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1201.4151

i

despite much recent progress @xcite , background - independent approaches to quantum gravity face several open challenges . these concern : ( i ) the definition of the quantum dynamics of the fundamental degrees of freedom of spacetime that they identify , and the full control over it ; ( ii ) the recovery of an effective description in terms of a smooth spacetime and geometry , once the dynamics is somehow defined , and in particular when the fundamental degrees of freedom are not continuous geometric data ; ( iii ) the contact with the effective dynamics of general relativity and quantum field theory , and with phenomenology . an example is given by loop quantum gravity ( lqg ) , a background - independent framework aiming to quantize the gravitational degrees of freedom in a nonperturbative way @xcite . to this purpose , a canonical quantization scheme is employed where the constraints are written in terms of the densitized triad and of the ashtekar barbero connection . the end result at the kinematical level is a hilbert space of ( spin network ) states associated with graphs embedded in the spatial manifold and labeled by algebraic data ( lorentz group elements or corresponding representation labels ) . as in any canonical scheme , while geometry is fully dynamical , the topology of the universe is fixed by construction , at least at the beginning . in general , however , one may ask whether it is possible to build a quantum theory inclusive of topology change or , in other words , if one can envisage an interacting multiverse scenario obeying a set of quantum rules . since the degrees of freedom of a single universe are already fields , eventually to be quantized , such a scenario is sometimes said to be one of `` third quantization . '' this can be achieved , at least at a formal level , by defining a field theory over the space of geometries , for given spatial topology @xcite . other ( albeit inconclusive ) arguments from a canonical quantum gravity perspective in favor of going to a `` third - quantization '' setting were also offered in @xcite . beside the issue of topology change , the main difficulty faced by the lqg approach is the complete definition of the quantum dynamics and the proof that the resulting theory leads back to einstein s gravity in an appropriate limit . a tentative but complete definition of the quantum dynamics of spin network states is obtained , via spin foam models @xcite ( a covariant definition of lqg dynamics ) , by embedding lqg states into the larger framework of group field theories ( gft ) @xcite , in turn strictly related to tensor models @xcite . these are quantum field theories on group manifolds whose states are indeed spin networks and whose feynman amplitudes are spin - foam models . this embedding has several advantages , from the lqg point of view . first of all , as said , it provides a complete definition of the quantum dynamics . second , it defines such dynamics as the superposition of interaction processes ( creation and annihilation ) of spin - network vertices , forming complexes of arbitrary topology , such that topology is naturally made dynamical ; it provides , in other words , a sort of local field - quantization scheme @xcite . third , the field - theory framework offers powerful mathematical and conceptual tools for tackling the issue of the continuum limit and of the extraction of effective dynamics for better contact with phenomenology . in doing so , however , one has to abandon the familiar framework of canonical quantization of a classical ( and local ) field theory of gravitation , and is forced to face new types of conceptual and mathematical difficulties . this program is just as ambitious as the original lqg one , if not more , and is difficult to realize in a complete and rigorous way , despite many recent advances . toy models inspired by the full theory then become very important . in fact , they fulfill three main purposes : ( i ) they offer a simplified testing ground for ideas and techniques developed in the full theory ; ( ii ) as such , they also have an important pedagogical value ; ( iii ) they may represent , in principle , an effective , approximate framework to which the full quantum dynamics may reduce , in some limit , and thus they may be directly applicable to phenomenological studies . obviously , due to their simplicity , one should be cautious in interpreting the result obtained in the context of such toy models as truly physical , and their validity can be assessed only once the relation between toy model and full theory has been understood . an important type of simplified scenario has been developed in the context of lqg , in a symmetry - reduced setting of interest for cosmology . in fact , in order to understand certain features of loop quantum gravity , one often resorts to a minisuperspace model , loop quantum cosmology ( lqc ) , where degrees of freedom are drastically reduced @xcite . in a pure friedmann robertson walker ( frw ) universe filled with a massless free scalar field , the classical and quantum dynamics of the universe as a whole can be described by the same formalism used for a free particle . in particular , the path integral is well defined @xcite and two - point correlation functions admit the usual classification @xcite . by now , a wealth of interesting results have been obtained in this context @xcite . given this analogy with the free particle , it is all the more natural to ask oneself if one can construct a sensible `` interaction '' among frw universes and , once this is done , to change the interpretation of the two - point function from particle transition amplitude to field propagator as in the usual field quantization . one would then write down a field theory on minisuperspace , to obtain a field - quantized lqc framework . another way to see the same field theory would then be as a toy model for group field theory , in which many of the difficult features of the latter are absent due to the global nature of the formulation and to the simplification provided by symmetry reduction , but where some ideas and techniques can still be applied . as with any toy model , one would then use it as a pedagogical testing ground and keep it available as a possible effective description of the full theory . we propose such a field theory for ( loop ) quantum cosmology in this paper , with the above motivations . the presentation is organized as follows . we review some basic features of lqc in sec . [ loqc ] , but the wheeler dewitt case is also easily recovered . in sec . [ gufc ] the field theory is defined by promoting the quantum hamiltonian constraint to the kinetic operator of a ( real ) scalar field @xmath0 on minisuperspace . we analyze the relation between different kinetic operators and the gauge choice , with particular focus on exactly solvable free theories . we discuss the various possible choices for the interaction term . following this general definition , we move on to analyze some consequences of the formalism . we analyze the free propagator of the theory first , corresponding to the evolution of a single universe ( sec . [ fft ] ) . we show how the embedding into a field - theory setting has immediate interesting consequences also for the single - universe dynamics . then , we consider how the presence of interactions affects this single - universe evolution . approximating the interaction as a mean - field term , we find an effective equation linear in the field @xmath0 , correcting the hamiltonian quantum constraint equation by an extra term ( sec . [ mfa ] ) . the latter mixes , in general , gravitational and matter degrees of freedom , and its exact form depends on the chosen initial interaction as well as on the mean - field configuration considered . we conclude with a discussion of other possible applications of this formalism .

following the idea of a field quantization of gravity as realized in group field theory , we construct a minisuperspace model where the wavefunction of canonical quantum cosmology ( either wheeler dewitt or loop quantum cosmology ) is promoted to a field , the coordinates are minisuperspace variables , the kinetic operator is the hamiltonian constraint operator , and the action features a nonlinear and possibly nonlocal interaction term . we discuss free - field classical solutions , the quantum propagator , and a mean - field approximation linearizing the equation of motion and augmenting the hamiltonian constraint by an effective term mixing gravitational and matter variables . depending on the choice of interaction ,

following the idea of a field quantization of gravity as realized in group field theory , we construct a minisuperspace model where the wavefunction of canonical quantum cosmology ( either wheeler dewitt or loop quantum cosmology ) is promoted to a field , the coordinates are minisuperspace variables , the kinetic operator is the hamiltonian constraint operator , and the action features a nonlinear and possibly nonlocal interaction term . we discuss free - field classical solutions , the quantum propagator , and a mean - field approximation linearizing the equation of motion and augmenting the hamiltonian constraint by an effective term mixing gravitational and matter variables . depending on the choice of interaction , this can reproduce , for example , a cosmological constant , a scalar - field potential , or a curvature contribution .

1510.02305

i

consider a communication network , which is a finite directed acyclic multigraph with each edge representing a noiseless transmission channel of unit capacity . there is a set of source nodes , each of which independently generates a set of data symbols belonging to a base field and transmits the data symbols simultaneously along the network . there is another set of receiver nodes , each of which attempts to recover a certain subset of source data symbols . a network is _ linearly solvable _ over a base field gf(@xmath1 ) if there is a linear network coding scheme , which encodes every outgoing symbol of a node as a gf(@xmath1)-linear combination of the incoming data symbols to this node , so that every receiver can recover its desired source data symbols at the same time . the algebraic structure of a base field is closely related to the linear solvability of a network . when a network is ( single - source ) multicast , _ i.e. _ , every receiver attempts to recover the same set of data symbols transmitted from a unique source , the fundamental theorem of linear network coding @xcite guarantees the existence of a linear solution when the base field is sufficiently large . since the connection of field size on the linear solvability of a multicast network was revealed , there have been extensive studies on the field size requirement and efficient construction of a linear solution [ 2 - 12 ] . among them , the best known explicit sufficient condition for a multicast network to be linearly solvable over gf(@xmath1 ) is @xmath16 , where @xmath17 is the number of receivers @xcite@xcite . moreover , the combination networks @xcite@xcite constitute the only investigated class of multicast networks with an explicit linear solvability characterization that matches algebraic identities of gf(@xmath1 ) and topological parameters of the network . specifically , an @xmath18-combination network , as depicted in fig . [ fig : combination_network ] , is known to be linearly solvable over gf(@xmath1 ) if and only if @xmath19 , where @xmath20 is the topological parameter for the number of layer-3 nodes . without the restriction of considering a multicast network , more algebraic identities inherited in a field other than size have been revealed to affect the linear solvability of a network . in @xcite , two networks were designed , one of which is linearly solvable only over a field with even characteristic , while the other is linearly solvable only over a field with odd characteristic . it is further unveiled in @xcite that for an arbitrary finite or co - finite set @xmath21 of prime numbers , a network can be constructed such that the network is linearly solvable over some field of characteristic @xmath7 for every @xmath7 in @xmath21 , whereas it is not linearly solvable over any field whose characteristic is not in @xmath21 . meanwhile , whether a set of polynomials with integer coefficients has common roots is connected to a network linear solvability problem . specifically , a general method was proposed in @xcite to associate a polynomial set with a network , such that the polynomials have common roots over a field if and only if the corresponding network has a linear solution over the same field . in @xcite , analogous equivalence was also established between the existence of common roots over a finite field for a polynomial set and the existence of a linear solution over the same field for an associated _ sum - network _ , in which all receivers attempt to recover the sum of all source data symbols . however , when attention is only paid to multicast networks , these results are not applicable anymore . first , since a multicast network is linearly solvable over a large enough field , it is linearly solvable over some field of characteristic @xmath7 for every prime @xmath7 . moreover , the general method proposed in @xcite and @xcite can not necessarily construct a multicast network from a polynomial set . if the field size were the only algebraic factor that affects the linear solvability of a multicast network , then all multicast networks would conceivably share a property @xmath22 , where @xmath23 refers to the minimum field size for the existence of a linear solution and @xmath24 refers to the maximum field size for the non - existence of a linear solution ( @xmath24 is set to 1 if the network is linearly solvable over every field . ) the first few exemplifying networks with the special property @xmath25 were not discovered until a rather recent work @xcite . as pointed out in @xcite , there are fundamental reasons underlying the intriguing observation that @xmath25 is possible for multicast networks : not only the field size @xmath1 , but also _ the order of proper subgroups in the multiplicative group _ gf(@xmath1)@xmath26 = gf(@xmath1)@xmath27 of gf(@xmath1 ) plays an important role in the linear solvability over gf(@xmath1 ) . however , even though it was further observed in @xcite that all these newly designed networks with @xmath25 share a common topological structure , the intrinsic impact of multiplicative subgroup orders of a field on the linear solvability of multicast networks remains unclear . in this paper , we explicate in depth how multiplicative subgroups in gf(@xmath1 ) can affect the linear solvability of multicast networks over gf(@xmath1 ) , as outlined as follows . * in sec . [ sec : general_framework ] , we construct a general class @xmath0 of layered multicast networks , which subsumes all multicast networks presented in @xcite with @xmath28 as special instances . after deriving a generalized cauchy - davenport theorem , we proceed to construct an explicit formula for the linear solvability of networks in @xmath0 over gf(@xmath1 ) . besides the topological parameters of @xmath0 , the concise formula involves the associated coset numbers of a multiplicative subgroup in gf(@xmath1 ) . it turns out to be the first that matches the topological structure of a multicast network and algebraic identities of a field other than size . * in sec . [ sec : subgroup_order_criterion ] , based on the general characterization of @xmath0 , we further formulate a subgroup order criterion for a pair of finite fields . as long as ( gf(@xmath1 ) , gf(@xmath2 ) ) satisfies the subgroup order criterion , where @xmath3 is possible , we can establish an instance in @xmath0 that is linearly solvable over gf(@xmath1 ) but not over gf(@xmath2 ) . [ sec : establish_instances ] , as an application of the subgroup order criterion , we are able to establish _ infinitely many _ multicast networks in @xmath0 with @xmath28 . as intriguing instances , for every @xmath29 , we can establish a multicast network with @xmath30 and @xmath31 for some @xmath32 . this proves that @xmath33 can tend to infinity for a multicast network . moreover , for _ arbitrary _ distinct primes @xmath7 and @xmath8 , there are _ infinitely many _ @xmath9 and @xmath10 such that an instance in @xmath0 can be found to be linearly solvable over gf(@xmath11 ) but _ not _ over gf(@xmath12 ) with @xmath13 . [ sec : construction_network_prescribed_q_min ] , based on the general characterization of @xmath0 , a new procedure is proposed to construct a multicast network with @xmath23 equal to any prescribed prime power @xmath15 , and the constructed network has a smaller size compared with the @xmath34-combination network , which has @xmath35 too . to summarize , we systematically develop a framework to reflect the intrinsic impact of multiplicative subgroups on the linear solvability of a multicast network . a number of new results on the comparison between @xmath23 and @xmath24 are also subsequently deduced . our findings suggest that a `` matching '' between the algebraic structure of a base field and the topological structure of a multicast network is necessary for the existence of a linear solution to the multicast network .

for a ( single - source ) multicast network , the size of a base field is the most known and studied algebraic identity that is involved in characterizing its linear solvability over the base field . in this paper , we design a new class of multicast networks and obtain an explicit formula for the linear solvability of these networks , which involves the associated coset numbers of a multiplicative subgroup in a base field . the concise formula turns out to be the first that matches the topological structure of a multicast network and algebraic identities of a field other than size . in particular , i ) for every , an instance in can be found linearly solvable over gf( ) but _ not _ over gf( ) , and ii ) for arbitrary distinct primes and , there are infinitely many and such that an instance in can be found linearly solvable over gf( ) but _ not _ over gf( ) with . on the other hand , the construction of also leads to a new class of multicast networks with nodes and edges , where is the minimum field size for linear solvability of the network . network coding , multicast , linear solution , multiplicative subgroup , coset , generalized cauchy - davenport theorem .

for a ( single - source ) multicast network , the size of a base field is the most known and studied algebraic identity that is involved in characterizing its linear solvability over the base field . in this paper , we design a new class of multicast networks and obtain an explicit formula for the linear solvability of these networks , which involves the associated coset numbers of a multiplicative subgroup in a base field . the concise formula turns out to be the first that matches the topological structure of a multicast network and algebraic identities of a field other than size . it further facilitates us to unveil _ infinitely many _ new multicast networks linearly solvable over gf( ) but not over gf( ) with , based on a subgroup order criterion . in particular , i ) for every , an instance in can be found linearly solvable over gf( ) but _ not _ over gf( ) , and ii ) for arbitrary distinct primes and , there are infinitely many and such that an instance in can be found linearly solvable over gf( ) but _ not _ over gf( ) with . on the other hand , the construction of also leads to a new class of multicast networks with nodes and edges , where is the minimum field size for linear solvability of the network . network coding , multicast , linear solution , multiplicative subgroup , coset , generalized cauchy - davenport theorem .

0707.2927

i

an _ extremal element _ of a lie algebra @xmath4 over a field @xmath1 of characteristic unequal to @xmath2 is an element @xmath5 for which @xmath6\subseteq kx$ ] . a _ sandwich element _ is an @xmath7 satisfying the stronger condition @xmath6=0 $ ] . the definition of extremal elements in characteristic @xmath2 is more involved , which is one reason for restricting ourselves to characteristics unequal to @xmath2 here . extremal elements and sandwich elements play important roles in both classical and modern lie algebra theory . in complex simple lie algebras , or their split analogues over other fields , extremal elements are precisely the elements that are long - root vectors relative to some maximal torus . sandwich elements are used in the classification of simple lie algebras in small characteristics @xcite ; they occur in the modular lie algebras of cartan type , such as the witt algebras . sandwich elements were originally introduced in relation with the restricted burnside problem @xcite . an important insight for the resolution of this problem is the fact that a lie algebra generated by finitely many sandwich elements is necessarily finite - dimensional . while this fact was first only proved under extra assumptions , in @xcite it is proved in full generality . we will use this result in what follows . the prominence of extremal elements in the work of kostrikin and zelmanov and in modular lie algebra theory led to the natural problem of describing all lie algebras generated by a fixed number of extremal elements @xcite . example suppose that we want to describe all lie algebras @xmath4 generated by two extremal elements @xmath8 and @xmath9 . since @xmath10 $ ] is a scalar multiple @xmath11 of @xmath8 and @xmath12=-[y,[y , x]]$ ] is a scalar multiple @xmath13 of @xmath9 , @xmath14 is spanned by @xmath15 $ ] . there may be linear dependencies between these elements , but let us assume that they are linearly independent . then @xmath16=[y,[x,[x , y]]]=[[y , x],[x , y]]+[x,[y,[x , y]]]=0-b[x , y]=b[y , x],\ ] ] and since we have assumed that @xmath17 \neq 0 $ ] , we find that @xmath18 . hence three - dimensional lie algebras with a distinguished pair of extremal generators are parameterised by the single number @xmath19 . moreover , all algebras with @xmath20 are mutually isomorphic and isomorphic to the split simple lie algebra of type @xmath21 , while the algebra with @xmath22 is nilpotent and isomorphic to the three - dimensional heisenberg algebra . this is a prototypical example of our results . the next smallest case of three generators is treated in @xcite and also by our results below . there the generic lie algebra is split of type @xmath23 and more interesting degenerations exist . we now generalise and formalise this example to the case of more generators , where we also allow for the flexibility of prescribing that certain generators commute . thus let @xmath0 be a finite simple graph without loops or multiple edges . let @xmath24 be the vertex set of @xmath0 and denote the neighbour relation by @xmath25 . fixing a field @xmath1 of characteristic unequal to @xmath2 , we denote by @xmath26 the quotient of the free lie algebra over @xmath1 generated by @xmath24 modulo the relations @xmath27=0 \text { for all } x , y \in \pi \text { with } x\not \sim y.\ ] ] so @xmath26 depends both on @xmath0 and on @xmath1 , but we will not make this dependence explicit in the notation . we write @xmath28 for the space of all @xmath1-linear functions @xmath29 . for every @xmath30 , also written @xmath31 , we denote by @xmath32 the quotient of @xmath26 by the ideal @xmath33 generated by the ( infinitely many ) elements @xmath34-f_x(y)x,\ x \in \pi , y \in { \mathcal{f}}.\ ] ] by construction @xmath32 is a lie algebra generated by extremal elements , corresponding to the vertices of @xmath0 , which commute when they are not connected in @xmath0 . the element @xmath35 is a parameter needed to express the extremality of @xmath36 . if @xmath0 is not connected , then both @xmath26 and @xmath32 naturally split into direct sums over all connected components of @xmath0 , so it is no restriction to assume that @xmath0 is connected ; we will do so throughout this paper . in the lie algebra @xmath37 the elements of @xmath24 map to sandwich elements , hence by @xcite this lie algebra is finite - dimensional . for general @xmath30 it turns out that @xmath38 ; see @xcite or the proof of theorem [ thm : variety ] below . it is therefore natural to focus on the lie algebras @xmath32 of the maximal possible dimension @xmath39 . this leads us to define the set @xmath40 the parameter space for all maximal - dimensional lie algebras of the form @xmath32 . example in the two - generator case above @xmath0 is the graph with two vertices joined by an edge . the sandwich algebra @xmath37 is the three - dimensional heisenberg algebra , and the condition that @xmath41 corresponds to our assumption above that @xmath15 $ ] be linearly independent . this linear independence forced the parameters @xmath19 and @xmath42 to be equal . here @xmath3 is the affine line with coordinate @xmath19 . all lie algebras corresponding to points @xmath20 are mutually isomorphic . our first main result is that @xmath3 carries a natural structure of an affine algebraic variety . to specify this structure we note that @xmath43 is a homogeneous ideal relative to the natural @xmath44-grading that @xmath26 inherits from the free lie algebra generated by @xmath24 . [ thm : variety ] the set @xmath3 is naturally the set of @xmath1-rational points of an affine variety of finite type defined over @xmath1 . this variety can be described as follows . fix any finite - dimensional homogeneous subspace @xmath45 of @xmath26 such that @xmath46 . then the restriction map @xmath47 maps @xmath3 injectively onto the set of @xmath1-rational points of a closed subvariety of @xmath48 . this yields a @xmath1-variety structure on @xmath3 which is independent of the choice of @xmath45 . we prove this theorem in section [ sec : moduli ] . in section [ sec : sandwich ] we first derive some relations between the sandwich algebra @xmath37 and the positive part of the _ complex _ kac - moody algebra of type @xmath0 . then we determine @xmath37 explicitly in the case where @xmath0 is a simply laced dynkin diagram of finite or affine type ; by this we mean any of the diagrams in figure [ fig : diagrams ] without or with vertex @xmath49 , respectively . see theorems [ thm : finite ] and [ thm : affine ] . in section [ sec : parameter ] we study the variety @xmath3 . after some observations for general @xmath0 , we again specialise to the diagrams of figure [ fig : diagrams ] . for these we prove that @xmath3 is an affine space , and that for @xmath50 in an open dense subset of @xmath3 the lie algebra @xmath32 is isomorphic to a fixed lie algebra ; see theorems [ thm : genericfinite ] and [ thm : genericaffine ] . the latter of these theorems is our second main result , and we paraphrase it here . let @xmath0 be any of the simply laced dynkin diagrams of affine type in figure [ fig : diagrams ] , let @xmath51 be the finite - type diagram obtained by removing vertex @xmath49 from @xmath0 , and let @xmath52 be the edge set of @xmath0 . then @xmath3 is isomorphic to the affine space of dimension @xmath53 over @xmath1 , and for @xmath50 in an open dense subset of @xmath3 the lie algebra @xmath32 is isomorphic to the chevalley algebra of type @xmath51 . by the _ chevalley algebra of type @xmath51 _ we mean the lie algebra obtained by tensoring a certain @xmath54-form of the complex simple lie algebra of type @xmath51 with the field @xmath1 ; see subsection [ ssec : finitesandwich ] for details . this lie algebra is often simple , but not always ; see ( * ? ? ? * chapter 4 ) and @xcite . we conclude the paper with remarks on applications and related work in section [ sec : notes ] .

for any finite graph and any field of characteristic unequal to we construct an algebraic variety over whose-points parameterise-lie algebras generated by extremal elements , corresponding to the vertices of the graph , with prescribed commutation relations , corresponding to the non - edges . after that , we study the case where is a connected , simply laced dynkin diagram of finite or affine type . we prove that is then an affine space , and that all points in an open dense subset of parameterise lie algebras isomorphic to a single fixed lie algebra . if is of affine type , then this fixed lie algebra is the split finite - dimensional simple lie algebra corresponding to the associated finite - type dynkin diagram .

for any finite graph and any field of characteristic unequal to we construct an algebraic variety over whose-points parameterise-lie algebras generated by extremal elements , corresponding to the vertices of the graph , with prescribed commutation relations , corresponding to the non - edges . after that , we study the case where is a connected , simply laced dynkin diagram of finite or affine type . we prove that is then an affine space , and that all points in an open dense subset of parameterise lie algebras isomorphic to a single fixed lie algebra . if is of affine type , then this fixed lie algebra is the split finite - dimensional simple lie algebra corresponding to the associated finite - type dynkin diagram . this gives a new construction of these lie algebras , in which they come together with interesting degenerations , corresponding to points outside the open dense subset . our results may prove useful for recognising these lie algebras .

1306.3706

i

in recent years , statisticians , scientists and engineers are increasingly analyzing enormous data sets . when data sets grow sufficiently large , computational costs may play a major role in the analysis , potentially constraining our choice of methodology or the number of data points we can afford to process . computational savings can translate directly to statistical gains if they : enable us to experiment with and prototype a variety of models , instead of trying only one or two , allow us to refit our models more often to adapt to changing conditions , allow for cross - validation , bagging , boosting , bootstrapping or other computationally intensive statistical procedures or open the door to using more sophisticated statistical techniques on a compressed data set . @xcite discuss the tradeoffs arising when we adopt this point of view . one simple manifestation of these tradeoffs is that we may run out of computing resources before we run out of data , in effect making the sample size @xmath4 a function of the efficiency of our fitting method . class imbalance is pervasive in modern classification problems and has received a great deal of attention in the machine learning literature [ @xcite ] . it can come in two forms : : : _ marginal imbalance_. one of the classes is quite rare ; for instance , @xmath5 . such imbalance typically occurs in data sets for predicting click - through rates in online advertising , detecting fraud or diagnosing rare diseases . : : _ conditional imbalance_. for most values of the features @xmath6 , the response @xmath7 is very easy to predict ; for instance , @xmath8 but @xmath9 . for example , such imbalance might arise in the context of email spam filtering , where well - trained classifiers typically make very few mistakes . both or neither of the above may occur in any given data set . the machine learning literature on class imbalance usually focuses on the first type , but the second type is also common . if , for example , our data set contains one thousand or one million negative examples for each positive example , then many of the negative data points are in some sense redundant . typically in such problems , the statistical noise is primarily driven by the number of representatives of the rare class , whereas the total size of the sample determines the computational cost . if so , we might hope to finesse our computational constraints by subsampling the original data set in a way that enriches for the rare class . such a strategy must be implemented with care if our ultimate inferences are to be valid for the full data set . this article proposes one such data reduction scheme , local case - control sampling , for use in fitting logistic regression models . the method requires one parallelizable scan over the full data set and yields a potentially much smaller subsample containing roughly half of the information found in the original data set . the simplest way to reduce the computational cost of a procedure is to subsample the data before doing anything else . however , uniform subsampling from an imbalanced data set is inefficient since it fails to exploit the unequal importance of the data points . case - control sampling sampling uniformly from each class but adjusting the mixture of the classes is a more promising approach . this procedure originated in epidemiology , where the positive examples ( cases ) are typically diseased patients and negative examples ( controls ) are disease - free [ @xcite ] . often , an equal number of cases and controls are sampled , resulting in a subsample with no marginal imbalance , and costly measurements of predictor variables are only made for selected patients [ @xcite ] . this method is useful in our context as well , since a logistic regression model fitted on the subsample can be converted to a valid model for the original population via a simple adjustment to the intercept [ @xcite ] . however , standard case - control sampling still may not make most efficient use of the data . for instance , it does nothing to exploit conditional imbalance in a data set that is marginally balanced . even with some marginal imbalance , a control that looks similar to the cases is often more useful for discrimination purposes than one that is obviously not a case . we propose a method , local case - control sampling , which attempts to remedy imbalance _ locally _ throughout the feature space . given a pilot estimate @xmath10 of the logistic regression parameters , local case - control sampling preferentially keeps data points for which @xmath7 is surprising given @xmath6 . specifically , if @xmath11 , we accept @xmath12 with probability @xmath13 , the @xmath14 residual of the pilot model . in the presence of extreme marginal or conditional imbalance , these errors will generally be quite small and the subsample can be many orders of magnitude smaller than the full data set . just as with case - control sampling , we can fit our model to the subsample and make an equally simple correction to obtain an estimate for the original data set . when the logistic regression model is correctly specified and the pilot is consistent and independent of the data , the asymptotic variance of the local case - control estimate is exactly twice the variance of a logistic regression fit on the ( potentially much larger ) full data set . this factor of two improves to @xmath1 if we accept with probability @xmath15 and weight accepted points by a factor of @xmath16 . for example , if @xmath17 then the variance of the subsampled estimate is only 20% greater than the variance of the full - sample mle . the subsample we take with @xmath2 is no more than @xmath18 times larger than the subsample for @xmath19 , and for data sets with large imbalance is roughly @xmath3 times as large . local case - control sampling also improves on the bias of standard case - control sampling . when the logistic regression model is misspecified , case - control sampling is in general inconsistent for the risk minimizer in the original population . by contrast , local case - control sampling is always consistent given a consistent pilot , and is also asymptotically unbiased when the pilot is . sections [ secsimulations ] and [ secspam ] present empirical results demonstrating the advantages of our approach in simulations and on the yahoo ! webspam data set . our setting is that of predictive classification : we are given @xmath4 independent and identically distributed observations , each consisting of predictors @xmath20 and a binary response @xmath21 , with joint probability measure @xmath22 . for our purposes , we assume the predictors are mapped into some real covariate vector space , so that @xmath23 our aim is to learn the function @xmath24 or equivalently to learn @xmath25 which could be infinite for some @xmath26 . a linear logistic regression model assumes @xmath27 is linear in @xmath26 ; that is , @xmath28 where @xmath29 . this is less of a restriction than it might seem , since @xmath26 may represent a very large basis expansion of some smaller set of `` raw '' features . nevertheless , in the real world , @xmath27 is unlikely to satisfy our parametric model for any given basis @xmath26 . when the model is misspecified , we can still view logistic regression as an m - estimator with convex loss equal to the negative log - likelihood for a single data set : @xmath30 as an m - estimator , under general conditions logistic regression in large samples will converge to the minimizer of the population risk @xmath31 [ @xcite ] . that is , @xmath32 converges to the population maximizer of the expected log - likelihood @xmath33.\end{aligned}\ ] ] if @xmath34 for some @xmath35 , then @xmath36 ; otherwise @xmath37 is the best linear approximation to @xmath27 in the sense of ( [ eqnpoprisk ] ) . for a misspecified model , @xmath38 can not possibly converge to @xmath27 no matter what sampling scheme or estimation procedure we use , or how much data we obtain . consistency , then , will mean that @xmath39 . model misspecification is ubiquitous in real - world applications of regression methods . for reasons of exposition , the misspecification always takes a simple form in our simulations , for example , in example [ ex1 ] there are two binary predictors , and we would have correct specification if only we added one interaction but in the real world it usually is neither possible nor even desirable to expand the feature basis until the model is correctly specified . for instance , if @xmath40 , then there are @xmath41500,500 quadratic terms . even if we included all those terms as features , we would still be missing cubic terms , quartic terms , and so on . some kinds of misspecification are milder than others , and some are easier to find and fix than others . seeking better - specified models ( without adding too much model complexity ) is a worthy goal , but realistically perfect specification is unattainable . our goal , then , is to speed up computation while still obtaining a good estimate of @xmath0 , the population logistic regression parameters . as we will see , standard case - control sampling achieves the first goal , but may fail at the second . recent years have seen substantial work on classification in imbalanced data sets . see @xcite and @xcite for surveys of machine learning efforts on this problem . many of the methods proposed involve some form of undersampling the majority class , oversampling the minority class , or both . @xcite examined the limit of marginally imbalanced logistic regression and proved it is equivalent to fitting an exponential family model to the minority class . one recurring theme is to preferentially sample negative examples that lie near positive examples in feature space . for example , @xcite propose selecting majority - class examples whose average distance to its three nearest minority examples is smallest . our method has a similar flavor since the probability of sampling a negative example @xmath42 is @xmath43 , which is large when the features @xmath26 are similar to those characteristic of positive examples . our proposal lies more in the tradition of the epidemiological case - control sampling literature . in particular , case - control sampling within several categorical strata has been studied by @xcite . typically , the strata are based on easy - to - measure screening variables available for a wide population , with more laborious - to - collect variables being measured on the sampled subjects . @xcite discuss survey calibration methods for efficient regression in two - stage sampling schemes , which are interesting but too computationally intensive for our purposes here .

for classification problems with significant class imbalance , subsampling can reduce computational costs at the price of inflated variance in estimating model parameters . we propose a method for subsampling efficiently for logistic regression by adjusting the class balance locally in feature space via an accept reject scheme . the method is simple and requires one parallelizable scan over the full data set . the factor of two improves to if we multiply the baseline acceptance probabilities by ( and weight points with acceptance probability greater than 1 ) , taking roughly times as many data points into the subsample . experiments on simulated and real data show that our method can substantially outperform standard case - control subsampling .

for classification problems with significant class imbalance , subsampling can reduce computational costs at the price of inflated variance in estimating model parameters . we propose a method for subsampling efficiently for logistic regression by adjusting the class balance locally in feature space via an accept reject scheme . our method generalizes standard case - control sampling , using a pilot estimate to preferentially select examples whose responses are conditionally rare given their features . the biased subsampling is corrected by a post - hoc analytic adjustment to the parameters . the method is simple and requires one parallelizable scan over the full data set . standard case - control sampling is inconsistent under model misspecification for the population risk - minimizing coefficients . by contrast , our estimator is consistent for provided that the pilot estimate is . moreover , under correct specification and with a consistent , independent pilot estimate , our estimator has exactly twice the asymptotic variance of the full - sample even if the selected subsample comprises a miniscule fraction of the full data set , as happens when the original data are severely imbalanced . the factor of two improves to if we multiply the baseline acceptance probabilities by ( and weight points with acceptance probability greater than 1 ) , taking roughly times as many data points into the subsample . experiments on simulated and real data show that our method can substantially outperform standard case - control subsampling .

math0310364

i

a geometrically finite hyperbolic surface is a topologically finite , complete riemannian surface of constant curvature -1 . in this paper we study the selberg zeta function , @xmath1 ( see [ zeta.sec ] for the definition ) , associated to the length spectrum of a geometrically finite surface @xmath2 . when @xmath2 has infinite area , the discrete spectrum of the positive laplacian @xmath3 will be finite and possibly empty . the appropriate spectral invariants at this level of generality are the _ scattering resonances _ or simply _ resonances_. these are the poles of the meromorphic continuation of the resolvent @xmath4 , with multiplicities ( see [ scatt.sec ] for the definition ) . building upon our previous results and the work of guillop and zworski @xcite , we exhibit an explicit connection ( theorem [ thm.zeta2 ] ) between selberg s zeta function and a hadamard product over the resonance set . this yields the following : [ thm.zeta ] let @xmath5 denote the number of cusps of @xmath2 , and let @xmath6 denote the euler characteristic of @xmath2 . the selberg zeta function @xmath7 extends to a meromorphic function of order two , with a divisor that can be divided into spectral and topological components : the spectral zeros of @xmath8 are given by the resonance set @xmath9 ( with multiplicities ) . in addition , @xmath1 has topological zeros at @xmath10 for @xmath11 , of order @xmath12 , and topological poles of order @xmath5 at @xmath13 . in the full geometrically finite context , the meromorphic continuation of @xmath7 to @xmath14 was proven by guillop @xcite . if @xmath2 has finite area , then meromorphic continuation with the order bound and divisor as given here can be deduced from the selberg trace formula ( see @xcite ) . in the convex co - compact case ( @xmath15 ) patterson - perry @xcite established the order bound , using thermodynamic formalism of ruelle and fried , and also computed the divisor . the thermodynamic formalism does not extend to the case @xmath16 . to prove that theorem [ thm.zeta ] holds for all geometrically finite @xmath2 requires a different approach from these previous results . as in the compact case , one can exploit knowledge of the divisor of the zeta function to link the resonance set to the length spectrum . [ cor.finite.euler ] the resonance set determines the length spectrum of @xmath2 , the euler characteristic , @xmath6 , and the number of cusps , @xmath5 . the length spectrum determines @xmath6 and @xmath5 up to a finite number of possibilities . the length spectrum , @xmath6 , and @xmath5 , together determine the resonance set . at present , we do not know whether the length spectrum ( by way of @xmath8 ) determines @xmath6 and @xmath5 when @xmath2 has infinite area and @xmath17 . that is , we can not rule out the possibility that the resonance set contributes to the multiplicity of each of the zeros of @xmath8 that lie in @xmath18 . however , if @xmath2 has finite area , the resonances are confined to a vertical strip , and hence for sufficiently negative @xmath19 , the multiplicity of the zero of @xmath8 equals @xmath12 . and in the convex co - compact case ( infinite area with no cusps ) , martin olbrich has pointed out to us that corollary 6.9 of @xcite , in conjunction with the theory of @xcite , shows that the multiplicity of resonances at @xmath20 is either zero ( for non - elementary groups ) or two ( for elementary groups ) . thus , in the convex co - compact case @xmath6 can be recovered as @xmath21 for any @xmath22 . by combining corollary [ cor.finite.euler ] with methods of teichmller theory we obtain the following application : [ finiteness ] let @xmath2 be a complete , geometrically finite hyperbolic surface of infinite area . then the length spectrum of @xmath2 determines @xmath2 up to finitely many possibilities . in particular , the resonance set determines @xmath2 up to finitely many possibilities . our proof of the first claim in theorem [ finiteness ] requires theorem [ thm.zeta ] . indeed , we do not know how to obtain bounds on the euler characteristic without using scattering theory . if @xmath2 is connected , then there exists a finitely generated , torsion free , discrete subgroup , @xmath23 , of @xmath0 , unique up to conjugation , such that @xmath2 is the quotient of the hyperbolic upper half - plane , @xmath24 , by @xmath23 acting as mbius transformations . conversely , any such quotient is a geometrically finite hyperbolic surface . moreover , the length spectrum of @xmath2 equals ( twice ) the set @xmath25 ( including multiplicities ) . thus , from theorem [ finiteness ] we have [ discrete ] let @xmath23 be a finitely generated , torsion - free , discrete subgroup of @xmath0 of co - infinite area . then the set of traces @xmath26 ( including multiplicities ) determines the conjugacy class of @xmath23 up to finitely many possibilities . theorem [ finiteness ] is due to mckean @xcite in the case that @xmath2 is compact and to mller @xcite in the case that @xmath2 has finite area . corollary [ discrete ] is also due to mckean @xcite in the case that @xmath23 is cocompact . theorem [ thm.zeta ] is proven by linking the zeta function to the scattering theory of @xmath27 . for geometrically finite @xmath2 scattering theory can be set up ( following the approach of guillop and zworski @xcite ) using a decomposition of the form @xmath28 where @xmath29 is a compact surface , each @xmath30 , @xmath31 , is a funnel , and each @xmath32 , @xmath33 is a cusp ( end ) . this is illustrated in figure [ gfsurface ] . ( for an exact description of funnels and cusps , see [ sec.model ] . ) the boundary of @xmath29 consists of @xmath34 closed geodesics ( uniquely determined ) and @xmath35 horocycles ( the choice of which is not unique ) along which @xmath29 is glued to the funnel and cusp ends , respectively . the ideal boundary @xmath36 is the disjoint union of @xmath5 ideal points and @xmath37 circles . [ gfsurface ] ] the scattering operator @xmath38 , described in greater detail in [ scatt.sec ] , can be viewed as a map from @xmath39 to itself , where a `` smooth function '' on @xmath36 is interpreted as an element of @xmath40 . because the funnels are isometric to standard models ( see [ sec.model ] ) , one can define relative scattering from the disjoint union @xmath41 to @xmath2 . on each funnel end , the laplacian @xmath42 carries dirichlet boundary conditions on the closed geodesic which joins @xmath43 to @xmath29 . the scattering operator for @xmath44 , denoted @xmath45 , is a direct sum of scattering operators for the dirichlet laplacians @xmath42 , and acts on @xmath46 . if we set @xmath47 ( where @xmath48 is the identity map on @xmath14 ) then @xmath49 acts on @xmath50 and the relative scattering operator @xmath51 is determinant class . thus one has a relative scattering determinant : @xmath52 in @xcite , guillop and zworski show that @xmath53 extends to the quotient of analytic functions of order at most four and compute its divisor in terms of the resonance set . our proof of theorem [ thm.zeta ] in [ zeta.sec ] involves establishing the connection between @xmath1 and @xmath54 , and then exploiting this information about @xmath54 . in the course of the proof we will improve the order bound on @xmath54 from four to two . * acknowledgments*. we are grateful to reza chamanara and jeffrey mcgowan for helpful conversations , and to martin olbrich for point out to us his results in @xcite . we also thank peter sarnak for encouraging us to remove the unnecessary geometric assumptions used in @xcite .

for hyperbolic riemann surfaces of finite geometry , we study selberg s zeta function and its relation to the relative scattering phase and the resonances of the laplacian . as an application we show that the conjugacy class of a finitely generated , torsion - free , discrete subgroup of is determined by its trace spectrum up to finitely many possibilities , thus generalizing results of mckean and mller to groups which are not necessarily cofinite .

for hyperbolic riemann surfaces of finite geometry , we study selberg s zeta function and its relation to the relative scattering phase and the resonances of the laplacian . as an application we show that the conjugacy class of a finitely generated , torsion - free , discrete subgroup of is determined by its trace spectrum up to finitely many possibilities , thus generalizing results of mckean and mller to groups which are not necessarily cofinite .

1008.0519

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from 2mass k@xmath36 , ( j@xmath11k)@xmath36 color - magnitude diagrams we found two distinct rc populations towards the galactic bulge , separated by @xmath37k@[email protected] magnitudes at latitude @xmath52 , and co - existing in fields over 13@xmath1 in longitude and 20@xmath1 in latitude . the presence of the two rc populations is particularly obvious at a galactic latitude of @xmath118@xmath1 . we detect the individual rc populations over a @xmath220@xmath10520@xmath1 area of sky , roughly symmetric about the galactic center . thus , the two populations cover essentially the entire galactic bulge / bar region ; however , the 2mass data do not probe the rc within @xmath25@xmath1 of the galactic plane , due to confusion limitations . we also find that the faint rc is dominant on the negative longitude side of the galactic center , while the bright rc is the principal population at positive longitudes . based on the age and metallicity sensitivity of the rc , predicted by theoretical stellar isochrones of the teramo group ( pietrinferni et al . 2004 ) , we find that the two rcs can not be due to any allowed combination of age or [ fe / h ] in the galactic bulge / bar at a single distance . number statistics firmly rule - out the possibility that the two luminosity peaks are due to rc plus red giant branch bump ( rgb bump ) or asymptotic giant branch bump ( agb bump ) . heretofore unknown stellar evolution effects are also militated against , due to the change in number ratio between positive and negative longitude sides of the galactic bulge . the obvious , and natural , explanation is that the faint and bright rc peaks reflect different distances of two populations . at ( @xmath3)=(@xmath4 ) we estimate distances of 6.5 and 8.8@xmath106 kpc for the two rcs , based on m@[email protected] , computed with theoretical offsets from the observed 47 tuc k - band absolute magnitude , and assuming an age of 12 gyr and a mean [ fe / h][email protected] dex . figure [ distdot-8 ] and [ fig - x ] should be consulted for distances at other locations . we note that these distances are based on the assumption of a bulge age equal to that of 47 tuc , near 12 gyr ; our estimated distances increase for younger bulge ages . radial velocity data , particularly from the brava survey ( howard et al . 2008 , 2009 ) , are consistent with the faint rc population being more distant than the bright rc population . proper motion results from sumi et al . ( 2003 ) and v07 are also generally consistent with a distance interpretation . an immediate conclusion is that these two co - existing populations are not consistent with the body of a bar , since the line of sight through a bar should occur at one distance , not two . other evidence against these populations being in the body of a bar is the fact that both populations mostly exist at fixed distances , independent of longitude ; in particular , the foreground rc peak k@xmath36 magnitude is unchanged over 9@xmath1 in longitude . this is in stark contrast to a bar , which other studies have claimed is pointed almost directly towards us , tilted by only @xmath220@xmath1 to the line of sight . for a bar tilted at 20@xmath1 the distance change over 9@xmath1 in longitude is expected to exceed 2kpc , or roughly 0.7 magnitudes . when we plot distance of the peaks versus longitude for latitude @xmath52 , and add approximate number counts , we find that the faint and bright rc counts peak near longitudes @xmath107 and @xmath54 , respectively . a line joining these two maxima is tilted , approximately 20@xmath1 to the line of sight and crosses the @xmath70 line within 1@xmath71 of the distance to the galactic center , projected onto the @xmath52 plane ; this intersection point is roughly midway between the foreground and background populations . as a function of latitude our luminosity plots show that the two rc peaks are closer in distance at lower galactic latitudes ( i.e. , toward the plane ) , mostly because of an increase in brightness of the background rc . the two rcs appear to merge around @xmath108 . when we make a crude correction to the rc m@xmath35 for the metallicity gradient with latitude along the minor axis , by zoccali et al . ( 2008 ) , we found that a vertical cut through the bulge / bar , at @xmath103 ( from @xmath109 to @xmath110 , i.e. , as viewed from the side ) shows an x - shaped morphology . the center of the x - shape occurs near [email protected] kpc , which is equal to , within the uncertainties , the galactic center distance of ghez et al ( 2008 ) , at [email protected] kpc . better agreement between these two centers would be obtained if the bulge is @xmath223 gyr younger than 47 tuc , or if the metallicity difference between bulge and 47 tuc rc stars is larger than we assume , or if there is an unspecified systematic error in the theoretical isochrones . the rc peak counts above the troughs in the double peaked rc luminosity functions gave a minimum estimate for the fraction of stars in the x versus stars in a broader rc distribution , perhaps from a bar or spheroidal component . this is a minimum because it is possible that the troughs are partly , or entirely , due to the overlapping wings of the distributions of the foreground / background rc components . we found that at ( @xmath3)=(@xmath4 ) the foreground / background rc x components are at least 20% of the total rc population ; at ( @xmath3)=(@xmath111 ) the foreground / background x - components must be close to 100% of the total rc ; on the other hand , near @xmath112 the estimated fraction due to foreground / background components ranges from almost negligible to 100% of the population . our analysis of the 2mass data contains no compelling direct evidence of a galactic bar , although one may be present in a smooth rc population between the rc peaks . as noted above , at ( @xmath3)=(@xmath111 ) , the 2mass data show no room for a bar . detailed modelling will be required to constrain a bar population in this data . it seems reasonable , however , that the x - shape may merge into a short bar at latitudes below @xmath113 . we note that the radial velocity data of the brava survey shows an increased dispersion near @xmath114 , as expected from a bar . detailed models of these radial velocities by shen et al . ( 2010 ) provide excellent fits with a pure bar structure ; they found that a spheroidal component can be no more than 8% of the total . our finding that at least 20% of rc stars are in the x - shape component at @xmath52 , rather than a bar , may be in conflict with the 8% limit of shen et al . ( 2010 ) ; however , their limit refers to a spherical distribution , rather than arcs at the ends of a bar , so it is possible that no conflict exists . a reasonable , qualitative , model to explain the observations presented here is that the high - latitude bulge ( above @xmath1156@xmath1 ) is dominated by the vertical extent of stars near the foreground and background ends of a galactic bar . in this scenario stars extend in all directions from the ends of the bar , i.e. , in both longitude and latitude . in longitude the stars occupy arcs , or partial orbits , emanating from the bar ends . for this reason , a line connecting the peaks in the foreground/ background rc populations trace the direction of the bar , tilted at @xmath220@xmath1 to the line of sight , assuming that there is no significant rotational lag . the axis of rotation of the bar is marked by the point where this line intersects with @xmath70 . in figure [ fig - x ] this point is located at 7.7 kpc from the sun ; it happens to lie roughly midway between the two rcs and within 1@xmath71 of the galactic center distance . this arrangement should produce symmetry in the mean radial velocity curve , similar to the brava observations . because the bar is almost end - on , the stars in the arcs , near the bar ends , spread almost tangentially to the line of sight , and thus appear at a nearly constant distance , as observed . however , at large angular distance , in longitude , from the bar ends , the stars along the arcs / orbits are closer to the galactic center distance , also observed . since stars emanating from the bar ends do so in all directions , they appear as two roughly circular areas , and take - on a peanut appearance when viewed nearly end - on to the bar ; a slight flattening of the peanut may be expected due to the gravitational potential in the latitude direction . when viewed from the side , the structure takes on an x - shape morphology , as the stars emanating from the bar ends merge into the bar . in this case the main component of the bar resides at low galactic latitudes , and perhaps for this reason the end components are relatively easy to detect at high latitude . an alternative description is that there is only an x - shaped component which extends to high latitudes , but without connecting to a bar . as before , since we are looking almost end - on to the axis of the x , the foreground and background appear at nearly constant distances . another , alternative , explanation of the observed two rc components is that we have detected concentrations in the thick disk near the ends of the bar . if correct , this scenario might naturally explain the similarity of the composition of thick disk and bulge red giant stars found by alves - brito et al . ( 2010 ) . if our two populations represent extensions from the ends of a bar , then their velocities should show cylindrical rotation , similar to bar models , but with a distribution not considered by shen et al . one major difficulty is that our results suggest a end - to - end bar length of @xmath22.5 kpc , whereas the best fit model of shen et al . ( 2010 ) indicates a bar half - length of 4kpc . x - shape , boxy , and peanut morphologies of extra - galactic bulges are well known phenomena , e.g. , ic 4767 , ngc 128 , ngc 4469 , ic 2531 ( e.g. , whitmore & bell 1988 ; bureau et al . 2006 . ) the x - shapes are relatively minor components of extra - galactic bulges , and significant image processing is usually required to highlight them , in stark contrast to the strong signal we find here for the galaxy . n - body simulations of isolated disk galaxies and the growth of bars ( e.g. , patsis et al . 2002 ; athanassoula 2005 ) find x - shape morphologies resulting from bar growth . in particular , the * x1v1 * resonance mode of patsis et al . ( 2002 ) appears very similar to our results for the galaxy . if these predictions are correct , the implication is that the bulge x - shape resulted from bar instabilities , and that the galaxy contains a pseudo - bulge , and a bar , rather than a classical bulge . x - morphologies also result from mergers , although bars are still involved : mihos et al . ( 1995 ) performed numerical simulations of satellite accretion by s0 galaxies , and found that prograde accretion induced the formation of bars , which subsequently buckled and produced x structures . since the formation of a bar from an isolated disk galaxy relies on interactions between its disk and halo , the overlap between these two mechanisms is greater than at first apparent . typical timescales for pseudo - bulge growth is @xmath21 gyr ( e.g. , athanassoula 2008 ) , which may be problematic for a rapid formation timescale for the galactic bulge ( e.g. , ballero et al . 2007 ) , as implied by the enhanced [ mg / fe ] ratios ( e.g. , fulbright et al . 2007 ; lecureur et al . 2007 ; mcwilliam & rich 1994 ) ; although , both the timescale and the mg enhancements have associated uncertainties which may bring them into rough agreement . another difficulty is that pseudo - bulges / bars are thought to result in well mixed orbits , such that metallicity gradients may not be expected ( zoccali 2010 ; howard et al . 2009 . ) however , the bulge metallicity gradient is well - known ( zoccali et al . 2008 ) , and the metallicities correlate with kinematic properties . it is possible that the gradient simply results from the super - position of two populations ( zoccali et al . 2010 ; babusiaux et al . 2010 ; hill et al . howard et al . ( 2009 ) speculated that dissipative processes during bar formation might explain the metallicity gradient . while an x - shape bulge qualitatively explains the 2mass photometry and much of the published kinematic data , some literature results are problematic . the @xmath1135 km / s radial velocity difference between bright and faint rc found by rws09 is difficult to understand ; asymmetric bar rotation could explain it , but such an asymmetry in velocities is not obvious in the brava data . also , the 0.4 dex difference in [ fe / h ] between @xmath116 and @xmath117 bulge components , found by rw09 , suggests an accreted system , rather than a bar or symmetric x - component ; however , provisional metallicities from the brava spectra , by koch ( 2010 , private communication ) do not support an [ fe / h ] asymmetry . perhaps the most important difficulty is that the x - shape is evident in our 2mass data when the dependence of mean [ fe / h ] on galactic latitude is taken into account , without the metallicity corrections the morphology is more k - shaped . this is a problem because the expectation is that the bar should not possess a metallicity gradient , and while x - morphologies are well known in external galaxies , we are unaware of k - morphologies . the final piece of confusing evidence is the lack of difference between the mean proper motions of bright and faint rc stars in the data of vieira et al . ( 2007 ) . these apparent inconsistencies demand that future work on the kinematics and composition of rc bulge stars be pursued for a more complete picture of the galactic bulge region . it is particularly important to verify whether there is a metallicity gradient with latitude in the rc x - populations ; if so , this would support the idea of howard et al ( 2009 ) , that the bar formed with dissipation ; if not , then the bulge is k - shaped . if the gradient simply reflects a changing ratio between two populations with different mean [ fe / h ] , then composition studies of these two populations would be interesting and useful . a definitive investigation of the possible metallicity asymmetry with longitude proposed by rangwala & williams ( 2009 ) is also necessary . extant age studies by zoccali et al . ( 2003 ) and clarkson et al . ( 2008 ) should be supplemented with a photometric survey for the ages of bulge stars at @xmath40=@xmath118 , in order to completely rule - out the possibility that age is responsible for the luminosity difference between the two rcs at this latitude . other important future work includes detailed modelling of the 2mass data , in order to determine the relative proportions of a bar and the foreground / background rc components . a high spatial resolution infrared photometric survey , such as the vista variable in the va lctea survey ( vvv , minniti et al . 2010 ) , covering roughly the same longitudinal extent as the current work , but going to much smaller latitudes will help to determine whether the x - shape continues to the galactic center , or joins the bar . analysis of the premilinary vvv catalogues is ongoing ( saito et al . 2010 , in preparation ) . independent verification of the foreground / background distances , at high latitude , with standard candles , such as miras , strong - lined rr lyrae stars , and eclipsing binaries , will also be very helpful . after this paper was submitted for publication nataf et al . ( 2010 ) reported a split rc toward the galactic bulge . their findings are broadly consistent with the results of this paper ; in particular , they find very nearly equal ( v@xmath11i ) colors , consistent with similar metallicity for the two rc components . part of this work was performed by am while visiting the kitp in santa barbara , which is supported under nsf grant pho5 - 51164 . am thanks andreas koch for a useful conversation . mz acknowledges the fondap center for astrophysics 15010003 , the basal center for astrophysics and associated technologies pfb-06 , fondecyt regular 1085278 , and the mideplan milky way millennium nucleus p07 - 021-f . alonso , a. , arribas , s. , & martinez - roger , c. 1999 , a&as , 140 , 261 alves , d.r . 2000 , , 539 , 732 alves - brito , a. , melndez , j. , asplund , m. , ramrez , i. , yong , d. 2010 , a&a , 513 , 35 athanassoula , e. 2005 , , 358 , 1477 athanassoula , e. 2008 , in `` formation and evolution of galaxy bulges '' , iau symposium , volume 245 , p. 93 - 102 , eds . martin bureau , e. athanassoula & beatriz barbuy ( cambridge : cup ) mcwilliam , a. , fulbright , j. , & rich , r.m . 2010 , in `` chemical abundances in the universe : connecting first stars to planets '' , iau symposium , volume 265 , p. 279 - 284 , katia cunha , monique spite & beatriz barbuy ( cambridge : cup ) nishiyama , s. , nagata , t. et al . 2006 , journal of physics : conference series , volume 54 , proceedings of `` the universe under the microscope - astrophysics at high angular resolution '' , eds . rainer schoedel , andreas eckart , susanne pfalzner & eduardo ros , pp . 62 - 66 ( 2006 ) . zoccali , m. 2010 , in `` chemical abundances in the universe : connecting first stars to planets '' , iau symposium , volume 265 , p. 271 - 278 , eds . katia cunha , monique spite & beatriz barbuy ( cambridge : cup )

from 2mass infra - red photometry we find two red clump ( rc ) populations co - existing in fields toward the galactic bulge at latitudes.5 , ranging over in longitude and 20 in latitude . alternatively , the x , or double funnel , may continue to the galactic center . from the sun this would appear peanut / box shaped , but x - shaped when viewed tangentially . 0.5 in

from 2mass infra - red photometry we find two red clump ( rc ) populations co - existing in fields toward the galactic bulge at latitudes.5 , ranging over in longitude and 20 in latitude . these rc peaks indicate two stellar populations separated by.3 kpc ; at ()=( ) the two rcs are located at 6.5 and 8.8.2 kpc . the double - peaked rc is inconsistent with a tilted bar morphology . most of our fields show the two rcs at roughly constant distance with longitude , also inconsistent with a tilted bar ; however , an underlying bar may be present . stellar densities in the two rcs changes dramatically with longitude : on the positive longitude side the foreground rc is dominant , while the background rc dominates negative longitudes . a line connecting the maxima of the foreground and background populations is tilted to the line of sight by , similar to claims for the tilt of a galactic bar . the distance between the two rcs decreases towards the galactic plane ; seen edge - on the bulge is x - shaped , resembling some extra - galactic bulges and the results of n - body simulations . the center of this x is consistent with the distance to the galactic center , although better agreement would occur if the bulge is 23 gyr younger than 47 tuc . our observations may be understood if the two rc populations emanate , nearly tangentially , from the galactic bar ends , in a funnel shape . alternatively , the x , or double funnel , may continue to the galactic center . from the sun this would appear peanut / box shaped , but x - shaped when viewed tangentially . 0.5 in

1112.0496

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the fusion of carbon nuclei is an important reaction in the description of type ia supernovae and other astronomical events in the cosmos like the superburst of an accreting neutron star . there is , unfortunately , a very large uncertainty in the predicted fusion cross sections for @xmath1c+@xmath1c that are needed in a stellar environment @xcite , partly because the cross sections are difficult to measure down to the low energies that are of interest , and partly because the data contain strong resonance structures that make it difficult to extrapolate the measured cross sections to low energy . in order to get some constraints on the extrapolation it is instructive to analyze the existing fusion data for @xmath0c+@xmath0c by trentalange et al . @xcite and charvet et al . @xcite , and also of the @xmath1c+@xmath0c data by dayras et al . @xcite and notani et al . @xcite , because these data do not exhibit strong resonance features . the @xmath0c+@xmath0c fusion data @xcite are analyzed by coupled - channels calculations that include couplings to the one - phonon quadrupole and octupole excitations in projectile and target , as well as mutual excitations of these states . the effect of one - neutron transfer , which can proceed to the @xmath1c+@xmath2c mass partition with positive @xmath3 value is also studied , and so is the effect of the neutron exchange with zero q value in @xmath1c+@xmath0c reactions . the fusion cross sections are determined by ingoing - wave - boundary conditions ( iwbc ) that are imposed at the minimum of the pocket in the entrance channel potential . the calculated cross sections defined this way are fairly smooth as functions of the center - of - mass energy and they are well suited to analyze the fusion data for @xmath0c+@xmath0c @xcite and @xmath1c+@xmath0c @xcite . the @xmath1c+@xmath1c fusion data , on the other hand , contain a lot of structures or resonances , and it is beyond the scope of this investigation to try to reproduce these data in detail . it is , however , of great interest to see how the calculated cross sections , obtained from the iwbc , compare to the data and how they possibly can be used to put constraints on the extrapolation to very low energies . the coupled equations are solved using either a standard woods - saxon @xcite or the m3y+repulsion , double - folding potentials @xcite . the coupled - channels effects on the calculated fusion cross sections are relatively modest compared to the large enhancement of several orders of magnitude that are commonly seen in calculations of heavy - ion fusion reactions . for the carbon systems , the enhancement of fusion compared to the no - coupling limit is typically a factor of 2 at energies far below the coulomb barrier . however , it is a challenge to explain the data in detail because the coupled - channels calculations are sensitive to channels that have rather high excitation energies . one problem is that these channels are closed at low beam energies but that problem can be solved by imposing the correct , decaying state boundary conditions at large distances between the reacting nuclei @xcite . another problem is that the nuclear structure of high - lying states is sometimes very uncertain . the m3y+repulsion potential was introduced in ref . @xcite to explain the hindrance that has been observed in the fusion of many heavy - ion systems at very low energies @xcite . the hindrance phenomenon was first observed as a suppression at very low energies of @xmath4ni+@xmath5y fusion data compared to calculations that were based on a standard woods - saxon potential @xcite . a simple explanation of the phenomenon is the existence of a shallow pocket in the entrance channel potential which forces the fusion cross section to vanish as the center - of - mass energy approaches the minimum energy of the pocket @xcite . important issues have been whether the fusion hindrance also occurs in light - ion systems , and how it will affect the extrapolation of measured cross section to the low energies that are of interest to astrophysics @xcite . the evidence for quasi - molecular resonances , observed in the elastic scattering @xcite and fusion reactions @xcite of @xmath1c+@xmath1c are usually explained as resonances in a shallow two - body potential between the reacting nuclei . we will show that the analysis of the @xmath0c+@xmath0c and @xmath1c+@xmath0c fusion data supports the idea of a fusion hindrance and the existence of a shallow pocket in the entrance channel potential . the fusion cross sections reported in refs . @xcite are based on measurements of the characteristic @xmath6-rays emitted from some of the evaporation residues that are produced , mostly those associated with the proton , neutron , and @xmath7 decay . the total fusion cross sections were obtained with the aid of statistical model calculations which is a major source of the systematic uncertainty . the systematic error is difficult to estimate but it can be quite large . for example , the systematic error of the absolute cross sections quoted in ref . @xcite is @xmath815% . since some of the systematic error concerns the overall normalization of the measured cross sections , and not so much the shape ( i. e. , the energy dependence of the cross section ) , it is of interest to adopt an adjustable overall normalization when analysing the data . the ingredients of the coupled - channels calculations are presented in the next section . the analysis of the @xmath0c+@xmath0c fusion data is discussed in section iii , where the optimum parameters of the m3y+repulsion potential , such as the radius parameter of @xmath0c and the diffuseness associated with the repulsive term , are determined . these parameters are used together with a parameterization of the experimental density of @xmath1c as input to the coupled - channels calculations of the fusion cross sections for @xmath1c+@xmath0c and @xmath1c+@xmath1c which are presented in sections iv and v , respectively . the systematics of the low - energy fusion cross sections for the 3 systems of carbon isotopes is discussed in section vi . finally , the conclusions of this work are presented in section vii .

fusion data forc+c ,c+c andc+c are analyzed by coupled - channels calculations that are based on the m3y+repulsion , double - folding potential . quadrupole and octupole transitions to low - lying states in projectile and target are included in the calculations , as well as mutual excitations of these states . the effect of one - neutron transfer is also considered but the effect is small in the measured energy regime .

fusion data forc+c ,c+c andc+c are analyzed by coupled - channels calculations that are based on the m3y+repulsion , double - folding potential . the fusion is determined by ingoing - wave - boundary conditions ( iwbc ) that are imposed at the minimum of the pocket in the entrance channel potential . quadrupole and octupole transitions to low - lying states in projectile and target are included in the calculations , as well as mutual excitations of these states . the effect of one - neutron transfer is also considered but the effect is small in the measured energy regime . it is shown that mutual excitations to high - lying states play a very important role in developing a comprehensive and consistent description of the measurements . thus the shapes of the calculated cross sections forc+c andc+c are in good agreement with the data . the fusion cross sections forc+c determined by the iwbc are generally larger than the measured cross sections but they are consistent with the maxima of some of the observed peak cross sections . they are therefore expected to provide an upper limit for the extrapolation into the low - energy regime of interest to astrophysics .

1311.1206

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both star formation and stellar feedback play crucial roles in galaxy evolution . star formation leads to stellar feedback , which in turn is assumed to regulate the star formation rate and prevent galaxies from turning all their gas into stars over less than a @xmath0 ; moderate star formation rates are implied from low and high redshift observations . in contrast , pure hydrodynamic simulations and semi - analytic models of galaxy formation tend to predict high gas densities within galaxies . these densities cause the gas to cool very efficiently and supersede the density threshold required for star formation @xcite . for purely hydrodynamic simulations , unrealistically strong stellar feedback is often necessary to regulate the star formation rate in galaxies @xcite . several approaches have been attempted to regulate simulations of star formation during galaxy evolution . within the framework of pure hydrodynamics , the most basic feedback mechanism is usually thermal feedback , or injection of some fraction of the supernova energy into the gas . since this energy is injected into dense , cold gas it cools efficiently and typically has small overall effect on galaxy evolution @xcite . momentum feedback is added by explicitly injecting momentum to the material that surrounds star forming regions @xcite . while the efficiency of such models is slightly better than thermal feedback @xcite , the ejected gas is almost always highly supersonic and kinteic energy is converted to thermal radiation very efficiently through shocks , unless the feedback is injected effectively over very large volumes @xcite . attempts have also been made to inject the energy into a warm component of a two - phase gas , effectively delaying the cooling until energy is transferred from the diffuse gas to a denser gas which immediately cools @xcite . an additional technique sometime used has been to increase the efficiency of the feedback by releasing the feedback energy in bursts rather than spread out @xcite . however , in all these recipes , the cooling of a parcel of gas near the plane of the disc still occurs with isobaric boundary conditions set by the weight of the atmosphere on top of it . once the gas cools , it contracts over a crossing time to regain its pressure . since the cooling rate scales as @xmath1 ( where @xmath2 is the gas density ) this leads to runway cooling . in order to increase the efficiency of supernova feedback , the cooling of injected energy is artificially delayed @xcite , and momentum feedback efficiency in enhanced by preventing it from interacting with its immediate , dense environment @xcite . in this paper we revisit the conjecture that non - thermal processes contribute to the total pressure . with this additional pressure , gas can reach hydrostatic equilibrium with a considerably lower gas density that naturally predicts lower star formation rate and bypasses the need for unrealistic supernovae feedback . the non - thermal pressure does not depend on the temperature of the gas , and the gas cools isochorically rather than isobarically , further stabilizing the gas . we develop a simple , easy to use , parametric model which allows to study ( analytically and in simulations ) the effect of such non - thermal components on galaxy formation . the enhanced star formation problem is closely related to the general complication of modeling the interstellar medium ( ism ) gas . while the use of a standard , purely thermodynamic equation of state of an ideal gas is justified outside of galaxies in the igm , it becomes less appropriate to use in haloes ( haloes of galaxy clusters exhibit non - negligible magnetic fields ) and even more so in the ism of galaxies . this gas is highly multiphased , and consists of cold , warm and hot gas arranged within atomic and molecular clouds , filaments , and bubbles @xcite . complicated chemistry and dynamics , as well as radiation fields at multiple wavelengths affect the behaviour and interrelation between the different phases . moreover , the dynamics of the gas are strongly affected by non - thermal components , namely turbulence , magnetic fields and cosmic rays ( cr ) . stars , through their formation , evolution and destruction pump energy into the ism by stirring turbulence , emitting high energy particles ( cosmic rays ) and releasing radiation that heats and drives the gas @xcite . gravitational energy also powers turbulence @xcite and heats the gas ( see however * ? ? ? turbulence , could , in principle be accurately followed by pure hydrodynamics . however , modeling turbulence requires high resolution and realistic driving of the turbulence which is still an open question @xcite . the detailed modeling of these effects is the subject of intensive ongoing efforts ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? all these physical phenomena are determind by the relatively small ( @xmath3pc ) scale of observed giant molecular clouds ( gmcs ) , large eddies of turbulence @xcite , and tangled magnetic fields . it is prohibitively challenging to include all the aforementioned effects and small scales in cosmological simulations . effective equations of state for star forming gas are thus constructed , directly pressurizing the thermal component of the gas @xcite . an equation of state for subgrid turbulence has been proposed in @xcite . @xcite proposed an effective eos directly related to star formation rates , and @xcite incorporated turbulent pressure as a sub - grid model of the various phases motivated by @xcite . @xcite showed that sub - grid models for supersonic turbulence can have a large affect on dwarf galaxies . while these attempts artificially pressurize the gas ( as we propose below ) they do not prevent the gas from over - cooling isobarically and still require the unrealistically strong feedback described above . recent attempts @xcite to simulate the effects of cosmic ray pressure on star formation and winds in galaxies have shown that they are able to drive significant outflows and could be efficient in regulating the star formation of star forming galaxies . @xcite separately implemented a two fluid approximation for cosmic rays and gas for single - galaxy simulations , and propagated the cr as a diffusive component with constant diffusion coefficient . @xcite used an mhd code to simulate anisotropic diffusion that preferentially diffuses crs along magnetic fields . in all three implementations the crs are found to drive winds in some cases . these results are encouraging but the simulations are of single , ideal galaxies and the suggested methods can not be easily extended to the evolution phases of the galaxies and cosmological initial conditions . their physics is too detailed and the required resolutions are too fine to be practical in full hydrodynamic cosmological simulations . to date , no simulations exists that include all the physical processes which are important for pressurizing the gas , and no cosmological - scale simulation will likely be able to simulate all this physics in the forthcoming future . in view of all these complications , here we follow a simpler , alternative avenue , and construct an effective eos that mimics some of the main observed characteristics of the non - thermal pressure components . first , we take advantage of the fact that a scale separation roughly exists between the parsec - scale phenomena discussed in the previous paragraph and that of typical observed scales for vertical scale heights of discs which range between @xmath4 for the milky way @xcite and quiescent star forming galaxies to @xmath5 for starbursting high redshift galaxies @xcite . this implies that in the context of galaxy formation simulations , it should be sufficient to resolve the ism on scales much larger than @xmath61 parsec in order to reproduce galactic discs with realistic characteristics . correspondingly we use a coarse grained , effective modeling of the ism . our model bridges the gap between the parsec and kilo - parsec scales . second , we complement this scale separation with an effective equation of state that is straightforward to use . in principle , one can attempt to apply a rigorous treatment of the physical processes that constitute the non - thermal physics as sub - grid models . however , even if realistic such modules were constructed , there remains the problem of stipulating physically - consistant initial conditions : how to seed magnetic fields , how and when cosmic rays are generated and accelerated , and what drives turbulence and precisely on which scales . in addition , one has to relate the initial conditions to star formation which most likely drives these effects . hence , we opt for a simple , easy to use , pressure - density relation for the non - thermal eos which we develop below . a key feature of our implementation is the calibration of the eos by the observed relation between the fir radiation - a star formation indicator - and the radio radiation , which constrains the joint energy content of cosmic rays and magnetic fields . this novel approach provides a quantitative relation between the gas density and the magnitude of the non - thermal components . additional physical assumptions which are required to complete the specific parametrization of the effective eos are then limited to factors of order unity , rather than being arbitrary . the structure of this paper is as follows . in [ sec : eos ] we describe possible modifications for the equation of state of the gas that manifest some important aspects of the non - thermal pressure components . in [ sec : toymodel ] we demonstrate the effectivenes of such modified eos and and the importance of the non - thermal conmponents in general , using simple calculations of a point ( single - cell ) model , focusing on the regaulation of the overall star formation rate . [ sec : ramses ] describes the incorporation of the model into the hydrodynamic code _ ramses _ @xcite and the setup and results of isolated galaxy simulations with and without our model . in [ sec : summary ] we summarize and discuss our results .

we treat three non - thermal components - turbulence , magnetic fields and cosmic rays - and effectively parametrize their amplitude . then , the non - thermal pressure model is incorporated into _ we conclude that the non - thermal pressure can prolong the star formation epoch and achieve consistency with observations without invoking artificially strong stellar feedback . [ firstpage ] galaxies : evolution hydrodynamics cosmic rays ism : general ism : magnetic fields

galaxy evolution depends strongly on the physics of the interstellar medium ( ism ) . motivated by the need to incorporate the properties of the ism in cosmological simulations we construct a simple method to include the contribution of non - thermal components in the calculation of pressure of interstellar gas . in our method we treat three non - thermal components - turbulence , magnetic fields and cosmic rays - and effectively parametrize their amplitude . we assume that the three components settle into a quasi - steady - state that is governed by the star formation rate , and calibrate their magnitude and density dependence by the observed radio - fir correlation , relating synchrotron radiation to star formation rates of galaxies . we implement our model in single cell numerical simulation of a parcel of gas with constant pressure boundary conditions and demonstrate its effect and potential . then , the non - thermal pressure model is incorporated into _ ramses _ and hydrodynamic simulations of isolated galaxies with and without the non - thermal pressure model are presented and studied . specifically , we demonstrate that the inclusion of realistic non - thermal pressure reduces the star formation rate by an order of magnitude and increases the gas depletion time by as much . we conclude that the non - thermal pressure can prolong the star formation epoch and achieve consistency with observations without invoking artificially strong stellar feedback . [ firstpage ] galaxies : evolution hydrodynamics cosmic rays ism : general ism : magnetic fields

1311.1206

i

on galactic scales , the interstellar medium exists at quasi - static pressure that is required to support the atmosphere above it . in an equilibrium configuration , loss of pressure due to cooling processes is balanced by heating , which for typical disc galaxies at low redshifts , is dominated by stellar feedback . stellar feedback , through its dependence on star formation rate is related to the ism gas density . it is a well - known result that when only thermal pressure is considered in simulations , the resulting ism density constrained by the pressure and heating - cooling equilibria leads to relatively large star formation rates and short ( @xmath207myr ) gas depletion time scales . this is considerably faster than depletion times of @xmath208 inferred from observations @xcite . in this work we consider the contribution of non - thermal pressure components to this picture . non - thermal pressure consists of turbulence , cosmic rays and magnetic fields , and we examine their impact in an effective model . current cosmological simulations generally do not include the later two , and do not always resolve turbulence . we demonstrate that non - thermal pressure components can be instrumental in solving the depletion time discrepancy in two respects : they reduce the quasi - steady state density and the corresponding star formation rates and cooling times , and they stabilize the gas by adding longer relaxation times in cases where star formation flickers on and off . the regulating effect has been shown previously for cosmic rays @xcite and turbulence @xcite but the two were not considered together and in any case were not yet formulated in a way which is applicable to large scale cosmological simulations . to test our assumptions we construct a simplified physical model for which all the non - thermal components achieve a steady state that is solely a function of density . while simplistic , the advantage of such an approach is that it is readily applicable in numerical simulations . furthermore , we calibrate this density dependence by using the observed relations between the star formation rate for various galaxy observations and the synchrotron radiation , so that the magnitude of the effect is reasonably constrained . to study its effect we first implement it into a single - zone numerical model that traces the evolution of a parcel of star forming gas with varying physical conditions under isobaric boundary conditions that mimic the pressure confinement of the gas by the atmosphere around and above it . using this mode we find that for a given , realistic , thermal feedback the depletion times naturally grow from @xmath209 to @xmath210 in better agreement with observations , and that the coarse grained density of the gas is reduced by several orders of magnitude . then , the model is implemented into the hydrodynamical code _ ramses _ and we present three simulations of the same isolated milky - way - like galaxy with three different physical models . in the first we use some of the `` standard '' recipes generally used for feedback . using that model reduces star formation by blowing the gas to high temperatures and low densities , and expelling it from the galaxy . then , to demonstrate the problem we deliberately turn off one of the key feedback components - the delayed cooling and show that gas cools and accumulates at the numerical pressure floor with high densities that cause a depletion of the stars within @xmath211 - at odds with observed depletion times of @xmath212 in the third simulation we introduce the non - thermal model calibrated by the observed radio fir relation . for that model , the gas remains pressurized at intermediate temperatures and densities , reducing the non - physical low density gas of the first model , and the non - physical high density star forming gas of the second model . this model is effective in regulating star formation for a long period of time ( @xmath213 ) without blowing the gas out of the galaxy altogether . this work is a natural first step in incorporating non - thermal pressure components in galactic scale simulation . the next steps can , and should , pursue several avenues of research . the first is to better model the various non - thermal components , their internal interaction and their interaction with the thermal component and star formation . this would relax the assumption of equipartition , and replace the observational constraints with more physically motivated ones . for this step , calibration against results from ism - scale hydrodynamic simulations will be beneficial . a different avenue to pursue , in tandem or separately , is to use our _ ramses _ patch to run large scale numerical simulations and to demonstrate its effect and applicability on cosmic scales . cosmological simulations today generally do not include the magnetic fields and cosmic rays , and do not always resolve turbulence , and our approach allows to circumvent this difficulty by using a simple effective parametrization . such simulations will naturally include realistic boundary conditions for the ism , namely the halo gas , and allow us to study its interactions with the ism and its effect on winds . ultimately , once large scale cosmological simulations are possible with all the necessary physics , a toy sub - grid model can also be calibrated directly to those simulations and used as a cheaper approximation for them .

we assume that the three components settle into a quasi - steady - state that is governed by the star formation rate , and calibrate their magnitude and density dependence by the observed radio - fir correlation , relating synchrotron radiation to star formation rates of galaxies . we implement our model in single cell numerical simulation of a parcel of gas with constant pressure boundary conditions and demonstrate its effect and potential . ramses _ and hydrodynamic simulations of isolated galaxies with and without the non - thermal pressure model are presented and studied . specifically , we demonstrate that the inclusion of realistic non - thermal pressure reduces the star formation rate by an order of magnitude and increases the gas depletion time by as much .

galaxy evolution depends strongly on the physics of the interstellar medium ( ism ) . motivated by the need to incorporate the properties of the ism in cosmological simulations we construct a simple method to include the contribution of non - thermal components in the calculation of pressure of interstellar gas . in our method we treat three non - thermal components - turbulence , magnetic fields and cosmic rays - and effectively parametrize their amplitude . we assume that the three components settle into a quasi - steady - state that is governed by the star formation rate , and calibrate their magnitude and density dependence by the observed radio - fir correlation , relating synchrotron radiation to star formation rates of galaxies . we implement our model in single cell numerical simulation of a parcel of gas with constant pressure boundary conditions and demonstrate its effect and potential . then , the non - thermal pressure model is incorporated into _ ramses _ and hydrodynamic simulations of isolated galaxies with and without the non - thermal pressure model are presented and studied . specifically , we demonstrate that the inclusion of realistic non - thermal pressure reduces the star formation rate by an order of magnitude and increases the gas depletion time by as much . we conclude that the non - thermal pressure can prolong the star formation epoch and achieve consistency with observations without invoking artificially strong stellar feedback . [ firstpage ] galaxies : evolution hydrodynamics cosmic rays ism : general ism : magnetic fields

1601.06821

i

two closed riemannian manifolds @xmath1 and @xmath2 are said to be _ isospectral _ if the laplace beltrami operators acting on functions on @xmath1 and on @xmath2 have the same spectrum , equivalently if the spectral zeta functions @xmath9 and @xmath10 are equal . the manifolds are said to be _ strongly isospectral _ if the spectra of every natural self - adjoint elliptic differential operator on @xmath1 and @xmath2 agree . in particular , if @xmath1 and @xmath2 are strongly isospectral , then the laplace de rham operators acting on differential @xmath11-forms on the respective manifold have the same spectrum , equivalently the zeta functions @xmath12 and @xmath13 encoding those spectra are equal . following kac s famous question `` can one hear the shape of a drum ? '' @xcite , the following broad questions have received a lot of attention @xcite . [ q:1 ] which isometry invariants of closed riemannian manifolds are isospectral invariants ? which ones are strongly isospectral invariants ? for example , two riemannian manifolds that are strongly isospectral have the same dimension , volume and betti numbers . by analogy with the number theoretic notion of so - called arithmetically equivalent number fields , sunada @xcite proposed a general construction showing that there exist riemannian manifolds that are strongly isospectral , but are not even homeomorphic . let us briefly recall sunada s construction . let @xmath14 be a @xmath0-covering of closed manifolds , where @xmath0 is a finite group . let @xmath15 , @xmath16 be two subgroups of @xmath0 with the property that there is an isomorphism @xmath17\cong \c[g / u_2]$ ] of linear permutation representations of @xmath0 , or equivalently that for every conjugacy class @xmath18 of @xmath0 we have @xmath19 . in this case we say that the formal linear combination @xmath20 is a @xmath0-relation . sunada proves that then the intermediate coverings @xmath21 and @xmath22 are strongly isospectral . in this paper , we want to take the analogy with number fields further . the cheeger - mller theorem @xcite , proving a conjecture of ray and singer @xcite , gives a special value formula for the spectral zeta functions of a riemannian manifold , somewhat analogous to the analytic class number formula for the dedekind zeta function of a number field @xcite . it implies that if @xmath1 and @xmath2 are strongly isospectral riemannian manifolds , then @xmath23 where , for a riemannian manifold @xmath24 , @xmath25 is the covolume of the lattice @xmath26 in the vector space @xmath27 with respect to a certain canonical inner product ( see notation [ not : pairings ] [ not : reg ] ) . the study of torsion homology and regulators of manifolds has attracted a lot of attention recently , see for instance @xcite . in light of equation ( [ eq : cheegermuller ] ) , a natural instance of question [ q:1 ] is whether , for @xmath1 and @xmath2 as above , an integer @xmath28 , and a prime number @xmath8 , we must necessarily have @xmath29=\#h_i(m_2,\z)[p^\infty],\end{aligned}\ ] ] where @xmath30 $ ] denotes the @xmath8-primary torsion subgroup . here , we answer this and similar questions in the context of sunada s construction . more broadly , we study homological properties of covering manifolds in @xmath0-relations , analogous to an old and fruitful line of research in number theory . for example , the following basic result is analogous to results of @xcite on class groups of number fields . [ prop : intro1 ] let @xmath14 be a @xmath0-covering of closed riemannian manifolds , let @xmath31 be a @xmath0-relation , and let @xmath32 be an integer . then for all primes @xmath8 that do not divide @xmath33 , we have @xmath34 \cong h_i(x / u_2,\z)[p^\infty].\ ] ] we actually prove a more general result , see theorem [ thm : nomackeyfunctor ] . in providing counter examples , we have concentrated on closed hyperbolic @xmath7-manifolds , a class of manifolds that plays an important rle in other branches of geometry , as well as in number theory . the following is a result in the opposite direction to that of proposition [ prop : intro1 ] . [ prop : intro2 ] let @xmath6 be a prime number . then there exist strongly isospectral closed hyperbolic @xmath7-manifolds @xmath1 and @xmath2 such that @xmath35\neq \#h_1(m_2,\z)[p^\infty].\ ] ] moreover , if @xmath36 , then there exist @xmath7-manifolds @xmath1 and @xmath2 as above with @xmath35=1,\;\;\text { and } \;\;\#h_1(m_2,\z)[p^\infty]=p.\ ] ] note that hyperbolic @xmath7-manifolds have torsion - free homology in degrees @xmath37 , @xmath38 , and @xmath7 , so in the situation of proposition [ prop : intro2 ] , equation ( [ eq : chmtors ] ) is trivially satisfied in those degrees . it is already known that for every prime @xmath8 , there exist strongly isospectral closed riemannian @xmath39-manifolds @xmath1 and @xmath2 such that @xmath40\neq \#h_1(m_2,\z)[p^\infty]$ ] . this is shown in @xcite , using the following two results . firstly , every finite group can be realised as the fundamental group of a closed smooth 4-manifold @xcite . secondly , for every prime @xmath8 , there exists a finite @xmath8-group @xmath0 with a @xmath0-relation @xmath31 , where the abelianisations of @xmath15 and @xmath16 have different orders ( * ? ? ? * , example 3 ) . at the other extreme , if @xmath24 is a closed oriented 2-manifold , then @xmath41 is torsion - free . riemannian @xmath7-manifolds present an in - between case . it follows from the elliptisation theorem , as proven by perelman @xcite , and from a theorem of ikeda @xcite , that any two isospectral @xmath7-manifolds with finite fundamental groups are homeomorphic . in particular , sunada s construction for 4-manifolds can not possibly generalise to @xmath7-manifolds . instead , we prove proposition [ prop : intro2 ] by a judicious choice of @xmath0-relation @xmath20 , a different one for each prime @xmath8 , and then by performing a computer search among @xmath0-coverings of hyperbolic @xmath7-manifolds and computing the resulting torsion homology of the intermediate coverings corresponding to @xmath15 and @xmath16 . nevertheless , we believe that the evidence of proposition [ prop : intro2 ] justifies the following conjecture in @xcite using techniques introduced in this paper , see in particular proposition [ prop : impliesconj ] ] . [ conj : intro ] for every prime number @xmath8 , there exist strongly isospectral closed hyperbolic @xmath7-manifolds @xmath1 and @xmath2 such that @xmath40\neq \#h_1(m_2,\z)[p^\infty]$ ] . it follows from equation ( [ eq : cheegermuller ] ) that if @xmath1 and @xmath2 are strongly isospectral , then @xmath42 can be expressed in terms of torsion homology of the two manifolds , and in particular is a rational number . moreover , if @xmath1 and @xmath2 arise from sunada s construction , then it follows from proposition [ prop : intro1 ] that the numerator and denominator of this rational number are only divisible by prime divisors of @xmath33 . like in the case of torsion , it would be interesting to understand the quotient of regulators in each degree separately . in this direction we have the following result , which we now state for arbitrary @xmath0-relations ( see definition [ defn : relation ] ) . below , for a group @xmath43 we write @xmath44 $ ] where @xmath45 $ ] denotes the derived subgroup of @xmath43 . [ thm : intro3 ] let @xmath14 be a @xmath0-covering of closed oriented riemannian @xmath46-manifolds , and let @xmath47 , @xmath48 be a @xmath0-relation . then : 1 . for every integer @xmath32 , the product @xmath49 is a rational number that is a product of powers of prime divisors of @xmath33 ; 2 . [ item : regcompq ] for every @xmath50 , the class of @xmath51 in @xmath52 only depends on the isomorphism class of the @xmath0-module @xmath53 ; 3 . [ item : regcompz ] if @xmath8 is a prime number that does not divide @xmath54 for any @xmath55 , then the @xmath8-part of @xmath56 only depends on the isomorphism class of the @xmath0-module @xmath57 . the representation theoretic invariant that appears in parts ( [ item : regcompq ] ) and ( [ item : regcompz ] ) of the theorem was first introduced by dokchitser dokchitser in the context of regulators of elliptic curves ( see e.g. @xcite ) , and is called the regulator constant of the representation with respect to the given @xmath0-relation . we actually relate the quotients of regulators to regulator constants in greater generality , namely without the hypothesis on @xmath8 in part ( [ item : regcompz ] ) of theorem [ thm : intro3 ] . but in general , there is an `` error term '' , which measures the failure of @xmath0-descent for homology in the covering see theorem [ thm : regcomparison ] for the precise statement , but see also remark [ rmrk : noclue ] . vignras @xcite introduced another well - known construction of isospectral manifolds , of a more arithmetic flavour . in @xcite , calegari and venkatesh study torsion homology in pairs of manifolds that are of a similar nature to the examples of vignras , called `` jacquet langlands pairs '' , that are not isospectral but whose spectra are closely related . they raise , in ( * ? ? ? * section 7.10 ) , the question of giving algebraic interpretations to the quotients of torsion homology and regulators separately , analogously to theorem [ thm : intro3 ] . it would also be interesting to investigate the same type of questions in the context of the vignras examples . in section [ sec : examples ] , we give examples of how theorem [ thm : intro3 ] allows one to deduce concrete information about the @xmath58$]-module structure of @xmath59 from the torsion homology of quotients of @xmath60 . it also allows one to formulate a possible representation theoretic line of attack on conjecture [ conj : intro ] , such as the following result , which will be proved as corollary [ cor : impliesconj ] . [ prop : impliesconj ] let @xmath8 be an odd prime , let @xmath61 , and define the subgroups @xmath62 of @xmath0 . suppose that there exists a @xmath0-covering @xmath14 of closed hyperbolic @xmath7-manifolds such that the difference of the betti numbers @xmath63 is odd . then conjecture [ conj : intro ] holds for @xmath8 . to fully appreciate the representation theoretic nature of proposition [ prop : impliesconj ] , see the full statement of corollary [ cor : impliesconj ] . the structure of the paper is as follows . in section [ sec : brauer ] we recall the formalism of burnside groups , representation groups , and @xmath0-relations , and the definition of regulator constants . in section [ sec : manifolds ] we investigate the behaviour of homological invariants of riemannian manifolds in @xmath0-relations , and prove proposition [ prop : intro1 ] and theorem [ thm : intro3 ] . in section [ sec : regconst ] we compute the regulator constants of the rational representations of @xmath64 for odd primes @xmath8 , and of @xmath65 , with respect to certain @xmath0-relations . these computations will allow us to deduce concrete information about the @xmath58$]-module structure of the rational homology of @xmath0-coverings from the torsion homology of intermediate coverings . together with theorem [ thm : regcomparison ] , these calculations will also imply proposition [ prop : impliesconj ] . we prove proposition [ prop : intro2 ] by a direct computer search , and section [ sec : calc ] is devoted to the methods and algorithms used to that end . in section [ sec : examples ] we have collected some interesting examples , illustrating the various phenomena that we investigate here . all our riemannian manifolds will be assumed to be finite - dimensional , oriented , and closed . by an automorphism of a riemannian manifold we mean an orientation preserving diffeomorphism from the manifold to itself that is a local isometry . by a hyperbolic @xmath7-manifold , we mean a quotient of the hyperbolic @xmath7-space by a discrete subgroup of orientation preserving isometries . if @xmath8 is a prime number , we will write @xmath66 for the localisation of @xmath67 at @xmath8 , i.e. the subring @xmath68 of @xmath69 . for a rational number @xmath70 , with @xmath71 and @xmath72 , the @xmath8-adic valuation of @xmath73 is defined to be @xmath74 . when @xmath24 is a @xmath67-module , we write @xmath75 for the torsion submodule of @xmath24 and @xmath76 for the torsion - free quotient of @xmath24 . we would like to thank nicolas bergeron , nathan dunfield , derek holt , emilio lauret , and karen vogtmann for helpful discussions , and the anonymous referee for suggestions that improved the exposition .

such manifolds have the same dimension and the same volume , and their rational homology groups are isomorphic . here the cheeger mller theorem implies that a certain product of orders of torsion homology and of regulators for agrees with that for . further , we prove that the-torsion in the homology of is isomorphic to that of for all primes . for , we give examples of pairs of strongly isospectral hyperbolic-manifolds for which the-torsion homology differs , and we conjecture such examples to exist for all primes .

given a finite group , a-covering of closed riemannian manifolds , and a so - called-relation , a construction of sunada produces a pair of manifolds and that are strongly isospectral . such manifolds have the same dimension and the same volume , and their rational homology groups are isomorphic . here , we investigate the relationship between their integral homology . the cheeger mller theorem implies that a certain product of orders of torsion homology and of regulators for agrees with that for . we exhibit a connection between the torsion in the integral homology of and on the one hand , and the-module structure of integral homology of the covering manifold on the other , by interpreting the quotients representation theoretically . further , we prove that the-torsion in the homology of is isomorphic to that of for all primes . for , we give examples of pairs of strongly isospectral hyperbolic-manifolds for which the-torsion homology differs , and we conjecture such examples to exist for all primes . -0.8 cm

1607.04430

i

let @xmath1 , @xmath2 , be independent and identically distributed random vectors , and assume that the cumulative distribution function , @xmath0 , of @xmath3 is continuous . by sklar s theorem ( sklar @xcite ) , there exists a unique copula , @xmath4 , such that @xmath5 where @xmath6 is the @xmath7th marginal distribution function of @xmath0 . for @xmath2 and @xmath8 , let @xmath9 be the rank of @xmath10 among @xmath11 ; namely , @xmath12 the vector of ranks is denoted by @xmath13 . the basic nonparametric estimator for the copula @xmath4 is the empirical copula ( deheuvels @xcite ) , and we use the version given by @xmath14^d.\ ] ] slightly different definitions are employed for instance in fermanian et al . @xcite and tsukahara @xcite , but the supremum distance between these empirical copula variants is at most @xmath15 . in section [ sec : simul ] , we will also use the so - called empirical checkerboard copula ; see . let @xmath16 be independent random variables which are uniformly distributed on the unit interval and are independent of the sample @xmath17 . for each @xmath8 , let @xmath18 be the order statistics based on @xmath19 . define @xmath20 one of the authors conceived that @xmath21 could be interpreted as a sample from some version of the empirical copula . although this was not quite correct , we found that picking one vector randomly from these @xmath22 vectors is equivalent to sampling from a smoothed version of the empirical copula , which we call the _ empirical beta copula _ in view of the marginal distribution of uniform order statistics . this idea may be regarded as baker @xcite s copula construction based on uniform order statistics , where @xmath23-tuples of order statistics are determined by the rank vectors @xmath24 , and the probability of choosing among them is equal to @xmath25 . the empirical beta copula arises as a particular case of the empirical bernstein copula when the degrees of all bernstein polynomials are set equal to the sample size . the bernstein copula and the empirical bernstein copula are introduced in sancetta and satchell @xcite , and the asymptotic behavior of the latter is studied in janssen et al . we give necessary and sufficient conditions for the empirical bernstein copula to be a genuine copula , conditions which hold for the empirical beta copula . we show asymptotic results for the empirical bernstein copula process under weaker assumptions than those in janssen et al . @xcite . the advantages of the empirical beta copula are that it is a genuine copula , that it does not require the choice of a smoothing parameter , and that simulating random samples from it is straightforward . furthermore , the corresponding empirical process converges weakly to a gaussian process under standard smoothness conditions on the underlying copula . for small samples , the empirical beta copula outperforms the empirical copula both in terms of bias and variance . compared with the empirical bernstein copula with polynomial degrees as suggested in janssen et al . @xcite , the empirical beta copula is still more accurate for several copula models , especially in terms of the bias . the paper is organized as follows . in section [ sec : cop ] , we define the empirical beta copula and prove its relation to the empirical bernstein copula together with some auxiliary results for the bernstein transformation . in section [ sec : asym ] , we formulate and prove the asymptotic results for the empirical bernstein copula process . a monte carlo simulation study is presented in section [ sec : simul ] , and we conclude the paper with some remarks in section [ sec : concl ] . some additional proofs are given in the appendix .

this idea can be regarded as a variant on baker s [ j. multivariate anal . the latter turns out to be a particular case of the empirical bernstein copula , the degrees of all bernstein polynomials being equal to the sample size . necessary and sufficient furthermore , the empirical process based on the empirical bernstein copula is shown to be asymptotically the same as the ordinary empirical copula process under assumptions which are significantly weaker than those given in janssen , swanepoel and veraverbeke [ j. stat . plan . infer . 142 ( 2012 ) a monte carlo simulation study shows that the empirical beta copula outperforms the empirical copula and the empirical checkerboard copula in terms of both bias and variance . compared with the empirical bernstein copula with the smoothing rate suggested by janssen et al . , its finite - sample performance is still significantly better in several cases , especially in terms of bias . _ ams 2010 subject classifications _ : 62g20 , 62g30 , 62h12 + _ keywords _ : copula , empirical copula , bernstein polynomial , empirical bernstein copula , checkerboard copula

given a sample from a multivariate distribution , the uniform random variates generated independently and rearranged in the order specified by the componentwise ranks of the original sample look like a sample from the copula of . this idea can be regarded as a variant on baker s [ j. multivariate anal . 99 ( 2008 ) 23122327 ] copula construction and leads to the definition of the empirical beta copula . the latter turns out to be a particular case of the empirical bernstein copula , the degrees of all bernstein polynomials being equal to the sample size . necessary and sufficient conditions are given for a bernstein polynomial to be a copula . these imply that the empirical beta copula is a genuine copula . furthermore , the empirical process based on the empirical bernstein copula is shown to be asymptotically the same as the ordinary empirical copula process under assumptions which are significantly weaker than those given in janssen , swanepoel and veraverbeke [ j. stat . plan . infer . 142 ( 2012 ) 11891197 ] . a monte carlo simulation study shows that the empirical beta copula outperforms the empirical copula and the empirical checkerboard copula in terms of both bias and variance . compared with the empirical bernstein copula with the smoothing rate suggested by janssen et al . , its finite - sample performance is still significantly better in several cases , especially in terms of bias . _ ams 2010 subject classifications _ : 62g20 , 62g30 , 62h12 + _ keywords _ : copula , empirical copula , bernstein polynomial , empirical bernstein copula , checkerboard copula

gr-qc0201040

i

coalescing neutron star ( ns ) binaries are likely to be one of the most important sources of gravitational radiation for the ground - based laser - interferometer detectors in ligo @xcite , virgo @xcite , geo600 @xcite , and tama @xcite . these interferometers are most sensitive to gw signals in the frequency range from about @xmath5 to @xmath6 , which corresponds to the last several thousand orbits of the inspiral . during this period , the binary orbit is decaying very slowly , with the separation @xmath7 and phase @xmath8 following the standard theoretical treatment for the inspiral of two point masses ( see , e.g. , @xcite ) . theoretical templates for the corresponding quasi - periodic gw signals covering an appropriate range of values for the ns masses , as well as orbital phases and inclination angles , can be calculated to great precision ( see , e.g. , @xcite and references therein ) , and template matching techniques can therefore be used to extract signals from noisy interferometer data . when the binary separation @xmath7 has decreased all the way down to a few ns radii , the system becomes dynamically unstable @xcite and the two stars merge hydrodynamically in @xmath9ms . the characteristic gw frequency of the final burst - like signal is @xmath10 , outside the range accessible by current broadband detectors . these gw signals may become detectable , however , by the use of signal recycling techniques , which provide increased sensitivity in a narrow frequency band @xcite . these techniques are now being tested at geo600 , and will be used by the next generation of ground - based interferometers . point - mass inspiral templates break down during the final few orbits . instead , 3d numerical hydrodynamic calculations are required to describe the binary merger phase and predict theoretically the gw signals that will carry information about the ns equation of state ( eos ) . the first hydrodynamic calculations of binary ns mergers in newtonian gravity were performed by nakamura , oohara and collaborators using a grid - based , eulerian finite - difference code @xcite . rasio and shapiro ( @xcite , hereafter rs ) later used lagrangian sph calculations to study both the stability properties of close ns binaries and the evolution of dynamically unstable systems to complete coalescence . since then , several groups have performed increasingly sophisticated calculations in newtonian gravity , exploring the full parameter space of the problem with either sph @xcite or the eulerian , grid - based piecewise parabolic method ( ppm ) @xcite , and focusing on topics as diverse as the gw energy spectrum @xcite , the production of r - process elements @xcite , and the neutrino emission as a possible trigger for gamma - ray bursts @xcite . some of these newtonian calculations have included terms to model approximately the effects of the gravitational radiation reaction @xcite . the first calculations to include the lowest - order ( 1pn ) corrections to newtonian gravity , as well as the lowest - order dissipative effects of the gravitational radiation reaction ( 2.5pn ) were performed by shibata , oohara , and nakamura , using an eulerian grid - based method @xcite . more recently , the authors ( @xcite , hereafter paper 1 and paper 2 , respectively ) , as well as ayal et al.@xcite , have performed pn sph calculations , using the pn hydrodynamics formalism developed by blanchet , damour , and schfer ( @xcite , hereafter bds ) . these calculations have revealed that the addition of 1pn terms can have a significant effect on the results of hydrodynamic merger calculations , and on the theoretical predictions for gw signals . for example , paper 1 showed a comparison between two calculations for initially corotating , equal - mass binary ns systems . in the first calculation , radiation reaction effects were included , but no 1pn terms were used , whereas a complete set of both 1pn and 2.5pn terms were included in the second calculation . it was found that the inclusion of 1pn terms affected the evolution of the system both prior to merger and during the merger itself . the final inspiral rate of the pn binary just prior to merger was much more rapid , indicating that the orbit became dynamically unstable at a greater separation . additionally , the gw luminosity produced by the pn system showed a series of several peaks , absent from the newtonian calculation . most previous hydrodynamic calculations of binary ns mergers have assumed corotating initial conditions , and many modeled the stars as initially spherical . however , real binary ns are unlikely to be described well by such initial conditions . just prior to contact , tidal deformations can be quite large and the stars can have very nonspherical shapes @xcite . nevertheless , because of the very low viscosity of the ns fluid , the tidal synchronization timescale for coalescing ns binaries is expected to always be longer than the orbital decay timescale @xcite . therefore , a corotating state is unphysical . in addition , at large separation , the ns in these systems are expected to be spinning slowly ( see , e.g. , @xcite ; observed spin periods for radio pulsars in double ns systems are @xmath11ms , much longer than their final orbital periods ) . thus , the fluid in close ns binaries should remain nearly _ irrotational _ ( in the inertial frame ) and the stars in these systems can be described approximately by irrotational riemann ellipsoids @xcite . in paper 2 , we showed that nonsynchronized initial conditions can lead to significant differences in the hydrodynamic evolution of coalescing binaries , especially in the amount of mass ejected as a result of the rotational instability that develops during the merger . there are many difficulties associated with numerical calculations of binary mergers with irrotational initial conditions , especially in pn gravity . foremost of these is the difficulty in preparing the initial , quasi - equilibrium state of the binary system . in synchronized binaries , the two stars are at rest in a reference frame which corotates with the system , and thus relaxation techniques can be used to construct accurately the hydrostatic equilibrium initial state in this corotating frame ( see @xcite and paper 1 ) . for irrotational systems , no such frame exists in which the entire fluid would appear to be in hydrostatic equilibrium . instead , one must determine self - consistently the initial velocity field of the fluid in the inertial frame . otherwise ( e.g. , when simple spherical models are used ) , spurious oscillations caused by initial deviations from equilibrium can lead to numerical errors . this is especially of concern in pn gravity , where the strength of the gravitational force at any point in the ns contains terms proportional to the gravitational potential and pressure at 1pn order . another serious problem for hydrodynamic calculations with irrotational initial conditions is the issue of spatial resolution and numerical convergence . as was first pointed out in rs2 , in a frame corotating with the binary orbital motion , irrotational stars appear counterspinning so that , when they first make contact during the coalescence , a vortex sheet is formed along the interface . this tangential discontinuity is kelvin - helmholtz unstable at all wavelengths@xcite . the sheet is expected to break into a turbulent boundary layer which propagates into the fluid and generates vorticity through dissipation on small scales . how well this can be handled by 3d numerical calculations with limited spatial resolution is unclear . the outline of our paper is as follows . section ii presents a summary of our numerical methods , including a brief description of our pn sph code , and an explanation of the method used to construct irrotational initial conditions in pn gravity . additionally , we give the parameters and assumptions for all new calculations discussed in this paper . section iii presents our numerical results based on sph calculations for several representative binary systems , all with irrotational initial configurations , but varying mass ratios and ns eos . to test our numerical methods , we also study the effects of changing the initial binary separation and the numerical resolution . section iv presents gw energy spectra calculated from the runs in this and previous papers , as well as a discussion of how the measurement of spectral features could constrain the ns eos . a summary of our pn results and possible directions for further research , including the possibility of fully relativistic sph calculations , are presented in section v.

