u03d5i\u232a} is a UMEB in M.Let d(d\u2032 \u2212 1)-member UMEBs in In this section, we will establish a flexible method to construct Md\u00d7d\u2032 be the Hilbert space of all d \u00d7 d\u2032 complex matrices. If V is a subspace of Md\u00d7d\u2032 such that each matrix in V is a d \u00d7 d\u2032 matrix ignoring d entries which occupy different rows and N columns with N < d, then there exists a d(d\u2032 \u2212 1)-member MEB in V, as well as a d(d\u2032 \u2212 1)-member UMEB in Md\u00d7d\u2032.Let d entries in V only occupy the former N columns. Let bi, i = 0, 1, ..., d \u2212 1, denote the column number of the ignored element in the i-th row. Obviously, bi+1 \u2212 bi = 0 or 1.Without loss of generality, we can always assume the ignored Denoted(d\u2032 \u2212 1) pure states in p\u2009\u2295\u2009d\u2032m denotes (p + m) mod d\u2032.We can construct V.Next, we prove that all the states in  constitu(i) Maximally entangled.C = 0 for any m, it is obvious that tmj \u2260 tm\u2032j for m \u2260 m\u2032.If C = 1 for some m \u2260 0, from the definition of tmj one has tm\u22121, j \u2260 bm\u22121. Note that bm \u2212 bm\u22121 = 0 or 1, then tm\u22121, j = bm\u22121 \u2295d\u2032 1.If tmj, we also have C = 0 for k \u2260 m \u2212 1. Hencep \u2212 m) mod d\u2032. In particular,From the definition of Thentmj \u2260 tm\u2032j for m \u2260 m\u2032. Namely, the states |\u03d5\u2032j,n\u232a in  Orthogonality.tmj\u232a = |tmj\u2032\u232a if and only if j = j\u2032.We first show that |tmj = tmj\u2032 for j = j\u2032. If j \u2260 j\u2032, without loss of generality, let tmj \u2260 tmj\u2032 when tm\u22121,j \u2260 tm\u22121,j\u2032. Otherwise, from the definition of tmj we have C\u2009=\u20091. Note that C = 1, as proved in (i). Therefore, tm\u22121, j\u2032 = bm\u22121, which contradicts to the definition of tmj. Furthermore, tmj \u2260 tmj\u2032 when tj0 \u2260 tj\u20320. Therefore,Obviously, d(d\u2032 \u2212 1) states {|\u03d5j, n\u232a} in |\u03d5j\u2032\u232a, we get the following MEB V, i.e., another UMEB in C5\u2009\u2297\u2009C6:By inverse unitary transformation |Actually both  and 10)10) are UConstructing a UMEB in V1 and V2 by elementary transformation respectively:One can easily get the following simple formations Then following Theorem 1 we can construct the following UMEBs V1 and V2, respectively,By inverse transformation \u03d5j\u232a}\u2009\u222a\u2009{|\u03c8j\u232a} constitutes a UMEB in V in -member UMEBs in r = 1, 2, \u2026, d \u2212 1, that is to say, it presents d \u2212 1 constructions of UMEB in In this section, we construct UMEBs consisting of fewer elements in d(d\u2032 \u2212 r)-member UMEB in Cd\u2009\u2297\u2009Cd\u2032:j = 0, 1,\u2026, s \u2212 1; n = 0, 1 \u2026, d \u2212 1.Let \u03d5l,j,n\u232a in ((i) It is obvious that |,j,n\u232a in  are all (ii) Orthogonality,M1 the d \u2297 (d\u2032 \u2212 n) matrix space, a subspace of Md\u00d7d\u2032. Since the number of {|\u03d5l,j,n\u232a} in  Denote j,n\u232a} in  equals tUMEBs in Obviously, 10 = 4 + 5 + 1 or 10 = 4 + 4 + 2. According to Theorem 2, we can construct the following 27-number UMEB  and 24-nCd\u2009\u2297\u2009Cd\u2032, which is more than all the previous numbers. For example, the 27-number UMEB -member UMEB in d(d \u2212 r)-number UMEBs in r = 1, 2, \u2026, d \u2212 1. Namely, we have presented more than d \u2212 1 constructions of UMEBs in 21 and ref.20. We have also shown 27-number UMEB and 24-number UMEB in We have provided new constructions of unextendible maximally entangled bases in arbitrary bipartite spaces"}
{"text": "Science China Information Sciences, 2011, 1596-1607 by Pan et al.Spiking neural P systems are a new candidate in spiking neural network models. By using neuron division and budding, such systems can generate/produce exponential working space in linear computational steps, thus provide a way to solve computational hard problems in feasible  time with a \u201ctime-space trade-off\u201d strategy. In this work, a new mechanism called neuron dissolution is introduced, by which redundant neurons produced during the computation can be removed. As applications, uniform solutions to two NP-hard problems: SAT problem and Subset Sum problem are constructed in linear time, working in a deterministic way. The neuron dissolution strategy is used to eliminate invalid solutions, and all answers to these two problems are encoded as indices of output neurons. Our results improve the one obtained in  Each neuron has a certain number of spikes and rules. Spikes can evolve through application of rules. Since SN P systems were proposed, they become a rapid developing area of membrane computing i, where, i \u2208 H, E is a regular expression over a, c \u2265 1, p \u2265 1, c \u2265 p, d \u2265 0. If E = ac, the firing rule is simply written as i. If d = 0, the firing rule is simply written as [E/ac \u2192 ap]i. If E = ac and d = 0, the firing rule is simply written as [ac \u2192 ap]i;forgetting rule [E/as \u2192 \u03bb]i, where, i \u2208 H, E is a regular expression over a, s \u2265 1. If E = as, the forgetting rule is simply written as [as \u2192 \u03bb]i;neuron division rule [E]i \u2192 j || k, where, i, j, k \u2208 H, E is a regular expression over a;neuron dissolution rule [E]i \u2192 \u03b4, where, i \u2208 H, E is a regular expression over a, object \u03b4 represents that neuron \u03c3i is dissolved;in, out \u2286 H represent the input and output neurons of \u03a0, respectively.syn shows the initial structure of the system and guides how to establish new synapses when new neurons are established.The synapse dictionary \u03c3i has h spikes, and ah \u2208 L(E), h \u2265 c, the firing rule i can be applied. c spikes are consumed (h \u2212 c spikes remain in neuron \u03c3i.), and p spikes are emitted after d time units (steps). If d = 0, p spikes are emitted immediately; if d = 1, p spikes are emitted at the next step; if this firing rule is applied at step t and d \u2265 1, p spikes are emitted at step t + d. Neuron \u03c3i is closed at steps t, t + 1, t + 2, \u2026, t + d \u2212 1, which means no rule will be applied and no spike will be received in this period. At step t + d, neuron \u03c3i becomes open again, and can receive new spikes. Once these p spikes are emitted from neuron \u03c3i, they reach each neuron \u03c3j which has a synapse going from neuron \u03c3i to neuron \u03c3j and is open. The spikes sent to a closed neuron are lost.If neuron \u03c3i has h spikes, and ah \u2208 L(E), h \u2265 s, the forgetting rule [E/as \u2192 \u03bb]i can be applied. s spikes are consumed immediately.If neuron \u03c3i has h spikes, and ah \u2208 L(E), and (2). no synapse , , ,  exists in the system, the neuron division rule [E]i \u2192 j || k can be applied. All h spikes in neuron \u03c3i are consumed and neuron \u03c3i is divided into two neurons \u03c3j and \u03c3k. No spike is in neurons \u03c3j and \u03c3k at this moment. The labels of the two generated neurons can be different or the same, and the labels of the two generated neurons can be different from or the same with the label of their father neuron \u03c3i, too. The new generated neurons inherit the synapses of their father neuron \u03c3i. That is to say, if there is a synapse  going from neuron \u03c3i to neuron \u03c3g, two synapses  and  are established after the division rule is applied; if there is a synapse  going from neuron \u03c3g to neuron \u03c3i, two synapses  and  are established after the division rule is applied. In addition to inheritance of synapses, new generated neurons also have synapses provided by the synapse dictionary syn. Synapses not existing in the synapse dictionary syn may appear because of inheritance of synapses. The condition (2) avoids the situation that the start and the end of a synapse are the same neuron. For example, if synapse  exists in the system, synapses ,  will appear which is not permitted.If (1). neuron \u03c33. Considering the two conditions mentioned in the above paragraph, 1). Neuron \u03c33 has one spike a and the regular expressions of both two division rules are exactly {a}, where a \u2208 {a}. Therefore, both of these two division rules meet the condition (1). 2). For rule [a]3 \u2192 2 || 3, the label 3 of the father neuron \u03c33 corresponds to i in the normalization rule, and the label 2 and 3 of the two new neurons \u03c32 and \u03c33 corresponds to j and k in the normalization rule. Synapses , , ,  cannot exist in the system. That is to say, , , ,  cannot exist in this system. However, a synapse  is in this system, therefore rule [a]3 \u2192 2 || 3 cannot be applied. Only rule [a]3 \u2192 3 || 4 meet the two conditions. The spike a in neuron \u03c33 is consumed, neuron \u03c33 is divided into two neurons \u03c33 and \u03c34, and two synapses ,  going from neuron \u03c32 to these two new neurons are established because there is a synapse going from neuron \u03c32 to the father neuron \u03c33 of the two new neurons (the inheritance of synapses). Because rules in this system are related to the labels of neurons, the new neuron \u03c33 contains these two rules. The system is changed to a]3 \u2192 3 || 4.A simple example shown in \u03c3i has h spikes, and ah \u2208 L(E), the neuron dissolution rule [E]i \u2192 \u03b4 can be applied. All h spikes in neuron \u03c3i are consumed and neuron \u03c3i is dissolved. All synapses going from/to neuron \u03c3i are dissolved, too.If neuron \u03c31 has one spike a and the regular expression of the dissolution rule is exactly {a}, where a \u2208 {a}. Then rule [a]1 \u2192 \u03b4 is applied and the system is changed to \u03c31 is dissolved, and synapse  connected with neuron \u03c31 is also dissolved.).A simple example shown in \u03c3i can be applied, this rule must be applied; if two or more rules in neuron \u03c3i can be applied, one of these rules is applied non-deterministically. Rules are applied in a sequential manner in each neuron and in parallel between neurons.At each step, if only one rule in neuron configuration of the system is described by the synapses connections, the spikes number in each neuron, and the state of each neuron (open or closed). By applying rules, the configuration is transformed from one to the next one. The transition sequence starting from the initial configuration is called a computation, and a computation halts if it reaches a configuration where all neurons are open and no rule can be applied.The IX, \u0398X both in a semi-uniform way and in a uniform way, where IX is a language over a finite alphabet and \u0398X is a total boolean function over IX (The elements in IX are instances.). In the semi-uniform way, a specified SN P system is constructed for each instance of a decision problem, in which the instance parameters are embedded in the SN P system. In the uniform way, a SN P system is constructed for all instances of a decision problem, in which the different instances parameters enter the SN P system as input spikes. The uniform solutions are preferred because they only relate to the structure of a problem.SN P systems can used to solve the decision problem r \u2265 1, ij \u2265 0 for each 1 \u2264 j \u2264 r, which means ij spikes enter the system through input neuron \u03c3in at step j. Specially, ij = 0 means no spike enters the system at step j.The input of a SN P system is a spike train X = {x1, x2, \u2026, xn}, a literal li is xi or \u00acxi for 1 \u2264 i \u2264 n. A clause Ci is a disjunction of literals Ci = ln1 \u2228 ln2 \u2228 \u2026 \u2228 lnr, 1 \u2264 r \u2264 n. A conjunctive normal form  is a conjunction of clauses C1 \u2227 C2 \u2227 \u2026 \u2227 Cm. An assignment is a mapping X \u2192 {0, 1} from each variable xi to its value . For example, X = {x1, x2, x3}, the conjunctive normal form is (x1 \u2228 \u00acx2) \u2227 (x1 \u2228 x3). The x1 \u2228 \u00acx2 and x1 \u2228 x3 are the two clauses. The first clause contains two literals x1 and \u00acx2, and the second clause contains two literals x1 and x3. If an assignment of x1, x2, \u2026, xn can be found, which makes at least one literal true in each clause and then makes all m clauses true, this SAT problem is satisfiable. Otherwise, this SAT problem is unsatisfiable i, i = 1, 23[a(a2)+/a2 \u2192 a]d[a \u2192 a]inxi, i = 1, 2, \u2026, n[a2 \u2192 a2]inxi, i = 1, 2, \u2026, n[a3 \u2192 a3]inxi, i = 1, 2, \u2026, n[a \u2192 a]Cxi1, i = 1, 2, \u2026, n[a3 \u2192 a]Cxi1, i = 1, 2, \u2026, n[a \u2192 a]Cxi0, i = 1, 2, \u2026, n[a2 \u2192 a]Cxi0, i = 1, 2, \u2026, n[forgetting rule:a2 \u2192 \u03bb]2[a2 \u2192 \u03bb]Cxi1, i = 1, 2, \u2026, n[a3 \u2192 \u03bb]Cxi0, i = 1, 2, \u2026, n[a \u2192 \u03bb]ot1t2\u2026tn, t1, t2, \u2026, tn = 0, 1[a2 \u2192 \u03bb]ot1t2\u2026tn, t1, t2, \u2026, tn = 0, 1[\u2026an \u22121 \u2192 \u03bb]ot1t2\u2026tn, t1, t2, \u2026, tn = 0, 1[neuron division rule:a]0 \u2192 o1 || o0[a]ot1 \u2192 ot11 || ot10, t1 = 0, 1[a]ot1t2 \u2192 ot1t21 || ot1t20, t1, t2 = 0, 1[\u2026a]ot1t2\u2026tn\u22121 \u2192 ot1t2\u2026tn\u221211 || ot1t2\u2026tn\u221210, t1, t2, \u2026, tn\u22121 = 0, 1[neuron dissolution rule:an]ot1t2\u2026tn \u2192 \u03b4, t1, t2, \u2026, tn = 0, 1.0 \u2192 o1 || o0 is applied to generate neurons \u03c3o1 and \u03c3o0, which means an assignment in regard to x1 has two choices: 1 or 0. Synapses  and  are established through the inheritance of synapse , and synapses  and  are established through synapse dictionary syn. Synapse  establishes a channel between the input and the assignment including x1 = 1; synapse  establishes a channel between the input and the assignment including x1 = 0. At the same time, auxiliary neuron \u03c32 has one spike, rule a \u2192 a is applied and one spike is emitted to neuron \u03c31; auxiliary neuron \u03c33 has one spike, rule a \u2192 a; 2n \u2212 1 is applied and one spike will be emitted to neurons \u03c32 and \u03c3d at step 2n. The system after step one is shown in \u03c31 has one spike, the firing rule a \u2192 a is applied, and one spike is emitted to neurons \u03c32, \u03c3o1 and \u03c3o0.At step two, neuron \u03c3o1 and \u03c3o0 has one spike, the division rule [a]ot1 \u2192 ot11 || ot10 is applied to generate neurons \u03c3o11, \u03c3o10, \u03c3o01 and \u03c3o00, which means an assignment in regard to x1 and x2 has four choices: 11, 10, 01, 00. Synapses , ,  and  are established through the inheritance of synapses , ; synapses , ,  and  are established through the inheritance of synapses , ; synapses , , S and  are established through synapse dictionary syn. Synapses  and  establish channels between the input and the assignments including x1 = 1; synapses  and  establish channels between the input and the assignments including x1 = 0; synapses  and  establish channels between the input and the assignments including x2 = 1; synapses  and  establish channels between the input and the assignments including x2 = 0. At the same time, auxiliary neuron \u03c32 has one spike, rule a \u2192 a is applied and one spike is emitted to neuron \u03c31. The system after step three is shown in At step three, each of neurons n \u2212 1, 2n neurons labeled ot1t2\u2026tn are generated. The system after step 2n \u2212 1 is shown in Similar process repeats. At step 2n, each neuron \u03c3ot1t2\u2026tn receives one spike emitted from neuron \u03c31 which will be deleted at the next step by the forgetting rule [a \u2192 \u03bb]ot1t2\u2026tn. Neuron \u03c32 receives two spikes , the forgetting rule a2 \u2192 \u03bb is applied at step 2n + 1, and no spike will be emitted to neuron \u03c31 later. At the same time, neuron \u03c3d receives one spike emitted from \u03c33. The system after step 2n is shown in At step 2Input Stage: At step 2n + 1, the first clause of the conjunctive normal form expression enters the system through input neurons \u03c3inxi, i = 1, 2, \u2026, n. The literal in regard to x1 enters neuron \u03c3inx1; the literal in regard to x2 enters neuron \u03c3inx2; \u2026 the literal in regard to xn enters neuron \u03c3inxn. At the same time, one spike is emitted to neuron \u03c3inxi from neuron \u03c3d.n + 2, the spikes in neuron \u03c3inxi are replicated, and are emitted to neurons \u03c3Cxi1 and \u03c3Cxi0.At step 2n + 3, different rules are applied according to the number of spikes in neurons \u03c3Cxi1 and \u03c3Cxi0.At step 2\u03c3Cxi1:For neuron \u03c3Cxi1, which means neither xi nor \u00acxi is in the clause, rule a \u2192 a is applied. One spike is emitted to neurons having synapses going from neuron \u03c3Cxi1 to them. It aims to show that xi = 1 makes no contribution to let the clause true.If one spike is in neuron \u03c3Cxi1, which means xi is in the clause, rule a2 \u2192 \u03bb is applied. These two spikes are deleted. It aims to show that xi = 1 makes contribution to let the clause true.If two spikes are in neuron \u03c3Cxi1, which means \u00acxi is in the clause, rule a3 \u2192 a is applied. One spike is emitted to neurons having synapses going from neuron \u03c3Cxi1 to them. It aims to show that xi = 1 makes no contribution to let the clause true.If three spikes are in neuron \u03c3Cxi0:For neuron \u03c3Cxi0, which means neither xi nor \u00acxi is in the clause, rule a \u2192 a is applied. One spike is emitted to neurons having synapses going from neuron \u03c3Cxi0 to them. It aims to show that xi = 0 makes no contribution to let the clause true.If one spike is in neuron \u03c3Cxi0, which means xi is in the clause, rule a2 \u2192 a is applied. One spike is emitted to neurons having synapses going from neuron \u03c3Cxi0 to them. It aims to show that xi = 0 makes no contribution to let the clause true.If two spikes are in neuron \u03c3Cxi0, which means \u00acxi is in the clause, rule a3 \u2192 \u03bb is applied. These three spikes are deleted. It aims to show that xi = 0 makes contribution to let the clause true.If three spikes are in neuron Satisfiability Stage: Each neuron \u03c3ot1t2\u2026tn receives zero or more spikes at step 2n + 3. If one neuron \u03c3ot1t2\u2026tn receives n spikes which means the clause contains n literals that make no contribution to let the clause true, the dissolution rule [an]ot1t2\u2026tn \u2192 \u03b4 is applied at step 2n + 4 to dissolve this neuron . Otherwise, at least one literal is true in this assignment and this assignment is reserved to check the next clause.m clauses are in a SAT problem, the satisfiability checking stage lasts for m + 3 steps. If some neurons \u03c3ot1t2\u2026tn are still in the system at step 2n + m + 3, the labels of these neurons \u03c3ot1t2\u2026tn are all solutions to this SAT problem, i.e., this SAT problem is satisfiable. Otherwise, this SAT problem is unsatisfiable.Due to SAT problem can be solved in linear time, and all solutions can be obtained through this system.It can be seen that any Some steps comparison results between our solution and other solutions, which use the neuron division to solve the NP-complete problems, are shown in SAT : (x1 \u22c1 x2) \u22c0 (\u00acx2 \u22c1 x3) \u22c0 (\u00acx1 \u22c1 x2 \u22c1 x3), the DDSN P system \u03a03,3 is used to solve it. After 12 computational steps, neurons \u03c3o111, \u03c3o101 and \u03c3o011 are remaining which shows that {x1 = true, x2 = true, x3 = true}, {x1 = true, x2 = false, x3 = true} and {x1 = false, x2 = true, x3 = true} are all solutions to this SAT problem.Considering a SAT problem The SN P system with neuron division and budding and the SN P system with neuron division need 21 steps and 26 steps to judge this problem has solutions, respectively, while our DDSN P system need only 12 steps.n, m \u2264 50) are solved using the three systems in SAT problems with different sizes  denotes the set of all instances of the Subset Sum problem having n integers. In this section, a uniform solution working in a deterministic way is constructed by DDSN P system, which can solve all SubsetSumproblem (n) problems in linear time.a0 \u22c5)n2 is introduced into the front of each spike train.An integer is represented by corresponding number of spikes. In order to generate necessary workspace before computing, a spike train (SubsetSumproblem (n) ;syn = {, , , , , }ini, di1), , |i = 1, 2, \u2026, n}\u22c3{}\u22c3{(1)|i = 2, 3, \u2026, n};\u22c3{;firing rule:a \u2192 a]i, i = 1, 23[a3(a3)+/a3 \u2192 a3]ini, i = 1, 2, \u2026, n[a3 \u2192 a]ini, i = 1, 2, \u2026, n[a \u2192 a]di1, i = 1, 2, \u2026, n[a3 \u2192 a3]di2, i = 1, 2, \u2026, n[an \u2192 a]4[a(a2)+/a2 \u2192 a2]s[a \u2192 a]s[forgetting rule:a2 \u2192 \u03bb]2[a3 \u2192 \u03bb]di1, i = 1, 2, \u2026, n[a \u2192 \u03bb]di2, i = 1, 2, \u2026, n[a2(a3)+/a5 \u2192 \u03bb]ot1t2\u2026tn, t1, t2, \u2026, tn = 0, 1[a \u2192 \u03bb]ot1t2\u2026tn, t1, t2, \u2026, tn = 0, 1[neuron division rule:a]0 \u2192 o1 || o0[a]ot1 \u2192 ot11 || ot10, t1 = 0, 1[a]ot1t2 \u2192 ot1t21 || ot1t20, t1, t2 = 0, 1[\u2026a]ot1t2\u2026tn\u22121 \u2192 ot1t2\u2026tn\u221211 || ot1t2\u2026tn\u221210, t1, t2, \u2026, tn\u22121 = 0, 1[neuron dissolution rule:a(a3)+]ot1t2\u2026tn \u2192 \u03b4, t1, t2, \u2026, tn = 0, 1[a2]ot1t2\u2026tn \u2192 \u03b4, t1, t2, \u2026, tn = 0, 1.0 \u2192 o1 || o0 is applied to generate neurons \u03c3o1 and \u03c3o0, which means one subset in regard to x1 has two choices: x1 is included in this subset (represent by 1) and x1 is not included in this subset (represent by 0). Synapses  and  are established through the inheritance of synapse ; synapses  and  are established through the inheritance of synapse ; the synapse  is established through synapse dictionary syn. The synapse between neurons d12 and o1 establishes a channel between the input and the subset having x1. At the same time, auxiliary neuron \u03c32 has one spike, rule a \u2192 a is applied and one spike is emitted to neuron \u03c31; auxiliary neuron \u03c33 has one spike, rulea \u2192 a; 2n \u2212 1 is applied and one spike will be emitted to neurons \u03c32 at step 2n. The system after step one is shown in \u03c31 has one spike, the firing rule a \u2192 a is applied, and one spike is emitted to neurons \u03c32, \u03c3o1 and \u03c3o0.At step two, neuron \u03c3o1 and \u03c3o0 has one spike, the division rule [a]ot1 \u2192 ot11 || ot10 is applied to generate neurons \u03c3o11, \u03c3o10, \u03c3o01 and \u03c3o00, which means one subset in regard to x1 and x2 has four choices: x1x2 are included in this subset (represent by 11), x1 is included in this subset and x2 is not included in this subset (represent by 10), x1 is not included in this subset and x2 is included in this subset (represent by 01), and x1x2 are not included in this subset (represent by 00). Synapses , , , , , , , ,  and  are established through the inheritance of synapse , , ,  and ; synapses  and  are established through synapse dictionary syn. Synapses  and  establish channels between the input and the subsets having x1; synapses  and  establish channels between the input and the subsets having x2. At the same time, auxiliary neuron \u03c32 has one spike, rule a \u2192 a is applied and one spike is emitted to neuron \u03c31. The system after step three is shown in At step three, each of neurons n \u2212 1, 2n neurons labeled ot1t2\u2026tn are generated. The system after step 2n \u2212 1 is shown in Similar process repeats. At step 2n, each neuron \u03c3ot1t2\u2026tn receives one spike emitted from neuron \u03c31 which will be deleted at the next step by the forgetting rule [a \u2192 \u03bb]ot1t2\u2026tn. Neuron \u03c32 receives two spikes , the forgetting rule a2 \u2192 \u03bb is applied at step 2n + 1, and no spike will be emitted to neuron \u03c31 later. The system after step 2n is shown in At step 2Input Stage: At step 2n + 1, x1, x2, \u2026, xn enter the system through input neurons \u03c3ini. 3x1 + 3 spikes (ax1+33) enter neuron \u03c3in1; 3x2 + 3 spikes (ax2+33) enter neuron \u03c3in2;\u2026 3xn + 3 spikes (axn+33) enter neuron \u03c3inn.n + 2, the firing rule a3(a3)+/a3 \u2192 a3 is applied, and three spikes are replicated and are emitted to neurons \u03c3di1 and \u03c3di2. Spikes in neuron \u03c3di1 are forgotten, and spikes in neuron \u03c3di2 are emitted to these \u03c3ot1t2\u2026,tn having synapses going from neuron \u03c3di2 to them (These neurons represent the subsets having the integer xi.) at step 2n + 3. This process repeats until only 3 spikes are in neuron \u03c3ini.At step 2n + xi + 2, the firing rule a3 \u2192 a is applied, and one spike is replicated and is emitted to neurons \u03c3di1 and \u03c3di2.At step 2n + xi + 3, the spike in neuron \u03c3di1 is emitted to neuron \u03c34 showing that all spikes in neuron \u03c3ini have been passed to neurons \u03c3ot1t2\u2026,tn having synapses going from neuron \u03c3di2 to them. The spike in neuron \u03c3di2 is forgotten. Up to this step, 3xi spikes are emitted to neurons \u03c3ot1t2\u2026,tn which represent the subsets having the integer xi.At step 2\u03c3ini are passed to neurons \u03c3ot1t2\u2026tn at step 2n + xmax + 3, neuron \u03c34 receives n spikes, and one spike is emitted to neuron \u03c3s at step 2n + xmax + 4. Up to this step, the number of spikes in neurons \u03c3ot1t2\u2026,tn is When all input spikes in neurons Checking Stage: At step 2n + xmax + 5, 2s + 1 spikes are in neuron \u03c3s, the firing rule a(a2) + /a2 \u2192 a2 is applied, two spikes are emitted to neurons \u03c3ot1t2\u2026tn. This process lasts for S circles.n + xmax + s + 4, only one spike is in neuron s, and this spike is emitted to neurons \u03c3ot1t2\u2026tn.At step 2\u03c3ot1t2\u2026tn.There are three rule execution situations in neurons \u03c3ot1t2\u2026tn initially. 2 spikes are emitted to this neuron from neuron \u03c3S, then forgetting rule a2(a3)+/a5 \u2192 \u03bb can be applied with 5 spikes consumed. The number of spikes decreases to S times, and all spikes in neuron \u03c3ot1t2\u2026tn are consumed. At this step, the last one spike is emitted to this neuron from neuron \u03c3S, forgetting rule a \u2192 \u03bb can be applied to consume this spike.\u03c3ot1t2\u2026tn initially. 2 spikes are emitted to this neuron from neuron \u03c3S, then forgetting rule a2(a3)+/a5 \u2192 \u03bb can be applied with 5 spikes consumed. The number of spikes decreases to \u03c3ot1t2\u2026tn are consumed. At this step, two spikes are emitted to this neuron from neuron \u03c3S, and neuron dissolution rule [a2]ot1t2\u2026tn \u2192 \u03b4 is applied to dissolve this neuron.\u03c3ot1t2\u2026tn initially. 2 spikes are emitted to this neuron from neuron \u03c3S, then forgetting rule a2(a3)+/a5 \u2192 \u03bb can be applied with 5 spikes consumed. The number of spikes decreases to S times, and \u03c3S is emitted to this neuron, and dissolution rule [a(a3)+]ot1t2\u2026tn \u2192 \u03b4 is applied to dissolve this neuron.\u03c3ot1t2\u2026tn are still in the system after step 2n + xmax + s + 5, the labels of these neurons \u03c3ot1t2\u2026tn are all solutions to this Subset Sum problem.If some neurons SubsetSumproblem (n) can be solved in linear time, and all solutions can be obtained through this system.It can be seen that any k means that all x1, \u2026, xn, S can be transformed into k-bit binary numbers.Some steps comparison results between our solution and other solutions, which use the non-deterministic method to solve the NP-complete problems, are shown in n \u2212 1)-times computations should be processed. Although the time complexity of each computation is a constant, the whole time complexity cannot be a polynomial of n. The proposed DDSN P system can solve the Subset Sum problem in a linear time, which improves the computational efficiency.The conventional methods use the nondeterminism of SN P systems to solve Subset Sum problem, which means that whether a random combination of integers is one of the solutions or not can be checked by one computational process. These SN P systems can only judge whether a certain subset is the answer or not, but cannot search all solution space to judge whether a Subset Sum problem has solutions. Even if we let all combinations be traversed artificially to determine whether a Subset Sum problem has solutions or not, (2Subset Sum problem (4): X = {1, 2, 3, 4}, S = 5, the DDSN P system \u03a04 is used to solve it. After 22 computational steps, neurons \u03c3o0110 and \u03c3o1001 are remaining which shows that {2, 3} and {1, 4} are all solutions to this Subset Sum problem. Methods proposed in [Considering a Subset Sum problem posed in , 20, 23 X = {1, 2, \u2026, n}, S = 5 are solved using the five systems in A series of Subset Sum problems: The new mechanism called neuron dissolution is introduced into the framework of SN P systems in this work. By this mechanism, redundant neurons can be dissolved immediately. The computational resources can be saved, which means more work can be done using the same resources, or the same work can be done using less resources. We also proved that this new variant of SN P system can obtain all solutions to NP-complete problems , such as SAT problem and the Subset Sum problem, in linear time, which enhances the application fields of SN P systems such as the register allocation problem.This work provides a new thought of storing information in SN P systems, which can be used to store other information. The dissolution rule can be used to many situations to decrease the space complexity of a SN P system. This variant of SN P system can be used to solve other NP-complete problems and application problems. It is also an attractive direction to introduce other biological phenomena into SN P systems to reduce computational resources and enhance computational space efficiency."}
{"text": "Scientific Reports3: Article number: 1627; 10.1038/srep01627 published online: 04092013; updated: 02022018.The original version of this Article contained an error in the spelling of the author Yangchao Shen, which was incorrectly given as Shen Yangchao.This error has now been corrected in the PDF and HTML versions of the Article.ViVj\u232a terms in Table 1 were omitted from the calculation of In addition, the \u2329should read:As a result, in the Abstract,5 random numbers that are guaranteed to have 5.2\u2009\u00d7\u2009104 bits of minimum entropy with a 99% confidence level.\u201d\u201cIn our experiment, we generate 1\u2009\u00d7\u200910should read:5 random numbers that are guaranteed to have 2.4\u2009\u00d7\u2009104 bits of minimum entropy with a 99% confidence level.\u201d\u201cIn our experiment, we generate 1\u2009\u00d7\u200910In the Results section, under subheading \u2018Random number results\u2019,\u03b4 does not have any noticable influence on the bound of min-entropy. Here we used the thresholds of KCBS violations \u201cAs shown in Table 1, we observe the expectation should read:\u03b4 does not have any noticeable influence on the bound of min-entropy. Here we used the thresholds of KCBS violations \u201cAs shown in Table 1, we observe the expectation In the title of Table 1,\u201cOur experimental test clearly shows the violation of the extended inequality (3) with 31 \u03c3\u201dshould read:\u201cOur experimental test clearly shows the violation of the extended inequality (3) with 18 \u03c3\u201dViVj\u232a in Table 1 for the n\u2009=\u20092\u2009\u00d7\u2009105 with the new biased distribution parameter \u03b1\u2009=\u200912 in order to observe the net randomness. Therefore, the contents of the paper related to the biased choice of measurement settings should be corrected as follows.Moreover, the presented data for the biased choice of measurement settings does not show the net randomness after including the terms \u2329In the Results section, under subheading \u2018Random number results\u2019,P (V1)\u2009=\u20091\u2009\u2212\u20094q, P (V2)\u2009=\u2009P (V3)\u2009=\u2009P (V4)\u2009=\u2009P (V5)\u2009=\u2009q, and q\u2009=\u2009\u03b1n\u22121/2 with \u03b1\u2009=\u20096 and n\u2009=\u2009105. We observe basically the same behavior of the min-entropy for the generated stream except for a slightly smaller bound due to the non-uniform setting. We get the min-entropy bound 5 rounds with violation of 4) exceeds the input entropy (1.14\u2009\u00d7\u2009104), and we obtain 2.1\u2009\u00d7\u2009103 net random bits.\u201d\u201cWe also generate random bits with a biased choice of measurement settings, where should read:P (V1)\u2009=\u20091\u2009\u2212\u20094q, P (V2)\u2009=\u2009P (V3)\u2009=\u2009P (V4)\u2009=\u2009P (V5)\u2009=\u2009q, and q\u2009=\u2009\u03b1n\u22121/2 with \u03b1\u2009=\u200912 and n\u2009=\u20092\u2009\u00d7\u2009105. We observe basically the same behavior of the min-entropy for the generated stream except for a slightly smaller bound due to the non-uniform setting. We get the min-entropy bound 5 rounds with violation of 4) exceeds the input entropy (3.28\u2009\u00d7\u2009104), and we obtain 6.8\u2009\u00d7\u2009103 net random bits.\u201d\u201cWe also generate random bits with a biased choice of measurement settings, where In the legend of Figure 4,P(Vi)\u2009=\u20091/5 and (c) a biased distribution with P (V1)\u2009=\u20091\u2009\u2212\u20094q, P (V2)\u2009=\u2009P (V3)\u2009=\u2009P (V4)\u2009=\u2009P (V5)\u2009=\u2009q, where q\u2009=\u20096(100000)\u22121/2 with the probablity of errors \u201c(a)(c)The min-entropy should read:P(Vi)\u2009=\u20091/5 and (c) a biased distribution with P (V1)\u2009=\u20091\u2009\u2212\u20094q, P (V2)\u2009=\u2009P (V3)\u2009=\u2009P (V4)\u2009=\u2009P (V5)\u2009=\u2009q, where q\u2009=\u200912(200000)\u22121/2 with the probability of errors \u201c(a)(c)The min-entropy Figures 4 and 5 based on the corrections of the In addition, this Article contains typographical errors in the Results section, under subheading \u2018The KCBS inequality\u2019.v1\u232a\u2009=\u2009|1\u232a, |v2\u232a\u2009=\u2009|2\u232a, |v3\u232a\u2009=\u2009R1  |v1\u232a, |v4\u232a\u2009=\u2009R2  |v2\u232a, |v5\u232a\u2009=\u2009R1  |v3\u232a and \u03b3\u2009=\u200951.83\u00b0 and R1,2 denote the rotation operations between |1\u232a to |3\u232a and between |2\u232a to |3\u232a, respectively.\u201d\u201cHere |should read:v1\u232a\u2009=\u2009|1\u232a, |v2\u232a\u2009=\u2009|2\u232a, |v3\u232a\u2009=\u2009R1\u22121  |v1\u232a, |v4\u232a\u2009=\u2009R1\u22121  R2  |v2\u232a, |v5\u232a\u2009=\u2009R1\u22121  |v3\u232aR2\u22121  |v3\u232a and \u03b3\u2009=\u2009103.68\u00b0 and R1,2 denote the rotation operations between |1\u232a to |3\u232a and between |2\u232a to |3\u232a, respectively.\u201d\u201cHere |In the legend of Figure 1,R1 and R2 represent the coherent rotations between |1\u232a to |3\u232a and between |2\u232a to |3\u232a, respectively, where \u03b8\u2009=\u200941.97\u00b0 and \u03d5\u2009=\u200964.09\u00b0. The sequence starts from |3\u232a state (black filled circle) after optical pumping. (c)\u2013(g) The pulse sequences for the measurement configurations (c) A1A2, (d) A2A3, (e) A3A4, (f) A4A5, (g) A5A\u20191, where \u03b3\u2009=\u200951.84\u00b0.\u201d\u201c(b) The pulse sequence to prepare should read:R1 and R2 represent the coherent rotations between |1\u232a to |3\u232a and between |2\u232a to |3\u232a, respectively, where \u03b8\u2009=\u200983.94\u00b0 and \u03d5\u2009=\u2009128.18\u00b0. The sequence starts from |3\u232a state (black filled circle) after optical pumping. (c)\u2013(g) The pulse sequences for the measurement configurations (c) A1A2, (d) A2A3, (e) A3A4, (f) A4A5, (g) A5A\u20191, where \u03b3\u2009=\u2009103.68\u00b0.\u201d\u201c(b) The pulse sequence to prepare"}
{"text": "Residence\u201d should have appeared as \u201c3.1.3. Residence.\u201d \u201c3.1.1. Socioeconomic risk factors\u201d should have appeared as \u201c3.1.4. Socioeconomic risk factors.\u201dIn the article, \u201cIs rotavirus still a major cause for diarrheal illness in hospitalized pediatric patients after rotavirus vaccine introduction in the Saudi national immunization program?\u201d,"}
{"text": "A novel aspect of the work is the introduction of a recommended distribution, incrementally learned from the data, to optimally refine the inferred network. Unlike existing system identification techniques, this \u201cactive learning\u201d method automatically focuses its attention on key undiscovered areas of the network, instead of targeting global uncertainty indicators like parameter variance. We show how active learning leads to faster inference while, at the same time, provides confidence intervals for the network parameters. We present simulations on artificial small-world networks to validate the methods and apply the method to real data. Analysis of frequency of motifs recovered show that cortical networks are consistent with a small-world topology model.Understanding how groups of neurons interact within a network is a fundamental question in system neuroscience. Instead of passively observing the ongoing activity of a network, we can typically perturb its activity, either by external sensory stimulation or directly via techniques such as two-photon optogenetics. A natural question is how to use such perturbations to identify the connectivity of the network efficiently. Here we introduce a method to infer sparse connectivity graphs from  A fundamental question of system neuroscience is how large groups of neurons interact, within a network to perform computations that go beyond the individual ability of each one. One hypothesis is that the emergent behavior in neural networks results from their organization into a hierarchy of modular sub-networks, or motifs, each performing simpler computations than the network as a whole .To test this hypothesis and to understand brain networks in general we need to develop methods that can reliably measure network connectivity, detect recurring motifs, elucidate the computations they perform, and understand how these smaller modules are combined into larger networks capable of performing increasingly complex computations.in-vivo, two-photon imaging data. Advances in two-photon imaging are giving us the first look at how large ensembles of neurons behave in-vivo during complex behavioral tasks  and E[yc|Ri = 0]. The dependence on the \u03ba parameter is not easy to discern at a glance, but we can observe how the Z-score behaves for various combinations of spiking rate parameters and \u03ba values.Using this approximation, we can gain some intuition on how the Z-score will behave under certain conditions. In particular, we see that the Z-score depends approximately linearly on the number of observed interactions, it also depends on what is essentially a ratio between \u03ba as a function of the fractional rate \u03ba. We decided to choose \u03ba = 10 for all experimental results since it seemed to provide a nice middle ground between dampening the Z-score for low spiking rate inhibitions and not giving disproportionately large Z-scores for small excitatory rates.Numerical issues arise from picking very large values of We analyze the behaviour of ill-conditioned but non-singular Fisher matrices, where there is a potentially large difference between the largest and smallest eigenvalues present in the Fisher information matrix, and show how the Wald test present in the model selection strategy copes with this issue. We use this analysis as we take the limit of the smallest eigenvalue going to 0 to illustrate what this test does for singular matrices.Overall, we provide a link between the smallest eigenvalue of the Fisher information matrix, and the largest observed parameter variance we obtain from the diagonal elements of the inverse Fisher information matrix. From this we finally conclude that regressor subsets that produce nearly singular matrices are consistently rejected from our model selection process.A, if \u03bb is an eigenvalue of A, then \u03bb\u22121 is an eigenvalue of A\u22121. So the inverse of a matrix with a very small eigenvalue has a correspondingly large eigenvalue.We first note that, for any non-singular matrix 1 \u2265 \u03bb2 \u2265 \u2026 \u2265 \u03bbr \u2265 0, where r is the number of rows (regressors) of the matrix. The corresponding eigenvalues of the inverse Fisher matrix are ordered: We also note that the Fisher information matrix is positive (semi) definite, we can thus sort its eigenvalues in descending order and write \u03bbxi is the i-th largest entry in the diagonal elements in the inverse Fisher matrix, the equality holds exactly for i = r. These values are the fisher variances of the regressors in Since the inverse Fisher information is symmetric, we can use the Schur-Horn theorem to see that its diagonal elements are majorized by its eigenvalues. By definition of majorization, this means that:From this, we can easily see that the largest observed fisher variance is at leastPAc yielding an ill-conditioned Fisher information matrix will be rejected.From this we can conclude that as the smallest value of the Fisher information matrix goes to 0, at least one regressor\u2019s Fisher variance grows arbitrarily large. Relating this back into Eqs \u03bd used for the random subset selection step of the elastic-forward model selection method. The \u03bd parameter is tested in the 0.5 to 0.9 ranges. This is compared against the baseline oracle lasso method and model selection where no random subset selection is performed (no-subset).We evaluate the effect of varying the sample fraction F1, precision, and recall performance metrics. Results are shown in All methods are evaluated on samples drawn from simulated network SW1CL using the \u03bd parameters perform similarly for all but the smallest sample sizes. Additionally, the use of no-subset model selection had a large performance decrease for small sample sizes when compared to all \u03bd parameters, but similar performance on larger sample sizes.We can observe from A natural question that may arise is if the Poisson GLM model can still capture directed interactions between neurons when the data does not come from a Poisson distribution.To that effect, we repeated the simulated experiments using the same connectivity matrices defined for networks SW1CL and SW3CL, but this time, the spiking activity was obtained using a Leaky Integrate and Fire model .vc is the membrane potential of neuron c, and sc its corresponding spike train.The equations of the model can be summarized asW and H are the inter-neuron and stimuli-neuron connectivity matrices respectively. The parameter h(t) represent the influence kernel, and depends on the synaptic density (a) and kernel delay (tD). b(t) is the direct current parameter.b(t) from a uniform distribution (b(t) \u223c U), where b was chosen such that the average spiking rate of each neuron is the same as in the original Poisson GLM simulations. The W and H connectivity matrices were linearly scaled with respect to the original SW1CL network to preserve the conditional spiking rates. For these simulations, we set a = 1.5 and tD = 2.For our simulations we sampled From Algorithm 1 Elastic-forward selectionRequire:Xc sequence of observations of neuron c\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 k maximum number of regressors to add per step\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 \u03b3 minimum allowed regressor p-value\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 r\u2032 = {} \u2003\u2003\u25b7 Initialize set of active regressors to empty set (only bias is included).\u2003repeat\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 Both sub routines are explained in algorithms 3 and 4if {ji} = \u2205 then\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 No suitable candidate foundreturn\u2003\u2003\u2003end if\u2003\u2003\u2003\u2003success = Falsen = |{ji}|\u2003\u2003rbest = r\u2032\u2003\u2003best = BICr\u2032\u2003\u2003BICwhile not sucess do\u2003\u2003r\u2021 = r\u2032 + {ji}[: n]\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 Update best set found so far using Algorithm 3ifmax \u2264 \u03b3 and BICr\u2021 \u2264 BICbestthen\u2003\u2003\u2003rbest = r\u2021\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003end if\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 Already found best set in the descending sequence, update and exit loopif BICr\u2021 \u2265 BICbest and rbest \u2260 r\u2032 then\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003success = Truer\u2032 = rbest\u2003\u2003\u2003\u2003end if\u2003\u2003\u2003n = n \u2212 1\u2003\u2003\u2003end while\u2003\u2003until True\u2003Algorithm 2 Iterative Active LearningRequire:X, I, k, \u03b3, \u03b2X Initial observations for all neurons\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 I Initial applied stimuli\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 k maximum number of regressors to add per step\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 \u03b3 minimum allowed regressor p-value\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 \u03b2 active learning smoothing parameter\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 forl \u2208 interventions do\u2003X and I as defined for Eq (3)\u2003\u2003forc \u2208 neurons do\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 Perform model selection using Algorithm 4\u2003\u2003\u2003c\u2003\u2003\u2003\u25b7 Get log-likelihood difference of considering all regressors that are not parent edges of neuron \u2003\u2003\u2003forr \u2208 Rnonparent(c)do\u2003\u2003\u2003Lr, c \u2190 Compute Loglikelihood difference \u2003\u2003\u2003\u25b7 Use \u2003\u2003\u2003\u2003end for\u2003\u2003\u2003end for\u2003\u2003\u2003\u25b7 Compute the impact and score of each stimuli according to previous observationsfors \u2208 stimulido\u2003\u2003forc \u2208 neuronsdo\u2003\u2003\u2003IMs,c \u2190 Compute impact of stimuli s on neuron c \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 Use Eq (14)\u2003\u2003\u2003\u2003end for\u2003\u2003\u2003forsi \u2208 stimulido\u2003\u2003\u2003s on stimuli si \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 Use Eq (17)\u2003\u2003\u2003\u2003end for\u2003\u2003\u2003SCs \u2190 Compute score of stimuli s \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7 Use Eqs  \u2190 Acquire Samples \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u25b7Acquire samples of X and I drawing I with probability Pl+1\u2003\u2003[0: k]\u2003{returnr\u2020 = r + {ji}, {ji}"}
{"text": "It is desirable to implement an efficient quantum information process demanding fewer quantum resources. We designed two compact quantum circuits for determinately implementing four-qubit Toffoli and Fredkin gates on single-photon systems in both the polarization and spatial degrees of freedom (DoFs) via diamond nitrogen-vacancy (NV) centers in resonators. The gates are heralded by the electron spins associated with the diamond NV centers. In contrast to the ones with one DoF, our implementations reduce the quantum resource and are robust against the decoherence. Evaluations of fidelities and efficiencies of our gates show that our schemes may be implemented with current technology. Encoding qubits in multiple degrees of freedom (DoFs) increases the capacity of the quantum channel, reduces the quantum resources, and are robust against the decoherence caused by the environments. Single photon is nowadays recognized as an excellent candidate for quantum information process in multiple DoFs as it carries many qubit-like DoFsms) even at room temperature32384345et al.39314849Isolated electron-spin in diamond nitrogen-vacancy (NV) center has been recognized as an unique and promising candidate for solid-state quantum computing. The appeal feature of the diamond NV center is its urtralong coherence time  gates acting on the target qubits, i.e., UToffoli\u2009=\u2009H. CCCPF. H. Here, we employ the NV-cavity entangled platform to overcome the weak interactions between two single photons involved in the Toffoli gate. As shown in ms\u232a\u2009=\u2009|\u00b11\u232a and |ms\u232a\u2009=\u2009|0\u232a  are the ground triple states of the diamond NV center. In absence of a magnetic field, |\u00b11\u232a and |0\u232a are split with 2.87\u2009GHz due to the spin-spin interaction, and the corresponding transitions, |0\u232a\u2009\u2194\u2009|\u00b11\u232a, are in the microwave regime. The six electron excited states includeE1\u232a\u2009=\u2009|E\u2212\u232a|\u22121\u232a\u2009\u2212\u2009|E+\u232a|+1\u232a, |E2\u232a\u2009=\u2009|E\u2212\u232a|\u22121\u232a\u2009+\u2009|E+\u232a|+1\u232a, |Ex\u232a\u2009=\u2009|X\u232a|0\u232a, |Ey\u232a\u2009=\u2009|Y\u232a|0\u232a, |A1\u232a\u2009=\u2009|E\u2212\u232a|+1\u232a\u2009\u2212\u2009|E+\u232a|\u22121\u232a, and |A2\u232a\u2009=\u2009|E\u2212\u232a|+1\u232a\u2009+\u2009|E+\u232a|\u22121\u232a. |E\u00b1\u232a, and |X\u232a, |Y\u232a are orbital states with angular momentum projections \u00b11 and 0, respectively. The energy gap arising from the spin-spin and spin-orbit interactions protects state |A2\u232a against small strain and magnetic filed. The transitions, |\u00b1\u232a\u2009\u2192\u2009|A2\u232a, are driven by the resonator optical pluses with \u03c3\u2212 and \u03c3+-polarized at 637 nm, respectively  gate up to two Hadamard (vely see . Therefo\u03c3+-polarized photon via port a (port c) senses a cold cavity, and departs through port b (d) with phase shift \u03c3\u2212-polarized photon via port a (port c) senses a hot cavity, and leaves from the cavity via port d (b) with phase shift \u03c3\u2212-polarized photon senses a cold cavity and it is transmitted, while the \u03c3+-polarized photon senses the hot cavity and it is reflected. The reflection and transmission coefficients of the NV-cavity system can be obtained by solving the Heisenberg equations for the cavity mode a with frequency \u03c9c and the diamond NV lowering operator \u03c3\u2212 with frequency \u03c90, that isCompared with the coupling of an unconfined diamond NV center, trapping a diamond NV center into an optical cavity enhances the coupling between the incident photon and the diamond NV center. Nowadays, various of optical microcavities, such as photonic crystal cavity\u03ba and \u03b3/2 are the decay rates of the cavity field and the diamond NV dipole, respectively. \u03bas/2 is the cavity intrinsic loss rate (side leakage). g is the coupling rate between the diamond NV center and the resonator. \u03c3z is the pauli operator. h and f are the noise operators. The input field is connected with the output filed by aking the weak excitation limitation, i.e., <\u03c3z>\u2009\u2248\u2009\u22121, through the operation, the transmission and reflection coefficients of the NV-cavity system can be respectively written as5758Here, \u03c9c\u2009=\u2009\u03c90\u2009=\u2009\u03c9p, if the Purcell factor g2/(\u03ba\u03b3)\u2009\u226b\u20091 with \u03bas\u2009\u2248\u20090 is ignored, the cavity mode of the hot cavity is reflected, i.e., r(\u03c9p)\u2009\u2192\u20091 and t(\u03c9p)\u2009\u2192\u20090. If \u03bas/\u03ba\u2009\u226a\u20091, the cavity mode of the bare cavity is transmission, i.e., r0(\u03c9p)\u2009\u2192\u20090 and t0(\u03c9p)\u2009\u2192\u2009\u22121. That is, the input-output relations between the single photon and NV center can be summarized asUnder the resonator condition R, a\u232a and |L, d\u232a with sz\u2009=\u2009\u22121, |L, a\u232a and |R, d\u232a with sz\u2009=\u2009+1, the polarization of the photon is defined with respect to the z axis (the diamond NV center axis).Here, |The quantum circuit we designed, as shown in \u03b11, \u03b12, \u03b21, \u03b22, \u03b31, \u03b32, \u03b41, and \u03b42 are arbitrary complex parameters satisfying |\u03b11|2\u2009+\u2009|\u03b12|2\u2009=\u20091, |\u03b21|2\u2009+\u2009|\u03b22|2\u2009=\u20091, |\u03b31|2\u2009+\u2009|\u03b32|2\u2009=\u20091, and |\u03b41|2\u2009+\u2009|\u03b42|2\u2009=\u20091. The indices 1 and 2 of R (or L) denote the R- (or L-) polarized wave-packets of the photon 1 and photon 2, respectively. a1 (b1) and a2 (b2) represent photon 1 (photon 2) emitted from the spatial modes 1 and 2, respectively.where 1, to depart the wave-packets, R and L, of photon 1 emitted from the spatial mode a2., i.e., the R1-component is transmitted and the L1-component is reflected. The R1-component does not interact with the diamond NV center and arrives at CPBS2 directly. Before and after the L1-component interacts with the block composed of CPBS3, NV, and CPBS4, the two single-qubit rotations Ry(\u2212\u03c0/2) and Ry(\u03c0/2) are respectively applied on photon 1 simultaneously with two Hadamard operations Hes applied on the diamond NV center. Subsequently, the R1-component and L1-component mix at CPBS2. The operations (CPBS1\u2009\u2192\u2009Ry(\u2212\u03c0/2), He\u2009\u2192\u2009CPBS3\u2009\u2192\u2009NV\u2009\u2192\u2009CPBS4\u2009\u2192\u2009Ry(\u03c0/2), He\u2009\u2192\u2009CPBS2) transform the system composed of the photon 1, photon 2, and diamond NV center from the initial state |\u03c80\u232a into |\u03c81\u232a. Here,In He can be achieved by employing \u03c0/2 pulse and it completes the following transformationsThe Hadamard transformation Ry(\u00b1\u03c0/2) can be achieved by employing two half-wave plates oriented at 45\u00b0 and 22.5\u00b0, respectively. In the standard basis {|R\u232a, |L\u232a}, Ry(\u00b1\u03c0/2) can be written asThe single-qubit rotations b2 is injected into the block. Before and after photon 2 interacts with the block, Ry(\u2212\u03c0/2)\u2009\u2192\u2009CPBS3\u2009\u2192\u2009NV\u2009\u2192\u2009CPBS4\u2009\u2192\u2009Ry(\u03c0/2)) transform |\u03c81\u232a intoNext, photon 2 in the spatial mode \u03c0 on the photon emitted from the spatial mode b2, i.e., |R2\u232a|b2\u232a\u2009\u2192\u2009\u2212|R2\u232a|b2\u232a and |L2\u232a|b2\u232a\u2009\u2192\u2009\u2212|L2\u232a|b2\u232a. If the diamond NV in the state \u03c0 and \u03c3z\u2009=\u2009|R\u232a\u2329R|\u2009\u2212\u2009|L\u232a\u2329L|, are performed on the photons emitted from the spatial modes b2 and a2, respectively. After the above operations, the state of the system composed the two photons becomesThird, we measure the electron spin state of the diamond NV center in the basis L1\u232a|a2\u232a|L2\u232a|b2\u232a, and has no change otherwise. Therefore, a two-photon four-qubit Toffoli gate which performs a NOT operation on the spatial (or polarization) state of the photon 2 depending on the states of other qubits can be achieved by the scheme depicted by It is obvious that the quantum circuit shown in The Fredkin gate swaps the states of the two target qubits conditional on the states of the control qubits. The quantum circuit depicted by a2 into the block composed of CPBS3, phase shifters 4. Before and after the photon interacts with such block, Hadamard operations, Hes applied on the diamond NV centers. The above operations  part with coefficient r0 (t) of incident photon in the cold (hot) cavity. Therefore, in the realistic case, the optical transition rules of the incident photon described by R, a\u232a|+1\u232a\u2009\u2192\u2009r|L, d\u232a|+1\u232a\u2009+\u2009t|R, b\u232a|+1\u232a, |R, a\u232a|\u22121\u232a\u2009\u2192\u2009t0|R, b\u232a|\u22121\u232a\u2009+\u2009r0|L, d\u232a|\u22121\u232a, |L, c\u232a|+1\u232a\u2009\u2192\u2009r|R, b\u232a|+1\u232a\u2009+\u2009t|L, d\u232a|+1\u232a, |L, c\u232a|\u22121\u232a\u2009\u2192\u2009t0|L, d\u232a|\u22121\u232a\u2009+\u2009r0|R, b\u232a|\u22121\u232a, |R, c\u232a|+1\u232a\u2009\u2192\u2009t0|R, d\u232a|+1\u232a\u2009+\u2009r0|L, b\u232a|+1\u232a, |R, c\u232a|\u22121\u232a\u2009\u2192\u2009r|L, b\u232a|\u22121\u232a\u2009+\u2009t|R, d\u232a|\u22121\u232a, |L, a\u232a|+1\u232a\u2009\u2192\u2009t0|L, b\u232a|+1\u232a\u2009+\u2009r0|R, d\u232a|+1\u232a, and |L, a\u232a|\u22121\u232a\u2009\u2192\u2009r|R, d\u232a|\u22121\u232a\u2009+\u2009t|L, d\u232a|\u22121\u232a. It is note that the above undesirable parts will degrade the performance of the emitter. Hence, in order to quality the performances of our schemes, we should investigate the average fidelity, which is defined as \u03c8r\u232a and |\u03c8i\u232a are the normalized output states of the system in the realistic and idealistic cases, respectively. Therefore, the average fidelity of our Toffoli or Fredkin gate, The key element of our schemes is the NV-cavity emitter. In the idealistic case, the optical transition rules of the emitter is perfect, however, the phase and amplitude of the incident photon are not perfect in experiment, and they are the functions of the \u03b1\u2009=\u2009\u03b11, sin\u2009\u03b1\u2009=\u2009\u03b12, cos\u2009\u03b2\u2009=\u2009\u03b21, sin\u2009\u03b2\u2009=\u2009\u03b22, cos\u2009\u03b3\u2009=\u2009\u03b31, sin\u2009\u03b3\u2009=\u2009\u03b32, and cos\u2009\u03b4\u2009=\u2009\u03b41, sin\u2009\u03b4\u2009=\u2009\u03b42 are taken for g2/\u03ba\u03b3 and \u03bas/\u03ba are plotted in where cos\u2009noutput/ninput, i.e., the yield of the incident photons. noutput and nintput are the numbers of gate\u2019s output and input photons, respectively. The average efficiencies of our Toffoli and Fredkin gates, Efficiency is another powerful and practical tool for qualifying the performance of the setup. In experiment, the efficiency is defined as \u03b1\u2009=\u2009\u03b11, sin\u2009\u03b1\u2009=\u2009\u03b12, cos\u2009\u03b2\u2009=\u2009\u03b21, sin\u2009\u03b2\u2009=\u2009\u03b22, cos\u2009\u03b3\u2009=\u2009\u03b31, sin\u2009\u03b3\u2009=\u2009\u03b32, and cos\u2009\u03b4\u2009=\u2009\u03b41, sin\u2009\u03b4\u2009=\u2009\u03b42. By calculating, one can find thatwhere cos\u2009g2/\u03ba\u03b3 and \u03bas/\u03ba are presented in The date shows g2/\u03ba\u03b3 and the lower \u03bas/\u03ba, the higher fidelity and efficiency. The high fidelities and efficiencies of our Toffoli and Fredkin gates can be achieved. For example, in the case \u03bas/\u03ba\u2009=\u20090.1 and g2/(\u03ba\u03b3)\u2009=\u20092.4, \u03bas/\u03ba\u2009=\u20091 and g2/(\u03ba\u03b3)\u2009=\u20092.4, From A2\u232a and other excited states, signal to noise in the ZPL channel, imperfect spin initialization and detection. These imperfections degrade the fidelities and efficiencies of our two gates, and the imperfections which are due to the technical imperfection will be improved with further technical advances. Given the current technology, our schemes are more efficient than the traditional ones with one DoF, the synthesis-based ones which are rely on CNOT gates and single rotations, and the parity-check-based ones, and they may be possible in experiment.Our schemes are deterministic if the unavoidable effect of photon loss and imperfections are neglected. Our schemes in practice working with photon loss, non-ideal single-photon sources, unbalanced CPBSs and BSs, timing jitter, spin flips during the optical excitation, the mix between state |How to cite this article: Wei, H.-R. and Zhu, P.-J. Implementations of two-photon four-qubit Toffoli and Fredkin gates assisted by nitrogen-vacancy centers. Sci. Rep. 6, 35529; doi: 10.1038/srep35529 (2016)."}
{"text": "The appendices are provided below.In the article \u201cCan we reduce negative blood cultures with clinical scores and blood markers? Results from an observational cohort study\u201d,Appendix 1: Clinical scores for prediction of positive blood culturesShapiro score . This score includes \u201cmajor criteria\u201d  and \u201cminor criteria\u201d . According to the original study, a blood culture is indicated if at least one major criterion or two minor criteria are present. In addition to this cut-off, we categorized each patient on the basis of a sum score of 0\u201315 points derived by assigning one point per \u201cminor criterion\u201d and 2 points per \u201cmajor criterion\u201d.Lee score [5]. Lee and colleagues found systolic blood pressure\u200a<\u200a90\u200amm Hg, heart rate\u200a>\u200a125\u200a/minute, body temperature <\u200a35\u00b0C or >\u200a40\u00b0C, WBC\u200a<\u200a4\u200a\u00d7\u200a109/L or\u200a>\u200a12\u200a\u00d7\u200a109/L, platelets\u200a<\u200a130\u200a\u00d7\u200a109/L, albumin\u200a<\u200a3.3\u200ag/dL or 33\u200ag/L, and CRP\u200a>\u200a17\u200amg/dL or 170\u200amg/L to accurately predict bacteremia in patients with CAP. As proposed by the authors, we assigned five points for body temperature <35\u00b0C or >40\u00b0C, three points each for hypotension, tachycardia, or CRP\u200a>\u200a17\u200amg/dL, and 2 points each for WBC\u200a<\u200a4\u200a\u00d7\u200a109/L or > 12\u200a\u00d7\u200a109/L, or albumin\u200a<\u200a3.3\u200ag/dL. The overall risk score was regarded as the sum of the calculated points, with the following risk categories being recognized: low risk: \u2264 5 points, moderate risk: \u22656\u201310 points, and high risk: \u226511 points.SIRS criteria [6]. As suggested by Jones and Lowes, who proposed that systemic inflammatory response syndrome (SIRS) could serve as a predictor of bacteremia, one point was assigned to each SIRS criterion: temperature >38.3\u00b0C or <36\u00b0C, heart rate\u200a\u2265\u200a90\u200a/min, tachypnea with a respiratory rate\u200a\u2265\u200a20 breaths/min, WBC\u200a>\u200a12\u200a\u00d7\u200a109/L, or normal WBC with \u226510% immature (band) forms.Metersky score [7]. This score includes antibiotic treatment, liver disease, systolic blood pressure\u200a<\u200a90\u200amm Hg, temperature\u200a<\u200a35\u00b0C or \u226540\u00b0C, pulse\u200a\u2265\u200a125\u200a/min, BUN\u200a\u2265\u200a30\u200amg/dL or urea \u2265 10.7\u200ammol/L, sodium\u200a<\u200a130\u200ammol/L, and WBC\u200a<\u200a5\u200a\u00d7\u200a109/L or >20\u200a\u00d7\u200a109/L. As recommended in the original publication, the risk of bacteremia was evaluated according to the number of predictors present: low-risk-group with zero predictors and prior antibiotics, moderate-risk group with zero predictors and no prior antibiotics or with one predictor and prior antibiotics, and high-risk group with one predictor and no prior antibiotics or with two or more predictors.Tokuda score I and II [8]. For Tokuda score I, the presence of chills, pulse\u200a>\u200a120\u200a/min, and high risk infective site was evaluated and patients were classified as follows with regard to their risk for bacteremia: low risk (no chills and pulse\u200a<\u200a120\u200a/min or chills and physicians\u2019 diagnosis of low risk infective site), intermediate risk (no chills and pulse\u200a>\u200a120\u200a/min), and high risk (chills and physicians\u2019 diagnosis of high risk infective site) as proposed in the original article. The Tokuda score II was calculated similarly, except that the variable pulse was replaced by CRP and the following risk groups were recognized: low risk (no chills and CRP\u200a<\u200a10\u200amg/dL or 100\u200amg/L or chills and physicians\u2019 diagnosis of low risk infective site), intermediate risk (no chills and CRP\u200a>\u200a10\u200amg/dL or 100\u200amg/L), and high risk (chills and physicians\u2019 diagnosis of high risk infective site).Appendix 1: Clinical scores for prediction of positive blood culturesAppendix 2: Microorganisms isolated from blood culturesAppendix 3: Characteristics of patients with true positive blood cultures and PCT\u200a<\u200a0.25\u200a\u03bcg/L"}
{"text": "In the article titled \u201cEpstein-Barr Virus-Induced Gene 3 (EBI3) Blocking Leads to Induce Antitumor Cytotoxic T Lymphocyte Response and Suppress Tumor Growth in Colorectal Cancer by Bidirectional Reciprocal-Regulation STAT3 Signaling Pathway\u201d , there wThe title should be corrected to \u201cEpstein-Barr Virus-Induced Gene 3 (EBI3) Blocking Leads to Induction of Antitumor Cytotoxic T Lymphocyte Response and Suppression of Tumor Growth in Colorectal Cancer by Bidirectional Reciprocal-Regulation STAT3 Signaling Pathway.\u201dIn the last sentence of the Introduction, \u201csuppress\u201d should be corrected to \u201cpromote.\u201dIn the Materials and Methods, in Section\u2009\u20092.3 \u201cBalb/c\u201d should be corrected to \u201cBALB/c,\u201d in Section\u2009\u20092.4 \u201cBalb/c\u201d should be corrected to \u201cBALB/c\u201d and \u201cRPMI1640\u201d should be corrected to \u201cRPMI 1640,\u201d and in Section\u2009\u20092.9 \u201c1x PBS\u201d should be corrected to \u201c1 \u00d7 PBS.\u201dIn the legend of Figure\u2009\u20092 \u201cBalb/c\u201d should be corrected to \u201cBALB/c\u201d and \u201cRPMI1640\u201d to \u201cRPMI 1640.\u201d"}
{"text": "L2][BF4]2\u22c5Me2CO  is high\u2010spin at room temperature, and undergoes an abrupt, hysteretic spin\u2010crossover at T1/2=137\u2005K (\u0394T1/2=14\u2005K) that proceeds to about 50\u2009% completeness. This is associated with a crystallographic phase transition, from phase\u20051  to phase\u20052 . The cations associate into chains in the crystal through weak intermolecular \u03c0\u22c5\u22c5\u22c5\u03c0 interactions. Phase\u20052 contains a mixture of high\u2010spin and low\u2010spin molecules, which are grouped into triads along these chains. The perchlorate salt [FeL2][ClO4]2\u22c5Me2CO also adopts phase\u20051 at room temperature but undergoes a different phase transition near 135\u2005K to phase\u20053  without a change in spin state.Crystalline [Fe The structural chemistry of spin\u2010crossover (SCO) compoundsCrystallographic symmetry breaking during SCO is observed in a number of materials.2]2+ ,Z\u2032) observed in a molecular compound.As part of our continuing investigations of complexes derived from 2\u22c56\u2009H2O or Fe[ClO4]2\u22c56\u2009H2O in acetone afforded crystalline [FeL2][BF4]2\u22c5Me2CO (4]2\u22c51[BFMe2CO) and [FeL2][ClO4]2\u22c5Me2CO (4]2\u22c512\u22c51[BF4]2} assemblies, through N\u2212H\u22c5\u22c5\u22c5F hydrogen bonds between the acetamido substituents and BF4\u2212 ions . The loss of the crystallographic c glide and inversion center in phase\u20052, together with its unit cell volume expansion, generates 24\u2005unique molecules in its asymmetric unit which are labelled \u2018A\u2032\u2013\u2018X\u2032  spin\u2010state patterning. The four mixed\u2010spin molecules are well\u2010separated from each other in the lattice  and \u03c6 \u2010Fe(1)\u2010N(22) in Figure\u2005Dd2 symmetric complex gives \u03b8=90\u00b0 and \u03c6=180\u00b0. Most low\u2010spin [Fe(bpp)2]2+ derivatives approach these values, but high\u2010spin complexes show much more variation. In practice, high\u2010spin complexes deviating more strongly from the ideal values of \u03b8 and \u03c6 are less likely to transform to their low\u2010spin state upon cooling.The presence or absence of SCO in solid, high\u2010spin [Fe(bpp)s N(2)\u2010Fe\u2010N22) in in2]2+ d\u03b8 versus \u03c6 relationship, which is not usual in plots of this type.Notably, nine of the ten high\u2010spin cations in phase\u20052 have a more distorted coordination geometry than the high\u2010spin molecule in phase\u20051, which could explain why they remain high\u2010spin at low temperatures Figure\u2005. Interes4]2\u22c512\u22c512\u22c51[ClOMe2CO indeed remains predominantly high\u2010spin between 5\u2013300\u2005K. However, an abrupt reduction of \u03c7MT from 3.3 to 3.0\u2005cm3\u2009mol\u22121\u2005K occurs reproducibly near 145\u2005K, close to the crystallographic phase transition temperature  should be addressed to the authors.SupplementaryClick here for additional data file."}
{"text": "It also studies k-conjugacy of partial arrays.Research in combinatorics on words goes back a century. Berstel and Boasson introduced the partial words in the context of gene comparison. Alignment of two genes can be viewed as a construction of two partial words that are said to be compatible. In this paper, we examine to which extent the fundamental properties of partial words such as compatbility and conjugacy remain true for partial arrays. This paper studies a relaxation of the compatibility relation called  The stimulus for recent works on combinatorics is the study of biological sequences such as DNA and protein that play an important role in molecular biology [The genetic information in almost all organisms is carried by molecules of DNA. A DNA molecule is a quite long but finite string of nucleotides of 4 possible types:  biology \u20133. Seque biology , 4 in or biology  considerA of size  over \u03a3, a finite alphabet, is partial function A : Z+2 \u2192 \u03a3, where Z+ is the set of all positive integers. In this paper, we extend the combinatorial properties of partial words to partial arrays. Also, this paper studies a relation called k-compatibility where a number of insertions and deletions are allowed as well as k-mismatches. The conjugacy result [k-Conjugacy of partial arrays is discussed.Partial array y result  which waIn this section, we give a brief overview of partial words .u of length n over A, a nonempty finite alphabet, is partial map u : {1,2,\u2026, n} \u2192 A. If 1 \u2264 i \u2264 n, then i belongs to the domain of u (denoted by Domain(u)) in the case where u(i) is defined, and i belongs to the set of holes of u (denoted by Hole(u)), otherwise.Partial word A with an empty set of holes.A word \u201310 is a u be a partial word of length n over A. The companion of u (denoted by u\u25ca) is map u\u25ca : {1,2,\u2026, n} \u2192 A \u222a {\u25ca} defined byLet u\u25ca = ba\u25caab\u25ca is the companion of the partial word. The length of the partial word is 6. D(u) = {1,2, 4,5}. H(u) = {3,6}.The symbol \u25ca is viewed as a \u201cdo not know\u201d symbol. Word u and v be two partial words of length n. Partial word u is said to be contained in partial word v (denoted by u \u2282 v), if Domain(u) \u2282 Domain(v) and u(i) = v(i) for all i \u2208 Domain(u). Partial words u and v are called compatible (denoted by u\u2191v), if there exists partial word w such that u \u2282 w and v \u2282 w (in which case we define u\u2228v by u \u2282 u\u2228v and v \u2282 u\u2228v and Domain(u\u2228v) = Domain(u) \u222a Domain(v)). As an example, u\u25ca = aba\u25ca\u25caa and v\u25ca = abab\u25caa.Let (i)Multiplication: If u\u2191v and x\u2191y, then ux\u2191vy.(ii)Simplification: If ux\u2191vy and |u| = |v|, then u\u2191v and x\u2191y.(iii)Weakening: If u\u2191v and w \u2282 u, then w\u2191v.The following rules are useful for computing with partial words:u, v, x, y be partial words such that ux\u2191vy. (i)u | \u2265|v|, then there exist partial words w, z such that u = wz, v\u2191w, and y\u2191zx.If |(ii)u | \u2264|v|, then there exist partial words w, z such that b = wz, v\u2191w, and x\u2191zy.If |Let u and v are called conjugate, if there exist partial words x and y such that u \u2282 xy and v \u2282 yx.Two partial words u and v are called k-conjugate, if there exist nonnegative integers k1, k2 whose sum is k and partial words x and y such that u\u2282k1xy and v\u2282k2yx.Two partial words This section is devoted to review the basic concepts on partial arrays .A of size  over \u03a3, a nonempty set or an alphabet, is partial function A : Z+2 \u2192 \u03a3, where Z+ is the set of all positive integers. For 1 \u2264 i \u2264 m, 1 \u2264 j \u2264 n, and if A is defined, then we say that  belongs to the domain of A  \u2208 D(A)). Otherwise, we say that  belongs to the set of holes of A  \u2208 H(A)).Partial array An array  over \u03a3 iA is a partial array of size  over \u03a3, then the companion of A (denoted by A\u25ca) is total function A\u25ca : Z+2 \u2192 \u03a3 \u222a {\u25ca} defined byIf A \u2192 A\u25ca allows defining the catenation of two partial arrays in a trivial way.The bijectivity of map A of size , where D(A) = {, , , , , , }, H(A) = {, }.Partial array LetA and B are two partial arrays of equal size, then A is contained in B denoted by A \u2282 B if D(A)\u2286D(B) andIf A and B are said to be compatible denoted by A\u2191B, if there exists partial array C such that A \u2282 C and B \u2282 C.Partial arrays The rules mentioned for partial words are also true for partial arrays.A, \u2009B, \u2009X, \u2009Y be partial arrays.(i)Multiplication: If A\u2191B and X\u2191Y, then AX\u2191BY either by column catenation or by row catenation.(ii)Simplification: If AX\u2191BY either by column catenation or by row catenation with A and B being of same size, then A\u2191B and X\u2191Y.(iii)Weakening: If A\u2191B and C \u2282 A, then C\u2191B.Let A, \u2009B, \u2009X, \u2009Y be partial arrays such that AX\u2191BY, either by column catenation or by row catenation. (i)A\u2265 order of B, then there exist partial arrays C, Z such that A = CZ, B\u2191C, and Y\u2191ZX.If order of (ii)A\u2264 order of B, then there exist partial arrays C, Z such that B = CZ, A\u2191C, and X\u2191ZY.If order of Let A and B are two partial arrays of same size and k is nonnegative integer, then A is said to be k-contained in B denoted by A\u2282kB if D(A) \u2282 D(B) and there exists subset E of D(A) of cardinality k called the error set such that If A and B are two partial arrays of same order and k is a nonnegative integer, then A and B are called k-compatible denoted by A\u2191kB if there exist partial array Z and nonnegative integers k1, \u2009k2 such that(i)A\u2282k1Z with error set E1;(ii)B\u2282k2Z with error set E2;(iii)E1\u2229E2 = \u03d5;(iv)k1 + k2 = k.If E1 = {, }, E2 = {} and k1 = 2, k2 = 1\u21d2k = 3; that is, A\u21913B.A and B are k-compatible, if there exists subset E of D(A)\u2229D(B) of cardinality k called the error set such that (i)A = B\u2200 \u2208 D(A)\u2229D(B)\u2216E;(ii)A \u2260 B\u2200 \u2208 E.If A and B are arrays, then A\u2191\u2218B means A = B. We sometimes use notation A\u2191k\u2264B, if set E has cardinality \u2264k.Equivalently, Multiplication. If A\u2191k1B and X\u2191k2Y, then AX\u2191k1+k2BY where A, \u2009B, \u2009X, and Y are partial arrays and k1, \u2009k2 are nonnegative integers, using column catenation.AX\u21916+7BY.Simplification. If AX\u2191kBY and order of A is equal to order of B, then A\u2191k1B and X\u2191k2Y for some k1, \u2009k2, satisfying k1 + k2 = k.AX\u21918BY\u21d2A\u21915B and X\u21913Y with 5 + 3 = 8.Weakening. If A\u2191kB and Z \u2282 A, then Z\u2191k\u2264B.Z\u2191\u22647B with k = 7.A and B be partial arrays of orders a \u00d7 b and a \u00d7 c, respectively. If there exist array Z of order a \u00d7 d and integers k1, \u2009k2, \u2009m, and n such that A\u2282k1Zm with error set E1 and B\u2282k2Zn with error set E2, then there exist integers p and q such that Ap\u2191k\u2264Bq withE1\u2229E2 = \u03d5, then Ap\u2191kBq.Let A and B be partial arrays of a \u00d7 b and a \u00d7 c, respectively. Let array z of order a \u00d7 d exist such that, by using column catenation, A\u2282k1Zm and B\u2282k2Zn for some integers k1, \u2009k2, \u2009m, and n. Let E1 be the error set of cardinality k1 such that A = Zm for all  \u2208 D(A)\u2216E1 and A \u2260 Zm for all  \u2208 E1 and E2 be the error set of cardinality k2 such that B = Zn for all  \u2208 D(B)\u2216E2 and B \u2260 Zn for all  \u2208 E2. We have An\u2282nk1Zmn with error set E1 of cardinality nk1 and Bm\u2282mk2Zmn with error set E2 of cardinality mk2.Let i, j)\u2264 and Zmn = a for some letter a. There are 4 possibilities.Let \u2264 \u2209 E1 and  \u2209 E2, then An\u2208{\u25ca, a} and Bm\u2208{\u25ca, a}. It does not give any error, when we align An with Bm.Case 2. If  \u2209 E1 and  \u2208 E2, then An\u2208{\u25ca, a} and Bm = b for some b \u2260 a. It gives an error in the alignment of An with Bm only when An = a or when  \u2208 D(A).Case 3. If  \u2208 E1 and  \u2208 E2, then Bm\u2208{\u25ca, a} and An = b for some b \u2260 a. It gives an error in the alignment of An with Bm only when Bm = a or when  \u2208 D(B).Case 4. If  \u2208 E1 and  \u2208 E2, then An = b for some b \u2260 a and Bm = c for some c \u2260 a. It gives an error in the alignment of An with Bm only when b \u2260 c.E1\u2229E2 = \u03d5 then An\u2191kBm with k = \u2016(D(a)\u2229E2)\u222a(D(B)\u2229E1\u2016 and E1\u2229E2 \u2260 \u03d5 then An\u2191k\u2264Bm.Therefore, if A\u22824Z3 with error set E1 = {, , , }, and\u2009\u2009B\u22822Z2 with error set E2 = {, }.\u2009k = 6:(i)D(A) = {, , , , , ,}.\u2009D(B) = {, , , , }(ii)D(A) = {, , , ,\u2009, , , , , , ,\u2009, , }.\u2009D(B) = {, , , ,\u2009, , , , , , ,\u2009, , , }(iii)E1 = {, , , , ,\u2009, , }.\u2009E2 = {, , , , ,\u2009}.\u2009E1\u2229E2 \u2260 \u03d5.\u2009k = \u2016(D(A)\u2229E2) \u222a (D(B)\u2229E1)\u2016 = \u2016, , , , ,\u2009, , , , , , ,\u2009, )\u2229, , , , ,\u2009))\u222a, , , , , , ,\u2009, , , , , ,\u2009, )\u2229, , , , , ,\u2009, ))\u2016 = \u2016, , ,\u2009, , , , , \u2016.\u2009k = 9:We have A2\u2191\u22649B3(A2\u21916B3).A and B of same order are called conjugate if there exist partial arrays X and Y such that A \u2282 XY and B \u2282 YX using row catenation or column catenation.Two partial arrays 0-conjugacy on partial arrays with same order is trivially reflexive and symmetric but not transitive.A \u2282 XY and B \u2282 YX showing that A and B are conjugate similarly and, by taking B \u2282 X\u2032Y\u2032 and C \u2282 Y\u2032X\u2032 showing that B and C are conjugate. But A and C are not conjugate.By taking That is, conjugate relation is not transitive.A and B be nonempty partial arrays of same size. If A and B are conjugate, then there exists partial array C such that AC\u2191CB, either by column catenation or by row catenation.Let A and B be nonempty partial arrays of same order. Suppose A and B are conjugate and let X, \u2009Y be partial arrays such that A \u2282 XY and B \u2282 YX either by column catenation or by row catenation; then AX \u2282 XYX and XB \u2282 XYX. So, for C = X, we have AC\u2191CB.Let A and B of same order are k-conjugate, if there exist nonnegative integers k1k2 whose sum is k and partial arrays X and Y such that A\u2282k1XY and B\u2282k2YX with row or column catenation.Two partial arrays A and B be nonempty partial arrays of same order. If A and B are k-conjugate, then there exists partial array Z such that AZ\u2191k\u2264ZB.Let A, \u2009B be two partial arrays of same order. Supposing that A and B are k-conjugate, then, by definition, there exist nonnegative integers k1, k2 whose sum is k and partial arrays X and Y such that A\u2282k1XY with error set E1 and B\u2282k2YX with error set E2 using row catenation or column catenation accordingly.Let AX\u2282k1XYX with error set E1 and XB\u2282k2XYX with error set E2\u2032 = {/ \u2208 E2} or E2\u2032 = {/ \u2208 E2} according to row or column catenation and so, for Z = X, we have AZ\u2191k\u2264ZB.Then, Given A\u22823XY and B\u22822YX, k = k1 + k2 = 5.There exist AZ\u2191\u22645ZB.There exist k-compatibility and k-conjugacy of partial arrays. We prove that, given partial arrays A, \u2009B and integers p, \u2009q satisfying |A|p = |B|q, we find k such that Ap\u2191kBq. Also, there exists partial array Z such that AZ\u2191k\u2264ZB.Motivated by compatibility and conjugacy properties of partial words, we define the conjugacy of partial array and derive the compatibility properties of partial arrays. By giving relaxation to the compatibility relation, we discuss"}
{"text": "Scientific Reports7: Article number: 4005410.1038/srep40054; published online: 01102017; updated: 03142017This Article contains typographical errors in the Results section under subheading \u2018ISs fragment the up states in the opposite hemisphere\u2019.\u201cThe presence of ISs in Hem 2 was associated with a change of the statistics of the slow-wave oscillations in Hem 1; in particular, as shown in Fig. 3c,d, the frequency of up states increased from 0.70\u2009\u00b1\u20090.06\u2009Hz to 0.81\u2009\u00b1\u20090.06\u2009Hz , while their duration decreased from 0.74\u2009\u00b1\u20090.04\u2009s to 0.64\u2009\u00b1\u20090.03\u2009s \u201d.should read:\u201cThe presence of ISs in Hem 2 was associated with a change of the statistics of the slow-wave oscillations in Hem 1; in particular, as shown in Fig. 3c,d, the frequency of up states increased from 0.57\u2009\u00b1\u20090.03\u2009Hz to 0.64\u2009\u00b1\u20090.02\u2009Hz , while their duration decreased from 0.74\u2009\u00b1\u20090.04\u2009s to 0.64\u2009\u00b1\u20090.03\u2009s \u201d."}
{"text": "Its general properties have been used to study the behavior of phylogenetic tree shape indices under the probability distribution it defines. But the explicit formulas provided by Ford for the probabilities of unlabeled trees and phylogenetic trees fail in some cases. In this paper we give correct explicit formulas for these probabilities.Ford's  In particular, we shall call the image of a cladogram under v in a tree T, \u03baT(v) is its number of descendant leaves. For every internal node v in an ordered tree T, with children v1\u227avv2, its numerical split is the ordered pair NST(v) = (\u03baT(v1), \u03baT(v2)). If, instead, T is unordered and if child(v) = {v1, v2} with \u03baT(v1) \u2a7d \u03baT(v2), then NST(v) = (\u03baT(v1), \u03baT(v2)). In both cases, the multiset of numerical splits of T is NS(T) = {NST(v)\u2223v \u2208 Vint(T)}. For instance, if T is the cladogram depicted in Let us introduce some more notations. For every node  symmetric branch point in a tree T is an internal node v such that if v1 and v2 are its children, then the subtrees Tv1 and Tv2 of T rooted at them have the same shape. For instance, the symmetric branch points in the cladogram depicted in AT and T\u2032 on \u03a3 and \u03a3\u2032, respectively, with \u03a3\u2229\u03a3\u2032 = \u2205, their root join is the cladogram T\u22c6T\u2032 on \u03a3 \u222a \u03a3\u2032 obtained by connecting the roots of T and T\u2032 to a (new) common root r; see T, T\u2032 are ordered cladograms, T\u22c6T\u2032 is ordered by inheriting the orderings on T and T\u2032 and ordering the children of the new root r as rT\u227arrT\u2032. If T and T\u2032 are tree shapes, a similar construction yields a tree shape T\u22c6T\u2032; if they are moreover ordered, then T\u22c6T\u2032 becomes an ordered tree shape as explained above.Given two cladograms \u03b1-model [n\u2a7e1, a family of probability density functions P\u03b1,n\u2217) into two arcs  and  and then adding a new arc from the inserted node w to a new leaf) or to a new root . The value of P\u03b1,n\u2217)((T\u2217) for T\u2217 \u2208 \ud835\udcafn\u2217 is determined through all possible ways of constructing cladograms with shape T\u2217 in this way. More specifically,(1)T1 and T2 denote, respectively, the only cladograms in \ud835\udcaf1 and \ud835\udcaf2, let P\u03b1,1\u2032(T1) = P\u03b1,2\u2032(T2) = 1;if (2)m = 3,\u2026, n, let Tm \u2208 \ud835\udcafm be obtained by adding a new leaf labeled m to Tm\u22121. Then for every (3)n of leaves is reached, the probability of every tree shape Tn\u2217 \u2208 \ud835\udcafn\u2217 is defined as When the desired number It is well known  that eve\ud835\udcaf5 with the same shape and how their probability P\u03b1,5\u2032 is built using the recursion in Step (2). If we generate all cladograms in \ud835\udcaf5 with this shape, we compute their probabilities P\u03b1,5\u2032, and then we add up all these probabilities, we obtain the probability P\u03b1,5\u2217)/(4 \u2212 \u03b1); cf. [For instance, 2(1 \u2212 \u03b1)/ \u2212 \u03b1; cf.P\u03b1,n\u2217)T \u2208 \ud835\udcafn, if \u03c0(T) = T\u2217 \u2208 \ud835\udcafn\u2217 and it has k symmetric branch points, thenFor every \u03c0\u22121(T\u2217)| = n!/2k To \u2208 \ud835\udcaa\ud835\udcafn, if \u03c0o(To) = T \u2208 \ud835\udcafn, thenFor every \u03c0o\u22121(T)| = 2n\u22121 , v \u2208 Vint(T\u2217)).because |(iii)To\u2217 \u2208 \ud835\udcaa\ud835\udcafn\u2217, if \u03c0o,\u2217(To\u2217) = T\u2217 \u2208 \ud835\udcafn\u2217 and it has k symmetric branch points, thenFor every \u03c0o,\u2217\u22121(T\u2217)| = 2n\u22121\u2212k , v \u2208 Vint(T\u2217), those differing only on the orderings on the children of the k symmetric branch points are actually the same ordered tree shape).because |Once P\u03b1,no,\u2217)n of probabilities of ordered tree shapes satisfies the useful Markov branching recurrence  = 1 and, for every n\u2a7e2, \u0393\u03b1(n) = (n \u2212 1 \u2212 \u03b1) \u00b7 \u0393\u03b1(n \u2212 1).The family  givm < n and for every Tm\u2217 \u2208 \ud835\udcaa\ud835\udcafm\u2217 and Tn\u2212m\u2217 \u2208 \ud835\udcaa\ud835\udcafn\u2212m\u2217, For every 0 < P\u03b1,1o,\u2217), for every n\u2a7e1 and T \u2208 \ud835\udcafn.Our first result is an explicit formula for T \u2208 \ud835\udcafn, its probability under the \u03b1-model is For every T \u2208 \ud835\udcafn, let To be any ordered cladogram such that \u03c0o(To) = T, and let To\u2217 = \u03c0\u2217(To) \u2208 \ud835\udcaa\ud835\udcafn\u2217 and T\u2217 = \u03c0(T) = \u03c0o,\u2217(To\u2217). If T\u2217 has k symmetric branch points, then, by , we denote its children by v1 and v2, then v \u2208 Vint(T)\u2216{rT}, the term \u0393\u03b1(\u03baT(v)) appears twice in this product: in the denominator of the factor corresponding to v itself and in the numerator of the factor corresponding to its parent. Therefore, all terms \u0393\u03b1(\u03baT(v)) in this product vanish except \u0393\u03b1(\u03baT(rT)) = \u0393\u03b1(n) (that appears in the denominator of its factor) and every \u0393\u03b1(\u03baT(v)) = \u0393\u03b1(1) = 1 with v, a leaf. Thus, Given then, by , 6), an, anT \u2208 \ud835\udcafthen, by , (12)P\u03b1,T \u2208 \ud835\udcafn, then k is the number of symmetric branching points in T and P\u03b1,n(T) becomesm is the number of internal nodes whose children have different numbers of descendant leaves. This formula does not agree with the one given in k + m < n \u2212 1. The first example of a cladogram with this property  is the cladogram ProblsAlpha.pdf in https://github.com/biocom-uib/prob-alpha)\u03b1 = 1/2, Ford's model is equivalent to the uniform model, where every cladogram in \ud835\udcafn has the same probability\u03b1 = 0, Ford's model gives rise to the Yule model [T \u2208 \ud835\udcafn is Ford states (see [) becomesP\u03b1,nT=2k+stly, by , \u00a73.12, le model , where t\ud835\udcaf8 the probabilities computed with ProblsAlpha.pdf), these probabilities do not add up 1.As a second reason, which can be checked using a symbolic computation program, let us mention that if we take expression  as the pCombining T\u2217 \u2208 \ud835\udcafn\u2217 with k symmetric branch points, For every This formula does not agree, either, with the one given in [P\u03b1,n)n satisfies the following Markov branching recurrence.The family of density mappings n does not satisfy any Markov branching recurrence; that is, there does not exist any symmetric mapping Q : \u2124+ \u00d7 \u2124+ \u2192 \u211d such that, for every k, l\u2a7e1 and for every Tk \u2208 \ud835\udcafk\u2217 and Tl \u2208 \ud835\udcafl\u2217, m leaves and k symmetric branch points, for instance, the tree shapes in \ud835\udcaf6\u2217 depicted in Tm\u2217\u22c6Tm\u2217 \u2208 \ud835\udcafm2\u2217 has 2k + 1 symmetric branch points and therefore k symmetric branch points and therefore q\u03b1 \u2260 2q\u03b1. This shows that there does not exist any well-defined, single real number Q such that Tm1,\u2217, Tm2,\u2217 \u2208 \ud835\udcafm\u2217.Against what is stated in , (P\u03b1,n"}
{"text": "Scientific Reports7:12636; doi:10.1038/s41598-017-13049-0; Article published online 03 October 2017The original HTML version of this Article contained errors in Table 1, where text was inadvertently inserted on three occasions.AGATCTYCACGAGCCCTATCAGGTGTCTTT Bgl II\u201d and now reads \u201cGAAGATCTYCACGAGCCCTATCAGGTGTCTTT Bgl II\u201d.For the Target gene \u2018CP hairpin sense strand\u2019 with Primer name \u2018CP-RNAi-P2\u2019, the Primer sequence (5\u2032-3\u2032) incorrectly read \u201cGAcomment=\u201d Underline the selected text \u201cGTCGACTCAACGCCGGAACTAGTGGAACTT Sal I\u201d and now reads \u201cGCGTCGACTCAACGCCGGAACTAGTGGAACTT Sal I\u201d.For the Target gene \u2018CP hairpin reverse strand\u2019 with Primer name \u2018CP-RNAi-P3\u2019, the Primer sequence (5\u2032-3\u2032) incorrectly read \u201cGCcomment=\u201d Underline the selected text \u201ccomment=\u201d Underline the selected text \u201cGGATCCTCACGAGCCCTATCAGGTGTCTTT BamH I\u201d and now reads \u201cCGGGATCCTCACGAGCCCTATCAGGTGTCTTT BamH I\u201d.For the Target gene \u2018CP hairpin reverse strand\u2019 with Primer name \u2018CP-RNAi-P4\u2019, the Primer sequence (5\u2032-3\u2032) incorrectly read \u201cCGcomment\u2009=\u201d\u2009Underline the selected text \u201ccomment\u2009=\u201d\u2009the selected text\u201ccomment\u2009=\u201d Underline the selected text \u201cThese errors have been corrected in the HTML version of the Article; the PDF version was correct from the time of publication."}
{"text": "In the article titled \u201cIntraoperative Contrast Enhanced Ultrasound Evaluates the Grade of Glioma\u201d , there wIn Section\u2009\u20092.7.3 (Expression of VEGF), \u201c41% to 75% was (+)\u201d should be corrected to \u201c41% to 75% was (+ +).\u201dThe age ranges were incorrectly reported in Section\u2009\u20093.1 (The Basic Condition of the Patients) as \u201c20 to 69 years with a mean age of 47.9 \u00b1 11.4\u201d and should be corrected to \u201c18 to 69 years with a mean age of 45 \u00b1 12.8 years.\u201d"}
{"text": "The statement in the Funding section is incorrect. The correct funding information is as follows: This work was supported by grants from the Hungarian Scientific Research Fund OTKA PD104878 and K109249, and by T\u00c1MOP 4.2.2.A\u201011/1/KONV\u20102012\u20100025, T\u00c1MOP\u20104.2.2/B\u201010/1\u20102010\u20100024, and T\u00c1MOP\u20104.2.4.A/2\u201011\u20101\u20102012\u20100001 projects, and by the University of Debrecen (RH/751/2015)."}
{"text": "Scientific Reports6: Article number: 23946; 10.1038/srep23946 published online: 04042016; updated: 01192017.\u221210\u2009Pa\u22121)\u2019 is incorrectly given as \u2018Acoustic compressibility (TPa\u22121)\u2019. The correct This Article contains errors. In The legend of Figure 3 contains errors.\u22121 for MCF7 and 4.3\u2009\u00b1\u20090.2\u2009\u2009TPa\u22121 for MDA\u2019.\u2018Cellular acoustic compressibility versus cell size: MCF7 and MDA show a very similar cell size, 17.3\u2009\u00b1\u20091.0\u2009\u03bcm, but different acoustic compressibility, 3.8\u2009\u00b1\u20090.3\u2009TPashould read:\u221210\u2009Pa\u22121 for MCF7 and 4.3\u2009\u00b1\u20090.2\u200910\u221210\u2009Pa\u22121 for MDA\u2019.\u2018Cellular acoustic compressibility versus cell size: MCF7 and MDA show a very similar cell size, 17.3\u2009\u00b1\u20091.0\u2009\u03bcm, but different acoustic compressibility, 3.8\u2009\u00b1\u20090.3\u200910\u221210\u2009Pa\u22121)\u2019 is incorrectly given as \u2018Acoustic compressibility (TPa\u22121)\u2019. The correct In In the Results section under the subheading \u2018Determination of cellular acoustic compressibility and optical deformability on the same cell\u2019,\u22121)\u201d.\u201cThe results obtained here indicate that if one uses an OD threshold-value of \u224811%, the cell identification based on OD introduces an error of 25%; while this error drops to 12% by using AC (threshold\u2009\u2248\u20094.07\u2009TPashould read:\u221210\u2009Pa\u22121).\u201d\u201cThe results obtained here indicate that if one uses an OD threshold-value of \u224811%, the cell identification based on OD introduces an error of 25%; while this error drops to 12% by using AC (threshold\u2009\u2248\u20094.07 10"}
{"text": "Upon publication of the original article , it was The second line of the subsection \u201cRead alignment\u201d currently reads \u201c--outSJfilterOverhangMin 12 12 12\u201d and should instead be \u201c--outSJfilterOverhangMin 12 12 12 12\u201d."}
{"text": "Scientific Reports6: Article number: 21259; 10.1038/srep21259 published online: 02162016; updated: 01292018.This Article contains typographical errors. In Table 1, the \u2018Yield per plant (g)\u2019 for PAT11 (NE) \u201c23.09\u2009\u00b1\u20090.92\u201d should read \u201c23.09\u2009\u00b1\u20098.11\u201d. In the Methods section under subheading \u2018PCR analysis of transgenic rice\u2019,2+ plus), 0.4\u2009\u03bcL 10\u2009mM dNTP, 0.3\u2009\u03bcL 10\u2009\u03bcM RePAT-F (5\u2032-GGATCCAGACTCACTCTGAG-3\u2032), 0.3\u2009\u03bcL 10\u2009\u03bcM RePAT-R (5\u2032-GCATGCGGTGGACACGCTGG-3\u2032) and 1\u2009U Taq DNA polymerase in a total volume of 20\u2009\u03bcL, and under the conditions of 94\u2009\u00b0C for 5\u2009min, then 30 cycles of 94\u2009\u00b0C for 30\u2009s, 58\u2009\u00b0C for 30\u2009s, 72\u2009\u00b0C for 30\u2009s, and finally 72\u2009\u00b0C for 8\u2009min.\u201d\u201cPCR assay was performed in a mixture containing 50 ng rice genomic DNA or 1\u2009ng plasmid DNA, 2\u2009\u03bcL 10\u2009\u00d7\u2009PCR buffer (Mgshould read:2+ plus), 0.4\u2009\u03bcL 10\u2009mM dNTP, 0.3\u2009\u03bcL 10\u2009\u03bcM RePAT-F (5\u2032-TTCCTTAAAGCGAAAACCCC-3\u2032), 0.3\u2009\u03bcL 10\u2009\u03bcM RePAT-R (5\u2032-ACGATCCTGAAGCCGAACCT-3\u2032) and 1\u2009U Taq DNA polymerase in a total volume of 20\u2009\u03bcL, and under the conditions of 94\u2009\u00b0C for 5\u2009min, then 30 cycles of 94\u2009\u00b0C for 30\u2009s, 58\u2009\u00b0C for 30\u2009s, 72\u2009\u00b0C for 30\u2009s, and finally 72\u2009\u00b0C for 8\u2009min.\u201d\u201cPCR assay was performed in a mixture containing 50 ng rice genomic DNA or 1\u2009ng plasmid DNA, 2\u2009\u03bcL 10\u2009\u00d7\u2009PCR buffer (Mg"}
{"text": "AbstractHemiptera, Phylloxeridae) is presented. Six family-group names are listed, three being synonyms. Thirty-five genus-group names, of which six are subjectively valid, are presented with their type species, etymology, and grammatical gender. Ninety-four species-group names are listed, of which 73 are considered subjectively valid. This is the last group of Aphidomorpha to be catalogued, bringing the list of valid extant species to 5,218.A taxonomic and nomenclatural catalog of the phylloxerids ( Phylloxeridae is a small family of Hemiptera, closely related to Adelgidae and Aphididae. Little is known of the biology of most of the family\u2019s 69 species, although that of the economically important grape phylloxeran, Daktulosphairavitifoliae (Fitch), has been studied in detail. Most species of phylloxerid feed on species of Juglandaceae or Fagaceae, with a large number forming galls on North American hickories (Carya spp.). Host alternation exists within the family  .rhyncha) . Whereasal times  and the recently , the Physil taxa , the entIn this catalog, we present six family-group, 35 genus-group, and 94 species-group names of extant phylloxerids. The family-group names include two valid subfamilies and two valid tribes and three subjective synonyms. The genus-group names include six valid names, 21 junior subjective synonyms, three junior objective synonyms, three junior homonyms, and two unavailable names. The species-group names include four subspecies , 14 subjective synonyms, one junior primary homonym, two nomina dubia, and four unavailable names.Phylloxeridae in English is usually pronounced with the accent on the third syllable. However, the name of its type genus, Phylloxera, is often pronounced with an accent on the second. Because the e of x\u0113r\u00f3s is an eta, the word made from it, once written in Roman letters and given Latin endings, must be considered to have a long e. The penultimate syllable of a Latin word must be accented when it contains such a long vowel and it is a fixed principle that the accentuation of Latin words is to be kept when they are borrowed into English. Therefore, strictly-speaking, only accentuation of the third syllable of Phylloxera is historically justified.The name DaktulosphairaDactylosphaera .Also as with other recent catalogs of groups of included . Where oPageBreakPHYLLOXERIDAEHerrich-Schaeffer 1854PHYLLOXERINAEHerrich-Schaeffer 1854Subfamily ACANTHOCHERMESINITribe AcanthochermesiniOriginal spelling. AcanthochermesKollar 1848Type genus. ACANTHOCHERMESAcanthochermesquercusType species. Chermes [Hemiptera]Etymology. Greek \u00e1kantha \u2018thorn\u2019 + Gender. MasculinequercusAcanthochermes)balbianii (Phylloxera)=albianii : 782 (PhsimiliquercusAcanthochermes)PHYLLOXERINITribe Original spelling. PhylloxeridenPhylloxeraBoyer de Fonscolombe, 1834Type genus. =DACTYLOSPHAER\u2013 DactylosphaeridaeOriginal spelling. DactylosphaeraShimer, 1867Type genus. =MORITZIELL\u2013 MoritzielliniOriginal spelling. MoritziellaB\u00f6rner 1908bType genus. =VACUN\u2013 Original spelling. VacunidenVacunavon Heyden 1837Type genus. APHANOSTIGMAPhylloxerapiriType species. Etymology. Greek aphan\u1e17s \u2018invisible\u2019 + -o + Greek stigma \u2018spot\u2019 [pterostigma]Gender. NeuterCINACIUM=CinaciumiaksuienseType species. Etymology. Japanese Kinako \u2018soybean flour\u2019 + -iumGender. Neuteriaksuiense (Cinacium)ksuiense : 473 (Cipiri (Phylloxera)piri : 119 tifoliae : 862 (Pepemphigoides (Phylloxera)=higoides : 1246 (Ppervastatrix (Peritymbiavitifoliae (Fitch))=astatrix : 4 (subsvastatrix (Planchon in Rhizaphis)=vitisana (Peritymbia)=vitisana : 109 (Pevitisviniferae (Phylloxera) nomen nudum=iniferae : 337 (Phvulpinae (Viteusvitifoliae (Fitch))=vulpinae : 213 (suFOAIELLAPhylloxeradanesiiType species. Etymology. (Anna) Fo\u00e0 [Italian entomologist] + -i + ella [diminutive suffix]Gender. FeminineBoerneriaPeritymbiaWestwood 1869Note. Replacement name for PageBreakBOERNERIA=PhylloxeradanesiiType species. Etymology. (Carl) B\u00f6rner [German entomologist] + -iaGender. FeminineBoerneriaCollembola) and BoerneriaCollembola). Replaced by FoaiellaB\u00f6rner 1909bNote. Junior homonym of danesii (Phylloxera)danesii : 431  [Russian entomologist] + -iaGender. Feminineulmifoliae (Aphanostigma)mifoliae : 144  Moritz [German entomologist] + -i + ella [diminutive suffix]Gender. FeminineNOTABILIA=PhylloxeranotabilisType species. notabilis \u2018remarkable, sizeable\u2019, inflected in the neuter pluralEtymology. Latin Gender. NeuterPARAMORITZIELLA Grassi in =PhylloxeracaryaefoliaeType species. MoritziellaEtymology. Greek par\u00e1 \u2018beside\u2019 + Gender. FemininePARAPERGANDEA=PhylloxeracaryaevenaeType species. PergandeaEtymology. Greek par\u00e1 \u2018beside\u2019 + Gender. FemininePARAPHYLLOXERA Grassi in =VacunaglabraType species. PhylloxeraEtymology. Greek par\u00e1 \u2018beside\u2019 + Gender. FemininePARTHENOPHYLLOXERA Grassi in =ParthenophylloxerailicisType species. PhylloxeraEtymology. Greek parth\u00e9nos \u2018girl, virgin\u2019 + Gender. FemininePERGANDEA=DactylosphaeraconicaType species. Etymology. (Theodore) Pergande [American entomologist] + -aGender. FemininePergandeaHymenoptera). Described as subgenus of DactylosphaeraShimer 1867Note. Junior homonym of PHYLLOXERELLA Grassi in =PhylloxerellaconfusaType species. Phylloxera + -ella [diminutive suffix]Etymology. Gender. FemininePHYLLOXEROIDES Grassi in =PhylloxeraitalicaType species. Phylloxera + Greek \u2013\u014d(i)d\u0113s \u2018resembling\u2019Etymology. Gender. MasculinePSYLLOPTERA=PsyllopteraquercinaType species. Psylla [Hemiptera: Psyllidae] + Greek pter\u00e1 \u2018wings\u2019Etymology. Gender. FeminineRHANIS=Type species. NonePageBreakEtymology. Greek rhan\u00eds \u2018drop (of a liquid)\u2019Gender. FeminineVacunaRhanisColeoptera)Note. Unavailable, described in synonymy with TROITZKYA=DactylosphaeracaryaesemenType species. Etymology. (Nikolay Nikolaevich) Troitzky [Russian entomologist] + -aGender. FeminineVACUNA=VacunacoccineaType species. Vacuna [minor goddess of ancient Italy]Etymology. Latin Gender. FeminineXEROPHYLLA=PemphiguscaryaecaulisType species. Etymology. Greek x\u0113r\u00f3s \u2018dry\u2019 + Greek ph\u00fdllon \u2018leaf\u2019Gender. FemininecaryaeavellanaPhylloxera)caryaecaulis (Pemphigus)aecaulis : 859 (Pecaryaemagna (Dactylosphaera)=yaemagna : 391 caryaefoliaePhylloxera)caryaeglobuliPhylloxera)hemisphericum (Dactylosphaera)=phericum : 387 (DacaryaegummosaPhylloxera)caryaepilula (Xerophylla) nomen nudumaepilula : 283 (XecaryaerenPhylloxera) original spelling caryaereniformis but caryaeren in prevailing usage (ICZN Article 33.3.1)caryaescissaPhylloxera)caryaesemen (Dactylosphaera) specific epithet first used by yaesemen : 392 (Dacaryaesepta (yaesepta caryaesepta (Dactylosphaera)subspecies yaesepta : 389 (DaperforansPhylloxeracaryaesepta (subspecies yaesepta )caryaevenae (Pemphigus)yaevenae : 444 (Pecastaneae (Chermes)astaneae : 106 (Chcastaneivora (Moritziella)aneivora : 400 (Mococcinea (Vacuna)coccinea : 289  nomen nudum=globifera (Rhanis) unavailable, described in synonymy with Vacunacoccineavon Heyden 1837=lobifera : 289 (RhPageBreakrutilaPhylloxera)=confusa Grassi in Phylloxera)conica (Dactylosphaera)conica : 390 ibericaPhylloxera)=lichtensteiniiPhylloxera)=davidsoniPhylloxera)deplanataPhylloxera)depressa (Dactylosphaera)depressa : 390 (DadevastatrixPhylloxera)foaePhylloxera)foveata (Dactylosphaera)foveata : 393 (DafoveolaPhylloxera)fraxiniPhylloxera) nomen dubium, only the gall was described and it is probably not a phylloxeridgeorgianaPhylloxera)glabra (Vacuna)glabra : 291 (VapunctataPhylloxera) original name bipunctatum but punctata in prevailing usage (ICZN Article 33.3.1)=globosa (globosa coniferum (Dactylosphaera)subspecies oniferum : 397 (Daglobosa (Dactylosphaera)subspecies globosa : 2 (Dactilicis (Grassi in Parthenophylloxera)intermediaPhylloxera)italica (Grassi in Phylloxeroides)kunugiPhylloxera)minima (Dactylosphaera)minima : 391 (DanotabilisPhylloxera)perniciosaPhylloxera)pictaPhylloxera)pilosulaPhylloxera)quercetiPhylloxera)quercina (Psylloptera)quercina : 85 (PsyspinulosaPhylloxera)=quercusPhylloxera)florentinaPhylloxera)=scutiferaPhylloxera) nomen dubium; Phylloxeraquercus Boyer de Fonscolombe except that scutifera was \u201cslightly larger and darker\u201d; he also drew a scale-like structure  that is not of phylloxerid origin, suggesting his description included a mixture of species=signoretiPhylloxera)=reticulataPhylloxera)PageBreakrileyiPhylloxera)rimosalisPhylloxera)russellaePhylloxera)similansPhylloxera)spiniferaPhylloxera)spinosa (Dactylosphaera)spinosa : 397 (DaspinuloidesPhylloxera)stanfordianaPhylloxera)stellataPhylloxera)subelliptica (Dactylosphaera)lliptica : 389 (DasymmetricaPergande 1904purpureaPhylloxerasymmetricasubspecies symmetricaPhylloxera)subspecies vasculosaPhylloxerasymmetricasubspecies texanaPhylloxera)tuberculiferaPhylloxera)PHYLLOXERININAESubfamily PhylloxerininiOriginal spelling. PhylloxerinaB\u00f6rner 1908aType genus. PHYLLOXERINAPhylloxerasalicisType species. Phylloxera + Latin -ina \u2018in relation to\u2019Etymology. Gender. FeminineGUERCIOJA=ChermespopuliType species. Etymology. (Giacomo Del) Guercio [Italian entomologist] + -jaGender. FeminineLAUFFERELLA=ChermespopuliType species. Etymology. (Jorge) Lauffer [German entomologist] + -ella [diminutive suffix]Gender. FemininePseudochermesGuerciojaMordvilko 1909Note. Replacement name for PSEUDOCHERMES=ChermespopuliType species. Chermes [Hemiptera]Etymology. Greek pseudo- \u2018untrue\u2019 + Gender. MasculinePseudochermes Nitsche in Hemiptera: Cryptococcidae). Replaced by LauferellaLindinger 1933Note. Junior homonym of PageBreakcapreaePhylloxerina)daphnoidisPhylloxerina)moniliferae (Guercioja) new name for ChermespopuliPhylloxerinapopularia (Pergande)iliferae : 696 (Gupopuli (Chermes) junior primary homonym of Phylloxerinapopuli (=populi : 269 (Chapopuli nyssae (Phylloxera)nyssae : 269 (Phpopularia (Phylloxera)opularia : 266 (Phpopuli (Chermes)populi : 81,83 (prolifera (Phylloxera)rolifera : 16 salicis : 616 salicola : 267 (PhACANTHAPHISPhylloxeraACANTHOCHERMESPhylloxerinae, AcanthochermesiniAPHANOSTIGMAPhylloxerinae, PhylloxerinibalbianiiAcanthochermesquercusbipunctatapunctataBOERNERIAFoaiellacapreaePhylloxerinacaryaeavellanaPhylloxeracaryaecaulisPhylloxeracaryaefallaxPhylloxeracaryaefoliaePhylloxeracaryaeglobuliPhylloxeracaryaegummosaPhylloxeracaryaemagnaPhylloxeracaryaecauliscaryaepilulaPhylloxeracaryaerenPhylloxeracaryaereniformiscaryaerencaryaescissaPhylloxeracaryaesemenPhylloxeracaryaeseptaPhylloxeracaryaevenaePhylloxeracastaneaePhylloxeracastaneivoraPhylloxeraCINACIUMAphanostigmacoccineaPhylloxeraconfusa Grassi in PhylloxeraconicaPhylloxeraPageBreakconiferumPhylloxeraglobosacorticalisPhylloxeraDACTYLOSPHAERAPhylloxeraDAKTULOSPHAIRAPhylloxerinae, PhylloxerinidanesiiFoaielladaphnoidisPhylloxerinadavidsoniPhylloxeradeplanataPhylloxeradepressaPhylloxeradevastatrixPhylloxeraescorialensisPhylloxeracoccineaEUPHYLLOXERAPhylloxeraflorentinaPhylloxeraquercusfoaePhylloxeraFOAIELLAPhylloxerinae, PhylloxerinifoveataPhylloxerafoveolaPhylloxerafraxiniPhylloxerageorgianaPhylloxeraglabraPhylloxeraglobiferaPhylloxeracoccineaglobosaPhylloxeraGUERCIOJAPhylloxerinahemisphericumPhylloxeracaryaeglobuliHYSTRICHIELLAPhylloxeraiaksuienseAphanostigmaibericaPhylloxeracorticalisilicis Grassi in PhylloxeraintermediaPhylloxeraitalica Grassi in PhylloxerakunugiPhylloxeraLAUFFERELLAPhylloxerinalichtensteiniiPhylloxeracorticalisMICRACANTHAPHIS Grassi in PhylloxeraminimaPhylloxeramoniliferaePhylloxerinaMORITZIELLAPhylloxeraNOTABILIAPhylloxeranotabilisPhylloxeranyssaePhylloxerinaOLEGIAPhylloxerinae, PhylloxeriniPARAMORITZIELLA Grassi in PhylloxeraPARAPERGANDEAPhylloxeraPARAPHYLLOXERA Grassi in PhylloxeraPageBreakPARTHENOPHYLLOXERA Grassi in PhylloxerapemphigoidesDaktulosphairavitifoliaeperforansPhylloxeracaryaeseptaPERGANDEAPhylloxeraPERITYMBIADaktulosphairaperniciosaPhylloxerapervastatrixDaktulosphairavitifoliaePHYLLOXERAPhylloxerinae, PhylloxeriniPHYLLOXERELLA Grassi in PhylloxeraPHYLLOXERINAPhylloxerininaePHYLLOXEROIDES Grassi in PhylloxerapictaPhylloxerapilosulaPhylloxerapiriAphanostigmapopulariaPhylloxerinapopuliPhylloxerinapopuliPhylloxerinamoniliferaeproliferaPhylloxerinaPSEUDOCHERMESPhylloxerinaPSYLLOPTERAPhylloxerapunctataPhylloxeraglabrapurpureaPhylloxerasymmetricaquercetiPhylloxeraquercinaPhylloxeraquercusPhylloxeraquercusAcanthochermesreticulataPhylloxeraRHANISPhylloxeraRHIZAPHIS Planchon in DaktulosphairaRHIZOCERADaktulosphairarileyiPhylloxerarimosalisPhylloxerarussellaePhylloxerarutilaPhylloxeracoccineasalicisPhylloxerinasalicolaPhylloxerinascutiferaPhylloxeraquercussignoretiPhylloxeraquercussimilansPhylloxerasimiliquercusAcanthochermesspiniferaPhylloxeraspinosaPhylloxeraspinuloidesPhylloxeraspinulosaPhylloxeraquercinaPageBreakstanfordianaPhylloxerastellataPhylloxerasubellipticaPhylloxerasymmetricaPhylloxeratexanaPhylloxeraTROITZKYAPhylloxeratuberculiferaPhylloxeraulmifoliaeOlegiaVACUNAPhylloxeravasculosaPhylloxerasymmetricavastatrix Planchon in DaktulosphairavitifoliaeVITEUSDaktulosphairavitifoliaeDaktulosphairavitisviniferaeDaktulosphairavitifoliaevitisanaDaktulosphairavitifoliaevulpinaeDaktulosphairavitifoliaeXERAMPELUSDaktulosphairaXEROPHYLLAPhylloxera"}
{"text": "Research with structured Electronic Health Records (EHRs) is expanding as data becomes more accessible; analytic methods advance; and the scientific validity of such studies is increasingly accepted. However, data science methodology to enable the rapid searching/extraction, cleaning and analysis of these large, often complex, datasets is less well developed. In addition, commonly used software is inadequate, resulting in bottlenecks in research workflows and in obstacles to increased transparency and reproducibility of the research. Preparing a research-ready dataset from EHRs is a complex and time consuming task requiring substantial data science skills, even for simple designs. In addition, certain aspects of the workflow are computationally intensive, for example extraction of longitudinal data and matching controls to a large cohort, which may take days or even weeks to run using standard software. The rEHR package simplifies and accelerates the process of extracting ready-for-analysis datasets from EHR databases. It has a simple import function to a database backend that greatly accelerates data access times. A set of generic query functions allow users to extract data efficiently without needing detailed knowledge of SQL queries. Longitudinal data extractions can also be made in a single command, making use of parallel processing. The package also contains functions for cutting data by time-varying covariates, matching controls to cases, unit conversion and construction of clinical code lists. There are also functions to synthesise dummy EHR. The package has been tested with one for the largest primary care EHRs, the Clinical Practice Research Datalink (CPRD), but allows for a common interface to other EHRs. This simplified and accelerated work flow for EHR data extraction results in simpler, cleaner scripts that are more easily debugged, shared and reproduced. R R  \"SELECT * FROM clinical WHERE practid == 255\"\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003## [1] \"SELECT * FROM clinical WHERE practid == 255 AND medcodes in\"rEHR provides convenience functions for these common situations. The functions run a select_events query and then group by patient id and selects only the earliest/latest event for each patient:Frequently, users need to find the first clinical event for a given patient (e.g. to identify dates of diagnosis of chronic diseases) or the most recent clinical event (e.g. to identify if a drug therapy has been prescribed within a certain time period). first_DM <- \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003last_DM <- \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003##\u2003\u2003patid\u2003\u2003eventdate\u2003medcode## 1\u2003 1004\u20032007-12-25\u2003\u2003\u2003\u2003351## 2\u2003 1005\u20032004-08-31\u2003\u2003\u2003\u2003351## 3\u2003 1008\u20032002-03-02\u2003\u2003\u2003\u2003351## 4\u2003 1010\u20032014-04-11\u2003\u2003\u2003\u2003351## 5\u2003 1012\u20032012-05-28\u2003\u2003\u2003\u2003351## 6\u2003 1015\u20032008-08-16\u2003\u2003\u2003\u2003351##\u2003\u2003patid\u2003\u2003eventdate\u2003medcode## 1\u2003 1004\u20032007-12-25\u2003\u2003\u2003\u2003351## 2\u2003 1005\u20032009-03-09\u2003\u2003\u2003\u2003351## 3\u2003 1008\u20032002-03-02\u2003\u2003\u2003\u2003351## 4\u2003 1010\u20032014-04-11\u2003\u2003\u2003\u2003351## 5\u2003 1012\u20032013-02-14\u2003\u2003\u2003\u2003351## 6\u2003 1015\u20032013-08-17\u2003\u2003\u2003\u2003273select_by_year function provides a simple interface to extract longitudinal data. On posix-compliant computers , this function can make use of parallel processes to select data for different years concurrently, greatly accelerating the extraction process on multicore machines. The function runs a series of selects over a year range and collects in a list of dataframes.Researchers will often want to extract data over a range of different time-points, for example they may want to calculate the prevalence of a condition and how this changes through time. When working with flat text files, this must be done with a complex nested loop that is both slow and error-prone. The cores must be set to 1 and the open database connection must be set with db. This is because the use of parallel::mclapply means that new database connections need to be started for each fork and temporary files are only available inside the same connection.The function applies a database select over a range of years and outputs as a list or a dataframe. Either a database object or a path to a database file can be supplied. If multiple cores are being used (i.e. cores > 1), a path to a database file must be used because the same database connection cannot be used across threads. In this case, a new database connection is made with every fork. Note that when working with temporary tables, columns argument is a character vector of column names to be selected. The individual elements can be of arbitrary length. This means it is possible to insert SQL clauses e.g. \u201cDISTINCT patid\u201d.Queries can be made against multiple tables, assuming that the columns being extracted are present in all tables. The year_range argument to specify the years to select data for. Selection is done according to the function passed to the selector_fn argument. select_events is the default but first_events and last_events can also be used, as well as custom selection functions. The where argument works in the same way as in select_events except that year-start and year-end criteria can be added as \u2018STARTDATE\u2019 and \u2018ENDDATE\u2019. These are translated to the correct year- start and end dates. Different start and end dates can be specified by supplying a function to the year_fn argument. This function must accept a single year argument and return a list with two elements\u2014\u201cstartdate\u201d and \u201cenddate\u201d, each of which must be date characters in posix format (i.e. \u201c%Y-%m-%d\u201d). Three functions are provided to define years  and a convenience function, build_date_fn is provided to which users can supply lists of year offsets, months and days for year- start and end to return a function that can be supplied as the year_fn argument. Finally the user can set the as_list argument to determine whether data from each year is returned as a separate list element or as a single data frame.A numeric vector of years is passed to the ramework , and qofTo show the utility of the package we demonstrate how one might extract an incident and prevalent cohort of diabetes patients from the simulated example data. Prevalent events for a chronic condition are selected by the earliest diagnostic event prior to the end of the time period in question. The denominator for the calculation of the prevalence is the total number of patients registered at that time point.registered_patients <- \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003## Using open database connection## Classes 'tbl_df', 'tbl' and 'data.frame': 1005 obs. of 8 variables:##\u2003$ patid\u2003\u2003\u00a0\u00a0:\u2003int\u2003\u20031001 1002 2002 3002 4002 1003 2003 1004 2004 \u2026##\u2003$ practid\u2003\u00a0\u00a0:\u2003int\u2003\u20031 2 2 2 2 3 3 4 4 4 \u2026##\u2003$ gender\u2003\u00a0\u00a0:\u2003int\u2003\u20031 1 1 1 0 0 1 0 1 1 \u2026##\u2003$ yob\u2003\u2003\u2003:\u2003num\u20031989 1942 1965 1959 1932 \u2026##\u2003$ crd\u2003\u2003\u2003:\u2003chr\u2003\u00a0\u00a0\"1998-03-22\" \"2003-07-10\" \"1997-10-15\" \u2026##\u2003$ tod\u2003\u2003\u2003:\u2003chr\u2003\u00a0NA NA NA NA \u2026##\u2003$ deathdate\u00a0:\u2003chr\u2003\u00a0NA NA NA NA \u2026##\u2003$ year\u2003\u2003\u00a0\u00a0:\u2003int\u2003\u00a02008 2008 2008 2008 2008 2008 2008 2008 2008 \u2026####\u20032008\u20032009\u20032010\u20032011\u20032012##\u2003\u00a0189\u2003\u2003195\u2003\u00a0201\u2003\u00a0206\u2003\u00a0\u00a0214select_by_year returns a dataframe in long form, with a year column for the longitudinal component. Next we collect the incident cases, which are those patients with first diagnoses at any point before the end of the year in question, plus the dates for the first diagnoses. In this case we include events matching our list of diabetes clinical codes in either clinical or referral files. Because we only want the first diagnosis dates we set the selector_fn argument to first_events:Notice that incident_cases <- \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003## Using open database connection## Classes 'tbl_df', 'tbl' and 'data.frame': 262 obs. of 5 variables:##\u2003$ patid\u2003\u2003\u00a0:\u2003int\u20031004 1005 1008 1015 1025 1035 1037 1038 1043 \u2026##\u2003$ eventdate\u00a0:\u2003chr\u2003\"2007-12-25\" \"2004-08-31\" \"2002-03-02\" \u2026##\u2003$ medcode\u00a0\u00a0:\u2003int\u2003351 351 351 351 351 293 277 273 351 257 \u2026##\u2003$ table\u2003\u2003:\u2003chr\u2003\"Clinical\" \"Clinical\" \"Clinical\" \"Clinical\" \u2026##\u2003$ year\u2003\u2003\u00a0:\u2003int\u20032008 2008 2008 2008 2008 2008 2008 2008 2008 \u2026dplyr package is imported to the namespace when the rEHR package is loaded. This simplifies and accelerates merging operations, using left_join from the dplyr package in the example below, and is an important part of the rEHR workflow:Note that in this case extra columns have been added for both year and table, to identify the table the event was found in. Because events were taken from more than one table , the incident_cases dataframe should be sorted and duplicates removed to ensure that only the first events are kept. The two datasets are then merged to give the dataset from which the denominators and numerators can be calculated. The ## All patients are kept )prevalence_dat <- ## Remove duplicates across clinical and referral tables:incident_cases %>%\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003ungroup -> incident_casesprev_terms and prev_totals. prev_terms adds logical columns for membership of incidence and prevalence denominators as well as a column for the contribution of the individual to that year\u2019s followup time. prev_totals summarises this information to calculate the denominators and numerators for prevalence and incidence, according to the users\u2019 grouping factors. The criteria for membership of the incidence and prevalence numerators and denominators as well as for followup time are shown in Prevalence and incidence can be calculated by the built-in functions An example in the use of these functions is provided below:prevalence_dat <- totals <- totals$prevalence$year_counts## Source: local data frame [5 x 4]####\u2003\u2003\u2003year\u2003numerator\u2003denominator\u2003prevalence##\u20031\u20032008\u2003\u2003\u2003\u2003\u200331\u2003\u2003 174.6721\u2003\u200317.74754##\u20032\u20032009\u2003\u2003\u2003\u2003\u200335\u2003\u2003 179.3785\u2003\u200319.51181##\u20033\u20032010\u2003\u2003\u2003\u2003\u200341\u2003\u2003 183.1403\u2003\u200322.38721##\u20034\u20032011\u2003\u2003\u2003\u2003\u200350\u2003\u2003 185.4182\u2003\u200326.96607##\u20035\u20032012\u2003\u2003\u2003\u2003\u200355\u2003\u2003 191.5838\u2003\u200328.70806totals$incidence$year_counts## Source: local data frame [5 x 4]####\u2003\u2003\u2003year\u2003numerator\u2003denominator\u2003incidence##\u20031\u20032008\u2003\u2003\u2003\u2003\u2003 4\u2003\u2003\u2003143.9014\u2003 2.779680##\u20032\u20032009\u2003\u2003\u2003\u2003\u2003 3\u2003\u2003\u2003144.4983\u2003 2.076149##\u20033\u20032010\u2003\u2003\u2003\u2003\u2003 4\u2003\u2003\u2003142.2806\u2003 2.811345##\u20034\u20032011\u2003\u2003\u2003\u2003\u2003 7\u2003\u2003\u2003135.5893\u2003 5.162648##\u20035\u20032012\u2003\u2003\u2003\u2003\u2003 5\u2003\u2003\u2003137.4675\u2003 3.637224Here we see that, in our simulated dataset, we have a diabetes prevalence of 17.7% in 2008 raising to 28.7% in 2012 and an incidence of 2.8% in 2008 increasing to 3.6% in 2012.In this section we demonstrate how to convert the longitudinal data from the previous section to a cohort dataset suitable for survival analysis and also illustrate algorithms to match controls to cases and to cut cohort data by time-varying covariates.build_cohort function. This returns a dataset with a single row for each patient and includes only patients in the numerator or denominator for whichever cohort type is chosen . Columns are added for start and end dates and for start and end times as integer differences from the cohort start date. A binary column is added to indicate membership of the case group. All patients with start dates greater than their end dates are removed from the dataset. The diagnosis_start argument is used to include the diagnosis date in the definition of the start dates for the patients. If it is not required for the diagnosis date to be included in the start date definition, this argument can be set to NULL. Here, we will first merge in practice data (i.e. dates for when practices are deemed to be up to standard) and then construct the cohort:One of the most common uses of EHR data in research is to build cohorts for survival analyses. The longitudinal data in the previous section is easily converted to survival cohort format using the practices <- prevalence_dat <- cohort <- \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003, The cohort is now ready for analysis, e.g. with a relatively simple proportional hazards regression model that only includes gender and exposure as predictors:## Add a logical column for death during cohortcohort$death <- \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003######\u2003\u2003\u2003\u2003\u2003 coef\u2003exp(coef)\u2003se(coef)\u2003\u2003\u2003z\u2003\u2003 p##\u2003gender\u20030.506\u2003\u2003 1.659\u2003\u20030.837\u2003\u00a0\u00a00.605\u20030.55##\u2003case\u2003\u2003-0.645\u2003\u2003 0.524\u2003\u20031.081\u2003-0.597\u20030.55#### Likelihood ratio test = 0.81 on 2 df, p = 0.667 n = 199,## number of events = 7rEHR package provides three methods for matching cases to controls:Matching cases to controls is an important pre-analysis step. The Incidence density matching (IDM)Exact matchingMatching on a dummy index date sourced from consultation filesget_matches function. With IDM, controls are selected for a particular case at the time of diagnosis (or other event such as death) from other members of the cohort who, at that time, do not have the diagnosis. The IDM sampling procedure allows the same patient to be selected as a control for more than one case, thus providing a full set controls for each case while still producing unbiased estimates of risk ####\u2003 id\u2003start\u2003 end\u2003 event\u2003 drug_1\u2003drug_2\u2003 drug_3_start\u2003drug_3_stop\u2003stage_1## 1\u20031\u2003\u2003 0\u20031000\u2003\u2003\u20030\u2003\u2003\u2003NA\u2003\u2003 NA\u2003\u2003\u2003\u2003\u2003 110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003\u2003300## 2\u20032\u2003\u2003 0\u2003 233\u2003\u2003\u20031\u2003\u2003\u2003NA\u2003\u2003 234\u2003\u2003\u2003\u2003\u2003 110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003\u2003NA## 3\u20032\u2003 234\u2003 689\u2003\u2003\u20031\u2003\u2003\u2003NA\u2003\u2003 234\u2003\u2003\u2003\u2003\u2003 110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003\u2003NA## 4\u20033\u2003\u2003 0\u2003 553\u2003\u2003\u20030\u2003\u2003\u2003NA\u2003\u2003 554\u2003\u2003\u2003\u2003\u2003 111\u2003\u2003\u2003\u2003\u2003400\u2003\u2003\u2003NA## 5\u20033\u2003 554\u20031000\u2003\u2003\u20030\u2003\u2003\u2003NA\u2003\u2003 554\u2003\u2003\u2003\u2003\u2003 111\u2003\u2003\u2003\u2003\u2003400\u2003\u2003\u2003NA## 6\u20034\u2003\u2003 0\u2003 122\u2003\u2003\u20031\u2003\u2003\u2003340\u2003\u2003 123\u2003\u2003\u2003\u2003\u2003 109\u2003\u2003\u2003\u2003\u2003400\u2003\u2003\u2003NA## Variables not shown: stage_2 (dbl), drug_1_state (dbl),## drug_2_state (dbl)## Binary covariates:tv_out3 <- \u2003\u2003\u2003 id, drug_3_state)tv_out3 <- \u2003\u2003\u2003 id, drug_3_state)## Source: local data frame [6 x 11]####\u2003 id\u2003start\u2003\u2003end\u2003event\u2003drug_1\u2003drug_2\u2003drug_3_start\u2003drug_3_stop\u2003stage_1## 1\u20031\u2003\u2003 0\u2003\u2003109\u2003\u2003\u20030\u2003\u2003 NA\u2003\u2003\u2003NA\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003300## 2\u20031\u2003 110\u2003\u2003399\u2003\u2003\u20030\u2003\u2003 NA\u2003\u2003\u2003NA\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003300## 3\u20031\u2003 400\u2003 1000\u2003\u2003\u20030\u2003\u2003 NA\u2003\u2003\u2003NA\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003300## 4\u20032\u2003\u2003 0\u2003\u2003109\u2003\u2003\u20031\u2003\u2003 NA\u2003\u2003\u2003234\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003NA## 5\u20032\u2003 110\u2003\u2003399\u2003\u2003\u20031\u2003\u2003 NA\u2003\u2003\u2003234\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003NA## 6\u20032\u2003 400\u2003\u2003689\u2003\u2003\u20031\u2003\u2003 NA\u2003\u2003\u2003234\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003NA## Variables not shown: stage_2 (dbl), drug_3_state (dbl)## incremental covariates:inc_1 <- \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003inc_1 <- \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003## Source: local data frame [6 x 11]####\u2003 id\u2003start\u2003 end\u2003event\u2003drug_1\u2003drug_2\u2003drug_3_start\u2003drug_3_stop\u2003stage_1## 1\u20031\u2003\u2003 0\u2003 299\u2003\u2003\u20030\u2003\u2003 NA\u2003\u2003\u2003NA\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003 300## 2\u20031\u2003300\u2003 449\u2003\u2003\u20030\u2003\u2003 NA\u2003\u2003\u2003 NA\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003 300## 3\u20031\u2003450\u20031000\u2003\u2003\u20030\u2003\u2003 NA\u2003\u2003\u2003NA\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003 300## 4\u20032\u2003\u2003 0\u2003 689\u2003\u2003\u20031\u2003\u2003 NA\u2003\u2003\u2003234\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003 NA## 5\u20033\u2003\u2003 0\u20031000\u2003\u2003\u20030\u2003\u2003 NA\u2003\u2003\u2003554\u2003\u2003\u2003\u2003\u2003111\u2003\u2003\u2003\u2003\u2003400\u2003\u2003 NA## 6\u20034\u2003\u2003 0\u2003 874\u2003\u2003\u20031\u2003\u2003340\u2003\u2003\u2003123\u2003\u2003\u2003\u2003\u2003109\u2003\u2003\u2003\u2003\u2003400\u2003\u2003 NA## Variables not shown: stage_2 (dbl), disease_stage (dbl)## Chaining combinations of the above using %>%tv_test %>%\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003head## Source: local data frame [6 x 14]####\u2003 id\u2003start\u2003 end\u2003event\u2003drug_1\u2003drug_2\u2003drug_3_start\u2003 drug_3_stop\u2003 stage_1## 1\u20031\u2003\u2003 0\u2003 109\u2003\u2003\u20030\u2003\u2003 NA\u2003\u2003\u2003NA\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003\u2003300## 2\u20031\u2003 110\u2003 299\u2003\u2003\u20030\u2003\u2003 NA\u2003\u2003\u2003NA\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003\u2003300## 3\u20031\u2003 300\u2003 399\u2003\u2003\u20030\u2003\u2003 NA\u2003\u2003\u2003NA\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003\u2003300## 4\u20031\u2003 400\u2003 449\u2003\u2003\u20030\u2003\u2003 NA\u2003\u2003\u2003NA\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003\u2003300## 5\u20031\u2003 450\u20031000\u2003\u2003\u20030\u2003\u2003 NA\u2003\u2003\u2003NA\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003\u2003300## 6\u20032\u2003\u2003 0\u2003 109\u2003\u2003\u20031\u2003\u2003 NA\u2003\u2003\u2003234\u2003\u2003\u2003\u2003\u2003110\u2003\u2003\u2003\u2003\u2003400\u2003\u2003\u2003NA## Variables not shown: stage_2 (dbl), drug_1_state (dbl), drug_2_state## (dbl), drug_3_state (dbl), disease_stage (dbl)In this section we briefly discuss some miscellaneous functions provided in the package.R implementation of this methodology is now part of the rEHR package.An important part of EHR analyses is the construction of lists of clinical codes to define conditions, comorbidities and other clinical entities of interest to the study . We haveMedicalDefinition, containing the terms to be searched for in the lookup tables. MedicalDefinition objects can be instantiated from terms defined within R or imported from a csv file. The constructor function can be provided with lists of: terms, codes , tests (test search terms), drugs (drug search terms), drugcodes (drug product codes). Within the individual argument lists, vectors of length > 1 are searched for together , in any order. Different vectors in the same list are searched for separately . Placing a \u201c-\u201d character at the start of a character vector element excludes that terms from the search. Providing NULL to any of the arguments means that this element will not be searched for. Underscores are treated as spaces. When searching for codes, a range of clinical codes can be searched for by providing two codes separated by a hyphen. e.g \u201cE114-E117z\u201d.Building draft lists of clinical codes is a two-stage process: First, the search is defined by instantiating an object of class ## Example construction of a clinical code listdef <- \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003MedicalDefinition objects using the import_definitions(input_file = \"path/to/file.csv\") function.Code lists can be defined in a csv file with format as shown in MedicalDefinition objects are then used to run searches against lookup tables provided with EHRs via the build_definition_lists function:The ## Use fileEncoding = \"latin1\" to avoid issues with non-ascii charactersmedical_table <- drug_table <- draft_lists <- cprd_uniform_hba1c_values function accepts a single argument of a dataframe in the CPRD \u201cAdditional\u201d table form containing only entity types for HbA1C and Fructosamine and converts any HbA1C and fructosamine values to a common mmol/mol scale. Once this conversion has taken place, the function also removes obvious mis-coding errors that are far outside the possible range. A dataframe is returned with an extra column hba1c_score.HbA1C tests for glycated haemoglobin are one of the best recorded clinical tests in UK primary care databases, to a large extent because of testing being incentivised under the UK Quality and Outcomes Framework pay-for-performance scheme ; 25. HowR expertise. We have provided the to_stata function to export dataframes to stata dta format. This function compresses a dataframe to reduce file size in the following ways:Sometimes researchers may need to share data with others in the same group who may not have date_fields argument) are converted to integer days from 1960-01-01 to avoid compatibility issues between R and Stata. An alternative origin can be set with the origin argument.Date variables (as specified by the integer_fields are converted from numeric to integer.Fields specified in the stata13 boolean argument indicates whether files should be stored in Stata13 format (Using readstata13::savedta13) or in Stata 12 compatible format (using foreign::write.dta). The former includes a further compression step, similar to the compress command in Stata.the R dataframes. For example, extractions of clinical events for a common condition such as diabetes or asthma will require the extraction of millions of rows of data. These may be easily stored as temporary database tables. This is also useful if you are working with a protected database that you only have read-only access to. The rEHR package has a suite of functions to deal with temporary database tables:The size of EHR databases may require keeping intermediate data extractions as database tables, rather than as in-memory temp_table is used to construct temporary tables and is illustrated in section 3append_to_temp_table appends rows to a temporary table based on a specified select statementto_temp_table exports a dataframe to a temporary database tabledrop_temp_table checks if a temporary table exists and then deletes if it doesdrop_all_temp_tables drops all temporary tables from the databaseselect_by_year) can only be used in the single core mode (i.e. set cores = 1) since multicore processes open up multiple parallel connections.Note that temporary tables are only associated with the currently open database connection. This means that functions capable of parallel processing  obtained under licence from the UK Medicines and Healthcare products Regulatory Agency. However, the interpretation and conclusions contained in this paper are those of the authors alone. The study was approved by the independent scientific advisory committee (ISAC) for CPRD research (reference number: 16_115R). No further ethics approval was required for the analysis of the data."}