gravitational wave ( gw ) signals from coalescing binary neutron stars may soon become detectable by laser - interferometer detectors . these are the most physically realistic initial conditions just prior to merger , since the neutron stars in these systems are expected to be spinning slowly at large separation , and the viscosity of neutron star matter is too small for tidal synchronization to be effective at small separation . however , the large shear that develops during the merger makes irrotational systems particularly difficult to study numerically in 3d . in addition , in pn gravity , accurate irrotational initial conditions are much more difficult to construct numerically than corotating initial conditions . here we describe a new method for constructing numerically accurate initial conditions for irrotational binary systems with circular orbits in pn gravity . we then compute the 3d hydrodynamic evolution of these systems until the two stars have completely merged , and we determine the corresponding gw signals . we present results for systems with different binary mass ratios , and for neutron stars represented by polytropes with or . compared to mergers of corotating binaries we find that pn effects lead to clearly identifiable features in the gw energy spectrum of binary neutron star mergers , which may yield important information about the nuclear equation of state at extreme densities .

gravitational wave ( gw ) signals from coalescing binary neutron stars may soon become detectable by laser - interferometer detectors . using our new post - newtonian ( pn ) smoothed particle hydrodynamics ( sph ) code , we have studied numerically the mergers of neutron star binaries with irrotational initial configurations . these are the most physically realistic initial conditions just prior to merger , since the neutron stars in these systems are expected to be spinning slowly at large separation , and the viscosity of neutron star matter is too small for tidal synchronization to be effective at small separation . however , the large shear that develops during the merger makes irrotational systems particularly difficult to study numerically in 3d . in addition , in pn gravity , accurate irrotational initial conditions are much more difficult to construct numerically than corotating initial conditions . here we describe a new method for constructing numerically accurate initial conditions for irrotational binary systems with circular orbits in pn gravity . we then compute the 3d hydrodynamic evolution of these systems until the two stars have completely merged , and we determine the corresponding gw signals . we present results for systems with different binary mass ratios , and for neutron stars represented by polytropes with or . compared to mergers of corotating binaries , we find that irrotational binary mergers produce similar peak gw luminosities , but they shed almost no mass at all to large distances . the dependence of the gw signal on numerical resolution for calculations performed with sph particles is extremely weak , and we find excellent agreement between runs utilizing and sph particles ( the largest sph calculation ever performed to study such irrotational binary mergers ) . we also compute gw energy spectra based on all calculations reported here and in our previous works . we find that pn effects lead to clearly identifiable features in the gw energy spectrum of binary neutron star mergers , which may yield important information about the nuclear equation of state at extreme densities .

gr-qc0201040

r

the coalescence process for binary ns systems is essentially the same qualitatively whether they are initially synchronized or irrotational . prior to merger , both ns show tidal elongation as well as the development of a tidal lag angle @xmath70 , as noted in papers 1 and 2 , created as the ns continuously try to maintain equilibrium while the coalescence timescale gets shorter and shorter . the inner edge of each ns rotates forward relative to the binary axis , and the outer edge of each ns rotates backward . we define @xmath70 to be the angle in the horizontal plane between the axis of the primary moment of inertia of each ns and the axis connecting the centers of mass of the respective ns . for equal - mass systems , the angle is the same for both ns . for binaries with @xmath71 we always find a larger lag angle for the secondary than the primary . mass shedding through the outer lagrange points is suppressed in irrotational binaries , but the formation of a differentially rotating remnant is quite similar . that said , it is important to understand the different computational challenges presented by irrotational systems . to demonstrate this , in fig . [ fig : dvirr ] we show the evolution of run e1 , with a @xmath1 eos , a mass ratio @xmath72 , and an initial separation @xmath30 . it is in all ways similar to run b1 , except that the ns start from an irrotational configuration . rather than plot sph particle positions , we instead show the density contours of the matter in the orbital plane , overlaying the velocity of the material in the corotating frame , defined by taking a particle - averaged tangential velocity , such that @xmath73_i}{\sum_im_i(r_{cyl})_i } , \label{eq : omegac}\ ] ] where the cylindrical radius is defined as @xmath74 . in synchronized binaries , the material maintains only a small velocity in the corotating frame prior to first contact . in contrast , for irrotational binaries material on the inner edge of each ns is counterspinning in the corotating frame , and thus we see a large discontinuity in the tangential velocity when first contact is made . this surface layer , initially at low density , is kelvin - helmholtz unstable to the formation of turbulent vortices on all length scales . meanwhile , material on the outer edge of each ns has less angular momentum in an irrotational binary configuration than in a synchronized one . thus , there is less total angular momentum in the system , and mass shedding is greatly suppressed , as we will discuss further in sec . [ sec : irrotq1 ] . at late times , though , the remnant rotates differentially , with the same profile seen in the synchronized case , namely @xmath75 attaining its maximum value at the center of the remnant , and decreasing as a function of radius . in table [ table : gw ] , we show some basic quantities pertaining our gw results , as well as the initial time for all runs , and the tidal lag angle @xmath70 which existed at the moment of first contact . the negative values of @xmath76 merely reflect the fact that our runs require more than 20 dynamical times before reaching peak gw luminosity . lag angles @xmath70 are listed for the primary and secondary , respectively , in systems with @xmath77 . each gw signal we compute typically shows an increasing gw luminosity as the stars approach contact , followed by a peak and then a decline as the ns merge together . most runs then show a second gw luminosity peak of smaller amplitude . for all of our runs we list the maximum gw luminosity @xmath78 and maximum gw amplitude @xmath79 , where @xmath80 , @xmath81 , and @xmath82 are defined by eqs . ( 23)-(25 ) of paper 1 ( see also eqs . [ eq : hpl ] and [ eq : hcr ] below ) . quantities referring to the first and second luminosity peaks are denoted `` 1 '' and `` 2 '' , respectively . in addition , we show the time @xmath83 at which the second peak occurs . we continued several of our runs to late times to study the full gw signal produced during the coalescence , as well as to study the properties of the merger remnants that may form in these situations . for each of these runs , we list several of the basic parameters of the merger remnant in table [ table : final ] , using the values computed for the remnant at @xmath84 . we identify the remnant mass @xmath85 , defining the edge of the remnant by a density cut @xmath86 , as well as the gravitational mass of the remnant in the pn runs , where the gravitational mass , which differs from the rest mass , is given by @xmath87 ( see paper 1 for more details ) . additionally , we list the kerr parameter @xmath88 , central and equatorial values of the angular velocity , @xmath89 and @xmath90 , the semi - major axis @xmath34 and ratios of the equatorial and vertical radii @xmath91 and @xmath92 , and the ratio of the principal moments of inertia @xmath93 . to study the effect of the eos on the evolution of irrotational ns binaries , we calculated mergers of both @xmath1 and @xmath0 polytropes ( runs e1 and f1 , respectively ) . a comparison of the binary separations as a function of time , along with the dimensionless gw luminosities and amplitudes are shown in fig . [ fig : gweos ] . immediately apparent is a difference in the location of the dynamical stability limit found in the two calculations . the orbit of the binary system containing ns with a softer eos ( run f1 ) remains stable at separations where the binary system with the stiffer eos has already begun to plunge inward dynamically . we see that , as in the synchronized case presented in paper 2 , the peak gw luminosity in dimensionless form is larger for the softer eos , but after a secondary luminosity peak the remnant relaxes toward a spheroidal , non - radiating configuration ( with essentially no emission whatsoever after @xmath94 ) . we also note that the secondary peak occurs sooner after the primary peak , by a factor of @xmath95 . even though they are presumed to be unphysical , calculations started from a synchronized initial condition make up much of the body of work performed to date on the binary ns coalescence problem . noting this , we compare our irrotational run e1 , with parameters detailed above , to one similar in every respect but started with a synchronized initial condition , our run b1 . in the top panel of fig . [ fig : gwspin ] , we see that the inspiral tracks do not align particularly well . synchronized binaries contain more total energy , and are thus less dynamically stable than irrotational ones , leading to a more rapid inspiral , even before the stability limit is reached . additionally , the binary separation `` hangs up '' earlier , at a separation @xmath96 , indicating the onset of mass shedding , as matter begins to expand radially outward from the system . the middle and bottom panels of the figure show the gw signals and luminosity , respectively , for the two runs . while the initial luminosity peaks are similar for both runs , both in amplitude and morphology , the secondary peaks are vastly different . the secondary peak for the synchronized system is of considerably greater magnitude than in the irrotational case , and delayed relative to it . we conclude that , while the adiabatic index seems to be the dominant factor in determining the gw signal during the merger itself , the initial velocity profiles of the ns play a key role in the evolution of the remnant , as well as affecting the orbital dynamics during inspiral . to better understand the features found in the gw signals of these calculations , particle plots for runs e1 , b1 , and f1 are shown in fig . [ fig : xyeos ] . comparing the leftmost panels , we see that at @xmath28 , when the gw luminosity peaks , the mass configurations are qualitatively similar , although more low - density material is seen at the edges of the newly forming remnant in the run with the @xmath0 eos , conforming to the general density profile expected for a softer eos . more subtle is the greater extension of the matter found in the synchronized run . since material on the outer edge of each ns has greater angular momentum when a synchronized initial condition is chosen , the calculation shows that such an initial condition allows for greater efficiency at channeling material outward during the final moments of inspiral . this difference is made abundantly clear by a comparison of the calculations at @xmath97 , shown in the center panels . we see extensive mass shedding from the synchronized run , much less from the irrotational runs . there is significantly more mass shedding from the run with the softer eos , but most of the material remains extremely close to the remnant . finally , by @xmath98 , we see that the softer eos produces a nearly spherical remnant , whereas the calculations with a stiffer choice of eos produce remnants which are clearly ellipsoidal , and will continue to radiate gws for some time , albeit at a much lower amplitude than at the peak . the strong influence of the choice of both eos and initial velocity profile on the final state of the remnant is shown in fig . [ fig : finaleos ] for the same three runs . in the top panel , we see the angular velocity profiles of the remnants at @xmath84 . we see that the choice of eos plays an important role near the center of the remnant , but at @xmath99 , the velocity profiles are essentially identical . the pattern holds as well for the mass profiles , which are shown in the bottom panel . the softer eos leads to a more centrally condensed remnant , as we would expect , but the remnants formed in both irrotational calculations contain virtually all the system mass within @xmath96 , with no more than @xmath55 escaping to larger radii . there is slightly more mass shedding past @xmath100 for the softer choice of eos , since more of the low - density material originally found at the edges of the ns is shed through the outer lagrange points of the system . by comparison , the synchronized run sheds almost @xmath101 of the total system mass past @xmath100 , even though the angular velocity profile at small radii is nearly the same as for the irrotational run with the same choice of eos . even though all well - measured ns masses in relativistic binary pulsars appear roughly consistent with a single ns mass @xmath102 @xcite , it is important to consider cases where the two ns have somewhat different masses . the roles played by the primary and secondary in unequal - mass binary mergers are remarkably different compared to the picture developed above for equal - mass systems . in paper 2 , we found that the primary ns in the system remains virtually undisturbed , simply settling at the center of the newly formed merger remnant . the secondary is tidally disrupted prior to merger , forming a single thick spiral arm . most of the material originally located in the secondary eventually forms the outer region of the merger remnant , but a significant amount of material is shed to form a thick torus around the central core . in fig . [ fig : xyq8 ] , we show particle plots for runs e2 and f2 , with a @xmath1 and @xmath0 eos , respectively , both with mass ratio @xmath61 . in the leftmost panels , at @xmath28 , we see that the secondary , located on the left , is tidally disrupted as it falls onto the primary . for the softer , @xmath0 eos ( run f2 ) , we see a greater extension of the secondary immediately prior to merger , as well as early mass shedding from the surface of the primary , as material from the secondary essentially blows it off the surface of the newly forming remnant . this process continues , so that by @xmath97 ( center panels ) , the secondary has begun to shed a considerable amount of mass in a single spiral arm which wraps around the system . much like in the equal - mass case , the spiral arm is much broader for the softer eos . mass loss from the primary is greatly reduced in the system with the stiffer eos , with only a scattering of particles originally located in the primary lifted off the surface . finally , by @xmath98 ( right panels ) , we see that the spiral arm has in both cases begun to dissipate , leaving a torus around the merger remnant containing approximately @xmath103 of the total system mass . although the merger process is significantly different for equal - mass and unequal - mass binaries , the details of the inspiral phase are reasonably similar . in particular , the evolution of the binary separation for runs with @xmath61 , shown in the top panel of fig . [ fig : gwq8 ] is roughly similar to what was found in fig . [ fig : gweos ] for binaries with @xmath72 . for both choices of the eos , the dynamical stability limit is located at approximately the same separation for both @xmath61 and @xmath72 binaries , although prior to the onset of instability the more massive equal - mass binaries show a more rapid stable inspiral . as we found before , the dynamical stability limit occurs further inward for the softer choice of the ns eos . in paper 2 , we found that the scaling of the maximum gw amplitude and luminosity as a function of the system mass ratio followed a steeper power law in synchronized binaries than would be predicted by newtonian point - mass estimates . newtonian physics predicts that @xmath104 and @xmath105 for merging binary systems . rs found the scaling in their numerical calculations to roughly follow empirical power law relationships given by @xmath106 and @xmath107 . the discrepancy results from the unequal role played by the two components during the final moments before plunge . the primary , which remains relatively undisturbed , contributes rather little to the gw signal , especially during the final moments before coalescence . thus , the gw power is reduced as the mass ratio is decreased . similar results were found in paper 2 for pn calculations of synchronized binaries , especially a steeper decrease in the gw luminosity as a function of the mass ratio for binaries with a soft eos . in the middle and bottom panels of fig . [ fig : gwq8 ] we show the gw signals and luminosity , respectively , for runs e2 and f2 . we find the same strong decrease in the gw power , especially for the softer @xmath0 eos . we expect that there should be a strong observational bias towards detecting mergers of equal - mass components , especially if the eos is softer . all our results presented above for equal - mass binaries used an initial binary separation @xmath30 , in part for the sake of comparison with previous calculations for synchronized binaries . this choice agrees with the standard approach of utilizing the largest possible separation for which the calculation can be performed using a reasonable amount of computational resources . this also has the advantage that any small deviations from equilibrium will generally be damped away before the ns actually make contact . it also allows for the best determination of the dynamical stability limit , however , a large initial separation can create problems in the case of irrotational binaries , since numerical shear viscosity inherently present in sph codes can lead to some degree of tidal synchronization of the ns during the inspiral phase . thus , by the time the merger takes place , the ns will no longer be completely irrotational . to study this effect , we calculated mergers for equal - mass ns with a @xmath1 eos starting at initial separations of both @xmath30 ( the aforementioned run e1 ) and @xmath37 ( run t2 ) . in the top panel of fig . [ fig : rvstsep0 ] , we show the binary separation as a function of time for both runs , noting the good agreement throughout . at the very end , during the merger itself , we do see the beginning of a slight discrepancy , attributable in large part to greater mass shedding in the calculation started at greater separation . this is similar to what was seen in sec . [ sec : irrotq1 ] , where run b1 , which had greater spin angular momentum , showed greater mass shedding , but the effect is greatly reduced in magnitude here since the ns in run t2 are nowhere near complete synchronization at the moment of first contact . in the bottom panel of the figure , we plot the ratio of the net spin angular momentum of the ns about their own centers of mass to the total angular momentum of the binary system , as a function of time . we see that the ns do gradually acquire a rotation pattern which corresponds to the direction of corotation , although there is nowhere near enough time to synchronize the binary . the effect is greatly enhanced immediately prior to merger in both cases , as the ns develop tidal lag angles and become distorted . by the time the binary initially started from @xmath30 reaches a separation of @xmath108 , the net angular momentum around each ns around its center of mass is equal to approximately @xmath109 the value we would expect should the binary be synchronized . this difference persists throughout the inspiral phase when the two calculations are compared . in fig . [ fig : gwsep0 ] , we compare the gw signals and luminosities for the two runs . we find excellent agreement between the two waveforms , both in amplitude and in phase . both runs show the modulated , damped gw luminosity which is characteristic of all runs we have computed using pn gravity . there is a slight difference in the amplitude of the signal during the second gw luminosity peak , but we expect such differences to be minor in light of such issues as the uncertainty in the equation of state and the larger problem of a proper relativistic treatment of gravitation . to focus on the effect of varying @xmath60 on the final results , we show the final mass and angular velocity profiles of the remnants for the two calculations in fig . [ fig : finsep0 ] . the results are in good agreement , although we see that the greater spin angular momentum of the run started at greater initial separation leads to approximately three times as much mass being deposited in a halo which surrounds the remnant while remaining gravitationally bound to it . in both cases , however , the total mass in the halo is less than @xmath55 of the total system mass . the inner region of the remnant in the run started from @xmath30 actually spins slightly slower than in the run started further inward , even though the ns have a greater spin angular momentum at the moment of first contact , but only because angular momentum transport outward was marginally more efficient in this case . we conclude that the choice of initial separation plays very little role in determining the results of our calculations , so long as we start from an initial separation @xmath110 . as discussed in sec . [ sec : summary ] , 3d calculations with limited spatial resolution could lead to gw signals which are dependent upon the number of particles used . to test this , gw signals computed from ns merger calculations , we performed runs t1 , t2 , and t3 , which all have equal - mass ns , start from the same initial separation @xmath37 , and use a @xmath1 eos , but vary by two orders of magnitude in the number of sph particles used , from run t1 with @xmath111 to run t3 with @xmath112 sph particles . although we could have used @xmath30 as an initial separation , we felt that the smaller initial separation was justified given the large computational overhead required to do a calculation with a million sph particles . to the best of our knowledge , run t3 is the largest and highest resolution sph calculation of irrotational binary ns coalescence to date . a comparison of the gw signals in both polarizations , as well as the gw luminosities , is shown in fig . [ fig : gwnp ] . we see that the lowest resolution run t1 produces a gw signal clearly different than higher resolution runs t2 and t3 . the difference is due in part to initial oscillations of the ns about quasi - equilibrium . such oscillations are greatly reduced by increasing the number of sph particles . since all three calculations started with the same approximate ellipsoidal models ( see sec . [ sec : initconf ] ) , we conclude that the amplitudes of the initial fluctuations result primarily from numerical noise . the two runs with higher resolution show good agreement from beginning to end , producing nearly identical gw luminosities . reassuringly , the gw signals also remain in phase throughout , indicating that calculations with @xmath2 can indeed model well these irrotational binary mergers . to estimate the precision which can be achieved for these numerical resolutions , we show in table [ table : gravnp ] a comparison of gw quantities computed at various times for each of the three runs . at @xmath113 , @xmath28 , and @xmath97 , we show the total gw strain @xmath114 , and the instantaneous frequency of the gws @xmath115 , defined by the relations @xmath116 we see that for the two high resolution runs , no quantity shown varies by more than about @xmath117 , whereas the difference is more than @xmath118 in the computed gw strains at late times between our highest and lowest resolution runs . the vortices forming at the surface of contact are shown in detail in figs . [ fig : dvnp1 ] and [ fig : dvnp2 ] . density contours in the orbital plane are overlaid with velocity vectors , which are plotted in the corotating frame of the binary , as defined by eq . [ eq : omegac ] . the upper left panels show the evolution of run t1 , with the bottom left and right panels representing runs t2 and t3 , respectively . in fig . [ fig : dvnp1 ] , we show the state of the three runs at @xmath28 . immediately apparent is that vortices have formed to the largest extent in the lowest resolution run , whereas in the higher resolution run there is little sign of particles mixing , except at a large distance along the vortex sheet from the center of the newly forming remnant . by @xmath119 , shown in fig . [ fig : dvnp2 ] , we see that there is a slight difference between the high resolution calculations with regard to the direction of the material flowing along the vortex sheet . in the highest resolution run , the streams of material flow nearly in a straight line from one vortex to the other , whereas in the middle and lowest resolution runs , there is a larger region of material which is accelerated toward the very center of the remnant . overall , though , there is excellent agreement between the two highest resolution calculations . the agreement of the gw signals calculated from runs t2 and t3 lead us to believe that the small - scale differences seen in the matter near the vortex sheet does not carry over into the bulk of the mass , responsible for the gw emission . essentially , the quadrupole moment at any given instant is most sensitively dependent upon the orientation of the densest regions at the cores of the respective ns , which are unaffected by the small - scale motion in the vortex sheet . the infall of the cores is driven by dynamical instability , leading them to plunge inward and merge , disrupting the vortex sheet and leading to the formation of the merger remnant .

, we find that irrotational binary mergers produce similar peak gw luminosities , but they shed almost no mass at all to large distances . the dependence of the gw signal on numerical resolution for calculations performed with sph particles is extremely weak , and we find excellent agreement between runs utilizing and sph particles ( the largest sph calculation ever performed to study such irrotational binary mergers ) .

gravitational wave ( gw ) signals from coalescing binary neutron stars may soon become detectable by laser - interferometer detectors . using our new post - newtonian ( pn ) smoothed particle hydrodynamics ( sph ) code , we have studied numerically the mergers of neutron star binaries with irrotational initial configurations . these are the most physically realistic initial conditions just prior to merger , since the neutron stars in these systems are expected to be spinning slowly at large separation , and the viscosity of neutron star matter is too small for tidal synchronization to be effective at small separation . however , the large shear that develops during the merger makes irrotational systems particularly difficult to study numerically in 3d . in addition , in pn gravity , accurate irrotational initial conditions are much more difficult to construct numerically than corotating initial conditions . here we describe a new method for constructing numerically accurate initial conditions for irrotational binary systems with circular orbits in pn gravity . we then compute the 3d hydrodynamic evolution of these systems until the two stars have completely merged , and we determine the corresponding gw signals . we present results for systems with different binary mass ratios , and for neutron stars represented by polytropes with or . compared to mergers of corotating binaries , we find that irrotational binary mergers produce similar peak gw luminosities , but they shed almost no mass at all to large distances . the dependence of the gw signal on numerical resolution for calculations performed with sph particles is extremely weak , and we find excellent agreement between runs utilizing and sph particles ( the largest sph calculation ever performed to study such irrotational binary mergers ) . we also compute gw energy spectra based on all calculations reported here and in our previous works . we find that pn effects lead to clearly identifiable features in the gw energy spectrum of binary neutron star mergers , which may yield important information about the nuclear equation of state at extreme densities .

1405.4328

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the unprecedented level of control in ultra - cold atom experiments has allowed for the realization of paradigmatic condensed matter models @xcite . in these systems the inter - particle interactions can be tuned by varying the scattering length through feshbach resonances and the atoms can be trapped in various geometries @xcite . such a flexibility makes ultra - cold atoms an almost ideal candidate for the study of strongly correlated many - body quantum systems as well as a playground for emerging new states of matter . these , in particular , include paired superfluids . recent experimental success in trapping ultra - cold bosonic atomic mixtures @xcite has rendered the study of pairing between components very timely . the impressively rapid experimental progress towards controlling polar molecules @xcite gives hope for accessing quantum many body systems with long range and anisotropic interaction in a very near future @xcite . a prominent example of bosonic systems currently available experimentally consists of an array of coupled one - dimensional tubes , with the interaction between tubes provided by dipolar forces . in the absence of the inter - tube tunneling , this system can be relevant to multi - component atomic mixtures . in general , when such a tunneling is finite , it represents coupled spin chains , which is one of the central topics in low - dimensional condensed matter physics @xcite . [ fig : setup ] of them are shown ) ; dashed arrows indicate the intra - plane @xmath1 and the inter - plane interactions @xmath2 . , scaledwidth=35.0% ] in the present work we discuss possible phases in a system of hardcore bosons confined to a stack of @xmath3 one - dimensional lattices tubes ( see fig . [ fig : setup ] ) . bosons in neighboring tubes interact via inter - tube interaction ( either nearest - neighbor or dipole - dipole ) , with the inter - tube tunneling suppressed ( this can be achieved experimentally with a deep optical lattice potential along the direction perpendicular to the tubes ) . our focus is on quantum many - body phases of self - assembled chains of molecules from different tubes @xcite . in previous theoretical studies , mostly variational methods have been used . in ref . @xcite the multilayered system has been mapped to a model amenable to classical monte carlo technique , and it has been shown that bosons in a stack of one - dimensional tubes can form superfluids of multi - atomic complexes chain superfluids ( csf ) @xcite , each chain consisting of one molecule from each tube , and there is , in general , a threshold for the csf formation . an interesting opportunity for the emergence of exotic parafermions , as a generalization of majorana fermions , in layered systems has been proposed in refs . @xcite and tested by monte carlo simulations @xcite . it was also suggested that in two parallel 1d lattices with no inter - tube tunneling an exotic superfluid , consisting of @xmath4 molecules from one tube and @xmath5 ones form the other , should also be possible to realize @xcite . in contrast to what was previously done , here we study the actual quantum hamiltonian of hard - core bosons by means of the _ ab initio _ path integral quantum monte carlo ( qmc ) simulations ( in continuous time ) using a multi - worm algorithm @xcite an extension of the worm algorithm @xcite and its two - worm modification @xcite . as we mentioned above , our algorithm is equally relevant to atomic mixtures and coupled spin chains . we will show that csf and other phases can be induced by infinitesimally small inter - layer interaction . our study is a first step toward _ ab - initio _ simulations of more involved cases including spin ladders and polar molecules with inter - tube tunneling . using this algorithm it should be possible to provide accurate recommendations for the experimental realizations of the complex dipolar phases .

hard - core dipolar bosons trapped in a parallel stack of 1d optical lattices ( tubes ) can develop several phases made of composites of particles from different tubes : superfluids , supercounterfluids and insulators as well as mixtures of those . bosonization analysis shows that these phases are threshold - less with respect to the dipolar interaction , with the key control knob " being filling factors in each tube , provided the inter - tube tunneling is suppressed . the effective _ ab - initio _ quantum monte carlo algorithm capturing these phases is introduced and some results are presented .

hard - core dipolar bosons trapped in a parallel stack of 1d optical lattices ( tubes ) can develop several phases made of composites of particles from different tubes : superfluids , supercounterfluids and insulators as well as mixtures of those . bosonization analysis shows that these phases are threshold - less with respect to the dipolar interaction , with the key control knob " being filling factors in each tube , provided the inter - tube tunneling is suppressed . the effective _ ab - initio _ quantum monte carlo algorithm capturing these phases is introduced and some results are presented .

1301.6674

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as of november , 2012 , more than 800 extra - solar planetary systems have been detected , and @xmath8 additional candidate systems from the _ kepler _ mission are waiting to be confirmed @xcite . one of the primary goals of the ongoing radial velocity ( rv ) and transit surveys is to identify a terrestrial mass planet ( @xmath9m@xmath10 ) in the so - called habitable zone ( hz ) , which is traditionally defined as the circumstellar region in which a terrestrial - mass planet with a @xmath0-@xmath1-n@xmath11 atmosphere can sustain liquid water on its surface @xcite . several potential hz planet candidates have already been detected , @xcite and it is expected that this number will greatly increase as time passes @xcite . in the near future we may be able to study habitable planets orbiting nearby m stars . these planets are relatively close to their parent stars , leading to shorter orbital periods and an increase in the probability of a transit . nasa s _ james webb space telescope _ ( jwst ) , scheduled to launch in 2018 , is considered to be marginally capable of obtaining a transit spectrum of an earth - like planet orbiting a late m dwarf @xcite . several other surveys are either underway ( * ? ? ? * mearth ) or getting ready to be commissioned ( * ? ? ? * hpf ) in an attempt to discover rocky planets in the hzs of low mass stars . the hz limits that were cited in many recent discoveries were obtained from 1-d radiative - convective , cloud - free climate model calculations by @xcite . for our sun , these authors estimated the boundaries of the hz to be @xmath12 au for the inner edge and @xmath13 au for the outer edge . these values represent the `` water loss '' and `` maximum greenhouse '' limits , respectively . other , less conservative limits for the inner edge are the `` runaway greenhouse '' and `` recent venus '' limits . the latter estimate is empirical , based on the inference that venus has not had liquid water on its surface for at least the last 1 billion years @xcite . for the outer edge , there is a corresponding `` early mars '' empirical estimate , based on the inference that mars did have liquid water on its surface 3.8 billion years ago . ( the `` 1st @xmath0 condensation '' limit of @xcite , should now be disregarded , as it has been shown that @xmath0 clouds generally warm a planet s climate @xcite ) . some studies have investigated the effects of clouds on planetary emission spectra of earth - like planets in a 1d model @xcite , while others studied the habitability of specific systems , particularly gl 581 , in 1d @xcite and 3d @xcite . several other studies @xcite parameterized these results to estimate relationships between hz boundaries and stellar parameters for stars of different spectral types . although these studies provided useful estimates of the hz width , the @xcite model has become outdated , for several reasons : 1 . @xcite used ` band models ' for @xmath1 and @xmath0 absorption in the thermal - infrared . these coefficients were considered valid up to @xmath14 k. these coefficients were later replaced @xcite by coefficients generated using the correlated-_k _ technique @xcite . a line - by - line ( lbl ) radiative transfer model , in this case lblrtm @xcite , was used to generate detailed spectra for @xmath1 and @xmath0 at a variety of different temperatures and pressures . once the detailed spectra were calculated , separate broad - band k - coefficients for both @xmath1 and @xmath0 were generated by r. freedman using standard procedures . but these coefficients were only derived for temperatures @xmath15 k and should therefore underestimate thermal - ir absorption in warm , moist greenhouse atmospheres . ( this prediction was verified by direct experimentation with that model . ) furthermore , the coefficients adopted by @xcite and used in subsequent climate modeling studies by the kasting research group were obtained using hitran 1996 database and had not been updated since then . recent studies @xcite have pointed that the @xcite model may have significantly overestimated absorption of thermal - ir radiation by collision - induced absorption ( cia ) bands of @xmath0 , which may affect the outer edge of the hz . the @xcite calculations spanned stellar effective temperatures from @xmath5 k to @xmath16 k , corresponding approximately to stellar classes f0 to m0 . stellar effective temperature affects the hz boundaries because the radiation from f stars is bluer relative to that from the sun , whereas the radiation from k and m stars is redder , and this affects calculated planetary albedos . the hz limits from @xcite model do not include m stars with effective temperatures lower than @xmath16 k. as pointed out above , such stars are promising candidates for current observational surveys because their hzs are closer to the star . therefore , potential rocky planets in the hzs will have shorter orbital periods and higher probability of transit . in this paper we address all the above major issues with the goal of deriving new , improved estimates for the boundaries of the hz . the outline of the paper is as follows : in [ model ] we describe our 1-d cloud - free climate model , corresponding model updates and model validation with other studies . in [ results ] we present results from our climate model and discuss various hz limits for our earth . [ fgkm ] presents hz boundaries around f , g , k and m spectral stellar spectral types , then provides a generalized expression to calculate hz boundaries and compares these boundaries with previous studies . we discuss the implications of these new results for currently known exoplanet planetary systems in [ discussion ] and present our conclusions in [ conclusions ] .

identifying terrestrial planets in the habitable zones ( hzs ) of other stars is one of the primary goals of ongoing radial velocity and transit exoplanet surveys and proposed future space missions . most current estimates of the boundaries of the hz are based on 1-d , cloud - free , climate model calculations by . however , this model used band models which were based on older hitran and hitemp line - by - line databases . additional calculations are performed for stars with effective temperatures between k and k , and the results are presented in parametric form , making them easy to apply to actual stars .

identifying terrestrial planets in the habitable zones ( hzs ) of other stars is one of the primary goals of ongoing radial velocity and transit exoplanet surveys and proposed future space missions . most current estimates of the boundaries of the hz are based on 1-d , cloud - free , climate model calculations by . however , this model used band models which were based on older hitran and hitemp line - by - line databases . the inner edge of the hz in model was determined by loss of water , and the outer edge was determined by the maximum greenhouse provided by a atmosphere . a conservative estimate for the width of the hz from this model in our solar system is 0.95 - 1.67 au . here , an updated 1-d radiative - convective , cloud - free climate model is used to obtain new estimates for hz widths around f , g , k and m stars . new and absorption coefficients , derived from the hitran 2008 and hitemp 2010 line - by - line databases , are important improvements to the climate model . according to the new model , the water loss ( inner hz ) and maximum greenhouse ( outer hz ) limits for our solar system are at au and au , respectively , suggesting that the present earth lies near the inner edge . additional calculations are performed for stars with effective temperatures between k and k , and the results are presented in parametric form , making them easy to apply to actual stars . the new model indicates that , near the inner edge of the hz , there is no clear distinction between runaway greenhouse and water loss limits for stars with t k which has implications for ongoing planet searches around k and m stars . to assess the potential habitability of extrasolar terrestrial planets , we propose using stellar flux incident on a planet rather than equilibrium temperature this removes the dependence on planetary ( bond ) albedo , which varies depending upon the host star s spectral type . we suggest that conservative estimates of the hz ( water loss and maximum greenhouse limits ) should be used for current rv surveys and _ kepler _ mission to obtain a lower limit on , so that future flagship missions like _ tpf - c _ and _ darwin _ are not undersized . our model does not include the radiative effects of clouds ; thus , the actual hz boundaries may extend further in both directions than the estimates just given . 2oh_2o 2h_2 2co_2 4ch_4

1301.6674

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in the subsections that follow , we estimate hz boundaries around a star similar to our sun . we first compare results from our model using hitran and hitemp databases , estimate hz limits for non - earth - like planets and discuss the effect of clouds on the hz boundaries . the inner edge of the hz is calculated by increasing the surface temperature of a fully saturated `` earth '' model from @xmath77 k up to @xmath78 k. the effective solar flux @xmath79 , which is the value of solar constant required to maintain a given surface temperature , is calculated from the ratio between the _ net _ outgoing ir flux @xmath80 and the _ net _ incident solar flux @xmath81 , both evaluated at the top of the atmosphere . the total flux incident at the top of the atmosphere is taken to be the present solar constant at earth s orbit @xmath59 wm@xmath60 . the planetary albedo is calculated as the ratio between the upward and downward solar fluxes . the calculated radiative fluxes , planetary albedo and water vapor profile for various surface temperatures are shown in fig . [ earthfluxes ] . absorption coefficients derived from the hitemp 2010 database , overlaid by bps formalism @xcite , were used in generating these results . [ fir ] shows that @xmath80 increases with surface temperature and then levels out at 291 wm@xmath60 , as the atmosphere becomes opaque to infrared radiation at all wavelengths closely matches with the value from fig . 4.37 of @xcite for a planet saturated with pure water vapor atmosphere and with a surface gravity of @xmath82ms@xmath60 . ] . beyond 2000 k , @xmath80 increases again as the lower atmosphere and surface begin to radiate in the visible and near - ir , where the water vapor opacity is low . @xmath81 initially increases as a consequence of absorption of near - ir solar radiation by @xmath1 . it then decreases to a constant value ( @xmath83 wm@xmath60 ) at higher temperatures as rayleigh scattering becomes important . planetary albedo ( fig . [ albp ] ) provides an alternative way of understanding this behavior . it goes through a minimum at a surface temperature of 400 k , corresponding to the maximum in @xmath81 , and then flattens out at a value of 0.193 . the inner edge of the hz for our sun can be calculated from fig . the behavior of @xmath80 and @xmath81 causes @xmath79 to increase initially and then remain constant at higher temperatures . two limits for the ihz boundary can be calculated . the first one is the `` moist greenhouse '' ( or water - loss ) limit which is encountered at a surface temperature of @xmath84 k when @xmath85 . at this limit , the water vapor content in the stratosphere increases dramatically , by more than an order of magnitude , as shown in fig . this is the relevant ihz boundary for habitability considerations , although it should be remembered that the actual inner edge may be closer to the sun if cloud feedback tends to cool the planet s surface , as expected . inventory assumed here is equal to the amount of water in earth s oceans @xmath86 grams . this amounts to @xmath87 atoms per cm@xmath60 . once the stratosphere becomes wet , water vapor photolysis releases hydrogen which can escape to space by diffusion limited escape rate . the time scale for water loss approaches the age of the earth when the mixing ratio of water is @xmath89 , which happens at a surface temperature of 340 k. ] the orbital distance corresponding to the cloud - free water loss limit is @xmath90 au for an earth - like planet orbiting the sun . the second ihz limit is the runaway greenhouse at which the oceans evaporate entirely . the limiting @xmath79 from fig . [ seff ] is 1.06 which corresponds to a distance of 0.97 au . both calculated ihz limits are significantly farther from the sun than the values found by @xcite ( @xmath12 au for the water - loss limit and @xmath91 au for the runaway greenhouse ) . the difference is caused by increased atmospheric absorption of incoming solar radiation by @xmath1 in the new model . as pointed out by @xcite , a third estimate for the ihz boundary can be obtained from radar observations of venus by magellan spacecraft , which suggest that liquid water has been absent from the surface of venus for at least 1 gyr @xcite . the sun at that time was @xmath92 of the present day luminosity , according to standard stellar evolutionary models ( * ? ? ? * ; * ? ? ? * see table 2 ) . the current solar flux at venus distance is @xmath93 times that of earth . therefore , the solar flux received by venus at that time was @xmath94 times that of earth . this empirical estimate of the ihz edge corresponds to an orbital distance of @xmath95 au for the present day . note that this distance is greater than venus orbital distance of 0.72 au because the constraint of surface water was imposed at an earlier time in the planet s history . in fig . [ bps ] we show @xmath80 as a function of surface temperature ( similar to fig . [ fir ] ) . we wish to compare the outgoing ir calculated from hitran & hitemp databases with and without overlaying the continuum absorption . [ bps ] shows two significant differences : 1 . the limiting value of @xmath80 which leads to a runaway greenhouse happens at a much higher value ( 440 wm@xmath60 , black & green curves ) when the bps @xmath1 continuum formalism is not implemented , and at a lower @xmath80 ( 291 wm@xmath60 , red & blue curves ) when the bps continuum is included in our model . the continuum is based on measurements of absorption in the water vapor window regions ( i.e @xmath96 @xmath25 and @xmath97 @xmath25 ) . at high temperatures , the contribution of the continuum absorption in these window regions becomes significant , and this , in turn , decreases the outgoing ir flux . the moist - greenhouse ( water loss ) limit moves much closer to the sun ( to 0.87 au ) when continuum absorption is not included , as compared to 0.99 au when it is included in our model . this is a direct consequence of the differences in @xmath80 described above : when @xmath80 increases with the continuum turned off , @xmath79 ( ratio of @xmath80 to @xmath81 ) increases and the ihz distance @xmath98 decreases . the result can be understood physically by noting that in the model where the continuum is absent , the planet needs more effective solar flux to maintain a given surface temperature because more thermal - ir radiation leaks away into space ; hence , the ihz boundary must move inward . + a similar change can be seen in the runaway greenhouse limit : the ` no bps ' model transitions to runaway at a higher @xmath79 than does the ` with bps ' model . the corresponding runaway greenhouse limit changes from 0.97 au ( with continuum absorption ) to 0.76 au ( without continuum absorption ) . fig . [ bps ] also shows that the upturn in @xmath80 beyond 800 k happens at lower surface temperatures when continuum absorption is not included . it should be remembered that this upturn in @xmath80 happens because , as the surface warms , the region in the troposphere over which the temperature profile follows a dry adiabat expands upward , while the moist convective layer in the upper troposphere becomes thinner . eventually , when the moist convective region ( the cloud layer ) begins thin enough , radiation emitted from the dry adiabatic portion of the atmosphere begins to escape to space . the dry adiabatic lapse rate is steeper than the moist adiabatic lapse rate by about a factor of 9 ( @xmath99 k / km vs. 1.1 k / km ) ; hence , the emitted radiation flux is much higher . this can be understood from the integrated form of schwarzchild s equation , which shows that the emitted flux is proportional to the temperature gradient . ( see , e.g. , eq . a4 in @xcite ) unlike @xcite , we find that the emitted flux increases at all thermal - ir wavelengths shorter than 4 @xmath100 m . the amount of visible radiation emitted remains negligible for surface temperatures of 2200 k or below . without the continuum there are fewer lines to cause absorption in these thermal - ir bands and hence a lower temperature would suffice to cause the upturn . [ bps ] also shows that the model that includes both hitemp & continuum ( red curve ) is the one that absorbs the most outgoing ir radiation ( which is the one that was used to derive inner hz limits in [ ihz ] ) . in determining the outer edge of the hz , the surface temperature of an earth - like planet with 1-bar n@xmath11 atmosphere was fixed at 273 k and the atmospheric @xmath0 partial pressure , @xmath101 , was varied from 1 to 35 bars ( the saturation vapor pressure for @xmath0 at that temperature ) . the stratospheric temperature was chosen as follows : the model atmosphere ( mars - like planet ) in which the onset of @xmath0 condensation occurs has a cold - trap temperature of 154 k at an altitude where the ratio of the saturation vapor pressure to the ambient pressure is unity . we replace the temperature profile above this altitude with a constant temperature of 154 k. this allows us to calculate the solar flux ( @xmath79 ) required to maintain a global mean surface temperature of 273 k as explained in [ ihz ] . our working hypothesis is that atmospheric @xmath0 would accumulate as these planets cooled because of the negative feedback provided by the carbonate - silicate cycle . results from our model calculations are shown in fig . [ outerfluxes ] . ) from hitemp and hitran databases , with ( blue & red curves ) and without ( black & green curves ) continuum absorption . @xmath80 is lower in the with bps case for both the databases because the continuum absorption becomes significant in the water vapor window regions.,scaledwidth=92.0% ] the incident solar ( @xmath81 ) and outgoing ir ( @xmath80 ) fluxes are shown in fig . [ firco2 ] . @xmath80 decreases initially as @xmath0 partial pressure is increased ; this is an indication of greenhouse effect of @xmath0 . at @xmath99 bars , @xmath80 asymptotically approaches a constant value as the atmosphere becomes optically thick at all infrared wavelengths . @xmath81 decreases monotonically with increases in @xmath0 partial pressure as a result of increased rayleigh scattering . correspondingly , the planetary albedo increases to high values at large @xmath0 partial pressures , as shown in fig . [ albpco2 ] . the solar and ir fluxes , acting in opposite directions , create a minimum of @xmath102 at a @xmath0 partial pressure of @xmath103 bar ( fig . [ seffco2 ] ) , corresponding to a distance @xmath104 au . this defines the maximum greenhouse limit on the outer edge of the hz . by comparison , @xcite model predicted @xmath105 au for the maximum greenhouse limit . as emphasized earlier , radiative warming by @xmath0 clouds is neglected here , even though they should be present in this calculation . therefore , our ohz limit should be considered as a conservative estimate , that is , the real outer edge is probably farther out . as with the inner edge model , a more optimistic empirical limit on the ohz can be estimated based on the observation that early mars was warm enough for liquid water to flow on its surface @xcite . assuming the dried up riverbeds and valley networks on martian surface are 3.8 gyr old , the solar luminosity at that time would have been @xmath106 of the present value ( see eq.(1 ) in @xcite and table 2 in @xcite ) . the present - day solar flux at mars distance is @xmath107 times that of earth . therefore , the solar flux received by mars at 3.8 gyr was @xmath108 times that of earth . the corresponding ohz limit today , then , would be @xmath109 au . note that this distance exceeds the maximum greenhouse limit of 1.70 au estimated above indicating that to keep early mars wet , additional greenhouse gases other @xmath0 and @xmath1 may be required . in fact , @xcite show that a 3-bar atmosphere containing 90 percent @xmath0 and 10 percent h@xmath11 could have raised the mean surface temperature of early mars above the freezing point of water . the warming is caused by the collision - induced absorption due to foreign - broadening by molecular hydrogen . it should be acknowledged that some authors ( e.g. , @xcite ) do not agree that early mars must have been warm ; however , in our view , these cold early mars models do not produce enough rainfall to explain valley formation @xcite . we summarize various cloud - free hz boundary estimates for earth in table [ table1 ] . although we updated our radiative transfer model to incorporate new absorption coefficients , this by itself may not yield a significantly better estimate for the width of the habitable zone . the reason is that it is widely acknowledged that the hz boundaries will be strongly influenced by the presence of clouds . @xmath1 clouds should move the inner edge inwards @xcite because their contribution to a planet s albedo is expected to outweigh their contribution to the greenhouse effect . ( a dense @xmath1 atmosphere is already optically thick throughout most of the thermal - ir , so adding clouds has only a small effect on the outgoing ir radiation . ) conversely , @xmath0 ice clouds are expected to cause warming in a dense co2 atmosphere because they backscatter outgoing thermal - ir radiation more efficiently than they backscatter incoming visible / near - ir radiation @xcite . one can demonstrate the nature of these cloud influences using 1-d models , as was done in @xcite . making quantitative statements is difficult , however , because the warming or cooling effect of clouds depends on a host of parameters , including their heights , optical depths , particle sizes , and most importantly , fractional cloud coverage . @xcite obtained as much as 70 degrees of warming out of an optical depth 10 @xmath0 cloud with @xmath110 cloud cover , but that warming dropped by 30 degrees if fractional cloud cover was reduced to @xmath58 . realistic fractional cloud cover for condensation clouds is closer to @xmath111 , because such clouds tend to form on updrafts , and approximately half the air in the troposphere is rising at any one time while the other half is descending . the best way to incorporate cloud effects in a climate calculation is to use a 3-d general circulation model ( gcm ) . attempts were made to explain warm early mars using such 3-d models @xcite but none have yet succeeded . one can , however , do significantly better than in our 1-d model , and so further research in this area is warranted @xcite . model & moist & runaway & recent venus & maximum & early mars + & greenhouse & greenhouse & & greenhouse & + this paper & 0.99 au & 0.97 au & 0.75 au & 1.70 au & 1.77 au + & & & & & + @xcite & 0.95 au & 0.84 au & 0.75 au & 1.67 au & 1.77 au + [ table1 ] in table [ table2 ] , we show the effect of surface gravities on the hzs of two planets . these planetary gravities were selected to encompass the mass range from mars ( gravity of 3.73 ms@xmath60 ) to a roughly 10 m@xmath10 super - earth ( gravity of 25 ms@xmath60 ) . both planets were assumed to have a 1 bar background n@xmath11 atmosphere . this may be unrealistic because proportionately more nitrogen is put on the smaller planet than the larger one ; however , this allows direct comparison with @xcite . table [ table2 ] shows that the habitability limits move slightly outward for a mars - sized planet and inward for a super - earth . this is because the column depth is larger for a mars - sized planet , which increases the greenhouse effect ( at the inner edge ) and albedo ( at the outer edge ) . since the inner edge moves closer to the star for the super - earth planet , while the outer edge changed little , we can conclude that , for a given surface pressure , larger planets have somewhat wider habitable zones than do small ones . model & moist & runaway & maximum + & greenhouse & greenhouse & greenhouse + mars - sized planet@xmath112 & 1.035 au & 1.033 au & 1.72 au + & & & + earth & 0.99 au & 0.97 au & 1.70 au + & & & + super - earth@xmath113 & 0.94 au & 0.92 au & 1.67 au + & & & + @xmath114 bar@xmath115 & * 1.00 au * & * 0.97 au * & + & & & + @xmath116 bar & 1.02 au & 0.97 au & + & & & + @xmath117 bar & 1.02 au & 0.97 au & + & & & + @xmath118 bar & 0.99 au & 0.97 au & + @xmath112 surface gravity = 3.73 m.s@xmath60 @xmath113 surface gravity = 25 m.s@xmath60 @xmath115 @xmath119 bar for our standard earth model . note that these @xmath0 pressures are not actual partial pressures ; rather , they represent the surface pressure that would be produced if this amount of @xmath0 were placed in the atmosphere by itself . the 330 ppmv of @xmath0 in our standard 1-bar atmosphere would produce a surface pressure of @xmath120 bar if the rest of the atmosphere was not present . when lighter gases such as n@xmath11 and o@xmath11 are present , they increase the atmospheric scale height and cause @xmath0 to diffuse upward , thereby lowering its partial pressure at the surface . [ table2 ] we also performed sensitivity tests on the inner edge of the hz by varying the amount of atmospheric @xmath0 ( the outer edge calculation already factors in this change in @xmath0 ) . it is quite possible that some terrestrial planets may have varying amount of @xmath0 because of different silicate weathering rates . as shown in table . [ table2 ] , changes in @xmath101 would not change the runaway greenhouse limit , as it is reached in an @xmath1-dominated atmosphere . the moist greenhouse limit does change , as an increase in @xmath101 increases the surface temperature , and hence facilitates water loss . the maximum destabilization occurs at a @xmath121 bar approximately 10 times the present terrestrial @xmath101 level ( the critical distance , shown in bold in table [ table2 ] , is 1.00 au ) . this suggests that a 10-fold increase in @xmath0 concentration relative to today could push earth into a moist greenhouse state ( assuming a fully saturated atmosphere ) . by contrast , the maximum destabilization occurred at @xmath122 times the present @xmath0 level in @xcite . at larger @xmath101 values the increase in surface pressure outstrips the increase in the saturation vapor pressure of water , so the atmosphere becomes more stable against water loss @xcite . we conclude that planets with few tenths of a bar of @xmath101 have narrower hzs than planets like earth on which @xmath101 is maintained at lower values by the carbonate - silicate cycle .

a conservative estimate for the width of the hz from this model in our solar system is 0.95 - 1.67 au . here , an updated 1-d radiative - convective our model does not include the radiative effects of clouds ; thus , the actual hz boundaries may extend further in both directions than the estimates just given .

identifying terrestrial planets in the habitable zones ( hzs ) of other stars is one of the primary goals of ongoing radial velocity and transit exoplanet surveys and proposed future space missions . most current estimates of the boundaries of the hz are based on 1-d , cloud - free , climate model calculations by . however , this model used band models which were based on older hitran and hitemp line - by - line databases . the inner edge of the hz in model was determined by loss of water , and the outer edge was determined by the maximum greenhouse provided by a atmosphere . a conservative estimate for the width of the hz from this model in our solar system is 0.95 - 1.67 au . here , an updated 1-d radiative - convective , cloud - free climate model is used to obtain new estimates for hz widths around f , g , k and m stars . new and absorption coefficients , derived from the hitran 2008 and hitemp 2010 line - by - line databases , are important improvements to the climate model . according to the new model , the water loss ( inner hz ) and maximum greenhouse ( outer hz ) limits for our solar system are at au and au , respectively , suggesting that the present earth lies near the inner edge . additional calculations are performed for stars with effective temperatures between k and k , and the results are presented in parametric form , making them easy to apply to actual stars . the new model indicates that , near the inner edge of the hz , there is no clear distinction between runaway greenhouse and water loss limits for stars with t k which has implications for ongoing planet searches around k and m stars . to assess the potential habitability of extrasolar terrestrial planets , we propose using stellar flux incident on a planet rather than equilibrium temperature this removes the dependence on planetary ( bond ) albedo , which varies depending upon the host star s spectral type . we suggest that conservative estimates of the hz ( water loss and maximum greenhouse limits ) should be used for current rv surveys and _ kepler _ mission to obtain a lower limit on , so that future flagship missions like _ tpf - c _ and _ darwin _ are not undersized . our model does not include the radiative effects of clouds ; thus , the actual hz boundaries may extend further in both directions than the estimates just given . 2oh_2o 2h_2 2co_2 4ch_4

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a straightforward application of the calculations presented in the previous sections is to apply them to currently known exoplanetary systems . [ observed ] shows various habitable zone boundaries ( expressed in terms of effective stellar flux ) as a function of stellar @xmath129 . the orbital parameters of the planets and stellar characteristics were obtained from exoplanets.org @xcite . the green - shaded habitable region is bounded by the moist greenhouse limit ( inner edge ) and the maximum greenhouse limit ( outer edge ) . several currently known terrestrial mass exoplanets that have been proposed to be in the hz by various studies are also shown . an important insight that can be obtained from this figure ( and one that can not be seen in the hz distance plot , fig . [ selsisdis ] ) is that a terrestrial mass planet that lies within the two vertical dashed - lines in the green shaded region is in the hz irrespective of the type of star it is orbiting . the corresponding flux boundaries for which a rocky planet is ` definitely ' in the hz are @xmath143 and @xmath144 . currently , two exoplanets fall within this region , hd 40307 g @xcite and gl 581 g @xcite . the detection of planets and orbital parameters for gl581 is complicated by the low amplitudes of the signal , stellar activity , and possible red noise . we have included a ` ? ' for gl 581 system of planets in the plot , indicating that there is an active ongoing discussion in the literature about the number of planets in this system and their exact orbital parameters @xcite . furthermore , for stars with t@xmath6 k , there is no clear distinction between runaway greenhouse and the moist - greenhouse limits . the reason is that for these stars , there are more photons available in the ir part of the spectrum , where @xmath1 is a good absorber . therefore a planet with @xmath1 dominated atmosphere quickly goes into runaway once it reaches the moist - greenhouse limit . note that another suggested hz candidate planet , hd 85512b @xcite , receives more than 5 times the stellar flux received by our earth , placing it even beyond the most liberal ( ` recent venus ' ) estimate of the inner edge . hence , it is very likely that this planet is not in the hz of its star . a question of importance to the exoplanet community is which hz limits to choose when identifying potentially habitable planets . for current rv surveys and _ kepler _ mission one should use the most conservative limits ( moist greenhouse and maximum greenhouse ) , because this will give a lower limit on @xmath7 , the fraction of sun - like stars that have at least one planet in the habitable zone @xcite . if one is interested in designing a future flagship mission , such as _ terrestrial planet finder _ ( tpf ) or _ darwin _ , then using these conservative limits ( which results in a lower limit on @xmath7 ) ensures that the telescope is not undersized . if , however , one was analyzing data obtained from such a telescope , the most optimistic limits ( recent venus and early mars ) should be used because one would not want to miss out on any potentially habitable planets . in fig . [ exoplanetsplot ] , we show the incident stellar flux as a function of planetary mass for the currently known exoplanets . the masses are obtained from exoplanets.org when available . also shown are habitable zone flux boundaries calculated from eq.([hzeq ] ) for terrestrial mass planets ( @xmath128m@xmath10 - @xmath82m@xmath10 ) . for the outer box ( light grey ) , the upper bound on the flux is taken to be the moist greenhouse limit for a star with @xmath145 k and the lower bound is the maximum greenhouse limit for a star with @xmath127 k. these are indicated by the diagonally opposite points on the green shaded region in fig . [ observed ] . for the inner box ( dark grey ) , the flux limits are the dashed lines in the green shaded region of fig . [ observed ] . the significance of this plot is that terrestrial planets in the inner box must be in the hz , irrespective of the stellar spectral type . mars , if it were more massive , would be in the hz around any main sequence star with @xmath146 k. for planets that are outside the dark grey region , but inside the light grey ( for example , earth ) one needs to know the host star s spectral type ( or @xmath129 ) to determine if that planet is in the hz . [ exoplanetsplot ] combines observable stellar and planetary parameters to further constrain hz boundaries for extrasolar planets . many of the currently known exoplanets have non - zero eccentricities , which can carry some of them ( and their possible moons ) in and out of the hz . the incident stellar flux on these eccentric planets has extreme variations between periastron and apoastron ( @xmath147^{2}$ ] ) . @xcite show that , provided that an ocean is present to act as a heat capacitor , it is primarily the time - averaged flux @xmath148 that affects the habitability over an eccentric orbit @xcite . mathematically : @xmath149 here , @xmath79 is the effective flux from circular orbit ( eq.([hzeq ] ) ) . planets with high orbital eccentricities ( @xmath150 ) have higher average orbital flux . this may help eccentric planets near the outer edge of the hz maintain habitable conditions . however , obliquity variations can influence the geographical distribution of irradiation @xcite and may change habitable conditions . earth itself appears to be perilously close to the moist greenhouse limit ( @xmath85 , blue filled circle in fig . [ observed ] ) . however , this apparent instability is deceptive , because the calculations do not take into account the likely increase in earth s albedo that would be caused by water clouds on a warmer earth . furthermore , these calculations assume a fully saturated troposphere that maximizes the greenhouse effect . for both reasons , it is likely that the actual hz inner edge is closer to the sun than our moist greenhouse limit indicates . note that the moist greenhouse in our model occurs at a surface temperature of 340 k. the current average surface temperature of the earth is only 288 k. even a modest ( 5 - 10 degree ) increase in the current surface temperature could have devastating affects on the habitability of earth from a human standpoint . consequently , though we identify the moist greenhouse limit as the inner edge of the habitable zone , habitable conditions for humans could disappear well before earth reaches this limit . additional uncertainty about habitability of planets around late - k and m stars . ( @xmath151 k ) comes from the fact that planets within the hzs of these stars are expected to be tidally locked @xcite . if the planet s orbital eccentricity is small , this can result in synchronous rotation , in which one side of a planet always faces the star ( as the moon does to the earth ) . climates of synchronously rotating planets are not well approximated by 1-d , globally averaged models . previous work has shown that such planets may indeed be habitable @xcite ; however , systematic exploration of synchronously rotating planets in different parts of the hz has not been attempted . even before doing these calculations , we can predict that planets near the outer edge of the hz , with their expected dense @xmath0 atmospheres , should be more effective at transporting heat around to their night sides , and hence should have a better chance of being habitable . given that survey like hpf and carmenes will specifically target mid - late m dwarfs , our future work will include estimating the hz boundaries of individual targets in detail . ongoing work by our team @xcite is yielding low resolution nir spectra from the infrared telelescope facility ( irtf ) to be used to derive stellar metallicities as well as yielding more realistic flux distributions and temperatures for use in the modeling . we anticipate having this information for @xmath152 650 m dwarfs drawn from the j@xmath153 @xcitecatalog , and have applied for time to observe @xmath152 300 more . estimates of luminosities will be derived using photometric and spectroscopic distances for now ( in cases where parallax measurements are absent ) , but eventually gaia @xcite will yield very precise parallaxes ( and by extension precise luminosities ) for all these target stars . recent discoveries by both the _ kepler _ mission and rv surveys have shown that planets can exist in stable orbits around multiple star systems @xcite . the hzs of these stars could potentially host terrestrial planets , which are at the threshold of current detection techniques . indeed , the discovery of kepler 47c @xcite which is 4.6 times the size of the earth s radii in the hz is a step closer to discovering rocky planets in the hz of multiple star systems . @xcite have recently published a possible detection of a 1.1 earth mass ( minimum ) planet in a 3.236 day orbit around @xmath154 centauri system . so , this system should now be a prime target for further observations to discover habitable planets . formation of dynamically stable terrestrial planets in the hzs of multiple star systems has been studied before @xcite and several studies estimated hz boundaries around these types of systems @xcite using @xcite model . our updated model results from fig . [ observed ] or eqs.([hzeq ] ) and ( [ dhz ] ) could change these estimates significantly . our new model results could also directly affect estimates of @xmath155 . recent analysis of _ kepler _ data @xcite and rv surveys @xcite concluded that @xmath156 . these values were based either on the @xcite model @xcite or the @xcite results @xcite . our new hz limits could impact these estimates significantly . in particular , there are large differences between @xcite calculations and our model results for low mass stars . the estimate of @xmath155 by @xcite is obtained by using @xcite relationships for planets orbiting m - stars . thus , this value may need to be re - evaluated . . the boundaries of the green - shaded region are determined by the moist - greenhouse ( inner edge , higher flux values ) & maximum greenhouse ( outer edge , lower flux values ) . a planet that receives stellar flux bounded by the two dashed vertical lines is in the hz irrespective of the stellar type . some of the currently known exoplanets that are thought to be in the hz by previous studies are also shown . the ` ? ' for gl 581 and tau ceti system of planets imply that there is an ongoing discussion about their existence . for stars with t@xmath6 k , there is no clear distinction between runaway and moist - greenhouse limit . ] m@xmath10 - @xmath82m@xmath10 , y - axis limits ) within the dark shaded region are in the hz irrespective of the stellar type . for those planets in the light grey region , one needs to know the stellar type to determine if they are in the hz . for example , the earth would not be habitable if it received its current incoming amount of energy from a cooler - type star , but it is ( obviously ) habitable in its current orbit around a g - type star . for planets such as this , the star s energy distribution matters when considering habitability . ]

this removes the dependence on planetary ( bond ) albedo , which varies depending upon the host star s spectral type . we suggest that conservative estimates of the hz ( water loss and maximum greenhouse limits ) should be used for current rv surveys and _ kepler _ mission to obtain a lower limit on , so that future flagship missions like _ tpf - c _ and _ darwin _ are not undersized .

identifying terrestrial planets in the habitable zones ( hzs ) of other stars is one of the primary goals of ongoing radial velocity and transit exoplanet surveys and proposed future space missions . most current estimates of the boundaries of the hz are based on 1-d , cloud - free , climate model calculations by . however , this model used band models which were based on older hitran and hitemp line - by - line databases . the inner edge of the hz in model was determined by loss of water , and the outer edge was determined by the maximum greenhouse provided by a atmosphere . a conservative estimate for the width of the hz from this model in our solar system is 0.95 - 1.67 au . here , an updated 1-d radiative - convective , cloud - free climate model is used to obtain new estimates for hz widths around f , g , k and m stars . new and absorption coefficients , derived from the hitran 2008 and hitemp 2010 line - by - line databases , are important improvements to the climate model . according to the new model , the water loss ( inner hz ) and maximum greenhouse ( outer hz ) limits for our solar system are at au and au , respectively , suggesting that the present earth lies near the inner edge . additional calculations are performed for stars with effective temperatures between k and k , and the results are presented in parametric form , making them easy to apply to actual stars . the new model indicates that , near the inner edge of the hz , there is no clear distinction between runaway greenhouse and water loss limits for stars with t k which has implications for ongoing planet searches around k and m stars . to assess the potential habitability of extrasolar terrestrial planets , we propose using stellar flux incident on a planet rather than equilibrium temperature this removes the dependence on planetary ( bond ) albedo , which varies depending upon the host star s spectral type . we suggest that conservative estimates of the hz ( water loss and maximum greenhouse limits ) should be used for current rv surveys and _ kepler _ mission to obtain a lower limit on , so that future flagship missions like _ tpf - c _ and _ darwin _ are not undersized . our model does not include the radiative effects of clouds ; thus , the actual hz boundaries may extend further in both directions than the estimates just given . 2oh_2o 2h_2 2co_2 4ch_4

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we have obtained new estimates for hzs around f , g , k and m main - sequence stars by ( 1 ) updating @xmath1 and @xmath0 absorption coefficients in the @xcite 1-d radiative - convective cloud - free climate model with the most recent lbl databases : hitran 2008 and hitemp 2010 , ( 2 ) correcting the @xmath1 rayleigh scattering coefficient , and ( 3 ) updating @xmath0 collision - induced absorption coefficients . these changes affect the inner and outer edges of the hz , respectively . our revised model predicts that the moist greenhouse limit for our sun , which defines the inner edge of the hz , is at 0.99 au . the outer edge of the hz , where gaseous @xmath0 produces its maximum greenhouse effect , is at 1.70 au . although it appears that earth is perilously close to the inner hz edge , in reality , cloud feedback and low upper tropospheric relative humidity act to stabilize earth s climate . theoretical studies and observational surveys that depend on these limits should use the updated values . we have also estimated hz boundaries for m stars with @xmath129 as low as @xmath4 k , which are primary targets for ongoing surveys such as _ habitable zone planet finder _ and _ mearth _ to discover potential habitable planets . we also showed that the effective stellar flux provides a better criterion in determining the hz limits , than equilibrium temperature . accordingly we have derived a generalized expression to calculate these fluxes for stars of different spectral types . our results show that some of the extrasolar planets that were previously thought to be within the hz may not be in that region . thus , our hz estimates can be used to narrow the target list for eventual characterization missions , such as _ jwst _ , to identify potential biomarkers on habitable planets . author contribution : r.k & r.r contributed equally to this work . an interactive webpage to obtain hzs and a fortran code is available here : http://depts.washington.edu/naivpl/content/hz-calculator . the fortran code is also available in the electronic version of the paper . the authors are especially grateful to david crisp for his invaluable comments , suggestions , and for answering our radiative transfer questions during the preparation of this work . the authors would like to thank the referee robin wordsworth for his constructive comments which improved the manuscript . note added in proof : r. wordsworth points out that his model uses increased vertical resolution in the lower atmosphere and that his results were found to be insensitive to further increases in resolution . all such calculations should be tested to determine whether they are robust to this issue . we also thank colin goldblatt , david paynter , richard freedman , itay halevy , eli mlawer and martin cohen ( u.c berkeley ) for their helpful discussions . the authors acknowledge the research computing and cyberinfrastructure unit of information technology services at the pennsylvania state university for providing advanced computing resources and services that have contributed to the research results reported in this paper . http://rcc.its.psu.edu [ http://rcc.its.psu.edu ] . this work was also facilitated through the use of advanced computational , storage , and networking infrastructure provided by the hyak supercomputer system , supported in part by the university of washington escience institute . this research has made use of the exoplanet orbit database and the exoplanet data explorer at exoplanets.org . r. k , r. r , j.f.k and sddg gratefully acknowledge funding from nasa astrobiology institute s virtual planetary laboratory lead team , supported by nasa under cooperative agreement nnh05zda001c , and the penn state astrobiology research center . v.e . acknowledges the support of the itaac project ( impact du trafic arien sur latmosphre et le climat ) , funded by the fondation sciences et technologies pour laoronautique et lespace ( stae ) , toulouse , france , within the rseau thmatique de recherche avance(rtra ) , and support from the european research council ( starting grant 209622 : e3arths ) . sm acknowledges support from nsf ast1006676 , ast1126413 , psarc , and the nasa nai . the center for exoplanets and habitable worlds is supported by the pennsylvania state university , the eberly college of science , and the pennsylvania space grant consortium . sddg was also supported by the oak ridge associated universities nasa postdoctoral management program , and did much of his work on this project while in residence at nasa headquarters . kasting , j. f. 2011b . joint meeting of the exoplanet and cosmic origins program analysis groups ( exopag and copag ) , april 26 , 2011 , baltimore , md , http://exep.jpl.nasa.gov/exopag/exopagcopagjointmeeting/ pepe , f. , mayor , m. , & lovis , c. et al . in : a. sozzetti , m.g . lattanzi , a.p . boss ( eds . ) , the astrophysics of planetary systems : formation , structure , and dynamical evolution , iau symp . 276 , p. 13

, cloud - free climate model is used to obtain new estimates for hz widths around f , g , k and m stars . new and absorption coefficients , derived from the hitran 2008 and hitemp 2010 line - by - line databases , are important improvements to the climate model . according to the new model , the water loss ( inner hz ) and maximum greenhouse ( outer hz ) limits for our solar system the new model indicates that , near the inner edge of the hz , there is no clear distinction between runaway greenhouse and water loss limits for stars with t k which has implications for ongoing planet searches around k and m stars . to assess the potential habitability of extrasolar terrestrial planets , we propose using stellar flux incident on a planet rather than equilibrium temperature

identifying terrestrial planets in the habitable zones ( hzs ) of other stars is one of the primary goals of ongoing radial velocity and transit exoplanet surveys and proposed future space missions . most current estimates of the boundaries of the hz are based on 1-d , cloud - free , climate model calculations by . however , this model used band models which were based on older hitran and hitemp line - by - line databases . the inner edge of the hz in model was determined by loss of water , and the outer edge was determined by the maximum greenhouse provided by a atmosphere . a conservative estimate for the width of the hz from this model in our solar system is 0.95 - 1.67 au . here , an updated 1-d radiative - convective , cloud - free climate model is used to obtain new estimates for hz widths around f , g , k and m stars . new and absorption coefficients , derived from the hitran 2008 and hitemp 2010 line - by - line databases , are important improvements to the climate model . according to the new model , the water loss ( inner hz ) and maximum greenhouse ( outer hz ) limits for our solar system are at au and au , respectively , suggesting that the present earth lies near the inner edge . additional calculations are performed for stars with effective temperatures between k and k , and the results are presented in parametric form , making them easy to apply to actual stars . the new model indicates that , near the inner edge of the hz , there is no clear distinction between runaway greenhouse and water loss limits for stars with t k which has implications for ongoing planet searches around k and m stars . to assess the potential habitability of extrasolar terrestrial planets , we propose using stellar flux incident on a planet rather than equilibrium temperature this removes the dependence on planetary ( bond ) albedo , which varies depending upon the host star s spectral type . we suggest that conservative estimates of the hz ( water loss and maximum greenhouse limits ) should be used for current rv surveys and _ kepler _ mission to obtain a lower limit on , so that future flagship missions like _ tpf - c _ and _ darwin _ are not undersized . our model does not include the radiative effects of clouds ; thus , the actual hz boundaries may extend further in both directions than the estimates just given . 2oh_2o 2h_2 2co_2 4ch_4

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a group is _ virtually free _ if it has a free subgroup of finite index . by stalling s ends theorem , @xmath0 has _ more than one end _ if and only if there exists a @xmath0-tree with finite edge stabilizers such that no vertex is stabilized by @xmath0 . the group @xmath0 is _ accessible _ if there exists a @xmath0-tree with finite edge stabilizers , and each vertex stabilizer having at most one end . the group @xmath0 is _ context - free _ if for every finite generating set @xmath74 of @xmath39 the set @xmath83}\text { sequence in } x^{\pm 1 } : \pi x_{[1\uparrow n]}=1\}\ ] ] is a context - free language . see review [ rev : lang ] for a complete definition . in 1983 , muller and schupp ( * ? ? ? * theorem iii ) showed that @xmath0 is context - free and accessible if and only if @xmath0 is virtually free . by ( * theorem i ) , a group is context - free if every closed path in the cayley graph is @xmath84-triangulable . in particular it follows that being context - free can be expressed in terms of the cayley graph and that context - free groups are finitely presentable . in 1985 , dunwoody @xcite proved that finitely presented groups are accessible , and so the accessibility hypothesis of ( * ? ? ? * theorem iii ) can be removed . in this article we investigate how to characterize virtual freeness of a group in terms of its cayley graph . using an accessiblility result ( theorem [ t : dun ] ) which is implicit in an earlier paper of dunwoody @xcite of 1979 we obtain new proofs of several results which originally used the accessibility of finitely presented groups . let us list these results . let @xmath2 be the cayley graph of @xmath39 then @xmath0 is virtually free if and only if one of the following hold : 1 . there exists @xmath85 such that every closed path in @xmath2 is @xmath86-triangulable ( see ( * ? ? ? * theorems i and iii ) ) . @xmath2 is quasi - isometric to a cayley graph of a free group ( see @xcite ) . @xmath0 is finitely presentable and @xmath87 ( see ( * ? ? ? * cor 1.2 ) , ( * ? ? ? * thm 2 ) or ( * ? ? ? * cor g ) ) . 4 . @xmath2 admits a uniform spanning tree ( @xcite ) . the ends of @xmath2 have finite diameter ( @xcite ) . @xmath2 has finite strong tree - width ( @xcite ) . the original proofs of ( 1)-(3 ) directly invoke the accesibility of finitely presentable groups . the proofs of ( 4)-(6 ) invoke ( 1 ) . as ( 1 ) is equivalent to @xmath0 being context - free , we found two other characterizations of @xmath0 being virtually free : 1 . there exists a generating set of @xmath0 and @xmath88 such that every @xmath84-locally geodesic in @xmath0 is a geodesic ( @xcite ) 2 . there exists a generating set of @xmath0 such that @xmath0 admits a ( regular ) broomlike combing with respect to the generating set(@xcite ) . in section [ s : dun ] , we will review the classical geometric characterization of @xmath0 being virtually free in terms of actions on trees . we will use an accessibility result of dunwoody of 1979 , implicit in @xcite . in section [ s : graphs ] , we show the equivalence of many conditions for graphs . using results of section [ s : dun ] , we will show that the cayley graph of @xmath0 satisfies any of these conditions if and only if @xmath0 is virtually free . in particular , we give new proofs of ( 1),(2),(4)-(6 ) . at the end of the section , we recall the definition of a context - free group in review [ rev : lang ] , and we make some remarks about the connection of the @xmath86-triangulations with the characterizations ( 7 ) and ( 8) . in section [ s : adim ] , we review the concept of asymptotic dimension . we use ( 1 ) to give a proof of ( 3 ) .

in 1985 , dunwoody showed that finitely presentable groups are accessible . dunwoody s result was used to show that context - free groups , groups quasi - isometric to trees or finitely presentable groups of asymptotic dimension 1 are virtually free . using another theorem of dunwoody of 1979 , we study when a group is virtually free in terms of its cayley graph , and we obtain new proofs of the mentioned results and others previously depending on these . _ 2000 mathematics subject classification . _ primary : 20f65 ; secondary : 05c25 _ key words . _ virtually free group , accessibility , cayley graph , asymptotic dimension .

in 1985 , dunwoody showed that finitely presentable groups are accessible . dunwoody s result was used to show that context - free groups , groups quasi - isometric to trees or finitely presentable groups of asymptotic dimension 1 are virtually free . using another theorem of dunwoody of 1979 , we study when a group is virtually free in terms of its cayley graph , and we obtain new proofs of the mentioned results and others previously depending on these . _ 2000 mathematics subject classification . _ primary : 20f65 ; secondary : 05c25 _ key words . _ virtually free group , accessibility , cayley graph , asymptotic dimension .

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distances to other galaxies obtained by observations of cepheid variables with the hubble space telescope ( hst ) lie at the core of most recent efforts to determine the extragalactic distance scale . the small observed scatter in the relationship between cepheids pulsation periods and luminosities , their large numbers ( which allow many independent measures of the distance to a galaxy ) , and the simplicity of the basic physics underlying their variability all have made them uniquely suitable for this purpose . as a consequence of their integral role in establishing the distance scale , however , any changes in the calibration and application of the cepheid period luminosity ( p l ) relationship will affect many other secondary methods of distance measurement . any improvements in techniques for obtaining cepheid distances are thus extremely valuable ; but this also means that any changes in this vital link should be well scrutinized if they are to be adopted . in a paper describing the final results of the hst key project on the extragalactic distance scale ( henceforward , the key project ) , freedman ( 2001 ) make a number of refinements to the techniques used in earlier papers in the series based upon newly - available information . first and foremost , microlensing experiments ( e.g. , udalski 1999 ) and dedicated efforts ( sebo 2001 ) have greatly enlarged the set of calibrating lmc cepheids beyond what had been studied when the initial key project p l relation was determined ( madore & freedman 1991 ) . the resulting samples have revealed a modest correction to the previously adopted p l slope for @xmath5 observations ; the @xmath6band slope remains unchanged . in the key project methodology , cepheid magnitudes are corrected for extinction using their observed color excess @xmath7 , where @xmath8 is the expected color of an unreddened cepheid of given period based upon the lmc calibrated p l relations . this procedure is sensitive to the reddening , since @xmath9 ) . an error in the p l slope in @xmath5 thus propagates into a larger , period dependent error in the true distance modulus ; in this case , @xmath10 . because in more distant galaxies only brighter , longer - period cepheids are observable , this generally results in a distance - dependent revision to hst cepheid distances which in extreme cases can reach -0.20 mag . further details are given in freedman et al . ( 2001 ) . second , our understanding of charge transfer efficiency and related effects in the wfpc2 ccds has greatly improved in recent years ( e.g. , stetson 1998 , dolphin 2000 ) , motivating revisions in the hill ( 1998 ) photometric zero points used in most earlier key project papers . freedman adopt the calibration of stetson ( 1998 ) , which results in a -0.02 mag mean adjustment in @xmath6 and -0.04 mag in @xmath5 from the hill `` long '' zero points . carrying this through the key project procedures for obtaining reddening - corrected distance moduli , they apply a net correction of -0.07@xmath11 mag to distances obtained using the hill `` long '' zero point ( where the error adopted reflects a conservative estimate of systematic differences among recent calibrations ) . the third change made in the revised key project procedures is the adoption of cepheid distance moduli adjusted for metallicity effects as standard . the typical metallicity of cepheids in the lmc with which the p l relation is calibrated differs by @xmath12 dex from that in many of the fields observed in the course of the key project . metallicity differences may produce measurable differences in the colors and magnitudes of cepheids in those fields from ones found in the lmc ; unfortunately , neither theoretical calculations ( as those of alibert 1999 or musella 1999 ) nor observations ( cf . kennicutt 1998 , sasselov et al . 1997 , kochanek 1997 , or nevalainen & roos 1998 ) have provided a definitive determination of the magnitude , or even sign , of this effect . freedman ( 2001 ) adopt a correction to cepheid magnitudes of @xmath13 mag / dex ( based upon current observational results ) as standard . in previous key project work , distance moduli uncorrected for metallicity effects were primarily used , though results for a correction of -0.24 mag / dex were also given . in their error budget , freedman estimate the potential systematic error in a typical key project cepheid distance measurement due to corrections for differences in metallicity from the lmc to be @xmath14 mag . even with these revisions , important uncertainties remain in the cepheid - calibrated distance scale . foremost among these is the distance to the large magellanic cloud ( lmc ) , using which the key project period luminosity relation has been calibrated . a wide variety of measurements of the distance to the lmc have been performed in the past few years . many of these studies disagree with each other statistically , spanning roughly 0.5 mag in distance modulus . freedman ( 2001 ) adopt @xmath15 mag as the distance modulus of the lmc ; the uncertainty in the lmc distance modulus leads directly to a systematic uncertainty in the cepheid distance scale of @xmath16 mag . after the uncertainties in the lmc distance and metallicity corrections , the next most significant contribution to the key project systematic error budget is the difficulty in determining zero points for wfpc2 photometry , which is estimated to lead to a @xmath17 mag uncertainty in hst cepheid distance moduli ; a variety of other systematic effects could enter at lower levels . given the remaining uncertainties , it is worthwhile to test the revised key project distance scale using a galaxy with a well - known , primary distance and with a metallicity typical of galaxies observed in the course of the key project . to be useful , such a test requires that cepheids be observed with the same instruments , filters , and parameter measurement techniques as used for objects in the key project sample . such a test has already been applied to the original key project distance scale based upon observations of cepheids in ngc 4258 ( maoz 1999 ) . the spiral galaxy ngc 4258 ( sabbc , @xmath18 mag ) presents a unique opportunity for such a test because of the precision with which its distance has been measured in a manner independent of the conventional ladder of astronomical distance scales ( freedman 1998 ) . furthermore , its metallicity and distance are similar to those of typical targets of hst cepheid programs . the distance to ngc 4258 , @xmath19 mpc ( corresponding to distance modulus @xmath20 mag ) , has been determined using its apparently simple , keplerian circumnuclear disk delineated by line - emitting water masers that orbit a supermassive black hole at its center ( miyoshi 1995 , maoz 1995 ) . this disk was discovered by vlbi observations of water maser emission from the central region of the galaxy ( miyoshi 1995 ) . it is about @xmath21 mas in diameter , @xmath22 mas in thickness , rotates at speeds of @xmath23 , and is viewed by us from nearly edge - on . most remarkably , the rotation curve of the maser sources is keplerian to high precision ( @xmath24 ) , which provides very strong evidence for a supermassive black hole at the galaxy center ( maoz 1995 ) . the high angular resolution ( @xmath25mas ) and spectral resolution ( @xmath26 ) of vlba observations ( moran 1995 ) allow a precise definition of the disk structure and kinematics . combining the observed rotation velocities with the measured centripetal acceleration in the disk ( greenhill 1995 ) or with the observed proper motions of the maser sources ( herrnstein 1999 ) permits independent measurements of the physical size of the disk ; comparing these to its observed angular extent yields the distance to the galaxy via simple geometry . these distance measurements are subject to only small uncertainties . the differential systematic error in the maser positions is @xmath27 mas ( moran 1995 ) , translating into a relative distance error smaller than @xmath28 . the distance to the masing disk scales with the disk s inclination as @xmath29 ; since the disk is viewed nearly edge - on , @xmath30 ( herrnstein 1996 ) , an error even as high as @xmath31 would introduce a distance error of only @xmath32 . relativistic corrections due to gravitational redshift and transverse doppler shift have been taken into account and are much smaller . the disk is slightly warped , but the distance determination is not sensitive to the warp model adopted . the warp contributes , though , to the small uncertainty in the disk inclination mentioned above . the total estimated uncertainty in this distance is @xmath33 mpc if the disk is presumed to be circular ( which it appears to be to better than @xmath28 ) . if nonzero eccentricities are allowed , the uncertainty increases to @xmath34 mpc ( we adopt this more conservative value for all further discussions ) . the direct , geometric methods used are believed to have minimal unknown systematic uncertainties . the two routes to a distance ( proper motions and accelerations ) yield results in agreement with each other to @xmath32 . as described in maoz ( 1999 ) , we have therefore obtained and analyzed hst observations of ngc 4258 with the intention of testing , and potentially of better determining , the zero point of the cepheid p - l relation . in that work , we used the then - standard key project methodologies and calibrations , and found a @xmath35 discrepancy between the hst cepheid distance and that obtained from studies of the masing disk . in this paper , we perform a similar test after obtaining a cepheid distance using the revised key project procedures , allowing an evaluation of the new techniques . we also provide herein full information ( locations , light curves , finding charts , etc . ) for the cepheids described in maoz 1999 . we present the observations , the reduction of the data and the search for cepheids in @xmath36 , the resulting derivation of a cepheid distance in @xmath37 , and the implications in @xmath38 .

in a previous paper ( maoz 1999 ) , we reported a hubble space telescope ( hst ) cepheid distance to the galaxy ngc 4258 obtained using the calibrations and methods then standard for the key project on the extragalactic distance scale . here , we reevaluate the cepheid distance using the revised key project procedures described in freedman ( 2001 ) . these revisions alter the zero points and slopes of the cepheid period - luminosity ( p - l ) relations derived at the large magellanic cloud ( lmc ) , the calibration of the hst wfpc2 camera , and the treatment of metallicity differences . we also provide herein full information on the cepheids described in maoz 1999 . using the refined key project techniques and calibrations , we also test the alternative cepheid p - l relations of feast ( 1999 ) , which yield more discrepant results .

in a previous paper ( maoz 1999 ) , we reported a hubble space telescope ( hst ) cepheid distance to the galaxy ngc 4258 obtained using the calibrations and methods then standard for the key project on the extragalactic distance scale . here , we reevaluate the cepheid distance using the revised key project procedures described in freedman ( 2001 ) . these revisions alter the zero points and slopes of the cepheid period - luminosity ( p - l ) relations derived at the large magellanic cloud ( lmc ) , the calibration of the hst wfpc2 camera , and the treatment of metallicity differences . we also provide herein full information on the cepheids described in maoz 1999 . using the refined key project techniques and calibrations , we determine the distance modulus of ngc 4258 to be mag ( unique to this determination ) mag ( systematic uncertainties in key project distances ) , corresponding to a metric distance of mpc and 1.2 from the maser distance of mpc . we also test the alternative cepheid p - l relations of feast ( 1999 ) , which yield more discrepant results . additionally , we place weak limits upon the distance to the lmc and upon the effect of metallicity in cepheid distance determinations .

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in recent years , there has been a dramatic growth in interest in the physics of nonequilibrium quantum systems , driven in large part by advances in experimental atomic physics , in particular in the area of optical lattices @xcite . the precision time - dependent control and observation of quantum effects afforded by these experiments , together with the high degree of isolation of the system from the environment , have invigorated the theoretical study of the time evolution of isolated many - body quantum systems . the predictions of previously abstract lines of theoretical inquiry into the mechanisms by which thermal behavior emerges from purely unitary time evolution , and the role of conservation laws and integrability in such dynamics , can now be directly compared with empirical evidence acquired in experimental laboratories . several theoretical studies into the quantum origins of thermalization in isolated nonintegrable systems have found that , away from the edges of the spectrum , few - body observables relax to the predictions of conventional statistical ensembles @xcite , a phenomenon which has been connected @xcite to the emergence of quantum chaos @xcite . in addition , it is now well established that the highly constrained dynamics of integrable systems can in fact give rise to the relaxation of few - body observables , and that the equilibrium values of these quantities are in many cases described by the generalized gibbs ensemble ( gge ) @xcite . the gge is constructed by maximizing the many - body entropy @xcite while constraining the mean values of all integrals of motion to their expectation values in the initial state @xmath0 of the system . the density matrix in the gge takes a gaussian form similar to that of the grand - canonical ensemble , and can be written as @xcite @xmath1 where @xmath2 $ ] is the gge partition function , the @xmath3 are the conserved integrals of motion , and the @xmath4 are the corresponding lagrange multipliers , which are determined by the constraints @xmath5=\langle\psi_i|\hat i_s|\psi_i\rangle$ ] . the significance of the fact that the gge provides an accurate description of observables following relaxation can be seen by contrasting its predictions with those of the `` diagonal ensemble '' ( de ) @xcite : for any initial state @xmath0 , the time evolution of an observable @xmath6 under a time - independent ( integrable or nonintegrable ) hamiltonian @xmath7 can be written as @xmath8 where @xmath9 , @xmath10 , @xmath11 , @xmath12 , and @xmath13 . the infinite - time average of the observable is therefore given by @xmath14 which defines the expectation value of @xmath6 in the de @xcite . the de involves as many constraints as the dimension of the many - body hilbert space ( the overlaps of the initial state with the eigenstates of @xmath7 ) , which grows exponentially with system size . by contrast , for models that can be mapped to noninteracting hamiltonians , the gge involves a number of constraints that is only polynomially large in the size of the system @xcite . it may , therefore , appear surprising that the predictions of the gge for expectation values of observables can agree with those of the de . from a many - body perspective , the success of the gge can be understood as follows @xcite : the eigenstates of a given integrable hamiltonian with similar distributions of conserved quantities have similar expectation values of few - body observables ( with the differences vanishing in the thermodynamic limit ) . furthermore , the majority of the states that contribute to the de have a distribution of conserved quantities similar ( in a coarse - grained sense ) to that of the initial state , and this is also the case for the states that contribute most strongly to the gge . these facts imply that differences between the weights in the de and the gge are irrelevant and both ensembles will produce the same results for few - body observables in the thermodynamic limit @xcite . this scenario can be viewed as a generalization of the eigenstate thermalization hypothesis @xcite , and has been explored in ref . @xcite for describing observables after relaxation by means of a single representative state . interestingly , it has been recently shown that in an integrable system , in the presence of localization , the gge can fail to describe observables after relaxation @xcite . this effect , which parallels the breakdown of eigenstate thermalization in nonintegrable disordered lattice systems in the presence of localization @xcite , has been related to the localized behavior of the underlying system of noninteracting particles to which some integrable models can be mapped @xcite . here we gain further insights into this phenomenon by undertaking single - particle and many - body analyses of noninteracting spinless fermions and hard - core bosons . we study the dynamics of noninteracting fermions within the aubry - andr model previously studied in ref . @xcite for hard - core bosons , and show that although the fermion momentum distribution equilibrates in the localized regime , it fails to equilibrate in the delocalized one . this should be contrasted with the density profiles , which exhibit the opposite behavior , equilibrating in the delocalized regime , but not in the localized regime @xcite . we find that , whenever observables do equilibrate to their gge values , they do so in a manner consistent with the gaussian equilibration scenario of ref . furthermore , we relate the failure of a given quantity to equilibrate in a certain regime to the behavior of the single - particle wave functions , as discussed in refs . @xcite . for hard - core bosons , we connect the results of ref . @xcite , in which the gge was shown to fail to describe the momentum distribution function after relaxation in the localized regime , to the single - particle results for fermions . in addition , for both hard - core bosons and spinless fermions , we study the density profiles and momentum distribution functions in the many - body eigenstates of the appropriate hamiltonians . we focus in particular on the behavior of these quantities in the many - body eigenstates that contribute to the de , to the microcanonical ensemble ( me ) , and to the microcanonical version of the gge , i.e. , the generalized microcanonical ensemble ( gme ) @xcite . we find indications that single - particle real - space localization in the localized phase and momentum - space localization in the delocalized phase lead to a distinctive behavior of the many - body eigenstate expectation values of the density and the momentum distribution of the fermions , respectively . no similar effect is detected in the many - body eigenstate expectation values of the momentum distribution of the hard - core bosons . this paper is organized as follows . in sec . [ sec : model ] , we introduce the models , quench protocols , and observables studied in later sections . we also review the statistical ensembles utilized to describe observables after relaxation . the time evolution of spinless fermions following a quench is studied in sec . [ sec : fermions ] . specifically , we examine the relaxation dynamics and time fluctuations of one - body observables , as well as properties of the single - particle eigenstates that help us understand the observed out - of - equilibrium behavior . section [ sec : hcbs ] is devoted to the study of one - particle observables in the many - body eigenstates of the bosonic and fermionic hamiltonians . in sec . [ sec : summary ] , we summarize our results and present our conclusions .

in general , isolated integrable quantum systems have been found to relax to an apparent equilibrium state in which the expectation values of few - body observables are described by the generalized gibbs ensemble . however , recent work has shown that relaxation to such a generalized statistical ensemble can be precluded by localization in a quasiperiodic lattice system . here we undertake complementary single - particle and many - body analyses of noninteracting spinless fermions and hard - core bosons within the aubry - andr model to gain insight into this phenomenon .

in general , isolated integrable quantum systems have been found to relax to an apparent equilibrium state in which the expectation values of few - body observables are described by the generalized gibbs ensemble . however , recent work has shown that relaxation to such a generalized statistical ensemble can be precluded by localization in a quasiperiodic lattice system . here we undertake complementary single - particle and many - body analyses of noninteracting spinless fermions and hard - core bosons within the aubry - andr model to gain insight into this phenomenon . our investigations span both the localized and delocalized regimes of the quasiperiodic system , as well as the critical point separating the two . considering first the case of spinless fermions , we study the dynamics of the momentum distribution function and characterize the effects of real - space and momentum - space localization on the relevant single - particle wave functions and correlation functions . we show that although some observables do not relax in the delocalized and localized regimes , the observables that do relax in these regimes do so in a manner consistent with a recently proposed gaussian equilibration scenario , whereas relaxation at the critical point has a more exotic character . we also construct various statistical ensembles from the many - body eigenstates of the fermionic and bosonic hamiltonians and study the effect of localization on their properties .

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we have studied the dynamics of the momentum distribution function of noninteracting spinless fermions following quenches to the delocalized , localized , and critical regimes of a quasiperiodic lattice system . we found that although the time - averaged value of this observable agrees exactly with the predictions of the gge in all three regimes , it does not exhibit relaxation after a quench in the delocalized phase . this is complementary to the failure of the on - site density of hard - core bosons ( and therefore the noninteracting fermion system considered here ) to equilibrate in the localized regime that was previously observed in ref . these behaviors can be understood in terms of localization of the single - particle eigenstates of the fermion model , in momentum space for the delocalized regime , and in real space for the localized regime , as discussed in refs . this analysis also helps us understand the previously observed failure of the gge to describe the momentum distribution functions of hcbs after relaxation in the localized regime @xcite as a consequence of nonvanishing time fluctuations of one - particle fermionic correlations that persist in the thermodynamic limit . we found that in the delocalized and localized regimes , the sf observables that do exhibit relaxation to the gge the density in the former case , and the momentum distribution function in the latter do so in a manner consistent with the gaussian equilibration picture of campos venuti and zanardi @xcite . in quenches to the critical point of the aubry - andr model , we observed that both the density and the momentum distribution function of sfs exhibit equilibration to the gge , but that the decay of the time fluctuations of these quantities with increasing system size is slower than that predicted by the conjecture of ref . @xcite . we also studied the expectation values of one - body observables in the many - body eigenstates of the sf and hcb hamiltonians , comparing results for the diagonal , microcanonical , and generalized microcanonical ensembles . we found a clear distinction between the predictions of the me for the expectation values and those of the gme . the differences between the expectation values in the former ensemble and the de were consistently larger ( and were in fact greater than 10% in all cases except for the momentum distributions in quenches to the localized phase ) and indications were found that these differences approach nonzero values as the system size is increased toward the thermodynamic limit . the predictions of the gme were found to be much closer to those of the de and , for the system sizes studied , the differences between the expectation values in these two ensembles were observed in most cases to decrease with increasing system size . our study indicates that the single - particle localization in momentum space in the delocalized regime and in real space in the localized regime that precludes relaxation of the corresponding observables to the gge also leads to finite thermodynamic - limit variances of momentum and site occupations , respectively , in the many - body eigenstates of the sf hamiltonian . by contrast , the failure of the gge to describe the momentum distribution of hcbs after relaxation in the localized phase was not found to be associated with any corresponding saturation of the variance of momentum occupations in eigenstates of the hcb hamiltonian with increasing system size . because of finite - size effects , we did not find clear indications of the behavior of the maximum differences between expectation values of the density and momentum distributions in distinct eigenstates contributing to the various ensembles as the system size is increased . it would be particularly important to understand whether the values of observables in the individual eigenstates constituting the gme approach their average values in this ensemble with increasing system size , in order to clarify whether generalized eigenstate thermalization occurs in the delocalized regime which would explain why the gge works there for describing the momentum distribution functions of hcbs after relaxation and whether generalized eigenstate thermalization fails ( as we expect ) in the localized regime , where the gge fails . it would also be interesting to see how the addition of nearest neighbor interactions @xcite , which break integrability , modify our findings for both sfs and hcbs . this work was supported by the office of naval research ( k.h . , l.f.s . , and m.r . ) , by nsf grant no . dmr-1147430 ( l.f.s . ) , by arc discovery project grant no . dp110101047 ( t.m.w . ) , and partially under kitp nsf grant no . phy11 - 25915 ( l.f.s . , t.m.w . , and m.r .

we show that although some observables do not relax in the delocalized and localized regimes , the observables that do relax in these regimes do so in a manner consistent with a recently proposed gaussian equilibration scenario , whereas relaxation at the critical point has a more exotic character . we also construct various statistical ensembles from the many - body eigenstates of the fermionic and bosonic hamiltonians and study the effect of localization on their properties .

in general , isolated integrable quantum systems have been found to relax to an apparent equilibrium state in which the expectation values of few - body observables are described by the generalized gibbs ensemble . however , recent work has shown that relaxation to such a generalized statistical ensemble can be precluded by localization in a quasiperiodic lattice system . here we undertake complementary single - particle and many - body analyses of noninteracting spinless fermions and hard - core bosons within the aubry - andr model to gain insight into this phenomenon . our investigations span both the localized and delocalized regimes of the quasiperiodic system , as well as the critical point separating the two . considering first the case of spinless fermions , we study the dynamics of the momentum distribution function and characterize the effects of real - space and momentum - space localization on the relevant single - particle wave functions and correlation functions . we show that although some observables do not relax in the delocalized and localized regimes , the observables that do relax in these regimes do so in a manner consistent with a recently proposed gaussian equilibration scenario , whereas relaxation at the critical point has a more exotic character . we also construct various statistical ensembles from the many - body eigenstates of the fermionic and bosonic hamiltonians and study the effect of localization on their properties .

astro-ph9903336

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recent progress in high redshift observations has made it possible to estimate the global star formation history of the universe to redshifts as high as @xmath5 . although the evolution of the star formation rate ( sfr ) is relatively well established at lower redshifts ( @xmath6 ) , there is still substantial uncertainty about the @xmath7 era . a study of lyman break galaxies in the hubble deep field shows a decline of the sfr towards high @xmath8 , indicating a peak in the sfr around @xmath9 ( madau et al . 1996 ) . on the other hand , a recent census of @xmath10 galaxies , detected in ly@xmath11 emission in a @xmath12 times wider field , suggests that the sfr is either constant , or increasing from @xmath13 to @xmath14 ( steidel 1999 ) . at present , these uncertainties render extrapolations of the observed sfr to still higher ( @xmath15 ) redshifts unreliable . we do know , however , that some stellar activity must have taken place at @xmath16 . the presence of heavy elements observed in the high redshift ly@xmath11 forest ( tytler , cowie ) , as well as the reionization of the intergalactic medium ( igm ) , require sources of metals and ionizing photons in addition to the currently known population of galaxies and quasars . such ultra high redshift objects are indeed expected in the current best fit cosmological cold dark matter ( cdm ) models , which predict that the first virialized halos appear at redshifts as high as @xmath17 . it is natural to identify these small halos as the sites where the first stars or massive black holes are born , forming the first generation of `` mini galaxies '' or `` mini quasars '' . the first collapsed halos have masses near the cosmological jeans mass , @xmath18 , implying a virial temperature of a few hundred k. in order to form either stars or black holes , the gas in these halos must be able to cool efficiently . at this low temperature , the only mechanism in the chemically simple primordial gas that satisfies this requirement is collisional excitations of @xmath1 molecules . at still higher temperatures ( @xmath19k ) and masses ( @xmath20 ) , efficient cooling is enabled by atomic hydrogen . the presence or absence of @xmath1 molecules in the first collapsed halos therefore determines whether these small halos can form any stars or quasars , before larger halos dominate the collapsed baryonic fraction . as a consequence , the @xmath1 abundance determines when the `` dark age '' ends : if there is sufficient @xmath1 inside the first small halos , the dark age ends earlier ; if @xmath1 is absent , it ends at the later redshift when the larger scales collapse . the fraction of @xmath1 molecules in the post recombination igm is @xmath21 . in the absence of a background radiation field , this fraction rises to @xmath22 at the dense central regions of collapsed halos ( haiman et al . 1996 ; tegmark et al . 1997 ) , high enough to satisfy the cooling criterion . however , @xmath1 molecules are fragile , and are photo - dissociated efficiently by a low intensity uv radiation . in a previous paper ( haiman et al . 1997 , hereafter hrl97 ) , we showed that the flux necessary for @xmath1 photo dissociation is several orders of magnitude smaller than the flux needed to reionize the universe . based on this result , we conjectured that the @xmath1 abundance in the first collapsed halos was suppressed soon after a small number of these halos formed stars or turned into mini quasars . this negative feedback would imply that the bulk of the first mini galaxies or mini quasars would reside in dark halos with masses at least @xmath23 , and that the `` dark age '' would end only at the typical collapse redshift of these halos . the first condensations at the @xmath24 times smaller jeans mass would then remain neutral , gravitationally confined gas clumps , until they merge into larger systems , or are photo evaporated by the uvb established after reionization ( barkana & loeb 1998 ) . a small fraction of these clumps is expected to survive mergers and photo evaporation until @xmath25 ; based on their column density and abundance , abel & mo ( 1998 ) have identified this survived fraction with the lyman limit systems observed near @xmath26 . a direct observational study of the first collapsed halos will be feasible in the future with the next generation space telescope ( ngst ) : with its @xmath27njy imaging sensitivity , ngst will be able to detect a @xmath28 halo with an average star formation efficiency at @xmath29 ( smith & koratkar 1998 ) . in this paper , we study the conjecture of hrl97 in a cosmological setting , by quantifying the radiative feedback of the first collapsed halos on their @xmath30 abundance , as the early uvb is established in press schechter models . the two main questions we address are : _ ( 1 ) how does the @xmath1 feedback effect the typical masses and collapse redshifts of the first halos _ ; and _ ( 2 ) could the first collapsed dark halos , with masses near the jeans mass , contribute to the early uvb , and to the reionization of the universe ? _ in hrl97 , we concluded that the negative feedback suppresses the fragmentation and collapse of small halos . in the present paper , we confirm this basic conclusion , although we find that it depends sensitively on the spectrum of the early sources . our calculations improve hrl97 in several ways . we solve the coupled evolution of the formation of new halos ( using the press schechter formalism ) , the build up of the early uvb , and the evolution of the @xmath1 abundance in the igm and inside each collapsed halo ; we explore the sensitivity of our conclusions on the spectral shape , specifically the x ray to uv flux ratio ; we replace our assumption of homogeneous slabs by the centrally condensed profiles expected for virialized halos ; and we employ a more realistic star formation criterion . this paper is organized as follows . in 2 , we quantify the spectrum of the early uvb after processing through the intergalactic h and @xmath1 . in 3 , we compute the @xmath1 abundance in virialized halos under this uvb , and define our criterion for star formation . in 4 , we compute the evolution of the uvb coupled with the @xmath1feedback , and present these results in 5 . in 6 , we discuss how these results are modified in the presence of early mini quasars with spectra extending to the x rays . finally , 7 summarizes the main conclusions and implications of this work . throughout this paper , we adopt a background @xmath31cdm cosmology with a tilted power spectrum , @xmath32 ; however , our results are relevant in any model where the cosmic structure is built hierarchically from the `` bottom up '' . for convenience , we also adopt the following terminology : a `` small halo '' is defined to be a halo with virial temperature @xmath33k , or equivalently , with mass @xmath34^{-3/2}$ ] . conversely , a `` large halo '' is defined as a halo with virial temperature @xmath19k , or equivalently , with mass @xmath35^{-3/2}$ ] .

the uvb at energies below 13.6ev suppresses the formation of stars or black holes inside small halos , by photo dissociating their only cooling agent , molecular . we self consistently compute the buildup of the early uvb in press if the uv sources in the first collapsed halos are stars , then their uv flux suppresses further star quasars with hard spectra extending tokev , then their x rays balance the effects of the uvb , the negative feedback does not occur , and reionization can be caused by the small halos . fermilab - pub-99/041-a

in hierarchical models of structure formation , an early cosmic uv background ( uvb ) is produced by the small (k ) halos that collapse before reionization . the uvb at energies below 13.6ev suppresses the formation of stars or black holes inside small halos , by photo dissociating their only cooling agent , molecular . we self consistently compute the buildup of the early uvb in press schechter models , coupled with photo dissociation both in the intergalactic medium ( igm ) , and inside virialized halos . we find that the intergalactic has a negligible effect on the uvb , both because its initial optical depth is small ( ) , and because it is photo dissociated at an early stage . if the uv sources in the first collapsed halos are stars , then their uv flux suppresses further star formation inside small halos . this results in a pause in the buildup of the uvb , and reionization is delayed until larger halos (k ) collapse . if the small halos host mini quasars with hard spectra extending tokev , then their x rays balance the effects of the uvb , the negative feedback does not occur , and reionization can be caused by the small halos . fermilab - pub-99/041-a

astro-ph9903336

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in hierarchical models of structure formation , small objects appear at high redshift , and larger objects form later , with the total fraction of collapsed baryons monotonically increasing with time . the radiation output associated with the collapsed halos gradually builds up a cosmic uv background . of interest here is the background in the 11.1813.6 ev range , enclosing the lyman and werner ( lw ) bands of molecular @xmath1 . photons in this range do not ionize neutral h atoms , and can travel to a large fraction of the hubble length across the igm , and photo dissociate molecular @xmath1 both in the igm and inside other collapsed halos . since the presence of @xmath1 is a necessary condition for the formation of stars or a central black hole , this leads to a feedback that controls the fraction of collapsed baryons turned into luminous sources , and the evolution of the uvb . before addressing the feedback itself , in this section we quantify the spectral shape of the uvb in the relevant frequency range this is determined by the absorption of the flux by intergalactic h and @xmath1 . consider an observer at redshift @xmath36 , measuring the uvb at the local frequency @xmath37 , where @xmath38 . the photons collected at the frequency @xmath37 are arriving from sources located at @xmath39 , and have redshifted their original frequency @xmath40 to @xmath37 . if during its travel , the photon s frequency equals the frequency of any atomic lyman line , it is absorbed by the neutral igm , because the optical depth of the igm in these lines are exceedingly high , e.g. @xmath41 at @xmath42 . this statement applies to all lyman lines with @xmath43 . the ly@xmath11 absorption ( @xmath44 ) has only a small effect on the background , since most of the absorbed photons are simply re - emitted at the same frequency , except those converted to lower frequency photons by two photon decays . in any case , the ly@xmath11 resonance is at 10.2 ev , outside the frequency range of interest here . the very high lyman lines ( @xmath45 ) have small enough cross sections so that the total optical depth of the igm at these frequencies drops below unity . however , these resonances have a negligible contribution to the total absorption in the 11.8 - 13.6ev range . here we make the simple assumption that a `` dark screen '' blocks the view of all sources at redshifts above @xmath46 . the maximum redshift is given by @xmath47 where @xmath48 is the frequency of the lyman line closest from above to the observed frequency , @xmath37 . the processing of the early uvb by the neutral igm leads to a characteristic spectral shape , resembling a `` sawtooth '' , since the closer @xmath37 is to a resonance , the nearer the screen is to the observer . the resulting spectrum is shown in a specific model ( described below ) in figure [ fig : sawtooth ] . the redshift distribution of sources is important for the magnitude of this `` sawtoothing '' effect . if a large fraction of the sources contributing to the uvb are located near the observer , then the effect is smaller ; if the sources are spread out over a wider range of redshifts , then the effect is larger . in the hierarchical models , new halos form exponentially rapidly at high redshifts . at any given time , most of the sources are expected to have formed recently , i.e. at redshifts only slightly above that of the observer . since the formation rate of new halos decreases with redshifts , the sawtoothing will be increasingly pronounced at later times . this is illustrated in figure [ fig : sawtooth ] : the flux is suppressed by a larger average factor at @xmath49 than it is at @xmath50 . in addition to absorption by neutral h , the uvb may also be modulated by the intergalactic @xmath1 . the @xmath1 in the igm removes photons from the uvb when these molecules are photo dissociated , and decreases the rate of photo dissociations inside collapsed halos . it is important , therefore , to assess the magnitude of the modulation from the intergalactic @xmath1 . when a lw photon is absorbed by an @xmath1 molecule , there is on average a @xmath52% chance that it causes a photo dissociation of the molecule . this is the average probability that the molecule decays from its excited state to a continuum of two distinct h atoms ( solomon , cf . field et al . 1966 ; stecher & williams 1967 ) . we will assume here that in the other @xmath53 of absorptions , the lw photons is re - emitted at the same frequency , and there is no net effect . this is justified because the quadrupole transitions among the excited electronic states are much slower than decay back to the electronic ground state , and also because the small dipole moment that preferred the upper state in the absorption will preferentially populate the initial lower state in the decay ( herzberg 1950 ) . if the @xmath1 fraction in the igm is @xmath54 , then the total optical depth of the igm at redshift @xmath8 and frequency @xmath55 to @xmath1dissociations is given by @xmath56 where the sum is over the 76 lw lines of @xmath1 ; @xmath57 is the neutral h number density in the igm ; @xmath58 is the probability that absorption into the @xmath59 lw line is followed by a dissociative decay ; @xmath60 is the oscillator strength of the @xmath59 lw line ; @xmath61 ; @xmath62 is the line element in our @xmath31cdm cosmology ; and @xmath63 is the absorption line profile centered around the resonance frequency @xmath48 . we included the equilibrium ortho to para ratio of 3:1 of implicitly in equation ( [ eq : tauh2a ] ) by appropriately multiplying the oscillator strengths with 0.75 or 0.25 respectively . furthermore , we assume that the line shapes follow a voigt profile with thermal broadening by the igm temperature ( taken to be @xmath64k , although the total optical depth , and therefore our results , are insensitive to this choice ) . more information on the details of pre dissociation are given in the appendix . equation [ eq : tauh2a ] gives a formal sum total optical depth of the 76 lw resonances , each of which is at a different frequency . from the point of view of a photon traveling across the igm , these resonances would be encountered at different redshifts . during its lifetime , a uv photon can therefore be subjected only to a subset of the lw resonances : as we argued in 2.1 above , the maximum redshift interval a uv photon can travel before it is absorbed by a neutral h atom corresponds to the redshift between two neutral h resonances . a simple calculation of the optical depth using equation [ eq : tauh2a ] would result in an overestimate by a factor of a few ( depending on @xmath8 and @xmath55 ) of the true value relevant to the observed modulation of the uvb . to impose the fact that the uvb is _ simultaneously _ modulated both by the intergalactic h and @xmath1 , one needs to change the summation in equation [ eq : tauh2a ] from @xmath65(1,76 ) to @xmath65(@xmath66,@xmath67 ) , where @xmath66 is the lw resonance closest in frequency from above to @xmath55 , and @xmath67 is the lw resonance farthest above @xmath55 before encountering an atomic lyman line ( see figure [ fig : sawtooth ] and figure [ fig : lw ] for visualizations of the relative location of the h and @xmath1 lines ) . in figure [ fig : igmh2 ] , we show the resulting optical depth as a function of the observed frequency , for two different redshifts , @xmath49 and @xmath50 . as the figure shows , the optical depth is largest at @xmath412.1ev . this is because the largest frequency gap between the atomic lines occurs between ly@xmath68 and ly@xmath69 , enclosing the largest subset of the @xmath1 lw lines with significant contributions to @xmath1 dissociations ( see figures [ fig : sawtooth ] and [ fig : lw ] ) . figure [ fig : igmh2 ] also shows that the maximum optical depth is 0.35 and 0.13 for @xmath50 and @xmath49 , respectively ( assuming @xmath70 ) . interestingly , the characteristic features in figure [ fig : igmh2 ] depend only on the relative frequencies of atomic vs. molecular resonant absorption lines ( shown in figure [ fig : lw ] ) , i.e. only on atomic and molecular physics . the total optical depths are near , but somewhat below unity . this result has several consequences . it is useful here to draw an analogy with the ionization of neutral h. the total optical depth of the igm to h photo ionizations ( just above 13.6ev ) is @xmath71 ^ 3 $ ] , which justifies the simple picture of isolated hii regions ( `` strmgren spheres '' ) , each bounded by a sharp ionization front , and continuously expanding around each ionizing source . the situation with @xmath1 is different in two ways : the optical depth is less than unity , and is distributed among several lw lines , each of which is individually optically thin , and causes dissociations at a different redshift relative to each source . as a consequence , no sharp @xmath1 dissociation `` front '' exists . a schematic sketch of four distinct regions in the radial structure of an @xmath30 photo dissociation region is shown in figure [ fig : pdr ] . region i extends to the usual ionization front or strmgren radius . inside this region , the temperature is high ( @xmath72k ) , hydrogen is ionized , is photo dissociated , and the abundance tends to zero . region ii is defined to extend from @xmath73 to the radius @xmath74 , where the photodissociation time becomes longer than the hubble time . here @xmath75 is the photon number flux in the lyman werner bands per second . in this region the temperature is that of the igm , and the fractions are still low , but slowly rise to its background value of @xmath76 . since the optical depth in the lyman werner lines are very small , the dissociation front is significantly diluted . region iii is defined to extend from @xmath77 to the radius @xmath78^{-1.5}~{\rm mpc}$ ] where lyman @xmath69 shifts to lyman@xmath68 . in this region , the source contributes to the soft uv background , but its flux is diminished by absorption into the atomic lyman series . in region iv beyond @xmath79 , all photons in the lyman werner bands are absorbed by an atomic lyman line . hence , at such large scales , an individual source does not contribute to the uvb and intergalactic dissociation . although the maximum optical depths from the intergalactic @xmath1 are relatively small , the uvb flux can be suppressed by as much as @xmath80% . however , we are interested in the suppression of the @xmath1 photo dissociation rate when the uvb falls on a collapsed halo . this suppression is smaller than 50% , since the photo dissociation rate is given by a weighted average of the observed flux in the 11.1813.6ev range . we therefore define an `` effective '' optical depth @xmath81 , that directly characterizes the suppression of the @xmath1 photo dissociation rate , as follows : @xmath82 @xmath83 , \label{eq : dissoc } \end{array}\ ] ] @xmath84 . \label{eq : tauave}\ ] ] here @xmath85 is the uvb flux before processing by the intergalactic @xmath30 , and @xmath86 is the optical depth at this frequency shown by the solid lines in figure [ fig : igmh2 ] . when the sources are assumed to be distributed according to the press schechter model ( see below ) , the maximum reduction in the @xmath1 photo dissociation rate is 10% at @xmath50 and 4% at @xmath49 ( see the dashed lines in figure [ fig : igmh2 ] ) . note , however , that these numbers assume a constant @xmath70 . in reality ( as will be shown below ) , the intergalactic @xmath1 fraction drops sharply soon after the appearance of the first sources , before the uvb builds up to the level of causing any @xmath1 feedback inside the collapsed clumps . this is because the @xmath1 dissociation time in the igm , @xmath87 becomes shorter than the hubble time at @xmath88 , while the feedback on the dense clumps typically turns on only later , when the uvb reaches @xmath89 . in summary , we have shown that the intergalactic @xmath1 might cause a maximum of @xmath90 suppression of the uvb near 12 ev , but the overall effect on the @xmath1 photo dissociation rate is small : less than 10% , assuming a constant intergalactic @xmath1 fraction of @xmath91 . we will show below that the intergalactic @xmath1 fraction is substantially reduced , and therefore this small modulation disappears , by the redshifts of interest when the uvb builds up to the levels to photo dissociate @xmath30 molecules in dense collapsed clumps .

in hierarchical models of structure formation , an early cosmic uv background ( uvb ) is produced by the small (k ) halos that collapse before reionization . schechter models , coupled with photo dissociation both in the intergalactic medium ( igm ) , and inside virialized halos . we find that the intergalactic has a negligible effect on the uvb , both because its initial optical depth is small ( ) , and because it is photo dissociated at an early stage .

in hierarchical models of structure formation , an early cosmic uv background ( uvb ) is produced by the small (k ) halos that collapse before reionization . the uvb at energies below 13.6ev suppresses the formation of stars or black holes inside small halos , by photo dissociating their only cooling agent , molecular . we self consistently compute the buildup of the early uvb in press schechter models , coupled with photo dissociation both in the intergalactic medium ( igm ) , and inside virialized halos . we find that the intergalactic has a negligible effect on the uvb , both because its initial optical depth is small ( ) , and because it is photo dissociated at an early stage . if the uv sources in the first collapsed halos are stars , then their uv flux suppresses further star formation inside small halos . this results in a pause in the buildup of the uvb , and reionization is delayed until larger halos (k ) collapse . if the small halos host mini quasars with hard spectra extending tokev , then their x rays balance the effects of the uvb , the negative feedback does not occur , and reionization can be caused by the small halos . fermilab - pub-99/041-a

1208.1041

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consider an arbitrary point @xmath7 , with coordinates @xmath8 , in an undeformed body . then we deform the body and the same material point is named @xmath9 , with coordinates @xmath10 . the position of @xmath9 is uniquely determined by its initial position @xmath7 , that is , @xmath11 . the _ displacement _ vector @xmath12 is then defined as @xmath13 we distinguish between lower ( covariant ) and upper ( contravariant ) indices in order to have the freedom to change to arbitrary coordinates if necessary . the deformation and the stress state of an elastic body is , within _ linear _ elasticity theory , described by means of the strain tensor @xmath14 and the stress tensor @xmath15 . the strain tensor as well as the stress tensor are both symmetric , that is , @xmath16}:=\frac 12 ( \varepsilon_{ij } -\varepsilon_{ji})=0 $ ] and @xmath17}=0 $ ] , see love ( 1927)@xcite , landau & lifshitz ( 1986)@xcite , hausshl ( 2007)@xcite , marsden & hughes ( 1983)@xcite , and podio - guidugli ( 2000)@xcite ; thus , @xmath14 and @xmath15 both have 6 independent components . the strain tensor can be expressed in terms of the displacement vector via @xmath18 whereupon @xmath19 is the metric of the three - dimensional ( 3d ) euclidean background space and @xmath20 . in cartesian coordinates , we have @xmath21 , with @xmath22 . the stress tensor fulfills the momentum law @xmath23 here @xmath24 is the mass density , @xmath25 the body force density , and a dot denotes the time derivative . the constitutive law in linear elasticity for a homogeneous anisotropic body , the generalized hooke law , postulates a linear relation between the two second - rank tensor fields , the stress @xmath15 and the strain @xmath26 : @xmath27 the elasticity tensor @xmath28 has the physical dimension of a stress , namely _ force / area . _ hence , in the international system of units ( si ) , the frame components of @xmath28 are measured in _ pascal _ p , with p:=n / m@xmath29 . in 3d , a generic fourth - order tensor has 81 independent components . it can be viewed as a generic @xmath30 matrix . since @xmath14 and @xmath31 are symmetric , certain symmetry relations hold also for the elasticity tensor . thus , @xmath32}=0\,\qquad \longrightarrow\qquad c^{ij[kl]}=0\,.\ ] ] this is the so - called _ right minor symmetry_. similarly and independently , the symmetry of the stress tensor yields the so - called _ left minor symmetry _ , @xmath33}=0\,\qquad \longrightarrow\qquad c^{[ij]kl}=0\,.\ ] ] both minor symmetries , ( [ s1-sym ] ) and ( [ s2-sym ] ) , are assumed to hold simultaneously . accordingly , the tensor @xmath34 can be represented by a @xmath35 matrix with 36 independent components . the energy density of a deformed material is expressed as @xmath36 . when the hooke law is substituted , this expression takes the form @xmath37 the right - hand side of ( [ energy1a ] ) involves only those combinations of the elasticity tensor components which are symmetric under permutation of the first and the last pairs of indices . in order to prevent the corresponding redundancy in the components of @xmath28 , the so - called _ major symmetry _ , @xmath38 is assumed . therefore , the @xmath35 matrix becomes symmetric and only 21 independent components of @xmath28 are left over . as a side remark we mention that the components of a tensor are always measured with respect to a local frame @xmath39 ; here @xmath40 numbers the three linearly independent legs of this frame , the triad . dual to this frame is the coframe @xmath41 , see schouten ( 1989)@xcite and post ( 1997)@xcite . for the elasticity tensor , the components with respect to such a local coframe are @xmath42 . they are called the physical components of @xmath0 . for simplicity , we will not set up a frame formalism since it does nt provide additional insight in the decomposition of the elasticity tensor , that is , we will use coordinate frames @xmath43 in the rest of our article . because of ( [ s1-sym ] ) , ( [ s2-sym ] ) , and ( [ paircom ] ) , we have the following * definition : * a _ 4th rank tensor _ of type @xmath44 qualifies to describe anisotropic elasticity if \(i ) its physical components carry the dimension of force / area ( in si pascal ) , \(ii ) it obeys the left and right minor symmetries , \(iii ) and it obeys the major symmetry . it is then called elasticity tensor ( or elasticity or stiffness ) and , in general , denoted by @xmath28 . in secs . 2.1 and 2.2 we translate our notation into that of voigt , see voigt ( 1928)@xcite and love ( 1927)@xcite , and discuss the corresponding 21-dimensional vector space of all elasticity tensors , see also del piero ( 1979)@xcite . incidentally , in linear electrodynamics , see post ( 1962)@xcite , hehl & obukhov ( 2003)@xcite , and itin ( 2009)@xcite , we have a 4-dimensional constitutive tensor @xmath45 , with @xmath46 . surprisingly , this tensor corresponds also to a @xmath35 matrix , like @xmath28 in elasticity . the major symmetry is the same , the minor symmetries are those of an antisymmetric pair of indices . in sec . 2.3 , we turn first to an algebraic decomposition of @xmath28 that is frequently discussed in the literature : the elasticity tensor is decomposed into the sum of the two tensors @xmath47 and @xmath48l}$ ] , which are symmetric or antisymmetric in the middle pair of indices , respectively . we show that @xmath1 and @xmath2 fulfill the major symmetry but _ not _ the minor symmetries and that they can be further decomposed . accordingly , this reducible decomposition does not correspond to a direct sum decomposition of the vector space defined by @xmath0 , as we will show in detail . furthermore we show that the vector space of @xmath1 is 21-dimensional and that of @xmath2 6-dimensional . often , in calculation within linear elasticity , the tensors @xmath1 and @xmath2 emerge . they are auxiliary quantities , but due to the lack of the minor symmetries , they are _ not _ elasticities . consequently , they can not be used to characterize a certain material elastically . these quantities are placeholders that are not suitable for a direct physical interpretation . subsequently , in sec . 2.4 , we study the behavior of the physical components of @xmath0 under the action of the general linear 3d real group @xmath49 . the @xmath49 commutes with the permutations of tensor indices . this fact yields the well - known relation between the action of @xmath49 and the action of the symmetry group @xmath50 . without restricting the generality of our considerations , we will choose local coordinate frames @xmath43 for our considerations . * figure 1 . * _ a tensor @xmath51 of rank 4 in 3-dimensional ( 3d ) space has @xmath52 independent components . the 3 dimensions of our image represent this 81d space . the plane @xmath53 depicts the 21 dimensional subspace of all possible elasticity ( or stiffness ) tensors . this space is span by its irreducible pieces , the 15d space of the totally symmetric elasticity @xmath54 ( a straight line ) and the 6d space of the difference @xmath55 ( also depicted , i am sad to say , as a straight line). oblique to @xmath53 is the 21d space @xmath56 of the reducible @xmath1-tensor and the 6d space of the reducible @xmath2-tensor . the @xmath53 `` plane '' is the only place where elasticities ( stiffnesses ) are at home . the spaces @xmath56 and @xmath2 represent only elasticities , provided the cauchy relations are fulfilled . then , @xmath57 and @xmath56 and @xmath53 cut in the 15d space of @xmath58 . notice that the spaces m and c are intersecting exactly on s. _ in this way , we arrive at an alternative decomposition of @xmath0 into two pieces @xmath3 and @xmath4 , which is _ irreducible _ under the action of the @xmath49 . the known device to study the action of @xmath50 is provided by young s tableaux technique . for the sake of completeness , we will be briefly describe young s technique in the appendix . in an earlier paper , see hehl & itin ( 2002)@xcite , we discussed this problem already , but here we will present rigorous proofs of all aspects of this irreducible decomposition . it turns our that the space of the @xmath3-tensor is 15-dimensional and that of the @xmath4-tensor 6-dimensional . in sec . 2.5 , we compare the reducible @xmath6-decomposition of sec . 2.3 with the irreducible @xmath5-decomposition of sec . 2.4 and will show that the latter one is definitely to be preferred from a physical point of view . the formulas for the transition between the @xmath6- and the @xmath5-decomposition are collected in the propositions 9 and 11 . the irreducible 4th rank tensor @xmath59 can alternatively be represented by a symmetric 2nd rank tensor @xmath60 ( proposition 10 ) . we visualized our main results with respect to the reducible and the irreducible decompositions of the elasticity tensor in figure 1 , for details , please see sec.2.5 . the group @xmath49 , which we are using here , provides the basic , somewhat coarse - grained decomposition of the elasticity tensor . for finer types of irreducible decompositions under the _ orthogonal _ subgroup @xmath61 of @xmath49 , see , for example , rychlewski ( 1984)@xcite , walpole ( 1984)@xcite , surrel ( 1993)@xcite , xiao ( 1998)@xcite , and the related work of backus ( 1970)@xcite and baerheim ( 1993)@xcite . as a result of our presentation , the great number of invariants , which emerge in the latter case , can be organized into two subsets : one related to the @xmath3 piece and the other one to the @xmath4 piece if @xmath0 . having now the irreducible decomposition of @xmath0 at our command , sec . 3 will be devoted to the physical applications . in sec.3.1 , we discuss the cauchy relations and show that they correspond to the vanishing of one irreducible piece , namely to @xmath62 or , equivalently , to @xmath63 . as a consequence , the totally symmetric piece @xmath3 can be called the cauchy part of the elasticity tensor @xmath0 , whereas the @xmath4 piece measures the deviation from this cauchy part ; it is the non - cauchy part of @xmath0 . the reducible pieces @xmath1 and @xmath2 defy such an interpretation and are not useful for applications in physics . in this sense , we can speak of two kinds of elasticity , a cauchy type and a non - cauchy type . in sec . 3.2 , this picture is brought to the elastic energy and the latter decomposed in a cauchy part and its excess , the `` non - cauchy part '' . in other words , this distinction between two kinds of elasticity is also reflected in the properties of the elastic energy . a null lagrangians is such an lagrangian whose euler - lagrange expression vanishes , see crampin & saunders ( 2005)@xcite ; we also speak of a `` pure divergence '' . in sec . 3.3 , elastic null lagrangians are addressed , and we critically evaluate the literature . we show in contrast to a seemingly widely held view that for an arbitrary anisotropic medium there does nt exist an elastic null lagrangian . the expressions offered in the corresponding literature are worthless as null lagrangians , since they still depend on _ all _ components of the elasticity tensor . we collect these results in proposition 12 . in sec . 3.4 , we define for acoustic wave propagation the cauchy and non - cauchy parts of the christoffel ( or acoustic ) tensor @xmath64 ( @xmath65 unit wave covector , @xmath66 mass density ) . we find some interesting and novel results for the christoffel tensor ( , see propositions 13 and 14 ) . in sec . 3.5 , we investigate the polarizations of the elastic wave . we show that the longitudinal wave propagation is completely determined by the cauchy part of the christoffel tensor , see proposition 15 . in proposition 16 a new result is presented on the propagation of purely polarized waves ; we were led to these investigations by following up some ideas about the interrelationship of the symmetry of the elasticity tensor and the christoffel tensor in the papers of alshits and lothe ( 2004)@xcite and bna et al . ( 2004 , 2007 , 2010)@xcite . in sec . 4 , we investigate examples , namely isotropic media ( sec . 4.1 ) and media with cubic symmetry ( sec . modern technology allows modeling composite materials with their effective elastic properties , see tadmore & miller ( 2011)@xcite . irreducible decomposition of the elasticity tensor can be used as a guiding framework for prediction of certain features of these materials . as an example , we presented in proposition 17 a complete new class of anisotropic materials that allow pure polarizations to propagate in arbitrary directions , similarly as in isotropic materials . we use here tensor analysis in 3d euclidean space with explicit index notation , see sokolnikoff ( 1951)@xcite and schouten ( 1954 , 1989)@xcite . coordinate ( holonomic ) indices are denoted by latin letters @xmath67 ; they run over @xmath68 . since we allow arbitrary curvilinear coordinates , covariant and contravariant indices are used , that is , those in lower and in upper position , respectively , see schouten ( 1989)@xcite . summation over repeated indices is understood . we abbreviate symmetrization and antisymmetrization over @xmath69 indices as follows : @xmath70&:=&\frac{1}{p!}\{\sum \text{over all even perms . } - \sum \text{over all odd perms.}\}\,.\end{aligned}\ ] ] the levi - civita symbol is given by @xmath71 , for even , odd , and no permutation of the indices @xmath72 , respectively ; the analogous is valid for @xmath73 . the metric @xmath74 has euclidean signature . we can raise and lower indices with the help of the metric . in linear elasticity theory , tensor analysis is used , for example , in love ( 1927)@xcite , sokolnikoff ( 1956)@xcite , landau & lifshitz ( 1986)@xcite , and hausshl ( 2007)@xcite . for a modern presentation of tensors as linear maps between corresponding vector spaces , see marsden & hughes ( 1983)@xcite , podio - guidugli ( 2000)@xcite , and hetnarski & ignaczak ( 2011)@xcite . in hehl & itin ( 2002)@xcite , we denoted the elasticity moduli differently . the quantities of our old paper translate as follows into the present one : @xmath75 , @xmath76 , @xmath77 , and @xmath78 .

in linear anisotropic elasticity , the elastic properties of a medium are described by the fourth rank elasticity tensor . the decomposition of into a partially symmetric tensor and a partially antisymmetric tensors is often used in the literature . irreducible _ under the 3-dimensional general linear group . we show that the-decomposition is unique , irreducible , and preserves the symmetries of the elasticity tensor . the-decomposition fails to have these desirable properties and is such inferior from a physical point of view . various applications of the-decomposition are discussed : the cauchy relations ( vanishing of ) , the non - existence of elastic null lagrangians , the decomposition of the elastic energy and of the acoustic wave propagation . we provide explicit examples of the effectiveness of the-decomposition . a complete class of anisotropic media is proposed that allows pure polarizations in arbitrary directions , similarly as in an isotropic medium . _

in linear anisotropic elasticity , the elastic properties of a medium are described by the fourth rank elasticity tensor . the decomposition of into a partially symmetric tensor and a partially antisymmetric tensors is often used in the literature . an alternative , less well - known decomposition , into the completely symmetric part of plus the reminder , turns out to be _ irreducible _ under the 3-dimensional general linear group . we show that the-decomposition is unique , irreducible , and preserves the symmetries of the elasticity tensor . the-decomposition fails to have these desirable properties and is such inferior from a physical point of view . various applications of the-decomposition are discussed : the cauchy relations ( vanishing of ) , the non - existence of elastic null lagrangians , the decomposition of the elastic energy and of the acoustic wave propagation . the acoustic or christoffel tensor is split in a cauchy and a non - cauchy part . the cauchy part governs the longitudinal wave propagation . we provide explicit examples of the effectiveness of the-decomposition . a complete class of anisotropic media is proposed that allows pure polarizations in arbitrary directions , similarly as in an isotropic medium . _ key index words : anisotropic elasticity tensor , irreducible decomposition , cauchy relations , null lagrangians , acoustic tensor _

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many modern measuring devices use cmos or ccd solid - state photosensors to convert light into a digital signal . due to imperfections of photosensors such a conversion is not ideal and leads to noise in the measured signal . therefore , one can either estimate and reduce the impact of noise from the sensor , or simulate and predict the impact of noise on the images quality . the problem of noise estimation has been addressed by the emva1288 standard @xcite . the high - level simulation of noise in photosensors , however , is still an area of active research . the main problem is that photosensors are affected by many different sources of noise , some of which can not be modelled adequately using only gaussian noise . [ review:2r2]different numerical models of cmos @xcite and ccd @xcite sensors are reported in the literature . the photosensor models differ by the generality and the scope of the noise sources being included . high - level simulators @xcite are used for the evaluation of how different noise sources influence image quality @xcite , while low - level simulator models specific aspects of the solid - sate physics of a sensor @xcite . the thoroughness of noise modelling differs as well ; for example , in some models of the photosensor , the dark current shot noise is described by its mean value @xcite , while other models use more sophisticated statistical description of noise @xcite . [ review2jei : r1q1 - 2 ] sophisticated software integrated suits such as the image systems evaluation toolkit @xcite ( iset ) were developed recently . [ [ review:1r2 ] ] models of cmos photosensors : + + + + + + + + + + + + + + + + + + + + + + + + + + + + both high - level @xcite and low - level @xcite numerical models of cmos photosensors are reported in the literature . low - level simulators like medici @xcite , which are able to model the 2-d distributions of potential and carrier concentrations , are used for simulation of solid - state physics in cmos photosensors @xcite . some of the papers deal with the numerical modelling of colour images that allow end users to see the effects of the noise on the images @xcite . other papers focus on the specific types of noise ; for example , the analysis of temporal noise in cmos active pixels sensors can be found in @xcite , an autoregressive model of fixed - pattern noise ( fpn ) was presented in @xcite along with other fpn models @xcite . high - level models ( e.g. , @xcite ) often assume that the adc , sense node and source follower are completely linear , which , as we will show in this paper , is not always the case . [ [ models - of - ccd - photosensors ] ] models of ccd photosensors : + + + + + + + + + + + + + + + + + + + + + + + + + + + the models of ccd photosensors @xcite and their noise parameters estimation @xcite are generally simpler than models of cmos sensors . the model by flory @xcite describes the noise of the sensor electronics as a combination of shot noise and amplifiers noise . the ccd camera noise model by cox @xcite includes photon and electronic shot noise , dark - current noise and readout noise ; however the model suggests that noise is stationary . healey and kondepudy @xcite model camera noise with offset fixed - pattern noise ( fpn ) , photon and dark - current shot noise , read noise and photo response non - uniformity ( prnu ) . [ review2jei : r2_q2]an adequate model was proposed by costantini and ssstrunk @xcite , where noises was categorised in four main types : photon shot noise , photo response non uniformity , dark current shot noise and read noise . [ [ review:18r2 ] ] the purpose of the article : + + + + + + + + + + + + + + + + + + + + + + + + + + + this article provides a literature review of the noise models that are used in the simulations of ccd and cmos photosensors and give a working numerical model of a photosensor . the aim of this article is to summarise the advances in numerical simulations of photosensors and provide a matlab implementation of a ccd / cmos sensor model , which has been demonstrated to adequately describe noise properties of a hardware cmos photosensor . the experimental results for a hardware 5 t cmos photosensor are provided in the article as validation for the developed numerical simulator . the article addresses the issue of the lack of high - level models of photosensors . it is shown that the dark noise may have more complicated structure than is typically assumed and therefore requires a more sophisticated statistical description . the formulated high - level model of a photosensor allows the simulation of realistic noise effects on the images and can be useful for testing of image processing algorithms . [ [ software - implementation ] ] software implementation : + + + + + + + + + + + + + + + + + + + + + + + + the matlab - based software implementation of the high - level ccd / cmos photosensor model described in this article is freely available on the website ` https://code.google.com/p/highlevelsensorsim/ ` and also on email request . the software simulator written as a series of functions that call the corresponding sub - functions sequentially to add the noise with specified parameters . two example test files are supplied that run : ( 1 ) a * simple * sensor model , which is completely linear and where all noise are gaussian , and ( 2 ) an * advanced * model , which has v / v and v / e non - linearities , lognormal noise models and other specific noises . both test files use the same code : adjusting the parameters in the test file , a user can turn the noise sources on or off and trigger non - linearities . the matlab code has a thorough documentation , both as comments inside the m - files and rendered for web view @xcite in a browser ( see _ /help _ directory and _ index.html _ file inside ) . introduction _ _ noise sources in solid - state photosensors : _ _ from photon to charge : _ _ _ photon shot noise and prnu _ _ dark signal , dark current fpn and dark current shot noise _ _ source follower noise ( johnson noise , flicker ( 1/f ) noise , and random telegraph noise ) _ from charge to voltage : _ _ _ sense node reset noise ( ktc noise ) _ _ offset fpn _ _ gain non - linearities ( v/@xmath0 gain and v / v gain non - linearity ) _ _ correlated double sampling _ from voltage to digital numbers _ ( including differential and integral linearity errors ) _ simulation methodology _ _ model : from photon to charge : _ _ _ simulating photon shot noise and prnu _ _ converting photons to electrons _ _ simulating dark signal , dark current fpn and dark current shot noise _ _ simulating source follower noise _ model : from charge to voltage : _ _ _ simulating sense node reset noise ( ktc noise ) and offset fpn _ _ converting electrons to voltage _ _ simulating v/@xmath0 and v / v gain non - linearity _ model : from voltage to digital numbers _ _ _ simulating adc non - linearity _ _ simulating quantisation noise _ experimental validation of the photosensor model _ _ radiometric function _ _ photon transfer curve _ _ signal - to - noise ratio _ _ photo response non - uniformity _ _ noise spectrogram _ _ dark signal performance for different integration times : _ _ _ short integration time _ _ long integration time _ conclusion _ _

the model includes photo - response non - uniformity , photon shot noise , dark current fixed pattern noise , dark current shot noise , offset fixed pattern noise , source follower noise , sense node reset noise , and quantisation noise . experimental results that confirm the validity of the numerical model are provided . the paper addresses the issue of the lack of comprehensive high - level photosensor models that enable engineers to simulate realistic effects of noise on the images obtained from solid - state photosensors .

in many applications , such as development and testing of image processing algorithms , it is often necessary to simulate images containing realistic noise from solid - state photosensors . a high - level model of ccd and cmos photosensors based on a literature review is formulated in this paper . the model includes photo - response non - uniformity , photon shot noise , dark current fixed pattern noise , dark current shot noise , offset fixed pattern noise , source follower noise , sense node reset noise , and quantisation noise . the model also includes voltage - to - voltage , voltage - to - electrons , and analogue - to - digital converter non - linearities . the formulated model can be used to create synthetic images for testing and validation of image processing algorithms in the presence of realistic images noise . an example of the simulated cmos photosensor and a comparison with a custom - made cmos hardware sensor is presented . procedures for characterisation from both light and dark noises are described . experimental results that confirm the validity of the numerical model are provided . the paper addresses the issue of the lack of comprehensive high - level photosensor models that enable engineers to simulate realistic effects of noise on the images obtained from solid - state photosensors .

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the article provides a literature review of the noise models that are used for the simulations of noise in ccd and cmos photosensors . a high - level model of ccd and cmos photosensors based on a literature review is formulated in this paper . the experimental results for a hardware 5 t cmos photosensor are presented as a validation for the developed model of a photosensor . the formulated model of the cmos photosensor was compared with the data from a custom - made hardware photosensor for validation of the photosensor model . results of photon transfer curve ( ptc ) estimation show the importance of the non - linearity introduced in the model . using the models of v / v and the adc non - linearity allow to better match the properties of the hardware cmos sensor . it has been also demonstrated that the dark current fpn and in particular the dark `` spikes '' in the case of a long integration time has a more complicated distribution than previously discussed in the literature . the gamma distribution with a superimposed uniform and gaussian distributions are shown to be a reasonable approximation of the dark `` spikes '' for long integration time . the article summarises the efforts in numerical simulations of solid - state photosensors providing an extensive survey of literature on noise in photosensors . a high - level model of cmos / ccd sensors has been formulated in order to provide engineers with insight on the noise impact on images quality . using such a high - level photosensor model , one can simulate either ccd or cmos sensors by application of appropriate noise models . the formulated high - level simulation model can be used to synthesise realistically noised images for the development and testing of the image processing algorithms .

in many applications , such as development and testing of image processing algorithms , it is often necessary to simulate images containing realistic noise from solid - state photosensors . a high - level model of ccd and cmos photosensors based on a literature review is formulated in this paper . the formulated model can be used to create synthetic images for testing and validation of image processing algorithms in the presence of realistic images noise .

in many applications , such as development and testing of image processing algorithms , it is often necessary to simulate images containing realistic noise from solid - state photosensors . a high - level model of ccd and cmos photosensors based on a literature review is formulated in this paper . the model includes photo - response non - uniformity , photon shot noise , dark current fixed pattern noise , dark current shot noise , offset fixed pattern noise , source follower noise , sense node reset noise , and quantisation noise . the model also includes voltage - to - voltage , voltage - to - electrons , and analogue - to - digital converter non - linearities . the formulated model can be used to create synthetic images for testing and validation of image processing algorithms in the presence of realistic images noise . an example of the simulated cmos photosensor and a comparison with a custom - made cmos hardware sensor is presented . procedures for characterisation from both light and dark noises are described . experimental results that confirm the validity of the numerical model are provided . the paper addresses the issue of the lack of comprehensive high - level photosensor models that enable engineers to simulate realistic effects of noise on the images obtained from solid - state photosensors .

cond-mat0210467

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the following one - parameter deformations of the exponential and logarithm functions @xmath13 which reduce to the standard exponential and logarithm respectively as the real deformation parameter @xmath1 approaches to zero , has been introduced recently in ref . the above functions have many very interesting properties @xcite ( some being identical to the ones of the undeformed functions ) that permit to construct a statistical mechanics ( and thermodynamics ) which generalizes the standard boltzmann - gibbs one . this @xmath1-deformed statistical mechanics preserves unaltered the structure of the ordinary one and can be used to explain a very large class of experimentally observed phenomena described by distribution functions exhibiting power law tails . the areas where this formalism can be applied include among others , low and high energy physics , astrophysics , econophysics , geology , biology , mathematics , information theory , linguistics , etc . @xcite . in ref . @xcite it has been shown that the statistical distribution @xmath14\ , ) \ , \label{i3}\end{aligned}\ ] ] which generalizes the boltzmann - gibbs distribution , can be obtained also by maximizing , after properly constrained , the entropy @xmath15 which reduces to the standard entropy @xmath6 as the deformation parameter approaches to zero . the coefficient @xmath8 , which also absorbs the boltzmann constant @xmath16 , depends on a free parameter @xmath9 ( see eq . ( 65 ) of ref . @xcite ) which remains to be determined together with the parameter @xmath3 which contains the information about the temperature @xmath17 of the system . a first question which arises naturally is if it is possible and how to find any criterion which allows us to fix the parameters @xmath3 and @xmath9 or at least express these in terms of the deformation parameter @xmath1 , in order to reduce the free parameters of the theory . a second question regards the properties of the entropy @xmath10 . it is well known that the boltzmann - shannon entropy @xmath6 is concave , additive and extensive . we know that @xmath10 is concave with respect to the variable @xmath5 , but what happens about its additivity and extensivity ? more in general , beside the boltzmann - shannon entropy other concave , additive and extensive entropies do exist ? a third question is related to the physical mechanism which originates the deformation introduced by the parameter @xmath1 . in other words , does a more fundamental theory exist where this deformation emerges , or is it simply a purely mathematical tool ? a fourth question is if it is possible to extend the theory originally developed in the framework of a classical kinetics to treat statistical systems in the context of a relativistic kinetics . a fifth question regards the deformation parameter @xmath1 . this parameter will continue to remain free or is it possible to determine its value self consistently within the theory ? the present paper deals with the above questions and its purpose is double . firstly , we will show that @xmath10 is the unique existing , concave , additive and extensive entropy , beside the boltzmann - shannon entropy . as we will see these properties of @xmath10 are sufficient to determine unequivocally the values of parameters @xmath3 and @xmath9 . secondly , we will show that the deformation introduced by @xmath1 is a purely relativistic effect and then we will explain the deformation mechanism within the einstein special relativity . then , we will formulate a relativistic @xmath1-kinetics and we will calculate the value of @xmath1 . finally , in order to test the predictability and correctness of theory here proposed we will consider two sets of experimental data . in particular we will analyze the cosmic rays spectrum which spans 13 decades in energy and 33 decades in particle flux that , as it is widely known , violates the boltzmann - gibbs statistics . as we will see , we have a high quality agreement between the theory and the observed data . the paper is organized as it follows . in sect . ii , generalizing the approach proposed in ref.s @xcite , we introduce a class of one parameter deformed structures and study their mathematical properties . in sect . iii , within this context , starting from the jaynes maximum entropy principle we consider the most general class of deformed statistical mechanics preserving the main features of the standard boltzmann - gibbs one . in sect . iv , we show that the entropy @xmath10 introduced in ref . @xcite is the only one existing beside the boltzmann - shannon entropy @xmath6 which is simultaneously concave , additive and extensive . then the statistical mechanics and thermodynamics based on @xmath10 can be viewed as a natural extension of the boltzmann - gibbs one , recovering this last as the deformation parameter @xmath1 approaches to zero . in sect . v , we consider the mean properties of @xmath1-exponential and @xmath1-logarithm which have a fundamental role in the formulation of the new statistical mechanics . in sect . vi , we extend the formalism to a time dependent and relativistic context . in particular after introducing the relativistic @xmath1-kinetic evolution equation we study its stationary state and prove the h - theorem . in sect . vii , we explain the origin of the @xmath1-deformation and show that it emerges naturally within the einstein special relativity . in sect . viii , we propose an approach which permits to determine within the special relativity the value of the parameter @xmath1 . in sect . ix , we compare two sets of experimental data with the predictions of the present theory . finally in sect . x , some concluding remarks are reported .

in ref . [ physica a * 296 * , 405 ( 2001 ) ] , starting from the one parameter deformation of the exponential function , a statistical mechanics has been constructed which reduces to the ordinary boltzmann - gibbs statistical mechanics as the deformation parameter approaches to zero . we show that is the unique existing entropy obtained by a continuous deformation of and preserving unaltered its fundamental properties of concavity , additivity and extensivity . these properties of permit to determine unequivocally the values of the above mentioned parameters and . subsequently , we explain the origin of the deformation mechanism introduced by and show that this deformation emerges naturally within the einstein special relativity . furthermore , we extend the theory in order to treat statistical systems in a time dependent and relativistic context . the novel statistical mechanics here presented preserves unaltered the mathematical and epistemological structure of the ordinary statistical mechanics and is suitable to describe a very large class of experimentally observed phenomena in low and high energy physics and in natural , economic and social sciences . finally , in order to test the correctness and predictability of the theory , as working example we consider the cosmic rays spectrum , which spans 13 decades in energy and 33 decades in flux , finding a high quality agreement between our predictions and observed data .

in ref . [ physica a * 296 * , 405 ( 2001 ) ] , starting from the one parameter deformation of the exponential function , a statistical mechanics has been constructed which reduces to the ordinary boltzmann - gibbs statistical mechanics as the deformation parameter approaches to zero . the distribution obtained within this statistical mechanics shows a power law tail and depends on the non specified parameter , containing all the information about the temperature of the system . on the other hand , the entropic form , which after maximization produces the distribution and reduces to the standard boltzmann - shannon entropy as , contains the coefficient whose expression involves , beside the boltzmann constant , another non specified parameter . in the present effort we show that is the unique existing entropy obtained by a continuous deformation of and preserving unaltered its fundamental properties of concavity , additivity and extensivity . these properties of permit to determine unequivocally the values of the above mentioned parameters and . subsequently , we explain the origin of the deformation mechanism introduced by and show that this deformation emerges naturally within the einstein special relativity . furthermore , we extend the theory in order to treat statistical systems in a time dependent and relativistic context . then , we show that it is possible to determine in a self consistent scheme within the special relativity the values of the free parameter which results to depend on the light speed and reduces to zero as recovering in this way the ordinary statistical mechanics and thermodynamics . the novel statistical mechanics here presented preserves unaltered the mathematical and epistemological structure of the ordinary statistical mechanics and is suitable to describe a very large class of experimentally observed phenomena in low and high energy physics and in natural , economic and social sciences . finally , in order to test the correctness and predictability of the theory , as working example we consider the cosmic rays spectrum , which spans 13 decades in energy and 33 decades in flux , finding a high quality agreement between our predictions and observed data .

1512.07083

c

in this paper we have shown how to use the information collected from a set of stabilizer measurements in order to extract information about a unitary error process acting on a graph state . since the stabilizer measurements are non - local operations , each of them depends on the errors that act on several different sites of the graph state . therefore the measurement outcomes of these stabilizers are correlated . we have studied in depth the case in which the local unitary errors are given by rotations aligned to one of the cartesian axes . this condition is needed to make the problem solvable , i.e. , we match the number of parameters to be reconstructed to the number of measured parameters ; moreover this restriction allows us to find efficient reconstruction methods . a surprising result arising from the fact that the stabilizer measurement outcomes are correlated is that in some circumstances it is not possible to reconstruct all the parameters of the unitary error even under the assumption that we can measure the stabilizers an arbitrary number of times . the possibility of completely reconstructing the error turns out to depend non - trivially on the connectivity structure of the graph state . specifically , in order to determine the intensity of local unitary rotations around the @xmath0 axis , the adjacency matrix @xmath109 associated to a graph state @xmath110 has to be a full - rank matrix , whereas to reconstruct rotations around the @xmath1 axis the matrix @xmath272 has to be full - rank , while rotations around the @xmath2 axis can always be reconstructed . we have ascertained for some classes of graph states whether it is possible to reconstruct all the errors . hence we have studied the properties of the adjacency matrices of _ linear chains _ ( with open and closed boundary conditions ) , _ square lattices _ in arbitrary number of dimensions , and ghz states ( these represented either as _ fully - connected _ graphs or as _ star _ graphs ) . finally we have shown that the performed reconstruction is resistant to various kinds of imperfections . in particular we have investigated the effect of a finite number of measurements of stabilizers , of the misalignment of the rotation axis and of the action of a depolarizing noise alongside the action of the unitary channel . in each of these cases , as long as the imperfections are small , the precision in the determination of the unitary channel is only mildly affected .

in the context of measurement - based quantum computation a way of maintaining the coherence of a _ graph state _ is to measure its stabilizer operators . aside from performing quantum error correction , it is possible to exploit the information gained from these measurements to characterize and then counteract a coherent source of errors ; that is , to determine all the parameters of an _ error channel _ that applies a fixed but unknown unitary operation to the physical qubits . the possibility of reconstructing the channels at all qubits depends non - trivially on the topology of the graph state .

in the context of measurement - based quantum computation a way of maintaining the coherence of a _ graph state _ is to measure its stabilizer operators . aside from performing quantum error correction , it is possible to exploit the information gained from these measurements to characterize and then counteract a coherent source of errors ; that is , to determine all the parameters of an _ error channel _ that applies a fixed but unknown unitary operation to the physical qubits . such a channel is generated , e.g. , by local stray fields that act on the qubits . we study the case in which each qubit of a given _ graph state _ may see a different error channel and we focus on channels given by a rotation on the bloch sphere around either the , or axis , for which analytical results can be given in a compact form . the possibility of reconstructing the channels at all qubits depends non - trivially on the topology of the graph state . we prove via perturbation methods that the reconstruction process is robust and supplement the analytic results with numerical evidence .

q-bio0702012

i

extensive interest has recently been devoted to the understanding of molecular motors , which play pivotal roles in cellular processes by performing mechanical work using the energy - driven conformational changes . kinesin , myosin , f@xmath2-atpase , groel , rna polymerase , and ribosome belong to a group of biological machines that undergoes a series of conformational changes during the mechanochemical cycle where the molecular conformation is directly coupled to the chemical state of the ligand . although substantial progress has been achieved in understanding the underlying physical principles that govern molecular motors during the last decade , major issues still remain to be resolved . specifically , some of the outstanding questions are as follows : ( a ) how is the chemical energy converted into mechanical work ? ( b ) how is the directionality of the molecular movement determined ? ( c ) how is the molecular movement coordinated or regulated ? several biochemical experiments have quantified the kinetic steps @xcite , single molecule experiments using optical tweezers have measured the mechanical response of individual molecular motors @xcite , and an increasing number of crystal structures have provided glimpses into the mechanisms of molecular motors @xcite . these experimental evidences , however , are not sufficient to fully address all the questions above . for example , little is known not only about the structural details of each chemical state but also about the kinetic pathways connecting them . hence , if feasible , a computational strategy using the coordinates from x - ray and/or nmr structures can shed light on the allosteric dynamics of molecular motors . although some initial numerical studies @xcite have proceeded towards addressing issue ( a ) for a few cases where both open and closed structures are _ explicitly known _ , no previous attempt has been made to answer issue ( c ) . in this paper we investigate this question in the context of the conventional kinesin where the mechanochemical coordination of the motor movement is best manifested among the motor proteins . one of the experimentally best studied molecular motors is the conventional kinesin ( kinesin-1 ) @xcite , a relatively small sized motor protein that transports cellular materials by walking in an unidirectional hand - over - hand manner along the microtubule ( mt ) filaments . compared to other motor proteins involved in material transport such as myosin and dynein , the conventional kinesin has a remarkable processivity , which can travel about a hundred ( @xmath38.2 @xmath4 ) steps without being dissociated from the mt . the mechanochemical cycle conjectured from experiments suggests that there must be a dynamic coordination between the two motor domains in order to achieve such high processivity . the quest to identify the origin of this dynamic coordination has drawn extensive attention among the kinesin community . since hancock and howard @xcite first hypothesized that the `` internal strain '' was needed for processivity , the strain - dependent mechanochemistry became a popular subject in kinesin studies @xcite . with the aid of optical tweezers , guydosh and block recently revisited this issue by monitoring the real - time kinesin dynamics in the presence of atp and @xmath5 , a tight binding atp analog @xcite . they discovered that , when @xmath5 was bound to the kinesin , the pause - time of the step increased substantially and that the normal step was restored only after the obligatory backstep . this suggests that @xmath5 is released only when the head bound with @xmath5 becomes the leading head ( l ) . supported by this observation , they advocated a kinetic model in which the rearward strain via the neck - linker facilitates the release of the ligand from the l @xcite . stated differently , the binding of the ligand to the l is inhibited because the rearward strain constitutes an unfavorable environment for the atp binding sites of the l. in the present study , we focus on the elucidation of the structural origin of the coordinated motion in kinesin by adopting a simple computational strategy . better straightforward evidence of the regulation on the nucleotide binding site can be obtained when a structure in which both kinesin heads are simultaneously bound to the mt binding site is determined . such a structure will allow us to identify the structural differences between the leading ( l ) and the trailing ( t ) head . to date , this structure , however , has not yet been reported . the only available structures include an isolated kinesin-1 without the mt @xcite , an isolated single - headed kinesin - like ( kif1a ) with various ligand states @xcite , and a single kif1a bound to the tubulin - dimer binding site @xcite . therefore , we utilized existing protein data bank ( pdb ) structures and manually built a model system of the two - headed kinesin molecule with both heads bound to the tubulin binding sites ( see fig.2 and legend ) . this model was used to generate an ensemble of structures via simulations . a direct comparison between the l and t equilibrium structures shows that the tension built on the neck - linker induces the disruption of the nucleotide binding site of the l , which directly supports inferences from experimental observations @xcite .

we test this hypothesis by explicitly modeling a-based kinesin structure whose both motor domains are bound on the tubulin binding sites . also , using an argument based on polymer theory , we estimate the internal tension built on the neck - linker to be pn . both of these conclusions support the experimental hypothesis . = 22pt

in the presence of atp , kinesin proceeds along the protofilament of microtubule by alternated binding of two motor domains on the tubulin binding sites . since the processivity of kinesin is much higher than other motor proteins , it has been speculated that there exists a mechanism for allosteric regulation between the two monomers . recent experiments suggest that atp binding to the leading head domain in kinesin is regulated by the rearward strain built on the neck - linker . we test this hypothesis by explicitly modeling a-based kinesin structure whose both motor domains are bound on the tubulin binding sites . the equilibrium structures of kinesin on the microtubule show disordered and ordered neck - linker configurations for the leading and the trailing head , respectively . the comparison of the structures between the two heads shows that several native contacts present at the nucleotide binding site in the leading head are less intact than those in the binding site of the rear head . the network of native contacts obtained from this comparison provides the internal tension propagation pathway , which leads to the disruption of the nucleotide binding site in the leading head . also , using an argument based on polymer theory , we estimate the internal tension built on the neck - linker to be pn . both of these conclusions support the experimental hypothesis . = 22pt

q-bio0702012

r

* mechanochemical cycle of kinesin : * we begin by reviewing the mechanochemical cycle of kinesin molecule on the mt to clarify the importance of dynamic coordination between the two motor domains for kinesin processivity . recent experiments using laser optical tweezers ( lot ) , cryo - electron microscopy , electron paramagnetic resonance , and fret , as well as the crystal structures at various states @xcite provide glimpses into the structural and dynamical details of how the kinesin molecule walks on the microbubule filaments . depending on the nucleotide state at the binding site , both the motor domain structure and the binding interface between kinesin and mt are affected . in particular , a minor change of the motor domain coupled to the nucleotide is amplified to a substantial conformational change of the neck linker between the ordered and the disordered state . experimental studies strongly suggest the mechanochemical cycle shown in fig.1 @xcite . the mechanical stepping cycle of kinesin initiates with the binding of atp to the empty kinesin head strongly bound to the mt [ @xmath6 . docking of the neck linker to the neck - linker binding motif on the leading head ( x in fig.1 ) propels the trailing head ( y in fig.1 ) in the ( + ) -direction of mt , which leads to an 8 @xmath4 mechanical step [ @xmath7 . the interaction with the mt facilitates the dissociation of adp from the catalytic core of l [ @xmath8 . atp is hydrolyzed and produces @xmath9 state for the t [ @xmath10 . when @xmath11 is released and the trailing head is unbound from the mt , the half - cycle is completed [ @xmath12 . the mechanical step is achieved in a hand - over - hand fashion by alternating the binding of the two motor domains ( x and y in fig.1 ) to the mt @xcite . high processivity of the kinesin requires this kinetic cycle to be stable ( remain within the yellow box in fig.1 ) . a premature binding of the atp to the leading head in the state of @xmath13 should be prevented , i.e. , the condition @xmath14/(k_r^{(iii)}+k_{diss}^{(iii)})$ ] and @xmath15/(k_r^{(iv)}+k_{diss}^{(iv)})\rightarrow 0 $ ] should be satisfied in fig.1 ( see supporting information for the master equation describing the kinetic cycle ) . atp binding to the ( iii ) or ( iv ) states can destroy the mechanochemical cycle of the kinesin . the binding of atp on the leading head should be suppressed before the @xmath16-@xmath11 is released from the t. otherwise , both heads become adp - bound states , which have a weak binding affinity to the mt , and that leads to dissociation from the mt . since the kinesin has a high processivity compared to other molecular motors , effective communication is required between the two heads regarding the chemical state of each of the partner motor domains . + * two - headed kinesin bound to the microtubule : * in the absence of interactions with the mt , the individual kinesin monomers fold into identical conformations . to achieve its biological function , however , folding into the native structure alone is not sufficient . coupled with the nucleotide and the mt , the two kinesin monomers in the dimeric complex need to alternate the acquisition of the native structure in a time - coordinated fashion for the uni - directional movement . the currently available three - dimensional structure ( pdb i d : 3kin , structure 2 in fig.2-a ) , in which each monomer is in its native state , does not provide such a dynamic picture by failing in fulfilling the geometrical requirement of simultaneous bindings of both motor domain to the adjacent tubulin binding sites that have an 8-nm gap . the inspection of 3-d structure suggests that a substantial increase of the distance between the two motor domains can be gained by breaking a few contacts associated with the neck - linker ( @xmath17 , @xmath18 ) and the neck - linker binding site on the motor domain ( @xmath19 ) . to this end , we manipulated the 3-d structure of 3kin around the neck - linker of the l and created a temporary structure whose two heads bind to the mt binding sites simultaneously . both l and t have energetic biases towards the identical native fold but the interactions with the tubulin binding sites adapt the dimeric kinesin structure into a different minimum structure , which is not known _ a priori_. we performed simulations ( see methods ) to relax this initial structure and to establish the thermal equilibrium ensemble of the kinesin molecule on mt ( see fig.3-a ) . transient dimeric kinesin conformations corresponding to the steps ( iii ) and ( iv ) during the cycle ( fig.1 ) allow us to investigate the structural deviation between l and t of kinesin molecule . this simple computational exercise can confirm or dismiss the experimental conjecture regarding whether the mechanochemical strain significantly induces regulation on the nucleotide binding site and also if it occurs in the l. + * catalytic core of the leading head is less native - like on the mt : * since the nucleotide binding and release dynamics is sensitively controlled by the kinesin structure , we assume that the nucleotide molecule has an optimal binding affinity to the kinesin motor domain in the native structure . for function there is a need to understand how the native structure of the kinesin motor domain is perturbed under the different topological constraints imposed on the dimeric kinesin configuration by interacting with the mt . the equilibrium ensemble of the structures shows that the neck - linker is in the docked state for the t but undocked for l. in comparison to the native structure , the overall shape of the nucleotide binding pocket in the t is more preserved . as long as the mt constrains the two heads 8-nm apart , this configuration is dominant in the thermal ensemble ( see fig.3-a ) . _ global shape comparison : _ there are in principle multiple ways to quantitatively compare the two motor domain structures . to assess the structural differences , the radius of gyration ( @xmath20 ) of the two motor domain structures from the equilibrium ensemble are computed ( see fig.3-b ) . because the neck - linker and the neck - helix adopt different configurations relative to the motor domain in each monomer , we perform a @xmath21 analysis for the motor domains only ( residue 2 - 324 ) . the @xmath21 distributions show that the l is slightly bigger than the t both in the size ( @xmath22 ) and in the dispersion ( @xmath23 ) . meanwhile the @xmath21 for the native state ( 3kin ) is @xmath24 . clearly , the sizes of both of the heads in the thermal ensemble are expanded at @xmath25 as compared to the native structure . the size alone does not tell much about the difference between the structures . the rmsd relative to the native structure and between the two motor domains ( residues 2 - 324 ) computed over the equilibrium ensemble gives @xmath26 , @xmath27 , @xmath28 , where @xmath29 is the rmsd between conformations x and y. if the @xmath30 helix is excluded from the rmsd calculation of motor domain ( residues 2 - 315 ) , then @xmath31 , @xmath32 , and @xmath33 . the rmsd analysis shows that the @xmath30 helix significantly contributes more for the deviation of the leading head from its native state than the t. additional detailed comparisons with respect to the native state can be made using the structural overlap function of @xmath34 pair , @xmath35 , which is defined as @xmath36 where @xmath37 if @xmath38 , otherwise @xmath39 . @xmath40 is the distance of @xmath34 pair in native state , where @xmath41 . by setting @xmath42 values identical in both heads ( i.e. , both heads have the same native state ) , we compute the @xmath35 values for the trailing and the leading heads , respectively . the relative difference of the @xmath35 value between t and l , @xmath43 is defined by @xmath44 which quantitatively measures the structural difference of the two heads . based on the @xmath43 value ( fig.3-c ) , the distances between the mt binding motif of the t ( l11 , l12 , @xmath45 , @xmath46 ) and other secondary structure units ( @xmath47 , @xmath48 , @xmath49 , @xmath50 , @xmath51 , @xmath52 ) are 50 % more native - like than in the l. _ conserved native contacts in trailing head reveals the strain propagation pathway in leading head : _ a direct measure of similarity to the native structure is the fraction of native contacts preserved in the thermal ensemble @xcite . since we assume that atp affinity is optimized in the native state , we can readily assess the quality of the structure using this measure . we quantify the nativeness of a pair using @xmath53 , where @xmath54 if @xmath55 , @xmath56 residues are in contact at the native state ( @xmath57 ) , and @xmath58 otherwise . @xmath59 ( with @xmath60 or @xmath61 ) is obtained by averaging over the thermal ensemble . when @xmath59 is averaged over all the native pairs , the average fraction of native contacts , @xmath62 is calculated as @xmath63 where @xmath64 is the total number of native pairs . for the t and l conformations , @xmath65 and @xmath66 , respectively . the relative difference of native contacts between the two kinesin heads at the pair level , @xmath67 , is quantified similarly to eq.[eqn : chi ] as @xmath68 in fig.4 , @xmath67 is color - coded based on its value . as expected from the equilibrium ensemble , conspicuous differences are found around the structural motifs having direct contacts with neck - linker , giving @xmath69 . quantitative inspection of the other contacts is illustrated in the structure . we color the kinesin head structure based on the @xmath67 value . the residue pairs are colored in magenta if @xmath70 , red if @xmath71 , where the positive @xmath67 signifies that the native contacts in trailing head are more intact . the residue pairs are colored in light - blue if @xmath72 , blue if @xmath73 . more intact contacts , when the trailing and the leading head are compared , are visualized by yellow line in fig.5-b . our analysis not only shows that there is higher probability of the formation of native contacts present in the t in comparison to the l , but also suggests how the tension is propagated towards the nucleotide binding site to disrupt the nativeness of the nucleotide binding pocket in the leading head . as expected , a dense network of intact contacts are found between the neck - linker ( @xmath18 ) and the neck - linker binding motif ( @xmath19 ) . this network continues along the @xmath52 helix , perturbing @xmath74 , @xmath75 , @xmath45 , and finally reaches the nucleotide binding site ( see si fig.6 for the nomencaltures of the secondary structures ) . it is surprising that the disruptions of native contacts are found particularly in the nucleotide binding site , which is believed to be the trigger point for the allosteric transition . all the important nucleotide binding motifs ( p - loop , switch-1 , switch-2 , and n4 ) are recognized by our simulational analysis using a nonlinear - hamiltonian ( see supporting information for the comparison with linear - harmonic potential represented as gaussian network model ) . + * estimate of the tension in the neck - linker : * the deformation of the leading motor domain is caused by the internal tension in the neck linker . the tension on the neck linker is estimated using the force ( @xmath76 ) versus extension ( @xmath77 ) relationship of a worm - like chain model @xcite , @xmath78 , \label{eqn : wlc}\ ] ] where @xmath79 is the persistence length of the polymer and @xmath61 is the contour length . @xmath80 for the 15-amino acid neck - linker ( residue from 324 to 338 ) @xmath81 , and in the equilibrium ensemble of structures , @xmath82 @xmath4 . assuming that @xmath83 @xmath4 @xcite for this segment , we estimate a tension @xmath84 @xmath85 . by integrating eq.[eqn : wlc ] for @xmath86 @xmath4 . the tensional energy stored in the neck - linker is obtained , which is 17 @xmath87 . about 20% of the atp hydrolysis energy ( @xmath88 ) is stored in the neck - linker and directly perturbs the nucleotide binding site of the l , whereas mechanical action to the t is dissipated through the dense network of contacts formed between the neck - linker ( @xmath89 ) and the neck - linker binding site ( @xmath90 ) . for a given extension @xmath91 , when the length of the neck - linker is varied by @xmath92 , the variation in the length of the neck - linker can affect the effective tension as @xmath93(\delta l / l)$ ] . for a given @xmath94 , the resulting tension change may be significant depending on the value of the extension @xmath77 . experimentally , the kinesin dynamics has recently been studied by varying the linker length by introducing a spacer composed of amino acids @xcite . since @xmath95 @xmath4 for the insertion of a single amino acid , the lengthening of the linker leads to a reduction of the tension by @xmath96 . in light of the force values controlling the kinesin dynamics in lot experiments , which is @xmath97 @xmath85 and 7 @xmath85 is the stall force , a value of @xmath98 can be significant . according to hackney _ et . _ s experimental analysis the processivity is reduced by @xmath99-fold when a single amino acid is inserted or deleted , and 6 or 12 additional amino acids resulted in to a 3@xmath1004 fold reduction in the kinetic processivity @xcite .

in the presence of atp , kinesin proceeds along the protofilament of microtubule by alternated binding of two motor domains on the tubulin binding sites . since the processivity of kinesin is much higher than other motor proteins , it has been speculated that there exists a mechanism for allosteric regulation between the two monomers . recent experiments suggest that atp binding to the leading head domain in kinesin is regulated by the rearward strain built on the neck - linker . the equilibrium structures of kinesin on the microtubule show disordered and ordered neck - linker configurations for the leading and the trailing head , respectively . the comparison of the structures between the two heads shows that several native contacts present at the nucleotide binding site in the leading head are less intact than those in the binding site of the rear head . the network of native contacts obtained from this comparison provides the internal tension propagation pathway , which leads to the disruption of the nucleotide binding site in the leading head .

in the presence of atp , kinesin proceeds along the protofilament of microtubule by alternated binding of two motor domains on the tubulin binding sites . since the processivity of kinesin is much higher than other motor proteins , it has been speculated that there exists a mechanism for allosteric regulation between the two monomers . recent experiments suggest that atp binding to the leading head domain in kinesin is regulated by the rearward strain built on the neck - linker . we test this hypothesis by explicitly modeling a-based kinesin structure whose both motor domains are bound on the tubulin binding sites . the equilibrium structures of kinesin on the microtubule show disordered and ordered neck - linker configurations for the leading and the trailing head , respectively . the comparison of the structures between the two heads shows that several native contacts present at the nucleotide binding site in the leading head are less intact than those in the binding site of the rear head . the network of native contacts obtained from this comparison provides the internal tension propagation pathway , which leads to the disruption of the nucleotide binding site in the leading head . also , using an argument based on polymer theory , we estimate the internal tension built on the neck - linker to be pn . both of these conclusions support the experimental hypothesis . = 22pt

hep-ph0606002

c

in this article , we analyze the vertices @xmath0 , @xmath1 , @xmath2 and @xmath3 within the framework of the light - cone qcd sum rules approach in an unified way . the strong coupling constants @xmath4 and @xmath5 are important parameters in evaluating the charmonium absorption cross sections in searching for the quark - gluon plasmas , our numerical values of the @xmath4 and @xmath5 are compatible with the existing estimations although somewhat smaller , the @xmath6 symmetry breaking effects are very large , about @xmath7 , the approximation of the @xmath6 symmetry @xmath249 is not suitable @xcite . for the scalar mesons @xmath8 and @xmath9 , we take the point of view that they are the conventional @xmath10 and @xmath11 meson respectively , and calculate the strong coupling constants @xmath250 and @xmath13 within the framework of the light - cone qcd sum rules approach . the numerical values of the scalar-@xmath14 and -@xmath15 coupling constants @xmath12 and @xmath13 are compatible with the existing estimations although somewhat smaller , the large values support the hadronic dressing mechanism . just like the scalar mesons @xmath34 and @xmath33 , the scalar mesons @xmath8 and @xmath9 may have small @xmath10 and @xmath11 kernels of typical @xmath10 and @xmath11 mesons size respectively . the strong coupling to virtual intermediate hadronic states ( or the virtual mesons loops ) can result in smaller mass than the conventional scalar mesons @xmath10 and @xmath11 in the constituent quark models , enrich the pure states @xmath10 and @xmath11 with other components . the @xmath251 and @xmath26 may spend part ( or most part ) of they lifetime as virtual @xmath252 and @xmath253 states . furthermore , we study the dependence of the strong coupling constants @xmath4 and @xmath5 on the non - perturbative parameter @xmath18 of the twist-2 @xmath19 meson light - cone distribution amplitude . the values of the strong coupling constants @xmath237 and @xmath5 are more sensitive to the @xmath18 comparing with the @xmath226 and @xmath17 . the largest uncertainties come from the uncertainties of the @xmath18 , they are the ideal channels to determine the parameter directly from the experimental data . once the experimental data for the values of the strong coupling constants @xmath237 and @xmath5 are available , powerful constraints can be put on the range of the parameter @xmath18 .

in this article , we analyze the vertices , , and within the framework of the light - cone qcd sum rules approach in an unified way . the strong coupling constants and are important parameters in evaluating the charmonium absorption cross sections in searching for the quark - gluon plasmas , our numerical values of the and are compatible with the existing estimations although somewhat smaller , the symmetry breaking effects are very large , about . for the charmed scalar mesons and , we take the point of view that they are the conventional and mesons respectively , and calculate the strong coupling constants and with the vector interpolating currents . the numerical values of the scalar- and - coupling constants and are compatible with the existing estimations , the large values support the hadronic dressing mechanism . furthermore , we study the dependence of the four strong coupling constants , , and on the non - perturbative parameter of the twist-2 meson light - cone distribution amplitude . + z. g. wang , s. l. wan + department of physics , north china electric power university , baoding 071003 , p. r. china + department of modern physics , university of science and technology of china , hefei 230026 , p. r. china + pacs numbers : 12.38.lg ; 13.25.jx ; 14.40.cs * key words : * strong coupling constants , light - cone qcd sum rules

in this article , we analyze the vertices , , and within the framework of the light - cone qcd sum rules approach in an unified way . the strong coupling constants and are important parameters in evaluating the charmonium absorption cross sections in searching for the quark - gluon plasmas , our numerical values of the and are compatible with the existing estimations although somewhat smaller , the symmetry breaking effects are very large , about . for the charmed scalar mesons and , we take the point of view that they are the conventional and mesons respectively , and calculate the strong coupling constants and with the vector interpolating currents . the numerical values of the scalar- and - coupling constants and are compatible with the existing estimations , the large values support the hadronic dressing mechanism . furthermore , we study the dependence of the four strong coupling constants , , and on the non - perturbative parameter of the twist-2 meson light - cone distribution amplitude . + z. g. wang , s. l. wan + department of physics , north china electric power university , baoding 071003 , p. r. china + department of modern physics , university of science and technology of china , hefei 230026 , p. r. china + pacs numbers : 12.38.lg ; 13.25.jx ; 14.40.cs * key words : * strong coupling constants , light - cone qcd sum rules

1503.07259

i

consider a sequence of probability measures @xmath2 on a common measurable space @xmath3 ; we assume that the probabilities have common dominating finite - measure @xmath4 and write the densities w.r.t . @xmath4 as @xmath5 . in particular , for some known @xmath6 , we let @xmath7 where the normalizing constant @xmath8 may be unknown . the context of interest is when the sequence of densities is associated to an ` accuracy ' parameter @xmath9 , with @xmath10 as @xmath11 with @xmath12 . this set - up is relevant to the context of discretised numerical approximations of continuum fields , as we will explain below . the objective is to compute : @xmath13 : = \int_e g(u)\eta_\infty(u)du\ ] ] for potentially many measurable @xmath14integrable functions @xmath15 . in practice one can not treat @xmath16 and must consider these distributions with @xmath17 . problems involving numerical approximations of continuum fields are discretized before being solved numerically . finer - resolution solutions are more expensive to compute than coarser ones . such discretizations naturally give rise to hierarchies of resolutions via the use of nested meshes . successive solution at refined meshes can be utilized to mitigate the number of necessary solves for the finest resolutions . for the solution of linear systems , the coarsened systems are solved as pre - conditioners within the framework of iterative linear solvers in order to reduce the condition number , and hence the number of necessary iterations at the finer resolution . this is the principle of multi - grid methods . for monte carlo methods , as in the context above , a telescoping sum of associated differences at successive refinement levels can be utilized . this is so that the bias of the resulting multilevel estimator is determined by the finest level but the variance of the estimators of the differences decays . the reduction in the variance at finer levels implies that the number of samples required to reach a given error tolerance is also reduced with increasing resolution . this procedure is then optimized to balance the extra per - sample cost at the finer levels . overall one can obtain a method with smaller computational effort to reach a pre - determined error than applying a standard monte carlo method immediately at the finest resolution @xcite . multi level monte carlo ( mlmc ) @xcite ( see also @xcite ) methods are such that one typically sets an error threshold for a target expectation , and then sets out to attain an estimator with the prescribed error utilizing an optimal allocation of monte carlo resources . within the context of @xcite , the continuum problem is a stochastic differential equation ( sde ) or pde with random coefficients , and the target quantity is an expectation of a functional , say @xmath18 , of the parameter of interest @xmath19 , over an ideal measure @xmath20 that avoids discretisation . the levels are a hierarchy of refined approximations of the function - space , specified in terms of a small resolution parameter say @xmath9 , for @xmath21 , thus giving rise to a corresponding sequence of approximate laws @xmath22 . the method uses the telescopic sum @xmath23 = { \mathbb{e}}_{\eta_0}[g(u ) ] + \sum_{l=1}^l \{{\mathbb{e}}_{\eta_l}[g(u)]-{\mathbb{e}}_{\eta_{l-1}}[g(u)]\}\ ] ] and proceeds by coupling the consecutive probability distributions @xmath24 , @xmath25 . thus , the expectations are estimated via the standard unbiased monte carlo averages @xmath26 where @xmath27 are i.i.d . samples , with marginal laws @xmath24 , @xmath22 , respectively , carefully constructed on a joint probability space . this is repeated independently for @xmath28 . the overall multilevel estimator will be @xmath29 under the convention that @xmath30 . a simple error analysis gives that the mean squared error ( mse ) is @xmath31 \}^2 = \underbrace{{\mathbb{e}}\ { \hat{y}_{l,{\rm multi}}- { \mathbb{e}}_{\eta_{l } } [ { g}(u)]\}^2}_{\rm variance } + \underbrace{\{{\mathbb{e}}_{\eta_l } [ { g}(u ) ] - { \mathbb{e}}_{\eta_{\infty } } [ g(u)]\}^2}_{\rm bias}\ . \label{eq : mse}\ ] ] one can now optimally allocate @xmath32 to minimize the variance term @xmath33 for fixed computational cost @xmath34 , where @xmath35 is the variance of @xmath36 $ ] and @xmath37 the computational cost for its realisation . using lagrange multipliers for the above constrained optimisation , we get the optimal allocation of resources @xmath38 . in more detail , the typical chronology is that one targets an mse , say @xmath39 , then ( i ) given a characterisation of the bias as an order of @xmath9 , one determines @xmath40 , @xmath41 , for some integer @xmath42 , and chooses a horizon @xmath43 such that the bias is @xmath44 and ( ii ) given a characterisation of @xmath35 , @xmath37 as some orders of @xmath9 , one optimizes the required samples @xmath45 needed to give variance @xmath44 . thus , a specification of the bias , variance and computational costs as functions of @xmath9 is needed . as a prototypical example of the above setting @xcite , consider the case @xmath46 with @xmath47 being the terminal position of the solution @xmath48 of a sde and @xmath22 is the distribution of @xmath47 under the consideration of a numerical approximation with time - step @xmath49 . the laws @xmath24 , @xmath22 can be coupled via use of the same driving brownian path . invoking the relevant error analysis for sde models , one can obtain ( for @xmath20 , @xmath50 , and defined on the common probability space ) : * weak error @xmath51| = { \mathcal{o}}(h_l^\alpha)$ ] , providing the bias @xmath52 , * strong error , @xmath53 , giving the variance @xmath54 , * computational cost for a realisation of @xmath55 , @xmath56 , for some constants @xmath57 related to the details of the discretisation method . the standard euler marayuma method for solution of sde gives the orders @xmath58 . assuming a general context , given such rates for bias , @xmath35 and @xmath37 , one proceeds as follows . recall that @xmath59 , for some integer @xmath42 . then , targeting an error tolerance of @xmath60 and letting @xmath61 , one has @xmath62 , as in @xcite . using the optimal allocation @xmath38 , one finds that @xmath63 . taking under consideration a target error of size @xmath64 , one sets @xmath65 , with @xmath66 chosen to control the total error for increasing @xmath43 . thus , for the resulted estimator in ( [ eq : multi])-([eq : mse ] ) , we have : @xmath67 to have a variance of @xmath39 , one sets @xmath68 , so @xmath66 may or may not depend on @xmath60 depending on whether this sum converges or not ( recalling that @xmath69 ) . in the case of euler - marayuma , for example , @xmath70 , @xmath71 , and the cost is @xmath72 , versus @xmath73 using a single level with mesh - size @xmath74 . if @xmath75 , corresponding for instance to the milstein method , then the cost is @xmath76 . the latter is the cost of obtaining the given level of error for a scalar random variable , and is therefore optimal . the worst scenario is when @xmath77 . in this case it is sufficient to set @xmath78 to make the variance @xmath39 , and then the number of samples on the finest level is given by @xmath79 whereas the total algorithmic cost is @xmath80 , where @xmath81 . in this case , one can choose the largest value for the bias , @xmath82 , so that @xmath83 and the total cost , @xmath84 , is dominated by this single sample . see @xcite for more details . it is important to note that the realizations @xmath85 , @xmath86 for a given increment must be coupled to obtain decaying variances @xmath35 . in the case of an sde driven by brownian motion one can simply simulate the driving noise on level @xmath87 and then upscale it to level @xmath88 by summing elements of the finer path @xcite . for the case of a pde forward model relying on uncertain input the scenario is quite similar @xcite . for example , in the case that the input is of fixed dimension and the levels arise due to discretization of the forward map alone within a finite element context , one would use the same realization of the input on two separate meshes for a pairwise - coupled realization . note that in the more general context of pde , it is natural to decompose @xmath89 , where @xmath90 is the spatio - temporal dimension of the underlying continuum . in particular , the number of degrees of freedom of a @xmath91dimensional field approximated on a mesh of diameter @xmath9 is given by @xmath92 . then , the forward solve associated to the evaluation of @xmath93 may range from linear ( @xmath94 ) to cubic ( @xmath95 ) in the number of degrees of freedom . for example , the solution of an sde or a sparse matrix - vector multiplication give @xmath94 , a dense matrix - vector multiplication would give @xmath96 , and direct linear solve by gaussian elimination would give @xmath95 . the present work will focus on the case of an inverse problem with fixed - dimensional input . indeed the difficulty arises here because we only know how to _ evaluate _ ( up - to a constant ) the target density at any given level , and can not directly obtain independent samples from it . there exist many approaches to solving such problem , for example one can review the recent works @xcite which use markov chain monte carlo ( mcmc ) methods in the multilevel framework . in this article a more natural and powerful formulation is considered , related with the use of sequential monte carlo approaches . sequential monte carlo ( smc ) methods are amongst the most widely used computational techniques in statistics , engineering , physics , finance and many other disciplines . in particular smc samplers @xcite are designed to approximate a sequence @xmath97 of probability distributions on a common space , whose densities are only known up - to a normalising constant . the method uses @xmath98 samples ( or particles ) that are generated in parallel , and are propagated with importance sampling ( often ) via mcmc and resampling methods . several convergence results , as @xmath99 grows , have been proved ( see e.g. @xcite ) . smc samplers have also recently been proven to be stable in certain high - dimensional contexts @xcite . current state of the art for the analysis of smc algorithms include the work of @xcite . in this work , the method of smc samplers is perfectly designed to approximate the sequence of distributions , but as we will see , implementing the standard telescoping identity of mlmc requires some ingenuity . in addition , in order to consider the benefit of using smc , one must analyze the variance of the estimate ; in such scenarios this is not a trivial extension of the convergence analysis previously mentioned . in particular , one must very precisely consider the auto - covariance of the smc approximations and consider the rate of decrease of this quantity as the time - lag between smc approximations increases . such a precise analysis does not appear to exist in the literature . we note that our work , whilst presented in the context of pdes , is not restricted to such scenarios and , indeed can be applied in almost any other similar context ( that is , a sequence of distributions on a common space , with increasing computational costs associated to the evaluation of the densities which in some sense converge to a given density ) ; however , the potential benefit of doing so , may not be obvious in general . this article is structured as follows . in section [ sec : set_up ] the ml identity and smc algorithm are given . in section [ sec : complex ] our main complexity result is given under assumptions and their implications are discussed . in section [ sec : ip ] we give a context where the assumptions of our theoretical results can be verified . in section [ sec : numerics ] our approach is numerically demonstrated on a bayesian inverse problem . section [ sec : complex ] and the appendix provide the proofs of our main theorem .

in this article we consider the approximation of expectations w.r.t . probability distributions associated to the solution of partial differential equations ( pdes ) ; this scenario appears routinely in bayesian inverse problems . in practice , one often has to solve the associated pde numerically , using , for instance finite element methods and leading to a discretisation bias , with the step - size level . in addition , the expectation can not be computed analytically and one often resorts to monte carlo methods . in the context of this problem , it is known that the introduction of the multilevel monte carlo ( mlmc ) method can reduce the amount of computational effort to estimate expectations , for a given level of error . this is achieved via a telescoping identity associated to a monte carlo approximation of a sequence of probability distributions with discretisation levels . in many practical problems of interest sampling of the associated sequence of probability distributions . it is shown that under appropriate assumptions , the attractive property of a reduction of the amount of computational effort to estimate expectations , for a given level of error , can be maintained within the smc context . the approach is numerically illustrated on a bayesian inverse problem . multilevel monte carlo , sequential monte carlo , bayesian inverse problems . : 65c30 , 65y20 .

in this article we consider the approximation of expectations w.r.t . probability distributions associated to the solution of partial differential equations ( pdes ) ; this scenario appears routinely in bayesian inverse problems . in practice , one often has to solve the associated pde numerically , using , for instance finite element methods and leading to a discretisation bias , with the step - size level . in addition , the expectation can not be computed analytically and one often resorts to monte carlo methods . in the context of this problem , it is known that the introduction of the multilevel monte carlo ( mlmc ) method can reduce the amount of computational effort to estimate expectations , for a given level of error . this is achieved via a telescoping identity associated to a monte carlo approximation of a sequence of probability distributions with discretisation levels . in many practical problems of interest , one can not achieve an i.i.d . sampling of the associated sequence of probability distributions . a sequential monte carlo ( smc ) version of the mlmc method is introduced to deal with this problem . it is shown that under appropriate assumptions , the attractive property of a reduction of the amount of computational effort to estimate expectations , for a given level of error , can be maintained within the smc context . the approach is numerically illustrated on a bayesian inverse problem . multilevel monte carlo , sequential monte carlo , bayesian inverse problems . _ ams subject classification _ : 65c30 , 65y20 .

1407.7746

i

the application of renormalization techniques to quantum gravity theories has always been problematic . this is due to the inherent tension between the notion of scale - dependent physics of the renormalization group on the one side , and the role of background - independence of general relativity on the other side . the renormalization group is a tool for dealing with a phenomenon that is inherent to many physical systems with a very large number of degrees of freedom , which are nonlinear , i.e. exhibit nontrivial self - interaction . to describe such systems , it is convenient to not consider all degrees of freedom on an equal footing , but rather to order them according to some hierarchy . this way one considers only a finite and manageable amount of the system at a time . this hierarchy often runs along a length scale , introducing a notion of `` smaller '' and `` larger '' degrees of freedom . often the dynamics on different scales are related to each other , in that collective effects of smaller degrees of freedom combine to larger ones . the non - linearity of the interaction between the small degrees of freedom influences the dynamics of the system on larger scales . this leads to a scale - dependence of the effective actions for the system , and this `` running of coupling constants '' is a key concept in wilson s approach to the renormalization group @xcite . on the other hand , one of the key concepts of general relativity is its general covariance , which makes the theory background - independent . in particular , diffeomorphisms of the space - time manifold act as gauge symmetries . arguably , the corresponding quantum theory should possess the same symmetries . this makes an easy distinction of `` small '' and `` large '' degrees of freedom difficult , because there is no external geometry to measure any length with . rather , because geometry itself is dynamical , spatial dimensions become observables , rather than parameters . the length scales themselves become degrees of freedom , making the applicability of usual renormalization techniques to quantum gravity theories difficult . + in this article , we will consider the tentatively background - independent spin foam model approach to quantum gravity , which has been very successful in describing a path integral for quantum gravity in terms of quantized geometry @xcite , @xcite , see also @xcite for a review . a crucial ingredient to spin foam models is an abstract discretization of space - time . we will show how working with _ embedded _ , rather than abstract , discretizations provides not only a natural notion of continuum limit , it also results in a conceptually clear analogue of `` scale '' in these models , as well as a mathematically well - defined notion of renormalization group flow for the background - independent context . [ [ the - plan - of - the - paper ] ] the plan of the paper : + + + + + + + + + + + + + + + + + + + + + + in section [ sec : holonomysfm ] we will discuss an approach to spin foam models based on the kkl - formalism @xcite , which works with embedded , rather than abstract , discretizations of space - time . as a result , there is a natural notion of continuum limit in the sense of a projective family . in order to construct the continuum path integral measure , the partial measures need to satisfy a strong condition , called _ cylindrical consistency_. we show how these conditions are precisely the analogue of wilson s rg flow equation in the background - independent context , and discuss the physical interpretation in terms of configurations and measurements . in section [ sec : diffinvariance ] we will demonstrate that the condition for the continuum path integral measures to be invariant under space - time diffeomorphisms , can be posed naturally within this context , and show how this is a very strong condition on the partial measures defined at each scale . in section [ sec : example ] we will discuss an easy example , where the rg flow can be solved completely and exactly , and demonstrate how the condition of diffeomorphism - invariance severely restricts the set of solutions , most of which spontaneously break background - independence . in section [ sec : approximations ] we will discuss several suggestions for how to derive approximate solutions in more complicated cases , and give an example with a quartic interaction term , which can be treated with these methods . while appendix [ app : properties ] provides mathematical details for some of the constructions , in appendix [ app : canonical ] the connection to refined algebraic quantization techniques is discussed , to make contact to the canonical framework in the case that space - time has a boundary .

in this article we discuss an implementation of renormalization group ideas to spin foam models , where there is no a priori length scale with which to define the flow . in the context of the continuum limit of these models , we show how the notion of cylindrical consistency of path integral measures gives a natural analogue of wilson s rg flow equations for background - independent systems . we discuss the conditions for the continuum measures to be diffeomorphism - invariant , and consider both exact and approximate examples .

in this article we discuss an implementation of renormalization group ideas to spin foam models , where there is no a priori length scale with which to define the flow . in the context of the continuum limit of these models , we show how the notion of cylindrical consistency of path integral measures gives a natural analogue of wilson s rg flow equations for background - independent systems . we discuss the conditions for the continuum measures to be diffeomorphism - invariant , and consider both exact and approximate examples .

1303.6863

i

in 1916 schwarzschild introduced the first exact solution of the vacuum einstein equations in four dimensions @xcite . this unique solution describes a spherically symmetric black hole . about fifty years later , in 1963 , the solution of a rotating black hole was discovered by kerr @xcite . if a compact dimension is added to the schwarzschild metric , it describes a five - dimensional black string . analogously the kerr spacetime turns into the spacetime of a rotating black string if a compact dimension is added . the idea of compact dimensions came up in the 1920s after kaluza suggested to add a fifth dimension to general relativity @xcite . this was one of the first attempts to unify gravity and the electromagnetic forces . in 1926 klein proposed a physical interpretation @xcite : the added dimension is compact , i.e. , it is curled up in itself and has a certain length , which is too small to be observed . but still the theory of kaluza and klein could not explain the weakness of gravity in comparison to the electromagnetic forces . in the 60s and 70s the string theory was developed . here the existence of higher dimensions is essential for the internal consistency of the theory , mostly they are also assumed to be compact . + the equations of motion for test particles in the schwarzschild spacetime were solved in 1931 by hagihara @xcite in terms of the elliptic weierstra @xmath0- , @xmath1- and @xmath2-functions . in the taub - nut @xcite , reissner - nordstrm @xcite and myers - perry @xcite spacetime the equations of motion were also solved in terms of the elliptic weierstra functions . the integration of geodesics was advanced in the papers of hackmann and lmmerzahl @xcite . they integrated the geodesics in the four - dimensional schwarzschild - de sitter space - time analytically in terms of the hyperelliptic @xmath3- and @xmath1-functions . the mathematical problem is based on the jacobi inversion problem which can be solved if restriced to the @xmath3-divisor . this method was also applied to find the analytical solution of the equations of motion in higher dimensional schwarzschild , schwarzschild-(anti)de sitter , reissner - nordstrm and reissner - nordstrm -(anti ) de sitter spactime @xcite as well as in kerr-(anti ) de sitter spacetimes @xcite and in higher dimensional myers - perry spacetime @xcite . recently , the geodesics equations in special cases were solved analytically in the singly spinning black ring spacetime @xcite and in the ( charged ) doubly spinning black ring spacetime @xcite . + the test particle motion in different spacetimes containing black strings was studied in @xcite . in @xcite and @xcite the analytical solutions of the equations of motion in the schwarzschild and kerr spacetime pierced by black string were presented . furthermore , the geodesic motion of test particles in field theoretical cosmic string spacetimes were investigated , namely abelian - higgs strings @xcite , two interacting abelian - higgs strings @xcite and cosmic superstrings @xcite . + in this paper we will consider static and rotating black strings , which can be obtained by adding an extra compact dimension to the schwarzschild and kerr metric . the geodesic motion in the ordinary schwarzschild spacetime and the kerr spacetime has already been analyzed in detail , however , when a compact dimension is added new behaviour occurs which is studied in this article . we present the analytic solutions of the geodesic equations in terms of the elliptic weierstra functions and discuss the corresponding orbits . in the first part the non - rotating black string is studied and compared to the schwarzschild black hole . in the second part the rotating black string is analysed and compared to the kerr black hole .

if a compact dimension is added to the four - dimensional schwarzschild or kerr spacetime , the new five - dimensional metric describes a ( rotating ) black string . the geodesics in the schwarzschild and kerr spacetime have been studied in great detail , however , when a compact dimension is added new behaviour occurs . we present the analytical solutions of the geodesic equations and discuss the possible orbits .

in this article we study the geodesic motion of test particles and light in the five - dimensional ( rotating ) black string spacetime . if a compact dimension is added to the four - dimensional schwarzschild or kerr spacetime , the new five - dimensional metric describes a ( rotating ) black string . the geodesics in the schwarzschild and kerr spacetime have been studied in great detail , however , when a compact dimension is added new behaviour occurs . we present the analytical solutions of the geodesic equations and discuss the possible orbits . the motion in the ordinary four - dimensional schwarzschild and kerr spacetime is compared to the motion in the ( rotating ) black string spacetime .

1303.6863

c

in this article we presented the complete set of analytical solutions of the geodesic equations of test particles and light in the static and the rotating black string spacetime . the static and the rotating black string metric are obtained by adding a compact dimension to the schwarzschild and kerr metric . the geodesic equations can be solved in terms of the elliptic @xmath0- , @xmath1- and @xmath2-functions . using effective potential techniques and parametric diagrams , possible types of orbits were derived . in the static case bos , eos and tos are possible , while in the rotating case bos , mbos , eos , teos , cteos , tros and tos are possible . in contrast to the ordinary four - dimensional schwarzschild and kerr spacetime , bound orbits of light are possible both in the static and the rotating black string spacetime . + hereby the analytic solutions do not only give a proof of the existence of bound orbits of light in the black string spacetime . they also present a usefull tool to calculate the exact orbits and their properties , including observables like the periastron shift of bound orbits , the light deflection of escape orbits , the deflecton angle and the lense - thirring effect . for the calculation of the observables analogous formulas to those given in @xcite can be used . observables can later be compared to observations . + the black strings considered here are so - called uniform black strings , since there is no dependence of the extra dimension . however , uniform black strings are subject to the gregory - laflamme instability @xcite , which is associated with the emergence of a branch of non - uniform black strings , whose horizon size is not constant w.r.t . the compact direction , but depends on the compact coordinate @xcite . at the endpoint of this nonuniform black string branch a horizon topology changing transition should be encountered @xcite . to gain a better understanding of this transition , it is essential to solve the geodesic equations . however , for the construction of nonuniform black strings no analytic techniques are available , and numerical techniques have to be employed . likewise , the geodesic equations must be solved numerically . it is therefore of high relevance , to have analytic solutions of the geodesic equations available as a testbed . moreover , it will be very interesting to see , how the set of analytic solutions changes , as the black strings become more and more nonuniform . also non - uniform rotating and charged black strings have been considered in @xcite and @xcite whose geodesics would be very interesting to study . + it is also interesting to consider higher dimensional ( @xmath264 ) schwarzschild spacetimes , here stable bound orbits are no longer possible @xcite . one may wonder wether bound orbits of light become possible if a compact dimension is added to the metric . but it can be shown that unlike in the ( rotating ) black string spacetime in five dimensions , stable bound orbits of light are not possible if a compact dimension is added to higher dimensional schwarzschild spacetimes . + moreover the bound orbits of light appear neither in the spacetime of an abelian - higgs string @xcite nor in the spacetime of two interacting abelian - higgs strings @xcite . there escape orbits are the only possibility for massless particles . for cosmic superstrings bound orbits of light are possible in melvin spacetimes but not in asymptotically conical spacetimes @xcite .

in this article we study the geodesic motion of test particles and light in the five - dimensional ( rotating ) black string spacetime . the motion in the ordinary four - dimensional schwarzschild and kerr spacetime is compared to the motion in the ( rotating ) black string spacetime .

in this article we study the geodesic motion of test particles and light in the five - dimensional ( rotating ) black string spacetime . if a compact dimension is added to the four - dimensional schwarzschild or kerr spacetime , the new five - dimensional metric describes a ( rotating ) black string . the geodesics in the schwarzschild and kerr spacetime have been studied in great detail , however , when a compact dimension is added new behaviour occurs . we present the analytical solutions of the geodesic equations and discuss the possible orbits . the motion in the ordinary four - dimensional schwarzschild and kerr spacetime is compared to the motion in the ( rotating ) black string spacetime .