{"text": "Integrins are adhesion receptors on the cell surface that enable cells to respond to their environment. Most integrins are heterodimers, comprising \u03b1 and \u03b2 type I transmembrane glycoprotein chains with large extracellular domains and short cytoplasmic tails. Integrins deliver signals through multiprotein complexes at the cell surface, which interact with cytoskeletal and signaling proteins to influence gene expression, cell proliferation, morphology, and migration. Integrin expression on \u03b3\u03b4 T cells (\u03b3\u03b4Tc) has not been systematically investigated; however, reports in the literature dating back to the early 1990s reveal an understated role for integrins in \u03b3\u03b4Tc function. Over the years, integrins have been investigated on resting and/or activated peripheral blood-derived polyclonal \u03b3\u03b4Tc, \u03b3\u03b4Tc clones, as well as \u03b3\u03b4 T intraepithelial lymphocytes. Differences in integrin expression have been found between \u03b1\u03b2 T cells (\u03b1\u03b2Tc) and \u03b3\u03b4Tc, as well as between V\u03b41 and V\u03b42 \u03b3\u03b4Tc. While most studies have focused on human \u03b3\u03b4Tc, research has also been carried out in mouse and bovine models. Roles attributed to \u03b3\u03b4Tc integrins include adhesion, signaling, activation, migration, tissue localization, tissue retention, cell spreading, cytokine secretion, tumor infiltration, and involvement in tumor cell killing. This review attempts to encompass all reports of integrins expressed on \u03b3\u03b4Tc published prior to December 2017, highlights areas warranting further investigation, and discusses the relevance of integrin expression for \u03b3\u03b4Tc function. Although much was known about integrins on lymphocytes as early as 1990 -induced adhesion via integrin activation; while \u03b12\u03b21 was required for collagen binding, FN binding relied on both \u03b14\u03b21 and \u03b15\u03b21. Most polyclonal \u03b3\u03b4Tc only expressed \u03b14\u03b21, whereas individual clones showed variation attributed to extended culturing and selection during cloning . While no \u03b13, \u03b1v, or \u03b23 expression was observed, less than 30% expressed \u03b11, \u03b12, or \u03b15 chains. CD8 cloning , corrobo cloning . Admitte cloning .+CD4\u2212CD8\u2212 \u03b3\u03b4Tc, and lack of \u03b13 or \u03b16 was confirmed. Activated CD25hi \u03b3\u03b4Tc bound FN better than resting CD25low \u03b3\u03b4Tc, mediated mostly by \u03b14 and partly by \u03b15. Culturing cells on immobilized anti-\u03b3\u03b4 TCR antibodies together with FN enhanced proliferation and increased CD25 expression, suggesting both signaling and adhesion roles for \u03b14 and \u03b15 integrins. While \u03b3\u03b4Tc adhesion required activation through the TCR, surface levels of \u03b14 and \u03b15 remained unaltered  to endothelial cells, fibroblasts, and epithelial cells independent of activation. Both \u03b1\u03b2Tc and \u03b3\u03b4Tc required CD11a/CD18 and \u03b14\u03b21 to bind endothelial cells, whereas CD11a/CD18-ICAM-1 interaction facilitated adherence to fibroblasts and epithelial cells. Phorbol dibutyrate treatment of PBMCs and cytokine stimulation of monolayers greatly enhanced T cell adhesion, correlated with their expression of CD11a/CD18 and \u03b14\u03b21 . CD11a, \u2212CD4\u2212 populations co-expressing \u03b3\u03b4TCR and CD8; cervical \u03b3\u03b4Tc (>20%) also expressed CD11c. \u03b11\u03b21 and \u03b14\u03b27 were co-expressed on CD11c+CCR7\u2212CD4\u2212 T cells, of which \u03b3\u03b4Tc were a part, but unfortunately not specifically analyzed. CD11c expression was associated with T cell homing and activation, and interferon \u03b3 (IFN\u03b3) secretion in a fraction of (E)-4-hydroxy-3-methyl-but-2-enyl pyrophosphate-stimulated \u03b3\u03b4Tc , was found on human \u03b3\u03b4Tc intraepithelial lymphocytes (IELs). While peripheral blood T cells did not express much \u03b1E\u03b27 the authors posited its upregulation after T cells extravasate in the lamina propria, since \u03b1E\u03b27 expression positively correlated with nearer proximity to epithelium . IL-2 an avidity . On \u03b1\u03b2Tc avidity . Peters oduction .in vivo, and the importance of this interaction for V\u03b41 activation and localization  and fibroblast cells more tightly , confirmand \u03b15\u03b21 .V\u03b41 predominance has been reported in tumor infiltrating lymphocytes from lung , colon , renal cCD11a/CD18 facilitates effector-target cell conjugation . This in\u2212 V\u03b42 cells\u2019 demise ; this inferred HIV envelope glycoprotein susceptibility resulting in CD4\u2019 demise . While V\u2019 demise .+ T cells  expression on peripheral blood T cells increased with age, leveling out later in life. \u03b3\u03b4Tc expressed more CD11b than \u03b1\u03b2Tc across all ages; and while not shown, CD11b was thought important for migration to spleen and liver, and to indicate antigen-specific memory T cells . Later s T cells , 65, but T cells . A longi+ and CD8+ \u03b1\u03b2Tc; however, their role in \u03b3\u03b4Tc development remains unknown , murine \u03b3\u03b4Tc differentially expressed \u03b22 integrins and produced more IFN\u03b3 and tumor necrosis factor \u03b1 in lymph nodes, spleen, and spinal cord compared to \u03b1\u03b2Tc . At basemulation . While ne to PMA . Indeed,e to PMA . In a muoduction . Adoptivoduction . Itgax, he mouse . Murine 1c+ \u03b3\u03b4Tc .\u2212/\u2212 hybridoma line transfected with CD3\u03b6 fusion proteins, VNR- but not TCR-engagement by ligand was required in conjunction with CD3\u03b6 chain signaling for IL-2 production ; its expression on splenic T cells was only observed after a minimum of 1\u2009week of stimulation . VNR expoduction . VNR recoduction . While hoduction , VNR hasoduction .+ \u03b3\u03b4Tc expressed \u03b14\u03b27 that enabled their mobilization by CCL25 in inflamed tissue, which in turn modulated IL-17 levels  versus cows (\u03b1\u03b2)  on the lamina propria . In a mo7 levels . Blockin induced . Bovine Prevalence of \u201cinflammatory\u201d \u03b3\u03b4Tc (i\u03b3\u03b4Tc) co-expressing high levels of gut-homing \u03b14\u03b27 and \u03b1E\u03b27 correlated with disease severity in both spontaneous and induced murine colitis models. Cytotoxicity, cytokine production, and NK cell receptor genes were upregulated on i\u03b3\u03b4Tc compared to other \u03b3\u03b4Tc subsets (expressing \u03b14\u03b27 or \u03b1E\u03b27) isolated from mesenteric lymph nodes in induced colitis, suggesting profound functional relevance of integrin co-expression on these cells .Salmonella typhimurium or Toxoplasma gondii, drastically reducing pathogen translocation and emphasizing the ability of \u03b1E\u03b27 to limit \u03b3\u03b4Tc IEL migration and impact host defense against infection (\u2212CD8\u2212 T cells expressing \u03b1E\u03b27 in mesenteric lymph nodes; in IEL, significantly increased CD62L\u2212 \u03b1E\u03b27-expressing CD4\u2212CD8\u2212 cells were observed, hypothesized to result from enhanced homing and retention, respectively (In \u03b1E\u03b27-knockout mice, \u03b3\u03b4Tc IEL migration within the intraepithelial compartment was enhanced  and remanfection . In a stectively .T cells use classical cell biological pathways in new ways . Thus, uGS reviewed the literature, wrote the manuscript, and designed both the table and the figure.The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest."}
{"text": "R0 is given by linear next infection operator, which determined the dynamic behaviors of system. We obtain that when R0 < 1, the disease-free periodic solution is globally asymptotically stable and when R0 > 1 by Poincar\u00e9 map we obtain that disease is uniformly persistent. Numerical simulations support the results and sensitivity analysis shows effects of parameters on R0, which provided references to seek optimal measures to control the transmission of lymphatic filariasis.In this paper a mosquito-borne parasitic infection model in periodic environment is considered. Threshold parameter  Lymphatic filariasis is a parasitic disease caused by filarial nematode worms and is a mosquito-borne disease that is a leading cause of morbidity worldwide. Lymphatic filariasis affects 120 million humans in tropical and subtropical areas of Asia, Africa, the Western Pacific, and some parts of the Americas .W. bancrofti parasites, which account for 90% of the global disease burden, dwell in the lymphatic system, where the adult female worms release microfilariae (mf) into the blood. Mf are ingested by biting mosquitoes as a blood meal of a mosquito, through several developmental stages, that is, first into immature larvae and then L3 larvae. Infective stage larvae L3 actively escape from the mosquito mouthparts entering another human host at the next blood meal through skin Sm\u2217(t). So, for any small \u03f5 existing a t0, for all t \u2265 t0 we havet\u2192\u221esup\u2061Nm(t) \u2264 (Mm\u2217 + \u03f5)\u0394, where \u0394 = supt>0\u2009(r(t) + \u03bc2(t))/inft>0\u03bc2(t). For \u03f5 small enough, Nm(t) \u2264 Mm\u2217\u0394.by the comparison principle and R0 < 1, the disease-free periodic solution (Sh\u2217(t), 0, Sm\u2217(t), 0) is globally asymptotically stable. And if R0 > 1, it is unstable.If R0 < 1, (Sh\u2217(t), 0, Sm\u2217(t), 0) is locally stable. Next we prove that when R0 < 1 the disease-free solution (Sh\u2217(t), 0, Sm\u2217(t), 0) has global attractivity.By R0 < 1 and by (iii) of \u03c1(\u03a6F\u2212V(\u03c9)) < 1. So there exists a small enough constant \u03b51 > 0 such that \u03c1(\u03a6F\u2212V+\u03b51N(\u03c9)) < 1, where \u03b51 > 0 there exists t1 > 0 such that Sh(t) \u2264 Sh\u2217(t) + \u03b51 and Sm(t) \u2264 Sm\u2217(t) + \u03b51, so for all t > t1 we have\u03c9-periodic solution v1(t) such that J(t) \u2264 eptv1(t), where J(t) = (Ih(t), Im(t))T and p = (1/\u03c9)ln\u2061\u03c1(\u03a6F\u2212V+\u03b51N(\u03c9)) < 0. Then limt\u2192\u221eJ(t) = 0; that is, limt\u2192\u221eIh(t) = 0 and limt\u2192\u221eIm(t) = 0.When Sh(t), Sm(t), we getMoreover, from the equations of f system  is globaX and X0 are positively invariant, and \u2202X0 is also a relatively closed set in X.DefineWe have 3\u2202X0=X\u2216X0=P : X \u2192 X be the Poincar\u00e9 map associated with system  is the unique solution of system  = x0. P is compact for the continuity of solutions of system  for all m > 1 and P1 = P. Now, prove Sh, 0, Sm, 0) : Sh > 0, Sm \u2265 0}\u2286M\u2202.We further define M\u2202\u2216{ : Sh > 0, Sm \u2265 0} \u2260 \u2205, then there exists at least a point  \u2208 M\u2202 satisfying Ih0 > 0 or Im0 > 0. We consider two possible cases.If Ih0 = 0 and Im0 > 0, then it is clear that from system (Im(t) \u2265 0 for any t > 0. From the second equation of system  \u2265 f system  and Sh >Im0 = 0 and Ih0 > 0, then Ih(t) = Ih0e0t[\u03bc1(\u03c4)+\u03c5(\u03c4)]d\u03c4\u2212\u222b > 0. From the third equation of system , Ih(t), Sm(t), Im(t))\u2209\u2202X0, so  \u2209 M\u2202. This leads to a contradiction; there exists one fixed point E0 = (Sh\u2217(t), 0, Sm\u2217(t), 0) of P in M\u2202.If R0 serves as a threshold parameter for the extinction and the uniform persistence of the disease.In the following, we will discuss the uniform persistence of the disease, and R0 > 1, then system ) > 1. For an arbitrary small constant \u03b7 > 0, that \u03c1(\u03a6F\u2212V\u2212\u03b7N(\u03c9)) > 1, N(t) is the same as in H2), we obtain any small enough \u03b5 > 0, \u222b0\u03c9[r(t) \u2212 \u03b1(t)\u03b5]dt > 0. Consider perturbed equations\u03c9-periodic solutions S\u03b5h\u2217(t) and S\u03b5m\u2217(t). For the continuity of solutions with respect to \u03b5, and for \u03b7 > 0 there exists \u03b51 > 0 for all t \u2208 ; thus we haveFrom mma 2 in  and Lemmmma 2 in , we obtaequationsS\u03b5h\u2032t=\u039bt\u2212\u2212\u03bc1tS\u03b5ht,S\u03b5m\u2032t=rtSx0 =  \u2208 X0, according to the continuity of the solution with respect to the initial condition; there exists \u03b4 for given \u03b51, for all x0 \u2208 X0 with \u2016x0 \u2212 E0\u2016 < \u03b4; it follows \u2016u \u2212 u\u2016 < \u03b51 for all t \u2208 .Denote x0 \u2208 X0. Without loss of generality, we can assume thatt \u2265 0,\u2009\u2009t = m\u03c9 + t\u2032, where t\u2032 \u2208  and m = [t/\u03c9] is the greatest integer less than or equal to t/\u03c9, so we haveIh(t) \u2264 \u03b51 and 0 \u2264 Im(t) \u2264 \u03b51 for all t \u2265 0. Then from the first and third equations of  such that (J(t) = v2(t)ep1t, where p1 = (1/\u03c9)ln\u2061(\u03c1(\u03a6F\u2212V\u2212\u03b7N(\u03c9))). For \u03c1(\u03a6F\u2212V\u2212\u03b7N(\u03c9)) > 1, Following, we prove4limm\u2192\u221esupConsider ; there eh\u2217t\u2212\u03b7.By  and 48)(41)limm\u2192y system:Iht\u00af=\u03b21tSM\u2202\u2216{ : Sh > 0, Sm \u2265 0} = \u2205 and {M1} is globally attractive in M\u2202, and all orbit in M\u2202 converges to {M1}. By R0. It wag system :53)dwdtdwdtR0. IR0. We choose parameters \u039b(t) = 0.6 + 0.4sin\u2061(2\u03c0t/12), \u03bc1(t) = 0.5 + 0.1sin\u2061(2\u03c0t/12), \u03bc2(t) = 0.8 + 0.1sin\u2061(2\u03c0t/12), \u03b21(t) = 0.6 + 0.1sin\u2061(2\u03c0t/12), \u03b22(t) = 0.7 + 0.1sin\u2061(2\u03c0t/12), \u03b11(t) = 0.2 + 0.1sin\u2061(2\u03c0t/12), \u03c5(t) = 0.02 + 0.03sin\u2061(2\u03c0t/12), r(t) = 0.5 + 0.4sin\u2061(2\u03c0t/12), K(t) = 0.9 + 0.3sin\u2061(2\u03c0t/12). By numerical calculations, we obtain R0 = 0.9243 < 1; then the disease will be extinct; see \u03b21(t) = 0.9 + 0.1sin\u2061(2\u03c0t/12), \u03b22(t) = 1.2 + 0.1sin\u2061(2\u03c0t/12), then R0 = 1.4662 > 1; the disease is permanent; see Sh, Ih) and  are in Figures Firstly, by the means of the software Matlab we compute \u03b21(t), \u03b22(t), K(t), and \u03b11(t), we fix all parameters as in \u03b201 = (1/12)\u222b012\u03b21(t)dt,\u2009\u2009\u03b202 = (1/12)\u222b012\u03b22(t)dt,\u2009\u2009k0 = (1/12)\u222b012K(t)dt, and \u03b10 = (1/12)\u222b012\u03b11(t)dt.In order to perform sensitivity analysis of parameters k0 and \u03b10 on R0. From \u03b10, R0 decreases, and the gradient also decreases, so this strengthens the psychological hint of susceptible human individuals to be benefit for the extinction of the disease. In k0 the sensitivity of R0 increases. That is to say, the carrying capacity of environment for mosquito is bigger and the disease is widespread more easily, so decreasing the circumstance fit survival for mosquitoes, such as contaminated pool or puddle and household garbage, is a necessary method for the extinction of disease.We first fix other parameters and detect the effect of parameters of \u03b210 and \u03b220 on R0; in R0 may be less than 1 when \u03b210 and \u03b220 are small; the smaller \u03b220 the more sensitive the effect on R0.Next, we consider the combined influence of parameters R0 is affected by \u03b210 and k0; with the increasing of k0 the sensitivity of R0 increases; if we fix \u03b210 as a constant the case will be similar to \u03b210 on the sensitivity of R0, so in the season in which temperature and humidity are more beneficial for mosquito population to give birth and propagate taking measures to avoid more bites is necessary.In R0; when R0 < 1 disease-free periodic solution is globally asymptotically stable and when R0 > 1 disease is uniformly persistent. We also give some numerical simulations which support the results we prove, confirming that R0 serves as a threshold parameter. Sensitivity analysis show effects of parameters on R0, which contribute to providing a decision support framework for determining the optimal coverage for the successful prevention programme.In this paper, we have studied the transmission of lymphatic filariasis; lymphatic filariasis is a mosquito-borne parasitic infection that occurs in many parts of the developing world. In order to systematically investigate the impact that vector genus-specific dependent processes may have on overall lymphatic filariasis transmission, we, according to the nature characteristic of lymphatic filariasis and considering the logistic growth in periodic environments of mosquito, model the transmission of lymphatic filariasis. The dynamic behavior of system  is deter"}
{"text": "Flabellinidae underwent a major revision and 17 new genera were proposed. The Abstract should read \u201c17 new genera\u201d.In our recently published study the traditional Borealia, Occidentella, and Orientella with Borealia Maksimova, 1977, an extinct trilobite, Occidentella Hoffmann, 1929, a junior synonym of Onchidella, a pulmonate slug, and Orientella Repina & Okuneva, 1969, another extinct trilobite. In a stroke of irony the two trilobite names were published in little known Russian journals, with Russian descriptions. To avoid homonymy, throughout the text, figures, and all supplementary materials replacement names for these three genera are as follows: Borealea Korshunova et al., nom. n., Occidenthella Korshunova et al., nom. n., and Orienthella Korshunova et al., nom. n. We are grateful to Jimmy Gaudin for alerting us to this unfortunate circumstance. Other corrections are as follows:Immediately after publication our attention was drawn to homonymy of three generic names: On p. 14 \u201c\u2026the same region our\u2026\u201d should be \u201c\u2026the same region as our\u2026\u201dOn p. 16 \u201c\u2026additional rows of of small\u2026\u201d should be \u201c\u2026additional rows of small\u2026\u201dParacoryphellaignicrystalla should be \u201c\u2026rachidian tooth with up to 15 denticles...\u201dOn p. 17 In the diagnosis of On p. 18 \u201c\u2026cusp of nealy 1/3 of the tooth\u2026\u201d should be \u201c\u2026cusp of nearly 1/3 of the tooth\u2026\u201dCoryphellapolaris Voldochenko, 1946\u2026\u201d should be \u201c\u2026Coryphellapolaris Volodchenko, 1946\u2026\u201d; \u201c\u2026\u2026\u201d should be \u201c\u2026\u2026\u201d; \u201c\u2026\u2026\u201d should be \u201c\u2026\u2026\u201dOn p. 19 \u201c\u2026Coryphellaverrucosa and Itaxiafalklandica should be 17.2%.On p. 26 Table 1. In fifth column: minimum uncorrected p-distances between On p. 27 \u201c\u2026original description in Verril, 1880\u2026\u201d should be \u201c\u2026original description in Verrill, 1880\u2026\u201d; \u201c\u2026detailed redescripton in Kuzirian\u2026\u201d should be \u201c\u2026detailed redescription in Kuzirian\u2026\u201dFjordiachriskaugei should be \u201c...orange-brown to reddish brown, sometimes blackish\u2026\u201dOn p. 31 In the diagnosis of Fjordialineata should be: \u201c\u2026orange-brown to salmon, bright red, and dark brown, sometimes almost black...\u201dOn p. 33 In the diagnosis of Guleniamonicae should be \u201c...delineated from central cusp...\u201dOn p. 35 In the diagnosis of Guleniaorjani should be \u201c\u2026delineated from central cusp...\u201dOn p. 37 In the diagnosis of F.funeka and F.ishitana is not\u2026\u201d should be \u201c\u2026inclusion of F.funeka and F.ischitana is not\u2026\u201d; \u201c\u2026and \u201cPiseinotecus\u201d gabienerei must be\u2026\u201d should be \u201c\u2026and \u201cPiseinotecus\u201d gabienerei must be\u2026\u201dOn p. 52 \u201c\u2026inclusion of Paraflabellinaishitana\u2026\u201d should be \u201c\u2026related to Paraflabellinaischitana\u2026\u201dOn p. 53 \u201c\u2026related to On p. 54 \u201c\u2026 (pleuroproctic in higher acleiproctic position)...\u201d should be \u201c\u2026(pleuroproctic in higher acleioproctic position)\u2026\u201dOn p. 57 \u201c\u2026on distinct elonagate elevations\u2026\u201d should be \u201c\u2026on distinct elongate elevations...\u201d; \u201c\u2026 (pleuroproctic in higher cleiproctic position)\u2026\u201d should be \u201c\u2026 (pleuroproctic in higher acleioproctic position)\u2026\u201dU.nihonrossija should be \u201c\u2026.apical parts of cerata without white pigment... rachidian tooth with up to seven denticles, central cusp without small denticles, double proximal receptaculum seminis\u2026\u201d.On p. 58 In the diagnosis of On p. 66 \u201c\u2026due to the lack of the morphlogical description\u2026\u201d should be \u201c\u2026due to the lack of a morphological description\u2026\u201dOn p. 68 \u201c\u2026always posses oral glands\u2026\u201d should be \u201c\u2026always possess oral glands\u2026\u201dOn p. 74 \u201c\u2026reduction of the triserial radula into a unserial one\u2026\u201d should be \u201creduction of the triserial radula into a uniserial one\u2026\u201dChlamyllaborealisborealis Bergh, 1886On p. 79 Figure 3. Should be On p. 91 \u201c\u201d should be \u201c\u201dChlamyllaatypica should read Chlamyllaborealisborealis Bergh, 1886. Localities for Himatinatrophina  GQ292023 and Microchlamyllaamabilis  GQ292022 should be \u201cWA, USA\u201d.Supplementary material 2."}
{"text": "In vivo quantitative ultrasound image analysis of femoral subchondral bone in knee osteoarthritis\u201d [P value was corrected from \u201cP = 0.000\u201d to more appropriate \u201cP < 0.001.\u201d In the article titled \u201cthritis\u201d , there w"}
{"text": "Nature Communications 10.1038/s41467-018-03958-7; published online 26 April 2018Correction to: 2-rich Martian atmosphere (\u03bc\u2642\u2009=\u20091.3\u2009\u00d7\u200910\u22125\u2009N\u2009s\u2009m\u22122) at 6.9\u2009mbar and 210\u2009K, which gives \u03c1\u2642\u2009=\u20091.6\u2009\u00d7\u200910\u221212\u2009kg\u2009m\u22123.\u2019 The correct version states \u2018\u03c1\u2642\u2009=\u20091.6\u2009\u00d7\u200910\u22122\u2009kg\u2009m\u22123\u2019 in place of \u2018\u03c1\u2642\u2009=\u20091.6\u2009\u00d7\u200910\u221212\u2009kg\u2009m\u22123\u2019. This has been corrected in both the PDF and HTML versions of the Article.The original version of this Article contained an error in the last sentence of the second paragraph of the \u2018Atmospheric rarefaction effects\u2019 section of the Results, which incorrectly read \u2018The other one emulates the rarefied, CO"}
{"text": "Scientific Reports6: Article number: srep3636410.1038/srep36364; published online: 11032016; updated: 03302017P. arionides\u2019, \u201cChu-bu\u201d should read \u201cChubu\u201d.This Article contains typographical errors in Table 1. For Location E, the \u2018L1\u2019 value \u201c2 (0%)\u201d should read \u201c2 (100%)\u201d. Additionally, the Region for \u2018"}
{"text": "Additionally, the 4th line from the bottom incorrectly reads \u201c9\u67088\u65e5.\u201d It should read \u201c9/8\u201d in the 1st line from the bottom. The correct version of"}
{"text": "This paper considers a high-dimensional stochastic SEIQR (susceptible-exposed-infected-quarantined-recovered) epidemic model with quarantine-adjusted incidence and the imperfect vaccination. The main aim of this study is to investigate stochastic effects on the SEIQR epidemic model and obtain its thresholds. We first obtain the sufficient condition for extinction of the disease of the stochastic system. Then, by using the theory of Hasminskii and the Lyapunov analysis methods, we show there is a unique stationary distribution of the stochastic system and it has an ergodic property, which means the infectious disease is prevalent. This implies that the stochastic disturbance is conducive to epidemic diseases control. At last, computer numerical simulations are carried out to illustrate our theoretical results. Mathematical models for differential equations have been widely applied in various fields \u20137. Speci\u03bc is the outflow rate corresponding to natural death and emigration. Since the quarantine process, using the standard incidence \u03b2 is the transmission coefficient between susceptible individuals and infected individuals. \u03bb is the quarantine rate of infected individuals, \u03c6 is the recovery rate of quarantined individuals, and \u03b51 and \u03b52 stand for the rate of disease-related death of infected and quarantined individuals, respectively. \u03b3 is the recovery rate of infected individuals. Furthermore, all the parameters are positive and the region R0, which determines disease extinction or permanence, whereIn fact, the main meaning of the research of infectious disease dynamics is to make people more comprehensively and deeply understand the epidemic regularity of infectious disease; then more effective control strategies are adopted to provide better theoretical support for the prevention and control of epidemics. To this end, many mathematical biology workers considered more realistic factors in the course of the study, such as population size change, migration, cross infection, and other practical factors. In the course of epidemics and outbreaks of infectious diseases, people always take various measures to control the epidemic in order to minimize the harm of epidemic diseases. Quarantine is one of the important means to prevent and control epidemic diseases; it has been used to control contagious diseases with some success. Specifically, during the severe acute respiratory syndrome (SARS) outbreak in 2002, remarkable results were also achieved. Among them, mathematical models have been used to investigate their impact on the dynamics of infectious diseases under quarantine \u201322, whicS\u02d9=\u039b\u2212\u03b2SINS\u02d9=\u039b\u2212\u03b2SINN(t) = S(t) + E(t) + I(t) + Q(t) + R(t), \u03b4\u2009\u2009(0 \u2264 \u03b4 < 1) is the vaccine coverage rate, p\u2009\u2009(0 \u2264 p \u2264 1) is the vaccine efficacy, and \u03b1 is the rate at which the exposed individuals become infected individuals. Other parameters are the same as in system \u2223S \u2265 0, E \u2265 0, I \u2265 0, Q \u2265 0, R \u2265 0, S + E + I + Q + R \u2264 \u039b/\u03bc} is a positively invariant set of system  = (\u039b/(\u03b4p + \u03bc), 0,0, 0, \u03b4p\u039b/\u03bc(\u03b4p + \u03bc)) which is globally asymptotically stable in the region D. That means the epidemic diseases will die out and the total individuals will become the susceptible and recovered individuals.When , system  has a unR1 > 1 holds, system  in the region D, which means the epidemic diseases will persist.When , system  has a unIn the real world, with the development of modern medicine, vaccination has become an important strategy for disease prevention and control in addition to quarantine, and numerous scholars have investigated the effect of vaccination on disease \u201330. For S(t), E(t), I(t), Q(t), and R(t). Then corresponding to system (Bi(t)\u2009\u2009 is the mutually independent standard Wiener process with Bi(0) = 0 a.s. \u03c3i(t)\u2009\u2009 is a continuous and bounded function for any t \u2265 0 and \u03c3i2(t)\u2009\u2009 are the intensities of Wiener processes.In the natural world, deterministic model is not enough to describe the species activities. Sometimes, the species activities may be disturbed by uncertain environmental noises. Consequently, some parameters should be stochastic \u201340. Thero system , a stoch(P1) Under what parameter conditions, will the disease die out?(P2) Under what conditions, will system  have a uIn this paper, we are mainly concerned with two interesting problems as follows:\u03a9, \u2131, {\u2131}t\u22650, \u2119) be a complete probability space with a filtration {\u2131t}t\u22650 satisfying the usual conditions . Further Bi(t)\u2009\u2009 is defined on the complete probability space.Throughout this paper, let , \u211d+5 = {x =  \u2208 \u211d5 : xi > 0\u2009\u2009}.x(t)\u2208. For each integer k \u2265 k0, let us define the stopping time as follows:\u2205 = \u221e (\u2205 represents the empty set). Evidently, \u03c4k is strictly increasing when k \u2192 \u221e. Let \u03c4\u221e = limk\u2192\u221e\u03c4k; thus \u03c4\u221e \u2264 \u03c4e a.s. So we just need to show that \u03c4\u221e = \u221e a.s. If \u03c4\u221e = \u221e is untrue, then there exist two constants T > 0 and \u03c2 \u2208  such that \u2119{\u03c4\u221e \u2264 T} > \u03c2. Thus there exists k1 \u2265 k0\u2009\u2009(k1 \u2208 N+) such thatC2-function M0 is a positive constant. Hence\u03c4k\u2227T and then taking the expectation, we have\u03a9k = {\u03c4k \u2264 T}, k \u2265 k1 and by (P(\u03a9k) \u2265 \u03c2. Notice that, for every \u03c9 \u2208 \u03a9k, there exists S, E, I, Q, or R which equals either 1/k or k. Thus\u03a9k(\u03c9) is the indicator function of \u03a9k(\u03c9). Let k \u2192 \u221e, which implies \u03c4\u221e = \u221e. This completes the proof of nt. HencedV^\u2264M0dt+such thatP\u03c4k\u2264T\u2265\u03c2,kr k. ThusV^S\u03c4k,\u03c9,EIn this section, we mainly explore the parameter conditions for extinction of the disease in system . Before S(0), E(0), I(0), Q(0), R(0)) \u2208 \u211d+5, the solution (S(t), E(t), I(t), Q(t), R(t)) of the system (\u03bc > (1/2)(\u03c312\u2228\u03c322\u2228\u03c332\u2228\u03c342\u2228\u03c352) holds, thenFor any given initial value (The proof of \u03bc > (1/2)(\u03c312\u2228\u03c322\u2228\u03c332\u2228\u03c342\u2228\u03c352). For any given initial value (S(0), E(0), I(0), Q(0), R(0)) \u2208 \u211d+5, ifLet V0 byt and dividing by t on both sides of  as a time-homogeneous Markov process in \ud835\udd3cn \u2282 \u211dn, which is described by the stochastic differential equation\ud835\udd3cn stands for a n-dimensional Euclidean space. The diffusion matrix takes the following form:Assume U \u2282 \ud835\udd3cn with regular boundary \u0393 such that U and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix In the domain x \u2208 \ud835\udd3cn\u2216U, the mean time \u03c4 at which a path issuing from x reaches the set U is finite, and supx\u2208\u0398\ud835\udd3cx\u03c4 < \u221e for every compact subset \u0398 \u2282 \ud835\udd3cn.If Assume that there is a bounded domain X(t) has a stationary distribution \u03c0(\u00b7). Furthermore, when f(\u00b7) is a function integrable with respect to the measure \u03c0, then x \u2208 \ud835\udd3cn.When F is uniformly elliptical in any bounded domain H; hereZ such thatU and a nonnegative C2-function V such that \u2200x \u2208 \ud835\udd3cn\u2216U, LV(x) < 0.To demonstrate Using S(0), E(0), I(0), Q(0), R(0)) \u2208 \u211d+5. If\u03c0(\u00b7) and it has ergodic property.For any given initial value  are positive constants satisfying the following conditions: \u03a5 > 0 large enough such that Uw =  \u00d7  \u00d7  \u00d7  \u00d7 . Since \u211d+5 which is the minimum point of C2-function V: \u211d+5 \u2192 \u211d+ by M = \u039b + \u0393 + \u03b4p + \u03b1 + \u03c6 + \u03b52 + 4\u03bc + \u03c312/2 + \u03c322/2 + \u03c342/2 + \u03c352/2.Define a D: \u03f5 is a sufficiently small constant satisfying the following conditions:\u2112V for any  \u2208 \u211d+5\u2216D.Next let us consider the following compact subset Case I. If  \u2208 D1, (R) \u2208 D1,  and 48)Case I. ICase II. If  \u2208 D2, (R) \u2208 D2,  and 49)Case II. Case III. If  \u2208 D3, it follows from (ows from  and 50)Case III.Case IV. If  \u2208 D4, (R) \u2208 D4,  and 51)Case IV. Case V. If  \u2208 D5, it follows from (ows from  and 52)Case V. ICase VI. If  \u2208 D6, (R) \u2208 D6,  and 53)Case VI. Case VII. If  \u2208 D7, (R) \u2208 D7,  and 54)Case VII.Case VIII. If  \u2208 D8, it follows from (ows from  and 55)Case VIIICase IX. If  \u2208 D9, (R) \u2208 D9,  and 56)Case IX. Case X. If  \u2208 D10, it follows from  and it has ergodic property. The proof of ows from  and 54)Case X. IR\u2217 > 1 holds, then system , the expression of R\u2217 coincides with the basic reproduction number R1 of system (\u03c312\u2228\u03c322\u2228\u03c332\u2228\u03c342\u2228\u03c352) and When R\u2217 = (\u03bc(1 \u2212 \u03b4p)\u03b2\u03b1)/((\u03b4p + \u03bc + \u03c312/2)(\u03b1 + \u03bc + \u03c322/2)(\u03bb + \u03b51 + \u03b3 + \u03bc + \u03c332/2)) > 1 holds, then system (\u03c0(\u00b7) and it has ergodic property.When n system  has a unThis paper studies the stochastic SEIQR epidemic model with quarantine-adjusted incidence and imperfect vaccination and obtains two thresholds which govern the extinction and the spread of the epidemic disease. Firstly, the existence of a unique positive solution of system  with any\u03b6i,k\u2009\u2009 stands for N distributed independent random variables and time increment \u0394t > 0.To illustrate the results of the above theorems, we next carry out some numerical simulations by the Matlab software. Let us consider the following discretization equations of system :(73)Sk+1S(0) = E(0) = I(0) = Q(0) = R(0) = 0.3, \u039b = 2, \u03bc = 0.55, \u03b2 = 0.25, \u03b3 = 0.2, \u03b1 = 2.5, \u03b4 = 0.5, p = 0.5, \u03b51 = 0.1, \u03b52 = 0.1, \u03bb = 0.18, \u03c6 = 0.15, \u03c31 = 0.15, \u03c32 = 1, \u03c33 = 1, \u03c34 = 0.5, \u03c35 = 0.25, and \u0394t = 0.01. ThenIn S(0) = E(0) = I(0) = Q(0) = R(0) = 0.5, \u039b = 0.3, \u03bc = 0.1, \u03b2 = 1.5, \u03b3 = 0.2, \u03b1 = 0.3, \u03b4 = 0.1, p = 0.2, \u03b51 = 0.05, \u03b52 = 0.05, \u03bb = 0.18, \u03c6 = 0.15, \u03c31 = \u03c32 = \u03c33 = \u03c34 = \u03c35 = 0.03, and \u0394t = 0.01. Then \u03c0(\u00b7) and it has ergodic property. In S(0) = E(0) = I(0) = Q(0) = R(0) = 0.1, \u039b = 0.2, \u03bc = 0.55, \u03b2 = 2.65, \u03b3 = 0.2, \u03b1 = 2.5, \u03b4 = 0.1, p = 0.1, \u03b51 = 0.1, \u03b52 = 0.1, \u03bb = 0.18, \u03c6 = 0.15, and \u0394t = 0.01.In \u03c3i = 0\u2009\u2009; then In \u03c31 = 0.15, \u03c32 = 1, \u03c33 = 1, \u03c34 = 0.5, \u03c35 = 0.25. Obviously, Synchronously, in"}
{"text": "This study addresses the question of whether native Mandarin Chinese speakers process and comprehend subject-extracted relative clauses (SRC) more readily than object-extracted relative clauses (ORC) in Mandarin Chinese. Presently, this has been a hotly debated issue, with various studies producing contrasting results. Using two eye-tracking experiments with ambiguous and unambiguous RCs, this study shows that both ORCs and SRCs have different processing requirements depending on the locus and time course during reading. The results reveal that ORC reading was possibly facilitated by linear/temporal integration and canonicity. On the other hand, similarity-based interference made ORCs more difficult, and expectation-based processing was more prominent for unambiguous ORCs. Overall, RC processing in Mandarin should not be broken down to a single ORC (dis)advantage, but understood as multiple interdependent factors influencing whether ORCs are either more difficult or easier to parse depending on the task and context at hand. When comparing sentence processing strategies between languages, several cross-linguistic differences have been observed, making it unclear whether strategies differ across languages. Occasionally, competing models make dichotomous predictions for a certain language, for example, when processing relative clauses in Mandarin Chinese (henceforth \u201cMandarin\u201d). In Mandarin, past studies are divided on their support for different contending models. In the present study, we employ eye-tracking to empirically investigate several relative clause processing models within different contexts to explore their interrelationships. First, we briefly introduce the topic of relative clauses, followed by several available processing accounts, and then discuss how these models might function in Mandarin.Example of a Mandarin Chinese relative clause: SRC / ORCtiZh\u01d0z\u00e9 Sh\u00eczh\u01ceng / Sh\u00eczh\u01ceng Zh\u01d0z\u00e9 ti\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0De] J\u00eczh\u011bi\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0C\u01ceif\u01ceng-Le\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0LiN\u00e0[ti criticize mayor / mayor criticize\u00a0\u00a0\u00a0\u00a0ti REL] reporteri interview-ASP Li Na[i [who ti criticized the mayor/the mayor criticized ti] interviewed Li Na.\u2019\u2018The reporterwh-movement and filler-gap dependencies . Furthermore, the gender of the nouns was controlled such that male and female names were distributed equally. Animacy has been a well-known issue for RC processing in Mandarin with ORCs having the preference of having an inanimate head and animate RC noun whereas SRCs are preferred to have an inanimate RC noun and animate head. Within animate-animate contexts, however, subject-modified SRCs appeared to be more frequent in comparison to subject-modified ORCs  Li\u00e1ngYu\u00e1niG\u0101ngc\u00e1i\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Ch\u00edd\u00e0o Le[ti\u00a0\u00a0invited Li Fang / Li Fang invited ti REL] Liang Yuan\u00a0\u00a0just.now\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0late ASP[\u2018Liang Yuan [who invited Li Fang / Li Fang invited] was late just now.\u2019Example of Implausible RC DistractorZh\u0101ng W\u011bi Ch\u012b Le De] Li Qi\u00e1ng Y\u01d0j\u012bng Hu\u00ed Ji\u0101 Le[[Zhang Wei ate REL] Li Qiang already returned\u2018Li Qiang who Zhang Wei ate already went home.\u2019Experiment 1 involved exposing participants to two different tasks. Each task was done in a separate session and half of the participants first took one task before the other. In both tasks, items were counterbalanced such that no participant would see the same item twice within a single task, nor would they see an identical item between tasks. Stimulus sentences were displayed horizontally on the centre left of a 17-inch Mitsubishi LCD monitor at a distance of 70 cm from the head and chin rest mount. All characters were displayed in Chinese MingLiU 30pt. At this distance, each character subtended a visual angle of 2.5\u00b0. Eye-movements were recorded using an EyeLink 1000 Core System. Prior to the experiment, participants were instructed in Mandarin that they would be reading Mandarin sentences displayed one at a time on a computer monitor, and were given the opportunity to ask questions about the procedure. Prior to each session, the camera was calibrated by a 9-point calibration method and subsequent validation. Calibration was periodically repeated throughout each session after block sessions (eight items).For the plausibility task, participants were instructed in Mandarin to read each sentence naturally and judge if the sentence meaning was plausible, that is, if the actions or ideas depicted would be able to exist in a real world, everyday setting. If the sentence meaning was plausible, they were instructed to press a button on a gamepad labelled \u201cTrue\u201d; conversely, if the sentence meaning was not plausible, they were instructed to press a button labelled \u201cFalse\u201d. Participants were instructed to read and judge each sentence within eight seconds. After pressing either button, the stimulus was immediately removed from the screen. Reading times were measured from the onset of the stimulus to the button press event. Eight practice trials were given to ensure participants understood the task.Li invite Liang?). Reading times were measured from the onset of the stimuli to this button press event. For the question, participants had up to eight seconds to answer. When answering, participants were instructed to press the \u201cTrue\u201d button for correct or true probes or the \u201cFalse\u201d button for incorrect or false probes. Eight practice trials were given to ensure participants understood the task. For both tasks, since reading times were measured from the onset of stimuli until the button response events  reading times and eye-movements are comparable between the two tasks.For the verification task, only minor changes were made to the procedure. Participants were instructed to read each sentence naturally and that after reading the sentence a comprehension question would appear. Again, participants were asked to read each sentence within eight seconds. When they were finished reading the sentence, participants were instructed to press a button that would replace the sentence with a comprehension/verification question , and go-past time, the combined RT for an interest region  before it is exited to the right  for the first time including any regressive readings out of the region to the left . Go-past times are thus greater than or equal to first-pass times for a region. Regression-in and regression\u2013out  proportion measures, the total reading time of the sentence and accuracy are also reported. While accuracy for the plausibility task denoted whether the participant accurately judged the experimental RCs as plausible, accuracy for the verification task indicated whether the participant accurately judged the probe to be true/correct or false/incorrect. The interest regions for analyses were the sentence, the RC structure , the relativizer (DE), the head noun (N2), the adverb (ADV), and the matrix verb (V2). Prior to the analyses, eye-fixations were first treated. Fixations below 80 ms were merged into a neighbouring fixation, and the remaining fixations under 80 ms and those exceeding 1000 ms were removed (523 fixations or 2.91%). Refer to The earliest reading time measure reported here is lme4 package (ti beat\u00a0\u00a0\u00a0\u00a0waiter\u00a0\u00a0\u00a0\u00a0/\u00a0\u00a0\u00a0\u00a0waiter beat\u00a0\u00a0\u00a0\u00a0ti\u00a0\u00a0more.than.once REL]that.one [last.night N2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0V2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0N3\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0G\u00f9k\u00e8i\u00a0\u00a0\u00a0\u00a0T\u012bngshu\u014dgu\u00f2\u00a0\u00a0\u00a0\u00a0L\u01ceob\u01cen\u00a0\u00a0\u00a0\u00a0B\u00ecngqi\u011b\u00a0\u00a0\u00a0\u00a0J\u00ecd\u00e9\u00a0\u00a0\u00a0\u00a0T\u0101\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0customeri\u00a0\u00a0heard.of\u00a0\u00a0\u00a0\u00a0boss\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0and\u00a0\u00a0\u00a0\u00a0remember him\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2018(The) customer [who beat up the waiter / the waiter beat up last night] has heard of the boss and remembers him.\u2019The procedure was similar to Experiment 1. All characters were displayed in simplified Chinese SimSun 22pt font, a visual angle of 1.8\u00b0. The font and size were changed to better fit the longer stimuli used in Experiment 2. Here, participants now had a maximum of 12 seconds to read the sentence and press any button when they were finished reading to replace the sentence with the question. Participants still had a maximum time of eight seconds to answer the verification/comprehension probe. The increase in allotted time also accommodated for the increased length of the items.Eye-fixations were treated following the same procedure as Experiment 1 which resulted in the removal of 1,963 fixations or 7.34%. The same LME methods were used as in Experiment 1. RC condition (ORC vs. SRC) and determiner type (Empty vs. DCL) were considered as fixed effects, and subject and item composed the random effects. If interaction of condition:type was significant, a pairwise analysis was conducted. Data trimming for each model resulted in the removal of 1.68% of the data. Refer to Tables D-H for means and standard errors and LME results within Accuracy. The analysis on the accuracy for the verification probes revealed no significant differences for RC condition (p = .516), determiner type (p = .920) or condition:type interaction (p = .531). The mean scores were rather close between items.Total reading time of the sentence. For the reading of the sentence, while both RC condition (p < .001) and determiner type (p < .01) were significant, interaction was not (p = .800). The general pattern of results revealed that the ORC condition had longer RTs than the SRC condition and that the DCL type had longer RTs compared to the Empty type.First-pass RT. For the RC condition (p = .068), even though ORCs were read quicker than SRCs, the result was not significant. For determiner type (p < .001), the Empty type had significantly longer RTs than the DCL type. Interaction was not significant (p = .428).Re-reading Time. In contrast to first-pass RT, ORC re-reading time was significantly longer than SRC re-reading time at this later stage of processing (p < .001). For determiner type, while the Empty type had longer RTs in comparison to the DCL type, the difference did not reach significance (p = .063). There was still no effect of interaction (p = .504).Go-past Time. While there was no significant difference between RC conditions (p = .129), there was a significant difference in determiner type (p < .05) showing unsurprisingly that the DCL type had longer RTs than the Empty type since the DCL type items had one additional region compared with the Empty type items. During this stage, there was a significant effect of interaction (p < .001). The pairwise comparison revealed that ORC:DCL had significantly longer RTs than SRC:DCL (p < .001). While ORC:Empty had the lowest RTs, it was not significantly faster than SRC:Empty in the pairwise analysis.Regression-out. The RC condition (p < .01) and determiner type (p < .01) revealed that ORCs were more likely to have a regression out than SRCs, and the DCL type was more likely than the Empty type. Again, there was a significant effect interaction showing that ORC:DCL was more likely to regress out than SRC:DCL (p < .001). Consequently, it appears that ORC:DCL was driving the effects for this measure.Regression-in. While RC condition (p = .299) was not significant, determiner type (p < .001) demonstrated that the Empty type was more likely to have a regression made back into the RC in comparison to the DCL type. Interaction was not significant (p = .141).First-pass RT. At the first-pass reading of the frequency phrase, there were no differences between RC conditions (p = .179), but determiner type (p < .05) demonstrated that the Empty type had longer RTs compared to the DCL type. Interaction was not significant (p = .075).Re-reading Time. RC condition (p = .146) and interaction (p = .368) did not show significant differences during re-reading. Again, determiner type (p < .05) revealed that the Empty type had significantly longer RTs compared to the DCL type sentences.Go-past Time. RC condition (p = .844) was still not significant during go-past time, while determiner type (p < .05) still demonstrated that the Empty type had longer RTs compared to the DCL type. Interaction of condition:type was significant (p < .05). However, this only demonstrated that ORC:Empty had significantly longer go-past RTs than ORC:DCL (p < .01).Regression-out. RC condition (p = .514), determiner type (p = .680) and interaction (p = .540) revealed no significant differences.Regression-in. RC condition (p = .131) was not significant, but determiner type (p < .05) revealed that the Empty type was more likely to have a regression back into the frequency phrase than the DCL type. There was a significant effect for interaction (p < .01), demonstrating that ORC:DCL was less likely to have a regression back into the phrase than SRC:DCL (p < .05).First-pass RT. For the RC condition (p < .05), it was shown that ORCs had significantly longer RTs than SRCs. Neither determiner type (p = .554) nor interaction (p = .415) at the relativizer were significant.Re-reading Time. In later re-reading times, the RC condition (p = .543) was no longer significant. However, determiner type (p < .01) indicated that the Empty type had longer RTs than the DCL type. Interaction was not significant (p = .311).Go-past Time. Only the RC condition (p < .05) revealed a significant difference in RTs, showing that ORCs as a whole had longer RTs in comparison to SRCs. There was no significance for determiner type (p = .103) and interaction (p = .640).Regression-out. The RC condition (p = .230), determiner type (p = .791) and interaction (p = .617) did not reveal any significant differences.Regression-in. For the RC condition (p < .01), ORCs were significantly more likely to have a regression back into the relativizer than SRCs (p < .01). However, determiner type (p = .670) was not significant. While there was a significant interaction effect found (p < .05), it only indicated that ORC:Empty was more likely to have a regression back into the relativizer than SRC:Empty (p < .01), despite both ORCs having higher regression-in means than their SRC counterparts.First-pass RT. RC condition (p = .497), determiner type (p = .578) and interaction (p = .778) revealed no significant differences during first-pass reading.Re-reading Time. For fixations made after first-pass, RC condition (p < .05) demonstrated that ORCs had longer RTs than SRCs, and determiner type (p < .05) revealed that the Empty type had longer RTs compared to DCL type items. Interaction did not show significant differences (p = .806).Go-past Time. While the RC condition (p = .692) and interaction (p = .340) were not significant, the determiner type (p < .05) showed that the Empty type items required longer RTs before moving on to the matrix clause verb.Regression-out. The RC condition (p < .05) revealed that the ORC condition was significantly more likely to make a regression out of the head noun back into previous parts of the sentence in comparison to SRCs. Determiner type (p = .624) was not significant. However, interaction (p < .05) was significant and demonstrated that ORC:Empty was significantly more likely to make a regression out of the head than SRC:Empty (p < .05).Regression-in. Only the RC condition (p < .05) was significant showing that ORCs were more likely to have a regression back into the head from later parts of the matrix clause. Determiner type (p = .911) and interaction (p = .938) were not significant.First-pass RT. RC condition (p = .955), determiner type (p = .302) and interaction (p = .728) revealed no significant differences during first-pass reading.Re-reading Time. For the RC condition (p < .01), ORCs had significantly longer RTs than SRCs, whereas determiner type (p = .440) was not significant. While interaction (p < .05) was significant, the pairwise analysis revealed that ORC:Empty only had significantly longer RTs than SRC:Empty (p < .01).Go-past Time. While RC condition (p = .209) and interaction (p = .967) were not significant, determiner type (p < .01) indicated that the Empty type sentences had significantly longer RTs in comparison to DCL sentences.Regression-out. RC condition (p = .414) and interaction (p = .921) were not significant; determiner type (p = .061) also revealed no significance even though the Empty sentences had a higher likelihood to regress out than DCL sentences.Regression-in. RC condition (p = .567), determiner type (p = .377) and interaction (p = .748) revealed no significant effects.First-pass RT. For the RC condition (p = .058), no significant differences were found. Also, neither determiner type (p = .136) nor interaction (p = .417) indicated significant differences during first-pass reading.Re-reading Time. RC condition (p = .643), determiner type (p = .997) and interaction (p = .863) revealed no significant effects.Go-past Time. RC condition (p = .562), determiner type (p = .921) and interaction (p = .377) revealed no significant effects.Regression-out. RC condition (p = .405), determiner type (p = .264) and interaction (p = .295) revealed no significant effects.Regression-in. RC condition (p = .912), determiner type (p = .537) and interaction (p = .886) revealed no significant effects.Next, we present the additional analyses as described above. Refer to Tables G and H for means, standard errors and LME results in First-pass RT. There was a significant effect for the RC condition (p < .01) showing that ORCs were read faster than SRCs. Determiner type (p < .001) was also significant and revealed that the Empty type sentences had longer RTs during first-pass reading compared to DCL sentences. Interaction (p = .069), however, did not reach the significance threshold.Re-reading Time. The RC condition (p < .001) and determiner type (p < .05) were both significant which demonstrated that ORC conditions had significantly longer RTs than SRCs and the Empty type items had significantly longer RTs compared to DCL items. Interaction (p = .993) was not significant.Go-past Time. The RC condition (p = .069) did not reveal a significant difference between ORCs and SRCs. Determiner type (p = .151) was also not significant. Interaction (p < .001) of condition:type was significant and demonstrated contrasting effects for the ORC types. This interaction showed that the ORC:DCL condition had significantly longer RTs than SRC:DCL (p < .001). On the other hand, it was revealed that the ORC:Empty condition had significantly faster RTs than the SRC:Empty (p < .05) condition.Regression-out. The RC condition (p < .001), determiner type (p < .001) and interaction (p < .001) were all significant. It was shown that the ORCs conditions and DCL types were significantly more likely to regress out than their counterparts. However, the pairwise analysis indicated that it was only the ORC:DCL condition which was significantly more likely to regress out of the RC structure than SRC:DCL (p < .001).Regression-in. While the RC condition (p = .097) was unable to reveal significant differences between conditions, determiner type (p < .001) and interaction (p < .01) were both significant. While the Empty type was significantly more likely to have a regression back into the RC structure, the pairwise analysis revealed that, opposite to regression-out, it was only ORC:Empty which was more likely to have a regression made back into the RC structure in comparison to SRC:Empty (p < .01).First-pass RT. While the RC condition (p = .640) and determiner type (p = .216) were not significant, interaction (p < .01) was significant. However, from the pairwise analysis, it was only revealed that ORC:Empty was significantly faster than SRC:Empty (p < .05).Re-reading Time. The RC condition (p < .001) revealed that ORCs had significantly longer re-reading times than SRCs, and determiner type (p < .01) showed that the Empty sentences had significantly longer RTs compared to DCL sentences. Interaction (p = .271) was not significant.Go-past Time. The RC condition (p < .01), determiner type (p < .001) and interaction (p < .01) were all significant. As with re-reading time, ORCs had significantly longer RTs compared to SRCs, and the Empty sentences had significantly longer RTs in comparison to their DCL counterparts. In contrast to first-pass time, the pairwise analysis showed that ORC:Empty now had significantly longer go-past times in comparison to SRC:Empty (p < .001).Regression-out. For the RC condition (p = .087), ORCs only had a trending likelihood of regressing out of the matrix clause in comparison with SRCs. However, determiner type (p < .01) revealed that the Empty type items were significantly more likely to regress out than DCL type items. Interaction (p < .01) was significant, and similar to go-past time, the pairwise analysis indicated that ORC:Empty was significantly more likely to have a regression out of the matrix clause in comparison to SRC:Empty (p < .001).Regression-in. RC condition (p = .661), determiner type (p = .088) and interaction (p = .942) revealed no significant differences between conditions and types.In contrast to Experiment 1, Experiment 2 clearly showed that ORCs were more difficult to process than SRCs. Nonetheless, the results also indicated that multiple processing factors were involved in the processing of Mandarin RCs revealing both ORC advantages and disadvantages: Canonicity (ORC facilitation), expectation (ORC disadvantage), and perhaps similarity interference (ORC disadvantage) as well.While integration resources were not directly supported in Experiment 2, evidence of canonicity was nevertheless present for both unambiguous and ambiguous ORC items during early RTs within both RC regions. Additionally, J\u00e4ger et al. (refer to Table 13 in ) also apIn addition to canonicity effects, the initial benefit for ORCs may also loosely provide indirect support for linear/temporal metrics of integration. However, integration was not directly supported at the relativizer or head noun which we attribute to antilocality effects. In other words, with the introduction of syntactic cues , there would be greater expectation or anticipation  for the For expectation-based effects, the general pattern of results observed in J\u00e4ger et al.  was replSimilarity-based interference was again hinted at by the indication of ORC difficulty at the matrix clause. Since J\u00e4ger et al.  also fouIn summary, while canonicity facilitated ORCs early on with indirect support for linear/temporal integration, the influences of expectation-based processing later reversed this within the RC. At the matrix clause, similarity-based interference was also observed to be a potential factor responsible for increasing ORC difficulty. In all, the reading of these sentences was seen to be influenced by multiple factors of processing.In this study, we sought out to determine which Mandarin relative clause structures are more demanding to process. We investigated the reading of ambiguous RCs as well as unambiguous RCs using eye-tracking. More specifically, we aimed to determine how the initial clause type ambiguity and processing factors such as canonicity, expectation, integration and similarity-based interference influence the reading of Mandarin sentences containing RCs. The results of Experiment 1 revealed that ambiguous ORCs were generally easier to process than SRCs, regardless of task design supporting canonicity, expectation, storage and integration-based effects. Yet, in the long run, ORCs became more difficult to process at the matrix clause, a result which may provide support for similarity-based interference as well as accounts on animacy preferences in Mandarin RC processing. The results of Experiment 2 revealed that canonicity and possibly locality facilitated the early readings of the ORC within the relative clause. Also, ambiguous ORCs remained easier to process compared to SRCs longer than unambiguous ORCs. ORCs were still more difficult during later RTs within the RC and matrix clause as explained by expectation-based processing and similarity interference. Experiment 2, however, did not provide direct evidence supporting linear/temporal integration-based models at the relativizer or head noun. This was possibly due to antilocality effects or due to the inclusion of the frequency phrase in items used in Experiment 2, given the irregular position of the phrase for ORCs.One particular framework of processing and cognitive behaviour can support the findings of this study, that is, Lewis and Vasishth\u2019s ,38 activWe view canonicity as a top-down mechanism based upon a coarsely-tuned account of a language\u2019s structural or thematic regularities. While expectation and anticipatory effects may be more dependent on fine-tuned structural and collocational frequencies, canonicity can influence processing even for less frequent structures based solely on regularities of the language. This interpretation would therefore differ from and supplement previous notions of canonicity which have been based upon both statistical frequency and regularity . We findIn contrast with canonicity, as syntactic cues which helped give an RC interpretation were introduced into the sentence , anticipatory processes greatly influenced the processing for the more frequent SRC structure. This caused SRCs to be processed more easily than ORCs at the relativizer and during later reading times for the RC  and RC structure  in Experiment 2. However, we understand this greater expectation or anticipation for the SRC structure to be an antilocality effect. We believe this effect could have possibly prevented the observation of a linear or temporally defined integration metric at the relativizer and head noun. As mentioned above, locality is a constraint on the reactivation of an item from memory. In general, after the initial activation of an item, the activation level will begin to decay, and the more distant a gap is to its filler, the greater the decay will be. Since ORCs would have less activation decay due to the gap and filler being more local defined by either a linear or temporal metric, ORCs should be easier to process when integrating filler-gap dependencies. This was clearly supported by the results of Experiment 1. Experiment 2, on the other hand, only was able to support effects of locality beyond the scope of the specific loci of integration in Mandarin Chinese. If we consider that locality does influence processing, then the fact that the results of Experiment 2 conflict with ORC locality is best explained by antilocality effects, rather than a structural-phrase integration metric. Lastly, there was partial evidence supporting a similarity-based interference when the matrix verb needed to retrieve its subject  from memory. This was indicated by the ORC difficulty found at the matrix clause for both experiments and all ORC types. We believe that similarity-based interference provides the most suitable explanation for the ORC difficulty here. The difficulty for ORCs at the matrix clause verb can be explained by the proactive interference of the ORC relative clause subject on the activation level of the matrix clause subject. On the other hand, the SRC relative clause object should not lower the activation level of the matrix clause subject. Thus, during the retrieval of the subject at the matrix verb, ORCs should have greater processing difficulty compared to SRCs.In summary, the results seem to be compatible with activation-based constraints on processing showing multiple influences on sentence processing. In the current study, we limited these to more global interpretations on sentence processing; as such, see Vasishth and Lewis ,38 and cThe current study is not without limitation and there are several issues left to be addressed. Both experiments potentially involved issues since animacy, passivation and object-modification were not addressed as independent factors. Consequently, the current study is somewhat limited in its overall interpretability. One issue, for example, is that the current study cannot dissociate semantics and syntax for canonical order effects. Yet, considering that ORCs are preferred to include the passive marker, the thematic canonicity of agent-to-patient may admittedly have a greater influence on processing compared to grammatical SVO word order.In the current study, while subject/object asymmetry was only investigated for RC processing in Mandarin, subject biases have also been observed within other structures as well. For instance, Simpson, Wu, and Li  using a Concerning canonical order facilitation, while the current study found clear benefits of canonicity at the RC structure for both experiments, it is still unclear what role statistical frequency can be attributed to for the items with attenuated clause type ambiguity. Hsiao and MacDonald  found thA notable issue of this study was that the frequency phrase in Experiment 2 still acted as a syntactic cue to help attenuate ambiguity. Thus, the items lacking the Det+Cl were still less ambiguous than the items of Experiment 1. Furthermore, the position of the frequency phrase is unnatural for the ORC condition. Consequently, the difficulty found at the relative clause or head noun for ORCs during later RTs may be attributed in some part to the unnaturalness of the frequency phrase for ORCs. Since the phrase is not in a canonical position for ORCs, it may also be the case that semantics rather than word order may have been facilitating ORCs during early RTs at the full RC region. Future studies using eye-tracking should further address the issue of semantics and also address the frequency phrase as experimental factors to determine its influence inside the RC and at head noun.In a similar vein, since the Det+Cl can either appear prior or after the RC, it may be best to compare such a design to determine the influence of modification position on the processing of the head noun using eye-tracking. In fact, previous research  has alreAn additional issue was that object-modified RCs were not addressed in this study. Considering that in situ object-modified RCs are not preferred, we believe future studies should follow Lin and Garnsey  and invet = 5.14, p < .001], neither item modification  nor the interaction of the two  were significantly different. It was found for both the modified and unmodified items, SRCs  were rated significantly higher than ORCs . In J\u00e4ger et al.  argued for an overall SRC advantage, they allowed that random variability may possibly contribute to this appropriate inconsistency to some extent. If there is random variability, then the possible contributing factors should be determined. It is possible that differences in experimental items, the method or the number of cues to attenuate temporary ambiguity, the experimental methodology , and participant-pools  may all contribute to the random variability. For instance, there have been many studies using Gibson and Wu\u2019s (27) items and unambiguous design ,20,28, bAlthough the previous studies and the current study used native Mandarin speakers (with the majority recruiting participants originating from either Mainland China or Taiwan), there is still variability among regional dialects of Mandarin. For instance, even among similar P\u01d4t\u014dnghu\u00e0 and Gu\u00f3y\u01d4 standard dialects of Mandarin , there are differences in grammar, phonology, and vocabulary. As such, it may be of empirical interest for future studies to assess the influence of dialect.In an effort to further previous eye-tracking studies that used either ambiguous relative clauses in Mandarin or syntactic cues to attenuate ambiguity, the current study shows that canonicity and linear/ temporal-based integrations metrics support an ORC advantage. However, these effects are more prominent when the structure of the RC is initially ambiguous. As such, we also show that as additional syntactic cues are given, the more likely, quickly and severely the expectations generated from the structural frequency will impact the processing of object-relative clauses. We view this as an antilocality effect. In addition, we also show evidence for a similarity-based interference within the matrix clause regardless of ambiguity. We argue along the lines of Vasishth and Lewis ,38 that At Nagoya University, Japan ethic committees are operated separately from the main institution within each graduate school; however, not all graduate schools have a committee. Since the Graduate School of Languages and Cultures at Nagoya University, Japan did not have an ethics committee at the time of the study, approval from such a committee could not be obtained. Instead, this research was approved by the faculty of the Graduate School of Languages and Cultures at Nagoya University, Japan which adheres to the Declaration of Helsinki for research using human subjects. In the current study, all participants first signed the informed written consent form prior to participating in the study and received monetary compensation at the end of their session. All personal information collected from participants was stored in a secured location, and participants were given pseudonyms for data analysis purposes. Participants were not subject to harm and could only experience mild discomfort from prolong seating and reading. Lastly, we declare that each of the participating authors did not have any conflict of interest during the completion of the study.The following sentences are all the ORC experimental items from Experiment 1. The SRC sentence condition is given only for the first item. The interest regions are designated between asterisk marks .1ORC: *\u674e\u82b3\u9080\u8bf7*\u7684*\u6881\u5a9b*\u521a\u624d*\u8fdf\u5230\u4e86*Li F\u0101ng y\u0101oq\u01d0ng de Li\u00e1ng Yu\u00e1n g\u0101ngc\u00e1i ch\u00edd\u00e0oleLi Fang invited REL Liang Yuan just late\u2018LiangYuan who LiFang invited was late just now.\u20191SRC: *\u9080\u8bf7\u674e\u82b3*\u7684*\u6881\u5a9b*\u521a\u624d*\u8fdf\u5230\u4e86*y\u0101oq\u01d0ng Li F\u0101ng de Li\u00e1ng Yu\u00e1n g\u0101ngc\u00e1i ch\u00edd\u00e0oleinvited Li Fang REL Liang Yuan just late\u2018LiangYuan who invited LiFang was late just now.\u20192*\u738b\u78ca\u8054\u7cfb*\u7684*\u5f20\u8273*\u521a\u624d*\u8fdb\u95e8\u4e86*W\u00e1ngL\u011bi li\u00e1nx\u00ec de Zh\u0101ng Y\u00e0n g\u0101ngc\u00e1i j\u00ecnm\u00e9nleWang Lei contact REL Zhang Yan just entered\u2018Zhang Yan who Wang Lei contacted just entered the door.\u20193*\u738b\u8273\u8d70\u8bbf*\u7684*\u738b\u4f1f*\u524d\u5929*\u53c2\u8d5b\u4e86*W\u00e1ng Y\u00e0n z\u01d2uf\u01ceng de W\u00e1ng W\u011bi qi\u00e1nti\u0101n c\u0101ns\u00e0ileWang Yan visited REL Wang Wei day.before.yesterday entered.competition\u2018Wang Wei who Wang Yan visited entered the competition the day before yesterday.\u20194*\u5f20\u4f1f\u52fe\u7ed3*\u7684*\u674e\u5f3a*\u53bb\u5e74*\u5165\u72f1\u4e86*Zh\u0101ng W\u011bi g\u014duji\u00e9 de L\u01d0 Qi\u00e1ng q\u00f9ni\u00e1n r\u00f9y\u00f9leZhang Wei conspired REL Li Qiang last.year jailed\u2018LiQiang who ZhangWei conspired with went to jail last year.\u20195*\u6768\u654f\u8f85\u5bfc*\u7684*\u6768\u660a*\u4eca\u5929*\u5c31\u4efb\u4e86*Y\u00e1ng M\u01d0n f\u01d4d\u01ceo de Y\u00e1ng H\u00e0o j\u012bnti\u0101n ji\u00f9r\u00e8nleYang Min mentor REL Yang Hao today inducted\u2018Yang Hao who Yang Min mentored was inducted today.\u20196*\u738b\u5f3a\u5173\u6ce8*\u7684*\u738b\u6d01*\u4e0a\u5468*\u83b7\u80dc\u4e86*W\u00e1ng Qi\u00e1ng gu\u0101nzh\u00f9 de W\u00e1ng Ji\u00e9 sh\u00e0ngzh\u014du hu\u00f2sh\u00e8ngleWang Qiang follow.with.interest REL Wang Jie last.week won\u2018Wang Jie who Wang Qiang follows with interest won last week.\u20197*\u738b\u971e\u62e5\u62a4*\u7684*\u9a6c\u8d85*\u4eca\u5929*\u4e0b\u53f0\u4e86*W\u00e1ng Xi\u00e1 y\u014dngh\u00f9 de M\u01ce Ch\u0101o j\u012bnti\u0101n xi\u00e0t\u00e1ileWang Xia supports REL Ma Chao today resigned\u2018Ma Chao who Wang Xia supports resigned today.\u20198*\u674e\u4e3d\u62a2\u6551*\u7684*\u7f57\u82f1*\u524d\u5929*\u727a\u7272\u4e86*L\u01d0 L\u00ec qi\u01cengji\u00f9 de Lu\u00f3 Y\u012bng qi\u00e1nti\u0101n x\u012bsh\u0113ngleLi Li saved REL Luo Ying day.before.yesterday sacrificed\u2018Luo Ying who Li Li saved sacrificed his life the day before yesterday.\u20199*\u674e\u660a\u5bfb\u627e*\u7684*\u738b\u521a*\u5df2\u7ecf*\u901d\u4e16\u4e86*L\u01d0 H\u00e0o x\u00fanzh\u01ceo de W\u00e1ng G\u0101ng y\u01d0j\u012bng sh\u00ecsh\u00ecleLi Hao search REL Wang Gang already passed.away\u2018Wang Gang who Li Hao is searching for has passed away already.\u201910*\u738b\u519b\u5173\u5fc3*\u7684*\u674e\u5a67*\u53bb\u5e74*\u7ed3\u5a5a\u4e86*W\u00e1ng J\u016bn gu\u0101nx\u012bn de L\u01d0 J\u00ecng q\u00f9ni\u00e1n ji\u00e9h\u016bnleWang Jun cares.about REL Li Jing last year married\u2018Li Jing who Wang Jun cares about married last year.\u201911*\u738b\u82b3\u63a8\u8350*\u7684*\u5f20\u5f3a*\u521a\u624d*\u4e0a\u53f0\u4e86*W\u00e1ng F\u0101ng tu\u012bji\u00e0n de Zh\u0101ng Qi\u00e1ng g\u0101ngc\u00e1i sh\u00e0ngt\u00e1ileWang Fang recommended REL Zhang Qiang just.now appear.on.stage\u2018Zhang Qiang who Wang Fang recommended appeared on the stage just now.\u201912*\u674e\u660e\u8058\u8bf7*\u7684*\u5218\u68a6*\u53bb\u5e74*\u8f6c\u884c\u4e86*L\u01d0 M\u00edng p\u00ecnq\u01d0ng de Li\u00fa M\u00e8ng q\u00f9ni\u00e1n zhu\u01cenh\u00e1ngleLi Ming hired REL Liu Meng last.year switched.profession\u2018Liu Meng who Li Ming hired switched to another profession last year.\u201913*\u738b\u831c\u63d0\u53ca*\u7684*\u674e\u660e*\u53bb\u5e74*\u642c\u5bb6\u4e86*W\u00e1ng Qi\u00e0n t\u00edj\u00ed de L\u01d0 M\u00edng q\u00f9ni\u00e1n b\u0101nji\u0101leWang Qian mentioned REL Li Ming last.year moved\u2018Li Ming who Wang Qian mentioned moved last year.\u201914*\u674e\u8273\u91c7\u8bbf*\u7684*\u738b\u4e3d*\u6628\u5929*\u81ea\u6740\u4e86*L\u01d0 Y\u00e0n c\u01ceif\u01ceng de W\u00e1ng L\u00ec zu\u00f3ti\u0101n z\u00ecsh\u0101leLi Yan interviewed REL Wang Li yesterday committed.suicide\u2018Wang Li who Li Yan interviewed committed suicide yesterday.\u201915*\u674e\u660e\u8d44\u52a9*\u7684*\u5218\u6d0b*\u53bb\u5e74*\u7834\u4ea7\u4e86*L\u01d0 M\u00edng z\u012bzh\u00f9 de Li\u00fa Y\u00e1ng q\u00f9ni\u00e1n p\u00f2ch\u01cenleLi Ming funded REL Liu Yang last.year bankrupted\u2018Liu Yang who Li Ming funded went bankrupt last year.\u201916*\u5468\u5a55\u63d0\u62d4*\u7684*\u5f20\u68a6*\u4e0a\u5468*\u8fdd\u7eaa\u4e86*Zh\u014du Ji\u00e9 t\u00edb\u00e1 de Zh\u0101ng M\u00e8ng sh\u00e0ngzh\u014du w\u00e9ij\u00ecleZhou Jie promoted REL Zhang Meng last.week broke.rules\u2018Zhang Meng who Zhou Jie promoted broke the rules last week.\u201917*\u5f20\u6770\u8d1f\u8d23*\u7684*\u5218\u9e4f*\u53bb\u5e74*\u8f9e\u804c\u4e86*Zh\u0101ng Ji\u00e9 f\u00f9z\u00e9 de Li\u00fa P\u00e9ng q\u00f9ni\u00e1n c\u00edzh\u00edleZhang Jie in.charge.of REL Liu Peng last.year resigned\u2018Liu Peng who Zhang Jie was in charge of resigned last year.\u201918*\u738b\u9759\u5f55\u7528*\u7684*\u674e\u83b2*\u4eca\u5929*\u52a0\u73ed\u4e86*W\u00e1ng J\u00ecng l\u00f9y\u00f2ng de L\u01d0 Li\u00e1n j\u012bnti\u0101n ji\u0101b\u0101nleWang Jing employed REL Li Lian today worked.overtime\u2018Li Lian who Wang Jing employed worked overtime today.\u201919*\u738b\u9759\u4fe1\u4efb*\u7684*\u674e\u6d9b*\u4eca\u5929*\u7f3a\u5e2d\u4e86*W\u00e1ng J\u00ecng x\u00ecnr\u00e8n de L\u01d0 T\u0101o j\u012bnti\u0101n qu\u0113x\u00edleWang Jing trust REL Li Tao today absent\u2018Li Tao who Wang Jing trusts is absent today.\u201920*\u738b\u9633\u57f9\u517b*\u7684*\u5468\u6d01*\u53bb\u5e74*\u638c\u6743\u4e86*W\u00e1ng Y\u00e1ng p\u00e9iy\u01ceng de Zh\u014du Ji\u00e9 q\u00f9ni\u00e1n zh\u01cengqu\u00e1nleWang Yang mentored REL Zhou Jie last.year in.power\u2018Zhou Jie who Wang Yang mentored came into power last year.\u201921*\u5218\u521a\u601d\u5ff5*\u7684*\u9ad8\u519b*\u6628\u5929*\u751f\u75c5\u4e86*Li\u00fa G\u0101ng s\u012bni\u00e0n de G\u0101o J\u016bn zu\u00f3ti\u0101n sh\u0113ngb\u00ecngleLiu Gang missed REL Gao Jun yesterday sick\u2018Gao Jun who Liu Gang missed was sick yesterday.\u201922*\u5f20\u6d9b\u62db\u5f85*\u7684*\u674e\u8273*\u53bb\u5e74*\u9000\u4f11\u4e86*Zh\u0101ng T\u0101o zh\u0101od\u00e0i de Li Y\u00e0n q\u00f9ni\u00e1n tu\u00ecxi\u016bleZhang Tao entertained REL Li Yan last.year retired\u2018Li Yan who Zhang Tao entertained retired last year.\u201923*\u5468\u8d85\u6307\u5bfc*\u7684*\u674e\u52c7*\u521a\u624d*\u51fa\u4e8b\u4e86*Zh\u014du Ch\u0101o zh\u01d0d\u01ceo de L\u01d0 Y\u01d2ng g\u0101ngc\u00e1i ch\u016bsh\u00ecleZhou Chao tutored REL Li Yong just.now accident\u2018Li Yong who Zhou Chao tutored had an accident just now.\u2019\u2018Something bad happened to Li Yong who Zhou Chao tutored just now.\u201924*\u738b\u82b3\u683d\u57f9*\u7684*\u5f20\u9759*\u4e0a\u5468*\u53bb\u4e16\u4e86*W\u00e1ng F\u0101ng z\u0101ip\u00e9i de Zh\u0101ng J\u00ecng sh\u00e0ngzh\u014du q\u00f9sh\u00ecleWang Fang mentored REL Zhang Jing last.week died.\u2018Zhang Jing who Wang Fang mentored died last week.\u201925*\u738b\u5a1f\u652f\u6301*\u7684*\u674e\u52c7*\u6628\u5929*\u83b7\u5956\u4e86*W\u00e1ng Ju\u0101n zh\u012bch\u00ed de L\u01d0 Y\u01d2ng zu\u00f3ti\u0101n hu\u00f2ji\u01cengleWang Juan support REL Li Yong yesterday won.award\u2018Li Yong who Wang Juan supports won an award yesterday.\u201926*\u5f20\u8d85\u5bb4\u8bf7*\u7684*\u738b\u5a55*\u521a\u624d*\u9053\u6b49\u4e86*Zh\u0101ng Ch\u0101o y\u00e0nq\u01d0ng de W\u00e1ng Ji\u00e9 g\u0101ngc\u00e1i d\u00e0oqi\u00e0nleZhang Chao entertained REL Wang Jie just.now apologized\u2018Wang Jie who Zhang Chao entertained apologized just now.\u201927*\u5f20\u9759\u902e\u6355*\u7684*\u674e\u9e4f*\u90a3\u5929*\u53d7\u4f24\u4e86*Zh\u0101ng J\u00ecng d\u00e0ib\u01d4 de L\u01d0 P\u00e9ng n\u00e0ti\u0101n sh\u00f2ush\u0101ngleZhang Jing arrested REL Li Peng that.day injured\u2018Li Peng who Zhang Jing arrested was injured that day.\u201928*\u5218\u6770\u8d4f\u8bc6*\u7684*\u9ad8\u71d5*\u6700\u7ec8*\u53d7\u9a97\u4e86*Li\u00fa Ji\u00e9 sh\u01cengsh\u00ed de G\u0101o Y\u00e0n zu\u00eczh\u014dng sh\u00f2upi\u00e0nleLiu Jie admire REL Gao Yan eventually deceived.\u2018Gao Yan who Liu Jie admires was deceived eventually.\u201929*\u738b\u5f3a\u57f9\u80b2*\u7684*\u5f20\u52c7*\u4eca\u5e74*\u521b\u4e1a\u4e86*W\u00e1ng Qi\u00e1ng p\u00e9iy\u00f9 de Zh\u0101ng Y\u01d2ng j\u012bnni\u00e1n chu\u00e0ngy\u00e8leWang Qiang mentored REL Zhang Yong this.year started.business\u2018Zhang Yong who Wang Qiang mentored started a business this year.\u201930*\u738b\u9759\u7167\u987e*\u7684*\u674e\u5a1c*\u4e0a\u5468*\u6350\u6b3e\u4e86*W\u00e1ng J\u00ecng zh\u00e0og\u00f9 de L\u01d0 N\u00e0 sh\u00e0ngzh\u014du ju\u0101nku\u01cenleWang Jing take.care REL Li Na last.week made.donation\u2018Li Na who Wang Jing takes care of made a donation last week.\u201931*\u738b\u731b\u670d\u52a1*\u7684*\u674e\u521a*\u4eca\u5e74*\u5165\u9009\u4e86*W\u00e1ng M\u011bng f\u00faw\u00f9 de L\u01d0 G\u0101ng j\u012bnni\u00e1n r\u00f9xu\u01cenleWang Meng served REL LI Gang this.year selected\u2018Li Gang who Wang Meng served was selected this year.\u201932*\u674e\u831c\u62c5\u5fc3*\u7684*\u9648\u5029*\u4eca\u5929*\u4f4f\u9662\u4e86*\u2003L\u01d0 Qi\u00e0n d\u0101nx\u012bn de Ch\u00e9n Qi\u00e0n j\u012bnti\u0101n zh\u00f9yu\u00e0nleLi Qian worried.about REL Chen Qian today hospitalized\u2018Chen Qian who Li Qian worried about is in the hospital today.\u2019The following sentences are all the ORC experimental items from Experiment 2. The SRC sentence condition is given only for the first item. For the unambiguous condition, the determiner + classifier region which is the first region for all of the sentences was removed. The interest regions are designated between asterisk marks .1. ORC: *\u90a3\u4e2a*\u6628\u665a*\u670d\u52a1\u751f\u63cd\u4e86*\u4e00\u987f*\u7684*\u987e\u5ba2*\u542c\u8bf4\u8fc7*\u8001\u677f*\u5e76\u4e14\u8bb0\u5f97\u4ed6\u3002*n\u00e0g\u00e8 zu\u00f3w\u01cen f\u00faw\u00f9sh\u0113ng z\u00f2ule y\u012b d\u00f9n de g\u00f9k\u00e8 t\u012bngshu\u014dgu\u00f2 l\u01ceob\u01cen b\u00ecngqi\u011b j\u00ecde t\u0101that.one last.night waiter beat more.than.once REL customer have heard of boss and remember him\u2018The customer who the waiter beat up last night has heard of the boss and remembers him.\u20191. SRC: *\u90a3\u4e2a*\u6628\u665a*\u63cd\u4e86\u670d\u52a1\u751f*\u4e00\u987f*\u7684*\u987e\u5ba2*\u542c\u8bf4\u8fc7*\u8001\u677f*\u5e76\u4e14\u8bb0\u5f97\u4ed6\u3002*n\u00e0g\u00e8 zu\u00f3w\u01cen z\u00f2ule f\u00faw\u00f9sh\u0113ng y\u012b d\u00f9n de g\u00f9k\u00e8 t\u012bngshu\u014dgu\u00f2 l\u01ceob\u01cen b\u00ecngqi\u011b j\u00ecde t\u0101that.one last.night beat waiter more.than.once REL customer have.heard.of boss and remember him\u2018The customer who beat up the waiter last night has heard of the boss and remembers him.\u20192. *\u90a3\u8f86*\u4e0b\u5348*\u6469\u6258\u8f66\u8ffd\u4e86*\u5f88\u4e45*\u7684*\u8f7f\u8f66*\u53d1\u73b0\u4e86*\u8bb0\u8005*\u6240\u4ee5\u505c\u4e86\u4e0b\u6765\u3002*n\u00e0li\u00e0ng xi\u00e0w\u01d4 m\u00f3tu\u014dch\u0113 zhu\u012ble h\u011bnji\u01d4 de ji\u00e0och\u0113 f\u0101xi\u00e0nle j\u00eczh\u011b su\u01d2y\u01d0 t\u00edngle xi\u00e0l\u00e1ithat.car afternoon motorcycle chased long.time REL car found reporter so stopped\u2018The car that the motorcycle chased for a long time in the afternoon found the reporter so it stopped.\u20193. *\u90a3\u4e2a*\u4eca\u5929*\u7537\u5b69\u6253\u4e86*\u51e0\u6b21*\u7684*\u5973\u5b69*\u770b\u5230\u4e86*\u6821\u957f*\u6240\u4ee5\u5047\u88c5\u8bfb\u4e66\u3002*n\u00e0g\u00e8 j\u012bnti\u0101n n\u00e1nh\u00e1i d\u01cele j\u01d0c\u00ec de n\u01dah\u00e1i k\u00e0nd\u00e0ole xi\u00e0ozh\u01ceng su\u01d2y\u01d0 ji\u01cezhu\u0101ng d\u00fash\u016bthat.one today boy hit several.times REL girl saw principal so pretended to read\u2018The girl who the boy hit several times today saw the principal and thus pretended to read.\u20194. *\u90a3\u8f86*\u5f53\u65f6*\u81ea\u884c\u8f66\u649e\u4e86*\u4e24\u6b21*\u7684*\u5409\u666e\u8f66*\u62e6\u4f4f\u4e86*\u8b66\u5bdf*\u5e76\u4e14\u8981\u6c42\u8c03\u67e5\u6e05\u695a\u3002*n\u00e0li\u00e0ng d\u0101ngsh\u00ed z\u00ecx\u00edngch\u0113 zhu\u00e0ngle li\u01cengc\u00ec de j\u00edp\u01d4ch\u0113 l\u00e1nzh\u00f9le j\u01d0ngch\u00e1 b\u00ecngqi\u011b y\u0101oqi\u00fa di\u00e0och\u00e1 q\u012bngchuthat.car then bike hit twice REL jeep stopped police and asked investigation clear\u2018The jeep that the bike hit twice then stopped the police and asked for a clear investigation.\u20195. *\u90a3\u4e2a*\u521a\u624d*\u7537\u5b69\u63a8\u4e86*\u4e00\u4e0b*\u7684*\u5987\u5973*\u5077\u4e86*\u5e97\u5458*\u5e76\u4e14\u6253\u4f24\u4e86\u5979\u3002*n\u00e0g\u00e8 g\u0101ngc\u00e1i n\u00e1nh\u00e1i tu\u012ble y\u012bxi\u00e0 de f\u00f9n\u01da t\u014dule di\u00e0nyu\u00e1n b\u00ecngqi\u011b d\u01cesh\u0101ngle t\u0101that.one just.now boy pushed a.bit REL woman stole clerk and wounded her\u2018The woman who the boy pushed just now stole from the clerk and wounded her.\u20196. *\u90a3\u4e2a*\u4e0a\u4e2a\u6708*\u7537\u5b69\u9080\u8bf7\u4e86*\u51e0\u6b21*\u7684*\u5973\u5b69*\u8ba4\u8bc6*\u738b\u8001\u5e08*\u56e0\u4e3a\u4e0a\u8fc7\u5979\u7684\u8bfe\u3002*n\u00e0g\u00e8 sh\u00e0ngg\u00e8yu\u00e8 n\u00e1nh\u00e1i y\u0101oq\u01d0ngle j\u01d0c\u00ec de n\u01dah\u00e1i r\u00e8nshi w\u00e1ngl\u01ceosh\u012b y\u012bnw\u00e8i sh\u00e0nggu\u00f2 t\u0101 de k\u00e8that.one last.month boy invited several.times REL girl knows teacher.WANG because went to her class\u2018That girl who the boy invited several times last month knows teacher Wong because she went to her class.\u20197. *\u90a3\u6761*\u53bb\u5e74*\u4e3b\u4eba\u6551\u4e86*\u597d\u51e0\u6b21*\u7684*\u72d7*\u559c\u6b22*\u5c0f\u7537\u5b69*\u6240\u4ee5\u5f88\u5174\u594b\u3002*n\u00e0ti\u00e1o q\u00f9ni\u00e1n zh\u01d4r\u00e9n ji\u00f9le h\u01ceoj\u01d0c\u00ec de g\u01d2u x\u01d0huan xi\u01ceon\u00e1nh\u00e1i su\u01d2y\u01d0 h\u011bn x\u012bngf\u00e8nthat.animal last.year master saved several.times REL dog like little.boy so it be very excited\u2018The dog that the master saved several times last year likes the little boy so it was very excited.\u20198. *\u90a3\u4e2a*\u521a\u624d*\u804c\u4e1a\u9009\u624b\u63a8\u4e86*\u4e00\u4e0b*\u7684*\u4e1a\u4f59\u9009\u624b*\u9a82\u4e86*\u88c1\u5224*\u800c\u4e14\u5a01\u80c1\u4e86\u4ed6\u3002*n\u00e0g\u00e8 g\u0101ngc\u00e1i zh\u00edy\u00e8 xu\u01censh\u01d2u tu\u012ble y\u012bxi\u00e0 de y\u00e8y\u00fa xu\u01censh\u01d2u m\u00e0le c\u00e1ip\u00e0n \u00e9rqi\u011b w\u0113ixi\u00e9le t\u0101that.one just.now professional.player pushed a.bit REL amateur scolded referee and threatened him\u2018The amateur who the professional player pushed a bit just now scolded the referee and threatened him.\u20199. *\u8fd9\u4e2a*\u4e0a\u4e2a\u6708*\u6740\u624b\u76d1\u89c6\u4e86*\u4e00\u6bb5\u65f6\u95f4*\u7684*\u4fa6\u63a2*\u8ba8\u538c*\u5f53\u5730\u4eba*\u6240\u4ee5\u6ca1\u6709\u5bfb\u6c42\u5e2e\u52a9\u3002*zh\u00e8ge sh\u00e0ngg\u00e8yu\u00e8 sh\u0101sh\u01d2u ji\u0101nsh\u00ecle y\u012bdu\u00e0n sh\u00edji\u0101n de zh\u0113nt\u00e0n t\u01ceoy\u00e0n d\u0101ngd\u00ecr\u00e9n su\u01d2y\u01d0 m\u00e9iy\u01d2u x\u00fanqi\u00fa b\u0101ngzh\u00f9this.one last.month killer watched a.while REL detective hated local so did.not ask for help\u2018The detective who the killer watched for a while last month hated the locals so he did not ask for help.\u201910. *\u90a3\u4f4d*\u6700\u8fd1*\u623f\u4e1c\u62b1\u6028\u4e86*\u597d\u591a\u6b21*\u7684*\u4f4f\u6237*\u627e\u4e86*\u5f8b\u5e08*\u800c\u4e14\u6253\u7b97*\u8d77\u8bc9\u3002*n\u00e0w\u00e8i zu\u00ecj\u00ecn f\u00e1ngd\u014dng b\u00e0oyu\u00e0nle h\u01ceodu\u014dc\u00ec de zh\u00f9h\u00f9 zh\u01ceole l\u01dcsh\u012b \u00e9rqi\u011b d\u01cesu\u00e0n q\u01d0s\u00f9that.person recently landlord complained many.times REL tenant found lawyer and intended sue\u2018The tenant who the landlord complained about many times recently found a lawyer and intends to sue.\u201911. *\u90a3\u4e2a*\u4e0a\u4e2a\u6708*\u6559\u7ec3\u9a82\u4e86*\u4e00\u987f*\u7684*\u7403\u5458*\u7231\u4e0a\u4e86*\u5973\u6b4c\u661f*\u8fd8\u9001\u5979\u793c\u7269\u3002*n\u00e0g\u00e8 sh\u00e0ngg\u00e8yu\u00e8 ji\u00e0oli\u00e0n m\u00e0le y\u012bd\u00f9n de qi\u00fayu\u00e1n \u00e0ish\u00e0ngle n\u01da g\u0113x\u012bng h\u00e1i s\u00f2ng t\u0101 l\u01d0w\u00f9that.one last.month coach scolded more.than.once REL player fell.love.with female singer and sent her gift\u2018The player who the coach scolded last month fell in love with a female singer and sent her a gift.\u201912. *\u90a3\u4f4d*\u4ee5\u524d*\u6307\u6325\u5bb6\u5d07\u62dc\u4e86*\u5f88\u4e45*\u7684*\u4f5c\u66f2\u5bb6*\u7ed3\u8bc6\u4e86*\u5c0f\u63d0\u7434\u624b*\u5e76\u4e14\u4e24\u4eba\u5e38\u89c1\u9762\u3002*n\u00e0w\u00e8i y\u01d0qi\u00e1n zh\u01d0hu\u012bji\u0101 ch\u00f3ngb\u00e0ile h\u011bnji\u01d4 de zu\u00f2q\u01d4ji\u0101 ji\u00e9sh\u00ecle xi\u01ceot\u00edq\u00ednsh\u01d2u b\u00ecngqi\u011b li\u01ceng r\u00e9n ch\u00e1ng ji\u00e0nmi\u00e0nthat.person before conductor respected long.time REL composer met violinist and both meet often\u2018The composer who the conductor respected for a long time in the past met a violinist and they meet often.\u201913. *\u8fd9\u4e2a*\u53bb\u5e74*\u7535\u89c6\u53f0\u6279\u8bc4\u4e86*\u51e0\u6b21*\u7684*\u5973\u6f14\u5458*\u5f88\u6b23\u8d4f*\u91d1\u57ce\u6b66*\u56e0\u4e3a\u4ed6\u4e2a\u6027\u5766\u7387\u3002*zh\u00e8ge q\u00f9ni\u00e1n di\u00e0nsh\u00ect\u00e1i p\u012bp\u00edngle j\u01d0c\u00ec de n\u01da y\u01cenyu\u00e1n h\u011bn x\u012bnsh\u01ceng j\u012bnch\u00e9ngw\u01d4 y\u012bnw\u00e8i t\u0101 g\u00e8x\u00ecng t\u01censhu\u00e0ithis.one last.year TV.station criticized several.times REL actress very.appreciate Jincheng Wu because his frank personality\u2018This actress who the TV station criticized several times last year appreciates Jincheng Wu very much because of his frank personality.\u201914. *\u90a3\u4f4d*\u4e0a\u4e2a\u6708*\u98de\u884c\u5458\u7ea6\u4e86*\u4e24\u6b21*\u7684*\u7a7a\u59d0*\u60f9\u6012\u4e86*\u7ecf\u7406*\u56e0\u4e3a\u5979\u5e38\u8fdf\u5230\u3002*n\u00e0w\u00e8i sh\u00e0ngg\u00e8yu\u00e8 f\u0113ix\u00edngyu\u00e1n yu\u0113le li\u01cengc\u00ec de k\u014dngji\u011b r\u011bn\u00f9le j\u012bngl\u01d0 y\u012bnw\u00e8i t\u0101 ch\u00e1ng ch\u00edd\u00e0othat.person last.month pilot ask out twice REL stewardess angered manager because she often late\u2018The stewardess who the pilot asked out twice last month angered the manger because she was often late.\u201915. *\u8fd9\u4f4d*\u4eca\u5929*\u5bfc\u6f14\u79f0\u8d5e\u4e86*\u591a\u6b21*\u7684*\u7537\u660e\u661f*\u6279\u8bc4\u4e86*\u5f71\u8bc4\u5bb6*\u5e76\u4e14\u8868\u793a\u5f88\u96be\u8fc7\u3002*zh\u00e8w\u00e8i j\u012bnti\u0101n d\u01ceoy\u01cen ch\u0113ngz\u00e0nle du\u014dc\u00ec de n\u00e1n m\u00edngx\u012bng p\u012bp\u00edngle y\u01d0ngp\u00edngji\u0101 b\u00ecngqi\u011b bi\u01ceosh\u00ec h\u011bn n\u00e1ngu\u00f2this.one today director praised many.times REL male star criticized critics and said he was very sad\u2018The male star who the director praised many times today criticized the critics and said he was very sad.\u201916. *\u90a3\u4f4d*\u6628\u5929*\u4f5c\u5bb6\u91c7\u8bbf\u4e86*\u4e24\u4e2a\u5c0f\u65f6*\u7684*\u8bb0\u8005*\u8d28\u7591\u4e86*\u53bf\u957f\u5019\u9009\u4eba*\u800c\u4e14\u626c\u8a00\u62a5\u590d\u3002*n\u00e0w\u00e8i zu\u00f3ti\u0101n zu\u00f2ji\u0101 c\u01ceif\u01cengle li\u01ceng g\u00e8 xi\u01ceosh\u00ed de j\u00eczh\u011b zh\u00edy\u00edle xi\u00e0n zh\u01ceng h\u00f2uxu\u01cenr\u00e9n \u00e9rqi\u011b y\u00e1ngy\u00e1n b\u00e0of\u00f9that.person yesterday writer interviewed two.hours REL reporter questioned county.magistrate candidate and threatened revenge\u2018The reporter who the writer interviewed for two hours yesterday questioned the county magistrate candidate and threatened revenge\u201917. *\u90a3\u4e2a*\u4eca\u65e9*\u72af\u4eba\u8ffd\u4e86*\u4e00\u9635*\u7684*\u5c0f\u72d7*\u55c5\u51fa*\u4e3b\u4eba*\u5e76\u4e14\u505c\u4e86\u4e0b\u6765\u3002*n\u00e0g\u00e8 j\u012bnz\u01ceo f\u00e0nr\u00e9n zhu\u012ble y\u012bzh\u00e8n de xi\u01ceo g\u01d2u xi\u00f9ch\u016b zh\u01d4r\u00e9n b\u00ecngqi\u011b t\u00edngle xi\u00e0l\u00e1ithat.one this.morning criminal chased a.while REL puppy sniffed.recognize the master and stopped\u2018The puppy who the prisoner chased a while this morning sniffed and recognized the mater and stopped.\u201918. *\u90a3\u4f4d*\u6628\u5929*\u90bb\u5c45\u6559\u8bad\u4e86*\u4e00\u756a*\u7684*\u5927\u5988*\u901a\u77e5*\u7ba1\u7406\u5458*\u7136\u540e\u8bc9\u4e86\u82e6\u3002*n\u00e0w\u00e8i zu\u00f3ti\u0101n l\u00ednj\u016b ji\u00e0oxunle y\u012b f\u0101n de d\u00e0m\u0101 t\u014dngzh\u012b gu\u01cenl\u01d0 yu\u00e1n r\u00e1nh\u00f2u sule k\u01d4that.person yesterday neighbor taught for.a.while REL aunt noticed administrator and complained\u2018The aunt who the neighbor taught a lesson yesterday noticed the administrator and complained.\u201919. *\u8fd9\u4f4d*\u53bb\u5e74*\u5916\u4ea4\u90e8\u8bbf\u95ee\u4e86*\u4e00\u6b21*\u7684*\u653f\u6cbb\u5bb6*\u652f\u6301*\u5916\u4ea4\u5b98*\u5e76\u4e14\u76f8\u4fe1\u4ed6\u3002*zh\u00e8w\u00e8i q\u00f9ni\u00e1n w\u00e0iji\u0101ob\u00f9 f\u01cengw\u00e8nle y\u012bc\u00ec de zh\u00e8ngzh\u00ec ji\u0101 zh\u012bch\u00ed w\u00e0iji\u0101ogu\u0101n b\u00ecngqi\u011b xi\u0101ngx\u00ecn t\u0101this.person last.year ministry.foreign.affairs visited once REL politician support diplomat and believed him.\u2018This politician who the Ministry of Foreign Affairs visited once last year supports the diplomat and believes him.\u201920. *\u90a3\u4e2a*\u4eca\u5e74*\u4f5c\u5bb6\u6279\u8bc4\u4e86*\u4e00\u756a*\u7684*\u8bc4\u8bba\u5bb6*\u6d3d\u8be2\u4e86*\u51fa\u7248\u5546*\u800c\u4e14\u5efa\u8bae\u4e86\u51fa\u7248\u5185\u5bb9\u3002*n\u00e0g\u00e8 j\u012bnni\u00e1n zu\u00f2ji\u0101 p\u012bp\u00edngle y\u012bf\u0101n de p\u00edngl\u00f9nji\u0101 qi\u00e0x\u00fanle ch\u016bb\u01censh\u0101ng \u00e9rqi\u011b ji\u00e0ny\u00ecle ch\u016bb\u01cen n\u00e8ir\u00f3ngthat.one this.year writer criticized for.a.while REL critic consulted publisher and suggested publication.content\u2018The critic who the writer criticized for a while this year consulted the publisher and suggested the content for publication.\u201921. *\u90a3\u4f4d*\u4e0b\u5348*\u5b66\u751f\u79f0\u8d5e\u4e86*\u591a\u6b21*\u7684*\u8001\u5e08*\u8ba4\u51fa*\u5bb6\u957f*\u7136\u540e\u6253\u62db\u547c\u3002*n\u00e0w\u00e8i xi\u00e0w\u01d4 xu\u00e9sh\u0113ng ch\u0113ngz\u00e0nle du\u014dc\u00ec de l\u01ceosh\u012b r\u00e8nch\u016b ji\u0101zh\u01ceng r\u00e1nh\u00f2u d\u01cezh\u0101oh\u016bthat.person afternoon student praised several.times REL teacher recognized parents and.then greeted\u2018The teacher who the student praised several times this afternoon recognized the parents and greeted them.\u201922. *\u90a3\u4f4d*\u4e0a\u5468*\u62a4\u58eb\u8bf7\u6559\u4e86*\u4e00\u6b21*\u7684*\u8425\u517b\u5e08*\u770b\u4e86*\u533b\u751f*\u800c\u4e14\u786e\u5b9a\u75c5\u5f81\u3002*n\u00e0w\u00e8i sh\u00e0ngzh\u014du h\u00f9sh\u00ec q\u01d0ngji\u00e0ole y\u012bc\u00ec de y\u00edngy\u01cengsh\u012b k\u00e0nle y\u012bsh\u0113ng \u00e9rqi\u011b qu\u00e8d\u00ecng b\u00ecngzh\u0113ngthat.person last.week nurse consulted once REL nutritionist went.to.see doctor and identified symptoms\u2018That nutritionist who the nurse consulted once last week went to see a doctor and identified symptoms.\u201923.*\u8fd9\u4f4d*\u4e0a\u4e2a\u6708*\u65b0\u540c\u4e8b\u4ecb\u7ecd\u4e86*\u51e0\u6b21*\u7684*\u5458\u5de5*\u8bf4\u670d\u8fc7*\u4e0a\u53f8*\u7136\u540e\u8f6c\u4e86\u90e8\u95e8\u3002*zh\u00e8w\u00e8i sh\u00e0ngg\u00e8yu\u00e8 x\u012bn t\u00f3ngsh\u00ec ji\u00e8sh\u00e0ole j\u01d0c\u00ec de yu\u00e1ng\u014dng shu\u014df\u00fagu\u00f2 sh\u00e0ngsi r\u00e1nh\u00f2u zhu\u01cenle b\u00f9m\u00e9nthis.person last.month new.colleague introduced several.times REL employee persuaded boss and.then transferred department\u2018This employee who the new colleague introduced several times last month persuaded the boss and then transferred to another department.\u201924. *\u8fd9\u4f4d*\u4e0a\u5348*\u59bb\u5b50\u966a\u4e86*\u8bb8\u4e45*\u7684*\u4e08\u592b*\u95ee\u4e86*\u65c5\u884c\u793e*\u7136\u540e\u51b3\u5b9a\u89c4\u5212\u65c5\u884c\u3002*zh\u00e8w\u00e8i sh\u00e0ngw\u01d4 q\u012bzi p\u00e9ile x\u01d4ji\u01d4 de zh\u00e0ngf\u016b w\u00e8nle l\u01dax\u00edngsh\u00e8 r\u00e1nh\u00f2u ju\u00e9d\u00ecng gu\u012bhu\u00e0 l\u01dax\u00edngthis.person morning wife accompany long.time REL husband inquire travel.agency and.then decide plan trip\u2018The husband who his wife accompanied a long time this morning inquired the travel agency and then decided to plan a trip.\u201925. *\u90a3\u4e2a*\u4e0a\u5348*\u5e97\u5458\u8be2\u95ee\u4e86*\u4e00\u756a*\u7684*\u5973\u5b69*\u627e\u5230*\u5988\u5988*\u7136\u540e\u56de\u4e86\u5bb6\u3002*n\u00e0g\u00e8 sh\u00e0ngw\u01d4 di\u00e0nyu\u00e1n x\u00fanw\u00e8nle y\u012b f\u0101n de n\u01dah\u00e1i zh\u01ceod\u00e0o m\u0101m\u0101 r\u00e1nh\u00f2u hu\u00edle ji\u0101that.one morning clerk asked for.a.while REL girl found mom and.then returned home\u2018The girl who the clerk asked about in the morning found the mom and then returned home.\u201926. *\u8fd9\u4e2a*\u53bb\u5e74*\u65b0\u5a18\u606d\u559c\u4e86*\u51e0\u6b21*\u7684*\u5546\u4eba*\u543b\u4e86*\u5973\u513f*\u5e76\u4e14\u4e3e\u529e\u559c\u5bb4\u3002*zh\u00e8ge q\u00f9ni\u00e1n x\u012bnni\u00e1ng g\u014dngx\u01d0le j\u01d0c\u00ec de sh\u0101ngr\u00e9n w\u011bnle n\u01da'\u00e9r b\u00ecngqi\u011b j\u01d4b\u00e0n x\u01d0y\u00e0nthis.one last.year bride congratulated several.times REL businessman kissed daughter and held wedding banquet\u2018This businessman who the bride congratulated several times last year kissed the daughter and held a wedding banquet.\u201927. *\u8fd9\u4f4d*\u6628\u665a*\u79d1\u5b66\u5bb6\u76f8\u4fe1\u4e86*\u4e00\u65f6*\u7684*\u64ad\u62a5\u5458*\u8054\u7cfb*\u7269\u7406\u4e13\u5bb6*\u5e76\u4e14\u8c03\u67e5\u4e86\u771f\u76f8\u3002*zh\u00e8w\u00e8i zu\u00f3w\u01cen k\u0113xu\u00e9ji\u0101 xi\u0101ngx\u00ecnle y\u012bsh\u00ed de b\u014db\u00e0oyu\u00e1n li\u00e1nx\u00ec w\u00f9l\u01d0 zhu\u0101nji\u0101 b\u00ecngqi\u011b di\u00e0och\u00e1le zh\u0113nxi\u00e0ngthis.person last.night scientist believed momentarily REL announcer contacted physician and investigated truth\u2018This announcer who the scientist believed momentarily last night contacted a physician and investigated the truth.\u201928. *\u90a3\u4e2a*\u524d\u5929*\u5c0f\u8bf4\u5bb6\u53d6\u60a6\u4e86*\u8bb8\u4e45*\u7684*\u6b4c\u5267\u5bb6*\u60f3\u5230*\u5927\u5b66\u6559\u6388*\u6240\u4ee5\u8054\u7edc\u4e86\u3002*n\u00e0g\u00e8 qi\u00e1nti\u0101n xi\u01ceoshu\u014dji\u0101 q\u01d4yu\u00e8le x\u01d4ji\u01d4 de g\u0113j\u00f9ji\u0101 xi\u01cengd\u00e0o d\u00e0xu\u00e9 ji\u00e0osh\u00f2u su\u01d2y\u01d0 li\u00e1nlu\u00f2lethat.one day.before.yesterday novelist flatter long.time REL opera.singer thought.of university.professor so contacted\u2018The opera singer who the novelist flattered for a long time the day before yesterday thought of the university professor so she contacted him.\u201929. *\u8fd9\u4e2a*\u521a\u624d*\u59d1\u59d1\u63a5\u4e86*\u4e00\u6b21*\u7684*\u53d4\u53d4*\u9001\u4e86*\u4f2f\u4f2f*\u5e76\u4e14\u4e00\u8d77\u5750\u8f66\u53bb\u3002*zh\u00e8ge g\u0101ngc\u00e1i g\u016bg\u016b ji\u0113le y\u012bc\u00ec de sh\u016bshu s\u00f2ngle b\u00f3bo b\u00ecngqi\u011b y\u012bq\u01d0 zu\u00f2ch\u0113 q\u00f9this.one just.now aunt picked up once REL the.younger.uncle sent the.older.uncle and went by car\u2018The younger uncle who the aunt picked up once just now sent off the older uncle and went by car together.\u201930. *\u8fd9\u4e2a*\u4eca\u665a*\u5ba2\u4eba\u8fce\u63a5\u4e86*\u591a\u6b21*\u7684*\u7537\u751f*\u9047\u89c1*\u4e00\u4f4d\u5973\u5b69*\u7136\u540e\u8981\u4e86\u7535\u8bdd\u53f7\u7801\u3002*zh\u00e8ge j\u012bnw\u01cen k\u00e8r\u00e9n y\u00edngji\u0113le du\u014dc\u00ec de n\u00e1nsh\u0113ng y\u00f9ji\u00e0n y\u012bw\u00e8i n\u01dah\u00e1i r\u00e1nh\u00f2u y\u00e0ole di\u00e0nhu\u00e0 h\u00e0om\u01cethis.one tonight guest greeted many.times REL boy met one.girl and.then asked for phone.number\u2018This boy who the guest greeted many times tonight met a girl and then asked for her phone number.\u201931. *\u90a3\u4e2a*\u521a\u521a*\u5973\u540c\u5b66\u9080\u8bf7\u4e86*\u4e00\u6b21*\u7684*\u7537\u751f*\u62d2\u7edd*\u597d\u670b\u53cb*\u5e76\u4e14\u79bb\u5f00\u4e86\u6559\u5ba4\u3002*n\u00e0g\u00e8 g\u0101ngg\u0101ng n\u01da t\u00f3ngxu\u00e9 y\u0101oq\u01d0ngle y\u012bc\u00ec de n\u00e1nsh\u0113ng j\u00f9ju\u00e9 h\u01ceop\u00e9ngy\u01d2u b\u00ecngqi\u011b l\u00edk\u0101ile ji\u00e0osh\u00ecthat.one just.now female.classmate invited once REL boy refused good.friend and left classroom\u2018The boy who the female classmate invited once just now refused a good friend and left the classroom.\u201932. *\u90a3\u4f4d*\u6628\u5929*\u4eb2\u621a\u62dc\u8bbf\u4e86*\u4e24\u6b21*\u7684*\u751f\u610f\u4eba*\u649e\u5230*\u9752\u6885\u7af9\u9a6c*\u5e76\u4e14\u611f\u5230\u8bb6\u5f02\u3002*n\u00e0w\u00e8i zu\u00f3ti\u0101n q\u012bnqi b\u00e0if\u01cengle li\u01cengc\u00ec de sh\u0113ngy\u00ecr\u00e9n zhu\u00e0ngd\u00e0o q\u012bngm\u00e9izh\u00fam\u01ce b\u00ecngqi\u011b g\u01cend\u00e0o y\u00e0 y\u00ecthat.person yesterday relatives visited twice REL businessman hit childhood friend and felt surprised\u2018The businessman who the relatives visited twice yesterday hit the childhood friend and felt surprised.\u2019S1 TablesWithin S1 Tables, Tables A-to-H are provided. Tables A to C detail the means and LME models for Experiment 1. The remainder of the tables detail the means and LME models for Experiment 2.(PDF)Click here for additional data file.S1 Data(XLSX)Click here for additional data file."}
{"text": "In this paper, the dynamical behaviors for a stochastic SIRS epidemic model with nonlinear incidence and vaccination are investigated. In the models, the disease transmission coefficient and the removal rates are all affected by noise. Some new basic properties of the models are found. Applying these properties, we establish a series of new threshold conditions on the stochastically exponential extinction, stochastic persistence, and permanence in the mean of the disease with probability one for the models. Furthermore, we obtain a sufficient condition on the existence of unique stationary distribution for the model. Finally, a series of numerical examples are introduced to illustrate our main theoretical results and some conjectures are further proposed. As is well known, transmissions of many infectious diseases are inevitably affected by environment white noise, which is an important component in realism, because it can provide some additional degrees of realism compared to their deterministic counterparts. Therefore, in recent years, stochastic differential equation (SDE) has been used widely by many researchers to model the dynamics of spread of infectious disease (see \u20135 and th\u03b2SI is frequently used (see \u039b/(dS(dR + \u03b5) + pdR). Applying the Lyapunov function method and the theory of persistence for dynamical systems, we can prove that, when R0 < 1, model  and, when R0 > 1, model  and disease I is permanent.It is well known that the basic reproduction number for model  is defin1, model  has a gl1, model  has a un\u03b2, dS, dI, and dR in model (i\u2009dt (0 \u2264 i \u2264 3) may be approximated by a normal distribution with zero mean and variance \u03c3i2dt (0 \u2264 i \u2264 3), respectively. That is, i\u2009dt may correlate with each other, we represent them by l-dimensional Brownian motion B(t) = (B1(t),\u2026, Bl(t)) as follows: \u03c3ij are all nonnegative real numbers. Therefore, model  = S, g(I) = I, and p = q = 0 has been investigated by Yang and Mao in  by \u2329h(t)\u232a = (1/t)\u222b0th(s)ds.For any function H1) and (H2) hold. Let (S(t), I(t), R(t)) be any positive solution of model (\u03c6(t) is defined for all t \u2265 0 satisfying limt\u2192\u221e\u03c6(t) = 0.Assume that  is defined byt\u2192\u221e\u03c6(t) = 0\u2009\u2009a.s.Taking integration from 0 to or model , we get , we havedR+\u03b5St\u2212S0ion from  with whiH1) and (H2) hold and \u03c3j1 = \u03c3j2 = \u03c3j3 = 0\u2009\u2009(1 \u2264 j \u2264 l). Then, for any solution (S(t), I(t), R(t)) of system (S(0), I(0), R(0)) \u2208 R+3, one has \u03bc = min\u2061{dS, dI, dR}.Assume that  = S(t) + I(t) + R(t), from model  and (H2) hold, \u03c3j1 = \u03c3j2 = \u03c3j3 = 0\u2009\u2009(1 \u2264 j \u2264 l), dS = dR, and dI = dS + \u03b1 with constant \u03b1 \u2265 0. Then, for any solution (S(t), I(t), R(t)) of model (S(0), I(0), R(0)) \u2208 R+3, one has Assume that (of model  with (S(N(0) = S(0) + I(0) + R(0). From the third equation of model (y(t) = \u222b0tedS + \u03b5)s(S(s)ds; then y(t), we obtain Since of model  we have e obtain yt=e\u2212pt\u039b1erefore, St=\u039b1\u2212qdStituting \u201323). Th. Th(24)ddS \u2260 dR in model , \u03c3r(x) = (\u03c3r1(x), \u03c3r2(x),\u2026, \u03c3rn(x)), and Br(t)\u2009\u2009(1 \u2264 r \u2264 m) are standard Brownian motions defined on the above probability space. The diffusion matrix is defined byV(x), we define Consider the following D in Rn with a regular  boundary such that(i)i = 1,2,\u2026, n and positive constant \u03b7 > 0 such that aii(x) \u2265 \u03b7 for all x \u2208 D;there exist some (ii)V(x) : Dc \u2192 R such that V(x) is second-order continuously differentiable function and that, for some \u03b8 > 0, LV(x)\u2264\u2212\u03b8 for all x \u2208 Dc, where Dc = Rn\u2216D.there exists a nonnegative function Assume that there is a bounded open subset \u03c0. That is, if function f is integrable with respect to the measure \u03c0, then for all x0 \u2208 RnThen  has a unTo study the permanence in mean with probability one of model  we need Y \u2208 C and Z \u2208 C satisfy limt\u2192\u221e(Z(t)/t) = 0\u2009\u2009a.s. If there is T > 0 such thatt \u2265 T, thenAssume that functions For the convenience of following statements, we denote H1) and (H2) hold. If one of the following conditions holds: (a)\u03c302f(S0)g\u2032(0) \u2264 \u03b2 + \u2211j=1l\u03c3j0\u03c3j2,(b)\u03c30 > 0\u2009\u2009and\u2009\u2009(\u03b2 + \u2211j=1l\u03c3j0\u03c3j2)2/2\u03c302 \u2212 (dI + \u03b3 + (1/2)\u03c322) < 0,then, for any initial value (S(0), I(0), R(0)) \u2208 R+3, one has I is stochastically extinct exponentially with probability one. Moreover, Assume that  and Applying It\u00f4's formula to model  leads toAssume that condition (b) holds. Sincehen from (43)ln\u2061It\u03f5 > 0 such that \u03b2 + \u2211j=1l\u03c3j0\u03c3j2 \u2265 g\u2032(0)f(\u03f5)\u03c302. We compute thatAssume that condition (a) holds. Choose constant \u03c302 = 0, which implies \u03c3j0 = 0\u2009\u2009(1 \u2264 j \u2264 l), we have from , from (H2), we can obtain f\u2032(\u03be)(S \u2212 S0) \u2264 f\u2032(S0)(S \u2212 S0). Hence, we haveWhen ave from (47)fx\u03b2fSrding to , 40), a, a\u03c302 = , we havefS\u2264fS0+f\u2032ce, from  and Lemm\u03c302 \u2260 0, from (F(u) is a monotone increasing for u \u2208  and monotone decreasing for u \u2208 /dS(dS + \u03b5 + p).Firstly, when U(I) = \u222bI(0)I(t)(1/g(I))dI; similarly to above proof of S2 = S02 + 2S0(S \u2212 S0)+(S \u2212 S0)2, we further havea + b)2 \u2264 2(a2 + b2), it follows that H(t), we easily have limt\u2192\u221e\u2329H2(t)\u232a = 0. By (S(t) + I(t) + R(t) \u2264 \u039b/dS\u2009\u2009a.s. for all t \u2265 0. Hence,t\u2192\u221e\u03a6(t) = 0. Therefore, taking t \u2192 \u221e in (Let tituting  and 21)U(I) = \u222bIther haveUIt\u2265\u03b2St\u2212dpression  of H(t), = 0. By , without0It.From , 89), a, aU(I) =lly have G2t\u2264M0It.t.From (91t\u222b0tIrdrIt.From (1t\u222b0tIrdrt \u2192 \u221e in  holds. TA(x) = h(x)hT(x), where x = ,aii(x)\u2009\u2009 the diagonal elements of matrix A(x). We have aii(x) = \u2211j=1lhij2(x).In this section, we discuss the stationary distribution of model  by usingH1) holds, f(S) = S, and there is a constant \u03c1 > 0 such that aii(x) > \u03c1, for any x \u2208 R+3 and i = 1,2, 3, \u03b3 > p, dI > dS, and \u03b3(dS + dR) > p(dI + dR). If R0 > 1 and S\u2217, I\u2217, R\u2217) is the unique endemic equilibrium of model (Assume that (> 1 and 9dS+p\u03b3dS+dg(I), the Lyapunov function structured in the following is different from that given in [We here use the Lyapunov function method to prove this theorem. The proof given here is similar to Theorem 5.1 in . But, dugiven in .V(x) and compact set K \u2282 R+3 such that LV(x)\u2264\u2212C for some C > 0 and x \u2208 R+3/K.By x =  \u2208 R+3. Define the function LV1(x), we have Denote LV2(x), we have Define the function LV3(x), we get Define the functionLV4(x), we get Define the functionR+3. Hence, we can easily obtain that there exists a constant C > 0 and a compact set K of R+3 such that, for any x \u2208 R+3/K, Define the Lyapunov function for model  as follohen from , 101), , 106)Vxhen from , and  = I/(1 + \u03c9I2), where \u03c9 is a positive constant. It is easy to verify that assumption (H1) holds. By Milstein's higher-order method [\u03beji\u2009\u2009 are N-distributed independent Gaussian random variables and \u0394t > 0 is time increment.Throughout the following numerical simulations, we choose r method , 30, we f(S) = S/(1 + 0.2S), \u039b = 1.85, q = 0.52, \u03b2 = 0.52, p = 0.24, \u03b5 = 0.2, \u03b3 = 0.3, \u03c9 = 2, dS = 0.4, dI = 0.21, dR = 0.3, \u03c301 = 0.15, \u03c302 = 0.99, \u03c311 = 0.23, \u03c312 = 0.17, \u03c321 = 0.14, \u03c322 = 0.72, \u03c331 = 0.47, and \u03c332 = 0.93. By computing, we obtain \u03c302f(S0)g\u2032(0)\u2212(\u03b2 + \u2211j=12\u03c3j0\u03c3j2) = 0.3442 > 0, and (\u03b2 + \u2211j=12\u03c3j0\u03c3j2)2/2\u03c302 \u2212 (dI + \u03b3 + (1/2)\u03c322) = 0.005 > 0. This shows that conditions (a) and (b) of I(t) of model  holds. The disease I(t) in model (Assume (in model  is stochf(S) = S/(1 + 1.5S), \u039b = 3, q = 0.2, \u03b2 = 2.1, p = 0.3, \u03b5 = 0.8, \u03b3 = 0.1, \u03c9 = 2, dS = 0.5, dI = 0.8, dR = 0.4, \u03c301 = 0.8, \u03c302 = 1.2, \u03c311 = 0.3, \u03c312 = 0.75, \u03c321 = 0.45, \u03c322 = 0.8, \u03c331 = 0.8, and \u03c332 = 0.3. By computing, we obtain I(t) of model  holds. The disease I(t) in model (Assume (in model  is stochf(S) = S/(1 + 0.1S), \u039b = 1.2, q = 0.5, \u03b2 = 1.5, p = 0.9, \u03b5 = 1.1, \u03b3 = 0.9, \u03c9 = 2, dS = 0.6, dI = 0.35, dR = 0.4, \u03c301 = 0.4, \u03c302 = 0.2, \u03c311 = 0.1, \u03c312 = 0.45, \u03c321 = 0.2, \u03c322 = 0.1, \u03c331 = 0.2, and \u03c332 = 0.3. By computing, we obtain I(t) of model (I(t).In model , we takeH1) holds. The disease I(t) in model (Assume (in model  is stochf(S) = S, \u039b = 0.67, q = 0.02, \u03b2 = 1.7, p = 0.05, \u03b5 = 3, \u03b3 = 0.99, \u03c9 = 4, dS = 0.29, dI = 0.53, dR = 0.39, \u03c301 = 0.025, \u03c302 = 0.02, \u03c311 = 0.0121, \u03c312 = 0.01, \u03c321 = 0, \u03c322 = 0, \u03c331 = 0.02, and \u03c332 = 0.01. By computing, we obtain that the basic reproduction number for deterministic model  = . Furthermore, we can verify that there is a constant \u03c1 > 0 such that aii(x) > \u03c1 for any x \u2208 R+3\u2009\u2009, dI\u2009 \u2212 \u2009dS = 0.24 > 0, \u03b3\u2009 \u2212 \u2009p = 0.94 > 0, \u03b3(dS + dR) \u2212 p(dI + dR) = 0.6272 > 0, and S\u2217 = 1.4230, I\u2217 = 0.3845, and R\u2217 = 0.1372.In model , we takef(S) = S/(1 + 0.4S), \u039b = 2.5, q = 0.5, \u03b2 = 1.4, p = 0.7, \u03b5 = 0.9, \u03b3 = 0.51, \u03c9 = 1.89, dS = 0.7, dI = 0.45, dR = 0.58, \u03c301 = 0.4, \u03c302 = 0.2, \u03c311 = 0.21, \u03c312 = 0.1, \u03c321 = 0.1, \u03c322 = 0.24, \u03c331 = 0.2, and \u03c332 = 0.1. By computing, we obtain that the basic reproduction number for deterministic model  = . Furthermore, we can verify that there is not a constant \u03c1 > 0 such that aii(x) > \u03c1 for any x \u2208 R+3 and i = 1,2, 3, dI \u2212 dS = \u22120.25 < 0, \u03b3 \u2212 p = \u22120.19 < 0, \u03b3(dS + dR) \u2212 p(dI + dR) = \u22120.0682 < 0 and In model , we takeH1) holds. Model  \u2261 S and (1) \u03c3j0 = 0\u2009\u2009(1 \u2264 j \u2264 l) or (2) \u03c3j1 = \u03c3j2 = \u03c3j3 = 0\u2009\u2009(1 \u2264 j \u2264 l). However, for the general model (f(S) \u2260 S and \u2260\u2009\u2009(1 \u2264 j \u2264 l), whether we also can establish similar results still is an interesting open problem.The stochastic persistence and permanence in the mean of the disease for model  are estaal model , particu\u03b2f(S(t)) and f2(S(t))g\u2032(I(t)). If we may get v1 and v2 are two positive constants; then the following perfect result may be established.In fact, under the above case, from the proofs of Theorems H1) holds. If I in model (Assume that (in model  is stochf(S) is a nonlinear function. The best perfect result on the stationary distribution is to prove that model (Another important open problem is about the existence of stationary distribution of model , that isat model  possesseHowever, the numerical examples given in"}
{"text": "Scientific Reports 10.1038/s41598-017-00198-5, published online 13 March 2017Correction to: This Article contains an error in the Methods section under subheading \u2018Data analysis\u2019.E[\u03bbij|Xij])\u2009=\u2009ln(workerij)\u2009+\u2009\u03b20\u2009+\u2009bi0\u2009+\u2009\u03b21new criteriai\u2009+\u2009\u03b22working hourij\u2009+\u2009\u03b23salaryij\u2009+\u2009\u03b24unemployment rateij\u2009+\u2009\u03b25yeari\u201d\u201cln(should read:E[Yij|Xij])\u2009=\u2009ln(workerij)\u2009+\u2009\u03b20\u2009+\u2009b0i\u2009+\u2009\u03b21new criteriai\u2009+\u2009\u03b22working hourij\u2009+\u2009\u03b23salaryij\u2009+\u2009\u03b24unemployment rateij\u2009+\u2009\u03b25yeari\u201d\u201cln("}
{"text": "Scientific Reports6: Article number: 3827610.1038/srep38276; published online: 12062016; updated: 01252017This Article contains a typographical error. In the Results section,\u03b5o(497\u2009nm) of 50294\u2009\u00b1\u2009259\u2009M\u22121\u2009cm\u22121 for Ca2+-saturated GCaMP6f and 1564\u2009\u00b1\u20091140\u2009M\u22121\u2009cm\u22121 for Ca2+-saturated GCaMP6fu, brightness values of 29673 and 7197\u2009M\u22121\u2009cm\u22121 were obtained for GCaMP6f and GCaMP6fu, respectively.\u2019\u2018With measured molar extinction coefficients should read:\u03b5o(497\u2009nm) of 50294\u2009\u00b1\u2009259\u2009M\u22121\u2009cm\u22121 for Ca2+-saturated GCaMP6f and 15646\u2009\u00b1\u20091140\u2009M\u22121\u2009cm\u22121 for Ca2+-saturated GCaMP6fu, brightness values of 29673 and 7197\u2009M\u22121\u2009cm\u22121 were obtained for GCaMP6f and GCaMP6fu, respectively\u2019.\u2018With measured molar extinction coefficients"}
{"text": "There are a number of errors in Table 1. The \u201cApplication\u201d value for \u201cP38MAPK\u201d in line 4 should read \u201cWB\u201d. The \u201cDilution\u201d value for \u201cMAPKAPK2\u201d in line 13 should read \u201c1:1000(WB)\u201d. The \u201cDilution\u201d value for \u201cHSP27\u201d in line 15 should read \u201c1:1000\u201d. Please view the correct In Fig 5, Fig 5C shows the incorrect image under the \u201cAF 1.0 \u03bcM\u201d header for \u201c0h\u201d. Please view the correct"}
{"text": "PP2A is composed of a scaffolding subunit (A), a catalytic subunit (C) and a regulatory subunit (B) that is classified into four families including B, B\u2032, B\u2032\u2032 and B\u2032\u2032\u2032/striatin. Here, we found that a distinct PP2A complex regulates NF\u2010\u03baB signalling by dephosphorylation of IKK\u03b2, I\u03baB\u03b1 and RelA/p65. The PP2A core enzyme AC dimer and the holoenzyme AB\u2032\u2032\u2032C trimer dephosphorylate IKK\u03b2, I\u03baB\u03b1 and RelA, whereas the ABC trimer dephosphorylates I\u03baB\u03b1 but not IKK\u03b2 and RelA in cells. In contrast, AB\u2032C and AB\u2032\u2032C trimers have little effect on dephosphorylation of these signalling proteins. These results suggest that different forms of PP2A regulate NF\u2010\u03baB pathway signalling through multiple steps each in a different manner, thereby finely tuning NF\u2010\u03baB\u2010 and IKK\u03b2\u2010mediated cellular responses. I\u03baB, inhibitor of kappa BIKK\u03b2, inhibitor of kappa B kinase \u03b2IKK\u03b2KN, kinase\u2010negative IKK\u03b2 mutantNF\u2010\u03baB, nuclear factor kappa BPPPs, protein phosphatasesNuclear factor kappa B (NF\u2010\u03baB) is a critical transcription factor that regulates many cellular and organismal processes including immune and inflammatory responses, cellular growth and cell survival. The NF\u2010\u03baB signalling pathway is regulated by the phosphorylation of several proteins including inhibitor of kappa B (I\u03baB) kinase \u03b2 (IKK\u03b2), an inhibitory protein I\u03baB\u03b1 and an essential subunit of NF\u2010\u03baB RelA/p65 Aberrant activation of NF\u2010\u03baB is linked to various diseases such as inflammatory disorders and cancer. Thus, there are numerous regulatory mechanisms at multiple levels to ensure the tight control of NF\u2010\u03baB activity Among these phosphatases, PP2A is the most abundant, constituting approximately 1% of total cellular proteins A previous RNAi screen revealed that PP2A plays an important role in the regulation of NF\u2010\u03baB signalling and identified the core enzyme subunits, A\u03b1, A\u03b2, C\u03b1 and C\u03b2, as negative regulators of the NF\u2010\u03baB signalling pathway in mouse astrocytes m L\u2010glutamine at 37 \u00b0C.cDNA\u2010encoding A\u03b1, A\u03b2, C\u03b1, C\u03b2, B\u03b1, B\u03b2, B\u03b3, B\u03b4, B\u2032\u03b1, B\u2032\u03b2, B\u2032\u03b3, B\u2032\u03b4, B\u2032\u03b5, B\u2032\u2032\u03b2, Strn and Strn3 subunits of PP2A were amplified from a human cDNA library by PCR. The cDNAs were inserted into pRK\u2010HA and pRK\u2010Flag expression vectors. Expression plasmids encoding IKK\u03b2, kinase\u2010negative IKK\u03b2 mutant (IKK\u03b2KN) and I\u03baB\u03b1 have been reported previously \u22121 TNF\u2010\u03b1 for 18 h. After lysing cells with buffer from the assay system, luciferase activity was analysed following the manufacturer's instructions.Nuclear factor kappa B activity was estimated by Dual\u2010Luciferase Reporter Assay System (Promega) using pNF\u2010\u03baB and pRL\u2010TK Luciferase reporter plasmids. After transfection of 0.01 \u03bcg NF\u2010\u03baB reporter plasmids with or without 0.01 \u03bcg expression plasmids of various PP2A subunits, IKK\u03b2, I\u03baB\u03b1 and/or RelA into GP2\u2010293 cells in collagen\u2010coated 96\u2010well dishes, cells were incubated in the presence or absence of 50 ng\u00b7mLm Tris/Cl, pH 7.5, 150 mm NaCl, 1 mm EGTA, 10 mm MgCl2, 60 mm \u03b2\u2010glycerophosphate, 1 mm Na3VO4, 1 mm 4\u2010amidino phenyl methyl sulfonyl fluoride, 50 KIU\u00b7mL\u22121 aprotinin, 20 \u03bcg\u00b7mL\u22121 pepstatin, 20 \u03bcg\u00b7mL\u22121 leupeptin, 2 mm DTT and 1% Triton X\u2010100. After centrifugation at 16 000 \u00d7 g for 20 min at 4 \u00b0C, the supernatants were used as cell lysates. The cell lysates were subjected to SDS/PAGE and then gel\u2010separated proteins were transferred to PVDF membranes (Millipore) and subjected to immunoblotting using the SuperSignal West Pico Chemiluminescence System (Pierce). Antibodies used were as follows: anti\u2010Flag (M2) , anti\u2010HA , anti\u2010I\u03baB\u03b1 , anti\u2010RelA/p65 , anti\u2010phospho\u2010I\u03baB\u03b1 , anti\u2010phospho\u2010RelA/p65(S468) , anti\u2010phospho\u2010RelA/p65(S536) , anti\u2010PP2A , anti\u2010PP2A , anti\u2010IKK\u03b2  and anti\u2010phospho\u2010IKK\u03b1/\u03b2 .GP2\u2010293 cells in collagen\u2010coated 12\u2010well dishes were transfected with 0.2 \u03bcg expression plasmids of various PP2A subunits, IKK\u03b2, I\u03baB\u03b1 and/or RelA. After 24\u2010h transfection, cells were washed with PBS and solubilized with buffer A consisting of 20 mvia anti\u2010Flag (M2) agarose beads (Sigma). IKK\u03b2 proteins were eluted in a buffer containing 20 mm Tris/Cl, pH 7.5, 150 mm NaCl, 2 mm DTT, 1% Triton X\u2010100 and 3 \u00d7 Flag peptides (Sigma) from anti\u2010Flag\u2010agarose beads with ultra\u2010free Centrifugal Filter Units (Millipore). For the dephosphorylation assay of I\u03baB\u03b1 and RelA, cells were transfected with plasmids encoding Flag\u2010RelA, HA\u2010I\u03baB\u03b1RR and IKK\u03b2EE, and then phosphorylated I\u03baB\u03b1RR and RelA protein complexes were purified from cell lysates via anti\u2010Flag (M2) agarose beads. PP2A proteins were purified from GP2\u2010293 cells following transfection with plasmids encoding the Flag\u2010tagged A\u03b1 subunit and HA\u2010tagged C\u03b1 subunit, together with or without HA\u2010tagged B subunits including B\u03b1, B\u03b2, B\u03b3, B\u03b4, B\u2032\u03b1, B\u2032\u03b2, B\u2032\u03b3, B\u2032\u03b4, B\u2032\u03b5, B\u2032\u2032\u03b2, Strn and Strn3. PP2A core enzyme composed of A\u03b1 and C\u03b1, and PP2A holoenzyme composed of A\u03b1, C\u03b1 and various B subunits were purified from cell lysates via anti\u2010Flag (M2) agarose beads with ultra\u2010free Centrifugal Filter Units in the previously described buffer. Phosphorylated IKK\u03b2 proteins (0.001 \u03bcg) or protein complexes (0.001 \u03bcg) of phosphorylated I\u03baB\u03b1RR and RelA were incubated with purified PP2A (0.001 \u03bcg) for 30 min at 37 \u00b0C, and then reaction mixtures were subjected to SDS/PAGE. Gel\u2010separated proteins were transferred to PVDF membranes and subjected to immunoblotting using anti\u2010phospho\u2010IKK\u03b1/\u03b2, anti\u2010phospho\u2010I\u03baB\u03b1 and anti\u2010phospho\u2010RelA antibodies, respectively.For the dephosphorylation assay of IKK\u03b2, GP2\u2010293 cells in 10\u2010cm dishes were transfected with expression plasmids encoding Flag\u2010IKK\u03b2, and then phosphorylated IKK\u03b2 was recovered from cell lysates PP2A is composed of three subunits scaffolding (A), regulatory (B) and catalytic (C) subunit Fig. A. Cells To investigate the involvement of the PP2A in regulation of NF\u2010\u03baB signalling, we constructed expression plasmids encoding each subunit of the B, B\u2032\u2032, B\u2032\u2032 and B\u2032\u2032\u2032/Strn family (Fig. We next investigated the effects of distinct B subunits on phosphorylation of the IKK\u03b2 activation loop. Cells were transfected with expression plasmids encoding IKK\u03b2 and plasmids encoding A\u03b1, C\u03b1 and each B subunit, and phosphorylation of the activation loop was investigated by immunoblotting. Expression of the AC core enzyme markedly suppressed IKK\u03b2 phosphorylation in a phosphatase activity\u2010dependent manner Fig. A. The ACTo reveal the effects of each B subunit on the NF\u2010\u03baB/I\u03baB\u03b1 complex, we transfected cells with expression plasmids encoding I\u03baB\u03b1RR, RelA and IKK\u03b2EE, together with plasmids encoding A\u03b1, C\u03b1 and each B subunit. Then, the effects of PP2A on IKK\u03b2EE\u2010mediated phosphorylation of I\u03baB\u03b1RR and RelA were investigated by immunoblotting. Expression of the AC core enzyme suppressed phosphorylation of I\u03baB\u03b1RR and RelA in a phosphatase activity\u2010dependent manner Fig. C. AC, ABThen, we investigated the effects of distinct B subunits on phosphorylation of endogenous NF\u2010\u03baB signalling proteins in TNF\u2010\u03b1\u2010stimulated cells. Cells were transfected with expression plasmids of PP2A, and then stimulated with TNF\u2010\u03b1. AC and AB\u2032\u2032\u2032C suppressed TNF\u2010\u03b1\u2010induced phosphorylation of IKK\u03b2 and RelA, whereas ABC, AB\u2032C and AB\u2032\u2032C had little effect Fig. E. AC, ABin vitro. Expression plasmids encoding Flag\u2010tagged A\u03b1, HA\u2010tagged C\u03b1 and HA\u2010tagged B subunits were transfected in cells, and PP2A complexes of various isoforms of B subunit family were purified from cells by using Flag\u2010agarose beads (Fig. in vitro, AC and AB\u2032\u2032\u2032C preferentially dephosphorylate IKK\u03b2 and RelA, and AC, ABC and AB\u2032\u2032\u2032C preferentially dephosphorylate I\u03baB\u03b1 in vivo, suggesting that intracellular mechanisms may relate to the preference of AC, ABC and AB\u2032\u2032\u2032C to dephosphorylate IKK\u03b2, I\u03baB\u03b1 and RelA.Dephosphorylation of IKK\u03b2, I\u03baB\u03b1RR and RelA was analysed by using purified PP2A enzyme ads Fig. A. Phosphads Fig. B. Followads Fig. C. Then, ads Fig. D. Followads Fig. E. These in vitro and in vitro, we investigated intracellular mechanisms of the preference of AC, ABC, AB\u2032\u2032C and AB\u2032\u2032\u2032C to dephosphorylate NF\u2010\u03baB signalling proteins. The effects of B subunits on the interaction between PP2A and IKK\u03b2 were analysed by immunoprecipitation assay. Cells were transfected with plasmids encoding Flag\u2010tagged C\u03b1D85N, HA\u2010tagged A\u03b1, HA\u2010tagged B subunits and HA\u2010tagged IKK\u03b2. PP2A complexes were immunoprecipitated by using Flag\u2010agarose beads from cells, and the association of IKK\u03b2 proteins to the PP2A complexes was analysed by immunoblotting (Fig. To resolve the discrepancies in the dephosphorylation of IKK\u03b2 and RelA between in vitro and in vivoin vivoMany studies have revealed important and complicated roles of PP2A in IKK\u03b2 regulation. For example, it has been reported that PP2A suppresses IKK\u03b2 activity NF\u2010\u03baB signalling is not uniformly regulated by PP2A but rather is subjected to specific regulation by distinct PP2A complexes at multiple steps. In contrast to AC and AB\u2032\u2032\u2032C, which dephosphorylate IKK\u03b2, I\u03baB\u03b1 and RelA, ABC dephosphorylates I\u03baB\u03b1 but not IKK\u03b2 and RelA in cells. These results indicate that ABC suppresses NF\u2010\u03baB activity without inhibition of IKK\u03b2. A recent study revealed that IKK\u03b2 not only activates NF\u2010\u03baB through the phosphorylation of I\u03baB\u03b1 but also regulates many cellular functions by phosphorylating various proteins in an NF\u2010\u03baB\u2010independent manner via A subunits of PP2A in the complexes. STRIPAK complexes have a critical role in protein dephosphorylation and act as important regulators of multiple vital signalling pathways, including the Hippo pathway, mitogen\u2010activated protein kinases and cytoskeleton remodelling. Recent studies suggest that the dysregulation of STRIPAK complexes correlates with human diseases including cancer Among the four classes of B subunit families, the B\u2032\u2032\u2032/Strn family proteins only facilitate dephosphorylation of three proteins including IKK\u03b2, I\u03baB\u03b1 and RelA. Members of the B\u2032\u2032\u2032/Strn family are evolutionarily conserved and have critical roles in biological processes such as development and cell growth PP2A is a confirmed tumour suppressor protein that is genetically altered or functionally inactivated in many cancers HK and YT designed and performed the experiments, interpreted the study, and wrote the paper. KO, MK, YN, ET and TH performed the experiments."}
{"text": "Scientific Reports6: Article number: 38356; 10.1038/srep38356 published online: 12062016; updated: 12222017.This Article contains errors in the legend of Figure 3.c\u2009=\u2009\u03bc|a\u232a\u2009+\u2009\u03bd|b\u232a\u03c8should read:c\u2009=\u2009\u03bc|0\u232a\u2009+\u2009\u03bd|1\u232a\u03d5In addition,a\u2009=\u2009\u03bc|a\u232a\u2009+\u2009\u03bd|b\u232a\u03d5should read:a\u2009=\u2009\u03bc|a\u232a\u2009+\u2009\u03bd|b\u232a\u03c8"}
{"text": "AbstractLeptomias Faust, 1886 occurring in the Sichuan Province of China, including the description of a new species, Leptomiasverticalis Ren, Zhang & Song, sp. n. from Jiulong County, Southwest Sichuan. New locality data and remarks for the other eleven species, a key to and distribution map of all twelve Sichuan species are provided. Leptomiaschenae Alonso-Zarazaga & Ren is transferred to Geotragus Schoenherr, 1845, where its valid name is G.granulatus , comb. n. in application of Art. 59.4. Structural details of Leptomiasverticalis and Geotragusgranulatus are illustrated.An account is given of the twelve species of  Leptomias Faust, 1886 is a diverse genus of flightless weevils  in the subfamily Entiminae, with a centre of distribution in China, India, Nepal, Afghanistan and Myanmar. It differs from related genera by having (i) the metanepisternum completely separated from the metaventrite and (ii) the upper edge of the scrobes directed towards the lower margin of the eye  does not belong in Leptomias but instead to the genus Geotragus Schoenherr, 1845. With the transfer of this species out of Leptomias effected and the new species described in this paper, the number of current species of Leptomias remains at 159 and that of species recorded from China at 89. The new species and the new combination are here documented and illustrated, along with a key to the species present in Sichuan. Moreover, species documented in Sichuan are usually described in Chinese and it might be difficult for most people to obtain information of their distribution.Ninety per cent of the Institute of Zoology, Chinese Academy of Sciences, Beijing, China (IZCAS); Forschungsmuseum Alexander Koenig, Bonn, Germany; Natural History Museum, London, UK; Naturkundemuseum, Berlin, Germany; Senckenberg Naturforschendes Museum, Frankfurt am Main, Germany; Senckenberg Naturhistorische Sammlungen, Dresden, Germany. The types of the new species are deposited in IZCAS.All specimens, including types, examined for this study are located in the following collections: Specimens were dissected after soaking them in soapy water overnight, for cleaning and softening, and the dissected parts were placed in a cold 10 % NaOH solution for 20 hours to macerate the soft tissues. After dissection, all parts were photographed and stored in glycerine in microvials pinned beneath the specimen from which they were dissected.The morphological terminology used in this study mainly follows Ren (2013). Measurements were made using an ocular micrometre as follows: standard length \u2013 in dorsal view from anterior margin of thorax to apex of elytra along midline; pronotal length \u2013 in dorsal view from anterior margin to base along midline; pronotal width \u2013 in dorsal view across widest part; elytral length \u2013 in dorsal view along suture of elytra from base to apex; elytral width \u2013 in dorsal view across widest part; rostral length \u2013 in lateral view in a straight line from apex to anterior margin of eye; rostral width \u2013 in dorsal view across base of rostrum. Measurements are made in millimetres.All observations and dissections were performed using a Nikon SMZ1500 stereo microscope. The habitus photographs were taken with a MP-E 65 macro lens mounted on a CANON EOS700D digital camera. Other photographs were taken with a CCD Qimagine MicroPublisher 5.0 RTV camera mounted on a Zeiss SteREO Discovery V.12 microscope. Extended-focus images were generated with Auto-Montage Pro 5.03.0061 and edited with Adobe Photoshop CS 14.0 if required.verbatim, with pinyin romanisation and comments in square brackets if labels are in Chinese; labels are separated by semicolons and lines by slashes.Label data are given Taxon classificationAnimaliaColeopteraCurculionidaeRen, Zhang & Songsp. n.http://zoobank.org/02A25235-8C44-4C3D-9270-AAD7BB9E9EC6L.ochrolineatus Chen, 1987 but differs by the following characters: elytra in lateral view abruptly sloping posteriorly, dorsal edge of slope of declivity straight, almost parallel to anterior margin; elytra in dorsal view at apical 1/3 with symmetrical crescent-shaped dark brown patches.This new species resembles Measurements (mm): Standard length: 8.00; pronotal length: 3.00; pronotal width: 3.00; elytral length: 5.00; elytral width: 3.20; rostral length: 1.30; rostral width: 1.20.Holotype, male. Habitus and colour : Standard length: 8.20; pronotal length: 3.20; pronotal width: 3.20; elytral length: 5.00; elytral width: 3.30; rostral length: 1.35; rostral width: 1.20; inner margin of protibiae with eleven blunt teeth (apex worn out), mesotibiae with ten small sharp teeth.Female paratypes. Measurements (in mm): Standard length: 10.40\u201311.30; pronotal length: 3.00\u20133.20; pronotal width: 3.10\u20133.20; elytral length: 6.00\u20136.70; elytral width: 3.70\u20133.90; rostral length: 1.29\u20131.40; rostral width: 1.28\u20131.32. Pronotum with anterior and posterior margins not truncate, slightly curved; greatest width just behind midpoint. Elytra much longer and wider than in male; in lateral view with posterior declivity straight and overhanging elytral apex; ventrite 5 : \u56db\u5ddd\u4e5d\u9f99\u53bf\u5357 [S\u00ecchu\u0101n J\u01d0ul\u00f3ngxi\u00e0n n\u00e1n] J08 / 2120m \u6838\u6843\u6797 [H\u00e9t\u00e1ol\u00edn] \u676f\u8bf1 [b\u0113iy\u00f2u] / 2001.VII.9\u201312 \u4e8e\u6653\u4e1c [Y\u00fa Xi\u01ceod\u014dng] / \u4e2d\u79d1\u9662\u52a8\u7269\u6240 ; : IOZ(E) 1965001. Paratypes : 1 \u2642: : same data as holotype except IOZ(E) 1965002; 2 \u2640: : same data as holotype except IOZ(E) 1965003, IOZ(E) 1965004; 2 \u2640: : same data as holotype except J09 / 2200m \u9752\u5188\u704c\u4e1b [Q\u012bngg\u0101ng Gu\u00e0nc\u00f3ng] and IOZ(E) 1965005, IOZ(E) 1965006.The specific epithet refers to the straight declivity of the elytra in lateral view.Sichuan (China).Taxon classificationAnimaliaColeopteraCurculionidaeChao, 1981Leptomiaselongitus Chao, 1981. Insects of Xizang 1: 543, pl. III\u201319.Holotype, \u2642: (white): \u897f\u85cf  \u8292\u5eb7  / 3800 \u516c\u5c3a  / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1976.VI.9 [handwritten] / \u91c7\u96c6\u8005: \u97e9\u5bc5\u6052 ; : HOLOTYPE; : IOZ(E) 906241. Paratypes: 1 \u2640: same data as holotype except ALLOTYPE printed on sea-green paper and IOZ(E) 906242. 1 \u2640: same data as holotype except PARATYPE printed on yellow paper and IOZ(E) 906243. 4 \u2640, 2 \u2642: same data as holotype except locality \u8292\u5eb7\u76d0\u4e95 , 2700 \u516c\u5c3a , collecting date 1976.VI.3, PARATYPE printed on yellow paper and IOZ(E) 906244\u2013IOZ(E) 906249. 1 \u2640: (white): \u897f\u85cf\u8292\u5eb7\u4ec0\u8349  / 19 [printed] 76 [handwritten] \u5e74  7 [handwritten] \u6708  15 [handwritten] \u65e5  / \u91c7\u96c6\u8005  2600 \u7c73  / \u4e2d\u56fd\u79d1\u5b66\u9662 ; : PARATYPE; : IOZ(E) 906250. 2\u2640: ditto, IOZ(E) 906251, IOZ(E) 906252.1 \u2642: : 3 / \u5f97\u8363\u9c7c\u6839 [D\u00e9r\u00f3ng Y\u00fag\u0113n] / \u9ad8\u5c71\u65b0 [G\u0101osh\u0101nx\u012bn] / 3700 \u516c\u5c3a [G\u014dngch\u01d0], \u4f55\u591a\u5409 [H\u00e9 Du\u014dj\u00ed] 80.6.8; : IOZ(E) 906253. 1\u2640: ditto, IOZ(E) 906254.Leptomiaselongitus is known from the province of Sichuan (Derong) and Xizang (Mangkang). It is narrowly distributed in the southwest of Sichuan : 58.PageBreak\u8005: \u9648\u5143\u6e05  and IOZ(E) 906860. 1 \u2640: (white): \u56db\u5ddd\u5b9d\u5174  / \u7857\u789b  2200\u20132700m [printed] / 1963.VI. [printed] 25 [handwritten] / \u5929\u6d25\u81ea\u7136\u535a\u7269\u9986 ; (white): \u91c7\u96c6\u8005: \u718a\u6c5f ; : IOZ(E) 906836. 2 \u2640: ditto, IOZ(E) 906837, IOZ(E) 906830.1 \u2642: (white): \u56db\u5ddd  \u5eb7\u5b9a\u74e6\u65b0  / \u6c9f  1450m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VI\u201322 [handwritten] / \u91c7\u96c6\u8005 \u738b\u66f8\u6c38; : IOZ(E) 906879. 1 \u2642: (white): \u56db\u5ddd  \u5eb7\u5b9a  / 2600m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.V\u201330 [handwritten] / \u91c7\u96c6\u8005 \u738b\u66f8\u6c38 ; : IOZ(E) 906854. 1 \u2642: same data as 906854 except collecting date 1983.VI\u201325, \u91c7\u96c6Leptomiasfoveicollis is widely distributed in Sichuan , Liaoning (Changtu) and Heilongjiang (Haerbing). From southwest to northeast of China, this species has a wide distribution range. Leptomiasfoveicollis is widely distributed in the central-western region of Sichuan : 409, fig. 7.Holotype, \u2642: (white): \u56db\u5ddd  \u9a6c\u5c14\u5eb7  / 2500m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VIII.17 [handwritten] / \u91c7\u96c6\u8005 \u738b\u66f8\u6c38 ; : HOLOTYPE; : IOZ(E) 905588.Leptomiasglobosus is an endemic species of China which recorded from Maerkang, central region of Sichuan  Insects of the Hengduan Mountains Region 2: 843, fig. 10.PageBreak905925. Paratypes: 1 \u2640: same data as holotype except ALLOTYPE printed on sea-green paper and IOZ(E) 905926. 5 \u2642, 3 \u2640: same data as holotype except PARATYPE printed on yellow paper and IOZ(E) 905927\u2013905934. 1 \u2642: (white): \u56db\u5ddd  \u6cf8\u5b9a\u78e8\u897f\u6d77  / \u87ba\u6c9f 1550m  / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1982.IX.16 [handwritten] / \u91c7\u96c6\u8005 \u738b\u66f8\u6c38 ; : PARATYPE; : IOZ(E) 905935. 1 \u2640: ditto, IOZ(E) 905936. 1 \u2640: same data as holotype except PARATYPE printed on yellow paper, 1500m, \u91c7\u96c6\u8005: \u5f20\u5b66\u5fe0  and IOZ(E) 905937. 1 \u2642: ditto, IOZ(E) 905938.Holotype, \u2642: (white): \u56db\u5ddd  \u6cf8\u5b9a\u78e8\u897f  / 1650m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VI.20 [handwritten] / \u91c7\u96c6\u8005: \u9648\u5143\u6e05 ; : HOLOTYPE; : IOZ(E) PageBreakLeptomiasmoxiensis is also an endemic species of China. It is recorded only from Luding, the central region of Sichuan : 397\u2013398, fig. 2.Holotype, \u2642: (white): \u897f\u85cf\u8292\u5eb7  / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1977.9.15 [handwritten] / \u91c7\u96c6\u8005  \u674e\u7ee7\u5747 ; : HOLOTYPE; : IOZ(E) 906290. Paratypes: 1 \u2640: same data as holotype except PARATYPE printed on yellow paper and IOZ(E) 906291.1 \u2642: (white): \u56db\u5ddd  \u4e61\u57ce  / 2900m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VI.28 [handwritten] / \u91c7\u96c6\u8005: \u5f20\u5b66\u5fe0 ; : IOZ(E) 906309. 2 \u2642: ditto, IOZ(E) 906310, IOZ(E) 906311. 1 \u2640: same data as 906309 except collecting date 1982.VI.17, \u91c7\u96c6\u8005: \u738b\u66f8\u6c38  and IOZ(E) 906293. 2 \u2640: ditto, IOZ(E) 906294, IOZ(E) 906296.Leptomiasnubilus is recorded from Sichuan (Xiangcheng) and Xizang (Mangkang). Xiangcheng is located in the southwest region of Sichuan : 406\u2013407, fig. 4.PageBreak905521. 8 \u2640, 4 \u2642: same data as holotype except PARATYPE printed on yellow paper and IOZ(E) 905522\u2013905528, IOZ(E) 905531, IOZ(E) 905537, IOZ(E) 905540\u2013905542. 3 \u2640, 6 \u2642: same data as holotype except PARATYPE printed on yellow paper, collecting date 1983.VII.4 and IOZ(E) 905529, IOZ(E) 905530, IOZ(E) 905532\u2013905536, IOZ(E) 905538, IOZ(E) 905539.Holotype, \u2642: (white): \u56db\u5ddd  \u5fb7\u683c  / 3200m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VII.6 [handwritten] / \u91c7\u96c6\u8005: \u9648\u5143\u6e05 ; : HOLOTYPE; : IOZ(E) 905520. Paratypes: 1 \u2640: same data as holotype except ALLOTYPE printed on sea-green paper and IOZ(E) Leptomiasochrolineatus is endemic to China and collected only from Sichuan (Dege). It is narrowly distributed in the northwest region of Sichuan : 404\u2013405, fig. 1.PageBreakprinted on yellow paper, collecting date 1981.VII.28 and IOZ(E) 905450, IOZ(E) 905456, IOZ(E) 905459, IOZ(E) 905460, IOZ(E) 905464, IOZ(E) 905475, IOZ(E) 905479, IOZ(E) 905484. 1 \u2642, 2 \u2640: same data as holotype except PARATYPE printed on yellow paper, collecting date 1981.VII.27 and IOZ(E) 905451, IOZ(E) 905469, IOZ(E) 905470. 1 \u2642, 1\u2640: same data as holotype except PARATYPE printed on yellow paper, collecting date 1981.VII.27, IOZ(E) 905453, IOZ(E) 905455 and with  \u91c7\u96c6\u8005 \u738b\u66f8\u6c38 . 1 \u2642, 1 \u2640: same data as 905455 except collecting date 1981.VII.28 and IOZ(E) 905457, IOZ(E) 905480. 1 \u2642, 1 \u2640: same data as holotype except PARATYPE printed on yellow paper, collecting date 1981.VII.27, IOZ(E) 905476, IOZ(E) 905487 and with  \u91c7\u96c6\u8005 \u5f20\u5b66\u5fe0 . 1 \u2642, 1 \u2640: same data as holotype except PARATYPE printed on yellow paper, IOZ(E) 905486, IOZ(E) 905490 and with  \u91c7\u96c6\u8005 \u738b\u66f8\u6c38 . 1 \u2640: (white): \u4e91\u5357  \u7ef4\u897f\u767d\u6d4e\u6c4e  / 1780m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1981.VII.12 [handwritten] / \u91c7\u96c6\u8005:  \u5ed6\u7d20\u67cf  / : PARATYPE; : IOZ(E) 905504. 1 \u2640: (white): \u4e91\u5357  \u7ef4\u897f\u767d\u6d4e\u6c4e  / 1780m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1981.VII.10 [handwritten] / 19\u53f7  / \u91c7\u96c6\u8005 \u738b\u66f8\u6c38  / : PARATYPE; : IOZ(E) 905505. 1 \u2640: (white): \u56db\u5ddd  \u5fb7\u683c  / 3200m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; 1983.VII.4 [handwritten] / \u91c7\u96c6\u8005: \u9648\u5143\u6e05 ; : PARATYPE; : IOZ(E) 905506.Holotype, \u2642: (white): \u4e91\u5357  \u7ef4\u897f\u6500\u5929  / \u9601  2500m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1981.VII.24 [handwritten] / \u91c7\u96c6\u8005:  \u5ed6\u7d20\u67cf  / : HOLOTYPE; : IOZ(E) 905433. Paratypes: 1 \u2640: same data as holotype except ALLOTYPE printed on sea-green paper and IOZ(E) 905434. 14 \u2642, 5 \u2640: same data as holotype except PARATYPE printed on yellow paper and IOZ(E) 905435\u2013905437, IOZ(E) 905448, IOZ(E) 905473, IOZ(E) 905482, IOZ(E) 905483, IOZ(E) 905485, IOZ(E) 905488, IOZ(E) 905489, IOZ(E) 905491\u2013905493, IOZ(E) 905495, IOZ(E) 905497\u2013905500, IOZ(E) 905503. 7 \u2642, 4 \u2640: same data as holotype except PARATYPE printed on yellow paper,  \u91c7\u96c6\u8005 \u5f20\u5b66\u5fe0  and IOZ(E) 905438, IOZ(E) 905442, IOZ(E) 905447, IOZ(E) 905452, IOZ(E) 905472, IOZ(E) 905477, IOZ(E) 905478, IOZ(E) 905481, IOZ(E) 905494, IOZ(E) 905501, IOZ(E) 905502. 12 \u2642, 5 \u2640: same data as holotype except PARATYPE printed on yellow paper, collecting date 1981.VII.26, IOZ(E) 905439\u2013905441, IOZ(E) 905443, IOZ(E) 905446, IOZ(E) 905449, IOZ(E) 905454, IOZ(E) 905458, IOZ(E) 905462, IOZ(E) 905463, IOZ(E) 905465\u2013905468, IOZ(E) 905471, IOZ(E) 905474, IOZ(E) 905496 and with \u5bc4\u4e3b: \u9ed1\u6843 . 1 \u2642, 2 \u2640: same data as holotype except PARATYPE printed on yellow paper, collecting date 1981.VII.26, IOZ(E) 905444, IOZ(E) 905445, IOZ(E) 905461 and with  \u91c7\u96c6\u8005 \u738b\u66f8\u6c38 . 4 \u2642, 4 \u2640: same data as holotype except PARATYPE Leptomiassublongicollis is recorded from Sichuan (Dege)  Fig.  and YunnTaxon classificationAnimaliaColeopteraCurculionidaeHeteromiasthibetanus Faust, 1888. Stett. Entomol. Zeit, 49(7\u20139): 285\u2013286.Leptomiasthibetanus (Faust): 1 \u2642: Thibet / Oayrollv ; Coll J. Faust / Ankauf 1900 ; Type ; Staatl. Museum f\u00fcr / Tierkunde Dresden .PageBreak\u9662 ; 1983.VI.10 [handwritten] / \u91c7\u96c6\u8005: \u9648\u5143\u6e05 ; : IOZ(E) 906817. 1 \u2640: (white): \u56db\u5ddd  \u7ea2\u539f  / 3500m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; 1983.VIII.27 [handwritten] / \u91c7\u96c6\u8005:  \u725b\u6625\u6765 ; : IOZ(E) 906819. 1 \u2642: (white): \u56db\u5ddd  \u8d21\u560e\u5c71  / \u71d5\u5b50\u6c9f 2500m  / \u4e2d\u56fd\u79d1\u5b66\u9662 ; 1983.VI.8 [handwritten] / \u91c7\u96c6\u8005: \u9648\u5143\u6e05 ; : IOZ(E) 906823. 1 \u2642: (white): \u56db\u5ddd  \u6cf8\u5b9a\u65b0\u5174  / 1900m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; 1983.VI.12 [handwritten] / \u91c7\u96c6\u8005: \u9648\u5143\u6e05 ; : IOZ(E) 906825.1 \u2640: (white): \u56db\u5ddd  \u8d21\u560e\u5c71  / \u71d5\u5b50\u6c9f 2500m  / \u4e2d\u56fd\u79d1\u5b66Leptomiasthibetanus is known from Sichuan  and Xizang. It is widely distributed from the central to northwest region of Sichuan : 405, fig. 2.Holotype, \u2642: (white): \u56db\u5ddd  \u5fb7\u683c  / 3200m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; 1983.VII.4 [handwritten] / \u91c7\u96c6\u8005: \u9648\u5143\u6e05 ; : HOLOTYPE; : IOZ(E) 905414. Paratypes: 1 \u2640: same data as holotype except ALLOTYPE printed on sea-green paper and IOZ(E) 905415. 6 \u2642, 3 \u2640: same data as holotype except PARATYPE printed on yellow paper and IOZ(E) 905416\u2013905424.Leptomiasvarians is an endemic species of China, collecting from Sichuan (Dege)  Fig. . L.variPageBreakTaxon classificationAnimaliaColeopteraCurculionidaeChen, 1992Leptomiaswenchuanensis Chen, 1992. In Chen S (Ed) Insects of the Hengduan Mountains Region 2: 843, fig. 11.PageBreakprinted] \u67f4\u6000\u6210 ; (white): \u56db\u5ddd  \u6c76\u5ddd  / \u6620\u79c0  / 900m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; : PARATYPE; : IOZ(E) 905922. 1 \u2640: (white): \u56db\u5ddd  \u9a6c\u5c14\u5eb7  / 2900m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VIII.18 [handwritten] / \u91c7\u96c6\u8005:  \u5f20\u5b66\u5fe0 ; : PARATYPE; : IOZ(E) 905923. 1 \u2640: (white): \u56db\u5ddd  \u9a6c\u5c14\u5eb7  / \u68a6\u7b14\u5c71  / 4000m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VIII.19 [handwritten] / \u91c7\u96c6\u8005: \u738b\u66f8\u6c38 ; : PARATYPE; : IOZ(E) 905924.Holotype, \u2642: (white): \u56db\u5ddd  \u6c76\u5ddd\u5367\u9f99  / 1920m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VII.24 [handwritten] / \u91c7\u96c6\u8005 \u738b\u66f8\u6c38 ; : HOLOTYPE; : IOZ(E) 905907. Paratypes: 1 \u2640: (white): \u56db\u5ddd  \u6c76\u5ddd  / \u5367\u9f99  1920m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VII.24 [handwritten] / \u91c7\u96c6\u8005 \u738b\u66f8\u6c38 ; : ALLOTYPE; : IOZ(E) 905908. 2 \u2642: same data as allotype except PARATYPE printed on yellow paper and IOZ(E) 905909, IOZ(E) 905910. 3 \u2640: same data as holotype except PARATYPE printed on yellow paper, collecting date 1983.VII.25, 1780m and IOZ(E) 905411, IOZ(E) 905413, IOZ(E) 905414. 1 \u2642: (white): \u56db\u5ddd  \u6c76\u5ddd  / \u5367\u9f99  1780m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VII.25 [handwritten] / \u91c7\u96c6\u8005 \u738b\u66f8\u6c38 ; : PARATYPE; : IOZ(E) 905912. 1 \u2640: ditto, IOZ(E) 905415. 1 \u2642: same data as 905912 except 1800m, \u91c7\u96c6\u8005 \u5f20\u5b66\u5fe0  and IOZ(E) 905416. 1 \u2640: (white): \u56db\u5ddd  \u5367\u9f99  / 2700m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VIII.9 [handwritten] / \u91c7\u96c6\u8005  \u725b\u6625\u6765 ; : PARATYPE; : IOZ(E) 905917. 1 \u2640: (white): \u56db\u5ddd  \u6c76\u5ddd  / 1300m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.IX.13 [handwritten] / \u91c7\u96c6\u8005: \u5f20\u5b66\u5fe0 ; : PARATYPE; : IOZ(E) 905918. 1 \u2640: (white): \u56db\u5ddd  \u6c76\u5ddd  / \u4e09\u5723\u6c9f 2500m  / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VIII.6 [handwritten] / \u91c7\u96c6\u8005: \u67f4\u6000\u6210 ; : PARATYPE; : IOZ(E) 905919. 1 \u2640: (white): \u56db\u5ddd  \u6c76\u5ddd\u6728\u6c5f\u576a [W\u00e8nchu\u0101n M\u00f9ji\u0101ngp\u00edng handwritten] / 1200m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VIII.8 [handwritten] / \u91c7\u96c6\u8005: \u67f4\u6000\u6210 ; : PARATYPE; : IOZ(E) 905920. 1 \u2640: (white): \u56db\u5ddd  \u6c76\u5ddd\u6620 \u79c0 [W\u00e8nchu\u0101n Y\u00ecngx\u00ecu handwritten] / 900\u20131000m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VIII.1 [handwritten] / \u91c7\u96c6\u8005: \u5f20\u5b66\u5fe0 ; : PARATYPE; : IOZ(E) 905921. 1 \u2640: (white): 1983.VIII.3 [handwritten] / \u91c7\u96c6\u8005  / 4200m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VII.13 [handwritten] / \u91c7\u96c6\u8005:  \u725b\u6625\u6765 ; : PARATYPE; : IOZ(E) 905958.Holotype, \u2642: (white): \u56db\u5ddd  \u5eb7\u5b9a  / 4200m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VII.13 [handwritten] / \u91c7\u96c6\u8005: \u9648\u5143\u6e05 ; : HOLOTYPE; : IOZ(E) 905953. Paratypes: 1 \u2640: same data as holotype except ALLOTYPE printed on sea-green paper and IOZ(E) 905954. 2 \u2642, 2 \u2640: same data as holotype except PARATYPE printed on yellow paper and IOZ(E) 905955, IOZ(E) 905957, IOZ(E) 905959, IOZ(E) 905961. 1 \u2642: (white): \u56db\u5ddd  \u5eb7\u5b9a\u6298\u591a  / \u5c71 \u57ad\u53e3 4200m  / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VII.13 [handwritten] / \u91c7\u96c6\u8005 \u738b\u66f8\u6c38 ; : PARATYPE; : IOZ(E) 905956. 1 \u2642: ditto, IOZ(E) 905960. 1 \u2642: (white): \u56db\u5ddd  \u5eb7\u5b9a  / 3100m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VI.24 [handwritten] / \u91c7\u96c6\u8005  \u9648\u5143\u6e05 ; : PARATYPE; : IOZ(E) 905962. 1 \u2640: ditto, IOZ(E) 905963. 1 \u2640: (white): \u56db\u5ddd  Leptomiaszheduoshanensis is another endemic species of China which recorded from Sichuan (Kangding). It is narrowly distributed in the central region of Sichuan comb. n.Leptomiasgranulatus Chao, 1980. Entomotaxonomia 2(1): 29.LeptomiaschenaeColeoptera 8: 89, 396 (replacement name for secondary homonymy). Alonso-Zarazaga & Ren, 2013. Catalogue of Palaearctic Geotragus is G.granulatus, not G.chenae, because of Art. 59.4 of the Code: \u201c59.4. Reinstatement of junior secondary homonyms rejected after 1960. A species-group name rejected after 1960 on grounds of secondary homonymy is to be reinstated as valid by an author who considers that the two species-group taxa in question are not congeneric, unless it is invalid for some other reason.\u201dThe correct name for this species under PageBreakshort and stout, exceeding anterior margin of eyes but not surpassing middle of eyes. Funicles with desmomere 1 elongate clavate, apical stout, 1.70\u00d7 longer than desmomere 2, distinctly wider than 2. Prementum with four setae. Prothorax transverse, sides evenly rounded, broadest behind middle, pronotum with extremely shallow, fine, incomplete, median longitudinal groove. Elytral interstriae slightly elevated, unequal in width, without tubercles. Proventriculus : \u56db\u5ddd\u5eb7\u5b9a [S\u00ecchu\u0101n K\u0101ngd\u00ecng] / 2400\u20132700 \u516c\u5c3a  / \u4e2d\u56fd\u79d1\u5b66\u9662 [Zh\u014dnggu\u00f3 K\u0113xu\u00e9yu\u00e0n]; (white): 1963.VII.28 [handwritten] / \u91c7\u96c6\u8005 \u5f20\u5b66\u5fe0 ; : HOLOTYPE; : IOZ(E) 906893.PageBreakPageBreakPageBreak29.68912 / E 102.06859 / \u4e2d\u56fd\u79d1\u5b66\u9662 [Zh\u014dnggu\u00f3 K\u0113xu\u00e9yu\u00e0n]; : IOZ(E) 1506001\u20131506006, IOZ(E) 1506022, IOZ(E) 1506032, IOZ(E) 1506033, IOZ(E) 1506052\u20131506057, IOZ(E) 1506063, IOZ(E) 1506064. 1 \u2642, 2 \u2640: same data as 1506064 except \u548c\u5e73\u7ec4 [H\u00e9p\u00edngz\u01d4], 1845m, 2011.VI.15, N 29.64937, E 102.09164, IOZ(E) 1506027, IOZ(E) 1506031 and IOZ(E) 1506062. 3 \u2642, 5 \u2640: same data as 1506064 except \u706b\u8349\u576a [Hu\u01d2c\u01ceop\u00edng], 2116m, 2011.VI.30, N 29.512675, E 102.133512, IOZ(E) 1506035, IOZ(E) 1506040\u20131506045 and IOZ(E) 1506061. 1 \u2642: (white): \u56db\u5ddd  \u6cf8\u5b9a\u78e8\u897f  / 1650m [handwritten] / \u4e2d\u56fd\u79d1\u5b66\u9662 ; (white): 1983.VI.20 [handwritten] / \u91c7\u96c6\u8005: \u9648\u5143\u6e05 ; : IOZ(E) 907104. 1 \u2642: same data as 907104 except \u91c7\u96c6\u8005 \u738b\u66f8\u6c38  and IOZ(E) 907106. 1 \u2640: same data as 907104 except 1500m, \u91c7\u96c6\u8005: [C\u01ceij\u00edzh\u011b] \u5f20\u5b66\u5fe0  and IOZ(E) 907009. 1 \u2640: (white): same data as 907104 except collecting date 1983.VI.17, and IOZ(E) 907010. 6 \u2642, 7 \u2640: same data as 907104 except \u56db\u5ddd\u6cf8\u5b9a , 1800m, 1983.VI.14 and IOZ(E) 906944\u2013906956.8 \u2642, 9 \u2640: : \u56db\u5ddd\u7518\u5b5c\u5dde\u6cf8\u5b9a\u53bf [S\u00ecchu\u0101n G\u0101nz\u012b Zh\u014du L\u00fad\u00ec Xi\u00e0n] / \u6298\u7530\u575d [Zh\u00e9ti\u00e1nb\u00e0] 2110m / 2011.VII.03 / \u4e2d\u56fd\u79d1\u5b66\u9662 [Zh\u014dnggu\u00f3 K\u0113xu\u00e9yu\u00e0n]; : leg. \u5f20\u534e\u5eb7 [Zh\u0101ng Hu\u00e1k\u0101ng] / N Geotragusgranulatus mainly occurs northeast and east of Gongga Mountain, which is the highest mountain in Sichuan province, China.Leptomias occurring in Sichuan Province, accounting for 14 % of the species presently known from China. Seven of them appear to be endemic to an area that stretches from Dege County to Wenchuan County (Fig. Leptomias diversity in China. The species occur in Sichuan at elevations between 900 and 4200 m, in a geographical rectangle delimited by 31\u00b048.600'N 98\u00b034.120'W and 31\u00b027.600'N 103\u00b036.600'W. Geotragus is recorded for the first time from Sichuan, which also presents a new northern-most record for the genus. Geotragus currently comprises 13 species, six of which occur in China.There are 12 species of nty Fig. . Seven e"}
{"text": "This significantly improves the previously best upper bound of 1.6667n by Fomin et\u00a0al. [STOC 2016] and 1.6740n by Gaspers and Mnich . Our new upper bound almost matches the best\u2010known lower bound of 21n/7\u22481.5448n, due to Gaspers and Mnich. Our proof is algorithmic, and shows that all minimal FVS of tournaments can be enumerated in time O(1.5949n).We study feedback vertex sets (FVS) in tournaments, which are orientations of complete graphs. As our main result, we show that any tournament on  This problem belongs to Karp's original list of 21 NP\u2010hard problems\u00a0The minimum fvs problem remains NP\u2010hard even in tournaments\u00a0T is a digraph with exactly one arc between any two of its vertices. Various approaches have been suggested to solve the minimum fvs problem on tournaments, including approximation algorithms\u00a0M(T) of minimal FVS in\u00a0T. Therefore, using this approach, the complexity of the minimum fvs problem in tournaments is within a polynomial factor of the maximum of\u00a0M(T) over all n\u2010vertex tournaments, which we denote by M(n).The M(n) was Moon\u00a01.4757n\u2264M(n)\u22641.7170n. This was improved by Gaspers and Mnich\u00a01.5448n\u2264M(n)\u22641.6740n. Very recently, an improvement on the upper bound was made by Fomin et\u00a0al.\u00a0M(n)\u22641.6667n. The problem of exactly determining M(n) was explicitly posed by Woeginger\u00a0The first one to provide nontrivial bounds on 1.1M(n). Our main combinatorial result is as follows:Theorem 1n has at most M(n)\u22641.5949n minimal FVS.Any tournament of order In this article, we make significant progress on establishing better bounds for\u00a0M(n) is attained by regular tournaments. For regular tournaments, we show an upper bound on M(n) that matches the lower bound:Theorem 2n has at most 21n/7 minimal FVS, and this is sharp: some regular tournament of order n has exactly 21n/7 minimal FVS.Any regular tournament of order We also consider regular tournaments , because the best\u2010known lower bound on M(n):The following Table n of nodes in the input tournament T. Starting with\u00a0T, they consider a vertex v with maximum out\u2010degree \u0394, and depending on the value of \u0394 and neighbors of\u00a0v, they construct subtournaments by deleting distinct vertices, such that each maximal transitive vertex set of T is contained in at least one subtournament. Applying the induction hypothesis to the subtournaments then implies their upper bound.Our proof of Theorem M for the maximum number of maximal transitive vertex sets in a tournament of order n containing a fixed set of\u00a0k vertices, and we will show that M\u22641.5949n\u2212k for all 0\u2264k\u2264n. A similar approach has been used by Gupta et\u00a0al.\u00a0r\u2010regular induced subgraphs in undirected graphs.Here, we use a refined technique, that yields upper bounds on the number of inclusion\u2010maximal vertex sets with certain properties. Namely, in addition to deleting vertices to generate subtournaments, we also keep fixed vertex sets. Within these subtournaments we only consider maximal transitive vertex sets that contain all the fixed vertices. We introduce a new function n can be listed in time O(1.5949n). Second, using an algorithm by Gaspers and Mnich\u00a0Corollary 1T of order n, all its minimal FVS can be listed in time M(T)\u00b7nO(1)=O(1.5949n) with polynomial delay and in polynomial space.Given any tournament Our combinatorial result has algorithmic consequences. First, our proof of Theorem\u00a0v with in\u2010degree 0 in a subtournament induced by a maximal transitive vertex set. Brandt et\u00a0al.\u00a0NP\u2010complete, a feasible approach to determine the Banks set is to enumerate all minimal FVS. For this purpose, Brandt et\u00a0al.\u00a0Enumerating the minimal FVS in tournaments has several interesting applications. For example, Banks\u00a02tournamentT= is a directed graph with exactly one edge between each pair of vertices. We denote the set of all tournaments with n vertices by Tn. A feedback vertex set (FVS) of T is a set F\u2286V(T) such that T\u2212F is free of (directed) cycles, where T\u2212F is the induced subgraph of\u00a0T after removing all vertices in F. An FVS is minimal if none of its proper subsets is an FVS.A M(T) the number of minimal FVS in a tournament T, and definen.Denote by T= be a tournament. For a set V\u2032\u2286V, let T[V\u2032] be the subtournament of\u00a0T induced by\u00a0V\u2032. For each v\u2208V, let N\u2212(v)={u\u2208V|\u2208A} and let N+(v)={u\u2208V|\u2208A}. We write v\u2192u if u\u2208N+(v) and call\u00a0v a predecessor of u and u a successor of v. For each v\u2208V, its in\u2010degree is d\u2212(v)=|N\u2212(v)| and its out\u2010degree is d+(v)=|N+(v)|; call Tregular if all its vertices have the same out\u2010degree. Let \u0394+(T) denote the maximum out\u2010degree over all vertices of T. Further,\u00a0T is strong if there is a directed path from v to u for each pair of vertices v,u\u2208V; let\u00a0Tn\u2605 denote the set of strong tournaments of order n. Note that any tournament can uniquely be decomposed into strong subtournaments S1,\u22ef,Sr such that v\u2192u for all v\u2208V(Si), u\u2208V(Sj) for all i<j.Observation 1T, we obtain M(T)=M(S1)\u00b7\u22ef\u00b7M(Sr).For any tournament Let M(n) from above by \u03b2n for some \u03b2 by considering strong tournaments of every order n.Therefore, we can bound Lemma 1In a tournament, any vertex contained in a cycle is contained in a directed triangle.Our proofs will use the following well\u2010known observation about cycles in tournaments:v1,\u22ef,v\u2113 be a shortest cycle containing v1 with \u2113>3, vi\u2192vi+1 for all i\u2208{1,\u22ef,\u2113\u22121} and v\u2113\u2192v1. Depending on the orientation of the arc between v1 and\u00a0v3, either v1,v2,v3 form a triangle or v1,v3,v4,\u22ef,v\u2113 is a shorter cycle containing\u00a0v1. \u25aaLet Henceforth, throughout the article by \u201ctriangle\u201d we always mean \u201cdirected triangle.\u201dtransitive if its induced subtournament is acyclic. Thus, a vertex set is a maximal transitive vertex set if and only if its complement is a minimal FVS. Instead of counting minimal FVS, we count maximal transitive vertex sets. The next property of maximal transitive vertex sets was already used by Moon\u00a0Lemma 2T, M(T)\u2264\u2211v\u2208V(T)M(d+(v)).For any tournament We call a vertex set W of T has a vertex v with in\u2010degree\u00a00 in T[W]. Hence, W is also a maximal transitive vertex set in T[N+(v)\u222a{v}]; this yields the bound. \u25aaAny maximal transitive vertex set M(T) in terms of a recurrence relation, in particular in combination with the next lemma that extends Lemma\u00a0Lemma 3n\u2208N and let T\u2208Tn\u2605. Then either T is regular, or for any d\u2208N at most 2d vertices in T have out\u2010degree at least n\u2212d\u22121.Let Lemma V\u223c be the set of vertices in T with out\u2010degree at least n\u2212d\u22121. Then any vertex in V\u223c has in\u2010degree at most d. Hence,Let V\u223c\u2260\u2205, for otherwise the statement of the lemma holds. We distinguish two cases.We may suppose that V\u223c\u2260V(T). Then, since T is strong and V\u223c\u2260\u2205, there is some arc from V(T)\u2216V\u223c to V\u223c. There are |V\u223c|2 arcs between vertices in V\u223c. Therefore, \u2211v\u2208V\u223c|N\u2212(v)|\u2265|V\u223c|2+1. Combining this inequality with\u00a0d\u2208N yields |V\u223c|\u22642d.Consider first the case that V\u223c=V(T). We may suppose that T is not regular, for otherwise the statement of the lemma holds. Note that not every vertex of V\u223c=V(T) can have in\u2010degree exactly d, since\u00a0T is not regular. Hence, some vertex in V\u223c has in\u2010degree at most d\u22121. Consequently,|V\u223c|2 arcs between vertices in V\u223c. Thus, \u2211v\u2208V\u223c|N\u2212(v)|\u2265|V\u223c|2. Combining these two inequalities and solving for d\u2208N yields |V\u223c|\u22642d. \u25aaSecond, consider the case that d vertices of out\u2010degree at least n\u2212d\u22121, as witnessed for instance by the triangle and d=1.We remark that a regular tournament may have more than 23M(n) of minimal FVS in any tournament of order\u00a0n is bounded from above by 1.5949n. For this purpose, for a tournament T and V\u2032\u2286V(T) let M be the number of maximal transitive vertex sets in T that contain all vertices in V\u2032. Also, letM(n)=M.In this section, we show that the maximum number Example.To clarify the definition, we compute M. Precisely, we show that M=2. There are two nonisomorphic tournaments for n=3:T1 is acyclic and thus has only a single maximal transitive vertex set, V(T1). Thus, M=1 for all v\u2208V(T1). The tournament T2 has three maximal transitive vertex sets, each consisting of exactly two vertices. Thus, each vertex of T2 is contained in exactly two maximal transitive vertex sets. This yields M=2 for all v\u2208V(T2). Summarizing, we get M=2.The tournament \u03b2=1.5949. We will show that M\u2264\u03b2n\u2212k for all n\u2208N and k\u2208{0,\u22ef,n}. To this end, ideally we would like to prove the following statements:(I)It holdsM\u2264\u03b2n\u2212kfor alln\u2265k>0.(II)It holdsM\u2264\u03b2n.Unfortunately, we are unable to do prove these directly. The reason is that our proof of Statement (I) for a fixed pair  with n\u2265k>0depends on the validity of Statement (II) for values n\u223c<n. Vice versa, our proof of the validity of Statement\u00a0(II) for fixed n\u2208N depends on the validity of Statement (I).Henceforth, fix Lemma 4n\u2208N. If M(n\u223c)\u2264\u03b2n\u223c and M\u2264\u03b2n\u223c\u2212k\u223c holds for all 0<k\u223c\u2264n\u223c<n, then M\u2264\u03b2n\u2212k for 0<k\u2264n.Let We will therefore establish the following two lemmas:Lemma 5n\u2208N. If M(n\u223c)\u2264\u03b2n\u223c, M\u2264\u03b2n\u223c\u2212k\u223c and M\u2264\u03b2n\u2212k\u223c for all 0<k\u223c\u2264n\u223c<n, then M(n)\u2264\u03b2n.Let The proof of Lemma\u00a0The proof of Lemma\u00a0We are ready to prove Theorem\u00a0Proof of Lemma 1n\u2208N, it holds M(n)\u22641.5949n. Clearly, M(1)\u22641\u22641.5949 and M\u22641\u22641.59491\u2212k for all k\u2208{0,1}. This yields our induction hypothesis. Lemma M(n) for all n\u2208N. \u25aaWe show that for all 4 be a minimum counterexample, that is, T is a tournament and V\u2032\u2286V(T) such that |V(T)|\u2212|V\u2032| is minimum and M>\u03b2|V(T)|\u2212|V\u2032|. Throughout this section, write n=|V(T)| and k=|V\u2032|>0.In this section, we prove Lemma M\u2264\u03b2n\u2212k for each of them; this yields the desired contradiction (and hence the truth of the statement of the lemma). In each case, we will use the minimality of  to bound M from above.Case 1:V\u2032 form a triangle.Three vertices in M=0\u2264\u03b2n\u2212k.Then, as no transitive vertex set contains all of these three vertices, Case 2:V\u2032 form a triangle with some vertex v\u2208V(T)\u2216V\u2032.Two vertices in V\u2032 does not contain v. Hence,Any transitive vertex set that contains all vertices in Case 3:v\u2208V\u2032 that is not contained in any cycle of T.There is a vertex W\u2287V\u2032 is a maximal transitive vertex set of T if and only if W\u2216{v}\u2287V\u2032\u2216{v} is a maximal transitive vertex set of T\u2212v. This yieldsThen, a set We will distinguish several cases and show that Remark 1M(n\u223c)<\u03b2n\u223c for n\u223c<n. The reason is that possibly V\u2032\u2216{v}=\u2205, in which case k\u22121=0 and we need that M\u2264\u03b2n\u22121.We remark that it is this case where we rely on the validity of Lemma  to which Cases 1\u20133 do not apply.Observation 2, then (i) any vertex of V\u2032 is contained in at least one triangle \u2264\u03b20n\u2212k for \u03b20=1.6181 (under the conditions imposed by the lemma). By\u00a0 Observation v\u2208V\u2032 that forms a triangle with two vertices w1,w2\u2209V\u2032. Any maximal transitive vertex set W\u2287V\u2032 (and thus containing v) cannot contain both w1 and w2. Therefore, w1\u2208W implies w2\u2209W and we get\u03b20n for \u03b20=1.6181.We remark that with Case 1\u20133 we can already show a bound of \u03b20=1.6181 to \u03b2=1.5949.The subsequent cases allow us to improve Case 4:w\u2209V\u2032 that is contained in two distinct triangles, both of which contain a vertex from V\u2032 (possibly shared by both triangles). Then we are in one of two cases, where vertices in V\u2032 are circled:There is a vertex , be distinct triangles containing w, such that v1,v2\u2208V\u2032 where possibly v1=v2. Let W be a maximal transitive vertex set of T containing\u00a0V\u2032. Then either w\u2209W or w\u2208W. Clearly, if w\u2208W then u1,u2\u2209W. We therefore have\u03b2n\u2212k, since \u03b2\u22651.4656.Case 5:v\u2208V\u2032 and w1,w2\u2208V(T)\u2216V\u2032 that form a triangle, such that\u00a0w1 also belongs to triangles , for some u1,u2,u3\u2208V(T)\u2216{v,w2}.There are vertices Let u1,u2,u3\u2208V(T)\u2216V\u2032, as otherwise Case 2 or Case 4 would apply. Any transitive vertex set W\u2287V\u2032 either contains w1 or not. If w1\u2208W then\u00a0w2\u2209W. Moreover, w1\u2208W implies that either u2\u2209W, or u2\u2208W but u1,u3\u2209W. Thus,\u03b2n\u2212k, since \u03b2\u22651.5702.Then we can assume that . Then some vertex v0\u2208V\u2032 forms a triangle with some w1,w2\u2208V(T)\u2216V\u2032, as Cases 1\u20133 do not apply. For i=1,2, let \u0394i be the set of triangles ti= that are disjoint from w3\u2212i and for which T is acyclic for all v\u2032\u2208V\u2032. Consequently, all triangles in \u03941\u222a\u03942 are disjoint from V\u2032, as Case\u00a04 does not apply. Further, all triangles in \u0394i are pairwise edge\u2010disjoint (as Case 5 does not apply), and therefore intersect only in\u00a0wi.Henceforth, we assume that Cases 1\u20135 do not apply to M, we again distinguish the maximal transitive vertex sets that contain w1 or w2, from those that do not contain either of them. Let W be a maximal transitive vertex set of T containing V\u2032.To prove an upper bound on w1,w2\u2209W. Then, T[W\u222a{wi}] contains a cycle for i=1,2, by maximality of W. Thus, by Lemma t= for some z1,z2\u2208W. We have that t\u2208\u0394i, since z1,z2 do not form a triangle with any v\u2032\u2208V\u2032 as z1,z2\u2208W. Thus, those W with w1,w2\u2209W can be partitioned into |\u0394i| classes, where the r\u2010th class contains the sets W that contain the two vertices of the r\u2010th triangle in \u0394i.First consider that \u03941\u222a\u03942. Consider two triangles tir=,tis=\u2208\u0394i:To use this argument effectively, we need some further observations about the relation among triangles in wi are pairwise edge\u2010disjoint (as Case 5 does not apply), the edge between\u00a0uir and vis has to be directed from vis to uir; else, wi,uir,vis would form a triangle that is not edge\u2010disjoint from the triangle wi,uir,vir. Likewise, the edge between\u00a0uis and vir has to be directed from\u00a0vir to uis. Ignoring symmetries obtained by swapping the roles of tir and tis, there are only two possibilities how the two remaining edges  can be oriented:Since all triangles that contain Case A, and to the situation in the right figure as Case B. Note that in Case A,  and  form triangles; while in Case\u00a0B, triangles are formed by  and .Observation 3uis,vis\u2208W implies that uir,vir\u2209W. In Case B, uir,vir\u2208W implies that vis\u2209W; and uis,vis\u2208W implies that uir\u2209W.In Case A, We refer to the situation in the left figure as tir=\u2208\u0394i let Vtir be the set of vertices that are excluded from those W with uir,vir\u2208W due to Observation Thus, for each 1 and \u03942 are vertex\u2010disjoint. Therefore, for each tir\u2208\u0394i, every vertex in Vtir is not contained in any triangle of\u00a0\u03943\u2212i. This implies that for any pair of triangles t1\u2208\u03941,t2\u2208\u03942 the sets Vt1,Vt2 are disjoint. Altogether, this means that we can bound the number of maximal transitive vertex sets W\u2287V\u2032 not containing w1,w2 from above byIn Lemma\u00a0i\u2208{1,2}. Let ti1,\u22ef,ti|\u0394i| be an ordering of the triangles in \u0394i such that |Vtir|\u2264|Vtis| for 1\u2264r<s\u2264|\u0394i|. Then for any pair r,s\u2208{1,\u22ef,\u0394i} with r\u2260s, Observation r<s, since \u03b2\u22651, we getr<s,|Vtir|=2(r\u22121) for all r=1,\u22ef,|\u0394i|. We obtain(\u03b22\u03b22\u22121)\u00b7(\u03b22\u03b22\u22121)=\u03b24(\u03b22\u22121)2.Thus, our goal is now to bound (\u22c6). Fix 1 is disjoint from every triangle in\u00a0\u03942.Lemma 6v0,w1,w2,\u03941, and \u03942 be defined as before. Then any triangle in\u00a0\u03941 is vertex\u2010disjoint from every triangle in \u03942.Let Let us now prove that indeed any triangle in\u00a0\u0394V\u2032 is a transitive set, as Case 1 does not apply. Thus, the vertices in\u00a0V\u2032 admit a topological order such that vx\u2032\u2192vy\u2032 for all vx\u2032,vy\u2032\u2208V\u2032 with x>y. Second, for each vertex z\u2208V(T)\u2216V\u2032 the set V\u2032\u222a{z} is a transitive set, as Case 2 does not apply. Therefore, the vertices of V(T)\u2216V\u2032 can be partitioned into layers Z1,\u22ef,Z\u2113 such that for each z\u2208Zr, z\u2192vs\u2032 if and only if s<r.First note that i=1,2, the vertices of any triangle \u2208\u0394i all belong to the same layer. This implies in particular that for i=1,2, all vertices in triangles of \u0394i belong to the same layer. Since w1 and\u00a0w2 are in different layers , this shows that any triangle in \u03941 is vertex\u2010disjoint from any triangle in\u00a0\u03942.We claim that for i\u2208{1,2} and let \u2208\u0394i be a triangle with wi\u2192uir,uir\u2192vir,vir\u2192wi. Suppose that uir\u2208Zu,vir\u2208Zv,wi\u2208Zw for some u,v,w\u2208{1,\u22ef,l}. So we must show that u=v=w to prove the claim.To show the claim, let u<w then wi,uir,vu\u2032 form a triangle, contradicting that Case 4 does not apply. If v>w then wi,vir,vv\u2032 form a triangle, again contradicting that Case 4 does not apply. Hence, v\u2264w\u2264u holds. If v<u then uir,vir,vv\u2032 form a triangle, contradicting the definition of \u0394i. So indeed u=v=w, and the claim holds. \u25aaIf W\u2287V\u2032 that contain exactly one of w1,w2 . Overall, if Cases 1\u20135 do not apply, with the obtained bound on (\u22c6), by\u00a0\u03b2n\u2212k, since \u03b2\u22651.5703.To complete the proof of Lemma This completes the proof of Lemma 5M(n)\u22641.5949n for all n\u2208N. For sake of contradiction, suppose that the statement of the lemma is not true. Let T\u2208Tn be a counterexample with minimum number of vertices. By Observation T is strong. In particular, any vertex of T has out\u2010degree at most n\u22122.We will show by induction that T satisfies the following stronger restriction:T is regular, or for any d\u2208N at most 2d vertices have out\u2010degree at least n\u2212d\u22121.First suppose that T is regular, then any vertex has out\u2010degree exactly (n\u22121)/2. Using that M(T)\u2264\u2211v\u2208V(T)M(d+(v)) by Lemma n\u226511 we obtain 21n/7>n\u00b71.5949n\u221212, and so any regular tournament with at least\u00a011 vertices has at most 21n/7 maximal transitive vertex sets. For n\u22649, the inequality M(n)\u226421n/7 was shown explicitly by Gaspers and Mnich\u00a0If d\u2208N, at most 2d vertices have out\u2010degree at least n\u2212d\u22121. Let v1,\u22ef,vn be a labeling of the vertices of T such that d+(v1)\u2265d+(v2)\u2265\u22ef\u2265d+(vn). Then\u00a0Lemma d\u2208N,d=i\u221212 implies i>2d, and we obtainSo we may assume that for any \u03b2n since \u03b2\u22651.5462. This completes the analysis of tournaments T satisfying\u00a0It follows thatM(T) in each of the four cases \u0394+(T)=n\u2212i for i=2,3,4,5. In each case, we bound M(T) via M(n\u223c) for values n\u223c<n and the result of\u00a0Lemma branching vectorb=, where t is the number of branches we consider of how a particular maximal transitive vertex set W of T can look like. Each number bi,i=1,\u22ef,t is a positive integer that is the sum of \u0394n and \u0394k, where \u0394n is the number of vertices by which the tournament order decreases in that branch and \u0394k is the number of vertices by which the parameter\u00a0kincreases. It might be that one of \u0394n and \u0394k is equal to zero in some branch, but the sum bi=\u0394n+\u0394k is always positive. We will show that M(T)\u2264\u2211i=1t\u03b2n\u2212bi\u2264\u03b2n in each case.We will now work toward removing assumption\u00a0M(T), we classify all maximal transitive vertex sets of T. Let W be a maximal transitive vertex set of T. We branch on carefully selected vertices whether they belong to\u00a0W or not. These choices either exclude certain other vertices from W (by acyclicity of W) and thus yield \u0394n>0; they force certain other vertices to be included into W, based on the following observation:Observation 4W be a maximal transitive vertex set of T, and let v be a vertex of T. If v\u2209W then at least one predecessor of v in T belongs to W. Equivalently, if no predecessor of v belongs to W, then v\u2208W.Let To bound of v. These choices and their implications will be depicted by case trees that also show the pair  for each branch.Case 1:\u0394+(T)=n\u22122v\u2605 be a vertex with maximum out\u2010degree and unique predecessor b. We distinguish the following subcases.Case 1.1:d+(b)=n\u22122b\u223c be the only predecessor of b. Let W be a maximal transitive vertex set of T.Let b\u2209W, we have that v\u2605\u2208W by applying Observation\u00a0v=v\u2605. We further have that b\u223c\u2208W by applying Observation\u00a0v=b.When W can be categorized as follows, with branching pairs :This shows that b=, which solves to 1.4656.This yield a branching vector Case 1.2:d+(b)\u2264n\u22123W be a maximal transitive vertex set of T. As in Case\u00a01.1, at least one of v\u2605,b belongs to W, by Observation v\u2605,b\u2208W then no predecessor of b belongs to\u00a0W, as T[W] is acyclic. Since b has at least two predecessors, these observations yield:Let Let This yields a branching vector of , which solves to 1.5538.\u03b2\u22651.5538.This completes the analysis of Case 1, where we used that Case 2:\u0394+(T)=n\u22123V\u2605 be the set of vertices with maximum out\u2010degree.Let Lemma 7v\u2605\u2208V\u2605 has two predecessors of out\u2010degree exactly n\u22123 or two predecessors of out\u2010degree at most n\u22124, or V\u2605 induces a triangle.Either some vertex We will apply this observation with various choices for V\u2605 contains at most four vertices.By Lemma V\u2605 contains a single vertex v\u2605 then its two predecessors are outside\u00a0V\u2605, and thus have out\u2010degree at most n\u22124.If V\u2605 consists of two vertices v1\u2605,v2\u2605 then assume v1\u2605\u2192v2\u2605. Thus, the two predecessors of\u00a0v1\u2605 are outside\u00a0V\u2605, and thus have out\u2010degree at most n\u22124.If V\u2605 consists of three vertices v1\u2605,v2\u2605,v3\u2605 then we may assume that V\u2605 does not induce a triangle . Thus, we can assume that vi\u2605\u2192vj\u2605 for 1\u2264i<j\u22643. Thus, the two predecessors of v1\u2605 are outside V\u2605, and thus have out\u2010degree at most n\u22124.If V\u2605 consists of four vertices. Then the subtournament T[V\u2605] has exactly six arcs. As each vertex in V\u2605 has two incoming arcs, this implies that for some vertex in V\u2605 its two predecessors are also in V\u2605 and therefore have out\u2010degree n\u22123. \u25aaFinally, assume that v\u2605\u2208V\u2605 has two predecessors with out\u2010degree exactly n\u22123 (Case 2.1),some v\u2605\u2208V\u2605 has two predecessors with out\u2010degree at most n\u22124 (Cases 2.2\u20132.7),or some T has exactly three vertices with out\u2010degree n\u22123 that form a triangle (Case 2.8).or Therefore, we can distinguish whetherN\u2212(v\u2605)={b1,b2} and b1\u2192b2.Case 2.1:d+(b1)=n\u22123 and d+(b2)=n\u22123b1 and b2 have at most one common predecessor. Let N\u2212(b1)={u1,u2}.Case 2.1.1:N\u2212(b1)\u2229N\u2212(b2)=\u2205.Then N\u2212(b2)={b1,u3} for some u3\u2209N\u2212(b1). Note that if v\u2605\u2209W,b2\u2208W, then u3\u2208W.  Then we branch as follows:Let Case 2.1.2N\u2212(b1)\u2229N\u2212(b2)={u2}.This yields a branching vector of  that solves to 1.5748.In Cases 2.1\u20132.7, let Then we branch as follows:Case 2.2:d+(b1)\u2264n\u22124, d+(b2)\u2264n\u22124 and b1,b2 share no common predecessor.b1 forms a triangle with b1,b2 and therefore any maximal transitive vertex set containing both b1,b2 cannot contain any predecessor of\u00a0b1. Then we branch as follows:Any predecessor of This yields a branching vector of  that solves to 1.5923.Case 2.3:d+(b1)\u2264n\u22124, d+(b2)=n\u22124 and one predecessor of b2 is a predecessor of\u00a0b1N\u2212(b1)={w1,...,wl} with l\u22653 and let N\u2212(b2)={b1,u,w1}, so that w1 is the unique predecessor common to b1,b2.Case 2.3.1:wi\u2192u for some i\u22082,...,l.Then {u,b2,wi} induces a triangle.Let Then we branch as follows:Case 2.3.2:u\u2192wi for all i\u22082,\u2026,l{u,wi,b1} induces a triangle for all i\u22082,\u2026,l.Then Then we branch as follows:This yields a branching vector of  that solves to 1.5780.This yields a branching vector of  that solves to 1.5772.Case 2.4:d+(b1)=n\u22124, d+(b2)=n\u22124 and all predecessors of b2 other than b1 are predecessors of\u00a0b1u be the only predecessor of b1 that is not a predecessor of b2. Then we branch as follows:Let This yields a branching vector of  that solves to 1.5912.Case 2.5:d+(b1)\u2264n\u22125,d+(b2)=n\u22124 and all predecessors of b2 are predecessors of\u00a0b1Then we branch as follows:This yields a branching vector of  that solves to 1.5820.Case 2.6:d+(b1)=n\u22124 and d+(b2)\u2264n\u22125b1,b2 have no common predecessor in Case 2.2. Hence, b1,b2 have either one, two, or three common predecessors.Case 2.6.1:b1 and b2 have exactly one common predecessor.Then we branch as follows:We already considered that Case 2.6.2:b1 and b2 share exactly two predecessors.sb2<n\u22125, or b2 has exactly one predecessor that is not a predecessor of b1. Let u1 be the only predecessor of b1 that is not a predecessor of b2. Note that any maximal transitive vertex set that contains n,b2 but not b1 has to contain u1.Then either For sb2<n\u22125 we branch as follows:This yields a branching vector of  that solves to 1.5923.This yields a branching vector of  that solves to 1.5817.sb2=n\u22125 and u2 being the only predecessor of b2 that is not a predecessor of b1, we branch as follows:For Case 2.6.3:b1 are predecessors of b2.All predecessors of v\u2605,b2 also contains b1. Then we branch as follows:Then any maximal transitive vertex set containing This yields a branching vector of  that solves to 1.5861.This yields a branching vector of  that solves to 1.5702.Case 2.7:d+(b1)\u2264n\u22125 and d+(b2)\u2264n\u22125Then we branch as follows:This yields a branching vector of  that solves to 1.5912.Case 2.8:n\u22123 and they form a triangle.There are exactly three vertices with score This yields a branching vector of  that solves to 1.4736.v1,v2,v3 and assume v1\u2192v2, v2\u2192v3, and v3\u2192v1.Call these vertices v1,v2,v3 have distinct predecessors, and let bi be the predecessor of\u00a0vi.Suppose that vi,vi+1,bi form a triangle and that bi+1 is the only predecessor of\u00a0vi+1 other than vi. Therefore, we branch as follows:Note that This yields a branching vector of  that solves to 1.5903.v1,v2 have a common predecessor b and b3 is the predecessor of b3.Now suppose that v3 is the only vertex where the orientations of the arcs incident to v1 differ from those incident to v2, any maximal transitive vertex set that does not contain v3 either contains both v1,v2 or none of them. We branch as follows:Since This yields a branching vector of  that solves to 1.5702.b, any maximal transitive vertex set that contains some vertex of v1,v2,v3 contains at exactly two of them. We branch as follows:If all three vertices have a common predecessor Case 3:\u0394+(T)=n\u22124V\u2605 be the set of vertices with out\u2010degree n\u22124. Since T is strong, some vertex v\u2605\u2208V\u2605 has a predecessor in V(T)\u2216V\u2605. Let b1,b2,b3 be the predecessors of\u00a0v\u2605. For i=1,2,3 let s\u223ci=|N\u2212(bi)\u2216{b1,b2,b3}|; assume, without loss of generality, that s\u223c1\u2265s\u223c2\u2265s\u223c3.Let v\u2605 we have |N\u2212(b1)|+|N\u2212(b2)|+|N\u2212(b3)|\u226510 and hence s\u223c1+s\u223c2+s\u223c3\u22657. Note that |N\u2212(bi)|\u22653 for i=1,2,3 and |N\u2212(bj)|\u22654 for some j\u2208{1,2,3}. Therefore s\u223c1\u22653, s\u223c2\u22652, s\u223c3\u22651.By the choice of s\u223c1,s\u223c2,s\u223c3.Case 3.1:s\u223c3=1If s\u223c3=1, then either s\u223c2\u22653 or s\u223c1\u22654. Let {u}=N\u2212(b3)\u2216{b1,b2,b3}. Then we branch as follows:We will estimate bounds for different values of Case 3.2:s\u223c1\u22654 or s\u223c2\u22653, s\u223c3\u22652We branch as follows:This yields a branching vector of  that solves to 1.5812.This yields a branching vector of  that solves to 1.5949.s\u223c2\u2265s\u223c3, we can now assume that s\u223c1=3,s\u223c2=2,s\u223c3=2.Case 3.3:s\u223c1=3,s\u223c2=2,s\u223c3=2 and all predecessors of b2 except the bi are predecessors of b3We branch as follows:Since Case 3.4:s\u223c1=3,s\u223c2=2,s\u223c3=2, and b2,b3 have exactly one common predecessor u1 in V(T)\u2216{b1,b2,b3}W be a maximal transitive vertex set, u2\u2260u1 be the other predecessor of b2 and u3\u2260u1 be the one of\u00a0b3. If v\u2605\u2208W but b2,b3\u2209W, then either u1\u2208W or u2,u3\u2208W, as otherwise b2 or b3 could be added to\u00a0W. Therefore, we branch as follows:Let This yields a branching vector of  that solves to 1.5516.This yields a branching vector of  that solves to 1.5793.b2,b3 have no common predecessor except possibly\u00a0b1.Case 3.5:s\u223c1=3,s\u223c2=2,s\u223c3=2, b2,b3 have no common predecessor in V(T)\u2216{b1,b2,b3} and two predecessors of b1 in V(T)\u2216{b1,b2,b3} are predecessors of\u00a0bi for some i=2,3b3=bi and let u1 be the only predecessor of\u00a0b1 that is not a predecessor of b3. Since s\u223c1=3, there is at least one predecessor of\u00a0b2 that is not a predecessor of b1. We branch as follows:Assume, without loss of generality, that Henceforth, we can assume that Case 3.6:s\u223c1=3,s\u223c2=2,s\u223c3=2, b2,b3 have no common predecessor in V(T)\u2216{b1} and both b2 and b3 share at most one predecessor of b1.b1 shares no predecessor in V(T)\u2216{b1,b2,b3} with b2 and none with b3, orb1 shares one predecessor with each of b2 and b3, orb1 shares a predecessor with one of b2,b3 but none with the other; without loss of generality, let b1 share a predecessor with b3.There are three cases to consider:b1,b2, and b3 are forming a triangle.Case 3.6.1:b1 shares one predecessor in V(T)\u2216{b1,b2,b3} with b2 and one with b3.b1,b2, and b3 do not form a triangle.w1 the common predecessor of b1 and b2 and let w2 be the common predecessor of b1 and b3. For i=1,2,3 let ui be the predecessor of bi that is neither w1 nor w2. We branch as follows:Let This yields a branching vector of  that solves to 1.5828.Case 3.6.2:b1 shares one predecessor in V(T)\u2216{b1,b2,b3} with b2 and one with b3.b1,b2, and b3 form a triangle.u1,u2,u3,w1,w2 be as in Case 3.6.1. Then we branch as follows:Let This yields a branching vector of  that solves to 1.5805.Case 3.6.3:b1 shares no predecessor in V(T)\u2216{b1,b2,b3} with b2 and one with b3.b1,b2, and b3 do not form a triangle.u3 the only predecessor of b3 that is not a predecessor of b1. Then we branch as follows:Let This yields a branching vector of  that solves to 1.5912.Case 3.6.4:b1 shares no predecessor in V(T)\u2216{b1,b2,b3} with b2 and one with b3.b1,b2, and b3 form a triangle.u3 be as defined in Case 3.6.3. Then we branch as follows:Let \u00a0This yields a branching vector of  that solves to 1.5889.Case 3.6.5:b1 shares no predecessor in V(T)\u2216{b1,b2,b3} with b2 and none with b3.b1,b2, and b3 do not form a triangle.s\u223c2=s\u223c3=2 but |N\u2212(bi)|\u2264n\u22124 for i=2,3, there is a vertex bj\u2260bi, j=1,2,3 such that bj\u2192bi. As the vertices b1,b2, and b3 form a transitive set, we have b1\u2192b2 and b1\u2192b3. Let u1,u2,u3 the predecessors of b1. Note that b2,b3,v\u2605 and each ui for i=1,2,3 form a cycle with b1. Therefore, if W contains some\u00a0ui, it can not contain b2,b3, or v\u2605. If it contains b2,b3, or v\u2605, it can not contain any\u00a0ui. Thus, we branch as follows:Since \u00a0This yields a branching vector of  that solves to 1.5437.Case 3.6.6:b1 shares no predecessor in V(T)\u2216{b1,b2,b3} with b2 and none with b3.We will split each of the cases into two subcases where we consider if This yields a branching vector of  that solves to 1.5904.This yields a branching vector of  that solves to 1.5397.b1,b2, and b3 form a triangle. Assume without loss of generality that b1\u2192b2,b2\u2192b3 and b3\u2192b1. Therefore all predecessors of b1 form a triangle with b1 and b2 and all predecessors of b3 form a triangle with b3 and b1. We branch as follows:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0This yields a branching vector of  that solves to 1.5800.Case 4:\u0394+(T)=n\u22125Let V\u2605 be the set of vertices with maximum out\u2010degree; Since T is strong, there is a vertex v\u2605 that has a predecessor in V(T)\u2216V\u2605.This completes the analysis of Case 3.N\u2212(v\u2605)={b1,b2,b3,b4}. By the choice of v\u2605, the vertices b1,b2,b3,b4 together have at least 17 incoming arcs. Exactly 6 of these arcs are incident to two of b1,b2,b3,b4. The remaining 11 arcs are incoming from vertices in V(T)\u2216{b1,b2,b3,b4}. For i=1,\u22ef,4 let s\u223ci be the number of arcs incoming to\u00a0bi from V(T)\u2216{b1,b2,b3,b4}, i.e. s\u223ci=|N\u2212(bi)\u2216{b1,b2,b3,b4}|. Assume, without loss of generality, that s\u223c1\u2265s\u223c2\u2265s\u223c3\u2265s\u223c4.Observation 5s\u223c4\u22651 and s\u223c3\u22652. If s\u223c2=2, then s\u223c3=s\u223c4=2.It holds, Let |N\u2212(bi)|\u22654 for all i=1,2,3,4, we get |N\u2212(bi)\u2216{b1,b2,b3,b4}|\u22651. There is at most one vertex with with three incoming arcs from b1,b2,b3,b4. This yields s\u223c3\u22652.Since s\u223c2=s\u223c3=2 and s\u223c4=1. If s\u223c4=1, then b1\u2192b4, b2\u2192b4, and b3\u2192b4. Since s\u223c3=2, b1\u2192b3, and b2\u2192b3. But then we get s\u223c2\u22653 that is a contradiction. \u25aaAssume that ; see Table M(T) in terms of pairs  for each of these six vectors in Table This leaves exactly six feasible vectors s\u223c1,s\u223c,s\u223c3,s\u223c4;s\u223c1,s\u223c,s\u223c3,s\u223c4;W that contains v\u2605,bi cannot contain any predecessor of bi in V(T)\u2216{b1,b2,b3,b4,v\u2605}.To this end, note that any maximal transitive vertex set \u03b2n since \u03b2\u22651.5612.This yields for four of the casesM(T) is bounded by\u03b2n since \u03b2\u22651.5691.For the remaining two cases, This last case completes the proof of Lemma 6M(n) of minimal FVS in n\u2010vertex tournaments, to 1.5448n\u2264M(n)\u22641.5949n. It remains to determine the growth of M(n) exactly\u2014Gaspers and Mnich\u00a0M(n)\u226421n/7\u22481.5448n for all n\u2208N, and we repose this conjecture here.In this article, we narrowed the gap between the lower and upper bounds for the maximum number cn for some constant c<2, on the number of minimal FVS in general directed graphs. As far as we know, currently only a bound of 2n/n is known, implied by Sperner's Lemma.In a different direction, it would be interesting to prove nontrivial upper bounds of the form"}
{"text": "E\u00d7B \u201cpolarization\u201d drift.The relativistic trajectory of a charged particle driven by the Lorentz force is different from the classical one, by velocity-dependent relativistic acceleration term. Here we show that the evolution of optical polarization states near the polarization singularity can be described in analogy to the relativistic dynamics of charged particles. A phase transition in parity-time symmetric potentials is then interpreted in terms of the competition between electric and magnetic \u2018pseudo\u2019-fields applied to polarization states. Based on this Lorentz pseudo-force representation, we reveal that zero Lorentz pseudo-force is the origin of recently reported strong polarization convergence to the singular state at the exceptional point. We also demonstrate the deterministic design of achiral and directional eigenstates at the exceptional point, allowing an anomalous linear polarizer which operates orthogonal to forward and backward waves. Our results linking parity-time symmetry and relativistic electrodynamics show that previous PT-symmetric potentials for the polarization singularity with a chiral eigenstate are the subset of optical potentials for the  V(x)\u2009=\u2009V*(\u2212x), providing physical observables even for non-equilibrium systems. The existence of real observables in PT-symmetric complex potentials has opened the field of non-Hermitian quantum mechanics6616n(x)\u2009=\u2009n*(\u2212x), the physics of PT-symmetric potentials has been applied to the exotic control of light flows1791320222325272526293031With the universal existence of open systems1346891011121314\u1ebc\u2009=\u2009mc2/[1\u2009\u2212\u2009(v/c)2]1/2, shows that observers in different frames will see different values of total energy for moving charged particles of velocity v. We note that this non-equilibrium condition results in non-Hermitian form of Hamiltonians, the necessary condition for the achievement of PT symmetry. In the context of the multidisciplinary realization of PT symmetry891011Meanwhile, it is known that relativistic electrodynamicsE\u00d7B drift  of charged particlesE\u00d7B \u201cpolarization\u201d drift of light. The phase transition in PT-symmetric potentials6161525263738Inspired by the polarization equation of motion from the Schr\u00f6dinger-like form of Maxwell\u2019s equationsz-axis of the electrically anisotropic material, with the unity permeability (\u03bc\u2009=\u20091). For the later discussion, we express the arbitrary permittivity tensor in the x-y plane, in terms of Pauli matrices26\u03c31\u20133 as \u03b5\u2009=\u2009(\u03b52\u03c31\u2009+\u2009\u03b53\u03c32\u2009+\u2009\u03b51\u03c33), orConsider the planewave propagating along the \u03b5o(z) and \u03b51\u22123(z) have slowly-varying complex values, \u03c31\u2009=\u2009, \u03c32\u2009=\u2009, and \u03c33\u2009=\u2009. By applying the spin (x\u2009\u00b1\u2009iy)/21/2 basesd\u03c8e/dz\u2009=\u2009Hs\u00b7\u03c8e with the temporal-like z-axis\u03c8e\u2009=\u2009T is the electric field amplitude of (\u00b1) optical spin waves ((+) for right-circular polarization (RCP) (x\u2009+\u2009iy)/21/2, and (\u2212) for left-circular polarization (LCP) (x\u2009\u2212\u2009iy)/21/2), and Hs is the traceless Hamiltonian expressed with Pauli matrices36Hs\u2009=\u2009(\u03b51\u03c31\u2009+\u2009\u03b52\u03c32\u2009+\u2009\u03b53\u03c33)/(i\u03bb) for \u03bb\u2009=\u20092\u03b5o1/2/k0 and the free-space wavenumber k0. If we assign the symmetry axis between x and y axes, the condition of PT-symmetric potentials15\u03b5o, \u03b52, and \u03b53, and imaginary-valued \u03b51. Note that imaginary-valued \u03b51 corresponds to the linear dichroism42\u03b52 represents the birefringence. \u03b53 represents the magneto-optical change of plasma permittivity induced by an external static magnetic field\u03b53\u2009=\u20090.where Sj\u2009=\u2009\u03c8e\u2020\u00b7\u03c3j\u00b7\u03c8e . The change of the SOP can then be expressed in view of light-matter interactionsd\u03c8e/dz\u2009=\u2009Hs\u00b7\u03c8e and its conjugate form d\u03c8e\u2020/dz\u2009=\u2009\u03c8e\u2020\u00b7Hs\u2020, which leads to the Lorentz pseudo-force equation of motion for the SOPIn this representation, the SOP of light is described by the Stokes parametersSn\u2009=\u2009T/S0 is the pseudo-velocity of the \u2018hypothetical\u2019 charged particle corresponding to the SOP of light, and E\u2009=\u20092\u00b7ImT/\u03bb and B\u2009=\u2009\u22122\u00b7ReT/\u03bb are the electric and magnetic pseudo-field in relation to imaginary- and real-parts of the permittivity, respectively. Note that 20\u2202t\u03b2\u2009=\u2009E\u2009+\u2009\u03b2\u00d7B \u2212(\u03b2\u00b7E)\u03b2. In this representation of optical polarization states, the acceleration of optical SOP comes from the Lorentz pseudo-force, F\u2009~\u2009dSn/dz Sn corresponds to the Joule effect\u03b2\u00b7E)\u03b2 in the relativistic equation of motion.where ~\u2009dSn/dz . The fir\u03b51, real \u03b52, and zero \u03b53). While \u03b52 of the birefringence derives the circulating acceleration on the Poincar\u00e9 sphere , PT-symmetric potentials naturally satisfy the ideal E\u00d7B drift34/\u03bb, Sn\u00b7E  providesE\u00d7B drift34\u0394\u03b5eig for the Hamiltonian equation d\u03c8e/dz\u2009=\u2009Hs\u00b7\u03c8e, as a function of the imaginary potential . First, before the EP where eigenvalues are real and non-degenerate (\u03b5i\u2009<\u2009\u03b5c), the magnetic pseudo-field is larger than the electric pseudo-field, resulting in the counter-directive acceleration of SOP (lower panels in d point in \u03b5i\u2009=\u2009\u03b5c), the equal magnitude of E\u2009=\u20092\u00b7\u03b5i\u00b7e1/\u03bb and B\u2009=\u2009\u22122\u03b5c\u00b7e2/\u03bb fields derives the suppression of total Lorentz pseudo-force on the southern Poincar\u00e9 sphere, especially with the zero net force at the south pole Sn\u2009=\u20092\u00b7\u03b5i\u00b7e1/\u03bb\u2009+\u2009e3\u2009\u00d7\u20092\u03b5c\u00b7e2/\u03bb\u2009=\u20090, \u03b5i\u2009>\u2009\u03b5c), the strong electric pseudo-field dominates the motion equation of the SOP, with the co-directive force (lower panels in B-dominant (before the EP), (ii) B\u2009=\u2009E (at the EP) and (iii) E-dominant regime (after the EP). It is worth mentioning that the stable point with the stationary polarization can also be obtained at the north pole by changing the sign of \u03b5c (converting the fast and slow axes for the birefringence) or \u03b5i (converting the gain and loss axes for the linear dichroism), allowing perfect RCP chirality. In terms of this Lorentz pseudo-force representation of SOP, we also note that PT-symmetric potentials15\u03b52 and imaginary-valued \u03b51 are the special case of the E\u00d7B drift with specific field vectors E\u2009=\u20092\u00b7\u03b5i\u00b7e1/\u03bb and B\u2009=\u2009\u22122\u03b5c\u00b7e2/\u03bb, implying the existence of unconventional polarization singularity at other SOPs  which will be discussed later.Based on the Lorentz pseudo-force equation of 616generate , \u03b5i\u2009<\u2009\u03b5cE\u00d7B drift (B|\u2009>\u2009|E|), the SOP for each spin simply rotates around the B field following the Sn\u00d7B of E\u00d7B drift34E\u00d7B axis (S3 axis) is thus reproduced by the slow evolution of SOPs near the Sn\u2009=\u2009\u2212e3 on the Poincar\u00e9 sphere , the (+) spin state converges to the (\u2212) spin when the state approaches the south pole through the gyration by the magnetic pseudo-field  and linear dichroism42Im[\u03b5x]\u2009\u2260\u2009Im[\u03b5y] for different x- and y-dissipation). The spin-based Hamiltonian equation d\u03c8e/dz\u2009=\u2009Hs\u00b7\u03c8e, for slowly-varying \u03b5x\u2009=\u2009\u03b5o\u2009+\u2009\u0394\u03b5(z) and \u03b5y\u2009=\u2009\u03b5o\u2009\u2212\u2009\u0394\u03b5(z) and constant \u03c7\u2009=\u2009\u03c7o with real-valued \u03b5o and \u03c7o, derives the Hamiltonian Hs for general chiral materials, in the form ofExtending the special case of the k2\u2009=\u2009\u03c92\u00b7(\u03bc0\u03b5o\u2009+\u2009\u03c7o2) (see \u03c7(z)). The Pauli expressionHs\u2009=\u2009a1\u03c31\u2009+\u2009a2\u03c32\u2009+\u2009a3\u03c33 has the coefficients of a1\u2009=\u2009\u2212(ik/2)\u00b7(\u0394\u03b5/\u03b5o), a2\u2009=\u2009\u2212(\u03c9\u03c7/2)\u00b7(\u0394\u03b5/\u03b5o), and a3\u2009=\u2009i\u03c9\u03c7. The electric and magnetic pseudo-fields for and \u03c1\u2009=\u2009\u0394\u03b5/\u03b5o.with the degree of electrical anisotropy E and B driving the SOP are strongly dependent on the type of the anisotropy: birefringence  or linear dichroism (imaginary \u03c1) both satisfying the condition of the E\u00d7B drift (E\u22a5B). For the case of birefringence with E\u2009=\u2009\u2212\u03c9\u03c7o\u03c1\u00b7e2 and B\u2009=\u2009\u2212k\u03c1\u00b7e1\u2009+\u20092\u03c9\u03c7o\u00b7e3, the pseudo-field satisfies |E|\u2009<\u2009|B| in most cases and the condition of |E|\u2009\u2265\u2009|B| enforces \u03b5o\u2009<\u2009\u22124\u03c7o2/(\u03bc0\u00b7\u03c12) and thus prohibits the existence of propagating waves at the EP. It is interesting to note that this restriction proves the necessity of complex potentials for obtaining the singularity; therefore we employ linear dichroism for achieving the EP for the propagating wave, by fulfilling the condition of |B|\u2009=\u2009|E|.\u03c1\u2009=\u2009i\u00b7\u03c1i) chiral materials, which derive pseudo-fields of E\u2009=\u2009k\u03c1i\u00b7e1 and B\u2009=\u2009\u2212\u03c9\u03c7o\u03c1i\u00b7e2\u2009+\u20092\u03c9\u03c7o\u00b7e3 with the EP condition of \u03c1i2\u2009=\u20094\u03c7o2/(\u03bc0\u03b5o) for |B|\u2009=\u2009|E|. The PT-symmetry-like phase transition , the linear-polarizing functionality in the structure of This directionality with achiral designer eigenstate at the EP allows the implementation of unconventional polarizers based on the polarization convergence at the EP. E\u00d7B drift toward a single direction for every initial velocity vectors.In summary, we found the link between the seemingly unrelated fields of PT symmetry optics and relativistic electrodynamics. This reinterpretation of PT symmetry brings insight to the singularity in polarization space, broadening the class of parity-time symmetric Hamiltonians in vector wave equations47485153How to cite this article: Yu, S. et al. Acceleration toward polarization singularity inspired by relativistic E\u00d7B drift. Sci. Rep.6, 37754; doi: 10.1038/srep37754 (2016).Publisher\u2019s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations."}
{"text": "Scientific Reports6: Article number: 37077; 10.1038/srep37077 published online: 11282016 updated: 01162017.This Article contains a typographical error. In the Results section, under the subheading \u201cSiC Sensor\u201d\u201cThese intermediate coupled pair states can be described by the singlet-triplet basis: the symmetric triplet states, T+\u2009=\u2009|\u2191\u2191\u232a, T0\u2009=\u2009(|\u2191\u2193\u232a\u2009+\u2009|\u2193\u2191\u232a)/, and T\u2212\u2009=\u2009|\u2193\u2193\u232a, each having spin angular momentum S\u2009=\u20091 with ms\u2009=\u2009+1, 0, \u22121 respectively or the anti-symmetric singlet state, S0\u2009=\u2009(|\u2191\u2193\u232a\u2009+\u2009|\u2193\u2191\u232a)/ which has spin angular momentum S\u2009=\u20090 with ms\u2009=\u20090\u201d.should read:\u201cThese intermediate coupled pair states can be described by the singlet-triplet basis: the symmetric triplet states, T+\u2009=\u2009|\u2191\u2191\u232a, T0\u2009=\u2009(|\u2191\u2193\u232a\u2009+\u2009|\u2193\u2191\u232a)/, and T\u2212\u2009=\u2009|\u2193\u2193\u232a, each having spin angular momentum S\u2009=\u20091 with ms\u2009=\u2009+1, 0, \u22121 respectively or the anti-symmetric singlet state, S0\u2009=\u2009(|\u2191\u2193\u232a\u2009\u2212\u2009|\u2193\u2191\u232a)/ which has spin angular momentum S\u2009=\u20090 with ms\u2009=\u20090\u201d."}
{"text": "There are two errors in the \u201cmiR-200/182 expression was induction in HSV-1 positive areas of the brain\u201d section of the Results. \u201cFig B in S1 File\u201d should be listed as \u201cFig D in S1 File\u201d and \u201cFig C in S1 File\u201d should be listed as \u201cFig E in S1 File\u201d.in situ hybridization in brain tissue\u201d section of the Results. In the first paragraph, \u201cFig D in S1 File\u201d should be listed as \u201cFig B in S1 File\u201d. In the second paragraph, \u201cFig E in S1 File\u201d should be listed as \u201cFig C in S1 File\u201d.There are two errors in the \u201cInduction of miR-200/182 expression visualized by Panel C in"}
{"text": "H-parallel case with a fixed incident wavelength. We denote \u03b7 > 0 as the typical size of the complex structure and obtain the effective equations by letting \u03b7 \u2192 0. For metallic permittivities with negative real part, plasmonic waves can be excited on the surfaces of the slits. For the waves to be in resonance with the height of the metallic layer, the corresponding results can be perfect transmission through the layer.We investigate the transmission properties of a metallic layer with narrow slits. We consider (time-harmonic) Maxwell's equations in the  In , the reldone. In  where thdone. In \u201313.Two-scale convergence has proven to be a very efficient tool in homogenization theory while dealing with the problems where the underlying medium is heterogenous. The concept of two-scale convergence is first introduced by Nguetseng in   Two-scal\u03b3, \u03b3)\u00d7 and therefore by defining a suitable test function, they have shown that the first component of j0 vanishes and they obtain j0 = \u2009\u2009for\u2009\u2009x \u2208 R\u2009\u2009and\u2009\u2009y \u2208 Y\u2216\u03a3. This is clearly not the case in this paper as the metallic subpart \u03a3 is chosen to be sinusoidal along y2-axis due to the geometry of the metallic structure given by j0 and we end up having a different j0 compared to that in  and 1 \u2264 r, s \u2264 \u221e be such that 1/r + 1/s = 1. Assume that \u039e \u2208 {\u03a9, \u03a3\u03b7} and l \u2208 \u21150; then as usual Lr(\u039e) and Hl,r(\u039e) denote the Lebesgue and Sobolev spaces with their usual norms and they are denoted by \u2016\u00b7\u2016r and \u2016\u00b7\u2016l,r. For the sake of clarity if \u03d5 \u2208 Lr(\u039e), then\u03d5 \u2208 Hl,r(\u039e), then\u03b1 =  \u2208 \u2115n is a multi-index, |\u03b1 | = \u03b11 + \u03b12 + \u22ef+\u03b1n, and D\u03b1 = \u2202\u03b1||/\u2202x1\u03b11\u2202x2\u03b12 \u22ef \u2202xn\u03b1n. Similarly, \u03b8,r, and \u03b8 are the H\u00f6lder, real, and complex interpolation spaces, respectively, endowed with their standard norms; for definition confer ,,j0 and Lloc2(\u03a9) which implies that up to a subsequence By , it follThe Field outside of R. For u\u03b7 \u2192 \u2207U uniformly on compact subsets of The Field in the Metal Part of R. We note that |a\u03b7| \u2264 C\u03b72 in \u03a3\u03b7 and |a\u03b7| \u2264 C in R\u2216\u03a3\u03b7; therefore (j\u03b7\u2016L2(\u03a3\u03b7)2 \u2264 C\u03b72 and \u2016j\u03b7\u2016L2(R\u2216\u03a3\u03b7)2 \u2264 C. This implies R \u00d7 \u03a3. Moreover, j0 = j0(x) = \u2207U(x) for a.e.  \u2208 R \u00d7 (Y \u2212 \u03a3).herefore  gives \u2016jDivergence of j0. Due to boundedness assumption on u\u03b7, by (j\u03b7\u2016L2(\u03a9) \u2264 k2\u2009\u2009sup\u03b7>0\u2061\u2016u\u03b7\u2016L2(\u03a9) < \u221e, \u2200\u03b7 > 0. For \u0398 \u2208 C0\u221e(\u03a9) and \u03c8 \u2208 Cper\u221e(Y), we test (x)\u03c8(x/\u03b7) which givesC0\u221e(\u03a9) is arbitrary, \u222bY\u2207y\u03c8(y) \u00b7 j0\u2009\u2009dy = 0 for a.e. x \u2208 \u03a9 which implies \u2207y \u00b7 j0 = 0 for y \u2208 Y in distributional sense. This shows that y; that is, x only. u\u03b7, by  we have  we test  by \u0398(x)\u03c8U as shown in [\u03c8 \u2208 L2:\u2207\u00b7\u03c8 = 0, \u03c8 is Y-periodic, and \u03c8 = 0\u2009\u2009in\u2009\u2009\u03a3}. We choose a test function \u03c6\u2254\u0398(x)\u03c8(x/\u03b7), where \u0398 \u2208 C0\u221e and \u03c8 \u2208 \u039e. We use \u03c6\u2254\u2207\u00b7(\u0398(x)\u03c8(x/\u03b7)) = \u03c8(x/\u03b7)\u00b7\u2207\u0398(x) as \u2207y \u00b7 \u03c8(y) = 0. Then\u03c8(y)|\u03a3 = 0 and is nonvanishing in Y\u2216\u03a3 and by  = U(x), all these lead toj0 = \u2207U(x) for j0 = 0 for  \u2208 R \u00d7 \u03a3.Next we determine the relation between shown in . We defi\u03a3 and by  it impliProof of (i). To conclude this part, the arguments rely on that of [x \u2208 R. To show this, we define a function Yj0dy = \u222bYj(x)dy = j(x) and y \u00b7 j0 = 0 and \u2207y \u00b7 j1 = 0, and (iii) \u03c80(y) = 0, which shows that  that of . We consies that  holds goy=0 from , we have=0 from . To verify the claim, let us choose C0\u221e(\u03a9). Note that U \u2208 L2(\u03a9) and we find also that x \u2208 R and x \u2208 \u03a9\u2216R; that is, a(x) = \u03b1 for x \u2208 R and a(x) = 1 for x \u2208 \u03a9\u2216R.hen from , we have\u03a9\u2032 with R \u2282 \u03a9\u2032 \u2282 \u2282\u03a9, L2(\u03a9\u2032), then, up to a subsequence, L2(\u03a9\u2032).The proof of U and U; see U and j\u2192;Case 1. Let \u03d5 \u2208 C0\u221e(\u03a9) from ((\u03a9) from  we have from ((\u03a9) from  we have(The combination of  and 55)55) givesx \u2208 R. Due to their rectangular metallic gratings inside R, the component along x1 direction vanishes; that is, the first component of \u03b1\u2202x22U = \u2212k2\u03bceffU.Here we can compare our upscaled equation with the limit problem obtained in , especiaThe proof is devided into three steps which are demonstrated below.(i) Uniqueness of the Limit Problem. With a fixed incident field ui we will show that the limit problem  = 0\u2009\u2009for\u2009\u2009x \u2208 \u211d\u2009\u2009and\u2009\u2009y \u2208 Y. The main ingredient for this uniqueness result is Rellich's first lemma and the fact that aeff is real and \u03bceff has positive imaginary part. In fact aeff is identity and \u03bceff is 1 outside of R. Let us denote the surface of a sphere Br(0) of radius r by Sr\u2009\u2009(\u2254\u2202Br(0)), where r is chosen so large such that R \u2282 Br(0). Let r0 be such r; then by (Br0(0). Since  > 0, thenu = 0 in R. Since r0 is chosen arbitrarly, for every r from   compare .(ii) Convergence to the Limit Problem Assuming an Lloc2-Bound. Let the radius r0 > 0 be such that \u03a9\u2254Br0(0). We begin with the assumption that\u03b7 \u2192 0, we obtain that U(x) = u(x)\u03c7(x)|\u03a9\u2216R + Nw\u22121u(x)\u03c7(x)|R solves (tion thatt\u03b7\u2254\u222b\u03a9u\u03b721R solves .u\u03b7 and \u2207u\u03b7 are uniformly convergent on every compact subset of \u211d2\u2216R. Let us choose r < r0 such that R \u2282 \u2282Br(0)\u2282\u2282\u03a9. By [u\u03b7s = u\u03b7 \u2212 ui coincides on \u211d2\u2216Br(0) with its Helmholtz representation through values and derivatives of u\u03b7 \u2212 ui on \u2202Br(0). By the similar representation formula, using the values and derivatives of U \u2212 ui on \u2202Br(0), we can extend U into all of \u211d2 to a solution of the Helmholtz equation \u22072U = \u2212k2U outside of R. Thus this construction of U shows that U \u2212 ui satisfies the Sommerfeld radiation condition. The uniform convergence of u\u03b7 \u2192 U and \u2207u\u03b7 \u2192 \u2207U on \u2202Br(0) implies the uniform convergence of u\u03b7 and its derivatives on all compact subsets of exterior of R. Finally by uniqueness of the limit from part (i), \u03b7 \u2192 0 for the whole sequence. This shows that the Sommerfeld radiation condition holds for r = |x | \u2192 \u221e. which establishes (We only need to verify the radiation condition . By Lemmablishes .(iii) Boundedness of t\u03b7. In the previous step the limit problem is obtained assuming |\u03a9\u2216R + Nw\u22121v\u03c7(x)|R is the unique solution of .assuming  is true.assuming  holds trution of  and satiR, the gradient estimate . Then by Rellich compactness lemma \u222b\u03a9\u2216R|v\u03b7|2 \u2192 0 as \u03b7 \u2192 0. For inside of R, we use the estimate  = 1. Thus t\u03b7 has to be bounded.For outside of estimate  for v\u03b7 estimate  on u\u03b7 aBy Theorems R be \u211d \u00d7  for h > 0. We assume planar front of waves that reaches the metallic slab (\u2212h < x2 < 0) from above (x2 > 0). The incoming waves would be partially reflected and partially transmitted through the metallic structure. Before we proceed any further we define the following parameters:M = 1amplitude of the incident wave, where \u2009\u03b8\u2254 incident angle, where \u03b8 \u2208 \u2009T\u2254 complex amplitude and phase shift, where T \u2208 \u2102\u2009\u2009Ai, Bi\u2254 complex amplitudes in the structure, where Ai, Bi \u2208 \u2102, i = 1,2\u2009R\u2254 complex amplitude of the reflected wave, where R \u2208 \u2102Let the rectangle U of (\u03c4\u2254\u221a(Nw/\u03b1). We are yet to determine the coefficients Ai, Bi, R, and T by using the interface (x2 = 0 and x2 = \u2212h) and transmission conditions (see (We write the solution U of  as(67)whons (see ).The Transfer Matrix M. We will calculate a transfer matrix M which basically gives a transformation relation between the solutions on the upper boundary x2 = 0 and the lower boundary x2 = \u2212h. To be precise, we define a map M : \u21022 \u2192 \u21022  asx2 = 0 and x2 = \u2212h which we choose as  and , respectively. In short the matrix M maps the vector M can be expressed as a \u21022\u2009\u2009\u00d7\u2009\u20092 matrix. Now we determine the transfer matrix M where the two columns are obtained by M \u00b7 t and M \u00b7 t.Columns of M. To obtain the first column of M, we study a solution U of the effective system such that U|x2=0+ = 1 and U in the interval  is given by |x2=0+ = U|x2=0\u2212 = 1, \u03b1\u2202x1U|x2=0+ = \u03b1\u2202x1U|x2=0\u2212 = 0, and \u03b1\u2202x2U|x2=0+ = \u03b1\u2202x2U|x2=0\u2212 = 0. With the help of these conditions, we obtain A2 = B2 = 0 and (1/2)A1B1cos(\u03c4kx1) = 1 which gives U = cos(\u03c4kx2). With the help of similar transmission condition we obtaingiven by . By tranU|x2=\u2212h\u2212 = U|x2=\u2212h+ = cos\u2061(\u03c4kh), \u2202x2U|x2=\u2212h\u2212 = \u03b1\u2202x2U|x2=\u2212h+ = \u03b1\u03c4ksin\u2061(\u03c4kh), and \u2202x1U|x2=\u2212h\u2212 = \u03b1\u2202x1U|x2=\u2212h+ = 0. This gives first column of M as (cos\u2061(\u03c4kh), \u03b1\u03c4ksin\u2061(\u03c4kh))t. A similar computation by taking M \u00b7 t in account will yield the second column of M as (\u2212(\u03b1\u03c4k)\u22121sin\u2061(\u03c4kh), cos\u2061(\u03c4kh))t. Thus the required transfer matrix is given by\u03c4\u2254\u221a(Nw/\u03b1) and \u03b1 = 1 \u2212 (3/2)\u03b3.The Transmission Coefficient. After having the matrix M in hand, our next step is to calculate the transfer coefficient T. With the help of matrix M, we map the values x2 = 0+ to the values x2 = \u2212h\u2212; that is, (\u22121 + R))eikx1sin\u2061(\u03b8) will get mapped to )eikx1sin\u2061(\u03b8). In other words,T, we eliminate the unknown R. Now we follow a simple elimination technique shown in [v \u2208 \u21022 and w \u2208 \u21022 byv\u22a5 will result in the elimination of R from , where Nw is defined by the help of an eigenvalue problem in the metallic part \u03a3 and we also notice that T depends on the wave number k by the relation Nw = Nw(k). For a rather simple Nw, the graph of |T|2 against the wave number k is shown in figure\u2009\u20094 in [By , we havere\u2009\u20094 in .\u03b5r < 0; then Nw \u2208 . Also \u03b1 = 1 \u2212 (3/2)\u03b3 \u2208 , where 0 < \u03b3 < (1/2). This implies that the term ((\u03b1\u03c4/cos\u2061(\u03b8)) + (cos\u2061(\u03b8)/\u03b1\u03c4)) in (T| \u2264 1 and we get |T| = 1\u21d4cos\u2061(\u03c4kh) = 1. This corresponds to a resonance of the plasmon waves in the metallic structure ) with height h.Let us focus again on the case of a material that permits perfect plasmon waves, that is, of a lossless material with negative permittivity, /\u03b1\u03c4)) in  is great M of (\u03c4kh) = 1, sin\u2061(\u03c4kh) = 0 and we get the transfer matrix M = I, the Identity matrix, corresponding to perfect transmission.We see that this effect can also be deduced from the transfer matrix M of , since f"}
{"text": "Scientific Reports6: Article number: 3852910.1038/srep38529; published online: 12062016; updated: 05172017In Figure 1, the latitudes \u201880.5\u2009N\u2019 and \u201881.0\u2009N\u2019 were incorrectly given as \u201881.0\u2009N\u2019 and \u201881.5\u2009N\u2019 respectively. In addition, the scale between 0\u2009km and 40\u2009km was incorrectly given as between 0\u2009km and 80\u2009km. The correct Figure 1 appears below as"}
{"text": "PGC\u20101\u03b1 plays a central role in hepatic gluconeogenesis. We previously reported that alternative splicing of the PGC\u20101\u03b1 gene produces an additional transcript encoding the truncated protein NT\u2010PGC\u20101\u03b1. NT\u2010PGC\u20101\u03b1 is co\u2010expressed with PGC\u20101\u03b1 and highly induced by fasting in the liver. NT\u2010PGC\u20101\u03b1 regulates tissue\u2010specific metabolism, but its role in the liver has not been investigated. Thus, the objective of this study was to determine the role of hepatic NT\u2010PGC\u20101\u03b1 in the regulation of gluconeogenesis. Adenovirus\u2010mediated expression of NT\u2010PGC\u20101\u03b1 in primary hepatocytes strongly stimulated the expression of key gluconeogenic enzyme genes (PEPCK and G6Pase), leading to increased glucose production. To further understand NT\u2010PGC\u20101\u03b1 function in hepatic gluconeogenesis in\u00a0vivo, we took advantage of a previously reported FL\u2010PGC\u20101\u03b1\u2212/\u2212 mouse line that lacks full\u2010length PGC\u20101\u03b1 (FL\u2010PGC\u20101\u03b1) but retains a slightly shorter and functionally equivalent form of NT\u2010PGC\u20101\u03b1 (NT\u2010PGC\u20101\u03b1254). In FL\u2010PGC\u20101\u03b1\u2212/\u2212 mice, NT\u2010PGC\u20101\u03b1254 was induced by fasting in the liver and recruited to the promoters of PEPCK and G6Pase genes. The enrichment of NT\u2010PGC\u20101\u03b1254 at the promoters was closely associated with fasting\u2010induced increase in PEPCK and G6Pase gene expression and efficient production of glucose from pyruvate during a pyruvate tolerance test in FL\u2010PGC\u20101\u03b1\u2212/\u2212 mice. Moreover, FL\u2010PGC\u20101\u03b1\u2212/\u2212 primary hepatocytes showed a significant increase in gluconeogenic gene expression and glucose production after treatment with dexamethasone and forskolin, suggesting that NT\u2010PGC\u20101\u03b1254 is sufficient to stimulate the gluconeogenic program in the absence of FL\u2010PGC\u20101\u03b1. Collectively, our findings highlight the role of hepatic NT\u2010PGC\u20101\u03b1 in stimulating gluconeogenic gene expression and glucose production.The transcriptional coactivator  Blood glucose levels are maintained by a tight regulation of glucose uptake by peripheral tissues and glucose production in the liver. During prolonged fasting, liver induces the gluconeogenic pathway that synthesizes glucose from non\u2010carbohydrate precursors such as pyruvate, lactate, glycerol and alanine. This metabolic process is primarily controlled by key gluconeogenic enzymes such as PEPCK and G6Pase, the activities of which are regulated at the transcriptional level through hormonal and nutrient signals , which cooperatively promote the full activation of PEPCK and G6Pase gene expression   Fig.\u00a0. NT\u2010PGC\u2010\u03b1\u2212/\u2212 mice deficient in full\u2010length PGC\u20101\u03b1 (FL\u2010PGC\u20101\u03b1) have been described previously . FL\u2010PGC\u20101g for 5\u00a0min at 4\u00b0C, and then resuspended in DMEM/F\u201012 medium supplemented with 25\u00a0mmol/L glucose, 10% FBS, 15\u00a0mmol/L HEPES, 100\u00a0nmol/L dexamethasone, and penicillin/streptomycin. Hepatocytes were plated on collagen\u2010coated 12\u2010well plates and allowed to attach for 1\u00a0h at 37\u00b0C in a humidified incubator. Cells were washed once with DMEM medium containing 5\u00a0mmol/L glucose and incubated in the same medium supplemented with 10% FBS, 100\u00a0nmol/L dexamethasone, 1\u00a0nmol/L insulin, and penicillin/streptomycin for 4\u00a0h. The medium was then replaced with serum\u2010free DMEM medium supplemented with 5\u00a0mmol/L glucose, 10\u00a0nmol/L dexamethasone, 1\u00a0nmol/L insulin, and penicillin/streptomycin. After an overnight incubation, cells were cultured in the same medium containing 10% FBS. When mimicking a fasted condition, cells were treated with or without 1\u00a0\u03bcmol/L dexamethasone and 10\u00a0\u03bcmol/L forskolin for 2\u00a0h in serum\u2010free DMEM medium supplemented with 5\u00a0mmol/L glucose and penicillin/streptomycin.Primary mouse hepatocytes were isolated as described previously with slight modifications . Recombination of AdEasy\u2010GFP, AdEasy\u2010NT\u2010PGC\u20101\u03b1\u2010HA, or AdEasy\u2010PGC\u20101\u03b1\u2010HA with adenoviral gene carrier vector was performed by transformation into BJ5183\u2010AD\u20101 competent cells . Recombinant adenovirus was produced by transfecting recombinant adenoviral plasmids into HEK293 cells  and normalized to the total protein content determined from the whole cell extracts.\u03b1\u2010HA and Flag\u2010HNF4\u03b1 were co\u2010transfected into HEK293 cells using Fugene 6 . Immunoprecipitation assays were carried out as described previously by Chang et\u00a0al.  and normalized using Renilla luciferase activity. Data represent mean\u00a0\u00b1\u00a0SEM of at least three independent experiments.HEK293 cells were transiently transfected with pHNF4\u2010tk\u2010luc  at different time points .Total RNA from livers and hepatocytes was extracted and reverse\u2010transcribed as described previously . Protein concentration was determined using Bio\u2010Rad DC protein assay reagents according to the manufacturer's instructions.Whole cell extracts were prepared from tissues and hepatocytes by homogenization in lysis buffer   . PEPCK and G6Pase were upregulated in ob/ob mice concomitantly with elevated expression of PGC\u20101\u03b1 and NT\u2010PGC\u20101\u03b1  necessary for binding to nuclear receptors such as HNF4\u03b1 and GR but not the C\u2010terminal domain required for binding to FOXO1   or control adenovirus expressing GFP (Ad\u2010GFP). HNF4\u03b1 is abundantly expressed in HepG2 cells, but its transcriptional activity is very low due to low expression of the PGC\u20101\u03b1 gene  that targets the 31\u00a0bp insert arising from alternative splicing of the PGC\u20101\u03b1 gene . Commercial algorithms also predicted that this sequence is not effectively targeted to NT\u2010PGC\u20101\u03b1 mRNA.To assess whether NT\u2010PGC\u20101\u03b1 activity by Akt\u2010 and Cdc2\u2010like kinase 2 (CLK2)\u2010mediated phosphorylation    assay using a PGC\u20101\u03b1 antibody. The PGC\u20101\u03b1 antibody has been confirmed for its specificity to immunoprecipitate FL\u2010PGC\u20101\u03b1 and NT\u2010PGC\u20101\u03b1 (WT mice) and NT\u2010PGC\u20101\u03b1254 (FL\u2010PGC\u20101\u03b1\u2212/\u2212 mice)  in the G6Pase promoter (data not shown). This might be due to weak or temporal formation of FL\u2010PGC\u20101\u03b1\u2010 and/or NT\u2010PGC\u20101\u03b1\u2010associated transcriptional complex on this region.Elevated expression of NT\u2010PGC\u20101les Fig.\u00a0, neither\u03b1\u2212/\u2212 mice exhibited normal fasting blood glucose levels after a 24\u00a0h\u2010fast  can co\u2010activate PPAR\u03b1 .PGC\u20101ice Fig.\u00a0A. A 24\u00a0hice Fig.\u00a0A. The inice Fig.\u00a0B. In conice Fig.\u00a0C. In agr\u03b1 plays a key role in fasting\u2010induced gluconeogenesis by linking hormonal and nutrient signals to the transcription program in the liver. We previously reported that fasting\u2010inducible NT\u2010PGC\u20101\u03b1 as well as full\u2010length PGC\u20101\u03b1 (FL\u2010PGC\u20101\u03b1) are produced from the canonical promoter of PGC\u20101\u03b1 gene in the liver but no additional transcripts from the distal alternative promoter of PGC\u20101\u03b1 gene . Our results clearly demonstrate that NT\u2010PGC\u20101\u03b1 is sufficient to activate the gluconeogenic program in primary hepatocytes. We think that NT\u2010PGC\u20101\u03b1 activates the promoters of PEPCK and G6Pase genes through interaction with HNF4\u03b1 and GR, but not FOXO1. Indeed, our ChIP analyses demonstrated that hepatic NT\u2010PGC\u20101\u03b1254 is enriched at the PEPCK and G6Pase promoters containing binding sites for HNF4\u03b1 and GR in FL\u2010PGC\u20101\u03b1\u2212/\u2212 mice. The in\u00a0vitro studies further confirmed that NT\u2010PGC\u20101\u03b1 interacts with HNF4\u03b1 and increases HNF4\u03b1\u2010mediated transcription of a reporter gene. HNF4\u03b1 plays a crucial role in the regulation of PEPCK and G6Pase genes  exhibited normal expression of C/EBP\u03b2, PEPCK and G6Pase in the fed and fasted states (Figs.\u00a0\u03b1254 was recruited to the PEPCK and G6Pase promoters with concomitant increase in fasting\u2010induced PEPCK and G6Pase gene expression, indicating that NT\u2010PGC\u20101\u03b1254 is sufficient to stimulate gluconeogenic gene expression in the absence of FL\u2010PGC\u20101\u03b1. Moreover, loss of FL\u2010PGC\u20101\u03b1 function might be compensated by elevated levels of NT\u2010PGC\u20101\u03b1254 protein in FL\u2010PGC\u20101\u03b1\u2212/\u2212 mice. In agreement with normal expression of PEPCK and G6Pase genes, FL\u2010PGC\u20101\u03b1\u2212/\u2212 mice had normal fasting blood glucose levels and efficiently produced glucose from pyruvate during a pyruvate tolerance test. Previously, Burgess et\u00a0al. reported conflicting results regarding the fasting response in FL\u2010PGC\u20101\u03b1\u2212/\u2212 mice. Six\u2010week\u2010old FL\u2010PGC\u20101\u03b1\u2212/\u2212 mice exhibited reduced hepatic glucose production in response to fasting (Burgess et\u00a0al. \u03b1\u2212/\u2212 mice may not generate sufficient energy to support gluconeogenesis since rapid growth and maturation are also occurring in this period. It has been shown that FL\u2010PGC\u20101\u03b1\u2212/\u2212 mice at 4\u20135\u00a0weeks of age exhibit temporary cold intolerance, whereas adult FL\u2010PGC\u20101\u03b1\u2212/\u2212 mice are able to defend against cold (Leone et\u00a0al. \u03b1\u2212/\u2212 mice after a 24\u00a0h fast and no alteration in hepatic gene expression profiles of fatty acid oxidation, mitochondrial oxidative phosphorylation, and lipogenic genes.Adenovirus\u2010mediated PGC\u20101\u03b1 isoforms in the liver? Although this study shows that NT\u2010PGC\u20101\u03b1 is recruited to the same target genes regulated by FL\u2010PGC\u20101\u03b1, it does not address the relative contributions of NT\u2010PGC\u20101\u03b1 and FL\u2010PGC\u20101\u03b1 to target gene expression. We speculate that the timing of their action may be different during the course of fasting because protein stabilities of FL\u2010PGC\u20101\u03b1 and NT\u2010PGC\u20101\u03b1 are differentially regulated (Zhang et\u00a0al. \u03b1 and FL\u2010PGC\u20101\u03b1 may regulate a subset of isoform\u2010specific target genes in response to fasting. Genome\u2010wide occupancy analysis will be needed to elucidate whether there is differential recruitment to genomic sites between NT\u2010PGC\u20101\u03b1 and FL\u2010PGC\u20101\u03b1. Our study also revealed that NT\u2010PGC\u20101\u03b1 and FL\u2010PGC\u20101\u03b1 activities are differentially regulated by insulin. Given that NT\u2010PGC\u20101\u03b1 gene expression is elevated in diabetic livers in ob/ob mice and that NT\u2010PGC\u20101\u03b1\u2010mediated gluconeogenic gene expression is not suppressed by insulin, it is possible that NT\u2010PGC\u20101\u03b1 overexpression is implicated in the pathogenesis of hyperglycemia. However, in normal physiological conditions, insulin would indirectly repress NT\u2010PGC\u20101\u03b1 activity by inhibiting glucagon\u2010induced expression of PGC\u20101\u03b1 gene.What are the physiological reasons for co\u2010expression of two different PGC\u20101\u03b1 is a functional transcriptional co\u2010activator promoting gluconeogenic gene expression. Here, we propose that gluconeogenesis driven by NT\u2010PGC\u20101\u03b1, along with PGC\u20101\u03b1, contributes to the elevated blood glucose level in response to fasting.Collectively, this study demonstrates, for the first time, that hepatic NT\u2010PGC\u20101None declared."}
{"text": "The values are calculated by employing the triple \u03b6 basis sets of the Slater type at the DFT level. 1J are calculated modeled by MeSeSeMe (1a), which shows the typical torsional angular dependence on \u03d5(CMeSeSeCMe). The dependence explains well the observed 1Jobsd  of small values (\u2264 64 Hz) for RSeSeR\u2032 (1) (simple derivatives of 1a) and large values (330\u2013380 Hz) observed for 4-substituted naphto-1,2-diselenoles (2) which correspond to symperiplanar diselenides. 1J  becomes larger as the electron density on Se increases. The paramagnetic spin-orbit terms contribute predominantly. The contributions are evaluated separately from each MO (\u03c8i) and each \u03c8i \u2192 \u03c8a transition, where \u03c8i and \u03c8a are occupied and unoccupied MO's, respectively. The separate evaluation enables us to recognize and visualize the origin and the mechanism of the couplings.Nuclear couplings for the Se-Se bonds,  J) provide highly important information around coupled nuclei, containing strongly bonded and weakly interacting states, since the values depend on the electron distribution between the nuclei -1,2-diselenole : Y = H (a)  contribute to 1JTL ? 1JPSO  and 1JSD+FC  are plotted versus 1JTL , although not shown. The correlations are given in  and 1JSD+FC  contribute 65% and 35% to 1JTL , respectively, irrespective of the \u03d5(CSeSeC) values:How do given in  and 3),,1JPSO 1J show the torsional angular dependence? What orbitals and transitions contribute to the dependence? 1JPSO  is analyzed next.Why does 1JPSO  is discussed by analyzing the contributions separately from each \u03c8i and each \u03c8i \u2192 \u03c8a transition.\u03d5 dependence of 1JPSO  contributed from \u03c81\u2013\u03c843, \u03c81\u2013\u03c838, \u03c839\u2013\u03c843, \u03c839, \u03c840, \u03c841, \u03c842, and \u03c843. The contribution from \u03c839\u2013\u03c843 to 1JPSO  is large, whereas that from \u03c81\u2013\u03c838 is small, although not shown. The plot of the contributions from \u03c839\u2013\u03c843 (y) versus those from \u03c81\u2013\u03c843 (x) provides an excellent correlation (y = 0.976x + 37.3 : r2 = 0.9999).\u03c839, \u03c840, \u03c841, \u03c842, and \u03c843 and \u03c839\u2013\u03c841, \u03c842\u2013\u03c843, and \u03c839\u2013\u03c843. Contributions from \u03c842 and \u03c843 exchange with each other at \u03d5 \u2248 90\u00b0. Those of \u03c839 and \u03c840 do at \u03d5 \u2248 135\u00b0  are very large at 0\u00b0 and 180\u00b0  will be clarified by analyzing the contributions from \u03c842 and \u03c843 at 0\u00b0 and 180\u00b0. The mechanism would be complex at 90\u00b0, since the small magnitude is the results of the total contributions from \u03c839\u2013\u03c843.Magnitudes of the contributions from and 180\u00b0 , althoug\u03c842 \u2192 \u03c844 and \u03c843 \u2192 \u03c844 transitions at both \u03d5 = 0\u00b0 and 180\u00b0 which are shown in \u03c842(HOMO-1), \u03c843(HOMO), and \u03c844(LUMO) are \u03c0(Se\u2013Se), \u03c0*(Se\u2013Se), and \u03c3*(Se\u2013Se), respectively, at \u03d5 = 0\u00b0 and 180\u00b0. \u03c842(HOMO-1) is essentially the same as \u03c843(HOMO) at \u03d5 = 90\u00b0. \u03c842 and \u03c843 at \u03d5 = 90\u00b0 are also drawn in \u03c842 and \u03c843 interconvert with each other. Contrary to the case of \u03d5 \u2248 0 and 180\u00b0, all of \u03c839\u2013\u03c843 contribute to 1JPSO  at \u03d5 \u2248 90\u00b0. Contributions from the \u03c842 \u2192 \u03c844 and \u03c843 \u2192 \u03c844 transitions to 1JPSO  at 90\u00b0 are almost cancelled by those from the \u03c839 \u2192 \u03c844, \u03c840 \u2192 \u03c844, and \u03c841 \u2192 \u03c844 transitions. In addition, both 1JSD  and 1JFC  substantially contribute at \u03d5 \u2248 90\u00b0. Consequently, it is difficult to specify a few orbitals, together with the transitions, which control 1J at \u03d5 \u2248 90\u00b0. The character of \u03c844 [LUMO: \u03c3*(Se\u2013Se)] does not change so much depending on \u03d5. Therefore, the behavior of \u03c839\u2013\u03c843 must be mainly responsible for the \u03d5 dependence in 1J  and helps us to understand the mechanism, especially at \u03d5 = 0\u00b0 and 180\u00b0.1JPSO , next extension is to clarify 1J on the basis of the MO theory.After elucidation of the mechanism for 1JTL  values, together with JPSO, 1JSD , 1JFC , and 1JSD+FC . Qn(Se))  into \u03c867 [HOMO: \u03c0*(Se\u2013Se)] and \u03c866 [HOMO-1: \u03c0(Se\u2013Se)] at the singlet state. The MO presentation in \u03c842 \u2192 \u03c844 and \u03c843 \u2192 \u03c844 transitions in 1JPSO  at \u03d5 = 0\u00b0 in \u03c867 (2a) and \u03c866 (2a) contain the \u03c0(Nap) character. Large 1JPSO  and small 1Jobsd  are well understood by the \u03d5 dependence in the calculated 1J values.J) provide highly important information around coupled nuclei, containing strongly bonded and weakly interacting states. The 1J values are analyzed as the first step to investigate the nature of the bonded and nonbonded interactions between the Se atoms through nJ. QC calculations are necessary for the analysis and the interpretation of the J values with physical meanings. Calculated nJTL are composed of the contributions from nJDSO, nJPSO, nJSD, and nJFC. The decomposition helps us to consider the mechanisms of the spin-spin couplings, which are closely related to the electronic structures of compounds. Main contributions are evaluated separately from each \u03c8i and each \u03c8i \u2192 \u03c8a transition, where \u03c8i and \u03c8a are occupied and unoccupied MO's, respectively.Nuclear spin-spin coupling constants  is calculated modeled by MeSeSeMe (1a), which shows the typical torsional angular dependence of \u03d5(CMeSeSeCMe). The dependence explains well 1Jobsd  of small values for RSeSeR\u2032 (1) and large values for 4-Y-1,8-Se2C10H5 (2) which correspond to symperiplanar diselenides. 1JTL  are confirmed to be controlled by Qn(Se). 1JTL  are demonstrated to be smaller when Qn(Se) becomes larger, experimentally and theoretically. The PSO terms contribute predominantly to 1J. The contributions are analyzed separately from each \u03c8i and each \u03c8i \u2192 \u03c8a transition. The MO description of each transition enables us to recognize and visualize clearly the origin and the mechanisms of the indirect nuclear spin-spin couplings. Important properties of molecules, such as electronic structures, will be clarified by elucidating the mechanisms of the spin-spin couplings on the basis of the MO theory."}
{"text": "The second of two targets (T2) embedded in a rapid serial visual presentation\t\t\t\t\t(RSVSVP) is often missed even though the first (T1) is correctly reported\t\t\t\t\t. The rate of correct T2 identification is quite high,\t\t\t\t\thowever, when T2 comes immediately after T1 (lag-1 sparing). This study\t\t\t\t\tinvestigated whether and how non-target items induce lag-1 sparing. One T1 and\t\t\t\t\ttwo T2s comprising letters were inserted in distractors comprising symbols in\t\t\t\t\teach of two synchronised RSVSVPs. A digit (dummy) was presented with T1 in\t\t\t\t\tanother stream. Lag-1 sparing occurred even at the location where the dummy was\t\t\t\t\tpresent (Experiment 1). This distractor-induced sparing effect was also obtained\t\t\t\t\teven when a Japanese katakana character (Experiment 2) was used as the dummy.\t\t\t\t\tThe sparing effect was, however, severely weakened when symbols (Experiment 3)\t\t\t\t\tand Hebrew letters (Experiment 4) served as the dummy. Our findings suggest a\t\t\t\t\ttentative hypothesis that attentional set for item nameability is\t\t\t\t\tmeta-categorically created and adopted to the dummy only when the dummy is\t\t\t\t\tnameable. That is, only an item that matches the attentional set can be\t\t\t\tprocessed further. In the TLC model, it is assumed that the observers initially\t\t\t\tadopt the attentional set for a target category in an endogenous manner. The\t\t\t\tattentional set requires periodic maintenance signals from the central executive in\t\t\t\tthe higher brain regions . That is, one stream had T1 and T2, and the\t\t\t\tother had the dummy T1 and T2. In this situation, the dummy T1 category should be\t\t\t\tconfigured as one of the distractor categories. Moreover, previous studies suggested\t\t\t\tthat lag-1 sparing did not occur when T1 and T2 locations were different \t\t\t\t\t\twith a resolution of 1024 \u00d7 768 pixels and a vertical refresh rate\t\t\t\t\t\tof 75 Hz. A viewing distance of 60 cm was maintained with a head-and-chin\t\t\t\t\t\trest. A PC/AT-compatible computer controlled the presentation of stimuli and\t\t\t\t\t\tcollection of data. Stimuli and experiments were programmed in Delphi 6\t\t\t\t\t\t(Borland Software Corporation). In every trial, from a set of letters of the\t\t\t\t\t\tEnglish alphabet excluding \u201cI\u201d,\t\t\t\t\t\t\u201cO\u201d, \u201cQ\u201d, and\t\t\t\t\t\t\u201cZ\u201d, three uppercase letters, all different, were\t\t\t\t\t\trandomly chosen as targets. Ten keyboard symbols served as distractors\t\t\t\t\t\t. The dummy T1 was\t\t\t\t\t\tan Arabic digit. Each item subtended a visual angle of around 1\u00b0 in\t\t\t\t\t\theight. The luminance of these items was 2.5 cd/m2 against a background with\t\t\t\t\t\ta luminance of 98.5 cd/m2. The stimulus display comprised a fixation cross\t\t\t\t\t\tat the centre of the screen and two synchronised RSVP streams to the left\t\t\t\t\t\tand right of the fixation cross. T1 was one of the three targets that\t\t\t\t\t\tappeared in one of the two streams, and the T2s of the remaining two targets\t\t\t\t\t\twere simultaneously presented in both streams. The dummy T1 was presented\t\t\t\t\t\tsimultaneously with T1 but in another stream. In a trial in which the dummy\t\t\t\t\t\tT1 was absent, a distractor item  was inserted instead. The\t\t\t\t\t\tdummy presence/absence was equally probable. The centre-to-centre distance\t\t\t\t\t\tbetween the two streams subtended a visual angle of 3.4\u00b0.The observers were individually tested in a dark room. F = 14.3, MSE = 226.8,\t\t\t\t\t\tp < .005, and Lag, F = 16.3,\t\t\t\t\t\tMSE = 97.0, p < .0001. It also\t\t\t\t\trevealed significant interactions between Dummy and Lag, F = 2.8, MSE = 49.91, p = .04, between T2\t\t\t\t\tlocation and Lag, F = 7.5, MSE = 68.77,\t\t\t\t\t\tp = .0002, and among the three factors,\t\t\t\t\tF = 2.7, MSE = 83.18,\t\t\t\t\t\tp = .04. The main effect of Dummy, however, was not\t\t\t\t\tsignificant, F = 0.6, p = .46 Moreover,\t\t\t\t\tthe interaction between Dummy and T2 location was not significant,\t\t\t\t\t\tF = 0.6, p = .48 The tests of the\t\t\t\t\tsimple effects, based on the significant interaction among the three factors,\t\t\t\t\trevealed significant simple-simple main effects of Lag in the present-consistent\t\t\t\t\tcondition, F = 8.3, p < .0001,\t\t\t\t\tpresent-inconsistent condition, F = 7.2,\t\t\t\t\t\tp < .0001, absent-consistent condition,\t\t\t\t\t\tF = 8.8, p < .0001, and\t\t\t\t\tabsent-inconsistent condition, F = 8.7,\t\t\t\t\t\tp < .0001. Moreover, a simple-simple main effect of\t\t\t\t\tDummy was found in the inconsistent condition at lag 1, F = 10.0, p = .002.A three-way analysis of variance (ANOVA) on T2 performance with three\t\t\t\t\twithin-subject factors showed significant main effects of T2 location,\t\t\t\t\t\tt(144) = 4.52, p < .0001;\t\t\t\t\t\tt(144) = 4.23, p < .0001;\t\t\t\t\t\tt(144) = 5.05, p < .0001,\t\t\t\t\trespectively. The identification rate of T2 at lag 1, however, was not\t\t\t\t\tsignificantly different from that at lag 2 in the absent-inconsistent condition,\t\t\t\t\t\tt(144) = 0.72, p = .47.Multiple comparisons using Ryan\u2019s method ,2 based t-test\t\t\t\t\trevealed that the correct rate for T1 identification was significantly lower\t\t\t\t\twhen T2 appeared at lag 1 in the present condition than when it appeared in the\t\t\t\t\tabsent condition, t(9) = 3.04, p = .01.Additionally, the correct rate for T1 identification was analysed to confirm\t\t\t\t\tcompetition between T1 and the dummy item. A two-tailed In Experiment 1, we found that lag-1 sparing was observed for both T2s\t\t\t\t\tconcurrently. Specifically, lag-1 sparing occurred at a location different from\t\t\t\t\tthe T1 location only when the dummy T1 was presented. On the other hand, at the\t\t\t\t\tT1 location, robust lag-1 sparing occurred regardless of the presence/absence of\t\t\t\t\tthe dummy item. At the location different from the T1 location, lag-1 sparing\t\t\t\t\tdid not occur when the dummy T1 was not presented, consistent with previous\t\t\t\t\tstudies showing that lag-1 sparing occurred only when a common location was\t\t\t\t\tshared by T1 and T2 . The comAdditionally, performance at a location different from the T1 location was\t\t\t\t\tseverely impaired at lag 1 when the dummy item was absent, even though\t\t\t\t\tperformance at the T1 location was quite high. These results support the notion\t\t\t\t\tthat the T2 item in each stream was processed as a part of a discrete\t\t\t\t\tattentional episode established at each stimulus location. That is, each stream\t\t\t\t\tof RSVP seems to be filtered by an attentional set that can be split into\t\t\t\t\tmultiple locations and works independently .Not all the results of this experiment, however, can be explained by\t\t\t\t\tmultidimensional attentional setting  based onOne might argue that the results of Experiment 1 reflect the adoption of an\t\t\t\t\tattentional set configured for an alphanumeric category, which is a\t\t\t\t\tmeta-category of letters and digits. That is, it was possible that the observers\t\t\t\t\tin Experiment 1 adopted the attentional set that corresponds to a category\t\t\t\t\tincluding both letters and digits all together, namely, alpha-numerals. Thus, an\t\t\t\t\talphanumeric attentional set might be applied to both the dummy and T2,\t\t\t\t\tresulting in conventional lag-1 sparing. The next experiment examined this\t\t\t\t\tpossibility by introducing a new category, which is not included in the\t\t\t\t\talphanumeric category, as a dummy category.This experiment aimed at testing whether the adoption of an alphanumeric attentional\t\t\t\tsetting produced lag-1 sparing as in the first experiment. In Experiment 2, a new\t\t\t\tcategory, Japanese katakana, was used as the category of the dummy T1. This category\t\t\t\twas quite familiar to the Japanese observers employed in this experiment and was not\t\t\t\tincluded in the alphanumeric category. If the results of Experiment 1 were a product\t\t\t\tof alphanumeric attentional setting, lag-1 sparing should not occur even when the\t\t\t\tdummy T1 of Japanese katakana was presented.Eleven Japanese students from Kyushu University, including one of the authors\t\t\t\t\t\t(Y.Y.), participated in this experiment. Except for Y.Y., they were unaware\t\t\t\t\t\tof the purpose of the experiment. All of them reported normal or\t\t\t\t\t\tcorrected-to-normal eyesight.The apparatus, stimuli, and procedure were identical to those in Experiment 1\t\t\t\t\t\texcept that, instead of digits, 10 Japanese katakana characters,\t\t\t\t\t\t\u201c\u30a2\u201d (a),\t\t\t\t\t\t\u201c\u30a4\u201d (i),\t\t\t\t\t\t\u201c\u30a6\u201d (u),\t\t\t\t\t\t\u201c\u30a8\u201d (e),\t\t\t\t\t\t\u201c\u30aa\u201d (o),\t\t\t\t\t\t\u201c\u30ab\u201d (ka),\t\t\t\t\t\t\u201c\u30ad\u201d (ki),\t\t\t\t\t\t\u201c\u30af\u201d (ku),\t\t\t\t\t\t\u201c\u30b1\u201d (ke), and\t\t\t\t\t\t\u201c\u30b3\u201d (ko), were introduced as dummies. The\t\t\t\t\t\tobservers were asked to ignore Japanese katakana.F = 11.5, MSE =\t\t\t\t\t143.03, p = .007, T2 location, F =\t\t\t\t\t10.0, MSE = 276.52, p = .01, and Lag,\t\t\t\t\t\tF = 11.3, MSE = 327.02,\t\t\t\t\t\tp < .0001. Significant interactions between Dummy\t\t\t\t\tand Lag, F = 4.4, MSE = 161.61,\t\t\t\t\t\tp = .005, between T2 location and Lag,\t\t\t\t\tF = 6.2, MSE = 205.49,\t\t\t\t\t\tp = .0006, and among the three factors,\t\t\t\t\tF = 2.9, MSE = 114.42, p = .03, were\t\t\t\t\tobtained. The interaction between Dummy and T2 location, F = 0.1, p = .80, was not significant. Tests of the simple\t\t\t\t\teffects, based on significant interactions among the three factors, revealed\t\t\t\t\tsignificant simple-simple main effects of Lag in the present-consistent\t\t\t\t\tcondition, F = 12.7, p < .0001,\t\t\t\t\tpresent-inconsistent condition, F = 6.4,\t\t\t\t\t\tp = .0001, and absent-consistent condition,\t\t\t\t\t\tF = 10.2, p < .0001, but\t\t\t\t\tnot in the absent-inconsistent condition, F = .40,\t\t\t\t\t\tp = .81. Moreover, simple-simple main effects of Dummy were\t\t\t\t\tfound in the inconsistent condition at lag 1, F = 6.6,\t\t\t\t\t\tp = .01, and at lag 3, F = 10.9,\t\t\t\t\t\tp = .001.t(160) = 5.40, p\t\t\t\t\t< .0001, present-inconsistent condition, t(160) = 3.12,\t\t\t\t\t\tp = .002, and absent-consistent condition,\t\t\t\t\t\tt(160) = 3.64, p = .0004. A post hoc\t\t\t\t\t\tt-test did not reveal a significant difference between the\t\t\t\t\tperformances at lag 1 and lag 2 in the absent-inconsistent condition,\t\t\t\t\t\tt(10) = 0.44, p = .67.Multiple comparison tests using Ryan\u2019s method, based on the\t\t\t\t\tsimple-simple main effect of Lag, indicated that the correct identification rate\t\t\t\t\tof T2 at lag 1 was significantly higher than that at lag 2 in the\t\t\t\t\tpresent-consistent condition, t-test did not reveal any difference between the\t\t\t\t\tcorrect identification rates of T1 when T2 appeared at lag 1 in the present\t\t\t\t\tcondition and when T2 appeared in the absent condition, t(10) =\t\t\t\t\t0.13, p = .90.A two-tailed In this experiment, as well as in Experiment 1, lag-1 sparing with Japanese\t\t\t\t\tkatakana as the dummy T1 was clearly observed. The involvement of an\t\t\t\t\talphanumeric attentional setting for both letters and digits can still explain\t\t\t\t\tlag-1 sparing observed in Experiment 1, but cannot explain the results of\t\t\t\t\tExperiment 2.Why did lag-1 sparing occur even though no attentional set was configured for the\t\t\t\t\tdummy T1? As a straightforward interpretation suggests, it is likely that the\t\t\t\t\tdummy T1 erroneously served as the actual T1, leading to the lag-1 sparing of\t\t\t\t\tthe trailing T2. In this interpretation, an attentional set for targets or an\t\t\t\t\tattentional set for distractors, related to active selection or active\t\t\t\t\trejection, would be involved in this erroneous selection. The cognitive system\t\t\t\t\tseemed mistakenly to select the dummy T1 because the dummy category (digits or\t\t\t\t\tJapanese katakana) was similar to the target category (letters) or because the\t\t\t\t\tdummy category was different from the distractor category (symbols) that made up\t\t\t\t\tthe majority of RSVP streams and consequently was not rejected.This experiment examined whether a dummy item belonging to a category which was\t\t\t\tsimply different from a distractor category led to lag-1 sparing. In Experiment 3,\t\t\t\tthe categories of dummy items and distractors used in Experiment 1 were reversed\t\t\t\t. Despite\t\t\t\tthe categorical reversal, the dummy and target categories were still clearly\t\t\t\tseparated although the difference between the dummy and distractor categories\t\t\t\tremained unchanged from that in Experiment 1. Lag-1 sparing would occur when a dummy\t\t\t\tsymbol item was presented if mere categorical difference between the dummy and\t\t\t\tdistractor was the decisive factor.Fourteen students from Kyushu University participated, and none of the\t\t\t\t\t\tstudents were aware of the purpose of the experiment. All reported normal or\t\t\t\t\t\tcorrected-to-normal eyesight.The fundamental aspects of the apparatus, stimuli, and procedure were\t\t\t\t\t\tidentical to those in Experiment 1, with the following exceptions: The dummy\t\t\t\t\t\tT1 category was changed to symbols, and the category of the distractors was\t\t\t\t\t\tchanged to digits. The observers were asked to ignore the symbols.F = 16.9,\t\t\t\t\t\tMSE = 405.87, p = .001, and Lag,\t\t\t\t\t\tF = 7.2, MSE = 226.64,\t\t\t\t\t\tp = .0001. It also revealed significant interactions\t\t\t\t\tbetween T2 location and Lag, F = 28.0,\t\t\t\t\t\tMSE = 137.69, p < .0001, and among\t\t\t\t\tthe three factors, F = 3.9, MSE = 87.7,\t\t\t\t\t\tp = .007. The main effect of Dummy was not significant,\t\t\t\t\t\tF = 0.4, p = .53. Moreover, the interactions between\t\t\t\t\tDummy and T2 location, F = 0.02, p =\t\t\t\t\t.90, and between Dummy and Lag, F = 1.2,\t\t\t\t\t\tp = .33, were not significant. Tests of the simple effects,\t\t\t\t\tbased on the significant interaction among the three factors, revealed\t\t\t\t\tsignificant simple-simple main effects of Lag in the present-consistent\t\t\t\t\tcondition, F = 10.9, p < .0001,\t\t\t\t\tpresent-inconsistent condition, F = 4.5,\t\t\t\t\t\tp = .002, absent-consistent condition,\t\t\t\t\tF = 13.3, p < .0001, and\t\t\t\t\tabsent-inconsistent condition, F = 15.8,\t\t\t\t\t\tp < .0001. Moreover, simple-simple main effects of\t\t\t\t\tDummy were found in the inconsistent condition at lag 1, F = 4.2, p = .04, and lag 5, F =\t\t\t\t\t4.5, p = .04.t(208) = 4.15, p < .0001,\t\t\t\t\tand the absent-consistent condition, t(208) = 5.75,\t\t\t\t\t\tp < .0001. The difference between the correct\t\t\t\t\tidentification rate of T2 at lag 1 and at lag 2, however, did not reach\t\t\t\t\tsignificance in the present-inconsistent condition, t(208) =\t\t\t\t\t0.70, p = .49, or absent-inconsistent condition,\t\t\t\t\t\tt(208) = 0.26, p = .79. That is, lag-1\t\t\t\t\tsparing was not observed in these inconsistent conditions.Multiple comparisons using Ryan\u2019s method, based on the simple-simple\t\t\t\t\tmain effect of Lag, indicated that the correct identification rate of T2 at lag\t\t\t\t\t1 was significantly higher than that at lag 2 in the present-consistent\t\t\t\t\tcondition, t-test did not reveal any difference between the\t\t\t\t\tcorrect identification rates of T1 when T2 appeared at lag 1 in the present\t\t\t\t\tcondition and when T2 appeared in the absent condition, t(13) =\t\t\t\t\t0.14, p = .89.A two-tailed In this experiment, lag-1 sparing was attenuated even when the dummy T1 was\t\t\t\t\tpresent. Specifically, although T2 performance at lag 1 was not significantly\t\t\t\t\thigher than that at lag 2 in the inconsistent condition, T2 performance at lag 1\t\t\t\t\tin the Dummy-present condition was higher than in the Dummy-absent condition.\t\t\t\t\tMoreover, the analysis of T1 performance suggests that the dummy T1 symbol did\t\t\t\t\tnot impair the performance for the actual T1. These results suggest that lag-1\t\t\t\t\tsparing largely depends on potential common properties between the dummy and\t\t\t\t\ttarget categories rather than on the categorical difference between the dummy\t\t\t\t\tand distractor categories.What then was the common property of the dummy and target categories producing\t\t\t\t\tlag-1 sparing in this study? A tentative answer to this question is the\t\t\t\t\tnameability of items. Names of items were different among the categories used in\t\t\t\t\tthe previous experiments. For example, an item belonging to letter, digit, and\t\t\t\t\tJapanese katakana categories is easy to name. Such easy-to-name dummy items\t\t\t\t\tmight have produced lag-1 sparing in Experiments 1 and 2. Symbols such as\t\t\t\t\t\u201c$\u201d, \u201c#\u201d, and\t\t\t\t\t\u201c!\u201d, however, are difficult to name. The difficult-to-name\t\t\t\t\tdummy items might not have produced lag-1 sparing in Experiment 3. If\t\t\t\t\teasy-to-name items were preferentially treated by the cognitive system, the\t\t\t\t\tattentional set would be erroneously adopted for such items, resulting in lag-1\t\t\t\t\tsparing.Experiment 4 was performed to determine whether lag-1 sparing with the dummy items\t\t\t\tdepended on item nameability. We used Hebrew alphabet letters as dummy categories,\t\t\t\tRoman alphabet letters and symbols as target categories, and digits as the\t\t\t\tdistractor category. Data were collected from Japanese students who knew the shape\t\t\t\tof Hebrew alphabet letters, but could not name an individual letter. If item\t\t\t\tnameability underlies dummy-driven lag-1 sparing, no lag-1 sparing with the dummy\t\t\t\titem from Hebrew letters would be observed because Japanese participants could not\t\t\t\tname them.Twelve Japanese adults participated in this experiment. All of them were\t\t\t\t\t\tunaware of the purpose of the experiment and reported normal or\t\t\t\t\t\tcorrected-to-normal eyesight.This experiment was similar to Experiment 1 except for the following. First,\t\t\t\t\t\tcategories of targets, dummies, and distractors were changed. Ten Roman\t\t\t\t\t\talphabet letters (\u201cA\u201d to \u201cK\u201d\t\t\t\t\t\texcluding \u201cI\u201d) or 10 symbols used in the previous\t\t\t\t\t\texperiments were employed as the targets. Digits served as the distractors.\t\t\t\t\t\tTen Hebrew alphabet letters, \u201c\u05d0\u201d ,\t\t\t\t\t\t\u201c\u05d1\u201d (bet),\t\t\t\t\t\t\u201c\u05d2\u201d (gimel),\t\t\t\t\t\t\u201c\u05d3\u201d ,\t\t\t\t\t\t\u201c\u05d4\u201d (he),\t\t\t\t\t\t\u201c\u05d6\u201d (zayin),\t\t\t\t\t\t\u201c\u05d7\u201d (chet),\t\t\t\t\t\t\u201c\u05d8\u201d (tet),\t\t\t\t\t\t\u201c\u05dc\u201d (lamed), and\t\t\t\t\t\t\u201c\u05e9\u201d (shin), were introduced as dummies. A\t\t\t\t\t\tpre-experiment questionnaire revealed that none of the observers knew Hebrew\t\t\t\t\t\tat all, and the observers were asked to ignore the Hebrew letters within the\t\t\t\t\t\tRSVP streams. Second, only lags 1, 2, and 5 were used. Thus, each observer\t\t\t\t\t\tperformed 120 trials with two experimental blocks including two\t\t\t\t\t\ttarget-category conditions . Each block contained\t\t\t\t\t\t2 dummy conditions (present or absent) \u00d7 2 T1 location conditions\t\t\t\t\t\t(right or left) \u00d7 3 lag conditions  \u00d7 5\t\t\t\t\t\treplications. In each block, the trial order was randomised. The order of\t\t\t\t\t\tthe blocks was counterbalanced across observers.F = 4.3,\t\t\t\t\t\t\tMSE = 383.17, p = .03. It also\t\t\t\t\t\trevealed significant interactions between Dummy and T2 location,\t\t\t\t\t\t\tF = 14.3, MSE = = 402.29,\t\t\t\t\t\t\tp = .003, between T2 location and Lag,\t\t\t\t\t\t\tF = 9.3, MSE = 382.93,\t\t\t\t\t\t\tp = .001, and among the three factors,\t\t\t\t\t\t\tF = 4.1, MSE = 227.80,\t\t\t\t\t\t\tp = .03. The main effects of Dummy,\t\t\t\t\t\tF = 4.7, p = .05, and T2 location,\t\t\t\t\t\t\tF = 3.5, p = .09, were\t\t\t\t\t\tmarginally significant. An interaction between Dummy and Lag,\t\t\t\t\t\t\tF = 0.2, p = .79, was not\t\t\t\t\t\tsignificant. Tests of simple effects based on the interaction between Dummy\t\t\t\t\t\tand T2 location revealed a significant simple main effect of Dummy in the\t\t\t\t\t\tconsistent condition, F = 18.8, p =\t\t\t\t\t\t.0003. Tests of the simple effects based on the significant interaction\t\t\t\t\t\tamong the three factors revealed significant simple-simple main effects of\t\t\t\t\t\tLag in the present-consistent condition, F = 3.2,\t\t\t\t\t\t\tp = .05, absent-consistent condition,\t\t\t\t\t\t\tF = 8.3, p = .0005, and\t\t\t\t\t\tabsent-inconsistent condition, F = 4.6,\t\t\t\t\t\t\tp = .01, but not in the present-inconsistent condition,\t\t\t\t\t\t\tF = 1.4, p = .26. Multiple\t\t\t\t\t\tcomparisons using Ryan\u2019s method, based on the simple-simple main\t\t\t\t\t\teffect of Lag, indicated that the correct identification rate of T2 at lag 1\t\t\t\t\t\twas no different from that at lag 2 in the absent-inconsistent condition,\t\t\t\t\t\t\tt(88) = 1.00, p = .32. A post hoc\t\t\t\t\t\t\tt-test did not reveal a significant difference in the\t\t\t\t\t\tperformance at lag 1 and lag 2 in the present-inconsistent condition,\t\t\t\t\t\t\tt(11) = 1.20, p = .25. Moreover, a\t\t\t\t\t\tsignificant simple-simple main effect of Dummy was acknowledged in the\t\t\t\t\t\tinconsistent condition at lag 1, F = 7.8,\t\t\t\t\t\t\tp = .007. Additionally, a t-test\t\t\t\t\t\trevealed that T1 performance was significantly lower when T2 appeared in the\t\t\t\t\t\tpresent-consistent condition than when it appeared in the absent-consistent\t\t\t\t\t\tcondition, t(11) = 2.32, p = .04.A three-way ANOVA on T2 performance with three within-subject factors \t\t\t\t\t\tshowed a significant main effect of Lag, F = 0.02, p = .98, separate\t\t\t\t\t\tone-way ANOVAs on T2 performance with Lag as a factor were performed. As a\t\t\t\t\t\tresult, significant main effects in the present-consistent condition,\t\t\t\t\t\t\tF = 3.7, p = .04, and\t\t\t\t\t\tabsent-consistent condition, F = 4.4,\t\t\t\t\t\t\tp = .02, were found. The main effects, however, in the\t\t\t\t\t\tabsent-inconsistent condition, F = 0.1,\t\t\t\t\t\t\tp = .93, and present-inconsistent condition,\t\t\t\t\t\t\tF = 1.0, p = .38, were not\t\t\t\t\t\tsignificant. Post hoc t-tests did not reveal a significant\t\t\t\t\t\tdifference in the performance at lag 1 and lag 2 in the present-inconsistent\t\t\t\t\t\tcondition, t(11) = 0.33, p = .75, and\t\t\t\t\t\tabsent-inconsistent condition, t(11) = 0.73,\t\t\t\t\t\t\tp = .48. Moreover, the difference in the performance at\t\t\t\t\t\tlag 1 between the present and absent conditions was not significant,\t\t\t\t\t\t\tt(11) = 1.17, p = .27. Furthermore, T2\t\t\t\t\t\tperformance averaged across lags in the absent-consistent condition was\t\t\t\t\t\tmarginally significantly higher than that in the present-consistent\t\t\t\t\t\tcondition, t(11) = 2.16, p = .05.\t\t\t\t\t\tAdditionally, a t-test revealed that T1 performance was\t\t\t\t\t\tsignificantly lower when T2 appeared in the present-consistent condition\t\t\t\t\t\tthan when it appeared in the absent-consistent condition,\t\t\t\t\t\tt(11) = 2.66, p = .02.Because a three-way ANOVA on T2 performance with three within-subject factors\t\t\t\t\t\tdid not show a significant interaction among the three factors,\t\t\t\t\t\t\tThe results showed that lag-1 sparing with the dummy item was weakened in the\t\t\t\t\tRoman alphabet condition and disappeared in the symbol condition when a Hebrew\t\t\t\t\talphabet letter, which was not nameable by the Japanese observers who\t\t\t\t\tparticipated in this experiment, was employed as the dummy item. The results are\t\t\t\t\tconsistent with the prediction that item nameability strongly influences\t\t\t\t\tdummy-driven lag-1 sparing. This idea is compatible with the present results in\t\t\t\t\tthat weak or no sparing effect was found in this experiment because Hebrew and\t\t\t\t\tsymbols were not nameable. Moreover, the results in the symbol condition suggest\t\t\t\t\tthat a mere categorical difference between the dummy and distractor categories\t\t\t\t\tdoes not determine lag-1 sparing.An unexpected finding in this experiment was that T2 performance in the stream\t\t\t\t\tconsistent with the actual T1 dropped when the dummy item was presented. This\t\t\t\t\twas not a general tendency in the previous experiments in this study. Hence,\t\t\t\t\tthis finding seems to be a stimulus-specific one. In Experiment 4, Hebrew\t\t\t\t\talphabet characters were employed as dummy items, and Japanese observers did not\t\t\t\t\tknow these characters. We surmise that quite unfamiliar items like Hebrew\t\t\t\t\tcharacters increased overall processing cost, affecting processing of the actual\t\t\t\t\tT1 and trailing T2. This is beyond the scope of the present study, but we may\t\t\t\t\texamine this issue in future research.The present study found that a non-target item in neither a target nor distractor\t\t\t\tcategory can elicit lag-1 sparing. Experiment 1 showed that a dummy T1 (digits) not\t\t\t\tbelonging to a target category  produced lag-1 sparing. Moreover, in\t\t\t\tExperiment 2, it was demonstrated that a dummy item from Japanese katakana caused\t\t\t\tlag-1 sparing for the following T2 of Roman alphabet letters, suggesting that\t\t\t\tdummy-based lag-1 sparing occurs beyond an alphanumeric attentional setting.\t\t\t\tAdditionally, Experiment 3 showed that a dummy item from symbols did not cause\t\t\t\trobust lag-1 sparing, suggesting that the mere presence of the dummy item at the\t\t\t\ttemporal location of T1 does not explain dummy-based lag-1 sparing. Finally, the\t\t\t\tresults of Experiment 4 suggest that nameability of the dummy item was related to\t\t\t\tdummy-driven lag-1 sparing. Categories such as the Roman alphabet, Japanese katakana\t\t\t\tletters, and digits were nameable whereas symbols and the Hebrew alphabet were not\t\t\t\tnameable by observers in the present experiments. Our findings suggest that\t\t\t\tattentional set for item nameability is meta-categorically created and adopted to\t\t\t\tthe dummy T1 only when the dummy T1 is nameable. The idea of a meta-categorical\t\t\t\tattentional set for nameability is consistent with almost all the results in this\t\t\t\tstudy.How can the cognitive system differentiate target from distractors if the\t\t\t\tmeta-categorical attentional set is actually adopted? Simple assumptions about the\t\t\t\tmeta-categorical attentional setting for nameability cannot explain why in\t\t\t\tExperiment 4 the observers could differentiate letter targets from the digit\t\t\t\tdistractors. Here we assume two-stage filtering, as illustrated in The results of a previous study  might alSD = 115.8), 421.7 (SD = 100.5), 545.9\t\t\t\t\t(SD = 105.0), 711.5 (SD = 98.8), and 479.9\t\t\t\t\t(SD = 388.2), respectively. The results of statistical\t\t\t\t\tcomparisonsOne might argue that the present results stemmed from an artefact involving a\t\t\t\tlow-level visual feature of the stimuli used in the experiments. Maki, Bussard,\t\t\t\tLopez, and Digby  showed t\t\t\t\tSD = 88.2, resAn account based on feature dissimilarity between the dummy and distractors may,\t\t\t\thowever, explain the present results that cannot be explained by item nameability.\t\t\t\tThese results were the higher T2 performance at lag 1 in the dummy-present condition\t\t\t\tcompared with that in the dummy-absent condition in Experiment 3 and the similar\t\t\t\tdifference in the Roman alphabet condition of Experiment 4. Maki et al.  showed tOther than the filtering dependent on attentional set, an attentional mechanism may\t\t\t\texplain the lag-1 sparing with the dummy item. In a previous study, Potter et al.\t\t\t\t\t suggesteA meta-categorical setting is not an irrational idea. Previous studies have suggested\t\t\t\tthat the character style or the type style is also the subject of an attentional\t\t\t\tset. For example, reported findings show that an attentional set was adopted for\t\t\t\tuppercase words inserted in the RSVP of lowercase words . Additio"}