1105.2630

r

we briefly recall the double null foliation formalism , see @xcite for more precise definitions . we use @xmath37 to denote the underlying space - time and use @xmath38 to denote the background metric . we assume that @xmath39 is spanned by a double null foliation generated by two optical functions @xmath40 and @xmath41 and they increase towards the future , @xmath42 and @xmath43 . we use @xmath44 / @xmath45 to denote the outgoing / incoming null hypersurfaces generated by the level surfaces of @xmath40 / @xmath41 . we use @xmath46 to denote the space - like two surface @xmath47 . we denote by @xmath48 the region of @xmath44 defined by @xmath49 ; similarly , we can define @xmath50 . the shaded region on the right represents the domain @xmath51 with @xmath42 . the function @xmath40 is in fact defined from @xmath52 to @xmath29 . when @xmath53 , this part of @xmath54 is assumed to be a flat light cone in minkowski space with vertex located at @xmath55 . we use @xmath56 to measure the maximal radius of the flat part of @xmath54 . in @xcite , the trapped surface forms at @xmath57 and @xmath58 . let @xmath59 be the null geodesic generators of the double null foliation and we define the lapse function @xmath60 by @xmath61 . the normalized null pair @xmath62 is defined by @xmath63 . on sphere @xmath46 we choose an arbitrary orthonormal frame @xmath64 . we call @xmath65 a _ null frame_. to denote an index from @xmath66 to @xmath67 and latin letters @xmath68 to denote an index from @xmath66 to @xmath69 . we also @xmath70 to denote the null direction either @xmath11 or @xmath12 . repeated indices are always understood as taking sums ] we use @xmath71 to denote levi - civita connection defined by the metric @xmath38 and we define the _ connection coefficients _ as follows , @xmath72 where @xmath73 . on @xmath74 , @xmath75 is the induced connection ; @xmath27 and @xmath28 are the projections to @xmath74 of the covariant derivatives @xmath76 and @xmath77 . given a weyl field @xmath78 , we introduce its null decomposition with respect to the given null frame , @xmath79 where @xmath80 is the space - time hodge dual of @xmath78 . when @xmath78 is the weyl curvature tensor , we use @xmath81 to denote its null components . we recall the null structure equations for the einstein vacuum space - times ( see @xcite ) . the originally einstein field equations are @xmath82 where @xmath83 is the ricci curvature of the underlying space - times . we express this tensorial equations by using the null frame , by definition , this yields the null structure equations . we only list the transport type null structure equations which are relevant to the current work . @xmath84 @xmath85 @xmath86 @xmath87 @xmath88 @xmath89 @xmath90 @xmath91 @xmath92 we also express the bianchi equations relative to the null frame to derive null bianchi equations . by @xmath93 . ] @xmath94 @xmath95 @xmath96 @xmath97 @xmath98 @xmath99 @xmath100 @xmath101 @xmath102 @xmath103 we review our scheme for energy estimates on weyl fields , see @xcite for the original resource . assume a weyl field @xmath104 and its hodge dual @xmath105 solve following divergence equations with source terms @xmath106 @xmath107 and @xmath108 are called _ weyl currents_. for vacuum , the curvature tensor @xmath6 is a weyl field with zero currents @xmath109 the bel - robinson tensor @xmath110_{\alpha\beta\gamma\delta}$ ] associated to @xmath104 is defined as follows for @xmath110 $ ] , @xmath111 for @xmath112 , @xmath113 for @xmath114 , @xmath115 for @xmath116 , ... , if there is no confusion in the context . ] @xmath117_{\alpha\beta\gamma\delta}=w_{\alpha\mu\gamma\nu}w_{\beta}{}^{\mu}{}_{\delta}{}^{\nu}+\wstar_{\alpha\mu\gamma\nu}{}\wstar_{\beta}{}^{\mu}{}_{\delta}{}^{\nu}.\ ] ] it is fully symmetric and traceless in all pair of indices . moreover , it satisfies the dominant energy condition which allows one to recover estimates for weyl field @xmath78 . in pratical terms , this condition can be expressed by formulas , @xmath118 we also list other null components of @xmath119 for future use , @xmath120 in view of , @xmath119 enjoys the following divergence equations @xmath121 given vector fields @xmath122 , @xmath123 and @xmath124 , we define the current @xmath125(x , y , z)_{\alpha}$ ] associated to @xmath122 , @xmath123 , @xmath124 and @xmath78 by @xmath126(x , y , z)_{\alpha } = q_{\alpha\beta\gamma\delta}x^\beta y^\gamma z^\delta.\ ] ] thus , @xmath127 where @xmath128 is the deformation tensor of @xmath122 defined by @xmath129 and @xmath130 we integrate on @xmath131 to derive the fundamental energy identity and @xmath12 are corresponding normals of the null hypersurfaces @xmath44 and @xmath45 . ] @xmath132 we list non - zero components of deformation tensors of @xmath11 and @xmath12 as well as the non - zero component of @xmath133 and @xmath133 for future use and @xmath134 in order to avoid @xmath135 and @xmath136 which do not have certain @xmath137 estimates , see @xcite for details . ] : @xmath138 we consider one derivative of curvature @xmath6 in a null direction @xmath70 . one easy but important observations is that @xmath139 is still a weyl field . we can commute @xmath140 with to derive commutes with hodge @xmath141 operator . ] @xmath142 where @xmath143 and @xmath144 we briefly recall the notions of _ signature _ and _ scale _ introduced by klainerman and rodnianski in @xcite . let @xmath145 be either a null component of curvature or a connection coefficient , we use @xmath146 , @xmath147 and @xmath148 to denote the number of times @xmath149 , respectively @xmath150 and @xmath134 appearing in the definition of @xmath145 . the _ signature _ of @xmath145 , @xmath151 , and the _ scale _ of @xmath145 , @xmath152 , are defined as @xmath153 we list the signatures and scales for all connection coefficients and curvature components , [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] we impose following rules on signatures , @xmath154 we define the _ scale invariant norms _ for @xmath145 . along null hypersurfaces @xmath155 or @xmath156 , @xmath157 on a two dimensional surface @xmath46 , @xmath158 those norms are obviously related by formulas , @xmath159 those scale invariant norms come up naturally with a small parameter @xmath29 . roughly speaking , it captures the _ smallness _ of the non - linear interaction . we have hlder s inequality in scale invariant form , @xmath160 similar estimates hold along null hypersurfaces . [ gain or not gain ] the rule of thumb for treating the nonlinear terms is , whenever one has a product of two terms , gains a @xmath30 . we do have cases that does not gain any power in @xmath29 . in fact , if @xmath161 is a bounded ( in usual sense ) scalar function ( say , bounded by a universal constant ) , the best we can hope is @xmath162 . in particular , in this paper , for @xmath161 can be @xmath163 where @xmath164 , we have to pay special attentions to the appearance of @xmath165 , see @xcite for more detailed descriptions . we introduce a family of scale invariant norms for connection coefficients where @xmath166 or @xmath167 , ] @xmath168 as well as for curvature components , @xmath169 finally , we introduce total norms . we define @xmath170 and @xmath171 ; similarly , we can define supremum norms for other scale invariant norms . the total norms are defined as follows , @xmath172 we use @xmath173 and @xmath174to denote total norms on the initial hypersurface @xmath54 . in the above definitions , all norms are scale invariant except for @xmath175 , @xmath176 , @xmath177 , @xmath178 and @xmath179 . those terms are understood to cause a loss of @xmath22 . notice also @xmath113 on incoming hypersurfaces @xmath45 is scale invariant . by abuse of language , we call those terms _ anomalies _ or _ anomalous _ if they cause a loss of @xmath180 in @xmath181 norm . notice also all the connection coefficients are not anomalous in @xmath182 norms . inspired by this , we use @xmath183 ( g ` for good ) to denote some ( good ) connection coefficient in @xmath184 and @xmath185 to denote an arbitrary connection coefficient . we use @xmath186 to denote some curvature component in @xmath187 and @xmath188 to denote an arbitrary connection coefficient . we shall also put a g ' as a lower index to other quantities in order to indicate that this quantity is not anomalous . for example , we can write @xmath189 because @xmath190 . for einstein equations with characteristic data prescribed on @xmath191 ( where the data is trivial ) and @xmath54 , we can freely specify the conformal geometry on @xmath54 . in other words , we can specify @xmath192 freely along @xmath54 to fix an initial data for the evolutionary problem . we remark that , in contrast to the case where the initial data is give on a space - like hypersurface , there is no constraints and the data can be freely specified . we require the initial data @xmath193 subject to the following norm is finite , @xmath194,\end{aligned}\ ] ] i.e. our main assumption on the initial data is the following short pulse ansatz , @xmath195 notice that the size of @xmath192 can be as large as @xmath22 so the problem is far away from small data regime . the ansatz was introduced by klainerman and rodnianski in @xcite . this initial data set is larger than those of christodoulou s original short pulses . in fact , allows more components than those of christodoulou s to be as large as @xmath22 . in this ansatz , we allow @xmath75 behaves as @xmath22 ; in christodoulou s ansatz , @xmath75 behaves as @xmath66 . under the ansatz , with the help of null structure equations , we can easily derive the following estimates on initial outgoing surfaces @xmath54 , under the ansatz , along the initial outgoing hypersurface @xmath54 , if @xmath29 is sufficiently small , there is a constant @xmath196 depending only on @xmath197 , such that @xmath198 thanks to this proposition , we shall replace by . and we omit the proof and refer the reader to @xcite or chapter 2 of @xcite . the next proposition claims that we can control connection coefficients provided that we have bound on curvatures . this is _ theorem a _ in @xcite and we omit the proof . [ theorema ] assume that @xmath199 , @xmath200 and @xmath201 are finite in @xmath202 . then there is a constant @xmath203 depending only on @xmath204 and @xmath201 such that to denote a constant depending only on @xmath204 and @xmath201 . ] @xmath205 we now state our main theorem . this is the following propagation estimates , which asserts that if holds on initial hypersurface @xmath54 , thus on the whole @xmath202 we can bound the curvature norms @xmath200 and @xmath201 by a function depending only on initial data . [ theoremc ] assume the short pulse ansatz hence , thus if @xmath29 is sufficiently small , we have @xmath206 where @xmath196 is a constant depending only on the size of the initial data . the main consequence of our estimates is the following global existence theorem , [ theoremd ] given initial data @xmath192 satisfying , if @xmath29 is sufficiently small , we can construct an unique solution of the einstein vacuum equations @xmath207 on the whole region @xmath202 . once we have the main estimates , the proof of the theorem is a typical continuity argument . we refer the reader to chapter 16 of @xcite for the detailed proof . we shall not repeat this argument . we want to remark that the formation of trapped surfaces can not be derived from the initial data ansatz . we have to add two more modifications to make sure that @xmath54 is free of trapped surface and @xmath208 is a trapped surface . more precisely , in addition to , we also assume that @xmath209 for sufficiently small @xmath210 such that @xmath211 and we also assume @xmath192 satisfies , @xmath212 where @xmath213 is a universal constant and @xmath56 measure the maximal radius of the flat part of @xmath54 . these condition guarantees the dynamical formation of trapped surfaces , we refer the readers to @xcite for details . we use a bootstrap argument to prove the * main estimates*. to be more precisely , we assume that @xmath200 and @xmath201 are finite . this assumption holds initially near @xmath54 . our main task is to carry out the following estimates under this assumption , @xmath214 thus , if @xmath29 is sufficiently small , we have @xmath215 this yields the * main estimates * . the derivation for consists of two steps . the first step is to derive energy estimates for curvature components ; the second step is to derive energy estimates for one derivatives of curvature components . the first step is done in @xcite : [ energy estimates for curvature ] if @xmath29 is sufficiently small , we have @xmath216 as we explained in the introduction , our main target is the second step , i.e. to derive energy estimates for one derivatives of curvature components in the form of without constructing rotational vector fields @xmath217 s . this is done in the rest of the paper .

the novelty of the paper is that , we completely avoid using rotation vector fields to establish the global existence theorem of the solution . more precisely , we use only canonical null directions as commutators to derive energy estimates at the level of one derivatives of null curvature components . we show that , thanks to the special cancelations coming from the null structure of non - linear interactions , desirable estimates on curvatures can be derived under the short pulse ansatz due to klainerman and rodnianski ( which is originally discovered by christodoulou ) .

in this note , we study energy estimates for einstein vacuum equations in order to prove the formation of black holes along evolutions . the novelty of the paper is that , we completely avoid using rotation vector fields to establish the global existence theorem of the solution . more precisely , we use only canonical null directions as commutators to derive energy estimates at the level of one derivatives of null curvature components . we show that , thanks to the special cancelations coming from the null structure of non - linear interactions , desirable estimates on curvatures can be derived under the short pulse ansatz due to klainerman and rodnianski ( which is originally discovered by christodoulou ) .

1201.0083

i

in this paper , we consider multistopping problems for discrete time sequences @xmath2 . in comparison to the usual stopping problem , there are @xmath1 stops @xmath3 allowed . the aim is to determine these stopping times in such a way that @xmath4 = e[x_{t_1}\vee\cdots \vee x_{t_m } ] = \mathrm{sup}.\ ] ] thus , the gain of a stopping sequence @xmath5 is the expected maximal value of the @xmath1 choices @xmath6 . in the case @xmath7 , this stopping problem reduces to the classical moser problem [ @xcite ] . we will see that optimal @xmath1-stopping times exist and are determined by a recursive description . our aim is to obtain _ explicit _ approximative solutions of the @xmath1-stopping problem in ( [ eq:1.1 ] ) under general distributional conditions . in particular , we do not assume that the random variables @xmath8 are independent or identically distributed or are even of specific i.i.d . form with @xmath9 as assumed in several papers in the literature . our basic assumption is convergence of the imbedded planar point process ( [ eq:1.2 ] ) of rescaled observations to some poisson point process @xmath10 in the plane , @xmath11 here @xmath12 is a nomalization of the @xmath8 induced typically from the central limit theorem for maxima respectively related point process convergence results . our aim is to prove that under some regularity conditions the optimal @xmath1-stopping problem of @xmath0 can be approximated by a suitable formulated @xmath1-stopping problem for the continuous time poisson process @xmath10 which serves as a limiting model for the discrete time model . furthermore , we want to show that the stopping problem in the limit model can be solved in explicit form . the solution is described by an increasing sequence of stopping curves with their related threshold stopping times . these curves solve usual one - stopping problems for transformed poisson processes and are characterized by differential equations of first order , which can be solved either in exact form or numerically . the solution for the limit model also allows us to construct approximative optimal stopping times for the discrete time model . we apply this approach in detail to the @xmath1-stopping of sequences @xmath13 _ with discount _ and observation costs and i.i.d . sequences @xmath14 . it has been observed in several papers in the literature that optimal stopping may have an easier solution in a related form for a poisson number of points or for imbedded homogeneous poisson processes as for instance in the classical house selling problem or in best choice problems . for @xmath7 [ see , e.g. , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite ] . for general reference , we refer to @xcite , chapter 2 . for @xmath15 , multistopping problems were introduced in @xcite who derived some structural results corresponding roughly to theorem [ theo:2.2 ] ; compare also some extensions in @xcite . the two stopping problem has been considered in the case of poissonian streams in @xcite . in this paper , differential equations were derived corresponding to those for the one - stopping problems in @xcite , @xcite and @xcite . multiple buying selling problems were studied in @xcite based on a vector valued formulation with pay - off given by the sum of the @xmath1-choices instead of the max as in ( [ eq:1.1 ] ) ; see also the extension in @xcite . in @xcite ( @xcite ) the case of 2-stopping problems for i.i.d . sequences was treated based on the approximative approach in @xcite ( @xcite ) . the results in this paper were rederived in @xcite ( @xcite ) and in @xcite . in case @xmath7 based on this approximation for several classes of independent and dependent sequences optimal solutions have been found in explicit form [ see @xcite(@xcite ) and @xcite ] . the present paper establishes an extension of the approximative approach as described above to @xmath1-stopping problems as in ( [ eq:1.1 ] ) . it is based on the dissertation of @xcite to which we refer for some technical details in the proofs . the program to establish this approximation approach in general is based on the following steps . in section [ sec:2 ] , we formulate the necessing recursive characterization of the optimal solutions of the @xmath1-stopping problem corresponding to bellman s optimality equation . section [ sec:3 ] is devoted to solve the @xmath1-stopping problem for the limit model of an inhomogeneous poisson process . a particular difficulty arises from the fact that in the limit model the intensity function is typically infinite along a lower boundary curve , in consequence , known stationary markovian techniques as for homogenous poisson processes do not apply . the main result , theorem [ theo:3.3 ] , shows that the optimal @xmath1-stopping problem can be reduced to @xmath1 1-stopping problems for transformed poission processes . the optimal stopping curves are characterized by a sequence of differential equations of first order . in section [ sec:4 ] , we are able to derive explicit solutions for some classes of differential equations , as appearing in the description of the optimal stopping curves . this part is based on developments in @xcite for the case @xmath7 . section [ sec:5 ] gives the basic approximation theorem ( theorem [ theo:5.2 ] ) allowing to approximate the finite discrete problems by @xmath1-stopping in the limit model . the proof of this result needs to develop a new technique . it also uses essentially the extension of the convergence of multiple stopping times in proposition [ prop:5.1 ] in @xcite for @xmath7 to @xmath16 . we restrict our presentation to the essential new part of this proof . finally in section [ sec:6 ] we obtain as application solutions in _ explicit _ form for optimal @xmath1-stopping problems for sequences @xmath13 with @xmath14 i.i.d . and with discount and observation costs @xmath17 , @xmath18 . it is remarkable that we get detailed results including the asymptotic constants as well as approximative optimal stopping sequences in explicit form . our aim is to extend these results in subsequent papers to further classes of stopping problems as to selection problems , to the sum cost case as well as to some classes of dependent sequences . it seems also possible as done in the case @xmath7 , to extend this approach to the case where cluster processes arise in the limit .

in this paper , we consider multistopping problems for finite discrete time sequences . -stops are allowed and the aim is to maximize the expected value of the best of these stops . the random variables are neither assumed to be independent not to be identically distributed . the basic assumption is convergence of a related imbedded point process to a continuous time poisson process in the plane , which serves as a limiting model for the stopping problem . the optimal-stopping curves for this limiting model are determined by differential equations of first order . a general approximation result is established which ensures convergence of the finite discrete time-stopping problem to that in the limit model . this allows the construction of approximative solutions of the discrete time-stopping problem . in detail , the case of i.i.d . sequences with discount and observation costs is discussed and explicit results are obtained . . .

in this paper , we consider multistopping problems for finite discrete time sequences . -stops are allowed and the aim is to maximize the expected value of the best of these stops . the random variables are neither assumed to be independent not to be identically distributed . the basic assumption is convergence of a related imbedded point process to a continuous time poisson process in the plane , which serves as a limiting model for the stopping problem . the optimal-stopping curves for this limiting model are determined by differential equations of first order . a general approximation result is established which ensures convergence of the finite discrete time-stopping problem to that in the limit model . this allows the construction of approximative solutions of the discrete time-stopping problem . in detail , the case of i.i.d . sequences with discount and observation costs is discussed and explicit results are obtained . . .

1509.07015

i

let @xmath8 be the following singularly perturbed laguerre weight @xmath9 with @xmath10 the hankel determinant is defined as @xmath11=\det(\mu_{i+j})_{i , j=0}^{k-1},\ ] ] where @xmath12 is the @xmath13-th moment of @xmath8 , namely , @xmath14 note that when @xmath15 , the integral in the above formula is convergent so that the hankel determinant @xmath16=d_k[w(x;t)]$ ] in is well - defined . moreover , it is well - known that the hankel determinant can be expressed as @xmath17 = \prod_{j=0}^{k-1 } \gamma_{j , n}^{-2}(t);\]]see @xcite , where @xmath18 is the leading coefficient of the @xmath5-th order polynomial orthonormal with respect to the weight function in . or , let @xmath19 be the @xmath5-th order monic orthogonal polynomial , then @xmath18 appears in the following orthogonal relation @xmath20 for fixed @xmath3 . moreover , the monic orthogonal polynomials @xmath19 satisfy a three - term recurrence relation as follows : @xmath21with @xmath22 and @xmath23 , where the appearance of @xmath3 and @xmath4 in the coefficients indicates their dependence on @xmath3 and the parameter @xmath4 in the varying weight . in this paper , however , we will focus on the case when @xmath1 . since all the above integrals on @xmath24 become divergent for negative @xmath4 , we need to deform the integration path from the positive real axis to certain curves in the complex plane . consequently , the orthogonality will be converted to the _ non - hermitian orthogonality _ in the complex plane . more precisely , let us define the following new weight function on @xmath25 : @xmath26where @xmath27 is a complex constant , the curves @xmath28 , @xmath29 and @xmath30 ; see figure [ contour - ortho ] , @xmath31 being a positive constant . the potential is defined in the cut plane @xmath32 $ ] as @xmath33 the orthogonality relation now takes the form @xmath34 . , width=340 ] with the weight function @xmath35 given in , the corresponding hankel determinant @xmath36 $ ] in is well - defined . however , since @xmath35 is not positive on @xmath37 , the orthogonal polynomials @xmath38 in may not exist for some @xmath5 , and only makes sense if all polynomials @xmath39 for @xmath40 exist . it is worth mentioning that as part of our results , we will show that there exists a @xmath41 , such that @xmath42 exists for @xmath3 large enough and @xmath43 ; cf . section [ sec - rhp - ops ] . the recurrence relation still makes sense for such @xmath4 if all of @xmath44 , @xmath19 and @xmath45 exist . note that in the literature , the polynomials with non - hermitian orthogonality have been studied in several different contexts ; see for example @xcite , where the cubic and quartic potentials are considered . one of the main motivations of this paper comes from the wigner time - delay in the study of quantum mechanical scattering problem . to describe the electronic transport in mesoscopic ( coherent ) conductors , wigner @xcite introduced the so - called time - delay matrix @xmath46 ; see also eisenbud @xcite and smith @xcite . the eigenvalues @xmath47 of @xmath46 , called the proper delay times , are used to describe the time - dependence of a scattering process . the joint distribution of the inverse proper delay time @xmath48 was found , by brouwer et al.@xcite , to be @xmath49 then the probability density function of the average of the proper time delay , namely the wigner time - delay distribution , is defined as @xmath50 the moment generating function is the laplace transformation of the wigner time - delay distribution @xmath51 which is closely related to the hankel determinant as follows : @xmath52}{d_n[w(x;0)]}.\ ] ] recently , texier and majumdar @xcite studied the wigner time - delay distribution by using a coulomb gas method . they showed that @xmath53\}\qquad \textrm{for large } n,\ ] ] where @xmath54 is the unique minimizer for an energy problem with the external field @xmath55 in , and @xmath56 $ ] is the minimum energy . moreover , the density @xmath57 is computed explicitly in @xcite , namely , @xmath58,~ \mbox{with}~0<a < b,~c = t/\sqrt{ab}.\ ] ] here positive @xmath59 and @xmath60 are independent of @xmath61 and implicitly determined by @xmath4 as follows : @xmath62 one may notice that @xmath54 is a probability measure on @xmath63 $ ] as long as @xmath64 is non - negative . since @xmath64 is a continuous function of @xmath4 , we see that @xmath57 in is non - negative for @xmath65 , where @xmath66 is the critical value of @xmath4 corresponding to the case @xmath67 ; see theorem [ theorem : asymptotic of hankel ] . it is very interesting to observe that , for this @xmath41 , we have @xmath68 and @xmath69 where a phase transition emerges at the left endpoint @xmath70 . here the critical values @xmath71 , @xmath72 and @xmath73 are explicitly given in and . it is also interesting to look at our problem from another point of view . due to the term @xmath74 in the exponent of , we may consider the origin as an essential singular point of the weight function . in recent years , orthogonal polynomials whose weights possess essential singularities have been studied extensively . for example , chen and its @xcite consider orthogonal polynomials associated with the weight @xmath75 they show that , for fixed degree @xmath3 , the recurrence coefficient satisfies a particular painlev iii equation with respect to the parameter @xmath4 , and the hankel determinant of fixed size @xmath76 $ ] equals to the isomonodromy @xmath6-function of the painlev iii equation with parameters depending on @xmath3 . the matrix model and hankel determinants @xmath36 $ ] associated with the weight in were also encountered by osipov and kanzieper @xcite in bosonic replica field theories . later , the large @xmath3 asymptotics of the hankel determinants @xmath76 $ ] associated with the weight function in is studied by the current authors in @xcite and @xcite . for @xmath77 $ ] , the asymptotics of the hankel determinants are derived and expressed in terms of certain painlev iii transcendents . the asymptotics of the recurrence coefficients are also obtained therein . in the case of the gaussian weight perturbed by essential singularity @xmath78 the double scaling limit of the hankel determinants are also characterized in terms of painlev iii transcendents by brightmore et al.in @xcite . recently , atkin , claeys and mezzadri @xcite extend the results to the case of laguerre and gaussian weight perturbed by a pole of higher order at the origin , they obtain the double scaling asymptotics of the hankel determinants in terms of a hierarchy of higher order analogs to the painlev iii equation . the main objective of this paper is to study the hankel determinant @xmath16 $ ] with respect to the weight in the region @xmath1 . first , for fixed degree @xmath5 , we will show that the recurrence coefficient @xmath79 satisfies a painlev iii equation , and the hankel determinant @xmath16 $ ] equals to the isomonodromy @xmath6-function of the painlev iii equation . then , we will derive the double scaling limit of the hankel determinant @xmath36 $ ] , the recurrence coefficients and leading coefficients of the associated orthogonal polynomials . our results are described in terms of a certain tronque solution of the painlev i equation . to state our results , we need certain special solutions to the painlev i equation @xmath80 the reader is referred to ( * ? ? ? * ch.32 ) for properties of the painlev i equation , as well as the other painlev equations . in @xcite , kapaev formulates the following model riemann - hilbert ( rh , for short ) problem for @xmath81 , associated with the painlev i equation . this model rh problem will play a crucial role later in the construction of a local parametrix in the steepest descent analysis . * @xmath82 is analytic for @xmath83 , where @xmath84 are illustrated in figure [ contour - p1-model ] . + associated with the painlev i equation , width=283 ] * let @xmath85 denote the limiting values of @xmath82 as @xmath86 tends to the contour @xmath87 from the left and right sides , respectively . then , @xmath82 satisfies the following jump conditions @xmath88 \left ( \begin{array}{cc } 1 & 0 \\ i & 1 \\ \end{array } \right ) , & z \in \gamma_k , \ k=\pm 2 ; \\[.4 cm ] \left ( \begin{array}{cc } 0 & -i\\ -i&0 \\ \end{array } \right ) , & z\in \gamma^ * , \end{array}\right .\ ] ] where @xmath89 and @xmath90 , with @xmath27 being a complex constant . * as @xmath91 , @xmath82 satisfies the asymptotic condition @xmath92 for @xmath93 , where @xmath94 @xmath95 and @xmath96 are the pauli matrices @xmath97 it is known that , for each @xmath98 , @xmath99 is a solution of the painlev i equation . as a consequence , the above rh problem for @xmath82 has a solution if and only if @xmath100 is not a pole of @xmath101 . due to the meromorphic property of the painlev i transcendents , one also see that the solution of the above rh problem for @xmath82 is meromorphic in the parameter @xmath100 . moreover , it is shown in kapaev @xcite that @xmath101 is the so - called _ tronque _ solution of painlev i whose asymptotic behavior is given by @xmath102\left ( 1+o(z^{-\frac 38})\right ) \ ] ] as @xmath103 and @xmath104 $ ] . here @xmath105 is the _ tritronque _ solution satisfying @xmath106 \quad \textrm{as } z \to \infty , \ -\frac{\pi}{5}<\arg z < \frac{7\pi}{5},\ ] ] where the coefficients @xmath107 can be determined recursively ; see for example joshi and kitaev @xcite . the solution @xmath108 will appear in our main results below . we mention several known facts about the coefficients in in addition to . for example , the explicit formulas of @xmath109 and @xmath110 are given in @xcite as @xmath111 where @xmath112 is the hamiltonian of painlev i. first of all , when the degree @xmath5 is fixed , we show that the recurrence coefficient @xmath113 satisfies a particular painlev iii equation with certain initial conditions . moreover , we prove that the hankel determinant @xmath114 $ ] is related to the @xmath6-function of the painlev iii equation . similar results for the weight in been obtained by chen and its @xcite . [ theorem : hankel as tau function ] for fixed non - negative integer @xmath5 , let @xmath79 be the recurrence coefficient in , and @xmath115 then @xmath116 satisfies the following painlev iii equation @xmath117 with the initial conditions @xmath118 , @xmath119 . moreover , we have @xmath120=\mathrm{const } \cdot \tau(t)\ ; e^ { { n^2t}/{2}}\ ; t^ { { k(k+n)}/{2}},\ ] ] where @xmath121 is the jimbo - miwa - ueno isomonodromy @xmath6-function of the above painlev iii equation . next , we let @xmath7 and consider the double scaling limit when @xmath122 and @xmath123 simultaneously . we show that the asymptotics of the hankel determinant @xmath124 $ ] associated with the weight in can be expressed in terms of the tronque solution @xmath101 of painlev i equation given in . [ theorem : asymptotic of hankel ] let the constants @xmath71 , @xmath72 and @xmath73 be defined as @xmath125and @xmath126 for @xmath127 and @xmath128 in a way such that @xmath129 remains bounded . suppose @xmath130 is fixed and @xmath131 is not a pole of the tronque solution @xmath101 , then an asymptotic approximation of the logarithmic derivative of the hankel determinant @xmath132 $ ] associated with the weight function is given by @xmath133 .5 cm we would also derive the double scaling limit of the recurrence coefficients and the leading coefficients of the orthonormal polynomials . [ theorem : asymptotic of recurrence coff ] under the same conditions as in the previous theorem , the monic polynomial @xmath42 defined in exists for large enough @xmath3 and @xmath4 close to @xmath71 . moreover , we have the asymptotics of the recurrence coefficients @xmath134 @xmath135 and @xmath136 where @xmath137 is the hamiltonian of painlev i given in . it is well - known that the tronque solutions of painlev i are meromorphic functions and possess infinitely many poles in the complex plane . therefore , to make the results valid in the above theorems , we require the @xmath131 in is bounded away from the poles of @xmath101 . recently , through a more delicate _ triple scaling limit _ , bertola and tovbis @xcite successfully obtain the asymptotics near the poles of @xmath101 . similar results near the poles of @xmath101 might be derived by using their ideas in @xcite . however , we do not pursue that part . instead , we focus on the main task of the present paper to demonstrate that the painlev i asymptotics can also occur for the weight with negative @xmath4 . the rest of the paper is arranged as follows . in section [ finite determinants ] , we provide a rh problem for the orthogonal polynomials with respect to the weight . a transformed version of the solution is shown to fulfill a lax pair , which is closely related to the painlev iii equation . several differential identities are stated and justified . theorem [ theorem : hankel as tau function ] is also proved in this section . section [ sec - e - measure ] is devoted to the determination of equilibrium measures , involving a positive measure and a signed measure . in section [ sec - rh - analysis ] , we carry out a nonlinear steepest descent analysis of the rh problem for the orthogonal polynomials . particular attention will be paid to the construction of the local parametrix at the critical endpoint @xmath138 , where the painlev i transcendents are involved . then , the proofs of theorems [ theorem : asymptotic of hankel ] and [ theorem : asymptotic of recurrence coff ] are given in the last section , section [ sec - proofs ] .

in this paper , we consider polynomials orthogonal with respect to a varying perturbed laguerre weight for and on certain contours in the complex plane . when the parameters , and the degree are fixed , the hankel determinant for the singular complex weight is shown to be the isomonodromy-function of the painlev iii equation . when the degree , is large and is close to a critical value , inspired by the study of the wigner time delay in quantum transport , we show that the double scaling asymptotic behaviors of the recurrence coefficients and the hankel determinant are described in terms of a boutroux tronque solution to the painlev i equation . 2010 _ mathematics subject classification _ : primary 33e17 , 34m55 , 41a60 . _ keywords and phrases _ : asymptotics ; hankel determinants ; painlev i equation ; painlev iii equation ; riemann - hilbert approach .

in this paper , we consider polynomials orthogonal with respect to a varying perturbed laguerre weight for and on certain contours in the complex plane . when the parameters , and the degree are fixed , the hankel determinant for the singular complex weight is shown to be the isomonodromy-function of the painlev iii equation . when the degree , is large and is close to a critical value , inspired by the study of the wigner time delay in quantum transport , we show that the double scaling asymptotic behaviors of the recurrence coefficients and the hankel determinant are described in terms of a boutroux tronque solution to the painlev i equation . our approach is based on the deift - zhou nonlinear steepest descent method for riemann - hilbert problems . 2010 _ mathematics subject classification _ : primary 33e17 , 34m55 , 41a60 . _ keywords and phrases _ : asymptotics ; hankel determinants ; painlev i equation ; painlev iii equation ; riemann - hilbert approach .

1608.07728

i

quantum key distribution protocols allow two users , alice ( @xmath0 ) and bob ( @xmath1 ) , to establish a shared secret key secure against an all - powerful adversary eve ( @xmath2 ) who is bounded only by the laws of physics - an end unattainable through classical means alone . several such protocols have been developed since the original bb84 @xcite ( the reader is referred to @xcite for a general survey ) and many of them include rigorous proofs of unconditional security . such a proof of security generally involves determining a bound on the protocol s key - rate ( to be defined shortly , though roughly speaking , it is the ratio of secret key bits to qubits sent ) as a function of the observed noise in the quantum channel . in this paper , we consider several qkd protocols , and derive key - rate expressions based on multiple channel statistics . furthermore , our key - rate bounds will utilize statistics from mismatched measurements ; that is to say , those measurement outcomes where a and b s choice of bases are incompatible - events which are typically discarded by the protocol specification ( there are exceptions as we mention next section ) . in fact , by using these mismatched measurement results , the key - rate bounds we derive demonstrate that many of the protocols we consider here can actually tolerate higher levels of noise than previously thought . thus , the primary contributions of this paper are two - fold . first , we derive a general approach to deriving key - rate expressions , in the asymptotic scenario , for a wide - range of discrete variable qkd protocol utilizing all possible channel statistics , including mismatched measurement results . secondly , by applying this technique to several , very different protocols ( including a limited - resource bb84 @xcite , an extended b92 @xcite , and a two - way semi - quantum protocol @xcite ) , we not only derive new key - rate expressions applicable to arbitrary , possibly asymmetric quantum channels , but , in many cases , our new bounds are substantial improvements over previous work with these protocols . along the way , we will also use our method to investigate optimal qkd protocols for asymmetric channels . we are not the first to consider the use of mismatched measurement outcomes for quantum key distribution . indeed , in the 1990 s barnett et al . @xcite showed that mismatched measurement results may be used to better detect an eavesdropper using an intercept - and - resend attack . in @xcite , mismatched measurement bases were applied to the four - state and six - state bb84 protocols @xcite . this method was shown to improve the key rate ( as determined by the devetak - winter equation in @xcite ) for certain quantum channels , namely the amplitude damping channel and rotation channel . in @xcite , mismatched measurement results were actually used to distill a raw key ( as opposed to being used only for channel tomography ) - a modified bb84 protocol was adopted and this method was shown to improve the key rate for certain channels . in @xcite , a modified four - state , two - basis bb84 was used where the first basis was the standard computational @xmath3 basis ( @xmath4 ) , while the second consisted of states @xmath5 and @xmath6 where @xmath7 . the authors of that work showed that for small @xmath8 , mismatched measurement bases can still be used to gain good channel estimates while also allowing @xmath0 and @xmath1 to use mismatched measurement bases to distill their key ( since , for @xmath8 small , even with differing bases , their measurement results will be nearly correlated ) . mismatched measurements were used in @xcite in order to get better channel statistics for a single - state semi - quantum protocol first introduced in @xcite . though single - state semi - quantum protocols utilize two - way quantum channels , they admit many simplifications which ease their security analysis . in this paper , we consider a multi - state semi - quantum protocol ( which are more difficult to analyze ) and show mismatched measurements improve its key - rate . in @xcite , it was proven , using mismatched measurement bases , that the three - state bb84 protocol from @xcite has a key rate equal to that of the full four state bb84 protocol assuming a symmetric attack . also a four - state protocol using three bases has a key rate equal to that of the full six - state bb84 protocol . in this paper , we will arrive at the same conclusion , though using a more information theoretic argument . however , we will also analyze other protocols , adapt our approach to two - way quantum channels , consider an optimized qkd protocol , and , for all considered protocols , derive new key - rate expressions suitable for asymmetric channels . our method , building off of our conference paper in @xcite ( where we only considered a three - state bb84 protocol ) , is very general and applicable to multiple qkd protocols , both one - way and , as we will demonstrate , two - way ( those which utilize a two - way quantum channel allowing a qubit to travel from @xmath0 to @xmath1 , then return to @xmath0 thus passing through the adversary twice ) . after an introduction to our notation , we will first explain the parameter estimation method and our technique . we will then apply it to variants of the bb84 protocol , confirming the results of @xcite mentioned above ( though also deriving key - rate expressions for arbitrary channels ) . we will consider the extended b92 protocol @xcite and derive an improved key - rate bound for it . we will then use our method to consider an `` optimal '' qkd protocol . finally , we will analyze a multi - state semi - quantum protocol from @xcite which relies on a two - way quantum channel . this new proof of security will derive a more optimistic bound on the key rate expression than the one previously constructed in @xcite ( the latter did not use mismatched measurement bases ) . we now introduce some notation we will use . let @xmath9 be the shannon entropy function , namely : @xmath10 where all logarithms in this paper are base two unless otherwise specified . we will occasionally use the notation @xmath11 to mean @xmath12 . we denote by @xmath13 the binary shannon entropy function , namely : @xmath14 . we write @xmath15 to mean the von neumann entropy of the density operator @xmath16 . if @xmath16 is finite dimensional ( and all systems in this paper are finite dimensional ) , then let @xmath17 be its eigenvalues . in this case @xmath18 . if @xmath16 acts on a bipartite hilbert space @xmath19 we will often write @xmath20 . when we write @xmath21 we mean the result of tracing out @xmath1 s portion of @xmath20 ( that is , @xmath22 ) . similarly for @xmath23 and for systems with three or more subspaces . given density operator @xmath20 we will write @xmath24 to mean @xmath25 and @xmath26 to mean @xmath27 . we denote by @xmath28 to be the conditional von neumman entropy defined as @xmath29 . if the context is clear , we will forgo writing the `` @xmath16 '' subscript . when we talk about qubits , we will often refer to the @xmath3 , @xmath30 , and @xmath31 bases , the states of which we denote : @xmath32 , @xmath33 , and @xmath34 , where : @xmath35

in this paper , we derive key - rate expressions for several different quantum key distribution protocols . our key - rate equations utilize multiple channel statistics , including those gathered from mismatched measurement bases - i.e. , when alice and bob choose incompatible bases . in particular , we will consider a limited - resource form of bb84 , an extended b92 , and a two - way semi - quantum protocol . for the last two protocols , we demonstrate that their tolerance to noise is higher than previously thought . along the way , we will also consider an optimal qkd protocol for various quantum channels . finally , all the key - rate expressions which we derive in this paper are applicable to any arbitrary , not necessarily symmetric , quantum channel .

in this paper , we derive key - rate expressions for several different quantum key distribution protocols . our key - rate equations utilize multiple channel statistics , including those gathered from mismatched measurement bases - i.e. , when alice and bob choose incompatible bases . in particular , we will consider a limited - resource form of bb84 , an extended b92 , and a two - way semi - quantum protocol . for the first protocol , we will show it has the same tolerance to noise as the full bb84 ( a result already known , however we provide an alternative , more information theoretic , proof ) . for the last two protocols , we demonstrate that their tolerance to noise is higher than previously thought . along the way , we will also consider an optimal qkd protocol for various quantum channels . finally , all the key - rate expressions which we derive in this paper are applicable to any arbitrary , not necessarily symmetric , quantum channel .

1504.01377

i

the study of the structure of finite dimensional algebras has gained importance in recent times for it may be possible to find presumably new examples of subfactors of a hyper finite @xmath4-factor along the lines of @xcite . the partition algebras are introduced by v. jones in @xcite and martin in @xcite and they are studied intensively by martin in @xcite . the cellularity of many subalgebras of partition algebras like tanabe algebras , party algebras , algebra of @xmath3-relations and signed partition algebras are studied in detail in @xcite , @xcite , @xcite and @xcite respectively . the question of semisimplicity of these algebras reduces to the nondegeneracy of gram matrices of such algebras . in this connection , the gram matrices of partition algebras , algebra of @xmath3-relations and signed partition algebras are studied in @xcite which are realized as direct sum of block sub matrices . the diagonal entry of the block submatrix is a product of @xmath5 quadratic polynomials and @xmath6 linear polynomials in the case of algebra of @xmath3-relations and signed partition algebras and it is a product @xmath7 linear polynomials in the case of partition algebras . in this paper , we introduce symmetric diagram matrices @xmath0 of size @xmath1 whose entries are @xmath2 based on the diagrams . we compute the eigenvalues of symmetric diagram matrices using elementary row and column operations by induction . as a byproduct , we compute the eigenvalues of gram matrices of a larger class of diagram algebras like the signed partition algebras , partition algebras and algebra of @xmath3 relations where the gram matrix of any diagram algebra is a direct product of symmetric diagram matrices .

in this paper , we introduce symmetric diagram matrices of size whose entries are . we compute the eigenvalues of symmetric diagram matrices using elementary row and column operations inductively . as a byproduct , we obtain the eigenvalues of gram matrices of a larger class of diagram algebras like the signed partition algebras , algebra of relations and partition algebras . * keywords : * eigenvalues , symmetric diagram matrices , gram matrices , partition algebras , signed partition algebras and algebra of-relations .

in this paper , we introduce symmetric diagram matrices of size whose entries are . we compute the eigenvalues of symmetric diagram matrices using elementary row and column operations inductively . as a byproduct , we obtain the eigenvalues of gram matrices of a larger class of diagram algebras like the signed partition algebras , algebra of relations and partition algebras . * keywords : * eigenvalues , symmetric diagram matrices , gram matrices , partition algebras , signed partition algebras and algebra of-relations . * mathematics subject classification(2010 ) . * 15b99 , 16z05 .

1503.00413

i

let @xmath4 be a smooth and compact riemannian manifold without boundary . we denote @xmath5 the laplace - beltrami operator on @xmath6 . an eigenfunction @xmath7 of @xmath8 satisfies @xmath9 with @xmath10 its eigenfrequency . in 1988 sogge @xcite proved that @xmath11 here , @xmath12 is independent of @xmath10 . in 2007 koch , tataru , and zworski @xcite extended this result to quasimodes , i.e. approximate eigenfunctions in the sense that @xmath13 in fact , their result holds for laplace - like semiclassical pseudodifferential operators . in this paper , we investigate bilinear eigenfunction estimates . that is , for two eigenfunctions @xmath7 and @xmath14 , we estimate @xmath15 in terms of their eigenfrequencies . one can of course use hlder s inequality and sogge s @xmath0 _ linear _ eigenfunction estimates to prove a upper bound of @xmath15 . for example , on a riemannian surface ( i.e. two dimensional riemannian manifold ) , let @xmath7 and @xmath14 be two @xmath16-normalized eigenfunctions with eigenvalues @xmath17 . then @xmath18 or with a different pair of hlder indices ( among other possible choices ) @xmath19 the second bound @xmath20 does not depend on the higher frequency @xmath21 , but is not necessarily better than the first bound @xmath22 ( given , say , @xmath23 ) . however in 2005 , burq , grard , and tzvetkov @xcite proved that @xmath24 this estimate is clearly better than both the two previous bounds . moreover , they showed that @xmath25 bound is _ sharp _ on @xmath26 ( see more discussion on the sharpness of bilinear eigenfunction estimates in section [ sec : sharpness ] ) . this improvement is crucial in burq , grard , and tzvetkov s investigation of nonlinear dispersive equations on manifolds @xcite . they used to study the well - posedness of cauchy problems involving nonlinear schdinger equations on compact riemannian surfaces . in particular , they obtained the _ critical _ well - posedness regularity for cubic schdinger equations on @xmath26 . in fact , burq , grard , and tzvetkov @xcite proved for spectral clusters . subsequently , they @xcite generalised to higher dimensions as follows . [ thm : bgt ] let @xmath4 be an @xmath2-dimensional compact manifold and @xmath27 . write @xmath28 . then for all @xmath29 and @xmath30 , we have @xmath31 moreover , all these estimates are sharp ( modulo the @xmath32 loss in dimension three ) . note that if @xmath7 is an eigenfunction with eigenfrequency @xmath10 , then @xmath33 . so theorem [ thm : bgt ] in particular applies to eigenfunctions . the bounds in theorem [ thm : bgt ] depend _ only _ on the lower eigenfrequency @xmath10 . in this paper , we generalise theorem [ thm : bgt ] to @xmath0 bilinear eigenfunction estimates for @xmath34 . in this case , the bound depends on both of the eigenfrequencies . we are able to derive a full range of sharp ( modulo some log loss when @xmath35 ) estimates in all dimensions . the two main themes that arise from our results are : 1 . the sharp bound for the @xmath0 norm of the product of two eigenfunctions is better than that obtained by @xmath36 2 . in the sharp bound , the higher eigenfrequency has a smaller exponent than the lower eigenfrequency . the extreme examples of this is @xmath16 bilinear estimates , the exponent of the higher frequency is @xmath37 . from a technical perspective it is just as easy to work with quasimodes ( rather than exact eigenfunctions ) . so in this paper we prove the @xmath0 bilinear estimates for quasimodes . to this end , it is convenient to work in the semiclassical setting . denote @xmath38 the semiclassical parameter . then an eigenfunction @xmath7 satisfies @xmath39 , where @xmath40 we write @xmath41 as a semiclassical pseudodifferential operator with symbol @xmath42 and study @xmath43 quasimodes as given by the following definition . [ hquasimodes ] a family @xmath44 ( @xmath45 ) is said to be an @xmath43 quasimode of a semiclassical pseudodifferential operator @xmath41 if @xmath46 in the semiclassical framework , we can treat quasimodes of semiclassical pseudodifferential operators that are similar to @xmath47 in the same fashion as quasimodes of the laplacian . we call such operators laplace - like ( see definition [ laplacelike ] ) . our main theorem states that [ thm : bilinear ] assume that @xmath42 is a laplace - like smooth symbol . let @xmath48 and @xmath49 be two families of @xmath50 and @xmath51 quasimodes of @xmath41 and @xmath52 , respectively . suppose that @xmath53 for @xmath54 and both @xmath48 and @xmath49 admit localisation property ( see definition [ localised ] ) . then @xmath55 where for @xmath56 , @xmath57 for @xmath58 , @xmath59 , @xmath60 and @xmath61 moreover , all these estimates are sharp ( modulo the @xmath32 loss in the case @xmath62 ) . we point out that @xmath63 considered in theorem [ thm : bgt ] is an @xmath43 quasimode of @xmath47 with @xmath38 . ( see section [ sec : qm ] . ) similarly , @xmath64 is an @xmath65 quasimode of @xmath66 with @xmath67 . therefore , the @xmath0 bilinear spectral cluster estimates follow as a simple consequence of the above theorem . moreover , they are also sharp . [ thm : efnbilinear ] suppose that @xmath68 . then @xmath69 where @xmath70 is as given in theorem [ thm : bilinear ] . in particular , let @xmath7 and @xmath14 be two @xmath16-normalized eigenfunctions with eigenvalues @xmath71 and @xmath72 . then @xmath73 moreover , the estimates are sharp on the sphere . an important application of the bilinear eigenfunctions estimates lies in the study of nonlinear dispersive equations on riemannian manifolds . consider the cauchy problem involving cubic ( focusing or defocusing ) nonlinear schrdinger equation on a compact riemannian manifold @xmath6 : @xmath74 there is a long history in studying the well / ill - posedness of this system , see e.g. burq , grard , and tzvetkov @xcite and the references therein . denote @xmath75 the sobolev space on @xmath6 . burq , grard , and tzvetkov ( * ? ? ? * theorem 1 ) proved that is locally well - posed in @xmath76 for @xmath77 on the two - sphere @xmath26 . their argument combines the @xmath16 bilinear eigenfunction estimate with the eigenvalue distribution on @xmath26 . on the questions of ill - posedness , their earlier work @xcite showed that is ill - posed in @xmath76 for @xmath78 . therefore , @xmath79 is the critical regularity on @xmath26 . we express this by saying that the critical threshold @xmath80 , following the notation in @xcite . it is only on special manifolds ( such as the sphere ) that such critical regularity threshold is known . another known example is the torus . bourgain @xcite and burq , grard , and tzvetkov @xcite , showed that @xmath81 where @xmath82 is the two - dimensional torus . it remains _ open _ to find the critical threshold on general riemannian manifolds , and to relate it to geometry of the manifold . for more information about nonlinear schrdinger equations on riemannian manifolds , we refer to @xcite and the references therein . burq , grard , and tzvetkov ( * ? ? ? * theorem 3 ) also proved the following @xmath16 trilinear eigenfunction estimates in dimension two and three . let @xmath4 be an @xmath2-dimensional compact manifold and @xmath27 . given @xmath83 and @xmath84 , we have @xmath85 here , @xmath86 $ ] ; @xmath87 depends on @xmath88 and @xmath6 ; but @xmath12 and @xmath87 are independent of @xmath10 , @xmath21 , and @xmath89 . we point out that when @xmath90 our @xmath91 bilinear estimate in theorem [ thm : bilinear ] can be derived from the above @xmath16 trilinear one . see also the recent work of yang @xcite on @xmath16 multilinear eigenfunction estimates and their applications to a more general class of nonlinear schrdinger equations . we remark that the bilinear eigenfunction estimates are sharp on the sphere . however , they are far from optimal in the case of torus . for example , on the two dimensional torus @xmath82 , given two eigenfunction @xmath7 and @xmath14 with eigenvalues @xmath71 and @xmath72 , @xmath92 where @xmath12 is independent of @xmath10 and @xmath21 . this follows the classical result of zygmund @xcite that @xmath93 . we organise the paper as follows . in section [ sec : loc ] , we use semiclassical analysis and localisation to reduce the problem to a local one . in section [ sec : bilinear ] , we prove theorem [ thm : bilinear ] . in section [ sec : sharpness ] , we show that the @xmath0 bilinear estimates are sharp by constructing appropriate quasimodes and spherical harmonics . section [ sec : sa ] is an appendix containing the basic semiclassical analysis on which this paper relies . throughout this paper , @xmath94 ( @xmath95 ) means @xmath96 ( @xmath97 ) for some constant @xmath98 independent of @xmath10 or @xmath99 ; @xmath100 means @xmath94 and @xmath101 ; the constants @xmath98 and @xmath12 may vary from line to line .

in this paper , we investigate the bilinear quasimode estimates on compact riemannian manifolds . we obtain results in the full range on all-dimensional manifolds with . this in particular implies the bilinear eigenfunction estimates . we further show that all of these estimates are sharp by constructing various quasimodes and eigenfunctions that saturate our estimates .

in this paper , we investigate the bilinear quasimode estimates on compact riemannian manifolds . we obtain results in the full range on all-dimensional manifolds with . this in particular implies the bilinear eigenfunction estimates . we further show that all of these estimates are sharp by constructing various quasimodes and eigenfunctions that saturate our estimates .

1603.07358

i

the problem of computing matrix exponentials arises in many theoretical and practical problems . numerous methods have been developed to efficiently compute @xmath7 or its product with a vector @xmath8 , where @xmath3 is an @xmath9 complex matrix and @xmath2 . we refer to the classical paper @xcite of moler and van loan for a survey of a general theory and numerical methods for matrix exponentials . for matrix exponential problems involving a large and sparse matrix @xmath3 , it is usually the product of the exponential with a vector that is of interest . this arises , for example , in solving the initial value problem ( @xcite ) @xmath10 see @xcite for some other applications . a large number of matrix exponential problems concern a _ positive definite _ @xmath3 ( i.e. @xmath11 is hermitian positive definite ) , which defines a stable dynamical system with a solution converging to a steady state . another important class of problems involve a skew - hermitian matrix @xmath3 ( i.e. @xmath12 with @xmath13 being hermitian ) , for which has a norm - conserving solution . such systems can be used to model a variety of physical problems where certain quantities such as energy are conserved . for example , a spectral method for solving the time - dependent schrdinger equation modeling n electrons leads to with a skew - hermitian matrix ; see @xcite . while we will study a general non - hermitian @xmath3 , we are particularly interested in these two important classes of problems , where stronger theoretical results can be derived . the krylov subspace methods are a powerful class of iterative algorithms for solving many large scale linear algebra problems . initially introduced by gallopoulos and saad @xcite , they have also become a popular method for approximating @xmath14 where @xmath15 is a fixed parameter typically representing a time step . for the ease of notation , we will assume throughout that @xmath16 . a comprehensive theory has been developed in the literature with error bounds demonstrating convergence of the krylov subspace methods and its relation to certain properties of the matrix . for example , earlier results in @xcite relate convergence of the krylov subspace methods to the norm of the matrix @xmath17 . more refined error bounds have later been derived , that provide sharper estimates of the errors by considering additional spectral information such as enclosing regions of the field of values of @xmath3 or positive definiteness of @xmath3 ; see @xcite and the references contained therein . for a real symmetric positive definite matrix @xmath3 , it has been shown in a recent work @xcite that the speed of convergence is also determined by the condition number of @xmath3 as in the conjugate gradient method . for positive definite matrices that are not necessarily hermitian , stronger convergence bounds have also been obtained in @xcite in terms of the field of values . however , most of these bounds are derived by assuming the field of values lying in a certain pre - defined region , which are not easy to apply or interpret . there is an inherited theoretical difficulty in quantitatively characterizing the influence on the convergence by the field of values , a two dimensional object . this issue also arises in the theory of the krylov subspace methods for solving linear systems . in this paper , we study the relation between the convergence of the krylov subspace methods and the field of values through its bounding rectangle @xmath18\times[-c , c]$ ] where @xmath19 , @xmath20 ( the smallest and the largest eigenvalue of the hermitian part of @xmath3 ) and @xmath21 ( the largest eigenvalue in absolute value of the skew - hermitian part of @xmath3 ) . with this approach , new _ a priori _ error bounds will be derived in terms of @xmath22 , @xmath23 and @xmath24 . simplified bounds will be presented for non - hermitian positive definite matrices and skew - hermitian matrices , which relate the speed of convergence to the size and the shape of the rectangular region . in particular , our bounds explain an interesting observed convergence behavior where the error may first stagnate for certain iterations before it starts to converge . numerical examples will be presented to demonstrate the behavior of the new error bounds . in developing our _ a priori _ error bounds , we also derive a new _ a posteriori _ error bound that is shown to provide a sharp and computable estimate of the error . the main technique used in deriving new _ a priori _ error bounds is the same as in the literature @xcite by constructing faber polynomial approximation of the exponential function in a region containing the field of values . the novelty in this work is to use the jacobi elliptic functions to construct a conformal mapping for the rectangular region that tightly encloses the fields of value and to show that this highly complicated mapping can be simplified to yield some simple final bounds . the paper is organized as follows . in section [ sec : preliminaries ] , we first present some preliminaries about the faber polynomial approximation and the jacobi elliptic functions . in section [ sec : aposteriori ] , we present a new _ a posteriori _ error bound , which relates the convergence to the decay properties of functions of banded matrices . to study this decay behavior , we construct a conformal mapping in section [ sec : conformal_mapping ] and present our new _ a priori _ error bound in section [ sec : apriori_nonhermitian ] . in section [ sec : apriori_skewhermitian ] , we apply the same idea on skew - hermitian matrices and derive simpler _ a priori _ bounds . numerical examples are presented in section [ sec : numerical_examples ] and some concluding remarks in section [ sec : concluding_remarks ] .

the _ a priori _ error bounds relate the convergence to , ( the smallest and the largest eigenvalue of the hermitian part of ) and ( the largest eigenvalue in absolute value of the skew - hermitian part of ) , which define a rectangular region enclosing the field of values of . in particular , our bounds explain an observed superlinear convergence behavior where the error may first stagnate for certain iterations before it starts to converge . numerical examples are given to demonstrate the theoretical bounds .

in this paper , we present new _ a posteriori _ and _ a priori _ error bounds for the krylov subspace methods for computing for a given and , where is a large sparse non - hermitian matrix . the _ a priori _ error bounds relate the convergence to , ( the smallest and the largest eigenvalue of the hermitian part of ) and ( the largest eigenvalue in absolute value of the skew - hermitian part of ) , which define a rectangular region enclosing the field of values of . in particular , our bounds explain an observed superlinear convergence behavior where the error may first stagnate for certain iterations before it starts to converge . the special case that is skew - hermitian is also considered . numerical examples are given to demonstrate the theoretical bounds .

1009.4970

i

randomized load balancing , where a job is assigned to a server from a small subset of randomly chosen servers , is very simple to implement , and can surprisingly deliver better performance ( for example reducing collisions , waiting times , backlogs ) in a number of applications , such as data centers , hash tables , distributed memory machines , path selection in networks , and task assignment at web servers . one useful model extensively used to study randomized load balancing schemes is the supermarket model . in the supermarket model , a key result by vvedenskaya , dobrushin and karpelevich @xcite indicated that when each poisson arriving job is assigned to the shortest one of @xmath0 randomly chosen queues with exponential service times , the equilibrium queue length can decay doubly exponentially in the limit as the population size @xmath1 , and the stationary fraction of queues with at least @xmath2 customers is @xmath3 , which indicates a substantially exponential improvement over the case for @xmath4 , where the tail of stationary queue length in the corresponding m / m/1 queue is @xmath5 . at the same time , the exponential improvement is also illustrated by another key work in which luczak and mcdiarmid @xcite studied the maximum queue length in the supermarket model with poisson arrivals and exponential service times . the distributed load balancing strategies in which individual job decisions are based on information on a limited number of other processors , have been studied by analytical methods in eager , lazokwska and zahorjan @xcite and by trace - driven simulations in zhou @xcite . based on this , the supermarket models can be developed by using either queueing theory or markov processes . most of recent research deals with a simple supermarket model with poisson arrivals and exponential service times by means of density dependent jump markov processes . the methods used in the recent literature are based on determining the behavior of the supermarket model as its population size grows to infinity , and its behavior is naturally described as a system of differential equations whose fixed point leads to a closed - form solution with a doubly exponential structure . readers may refer to , such as , azar , broder , karlin and upfal @xcite , vvedenskaya , dobrushin and karpelevich @xcite and mitzenmacher @xcite . certain generalizations of the supermarket models have been explored , for example , in studying simple variations by mitzenmacher and vcking @xcite , mitzenmacher @xcite , vcking @xcite , mitzenmacher , richa , and sitaraman @xcite and vvedenskaya and suhov @xcite ; in discussing load information by mirchandaney , towsley , and stankovic @xcite , dahlin @xcite and mitzenmacher @xcite ; and in mathematical analysis by graham @xcite , luczak and norris @xcite and luczak and mcdiarmid @xcite . using fast jackson networks , martin and suhov @xcite , martin @xcite , suhov and vvedenskaya @xcite studied supermarket mall models , where each node in a jackson network is replaced by @xmath6 parallel servers , and a job joins the shortest of @xmath7 randomly chosen queues at the node to which it is directed . for non - poisson arrivals or for non - exponential service times , li , lui and wang @xcite discussed the supermarket model with poisson arrivals and ph service times , and indicated that the fixed point decreases doubly exponentially , where the stationary phase - type environment is shown to be a crucial factor . bramson , lu and prabhakar @xcite provided a modularized program based on ansatz for treating the supermarket model with poisson arrivals and general service times , and li @xcite further discussed this supermarket model by means of a system of integral - differential equations , and illustrated that the fixed point decreases doubly exponentially and that the heavy - tailed service times do not change the doubly exponential solution to the fixed point . for the ph distribution , readers may refer to neuts @xcite and li @xcite . the map is a useful mathematical model , for example , for describing bursty traffic , self similarity and long - range dependence in modern computer networks , e.g. , see adler , feldman and taqqu @xcite . for detail information of the map , readers may refer to chapter 5 in neuts @xcite , lucantoni @xcite , chapter 1 in li @xcite , and three excellent overviews by neuts @xcite , chakravarthy @xcite and cordeiro and kharoufeh @xcite . in computer networks , andersen and nielsen @xcite applied the map to describe long - range dependence , and yoshihara , kasahara and takahashi @xcite analyzed self - similar traffic by means of a markov - modulated poisson process . it is interesting to answer whether or how non - poisson arrivals or non - exponential service times can disrupt doubly exponential solutions to the fixed points in supermarket models . to that end , this paper studies a supermarket model with maps and ph service times , and shows that there still exists a doubly exponential solution to the fixed point . the main contributions of the paper are threefold . the first one is to provide a novel matrix - analytic approach to study the supermarket model with maps and ph service times . based on density dependent jump markov processes , the supermarket model is described as a system of differential vector equations whose fixed point has a closed - form solution with a doubly exponential structure . the second one is to obtain a crucial result that the fixed point can be decomposed into the product of two factors inflecting arrival information and service information , which indicates that the doubly exponential solution to the fixed point can exist extensively , but it is not always unique for more general supermarket models . the third one is to analyze exponential convergence of the current location of the supermarket model to its fixed point . not only does the exponential convergence indicate the existence of the fixed point , but it also shows that such a convergent process is very fast . to study the limit behavior of the supermarket model as its population size goes to infinity , this paper applies the kurtz theorem to study density dependent jump markov process given in the supermarket model with maps and ph service times , which leads to the lipschitz condition under which the fraction measure of the supermarket model weakly converges the system of differential vector equations . the remainder of this paper is organized as follows . in section 2 , we first describe a supermarket model with maps and ph service times . then the supermarket model is described as a systems of differential vector equations in terms of density dependent jump markov processes . in section 3 , we first introduce a fixed point of the system of differential vector equations , and set up a system of nonlinear equations satisfied by the fixed point . then we provide a closed - form solution with a doubly exponential structure to the fixed point , and show that the fixed point can be decomposed into the product of two factors inflecting arrival information and service information . in section 4 , we provide an important observation in which the doubly exponential solution to the fixed point is not always unique for more general supermarket models . in section 5 , we study exponential convergence of the current location of the supermarket model to its fixed point . in section 6 , we apply the kurtz theorem to study density dependent jump markov process given in the supermarket model with maps and ph service times , which leads to the lipschitz condition under which the fraction measure of the supermarket model weakly converges the system of differential vector equations . some concluding remarks are given in section 7 .

. 0.5 cm * keywords : * randomized load balancing , supermarket model , matrix - analytic approach , doubly exponential solution , density dependent jump markov process , markovian arrival process ( map ) , phase - type ( ph ) distribution , fixed point , exponential convergence , lipschitz condition .

in this paper , we provide a novel matrix - analytic approach for studying doubly exponential solutions of randomized load balancing models ( also known as _ supermarket models _ ) with markovian arrival processes ( maps ) and phase - type ( ph ) service times . we describe the supermarket model as a system of differential vector equations by means of density dependent jump markov processes , and obtain a closed - form solution with a doubly exponential structure to the fixed point of the system of differential vector equations . based on this , we show that the fixed point can be decomposed into the product of two factors inflecting arrival information and service information , and further find that the doubly exponential solution to the fixed point is not always unique for more general supermarket models . furthermore , we analyze the exponential convergence of the current location of the supermarket model to its fixed point , and apply the kurtz theorem to study density dependent jump markov process given in the supermarket model with maps and ph service times , which leads to the lipschitz condition under which the fraction measure of the supermarket model weakly converges the system of differential vector equations . this paper gains a new understanding of how workload probing can help in load balancing jobs with non - poisson arrivals and non - exponential service times . 0.5 cm * keywords : * randomized load balancing , supermarket model , matrix - analytic approach , doubly exponential solution , density dependent jump markov process , markovian arrival process ( map ) , phase - type ( ph ) distribution , fixed point , exponential convergence , lipschitz condition .

1009.4970

c

in this paper , we provide a novel matrix - analytic approach for studying doubly exponential solutions of the supermarket models with maps and ph service times . we describe the supermarket model as a system of differential vector equations , and obtain a closed - form solution with a doubly exponential structure to the fixed point of the system of differential vector equations . based on this , we shows that the fixed point can be decomposed into the product of two factors inflecting arrival information and service information , and indicate that the doubly exponential solution to the fixed point is not always unique for more general supermarket models . furthermore , we analyze the exponential convergence of the current location of the supermarket model to its fixed point , and apply the kurtz theorem to study density dependent jump markov process given in the supermarket model with maps and ph service times , which leads to the lipschitz condition under which the fraction measure of the supermarket model weakly converges the system of differential vector equations . therefore , we gain a new and crucial understanding of how the workload probing can help in load balancing jobs with either non - poisson arrivals or non - exponential service times . our approach given in this paper is useful in the study of load balancing in data centers and multi - core servers systems . we expect that this approach will be applicable to the study of other randomized load balancing schemes , for example , analyzing a renewal arrival process or a general service time distribution , discussing retrial service discipline and processor - sharing discipline , and studying supermarket networks .

in this paper , we provide a novel matrix - analytic approach for studying doubly exponential solutions of randomized load balancing models ( also known as _ supermarket models _ ) with markovian arrival processes ( maps ) and phase - type ( ph ) service times . we describe the supermarket model as a system of differential vector equations by means of density dependent jump markov processes , and obtain a closed - form solution with a doubly exponential structure to the fixed point of the system of differential vector equations . based on this , we show that the fixed point can be decomposed into the product of two factors inflecting arrival information and service information , and further find that the doubly exponential solution to the fixed point is not always unique for more general supermarket models . furthermore , we analyze the exponential convergence of the current location of the supermarket model to its fixed point , and apply the kurtz theorem to study density dependent jump markov process given in the supermarket model with maps and ph service times , which leads to the lipschitz condition under which the fraction measure of the supermarket model weakly converges the system of differential vector equations . this paper gains a new understanding of how workload probing can help in load balancing jobs with non - poisson arrivals and non - exponential service times

in this paper , we provide a novel matrix - analytic approach for studying doubly exponential solutions of randomized load balancing models ( also known as _ supermarket models _ ) with markovian arrival processes ( maps ) and phase - type ( ph ) service times . we describe the supermarket model as a system of differential vector equations by means of density dependent jump markov processes , and obtain a closed - form solution with a doubly exponential structure to the fixed point of the system of differential vector equations . based on this , we show that the fixed point can be decomposed into the product of two factors inflecting arrival information and service information , and further find that the doubly exponential solution to the fixed point is not always unique for more general supermarket models . furthermore , we analyze the exponential convergence of the current location of the supermarket model to its fixed point , and apply the kurtz theorem to study density dependent jump markov process given in the supermarket model with maps and ph service times , which leads to the lipschitz condition under which the fraction measure of the supermarket model weakly converges the system of differential vector equations . this paper gains a new understanding of how workload probing can help in load balancing jobs with non - poisson arrivals and non - exponential service times . 0.5 cm * keywords : * randomized load balancing , supermarket model , matrix - analytic approach , doubly exponential solution , density dependent jump markov process , markovian arrival process ( map ) , phase - type ( ph ) distribution , fixed point , exponential convergence , lipschitz condition .

1505.04676

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evolutionary game theory ( egt ) has been proven to be a suitable mathematical framework to model biological and social evolution whenever the success of an individual depends on the presence or absence of other strategies @xcite . egt was introduced in 1973 by smith and price @xcite as an application of classical game theory to biological contexts , and has since then been widely and successfully applied to various fields , not only biology itself , but also ecology , population genetics , and computational and social sciences @xcite . in these contexts , the payoff obtained from game interactions is translated into reproductive fitness or social success @xcite . those strategies that achieve higher fitness or are more successful , on average , are favored by natural selection , thereby increase in their frequency . equilibrium points of such a dynamical system are the compositions of strategy frequencies where all the strategies have the same average fitness . biologically , they predict the co - existence of different types in a population and the maintenance of polymorphism . as in classical game theory with the dominant concept of nash equilibrium @xcite , the analysis of equilibrium points in random evolutionary games is of great importance because it allows one to describe various generic properties , such as the overall complexity of interactions and the average behaviours , in a dynamical system . understanding properties of equilibrium points in a concrete system is important , but what if the system itself is not fixed or undefined ? analysis of random games is insightful for such scenarios . to this end , it is ambitious and desirable to answer the following general questions : * how are the equilibrium points distributed ? how do they behave when the number of players and strategies change ? * mathematical analysis of equilibrium points and their stability in a general ( multi - player multi - strategy ) evolutionary game is challenging because one would need to cope with systems of multivariate polynomial equations of high degrees ( see section [ sec : pre ] for more details ) . nevertheless , some recent attempts , both through numerical and analytical approaches , have been made . one approach is to study the probabilities of having a concrete number of equilibria , whether all equilibrium points or only the stable ones are counted , if the payoff entries follow a certain probability distribution @xcite . this approach has the advantage that these probabilities provide elaborate information on the distribution of the equilibria . however , it consists of sampling and solving of a system of multivariate polynomial equations ; hence is restricted , even when using numerical simulations , to games of a small number of players and/or small number of strategies : it is known that it is impossible to ( analytically ) solve an algebraic equation of a degree greater than @xmath5 @xcite . another possibility is to analyze the attainability of the patterns and the maximal number of evolutionarily stable strategies ( ess ) @xcite , revealing to some extent the complexity of the interactions . this line of research has been paid much attention in evolutionary game theory and other biological fields such as population genetics @xcite . more recently , in @xcite , the authors investigate the expected number of internal equilibria in a multi - player multi - strategy random evolutionary game where the game payoff matrix is generated from normal distributions . by connecting egt and random polynomial theory , they describe a computationally implementable formula of the mean number of internal equilibrium points for the general case , lower and upper bounds the multi - player two - strategy random games , and a close - form formula for the two - player multi - strategy games . in this paper , we address the aforementioned questions , i.e. , of analysing distributions and behaviours of the internal equilibria of a random evolutionary game , in an _ average _ manner . more specifically , we first analyse the expected density of internal equilibrium points , @xmath6 , i.e. the expected number of such equilibrium points per unit length at point @xmath7 , in a @xmath0-player @xmath1-strategy random evolutionary game where the game payoff matrix is generated from a normal distribution ( for short , normal evolutionary games ) . here the parameter @xmath8 , with @xmath9 , denotes the ratio of frequency of strategy @xmath10 to that of strategy @xmath1 , respectively ( more details in section [ sec : pre ] ) . in such a random game , we then analyse the expected number of internal equilibria , @xmath3 , and , as a result , characterize the expected number of internal _ stable _ equilibria , @xmath11 . we obtain both quantitative ( asymptotic formula ) and qualitative ( monotone properties ) results of @xmath2 and @xmath3 , as functions of the ratios , @xmath7 , the number of players , @xmath0 , and that of strategies , @xmath1 . to obtain these results , we develop further the connection between egt and random polynomial theory explored in @xcite , and more importantly , establish appealing ( previously unexplored ) connections to the well - known classes of polynomials , the bernstein polynomials and legendre polynomials . in contrast to the direct approach used in @xcite , our approach avoids sampling and solving a system of multivariate polynomial equations , thereby enabling us to study games with large numbers of players and/or strategies . we now summarise the main results of the present paper . the main analytical results of the present paper can be summarized into three categories : asymptotic behaviour of the density function and the expected number of ( stable ) equilibria , a connection between the density function with the legendre polynomials , and monotonic behaviour of the density function . in addition , we provide numerical results and illustration for the general games when both the numbers of players and strategies are large . to precisely describe our main results , we introduce the following notation regarding asymptotic behaviour of two given functions @xmath12 and @xmath13 @xmath14 note that throughout the paper we sometimes put arguments of a function as subscripts . for instance , the expected density of internal equilibrium points , @xmath6 , besides @xmath7 , is also analyzed as a function of @xmath1 and @xmath0 . we will explicitly state which parameter(s ) is being varied whenever necessary to avoid the confusion . the main results of the present paper are the following . as described above , @xmath6 denotes the expected number of internal equilibrium points per unit length at point @xmath7 , in a @xmath0-player @xmath1-strategy random evolutionary game where the game payoff matrix is generated from a normal distribution ; @xmath3 the expected number of internal equilibria ; and @xmath11 the expected number of internal stable equilibria . the formal definitions of these three functions are given in section [ sec : pre ] . in theorem [ theo : concentration ] , we prove the following asymptotic behaviour of @xmath15 for all @xmath16 : @xmath17 . we also prove that @xmath15 is always bounded from above and @xmath18 . 2 . in theorem [ theo : behavior of e2 ] , we prove a novel upper bound for the expected number of multi - player two - strategy random games , @xmath19 and obtain its limiting behaviour : @xmath20 . this upper bound is sharper than the one obtained in ( * ? ? ? * theorem 2 ) , which is , @xmath21 . these results lead to two important corollaries . first , we obtain a sharper bound for the expected number of stable equilibria , @xmath22 , and the corresponding limit , @xmath23 , see corollary [ cor : expected stable equi ] . the second corollary , corollary [ cor : expected zeros bernstein ] , is mathematically significant , in which we obtain lower and upper bounds and a limiting behaviour of the expected number of real zeros of a random bernstein polynomial . 3 . in theorem [ theo : fd interm of pd ] , we establish an expression of @xmath15 in terms of the legendre polynomial and its derivative . 4 . in theorem [ theo : fd interms of pd and pd-1 ] , we express @xmath15 in terms of the legendre polynomials of two consecutive order . 5 . in theorem [ theo : fd / d decreases ] , we prove that @xmath24 is a decreasing function of @xmath0 for any given @xmath25 . consequently , @xmath26 and @xmath27 are decreasing functions of @xmath0 . 6 . in proposition [ prop : condition for f increase ] , we provide a condition for @xmath15 being an increasing function of @xmath0 for any given @xmath25 . we conjecture that this condition holds true and support it by numerical simulation . . in theorem [ theo : en2 ] , we provide an upper bound for @xmath28 . we also make a conjecture for @xmath6 and @xmath3 in the general case ( @xmath29 ) . we offer numerical illustration for our main results in section [ sec : simulation ] . the density function @xmath6 provides insightful information on the distribution of the internal equilibria : integrating @xmath6 over any interval produces the expected number of real equilibria on that interval . in particular , the expected number of internal equilibria @xmath3 is obtained by integrating @xmath6 over the positive half of the space . theorem [ theo : fd / d decreases ] and proposition [ prop : condition for f increase ] , which are deduced from theorems [ theo : fd interm of pd ] and [ theo : fd interms of pd and pd-1 ] , are qualitative statements , which tell us _ how _ the expected number of internal equilibria per unit length @xmath30 in a @xmath0-player two - strategy game changes when the number of players @xmath0 increases . theorem [ theo : concentration ] quantifies its behaviour showing that @xmath30 is approximately ( up to a constant factor ) equal to @xmath4 . the function @xmath31 , as seen in theorem [ theo : concentration ] , certainly satisfies the properties that @xmath32 increases but @xmath33 decreases . thus , it strengthens theorem [ theo : fd / d decreases ] and further supports conjecture [ cojecture : fd increases ] . theorem [ theo : behavior of e2 ] is also a quantitative statement which provides an asymptotic estimate for the expected number of internal ( stable ) equilibria . furthermore , it is important to note that the expected number of real zeros of a random polynomial has been extensively studied , dating back to 1932 with block and plya s seminal paper @xcite ( see , for instance , @xcite for a nice exposition and @xcite for the most recent progress ) . therefore , our results , in theorems [ theo : behavior of e2 ] , [ theo : fd interm of pd ] and [ theo : fd interms of pd and pd-1 ] , provide important , novel insights within the theory of random polynomials , but also reveal its intriguing connections and applications to egt . the rest of the paper is structured as follows . in section [ sec : pre ] , we recall relevant details on egt and random polynomial theory . section [ sec : 2d games ] presents full analysis of the expected density function and the expect number of internal equilibria of a multi - player two - strategy game . the results on asymptotic behaviour and on the connection to legendre polynomials and its applications are given in sections [ subsec : asymptotic ] and [ subsec : connection ] , respectively . in section [ sec : general games ] , we provide analytical results for the two - player multi - strategy game and numerical simulations for the general case . therein we also make a conjecture about an asymptotic formula for the density and the expected number of internal equilibria in the general case . in section [ sec : conclusion ] , we sum up and provide future perspectives . finally , some detailed proofs are presented in the appendix .

in this paper , we study the distribution and behaviour of internal equilibria in a-player-strategy random evolutionary game where the game payoff matrix is generated from normal distributions . the study of this paper reveals and exploits interesting connections between evolutionary game theory and random polynomial theory . the main contributions of the paper are some qualitative and quantitative results on the expected density , , and the expected number , , of ( stable ) internal equilibria . we also make a conjecture for games with more than two strategies . thirdly , we provide numerical simulations for our analytical results and to support the conjecture . as consequences of our analysis , some qualitative and quantitative results on the distribution of zeros of a random bernstein polynomial are also obtained .

in this paper , we study the distribution and behaviour of internal equilibria in a-player-strategy random evolutionary game where the game payoff matrix is generated from normal distributions . the study of this paper reveals and exploits interesting connections between evolutionary game theory and random polynomial theory . the main contributions of the paper are some qualitative and quantitative results on the expected density , , and the expected number , , of ( stable ) internal equilibria . firstly , we show that in multi - player two - strategy games , they behave asymptotically as as is sufficiently large . secondly , we prove that they are monotone functions of . we also make a conjecture for games with more than two strategies . thirdly , we provide numerical simulations for our analytical results and to support the conjecture . as consequences of our analysis , some qualitative and quantitative results on the distribution of zeros of a random bernstein polynomial are also obtained . example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore

1203.0750

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sample path properties of stochastic processes have been deeply studied for a long time , starting with the works of kolmogorov , lvy and others on the modulus of continuity and laws of the iterated logarithm of the brownian motion . since the late 1960s , these results were extended to general gaussian processes , while a finer study of the local properties of these sample paths was carried out ( we refer to berman @xcite , dudley @xcite , orey and pruit @xcite , orey and taylor @xcite and strassen @xcite , for the early study of gaussian paths and their rare events ) . among the large literature dealing with fine analysis of regularity , hlder exponents continue to be widely used as a local measure of oscillations ( see @xcite for examples of recent works in this area ) . two different definitions , called local and pointwise hlder exponents , are usually considered for a stochastic process @xmath1 , depending whether the increment @xmath2 is compared with a power @xmath3 or @xmath4 inside a ball @xmath5 when @xmath6 . as an example , with probability one , the local regularity of fractional brownian motion @xmath7 is constant along the path : the pointwise and local hlder exponents at any @xmath8 are equal to the self - similarity index @xmath9 ( e.g. see @xcite ) . this field of research is also very active in the multiparameter context and a non - exhaustive list of authors and recent works in this area includes ayache @xcite , dalang @xcite , khoshnevisan @xcite , lvy - vhel @xcite , xiao @xcite . as an extension to the multiparameter one , the set - indexed context appeared to be the natural framework to describe invariance principles studying convergence of empirical processes ( e.g. see @xcite ) . the understanding of set - indexed processes and particularly their regularity is a more complex issue than on points of @xmath10 . the simple continuity property is closely related to the nature of the indexing collection . as an example , brownian motion indexed by the lower layers of @xmath11 ^ 2 $ ] ( i.e. the subsets @xmath12 ^ 2 $ ] such that @xmath13\subseteq a$ ] for all @xmath14 ) is discontinuous with probability one ( we refer to @xcite or @xcite for the detailed proof ) . as a matter of fact , necessary and sufficient conditions for the sample path continuity property were investigated , starting with dudley @xcite who introduced a sufficient condition on the metric entropy of the indexing set , followed by fernique @xcite who gave a necessary conditions in the specific case of stationary processes on @xmath10 . talagrand gave a definitive answer in terms of majorizing measures @xcite ( see @xcite or @xcite for a complete survey and also @xcite for a lil and lvy s continuity moduli for set - indexed brownian motion ) . the question was left open so far concerning the exact hlder regularity of set - indexed processes . a formal set - indexed setting has been introduced by ivanoff and merzbach in order to study standard issues of stochastic processes , such as martingale and markov properties ( see @xcite ) . in this framework , an _ indexing collection _ @xmath15 is a collection of subsets of a measure space @xmath16 , which is assumed to satisfy certains properties such as stability by intersection of its elements . section [ sec : ho - cont ] of the present paper uses these properties , instead of conditions on the metric entropy , to derive a kolmogorov - like criterion for hlder - continuity of a set - indexed process . the collection of sets @xmath15 is endowed with a metric @xmath17 and a nested sequence @xmath18 of finite subcollections of @xmath15 such that each element of @xmath15 can be approximated as the decreasing limit ( for the inclusion ) of its projections on the @xmath19 s . we consider a supplementary assumption ( @xmath20 ) on @xmath21 and @xmath17 which impose that : 1 ) the distance from any @xmath22 to @xmath19 can be related to the cardinal @xmath23 , roughly by @xmath24 , where @xmath25 is called the _ discretization exponent _ of @xmath26 ; and 2 ) a minimality condition on the class @xmath27 that is verified in most cases . this is discussed in section [ sec : ho - cont ] , together with the links between our assumption and entropic conditions of previous works . we prove in theorem [ kolmth ] : if @xmath28 is a set - indexed process and @xmath29 are positive constants such that @xmath30\leq k\ { d_{\mathcal{a}}}(u , v)^{{q_{\underline{\mathcal{a}}}}+\beta } $ ] for all @xmath31 , then for all @xmath32 , there exist a random variable @xmath33 and a constant @xmath34 such that almost surely @xmath35 , the increments of a set - indexed process @xmath28 are defined on the class @xmath36 of sets @xmath37 where @xmath38 by the _ inclusion - exclusion formula _ @xmath39 this definition extends the notion of rectangular increments for multiparameter processes . for instance , quantities like @xmath40 } \mathbf{b } = \mathbf{b}_{\mathbf{v } } - \mathbf{b}_{(u_1,v_2 ) } - \mathbf{b}_{(v_1,u_2 ) } + \mathbf{b}_{\mathbf{u}}$ ] , where @xmath41 and @xmath42 is the brownian sheet , were proved to be useful to derive geometric sample path properties of the process ( see e.g. some of the works of dalang and walsh @xcite ) . let us notice that some processes can satisfy an increment stationarity property with respect to these increments while they do not for quantities @xmath0 . moreover , this inclusion - exclusion principle is very useful when it comes to martingale and markov properties . according to this definition , another way to express the hlder - continuity of @xmath43 is @xmath44 , for @xmath45 . this question is clarified in section [ sec : kolm ] . the purpose of hlder exponents is the ( optimal ) localization of the hlder - continuity concept . following the previous discussion , the first definition for local and pointwise hlder exponents is based on the comparison between @xmath46 and a power @xmath47 or @xmath4 in a ball @xmath48 around @xmath49 when @xmath6 . another definition compares @xmath50 for @xmath51 in @xmath36 with @xmath52 and @xmath53 for each @xmath54 , to a power @xmath55 when @xmath6 . as in the real - parameter setting , these two kinds of exponents , precisely defined in sections [ sec : hoexp ] and [ sec : hocl ] , provide a fine knowledge of the local behaviour of the sample paths . in section [ sec : flow ] , the different hlder exponents are linked to the hlder regularity of projections of the set - indexed process on increasing paths . the _ pointwise continuity _ has been introduced in the multiparameter setting in @xcite and in the set - indexed setting in @xcite as a weak form of continuity . in this definition , the _ point mass jumps _ are the only kind of discontinuity considered . without any supplementary condition on the indexing collection , the set - indexed brownian motion satisfies this property , even on lower layers where it is not continuous . in section [ sec : hopc ] , we define the pointwise continuity hlder exponent of a pointwise continuous process @xmath43 by a comparison between @xmath56 with a power @xmath57 when @xmath58 , where @xmath59 is a decreasing sequence of elements in @xmath60 which converges to @xmath61 . in the gaussian case , we prove in section [ sec : as - exp ] that the different aforementioned hlder exponents admit almost sure values . assumption ( @xmath20 ) is the key to extend this result from the multiparameter to the set - indexed setting . moreover these almost sure values can be obtained uniformly on @xmath15 for the local exponent . however , this a.s . result can not be obtained for the pointwise exponent ( even for multiparameter processes ) . nevertheless , we proved that it holds for the set - indexed fractional brownian motion ( defined in @xcite ) in section [ sec : appl ] , thus improving on a result in the multiparameter case @xcite . as this requires some specific extra work , we believe that the uniform almost sure result might not be true for the pointwise exponent of any gaussian process . finally , we also applied our results to the set - indexed ornstein - uhlenbeck ( siou ) process @xcite , for which all exponents are almost surely equal to @xmath62 at any set @xmath22 .

the first key result is a kolmogorov - like hlder - continuity theorem derived from the approximation of the indexing collection by a nested sequence of finite subcollections . then , the localization of these properties leads to various definitions of hlder exponents . in the case of gaussian processes , almost sure values are proved for these exponents , uniformly along the sample paths . as an application , the local regularity of the set - indexed fractional brownian motion and the set - indexed ornstein - uhlenbeck process are proved to be constant , with probability one . _ ams classification _ : 60g60 , 60g17 , 60g15 , 60g22 , 60j65 . _ key words _ : fractional brownian motion , gaussian processes , hlder exponents , kolmogorov criterion , local regularity , random fields , sample path properties , multiparameter and set - indexed processes .

in this paper , we study the hlder regularity of set - indexed stochastic processes defined in the framework of ivanoff - merzbach . the first key result is a kolmogorov - like hlder - continuity theorem derived from the approximation of the indexing collection by a nested sequence of finite subcollections . increments for set - indexed processes are usually not simply written as , hence we considered different notions of hlder - continuity . then , the localization of these properties leads to various definitions of hlder exponents . in the case of gaussian processes , almost sure values are proved for these exponents , uniformly along the sample paths . as an application , the local regularity of the set - indexed fractional brownian motion and the set - indexed ornstein - uhlenbeck process are proved to be constant , with probability one . _ ams classification _ : 60g60 , 60g17 , 60g15 , 60g22 , 60j65 . _ key words _ : fractional brownian motion , gaussian processes , hlder exponents , kolmogorov criterion , local regularity , random fields , sample path properties , multiparameter and set - indexed processes .

1005.2934

i

inflation proved to be very successful both theoretically @xcite and observationally @xcite as a theory of early universe . the simplest models of inflation consist of a scalar field which is minimally coupled to gravity . a period of acceleration expansion is obtained if the potential is flat enough to allow for the inflaton field to slowly roll towards its minimum . with sufficient tunings in the parameters of the model , one can achieve 60 number of e - foldings or more to solve the horizon and the flatness problems of the standard cosmology . hybrid inflation @xcite is an interesting model of inflation containing two scalar fields , the inflaton field and the waterfall field . in linde s original hybrid inflation , the energy density during inflation is dominated by the vacuum while the inflaton is slowly rolling . the waterfall field is very heavy compared to the hubble expansion rate during inflation , @xmath0 , and it quickly rolls to its instantaneous minimum . the potential has the property that once the inflaton field reaches a critical value , @xmath1 , the waterfall field becomes tachyonic triggering an instability and inflation ends quickly thereafter and the systems settles down into its global minimum . usually it is assumed that the waterfall field does not play any role in curvature perturbations during inflation and during phase transition . in this picture one basically borrows the technics and the results of single field inflationary models . that is , the super - horizon curvature perturbations , once they leave the hubble radius , are frozen and remain unchanged until they re - enter the hubble radius at a later time , such as at the time of cmb decouplings . here we would like to examine this picture more closely . we would like to see if hybrid inflation is safe under large scale curvature perturbations during the waterfall phase transition . if one considers only the classical evolution of the system , we show that during the phase transition the highly - blue tilted entropy perturbations can induce large blue - tilted curvature perturbations on super - horizon scales which can completely dominate over the original adiabatic curvature perturbations . however , we show that the quantum back - reactions of the waterfall field inhomogeneities produced during the phase transition become important before the classical back - reactions become relevant . we demonstrate that the cumulative quantum back - reactions of the short - wavelength inhomogeneities are so strong that they uplift the tachyonic instability of the entropy perturbations and the curvature perturbations freezes . ideas similar to this line of thought , studying the amplifications of large scale curvature perturbations during preheating , were studied in @xcite-@xcite and more recently in @xcite . the paper is organized as follows . in section [ hybrid ] we review the basics of hybrid inflation and obtain the background evolutions of the inflaton and the waterfall fields . in section [ entropy ] the entropy perturbations and in section [ curvature ] their effects on curvature perturbations are studied . in section [ back ] the classical non - linear back - reactions as well as the quantum mechanical back - reactions are calculated and are compared to each other . brief conclusions and discussions are followed in section [ conclusions ] . while our work was finished the work by lyth @xcite appeared which has overlaps with our results . see also @xcite which appeared shortly after our work .

if one considers only the classical evolution of the system we show that the highly blue - tilted entropy perturbations induce highly blue - tilted large scale curvature perturbations during the waterfall phase transition which dominate over the original adiabatic curvature perturbations . however , we show that the quantum back - reactions of the waterfall field inhomogeneities produced during the phase transition dominate completely over the classical back - reactions . this indicates that the standard hybrid inflation model is safe under large scale curvature perturbations during the waterfall phase transition .

in this paper the possibility of generating large scale curvature perturbations induced from the entropic perturbations during the waterfall phase transition of standard hybrid inflation model is studied . we show that whether or not appreciable amounts of large scale curvature perturbations are produced during the waterfall phase transition depend crucially on the competition between the classical and the quantum mechanical back - reactions to terminate inflation . if one considers only the classical evolution of the system we show that the highly blue - tilted entropy perturbations induce highly blue - tilted large scale curvature perturbations during the waterfall phase transition which dominate over the original adiabatic curvature perturbations . however , we show that the quantum back - reactions of the waterfall field inhomogeneities produced during the phase transition dominate completely over the classical back - reactions . the cumulative quantum back - reactions of very small scales tachyonic modes terminate inflation very efficiently and shut off the curvature perturbations evolution during the waterfall phase transition . this indicates that the standard hybrid inflation model is safe under large scale curvature perturbations during the waterfall phase transition .

1005.2934

c

in this paper the possibility of producing large scale curvature perturbations induced from the entropy perturbations during the waterfall phase transition in hybrid inflation are studied . we have shown that whether or not appreciable amounts of large scale curvature perturbations are produced depend crucially on the competition between classical and quantum mechanical back - reactions to terminate inflation . if one considers only the classical back - reaction effects , one obtains a significant large scale curvature perturbations which completely dominate over the initial curvature perturbations . the induced large scale curvature perturbations would be highly blue - tilted with @xmath222 as in @xcite . however , we have shown that the quantum - mechanical back - reactions of the waterfall field inhomogeneities produced during the phase transition dominate before the classical - back - reaction becomes important . in the hartree approximation we found that the quantum back - reactions shuts off the classical tachyonic instability very efficiently terminating inflation as well as curvature perturbations evolutions quickly after phase transition we have shown that the main contribution to quantum back - reactions comes from the cumulation of the very small scales inhomogeneities , modes which are tachyonic during the phase transition but remain sub - horizon during entire inflationary period . we also made the interesting observation that in standard hybrid inflation where the waterfall field rapidly freezes to @xmath185 at the background level , one can use @xmath223 as the effective classical trajectory . in summary , the quantum back - reactions have two crucial effects . first they determine the end of inflation given by eq . ( [ qtoc ] ) and eq . ( [ eoi ] ) . second , as mentioned above , they provide the effective classical trajectory via @xmath183 . although we have presented the analysis here only for the standard hybrid inflation , but we believe that this picture also holds for other models of inflation where there are sharp phase transitions at the end of inflation . this includes models of brane inflation where inflation ends abruptly due to tachyon formation once the distance between the brane and anti - brane reaches a critical value . however , it would be interesting to see what happens in models of inflation , such as in double inflation @xcite where there is a mild phase transition in fields evolution during early stages of inflation . since the difference @xmath224 in eq . ( [ final ] ) is not very large , an appreciable amount of curvature perturbations can be created even when the quantum mechanical back - reactions are taken into account . this in turn can produce features in curvature perturbations such as in models @xcite . it would be interesting to see the observational effects of the phase transitions during inflation as considered e.g. in @xcite . * acknowledgments * we would like to thank razieh emami , jinn - ouk gong , david lyth , mohammad hossein namjoo , david wands and specially m. sasaki for useful discussions and correspondences . we specially thank bruce bassett for many insightful discussions and for bringing refs . @xcite into our attention which initiated this work . a. a. a. would like to thank ipm and `` bonyad nokhbegan iran '' for partial support .

in this paper the possibility of generating large scale curvature perturbations induced from the entropic perturbations during the waterfall phase transition of standard hybrid inflation model is studied . we show that whether or not appreciable amounts of large scale curvature perturbations are produced during the waterfall phase transition depend crucially on the competition between the classical and the quantum mechanical back - reactions to terminate inflation .

in this paper the possibility of generating large scale curvature perturbations induced from the entropic perturbations during the waterfall phase transition of standard hybrid inflation model is studied . we show that whether or not appreciable amounts of large scale curvature perturbations are produced during the waterfall phase transition depend crucially on the competition between the classical and the quantum mechanical back - reactions to terminate inflation . if one considers only the classical evolution of the system we show that the highly blue - tilted entropy perturbations induce highly blue - tilted large scale curvature perturbations during the waterfall phase transition which dominate over the original adiabatic curvature perturbations . however , we show that the quantum back - reactions of the waterfall field inhomogeneities produced during the phase transition dominate completely over the classical back - reactions . the cumulative quantum back - reactions of very small scales tachyonic modes terminate inflation very efficiently and shut off the curvature perturbations evolution during the waterfall phase transition . this indicates that the standard hybrid inflation model is safe under large scale curvature perturbations during the waterfall phase transition .

0912.0065

i

the ground - based gravitational waves detectors ligo , virgo , and geo 600 ( @xcite , @xcite , @xcite ) are rapidly improving in sensitivity . by 2015 , advanced versions of these detectors should be taking data with a design sensitivity approximately 10 times greater than the previous generation , and the probed volume will grow by a factor of about a thousand . such improvements in detector sensitivity mean that the first gravitational - wave signature of a compact - binary coalescence ( cbc ) could be detected in the next few years ( see for example @xcite ) . among the expected signals , a special role is covered by inspiralling compact binaries . this follows from the ability to model the phase and amplitude of the signals quite accurately and consequently to maximize the signal - to - noise ratio ( snr ) by using matched filtering techniques . matched filters also provide a maxim likelihood estimation of the waveform parameters such as component masses or time of coalescence . the choice of the mles as reference estimators is also motivated by the fact that if an unbiased estimator that attain the crlb exists , is the mle@xcite . + + the existing gw frequentist literature ( @xcite .. @xcite ) evaluates the mle accuracy in two ways : ( a ) analytically by calculating the so called fisher information matrix ( fim ) or equivalently the cramer rao lower bound ( crlb ) which is the square root of the diagonal elements of the inverse fim , and ( b ) numerically by performing monte carlo simulations . + the fim was derived analytically in ( @xcite,@xcite,@xcite ) using newtonian waveforms , extended to second - order post - newtonian ( @xcite,@xcite ) and recently revisited up to 3.5pn ( @xcite,@xcite ) . in ( @xcite),(@xcite ) the authors calculate the crlb for the three _ standard _ binary systems ( nns , nbh , bbh ) , and show how the errors change when the successive different pn orders are taken into account . they consider initial ligo , advanced ligo and virgo noises . they also considers pn corrections to the _ amplitude_. + monte carlo simulations were performed , for example in ( @xcite , @xcite ) , for the lowest pn orders , where it is also suggested that the inclusion of the higher pn orders would be computationally expensive . more recent monte carlo simulations with 3.5 pn waveforms are described in ( @xcite ) . + we did not try to compare the uncertainties derived here to other existing papers ( especially from the 90 ) since different parameter sets , noise spectra ( even for the same antenna ) and pn terms were used . for example in ( @xcite ) a comparison between the crlb and other bounds is done for a waveform at the 0th pn order . this work also uses different conventions on the waveform spectrum than more recent literature . in ( @xcite ) phasing is extended to the 1.5pn order . the spin parameters are taken in account but the noise spectrum for ligo is different than the currently used design noise curves . in ( @xcite ) , ( @xcite ) , ( @xcite ) the 2pn wave was used . in the work ( @xcite ) interesting observations are made about the fluctuation of the parameters variance with the pn orders , analyzing both the case of spin and spinless systems . the fluctuations of the variance in the spinless case is also stressed in ( @xcite ) . + + the crlb is a convenient tool to approximate the accuracies in large snrs and to obtain error bounds for unbiased estimators . unfortunately , for low snrs ( below 20 ) where the first detections might emerge , the crlb can grossly underestimate the errors @xcite,@xcite,@xcite,@xcite,@xcite . the reason is that with non linear data models and ( or ) with non gaussian noises the crlb depends only on the curvature of the likelihood function around the true value of the parameter . + + in this paper we apply a recently derived analytical tool to better predict an mle accuracy and to establish necessary conditions on the signal - to - noise ratio ( snr ) for the mle error to attain the crlb . explicitly , within the frequentist framework , for arbitrary probability distribution of the noise , expansions of the bias and the covariance of a mle in inverse powers of the snr are discussed . the first order of the expansion of the variance is the inverse of the fim . by requiring that the second order covariance is smaller , or much smaller , than the first order this approach predicts a necessary snr to approximate the error with the crlb . the higher order of the expansions are determinant in the low snr regime where the inverse fim underestimates the error . we compared our the errors computed using the first two orders of the expansions to the monte carlo simulations in ( @xcite ) . we observed the first two orders of the expansions get error predictions closer than the crlb to what is observed in the numerical simulations . in ( @xcite ) the simulations are related to the fim to estabish ranges of snr where the crlb describe the error . our expansions predict the same snr range of validity for the crlb . + the expansions are sensitive to the side lobes of the likelihood function because they make use of higher order derivatives than the second one ( which is only sensitive to the curvature of the main lobe ) . the methodology also provides new insight on the relationship between waveform properties , snr , dimension of the parameter space and estimation errors . for example the timing match filtering accuracy achieves the crlb only if the snr is larger than the kurtosis of the gravitational wave spectrum and the necessary snr is much larger if the other physical parameters are unknown . more specifically the mle of the arrival time for ns - ns binary signals might require an snr equal to 2 with the time as the only parameter or 15 when all the other parameters are unknown . these results are important to understand the domain of validity of recent papers like @xcite that defines @xmath0 confidence regions in direction reconstruction with time triangulation . the regions discussed in @xcite for snr smaller than 10 are based on timing mles , with the arrival time being the only unknown parameter , and the time uncertainties quantified by the crlb . + we also note that @xcite , using a formalism introduced in @xcite,@xcite , describes a consistency criterion , different from the condition derived in this paper , for the validity of the crlb that , if applied to a 4pp compact binary signal computed with a 2pn expansion and @xmath1 , requires an snr of at least 10 . at the time of the writing of this paper we established with m.vallisneri that the equation ( [ varmatrixsimplified ] ) of this paper becomes , in the one parameter case and colored gaussian noise , equivalent to equation ( 60 ) in ( @xcite ) or ( a35 ) in ( @xcite ) . a comparison for the gaussian noise and multi parameter case is object of current work , while a comparison for arbitrary noise is not possible because ( @xcite ) and ( @xcite ) use from the beginning of their derivations gaussian noises . the explicit calculations shown here for different gws are also not performed in ( @xcite ) and ( @xcite ) . + + in section [ expansions ] we present the explicit expressions of the expansions of the bias and the covariance matrix for arbitrary noise and size of the parameter space . in section [ match ] we explain how the expansion can be evaluated for signals in additive colored gaussian noise . in section [ waveformsection ] we describe the post - newtonian inspiral waveform used for the examples , describe the parameter space and the initial and advanced ligo noises . in section [ onedim ] we study the one - dimensional parameter space results when only one parameter at a time is considered unknown . in section [ fullparspace ] we present the results for full parameter space with the initial and advanced ligo noises . we also compare our results with published results from monte carlo simulations . in section [ conclusions ] we present some conclusions and in the appendix we describe the derivation of the expansions as well as the relationship of this method with the statistics literature .

we evaluate the expansions , for the first time , for gw signals in noises of gw interferometers . we also compare the error estimates using the first two orders of the expansions with existing numerical monte carlo simulations . the first two orders of the covariance allows to get error predictions closer to what is observed in numerical simulations than the crlb . for example the timing match filtering can achieve the crlb only if the snr is larger than the kurtosis of the gravitational wave spectrum and the necessary snr is much larger if other physical parameters are also unknown .

in this paper we apply to gravitational waves ( gw ) from the merger phase of binary systems a recently derived frequentist methodology to calculate analytically the error for a maximum likelihood estimate ( mle ) of physical parameters . we use expansions of the covariance and the bias of a mle estimate in terms of inverse powers of snrs where the square root of the first order in the covariance expansion is the cramer rao lower bound ( crlb ) . we evaluate the expansions , for the first time , for gw signals in noises of gw interferometers . the examples are limited to a single , optimally oriented , interferometer . we also compare the error estimates using the first two orders of the expansions with existing numerical monte carlo simulations . the first two orders of the covariance allows to get error predictions closer to what is observed in numerical simulations than the crlb . the methodology also predicts a necessary snr to approximate the error with the crlb and provides new insight on the relationship between waveform properties , snr , dimension of the parameter space and estimation errors . for example the timing match filtering can achieve the crlb only if the snr is larger than the kurtosis of the gravitational wave spectrum and the necessary snr is much larger if other physical parameters are also unknown .