{"text": "Throughout the captions of Figure S2 and Figure S3 all instances of \u201csigma_syn/sigma_ant\u201d should read \u201cN_ant_double/N_syn_double.\u201d"}
{"text": "Recently, in the compressed sensing framework we found that a two-dimensional interior region-of-interest (ROI) can be exactly reconstructed via the total variation minimization if the ROI is piecewise constant . Here we present a general theorem charactering a minimization property for a piecewise constant function defined on a domain in any dimension. Our major mathematical tool to prove this result is functional analysis without involving the Dirac delta function, which was heuristically used by Yu and Wang (2009). While in general an interior region-of-interest (ROI) cannot be uniquely reconstructed from projection data only associated with lines through the ROI , 2, in td > 0, let \u03a9 \u2282 \u211dd be a d-dimensional open bounded set and denote its boundary by \u2202\u03a9. Then the space of functions of bounded variation is\u03c5||L1(\u03a9) is the integral of |\u03c5(x)| over \u03a9, and\u03c5 \u2208 BV(\u03a9). Here C01(\u03a9) is the space of continuously differentiable functions that vanish on \u2202\u03a9, and for \u03d5 = T \u2208 C01(\u03a9)d, div\u2009\u03d5 = \u2211i = 1d\u2202\u03d5i/\u2202xi. The Sobolev spaceFor piecewise constant or piecewise smooth functions, it is natural to use the space of functions of bounded variation  to captuC1 boundary, and it is decomposed into a union of a finite number of subsets with disjoint interiorsm has a piecewise C1 boundary. The unit outward normal vector on \u2202\u03a9m is denoted by \u03c5m. Denote i and j between 1 and M. We write meas(\u0393i j) for the (d \u2212 1)-dimensional measure of \u0393i j; it is the area of \u0393i j for d = 3, and the length of \u0393i j for d = 2. The symbol \u2211i<j will refer to a summation for those i and j with a nonempty \u0393i j in the range 1 \u2264 i < j \u2264 M. The main result of this note is the following. We that assume \u03a9 has a piecewise f be a piecewise constant function corresponding to the decomposition (Let position :f(x) = cm \u2208 \u211d for x \u2208 \u03a9m, 1 \u2264 m \u2264 M. Then we haveConsequently, we have the minimization property\u03d5 \u2208 C01(\u03a9)d,g(x) = 0 in \u222b\u03a9|\u222b\u03a9|\u03d5 \u2208 C) = 0 in  follows \u03b5 > 0, define two open subsetsx, D) = min\u2009{|x \u2212 y| : y \u2208 D} is the distance between x and a closed set D. Obviously, for some constant c > 0,B\u03b4 is the ball of radius \u03b4 centered at the origin, \u03b7\u03b4(x) = \u03b7(x/\u03b4)/\u03b4d, and\u03d5\u03b5,\u03b4(x)| \u22641 for x \u2208 \u03a9, and for \u03b4 sufficiently small, \u03d5\u03b5,\u03b4(x) \u2208 C0\u221e(\u03a9)d. Moreover, as \u03b4 \u2192 0, \u03d5\u03b5,\u03b4(x) converges uniformly to \u03c8\u03b5(x) for x \u2208 \u03a9\\(\u03a9\u03b5/2\u2202 \u22c3 \u03a9\u03b5/2\u0393). Thus from \u222b\u03a9|ombining  and 21)g\u2208C1(\u03a9\u0305).d, and1\u03b7(x)={c0eg \u2208 W1,1(\u03a9), we use the density ofW1,1(\u03a9) [n \u2192 \u221e in . For W1,1(\u03a9)  and choo\u221e.Since  defines n \u2192 \u221e in  for gn\u2282Ce obtain  for g \u2208 Br0 \u2282 \u211d2 be a disk of radius r0 centered at the origin. Consider a piecewise constant, radial function f(r) defined on Br0 such that it has a jump jm \u2208 \u211d at rm, 1 \u2264 m \u2264 M, where 0 < r1 < \u22ef < rM < r0. Then by  to achieve specific properties for satisfactory recovery of an underlying signal, rather than imposed by a specific detector arrangement as in limited data tomography. However, in a broad sense, compressed sensing can be interpreted as achieving better reconstruction from less data relative to the common practice. Hence, while the current name is not far off, an alternative phrase for our approach can be \u201ctotal variation minimization-based interior tomography.\u201d In conclusion, we have extended the total variation minimization property of a piecewise function from two-dimensions to any dimensionality in the Sobolev space, which can be used for exact reconstruction of any piecewise function on an ROI by minimizing its total variation under the constraint of the truncated projection data through the ROI. Previously, we implemented an alternating iterative reconstruction algorithm to minimize the total variation, which is time-consuming and needs improvement. Under the guidance of the theoretical finding presented here, we are working to develop a multidimensional ROI reconstruction algorithm for better performance. Clearly, major efforts are still needed in this direction."}
{"text": "Transcriptional and postranslational regulation of the cell cycle has been widely studied. However, there is scarce knowledge concerning translational control of this process. Several mammalian eukaryotic initiation factors (eIFs) seem to be implicated in controlling cell proliferation. In this work, we investigated if the human eIF3f expression and function is cell cycle related.The human eIF3f expression has been found to be upregulated in growth-stimulated A549 cells and downregulated in G0. Western blot analysis and eIF3f promotor-luciferase fusions revealed that eIF3f expression peaks twice in the cell cycle: in the S and the M phases. Deregulation of eIF3f expression negatively affects cell viability and induces apoptosis.The expression pattern of human eIF3f during the cell cycle confirms that this gene is cell division related. The fact that eIF3f expression peaks in two cell cycle phases raises the possibility that this gene may exert a differential function in the S and M phases. Our results strongly suggest that eIF3f is essential for cell proliferation. Saccharomyces cerevisiae, and the highest in humans (thirteen subunits from eIF3a to eIF3m) , mapped on chromosome 11p15.4, we used the 11:7947867-11:7948866 fragment to generate a plasmid, called pF11P1K, that encompasses 1 kb of genomic DNA corresponding to the promoter . For the intronless copy [NCBI removed record XM_290345], mapped on chromosome 2p16.1, we used the 2:58331072-2:58332078 fragment to generate a plasmid, called pF02P1K, that also encompasses 1 kb of genomic DNA corresponding to the promoter region of the eIF3f intronless copy . The pMSG plasmid with the leaky inducible (dexamethasone) MMTV LTR promoter was used to generate eIF3f gene expression vector, called pFH11S, and eIF3f antisense vector, called pFH11A. Total RNA was extracted from A549 cells with TRIzol, according to the manufacturer's instructions. eIF3f gene sequence [NCBI Entrez Gene 86652: Locus NP_003745] was amplified by RT-PCR using forward primer TTCTCGACAAGATGGCCACACCGGCG and reverse primer TCACAGGTTTACAAGTTTTTCATTGAG. Amplification product was subcloned in the pBluescript SK(-) vector (plasmid construct pSK11F) and orientated into the pMSG plasmid to generate the sense and antisense constructs. All constructs were verified by sequencing . The verified antisense targeting sequence of eIF3f is 5'---ttctcgacaagATGgccacaccggcggtaccagtaagtgctcctccggccacgccaaccccagtcccggcggcggccccagcctcagttccagcgccaacgccagcaccggctgcggctccggttcccgctgcggctccagcctcatcctcagaccctgcggcagcagcggctgcaactgcggctcctggccagaccccggcctcagcgcaagctccagcgcagaccccagcgcccgctctgcctggtcctgctcttccagggcccttccccggcggccgcgtggtcaggctgcacccagtcattttggcctccattgtggacagctacgagagacgcaacgagggtgctgcccgagttatcgggaccctgttgggaactgtcgacaaacactcagtggaggtcaccaattgcttttcagtgccgcacaatgagtcagaagatgaagtggctgttgacatggaatttgctaagaatatgtatgaactgcataaaaaagtttctccaaatgagctcatcctgggctggtacgctacgggccatgacatcacagagcactctgtgctgatccacgagtactacagccgagaggcccccaaccccatccacctcactgtggacacaagtctccagaacggccgcatgagcatcaaagcctacgtcagcactttaatgggagtccctgggaggaccatgggagtgatgttcacgcctctgacagtgaaatacgcgtactacgacactgaacgcatcggagttgacctgatcatgaagacctgctttagccccaacagagtgattggactctcaagtgacttgcagcaagtaggaggggcatcagctcgcatccaggatgccctgagtacagtgttgcaatatgcagaggatgtactgtctggaaaggtgtcagctgacaatactgtgggccgcttcctgatgagcctggttaaccaagtaccgaaaatagttcccgatgactttgagaccatgctcaacagcaacatcaatgaccttttgatggtgacctacctggccaacctcacacagtcacagattgcactcaatgaaaaacttgtaaacctgTGA---3'.et al. [Design and production of siRNAs against eIF3f was carried out according to Kittler et al. . Brieflyet al.  to selecctccagaacggccgcatgagcatcaaagcctacgtcagcactttaatgggagtccctgggaggaccatgggagtgatgttcacgcctctgacagtgaaatacgcgtactacgacactgaacgcatcggagttgacctgatcatgaagacctgctttagccccaacagagtgattggactctcaagtgacttgcagcaagtaggaggggcatcagctcgcatccaggatgccctgagtacagtgttgcaatatgcagaggatgtactgtctggaaaggtgtcagctgacaatactgtgggccgcttcctgatgagcctggttaaccaagtaccgaaaatagttcccgatgactttgagaccatgctcaacagcaacatcaatgaccttttgatggtgacctacctg---3'.in vitro transcription of the eIF3f fragment sequence with T7 and T3 RNA polymerases. siRNAs against eIF3f were obtained by digestion of dsRNA with RNase III, generating a mixture of siRNAs from 25 to 30 pb. Scrambled siRNAs were obtained by in vitro transcription of the MCS of vector pSL301 with SP6 and T7 RNA polymerases and further digestion with RNase III. siRNAs were cleaned with Q-sepharose columns.The amplified sequence was cloned into the pGEM plasmid to generate the pF11SIR vector. dsRNA were obtained by Cells were accumulated in G0 by serum deprivation (0.1% FBS) for 48 hours . SynchroTotal RNA was extracted from A549 cells with TRIzol, according to the manufacturer's instructions. The resulting RNA pellet was air-dried, resuspended in nuclease-free water, and treated with amplification-grade DNase I. RNA was quantified using a spectrophotometer ; A260/280 readings between 1.8 and 2.0 were used to ensure purity. Total RNA was also quantified with a VersaFluor fluorometer  using Quant-iT\u2122 RiboGreen RNA Assay Kit, according to the manufacturer's instructions. Agarose/formaldehyde denaturing gel electrophoresis, as described by Sambrook and Russel , was per4, 4 mM EGTA, 1% Triton X-100 and 1 mM DTT) containing a protease inhibitor mix. 15 \u03bcg of protein were separated per lane by 10% SDS-PAGE gel electrophoresis. After electrophoretic transfer of proteins to a polyvinylidene difluoride (PVDF) membrane, eIF3f, Cyclin B1 or caspase-9 specific bands were detected by reaction with a rabbit antibody against eIF3f, a mouse antibody against caspase-9, followed by horseradish peroxidase-conjugated anti-rabbit or anti-mouse secondary antibody, respectively. Bands were visualized by enhanced chemiluminescence (ECL). Immunoblots were stripped using mild antibody stripping solution and re-probed with a mouse anti-tubulin antibody. For quantification purposes, densitometric measurements were performed using the Kodak Digital Science Electrophoresis Documentation and Analysis System 100 for Windows . All eIF3f values were normalized to tubulin levels.For Western blot analysis, cells were scraped from dishes and cellular protein extracts were prepared by homogenization in ice cold lysis buffer  was 2 \u03bcg, and 3 \u03bcg for the siRNAs. Transfection times varied according to each experiment. Cells were harvested 8, 24 and 48 hours after transient transfection with siRNAs, eIF3f sense and antisense constructs, and control plasmids. For transfections with pMSG and derived constructs, 0.1 mM dexamethasone was added to the culture medium. For luciferase activity assays with cells in G1, cell cultures were transiently transfected with the promoter plasmid constructs for 20 hours, then arrested in M phase with nocodazole for 24 hours, and finally released from nocodazole to harvest cells 4 hours later (G1 cells). For luciferase activity assays with cells in M or S, first cell cultures were transiently transfected with the promoter plasmid constructs for 24 hours, then these cultures were arrested with nocodazole or hydroxyurea for 24 hours, to finally harvest transfected cells in M or S phase, respectively. To determine luciferase activity in G0, cell cultures were transiently transfected with the promoter plasmid constructs for 6 hours, and then cultures were accumulated in G0 by serum deprivation for 48 hours after which transfected cells were harvested for luciferase activity assays.4, 4 mM EGTA, 1% Triton X-100 and 1 mM DTT). Each clarified cell lysate was first assayed for total protein, using the Bio-Rad Protein Assay, according to the manufacture's instructions . After adjusting samples for equivalent protein concentration in lysis buffer, samples were assayed for luciferase activity as described by Sambrook and Russel [Promoter activity was determined by measuring the luciferase activities of cell lysates 48 hours after transfection with empty pGL3, pF11P1K or pF02P1K plasmids. Briefly, cells were lysed in ice cold lysis buffer  and collected. Floating cells were also recovered and included with the formerly adherent cells for cell viability with the trypan blue exclusion method. Percent of cell viability was determined microscopically  by counting triplicate samples of 200 to 400 cells in a hemotocytometer chamber.Adherent cells were washed twice with PBS and stained with a mixture of 4 \u03bcg/ml acridine orange and 4 \u03bcg/ml ethidium bromide in PBS  for 5 miAll experiments were repeated at least three times. Results of multiple experiments are expressed as mean \u00b1 standard error (SE). Analysis of Student's t test was used to assess the differences between means. A P < 0.05, with respect to control experiments, was accepted as statistically significant.The authors declare that they have no competing interests.AEHM participated in the design of the study, carried out the molecular genetic and cellular studies, performed the statistical analysis and drafted the manuscript. MAPG conceived the study and participated in its design and coordination and helped to draft the manuscript. All authors read and approved the final manuscript.We would like to thank Y.E. Guzm\u00e1n-Infante, A. Mendoza-Pineda, D.A. Bola\u00f1os-Cornejo and A. Aguilera-M\u00e9ndez for technical assistance. We would also like to thank S. Z\u00e1rate for critically reading the manuscript. A.E. Higareda-Mendoza was a recipient of a CONACyT Doctoral Scholarship. This research was supported by CIC Research Programs 2.15 and 16.3 from the Universidad Michoacana de San Nicol\u00e1s de Hidalgo, and the Mixed Fund CONACyT-Goverment of the State of Michoacan MICH-2009-C05-116168.Original Western blots for the data used in Figure Click here for fileOriginal Western blots for the data used in Figure Click here for fileOriginal Western blots for the eIF3f and tubulin data used in Figure Click here for fileOriginal Western blots for the Cyclin B1 data used in Figure Click here for fileOriginal Northern blots for the data used in Figure Click here for fileOriginal Northern blots for the data used in Figure Click here for fileOriginal Western blots for the data used in Figure Click here for fileAmplified micrographs for the data used in Figure Click here for fileOriginal Western blots for the data used in Figure 6Aand Click here for fileAmplified micrographs for the data used in Figure Click here for file"}
{"text": "Even though the commonality of the observed genomic rates remains unexplained, it implies that mutation rates in unstressed microbes reach values that can be finely tuned by evolution. An insight originating in the 1920s and maturing in the 1960s proposed that the genomic mutation rate would reflect a balance between the deleterious effect of the average mutation and the cost of further reducing the mutation rate. If this view is correct, then increasing the deleterious impact of the average mutation should be countered by reducing the genomic mutation rate. It is a common observation that many neutral or nearly neutral mutations become strongly deleterious at higher temperatures, in which case they are called temperature-sensitive mutations. Recently, the kinds and rates of spontaneous mutations were described for two microbial thermophiles, a bacterium and an archaeon. Using an updated method to extrapolate from mutation-reporter genes to whole genomes reveals that the rate of base substitutions is substantially lower in these two thermophiles than in mesophiles. This result provides the first experimental support for the concept of an evolved balance between the total genomic impact of mutations and the cost of further reducing the basal mutation rate.Rates of spontaneous mutation have been estimated under optimal growth conditions for a variety of DNA-based microbes, including viruses, bacteria, and eukaryotes. When expressed as genomic mutation rates, most of the values were in the vicinity of 0.003\u20130.004 with a range of less than two-fold. Because the genome sizes varied by roughly 10 Spontaneous mutations are key drivers of evolution and disease. In microbes, most mutations are deleterious, some are neutral (without significant impact), and a few are advantageous. Because deleterious mutations reduce fitness, there should be constant selection for antimutator mutations that reduce rates of spontaneous mutation. However, such reductions are necessarily achieved at some cost. Therefore, a mutation rate should converge evolutionarily on a value that reflects this trade-off. For DNA microbes, the observed genomic mutation rate is remarkably (and mysteriously) invariant, in the neighborhood of 0.003\u20130.004, with a range of less than two-fold despite huge variation per average base pair in organisms with a wide diversity of life histories. Would an environmental condition that increased the average deleterious impact of a mutation be balanced by additional investments in antimutator mutations? It is widely observed that many mutations with mild impacts become strongly deleterious at higher temperatures, so mutation rates were measured in two thermophiles, a bacterium and an archaeon. Remarkably, both displayed average mutation rates reduced by about five-fold from the characteristic mesophilic value, most of the decrease reflecting a 10-fold reduction in the rate of base substitutions. It has become increasingly clear that the basal rate of spontaneous mutation per genome per replication is remarkably invariant in DNA microbes: using a classical correction factor for estimating the ratio of all base-pair substitutions (BPSs) to detected base-pair substitutions, genomic mutation rates (mutations per genome per replication) vary by less than twofold while genome sizes vary by \u22486,000-fold . Thus, whypothesis of dangerous missense). This simple prediction was supported by the observation that missense mutations accumulated to a lesser extent (compared to synonymous mutations) in thermophiles than in mesophiles during the course of molecular evolution , implying stronger purifying selection in thermophiles If the Kimura conjecture is correct, then increasing the average deleterious impact of a spontaneous mutation  would lower the rate of mutation, at least on an evolutionary time scale. The concept of an equilibrium basal mutation rate is difficult to test in a laboratory context because any imposed resetting of the equilibrium would probably require numbers of generations large even by microbial standards, and is difficult to test convincingly because only one or a few habitats could be explored. However, it has recently proven possible to test the concept by examining a natural evolutionary experiment, life at high temperatures. Those who gather mutants for fun or profit have often observed that the most common class of mutations is to temperature sensitivity, indicating that many missense mutations are well tolerated at the standard growth temperature but become much more deleterious, often to the point of lethality, at a temperature only 5\u00b0C\u201310\u00b0C higher. This widespread anecdotal observation implies that macromolecular stability becomes increasingly dependent on structural integrity as temperatures rise, a reasonable conjecture in keeping with the considerable constraints observed in the proteins of thermophilic microbes e.g., . It is tThe first phase of determining genomic mutation rates involves measuring a mutation frequency, converting the frequency to a rate, and taking precautions to exclude or take into account the impact of perturbations such as differential growth rates of mutants versus wild type and delayed expression of the mutant phenotype. In addition to measuring rates, it is crucial to identify the kinds of mutations that arise in order to exclude biases due to massive mutational hotspots or to bizarre classes of mutations. The typical result is a rate for a mutation-reporter gene, which is then extrapolated to the whole genome provided that the spectrum of mutations is fairly ordinary. However, there is a substantial problem here: while most indels are detected, most BPSs fail to produce a phenotypic change detectable in the laboratory. One must therefore estimate their full frequencies. (An exception is the still rare case that mutation detection is achieved with phenotype-blind genomic DNA sequencing.) Two methods have been applied. Both make the reasonable assumption that almost all indels and chain-termination (CT) BPSs are detected with high efficiency in protein-coding sequences.  The first method was based in part on the average relative frequencies of CT and non-CT BPSs in a handful of spectra and provided a correction factor for base substitutions of 4.726 E. coli lacI gene, where \u223c72% of all mutations are indels arising at a stretch of 13 BPSs consisting of 3.25 repeats of a tetramer The other major barrier to accurate extrapolation from a mutation-reporter gene to the whole genome becomes manifest when sequencing reveals a major hotspot. Mutation rates at particular sites vary greatly, but most mutational spectra display a range of site-specific numbers of mutations ranging from 1 to hotspots with from several percent to even a quarter of the whole collection. The impact of a hotspot containing a quarter of all the mutations is modest, but some genes contain single hotspots bearing the large majority of mutations; the classic example is the Sulfolobus acidocaldariusThermus thermophilusS. acidocaldarius, BPSs were a smaller fraction of the spectrum than in mesophiles, and this observation prompted the hypothesis of dangerous missense. Note, however, that if greater fractions of missense mutations are phenotypically detectable in thermophiles than in mesophiles, then the historical method of correcting for undetected BPSs becomes inappropriate when based on mesophiles. It is therefore advisable to resort exclusively to the CT method for estimating total BPS rates, which is the central result for this report.All informative microbial mutation rates obtained before 2000 were for mesophilic species, but rates and spectra are now available for two genes in each of two very different thermophiles, the crenarchaeon lacZ\u03b1 equivalent), sometimes based on the same sources as for p values are 0.018 for both the total mutation rate and its BPS component, and 0.27 for the indel values that include the hotspots.Genomic mutation rates have long been suspected to evolve as a balance between the deleterious impact of the average mutation and the cost of further reducing the mutation rate. A test of this conjecture on the evolutionary scale could consist of estimating mutation rates in organisms whose environment increases the impact of the average mutation. Because many base substitutions do greater harm at higher temperatures, thermophiles were suitable candidate organisms. For both a bacterium and an archaeon, the thermophiles display sharply reduced rates of base pair substitutions compared to the typical mesophile.37\u201338\u201341\u201350\u201350\u201351\u201368\u201369, providing no hint of a role for this variable, as also noted in the earlier molecular-evolution study The lower mutation rates in thermophiles are likely to reflect their higher optimal growth temperatures. There is no obvious hint of a particular aspect of life history other than temperature that sets the two thermophiles apart from the mesophiles. The %(G+C) values for the ten organisms in p value of 0.27 for these data. One candidate explanation for this difference is that the reduction in BPS rates is achieved by the accumulation of modifiers selected to target BPS mutagenesis but at most incidentally targeting indel mutagenesis. Because single-base additions and deletions tend to be the large majority of indels in mesophiles  and are similarly frequent in thermophiles , these small indels must be the main targets of antimutagenic modifiers acting on indels generally. Both single-base indels and BPSs result from errors of insertion followed by failures of proofreading and DNA mismatch repair in well studied model organisms such as E. coli and S. cerevisiae, but little is known about the sources of spontaneous mutations in S. acidocaldarius and T. thermophilus.The hypothesis of dangerous missense predicts that BPS rates will be reduced in thermophiles but does not speak directly to indel rates. However, indel rates are also reduced, although less strongly than are BPS rates and with a Are there likely to be other outliers with informative deviations from the mutational pattern that is consistently displayed among the mesophilic microbes examined to date with respect to either the mutation rate or the BPS:indel ratio?Mutations to cold sensitivity are rarely reported and are anecdotally described as difficult to discover. If they are indeed rare, perhaps fewer missense mutations produce mutant phenotypes in psychrophiles than in mesophiles. One evolutionary consequence might then be a relaxation to a higher spontaneous rate of BPS mutation, perhaps with little effect on the rate of indel mutation.+ and H+, respectively, compared to other microbes. These ionic environments might be unusually stressful to mutants carrying missense mutations, resulting in adjustments to their mutational patterns in the same direction as seen for thermophiles. Although without significance because of sampling constraints, S. acidocaldarius than to the non-acidophile T. thermophilus. Unfortunately, an attempt to characterize mutation in the halophilic archaeon Haloferax volcanii failed, probably because this mesophile is highly polyploid Because of incomplete buffering against the impacts of their environments, halophiles and acidophiles experience relative high internal concentrations of NaOenococcus oeni, used in wine making to convert malic acid to lactic acid, lacks the usual bacterial DNA mismatch repair (MMR) system and has a high mutation rate as judged by mutations conferring resistance to rifampin and erythromycin, as does Oenococcus kitaharaeThe lactic acid bacterium S. acidocaldarius, lack all known bacterial MMR genes, but S. acidocaldarius, at least, displays an antimutator phenotype compared with mesophiles. How can this be? In Escherichia coli, the mutation rate per average base pair \u22488\u00d710\u221210 .T\u200a=\u200athe measured mutation rate at T, corrected where necessary for mutants expressing the characteristic phenotype but revealed by sequencing to lack mutations in the reporter gene, but not corrected for mutants with two or more mutations (which are infrequent and sometimes absent). In many cases, \u03bcT\u200a=\u200af/ln(\u03bcTN) where f\u200a=\u200athe measured mutation frequency for the given target, N\u200a=\u200athe final population size, and the median \u03bcT over several cultures is used \u03bcM\u200a=\u200anumber of sequenced mutants\u200a=\u200aB+I, where B\u200a=\u200anumber of BPS mutants and I\u200a=\u200anumber of indel mutants, the latter also including complex mutants  regardless of their components.b\u200a=\u200a[\u03bcT corrected upwards by (I+4.726B)/M]/T\u200a=\u200a(I+4.726B)( \u03bcT/MT). The genomic mutation rate \u03bcg\u200a=\u200aG\u03bcb.For the \u201chistorical\u201d method, we correct for undetected BPSs by multiplying the number of detected BPSs by 4.726 g(I) is calculated as above ignoring the BPS component, B becomes CTB\u200a=\u200anumber of mutations to a chain-terminating codon , and P\u200a=\u200anumber of possible mutational pathways to a CT mutation within T . Then the BPS genomic mutation rate \u03bcg(B)\u200a=\u200a\u03bcT (3CTB/MP)G. The total genomic rate \u03bcg\u200a=\u200a\u03bcg(I)+\u03bcg(B).For the \u201cCT\u201d method, the indel genomic mutation rate \u03bcG\u200a=\u200a6407. This system is unique among popular mutation reporters. It consists of an E. coli lacZ\u03b1 transgene embedded in the single-stranded DNA of the M13 genome and carrying both an upstream regulatory region and the beginning of the lacZ gene. Because thousands of mutants have been sequenced, it has become apparent which mutations are detectable when present singly and which are not BT\u200a=\u200a245) and for single-base indels (T\u00b11\u200a=\u200a177) are thus well defined, and we further assume that the infrequent larger indels are fully detectable (LT\u200a=\u200a239). The measured mutation frequency f was 5.86\u00d710\u22124M\u200a=\u200a117, B\u200a=\u200a67, I\u00b11\u200a=\u200a11 and LI\u200a=\u200a39. Assuming that virtually all replication occurs by a rolling circle mechanism, the mutation rate is calculated as for RNA viruses, \u03bc\u200a=\u200af/2c where c is the number of consecutive cycles of infection 13 pfu) were added to l L of medium containing E. coli cells diluted from an overnight culture to about 107 cells/ml, so that the multiplicity of infection was about 103. The input of infected cells from the plaque was \u2264108, so that the input concentration of infected cells \u200a=\u200a108/103/ml\u200a=\u200a105/ml, that is, no more than 105/107\u200a=\u200a0.01 of all cells. c\u22482.5 in the plaque +1 in the liquid culture \u200a=\u200a3.5. Then \u03bcb\u200a=\u200a(f/2c)\u03a3(proportion of mutations of type i\u200a=\u200aiN/117)(1/iT)\u200a=\u200a(5.86\u00d710\u22124/7)[(3\u00d767/117)(1/245)+(11/117)(1/177)+(39/117)(1/239)]\u200a=\u200a7.48\u00d710\u22127. \u03bcg(B)\u200a=\u200a(f/2c)(3\u00d7proportion of BPSs)(G/BT)\u200a=\u200a(5.86\u00d710\u22124/7)(3\u00d767/117)(6407/245)\u200a=\u200a0.00376, \u03bcg(I+L)\u200a=\u200a(f/2c)[(proportion of \u00b11 indels)(1/T\u00b11)+(proportion of larger indels)(1/LT)](G)\u200a=\u200a(5.86\u00d710\u22124/7)[(11/117)(1/177)+(39/117)(1/239)](6407)\u200a=\u200a0.00103, and \u03bcg\u200a=\u200a\u03bcg(B)+\u03bcg(I+L)\u200a=\u200a0.00479.G\u200a=\u200a4.850\u00d7104. For the cII gene, T\u200a=\u200a294, f\u200a=\u200a5.36\u00d710\u22125, and, for 93 mutants, B\u200a=\u200a55 (CTB\u200a=\u200a8 and P\u200a=\u200a35) and I\u200a=\u200a38 (T\u200a=\u200af/ln(\u03bcT N)\u200a=\u200a6.10\u00d710\u22126. Using the historical method, \u03bcb\u200a=\u200a(6.10\u00d710\u22126)(38+55\u00d74.726)(1/93)(1/294)\u200a=\u200a6.64\u00d710\u22128 and \u03bcg\u200a=\u200a0.00304. \u03bcg(I)\u200a=\u200a(6.10\u00d710\u22126)(38/93)(4.85\u00d7104/294)\u200a=\u200a0.00041. Using the CT method, \u03bcg(B)\u200a=\u200a(6.10\u00d710\u22126)(8/93)(3\u00d74.85\u00d7104/35)\u200a=\u200a0.00218 and \u03bcg\u200a=\u200a\u03bcg(I)+\u03bcg(B)\u200a=\u200a0.00259.G\u200a=\u200a1.523\u00d7105. For the tk gene, T\u200a=\u200a1131, f\u200a=\u200a6\u00d710\u22125, N\u200a=\u200a(0.3\u201320)\u00d7108 and, for 67 mutants, B\u200a=\u200a22 (CTB\u200a=\u200a5 and P\u200a=\u200a90) and I\u200a=\u200a45 T\u200a=\u200af/ln(\u03bcTN) and \u03bcT\u200a=\u200af/2c (with c\u200a=\u200a2), respectively. The corresponding values are \u03bcT\u200a=\u200a8.40\u00d710\u22126 and 10\u00d710\u22126, giving a mean of 9.20\u00d710\u22126. Using the historical method, \u03bcb\u200a=\u200a(9.20\u00d710\u22126)(45+22\u00d74.726)(1/67)(1/1131)\u200a=\u200a1.81\u00d710\u22128 and \u03bcg\u200a=\u200a0.00275. \u03bcg(I)\u200a=\u200a(9.20\u00d710\u22126)(45/67)(1.52\u00d7105/1131)\u200a=\u200a0.00083. Using the CT method, \u03bcg(B)\u200a=\u200a(9.20\u00d710\u22126)(5/67)(3\u00d71.52\u00d7105/90)\u200a=\u200a0.00348 and \u03bcg\u200a=\u200a\u03bcg(I)+\u03bcg(B)\u200a=\u200a0.00432.See G\u200a=\u200a1.689\u00d7105. For the rI gene, T\u200a=\u200a294, \u03bcT\u200a=\u200a2.82\u00d710\u22126 and, for 66 mutants, B\u200a=\u200a34 (CTB\u200a=\u200a6 and P\u200a=\u200a43) and I\u200a=\u200a32 b\u200a=\u200a(2.82\u00d710\u22126)(32+34\u00d74.726)(1/66)(1/294)\u200a=\u200a2.80\u00d710\u22128 and \u03bcg\u200a=\u200a0.00473. \u03bcg(I)\u200a=\u200a(2.82\u00d710\u22126)(32/66)(1.689\u00d7105/294)\u200a=\u200a0.00079. Using the CT method, \u03bcg(B)\u200a=\u200a(2.82\u00d710\u22126)(6/66)(3\u00d71.69\u00d7105/43)\u200a=\u200a0.00302 and \u03bcg\u200a=\u200a\u03bcg(I)+\u03bcg(B)\u200a=\u200a0.00381.G\u200a=\u200a2.127\u00d7106. For the pyrEF genes, T\u200a=\u200a1326, \u03bcT\u200a=\u200a3.21\u00d710\u22127 and, for 73 mutants, B\u200a=\u200a19 (CTB\u200a=\u200a2 and P\u200a=\u200a103) and I\u200a=\u200a54 (or 18 without the hotspot) g(I)\u200a=\u200a(3.21\u00d710\u22127)(54/73)(2.127\u00d7106/1326)\u200a=\u200a0.000381. Using the CT method, \u03bcg(B)\u200a=\u200a(3.21\u00d710\u22127)(2/73)(3\u00d72.127\u00d7106/103)\u200a=\u200a0.000545. \u03bcg\u200a=\u200a\u03bcg(I)+\u03bcg(B)\u200a=\u200a0.000926 (or 0.000672 without the indel hotspot).G\u200a=\u200a2.226\u00d7106. For the pyrEF genes, T\u200a=\u200a1240, \u03bcT\u200a=\u200a3.37\u00d710\u22127 and, for 108 mutants, B\u200a=\u200a13 (CTB\u200a=\u200a1 and P\u200a=\u200a184) and I\u200a=\u200a95 (or 46 without the hotspot) g(I)\u200a=\u200a(3.37\u00d710\u22127)(95/108)(2.226\u00d7106/1240)\u200a=\u200a0.000532. Using the CT method, \u03bcg(B)\u200a=\u200a(3.37\u00d710\u22127)(1/108)(3\u00d72.226\u00d7106/184)\u200a=\u200a0.000113. \u03bcg\u200a=\u200a\u03bcg(I)+\u03bcg(B)\u200a=\u200a0.000645 (or 0.000371 without the indel hotspot).G\u200a=\u200a4.639\u00d7106. For the lacI gene, \u03bcT\u200a=\u200a6.043\u00d710\u22127 (excluding 10 IS insertions) T\u200a=\u200a1083 and, for 721 mutants, B\u200a=\u200a80 (CTB\u200a=\u200a24 and P\u200a=\u200a110) and I\u200a=\u200a641 (or 116 without the indel hotspot) b\u200a=\u200a(6.043\u00d710\u22127)(641+80\u00d74.726)(1/721)(1/1083)\u200a=\u200a7.89\u00d710\u221210 and \u03bcg\u200a=\u200a0.00366. \u03bcg(I)\u200a=\u200a(6.043\u00d710\u22127)(641/721)(4.639\u00d7106/1083)\u200a=\u200a0.00230. Using the CT method, \u03bcg(B)\u200a=\u200a(6.043\u00d710\u22127)(24/721)(3\u00d74.639\u00d7106/110)\u200a=\u200a0.00255 and \u03bcg\u200a=\u200a\u03bcg(I)+\u03bcg(B)\u200a=\u200a0.00485 (or 0.002961 without the indel hotspot).G\u200a=\u200a1.246\u00d7107 and calculations are as above.URA3 gene, T\u200a=\u200a804, P\u200a=\u200a123, and four sets of values are available. For the first T\u200a=\u200a2.77\u00d710\u22128 and, for 106 mutants, B\u200a=\u200a89 (CTB\u200a=\u200a39) and I\u200a=\u200a17; for the historical method, \u03bcb\u200a=\u200a1.42\u00d710\u221210 and \u03bcg\u200a=\u200a0.00177; \u03bcg(I)\u200a=\u200a0.00007; for the CT method, \u03bcg(B)\u200a=\u200a0.00310; and \u03bcg\u200a=\u200a0.00317. For the second T\u200a=\u200a6.25\u00d710\u22128 and, for 20 mutants, B\u200a=\u200a15 (CTB\u200a=\u200a4) and I\u200a=\u200a5; for the historical method, \u03bcb\u200a=\u200a2.95\u00d710\u221210 and \u03bcg\u200a=\u200a0.00368; \u03bcg(I)\u200a=\u200a0.00024; for the CT method, \u03bcg(B)\u200a=\u200a0.00380; and \u03bcg\u200a=\u200a0.00404. For the third T\u200a=\u200a3.50\u00d710\u22128 and, for 106 mutants, B\u200a=\u200a89 (CTB\u200a=\u200a39) and I\u200a=\u200a17; for the historical method, \u03bcb\u200a=\u200a1.56\u00d710\u221210 and \u03bcg\u200a=\u200a0.00195; \u03bcg(I)\u200a=\u200a0.00017; for the CT method, \u03bcg(B)\u200a=\u200a0.00022; and \u03bcg\u200a=\u200a0.00038. For the fourth T\u200a=\u200a4.75\u00d710\u22128 and, for 106 mutants, B\u200a=\u200a89 (CTB\u200a=\u200a39) and I\u200a=\u200a17; for the historical method, \u03bcb\u200a=\u200a2.37\u00d710\u221210 and \u03bcg\u200a=\u200a0.00295; \u03bcg(I)\u200a=\u200a0.00014; for the CT method, \u03bcg(B)\u200a=\u200a0.00446; and \u03bcg\u200a=\u200a0.00460. The respective averages are, for the historical method, \u03bcb\u200a=\u200a2.08\u00d710\u221210 and \u03bcg\u200a=\u200a0.00259; \u03bcg(I)\u200a=\u200a0.00015; and, for the CT method, \u03bcg(B)\u200a=\u200a0.00289 and \u03bcg\u200a=\u200a0.00305.For the CAN1 gene, T\u200a=\u200a1773, P\u200a=\u200a226, and three sets of values are available. For the first T\u200a=\u200a2.77\u00d710\u22127 and, for 20 mutants, B\u200a=\u200a11 (CTB\u200a=\u200a1) and I\u200a=\u200a9; for the historical method, \u03bcb\u200a=\u200a5.18\u00d710\u221210 and \u03bcg\u200a=\u200a0.00645; \u03bcg(I)\u200a=\u200a0.00095; for the CT method, \u03bcg(B)\u200a=\u200a0.00249; and \u03bcg\u200a=\u200a0.00344. For the second T\u200a=\u200a3.01\u00d710\u22127 and, for 23 mutants, B\u200a=\u200a17 (CTB\u200a=\u200a5) and I\u200a=\u200a6; for the historical method, \u03bcb\u200a=\u200a4.13\u00d710\u221210 and \u03bcg\u200a=\u200a0.00514; \u03bcg(I)\u200a=\u200a0.00036; for the CT method, \u03bcg(B)\u200a=\u200a0.00701; and \u03bcg\u200a=\u200a0.00737. For the third T\u200a=\u200a1.52\u00d710\u22127 and, for 227 mutants, B\u200a=\u200a150 (CTB\u200a=\u200a70) and I\u200a=\u200a77 (including 13 complex mutations); for the historical method, \u03bcb\u200a=\u200a2.97\u00d710\u221210 and \u03bcg\u200a=\u200a0.00370; \u03bcg(I)\u200a=\u200a0.00036; for the CT method, \u03bcg(B)\u200a=\u200a0.00775; and \u03bcg\u200a=\u200a0.00811. The respective averages are, for the historical method, \u03bcb\u200a=\u200a4.09\u00d710\u221210 and \u03bcg\u200a=\u200a0.00510; \u03bcg(I)\u200a=\u200a0.00056; and, for the CT method, \u03bcg(B)\u200a=\u200a0.00575 and \u03bcg\u200a=\u200a0.00631.For the URA3 plus three CAN1 values (sum\u00f77) are: for the historical method, \u03bcb\u200a=\u200a2.94\u00d710\u221210 and \u03bcg\u200a=\u200a0.00366; \u03bcg(I)\u200a=\u200a0.00033; and, for the CT method, \u03bcg(B)\u200a=\u200a0.00412 and \u03bcg\u200a=\u200a0.00444.The averages of the four G\u200a=\u200a1.252\u00d7107, values are from ura4 gene, T\u200a=\u200a795, \u03bcT\u200a=\u200a4.56\u00d710\u22128 and, for 39 mutants, B\u200a=\u200a22  and I\u200a=\u200a17; for the historical method, \u03bcb\u200a=\u200a1.78\u00d710\u221210 and \u03bcg\u200a=\u200a0.00223; \u03bcg(I)\u200a=\u200a0.00031; for the CT method, \u03bcg(B)\u200a=\u200a0.00189; and \u03bcg\u200a=\u200a0.00221. For the ura5 gene, T\u200a=\u200a648, \u03bcT\u200a=\u200a8.44\u00d710\u22128 and, for 49 mutants, B\u200a=\u200a34  and I\u200a=\u200a15; for the historical method, \u03bcb\u200a=\u200a4.67\u00d710\u221210 and \u03bcg\u200a=\u200a0.00585; \u03bcg(I)\u200a=\u200a0.00050; for the CT method, \u03bcg(B)\u200a=\u200a0.00337; and \u03bcg\u200a=\u200a0.00387. The average values for the two genes are: for the historical method, \u03bcb\u200a=\u200a3.23\u00d710\u221210 and \u03bcg\u200a=\u200a0.00404; \u03bcg(I)\u200a=\u200a0.00041; for the CT method, \u03bcg(B)\u200a=\u200a0.00263; and \u03bcg\u200a=\u200a0.00304.G\u200a=\u200a3.804\u00d7107. Using the old mutation data (see ad-3AB, \u03bcb\u200a=\u200a4.11\u00d710\u221211 and \u03bcg\u200a=\u200a0.00172. For mtr, \u03bcb\u200a=\u200a9.15\u00d710\u221211 and \u03bcg\u200a=\u200a0.00383. The average values are \u03bcb\u200a=\u200a6.63\u00d710\u221211 and \u03bcg\u200a=\u200a0.00278.data see , for ad-"}
{"text": "In a landmark paper, Nadeau and Taylor  formulat Rearrangements are genomic \u201cearthquakes\u201d that change the chromosomal architectures. The fundamental question in molecular evolution is whether there exist \u201cchromosomal faults\u201d (rearrangement hotspots) where rearrangements are happening over and over again. The random breakage model (RBM) postulates that rearrangements are \u201crandom,\u201d and thus there are no rearrangement hotspots in mammalian genomes. RBM was proposed by Susumo Ohno in 1970 and later was formalized by Nadeau and Taylor in 1984. It was embraced by biologists from the very beginning due to its prophetic prediction power, and only in 2003 was refuted by Pevzner and Tesler, who suggested an alternative fragile breakage model (FBM) of chromosome evolution. However, the rebuttal of RBM caused a controversy, and in 2004, Sankoff and Trinh gave a rebuttal of the rebuttal of RBM. This led to a split among researchers studying chromosome evolution: while most recent studies support the existence of rearrangement hotspots, others feel that further analysis is needed to resolve the validity of RBM. In this paper, we develop a theory for analyzing complex rearrangements (including transpositions) and demonstrate that even if transpositions were a dominant evolutionary force, there are still rearrangement hotspots in mammalian genomes. In 1970, Susumu Ohno came up with two fundamental models of chromosome evolution that were subject to many controversies in the last 35 years ,3. The oRearrangements are genomic \u201cearthquakes\u201d that change the chromosomal architectures. The fundamental question in molecular evolution is whether there exist \u201cchromosomal faults\u201d (rearrangement hotspots) where rearrangements are happening over and over again. RBM postulates that rearrangements are \u201crandom,\u201d and thus there are no rearrangement hotspots in mammalian genomes.synteny blocks are constructed correctly, and (2) chromosomal architectures mainly evolve by the \u201cstandard\u201d rearrangement operations , then every evolutionary scenario for transforming the mouse genome into the human genome must have a very large number of breakpoint re-uses. This result implies that the same regions of the genome are being broken over and over again in the course evolution (rearrangement hotspots), a contradiction to RBM . Consequently, Pevzner and Tesler  is the permutationInitially, genome rearrangements were modeled by a combinatorial problem of sorting by reversals, as described below. The order of genes in two organisms is represented by permutations \u03c1 has the effect of reversing the order of i\u03c0\u03b9+1\u2026\u03c0j\u03c0 and transforming 1\u2026\u03c0i-1\u03c0i\u2026\u03c0j\u03c0j+1\u2026\u03c0n\u03c0 into \u03c0\u22c5\u03c1 = 1\u2026\u03c0i-1\u03c0j\u2026\u03c0i\u03c0j+1\u2026\u03c0n\u03c0. Given permutations \u03c0 and \u03c3, the reversal distance problem is to find a series of reversals 1\u03c1,2\u03c1,\u2026,t\u03c1 such that 1\u00b7\u03c12 ...\u00b7\u03c1t = \u03c3\u03c0\u00b7\u03c1 and t is minimal. We call t the reversal distance between \u03c0 and \u03c3. Sorting \u03c0 by reversals is the problem of finding the reversal distance d(\u03c0) between \u03c0 and the identity permutation (12...n).The reversal 1\u03c02\u2026\u03c0n\u03c0 = \u03c0 by adding 0\u03c0 = 0 and n+1\u03c0 = n + 1. We call a pair of elements , 0 \u2264 i \u2264 n, of \u03c0 an adjacency if i - \u03c0i+1||\u03c0 =1, and a breakpoint if i|\u03c0 \u2212 i+1|\u03c0 > 1. It is easy to see that d(\u03c0) \u2265 b(\u03c0) / 2, where b(\u03c0) is the number of breakpoints in \u03c0. However, the estimate of reversal distance in terms of breakpoints is very inaccurate. Bafna and Pevzner  changes both the order and the signs of the elements within that fragment. We are interested in the minimum number of reversals d(\u03c0) required to transform a signed permutation \u03c0 into the identity signed permutation (+1+2...+n).Finding a maximal cycle decomposition is a difficult problem. Fortunately, in the more biologically relevant case of d(\u03c0) \u2265 n +1 \u2212 c(\u03c0) approximates the reversal distance extremely well. Hannenhalli and Pevzner [h(\u03c0) is the number of hurdles in \u03c0.The concept of a breakpoint graph extends naturally to signed permutations by mimicking every directed element by two undirected elements, which substitute for the tail and the head of the directed element . For sig Pevzner  showed t\u03c0 and \u0393, we are interested in a most parsimonious scenario of evolution of \u03a0 into \u0393 . We assume that \u03a0 and \u0393 contain the same set of genes.In the model of the multichromosomal genomes we consider, every gene is represented by an integer whose sign (\u201c+\u201d or \u201c\u2013\u201d) reflects the direction of the gene. A chromosome is defined as a sequence of genes, while a genome is defined as a set of chromosomes. Given two genomes \u03a0 be a multichromosomal genome. Every chromosome \u03c0 in \u03a0 can be viewed either from left to right ) or from right to left ), leading to two equivalent representations of the same chromosome . The four most common elementary rearrangement events in multichromosomal genomes are reversals, translocations, fusions, and fissions, defined below.Let \u03c0 = 1\u2026\u03c0n\u03c0 be a chromosome and 1 \u2264 i \u2264 j \u2264 n. A reversal \u03c1 on a chromosome \u03c0 rearranges the genes inside \u03c0 = 1\u2026\u03c0i-1\u03c0\u03b9\u2026\u03c0j\u03c0j+1\u2026\u03c0n\u03c0 and transforms \u03c0 into 1\u2026\u03c0i-1\u03c0 \u2212 j\u2026\u03c0\u2212i\u03c0j+1\u2026\u03c0n\u03c0. Let \u03c0 = 1\u2026\u03c0n\u03c0 and \u03c3 = \u03b9\u2026\u03c3m\u03c3 be two chromosomes and 1 \u2264 i \u2264 n +1, 1 \u2264 j \u2264 m + 1. A translocation \u03c1 exchanges genes between chromosomes \u03c0 and \u03c3 and transforms them into chromosomes 1\u2026\u03c0i-1\u03c3j\u2026\u03c3m\u03c0 and \u03b9\u2026\u03c3j-1\u03c0i\u2026\u03c0n\u03c3 with (i \u2013 1) + (m \u2013 j + 1) and (j \u2013 1) + (n \u2013 i + 1) genes, respectively. We denote as \u03a0\u00b7\u03c1 the genome obtained from \u03a0 as a result of a rearrangement  \u03c1. Given genomes \u03a0 and \u0393, the genomic sorting problem is to find a series of reversals and translocations 1\u03c1,2\u03c1,\u2026,t\u03c1 such that 1\u00b7\u03c12\u00b7\u2026\u00b7\u03c1t\u03a0\u00b7\u03c1 = \u0393 and t is minimal. We call t the genomic distance between \u03a0 and \u0393. The genomic distance problem is the problem of finding the genomic distance d between \u03a0 and \u0393.Let \u03c1 concatenates the chromosomes \u03c0 and \u03c3, resulting in a chromosome 1\u2026\u03c0n\u03c31\u2026\u03c3m\u03c0 and an empty chromosome \u00d8. This special translocation, leading to a reduction in the number of (nonempty) chromosomes, is known in molecular biology as a fusion. The translocation \u03c1 for 1 < i < n \u201cbreaks\u201d a chromosome \u03c0 into two chromosomes: (1\u2026\u03c0i-1\u03c0) and (i\u2026\u03c0n\u03c0). This translocation, leading to an increase in the number of (nonempty) chromosomes, is known as a fission.A translocation Text S1(123 KB PDF)Click here for additional data file."}
{"text": "In the original publication of this article , some er\u201cCreating, umil/L\u201d should be \u201cCreatine, umol/L\u201d\u201cGlucose, nmlo/L\u201d should be \u201cGlucose, nmol/L\u201d\u201cUrea, nmo/L\u201d should be \u201cUrea, nmol/L\u201dThe original publication has been corrected."}
{"text": "Kunn\u0101\u0161\u0101) of \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b, a ninth-century physician and student of \u1e24unayn b. Is\u1e25\u0101q. The seven books of the handbook appear to follow the model of Paul of Aegina\u2019s Pragmateia both in composition and content. The actual significance of the handbook in the history of Syriac and Arabic medicine is yet to be assessed, but there can be no doubt that it will be a pivotal source that illustrates the development of Syriac medicine during a period of four centuries at the moment when it was being translated to lay the foundations of the nascent medical tradition in Arabic.A little-known thirteenth-century manuscript preserved in Damascus contains by far the largest Syriac medical work that has survived till today. Despite the missing beginning, a preliminary study of the text allows us to argue that it is the medical handbook (entitled  What we find on those supplementary folios  seems to be a fragment from an independent Syriac medical manuscript of a later date. The text there is divided into sections that deal with anatomy, the preparation of theriac \u201caccording to the opinion of Galen\u201d and \u201cIndian drugs\u201d.The unique manuscript preserved in the library of the Syriac Orthodox Patriarchate near Damascus is not totally unknown but unfortunately it has never received a proper description, and due to its inaccessibility it remained out of scholarly attention and inquiry.sop 238, ff. 431\u2013435) from the missing part can now be found attached to the end of the text. Those five folios contain the table of contents (damaged) as well as the very beginning of the original text (first chapter of M\u0113mr\u0101 i) and a page from the introduction where the author traces the history of the art of medicine and healing back to Asclepius.The original codex is damaged and lacks some folios. Especially noticeable is the absence of the opening part (the text begins in the middle of the second quire) although five folios  that follows the text, the manuscript was completed on September 30, ag 1535 [= 1224 ce] by deacon Basil the son of Rabban Yo\u1e25annan, an archpriest of Melitene.ce an additional important Syriac manuscript with works on secular subjects that has survived in modern apographs.ce a famous illustrated Greek manuscript preserved in the Gennadius Library (Athens).6According to the colophon  and Severus, a metropolitan of Syria and the future patriarch Ignatius Aphrem Barsoum (1887\u20131957).The medical text is written on paper in a very condensed ductus of Serto which makes reading the manuscript a serious challenge despite its overall good condition. The manuscript contains a large number of notes left by its readers that can shed light on its history. Particularly, the notes on ff. 402Finally, the presence of the Armenian quire signatures and Arabic glosses hint at wide circulation of the manuscript.22.1Kunn\u0101\u0161\u0101, as it features, for example, in the opening and closing rubrics of some of the Memr\u0113. Considering that the term Kunn\u0101\u0161\u0101 was regularly applied to medical handbooks, it is natural to see it as the title of the magnificent work before us.Due to the loss of the beginning of the manuscript we are bereft of priceless information about the title of the work and its author. Nevertheless, while going through the text one\u2019s attention is regularly drawn to the term sop 238, f. 304v): \u071f\u0718\u0722\u072b\u0710 \u0715\u0725\u0720 \u0725\u0308\u0720\u072c\u0710 \u0715\u071f\u0718\u072a\u0308\u0717\u0722\u0710 \u0718\u072b\u0718\u0718\u0715\u0725\u071d\u0308\u0717\u0718\u0722 \u0718\u0710\u0723\u071d\u0718\u072c\u0717\u0718\u0722Moreover, we cannot exclude the possibility that an extended version of that title featured in the work\u2019s original title, that could be, following the practice of that time, quite verbose. For instance, we find at the end of M\u0113mr\u0101 iv . 106 chapters. Hygiene, Diet, Materia MedicaM\u0113mr\u0101 i , rules for the preparation of compounds.Complaints of pregnant women, regimen (\u0715\u0718\u0712\u072a\u0710) of children, middle-aged, and old people, air, water, wine, regimen fitting to the different seasons, fatigue, purgation, bath (\u0721\u0723\u071a\u0718\u072c\u0710), food, sop 238, ff. 92v\u2013151v). 47 Chapters. Local Ailments\u2014HeadM\u0113mr\u0101 ii , phrenitis (\u0726\u072a\u0717\u0722\u071d\u071b\u071d\u0723), madness (\u0721\u0710\u0722\u071d\u0710), delirium (\u0728\u0712\u072a\u0710), lethargy (\u0720\u071d\u072c\u0710\u072a\u0713\u0718\u0723), epilepsy (\u0717\u0726\u071d\u0720\u071d\u0721\u0726\u0723\u071d\u0710), insanity (\u072b\u071b\u0718\u072a\u0718\u072c\u0710), apoplexy (\u0710\u0726\u0718\u0726\u0720\u071d\u071f\u0723\u071d\u0710), loss of memory, melancholy (\u0721\u0717\u0720\u0710\u0722\u071f\u0718\u0720\u071d\u0710), spasm (\u0721\u0729\u071d\u0723\u0718\u072c\u0710), numbness (\u072c\u0722\u0718\u0712\u0718\u072c\u0710), trembling (\u072a\u0725\u0720\u0710), afflictions of the hair, alopecia (\u072c\u0725\u0720\u0718\u072c\u0710), afflictions of the skin, lice (\u0729\u0720\u0721\u0308\u0710), nits (\u0722\u0712\u0308\u0710), love-sick people.sop 238, ff. 152r\u2013211r). 81 Chapters. Local Ailments\u2014Eyes, Ears, ChestM\u0113mr\u0101 iii (Pus that forms in the eyes (\u071b\u0726\u072a\u0308\u0710 \u0715\u0712\u0725\u071d\u0308\u0722\u0710), ophthalmia (\u0715\u0713\u0722\u0710), eye swelling (\u202e\u0722\u0726\u071d\u071a\u0718\u072c\u0710 \u0712\u0725\u071d\u0308\u0722\u0710\u202c\u200e), itching in the eyes (\u071a\u071f\u072c\u0710 \u0715\u0712\u0725\u071d\u0308\u0722\u0710), inflammation in the eyes (\u0726\u0720\u0717\u0713\u0721\u0718\u0722\u071d), eyelids (\u072c\u0720\u071d\u0308\u0726\u0710) disorders; diseases of sclera (\u071f\u0718\u072c\u071d\u0722\u0710), ears, nostrils (\u0722\u071a\u071d\u072a\u0308\u0710), teeth, tongue, mouth, chest; warts (\u0715\u0729\u0720\u0718\u0308\u0722\u0710), cough (\u072b\u0725\u0720\u0710), inflammation of the lungs (\u0726\u0717\u072a\u071d\u0726\u0720\u0721\u0718\u0722\u071d\u0710), spitting blood (\u072a\u0729\u0729 \u0715\u0721\u0710), wasting (\u0726\u072c\u071d\u0723\u071d\u0723), pleurisy (\u0715\u0729\u072a\u072c\u0710), heart diseases, swoon (\u0725\u0726\u072c\u0710), humpback (\u071f\u0718\u0723\u072c\u0722\u0718\u072c\u0710).sop 238, ff. 211v\u2013304v). 67 Chapters. Local Ailments\u2014Abdomen, GenitalsM\u0113mr\u0101 iv (On stomach diseases (\u0725\u0720 \u071a\u0308\u072b\u0710 \u0715\u0721\u072c\u0729\u071d\u0721\u071d\u0722 \u0712\u0710\u0723\u071b\u0718\u0721\u071f\u0710), bulimia (\u0712\u0718\u0720\u071d\u0721\u0718\u0723), vomiting (\u072c\u071d\u0718\u0712\u0710), hiccups (\u0726\u0718\u071f\u072c\u0710), cholera (\u071f\u0718\u0720\u0717\u072a\u0710), jaundice (\u071d\u072a\u0729\u0722\u0710), diarrhoea (\u0715\u071d\u0720\u0710), stomach spasm (\u0721\u0725\u0723\u0710), dysentery (\u0715\u0718\u0723\u0722\u071b\u072a\u071d\u0710), intestinal obstruction (\u0710\u071d\u0720\u0717\u0710\u0718\u0723), worms (\u072b\u0718\u072b\u0308\u0720\u0710); diseases of the kidneys (\u071f\u0718\u0720\u071d\u0308\u072c\u0710), bladder (\u072b\u0720\u0726\u0718\u071a\u072c\u0710), genitals (\u0713\u0712\u072a\u0718\u072c\u0710), flow of semen (\u0715\u0718\u0712\u0710 \u0715\u0719\u072a\u0725\u0710), sexual desire, disorders of menstruation (\u071f\u0723\u0726\u0710), gout (\u0726\u0718\u0715\u0710\u0713\u072a\u0710), diseases of the testicles (\u0710\u072b\u0308\u071f\u0710), haemorrhoids (\u0723\u0718\u072a\u0308\u0713\u0710).sop 238, ff. 305r\u2013358v). Some 90 Chapters.M\u0113mr\u0101 v , synuchos (\u202e\u0723\u0718\u0722\u0718\u071f\u0718\u0723\u202c\u200e), skin inflammation (\u202e\u0721\u072b\u072a\u0710\u202c\u200e), pustules (\u202e\u071a\u0721\u071b\u0710\u202c\u200e), tertian fever (\u202e\u071b\u072a\u071d\u071b\u0718\u0723\u202c\u200e), quartan fever (\u202e\u071b\u0710\u071b\u0710\u072a\u071b\u0717\u0718\u0723\u202c\u200e), semitertian fever (\u202e\u0717\u0721\u071d\u071b\u072a\u071d\u0718\u0723\u202c\u200e), compound fevers (\u202e\u0710\u072b\u072c\u0718\u0308\u072c\u0710 \u0721\u072a\u0308\u071f\u0712\u072c\u0710\u202c\u200e), ailment of ulcers (\u202e\u072b\u0718\u071a\u0722\u0710\u202c\u200e), itching (\u202e\u071a\u071f\u071f\u0710\u202c\u200e), cancer (\u202e\u0723\u072a\u071b\u0722\u0710\u202c\u200e), hernia (\u202e\u072c\u0720\u071a\u0710\u202c\u200e), hardened swelling (\u202e\u071b\u072a\u0722\u0718\u072c\u0710\u202c\u200e), tumour (\u202e\u072a\u0718\u0712\u0720\u0710\u202c\u200e), tetter (\u202e\u071a\u0719\u0719\u071d\u072c\u0710\u202c\u200e), leprosy (\u202e\u0710\u072a\u071d\u0722\u0718\u072c\u0710\u202c\u200e); different forms of prognostication, including urine (\u202e\u072c\u0726\u072b\u0718\u072a\u072c\u0710\u202c\u200e).sop 238, ff. 358r\u2013361r). [In 94 Chapters]. Lethal Poisons (\u202e\u0723\u0721\u0721\u0308\u0722\u0710 \u0729\u071b\u0718\u0308\u0720\u0710\u202c\u200e)M\u0113mr\u0101 vi . 25 Chapters. Compound Drugs, SurgeryM\u0113mr\u0101 vii  and hiera (\u202e\u0710\u071d\u0717\u072a\u0710\u202c\u200e) and pays much attention to the preparation of innumerable sorts of pills (\u202e\u071f\u071f\u0308\u0720\u072c\u0710\u202c\u200e), gargles (\u202e\u0725\u0718\u072a\u0725\u072a\u0710\u202c\u200e), bandages (\u202e\u0725\u0728\u0712\u0710\u202c\u200e), compresses (\u202e\u0710\u0723\u0726\u0720\u0722\u071d\u0710\u202c\u200e), elixirs (\u202e\u071f\u0723\u071d\u072a\u071d\u0722\u202c\u200e) and collyria (\u202e\u072b\u071d\u0308\u0726\u0710\u202c\u200e). Special chapters are devoted to the ailments of the teeth and eyes. The M\u0113mr\u0101 ends with two auxiliary chapters on measures and substitute drugs.Kunn\u0101\u0161\u0101 reminds us of the medical compendia and handbooks that are known from the late antique and early Islamic periods.Kunn\u0101\u0161\u0101 covers all the standard subjects of the medical handbooks: regimen and materia medica (M\u0113mr\u0101 i), diseases pertaining to a particular body part are presented by and large following the principle from head to toe , fevers and external diseases (M\u0113mr\u0101 v), poisons (M\u0113mr\u0101 vi), compound drugs (M\u0113mr\u0101 vii). What strikes one perhaps most of all is the author\u2019s splendid command of pharmacology  for the treatment of each diseases is replete with various recipes.Both in its composition and content the Pragmateia of Paul of Aegina (7th c.) is most likely to have been the model of this Kunn\u0101\u0161\u0101. Indeed, at first glance the two works appear to have much in common. For example, each of the two handbooks is divided into seven parts that begin with complaints of pregnant women and end with measures and substitute drugs; each chapter first presents the aetiology and symptoms of an ailment and afterwards its treatment. Nevertheless, close comparison demonstrates also considerable differences. Thus, whilst Paul\u2019s Book Three  relates to Memr\u0113 ii, iii and iv, the other Memr\u0113 usually cover the material from more than one Book . Furthermore, whereas some material of Paul\u2019s Pragmateia is not present whatsoever , in other subjects the Kunn\u0101\u0161\u0101 demonstrates a higher level of detail and proficiency . It is therefore may well be that the text of Paul\u2019s Pragmateia was mediated through another source. The relationship between Paul\u2019s compendium and the Kunn\u0101\u0161\u0101 merits further research because, as we shall see below, the influence of the Pragmateia can be traced not only on the level of composition but also in content.However, among the available manuals the 2.3Kunn\u0101\u0161\u0101. Thus we come across references and citations attributed to Hippocrates, Dioscorides, Rufus of Ephesus, Galen, Philagrius, Oribasius, Aetius of Amida, Alexander of Tralles, and Paul of Aegina. Some names escape straightforward identification, for example Theodoretus and Theophilus. As for the native Syriac authors, only two names have been noticed, \u1e24unayn b. Is\u1e25\u0101q and Ab\u016b Zakar\u012by\u0101\u02be Y\u016b\u1e25ann\u0101 b. M\u0101sawaih. It goes without saying that the explicit indications are nothing more than just the tip of the iceberg and it should be possible to detect the principle sources of the Kunn\u0101\u0161\u0101. That issue will undoubtedly be one of the central questions on the agenda for further research. Nevertheless, already based on a brief acquaintance with the text it is possible to argue that Paul of Aegina occupies a very prominent position. A reading of selected chapters of the Kunn\u0101\u0161\u0101 shows the wide-ranging influence of Paul\u2019s Pragmateia.The names of medical authorities are sparingly scattered in the text of the sop 238, ff. 124v\u2013128r) that, as we shall see shortly, is a faithful version of the opening section on melancholy in Paul\u2019s Pragmateia.By way of example, let me provide a citation from the beginning of a seven-page long chapter on melancholy 10[1] \u1f29 \u03bc\u03b5\u03bb\u03b1\u03b3\u03c7\u03bf\u03bb\u1f77\u03b1 \u03c0\u03b1\u03c1\u03b1\u03c6\u03c1\u03bf\u03f2\u1f7b\u03bd\u03b7 \u03c4\u1f77\u03f2 \u1f10\u03f2\u03c4\u03b9\u03bd \u1f04\u03bd\u03b5\u03c5 \u03c0\u03c5\u03c1\u03b5\u03c4\u03bf\u1fe6 [2] \u1f10\u03c0\u1f76 \u03bc\u03b5\u03bb\u03b1\u03b3\u03c7\u03bf\u03bb\u03b9\u03ba\u1ff7 \u03bc\u1f71\u03bb\u03b9\u03f2\u03c4\u03b1 \u03c7\u03c5\u03bc\u1ff7 \u03b3\u03b9\u03bd\u03bf\u03bc\u1f73\u03bd\u03b7 \u03ba\u03b1\u03c4\u03b5\u03b9\u03bb\u03b7\u03c6\u1f79\u03c4\u03b9 \u03c4\u1f74\u03bd \u03b4\u03b9\u1f71\u03bd\u03bf\u03b9\u03b1\u03bd, [3] \u03c0\u03bf\u03c4\u1f72 \u03bc\u1f72\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03c0\u03c1\u03c9\u03c4\u03bf\u03c0\u03b1\u03b8\u03bf\u1fe6\u03bd\u03c4\u03bf\u03f2 \u03c4\u03bf\u1fe6 \u1f10\u03b3\u03ba\u03b5\u03c6\u1f71\u03bb\u03bf\u03c5, [4] \u03c0\u03bf\u03c4\u1f72 \u03b4\u1f72 \u03c4\u1ff7 \u1f45\u03bb\u1ff3 \u03f2\u03c5\u03bc\u03bc\u03b5\u03c4\u03b1\u03b2\u03b1\u03bb\u03bb\u03bf\u03bc\u1f73\u03bd\u03bf\u03c5 \u03f2\u1f7d\u03bc\u03b1\u03c4\u03b9\u0387 [5] \u03ba\u03b1\u1f76 \u03c4\u03c1\u1f77\u03c4\u03bf\u03bd \u03b4\u1f72 \u03bc\u03b5\u03bb\u03b1\u03b3\u03c7\u03bf\u03bb\u1f77\u03b1\u03f2 \u03b5\u1f36\u03b4\u1f79\u03f2 \u1f10\u03f2\u03c4\u03b9\u03bd, \u1f43 \u03c6\u03c5\u03f2\u1ff6\u03b4\u1f73\u03f2 \u03c4\u03b5 \u03ba\u03b1\u1f76 \u1f51\u03c0\u03bf\u03c7\u03bf\u03bd\u03b4\u03c1\u03b9\u03b1\u03ba\u1f78\u03bd \u03ba\u03b1\u03bb\u03bf\u1fe6\u03f2\u03b9\u03bd, [6] \u1f10\u03c0\u1f76 \u03c6\u03bb\u03b5\u03b3\u03bc\u03bf\u03bd\u1fc7 \u03c4\u1ff6\u03bd \u03c0\u03b5\u03c1\u1f76 \u03c4\u1f78\u03bd \u03f2\u03c4\u1f79\u03bc\u03b1\u03c7\u03bf\u03bd \u1f51\u03c0\u03bf\u03c7\u03bf\u03bd\u03b4\u03c1\u1f77\u03c9\u03bd \u03f2\u03c5\u03bd\u03b9\u03f2\u03c4\u1f71\u03bc\u03b5\u03bd\u03bf\u03bd [7] \u03c0\u03bf\u03c4\u1f72 \u03bc\u1f72\u03bd \u03b1\u1f54\u03c1\u03b1\u03f2 \u03c4\u03b9\u03bd\u1f70\u03f2 \u03bc\u03bf\u03c7\u03b8\u03b7\u03c1\u1f71\u03f2, [8] \u03c0\u03bf\u03c4\u1f72 \u03b4\u1f72 \u03ba\u03b1\u1f76 \u03c4\u1fc6\u03f2 \u03bf\u1f50\u03f2\u1f77\u03b1\u03f2 \u03c4\u03bf\u1fe6 \u03c7\u03c5\u03bc\u03bf\u1fe6 \u03bc\u1f73\u03c1\u03bf\u03f2 \u1f00\u03bd\u03b1\u03c0\u03b5\u03bc\u03c0\u1f79\u03bd\u03c4\u03c9\u03bd \u03c0\u03c1\u1f78\u03f2 \u03c4\u1f78\u03bd \u1f10\u03b3\u03ba\u1f73\u03c6\u03b1\u03bb\u03bf\u03bd.[1] Melancholy is a disorder of the intellect without fever, [2] occasioned mostly by a melancholic humour seizing the understanding; [3] sometimes the brain being primarily affected, [4] and sometimes it being altered together with the entire of the body. [5] And there is a third type called the flatulent and hypochondriac, [6] occasioned by inflammation of one of the parts in the hypochondria adjoining to the stomach, [7] by which sometimes noxious vapours, [8] and sometimes a part of the substance of the humour, is transmitted to the brain.Kunn\u0101\u0161\u0101, M\u0113mr\u0101 ii.26 11\u0706[1] \u0721\u0717\u0720\u0710\u0722\u071f\u0718\u0720\u071d\u0710 \u0717\u071f\u071d\u0720 \u0710\u071d\u072c\u071d\u0717\u0307 \u0706 \u0722\u071f\u071d\u0722\u0710 \u0715\u0721\u0718\u071a\u0710 \u0715\u072a\u072b\u0710 \u0706 \u0715\u0720\u0717 \u0722\u0307\u0726\u0729 \u071a\u0718\u0712\u0720\u0710 \u0715\u071a\u0718\u072b\u0712\u0710 \u0718\u0710\u0712\u071d\u0715\u0718\u072c \u0717\u0718\u0722\u0710 . \u0715\u0712\u0720\u0725\u0715 \u0710\u072b\u072c\u0710 \u0717\u0307\u0718\u071d\u0710 \u0707 [2] \u202d\u202c \u0721\u0323\u0722 \u071f\u0718\u0721\u0718\u0723 \u0721\u0717\u0720\u0710\u0722\u071f\u0718\u0720\u071d\u0729\u071d\u0710 \u0715\u0720\u0307\u0712\u071f \u0720\u072c\u072a\u0725\u071d\u072c\u0710 . [3] \u202d\u202c \u071f\u0715 \u0712\u0719\u0712\u0722 \u0721\u0307\u0722 \u0717\u0718 \u0721\u0718\u071a\u0710 \u0729\u0715\u071d\u0721\u0718\u072c \u071a\u072b\u0718\u072b\u0718\u072c\u0710 \u0722\u071a\u072b . [4] \u202d\u202c \u0712\u0719\u0712\u0722 \u0715\u071d\u0722 \u071f\u0715 \u0725\u0721 \u071f\u0720\u0717 \u0726\u0713\u072a\u0710 \u0706 \u0721\u0710 \u0715\u071d\u072c\u071d\u072a \u0725\u0720\u0718\u0712\u202d\u202c\u0706accompanied by an injury of the intellect and insanity without fever, [2] occasioned by a melancholic humour seizing the understanding; [3] sometimes the brain being primarily affected, [4] and sometimes it being altered together with the entire body when that humour abundantly exceeds. [5] And there is a third type called the flatulent and hypochondriac,are secreted from thick and phlegmatic humour and are transmitted to the brain, [8] and sometimes a part of the substance of the humour ascends.[1] Melancholy is a disorder of the brain Kunn\u0101\u0161\u0101 demonstrates that its author basically reproduced the key passage from Paul of Aegina\u2019s discussion of melancholy while providing it with just a few additions that were apparently necessary to ensure the correct interpretation of the text. We should not exclude, however, a possibility that both the text of Paul\u2019s treatise and the interpolations might go back to an intermediary source that is of course already lost. Nevertheless, a comprehensive study of Kunn\u0101\u0161\u0101 will help to clarify the issue. A comparison with other relevant sources, such as the Small Compendium in seven books of Y\u016b\u1e25ann\u0101 b. Sar\u0101biy\u016bn, may also shed light on the background and sources of Kunn\u0101\u0161\u0101 .14A comparison of the original Greek with the version found in the Thus paragraph [1] enumerates other possible effects that may coincide with a disorder of the brain. The second type of melancholy [4] may embrace the entire body, and for that reason the Syriac author elucidates the point by stressing that such a condition is possible \u201cwhen that humour abundantly exceeds.\u201d And finally a description of the third type [7] is provided with the helpful precision that behind the inflamed hypochondria that produces the vapours stands a \u201cthick and phlegmatic humour.\u201dPragmateia of Paul of Aegina was translated into Syriac in the eighth or ninth century and the Syriac version was used by at least two Syriac authors, Y\u016b\u1e25ann\u0101 b. Sar\u0101biy\u016bn (9th c.) and Bar Bahl\u016bl (10th c.).Lexicon of Bar Bahl\u016bl. For that reason the text of the Kunn\u0101\u0161\u0101 will contribute greatly to our better awareness of the Syriac version of the Pragmateia and will help to comprehend its significance for the Syriac medical tradition.16As demonstrated by Peter E. Pormann, the Kunn\u0101\u0161\u0101 contains a large number of quotations from Greek sources. Since the majority of those translations have vanished for good, it can deservedly be considered an invaluable treasure chest for research into the Syriac translations of Classical medical literature. Particularly, it is of utmost importance for the ninth-century translations that, as it seems, were readily available to \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b. By way of example, it is worth analyzing a quotation from Hippocrates\u2019 Aphorisms, the Syriac version of which happens to survive.As mentioned earlier, the ii.47 (iv. 482.17\u201318 Littr\u00e9)Aph. \u03a0\u03b5\u03c1\u1f76 \u03c4\u1f70\u03f2 \u03b3\u03b5\u03bd\u1f73\u03c3\u03b9\u03b1\u03f2 \u03c4\u03bf\u1fe6 \u03c0\u1f7b\u03bf\u03c5 \u03bf\u1f31 \u03c0\u1f79\u03bd\u03bf\u03b9 \u03ba\u03b1\u1f76 \u03bf\u1f31 \u03c0\u03c5\u03c1\u03b5\u03c4\u03bf\u1f76 \u03be\u03c5\u03bc\u03b2\u03b1\u1f77\u03bd\u03bf\u03c5\u03c3\u03b9 \u03bc\u1fb6\u03bb\u03bb\u03bf\u03bd, \u1f22 \u03b3\u03b5\u03bd\u03bf\u03bc\u1f73\u03bd\u03bf\u03c5.Pains and fevers occur rather at the formation of pus than when it is already formed.Kunn\u0101\u0161\u0101, M\u0113mr\u0101 v \u0706\u0710\u0726 \u0710\u071d\u0726\u0726\u0718\u0729\u072a\u0710\u071b\u071d\u0723 \u0717\u071f\u0722\u0710 \u0715\u0712\u0721\u0718\u0720\u0715\u0710 \u0720\u0721 \u0715\u0721\u0718 \u0713\u0720\u0710 \u0713\u0715\u072b\u071d\u0722 \u071f\u0710\u0712\u0308\u0710 \u0718\u0710\u072b\u072c\u0718\u0308\u072c\u0710 \u071d\u072c\u071d\u072a \u0721\u0722 \u0721\u0710 \u0715\u0710\u072c\u071d\u0720\u0715.\u0706A1\u0706And also Hippocrates: Pains and fevers occur rather at the generation of pus than when it is already generated.Une version syriaque, p. 9 lines 1\u20132): \u0706\u0712\u0717\u0718\u071d\u0710 \u0715\u0721\u0718\u0713\u0720\u0710 \u071f\u0710\u0712\u0308\u0710 \u0718\u0710\u072b\u072c\u0718\u0308\u072c\u0710 \u071d\u072c\u071d\u072a\u0713\u0715\u072b\u071d\u0722 \u0721\u0323\u0722 \u0721\u0710 \u0715\u071f\u0715 \u0717\u0323\u0718\u0710.\u0706A2\u0706Pains and fevers occur rather at the production of pus than when it is already produced.An anonymous Syriac translation  are rendered employing Syriac verb Aphorisms made by his master \u1e24unayn b. Is\u1e25\u0101q. Intriguingly, such a conclusion urges the reappraisal of a widely accepted assertion of Rainer Degen, who argued that the translation edited by Pognon should be attributed to \u1e24unayn.Kunn\u0101\u0161\u0101\u2019s potential significance for the study of the Syriac translations of the Aphorisms, an area of inquiry that has generated a stream of publications in recent years.19As far as we know, \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b was not active as a translator from GreekKunn\u0101\u0161\u0101 one should not overlook immediate input by its author, who regularly adds his own opinion after presenting the (probably borrowed) convictions of others. Since that characteristic appears most often in the discussion of disease treatments, we should attribute such personal remarks and comments to the author\u2019s first-hand experience as a practitioner.Besides the written medical works that establish the general framework of the 3Kunn\u0101\u0161\u0101 is damaged and lacks the very beginning of the treatise that must have contained an indication of both the author and the title. However, we are fortunate to have a few pages from that opening part of the treatise that were bound at the end of the treatise. Besides the table of contents, we find there also a small fragment from the introduction to the Kunn\u0101\u0161\u0101 that ends with the following rubric :As mentioned earlier, the unique manuscript of the \u072b\u0720\u0721 \u0721\u0726\u0729\u0712\u072a\u0718\u071a\u0710 \u0715\u072a\u0712\u0722 \u071d\u072b\u0718\u0725 \u0712\u072a \u0725\u0720\u071d \u0710\u0723\u071d\u0710 \u0721\u071d\u072c\u072a\u0710 \u072b\u0307\u0718\u0710 \u0720\u0729\u0718\u0308\u0720\u0723\u0710The introduction of Rabban \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b, an excellent physician who is worthy of praise, is completed.mappaq b-r\u016b\u1e25\u0101  indicates a special type of introduction that corresponds to the Greek prooemium and that was common in Syriac scholarly literature.Kunn\u0101\u0161\u0101 should be attributed to \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b or in Arabic \u02bf\u012as\u0101 b. \u02bfAl\u012b, who is known as a personal physician of Caliph al-Mu\u02bftamid (r. 870\u2013892), a disciple of \u1e24unayn b. Is\u1e25\u0101q and the author of the Syriac-Arabic Lexicon that is extant in multiple manuscript copies.Lexicon, has to be identified \u201cwith physician \u02bf\u012as\u0101 b. \u02bfAl\u012b, who lived in the second half of the ninth century and was a student of \u1e24unayn b. Is\u1e25\u0101q.\u201d23The expression \u02bfUy\u016bn al-anb\u0101\u02be f\u012b \u1e6dabaq\u0101t al-a\u1e6dibb\u0101\u02be, while extolling \u02bf\u012as\u0101 b. \u02bfAl\u012b\u2019s excellence in the science of medicine, mentions that he produced many works (ta\u1e63\u0101n\u012bf) among which two are particularly mentioned, the Book of the benefits that can be obtained from the parts of animals  and the Book of Poisons  in two parts (m\u0101q\u0101lat\u0101n). Whereas the former work is extant but remains so far unpublished,26If this is indeed so, one wonders if there is any positive evidence of the medical works of \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b. Unfortunately, despite recurrent mention in the Syriac sources of \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b as physician (\u0710\u0723\u071d\u0710), apart for a fragment (on which see below) none of his works in Syriac is extant.Kunn\u0101\u0161\u0101 and the literary output of \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b? It seems that there is one point of pivotal significance. Namely, Ibn Ab\u012b U\u1e63aibi\u02bfa records that \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b was the author of a Book of Poisons that was composed in two parts. As we have seen, the Kunn\u0101\u0161\u0101 contains two Memr\u0113 that deal with poisons. It is not the Memr\u0113 themselves that are important now, though, but an introductory rubric that opens M\u0113mr\u0101 vi. In it the scribe apologizes for reproducing the original M\u0113mr\u0101 only partially because \u201cthe author composed two other Memr\u0113 about poisons\u201d . It is thus very tempting to see in those \u201ctwo Memr\u0113\u201d on the subject of poison a treatise that is recorded by Ibn Ab\u012b U\u1e63aibi\u02bfa with attribution to \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b.With all that information in hand, is there a chance to establish a connection between the Kunn\u0101\u0161\u0101\u2019s author with \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b is provided by the only medical piece by him that has survived. Namely, manuscript Vat. sir. 217, ff. 226v\u2013227v contains a brief fragment reading as follows:27Corroborative evidence for identifying the \u072c\u0718\u0712 \u071f\u072c\u0712\u071d\u0722\u0722 \u0729\u0720\u071d\u0720 \u0721\u0722 \u071f\u072c\u0712 \u072a\u0712\u0722 \u071d\u072b\u0718\u0725 \u0712\u072a \u0725\u0720\u071d \u072c\u0720\u0721\u071d\u0715 \u072a\u0712\u0722 \u071a\u0718\u0722\u071d\u0722 \u072a\u071d\u072b \u0710\u0723\u0718\u0308\u072c\u0710. \u0717\u0307\u0718 \u0715\u0725\u0720 \u0710\u0723\u071d\u0718\u072c\u0710 \u072c\u0721\u071d\u0717\u0710\u071d\u072c \u0718\u071d\u072c\u071d\u072a\u0710 \u0712\u071a\u072c\u071d\u072c\u0718\u072c\u0710 \u0721\u0722 \u0715\u0721\u071d\u071d\u0308\u0718\u0717\u071dWe also write a fragment from the book on medicine of Rabban \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b, a disciple of Rabban \u1e24unayn, the head of the physicians. [It is] admirable and more precise than the similar [works].Lexicon of \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b could certainly fit well as a possible source of the fragmentLexicon they are dispersed among many other words, and secondly, the content of the entries is slightly different. With the availability of the new source, one may wonder whether the fragment was drawn from the Kunn\u0101\u0161\u0101. Indeed, there is an exact correspondence between the inventory and the list of the simples present in the first M\u0113mr\u0101. However the texts are not absolutely identical and in order to understand the difference, I would like to make a brief digression about the composition of the list of simples in the Kunn\u0101\u0161\u0101.The text was first mentioned in the eighteenth-century catalogue description of the manuscriptSimple Drugs . Considering the vast size of Galen\u2019s text, the obvious solution was to make an abridgement that, however, retains the drugs\u2019 original order following the Greek alphabet. Taking advantage of the available Syriac translation of the Simple Drugs, \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b added his own touch to the list. Namely, the Syriac version of the Simple Drugs must have already contained the Syriac equivalents of the Greek drug-names embedded in the text whereas the original Greek terms were reproduced in transliteration. Apparently, \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b was trying to keep up with developments in the field of medicine, particularly with the growing role of Arabic, and for that reason it is quite natural that he decided to increase the usefulness of the list by providing it with Arabic equivalents.While aiming to present the simples and their healing properties \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b could do no better than to exploit the relevant part of Galen\u2019s Kunn\u0101\u0161\u0101 and the fragment, because the list covers only the drug names from alpha through delta and thus it is possible that it was copied not from the original treatise but from an intermediary version. Whatever the case may be, it is the actual content of the fragment that is worthy of notice. What we find in it is exclusively the Greek terms and their Syriac and Arabic equivalents.\u012a\u0161\u014d\u02bf bar \u02bfAl\u012b\u2019s efforts to render Galen\u2019s catalogue of drugs more practical and user-friendly were undoubtedly rewarded and the fragment in the Vatican manuscript offers good proof for that. We cannot be sure that there is a direct relationship between the sop 238, f. 39v). The Lexicon and the Vatican fragment provide only the drugs\u2019 names.To illustrate the situation let us look at the first three terms in Galen\u2019s catalogue. Galen\u2019s list begins with discussions of \u1f00\u03b2\u03c1\u1f79\u03c4\u03bf\u03bd\u03bf\u03bd, \u1f04\u03b3\u03bd\u03bf\u03f2 and \u1f04\u03b3\u03c1\u03c9\u03c3\u03c4\u03b9\u03f2. The presentation of the drugs\u2019 properties is quite lengthy and occupies eight, three and one pages in K\u00fchn\u2019s edition respectively.Lexicon with the Kunn\u0101\u0161\u0101 we observe the following difference: The Lexicon\u2019s entries are more condensed and omit the reference to the informants (L1); the Syriac equivalents feature more as an exception (L1), whereas normally we find only the Arabic ones (L2 and L3) that are the primary focus of the Lexicon. The Kunn\u0101\u0161\u0101\u2019s approach is different for it tends to provide both the Syriac and Arabic equivalent (with the exception of L2 where only the Syriac one is indicated). The same procedure is displayed by the Vatican fragment that indicates both the Syriac and Arabic equivalents (with the exception of L2). There thus can be no doubt that the Vatican fragment is based on the Kunn\u0101\u0161\u0101 and not on the Lexicon.First of all, if we compare the Simple Drugs.\u0161un\u0101y\u0101 must have featured in \u1e24unayn\u2019s translation of the Simple Drugs.37In terms of possible sources that supplied the necessary information, it can be noted that the Syriac equivalents in L1 and L3 were introduced already by Sergius of R\u0113\u0161\u02bfain\u0101 in his translation of the Kunn\u0101\u0161\u0101 and the fragment preserved in the Vatican manuscript, we are justified in relying on the authorial attribution of the latter to \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b. After all, the extolling epithet (\u201cadmirable and thoroughly precise\u201d) can by all means be applied to the text of the Kunn\u0101\u0161\u0101.Having clarified the relationship between the Kunn\u0101\u0161\u0101. In particular, we find there  yet another inventory, this time of simples that are classified according to their primary qualities . The classification is followed by a list of substitute drugs . Both lists are present in the Kunn\u0101\u0161\u0101. The classification of the simples is present in M\u0113mr\u0101 i.70\u201385  whereas the substitute drugs appear at the very end of the treatise, M\u0113mr\u0101 vii.25 . This disconnected presence of the three fragments supports the assumption that they may not depend directly on the Kunn\u0101\u0161\u0101 but rather on an intermediary medical compilation that contained selected material from the Kunn\u0101\u0161\u0101. That issue requires further study.38It is worth noting in passing that the same Vatican manuscript also contains other fragments that match the text of the It is hard to overestimate the significance of a thirteenth-century medical manuscript preserved in the Syriac Orthodox Patriarchate. Even in its damaged form, the unique manuscript preserves the most extensive and voluminous Syriac medical treatise known to date and will undoubtedly change the entire field of Syriac medicine.40Kunn\u0101\u0161\u0101) that was written by \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b, who was a disciple of \u1e24unayn b. Is\u1e25\u0101q, a personal physician of Caliph al-Mu\u02bftamid (r. 870\u2013892) and the author of the Syriac-Arabic Lexicon. This identification is supported by a fragment from the Kunn\u0101\u0161\u0101 preserved in a Vatican manuscript with an explicit attribution to \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b as well as by a reference in the Kunn\u0101\u0161\u0101 itself stating that its author also produced a treatise about poisons in two books. It is that treatise of \u012a\u0161\u014d\u02bf bar \u02bfAl\u012b that was known in the Arabic medical tradition. A lack of references to the text in Arabic medical works is likely to be indicative of the fact that it was never translated into Arabic.41A preliminary study of the text allows it to be identified as a medical handbook  of \u02bfAl\u012b b. Rabban al-\u1e6cabar\u012b and the Book of Treasure .In fact, in the case of the Kunn\u0101\u0161\u0101 will prove a document of major significance with regard to the availability of medical works in ninth-century Baghdad. Citations that range from Hippocrates to \u1e24unayn show the Kunn\u0101\u0161\u0101 to be an indispensable witness to both Syriac translations of the classical Greek sources and indigenous Syriac medical works whose main body is lost.Kunn\u0101\u0161\u0101 differs from Sergius\u2019 translation it becomes an important witness for the later translation history of the Simple Drugs in the Syriac tradition.Undoubtedly, the Kunn\u0101\u0161\u0101 is, however, not lacking in serious difficulties, the least of which is a small tight handwriting that presents a challenge even to reading the text. The main obstacle is posed by the underdevelopment of the field of Syriac medicine and particularly by our meagre awareness concerning the translation techniques employed by the eighth- and ninth-century translators and the lexicography of such areas of medicine as anatomy, therapeutics, nosology and pharmacology.Kunn\u0101\u0161\u0101 will be uncritical and premature.A study of the"}
{"text": "N\u202f=\u202f2086), we investigated preferences for consequentialist vs. non-consequentialist social partners endorsing instrumental harm or impartial beneficence and examined how such preferences varied across different types of social relationships. Our results demonstrate robust preferences for non-consequentialist over consequentialist agents in the domain of instrumental harm, and weaker \u2013 but still evident \u2013 preferences in the domain of impartial beneficence. In the domain of instrumental harm, non-consequentialist agents were consistently viewed as more moral and trustworthy, preferred for a range of social roles, and entrusted with more money in economic exchanges. In the domain of impartial beneficence, preferences for non-consequentialist agents were observed for close interpersonal relationships requiring direct interaction  but not for more distant roles with little-to-no personal interaction . Collectively our findings demonstrate that preferences for non-consequentialist agents are sensitive to the different dimensions of consequentialist thinking and the relational context.Previous work has demonstrated that people are more likely to trust \u201cdeontological\u201d agents who reject harming one person to save many others than \u201cconsequentialist\u201d agents who endorse such instrumental harms, which could explain the higher prevalence of non-consequentialist moral intuitions. Yet consequentialism involves endorsing not just instrumental harm, but also impartial beneficence, treating the well-being of every individual as equally important. In four studies  or help  for the greater good\u2022Non-consequentialist agents who refuse to harm for the greater good are consistently preferred across a range of measures\u2022Consequentialist agents who help for the greater good are perceived as worse friends and spouses, but not political leaders In contrast, when someone rejects inflicting harm on an innocent they are said to be making a \u201cnon-consequentialist\u201d, or \u201cdeontological\u201d judgment in line with deontological ethical theories  judgment in line with consequentialist ethical theories . These ties e.g.  positingSuch dilemmas capture our imagination not just because they force an internal moral conflict, but because we recognize the reputational consequences that these impossible decisions might have for those who make them. Recent research has shown that agents who make consequentialist judgments in sacrificial dilemmas are seen as less moral, trustworthy and warm, chosen less frequently as social partners, and trusted less in economic exchanges e.g. . Such prinstrumental harm\u201d). At the core of utilitarianism is the idea of impartial beneficence, that we must impartially maximise the well-being of all sentient beings on the planet in such a way that \u201c[e]ach is to count for one and none for more than one\u201d  to save one's own child over, e.g., two strangers' children, and are incorporated into many forms of deontological ethics ; several distinct dimensions along which the agent's character could be perceived ; the different social roles in which the agent would be preferred ; and the different processes or motivations perceived to influence the agent's moral decision .22.12.1.1https://osf.io/yuv2m/.We report all measures,2.1.2For all studies, relevant ethical guidelines were followed and the research was approved through University of Oxford's Central University Research Ethics Committee, with the reference number MS-IDREC-C1-2015-098.2.1.3N\u202f=\u202f4) or failed a simple comprehension check asking them to indicate the judgment their partner made in the dilemma (N\u202f=\u202f5). This left us a final sample of 192 participants . Our sample size was determined through an a priori power analysis (see supplementary methods for details) and a sensitivity power analysis for our main ANCOVA analysis, assuming an \u03b1 of 0.05 and power of 0.80, indicated that the minimum effect size we had power to detect was a small-to-medium effect of f\u202f=\u202f0.20.We recruited 201 participants via MTurk, and paid them $1.00 for their time. Participants were excluded from completing the survey if they had participated in related studies by us in the past, and were excluded from analysis if took the survey more than once , where participants were asked to report their perceptions of a protagonist who made either a characteristically consequentialist or non-consequentialist decision in an impartiality dilemma. We included two different dilemmas to test the generality of any effects we observed and to demonstrate that findings were not specific to the one particular instantiation of the underlying impartiality dilemma. Given that the pattern of results was broadly the same2.1.5Participants read one of two2.1.6Participant Moral Judgment in the moral dilemmas was measured using three questions. First, participants were asked to make a binary judgment about what they thought the protagonist in the dilemma should have done (e.g. \u201cVolunteer to help build the houses\u201d vs. \u201cSpend the time cheering up her mother\u201d). Second and third, participants were asked to indicate how morally wrong they thought it would be to make the consequentialist (e.g. volunteer) and non-consequentialist (e.g. spend the time with the mother) decisions on a 1\u20137 scale .Participant Consequentialist Tendencies were measured using the Oxford Utilitarianism Scale  - consists of 5 items (\u03b1\u202f=\u202f0.70) that all tap endorsement of the impartial maximisation of the greater good even at the cost of personal self-sacrifice, such as \u201cIf the only way to save another person's life during an emergency is to sacrifice one's own leg, then one is morally required to make this sacrifice\u201d. The second subscale - Instrumental Harm (OUS-IH) - consists of 4 items (\u03b1\u202f=\u202f0.78) that tap a willingness to cause harm in order to bring about the greater good, including \u201cIt is morally right to harm an innocent person if harming them is a necessary means to helping several other innocent people\u201d.ale OUS: , consistCharacter Ratings were measured with seven questions in which participants rated on a 1\u20137 scale how they perceived the protagonist in the story to be in terms of how moral , trustworthy , loyal , reliable , warm or cold , competent , and capable  they thought the protagonist in the story to be.Role Suitability were measured with four questions in which participants rated on 1\u20137 scale how good a partner they thought the protagonist in the story would be in four types of social roles: as a friend, as a spouse, as a boss, and as a political leader, specifically President of the United States .Perceived Motivations of the protagonist's decision in the dilemma was measured through two items where participants indicated how much they thought the protagonist was driven by \u201caltruistic, empathic motives\u201d, and how much they thought the protagonist was driven by \u201cstrategic, reasoned motives\u201d .2.1.7U test . Means, standard deviations, p-values from the Mann-Whitney U tests, and effect sizes can be seen in Our primary measure of interest, like other recent studies e.g. , was howFor completeness, we also report in the supplementary materials a series of 2\u202f\u00d7\u202f2 ANOVAs in which we entered both participant consequentialist or non-consequentialist judgment and protagonist consequentialist or non-consequentialist decision. However, given that we were primarily interested in how the protagonist in the story was perceived overall depending on her decision, not how she was perceived differently by participants who themselves endorsed either option, in the interests of clarity and conciseness we have chosen to report these analyses in the supplementary materials.2.22.2.1Overall, most participants endorsed the characteristically non-consequentialist option of helping the family member over the characteristically consequentialist option of helping the greater number of strangers (63%), but this was more pronounced in the spending money dilemma (81%) than the spending time dilemma (55%).While participants did not think that either action was morally wrong, participants who reported higher consequentialist tendencies thought it would be more wrong to help the family number over the greater number of strangers. In the interests of space, we report all further results on participant judgments in the supplementary materials.2.2.2F\u202f=\u202f7.23, p\u202f=\u202f.008, \u03b7p2\u202f=\u202f0.04; U\u202f=\u202f3781, p\u202f=\u202f.026, d\u202f=\u202f0.37), reliable \u202f=\u202f17.36, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.08; U\u202f=\u202f3341, p\u202f<\u202f.001, d\u202f=\u202f0.59), loyal \u202f=\u202f128.23, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.40; U\u202f=\u202f1333, p\u202f<\u202f.001, d\u202f=\u202f1.61), and warm \u202f=\u202f32.76, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.15; U\u202f=\u202f2730, p\u202f<\u202f.001, d\u202f=\u202f0.81). There were, however, no differences between consequentialist and non-consequentialist agents in terms of perceived morality \u202f=\u202f0.00, p\u202f=\u202f.97, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f4473, p\u202f=\u202f.72, d\u202f=\u202f0.02), competence \u202f=\u202f3.35, p\u202f=\u202f.069, \u03b7p2\u202f=\u202f0.02; U\u202f=\u202f4023, p\u202f=\u202f.11, d\u202f=\u202f0.25), or capability \u202f=\u202f1.60, p\u202f=\u202f.21, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f4235, p\u202f=\u202f.31, d\u202f=\u202f0.17).When the protagonist in the dilemma decided to help the family member over the greater number of strangers, they were perceived as significantly more trustworthy \u202f=\u202f36.65, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.16; U\u202f=\u202f2519, p\u202f<\u202f.001, d\u202f=\u202f0.87) and spouse \u202f=\u202f46.20, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.20; U\u202f=\u202f2417, p\u202f<\u202f.001, d\u202f=\u202f0.97), but there was no difference in suitability as a boss \u202f=\u202f0.00, p\u202f=\u202f1.00, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f4545, p\u202f=\u202f.88, d\u202f=\u202f0.00), and the protagonist was actually seen to make a better political leader if they made the characteristically consequentialist decision \u202f=\u202f8.62, p\u202f=\u202f.004, \u03b7p2\u202f=\u202f0.04; U\u202f=\u202f3471, p\u202f=\u202f.002, d\u202f=\u202f0.42).When the protagonist made the characteristically non-consequentialist decision to help the family member over the greater number of strangers they were expected to make a better friend \u202f=\u202f39.95, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.17; U\u202f=\u202f2409, p\u202f<\u202f.001, d\u202f=\u202f0.91, but there was no difference between agents in perceptions of empathic, altruistic motives, F\u202f=\u202f0.07, p\u202f=\u202f.80, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f4544, p\u202f=\u202f.87, d\u202f=\u202f0.03.Participants thought the consequentialist protagonist was driven more by strategic, reasoned motives than the non-consequentialist protagonist, 2.4Results from Study 1 using impartiality dilemmas were less consistent than the unequivocal preference for the non-consequentialist in sacrificial dilemmas we have seen in previous work , but oveCertain limitations of this study should be noted, however. First, Study 1 relied on character ratings of the protagonist and did not measure participants' actual behaviour. Although important work on this topic has been conducted without measuring behaviour e.g. , we thin3In Study 2 we looked again at perceptions of consequentialist agents endorsing impartial beneficence, this time adding an economic Trust Game to establish whether the partner preference has real behavioural consequences in a cooperation market, and turned back to examining a third-party judge, as in the majority of this work . We also3.13.1.1https://osf.io/prm3a/). We report all measures,https://osf.io/bdev3/.Our design, hypotheses, and analysis plan were all pre-registered at the Open Science Framework (3.1.2N\u202f=\u202f6) or failed a simple comprehension check asking them to indicate the judgment their partner made in the dilemma (N\u202f=\u202f41). This left us with a final sample of 953 participants . Our sample size was determined through an a priori power analysis (see supplementary methods for details) and a sensitivity power analysis for our main 2\u202f\u00d7\u202f2 ANCOVA analysis, assuming an \u03b1 of 0.05 and power of 0.80, indicated that the minimum effect size we had power to detect was a small effect of f\u202f=\u202f0.09.In accordance with the pre-registration, we recruited 1000 participants via MTurk. Participants were excluded from completing the study if they had participated in related studies by us in the past, and were excluded from analysis if they did not complete the study in full . We then determined bonuses for participants within each condition by looking at what percentage the corresponding second mover said they would return and paid participants accordingly.3.1.3We had a 2  between-subjects design, with two dilemmas in each category being used as variations of the experimental materials. Participants read one of four dilemmas and rated a partner (\u201cagent\u201d) who read the same dilemma and made either a consequentialist or non-consequentialist judgment and justification ; and second, as a continuous measure on an eleven-point scale of how wrong participants thought it would be to perform the consequentialist action instead of the non-consequentialist one .Partner Preference was measured by a single question in which participants were asked to indicate what kind of partner they would have preferred to play the TG with if they had a choice: someone who made a consequentialist or non-consequentialist judgment.Role Suitability, like in Study 1, was measured with four questions in which participants rated on 1\u20137 scale how good a partner they thought the protagonist in the story would be in four types of social roles: as a friend, as a spouse, as a boss, and as political leader, i.e. President of the United States .Character Ratings were measured with seven questions in which participants rated on 1\u20137 scale how moral, trustworthy, loyal, warm or cold, sociable, competent, and capable they thought the agent to be. Ratings of how sociable  and warm or cold  participants thought the agent was were combined into an overall score of warmth (\u03b1\u202f=\u202f0.76); and ratings of how competent  and capable  participants thought the agent was were combined into an overall score of competence (\u03b1\u202f=\u202f0.88). Ratings of how moral  and trustworthy  participants thought the agent were combined into an overall score of morality (\u03b1\u202f=\u202f0.85). We analyzed loyalty separately because in the impartiality dilemmas but not the sacrificial ones the non-consequentialist action strongly involved loyalty (to one's parents). We created these composite scores both in the interests of space and because, given they are intended to measure the same construct, we expected \u2013 and found - no differences between the results for each. Nonetheless, full results using the individual items can be seen in the supplementary results.3.1.6U tests . All Ms and SDs as a function of dilemma type and agent judgment, along with the p-values from a Mann-Whitney U test and effect size of the difference between a consequentialist and non-consequentialist agent for each type of dilemma can be seen in As outlined in the pre-registration, our primary analysis of interest was on the main effects of agent consequentialist or non-consequentialist judgment in both the sacrificial and impartiality dilemmas, and whether there was an interaction effect between them such that the effect was stronger for one type of dilemma. Across Studies 2\u20134, we used a 2\u202f\u00d7\u202f2 ANCOVA  to look at the interactive effect of dilemma type and agent judgment and explore our main effects while controlling for any variance caused by participants' own judgments. Because the data was again not normally distributed we complemented this ANCOVA with non-parametric Mann-Whitney In accordance with the pre-registration, in the supplementary materials we also report (1) main effects of agent judgment for each dilemma separately, and (2) results from a 2\u202f\u00d7\u202f2\u202f\u00d7\u202f2 ANOVA in which we entered participant moral judgment as a fixed factor (instead of a covariate). As can be seen in the supplementary materials, results were consistent across the two dilemmas in each category, and we only found significant 3-way interactions in four of the eleven DVs .3.23.2.1r\u202f=\u202f0.22 p\u202f<\u202f.001), and higher scores on the Instrumental Harm subscale predicted lower wrongness judgments of the consequentialist action in the sacrificial dilemmas  .The majority of participants endorsed the non-consequentialist option in the sacrificial dilemmas (69%), rejecting the sacrifice of one to save the lives of a greater number. This was the same for the footbridge (69%) and vaccine (69%) variants. Similarly, most participants endorsed the non-consequentialist option in the impartiality dilemmas (70%), endorsing helping a family member over impartially helping a greater number. This was the same for both dilemmas but was more pronounced in the spending money (83%) than the spending time variant (55%). Higher scores on the Impartial Beneficence subscale of the OUS predicted lower wrongness judgments of the consequentialist action in the impartiality dilemmas \u202f=\u202f74.69, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.07; U\u202f=\u202f147,829, p\u202f<\u202f.001, d\u202f=\u202f0.56, and a significant interaction of dilemma type and agent judgment on how moral participants perceived the agent to be, F\u202f=\u202f61.30, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.06. While a non-consequentialist was seen as more moral than the consequentialist in the sacrificial dilemmas, F\u202f=\u202f138.84, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.22; U\u202f=\u202f46,702, p\u202f<\u202f.001, d\u202f=\u202f1.04, there was no difference in perceived morality of non-consequentialist vs. consequentialist agents in the impartiality dilemmas, F\u202f=\u202f0.33, p\u202f=\u202f.56, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f27,382, p\u202f=\u202f.56, d\u202f=\u202f0.05.We first looked at character ratings see . For perF\u202f=\u202f52.71, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.05; U\u202f=\u202f143,077, p\u202f<\u202f.001, d\u202f=\u202f0.47, and a significant interaction of dilemma type and agent judgment on how warm and sociable participants perceived the agent to be, F\u202f=\u202f59.01, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.06. While a non-consequentialist was seen as warmer than the consequentialist in the sacrificial dilemmas, F\u202f=\u202f119.95, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.20; U\u202f=\u202f45,840, p\u202f<\u202f.001, d\u202f=\u202f0.98, there was no difference in the impartiality dilemmas, F\u202f=\u202f0.08, p\u202f=\u202f.77, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f25,980, p\u202f=\u202f.68, d\u202f=\u202f0.03.For perceived warmth, the ANCOVA revealed the predicted main effect of agent judgment, F\u202f=\u202f7.53, p\u202f=\u202f.001, \u03b7p2\u202f=\u202f0.01; U\u202f=\u202f125,545, p\u202f=\u202f.004, d\u202f=\u202f0.18, and a significant interaction effect of agent judgment and dilemma judgment, F\u202f=\u202f3.24, p\u202f=\u202f.006, \u03b7p2\u202f=\u202f0.00. While a non-consequentialist was seen as more competent than the consequentialist in the sacrificial dilemmas, F\u202f=\u202f10.10, p\u202f=\u202f.002, \u03b7p2\u202f=\u202f0.02; U\u202f=\u202f34,981, p\u202f=\u202f.002, d\u202f=\u202f0.29, there was no difference in the impartiality dilemmas, F\u202f=\u202f0.46, p\u202f=\u202f.50, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f27,898, p\u202f=\u202f.34, d\u202f=\u202f0.06.For perceived competence, the ANCOVA showed the predicted main effect of agent judgment, F\u202f=\u202f283.97, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.23; U\u202f=\u202f174,248, p\u202f<\u202f.001, d\u202f=\u202f1.07, and there was no significant interaction of dilemma type and agent judgment, F\u202f=\u202f0.27, p\u202f=\u202f.61, \u03b7p2\u202f=\u202f0.00. For both dilemma types, an agent who made a non-consequentialist judgment was seen as more loyal than one who made a consequentialist judgment. This was the case both in the sacrificial dilemmas F\u202f=\u202f128.14, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.21; U\u202f=\u202f45,614, p\u202f<\u202f.001, d\u202f=\u202f1.01, and the impartiality dilemmas, F\u202f=\u202f159.22, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.26; U\u202f=\u202f41,528, p\u202f<\u202f.001, d\u202f=\u202f1.15.Finally, we turned to the single item of perceived loyalty. The ANCOVA revealed the predicted main effect of agent judgment, Overall, then, across all the character ratings there were significant main effects such that the non-consequentialist was seen as more moral, warmer, more competent, and more loyal see . With th3.2.3F\u202f=\u202f3.80, p\u202f=\u202f.052, \u03b7p2\u202f=\u202f0.00, and though there was no main effect of agent judgment when using the ANCOVA controlling for participant wrongness, F\u202f=\u202f3.18, p\u202f=\u202f.075, \u03b7p2\u202f=\u202f0.00, there was a significant main effect in the non-parametric test , U\u202f=\u202f121,638, p\u202f=\u202f.043, d\u202f=\u202f0.12. Looking at simple effects, we found that participants transferred significantly more to a non-consequentialist agent than a consequentialist agent in the sacrificial dilemmas, F\u202f=\u202f7.22, p\u202f=\u202f.007, \u03b7p2\u202f=\u202f0.001; U\u202f=\u202f34,431, p\u202f=\u202f.005, d\u202f=\u202f0.24, but there was no difference in the impartiality dilemmas, F\u202f=\u202f0.01, p\u202f=\u202f.91, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f26,593, p\u202f=\u202f.098, d\u202f=\u202f0.01 \u202f=\u202f12.02, p\u202f=\u202f.57, \u03b7p2\u202f=\u202f0.01, and no main effect of agent judgment, F\u202f=\u202f0.33, p\u202f=\u202f.57, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f111,551, p\u202f=\u202f.49, d\u202f=\u202f0.04. While the interaction was non-significant, simple effects revealed that a non-consequentialist agent was predicted to return more than the consequentialist in the sacrificial dilemmas, F\u202f=\u202f8.53, p\u202f=\u202f.004, \u03b7p2\u202f=\u202f0.002; U\u202f=\u202f34,822, p\u202f=\u202f.002, d\u202f=\u202f0.26, but the consequentialist was predicted to return more than the non-consequentialist in the impartiality dilemmas, F\u202f=\u202f3.99, p\u202f=\u202f.046, \u03b7p2\u202f=\u202f0.001; U\u202f=\u202f23,683, p\u202f=\u202f.036, d\u202f=\u202f\u22120.19.For predicted returns, surprisingly an ANCOVA revealed no significant interaction of dilemma type and agent judgment on how much participants expected their partner to return, 3.2.4B\u202f=\u202f\u22120.70, SE\u202f=\u202f0.14, Z\u202f=\u202f\u22125.16, p\u202f<\u202f.001, with chi-square analyses revealing that while most participants (70%) preferred the non-consequentialist in the sacrificial dilemmas, x2(1)\u202f=\u202f75.71, p\u202f<\u202f.001, there was no difference in preference for a non-consequentialist (53%) or consequentialist (47%) in the impartial dilemmas, x2(1)\u202f=\u202f1.95, p\u202f=\u202f.16.Next, we looked at which type of person the participant would have liked to play the TG with, if they'd had a choice: the consequentialist or non-consequentialist. A logistic regression revealed a significant effect of dilemma type, 3.2.5F\u202f=\u202f137.60, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.13; U\u202f=\u202f68,383, p\u202f<\u202f.001, d\u202f=\u202f0.76, along with a significant interaction of dilemma type and agent judgment on how good a friend participants thought the agent would be, F\u202f=\u202f16.78, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.02. In both dilemma types, when the agent made a non-consequentialist judgment they were expected to make a better friend, but this was stronger for the sacrificial dilemmas, F\u202f=\u202f123.59, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.20; U\u202f=\u202f45,524, p\u202f<\u202f.001, d\u202f=\u202f1.00, than for the impartiality dilemmas, F\u202f=\u202f29.66, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.06; U\u202f=\u202f33,921, p\u202f<\u202f.001, d\u202f=\u202f0.50.Next, we looked at perceived suitability for different roles see . ConsideF\u202f=\u202f133.71, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.12; U\u202f=\u202f69,310, p\u202f<\u202f.001, d\u202f=\u202f0.75, along with a significant interaction of dilemma type and agent judgment on how good a friend participants thought the agent would be, F\u202f=\u202f10.84, p\u202f=\u202f.001, \u03b7p2\u202f=\u202f0.01. In both dilemmas, when the agent made a non-consequentialist judgment they were expected to make a better spouse, but this was stronger for the sacrificial dilemmas, F\u202f=\u202f112.39, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.19; U\u202f=\u202f44,716, p\u202f<\u202f.001, d\u202f=\u202f0.95, than for the impartiality dilemmas, F\u202f=\u202f33.59, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.07; U\u202f=\u202f34,319, p\u202f<\u202f.001, d\u202f=\u202f0.54.For perceived suitability as a spouse, an ANCOVA revealed the predicted main effect of agent judgment, F\u202f=\u202f22.95, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.02; U\u202f=\u202f133,232, p\u202f<\u202f.001, d\u202f=\u202f0.32 and a significant interaction of dilemma type and agent judgment on how good a boss participants thought the agent would be, F\u202f=\u202f18.12, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.02. When the agent made a non-consequentialist judgment in the sacrificial dilemmas they were expected to make a better boss, F\u202f=\u202f39.98, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.08; U\u202f=\u202f39,299, p\u202f<\u202f.001, d\u202f=\u202f0.57, but there was no difference for the impartiality dilemmas F\u202f=\u202f0.15, p\u202f=\u202f.70, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f27,519, p\u202f=\u202f.49, d\u202f=\u202f0.04.For suitability as a boss, an ANCOVA revealed the predicted main effect of agent judgment, F\u202f=\u202f12.81, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.01, but no main effect of agent judgment, F\u202f=\u202f0.01, p\u202f=\u202f.92, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f111,573, p\u202f=\u202f.65, d\u202f=\u202f0.00. Looking at simple effects, in the sacrificial dilemmas a non-consequentialist agent was expected to make a better political leader, F\u202f=\u202f5.54, p\u202f=\u202f.019, \u03b7p2\u202f=\u202f0.01; U\u202f=\u202f33,389, p\u202f=\u202f.040, d\u202f=\u202f0.21, but in the impartiality dilemmas it was a consequentialist agent who was expected to make a better political leader, F\u202f=\u202f7.55, p\u202f=\u202f.006, \u03b7p2\u202f=\u202f0.002; U\u202f=\u202f22,423, p\u202f=\u202f.003, d\u202f=\u202f\u22120.26.For suitability as a political leader, an ANCOVA revealed a significant interaction of dilemma type and agent judgment on how good a political leader participants thought the agent would be, Overall, results showed that a non-consequentialist agent in the sacrificial dilemmas was seen to make a better friend, boss, spouse, and political leader than the agent who made a consequentialist judgment see . In the 3.3In Study 2 we had two key aims: first, to investigate how participants perceived and interacted with someone who made a consequentialist or non-consequentialist judgment in impartiality dilemmas; and second, to replicate previous work demonstrating partner preference in sacrificial dilemmas in a pre-registered study with new measures and a different dilemma.In the sacrificial dilemmas we successfully replicated and extended previous findings, showing that people who make non-consequentialist judgments in sacrificial dilemmas are perceived more positively than those who make a consequentialist judgment. Non-consequentialists received more transfers in a Trust Game, were seen as more moral and trustworthy, and when given a choice participants indicated they would rather play with a non-consequentialist than a consequentialist. Moreover, we found that this preference for non-consequentialists extended to character ratings of warmth, competence, and loyalty, and that non-consequentialists were seen to make a better friend, a better boss, a better spouse, and a better political leader. In every single dependent measure, a person making a non-consequentialist judgment in a sacrificial dilemma was preferred.In the impartiality dilemmas, the picture was much less consistent. The consequentialist in the impartial dilemmas was preferred on two measures: they were expected to return more in the TG, and they were thought to make a better political leader. The non-consequentialist was preferred on three measures: they were expected to be more loyal and make a better friend and spouse. For most measures, however, there were no differences between the consequentialist and non-consequentialist: they were seen as equally moral, warm, and competent; they were thought to make an equally good boss. While impartial consequentialists were expected to make a better political leader, they were thought to make a worse friend and spouse; and while they were expected to return more in a TG, this did not translate into actual increased transfers by participants, nor greater selection of the consequentialist for a future TG.One potential limitation is that while 83% of participants endorsed the non-consequentialist judgment in the spending money variant, only 55% of participants did so in the spending time variant. Moreover, in this study \u2013 and in contrast to Study 1 \u2013 we did not find the expected negative correlation between OUS impartial beneficence scores and perceived wrongness of the consequentialist action in the spending time dilemma. This suggests that in this study participants themselves did not actually perceive the spending time dilemma to be one in which one action is consequentialist, and if this is the case, one must be careful about inferring too much about how people think about \u201cconsequentialist\u201d judgers in the same dilemma .4In Study 3, we sought to replicate and extend the results in Study 2 using a streamlined set of moral judgment scenarios and a Prisoner's Dilemma PD:  instead 4.14.1.1https://osf.io/dfz2j/) as part of the Pre-Registration Challenge. We report all measures, manipulations, and exclusions in this study. All data, analysis code, and experiment materials are available for download at: https://osf.io/v6z53/.Our design, hypotheses, and analysis plan were all pre-registered at the Open Science Framework (4.1.2N\u202f=\u202f5), or failed a simple comprehension check asking them to indicate the decision their partner made in the dilemma (N\u202f=\u202f12), leaving us with a final sample of 485 participants . Our sample size was determined through an a priori power analysis (see supplementary methods for details) and a sensitivity power analysis for our main 2\u202f\u00d7\u202f2 ANCOVA analysis, assuming an \u03b1 of 0.05 and power of 0.80, indicated that the minimum effect size we had power to detect was a small effect of f\u202f=\u202f0.13. All participants were paid $1.80 for participating and were again paid bonuses depending on their decision in the game. To calculate bonuses, we selected the decision (to cooperate or defect) of the first participant who met the criteria for each condition  and then matched participants accordingly (see supplementary file for more information).In accordance with the pre-registration, 498 participants completed the survey online via MTurk. Participants were excluded from completing the survey if they had participated in related studies by us in the past, and were excluded from analysis if they completed the survey more than once \u202f\u00d7\u202f2  between-subjects experimental design. The study was identical to Study 2, except we used two dilemmas  instead of four, and used a PD instead of a TG.4.1.4https://osf.io/v6z53). Participants were required to complete four comprehension questions about the payoffs to each player depending on both player's choices in an example matrix, and they had to answer these correctly in order to move to the next page. After completing this, participants entered the main part of the study where they were told what decision their partner made in the moral dilemma, and made their choice either to cooperate (X) or defect (Y) in the main PD. For the main PD, if both players cooperated they received 3 points each; if both defected they received 1 point each; and if one defected and one cooperated, the defector received 5 points and the cooperator received 0 points.Like in Study 2, participants were first introduced to the moral dilemma and asked to make their own judgments. Next, participants were introduced to the PD and received extensive training about the structure of the game based on the procedure of 4.1.5Ms and SDs as a function of dilemma type and agent judgment, as well as p-values from a Mann-Whitney U test, and effect sizes. In accordance with the pre-registration, for the supplementary materials we again ran analyses looking at a 2\u202f\u00d7\u202f2\u202f\u00d7\u202f2 ANOVA in which we entered participant moral judgment as a fixed factor (instead of a covariate). There were, however, no significant 3-way interactions, further justifying our focus on the ANCOVA.The analysis plan for this study was identical to that in Study 2. See 4.24.2.1The majority of participants endorsed the non-consequentialist option in the sacrificial dilemma (68%), rejecting the sacrifice of one to save the lives of a greater number. Similarly, most participants endorsed the non-consequentialist option in the impartiality dilemma (79%), endorsing helping a family member over impartially helping a greater number. As for the previous two studies, further results looking at participant judgment can be found in the supplementary materials.4.2.2F\u202f=\u202f84.19, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.15; U\u202f=\u202f41,514, p\u202f<\u202f.001, d\u202f=\u202f0.79, and a significant interaction of dilemma type and agent judgment on how moral participants perceived the agent to be, F\u202f=\u202f72.68, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.13. While a non-consequentialist was seen as more moral than the consequentialist in the sacrificial dilemma, F\u202f=\u202f142.20, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.37; U\u202f=\u202f13,230, p\u202f<\u202f.001, d\u202f=\u202f1.47, there was no difference in the impartiality dilemma, F\u202f=\u202f0.11, p\u202f=\u202f.74, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f7080, p\u202f=\u202f.65, d\u202f=\u202f0.03.We first looked at character ratings see . For theF\u202f=\u202f87.88, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.15; U\u202f=\u202f42,494, p\u202f<\u202f.001, d\u202f=\u202f0.83, and a significant interaction of dilemma type and agent judgment on how warm and sociable participants perceived the agent to be, F\u202f=\u202f50.61, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.10. While a non-consequentialist was seen as warmer than the consequentialist in the sacrificial dilemmas, F\u202f=\u202f140.09, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.36; U\u202f=\u202f13,218, p\u202f<\u202f.001, d\u202f=\u202f1.47, there was no difference in the impartiality dilemmas, F\u202f=\u202f2.03, p\u202f=\u202f.16, \u03b7p2\u202f=\u202f0.01; U\u202f=\u202f7700, p\u202f=\u202f.097, d\u202f=\u202f0.18.For perceived the two items measuring warmth (\u03b1\u202f=\u202f0.74), the ANCOVA revealed the predicted main effect of agent judgment, F\u202f=\u202f23.96, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.05; U\u202f=\u202f36,630, p\u202f<\u202f.001, d\u202f=\u202f0.45, but no interaction between agent judgment and dilemma type, F\u202f=\u202f0.28, p\u202f=\u202f.60, \u03b7p2\u202f=\u202f0.00. The non-consequentialist was seen as more competent in both the sacrificial, F\u202f=\u202f13.57, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.05; U\u202f=\u202f9895, p\u202f<\u202f.001, d\u202f=\u202f0.47, and impartiality dilemmas, F\u202f=\u202f10.27, p\u202f=\u202f.002, \u03b7p2\u202f=\u202f0.04; U\u202f=\u202f8291, p\u202f=\u202f.005, d\u202f=\u202f0.41.For the two items measuring perceived competence (\u03b1\u202f=\u202f0.89), the ANCOVA showed the predicted main effect of agent judgment, F\u202f=\u202f259.78, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.35; U\u202f=\u202f49,147, p\u202f<\u202f.001, d\u202f=\u202f1.44, and a non-significant but marginal interaction between agent judgment and dilemma type, F\u202f=\u202f3.67, p\u202f=\u202f.056, \u03b7p2\u202f=\u202f0.001. The non-consequentialist was seen as more loyal in both the sacrificial, F\u202f=\u202f149.13, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.38; U\u202f=\u202f13,265, p\u202f<\u202f.001, d\u202f=\u202f1.52, and impartiality dilemmas, F\u202f=\u202f110.38, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.32; U\u202f=\u202f11,116, p\u202f<\u202f.001, d\u202f=\u202f1.35, though the effect sizes were slightly larger in the sacrificial dilemma.Finally, we turned to the single item of perceived loyalty. The ANCOVA showed the predicted main effect of agent judgment, 4.2.3B\u202f=\u202f\u22121.56, SE\u202f=\u202f0.62, Z\u202f=\u202f\u22122.51, p\u202f=\u202f.012, and a marginally significant interaction of agent judgment and dilemma type, B\u202f=\u202f\u22120.86, SE\u202f=\u202f0.46, Z\u202f=\u202f\u22121.88, p\u202f=\u202f.061. These interaction effects were supplemented by a significant main effect of agent judgment, B\u202f=\u202f1.19, SE\u202f=\u202f0.33, Z\u202f=\u202f3.60, p\u202f<\u202f.001, such that across dilemmas and regardless of participant's own judgments, there was more cooperation extended to the non-consequentialist (73%) than the consequentialist (63%) agent. There was, however, no main effect of participant judgment, B\u202f=\u202f0.08, SE\u202f=\u202f0.45, Z\u202f=\u202f0.17, p\u202f=\u202f.86, suggesting that overall, consequentialist participants were roughly as equal to cooperate (77%) as non-consequentialist participants were (64%) - though note that this difference was significant when looking at simple effects with a chi-square analysis ignoring the effects of agent judgment and dilemma type, x2(1)\u202f=\u202f5.79, p\u202f=\u202f.016, with consequentialists more likely to cooperate.In order to assess the effects of agent judgment, dilemma type, and participant judgment in the dilemma on the likelihood that participants made cooperative decisions in the PD we conducted a 2\u202f\u00d7\u202f2\u202f\u00d7\u202f2 logistic regression model. This model revealed no significant three-way interaction between agent, dilemma, and participant judgment, only a significant interaction of agent judgment and participant judgment, x2(1)\u202f=\u202f11.48, p\u202f<\u202f.001, while consequentialist participants were no more likely to cooperate with a consequentialist (81%) or non-consequentialist (72%) agent, x2(1)\u202f=\u202f1.25, p\u202f=\u202f.26  than a consequentialist one (55%), \u202f.26 see . This, wx2(1)\u202f=\u202f6.73, p\u202f=\u202f.009. In contrast, in the impartiality dilemma there was equal cooperation extended towards the consequentialist (71%) and non-consequentialist (72%) agents, x2(1)\u202f=\u202f0.02, p\u202f=\u202f.88.Finally, while the interaction between agent judgment and dilemma type was only marginally significant, we looked at simple effects regardless and found that across both consequentialist and non-consequentialist participants, people were more likely to cooperate with the non-consequentialist (73%) than the consequentialist agent (57%) in the sacrificial dilemma, 4.2.4B\u202f=\u202f\u22120.28, SE\u202f=\u202f0.19, Z\u202f=\u202f\u22121.48, p\u202f=\u202f.14 with chi-square analyses revealing that across the dilemmas most participants (65%) preferred to play with a non-consequentialist over a consequentialist (35%). This was the case both in the sacrificial dilemma, with 68% preferring the non-consequentialist, x2(1)\u202f=\u202f33.12, p\u202f<\u202f.001; and the impartiality dilemma, with 62% preferring the non-consequentialist, x2(1)\u202f=\u202f12.93, p\u202f<\u202f.001.Next, we looked at which agent the participant would have liked to play the PD with, if they'd had a choice. A logistic regression revealed no overall effect of dilemma type, 4.2.5F\u202f=\u202f97.14, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.17; U\u202f=\u202f42,377, p\u202f<\u202f.001, d\u202f=\u202f0.87, along with a significant interaction of dilemma type and agent judgment on how good a friend participants thought the agent would be, F\u202f=\u202f26.17, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.05. In both dilemmas when the agent made a non-consequentialist judgment they were expected to make a better friend, but this effect was stronger for the sacrificial dilemma, F\u202f=\u202f117.38, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.19; U\u202f=\u202f12,759, p\u202f<\u202f.001, d\u202f=\u202f1.34, than for the impartiality dilemma, F\u202f=\u202f10.40, p\u202f=\u202f.001, \u03b7p2\u202f=\u202f0.04; U\u202f=\u202f8093, p\u202f=\u202f.014, d\u202f=\u202f0.39.Next, we looked at perceived suitability for different roles see . For perF\u202f=\u202f124.95, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.21; U\u202f=\u202f44,134, p\u202f<\u202f.001, d\u202f=\u202f0.99, along with a significant interaction of dilemma type and agent judgment on how good a spouse participants thought the agent would be, F\u202f=\u202f18.36, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.04. In both dilemmas when the agent made a non-consequentialist judgment they were expected to make a better spouse, but this was stronger for the sacrificial dilemma, F\u202f=\u202f119.82, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.33; U\u202f=\u202f12,813, p\u202f<\u202f.001, d\u202f=\u202f1.35, than for the impartiality dilemma, F\u202f=\u202f22.69, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.09; U\u202f=\u202f9012, p\u202f<\u202f.001, d\u202f=\u202f0.60.For perceived suitability as a spouse, an ANCOVA revealed the predicted main effect of agent judgment, F\u202f=\u202f47.36, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.09; U\u202f=\u202f38,815, p\u202f<\u202f.001, d\u202f=\u202f0.63, and a significant interaction of dilemma type and agent judgment on how good a boss participants thought the agent would be, F\u202f=\u202f15.50, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.03. When the agent made a non-consequentialist judgment they were expected to make a better boss, but this was stronger again for the sacrificial dilemma F\u202f=\u202f51.14, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.17; U\u202f=\u202f11,316, p\u202f<\u202f.001, d\u202f=\u202f0.90, than for the impartiality dilemma, F\u202f=\u202f4.68, p\u202f=\u202f.032, \u03b7p2\u202f=\u202f0.02; U\u202f=\u202f7904, p\u202f=\u202f.036, d\u202f=\u202f0.28.For suitability as a boss, an ANCOVA revealed the predicted main effect of agent judgment, F\u202f=\u202f8.64, p\u202f=\u202f.003, \u03b7p2\u202f=\u202f0.02; U\u202f=\u202f33,156, p\u202f=\u202f.013, d\u202f=\u202f0.29, and a significant interaction of dilemma type and agent judgment on how good a political leader participants thought the agent would be, F\u202f=\u202f8.61, p\u202f=\u202f.004, \u03b7p2\u202f=\u202f0.02. When the agent made a non-consequentialist judgment in the sacrificial dilemma they were expected to make a better political leader, F\u202f=\u202f14.69, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.06; U\u202f=\u202f9693, p\u202f<\u202f.001, d\u202f=\u202f0.49, but there was no difference in the impartiality dilemma, F\u202f=\u202f0.00, p\u202f=\u202f.99, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f6702, p\u202f=\u202f.77, d\u202f=\u202f0.01.Finally for perceived suitability as a political leader, i.e. President of the United States, an ANCOVA revealed the predicted main effect of agent judgment, 4.3In the context of a sacrificial dilemma, we successfully replicated and extended previous findings. We found that on every single dependent measure, people who made non-consequentialist judgments in a sacrificial dilemma were perceived more positively than those who made a consequentialist judgment. As in Study 2, the pattern of results for the impartiality dilemma was more nuanced, though a definite pattern of preference for the non-consequentialist was observed. Out of 10 dependent measures, the non-consequentialist was preferred in six measures and there was no preference in four measures. The non-consequentialist was thought to be more competent and loyal; thought to make a better friend, spouse, and boss; and preferred as a future partner in a PD. There was no preference, however, in perceptions of morality or warmth, perceived suitability as a political leader, or cooperation extended in a PD.Our results are equivocal as to whether non-consequentialist or consequentialist participants were actually more cooperative. While neither non-consequentialist nor consequentialist participants reliably cooperated more overall, we did find that while non-consequentialist participants selectively cooperated with non-consequentialist agents, consequentialist participants cooperated equally with both agents. We find no evidence, then, that non-consequentialists are systematically more cooperative than their consequentialist counterparts. That said, it is important to recognize two key caveats. The first caveat is that behaviour in the PD requires both the goal of cooperation and the expectation that others will cooperate , or in o5In our final study, we wanted to address two potential concerns with the previous studies. The first concern is that there were differences in how much the agents expressed conflict, or recognized competing moral arguments against their decision. The non-consequentialist agent in all dilemmas recognized the conflict by noting they could bring about better consequences, but thought that other moral concerns were more important. Similarly in the impartiality dilemma the consequentialist agent briefly acknowledged the conflicting deontological duties , but in the sacrificial dilemma the consequentialist agent made no mention of the conflicting duties . It is possible, then, that the strong preference for the non-consequentialist we observed in the sacrificial but not impartial dilemmas was partly driven by the fact that the consequentialist in the sacrificial dilemmas expressed no awareness of moral conflict, but the consequentialist in the impartiality dilemma did . Second and third, participants indicate separately how much they thought the agent's decision was driven by strategic and altruistic motives .The second concern we wanted to address was that in Studies 2\u20133 we collected data on perceived motives of the agent, but did not report the results in the main manuscript because of issues with the wording of the question (see Footnote 6). We had asked participants to indicate on a 11-point scale how much they thought the agent's decision was driven \u201cmore by strategic, reasoned motives versus more empathic, altruistic motives?\u201d. However, on reflection this measure was sub-optimal both because it conflates reasoned and strategic motives  and forces participants to select one side, when it is possible that participants thought the action is both more strategic or reasoned 5.15.1.1https://osf.io/xr428/) as part of the Pre-Registration Challenge. We report all measures, manipulations, and exclusions in this study. All data, analysis code, and experiment materials are available for download at: https://osf.io/zf2dp/.Our design, hypotheses, and analysis plan were all pre-registered at the Open Science Framework (5.1.2N\u202f=\u202f2), or failed a simple comprehension check asking them to indicate the decision their partner made in the dilemma (N\u202f=\u202f41), leaving us with a final sample of 456 participants . Our sample size was determined through an a priori power analysis (see supplementary methods for details) and a sensitivity power analysis for our main 2\u202f\u00d7\u202f2 ANCOVA analysis, assuming an \u03b1 of 0.05 and power of 0.80, indicated that the minimum effect size we had power to detect was a small effect of f\u202f=\u202f0.13. All participants were paid $1.20 for participating, in accordance with an hourly US minimum wage of $7.25 and the survey taking approximately 10\u202fmin.In accordance with the pre-registration, 500 participants completed the survey online via MTurk. Participants were excluded from completing the survey if they had participated in related studies by us in the past, and were excluded from analysis if they completed the survey more than once \u202f\u00d7\u202f2  between-subjects experimental design, and the procedure was the same as in Study 2, except without an economic game and the addition of the three motives questions outlined in the introduction to this study.5.1.4Ms and SDs as a function of dilemma type and agent judgment, as well as p-values from a Mann-Whitney U test, and effect sizes. Again, see supplementary results for analyses looking at a 2\u202f\u00d7\u202f2\u202f\u00d7\u202f2 ANOVA in which we entered participant moral judgment as a fixed factor instead of a covariate, though like in the previous studies there were no significant 3-way interactions.The analysis plan for this study was identical to that in Studies 2\u20133. See 5.25.2.1The majority of participants endorsed the non-consequentialist option in the sacrificial dilemmas (59%), rejecting the sacrifice of one to save the lives of a greater number. Similarly, most participants endorsed the non-consequentialist option in the impartiality dilemmas (85%), endorsing helping a family member over impartially helping a greater number. Again, further results looking at participant judgment can be found in the supplementary materials.5.2.2F\u202f=\u202f34.29, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.07; U\u202f=\u202f33,548, p\u202f<\u202f.001, d\u202f=\u202f0.54, and a significant interaction of dilemma type and agent judgment on how moral participants perceived the agent to be, F\u202f=\u202f8.54, p\u202f=\u202f.004, \u03b7p2\u202f=\u202f0.02. The non-consequentialist was seen as more moral than the consequentialist in both the sacrificial dilemma, F\u202f=\u202f32.58, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.13; U\u202f=\u202f9165, p\u202f<\u202f.001, d\u202f=\u202f0.74, and the impartiality dilemma, F\u202f=\u202f5.27, p\u202f=\u202f.023, \u03b7p2\u202f=\u202f0.02; U\u202f=\u202f7397, p\u202f=\u202f.050, d\u202f=\u202f0.29, though the effect was stronger in the sacrificial dilemma.We first looked at character ratings see . For theF\u202f=\u202f23.33, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.05; U\u202f=\u202f31,901, p\u202f<\u202f.001, d\u202f=\u202f0.42, and a significant interaction of dilemma type and agent judgment on how warm and sociable participants perceived the agent to be, F\u202f=\u202f12.59, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.03. While a non-consequentialist was seen as warmer than the consequentialist in the sacrificial dilemma, F\u202f=\u202f32.19, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.12; U\u202f=\u202f8996, p\u202f<\u202f.001, d\u202f=\u202f0.74, there was no difference in the impartiality dilemma, F\u202f=\u202f0.95, p\u202f=\u202f.33, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f6795, p\u202f=\u202f.47, d\u202f=\u202f0.11.For the two items measuring perceived warmth (\u03b1\u202f=\u202f0.77), the ANCOVA revealed the predicted main effect of agent judgment, F\u202f=\u202f1.99, p\u202f=\u202f.16, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f28,134, p\u202f=\u202f.12, d\u202f=\u202f0.13, and no interaction between agent judgment and dilemma type, F\u202f=\u202f1.52, p\u202f=\u202f.22, \u03b7p2\u202f=\u202f0.00. Contrary to predictions, there was no difference in the sacrificial dilemma, F\u202f=\u202f0.00, p\u202f=\u202f.95, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f6473, p\u202f=\u202f.99, d\u202f=\u202f0.01, but the non-consequentialist was seen as more competent in the impartiality dilemma, F\u202f=\u202f4.48, p\u202f=\u202f.035, \u03b7p2\u202f=\u202f0.02; U\u202f=\u202f7488, p\u202f=\u202f.032, d\u202f=\u202f0.27.For the two items measuring perceived competence (\u03b1\u202f=\u202f0.89), the ANCOVA showed no main effect of agent judgment, F\u202f=\u202f122.74, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.21; U\u202f=\u202f39,389, p\u202f<\u202f.001, d\u202f=\u202f1.01, and a significant interaction between agent judgment and dilemma type, F\u202f=\u202f4.78, p\u202f=\u202f.029, \u03b7p2\u202f=\u202f0.01. The non-consequentialist was seen as more loyal in both the sacrificial, F\u202f=\u202f37.28, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.14; U\u202f=\u202f9081, p\u202f<\u202f.001, d\u202f=\u202f0.77, and impartiality dilemmas, F\u202f=\u202f92.72, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.29; U\u202f=\u202f10,403, p\u202f<\u202f.001, d\u202f=\u202f1.24, though the effect was stronger in the impartiality dilemma \u202f=\u202f55.12, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.11; U\u202f=\u202f35,363, p\u202f<\u202f.001, d\u202f=\u202f0.68, and no interaction effect, F\u202f=\u202f1.13, p\u202f=\u202f.29, \u03b7p2\u202f=\u202f0.00. In both dilemmas when the agent made a non-consequentialist judgment they were expected to make a better friend \u202f=\u202f34.59, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.13; U\u202f=\u202f9094, p\u202f<\u202f.001, d\u202f=\u202f0.77; impartiality dilemma: F\u202f=\u202f21.10, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.09; U\u202f=\u202f8494, p\u202f<\u202f.001, d\u202f=\u202f0.58).Next, we looked at perceived suitability for different roles see . For perF\u202f=\u202f48.53, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.10; U\u202f=\u202f34,817, p\u202f<\u202f.001, d\u202f=\u202f0.64, but no significant interaction, F\u202f=\u202f0.02, p\u202f=\u202f.90, \u03b7p2\u202f=\u202f0.00. The non-consequentialist agent was thought to make a better spouse in both the sacrificial, F\u202f=\u202f21.81, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.09; U\u202f=\u202f8589, p\u202f<\u202f.001, d\u202f=\u202f0.60, and the impartiality dilemma, F\u202f=\u202f26.83, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.11; U\u202f=\u202f8706, p\u202f<\u202f.001, d\u202f=\u202f0.67.For perceived suitability as a spouse, an ANCOVA revealed the predicted main effect of agent judgment, F\u202f=\u202f3.58, p\u202f=\u202f.059, \u03b7p2\u202f=\u202f0.01; U\u202f=\u202f27,809, p\u202f=\u202f.17, d\u202f=\u202f0.17, and no interaction of dilemma type and agent judgment, F\u202f=\u202f0.40, p\u202f=\u202f.53, \u03b7p2\u202f=\u202f0.00. Similarly for perceived suitability as a political leader, an ANCOVA revealed no main effect of agent judgment, F\u202f=\u202f1.81, p\u202f=\u202f.018, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f23,441, p\u202f=\u202f.071, d\u202f=\u202f0.14, and no interaction of dilemma type and agent judgment, F\u202f=\u202f0.04, p\u202f=\u202f.84, \u03b7p2\u202f=\u202f0.00.For suitability as a boss, an ANCOVA revealed no main effect of agent judgment, 5.2.4F\u202f=\u202f65.63, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.13; U\u202f=\u202f15,222, p\u202f<\u202f.001, d\u202f=\u202f0.74, and a significant interaction of dilemma type and agent judgment, F\u202f=\u202f41.39, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.08. In the sacrificial dilemma the non-consequentialist was thought as being more driven by emotion and the consequentialist by reason, F\u202f=\u202f118.54, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.34; U\u202f=\u202f2185, p\u202f<\u202f.001, d\u202f=\u202f1.42, but there was no difference in the impartiality dilemma, F\u202f=\u202f1.19, p\u202f=\u202f.28, \u03b7p2\u202f=\u202f0.01; U\u202f=\u202f5724, p\u202f=\u202f.15, d\u202f=\u202f0.15.First looking at perceptions of whether the agents' decision was driven more by emotion or reason on binary scale, we observed a significant main effect of agent judgment, F\u202f=\u202f107.04, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.19; U\u202f=\u202f13,549, p\u202f<\u202f.001, d\u202f=\u202f0.92, and a significant interaction of dilemma type and agent judgment, F\u202f=\u202f74.39, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.14. In the sacrificial dilemma the consequentialist was thought as being more driven by strategic motives than the non-consequentialist, F\u202f=\u202f227.48, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.50; U\u202f=\u202f1235, p\u202f<\u202f.001, d\u202f=\u202f2.04, but there was no difference in the impartiality dilemma, F\u202f=\u202f1.14, p\u202f=\u202f.29, \u03b7p2\u202f=\u202f0.01; U\u202f=\u202f5859, p\u202f=\u202f.23, d\u202f=\u202f0.16 \u202f=\u202f17.96, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.04; U\u202f=\u202f20,562, p\u202f<\u202f.001, d\u202f=\u202f0.40, and a significant interaction of dilemma type and agent judgment, F\u202f=\u202f11.73, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.03. In the sacrificial dilemma there was no difference in how much either agent was thought to be influenced by altruistic motives, F\u202f=\u202f0.33, p\u202f=\u202f.56, \u03b7p2\u202f=\u202f0.00; U\u202f=\u202f6421, p\u202f=\u202f.93, d\u202f=\u202f0.07, but in the impartiality dilemma the consequentialist was thought to be more driven by altruistic motives, F\u202f=\u202f32.43, p\u202f<\u202f.001, \u03b7p2\u202f=\u202f0.13; U\u202f=\u202f3866, p\u202f<\u202f.001, d\u202f=\u202f0.76.Finally, we looked at how much the agent's decision was thought to be influenced by altruistic motives. We observed a significant main effect of agent judgment, 5.3In Study 4 we conducted another pre-registered investigation of perceptions of non-consequentialist and consequentialist agents in both sacrificial and impartiality dilemmas, adding three new questions to assess perceptions of the agent's motives and addressing potential concerns with the manipulation of the agent's judgment.First, we saw that the consequentialist was seen as more influenced by strategic motives in the sacrificial dilemma, but more by altruistic motives in the impartiality dilemma. And while in the sacrificial dilemma the non-consequentialist was thought as being more driven by emotion and the consequentialist by reason, there was no difference in the impartiality dilemmas for perceptions of being driven by reason or emotions.Second, we addressed the potential concern with the previous studies that the consequentialist expressed awareness of conflict for the impartiality, but not sacrificial, dilemma. When ensuring that all agents expressed no recognition of competing moral reasons, we broadly replicated the findings of Study 3, with some minor differences. Like in Study 3, in the impartiality dilemma the non-consequentialist was seen as more competent, more loyal, and expected to make a better friend and spouse. However, while in Study 3 the non-consequentialist was thought to make a better boss, there was no difference in Study 4; and while in Study 3 there was no difference in perceived morality, in Study 4 the non-consequentialist was seen as more moral. In neither study was there a difference in perceptions of warmth or suitability as a political leader. It seems clear, then, that our pattern of results cannot be explained simply by the agent's (lack of) expression of moral conflict: regardless of whether they expressed conflict or not, the non-consequentialist in the impartiality dilemma tended to be favored for direct, interpersonal roles. This mirrors our previous work showing that expressing internal conflict through reported emotional difficulty when making a consequentialist decision reduces, but does not fully eliminate, the preference for a non-consequentialist over a consequentialist . It will6Much work over the last decade has focused on the psychological processes underlying judgments about whether it is moral to sacrifice one innocent person to save a greater number of people. In recent years, befitting the fundamentally social role of moral judgments, researchers have begun to consider the social consequences of these judgments: how are consequentialist individuals who endorse harming for the greater good perceived? Previous work has shown that the judgment a person makes in these sacrificial dilemmas influences how moral, warm, and competent they are perceived to be, and even how much cooperation is extended towards them in economic games e.g. . But whiinstrumental harm\u201d, which can be theoretically and empirically distinguished from impartiality dilemmas that tap the more positive, impartial welfare-maximising dimension  of consequentialist theories and consequentialist tendencies in ordinary people. Because this previous work on perception of consequentialist agents has focused almost exclusively on sacrificial dilemmas, it has remained unknown whether the preference for non-consequentialists over consequentialists operates similarly in impartiality dilemmas in which someone faces the decision to help someone close to them or a greater number of strangers.As outlined in the two-dimensional model of utilitarian psychology , judgmenIn four studies, we investigated perceptions of consequentialist and non-consequentialist decision makers in both sacrificial dilemmas tapping instrumental harm, and impartiality dilemmas tapping impartial beneficence. Pre-registering our analyses and predictions, we included the most comprehensive range of dependent measures used in this literature to date, using different economic games (the Trust Game and the Prisoner's Dilemma); examining the distinct dimensions along which the agent's character could be perceived ; exploring the different processes or motivations perceived to influence the agent's moral decision ; and considering the different social roles and relationships in which the agent would be preferred .6.1In the domain of sacrificial harm, our findings strongly confirm previous work highlighting the cost of being consequentialist. We show in three studies that non-consequentialists were consistently preferred over consequentialists. We argue that this is perfectly explicable on a partner choice account of non-consequentialist moral intuitions . The con6.2What about impartiality dilemmas? As discussed in the introduction, how people are perceived in the domain of impartial beneficence remains largely unknown, because almost all previous work has focused on sacrificial dilemmas. The pattern of results in the impartiality dilemmas was much more nuanced than the unequivocal preference for the non-consequentialist in the sacrificial dilemmas, though simply counting overall results suggest that the non-consequentialist has the edge over the consequentialist in the domain of impartial beneficence too: overall, we saw few cases where the consequentialist was preferred, typically seeing either a null effect or that the non-consequentialist was favored. Most interestingly, our results using the impartiality dilemmas appear to suggest a predictable pattern of when non-consequentialists are preferred \u2013 and when they are not.Impartial consequentialists were consistently disfavored for roles involving a direct interpersonal relationship, even if they were not explicitly rated as being deficient in morality and warmth. In all four studies, the impartial consequentialist was thought to be less loyal and thought to make a worse friend and spouse, even if they were not always explicitly rated as being deficient in morality and warmth. And in Study 3 - but not Study 2 - the impartial consequentialist was also disfavored as a future partner in a Prisoner's Dilemma and was thought to make a worse boss. This makes sense given the theoretical basis we draw on in the introduction: consequentialism's requirement for the impartial maximisation of welfare is often inconsistent with the nature of special relationships like friendship and familial duties that are a fundamental part of common-sense morality . When weWhile it makes sense for non-consequentialists to be favored for direct, interpersonal relationships, it is much more reasonable \u2013 even preferable - to favor a consequentialist for distant, impersonal roles like a political leader, and this what we what found in Studies 1 and 2. The job of an effective political leader can be plausibly described as to make constituents better off, and part of this requires acting impartially to not favor one's own self-interest or the interest of one's immediate family. Indeed, recent work has shown that people do not endorse efficient maximisation in charitable giving unless one is in a position of responsibility, like a political leader . Of courOverall, then, while the pattern of results is weaker than for sacrificial dilemmas, a tentative conclusion can be drawn: there may be some costs of being consequentialist in the domain of impartial beneficence, and this is especially manifested when considering suitability and desirability for direct, personal relationships.6.3harm for the greater good, compared to a non-consequentialist agent who refuses to help for the greater good. There are of course important differences \u2013 both theoretical and psychological - between instrumentally killing someone to achieve the greater good and impartially maximising welfare by privileging strangers over our family  and their preference for non-consequentialists in interpersonal interactions (c.f. the other results) cancelled each other out. When playing a game in which the implicit social contract is stronger (for example through players knowing each other), it is possible that we would see a preference for the non-consequentialist.First, we acknowledge that the anonymous economic game context may not be ideal to study partner preferences in the context of impartial beneficence: the consequentialist agents explicitly indicated they thought it better to impartially maximise welfare even with strangers, and participants were themselves strangers to the agent. And while they seemed to understand that they stood to benefit from the agent's impartial beneficence in an anonymous economic exchange, they Second, even if participants did not cooperate more with either agent in the anonymous economic games, they did consistently rate the non-consequentialist as being a better friend and spouse, in line with previous evidence that non-consequentialists in sacrificial dilemmas are favored as long-term mating partners . Given that most people do tend to be more prosocial towards their close friends and family, if in real life non-consequentialists were preferred as friends and spouses, this would also lead to the same partner choice mechanisms occurring.6.5Finally, our work has practical implications concerning how in everyday life groups and individuals who advocate a more impartial, welfare-maximising consequentialist approach to moral decisions \u2013 such as the \u201ceffective altruism\u201d movement \u2013 might expect to be perceived, and how this might limit their advocacy. Peter Singer is the most influential living utilitarian thinker, and in part because of his utilitarian consequentialist moral views is a leading proponent of effective altruism: a movement built around the idea of using reason and evidence to find the best ways to help others. In particular, effective altruism is concerned with using one's resources to have the most impact in helping others, and many effective altruists have pledged to give at least 10% of their income to cost-effective charities . In prac"}
{"text": "There are at least three errors in the manuscript, they are as follows:Models with time-varying parameters\u201d paragraph after equation (4), for the sentences- In the \u201c\u03b5i(t) describe the clustering of the population  but can also describe a reduction in the population due to voluntary avoidance behavior or social distancing. However due to the absence of structural identifiability properties  it should be very difficult to estimate simultaneously both \u03b2(t) and \u03b5i(t).\u201d\u201c\u03b5i(t)\u201d must be replaced by \u201c\u03b5i(t)\u201d in accordance with the rest of the manuscript. Then these sentences become:\u201c\u03b5i(t) describe the clustering of the population  but can also describe a reduction in the population due to voluntary avoidance behavior or social distancing. However due to the absence of structural identifiability properties  it should be very difficult to estimate simultaneously both \u03b2(t) and \u03b5i(t).\u201d\u201c\u03b2(t) must be read as:- In the Fig 3 caption the equation of \u03b2(t) = \u03b20.(1 + \u03b21 sin(2\u03c0t/365+2\u03c0\u03d5) + \u03b22 sin(2\u03c0t/(3 365)+2\u03c0\u03d5) + \u03b23 sin(2\u03c0t/(0.5 365)+2\u03c0\u03d5))\u201cand not:\u03b2(t) = \u03b20.(1 + \u03b21 sin(2\u03c0t/365+2\u03c0\u03d5) + \u03b22 sin(2\u03c0t/(3 365)+2\u03c0\u03d5)) + \u03b23 sin(2\u03c0t/(0.5 365)+2\u03c0\u03d5))\u201cSIRS model\u201d paragraph after equation (5), the sentence- In the \u201c\u03c3, the reporting rate \u03c1 and the initial conditions \u03b3(0) by \u03b2(0) but to reduce problems linked to practical non-identifiability materialized by correlation between some estimates, informative priors were used for \u03c1 .\u201d\u201cInitially non-informative priors were used for the volatility must be replaced (as in the uncorrected proofs) by:\u03c3, the reporting rate \u03c1 and the initial conditions (S(0), I(0), \u03b2(0)) to reduce problems linked to practical non-identifiability materialized by correlation between some estimates, informative priors were used for \u03c1 .\u201d\u201cInitially non-informative priors were used for the volatility"}
{"text": "Nature Communications 10.1038/ss41467-019-08723-y, published online 15 February 2019.Correction to: \u03c1c\u2009=\u20090.075\u2009m\u22122 within a specified domain31. Four connectors representing the fibers are defined for each cross-link based on a random-weighted sampling process with uniform orientation distribution and a distribution in length resembling the shape of a Poisson distribution with mean Lc\u2009=\u200910\u2009m, if not specified otherwise.\u2019 The correct version states \u2018\u03c1c\u2009=\u20090.075\u2009\u03bcm\u22122\u2019 in place of \u2018\u03c1c\u2009=\u20090.075\u2009m\u22122\u2019 and \u2018Lc\u2009=\u200910\u2009\u03bcm\u2019 rather than \u2018Lc\u2009=\u200910\u2009m\u2019.The original version of this Article contained errors in the third and fourth sentences of the \u2018Computational material models\u2019 section of the \u201cMethods\u201d, which incorrectly read \u2018The model is generated by randomly placing cross-links at a density of \u03c1c\u2009=\u20095\u2009\u00d7\u200910\u22124\u2009m\u22123 within the domain of a representative volume element (RVE) of the membrane. Fibers were then defined by four random connections between these crosslinks, uniformly distributed within the membrane plane and sampled from a distribution27p leading to a von-Mises distributed out-of-plane angle \u03c6, controlled by the concentration parameter \u03b2\u2009=\u20093, and Poisson-like distributed fiber lengths with mean Lc\u2009=\u200910\u2009m.\u2019 The correct version states \u2018\u03c1c\u2009=\u20095\u2009\u00d7\u200910\u22124\u2009\u03bcm\u22123\u2019 instead of \u2018\u03c1c\u2009=\u20095\u2009\u00d7\u200910\u22124\u2009m\u22123\u2019, and \u2018Lc\u2009=\u200910\u2009\u03bcm\u2019 instead of \u2018Lc\u2009=\u200910\u2009m\u2019.The second and third sentences of the fourth paragraph of the same section originally incorrectly read \u2018The network was generated by seeding cross-links at a density This has been corrected in both the PDF and HTML versions of the Article."}
{"text": "This report re-introduces the Ramachandran number (\u211b) as a residue-level structural metric that could simply the life of anyone contending with large numbers of protein backbone conformations . Previously, the Ramachandran number (\u211b) was introduced using a complicated closed form, which made the Ramachandran number difficult to implement. This report discusses a much simpler closed form of \u211b that makes it much easier to calculate, thereby making it easy to implement. Additionally, this report discusses how \u211b dramatically reduces the dimensionality of the protein backbone, thereby making it ideal for simultaneously interrogating large numbers of protein structures. For example, 200 distinct conformations can easily be described in one graphic using \u211b (rather than 200 distinct Ramachandran plots). Finally, a new Python-based backbone analysis tool\u2014B Proteins are a class of biomolecules unparalleled in their functionality . A naturi can be almost completely described by only two dihedral angles: \u03d5i and \u03c8i  it serves as a map for structural \u201ccorrectness\u201d , since mWhile the Ramachandran plot has been useful as a measure of protein backbone conformation, it is not popularly used to assess structural dynamism and transitions . This is because of the two-dimensionality of the plot: describing the behavior of every residue involves tracking its position in two-dimensional  space. For example, a naive description of positions of a peptide in a Ramachandran plot  needs moBeyond Ramachandran plots, tracking changes in a protein trajectory is either overly detailed or overly holistic: an example of an overly detailed study is the tracking of exactly one or a few atoms over time ; an example of a holistic metric is the radius of gyration . With our understanding of protein dynamics undergoing a new renaissance\u2014especially due to intrinsically disordered proteins and allostery\u2014having hypothesis-agnostic yet detailed (residue-level) metrics of protein structure has become even more relevant. But unfortunately, there has been no single compact descriptor of protein structure. This impedes the na\u00efve or hypothesis-free exploration of new trajectories/ensembles.number [\u211b or simply \u211b]\u2014with little loss of information , these pictograms are named multi-angle pictures (MAPs). BackMAP is presently available on GitHub (https://github.com/ranjanmannige/BackMAP).It has recently been shown that the two Ramachandran backbone parameters  may be conveniently combined into a single number\u2014the Ramachandran ormation . In a prormation . This reormation , and furThe Ramachandran number is both an idea and an equation. Conceptually, the Ramachandran number (\u211b) is any closed form that collapses the dihedral angles \u03d5 and \u03c8 into one structurally meaningful number . Mannigemin, \u03d5max) and \u03c8 \u2208 \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2002\u2003\u2003\u2002\u2002\u2002\u2002\u2002\u2002\u2002\u200213\u2003# Finally, creating (but not showing) the graph\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2002\u2002\u2002\u2002\u2002\u2003\u200214\u2003backmap.draw_xyz\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2002\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u2003\u2003\u2003\u2003\u2002\u2002\u2002\u2002\u2003\u2003\u2003\u2002\u2009\u200218\u2003# Now, we display the graph:\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2002\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u2003\u2002\u2002\u2002\u2002\u2003\u2003\u2003\u2003\u2003\u2002\u200219\u2003plt.show # ... one can also use plt.savefig to save to file\u2003\u2002\u2009\u2002\u2003\u200320The code above results in D1) and previous structure (D\u22121) within the trajectory, etc.Additionally, by changing how one assigns values to \u201cX\u201d and \u201cY,\u201d one can easily construct and draw other types of graphs such as time-resolved histograms, per-residue fluctuations when compared to the first .for chain in grouped_data.keys:\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u200311\u2003\u2003models, residues, Rs = grouped_data[chain]\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u200312\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u200313\u2003\u2003'Begin custom code'\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u200314\u2003\u2003X = ; Y=; Z=; # Will set X=model, Y=R, Z=P(R)\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u200315\u2003\u2003# Bundling the three lists into one 2d array\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2009\u200316\u2003\u2003new_data =  np.array))\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u200317\u2003\u2003# Getting all R values, model by model\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u200318\u2003\u2003for m in sorted): # column 0 is the model column\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u200319\u2003\u2003\u2003# Getting all Rs for that model #\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u200320\u2003\u2003\u2003current_rs = new_data[np.where] # column 2 contains R\u2003\u2003\u2003\u2002\u200921\u2003\u2003\u2003# Getting the histogram\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u200322\u2003\u2003\u2003a,b = np.histogram)\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2009\u2009\u2009\u200323\u2003\u2003\u2003max_count = float(np.max(a))\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u200324\u2003\u2003\u2003for i in range(len(a)):\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u200325\u2003\u2003\u2003\u2003X.append(m); Y.append((b[i]+b[i+1])/2.0); Z.append(a[i]/float(np.sum(a)));\u2002\u2003\u2002\u2009\u200226\u2003\u2003'End custom code'\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u200327\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u200228\u2003\u2003# Finally, creating (but not showing) the graph\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u200329\u2003\u2003backmap.draw_xyz\"\u2003\u2003\u2003\u2003\u2002\u2009\u200231\u2003\u2003\u2003\u2003,cmap = 'Greys', ylim=)\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u200332\u2003\u2003plt.yticks)\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2003\u200333\u2003\u2003# Now, we display the graph:\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u200234\u2003\u2003plt.show # ... one can also use plt.savefig to save to file\u2003\u2003\u2003\u2003\u2003\u2003\u2002\u2009\u2002\u2003\u2003\u2003\u2003\u2003\u200335Other types of graphs can be easily created by modifying part three of the code above. For example, the following code creates histograms of The code above results in ackMAP provides two new colormaps: \u201cChirality\u201d , and \u201cSecondaryStructure\u201d . Right twisting backbones are shown in red; left twisting backbones are shown in blue. ackMAP do. In case it is known that the protein backbone accesses non-traditional regions of the Ramachandran plot, a four-color schematic will be needed (see below for more discussions).Aside from the general color maps (cmaps) that exist in matplotlib , BackMAP can be used as a stand alone package by running \u201c> python -m backmap -pdb <pdb_dir_or_file>.\u201d The sectons below describe the expected outputs and how they may be interpreted.BIn particular, each column in As compared to the conformationally stable protein above, protein 2fft is much more flexible. The diverse states occupied by each residue  and 11D Yet, interestingly, s; The Ramachandran number increases in value from the bottom left of the Ramachandran plot to the top right in sweeps that are parallel to the negative sloping diagonal. As discussed in L helix region becomes relatively prominent). On the other hand, prolines are known to form polyproline-II helices  the positively sloped diagonal i.e.,-signed\u201d within the command line implementation or by adding \u201csigned=True\u201d when making backmap.R calls.To accomodate the situation where achiral backbones are expected , an additional Ramachandran number\u2014the s, s easily distinguishes \u03b1D from ppII.As an example of the utility of \u211bs, while useful, would be important in very limited scenarios, as more than 96% of the amino acids in the PDB occupy the upper-left region of the Ramachandran plot (with 3% of \u201crule breakers\u201d contributed mostly by glycines).Note that the signed \u211backMAP) has been provided in an online GitHub repository to promote the utility of \u211b as a universal metric.A simpler Ramachandran number is reported\u2014http://scop.berkeley.edu/downloads/pdbstyle/pdbstyle-sel-gs-bib-40-2.06.tgz, contains 13,760 three-dimensional protein conformations (one domain per conformation) with lower than 40% sequence identity. Secondary structure annotations were assigned using the DSSP algorithm (Statistics about single amino acid conformations and secondary structures (excepting ppII) were derived from the Structural Classification of Proteins or SCOPe website  describe single conformations and the last two  describe ensembles. \u211b-based MAPs were created for each structure The output of this command line implementation were used in D1), and deviation in structure compared to the previous conformation in the trajctory (D\u22121). For any residue r at time t, these equations can be described as follows:In order to describe changes in structure, this report uses two metrics for structural deviation: deviation in present structure when compared to the first conformation in the trajctory  and \u03d5 \u2208  rounds x to the closest integer value, \u03c3 is a scaling factor, discussed below, and \u03bb is the range of an angle in degrees . Effectively, this equation does the following: (1) It divides up the Ramachandran plot into (360\u00b0 \u03c31/\u00b0)2 squares, where \u03c3 is a user-selected scaling factor that is measured in reciprocal degrees  to the integer value (R\u2124) assigned to the divvied-up square that they it exists in. Further discussions on this process are available in the previous publication  or [\u2212\u03c0, \u03c0),\u03c3s indicate higher accuracy, Conformation of this limit is shown numerically in Assuming, a different range ), the Ramachandran number in that frame of reference will be"}
{"text": "AbstractNA In our recent publication , there are errors in Table 1 (Column: GenBank accession numbers). The corrections are as follows:Page 125:MN453513\u201dAccession number of ZMKU AM 01442: \u201cMN453508\u201d should be \u201cMN453514\u201dAccession number of ZMKU AM 01446: \u201cMN453509\u201d should be \u201cMN453515\u201dAccession number of ZMKU AM 01451: \u201cMN453510\u201d should be \u201cMN453516\u201dAccession number of ZMKU AM 01467: \u201cMN453511\u201d should be \u201cMN453517\u201dAccession number of ZMKU AM 01475: \u201cMN453512\u201d should be \u201cMN453518\u201dAccession number of ZMKU AM 01479: \u201cMN453513\u201d should be \u201cMN453519\u201dAccession number of ZMKU AM 01485: \u201cMN453514\u201d should be \u201cMN453520\u201dAccession number of ZMKU AM 01493: \u201cMN453515\u201d should be \u201cPageBreakMN453521\u201dAccession number of ZMKU AM 01498: \u201cMN453516\u201d should be \u201cMN453522\u201dAccession number of ZMKU AM 01503: \u201cMN453517\u201d should be \u201cMN453526\u201dAccession number of ZMKU AM 01516: \u201cMN453518\u201d should be \u201cMN453527\u201dAccession number of ZMKU AM 01520: \u201cMN453519\u201d should be \u201cPage 126:MN453508\u201dAccession number of ZMKU AM 01418: \u201cMN453520\u201d should be \u201cMN453509\u201dAccession number of ZMKU AM 01423: \u201cMN453521\u201d should be \u201cMN453510\u201dAccession number of ZMKU AM 01425: \u201cMN453522\u201d should be \u201cMN453511\u201dAccession number of ZMKU AM 01426: \u201cMN453523\u201d should be \u201cPage 127:MN453512\u201dAccession number of ZMKU AM 01430: \u201cMN453524\u201d should be \u201cMN453523\u201dAccession number of ZMKU AM 01507: \u201cMN453525\u201d should be \u201cMN453524\u201dAccession number of ZMKU AM 01509: \u201cMN453526\u201d should be \u201cMN453525\u201dAccession number of ZMKU AM 01511: \u201cMN453527\u201d should be \u201c"}
{"text": "In Abstract, the term \u201c(DEP)\u201d should be \u201c(PED).\u201d\u00b5m, NS; M12: 312\u2009\u00b1\u200984\u2009\u00b5m, NS; M24: 312\u2009\u00b1\u200912.3, NS)\u201d should be corrected to \u201cChange of CRT did not reach the significant difference at any time point of the study .\u201dIn Results, \u201cChange of CRT did not reach the significant difference at any time point of the study  Refractory to Ranibizumab\u201d , there w"}
{"text": "Sympathetic nerve innervation is required for beigeing in white fatQiang Cao, Jia Jing, Xin Cui, Hang Shi, Bingzhong XuePhysiol Rep, 7 (6), 2019, e14031.https://doi.org/10.14814/phy2.14031In Xue et al. (2019), Figures 2A and 4A has labeling errors.In Figure 2A, group 3 that was originally labeled as \u201ciBAT\u2010C & iWAT\u20106\u2010OHDA,\u201d should be corrected as \u201ciBAT\u20106\u2010OHDA & iWAT\u2010C\u201d.In Figure 4A, group 3 that was originally labeled as \u201ciBAT\u2010C & iWAT\u20106\u2010OHDA,\u201d should be corrected as \u201ciBAT\u20106\u2010OHDA & iWAT\u2010C\u201d.The authors apologize for the errors."}
{"text": "In the Abstract and the Introduction sections, \u201cinfection-free\u201d should be corrected to \u201cuninfected.\u201dR0=(k\u03b4/pu)x0, R0=(k\u03b4/pu)x0 .\u201d\u201cWe can denote the basic reproduction number of the HIV virus for the model (3) as E0 near;\u201d should be corrected to \u201cProof. We consider linear system of the model (3) at E0.\u201dThe sentence \u201cProof. We consider linear system of the model (3) in R0 < 1, pu \u2212 k\u03b4x0 \u2260 0. Therefore, \u03bb=0 is not root of (9). If (8) has pure imaginary root \u03bb=i\u03c9(\u03c9 > 0) for some \u03c4 > 0, substituting it into (8) and separating the real and imaginary parts,\u201d should be corrected to \u201cWhen R0 < 1, pu \u2212 k\u03b4x0 \u2260 0. Therefore, \u03bb=0 is not root of (9). If (9) has pure imaginary root \u03bb=i\u03c9(\u03c9 > 0) for some \u03c4 > 0, substituting it into (9) and separating the real and imaginary parts.\u201dThe sentence \u201cWhen \u03c4kn)( > 0 such that (23) has pure imaginary \u03bb=i\u03c9k,\u201d should be corrected to \u201cThen, we get the corresponding \u03c4kn)( > 0 such that (19) has pure imaginary \u03bb=i\u03c9k.\u201dThe sentence \u201cThen, we get the corresponding b2\u03c92+b32=(\u03c93 \u2212 a2\u03c9)2+(a1\u03c92 \u2212 a3)2\u201d should be corrected to \u201cFrom (22), we obtain b22\u03c92+b32=(\u03c93 \u2212 a2\u03c9)2+(a1\u03c92 \u2212 a3)2.\u201dThe sentence \u201cFrom (19), we obtain Equation (L) should be corrected as follows:In the Local and Global Stability of the Equilibria section, there were the following errors:(iii) In the Simulations and Conclusions section, the sentence \u201cBy direct calculations, we get that (19) has a positive root;\u201d should be corrected to \u201cBy direct calculations, we get that (24) has a positive root.\u201d(iv) The formats of references 2, 14, 25, 26, 27, and 29 were updated.In the article titled \u201cModeling Inhibitory Effect on the Growth of Uninfected T Cells Caused by Infected T Cells: Stability and Hopf Bifurcation\u201d , there wThis has been corrected in place."}
{"text": "The authors are retracting this article  after anFigure\u00a01C, upper panel: areas of the image labelled as \u201cMock\u201d and of that shown as \u201cScramble\u201d are duplicated.Figure\u00a01D: the image labelled as \u201cScramble, 24\u00a0h\u201d and that shown as \u201csiNFAT5, 48\u00a0h\u201d are duplicated.Figure\u00a03A: areas of the image labelled as \u201cInvaded, siCtrl\u201d and that shown as \u201cBefore Scraping, siS100A4\u201d are duplicated."}
{"text": "VaccinationCoverageforSelectedVaccinesandExemptionRatesAmongChildreninKindergarten\u2014UnitedStates,2017\u201318SchoolYear,\u201d on page 1121, an incorrect figure was published. The corrected figure follows.In the report \u201c"}
{"text": "The number given as \u201c2.3%\u201d should have been \u201c2.6%.\u201d This article has been corrected.1In the Original Investigation titled \u201cAntipsychotic Treatment Among Youths With Attention-Deficit/Hyperactivity Disorder,\u201d"}
{"text": "Scientific Reports 10.1038/s41598-018-33381-3, published online 11 October 2018Correction to: This Article contains an incomplete equation in the Methods section, under the subheading \u2018Statistical models\u2019.\u03c2, \u03c4\u03b3 and \u03c4\u03c1. f\u03c2 was further multiplied by \u03c6 and 1\u2009\u2212\u2009\u03c6 to obtain the proportion of variance explained by \u03c5 and v, respectively\u201d.\u201cThe proportion of marginal variance explained by each component of the RR and lethality models was given by:should read:\u03c2, \u03c4\u03b3 and \u03c4\u03c1. f\u03c2 was further multiplied by \u03c6 and 1\u2009\u2212\u2009\u03c6 to obtain the proportion of variance explained by \u03c5 and v, respectively\u201d.\u201cThe proportion of marginal variance explained by each component of the RR and lethality models was given by:"}
{"text": "Figure 5, and we have corrected data in the result sections. First, in the results section , there is an error in the legend box. (Column 1 as \u201cPeriod\u201d), the label \u201c1\u201d should be amended a \u201c1=1989\u201393\u201d. It is not, \u201c1=1994\u201393\u201dThere were three errors in Figure ), there are errors in the legend box (Column 1 as \u201cPeriod\u201d), the label \u201c1\u201d should be amended a \u201c1=1989\u201393\u201d. It is not, \u201c1=1994\u201393\u201d. The column 2 as \u201cNumber\u201d, the number for \u201c1=1989\u201393\u201d, \u201c2=1994\u201398\u201d, \u201c3=1999\u201303\u201d, \u201c4=2004\u201308\u201d, \u201c5=2009\u201313\u201d should be amended as 564, 557, 904, 857, and 817, respectively.Second, in the results section , the Survival curve was duplicated from Figure , it should be changed to the correct figure.Third, in the results section (page 203, Figure 5 is provided below. The authors apologize for these errors.The correct"}
{"text": "Muscle protein synthesis and muscle net balance plateau after moderate protein ingestion in adults. However, it has been suggested that there is no practical limit to the anabolic response of whole-body net balance to dietary protein. Moreover, limited research has addressed the anabolic response to dietary protein in adolescents. The present study determined whether whole-body net balance plateaued in response to increasing protein intakes during post-exercise recovery and whether there were age- and/or sex-related dimorphisms in the anabolic response.\u2212\u20091\u00b7h\u2212\u20091) as crystalline amino acids modeled after egg protein. Steady-state protein kinetics were modeled noninvasively with oral L-[1-13C]phenylalanine. Breath and urine samples were taken at plateau to determine phenylalanine oxidation and flux (estimate of protein breakdown), respectively. Whole-body net balance was determined by the difference between protein synthesis (flux \u2013 oxidation) and protein breakdown. Total amino acid oxidation was estimated from the ratio of urinary urea/creatinine.Thirteen adults  and 14 adolescents  performed ~\u20091\u00a0h variable intensity exercise  prior to ingesting hourly mixed meals that provided a variable amount of protein , indicating an anabolic plateau. Net balance was maximized at ~\u20090.15, 0.12, 0.12, and 0.11\u00a0g protein\u00b7kg\u2212\u20091\u00b7h\u2212\u20091 in M, F, AM, and AF, respectively. When collapsed across age, the y-intercept  was greater (overlapping CI did not contain zero) in adolescents vs. adults. Urea/creatinine excretion increased linearly  across the range of protein intakes. At plateau, net balance was greater (P\u2009<\u20090.05) in AM vs. M.Mixed model biphasic linear regression explained a greater proportion of net balance variance than linear regression (all, Our data suggest there is a practical limit to the anabolic response to protein ingestion within a mixed meal and that higher intakes lead to deamination and oxidation of excess amino acids. Consistent with a need to support lean mass growth, adolescents appear to have greater anabolic sensitivity and a greater capacity to assimilate dietary amino acids than adults. With a target of n\u2009=\u200942 unique intakes, adults were randomly assigned to consume a protein intake from seven pre-defined ranges . However, due to time constraints and scheduling conflicts participants completed a minimum of five and a maximum of eight trials. In addition, based on an interim analysis, some participants were additionally randomized to intakes 0.188\u00a0g/kg/h to better define the breakpoint. With a target of n\u2009=\u200942 unique intakes, adolescents were randomly assigned to consume a protein intake from six pre-defined ranges  with all participants completing six trials. Adult female participants completed trials during the predicted luteal phase, which was defined as the second half of the menstrual cycle. Following an overnight fast, participants consumed a protein-free liquid carbohydrate beverage (1\u00a0g\u00b7kg\u2212\u20091) as a 1:1 ratio of maltodextrin  and sports drink powder  before reporting to the laboratory. The purpose of the beverage was to help replenish overnight fasted losses of liver glycogen and to provide exogenous carbohydrate to fuel the exercise stimulus. Approximately 1\u00a0h after carbohydrate ingestion, participants completed the LIST according to a previously described exercise protocol [Each participant performed 5\u20138 metabolic trials consisting of two components: a modified version of the Loughborough Intermittent Shuttle Test (LIST) and a subsequent 8-h postprandial period with a variable protein intake  and fat (0.08\u20130.18\u00a0g\u00b7kg\u2212\u20091\u00b7h\u2212\u20091) with 0.42\u00a0g\u00b7kg\u2212\u20091\u00b7h\u2212\u20091 of carbohydrate per meal. Protein was provided as crystalline amino acids  modeled after the composition of egg protein [\u2212\u20091 Na13CO2 and 1.86\u00a0mg\u00b7kg\u2212\u20091\u00a0L-[1-13C]phenylalanine  [\u2212\u20091\u00b7h\u2212\u20091\u00a0L-[1-13C]phenylalanine to maintain isotopic steady state and a constant oral infusion of phenylalanine. Excess tyrosine (3.33\u00a0mg\u00b7kg\u2212\u20091\u00b7h\u2212\u20091) was provided to ensure metabolic partitioning of the phenylalanine carboxyl group into either synthesis or oxidation [13C]phenylalanine has been previously shown to produce the isotopic steady-state condition required for amino acid flux and oxidation determination within an 8-h postprandial period phenylalanine enrichment by liquid chromatography-tandem mass spectrometry, respectively, as previously described phenylalanine produced isotopic steady states in urine, plasma, and breath CO2 similar to steady states produced by intravenous phenylalanine tracer models [13C]phenylalanine tracer yields similar enrichments of [13C] phenylalanine in urine as compared to plasma, which subsequently obviates the necessity of concomitant blood sampling [13C]phenylalanine is an acceptable oral tracer in studies of amino acid flux and oxidation using urinary (rather than plasma) tracer enrichment, which makes it ideally suited for use in vulnerable populations such as children and adolescents.Oral amino acid tracers have been used to model whole-body protein metabolism noninvasively in adults  and adolr models , 33. Fursampling . Thus, t13C]phenylalanine in urine (as an estimate of plasma enrichment) [13CO2 breath enrichment was calculated using standard equations developed by Matthews et al. [\u2212\u20091\u00b7h\u2212\u20091) was measured from L-[1-13C]phenylalanine tracer dilution in the urinary pool at isotopic steady state and calculated as follows:The isotopic enrichment of L-[1-ichment)  and 13COs et al.  and Roses et al. . Phenyla13C] phenylalanine ingested (\u03bcmol\u00b7kg\u2212\u20091\u00b7h\u2212\u20091); I is the rate of L-phenylalanine ingested (\u03bcmol\u00b7kg\u2212\u20091\u00b7h\u2212\u20091); Ei and Eu are the isotopic enrichments as mole fractions of the test drink and urinary phenylalanine, respectively, at isotopic plateau.Where i is the rate of L-phenylalanine enrichment as follows:The rate of phenylalanine oxidation  was calculated as the difference between protein synthesis and protein breakdown.Protein breakdown was estimated by the rate of appearance of phenylalanine in the urine and protein synthesis was calculated as the difference between phenylalanine flux and oxidation (see above) . Whole-br2 of the linear regression mixed model (described below) was compared to the r2 of the biphasic linear regression mixed model (described below), with the highest r2 identifying the preferred model. The linear mixed model used participants as a random variable using PROC MIXED  to determine the r2. If the data conformed to a biphasic model, breakpoint analysis of the net balance data using a biphasic linear regression mixed model  was performed in agreement with previous studies [To determine whether the relationship between protein intake and whole-body net balance was better explained by linear or biphasic regression, the  studies , 36, 37 whereby the null hypothesis was rejected if the interval did not contain zero . To estaP\u2009\u2264\u20090.05.whereby the null hypothesis was rejected if the interval did not contain zero. Linear correlation  was used to identify a relationship between urea/creatinine excretion and protein intake. The whole-body net balance breakpoint was identified for each analysis and a two-way ANOVA  was used to identify main effects of age, sex, and age \u00d7 sex interactions for the mean net balance values above the breakpoint to assess between-group differences. Where significant interactions were identified, a Bonferroni corrected t-test  was used to compare net balance means above the breakpoint. Significance was set at \u2212\u20091\u00b7h\u2212\u20091) and adolescents (0.12\u2009\u00b1\u20090.01\u00a0g\u00b7kg\u2212\u20091\u00b7h\u2212\u20091). The y-intercept  was greater  in AM (1.2\u2009\u00b1\u20090.06\u00a0mg\u00b7kg\u2212\u20091\u00b7h\u2212\u20091) when compared to M (0.46\u2009\u00b1\u20090.09\u00a0mg\u00b7kg\u2212\u20091\u00b7h\u2212\u20091), F (0.35\u2009\u00b1\u20090.07\u00a0mg\u00b7kg\u2212\u20091\u00b7h\u2212\u20091), and AF (0.67\u2009\u00b1\u20090.05\u00a0mg\u00b7kg\u2212\u20091\u00b7h\u2212\u20091). When collapsed across age, adolescents (0.87\u2009\u00b1\u20090.04\u00a0mg\u00b7kg\u2212\u20091\u00b7h\u2212\u20091) had a greater (overlapping CI did not contain zero) y-intercept vs. adults (0.44\u2009\u00b1\u20090.07\u00a0mg\u00b7kg\u2212\u20091\u00b7h\u2212\u20091).Mixed model biphasic linear regression explained a greater proportion of net balance variance than linear regression Table\u00a0, indicat\u2212\u20091\u00b7h\u2212\u20091 in M, F, AM, and AF, respectively, with no differences  in the breakpoint intake. When collapsed across age, the breakpoint protein intake was not different (overlapping CI contained zero) between adults (0.16\u2009\u00b1\u20090.03\u00a0g\u00b7kgFFM\u2212\u20091\u00b7h\u2212\u20091) and adolescents (0.13\u2009\u00b1\u20090.02\u00a0g\u00b7kgFFM\u2212\u20091\u00b7h\u2212\u20091). The y-intercept was greater  in AM (1.3\u2009\u00b1\u20090.08\u00a0mg\u00b7kgFFM\u2212\u20091\u00b7h\u2212\u20091) when compared to M (0.54\u2009\u00b1\u20090.1\u00a0mg\u00b7kgFFM\u2212\u20091\u00b7h\u2212\u20091), F (0.33\u2009\u00b1\u20090.06\u00a0mg\u00b7kgFFM\u2212\u20091\u00b7h\u2212\u20091), and AF (0.77\u2009\u00b1\u20090.04\u00a0mg\u00b7kgFFM\u2212\u20091\u00b7h\u2212\u20091). When collapsed across age, adolescents (1.1\u2009\u00b1\u20090.05\u00a0mg\u00b7kgFFM\u2212\u20091\u00b7h\u2212\u20091) had greater (overlapping CI did not contain zero) y-intercept vs. adults (0.45\u2009\u00b1\u20090.08\u00a0mg\u00b7kgFFM\u2212\u20091\u00b7h\u2212\u20091).When normalized to FFM, whole-body net balance increased up to a plateau in all groups at a protein intake corresponding to 0.17\u2009\u00b1\u20090.05, 0.14\u2009\u00b1\u20090.04, 0.14\u2009\u00b1\u20090.04, and 0.12\u2009\u00b1\u20090.02\u00a0g\u00b7kgFFMr\u2009\u2265\u20090.76; P\u2009<\u20090.01) with protein intake in all groups  increased linearly  and an age \u00d7 sex interaction (P\u2009=\u20090.04) for whole-body net balance at plateau  had a greater (P\u2009<\u20090.05) net balance at plateau than M (~\u20091.9\u00a0mg\u00b7kg\u2212\u20091\u00b7h\u2212\u20091) with no further differences (P\u2009>\u20090.05) between groups. When normalized to FFM, there was a main effect age (P\u2009=\u20090.04), sex (P\u2009=\u20090.02), and an age \u00d7 sex interaction (P\u2009<\u20090.01) for whole-body net balance at plateau  had a greater (P\u2009<\u20090.05) net balance at plateau than M (~\u20092.2\u00a0mg\u00b7kgFFM\u2212\u20091\u00b7h\u2212\u20091) with no further differences (P\u2009>\u20090.05) between groups.There was a main effect of age  alone [\u2212\u20091 in isolation [\u2212\u20091 [\u2212\u20091 [\u2212\u20091\u00b7h\u2212\u20091, respectively, and would fall within the intake ranges (0.02\u20130.25\u00a0g\u00b7kg\u2212\u20091\u00b7h\u2212\u20091) in the present study. With such a limited number of protein intakes, which in our hands would represent single intakes below and above the breakpoint, respectively, we argue that it is challenging to draw clear inferences as to the potential saturation of whole-body net balance. In contrast, the repeated design of the present IAAO study permitted us to investigate a greater dynamic range of protein intakes that resulted in a clear breakpoint in whole-body net balance in all groups studied. Given that our adult males plateaued after the ingestion of ~\u20090.15\u00a0g\u00b7kg\u2212\u20091\u00b7h\u2212\u20091, it is possible that had others [\u2212\u20091 bolus or 0.23\u00a0g\u00b7kg\u2212\u20091\u00b7h\u2212\u20091) a similar plateau in whole-body anabolism would have been apparent. Alternatively, bolus protein ingestion more robustly stimulates whole-body protein synthesis [The post-exercise consumption of dietary protein is important for recovery and growth  as it would help replenish exercise-induced amino acid oxidative losses  and prov1) alone \u201312. This1) alone . This thsolation  or when solation , whereasion [\u2212\u20091 , 11 and \u2212\u20091 [\u2212\u20091  when con\u2212\u20091 [\u2212\u20091 , 11 and \u2212\u20091 [\u2212\u20091 . Metabol\u2212\u20091 [\u2212\u20091 \u201346 would\u2212\u20091 [\u2212\u20091 , 48 suchynthesis , 50, whiynthesis , and whoynthesis  are suppynthesis , we beliynthesis , would pIt has been suggested that the level at which protein intake becomes excessive is reflected by an increase in amino acid oxidation with graded protein intakes . When di\u2212\u20091 (~\u200996\u00a0mg\u00b7h\u2212\u20091) in females and 3.8\u00a0g\u00b7d\u2212\u20091 (~\u2009150\u00a0mg\u00b7h\u2212\u20091) in males during this rapid growth period [During the pubertal growth spurt, a healthy normally developing child has a ~\u20093-fold increase in growth velocity compared to the pre-pubertal period (<\u200910 y) , wherebyh period . Moreoveh period  and may h period , which sh period . Given th period , we specUnlike resistance exercise, variable intensity exercise is generally not associated with marked growth or muscle hypertrophy in weight-stable adults . In contWe report that whole-body net balance plateaued in response to increasing protein intakes after a bout of variable intensity exercise, which is in agreement with our hypothesis and suggests that the anabolic response to post-exercise protein ingestion has a practical limit in both adults and adolescents. Importantly, we did not observe any statistical differences in the breakpoint between sexes within an age group, suggesting that postprandial protein recommendations to maximize whole-body net balance after exercise are primarily influenced by total body and fat-free mass. In further agreement with our hypothesis, the greater whole-body net balance at very low and optimal protein intakes in adolescents compared to adults highlight that active youth during the pubertal growth spurt have both a greater anabolic sensitivity and anabolic potential to mixed meal dietary protein ingestion during recovery from exercise."}
{"text": "Nature Communications 10.1038/s41467-019-11595-x, published online 9 August 2019.Correction to: aAssuming a tangential velocity of 150\u2013300\u2009km\u22121 from the proper motions of ref. 22\u2019 The correct version states \u2018150\u2013300\u2009km\u2009s\u22121\u2019 in place of \u2018150\u2013300\u2009km\u22121\u2019.The original version of this Article contained an error in last sentence of the legend to Table 4, which incorrectly read \u2018\u22126\u2009M\u2299\u2009y\u22121)\u2019, \u2018\u22125\u2009M\u2299\u2009y\u22121)\u2019 and \u2018\u22124\u2009M\u2299\u2009y\u22121\u2009km\u22121)\u2019, instead of the correct \u2018\u22126\u2009M\u2299\u2009yr\u22121)\u2019, \u2018\u22125\u2009M\u2299\u2009yr\u22121)\u2019 and \u2018\u22124\u2009M\u2299\u2009yr\u22121\u2009km\u2009s\u22121)\u2019, respectively.The original version also contained errors in Table 5, in which the headings of the fifth, sixth and seventh rows incorrectly read, \u2018aAssuming a tangential velocity of 150\u2013300\u2009km\u22121 from the proper motions of ref. 22\u2019 The correct version states \u2018150\u2013300\u2009km\u2009s\u22121\u2019 in place of \u2018150\u2013300\u2009km\u22121\u2019.The last sentence of the legend to Table 5 originally incorrectly read \u2018S5.8\u2009=\u20090.794\u2009\u00b1\u20090.03\u2009mJy, \u03bd\u2009=\u20095.8\u2009GHz, vj\u2009=\u2009600\u2009\u00b1\u2009100 km\u22121, \u03b80\u2009=\u200952.3\u00b0\u2009\u00b1\u20094.4\u00b0, D\u2009=\u20092.2\u2009kpc, T\u2009=\u200910,000\u2009K, the resulting ionised mass-loss rate is 1.81\u2009\u00b1\u20090.33\u2009\u00d7\u200910\u22126\u2009M\u2299\u2009yr\u22121, consistent with ref. 22.\u2019 The correct version states \u2018vj\u2009=\u2009600\u2009\u00b1\u2009100\u2009km\u2009s\u22121\u2019 in place of \u2018vj\u2009=\u2009600\u2009\u00b1\u2009100\u2009km\u22121\u2019.The original version also contained an error in the third sentence of the first paragraph of the \u2018Ionised mass-loss rate on source\u2019 section of the Methods, which incorrectly read \u2018From our observations we obtain the following parameters for core B: This has been corrected in both the PDF and HTML versions of the Article."}
{"text": "The \u201cuniversality\u201d refers to the fact that this result is model\u2010free, that is, not dependent on an underlying stochastic process. We extend Cover's theorem to the setting of stochastic portfolio theory: the market portfolio is taken as the num\u00e9raire, and the rebalancing rule need not be constant anymore but may depend on the current state of the stock market. By fixing a stochastic model of the stock market this model\u2010free result is complemented by a comparison with the num\u00e9raire portfolio. Roughly speaking, under appropriate assumptions the asymptotic growth rate coincides for the three approaches mentioned in the title of this paper. We present results in both discrete and continuous time.Cover's celebrated theorem states that the long\u2010run yield of a properly chosen \u201cuniversal\u201d portfolio is almost as good as that of the best  This enables us to analyze strategies which depend on the market weights, and the performance of relative wealth with respect to the market portfolio.In this paper, we work under the setting of SPT. Namely, the market portfolio is taken as the benchmark, or \u201cnum\u00e9raire,\u201d so that the primary assets are the market weights which take values in the open 1.1.1M\u2010Lipschitz portfolio maps denoted by LM. Each element of LM maps the market weights to long\u2010only portfolio weights in \u0394\u00afd t=0\u221e the relative wealth process corresponding to a portfolio strategy(\u03c0t)t=1\u221e, we are interested in comparing the asymptotic growth ratesbest retrospectively chosen portfolio at time T in the class L:=\u22c3M=1\u221eLM ;the analog of Cover's universal portfolio whose relative wealth process (Vt(\u03bd))t=0\u221e is defined in L with full support on each LM);the log\u2010optimal portfolio among the class of long\u2010only strategies, whose relative wealth process is denoted by (V^t)t=0\u221e.the Denoting by Theorem 1.1(\u03bct)t=0\u221e be a time\u2010homogenous ergodic Markov process in discrete time describing the dynamics of the market weights. ThenLet The first two portfolios can be compared in a model\u2010free way  is asymptotically as good as the best one chosen with hindsight, and the log\u2010optimal portfolio constructed with full knowledge of the underlying process.Intuitively, this theorem says that a suitable 1.1.2functionally generated portfolios  is correctly specified, model parameters cannot be estimated precisely and always come with a confidence interval. So, in practice, the As for the original theorems of Cover and Jamshidian, a valid criticism is of course that we only establish asymptotic equality on a first\u2010order log\u2010return basis. As such, a lot of important information is lost in the limit. However, one cannot expect to obtain any information on higher order terms unless further quantitative assumptions are made on the considered models. Cover's aim and also the goal of the present paper is to be as model\u2010free as possible.The remainder of the paper is organized as follows. In Section\u00a02For expositional simplicity, time is discrete in this section.2.1Cover's insight reveals that the \u201cwisdom of hindsight\u201d does not give significant advantages over a properly chosen \u201cuniversal\u201d portfolio constructed using only historical and current prices of the assets. The relevant optimality criterion here is the asymptotic growth rate of the portfolio.T\u2208N and think of an investor who at time T looks back which stock she should have bought at time t=0 . There is an obvious solution: pick i\u2208{1,\u22ef,d} which maximizes the normalized logarithmic returnt=0 instead of t=T. Here is the remedy T which tends to zero as T\u2192\u221e. Hence this buy\u2010and\u2010hold portfolio, which corresponds to a universal portfolio in the sense of Cover, has asymptotically the same normalized logarithmic return as the\u2014only retrospectively known\u2014best performing stock.Let us sketch this\u2014at first glance surprising\u2014result in a particularly easy setting \u2208\u0394\u00afd, that is, bj\u22650 and \u2211j=1dbj=1. The value of the corresponding constant rebalanced portfolio (Vt(b))t=0\u221e starting at V0(b)=1 is defined by holding throughout the proportion bj of the current wealth in stock j, so that V0(b)=1 ands=((stj)j=1d)t=0\u221e\u2282d of the stocks.Instead of these \u201cpure\u201d investments, Cover considered a more ambitious setting, namely, all T and define the quantity VT\u2217 bys=t=0T. Again, the idea is that, with hindsight, that is, knowing t=0T, one considers the best weight b\u2208\u0394\u00afd which attains the maximum (VT\u2217)T=0\u221e as T\u2192\u221e, uniformly over all price paths. For this reason, the portfolio is said to be universal. In order to do so, let \u03bd be a probability measure on \u0394\u00afd which replaces the previous uniform distribution over the d stocks. The universal portfolio is built by investing at time 0 the portion d\u03bd(b) of initial capital in the constant rebalanced portfolio V(b) and by subsequently following the constant rebalanced portfolio process (Vt(b))t=0T. The explicit formula for the wealth isVt(b) is defined by universal portfolio is given by the wealth\u2010weighted averageFix again Theorem 2.1\u0394\u00afd with full support. Thenall trajectories s=t=0\u221e for which there are constants 0<c\u2264C<\u221e such thatCover, : Let \u03bd bLet us now recall Cover's celebrated result:Remark 2.2 distribution on \u0394d  be the set of probability measures on \u0394\u00afd. For each \u03bc\u2208M1(\u0394\u00afd), consider the value \u222b\u0394\u00afdVT(b)(s)d\u03bc(b) of the mixture portfolio with initial measure \u03bc. Note that the constant rebalanced portfolio VT(b) corresponds to the case where \u03bc is the point mass at b. It is easy to see thatVT\u2217(s) is defined by Let 2.2 denote the market capitalizations of the stocks rather than their prices. Then we define the vector of market weights\u2208\u0394d bymarket portfolio  as the num\u00e9raire t=0\u221e, expressed in units of the market portfolio and starting at V0=1, is obtained by the following recursive relation:(\u03c0t)t=1\u221e, where the portfolio weight \u03c0t is used over the time interval . In this paper, all trading strategies are fully invested in the equity market, that is, the portfolio weights sum to 1 for all t. In particular, the strategies do not lend or borrow money. Henceforth, all wealth processes are measured in units of the market portfolio.The portfolio maps. These are (Borel) measurable functions\u03bct= the weights (\u03c0(\u03bct)=(\u03c01(\u03bct),\u22ef,\u03c0d(\u03bct)) according to which an agent distributes current wealth among the d stocks at time t. The constant rebalanced portfolio strategies considered by Cover correspond to the constant functions \u03c0:\u0394d\u2192\u0394\u00afd.We will focus on trading strategies defined by (deterministic) constant rebalanced portfolios to certain families of portfolio maps. First, we note that Cover's and Jamshidian's definition of a universal portfolio as in G denote some appropriate space of portfolio maps, B(G) its Borel \u03c3\u2010algebra and \u03bd some probability measure on G.Definition 2.4). Then, the corresponding universal portfolio at time t is given by the wealth\u2010weighted averageLet \u03bd be a probability measure on In this paper, we extend Cover's theory of \u03c0\u03bd is given byFrom 2.3S=t=0\u221e and the corresponding relative market capitalizations \u03bc=t=0\u221e are now assumed to be stochastic processes defined on a filtered probability space t=0\u221e,P).To define the log\u2010optimal portfolio, we consider a probabilistic setting. The stock price process T, this portfolio is by definition the maximizer of the expected logarithmic growth rate(\u03c0t)t=1T. Under mild assumptions on the process a unique optimizer exists; see, for example, Becherer  has the form \u03c0t=\u03c0(\u03bct\u22121), where \u03c0:\u0394d\u21a6\u0394\u00afd as in \u03c0^.To connect the log\u2010optimal portfolio with universal portfolios in the sense of Definition\u00a0The Markovian assumption can be motivated by the stability of capital distributions of equity markets t=0\u221e with values in \u0394d. We consider as far as possible a model\u2010free approach, but will introduce a probabilistic setting when the log\u2010optimal portfolio is involved.Throughout this section, we work in discrete time and assume that the market weights are described by a 3.1We start by defining rigorously, in the present setting, the three portfolios introduced in Sections 3.1.1T a strategy which is optimal within a certain class of strategies, in our case portfolio maps \u03c0:\u0394d\u2192\u0394\u00afd. A moment's reflection reveals that it does not make sense to allow to choose among all measurable functions \u03c0:\u0394d\u2192\u0394\u00afd. Indeed, there is no restriction to choose \u03c0 such that \u03c0(\u03bct)=ej(t), where j(t)\u2208{1,\u22ef,d} maximizes \u03bct+1j/\u03bctj. This is asking for too much clairvoyance and does not allow for meaningful results \u2212\u03c0(y)\u22251\u2264L\u2225x\u2212y\u22251, x,y\u2208\u0394d.For However, it does make sense  to restrict to more regular trading strategies. In particular, we work with the following set of Remark 3.2LM of M\u2010Lipschitz functions \u03c0:\u0394d\u2192\u0394\u00afM\u22121d is a compact metric space with respect to the topology of uniform convergence induced by the norm \u2225\u03c0\u2225\u221e=sup{\u2225\u03c0(x)\u22251:x\u2208\u0394d}.The set Remark 3.3Instead of Lipschitz functions we could just as well consider other compact function spaces, for example, H\u00f6lder spaces equipped with a proper norm. This is done in the context of functionally generated portfolios in Section\u00a0Definition 3.4(\u03bct)t=0T\u2208(\u0394d)T+1, we defineFor a given trajectory The retrospectively chosen best performing portfolio among the above Lipschitz maps is defined as follows:\u03c0\u21a6VT\u03c0, there exists an optimizer \u03c0\u2217,M\u2208LM (not necessarily unique) such that VT\u2217,M=VT\u03c0\u2217,M, thus the sup above can be replaced by max\u2009.By compactness (see Remark\u00a03.1.2predictable process \u03c0M=(\u03c0tM)t=1\u221e, that is, one which depends only on the history of the market weights, such that the performance of (Vt\u03c0M)t=0\u221e is asymptotically as good as that of t=0\u221e. This can be achieved by the universal portfolio introduced in Definition\u00a0G is now LM as in Definition\u00a0LM is a compact metric space, we may find a (Borel) probability measure \u03bd on  with full support; this will be essential for establishing an analog to Theorem\u00a0Our aim is to find a 3.1.3\u03bc=(\u03bct)t=0\u221e is a time\u2010homogeneous Markov process  log\u2010optimal portfolio, we assume that )x\u2208\u0394d, that is, for all Borel sets A\u2286\u0394\u00afd we have P[\u03bct+1\u2208A|Ft]=\u03f1. For further notions concerning ergodic Markov processes, we refer to Eberle  of \u03bct+1. We therefore choose \u03c0^(x)\u2208\u0394\u00afd as the maximizer\u03c0^(\u00b7) can be chosen to be measurable . For x\u2208\u0394d, define the number L(x) as the value of the optimization problem \u03c0(x)=x (which corresponds to the market portfolio) we clearly have L(x)\u22650 for each x\u2208\u0394d. We obtain the a.s.\u00a0relationV^=(V^t)t=0\u221e denotes the long\u2010only log\u2010optimal wealth process V\u03c0^ defined by the portfolio map \u03c0^ via Assumption 3.6Using the above notation we assume thatThe Theorem 3.7\u03bc0\u2208\u0394d,L1.Under Assumptions \u03c0:\u0394d\u2192\u0394\u00afd be any measurable portfolio map such that0, thatL1.More generally, let Applying Birkhoff's ergodic theorem for discrete time Markov processes t=0\u221e in \u0394d, we haveFix We are now ready to compare the asymptotic performance of the three approaches. We first establish an analog of Theorem\u00a0LM is compact and \u03bd has full support, it is not difficult to see that for any \u03b7>0, there exists \u03b4>0 such that every \u03b7\u2010neighborhood of a point \u03c0\u2208LM has \u03bd\u2010measure bigger than \u03b4.The inequality \u201c\u2265\u201d is obvious. For the reverse inequality, we follow the argument of Blum and Kalai . As LM i(\u03bct)t=0\u221e in \u0394d be given. For a fixed time T, let \u03c0\u2217,M\u2208LM be an optimizer of \u03c0M\u2208LM with \u2225\u03c0M\u2212\u03c0\u2217,M\u2225\u221e<\u03b7, that is, such that, for every x\u2208\u0394d we have \u2225\u03c0M(x)\u2212\u03c0\u2217,M(x)\u22251=\u2211j=1d|\u03c0M(x)j\u2212\u03c0\u2217,M(x)j|<\u03b7.Let a trajectory \u03b7>0 small enough so that \u03b1=\u03b7Md<1 and define, for x\u2208\u0394d,\u03c0\u223c maps \u0394d into \u0394\u00afd.Choose Using \u03b5>0. Choosing \u03b7>0 sufficiently small we can make \u03b1=\u03b7Md small enough such that the final term is bigger than \u2212\u03b5. Summing up, we have\u2225\u03c0M\u2212\u03c0\u2217,M\u2225\u221e<\u03b7.Fix B=B\u03b7 the \u2225\u00b7\u2225\u221e\u2010ball with radius \u03b7 in LM which has \u03bd\u2010measure at least \u03b4>0, where \u03b4 only depends on \u03b7. As each element \u03c0M of B satisfies T to infinity and letting \u03b5 to zero. \u25a1Denote by LM for a fixed M. In Corollary\u00a0L=\u22c3MLM.Theorem 3.10\u03a9=(\u0394d)N be the canonical path space equipped with its natural filtration and a probability measure P. Define \u03bc=(\u03bct)t=0\u221e to be the canonical process, that is, \u03bct(\u03c9)=\u03c9t, which takes values in \u0394d and satisfies Assumptions M>0 be a fixed Lipschitz constant for the space LM. Consider the following objects that are defined for each trajectory (\u03bct)t=0\u221e:(i)T\u2208N the portfolio map \u03c0\u2217,M\u2208LM as well as the corresponding wealth VT\u2217,M:=VT\u03c0\u2217,M as in Define for each (ii)LM with full support and consider the wealth process of the universal portfolio (VtM(\u03bd))t=0\u221e as of Fix a probability measure \u03bd on (iii)\u03c0\u2208LM by(V^tM)t=0\u221e=(Vt\u03c0^M)t=0\u221e via Define the log\u2010optimal portfolio among the portfolio maps Then, we have P\u2010a.s.L\u03c0 is given in all trajectories (\u03bct)t=0\u221e in \u0394d.Let Note that in Theorem\u00a0\u03c0^M is well\u2010defined; simply use the compactness of LM with respect to \u2225\u00b7\u2225\u221e t=0\u221e in \u0394d was shown in Theorem\u00a0That the first equality in T\u2208N, we obviously haveP\u2010a.s.(V^tM)t=0\u221e as the log\u2010optimizer within the class LM, we haveLM.For each fixed 1Tlog(VTM(\u03bd)) is bounded from below, see, e.g., lim infT\u2192\u221e1Tlog(VTM(\u03bd)) is P\u2010a.s.\u00a0constant and equal to limT\u2192\u221e1Tlog(V^TM). This completes the proof of the theorem. \u25a1Combining now M to infinity in the following way. For M=1,2,3,\u22ef choose a measure \u03bdM on LM with full support. Define \u03bd=\u2211M=1\u221e2\u2212M\u03bdM and the wealth of the universal portfolio V(\u03bd) as in L=\u22c3M=1\u221eLM. Recall that (V^t)t=0\u221e is the wealth process of the (long\u2010only) log\u2010optimal portfolio Corollary 3.11P\u2010a.s.L is defined in Under the assumptions of Theorem\u00a0Next we will send M\u2192\u221e in VT(\u03bd)\u22652\u2212MVTM(\u03bdM) for every M, so we have by Theorem\u00a0MLetting P\u2010a.s. As by Lemma\u00a0(Vt(\u03bd)V^t)t=0\u221e is a nonnegative supermartingale, it converges P\u2010a.s. to a finite limit as t\u2192\u221e. This in turn implies \u25a1Now the corollary is proved ifLemma 3.12(Vt(\u03bd)V^t)t=0\u221e is a nonnegative supermartingale.The process \u03c0:\u0394d\u2192\u0394\u00afd, (Vt\u03c0V^t)t=0\u221e is a nonnegative supermartingale. Indeed, by Lemma\u00a0(Vt(\u03bd)V^t)t=0\u221e,\u25a1First note that for any Lemma 3.13\u03c0^ be given by \u03c0:\u0394d\u2192\u0394\u00afd and every x\u2208\u0394d,Let Here we establish the supermartingale property used in the previous proof.\u03b1\u2208 and define \u03c0\u03b1=\u03b1\u03c0+(1\u2212\u03b1)\u03c0^. Then by the (long only) log\u2010optimality of \u03c0^ we have for every x\u2208\u0394d\u03c0\u03b1\u2265(1\u2212\u03b1)\u03c0^. By Fatou's lemma, we therefore have\u25a1We proceed as in the proof of Becherer  denotes the H\u00f6lder space of 2\u2010times continuously differentiable functions from \u0394\u00afd\u2192R whose derivatives are \u03b1\u2010H\u00f6lder continuous. That is,k denoting a multi\u2010index in N2. For \u03b1=0, the second term in this norm is left away. Note that G is only defined on the simplex \u0394d. In order that the partial derivatives are well\u2010defined, we assume that each G is extended to an open neighborhood of \u0394d such that G(x)=G(x\u2032), where x\u2032 is the orthogonal projection of x onto \u0394d. The choice of the extension is irrelevant.We consider the following set of concave functions. For some fixed Lemma 4.1M,\u03b1>0, the set GM,\u03b1 is compact with respect to \u2225\u00b7\u2225C2,0.For any Here is an analytical lemma whose proof is given in the appendix.GM,\u03b1, we associate now the set of functionally generated portfolios FGM,\u03b1 in the spirit of Fernholz n=1\u221e of  along (Tn) in the sense of F\u00f6llmer  and \u03c0G be defined as in (\u03bct)t\u22650 be a continuous path satisfying Assumption V\u03c0G satisfiesg(dt)=\u221212G(\u03bct)\u2211i,jDijG(\u03bct)dt.Let Using 4.2We again consider (i) the best retrospectively chosen portfolio, (ii) the universal portfolio, and (iii) the log\u2010optimal portfolio. To define the log\u2010optimal portfolio, we will restrict to a specific stochastic model introduced in Section\u00a04.2.1FGM,\u03b1 and a given continuous path (\u03bct)t\u22650 satisfying Assumption M,\u03b1>0 fixed, we defineWe consider the set of functionally generated portfolios Lemma 4.4T,M,\u03b1>0 be fixed and (\u03bct)t\u22650 be a continuous path satisfying Assumption G\u21a6VTG where VTG is given by G\u21a6VTG is continuous from  to R.Let We first prove that an optimizer exists by establishing the following continuity property whose proof can be found in the appendix.Proposition 4.5T be fixed and (\u03bct)t\u22650 be a continuous path satisfying Assumption VT\u2217,M,\u03b1 be defined by GT\u2217\u2208GM,\u03b1and in turn a portfolio \u03c0T\u2217 generated by GT\u2217 such thatLet  as shown in Lemma\u00a0\u25a1This is simply a consequence of continuity as proved in Lemma\u00a04.2.2m be a Borel probability measure on . Consider the map\u03c0G is given by  a Borel probability measure \u03bd via the pushforward \u03bd=F\u2217m. As in Definition\u00a0Remark 4.6More precisely, we need to verify that the universal portfolio still allows for pathwise integration and that the value of the portfolio  is given by the right\u2010hand side of To define the analog of Cover's/Jamshidian's portfolio in the present setting, let 4.2.3\u03bc=t\u22650 follows a time\u2010homogeneous Markovian It\u00f4 diffusion, defined on t\u22650,P) with values in \u0394d, given by\u00b7 denotes the matrix square root, W is a d\u2010dimensional Brownian motion, \u03bb is a Borel measurable function from \u0394d\u2192Rd, and c is a Borel measurable function from \u0394d\u2192S+d, satisfying\u0394d. Note that (\u03bct)t\u22650 given by structure condition \u03bb(\u03bcs)ds). This structural condition characterizes the condition of \u201cno unbounded profit with bounded risk\u201d (NUPBR) in the case of continuous semimartingales .In this setting, the proportions of current (relative) wealth invested in each of the assets are described by processes \u03c0 in the following set:V\u03c0 and V\u03b8 for \u03c0,\u03b8\u2208\u03a0. Using for every\u03c0\u2208\u03a0 if we choose \u03b8\u2208\u03a0 such that\u03b8/\u03bc to ordinary portfolio weights via\u00a0Fernholz and Karatzas ]\u2264E[log(V^T)] for all \u03c0\u2208\u03a0.Next we consider the log\u2010optimal portfolio defined by\u00a0Karatzas , the genBy all strategies in \u03a0. In the sequel, we shall mainly consider suprema taken over smaller sets, in particular over FGM,\u03b1. Note that in this case the optimizer will still be a function of the market weights due to the Markov property of (\u03bct)t\u22650.So far we have optimized over Proposition 4.7(\u03bct)t\u22650 be of the form G, that is,Let In this context, let us also answer the question of when the log\u2010optimal portfolio is functionally generated. This is needed to relate its asymptotic growth rate to the one of the best retrospectively chosen portfolio and the universal portfolio.\u25a1The assertion follows from expression\u00a04.2.4Assumption 4.8\u0394d.The process \u03bc as given in\u00a0limT\u2192\u221e1TlogVT\u03c0. For the precise notion of ergodicity in continuous time, we refer to Eberle (With this assumption we derive an expression of the asymptotic growth rate o Eberle .Hd.Theorem 4.9(i)\u03c0:\u0394d\u2192Hd be any (\u03f1\u2010measurable) portfolio map such that0, thatLet (ii)L:=12\u222b\u0394d\u03bb\u22a4(x)c(x)\u03bb(x)\u03f1(dx)<\u221e. Then, for \u03f1\u2010a.e.\u00a0starting value \u03bc0, it holds thatAssume that Under Assumption\u00a0In the following theorem, we consider portfolio maps which are not necessarily long\u2010only, but can take values in the hyperplane Lemma 4.10M be a continuous local martingale such thatlimT\u2192\u221e1TMT=0, P\u2010a.s.Let The proof of Theorem\u00a0Proof of Theorem 4.9logVT\u03c0 reads as(loglogT)/T, therefore yields Condition\u00a0P\u2010a.s. ) and thus assertion (i).Let us start by proving statement (i). By\u00a0logV^T simplifies to\u25a1Concerning statement (ii), note from 4.3Theorem 4.11M,\u03b1>0 be fixed and let (\u03bct)t\u22650 be a continuous path satisfying Assumption i\u2208{1,\u2026,d}m on GM,\u03b1 with full support and set \u03bd=F\u2217m with F defined in V\u2217,M,\u03b1 and VM,\u03b1(\u03bd) are defined in Let As in discrete time, we will establish asymptotic equality of the growth rates of all three portfolio types introduced in Section\u00a0m has full support and GM,\u03b1 is compact, we have that, for \u03b7>0 there exists some \u03b4>0, such that every \u03b7\u2010neighborhood of a point G\u2208GM,\u03b1 has m\u2010measure bigger than \u03b4.The inequality \u201c\u2265\u201d is obvious. For the converse inequality, we proceed similarly as in the previous section (using only generating functions). As T\u22651 and denote by GT\u2217 the optimizer as of Proposition\u00a0G such \u2225G\u2212GT\u2217\u2225C2,0\u2264\u03b7. Then it follows from \u03b5>0 and note that by assumption T\u21a61TT on T can be bounded by some constant. Therefore, we can choose \u03b7 sufficiently small such that KT\u2264\u03b5 for all T\u22651.Let B=B\u03b7(GT\u2217) the \u2225\u00b7\u2225C2,0\u2010ball with radius \u03b7 in GM,\u03b1 which has m\u2010measure at least \u03b4>0, where \u03b4 only depends on \u03b7. We then may estimate using Jensen's inequality and\u00a0T\u2192\u221e for any given \u03b5 (which determines \u03b7 and in turn \u03b4) yields the assertion. \u25a1Denote by FGM,\u03b1 and suppose henceforth that (\u03bct)t\u22650 is of the form V^M,\u03b1 by V^M,\u03b1=V\u03c0^M,\u03b1, whenever \u03c0^M,\u03b1 is well\u2010defined. As\u03c0^M,\u03b1 as optimizer for all T>0, V^M,\u03b1 corresponds to the log\u2010optimal portfolio among functionally generated portfolios with generating function in GM,\u03b1.Theorem 4.12M,\u03b1>0 be fixed and let (\u03bct)t\u22650 be a stochastic process of the form m on GM,\u03b1 with full support and set \u03bd=F\u2217m with F defined in V^TM,\u03b1 denotes the log\u2010optimal portfolio among FGM,\u03b1\u2010maps defined via V\u2217,M,\u03b1 and VM,\u03b1(\u03bd) are defined pathwise in Let To compare the asymptotic performance with that of the log\u2010optimal portfolio, we optimize over portfolio maps in \u03c0^M,\u03b1 is well\u2010defined. Indeed, the map to R. This together with compactness of GM,\u03b1 with respect to \u2225\u00b7\u22252,0 imply the well\u2010definedness of \u03c0^M,\u03b1.We first note that G imply the assumptions of the ergodic theorem t\u22650 as log\u2010optimizer within the class FGM,\u03b1VTM,\u03b1(\u03bd) is given by FGM,\u03b1.Due to P\u2010a.s.\u00a0constant and equal to limT\u2192\u221e1Tlog. Hence the assertion is proved. \u25a1Combining now M on \u03b1. Setting \u03b1=1M we choose for M=1,2,3,\u22ef a measure mM on GM,1M with full support. Define m=\u2211M=1\u221e2\u2212MmM and the process V(\u03bd) byAs in the previous section, we can formulate a result not depending explicitly on the constant Corollary 4.13(\u03bct)t\u22650 be a stochastic process of form c satisfy G^\u2208C2(\u0394\u00afd). Then we have P\u2010a.s.Let In order to compare the performance with the one of the global log\u2010optimal portfolio, whenever it is functionally generated, we combine the above results with Proposition\u00a0L is well\u2010defined due to \u03b5>0, there exists some M>0 and some function G\u2208GM,1M such thatG\u21a6VG as asserted in Lemma\u00a0G\u2208GM,1M close enough with respect to the \u2225\u00b7\u2225C2,0 to the optimizing function G^\u2208C2(\u0394\u00afd) whose generated portfolio yields V^ due to (Vt(\u03bd)V^t)t\u22650 is a nonnegative supermartingale. It converges P\u2010a.s.\u00a0to a finite limit as t\u2192\u221e. This in turn implies \u25a1Note first that C2\u2010functionally generated portfolios without imposing the drift condition in Proposition\u00a0VTfun the wealth process of the log\u2010optimal portfolio among concave C2\u2010functionally generated portfolios, that is, \u03c0fun is defined as in arg max over all concave C2\u2010functionally generated portfolios.Corollary 4.14(\u03bct)t\u22650 be a stochastic process of form Let Finally, a similar result can be obtained by restricting the log\u2010optimal portfolio to the class of lim inf because the supermartingale argument from the proof of Corollary\u00a0\u25a1The proof is the same as the first part of Corollary"}
{"text": "This article explores how probabilistic programming can be used to simulate quantum correlations in an EPR experimental setting. Probabilistic programs are based on standard probability which cannot produce quantum correlations. In order to address this limitation, a hypergraph formalism was programmed which both expresses the measurement contexts of the EPR experimental design as well as associated constraints. Four contemporary open source probabilistic programming frameworks were used to simulate an EPR experiment in order to shed light on their relative effectiveness from both qualitative and quantitative dimensions. We found that all four probabilistic languages successfully simulated quantum correlations. Detailed analysis revealed that no language was clearly superior across all dimensions, however, the comparison does highlight aspects that can be considered when using probabilistic programs to simulate experiments in quantum physics. Probabilistic models are used in a broad swathe of disciplines ranging from the social and behavioural sciences, biology, the physical and computational sciences, to name but a few. At their very core, probabilistic models are defined in terms of random variables, which range over a set of outcomes that are subject to chance. For example, a measurement on a quantum system is a random variable. By performing the measurement, we record the outcome as the value of the random variable. Repeated measurements on the same preparation allow determining the probability of each outcome. Probabilistic programming offers a convenient way to express probabilistic models by unifying techniques from conventional programming such as modularity, imperative or functional specification, as well as the representation and use of uncertain knowledge. A variety of probabilistic programming languages (PPLs) have been proposed , i \u2208 {0, 1}. In a similar way, variables B0 and B1 are introduced. Therefore, the preceding four pairwise distributions can be represented as the the grid of sixteen probabilities depicted in The basis of the EPR experiment is two systems The goal of an EPR experiment is to empirically determine whether quantum particles are entangled. We will not go into the details of what entanglement is, but rather focus on showing how statistical correlations between variables determine the presence of entanglement. Entanglement is determined if any of the following inequalities is violated.A, B) of quantum particles to joint measurements. More specifically, each such pair is measured in one of the four measurement conditions represented by the grid of probabilities depicted in For historical reasons, the set of four inequalities have become known as the Clauser-Horn-Shimony-Holt (CHSH) inequalities . The datA0, A1, B0, B1, the maximum value that can be computed by any of the inequalities happens to be 2. This is why the boundary of violation in the inequalities is 2 as it demarcates the boundary which standard statistical correlations cannot transcend. This fact presents a challenge for a PPL, which is based on standard probability theory. How can a PPL be developed to simulate non-classical quantum correlations?The maximum possible violation of the CHSH inequalities is 4, i.e., three pairs of variables are maximally correlated (= 1) and the fourth is maximally anti-correlated (= -1). However, if the experiment is modelled by a joint probability distribution across the four variables P, e.g., entangled quantum particles, is to be studied. An experimental design is devised in which P is examined in the four experimental conditions called \u201cmeasurement contexts\u201d. A measurement context Mi, 1 \u2264 i \u2264 4 is designed to study P from a particular experimental perspective. For example, one measurement context corresponds to X = 0 and Y = 1 which yields probabilities over the four possible outcomes of joint measurements of A and B. We will denote these outcomes as {00|01, 01|01, 10|01, 11|01}. For example, 00|01 denotes the outcome A = 0, B = 0 in the measurement context M2 = {X = 0, Y = 1}.P. Composition offers the distinct advantage of allowing experimental designs to be theoretically underpinned by hypergraphs in a modular way , ], , ]],\u2003\u2003\u2003, ], , ]]]The next step involves producing four hyperedges to represent the four explicit joint measurement contexts on both systems. Two variables are given to assist with computing this result.edge_aH[0]:edge_bH[1]:\u2003Thereafter, each hyperedge is defined as the combined sets produced by the following expression.fr_edge = vertex_aedge_a:vertex_bedge_b:\u2003fr_edge.append\u2003\u2003\u2003fr_edges.append(fr_edge)m) are all members of set M, where M are the possible measurement choices for the scenario (in this case two), the second (n) being the other possible measurement choice, and the last variable o being all edges from the hypergraph associated with the measurement choice.The last step involves calculating the hyperedges of both systems as dependent on the other. To achieve this programmatically, three variables are declared. The first :mc_i =1-mc)\u2003edgeH[mc]:\u2003o, in some selected hypergraph, a second variable j is declared as two possible values. For each possibility, a hyperedge is then defined as the variable k, being the combination of all vertices in o given to a function that produces the hyperedge.For each j:fr_edge = \u2003i):\u2003s, q, r, u, t, and v.The mentioned function computes the hyperedge by declaring single-use variables edge_b = H[mc_i][i]vertex_a = edge[(i-j)]vertex_b = edge_b[0]vertex_c = edge_b[1]vertices_a = [vertex_a[0], vertex_b[0],\u2003vertex_a[1], vertex_b[1]]\u2003vertices_b = [vertex_a[0], vertex_c[0],\u2003vertex_a[1], vertex_c[1]]\u2003Thereafter, a set is constructed by use of its variables, and portions of the desired hyperedges are iteratively returned.Upon calculation of the last step of the process, the hyperedges corresponding to the measurement choices of both EPR systems as dependent on the other are produced, totaling the necessary constraints described in binary format.fr_edge.append\u2003\u2003\u2003fr_edge.append\u2003\u2003\u2003fr_edges.append(fr_edge)\u2003fr_edgesa, b, x, and y that were previously mentioned as specifying one of the 16 possible vertices. These values are restricted to binary outcomes by means of specifying Bernoulli distributions for which the sampler runs the process. The experiment is fixed such that each vertex has an equal possibility of being sampled as any other vertex, and results may only be discounted if they do not comply with specified input correlations. Upon selecting a vertex at a step in the iterative process, the array index associated with the binary representation is incremented by one, via the use of the vertex mapping function. Simultaneously, another array representing the hyperedges of the FR product is also incremented by one at all indexes associated with the hyperedges containing the said vertex. The iterative process only exits when the sum of the global distribution is equivalent to the desired number of iterations. Thereafter, each vertex is normalised by the sum of the values in the corresponding array of hyperedges in which its associated vertex is contained, and is multiplied by 3 (for reflection of the number of associated hyperedges). A visualisation of the hyperedges associated with the vertex at index 00|00 of the global distribution can be seen in To compute the four pairwise distributions at the basis of the EPR experiment :hyperedges = foulis_randall_product\u2003hyperedges_tallies = zeros(12)\u2003global_distribution = zeros(16)\u2003 < N:\u2003with pm.Model:\u2003\u2003pm.Uniform\u2003\u2003\u2003pm.Bernoulli\u2003\u2003\u2003pm.Bernoulli\u2003\u2003\u2003pm.Bernoulli\u2003\u2003\u2003pm.Bernoulli\u2003\u2003\u2003S = pm.sample\u2003\u2003\u2003c = S.get_values\u2003\u2003\u2003a = S.get_values\u2003\u2003\u2003b = S.get_values\u2003\u2003\u2003x = S.get_values\u2003\u2003\u2003y = S.get_values\u2003\u2003\u2003i:\u2003\u2003:\u2003\u2003\u2003edgeget_hyperedges:\u2003\u2003\u2003\u2003\u2003hyperedges_tallies[edge] += 1\u2003\u2003\u2003\u2003\u2003global_distribution[\u2003\u2003\u2003\u2003get_vertex] += 1\u2003\u2003\u2003\u2003\u2003z = \u2003a, b, x, yproduct:\u2003summed_tally = ))\u2003\u2003\u2003global_distribution[get_vertex] /= summed_tally\u2003\u2003global_distribution *= 3\u2003global_distribution\u2003Given below in numpyzeros, array, fliplr,itertoolsproductpymc3 as pmget_vertex:((x*8)+(y*4))+(b+(a*2))\u2003get_hyperedges:l = \u2003idx, e(H):\u2003ne:\u2003\u2003l.append(idx)\u2003\u2003\u2003l\u2003foulis_randall_product:fr_edges = \u2003H = , ], , ]],\u2003, ], , ]]]\u2003\u2003\u2003\u2003edge_aH[0]:\u2003edge_bH[1]:\u2003\u2003fr_edge = \u2003\u2003\u2003vertex_aedge_a:\u2003\u2003\u2003vertex_bedge_b:\u2003\u2003\u2003\u2003fr_edge.append\u2003\u2003\u2003\u2003\u2003\u2003fr_edges.append(fr_edge)\u2003\u2003\u2003mc:\u2003mc_i = abs(1-mc)\u2003\u2003edgeH[mc]:\u2003\u2003j:\u2003\u2003\u2003fr_edge = \u2003\u2003\u2003\u2003i):\u2003\u2003\u2003\u2003edge_b = H[mc_i][i]\u2003\u2003\u2003\u2003\u2003vertex_a = edge[(i-j)]\u2003\u2003\u2003\u2003\u2003vertex_b = edge_b[0]\u2003\u2003\u2003\u2003\u2003vertex_c = edge_b[1]\u2003\u2003\u2003\u2003\u2003vertices_a = [\u2003\u2003\u2003\u2003\u2003vertex_a[0], vertex_b[0],\u2003\u2003\u2003\u2003\u2003\u2003vertex_a[1], vertex_b[1]]\u2003\u2003\u2003\u2003\u2003\u2003vertices_b = [\u2003\u2003\u2003\u2003\u2003vertex_a[0], vertex_c[0],\u2003\u2003\u2003\u2003\u2003\u2003vertex_a[1], vertex_c[1]]\u2003\u2003\u2003\u2003\u2003\u2003fr_edge.append\u2003\u2003\u2003\u2003\u2003\u2003fr_edge.append\u2003\u2003\u2003\u2003\u2003\u2003fr_edges.append(fr_edge)\u2003\u2003\u2003\u2003fr_edges\u2003generate_global_distribution:hyperedges = foulis_randall_product\u2003hyperedges_tallies = zeros(12)\u2003global_distribution = zeros(16)\u2003 < N:\u2003with pm.Model:\u2003\u2003pm.Uniform\u2003\u2003\u2003pm.Bernoulli\u2003\u2003\u2003pm.Bernoulli\u2003\u2003\u2003pm.Bernoulli\u2003\u2003\u2003pm.Bernoulli\u2003\u2003\u2003S = pm.sample\u2003\u2003\u2003c = S.get_values\u2003\u2003\u2003a = S.get_values\u2003\u2003\u2003b = S.get_values\u2003\u2003\u2003x = S.get_values\u2003\u2003\u2003y = S.get_values\u2003\u2003\u2003i:\u2003\u2003:\u2003\u2003\u2003edgeget_hyperedges:\u2003\u2003\u2003\u2003\u2003hyperedges_tallies[edge] += 1\u2003\u2003\u2003\u2003\u2003global_distribution[\u2003\u2003\u2003\u2003get_vertex] += 1\u2003\u2003\u2003\u2003\u2003z = \u2003a, b, x, yproduct:\u2003summed_tally = ))\u2003\u2003\u2003global_distribution[get_vertex] /= summed_tally\u2003\u2003global_distribution *= 3\u2003global_distribution\u2003@model macro. To obtain randomly sampled non-negative values for a Bernoulli distribution, the model requires the declaration of a uniform Beta prior, invoked with the Beta method. Then Bernoulli distributions are declared with a Bernoulli method, once again accompanied by probabilities describing sampling biases for later generated distributions, as well as a Uniform distribution.Like PyMC3, Turing.jl isolates operations on random variables to a single model with the use of the generate_global_distribution method of sample function invokes the model, a step-method, as well as the number of desired iterations. In this case, the Sequential Monte Carlo sampling (abbreviated to SMC) has been applied.In the foulis_randall_product method, before returning the result.To obtain the trace of a distribution in the model, the output must be indexed with the precession of a colon. In the example, the results are retrieved, tallied, and normalised by means of the TuringDistributionsfr_edgesArray{Array{Array{Float64}}}(0)\u2003H,],,]],\u2003,],,]]]\u2003\u2003\u2003\u2003\u2003i1(H[1])[1]\u2003j1(H[2])[1]\u2003\u2003fr_edgeArray{Array{Float64}}(0)\u2003\u2003\u2003k1(H[1][i])[1]\u2003\u2003\u2003l1(H[1][j])[1]\u2003\u2003\u2003\u2003append!\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003append!\u2003\u2003\u2003\u2003\u2003\u2003mc12\u2003mc_i(3mc)\u2003\u2003k1(H[mc])[1]\u2003\u2003j12\u2003\u2003\u2003fr_edge = Array{Array{Float64}}(0)\u2003\u2003\u2003\u2003i1(H[mc][k])[1]\u2003\u2003\u2003\u2003edge_bH[mc_i][i]\u2003\u2003\u2003\u2003\u2003vertex_aH[mc][k][(ij)1]\u2003\u2003\u2003\u2003\u2003vertex_bedge_b[1]\u2003\u2003\u2003\u2003\u2003vertex_cedge_b[2]\u2003\u2003\u2003\u2003\u2003vertices_a[vertex_a[1], vertex_b[1],\u2003\u2003\u2003\u2003\u2003vertex_a[2], vertex_b[2]]\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003vertices_b[vertex_a[1], vertex_c[1],\u2003\u2003\u2003\u2003\u2003vertex_a[2], vertex_c[2]]\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003this_edge_bArray{Float64}(0)\u2003\u2003\u2003\u2003\u2003append!\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003append!\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003append!\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003fr_edges\u2003((x8)(y4))(b(a2))1\u2003l\u2003i1(H)[1]\u2003\u2003\u2003append!\u2003\u2003\u2003\u2003\u2003\u2003l\u2003@modelz\u2003a(0.5)\u2003b(0.5)\u2003x(0.5)\u2003y(0.5)\u2003c\u2003hyperedges\u2003hyperedges_tallies(12)\u2003global_distribution(16)\u2003 < N\u2003r)\u2003\u2003ar\u2003\u2003br\u2003\u2003xr\u2003\u2003yr\u2003\u2003cr\u2003\u2003i1N\u2003\u2003(c[i] < constraints[x[i]1][y[i]1][a[i]1][b[i]1])\u2003\u2003\u2003I\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003associated_hyperedges\u2003\u2003\u2003\u2003j1(associated_hyperedges)[1]\u2003\u2003\u2003\u2003hyperedges_tallies[\u2003\u2003\u2003\u2003\u2003associated_hyperedges[j]]1\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003global_distribution1\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003a01, b01, x01, y01\u2003summed_amount0\u2003\u2003I\u2003\u2003\u2003\u2003\u2003\u2003associated_hyperedges\u2003\u2003edge_index1(associated_hyperedges)[1]\u2003\u2003summed_amounthyperedges_tallies[edge_index]\u2003\u2003\u2003\u2003\u2003global_distributionsummed_amount\u2003\u2003\u2003global_distribution3\u2003Flip method, generating a Bernoulli distribution on which the If method can then associate the results to desired values, or perpetuation of other methods.To achieve a joint-probability distribution on a measurement context of random variables, Figaro\u2019s syntactic elements reveal fundamental differences in its approach. A class is the advised object for the purpose of declaring a model. For each random variable in the model, a probability bias is applied to Figaro\u2019s GenerateGlobalDistribution method, it can be seen that after initialising the FR product , the sampling process is called by means of start, stop, and kill chains applied on the algorithm object. On the preceding line, the MetropolisHastings method implies the Metropolis-Hastings step-method will be used for the sampling process, and the outcomes of the model class will be considered. For more complex experiments, the ProposalScheme may be modified, however not in this case. The sampleFromPosterior sub-method chained to calls on each variable compile the required distributions on execution. The take sub-method chained to the sampling methods are used to declare the number of outcomes retrieved from the sampler. This aspect is consequent of sampler delivering results via Stream primitives, a resource-efficient consideration ensuring that only required data is evaluated. Thereafter, the proceeding code tallies the indexes of the globalDistribution array, and normalises the results.In com.cra.figaro.algorithm.sampling._com.cra.figaro.language._com.cra.figaro.library.compound.Ifcom.cra.figaro.library.atomic.continuous.UniformMain {: Array[Array[Array]] = {\u2003foulisRandallEdges = Array[Array[Array]] \u2003\u2003hypergraphs = Array, Array),\u2003\u2003\u2003Array, Array)),\u2003\u2003\u2003\u2003\u2003\u2003\u2003Array, Array),\u2003\u2003\u2003Array, Array)))\u2003\u2003\u2003\u2003\u2003\u2003\u2003(edgeA <- hypergraphs) {\u2003\u2003(edgeB <- hypergraphs) {\u2003\u2003\u2003foulisRandallEdge = Array[Array]\u2003\u2003\u2003\u2003(vertexA <- edgeA) {\u2003\u2003\u2003\u2003(vertexB <- edgeB) {\u2003\u2003\u2003\u2003\u2003foulisRandallEdge ++= Array)\u2003\u2003\u2003\u2003\u2003\u2003\u2003}\u2003\u2003\u2003\u2003\u2003\u2003}\u2003\u2003\u2003\u2003\u2003foulisRandallEdges ++= Array\u2003\u2003\u2003\u2003\u2003}\u2003\u2003\u2003\u2003}\u2003\u2003\u2003(measurementChoice <-to){\u2003\u2003\u2003measurementChoiceInverse =\u2013 measurementChoice\u2003\u2003\u2003\u2003(edge <- hypergraphs(measurementChoice)) {\u2003\u2003\u2003\u2003(j <-to) {\u2003\u2003\u2003\u2003\u2003foulisRandallEdge = Array[Array]\u2003\u2003\u2003\u2003\u2003\u2003(i <- edge.indices) {\u2003\u2003\u2003\u2003\u2003\u2003edgeB = hypergraphs(measurementChoiceInverse)(i)\u2003\u2003\u2003\u2003\u2003\u2003\u2003vertexA = edge(Math.abs(i-j))\u2003\u2003\u2003\u2003\u2003\u2003\u2003vertexB = edgeB\u2003\u2003\u2003\u2003\u2003\u2003\u2003vertexC = edgeB\u2003\u2003\u2003\u2003\u2003\u2003\u2003verticesA = Array\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003verticesB = Array\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003foulisRandallEdge ++= Array(\u2003\u2003\u2003\u2003\u2003\u2003\u2003Array(\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003verticesA(measurementChoice),\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003verticesA(measurementChoiceInverse),\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003verticesA(measurementChoice+),\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003verticesA(measurementChoiceInverse+)))\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003foulisRandallEdge ++= Array(\u2003\u2003\u2003\u2003\u2003\u2003\u2003Array(\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003verticesB(measurementChoice),\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003verticesB(measurementChoiceInverse),\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003verticesB(measurementChoice+),\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003verticesB(measurementChoiceInverse+)))\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003}\u2003\u2003\u2003\u2003\u2003\u2003foulisRandallEdges ++= Array\u2003\u2003\u2003\u2003\u2003\u2003}\u2003\u2003\u2003\u2003\u2003}\u2003\u2003\u2003\u2003}\u2003\u2003\u2003foulisRandallEdges\u2003\u2003\u2003}\u2003Model {\u2003= Array[Element]\u2003\u2003(i <-to){\u2003\u2003: += If\u2003\u2003\u2003}\u2003\u2003: += Uniform\u2003\u2003}\u2003:= {\u2003((x*)+(y*))+(b+(a*))\u2003\u2003}\u2003: Array = {\u2003l = Array\u2003\u2003(i <- H.indices) {\u2003\u2003(H(i).deep.contains(n.deep)) {\u2003\u2003\u2003l: += i\u2003\u2003\u2003\u2003}\u2003\u2003\u2003}\u2003\u2003l\u2003\u2003}\u2003: = {\u2003\u2003hyperedges = FoulisRandallProduct\u2003\u2003hyperedgesTallies = Array.fill{}\u2003\u2003globalDistribution = Array.fill{}\u2003\u2003 {\u2003\u2003model =Model\u2003\u2003\u2003algorithm = MetropolisHastings\u2003\u2003\u2003algorithm.start\u2003\u2003\u2003algorithm.stop\u2003\u2003\u2003algorithm.kill\u2003\u2003\u2003a = algorithm.sampleFromPosterior.take(N).toArray\u2003\u2003\u2003\u2003b = algorithm.sampleFromPosterior.take(N).toArray\u2003\u2003\u2003\u2003x = algorithm.sampleFromPosterior.take(N).toArray\u2003\u2003\u2003\u2003y = algorithm.sampleFromPosterior.take(N).toArray\u2003\u2003\u2003\u2003c = algorithm.sampleFromPosterior.take(N).toArray\u2003\u2003\u2003\u2003(i <-until N) {\u2003\u2003\u2003val x_x = x(i).toInt\u2003\u2003\u2003val y_y = y(i).toInt\u2003\u2003\u2003val a_a = a(i).toInt\u2003\u2003\u2003val b_b = b(i).toInt\u2003\u2003\u2003(c(i) < constraints(x_x)(y_y)(a_a)(b_b)) {\u2003\u2003\u2003)) {\u2003\u2003\u2003\u2003\u2003hyperedgesTallies(edge) +=\u2003\u2003\u2003\u2003\u2003\u2003}\u2003\u2003\u2003\u2003globalDistribution) +=\u2003\u2003\u2003\u2003}\u2003\u2003\u2003}\u2003\u2003}\u2003 {\u2003summedAmount =\u2003\u2003associatedHyperedges = GetHyperedges)\u2003\u2003\u2003(edgeIndex <- associatedHyperedges.indices) {\u2003\u2003summedAmount += hyperedgesTallies(edgeIndex)\u2003\u2003\u2003}\u2003\u2003globalDistribution) =\u2003\u2003globalDistribution) / summedAmount\u2003\u2003\u2003}\u2003\u2003globalDistribution\u2003\u2003}\u2003}def containers; or can be flexibly given implicitly, encouraging the use of stochastic functions to specify probabilistic models. Inside a container, probabilities of random variables are specified first. As Pyro is built upon PyTorch, explicit values match PyTorch types, in this case resulting in Tensor type values.Pyro\u2019s context management is integrated into Python\u2019s generate_global_distribution method, Pyro\u2019s atomic sampling capabilities allow for the requirement of fewer syntactic constructs to communicate the sampling process. Each sample call accepts a distribution, in this case either Bernoulli or Uniform. Upon sampling, the outcomes are tallied and normalised, before presenting the result.As can be seen in pyrosampletorchTensortorch.autogradVariablenumpyzeros, array, fliplr,itertoolsproductpyro.distributionsBernoulli, Uniformget_vertex:((x*8)+(y*4))+(b+(a*2))\u2003get_hyperedges:l = \u2003idx, e(H):\u2003ne:\u2003\u2003l.append(idx)\u2003\u2003\u2003l\u2003foulis_randall_product:fr_edges = \u2003H = , ], , ]],\u2003, ], , ]]]\u2003\u2003\u2003\u2003edge_aH[0]:\u2003edge_bH[1]:\u2003\u2003fr_edge = \u2003\u2003\u2003vertex_aedge_a:\u2003\u2003\u2003vertex_bedge_b:\u2003\u2003\u2003\u2003fr_edge.append\u2003\u2003\u2003\u2003\u2003\u2003fr_edges.append(fr_edge)\u2003\u2003\u2003mc:\u2003mc_i =(1-mc)\u2003\u2003edgeH[mc]:\u2003\u2003j:\u2003\u2003\u2003fr_edge = \u2003\u2003\u2003\u2003i):\u2003\u2003\u2003\u2003edge_b = H[mc_i][i]\u2003\u2003\u2003\u2003\u2003vertex_a = edge[(i-j)]\u2003\u2003\u2003\u2003\u2003vertex_b = edge_b[0]\u2003\u2003\u2003\u2003\u2003vertex_c = edge_b[1]\u2003\u2003\u2003\u2003\u2003vertices_a = [\u2003\u2003\u2003\u2003\u2003vertex_a[0], vertex_b[0],\u2003\u2003\u2003\u2003\u2003\u2003vertex_a[1], vertex_b[1]]\u2003\u2003\u2003\u2003\u2003\u2003vertices_b = [\u2003\u2003\u2003\u2003\u2003vertex_a[0], vertex_c[0],\u2003\u2003\u2003\u2003\u2003\u2003vertex_a[1], vertex_c[1]]\u2003\u2003\u2003\u2003\u2003\u2003fr_edge.append\u2003\u2003\u2003\u2003\u2003\u2003fr_edge.append\u2003\u2003\u2003\u2003\u2003\u2003fr_edges.append(fr_edge)\u2003\u2003\u2003\u2003fr_edges\u2003generate_global_distribution:hyperedges = foulis_randall_product\u2003hyperedges_tallies = zeros(12)\u2003global_distribution = zeros(16)\u2003 < N:\u2003a =))))\u2003\u2003b =))))\u2003\u2003x =))))\u2003\u2003y =))))\u2003\u2003value =),\u2003\u2003Variable(Tensor([1.0])))))\u2003\u2003\u2003:\u2003\u2003edgeget_hyperedges:\u2003\u2003\u2003hyperedges_tallies[edge] += 1\u2003\u2003\u2003\u2003global_distribution[get_vertex] += 1\u2003\u2003\u2003z = \u2003a, b, x, yproduct:\u2003summed_tally = ))\u2003\u2003\u2003global_distribution[get_vertex] /= summed_tally\u2003\u2003global_distribution *= 3\u2003global_distribution\u2003A and B in the four measurement contexts are specified. For example, the following code fragments A and B to be maximally correlated in three measurement contexts and maximally anti-correlated in the fourth. With these input correlations, maximum violation of the CHSH inequalities would be expected, essentially simulating a PR box ,\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003]\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003p = generate_global_distributionconstraints, ],, ]],\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003, ],\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003, ]]]\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003pconstraints = Array, Array),\u2003Array, Array)),\u2003\u2003\u2003\u2003\u2003Array, Array),\u2003Array, Array)))\u2003\u2003\u2003\u2003\u2003P = GenerateGlobalDistributionFor each of the four PPLs, code is specified equality:f1:\u2003((2 * (p[v1] + p[v2])) \u2212 1)\u2003\u2003f2:\u2003(p[v1] + p[v2]) \u2212 (p[v3] + p[v4])\u2003\u2003delta = 0.5 *  \u2212 f1) +  \u2212 f1) +\u2003\u2003 \u2212 f1) +  \u2212 f1))\u2003\u20032 * (1 + delta) >=) + ) +\u2003\u2003) + ))\u2003\u2003tests = \u2003\u2003((2(p[v1]p[v2]))1)\u2003\u2003\u2003\u2003(p[v1]p[v2])(p[v3]p[v4])\u2003\u2003\u2003delta0.5)) +\u2003\u2003)))\u2003\u2003(2(1delta)) >=)) +\u2003\u2003)))\u2003\u2003tests = \u2003(v1:v2:v3:v4: Int):= {(v1:v2:):= {\u2003Math.abs((* (p(v1) + p(v2))) \u2212)\u2003\u2003}\u2003(v1:v2:v3:v4:):= {\u2003(p(v1) + p(v2)) \u2212 (p(v3) + p(v4))\u2003\u2003}\u2003delta =* (\u2003(f1 \u2212 f1) + (f1 \u2212 f1) +\u2003\u2003(f1 \u2212 f1) + (f1 \u2212 f1))\u2003\u2003(* (+ delta)) >= Math.abs(\u2003(v1*f2) + (v2*f2) +\u2003\u2003(v3*f2) + (v4*f2))\u2003\u2003}tests = ArrayUpon conducting simulations using the input correlations given above, the predicted maximum violation of the CHSH inequalities were observed for all four PPLs specified above.We conducted several experiments to compare the numerical accuracy and execution time of the different implementations.For all PPLs, statistical outputs confirming the success of the FR product  are given with an acceptable margin of error. While this is consequent of more than a single factor, it is perceived that the largest contributor to accuracy is the computation of random values for each PPL. What can be observed is that, with larger sample sizes (bearing more perfectly random distributions), that the margin of error decreases, as can be seen below. This is typical, as the experiment design\u2019s normalisation process is dependent on the even spread of tallies across the global distribution, and more specifically, the degree to which the sampler is random. It should also be considered that the various PPLs apply data types that round values for a loss of statistical precision where it may serve meaning. For example, while a single value may lose a minute portion of its whole beyond the decimal point (due to automatic rounding), when calculating a handful of these values per iteration of some few thousand iterations, the difference becomes observable.From observing the results in Another statistic that has been observed is the compilation time of each PPL, which typically increases with number of iterations. For relativity of results, it should be noted that all non-accelerated tests of this kind were executed within a Bash execution terminal, on a Macintosh operating system, bearing a 45nm \u201cPenryn\u201d 2.4 GHz Intel \u201cCore 2 Duo\u201d processor, and 4 GB of SDRAM, whereas all accelerated tests were executed within a bash execution terminal of Amazon Web Services Linux (2nd distribution). The specification of the \u2018Elastic-Compute Cloud\u2019 on which the Linux distribution executed was a 2018 \u201cp2.xlarge\u201d 2.7 GHz Intel Broadwell processor, with 61 GB of SDRAM. The instance also provides an NVIDIA GK210 GPU multi-vCPU (count of 4) processor, and 12 GB of GPU RAM.In Recall that the challenge posed was how to develop probabilistic program which can simulate quantum correlations in an EPR experiment. The solution adopted was to program a hypergraph formalism to underpin the simulations. This formalism is modular where the FR product of the modules is used to impose the no-signalling constraint. In execution, all four PPLs successfully simulated an EPR experiment producing quantum correlations. Therefore, we conclude that the hypergraph formalism has been shown to be a promising basis for such simulations. In addition, the hypergraph formalism is also rendered into program syntax in a fairly straightforward way. However, the formalism does pose a challenge with respect to the accessibility criterion of the PPLs. The challenge is due to an inherent ambiguity present in the composite hypergraph produced by the FR product, which has 12 edges in the EPR experiment. Four of those edges represent \u201cactual\u201d measurement contexts  for the case of the experimentation. Thus it wouldn\u2019t be perceived that this is a determinant factor.In experimentation, we found that Pyro provided the syntactic constructs needed to neatly describe its processes in fewer procedures than those of the others. While PyMC3\u2019s origins in Python also made it an expressive alternative, the excessive nomenclature surrounding the declarations of methods and data-types for both Turing.jl and Figaro convoluted their descriptions. While in comparison to the other PPLs, Figaro\u2019s accuracy is inferior, it could be argued that the sample iterations describe an experimental setting that does not consider improving float precision. Coupled with the trend of Figaro\u2019s improvement in its number of iterations, and the measure of accuracy between PPLs may converge. The same cannot be said for the time complexity of the EPR experimentation, where it was observed that PyMC3\u2019s compilation grew substantially with the number of sample iterations being executed. Still, in instances where accuracy is a key factor and limitations are perceived in Pyro\u2019s functionality, PyMC3 would be the suitable alternative.Probabilistic programming offers new possibilities for quantum physicists to specify and simulate experiments, such as the EPR experiment illustrated in this article. This is particularly relevant for experiments requiring advanced statistical inference on unknown parameters, especially in the case of techniques that involve large amounts of data and computational time. Furthermore, probabilistic machine learning models that are conveniently expressed in probabilistic programming languages can advance our understanding of the underlying physics of the experiments.It is important to note that the benefits of probabilistic programming are not restricted to experiments involving the analysis of quantum correlations. Since any probabilistic programming language is based on random variables, we can ask the question what exactly is a random variable in quantum physics. Focusing on a single measurement context, due to the normalization constraint, we can think of the measurement context as a  probability distribution over random variables, which describes the measurement outcomes. The probability distribution is a normalized measure over the sigma algebra defined by the outcomes. This measure is defined via Born\u2019s rule, that is, the quantum state is embedded in the measure. An EPR experiment is essentially a state preparation protocol where deterministic operations are embedded in the quantum state (the unitary operations leading to its preparation), followed by the measurement, which results in stochastic outcomes. More generally, we can think of a larger system where we only measure a subsystem. This leads to quantum channels, which are described by completely positive trace preserving maps. A quantum channel, however, must be deterministic in the sequence of events, and, for instance, a measurement choice at a later time step cannot depend on the outcome of a previous measurement. We must factor in such classical and quantum memory effects, as well as the potentially indeterminate causal order of events. The quantum comb , 23 or p"}