0912.0065

c

in this paper we applied a recently derived statistical methodology to gravitational waves generated by the inspiral phase of binary mergers and for noise spectral densities of gravitational wave interferometers . explicitly we computed the first two orders of mle expansions of bias and covariance matrix to evaluate mle uncertainties . we also compared the improved error estimate with existing numerical estimates . the value of the second order of the variance expansions allows to get error predictions closer to what is observed in numerical simulations than the inverse of the fim . the condition where the second order covariance is negligible with respect to the first order predicts correctly the necessary snr to approximate the error with the crlb and provides new insight on the relationship between waveform properties snrs and estimation errors . future applications include imr waveforms , network of detectors and source location estimation .

in this paper we apply to gravitational waves ( gw ) from the merger phase of binary systems a recently derived frequentist methodology to calculate analytically the error for a maximum likelihood estimate ( mle ) of physical parameters . the methodology also predicts a necessary snr to approximate the error with the crlb and provides new insight on the relationship between waveform properties , snr , dimension of the parameter space and estimation errors .

in this paper we apply to gravitational waves ( gw ) from the merger phase of binary systems a recently derived frequentist methodology to calculate analytically the error for a maximum likelihood estimate ( mle ) of physical parameters . we use expansions of the covariance and the bias of a mle estimate in terms of inverse powers of snrs where the square root of the first order in the covariance expansion is the cramer rao lower bound ( crlb ) . we evaluate the expansions , for the first time , for gw signals in noises of gw interferometers . the examples are limited to a single , optimally oriented , interferometer . we also compare the error estimates using the first two orders of the expansions with existing numerical monte carlo simulations . the first two orders of the covariance allows to get error predictions closer to what is observed in numerical simulations than the crlb . the methodology also predicts a necessary snr to approximate the error with the crlb and provides new insight on the relationship between waveform properties , snr , dimension of the parameter space and estimation errors . for example the timing match filtering can achieve the crlb only if the snr is larger than the kurtosis of the gravitational wave spectrum and the necessary snr is much larger if other physical parameters are also unknown .

1004.1247

i

due to their proximity the magellanic clouds ( mcs ) offer an excellent opportunity to study their spatially resolved star formation ( sf ) histories . sf can be triggered by several mechanisms such as , e.g. , the self - induced gravitational collapse of a molecular cloud , tidal shocking , a turbulent interstellar medium , or cloud - cloud collisions ( e.g. , * ? ? ? the mcs and the milky way ( mw ) are interacting with each other . the formation of star clusters younger than @xmath31 gyr in the mcs was probably triggered by interactions of the galaxies with each other and with the mw ( e.g. , * ? ? ? star clusters may be produced through strong shock compressions induced by close encounters of their host galaxies , which causes enhanced star formation . conversely , the star formation rate decreases again once the galaxies recede from each other . repeated encounters then lead to episodic cluster formation . in the mcs , a correlation between young star clusters and putative close encounters with each other and mw has been suggested by , e.g. , ( * ? ? ? * g95 ) , ( * ? ? ? * pu00 ) , and ( * ? ? ? * c06 ) . strong tidal perturbations induced by the encounters could also have triggered the formation of clusters ( e.g. , * ? ? ? * ) in the mcs . possible orbits of the small magellanic cloud ( smc ) , the large magellanic cloud ( lmc ) , and the mw have been modeled by several authors ( e.g. , * ? ? ? they found that it was difficult to keep the clouds bound to each other for more than 1 gyr in the past . the lmc and the smc have been part of a triple system together with the milky way since at least 1 gyr ( e.g. , * ? ? ? * kallivayalil et al . 2006a / b ) . it is possible that the clouds are not a bound system and that they are making their first passage close to the mw . interestingly , the cluster formation histories of the lmc and smc show large differences . in the lmc , two main epochs of cluster formation ( e.g. , * ? ? ? * ) have been observed that are separated by an age gap of about 4 - 9 gyr ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , in which no star clusters have formed . the two epochs of pronounced cluster formation occurred @xmath49 gyr ago and @xmath53 - 4 gyr ago . in the lmc , a few globular clusters are found that are as old as the oldest galactic globulars @xcite . during the past @xmath54 gyr , star clusters have been forming continuously until the present day . in contrast , the star clusters in the smc cover a wide range of ages and continued to form over at least the last @xmath510.5 gyr ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) to the present day . interestingly , in the smc the cluster formation history appears to have started with a delay since the smc formed its first and only globular cluster , ngc121 , 2 - 3 gyr later than the lmc or the mw ( * ? ? ? * and references therein ) . the lmc contains about @xmath54200 star clusters , while in the smc @xmath5770 star clusters have formed ( and survived ) . the cluster census is probably still incomplete , missing small and faint clusters that are yet to be detected . ongoing and prospective space - based observations may further increase the number of known star clusters . the most recent catalog cross - correlating all known objects of the lmc , smc , and the magellanic bridge region was published by @xcite ( b08 ) . however , the cluster sample still is highly incomplete as pointed out by the authors . only for a few clusters in b08 s catalog , ages have been determined . for young smc clusters , @xcite ( pu99 ) used isochrone fitting on data from the optical gravitational lensing experiment ( ogle ii ; * ? ? ? * ) to determine ages for 93 clusters . @xcite compiled ages for 306 binary cluster candidates in the lmc from a variety of literature sources ranging from multiwavelength integrated light studies to isochrone fitting to resolve clusters . @xcite ( rz05 ) made use of integrated colors and derived ages for 200 clusters . c06 determined ages of 164 associations and 311 star clusters based on data from the ogle using isochrone fitting . their sample is the largest available catalog with smc cluster ages . ages for young lmc clusters have been provided by g95 based on integrated colors ( 624 objects ) and by pu00 using isochrone fitting applied to ogle - ii data ( @xmath5600 clusters ) . luminosities have been published for 204 smc star clusters by rz05 measuring integrated colors from the magellanic clouds photometric survey ( mcps ) . @xcite ( b96 ) published integrated photometry of 624 lmc star clusters that was based on observations carried out at the 0.61-m telescope at ctio in chile and at the 2.15-m casleo telescope in argentina . in the present study we increase the number of age - dated young lmc and smc star clusters and calculate v - band luminosities . we aim at improving the understanding of the cluster age distribution of these two irregular galaxies and present spatial distribution maps of the star clusters in both galaxies . to achieve this goal , we make use of ground - based data of the magellanic clouds photometric surveys ( mcpss ) @xcite . in the next section the observations and data reduction are described . in @xmath6 [ sec : metunddistmod ] the distances , reddenings , and metallicities of both the smc and the lmc are given . in @xmath6 [ sec : agedist ] the clusters age distribution , spatial distribution , and dissolution effects are discussed and in @xmath6 [ sec : youngies_lum ] the correlation between the cluster luminosities and age / radius is derived .

in this paper we discuss the age and spatial distribution of young ( age gyr ) smc and lmc clusters using data from the magellanic cloud photometric surveys . ages of 324 and 1193 populous star clusters in the small and the large magellanic cloud have been determined fitting padova and geneva isochrone models to their resolved color - magnitude diagrams . we present the spatially resolved recent star formation history of both clouds based on young star clusters . the first peak may have been triggered by a close encounter between the smc and the lmc . in both galaxies the youngest clusters

in this paper we discuss the age and spatial distribution of young ( age gyr ) smc and lmc clusters using data from the magellanic cloud photometric surveys . luminosities are calculated for all age - dated clusters . ages of 324 and 1193 populous star clusters in the small and the large magellanic cloud have been determined fitting padova and geneva isochrone models to their resolved color - magnitude diagrams . the clusters cover an age range between 10 myr and 1 gyr in each galaxy . for the smc a constant distance modulus of = 18.90 and a metallicity of z = 0.004 were adopted . for the lmc , we used a constant distance modulus of = 18.50 and a metallicity of z = 0.008 . for both galaxies , we used a variable color excess to derive the cluster ages . we find two periods of enhanced cluster formation in both galaxies at 160 myr and 630 myr ( smc ) and at 125 myr and 800 myr ( lmc ) . we present the spatially resolved recent star formation history of both clouds based on young star clusters . the first peak may have been triggered by a close encounter between the smc and the lmc . in both galaxies the youngest clusters reside in the supergiant shells , giant shells , the inter - shell regions , and toward regions with a high h content , suggesting that their formation is related to expansion and shell - shell interaction . most of the clusters are older than the dynamical age of the supergiant shells . no evidence for cluster dissolution was found . computed v band luminosities show a trend for fainter magnitudes with increasing age as well as a trend for brighter magnitudes with increasing apparent cluster radii .

1004.1247

i

we have presented ages and luminosities of 324 and 1193 populous smc and lmc star clusters , respectively . an age range of @xmath59 myr to 1 gyr was covered based on isochrone fitting to resolved color - magnitude diagrams in both galaxies . using only cluster ages derived in this study , we find two maxima of enhanced cluster formation for both galaxies which appear to be correlated . in the smc , the peaks are found at @xmath5160 myr and @xmath5630 myr , and in the lmc at @xmath5125 myr and @xmath5800 myr . model calculations predict that the last close encounter between lmc and smc occurred around 100 - 200 myr ago . during a close encounter , the star formation is expected to be enhanced . therefore , the first peaks in the cluster age distributions could have been triggered by this tidal interaction . extending our samples with cluster ages derived by c06 we find a third pronounced period of enhanced cluster formation in the smc at around 8 myr . we find the same in the lmc combining our sample with the one of pu00 . these peaks are only visible if we extend our sample with objects classified as associations , objects which did not or could not reach higher ages because they dissolve too quickly . the youngest objects in both galaxies are associated with super giant shells , giant shells , the inter - shell region , and with hii regions . their formation is probably related to shell expansion and shell interaction . in the spatial distribution of the clusters younger than @xmath516 myr the two smc shells are clearly visible . the older objects are widely spread across the entire smc main body , but show a concentration in the western part of the galaxy . in the lmc , the youngest objects are concentrated in 30 doradus , sgs 11 ( lmc 4 ) , and in the giant shells located in the western part and in the bar region . the older lmc clusters are mostly distributed along the bar and along the rim . one can see nicely how star cluster formation propagated along the lmc bar . we find no indication for propagating star cluster formation in the sgss in either lmc or smc . most of the lmc star clusters are older than the dynamical ages of the sgss and therefore may have formed in shells , which already have dissolved and can not be detected at the present day . no obvious dissolution effects were found for mcs star clusters younger than @xmath51 gyr . it is quite difficult to ascertain a real absence of cluster dissolution using this study . two biases may play a major role : 1 . infant mortality can not be accounted for , because very young star clusters and ob - associations are not included in our sample ; and 2 . cluster dissolution processes for clusters older than @xmath51 gyr , because we did not age - date clusters in this age range . within the time period considered here - 10 myr to 1 gyr - we do not find evidence for cluster dissolution . in both galaxies , the clusters become fainter with increasing age . the very massive hot stars , which are still present in the young star clusters and contribute most of the light , become fainter and redder with increasing age and so do the star clusters . this trend can be seen in both the lmc and the smc . the total cluster luminosity increases with increasing radius due to a larger number of stars within the cluster radius . we thank the anonymous referee for extremely useful suggestions to improve our paper . we gratefully acknowledge support by the swiss national science foundation through grant number 200020 - 105260 and 200020 - 113697 . andreas koch acknowledges support by an stfc postdoctoral fellowship .

reside in the supergiant shells , giant shells , the inter - shell regions , and toward regions with a high h content , suggesting that their formation is related to expansion and shell - shell interaction . most of the clusters are older than the dynamical age of the supergiant shells . computed v band luminosities show a trend for fainter magnitudes with increasing age as well as a trend for brighter magnitudes with increasing apparent cluster radii .

in this paper we discuss the age and spatial distribution of young ( age gyr ) smc and lmc clusters using data from the magellanic cloud photometric surveys . luminosities are calculated for all age - dated clusters . ages of 324 and 1193 populous star clusters in the small and the large magellanic cloud have been determined fitting padova and geneva isochrone models to their resolved color - magnitude diagrams . the clusters cover an age range between 10 myr and 1 gyr in each galaxy . for the smc a constant distance modulus of = 18.90 and a metallicity of z = 0.004 were adopted . for the lmc , we used a constant distance modulus of = 18.50 and a metallicity of z = 0.008 . for both galaxies , we used a variable color excess to derive the cluster ages . we find two periods of enhanced cluster formation in both galaxies at 160 myr and 630 myr ( smc ) and at 125 myr and 800 myr ( lmc ) . we present the spatially resolved recent star formation history of both clouds based on young star clusters . the first peak may have been triggered by a close encounter between the smc and the lmc . in both galaxies the youngest clusters reside in the supergiant shells , giant shells , the inter - shell regions , and toward regions with a high h content , suggesting that their formation is related to expansion and shell - shell interaction . most of the clusters are older than the dynamical age of the supergiant shells . no evidence for cluster dissolution was found . computed v band luminosities show a trend for fainter magnitudes with increasing age as well as a trend for brighter magnitudes with increasing apparent cluster radii .

1303.4268

i

consider an asset price process @xmath0 with @xmath1 , paying no dividend , defined on a complete filtered probability space @xmath2 with a given risk - neutral measure @xmath3 , and assume that interest rates are zero . in the black - scholes - merton ( bsm ) model , the dynamics of the logarithm of the asset price are given by @xmath4 , where @xmath5 represents the instantaneous volatility and @xmath6 is a standard brownian motion . the no - arbitrage price of the call option at time zero is then given by the famous bsm formula @xcite : @xmath7 , with @xmath8 , where @xmath9 is the standard normal distribution function . for a given market price @xmath10 of the option at strike @xmath11 and maturity @xmath12 , the spot implied volatility @xmath13 is the unique solution to the equation @xmath14 . for any @xmath15 and @xmath16 , we define as in @xcite a forward - start option with forward - start date @xmath17 , maturity @xmath12 and strike @xmath11 as a european option with payoff @xmath18 where we define @xmath19 pathwise . by the stationary increment property , its value is simply @xmath20 in the bsm model . for a given market price @xmath21 of the option at strike @xmath11 , forward - start date @xmath17 and maturity @xmath12 , the forward implied volatility smile @xmath22 is then defined ( see also @xcite ) as the unique solution to @xmath23 . the forward smile is a generalisation of the spot implied volatility smile , and the two are equal when @xmath24 . asymptotics of the spot implied volatility surface have received a large amount of attention over the past decade . these results have helped shape calibration methodologies based on arbitrage - free approximation of the spot smile in a large variety of models . small - maturity asymptotics have been studied by berestycki - busca - florent @xcite using pde methods for continuous time diffusions and by henry - labordre @xcite using heat kernel expansions . forde et al . @xcite and jacquier et al . @xcite derived small and large - maturity asymptotics in the heston model and affine stochastic volatility models with jumps using large deviations and saddlepoint methods . wing asymptotics ( as @xmath25 ) have been studied by lee @xcite and further extended by benaim and friz @xcite and in @xcite . fouque et al . @xcite have used perturbation techniques in order to study slow and fast mean - reverting stochastic volatility models . small - noise expansions have been studied by osajima @xcite and takahashi @xcite using watanabe expansions and in deuschel et al . @xcite using laplace method on wiener space . see @xcite for a general overview of implied volatility asymptotics in stochastic volatility models . in exponential lvy models implied volatilities of out - of - the - money options explode as the maturity tends to zero , while the implied volatility for at - the - money options converges to the volatility of the diffusion component as the maturity tends to zero . small - maturity asymptotics for models with jumps ( including lvy processes ) have been investigated in @xcite . however , these asymptotics do not provide any information on the forward smile or forward - start - based payoffs , such as cliquets and forward - start options and the literature on this topic is sparse . glasserman and wu @xcite introduced different notions of forward volatilities for forecasting purposes , keller - ressel @xcite studies a very specific type of asymptotic ( when the forward - start date @xmath17 becomes large ) , and empirical results have been carried out in @xcite . in @xcite the authors empirically studied the forward smile in sato models and ran comparisons with a suite of models including heston and local volatility models for forward smile sensitive products such as cliquets . recently , small and large - maturity forward smile asymptotics were derived in @xcite for a general class of models including the heston model . however , the results in @xcite only apply to the so - called diagonal small - maturity regime , i.e. the behaviour ( as @xmath26 tends to zero ) of the process @xmath27 . the conjecture , stated there , is that for fixed @xmath28 the heston forward smile ( corresponding to @xmath29 ) explodes to infinity ( except at - the - money ) as @xmath12 tends to zero . in this paper we confirm this conjecture and give a high - order expansion for the forward smile . the main result ( theorem [ theorem : fwdsmilesmalltime ] ) is that the small - maturity heston forward smile explodes according to the following asymptotic : @xmath30 for @xmath31 and @xmath28 as @xmath12 tends to zero . here @xmath32 and @xmath33 are even continuous functions ( over @xmath34 ) with @xmath35 and independent of the heston correlation . in the at - the - money case ( @xmath36 ) a separate model - independent analysis ( lemma [ lem : modelindatm ] and theorem [ theorem : fwdatmsmalltime ] ) shows that the small - maturity limit is well defined and @xmath37 holds for any well - behaved diffusion where @xmath38 is the instantaneous variance at time @xmath17 . this exploding nature is consistent with empirical observations in @xcite and the diagonal small - maturity asymptotic from @xcite . the paper is structured as follows . in section [ sec : fwdtimescales ] we introduce the notion of a forward time - scale and characterise it in the heston model . in section [ sec : smfwdstartasymp ] we state the main result on small - maturity asymptotics of forward - start options in the heston model . section [ sec : smfwdsmileasymp ] tackles the forward implied volatility asymptotics : section [ sec : otmasymp ] translates the results of section [ sec : smfwdstartasymp ] into out - of - the - money forward smile asymptotics , and section [ sec : atmfwdsmile ] presents a model - independent result for the at - the - money forward implied volatility . section [ sec : smhnumerics ] provides numerical evidence supporting the asymptotics derived in the paper and the main proofs are gathered in section [ sec : proofssmalltau ] . * notations : * @xmath39 shall always denote expectation under a risk - neutral measure @xmath40 given a priori . we shall refer to the standard ( as opposed to the forward ) implied volatility as the spot smile and denote it @xmath41 . the forward implied volatility will be denoted @xmath42 as above . in this paper we will always assume that forward - start date is greater than zero ( @xmath28 ) unless otherwise stated .

in this paper we investigate the asymptotics of forward - start options and the forward implied volatility smile in the heston model as the maturity approaches zero . we prove that the forward smile for out - of - the - money options explodes and compute a closed - form high - order expansion detailing the rate of the explosion . furthermore the result shows that the square - root behaviour of the variance process induces a singularity such that for certain parameter configurations one can not obtain high - order out - of - the - money forward smile asymptotics . in the at - the - money case a separate model - independent analysis shows that the small - maturity limit is well defined for any it diffusion .

in this paper we investigate the asymptotics of forward - start options and the forward implied volatility smile in the heston model as the maturity approaches zero . we prove that the forward smile for out - of - the - money options explodes and compute a closed - form high - order expansion detailing the rate of the explosion . furthermore the result shows that the square - root behaviour of the variance process induces a singularity such that for certain parameter configurations one can not obtain high - order out - of - the - money forward smile asymptotics . in the at - the - money case a separate model - independent analysis shows that the small - maturity limit is well defined for any it diffusion . the proofs rely on the theory of sharp large deviations ( and refinements ) and incidentally we provide an example of degenerate large deviations behaviour .

1702.06290

i

skein modules were independently introduced in 1987 by przytycki @xcite and turaev @xcite . they generalize knot polynomials in @xmath5 to knot polynomials in arbitrary 3-manifolds . the essence is that skein modules are quotients of free modules over ambient isotopy classes of links in 3-manifolds by properly chosen local ( skein ) relations . let @xmath6 be an oriented @xmath7-manifold , @xmath8 $ ] , @xmath9 the set of all oriented links in @xmath6 up to ambient isotopy in @xmath6 and let @xmath10 be the submodule of @xmath11 generated by the skein expressions @xmath12 , where @xmath13 , @xmath14 and @xmath15 comprise a conway triple represented schematically by the illustrations in figure [ skein ] . locally.,width=163 ] for convenience we allow the empty knot , @xmath16 , and add the relation @xmath17 , where @xmath18 denotes the trivial knot . then the _ homflypt skein module _ of @xmath6 is defined to be : @xmath19,u^{-1 } l_{+ } -ul_{- } -zl{}_{0 } \right)={\raise0.7ex\hbox{$ r\mathcal{l } $ } \!\mathord{\left/ { \vphantom { r\mathcal{l } s } } \right . \kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$ s $ } } .\ ] ] skein modules of @xmath7-manifolds have become very important algebraic tools in the study of @xmath7-manifolds , since their properties renders topological information about the @xmath7-manifolds . unlike the kauffman bracket skein module , the homflypt skein module of a @xmath7-manifold , also known as _ conway skein module _ and as _ third skein module _ , is very hard to compute and very little is known so far . more precisely , @xmath20 $ ] , where the empty link is a generator of the module ( @xcite , @xcite ) . also , for the solid torus st , @xmath3 is a free , infinitely generated @xmath21$]-module isomorphic to the symmetric tensor algebra @xmath22 , where @xmath23 denotes the conjugacy classes of non trivial elements of @xmath24 ( @xcite , @xcite , @xcite , @xcite ) . further , let @xmath25 denote a surface . then @xmath26 is an algebra which , as an @xmath27 module , is a free module isomorphic to the symmetric tensor algebra , @xmath28 , where @xmath29 denotes the conjugacy classes of nontrivial elements of @xmath30 ( @xcite ) . moreover , @xmath31 is freely generated by standard oriented unlinks as presented in figure [ mroc ] ( @xcite ) and @xmath32 is freely generated by the empty link over a properly chosen ring ( @xcite ) . finally , @xmath33 is isomorphic to @xmath34 modulo torsion , where @xmath35 are oriented 3-manifolds and @xmath36 their connected sum ( @xcite ) . in @xcite the homflypt skein module of the lens spaces @xmath0 is computed using diagrammatic method . the diagrammatic method could in theory be generalized to the case of @xmath37 , but the diagrams become even more complex to analyze and several induction arguments fail . .,width=124 ] in @xcite the most generic analogue of the homflypt polynomial , @xmath2 , for links in the solid torus @xmath38 has been derived from the generalized hecke algebras of type @xmath39 , @xmath40 , via a unique markov trace constructed on them . this algebra was defined by lambropoulou in @xcite and is related to the knot theory of the solid torus , the artin group of coxeter group of type b , @xmath41 , and to the affine hecke algebra of type a. the lambropoulou invariant @xmath2 recovers the homflypt skein module of st , @xmath3 , and is appropriate for extending the results to the lens spaces @xmath42 , since the combinatorial setting is the same as for @xmath38 , only the braid equivalence includes the braid band moves ( shorthanded to _ bbm _ ) , which reflect the surgery description of @xmath42 . for the case of @xmath0 , in order to extend @xmath2 to an invariant of links in @xmath0 we need to solve an infinite system of equations resulting from the braid band moves . namely we force : @xmath43 for all @xmath44 and for all possible slidings of @xmath45 . the above equations have particularly simple formulations with the use of a new basis , @xmath4 , for the homflypt skein module of @xmath38 , that we give in @xcite . this basis was predicted by przytycki and is crucial in this paper , since _ bbm _ s are naturally described by elements in this basis . in order to show that the set @xmath4 is a basis for @xmath46 , we started in @xcite with the well - known basis of @xmath46 , @xmath47 , discovered independently in @xcite and @xcite with diagrammatic methods , and a basis @xmath48 of the algebra @xmath40 and we followed the steps below : * an ordering relation in @xmath47 is defined and it is shown that the set is totally ordered . * elements in @xmath47 are converted to linear combinations of elements in the new set @xmath4 as follows : * elements in @xmath47 are first converted to elements in the linear basis of @xmath49 , @xmath48 . * using conjugation , the gaps appearing in the indices of the looping generators in the monomials in @xmath50 are managed . * using conjugation , the exponents of the looping generators are ordered . * using conjugation and stabilization moves , the ` braiding tails ' are removed from the above monomials and thus , the initial elements in @xmath50 are converted to linear combination of elements in @xmath4 . * finally , the sets @xmath47 and @xmath4 are related via a block diagonal matrix , where each block is an infinite lower triangular matrix . * the diagonal elements in the above matrix are invertible , making the matrix invertible and thus , the set @xmath4 is a basis for @xmath46 . the new basis is appropriate for computing the homflypt skein module of the lens spaces @xmath42 in general . note that @xmath46 plays an important role in the study of homflypt skein modules of arbitrary c.c.o . @xmath7-manifolds , since every c.c.o . @xmath7-manifold can be obtained by surgery along a framed link in @xmath5 with unknotted components . the family of the lens spaces , @xmath42 , comprises the simplest example , since they are obtained by rational surgery on the unknot . equations ( [ eqbbm ] ) are very controlled in the algebraic setting , because , as shown in @xcite , they can be performed only on elements in @xmath4 . this is shown by following the technique developed in @xcite . the difference lies in the fact that here we deal with elements in @xmath4 and at the same time with their result after the performance of a _ bbm _ and we keep track of how _ bbm _ s affect the steps described above . more precisely , in @xcite we followed the steps below : * equations ( [ eqbbm ] ) boil down by linearity to considering only words in the canonical basis @xmath51 of the algebra @xmath52 . * for words in @xmath53 equations of the form @xmath54 are obtained , where @xmath55 is the result of the performance of a braid band move on the _ first _ moving strand of the closed braid @xmath56 , and @xmath57 . * then , elements in @xmath53 are expressed to elements in the linear basis @xmath48 of @xmath52 and it is shown that the equations for words in @xmath53 are equivalent to equations of the form @xmath58 , where @xmath59 . * a set @xmath60 is then introduced , consisting of monomials in the looping generators @xmath61 s with no gaps in the indices ( as in @xmath4 ) but not necessarily ordered exponents . * equations for words in @xmath50 are now reduced to equations obtained from elements in the @xmath52-module @xmath60 , where the braid band moves are performed on _ any moving strand_. * equations of the form @xmath62 are now obtained , where @xmath63 is the result of the performance of a braid band move on the @xmath64 moving strand of the closed braid @xmath65 , and @xmath66 an element in the augmented set @xmath60 followed by a ` braiding tail ' . * using conjugation , the exponents then become in decreasing order and equations obtained from elements in the @xmath52-module @xmath60 by performing _ s on all moving strands are reduced to equations for words in @xmath52-module @xmath4 by performing _ s on all moving strands . * the ` braiding tails ' from elements in the @xmath52-module @xmath4 are now eliminated and it is shown that equations for words in the @xmath52-module @xmath4 by performing braid band moves on any strand , are now reduced to equations obtained from elements in the basis @xmath4 of @xmath3 by performing braid band moves on every moving strand : @xmath67 in @xcite we elaborate on the infinite system . the importance of our approach is that it can shed light on the problem of computing skein modules of arbitrary c.c.o . @xmath7-manifolds , since any @xmath7-manifold can be obtained by surgery on @xmath5 along unknotted closed curves . indeed , one can use our results in order to apply a braid approach to the skein module of an arbitrary c.c.o . the main difficulty of the problem lies in selecting from the infinitum of band moves ( or handle slide moves ) some basic ones , solving the infinite system of equations and proving that there are no dependencies in the solutions . note that the computation of @xmath1 is equivalent to constructing all possible analogues of the homflypt or 2-variable jones polynomial for knots and links in @xmath0 , since the linear dimension of @xmath1 means the number of independent homflypt - type invariants defined on knots and links in @xmath0 . the paper is organized as follows : in [ basics ] we recall the setting and the essential techniques and results from @xcite . more precisely , we describe braid equivalence for knots and links in @xmath0 and we present a sharpened version of the reidemeister theorem for links in @xmath0 . we also provide geometric formulations of the braid equivalence via mixed braids in @xmath5 using the @xmath68-moves and the braid band moves and give algebraic formulations in terms of the mixed braid groups @xmath69 . in [ solidtorus ] we present results from @xcite and @xcite . more precisely , we recover the homflypt skein module of the solid torus st , @xmath3 , via algebraic techniques and we present a new basis for @xmath3 , @xmath4 , from which the braid band moves are naturally described . the aim of this section is to set a homogeneous ground in computing skein modules of c.c.o . @xmath7-manifolds in general via algebraic means . in 3 we derive the relation between @xmath3 and @xmath1 and show that in order to compute @xmath1 we only need to consider elements in the basis @xmath4 of @xmath3 and impose on the lambropoulou invariant @xmath2 relations coming by performing all possible braid band moves on elements in @xmath4 .

our starting point is the knot theory of the solid torus st and the lambropoulou invariant , , for knots and links in st , the universal analogue of the homflypt polynomial in st . the relation between and is established in and it is shown that in order to compute , it suffices to solve an infinite system of equations obtained by performing all possible braid band moves on elements in the basis of , , presented in . the solution of this infinite system of equations is very technical and is the subject of a sequel paper .

in this paper we present recent results toward the computation of the homflypt skein module of the lens spaces , , via braids . our starting point is the knot theory of the solid torus st and the lambropoulou invariant , , for knots and links in st , the universal analogue of the homflypt polynomial in st . the relation between and is established in and it is shown that in order to compute , it suffices to solve an infinite system of equations obtained by performing all possible braid band moves on elements in the basis of , , presented in . the solution of this infinite system of equations is very technical and is the subject of a sequel paper .

1304.3947

c

in this paper we have studied the breakdown of normal hyperbolicity and its consequences for quantities related to reaction dynamics ; in particular , the dividing surface , the flux through the dividing surface , and the gap time distribution . our approach is to study these questions using simple , two degree - of - freedom hamiltonian models for which calculations for the different geometrical and dynamical quantities can be carried out exactly . for our examples , we showed that resonances ( homoclinic orbits ) within the normally hyperbolic invariant manifold may , or may not , lead to ` loss of normal hyperbolicity ' . moreover , we showed that for our examples the onset of such resonances results in a change in topology of the dividing surface , but it does not affect our ability to define a dividing surface ( ds ) , and that the flux through the ds varies continuously with energy , even as the energy is varied in such a way that normal hyperbolicity is lost . for both our examples we have shown that the gap time distribution exhibits a signature singularity at energies corresponding to emergence of a homoclinic orbit in the ds , but these singularities are not associated with loss of normal hyperbolicity . fm , pc , and sw acknowledge the support of the office of naval research ( grant no . n00014 - 01 - 1 - 0769 ) and the leverhulme trust .

in this paper we study the breakdown of normal hyperbolicity and its consequences for reaction dynamics ; in particular , the dividing surface , the flux through the dividing surface ( ds ) , and the gap time distribution . our approach is to study these questions using simple , two degree - of - freedom hamiltonian models where calculations for the different geometrical and dynamical quantities can be carried out exactly . for our examples , we show that resonances within the normally hyperbolic invariant manifold may , or may not , lead to a ` loss of normal hyperbolicity ' . moreover , we show that the onset of such resonances results in a change in topology of the dividing surface , but does not affect our ability to define a ds . the flux through the ds varies continuously with energy , even as the energy is varied in such a way that normal hyperbolicity is lost . for our examples the gap time distributions exhibit singularities at energies corresponding to the existence of homoclinic orbits in the ds , but these singularities are not associated with loss of normal hyperbolicity .

in this paper we study the breakdown of normal hyperbolicity and its consequences for reaction dynamics ; in particular , the dividing surface , the flux through the dividing surface ( ds ) , and the gap time distribution . our approach is to study these questions using simple , two degree - of - freedom hamiltonian models where calculations for the different geometrical and dynamical quantities can be carried out exactly . for our examples , we show that resonances within the normally hyperbolic invariant manifold may , or may not , lead to a ` loss of normal hyperbolicity ' . moreover , we show that the onset of such resonances results in a change in topology of the dividing surface , but does not affect our ability to define a ds . the flux through the ds varies continuously with energy , even as the energy is varied in such a way that normal hyperbolicity is lost . for our examples the gap time distributions exhibit singularities at energies corresponding to the existence of homoclinic orbits in the ds , but these singularities are not associated with loss of normal hyperbolicity .

hep-th0509043

i

the idea that the universe is comprised of any number of extra dimensions , in addition to the three non - compact spatial dimensions that we observe , goes back all the way to the work of kaluza and klein in the 1920 s @xcite , where extra dimensions were first proposed as a manner in which one could unify particle interactions with gravity . since then , theories with extra dimensions have evolved from being a mathematical curiosity , albeit one with remarkable consequences for particle physics @xcite , to a theoretical necessity with the advent of string theory . since the famous anomalies cancellation calculation of green and schwarz @xcite , the challenge has remained for cosmologists and phenomenologists alike to account for the presence of extra dimensions in physics . from a pessimist s point of view , one might view these extra dimensions as an unwanted surplus that has to be swept under the rug at presently accessible energies . in its most unimaginative form , one could view this as the gist of the moduli problem . from an optimist s point of view however , one could view the requirement of extra dimensions for the consistency of string theory as a new window of opportunity , through which one might be able to resolve some of the outstanding problems of particle physics and cosmology . taken as an essential part of the radically new picture of space - time that string theory offers us , extra dimensions , with strings and branes of various dimensions propagating through them , could offer us a potentially non - anthropic explanation for the observed dimensionality of space - time @xcite . in addition to offering us potentially non - singular cosmologies @xcite , they could very well be at the heart of the resolution of the dark matter problem @xcite . recently , several alternative stringy cosmologies have emerged as possible successors to the standard big bang model of the universe , such as the pre - big bang model @xcite , brane / string gas cosmology @xcite , and the cyclic / ekpyrotic scenarios @xcite , all of which have the ultimate aim of becoming complete and testable models of the early universe . a central question that arises in determining whether or not these models of the early universe reproduce observations is , how metric fluctuations in the presence of extra dimensions evolve in the backgrounds that these cosmologies propose . the importance of understanding this question can not be understated in the context of the abundance of available experimental data , against which we must compare our eventual predictions . for instance , in the ekpyrotic / cyclic scenario the effects of branes colliding in a 5-dimensional bulk are explored key to the success of this program is being able to follow the evolution of metric fluctuations through the bounces of the extra dimension . the eventual goal of this is to be able to explain the observed inhomogeneities of the cosmic microwave background as being seeded by radion fluctuations generated through the motion of the branes towards each other , rather than through the quantum fluctuations of some putative inflaton field . similarly , in the context of the pre - big bang scenario , where an expanding and a contracting phase of the universe s evolution are naturally related to each other by the scale factor duality symmetry inherent to low energy effective string theory , a universe undergoing a big bang - big crunch transition is naturally implemented . in this context , the evolution of metric fluctuations through the bounce becomes a question of central importance in understanding the way the big bang phase of our universe s evolution happened the way it did . as it turns out , although at first seemingly unrelated , an associated and as yet unexamined issue is how any tentative solution of the moduli problem will modify the spectrum of the metric fluctuations which we observe . given that the universe is likely to contain many extra dimensions whose shape and volume moduli are dynamical in the early universe , one might expect that the dynamics of these moduli fields will lead to a highly nontrivial modification of any initial pre - stabilization spectrum . it is the goal of this report to show that , in the context of a 5-d universe where an extra dimension undergoes non - singular bounces , or is en route to stabilization , the final spectrum for the bardeen potential corresponding to ( long wavelength ) fluctuations of the scale factor for the non - compact dimensions , is identical to the initial spectrum of the five - dimensional fluctuations ( in a sense which will be specified later ) . specifically , if we had an initial scale invariant spectrum for the five - dimensional fluctuations , we would end up with a scale invariant spectrum for the bardeen potential after the bounce , or after the transients of the stabilization had settled down . furthermore , we show that the variable corresponding to metric fluctuations of the compact dimensions decays , which corresponds to the stability of the radion degree of freedom to fluctuations in our non - singular setup . one application of our work concerns the evolution of fluctuations in ekpyrotic / cyclic type models . in the context of a four space - time dimensional effective field theory toy model of this scenario , the dynamics of perturbations has been investigated in detail . the initial analyses @xcite yielded the result that a scale - invariant spectrum before the bounce transfers to a scale - invariant spectrum after the bounce . these analyses , however , were questioned in @xcite . a serious complicating factor turned out to be the fact that the proposed background evolution was singular , thus requiring the use of `` matching conditions '' ( such as those derived in @xcite in the context of an expanding cosmology undergoing a sudden phase transition ) to compute the post - bounce spectrum of fluctuations . as discussed in @xcite , there is a very sensitive dependence of the final result on the choice of the matching surface . independently , there has been recent work on the evolution of fluctuations through a nonsingular bounce in four space - time dimensional cosmologies , in which the bounce is constructed by adding extra terms to the standard lagrangian . these analyses @xcite yield results showing a sensitive dependence on the nature of the bounce . the bottom line of this work is that a correct analysis of fluctuations in the ekpyrotic / cyclic scenario needs to be done in a five space - time dimensional context , a context in which the nature of the bounce is unambiguous . a first important step in this direction was taken in @xcite , confirming the result that a scale - invariant spectrum passes through the bounce ( in the five - dimensional context , the bounce means that the radius of the extra spatial dimension bounces , not that the four space - time dimensional scale factor bounces ) without change in the spectral index . however , the analysis of @xcite was done in the context of a singular background and assuming specific matching conditions for fluctuations applied at a point when the perturbations in fact blow up . thus , the results are open to doubt . in this work , we study the transfer of fluctuations through a cosmology in which two boundary branes approach each other and bounce without encountering a singularity ( see also @xcite for previous work done in the context of a particular nonsingular ekpyrotic - type bounce proposed in @xcite ) . we begin by introducing a non - singular bouncing model of a 5-dimensional universe where one dimension is compactified on a circle or an orbifold . as we shall see shortly in detail , our non - singular background is obtained by considering the effects of gas composed of massless string modes on the dynamics of space - time , in the context of a consistent low energy effective theory limit , or ` massless background ' , of string theory . we then set up the framework for studying cosmological perturbations in this model and study how these transfer through the various bounces that the extra dimension undertakes . although our setup is seemingly specific to string gas cosmology , the essence of our framework is that we have a non - singular bounce / stabilization mechanism that is affected by degrees of freedom that become massless at a certain point . hence , the hope is that the results obtained here can be generalized to other settings , an issue we will discuss in detail when we consider applications to different approaches to stringy cosmology . we now commence our paper with a few preliminaries .

in this report , we study within the context of general relativity with one extra dimension compactified either on a circle or an orbifold , how radion fluctuations interact with metric fluctuations in the three non - compact directions . we find that the metric fluctuations transfer undisturbed through the bounces or through the transients of the pre - stabilization epoch . our background is obtained by considering the effects of a gas of massless string modes in the context of a consistent massless background ( or low energy effective theory ) limit of string theory . we discuss applications to various approaches to early universe cosmology , including the ekpyrotic / cyclic universe scenario and string gas cosmology .

in this report , we study within the context of general relativity with one extra dimension compactified either on a circle or an orbifold , how radion fluctuations interact with metric fluctuations in the three non - compact directions . the background is non - singular and can either describe an extra dimension on its way to stabilization , or immediately before and after a series of non - singular bounces . we find that the metric fluctuations transfer undisturbed through the bounces or through the transients of the pre - stabilization epoch . our background is obtained by considering the effects of a gas of massless string modes in the context of a consistent massless background ( or low energy effective theory ) limit of string theory . we discuss applications to various approaches to early universe cosmology , including the ekpyrotic / cyclic universe scenario and string gas cosmology .

0903.0667

i

one of the primary goals of the large hadron collider ( lhc ) , a proton - proton collider with a center of mass energy @xmath2 tev , will be to investigate the sector responsible for the breaking of the electroweak symmetry . in the standard model ( sm ) , a single higgs doublet is responsible for the spontaneous breakdown of the @xmath3 gauge group to @xmath4 . the coupling constants of the sole physical higgs scalar to the rest of the sm particles are completely determined by their masses , and consequently there is little guesswork involved in determining the most promising channels @xcite in which one might hope to discover such a scalar . for a relatively light ( @xmath5 ) sm higgs boson , those channels are @xmath6 and @xmath7 , while for an intermediate - mass ( @xmath8 ) higgs , the single most promising channel is the weak - boson fusion ( wbf ) @xcite process @xmath9 @xcite . for a heavier higgs , with @xmath10 , the most relevant channels are @xmath11 and @xmath12 , with the higgs produced via either gluon fusion or wbf @xcite . in models where the higgs sector differs significantly from that of the standard model , however , the situation can change dramatically . this is true even in cases where the low - energy effective theory describing a given model at the weak scale contains only a single , light , @xmath0-even higgs scalar . indeed , at low energies , many models with extended higgs sectors have effective descriptions that are `` standard - model - like '' in the sense that they contain a single light higgs boson , but one whose couplings to the standard model fermions and gauge bosons differ potentially significantly from those of a sm higgs . such discrepancies , in turn , can translate into vast differences in lhc phenomenology : some ( or , in severe cases , even all ) of the standard detection channels for a sm higgs may disappear as a result of such modifications , while others , related to processes buried beneath background in the sm , may become crucial for discovery . one set of channels which are not terribly significant for the discovery of a sm higgs , but could become so in models with modified higgs sectors , consists of those involving direct decays of the higgs boson to a pair of high-@xmath13 leptons . in the sm , a light higgs boson ( with mass @xmath14 gev ) decays predominantly into @xmath15 , and the ratio @xmath16 ( where @xmath17 ) is roughly proportional to @xmath18 , due to the fact that in the sm , the same higgs doublet is responsible for giving mass to both quarks and leptons . consequently , attention has been focussed predominately on processes in which the higgs boson decays to a tau pair ( with a branching ratio of about 10% ) , and in particular on the weak - boson fusion process @xmath9 . this is the only process particularly relevant for sm higgs discovery in which the higgs decays directly to leptons , though it is now regarded as one of the most promising discovery channels for a sm higgs in the intermediate mass region @xcite . searching for the higgs in the @xmath19 and @xmath20 channels is more difficult , due to a combination of factors , including enhanced sm backgrounds and suppressed signal cross - sections . by contrast , processes in which a sm higgs boson decays into first- or second - generation leptons are generally assumed to be irrelevant for discovery . this is because under the assumption of yukawa - coupling universality among the lepton generations ( an assumption we will be making throughout the present work ) , the small size of @xmath21 compared to @xmath22 results in @xmath23 being roughly two orders of magnitude smaller than @xmath24 , with @xmath25 nearly three orders of magnitude smaller still . consequently , the rates for processes involving @xmath26 and @xmath27 are extremely suppressed relative to those involving tau pairs , both in the sm and in most simple extensions of the higgs sector . on the other hand , there are strong motivations for considering processes of this sort at the lhc . experimentally , a signal involving a pair of high-@xmath28 muons or electrons will be easy to identify , as the muon- and electron - identification efficiencies at each of the lhc detectors are each greater than 90%@xcite . furthermore , once a higgs boson is discovered in these channels , its mass could be readily reconstructed with high precision . such channels could also be of use in determining the higgs yukawa couplings to leptons . two - higgs - doublet models ( 2hdm ) , which stand as perhaps the simplest , most tractable example of a non - minimal electroweak - symmetry - breaking sector , provide a useful context in which to study the role of leptonic higgs - decay processes . these models arise in a number of beyond - the - standard - model contexts from supersymmetry to little higgs scenarios @xcite and have a rich phenomenology , many of whose consequences for lhc physics are still being uncovered . in general , 2hdm can be categorized according to how the higgs doublets couple to the sm quarks and leptons . in what has become known as a type i 2hdm , one doublet is responsible for the masses of both quarks and leptons , while the other decouples from the fermions entirely . in a type ii 2hdm , one higgs doublet couples to the up - type quark sector , while the other higgs doublet couples to both the down - type quark sector and the charged leptons as is the case , for example , in the minimal supersymmetric standard model ( mssm ) . in both of these standard scenarios , the leptonic branching ratios for a light higgs do not differ much from their sm values throughout most of parameter space coupling is suppressed due to radiative corrections @xcite , and consequently @xmath29 becomes large . ] , since the same doublet gives masses to both the bottom quark and the charged leptons . one interesting alternative possibility , which will be the primary focus of the present work , is a 2hdm scenario in which one higgs doublet couples exclusively to ( both up- and down - type ) quarks , while the other couples exclusively to leptons a scenario which we will henceforth dub the leptophilic two - higgs - doublet model ( l2hdm ) . this model has been discussed previously in the literature in relation to its effect on higgs branching fractions and decay widths @xcite , flavor physics @xcite , and potential implications for neutrino phenomenology @xcite and dark matter studies @xcite . some analyses of the lhc phenomenology of the model were presented in ref . @xcite , which focused on the non - decoupling region of the parameter space where additional physical higgs scalars are light . in this work , we discuss the leptonic decays of the lightest @xmath0-even higgs scalar in the l2hdm at the lhc . in particular , we examine the discovery potential in a decoupling regime in which only one light scalar , which resembles the sm higgs , appears in the low - energy effective description of the model . we begin in section [ sec : param ] by presenting the model and reviewing how the coupling structure of the lightest neutral higgs particle is modified from that of a sm higgs . in section [ sec : bounds ] , we discuss the applicable experimental constraints from flavor physics , direct searches , etc . and show that they still permit substantial deviations in the couplings between the higgs boson and the other sm fields away from their standard - model values . in section [ sec : brxsec ] , we discuss the implications of such modifications on the higgs branching ratios and production rates . in section [ sec : lhcsignatures ] , we discuss potential higgs discovery channels in which the higgs boson decays directly into a pair of charged leptons , and in section [ sec : combined ] , we calculate the discovery potential for a light , leptophilic higgs using the combined results from all of these leptonic channels . in section [ sec : conclusion ] we conclude .

in this work , we examine a two - higgs - doublet extension of the standard model in which one higgs doublet is responsible for giving mass to both up- and down - type quarks , while a separate doublet is responsible for giving mass to leptons . we examine both the theoretical and experimental constraints on the model and show that large regions of parameter space are allowed by these constraints in which the effective couplings between the lightest neutral higgs scalar and the standard - model leptons are substantially enhanced . we investigate the collider phenomenology of such a `` leptophilic '' two - higgs - doublet model and show that in cases where the low - energy spectrum contains only one light ,-even scalar , a variety of collider processes essentially irrelevant for the discovery of a standard model higgs boson ( specifically those in which the higgs boson decays directly into a charged - lepton pair ) can contribute significantly to the discovery potential of a light - to - intermediate - mass ( gev ) higgs boson at the lhc .

in this work , we examine a two - higgs - doublet extension of the standard model in which one higgs doublet is responsible for giving mass to both up- and down - type quarks , while a separate doublet is responsible for giving mass to leptons . we examine both the theoretical and experimental constraints on the model and show that large regions of parameter space are allowed by these constraints in which the effective couplings between the lightest neutral higgs scalar and the standard - model leptons are substantially enhanced . we investigate the collider phenomenology of such a `` leptophilic '' two - higgs - doublet model and show that in cases where the low - energy spectrum contains only one light ,-even scalar , a variety of collider processes essentially irrelevant for the discovery of a standard model higgs boson ( specifically those in which the higgs boson decays directly into a charged - lepton pair ) can contribute significantly to the discovery potential of a light - to - intermediate - mass ( gev ) higgs boson at the lhc .

1506.01729

i

understanding , characterizing and classifying novel states of matter is one of the main goal of the research in solid state physics . in particular , systems where the thermal or quantum fluctuations are able to melt long range order , the so - called spin liquids , draw a lot of attention since they generally go beyond nel paradigm @xcite . the pyrochlore lattice , made of corner - sharing tetrahedra is the archetype of a three - dimensional frustrated lattice and has proven during the last years to be a rich playground for studying such spin liquid states @xcite . among the variety of possible pyrochlore systems , the compound yb@xmath0ti@xmath0o@xmath1 has been presented as one of the possible realizations of a quantum spin liquid ground - state . more precisely , it has been proposed as a candidate for the quantum variant of the spin ice state observed in ho@xmath0ti@xmath0o@xmath1 and dy@xmath0ti@xmath0o@xmath1 . in these classical spin ices , the ising - like anisotropy of the magnetic moments along the local @xmath3 directions , combined with an effective ferromagnetic interaction induce a macroscopically degenerated ground state characterized by the local ice - rule @xcite ( in each tetrahedron , two spins point in and two spins point out ) . in yb@xmath0ti@xmath0o@xmath1 , the yb@xmath4 magnetic moment shows a weak xxz planar anisotropy perpendicular to the local @xmath3 directions @xcite . however , the strongly anisotropic interactions tensor @xcite , whose main component is ferromagnetic , induces a local constraint analog to the ice - rule . the static and dynamical magnetic correlations in yb@xmath0ti@xmath0o@xmath1 have been investigated by means of neutron scattering experiments . short range correlations settle around @xmath5 k , giving rise to rods of diffuse scattering along high symmetry directions @xcite . at lower temperature , a phase transition is observed in specific heat measurements around 0.2 k @xcite , but the critical temperature depends on the nature of the samples ( polycrystal or single crystal ) and their quality @xcite . this transition was shown to be first order @xcite , in both single crystals and powder samples but its nature remains debated . several studies ( including neutron scattering and muon spin relaxation measurements ) evidence the stabilization of an ordered ferromagnetic state , with a reduced static magnetic moment @xcite , while others do not @xcite . to understand the underlying microscopic mechanism leading to such a non - conventional ground - state , several studies attempted to determine the exchange couplings , trying to reproduce the experimental results . different sets of interactions were obtained , depending on the fitted quantity : diffuse scattering @xcite or field induced spin wave excitations @xcite . the latter parameters allow reproducing specific heat @xcite , and were further refined using terahertz spectroscopy measurements @xcite . these parameters place yb@xmath0ti@xmath0o@xmath1 in a `` splayed ferromagnetic '' phase , quite far from the `` quantum spin liquid '' phase of the theoretical phase diagram @xcite . since the precise nature of the elementary excitations generally depends on the ground - state itself , we propose in this article to address the unconventional magnetic properties of yb@xmath0ti@xmath0o@xmath1 through a detailed study of the spin dynamics on a single crystal in the zero field correlated phase below 2 k , which has been little explored up to now @xcite . in section [ sec : exp ] , we first present macroscopic measurements performed on a piece of our single crystal , which show evidence for a transition at 175 mk . by means of inelastic neutron scattering , we show that the excitations spectrum is a continuum characterized by a very broad and almost flat dynamical response extending up to @xmath2 mev , and coexisting or not with a quasi - elastic response depending on the wave - vector . these excitations do not depend on the temperature below the stabilization of short - range correlations ( i.e. 2 k ) , and , in particular , entering the ordered phase does not affect the spectra . then , in section [ sec : num ] , we compare these inelastic data with classical calculations combining monte carlo and spin dynamics simulations . we show that , although the qualitative features of the spectra are reproduced , calculations predict an energy scale twice smaller than the experimental observations . these calculations are based on our determination of a reliable and robust set of exchange couplings , able to reproduce both the spin - wave spectrum of the field polarized phase as well as the diffuse scattering in zero field . the obtained couplings do place yb@xmath0ti@xmath0o@xmath1 in a ferromagnetic phase , yet very close to the boundary with an antiferromagnetic phase . then , although yb@xmath0ti@xmath0o@xmath1 is not a canonical quantum spin liquid as initially proposed , we propose that the proximity of competing ferromagnetic and antiferromagnetic correlations restore strong quantum fluctuations@xcite and is at the origin of the anomalous static and dynamical behaviors observed in our experiments .

in particular , the simulations predict an energy scale twice smaller than the experimental observations . ybtio lies at the border between a ferro and an antiferromagnetic phase . these results suggest that the unconventional ground state of ybtio is governed by strong quantum fluctuations arising from the competition between those phases

in this work , we show that the zero field excitation spectra in the quantum spin ice candidate pyrochlore compound ybtio is a continuum characterized by a very broad and almost flat dynamical response which extends up to mev , coexisting or not with a quasi - elastic response depending on the wave - vector . the spectra do not evolve between 50 mk and 2 k , indicating that the spin dynamics is only little affected by the temperature in both the short - range correlated and ordered regimes . although classical spin dynamics simulations qualitatively capture some of the experimental observations , we show that they fail to reproduce this broad continuum . in particular , the simulations predict an energy scale twice smaller than the experimental observations . this analysis is based on a careful determination of the exchange couplings , able to reproduce both the zero field diffuse scattering and the spin wave spectrum rising in the field polarized state . according to this analysis , ybtio lies at the border between a ferro and an antiferromagnetic phase . these results suggest that the unconventional ground state of ybtio is governed by strong quantum fluctuations arising from the competition between those phases . the observed spectra may correspond to a continuum of deconfined spinons as expected in quantum spin liquids .

1506.01729

c

in this work , we show that the short range correlations that develop below 2 k in yb@xmath0ti@xmath0o@xmath1 come along with non - conventional excitations : no well defined spin waves are observed in both the short range correlated and long range ordered regimes ; rather , excitations spectra are characterized by a very broad and almost flat dynamical response which extends up to @xmath2 mev and does not evolve with temperature below 2 k , sometimes coexisting with a quasi - elastic response depending on the wave - vector . based on monte carlo spin dynamics and rpa calculations , we determine a new set exchange couplings that allows to reproduce both the diffuse scattering in the short range correlated regime and the spin wave spectra observed in the field polarized phase . this careful determination of the exchange tensors places yb@xmath0ti@xmath0o@xmath1 in the close vicinity to a fm / afm phase boundary , and , in any case , far from the canonical quantum spin liquid state . this location points out the possible role of quantum fluctuations arising from these two competing phases . we show further that conventional rpa fails to reproduce the experimental excitation spectra at low temperature . spin dynamics simulations performed in the short range correlated regime reproduce however some features of the excitation spectra but lead to an energy bandwidth which is twice smaller than the experimental observations . we speculate that quantum fluctuations between fm and afm phases govern the spin dynamics in yb@xmath0ti@xmath0o@xmath1 and especially that the observed spectra rather correspond to a continuum of deconfined spinons as expected in quantum spin liquids , than simple paramagnons . we acknowledge m. gingras and l. jaubert for fruitful discussions . we thank c. paulsen for allowing us to use his squid dilution magnetometers . we also thank ph . boutrouille , ( cryogeny group at llb ) for his technical help while setting up he dilution fridge .

in this work , we show that the zero field excitation spectra in the quantum spin ice candidate pyrochlore compound ybtio is a continuum characterized by a very broad and almost flat dynamical response which extends up to mev , coexisting or not with a quasi - elastic response depending on the wave - vector . this analysis is based on a careful determination of the exchange couplings , able to reproduce both the zero field diffuse scattering and the spin wave spectrum rising in the field polarized state . according to this analysis , . the observed spectra may correspond to a continuum of deconfined spinons as expected in quantum spin liquids .

in this work , we show that the zero field excitation spectra in the quantum spin ice candidate pyrochlore compound ybtio is a continuum characterized by a very broad and almost flat dynamical response which extends up to mev , coexisting or not with a quasi - elastic response depending on the wave - vector . the spectra do not evolve between 50 mk and 2 k , indicating that the spin dynamics is only little affected by the temperature in both the short - range correlated and ordered regimes . although classical spin dynamics simulations qualitatively capture some of the experimental observations , we show that they fail to reproduce this broad continuum . in particular , the simulations predict an energy scale twice smaller than the experimental observations . this analysis is based on a careful determination of the exchange couplings , able to reproduce both the zero field diffuse scattering and the spin wave spectrum rising in the field polarized state . according to this analysis , ybtio lies at the border between a ferro and an antiferromagnetic phase . these results suggest that the unconventional ground state of ybtio is governed by strong quantum fluctuations arising from the competition between those phases . the observed spectra may correspond to a continuum of deconfined spinons as expected in quantum spin liquids .

1205.3340

i

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` if we have to go on with these damned quantum jumps , + then i m sorry that i ever got involved . '' + _ e. schrdinger _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ when two ( or more ) quantum systems interact , for example in a scattering process , a correlation arises among them ; this correlation lasts in time and can be exploited to collect information about , or even modify , one of the systems without directly acting on it , no matter how far it is . this phenomenon , which is called _ entanglement _ and has no analogoue in classical theory , grossly violates the locality principle , as was first highlighted by einstein , podolski and rosen in @xcite . purpose of the authors was to demonstrate that the existence of entanglement is incompatible with the completeness of quantum theory , but their thesis was proved to be wrong in many occasions over time , first of all by john s. bell in his 1964 article `` _ _ on the einstein - poldolsky - rosen paradox _ _ '' @xcite . bell showed that any theory reproducing the predictions of quantum mechanics has to violate locality . quantum theory is thus a complete non - local theory , with entanglement as an inevitable consequence of its principles . the article by epr , against the scopes of the authors , laid the bases for the birth of quantum information theory , the purpose of which is to study the nature of entanglement and the way to exploit it . let us consider a state in a composite quantum system . it may seem reasonable to suppose that this state has a certain fixed amount of entanglement , or in other words , the system has a fixed amount of quantum correlation within . however , it has been proven , as we are going to explain in more detail in the present work , that the quantity and even the existence of entanglement for a certain state depends on the choice of the factorization of the algebra of observables , i.e. on the choice of the physical observables used to describe it . as an example , a composite system consisting of a couple of spins could be described in terms of total spin , and in this factorization no entanglement will appear : the system would not even look `` composite '' . the possibility for entanglement to be `` created '' and `` destroyed '' is maybe its most fascinating feature , and is the basis for many of the recent achievements in quantum information and quantum computing , such as the protocols of quantum cryptography and quantum teleportation , or the superdense coding . the questions `` when '' and `` how '' it is possible to manipulate entanglement are thus of main importance . the first part of the present work will be devoted to a summary of the necessary basic concepts to deal with quantum mechanics and , in particular , with quantum information theory , such as the hilbert space , the tensor product space , the density operator , and so on . + then , in * chapter [ chap2 ] * , we will start the central part of our exposition , treating the case of pure states ; it will be shown how it is always possible to make any pure state appear either entangled or not ( in the latter case we talk of _ separable _ or _ factorized _ states ) . + the possibility for a state to be described in terms of a wavefunction is strictly connected with the possibility to make it appear entangled ( or not ) . in * chapter [ chap3 ] * we will thus consider the more general case of mixed states , and we will see that the situation here is more complex : we will demonstrate that it is still always possible to represent any state as separable , but we will also see examples of _ absolutely separable _ states , i.e. states which appear separable in every factorization ; even if it is not always possible to make a mixed state appear entangled , we will introduce two classes of states for which this possibility exists . the chapter will be closed by two complements , a basic exposition of the protocol of quantum teleportation , and a description of the geometry of physical states for the case of two qubits , i.e. for @xmath0 dimensional composite system . + we will then draw our conclusions , briefly summarizing the results here exposed . the present work includes , in its last pages , a short list of references to the literature .

in this work we will examine how a given quantum state can appear to be either entangled or separable depending on the choice of the factorization of the algebra of observables of a composite multi - partite quantum system . we will focus our attention on mixed states here . for pure states we will discuss this statement and investigate under what conditions a mixed state can be represented as entangled . in the first part of this work , we will introduce some basic concepts necessary for the following discussion , such as the hilbert space , the tensor product structure , the qubit and the density operator . in the central part , some possible constraints under wich a quantum state can be represented as entangled are discussed : * for pure states no further restrictions are necessary and we will summarize the relevant results reported in the literature recently . in short , depending on the choice of observables , a pure quantum state of a composite system can be demonstrated to be either entangled or separable . in this sense , entanglement is not an invariant property of a given state , despite its fundamental role concerning properties of multi - partite systems ( such as in the famous einstein - podolsky - rosen considerations ) . * for mixed states the situation is more complex and only partial results are known . however , it is not known in general how much `` mixedness '' is admissible , such that a state can still be represented as entangled . we will also describe and discuss the geometry of quantum physical states for the case of two qubits . we will close with our conclusions , an outlook on possible extensions of the results , and a list of references to the literature . thus , entanglement ( or , to the contrary , separability ) of a given quantum state is not uniquely determined by properties of the state , but may depend on the choice of the factorization of the algebra of observables . in the present work , we will distinguish in particular the cases of pure and mixed states . for mixed states , the situation is more complex and only partial results are known : while it is always possible to choose a factorization such that a state appears separable , a general criterion to determine whether a state can be represented as entangled is not known . these results will be illustrated by several examples , the phenomenon of quantum teleportation , and the geometry of the states of two qubits . 1.5 cm * * submitted to : + dipartimento di fisica , universit di pisa defended on : + 29 february 2012

in this work we will examine how a given quantum state can appear to be either entangled or separable depending on the choice of the factorization of the algebra of observables of a composite multi - partite quantum system . we will focus our attention on mixed states here . for pure states it is always possible to find both a factorization such that the state appears entangled and another factorization where it appears separable . on the other hand , for a mixed state it is not always possible to find a factorization such that it appears maximally entangled . we will discuss this statement and investigate under what conditions a mixed state can be represented as entangled . in the first part of this work , we will introduce some basic concepts necessary for the following discussion , such as the hilbert space , the tensor product structure , the qubit and the density operator . in the central part , some possible constraints under wich a quantum state can be represented as entangled are discussed : * for pure states no further restrictions are necessary and we will summarize the relevant results reported in the literature recently . in short , depending on the choice of observables , a pure quantum state of a composite system can be demonstrated to be either entangled or separable . in this sense , entanglement is not an invariant property of a given state , despite its fundamental role concerning properties of multi - partite systems ( such as in the famous einstein - podolsky - rosen considerations ) . * for mixed states the situation is more complex and only partial results are known . indeed , it is always possible to choose a factorization such that a state appears separable , as we shall demonstrate . however , it is not known in general how much `` mixedness '' is admissible , such that a state can still be represented as entangled . this will be investigated in more detail in our work . the general statements at which we arrive will be applied to some illustrative examples and to the well known phenomenon of quantum teleportation , an entanglement feature of great interest in quantum information theory . we will also describe and discuss the geometry of quantum physical states for the case of two qubits . we will close with our conclusions , an outlook on possible extensions of the results , and a list of references to the literature . + dipartimento di fisica , universit di pisa + largo pontecorvo 3 , 56127 pisa , italy in quantum systems , entanglement corresponds to nonclassical correlation of nonlocal observables . thus , entanglement ( or , to the contrary , separability ) of a given quantum state is not uniquely determined by properties of the state , but may depend on the choice of the factorization of the algebra of observables . in the present work , we expose and systematize some recently reported results about the possibility to represent a single quantum state as either entangled or separable . we will distinguish in particular the cases of pure and mixed states . for pure states , it has been shown that observables can always be constructed such that any state has any amount of entanglement possible . for mixed states , the situation is more complex and only partial results are known : while it is always possible to choose a factorization such that a state appears separable , a general criterion to determine whether a state can be represented as entangled is not known . these results will be illustrated by several examples , the phenomenon of quantum teleportation , and the geometry of the states of two qubits . 1.5 cm * * submitted to : + dipartimento di fisica , universit di pisa defended on : + 29 february 2012

0907.4587

i

the symmetry of the pair wavefunction provides important clues to the mechanism of high temperature superconductivity . phase sensitive experiments have shown that the dominant component of the order parameter exhibits @xmath8 symmetry . @xcite in cuprates of the yba@xmath0cu@xmath1o@xmath9-type with orthorhombic crystal symmetry , an @xmath7wave admixture resulting in an anisotropy of the superconducting ( sc ) energy gap , @xmath10 , between the @xmath5 and @xmath6axes is expected . a more general trend of increasing @xmath7-wave admixture with increasing doping level has also been reported . @xcite in yba@xmath0cu@xmath1o@xmath11 ( ybco@xmath11 ) , such an admixture has been confirmed by angle - resolved photoemission spectroscopy ( arpes ) @xcite and flux measurements on josephson junctions . @xcite however , the interpretation of the arpes data is complicated because the surface properties of ybco@xmath11 may differ from those of the bulk . @xcite the phase - sensitive measurements , on the other hand , have yielded only lower bounds ( 9% ) to the @xmath7-wave admixture . in addition , phenomenological calculations based on the two - dimensional hubbard model @xcite have demonstrated that the in - plane anisotropy of the spin fluctuation spectrum determined by inelastic neutron scattering on detwinned ybco @xcite is consistent with a small isotropic @xmath7-wave admixture to the @xmath8-pairing symmetry . however , the same calculations show that the dispersion and spectral weight of the spin fluctuations are also influenced by anisotropic hopping parameters that are difficult to determine independently . moreover , different interpretations of the neutron data have also been proposed @xcite . further experiments are therefore required to conclusively establish the magnitude of the @xmath7-wave contribution to the sc gap in ybco@xmath11 . inelastic light ( raman ) scattering is a powerful probe of electronic and lattice vibrational excitations in high - t@xmath12 superconductors . @xcite both electronic and phononic raman scattering have been applied to study the superconducting gap anisotropy in ybco@xmath11 . the anisotropy can be inferred from the energies of gap features of the electronic continuum in various polarization geometries , @xcite but the accuracy of this method is limited by overlap with raman - active phonons . possible manifestations of the gap anisotropy and an anisotropic electron - phonon coupling in the lineshape of a raman - active out - of - plane vibration of the in - plane oxygen atoms of @xmath13 symmetry have also been investigated . @xcite however , the analysis of these data has been contested , @xcite and a quantitative estimate of the gap anisotropy has not been extracted from them . the present work was in part motivated by a recent theoretical study that yielded quantitative predictions for the electronic raman continua in a @xmath14-wave superconductor . @xcite in order to enable a detailed comparison with these predictions , we have performed raman scattering measurements of the temperature dependence of the electronic continua and phonon modes in twin - free , slightly overdoped ybco@xmath2 and moderately overdoped y@xmath3ca@xmath4ba@xmath0cu@xmath1o@xmath2 ( henceforth ybco@xmath2:ca ) single crystals . we found that interference between the electronic and phononic scattering channels imposed severe limitations on our capability to extract information about the gap anisotropy from the electronic continuum alone , in accordance with previous work . @xcite we have therefore employed a phenomenological model that treats the electronic and phononic contributions to the raman signal on equal footing . this formalism yields predictions for the energy , intrinsic linewidth , and fano parameters of phonons coupled to the electronic continuum as a function of temperature . from a comparison to the experimentally determined lineshapes of two phonon modes , the @xmath13 vibration of the in - plane oxygen at 340 @xmath15 as well as an apical - oxygen vibration at 501 @xmath15 , we extract an upper bound of 20% on the @xmath7-wave admixture to the sc energy gap . this is consistent with a 10% to 15% admixture of @xmath7-wave to the @xmath8 pair - breaking peak observed in the electronic continuum in the superconducting state , and it coincides with the lower bound extracted from the phase - sensitive measurements @xcite . similar effects have been recently reported for an organic superconductor with the same point group as yba@xmath0cu@xmath1o@xmath11 ( @xmath16 ) . in this case , however , the analysis led to the conclusion that only an isotropic @xmath7-wave state is present . @xcite this paper is organized as follows . in sec . [ spectra ] we describe experimental details and discuss the raw raman spectra . an introduction to our theoretical model is given in sec . [ model ] . in particular , we focus on a simultaneous description of both , the electronic raman response and the phonon anomalies above and below @xmath17 . based on this description we are able to analyze the superconductivity - induced changes in the phonon lineshape in sec . [ phonons ] . a summary of our results and a comparison to prior work is contained in sec . [ conclusions ] .

the lineshapes of several phonon modes involving plane and apical oxygen vibrations exhibit pronounced anisotropies with respect to the incident and scattered light field configurations . based on a theoretical model that takes both electronic and phononic contributions to the raman spectra into account we argue that the raman spectra are consistent with anwave admixture with an upper limit of 20 percent .

inelastic light ( raman ) scattering has been used to study electronic excitations and phonon anomalies in detwinned , slightly overdoped ybacuo and moderately overdoped ycabacuo single crystals . in both samples modifications of the electronic pair - breaking peaks when interchanging the- and-axis were observed . the lineshapes of several phonon modes involving plane and apical oxygen vibrations exhibit pronounced anisotropies with respect to the incident and scattered light field configurations . based on a theoretical model that takes both electronic and phononic contributions to the raman spectra into account , we attribute the anisotropy of the superconductivity - induced changes in the phonon lineshapes to a smallwave admixture to the pair wave function . our theory allows us to disentangle the electronic raman signal from the phononic part and to identify corresponding interference terms . we argue that the raman spectra are consistent with anwave admixture with an upper limit of 20 percent .

astro-ph0110054

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the models used to account for the infrared magnitude - redshift relation of luminous radio galaxies are sensitive to the assumptions made concerning the evolution of their stellar content , as well as the choice of cosmological model . galaxies at a particular redshift are observed at different cosmic epochs in different cosmological models and so their predicted spectra at a given redshift can vary quite dramatically . accurate modelling of the evolution of galaxy spectra with cosmic time is required if the infrared k magnitude - redshift relation is to be used to constrain the range of acceptable world models . in this paper the spectral evolution of galaxies is modelled for different plausible cosmological models using the most recent spectral synthesis codes of bruzual and charlot ( 2001 ) , gissel2000 , and these results compared with the observational data . even for the simplest cases , numerous cosmological and evolutionary effects need to be taken into account and it is often not intuitively obvious how changes in the assumptions made change the predicted relations one of the objectives of this paper is to provide these insights . although our prime concern is the k - z relation for radio galaxies , the considerations of this paper are relevant to all such cosmological studies . the radio galaxies associated with the most powerful extragalactic radio sources seem to form a remarkably uniform class of galaxy out to large redshifts . following the early work of lilly and longair ( 1984 ) , best _ et al . _ ( 1998 ) carried out a much more detailed analysis of the photometric properties of a complete sample of 3cr radio galaxies , the brightest radio galaxies in the northern sky selected at 178 mhz ( laing _ et al . they modelled the k - z relation for world models with @xmath7 and @xmath8 , in both cases assuming @xmath9 . these models included passive evolution of the stellar populations of the galaxies with formation redshifts of 3 , 5 and 20 . the passive evolution models in the @xmath8 universe provided a reasonable fit to the observations , whereas those with @xmath10 gave a poor fit . the models with low redshifts of formation , @xmath11 , resulted in more luminous galaxies than those observed at @xmath12 , suggesting that the stellar populations of the galaxies must have formed at @xmath13 . eales and rawlings ( 1996 ) and eales _ et al . _ ( 1997 ) determined the k - z relation for a complete sample of 6c radio galaxies which were selected at 151 mhz and are about six times fainter in flux density than the 3cr sample ( eales 1985 ) . the 6c host galaxies have similar k - luminosities to those of the 3cr galaxies at @xmath14 , but are about 0.6 magnitudes fainter at larger redshifts . nevertheless , the high redshift 6c galaxies are significantly brighter than expected in world models in which no evolution of the stellar populations of the galaxies is assumed . the objectives of this paper are to review the current status of the k - z relation for 3cr and 6c luminous radio galaxies ( section 2 ) and then to carry out a more extensive analysis of the resulting k - z relation for a wider range of plausible world models than those considered by best _ ( 1998 ) and eales _ et al . _ ( 1997 ) , including models with a non - zero cosmological constant . the predicted evolution of the stellar populations of the galaxies is derived from the most recent spectral synthesis codes of bruzual & charlot ( 1993 , 2001 ) , gissel2000 ; a wide range of stellar synthesis models is considered , as well as the effects of changes in metallicity . the details of the different cosmological models , the spectral synthesis codes and procedures used to predict the k - z relations are described in section 3 . the results of these calculations are described in section 4 , indicating which parameters have the greatest impact upon the form of the predicted k - z relation . also included in this section are the results of varying the star formation history and metallicity in plausible ways . most massive galaxies at small redshifts have high metallicities and colour - magnitude relations show that more massive galaxies are redder in colour , suggesting a mass - metallicity correlation ( see , for example , arimoto & kodama , 1996 ) . on the other hand , at high redshifts , we might expect a lower metal content , as suggested by the analyses of pei _ et al . _ ( 1999 ) and pettini _ et al . _ ( 1997 ) . in section 5 the results of these analyses are discussed and our conclusions are summarised in section 6 . throughout the paper , @xmath15 km s@xmath16 mpc@xmath16 is adopted , unless otherwise stated .

infrared magnitude - redshift relations ( k - z relations ) for the 3cr and 6c samples of radio galaxies are presented for a wide range of plausible cosmological models , including those with non - zero cosmological constant . 1to 0pt#1 [ firstpage ] galaxies active galaxies radio galaxies galaxy evolution cosmological parameters .

infrared magnitude - redshift relations ( k - z relations ) for the 3cr and 6c samples of radio galaxies are presented for a wide range of plausible cosmological models , including those with non - zero cosmological constant . variations in the galaxy formation redshift , metallicity and star formation history are also considered . the results of the modelling are displayed in terms of magnitude differences between the models and no - evolution tracks , illustrating the amount of k - band evolution necessary to account for the observational data . given a number of plausible assumptions , the results of these analyses suggest that : ( i ) cosmologies which predict ( where denotes the current age of the universe ) can be excluded ; ( ii ) the star formation redshift should lie in the redshift interval , values towards the lower end of the range being preferred in cosmologies with larger values of ; ( iii ) the einstein - de sitter model ( ) provides a reasonable fit to the data ; ( iv ) models with finite values of can provide good agreement with the observations only if appropriate adjustments of other parameters such as the galaxy metallicities and star - formation histories are made . without such modifications , even after accounting for stellar evolution , the high redshift radio galaxies are more luminous ( that is , more massive ) than those nearby in models with finite , including the favoured model with . for cosmological models with larger values of , the conclusions are the same regardless of whether any adjustments are made or not . the implications of these results for cosmology and models of galaxy formation are discussed . # 1to 0pt#1 [ firstpage ] galaxies active galaxies radio galaxies galaxy evolution cosmological parameters .

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it is simplest to illustrate the effects of changing the parameters of the cosmological models by subtracting the predicted k - z track for the chosen cosmology from the corresponding predictions of the @xmath65 , @xmath66 model : @xmath67 is defined as @xmath68 so that negative values of @xmath67 mean that a galaxy of a given absolute luminosity would be observed to have a fainter apparent magnitude . the results for the cosmologies listed in table 1 are shown in fig . [ fig : 4 ] and tabulated in table 2 . it is instructive to distinguish those contributions to @xmath69 due to the choice of cosmological model and those due to the evolving spectrum of the galaxy . the latter can be thought of as the k - correction including the effects of the evolution of the stellar population . following the notation of longair ( 1998 ) , the flux density of the galaxy is given by @xmath70}{4\pi d_i^2(z)(1 + z)}\end{aligned}\ ] ] where @xmath71 labels the choice of cosmological model and @xmath72 is a distance measure which is related to the standard luminosity distance @xmath73 by @xmath74 . the luminosities of the modelled galaxies in the different cosmologies are normalised by varying their stellar mass , such that at zero redshift they have k - band luminosities @xmath75 matching the observed k - z relation . therefore if @xmath76 $ ] is the luminosity of the galaxy per solar mass , @xmath77 = l^{\prime}[\nu_0(1 + z),t_i(z ) ] \times \frac{l_{\rm{k(z=0)}}}{l^{\prime}[\nu_0,t_i(0)]}\ ] ] the ratio of the flux densities in the _ i_th model to that in the fiducial model , for the same observed flux density at small redshifts is given by @xmath78}{l^{\prime}[\nu_0 , t_i(0)]}\times \frac{l^{\prime}[\nu_0 , t_0(0)]}{l^{\prime}[\nu_0(1 + z ) , t_0(z ) ] } \times \frac{d_0 ^ 2(z)}{d_i^2(z)}\ ] ] thus , the magnitude difference associated with the choice of cosmological model is @xmath79,\ ] ] while that associated with observing different parts of the evolving spectral energy distribution at different cosmic times is @xmath80}{l^{\prime}[\nu_0 , t_i(0)]}\times \frac{l^{\prime}[\nu_0 , t_0(0)]}{l^{\prime}[\nu_0(1 + z ) , t_0(z)]}\right]\ ] ] the total change in magnitude is given by @xmath81 . table 2 shows these differences for the models listed in table 1 . cc@=lc@[email protected]@[email protected]@[email protected]@[email protected]@[email protected]@[email protected]@[email protected] & & & & & & & & + & & & & & & & & + @xmath82 & & & & & & & & + & @xmath83 & 0.00 & @xmath83 & 0 & 00 & @xmath83 & 0 & 00 & @xmath83 & 0 & 00 & @xmath83 & 0 & 00 & @xmath83 & 0 & 00 & @xmath83 & 0 & 00 & @xmath83 & 0 & 00 + & @xmath84 & 0.00 & @xmath84 & 0 & 00 & @xmath84 & 0 & 00 & @xmath84 & 0 & 00 & @xmath84 & 0 & 00 & @xmath84 & 0 & 00 & @xmath84 & 0 & 00 & @xmath84 & 0 & 00 + & @xmath85 & 0.00&@xmath85 & 0 & 00&@xmath85 & 0 & 00&@xmath85 & 0 & 00&@xmath85 & 0 & 00&@xmath85 & 0 & 00&@xmath85 & 0 & 00&@xmath85 & 0 & 00 + & @xmath86 & 0.00 & @xmath86 & 0 & 00 & @xmath86 & 0 & 00 & @xmath86 & 0 & 00 & @xmath86 & 0 & 00 & @xmath86 & 0 & 00 & @xmath86 & 0 & 00 & @xmath86 & 0 & 00 + + @xmath87 & & & & & & & & + & @xmath83 & 0.59 & @xmath83 & 0 & 69 & @xmath83 & 1 & 11 & @xmath83 & 0 & 75 & @xmath83 & 0 & 77 & @xmath83 & 0 & 72 & @xmath83 & 0 & 90 & @xmath83 & 0 & 77 + & @xmath84 & 0.00 & @xmath84 & @xmath88 & 34 & @xmath84 & @xmath88 & 47 & @xmath84 & @xmath88 & 54 & @xmath84 & @xmath88 & 60 & @xmath84 & @xmath88 & 44 & @xmath84 & @xmath88 & 92 & @xmath84 & @xmath88 & 59 + & @xmath85 & 0.00 & @xmath85 & @xmath88 & 13 & @xmath85 & 0 & 00 & @xmath85 & @xmath88 & 11 & @xmath85 & @xmath88 & 02 & @xmath85 & @xmath88 & 04 & @xmath89 & @xmath88 & 21 & @xmath85 & @xmath88 & 07 + & @xmath86 & 0.00 & @xmath86 & @xmath88 & 47 & @xmath86 & @xmath88 & 47 & @xmath86 & @xmath88 & 65 & @xmath86 & @xmath88 & 62 & @xmath86 & @xmath88 & 48 & @xmath86 & @xmath90 & 13 & @xmath86 & @xmath88 & 66 + + @xmath91 & & & & & & & & + & @xmath83 & 0.85 & @xmath83 & 1 & 10 & @xmath83 & 1 & 24 & @xmath83 & 1 & 33 & @xmath83 & 1 & 12 & @xmath83 & 1 & 15 & @xmath83 & 1 & 54 & @xmath83 & 1 & 32 + & @xmath84 & 0.00 & @xmath84 & @xmath88 & 59 & @xmath84 & @xmath88 & 84 & @xmath84 & @xmath88 & 99 & @xmath84 & @xmath88 & 78 & @xmath84 & @xmath88 & 67 & @xmath84 & @xmath90 & 30 & @xmath84 & @xmath88 & 97 + & @xmath85 & 0.00 & @xmath85 & @xmath88 & 29 & @xmath85 & @xmath88 & 13 & @xmath85 & @xmath88 & 39 & @xmath85 & @xmath88 & 10 & @xmath85 & @xmath88 & 12 & @xmath85 & @xmath88 & 40 & @xmath85 & @xmath88 & 21 + & @xmath86 & 0.00 & @xmath86 & @xmath88 & 88 & @xmath86 & @xmath88 & 97 & @xmath86 & @xmath90 & 38 & @xmath86 & @xmath88 & 88 & @xmath86 & @xmath88 & 79 & @xmath86 & @xmath90 & 70 & @xmath86 & @xmath90 & 18 + + @xmath92 & & & & & & & & + & @xmath83 & 1.00 & @xmath83 & 1 & 43 & @xmath83 & 1 & 68 & @xmath83 & 1 & 88 & @xmath83 & 1 & 49 & @xmath83 & 1 & 46 & @xmath83 & 1 & 98 & @xmath83 & 1 & 77 + & @xmath84 & 0.00 & @xmath84 & @xmath88 & 77 & @xmath84 & @xmath90 & 13 & @xmath84 & @xmath90 & 37 & @xmath84 & @xmath88 & 86 & @xmath84 & @xmath88 & 82 & @xmath84 & @xmath90 & 48 & @xmath84 & @xmath90 & 24 + & @xmath85 & 0.00 & @xmath85 & @xmath88 & 72 & @xmath85 & @xmath88 & 42 & @xmath85 & @xmath88 & 85 & @xmath85 & @xmath88 & 42 & @xmath85 & @xmath88 & 37 & @xmath85 & @xmath88 & 81 & @xmath85 & @xmath88 & 39 + & @xmath86 & 0.00 & @xmath86 & @xmath90 & 49 & @xmath86 & @xmath90 & 55 & @xmath86 & @xmath93 & 22 & @xmath86 & @xmath90 & 28 & @xmath86 & @xmath90 & 19 & @xmath86 & @xmath93 & 29 & @xmath86 & @xmath90 & 63 + fig . 4 shows that galaxies are expected to be fainter at large redshifts in all the models considered as compared with the model with @xmath8 , @xmath9 ; this means that for the same observed k - magnitude , the galaxies would have to be intrinsically more luminous . there are several effects which contribute to this decrease . the most important of these is the fact that , for a given redshift , galaxies are observed at greater radial co - moving coordinate distances as compared with the model with @xmath8 , @xmath94 . in addition , the cosmic times at which the spectral energy distributions are observed are significantly different . what is important , however , are the relative epochs at which the galaxies are observed . it turns out that , in the passively evolving models , the relative changes in observed luminosity ( @xmath95 ) are significantly less than the effects of the cosmological model ( @xmath96 ) . this can be understood from table 2 which shows the ages and distance coordinates at which galaxies at different redshifts are observed in the different cosmologies listed in table 1 assuming @xmath15 km s@xmath16 mpc@xmath16 ; included in this table are the contributions of @xmath97 , the total change in k - magnitude @xmath98 and the difference between them @xmath95 , which represents the contribution of the changing spectral energy distribution of the galaxies with cosmic epoch . table 2 demonstrates quantitatively the important point that , at all redshifts , the choice of cosmological model is always much more important than the evolution corrections within the context of passively evolving models for the stellar populations of the galaxies , so far as the _ differences _ from the fiducial model are concerned . it is interesting that it is possible to reproduce quite accurately the results of the exact computations by interpolating the luminosity changes at the appropriate redshifted wavelength in the template spectra shown in fig . it is found that this approach generally produces results accurate to about 0.1 mag , which is normally accurate enough to draw useful conclusions . the value of this approach is that it enables the investigator to use fig . 3(a ) to work out quite precisely the effects of the k - corrections for any other world model with any given value of hubble s constant and for alternative assumptions relative to the template spectral evolution shown in fig . poggianti ( 1997 ) provides tables of k - corections and evolutionary corrections for observations in a number of specific filters , assuming an @xmath99 cosmology . we have compared these results with those of our two most similar cosmologies , i.e. @xmath100 , @xmath9 and @xmath101 , @xmath9 . despite a different set of spectral synthesis models , the evolutionary corrections for an elliptical galaxy formed with an e - folding time of 1 gyr given by poggianti are in good general agreement with our own , suggesting that the choice of stellar synthesis code does not have a large impact on the evolutionary corrections determined . the k corrections given by poggianti for their single cosmology in the k waveband also agree well with our own results . considering first the range of models shown in fig . 4 , it is evident that only the passively evolving @xmath102 model provides a reasonable fit to the data . the error bars shown in the figure represent the standard errors in the mean of the k - magnitude of the galaxies , within the given redshift intervals , and are less than the standard deviations shown in fig . 1 by a factor of @xmath103 , where @xmath104 . for illustrative purposes , a quantitative comparison between four of the models plotted in fig . [ fig : 4 ] is given in table 3 . these are the models with @xmath105 ; @xmath106 ; @xmath107 ; and @xmath108 . in the first part of the table , the magnitude difference , @xmath67 , between the data points and the no - evolution tracks at redshifts @xmath109 and 2.0 are listed , and in the second the difference in @xmath67 between the data points and the passive evolution models of fig . [ fig : 4 ] are shown . these data are also included as the first two entries in table 4 . the k - z relations have been normalised to the same k - magnitude at low redshift , for both the passive evolution and the no - evolution tracks since the mean magnitudes of the galaxies at @xmath110 are consistent with a very well - defined linear k - z relation , as can be seen from fig . 1 . positive values of @xmath67 indicate that the predicted magnitudes correspond to brighter galaxies than are needed to account for the data . for example , the @xmath111 model results in slightly brighter tracks than the mean of the data points at @xmath112 . the importance of these results is that , in general , except for the fiducial model , @xmath102 , the galaxies are predicted to be fainter at large redshifts than they are observed to be , even when passive evolution is taken into account . this is a somewhat unexpected result . first of all , according to the hierarchical clustering model , galaxies at large redshifts should be less luminous than those nearby since they contain fewer stars . the present results go in entirely the wrong direction . for the most extreme @xmath4 models the galaxies would need to contain about ten times more stars to account for these differences . second , it might be argued that this result would be found if there were a correlation between radio luminosity and the k - luminosity of the galaxy . in this case , the galaxies at large redshift would be expected to be larger than those at small redshifts . the analysis of mclure and dunlop ( 2000 ) showed , however , that the half - width diameters of the luminosity profiles of the radio galaxies are invariant as a function of redshift , suggesting that the galaxies have attained a similar evolutionary state so far as their total stellar populations are concerned . because of these somewhat surprising conclusions , the impact of changing the assumptions made in the fiducial model in plausible ways has been analysed in more detail . ccr@ @xmath113 lr@ @xmath113 lr@ @xmath113 l cosmological & sample & + model & & & & + @xmath114 , @xmath115 & 3cr data & 0.597 & 0.140 & 1.022 & 0.126 & 1.780 & 0.114 + & 6c data & 0.706 & 0.097 & 0.433 & 0.145 & 1.136 & 0.233 + & combined data & 0.646 & 0.089 & 0.742 & 0.115 & 1.222 & 0.210 + @xmath116 & combined data & 0.778 & 0.091 & 1.110 & 0.118 & 1.945 & 0.220 + @xmath117 & combined data & 0.712 & 0.091 & 1.076 & 0.119 & 1.925 & 0.220 + @xmath118 & combined data & 0.789 & 0.091 & 1.101 & 0.117 & 1.758 & 0.214 + & model & + @xmath119 & standard initial starburst & -0.177 & 0.089 & 0.162 & 0.115 & 0.336 & 0.210 + @xmath116 & standard initial starburst & -0.303 & 0.091 & -0.262 & 0.118 & -0.696 & 0.220 + @xmath117 & standard initial starburst & -0.352 & 0.091 & -0.310 & 0.119 & -0.799 & 0.220 + @xmath118 & standard initial starburst & -0.420 & 0.091 & -0.295 & 0.117 & -0.454 & 0.214 + lcccc section & & & @xmath67 from the observational data : & + & & z=0.5&z=1.0&z=2.0 + & standard passive evolution model : & -0.18 & 0.16 & 0.34 + & @xmath65 , @xmath120 , @xmath121 , @xmath122 & & & + & 1gyr starburst , @xmath123 . & & & + & change from standard passive evolution model & + & & z=0.5 & z=1.0 & z=2.0 + ( 4.1 ) & @xmath47 , @xmath120 & -0.13 & -0.42 & -1.03 + & @xmath47 , @xmath124 & -0.18 & -0.47 & -1.14 + & @xmath125 , @xmath126 & -0.24 & -0.46 & -0.79 + + ( 4.2 ) & @xmath127=2 & 0.10 & 0.27 & - + & @xmath127=3 & -0.00 & 0.10 & 0.57 + & @xmath127=5 & -0.02 & 0.04 & 0.14 + & @xmath127=10 & 0.00 & 0.00 & 0.00 + & @xmath127=20 & -0.04 & -0.05 & -0.14 + & @xmath127=100 & 0.00 & -0.02 & -0.12 + + ( 4.3 ) & @xmath128 & 0.00 & 0.00 & 0.00 + & @xmath129 & -0.02 & 0.04 & -0.01 + & @xmath130&-0.01 & -0.05 & 0.12 + + ( 4.4 ) & @xmath131 & 0.05 & 0.22 & 0.45 + & @xmath132 & 0.09 & 0.10 & 0.32 + & @xmath133 & 0.00 & 0.00 & 0.00 + & @xmath134 & 0.13 & -0.04 & -0.16 + + ( 4.5)&instantaneous sfr & -0.04 & -0.09 & -0.21 + & 1gyr constant rate & 0.00 & 0.00 & 0.00 + & exponential sfr & 0.05 & 0.07 & 0.05 + + ( 4.5)&extra starburst , 2.5% mass , 10@xmath135years old & 0.04 & 0.07 & 0.17 + & extra starburst , 2.5% mass , 10@xmath136years old & 0.32 & 0.28 & 0.22 + & extra starburst , 2.5% mass , 10@xmath137years old & 0.04 & 0.04 & 0.05 + + ( 4.5)&extra starburst , 10@xmath136years old , 0.5% mass & 0.07 & 0.06 & 0.05 + & extra starburst , 10@xmath136years old , 1.0% mass & 0.14 & 0.12 & 0.09 + & extra starburst , 10@xmath136years old , 2.5% mass & 0.32 & 0.28 & 0.22 + & extra starburst , 10@xmath136years old , 5.0% mass & 0.56 & 0.50 & 0.40 + the epoch at which the stellar populations were formed was allowed to vary between redshifts of 2 and 100 . the sense of the changes to the k - z relations are similar for all cosmological models . fig . [ fig : 5 ] shows the k - band magnitude differences of the models relative to the no - evolution track in the same cosmology for models with @xmath102 and @xmath138 . the magnitude differences relative to the fiducial model are listed in table 4 for @xmath102 . results have not been tabulated for models with @xmath138 , as the magnitude differences within these models were not dissimilar to those in the einstein de sitter cosmology . in fig . [ fig : 5 ] , the values of @xmath67 are plotted against cosmic time as a fraction of @xmath2 , where @xmath2 is the present age of the universe listed in table 1 for @xmath139 . in fig . [ fig : 5 ] , it is assumed that the galaxies have solar metallicity . all models have been normalised to the same magnitude at the lowest redshifts , at which the mean k - z relation is well defined . the models diverge at earlier times . as expected , if the epoch of star formation is less than the reference value @xmath140 the predicted tracks are more luminous at redshifts @xmath141 . the largest differences occur in the cases in which the star formation is assumed to begin at @xmath142 and 3 , these being less than that at which radio galaxies are known to exist . nonetheless , it is noteworthy that , unless the epoch of star formation were as unreasonably low as @xmath143 , changing the epoch of star formation does not result in a particularly large change in @xmath67 . the reason for this can be understood from fig . 3(a ) . by adopting 2.2 @xmath53 m as the wavelength at which the magnitude - redshift relation is determined , even at redshift @xmath92 , the changes in the luminosity of the galaxies at a rest wavelength of 500 nm are quite small and change monotonically with time indeed , the difference between observing the starburst at ages 0.32 to @xmath144 years amounts to less than a factor of three at 5000@xmath145 in the rest frame . this behaviour contrasts dramatically with what would be expected if the redshift - magnitude relation were determined in , say , the r waveband . then , fig . 3(a ) shows that the variations in luminosity at rest frame wavelengths of 1500@xmath145 would be very sensitive to even small amounts of ongoing star formation . these calculations illustrate the considerable advantages of carrying out studies of the redshift magnitude relation in the k - waveband the observations provide information more directly about the bulk star formation activity of the galaxies . as expected , as @xmath127 increases the predicted k - z relations become fainter at large redshifts and there is little difference between the tracks in both models if @xmath127 is greater than 10 . in the case of the model @xmath146 , the fit to the data can be improved considerably if the star formation redshift @xmath147 is adopted . values of @xmath67 for the examples given in fig . 5(a ) at redshifts of 0.5 , 1.0 and 2.0 are included in table 4 . intuitively , it would seem that the choice of hubble s constant would have a significant impact upon the form of the k - z relation since the cosmic time - scale is significantly different if , say , @xmath148 were are large as @xmath149 . then , all cosmic time - scales would be reduced by a factor of two , allowing much less time for the evolution of the stellar populations of the galaxies and also observing them at very much earlier times at large redshifts . these changes , as well as those for @xmath150 , are given in table 4 . in the cases of models with larger values for @xmath148 , an initial starburst of very short duration was adopted so that the galaxy is observed during the passive evolution phase ; the tabulated values in table 4 have included this change in star formation history . it is noteworthy that changing the value of hubble s constant has a remarkably small effect upon the predicted evolutionary tracks . the reasons for this can be understood from inspection of fig . as was emphasised in section 3.2 , so long as the analysis is restricted to redshifts less than 3 , the spectral evolution of the stellar energy distribution is monotonic with time and can be reasonably described by power - laws at different wavelengths from 0.5 to 2.2 @xmath53 m . thus , if the value of hubble s constant is changed , the stellar mass of the galaxies required to normalise the k - z relation at low redshift is changed , but the differences in the values of @xmath151 depend primarily upon the changing _ relative _ cosmic times observed at different redshifts . in general , these are quite small and so changing the value of @xmath148 does not greatly change the evolutionary corrections . if the redshift magnitude relation were determined at rest - frame uv wavelengths , this would certainly not be the case . the metallicity of the galaxies can significantly influence their evolutionary tracks . rather than attempting to adopt some arbitrary chemical history , the evolutionary codes have been run for metallicities of @xmath152 , @xmath153 , @xmath60 and @xmath154 . the results are displayed in fig . [ fig : 6 ] for the same two cosmologies discussed in section 4.2 and for @xmath155 . for the einstein de sitter cosmology , these results are also tabulated in table 4 . all four k - z tracks are normalised to the same magnitude scale as the fiducial model , rather than to the same magnitude at the present epoch . this procedure has been adopted so that at all times the variations due to changes in metallicity are clearly displayed . the sense of the changes is that , the lower the metallicity , the more luminous the galaxy at early epochs . it is apparent from fig . [ fig : 6](b ) , that assuming a lower metallicity at large redshifts improves the fit of the @xmath108 model as compared with the fiducial model . the lowest metallicity track is initially the brightest ( largest @xmath67 ) , and the @xmath154 track the faintest , but the tracks cross over at low redshift . the greatest variation between the tracks occurs at early epochs and , as we have seen from section 3.2 , it is at these young ages that the galaxies undergo the greatest amount of evolution . the early evolution of the spectral energy distribution is highly sensitive to the metallicity of the galaxy because , for a star of a given mass , changing the metallicity leads to a change in the age at which it moves from the main sequence onto the agb ; for lower values of @xmath156 the stars evolve more rapidly since there is less line blanketing and lower opacity . this has two consequences : firstly , the more rapid evolution means that lower metallicity stars were more luminous at early times ; secondly , the stars were also shorter lived and at later times there are fewer stars present in the older galaxies , which therefore appear fainter than a higher metallicity galaxy of a similar age . it is interesting that the divergence of the different metallicity tracks occurs when the universe was roughly half its present age , corresponding to the redshift at which the k - z relations for the 3cr and 6c samples begin to diverge . it is just conceivable that such an effect could account for the difference between these samples , but it would require those 6c galaxies which are less luminous than the 3cr galaxies to have higher metallicities . there is no obvious reason why greater radio luminosity should be correlated with lower metallicity of the stellar populations of the radio galaxies . the diagrams in fig . [ fig : 6 ] can be used to construct approximate evolving galaxy models in which the metallicities of the galaxies increase with cosmic epoch . such evolution tracks can be found by changing from one locus to another as the metallicity increases . the range of models shown in fig . [ fig : 6 ] illustrates how such changes would alter the predicted k - z relations . the results so far have involved an initial starburst during which the star formation rate has been assumed to be constant . it is important to consider also models in which there is ongoing star formation , and in particular those in which the occurrence of a radio source event might be associated with a major star formation event in the host galaxy . this could occur if , for example , the central black hole were fuelled following some form of galaxy merger . to understand the significance of these results , we need to look more closely at the origin of the k - band flux density . the k - band luminosity of evolving galaxies arises from two stellar populations . stars with mass m@xmath1572m@xmath29 evolve beyond core and shell helium burning to become asymptotic giant stars ; those with m@xmath1582m@xmath29 become red giant stars forming the old red giant branch seen , for example , in globular clusters . for the present study , we are mainly interested in the evolution of the overall stellar population of the galaxies over cosmic time - scales greater than @xmath50years . this age corresponds to the main sequence lifetime of a star of mass 2@xmath159 . therefore , at ages greater than @xmath50years , the evolution is dominated by stars evolving onto the standard red giant branch and the arguments given in section 3.2 can account for the decreasing luminosity of the galaxy . on the other hand , if there are epochs of rapid star formation , the bulk of the near - infrared luminosity is associated with stars on the asymptotic giant branch . for the most luminous stars , evolution onto the asymptotic giant branch takes place after a few times @xmath62years . figure [ fig : 3](b ) shows dramatically the large increase in k - luminosity of an instantaneous starburst when it is about @xmath63years old associated with the formation of agb stars . on the other hand , if the starburst is of longer duration , say @xmath64years as illustrated in figure [ fig : 3](c ) , the effect is somewhat diluted by the continued formation of all types of massive stars . considering first variations in the long - term star formation history of the galaxy populations , fig . [ fig : 7 ] displays three different types of star formation history : ( i ) the standard 1-gyr starburst , ( ii ) an instantaneous starburst of negligibly short duration , and ( iii ) a model in which the star formation rate decays exponentially with an e - folding time of @xmath50 years . there is very little difference between the three models , all three of which have been normalised to the same stellar mass at late cosmic times . the brightest model ( largest @xmath67 ) except at the very earliest times is the exponential star formation model , and the faintest the instantaneous burst model . these results are not surprising since , it is only the behaviour after @xmath50years which is relevant to the k - z relation for @xmath160 . once star formation ceases , the continued star formation of the exponential model , and sustained production of agb stars ( albeit in small numbers ) , is sufficient to make this model the brightest track at low redshift . the most evolved model , the instantaneous starburst , has the lowest luminosity at low redshift . these differences between the models are very small , as can be seen from the values given in table 4 . [ fig : 8 ] shows the standard model coupled with the seds of instantaneous starbursts at three different ages ( @xmath62 , @xmath63 and @xmath64 years old ) , the additional starburst accounting for 2.5% of the total stellar mass of the galaxy . the model with the additional burst occurring the longest time before observations , @xmath64 years , gives the least change from the predictions of the underlying model . similarly , the @xmath62 year old recent burst also results in only a slight difference at low redshift . in the first case , the additional young ( @xmath64 years old ) high mass stars have evolved beyond the agb , and in the second case ( @xmath62 years old ) they have not yet had enough time for their evolution to reach that stage . [ fig : 3 ] shows that the rest frame k - band flux of an evolved instantaneous burst at these ages is much closer to that for the underlying star formation model than the @xmath63 year old track . at high redshift , however , we need to consider a somewhat bluer rest - frame wavelength region . whilst there is a small difference for the @xmath64 year old model , the @xmath62 year old recent burst is slightly brighter still , which is seen in fig . [ fig : 8 ] . the model with the @xmath63 year old recent burst component shows the greatest change from the standard model , as expected from fig . [ fig : 3 ] . the @xmath67 values for these models given in table 4 are solely due to the additional recent burst component . the additional burst consistently brightens the k - z tracks at late times . the reason for the effect being most pronounced at small redshifts is that the luminosity of the starburst is assumed to be independent of redshift . since the k - luminosity increases in roughly power - law fashion with increasing redshift , the luminosity of the starburst is soon swamped by the majority population of red giant branch stars . [ fig : 9 ] displays the results for the same set of models as in fig . [ fig : 8 ] , but the mass fraction of the additional starburst is varied rather than its age . the tracks in this figure involve an additional @xmath63 year starburst and four mass fractions . if the mass of the starburst as a function of redshift remains constant , the overall shape of the k - z relation is flattened at late times and does not result in an improved fit to the observational data . some scatter may be introduced into the k - z relation , because the radio galaxies are observed at a range of ages from @xmath62 to @xmath64 years . in order for starbursts to change significantly the shapes of the k - z relations , the additional starburst mass would need to change with redshift . this is quite possible , as there is more gas available in any merger - induced starburst at high redshift . the same would be true if the starburst were radio jet induced ( e.g. rees 1989 ) , which provides a possible explanation of the alignment effect ( mccarthy _ et al . _ 1987 , chambers _ 1987 ) . figs . [ fig : 8 ] and 9 can be used to estimate how significantly the k - band luminosity could have changed if the magnitude of the starburst were to vary with cosmic epoch . just as in the case of evolving metallicity , so in the case of evolving starbursts , the evolutionary history can be estimated by moving from one track to another in figs . [ fig : 8 ] and 9 . for example , in fig . [ fig : 9 ] , if the early starbursts involved 5% of the galaxy mass , the evolutionary track would begin on the dotted line ; if this subsequently decreased to 2.5% and 1% , the track would change to the dot - dashed and dashed lines respectively . a 5% starburst by mass is the absolute maximum allowable given the constraints of the observed ultraviolet luminosities for the 3cr radio galaxies at @xmath38 ( e.g. best _ et al . _ 1998 ) . in this extreme case the magnitude increase in k is at most about 0.5 magnitudes .

variations in the galaxy formation redshift , metallicity and star formation history are also considered . the results of the modelling are displayed in terms of magnitude differences between the models and no - evolution tracks , illustrating the amount of k - band evolution necessary to account for the observational data . the implications of these results for cosmology and models of galaxy formation are discussed . #

infrared magnitude - redshift relations ( k - z relations ) for the 3cr and 6c samples of radio galaxies are presented for a wide range of plausible cosmological models , including those with non - zero cosmological constant . variations in the galaxy formation redshift , metallicity and star formation history are also considered . the results of the modelling are displayed in terms of magnitude differences between the models and no - evolution tracks , illustrating the amount of k - band evolution necessary to account for the observational data . given a number of plausible assumptions , the results of these analyses suggest that : ( i ) cosmologies which predict ( where denotes the current age of the universe ) can be excluded ; ( ii ) the star formation redshift should lie in the redshift interval , values towards the lower end of the range being preferred in cosmologies with larger values of ; ( iii ) the einstein - de sitter model ( ) provides a reasonable fit to the data ; ( iv ) models with finite values of can provide good agreement with the observations only if appropriate adjustments of other parameters such as the galaxy metallicities and star - formation histories are made . without such modifications , even after accounting for stellar evolution , the high redshift radio galaxies are more luminous ( that is , more massive ) than those nearby in models with finite , including the favoured model with . for cosmological models with larger values of , the conclusions are the same regardless of whether any adjustments are made or not . the implications of these results for cosmology and models of galaxy formation are discussed . # 1to 0pt#1 [ firstpage ] galaxies active galaxies radio galaxies galaxy evolution cosmological parameters .