{"text": "Arthonia ulleungdoensis Lee & Hur is described as a new lichen species from South Korea. The new species is distinguishable from Arthonia ruana A. Massal. by its large, rounded and non-punctiform apothecia, taller apothecial section, asci with fewer spores, and larger and permanently colorless spores. Molecular analyses employing mitochondrial small subunit (mtSSU) and RNA polymerase subunit II (RPB2) sequences strongly support Arthonia ulleungdoensis as a distinct species in the genus Arthonia. Overall, 22 Arthonia species are currently recorded in South Korea. A surrogate key is provided to assist in the identification of all 10 taxa of Arthonia/Arthothelium with muriform spores in Northeast Asia.  Arthonia is one of the least explored species in lichen taxonomy although the genus is comprised of about five hundred species worldwide [Arthonia and related genera is unclear, and many Arthonia species have synonyms for Arthothelium or other genera. Molecular phylogeny also showed that Arthonia species are arranged over genera and even families [Arthonia is the \u2018unstable structure\u2019 of the species. Many species in Arthonia are highly pioneering. They disperse rapidly but become sterile with a deformed structure after spore discharge and dispersal in a short term. Therefore, although Arthonia species are encountered in the field, they are useless for analysis in many cases by the old, unfilled and barren ascomata with no ascus and spore. The third reason for the demanding genus is a \u2018poor description\u2019 of previous references as old descriptions are insufficient to explain specific characteristics of microlichens in anatomy, such as genus Arthonia particularly. The deficient depiction in the past is due mainly to the poor quality of microscopes, and still discourages a comparison with other species/specimens.The genus orldwide . One of families . AnotherArthonia received much less attention in Asia compared to Europe and America, several Asian countries have actively studied Arthonia. In particular, in Asia, India has explored the genus the most since 2000 and dynamic studies such as local detection, new species/records discovery, diversity assessment and ethnolichenological survey for Arthonia were reported [Arthonia species which were detected locally [Arthonia since 2013 and many new species and records, including lichenized and lichenicolous fungi, were discovered [Arthonia species were reported recently from China [Although the genus reported ,4,5,6. T locally ,8,9. Souscovered ,11,12. Fom China ,15.Arthonia. A concentrated field study was achieved in Ulleung Island and Pohang, which are an eastern island and seashore region of South Korea, during the summer of 2017  and a compound microscope  and imaged using a software program  and an Axiocam ERc 5s camera  mounted on a Zeiss scope A1 microscope . The ascospores were investigated at 1000\u00d7 magnification in water. The length and width of the ascospores were measured and the range of spore sizes was shown with averages, standard deviation, and number of measured spores. Thin-layer chromatography (TLC) was performed using solvent systems A and C according to standard methods . Hand-cut sections of ascomata or thallus from all collected specimens were prepared for DNA isolation and DNA was extracted in line with the manufacturer\u2019s instructions . Two-way PCR amplification for the mitochondrial small subunit (mtSSU) and RNA polymerase subunit II (RPB2) genes was achieved using Bioneer\u2019s AccuPower PCR Premix  in 20-\u03bcL tubes and primers, mrSSU1 and mrSSU3R  and fRPBAll mtSSU and RPB2 sequences were aligned and edited manually using ClustalW in Bioedit . The bootstrap values were obtained in RAxML GUI 1.5 beta   using thArthonia/Arthothelium taxa to show the position of the new species in molecular phylogeny. The integrated tree shows that the new species is classified in the genus Arthonia . Arthonia ulleungdoensis exhibits important characteristics of the genus Arthonia, particularly A. sect. Arthonia, such as maculate ascomata, olive-brown ascomatal pigments, persistently colorless spores, and photophilous [Arthonia ulleungdoensis is identified as a unique species in both morphology and molecular phylogeny, being positioned in the genus Arthonia.A concatenated tree was produced from two different loci, which involved overall 70 sequences (35 sequences for each mtSSU and RPB2) . This coArthonia . Arthoniophilous , althougArthonia ulleungdoensis B.G.Lee & J.-S.Hur sp. nov.\u2003.Type: South Korea, Gyeongsangbuk-do, Ulleung-gun, Ulleung-eup, Ulleung forest trail, 37\u00b0 31.20\u2032 N; 130\u00b0 54.29\u2032 E, 300 m alt., on n = 6); bark layer hyaline, 10\u201315 \u03bcm; apothecial section 100\u2013130 \u03bcm thick; epihymenium brown to dark brown, 15\u201320 \u03bcm high; hymenium hyaline to light brown, 50\u201360 \u03bcm high, oil droplets near base of hymenium or in hypothecium; hypothecium brown to dirty brown, 25\u201335 \u03bcm high; paraphysoids septate, anticlinally arranged, 1\u20131.5 \u03bcm wide, somewhat branched at tips; tips swollen and pigmented, 1.5\u20132 \u03bcm wide. Asci wide to narrow clavate, 2-, 4- or 6-spored, 32\u201349 \u00d7 20\u201331 \u03bcm (n = 5); ascospores hyaline, muriform, 4- to 8-transverse and 0- to 3-longitudinal septa, without a gelatinous sheath, old spores with dark septum, 20.5\u201331 \u00d7 9\u201313 \u03bcm (length n = 28). Pycnidia not detected.Thallus corticolous, crustose, hypophloedal, white to gray, without bleaching; epinecral layer 15\u201318 \u03bcm; medulla indistinct; photobiont trentepohlioid, forming a distinct algal layer, parallel to the substratum, between the epinecral layer and the brown bark layer or under apothecia, often in top layer of the brown bark, 45\u201354 \u03bcm thick, cells irregular, globose to angular, single or in chains, 4.5\u20136 \u00d7 6\u20139 \u03bcm, looking somewhat shrunken and supposed to be a little larger originally. Apothecia rounded, erumpent, black, epruinose, with or without epinecral bark layer on the surface of apothecia, 0.79\u20132.63 \u00d7 0.72\u20131.97 mm .Acer takesimense. It is currently known by a specimen in the type collection . Distribution and ecology: This species occur on the bark of Etymology: This species epithet is named for the collection locality, South Korea.Arthonia ulleungdoensis can be confused with Arthonia ruana in having black and epruinose apothecia, brown epihymenium, hyaline to light brown hymenium, brown hypothecium, K+ olive green reaction on an apothecial section, and the presence of a trentepohlioid alga [vs. up to 1.6 mm), a taller apothecial section (100\u2013130 \u03bcm vs. 70\u201395 \u03bcm), taller hymenium (50\u201360 \u03bcm vs. 35\u201350 \u03bcm), 2- to 6-spored asci (vs. 8-spored asci), and larger spores (20.5\u201331 \u00d7 9\u201313 \u03bcm vs. 15\u201326 \u00d7 7\u201310.5 \u03bcm) without changing their colors when mature (vs. brown old spores) [Arthothelium norvegicum in having a white thallus, black and rounded apothecia without pruina, brownish epihymenium and hypothecium, colorless hymenium, K+ greenish apothecial section reaction, muriform spores, and the presence of a trentepohlioid alga [vs. 0.3\u20130.6 mm diam.), smaller spores (20.5\u201331 \u00d7 9\u201313 \u03bcm vs. 29\u201336 \u00d7 12\u201315 \u03bcm), and colorless spores when mature (vs. brown old spores) [Arthothelium fecundum Zahlbr. in having black, rounded and epruinose apothecia, dark-colored epihymenium and hypothecium, permanently colorless spores, similar ascospore size (20.5\u201331 \u00d7 9\u201313 \u03bcm vs. 25\u201327 \u00d7 9\u201311 \u03bcm), and the presence of a trentepohlioid alga [vs. up to 0.7 mm diam.), immersed thallus , no prothallus , 2- to 6-spored asci (vs. 8-spored asci), and I+ light blue hymenium (vs. I+ reddish hymenium) reaction [Notes: oid alga . However spores) . The newoid alga . However spores) . The newArthonia/Arthothelium with muriform spores from Northeast Asia (10 taxa): This key includes all Arthonia/Arthothelium species with muriform spores in Korea (4), Japan (4) and China (3). However, only one species, Arthothelium japonicum, is excluded from the key because its ascomata represent a perithecioid type [Arthonia/ Arthothelium species.Key to the species in oid type  and may Protococcus \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.Arthothelium collosporum1. Photobiont Trentepohlia \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 2- Photobiont absent or 2. Hypothecium dark-colored \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 3- Hypothecium colorless or pale brownish \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026... 5Arthothelium feduncum3. Ascospores over 10 transversely-septate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.- Ascospores less than 10 transversely-septate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 4Arthonia ulleungdoensis4. Ascospores permanently colorless but with dark septum when mature, 20.5\u201331 \u00d7 9\u201313 \u03bcm, 2\u20136-spored asci \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026- Ascospores brownish when mature, 15\u201326 \u00d7 7\u201310 um, 8-spored asci\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.Arthonia ruana (Arthothelium dispersum)Arthothelium punctatum5. Ascospores up to 10 transversely-septate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..- Ascospores less than 10 transversely-septate \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 6Arthothelium spectabile6. Thallus usually delimited by a brown line \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026- Thallus effuse without a delimited line \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 7Arthothelium pertenerum7. Ascospores smaller, 16\u201320 \u00d7 7 \u03bcm \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. - Ascospores larger, over 20 \u00d7 10 \u03bcm \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 88. Ascospores 19\u201327 \u00d7 10\u201315 \u03bcm, photobiont not seen, hypothecium colorlessArthothelium scandinavicum\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.Trentepohlia, hypothecium colorless or pale brownish \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026Arthothelium scandinavicum var. japonicum- Ascospores 28\u201336 \u00d7 12\u201314 \u03bcm, photobiont"}
{"text": "An implicit Euler finite\u2010volume scheme for a degenerate cross\u2010diffusion system describing the ion transport through biological membranes is proposed. The strongly coupled equations for the ion concentrations include drift terms involving the electric potential, which is coupled to the concentrations through the Poisson equation. The cross\u2010diffusion system possesses a formal gradient\u2010flow structure revealing nonstandard degeneracies, which lead to considerable mathematical difficulties. The finite\u2010volume scheme is based on two\u2010point flux approximations with \u201cdouble\u201d upwind mobilities. The existence of solutions to the fully discrete scheme is proved. When the particles are not distinguishable and the dynamics is driven by cross diffusion only, it is shown that the scheme preserves the structure of the equations like nonnegativity, upper bounds, and entropy dissipation. The degeneracy is overcome by proving a new discrete Aubin\u2013Lions lemma of \u201cdegenerate\u201d type. Numerical simulations of a calcium\u2010selective ion channel in two space dimensions show that the scheme is efficient even in the general case of ion transport. On a macroscopic level, the transport can be described by nonlinear partial differential equations for the ion concentrations  and the surrounding electric potential. A classical model for ion transport are the Poisson\u2013Nernst\u2013Planck equations\u00a0ui and fluxes ith ion species is governed by the equationsi\u2009=\u20091, \u2026, n, where u0=1\u2212\u2211i=1nui is the concentration (volume fraction) of the electro\u2010neutral solvent, Di\u2009>\u20090 is a diffusion coefficient, \u03b2\u2009>\u20090 is the  inverse thermal voltage, and ith species. Observe that we assumed Einstein's relation which says that the quotient of the diffusion and mobility coefficients is constant, and we call this constant 1/\u03b2. The electric potential is determined by the Poisson equation\u03bb2 is the  permittivity constant and f\u2009=\u2009f(x) is a permanent background charge density. Equations\u00a0In the general case, the evolution of the concentrations \u2202\u03a9 is supposed to consist of two parts, the insulating part \u0393N, on which no\u2010flux boundary conditions are prescribed, and the union \u0393D of boundary contacts with external reservoirs, on which the concentrations are fixed. The electric potential is prescribed at the electrodes on \u0393D. This leads to the mixed Dirichlet\u2013Neumann boundary conditionsu\u203ei1\u2264i\u2264n and \u03a6\u203e can be defined on the whole domain \u03a9. Finally, we prescribe the initial conditionsIn order to match experimental conditions, the boundary Aij(u)), defined by Aij(u)\u2009=\u2009Diui for i\u2009\u2260\u2009j and Aii(u)\u2009=\u2009Di(u0\u2009+\u2009ui) is not symmetric and not positive definite. It was shown in Burger and coworkers u\u2009=\u2009 and u0=1\u2212\u2211i=1nui, such that Bii\u2009=\u2009Diu0ui, Bij\u2009=\u20090 for i\u2009\u2260\u2009j provide a diagonal positive definite matrix, and wj are the entropy variables, defined byThe main mathematical difficulties of Equations\u00a0\u2202h/\u2202ui.We refer to , Lem. 7 u, \u03a6)\u2009\u21a6\u2009w to entropy variables can be inverted, giving u\u2009=\u2009u withThe entropy structure of ui is positive and bounded from above, that is,Then C\u2009>\u20090 depends on the Dirichlet boundary data. Because ofu01/2ui and u01/2. Since u0 may vanish locally, this does not give gradient bounds for ui, which expresses the degenerate nature of the cross\u2010diffusion system. As a consequence, the flux has to be formulated in the terms of gradients of u01/2ui and u01/2 only, namelyThis yields The challenge is to derive a discrete version of this formulation. It turns out that Di are the same and the drift term vanishes). This suggests to use the entropy variables as the unknowns, as it was done in our previous work ui for the numerical discretization.Our aim is to design a numerical approximation of Vi\u2009=\u2009\u2207u0\u00a0\u2212\u00a0\u03b2ziu0\u2207\u03a6 have the structure \u2207v\u2009+\u2009vF, where \u2207v is the diffusion term and vF is the drift term. We discretize V by using a two\u2010point flux aproximation with \u201cdouble\u201d upwind mobilities.We propose a backward Euler scheme in time and a finite\u2010volume scheme in space, based on two\u2010point approximations. The key observation for the numerical discretization is that the fluxes can be written on each cell in a \u201cdouble\u201d drift\u2010diffusion form, that is, both nonnegativity, which follows from a discrete minimum principle argument. It is well known that the maximum priciple generally does not hold for systems of equations. Therefore, it is not a surprise that the upper bound comes only at a price: We need to assume that the all diffusion coefficients Di are the same. Under this assumption, u0=1\u2212\u2211i=1nui solves a drift\u2010diffusion equation for which the (discrete) maximum principle can be applied. In order to prove that the scheme satisfies an entropy\u2010dissipation inequality and also to complete the convergence analysis of the scheme successfully, we need a stronger additional assumption: We assume that the drift terms, and therefore the coupling with the Poisson equation, can be neglected. This means that our main results are obtained for a simplified degenerate cross\u2010diffusion system, no more corresponding to the initial ion transport model but still of mathematical interest. Nevertheless, the scheme we propose can be applied to the full ion transport model, and this is done in the last section of this paper.Under certain assumptions, the structure of the equations is preserved on the discrete level. Because of the drift\u2010diffusion structure, we are able to prove that the scheme preserves the \u2202\u03a9. Mixed Dirichlet\u2013Neumann boundary conditions could be prescribed as well, but the proofs would become even more technical. The main results are as follows.Di\u00a0=\u00a0D for all i, we prove the existence of solutions to the fully discrete numerical scheme  is an open, bounded, polygonal domain with \u2202\u03a9\u2009=\u2009\u0393D\u00a0\u222a\u00a0\u0393N\u00a0\u2208\u00a0C0, 1, \u0393D\u00a0\u2229\u00a0\u0393N\u2009=\u2009\u00d8.Domain: H2T\u2009>\u20090, Di\u2009>\u20090, \u03b2\u2009>\u20090, and i\u2009=\u20091, \u2026, n.Parameters: H3Background charge: H4uiI\u2208L\u221e\u03a9, u\u203ei\u2208H1\u03a9 satisfy uiI\u22650, u\u203ei\u22650 and 1\u2212\u2211i=1nuiI\u22650, 1\u2212\u2211i=1nu\u203ei\u22650 in \u03a9 for i\u2009=\u20091, \u2026, n, and \u03a6\u203e\u2208H1\u03a9\u2229L\u221e\u03a9.Initial and boundary data: We summarize our general hypotheses on the data:A1\u2202\u03a9\u2009=\u2009\u0393N, that is, we impose no\u2010flux boundary conditions on the whole boundary.A2Di\u2009=\u2009D\u2009>\u20090 for i\u2009=\u20091, \u2026, n.The diffusion constants are equal, A3The drift terms are set to zero, \u03a6\u2009\u2261\u20090.For our main results, we need additional technical assumptions:Remark 1Di\u00a0=\u00a0D for all i, summing i\u00a0=\u00a01, \u2026, n gives(Discussion of the assumptions). Assumption (A1) is supposed for simplicity only. Mixed Dirichlet\u2013Neumann boundary conditions can be included in the analysis . For simplicity, we consider a uniform time discretization with time step \u25b3t\u2009>\u20090, and we set tk\u2009=\u2009k\u25b3t for k\u2009=\u20091, \u2026, N, where T\u2009>\u20090, t\u2009=\u2009T/N. The domain \u03a9 is discretized by a regular and admissible triangulation in the sense of [\ud835\udcaf of open polygonal convex subsets of \u03a9 , a family xKK\u2208\ud835\udcaf associated to the cells. The admissibility assumption implies that the straight line between two centers of neighboring cells xKxL\u203e is orthogonal to the edge \u03c3\u2009=\u2009K|L between two cells K and L. The condition is satisfied by, for instance, triangular meshes whose triangles have angles smaller than \u03c0/2 [For the definition of the numerical scheme for \u2130ext=\u2130extD\u222a\u2130extN. For given K\u2208\ud835\udcaf, we define the set K, which is the union of internal edges and edges on the Dirichlet or Neumann boundary, and we set We assume that the family of edges h\ud835\udcaf=supdiamK:K\u2208\ud835\udcaf. For \u03c3\u2009=\u2009K|L, we denote by d\u03c3\u2009=\u2009d the Euclidean distance between xK and xL, while for \u03c3\u2009=\u2009d. For a given edge m(\u03c3) denotes the Lebesgue measure of \u03c3.The size of the mesh is defined by \u03b6\u2009>\u20090 such that for all K\u2208\ud835\udcaf and We impose a regularity assumption on the mesh: There exists This hypothesis is needed to apply discrete functional inequalities see , 17 and \u210b\ud835\udcaf of piecewise constant functions is defined byIt remains to introduce suitable function spaces for the numerical discretization. The space H1 norm on this space is given byThe (squared) discrete H\u22121 norm is the dual norm with respect to the L2 scalar product,The discrete Then\u210b\ud835\udcaf,\u25b3t of piecewise constant in time functions with values in \u210b\ud835\udcaf,L2) normFinally, we introduce the space ui\u2208\u210b\ud835\udcaf with values u\u203ei,\u03c3 on the Dirichlet boundary . Then we introduceFor the numerical scheme, we introduce some further definitions. Let \u03c3\u2009=\u2009K|L, requiring that they vanish on the Neumann boundary edges, \u03c3\u2208\u2130ext,KN. Then the discrete integration\u2010by\u2010parts formula becomes for u\u2208\u210b\ud835\udcafThe numerical fluxes \u2202\u03a9\u2009=\u2009\u0393N, this formula simplifies toWhen 2.2K\u2208\ud835\udcaf and edges u0,KI=1\u2212\u2211i=1nui,KI and u\u203e0,\u03c3=1\u2212\u2211i=1nu\u203ei,\u03c3.We need to approximate the initial, boundary, and given functions on the elements K\u2208\ud835\udcaf, k \u2208{1, \u2026, N}, i\u2009=\u20091, \u2026, n, and ui,Kk\u22121\u22650 be given. Then the values ui,Kk are determined by the implicit Euler scheme\u2131i,K,\u03c3k are given by the upwind scheme\u03c4\u03c3 is defined in \ud835\udcb1i,K,\u03c3k is the \u201cdrift part\u201d of the flux,i\u2009=\u20091, \u2026, n. Observe that we employed a double upwinding: one related to the electric potential, defining u^0,\u03c3,ik, and another one related to the drift part of the flux, \ud835\udcb1i,K,\u03c3k. The potential is computed viaThe numerical scheme is as follows. Let u\u203ei,\u03c3 and \u03a6\u203e\u03c3 for \u03c3\u2208\u2130extD.We recall that the numerical boundary conditions are given by ui,\ud835\udcaf,\u25b3t, \u03a6\ud835\udcaf,\u25b3t the functions in \u210b\ud835\udcaf,\u25b3t associated to the values ui,Kk and \u03a6Kk, respectively. Moreover, when dealing with a sequence of meshes \ud835\udcafmm and a sequence of time steps (\u25b3tm)m, we set ui,m=ui,\ud835\udcafm,\u25b3tm, \u03a6m=\u03a6\ud835\udcafm,\u25b3tm.Remark 2u0,Kk and u0,\u03c3k are defined in ui,\u03c3k simplifies toWhen Assumptions (A1)\u2013(A3) hold, the numerical scheme simplifies toui,\u03c3k, the upwinding value does not depend on i anymore such thatIn the definition of \u2211i=1nui,\u03c3k from below in the proof of the discrete entropy inequality; see This property is needed to control the sum This formulation is needed in the convergence analysis.We denote by 2.3Theorem 1Let (H1)\u2013(H4) and (A2) hold. Then there exists a solution  to Schemesatisfyinguk\u2208\ud835\udc9f\u203eand, if the initial data lie in\ud835\udc9f, uk\u2208\ud835\udc9f. If additionally Assumptions (A1) and (A3) hold, the solution is unique.(Existence and uniqueness of solutions). Since our scheme is implicit and nonlinear, the existence of an approximate solution is nontrivial. Therefore, our first result concerns the well\u2010posedness of the numerical scheme.u0k=1\u2212\u2211i=1nuik is nonnegative. Indeed, summing i\u2009=\u20091, \u2026.n, we obtainAssumption (A2) is needed to show that \u2211i=1nDiui,Kk=D1\u2212u0,Kk, and we can apply the discrete minimum principle, which then implies an uik. This bound allows us to apply a topological degree argument; see u0k, which solves a discrete nonlinear equation, and then to show the uniqueness of uik for i\u2009=\u20091, \u2026, n by introducing a semimetric d for two solutions uk=u1k\u2026unk and vk=v1k\u2026vnk and showing that it is monotone in k, such that a discrete Gronwall argument implies that uk\u2009=\u2009vk.Under Assumption (A2), it follows that Theorem 2Let Assumptions (H1)\u2013(H4) and (A1)\u2013(A3) hold. Then the solution to Schemeconstructed in theoremsatisfies the discrete entropy inequalitywith the discrete entropyand the discrete entropy production. mptions H\u2013(H4) andThe second result shows that the scheme preserves a discrete version of the entropy inequality.\u2211i=1nui,\u03c3k. In the continuous case, this sum equals 1\u2009\u2212\u2009u0. On the discrete level, this identity cannot be expected since the value of ui,\u03c3k depends on the upwinding value; see \u2211i=1nui,\u03c3k\u22651\u2212u0,\u03c3k; see Section k is the discrete counterpart of Assumption (A3) is required to estimate the expression Theorem 3Let (H1)\u2013(H4) and (A1)\u2013(A3) hold and let\ud835\udcafmand (\u25b3tm) be sequences of admissible meshes and time steps, respectively, such thath\ud835\udcafm\u21920and \u25b3tm\u00a0\u2192\u00a00 asLet  be the solution toconstructed in theoremThen there exist functions u0, u\u2009=\u2009 satisfyinguxt\u2208\ud835\udc9f\u203e,where u is a weak solution towith \u0393N\u2009=\u2009\u2202\u03a9), that is, for all\u03d5\u2208C0\u221e)and i\u2009=\u20091, \u2026, n,(Convergence of the approximate solution). Let H\u2013(H4) andThe main result of this paper is the convergence of the approximate solutions to a solution to the continuous cross\u2010diffusion system.u0,m1/2. The difficult part is to show the strong convergence of u0,m1/2ui,m, since there is no control on the discrete gradient of ui, m. The idea is to apply a discrete Aubin\u2013Lions lemma of \u201cdegenerate\u201d type, proved in lemma\u00a0The compactness of the concentrations follows from the discrete gradient estimates derived from the entropy inequality 33.1i\u2009=\u20091, \u2026, n. We show that this truncation is, in fact, not needed if the initial data are nonnegative. In the following let (H1)\u2013(H4) hold.Lemma 4uik). Let  be a solution toThenui,Kk\u22650for allK\u2208\ud835\udcaf, k \u2208{1, \u2026, N}, and i\u2009=\u00a01, \u2026, n. IfuiI>0andu\u203ei>0then alsoui,Kk>0for allK\u2208\ud835\udcaf, k \u2208{1, \u2026, N}.. Let Assumption (A2) hold and let  be a solution toandThenu0,Kk\u22650for allK\u2208\ud835\udcaf, k\u00a0\u2208\u00a0{1, \u2026, N}. Ifu0I>0andu\u203ei>0then alsou0,Kk>0for allK\u2208\ud835\udcaf, k\u00a0\u2208\u00a0{1, \u2026, N}. hold. Then schemehas a solution  which satisfiesuKk\u2208\ud835\udc9f\u203efor allK\u2208\ud835\udcafand. k\u00a0=\u00a00, we have uK0\u2208\ud835\udc9f\u203e by assumption. The function \u03a60 is uniquely determined by scheme\u00a0uk\u00a0\u2212\u00a01, \u03a6k\u00a0\u2212\u00a01) with uKk\u22121\u2208\ud835\udc9f\u203e. Let n and the number of cells K\u2208\ud835\udcaf. For given K\u2208\ud835\udcaf and i\u00a0=\u00a01, \u2026, n, we define the function uK0, , ui, \u03c3, u\u03c30, , and u^0,\u03c3i are defined in F=Fi,Ki=1,\u2026,n,K\u2208\ud835\udcaf. Then u\u21a6Fi,Ku0=mKui,K\u2212ui,Kk\u22121/\u25b3t is affine.The function F\u2009=\u00a00 satisfies u\u2208\ud835\udc9f or F\u2009=\u00a00 with \u03c1\u00a0\u2208\u00a0 satisfies We have proved above that any solution to F\u2009=\u00a00 has the unique solution u\u00a0=\u00a0uk\u00a0\u2212\u00a01 and consequently, The equation We argue by induction. For uk to F\u2009=\u20090 satisfying uk\u2208\ud835\udc9f\u203e. Hence, uk solves the original scheme\u00a0We infer the existence of a solution Lemmas\u00a03.2The proof of Theorem Step 1: uniqueness for u0. If k\u2009=\u20090, the solution is uniquely determined by the initial condition. Assume that u0k\u22121 is given. Thanks to Assumptions (A2) and (A3), the sum of i\u2009=\u20091, \u2026, n gives an equation for u0k=1\u2212\u2211i=1nuik :u0 and v0 be two solutions to the previous equation and set w0: = u0\u2009\u2212\u2009v0. Then w0 solvesLet wK0, /D, sum over K\u2208\ud835\udcaf, and use discrete integration by parts We multiply this equation by y|y|\u2009\u2212\u2009z|z|)(y\u2009\u2212\u2009z)\u2009\u2265\u20090 for y, z\u2009\u21a6\u2009z|z|, that the third term is nonnegative, too. Consequently, the three terms must vanish and this implies that wK0, \u2009=\u20090 for all K\u2208\ud835\udcaf. This shows the uniqueness for u0.The first two terms on the right\u2010hand side are clearly nonnegative. We infer from the elementary inequality \u2009=\u20090, we find after resolving the recursion thatSince ui,K\u2113 are nonnegative and bounded by 1 for all K\u2208\ud835\udcaf, for all \u2113\u2009\u2265\u20090 and for all 1\u2009\u2264\u2009i\u2009\u2264\u2009n, it is clear that \u2211\u2113=1k\u03f5S3\u2113\u21920 when \u03f5\u2009\u2192\u20090. Then, we may perform the limit \u03f5\u2009\u2192\u20090 in the previous inequality yielding d\u03f5\u2009\u2192\u20090. A Taylor expansion as in Zamponi and J\u00fcngel [d\u03f5ukvk\u226518\u2211K\u2208\ud835\udcafmK\u2211i=1nui,Kk\u2212vi,Kk2. We infer that uk\u2009=\u2009vk, finishing the proof.As the densities 44.1Proof of Theorem 2logui,Kk,\u03f5/u0,Kk,\u03f5, where ui,Kk,\u03f5\u2254ui,Kk+\u03f5 for i\u00a0=\u00a00, \u2026, n. The regularization is necessary to avoid issues when the concentrations vanish. After this multiplication, we sum the equations over i\u00a0=\u00a01, \u2026, n and K\u2208\ud835\udcaf and use discrete integration by parts to obtainThe idea is to multiply h(u)\u2009\u2212\u2009h(v)\u2009\u2264\u2009h'(u)(u\u2009\u2212\u2009v) for all u, The convexity of y, z\u2009>\u20090. Then, by the Cauchy\u2013Schwarz inequality,In order to estimate the remaining terms, we recall two elementary inequalities. Let Inequality u0,Kk=1\u2212\u2211i=1nui,Kk in A2 to find thatWe use the definition of B1 by using the abbreviation ui,\u03c3k,\u03f5=ui,\u03c3k+\u03f5:We rewrite B11. Indeed, if u0,Kk\u2264u0,Lk, we have ui,\u03c3k=ui,Kk and we use the first inequality in u0,Kk>u0,Lk then ui,\u03c3k=ui,Lk and we employ the second inequality in We apply inequality B2. In view of Assumption (A3), Equation\u00a0Finally, we consider A2 such thatThe last expression cancels with A0, A1, B1, and A2\u2009+\u2009B2, we deduce from Putting together the estimates for \u03f5\u2009\u2192\u20090, we infer that Since the right\u2010hand side converges to zero as First, we prove 4.2h\ud835\udcaf and time step \u25b3t. The scheme provides uniform v\u2208\u210b\ud835\udcaf,\u25b3t byFor the proof of the convergence result, we need estimates uniform in the mesh size Lemma 7Let (H1)\u2013(H4) and (A1)\u2013(A3) hold. The solution u to schemesatisfies the following uniform estimates:where the constant C\u2009>\u20090 is independent of the mesh\ud835\udcafand time step size \u25b3t.(A priori estimates). k\u00a0=\u00a01, \u2026, N to obtainWe claim that estimates t\u2009=\u20090 is bounded independently of the discretization, we infer immediately the bound for u01/2 in \u210b\ud835\udcaf,\u25b3t. For the bound on u01/2ui in \u210b\ud835\udcaf,\u25b3t, we observe thatSince the entropy at time ui,Therefore, together with the k\u2009=\u20090, \u2026, N and using the estimates from the entropy inequality, we achieve the bound on u01/2ui.Then, summing over \u03d5\u2208\u210b\ud835\udcaf be such that \u03d51,\ud835\udcaf=1 and let k\u00a0\u2208\u00a0{1, \u2026, N} and i\u00a0\u2208\u00a0{1, \u2026, n}. We multiply the Scheme\u00a0K and we sum over K\u2208\ud835\udcaf. Using successively discrete integration by parts, the rewriting of the numerical fluxes ui, we computeIt remains to prove estimate i\u2009=\u20091, \u2026, n,\u0394t\u22121u0k\u2212u0k\u22121=\u2212\u0394t\u22121\u2211i=1nuik\u2212uik\u22121 follows from those for i\u2009=\u20091, \u2026, n, completing the proof.This shows that, for 5In this section, we establish the convergence of the sequence of approximate solutions, constructed in theorem 5.1TKL as the cell with the vertexes xK, xL and those of \u03c3. For TK\u03c3 as the cell with vertex xK and those of \u03c3. Then \u03a9 can be decomposed as\ud835\udca9K denotes the set of neighboring cells of K. The discrete gradient \u2207\ud835\udcaf,\u25b3tv on \u03a9T: = \u03a9\u2009\u00d7\u2009 for piecewise constant functions v\u2208\u210b\ud835\udcaf,\u25b3t is defined bynKL denotes the unit normal on \u03c3\u2009=\u2009K|L oriented from K to L. To simplify the notation, we set \u2207m\u2254\u2207\ud835\udcafm,\u25b3tm. The solution to the approximate scheme\u00a0um0, , um1, , \u2026, un, m.Lemma 8There exist functionsandsuch that, possibly for subsequences, as m\u2009\u2192\u2009\u221e,um0, ) is uniformly bounded in \u210b\ud835\udcaf,\u25b3t. Indeed, by the First, we claim that , as defined in lemma\u00a0We finish the proof of theorem \u03d5\u2208C0\u221e) and let supp\u03d5\u2282\u03a9\u203e\u00d70Nm\u22121\u25b3tm . For the limit, we follow the strategy used, for instance, in Chainais\u2010Hillairet and coworkers Let The convergence results of lemma\u00a0\u03d5Kk=\u03d5xKtk, we multiply Scheme\u00a0\u25b3tm\u03d5Kk\u22121 and sum over K\u2208\ud835\udcafm and k\u2009=\u20091, \u2026, Nm. Thenm from now on to simplify the notation,Next, setting Fi0(m)\u2009\u2212\u2009Fi(m)\u2009\u2192\u20090 as i\u2009=\u20091, 2, 3. Then, because of F10(m)\u2009+\u2009DF20(m)\u2009\u2212\u2009DF30(m)\u2009\u2192\u20090, which finishes the proof. We start by verifying that F10(m)\u2009\u2212\u2009F1(m)\u2009\u2192\u20090. For this, we rewrite F1(m) and F10(m), using \u03d5KN=0:The aim is to show that \u03d5 and the uniform ui, we find thatIn view of the regularity of u0,\u03c3k1/2=u0,Kk1/2+u0,\u03c3k1/2\u2212u0,Kk1/2, i.e.Using discrete integration by parts, the second integral becomesF20(m)\u2009=\u2009G1(m)\u2009+\u2009G2(m), whereFurthermore, we write F21(m)\u2009\u2212\u2009G1(m)\u2009\u2192\u20090, F22(m)\u2009\u2192\u20090, and G2(m)\u2009\u2192\u20090. This implies thatThe aim is to show that \u03d5, by taking the mean value over TKL,C\u2009>\u20090 only depends on \u03d5. It yieldsF21(m)\u2009\u2212\u2009G1(m)|\u2009\u2192\u20090 as First we notice that, due to the admissibility of the mesh and the regularity of F22(m) and G2(m). To this end, we remark that d\u03c3\u2264h\ud835\udcaf and hence, together with the regularity of \u03d5, and the Cauchy\u2010Schwarz inequality,It remains to analyze the expressions G2(m) can be estimated in a similar way.The term F30(m)\u2009\u2212\u2009F3(m)|\u2009\u2192\u20090. The is completely analogous to the previous arguments, sinceFinally, we need to show that |Fi0(m)\u2009\u2212\u2009Fi(m)|\u2009\u2192\u20090 for i\u2009=\u20091, 2, 3, and since F1(m)\u2009+\u2009DF2(m)\u2009\u2212\u2009DF3(m)\u2009=\u20090, the convergence u solves Summarizing, we have proved that |61/2\u2212) inside the channel region. These ions contribute to the permanent charge density f\u2009=\u2009\u2212uox/2 in the Poisson equation, but also to the total sum of the concentrations. We consider three further types of ions: calcium , sodium , and chloride . While the concentrations of these ion species satisfy the evolution equations\u00a0NA\u2009\u2248\u20096.022\u2009\u00d7\u20091023\u2009mol\u22121 is the Avogadro constant and utyp\u2009=\u20093.7037\u2009\u00d7\u20091025\u2009L\u22121 the typical concentration  and centers inside the channel region. On these discs, the oxygen concentration is set to Ek with respect to the stationary solution, whereui,K\u221e\u03a6\u221e is the steady state determined from the boundary data. Figure\u00a0L1 norms of the concentrations and electric potential decay with exponential rate. Interestingly, after some initial phase, the convergence is rather slow and increases after this intermediate phase. This phase can be explained by the degeneracy at u0\u2009=\u20090, which causes a small entropy production slowing down diffusion. Indeed, as shown in Gerstenmayer and J\u00fcngel The simulations suggest that the solution tends towards a steady state as \u03bb in the Poisson equation has the value \u03bb2\u2009=\u20094.68\u2009\u00d7\u200910\u22124. Therefore, the drift term is moderately convective (in the sense that the modulus of the electric field |\u2207\u03a6| is moderately large). Figure\u00a0L1 error for a smaller value of \u03bb\u2009=\u20092\u2009\u00d7\u200910\u22124. It turns out that the scheme is still entropy dissipative, but the entropy decay is much slower and the L1 error decreases with smaller rate.The scaled permittivity constant h\ud835\udcaf\u22480.01. This mesh is obtained from the coarse mesh by a regular refinement, dividing the triangles into four triangles of the same shape. The reference solution is compared to approximate solutions on coarser nested meshes. In Figure\u00a0L1 norm between the reference solution and the solutions on the coarser meshes at two fixed time steps k\u2009=\u200950 and k\u2009=\u20091,400 are plotted. We clearly observe the expected first\u2010order convergence in space.Since Assumptions (A1)\u2013A3) are not satisfied in our test case, the convergence result of theorem  are not"}
{"text": "Bdnf gene. Viable and fertile homozygote animals were generated, with the GFP signal marking neuronal cell bodies translating the Bdnf mRNA. Importantly, the distribution of immunoreactive BDNF remained unchanged, as exemplified by its accumulation in mossy fiber terminals in the transgenic animals. GFP-labeled neurons could be readily visualized in distinct layers in the cerebral cortex where BDNF has been difficult to detect with currently available reagents. In the hippocampal formation, quantification of the GFP signal revealed that <10% of the neurons do not translate the Bdnf mRNA at detectable levels, with the highest proportion of strongly labeled neurons found in CA3.While BDNF is receiving considerable attention for its role in synaptic plasticity and in nervous system dysfunction, identifying brain circuits involving BDNF-expressing neurons has been challenging. BDNF levels are very low in most brain areas, except for the large mossy fiber terminals in the hippocampus where BDNF accumulates at readily detectable levels. This report describes the generation of a mouse line allowing the detection of single brain cells synthesizing BDNF. A bicistronic construct encoding BDNF tagged with a P2A sequence preceding GFP allows the translation of BDNF and GFP as separate proteins. Following its validation with transfected cells, this construct was used to replace the endogenous  Bdnf mRNAs using GFP as a surrogate marker. The availability of these transgenic animals will also help in understanding the action of drugs such as ketamine that are thought to act by increasing Bdnf translation.BDNF is a highly conserved growth factor known to be essential for the function of the nervous system. Its very low abundance in the brain has retarded the development of drugs targeting BDNF-expressing neurons in disease-relevant brain areas. The present report describes a novel approach allowing the localization of single cells in the adult mouse brain actively translating Bdnf transcription between different brain regions and from one neuron to the next as long documented by in situ hybridization studies in the adult brain of mice, rats, and pigs  is a secreted growth factor required for the development and function of the nervous system . In humaand pigs . Given t neurons , differe neurons  suggest ding GFP . In addiFP :ttaattaagccaccatgaccatcctgtttctgaccatggtcatcagctacttcggctgcatgaaggccgctcccatgaaggaagtgaacgtgcacggccagggcaacctggcttatcctggcgtgcggacacacggcaccctggaatctgtgaacggccctagagctggcagcagaggcctgaccacaacaagcctggccgacaccttcgagcacgtgatcgaggaactgctggacgaggaccagaaagtgcggcccaacgaggaaaaccacaaggacgccgacctgtacaccagcagagtgatgctgagcagccaggtgcccctggaaccccctctgctgttcctgctggaagagtacaagaactacctggacgccgccaacatgagcatgagagtgcggagacacagcgacccagctagaagaggcgagctgagcgtgtgcgacagcatcagcgagtgggtcacagccgccgacaagaaaaccgccgtggacatgtctggcggcaccgtgaccgtgctggaaaaggtgccagtgtccaagggccagctgaagcagtacttctacgagacaaagtgcaaccccatgggctacaccaaagagggctgcagaggcatcgacaagagacactggaacagccagtgcagaaccacccagagctacgtgcgggccctgacaatggacagcaagaaaagaatcggctggcggttcatcagaatcgacaccagctgcgtgtgcaccctgaccatcaagagaggcagaggatccggcatggtgtctaagggggaggaactgttcaccggcgtggtgcccatcctggtggaactggatggcgacgtgaacggacacaagttcagcgtgtccggcgagggcgaaggcgacgccacatacggaaagctgaccctgaagttcatctgcaccaccggcaagctgcccgtgccttggcctaccctcgtgaccacactgacctacggcgtgcagtgcttcagcagataccccgaccatatgaagcagcacgacttcttcaagagcgccatgcccgagggctacgtgcaggaaagaaccatcttctttaaggacgacggcaactacaagaccagggccgaagtgaagttcgagggcgacaccctcgtgaacagaatcgagctgaagggcatcgacttcaaagaggacggcaacatcctgggccacaagctggagtacaactacaacagccacaacgtgtacatcatggccgacaagcagaaaaacggcatcaaagtgaacttcaagatccggcacaacatcgaggacggctccgtgcagctggccgaccactaccagcagaacacccctatcggcgacggccctgtgctgctgcctgacaaccactacctgagcacccagtccgccctgagcaaggaccccaacgagaagagggaccacatggtgctgctggaattcgtgaccgccgctggcatcaccctgggcatggacgagctgtacaaatgaggcgcgcc;Bdnf-Gfp : nls-Gfp (BamHI and AscI restriction sites are underlined): ggatccggcgccaccaatttcagcctgctgaaacaggccggcgacgtggaagagaaccctggccctccaaagaagaagcggaaggtcatggtgtccaagggcgaggaactgttcaccggcgtggtgcccatcctggtggaactggatggcgacgtgaacggccacaagttcagcgtgtccggcgagggcgaaggcgacgccacctatggcaagctgacactgaagttcatctgcaccaccggcaagctgcccgtgccttggcctaccctcgtgacaaccctgacctacggcgtgcagtgcttcagcagataccccgaccacatgaagcagcacgacttcttcaagagcgccatgcccgagggctacgtgcaggaacggaccatcttctttaaggacgacggcaactacaagaccagggccgaagtgaagttcgagggcgataccctcgtgaaccggatcgagctgaagggcatcgacttcaaagaggacggcaacatcctgggccacaagctggagtacaactacaacagccacaacgtgtacatcatggccgacaagcagaaaaacggcatcaaagtgaacttcaagatcaggcacaacatcgaggacggctccgtgcagctggccgaccactaccagcagaacacccccatcggagatggccccgtgctgctgcccgacaaccactacctgagcacacagagcgccctgtccaaggaccccaacgagaagagggaccacatggtgctgctggaatttgtgaccgccgctggcatcacactgggcatggacgagctgtacaagtgaggcgcgcc.P2a-Sv40The PacI/AscI restricted BDNF\u2013GFP fragment was ligated into the identically restricted pAAV plasmid . The BDNThe biosynthesis and secretion of tagged BDNF proteins were analyzed using HEK293 cells transfected with plasmids encoding wild-type (WT) BDNF, BDNF\u2013GFP, and BDNF-P2A-GFP. The enhanced version of GFP was used throughout. Cultures were maintained in Gibco DMEM supplemented with 10% FBS, 1% GlutaMAX and 1% nonessential amino acids . Transfections were performed in a six-well format using 2 \u03bcg of the indicated DNAs combined with 4 \u03bcl of Invitrogen Lipofectamine 2000 transfection reagent (Thermo Fisher Scientific) diluted within Gibco Opti-MEM medium (Thermo Fisher Scientific). Five hours after transfection, HEK293 cells were cultured in N2B27 medium consisting of equal volumes of Gibco Neurobasal medium and DMEM-F12 (Thermo Fisher Scientific), 1% B27 supplement (Thermo Fisher Scientific), 1% GlutaMAX, and 1% penicillin-streptomycin . BSA (Sigma-Aldrich) was used at a reduced concentration of 75 \u03bcg/ml to facilitate the analysis of the conditioned media by SDS-PAGE. BDNF levels were quantified in conditioned media, and brain lysates by ELISA .d-lysine (Sigma-Aldrich), cells were maintained in Neurobasal medium supplemented with 1% GlutaMAX supplement, 1% Penstrep, and 2% SM1 supplement (Stem Cell Technologies). Neurons were cultured for up to 12 d with 50% media changes performed three times weekly. Subsequent transfections were performed on E14.5 neurons at 5DIV using 0.5 \u03bcg of indicated DNAs and 1 \u03bcl of Lipofectamine 2000 (see above). Depolarization of E17.5 neurons at DIV11 was achieved by supplementing media with 1 mm 4-aminopyridine  for 24 h.Cortices of mice at embryonic day 14.5 (E14.5) for transfection and TrkB phosphorylation assays and E17.5 for immunostaining studies were collected in Hanks\u2019 buffered salt solution (Sigma-Aldrich) and trypsinized in 1 mg/ml trypsin (Worthington) for 20 min at 37\u00b0C. The reaction was then stopped using 1 mg/ml trypsin inhibitor (Sigma-Aldrich) before the addition of 1 mg/ml DNase I (Thermo Fisher Scientific) and gentle dissociation with a 5 ml serological pipette. Cells were then pelleted by centrifugation at 1400 rpm for 5 min and resuspended in DMEM supplemented with 2% FBS, 1% GlutaMAX, and 1% Penstrep. Three hours after plating into wells coated with poly-z-stack images .Twenty-four hours after transfection or treatment with 4-AP, neurons were briefly washed with PBS and fixed with 4% paraformaldehyde  for 15 min. After a 5 min permeabilization with PBS containing 0.1% Triton X-100 , cells were incubated for 1 h in blocking solution [3% donkey serum (Sigma-Aldrich) and 1% BSA in PBS-T] at room temperature (RT). Coverslips were then incubated overnight in primary antibodies diluted in blocking solution at the following concentrations: anti-BDNF mAb #9 7 \u03bcg/ml; , chickenm sodium orthovanadate (Sigma-Aldrich) to inhibit phosphatase activity and analyzed by SDS-PAGE for TrkB phosphorylation.The conditioned media of transfected HEK293 cells transfected with BDNF cDNAs were standardized to a BDNF concentration of 25 ng/ml after quantification using a BDNF ELISA . Before m Tris-HCl, 150 mm NaCl, 1 mm EDTA, 0.1% SDS, 0.2% sodium deoxycholate, and 1% Triton X-100) supplemented with phosphatase and protease inhibitor cocktail mixes, 10 \u03bcm phenanthroline monohydrate, 10 mm aminohexanoic acid, 10 \u03bcg/ml aprotonin, and 2 mm sodium orthovanadate . Lysates and conditioned media were centrifuged at 15,000 rpm to remove insoluble components before analysis by SDS-PAGE. Proteins were separated on 4\u201312% NuPAGE Bis-Tris gels (Invitrogen) and transferred to GE Healthcare Protran NC nitrocellulose membranes (Thermo Fisher Scientific) using a Trans-Blot semi-dry transfer unit (Bio-Rad). Membranes were subsequently blocked for 1 h in blocking solution [5% blotting-grade blocker (Bio-Rad) and 1% BSA in TBS containing 0.1% Tween ] and then probed overnight at 4\u00b0C with antibodies to \u03b2-actin , BDNF , BDNF propeptide , GFP , or phosphoTrkA (Tyr674/675)/TrkB  in blocking solution (1:2000). Following three 10 min washes in TBS-T, membranes were incubated at RT with HRP-conjugated anti-goat , anti-mouse, anti-rabbit  or anti-chicken  secondary antibodies within blocking solution (1:2000). After a further three 20 min washes in TBS-T, membranes were developed using WesternBright ECL HRP Substrate (Advansta). Densitometric analysis of all blots was performed using quantification functions on Bio-Rad ImageLab software. For blots requiring BDNF quantification, recombinant BDNF standards (Regeneron/Amgen) between 300 and 18.75 pg were run alongside to create calibration curves, as appropriate.Homogenized brain tissues, HEK293 cells, and cultured neurons were incubated for 20 min on ice in RIPA buffer (50 mBdnf knock-out (\u2212Bdnf/\u2212) animals were generated by crossing mice with two floxed Bdnf alleles  Act, 1986.  alleles  with mic alleles . Bdnf-P2sequence , the SV4sequence  and a GFated see C. After Three-month-old mice killed by pentobarbital injections were transcardially perfused with ice-cold PBS and 4% PFA, and their brains were removed and postfixed at RT for 4 h before cryoprotecting in 30% w/v sucrose solution at 4\u00b0C overnight. The following day, brains were embedded in OCT  compound and sectioned at 40 \u03bcm using a cryostat. Sections were blocked in blocking solution (3% donkey serum and 4% BSA in PBS-T) for 1 h before incubating overnight with mouse anti-BDNF (mAb #9) and chicken anti-GFP (1:1000). Sections were then washed three times for 10 min with PBS-T before incubating with Alexa Fluor 555 anti-mouse IgG and Alexa Fluor 488 anti-chicken IgY secondary antibodies  for 1 h at RT. After a final wash in PBS-T for 10 min, sections were incubated with DAPI diluted in PBS (1:4000) for 20 min and mounted onto precoated polylysine slides (VWR) with Dako fluorescence mounting media. Images of gross brain regions were acquired on a confocal microscope using a 20\u00d7 objective. For counts of GFP-positive nuclei, images were captured using a 63\u00d7 oil-immersion and then analyzed using FIJI  and CellBdnf-P2a-Gfp bodyweights, a Kruskal\u2013Wallis test was used with a Conover-Iman post hoc test for multiple comparisons. TrkB activation by BDNF fusion proteins was compared against that of BDNF-myc and analyzed using a one-sample t test. An adjusted p value (\u22640.0125) was considered significant after a Bonferroni correction for multiple comparisons. Differences in BDNF and GFP signal intensities in depolarized Bdnf-P2a-Gfp neurons were analyzed using a Student\u2019s t test. All results were expressed as the mean \u00b1 SE, and p \u2264 0.05 was considered to be significant unless otherwise stated.Data were analyzed using Microsoft Excel 2013 and RStudio software. For analysis of Bdnf constructs was first tested using transfected HEK293 cells. Constructs encoding unmodified BDNF, BDNF directly fused with GFP or separated from BDNF by a P2A sequence . In addition, the transgene did not measurably interfere with the fertility of the animals. Coronal brain sections of homozygous animals were then examined by confocal microscopy following perfusion, fixation, and staining with antibodies to GFP, BDNF, as well as nuclear staining with DAPI . Selective GFP labeling can be readily observed in distinct cortical layers, including layers 2, 5, and 6 as well as in distinct nuclei including the amygdala as well as all subdivisions of the hippocampal formation  35.5 \u00b1 2.11 ng/g WT cortex; and 45.4 \u00b1 6.38 ng/g for homozygote Bdnf-P2a-Gfp animals. The corresponding values for the hippocampus were 97.4 \u00b1 6.00 ng/g and 109.3 ng/g \u00b1 17.11 for WT and homozygous animals, respectively. To confirm that GFP is cleaved after the BDNF\u2013P2A sequence in vivo, we analyzed the lysates of cortices from wild-type, heterozygote, and Bdnf-P2a-Gfp homozygote animals by Western blot . Importantly, the distribution of the endogenous BDNF protein remains unchanged when comparing the staining of BDNF in the hippocampal formation  is reduced by \u223c50%. Caution should then be exerted when using comparatively large fusion constructs such as BDNF\u2013GFP as they would seem unlikely to efficiently activate TrkB. The results presented in Bdnf gene substitution with GFP directly coupled to the C terminal of BDNF . It is conceivable that GFP may have been cleaved from BDNF in the surviving animals as a functional cleavage site at the C terminal of BDNF has been noted following the isolation of BDNF from brain homogenates , possibly due to the selective inclusion of DAPI-positive cells in the granule cell and pyramidal cell layer. We also note that the results summarized in in situ hybridization studies in the rat (http://mouse.brain-map.org/gene/show/11850).The approach described here now opens the possibility to use the GFP signal to isolate and sort cells from the adult brain based on GFP signal intensity, thus allowing their individual profiling by RNAseq. Such results would help to inform the development of drugs selectively targeting these neurons and may deliver new clues as to the endogenous regulators of BDNF expression. Similar objectives could in principle also be reached by randomly isolating single cells from brain regions of interest without prior cell marking. As such data are indeed available for the adult mouse hippocampus , we comp the rat  and the Bdnf mRNA by allowing the selection of cells based on the intensity of the GFP signal. This should prove useful toward the development of new drugs aiming at selectively increasing the levels of BDNF in brain regions of interest, including rapidly acting depressants such as ketamine thought to act by increasing BDNF translation (In conclusion, the mouse line reported in this study should facilitate the detailed characterization of brain neurons actively translating the nslation ."}
{"text": "We prove several unique continuation results for biharmonic maps between Riemannian\u00a0manifolds. Moreover, let \u03d5:M\u2192N be a smooth map. One option of finding such maps is to find extrema of their energyharmonic mapsand are characterized by the vanishing of the so\u2010called tension field, that is\u03c4(\u03d5)\u2208\u0393(\u03d5*TN). Many results on harmonic maps have been obtained in the past decades, we refer to Finding interesting maps between Riemannian manifolds is one of the most challenging problems in modern Riemannian geometry. Suppose that biharmonic maps, which were first studied in \u03d5:M\u2192N, which is defined asbitension field, that is{ei},i=1,\u2026,m=dimM is a local orthonormal frame field tangent to M and \u0394 represents the Laplacian on \u03d5*TN. For recent results on biharmonic submanifolds we refer to Currently, there is a growing interest in a geometric variational problem that generalizes harmonic maps, the so\u2010called Theorem 1.1(Sampson) Let \u03d51,\u03d52:M\u2192N be two harmonic maps. If they agree on an open subset, then they are identical; and indeed the conclusion holds if \u03d51 and \u03d52 agree to infinitely high order at some point. In particular, a harmonic map which is constant on an open subset is a constant\u00a0map.In this article, we want to focus on one particular aspect regarding the qualitative behaviour of harmonic and biharmonic maps, namely the unique continuation property. For harmonic maps, the question of unique continuation was settled by Sampson Z\u2212\u2207Z and for the (rough) Laplacian on \u03d5*TN we use \u0394:=Trg(\u2207\u2207\u2212\u2207\u2207).Throughout this article, we will use the following sign conventions. For the Riemannian curvature tensor field, we use \u27e8\u00b7,\u00b7\u27e9 to indicate the Riemannian metrics on various vector bundles, and the same symbol \u2207 for the corresponding Riemannian\u00a0connections.In general, we will use the same symbol All manifolds are assumed to be connected and we will work only with smooth\u00a0objects.i,j,k for indices on the domain ranging from 1 to m and Greek indices \u03b1,\u03b2,\u03b3 for indices on the target which take values between 1 and n. When the range of the indices is from 1 to q, for some positive integer q, we will often denote them by a,b,c.Whenever we will make use of indices, we will use Latin indices We will use the Einstein summation convention, that is, repeated indices that are in the diagonal position indicate the\u00a0sum.2In this section, we will prove the results obtained in this\u00a0article.Our strategy of proof is to cleverly rewrite the biharmonic map equation such that we are effectively dealing with a second\u2010order problem to which we can apply the classical result from Aronszajn 2.1Theorem 2.1A be a linear elliptic second\u2010order differential operator defined on an open subset D of Rm. Let u= be functions in D satisfying the inequalityu=0 in an open subset of D, then u=0 throughout D.Let We recall the following ,\u00a0p.248T\u03d5:M\u2192N be a smooth map and let \u03c3 be a section in the pull\u2010back bundle \u03d5*TN. We consider a local chart  on M and a local chart  on N such that \u03d5(U)\u2282V. The section \u03c3 can be written as\u0394\u03c3 is given byui\u03b1=\u2202u\u03b1\u2202xi and (\u03d5\u03b2)\u03b2 is the corresponding expression for \u03d5 in local coordinates. Then, by a straightforward computation, we get , is empty, then, as M is connected, IntA=A=M, that is, \u03d5 is harmonic\u00a0everywhere.If the boundary of p0\u2208\u2202(IntA). Furthermore, let U be an arbitrary open subset containing p0. Clearly p0 does not belong to IntA and U\u2229IntA\u2260\u2205.Assume that there exists M\u2216A is open in M, p0 does not belong to M\u2216A and U\u2229(M\u2216A)\u2260\u2205.On the other hand, we havep0 includes an open subset where \u03c4(\u03d5) vanishes everywhere and an open subset where \u03c4(\u03d5)\u22600 at any\u00a0point.Thus, any open subset containing V be an open subset containing \u03d5(p0) and U an open subset containing p0 such that \u03d5(U)\u2282V. Assume that U and V are the domains of local charts. Consider an open subset D in M, containing p0, such that its closure in M is compact and contained in U. As we have seen, the set D contains an open subset where \u03c4(\u03d5) vanishes everywhere and an open subset where \u03c4(\u03d5)\u22600 at any\u00a0point.Let U, and so on D,Dgij, \u03d5\u03b1 and their derivatives, \u0393\u03b2\u03b1\u03b8 and their derivatives, are bounded on D. Applying Theorem\u00a0u=0, which implies that \u03c4(\u03d5) vanishes everywhere in D. This contradiction implies \u2202(IntA)=\u2205 and we end the proof.\u25a1Denote2.2In this subsection, we will prove Theorem\u00a0\u03b9:Sn\u2192Rn+1 and form the composite map \u03d5:=\u03b9\u2218\u03d5:M\u2192Rn+1.Lemma 2.3\u03d5:M\u2192Sn\u2282Rn+1 with the constant curvature metric, the equation for biharmonic maps acquires the formRicM denotes the Ricci tensor field on the domain manifold M and {ei},i=1,\u2026,m is an orthonormal frame\u00a0field.For To this end, we consider the inclusion map 1 the following formula holds true\u03c3\u2208\u0393(\u03d5*TN), then we can also think of \u03c3 as a section in the pull\u2010back bundle \u03d5*TRn+1 and the connections along \u03d5 and \u03d5 are related viaIt is well known that for a spherical target of constant curvature \u03b8(X)=\u27e8d\u03d5(X),\u03c4(\u03d5)\u27e9=\u27e8d\u03d5(X),\u0394\u03d5\u27e9, see By a direct calculation one finds that\u2207=\u2207\u03d5. To this end, we fix p\u2208M arbitrary and let {Xi},i=1,\u2026,m be a geodesic frame field around p. We calculate at the point p\u0394|d\u03d5|2 we again compute at pTrg\u22072d\u03d5=\u2212\u0394d\u03d5+d\u03d5(RicM), where \u0394 is the Hodge\u2013Laplacian acting on Rn+1\u2010valued one\u2010forms. Due to our sign convention, we have \u2212\u0394d\u03d5=d\u0394\u03d5=\u2207\u0394\u03d5 such that we findp\u25a1As a next step, we prove thatv=\u2207\u03d5 and w=\u0394\u03d5. In terms of a local chart  on M and {ea} denoting the canonical basis of Rn+1, a=1,\u2026,n+1, we can write v=\u03d5iadxi\u2297ea, where \u03d5ia=\u2202\u03d5a\u2202xi.Now, we define new variables Then any solution of \u03d51,\u03d52 and their corresponding new variables v1,v2,w1,w2. We set u:=y1\u2212y2. Note that the function u takes values in R(n+1)(m+2), which can be seen fromu as being defined on the image of U and ui(n+1)(m+1)+a=\u2202u(n+1)(m+1)+a\u2202xi.At this point we consider two biharmonic maps \u0394(w1\u2212w2) can be expressed as a sum of the components of u and their first order derivatives multiplied by terms that do not contain the components of u or their derivatives. This allows us to obtain the estimate for |\u0394(w1\u2212w2)|.Lemma 2.4D\u2282U be an open subset such that its closure in M is compact and included in U. Then, the following estimate holds on DC depends on D and the derivatives of \u03d51 and \u03d52 up to third\u00a0order.Let As a next step, we prove that D is assumed to be compact and both \u03d51,\u03d52 are smooth by assumption.\u25a1Using Proof of Theorem 1.4a,b=1,\u2026,(n+1)(m+2).Making use of u=0 on D. We finish the proof by denoting A:={p\u2208M:\u03d51(p)=\u03d52(p)} and using the same arguments as in the proof of Theorem 1.2.\u25a1Applying Theorem\u00a02.3In this section, we prove Theorem\u00a0\u03d5:M\u2192N and a local chart  on M and  on N such that \u03d5(U)\u2282V. In order to avoid any notational confusion the corresponding expression for \u03d5 in local coordinates will be denoted by \u03d5^, that isLemma 2.5\u03d5:M\u2192N be a biharmonic map. In terms of local coordinates, it satisfiesLet First, we consider a biharmonic map u\u03b8=\u03c4\u03b8(\u03d5)=\u0394\u03d5\u03b8+\u27e8d\u03d5\u03b1,d\u03d5\u03b2\u27e9\u0393\u03b1\u03b2\u03b8 in \u25a1Using the local form of the tension field v=\u03d5i\u03b1dxi\u2297e\u03b1, where \u03d5i\u03b1=\u2202\u03d5\u03b1\u2202xi and {e\u03b1} denotes the canonical basis of Rn.Now, as in the spherical case, we define new variablesThen, any solution of F3 is given byHere, \u03d51,\u03d52:M\u2192N and let  be a local chart on M and  a local chart on N such that \u03d51(U)\u2282V and also \u03d52(U)\u2282V.At this point, we consider two biharmonic maps u:=y1\u2212y2. Note that the function u takes values in Rn(m+2), which can be seen fromWe set |\u0394(w1\u2212w2)|, we need the image of V in Rn to be convex and with compact closure such that we can apply the mean value theorem for functions defined on the image of V. Then, when applying the mean value inequality, the standard norm of the differential of the function will be bounded on the image of V. So, first, we consider the image of V to be an open ball of radius \u03b5 in the Euclidean space Rn and then we shrink it to a ball of radius \u03b52. Accordingly, we consider a smaller domain U. Thus, we will make use of the mean value theorem for functions defined on the open ball of radius \u03b52 but which are also defined on the closed ball of radius \u03b52.In order to obtain the estimate for f:Bn(\u03b52)\u2192R we will apply the following inequalityy* belongs to the standard segment that joins y1 with y2.More precisely, for \u0394(w1\u2212w2) can be expressed as a sum of the components of u and their first\u2010order derivatives multiplied by terms that do not contain the components of u or their\u00a0derivatives.Then, as a next step, we prove that Lemma 2.6D\u2282U be an open subset such that its closure in M is compact and included in U. Then, the following estimate holds on DC depends on D and the derivatives of \u03d5^1 and \u03d5^2 up to third\u00a0order.Let We haveUsing The first term can be controlled as followsFor the second term we rewriteThe third contribution can be manipulated as followsRegarding the fourth term we findFor the fifth term we obtainThe sixth contribution can be estimated as\u25a1The seventh term can be estimated asProof of Theorem 1.4a,b=1,\u2026,n(m+2).Making use of u=0 on D. We finish the proof by denoting A:={p\u2208M:\u03d51(p)=\u03d52(p)} and using the same arguments as in the proof of Theorem\u00a0\u25a1Applying Theorem\u00a02.4Sn.In this section, we prove Theorem\u00a0Sn be the Euclidean unit sphere and denote by N and S the north and south pole, respectively. It is well known thatLet (ya) be local coordinates on Sn\u22121,a=1,\u2026,n\u22121. Then,  are local coordinates on Sn\u2216{N,S}.Let Sn are given by\u223c to indicate objects on Sn\u22121.In this geometric setup the Christoffel symbols on Sn\u22121 is given byThe equator  be a local chart on M and we denote the domain of the above local coordinates on Sn\u2216{N,S} by V. In addition, we assume that \u03d5(U)\u2282V.Now, let \u03d5 in local coordinates by \u03d5^, that is,We denote the corresponding expression for W is an open subset of U and \u03d5(W)\u2282Sn\u22121, that is, \u03d5(W)\u2282Sn\u22121\u2229V. Hence, in W we have \u03d5n=\u03c02. Now, define f:U\u2192R,f:=\u03d5n\u2212\u03c02. Clearly, f vanishes when restricted to W.Assume that D be an open subset of U such that its closure in M is compact and included in U, and W\u2282D\u2282U.Let Lemma 2.7f:U\u2192R defined above satisfies the following estimate on DThe function We will prove the following estimate:\u03942f=\u03942\u03d5n and we will use \u03b8=n in order to obtain the\u00a0estimates.We have |\u27e8\u2207\u0394\u03d5b,d\u03d5c\u27e9\u0393bcn|. We immediately obtainWe expand the first term as\u25a1We write the fourth term in the same way as the first one and, the only term we must estimate isProof of Theorem 1.6u and F. Applying Theorem\u00a0u=0 on D, that is, \u03d5 maps the whole of D into Sn\u22121.As in the proofs of Theorems\u00a0A:={p\u2208M:\u03d5(p)\u2208Sn\u22121} and using the same arguments as in the proof of Theorem\u00a0\u25a1We finish the proof by denoting 2.5 be a local chart on N such thatr denotes the dimension of the submanifold P. Moreover, we assume that the image of V in Rn is convex. For example, we can assume that it is the interior of an n\u2010cube with side lengths \u03b5/2 that lies inside an n\u2010cube with side lengths \u03b5.In this section, we prove Theorem\u00a0 be a local chart on M and assume that \u03d5(U)\u2282V. We denote the corresponding expression for \u03d5 in local coordinates by \u03d5^, that is,P is totally geodesic in N, we have on the intersection V\u2229PP\u0393\u03b1\u03b2\u03b8 denote the Christoffel symbols of the submanifold P for 1\u2a7d\u03b8,\u03b1,\u03b2\u2a7dr. On V\u2229P we also haveW is an open subset of U and \u03d5(W)\u2282P. Hence, on W we havea\u223c=1,\u2026,n\u2212r. Clearly, we have f:U\u2192Rn\u2212r and also f|W=0. Now, let D be an open subset of U such that its closure in M is compact and included in U, and W\u2282D\u2282U.In addition, let Lemma 2.8D, the function f:U\u2192Rn\u2212r defined above satisfies the following estimateC depends on the geometry of N.Restricted to We will prove the following estimate:b,c=1,\u2026,r. Since we have\u27e8\u2207\u0394\u03d5b,d\u03d5c\u27e9\u0393bcr+a\u223c. To this end we define a new function \u03d5^\u2032:U\u2192Rn by\u03d5\u2032:U\u2192N takes values in P. We note that, according to \u0393bcr+a\u223c=\u0393bcr+a\u223c(\u03d5)=0 on W, but at the points in D\u2216W the image of the map \u03d5 may not be in P such that \u0393bcr+a\u223c\u22600. However, on U, and thus also on D, we haveD\u25a1We make use of Proof of Theorem 1.7u and F. Applying Theorem\u00a0u=0 on D, that is, \u03d5 maps the whole of D into P.As in the proofs of Theorems\u00a0\u25a1We finish the proof using the same arguments as in the proof of Theorem\u00a0Remark 2.9\u03d5=\u03c8\u2218\u03d5 of a biharmonic map \u03d5 and a totally geodesic map \u03c8 is biharmonic. Indeed, for arbitrary maps \u03c8 and \u03d5 and for any \u03c3\u2208\u0393(\u03d5*TN) we have\u03c8 is totally geodesic we get \u2207X\u03d5d\u03c8(\u03c3)=d\u03c8(\u2207X\u03d5\u03c3), and then, by straightforward computations we find\u03d5 is biharmonic, \u03d5 is biharmonic\u00a0too.As to be expected, similar to the case of harmonic maps, the composition \u03c8 defines a regular, totally geodesic submanifold, we can give an alternative proof of the above fact, similar to the proof of Theorem\u00a0In the particular case when 2.6Theorem 2.10(Sampson) Assume that \u03d5:M\u2192N is a harmonic map, with q=\u03d5(p). Let S be a C2 hypersurface in N passing through q, at which point we assume that the second fundamental form is definite. If \u03d5 is not a constant mapping, then no neighbourhood of p is mapped entirely on the concave side of S.Another geometric application of the unique continuation property of harmonic maps was provided by Sampson [Besides the unique continuation property, the proof of this theorem makes use of the maximum principle. This powerful tool only exists for second\u2010order elliptic partial differential equations, not for fourth order ones, such that we cannot expect to find a generalization of Sampson's maximum principle for proper biharmonic\u00a0maps.\u03d5\u03b1 is biharmonic if and only ifIndeed, in \u03d5\u03b1 is harmonic if and only ifc1 and c2 are real\u00a0constants.The map \u03b1(t) admits a local extremum point at t0 such that \u03b1(t0)>0, then t0 is a minimum point. Indeed, consider t0\u2208I, I being an open interval of R, such that \u03b1(t)>0 on I. If t0 is an extremum point, harmonicity implies \u03b1\u2032\u2032(t0)=m\u03b1(t0)>0 and it follows that t0 is a minimum\u00a0point.We note that if \u03d5\u03b1 is biharmonic if and only ifc1, c2, c3 and c4 are real constants.The map Remark 2.11(1)\u03b1(t) that admit a local maximum point at t0 such that \u03b1(t0)>0. For example, we may consider\u03d5\u03b1) lies in the concave side of the m\u2010sphere Sm(1me) of radius 1me.In the biharmonic case, we have solutions (2)\u03b1(t) that admit a local minimum point at t0 such that \u03b1(t0)>0. For example, we haveWe also have solutions"}
{"text": "AbstractAnnamanumflavimaculatumsp. nov. is described and illustrated from Guizhou and Guangxi, China. Diagnosis for distinguishing the new species to its close congeners is presented and identification key to the genus is also updated. Annamanum Pic, 1925 is a large genus in the subfamily Lamiinae  with 30 described species , A.lunulatum , and A.magnum Holzschuh, 2017, this is the fourth Annamanum species recorded in the Leigongshan area.The genus  species  distribu species . Of thes species . With thSpecimens were collected by two collecting methods: net sweeping and six level Lindgren funnel traps  with 99% ethanol as lure. Collected specimens were pinned or glued on pinned paper cards. Labels were handwritten or printed in Chinese. Materials from Guizhou are preserved in the School of Life Sciences, Guizhou Normal University, Guiyang, Guizhou, China. Materials from Guangxi are preserved in Collection of Wen-Xuan Bi, Shanghai, China.Specimen examination and dissection were conducted under an AmScope SM-4TZ stereomicroscope. Adults were photographed with Canon EOS 6D digital camera equipped with EOS MP-E 65 lenses. Male genitalia were photographed with Olympus DP22 camera mounted on an Olympus SZX7 stereomicroscope.The collection acronyms used in the text are as follows:CBWX Collection Wen-Xuan Bi, Shanghai, China;GZNULSSchool of Life Sciences, Guizhou Normal University, Guiyang, China.Taxon classificationAnimaliaColeopteraCerambycidaeBF6CC0CF-7653-53CB-975F-C1349AD10D3Chttp://zoobank.org/C4EE8EDD-377D-4625-94DB-24C6FA355C94Queniao Tea Farm, Queniao Village, Leishan County, Guizhou Province, China.GZNULS)], HOLOTYPE / Annamanum / flavimaculatum / Shulin Yang [handwritten red label].Holotype male, glued on paper point, with genitalia in a separate microvial. Original label: \u201c\u4e2d\u56fd\u8d35\u5dde\u7701\u96f7\u5c71\u53bf\u65b9\u7965\u4e61\u96c0\u9e1f\u6751\u8336\u573a\uff0c2015\u5e746\u670818\u65e5\uff0c\u516d\u5c42\u6f0f\u6597\u8bf1\u6355\u5668\uff0c\u6768\u4e66\u6797\u91c7\u201d  (GZNULS); 1\u2640, original labels: \u201c\u4e2d\u56fd\u8d35\u5dde\u7701\u96f7\u5c71\u53bf\u96f7\u516c\u5c71\u56fd\u5bb6\u81ea\u7136\u4fdd\u62a4\u533a\uff0c2017\u5e748\u670827\u65e5\uff0c\u6768\u7ecd\u52c7\u91c7\u201d  (GZNULS); 1\u2640, Original label: \u201c\u4e2d\u56fd\u8d35\u5dde\u7701\u96f7\u5c71\u53bf\u65b9\u7965\u4e61\u96c0\u9e1f\u6751\u8336\u573a\uff0c2015\u5e746\u670818\u65e5\uff0c\u6768\u5149\u7956\u91c7\u201d  (GZNULS); 1\u2640, Original label: \u201c\u4e2d\u56fd\u8d35\u5dde\u7701\u96f7\u5c71\u53bf\u65b9\u7965\u4e61\u96c0\u9e1f\u6751\u8336\u573a\uff0c2019\u5e748\u67082\u65e5\uff0c\u6768\u4e66\u6797\u91c7\u201d  (GZNULS); 1\u2640, Original label: \u201c\u4e2d\u56fd\u8d35\u5dde\u7701\u96f7\u5c71\u53bf\u65b9\u7965\u4e61\u96c0\u9e1f\u6751\u8336\u573a\uff0c2019\u5e749\u670812\u65e5\uff0c\u6768\u4e66\u6797\u91c7\u201d  (GZNULS); 1\u2642, original labels: \u201c\u5e7f\u897f\u732b\u513f\u5c71\u8ff4\u9f99\u5bfa\uff0c1900 \u2013 1700 m, 2012.VII.23\uff0c\u6bd5\u6587\u70dc\u201d  (CBWX); 1\u2642, original labels: \u201c\u5e7f\u897f\u732b\u513f\u5c71\u4e09\u6c5f\u6e90\uff0c1950\u20132000 m, 2014.VII.30\uff0c\u5b8b\u6653\u5f6c\u201d  (CBWX); 1\u2640, original labels: \u201c\u5e7f\u897f\u4e34\u6842\u5e7f\u798f\u9876\uff0c1350 m, 2018.VII.2\u201d  (CBWX).Paratypes: 2\u2642\u2642, original labels: \u201c\u4e2d\u56fd\u8d35\u5dde\u7701\u96f7\u5c71\u53bf\u96f7\u516c\u5c71\u56fd\u5bb6\u7ea7\u81ea\u7136\u4fdd\u62a4\u533a\uff0c2012\u5e747\u670822\u65e5\uff0c2016\u5e747\u670813\u65e5\uff0c\u6768\u4e66\u6797\u91c7\u201d  , female . Measurements for specimens from Guanxi (Figure Head: black, frons generally densely punctured, vertex densely punctured, both frons and vertex covered with dense yellow hairs. Antennae of males exceed apex of elytra by six antennomeres; of females by five antennomeres; antennal tubercles strongly raised; scape and pedicel black, covered with long hairs, not erect but flat towards apex; cicatrix complete, narrow; rest of the antennomeres reddish brown, sparsely covered with white yellowish hairs; base and apex of each antennomere are covered with darker hairs. Eyes deeply emarginated; lower lobe twice as high as gena and one-fourth as wide as frons width between lower lobes of eyes. Labium with small sparse punctures and sparse long dark brown hairs. Mandibles with dense long yellow hairs at outer side and sparse hairs on the front. Thorax: Pronotum black with coarse granules, covered with yellow hairs, whose thickness varies among individuals; disk slightly raised; a small callus at each side of the apical margin, not extending beyond middle. Lateral spines strong, acute, slightly posteriorly and upwards curved. Sternum reddish brown, covered with dense yellow hairs. Scutellum covered with dense white yellowish pubescence, apex rounded. Mesosternal intercoxal process with a slightly projected antero-ventral tubercle (Figure Elytra: gradually tapered in male, less tapered in female, with irregular coarse granules obliquely protruding backwards and gradually smaller posteriorly. Basal fifth bulged between humeri and scutellum. Basal half black, intermingled with yellow and black hairs; black hairs forming a broad transverse black band at the middle, nearly reaching suture. Apical half reddish brown, without granules but with coarse small punctures, mostly covered by yellow hairs forming a large yellow marking with intermingled black hair dots; some of these dots are near suture on the anterior half of the yellow marking and weakly form a line which extends obliquely towards outer margin of elytron from middle of the marking; black dots larger in female. Apex nearly rounded, slightly truncated in inner half. Legs: with dense white yellowish pubescence; femora dark brown, clubbed, not cylindrical, with sparse small punctures; tibiae reddish brown. Abdomen: reddish brown, ventrites with white yellowish pubescence intermingled with sparse punctures. Pygidium shallowly truncated at apex, deeper at the middle in some males. The sexual dimorphism is not very conspicuous. Male genitalia (Figure Female genitalia (Figure e Figure  13.8\u201317.e Figure  17.5\u201320.e Figure . Elytra:a Figure , laterala Figure  moderateThe name refers to the yellow patch on the apical half of elytron.China: Guizhou, Guangxi."}
{"text": "A spreadsheet summary of these case histories are included along with a separate spreadsheet, by which maximum likelihood assessment was performed. These data transparently enable researchers to access case history input parameters and processing details, and to compare the case history processing protocols with the ones of different researchers (e.g.: \u201cThe influence of SPT procedures in soil liquefaction resistance evaluations.\u201d This data article provides a summary of seismic soil liquefaction triggering and non-triggering case histories, which were compiled, screened for data completeness and quality, and then processed for the development of triggering relationships proposed in \u201cSPT-based probabilistic and deterministic assessment of seismic soil liquefaction triggering hazard\u201d  Specifications TableValue of the data\u2022USA, Japan, Argentina, China, Philippines.This database presents a summary of 210 liquefaction and non-liquefaction case histories compiled from a number of earthquake events in \u2022This database can be used for the development of new probabilistic and deterministic seismic soil liquefaction triggering relationships.\u2022Additionally, a comparison among case histories and protocol details of similar databases \u2022Critical Depth Lower Bound (ft)\u2022Critical Depth (ft) =Critical depth for liquefaction\u2022Critical Depth (m) =Critical depth for liquefaction\u2022critical depth (ft) =Standard deviation of critical depth\u03c3\u2022Water Depth (ft)\u2022water table (ft) =Standard deviation of water table\u03c3\u20221(pcf) =Unit weight above ground water table\u03b3\u2022\u03b31 (pcf) =Standard deviation of \u03b31\u03c3\u20222 (pcf) =Unit weight below ground water table\u03b3\u2022\u03b32 (pcf) =Standard deviation of \u03b32\u03c3\u2022v (psf) =Vertical total stress\u03c3\u2022\u03c3v (psf) =Standard deviation of \u03c3v\u03c3\u2022v\u05f3 (psf) =Vertical effective stress\u03c3\u2022\u03c3v\u05f3 (psf) =Standard deviation of \u03c3\u03c3v\u05f3\u03c3\u2022\u03c3\u03c3\u05f3= Correlation coefficient between \u03c3v\u05f3 and \u03c3v\u03c1\u2022d=Shear mass modal participation factor, commonly referred to as cyclic shear stress reduction factorr\u2022rd=Standard deviation of\u03c3\u2022max (g)= Peak horizontal accelerationa\u2022amax=Standard deviation of amax\u03c3\u2022MwMagnitude for K\u2022Magnitude Scaling Factor (MSF)\u2022KMw=Standard deviation of KMw\u03c3\u2022\u03c3\u05f3v,Mw=Actual CSR, not normalized to a reference state of \u03c3\u05f3v = 1\u202fatm, Mw =7.5CSR\u2022CSR=Standard deviation of CSR\u03c3\u05f3v,Mw\u03c3\u2022\u03c3\u05f3v=100kPa,Mw=7.5= CSR normalized to \u03c3\u05f3v =1\u202fatm, Mw=7.5CSR\u2022CSRN=Standard deviation of CSR\u03c3\u05f3v=100kPa,Mw=7.\u03c3\u202250 (mm)=Median grain sizeD\u2022D50=Standard deviation of D50\u03c3\u2022FC %=Fines content\u2022FC %=Standard deviation of FC\u03c3\u2022Sampler (SS: standard sampler or SWL: sampler without liners)\u2022S= SPT correction for sampler configuration detailsC\u2022Rod length\u2022R= SPT correction factor for the rod lengthC\u2022Borehole diameter (mm)\u2022B= SPT correction for borehole diameterC\u2022Energy Ratio \u2022E= SPT correction for hammer energy ratio (ER)C\u2022Number of N values\u2022N (mean) =Field measured standard penetration resistance (blows/30\u202fcm)\u2022N =Standard deviation of N\u03c3\u2022N=Overburden correction factorC\u2022SPT Correction (Energy + Rod length)\u20221)60= Standard penetration test blowcount value corrected for overburden, energy, equipment and procedural factors.(N\u2022(N1)60 =Standard deviation of (N1)60\u03c3\u2022Reference\u2022s,40ft (fps) = Shear wave velocity for the upper 40 ftV*\u2022\u03c3=Correction for overburden stressK\u2022Mw=Correction for magnitude (duration) effectsK\u2022Mw=7.5,\u03c3\u05f3v=100kPA= CSR normalized to \u03c3\u05f3v =100\u202fkPa, Mw=7.5CSR\u20221,60,CS=Fines-corrected N1,60 valueNThe output table in each spreadsheet contains the following columns:1,60 and CSR graphs, a summary page including surface manifestation, location of the site and expertise comments, are given in Additionally, for each case history, a detailed presentation of the borehole data, maps, laboratory data, corrected NOn the basis of the compiled resulting database, the maximum likelihood assessment was performed by using the Microsoft Excel Software Tool: Solver. The spreadsheet, which presents MLE assessments, is presented in"}
{"text": "After eCPR, patients commonly present with a combined respiratory and metabolic acidosis . It is c2 .2 and pH with hospital survival in eCPR.The aim of the present study was to correlate arterial paCO2 and pH values were (mean\u2009\u00b1\u2009standard deviation) 38.3\u2009\u00b1\u20098.9\u2009mmHg/7.28\u2009\u00b1\u20090.14 (+\u20091\u2009h), 38.5\u2009\u00b1\u20098.5\u2009mmHg/7.30\u2009\u00b1\u20090.11 (+\u20093\u2009h), 38.72\u2009\u00b1\u20097.42\u2009mmHg/7.31\u2009\u00b1\u20090.11 (+\u20096\u2009h), 38.62\u2009\u00b1\u20097.26\u2009mmHg/7.34\u2009\u00b1\u20090.10 (+\u200912\u2009h), and 38.22\u2009\u00b1\u20095.62\u2009mmHg/7.38\u2009\u00b1\u20090.09 (+\u200924\u2009h), respectively. When comparing patients with paCO2 <\u200935, 35\u201345, and >\u200945\u2009mmHg, survival was statistically similar for all observed time points. There was however a highly significant association between hospital survival and pH when comparing groups with pH <\u20097.3, 7.3\u20137.4, and >\u20097.4 . There was however a significant linear correlation of maximum NSE and pH at 1, 3, and 6\u2009h after eCPR .As secondary endpoint and surrogate for neurological outcome, neuron-specific enolase (NSE) was analyzed. Maximum NSE measured within 72\u2009h after eCPR was 150.8\u2009\u00b1\u2009145.1\u2009\u03bcg/l (mean\u2009\u00b1\u2009standard deviation). When correlating maximum NSE with paCO2. Also elevated NSE as a marker for neural injury did correlate with pH but not with paCO2. Being a retrospective, observational, single-center study, inherent limitations and biases are to be presumed and findings are to be considered hypothesis generating. Until further data are available however, it might be reasonable to correct both respiratory and metabolic acidosis in eCPR patients.In this registry study, we found a strong correlation between hospital survival and arterial pH but no such correlation with paCO"}
{"text": "This paper defines a Sahlqvist fragment for relevant logic and establishes that each class of frames in the Routley-Meyer semantics which is definable by a Sahlqvist formula is also elementary, that is, it coincides with the class of structures satisfying a given first order property calculable by a Sahlqvist-van Benthem algorithm. Furthermore, we show that some classes of Routley-Meyer frames definable by a relevant formula are not elementary. Sahlqvist Correspondence Theorem is a celebrated result in modal logic ., 16.8, 1mutatis mutandis, the argument for the Sahlqvist Correspondence Theorem  can be ented in . Moreovebi-approximation semantics (which unfortunately is much more complicated than the Routley-Meyer framework). Other settings without boolean negation where Sahlqvist theorems have been obtained are positive modal logic POS must be the case. Hence, (1). So (1) and (2) are actually equivalent.Next we observe that one may assume that any unary predicate  appearing in POS, appears also in the antecedent of (1). For suppose not, that is, there a unary predicate P1,\u2026,Pn\u2200x1,\u2026,xk(Up\u2264(P1)\u2227\u2026\u2227Up\u2264(Pn)\u2227REL\u2227(DN)AT\u2227IMP \u2283\u00acNEG\u2228POS).(3)\u2003\u2200Observe that \u00acNEG \u2228POS is a positive formula.It is easy to see that (1) is equivalent to \u03c01(xi1),\u2026,\u03c0k(xik) are all the conjuncts of (DN)AT and IMP in the antecedent of (3) in which the unary predicate Pi occurs. Then if \u03c0j(xij) appears in one of the conjuncts in IMP, it must be a formula of the form \u2200yz(Rxijyz \u2227\u2203b(Ob \u2227 b \u2264 y) \u2283 Piz), in which case we define \u03c3(\u03c0j(xij)) = \u03bbu.(\u2203yz(Rxijyz \u2227\u2203b(Ob \u2227 b \u2264 y) \u2227 z \u2264 u)). On the other hand if \u03c0j(xij) appears in one of the conjuncts in (DN)AT we put \u03c3(\u03c0j(xij)) = \u03bbu.(xij \u2264 u) in case \u03c0j(xij) = Pixij and \u03c3(\u03c0j(xij)) = \u03bbu.(xij\u2217\u2217\u2264 u) in case \u03c0j(xij) = \u00ac\u00acPixij\u2217\u2217. Note that \u03c3 is a well-defined function and that for any \u03c0j(xij), if \u03c0j(xij)[w] then \u2200y(\u03c3(\u03c0j(xij)(y) \u2283 Piy)[w]. Next define \u03b4(Pi) = \u03bbu.(\u03c3(\u03c01(xi1))(u) \u2228\u2026 \u2228 \u03c3(\u03c0k(xik))(u)). Now, if (DN)AT[ww1\u2026wk] and IMP[ww1\u2026wk] then \u2200u(\u03b4(Pi)(u) \u2283 Piu)[ww1\u2026wk].Suppose Up\u2264(\u03b4(Pi)) holds for any Pi. This is so because each \u03c3(\u03c0j(xij))(u) is upward closed under \u2264, and any union of upward closed sets under \u2264 is also upward closed under \u2264. Moreover, by the reflexivity of \u2264, [\u03b4(P1)/P1\u2026\u03b4(Pn)/Pn](DN)AT and [\u03b4(P1)/P1\u2026\u03b4(Pn)/Pn]IMP will hold trivially. So, from \u03b4(P1)/P1\u2026\u03b4(Pn)/Pn]\u2200x1,\u2026,xk(Up\u2264(P1)\u2227\u2026\u2227Up\u2264(Pn)\u2227REL\u2227(DN)AT\u2227IMP \u2283\u00acNEG\u2228POS),(4)\u2003(\u00acNEG \u2228POS)).(5)\u2003\u2200Given that any unary predicate appearing in POS also appears in the antecedent of (4), (5) is a first order formula in the signature of Routley-Meyer frames, which contains R,\u2217, and O as the only non-logical symbols.At this point we should note that P1,\u2026,Pn be arbitrary and suppose further that ww1\u2026wk] and IMP[ww1\u2026wk], it must be the case that \u2200u(\u03b4(Pi)(u) \u2283 Piu)[ww1\u2026wk]. Hence, by Lemma 3, We have seen that (1) implies (5). All that is left is to show that (5) implies (1). Recall that (1) is equivalent to (3). Thus it suffices to prove that (5) implies (3). Assume (5), let The procedure in the above proof is better understood by working out some examples, which we will do next for the benefit of the reader.M is any model based on Rwyz and V on V (p) = {x : y \u2264 x} and V suffices to show that By the Sahlqvist-van Benthem algorithm, the following list of equivalences must hold: V such that By the Sahlqvist-van Benthem algorithm, x(x\u2217\u2217\u2264 x \u2227 x \u2264 x\u2217\u2217) when we consider validity with respect to all the worlds in O of the frame. This is not the same in general as \u2200x(x = x\u2217\u2217) which is the condition usually required to validate \u223c\u223c p \u2192 p. However, it is certainly the case that, using the construction from Theorem 5 . Suppose M is an arbitrary model based on Rwyz \u2203u1u2(Ru1u2y \u2227 Pu1 \u2227 Qu2) both hold, u1 \u2264 z and u2 \u2264 z, and, by the Hereditary Lemma, Pz and Qz as desired. Hence, V on V (p) = {x : u1 \u2264 x} and V (q) = {x : u2 \u2264 x} suffices to guarantee that y,z(Rxyz \u2283 x \u2264 z \u2227 y \u2264 z) by some manipulations.By the Sahlqvist-van Benthem algorithm, x,y,z(Ox \u2227 Rxyz \u2283\u2203u2u3(Ryu2u3 \u2227\u2203b(Ob \u2227 b \u2264 u2) \u2227 u3 \u2264 z)) when we consider correspondence with respect to the worlds in O of a given frame. This condition is actually equivalent to the condition \u2200x\u2203b(Ob \u2227 Rxbx) corresponding in  then, then \u2200y(Piy \u2283 \u03c3(\u03c0j(xij))(y))[w]. Next define \u03b4(Pi) = \u03bbu.(\u03c3(\u03c01(xi1))(u) \u2227\u2026 \u2227 \u03c3(\u03c0k(xik))(u)). Now, if (T)NAT[ww1\u2026wk] and IMP[ww1\u2026wk] then \u2200u(Piu \u2283 \u03b4(Pi)(u))[ww1\u2026wk]. The remainder of the proof is as before but using again the contrapositive formulation of Lemma 3 and noting that the intersection of a collection of upward closed sets under \u2264 is also upward closed under \u2264. \u25a1Finally, let p \u2192t)\u2227\u223ct \u2192\u223c p. Using our Sahlqvist-van Benthem algorithm, we obtain that the following equivalences hold: Up\u2264(P), Rwyz, \u2200u,v(Ryuv \u2227 Pu \u2283\u2203b(Ob \u2227 b \u2264 v)) and \u00ac\u2203b(Ob \u2227 b \u2264 y\u2217), it must be that \u2203u2(Ryz\u2217u2 \u2227\u00ac\u2203b(Ob \u2227 b \u2264 u2)). Thus if Pz\u2217, \u2203b(Ob \u2227 b \u2264 u2), which is a contradiction, so \u00acPz\u2217. Consequently M based on V (p) = {x : z\u2217\u2264 x}. Now, if Ryu1u2 and u1 \u2208 V (p), that is, z\u2217\u2264 u1, we that, by p3, Ryz\u2217u2, and hence, \u2203b(Ob \u2227 b \u2264 u2). This shows that Consider the dual relevant Sahlqvist implication (\ud835\udf03) defined by \u0398(\u03d5) = \ud835\udf03 \u2192 \u03d5, and applications of \u2228 where the disjuncts share no propositional variable in common.A relevant Sahlqvist formula is any formula built up from  relevant Sahlqvist implications, propositional variables, and negated propositional variables using \u2227, the operations on formulas \u0398 (for any propositional variable free relevant formula Every relevant Sahlqvist formula has a local first order correspondent onRoutley-Meyer frames.Immediate from Lemma 11, Lemma 16 and Lemma 9. The only thing the reader should note is that In this paper we have defined a fragment of relevant languages analogue to the Sahlqvist fragment of modal logic. We then went to establish that every class of Routley-Meyer frames definable by a formula in this fragment is actually elementary. This isolates a modest but remarkable collection of relevant formulas. We also showed that there are properties of Routley-Meyer frames definable by relevant formulas which are not first order axiomatizable."}
{"text": "Nature Communications 10.1038/s41467-019-10107-1, published online 21 May 2019.Correction to: The original version of this Article contained errors in Figs\u00a02 and 3. In Fig. 2b, the scale bar label \u2018500 \u00b5m\u2019 was originally incorrectly given as \u2018200 \u00b5m\u2019. In Fig.\u00a02d, the scale bar label \u2018500 \u00b5m\u2019 was originally incorrectly given as \u2018100 \u00b5m\u2019. In Fig.\u00a03c, the scale bar label \u2018200 \u00b5m\u2019 was originally incorrectly given as \u2018100 \u00b5m\u2019. This has been corrected in the PDF and HTML versions of the Article."}
{"text": "In oncology, the aim of dose-finding phase I studies is to find the maximum tolerated dose for further studies. The use of combinations of two or more agents is increasing. Several dose-finding designs have been proposed for this situation. Numerous publications have however pointed out the complexity of evaluating therapies in combination due to difficulties in choosing between different designs for an actual trial, as well as complications related to their implementation and application in practice.In this work, we propose R functions for Wang and Ivanova\u2019s approach. These functions compute the dose for the next patients enrolled and provide a simulation study in order to calibrate the design before it is applied and to assess the performance of the design in different scenarios of dose-toxicity relationships. This choice of the method was supported by a simulation study which the aim was to compare two designs in the context of an actual phase I trial: i) in 2005, Wang and Ivanova developed an empirical three-parameter model-based method in Bayesian inference, ii) in 2008, Yuan and Yin proposed a simple, adaptive two-dimensional dose-finding design. In particular, they converted the two-dimensional dose-finding trial to a series of one-dimensional dose-finding sub-trials by setting the dose of one drug at a fixed level. The performance assessment of Wang\u2019s design was then compared with those of designs presented in the paper by Hirakawa et al. (2015) in their simulation context.It is recommended to assess the performances of the designs in the context of the clinical trial before beginning the trial. The two-dimensional dose-finding design proposed by Wang and Ivanova is a comprehensive approach that yields good performances. The two R functions that we propose can facilitate the use of this design in practice. The study received regulatory approval in January 2013 and officially started in France in May 2013. The protocol was amended in 2014. The main modification concerned the dose-escalation method to apply the method published by Wang and Ivanova !=nd2 | dim(npts)[2]!=nd1)stop(\u201cVerify dimensions matrix npts (dimension nd2xnd1)\u201d)## ntoxif (is.matrix(ntox)==FALSE)stop(\u201cntox need to be matrix (dimension nd2xnd1)\u201d)if (dim(ntox)[1]!=nd2 | dim(ntox)[2]!=nd1)stop(\u201cVerify dimensions matrix ntox (dimension nd2xnd1)\u201d)## ntox<=nptstemp <- npts-ntoxif (sum(temp<0)!=0)stop(\u201cVerify matrix ntox and npts (Nb of toxicty greated than nb of patients)\u201d)\u2003\u2003# Compute the probabilities pijk\u2003pijk_model = function\u2003{\u2003\u2003pijk <- (1\u2014((1-ai)^alpha*(1-bj)^(Beta+Gamma*(1-ai))))\u2003\u2003return(pijk)\u2003}\u2003#likelihood is the joint conditional density of the observed responses (=f(D_k|theta) in the Ivanova\u2019s paper)\u2003likelihood = function\u2003{\u2003\u2003rst <- 1\u2003\u2003for(i in 1:nd1){\u2003\u2003\u2003for(j in 1:nd2){\u2003\u2003\u2003\u2003pijk <- pijk_model\u2003\u2003\u2003\u2003rst\u2003\u2003<- rst * pijk^ntox * (1-pijk)^\u2003\u2003\u2003}\u2003\u2003}\u2003\u2003return(rst)\u2003}\u2003\u2003#Estimation_pijk: to estimate the pijk\u2003\u2003 Estimation_pijk = function\u2003\u2003{\u2003\u2003\u2003#set.seed(55)\u2003\u2003\u2003alpha_samp <- rexp\u2003\u2003\u2003Beta_samp\u2003<- rexp\u2003\u2003\u2003Gamma_samp <- rexp\u2003\u2003\u2003pijk\u2003\u2003\u2003\u2003<- matrix\u2003\u2003\u2003num\u2003\u2003\u2003\u2003 <- numeric(nsamp)\u2003\u2003\u2003f_lik\u2003 \u2003\u2003<- likelihood\u2003\u2003\u2003for(j in 1:nd1)\u2003\u2003\u2003{\u2003\u2003\u2003\u2003for(k in 1:nd2)\u2003\u2003\u2003\u2003{\u2003\u2003\u2003\u2003\u2003pijk_samp = pijk_model\u2003\u2003\u2003\u2003\u2003pijk <- sum(pijk_samp*f_lik)/sum(f_lik)\u2003\u2003\u2003\u2003}\u2003\u2003\u2003}\u2003\u2003\u2003return(pijk)\u2003\u2003}\u2003\u2003# Recommended_dose: is the decision rule to recommend the next dose: the dose associated with a probability the closest to the target\u2003\u2003Recommended_dose = function\u2003\u2003{\u2003\u2003\u2003next_d1 <- NA\u2003\u2003\u2003next_d2 <- NA\u2003\u2003\u2003# Next assignement is to one of the combinations from: , , , , , , et \u2003\u2003\u2003next_assig <- list,nd1), cur_d2),\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 #\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 c,nd1), cur_d2),\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003#\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 c,\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003#\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 c,nd2)),\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003#\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 c,nd2)),\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003#\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 c,nd1), min,nd2)), #\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 c,nd1), min,nd2))) #\u2003\u2003\u2003next_assig\u2003\u2003\u2003\u2003\u2003\u2003<- next_assig[!duplicated(next_assig)]\u2003\u2003\u2003length_next_assig\u2003\u2003 <- length(next_assig)\u2003\u2003\u2003difference\u2003\u2003\u2003\u2003\u2003\u2003<- c\u2003\u2003\u2003for(ll in 1:length_next_assig)\u2003\u2003\u2003{\u2003\u2003\u2003\u2003difference\u2003\u2003<- c)\u2003\u2003\u2003}\u2003\u2003\u2003ind_min_dist <- which.min(difference)\u2003\u2003\u2003next_d1\u2003\u2003\u2003 <- next_assig[[ind_min_dist]][1]\u2003\u2003\u2003next_d2\u2003\u2003\u2003 <- next_assig[[ind_min_dist]][2]\u2003\u2003\u2003return)\u2003\u2003}\u2003\u2003colnames(npts) <-paste\u2003\u2003rownames(npts) <-paste\u2003\u2003colnames(ntox) <-paste\u2003\u2003rownames(ntox) <-pastepijk\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<- Estimation_pijkif(sum(npts)<ntot)\u2003\u2003\u2003{cur_d1\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <- Recommended_dose[[1]]cur_d2\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <- Recommended_dose[[2]]MTD_lv\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <- NA\u2003\u2003\u2003}if(sum(npts)==ntot)\u2003\u2003\u2003{difference\u2003\u2003\u2003\u2003\u2003\u2003 <- abs(pijk-target)MTD_comb\u2003\u2003\u2003\u2003 \u2003<- which, arr.ind = T)cur_d1\u2003\u2003\u2003\u2003\u2003\u2003 \u2003 <- MTD_comb[2]cur_d2\u2003\u2003\u2003\u2003\u2003\u2003 \u2003 <- MTD_comb[1]MTD_lv\u2003\u2003\u2003\u2003\u2003\u2003\u2003<- apply\u2003\u2003\u2003}print(\u201c- Current Dose agent A (cur_d1):\u201d)print(cur_d1)print(\u201c- Current Dose agent B (cur_d2):\u201d)print(cur_d2)print(\u201c- Number of patients treated at each dose level (npts):\u201d)print(npts)print(\u201c- Number of DLT observed at each dose level (ntox):\u201d)print(ntox)print(\u201c- Updated estimate of the MTD for each dose level of the agent B (MTD_lv):\u201d)print(MTD_lv)return)}ExampleThe example concerns a phase I trial evaluating two agents, Agent A with five dose levels and Agent B with three dose levels. Thirty patients will be included in the trial, by cohorts of 3 patients. The first cohort will be allocated to dose level . Assuming that no toxicity is been observed at this dose, the next dose for the next cohort of patients is given as follows:SBS_designTo apply our R function, the arguments will be set as:prior_1 \u2190 cprior_2 \u2190 cnsamp \u2190 1000target \u2190 0.3nd1 \u2190 5nd2 \u2190 3ntot \u2190 30# Data# matrix presenting the number of patients treated for each combinationnpts \u2190 matrix# matrix presenting the number of toxicities for each combinationntox \u2190 matrix# Cohort 1, cur_d1 \u2190 1 #agent Acur_d2 \u2190 1 #agent Bnpts \u2190 3SBS_design\u201cprior_1 must be a vector (length nd1)\u201d. If there is no problem and the parameters are considered correct, calculations can then be continued, with the following message: \u201c+ Parameters OK\u201d.The program first verifies consistencies in the parameters and error messages can be displayed. For example, if the length of prior_1 is superior or inferior to nd1, the program stops and signals that A summary of the number of patients treated at each dose level, the numbers of DLTs observed at each dose level and the next dose to be explored are then given.The function SIM_design is used to generate simulation replicates of a phase I trial using the two-dimensional dose-finding design developed by Wang and Ivanova, under a specified dose-toxicity configuration.Argumentsnd1: The number of dose levels of Agent And2: The number of dose levels of Agent Btox_true: A specified dose-toxicity configurationprior_1: A vector of initial guesses of toxicity probabilities associated with the doses for Agent Aprior_2: A vector of initial guesses of toxicity probabilities associated with the doses for Agent Btarget: The target DLT ratentot: The number of patients enrolledncoh_st: The number of patients enrolled in the cohort used for the starting rulencoh: The number of patients enrolled in the cohort used for the main designnsamp: The number of iteration needed for the MCMCnsim: The number of simulationsseed: Seed of the random number generatorValueresult_pt: Average number of patients treated at the test doses. If nsim = 1, this is a vector of length n indicating the doses assigned to the patients in the simulated trial.result_tox: Average number of toxicities seen at the test doses. If nsim = 1, this is a vector of length n indicating the toxicity outcomes of the patients in the simulated trial.result_MTD: Distribution of the MTD estimates. If nsim = 1, this is a single numerical value of the recommended MTD in simulated trial. If nsim = 1, this is a single numerical value of the recommended MTD in the simulated trial.result_MTD_lv: Distribution of the MTD estimates for each dose level of Agent B. If nsim = 1, this is a single numeric value of the recommended MTD for each dose level of Agent B, in the simulated trial.R codeSIM_design=function{print(\u201c*****************************************************************\u201d)printprint(\u201c*\u2003\u2003\u2003\u2003 in Discrete Dose Space of Wang K and Ivanova A\u2003\u2003\u2003\u2003\u2003\u2003*\u201d)print\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 \u2003\u2003 *\u201d)print(\u201c*****************************************************************\u201d)print(\u201c* *Institut Curie\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003*\u201d)print(\u201c* 25 rue d\u2019Ulm\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 *\u201d)print(\u201c* 75005\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 *\u201d)print(\u201c* France\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003*\u201d)print(\u201c*****************************************************************\u201d)print# Data Checkprint(\u201c+--------------------+\u201d)print(\u201c|\u2003 Data Check\u2003 |\u201d)print(\u201c+--------------------+\u201d)quote(\u201c\u2003\u201d)# For nd1 and nd2## nd1 et nd2 are numericif(is.numeric(nd1)==FALSE) stop(\u201cnd1 must be numeric\u201d)if(is.numeric(nd2)==FALSE) stop(\u201cnd2 must be numeric\u201d)## nd1 nd2 entier>0if(ceiling(nd1)-nd1!=0 | nd1<=0) stop(\u201cnd1 must be Integer (>0)\u201d)if(ceiling(nd2)-nd2!=0 | nd2<=0) stop(\u201cnd2 must be Integer (>0)\u201d)## nd1 >= nd2if(nd1 < nd2) stop(\u201cnd1 must be superior to nd2\u201d)# For prior_1 prior_2## For prior_1if(is.vector(prior_1)==FALSE)stop(\u201cprior_1 must be a vector (Numeric Argument)\u201d)if(length(prior_1)!=nd1)stop(\u201cprior_1 must be a vector (lengh nd1)\u201d)if(min(prior_1)<0 | max(prior_1)>1) stop(\u201cVerify probability prior_1 (<0 or >1)\u201d)## For prior_2if(is.vector(prior_2)==FALSE)stop(\u201cprior_2 must be a vector (Numeric Argument)\u201d)if(length(prior_2)!=nd2)stop(\u201cprior_2 must be a vector (lengh nd2)\u201d)if(min(prior_2)<0 | max(prior_2)>1) stop(\u201cVerify probability prior_2 (<0 or >1)\u201d)# For tox_true## tox_true is matrixif(is.matrix(tox_true)==FALSE)stop(\u201ctox_true must be a matrix (Numeric Argument)\u201d)## Dimension de la matriceif(dim(tox_true)[[1]]!=nd2 | dim(tox_true)[[2]]!=nd1)stop(\u201ctox_true must be a matrix (Dimension: nd2 x nd1)\u201d)## Proba of tox between 0 and 1if(min(tox_true)<0 | max(tox_true)>1) stop(\u201cVerify probability tox_true (<0 or >1)\u201d)# Target between 0 and 1if(min(target)<0 | max(target)>1) stop(\u201cVerify probability of tox (<0 or >1)\u201d)# For ntot, ncoh_st, ncoh## ntot, ncoh_st, ncoh, nsamp are numericif(is.numeric(ntot)==FALSE) stop(\u201cntot must be numeric\u201d)if(is.numeric(ncoh_st)==FALSE) stop(\u201cncoh_st must be numeric\u201d)if(is.numeric(ncoh)==FALSE) stop(\u201cncoh must be numeric\u201d)if(is.numeric(nsamp)==FALSE) stop(\u201cnsamp must be numeric\u201d)if(is.numeric(nsim)==FALSE) stop(\u201cnsim must be numeric\u201d)## ntot, ncoh_st, ncoh entier>0if(ceiling(ntot)-ntot!=0 | ntot<=0) stop(\u201cntot must be Integer (>0)\u201d)if(ceiling(ncoh_st)-ncoh_st!=0 | ncoh_st<=0) stop(\u201cncoh_st must be Integer (>0)\u201d)if(ceiling(ncoh)-ncoh!=0 | ncoh<=0) stop(\u201cncoh must be Integer (>0)\u201d)if(ceiling(nsim)-nsim!=0 | nsim<=0) stop(\u201cnsim must be Integer (>0)\u201d)print(\u201c+ Parameters OK\u201d)result_MTD\u2003\u2003\u2003\u2003<- matrixresult_tox\u2003\u2003\u2003\u2003\u2003<- matrixresult_pt\u2003\u2003\u2003\u2003\u2003 <- matrixset.seed(graine)result_MTD\u2003\u2003\u2003\u2003<- matrixresult_MTD_lv\u2003 \u2003 <- matrixresult_tox\u2003\u2003\u2003\u2003\u2003<- matrixresult_pt\u2003\u2003\u2003\u2003\u2003 <- matrixDesign_Wang_Ivanova = function{\u2003d1_max\u2003\u2003\u2003\u2003 <- 0\u2003d2_max\u2003\u2003\u2003\u2003 <- 0\u2003### Start-up Rule\u2003cur_d1\u2003\u2003\u2003\u2003<- 1\u2003cur_d2\u2003\u2003\u2003\u2003<- 1\u2003start_up_end <- FALSE\u2003###################Other used functions############\u2003\u2003# gene_tox: to generate toxicity data\u2003gene_tox = function\u2003{\u2003\u2003Z_tox = runif(ncoh)\u2003\u2003y_tox = as.numeric(Z_tox <= true_tox)\u2003\u2003return(y_tox)\u2003}\u2003#Compute the probabilities pijk\u2003pijk_model = function\u2003{\u2003\u2003pijk <- (1\u2014((1-ai)^alpha*(1-bj)^(Beta+Gamma*(1-ai))))\u2003\u2003return(pijk)\u2003}\u2003#likelihood is the joint conditional density of the observed responses (=f(D_k|theta) in the Ivanova\u2019s paper)\u2003likelihood = function\u2003{\u2003\u2003rst <- 1\u2003\u2003for(i in 1:nd1){\u2003\u2003\u2003for(j in 1:nd2){\u2003\u2003\u2003pijk <- pijk_model\u2003\u2003\u2003\u2003\u2003\u2003\u2003 rst\u2003\u2003<- rst * pijk^ntox * (1-pijk)^\u2003\u2003 \u2003}\u2003\u2003}\u2003\u2003return(rst)\u2003}\u2003#Estimation_pijk: to estimate the pijk\u2003Estimation_pijk = function\u2003{\u2003\u2003alpha_samp \u2003 <- rexp\u2003\u2003Beta_samp\u2003\u2003<- rexp\u2003\u2003if (Interaction==0) {Gamma_samp <- 0}\u2003\u2003else {Gamma_samp <- rexp}\u2003\u2003pijk\u2003\u2003\u2003\u2003\u2003<- matrix\u2003\u2003num\u2003\u2003\u2003\u2003 <- numeric(nsamp)\u2003\u2003f_lik\u2003\u2003\u2003\u2003<- likelihood\u2003\u2003for(j in 1:nd1)\u2003\u2003{\u2003\u2003\u2003for(k in 1:nd2)\u2003\u2003\u2003{\u2003\u2003\u2003\u2003pijk_samp = pijk_model\u2003\u2003\u2003\u2003pijk <- sum(pijk_samp*f_lik)/sum(f_lik)\u2003\u2003\u2003}\u2003\u2003}\u2003\u2003return(pijk)\u2003}\u2003### The start-up Rule\u2003start_up = function\u2003{y\u2003\u2003\u2003\u2003\u2003\u2003\u2003<- c\u2003npts\u2003\u2003\u2003\u2003\u2003\u2003<- ntox <- matrix\u2003colnames(npts) \u2003<-paste\u2003rownames(npts)\u2003<-paste\u2003colnames(ntox) \u2003<-paste\u2003rownames(ntox)\u2003<-paste\u2003n_rem\u2003\u2003\u2003\u2003\u2003 <- ntot\u2003while(start_up_end == FALSE && n_rem >= ncoh_st)\u2003{\u2003\u2003d1_stup <- cur_d1\u2003\u2003d2_stup <- cur_d2\u2003\u2003y_tox <- c\u2003\u2003n_rem <- n_rem-ncoh_st\u2003\u2003#agent 2 is in rows from 1 to nd2 beginning by the end, as in the paper\u2003\u2003npts <- npts + ncoh_st\u2003\u2003true_tox <- tox_true\u2003\u2003y_tox\u2003 <- gene_tox\u2003\u2003y\u2003\u2003\u2003 <- c)\u2003\u2003ntox <- ntox+sum(y_tox)\u2003\u2003# if no tox observed => escalate the first agent\u2003\u2003if(sum(y_tox) == 0)\u2003\u2003{\u2003\u2003\u2003if(cur_d1 < nd1)\u2003\u2003\u2003{\u2003\u2003\u2003\u2003d1_stup <- cur_d1+1\u2003\u2003\u2003}\u2003\u2003\u2003if(cur_d1 == nd1 && cur_d2 < nd2)\u2003\u2003\u2003{\u2003\u2003\u2003\u2003d1_stup <- cur_d1-2\u2003\u2003\u2003\u2003d2_stup <- cur_d2+1\u2003\u2003\u2003}\u2003\u2003\u2003if(cur_d1 == nd1 && cur_d2 == nd2)\u2003\u2003\u2003{\u2003\u2003\u2003\u2003start_up_end <- TRUE\u2003\u2003\u2003}\u2003\u2003}\u2003\u2003# if at least one tox is observed\u2003\u2003else\u2003\u2003{\u2003\u2003\u2003if(cur_d1 > 2 && cur_d2 < nd2)\u2003\u2003\u2003{\u2003\u2003\u2003\u2003d1_stup = cur_d1-2\u2003\u2003\u2003\u2003d2_stup = cur_d2+1\u2003\u2003\u2003}\u2003\u2003\u2003else\u2003\u2003\u2003{\u2003\u2003\u2003\u2003start_up_end <- TRUE\u2003\u2003\u2003}\u2003\u2003}\u2003\u2003cur_d1 <- d1_stup\u2003\u2003cur_d2 <- d2_stup\u2003}\u2003### Next dose\u2003pijk\u2003\u2003\u2003\u2003 <- Estimation_pijk\u2003dose_level_1 <- which.min)\u2003cur_d1 <- dose_level_1\u2003cur_d2 <- 1\u2003nused\u2003<- ntot-n_rem\u2003return)\u2003}\u2003# Recommended_dose: is the decision rule to recommend the next dose: the dose associated with a probability the closest to the target\u2003Recommended_dose = function\u2003{\u2003\u2003next_d1 <- NA\u2003\u2003next_d2 <- NA\u2003\u2003# Next assignement is to one of the combinations from: , , , , , , et \u2003\u2003next_assig <- list,nd1), cur_d2),\u2003\u2003\u2003\u2003\u2003\u2003 #\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 c,nd1), cur_d2),\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003#\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 c,\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003#\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 c,nd2)),\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003#\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 c,nd2)),\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003#\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 c,nd1), min,nd2)), #\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 c,nd1), min,nd2))) #\u2003\u2003next_assig\u2003\u2003\u2003\u2003\u2003\u2003<- next_assig[!duplicated(next_assig)]\u2003\u2003length_next_assig\u2003 \u2003 <- length(next_assig)\u2003\u2003difference\u2003\u2003\u2003\u2003\u2003\u2003<- c\u2003\u2003for(ll in 1:length_next_assig)\u2003\u2003{\u2003\u2003\u2003difference\u2003\u2003<- c)\u2003\u2003}\u2003\u2003ind_min_dist <- which.min(difference)\u2003\u2003next_d1\u2003\u2003 <- next_assig[[ind_min_dist]][1]\u2003\u2003next_d2\u2003\u2003 <- next_assig[[ind_min_dist]][2]\u2003\u2003return)\u2003}\u2003## nd1 et nd2 are numeric\u2003if(is.numeric(nd1)==FALSE) stop(\u201cnd1 must be numeric\u201d)\u2003if(is.numeric(nd2)==FALSE) stop(\u201cnd2 must be numeric\u201d)\u2003## nd1 nd2 entier>0\u2003if(ceiling(nd1)-nd1!=0 | nd1<=0) stop(\u201cnd1 must be Integer (>0)\u201d)\u2003if(ceiling(nd2)-nd2!=0 | nd2<=0) stop(\u201cnd2 must be Integer (>0)\u201d)\u2003## nd1 >= nd2\u2003if(nd1 < nd2) stop(\u201cnd1 must be superior to nd2\u201d)\u2003# For tox_true\u2003## tox_true is matrix\u2003if(is.matrix(tox_true)==FALSE)stop(\u201cfu must be a matrix (Numeric Argument)\u201d)\u2003## Proba of tox between 0 and 1\u2003if(min(tox_true)<0 | max(tox_true)>1) stop(\u201cVerify probability tox_true (<0 or >1)\u201d)\u2003rst_start_up <- start_up\u2003npts\u2003\u2003\u2003 <- rst_start_up$npts\u2003ntox\u2003\u2003\u2003 <- rst_start_up$ntox\u2003cur_d1\u2003\u2003<- rst_start_up$cur_d1\u2003cur_d2\u2003\u2003<- rst_start_up$cur_d2\u2003ntot\u2003\u2003\u2003 <- rst_start_up$ntot\u2003n_rem\u2003\u2003<- rst_start_up$n_rem\u2003nused\u2003 \u2003<- rst_start_up$nused\u2003y\u2003\u2003\u2003\u2003<- rst_start_up$y\u2003pijk\u2003 \u2003 <- rst_start_up$pijk\u2003### Two-Dimensional Trial\u2003for(i in 1:(n_rem %/% ncoh))\u2003{\u2003\u2003npts <- npts + ncoh\u2003\u2003true_tox\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<- tox_true\u2003\u2003y_tox\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<- gene_tox\u2003\u2003y\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<- c)\u2003\u2003ntox <- ntox+sum(y_tox)\u2003\u2003pijk\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<- Estimation_pijk\u2003\u2003next_RD\u2003\u2003 \u2003 \u2003\u2003\u2003\u2003\u2003\u2003 <- Recommended_dose\u2003\u2003cur_d1\u2003\u2003 \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <- next_RD[[1]]\u2003\u2003cur_d2\u2003\u2003 \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <- next_RD[[2]]\u2003}\u2003## target between 0 and 1\u2003if(min(target)<0 | max(target)>1) stop(\u201cVerify probability of tox (<0 or >1)\u201d)\u2003### MTD\u2003difference <- abs(pijk-target)\u2003MTD_comb <- which, arr.ind = T)\u2003MTD\u2003\u2003\u2003<- c)#c\u2003### MTD by level\u2003MTD_lv\u2003\u2003<- apply\u2003return)}ntot_used = 0for(i in 1:nsim){result = Design_Wang_Ivanovaresult_MTD[(nd2-result$MTD[2]+1),result$MTD[1]]\u2003\u2003<- (result_MTD[(nd2-result$MTD[2]+1),result$MTD[1]]+1)result_pt\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<- (result$npts+result_pt)result_tox\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <- (result$ntox+result_tox)MTD_lv\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <- as.numeric(result$MTD_lv)MTD_lv\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 <- ifelse(is.na(MTD_lv),0,MTD_lv)ntot_used\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<- ntot_used+sum(result$npts)ntot_mean\u2003\u2003\u2003\u2003\u2003 \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003<- ntot_used/i\u2003for(ii in 1:nd2)\u2003{\u2003\u2003if(MTD_lv[ii]!=0)\u2003\u2003{result_MTD_lv] <- \u2003\u2003}\u2003\u2003}}Rst_MTD\u2003\u2003\u2003\u2003 <- (result_MTD*100)/nsimRst_MTD_lv\u2003\u2003\u2003<- (result_MTD_lv*100)/nsimRst_pt\u2003\u2003\u2003 \u2003\u2003<- ((result_pt/nsim)/ntot_mean)*100Rst_tox\u2003\u2003\u2003\u2003\u2003 <- ((result_tox/nsim)/ntot_mean)*100# Imput Parametersprint(\u201c+--------------------------+\u201d)print(\u201c|\u2003Imput Parameters\u2003 |\u201d)print(\u201c+--------------------------+\u201d)print(paste(\u201c\u2003- Number of dose level agent A (nd1):\u201d,nd1))print(paste(\u201c\u2003- Number of dose level agent B (nd2):\u201d,nd2))print(paste(\u201c\u2003- Prior distribution agent A (prior_1):\u201d,paste(c(prior_1),collapse=\u201c\u2003\u201d)))print(paste(\u201c\u2003- Prior distribution agent B (prior_2):\u201d,paste(c(prior_2),collapse=\u201c\u2003\u201d)))print(paste(\u201c\u2003- Target DLT Rate (target):\u201d,target))print(paste(\u201c\u2003- Nb of patients enrolled (ntot):\u201d,ntot))print(paste(\u201c\u2003- Nb of patients enrolled in the cohort used for the starting rule (ncoh_st):\u201d,ncoh_st))print(paste(\u201c\u2003- Nb of patients enrolled in the cohort used for the main design (ncoh):\u201d,ncoh))print(paste(\u201c\u2003- Number of iteration needed for the MCMC (nsamp):\u201d,nsamp))print(paste(\u201c\u2003- Number of simulation (nsim):\u201d,nsim))print(\u201c+--------------------------+\u201d)print(\u201c|\u2003 Output Parameters\u2003|\u201d)print(\u201c+--------------------------+\u201d)print(\u201c\u2003- Average number of patients treated at the test doses in terms of percentage (Rst_pt):\u201d)print)print(\u201c\u2003- Average number of toxicities seen at the test doses in terms of percentage(Rst_tox):\u201d)print)print(\u201c\u2003- Distribution of the MTD estimates in terms of percentage(Rst_MTD):\u201d)print)print(\u201c\u2003- Distribution of the MTD estimates for each dose level of the agent B in terms of percentage (Rst_MTD_lv):\u201d)print)print(\u201c+--------------------------------------------------------------------+\u201d)print(\u201c| Send feed back via e-mail:\u2003 \u2003 \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003 |\u201d)print(\u201c|\u2003\u2003\u2014RAJOUTER TON MAIL\u2003 \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003|\u201d)print(\u201c+--------------------------------------------------------------------+\u201d)return)}ExampleFor example, for scenario 1, the parameters should be be fixed in the R function as:tox_true \u2190 matrix, byrow = T, nrow = 3)prior_1 \u2190 c # 6 dose levels of agent Aprior_2 \u2190 c #3 dose levels of agent Btarget \u2190 0.2ntot \u2190 54ncoh_st \u2190 2ncoh \u2190 3nsamp \u2190 1nd1 \u2190 6nd2 \u2190 3seed \u2190 2nsim \u2190 4000Interaction \u2190 0# Generate 4000 replicates of the two-dimensional dose-finding design of Wang and Ivanova with 54 subjectsresult_sc = SIM_designTo validate our R scripts, we ran the same scenarios (dose-toxicity relationships) proposed by Wang for the simulation study. The results were compared to those published in their paper. The results using R function were similar to those published by Wang.Using the R functions, the results on the percentage of DLTs observed, and a recommended MTD at the end of the trial can be also obtained. The MTD is defined as the dose combination associated with a probability of DLT closest to the target toxicity (results available on request).As stated by Hirakawa et al., thus far, there are no silver bullet designs to resolve the two-agent dose-finding problem. The operating characteristics of model-based dose-finding methods may be varied depending on prior toxicity probabilities, prior distributions, and number of patients included in trials. It is recommended to assess the performance of the designs in the context of the clinical trial before beginning the trial. The two-dimensional dose-finding design proposed by Wang and Ivanova is a comprehensive approach that yields good performances. The two R functions that we propose can facilitate the use of this design in practice. With the first function, the design can be applied to an actual phase I trial. With the second function, a simulation study can be run in order to calibrate the design before its application for a phase I trial, or in order to compare it with other designs. Both R functions are simple, and each part of the algorithm  can be modified depending on the requirements of the trial: i) the start-rule can be updated depending on the clinical context, ii) even if the design proposes to identify a set of MTDs, one MTD can be chosen from the set. For example, the MTD with a probability of DLT closest to the target toxicity can be chosen, or clinicians may prefer to choose to identify the dose for further trials depending on the clinical context."}
{"text": "Nature Communications; 10.1038/s41467-018-05698-0; published online: 09 Aug 2018Correction to: '', respectively.\u2019 The correct version states \u2018\u03c3\u2032\u2032\u2019 in place of \u2018\u03c3''\u2019.The original version of this Article contained an error in the seventh sentence of the second paragraph of the \u2018TA amplitudes for double pump-pulse excitation\u201d section of the Results, which incorrectly read \u2018The absorption cross-sections of the transitions from the single exciton state to the biexciton and triexciton states are denoted by \u03c3\u2032 and \u03c3'' are equal to \u03c3 or not. The black dotted curves in Fig.\u00a03b, c show the predicted results of the TA amplitudes by assuming identical cross-sections, i.e., \u03c3\u2009=\u2009\u03c3 \u2032\u2009=\u2009\u03c3''\u2009=\u2009.\u2019 The correct version states \u2018\u03c3\u2032 and \u03c3\u2032\u2032 are equal\u201d instead of \u2018\u03c3\u2032 and \u03c3'' are equal\u2019 and \u2018\u03c3\u2009=\u2009\u03c3\u2032\u2009=\u2009\u03c3\u2032\u2032\u2019 in place of \u2018\u03c3\u2009=\u2009\u03c3 \u2032\u2009=\u2009\u03c3''\u2009=\u2009\u2019.The ninth and tenth sentences of the same paragraph originally incorrectly read \u2018The most important issue of this work is to clarify whether the multiple exciton absorption cross-sections \u03c3\u2032 and \u03c3This has been corrected in both the PDF and HTML versions of the Article."}