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white dwarfs can be sources of soft ( @xmath3 0.4 kev ) x - ray emission if their atmospheres have high temperatures and low opacities : @xmath4 23,000 k for da white dwarfs with pure hydrogen atmospheres @xcite , 23,000 k @xmath5 54,000 k for da white dwarfs containing significant quantities of heavy elements in their atmospheres @xcite , and @xmath4 100,000 k for do and pg 1159 white dwarfs with helium - rich atmospheres @xcite . @xmath1 observations of such x - ray sources show soft spectra rising toward @xmath1 s low - energy limit at 0.1 kev . the do white dwarf kpd 0005 + 5106 (= wd0005 + 511 ) , with its spectrum peaking at 0.2 kev , appears to pose an exception and has been interpreted as having cool ( @xmath6 k ) coronal emission @xcite . _ no hard x - ray ( @xmath2 0.5 kev ) emission is expected from single white dwarfs . _ it is thus puzzling that wd2226@xmath0210210 by @xcite . ] , the 103,600 k dao white dwarf @xcite in the helix nebula , appears to be single @xcite , but has hard x - ray emission . @xmath1 observations of wd2226@xmath0210 have detected not only a soft spectral component as expected from the white dwarf s photosphere , but also a hard spectral component peaking at 0.8 kev @xcite . recent @xmath7 observations have confirmed that the hard x - ray emission is unresolved and coincident with the white dwarf ; however , the luminosity and variability of the hard x - ray emission is similar to that of a dme star @xcite . follow - up spectroscopic observations by @xcite detected variability in the h@xmath8 line profile of wd2226@xmath0210 , suggesting the presence of a companion . it is thus possible that wd2226@xmath0210 has an x - ray - emitting dme companion which is too faint and close to the white dwarf to be detected at visible wavelengths . based on the @xmath9 magnitude of wd2226@xmath0210 reported by @xcite and the @xmath10 , @xmath11 , and @xmath12 magnitudes from the 2mass survey ( see table 3 below ) , we estimate that the hypothetical companion must have a spectral type later than m5 - 6v . hard x - ray emission from white dwarfs may be used to infer the presence of a binary companion . indeed , @xcite found 9 da white dwarfs with hard x - ray emission using the @xmath1 all - sky survey , and all of these 9 white dwarfs have late - type ( f , g , k , and m ) companions . this has motivated us to search for other white dwarfs which exhibit hard x - ray emission . we have found that 94 white dwarfs cataloged by @xcite appear coincident with @xmath1 x - ray point sources in the wga catalog ( * ? ? ? * hereafter wgacat ) . to confirm the positional coincidence , we have downloaded the @xmath1 data , and compared these x - ray images with optical images of the white dwarfs . we have further extracted the @xmath1 spectra to examine whether hard x - ray emission is present . for white dwarfs associated with hard x - ray emission , we have used the literature and near - ir photometry to assess the existence of binary companions , and further investigate the origin of hard x - ray emission from the apparently single white dwarfs . this paper reports the results of our study . the sample of white dwarfs with hard x - ray emission and our method of analysis are described in section 2 , the binarity status of the white dwarfs with hard x - ray emission is reported in section 3 , and the origin and implications of the hard x - ray emission associated with white dwarfs is discussed in section 4 . a summary is given in section 5 .

twelve of these white dwarfs with hard x - ray emission are in known binary systems , in two of which the accretion of the close companion s material onto the white dwarf produces hard x - ray emission , and in the other ten of which the late - type companions coronal activity emits hard x - rays .

inspired by the hard x - ray emission from wd2226 , the central star of the helix nebula , we have made a systematic search for similar sources by correlating the white dwarf catalog of and the pspc point source catalog of . we find 76 white dwarfs coincident with x - ray sources at a high level of confidence . among these sources , 17 show significant hard x - ray emission at energies 0.5 kev . twelve of these white dwarfs with hard x - ray emission are in known binary systems , in two of which the accretion of the close companion s material onto the white dwarf produces hard x - ray emission , and in the other ten of which the late - type companions coronal activity emits hard x - rays . one apparently single white dwarf is projected near an agn which is responsible for the hard x - ray emission . the remaining four white dwarfs and two additional white dwarfs with hard x - ray emission appear single . the lack of near - ir excess from the apparently single white dwarfs suggests that either x - ray observations are more effective than near - ir photometry in diagnosing faint companions or a different emission mechanism is needed . it is intriguing that 50% of the six apparently single white dwarfs with hard x - ray emission are among the hottest white dwarfs . we have compared x - ray properties of 11 hot white dwarfs with different spectral types , and conclude that stellar pulsation and fast stellar winds are not likely the origin of the hard x - ray emission , but a leakage of the high - energy wien tail of emission from deep in the stellar atmosphere remains a tantalizing source of hard x - ray emission from hot do and dqzo white dwarfs . a complete survey using the entire pspc archive is needed to enlarge the sample of white dwarfs with hard x - ray emission . follow - up near - ir photometric observations are needed to verify the existence of late - type companions and high - resolution deep x - ray observations are needed to verify the positional coincidence and to study the x - ray spectral properties in order to determine the origin and nature of the hard x - ray emission .

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x - ray emission from white dwarfs is expected to be soft , but roughly 20% of x - ray sources associated with white dwarfs exhibit a hard x - ray component ( @xmath20.5 kev ) . what is the origin of hard x - ray emission associated with white dwarfs ? we initially speculated that all white dwarfs with hard x - ray emission possess binary companions , but our literature search reveals that 12 of the 17 white dwarfs with hard x - ray emission are known to be in binary or multiple systems , one is superposed by chance near a background agn , and the remaining four are apparently single white dwarfs . below we argue that the binary companions are indeed directly or indirectly responsible for the hard x - ray emission associated with white dwarfs in binary systems , use 2mass photometry to assess the existence of late - type companions , and discuss the hard x - ray emission from the hottest apparently single white dwarfs . two of our 12 white dwarfs in binary systems are known to accrete material from their companions : mira and the cataclysmic variable v3885 sgr . the accretion of the companion s material onto the surface of the white dwarf produces hard x - ray emission @xcite . of the ten non - accreting systems , six have one or two dme companions , two have a k2v companion , one has an f5iv - v companion , and one has a g5iii companion . these late - type companions either are known for their active coronae or have emission lines indicating coronal activity ; therefore , these companions are most likely responsible for the observed hard x - ray emission . in some cases , the origin of the hard x - ray emission has been unambiguously established to be the late - type companions by either positional coincidence ( e.g. , wd1633 + 572 ) or observations of eclipses ( e.g. , wd0347 + 171 ) . to confirm that these late - type companions are the source of the hard x - ray emission , we compare the hard x - ray luminosities and plasma temperatures of the binary systems with those expected from single late - type stars . we have fitted thin plasma emission models @xcite to the 0.52.4 kev portion of the spectra for the objects with sufficient counts in this energy band . the resultant plasma temperature @xmath26 and x - ray luminosity @xmath27 in the 0.52.4 kev band are given in table 2 . the @xmath27 of white dwarfs with late - type companions are completely consistent with those seen in dm and k stars , 10@xmath2810@xmath29 ergs s@xmath30 @xcite . the best - fit plasma temperatures are also in the range for stellar coronae . we further assess whether coronal activities of the late - type companions may be induced by binary interaction by correlating the projected binary separation , @xmath22 , with the x - ray luminosity ( see table 2 ) . no clear correlation is seen . this lack of correlation may be caused by the small number of binary systems in our sample and the wide range of physical parameters involved . the cataclysmic variable wd1944@xmath0421 ( v3885 sgr ) has the highest hard x - ray luminosity , which is also in accord with the expectation of accretion from a roche - lobe - filled companion . thus , the hard x - ray emission from white dwarfs in known binary systems can be explained by the presence of their companions . the remaining four white dwarfs associated with hard x - ray emission appear to be single . if these white dwarfs contain previously - unknown , late - type companions , near - ir photometry may reveal these companions ( e.g. , * ? ? ? we have obtained @xmath31 photometric measurements of white dwarfs with hard x - rays available in the second incremental data release of the two micron all sky survey ( 2mass ) , and listed them in table 3 , along with optical photometry from @xcite . in the case of wd1633 + 572 where the white dwarf and the binary dme companions are resolved by 2mass , separate entries are given . to enlarge the sample for comparison , we have added two additional white dwarfs with hard x - ray emission that are not from our survey : wd0005 + 511 (= kpd 0005 + 5106 ; its hard x - ray emission will be discussed in 4.3 ) and wd2226@xmath0210 (= the central star of the helix nebula ; its hard x - ray emission is described in 1 ) . clear near - ir excess is observed in the four known binary systems , but not in the apparently single white dwarfs . the lack of near - ir excess , in conjunction with known distances , places constraints on the possible companions of the white dwarfs . the constraint is more stringent for white dwarfs at smaller distances because it would be harder to hide a companion . the nearest apparently single white dwarf with hard x - ray emission is wd1134 + 300 at 15.3 pc . we estimate that its near - ir magnitudes can hide only a brown dwarf companion more than 3 mag fainter than an m7v star . therefore , wd1134 + 300 has no stellar - mass companions , and the agn projected within the pspc psf @xcite is solely responsible for the hard x - ray emission observed . two other white dwarfs with hard x - ray emission have known distances , wd1234 + 481 and wd2226@xmath0210 . their lack of near - ir excess indicates that wd1234 + 481 can hide a companion of spectral type m7v or later , and wd2226@xmath0210 later than m5 - 6v . if these two apparently single white dwarfs indeed have faint , late - type companions , then the hard x - ray emission detected by @xmath1 is more effective than the near - ir excess detected by 2mass in diagnosing the existence of a faint late - type companion . two of the apparently single white dwarfs associated with hard x - rays from our survey are among the hottest known : wd1159@xmath0034 and wd2226@xmath0210 . to compare these two hot white dwarfs to the one that has been suggested to possess a corona , kpd 0005 + 5106 , we have retrieved an archival @xmath1 pspc pointed observation ( rf200428n00 ) that was made with the boron filter for an exposure time of 5 ks . as shown in figure 3 , the pspc spectrum of kpd0005 + 5106 shows not only the soft atmospheric emission below 0.5 kev , but also hard x - ray emission near 1 kev . this hard x - ray emission was not detected previously by @xcite using the @xmath1 all - sky survey observation of exposure time 504 s. as the 5 ks pointed observation detected only 24@xmath326 counts in the 0.52.4 kev band , if the hard x - ray flux is constant , we expect only [email protected] hard x - ray counts in a 504 s exposure . the non - detection of the hard component by @xcite thus does not imply a temporal variation . the presence of hard x - ray emission from kpd0005 + 5106 requires a plasma temperature at least 10@xmath33 k , much higher than that suggested by @xcite , and may be responsible for photoionizing and producing the recombination lines observed @xcite . we have examined the 2mass @xmath31 photometric data for kpd 0005 + 5106 ( see table 3 ) , and find no evidence for a near - ir excess . thus , kpd 0005 + 5106 is another apparently single hot white dwarf with hard x - ray emission . while our statistical sample is extremely limited , it is intriguing that @xmath1450% of the apparently single white dwarfs with hard x - ray emission are among the hottest known white dwarfs : 120,000 k for wd0005 + 511 ( = kpd 0005 + 5106 ; * ? ? ? * ) , 140,000 k for wd1159@xmath0034 (= pg 1159@xmath0034 ; * ? ? ? * ) , and 103,600 k for wd2226@xmath0210 (= cspn of the helix ; * ? ? ? is it possible that the hard x - ray emission is the high - energy wien tail of the blackbody emission from deep in the stellar atmosphere ? it is beyond the scope of this paper to model the atmospheres of these white dwarfs and answer this question theoretically . instead , we will examine x - ray observations of the hottest white dwarfs to search for trends in their hard x - ray properties in order to gain insight into the origin of their hard x - ray emission . we first examine x - ray properties of pg 1159 stars . many pointed @xmath1 pspc observations of pg 1159 stars were made with the boron filter . these x - ray sources will be absent in the wgacat , and it is necessary to search the @xmath1 archive for pointed and serendipitous observations of pg 1159 stars . pg 1159 stars are divided into two groups : with and without planetary nebulae ( pns ) . @xmath1 observations of cataloged pns have been analyzed and reported by @xcite ; pspc observations of five pns with pg 1159 central stars are available . of these five , wd0726 + 133 ( in abell 21 ; pg 1159 ) and wd2333 + 301 ( in jn1 ; doz.3 ) are not detected ; wd2117 + 342j ( in mwp1 ; do ) is detected at a 40@xmath13 off - axis position in the pspc field - of - view ( this paper ) , but the poor psf does not allow us to assess accurately whether faint hard x - ray emission exists ; wd0044@xmath0121 ( in ngc246 ; pg 1159 ) and wd1821 + 643 ( in k1 - 16 ; doz.4 ) are centered in pointed pspc observations , but only soft ( @xmath30.5 kev ) x - ray emission from the white dwarf is detected . four pg 1159 stars without pns have pointed pspc observations : wd0122@xmath0753j (= rxj0122@xmath07521 ; do ) , wd1144 + 004 (= pg 1144 + 005 ; dqzo1 ) , wd1159@xmath0034 (= pg 1159@xmath0034 ; dqzo.4 ) , and wd1501 + 664 (= pg 1501 + 661 ; dz1 ) ; all four have been included in our survey listed in table 1 . soft x - ray emission is detected in these four pg 1159 stars without pns , but 3-@xmath16 detection of hard x - rays is obtained only for pg 1159 itself . we have further combed table 1 for hot white dwarfs similar to wd0005 + 511 or wd2226@xmath0210 , and find a hot do white dwarf , wd1522 + 662 , and a hot dao white dwarf , wd1957 + 225 at the center of the dumbbell nebula . these four hot do and dao white dwarfs and the above nine pg 1159 white dwarfs will be discuss below in more detail . the spectral type , stellar effective temperature , visual magnitude , and @xmath1 observations of these 13 hot white dwarfs are summarized in table 4 . to illustrate the spectral properties of the 11 hot white dwarfs whose x - ray emission has been detected by @xmath1 pspc observations , in figure 4 we present their soft x - ray images in the 0.10.4 kev band and the hard x - ray images in the 0.62.4 kev band . it is evident that hard x - ray emission is clearly detected only from wd0005 + 511 , wd1159@xmath0034 , and wd2226@xmath0210 , as we have concluded earlier , and possibly detected at a 2@xmath16 level from wd0122@xmath0753j . the spectral types of these four white dwarfs are do , dqzo.4 , dao , and do , respectively . below we divide the 11 hot white dwarfs detected in x - rays according to their spectral types , discuss the implications of their hard x - ray properties , and speculate on the possibility of a photospheric origin of the hard x - ray emission . * _ dao ( wd1957 + 225 and wd2226@xmath0210 ) : _ + hard x - ray emission is detected from wd2226@xmath0210 in the helix nebula , but not from wd1957 + 225 in the dumbbell nebula . as these two dao white dwarfs have similar effective temperatures , and as wd1957 + 225 is only 0.8 mag fainter but has a four times longer exposure time , the lack of hard x - ray emission from wd1957 + 225 signifies a real difference from wd2226@xmath0210 . furthermore , wd2226@xmath0210 shows temporal variations of its hard x - ray emission and h@xmath8 line profile , suggesting the existence of a late dme companion @xcite . therefore , there is no evidence indicating that deep photospheric emission is the origin of hard x - rays associated with dao white dwarfs . * _ do ( wd0005 + 511 , wd0122@xmath0753j , wd1522 + 662 , and wd2117 + 342j ) : _ + among these four do white dwarfs , wd0005 + 511 has an unambiguous detection of hard x - rays , and wd0122@xmath0753j has a possible detection of hard x - rays . wd1522 + 662 does not show hard x - ray emission , but the @xmath1 observation has only 4.7 ks exposure time and wd1522 + 662 is faint with @xmath34 = 16.4 mag . wd2117 + 342j is observed at a 40@xmath13 from the center of the pspc field - of - view and the poor psf prohibits a conclusive assessment of the existence of faint hard x - ray emission . it may be possible that faint hard x - ray emission can emerge from the hot , deep layer of the photospheres of do white dwarfs , but more detections are needed to confirm it . * _ dqzo ( wd1144 + 044 , wd1159@xmath0034 , and wd1821 + 643 ) : _ + wd1159@xmath0034 has hard x - ray emission . wd1144 + 044 has a non - detection , but it is slightly fainter than wd1159@xmath0034 and its exposure time is only half as long . wd1821 + 643 in the pn k1 - 16 is projected near a bright hard x - ray source , a cluster of galaxies surrounding the qso e1821 + 643 @xcite , so it is difficult to determine its hard x - ray properties accurately from figure 4 . we have examined an archival @xmath7 hetg observation of qso e1821 + 643 ( pi : c. r. canizares ; 101 ks ) ; in this observation wd1821 + 643 is detected and clearly resolved from qso e1821 + 643 , but no hard x - ray emission from wd1821 + 643 is seen . wd1159@xmath0034 and wd1821 + 643 are well - known pulsators @xcite , while wd1144 + 044 is not . it is thus unlikely that the hard x - ray emission is related to the stellar pulsation . * _ pg 1159 and dz1 ( wd0044@xmath0121 and wd1501 + 664 ) : _ + wd0044@xmath0121 is a pg 1159 star in the pn ngc246 ; wd1501 + 664 is classified as dz1 and is the hottest white dwarf @xcite . neither of these two hot white dwarfs show hard x - ray emission ; the high opacity of the h- and he - free atmosphere of wd1501 + 664 may be the culprit of its non - detection @xcite . the above sample of hot white dwarfs is very limited , but allows us to eliminate improbable origins of hard x - ray emission . for example , stellar pulsation is not likely to produce hard x - ray emission , as wd1159@xmath0034 is the only pulsator with hard x - ray emission . stellar winds are not likely to be the origin of hard x - ray emission either , as wd1159@xmath0034 has hard x - rays but no measurable past or ongoing mass loss @xcite , while appreciable mass loss but no hard x - ray emission has been detected from wd0044@xmath0121 and wd1821 + 643 @xcite . there leaves the tantalizing suggestion that hard x - rays may be emitted by hot do and dqzo white dwarfs . while it is necessary to model their atmospheres to understand theoretically whether hard x - rays from beneath the atmosphere may leak through , it is also necessary to obtain better high - resolution x - ray images to confirm that the hard x - ray emission is indeed associated with the white dwarfs ( as opposed to background objects projected in their vicinity ) and high - quality x - ray spectra for detailed spectral analysis .

the lack of near - ir excess from the apparently single white dwarfs suggests that either x - ray observations are more effective than near - ir photometry in diagnosing faint companions or a different emission mechanism is needed . it is intriguing that 50% of the six apparently single white dwarfs with hard x - ray emission are among the hottest white dwarfs . is needed to enlarge the sample of white dwarfs with hard x - ray emission .

inspired by the hard x - ray emission from wd2226 , the central star of the helix nebula , we have made a systematic search for similar sources by correlating the white dwarf catalog of and the pspc point source catalog of . we find 76 white dwarfs coincident with x - ray sources at a high level of confidence . among these sources , 17 show significant hard x - ray emission at energies 0.5 kev . twelve of these white dwarfs with hard x - ray emission are in known binary systems , in two of which the accretion of the close companion s material onto the white dwarf produces hard x - ray emission , and in the other ten of which the late - type companions coronal activity emits hard x - rays . one apparently single white dwarf is projected near an agn which is responsible for the hard x - ray emission . the remaining four white dwarfs and two additional white dwarfs with hard x - ray emission appear single . the lack of near - ir excess from the apparently single white dwarfs suggests that either x - ray observations are more effective than near - ir photometry in diagnosing faint companions or a different emission mechanism is needed . it is intriguing that 50% of the six apparently single white dwarfs with hard x - ray emission are among the hottest white dwarfs . we have compared x - ray properties of 11 hot white dwarfs with different spectral types , and conclude that stellar pulsation and fast stellar winds are not likely the origin of the hard x - ray emission , but a leakage of the high - energy wien tail of emission from deep in the stellar atmosphere remains a tantalizing source of hard x - ray emission from hot do and dqzo white dwarfs . a complete survey using the entire pspc archive is needed to enlarge the sample of white dwarfs with hard x - ray emission . follow - up near - ir photometric observations are needed to verify the existence of late - type companions and high - resolution deep x - ray observations are needed to verify the positional coincidence and to study the x - ray spectral properties in order to determine the origin and nature of the hard x - ray emission .

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we have correlated the recent white dwarf catalog by @xcite with the @xmath1 pspc point source catalog ( wgacat ) , and found 76 white dwarfs coincident with x - ray sources at a high level of confidence . we have further found that 17 of these sources show hard x - ray emission ( @xmath2 0.5 kev ) . two of these white dwarfs with hard x - ray emission accrete material from their companions , while the other ten have late - type companions that are known to have active coronae and emit hard x - rays . one white dwarf has an agn projected within the pspc psf contributing to the hard x - ray emission . the remaining four white dwarfs ( wd0339@xmath0451 , wd1159@xmath0034 , wd1234 + 481 , and wd1333 + 510 ) and two additional white dwarfs ( wd0005 + 511 and wd2226@xmath0210 ) with hard x - ray emission appear single . the lack of near - ir excess for wd1234 + 481 and wd2226@xmath0210 at known distances constrains the possible spectral types of the hidden companions to later than m7v and m5 - 6v , respectively . we suggest that hard x - ray emission may be more effective than near - ir photometry in diagnosing faint , late - type companions of white dwarfs , if the hard x - ray emission associated with the six apparently single white dwarfs indeed originates from hidden companions . it is intriguing that three of the six apparently single white dwarfs with hard x - ray emission have high stellar effective temperatures . we have searched the @xmath1 archive for observations of pg 1159 stars and examined the x - ray properties of a sample of 13 hot white dwarfs with different spectral types . comparisons among these hot white dwarfs lead to the following conclusions : ( 1 ) dao wd2226@xmath0210 most likely possesses a late dme companion which emits hard x - rays , ( 2 ) stellar pulsation can not be connected to the hard x - ray emission , ( 3 ) fast stellar winds are not likely to be the origin of hard x - ray emission , and ( 4 ) the high - energy wien tail of emission deep in the atmospheres of hot do and dqzo white dwarfs remains a tantalizing explanation for the hard x - ray emission observed . our statistics are limited by the incompleteness of the wgacat , which has been derived solely from pointed @xmath1 pspc observations made without filters , while many pspc observations of white dwarfs were made with a boron filter . a complete survey for white dwarfs with hard x - ray emission using the entire @xmath1 archive is needed to enlarge the sample . follow - up high - resolution deep x - ray observations with @xmath7 or _ xmm - newton _ are needed to confirm the positional coincidence of the white dwarfs and the x - ray sources , and to study the spectral properties in order to investigate the origin and nature of hard x - ray emission associated with white dwarfs . we thank the anonymous referee for making critical comments which helped improve our paper . we also thank j. liebert , r. napiwotzki , r. petre , and k. werner for reading the manuscript and making useful suggestions . this research has made use of the simbad database , operated at cds , strasbourg , france , and the digital sky survey produced at the space telescope science institute under u.s . government grant nag w-2166 . we have also used data product from the 2mass , which is a joint project of the university massachusetts and the infrared processing and analysis center / california institute of technology , funded by nasa and nsf . lllllrrrrr 0027@xmath0636 & ... & da1 & j0029.9@xmath06324 & 400160 & 2.7 & good & 4081 @xmath32 64 & 27 @xmath32 5 & 0 @xmath32 0 + 0037@xmath35312 & gd8 & da1 & j0039.8@xmath353132 & 201045 & 28.4 & good & 2248 @xmath32 47 & 106 @xmath32 10 & 65 @xmath32 8 + 0048@xmath0294 & focapsgp2:31 & da & j0051.2@xmath02910 & 700275 & 24.5 & u1,2 & 46 @xmath32 7 & 11 @xmath32 3 & 24 @xmath32 5 + 0050@xmath0332 & gd659 & da1 & j0053.2@xmath03300 & 200410 & 3.4 & good & 3787 @xmath32 62 & 27 @xmath32 5 & 3 @xmath32 2 + 0116@xmath0231 & gd695 & da3 & j0118.6@xmath02254 & 100376 & 17.0 & good & 50 @xmath32 7 & 5 @xmath32 2 & 1 @xmath32 1 + 0122@xmath0753j & & do & j0122.8@xmath07521 & 300369 & 5.6 & good & 3130 @xmath32 56 & 98 @xmath32 10 & 8 @xmath32 3 + 0131@xmath0163 & gd984 & da1@xmath35 dm & j0134.4@xmath01607 & 200485 & 0.9 & good & 633 @xmath32 25 & 4 @xmath32 2 & 0 @xmath32 0 + 0136@xmath35251 & pg0136@xmath35251 & dap1 & j0138.8@xmath352523 & 200539 & 2.1 & good & 550 @xmath32 24 & 5 @xmath32 2 & 0 @xmath32 0 + 0216@xmath0032 & vz cet & da+m2 - 7iii & j0219.3@xmath00258 & 201501 & 9.1 & good & 5 @xmath32 2 & 18 @xmath32 4 & 40 @xmath32 6 + 0304@xmath35154 & ... & dc & j0307.0@xmath351540 & 800104 & 7.7 & u1,2 & 48 @xmath32 7 & 11 @xmath32 3 & 5 @xmath32 2 + 0320@xmath0539 & lb1663 & da1.5 & j0322.2@xmath05345 & 800307 & 21.5 & good & 2534 @xmath32 50 & 14 @xmath32 3 & 1 @xmath32 1 + 0333@xmath0350 & ... & da & j0335.5@xmath03449 & 600127 & 18.0 & good & 1480 @xmath32 38 & 20 @xmath32 4 & 5 @xmath32 2 + 0339@xmath0451 & ... & da & j0341.4@xmath04500 & 900495 & 48.6 & good & 64 @xmath32 8 & 22 @xmath32 5 & 18 @xmath32 4 + 0347@xmath35171 & v471tau & da2@xmath35k2v & j0350.4@xmath351714 & 200107 & 31.8 & good & 6538 @xmath32 81 & 688 @xmath32 26 & 809 @xmath32 28 + 0416@xmath0550 & ... & da & j0417.1@xmath05457 & 600456 & 17.8 & good & 254 @xmath32 16 & 16 @xmath32 4 & 11 @xmath32 3 + 0425@xmath35168 & eggr37 & da2 & j0428.6@xmath351658 & 200083 & 2.9 & good & 33 @xmath32 6 & 3 @xmath32 2 & 0 @xmath32 0 + 0426@xmath35588 & gj169.1b & dc@xmath35m4 & j0431.1@xmath355859 & 201114 & 3.5 & good & 22 @xmath32 5 & 4 @xmath32 2 & 4 @xmath32 2 + 0429@xmath35176 & hz9 & [email protected] & j0432.4@xmath351744 & 200443 & 20.3 & good & 384 @xmath32 20 & 180 @xmath32 13 & 161 @xmath32 13 + 0443@xmath0037j & rej0443@xmath0034 & da & j0443.0@xmath00346 & 200997 & 10.1 & good & 14710 @xmath32 121 & 336 @xmath32 18 & 42 @xmath32 6 + 0446@xmath0789 & ... & da3 & j0443.6@xmath07851 & 201073 & 8.3 & good & 74 @xmath32 9 & 1 @xmath32 1 & 1 @xmath32 1 + 0518@xmath0105 & rej0521@xmath0102 & da2 & j0521.3@xmath01029 & 200830 & 5.3 & good & 370 @xmath32 19 & 1 @xmath32 1 & 0 @xmath32 0 + 0531@xmath0022 & rej0534@xmath0021 & da2 & j0534.3@xmath00213 & 200932 & 8.0 & good & 284 @xmath32 17 & 17 @xmath32 4 & 6 @xmath32 2 + 0548@xmath35000 & gd257 & da1 & j0550.6@xmath350005 & 200585 & 4.4 & good & 1404 @xmath32 38 & 18 @xmath32 4 & 0 @xmath32 0 + 0558@xmath0756 & ... & do & j0556.9@xmath07540 & 201245 & 18.5 & u2,3 & 10 @xmath32 3 & 5 @xmath32 2 & 13 @xmath32 4 + 0651@xmath0020 & gd80 & da1 & j0654.2@xmath00209 & 201363 & 0.4 & good & 70 @xmath32 8 & 1 @xmath32 1 & 0 @xmath32 0 + 0715@xmath0704j & rej0715 - 705 & da1 & j0715.2@xmath07025 & 400321 & 2.8 & good & 5439 @xmath32 74 & 27 @xmath32 5 & 0 @xmath32 0 + 0718@xmath0316 & incma & dao@xmath35 dm & j0720.7@xmath03146 & 300338 & 3.2 & good & 411 @xmath32 20 & 4 @xmath32 2 & 2 @xmath32 1 + 0734@xmath0143 & ... & da & j0736.6@xmath01428 & 500295 & 9.1 & u3 & 5 @xmath32 2 & 19 @xmath32 4 & 23 @xmath32 5 + 0736@xmath35053 & procyon b & da+f5iv - v & j0739.3@xmath350513 & 200437 & 3.8 & good & 11700 @xmath32 110 & 874 @xmath32 30 & 118 @xmath32 11 + 0800@xmath0477j & & da & j0800.4@xmath04745 & 400158 & 2.4 & good & 1333 @xmath32 37 & 42 @xmath32 6 & 0 @xmath32 0 + 0805@xmath35654 & pg0805@xmath35654 & da1 & j0809.6@xmath356518 & 700258 & 8.4 & u2 & 198 @xmath32 14 & 67 @xmath32 8 & 18 @xmath32 4 + 0824@xmath35288 & pg0824@xmath35289 & da@xmath35dc@xmath35m3 & j0827.0@xmath352844 & 201083 & 2.7 & good & 586 @xmath32 24 & 5 @xmath32 2 & 1 @xmath32 1 + 0839@xmath0528 & ic2391 kr1 & da3 & j0841.1@xmath05300 & 200501 & 22.9 & u2,3 & 16 @xmath32 4 & 14 @xmath32 4 & 30 @xmath32 6 + 0840@xmath35200 & lb1876 & da5 & j0842.8@xmath351951 & 200250 & 1.9 & u2 & 11 @xmath32 3 & 4 @xmath32 2 & 10 @xmath32 3 + 0841@xmath35033j & rej0841@xmath35032 & da1 & j0841.0@xmath350320 & 201362 & 0.3 & good & 182 @xmath32 13 & 1 @xmath32 1 & 0 @xmath32 0 + 0842@xmath35490 & hd74389b & da@xmath35a2v & j0845.8@xmath354852 & 200816 & 0.3 & good & 91 @xmath32 10 & 3 @xmath32 2 & 0 @xmath32 0 + 0903@xmath35166 & ... & ... & j0905.9@xmath351624 & 700385 & 6.2 & u2,3 & 21 @xmath32 5 & 21 @xmath32 5 & 12 @xmath32 3 + 0904@xmath35511 & pg0904@xmath35511 & da1.5 & j0907.7@xmath355058 & 800474 & 9.6 & good & 524 @xmath32 23 & 25 @xmath32 5 & 26 @xmath32 5 + 0916@xmath0197j & rej0916@xmath0194 & da & j0916.9@xmath01946 & 400162 & 2.9 & good & 1494 @xmath32 39 & 19 @xmath32 4 & 0 @xmath32 0 + 0937@xmath35505 & pg0937@xmath35505 & da1 & j0940.3@xmath355021 & 200957 & 4.1 & good & 239 @xmath32 16 & 0 @xmath32 0 & 0 @xmath32 0 + 0954@xmath35697 & pg0954@xmath35697 & da2.5 & j0958.4@xmath356928 & 600101 & 21.4 & u4 & 174 @xmath32 13 & 76 @xmath32 9 & 38 @xmath32 6 + 1010@xmath35064 & pg1010@xmath35065 & da1 & j1013.4@xmath350612 & 200540 & 6.8 & good & 204 @xmath32 14 & 5 @xmath32 2 & 2 @xmath32 1 + 1032@xmath35534j & rej1032@xmath35535 & da1 & j1032.1@xmath355330 & 900149 & 17.7 & good & 40710 @xmath32 202 & 335 @xmath32 18 & 81 @xmath32 9 + 1040@xmath35451 & pg1040@xmath35451 & da1 & j1043.5@xmath354454 & 201020 & 14.7 & good & 1787 @xmath32 42 & 66 @xmath32 8 & 28 @xmath32 5 + 1059@xmath35514j & & da & j1059.2@xmath355124 & 400159 & 3.2 & good & 6582 @xmath32 81 & 102 @xmath32 10 & 0 @xmath32 0 + 1109@xmath35244 & pg1109@xmath35244 & da1.5 & j1112.6@xmath352409 & 201365 & 0.2 & good & 46 @xmath32 7 & 0 @xmath32 0 & 0 @xmath32 0 + 1134@xmath35300 & gj433.1 & da2 & j1137.0@xmath352948 & 200091 & 33.9 & good & 327 @xmath32 18 & 59 @xmath32 8 & 51 @xmath32 7 + 1144@xmath35004 & ... & dqzo1 & j1146.5@xmath350012 & 201242 & 5.8 & good & 149 @xmath32 12 & 5 @xmath32 2 & 3 @xmath32 2 + 1159@xmath0034 & gwvir & dqzo.4 & j1201.7@xmath00345 & 701202 & 13.6 & good & 779 @xmath32 28 & 12 @xmath32 4 & 13 @xmath32 4 + 1213@xmath35528 & eguma & da4@xmath35dm2e & j1215.6@xmath355230 & 200953 & 2.6 & good & 249 @xmath32 16 & 68 @xmath32 8 & 84 @xmath32 9 + 1229@xmath35290 & ... & dc & j1231.7@xmath352848 & 201163 & 1.8 & u1,3 & 113 @xmath32 11 & 44 @xmath32 7 & 39 @xmath32 6 + 1234@xmath35481 & pg1234@xmath35481 & da1 & j1236.7@xmath354755 & 200578 & 2.5 & good & 2127 @xmath32 46 & 89 @xmath32 9 & 0 @xmath32 0 + 1254@xmath35223 & gd153 & da1 & j1257.0@xmath352201 & 132471 & 8.6 & good & 16673 @xmath32 130 & 95 @xmath32 10 & 0 @xmath32 0 + 1255@xmath35258j & hd112313 & cspn+g5iii & j1255.5@xmath352553 & 201514 & 18.8 & good & 90 @xmath32 9 & 29 @xmath32 5 & 39 @xmath32 6 + 1314@xmath35293 & hz43a & [email protected] & j1316.3@xmath352906 & 100308 & 21.5 & good & 1585270 @xmath32 1260 & 7312 @xmath32 86 & 120 @xmath32 11 + 1317@xmath35453 & gj2100 & da3.5 & j1319.2@xmath354505 & 900325 & 10.3 & u1,2 & 34 @xmath32 6 & 12 @xmath32 4 & 14 @xmath32 4 + 1325@xmath35581 & eggr358 & da7 & j1327.6@xmath355755 & 600458 & 18.1 & u2,4 & 247 @xmath32 16 & 35 @xmath32 6 & 44 @xmath32 7 + 1333@xmath35510 & pg1333@xmath35510 & da & j1335.2@xmath355049 & 800047 & 16.4 & good & 447 @xmath32 21 & 51 @xmath32 7 & 47 @xmath32 7 + 1403@xmath0077 & pg1403@xmath0077 & da1 & j1406.0@xmath00758 & 200528 & 4.8 & good & 345 @xmath32 19 & 9 @xmath32 3 & 1 @xmath32 1 + 1446@xmath35634j & ... & da1 & j1446.0@xmath356329 & 700975 & 4.2 & good & 467 @xmath32 22 & 5 @xmath32 2 & 2 @xmath32 1 + 1501@xmath35664 & rej1502@xmath35661 & dz1 & j1502.1@xmath356612 & 170001 & 43.2 & good & 357416 @xmath32 600 & 1689 @xmath32 41 & 12 @xmath32 4 + 1522@xmath35662 & ... & do & j1522.9@xmath356604 & 201240 & 4.7 & good & 742 @xmath32 27 & 13 @xmath32 4 & 1 @xmath32 1 + 1620@xmath0391 & eggr274 & da2 & j1623.5@xmath03913 & 200588 & 1.9 & good & 634 @xmath32 25 & 2 @xmath32 1 & 2 @xmath32 1 + 1631@xmath35781 & [email protected] & da1@xmath35dm4e & j1629.1@xmath357804 & 170154 & 37.5 & good & 125636 @xmath32 350 & 1355 @xmath32 37 & 350 @xmath32 19 + 1633@xmath35572 & gj630.1b & dq8@xmath35dm4e & j1634.3@xmath355709 & 200721 & 47.5 & good & 2397 @xmath32 49 & 1249 @xmath32 35 & 1038 @xmath32 32 + 1634@xmath0573 & hd149499b & doz1@xmath35k2ve & j1638.5@xmath05728 & 200773 & 1.4 & good & 240 @xmath32 16 & 145 @xmath32 12 & 146 @xmath32 12 + 1636@xmath35351 & pg1636@xmath35351 & da1.5 & j1638.4@xmath353500 & 201082 & 4.2 & good & 909 @xmath32 30 & 9 @xmath32 3 & 0 @xmath32 0 + 1641@xmath35387 & gd357 & da3 & j1643.1@xmath353840 & 201538 & 5.5 & u1 & 26 @xmath32 5 & 10 @xmath32 3 & 13 @xmath32 4 + 1650@xmath35406j & rej1650@xmath35403 & da1 & j1650.3@xmath354037 & 201080 & 2.2 & good & 55 @xmath32 7 & 1 @xmath32 1 & 0 @xmath32 0 + 1657@xmath35343 & pg1657@xmath35343 & da2 & j1658.8@xmath353418 & 201079 & 6.4 & good & 591 @xmath32 24 & 7 @xmath32 3 & 0 @xmath32 0 + 1658@xmath35440 & pg1658@xmath35440 & dap1 & j1659.8@xmath354400 & 201078 & 6.7 & good & 503 @xmath32 22 & 7 @xmath32 3 & 1 @xmath32 1 + 1659@xmath35442 & pg1659@xmath35442 & da & j1700.6@xmath354410 & 201078 & 6.7 & u1,2 & 22 @xmath32 5 & 7 @xmath32 3 & 7 @xmath32 3 + 1802@xmath35213 & gd372 & da4 & j1804.4@xmath352120 & 200940 & 13.7 & u2,3 & 35 @xmath32 6 & 21 @xmath32 5 & 14 @xmath32 4 + 1821@xmath35643 & dsdra & doz.4 & j1821.8@xmath356422 & 700948 & 2.0 & good & 135 @xmath32 12 & 9 @xmath32 3 & 20 @xmath32 5 + 1844@xmath0654 & ... & da & j1848.9@xmath06525 & 200941 & 16.0 & good & 757 @xmath32 28 & 126 @xmath32 11 & 64 @xmath32 8 + 1906@xmath0600 & ... & dc & j1910.8@xmath05958 & 300047 & 5.2 & u3 & 7 @xmath32 3 & 20 @xmath32 5 & 41 @xmath32 6 + 1910@xmath35047 & ... & da2 & j1912.5@xmath350452 & 400271 & 2.8 & u4 & 20 @xmath32 45 & 10 @xmath32 3 & 27 @xmath32 5 + 1944@xmath0421 & v3885sgr & cv & j1947.6@xmath04200 & 300232 & 10.4 & good & 345 @xmath32 19 & 448 @xmath32 21 & 699 @xmath32 26 + 1957@xmath35225 & ... & dao & j1959.6@xmath352243 & 900016 & 5.9 & good & 828 @xmath32 29 & 5 @xmath32 2 & 0 @xmath32 0 + 2013@xmath35400j & rej2013@xmath35400 & dao & j2013.1@xmath354002 & 400157 & 3.0 & good & 1166 @xmath32 34 & 17 @xmath32 4 & 9 @xmath32 3 + 2014@xmath0575 & rej2018@xmath0572 & da2 & j2018.8@xmath05721 & 200580 & 4.1 & good & 148 @xmath32 12 & 3 @xmath32 2 & 0 @xmath32 0 + 2020@xmath0425 & [email protected] & da & j2023.9@xmath04224 & 200488 & 0.9 & good & 71 @xmath32 8 & 2 @xmath32 1 & 0 @xmath32 0 + 2028@xmath35390 & gd391 & da2 & j2029.9@xmath353913 & 200412 & 2.2 & good & 46 @xmath32 7 & 0 @xmath32 0 & 0 @xmath32 0 + 2032@xmath35248 & hd340611 & da2.5 & j2034.3@xmath352504 & 200087 & 9.9 & good & 100 @xmath32 10 & 2 @xmath32 1 & 5 @xmath32 2 + 2034@xmath0275j & & da & j2034.9@xmath02734 & 201236 & 5.0 & good & 574 @xmath32 24 & 4 @xmath32 2 & 2 @xmath32 1 + 2052@xmath35466j & & do & j2052.6@xmath354639 & 200114 & 3.5 & u3 & 801 @xmath32 28 & 118 @xmath32 11 & 27 @xmath32 5 + 2056@xmath35033 & pg2056@xmath35033 & da1 & j2058.7@xmath350332 & 200955 & 5.6 & good & 37 @xmath32 6 & 2 @xmath32 1 & 1 @xmath32 1 + 2111@xmath35498 & gd394 & da1.5 & j2112.7@xmath355006 & 200427 & 0.4 & good & 207 @xmath32 14 & 1 @xmath32 1 & 0 @xmath32 0 + 2117@xmath35342j & v2027cyg & do & j2117.1@xmath353412 & 201512 & 26.6 & good & 6588 @xmath32 81 & 194 @xmath32 14 & 66 @xmath32 8 + 2153@xmath0419 & rej2156@xmath0414 & da & j2156.5@xmath04142 & 200487 & 0.7 & good & 397 @xmath32 20 & 2 @xmath32 1 & 0 @xmath32 0 + 2154@xmath0512 & gj841b & dq7@xmath35dm3 - 5e & j2157.7@xmath05059 & 600146 & 5.1 & good & 1851 @xmath32 43 & 606 @xmath32 25 & 706 @xmath32 27 + 2309@xmath35105 & gd246 & da1 & j2312.3@xmath351047 & 100578 & 10.3 & good & 36832 @xmath32 190 & 141 @xmath32 13 & 3 @xmath32 2 + 2321@xmath0549 & rej2324@xmath0544 & da & j2324.5@xmath05441 & 400166 & 4.1 & good & 1964 @xmath32 44 & 20 @xmath32 5 & 1 @xmath32 1 + 2357@xmath35296 & pg2357@xmath35296 & da1 & j0000.1@xmath352956 & 200535 & 3.8 & good & 36 @xmath32 6 & 2 @xmath32 1 & 1 @xmath32 1 + llllll 0216@xmath0032 & da+m2 - 7iii & [email protected] & 2.04 : & 0.5 & 29.9 + 0339@xmath0451 & da & ... & ... & ... & ... + 0347@xmath35171 & da2+k2v & [email protected] & 1.81 & 0.8 & 29.8 + 0429@xmath35176 & da2+dm4.5e & 21.58 & 1.91 & 0.6 & 28.6 + 0736@xmath35053 & da+f5iv - v & [email protected] & 1.20 & 0.2 & 27.3 + 1134@xmath35300 & da2+(agn ) & [email protected] & ... & ... & ... + 1159@xmath0034 & dqzo.4 & ... & ... & ... & ... + 1213@xmath35528 & da4+dm2e & [email protected] & 1.89 & 0.8 & 28.8 + 1234@xmath35481 & da1 & 6.3 & ... & ... & ... + 1255@xmath35258j & cspn+g5iii & [email protected] & @xmath20 . & 0.7 & 29.3 + 1314@xmath35293 & da1+dm3.5e & [email protected] & @xmath361.85 & 0.6 & 28.0 + 1333@xmath35510 & da & ... & ... & ... & ... + 1631@xmath35781 & da1+dm4e & 19 . & @xmath20 . & 0.8 & 28.7 + 1633@xmath35572 & dq8 + 2(dm3 - 4e ) & [email protected] & @xmath362.58 & 0.4 & 28.0 + 1634@xmath0573 & doz1+k2ve & [email protected] & @xmath361.69 & 0.7 & 29.3 + 1944@xmath0421 & cv & [email protected] & @xmath02.09 & 1.2 & 30.3 + 2154@xmath0512 & dq7 + 2(dm3 - 5e ) & [email protected] & @xmath362.66 & 0.8 & 29.0 + lllllll 0005 + 511 & do & 13.32 & 13.93 & 14.13 & 14.18 & j0008181 + 512316 + 0216@xmath0032 & da+m2 - 7iii & 11.32 & 1.06 & 1.89 & 2.44 & j2192081@xmath0025841 + 0429 + 176&da2+dm4.5e & 13.93 & 10.76 & 10.12 & 9.93 & j0432237 + 174502 + 1134 + 300 & da2+(agn ) & 12.50 & 12.95 & 13.04 & 13.13 & + 1159@xmath0034 & dqzo.4 & 14.87 & 15.58 & 15.87 & 15.78 & j1201459@xmath0034540 + 1234 + 481 & da1 & 14.42 & 14.99 & 15.07 & 15.12 & j1236451 + 475522 + 1631 + 781 & da1+dm4e & 13 & 11.00 & 10.28 & 10.15 & j1629102 + 780439 + 1633 + 572 & dq8 + & 14.99 & 14.09 & 14.08 & 14.07 & j1634216 + 571008 + & 2(dm3 - 4e ) & 12.87 & 8.50 & 8.04 & 7.77 & j1634204 + 570943 + 2226@xmath0210 & dao & 13.54 & 14.35 & 14.50 & 14.62 & j2229385@xmath0205013 + llrllr wd0005@xmath35511 & do & 120,000 & 13.32 & s , h & 5.0 + wd0044@xmath0121 & pg1159 & 150,000 & 11.84 & s & 11.4 + wd0122@xmath0753j & do & 180,000 & 15.4 & s , h ? & 5.6 + wd0726 + 133 & pg1159 & 140,000 & 15.99 & nd & 2.8 + wd1144@xmath35004 & dqzo1 & 150,000 & 15.10 & s & 5.8 + wd1159@xmath0034 & dqzo.4 & 140,000 & 14.84 & s , h & 13.6 + wd1501@xmath35664 & dz1 & 170,000 & 15.9 & s & 43.2 + wd1522@xmath35662 & do & 140,000 & 16.4 & s & 4.7 + wd1821@xmath35643 & doz.4 & 140,000 & 15.04 & s & 3.1 + wd1957@xmath35225 & dao & 108,600 & 14.2 & s & 19.9 + wd2117@xmath35342j & do & 170,000 & 13.16 & s & 26.6 + wd2226@xmath0210 & dao & 103,600 & 13.4 & s , h & 4.9 + wd2333 + 301 & doz.3 & 150,000 & 16.13 & nd & 4.2 +

we find 76 white dwarfs coincident with x - ray sources at a high level of confidence . among these sources , 17 show significant hard x - ray emission at energies 0.5 kev . one apparently single white dwarf is projected near an agn which is responsible for the hard x - ray emission . the remaining four white dwarfs and two additional white dwarfs with hard x - ray emission appear single . we have compared x - ray properties of 11 hot white dwarfs with different spectral types , and conclude that stellar pulsation and fast stellar winds are not likely the origin of the hard x - ray emission , but a leakage of the high - energy wien tail of emission from deep in the stellar atmosphere remains a tantalizing source of hard x - ray emission from hot do and dqzo white dwarfs . a complete survey using the entire pspc archive follow - up near - ir photometric observations are needed to verify the existence of late - type companions and high - resolution deep x - ray observations are needed to verify the positional coincidence and to study the x - ray spectral properties in order to determine the origin and nature of the hard x - ray emission .

inspired by the hard x - ray emission from wd2226 , the central star of the helix nebula , we have made a systematic search for similar sources by correlating the white dwarf catalog of and the pspc point source catalog of . we find 76 white dwarfs coincident with x - ray sources at a high level of confidence . among these sources , 17 show significant hard x - ray emission at energies 0.5 kev . twelve of these white dwarfs with hard x - ray emission are in known binary systems , in two of which the accretion of the close companion s material onto the white dwarf produces hard x - ray emission , and in the other ten of which the late - type companions coronal activity emits hard x - rays . one apparently single white dwarf is projected near an agn which is responsible for the hard x - ray emission . the remaining four white dwarfs and two additional white dwarfs with hard x - ray emission appear single . the lack of near - ir excess from the apparently single white dwarfs suggests that either x - ray observations are more effective than near - ir photometry in diagnosing faint companions or a different emission mechanism is needed . it is intriguing that 50% of the six apparently single white dwarfs with hard x - ray emission are among the hottest white dwarfs . we have compared x - ray properties of 11 hot white dwarfs with different spectral types , and conclude that stellar pulsation and fast stellar winds are not likely the origin of the hard x - ray emission , but a leakage of the high - energy wien tail of emission from deep in the stellar atmosphere remains a tantalizing source of hard x - ray emission from hot do and dqzo white dwarfs . a complete survey using the entire pspc archive is needed to enlarge the sample of white dwarfs with hard x - ray emission . follow - up near - ir photometric observations are needed to verify the existence of late - type companions and high - resolution deep x - ray observations are needed to verify the positional coincidence and to study the x - ray spectral properties in order to determine the origin and nature of the hard x - ray emission .

hep-th0008024

i

invariant functions under the transformations of a compact linear group ( hereafter abbreviated in clg ) @xmath0 , acting in an euclidean space @xmath1 , play an important role in physics , and the determination of their properties is often a basic problem to solve in many physical contexts , such as the determination of patterns of spontaneous symmetry breaking , the analysis of phase spaces and structural phase transitions ( landau s theory ) , covariant bifurcation theory , crystal field theory and so on . a @xmath0-invariant function @xmath13 , @xmath14 , takes on constant values along each orbit of @xmath0 , thus , if one has to analyze its properties , it is certainly more economical , and generally more effective , to think of it as a function defined in the orbit space @xmath7 of the action of @xmath0 in @xmath1 . in this way , it is possible to take fully into account the invariance properties of @xmath13 , while maintaining its regularity properties , but avoiding the troubles that could be met , for instance in the determination of the minima , owing to its degeneracy along the @xmath0-orbits . this approach to the study of the properties of a @xmath0-invariant function , obviously requires a detailed knowledge of the structure of the orbit space @xmath7 , which can be obtained as follows , using the methods of invariant theory @xcite . let @xmath15 be a minimal integrity basis ( hereafter abbreviated in mib ) of the ring @xmath16^g$ ] of polynomial invariants of @xmath0 . the mib defines an _ orbit map _ @xmath17 , mapping @xmath1 onto a semi - algebraic subset , @xmath18 , of @xmath8 , which provides a diffeomorphic image of @xmath7 ( see , for instance , @xcite , @xcite or @xcite and references therein ) . if @xmath13 is a @xmath0-invariant polynomial or @xmath19-function it can be written in terms of a polynomial or , respectively , @xmath19-function @xmath20 , in the form @xmath21 @xcite . the function @xmath20 has the same range as @xmath13 , but its domain is a faithful image of @xmath7 . the price to pay in the orbit space approach to the analysis of a @xmath0-invariant function is essentially twofold : 1 . mib s are sometimes difficult to determine . the domain of the associated function @xmath20 is not the whole euclidean space @xmath8 , but reduces to the semi - algebraic subset @xmath18 @xcite , not trivial to determine . as for the problem stated under item 2 , the following simple solution has been proposed . it has been shown @xcite that , if the relations among the elements of the mib s are known , the polynomial equalities and inequalities defining @xmath18 and its strata can be determined from the rank and positivity properties of a matrix @xmath10 , defined only in terms of the gradients of the elements of a mib . when there are no relations among the elements of the mib s , that is for compact _ coregular _ linear groups , the matrices @xmath10 play the role of inverse metric matrices , and the isomorphism classes of the orbit spaces of all these groups can be classified in terms of equivalence classes of matrices @xmath10 . a way to obtain the matrices @xmath10 generated by clg s , bypassing the actually insoluble problem of determining a mib for each group , has been suggested in @xcite . the idea is to use an axiomatic approach , that is to forget altogether the original definition of the matrices @xmath10 and to characterize them as far as possible through their structural properties ( _ allowable _ ) . the possibility of actually computing them is favored by the identification and convenient formalization of a set of sufficiently strong , but handy , analytic conditions , shared by all @xmath10 s . the less immediately evident of these conditions have been translated into a set of differential equations ( _ boundary conditions _ ) involving the elements of the matrices @xmath10 . in the case of coregular groups , it has been proved that , in the case of less than 5 basic invariants @xcite , the allowable @xmath22-matrices can be determined from the solution of a unique universal differential equation ( that here we shall call _ master equation _ ) , satisfying convenient initial conditions . the master equation involves as free parameters only the degrees @xmath11 of the basic polynomial invariants . the conditions defining allowable @xmath22-matrices for _ coregular _ clg s turn out to be so restrictive that , for each choice of a set of degrees @xmath23 , they turn out to select ( at least for @xmath24 ) only a finite number of equivalence classes of matrices @xmath10 . thus , even if the results obtained along these lines are still strongly incomplete , one may reasonably hope to be able to obtain in the future a classification of the orbit spaces of all compact coregular linear groups , even if the classification of these groups is not yet complete and/or the explicit form of the elements of the corresponding mib s is not known . in this paper , we shall shortly review the geometry of linear group actions and describe how the invariant theory may successfully be applied to determine the stratification of the orbit space of a compact linear group . then , we shall turn our attention on the axiomatic approach for the determination of the of clgs . we shall focus on new developments concerning the generated by groups in the class @xmath25 of non - coregular groups with only one independent relation among the @xmath26 elements of their mib s . we shall show that the relation among the elements of a mib is determined by one of the irreducible factors of the determinant of @xmath10 , which , along with @xmath10 , has to satisfy the master equation and some additional conditions , part of which can be put in the form of a subsidiary differential equation ( _ second order boundary conditions _ ) . the effectiveness of these conditions is tested in the simplest case of three basic invariants , by determining all the solutions and selecting those which lead to allowable . in fact , the conclusion of our analysis is that , leaving aside the trivial cases in which at least one of the invariants @xmath27 is linear , * there is only one monoparametric discrete family of allowable non - equivalent @xmath28 , @xmath29 , whose elements may be generated by groups in @xmath30 . the degrees of the @xmath31 s are @xmath32 , @xmath33 and , with a convenient choice of the @xmath31 s , the basic relation can be written in the form @xmath34 . * every allowable of the family is generated by at least a group @xmath35 . * if the action of the groups is restricted to the unit sphere @xmath36 of @xmath1 ( which is not essentially restrictive for what concerns the characterization of the orbit space ) , all the orbit spaces @xmath37 , @xmath35 turn out to be isomorphic . we shall present the matter in the following order . in section 2 we shall recall some known results concerning the characterization of orbit spaces ( see , for instance , @xcite ) in the approach . in section 3 we recall the derivation of the boundary conditions , of the master relation and of the additional conditions that must be satisfied in the case of coregular clg s . suitable conditions are subsequently derived for non - coregular groups , with particular attention to the case in which there is only one basic relation : this is the original part of the paper . in 4 we formalize our approach to the determination of the that could virtually be associated to actual non - coregular clg s with one independent relation among the basic invariants ; the boundary conditions and the master relation are presented as equations and the notions of proper and allowable solutions of the master equation are defined . in 5 we compute first all the proper and allowable solutions of the master equation and , after a further selection , we arrive at the determination of the equivalence classes of the of all the non - coregular clg s of class @xmath38 . the last part of the section is devoted to the problem of the correspondence between allowable and generating clg s . our conclusions agree with a result about locally smooth actions on manifolds with orbits of codimension @xmath39 ( * ? ? ? iv 8.2 , p. 206 ) . according to it , if the action of the group is restricted to the unit sphere @xmath40 of @xmath41 , then either @xmath42 is diffeomorphic to the unit interval @xmath43 $ ] , or @xmath42 is diffeomorphic to the @xmath44-sphere @xmath45 . the former case refers to coregular groups of class @xmath46 , the latter , both to non - coregular groups of class @xmath47 ( whose orbit space is isomorphic to some @xmath48 , @xmath49 , where @xmath50 acts on @xmath51 ) , and to coregular groups of class @xmath52 ( e.g. the linear groups @xmath53 acting in @xmath54 for @xmath55 ) . the fact that coregular and non - coregular groups may share orbit spaces belonging to the same isomorphism class is an intriguing fact that suggested us to treat the axiomatic approach stressing , whenever possible , analogies and differences between those two cases . it is worth noting that the effectiveness of our method does not depend on the dimension of the real vector space upon which @xmath0 acts , so it may be applied , in principle , to determine the orbit spaces of all non - coregular groups with only one independent relation among the basic polynomial invariants .

invariant functions under the transformations of a compact linear group acting in can be expressed in terms of functions defined in the orbit space of , i.e. as functions of a finite set of basic invariant polynomials , , which form an integrity basis ( ib ) for . the effectiveness of the method is tested for ib s fformed 3 ( independent ) basic invariants . the result is obtained through the computation of a metric matrix , which is defined only in terms of the scalar products between the gradients , and whose domain of semi - positivity is known to realize the orbit space of as a semi - algebraic variety in the space spanned by the variables . after a short review of the approach that recently enabled to solve the analogous problem for coregular groups with less than 5 basic invariants , we determine the matrices from the solutions of a universal differential equation ( _ master equation _ ) , which satisfy new convenient additional conditions , which fit for the non - coregular case . the master equation involves as free parameters only the degrees of the s . this approach bypasses the actual impossibility of explicitely determining a set of basic polynomial invariants for each group . our results may be relevant in physical contexts where the study of covariant or invariant functions is important , like in the determination of patterns of spontaneous symmetry breaking in quantum field theory , in the analysis of phase spaces and structural phase transitions ( landau s theory ) , in covariant bifurcation theory , in crystal field theory and so on .

invariant functions under the transformations of a compact linear group acting in can be expressed in terms of functions defined in the orbit space of , i.e. as functions of a finite set of basic invariant polynomials , , which form an integrity basis ( ib ) for . we develop a method to determine the isotropy classes of the orbit spaces of all the real linear groups whose ibs satisfy only one independent relation . the effectiveness of the method is tested for ib s fformed 3 ( independent ) basic invariants . the result is obtained through the computation of a metric matrix , which is defined only in terms of the scalar products between the gradients , and whose domain of semi - positivity is known to realize the orbit space of as a semi - algebraic variety in the space spanned by the variables . after a short review of the approach that recently enabled to solve the analogous problem for coregular groups with less than 5 basic invariants , we determine the matrices from the solutions of a universal differential equation ( _ master equation _ ) , which satisfy new convenient additional conditions , which fit for the non - coregular case . the master equation involves as free parameters only the degrees of the s . this approach bypasses the actual impossibility of explicitely determining a set of basic polynomial invariants for each group . our results may be relevant in physical contexts where the study of covariant or invariant functions is important , like in the determination of patterns of spontaneous symmetry breaking in quantum field theory , in the analysis of phase spaces and structural phase transitions ( landau s theory ) , in covariant bifurcation theory , in crystal field theory and so on . 3truemm _ keywords _ : geometric invariant theory , linear group actions , orbit spaces , non - coregular algebraic linear groups , spontaneous symmetry breaking + 1991 msc : 14d25 , 14l30 + pacs : 02.20.hj , 11.30.qc , 11.15ex , 61.50.ks 15.5 cm 20.5 cm # 1#2#3_#1^#2 # 1#2#3_#1^#2 # 1#2 # 1 , # 2 # 1#2#3_#1^#2 [ section ] [ section ] [ section ] [ section ] [ section ]

hep-th0605070

c

previous studies of defect formation during tachyon condensation have neglected the role played by bulk fields . it was argued @xcite that when bulk fields are included , the formation of topological defects is suppressed or prevented . the argument was that the energy of the desitter fluctuations is of the order the hubble constant during inflation , while the gradients of the fields along the compact dimensions have corresponding energies @xmath94 ; so there is not enough energy available for defect formation . we have constructed a toy model that captures the essential features of the tachyon condensation process when both the brane worldvolume gauge field and the bulk rr field are included . in this model the gradients of the bulk field are formed as a result of classical field evolution and the energy is provided by the tension of the brane - anti - brane pair . since this energy can also be described as the energy of the tachyon field placed at the top of the potential , in the toy model this corresponds to the potential energy of an abelian higgs field . we argued that inside a large enough area around a defect there is always enough energy in the form of potential energy of the scalar field such that there is no energetic constraint on the formation of topological defects . in terms of energetics , the presence of the bulk with a chern - simons coupling increases the energy cost of a defect . the increase is greatest if the chern - simons coupling is large and if the extra dimension is _ small_. however , the energy associated with a defect never exceeds the value in the global defect model , that is , the value it would take if there were no gauge field or if @xmath95 . the bulk field also changes the long range interactions between defects , so that their mutual attraction or repulsion shows power law , rather than exponential , dependence on separation . this speeds up the evolution of the defect network , so that defect - antidefect pairs find each other and annihilate more efficiently . again , the defect network evolution is never faster than in the global defect model , which is known to support a defect network after a symmetry breaking transition . another effect we observe in the lattice simulations of the toy model is the decrease of the magnetic flux unit through the vortex core as a function of the chern - simons coupling constant . this flux also acts as the charge density for the bulk field and if the chern - simons coupling becomes too large this charge density becomes very small , reducing the energy leakage from the brane to the bulk . one drawback of the toy model comes from using an abelian higgs field instead of a tachyon on the brane . we did this so that we could follow the evolution of the defects after they have formed while the tachyon develops singularities . at the same time the brane fields do not drop out of the dynamics @xcite once the defects have formed .

studies of this possibility have generally ignored the interaction of the brane fields with fields in the bulk ; recently it has been argued that interactions with bulk fields suppress or even eliminate defect formation . to investigate the impact of bulk fields on brane defect formation , we construct a toy model that captures the essential features of the tachyon condensation with bulk fields .

it has recently been realized that brane - antibrane annihilation ( a possible explanation for ending inflation ) may result in defect formation , due to the dynamics of the tachyon field . studies of this possibility have generally ignored the interaction of the brane fields with fields in the bulk ; recently it has been argued that interactions with bulk fields suppress or even eliminate defect formation . to investigate the impact of bulk fields on brane defect formation , we construct a toy model that captures the essential features of the tachyon condensation with bulk fields . we study the structure of defects in this toy model , and simulate their formation and evolution on the lattice . we find that , while the energetics and interactions of defects are influenced by the size of the extra dimension and the bulk - brane coupling , the bulk - brane coupling does not prevent the formation of a defect network . # 1#2 = # 1 # 1eq . ( [ # 1 ] ) ii

1407.4119

c

we have modeled the stellar age distributions of young clusters based on the density - dependent star formation model of @xcite . in this class of models , the star formation efficiency increases more rapidly in higher volume - density gas than in low volume - density gas . our model starts with starless molecular clumps in equilibrium , and we assume that a static configuration is retained throughout the star formation episode . for this reason , we limit the duration of star formation to five initial free - fall times of the clump . we also assume that the clump neither gains nor loses mass . therefore , the star formation rate is steadily decreasing with time since , as star - formation depletes the clump gas , the star formation efficiency per free - fall time is applied to an ever lower amount of gas on an ever longer time - scale . our results are equations describing the star formation rate evolution of a clump with given mass and radius ( eq . [ eq : sfr2 ] ) . we also provide analytical equations obtained for the specific case of an isothermal sphere of gas ( eqs . [ eq : sfrcpl ] and [ eq : cround ] ) . for each of our model clumps , we have presented the star formation history and the linear distributions of the star ages if the newly - formed cluster is observed 5myr after the onset of star formation . regarding the star age distribution , we have considered the _ intrinsic _ age distribution , where star ages are binned with a bin size of 0.1myr , and the _ measured _ age distribution , which accounts for the errors affecting individual star ages . that is , the measured age distribution is the intrinsic one convolved with a gaussian function . we have successively discussed the star formation history of the clump in relation to the clump properties ( mainly mean density and star formation efficiency per free - fall time ; see figs . [ fig : princ ] and [ fig : princb ] ) and how the star formation history translates into the cluster star age distribution ( figs . [ fig : ad ] and [ fig : adb ] ) . in our study of the clump star formation history , we have considered two regimes , depending on whether the star formation rate decreases by a factor of 2 during star formation or not . if the evolution is rapid enough for it to decrease by a factor of 2 , one can define the half - life time of the clump star formation rate , namely , the time needed for the star formation rate to decrease to half its initial value . this half - life time is a key - driver of the fwhm of the cluster intrinsic age distribution . if the star formation rate has decreased by less than a factor of two at the end of the star - formation episode , the half - life time of the star formation rate can not be defined , and the fwhm of the intrinsic age distribution is driven by the duration of star formation . we have embodied these two regimes with two distinct star formation efficiencies per free - fall time , @xmath1 and @xmath2 . when @xmath1 ( rapid evolution ) , the half - life time of the star formation rate is of the order of the free - fall time at the clump mean volume density . ( note that this result is valid for an isothermal sphere of gas . shallower density profiles lead to longer half - life times . ) therefore , the denser the cluster - parent clump , the shorter the half - life time of its star formation rate and the shorter the stellar age spread of the resulting cluster . caution should be taken , however , as the fwhm of the age distribution does not depend only on the half - life time of the star formation rate . firstly , star - age binning may prevent a short half - life time from being resolved in the intrinsic age distribution . this is the case encountered for our densest model clump ( model a ) . secondly , uncertainties in the age of individual stars inflate the measured age spread . in case of a very dense fastly - evolving clump , the measured age spread embodies mostly the star age uncertainty and loses memory of the short star formation rate half - life time . therefore , these two effects large age - bin size and age uncertainties weaken the contrast between the measured age spread for high- and low - density clumps ( see fig . [ fig : sum ] , top panel ) . the predicted effect that high - density clumps give rise to narrower age spreads seems to have been observed , with the fwhm of the stellar age distribution in the central regions of the high - density clusters ngc3603 yc and wd 1 found to be an order of magnitude smaller than for the orion nebula cluster ( see * ? ? ? * ; * ? ? ? for the central regions of the young starburst cluster wd 1 , @xcite derived a fwhm of 0.4myr . they estimated a photometric error contribution of 0.25 myr . assuming that the observed fwhm is the quadratic sum of the age error and of the intrinsic fwhm , we derive @xmath199myr . this value agrees well with what we found for clump b ( see top panel of fig . [ fig : sum ] ) . we therefore suggest that the central regions of wd 1 formed out of gas with a density of the order of @xmath200 if @xmath87 . matching a cluster age spread to the density of its parent clump can be done by using the bottom panel of fig . [ fig : sum ] . it provides the relation between the half - life time and the mean volume density of molecular clumps with an @xmath201-density profile when @xmath1 . observers can build on it to predict the relation between the clump mean density and the fwhm of the stellar age distribution they would measure for their own age - bin size and uncertainties in stellar ages . it can also be rescaled to another @xmath27-value : lower efficiencies lead to higher gas densities for given fwhms . if the star formation efficiency per free - fall time is lower than 0.1 , the evolution is slower . this implies that to achieve a given age spread @xmath119 , the free - fall time of the gas must be shorter ( to compensate for the slower @xmath0 ) and , therefore , the gas density must be higher . as an example , if an intrinsic age spread @xmath119 is achieved by a clump of mean density @xmath139 with @xmath36 , the same age spread is achieved by a clump of mean density @xmath202 if @xmath203 . but what if the star formation rate decreases so slowly that its half - life time can not be defined ? if the star formation efficiency per free - fall time is as low as @xmath2 , it takes about 10 clump free - fall times for the star formation rate to decrease to half its initial value ( again assuming an isothermal density profile ) . therefore , if the star formation duration is shorter than that , the half - life time can not be defined , and the fwhm of the intrinsic star age distribution is driven by the duration of the star formation episode @xmath120 ( e.g. bottom panel of fig . [ fig : ad ] ) . to estimate @xmath120 requires a detailed understanding of how long it takes for stellar feedback processes to terminate star formation inside the clump , which is beyond the scope of this paper . yet , we have shown that building on the invariance of the shape of the _ young _ cluster mass function can help us reach some conclusions . that the shape of the cluster mass function does not change with time at young ages points to a mass - invariant star formation efficiency and , therefore , to a duration of star formation equal to a given number of free - fall times for all cluster - forming clumps . therefore , in this case too , we reach the conclusion that the @xmath119 scales with the clump free - fall time @xmath3 , i.e. @xmath204 . again , denser clumps give rise to narrower star age distributions . the normalisation of the @xmath139-@xmath119 relation depends on how many free - fall times star formation lasts . we have also searched the literature for the age spreads and densities of observed star - forming regions . we find that current data support a scenario where the characteristic time - scale of star formation is of order @xmath4-@xmath5 , which is reminiscent of the earlier study by @xcite ( see fig . [ fig : litdat ] ) . note that the @xmath118-axis of fig . [ fig : litdat ] often refers to the observed fwhm of the stellar age distribution . in our model , it reflects the half - life time @xmath86 of the star formation rate . as such , the fwhm is not representative of the duration of star formation which can be significantly longer . the key conclusion of our work is that the observation in young clusters of stellar age spreads spanning 1-to-2 orders of magnitude @xcite does not require different cluster - formation mechanisms @xcite . this may simply reflect the wide range of volume densities among cluster - forming clumps . our work also suggests that the intrinsic star age spread in clusters may constitute a promising probe into the mean volume density of their gaseous progenitors , provided that the star formation efficiency per free - fall time is well - constrained . gp acknowledges support from the olympia - morata program of heidelberg university . this work was also supported by the sonderforschungsbereich sfb 881 `` the milky way system '' ( subproject b5 ) of the german research foundation ( dfg ) . we thank nicola da rio and rob jeffries for interesting discussions , and the referee for a thorough report .

we model the star formation history of molecular clumps in equilibrium by associating a star formation efficiency per free - fall time , , to their volume density profile . we focus on the case of isothermal spheres and we obtain the evolution with time of their star formation rate . our model predicts a steady decline of the star formation rate , which we quantify with its half - life time , namely , the time needed for the star formation rate to drop to half its initial value . given the uncertainties affecting the star formation efficiency per free - fall time , we consider two distinct values : and . when , the half - life time is of the order of the clump free - fall time , . as a result , the age distributions of stars formed in high - density clumps have smaller full - widths at half - maximum than those of stars formed in low - density clumps . when the star formation efficiency per free - fall time is 0.01 , the half - life time is 10 times longer , i.e. 10 clump free - fall times . we explore what happens if the duration of star formation is shorter than 10 , that is , if the half - life time of the star formation rate can not be defined . there , we build on the invariance of the shape of the young cluster mass function to show that an anti - correlation between the clump density and the duration of star formation is expected . we also discuss how the age - bin size and uncertainties in stellar ages affect our results .

it has recently been suggested that high - density star clusters have stellar age distributions much narrower than that of the orion nebula cluster , indicating a possible trend of narrower age distributions for denser clusters . we show this effect to likely arise from star formation being faster in gas with a higher density . we model the star formation history of molecular clumps in equilibrium by associating a star formation efficiency per free - fall time , , to their volume density profile . we focus on the case of isothermal spheres and we obtain the evolution with time of their star formation rate . our model predicts a steady decline of the star formation rate , which we quantify with its half - life time , namely , the time needed for the star formation rate to drop to half its initial value . given the uncertainties affecting the star formation efficiency per free - fall time , we consider two distinct values : and . when , the half - life time is of the order of the clump free - fall time , . as a result , the age distributions of stars formed in high - density clumps have smaller full - widths at half - maximum than those of stars formed in low - density clumps . when the star formation efficiency per free - fall time is 0.01 , the half - life time is 10 times longer , i.e. 10 clump free - fall times . we explore what happens if the duration of star formation is shorter than 10 , that is , if the half - life time of the star formation rate can not be defined . there , we build on the invariance of the shape of the young cluster mass function to show that an anti - correlation between the clump density and the duration of star formation is expected . we therefore conclude that , regardless of whether the duration of star formation is longer than the star formation rate half - life time , denser molecular clumps yield narrower star age distributions in clusters . published densities and stellar age spreads of young clusters and star - forming regions actually suggest that the time - scale for star formation is of order- . we also discuss how the age - bin size and uncertainties in stellar ages affect our results . we conclude that there is no need to invoke the existence of multiple cluster formation mechanisms to explain the observed range of stellar age spreads in clusters .

1309.7395

i

in a recent work , fowler and pisanski @xcite introduced the notion of the _ hl - index _ of a graph that is related to the homo - lumo separation studied in theoretical chemistry ( see also jakli , fowler , and pisanski @xcite ) . this is the gap between the highest occupied molecular orbital ( homo ) and lowest unoccupied molecular orbital ( lumo ) . in the hckel model @xcite , the energies of these orbitals are in linear relationship with eigenvalues of the corresponding molecular graph and can be expressed as follows . let @xmath0 be a graph of order @xmath3 , and let @xmath4 be the eigenvalues of its adjacency matrix . the eigenvalues occurring in the homo - lumo separation are the median eigenvalues @xmath5 and @xmath6 , where @xmath7 the _ hl - index _ @xmath8 of the graph @xmath0 is then defined as @xmath9 a simple unweighted graph @xmath0 is said to be _ subcubic _ if its maximum degree is at most @xmath10 . in @xcite it is proved that every subcubic graph @xmath0 satisfies @xmath11 and that if @xmath0 is bipartite , then @xmath12 . the following is the main result from @xcite . [ thm : cubic ] the median eigenvalues @xmath13 and @xmath14 of every subcubic graph @xmath0 are contained in the interval @xmath15 $ ] , i.e. , @xmath16 . this result is best possible since the heawood graph ( the bipartite incidence graph of points and lines of the fano plane ) has @xmath17 . the following conjecture was proposed in @xcite . [ conj : planar ] if @xmath0 is a planar subcubic graph , then @xmath18 . the conjecture has been verified for planar bipartite graphs in @xcite . in this paper we prove a surprising extension of @xcite and of conjecture [ conj : planar ] that holds for all bipartite subcubic graphs with a single exception of the heawood graph ( or disjoint union of copies of it ) . the following are our main results . [ thm : cubic bipartite ] let @xmath0 be a bipartite subcubic graph . if every connected component of @xmath0 is isomorphic to the heawood graph , then @xmath19 . in any other case , the median eigenvalues @xmath13 and @xmath14 are contained in the interval @xmath1 $ ] , i.e. , @xmath20 . theorem [ thm : cubic bipartite ] shows that the median eigenvalues @xmath21 and @xmath22 are small , but our proof can be tweaked to give much more a positive fraction of ( median ) eigenvalues lie in the interval @xmath1 $ ] . [ thm : cubic bipartite linearly many ] there is a constant @xmath23 such that for every bipartite subcubic graph @xmath0 , none of whose connected components is isomorphic to the heawood graph , all its eigenvalues @xmath24 , where @xmath25 , belong to the interval @xmath1 $ ] .

it is proved that the median eigenvalues of every connected bipartite graph of maximum degree at most three belong to the interval $ ] with a single exception of the heawood graph , whose median eigenvalues are . moreover , this surprising result has been motivated by the problem about homo - lumo separation that arises in mathematical chemistry . 2010 mathematics subject classification : 05c50

it is proved that the median eigenvalues of every connected bipartite graph of maximum degree at most three belong to the interval $ ] with a single exception of the heawood graph , whose median eigenvalues are . moreover , if is not isomorphic to the heawood graph , then a positive fraction of its median eigenvalues lie in the interval $ ] . this surprising result has been motivated by the problem about homo - lumo separation that arises in mathematical chemistry . 2010 mathematics subject classification : 05c50

1610.06458

i

several decades since the introduction of the optical fiber , channel capacity at high powers remains a vexing conundrum . existing achievable rates saturate at high powers because of linear multiplexing and treating the resulting interference as noise in network environments @xcite . furthermore , it is difficult to estimate the capacity via numerical simulations , because channel has memory . multi - user communication problem for ( an ideal model of ) optical fiber can be reduced to single - user problem using the nonlinear frequency - division multiplexing ( nfdm ) @xcite . this addresses deterministic distortions , such as inter - channel and inter - symbol interference ( signal - signal interactions ) . the problem is then reduced to finding the capacity of the point - to - point optical fiber set by noise . there are two effects in fiber that impact shannon capacity in point - to - point channels . ( 1 ) phase noise . nonlinearity transforms additive noise to phase noise in the channel . as the amplitude of the input signal tends to infinity , the phase of the output signal tends to a uniform random variable in the zero - dispersion channel ( * ? ? ? * section iv ) . as a result , phase carries finite information in the non - dispersive fiber . ( 2 ) multiplicative noise . dispersion converts phase noise to amplitude noise , introducing an effect which at high powers is similar to fading in wireless channels . importantly , the conditional entropy grows strongly with input signal . in this paper , we study the asymptotic capacity of a discrete - time periodic model of the optical fiber as the input power tends to infinity . the role of the nonlinearity in point - to - point discrete channels pertains to signal - noise interactions , captured by the conditional entropy . the main result is the following theorem , describing capacity - cost function in models with constant and non - constant loss ; see definition [ def : loss ] . consider the discrete - time periodic model of the nls channel described in section [ sec : mssfm ] , with non - zero dispersion . capacity is asymptotically rcl c(p)= ( p)+c , & non - constant loss , + p+c , & constant loss , where @xmath0 is dimension of the input signal space , @xmath11 is the average input signal power and @xmath16 . in particular , @xmath17 in finite - dimensional models . intensity modulation and direct detection ( photon counting ) is nearly capacity - achieving in the limit @xmath11 , where capacity is dominated by the first terms in @xmath9 expressions . [ thm : main ] from the theorem [ thm : main ] and ( * ? ? ? * theorem 1 ) , the asymptotic capacity of the dispersive fiber is much smaller than the asymptotic capacity of ( the discrete - time model of ) the zero - dispersion fiber , which is @xmath18 , @xmath19 . dispersion reduces the capacity , by increasing the conditional entropy . with @xmath0 dofs at the input , @xmath1 dofs are asymptotically lost to signal - noise interactions , leaving signal energy as the only useful dof for transmission . there are a finite number of dofs in all computer simulations and physical systems . however , as a mathematical problem , the following corollary holds true . capacity saturates to a constant @xmath19 in infinite - dimensional models , including the continuous - time model . [ cor : inf ] the power level where signal - noise interactions begin to appreciably impact the slope of the @xmath9 is not determined in this paper . numerical simulations indicate that the conditional entropy does not increase with input in the nonlinear fourier domain , for a range of power larger than the optimal power in wavelength - division multiplexing ( * ? ? ? * fig . 9 ( a ) ) . in this regime , signal - noise interactions are weak and the capacity is dominated by the ( large ) number @xmath13 in the theorem [ thm : main ] . a numerical estimation of the capacity of the point - to - point fiber at input powers higher than those in fig . [ fig : nfdm ] should reveal the impact of the signal - dependent noise on the asymptotic capacity . the contributions of the paper are presented as follows . the continuous - time model is discretized in section [ sec : mssfm ] . the main ingredient is a modification of the split - step fourier method ( ssfm ) that shows noise influence more directly compared with the standard ssfm . a _ unit _ is defined in the modified ssfm ( mssfm ) model that plays an important role throughout the paper . the mssfm and units simplify the information - theoretic analysis . theorem [ thm : main ] and corollary [ cor : inf ] are proved in section [ sec : proof1 ] . the main ingredient here is an appropriate partitioning of the dofs in a suitable coordinate system , and the proof that the achievable rate of one group of dofs is bounded in input . no assumption is made on input power in this first proof . theorem [ thm : main ] is proved again in section [ sec : proof2 ] by considering the limit @xmath11 , which adds further intuition . firstly , it is shown that , as the input @xmath8 , the action of the discrete periodic stochastic nonlinear schrdinger ( nls ) equation tends to multiplication by a random matrix ( with fixed probability distribution function ( pdf ) , independent of the input ) . as a result , perhaps counter - intuitively , as @xmath8 noise simplifies the nonlinear channel to a _ linear _ multiple - input multiple - output ( non - coherent ) fading channel . secondly , the asymptotic capacity is computed , without calculating the conditional pdf of the channel , entropies , or solving the capacity optimization problem . because of the multiplicative noise , the asymptotic rate depends only on the knowledge that whether channel random operator has any deterministic component . the conditional pdf merely modifies the bounded number @xmath13 in the theorem [ thm : main ] . note that we do not apply local analysis based on perturbation theories ( valid in the low power regime ) . the proof of the theorem [ thm : main ] , _ e.g. _ , the asymptotic loss of dofs , is based on a global analysis valid for any signal and noise ; see section [ sec : proof1 ] .

the main observation is that , nonlinearity introduces a multiplicative noise in the channel , similar to fading in wireless channels . is finite , so that carries finite information . this observation is used to show several results . it is shown that asymptotically as , , where is the dimension of the input signal space and is a bounded number . in particular , in finite - dimensional periodic models . secondly , it is shown that capacity saturates to a constant in infinite - dimensional models where . an expression is provided for the constant , by showing that , as the input , the action of the discrete periodic stochastic nonlinear schrdinger equation tends to multiplication by a random matrix ( with fixed distribution , independent of input ) . thus , perhaps counter - intuitively , noise simplifies the nonlinear channel at high powers to a _ linear _ multiple - input multiple - output fading channel . as signal - noise interactions gradually reduce the slope of the , to a point where increasing the input power returns diminishing gains .

it is shown that signal energy is the only available degree - of - freedom ( dof ) for fiber - optic transmission as the input power tends to infinity . with signal dofsat the input , dofs are asymptotically lost to signal - noise interactions . the main observation is that , nonlinearity introduces a multiplicative noise in the channel , similar to fading in wireless channels . the channel is viewed in the spherical coordinate system , where signal vector is represented in terms of its norm and direction . the multiplicative noise causes signal direction to vary randomly on the surface of the unit-sphere in , in such a way that the effective area of the support of does not vanish as . on the other hand , the surface area of the sphere is finite , so that carries finite information . this observation is used to show several results . firstly , let be the capacity of a discrete - time periodic model of the optical fiber with distributed noise and frequency - dependent loss , as a function of the average input power . it is shown that asymptotically as , , where is the dimension of the input signal space and is a bounded number . in particular , in finite - dimensional periodic models . secondly , it is shown that capacity saturates to a constant in infinite - dimensional models where . an expression is provided for the constant , by showing that , as the input , the action of the discrete periodic stochastic nonlinear schrdinger equation tends to multiplication by a random matrix ( with fixed distribution , independent of input ) . thus , perhaps counter - intuitively , noise simplifies the nonlinear channel at high powers to a _ linear _ multiple - input multiple - output fading channel . as signal - noise interactions gradually reduce the slope of the , to a point where increasing the input power returns diminishing gains . nonlinear frequency - division multiplexing can be applied to approach capacity in optical networks , where linear multiplexing achieves low rates at high powers .

1610.06458

c

the asymptotic capacity of the discrete - time periodic model of the optical fiber is characterized as a function of the input power in theorem [ thm : main ] . with @xmath0 signal dofsat the input , @xmath1 dofs are asymptotically lost , leaving signal energy as the only available dof for transmission . the appropriate input distribution is a log - normal pdf for the signal norm . signal - noise interactions limit the operation of the optical communication systems to low - to - medium powers .

it is shown that signal energy is the only available degree - of - freedom ( dof ) for fiber - optic transmission as the input power tends to infinity . with signal dofsat the input , dofs are asymptotically lost to signal - noise interactions . firstly , let be the capacity of a discrete - time periodic model of the optical fiber with distributed noise and frequency - dependent loss , as a function of the average input power .

it is shown that signal energy is the only available degree - of - freedom ( dof ) for fiber - optic transmission as the input power tends to infinity . with signal dofsat the input , dofs are asymptotically lost to signal - noise interactions . the main observation is that , nonlinearity introduces a multiplicative noise in the channel , similar to fading in wireless channels . the channel is viewed in the spherical coordinate system , where signal vector is represented in terms of its norm and direction . the multiplicative noise causes signal direction to vary randomly on the surface of the unit-sphere in , in such a way that the effective area of the support of does not vanish as . on the other hand , the surface area of the sphere is finite , so that carries finite information . this observation is used to show several results . firstly , let be the capacity of a discrete - time periodic model of the optical fiber with distributed noise and frequency - dependent loss , as a function of the average input power . it is shown that asymptotically as , , where is the dimension of the input signal space and is a bounded number . in particular , in finite - dimensional periodic models . secondly , it is shown that capacity saturates to a constant in infinite - dimensional models where . an expression is provided for the constant , by showing that , as the input , the action of the discrete periodic stochastic nonlinear schrdinger equation tends to multiplication by a random matrix ( with fixed distribution , independent of input ) . thus , perhaps counter - intuitively , noise simplifies the nonlinear channel at high powers to a _ linear _ multiple - input multiple - output fading channel . as signal - noise interactions gradually reduce the slope of the , to a point where increasing the input power returns diminishing gains . nonlinear frequency - division multiplexing can be applied to approach capacity in optical networks , where linear multiplexing achieves low rates at high powers .

1003.0236

i

our current understanding of cosmology and high energy physics leaves many questions unanswered . one of the most fundamental of these questions is why our universe has three large dimensions . this may be tied to the more general question of the overall shape and structure of the universe . in fact , it is possible that our universe was not always three dimensional or that other places outside of our observable universe have a different dimensionality . there are surely long - lived vacua where one or more of our three dimensions are compactified , since this does not even rely on the presence of extra - dimensions and indeed happens in the standard model @xcite . eternal inflation can provide a means to populate these vacua , and naturally leads to a highly inhomogeneous universe on very long length scales . further , it seems likely that these lower - dimensional vacua are at least as numerous as three dimensional ones since there are generally more ways to compactify a greater number of spatial dimensions . if we do indeed have a huge landscape of vacua ( e.g. @xcite ) then it seems all the more reasonable that there should be vacua of all different dimensionalities and transitions between them ( see e.g. @xcite ) . we will ignore the subtle issues of the likelihood of populating those vacua ( the measure problem " ) . instead we will focus on the possibility of observing such regions of lower dimensionality since surely such a discovery would have a tremendous effect on our understanding of cosmology and fundamental physics . our compact dimensions are generically unstable to decompactification @xcite . thus it seems possible that the universe began with all the dimensions compact ( the starting point in @xcite for example ) . in this picture our current universe is one step in the chain towards decompactifying all dimensions . of course , eternal inflation may lead to a very complicated history of populating different vacua , but in any case , it seems reasonable to consider the possibility that we came from a lower dimensional ancestor " vacuum . we will assume that prior to our last period of slow - roll inflation our patch of the universe was born in a transition from a lower dimensional vacuum . our universe then underwent the normal period of slow - roll inflation . for our signals to be observable we will assume that there were not too many more than the minimal number of efolds of inflation necessary to explain the cmb sky . this may be reasonable because this is very near a catastrophic boundary : large scale structures such as galaxies would not form if inflation did not last long enough to dilute curvature sufficiently @xcite . since achieving slow - roll inflation is difficult and the longer it lasts the more tuned the potential often is , there may be a pressure to be close to this lower bound on the length of inflation . we will actually use the energy density in curvature , @xmath0 , in place of the number of efolds of inflation . the observational bound requires that @xmath1 today ( this corresponds to @xmath2 efolds for high scale inflation ) . the existence of galaxies requires @xmath3 today ( corresponding to @xmath4 efolds if we use the bound from @xcite ) . thus @xmath0 may be close to the observational bound today . other , similar arguments have also been made for a relatively large curvature today @xcite . most signals of the presence of other vacua , e.g. bubble collisions @xcite , also rely on this assumption . these signals have also mostly been explored assuming that the other vacua are all 3 + 1 dimensional . while an important first step , this seems like a serious oversimplification . we find interesting differences in the case that our parent vacuum was lower dimensional . in particular , our universe can be anisotropic , with different spatial curvatures in the different directions . this anisotropic curvature dilutes exponentially during inflation , making the universe appear very isotropic at early times . however , this curvature ( @xmath0 ) grows at late times , leading to several observable effects . this anisotropic curvature sources an anisotropy in the hubble expansion rate , since the different dimensions expand at different rates . the most interesting signal is an anisotropy in the normal cmb curvature measurement . the angular size of a standard ruler " now appears to depend on the orientation of that ruler . in the cmb this shows up as unexpected correlations between modes of all angular sizes . unlike the normal curvature measurement , this anisotropic curvature measurement is not degenerate with the scale factor expansion history and is thus easier to measure . this anistropic curvature also leads to a significant quadrupolar anisotropy in the cmb which constrains the size of @xmath0 . there are possibly other observables from 21 cm measurements , direct measurements of the hubble expansion ( e.g. from supernovae ) , or from searches looking for nontrivial topology of the universe .

it seems generic to have vacua with lower dimensionality than ours . we consider the possibility that the observable universe originated in a transition from one of these vacua . further , this anisotropic curvature causes different dimensions to expand at different rates . this leads to other potentially observable signals including a quadrupolar anisotropy in the cmb which limits the size of the curvature .

it seems generic to have vacua with lower dimensionality than ours . we consider the possibility that the observable universe originated in a transition from one of these vacua . such a universe has anisotropic spatial curvature . this may be directly observable through its late - time effects on the cmb if the last period of slow - roll inflation was not too long . these affect the entire sky , leading to correlations which persist up to the highest cmb multipoles , thus allowing a conclusive detection above cosmic variance . further , this anisotropic curvature causes different dimensions to expand at different rates . this leads to other potentially observable signals including a quadrupolar anisotropy in the cmb which limits the size of the curvature . conversely , if isotropic curvature is observed it may be evidence that our parent vacuum was at least 3 + 1 dimensional . such signals could reveal our history of decompactification , providing evidence for the existence of vastly different vacua .