{"text": "We give a sufficient condition for a nonnegative integer list to be graphic based on its largest and smallest elements, length, and sum. This bound generalizes a result of Zverovich and Zverovich. A degree sequence is said to be graphic if it corresponds to the set of edge adjacency values of the nodes for some simple graph. In 1960, Erd\u0151s and Gallai gave a complete characterization of the set of\u02dd graphic degree sequences.We denote a finite list of nonnegative integers, Let \u03b1 =  be a nonincreasing degree sequence. Then the degree sequence \u03b1 is graphic if and only if the sum of \u03b1 is even and for each integer k where 1 \u2264 k \u2264 n,It was later shown by I. Zverovich and E. Zverovich that some degree sequences with bounded largest and smallest elements can be verified to be graphic based on their length.Let \u03b1 =  be a nonincreasing degree sequence of positive integers with even sum. Ifthen \u03b1 is graphic.There have been several extensions to the original result of Zverovich and Zverovich. These results have been either specializations, such as when the gaps between consecutive integers in the sequence are bounded , or by sLet \u03b1 =  be an integer sequence such that n \u2212 1 \u2265 \u03b11 \u2265 ... \u2265 \u03b1n \u2265 0 and with even sumwhere n\u03b11> s > n\u03b1n. Ifthen \u03b1 is graphic.1complement of a degree sequence \u03b1 =  is the sequence \u03b1 is graphic if and only if Before providing a proof of this result, we state some needed definitions and theorems. The \u03b1 majorizes a second sequence \u03b2 = , where both sequence have the same length and sum, if and only ifk from 1 to n.A second definition is that a sequence Majorization is a partial order over the set of degree sequences with identical sum. Our use of majorization stems from the following result.If the degree sequence \u03b1 is graphic and \u03b1 majorizes \u03b2, then \u03b2 is graphic.In addition, we use the following two results that reduce the number of Erd\u0151s-Gallai inequalities needed\u02dd to verify that a sequence is graphic. These are found in For the degree sequence \u03b1 = , define the set of indicessuch that i \u2208 if \u03b1i> \u03b1i+1. For For the degree sequence \u03b1 = , define the set of indiceswhere i \u2208 if \u03b1i \u2265 i \u2212 1. For We are now ready to prove \u03b11, \u03b1n, n, and s. We then define the sequence \u03b1\u2032 \u2208 \u03b11> \u03b3 \u2265 \u03b1n and s = p\u03b11 + \u03b3 + (n \u2212 p \u2212 1)\u03b1n. From the condition that n\u03b11> s > n\u03b1n, it follows that the sequence \u03b1\u2032 cannot be regular, i.e., \u03b11> \u03b1n. To show that the sequence \u03b1\u2032 exists, we construct it by first defining the value of \u03b3 as \u03b3 = \u03c4 + \u03b1n where \u03c4 is the solution to the congruence equation\u03c4 < \u03b11\u2212 \u03b1n. Since \u03b11> \u03b1n, then \u03c4 is well-defined. From definition of \u03c4, the value of \u03b3 is \u03b1n \u2264 \u03b3 < \u03b11. It follows from the congruence equation that s \u2212 n\u03b1n \u2212 \u03c4 = p(\u03b11 \u2212 \u03b1n) for some integer p. To show that the sequence < p < n. First, we rewrite the value for p as\u03c4 < \u03b11\u2212 \u03b1n, \u03b1n \u2264 ... \u2264 \u03b11, and < p < n and establishing that the sequence We begin by defining the set \u03b1\u2032 majorizes all the other sequences in \u03b2 \u2208 \u03b1\u2032 does not majorize \u03b2 then there would exist an index k where k = p + 1, since the first p values of \u03b10 are the maximum possible value \u03b11. If \u03b2 to be less than \u03b1n and violating its membership in the set n similar argument for remainder of the indices, establishing that \u03b1\u2032 must majorize all the members of \u03b1\u2032 is graphic in order to prove the result.We observe that the sequence \u03b1\u2032 that \u03b1\u2032 for our argument since We also make the assumption concerning k = p and k = p + 1 in order to prove that the sequence \u03b1\u2032 is graphic. First consider the case when p < \u03b1n. The resulting EG inequality for k = p is\u03b11> n\u22121, but we assumed that \u03b11 \u2264 n\u22121 ensuring that this inequality always holds. There is an identical argument for the case when k = p + 1 establishing that the EG inequalities always hold when p < \u03b1n.Now let us examine the Erd\u0151s-Gallai (EG) inequalities for\u03b1\u2032. We apply p \u2265 \u03b1n. By first multiplying both sides of the Inequality s\u2212n\u03b1n)(n\u03b11\u2212s)/(\u03b11\u2212\u03b1n)2, we algebraically manipulate this new inequality to derive the following equivalent expression,Now consider when the case when \u03b11 \u2265 n \u2212 \u03b1n \u2212 1 along with s \u2265 p\u03b11 + (n \u2212 p)\u03b1n, this new Inequality p\u03b11 \u2264 p(p \u2212 1) + \u03b3 + (n \u2212 p \u2212 1)\u03b1n, which is precisely the EG inequality for \u03b1\u2032 at k = p.Combining the facts that k = p + 1, we split the instance into two cases: p \u2265 \u03b3 and p < \u03b3. If p < \u03b3, then we use k such that \u03b1k \u2265 k \u2212 1, in order to prove that a sequence is graphic. Thus, if p < \u03b3, then satisfying the EG inequality for k = p already established that the sequence is graphic. Else, if p \u2265 \u03b3, then \u03b3 \u2264 2p \u2212 \u03b1n\u22121 and summing this inequality with the Inequality p\u03b11+\u03b3 \u2264 (p+1)p+(n\u2212(p+1))\u03b1n, which is the EG inequality for k = p+1. Therefore, the premise is established.For 2\u03b1 is graphic by This result generalizes the bound of Zverovich and Zverovich. We observe that for the right hand side of Inequality \u03b11, define \u03b1n = 2, n = \u03b11 + 1, and s = 4\u03b11 \u2212 2. These values satisfy the Inequality A equality, and so any sequence with these parameters is graphic. We notice that modifying any one of these values so that the inequality is no longer satisfied while keeping the other three fixed allows for non-graphic sequences. For example, if we increase \u03b11 by 1 or decrease n by 1, we have sequences, such as \u03b1n by 1 or increase the sum by 2 (in order to preserve an even sum), then we have sequences such as In general, this bound is sharp in the sense that there exist sequences that satisfy the inequality, but any modification to parameters that causes the inequality to be not satisfied allows for non-graphic sequences to be formed. For example, consider the following set of degree sequences: for a given value of \u03b11 \u2264 n \u2212 1 and this restriction cannot be relaxed. This is because there are sequences where \u03b11> n \u2212 1 but fulfill the inequality ). In contrast to Zverovich and Zverovich result, if that inequality is satisfied, it implies that \u03b11 \u2264 n\u22121 (proof of Theorem 1.2, In comparison to the Zverovich and Zverovich result, there is a slight difference in the requirements for applying the bound. This bound requires that \u03b11 = \u03b1n, but we can handle this case by using in conjunction the result that states if \u03b11 \u2212 \u03b1n \u2264 1, then \u03b1 is graphic (\u03b1i> 0. This is because zeros in a degree sequence do not change whether the entire sequence is graphic or not and can be discarded.It should be noted that this bound cannot be used for sets of regular sequences, i.e.,  graphic , Lemma 1Finally, this result has practical use when coupled with the usual Erd\u0151s-Gallai conditions for checking\u02dd if a sequence is graphic. Before a sequence can be tested using the EG conditions, it must first be sorted. During this sorting step, the values for largest and smallest indices, sequence length, and sequence sum can be easily extracted. Thus, this new condition may save a number of steps in graphic testing for certain sequences by verifying that a sequence is graphic without having to compute the full EG inequalities ."}
{"text": "Scientific Reports 10.1038/s41598-018-29880-y, published online 31 July 2018Correction to: This Article contains errors in Tables\u00a0In addition, in Tables\u00a0\u201cAbsolute\u201dshould read:\u201cMean/Median*\u201dAs a result, in the legends of Tables\u00a0\u201c\u201dshould read:\u201c\u201dThe correct Tables"}
{"text": "Amanita muscaria, has been regarded as a universal non\u2010selective GABA\u2010site agonist. Deletion of the AGABA receptor (ARGABA) \u03b4 subunit in mice (\u03b4KO) leads to a drastic reduction in high\u2010affinity muscimol binding in brain sections and to a lower behavioral sensitivity to muscimol than their wild type counterparts. Here, we use forebrain and cerebellar brain homogenates from WT and \u03b4KO mice to show that deletion of the \u03b4 subunit leads to a\u00a0>\u00a050% loss of high\u2010affinity 5 nM [3H]muscimol\u2010binding sites despite the relatively low abundance of \u03b4\u2010containing ARGABAs (\u03b4\u2010ARGABA) in the brain. By subtracting residual high\u2010affinity binding in \u03b4KO mice and measuring the slow association and dissociation rates we show that native \u03b4\u2010ARGABAs in WT mice exhibit high\u2010affinity [3H]muscimol\u2010binding sites (DK ~1.6\u00a0nM on \u03b14\u03b2\u03b4 receptors in the forebrain and ~1 nM on \u03b16\u03b2\u03b4 receptors in the cerebellum at 22\u00b0C). Co\u2010expression of the \u03b4 subunit with \u03b16 and \u03b22 or \u03b23 in recombinant (HEK 293) expression leads to the appearance of a slowly dissociating [3H]muscimol component. In addition, we compared muscimol currents in recombinant \u03b14\u03b23\u03b4 and \u03b14\u03b23 receptors and show that \u03b4 subunit co\u2010expression leads to highly muscimol\u2010sensitive currents with an estimated EC50 of around 1\u20132\u00a0nM and slow deactivation kinetics. These data indicate that \u03b4 subunit incorporation leads to a dramatic increase in ARGABA muscimol sensitivity. We conclude that biochemical and behavioral low\u2010dose muscimol selectivity for \u03b4\u2010subunit\u2010containing receptors is a result of low nanomolar\u2010binding affinity on \u03b4\u2010GABAARs.Muscimol, the major psychoactive ingredient in the mushroom  GABA\u2010site agonist. Here, we show that \u03b4 subunit incorporation leads to a dramatic increase in AGABA receptor (ARGABA) muscimol sensitivity. The biochemical and behavioral low\u2010dose muscimol selectivity for \u03b4 subunit\u2010containing receptors was because of low nanomolar\u2010binding affinity on \u03b14/6\u03b2\u03b4 ARGABAs. This paints a consistent picture in which extrasynaptic \u03b4\u2010ARGABAs are not only exquisitely sensitive to GABA, but also the GABA analogs gaboxadol and muscimol.Muscimol has been regarded as a universal non\u2010selective  GABAAR subunits are coded by 19 separate genes, \u03b11\u2010\u03b16, \u03b21\u2010\u03b23, \u03b31\u2010\u03b33, \u03b4, \u03b5, \u03c0, \u03b8, and \u03c11\u2010\u03c13 (Olsen and Sieghart AR complexes formed in the brain are of type \u03b1\u03b2\u03b32 (\u03b32\u2010GABAAR) with a likely subunit stoichiometry of 2(\u03b1):2(\u03b2):1(\u03b32)  reside in extra\u2010 and perisynaptic membranes where their high GABA sensitivity allows them to be activated by ambient [GABA] to mediate tonic inhibition of the nerve cell , cerebral cortex (\u03b14\u03b22/3\u03b4), hippocampal dentate gyrus granule cells (\u03b14\u03b22/3\u03b4), caudate\u2010putamen and in the nucleus accumbens (\u03b14\u03b23\u03b4)  is the major inhibitory neurotransmitter in vertebrate brain. The inhibitory action of GABA is mediated via ionotropic GABAARs are quite different from classical \u03b32\u2010GABAARs. \u03b4\u2010GABAARs have much higher affinity for GABA, are insensitive to classical benzodiazepines, show high sensitivity to neurosteroids and Zn2+  when compared to GABA. This is likely because of GABA being a partial agonist on these receptors  mice lose low dose THIP effects on tonic currents in neurons in brain slices and behavioral sensitivity to low doses of THIP  and \u03b4KO mouse brains and to several \u03b1\u03b2\u03b3 and \u03b1\u03b2\u03b4\u2010type recombinant GABAARs by measuring binding and unbinding kinetics. Subtraction of residual high\u2010affinity (5\u00a0nM) [3H]muscimol binding that is seen on abundant GABAAR subtypes in \u03b4KO mice from binding in WT membranes allowed us to isolate a native \u03b4\u2010GABAAR component. This isolated component showed very slow muscimol dissociation rate with an apparent KD  for muscimol of 1.6\u00a0nM for \u03b14\u03b2\u03b4 receptors in the forebrain and around 1\u00a0nM for \u03b16\u03b2\u03b4 receptors in the cerebellum. Recombinant \u03b14\u03b23\u03b4 receptors expressed in oocytes revealed a biphasic response to muscimol with the high\u2010muscimol affinity (slowly deactivating/dissociating) component showing an approximate EC50 of around 1\u20132\u00a0nM.In this study, we investigated high\u2010affinity (5\u00a0nM) muscimol (22\u00a0Ci/mmol) was purchased from PerkinElmer Life and Analytical Sciences . Unlabeled muscimol was from Sigma\u2010Aldrich . GABA was from Sigma\u2010Aldrich (Cat. No. A2129).muscimol (5\u00a0nM) was measured in assay buffer at 22\u00b0C in a total volume of 300\u00a0\u03bcL. Triplicate technical replicates of mouse forebrain (190\u2013215\u00a0\u03bcg protein), cerebellar (180\u2013210\u00a0\u03bcg protein) or HEK cell (92\u2013132\u00a0\u03bcg protein) membranes for each time point were incubated with shaking for various times (15\u00a0s\u201315\u00a0min) to measure association of the binding. Non\u2010specific binding was determined in the presence of 100\u00a0\u03bcM GABA. The incubation was terminated by filtration of the samples with a Brandel Cell Harvester  onto Whatman GF/B filters . The samples were rinsed twice with 4\u20135\u00a0mL of ice\u2010cold assay buffer. Filtration and rinsing steps took a total time of ~15\u00a0s. Air\u2010dried filters were immersed in 3\u00a0mL of Optiphase HiSafe 3 scintillation fluid  and radioactivity determined in a Wallac model 1410 liquid scintillation counter . The maximal binding disintegrations per minute (DPM) values (at 15\u00a0min in association) for recombinant studies with 5\u00a0nM [3H]muscimol were between 700 and 2500 DPMs of specific binding (background subtracted). In native membranes, the maximal DPM values were between 2500 and 3000 for WTs and 1300\u20131500 for \u03b4KOs. Mock transfection with pRK5 plasmid did not produce any specific binding over the background.The binding of [3H]muscimol binding, triplicate technical replicates of each sample of mouse brain or HEK cell membranes for each time point were first pre\u2010incubated at 22\u00b0C in a total volume of 300\u00a0\u03bcL for 15\u00a0min with 5\u00a0nM [3H]muscimol in the absence and presence of 100\u00a0\u03bcM GABA. The dissociation was then started by adding 100\u00a0\u03bcL of 400\u00a0\u03bcM or 100\u00a0\u03bcM (non\u2010specific binding) cold GABA to the incubation mixtures to reach a final 100\u00a0\u03bcM GABA concentration in all tubes. The tubes were mixed and incubations at 22\u00b0C were terminated at various time points (30\u00a0s \u2013 30\u00a0min) as described above. Dissociation of [3H]muscimol from recombinant receptors in HEK cell membranes was also measured at 0\u20134\u00b0C (on ice) to evaluate how fast [3H]muscimol dissociates from receptors while washing the filter with ice\u2010cold assay buffer during filtration.To measure dissociation of [3H]muscimol to WT and \u03b4KO mouse forebrain and cerebellar membranes was performed essentially as described by Uusi\u2010Oukari and Korpi (3H]muscimol (0.1\u201330\u00a0nM) at 0 to 4\u00b0C for 30\u00a0min in the absence and presence of 100\u00a0\u03bcM GABA determining the non\u2010specific binding. The incubations were terminated as described above.Saturation analysis of [nd Korpi . Triplic3H]muscimol to \u03b4\u2010GABAARs in WT animals, \u2018native \u03b4\u2010GABAARs\u2019, were calculated by subtracting the specific \u03b4KO binding values (binding to \u03b32\u2010GABAARs) from the corresponding WT values at each time point: native \u03b4\u2010GABAARs\u00a0=\u00a0WT\u2010 \u03b4KO. Because of the lack of low\u2010affinity binding and the relatively small number of time points in our assays, the binding curves fitted better in \u2018one binding site\u2019 model. However, varying fast and slow dissociation components are obvious in the graphs  according to manufacturer's instructions.Bmax and KD values were analyzed with Graph Pad Prism 7 software . Statistical significances between the groups were analyzed using unpaired t\u2010test and one\u2010way anova followed by Tukey's post hoc test Graph Pad (Graph Pad Prism 7). p\u2010values of <\u00a00.05 were considered significant. In this study, no sample calculation, assessment of data outliers and data normality were performed, and experiments were done unblinded.Association and dissociation curves for estimation of association and dissociation rate constants, and saturation binding for estimation of ARs to high\u2010affinity muscimol binding, we measured the time course of 5\u00a0nM [3H]muscimol binding to forebrain and cerebellar membranes from both wild\u2010 type and \u03b4KO mice. Deletion of the \u03b4 subunit led to >\u00a050% reduction of 5\u00a0nM [3H]muscimol binding at 22\u00b0C to both forebrain and cerebellar membranes when compared to WT mice  was around four times higher in the cerebellum when compared to forebrain both in WT as well as in the \u03b4KO mice . Binding of 5\u00a0nM [3H]muscimol to non\u2010\u03b4\u2010GABAARs in \u03b4KO forebrain was around 100 fmoles and about 300 fmoles (per mg membrane protein) in the cerebellum  of non\u2010\u03b4\u2010GABAARs were occupied by muscimol under our binding conditions  when compared to \u03b4\u2010GABAARs, which was surprising since faster muscimol association would contribute to higher muscimol affinity in \u03b4\u2010GABAARs. This slower muscimol association to \u03b4\u2010GABAARs is reflected in higher forebrain and cerebellar association rate constants (kon) of [3H]muscimol binding to \u03b4KOs than to WT mouse membranes .To better illustrate high\u2010affinity muscimol binding kinetics to \u03b4\u2010GABAved Figs\u00a0 and 3. B mostly \u03b3\u2010GABAARs  mostly \u03b3\u2010GABAARs 3H]muscimol unbinding for up to 30\u00a0min. A comparison of muscimol dissociation between WT and \u03b4KO animals shows that almost all of the slow muscimol dissociation is because of \u03b4\u2010GABAARs, with only a minor component present in both the forebrain and cerebellum of \u03b4KO animals, which is because of the high\u2010affinity muscimol binding to non\u2010\u03b4\u2010GABAARs  HmuscimolARs in \u03b4KO mice from binding to total GABAARs in WT mice, we were able to determine a KD value based on the equation\u00a0KD\u00a0=\u00a0koff/kon. The calculated KD value for \u03b4\u2010GABAARs in the fore(mid)brain (predominantly \u03b14\u03b2\u03b4) is 1.6\u00a0nM, and the KD for \u03b4\u2010GABAARs in the cerebellum (\u03b16\u03b2\u03b4) is 1.1\u00a0nM. Therefore, under our binding conditions [5\u00a0nM [3H]muscimol and 22\u00b0C], the majority of \u03b4\u2010receptors both in forebrain and cerebellum should be occupied at equilibrium.After subtraction of binding to non\u2010\u03b4\u2010GABAARs have low affinity and are therefore not occupied at 5\u00a0nM [3H]muscimol. Dissociation rate constants koff of [3H]muscimol binding were higher in \u03b4KOs than in WTs in both forebrain (p\u00a0<\u00a00.001) and cerebellar membranes (p\u00a0<\u00a00.05) (Table\u00a0t\u2010test) indicating faster [3H]muscimol dissociation in \u03b4KOs lacking \u03b4\u2010GABAARs. The Koff values of the calculated native \u03b4\u2010GABAARs in both forebrain and cerebellum were smaller than those of \u03b4KO indicating slower [3H]muscimol dissociation from \u03b4\u2010GABAARs than from \u03b32\u2010GABAARs  conformations which could also depend on subunit composition. We therefore decided to measure association and dissociation on selected recombinant receptor subtypes. As observed for native \u03b4\u2010GABAARs [3H]muscimol association at 22\u00b0C was much slower in \u03b16\u03b22\u03b4 receptors when compared to high\u2010affinity binding to \u03b11\u03b22\u03b32 and \u03b16\u03b22\u03b32 recombinant receptors .Measurements in native brain tissues have the advantage that we can measure native receptors. The disadvantages is that the fraction of \u03b4 receptors is variable . In addition, because of the low\u2010muscimol affinity of most \u03b32\u2010GABA3H]muscimol from \u03b16\u03b22, \u03b16\u03b22\u03b32 and especially \u03b11\u03b22\u03b32 receptor subtypes was very fast   Table\u00a0d. From a3H]muscimol dissociation kinetics at 22\u00b0C with unbinding at lower temperature (0\u00b0C) on selected \u03b32 and \u03b4\u2010GABAAR subtypes. At 0\u00b0C dissociation from \u03b16\u03b22\u03b4 and \u03b16\u03b22\u03b32 were significantly slower than from \u03b11\u03b22\u03b32 GABAARs with 70% of [3H]muscimol still remaining bound to \u03b16\u03b22\u03b4 subtype at 30\u00a0min after start of the dissociation (Table\u00a0p\u00a0<\u00a00.001). [3H]Muscimol dissociated also significantly slower from \u03b16\u03b22\u03b4 when compared to \u03b16\u03b22\u03b32 GABAARs , a difference that was also noted at 22\u00b0C , we decided to compare Hmuscimolsee Fig.\u00a0c. It canors Fig.\u00a0d.3H]muscimol binding sites as well as binding kinetics are in the same range as found in the literature , electrophysiological measurements are typically performed at 22\u00b0C and in rodent behavioral experiments receptors are studied at body temperature (37\u00b0C). Such temperature differences could have a major influence on binding affinities of GABA and GABA analogs. Also, the high\u2010affinity muscimol binding sites have been interpreted to represent desensitized or otherwise non\u2010functional high\u2010affinity conformations  of 1.1\u00a0nM in the cerebellum and 1.6\u00a0nM in fore/mid\u2010brain  apparent [3H]muscimol affinities (KD) for these non\u2010\u03b4\u2010GABAARs were also around 1\u00a0nM  and partly also from higher affinity \u03b16\u03b2\u03b32 receptors, while the extremely slow dissociation from \u03b4\u2010GABAARs allows the majority of muscimol to be retained as seen in autoradiographs , seems to be a plausible explanation. Another  possibility is that such high\u2010affinity binding to non\u2010\u03b4\u2010GABAARs is due to freezing, since at 22\u00b0C room temperature high\u2010affinity binding was lower when never\u2010frozen whole brain membranes were used  effects in \u03b4KO mice.We show here for the first time that co\u2010expression of the \u03b4 subunit leads to highly muscimol\u2010sensitive \u03b14\u03b23\u03b4 currents. Remarkably, the ECARs is challenging since they generally show biphasic GABA and THIP concentration response curves likely because of incomplete \u03b4 subunit incorporation into functional receptors  value \u22123.60 for GABA, whereas adding hydrophobic ring structures in muscimol (logP\u00a0=\u00a0\u22121.71) and THIP (logP\u00a0=\u00a0\u22120.81) . It seems therefore likely that GABA has the lowest BBB permeability, followed by muscimol and THIP. Given that THIP affinity for \u03b4\u2010GABAARs is lower when compared to muscimol  permeability usually correlates with lipid\u2010solubility and is therefore rather poor for highly water\u2010soluble molecules like GABA, muscimol and THIP. Consistent with a low BBB permeability it has been shown that only around 0.02% 1/5000) of peripherally injected [/5000 of A agonist binding sites in GABAARs are located at the two extracellular \u03b2+\u03b1\u2010 interfaces , without the \u03b4 subunit actually directly contributing to the GABA\u2010binding site. This implies that \u03b4 increases the GABA\u2010binding\u2010site affinity and slows muscimol dissociation in the \u03b2\u03b1\u03b4\u03b2\u03b1 pentamer allosterically. The reciprocal of dissociation rate constant, the drug\u2010target residence time \u03c4 (=\u00a01/koff), has been shown to often predict in\u00a0vivo efficacy better than binding affinity . For example, muscimol EC50 for \u03b11\u03b2\u03b32 GABAARs is\u00a0~1\u00a0\u03bcM, whereas GABA EC50 is ~100\u00a0\u03bcM  induces low nanomolar muscimol currents.It appears that in general the GABA analog muscimol is similar to GABA in many aspects, only that it shows about 100\u20131000 times higher affinity (with THIP having intermediate affinity) across the board for different GABAA analog THIP, which has been shown to be highly selective for \u03b4\u2010GABAARs  and NIH grant AA021213 to MW. The authors have no conflict of interest to declare."}
{"text": "We consider call option prices close to expiry in diffusion models, in an asymptotic regime (\u201cmoderately out of the money\u201d) that interpolates between the well\u2010studied cases of at\u2010the\u2010money and out\u2010of\u2010the\u2010money regimes. First and higher order small\u2010time moderate deviation estimates of call prices and implied volatilities are obtained. The expansions involve only simple expressions of the model parameters, and we show how to calculate them for generic local and stochastic volatility models. Some numerical computations for the Heston model illustrate the accuracy of our results. Today's price of the underlying, the spot value S0, is known and fixed. Discrete option data are available from the market, typically quoted in (Black\u2013Scholes) implied volatilities; see Figure Consider a European call option struck at C as t\u21930 exhibits very different behavior in the respective cases K>S0 (\u201cout\u2010of\u2010the\u2010money\u201d) and K=S0 (\u201cat\u2010the\u2010money\u201d). We argue that there is a significant asymptotic regime in between, namely,More specifically, small\u2010maturity approximations of option prices have been studied extensively in recent years. Starting with Carr and Wu , it was c for the normalized call price as a function of log\u2010moneyness\u00a0k=log(K/S0)c depends tacitly on S0, the (fixed) spot value.k=0. In the Black\u2013Scholes model, writing c=c BS  with volatility parameter \u03c3>0, we have the following ATM call price behavior\u03c30=v0>0),v0>0)\u03c3 imp  for the Black\u2013Scholes implied volatility with log\u2010moneyness\u00a0k and maturity\u00a0t.) Higher order terms in t will be model dependent. For instance, in the Heston case, with variance dynamics dV=\u2212\u03ba(V\u2212v\u00af)dt+\u03b7VdW, implied variance has the ATM expansiona(0) has no easy interpretation in terms of the model parameters.To put our results into perspective, we recall some well\u2010known facts on option price approximations close to expiry. We write and Nutz , the sam and Lee , and we k=0 to kt=o(t) amounts to what we dub \u201calmost ATM\u201d (AATM) regime.kt\u223ct\u03b2 is in the AATM regime if and only if \u03b2>1/2.) Again for generic semimartingale models with diffusive component and spot volatility \u03c30>0, it is easy to see from Caravenna and Corbetta (kt ceases to be o(t). Indeed, for kt=\u03b8t with constant factor \u03b8>0, we have, from Caravenna and Corbetta  stands for a Gaussian random variable with mean \u2212\u03b8 and variance\u00a0\u03c302. This, too, holds true in the stated semimartingale generality. In any case, the proof is based on the L\u00e9vy case with nonzero diffusity v0, and the result follows from comparison results, which imply that the difference is negligible to first order. For a thorough discussion of the regime k=O(t) in the  diffusion case, see Pagliarani and Pascucci Pascucci .log(1/t),t\u2010behavior of call prices described above and in fact\u2113(t), see Mijatovi\u0107 and Tankov =12k2/\u03c32 in the Black\u2013Scholes model. Similar results appear in the literature, with different levels of mathematical rigor, for other and/or generic diffusion models; see Berestycki, Busca, and Florent , which we call moderately out\u2010of\u2010the\u2010money (MOTM). We have a threefold interest in this MOTM regime,Throughout the paper, we reserve the term OTM for (i)reality of quoted (short\u2010dated) option prices, where strikes of option price data with acceptable bid\u2013ask spreads tend to accumulate \u201caround the money,\u201d as illustrated in Figure k=O(t\u03b2) for some \u03b2>0. There is no reason why quoted strikes should always be almost ATM (\u03b2>1/2), which effectively means an extreme concentration around the money; we are thus led to study the regime\u00a0First, it is related to the (ii)mathematical convenience. In contrast to the genuine OTM regime (large deviation regime) in which the rate function \u039b(k) is notoriously difficult to analyze\u2014often related to geodesic distance problems\u2014MOTM naturally comes with a quadratic rate function and, most remarkably, higher order expansions are always explicitly computable in terms of the model parameters. The terminology moderately OTM (MOTM) is in fact in reference to moderate deviations theory, which effectively interpolates between the central limit and large deviations regimes.The second reason is (iii)Finally, our third point is that MOTM expansions naturally involve quantities very familiar to practitioners, notably, spot (implied) volatility, implied volatility skew, and so on.fractional stochastic volatility models, see Forde and Zhang (first\u2010order moderate deviation principle (MDP) for diffusions under classical regularity assumptions from SDE theory. The main difference to the bulk of our results is that we develop higher order expansions, until Section\u00a0In the Black\u2013Scholes model, it is easy to check that we have the MOTM asymptoticsTheorems\u00a0 below asTheorems\u00a0; then, cnd Zhang  and Guennd Zhang . Guillinnd Zhang , who connd Zhang , the dynnd Zhang  contains(Xn)n\u22651 with finite exponential moments. Then the empirical means X^n:=n\u22121\u2211k=1nXk converge to zero , and this is quantified by an LDP according to Cram\u00e9r's classical theorem: P[X^n>x]=exp(\u2212I(x)n+o(n)) decreases exponentially as n\u2192\u221e for fixed x>0, governed by a rate function I(x)=supy\u2208R(yx\u2212logE[eyX1]). On the other hand, by the CLT, nX^n=n\u22121/2\u2211k=1nXk has a Gaussian limit law. Moderate deviations cover intermediate scalings, i.e., nanX^n with an\u21920 and nan\u2192\u221e. It turns out  This is sometimes called an MDP. Formally, an MDP is thus just a certain LDP with appropriate scaling and speed function. Still, the terminology is often useful because of the trichotomyTo round off the introduction, we briefly recall some background on moderate deviations. Consider the classical setting of a centered i.i.d. sequence Zeitouni . SeveralZeitouni  and refeS0=1.The rest of the paper is organized as follows. Section\u00a02general stochastic volatility model, i.e., a positive martingale (St)t\u22650 with dynamicsS0=1. We assume that the stochastic volatility process (\u03c3t)t\u22650 itself is an It\u00f4\u2010diffusion, started at some deterministic value \u03c30, called spot volatility. Recall that in any such stochastic volatility model, the local (or effective) volatility is defined byS\u223ct=St (in law) for all fixed times. See Brunick and Shreve  match in both models. Recall also Dupire's formula in this contextAssumption 2.1t>0, St has a continuous pdf K\u21a6q, which behaves asymptotically as follows for small time:K=ek in some neighborhood of\u00a0S0=1. The energy function\u00a0\u039b is smooth in some neighborhood of zero, with \u039b(0)=\u039b\u2032(0)=0. Moreover, limk\u21920\u03b3(k)=\u03b3(0)>0.For all We consider a d Shreve  for precAssumption 2.2t\u21930 and K\u2192S0=1, the local volatility function of\u00a0(St)t\u22650 converges to spot volatilityFor 0  here) and the Hessian of the energy function \u039b=\u039b(k),Theorem 2.3\u2113>0 varies slowly at zero and \u03b2\u2208.(i)The call price satisfies the moderate deviation estimate(ii), then the following moderate second\u2010order expansion holds trueIf we restrict \u03b2 to Under Assumptions\u00a0\u03c302, equal to \u03c3 imp 2, and implied variance skew S=\u2202\u2202k|k=0\u03c3 imp 2.with spot\u2010variance The latter assumption is fairly harmless  expansion\u03b8\u21a6\u03b82/2\u03c302, typical of moderate deviation problems.In particular, for c=exp(\u2212\u039b(k)/t(1+o(1))) yields a correct result, upon Taylor expanding\u00a0\u039b. Mind, however, that this needs a proof using the specifics of our situation, in light of the fact that validity of a large deviation principle does not automatically imply an MDP.In a nutshell, \u039b\u2032\u2032(0),\u039b\u2032\u2032\u2032(0),\u22ef appearing above are always computable from the initial values and the diffusion coefficients of the stochastic volatility model. This is in stark contrast to the OTM regime, where one needs the function \u039b(\u00b7), which is in general not available in closed form . We quote the following result on N\u2010factor models from Osajima  is Markov, started at  with \u03c3\u00af0\u2208RN\u22121 and \u03c3\u00af01>0, with stochastic volatility \u03c3\u2261\u03c31, where the generator has (nondegenerate) principal part \u2211aij\u2202ij in the sense that a\u22121 defines a Riemannian metric. ThenAssume that The quantities  Osajima  and refeT=1).\u25a1See Osajima .1/\u03b2 is not an integer, then ktm/t tends to infinity for m=\u230a1/\u03b2\u230b, of order t\u03b2\u230a1/\u03b2\u230b\u22121 (up to a slowly varying factor). If, on the other hand, 1/\u03b2 is an integer, then the last summand of the sum \u2211m=2\u230a1/\u03b2\u230b in\u00a0\u2113(t), which means that the following term log(1/t) may be asymptotically larger. The upper summation limit \u230a1/\u03b2\u230b thus ensures that no irrelevant ) terms are contained in the sum. Note that \u230a1/\u03b2\u230b=2 for \u03b2\u2208, and \u230a1/\u03b2\u230b\u22653 for \u03b2\u2208, and so\u00a0If Theorem 2.6\u039b\u2032\u2032(0)=\u03c30\u22122=v0\u22121, and that the Berestycki\u2013Busca\u2013Florent formula \u03c3 imp 2=k2/2\u039b(k) holds. Then the small\u2010time ATM implied variance skew and curvature, respectively, relate to\u00a0\u039b viaSuppose that\u00a0\u039b is a function with the properties required in Assumption\u00a0The passage from the derivatives of the energy function to ATM derivatives of the implied volatility in the short time limit is best conducted via the BBF formula that was proved in Berestycki et\u00a0al. . That s. In this\u25a1By the BBF formula and our smoothness assumptions on\u00a0\u039b,Proposition 3Proof of Theorem 2.3St satisfies q=\u2202KKC, we have, by Dupire's formula\u00a0Kt=ekt with kt\u21930 as stated, we apply Assumption\u00a0t\u21930,\u039b(0)=\u039b\u2032(0)=0, we havet, the integrand in\u00a0x=1, and by the Laplace method  if \u03b2\u2208, and that kt3/t dominates log(kt2t\u22123/2) if \u03b2\u2208. For m\u2208{2,3}, we calculatelog\u2113(t)=o(logt), and som=2 and \u03b2\u2208, and for m=3 and \u03b2\u2208, as desired.\u25a1As the density of\u00a0 Teugels , we knowlogc further.Inspecting the preceding proof, it is easy to see that we can expand Proof of Theorem 2.5ktm/t=o(1) for m\u2265\u230a1/\u03b2\u230b+1.\u25a1Taking logs in\u00a04Corollary 4.1kt=t\u03b2\u2113(t) with \u03b2\u2208 and \u2113>0 slowly varying. Then the implied volatility has the MOTM expansionUnder the assumptions of Theorem\u00a0L, V of Gao and Lee  resp. V=t1/2\u03c3 imp , the dimensionless implied volatility. Then corollary\u00a07.2 of Gao and Lee  in Bingham et\u00a0al., \u25a1We use our main result Theorem\u00a0 in conju and Lee  means L= and Lee  implies \u03b2\u2208, under very mild assumptions , one can compute the implied volatility expansion using the results of Gao and Lee  if \u03b2 is close to zero. Suppose, on the other hand, that \u03b2 is very close to\u00a012: Then the summands m>\u230a1/\u03b2\u230b=2 in\u00a0o(1)\u2010term of\u00a0We have no doubt that Corollary\u00a0 and Lee . Howevert\u226akt, and so the k\u2010term dominates the O(t)\u2010term, which in turn identifies the implied variance skew as\u221213\u03c304\u039b\u2032\u2032\u2032(0). We have thus arrived at an alternative proof of the skew representation\u00a0Corollary\u00a055.1C2 on . An expansion of the pdf\u00a0q of\u00a0St has been worked out in Gatheral, Hsu, Laurence, Ouyang, and Wang  This shows that Assumption\u00a0kt=\u03b8t\u03b2 and \u03b8>0S the  implied variance skew, and so the implied volatility skew is given by S/2\u03c30, which equals S/2\u03c3(1)=S/2\u03c3(S0) in model\u00a0local skew \u03c3\u2032(1)=\u03c3\u2032(S0) equals twice the implied volatility skew,Clearly, Assumption\u00a0and Wang . They asand Wang . Proposiand Wang  states tabord\u00e8re . Generic)2,and5.u0=\u03c35.22\u039b(k) is the squared geodesic distance from  to {:\u03c3>0} with K=ek. Theorem\u00a02.2 in Berestycki, Busca, and Florent \u22611) and the 3/2\u2010model (\u03bd(v)=v; see Lewis, L of the stochastic process , neglecting first\u2010order terms, can be written asD2f denotes the Hessian matrix of f, and the coefficient matrix g=(gij) is given byb1=g11|v=v0=v0 and b2=34\u2211i=12g1i\u2202ig11|v=v0=34\u03c1\u03b7v0\u03bd(v0). If we assume that the coefficients in k\u21920. Therefore, the quantities \u039b\u2032\u2032(0)=v0\u22121=\u03c30\u22122 and \u039b\u2032\u2032\u2032(0)=\u221232\u03c1\u03b7\u03bd(v0)/v02 can easily be computed, as well as the small\u2010time ATM implied variance skewNow we describe how the expressions appearing in the expansions from Theorem\u00a0 Theorem\u00a0 have ver5.3v\u00af,\u03ba,\u03b7>0, and d\u27e8W,Z\u27e9t=\u03c1dt with \u03c1\u2208. According to Forde and Jacquier (He is the (not explicitly available) Legendre transform of\u03c1\u00af2=1\u2212\u03c12.) This expansion impliesThis section contains an application of the results of Sections\u00a0Jacquier , the firSt, we get the density approximationk, where \u03c6t is the characteristic function of\u00a0Xt=logSt, and\u00a0p\u2217 and\u00a0U are defined on p.\u00a0693 of Forde et\u00a0al.  From\u00a0U(p\u2217(0))=U(0)=1, we see that the factor\u00a0\u03b3(k) from\u00a0k\u21920.The locally uniform density asymptotics\u00a0e et\u00a0al. .  in the numerator of\u00a0St, the analysis of which we have just described. Virtually the same saddle point approach can be applied to the numerator \u2202tC, yielding convergence of the quotient to\u00a0\u03c302.To verify Assumption\u00a0He is given by \u039b He (k)=supx{kx\u2212\u0393(x)} with maximizer x*=x\u2217(k). From general facts on Legendre transforms,x\u2217(0)=0, which implieskt=\u03b8t\u03b2 and \u03b8>0, we then obtain the MOTM call price estimate\u03b2\u2208, Theorem\u00a0t\u21930:We now calculate our MOTM asymptotic expansions for the Heston model. The Legendre transform\u00a0\u039b\u039b\u2032\u2032\u2032(0), we get the explicit expression S He =\u03b7\u03c1/2 for the skew. This agrees with Gatheral  However, Gao and Wang Assumption 6.1\u03b2\u2208, the rescaled mgf satisfiesFor all In this section, we discuss a different approach at small\u2010time moderate deviations. While yielding only first\u2010order results, its conditions are usually easy to check for models with explicit characteristic function. Assumptions\u00a0We expect that this assumption holds for diffusion models in considerable generality. It is easy to check that\u00a0q by a Freidlin\u2013Wentzell LD argument for\u00a0x sufficiently large. As for\u00a0x\u22480, because \u039b(x) increases with\u00a0|x|. Finally, Heuristically, Assumption\u00a0St is defined byp+(t) must grow faster than t\u03b2\u22121 as t\u21930. In the Heston model, e.g., the critical moment is of order p+(t)\u223c(const)/t\u226bt\u03b2\u22121 for small\u00a0t, as follows from inverting\u00a0p+(t)\u2261p+ does not depend on\u00a0t, and is finite for most models used in practice. Therefore, p+=\u221e, but it is easy to check that it does not satisfy\u00a0The critical moment of\u00a0Xt  in Theorem\u00a0call prices in Theorem\u00a0Theorem 6.2kt=\u03b8t\u03b2 with \u03b2\u2208 and \u03b8>0, we have a first\u2010order MD estimate for the cdf of\u00a0Xt:Under Assumption\u00a0After this discussion of Assumption\u00a0\u0393Z is finite and differentiable on\u00a0R, the G\u00e4rtner\u2013Ellis theorem (theorem\u00a02.3.6 in Dembo & Zeitouni (Zt)t\u22650 satisfies an LDP as t\u21930, with rate\u00a0at and good rate function\u00a0\u039bZ, the Legendre transform of \u0393Z. Trivially, \u039bZ is qquadratic, too:\u03b8>0. Applying the lower estimate of the LDP to  yields is finite for all t\u2208. Let v0=\u03c302>0. Then the following are equivalent(i)kt=\u2113(t)t\u03b2, with \u03b2\u2208 and \u2113>0 slowly varying at zero, it holds thatFor (ii)Under the assumptions of\u00a0(i), we haveLet As in the LD/OTM regime, first\u2010order cdf asymptotics translate readily into call price asymptotics. The proof of the following result is similar to Pham  by a mil\u03b5>0 and define k\u223ct=(1+\u03b5)kt. Thenk\u223ct.\u03b5\u21930 to get the desired lower bound for c.First assume\u00a0(i). Let p>1 and note that, by definition of\u00a0p\u21a6t\u2217(p), we have E[Stp+1]<\u221e for all t\u2208. Define S\u00aft=sup0\u2264u\u2264tSu for t\u22650. By Doob's inequality  has a finite pth moment,S, we thus conclude1/p+1/q=1 and apply H\u00f6lder's inequality.p\u2191\u221e, i.e., q\u21931. The same argument yields the lower bound of the implication (ii) \u27f9 (i). The remaining upper bound of (ii) \u27f9 (i) is shown very similarly to the lower bound of the implication (i) \u27f9 (ii).\u25a1As for the upper bound, we let kt=\u2113(t)t\u03b2 is replaced by t\u226akt\u226a1.In the light of the general MDP result by Gao and Wang  quoted a"}
{"text": "We show that an analogue of the ball-box theorem holds true for a class of corank 1, non-differentiable tangent subbundles that satisfy a geometric condition. We also we give examples of such bundles and give an application to dynamical systems. C1 contact bundles. Another place where sub-Riemannian geometry pops up in dynamical systems is in (p) = TpM, then such a bundle is called step 2, completely non-integrable at p.\u2113(\u22c5) denotes length with respect to the given metric g on \u0394. d\u0394 is called the sub-Riemannian distance. We let B\u0394 denote the ball of radius \ud835\udf16 around p with respect to the distance d\u0394. Given p \u2208 M and a coordinate neighbourhood U, coordinates z =  are called adapted at p for \u0394 if, p =\u20090, q is transverse to q \u2208 U. Given such coordinates and \u03b1 >\u20090, we define the coordinate weighted box as follows: ball-box theorem ))K1, K2, rms (see  to applyC1 bundles). Motivated by the discussion above, our proof is very geometric and uses Stokes\u2019 theorem as the main tool rather than classical ODE analysis. The techniques that we use are inspired from a proof of the Frobenius theorem given in . We denote this integral curve by t \u2192 etX(q).We call a continuous vector-field uniquely integrable if it is integrable and if X defined on U1 has Cr family of solutions if there exists some U2 \u2282 U1 and an \ud835\udf160 such that for all p \u2208 U2, there exists an integral curve passing through p with \ud835\udf16p \u2265 \ud835\udf160 so that this choice of solutions seen as maps from  \u00d7 U2 \u2192 U1 are Cr.We say that an integrable vector field Next proposition is almost a direct corollary of the Theorem 6.1  be continuous. Then, the ODEhasunique andC1solutionsy(t) = \u03b7,y(t0) = y0for all  \u2208 U, if for anyp \u2208 U, there exists a neighbourhoodUp, a non-singularn \u00d7 nmatrixA and a continuousn \u00d7 mmatrixC such that the 1 \u2212formshavecontinuous exterior derivative.z1,\u2026,zm). This generality is not needed for our purposes; therefore, we assume that m =\u20090 . For thC1 solutions above is equivalent to unique integrability of the extended vector field C1 family of integral curves which would be given by t \u2192 ) \u2282 U for t small enough. Let t, y) space. Then, this bundle is the intersection of the kernel of the 1-forms \u03b7i = dyi \u2212 fidt, i.e \u03b7i is a basis of sections for A simply asks that there must exist a basis of sections for C1 solutions. To prove Proposition 2.1, we will show that the conditions given there imply that each vector field Xi is equipped with a continuous exterior derivative. We have that \u03b1j = dxj for j\u2260i and \u03b1i = \u03b7 which is the 1-form given in the proposition. Note that \u03b7(Xi) =\u20090 since \u03b7 annihilates \u0394 which contains Xi. Moreover, by our choice of adapted coordinates, \u03b7\u2260dxi on some neighbourhood around p0. Now by assumption, \u03b7 has continuous exterior derivative and dxj has continuous exterior derivative 0, so Xi is uniquely integrable. \u25a1We start by translating the terminology of Theorem 2 to something more closer to ours. Note that the unique integrability of the non-autonomous ODE U is given, we can find \ud835\udf160 be such that for all k =\u20091,...n, t \u2264 \ud835\udf160, C1. We will impose some more conditions which will fix U once and for all for the rest of the paper. The conditions are as follows:k = {1,\u2026,n}For all, i, j \u2208{1,\u2026,n}p \u2208 U.For some, i, j \u2208{1,\u2026,n}p \u2208 U and for some constant B(n) >\u20090.For some, With this proposition, once a suitable neighbourhood Xk(0) = \u2202k and \u03b7 is non-involutive at 0.These are possible since \ud835\udf160 starting with choosing it small enough so that K2 is as in Theorem 1 and We will also do some modifications on Xk|\u221e\u2264\u20092 is that uniqueness and differentiablity of solutions is extended to n +\u20094-tuple compositions. That is for all ik \u2208{1,\u2026,n} and k =\u20091,\u2026,n +\u20094, one has that the following is well defined:C1 map with respect to q and \ud835\udf16 \u2264 \ud835\udf160 where K2 is as in Theorem 1.One consequence of this and the fact that |E(n),G(n) >\u20090 we fill shrink \ud835\udf160 so that \ud835\udf160 later on.In the remaining part of the proof, there will be places where for some constants \u03b3 :  \u2192 U with \ud835\udf16 \u2264 \ud835\udf160 that are either unit parameterized or satisfy t. Therefore, the reader should keep in mind that for such curves x be the projection to x coordinates.From now on, we will always be working with admissible curves Let\u03b3\u2113 :  \u2192 Ufor\u2113 =\u20091, 2 be two \u0394 \u2212admissiblecurves such that\u03b31(0) = \u03b32(0) = q, \u03a0x\u03b31(\u03b51) = \u03a0x\u03b32(\u03b52), Let\u03b5 = max{\u03b51, \u03b52} and\u03b2be the segment in the\u2202ydirection that connects\u03b31(\u03b51) to\u03b32(\u03b52). Let\u03b1\u2113be the 1-chains which are projections of\u03b3\u2113along\u2202yto \u0394qand assume that\u03b1\u2113 \u2282 U. Then, for any 2-chainP \u2282\u0394q \u2229 Uwhose boundary iswe have thatand\u03b31(\u03b51) = q1, \u03b32(\u03b52) = q2. Assume w.l.o.g that \u2202y direction. Since \u03b3\u2113 are admissible curves, we have that t a.e. for some piecewise C0 functions \u03b1\u2113 are integral curves of Zi starting at t =\u20090 and q which are inside \u0394q \u2229 U. In particular, U bounded by these 1-chains as follows: \u03b5\u2113] to U. Since \u03b1\u2113(s) and \u03b3\u2113(s) are piecewise C1 in the s variable, the domain of this map can be partitioned into smaller rectangles on which v\u2113 are differentiable and therefore whose images are two cells. Then for each \u2113 the images of v\u2113 become 2-chains which we denote as C\u2113. Note that v1 = v2 = q for all t. And also let the image of v2 be a curve \u03c4. Since v1 is \u03b2 \u22c5 \u03c4. It is also clear that v\u2113 = \u03b1\u2113(s) and v\u2113 = \u03b3\u2113(s). Then we orient these curves and Ci such that modulo some 1-chains whose image is q: \u2202C1 = \u03b31 \u2212 \u03b2 \u2212 \u03c4 \u2212 \u03b11, \u2202C2 = \u03c4 \u2212 \u03b32 + \u03b12.Denote U bounded by concatenating \u03b31, \u03b32 and \u03b2  in the right orientation so that \u2202\u0393 = \u03b2 \u2212 \u03b31 + \u03b32. Finally, also orient P so that \u2202P = \u03b11 \u2212 \u03b12. Then, \u0393,C1, C2 and P form a closed 2-chain CC \u2260\u20090 for all p \u2208 U and that Since Now using Eqs.\u00a0P satisfies some properties, we can get a more precise estimateIn the case Assume thatP, \u03b31, \u03b32, qare as in Proposition 2.4, P \u2282 B\ud835\udf160), area of P is less thanD(n)\u03b52and\u03b5 \u2264 F(n)\ud835\udf160for someD(n),G(n),F(n) >\u20090. Then for\ud835\udf160small enough, there exists a constantC(n) such thatMoreover, ifPis everywhere tangent toXi(q), Xj(q) (q fixed),andP is everywhere tangent to \u0394q, q has modulus of continuity \u03c9, we have that E(n) >\u20090. But |p \u2212 q|\u2264 G(n)\ud835\udf160. Choose \ud835\udf160 small enough so that \ud835\udf160 small enough so that P is everywhere tangent to Xi(q),Xj(q) for q is fixed. So in this case, we use the fact that (letting p \u2208 P) \ud835\udf160 small enough so that d\u03b7p(Xi(p),Xj(p))\u2260\u20090 and never changes sign then d\u03b7p(Xi(q),Xj(q)) never changes sign. Then choosing a parametrization P such that k = i, j, one gets\ud835\udf160 small enough so that c and Eq.\u00a0\u03b7(\u2202y) always has constant sign which wlog we can assume positive. So signSince \ud835\udf16, we will construct a certain n-dimensional manifold Xi, which is admissible and transverse to \u2202y direction. This is carried out in Section\u00a0The part of the Theorem 1 about smooth adapted coordinates is divided into two separate parts. First given \ud835\udf16|\u2264\u20092n\ud835\udf160, due to condition in Eq.\u00a0C1 function T\ud835\udf16 : n \u2192 VXi are uniquely integrable and since due to their form, an integral curve of Xi can intersect an integral curve of Xj for i\u2260j only once. Therefore, the image of T, which we denote as C1 manifold. Moreover, every point on it is obviously accessible. Finally, it can be given as a graph over , in fact due to the form of the vector fields T\ud835\udf16 = ) for some C1 function a. In particular, note that if  = T\ud835\udf16, then |x| = |t|. Also Given any |Letbeas above and |\ud835\udf16|\u2264\u20092n\ud835\udf160. Then,For anyp =  \u2208 Uwith |xi|\u2264 \ud835\udf16, there exists a uniquewiththe samex coordinates as p.For anyonehas that |y|\u2264|x|C(n)\u03c9x|2|.ti = xi one has T\ud835\udf16 = ). For the second, letting q is the end point of composition of flows tangent to Xi which have modulus of continuity \u03c9, total time of flow less than |x| and length less than 2|x|. Since the flow starts at the point 0, where Xk(0) = \u2202k, its rise in y coordinates is bounded by |x|C(n)\u03c9x|2|. \u25a1For the first item, taking p =  with |xi|\u2264 \ud835\udf16, p to the point q on x coordinate as p. For |\ud835\udf16|\u2264 \ud835\udf160, we define the boxes K1\ud835\udf162 (which are inside U by conditions on \ud835\udf160). Next, Lemma finalizes the proof of existence of smooth adapted coordinates in Theorem 2.Define for There exists an\ud835\udf160 >\u20090 andK1, K2 >\u20090 such thatfor all\ud835\udf16 \u2264 \ud835\udf160onehas,\ud835\udf160 that has been fixed before Eq.\u00a0\ud835\udf160) for all \ud835\udf16 \u2264 \ud835\udf160, one has V\ud835\udf16, we can travel with admissible curves of length less than or equal to \u2202y up to some uniform amount \ud835\udf16 and reach anywhere in \u03c4 :  \u2192 U with We start with the Now, we build the curves that allows us to travel horizontally from any \u03ba4 \u2218 \u03ba3 \u2218 \u03ba2 \u2218 \u03ba1, so that s with \u2113(s) = i, j. Also U and always lies on the \u2202y axis passing above q1. Moreover, by Eq.\u00a0n +\u20094 integral curves of some \u00b1 X\u2113 with integration times less than \ud835\udf160. Denoting the image of this curve as \ud835\udf16 \u2264 \ud835\udf160 such that setting There exists some Xi or Xj, we can go in the opposite direction along the \u2202y.By reverting either We let \u2202y) while the first item guarantees that we can travel more than |p \u2212 q1|. Then, by mean value theorem by reducing \ud835\udf16, we can reach any point on the \u2202y axis passing through q1 whose distance to q1 is less than Note first that these are admissible curves which have length less than \u03b32 = \u03ba3 \u2218 \u03ba4 (defined up to \u03b31(0) = \u03b32(0) = q and \u03a0x(\u03b31(\u03b51)) = \u03a0x(q1) = \u03a0x(\u03b32(\u03b52)) = \u03a0x(q2). Projection of \u03b3i along \u2202y to \u0394q gives us \u03b1i the boundary of a parallelogram P with sides\u2019 lengths less than \u03b1i contain q and |q|\u2264 \ud835\udf16, they are inside B \u2282 U  + \u2113(\u03b31) \u2264\u20092\ud835\udf16. Therefore, P \u2282 B\ud835\udf16). Also area of P is less than D(n)\ud835\udf162. So since \u03b31(\ud835\udf161) = q1, using Eq.\u00a0Now, it remains to show that The claim about opposite directions is an automatic consequence of Eq.\u00a0p =  \u2208 U such that d\u0394 \u2264 \ud835\udf16, then |x|\u2264\u20092n\ud835\udf16 and K2.To prove this inclusion, we need to prove that if d\u0394 \u2264 \ud835\udf16 means that there exists an admissible curve \u03b31 :  \u2192 0 that connects 0 to p such that \u2113(\u03b31) =\u20092\ud835\udf16. Since for all i, |xi|\u2264 \u2113(\u03b31) =\u20092\ud835\udf16 we get |x|\u2264\u20092n\ud835\udf16. So it remains to estimate The condition d\u03b7|\u221e appearing there is replaced by |d\u03b7|\u0394|\u221e which will be thanks to Proposition 2.4. We first state a lemma ) \u2282 U. Then, there exists a 2-chain cc such that\u2202(cc) = c, a(cc) \u2264 C(n)\u2113(c)2andcc \u2282 B) wherea(\u22c5) denotes Euclidean area.x|\u2264\u20092n\ud835\udf16, there exists a point q on q \u2212 p|\u2264 K24n2\ud835\udf162 for some K2. We can connect q to 0 by along \u03b32 and apply Proposition 2.4 to the curves \u03b31, \u03b32 again  = \u03b32(0) =\u20090 and \u03b31(\ud835\udf161) = p, \u03b32(\ud835\udf162) = q which have the same x coordinates. Then, Now since |\u03b31 and \u03b32 along \u2202y to \u03940 forms a closed piecewise C1 curve \ud835\udf16 +\u20094n\ud835\udf16 \u2264\u20096n\ud835\udf16. Since they also contain 0, these curves are in B which is in U by condition in Eq.\u00a0P that satisfies the conditions in Proposition 2.4 and Corollary 2.6. By Eq.\u00a02.4B) \u2282 B \u2282 U. So by Lemma 2.9, \u03b1 bounds a 2-cycle P \u2282 B \u2282 U \u2229\u03940 such that |P|\u2264 C(n)\ud835\udf162. Since \u03b31(\ud835\udf161) = p and \u03b32(\ud835\udf162) = q we have by Eq.\u00a0The projection of Now, it is easy to prove the rest using Lemmas 2.7 and 2.8. Note again that for x|\u2264\u20092n\ud835\udf16 and z \u2212 y|\u2264\u20094K2n2\ud835\udf162. But we know that First, we prove that Then, we prove C1 adapted coordinates. Note that for each \ud835\udf16, C1 surfaces given as the image of the map T\ud835\udf16 with |ti|\u2264 \ud835\udf16 and for \ud835\udf161 < \ud835\udf162, \u03d5 : V \u2192 U on some appropriately sized domain V \u2282 U such thatC1 diffeomorphism onto its image and takes each \ud835\udf16 \u2264\u20092n\ud835\udf160) to the x plane. It also maps X\u2113(0) to \u2202\u2113. This is again an adapted coordinate system. In particular in this adapted coordinate system x|\u2264 \ud835\udf16 and |y|\u2264 K\ud835\udf162. So we get Finally, we prove the statement about the existence of d2 =\u20090 and usual homotopy theory, but it is out of the scope of this paper. These will be consider in a separate paper but the interested reader can have a look at the preprint [In this section, we give some of the very basic properties of continuous exterior derivative preprint .To construct examples, we give an alternative characterization Lemma 3.1 which is proven in  -form inC0topology, which becomes the continuous exterior derivative of\u03b7.Although this construction is local, one can show that the usual glueing techniques can be used to produce global examples. Since we are only interested in local examples here, interested read should refer to . A classLetsuchthat eachaiis continuous and has continuous partial derivatives with respect to allxjforj\u2260i. Then,\u03b7has continuous exterior derivative.C1 functions (by mollification) C0 topology to ai, such that i\u2260j. Then, define \u03b7k converges in C0 topology to \u03b7 and d\u03b7k converges to d\u03b7 is the continuous exterior derivative of \u03b7 by Lemma 3.1. \u25a1By assumptions on regularity of the functions, we can find Now, we create an example from \u03b7 = ady + bdx + cdz. Then since a =\u20091. To create a non-integrable and non-differentiable example, we have to satisfy (cx \u2212 bz + byc \u2212 cyb)(p) >\u20090 at some point where \u03b7 is non-differentiable. Consider \u03b1, \u03b2 >\u20090, which gives \u03b7 (or any of its product with a differentiable function) is non-differentiable at x =\u20090,z =\u20090,y =\u20090 but \u03b7 \u2227 d\u03b7(0) = dx \u2227 dz \u2227 dy. Therefore, there exists a neighbourhood of 0 on which \u03b7 is defined but non-differentiable and yet satisfies the sub-Riemannian properties mentioned in this paper. To create a non-H\u00f6lder example out of this, we can replace for instance x =\u20090,z =\u20090,y =\u20090 and also non-integrable.We first consider the class of examples given by Lemma 3.2. We restrict our-self to It is possible to given an example in which non of the components are differentiable in any variable but it requires to prove some analytic properties of continuous exterior derivative. We state the example without proof but the interested reader can see Example 2.17 and Lemma 2.15 in  for the Consider the following 1 \u2212form In this subsection, we give an application to dynamical systems.M be a compact Riemannian manifold of dimension n +\u20091 and f : M \u2192 M is a C3 diffeomorphism. Assume moreover that there exists a continuous splitting Dfx. This splitting is called partially hyperbolic if there exists constants K, \u03bb\u03c3, \u03bc\u03c3 >\u20090 for \u03c3 = s, c, u such that \u03bcs \u2264\u20091, \u03bcs < \u03bbc, \u03bcc < \u03bbu, \u03bbu >\u20091 and for all \u03c3 = s, c, u, x \u2208 M and v\u03c3 \u2208 TxM with |v\u03c3| =\u20091 one has, f, the Dfx expands Eu and Es are uniquely integrable into what are called as unstable and stable manifolds. Given p \u2208 U, we denote the connected component of the stable and unstable manifold in U that contains p as Ec, Ecs = Ec \u2295 Es or Ecu = Ec \u2295 Eu maybe be non-integrable both in the case the bundles are continuous (see [Let ous (see ), a wellous (see ) and or ous (see ). The inous (see ). It is Ec and Es are C1 and center bunched, partially hyperbolic then Ecs is uniquely integrable. As far as we are aware, there is no general result on integrability of such continuous bundles that do not make any assumptions on the differentiable and topological properties of the manifold M  inside partially hyperbolic systems.A dynamical assumption that we will make is center bunching. A system is called center bunched if instance  where ceinstance  where a instance , the autinstance  where thtorus or  where thOur aim is to make one small step towards an integrability condition for continuous bundles that relies only on the constants. The only place where differentiability is required in the proof of the theorem in  is whereAssumef : M \u2192 Mis a diffeomorphism of a compact manifoldwhich admits a center bunched partially hyperbolicsplittingwhereAssume moreover thatadmitsa continuous exterior derivative. Then,Ecsis uniquely integrable with aC1foliation.\u03b7 \u2227 d\u03b7(p) >\u20090 where Ecs = ker(\u03b7) and d\u03b7 is the continuous exterior derivative. Then by Theorem 1, there exists a C1 adapted coordinate system and a neighbourhood U of p on which every point q on the local unstable manifold \u03ba(t) such that \u03ba(0) = p, \u03ba(\u2113(\u03ba)) = q and d \u2265 c|\u03ba|2. To show that the last condition can be achieved, we choose first a smooth adapted coordinate system at p for Esc so that the unstable bundle is very close to the \u2202y direction (since both are transverse to Esc this is possible). By choosing it close enough, we can make sure that when we pass to the C1 adapted coordinates using the transformation given in Eq.\u00a0\u2202y direction are still close enough so that in a small enough neighbourhood U and for any xi|\u2264 \u03b4|y| where C1 adapted coordinate system, for \ud835\udf16 small enough, we pick y| = K1\ud835\udf162. But the ball-box theorem tells us that there exists a length-parameterized admissible curve \u03ba such that \u2113(\u03ba) \u2264 \ud835\udf16, \u03ba(0) = p and \u03ba(\u2113(\u03ba)) = q. Since p =\u20090). Therefore, for some constant c, \u03b7 \u2227 d\u03b7 =\u20090 everywhere. Then, by the integrability theorem of Hartman in [Ecs integrates to a unique C1 foliation. \u25a1As in , one stam 4.1 of  are fullrtman in , this me"}
{"text": "Anatidae); herons and egrets (Ardeidae); terns (Sternidae); and gulls (Laridae). This article describes the level of lead in the most commonly studied tissue types: feathers, bones and the liver. The study also presents data concerning the concentration of lead in the eggs of water birds. The highest levels of lead pollution can be observed in China and Korea, related to their high level of industrialization. In Iran too, environmental lead pollution is high, likely due to the developed petrochemical industry. Lead pollution in Japan, as well as in Western European countries , seems to be much lower than in China, India or Iran. Nevertheless, the level of pollution in Europe is higher than satisfactory, despite the introduction of a number of bans related to, for example, the use of leaded petrol or lead-containing paints. Finally, the USA and Canada appear to be the areas with the lowest lead pollution, possibly due to their low population densities.Due to the ability of birds to travel long distances in the air, the potential feeding area of each individual is much larger than that of typical terrestrial animals. This makes birds a convenient indicator of environmental lead (Pb) pollution over large areas, in particular areas of inland and coastal waters. The aim of this study was to assess the concentrations of Pb in various organs of water birds from a variety of locations. The focus was on ducks, geese and swans ( The ubiquity and toxicity of lead (Pb) have it ranked as the second most dangerous environmental poison in the world ATSDR .Currently, Pb is still used in the production of batteries, fishing sinkers and bullets for firearms . Nevertheless, all these birds play essential roles in the functioning of the ecosystem and, due to their position in the trophic chain, they are susceptible to bioaccumulation of pollutants, including heavy metals. All individuals differ in body weight, metabolic rate, habitat, range and diet, and therefore, it can be expected that birds living in the same area will accumulate Pb in different amounts depending on these factors. Many of these birds can also live directly in or on the outskirts of cities . Moreover, many of them are hunted species, which makes them available for ecotoxicological studies (e.g. ducks).The aim of this study was to make a comparative analysis of Pb content in the liver, bones, eggs and feathers of birds living in the Northern Hemisphere on the basis of published data from the last 20\u00a0years (1998\u20132018). The available literature provided data on members of the families With the wealth of available information using wild bird species as biomonitors, it should be possible to assess the state of the entire aquatic ecosystem of the Northern Hemisphere in terms of Pb pollution over of the last 20\u00a0years.https://www.ncbi.nlm.nih.gov/pubmed/) was used to search for all articles on a given topic  and \u201dSternidae and Laridae: \u201c(Pb or lead) and (gull or larus or tern or sterna)\u201dArdeidae: \u201c(Pb or lead) and (heron or egret)\u201dThe PubMed search engine  chicks from Hongdo Island, Gyeongsangnam-do (Korea), feathers showed a lead concentration of 3.24\u2009\u00b1\u20091.75\u00a0\u03bcg/g dry weight (dw) (Kim and Oh Larus saundersi) in Dongtai, Jiangsu province (China), the concentration of lead in feathers was 2.05\u2009\u00b1\u20090.47\u00a0\u03bcg/g dw  population in the Hara Biosphere Reserve of Southern Iran, a country with a very active oil industry, the concentration of lead in feathers was 7.04\u00a0\u03bcg/g dw  collected in Midway, the measured concentration of lead was 0.289\u00a0\u00b1\u20090.063\u00a0\u03bcg/g dw (Burger and Gochfeld Sterna fuscata), the Pb level was 0.519\u00a0\u00b1\u20090.048\u00a0\u03bcg/g dw. In the same study, the concentrations of lead in grey-backed terns (Onychoprion lunatus) and white terns  were 0.942\u00a0\u00b1\u20090.312\u00a0\u03bcg/g dw and 1.380\u00a0\u00b1\u20090.693\u00a0\u03bcg/g dw, respectively. The levels of lead concentration among the brown noddy (Anous stolidus) and grey-backed tern (Onychoprion lunatus) populations nesting in Midway were 0.289\u00a0\u00b1\u20090.063\u00a0\u03bcg/g dw and 0.942\u00a0\u00b1\u20090.312\u00a0\u03bcg/g dw, respectively  from Long Island, NY (USA) was 4.10\u2009\u00b1\u20090.26\u00a0\u03bcg/g dw (Burger Leucophaeus pipixcan) in Agassiz National Wildlife Refuge, MN (USA), the concentration of lead was 2.86\u2009\u00b1\u20090.67\u00a0\u03bcg/g dw (Burger and Gochfeld Larus glaucescens) from the clean areas of the Aleutian Islands in Alaska (USA), the concentration of this heavy metal was 0.855\u00a0\u00b1\u20090.133\u00a0\u03bcg/g dw , the concentration of lead was 0.707\u2009\u00b1\u20090.131\u00a0\u03bcg/g dw  in the National Park of the Galician Atlantic Islands on the north-western coast of Spain measured 1\u00a0year later was 1.2\u00a0\u03bcg/g dw and in 2007 as low as 0.2\u00a0\u03bcg/g dw  chick feathers, the concentrations of lead were 0.298\u2009\u00b1\u20090.065\u00a0\u03bcg/g dw and 1.365\u2009\u00b1\u20090.518\u00a0\u03bcg/g dw, respectively  feathers ranged from 0.08\u2009\u00b1\u20090.01\u00a0\u03bcg/g dw in 2011 to 0.13\u2009\u00b1\u20090.01\u00a0\u03bcg/g dw in 2012  and the USA show lower levels of lead pollution. However, more research on European and American populations of Sternidae and Laridae is required to accurately assess the state of the environment there.The aforementioned numbers demonstrate that certain Asian countries  are the most highly polluted with lead, overall. The average feather lead levels in Anatidae indicate high levels of contamination in Korea (Table Anas poecilorhyncha) and white-fronted goose  from Gimpo, Gyeonggi-do (Korea), measured lead concentrations in feathers were 1.69\u2009\u00b1\u20091.54\u00a0\u03bcg/g dw and 1.96\u2009\u00b1\u20091.04\u00a0\u03bcg/g dw, respectively (Kim and Oh Cygnus cygnus) contained 3.64\u2009\u00b1\u20091.13\u00a0\u03bcg/g dw of lead  feathers from Baroghil valley (sparsely populated areas) and Soan valley (densely populated areas), the concentration of lead was 0.91\u2009\u00b1\u20090.03\u00a0\u03bcg/g dw and 1.97\u2009\u00b1\u20090.57\u00a0\u03bcg/g dw, respectively  showed higher lead levels of 1.19\u2009\u00b1\u20090.74\u00a0\u03bcg/g dw  and 2.34\u2009\u00b1\u20090.31\u00a0\u03bcg/g dw  , pintail ducks (Anas acuta) and greylag geese (Anser anser) of 2.02\u2009\u00b1\u20093.13\u00a0\u03bcg/g dw, 3.05\u2009\u00b1\u20093.51\u00a0\u03bcg/g dw and 0.81\u2009\u00b1\u20091.16\u00a0\u03bcg/g dw, respectively  was 0.530\u2009\u00b1\u20090.066\u00a0\u03bcg/g dw  was also low, at 0.488\u2009\u00b1\u20090.075\u00a0\u03bcg/g dw , the concentration of lead in the feathers of the common eider (Cygnus olor) caught at Keszthely Bay, Lake Balaton, Hungary, the level of lead in feathers was 1.11\u2009\u00b1\u20091.23\u00a0\u03bcg/g dw  feathers were 0.45\u00a0\u03bcg/g dw and 0.18\u00a0\u03bcg/g dw, respectively  and great egret  in a recent study were 4.55\u2009\u00b1\u20090.96\u00a0\u03bcg/g dw and 5.15\u2009\u00b1\u20094.62\u00a0\u03bcg/g dw, respectively (Table Nycticorax nycticorax), the lead concentration was 5.28\u2009\u00b1\u20092.22\u00a0\u03bcg/g dw  of 4.80\u2009\u00b1\u20090.67\u00a0\u03bcg/g dw  and little egrets (Egretta garzetta), the levels were 4.6\u2009\u00b1\u20090.4\u00a0\u03bcg/g dw and 4.4\u2009\u00b1\u20090.6\u00a0\u03bcg/g dw, respectively (Burger and Gochfeld Nycticorax nycticorax) nestlings was 9.1\u2009\u00b1\u20092.2\u00a0\u03bcg/g dw (Burger and Gochfeld Egretta garzetta), lead concentrations ranged from 0.8 to 4.4\u00a0\u03bcg/g dw, depending on where the feathers were collected  and the black-crowned night heron (Nycticorax nycticorax) were 4.2\u2009\u00b1\u20091.0\u00a0\u03bcg/g dw and 5.6\u2009\u00b1\u20090.7\u00a0\u03bcg/g dw, respectively  and grey heron (Ardea cinerea) nestlings from the city of Pyeongtaek in Gyeonggi-do (Korea), the concentrations of lead in feathers were 2.65\u2009\u00b1\u20090.76\u00a0\u03bcg/g dw and 2.05\u2009\u00b1\u20091.27\u00a0\u03bcg/g dw, respectively (Kim and Oh Nycticorax nycticorax) was 2.57\u2009\u00b1\u20091.49\u00a0\u03bcg/g dw (Kim and Oh Bubulcus ibis) in the areas of Lahore and Sialkot indicate a contamination level of 297\u2009\u00b1\u200911\u00a0\u03bcg/g dw and 286\u2009\u00b1\u200918\u00a0\u03bcg/g dw, respectively  were 32.5\u2009\u00b1\u200910.3\u00a0\u03bcg/g dw (Shorkot) and 43.1\u2009\u00b1\u200913.4\u00a0\u03bcg/g dw (Mailsi)  near the Chenab River and Ravi River were as high as 37.5\u2009\u00b1\u200910.7\u00a0\u03bcg/g dw and 76.5\u2009\u00b1\u20098.6\u00a0\u03bcg/g dw, respectively  near Rawal Lake reservoir site, not far from Islamabad City, the level of lead was 60.2\u2009\u00b1\u200920.7\u00a0\u03bcg/g dw  was 4.22\u00a0\u03bcg/g dw  and the black-crowned night heron (Nycticorax nycticorax) were 1.110\u2009\u00b1\u20090.309\u00a0\u03bcg/g dw and 0.671\u2009\u00b1\u20090.105\u00a0\u03bcg/g dw  in 1989 was 1.460\u2009\u00b1\u20090.765\u00a0\u03bcg/g dw and has systematically become lower since that time  and tricolored heron (Egretta tricolor), the concentrations of lead in feathers were 0.247\u00a0\u03bcg/g dw and 0.296\u00a0\u03bcg/g dw, respectively  and black-crowned night heron (Nycticorax nycticorax) populations, the concentrations of lead in feathers were 4.52\u00a0\u03bcg/g dw and 3.36\u00a0\u03bcg/g dw, respectively  and Enmedio Island (south coast of Spain), the levels of lead were 0.087\u2009\u00b1\u20090.097\u00a0\u03bcg/g dw and 0.462\u2009\u00b1\u20090.824\u00a0\u03bcg/g dw, respectively  have since the 1990s until now shown high levels of pollution (lead concentration above 2\u00a0\u03bcg/g dw). Opposite results are shown by studies conducted in North American and European countries. In the populations of Ardeidae in these countries, the concentration of lead in feathers has consistently been below 1\u00a0\u03bcg/g dw. However, further environmental studies are still needed to show the current and accurate state of the environment in these countries.Studies on Due to their high level of calcium in the form of phosphates, bones accumulate xenobiotics, including heavy metals. However, the level of heavy metals, including lead, in bones does not indicate a temporary state of the animal, but the average exposure throughout its whole lifetime. Therefore, toxicological studies of bone materials from animals allow for the analysis of the average state of the natural environment over several years prior to the collection of the studied material (Ha\u0107 and Krechniak Anas platyrhynchos) and spot-billed ducks (Anas poecilorhyncha) were measured at 10.6\u2009\u00b1\u200911.1\u00a0\u03bcg/g dw and 10.30\u2009\u00b1\u20096.94\u00a0\u03bcg/g dw, respectively , lead concentration was 0.93\u2009\u00b1\u20091.22\u00a0\u03bcg/g dw  and the black-crowned night heron (Nycticorax nycticorax) from Pyeongtaek, the levels of this heavy metal were 2.60\u2009\u00b1\u20091.11\u00a0\u03bcg/g dw and 4.71\u2009\u00b1\u20093.29\u00a0\u03bcg/g dw, respectively (Kim and Oh Egretta intermedia) and the little egret (Egretta garzetta), the lead concentrations were 1.17\u2009\u00b1\u20090.90\u00a0\u03bcg/g dw and 1.26\u2009\u00b1\u20091.36\u00a0\u03bcg/g dw, respectively , the concentrations of lead in the bones of mallards (Melanitta perspicillata) and wood ducks/carolina ducks (Aix sponsa) were 5.06\u00a0\u03bcg/g dw and 5.86\u00a0\u03bcg/g dw, respectively (Bagley and Locke Anas platyrhynchos) and the Canada goose (Branta canadensis), the concentrations were 13.3\u00a0\u03bcg/g dw and 2.66\u00a0\u03bcg/g dw, respectively (Bagley and Locke Bubulcus ibis) and laughing gulls (Larus atricilla) were 10.57\u2009\u00b1\u20095.12\u00a0\u03bcg/g dw and 9.24\u2009\u00b1\u20091.23\u00a0\u03bcg/g dw  and sandwich terns , the concentrations of lead in bones were 3.28\u2009\u00b1\u20091.69\u00a0\u03bcg/g dw and 1.49\u2009\u00b1\u20093.31\u00a0\u03bcg/g dw, respectively , in the 1960s, at the time when leaded petrol and paint were used, the concentrations of lead in the populations of surf scoters (Marmaronetta angustirostris) and white-headed ducks , the concentrations of lead in bones were 5.19\u00a0\u03bcg/g dw and 91.75\u00a0\u03bcg/g dw  in the north-western part of Poland around the city of Szczecin and the S\u0142o\u0144sk Waterfowl Reserve, the concentration of lead was 5.908\u2009\u00b1\u20096.70\u00a0\u03bcg/g dw and 1.574\u2009\u00b1\u20091.863\u00a0\u03bcg/g dw, respectively  from West Galveston Bay, TX (USA), the concentration of lead in livers was 18.0\u2009\u00b1\u200913.0\u00a0\u03bcg/g dw among males and 12.0\u2009\u00b1\u20098.0\u00a0\u03bcg/g dw among females  in the same area, the level of lead in the liver was 17.70\u2009\u00b1\u20092.47\u00a0\u03bcg/g dw , the concentration of lead in the liver was 1.433\u2009\u00b1\u20090.333\u00a0\u03bcg/g dw , the level of lead in the liver was 1.767\u2009\u00b1\u20090.667\u00a0\u03bcg/g dw  were 0.800\u2009\u00b1\u20090.166\u00a0\u03bcg/g dw and 0.833\u2009\u00b1\u20090.167\u00a0\u03bcg/g dw  and the thick-billed murre (Uria lomvia) in Baffin Bay (Canada), the concentrations of lead in the liver were 0.110\u2009\u00b1\u20090.043\u00a0\u03bcg/g dw and 0.303\u2009\u00b1\u20090.440\u00a0\u03bcg/g dw, respectively  was 0.057\u2009\u00b1\u20090.030\u00a0\u03bcg/g dw  livers from 1983 and 1992 show that lead levels in Baffin Bay (Canada) have been below 0.09\u00a0\u03bcg/g dw for a while , in the population of black-tailed gulls (Larus crassirostris), the level of lead in the liver was 0.022\u2009\u00b1\u20090.009\u00a0\u03bcg/g dw  was 5.1\u2009\u00b1\u20090.8\u00a0\u03bcg/g dw  from Hongdo Island and Rando Island (Korea), the concentrations of lead in the liver were 4.82\u2009\u00b1\u20091.80\u00a0\u03bcg/g dw and 3.71\u2009\u00b1\u20092.17\u00a0\u03bcg/g dw  and little egrets (Egretta garzetta) from the provinces of Gyeonggi-do, Chungcheongnam-do and Seoul city in South Korea, the concentrations of lead in the livers were 5.32\u2009\u00b1\u20092.01\u00a0\u03bcg/g dw and 4.19\u2009\u00b1\u20091.57\u00a0\u03bcg/g dw, respectively (Kim and Oh Ixobrychus eurhythmus), the concentration was 7.97\u2009\u00b1\u20094.38\u00a0\u03bcg/g dw (Kim and Oh Nycticorax nycticorax) and grey herons (Ardea cinerea), the concentrations of lead in livers were 4.43\u2009\u00b1\u20092.42\u00a0\u03bcg/g dw and 3.56\u2009\u00b1\u20091.93\u00a0\u03bcg/g dw, respectively (Kim and Oh Egretta intermedia) and little egrets (Egretta garzetta), the levels of lead were 2.98\u2009\u00b1\u20091.19\u00a0\u03bcg/g dw and 3.36\u2009\u00b1\u20091.29\u00a0\u03bcg/g dw, respectively  and the little egret (Egretta garzetta) from Nilgiris district, Tamil Nadu (India), the concentrations of lead in the livers were 13.26\u2009\u00b1\u20091.23\u00a0\u03bcg/g dw and 3.23\u2009\u00b1\u20091.60\u00a0\u03bcg/g dw  was 2.10\u2009\u00b1\u20090.43\u00a0\u03bcg/g dw  of 0.728\u2009\u00b1\u20090.368\u00a0\u03bcg/g dw  and spot-billed ducks (Anas poecilorhyncha), the concentrations of lead in the liver were 4.74\u2009\u00b1\u20092.92\u00a0\u03bcg/g dw and 4.61\u2009\u00b1\u20092.51\u00a0\u03bcg/g dw, respectively , lead contamination was 1.32\u2009\u00b1\u20092.61\u00a0\u03bcg/g dw , the median lead level was 2\u00a0\u03bcg/g dw (Nam and Lee Anas strepera) and common teals (Anas crecca) wintering on Miankaleh and Gomishan International Wetlands, located in the southern part of Caspian Sea (northern Iran), the concentrations of lead in the liver were 10.73 and 3.60\u00a0\u03bcg/g dw , the concentrations of lead were 3.87\u2009\u00b1\u20091.37\u00a0\u03bcg/g dw and 7.87\u2009\u00b1\u20093.33\u00a0\u03bcg/g dw  showed that the concentration of lead was 1.641\u00a0\u03bcg/g dw (Ayas and Kolankaya The use of the ia Table . AccordiAnser anser) the concentration of lead in livers was as high as 151.4\u2009\u00b1\u200995.8\u00a0\u03bcg/g dw  (5.8\u00a0\u03bcg/g dw), the common teal (Anas crecca) (2.2\u00a0\u03bcg/g dw), the eurasian wigeon (Anas penelope) (1.7\u00a0\u03bcg/g dw), the gadwall (Anas strepera) (1.6\u00a0\u03bcg/g dw), the mallard (Anas platyrhynchos) (17\u00a0\u03bcg/g dw), the northern pintail (Anas acuta) (18\u00a0\u03bcg/g dw), the northern shoveler (Anas clypeata) (3.3\u00a0\u03bcg/g dw), the red-crested pochard (Netta rufina) (1.6\u00a0\u03bcg/g dw) and the tufted duck (Aythya fuligula) (6.1\u00a0\u03bcg/g dw) (Mateo and Guitart Marmaronetta angustirostris) and white-headed ducks , lead concentrations were 1.5\u00a0\u03bcg/g dw and 54\u00a0\u03bcg/g dw, respectively , the levels of lead were 2.39\u00a0\u03bcg/g dw and 2.43\u00a0\u03bcg/g dw, respectively, while in northern shovelers (Anas clypeata) and common teals (Anas crecca), levels were 0.986\u00a0\u03bcg/g dw and 0.465\u00a0\u03bcg/g dw, respectively  was 0.77\u2009\u00b1\u20090.17\u00a0\u03bcg/g dw (Levengood Spatula discors) from the area of the Gulf of Mexico, TX (USA), the concentration of lead was 0.40\u00a0\u00b1\u20090.67\u00a0\u03bcg/g dw  was 0.830\u2009\u00b1\u20090.150\u00a0\u03bcg/g dw  1.7\u00a0\u03bcg/g dw, the brent goose (Branta bernicla) 4.3\u00a0\u03bcg/g dw, the canada goose (Branta canadensis) 1.7\u00a0\u03bcg/g dw, the mallard (Anas platyrhynchos) 3.0\u00a0\u03bcg/g dw, the snow goose (Chen caerulescens) 4.0\u00a0\u03bcg/g dw, the surf scoter (Melanitta perspicillata) 3.0\u00a0\u03bcg/g dw, the velvet scoter (Melanitta fusca) 2.7\u00a0\u03bcg/g dw and the wood duck/carolina duck (Aix sponsa) 5.0\u00a0\u03bcg/g dw  collected more recently in Chesapeake Bay, USA, the level of lead was 0.41\u2009\u00b1\u20090.29\u00a0\u03bcg/g dw, despite the collection of research material from military areas (Beyer and Day Somateria spectabilis) and spectacled eiders (Somateria fischeri) caught near Barrow in the Beaufort Sea in northern Alaska, the concentrations of lead in the liver were 0.13\u2009\u00b1\u20090.09\u00a0\u03bcg/g dw and 0.19\u2009\u00b1\u20090.27\u00a0\u03bcg/g dw, respectively  from Table Bay  and from Tern Island in Foxe Basin in the Arctic Circle, the level of lead in the livers of these birds was 0.088\u00a0\u03bcg/g dw and 0.181\u00a0\u03bcg/g dw, respectively , in the eggs of Canada geese (Branta canadensis), the concentration of lead measured in recent studies was 0.483\u2009\u00b1\u20090.108\u00a0\u03bcg/g dw and in eggs of mallards (Anas platyrhynchos), it was 0.186\u2009\u00b1\u20090.036\u00a0\u03bcg/g dw  from South Brother Island (New York Harbor) and Mill Rock (New York Harbor) contained 0.313\u2009\u00b1\u20090.072\u00a0\u03bcg/g dw and 0.837\u2009\u00b1\u20090.587\u00a0\u03bcg/g dw, respectively (Burger and Elbin Somateria sp.), in the Yukon-Kuskokwim Delta, Alaska (USA), the concentration of lead was significantly higher at 3.21\u2009\u00b1\u20090.15\u00a0\u03bcg/g dw  was only 0.306\u2009\u00b1\u20090.099\u00a0\u03bcg/g dw  from the areas around the industrialized cities of Pakistan\u2014Lahore and Sialkot\u2014showed a very high concentration of lead, 47\u2009\u00b1\u200913\u00a0\u03bcg/g dw and 41\u2009\u00b1\u20098\u00a0\u03bcg/g dw, respectively (Table Bubulcus ibis) eggshells was lower, at 5.40\u2009\u00b1\u20093.01\u00a0\u03bcg/g dw  egg contents were 0.89\u2009\u00b1\u20090.25\u00a0\u03bcg/g dw and 0.84\u2009\u00b1\u20090.54\u00a0\u03bcg/g dw  who spend their breeding period in the same uncontaminated areas were 0.37\u2009\u00b1\u20090.28\u00a0\u03bcg/g dw and 0.44\u2009\u00b1\u20090.33\u00a0\u03bcg/g dw , the degree of contamination was 0.13\u2009\u00b1\u20090.30\u00a0\u03bcg/g dw for the Islam Headworks and 0.58\u2009\u00b1\u20090.88\u00a0\u03bcg/g dw in the Trimmu Headworks  eggshells, lead concentration in the Trimmu Headworks was 1.9\u2009\u00b1\u20091.3\u00a0\u03bcg/g dw and in the Islam Headworks was 1.09\u2009\u00b1\u20090.83\u00a0\u03bcg/g dw  were 1.34\u2009\u00b1\u20091.25\u00a0\u03bcg/g dw and 1.44\u2009\u00b1\u20091.13\u00a0\u03bcg/g dw, respectively \u20140.172\u2009\u00b1\u20090.090\u00a0\u03bcg/g dw and 0.191\u2009\u00b1\u20090.094\u00a0\u03bcg/g dw, respectively  in a study from the early 1990s was 0.341\u00a0\u03bcg/g dw (Ayas and Kolankaya Ardea cinerea), the lead content in eggs and eggshell was 13.6\u2009\u00b1\u20093.2\u00a0\u03bcg/g dw and 3.6\u2009\u00b1\u20091.4\u00a0\u03bcg/g dw, respectively  was very low , the great egret  and the black-crowned night heron (Nycticorax nycticorax) were 88.5\u2009\u00b1\u200914.6\u00a0\u03bcg/g dw, 82.1\u2009\u00b1\u20096.9\u00a0\u03bcg/g dw and 84.9\u2009\u00b1\u20096.9\u00a0\u03bcg/g dw, respectively  and the little egret (Egretta garzetta) were very low, at the level of 0.007\u2009\u00b1\u20090.007\u00a0\u03bcg/g dw and 0.014\u2009\u00b1\u20090.015\u00a0\u03bcg/g dw  was 0.039\u2009\u00b1\u20090.011\u00a0\u03bcg/g dw (Burger and Gochfeld Nycticorax nycticorax) eggs on South Brother Island (New York Harbor) and Mill Rock (New York Harbor) were 0.370\u2009\u00b1\u20090.141\u00a0\u03bcg/g dw and 0.054\u2009\u00b1\u20090.034\u00a0\u03bcg/g dw, respectively , the concentration of lead in eggs of the black-crowned night heron  from Long Island, NY (USA) (Burger and Gochfeld Sterna hirundo) ranged from 0.022 to 0.528\u00a0\u03bcg/g dw, depending on the place where the eggs were collected and the breeding season. Lead concentrations in common tern eggs collected from Mike\u2019s Island in New Jersey (USA) in the years 2000 and 2002 were 0.100\u2009\u00b1\u20090.021\u00a0\u03bcg/g dw and 0.022\u2009\u00b1\u20090.004\u00a0\u03bcg/g dw, respectively (Burger and Gochfeld Larus argentatus) and the great black-backed gull (Larus marinus) were 0.273\u2009\u00b1\u20090.069\u00a0\u03bcg/g dw and 0.227\u2009\u00b1\u20090.075\u00a0\u03bcg/g dw (Burger Sterna hirundo) and the Forster\u2019s tern (Sterna forsteri), the concentrations were 0.164\u2009\u00b1\u20090.025\u00a0\u03bcg/g dw and 0.056\u2009\u00b1\u20090.007\u00a0\u03bcg/g dw, respectively (Burger Leucophaeus pipixcan) in Agassiz National Wildlife Refuge, MN (USA), the concentration of lead was 0.129\u2009\u00b1\u20090.016\u00a0\u03bcg/g dw (Burger and Gochfeld Larus argentatus) population study from Mill Rock in the New York/New Jersey Harbor Estuaries, where lead concentration in eggs in 2011 and 2012 was 0.040\u2009\u00b1\u20090.024\u00a0\u03bcg/g dw and 0.451\u2009\u00b1\u20090.095\u00a0\u03bcg/g dw, respectively (Burger and Elbin Larus marinus), the concentration of lead in eggs collected at this location was 0.138\u2009\u00b1\u20090.044\u00a0\u03bcg/g dw  during the first decade of the twenty-first century was 0.107\u2009\u00b1\u20090.029\u00a0\u03bcg/g dw . The lead content of the eggs of the glaucous-winged gull  from Hongdo Island (Korea) were 3.10\u2009\u00b1\u20091.36\u00a0\u03bcg/g dw and 0.92\u2009\u00b1\u20090.24\u00a0\u03bcg/g dw, respectively (Kim and Oh Larus saundersi) near the city of Dongtai, in the province of Jiangsu (China), the concentration of lead is as high as 79\u2009\u00b1\u200911\u00a0\u03bcg/g dw  in Hong Kong, the concentration of lead in egg contents in the end the twentieth century was 0.010\u2009\u00b1\u20090.021\u00a0\u03bcg/g dw  were 0.023\u2009\u00b1\u20090.015\u00a0\u03bcg/g dw and 0.061\u2009\u00b1\u20090.042\u00a0\u03bcg/g dw, respectively  egg contents and eggshells, lead concentrations were 0.286\u2009\u00b1\u20090.064\u00a0\u03bcg/g dw and 0.260\u2009\u00b1\u20090.049\u00a0\u03bcg/g dw  found near the mouth of the river, Pb levels were 3.70\u2009\u00b1\u20090.80\u00a0\u03bcg/g dw and 30.0\u2009\u00b1\u20098.8\u00a0\u03bcg/g dw  and the Mediterranean gull  revealed concentrations of 0.204\u00a0\u03bcg/g dw and 0.060\u00a0\u03bcg/g dw, respectively  in Axios Delta (Greece), lead concentration was 0.249\u00a0\u03bcg/g dw  in northern Poland, the concentration of lead ranged from 0.4\u2009\u00b1\u20090.1 to 0.7\u2009\u00b1\u20090.8\u00a0\u03bcg/g dw, depending on the egg collection site  on the eggs of the yellow-legged gull  are also heavily polluted with this heavy metal. Western Europe  is less polluted with lead in comparison. Nevertheless, the level of pollution in these European countries is greater than satisfactory, despite the introduction of a number of bans restricting the use of lead, for example, the use of leaded petrol or lead-containing paints. Lead does not decompose and so it can persist in the environment for many years after its use has ceased. Finally, the USA and Canada appear to be the areas with the lowest level of lead pollution. This may be due to the low population density in these countries and a high concern for the environment, encouraging the use of modern technologies that do not require the use of this element. Abandoning the use of lead in petrol has also been crucial in reducing environmental lead contamination. As a result, the current level of lead pollution in the USA is much lower than in the 1970s and 1980s. However, due to the fragmented nature of the data, it is difficult to determine any such a trend across Europe."}
{"text": "Coreference resolution is a challenging part of natural language processing (NLP) with applications in machine translation, semantic search and other information retrieval, and decision support systems. Coreference resolution requires linguistic preprocessing and rich language resources for automatically identifying and resolving such expressions. Many rarer and under-resourced languages (such as Lithuanian) lack the required language resources and tools. We present a method for coreference resolution in Lithuanian language and its application for processing e-health records from a hospital reception. Our novelty is the ability to process coreferences with minimal linguistic resources, which is important in linguistic applications for rare and endangered languages. The experimental results show that coreference resolution is applicable to the development of NLP-powered online healthcare services in Lithuania. Digital means of medical informatics, especially when applying natural language processing (NLP), are indispensable in the application of e-health and digitalization of medical records and processes . The useWith the development of Semantic Web technology, web information retrieval (IR) is changing towards meaning-based IR. The quality of retrieved documents relevant to the user also highly depends on the information extraction (IE) methods applied. In general, IE focuses on automatic extraction of structured information from the unstructured source. Standard document text preprocessing steps used in IE are lexical analysis, morphological analysis, and named entity recognition (NER), which can be complemented by coreference resolution and semantic annotation. The main issue here is the ambiguity and complexity of the natural language, thus making the progress in IE dependent on the evolution of the NLP techniques. While for widely used languages (such as English), the IE-related NLP research has already reached the levels of maturity and practical application on a massive scale  , but theNER when applied to biomedical texts is a critical step for developing decision support tools for smart healthcare. Examples for it are as follows: drug name recognition (DNR), which recognizes pharmacological substances from biomedical texts and classifies them for discovering drug-drug interactions , 13; bioOur previous work included the development of the semantic search framework  for answThe rest of the paper is structured as follows. In Machine-learning and rule-based approaches are efficient methods in semantic processing, especially when enhanced with external knowledge and coreference clues derived from the structured document, while often still performing better (in comparison with classic implementations) in coreference resolution when provided with ground truth mentions , while fIn general, the coreference resolution methods can be classified into knowledge rich and knowledge poor. Both methods require large resources such as semantic information, syntactic annotations, or preannotated corpora of hospital transcripts from a hospital reception. Under resourced, rarer languages, like Lithuanian, usually do not have such resources available.While up-to-date, no research has been performed to solve coreferences in Lithuanian, but many solutions have been proposed for other languages, mostly for English . Note thFor Latvian, the only solution is LVCoref . It is aFor Polish, rule-based Ruler  for scorFor Russian, RU-EVAL-2014  was an eFor Czech, coreferences are annotated in the tectogrammatical layer of Prague Dependency Treebank (PDT) and their first coreference resolution approach was rule based . At firsConsidering languages that are more-or-less grammatically similar to Lithuanian , we summarize the related work on Latvian  and Slavic languages such as Polish, Russian, and Czech in In summary, the rule-based solutions have the advantages of easier adaptability and provide comparable results when good training data are not available as is the case for Lithuania. Many of more advanced solutions cannot be fully adapted for rarer and under-resourced languages due to the lack of available linguistic resources, as is the case with Lithuanian language. For example, BART at the time supported 64 feature extractors, but due to lack of language-specific resources for the Polish language, only 13 could be utilized. The solutions that are not heavy on linguistic resources can be very useful for resource-poor languages in general.Coreference resolution (or anaphora) is an expression, the interpretation of which depends on another word or phrase presented earlier in the text (antecedent). For example, \u201cTom has a backache. He was injured.\u201d Here the words \u201cTom\u201d and \u201cHe\u201d refer to the same entity. Without resolving the relationship between these two structures, it would not be possible to determine why Tom has the backache, nor who was injured. In such cases, semantic information would be lost.Anaphoric objects are expressed with pronouns and cannot be independently interpreted without going back to its antecedent. In this work, such expressions are called coreferences, unless it is required to make a distinction. Usage of such expressions can vary depending on the type and the style of the text. Here we focus on texts from medical-related domains.The role of coreference resolution in the semantic search framework is to provide additional semantic information after named entity recognition before semantic annotation .What features of text, sentence, and word help us recognize the existence of coreference (they are specified in the package Concepts of Input Flow)What kind of text preprocessing is requiredWhat additional resources are required for resolution of certain type coreferences (they are specified in the package Database of Public Persons and Classification of Professions)In this chapter, a conceptualization of coreference resolution is presented. A given model, which is expressed as UML class diagram , specifiA text segmented into sentences and lexemesMorphological features of lexemes identifiedNamed entities recognizedFor example, from the model provided, it is clear that, before coreference resolution starts, it is important to preprocess text and obtain the following:Text preprocessing itself is not a task of coreference resolution, so it is out of the scope of this paper.It is worthy to mention that the model is quite abstract, language independent, and technology independent. Therefore, it is applicable not only for Lithuanian but for grammatically similar languages as well. Concepts of this model are used for the formalization of coreference resolution rules in the next section. The concepts are explained in more detail below.Text, Lexical_Unit, and Named_Entity. The concept Text assumes a textual document whose content should be analysed. Each test has an associated publication date, which is important for solving coreferences. Each text consists of at least one Lexical_Unit, which includes paragraphs, sentences, words, and punctuations, classified into the Sentence and Lexeme categories. Lexeme assumes lexical units such as words, punctuations, and numbers. Each lexeme is characterized by a lemma and a part of speech, and some of them (nouns and pronouns) by grammatical gender and number. The lexeme could be specialized by POS category: Noun, Pronoun, and Other_Part_Of_Speech. Special cases of Other_Part_Of_Speech are Comma and Conjunction, which are required for the description of conditions of some coreference resolution rules.The main concepts of coreference resolution are Named_Entity concept defines an object to whom pronouns or certain nouns can refer. NER algorithms usually recognize three types of entities: a person (Person_NE), an organization (Organization_NE), and a location (Location_NE). The named entities of a person type require special attention a person can be mentioned not only using pronouns but also using a position he/she holds (Position_Held) and a professional name (Profession). Additional information about a person could help resolving such coreferences more precisely. As an example, source of such information could be a Database of Public Person, which includes Known_Person\u2014a well-known person mentioned as Person_NE in the text. The output of a coreference resolution algorithm is a Coreference\u2014a relationship between coreferents. For each coreference, its type , subtype (relative pronoun and noun repetition), position , and group  are specified. Each referent refers at least to one coreferent (a concept Mention). Each Mention starts at a certain position in the text, is of a certain length, and fits at least one Lexeme. Some of them can fit a certain Named_Entity.A The decision table with guidelines for the application of the certain resolution algorithm is shown as A1: specific rules resolution algorithm for resolution of certain usage of pronounsA2: general pronoun resolution algorithm which focuses on the cases where pronouns refer to nouns (or noun phrases) that are recognized as named entities of \u201cperson\u201d classA3: PRA  resolution algorithm for resolution of nouns recognized as named entities and their repeated usage in the same textA4: HHS  resolution algorithm for resolution of nouns recognized as profession names including their synonyms and hypernyms/hyponymsA5: feature resolution algorithm for resolution of nouns that represent certain feature (at the moment only public position being held) of the named entity of a personFor resolution of a specific type of references, we propose the following algorithms:The coreference resolution starts from the sequential analysis of each lexeme looking for a certain type of pronoun and noun. Depending on identified features of lexeme, a decision about further analysis is taken. The decision table  summarizThe idea is that the pronoun-related coreferences should be solved first sequentially by checking the conditions C1, C2, and C3. Then a noun-related coreference resolution should start by sequentially checking the conditions C4, C6, and C7.First-order logic (FOL) formulas are employed to define the main conditions the algorithms should check when resolving coreferences. The concepts of the coreference resolution model  became tThe algorithms follow the grammar rules of the Lithuanian language which are based on the analysis of morphological features of lexemes and their order in the sentence and text. Examples of Lithuanian language sentences were translated into English as closely as possible. All proper names were changed to generic abbreviations to comply with GDPR.vyras [noun], kuris [pronoun] skund\u0117si nugaros skausmu.[LT] \u0160iandien buvo at\u0117j\u0119s man [noun] who [pronoun] had a backache came today.[EN] A vyras [noun], su [preposition] kuriuo [pronoun] aptar\u0117me nugaros skausm\u0105.[LT] \u0160iandien buvo at\u0117j\u0119s man [noun] with [preposition] whom [pronoun] we discussed a backache came today.[EN] A In some cases, there exists a rather rigid structure for pronoun usage and it can be easily defined by using specific rules, for example,Both examples are similar in their construction: [noun] [comma] [optional preposition] [specific pronoun]. In both cases, pronoun \u201ckuriuo\u201d refers to the noun \u201cvyras.\u201d In the first example, we do not have an optional preposition \u201csu,\u201d while we have it in the second one.A condition for the existence of such reference formally is defined as follows:For every sentence s of text t and for every \u201cRelative\u201d type pronoun p, which is contained in the sentence s and has a start position sp1, is of length ln1, follows comma c or follows prepositional lexeme l1, which follows comma c, and for every noun l2, which has a start position sp2, is of length ln2, precedes comma c, is of the same gender gand of the same number n as the pronoun p, the only one coreference relation r, which is resolved in text t, is of \u201cPronominal\u201d type, \u201cRelative\u201d subtype, \u201cBackward\u201d position and \u201cSingle\u201d group between the pronoun p and the noun n, its referent starts at position sp1 and has length ln1, and which fits only one lexeme p and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one lexeme l2, exists (Rule 1).\u2009 Rule 1: \u2200t, s, p, l1, c, l2, g, n, sp1, sp2, ln2. [Text(t) \u22c0 Sentence(s) \u22c0 consists_of \u22c0 Pronoun(p) \u22c0 contains \u22c0 has_type \u22c0 has_start_position \u22c0 has_length \u22c0 Comma(c) \u22c0  \u22c1 (Lexeme(l1) \u22c0 has_pos \u22c0 follows \u22c0 follows) \u22c0 Noun(l2) \u22c0 follows \u22c0 has_gender  \u22c0 has_gender  \u22c0 has_number \u22c0 has_number \u22c0 has_start_position \u22c0 has_length \u27f6 \u2203!r \u2203!m. [Coreference(r) \u22c0 resolved_in \u22c0 has_type \u22c0 has_subtype \u22c0 has_position \u22c0 has_ group \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 Mention(m) \u22c0 refers_to \u22c0 has_start_position \u22c0 has_length \u22c0 fits]]Tomo [noun], Lino [noun], Petro [noun] ir [conjunction] Egl\u0117s [noun], kuri\u0173 [pronoun] su\u017ealojimai atrod\u0117 pana\u0161\u016bs.[LT] Komisija nerado pana\u0161um\u0173 tarp Tom [noun], Linas [noun], Peter [noun] and [conjunction] Egl\u0117 [noun], who [pronoun] shared similar injuries.[EN] The committee did not find The relative pronoun might be plural and refer to multiple singular  nouns:In this case, a plural pronoun \u201ckuri\u0173\u201d is referring to four singular nouns that have different genders. The previous rule would not be able to solve such coreference. For this case, the construction would be: [noun] [comma] [noun] [comma] [noun] [conjunction] [noun] [comma] [optional preposition] [specific pronoun].For such case, a special condition must be defined:For every sentence s in text t and for every \u201cRelative\u201d type pronoun p of \u201cPlural\u201d number, which is contained in the sentence s and has a start position sp1, is of length ln1, follows comma c1 or follows prepositional lexeme l, which follows comma c1, and for every noun n1, which precedes comma c1, has a start position sp2, is of length ln2, follows conjunction j, and for every noun n2, which precedes conjunction j, has a start position sp3, is of length ln3, and for every existing noun n3, which follows comma c2, and for every existing noun n4, which precedes comma c2, has a start position sp4, is of length ln4, the only one coreference relation r, which is resolved in text t, is of \u201cPronominal\u201d type, \u201cRelative\u201d subtype, \u201cBackward\u201d position and \u201cMultiple\u201d group, its referent starts at position sp1 and has length ln1, fits only one lexeme p, refers to only one mention m1, which starts at position sp2, has length ln2, and fits noun n1, refers to only one mention m2, which starts at position sp3, has length ln3, and fits only one noun n2, and refers at least to one mention m3, which starts at position sp4, has length ln4, and fits noun n4, exists (Rule 2).\u2009 Rule 2: \u2200t, s, p, l, c1, n1, sp1, ln1, sp2, ln2, j, n2, sp3, ln3.[Text(t) \u22c0 Sentence(s) \u22c0 consists_of \u22c0 Pronoun(p) \u22c0 contains \u22c0 has_number \u22c0 has_type \u22c0 has_start_position \u22c0 has_length \u22c0 Comma(c1) \u22c0  \u22c1 (Lexeme(l) \u22c0 has_pos \u22c0 follows \u22c0 follows) \u22c0 Noun(n1) \u22c0 follows \u22c0 has_start_position \u22c0 has_length \u22c0 Conjunction(j) \u22c0 follows \u22c0 Noun(n2) \u22c0 follows \u22c0 has_start_position \u22c0 has_length \u22c0  \u22c0 Comma(c2) \u22c0 Noun(n4) \u22c0 follows \u22c0 follows \u22c0 has_start_position \u22c0 has_length) \u27f6 \u2203!r \u2203!m1 \u2203!m2 \u2203m3. [Coreference(r) \u22c0 resolved_in \u22c0 has_type \u22c0 has_subtype \u22c0 has_position \u22c0 has_ group \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 Mention(m1) \u22c0 refers_to \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 Mention(m2) \u22c0 refers_to \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 Mention(m3) \u22c0 refers_to \u22c0 has_start_position \u22c0 has_length \u22c0 fits]]\u2009 [PL] Dzisiaj przychodzi\u0142 m\u0119\u017cczyzna [noun], kt\u00f3ry [pronoun] skar\u017cy\u0142 si\u0119 na b\u00f3l plec\u00f3w.\u2009 [RU] \u0421\u0435\u0433\u043e\u0434\u043d\u044f \u043f\u0440\u0438\u0445\u043e\u0434\u0438\u043b \u043c\u0443\u0436\u0447\u0438\u043d\u0430 [noun], \u043a\u043e\u0442\u043e\u0440\u044b\u0439 [pronoun] \u0436\u0430\u043b\u043e\u0432\u0430\u043b\u0441\u044f \u043d\u0430 \u0431\u043e\u043b\u044c \u0432 \u0441\u043f\u0438\u043d\u0435.Though examples illustrating the certain case of coreference are given in Lithuanian and English only, rules for resolution of such coreferences could be applied for other languages as well. For example, Rule 1 could be successfully applied for coreference resolution in Polish or Russian languages. Let us take the same example of a sentence in Polish and Russian:We can see that a structure of the sentence (number and order of lexemes) is similar, a pronoun goes after the comma and it refers to a noun, and compatibility of morphological features  of noun and pronoun is retained. From the given example, we understand that the coreference relation between pronoun and noun exists and conditions for such existence are the same as specified in Rule 1.Jonas Jonaitis [person noun phrase] skambino \u012f registrat\u016br\u0105. Jis [pronoun] skund\u0117si galvos skausmu.[LT] Jonas Jonaitis [person noun phrase] called a reception. He [pronoun] complained about headache.[EN] This algorithm focuses on the cases where pronouns refer to nouns (or noun phrases) that are recognized as named entities of \u201cperson\u201d class by NER. The algorithm starts from the identification of not demonstrative pronoun. In a given example below, such a pronoun is in the second sentence\u2014\u201cJis\u201d (\u201cHe\u201d)If the pronoun is in the relative clause, the algorithm moves backwards analysing words going before the pronoun. In a given example, the pronoun is at the beginning of the sentence, so remaining parts of the sentence are not analysed, and the algorithm moves one sentence backwards.s1 before pronoun p:The conditions for the existence of such reference formally could be defined as three alternatives. The first one describes conditions for reference existing in the same sentence For each text's t sentence s1 and pronoun p not of Demonstrative type that is contained in sentence s1 and has gender g, number n, start position sp1 and length of ln1, and named entity e1 that is in the same sentence s1, is expressed by lexeme l, and has gender g, number n, start position sp2 and is of length ln2, and is before pronoun p (sp2 is lower than sp1), but closer to pronoun p than possible named entities e2 and e3 (sp2 higher than sp3 and sp4), the only one coreference relation r, which is resolved in text t, is of \u201cPronominal\u201d type, \u201cRelative\u201d subtype, \u201cBackward\u201d position and \u201cSingle\u201d group between the pronoun p and the named entity e1, its referent starts at position sp1 and has length ln1, and which fits only one pronoun p and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one named entity e1, exists (Rule 3).\u2009 Rule 3: \u2200t, s1, p, l, e1, g, n, sp1, ln1, sp2, ln2.[Text(t) \u22c0 Sentence(s1) \u22c0 consists_of \u22c0 Pronoun(p) \u22c0 contains \u22c0 \u00achas_type \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 has_length \u22c0 Person_NE(e1) \u22c0 includes \u22c0 Lexeme(l) \u22c0 expressed_by \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 has_length \u22c0 sp2<sp1 \u22c0 \u00ac \u22c0 includes \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 Person_NE(p3) \u22c0 includes \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 sp2>sp3 \u22c0 sp4>sp2)) \u27f6 \u2203!r \u2203!m. [Coreference(r) \u22c0 resolved_in \u22c0 has_type  \u22c0 has_subtype  \u22c0 has_position \u22c0 has_group \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 Mention(m) \u22c0 refers_to \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 fits]]p refers to the named entity in the previous sentence s2:The second alternative describes a case when a pronoun For each text's t sentence s1, s2, where s1 follows s2, and pronoun p not of Demonstrative type that is contained in sentence s1 and has gender g, number n, start position sp1 and length of ln1, and named entity e1 that is contained in sentence s2, is expressed by lexeme l, and has gender g, number n, start position sp2 and is of length ln2, and is closer to pronoun p than possible named entities e2 and e3 (sp2 higher than sp3 and sp4), the only one coreference relation r, which is resolved in text t, is of \u201cPronominal\u201d type, \u201cRelative\u201d subtype, \u201cBackward\u201d position and \u201cSingle\u201d group between the pronoun p and the named entity e1, its referent starts at position sp1 and has length ln1, and which fits only one pronoun p and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one named entity e1, exists (Rule 4).\u2009 Rule 4: \u2200t, s1, s2, p, l, e1, g, n, sp1, ln1, sp2, ln2.[Text(t) \u22c0 Sentence(s1) \u22c0 Sentence(s2) \u22c0 consists_of \u22c0 consists_of \u22c0 follows  \u22c0 Pronoun(p) \u22c0 contains \u22c0 \u00achas_type \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 has_length \u22c0 Person_NE(e1) \u22c0 includes \u22c0 Lexeme(l) \u22c0 expressed_by \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 has_length \u22c0 \u00ac \u22c0 includes \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 Person_NE(p3) \u22c0 includes \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 sp2>sp3 \u22c0 sp4>sp2)) \u27f6 \u2203!r \u2203!m. [Coreference(r) \u22c0 resolved_in \u22c0 has_type  \u22c0 has_subtype  \u22c0 has_position \u22c0 has_group \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 Mention(m) \u22c0 refers_to \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 fits]]p in the sentence s1 refers to the named entity in the sentence s3, preceding sentences s2 and s1:The third alternative describes a case when a pronoun For each text's t sentence s1, s2, s3, where s1 follows s2 and s2 follows s3, and pronoun p not of Demonstrative type that is contained in sentence s1 and has gender g, number n, start position sp1 and length of ln1, and named entity e1 that is contained in sentence s3, is expressed by lexeme l, and has gender g, number n, start position sp2 and is of length ln2, and is closer to pronoun p than possible named entities e2 and e3 (sp2 higher than sp3 and sp4), the only one coreference relation r, which is resolved in text t, is of \u201cPronominal\u201d type, \u201cRelative\u201d subtype, \u201cBackward\u201d position and \u201cSingle\u201d group between the pronoun p and the named entity e1, its referent starts at position sp1 and has length ln1, and which fits only one pronoun p and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one named entity e1, exists (Rule 5).\u2009 Rule 5: \u2200t, s1, s2, s3, p, l, e1, g, n, sp1, ln1, sp2, ln2.[Text(t) \u22c0 Sentence(s1) \u22c0 Sentence(s2) \u22c0 Sentence(s3) \u22c0 consists_of \u22c0 consists_of \u22c0 consists_of \u22c0 follows  \u22c0 follows  \u22c0 Pronoun(p) \u22c0 contains \u22c0 \u00achas_type \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 has_length \u22c0 Person_NE(e1) \u22c0 Lexeme(l) \u22c0 expressed_by \u22c0 includes \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 has_length \u22c0 \u00ac \u22c0 includes \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 Person_NE(e3) \u22c0 includes \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 sp2>sp3 \u22c0 sp4>sp2)) \u27f6 \u2203!r \u2203!m. [Coreference(r) \u22c0 resolved_in \u22c0 has_type  \u22c0 has_subtype  \u22c0 has_position \u22c0 has_group \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 Mention(m) \u22c0 refers_to \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 fits]]man [pronoun] teko analizuoti, padidinus skaitmenines paslaugas tik nedidel\u0117 dalis Lietuvos [location noun] ligonini\u0173 suma\u017eino etat\u0173 ar atleido darbuotojus, o tai l\u0117m\u0117 nema\u017e\u0105 papildom\u0105 ind\u0117l\u012f \u012f paslaugos kokyb\u0119\u201d teig\u0117 J. Jonaitis [person noun phrase].[LT] Pasteb\u0117tina, kad ligogin\u0117se apsilank\u0117 10 mln. pacient\u0173, nepaisant to, kad 2016 m. j\u0173 buvo 4% ma\u017eiau . Tai labiausiai l\u0117m\u0117 skaitmenini\u0173 paslaug\u0173 padidinimas: \u201eKiek I [pronoun] had analysed, only a small part of Lithuanian [location noun] hospitals have reduced their posts or dismissed employees, but digitization of services has led to a significant additional contribution to the quality of service,\u201d said J. Jonaitis [person noun phrase].[EN] It is noteworthy that the total hospital patient count has reached 10 million, even though in 2016 the number was less than 4% (around 9.5 million patients). This was influenced by the digitization of e-health services. \u201cAs far as Another example presents a case when a coreferent of the pronoun \u201cman\u201d  is in the following sentence:s1 after pronoun was mentioned:If the algorithm does not find any named entities moving backwards, it moves back to pronoun and proceeds forward. The algorithm continues moving forward until it locates \u201cJ. Jonaitis\u201d entity, which is recognized as a person. Since the gender of the pronoun \u201cman\u201d is ambiguous , only their grammatical numbers are compared. Both are singular; therefore, the algorithm picks \u201cJ. Jonaitis\u201d as a postcedent of the corefering object \u201cman.\u201d Conditions for the existence of such reference formally could be defined as two alternatives. The first one describes the conditions for reference existing in the same sentence For each text's t sentence s1 and pronoun p not of Demonstrative type that is contained in sentence s1 and has gender g, number n, start position sp1 and length of ln1, and named entity e1 that is in the same sentence s1, is expressed by lexeme l, and has gender g, number n, start position sp2 and is of length ln2, and is after pronoun p (sp2 is higher than sp1), but closer to pronoun p than possible named entities e2 and e3 (sp2 higher than sp3 and sp4), the only one coreference relation r, which is resolved in text t, is of \u201cPronominal\u201d type, \u201cRelative\u201d subtype, \u201cBackward\u201d position and \u201cSingle\u201d group between the pronoun p and the named entity e1, its referent starts at position sp1 and has length ln1, and which fits only one pronoun p and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one named entity e1, exists (Rule 6).\u2009 Rule 6: \u2200t, s1, p, l, e1, g, n, sp1, ln1, sp2, ln2.[Text(t) \u22c0 Sentence(s1) \u22c0 consists_of \u22c0 Pronoun(p) \u22c0 contains \u22c0 \u00achas_type \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 has_length \u22c0 Person_NE(e1) \u22c0 includes \u22c0 Lexeme(l) \u22c0 expressed_by \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 has_length \u22c0 sp1<sp2 \u22c0 \u00ac \u22c0 includes \u22c0 has_start_position \u22c0 Person_NE(e3) \u22c0 includes \u22c0 has_start_position \u22c0 sp2<sp3 \u22c0 sp4<sp2)) \u27f6 \u2203!r \u2203!m. [Coreference(r) \u22c0 resolved_in \u22c0 has_type  \u22c0 has_subtype  \u22c0 has_position \u22c0 has_group \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 Mention(m) \u22c0 refers_to \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 fits]]p refers to the named entity in the following sentence s4:The second alternative describes the case when the pronoun For each text's t sentence s1, s4, where s4 follows s1, and pronoun p not of Demonstrative type that is contained in sentence s1 and has gender g, number n, start position sp1 and length of ln1, and named entity e1 that is contained in sentence s2, is expressed by lexeme l, and has gender g, number n, start position sp2 and is of length ln2, and is closer to pronoun p than possible named entities e2 and e3 ( sp2 higher than sp3 and sp4 ), the only coreference relation r, which is resolved in text t, is of \u201cPronominal\u201d type, \u201cRelative\u201d subtype, \u201cBackward\u201d position and \u201cSingle\u201d group between the pronoun p and the named entity e1, its referent starts at position sp1 and has length ln1, and which fits only one pronoun p and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one named entity e1, exists (Rule 7).\u2009 Rule 7: \u2200t, s1, s2, p, l, e1, g, n, sp1, ln1, sp2, ln2.[Text(t) \u22c0 Sentence(s1) \u22c0 Sentence(s2) \u22c0 consists_of \u22c0 consists_of \u22c0 follows  \u22c0 Pronoun(p) \u22c0 contains \u22c0 \u00achas_type \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 has_length \u22c0 Person_NE(e1) \u22c0 includes \u22c0 Lexeme(l) \u22c0 expressed_by \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 has_length \u22c0 \u00ac \u22c0 includes \u22c0 has_start_position \u22c0 Person_NE(e3) \u22c0 includes \u22c0 has_start_position \u22c0 sp2<sp3 \u22c0 sp4<sp2)) \u27f6 \u2203!r \u2203!m. [Coreference(r) \u22c0 resolved_in \u22c0 has_type  \u22c0 has_subtype  \u22c0 has_position \u22c0 has_group \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 Mention(m) \u22c0 refers_to \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 fits]]The algorithm ignores demonstrative pronouns because they are often used to refer to entities that are not present in the written text. Such pronouns do not carry any additional semantic information and do not refer to any noun phrase. They are used mostly for syntactic reasons and due to that are usually ignored in coreference resolution.Tomaitis [named entity] pateko \u012f avarij\u0105. Po piet\u0173 Tomait\u012f [named entity] i\u0161ve\u017e\u0117 \u012f operacin\u0119.[LT] Tomaitis [named entity] got into a car accident. In the afternoon, Tomaitis [named entity] has been taken to a surgery room.[EN] This algorithm is based on exact  string matches and several rules for acronyms. Once a first named entity that can be matched with an initial named entity is found, then the algorithm stops to keep annotations simple: B\u2009\u27f6\u2009A, C\u2009\u27f6\u2009B and D\u2009\u27f6\u2009C. This allows the formation of the coreference chains linking all mentions of the same entity in a text that can be later reused for semantic analysis, for example,In this example, two mentions of the same entity are made: \u201cTomaitis\u201d and \u201cTomait\u012f.\u201d They are of different cases, but their lemmas are identical. A condition for the existence of such reference formally is defined as follows:For each text's t sentence s1 that includes named entity e1, that has start position sp1 and is of length ln1, which is expressed by lexeme l1 that has lemma l and for each same text's t sentence s2 that includes named entity e2, that has a start position sp1 and is of length ln1, which is expressed by lexeme l2 that has lemma l, the only one coreference relation r, which is resolved in text t, is of \u201cNominal\u201d type, \u201cRepetition\u201d subtype, \u201cIrrelevant\u201d position and \u201cSingle\u201d group between the noun n1 and the noun n2, its referent starts at position sp1 and has length ln1, and which fits only one noun n1 and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one noun n2, exists (Rule 8).\u2009 Rule 8: \u2200t, s1, s2, e1, e2, sp1, ln1, sp2, ln2.[Text(t) \u22c0 Sentence(s1) \u22c0 Sentence(s2) \u22c0 consists_of \u22c0 consists_of \u22c0 Named_Entity(e1) \u22c0 includes \u22c0 has_start_position \u22c0 has_length \u22c0 Named_Entity(e2) \u22c0 includes \u22c0 has_start_position \u22c0 has_length \u22c0 e1\u2260e2 \u22c0 (\u2203l1 \u2203l2 \u2203l.(Lexeme(l1) \u22c0 Lexeme(l2) \u22c0 expressed_by \u22c0 expressed_by \u22c0 has_lemma \u22c0 has_lemma) \u27f6 \u2203!r \u2203!m. [Coreference(r) \u22c0 resolved_in \u22c0 has_type  \u22c0 has_subtype  \u22c0 has_position \u22c0 has_group \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 fits \u22c0 Mention(m) \u22c0 refers_to \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 fits]]Acronym rules vary depending on the type of named entity .Gydytojai [noun referring to profession] skund\u017eiasi dideliu darbo kr\u016bviu. Chirurg\u0173 [noun referring to profession] darbo kr\u016bvis pats did\u017eiausias.[LT] Doctors [noun referring to profession] complain about heavy workload. Surgeons' [noun referring to profession] workload is the greatest.[EN] This algorithm is based on profession classification. It attempts to resolve the use of synonyms and hypernyms/hyponyms.The algorithm determines that \u201cDoctor\u201d in professions classification is a hyponym of \u201cSurgeon,\u201d they also agree in gender and number; therefore, the algorithm adds their pair to annotations. Conditions for the existence of such reference formally are defined as follows:For each text's t sentence s1 that has profession p1, which is either broader or narrower than profession p2, name v1 expressing noun n1, which has gender g, number m, start position sp1 and is of length ln1, and for each same text's t sentence s2 that has profession p2, which is either broader or narrower than profession p1, name v2 expressing noun n2, which has gender g, number m, start position sp2 and is of length ln2, the only one coreference relation r, which is resolved in text t, is of \u201cNominal\u201d type, \u201cHypernym_hyponym\u201d subtype, \u201cIrrelevant\u201d position and \u201cSingle\u201d group between the noun n1 and the noun n2, its referent starts at position sp1 and has length ln1, and which fits only one noun n1 and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one noun n2, exists (Rule 9).\u2009 Rule 9: \u2200t, s1, s2, n1, n2, sp1, ln1, sp2, ln2, v1, v2, p1, p2.[Text(t) \u22c0 Sentence(s1) \u22c0 Sentence(s2) \u22c0 consists_of \u22c0 consists_of \u22c0 Noun(n1) \u22c0 contains \u22c0 has_start_position \u22c0 has_length \u22c0 Noun(n2) \u22c0 contains \u22c0 has_start_position \u22c0 has_length \u22c0 n1\u2260n2 \u22c0 Profession(p1) \u22c0 Profession(p2) \u22c0 p1\u2260p2 \u22c0 Profession_Name(v1) \u22c0 Profession_Name(v2) \u22c0 express \u22c0 express \u22c0 describes \u22c0 describes \u22c0  \u22c1 broadens) \u22c0 has_gender \u22c0 has_gender \u22c0 has_number \u22c0 has_number \u27f6 \u2203!r \u2203!m. [Coreference(r) \u22c0 resolved_in \u22c0 has_type  \u22c0 has_subtype  \u22c0 has_position \u22c0 has_group \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 Mention(m) \u22c0 refers_to \u22c0 has_start_position \u22c0 has_length \u22c0 fits]]vadovu [noun referring to profession]. Deja, vyr. chirurgo [noun referring to profession] vykdoma operacija buvo nes\u0117kminga.[LT] J. Jonaitis buvo operacijos head surgeon [noun referring to profession] of operation. Unfortunately, the last operation of chief surgeon [noun referring to profession] was unsuccessful.[EN] From now J. Jonaitis was the An example of synonym is given as follows:Both \u201chead surgeon\u201d and \u201cchief surgeon\u201d are synonymous; therefore, the condition for the existence of such reference formally could be defined as follows:For each text's t sentence s1 that has a profession's p name v1, which is expressed by noun n1 that has gender g, number m, start position sp1 and is of length ln1, and for each same text's t sentence s2 that has same profession's p name v2 expressed by noun n2 that has gender g, number m, start position sp2 and is of length ln2, the only one coreference relation r, which is resolved in text t, is of \u201cNominal\u201d type, \u201cSynonym\u201d subtype, \u201cIrrelevant\u201d position and \u201cSingle\u201d group between the noun n1 and the noun n2, its referent starts at position sp1 and has length ln1, and which fits only one noun n1 and refers to only one mention m, which starts at position sp2, has length ln2, and fits only one noun n2, exists (Rule 10).\u2009 Rule 10: \u2200t, s1, s2, n1, n2, sp1, ln1, sp2, ln2, v1, v2, p, g, n.[Text(t) \u22c0 Sentence(s1) \u22c0 Sentence(s2) \u22c0 consists_of \u22c0 consists_of \u22c0 Noun(n1) \u22c0 contains \u22c0 has_start_position \u22c0 has_length \u22c0 Noun(n2) \u22c0 contains \u22c0 has_start_position \u22c0 has_length \u22c0 n1\u2260n2 \u22c0 Profession_name(v1) \u22c0 Profession_name(v2) \u22c0 Profession(p) \u22c0 express \u22c0 express \u22c0 describes \u22c0 describes \u22c0 has_gender \u22c0 has_gender \u22c0 has_number \u22c0 has_number \u22c0 n1\u2260n2 \u27f6 \u2203!r \u2203!m. [Coreference(r) \u22c0 resolved_in \u22c0 has_type  \u22c0 has_subtype \u22c0 has_position \u22c0 has_group \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 Mention(m) \u22c0 refers_to \u22c0 has_start_position \u22c0 has_length \u22c0 fits]][LT] K\u0105 rekomenduoja S. Suskelis [person noun phrase]? Aptarkime kardiologo [noun referring to held position] si\u016blom\u0105 gydymo plan\u0105.[EN] What does S. Suskelis [person noun phrase] recommend? Let's discuss treatment plan proposed by the cardiologist [noun referring to held position]. Here a noun \u201ccardiologist\u201d is selected, the algorithm moves backwards till it reaches \u201cS. Suskelis\u201d and checks the knowledge base if at the time of the publication of the medical record he has held the position of the cardiologist.. Since \u201che holds it\u201d the algorithm checks if \u201cS. Suskelis\u201d and \u201ccardiologist\u201d agree in gender and number. They agree, and their pair is added to annotation as a feature reference. A condition for the existence of such reference formally is defined as follows:This algorithm at the time attempts to resolve only those cases when a person is being referred to by his public post (feature) that he holds, other types of features are not currently resolved, for example,For each text's t sentence s1 that has known person k, who during publication date d had certain position h (publication date d is same or later than position h start date fd and same or earlier than position h end date td), mention as named entity e, that has a start position sp1 and is of length ln1, and for each same text's t sentence s2 mentioned noun n, that has a start position sp2 and is length ln2, which is mentioned after named entity e (noun n has a higher start position sp2 than named entity's sp1), whose lemma l matches with position's h lemma l, number is Singular and gender g matches known person's k gender g, the only one coreference relation r, which is resolved in text t, is of \u201cNominal\u201d type, \u201cFeature\u201d subtype, \u201cBackward\u201d position and \u201cSingle\u201d group between the noun n and the named entity e, its referent starts at position sp2 and has length ln2, and which fits only one noun n and refers to only one mention m, which starts at position sp1, has length ln1, and fits only one named entity e, exists (Rule 11).\u2009 Rule 11: \u2200t, s1, s2, n, k, h, l, d, fd, td, g, sp1, sp2, ln1, ln2.[Text(t) \u22c0 Sentence(s1) \u22c0 Sentence(s2) \u22c0 consists_of \u22c0 consists_of \u22c0 Person_NE(e) \u22c0 includes \u22c0 Noun(n) \u22c0 contains \u22c0 Known_Person(k) \u22c0 mentioned_as \u22c0 Position_held(h) \u22c0 holds \u22c0 has_lemma \u22c0 has_lemma \u22c0 has_publication_date \u22c0 has_from_date \u22c0 has_to_date \u22c0 fd\u2264d \u22c0 td\u2265d \u22c0 has_gender \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 has_start_position \u22c0 sp1<sp2 \u22c0 has_length \u22c0 has_length \u27f6 \u2203!r \u2203!m. [Coreference(r) \u22c0 resolved_in \u22c0 has_type  \u22c0 has_subtype  \u22c0 has_position \u22c0 has_ group \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 Mention(m) \u22c0 refers_to \u22c0 has_start_position \u22c0 has_length \u22c0 fits]]\u0161eimos gydytojas [noun referring to held position]? T. Tomaitis [person noun phrase] yra labai patyr\u0119s j\u016bs\u0173 \u0161eimos gydytojas.[LT] Koks yra mano family doctor [noun referring to held position]? T. Tomaitis [person noun phrase] is very experienced doctor assigned to you.[EN] Who is my In this case, it is also relevant to track if coreference is pointing backward or forward. We can rewrite the same example and switch a known person with his positions:sp1 is higher than sp2 and coreference has different position constant values:As a result, For each text's t sentence s1 that has known person k, who during publication date d had certain position h (publication date d is same or later than position h start date fd and same or earlier than position h end date td), mention as named entity e, that has a start position sp1 and is of length ln1, and for each same text's t sentence s2 mentioned noun n, that has a start position sp2 and is length ln2, which is mentioned after named entity e (noun n has a lower start position sp2 than named entity's sp1), whose lemma l matches with position's h lemma l, number is Singular and gender g matches known person's k gender g, the only one coreference relation r, which is resolved in text t, is of \u201cNominal\u201d type, \u201cFeature\u201d subtype, \u201cForward\u201d position and \u201cSingle\u201d group between the noun n and the named entity e, its referent starts at position sp2 and has length ln2, and which fits only one noun n and refers to only one mention m, which starts at position sp1, has length ln1, and fits only one named entity e, exists (Rule 12).\u2009 Rule 12: \u2200t, s1, s2, n, k, h, l, d, fd, td, g, sp1, sp2, ln1, ln2.[Text(t) \u22c0 Sentence(s1) \u22c0 Sentence(s2) \u22c0 consists_of \u22c0 consists_of \u22c0 Person_NE(e) \u22c0 includes \u22c0 Noun(n) \u22c0 contains \u22c0 Known_Person(k) \u22c0 mentioned_as \u22c0 Position_held(h) \u22c0 holds \u22c0 has_lemma \u22c0 has_lemma \u22c0 has_publication_date \u22c0 has_from_date \u22c0 has_to_date \u22c0 fd\u2264d \u22c0 td\u2265d \u22c0 has_gender \u22c0 has_gender \u22c0 has_number \u22c0 has_start_position \u22c0 has_start_position \u22c0 sp1>sp2 \u22c0 has_length \u22c0 has_length \u27f6 \u2203!r \u2203!m. [Coreference(r) \u22c0 resolved_in \u22c0 has_type  \u22c0 has_subtype  \u22c0 has_position \u22c0 has_ group \u22c0 has_start_position \u22c0 has_length \u22c0 fits \u22c0 Mention(m) \u22c0 refers_to \u22c0 has_start_position \u22c0 has_length \u22c0 fits]]The coreference resolution algorithms and rules presented in Coreference resolution for Lithuanian was implemented using Java programming language and JSON data format for annotation storage. But the proposed approach is not technology dependent, and for other languages, it can be implemented on any other platform.The evaluation was performed by analysing 100 articles that have been preannotated and are available in our Lithuanian Language Coreference Corpus , in addiF1 metrics. Recall R is the ratio of correctly resolved anaphoric expressions C to the total number of anaphoric expressions T. Precision P is the ratio of correctly resolved anaphoric expressions C to the number of resolved anaphoric expressions F. F1 is a harmonic mean of P and R:For evaluation, we used precision, recall, and Five experiments were performed with different combinations of coreferencing algorithms presented in The database of public persons must be constantly updated as new information becomes available. Otherwise, recall will get noticeably lower when annotating newer texts.In the case where plural pronouns and nouns are used, they are difficult to be identified because of many variations possible that often ignore grammatical compatibility rules.Note the following threats to validity of our results:Linking the named entity to the position held taking into account the date of the publication of the text is limited considering that the text might be published today but written about things that happened in the past. There are no tools, which can identify the timeframe of a certain part of the text.Medical entity recognition and coreferencing are difficult tasks in Lithuanian natural language processing (NLP). We proposed the coreference resolution approach for the Lithuanian language. The coreference resolution algorithm depends on morphological and named entity recognition (NER) annotations and preexisting databases. Due to the proposed approach being detached from specific implementation and rules being formalized, it would not be difficult to adapt it for grammatically similar languages. Our novelty is the ability to process coreferences with minimal linguistic resources, which are very important to consider in linguistic applications for under-resourced and endangered languages. While the proposed method provides encouraging results, when analysing transcribed medical records and other corpora, and they are comparable to the results achieved by other authors applying different resolution approaches on other languages, it has certain limitations: it is domain specific and is able to resolve only a subset of coreference types, while the relatively small dataset was used for experiments. Nevertheless, we hope that our method can contribute to the sustainable development of the NLP-powered online healthcare services in Lithuania."}