1003.0236

c

a universe produced as a result of bubble nucleation from an ancestor vacuum which has two large dimensions and one small , compact dimension is endowed with anisotropic curvature @xmath0 . such an anisotropic universe is also produced in the case when our 3 + 1 dimensional universe emerges from a transition from a 1 + 1 dimensional vacuum . in this case , depending upon the curvature of the compact dimensions , the resulting universe can have either positive or negative curvature along two dimensions , with the other remaining flat . the geometry of the equal time slices of the daughter universe are such that two of the directions are curved while the other dimension is flat . immediately after the tunneling event , the energy density of the universe is dominated by this anisotropic curvature @xmath0 . this curvature drives the curved directions to expand differently from the flat direction , resulting in differential hubble expansion @xmath42 between them . the expansion of the universe dilutes @xmath0 until it becomes small enough to allow slow roll inflation . at this time , the universe undergoes a period of inflation during which the curvature @xmath0 and the differential hubble expansion @xmath42 are exponentially diluted . however , during the epochs of radiation and matter domination , the curvature red shifts less strongly than either the radiation or the matter density . consequently , the fractional energy density @xmath0 in curvature grows with time during these epochs . this late time emergence of an anisotropic curvature @xmath0 also drives a late time differential hubble expansion @xmath42 in the universe . these late time , anisotropic warps of the space - time geometry are all proportional to the current fractional energy density in curvature , @xmath79 . they can be observed in the present epoch if inflation does not last much longer than the minimum number of efolds required to achieve a sufficiently flat universe ( @xmath235 efolds for high scale inflation ) . anisotropic curvature leads to the warping of the angular size of standard rulers . this warping is a function of both the angle and orientation of the ruler in the sky . consequently , this effect is immune to degeneracies from the expansion history of the universe since it affects rulers that are along the same line of sight but oriented differently . the cmb is also warped by the anisotropic curvature . in addition to the geometric warping , the differential hubble expansion @xmath42 also preferentially red shifts the energies of the cmb photons . this energy shift differentially changes the monopole temperature of the cmb giving rise to a quadrupole in the cmb . furthermore , since the anisotropic curvature is a late time effect , it affects all the modes that can be seen in the cmb . consequently , this effect leads to statistical anisotropy on all angular scales . this effect is different from other signatures of anisotropy considered in the literature @xcite . previous work has concentrated on the correlations that are produced due to the initial anisotropy in the universe at the beginning of inflation . since these modes are roughly stretched to the hubble size today , these initial anisotropies only affect the largest modes in the sky and are hence low l effects in the cmb . the late time anisotropy however warps the entire sky and leads to statistically robust correlations on all angular scales . the anisotropies in the pre - inflationary vacuum can however lead to other interesting signatures , for example in the gravitational wave spectrum @xcite . these signatures are an independent check of this scenario . anisotropies that affect all angular scales have also been previously considered @xcite . these required violations of rotational invariance during inflation and the anisotropy emerges directly in the primordial density perturbations . in our case , the density perturbations are isotropic and the anisotropy observed today is a result of a late time warp of the space - time . anisotropic curvature is already more stringently constrained than isotropic curvature . while isotropic curvature is bounded to be @xmath161 , it is difficult for anisotropic curvature to be much larger than @xmath160 without running afoul of current data , in particular , the size of the cmb quadrupole . since the measurement of curvature is ultimately limited by cosmic variance @xmath144 , there is a window between @xmath236 that can be probed by upcoming experiments , including planck . future cosmological measurements like the 21 cm experiments will significantly improve bounds on the curvature of the universe . a discovery of isotropic curvature would be evidence suggesting that our ancestor vacuum had at least three large space dimensions . on the other hand , a discovery of anisotropic curvature would be strong evidence for the lower dimensionality of our parent vacuum . the anisotropy produced from such a transition has a very specific form due to the symmetries of the transition . it leads to correlations only amongst certain modes in the cmb ( for example , only @xmath220 and @xmath221 ) . this distinguishes it from a generic anisotropic @xmath61 dimensional pre - inflationary vacuum which will generically have power in all modes . in these scenarios , it is also natural for the universe to have non - trivial topology . the existence of a non - trivial topological scale within our observable universe can be searched for using the classic circles in the sky " signal . if both the non - trivial topology and anisotropic curvature can be discovered , implying a period of inflation very close to the catastrophic boundary , it would be powerful evidence for a lower dimensional ancestor vacuum . a discovery of these effects would establish the existence of vacua vastly different from our own standard model vacuum , lending observational credence to the landscape .

such a universe has anisotropic spatial curvature these affect the entire sky , leading to correlations which persist up to the highest cmb multipoles , thus allowing a conclusive detection above cosmic variance . conversely , if isotropic curvature is observed it may be evidence that our parent vacuum was at least 3 + 1 dimensional .

it seems generic to have vacua with lower dimensionality than ours . we consider the possibility that the observable universe originated in a transition from one of these vacua . such a universe has anisotropic spatial curvature . this may be directly observable through its late - time effects on the cmb if the last period of slow - roll inflation was not too long . these affect the entire sky , leading to correlations which persist up to the highest cmb multipoles , thus allowing a conclusive detection above cosmic variance . further , this anisotropic curvature causes different dimensions to expand at different rates . this leads to other potentially observable signals including a quadrupolar anisotropy in the cmb which limits the size of the curvature . conversely , if isotropic curvature is observed it may be evidence that our parent vacuum was at least 3 + 1 dimensional . such signals could reveal our history of decompactification , providing evidence for the existence of vastly different vacua .

1206.6564

r

observational constraints suggest that turbulent magnetic power in inertial range turbulence in the solar wind consists of approximately 90% alfvn waves , and the remaining 10% of the power in some mixture of the compressible kinetic fast and slow waves @xcite . numerical simulations of compressible mhd turbulence suggest that the distribution of turbulent power in wavevector space is isotropic for the fast waves and critically balanced for the alfvn and slow waves @xcite . therefore , we believe the most realistic model of solar wind turbulence using synthetic spacecraft data is given by the lower right - hand plot in figure [ fig : ssd_nbpar ] . the analysis of the density - parallel magnetic field correlation using 10 years of _ wind _ spacecraft data is discussed in detail in a companion work @xcite , so we give here only a few brief details . the density - parallel magnetic field correlation @xmath178 is computed using measurements from the magnetic field investigation ( mfi ) @xcite and the three dimensional plasma ( 3dp ) experiment @xcite on the _ wind _ spacecraft in the unperturbed solar wind at 1 au during the years 1994 - 2004 . using 300-s intervals of ambient solar wind data ( corresponding to inertial range scales of approximately @xmath179 ) , the proton density and magnetic field measurements at 3 s cadence are decimated by a factor of 10 ( to 30 s cadence ) . magnetic field measurements are rotated to a field - aligned coordinate system , defined by the local mean field direction computed using 100-s windows , to compute @xmath116 , and proton density data @xmath180 is detrended over the same time intervals . the zero - lag cross - correlation @xmath151 is computed for 119,512 data intervals . a joint histogram of @xmath113 normalized in each @xmath31 bin is generated , and the peak histogram values and fwhm error bars are plotted on top of the theoretical synthetic spacecraft data plots of @xmath113 from mhd ( left ) and kinetic ( right ) theory in figure [ fig : money ] . for this direct comparison to the data , the theoretical results from the synthetic spacecraft data are computed in the same manner as the spacecraft measurements . in particular , the synthetic plasma volume is sampled at only @xmath181 uniformly spaced points along a trajectory of length @xmath182 , which corresponds to a range of scales @xmath183 . the smaller number of timesteps and shorter total sampled length leads to a larger standard deviation using 256 ensembles than the vlasov - maxwell results in [ sec : ssd_predict ] and to slight quantitative changes in the @xmath113 curves . the agreement between the @xmath113 from _ wind _ spacecraft data and synthetic data curve for @xmath141 in figure [ fig : money ] is striking , indicating that the observed correlation is consistent with a statistically negligible kinetic fast wave energy contribution for the large sample used in this study @xcite . we do note that a very small fraction of the intervals have @xmath184 , possibly indicating a minority population of fast waves . as discussed in our companion paper @xcite , this result has important consequences for the turbulent cascade of energy from large to small scales : since only the fast wave turbulent cascade is expected to nonlinearly transfer energy to whistler waves at @xmath185 , and the frequency mismatch between the fast and either the alfvn or slow waves should prevent non - linear coupling , our analysis suggests that there is little or no transfer of large scale turbulent energy through the inertial range down to whistler waves at small scales . comparison of the observationally measured @xmath186 in figure [ fig : money ] with both the mhd and kinetic results leads to another important conclusion of this study : the nature of the compressible fluctuations in the solar wind inertial range is not well modeled by mhd theory . this result is not surprising given that the solar wind is weakly collisional at inertial range scales , so a kinetic description is necessary to model accurately the compressible fluctuations . our study also enables us to constrain the wavevector distribution of slow wave energy by comparing the observational results with the three plots in the bottom row of figure [ fig : ssd_nbpar ] . the leftmost plot has an isotropic distribution of slow wave power and shows two features in the @xmath141 curve not seen in the observed data : ( 1 ) a bump at @xmath44 and ( 2 ) a slight increase in the correlation value at @xmath187 . these quantitative changes in the @xmath141 curve appear the same with or without the alfvn waves at @xmath188 ( compare the left plots in the third and fourth row of figure [ fig : ssd_nbpar ] ) , so this effect is not due to the isotropic distribution of alfvn waves . therefore , the slow waves appear to be anisotropically distributed , and may be well described by critical balance as suggested by theories of slow wave passive advection @xcite . however , the uncertainty of the wind measurements can not definitively rule out the possibility of an isotropic distribution for the slow mode . one of the key points of our companion work @xcite is the interpretation that the compressible fluctuations in the solar wind consist of a critically balanced distribution of kinetic slow wave fluctuations . previous analyses have generally dismissed the possibility of kinetic slow waves because , in an isotropic maxwellian plasma with warm ions , the collisionless damping via free - streaming along the magnetic field is strong @xcite . however , the damping rate of the slow waves for the nearly perpendicular wavevectors of a critically balanced distribution is proportional to the parallel component of the wavevector , @xmath189 . this feature can be seen in the right column of figure [ fig : polar ] , where the slow wave damping rate ( blue ) is zero at perpendicular wavevectors @xmath36 ( open circles ) and increases as the wavevector angle decreases . for exactly perpendicular wavevectors , the damping rate drops to zero this perpendicular limit of the slow wave corresponds to an undamped , non - propagating pressure - balanced structure ( pbs ) . it has been derived theoretically by @xcite and demonstrated numerically by @xcite that the alfvn wave dynamics advects and cascades the slow waves , so the energy cascade rate of the slow waves is related not to the slow wave frequency , but to the alfvn wave frequency . therefore , although the slow wave fluctuations at the high @xmath190 boundary of critical balanced distribution ( along @xmath191 ) may suffer strong collisionless damping , the more nearly perpendicular slow waves may be cascaded to smaller scales on the timescale of the alfvnic turbulence , while the collisionless damping of these modes remains weak . this could lead to a slow wave energy distribution that is _ more _ anisotropic than critical balance , possibly with @xmath192 . using synthetic spacecraft data , we can test whether this picture of a distribution of slow wave power that is more anisotropic than critical balance is consistent with the observational findings in figure [ fig : money ] . to do this , we repeat the analysis in the lower right plot of figure [ fig : ssd_nbpar ] but with a more anisotropic distribution of slow waves given by @xmath193 . the results of this test , shown in figure [ fig : icb - ecb ] , demonstrate that the @xmath141 curve does not change significantly for a more anisotropic slow wave distribution . therefore , the picture of the solar wind compressible fluctuations comprised of slow wave fluctuations that are more anisotropically distributed than critical balance is consistent with the observational data . one of the limitations of this investigation is that the vlasov - maxwell eigenfunctions are computed assuming isotropic maxellian equilibrium distributions . plasma measurements in the solar wind frequently find anisotropic temperature distributions and non - maxwellian tails at high energy @xcite . in particular , it is possible that mirror modes may contribute to the measured anti - correlation between the density and parallel magnetic field , although an analysis of 10 years of _ wind _ data show that only a small fraction of the solar wind measurements occupy the region of parameter space near the mirror instability threshold @xcite . how the density - parallel magnetic field correlation is altered by these conditions is an important issue to be addressed in future work .

it is demonstrated that mhd theory is inadequate to describe the compressible turbulent fluctuations and that the observed density - parallel magnetic field correlation is consistent with a statistically negligible kinetic fast wave energy contribution for the large sample used in this study . a model of the solar wind inertial range fluctuations these results imply that there is little or no transfer of large scale turbulent energy through the inertial range down to whistler waves at small scales .

kinetic plasma theory is used to generate synthetic spacecraft data to analyze and interpret the compressible fluctuations in the inertial range of solar wind turbulence . the kinetic counterparts of the three familiar linear mhd wave modes the fast , alfvn , and slow waves are identified and the properties of the density - parallel magnetic field correlation for these kinetic wave modes is presented . the construction of synthetic spacecraft data , based on the quasi - linear premise that some characteristics of magnetized plasma turbulence can be usefully modeled as a collection of randomly phased , linear wave modes is described in detail . theoretical predictions of the density - parallel magnetic field correlation based on mhd and vlasov - maxwell linear eigenfunctions are presented and compared to the observational determination of this correlation based on 10 years of__wind _ _ spacecraft data . it is demonstrated that mhd theory is inadequate to describe the compressible turbulent fluctuations and that the observed density - parallel magnetic field correlation is consistent with a statistically negligible kinetic fast wave energy contribution for the large sample used in this study . a model of the solar wind inertial range fluctuations is proposed comprised of a mixture of a critically balanced distribution of incompressible alfvnic fluctuations and a critically balanced or more anisotropic than critical balance distribution of compressible slow wave fluctuations . these results imply that there is little or no transfer of large scale turbulent energy through the inertial range down to whistler waves at small scales .

1206.6564

c

in this paper , we have presented the first application of kinetic plasma theory to analyze and interpret the compressible fluctuations in the inertial range of solar wind turbulence . this novel approach is motivated by the fact that the dynamics in the solar wind plasma is weakly collisional , a limit in which mhd theory , used in all previous related investigations , is formally invalid . investigation of the compressible fast and slow wave modes requires a kinetic description to resolve both the wave dynamics and the collisionless kinetic damping mechanisms . we identify quantitatively the linear kinetic wave modes of vlasov - maxwell theory that correspond to the linear mhd fast and slow wave modes , and verify that the general qualitative properties of the density - parallel magnetic field correlation @xmath126that fast waves are positively correlated , and slow waves are negatively correlated remain unchanged for the weakly collisional conditions of the solar wind plasma . we then describe the procedure used to generate synthetic spacecraft data used to interpret actual single - point spacecraft measurements of turbulence in the solar wind . we define and discuss the _ quasi - linear premise_that some properties of magnetized plasma turbulence can be understood by modeling the turbulence as a collection of randomly phased , linear waves upon which the method of synthetic spacecraft data is based . theoretical arguments for the validity of the quasi - linear premise are presented ; a review of supporting and conflicting evidence for the quasi - linear premise in the literature is presented in @xcite . we outline how the synthetic plasma data cubes are used to generate synthetic reduced time series of single - point measurements that can be analyzed using the same procedures as the actual spacecraft measurements . comparison of the results of the analyses of both the synthetic and actual spacecraft data enables novel physical interpretations of the solar wind turbulence measurements @xcite . next , we use the synthetic spacecraft data method to predict the characteristics of the density - parallel magnetic field correlation @xmath132 as a function of plasma @xmath31 in the presence of a turbulent spectrum of wave modes . comparison of these synthetic predictions of @xmath132 , based on both linear mhd and linear vlasov - maxwell eigenfunctions , to the results from an analysis of a 10-year data set of observations from the _ wind _ spacecraft leads to the following conclusions : 1 . the predicted @xmath132 and its dependence on plasma @xmath31 using linear mhd eigenfunctions is significantly different from the prediction using linear vlasov - maxwell eigenfunctions . only the prediction based on kinetic theory appears to agree with the spacecraft measurements , leading to the expected conclusion that mhd theory is inadequate to describe the compressible fluctuations in the weakly collisional solar wind . 2 . strong _ a posteriori _ evidence for the validity of the quasi - linear premise is provided by the striking agreement between the observationally determined @xmath132 over a very large statistical sample and the predicted @xmath132 based on synthetic spacecraft data . the observed @xmath132 computed in a companion work @xcite is consistent with a statistically negligible kinetic fast wave energy contribution for the large sample used in this study . note , however , that the our companion work also found that a very small fraction of the intervals have @xmath184 , possibly indicating a trace population of fast waves @xcite . the quantitative dependence of @xmath132 on the ion plasma beta @xmath31 provides evidence that the slow wave fluctuations are not isotropically distributed , but rather have an anisotropic distribution , that is possibly given by the condition of critical balance @xcite or that is more anisotropic than critical balance . in conclusion , our analysis using kinetic theory to interpret the compressible fluctuations motivates the following physical model of the turbulent fluctuations in the solar wind inertial range . in this model , the solar wind inertial range consists of a mixture of turbulent fluctuations , with 90% of the energy due to incompressible alfvnic fluctuations and the remaining 10% of the energy due to compressible slow wave fluctuations . the alfvnic turbulent power is distributed anisotropically in wave vector space according to critical balance . the turbulent alfvn wave dynamics advects and cascades the slow wave fluctuations to smaller scales at the alfvn wave frequency . the slow wave turbulent power may be either critically balanced , or more anisotropic than critical balance due to collisionless damping of the slow wave fluctuations . since only the fast wave turbulent cascade is expected to nonlinearly transfer energy to whistler waves at @xmath185 , and the frequency mismatch between the fast and either the alfvn or slow waves should prevent non - linear coupling , there is little or no transfer of large scale turbulent energy through the inertial range down to whistler waves at small scales . therefore , any whistler wave fluctuations at scales @xmath185 must be generated by some other process , _ e.g. _ , kinetic temperature anisotropy instabilities @xcite or kinetic drift instabilities driven by differential flow between protons and alpha particles @xcite . finally , the lack of statistically significant fast wave energy has important implications for efficient numerical modeling of solar wind turbulent fluctuations . this work demonstrates clearly the importance of a kinetic approach to model adequately the turbulent fluctuations , yet a general kinetic numerical treatment_e.g . _ , the particle - in - cell ( pic ) method in three spatial dimensions ( required for physically relevant modeling of the dominant nonlinear interactions in solar wind turbulence @xcite ) is too computationally costly to be presently feasible . fortunately , it is possible to perform kinetic numerical simulations of solar wind turbulence in three spatial dimensions using gyrokinetics , a rigorous , low - frequency , anisotropic limit of kinetic theory @xcite . in the derivation of the gyrokinetic equation , the crucial step is an averaging over the particle gyrophase , which leads to a theory with the following properties : the fast / whistler wave and the cyclotron resonances are discarded ; all finite larmor radius effects and collisionless dissipation via the landau resonance are retained ; and one of the dimensions of velocity in phase space is eliminated , reducing the particle distribution function from six to five dimensions . it has been previously pointed out that one can not rule out the contribution of fast wave or whistler wave physics to solar wind turbulence , and that that therefore gyrokinetics is an incomplete description of the turbulence @xcite . the novel observational analysis presented here suggests that the fast wave , in fact , does not play a statistically significant role in the turbulent dynamics of the inertial range , and that therefore a gyrokinetic approach sufficiently describes all important physical mechanisms in the solar wind inertial range . k. g. k. and g. g. h. thank jack scudder for inspiring discussions . the work has been supported by nsf career award ags-1054061 and nasa nnx10ac91 g .

kinetic plasma theory is used to generate synthetic spacecraft data to analyze and interpret the compressible fluctuations in the inertial range of solar wind turbulence . the kinetic counterparts of the three familiar linear mhd wave modes the fast , alfvn , and slow waves are identified and the properties of the density - parallel magnetic field correlation for these kinetic wave modes is presented . theoretical predictions of the density - parallel magnetic field correlation based on mhd and vlasov - maxwell linear eigenfunctions are presented and compared to the observational determination of this correlation based on 10 years of__wind _ _ spacecraft data . is proposed comprised of a mixture of a critically balanced distribution of incompressible alfvnic fluctuations and a critically balanced or more anisotropic than critical balance distribution of compressible slow wave fluctuations .

kinetic plasma theory is used to generate synthetic spacecraft data to analyze and interpret the compressible fluctuations in the inertial range of solar wind turbulence . the kinetic counterparts of the three familiar linear mhd wave modes the fast , alfvn , and slow waves are identified and the properties of the density - parallel magnetic field correlation for these kinetic wave modes is presented . the construction of synthetic spacecraft data , based on the quasi - linear premise that some characteristics of magnetized plasma turbulence can be usefully modeled as a collection of randomly phased , linear wave modes is described in detail . theoretical predictions of the density - parallel magnetic field correlation based on mhd and vlasov - maxwell linear eigenfunctions are presented and compared to the observational determination of this correlation based on 10 years of__wind _ _ spacecraft data . it is demonstrated that mhd theory is inadequate to describe the compressible turbulent fluctuations and that the observed density - parallel magnetic field correlation is consistent with a statistically negligible kinetic fast wave energy contribution for the large sample used in this study . a model of the solar wind inertial range fluctuations is proposed comprised of a mixture of a critically balanced distribution of incompressible alfvnic fluctuations and a critically balanced or more anisotropic than critical balance distribution of compressible slow wave fluctuations . these results imply that there is little or no transfer of large scale turbulent energy through the inertial range down to whistler waves at small scales .

1604.04754

i

lattice networks are frequently employed to describe the mechanical response of materials and structures that are discrete by nature at one or more length scales , such as 3d - printed structures , woven textiles , paper , or foams . for lattice networks representing fibrous microstructures for instance , individual fibres can be identified with one - dimensional springs or beams . further examples are the models of e.g. @xcite , @xcite , @xcite , and @xcite . the reason why lattice models may be preferred over conventional continuum theories and finite element ( fe ) approaches , is twofold . first , the meaning and significance of the physical parameters associated with individual interactions in the lattice networks is easy to understand , whereas the parameters in constitutive continuum models represent the small - scale mechanics only in a phenomenological manner . an example is the young s modulus or ultimate strength of a spring or beam ( fibre or yarn ) versus that of the network . second , the formulation and implementation of lattice models is generally significantly easier compared to that of alternative continuum models . large deformations , large yarn re - orientations , and fracture are for instance easier to formulate and implement ( cf . e.g. the continuum model of @xcite that deals with large yarn re - orientations ) . thanks to the simplicity and versatility of lattice networks , they are furthermore used for the description of heterogeneous cohesive - frictional materials . the reason is that discrete models can realistically represent distributed microcracking with gradual softening , implement material structure with inhomogeneities , capture non - locality of damage processes , and reflect deterministic or stochastic size effects . examples of the successful use of lattice models for such materials are given by @xcite , @xcite , @xcite , and @xcite . as lattice models are typically constructed at the meso- , micro- , or nano - scale , they require reduced - model techniques to allow for application - scale simulations . a prominent example is the quasicontinuum ( qc ) method , which specifically aims at discrete lattice models . the qc method was originally introduced for conservative atomistic systems by @xcite and extended in numerous aspects later on , see e.g. @xcite . subsequent generalizations for lattices with dissipative interactions ( e.g. plasticity and bond sliding ) were provided by @xcite . in principle , the qc is a numerical procedure that can deal with local lattice - level phenomena in small regions of interest , whereas the lattice model is coarse grained in the remainder of the domain . the aim of this contribution is to develop a qc framework that can deal with the initiation and subsequent propagation of damage and fracture in the underlying structural lattice model . because such a phenomenon tends to be a highly localized and rather unstable process , sensitive to local mesh details , the qc framework must fully refine in critical regions _ before _ any damage occurs in order to capture the physics properly , cf . [ sect : introduction : fig:1 ] . moreover , as the entire problem is evolutionary , the location of the fully resolved region must evolve as well , which requires an adaptive qc framework . several previous studies have also focused on adaptivity in qc methodologies , but they were always dealing with atomistic lattice models at the nano - scale , see e.g. @xcite , @xcite , @xcite . contrary to that , this contribution focuses on structural lattice networks for materials with discreteness at the meso - scale . the qc approach aimed for is schematically presented in fig . [ sect : introduction : fig:1 ] . the macro - scale fracture emerges as individual interactions failures at the lattice level . their damage leads to strain - softening and hence , the fracture process zone remains spatially localized . consequently , only the crack tip and the process zone have to be fully resolved . the displacement fluctuations elsewhere remain small , allowing for efficient interpolation and coarse graining . due to the spatial propagation of the crack front through the system of interest , available qc formulations need to be generalized to involve dissipation induced by damage , and an adaptive meshing scheme that includes a suitable marking strategy needs to be developed . the theoretical framework employed in this contribution is closely related to our variational qc formulation for hardening plasticity discussed in @xcite . the present work can be viewed as an extension towards lattices with localized damage and with an adaptive refinement strategy . in principle , the overall procedure is based on the variational formulation by @xcite , developed for _ rate - independent _ inelastic systems . this variational formulation employs at each time instant @xmath0 an incremental potential energy @xmath1 , that can be minimized with respect to the observable ( kinematic ) as well as the internal ( history , dissipative ) variables , i.e. @xmath2 where @xmath3 denotes a generalized state variable . hence , this formulation is different from the one employed in the virtual - power - based qc framework of @xcite , which is based on the virtual - power statement of the lattice model in combination with a coleman - noll procedure . note that earlier variational formulations for rate - independent systems can be found e.g. in @xcite , @xcite , @xcite , @xcite , and @xcite . the theoretical concepts of the variational formulation and its application to damaging lattice models are discussed in section [ sect : varform ] . after the incremental energy is presented for the full lattice system , two reduction steps can be applied to it in analogy to the standard qc framework , see e.g. @xcite , @xcite , @xcite . in the first step , _ interpolation _ constrains the displacements of all atoms according to the displacements of a number of selected representative atoms , or _ repatoms _ for short . this procedure reduces the number of degrees of freedom drastically . in the second step , only a small number of atoms is sampled to approximate the exact incremental energy @xmath1 , its gradients , and hessians , analogously to the numerical integration of fe technology . this step is referred to as _ summation _ , and it entails also a significant reduction of the number of internal variables . together , the two steps yield a reduced state variable @xmath4 and an approximate incremental energy @xmath5 . in section [ sect : qc ] , a more detailed discussion about qc techniques , adaptive modelling , marking strategy , and mesh refinement will be provided . consequences of the proposed mesh refinement strategy will be discussed also from the energetic point of view . the minimization problem provides governing equations presented in section [ sect : sol ] , where a suitable solution strategy is described . incremental energy minimization procedures often employ some version of the alternating minimization ( am ) strategy , see e.g. @xcite . the approach used here , however , minimizes the so - called _ reduced energy _ , i.e. the energy potential @xmath1 with eliminated internal variables , cf . @xcite or @xcite . as a result , the overall solution process simplifies and is more efficient compared to the am approach . in section [ sect : examples ] , the proposed theoretical developments are first applied to an l - shaped plate test . the force - displacement diagrams and crack paths predicted with the adaptive qc approach are compared to those predicted with full lattice computations . further , the energy consistency during the entire evolution is assessed and the errors in energies are discussed . the second numerical example focuses on the antisymmetric four - point bending test , described e.g. in @xcite . it demonstrates the ability of the adaptive qc scheme to predict nontrivial , curved crack paths . finally , this contribution closes with a summary and conclusions in section [ sect : conclusion ] .

lattice networks with dissipative interactions can be used to describe the mechanics of discrete micro- or meso - structures of materials such as 3d - printed structures and foams , or more generally heterogeneous materials . this contribution deals with the crack initiation and propagation in such materials and focuses on an adaptive multiscale approach that captures the spatially evolving fracture . lattice networks naturally incorporate non - locality , large deformations , and dissipative mechanisms taking place directly inside fracture zones . because the physically relevant length scales for fracture are significantly larger than those of individual interactions , discrete models are computationally expensive . the quasicontinuum ( qc ) method is a multiscale approach specifically constructed for discrete models . this method reduces the computational cost by fully resolving the underlying lattice only in regions of interest , while coarsening elsewhere . in this contribution , an incremental energy potential corresponding to the full lattice model with damageable interactions is constructed for engineering - scale predictions . subsequently , using a variational qc approach the incremental energy along with governing equations for the upscaled lattice network are derived . in order to deal with the spatially evolving fracture zone in an accurate , yet efficient manner , an adaptive scheme is proposed that utilizes the underlying lattice . implications induced by the adaptive procedure are discussed from the energy - consistency point of view , and theoretical considerations are demonstrated on two examples . lattice networks , quasicontinuum method , damage , adaptivity , variational formulation , multiscale modelling

lattice networks with dissipative interactions can be used to describe the mechanics of discrete micro- or meso - structures of materials such as 3d - printed structures and foams , or more generally heterogeneous materials . this contribution deals with the crack initiation and propagation in such materials and focuses on an adaptive multiscale approach that captures the spatially evolving fracture . lattice networks naturally incorporate non - locality , large deformations , and dissipative mechanisms taking place directly inside fracture zones . because the physically relevant length scales for fracture are significantly larger than those of individual interactions , discrete models are computationally expensive . the quasicontinuum ( qc ) method is a multiscale approach specifically constructed for discrete models . this method reduces the computational cost by fully resolving the underlying lattice only in regions of interest , while coarsening elsewhere . in this contribution , an incremental energy potential corresponding to the full lattice model with damageable interactions is constructed for engineering - scale predictions . subsequently , using a variational qc approach the incremental energy along with governing equations for the upscaled lattice network are derived . in order to deal with the spatially evolving fracture zone in an accurate , yet efficient manner , an adaptive scheme is proposed that utilizes the underlying lattice . implications induced by the adaptive procedure are discussed from the energy - consistency point of view , and theoretical considerations are demonstrated on two examples . the first one serves as a proof of concept , illustrates the consistency of the adaptive schemes , and presents errors in energies compared to the fully - resolved solution . the second one demonstrates the performance of the adaptive qc scheme for a more complex problem for which the fully resolved solution is not accessible . lattice networks , quasicontinuum method , damage , adaptivity , variational formulation , multiscale modelling

1509.01332

i

information - theoretic arguments often involve random coding techniques to prove the existence of good codes . usually , the codebook is constructed by randomly choosing the components of each codeword independently and identically from a judiciously chosen probability distribution . while this technique is powerful , the resulting codebooks do not exhibit any structure that may be of practical interest . one such desirable structure is linearity , which allows complexity reductions at the encoder and decoder by utilizing efficient algebraic processing techniques . further , in certain communication scenarios linear codes achieve higher rates than random code ensembles , as was shown by krner and marton @xcite for a distributed source coding problem . structured coding schemes have been widely studied in the literature , especially for communications in the presence of side information and in multi - terminal networks . for an overview of structured coding schemes we refer the reader to @xcite and the references therein . for communication in the wireless domain , structured codes can be obtained by choosing finite subsets of points from lattices @xcite . a lattice is an infinite discrete set of points in the euclidean space that are regularly arranged and are closed under addition . codes based on lattices , known as _ ( nested ) lattice codes _ or _ voronoi codes _ , are the analogues of linear codes in wireless communications . efficient lattice based strategies are known for a variety of communication scenarios , such as for achieving the capacity of the point - to - point additive white gaussian noise ( awgn ) channel @xcite , for dirty - paper coding @xcite , the wyner ziv problem @xcite and communication in relay networks @xcite , to name only a few . in this paper we present capacity - achieving lattice strategies for communication in gaussian broadcast channels where receivers have prior side information about the messages being transmitted . in particular , we assume that the transmitter is broadcasting @xmath4 message symbols @xmath5 from a finite field @xmath6 , of prime size @xmath0 , to all the receivers , and each receiver may have _ coded side information _ about the messages : the prior knowledge of the values of ( possibly multiple ) @xmath6-linear combinations of @xmath5 . the number of linear combinations available as side information and the coefficients of these linear combinations can differ from one receiver to the next . the capacity of this channel follows from the results of tuncel @xcite , where the achievability part requires the use of random gaussian codebooks . the broadcast channel with coded side information is an instance of a noisy wireless version of _ index coding _ index coding considers a noiseless broadcast link where each receiver demands a subset of the source messages and knows the values of some other subset as side information . the modification of the ( noiseless ) index coding problem in which the receivers have access to linear combinations of messages was studied recently in @xcite . the case of gaussian broadcast channel with coded side information is motivated by applications to multi - terminal communication networks . it is known that signal interference in wireless channels can be harnessed by decoding linear combinations of transmit messages instead of either treating interference as noise or decoding interference along with the intended message @xcite . when such a technique is used in a mutli - hop communication protocol , one encounters receivers that have coded side information obtained from transmissions in the previous phases . similarly , in a network that consists of both wired and wireless channels , the symbols received from wired links can be utilized as side information for decoding the wireless signals . if a linear network code is used in the wired part of the network , then the side information is in the form of linear combinations of the source messages . [ ex : relay_network ] consider a wireless network with two base stations @xmath7 and @xmath8 , that hold message symbols @xmath9 and @xmath10 , respectively . the base stations are required to broadcast @xmath9 and @xmath10 to four user nodes @xmath11 through the relay node @xmath12 , see fig . [ fig : relay_network ] . in the first phase of the protocol , @xmath7 and @xmath8 encode the data symbols @xmath9 and @xmath10 , and transmit the resulting codewords simultaneously . by using the decoding technique of compute - and - forward @xcite , @xmath13 reliably decodes some linear combination @xmath14 , @xmath15 , from the received noisy superposition of the two transmit signals . on the other hand , @xmath12 has a higher signal - to - noise ratio and successfully decodes both @xmath9 and @xmath10 by behaving as a multiple - access receiver . further , there is no signal interference at @xmath16 and @xmath17 , and these two nodes reliably decode @xmath9 and @xmath10 , respectively . we observe that the second phase of the protocol is a broadcast channel with coded side information at the receivers : the relay needs to jointly broadcast @xmath18 to four user nodes , the first three users @xmath19 have prior knowledge of the linear combinations @xmath20 , @xmath21 and @xmath14 , respectively , while the fourth user has no such side information . [ ex : overlay ] encodes @xmath5 and broadcasts wirelessly to the destination nodes @xmath22 in order to supplement the communication through the wired network . each destination receives ( possibly multiple ) @xmath6-linear combinations of the source symbols through a linear ( wireline ) network code and uses this information to decode the wireless signal broadcast by @xmath23.,width=288 ] assume a network of noiseless wired links in the form of a directed acyclic graph , where the source node @xmath23 desires to multicast @xmath4 independent messages @xmath24 to a set of destination nodes @xmath25 . the wireline network employs a traditional _ ( scalar ) linear network code _ @xcite , i.e. , the symbol transmitted on each outgoing edge of a node is an element of @xmath6 generated as a linear combination of the symbols received on its incoming edges . at every destination node @xmath26 , the decoder attempts to recover the @xmath4 message symbols from their @xmath6-linear combinations received on its incoming edges . recovery is possible if and only if the number of linearly independent equations available at @xmath27 is @xmath4 . it is known that the maximum number of linearly - independent equations that can be made available at @xmath27 is @xmath28 , where @xmath29 is the maximum number of edge - disjoint paths from @xmath23 to @xmath27 , see @xcite . it follows that multicasting is possible if and only if @xmath30 for every @xmath26 . now suppose there exist destination nodes with @xmath31 less than @xmath4 , i.e. , the communication demands are beyond the wireline network s capacity . a solution to meet the demands is to broadcast a wireless signal from the source to fill the capacity deficiency of the wired network , see fig . [ fig : overlay ] . at each destination , the @xmath6-linear combinations obtained from the wireline network serve as side information to decode the wireless broadcast signal . a special case of coded side information is the gaussian broadcast channel where each receiver has prior knowledge of the values of some subset of the @xmath4 messages . the known capacity - achieving coding schemes for this special case are based on random coding using i.i.d . ( independent and identically distributed ) codewords @xcite . constructions of binary codes for this channel were proposed in @xcite , explicit codes based on lattices were proposed in @xcite and codes based on quadrature amplitude modulation were constructed in @xcite . the objective of this paper is to prove that lattice codes achieve the capacity of gaussian broadcast channels with coded side information . we use the information - theoretic framework set by erez and zamir @xcite to this end . the proposed coding scheme uses an ensemble of construction a lattices and the decoding scheme involves _ algebraic binning _ @xcite where the receiver side information is used to expurgate the channel code and obtain a lower rate subcode . the set of linear equations available as side information may differ from one receiver to another , and hence , each receiver must employ a different binning scheme for the same channel code . the coding scheme ensures that the binning performed at each receiver produces a good lattice subcode of the transmitted code . following expurgation , each receiver decodes the channel output by minimum mean square error ( mmse ) scaling and quantization to an infinite lattice . the algebraic structure of the coding scheme facilitates the performance analysis by decomposing the original channel into multiple independent point - to - point awgn channels one corresponding to each receiver where each of the point - to - point awgn channels uses a lattice code for communication . unlike @xcite , where achievability in a point - to - point awgn channel was proved using error exponent analysis , we provide a direct proof based only on simple counting arguments . as a corollary to the main result , we obtain an alternative proof of the goodness of lattice codes in achieving the capacity of the point - to - point awgn channel . previous proofs of this result presented in @xcite also use ensembles of lattices obtained by applying construction a to random linear codes over a prime field @xmath6 ; see also @xcite . while @xcite used primes @xmath0 that were exponential in the code length @xmath1 , @xcite and @xcite improved this result to let @xmath0 grow as @xmath32 and @xmath33 , respectively . the corollary presented in this paper further improves these results by enabling a rate of growth of @xmath2 , for any choice of @xmath3 . the organization of this paper is as follows . we introduce the channel model in section [ sec : channel_model ] and review the relevant background on lattices and lattice codes in section [ sec : lattice_prelim ] . in section [ sec:3 ] , we state the main theorem , and describe the lattice code ensemble and encoding and decoding procedures . we prove the main theorem and state a few corollaries in section [ sec : proof_of_main ] , and finally , we discuss some concluding remarks in section [ sec : conclusion ] . _ notation : _ matrices and column vectors are denoted by bold upper and lower case letters , respectively . the symbol @xmath34 denotes the euclidean norm of a vector , and @xmath35 is the transpose of a matrix or a vector . the kronecker product of two matrices @xmath36 and @xmath37 is @xmath38 , @xmath39 is the @xmath40 identity matrix , and @xmath41 is the all zero matrix of appropriate dimension . the symbol @xmath42 denotes logarithm to the base @xmath43 and @xmath44 denotes logarithm to the base @xmath45 . the expectation operator is denoted by @xmath46 . the symbol @xmath47 denotes the elements in the set @xmath48 that do not belong to the set @xmath49 .

lattices possess elegant mathematical properties which have been previously used in the literature to show that structured codes can be efficient in a variety of communication scenarios , including coding for the additive white gaussian noise ( awgn ) channel , dirty - paper channel , wyner - ziv coding , coding for relay networks and so forth . following the approach introduced by erez and zamir , we show that lattice codes are optimal for the family of gaussian broadcast channels where the source transmits a set of common messages to all receivers and each receiver has _ coded side information _ , i.e. , prior information in the form of linear combinations of the messages . this channel model , which is an instance of the gaussian version of index coding , is motivated by applications to multi - terminal networks where the nodes may have access to coded versions of the messages from previous signal hops or through orthogonal channels . the known results on the capacity of this channel are based on random gaussian codebooks . additive white gaussian noise ( awgn ) , capacity , construction a , gaussian broadcast channel , index coding , lattice , side information , structured codes .

lattices possess elegant mathematical properties which have been previously used in the literature to show that structured codes can be efficient in a variety of communication scenarios , including coding for the additive white gaussian noise ( awgn ) channel , dirty - paper channel , wyner - ziv coding , coding for relay networks and so forth . following the approach introduced by erez and zamir , we show that lattice codes are optimal for the family of gaussian broadcast channels where the source transmits a set of common messages to all receivers and each receiver has _ coded side information _ , i.e. , prior information in the form of linear combinations of the messages . this channel model , which is an instance of the gaussian version of index coding , is motivated by applications to multi - terminal networks where the nodes may have access to coded versions of the messages from previous signal hops or through orthogonal channels . the known results on the capacity of this channel are based on random gaussian codebooks . the structured coding scheme proposed in this paper utilizes construction a lattices designed over prime finite fields , and _ algebraic binning _ at the decoders to expurgate the channel code and obtain good lattice subcodes , for every possible set of linear combinations available as side information . as a corollary , we show that lattice codes based on construction a can achieve the capacity of single - user awgn channels with the size of the prime field growing as a function of the code length as , for any fixed , which is the slowest yet reported in the literature . additive white gaussian noise ( awgn ) , capacity , construction a , gaussian broadcast channel , index coding , lattice , side information , structured codes .

1109.3475

i

the most common metric in coding theory is the hamming metric . in this paper we deal with another frequently used metric , so called lee metric . this metric was introduced in @xcite and @xcite for transmission of signals . the main motive power for studying codes in lee metric goes back to the seminal paper of golomb and welch @xcite . in this paper we will study the existence and will enumerate diameter perfect lee codes . first we recall some definitions and notation . as usual , let @xmath9 be the set of integers , @xmath10 denote the integers modulo @xmath11 and let @xmath12 be the @xmath2-fold cartesian product of a set @xmath13 throughout the paper we will use @xmath14 and @xmath15 for the abelian ( component - wise ) additive group on @xmath14 and @xmath15 as well . it will always be clear from the context whether we have in mind a set or the group on this set.because of the coding theory context the elements of @xmath14 and @xmath15 will also be called words . the lee distance ( = the manhattan distance , the zig - zag distance ) @xmath16 of two words @xmath17 @xmath18 is given by @xmath19 for @xmath20 and by @xmath21 @xmath22 for @xmath23 by @xmath24 we denote the lee sphere of radius @xmath25 centered at @xmath26 that is , @xmath27 . for @xmath28 with @xmath29 the double lee sphere @xmath30 of radius @xmath25 with its center comprising @xmath31,@xmath32 is the set @xmath33 . perfect error - correcting lee codes can be introduced in several ways . for the purpose of this paper it is convenient to define them through tilings . in order not to have to formulate the same statement / definition twice , once for @xmath15 and then for @xmath34 the symbol @xmath35 @xmath36stands for both of them ; that is , by @xmath35 we mean @xmath15 or @xmath37 we set @xmath38 and @xmath39 where the @xmath40-th coordinate equals @xmath41 let @xmath42 be a subset of @xmath35 . by a copy of @xmath42 we mean an image of @xmath42 under a linear distance preserving bijection on @xmath43 translations of @xmath42 form a special family of copies of @xmath44 we recall that by a translation of @xmath42 we understand the set @xmath45 @xmath46 where @xmath47 . as the sphere @xmath48 is symmetric , each copy of the sphere is its translation . each copy of the double - sphere @xmath49 can be obtained by a translation of the double - sphere @xmath50 for some @xmath51 a collection @xmath52 of copies of @xmath42 constitutes a tiling of @xmath35 by @xmath53 if @xmath54 and @xmath55 for all @xmath56 a tiling @xmath57 of @xmath58 by @xmath42 that consists of translations of @xmath42 can be described as @xmath59 where @xmath60 . @xmath36a tiling @xmath57 by translations of @xmath42 is called periodic if @xmath61 is periodic , and @xmath57 is called a lattice tiling if @xmath61 is a lattice ( = a subgroup ) of @xmath35 . let @xmath62 in @xmath35 such that , if @xmath63 then @xmath64 a set @xmath65 is called a perfect @xmath25-error - correcting lee code if @xmath66 is a tiling of @xmath67 if @xmath68 is a lattice then @xmath68 is called a linear code . for @xmath69 the perfect @xmath25-error - correcting lee code is denoted by @xmath70 while for @xmath71 it is denoted by @xmath72 the long - standing golomb - welch conjecture deals with the existence of @xmath70 codes . although there is a vast literature on the subject the conjecture is still far from being solved . in this paper we focus on diameter-@xmath73 perfect lee codes , which constitute a generalization of perfect error - correcting lee codes . ahlswede et al . , see @xcite , introduced diameter perfect codes for distance regular graphs . let @xmath74 be a metric space . then a set @xmath75 is a diameter-@xmath73 code if @xmath76 for any @xmath77 and a set @xmath78 is an anticode of diameter @xmath73 if @xmath79 for all @xmath80 each graph @xmath81 can be seen as a metric space . it is proved in @xcite that if @xmath81 is a distance regular graph , @xmath82 is a diameter-@xmath73 code in @xmath83 and @xmath84 is an anticode of diameter @xmath85 of the maximum size in @xmath81 then @xmath86 that is , the sphere packing bound applies to distance regular graphs as well . a diameter-@xmath73 code that attains the bound in the above inequality is called diameter-@xmath73 perfect code . this definition can be used for example in hamming scheme as the related graph is distance regular . etzion @xcite showed that the sphere packing bound applies to @xmath14 endowed with lee metric as well although the related graph is not distance regular ; analogically to @xcite he defined the diameter-@xmath73 perfect lee code as a diameter-@xmath73 code that attains the equality in the sphere packing bound . clearly , a definition through the sphere packing bound can not be extended to an infinite space . to deal with the case of @xmath15 etzion proved that if @xmath68 is a lattice in @xmath15 with the minimum distance @xmath73 and a volume @xmath87 then @xmath88 for any anticode of diameter @xmath89 then he called @xmath68 to be the diameter perfect code if @xmath68 attains the equality in the bound . as far as we know there is no formal definition of the diameter perfect lee code in @xmath90 in case when @xmath68 is not a lattice . now we provide a unified definition of a diameter-@xmath73 perfect lee code for both spaces @xmath14 and @xmath15 regardless whether the code is a linear one . let @xmath91 be a family of subsets of an underlying set @xmath92 then a set @xmath93 is called a transversal of @xmath94 if there is a bijection @xmath95 so that @xmath96 equivalently , @xmath97 is a transversal of @xmath98 if @xmath99 for each @xmath100 and @xmath101 for all @xmath102 let @xmath65 . then @xmath68 is a diameter-@xmath73 perfect lee code in @xmath35 if @xmath68 is a diameter-@xmath73 code , and there is a tiling @xmath103 of @xmath104 by the anticode of diameter @xmath85 of maximum size such that @xmath61 is a transversal of @xmath105 the diameter-@xmath73 perfect lee code in @xmath15 will be denoted by @xmath106 and in @xmath14 by @xmath107 . in other words , a diameter-@xmath73 code @xmath68 is perfect if there is a tiling @xmath57 by anticodes of diameter @xmath85 of the maximum size so that each tile in @xmath57 contains exactly one codeword of @xmath68 . it is not difficult to see that for @xmath14 and for lattice case in @xmath15 our definition of diameter perfect lee codes is equivalent to the definition due to etzion . any error - correcting perfect lee code is also a diameter perfect lee code . indeed , it is easy to see that , for @xmath73 even , the anticode of diameter @xmath108 of the maximum size is the lee sphere @xmath48 with @xmath109 thus , for @xmath73 odd , @xmath110 and @xmath111 codes are @xmath70 and @xmath112 codes where @xmath113 , respectively . it was proved in @xcite that , for @xmath73 odd , the anticode of diameter @xmath73 of maximum size is the double - sphere @xmath114 with @xmath115 we point out an advantage of our definition . suppose that we need to show the existence of a diameter-@xmath73 perfect lee code in @xmath116where @xmath73 is even . then all we need to do is to prove that there is a tiling @xmath117 of @xmath35 by the double - sphere @xmath114 of radius @xmath118 . indeed , choose a fixed tile @xmath119 in @xmath105 let @xmath120 be one of the two words forming the center of @xmath121we will prove that the set @xmath122 belongs to the center of @xmath123 and @xmath124 is even , @xmath125 is a distance-@xmath73 perfect lee code . clearly , @xmath126 is well defined as the center of @xmath123 consists of two words at distance @xmath127 that is , one of them at even and the other at odd distance from @xmath128 @xmath126 is a transversal of @xmath129 thus we need only to prove that @xmath130 for any @xmath131 . clearly , the distance of any two words in @xmath126 is even . this is obvious to see for @xmath132 it is not true in general for @xmath133 it is valid only in the case when @xmath0 is even . however , as the size of @xmath114 is even ( see e ) @xmath134 has to be even as well , which in turn implies @xmath0 is even . so for the distance of any two words in @xmath126 we have @xmath135 and since @xmath124 is even , we get @xmath136 we have proved : [ x]if there is a tiling of @xmath35 by double - spheres @xmath114 then there is a distance-@xmath73 perfect lee code in @xmath35 with @xmath137moreover , if @xmath138 is a tiling of @xmath139 by translations of @xmath114 such that the weight @xmath140 of @xmath141 is even for all @xmath142 then @xmath68 is a diameter-@xmath73 perfect lee code . the main goal of this paper is to study the existence and to enumerate diameter perfect lee codes . it turns out that some results on diameter perfect lee codes and on perfect error - correcting lee codes are similar but in some cases the results on the two types of codes are very much different . e.g. , it was proved in @xcite that there is only one , up to isomorphism , tiling of @xmath143 by lee spheres @xmath144 but we show in this paper that there are uncountable many tilings of @xmath143 by double lee sphere @xmath145 . the rest of the paper is organized as follows . golomb - welch conjecture claims that there are no @xmath70 codes for @xmath146 and @xmath147 the current state of this conjecture as well as its extension to @xmath110 codes and a further extension to perfect dominating sets is provided in section 2 . we defined perfect lee codes by means of tilings . in @xcite s. stein introduced an algebraic construction of lattice tilings based on group homomorphisms . he used this construction for tilings by crosses . this construction , and its variations occur in many papers , see e.g. @xcite . in section 3 a generalization of stein s construction is given . let @xmath42 be a subset of @xmath148 we state a necessary and sufficient condition for the existence of a lattice tiling of @xmath15 by @xmath42 in terms of a group homomorphism . we guess that this construction will turn useful outside the scope of this paper . the main results of this paper are stated in section 4 . it is believed , see section 2 , that , with only one exception of @xmath149 code , there are no @xmath110 codes , and also no @xmath111 codes for @xmath150 and @xmath151 all values of @xmath0 for which there exists a linear @xmath152 code ( = a linear @xmath153 code ) were determined in @xcite . we prove an analogous result for linear @xmath154 codes . as to the enumeration of all @xmath155 codes , etzion @xcite proved that there are uncountable many of them if @xmath2 is a power of @xmath156 we show that this is also the case for the other values of @xmath157 the situation with @xmath158 codes is very much different . it was proved in @xcite that there are uncountable many @xmath159 codes if @xmath8 is not a prime . however , on the other hand it was showed there that there is only one , up to isomorphism , @xmath160 code , and the authors conjectured that the same is true for each @xmath2 where @xmath8 is a prime . we note that the existence of uncountable many @xmath155 codes implies that there are uncountable many non - periodic @xmath155 codes as the total number of periodic @xmath155 codes is at most @xmath161 non - periodic @xmath155 codes were constructed by means of non - regular lattice tilings ( i.e. tilings that are not a face - to - face tilings ) of @xmath162 by a cluster of cubes centered at words of a double - sphere @xmath163 this is another example of a well know situation when to solve a problem we need to generalize it first . we note that results in section 4 answer in the affirmative questions ( 4 ) , ( 5 ) , and ( 8) raised by etzion in @xcite . question ( 6 ) from @xcite was answered in @xcite . finally , in section 5 an efficient decoding algorithm for linear perfect lee codes , both diameter and error - correcting , is designed . thanks to the representation of these codes by tilings constructed via a group homomorphism , the computation complexity of this algorithm is @xmath164 for @xmath165 codes , and its complexity is @xmath166 for @xmath167 codes . we recall that @xmath168 codes are believed not to exists for @xmath169 and @xmath151

lee codes have been intensively studied for more than 40 years . interest in these codes has been triggered by the golomb - welch conjecture on the existence of the perfect error - correcting lee codes . in this paper we produce diameter perfect lee codes by an algebraic construction that is based on a group homomorphism . we hope that this construction will turn out to be useful far beyond the scope of this paper . diameter codes , error - correcting codes , lee metric , perfect codes .

lee codes have been intensively studied for more than 40 years . interest in these codes has been triggered by the golomb - welch conjecture on the existence of the perfect error - correcting lee codes . in this paper we deal with the existence and enumeration of diameter perfect lee codes . as main results we determine all for which there exists a linear diameter- perfect lee code of word length over and prove that for each there are uncountable many diameter- perfect lee codes of word length over this is in a strict contrast with perfect error - correcting lee codes of word length over as there is a unique such code for and its is conjectured that this is always the case when is a prime . we produce diameter perfect lee codes by an algebraic construction that is based on a group homomorphism . this will allow us to design an efficient algorithm for their decoding . we hope that this construction will turn out to be useful far beyond the scope of this paper . diameter codes , error - correcting codes , lee metric , perfect codes .

1311.0028

i

studying the distribution of zeros of orthogonal polynomials is a classical theme that has been addressed in an enormous number of research papers . the sole reason for revisiting this topic here is the crucial role of _ legendre - gauss - lobatto ( lgl ) grids _ , formed by the zeros of corresponding orthogonal polynomials , for the development of efficient preconditioners for high order finite element and even spectral discretizations of pdes , see e.g. @xcite . in connection with elliptic pdes , which often possess very smooth solutions and thus render high order methods at least potentially extremely efficient , a key constituent is the fact that interpolation at lgl grids give rise to fully robust ( with respect to the polynomial degrees ) isomorphisms between high order polynomial spaces and low order finite element spaces on lgl grids . however , unfortunately , one quickly faces some serious obstructions to fully exploiting this remarkable potential of lgl grids when simultaneously trying to exploit the flexibility of _ discontinuous galerkin _ ( dg ) schemes , namely locally refined grids and locally varying polynomial degrees . indeed , as explained in @xcite , the essential source of the problems encountered then is the fact that lgl grids are _ not nested_. this affects the choice and analysis of suitable _ auxiliary spaces _ , when using the auxiliary space method , see e.g. @xcite as preconditioning strategy , as well as the efficient solution of the corresponding auxiliary problems . as a crucial remedy , certain hierarchies of nested _ dyadic _ grids have been introduced in @xcite which are _ associated _ in a certain sense with lgl grids . the term `` associated '' encapsulates a number of properties of such dyadic grids , some of which have been used and claimed in @xcite but will be proved here which is the central objective of this paper . the layout of the paper is as follows . after collecting , for the convenience of the reader , in section [ sect : ultraspherical ] some classical facts and tools used in the sequel , we formulate in section [ sect : main ] the main results of the paper . the first one , theorem [ thm : a ] , is concerned with the quasi - uniformity of lgl grids as well as with a certain notion of equivalence between lgl grids of _ different _ order . this is important for dealing with dg - discretizations involving locally varying polynomial degrees , see @xcite . the second one , theorem [ thm : b ] , concerns certain hierarchies of nested dyadic grids that are associated in a very strong sense with lgl grids . both theorems play a crucial role for the design and analysis of preconditioners for dg systems . section [ sect : proofa ] is devoted to the proof of theorem [ thm : a ] . this requires deriving a number of _ refined properties _ of lgl grids which to our knowledge can not be found in the literature . in particular , we need to revisit in section [ sect : cgl ] some close relatives namely _ chebyshev - gauss - lobatto ( cgl ) nodes _ since they have explicit formulae that help deriving sharp estimates . the central subject of section [ sect : dyadic ] is the generation of dyadic grids associated in a certain way with a given other grid as well as the analysis of the properties of these dyadic grids . in particular , the results obtained in this section lead to a specific hierarchy for which the properties claimed in theorem [ thm : b ] are verified . throughout the paper we shall employ the following notational convention . by @xmath0 we mean that the quantity @xmath1 can be bounded by a constant multiple of @xmath2 uniformly in the parameters @xmath1 and @xmath2 may depend on . likewise @xmath3 is equivalent to @xmath4 and @xmath5 means @xmath6 and @xmath7 .

legendre - gauss - lobatto ( lgl ) grids play a pivotal role in nodal spectral methods for the numerical solution of partial differential equations . they not only provide efficient high - order quadrature rules , but give also rise to norm equivalences that could eventually lead to efficient preconditioning techniques in high - order methods . unfortunately , a serious obstruction to fully exploiting the potential of such concepts is the fact that lgl grids of different degree are not nested . this affects , on the one hand , the choice and analysis of suitable _ auxiliary spaces _ , when applying the _ auxiliary space method _ as a principal preconditioning paradigm , and , on the other hand , the efficient solution of the auxiliary problems . as a central remedy , we consider certain nested hierarchies of dyadic grids of locally comparable mesh size , that are in a certain sense properly associated with the lgl grids . their actual suitability requires a subtle analysis of such grids which , in turn , relies on a number of refined properties of lgl grids . the central objective of this paper is to derive just these properties .

legendre - gauss - lobatto ( lgl ) grids play a pivotal role in nodal spectral methods for the numerical solution of partial differential equations . they not only provide efficient high - order quadrature rules , but give also rise to norm equivalences that could eventually lead to efficient preconditioning techniques in high - order methods . unfortunately , a serious obstruction to fully exploiting the potential of such concepts is the fact that lgl grids of different degree are not nested . this affects , on the one hand , the choice and analysis of suitable _ auxiliary spaces _ , when applying the _ auxiliary space method _ as a principal preconditioning paradigm , and , on the other hand , the efficient solution of the auxiliary problems . as a central remedy , we consider certain nested hierarchies of dyadic grids of locally comparable mesh size , that are in a certain sense properly associated with the lgl grids . their actual suitability requires a subtle analysis of such grids which , in turn , relies on a number of refined properties of lgl grids . the central objective of this paper is to derive just these properties . this requires first revisiting properties of close relatives to lgl grids which are subsequently used to develop a refined analysis of lgl grids . these results allow us then to derive the relevant properties of the associated dyadic grids .

1409.1108

i

the context of this paper is the enumeration of finite relational structures . a relational structure @xmath5 is _ embeddable _ in a relational structure @xmath6 , in notation @xmath7 , if @xmath5 is isomorphic to an induced substructure of @xmath6 . the embeddability relation is a quasi order . several significant properties of relational structures or classes of relational structures can be uniquely expressed in term of this quasi order . this is typically the case of hereditary classes : a class @xmath0 of structures is _ hereditary _ if it contains every relational structure which can be embedded in some member of @xmath0 . interesting hereditary classes abound . in the late forties , frass , following the work of cantor , hausdorff and sierpinski , pointed out the role of the quasi - ordering of embeddability and hereditary classes in the theory of relations ( see his book @xcite for an illustration ) . recent years have seen a renewed interest for the study of these classes , particularly those made of finite structures . many results have been obtained . some are about obstructions allowing to define these classes , others on the behavior of the function @xmath1 , the _ profile _ of @xmath0 which counts , for every integer @xmath2 , the number @xmath3 of members of @xmath0 defined on @xmath2 elements , isomorphic structures being identified . general counting results have been obtained , as well as precise results , for graphs , tournaments and ordered graphs ( see the survey @xcite ) . enumeration results on permutations , motivated by the _ stanley - wilf conjecture _ , solved by marcus and tards ( 2004 ) , fall also under this frame , an important fact due to cameron @xcite . indeed , to each permutation @xmath8 of @xmath9:=\{1 , \dots , n\}$ ] we may associate the relational structure @xmath10 , \leq , \leq_{\sigma})$ ] , that we call _ bichain _ , made of two linear orders on @xmath9 $ ] ( @xmath11 being the natural order on @xmath9 $ ] and @xmath12 the linear order defined by @xmath13 if and only if @xmath14 ) . as it turns out , the order defined on permutations and the embeddability between bichains coincide ( see subsection [ permutation - bichain ] for details and examples ) . in this paper , we show how some results obtained by albert and atkinson @xcite for classes of permutations extend to classes of ordered binary relational structures . we prove notably theorem [ theo : algebraic ] . for this purpose , we recall in section [ sect : basic ] some basic definitions of the theory of relations , we survey in section [ sect : bichains ] some results concerning classes of permutations and show how permutations are related to relational structures . then , we illustrate the role of indecomposable structures ( see section [ sect : indecomposable ] ) and of well quasi order ( see section [ sect : well - quasi - order ] ) in enumeration results . finally , in section [ sect : conjecture ] , we present a conjecture and a partial solution , a special case answering a question of albert and atkinson @xcite . our results have been presented at the international conference on discrete mathematics and computer science ( dimacos11 ) held in mohammedia , morocco , may-5 - 8 , 2011 , and at the international symposium on operational research ( isor11 ) , held in algiers , algeria , may 30-june 2 , 2011 @xcite . we are pleased to thanks the organizers of these meetings for their help .

is . many results about the behavior of the function have been obtained . albert and atkinson have shown that the generating series of several classes of permutations are algebraic . in this paper , we show how their results extend to classes of ordered binary relational structures ; putting emphasis on the notion of hereditary well quasi order , we discuss some of their questions and answer one . * ams subject classification : * 05c30 , 06f99 , 05a05 , 03c13 .

let be a class of finite combinatorial structures . the _ profile _ of is the function which counts , for every integer , the number of members of defined on elements , isomorphic structures been identified . the _ generating function of _ is . many results about the behavior of the function have been obtained . albert and atkinson have shown that the generating series of several classes of permutations are algebraic . in this paper , we show how their results extend to classes of ordered binary relational structures ; putting emphasis on the notion of hereditary well quasi order , we discuss some of their questions and answer one . * ams subject classification : * 05c30 , 06f99 , 05a05 , 03c13 . * keywords * : profile , well quasi - ordering , indecomposability , permutations .

1211.2309

i

let @xmath0 be a commutative ring . in this article we address the following questions : * question a : * can the classical morita theory of @xmath0-algebras be viewed as a `` homotopy theory '' ? * question b : * does the brauer group of @xmath0 admit a natural `` tensor categorification '' ? the classical morita theory @xcite defines an equivalence relation between @xmath0-algebras . namely , two @xmath0-algebras @xmath1 and @xmath2 are called _ morita equivalent _ if they have equivalent categories of representations , or equivalently , if there exist bimodules @xmath3 and @xmath4 and bimodule isomorphisms @xmath5 and @xmath6 . in order to view this classical notion from a homotopical perspective we will consider @xmath0-categories , categories enriched over @xmath0-modules ; see [ sec : preliminaries ] . every @xmath0-algebra is naturally a @xmath0-category with a single object and all the usual notions ( module , bimodule , tensor product , etc . ) extend automatically to this larger setting . in particular , the category @xmath7 of ( small ) @xmath0-categories is closed symmetric monoidal . our answer to * question a * is then the following : [ thm : main1 ] the category @xmath7 carries a combinatorial symmetric monoidal quillen model structure . the weak equivalences are the _ morita equivalences _ , the @xmath0-linear functors @xmath8 inducing an equivalence @xmath9 on the module categories ( definition [ defi : morita_eq ] ) , and the cofibrations are the @xmath0-linear functors that are injective on objects . moreover , two @xmath0-algebras become isomorphic in the associated homotopy category @xmath10 $ ] if and only if they are morita equivalent . based on the foundational work of brauer and others , auslander and goldman @xcite introduced the brauer group @xmath11 of a commutative ring @xmath0 in their study of separable algebras . concretely , @xmath11 consists of the morita equivalence classes of azumaya @xmath0-algebras with addition induced by the tensor product . any homomorphism @xmath12 of commutative rings gives naturally rise , by extension of scalars , to a map @xmath13 . when @xmath14 is a finite galois extension of a field @xmath15 , one moreover has a well - defined corestriction ( or transfer , or weil restriction ) map @xmath16 in the opposite direction ; see @xcite . now , let @xmath17 be an arbitrary abelian group . recall that a _ categorification of @xmath17 _ consists of an additive category @xmath18 with the property that its grothendieck group @xmath19 identifies with @xmath17 . in this vein we define a _ tensor categorification of @xmath17 _ ( or _ @xmath20-categorification _ ) to be a symmetric monoidal category @xmath21 with the property that its picard group @xmath22 identifies with @xmath17 . our answer to * question b * is then the following : [ thm : main2 ] the category @xmath23 of @xmath20-invertible objects in the homotopy category @xmath24 is a @xmath20-categorification of @xmath11 . given any ring homomorphism @xmath12 , we have a base - change @xmath20-functor @xmath25 such that @xmath26 . when @xmath27 is a finite galois extension of a field @xmath0 , we have moreover a corestriction @xmath20-functor @xmath28 such that @xmath29 . note that theorem [ thm : main2 ] not only conceptually @xmath20-categorifies the brauer group but moreover its functorial behavior . roughly speaking , it formalizes the equality @xmath30

let be a commutative ring . in this article we construct a symmetric monoidal quillen model structure on the category of small-categories which enhances classical morita theory . we then use it in order to obtain a natural tensor categorification of the brauer group and of its functoriality . ---

let be a commutative ring . in this article we construct a symmetric monoidal quillen model structure on the category of small-categories which enhances classical morita theory . we then use it in order to obtain a natural tensor categorification of the brauer group and of its functoriality . ---

1212.6735

i

a classical theorem of dirac @xcite states that every graph @xmath0 on @xmath13 vertices with minimum degree @xmath14 contains a hamiltonian cycle . in this paper , we generalise this result to edge - coloured graphs . an _ edge - coloured graph _ is a graph @xmath0 with an edge colouring @xmath15 of @xmath0 . we say that @xmath0 is _ properly coloured _ if no two adjacent edges of @xmath0 have the same colour . moreover , @xmath0 is said to be _ rainbow _ if all edges have distinct colours . we consider the following analogue of degree for edge - coloured graphs @xmath0 . given a vertex @xmath16 , the _ colour degree @xmath17 _ is the number of distinct colours of edges incident to @xmath3 . the _ minimum colour degree _ @xmath1 of an edge - coloured graph @xmath0 is the minimum @xmath17 over all vertices @xmath3 in @xmath0 . one elementary result of graph theory states that every graph @xmath0 with @xmath18 contains a cycle . however , for all @xmath19 , there exist edge - coloured graphs @xmath0 with @xmath20 that do not contain any properly coloured cycles . grossman and hggkvist @xcite gave a sufficient condition for the existence of properly coloured cycles in edge - coloured graphs with two colours , which does not depend on @xmath1 . later on , yeo @xcite extended the result to edge - coloured graphs with any number of colours . one natural generalisation of dirac s theorem is to determine the minimum colour degree threshold for the existence of a rainbow hamiltonian cycle . however , such thresholds do not exist for all @xmath21 . indeed , for every even integer @xmath22 , there exists a properly coloured complete graph @xmath23 on @xmath21 vertices using exactly @xmath24 colours . note that @xmath25 , but @xmath23 does not contain a rainbow hamiltonian cycle . in fact , for each @xmath26 , there is a properly coloured @xmath27 that does not contain any rainbow hamiltonian path ( see @xcite ) . hence , a better question would be to ask for the minimum colour degree threshold for the existence of a properly coloured hamiltonian cycle . the problem of finding properly coloured spanning subgraphs in edge - coloured complete graphs @xmath23 has been investigated by numerous researchers . bang - jensen , gutin and yeo @xcite proved that if @xmath23 contains a properly coloured @xmath5-factor , then @xmath23 also contains a properly coloured hamiltonian path . a graph @xmath0 is said to be a _ @xmath28-path - cycle _ if @xmath0 is a vertex - disjoint union of at most one path and a number of cycles . feng , giesen , guo , gutin , jensen and rafiey @xcite showed that @xmath23 contains a properly coloured hamiltonian path if and only if it contains a spanning properly coloured @xmath28-path - cycle . we define _ @xmath29 _ to be the maximum number of edges of the same colour incident to the same vertex . in other words , @xmath30 over all monochromatic subgraphs @xmath31 , where an edge - coloured graph @xmath0 is _ monochromatic _ if all edges have the same colour . notice that @xmath32 . bollobs and erds @xcite proved that if @xmath33 , then @xmath23 contains a properly coloured hamiltonian cycle . they further conjectured that @xmath34 suffices . their result was subsequently improved by chen and daykin @xcite , shearer @xcite and alon and gutin @xcite . in @xcite , the author showed that for any @xmath35 , every @xmath23 with @xmath36 contains a properly coloured hamiltonian cycle , provided @xmath21 is large enough . therefore , the conjecture of bollobs and erds is true asymptotically . for a survey regarding properly coloured subgraphs in edge - coloured graphs , we recommend chapter 16 of @xcite . let @xmath0 be an edge - coloured graph ( not necessarily complete ) . li and wang @xcite proved that @xmath0 contains a properly coloured path of length @xmath37 or a properly coloured cycle of length at least @xmath38 . in @xcite , the author improved this result by showing that @xmath0 contains a properly coloured path of length @xmath37 or a properly coloured cycle of length at least @xmath39 . in the same paper , the author showed that every connected edge - coloured graph @xmath0 contains a properly coloured hamiltonian cycle or a properly coloured path of length at least @xmath40 . furthermore , the author also conjectured the following . [ conj : path ] every connected edge - coloured graph @xmath0 contains a properly coloured hamiltonian cycle or a properly coloured path of length @xmath41 . if the conjecture is true , then every edge - coloured graph @xmath0 with @xmath42 contains a properly coloured hamiltonian cycle . in this paper , we prove the following weaker result that @xmath42 implies the existence of a properly coloured @xmath5-factor in @xmath0 . [ thm:2factor ] every edge - coloured graph @xmath0 with @xmath43 contains a properly coloured @xmath5-factor . we also show that if @xmath44 and @xmath45 is large enough , then @xmath0 does indeed contain a properly coloured hamiltonian cycle . furthermore , @xmath0 contains a properly coloured cycle of length @xmath10 for all @xmath46 . [ thm : pchc ] for any @xmath35 , there exists an integer @xmath7 such that every edge - coloured graph @xmath0 with @xmath47 and @xmath48 contains a properly coloured cycle of length @xmath10 for all @xmath46 . given a subgraph @xmath49 in @xmath0 , we write @xmath50 for the ( edge - coloured ) subgraph obtained from @xmath0 by deleting all edges in @xmath49 . for an edge - coloured graph @xmath0 , let _ @xmath51 _ be the minimum @xmath52 over all monochromatic subgraphs @xmath49 in @xmath0 . note that @xmath53 . note that for an edge - coloured complete graph @xmath23 , we have @xmath54 . we prove the following stronger statement , which implies theorem [ thm : pchc ] . [ thm : pchc2 ] for any @xmath35 , there exists an integer @xmath7 such that every edge - coloured graph @xmath0 with @xmath55 and @xmath48 contains a properly coloured cycle of length @xmath10 for all @xmath46 . we now outline the proof of theorem [ thm : pchc2 ] for the case when @xmath56 , i.e. the existence of a properly coloured hamiltonian cycle . the proof adapts the absorption technique introduced by rdl , ruciski and szemerdi @xcite , which was used to tackle hamiltonicity problems in hypergraphs . the proof is divided into two main steps . in the first step , we find ( by lemma [ lma : abscycle ] ) a small ` absorbing cycle ' @xmath57 in @xmath0 . the absorbing cycle @xmath57 has the property that , given any small number of vertex - disjoint properly coloured paths @xmath58 in @xmath0 with @xmath59 for each @xmath60 , there exists a properly coloured cycle @xmath61 in @xmath0 with @xmath62 . thus , we have reduced the problem to covering the vertex set @xmath63 with small number of vertex - disjoint properly coloured paths . we remove the vertices of @xmath57 from @xmath0 and let @xmath64 be the resulting graph . since @xmath57 is small , we may assume that @xmath65 for some small @xmath66 . then we find a properly coloured @xmath5-factor in @xmath64 using lemma [ lma:2factor1 ] such that every cycle has length at least @xmath67 . ( although theorem [ thm:2factor ] also implies that @xmath64 contains a properly coloured @xmath5-factor , there is no bound on the lengths of the cycles . ) hence @xmath68 can be covered by at most @xmath69 vertex - disjoint properly coloured paths @xmath58 . by the ` absorbing ' property of @xmath57 , there is a properly coloured cycle @xmath61 with @xmath70 . therefore , @xmath61 is a properly coloured hamiltonian cycle as required . the paper is organised as follows . in the next section , we set up some basic notation and give the extremal example for conjecture [ conj : path ] . section [ sec:2fact ] is devoted to finding properly coloured @xmath5-factors and contains the proof of theorem [ thm:2factor ] . the absorbing cycle is constructed in section [ sec : abscycle ] . finally , theorem [ thm : pchc2 ] is proved in section [ sec : proof ] .

let be an edge - coloured graph . the minimum colour degree of is the largest integer such that , for every vertex , there are at least distinct colours on edges incident to . we say that is properly coloured if no two adjacent edges have the same colour . in this paper , we show that every edge - coloured graph with contains a properly coloured-factor . furthermore , we show that for any there exists an integer such that every edge - coloured graph with and contains a properly coloured cycle of length for every .

let be an edge - coloured graph . the minimum colour degree of is the largest integer such that , for every vertex , there are at least distinct colours on edges incident to . we say that is properly coloured if no two adjacent edges have the same colour . in this paper , we show that every edge - coloured graph with contains a properly coloured-factor . furthermore , we show that for any there exists an integer such that every edge - coloured graph with and contains a properly coloured cycle of length for every . this result is best possible in the sense that the statement is false for .

1005.5713

i

relaxation processes in complex systems such as glass - forming liquids , polymeric melts , liquid crystals and even lipid membranes is a very active area of research @xcite . they have been studied using different techniques ranging from nmr and brillouin light scattering to dielectric spectroscopy @xcite . the variety of relaxation phenomena occurring in these systems are non - debye and therefore can not be characterized by a single relaxation time @xcite . these multiscale time relaxations suggest an accumulative effect of different elemental relaxation processes that can be associated with collective motions in the system @xcite . the average effects of these cooperative mechanisms observed in experiments can be accurately described by means of well known empirical relations such as the williams - watts function in time space , or through its conjugated frequency domain ( havrilak - nagami ) function . these functions , and many others that can be found in the literature @xcite , reflect the specific features of the micro- or mesoscopic relaxation dynamics of each system . the microscopic description of these features may involve the formulation of specific molecule - molecule interaction models that lead to particular results . in view of this , it seems important to have a general theoretical model able to cope with different relaxation processes that may describe , for example , the dielectric relaxation in diverse materials including solids , glass - forming liquids and polymer melts @xcite . these last systems may exhibit , for example , anomalous kerr effect which can be described in terms of models based on fractional fokker - planck equations ( ffpe ) @xcite . in this work we adopt a mesoscopic point of view for describing the relaxation of a system of non - interacting polar molecules inmersed in a time - dependent external electric field , because at this level of description it is still possible to formulate the mentioned general framework able to cope with different relaxation dynamics . in particular , we use the mesoscopic non - equilibrium thermodynamics ( mnet ) formalism in order to derive a non - markovian fokker - planck like equation ( nmfpe ) for the evolution of the probability distribution describing the orientation and position of the molecules . the mnet method uses the entropy produced by the system when it is driven out from equilibrium and is particularly useful for describing the stochastic evolution of systems at the mesoscopic scale @xcite . the advantage of this formalism lies in the fact that the nmfpe derived has a kinetic origin , and therefore there is no explicit assumption on the specific time correlation properties of the underlying gaussian stochastic process , contrary to what happens with equations derived from langevin formalisms @xcite . within this framework , linear relations between generalized forces and fluxes occurring in phase space are postulated , which allow us to introduce memory effects accounting for the non - instantaneous response of the system to the external perturbations . the effects of the non - markovian dynamics on a measurable property of the system like its complex dielectric susceptibility can then be analyzed in a systematic way by considering different memory kernels . in this work we will concentrate mainly on memory kernels decaying as power laws . we will show that these particular functions turn out to be important since they lead to the mentioned ffpe . in the section ii we will derive the nmfpe for a system of non - interacting polar molecules by using mnet . this equation describes both rotational diffusion and the diffusion of polarization , and we will show that it can be recast into a ffpe when proper memory kernels are considered . in section iii we will construct the equations for the first two moments of distribution . subsequently , in section iv we will present a perturbation procedure which allow for a solution of the obtained dynamic equations , when the system is subjected to an external harmonic electric field superposed to a large static electric field . consequently , we will obtain a closed expression for the non - homogeneous dielectric susceptibility of the system which incorporates non - markovian effects , and the effects of the three first moments of the distribution function . these effects will be analyzed systematically in section v by considering diverse limiting cases and memory functions . memory kernels decaying as power laws in time will be shown to generate ffpe and response functions of the cole - cole type @xcite . a comparison with recent experimental results is performed to validate the models derived . in section vi we will propose a model based on a theory of distribution of relaxation times which can be used to give an insight of the physical processes producing power law kernels and , accordingly , ffpe . finally , we will present some final remarks an discuss the limitations of our model in section vii .

mesoscopic non - equilibrium thermodynamics is used to formulate a model describing non - homogeneous and non - debye dielectric relaxation . the model is presented in terms of a fokker - planck equation for the probability distribution of non - interacting polar molecules in contact with a heat bath and in the presence of an external time - dependent electric field . the model is developed in terms of the evolution equations for the first two moments of the distribution function .

mesoscopic non - equilibrium thermodynamics is used to formulate a model describing non - homogeneous and non - debye dielectric relaxation . the model is presented in terms of a fokker - planck equation for the probability distribution of non - interacting polar molecules in contact with a heat bath and in the presence of an external time - dependent electric field . memory effects are introduced in the fokker - planck description through integral relations containing memory kernels , which in turn are used to establish a connection with fractional fokker - planck descriptions . the model is developed in terms of the evolution equations for the first two moments of the distribution function . these equations are solved by following a perturbative method from which the expressions for the complex susceptibilities are obtained as a functions of the frequency and the wave number . different memory kernels are considered and used to compare with experiments of dielectric relaxation in glassy systems . for the case of cole - cole relaxation , we infer the distribution of relaxation times and its relation with an effective distribution of dipolar moments that can be attributed to different segmental motions of the polymer chains in a melt .

1005.5713

c

a physical insight of the mechanism underlying the power - law behavior of the memory kernel in eq . ( [ frac002 ] ) can be elucidated by decomposing the corresponding spectrum in terms of elementary debye processes , that is , by appealing to the theory of distribution of relaxation times @xcite . for simplicity , our analysis will be focused on the description leading to cole - cole parallel susceptibility ( [ limiting004 ] ) with the use of the power law memory function , eq . ( [ frac002 ] ) . we will proceed by assuming the existence of a distribution of elementary processes with proper relaxation times and amplitudes , whose superposition generates a wider spectrum such that corresponding to the cole - cole function , eq . ( [ limiting004 ] ) . in this form , the cole - cole function can be recovered from a superposition of elementary debye processes through the relation @xcite @xmath299 where for simplicity in notation we have introduced the variable @xmath300 and used the definition @xmath301 in the second equality . the function @xmath302 is a continuous distribution of relaxation times having the explicit form @xmath303 in fact , the analysis can be simplified by considering a discrete expansion of eqs . ( [ limiting008 ] ) and ( [ limiting009 ] ) , that is , by approximating the cole - cole response law by a superposition of independent debye - like processes @xmath304 , distributed over a discrete space in the form @xmath305 where the amplitudes , @xmath306 , and relaxation times , @xmath307 , of the @xmath308-th process are given , respectively , by @xmath309 notice that the index @xmath308 in the summation of eq . ( [ limiting011 ] ) , quantifies the time - scale separation between the @xmath308-th elementary process and the process with the relaxation time @xmath310 , which defines the location of the maximum of the dielectric response function . those processes labeled by an index @xmath311 , evolve toward equilibrium in a characteristic time which is much larger than @xmath310 , while those with @xmath312 , decay much faster than the processes with relaxation time of the order of @xmath310 . [ figure006 ] illustrates the approximation of both the real and imaginary parts of the normalized cole - cole function , @xmath313^{2 } + \left ( \tau_{cc } \omega \right)^{2\alpha } \sin^{2 } \left ( \alpha \pi / 2\right ) } , \label{limiting011a}\ ] ] @xmath314^{2 } + \left ( \tau_{cc } \omega \right)^{2\alpha } \sin^{2 } \left ( \alpha \pi / 2\right ) } , \label{limiting011b}\ ] ] by the superposition of debye components over the discrete distribution of relaxation times given by eq . ( [ limiting011 ] ) with @xmath315 , and where the infinite sum in eq . ( [ limiting011 ] ) has been approximated by a sum over a finite interval ranging from @xmath316 to @xmath317 . ( [ limiting011 ] ) constitutes a very good approximation to the cole - cole function , reproducing very well the behavior of the latter over the whole frequency range considered . it is interesting to notice that a partial superposition of independent debye contributions in which elementary processes within a given range are suppressed , yields a two step relaxation dynamics manifested through the dielectric loss with well marked peaks as those observed experimentally @xcite . this elimination of elementary processes can be represented by non - uniform distributions of characteristic times @xcite . ) . the real and imaginary part of the cole - cole function , eq . ( [ limiting004 ] ) , are represented by the continuous curve . they correspond to @xmath318 . the superposition of debey components , eqs . ( [ limiting010 ] ) and ( [ limiting011 ] ) , is represented by the square symbols and it can be observed that it fits the cole - cole function over the whole range of frequencies . some of the debye components , those with indexes @xmath319 , are explicitly shown . ] these results can be used to give an insight of the physical mechanisms underlying the relaxation process if we consider eqs . ( [ limiting002 ] ) and ( [ limiting003 ] ) , and the definition of the static susceptibility @xmath320 . the amplitude of each elementary debye contribution in eq . ( [ limiting010 ] ) , can be written in terms of an effective numerical density of polar elements @xmath321 participating in the process with relaxation time @xmath307 , that is @xmath322 comparing eqs . ( [ limiting011 ] ) and ( [ limiting013 ] ) we obtain @xmath323 where @xmath324 , is the effective fraction of polar elements which contribute to process @xmath308 . in the limit of slow and fast processes , i.e. @xmath311 and @xmath312 , respectively , and in terms of the relaxation time @xmath307 , eq . ( [ limiting014 ] ) , @xmath325 can be recast into @xmath326 this result implies that the power law memory kernel ( [ frac002 ] ) and consequently the cole - cole function ( [ limiting004 ] ) , can be suitable for describing the dynamics of systems in which a distribution of relaxation times exist , such that the fraction of polar elements of the system participating in the processes with relaxation time @xmath307 , scales as @xmath327 in the range of fast processes , while in the range of slow processes it scales as @xmath328 . in addition , it should be noticed that the fraction of polar elements participating in a given relaxation process depends on the dimensionless parameter @xmath265 which in turn depends on the static applied field @xmath181 .

different memory kernels are considered and used to compare with experiments of dielectric relaxation in glassy systems . for the case of cole - cole relaxation , we infer the distribution of relaxation times and its relation with an effective distribution of dipolar moments that can be attributed to different segmental motions of the polymer chains in a melt .

mesoscopic non - equilibrium thermodynamics is used to formulate a model describing non - homogeneous and non - debye dielectric relaxation . the model is presented in terms of a fokker - planck equation for the probability distribution of non - interacting polar molecules in contact with a heat bath and in the presence of an external time - dependent electric field . memory effects are introduced in the fokker - planck description through integral relations containing memory kernels , which in turn are used to establish a connection with fractional fokker - planck descriptions . the model is developed in terms of the evolution equations for the first two moments of the distribution function . these equations are solved by following a perturbative method from which the expressions for the complex susceptibilities are obtained as a functions of the frequency and the wave number . different memory kernels are considered and used to compare with experiments of dielectric relaxation in glassy systems . for the case of cole - cole relaxation , we infer the distribution of relaxation times and its relation with an effective distribution of dipolar moments that can be attributed to different segmental motions of the polymer chains in a melt .

1104.1034

i

dependence among values of variables occurs everywhere in computer science ( databases , software engineering , knowledge representation , ai ) but also the social sciences ( human history , stock markets , etc . ) . in his monograph @xcite in 2007 vnnen introduced functional dependence into the language of first - order logic . functional dependence of the value of @xmath9 from the values of @xmath10 means that there exists a determinating function @xmath11 with @xmath12 , i.e. , the value of @xmath9 is completely determined by the values of @xmath13 alone . we denote this form of dependence ( or determination ) by the _ dependence atom _ @xmath14 . to examine dependence between situations , plays , worlds , events or observations we consider collections of these , so called _ teams_. for example , a database can be interpreted as a team . in this case @xmath15 means that in every record the value of the attribute @xmath9 is determined by the values of the attributes @xmath13 . in modal logic a team is a set of worlds in a kripke structure . here @xmath14 means that in every world of the team the value of the atomic proposition @xmath9 is determined by the propositions @xmath10 , i.e. , there is a fixed boolean function @xmath16 that determines the value of @xmath9 from the values of @xmath13 for all worlds in the team . in first - order logic @xmath17 means the same for a function @xmath18 where @xmath19 is the universe of a first - order structure . _ dependence logic _ @xcite is then defined by simply adding dependence atoms to usual first - order logic and _ modal dependence logic _ ( @xmath0 ) @xcite is defined by introducing dependence atoms to modal logic . besides the inductive semantics ( which we will use here ) vnnen also gave two equivalent game - theoretic semantics for @xmath0@xcite . sevenster showed that for singleton sets of worlds there exists a translation from @xmath0to plain modal logic @xcite . sevenster also showed that the satisfiability problem for @xmath0is @xmath20@xmath21-complete @xcite and lohmann and vollmer continued the complexity analysis of the satisfiability problem for @xmath0by systematically restricting the set of allowed modal and propositional operators and completely classifying the complexity for all fragments of @xmath0definable in this way @xcite . sevenster @xcite also introduced classical disjunction ( which is _ classical _ in a more set theoretic way of looking at the semantics ; cf . @xcite ) into the language of @xmath0 . in the following we always think of the version that includes both classical disjunction ( here denoted by @xmath22 ) as well as dependence disjunction when we write @xmath0 . the method of systematically classifying the complexity of logic related problems by restricting the set of operators allowed in formulae goes back to lewis who used this method for the satisfiability problem of propositional logic @xcite . recently it was , for example , used by hemaspaandra et al . for the satisfiability problem of modal logic @xcite and by lohmann and vollmer for the satisfiability problem of @xmath0@xcite . the motivation for this approach is that by systematically examining all fragments of a logic one might find a fragment which allows for efficient algorithms but still has high enough expressivity to be useful in practice . on the other hand , this systematic approach usually leads to insights into the sources of hardness , i.e. , the exact components of the logic that make satisfiability , model checking etc . hard . in this paper we transfer the method from satisfiability @xcite to model checking and classify the model checking problem for almost all fragments of @xmath0definable by restricting the set of allowed modal ( @xmath23 , @xmath7 ) and propositional ( @xmath24 , @xmath25 , @xmath22 , @xmath26 ) operators to an arbitrary subset of all operators . the model checking problem asks whether a given formula is true in a given team of a given kripke structure . for plain modal logic this problem is solvable in @xmath8as shown by clarke et al . a detailed complexity classification for the model checking problem over fragments of modal logic was shown by beyersdorff et al . @xcite ( who investigate the temporal logic @xmath27which contains plain modal logic as a special case ) . in the case of @xmath0it turns out that model checking is @xmath6-complete in general and that this still holds for several seemingly quite weak fragments of @xmath0 , e.g. , the one without modalities or the one where nothing except dependence atoms and @xmath7 is allowed ( first and fourth line in [ results - unbounded - dep ] ) . strangely , this also holds for the case where only the both disjunctions @xmath28 and @xmath22 are allowed and not even dependence atoms occur ( third line in [ results - unbounded - dep ] ) . furthermore it seems natural to not only restrict modal and propositional operators but to also impose restrictions on dependence atoms . one such restriction is to limit the arity of dependence atoms , i.e. , the number @xmath29 of variables @xmath13 by which @xmath9 has to be determined to satisfy the formula @xmath14 , to a fixed upper bound @xmath30 ( the logic is then denoted by @xmath31 ) . for this restriction model checking remains @xmath6-complete in general but for the fragment with only the @xmath7 operator allowed this does not hold any more ( seventh line in table [ results - bounded - dep ] ) . in this case either @xmath24 ( fourth line in table [ results - bounded - dep ] ) or @xmath25 ( sixth line in table [ results - bounded - dep ] ) is needed to still get @xmath6-hardness . we classify the complexity of the model checking problem for fragments of @xmath0with unbounded as well as bounded arity dependence atoms . we are able to determine the tractability of each fragment except the one where formulae are built from atomic propositions and unbounded dependence atoms only by disjunction and negation ( sixth line in table [ results - unbounded - dep ] ) . in each of the other cases we either show @xmath6-completeness or show that the model checking problem admits an efficient ( polynomial time ) solution . @xmath32 @xmath33 operator present @xmath34 operator absent @xmath35 complexity independent of operator @xmath36 @xmath33 operator present @xmath34 operator absent @xmath35 complexity independent of operator in table [ results - unbounded - dep ] we list our complexity results for the cases with unbounded arity dependence atoms and in table [ results - bounded - dep ] for the cases with an a priori bound on the arity . in these tables a + means that the operator is allowed , a - means that the operator is forbidden and a * means that the operator does not affect the complexity of the problem .

modal dependence logic ( ) was introduced recently by vnnen . it enhances the basic modal language by an operator . for propositional variables the atomic formula intuitively states that the value of is determined solely by those of . we show that model checking forformulae over kripke structures is-complete and further consider fragments ofobtained by restricting the set of allowed propositional and modal connectives . it turns out that several fragments , e.g. , the one without modalities or the one without propositional connectives , remain-complete . we also consider the restriction ofwhere the length of each single dependence atom is bounded by a number that is fixed for the whole logic . we show that the model checking problem for this boundedis still-complete while for some fragments , e.g. , the fragment with only , the complexity drops to . furthermore we almost completely classifiy the computational complexity of the model checking problem for all restrictions of propositional and modal operators for both unbounded as well as boundedwith both classical as well as dependence disjunction . this is the second arxiv version of this paper . it extends the first version by the investigation of the classical disjunction . a shortened variant of the first arxiv version was presented at sofsem 2012 . * acm subject classifiers : * f.2.2 complexity of proof procedures ; f.4.1 modal logic ; d.2.4 model checking * keywords : * dependence logic , modal logic , model checking , computational complexity

modal dependence logic ( ) was introduced recently by vnnen . it enhances the basic modal language by an operator . for propositional variables the atomic formula intuitively states that the value of is determined solely by those of . we show that model checking forformulae over kripke structures is-complete and further consider fragments ofobtained by restricting the set of allowed propositional and modal connectives . it turns out that several fragments , e.g. , the one without modalities or the one without propositional connectives , remain-complete . we also consider the restriction ofwhere the length of each single dependence atom is bounded by a number that is fixed for the whole logic . we show that the model checking problem for this boundedis still-complete while for some fragments , e.g. , the fragment with only , the complexity drops to . we additionally extendby allowing classical disjunction introduced by sevenster besides dependence disjunction and show that classical disjunction is always at least as computationally bad as bounded arity dependence atoms and in some cases even worse , e.g. , the fragment with nothing but the two disjunctions is-complete . furthermore we almost completely classifiy the computational complexity of the model checking problem for all restrictions of propositional and modal operators for both unbounded as well as boundedwith both classical as well as dependence disjunction . this is the second arxiv version of this paper . it extends the first version by the investigation of the classical disjunction . a shortened variant of the first arxiv version was presented at sofsem 2012 . * acm subject classifiers : * f.2.2 complexity of proof procedures ; f.4.1 modal logic ; d.2.4 model checking * keywords : * dependence logic , modal logic , model checking , computational complexity

1412.6521

i

consider a system at finite charge density and temperature . if momentum is conserved then a non - zero overlap between the electric current and the momentum operator leads to an infinite dc electrical conductivity . this feature is indeed observed in the simplest holographic model of such a system : the reissner - nordstrm - ads ( rn ) black brane . a natural way to build more realistic holographic models that allow momentum to dissipate is to modify the rn geometry such that translational invariance is broken explicitly along the boundary directions . this can be achieved by introducing spatially - dependent sources for ( neutral ) matter fields in the bulk . the backreaction of such sources typically leads to an inhomogeneous solution . examples include a spatially - periodic massive neutral scalar @xcite or chemical potential @xcite . in these studies the dc conductivity is indeed finite , in good agreement with the field theory calculation of @xcite , and the delta function is resolved into a drude peak of the form @xmath0 however , the study of such setups is technically involved because one must solve non - linear pdes . it is thus desirable to have a simpler holographic model that incorporates dissipation . in this way one may hope to uncover , in a computationally simple way , universal properties of field theories with holographic duals in which momentum can relax . one can obtain such a model with a homogeneous bulk geometry by exploiting the presence of a symmetry in the matter sector . for example , a set of massless scalar fields that are linear in the boundary coordinates were added to the usual einstein - maxwell system in @xcite . the shift symmetry of the scalars leads to a homogeneous bulk stress tensor . we shall refer to such scalars as axions . moreover , by choosing the axion configuration in a particular way it is also possible to render the geometry isotropic , allowing for the existence of an analytical black brane solution . the finite dc conductivity can be expressed analytically and , in four bulk dimensions , turns out to be independent of temperature . at finite chemical potential , a transition from a coherent ( i.e. drude - like ) metallic behaviour to an incoherent one as the parameter that controls the amount of momentum relaxation is increased was identified in this model by @xcite . a similar transition was observed in the thermal conductivity of this system at zero chemical potential in @xcite , wherein this feature was nicely explained in terms of the structure of the quasi - normal modes ( qnm ) of the black brane . there it was observed that the black brane conducts coherently when there is a purely - dissipative mode parametrically separated from the rest of the qnm . this model can be generalized in a number of ways . one is to introduce a further neutral scalar field , this time with a potential and couplings to the axions and maxwell field . we shall refer to such a scalar as a dilaton . momentum - dissipating black brane solutions to such theories were constructed in @xcite and referred to as ` q - lattices ' . the dc conductivity of these solutions is again finite and determined in terms of the horizon properties of the background @xcite . moreover , @xcite identified metallic and insulating phases , the latter of which being characterized by a vanishing dc conductivity at zero temperature . in the metallic phase , drude behaviour is observed for a large range of temperatures , while in the insulating phase the low frequency region is not described by drude physics . metal - insulator transitions were observed in similar models in @xcite . see @xcite for further studies of such models and also @xcite for alternative generalizations . a conceptually distinct approach was considered in @xcite , wherein the theory of massive gravity @xcite was employed to remove translational invariance on the boundary as a result of breaking ( a subset of ) diffeomorphisms in the bulk . the resulting solutions possess a finite dc conductivity for which an analytical expression was provided in @xcite adapting a calculation in @xcite . in @xcite , it was argued that there is a qualitative connection between massive gravity and a perturbative inhomogeneous lattice generated by a neutral scalar with spatially modulated boundary conditions . more precisely , the authors found that , at leading order in the lattice strength , the equations of motion that govern the conductivity are those of massive gravity if the radially - dependent graviton mass is proportional to the square of the neutral scalar . see @xcite for further studies of massive gravity in this context . the simplicity of these models makes them amenable to generalization , allowing us to conveniently study different field theory phenomena in setups in which translational invariance is broken . the goal of this paper is to study superconductivity in the presence of momentum relaxation , focusing on the axion model and a particular axion - dilaton model . to do so , we shall generalize the by now canonical model of holographic superconductivity of @xcite , consisting of an einstein - maxwell - charged scalar theory , by coupling it to the neutral scalar sectors of @xcite and @xcite . the condensation of the charged order parameter is dual to the appearance of charged scalar hair on the black brane ( for scale invariant boundary conditions that set the source of the dual operator to zero ) as we lower the temperature below a critical value @xmath1 . superconducting inhomogeneous lattices were constructed in @xcite in an einstein - maxwell - charged scalar system with spatially - modulated chemical potential . below the critical temperature , the optical conductivity behaves at low frequencies as a two - fluid model consisting of a pole in the imaginary part plus a normal drude component : @xmath2 note that the appearance of a pole in the imaginary part demands there must be a delta function in the real part with strength set by @xmath3 in order to satisfy the kramers - kronig relations . as the temperature is decreased below @xmath1 , it was observed that the parameters @xmath3 and @xmath4 grow rapidly while @xmath5 quickly decreases . in addition , other simplified models of momentum relaxation have been utilized in the study of holographic superconductors . the condensation of a charged scalar was studied in a massive gravity theory in @xcite . where @xmath6 is the reference metric . this is different to the massive gravity theory whose optical conductivity coincides with that of @xcite . ] their results qualitatively agree with those in @xcite ; namely , they find a two - fluid model with a drude normal component . in @xcite , a charged scalar was coupled to the model of @xcite , which only has one neutral massless scalar . there it was observed that the critical temperature decreases as the scale of momentum dissipation increases and that a pseudo gap appears in the optical conductivity in the direction in which the breaking of translational invariance is introduced . see @xcite for work on the dynamical formation of a condensate in the same model . more recently , @xcite studied a holographic superconductor in an axion - dilaton model as we also do below , although with a different mass for the dilaton . there it was observed that the critical temperature is reduced by the presence of momentum relaxation and that breaking the @xmath7 from the metallic phase results in a two - fluid model behaviour with drude normal component . this is not so in the insulating phase , for which the normal component is not well described by a drude peak . see also @xcite for an alternative approach . in this paper we construct superconducting solutions and observe the transition from all the unbroken phases we have considered ; namely , the exact solution of @xcite in the axion model and one choice of metallic and insulating solutions in the axion - dilaton model of @xcite . crucially , we first demonstrate that the broken phases are indeed thermodynamically preferred , confirming their physical relevance . our main result is the observation of a zero - frequency pole in the imaginary part of the electrical conductivity , confirming that our solutions are indeed superconducting . moreover , in cases in which the normal phase possesses a drude behaviour at low frequencies ( namely , in the axion model with small momentum dissipation parameter and the metallic solution in the axion - dilaton model ) , the broken phase can be described as a two - fluid model whose low - frequency conductivity is given by . we find that this can be explained by the structure of the lowest qnms : drude peaks arise when there is a purely - dissipative mode that is well - separated from the rest of the excitations , resembling the picture of @xcite . when this ceases to be true , we observe departures from the drude behaviour in the real part of the conductivity and thus the first term in must be modified . we study the temperature dependence of the parameters in for the broken phase of the metal of the axion - dilaton model . as we lower the temperature from @xmath1 , we find that @xmath4 first decreases slowly and then increases quickly , @xmath5 is monotonically decreasing reaching a small value , and @xmath3 monotonically increases approaching a constant . we find that @xmath3 exhibits the same behaviour in the broken phase of the insulator and also in the broken phase studied in the axion model . we also check the ferrell - glover - tinkham ( fgt ) sum rule @xcite , which relates ` missing ' spectral weight to the strength of the zero - frequency pole . we verify that the sum rule holds for the broken metallic phase and the broken phase in the axion model , as was also demonstrated for the inhomogeneous lattice of @xcite . this paper is organized as follows . in section [ sec : axion ] we study the axion model coupled to a charged scalar . we describe a general class of solutions which implement momentum relaxation and discuss the calculation of the optical conductivity for such configurations . after a review of the normal phase and its conductivity , we move on to explore the broken phase . in section [ sec : axion_dilaton ] we also follow this structure for a particular axion - dilaton model and study both metallic and insulating states . we conclude with a discussion of our results and comments on extensions and more general models in section [ sec : discussion ] .