{"text": "Following publication of this article  we foundKa\u2009\u2248\u20091\u2009\u00d7\u2009105\u00a0M\u22121 as well as a binding site size of n\u2009\u2248\u20092.5 base pairs for mitoxantrone. An unwinding angle of mitoxantrone-intercalation of \u03d1\u00a0\u2248\u00a016\u00b0 was determined. The original article has been updated.Using this method, we were able to estimate an equilibrium constant of association"}
{"text": "Panax ginseng, Rehmannia glutinosa, Angelica polymorpha, Atractylodes macrocephala, Glycyrrhiza uralensis, Ziziphus jujube, and Polygala tenuifolia to boost qi and blood circulation, strengthen the heart, and calm the spirit\u2014skillfully linking heart, spleen, kidney, qi, blood and brain as a whole to treat age-related dementia. The purpose of this review is to outline TCM concepts for the treatment of dementia and illustrated with a historical prescription for the treatment of the condition, with the hope that this description may lead to advances in its management.The Traditional Chinese Medicine (TCM) theory that \u201ckidneys give rise to marrow, and the brain is the sea of marrow\u201d has been a guide for the clinical application of kidney, qi and blood tonics for prevention and treatment of dementia and improvement in memory. As low resistance end-organs, both the brain and the kidneys are subjected to blood flow of high volumes throughout the cardiac cycle. Alzheimer\u2019s disease and vascular dementia are two common causes of dementia, and it is increasingly recognized that many older adults with dementia have both AD and vascular pathologies. The underlying molecular mechanisms are incompletely understood, but may involve atherosclerosis, vascular dysfunction, hypertension, type 2 diabetes, history of cardiac disease and possibly, kidney dysfuntion, leading to reduced erythropoietin production, anemia, brain energy deficit and slow excitotoxicity. During the Ming Dynasty, Zhang Jing-Yue used Qi Fu Yin (seven blessings decoction), comprising  Dementia is a progressive syndrome characterized by memory deficits, cognitive impairment, and deterioration in emotional control and social behavior , 2. The The prevalence of dementia increases exponentially with age, from 3.0% among those aged 65 to 74\u00a0years to 18.7% among 75 to 84\u00a0years old and 47.2% in individuals over the age of 85\u00a0years . An undeAccording to TCM, the brain and bone marrow are the outgrowths of the kidneys. In Lingshu Meridians, it is stated that \u201cat conception, essence is formed.\u201d After essence is formed, the brain and bone marrow are formed. The kidneys contain the essence, the essence sustains the marrow, and the marrow nourishes the brain. According to \u201cLingshu Discussion on Seas,\u201d humans have a marrow sea, a blood sea, a qi sea and a water/grain sea (stomach). This is the meaning of the four seas. Among the four seas in the human body, the marrow sea refers to the brain. According to the \u201cCategory Text\u201d Volume 9: where there is bone, there is marrow, and the brain has the most. Thus, all marrow is related to the brain, and the marrow sea refers to the brain. <\u300a\u7075\u67a2\u00b7\u7ecf\u8109\u300b\u66f0:\u201c\u4eba\u59cb\u751f,\u5148\u6210\u7cbe,\u7cbe\u6210\u800c\u8111\u9ad3\u751f\u3002\u201d \u80be\u85cf\u7cbe,\u7cbe\u5145\u9ad3,\u9ad3\u8363\u8111\u3002\u300a\u7075\u67a2\u00b7\u6d77\u8bba\u300b\u6b67\u4f2f\u66f0:\u201c\u4eba\u6709*\u9ad3\u6d77,\u6709\u8840\u6d77,\u6709\u6c14\u6d77,\u6709\u6c34\u8c37\u4e4b\u6d77,\u51e1\u6b64\u56db\u8005,\u4ee5\u5e94\u56db\u6d77\u4e5f\u3002\u201d\u4eba\u4f53\u56db\u6d77\u4e4b\u4e00,\u9ad3\u6d77\u5373\u6307\u8111\u3002\u300a\u7c7b\u7ecf\u300b\u5377\u4e5d\u6ce8:\u201c\u51e1\u9aa8\u4e4b\u6709\u9ad3,\u60df\u8111\u4e3a\u6700\u5de8,\u6545\u8bf8\u9ad3\u7686\u5c5e\u4e8e\u8111,\u800c\u8111\u4e3a\u9ad3\u4e4b\u6d77\u3002> , 11. ThiTCM views the body as a single entity and an individual\u2019s being depends on interactions between the different body parts. Qi, blood, and body fluids are essential for life. Qi is interpreted as \u201clife energy\u201d or \u201clife force.\u201d The Chinese character for \u201cqi\u201d (\u6c23) means air or gas and may have some of the characteristics of air or oxygen gas. Original qi originates from the kidneys, the site where \u201ccongenital essence,\u201d an essential and vital substance inherited from one\u2019s parents upon conception is stored. Pectoral qi is formed from the combination of inhaled fresh air from the lungs and food essence from the stomach and spleen. It permeates the blood vessels and moves outward during expiration and inward during inspiration. Nutritive qi supplies nourishment to the body. It comes from food and digestive activities and circulates via blood vessels. Protective qi is similar to the immune system and helps in the prevention of illnesses.Phlegm is a pathological substance generated by disturbance of body fluid that blocks qi. Substantial phlegm is visible such as sputum, whereas insubstantial phlegm is invisible. Lipids and lipoprotein metabolic disorders  are believed to contribute to phlegm in the blood, whereas platelet activation, thrombosis, endothelial injury, and atheromatous plaques contribute to blood stasis . Small-mThe TCM theory of \u201ckidneys give rise to marrow, and the brain is the sea of marrow\u201d implies a close relationship between the kidneys, the marrow, and the brain. Chronic renal failure is asymptomatic at first, until kidney function has decreased to less than 25% of normal. Patients then present with nocturia and anorexia, and raised serum levels of nitrogenous compounds such as urea and creatinine. Advanced renal failure causes significant impairment of all renal function and affects virtually all body systems, and causes change in urea, electrolytes, and other blood constituents. End-stage renal disease is the term used when more than 90% of renal function is lost and may be complicated by anemia, bleeding, bone disease, hypertension, congestive heart failure, digestive tract problems, and dementia. It is proposed that part of the renal-cerebral connection may be due to small vessel disease in both the kidneys and brain. There are many hemodynamic similarities between the vascular beds of these organs \u2014both kidAnemia is a common finding in patients with CKD. Analyses of baseline data from adults with chronic kidney disease indicate that the prevalence of cognitive impairment is higher among those with lower eGFR, and this association is independent of traditional vascular risk factors, but rather, related to anemia . The kidL [Epo and EpoR are also expressed in neurons of the CNS, including the neocortex, hippocampus, and hypothalamus, as well as dorsal root ganglia, and Schwann cells . Epo impL . In addiL . CollectL .CKD patients also show raised serum levels of nitrogenous wastes. Decreased cognitive function has been associated with the retention of uremic toxins, with performance improving with more intensive dialysis or kidney transplantation . The enz+/K+ ATPases that maintain resting potential in neurons. This results in depolarization, excessive calcium entry into neurons, activation of calcium dependent enzymes such as proteases, lipases, and endonucleases, and slow excitotoxicity [The TCM theory that memory and cognition are disordered if channels that connect the heart and brain are blocked by phlegm implies a close relationship between the cardiovascular system and the brain, a connection which is also highly emphasized in Western medicine. Heart disease is any condition that impairs cardiac function regardless of the specific modality that is affected . The funtoxicity . Reductitoxicity , but aretoxicity . They matoxicity , 52, buttoxicity \u201355. Furttoxicity , and redtoxicity . Dementitoxicity .Increasing evidence indicates that consumption of bioactive compounds or phytochemicals in herbs produces antioxidant, anti-inflammatory, and anti-carcinogenic effects fig. . Unlike Multi-herb formulas rather than single herbs are common in TCM medication. Each herb in a formula has a specific role\u2014sovereign (\u541b), minister (\u81e3), assistant (\u4f50), and courier (\u4f7f). Sovereign and minister herbs treat the main symptoms and have a major role in the formula. Assistant herbs assist the sovereign and minister herbs to treat the accompanying symptoms, or reduce the side effects of the major herbs. Courier herbs help to lead the other components to the affected area. Interactions between the herbs, such as mutual reinforcement, antagonism, or detoxification, help to determine the formula\u2019s therapeutic efficacy. The nature of the herbs, including the four properties  and the five tastes  as well as characteristics such as meridian-tropism, are taken into consideration by TCM physicians when formulating a prescription .Zhang Jing-Yue \u5f20\u666f\u5cb3 (c.1563\u20131640) had a great influence on the development of TCM, towards the end of the Ming Dynasty in China. He was born in Shaoxing County of Zhejiang Province, the birthplace of many of the country\u2019s most renowned scholars and writers. Zhang traveled with his father to Beijing, where he studied medicine, and became an imperial physician working for Emperor Wanli. Because he made good use of processed Rehmannia glutinosa, he was also called \u201cZhang Shudi.\u201d He was an outstanding medical scientist, the representative of the ancient Chinese medicine practice of \u201cwarm tonification,\u201d and his academic thought has a great influence on future generations. A chapter on dementia (\u75f4\u5446) in the Ming dynasty book \u300aJingyue Quanshu\u300b  describes how the collapse of original qi, together with the presence of impure qi in the meridians and heart orifices, can lead to the problem of dementia.In this chapter, Zhang Jing-Yue was of the opinion that \u201cdementia syndrome is characterized by lack of sputum, or stagnation, or failure, or anxiety, or suspicion, or panic, gradually leading to dementia. Speech and words are in reverse order and movement is sluggish. There is excessive sweating and depression. The symptoms may be extremely unusual and bizarre. The pulse is stringy or increased in frequency, large or small, and often changes. Persons have pathological qi in the heart and liver and biliary systems. The qi is unclear. If the body is strong and there is no reduction in food intake or other weaknesses, it is appropriate to treat using Fu Man Jian. It is the most stable and wonderful prescription. Nevertheless, some cases will recover and some will not, depending on the strength or weakness of stomach qi and original qi. Time is needed for recovery, which cannot be speeded up. Those who have this syndrome may be exacerbated by anxiety or depression leading to absent-mindedness and confusion. In this case, it is important to quickly support the good qi, mostly through the use of Qi Fu Yin or Da Bu Yan\u201d. < \u660e\u4ee3\u300a\u666f\u5cb3\u5168\u4e66\u300b\u5f20\u666f\u5cb3\u8ba4\u4e3a:\u201c\u75f4\u5446\u8bc1,\u51e1\u5e73\u7d20\u65e0\u75f0,\u800c\u6216\u4ee5\u90c1\u7ed3,\u6216\u4ee5\u4e0d\u9042,\u6216\u4ee5\u601d\u8651,\u6216\u4ee5\u7591\u8d30,\u6216\u4ee5\u60ca\u6050,\u800c\u6e10\u81f4\u75f4\u5446\u3002\u8a00\u8f9e\u98a0\u5012,\u4e3e\u52a8\u4e0d\u7ecf,\u6216\u591a\u6c57,\u6216\u5584\u6101,\u5176\u8bc1\u5219\u5343\u5947\u4e07\u602a,\u65e0\u6240\u4e0d\u81f3\u3002\u8109\u5fc5\u6216\u5f26\u6216\u6570,\u6216\u5927\u6216\u5c0f,\u53d8\u6613\u4e0d\u5e38\u3002\u6b64\u5176\u9006\u6c14\u5728\u5fc3\u6216\u809d\u80c6\u4e8c\u7ecf,\u6c14\u6709\u4e0d\u6e05\u800c\u7136\u3002\u4f46\u5bdf\u5176\u5f62\u4f53\u5f3a\u58ee,\u996e\u98df\u4e0d\u51cf,\u522b\u65e0\u865a\u8131\u7b49\u8bc1\u3002\u5219\u6089\u5b9c\u670d\u86ee\u714e\u6cbb\u4e4b\u3002\u6700\u7a33\u6700\u5999\u3002\u7136\u6b64\u8bc1\u6709\u53ef\u6108\u8005,\u6709\u4e0d\u53ef\u6108\u8005,\u4ea6\u5728\u4e4e\u80c3\u6c14\u5143\u6c14\u4e4b\u5f3a\u5f31,\u5f85\u65f6\u800c\u590d,\u975e\u53ef\u6025\u4e5f\u3002\u51e1\u6b64\u8bf8\u8bc1,\u82e5\u4ee5\u5927\u60ca\u731d\u6050,\u4e00\u65f6\u5076\u4f24\u5fc3\u80c6,\u800c\u81f4\u5931\u795e\u660f\u4e71\u8005\u3002\u6b64\u5f53\u4ee5\u901f\u6276\u6b63\u6c14\u4e3a\u4e3b,\u5b9c\u4e03\u798f\u996e,\u6216\u5927\u8865\u5143\u714e\u4e3b\u4e4b\u3002> Panax ginseng, Rehmannia glutinosa, Angelica polymorpha, Atractylodes macrocephala, Glycyrrhiza uralensis, Ziziphus jujube, and Polygala tenuifolia  is a mixture of seven herbs, ia Table .No.OrderAccording to one of three foundation books for TCM, Shennong Materia Medica, which is a compilation of oral traditions between 200 and 250 A.D., and attributed to a \u201cdivine farmer\u201d Shennong, said to have lived around 2800 B.C.: ginseng has a sweet taste, and mildly cold in nature. It is mainly used for supplementing the five internal organs. It calms the spirit, stabilizes the soul, reduces fear, dispels impure qi, brightens the eyes, increases happiness, and is beneficial for thinking and wisdom. Long-term consumption leads to lightening of the body and extension of years. Another name is Human Title or Ghost Cover. It grows in valleys. < \u795e\u519c\u672c\u8349\u7ecf:\u4eba\u53c2,\u5473\u7518\u5fae\u5bd2,\u4e3b\u8865\u4e94\u810f,\u5b89\u7cbe\u795e,\u5b9a\u9b42\u9b44,\u6b62\u60ca\u60b8,\u9664\u90aa\u6c14,\u660e\u76ee,\u5f00\u5fc3\u76ca\u667a\u3002\u4e45\u670d,\u8f7b\u8eab\u5ef6\u5e74\u3002\u4e00\u540d\u4eba\u8854,\u4e00\u540d\u9b3c\u76d6\u3002\u751f\u5c71\u8c37\u3002> Bencao Gangmu or Compendium of Materia Medica, a Chinese materia medica written by Shi-Zhen Li during the Ming dynasty in 1578 AD: ginseng supplements the five internal organs. It calms the spirit, stabilizes the soul, reduces fear, dispels impure qi, brightens the eyes, increases happiness, and is beneficial for thinking and wisdom. Long-term consumption leads to lightening of the body and extension of years. < \u672c\u8349\u7eb2\u76ee:\u8865\u4e94\u810f,\u5b89\u7cbe\u795e,\u5b9a\u9b42\u9b44,\u6b62\u60ca\u60b8,\u9664\u90aa\u6c14,\u660e\u76ee\u5f00\u5fc3\u76ca\u667a\u3002\u4e45\u670d\u8f7b\u8eab\u5ef6\u5e74\u3002> [According to \u4e45\u670d\u8f7b\u8eab\u5ef6\u5e74\u3002> . As can \u4e45\u670d\u8f7b\u8eab\u5ef6\u5e74\u3002> .Usage and dosage: for oral consumption, as a decoction 3\u201310\u00a0g. If using a high dose of 10\u201330\u00a0g, should be prepared first as a single decoction and poured into the final prescription herbs. It can also be ground as a powder 1\u20132\u00a0g; or paste; or soaked in liquor/wine, or made into pills. < \u7528\u6cd5\u7528\u91cf:\u5185\u670d:\u6c64\u5242,3-10\u514b,\u5927\u5242\u91cf10-30\u514b,\u5b9c\u53e6\u714e\u5151\u5165;\u6216\u7814\u672b,1-2\u514b;\u6216\u6577\u818f;\u6216\u6ce1\u9152;\u6216\u5165\u4e38\u6563\u3002>Panax ginseng: not to be consumed by persons with no deficiency, heat syndrome, or no weakness in qi. < \u5b9e\u8bc1\u3001\u70ed\u8bc1\u800c\u6b63\u6c14\u4e0d\u865a\u8005\u5fcc\u670d\u3002>Contraindications for According to Shen Nong\u2019s Herbal Classic: Rehmannia has a sweet and cold taste and is non-toxic. It is mainly used for treatment of fractures and muscle injury, internal injury, dispelling blood stasis, filling of the bone marrow, and enhancing muscle growth. As a decoction, it helps to remove the accumulation of cold and heat, and remove paralysis. The raw herb is better. Long-term consumption leads to lightening of the body and anti-aging. < \u795e\u519c\u672c\u8349\u7ecf:\u5730\u9ec4,\u7518\u3001\u5bd2,\u65e0\u6bd2\u3002\u4e3b\u6298\u8dcc\u7edd\u7b4b,\u4f24\u4e2d,\u9010\u8840\u75f9,\u586b\u9aa8\u9ad3,\u957f\u808c\u8089,\u4f5c\u6c64\u9664\u5bd2\u70ed\u79ef\u805a,\u9664\u75f9,\u751f\u8005\u5c24\u826f\u3002\u4e45\u670d\u8f7b\u8eab\u4e0d\u8001\u3002> According to Compendium of Materia Medica: it fills the bone marrow, enhances muscle growth, increases essence and blood, supplements the five internal organs after internal injury, unblocks the blood circulation, is beneficial to the ears and eyes, and blackens the beard and hair. < \u672c\u8349\u7eb2\u76ee:\u586b\u9aa8\u9ad3,\u957f\u808c\u8089,\u751f\u7cbe\u8840,\u8865\u4e94\u810f\u5185\u4f24\u4e0d\u8db3,\u901a\u8840\u8109,\u5229\u8033\u76ee,\u9ed1\u987b\u53d1\u3002> Usage and dosage: for oral consumption as a decoction 10\u201330\u00a0g. Or make into pills, pastes, or soak in wine/liquor < \u7528\u6cd5\u7528\u91cf:\u5185\u670d:\u714e\u6c64,10-30\u514b,\u6216\u5165\u4e38\u6563;\u6216\u6577\u818f;\u6216\u6d78\u9152\u3002>Contraindications for Rehmannia glutinosa: persons with weakness in stomach, qi stagnation, abundant sputum, abdominal distension, and loose stools < \u80c3\u865a\u5f31,\u6c14\u6ede\u75f0\u591a,\u8179\u6ee1\u4fbf\u6e8f\u8005\u5fcc\u670d\u3002>According to Shen Nong\u2019s Materia Medica: Angelica has a bitter taste. It is warm in nature and non-toxic. It is mainly used for treatment of cough, reversed flow of qi, warm malaria, cold and hot contacts in the skin, women with uterine bleeding and infertility, all sore ulcers; and sores. Boil the herb to drink. <\u795e\u519c\u672c\u8349\u7ecf:\u5f53\u5f52:\u82e6\u3001\u6e29,\u65e0\u6bd2\u3002\u4e3b\u54b3\u9006\u4e0a\u6c14,\u6e29\u7627,\u5bd2\u70ed\u6d17\u6d17\u5728\u76ae\u80a4\u4e2d,\u5987\u4eba\u6f0f\u4e2d\u7edd\u5b50,\u8bf8\u6076\u75ae\u75a1,\u91d1\u75ae\u3002\u716e\u6c41\u996e\u4e4b \u3002> Usage and dosage: fried black, grind fine powder, each time use of 9\u00a0g, add a cup of water, a little wine, and decoction. 6~12\u00a0g. < \u7528\u6cd5\u7528\u91cf:\u7092\u9ed1,\u5171\u7814\u7ec6\u672b,\u6bcf\u75289\u514b,\u6c34\u4e00\u676f,\u9152\u5c11\u8bb8,\u714e\u670d\u30026~12\u514b\u3002>Angelica polymorpha: excessive use causes tiredness, drowsiness, and other reactions. Stopping the medicine leads to disappearance of these symptoms. Allergic reaction: not suitable for persons with menorrhagia, bleeding tendency, yin deficiency, internal heat, and loose stools or diarrhea. Not to be consumed by persons with hot bleeding tendencies. To be consumed with caution by persons with wet fullness and bloating. < \u7528\u91cf\u8fc7\u5927\u6709\u75b2\u5026\u55dc\u7761\u7b49\u53cd\u5e94,\u505c\u836f\u6d88\u5931\u3002\u8fc7\u654f\u53cd\u5e94\u6708\u7ecf\u8fc7\u591a,\u6709\u51fa\u8840\u503e\u5411,\u9634\u865a\u5185\u70ed,\u5927\u4fbf\u6e8f\u6cc4\u8005\u5747\u4e0d\u5b9c\u670d\u7528\u3002\u70ed\u76db\u51fa\u8840\u60a3\u8005\u7981\u670d\u3002\u6e7f\u76db\u4e2d\u6ee1\u614e\u670d\u3002>Contraindications for There is a saying that \u201cginseng in the north and Atractylodes in the south,\u201d suggesting that Atractylodes is almost as highly regarded as ginseng in TCM. According to Shen Nong\u2019s Herbal Classic: Atractylodes macrocephala has a bitter taste and warm nature. It is mainly used for treatment of rheumatism, joint ache caused by wind and dampness, muscle paralysis, muscle spasm, and jaundice. It stops perspiration, dispels heat, helps digestion, and is used in decoctions. Long-term consumption leads to lightening of the body, extension of years, and satiety. Another name is mountain thistle, it grows in valleys. < \u795e\u519c\u672c\u8349\u7ecf:\u767d\u672f:\u5473\u82e6\u6e29\u3002\u4e3b\u98ce\u5bd2\u6e7f\u75f9\u6b7b\u808c,\u75c9\u75b8,\u6b62\u6c57,\u9664\u70ed,\u6d88\u98df,\u4f5c\u714e\u9975\u3002\u4e45\u670d\u8f7b\u8eab\u5ef6\u5e74,\u4e0d\u9965\u3002\u4e00\u540d\u5c71\u84df,\u751f\u5c71\u8c37\u3002> .Usage and dosage: 6~12\u00a0g.< \u7528\u6cd5\u7528\u91cf:6~12\u514b\u3002>Contraindications for Atractylodes macrocephala: not to be consumed by persons with Yin deficiency, qi stagnation, and nausea < \u9634\u865a\u71e5\u6e34,\u6c14\u6ede\u80c0\u95f7\u8005\u5fcc\u670d\u3002>Glycyrrhiza uralensis has a sweet taste and a neutral nature. It is mainly used to dispel cold and hot, impure qi in the five viscera  and six hollow organs , toughen tendons and bones, increase muscle mass, multiply strength, facilitate recovery after traumatic injury, and detoxification. Long-term consumption leads to lightening of the body and extension of years. < \u795e\u519c\u672c\u8349\u7ecf:\u7099\u7518\u8349:\u5473\u7518,\u5e73\u3002\u4e3b\u4e94\u810f\u516d\u8151\u5bd2\u70ed\u90aa\u6c14,\u575a\u7b4b\u9aa8,\u957f\u808c\u8089,\u500d\u529b,\u91d1\u75ae\u5c30,\u89e3\u6bd2\u3002\u4e45\u670d\u8f7b\u8eab\u5ef6\u5e74\u3002> [According to Shen Nong\u2019s Herbal Classic: \u4e45\u670d\u8f7b\u8eab\u5ef6\u5e74\u3002> Usage and dosage: 1.5~9\u00a0g. < \u7528\u6cd5\u7528\u91cf:1.5~9\u514b\u3002>Glycyrrhiza uralensis: not appropriate to combine with Jing Da Ji (Radix Euphorbiae), Yuan Hua (Lilac Daphne Flower Bud), and Gan Sui (Radix KanSui) together. < \u4e0d\u5b9c\u4e0e\u4eac\u5927\u621f,\u82ab\u82b1,\u7518\u9042\u540c\u7528\u3002>Contraindications for According to Shen Nong\u2019s Materia Medica, it is mainly used to treat heart and abdominal cold and heat, and aggregation of impure qi; aching, pain; and dampness arthralgia. Long-term consumption leads to calming of the five viscera , lightening of the body, and extension of years. < \u795e\u519c\u672c\u8349\u7ecf:\u4e3b\u5fc3\u8179\u5bd2\u70ed,\u90aa\u7ed3\u6c14\u805a,\u56db\u80a2\u9178\u75db\u6e7f\u75f9,\u4e45\u670d\u5b89\u4e94\u810f,\u8f7b\u8eab\u5ef6\u5e74\u3002> .According to Compendium of Materia Medica, the seed has a sweet taste and a neutral nature. The processed herb is used for treatment of biliary system weakness, insomnia, polydipsia, and sweating. Its raw form is used for treatment of heaty biliary system and accompanying drowsiness. It is also the drug for treatment of the diseased accupoints: liver meridian of foot\u2014Jueyin, and gallbladder meridian of foot\u2014Shaoyang. < \u672c\u8349\u7eb2\u76ee:\u201c\u5176\u4ec1\u7518\u800c\u6da6,\u6545\u719f\u7528\u7597\u80c6\u865a\u4e0d\u5f97\u7720,\u70e6\u6e34\u865a\u6c57\u4e4b\u8bc1;\u751f\u7528\u7597\u80c6\u70ed\u597d\u7720,\u7686\u8db3\u53a5\u9634\u3001\u5c11\u9633\u836f\u4e5f\u3002\u201d > Usage and dosage: put into the decoction, 9~15\u00a0g. < \u7528\u6cd5\u7528\u91cf:\u5165\u6c64\u5242,9~15\u514b\u3002>.Contraindications for Ziziphus jujube: to be consumed with caution by persons with impure smoldering fire and those with chronic diarrhea. Not to be used with Fang Ji (Radix Stephaniae tetrandrae). < \u51e1\u6709\u5b9e\u90aa\u90c1\u706b\u53ca\u60a3\u6709\u6ed1\u6cc4\u75c7\u8005\u614e\u670d\u3002\u6076\u9632\u5df1\u3002>.According to Shen Nong\u2019s Herbal Classic: Polygala has a bitter taste and a warm nature. It is mainly used to treat cough with dyspnea, internal injury, make up for deficiency, dispel impure qi, and relieve obstruction in the nine orifices . It is beneficial for wisdom, and makes one clever and not forgetful. It strengthens ambition and resolve, and multiples strength. Long-term consumption leads to lightening of the body and anti-aging. The name of the name of the leaf is Small Grass. Another name is Spine Wan (Lu De Ming Er Ya Yin Sheng cited amaranth). Another name is Spine Around . Another name is Fine Grass. It grows in valleys. < \u795e\u519c\u672c\u8349\u7ecf:\u8fdc\u5fd7:\u5473\u82e6\u6e29\u3002\u4e3b\u54b3\u9006,\u4f24\u4e2d,\u8865\u4e0d\u8db3,\u9664\u90aa\u6c14,\u5229\u4e5d\u7a8d,\u76ca\u667a\u6167,\u4f7f\u4eba\u806a\u660e,\u4e0d\u5fd8,\u5f3a\u5fd7\u500d\u529b\u3002\u4e45\u670d,\u8f7b\u8eab\u4e0d\u8001\u3002\u53f6\u540d\u5c0f\u8349,\u4e00\u540d\u68d8\u83c0(\u9646\u5fb7\u660e\u5c14\u96c5\u97f3\u4e49\u5f15\u4f5c\u82cb),\u4e00\u540d\u68d8\u7ed5(\u5fa1\u89c8\u4f5c\u8981\u7ed5),\u4e00\u540d\u7ec6\u8349\u3002\u751f\u5ddd\u8c37\u3002> According to compendium of Materia Medica: it has a bitter taste and warm nature and non-toxic. It is mainly used to treat forgetfulness. Take Polygala at the end and boil with water to consume < \u672c\u8349\u7eb2\u76ee:\u82e6\u3001\u6e29\u3001\u65e0\u6bd2\u3002\u300c\u4e3b\u6cbb\u300d\u5584\u5fd8\u75c7\u3002\u53d6\u8fdc\u5fd7\u4e3a\u672b,\u51b2\u670d\u3002> Usage and dosage: boil and consume: 3~9\u00a0g. < \u7528\u6cd5\u7528\u91cf:\u714e\u670d,3~9\u514b\u3002>Contraindications for Polygala tenuifolia: persons with gastritis and gastric ulcer should use with caution. < \u6709\u80c3\u708e\u53ca\u80c3\u6e83\u75a1\u8005\u614e\u7528\u3002>Panax ginseng , tonifying qi and lifting yang, is beneficial for happiness, thinking and wisdom, strengthens the spleen and nourishes the stomach < \u4eba\u53c2,\u8865\u6c14\u8865\u9633,\u5f00\u5fc3\u76ca\u667a,\u5065\u813e\u517b\u80c3 > [Atractylodes macrocephala , strengthens the spleen, reduces dampness, benefits qi, and helps in the circulation < \u767d\u672f,\u5065\u813e\u71e5\u6e7f,\u52a0\u5f3a\u76ca\u6c14\u52a9\u8fd0\u4e4b\u529b > [Rehmannia glutinosa  , Angelica polymorpha , Ziziphus jujube (assistant), and Polygala tenuifolia (assistant) boost the blood, calms the mind, and calms the spirit. < \u719f\u5730\u3001\u5f53\u5f52\u3001\u67a3\u4ec1(\u4f50)\u3001\u8fdc\u5fd7(\u4f50),\u8865\u8840\u5b81\u5fc3\u5b89\u795e\u3002> [Glycyrrhiza uralensis  is beneficial to qi, nourishes the heart, and harmonizes the different medicines, thus leading to filling of the qi and blood and calming of the mind and spirit, resulting in recovery from disease. < \u7099\u7518\u8349,\u76ca\u6c14\u548c\u4e2d\u517b\u5fc3,\u8c03\u548c\u8bf8\u836f,\u4ece\u800c\u4f7f\u6c14\u8840\u5145\u3001\u5fc3\u795e\u5b89\u5219\u75c5\u6108\u3002> [Qi Fu Yin is used for treatment of heart and qi deficiency, treatment of neurasthenia, and treatment of age-related dementia. It has effects on benefitting qi, supplementing the blood, nourishing the heart, and calming the spirit. < \u6cbb\u7597\u5fc3\u6c14\u865a,\u6cbb\u7597\u795e\u7ecf\u8870\u5f31,\u6cbb\u7597\u8001\u5e74\u6027\u75f4\u5446\u3002\u6709\u76ca\u6c14\u8865\u8840,\u517b\u5fc3\u5b81\u795e\u7684\u529f\u6548\u3002> , 67. Pan\u667a,\u5065\u813e\u517b\u80c3 > , 67 . At\u76ca\u6c14\u52a9\u8fd0\u4e4b\u529b > , 67. Reh\u8865\u8840\u5b81\u5fc3\u5b89\u795e\u3002> , 67 Glyc\u5fc3\u795e\u5b89\u5219\u75c5\u6108\u3002> , 67Persons who have heat syndrome and no weakness in qi should not consume. Women should stop using it during menstruation < \u5b9e\u8bc1\u3001\u70ed\u8bc1\u800c\u6b63\u6c14\u4e0d\u865a\u8005\u5fcc\u670d\u3002\u5987\u5973\u7ecf\u671f\u505c\u7528\u3002> , 67.None recorded to our knowledge.The following herbs are boiled in 400\u00a0cc of water and left to simmer until the volume reduces to 280\u00a0cc. This can be divided into 2 servings per day, to be taken on an empty stomach, when warm. <\u4e0a\u836f\u7528\u6c34400\u6beb\u5347,\u714e\u53d6280\u6beb\u5347,\u7a7a\u8179\u65f6\u6e29\u670d\u3002> . Review Panax ginseng, Rehmannia glutinosa, Angelica polymorpha, Atractylodes macrocephala, Glycyrrhiza uralensis, Ziziphus jujube, and Polygala tenuifolia to boost qi and blood circulation, strengthen the heart, and calm the spirit\u2014skillfully linking heart, spleen, kidney, qi, blood and brain as a whole to treat age-related dementia.Alzheimer\u2019s disease and vascular dementia are two common causes of dementia, and it is increasingly recognized that many older adults with dementia have both AD and vascular pathologies. The underlying molecular mechanisms are incompletely understood, but may involve atherosclerosis, vascular dysfunction, hypertension, type 2 diabetes, history of cardiac disease, and total homocysteine, and possibly, lack of erythropoietin formation by the kidney. During the Ming Dynasty, Zhang Jing-Yue used Qi Fu Yin (seven blessings decoction), comprising"}
{"text": "Although bone age plays a special role in determining the child\u2019s age, there are some variations in skeletal growth of different people. The aim of this study was to compare the bone age with chronological age of children aged 2\u201318 years old in order to recognize whether Greulich-Pyle (GP) method could be reliable for Iranian children? The standard radiograph of Left hand was taken in 40 healthy subjects, then the bone age was determined according to GP. Mean\u202f\u00b1\u202fSD bone ages were delayed 1.12\u202f\u00b1\u202f0.65, 0.82\u202f\u00b1\u202f1.34 and 0.10\u202f\u00b1\u202f0.51 years than the mean chronological ages in 2.99\u20135.99, 10\u201313.99 and 14\u201317.99 age group, respectively; and advanced \u22120.33\u202f\u00b1\u202f3.12 years in the 6\u20139.99 age group. In BMI levels <18.5, 18.5\u201324.9, 25\u201329.9 and \u226530, Mean\u202f\u00b1\u202fSD bone ages in males were delayed 2.25\u202f\u00b1\u202f0.21, 0.14\u202f\u00b1\u202f0.55, 0.87\u202f\u00b1\u202f0.41 and 4.05\u202f\u00b1\u202f0.70 years than the mean chronological ages, respectively. In BMI range of 18.5\u201324.9 and BMI\u202f\u2265\u202f30, Mean\u202f\u00b1\u202fSD bone age in females was delayed 0.50\u202f\u00b1\u202f0.49 and 0.45\u202f\u00b1\u202f0.63 years than the mean chronological ages, respectively. For BMI\u202f<\u202f18.5, Mean\u202f\u00b1\u202fSD bone age in females were advanced \u22120.40\u202f\u00b1\u202f2.69 years than mean chronological ages. Considering these differences, Iranian boys may have a different pattern of bone growth from GP standards. Specifications TableValue of the Protocol\u2022The data demonstrate that variations in skeletal growth in children aged 6\u20139.99 and 10\u201313.99 years old are greater than those evaluated by the Greulich and Pyle atlas, and that there are significant differences between bone and chronological ages.\u2022The categorizing groups based on BMD shows a great differences among our Mean\u202f\u00b1\u202fSD chronological and bone ages in those male subjects who are underweight (BMD\u202f<\u202f18.5) and obese (BMI\u202f\u2265\u202f30). The Mean\u202f\u00b1\u202fSD of difference of bone and chronological ages in male normal children (18.5\u202f\u2264\u202fBMI\u202f\u2264\u202f24.9) and overweight children (25\u202f\u2264\u202fBMI\u202f\u2264\u202f29.9) were 0.14\u202f\u00b1\u202f0.55 and 0.87\u202f\u00b1\u202f0.41 years, respectively.\u2022The Mean\u202f\u00b1\u202fSD difference of chronological and bone ages in female subjects that are underweight, normal, and obese is \u22120.40\u202f\u00b1\u202f2.69, 0.50\u202f\u00b1\u202f0.49 and 0.45\u202f\u00b1\u202f0.63, respectively.\u2022The variation in skeletal growth of Iranian children and children\u2019s group that the GP Atlas was taken could be related to environmental, economic, social, cultural and racial differences. Although this discrepancy is greater in boys than girls, so it may be affected by gender in Iran. Thus, GP Atlas could be more accurate to determine bone age in Iranian girls than boys.The Mean\u202f\u00b1\u202fSD bone and chronological ages, weight, height and BMI of subjects based on gender are presented. Although the Mean\u202f\u00b1\u202fSD BMI in boys was more than girls, the difference between Mean\u202f\u00b1\u202fSD chronological and bone age in boys was lower than that of girls. The difference of Mean\u202f\u00b1\u202fSD of chronological and bone ages in group 1 was more than other aged grouped , Fig. 1.The aim of this methodology is to determine the differences between bone age and chronological age in Iranian children which could lead more accurate estimation of bone age. To compare bone age with chronological age, 40 children were randomly selected. Subjects aged between 2 and 18 years living in Tehran City, Iran. Exclusion criteria were history of systemic diseases more than 1 month, history of chronic systemic diseases, history of hospitalization for more than a week, and people with height or weight above the percentile of 97% or below 3 percent percentile. After receiving informed consent from patients and their parents, radiograph of the left wrist and hand was taken. Based on GP , all radT-Test by SPSS.The normality of data was checked out by Shapiro\u2013Wilk Test. For each group, mean\u202f\u00b1\u202fSD of bone age, chronological age, weight, height and BMI (Body Mass Index) were determined; then data were analyzed using One-Sample Although GP method is reliable and known method, there are some variations in skeletal growth of the Iranian children and children\u2019s group that the GP Atlas was taken that could be related to environmental, economic, social, cultural and racial differences. Since this discrepancy is greater in boys than girls, it may be affected by gender in Iran. Therefore, GP Atlas could be more accurate to determine bone age in Iranian girls than boys."}
{"text": "These transitions do not require for their realization the energy-consumable anisotropic rotation of the amino group of A around the exocyclic C6-N6 bond. They are controlled by the non-planar transition states with quasi-orthogonal geometry (symmetry C1) joined by the single intermolecular (\u0422)N3H\u00b7\u00b7\u00b7N6(\u0410) H-bond . The Gibbs free energies of activation for these non-dissociative, dipole-active conformational transitions consist 7.33 and 7.81\u2009kcal\u2219mol\u22121, accordingly. Quantum-mechanical (QM) calculations in combination with Bader\u2019s quantum theory of \u201cAtoms in Molecules\u201d (QTAIM) have been performed at the MP2/aug-cc-pVDZ//B3LYP/6-311++G level of QM theory in the continuum with \u03b5\u2009=\u20094 under normal conditions.In this study it was theoretically shown that discovered by us recently (Brovarets\u2019  It was shown by NMR methods5 that Watson-Crick\u2009\u2194\u2009Hoogsteen breathing in DNA duplex containing A\u2219T rich region occurs via the switching of the Watson-Crick DNA base pair (bp) from the anti- to syn-conformation with the probability ~10\u22122 and represents one of the pathways for the reaction of formaldehyde with DNA10. Thorough calculations by the method of molecular dynamics indicate that \u0410\u00b7\u0422(WC)\u2009\u2194\u2009\u0410\u00b7\u0422(\u041d) transitions of actually bps and anti\u2009\u2194\u2009syn transitions of the A around the glycosidic bond are closely correlated processes, for which Gibbs free energy of activation is 10\u201311\u2009kcal\u2219mol\u22121 under normal conditions8.Spontaneous transition of the DNA base pairs from the Watson-Crick (WC) to Hoogsteen (H) configuration and 13. Comprehensive analysis of the current literature data showed that the nature of these biologically-important processes has not been investigated at all. Currently in the literature there is only one single theoretical work devoted to the study of the anti\u2009\u2194\u2009syn non-dissociative transitions in irregular pairs of nucleotide bases that do not have an exocyclic amino group in its composition14.Based on analysis of the microstructural nature of these transitions, it is quite logical to connect it with the analogical properties of the isolated DNA bps1), short-lived wobbled (w) conformers \u2013 \u0410\u00b7\u0422(wWC), \u0410\u00b7\u0422(wrWC), \u0410\u00b7\u0422(w\u041d) and \u0410\u00b7\u0422(wr\u041d) for each of the four classical \u0410\u00b7\u0422(WC) DNA bps \u2013 Watson-Crick \u0410\u00b7\u0422(WC), reverse Watson-Crick \u0410\u00b7\u0422(rWC), Hoogsteen \u0410\u00b7\u0422(\u041d) and reverse Hoogsteen \u0410\u00b7\u0422(r\u041d)11. It is known from the literature data, that these bps joined by different H-bonds are formed due to the rotation of the DNA base relative to the other on 180\u00b0 around: the (A)N1\u2013N3(T) axis for the reverse Watson-Crick \u0410\u00b7\u0422(rWC) or in other terms Donohue DNA\u00a0bp23; the (A)C9\u2032-N9 axis for the Hoogsteen A\u00b7T(H) bp30 and the (A)N7\u2013N3(T) axis for the reverse Hoogsteen A\u00b7T(rH) or in other terms Haschemeyer\u2013Sobell bp34.Recently, we have theoretically revealed novel high-energetic, significantly non-planar (symmetry CWC), \u0410\u00b7\u0422(wrWC), \u0410\u00b7\u0422(w\u041d) and \u0410\u00b7\u0422(wr\u041d) conformers have essentially non-planar structure joined by the two anti-parallel N6H/N6H\u2032\u00b7\u00b7\u00b7O4/O2 and N3H\u00b7\u00b7\u00b7N6 H-bonds . These specific intermolecular contacts involve pyramidalized A amino group, acting simultaneously as an acceptor and a donor of the H-bonding. The transition states (TSs) \u2013 TS\u0410\u00b7\u0422(WC)\u2194\u0410\u00b7\u0422(wWC), TS\u0410\u00b7\u0422(rWC)\u2194\u0410\u00b7\u0422(wrWC), TS\u0410\u00b7\u0422(\u041d)\u2194\u0410\u00b7\u0422(w\u041d) and TS\u0410\u00b7\u0422(r\u041d)\u2194\u0410\u00b7\u0422(wr\u041d) \u2013 of the dipole-active conformational transformations of the basic, plane-symmetric state of the classical \u0410\u00b7\u0422 DNA bps into the high-energetic, essentially non-planar wobbled bps and vice versa possess wobble structures (symmetry C1) and are joined by the N6H/N6H\u2032\u00b7\u00b7\u00b7O4/O2 and N3H\u00b7\u00b7\u00b7N6 H-bonds. The \u0410\u00b7\u0422(wWC), \u0410\u00b7\u0422(wrWC), \u0410\u00b7\u0422(w\u041d) and \u0410\u00b7\u0422(wr\u041d) conformers was found to be dynamically stable structures with short lifetime \u03c4\u2009=\u2009(1.4\u20133.9)\u2009ps. It was assumed that these conformational transitions are directly related to the thermally-driven fluctuational behavior of DNA \u2013 pre-melting and breathing7.It was found that revealed \u0410\u00b7\u0422(wWC), \u0410\u00b7\u0422(wrWC), \u0410\u00b7\u0422(w\u041d) and \u0410\u00b7\u0422(wr\u041d) control the \u0410\u00b7\u0422(wWC)/\u0410\u00b7\u0422(wrWC)\u2009\u2194\u2009\u0410\u00b7\u0422(w\u041d)/\u0410\u00b7\u0422(wr\u041d) conformational transitions. Moreover, in view of the recently discovered conformational transitions for the classical A\u00b7T DNA bps - \u0410\u00b7\u0422(WC)\u2009\u2194\u2009\u0410\u00b7\u0422(wWC), \u0410\u00b7\u0422(rWC)\u2009\u2194\u2009\u0410\u00b7\u0422(wrWC), \u0410\u00b7\u0422(\u041d)\u2009\u2194\u2009\u0410\u00b7\u0422(w\u041d) and \u0410\u00b7\u0422(r\u041d)\u2009\u2194\u2009\u0410\u00b7\u0422(wr\u041d)11, they are also intermediates of the biologically-important \u0410\u00b7\u0422(WC)/\u0410\u00b7\u0422(rWC)\u2009\u2194\u2009\u0410\u00b7\u0422(\u041d)/\u0410\u00b7\u0422(r\u041d) conformational transitions.In this work it was established for the first time that just-mentioned novel conformers \u0410\u00b7\u0422(wWC)\u2009\u2194\u2009\u0410\u2219\u0422(wH) and \u0410\u2219\u0422(wrWC)\u2009\u2194\u2009\u0410\u2219\u0422(wrH), and together with them conformational transition of the \u0410\u2219\u0422 DNA bps \u2013 \u0410\u2219\u0422(WC)/\u0410\u2219\u0422(rWC)\u2009\u2194\u2009\u0410\u2219\u0422(H)/\u0410\u2219\u0422(rH), does not require for their realization the rotation of the amino group of A around the exocyclic C6N6 bond35.Energetically favorable mechanism of the conformational pairwise transformation of the intermediates \u0410\u2219\u0422N3H\u00b7\u00b7\u00b7N6(\u0410) H-bond between the imino group of T and pyramidilized amino group of A. The Gibbs free energies of activation for these non-dissociative, dipole-active conformational transitions consist 7.33 and 7.81\u2009kcal\u2219molWC)\u2009\u2194\u2009\u0410\u2219\u0422(wH) and \u0410\u2219\u0422(wrWC)\u2009\u2194\u2009\u0410\u2219\u0422(wrH) \u2013 are realized via the anisotropic rotation of the amino group of A N6H/N6H\u2032\u00b7\u00b7\u00b7O4/O2(T) and (T)N3H\u00b7\u00b7\u00b7N6(A) H-bonds) around the exocyclic C6N6 bond. In TSs of these conformational transitions the pyramidality of the amino group of A significantly increases: this causes increase of the energy of the N3H\u00b7\u00b7\u00b7N6 H-bond and decrease of the energy of the intermolecular N6H/N6H\u2032\u00b7\u00b7\u00b7O4/O2 H-bond. The transitions states of these reactions \u2013 TScys\u0410\u00b7\u0422(wWC)\u2194\u0410\u00b7\u0422(w\u041d), TStrans\u0410\u00b7\u0422(wWC)\u2194\u0410\u00b7\u0422(w\u041d) and TScys\u0410\u00b7\u0422(wrWC)\u2194\u0410\u00b7\u0422(wr\u041d), TStrans\u0410\u00b7\u0422(wrWC)\u2194\u0410\u00b7\u0422(wr\u041d) \u2013 have close energy in corresponding conformational transformations . Thus, these TSs of the mutual conformational transformation of the wobble intermediates \u2013 \u0410\u2219\u0422(wWC)\u2009\u2194\u2009\u0410\u2219\u0422(wH) and \u0410\u2219\u0422(wrWC)\u2009\u2194\u2009\u0410\u2219\u0422(wrH) of the classical \u0410\u2219\u0422 DNA bps \u2013 \u0410\u2219\u0422(WC)/\u0410\u2219\u0422(rWC)\u2009\u2194\u2009\u0410\u2219\u0422(H)/\u0410\u2219\u0422(rH) \u2013 determine their conformational transformations.Two other mechanisms \u2013 the \u0410\u2219\u0422 of their mutual conformational transformations together with their harmonic vibrational frequencies at the B3LYP/6\u2013311++G level of theoryel \u2013 electronic energy, while Ecorr \u2013 thermal correction.The Gibbs free energy G for all structures has been received at the MP2/6-311++G level of theory by the formula:int have been obtained at the MP2/6-311++G level of theory as a difference between the BSSE-corrected72 electronic energy of the bp and electronic energies of the isolated bases.The electronic energies of interaction \u2206E78 has been applied for the analysis of the electron density distribution by AIMAll program package79, using wave functions calculated at the B3LYP/6-311++G level of theory. We considered the presence of the  bond critical point (BCP), a bond path between the donor and acceptor of the intermolecular contact and positive value of the Laplacian at this BCP (\u0394\u03c1\u2009>\u20090) as criteria for the existence of the H-bond or attractive van der Waals contact formation84.Bader\u2019s quantum theory of Atoms in Molecules (QTAIM)86 in TSs of the conformational transitions have been estimated by the Espinosa-Molins-Lecomte (EML) formula88:The energies of the attractive van der Waals contacts89:91.The energies of the conventional AH\u00b7\u00b7\u00b7B H-bonds have been calculated by the Iogansen\u2019s formula92.In this study the numeration for the DNA bases is generally accepted95, in particular in the base-pair recognition pocket of the high-fidelity DNA-polymerase46. At this, we have relied on the experience received in the previous works98 on the related topic and systems, in which the negligibly small impact of the stacking and sugar-phosphate backbone on the tautomerisation processes has been shown.In this study we have provided investigations at the basic, but sufficient level of the isolated H-bonded pairs of nucleotide bases, that adequately simulates the processes in real biological systems11 we have succeed to establish in the classical \u0410\u2219\u0422 DNA bps with Cs symmetry \u2013 Watson-Crick (WC), reverse Watson-Crick \u0410\u00b7\u0422(rWC), Hoogsteen \u0410\u00b7\u0422(\u041d) and reverse Hoogsteen \u0410\u00b7\u0422(r\u041d) DNA bps \u2013 novel high-energetic, dynamically-stable, mirror-symmetrical \u0410\u2219\u0422(wWC)R,L, \u0410\u2219\u0422(wH)R,L, \u0410\u2219\u0422(wrWC)R,L and \u0410\u2219\u0422(wrH)R,L conformational states. Their distinguished feature independently of the pair, in which they are realized, is significantly non-planar structure (\u04211 symmetry), caused by the pyramidal structure of the \u2265C6N6H2 amino fragment of the A DNA base, in which the amino group acts simultaneously as a donor and an acceptor of the specific intermolecular interaction with T through the two (\u0422)N3H\u00b7\u00b7\u00b7N6(A) and (A)N6H/N6H\u2032\u00b7\u00b7\u00b7O4/O2(T) H-bonds. Each of the four \u0410\u2219\u0422 Watson-Crick DNA bps transfers into the aforementioned conformer via two mirror-symmetric pathways through the TS\u0410\u2219\u0422(WC)\u2194\u0410\u2219\u0422(wWC)R,L, TS\u0410\u2219\u0422(rWC)\u2194\u0410\u2219\u0422(wrWC)R,L, TS\u0410\u2219\u0422(H)\u2194\u0410\u2219\u0422(wrH)R,L and TS\u0410\u2219\u0422(rH)\u2194\u0410\u2219\u0422(wrH)R,L (C1 symmetry). At this, the structures, which names differ from each other only by the subscripts R and L, are mirror-symmetrical, that is enantiomers. It is well known that enantiomers have identical scalar physico-chemical characteristics and differ only by the direction of the dipole moment.In our previous paperLet analyze the biological significance of these non-usual conformers of the classical \u0410\u2219\u0422 DNA bps.In this context it was fixed important result \u2013 these conformers are responsible for the two different WC/rWC\u2009\u2194\u2009H/rH mechanisms of the non-dissociative conformational transformation of the \u0410\u2219\u0422 DNA bps Fig.\u00a0\u20133.Figure\u0410\u2219\u0422(wWC)R,L\u2194\u0410\u2219\u0422(wH)R,L and TS\u0410\u2219\u0422(wrWC)R,L\u2194\u0410\u2219\u0422(wrH)R,L (C1 symmetry) with low values of imaginary frequency . Both of them are joined by the one-single intermolecular (T)N3H\u00b7\u00b7\u00b7N6(A) H-bond  between the imino group of T and pyramidilized amino group of A. In this case, conformational transformations of the \u0410\u2219\u0422 DNA bps are realized by the following non-dissociative scenario (each of them \u2013 by the mirror-symmetric pathways): \u0410\u2219\u0422(WC) (0.00)\u2009\u2194\u2009TS\u0410\u2219\u0422(WC)\u2194\u0410\u2219\u0422(wWC)R,L (7.13)\u2009\u2194\u2009\u0410\u2219\u0422(wWC)R,L (5.36)11\u2009\u2194\u2009TS\u0410\u2219\u0422(wWC)R,L\u2194\u0410\u2219\u0422(wH)R,L (7.33)\u2009\u2194\u2009\u0410\u2219\u0422(wH)R,L (5.35)\u2009\u2194\u2009TS\u0410\u2219\u0422(wH)R,L\u2194\u0410\u2219\u0422(H) (7.24)\u2009\u2194\u2009\u0410\u2219\u0422(H) (\u22120.44)11 and \u0410\u2219\u0422(rWC) (0.00)\u2009\u2194\u2009TS\u0410\u2219\u0422(rWC)\u2194\u0410\u2219\u0422(wrWC)R,L (7.26)\u2009\u2194\u2009\u0410\u2219\u0422(wrWC)R,L (5.97)11\u2009\u2194\u2009TS\u0410\u2219\u0422(wrWC)R,L\u2194\u0410\u2219\u0422(wrH)R,L (7.81)\u2009\u2194\u2009\u0410\u2219\u0422(wrH)R,L (5.79)\u2009\u2194\u2009TS\u0410\u2219\u0422(wrH)R,L\u2194\u0410\u2219\u0422(rH) (7.41)\u2009\u2194\u2009\u0410\u2219\u0422(rH) (\u22120.03)11. Notably, obtained energetic barriers are in good coincidence with the molecular-dynamic data for the \u0410\u2219\u0422(WC)\u2009\u2194\u2009\u0410\u2219\u0422(H) transition .First of these conformational transformations, which are the most energetically favorable mechanisms, are controlled by the soft TS\u0410\u2219\u0422(wWC)R,L\u2194\u0410\u2219\u0422(wH)R,L and TS\u0410\u2219\u0422(wrWC)R,L\u2194\u0410\u2219\u0422(wrH)R,L, which pairwise link the \u0410\u2219\u0422(wWC)R,L and \u0410\u2219\u0422(wH)R,L, \u0410\u2219\u0422(wrWC)R,L and \u0410\u2219\u0422(wrH)R,L conformers, are transition states of the WC/rWC\u2009\u2194\u2009H/rH conformational transformations of the classical \u0410\u2219\u0422 DNA bps.Herewith, some R structures transform into the other R structures, the same concerns L-structures. Saying in other words, pathways of these dipole-active conformational transformations are mirror-symmetric. In fact, the TS35 and is controlled by the TScys\u0410\u2219\u0422(wWC)R,L\u2194\u0410\u2219\u0422(wH)L,R, TStrans\u0410\u2219\u0422(wWC)R,L\u2194\u0410\u2219\u0422(wH)L,R and TScys\u0410\u2219\u0422(wrWC)R,L\u2194\u0410\u2219\u0422(wrH)L,R, TStrans\u0410\u2219\u0422(wrWC)R,L\u2194\u0410\u2219\u0422(wrH)L,R, that have non-planar structure (\u04211 symmetry) and quite high values of the imaginary frequencies (~252 i cm\u22121). These TSs are joined by the two anti-parallel intermolecular (\u0422)N3H\u00b7\u00b7\u00b7N6(A) and (A)N6H/N6H\u2032\u00b7\u00b7\u00b7O4/O2(T) H-bonds; notably, first of them is significantly stronger than the second one. The attractive O2\u00b7\u00b7\u00b7N7 and O4\u00b7\u00b7\u00b7N7 van der Waals contacts with weak energy  also participate in the stabilization of the TScys\u0410\u2219\u0422(wWC)R,L\u2194\u0410\u2219\u0422(wH)L,R and TScys\u0410\u2219\u0422(wrWC)R,L\u2194\u0410\u2219\u0422(wrH)L,R, accordingly.High-energetic mechanism of the WC/rWC\u2009\u2194\u2009H/rH conformational transitions of the \u0410\u2219\u0422 DNA bps is connected with anisotropic rotation of the amino group of A around the exocylic \u04216-N6 bondvice versa and WC/rWC\u2009\u2194\u2009H/rH conformational transitions of the classical \u0410\u2219\u0422 DNA bps occur in such a case :In this case, the R structures transform into the L structures and \u0410\u2219\u0422(WC)\u2194\u0410\u2219\u0422(wWC)R,L (7.13)\u2009\u2194\u2009\u0410\u2219\u0422(wWC)R,L (5.36)11\u2009\u2194\u2009TScys\u0410\u2219\u0422(wWC)R,L\u2194\u0410\u2219\u0422(wH)L,R (14.89)\u2009\u2194\u2009\u0410\u2219\u0422(wH)L,R (5.35)\u2009\u2194\u2009TS\u0410\u2219\u0422(wH)L,R\u2194\u0410\u2219\u0422(H) (7.24)\u2009\u2194\u2009\u0410\u2219\u0422(H) (\u22120.44)11;\u0410\u2219\u0422(WC) (0.00)\u2009\u2194\u2009TS\u0410\u2219\u0422(WC)\u2194\u0410\u2219\u0422(wWC)R,L (7.13)\u2009\u2194\u2009\u0410\u2219\u0422(wWC)R,L (5.36)11\u2009\u2194\u2009TStrans\u0410\u2219\u0422(wWC)R,L\u2194\u0410\u2219\u0422(wH)L,R (14.88)\u2009\u2194\u2009\u0410\u2219\u0422(wH)L,R (5.35)\u2009\u2194\u2009TS\u0410\u2219\u0422(wH)L,R\u2194\u0410\u2219\u0422(H) (7.24)\u2009\u2194\u2009\u0410\u2219\u0422(H) (\u22120.44)11;\u0410\u2219\u0422(WC) (0.00)\u2009\u2194\u2009TS\u0410\u2219\u0422(rWC)\u2194\u0410\u2219\u0422(wrWC)R,L (7.26)\u2009\u2194\u2009\u0410\u2219\u0422(wrWC)R,L (5.97)11\u2009\u2194\u2009TScys\u0410\u2219\u0422(wrWC)R,L\u2194\u0410\u2219\u0422(wrH)L,R (15.01)\u2009\u2194\u2009\u0410\u2219\u0422(wrH)L,R (5.79)\u2009\u2194\u2009TS\u0410\u2219\u0422(wrH)L,R\u2194\u0410\u2219\u0422(rH) (7.41)\u2009\u2194\u2009\u0410\u2219\u0422(rH) (\u22120.03)11 and\u0410\u2219\u0422(rWC) (0.00)\u2009\u2194\u2009TS\u0410\u2219\u0422(rWC)\u2194\u0410\u2219\u0422(wrWC)R,L (7.26)\u2009\u2194\u2009\u0410\u2219\u0422(wrWC)R,L (5.97)11\u2009\u2194\u2009TStrans\u0410\u2219\u0422(wrWC)R,L\u2194\u0410\u2219\u0422(wrH)L,R (15.00)\u2009\u2194\u2009\u0410\u2219\u0422(wrH)L,R (5.79)\u2009\u2194\u2009TS\u0410\u2219\u0422(wrH)L,R\u2194\u0410\u2219\u0422(rH) (7.41)\u2009\u2194\u2009\u0410\u2219\u0422(rH) (\u22120.03)11  level of QM theory in the continuum with \u03b5\u2009=\u20094 under normal conditions).\u0410\u2219\u0422(rWC) (0.00)\u2009\u2194\u2009TS101, stay planar.It should be noted that the orientation of the methyl group of the T DNA base does not alter in the course of all reactions of conformational transitions. At this, the heterocycles of the DNA bases, capable for the out-of-plane bendingSo, obtained by us results launch the conception of the \u201cmechanics\u201d of the non-dissociative WC/rWC\u2009\u2194\u2009H/rH conformational transformations of the classical \u0410\u2219\u0422 DNA bps.anti\u2009\u2194\u2009syn transition of A around the glycosidic bond  and reorganization of stacking and hydratation8. Simple comparison of the energetics, determining these processes, clearly indicates that the first two of them plays a leading role. This fact gives hope that obtained in this paper data are closely related to the nature of the \u0410\u2219\u0422(WC)\u2009\u2194\u2009\u0410\u2219\u0422(H) thermal fluctuation process, which occurs in DNA7. This conclusion can be verified, applying the newest methods of ab initio dynamics for the short fragments of DNA.Of course, in the composition of DNA these conformational transitions represent a self-consistent transformation of the bps, the 11, we offered novel non-dissociative mechanisms of the \u0410\u2219\u0422(WC)\u2009\u2194\u2009\u0410\u2219\u0422(H) and \u0410\u2219\u0422(rWC)\u2009\u2194\u2009\u0410\u2219\u0422(rH) conformational transitions, that do not require for their realization energy-consuming anisotropic rotation of the amino group of the A DNA base around the C6-N6 exocyclic bond. Figuratively speaking, at the transformation of the A base from the anti- to syn-conformation leading to the formation of the Hoogsteen \u0410\u2219\u0422(H) and reverse Hoogsteen \u0410\u2219\u0422(rH) bps, it dynamically relies as on the support on the T DNA base through the pyramidilized amino group of A, interacting with it in the TS region by one single (\u0422)N3H\u00b7\u00b7\u00b7N6(\u0410) H-bond.By applying developed by us novel ideas according the high-energetic conformers of the classical \u0410\u2219\u0422 DNA bps7, most likely occurs by the non-dissociative mechanism: A, rotating from the anti- to syn-configuration, interacts with T via the intermolecular H-bonds along the entire process of the conformational transformation.In the light of the obtained by us results, it could be suggested that the \u0410\u2219\u0422(WC)\u2009\u2194\u2009\u0410\u2219\u0422(H) conformational transition in DNA duplex, which was registered experimentally"}
{"text": "The entry for the column \u201cMSA principal city\u201d and row \u201cTdap \u22651 dose\u201d should have read \u201c88.8 (87.2 to 90.1). Entries for the column \u201cDifference between non-MSA and MSA principal city\u201d and row \u201cMenACWY, \u22651 dose\u201d should have read -7.4 (-10.0 to -4.7), row \u201cMenACWY, \u22652 dose\u201d should have read -12.0 (-19.5 to -4.6), row \u201cHPV, \u22651dose\u201d should have read -10.8 (-14.0 to -7.6), and row \u201cHPV UTD\u201d should have read -10.0 (-13.3 to -6.6). Entries for the column \u201cDifference between MSA nonprincipal city and principal city\u201d and row \u201cMenACWY, \u22651 dose\u201d should have read 0.1 (-2.0 to 2.2), row \u201cHPV, \u22651dose\u201d should have read -7.0 (\u00ad\u20119.6 to -4.4), and row \u201cHPV UTD\u201d should have read -5.5 (-8.3 to -2.6).In the report \u201cNational, Regional, State, and Selected Local Area Vaccination Coverage Among Adolescents Aged 13\u201317 Years \u2014 United States, 2017,\u201d on page 914, in Table 2, the entry for the column \u201cDifference\u201d and row \u201c\u22653 Hepatitis B doses\u201d should have read -2.6 (-4.8 to"}
{"text": "Scientific Reports 10.1038/s41598-019-47976-x, published online 08 August 2019Correction to: In the original version of the Article, the full error bars were omitted from Figure\u00a09A due a technical error during publication.In addition, in the Results and Discussion Section under the subheading \u2018Direct transfer of a qPCR protocol to the QS3D dPCR system\u2019,E. amylovora31 was transferred to the QS3D dPCR platform, using the same primer and probe concentrations and thermal cycling conditions as a first step to test the QS3D dPCR technology.\u201d\u201cA known qPCR protocol for now reads:E. amylovora31 was transferred to the QS3D dPCR platform, using the same primer and probe concentrations and thermal cycling conditions as a first step to test the QS3D dPCR technology.\u201d\u201cA known qPCR protocol for Finally, in the Legend of Table 2,cLog Copies mL\u22121\u2009=\u2009Log [copies \u03bcL\u22121rxn\u2009\u00d7\u20091/Drxn tube x1/EDNA Extr. X1/VolPl.Macerates]; i.e. Log (copies \u03bcL\u22121rxn\u2009\u00d7\u20091.79\u2009\u00d7\u2009103) for Apple, and Log (copies \u03bcL\u22121rxn\u2009\u00d7\u20094.81\u2009\u00d7\u2009103) for pear.\u201d\u201cnow reads:cLog Copies mL\u22121\u2009=\u2009Log [copies \u03bcL\u22121rxn\u2009\u00d7\u20091/Drxn tube\u2009\u00d7\u20091/EDNA Extr\u2009\u00d7\u20091/VolPl.Macerates]; i.e. Log (copies \u03bcL\u22121rxn\u2009\u00d7\u20091.79\u2009\u00d7\u2009103) for Apple, and Log (copies \u03bcL\u22121rxn\u2009\u00d7\u20094.81\u2009\u00d7\u2009103) for pear.\u201d\u201cThis has been corrected in both the PDF and HTML versions of the Article."}
{"text": "This work aims to discuss a predator-prey system with distributed delay. Various conditions are presented to ensure the existence and global asymptotic stability of positive periodic solution of the involved model. The method is based on coincidence degree theory and the idea of Lyapunov function. At last, simulation results are presented to show the correctness of theoretical findings. It is well known that the qualitative analysis of predator-prey models is an interesting mathematical problem and has received great attention from both theoretical and mathematical biologists \u20135. In paf1(u) = au, (II) u(t) represents the prey density at time t, c > 0 is the half-saturation constant, a > 0 denotes the search rate of the predator. Holling type II functional response is most typical of predators that specialized on one or a few prey  stands for the integer part of t, t \u2208  and t \u2260 0, 1, 2, \u22ef. The solution Following , 43 and t \u2208 =r\u00af2\u03b1.It follo))]=r\u00af1\u03b1,\u2211k=0\u03b1-1[))]=r\u00af2\u03b1.It follo(a) If x1(\u03b71) \u2265 x2(\u03b72), then it follows from  and and(a) I2r\u00af1\u03b1\u2254\u03c71,x1(k)\u2265x12r\u00af1\u03b1\u2254\u03c72.By (23),24)From  and (24)Thus by , 25) an an(a) If2r\u00af2\u03b1\u2254\u03c74,x2(k)\u2265x22r\u00af2\u03b1\u2254\u03c75.It follo(b) If x1(\u03b71) < x2(\u03b72), then it follows from  has no relation with \u03c1 \u2208 . Let M = max{\u03c73, \u03c76, \u03c79, \u03c712} + M0, where M0 > 0 which satisfies v = {v(k)} = {(x1(k), x2(k))T} of  and and(b) I2r\u00af2\u03b1\u2254\u03c77,x2(k)\u2265x22r\u00af2\u03b1\u2254\u03c78.By (31),32)From , we getThus by , 33) an an(b) Ifr\u00af2\u03b1\u2254\u03c710,x1(k)\u2265x1r\u00af2\u03b1\u2254\u03c711.It follo\u00af2\u03b1\u2254\u03c711.3It follov = {v(k)} \u2208 X: \u2016v\u2016 < M}, then \u03a9 is an open, bounded set in X and (a) of Lemma 1 is satisfied. When v \u2208 \u2202\u03a9 \u2229 KerL, v = {T} with \u2016v\u2016 = max{|x1|, |x2|} = M. Then\u03d5 = \u03bcQNv + (1 \u2212 \u03bc)Gv, \u03bc \u2208 , whereJ be the identity mapping, we haveLv = Nv has at least one solution in \u03b1 periodic solution in \u03b1 positive periodic solution of system  and (H2) are satisfied and furthermore suppose that there exist positive constants \u03bd, \u03c31and \u03c32such thatThen the positive \u03c9-periodic solution of system , y*(k))T} of (Define a function given by  and 45)V by V(N(f system  and 41)V by V(N(ondition  that \u2203 \u03f5k))T} of . The proRemark 1In },where r1Example 2 Consider the model as follows:r1(k) = 0.2 + cos k\u03c0, r2(k) = 0.1 + cos k\u03c0, m = 2, k1 = 0.2, k2 = 0.12, k3 = 0.24, k4 = 0.35,. So \u03c31 = 0.18, \u03c32 = 0.23, \u03bd = 0.04. Thus the conditions (H1)-(H2) and  and  of Theormx1(k)]}.where r1Based on the previous works and some biological meanings of predators and preys, we propose a new discrete delayed predator-prey system. By using the continuation theorem in coincidence degree theory, we present a set of sufficient conditions to ensure to ensure the existence of positive periodic solution of the discrete delayed predator-prey system. In addition, we also discussed the global asymptotic stability of positive periodic solution for the considered system. The obtained theoretical findings have important significance in biological ecology. Considering the effect of random factor, it is meaningful for us to deal with the dynamics of stochastic predator-prey system. This topic will be our future research direction.S1 Figx1(t) and the red line stands for x2(t).The blue line stands for (DOC)Click here for additional data file.S2 Fig(DOC)Click here for additional data file.S3 Figx1(t) and the red line stands for x2(t).The blue line stands for (DOC)Click here for additional data file.S4 Fig(DOC)Click here for additional data file."}
{"text": "Nature Communicationshttps://www.nature.com/articles/s41467-019-10342-6, published online 24 May 2019.Correction to: 2+/Au \u2192 CO2+ A\u2019, \u2018CO2+/Au \u2192 CO2+ B\u2019 and \u2018CO2+/Au \u2192 O2\u2013 C\u2019 instead of the correct panel a, b, c \u2018CO2+/Au \u2192 CO2+\u2019, \u2018CO2+/Au \u2192 O2+\u2019 and \u2018CO2+/Au \u2192 O2\u2013\u2019, respectively. This has been corrected in the PDF and HTML versions of the Article.The original version of this Article contained errors in Fig.\u00a01. The titles at the top of each panel a, b, c were incorrectly given as \u2018CO"}
{"text": "H be a transfer Krull monoid over a finite abelian group G . Then each nonunit a\u2208H can be written as a product of irreducible elements, say k is called the length of the factorization. The set L(a) of all possible factorization lengths is the set of lengths of a. It is classical that the system \u2112(H)\u00a0=\u00a0{L(a)\u2223a\u2208H} of all sets of lengths depends only on the group G, and a standing conjecture states that conversely the system \u2112(H) is characteristic for the group G. Let H\u2032 be a further transfer Krull monoid over a finite abelian group G\u2032 and suppose that \u2112(H)\u00a0=\u00a0\u2112(H\u2032). We prove that, if r\u2264n\u22123 or (r\u2265n\u22121\u22652 and n is a prime power), then G and G\u2032 are isomorphic.Let  Then each non-unit a\u2208H can be written as a product of atoms, and if H, then k is called the length of the factorization. The set L(a) of all possible factorization lengths is the set of lengths of a, and \u2112(H)\u00a0=\u00a0{L(a)\u2223a\u2208H} is called the system of sets of lengths of H (for convenience we set L(a)\u00a0=\u00a0{0} if a is an invertible element of H). Under a variety of noetherian conditions on H  all sets of lengths are finite. Sets of lengths  are a well-studied means for describing the arithmetic structure of monoids.Let H be a transfer Krull monoid over a finite abelian group G. Then, by definition, there is a weak transfer homomorphism \ud835\udf03:H\u2192\u212c(G), where \u212c(G) denotes the monoid of zero-sum sequences over G, and hence \u2112(H)\u00a0=\u00a0\u2112(\u212c(G)). We use the abbreviation \u2112(G)\u00a0=\u00a0\u2112(\u212c(G)). By a result due to Carlitz in 1960, we know that H is half-factorial ) if and only if |G|\u22642. Suppose that |G|\u22653. Then there is some a\u2208H with |L(a)|>1. If k,\u2113\u2208L(a) with k<\u2113 and m\u2208\u2115, then L(am)\u2283{km+\u03bd(\u2113\u2212k)\u2223\u03bd\u2208} which shows that sets of lengths can become arbitrarily large. Note that the system of sets of lengths of H depends only on the class group G. The associated inverse question asks whether or not sets of lengths are characteristic for the class group. In fact, we have the following conjecture .Conjecture 1.1.G be a finite abelian group with D(G)\u22654. If G\u2032 is an abelian group with \u2112(G)\u00a0=\u00a0\u2112(G\u2032), then G and G\u2032 are isomorphic.Let D(G)\u00a0=\u00a03, then we have \u2112(G) is studied with methods from Additive Combinatorics. In particular, zero-sum theoretical invariants (such as the Davenport constant or the cross number) and the associated inverse problems play a crucial role. Most of these invariants are well-understood only in a very limited number of cases  is known and the associated inverse problem is solved; however, if n is not a prime power and r\u22653, then the value of the Davenport constant Note if refer to , 9).r,n\u2208\u2115 and 2r<n\u22122 (this is done by Geroldinger and Zhong \u00a0=\u00a0{x\u2208\u2124\u2223a\u2264x\u2264b} the discrete, finite interval between a and b. If L\u2282\u2115 is a subset, then \u0394(L) denotes the set of (successive) distances of L  if and only if d\u00a0=\u00a0b\u2212a with a,b\u2208L distinct and \u2229L\u00a0=\u00a0{a,b}) and \u03c1(L)\u00a0=\u00a0supL\u2215minL denotes its elasticity \u00a0=\u00a01).Our notation and terminology are consistent with , 9. For r\u2208\u2115 and let r-tuple of elements of G. Then ei\u22600 for all i\u2208 and if for all i\u2208. Furthermore, G if it is independent and n\u2208\u2115, we denote by Cn an additive cyclic group of order n. Since r\u00a0=\u00a0r(G) is the rank of G and nr\u00a0=\u00a0exp(G) is the exponent of G.Let 2.1.H be a monoid with unit element 1\u00a0=\u00a01H\u2208H. An element a\u2208H is said to be invertible (or an unit) if there exists an element a\u2032\u2208H such that H will be denoted by H\u00d7, and we say that H is reduced if H\u00d7\u00a0=\u00a0{1}. The monoid H is said to be unit-cancellative if for any two elements a,u\u2208H, each of the equations au\u00a0=\u00a0a or ua\u00a0=\u00a0a implies that u\u2208H\u00d7. Clearly, every cancellative monoid is unit-cancellative.By a monoid, we mean an associative semigroup with unit element, and if not stated otherwise we use multiplicative notation. Let H is unit-cancellative. An element u\u2208H is said to be irreducible (or an atom) if u\u2209H\u00d7 and for any two elements a,b\u2208H, u\u00a0=\u00a0ab implies that a\u2208H\u00d7 or b\u2208H\u00d7. Let \ud835\udc9c(H) denote the set of atoms, and we say that H is atomic if every non-unit is a finite product of atoms. If H satisfies the ascending chain condition on principal left ideals and on principal right ideals, then H is atomic .If We say a monoid homomorphism \ud835\udf03:H\u2192B be a weak transfer homomorphism between atomic unit-cancellative monoids. It follows that for all a\u2208H, we have \u2112(H)\u00a0=\u00a0\u2112(B).Let H is a transfer Krull monoid if one of the following equivalent conditions holds:\ud835\udf03:H\u2192\u210b* for a commutative Krull monoid \u210b* .There exists a weak transfer homomorphism G0 of an abelian group.There exists a weak transfer homomorphism We say H is a transfer Krull monoid over G0. If G0 is finite, then H is said to be a transfer Krull monoid of finite type.If the second condition holds, then we say R is a transfer Krull domain (of finite type) if its monoid of cancelative elements R\u2219 is a transfer Krull monoid (of finite type).We say a domain Let In particular, commutative Krull monoids are transfer Krull monoids. Rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains are commutative Krull monoids with finite class group such that every class contains a prime divisor , 2, and v-noetherian nor completely integrally closed. To give a noncommutative example, let \ud835\udcaa be a holomorphy ring in a global field K, A a central simple algebra over K, and H a classical maximal \ud835\udcaa-order of A such that every stably free left R-ideal is free. Then H is a transfer Krull monoid over a ray class group of \ud835\udcaa , and If Lemma 2.4.A\u2208\u212c(G) with \u0394(supp(A))\u2260\u2205, the set \u0394(supp(A)), length at least 1, and bound M2.There exist constants Proof.See  and k(Vj)\u22641 for all j\u2208.Let Let Proof.G),exp(G)k(A)}\u2282L(AG)exp, we haveand hence1. Since {exp(\u0394(G0) | min\u0394(supp(A)) and min\u0394(G0)\u2265\u230aexp(G)\u22152\u230b, it follows thatSince min\u0394(G0)\u00a0=\u00a0min\u0394(supp(A))\u00a0=\u00a0exp(G)\u2212exp(G)k(A) and hence Therefore ming\u2208supp(A) such that B1\u2208\u212c(supp(A)). If there exists an atom A1 with k(A1)<1 such that A1 divides B1, then ord(h)\u00a0=\u00a0exp(G) and vh(A)\u00a0=\u00a01 for all h\u2208supp(A).Let supp(A) is not an LCN-set. Then there exists A1\u2208\ud835\udc9c(supp(A)) with k(A1)<1. It follows by 1 that h\u2208supp(A1) which implies that A1 | A, a contradiction. Thus supp(A) is an LCN-set and hence min\u0394(supp(A))\u2264|supp(A)|\u22122.2. Assume to the contrary that G0|\u00a0=\u00a0r(G)+1 and r(\u27e8G0\u27e9)\u00a0=\u00a0r(G). If h\u2208\u27e8G0\u2216{h}\u27e9, then r(\u27e8G0\u2216{h}\u27e9)\u00a0=\u00a0r(G). Otherwise let d\u00a0=\u00a0min{k\u2208\u2115\u2223kh\u2208\u27e8G0\u2216h\u27e9} and hence (G0\u2216{h})\u222a{dh} is also a minimal non-half-factorial LCN-set with r(\u27e8G0\u2216{h}\u27e9)\u00a0=\u00a0r(G).3. By Geroldinger and Zhong , say i\u00a0=\u00a01, such that k(U1)<1. Then exp(G)k(U1)<exp(G). Since 4. Assume to the contrary that there exists L(BG)exp\u2265exp(G)\u2113 that It follows by maxj\u2208, say j\u00a0=\u00a01, such that k(V1)>1. Then exp(G)k(U1)>exp(G). Since Assume to the contrary that there exists L(BG)exp\u2264exp(G)k that It follows by min3.First, we introduce new invariants which play a crucial role in the proof of Theorem 1.2 (see Proposition 3.5).Definition 3.1.d\u2208\u03941(G) and k\u2208\u2115. We defineLet \u03c1 the limit of Then \u03c1\u00a0=\u00a0\u03c1(G) if and only if G is not a cyclic group of order 4,6 or 10.It follows by Geroldinger and Zhong , say i\u00a0=\u00a01, such that t1\u2265M. Set I\u00a0=\u00a0{i\u2208\u2223ti\u2265M} and M thatSince minL(Bk) is an AAP with difference d and Bk, we obtainSince d | min\u0394(G0) and hence J\u2282 with |J|\u2265maxL(Bk)\u2212M|\ud835\udc9c(G)|D(G) such that Therefore Therefore\u03c1\u2264\u03c1*.By definition, we infer G0\u2282G with d | min\u0394(G0) and B\u2208\u212c(G0) such that \u03c1(L(B))\u00a0=\u00a0\u03c1*. Since supp(B)\u2282G0, we infer min\u0394(G0) divides min\u0394(supp(B)) and hence d divides min\u0394(supp(B)).2. Let M such that min\u0394(supp(B))\u2208\u0394(L(BM)). It follows by Lemma 3.2 and Lemma 2.2.6 thatBy Lemma 2.4, there exists a constant d divides min\u0394(supp(B)).Thus the assertion follows by k\u2208\u2115 such that kd\u2208\u03941(G), we have 3. For every Proposition 3.5.\u2112(G)\u00a0=\u00a0\u2112(G\u2032) for some finite abelian group G\u2032 and let d\u2208\u03941(G). Then \u03c1\u00a0=\u00a0\u03c1, and Suppose Proof.\u2112(G)\u00a0=\u00a0\u2112(G\u2032), it follows by definition that \u03c1\u00a0=\u00a0\u03c1 for every k\u2208\u2115 such that kd\u2208\u03941(G). By Lemma 3.4.3, we obtain Since Lemma 3.6.d\u2208\u03941(G). Then \u03c1*\u2265K. In particular, if d\u2208, then Let Proof.G0\u2282G with d dividing min\u0394(G0) and K(G0)\u00a0=\u00a0K. Then there exists A\u2208\ud835\udc9c(G0) such that k(A)\u00a0=\u00a0K\u22651. Sincewe haveSuppose d\u2208, we let n1. Set \u0394(G0)\u00a0=\u00a0d by Lemma 2.3.1.In particular, if Since Lemma 3.7.G0\u2282G be a subset with d | min\u0394(G0), where d\u2208\u03941(G) satisfies A\u2208\ud835\udc9c(G0) with k(A)>1, then s\u2208\u2115.Suppose nr is a prime power, then if n1 is not a prime power, then K>r.if In particular,Proof.t\u00a0=\u00a0r\u2212d and we start with the following claim.Let Claim A.B\u2208\u212c(G0) with g\u2208supp(B) and supp(B) is an LCN-set. Then there exists B0 is a product of atoms having cross number 1 and g\u2208supp(B0) such that BB0 is a product of atoms having cross number 1.Let Proof of Claim A.B\u2208\u212c(G0) with g\u2208supp(B) and supp(B) is an LCN-set, such that the assertion does not hold. Suppose |supp(B)| is minimal in all the counterexamples.Assume to the contrary that there exists a G1\u00a0=\u00a0supp(B). If for all g\u2208G1, ord(g)\u2223vg(B), then B is a product of atoms having cross number 1, a contradiction. Therefore there exits g\u2208supp(B), we infer Set i\u2208 and and Hi is an qi-group of rank r(Hi)\u2264r\u2212t\u00a0=\u00a0d which implies that there exists E|\u2264r(Hi)\u2264d such thatLet Therefore there exists Since \u0394(supp(Bi))\u00a0=\u00a00 and hence supp(Bi) is half-factorial. Therefore Bi is a product of atoms having cross number 1.Note that Since y\u2208\u2115 and i\u2208 and every g\u2208supp(C). Since |supp(C)|<|supp(B)|, it follows by the minimality of |supp(B)| that there exists C0 is a product of atoms having cross number 1 and g\u2208supp(C0), such that CC0 is a product of atoms having cross number 1. Let BB0 is a product of atoms having cross number 1, a contradiction to our assumption.Therefore A\u2208\ud835\udc9c(G0) be with k(A)>1. Then Lemma 2.6.2 implies that supp(A) is an LCN-set. Set Claim A implies that there exist atoms \u2113\u2208\u21150, such that d | min\u0394(G0), we infer s\u2208\u2115.Let Now we begin to prove the \u201cin-particular\u2019\u2019 parts.j\u2208 and every m\u2208\u2124, we denote by \u2225m\u2225j the least positive residue of m modulo 1. For every K, we have s\u2208\u2115. Let H be a subgroup of G withand let H with ord(e)\u00a0=\u00a0n1 and i\u2208, j\u2208. Set W\u2208\ud835\udc9c(G2), we haveandBy definition of We suppose that Since for each j\u2208, whenceThereforer\u22121 | min\u0394(G2) which implies that K\u2265K(G2). Let Since s\u2208\u2115, it follows that Since nr is a prime power, then K(G)<r by Lemma 2.1. It follows by 1. that s\u2208\u2115. Therefore 2. If n1 is not a prime power, then u\u22652 and K>1+r\u22121\u00a0=\u00a0r. \u25a13. If Proposition 3.8.s\u2208\u2115 be maximal such that G has a subgroup isomorphic to d\u2208\u03941(G) with d\u2265r, then If d\u2265nr\u22121, then \u03c1*\u00a0=\u00a0K.If r\u00a0=\u00a0nr\u22121\u22653. If p and k\u2208\u2115, then K\u00a0=\u00a0\u03c1*.Suppose Let Proof.d\u2265r, it follows by Lemma 2.5(items 3 and 4) that d>m(G) and 1. Since nr and let Let G0\u2282G with d | min\u0394(G0) be such that B\u2208\u212c(G0) with \u03c1(L(B))\u00a0=\u00a0\u03c1*. Setwhere k\u00a0=\u00a0minL(B), \u2113\u00a0=\u00a0maxL(B), and Let i\u2208 such that k(Ui)>1, then Lemma 2.6.2 implies that supp(Ui) is an LCN-set and hence min\u0394(supp(Ui))\u2264m(G). Sincewe get a contradiction to d>m(G). Therefore k(Ui)\u22641 for all i\u2208.If there exists i\u2208 such that k(Vi)<1, then Lemma 2.6.1 implies that i\u2208. It follows thatIf there exists Then G0\u2282G be such that d | min\u0394(G0) and A\u2208\ud835\udc9c(G0) such that k(A)<1, then Lemma 2.6.1 implies that A|\u22652. Thus G0 is an LCN-set. Let B\u2208\u212c(G0) such that which implies that \u03c1*\u00a0=\u00a0\u03c1(L(B))\u2264K.2. Let G0\u2282G be such that d | min\u0394(G0) and K(G0)\u00a0=\u00a0K. Then there exists an atom A\u2208\ud835\udc9c(G0) such that k(A)\u00a0=\u00a0K\u22651. Since and hence \u03c1*\u00a0=\u00a0K.Let r\u00a0=\u00a0nr\u22121\u22653 and we proceed to prove the following claim.3. Let Claim B.\u03c1*>K and let G0\u2282G be such that (r\u22121) | min\u0394(G0) and \u03c1(G0)>K.g\u2208G0 with ord(g)\u00a0=\u00a0nr such that \u2212g\u2208G0.There exists G2|\u00a0=\u00a0r. If there exists a\u2208 such that ag\u2208\u27e8G2\u27e9, then G2 is a basis of G.Let G, and Suppose Proof of Claim B.By Lemma 2.5.2, we infer that a. If G0 is an LCN-set, then for every B\u2208\u212c(G0), we havewhich implies that \u03c1(L(B))\u2264K. Therefore \u03c1(G0)\u2264K, a contradiction. Thus there exists A\u2208\ud835\udc9c(G0) such that k(A)<1. Lemma 2.6.1 implies that A\u00a0=\u00a0g(\u2212g) for some g\u2208G0 with ord(g)\u00a0=\u00a0nr. Hence {g,\u2212g}\u2282G0.b. Let E\u2282G2 be minimal such that there exists a\u2208 such that ag\u2208\u27e8E\u27e9 and let dgg\u2208\u27e8E\u27e9. Then there exists an atom V\u2208\ud835\udc9c(E\u222a{g}) with supp(V)|\u2264r+1. Let T\u2208\u2131(E). Thenr\u22121 | dg+\u2113\u22122. Therefore dg\u00a0=\u00a01 and Note that for each g)Ti, g\u2208\u27e8E\u27e9 which implies that dg\u00a0=\u00a01 by the minimality of dg. The minimality of E implies that supp(Ti)\u00a0=\u00a0E for each i\u2208. Then for every h\u2208E, ord(h)\u00a0=\u00a0nr and Suppose k((\u2212g)T1)<1, then Lemma 2.6.1 implies that T1\u00a0=\u00a0g, a contradiction. ThereforeIf E|\u2264|G2|\u2264r and r\u00a0=\u00a0nr\u22121, we have |E|\u00a0=\u00a0|G2|\u00a0=\u00a0r. Let G, Since |dg\u00a0=\u00a01 and V\u00a0=\u00a0gT and hence |T|\u22652. We infer g}\u222aE is an LCN-set and min\u0394({\u2212g}\u222aE)\u2264|E|\u22121. Since (r\u22121) | min\u0394({\u2212g}\u222aE) and |E|\u2264|G2|\u00a0=\u00a0r, we haveSuppose Let E1|\u00a0=\u00a0r+1 and hence It follows by Geroldinger and Zhong .Let k(V)>1, then Lemma 2.6.2 and  and every h\u2208G2, mg\u2209\u27e8G2\u2216{h}\u27e9. Note that nr\u22654. Let andare both atoms. Since r\u22121 | |I|\u22121 and r\u22121 | |I|. Hence r\u00a0=\u00a02, a contradiction to r\u22653.By the minimality of k(V)\u00a0=\u00a01 which implies that i\u2208. It follows by k(V1)>1 and  such that ag\u2208\u27e8G1\u2216{h}\u27e9. Since |G1\u2216{h}|\u00a0=\u00a0r, it follows by a and b that G such that Let t\u2208, i\u2208 and b that Assume to the contrary that there exists We only need to prove Since B\u2208\u212c(G0) such that \u03c1(L(B))\u00a0=\u00a0\u03c1(G0) and assume thatwhere k,\u2113\u2208\u2115 and Let k(Ui)\u22651 for all i\u2208 and k(Vj)\u22641 for all j\u2208. Since B by i\u2208 and each j\u2208.Note that j0\u2208 such that i0\u2208 such that j\u2208. If there exists i1\u2208 such that Since there must exist Therefore\u03c1(G0)>K. If there exists j1\u2208 such that a contradiction.It follows by Lemma 3.6 that andTo sum up, we obtainJ\u00a0=\u00a0{j\u2208\u2223Vj\u00a0=\u00a0g(\u2212g)}. ThenandLet Therefore We distinguish two cases to finish the proof.p and k\u2208\u2115 with pk\u22654. Then G and we obtainSuppose that Claim B thatIt follows by p and any k\u2208\u2115. Assume to the contrary that K<\u03c1*. Then Claim B implies that \u03c1<r. Since nr is not prime power, it follows by Lemma 3.7.3 that K>r, a contradiction. \u25a1Suppose that 4.G\u2032 is a further finite abelian group with From now on, we assume Lemma 4.1.\u2112(G)\u00a0=\u00a0\u2112(G\u2032) and d\u2208\u03941(G). ThenG is isomorphic to a subgroup of G\u2032, then G\u2245G\u2032.If Suppose Proof.D(G)\u00a0=\u00a0D(G\u2032). It follows from . Then d\u2265max{r,\u230anr\u22152\u230b}, Proposition 3.8.1 implies that Suppose that \u0394*(G) is an interval, we obtain that k be maximal such that there exists a subgroup H of G with k\u2032 be maximal such that there exists a subgroup H\u2032 of G\u2032 with r\u2265r+k\u2265nr\u22122 and 5. By 3, we have r\u2260r(G\u2032) and by symmetry, we can assume r<r(G\u2032). Thus nr is even, nr\u00a0=\u00a02r+2, and r(G\u2032)\u00a0=\u00a0r+1. Since \u0394*(G) and k\u2032\u2265r. Then G is a subgroup of G\u2032 which implies that G\u2245G\u2032 by Lemma 4.1.1, a contradiction.Assume to the contrary that Proof of Theorem 1.2.By definition of transfer Krull monoids, it follows that r\u2264n\u22123. If \u0394*(G) is not an interval, then [G\u2245G\u2032.1. Let \u0394*(G) is an interval. Then Theorem 4.2.5 implies that n\u00a0=\u00a0exp(G\u2032) and r\u00a0=\u00a0r(G\u2032). Therefore G\u2032 is isomorphic to a subgroup of G which implies that G\u2245G\u2032 by Lemma 4.1.1.Suppose n\u00a0=\u00a02, then G is an elementary two-group and the assertion follows by Geroldinger and Halter-Koch [n\u22653 is a prime power. Then Theorem 4.2.2 implies that r\u00a0=\u00a0r(G\u2032) and G is isomorphic to a subgroup of G\u2032 which implies that G\u2245G\u2032 by Lemma 4.1.1.2. If 5.D(G)\u22654 and Throughout this section, let Conjecture 5.1.r\u2265n\u22121. Then the following hold:s\u2208\u2115 with gcd\u00a0=\u00a01.Suppose C2 implies C1. By Lemma 3.7.1, we have s\u2208\u2115 and if n is a prime power, then C2 holds.It is easy to see that Proposition 5.2.G\u2032 be a finite abelian group with \u2112(G\u2032)\u00a0=\u00a0\u2112(G). If r\u2265n\u22121 and C1 holds, then G\u2245G\u2032.Let Proof.n is a prime power, then the assertion follows by Theorem 1.2.If n is not a prime power. Then r\u2265n\u22121\u22655. Let \u2112(G)\u00a0=\u00a0\u2112(G\u2032) and let s\u2208\u2115. If s\u2032\u2208\u2115. Thus s,n)\u00a0=\u00a01. Therefore G is isomorphic to a subgroup of G\u2032 which implies that G\u2245G\u2032 by Lemma 4.1.1.Suppose s,n)\u00a0=\u00a01, we have n\u00a0=\u00a0r+2, a contradiction.Suppose G\u2032 and there is known no group G\u2032 with Recall that Proposition 5.3.G\u2032 be a finite abelian group with \u2112(G\u2032)\u00a0=\u00a0\u2112(G). If r\u2265max{(u\u22121)n+1,n}\u22653 and K(G)\u00a0=\u00a0K*(G), then G\u2245G\u2032.Let Proof.u\u00a0=\u00a01, then the assertion follows by Theorem 1.2.2. Suppose u\u22652. Note r\u2265(\u03c9(n)\u22121)n+1, we have s\u2032\u2208\u2115. Therefore s,n)\u00a0=\u00a01. The assertion follows by Proposition 5.2.If"}
{"text": "Scientific Reports 10.1038/s41598-019-39429-2, published online 11 February 2019Correction to: The Article contains errors in Table 1, where the reported second column heading under Urea CD was incorrectly titled. Additionally, there are errors in four of the values listed under \u2018Urea Fluorescence\u2019 and \u2018Urea CD\u2019.The correct Table\u00a0Additionally, there is an error in Table 3, in which the value \u201c137\u2009\u00b1\u20092x\u201d under FXN L198C should read \u201c137\u2009\u00b1\u20092\u201d."}
{"text": "Correction to: BMC Bioinformaticshttps://doi.org/10.1186/s12859-019-2997-9Following publication of the original article , the autPage 9, first column:The paragrapho\u00a0bt(A): sec) by the local vascular network within/sec) by the local vascular network within A during the previous time interval t\u2009\u2212\u2009\u0394\u03c4\u2009\u2192\u2009t\u201d\u201c- Should be replaced witho_bt(A) : Oxygen supply rate (pmols/sec) by the local vascular network within A during the previous time interval t\u2009\u2212\u2009\u0394\u03c4\u2009\u2192\u2009t \u201d\u201c- gl_bt(A): Glucose supply rate \u2026 \u201d ,In the subsequent paragraph, i.e. the paragraph starting starting with the phrase \u201c - \u0394\u03c4\u2009\u2192\u2009t \u201d should be replaced with \u201c interval t\u2009\u2212\u2009\u0394\u03c4\u2009\u2192\u2009t \u201dthe phrase \u201c interval\u2212Page 11, second column:The equation ao(\u03b2) is increasing, \u03b2_ is actually \u2026 \u201d the \u201c\u03b2_ \u201d should be \u201c In the subsequent sentence, i.e. \u201cSince \u03b2\u2009_\u2009\u2009>\u2009\u03b22, we have that \u2026 \u201d the \u201c \u03b2_ \u201d should be \u201c In the subsequent paragraph (first bullet point), in the sentence \u201c If \u03b2\u2009_\u2009\u2009\u2264\u2009\u03b22, we have that for each \u03b2\u2009\u2208\u2009 it holds that ao(\u03b2)\u2009\u2265\u20090. \u201d all occurences of \u201c \u03b2_ \u201d should be \u201c In the subsequent paragraph (second bullet point), i.e. \u201c If ao(\u03b2) \u201d should be \u201c agl(\u03b2)\u00a0\u201d.In paragraph 11a, first line, \u201c Page 12, first column:\u03b2\u2009>\u2009\u03b22 or \u03b2\u2009>\u2009\u03b22 should be In the sentence \u201c Case 2.2.1. If \u03b2\u2009\u2264\u2009\u03b22, \u03b2\u2009\u2264\u2009\u03b22 should be In the sentence \u201c Case 2.2.2. If In the same paragraph, the mathematical expression Page 12, second column:In the sentence \u201c Again, we pick a random Page 14, second column:The equationsshould beThe equationsshould bePage 15, first column:The equationsshould bePage 15, second column:The equationsShould beThe second equation appearing in this column, i.e.Should be"}
{"text": "C\u2212H\u2219\u2219\u2219\u03c0 and N\u2212H\u2219\u2219\u2219\u03c0 interactions can have an important contribution for protein stability. However, direct measurements of these interactions in proteins are rarely reported. In this work, we combined the mutant cycle experiments and molecular dynamics (MD) simulations to characterize C\u2212H\u2219\u2219\u2219\u03c0 and N\u2212H\u2219\u2219\u2219\u03c0 interactions and their cooperativity in two model proteins. It is shown that the average C\u2212H\u2219\u2219\u2219\u03c0 interaction per residue pair is ~ \u22120.5\u2009kcal/mol while the N\u2212H\u2219\u2219\u2219\u03c0 interaction is slightly stronger. The triple mutant box measurement indicates that N\u2212H\u2219\u2219\u2219\u03c0\u2219\u2219\u2219C\u2212H\u2219\u2219\u2219\u03c0 and C\u2212H\u2219\u2219\u2219\u03c0\u2219\u2219\u2219C\u2212H\u2219\u2219\u2219\u03c0 can have a positive or negative cooperativity. MD simulations suggest that the cooperativity, depending on the local environment of the interactions, mainly arises from the geometric rearrangement when the nearby interaction is perturbed. Theoretical studies show that N\u2212H\u2219\u2219\u2219\u03c0, O\u2212H\u2219\u2219\u2219\u03c0 and C\u2212H\u2219\u2219\u2219\u03c0 can have very different optimum geometries, with the interaction strength order O\u2212H\u2219\u2219\u2219\u03c0\u2009>\u2009N\u2212H\u2219\u2219\u2219\u03c0\u2009>\u2009C\u2212H\u2219\u2219\u2219\u03c011. The S\u2212H\u2219\u2219\u2219\u03c0 interaction can be weaker9 or stronger12 than O\u2212H\u2219\u2219\u2219\u03c0, but is generally stronger than N\u2212H\u2219\u2219\u2219\u03c0 and C\u2212H\u2219\u2219\u2219\u03c012. The computational interaction energy of the indole-benzene dimer where the N\u2212H\u2219\u2219\u2219\u03c0 interaction exists can reach \u22125.2\u2009kcal/mol13. The computational interaction energies between benzene and CH4, NH3, H2S, and H2O are \u22121.4, \u22122.5, \u22122.9 and \u22123.0\u2009kcal/mol, respectively9. The computational binding energies between indole and CH4, NH3, H2S, and H2O are \u22122.0, \u22122.6, \u22124.9 and \u22123.6\u2009kcal/mol, respectively12. The importance of the S\u2212H\u2219\u2219\u2219\u03c0 and C\u2212H\u2219\u2219\u2219\u03c0 interactions in proteins has also been highlighted by their occurrence in the PDB database search15. The C\u2212H\u2219\u2219\u2219\u03c0 interaction has been observed directly in proteins by nuclear magnetic resonance (NMR) spectroscopy methods where the across C\u2212H\u2219\u2219\u2219\u03c0 J-coupling is detected16. Quantification of C\u2212H\u2219\u2219\u2219\u03c0 in calix[4]pyrrole receptors yields a magnitude of \u22121\u2009kcal/mol17. The C\u2212H\u2219\u2219\u2219\u03c0 interaction in benzene\u2212methane, ethane, propane, and butane, increases monotonically from \u22121.1 to \u22122.7\u2009kcal/mol20. The measurement of C\u2212H\u2219\u2219\u2219\u03c0 interactions in a cyclohexylalanine\u2212phenylalanine pair in the core of a synthetic peptide indicates that each C\u2212H\u2219\u2219\u2219\u03c0 contact can contribute about \u22120.7\u2009kcal/mol to peptide stability21. In real proteins, C\u2212H\u2219\u2219\u2219\u03c0 mainly occurs between an aliphatic side chain and an aromatic ring, or between two aromatic rings14. Although C\u2212H\u2219\u2219\u2219\u03c0 interactions are well documented in proteins1, direct measurements of C\u2212H\u2219\u2219\u2219\u03c0 and N\u2212H\u2219\u2219\u2219\u03c0 strength in proteins are scarce.X\u2212H\u2219\u2219\u2219\u03c0 interactions in biomolecules, where X can be C, N, O, or S are weak and attractive interactions between the X\u2212H component and aromatic groups. The high incidence in biomolecules makes X\u2212H\u2219\u2219\u2219\u03c0 interactions an important contributor to the structure and function, and has led to an increasing number of theoretical and experimental studies devoted to characterization of such interactions22. By forming networks of weak interactions that compete against the entropy of flexible polypeptides, proteins fold into their biologically functional three-dimensional structures23. As a part of the interaction network, how X\u2212H\u2219\u2219\u2219\u03c0 interactions coexist and cooperate in proteins is an important question. Only a few studies have addressed the X\u2212H\u2219\u2219\u2219\u03c0 cooperativity, mainly in small molecules. The cooperativity of C\u2212H\u2219\u2219\u2219\u03c0 interactions in small molecules is studied using molecular torsional balances24. The average C\u2212H\u2219\u2219\u2219\u03c0 interaction strength increases as more C\u2212H\u2219\u2219\u2219\u03c0 pairs are formed, suggesting a positive cooperativity. This is opposite to the findings of an earlier computational study where the negative cooperativity is concluded for the same complexes25. The C\u2212H\u2219\u2219\u2219\u03c0 and N\u2212H\u2219\u2219\u2219\u03c0 cooperativity in proteins remains largely unexplored.Another important issue about X\u2212H\u2219\u2219\u2219\u03c0 interactions is their cooperativity. Cooperativity is a central concept for understanding molecular recognition and supramolecular self-assembly26. SNase is an enzyme that hydrolyzes nucleotides in DNA or RNA. A stable mutant of SNase, \u0394\u2009+\u2009PHS, is selected as the test system27. It is found experimentally that the C\u2212H\u2219\u2219\u2219\u03c0 interaction on average is about \u22120.5\u2009kcal/mol and the N\u2212H\u2219\u2219\u2219\u03c0 interaction on average is about \u22120.6\u2009kcal/mol. N\u2212H\u2219\u2219\u2219\u03c0\u2026C\u2212H\u2219\u2219\u2219\u03c0 and C\u2212H\u2219\u2219\u2219\u03c0\u2026C\u2212H\u2219\u2219\u2219\u03c0 can have different cooperativities. Molecular dynamics (MD) simulations can reproduce N\u2212H\u2219\u2219\u2219\u03c0 and C\u2212H\u2219\u2219\u2219\u03c0 interactions and their cooperativities with reasonable accuracy. Geometric parameters that are important for C\u2212H\u2219\u2219\u2219\u03c0 and N\u2212H\u2219\u2219\u2219\u03c0 interactions are discussed. Their contribution to cooperativity is illustrated. With the combination of experimental and computational results, a better view of C\u2212H\u2219\u2219\u2219\u03c0, N\u2212H\u2219\u2219\u2219\u03c0 and their cooperativity is obtained.In this work, we attempt to measure the C\u2212H\u2219\u2219\u2219\u03c0 and N\u2212H\u2219\u2219\u2219\u03c0 interactions in protein GB3 and staphylococcal nuclease (SNase). GB3 is the third immunoglobulin binding domain of protein G, a model protein that has been extensively studiedBased on the X-ray crystal structures, a series of X\u2212H\u2219\u2219\u2219\u03c0 interactions can be identified in GB3 and \u0394\u2009+\u2009PHS . GB3 has five residue pairs that may form C\u2212H\u2219\u2219\u2219\u03c0 interactions, L5\u2212F30, T18\u2212F30, L5\u2212Y33, I7\u2212Y33, and T16\u2212Y33, and one residue pair N37\u2212Y33 that can form the N\u2212H\u2219\u2219\u2219\u03c0 interaction Fig.\u00a0. \u0394\u2009+\u2009PHSG of all proteins were derived from the denaturation curves. The values of [D]50%, m values for the wild type and mutant proteins are given in Supplementary Table\u00a028. The values of C\u2212H\u2219\u2219\u2219\u03c0 interactions are shown in Table\u00a0The folding free energies \u0394x and y in the X\u2212H\u2219\u2219\u2219\u03c0 pair to alanine residues to completely remove the interactions between the two. However, eliminating an aromatic residue in a protein core can be detrimental to protein stability. Instead, we only mutated the aromatic side chain (y) to a leucine (y\u2032) which is still hydrophobic but disrupts the X\u2212H\u2219\u2219\u2219\u03c0 interaction . For the X\u2212H component (x), conservative mutations are introduced (x\u2032) to remove the X\u2212H\u2219\u2219\u2219\u03c0 interaction and maintain the protein folding at the same time. These mutations may create residual pairwise side chain interactions in x\u2032y\u2032, xy\u2032, and x\u2032y. Furthermore, for a residue like leucine  which has two CH3 and one CH, it can form multiple C\u2212H\u2219\u2219\u2219\u03c0 interactions which complicate the interpretation of the experimental results. These problems can be solved with the assist of MD simulations.According to DMC, it is preferable to mutate the two side chains 29, Charmm2730, and Gromos53a631. The experimental C\u2212H\u2219\u2219\u2219\u03c0 and N\u2212H\u2219\u2219\u2219\u03c0 interaction energies were used as a benchmark to evaluate the accuracy of different force fields. The root mean square deviation (RMSD) between the experimental and predicted X\u2212H\u2219\u2219\u2219\u03c0 interactions was calculated:N is 31, the total number of measured residue pairs that form X\u2212H\u2219\u2219\u2219\u03c0 interactions, \u0394\u0394Gexp is the experimental X\u2212H\u2219\u2219\u2219\u03c0 interaction energy, and \u0394\u0394EMD is the calculated interaction energy. Charmm27 appears to perform better than the other two force fields. Its RMSD value is 0.27\u2009kcal/mol (after removing two apparent outliers), while the RMSDs of Amber99sb and Gromos53a6 are 0.41 and 0.47\u2009kcal/mol, respectively , and \u03c9 is the \u2220C\u2212H\u2212X angle  versus \u0394\u0394ECH3\u2219\u2219\u2219\u03c0 shows that the geometries with shorter dCX and larger \u03c9 have more negative interaction energies , although the absolute value of \u2206\u0394\u0394E is generally larger than that of \u2206\u0394\u0394Gcoop  and thus not pursued. As discussed above, the residual interactions caused by the experimental non-alanine mutations complicate the interpretation of \u2206\u0394\u0394Gcoop. To solve this problem, we rebuilt TMBs by mutating the three side chains, for example L25, F34, and V74 in L25\u2212F34\u2212V74, to alanines systematically in MD simulations. The cooperativity energy \u2206\u0394\u0394E\u2032 was calculated for the residue groups listed above with the same procedure  can vary from \u221240% (cooperative) to +60% (anticooperative)  distance when the aliphatic side chain of the second C\u2212H\u2219\u2219\u2219\u03c0 (or N\u2212H\u2219\u2219\u2219\u03c0) is mutated to alanine. In other words, removing the second C\u2212H\u2219\u2219\u2219\u03c0 (or N\u2212H\u2219\u2219\u2219\u03c0) interaction weakens the first C\u2212H\u2219\u2219\u2219\u03c0 (or N\u2212H\u2219\u2219\u2219\u03c0) interaction, suggesting a positive cooperativity. For 9 out of 10 groups, the distance change \u0394d predicts the cooperativity consistent with the interaction energy result  Fig.\u00a0. So it iult Fig.\u00a0, indicat33. However, measuring C\u2212H\u2219\u2219\u2219\u03c0 interactions in the protein interior using DMC can be challenging because removing an aromatic side chain can destabilize and even unfold the protein. In this work, we only mutate the aromatic residue to leucine which maintains the protein folding and removes the C\u2212H\u2219\u2219\u2219\u03c0 interaction. Two very stable proteins GB3 and \u0394\u2009+\u2009PHS were selected for the purpose. One caveat of the F or Y to L mutation is that residual interactions with leucine complicate the data interpretation. Molecular dynamics simulations were used to decompose the various contributions and help us focus on the C\u2212H\u2219\u2219\u2219\u03c0 interactions. The good agreement between experimental and computational interaction energies validates the procedure which provides important insights about the C\u2212H\u2219\u2219\u2219\u03c0 and N\u2212H\u2219\u2219\u2219\u03c0 interactions.DMC experiments are commonly used to measure residue\u2212residue interactions, such as salt bridges and hydrogen bonds21. It is likely that different interactions compete with each other in proteins so that the C\u2212H\u2219\u2219\u2219\u03c0 interaction of a specific residue pair is not in an optimum geometry. This is evident from the interaction energy landscape of methyl\u2212aromatic ring pair  calculations, the cooperativity energy of MMB is 0.74\u2009kcal/mol, indicating that C\u2212H\u2219\u2219\u2219\u03c0\u2026C\u2212H\u2219\u2219\u2219\u03c0 is anticooperative in this model, while the cooperativity energy of MBM is 0.03\u2009kcal/mol, suggesting that there is no cooperativity in this model. Similar to the result in the MD simulations, the geometric reorganization occurs in the MMB model where the two methanes compete for the binding site. No such competition exists in the MBM model where the cooperativity energy is close to zero. The QM calculations highlight the importance of geometric reorganization to cooperativity.Two simpler cooperativity models were built using two methane and one benzene molecules, with methanes on the same side (MMB) or opposite side (MBM) of the benzene. The cooperativity energies of MMB and MBM models were calculated at the MP2/aug-cc-pvtz leveldC(N)X and \u03c9. The experimental TMB analysis suggests that the cooperativity of X\u2212H\u2219\u2219\u2219\u03c0 interactions can be either positive or negative, depending on the local environment. The cooperativity trend is successfully captured by MD simulations where the cooperativity energy can reach ~ \u221240% to 60% of C\u2212H\u2219\u2219\u2219\u03c0 or N\u2212H\u2219\u2219\u2219\u03c0 interactions, highlighting its importance in proteins. The geometric rearrangement is the main cause for the cooperative interactions. It is worth noting that the C\u2212H\u2219\u2219\u2219\u03c0 and N\u2212H\u2219\u2219\u2219\u03c0 interactions and the cooperativity were only measured for two proteins GB3 and \u0394\u2009+\u2009PHS. More measurements will be needed to see whether the conclusions also hold for other proteins. But we expect that the mechanism behind the interactions is universal for all protein molecules.In this study, we measured the strength of C\u2212H\u2219\u2219\u2219\u03c0 and N\u2212H\u2219\u2219\u2219\u03c0 interactions in GB3 and SNase. The C\u2212H\u2219\u2219\u2219\u03c0 interaction is about 0.3 to \u22120.9\u2009kcal/mol whereas the N\u2212H\u2219\u2219\u2219\u03c0 interaction is about \u22120.2 to \u22120.9\u2009kcal/mol. The energy decomposition from MD simulations helps determine the C\u2212H\u2219\u2219\u2219\u03c0 and N\u2212H\u2219\u2219\u2219\u03c0 interactions for individual methyl\u2212aromatic and amino\u2212aromatic pairs and identify important geometric parameters E. coli strain BL21 (DE3) cells for protein expression. The purification procedure for GB3 and its variants has been described previously35. \u0394\u2009+\u2009PHS and its variants were purified using the same procedure as described by Shortle and Meeker36.The wild type and mutants of GB3 and \u0394\u2009+\u2009PHS were prepared with the PCR-based site-directed mutagenesis on vector pET-11b. These plasmids were transformed into the S is the measured Fluo340nm or Fluo348nm, \u03b1N and \u03b1U are the intercepts and \u03b2N and \u03b2U are the slopes of the Fluo340nm or Fluo348nm baselines at low (N) and high (U) denaturant concentrations, R is the Boltzmann constant, T is the temperature, [D] is the denaturant concentration, [D]50% is the denaturant concentration at which the protein is 50% denatured.All the denaturation measurements were performed using a HITACHI f-4600 fluorescence Spectrophotometer. Mixtures consisted of up to 6.0\u2009M GdnHCl and 50\u2009\u00b5M proteins  were incubated for 30\u2009min at 30\u2009\u00b0C. The signal intensity at 340\u2009nm for GB3 and 348\u2009nm for SNase was extracted and fitted using the following equation,38. Double mutant cycles were performed to quantify C\u2212H\u2219\u2219\u2219\u03c0 interactions and N\u2212H\u2219\u2219\u2219\u03c0 interactions in this work. To build the DMC, dozens of single and double mutants were prepared. Single mutants included L5V, I7V, T16A, T18A, N37A, F30L, Y33L, Y33F in GB3 and L25V, V74A, I92V, F34L in \u0394\u2009+\u2009PHS. Double mutants contained two substitutions, L5V\u2212F30L, L5V\u2212Y33L, I7V\u2212Y33L, T16A\u2212Y33F, T16A\u2212Y33L, T18A\u2212F30L, N37A\u2212Y33L and N37A\u2212Y33F in GB3, and L25V\u2212F34L, V74A\u2212F34L, I92V\u2212F34L in \u0394\u2009+\u2009PHS. The folding free energy for each mutant was determined from the denaturation curve monitored by fluorescence. The C\u2212H\u2219\u2219\u2219\u03c0 or N\u2212H\u2219\u2219\u2219\u03c0 interaction energy with the aromatic ring was then calculated using:Gxy, \u0394Gx\u2032y, \u0394Gxy\u2032, and \u0394Gx\u2032y\u2032 are the folding free energy for the wild type protein xy, single mutants x\u2032y and y\u2032x, and the double mutant x\u2032y\u2032, respectively. The symbols x and y denote the aliphatic and aromatic side chains in the C\u2212H\u2219\u2219\u2219\u03c0 or N\u2212H\u2219\u2219\u2219\u03c0 pair. This expression can be defined for both GB3 and \u0394\u2009+\u2009PHS proteins.Double mutant cycle (DMC), proposed by Fersht and co-workers, can eliminate the contribution of the secondary interactions and obtain accurate binding energy for the interaction between two residues39. Double mutants of GB3  and \u0394\u2009+\u2009PHS  were used to set TMBs. All of these double mutant proteins could be expressed except L5V-I7V of GB3. Triple mutants were prepared, including L5V-T16A-Y33L, L5V-T18A-F30L, I7V-T16A-Y33L, L5V-N37A-Y33L, I7V-N37A-Y33L, T16A-N37A-Y33L, and T16A-N37A-F33L for GB3, and L25V-V74A-F34L, L25V-F34L-I92V, and V74A-I92V-F34L for \u0394\u2009+\u2009PHS. These mutants were used to quantify the cooperativity in C\u2212H\u2219\u2219\u2219\u03c0\u2219\u2219\u2219C\u2212H\u2219\u2219\u2219\u03c0 interactions and C\u2212H\u2219\u2219\u2219\u03c0\u2219\u2219\u2219N\u2212H\u2219\u2219\u2219\u03c0 interactions. The folding free energy for each mutant was measured using the same method mentioned above. The cooperativity energy was then calculated using:y represents the aromatic residue, x and z represent nonaromatic residues, \u2206Gxyz, \u2206Gx\u2032yz, \u2206Gxy\u2032z, \u2206Gxyz\u2032, \u2206Gx\u2032y\u2032z, \u2206Gx\u2032yz\u2032, \u2206Gxy\u2032z\u2032, and \u2206Gx\u2032y\u2032z\u2032 are the folding free energy of the wild type protein xyz, single mutants x\u2032yz, xy\u2032z and xyz\u2032, double mutants x\u2032y\u2032z, x\u2032yz\u2032, xy\u2032z\u2032 and triple mutants x\u2032y\u2032z\u2032, respectively.Two double mutant cycles can be combined to produce a TMB, which can be used for quantification of cooperative effects. Extensive studies have been performed by Hunter and co-workers using triple mutant box experiments to evaluate cooperativity in non-covalent interactions40 with Amber99sb29, Charmm2730, or Gromos53a631 force fields. The structures of all variants of GB3 and \u0394\u2009+\u2009PHS were produced by FoldX41 with the protein backbone fixed. Each protein was solvated by adding 10.0\u2009\u00c5 TIP3P water42 (or SPC water when the Gromos53a6 force field was used) in a rectangular box, and counter ions were used to neutralize the system. 500,000 steps of energy minimization followed by 1\u2009ns MD simulation at constant pressure (1\u2009atm) and temperature (303\u2009K) were performed to equilibrate the system before the production running. Three 10\u2009ns MD production runs with different random starting velocities were performed with snapshots saved every 50\u2009ps which were then used in the data analysis and error estimation. All backbone heavy atoms are restrained in the equilibrium and production runs. Temperature was regulated by a modified Berendsen thermostat43 and pressure was controlled by the extended ensemble Parrinello-Rahman approach45. The long-range electrostatic interactions were evaluated by the Particle mesh Ewald method47. The nonbonded pair list cutoff was 10\u2009\u00c5 and the list was updated every 10\u2009fs. The LINCS algorithm48 was used to constrain all bonds linked to hydrogen in the protein, whereas the SETTLE algorithm49 was used to constrain bonds and angles of water molecules, allowing a time step of 2\u2009fs. In the energy decomposition analysis, only the interaction energy between the paired residues of C\u2212H\u2219\u2219\u2219\u03c0 or N\u2212H\u2219\u2219\u2219\u03c0 was calculated. The computational interaction energy \u0394\u0394E was calculated by,Exy, \u0394Ex\u2032y, \u0394Exy\u2032, and \u0394Ex\u2032y\u2032 are the x\u2212y interaction energy in the wild type protein, x\u2032\u2212y in the single mutant x\u2032y, x\u2212y\u2032 in the single mutant y\u2032x, and x\u2032\u2212y\u2032 in the double mutant x\u2032y\u2032, respectively. The symbols x and y are the same as those in Eq.\u00a0\u03b5 of 4.0 was used for electrostatic interaction energy calculations. The computational cooperativity energy \u0394\u0394\u0394E was calculated by,y represents the aromatic residue, x and z represent nonaromatic residues, \u2206Exyz, \u2206Ex\u2032yz, \u2206Exy\u2032z, \u2206Exyz\u2032, \u2206Ex\u2032y\u2032z, \u2206Ex\u2032yz\u2032, \u2206Exy\u2032z\u2032, and \u2206Ex\u2032y\u2032z\u2032 are the interaction energy of x\u2212y\u2212z, x\u2032\u2212y\u2212z, x\u2212y\u2032\u2212z, x\u2212y\u2212z\u2032, x\u2032\u2212y\u2032\u2212z, x\u2032\u2212y\u2212z\u2032, x\u2212y\u2032\u2212z\u2032, and x\u2032\u2212y\u2032\u2212z\u2032 in the wild type protein xyz, single mutants x\u2032yz, xy\u2032z and xyz\u2032, double mutants x\u2032y\u2032z, x\u2032yz\u2032, xy\u2032z\u2032 and triple mutants x\u2032y\u2032z\u2032, respectively.MD simulations were performed using the GROMACS 4.5 package50 level. The energy calculations were performed at the MP2/aug-cc-pvtz34 level. All the calculations were done using the Gaussian 09 software51.Two methane and one benzene molecules were built to model the cooperativity of C\u2212H\u2219\u2219\u2219\u03c0\u2219\u2219\u2219C\u2212H\u2219\u2219\u2219\u03c0. The geometries of the two models, MMB and MBM, were optimized at the MP2/6-31\u2009+\u2009GSupplementary information."}
{"text": "Highly significant and positive correlations at phenotypic level were observed in grain weight per hill (0.796), filled grains per panicle (0.702), panicles per hill (0.632), and tillers per hill (0.712) with yield per hectare, while moderate positive correlations were observed in flag leaf length to width ratio (0.348), days to flowering (0.412), and days to maturity (0.544). By contrast, unfilled grains per panicle (-0.225) and plant height (-0.342) had a negative significant association with yield per hectare. Filled grains per panicle (0.491) exhibited the maximum positive direct effect on yield followed by grain weight per hill (0.449), while unfilled grain per panicle (-0.144) had a negative direct effect. The maximum indirect effect on yield per hectare was recorded by the tillers per hill through the panicles per hill. Therefore, tillers per hill, filled grains per panicle, and grain weight per hill could be used as selection criteria for improving grain yield in rice.The associations among yield-related traits and the pattern of influence on rice grain yield were investigated. This evaluation is important to determine the direct and indirect effects of various traits on yield to determine selection criteria for higher grain yield. Fifteen rice genotypes were evaluated under tropical condition at five locations in two planting seasons. The experiment was laid out in a randomized complete block design with three replications across the locations. Data were collected on vegetative and yield components traits. The pooled data based on the analysis of variance revealed that there were significant differences ( Oryza sativa L.) is an important staple food that constitutes a dominant portion of a world standard diet. In spite of its position among the highly rated cereals, the geometric growth rate of the global population has called for improving the current yield for this extremely important cereal. Thus, several methods have been attempted by scientists to combat this perennial problem. Some researchers have attempted nutritional methods such as physiological methods, breeding, and pest and disease control [Rice , and three commercial varieties were used in this study. The commercial varieties of MR219, MR220, and MR253 were developed by the Malaysian Agricultural Research and Development Institute (MARDI) and officially released in 2001. These varieties were the first set of varieties to be developed by means of a direct seeding planting system. The emphasis was on the panicle characteristics, short life cycle (105\u2013111 days), fairly long but strong culms, and being tolerant to blast and bacterial leaf blight. VN121, VN124, and VN001 were developed by Vietnam Atomic Energy Institute (VINATOM), and the main characteristics of these varieties include good ratoon potential and semidwarf stature. Three mutant varieties were obtained from Bangladesh Institute of Nuclear Agriculture (BINA), namely, Binadhan4, Binadhan7, and Iratom. These varieties possess high-tillering capacity but are prone to lodging. Six advanced mutant lines  were promoted from preliminary studies of ion beam irradiation  with an 2, with subplot size of 4 m2 unit for each genotype in each replication. Optimum date for transplanting at each location was followed according to the farmer's schedule.The field trials were conducted in five locations in two different cropping seasons in peninsular Malaysia, namely, Kota Sarang Semut, Seberang Perai, Tanjung Karang, Sekinchan, and Serdang. The locations covered a wide range of environmental conditions differing in management practices (farmers' fields versus research field), water system (supplementary irrigation versus rainfed), and temperatures (moderate and warm climate conditions). The locations were chosen to represent major rice producing areas in Malaysia. In each environment, the experiment was laid out in a randomized complete block design with three replications. Plot size was 342 m\u22121. Phosphorus (in form of triple superphosphate) was applied at 15 days at the rate of 57kg per ha\u22121 and potassium (applied in form of Murate of potash) at 42kg ha\u22121.Fertilizer application was applied following the recommendation by Malaysian Agricultural Research and Development Institute (MARDI) as nitrogen was applied in form of urea in 30, 55, and 75 days after transplanting at 80, 12, and 20 kg per hectare. NPK fertilizer was also applied in triplicate starting on days 15, 55, and 75 after transplanting at the rate of 140, 107, and 50 kg ha\u22121) was estimated from the weight of threshed grains from all panicles in 1.5 \u00d7 1.5 m2, excluding border rows.Five hills were randomly selected for each genotype in each replication to record observations for plant height, flag leaf length to width ratio, days to 50% flowering, days to maturity, tillers per hill, panicles per hill, panicle length, filled grains per panicle, unfilled grains per panicle, 100-grain weight, and grain weight per hill. Yield in ton per hectare . Association of the various characters with yield per hectare was worked out at genotypic and phenotypic levels as described by Kashiani and Saleh . The pheEffects of Vegetative and Yield Component Variables on Yield per Hectarer112=P112+r12P212+r13P312+r14P412+r15P512+r16P612+r17P712+r18P812+r19P912+r110P1012+r111P1112\u2009r212=r21P112+P212+r23P312+r24P412+r25P512+r26P612+r27P712+r28P812+r29P912+r210P1012+r211P1112\u2009r312=r31P112+r32P212+P312+r34P412+r35P512+r36P612+r37P712+r38P812+r39P912+r310P1012+r311P1112\u2009r412=r41P112+r42P212+r43P312+P412+r45P512+r46P612+r47P712+r48P812+r49P912+r410P1012+r411P1112\u2009r512=r51P112+r52P212+r53P312+r54P412+P512+r56P612+r57P712+r58P812+r59P912+r510P1012+r511P1112\u2009r612=r61P112+r62P212+r63P312+r64P412+r65P512+P612+r67P712+r68P812+r69P912+r610P1012+r611P1112\u2009r712=r71P112+r72P212+r73P312+r74P412+r75P512+r76P612+P712+r78P812+r79P912+r710P1012+r711P1112\u2009r812=r81P112+r82P212+r83P312+r84P412+r85P512+r86P612+r87P712+P812+r89P912+r810P1012+r811P1112\u2009r912=r91P112+r92P212+r93P312+r94P412+r95P512+r96P612+r97P712+r98P812+P912+r910P1012+r911P1112\u2009r1012=r101P112+r102P212+r103P312+r104P412+r105P512+r106P612+r107P712+r108P812+r109P912+P1012+r1011P1112\u2009r1112=r111P112+r112P212+r113P312+r114P412+r115P512+r116P612+r117P712+r118P812+r119P912+r1110P1012+P1112.\u2009 The studied traits were further subdivided into two-stage relations: first-order components and second-order components. The first-order components include plant height, flag leaf length to width ratio, days to flowering, days to maturity, and tillers per hill. The second-order components are panicles per hill, panicle lengths, filled grains per panicle, unfilled grains per panicle, 100-grain weight, and grain weight per hill. The cause and effect relationships between the two components were worked out using additional simultaneous equations arranged in matrix notation as indicated in the equations below. Effects of First-Order Components on the Panicles per Hill, Panicle Lengths, Grains per Panicle, Unfilled Grains per Panicle, 100-Grain Weight, and Weight per HillPanicles per Hill\u2009r16\u2009\u2009=\u2009\u2009P16\u2009\u2009+\u2009 r12P26\u2009\u2009+\u2009 r13P36\u2009\u2009+\u2009 r14P46\u2009\u2009+\u2009 r15P56\u2009r26\u2009\u2009=\u2009 r21P16\u2009\u2009+\u2009\u2009P26\u2009\u2009+\u2009 r23P36\u2009\u2009+\u2009 r24P46\u2009\u2009+\u2009 r25P56\u2009r36\u2009\u2009=\u2009 r31P16\u2009\u2009+\u2009 r32P26\u2009\u2009+\u2009 P36 + r34P46\u2009\u2009+\u2009 r35P56\u2009r46\u2009\u2009=\u2009 r41P16\u2009\u2009+\u2009 r42P26\u2009\u2009+\u2009 r43P36\u2009\u2009+\u2009\u2009P46\u2009\u2009+\u2009 r45P56\u2009r56=r51P16\u2009\u2009+\u2009 r52P26\u2009\u2009+\u2009 r53P36\u2009\u2009+\u2009 r54P46\u2009\u2009+\u2009 P56.\u2009Panicle Lengths\u2009r17\u2009\u2009=\u2009\u2009P17\u2009\u2009+\u2009 r12P27\u2009\u2009+\u2009 r13P37\u2009\u2009+\u2009 r14P47\u2009\u2009+\u2009 r15P57\u2009r27\u2009\u2009=\u2009 r21P17\u2009\u2009+\u2009\u2009P27\u2009\u2009+\u2009 r23P37\u2009\u2009+\u2009 r24P47\u2009\u2009+\u2009 r25P57\u2009r37\u2009\u2009=\u2009 r31P17\u2009\u2009+\u2009 r32P27\u2009\u2009+\u2009\u2009P37\u2009\u2009+\u2009 r34P47\u2009\u2009+\u2009 r35P57\u2009r47\u2009\u2009=\u2009 r41P17\u2009\u2009+\u2009 r42P27\u2009\u2009+\u2009 r43P37\u2009\u2009+\u2009\u2009P47\u2009\u2009+\u2009 r45P57\u2009r57\u2009\u2009=\u2009 r51P17\u2009\u2009+\u2009 r52P27\u2009\u2009+\u2009 r53P37\u2009\u2009+\u2009 r54P47\u2009\u2009+\u2009 P57.\u2009Grains per Panicle\u2009r18\u2009\u2009=\u2009\u2009P18\u2009\u2009+\u2009 r12P28\u2009\u2009+\u2009 r13P38\u2009\u2009+\u2009 r14P48\u2009\u2009+\u2009 r15P58\u2009r28\u2009\u2009=\u2009 r21P18\u2009\u2009+\u2009\u2009P28\u2009\u2009+\u2009 r23P38\u2009\u2009+\u2009 r24P48\u2009\u2009+\u2009 r25P58\u2009r38\u2009\u2009=\u2009 r31P18\u2009\u2009+\u2009 r32P28\u2009\u2009+\u2009\u2009P38\u2009\u2009+\u2009 r34P48\u2009\u2009+\u2009 r35P58\u2009r48\u2009\u2009=\u2009 r41P18\u2009\u2009+\u2009 r42P28\u2009\u2009+\u2009 r43P38\u2009\u2009+\u2009\u2009P48\u2009\u2009+\u2009 r45P58\u2009r58\u2009\u2009=\u2009 r51P18\u2009\u2009+\u2009 r52P28\u2009\u2009+\u2009 r53P38\u2009\u2009+\u2009 r54P48\u2009\u2009+\u2009 P58.\u2009Unfilled Grains per Panicle\u2009r19\u2009\u2009=\u2009\u2009P19\u2009\u2009+\u2009 r12P29\u2009\u2009+\u2009 r13P39\u2009\u2009+\u2009 r14P49\u2009\u2009+\u2009 r15P59\u2009r29\u2009\u2009=\u2009 r21P19\u2009\u2009+\u2009\u2009P29\u2009\u2009+\u2009 r23P39\u2009\u2009+\u2009 r24P49\u2009\u2009+\u2009 r25P59\u2009r39\u2009\u2009=\u2009 r31P19\u2009\u2009+\u2009 r32P29\u2009\u2009+\u2009\u2009P39\u2009\u2009+\u2009 r34P49\u2009\u2009+\u2009 r35P59\u2009r49\u2009\u2009=\u2009 r41P19\u2009\u2009+\u2009 r42P29\u2009\u2009+\u2009 r43P39\u2009\u2009+\u2009\u2009P49\u2009\u2009+\u2009 r45P59\u2009r59\u2009\u2009=\u2009 r51P19\u2009\u2009+\u2009 r52P29\u2009\u2009+\u2009 r53P39\u2009\u2009+\u2009 r54P49\u2009\u2009+\u2009 P59.\u2009Unfilled Grains per Panicle\u2009r19\u2009\u2009=\u2009\u2009P19\u2009\u2009+\u2009 r12P29\u2009\u2009+\u2009 r13P39\u2009\u2009+\u2009 r14P49\u2009\u2009+\u2009 r15P59\u2009r29\u2009\u2009=\u2009 r21P19\u2009\u2009+\u2009\u2009P29\u2009\u2009+\u2009 r23P39\u2009\u2009+\u2009 r24P49\u2009\u2009+\u2009 r25P59\u2009r39\u2009\u2009=\u2009 r31P19\u2009\u2009+\u2009 r32P29\u2009\u2009+\u2009\u2009P39\u2009\u2009+\u2009 r34P49\u2009\u2009+\u2009 r35P59\u2009r49\u2009\u2009=\u2009 r41P19\u2009\u2009+\u2009 r42P29\u2009\u2009+\u2009 r43P39\u2009\u2009+\u2009\u2009P49\u2009\u2009+\u2009 r45P59\u2009r59\u2009\u2009=\u2009 r51P19\u2009\u2009+\u2009 r52P29\u2009\u2009+\u2009 r53P39\u2009\u2009+\u2009 r54P49\u2009\u2009+\u2009 P59.\u2009100-Grain Weight\u2009r110\u2009\u2009=\u2009\u2009P110\u2009\u2009+\u2009 r12P210\u2009\u2009+\u2009 r13P310\u2009\u2009+\u2009 r14P410\u2009\u2009+\u2009 r15P510\u2009r210\u2009\u2009=\u2009 r21P110\u2009\u2009+\u2009\u2009P210\u2009\u2009+\u2009 r23P310\u2009\u2009+\u2009 r24P410\u2009\u2009+\u2009 r25P510\u2009r310\u2009\u2009=\u2009 r31P110\u2009\u2009+\u2009 r32P210\u2009\u2009+\u2009\u2009P310\u2009\u2009+\u2009 r34P410\u2009\u2009+\u2009 r35P510\u2009r410\u2009\u2009=\u2009 r41P110\u2009\u2009+\u2009 r42P210\u2009\u2009+\u2009 r43P310\u2009\u2009+\u2009\u2009P410\u2009\u2009+\u2009 r45P510\u2009r510\u2009\u2009=\u2009 r51P110\u2009\u2009+\u2009 r52P210\u2009\u2009+\u2009 r53P310\u2009\u2009+\u2009 r54P410\u2009\u2009+\u2009 P510.\u2009100-Grain Weight\u2009r110\u2009\u2009=\u2009\u2009P110\u2009\u2009+\u2009 r12P210\u2009\u2009+\u2009 r13P310\u2009\u2009+\u2009 r14P410\u2009\u2009+\u2009 r15P510\u2009r210\u2009\u2009=\u2009 r21P110\u2009\u2009+\u2009\u2009P210\u2009\u2009+\u2009 r23P310\u2009\u2009+\u2009 r24P410\u2009\u2009+\u2009 r25P510\u2009r310\u2009\u2009=\u2009 r31P110\u2009\u2009+\u2009 r32P210\u2009\u2009+\u2009\u2009P310\u2009\u2009+\u2009 r34P410\u2009\u2009+\u2009 r35P510\u2009r410\u2009\u2009=\u2009 r41P110\u2009\u2009+\u2009 r42P210\u2009\u2009+\u2009 r43P310\u2009\u2009+\u2009\u2009P410\u2009\u2009+\u2009 r45P510\u2009r510\u2009\u2009=\u2009 r51P110\u2009\u2009+\u2009 r52P210\u2009\u2009+\u2009 r53P310\u2009\u2009+\u2009 r54P410\u2009\u2009+\u2009 P510.\u2009Grain Weight per Hill\u2009r111\u2009\u2009=\u2009P111\u2009\u2009+\u2009 r12P211\u2009\u2009+\u2009 r13P311\u2009\u2009+\u2009 r14P411\u2009\u2009+\u2009 r15P511r211\u2009\u2009=\u2009 r21P111\u2009\u2009+\u2009\u2009P211\u2009\u2009+\u2009 r23P311\u2009\u2009+\u2009 r24P411\u2009\u2009+\u2009 r25P511\u2009r311\u2009\u2009=\u2009 r31P111\u2009\u2009+\u2009 r32P211\u2009\u2009+\u2009\u2009P311\u2009\u2009+\u2009 r34P411\u2009\u2009+\u2009 r35P511\u2009r411\u2009\u2009=\u2009 r41P111\u2009\u2009+\u2009 r42P211\u2009\u2009+\u2009 r43P311\u2009\u2009+\u2009\u2009P411\u2009\u2009+\u2009 r45P511\u2009r511\u2009\u2009=\u2009 r51P111\u2009\u2009+\u2009 r52P211\u2009\u2009+\u2009 r53P311\u2009\u2009+\u2009 r54P411\u2009\u2009+\u2009 P51.\u2009Effects of Second-Order Components on Yield per Hectarer612\u2009\u2009=\u2009\u2009P612\u2009\u2009+\u2009 r67P712\u2009\u2009+\u2009 r68P812\u2009\u2009+\u2009 r69P912\u2009\u2009+\u2009 r610P1012\u2009\u2009+\u2009 r611P1112\u2009r712\u2009\u2009=\u2009 r76P612\u2009\u2009+\u2009\u2009P712\u2009\u2009+\u2009 r78P812\u2009\u2009+\u2009 r79P912\u2009\u2009+\u2009 r710P1012\u2009\u2009+\u2009 r711P1112\u2009r812\u2009\u2009=\u2009 r86P612\u2009\u2009+\u2009 r87P712\u2009\u2009+\u2009\u2009P812\u2009\u2009+\u2009 r89P912\u2009\u2009+\u2009 r810P1012\u2009\u2009+\u2009 r811P1112\u2009r912\u2009\u2009=\u2009 r96P612\u2009\u2009+\u2009 r97P712\u2009\u2009+\u2009 r98P812\u2009\u2009+\u2009\u2009P912\u2009\u2009+\u2009 r910P1012\u2009\u2009+\u2009 r911P1112\u2009r1012\u2009\u2009=\u2009 r106P612\u2009\u2009+\u2009 r107P712\u2009\u2009+\u2009 r108P812\u2009\u2009+\u2009 r109P912\u2009\u2009+\u2009\u2009P1012\u2009\u2009+\u2009 r1011P1112\u2009r1112\u2009\u2009=\u2009 r116P612\u2009\u2009+\u2009 r117P712\u2009\u2009+\u2009 r118P812\u2009\u2009+\u2009 r119P912\u2009\u2009+\u2009 r1110P1012\u2009\u2009+P1112.\u2009The data presented in In this study, the phenotypic and genotypic correlation coefficient of vegetative, yield, and yield component are separated for a clear understanding as shown in The correlation coefficient of phenotypic characters ranges from 0.026 to 0.818 while the genotypic character correlation coefficient ranges from 0.04 to 0.919. This indicates that there is higher magnitude at the genotypic level in most cases as compared with the corresponding phenotypic level. The yield per hectare shows a strong, positive, and highly significant correlation with grain weight per hill, tillers per hill, filled grains per panicle, and panicles per hill, respectively. Selection based on these four components is effective due to their equal contribution towards grain yield increment.Grain yield is considered as the artefact of all the contributory traits , and the correlation coefficient of these contributory factors with final yield are partitioned into direct and indirect effects as presented in The present study revealed a significant interrelationship among various vegetative and yield components. These traits define the limitation to yield per hectare and the component characters through the direct and indirect effects as a result of interrelationships between them. Therefore, the use of path coefficient analysis investigates the direct and indirect relationships among the component characters through the partitioning of correlation coefficients . The patThe vegetative traits were grouped as the first-order component which includes the following: plant height, flag leaf length to width ratio, days to flowering, days to maturity, and tillers per hill. The second-order component was regarded as yield component traits which were regarded as the principal yield determining factors in rice, and these comprise panicles per hill, panicle length, filled grains per panicle, unfilled grains per panicle, 100-grain weight, and grain weight per hill. The interrelation between these two components is presented in Tables The path of the influence of first-order component on panicles per hill  revealedThe path analysis of second-order component on yield was presented in The critical analysis of path coefficient analysis and partitioning of correlation reveals that tillers per hill and grain weight per hill possessed positive direct effect and positive association with yield per hectare. Selection for the improvement of grain yield can be efficient if it is based on tillers per hill and grain weight per hill because of their contribution directly towards grain yield. Breeders in these areas should, therefore, develop early maturing genotypes focusing on tillers per hill and grain weight per hill for improving the grain yield per plant in both rainfed and irrigated area."}