momentum relaxation can be built into many holographic models without sacrificing homogeneity of the bulk solution . in this paper we turn on a charged scalar field in order to explore the condensation of a charged scalar operator in the dual theories . after demonstrating that the relaxed superconductors we construct are thermodynamically relevant , we find that the finite dc electrical conductivity of the normal phase is replaced by a superfluid pole in the broken phase . moreover , when the normal phase possesses a drude behaviour at low frequencies , the optical conductivity of the broken phase at low frequencies can be described by a two - fluid model that is a sum of a drude peak and a superfluid pole , as was found recently for inhomogeneous holographic superconductors . we also study cases in which this low - frequency behavior does not hold . we find that the drude description is accurate when the retarded current - current correlator has a purely - dissipative pole that is well - separated from the rest of the excitations .

momentum relaxation can be built into many holographic models without sacrificing homogeneity of the bulk solution . in this paper we study two such models : one in which translational invariance is broken in the dual theory by spatially - dependent sources for massless scalar fields and another that features an additional neutral scalar field . we turn on a charged scalar field in order to explore the condensation of a charged scalar operator in the dual theories . after demonstrating that the relaxed superconductors we construct are thermodynamically relevant , we find that the finite dc electrical conductivity of the normal phase is replaced by a superfluid pole in the broken phase . moreover , when the normal phase possesses a drude behaviour at low frequencies , the optical conductivity of the broken phase at low frequencies can be described by a two - fluid model that is a sum of a drude peak and a superfluid pole , as was found recently for inhomogeneous holographic superconductors . we also study cases in which this low - frequency behavior does not hold . we find that the drude description is accurate when the retarded current - current correlator has a purely - dissipative pole that is well - separated from the rest of the excitations .

hep-lat9704011

i

numerical simulations of gluons on a space - time lattice provide at present the most reliable means of studying glueballs . glueball correlation functions are , however , notoriously difficult quantities to measure in monte carlo simulations : since the masses of these states are rather high and their creation operators have large vacuum fluctuations , the signal - to - noise ratio falls extremely rapidly as the separation between the source and sink is increased . because of this , reliable studies of glueballs on fine lattices have required prohibitively large computer resources . thus , the development of more efficient simulation techniques in lattice qcd is crucial to establishing a detailed description of glueballs and their interactions . the objective of this work is to examine the effectiveness of using an improved , anisotropic lattice action to reduce the computational effort needed to determine the glueball spectrum in quenched qcd . improved actions allow access to continuum physics on coarser lattices than possible using the simple wilson discretization . coarse lattice simulations are more efficient for several reasons : for a given physical volume , much fewer lattice sites are needed ; the alleviation of critical slowing down permits the faster generation of statistically independent gauge - field configurations ; glueball operator smearing is faster due to the decreased number of links and a decrease in the number of smearing iterations required ; glueball wavefunctions extend over much fewer lattice sites on a coarse lattice , making the variational technique far more effective when using a feasible number ( a dozen or so ) of basis operators . however , for glueball mass calculations , the coarseness of the temporal lattice spacing is a severe drawback . as the masses in lattice units of the states of interest are so large , the number of correlator time intervals which can be measured is reduced greatly @xcite . a straightforward solution to this problem which preserves the computational advantages of coarse lattices@xcite is to make use of anisotropic lattices in which the temporal spacing is much smaller than that in the spatial directions . this enables us to exploit the enhanced signal - to - noise of the correlation functions at smaller temporal separations . a natural scale for the temporal lattice spacing should be the inverse of the energy of the states of interest ; thus , for glueballs , a temporal cut - off larger than 1.5 gev allows resolution from accessible statistics of the correlator over a few time - slices . meanwhile , the scale for the spatial lattice should be set by the size of the wavefunction of the state ; a spatial grid separation in the range @xmath2 fm would seem reasonable . since we propose to use lattices in which the temporal lattice spacing is small , improvement of the discretization in this direction is not needed . thus , a lattice action which couples only nearest - neighbor time - slices can be used . the transfer matrix corresponding to such an action is hermitian and positive definite ; all of our effective masses must converge to their plateau values monotonically from above . this ensures the validity of variational techniques which minimize the effective masses at small temporal separations . such techniques are very effective in diminishing the excited - state contributions to the glueball correlation functions and are crucial for efficient extraction of ground - state masses . in this paper , we demonstrate the increased efficiency of glueball simulations using these actions on anisotropic lattices . we present results for the masses of three of the lighter su(3 ) glueball states , the scalar ( @xmath3 ) , the tensor ( @xmath0 ) , and the pseudovector ( @xmath1 ) . the masses of the first excited states in the scalar and tensor channels were also examined . ten simulations at lattice spacings ranging from 0.2 to 0.4 fm were performed , enabling reliable extrapolations to the continuum limit ( although the mass of the scalar glueball was somewhat problematic ) . the results are compared to previous simulation data obtained using the wilson action and we find that more accurate determinations of the tensor and pseudovector glueball masses have been achieved . a comparison of efficiencies is also made . lastly , finite volume effects are shown to be small . the new action used in our simulations is described in sec . [ sec : action ] . the details of the glueball simulations , including the construction of the glueball operators , the generation of the gauge - field configurations , and the analysis of the monte carlo data , are given in sec . [ sec : simulation ] . the hadronic scale @xmath4 is used to relate our results at different values of the coupling @xmath5 and the aspect ratio @xmath6 . the determination of this scale in terms of the lattice spacing using the static potential is outlined in sec . [ sec : r0as ] . [ sec : resdiscuss ] contains our results and discussion : the glueball mass measurements are presented in detail ; finite volume effects are studied ; extrapolations of the masses at finite spacing to the continuum limit are undertaken ; the conversion of our results into physical units is described ; and a comparison of efficiencies with wilson action simulations is made . our conclusions are given in sec . [ sec : conclusions ] , along with an outline of future work .

monte carlo results for the low - lying glueball spectrum using an improved , anisotropic action are presented . ten simulations at lattice spacings ranging from 0.2 to 0.4 fm and two different anisotropies have been performed in order demonstrate the advantages of using coarse , anisotropic lattices to calculate glueball masses .

monte carlo results for the low - lying glueball spectrum using an improved , anisotropic action are presented . ten simulations at lattice spacings ranging from 0.2 to 0.4 fm and two different anisotropies have been performed in order demonstrate the advantages of using coarse , anisotropic lattices to calculate glueball masses . our determinations of the tensor ( ) and pseudovector ( ) glueball masses are more accurate than previous wilson action calculations . epsf

astro-ph0101468

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in this paper we have quantified the effects of morphological evolution on the observed properties of early - type galaxies in clusters at @xmath73 . a manifestation of morphological evolution is the `` progenitor bias '' : at high redshift , the progenitors of the youngest present - day early - type galaxies drop out of the sample . in general , the observed evolution of the mean luminosity and color of early - type galaxies is slower than the true evolution of all progenitors of early - type galaxies , and the scatter in luminosity and color is smaller . if morphological evolution is strong the observed mean evolution is similar to that of a single age stellar population formed at @xmath25 , and the scatter is approximately constant with redshift . these results can be derived analytically ( cf . appendix [ equations.sec ] ) . in traditional models it is difficult to explain the low number fractions of early - type galaxies in distant clusters and the homogeneity and slow evolution of their stellar populations simultaneously . we have shown that the observed constant scatter in the cm relation and the fp and the slow evolution of the mean @xmath3 ratio are natural consequences of morphological evolution . observations of early - type galaxies in clusters at @xmath73 are well fitted by our models , and we can constrain the morphological transformation rate and the star formation histories of galaxies prior to transformation from the fits . the observed evolution of the early - type galaxy fraction constrains the transformation rate , the evolution of the mean @xmath3 ratio constrains the time of onset of star formation , and the scatter in the color - magnitude relation and the fundamental plane constrain the star formation histories of galaxies prior to transformation . the best fits are obtained if approximately 50% of present - day early - type galaxies were transformed from other morphological types at @xmath0 , star formation commenced at early times ( @xmath95 ) , and star formation was approximately constant prior to transformation . such gradual star formation histories are consistent with the low numbers of star burst galaxies seen in intermediate redshift clusters ( e.g. , abraham et al . 1996 , balogh et al . 1998 , ellingson et al . 2000 ) , although some star bursts could be concealed by dust ( poggianti et al . 1999 ) . the observed color and luminosity evolution of early - type galaxies can be corrected for the effects of progenitor bias . we have shown that for a given morphological transformation rate the progenitor bias is proportional to the scatter in @xmath3 ratio or color . the correction is maximal if the scatter in the cm relation and the fp at @xmath18 is entirely due to age variations between galaxies . this maximum correction to the evolution of rest frame @xmath118 is approximately @xmath147 , and the correction to the evolution of rest frame @xmath148 is approximately @xmath149 . these corrections can easily be computed for other rest frame bands using stellar population synthesis models . as an example , the worthey ( 1994 ) and bruzual & charlot ( 2000 ) models give @xmath150 for solar metallicity and salpeter ( 1955 ) imf . therefore , the correction to @xmath151 is @xmath152 . the corrections are upper limits , because variations in dust content and metallicity probably contribute to the scatter . on the other hand , as pointed out by , e.g. , trager et al . ( 2000 ) variations in metallicity and age may conspire to produce a low scatter . if this is confirmed the corrections may be larger than the values given here . because the mean luminosity and color evolution are underestimated , the mean age of the stars in early - type galaxies that is derived from them is overestimated in traditional models . we find that the mean luminosity weighted formation redshift of early - type galaxies can be as low as @xmath153 for @xmath6 and @xmath7 , or @xmath8 for @xmath6 and @xmath9 , if the scatter in the cm relation and the fp is entirely caused by age variations . this result is important , because previous studies which ignored the effects of progenitor bias found much earlier formation epochs of the stars in early - type galaxies ( e.g. , ellis et al . 1997 , stanford et al . 1998 , van dokkum et al . the redshift range 2 3 is of interest because it is observationally accessible . in particular , our results are consistent with the idea that ( some ) ly - break galaxies are the star forming building blocks of present - day early - type galaxies in clusters ( baugh et al . 1998 ) . in our main analysis we treated the evolution of early - type galaxies as one class , but we also explored a model which separates the two classes . in our model , only the s0 galaxies evolve morphologically from spiral galaxies , as advocated by dressler et al . the model predicts that colors and @xmath3 ratios of luminous s0 galaxies in nearby clusters are systematically offset from those of elliptical galaxies , in the sense that elliptical galaxies are redder and have higher @xmath3 ratios . this result is in conflict with studies of bright early - type galaxies in the cores of rich clusters , which generally find very little difference between elliptical and s0 galaxies ( e.g. , bower et al . 1992 , ellis et al . there are several ways to resolve this conflict : star formation may have started later in the elliptical galaxies , or most of the star formation in elliptical and s0 galaxies must have occurred at very early times , long before morphological transformation to early - type galaxy . alternatively , there might be a conspiracy between age and metallicity ( e.g. , trager et al . 2000 ) such that s0 galaxies have systematically higher metallicities than elliptical galaxies of the same luminosity . this can be tested by absorption line measurements ; current measurements do not indicate such an effect for cluster galaxies . finally , the separation between elliptical and s0 galaxies may not be very sharp . rix & white ( 1990 ) showed that many s0 galaxies will be classified as elliptical galaxies when not viewed edge - on . jrgensen and franx ( 1994 ) showed that the elliptical and s0 galaxies in coma can be modeled by a population of galaxies with a homogeneous distribution of bulge - to - total light ratio . hence , the galaxies which are classified as elliptical galaxies may also have evolved since @xmath21 . it is interesting to compare our results to predictions of semi - analytical galaxy formation models in cdm cosmologies ( white & frenk 1991 ; cole 1991 ) . because the morphologies of galaxies change quite rapidly in these models ( e.g. , baugh , cole , & frenk 1996 , kauffmann 1996 ) progenitor bias is an important ingredient . in particular , the models provide reasonable fits to the observed scatter in the cm relation ( kauffmann & charlot 1998 ; cole et al . 2000 ) and the evolution of the mean @xmath3 ratio of massive galaxies ( diaferio et al . 2000 ) , even though galaxies in these simulations have quite complex ( star ) formation histories . however , in these complex models it is difficult to isolate and study specific effects such as the progenitor bias . the main contribution of our work is that we use very simple models with only three free parameters . we show that the effects of progenitor bias can be constrained directly from the observations , and it is straightforward to reproduce our results . the main uncertainty in the present paper is the rate of morphological evolution . more high resolution , large field studies of distant clusters are needed to constrain the evolution of the early - type galaxy fraction better . detailed simulations are needed to determine the effects of infall on the observed properties and morphological mix of cluster galaxies ( e.g. , ellingson et al . furthermore , studies of the evolution of the mass function of rich clusters to @xmath21 are necessary to determine whether high redshift clusters are representative progenitors of present - day rich clusters . as pointed out by , e.g. , kauffmann ( 1995 ) significant evolution in the mass function of rich clusters may result in a `` progenitor bias '' at the cluster level . we thank the referee , richard ellis , for constructive comments which improved the presentation and clarity of the paper significantly . p. g. v. d. acknowledges support by nasa through hubble fellowship grant hf-01126.01 - 99a awarded by the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , for nasa under contract nas 5 - 26555 .

morphological and spectroscopic studies of high redshift clusters indicate that a significant fraction of present - day early - type galaxies was transformed from star forming galaxies at . on the other hand , the slow luminosity evolution of early - type galaxies and the low scatter in their color - magnitude relation indicate a high formation redshift of their stars . in this paper we construct models which reconcile these apparently contradictory lines of evidence , and we quantify the effects of morphological evolution on the observed photometric properties of early - type galaxies in distant clusters . furthermore , the scatter in age , and hence the scatter in color and luminosity , is approximately constant with redshift . these results are consequences of the `` progenitor bias '' : the progenitors of the youngest low redshift early - type galaxies drop out of the sample at high redshift . we construct models which reproduce the observed evolution of the number fraction of early - type galaxies in rich clusters and their color and luminosity evolution simultaneously . our modelling indicates that% of early - type galaxies were transformed from other galaxy types at , and their progenitor galaxies may have had roughly constant star formation rates prior to morphological transformation . the progenitor bias is a linear function of the scatter in the color - magnitude relation produced by age variations , and is maximal if the observed scatter is entirely due to age differences . after correcting the observed evolution of the mean ratio for the maximum progenitor bias we find that the mean luminosity weighted formation redshift of stars in early - type galaxies for and , and for and . the results are consistent with the idea that ( some ) ly - break galaxies are star forming building blocks of massive early - type galaxies in clusters .

morphological and spectroscopic studies of high redshift clusters indicate that a significant fraction of present - day early - type galaxies was transformed from star forming galaxies at . on the other hand , the slow luminosity evolution of early - type galaxies and the low scatter in their color - magnitude relation indicate a high formation redshift of their stars . in this paper we construct models which reconcile these apparently contradictory lines of evidence , and we quantify the effects of morphological evolution on the observed photometric properties of early - type galaxies in distant clusters . we show that in the case of strong morphological evolution the apparent luminosity and color evolution of early - type galaxies are similar to that of a single age stellar population formed at , irrespective of the true star formation history of the galaxies . furthermore , the scatter in age , and hence the scatter in color and luminosity , is approximately constant with redshift . these results are consequences of the `` progenitor bias '' : the progenitors of the youngest low redshift early - type galaxies drop out of the sample at high redshift . we construct models which reproduce the observed evolution of the number fraction of early - type galaxies in rich clusters and their color and luminosity evolution simultaneously . our modelling indicates that% of early - type galaxies were transformed from other galaxy types at , and their progenitor galaxies may have had roughly constant star formation rates prior to morphological transformation . the effect of the progenitor bias on the evolution of the mean ratio and color can be estimated . the progenitor bias is a linear function of the scatter in the color - magnitude relation produced by age variations , and is maximal if the observed scatter is entirely due to age differences . after correcting the observed evolution of the mean ratio for the maximum progenitor bias we find that the mean luminosity weighted formation redshift of stars in early - type galaxies for and , and for and . our analysis places the star formation epoch of early - type galaxies later than previous studies which ignored the effects of progenitor bias . the results are consistent with the idea that ( some ) ly - break galaxies are star forming building blocks of massive early - type galaxies in clusters . 50min 054ms1054 0t_0 keywords galaxies : evolution , galaxies : elliptical and lenticular , cd , galaxies : structure of , galaxies : clusters

1609.03045

i

one of the great opportunities offered by modern genomics is that phylogenetics applied on a genomic scale ( phylogenomics ) should be especially powerful for elucidating gene and genome evolution , relationships among species and populations , and processes of speciation and molecular evolution . however , a well - recognized hurdle is the sheer volume of genomic data that can now be generated relatively cheaply and quickly , but for which analytical tools are lagging . there is a major need to explore new approaches to undertake comparative genomic and phylogenomic studies much more rapidly and robustly than existing tools allow . principal component analysis is the most popular approach for reducing the dimension of multidimensional data sets . the problem of multidimensionality is acute in the rapidly growing area of phylogenomics , which can provide insight into relationships and evolutionary patterns of a diversity of organisms , from humans , plants and animals , to microbes and viruses . data sets consisting of collections of phylogenetic trees are challenging to analyse , due both to high dimensionality and the complexity of the space containing the data . multivariate statistical procedures such as outlier detection @xcite , clustering @xcite and multidimensional scaling @xcite have previously been applied to such data sets . however , principal component analysis is perhaps the most useful multivariate statistical tool for exploring high - dimensional data sets , due to its flexibility and its attractive properties . for example , @xcite and @xcite recently showed that principal component analysis automatically projects to the subspace where the global solution of k - means clustering lies , and so facilitates k - means clustering to find near - optimal solutions . although principal component analysis for data in @xmath2 can be defined in several different ways , the following description is natural for reformulating the procedure in tree - space . suppose we have data @xmath3 where @xmath4 for @xmath5 . for any set of @xmath1 points @xmath6 we can define @xmath7 so that @xmath8 is the hyperplane in @xmath2 containing @xmath9 . the orthogonal @xmath10 distance of any point @xmath11 from @xmath8 is denoted @xmath12 , and the sum of squared projected distances of the data @xmath13 onto the hyperplane is defined by @xmath14 then the @xmath0-th order principal component @xmath15 corresponds to a choice of @xmath16 which _ minimizes _ this sum . in @xmath2 , @xmath17 is the sample mean , @xmath18 is the line through the sample mean which minimizes the sum of squared projected distances , and so on for @xmath19 . although it is not explicit in the definition above , in @xmath2 the principal components are nested : @xmath20 this description of principal component analysis relies heavily on the vector space properties of @xmath2 : @xmath8 is defined as a linear combination of vectors and the procedure uses orthogonal projection . however , the space of phylogenetic trees with @xmath21 leaves is not an euclidean vector space . it follows that we can not directly apply classical principal component analysis to a data set consisting of phylogenetic trees . the set @xmath22 of all phylogenetic trees with @xmath21 leaves labelled @xmath23 is a so - called cat(0 ) space @xcite . this means that @xmath22 is a metric space with a unique geodesic ( shortest length path ) between any pair of points , with the geodesic computable in @xmath24 steps @xcite . amongst other properties , projection onto closed sets is well - defined in cat(0 ) spaces . the analogue of the zero - th order principal component is given by the _ frchet mean _ of the data @xmath25 . the frchet mean is a statistic which characterizes the central tendency of a distribution in arbitrary metric spaces . for any metric space @xmath26 equipped with metric @xmath27 , the frchet population mean , @xmath28 , with respect to distribution @xmath29 is defined by @xmath30 the discrete analogue , the weighted frchet mean of a sample @xmath31 with respect to a weight vector @xmath32 , is @xmath33 where the weights @xmath34 satisfy @xmath35 for @xmath36 . in any cat(0 ) space , @xmath37 is a well - defined unique point given data @xmath13 and weight vector @xmath32 . the definition of the zero - th order principal component @xmath17 in @xmath2 given above coincides with the definition of the frchet sample mean with weights @xmath38 in any cat(0 ) space . several algorithms for computing the frchet sample mean in @xmath22 have been developed @xcite and we review these later in section [ sec : fmalgorithms ] as they play an important role in our methodology . methods for constructing a principal geodesic in tree - space , an analogue of @xmath39 as defined above , have recently been developed . in @xcite , the approach involved firing geodesics from some mean tree . for each candidate geodesic @xmath40 , the sum of squared projected distances @xmath41 was computed and a greedy algorithm was used to adjust @xmath40 in order to mimimize @xmath41 . the geodesics considered were infinitely long , but these have the disadvantage that in some cases many such geodesics fit the data equally well . subsequent approaches therefore considered finitely long geodesic segments @xcite . the geodesic segment between two points @xmath42 is analogous to @xmath8 in equation @xmath43 with @xmath44 , except that the weights @xmath45 must constrained to be positive under the analogy . @xcite constrained the ends of the geodesic to be points in the sample @xmath13 and sought the corresponding geodesic @xmath40 which mimimizes @xmath41 , whereas @xcite did not restrict the geodesic and used a stochastic optimization algorithm to perform the minimization . in this paper we address two fundamental questions : ( i ) which geometric object most naturally plays the role of a @xmath0-th order principal component in tree - space ; and ( ii ) given such an object , how can we efficiently project data points onto the object ? our proposed solution is to replace the definition of @xmath46 given in equation @xmath43 with the locus of the weighted frchet mean of points @xmath9 in tree - space . specifically , suppose @xmath47 and define @xmath48 by @xmath49 where @xmath50 is the @xmath0-dimensional simplex of probability vectors @xmath51 and @xmath52 is the frchet mean of the points in set @xmath16 with weights @xmath53 . we call @xmath8 the _ locus of the frchet mean _ of @xmath16 . our choice of notation is intended to emphasize the analogy between the definition of @xmath8 in tree - space and the corresponding definition for @xmath2 in equation @xmath43 . the locus of the frchet mean is a type of _ minimal surface _ , as the following physical analogy suggests : imagine connecting a point @xmath54 to points @xmath55 by @xmath1 pieces of elastic . when the point @xmath56 is free to move , it will move under the action of the elastic into an equilibrium position in tree - space . we can imagine how this equilibrium point changes as the stiffness in the pieces of elastic is varied , which corresponds to varying @xmath57 . as the equilibrium point moves around it scans out a surface in tree - space . in euclidean space the locus of the frchet mean of some collection of points is a subset of a hyperplane . in tree - space , as we will show , the hyper - surface can be curved . surfaces of this kind have recently been studied by @xcite in the context of riemannian manifolds and other geodesic metric spaces . we discuss the relationship of the present paper to that work in section [ sec : discussion ] . our main theoretical results are as follows . first , when @xmath58 we derive a set of local implicit equations for @xmath8 . these allow us to derive conditions for @xmath8 to be locally flat , and also enable us to construct explicit realizations of @xmath8 in certain interesting cases . secondly , using the implicit equations we show the locus of the frchet mean @xmath8 in @xmath22 is locally @xmath0 dimensional for generic @xmath16 , and thus forms a suitable candidate for a @xmath0-th order principal component . third , we present an algorithm for projection onto @xmath8 which relies only on the cat(0 ) properties of @xmath22 . we demonstrate accuracy of the projection algorithm via a simulation study . the remainder of the paper is organized in the following way . section [ review ] reviews the fundamental concepts and results in phylogenetic tree - space : the construction of geodesics , calculation of the frchet mean and convex hulls . section [ sec : lfm ] studies the geometric features of the locus of the frchet mean , in particular establishing its dimension with theorem [ thm : lfm_dimension ] . we draw the reader s attention to section [ sec : lfm_example ] which contains an explicit example which illustrates geodesics in tree - space , convex hulls , and details of the calculation of dimension . section [ sec : projection_and_pca ] presents the algorithm for projecting sets of phylogenetic tree data onto the locus of the frchet mean of @xmath1 fixed trees and describes the algorithm used to fit these objects to the data . this section also contains a simulation study to test the effectiveness of the algorithms . in section [ sec : results ] we apply these methods to two data sets . the first is a set of gene trees coming from fish and tetrapods intended to investigate the relationship of coelacanth and lungfish to the tetrapods , and the second is a set of gene trees from apicomplexa . we make some concluding remarks in section [ sec : discussion ] .

most biological data are multidimensional , posing a major challenge to human comprehension and computational analysis . principal component analysis is the most popular approach to rendering two- or three - dimensional representations of the major trends in such multidimensional data . the problem of multidimensionality is acute in the rapidly growing area of phylogenomics . principal component analysis offers a means of quantifying variation and summarizing a collection of phylogenies by dimensional reduction . we propose a geometric object which represents a-th order principal component : the locus of the weighted frchet mean of points in tree - space , where the weights vary over the standard-dimensional simplex . combined with a stochastic optimization algorithm , this projection algorithm gives a procedure for constructing a principal component of arbitrary order in tree - space . simulation studies confirm these algorithms perform well , and they are applied to data sets of apicomplexa gene trees and the african coelacanth genome . the results enable visualizations of slices of tree - space , revealing structure within these complex data sets .

most biological data are multidimensional , posing a major challenge to human comprehension and computational analysis . principal component analysis is the most popular approach to rendering two- or three - dimensional representations of the major trends in such multidimensional data . the problem of multidimensionality is acute in the rapidly growing area of phylogenomics . evolutionary relationships are represented by phylogenetic trees , and very typically a phylogenomic analysis results in a collection of such trees , one for each gene in the analysis . principal component analysis offers a means of quantifying variation and summarizing a collection of phylogenies by dimensional reduction . however , the space of all possible phylogenies on a fixed set of species does not form a euclidean vector space , so principal component analysis must be reformulated in the geometry of tree - space , which is a cat(0 ) geodesic metric space . previous work has focused on construction of the first principal component , or principal geodesic . here we propose a geometric object which represents a-th order principal component : the locus of the weighted frchet mean of points in tree - space , where the weights vary over the standard-dimensional simplex . we establish basic properties of these objects , in particular that locally they generically have dimension , and we propose an efficient algorithm for projection onto these surfaces . combined with a stochastic optimization algorithm , this projection algorithm gives a procedure for constructing a principal component of arbitrary order in tree - space . simulation studies confirm these algorithms perform well , and they are applied to data sets of apicomplexa gene trees and the african coelacanth genome . the results enable visualizations of slices of tree - space , revealing structure within these complex data sets .

1609.03045

c

this paper presents three main innovations : ( i ) use of the locus of the frchet mean @xmath8 as an analogue of a principal component in tree - space , ( ii ) proof that @xmath8 has the desired dimension , and ( iii ) the geometric projection algorithm for projecting data onto @xmath8 . the locus of the frchet mean was first proposed as a geometric object for principal component analysis in tree - space by @xcite , though in @xcite pennec has made a similar proposal for analogues of principal component analysis in riemannian manifolds and other geodesic metric spaces . the barycentric subspaces of pennec correspond exactly to the surfaces @xmath8 considered in this paper . pennec s methodology , however , is principally based in the context of a riemannian manifold rather than in tree - space , though he points out the potential for generalization . there are substantial differences between pennec s analogue of principal component analysis on riemannian manifolds ( barycentric subspace analysis ) and the methodology presented in this paper . in particular , a key aim of barycentric subspace analysis is to produce _ nested _ principal components , in the sense of equation @xmath367 , while we do not make that restriction here . for example , if we consider a surface @xmath8 for @xmath310 then the only geodesics which are obviously contained in @xmath8 are the edges , and it is unappealing to restrict the principal geodesic to be one of these . our analysis has been restricted to data sets with relatively few taxa and to the construction of the first and second principal components . the algorithms presented in this paper scale linearly with respect to the number of data points @xmath368 , but run in polynomial time with respect to the number of taxa @xmath94 . however , by partitioning the data set for the geometric projection algorithm , parallel computer architectures can be employed and the speed - up is approximately proportional to the number of processors used . while the geometric projection algorithm runs relatively quickly , the calculations involved in searching for the optimal set of vertices @xmath16 can be very substantial . the experimental data sets in section [ sec : results ] took between 1 and 3 days to analyse , running on 4 processors each . for higher order components with @xmath369 , this computational burden will increase , and it is likely that finding a global minimum for @xmath337 will be more difficult . the figures in section [ sec : results ] show the potential for creating visualizations of the data which reveal meaningful biological structure . the pattern of projected points obtained for the experimental data sets we considered were very similar to results obtained via multidimensional scaling . however , multidimensional scaling is not capable of revealing the features of the data set which cause the observed variation . more information could be included in the graphical representation of our results , such as the distance of the data points from their projections , information about the principal geodesic , and the proximity of points to orthant boundaries . we presented the geometric projection algorithm without a proof of convergence , and used simulation to assess its accuracy . the algorithm is attractive in that it is defined entirely in terms of the geodesic structure on tree - space , and so it could be used on any geodesic metric space , including riemannian manifolds . the algorithm clearly deserves further investigation , and we aim to study its properties in a future publication .

however , the space of all possible phylogenies on a fixed set of species does not form a euclidean vector space , so principal component analysis must be reformulated in the geometry of tree - space , which is a cat(0 ) geodesic metric space . previous work has focused on construction of the first principal component , or principal geodesic . here we establish basic properties of these objects , in particular that locally they generically have dimension , and we propose an efficient algorithm for projection onto these surfaces .

most biological data are multidimensional , posing a major challenge to human comprehension and computational analysis . principal component analysis is the most popular approach to rendering two- or three - dimensional representations of the major trends in such multidimensional data . the problem of multidimensionality is acute in the rapidly growing area of phylogenomics . evolutionary relationships are represented by phylogenetic trees , and very typically a phylogenomic analysis results in a collection of such trees , one for each gene in the analysis . principal component analysis offers a means of quantifying variation and summarizing a collection of phylogenies by dimensional reduction . however , the space of all possible phylogenies on a fixed set of species does not form a euclidean vector space , so principal component analysis must be reformulated in the geometry of tree - space , which is a cat(0 ) geodesic metric space . previous work has focused on construction of the first principal component , or principal geodesic . here we propose a geometric object which represents a-th order principal component : the locus of the weighted frchet mean of points in tree - space , where the weights vary over the standard-dimensional simplex . we establish basic properties of these objects , in particular that locally they generically have dimension , and we propose an efficient algorithm for projection onto these surfaces . combined with a stochastic optimization algorithm , this projection algorithm gives a procedure for constructing a principal component of arbitrary order in tree - space . simulation studies confirm these algorithms perform well , and they are applied to data sets of apicomplexa gene trees and the african coelacanth genome . the results enable visualizations of slices of tree - space , revealing structure within these complex data sets .

astro-ph0003294

i

emission - line nebulae in galactic nuclei are generally considered to fall into three major categories : star - forming or nuclei , seyfert nuclei , and low - ionization nuclear emission - line regions , or liners . the formal divisions between these classes are somewhat arbitrary , as the observed emission - line ratios of nearby galactic nuclei fall in a continuous distribution between liners and seyfert nuclei and between liners and nuclei ( e.g. , * ? ? ? traditionally , liners have been defined as those nuclei having emission - line flux ratios which satisfy the relations [ ] @xmath33727/ [ ] @xmath35007 @xmath4 and [ ] @xmath36300/ [ ] @xmath35007 @xmath5 @xcite . it is possible to construct alternative but practically equivalent definitions , based on other line ratios , which can be applied to datasets that do not include the wavelengths of the [ ] , [ ] , or [ ] lines ( e.g. , * ? ? ? a sizeable minority of galactic nuclei has emission - line ratios which are intermediate between those of `` pure '' liners and those of typical regions powered by hot stars ; these galaxies would be classified as liners except that their [ ] @xmath36300 line strengths are too small in comparison with other lines to meet the formal liner criteria . objects falling into this category have been dubbed `` transition '' galaxies by @xcite , and although this nomenclature is somewhat ambiguous we adopt it here for consistency with the spectroscopic survey of @xcite . that survey defined the transition class in terms of the following flux ratios : [ ] @xmath35007/h@xmath6 @xmath7 3 , + 0.08 @xmath8 [ ] @xmath36300/h@xmath9 @xmath7 0.17 , + [ ] @xmath36583/h@xmath9 @xmath10 0.6 , + [ ] @xmath116716 , 6731 /h@xmath9 @xmath10 0.4 . @xcite have used the term `` weak- [ ] liners '' to refer to galaxies having [ ] @xmath36583/h@xmath9 @xmath12 ( typical of liners ) but which have [ ] @xmath36300/h@xmath9 @xmath13 . this category is essentially identical to the transition class of @xcite , and we will refer to these galaxies as transition objects in this paper . according to the survey results of @xcite , this transition class accounts for 13% of all nearby galaxies , making them about as numerous as seyfert nuclei . the hubble type distribution of transition galaxies is intermediate between that of liners , which are most common in e / s0/sa galaxies , and that of nuclei , which occur most often in hubble types later than sb @xcite . roughly @xmath14 of galaxies with hubble types ranging from s0 to sbc belong to the transition class . there is not a consensus , however , as to whether these transition objects should be regarded as star - forming nuclei , as accretion - powered active nuclei , or as composite objects powered by an agn and by hot stars in roughly equal proportion . there is a large body of literature on the subject of the excitation mechanism of liners which is relevant to the similar transition class . a variety of physical mechanisms has been proposed to explain the emission spectra of liners , including shocks , photoionization by a nonstellar ultraviolet ( uv ) and x - ray continuum , and photoionization by hot stars . ( see filippenko 1996 for a review . ) the possibility that liners ( and seyfert nuclei as well ) might be photoionized by starlight was raised by @xcite , who suggested that very hot ( @xmath15 k ) wolf - rayet ( w - r ) stars in a metal - rich starburst could give rise to an ionizing continuum with a nearly power - law shape in the extreme - uv . more recent atmosphere models have indicated substantially lower temperatures for w - r stars , however , casting doubt on the warmer hypothesis @xcite . subsequent photoionization models have attempted to explain liner and transition - type spectra as resulting from massive main - sequence stars . @xcite found that the spectra of weak- [ ] liners could be explained in terms of photoionization by o3o4 stars having effective temperatures of @xmath16 k , at ionization parameters of @xmath17 to @xmath18 . @xcite carried this line of argument farther , proposing that genuine liner spectra could be generated by early o stars with @xmath19 k , provided that a high - density component ( @xmath20 @xmath2 ) is present in the nlr ; the high densities are needed to boost the strengths of high critical - density emission lines , most notably [ ] @xmath36300 . similar conclusions were reached by @xcite , who explored the effects of absorption by ionized gas as a means to harden the effective ionizing spectrum . recent observations , particularly in the uv and x - ray bands , have provided convincing evidence that many liners are in fact agns , particularly the `` type 1 '' liners which have a broad component to the h@xmath9 emission line ( for a recent review see * ? ? ? the possibility has remained , however , that some liners and transition nuclei are powered entirely by bursts of star formation . an important shortcoming of the model calculations performed by @xcite and @xcite is that the ionizing continua used as input were those of single o - type stars ; these studies did not address the question of whether a liner or transition - type spectrum could result from the the _ integrated _ ionizing continuum of a young stellar cluster . compared with these single - star models , the contribution of late - o and b stars will soften the ionizing spectrum , making the emission - line ratios tend toward those of normal regions . w - r stars , on the other hand , will harden the ionizing spectrum during the period when these stars are present , roughly @xmath21 myr after the burst . another drawback of the o - star models is that they require the presence of stars with effective temperatures higher than are thought to occur in regions of solar or above - solar metallicity , in order to produce a liner or transition - type spectrum rather than an region spectrum . their applicability to galactic nuclei is therefore somewhat unclear . other mechanisms have been proposed for generating liner or transition - type spectra . shock excitation by supernova remnants in an aging starburst may give rise to some transition objects ; the nucleus of ngc 253 is a likely candidate for such an object @xcite . also , post - agb stars and planetary nebula nuclei will produce a diffuse ionizing radiation field which could be responsible for the very faint liner emission ( with h@xmath9 equivalent widths of @xmath22 ) observed in some ellipticals and spiral bulges @xcite . an alternate possibility is that the transition galaxies may simply be composite systems consisting of an active nucleus surrounded by star - forming regions . for a galaxy at a distance of 10 mpc , for example , a 2-wide spectroscopic aperture will include regions within 50 pc of the nucleus . galaxies having emission lines both from a liner nucleus and from surrounding star - forming regions , in roughly equal proportions , will appear to have a transition - type spectrum . this interpretation was advocated by @xcite as the most likely explanation for the majority of transition galaxies , and is consistent with the observed hubble type distribution for the transition class . other authors have similarly contended that transition galaxies are agn/ region composites , based on optical line - profile decompositions @xcite and near - infrared spectra @xcite . two of the 65 transition nuclei observed in the @xcite survey have a broad component to the h@xmath9 emission line , indicating the likely presence of an agn , and it is probable that many more transition nuclei contain obscured agns which were not detected in the optical spectra . on the other hand , radio observations do not appear to support the composite agn / starburst interpretation . in a vla survey of nearby galactic nuclei , nagar et al . ( 1999 ) find compact , flat - spectrum radio cores in more than 50% of liner nuclei , but in only 6% ( 1 of 18 ) of transition objects . this discrepancy suggests that the simple picture of an ordinary liner surrounded by star - forming regions may not apply to the majority of transition objects . recent results from the _ hubble space telescope _ ( _ hst _ ) have shed new light on the question of the excitation mechanism of transition nuclei . as shown by @xcite , the uv spectrum of the well - known transition nucleus in ngc 4569 over 1200 - 1600 is virtually identical to that of a w - r knot in the starburst galaxy ngc 1741 , indicating that o stars with ages of a few myr dominate the uv continuum . @xcite find that the nuclear star cluster in ngc 4569 is producing sufficient uv photons to ionize the surrounding narrow - line region , a key conclusion which provides fresh motivation to study stellar photoionization models . the brightness of the ngc 4569 nucleus , and the consequently high s / n observations that have been obtained , make it one of the best objects with which to study the transition phenomenon . the recent availability of the starburst99 model set @xcite has prompted us to reexamine the issue of ionization by hot stars in liners and transition nuclei . these models give predictions for the spectrum and luminosity of a young star cluster , for a range of values of cluster age , metal abundance , and stellar initial mass function ( imf ) properties . using the photoionization code cloudy @xcite in combination with the starburst99 model continua , we have calculated the expected emission - line spectrum of an region illuminated by a young star cluster , to test the hypothesis that some liners and transition nuclei may be powered by starlight . similar calculations have been performed by @xcite , but for the physically distinct case of metal - poor objects representing galaxies . other examples of photoionization calculations for regions using evolving starburst continua are presented by @xcite , @xcite , and @xcite .

motivated by the discovery of young , massive stars in the nuclei of some liner/ `` transition '' nuclei such as ngc 4569 , we have computed photoionization models to determine whether some of these objects may be powered solely by young star clusters rather than by accretion - powered active nuclei . while such features have not generally been detected to date , they could be revealed in observations having higher spatial resolution .

motivated by the discovery of young , massive stars in the nuclei of some liner/ `` transition '' nuclei such as ngc 4569 , we have computed photoionization models to determine whether some of these objects may be powered solely by young star clusters rather than by accretion - powered active nuclei . the models were calculated with the photoionization code cloudy , using evolving starburst continua generated by the the starburst99 code of . we find that the models are able to reproduce the emission - line spectra of transition nuclei , but only for instantaneous bursts of solar or higher metallicity , and only for ages of myr , the period when the extreme - ultraviolet continuum is dominated by emission from wolf - rayet stars . for clusters younger than 3 myr or older than 6 myr , and for models with a constant star - formation rate , the softer ionizing continuum results in an emission spectrum more typical of regions . this model predicts that wolf - rayet emission features should appear in the spectra of transition nuclei . while such features have not generally been detected to date , they could be revealed in observations having higher spatial resolution . demographic arguments suggest that this starburst model may not apply to the majority of transition nuclei , particularly those in early - type host galaxies , but it could account for some members of the transition class in hosts of type sa and later . the starburst models during the wolf - rayet - dominated phase can also reproduce the narrow - line spectra of some liners , but only under conditions of above - solar metallicity and only if high - density gas is present ( ) . this scenario could be applicable to some `` type 2 '' liners which do not show any clear signs of nonstellar activity .

astro-ph0003294

c

the model results are displayed in figures [ oratio][ohimetal ] . to compare the model outputs with the observed properties of a variety of galaxy types , we have used the emission - line data compiled by @xcite . this catalog has the advantages of a homogeneous classification system , small measurement aperture ( @xmath46 ) , and careful starlight subtraction to ensure accurate emission - line data . in order to reduce confusion and to keep the sample of comparison objects to a reasonable number , we included only objects with unambiguous classifications as , liner , transition , or seyfert . objects with borderline or ambiguous classifications , such as `` liner / seyfert , '' were excluded for clarity . the comparison sample was further reduced by excluding galaxies in which any of the emission lines h@xmath9 , h@xmath6 , [ ] @xmath35007 , [ ] @xmath36300 , [ ] @xmath36583 , or [ ] @xmath3@xmath36716,6731 was undetected or was flagged as having a large uncertainty in flux ( `` b '' or `` c '' quality flags ) . the measured line ratios are corrected for both galactic and internal reddening . figure [ oratio ] plots the ratio [ ] @xmath35007/h@xmath6against [ ] @xmath36300/h@xmath9 at a burst age of 4 myr , for the solar - metallicity model grids a , b , c , and d. the model results are plotted for a density of @xmath47 @xmath2 , as an approximate match to the density of @xmath48 @xmath2 measured for ngc 4569 @xcite . the diagram shows that the instantaneous burst models ( a and b ) are a good match to the line ratios of the transition nuclei , for log @xmath49 . the dusty models from grid b fall more centrally within the region defined to contain transition objects , but the models without dust still closely match transition nuclei having lower [ ] @xmath36300/h@xmath9 ratios . figures [ nratio ] and [ sratio ] show the corresponding diagrams for [ ] @xmath36583/h@xmath9 and for [ ] @xmath116716 , 6731/h@xmath9 , respectively . in both cases we find that the single - burst models span the region occupied by transition nuclei in the diagnostic diagrams . in the constant star - formation rate models ( c and d ) , the uv continuum remains softer than in the w - r - dominated phase of the instantaneous burst models , because of the ongoing formation of luminous o stars . as a result , the low - ionization emission lines are significantly weaker than in the instantaneous burst models at 4 myr . these constant star - formation rate sequences are a reasonable match to the region of nuclei in the diagram , and for [ ] /h@xmath9and [ ] /h@xmath9 the agreement with nuclei is improved at lower densities of @xmath50 @xmath2 , a value more typical of nuclei . these constant star - formation rate models are probably appropriate for galaxies having spatially extended , ongoing star formation in their nuclei . we note that the model curves shown in the figures should not be expected to follow the locus of nuclei in each plot , despite the fact that the models are generated with a starburst continuum . the range of line ratios observed in regions is primarily a sequence in metal abundance @xcite , while our models are shown as sequences in @xmath36 for a given metallicity and density . another point to note about the diagrams is that some of the transition galaxies fall outside the region nominally defined for transition objects , particularly in the [ ] /h@xmath9 ratio ( figure [ oratio ] ) . these galaxies were classified as transition objects by @xcite on the basis of meeting the majority of the classification criteria . similarly , some overlap can be seen in the diagrams between the regions occupied by liners and seyfert nuclei ; this again reflects the fact that galaxies span a continuous range in the values of these emission - line ratios . the variation of [ ] line strength as a function of density is shown in figure [ odensity ] for model grid a at @xmath51 myr . for the range of densities considered ( @xmath52 to @xmath53 @xmath2 ) , the models closely overlap the transition region in the diagram at log @xmath49 . introducing ism depletion and dust grains to the nebula primarily increases the [ ] /h@xmath9 ratio at low density . as expected , the [ ] /h@xmath9 ratio increases with @xmath39 up to densities of @xmath54 @xmath2 , while at densities approaching the critical density of the @xmath36300 transition ( @xmath55 @xmath2 ) this ratio saturates and begins to turn over . figure [ oage ] shows the [ ] /h@xmath9 ratio as a function of burst age ( at @xmath56 @xmath2 ) for model grid a , for ages of 2 to 6 myr . this diagram highlights the dramatic changes that w - r stars generate in the surrounding nebula . from 3 to 5 myr after the burst , when w - r stars dominate the uv continuum , the harder ionizing continuum boosts the strength of [ ] by an order of magnitude and the model sequences appear adjacent to the transition region , with relatively little evolution in the line ratios during this period . as the burst ages beyond 6 myr , the [ ] /h@xmath9 ratio continues to fall , and the emission - line strengths drop rapidly as the ionizing continuum softens and its luminosity decreases . at @xmath57 , the wr - dominated phase occurs slightly later , during time steps 4 , 5 , and 6 myr . the [ ] /h@xmath9 and [ ] /h@xmath9 ratios have a similar dependence on burst age , and are displayed in figures [ nage ] and [ sage ] , respectively ; these results are quite similar to the calculations presented by @xcite to illustrate the effects of the w - r continuum on the [ ] /h@xmath9 ratio . while the [ ] /h@xmath9 ratios in the models at @xmath58 are too low by @xmath59 dex to fit within the nominal transition region , they still closely match those transition nuclei having relatively low values for [ ] /h@xmath9 , and the low - ionization line strengths can be further enhanced by the inclusion of dust and depletion ( as in figure [ oratio ] ) . one puzzling aspect of figure [ oage ] is that for ages outside the range 35 myr , the models predict [ ] /h@xmath9 ratios too low to match the majority of nuclei . @xcite and @xcite discuss this same problem in the context of low - metallicity starburst galaxies . they propose that shocks generated by supernovae and stellar winds provide the additional [ ] emission , without making a significant contribution to the [ ] or [ ] line strengths . shocks could play a similar role in transition galaxies , as in the model of @xcite , although the lack of high - excitation uv line emission in transition nuclei is problematic for the shock hypothesis . a higher upper mass cutoff alleviates this problem to some extent , at least for very young bursts . increasing @xmath33 to 120 @xmath34 boosts [ ] /h@xmath9 by @xmath60 dex for ages of @xmath61 myr . due to the very short lifetimes of the highest - mass stars , however , the @xmath33 = 120 , 100 , and 70 @xmath34 model grids result in identical emission - line spectra from @xmath62 myr onward . metal abundance is an additional parameter which must be considered . the models displayed up to this point were all calculated for a solar abundance set , while the nuclei of early - type spirals are likely to have enhanced heavy - element abundances . as discussed by @xcite , the continuum shortwards of 228 is strongest in the high - metallicity models , because the increased mass - loss rates lead preferentially to the formation of w - r stars at high metal abundance . figure [ olometal ] plots the [ ] /h@xmath9 ratio for abundances of @xmath63 = 0.2 , 0.4 , 1 , and 2 @xmath41 , at a density of @xmath47 @xmath2 and @xmath51 myr . from this diagram , it is clear that solar or higher abundances are necessary to match the [ ] strengths of the transition nuclei ; at lower abundances the line ratios are a better match to those of the high - excitation ( low - metallicity ) nuclei . figure [ ohimetal ] displays the density dependence of the [ ] /h@xmath9 ratio for the @xmath64 model grid ; by comparison with the solar - metallicity model grid in figure [ odensity ] , the higher abundances result in a lower - excitation spectrum with enhanced [ ] emission , due to the harder extreme - uv continuum . it would be advantageous to compare the model results with a wider variety of emission lines . unfortunately , measurements of other optical emission lines are scarce for transition nuclei . the ho et al . survey did not include the [ ] @xmath33727 line , and there is no other homogeneous catalog of [ ] measurements for transition galaxies . to be consistent with a liner or transition - type classification , a model calculation must result in the flux ratio [ ] @xmath33727 / [ ] @xmath35007 @xmath65 . in fact , all of the models with log @xmath66 do satisfy this criterion . thus , any of our models which is consistent with the ho et al . liner or transition classification criteria is also consistent with the original @xcite criterion for the [ ] / [ ] ratio in liners . the relative strengths of uv lines such as ] @xmath32326 , ] @xmath31909 , and @xmath31549 can provide further diagnostics , but none of these lines is detected in ngc 4569 @xcite . the only other transition nucleus having _ hst _ uv spectra available is ngc 5055 , and its spectrum appears to be devoid of uv emission lines as well @xcite . we ran one additional model grid to test whether different model atmospheres for o stars would lead to different results . the starburst99continua were calculated using stellar atmosphere models compiled by @xcite , which are based on the @xcite model set for the massive stellar component . the recent costar model grid of @xcite , which includes non - lte effects , stellar winds , and line blanketing for o stars , makes dramatically different predictions for the ionizing spectra . as shown by @xcite , the costar models yield a luminosity in the he@xmath32 continuum which is four orders of magnitude greater than that predicted by the kurucz models , for the most massive o stars which dominate the uv luminosity at burst ages of @xmath67 myr . in the costar - based models the photon output of the cluster below 228 is essentially constant from 0 to 5 myr . to investigate the effects of this harder o - star continuum on the emission - line spectra , we ran a grid of models using the evolving starburst continua computed by @xcite with the costar atmospheres . model parameters were the same as for model grid a except that an upper mass limit of 120 @xmath34 was used . we find that using the costar atmospheres has a relatively minor effect on our results . in comparison with the starburst99-based models having @xmath33 = 120@xmath34 , the costar model grid yields an increase in the [ ] /h@xmath9 ratio of @xmath68 dex for @xmath69 myr , while [ ] /h@xmath9 and [ ] /h@xmath9 are essentially unaffected . during the period @xmath70 myr , the costar - based models result in an region spectrum , demonstrating that w - r stars are still required in order to generate liner or transition - type line ratios . the strength of the [ ] emission lines at 7291 and 7324 is often used as a diagnostic of dust and depletion , because in the absence of depletion these lines are predicted to be strong in photoionized gas ( e.g. , * ? ? ? * ; * ? ? ? ( the @xmath37291 line is a cleaner diagnostic since @xmath37324 is blended with [ ] @xmath37325 . ) however , for a 4 myr - old burst with nebular conditions of @xmath56 @xmath2 , @xmath58 , and an undepleted solar abundance set , our calculations yield a maximum prediction of only 0.2 for the ratio of [ ] @xmath3@xmath37291 , 7324 to h@xmath6 . only at very low ionization parameters ( @xmath71 ) does the [ ] emission become stronger than h@xmath6 . since h@xmath6 is only barely visible in the spectra of many transition objects ( prior to careful starlight subtraction , at least ) , typical observations may not have sufficient sensitivity to detect faint [ ] lines in these objects . high - quality spectra of liners do not show [ ] emission @xcite , indicating that ca is likely to be depleted onto grains in these objects , but similar data are not generally available for transition nuclei . if the [ ] lines are found to indicate a high level of depletion onto dust grains in transition nuclei , this would also provide a further argument against shock - heating models , as shocks will tend to destroy grains ( e.g. , * ? ? ? the results shown in the preceding figures demonstrate that the starburst models are in fact able to reproduce the major diagnostic emission - line ratios of transition nuclei with reasonable accuracy , during the period @xmath72 = 35 myr when w - r stars are present . for a density of @xmath45 @xmath2 and an age of 4 myr , the solar - metallicity models with and without depletion bracket the range of values observed in real transition nuclei for the line ratios [ ] /h@xmath9 , [ ] /h@xmath9 , and [ ] /h@xmath9 . we do not attempt to fine - tune a model to produce an exact match with the spectrum of ngc 4569 , but the basic solar - metallicity dust - free model at @xmath51 myr with @xmath73 @xmath2 and log @xmath74 closely fits the observed [ ] /h@xmath9 ratio , while overpredicting [ ] /h@xmath9 by @xmath60 dex and underpredicting [ ] /h@xmath9 by @xmath75 dex . we emphasize that the starburst models are only able to produce transition - type spectra for the case of an instantaneous burst ; that is , when the burst duration is shorter than the timescale for evolution of the most massive stars . multiple - burst populations can only yield a transition spectrum if the dominant population is @xmath0 myr old and the older or younger bursts do not contribute significantly to the ionizing photon budget . models with a constant star - formation rate produce region spectra at all ages , as the softer ionizing continua do not produce sufficient [ ] @xmath36300 emission in the surrounding region to match transition - type spectra . the parameter which is most important for determining the hardness of the ionizing continuum is the number ratio of w - r stars to o stars , which exceeds @xmath76 during the w - r - dominated phase in the starburst99 models at solar metallicity , and approaches or exceeds unity at @xmath57 . in the constant star - formation rate models at solar metallicity , the w - r / o ratio levels off at @xmath77 after about 4 myr . the compact size of the ngc 4569 nucleus is consistent with the requirement that the burst duration must be brief ( @xmath78 myr ) in order to generate a transition - type spectrum . the fwhm size of the starburst core in ngc 4569 is only @xmath79 pc . for such a burst to occur in @xmath78 myr would require a propagation speed for star formation of only @xmath79 km s@xmath24 . in fact , the typical velocities in the ngc 4569 nucleus are much greater than 10 km s@xmath24 : the [ ] @xmath36583 line has a velocity width of 340 km s@xmath24 @xcite . thus , the ngc 4569 nucleus could represent the result of a single , rapid burst of star formation . although our results suggest that transition galaxy spectra may be attributed to a starburst with a high w - r / o - star ratio , the demographics of transition nuclei and nuclei indicate that many transition galaxies are probably not formed by this mechanism . in the starburst99 models , the w - r - dominated phase in an instantaneous burst lasts for @xmath80 myr ( i.e. , 3 time steps in the calculations ) . an region surrounding an instantaneous burst will be visible for @xmath81 myr , after which the emission lines will fade rapidly ( e.g. , * ? ? ? * ) . thus , for an instantaneous burst population , the transition phase and the nucleus phase will have approximately equal lifetimes . if all nuclei consisted of instantaneous burst stellar populations with nebular conditions conducive to the formation of transition - type spectra , then nuclei and transition nuclei should be roughly equal in number . in reality , it is likely that a large fraction of star - forming nuclei contain multiple bursts of star formation and/or conditions of low density or low metallicity , so all star - forming nuclei should not be expected to evolve through a transition - type phase . although it is difficult to make specific predictions , it is probably safe to conclude that for a given hubble type , transition nuclei generated solely by starbursts should be considerably less numerous than ordinary nuclei . the statistics compiled by @xcite provide a basis for comparison . in early - type galaxies ( e and s0 ) , transition nuclei outnumber nuclei by a 3-to-1 margin . only for hubble types sb and later do nuclei begin to outnumber transition nuclei by a factor of 2 or more . the most straightforward interpretation of this trend is that in early - type host galaxies , the majority of transition nuclei are actually agn/ region composites , as proposed by @xcite and others . at intermediate and late hubble types , the population of transition nuclei may consist of both composite objects and `` pure '' starbursts evolving through the w - r - dominated phase . the presence of transition nuclei in a small fraction ( @xmath82 ) of elliptical galaxies @xcite presents a particularly intriguing problem . the ho et al . survey detected five transition nuclei in ellipticals but not a single case of an elliptical galaxy hosting an nucleus . given that the models which have been considered for transition nuclei involve star formation , either alone or in combination with an agn , this observation is rather puzzling . perhaps faint agns in elliptical nuclei can produce transition - type spectra without substantial star formation activity . four of the five transition nuclei found in ellipticals by @xcite have borderline or ambiguous spectroscopic classifications , however , so `` pure '' transition nuclei in ellipticals are evidently quite rare . given these results , one might expect to see transition - type emission spectra in some fraction of disk regions in spiral galaxies , but in fact such spectra are never found . single - burst models for disk region spectra are only compatible with observed line ratios for model ages of @xmath70 myr @xcite , as the harder ionizing spectrum after 3 myr makes the models overpredict the strengths of the low - ionization lines . bresolin et al . suggest that either current stellar evolution models are at fault , or that disk regions are disrupted before reaching an age of 3 myr , in which case the w - r phase would not be observed in the nebular gas . an alternate ( and perhaps more attractive ) possibility is that the majority of nuclei , as well as disk regions , are better described by the models with constant star - formation rate , or contain multiple bursts of star formation with an age spread of a few myr , which would result in a spectrum similar to the constant star - formation rate models . for understanding the physical nature of transition objects , the observational challenge is to search for any unambiguous signs of nonstellar activity . detection of broad h@xmath9 emission , or a compact source of hard x - ray emission with a power - law spectrum , would provide evidence for an agn component . high - resolution optical spectra ( from _ hst _ ) could provide a means to spatially resolve a central agn - dominated narrow - line region from the surrounding starburst - dominated component . since direct evidence for accretion - powered nuclear activity in transition nuclei is generally lacking , it should not be assumed that any given transition object actually contains an agn unless observations specifically support that interpretation . one further effect that should be considered in starburst models in the future is photoionization by the x - rays generated by the starburst . x - ray binaries and supernova remnants will provide high - energy ionizing photons , resulting in a spatially extended source of soft x - ray emission as observed in the nucleus of ngc 4569 @xcite , for example . ( the massive main sequence stars will contribute only a negligible amount to the total x - ray luminosity of a starburst ; see helfand & moran 1999 . ) photoionization by x - rays will naturally lead to an enhancement of the low - ionization forbidden lines , and this could contribute to the excitation of some transition galaxies . the strength of [ ] @xmath36300 is the key distinguishing factor between liners and transition nuclei , and matching the observed strength of this line is the major challenge of starburst models for liners . our calculations show that liner spectra can only be generated by the starburst99 clusters under a very specific and limited range of circumstances . model grids a and b , while matching the [ ] / h@xmath9 ratio of transition nuclei quite well , do not overlap at all with the main cluster of liners in figure [ oratio ] , even at high densities and even when depletion and dust grains are included . only grid g with @xmath57 is able to replicate the high [ ] / h@xmath9 ratios of most liners , and only during @xmath83 46 myr and at densities of @xmath39 @xmath84 @xmath2 . in agreement with previous models , we find that values of log @xmath85 @xmath86 to @xmath87 reproduce the observed [ ] / h@xmath9ratios of liners . however , at such high densities the models underpredict the strengths of [ ] and [ ] relative to h@xmath9 . single - zone models require @xmath88 @xmath2 to match the [ ] /h@xmath9 ratios of liners and @xmath89 @xmath2 for [ ] /h@xmath9 . agreement with liner spectra can be achieved with a simple two - zone model , in which high - density and low - density components are present , similar to the scenario proposed by @xcite . as an example , a two - component model constructed from grid a containing gas at ( @xmath90 @xmath2 , @xmath91 ) and at ( @xmath92 @xmath2 , @xmath93 ) produces emission - line ratios which are consistent with all the liner classification criteria of both the @xcite and @xcite systems , if the two density componenets are scaled so as to contribute equally to the total h@xmath6 luminosity . as a local comparison , observations of near - infrared emission indicate the presence of clouds having @xmath94 @xmath2 in the galactic center region @xcite , so it is plausible that other galactic nuclei may contain ionized gas at similarly high densities even in the absence of an observable agn . a starburst origin for some liner 2 nuclei would provide a natural explanation for the lower values of the x - ray / h@xmath9 flux ratio seen in these objects , in comparison with agn - like liner 1 nuclei @xcite . it seems unlikely , however , that many liners are generated by this starburst mechanism . about 15% of liners are known to have a broad component of the h@xmath9 emission line , indicating a probable agn @xcite . by analogy with the seyfert population , a much larger fraction of liners is likely to have broad - line regions which are either obscured along our line of sight , or are simply too faint to be detected in ground - based spectra against a bright background of starlight . many liners show signs of nuclear activity that can not be explained by stellar processes : compact flat - spectrum radio sources or jets , compact x - ray sources with hard power - law spectra , or double - peaked broad balmer - line emission , for example . as an increasing body of observational work supports the idea that many liners are in fact agns , there is less incentive to consider purely stellar models for their excitation . demographic arguments , similar to those given above for transition nuclei , can be applied for the liner population . since the liner phase only occurs for instantaneous bursts at high density and high metallicity , the starburst scenario implies that nuclei should be considerably more numerous than starburst - generated liners for a given hubble type . while liners are common in early - type hosts , nuclei are not found in elliptical hosts and are seen in fewer than 10% of s0 galaxies this disparity is a strong argument against a starburst origin for those liners in early - type galaxies . in later hubble types the situation is less clear , however . nuclei occur in @xmath95 of spirals of type sc and later , while liners occur in just 5% of these galaxy types @xcite . it is conceivable that some of the liners in intermediate to late - type hosts could have a starburst origin , and this issue could be resolved by further uv and x - ray observations in the future . interestingly , the ho et al . survey did not find any examples of broad h@xmath9 emission in liners or transition objects with hosts of type sc or later ; perhaps star formation plays a more prominent role than accretion - powered activity in these objects . while a few liners show spectral features of young stars in the uv @xcite , the quality of the observational data is poor in comparison with the ngc 4569 uv spectrum , and it is difficult to set meaningful constraints on the age of the young stellar population . ngc 404 is a possible candidate for a starburst - generated liner , but in its uv spectrum the p cygni features are weak in comparison with ngc 4569 , indicating either an older burst population or dilution by a featureless agn continuum @xcite . the liners having uv spectral features from massive stars may also host obscured agns which can be detected in other wavebands . for example , the uv continuum of the liner ngc 6500 appears to have its origin in hot stars @xcite , but observations of a parsec - scale radio jet unambiguously demonstrate that nonstellar activity is occurring as well @xcite . the starburst models presented here run into two obvious problems . first , w - r galaxies are almost never known to have liner or transition - type spectra . second , liners and transition nuclei almost never show w - r features in their spectra . is there any way to reconcile the starburst models with these facts ? w - r galaxies are identified by the appearance of the 4650 blend in their spectra ( e.g. , * ? ? ? since the formation of w - r stars is enhanced at high @xmath63 , the strength of this feature relative to h@xmath6 increases dramatically with metallicity , from @xmath44 at @xmath96 to @xmath97 at @xmath98 @xcite . however , in the nuclei of early - type spirals where high metallicities are expected , a nuclear starburst will be surrounded by the old stellar population of the galactic bulge , making the detection of the w - r bump extremely difficult @xcite . most of the currently known w - r galaxies are late - type spirals or irregular galaxies @xcite in which the w - r bump is visible against the nearly featureless starburst continuum . when the w - r bump is detected in galaxies , its amplitude above the continuum level is generally far smaller than that of h@xmath6 or even h@xmath99(e.g . . in most of the liner and transition galaxy spectra in the catalog of @xcite , however , h@xmath6 barely appears and h@xmath99 is too weak to be visible at all prior to continuum subtraction . even in a high - metallicity environment , where the total intensity of the w - r bump can be comparable to that of h@xmath6 , the amplitude of the w - r bump above the continuum will be much lower than that of h@xmath6 because the flux in the w - r feature is spread over @xmath100 . thus , the detection of w - r emission in galactic nuclei is strongly biased toward late - type , bulgeless galaxies . in late - type or dwarf irregular galaxies where the w - r bump is visible , the w - r / o - star ratio is expected to be much smaller owing to the lower metallicity , and the resulting softer ionizing spectrum will tend to produce an region spectrum rather than a transition object . the gas density as a function of hubble type may play a role as well ; in a study of nuclei , @xcite find a weak trend toward lower nebular densities in later - type host galaxies . observational detection of w - r features in transition nuclei is perhaps the clearest test of the starburst models , if sufficiently sensitive observations can be obtained . the uv spectrum of ngc 4569 is consistent with an age of @xmath101 myr , an age at which w - r stars are expected to be present . previous optical spectra have not revealed the 4650 w - r bump in ngc 4569 , but further observations with high s / n and small apertures would be worthwhile . the lack of @xmath31640 emission in the uv spectrum of ngc 4569 is potentially a more serious problem since the burst population should dominate at short wavelengths . the models of @xcite predict an equivalent width of at least 2 in the w - r - generated @xmath31640 line during the period 36 myr for an instantaneous burst of solar or higher metallicity , while the observed upper limit of @xmath102 erg s@xmath24 @xmath30 @xcite corresponds to an equivalent width limit of @xmath103 . it should be noted , however , that the @xmath31640 line lies at the extremely noisy blue end of the fos g190h grating setting , in a region where detection of emission or absorption features is difficult . two w - r galaxies may provide useful points of reference for the starburst models . ngc 3367 is classified by @xcite as an nucleus on the basis of its [ ] /h@xmath9 and [ ] /h@xmath9 ratios , although its [ ] /h@xmath9 ratio of 0.83 is more consistent with a liner or transition - type classification and its emission lines are markedly broader than those of typical nuclei . @xcite describe ngc 3367 as a starburst - dominated transition object ( see also * ? ? ? the 4650 w - r bump was noted by @xcite , who also suggested a liner/ classification and a composite source of ionization . as a borderline nucleus / transition object with clear evidence for w - r stars , this object deserves further study , to determine whether there is indeed an agn or whether the enhanced low - ionization emission may be the result of ionization by the w - r population . the electron density of 835 @xmath2 measured from the [ ] doublet @xcite is also noteworthy , as this is among the highest densities found for an nucleus in the ho et al . survey . another intriguing object is the nucleus of ngc 6764 , which has been classified variously as a seyfert , a liner , and a starburst by different authors ( see * ? ? ? this galaxy exhibits prominent emission in the 4650 w - r blend @xcite . a recent study by @xcite demonstrates that the narrow emission lines are consistent with a liner classification , but there are no unambiguous signs of nonstellar activity in the nucleus . @xcite find that the nucleus contains @xmath104 w - r stars , and that the overall properties of the object are consistent with ionization by the starburst alone , rather than by a starburst / agn composite . if this conclusion is confirmed by further observations , ngc 6764 could be considered the best candidate for a liner photoionized by a starburst during its w - r - dominated phase . @xcite derived an age of 910 myr for the starburst in ngc 6764 based on near - infrared emission - line diagnostics , but this age is inconsistent with the presence of w - r stars , at least for the case of an instantaneous burst . compared with typical liners and transition nuclei , conditions for detection of w - r spectral features in these two objects are perhaps more favorable . both host galaxies are of late hubble types ( sbc for ngc 3367 and sbbc for ngc 6764 ) in comparison with the majority of liners and transition nuclei , so the level of contamination by the surrounding old stellar population is relatively low . furthermore ( and partly as a result of this ) , the emission - line equivalent widths in these two nuclei are relatively high for liners or transition nuclei . if the emission - line spectra of ngc 3367 or ngc 6764 had been superposed on a luminous early - type spiral bulge , the w - r emission might never have been noticed . the detection of w - r emission in a liner does not automatically imply a purely starburst origin for the emission lines , of course , since starbursts and agns are often known to coexist . mrk 477 is a well - known example of a seyfert 2 galaxy having a large population of w - r stars in its nucleus @xcite . in the context of the starburst model , however , the crucial test is to search for additional examples of liners or transition nuclei which exhibit a high ratio of w - r to o stars but no signs of accretion - powered activity . the most important limitation of these calculations comes from the accuracy of the input continua . the conclusion that the starburst models are able to reproduce transition or liner spectra under some circumstances depends crucially on the presence of w - r stars to provide a hard and luminous ionizing continuum . unfortunately , the continuum shape and luminosity of w - r stars in the extreme - uv band are quite uncertain , particularly in the he@xmath32 continuum where stellar winds have a dramatic effect . several up - to - date reviews of the numerous difficulties involved in modeling w - r spectra can be found in the volume edited by @xcite . the starburst99 model grid uses the w - r atmospheres of @xcite , which are calculated for a pure helium composition , but more recent atmosphere models are beginning to include the effects of line - blanketing , as well as clumping and departures from spherical symmetry . as discussed by @xcite , the w - r / o - star ratio is also extremely model dependent , and may be revised in future generations of models . this would have a direct effect on the strength of the extreme - uv continuum and consequences for the nebular emission lines . furthermore , the starburst99 models neglect binary evolution , although this is more likely to affect the w - r / o ratio at low metallicity . since the results of the transition - object models are highly dependent on the most uncertain portion of the w - r spectrum , new photoionization calculations should be computed to assess the impact of different w - r evolution and atmosphere models in the future . the shape of the imf in starburst regions is a subject of some debate , and the possible variation of the imf with metallicity is of particular importance for galactic nuclei , which are likely to have @xmath105 . @xcite and @xcite suggested that @xmath33 should be lower in regions of higher metal abundance , but this issue has not been settled definitively . star - count observations demonstrate that the imf slope and @xmath33 do not appear to vary with metallicity @xcite , at least for @xmath106 . while nebular diagnostics in galaxies are generally consistent with a salpeter imf with @xmath107 @xmath34 at subsolar metallicity ( e.g. , * ? ? ? * ) , at high metallicity the observational situation is somewhat ambiguous . @xcite find that the mean stellar temperature in regions decreases significantly with increasing @xmath63 , and that the @xmath35876 / h@xmath6 ratios of regions at @xmath108 are more consistent with @xmath33 = 30 @xmath34 than with @xmath33 = 100 @xmath34 . such a low value for @xmath33 would pose serious difficulties for any starburst models of liners and transition nuclei , as the massive progenitors of w - r stars would not be present . counterbalancing this trend , the strong tidal forces , turbulence , and magnetic field strengths in galactic nuclei may act to raise the jeans mass and favor the formation of more massive stars @xcite . in the galactic center , there are stars with initial masses of @xmath109 @xmath34 @xcite , and one galactic center object ( the pistol star ) may have @xmath110 as high as 200250 @xmath34 @xcite . thus , the proposed trend toward lower values of @xmath33 at high metallicity in disk regions may not apply to galactic nuclei . detailed comparison of the uv spectra of galaxies such as ngc 4569 with starburst population synthesis models can provide useful constraints on the population of high - mass stars in nuclear starbursts . despite these uncertainties , these photoionization models have a major advantage compared with previous generations of w - r or o - star models for liners and transition nuclei , in that the starburst99 models with standard parameters are constructed to represent the actual stellar populations in starbursts , to the best of current knowledge . previous o - star models @xcite required the presence of hypothetical , unusually hot stars in order to explain liner or transition spectra , and they did not address the evolution of the young stellar population at all . the starburst models presented here provide a more plausible mechanism to generate a transition - type spectrum , even if this model may apply only to a relatively small fraction of the population of transition galaxies .

we find that the models are able to reproduce the emission - line spectra of transition nuclei , but only for instantaneous bursts of solar or higher metallicity , and only for ages of myr , the period when the extreme - ultraviolet continuum is dominated by emission from wolf - rayet stars . for clusters younger than 3 myr or older than 6 myr , and for models with a constant star - formation rate , the softer ionizing continuum results in an emission spectrum more typical of regions . this model predicts that wolf - rayet emission features should appear in the spectra of transition nuclei . demographic arguments suggest that this starburst model may not apply to the majority of transition nuclei , particularly those in early - type host galaxies , but it could account for some members of the transition class in hosts of type sa and later .

motivated by the discovery of young , massive stars in the nuclei of some liner/ `` transition '' nuclei such as ngc 4569 , we have computed photoionization models to determine whether some of these objects may be powered solely by young star clusters rather than by accretion - powered active nuclei . the models were calculated with the photoionization code cloudy , using evolving starburst continua generated by the the starburst99 code of . we find that the models are able to reproduce the emission - line spectra of transition nuclei , but only for instantaneous bursts of solar or higher metallicity , and only for ages of myr , the period when the extreme - ultraviolet continuum is dominated by emission from wolf - rayet stars . for clusters younger than 3 myr or older than 6 myr , and for models with a constant star - formation rate , the softer ionizing continuum results in an emission spectrum more typical of regions . this model predicts that wolf - rayet emission features should appear in the spectra of transition nuclei . while such features have not generally been detected to date , they could be revealed in observations having higher spatial resolution . demographic arguments suggest that this starburst model may not apply to the majority of transition nuclei , particularly those in early - type host galaxies , but it could account for some members of the transition class in hosts of type sa and later . the starburst models during the wolf - rayet - dominated phase can also reproduce the narrow - line spectra of some liners , but only under conditions of above - solar metallicity and only if high - density gas is present ( ) . this scenario could be applicable to some `` type 2 '' liners which do not show any clear signs of nonstellar activity .

astro-ph0003294

c

our primary conclusion is that for standard starburst parameters and for nebular conditions which may be typical of galactic nuclei , the starburst models are able to reproduce the important diagnostic emission - line ratios for liner/ transition galaxies , otherwise known as weak- [ ] liners . the key ingredient needed to generate a transition - type spectrum is a uv continuum dominated by w - r stars , a condition which occurs during @xmath111 35 myr after an instantaneous burst . a transition - type emission spectrum may thus be a phase in the evolution of some nuclear regions in which the ionizing continuum is generated by a single - burst stellar population . the models are also able to produce an [ ] / h@xmath9 ratio high enough to match liner spectra , but only for conditions of above - solar metallicity combined with the presence of high - density ( @xmath112 @xmath2 ) clouds . a sensitive search for w - r spectral features in transition nuclei would provide a test of this starburst scenario . this model may apply only to a small fraction of liners and transition nuclei ; many liners and some transition objects show clear signs of nonstellar activity , and the starburst models may not apply at all to objects in early - type host galaxies . further multiwavelength observations of transition nuclei will be of great utility for determining what fraction of them contain genuine active nuclei , and what fraction appear to be purely the result of stellar phenomena . research by a.j.b . is supported by a postdoctoral fellowship from the harvard - smithsonian center for astrophysics . this research was also supported financially by grant ar-07988.02 - 96a , awarded to j.c.s . by stsci , which is operated by aura for nasa under contract nas5 - 26555 . this work would not have been possible without the excellent software created and distributed by gary ferland and the cloudy team , and by claus leitherer and the starburst99 team . we also thank gary ferland for providing a helpful referee s report , claus leitherer for additional helpful comments on the manuscript , and daniel schaerer for supplying model starburst spectra in electronic form . line - ratio diagram of [ ] @xmath35007 / h@xmath6 against [ ] @xmath36300 / h@xmath9 , for model grids a ( solid line ) , b ( long - dashed line ) , c ( short - dashed line ) , and d ( dot - dashed line ) , at a burst age of 4 myr and @xmath56 @xmath2 . the input continuum has solar metallicity , imf power - law slope @xmath40 , and @xmath33 = 100 @xmath34 . the following description applies to this and all subsequent plots : the small squares along each model line correspond to the model grid points at log @xmath36 = @xmath113 , and @xmath37 , with @xmath36 increasing upward along the line . the points plotted represent galaxies from the ho et al . catalog , as follows : _ small circles _ : nuclei . _ squares : _ liner/ transition objects . ngc 4569 is represented by an open square . _ triangles : _ `` pure '' liners . _ crosses : _ seyfert nuclei . the dotted line encloses the region defined for transition nuclei according to the criteria of @xcite .

the starburst models during the wolf - rayet - dominated phase can also reproduce the narrow - line spectra of some liners , but only under conditions of above - solar metallicity and only if high - density gas is present ( ) . this scenario could be applicable to some `` type 2 '' liners which do not show any clear signs of nonstellar activity .

motivated by the discovery of young , massive stars in the nuclei of some liner/ `` transition '' nuclei such as ngc 4569 , we have computed photoionization models to determine whether some of these objects may be powered solely by young star clusters rather than by accretion - powered active nuclei . the models were calculated with the photoionization code cloudy , using evolving starburst continua generated by the the starburst99 code of . we find that the models are able to reproduce the emission - line spectra of transition nuclei , but only for instantaneous bursts of solar or higher metallicity , and only for ages of myr , the period when the extreme - ultraviolet continuum is dominated by emission from wolf - rayet stars . for clusters younger than 3 myr or older than 6 myr , and for models with a constant star - formation rate , the softer ionizing continuum results in an emission spectrum more typical of regions . this model predicts that wolf - rayet emission features should appear in the spectra of transition nuclei . while such features have not generally been detected to date , they could be revealed in observations having higher spatial resolution . demographic arguments suggest that this starburst model may not apply to the majority of transition nuclei , particularly those in early - type host galaxies , but it could account for some members of the transition class in hosts of type sa and later . the starburst models during the wolf - rayet - dominated phase can also reproduce the narrow - line spectra of some liners , but only under conditions of above - solar metallicity and only if high - density gas is present ( ) . this scenario could be applicable to some `` type 2 '' liners which do not show any clear signs of nonstellar activity .

cond-mat0001138

i

the phenomenon of multifractality consists in a scale dependence of critical exponents . it has been widely discussed in the literature for a wide range of issues , such as statistics of strange sets @xcite , diffusion limited aggregation @xcite , wavelet transforms @xcite , conformal invariance @xcite or statistical properties of critical wave functions of massless dirac fermions in a random magnetic field @xcite . the aim of our work is not only to describe a new model possessing multiscaling dependence , but also to show that the phenomenon of multifractality is related to local nonuniformity of the exponentially growing ( `` hyperbolic '' ) underlying `` target '' phase space , through an example of entangled random walks distribution in homotopy classes . indeed , to our knowledge , almost all examples of multifractal behavior for physical @xcite or more abstract @xcite systems share one common feature all target phase spaces have a noncommutative structure and are locally nonuniform . we believe that multiscaling is a much more generic physical phenomenon compared with uniform scaling , appearing when the phase space of a system possesses a hyperbolic structure with local symmetry breaking . such perturbation of local symmetry could be either regular or random from our point of view the details of the origin of local nonuniformity play a less significant role . we discuss below the basic features of multifractality in a locally nonuniform regular hyperbolic phase space . we show in particular that a multifractal behavior is encountered in statistical topology in the case of entangled ( or knotted ) random walks distribution in topological classes . the paper is organized as follows . in section [ sect:2 ] we consider a 2d @xmath0step random walk in a nonsymmetric array of topological obstacles and investigate the multiscaling properties of the `` target '' phase space for a set of specific topological invariants the `` primitive paths '' . the renormalization group computations of mean length of the primitive path , as well as return probabilities to the unentangled topological state are developed in section [ sect:3 ] . section [ sect:4 ] is devoted to the application of conformal methods to a geometrical analysis of multifractality in locally nonuniform hyperbolic spaces .

the comparison of the analytical rg it is shown explicitly by direct counting for the discrete version of the model and by conformal methods for the continuous version that multifractality occurs when local uniformity of the phase space ( which has an exponentially large number of states ) has been broken .

multifractal properties of the distribution of topological invariants for a model of trajectories randomly entangled with a nonsymmetric lattice of obstacles are investigated . using the equivalence of the model to random walks on a locally nonsymmetric tree , statistical properties of topological invariants , such as drift and return probabilities , have been studied by means of a renormalization group ( rg ) technique . the comparison of the analytical rg results with numerical simulations as well as with the rigorous results of p.gerl and w.woess demonstrates clearly the validity of our approach . it is shown explicitly by direct counting for the discrete version of the model and by conformal methods for the continuous version that multifractality occurs when local uniformity of the phase space ( which has an exponentially large number of states ) has been broken . = -0.25 in = 6.7 in =

1508.01557

i

in this paper , we introduce new finite difference schemes for the hamilton - jacobi equation @xmath4 and prove rates of convergence . the hamilton - jacobi equation appeared recently as the continuum limit of nondominated sorting , which is widely used in scientific and engineering contexts @xcite . let us briefly describe the connection . let @xmath5 be _ i.i.d . _ random variables on @xmath6 with continuous density @xmath7 . let @xmath8 denote the elements in @xmath9 that are coordinatewise minimal . the set @xmath8 is called the _ first pareto front _ of @xmath10 , and the elements of @xmath8 are called _ pareto optimal _ or _ nondominated_. the _ second pareto front _ , denoted @xmath11 , is the set of minimal elements from @xmath12 , and the @xmath13 _ pareto front _ is defined as @xmath14 the process of sorting the set @xmath10 into pareto fronts , or nondominated layers , is called _ nondominated sorting _ , and is widely used in multi - objective optimization ( see @xcite and references therein ) , with recent applications to machine learning @xcite . it turns out that nondominated sorting is equivalent to finding the longest chain in a partially ordered set , which has a long history in probability and combinatorics @xcite . it was shown in @xcite that the pareto fronts converge almost surely in the limit as @xmath15 to the level sets of the unique nondecreasing is _ nondecreasing _ if @xmath16 is nondecreasing for all @xmath17 . ] viscosity solution of . figure [ fig : demo ] gives an illustration of this continuum limit . in @xcite , a fast algorithm called _ pde - based ranking _ was proposed for approximate nondominated sorting of large datasets . the basic idea is to estimate the density @xmath7 from a ( relatively small ) subset of @xmath5 , and then use the numerical solution of the hamilton - jacobi equation as an approximation of nondominated sorting . it was shown in @xcite that pde - based ranking is substantially faster than nondominated sorting in relatively low dimensions , while maintaining high levels of sorting accuracy . the numerical scheme for used in @xcite is based on backward finite differences and can be solved very efficiently in a single pass . due to the fact that information flows along coordinate axes in the definition of nondominated sorting , this scheme is upwind ( or monotone ) , and convergence of the scheme was established in @xcite . although the scheme used in @xcite is simple and efficient , it suffers from poor accuracy , with formal rates of convergence on the order of @xmath1 for a grid with spacing @xmath18 in dimension @xmath0 . in this paper , we propose two new and highly efficient finite difference schemes for solving . both schemes have a formal accuracy of @xmath2 when the solution is smooth , and we prove the usual @xmath3 rates in the context of non - smooth viscosity solutions . these schemes can be used to increase the accuracy of pde - based ranking @xcite _ without _ increasing computational complexity . alternatively , with these highly accurate schemes we can afford to use a coarser grid resolution , and thus we can reduce the computational complexity of pde - based ranking while maintaining high accuracy . this is particularly important in applications of nondominated sorting @xcite , which will benefit from highly accurate and efficient algorithms for sorting massive datasets . we detail the new schemes and our main results in the next section . we pose the hamilton - jacobi equation on a compact domain as follows : @xmath19^n\\ u & = 0 & & \text{on } \gamma , \end{aligned}\right\}\ ] ] where @xmath20^n \setminus ( 0,1]^n$ ] and @xmath21 . notice we have modified by taking the positive parts of the partial derivatives . this is necessary to obtain existence of a viscosity solution of , and is a well - known issue with viscosity solutions on boundaries of domains ( see @xcite ) . we elaborate on this briefly in section [ sec : comparison ] . we should mention that there is no loss of generality in considering the domain @xmath22^n$ ] in . indeed , we can make a simple scaling argument to transform the domain of into @xmath23 $ ] for any @xmath24 . let @xmath18 , @xmath25 , and @xmath26 . for @xmath27 set @xmath28 . we recall the numerical scheme for from @xcite : @xmath29^n_h\\ u_h(x ) & = 0 & & \text{if } x \in \gamma_h , \end{aligned}\right\}\ ] ] where @xmath30^n_h \to { \mathbb{r}}$ ] is the numerical solution and @xmath31 the solution @xmath32 of can be solved efficiently in a single pass , which is reminiscent of the fast marching @xcite and fast sweeping @xcite algorithms . in dimension @xmath33 , the scheme is quadratic and can be solved in closed form @xmath34 in dimensions @xmath35 , the scheme can be solved via any iteration method , such as a bisection search . convergence of to the viscosity solution of was established in @xcite . while is optimal in terms of computational complexity on a fixed grid , its accuracy is at best @xmath1 . to see this , consider the special case of @xmath36 and @xmath37 . the solution @xmath38 is smooth on @xmath39^n$ ] , and has a gradient singularity on the boundary @xmath40 . we can use the comparison principle @xcite for to show that in general @xmath41^n } f^\frac{1}{n } \leq u(x ) \leq n(x_1\cdots x_n)^\frac{1}{n } \sup_{[0,1]^n } f^\frac{1}{n}.\ ] ] therefore , the gradient singularity on @xmath40 is typical for solutions of whenever @xmath42^n } f > 0 $ ] . let @xmath43 . by the concavity of @xmath44 @xmath45 on the other hand , if @xmath46 then @xmath47 therefore , for any @xmath48^n_h$ ] such that @xmath46 for some @xmath17 we have @xmath49 setting @xmath50 and invoking the comparison principle for yields @xmath51 letting @xmath52 we find that @xmath53 therefore , the scheme makes an error on the order of @xmath1 in the immediate vicinity of the boundary @xmath40 . since @xmath38 is generally not smooth , the best theoretical rate that one can prove in the context of viscosity solutions is typically strictly worse than the formal rate ( see @xcite ) . in section [ sec : numerics ] , we show numerical results indicating that the @xmath54 convergence rate of @xmath1 is typically observed in practice . since the slow convergence rate is caused by a singularity in the gradient of @xmath38 on @xmath40 , it is natural to look for a transformation of @xmath38 that removes this singularity . with this in mind , we set @xmath55 where @xmath38 is the nondecreasing viscosity solution of . when @xmath56 we have @xmath57 , which is lipschitz continuous ( in fact smooth ) on @xmath22^n$ ] . in general , we prove in lemma [ lem : vexun ] that @xmath58^n)$ ] whenever @xmath59^n)$ ] . furthermore , @xmath60 is a viscosity solution of @xmath61^n\\ v & = 0 & & \text{on } \gamma . \end{aligned}\right\}\ ] ] since @xmath60 is lipschitz continuous , it is reasonable to suspect that a numerical scheme for would have a better convergence rate than . we therefore propose the following finite difference scheme for : @xmath62^n_h\\ v_h(x ) & = 0 & & \text{if } x \in \gamma_h , \end{aligned}\right\}\ ] ] where @xmath63^n_h \to { \mathbb{r}}$ ] . we take @xmath64 to be the largest solution of at each @xmath48^n_h$ ] . it is interesting to note that when @xmath7 is constant , the solution of the scheme is @xmath65 , which is the _ exact _ solution of . the scheme can be solved efficiently in a single pass , similar to , and in dimension @xmath33 we have the closed form expression @xmath66 where @xmath67 in theorem [ thm : convpv ] , we prove convergence of when @xmath7 is continuous and nonnegative . our main result is the following convergence rate . [ thm : ratev ] suppose @xmath68^n)$ ] and @xmath69 . let @xmath70 be the solution of and let @xmath38 be the nondecreasing viscosity solution of . then @xmath71 and @xmath72_{1;[0,1]^n},\inf_{[0,1]^n}f\big)$ ] . notice that has a zeroth order term with the wrong sign for comparison principle arguments to hold . we can see this by observing that the method of vanishing viscosity takes the form @xmath73 since the standard proof of convergence rates for numerical approximations to viscosity solutions is based on the proof of the comparison principle @xcite , we can not directly apply these techniques to . our proof of theorem [ thm : ratev ] passes through an auxiliary problem , which actually suggests another numerical scheme for solving . based on our observation that @xmath37 is the viscosity solution of corresponding to @xmath36 , it is natural in more general settings to make the ansatz @xmath74 for some function @xmath75^n \to [ 0,\infty)$ ] . if this ansatz is correct , then @xmath76 would be a viscosity solution of @xmath77^n.\ ] ] it turns out that is well - posed within the class of bounded viscosity solutions _ without _ imposing a boundary condition . the boundary condition is actually encoded into the pde due to the fact that the term @xmath78 vanishes on @xmath79 . this suggests , for example , that we should expect @xmath80 . due to the degeneracy of the terms @xmath81 , there are in general infinitely many unbounded viscosity solutions of . we characterize @xmath76 from as the maximal bounded viscosity solution of , and we show in lemma [ lem : wlip ] that @xmath82^n)$ ] whenever @xmath59^n)$ ] . we propose the following numerical scheme for : @xmath83^n_h,\ ] ] where @xmath84^n_h \to [ 0,\infty)$ ] . here , we take @xmath85 to be the largest solution of at each @xmath86^n_h$ ] . we note that for @xmath86^n_h$ ] such that @xmath87 , the quantity @xmath88 is undefined . since this term appears in the form @xmath89 , its value is not used in the scheme . it is interesting to note that when @xmath7 is constant , the scheme gives the _ exact _ solution of . this scheme can be solved efficiently in a single pass , and in dimension @xmath33 the scheme can be solved in closed form @xmath90 where @xmath91 and @xmath92 since the zeroth order term in has the correct sign , we can use a modification of the standard convergence proof to establish the following convergence rate . [ thm : rate1 ] suppose that @xmath68^n)$ ] and @xmath93 . let @xmath94 be the solution of , and let @xmath76 be the maximal bounded viscosity solution of . then latexmath:[\[\label{eq : rate1 } where @xmath72_{1;[0,1]^n } , \inf_{[0,1]^n}f\big)$ ] . our proof of theorem [ thm : ratev ] proceeds by first showing ( in lemma [ lem : changeofvar ] ) that @xmath96 and then invoking theorem [ thm : rate1 ] . although both and have the same provable convergence rates , our numerical results presented in section [ sec : numerics ] suggest that in general has a better experimental convergence rate than . this can be explained by observing that @xmath38 can only have gradient singularities when transitioning from zero to a positive value . the transformation @xmath97 regularizes these gradient singularities _ anywhere _ in the domain @xmath22^n$ ] , and not just on the boundary @xmath40 . on the other hand , the transformation @xmath98 is designed only to capture singularities on the boundary @xmath40 . in section [ sec : numerics ] , we give an example of a discontinuous function @xmath7 for which exhibits a better convergence rate than for the reason outlined above . this paper is organized as follows . in section [ sec : comparison ] , we prove comparison principles for viscosity solutions of and . in section [ sec : conv ] , we use the barles - souganidis framework @xcite to prove convergence of the schemes and under the assumption that @xmath7 is continuous and nonnegative . in section [ sec : rates ] we prove theorems [ thm : ratev ] and [ thm : rate1 ] establishing rates of convergence for and when @xmath7 is positive and lipschitz . the proofs of the convergence rates require lipschitz estimates for the viscosity solutions of and . these are obtained in section [ sec : s3lip ] . in section [ sec : numerics ] , we show the results of numerical simulations comparing all three schemes .

non - dominated sorting arranges a set of points in-dimensional euclidean space into layers by repeatedly removing the coordinatewise minimal elements . it was recently shown that nondominated sorting of random points has a hamilton - jacobi equation continuum limit . the obvious numerical scheme for this pde has a slow convergence rate of . in this paper , we introduce two new numerical schemes that have formal rates of and we prove the usual theoretical rates

non - dominated sorting arranges a set of points in-dimensional euclidean space into layers by repeatedly removing the coordinatewise minimal elements . it was recently shown that nondominated sorting of random points has a hamilton - jacobi equation continuum limit . the obvious numerical scheme for this pde has a slow convergence rate of . in this paper , we introduce two new numerical schemes that have formal rates of and we prove the usual theoretical rates . we also present the results of numerical simulations illustrating the difference between the formal and theoretical rates .

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we conducted a multi - site campaign on the accreting pulsator sdss1610 - 0102 for a duration of 11 days using seven observatories around the globe in may 2007 . the photometric periods in our light curve were consistent with previous observations from 2004 and 2005 @xcite , indicating their long - term stability . the most interesting result from our multi - site campaign is the detection of a resolved evenly spaced pulsational triplet at 609s . the even spacing of the triplet suggests that it is induced by rotation , and the rotational period of 4.8 days derived from the spacing has strong implications for the transport of angular momentum and its long - term evolution . we investigated the possibility that variability of period and/or amplitude in a single frequency light curve could produce the observed triplet , and find that this is not a likely possibility . either the period of 4.8 days is a measure of the uniform rotation period for the entire star or it is suggestive of differential rotation in the star . in either case , the prospects of constraining rotation in an accreting white dwarf with asteroseismic techniques is immensely exciting . conducting similar multi - site campaigns on other accreting pulsators could help us form a picture of how angular momentum is exchanged in the interior of the white dwarf , and its significance from the perspective of binary evolution and stability . the rotation of the underlying white dwarf is also important for type ia supernovae ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? our spectroscopic measurement of the orbital period is [email protected] is consistent within uncertainties with the photometrically observed first harmonic of the orbital period at 41.5min . our second striking result is the detection of linear combination frequencies apparently caused by an interplay of the dominant pulsation mode at 609s and the first harmonic of the orbital period . such a physical interaction between nonradial pulsation and orbital motion has never been detected before for variable white dwarfs , and is perhaps suggestive of tides . a thorough investigation is left to a future theoretical paper . szkody and mukadam thank nsf for financially supporting this project through the grant ast-0607840 . dr . bildsten is grateful to nsf for the grant ast-0707633 . tovmassian and zharikov acknowledge the conacyt grant 45847/a for supporting the observations made at oan - spm . professors woudt and warner acknowledge research funding from the university of cape town and from the national research foundation . based on observations obtained with the apache point observatory 3.5-meter telescope , which is owned and operated by the astrophysical research consortium and on observations collected at the centro astronmico hispano alemn ( caha ) at calar alto , operated jointly by the max - planck institut fr astronomie and the instituto de astrofsica de andaluca ( csic ) . some of the observations reported in this paper were obtained with the southern african large telescope ( salt ) . along with added gaussian noise . this figure helps us understand the fluctuations we can expect in the measurements of period and amplitude from runs much shorter than the duration required to resolve the triplet , when fitting a single frequency to the data.[dm2],title="fig : " ] along with added gaussian noise . this figure helps us understand the fluctuations we can expect in the measurements of period and amplitude from runs much shorter than the duration required to resolve the triplet , when fitting a single frequency to the data.[dm2],title="fig : " ]

, we conducted a multi - site campaign on the accreting pulsator sdssj161033.64 - 010223.3 ( v386 ser ) using seven observatories located around the world in may 2007 over a duration of 11 days . we report the best fit periodicities here , which were also previously observed in 2004 , suggesting their underlying stability . the even nature of the triplet is suggestive of rotational splitting , implying an enigmatic rotation period of about 4.8 days . there are two viable alternatives assuming the triplet is real : either the period of 4.8 days is representative of the rotation period of the entire star with implications for the angular momentum evolution of these systems , or it is perhaps an indication of differential rotation with a fast rotating exterior and slow rotation deeper in the star . investigating the possibility that a changing period could mimic a triplet suggests that this scenario is improbable , but not impossible . using time - series spectra acquired in may 2009 , three of the observed photometric frequencies from our may 2007 campaign appear to be linear combinations of the 609s pulsation mode with the first harmonic of the orbital period at 41.5min . this is the first discovery of a linear combination between nonradial pulsation and orbital motion for a variable white dwarf .

nonradial pulsations in the primary white dwarfs of cataclysmic variables can now potentially allow us to explore the stellar interior of these accretors using stellar seismology . in this context , we conducted a multi - site campaign on the accreting pulsator sdssj161033.64 - 010223.3 ( v386 ser ) using seven observatories located around the world in may 2007 over a duration of 11 days . we report the best fit periodicities here , which were also previously observed in 2004 , suggesting their underlying stability . although we did not uncover a sufficient number of independent pulsation modes for a unique seismological fit , our campaign revealed that the dominant pulsation mode at 609s is an evenly spaced triplet . the even nature of the triplet is suggestive of rotational splitting , implying an enigmatic rotation period of about 4.8 days . there are two viable alternatives assuming the triplet is real : either the period of 4.8 days is representative of the rotation period of the entire star with implications for the angular momentum evolution of these systems , or it is perhaps an indication of differential rotation with a fast rotating exterior and slow rotation deeper in the star . investigating the possibility that a changing period could mimic a triplet suggests that this scenario is improbable , but not impossible . using time - series spectra acquired in may 2009 , we determine the orbital period of sdssj161033.64 - 010223.3 to be 83.8.9min . three of the observed photometric frequencies from our may 2007 campaign appear to be linear combinations of the 609s pulsation mode with the first harmonic of the orbital period at 41.5min . this is the first discovery of a linear combination between nonradial pulsation and orbital motion for a variable white dwarf .

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currently , the interpretation of the reflection continuum observed in the x - ray spectra of agn is the primary means to glean information regarding the accretion geometry and its radiative environment . it is therefore vital that the models of reflection , on which such conclusions are based , explore a number of different physical situations which may be relevant to accretion physics . motivated by the inhomogeneous nature of recent numerical simulations of radiation - dominated accretion disks , we have begun a study of examining the consequences of such density changes to the x - ray reflection spectrum . our first results , presented in the previous sections , considered reprocessing from a two - density model , where density steps or jumps were inserted into a uniform slab , and a slice from a pbi simulation . the new reflection spectra were then compared with ones calculated assuming a constant density atmosphere , as these are most frequently used in data analysis . we found that the density inhomogeneities can result in observationally important differences between the two cases . as in models with a hydrostatic atmosphere , the reflection spectra can no longer be described by a single ionization parameter , but exhibit features from a mixture of ionization states . however , while in the hydrostatic case the spectra can be adequately described as a diluted constant density model because of a diffuse hot scattering layer on the surface @xcite , the inhomogeneous models can show emission from both low and high @xmath21 material simultaneously , depending on the nature of the density change . this gives the spectra a complexity and richness that the hydrostatic models lacked . the magnitude and characterization of the change in the reflection spectrum depends on both the nature of the inhomogeneity and the strength of the incident x - rays . for the two - density models , where the calculations could be distinguished by the original ionization parameter @xmath21 of the uniform slab , the greatest impact on the fe k@xmath0and ly@xmath0 ews occurred when @xmath80 for @xmath81 . for larger ionization parameters , steps deeper into the gas could make an impact . this correlation is not expected to continue indefinitely however . for example , if an atmosphere had a drop in density at @xmath82 and it was illuminated to the extent that gas was ionized down to this depth , then the additional line emission or absorption caused by the inhomogeneity would be smeared out by compton scattering while escaping the layer . thus , a general conclusion seems to be that density inhomogeneities must be within 2 thomson depths of the surface to have any impact on the reflection spectrum . however , there must also be a lower - limit to the depth of any density change in the reflecting medium for it to be effective in altering the spectrum . in the extreme case of cold reflection ( e.g. , @xmath83 ) ionization effects are important only at @xmath84 from the surface . a density inhomogeniety at such a depth would change such a small amount of gas that it would have a negligible effect on the resulting spectrum . the results of the two - density models also indicate that relatively small changes in density ( say , @xmath85few ) can alter the reprocessed emission . again , this depends on the effective ionization parameter of the layer , but , as was seen in [ sect : bubbles ] , underdense regions beneath the surface of a moderately irradiated atmosphere can result in a substantial 6.7 keviron line . but , if that region was underdense but a large amount , such as 1001000 times lower , then it will be ionized to such an extent that it has very little effect on the outgoing spectrum . similarly , an overdense region below the surface may enhance the 6.4 kev fe k@xmath0line . analogous arguments also apply to the soft x - ray emission lines such as . how realistic are these requirements on the density inhomogeneities ? it certainly appears plausible , even likely , that accretion flows will naturally generate density contrasts greater than a few , especially in the radiation dominated regime where the photon bubble instability can enhance already existing clumpiness . what remains unknown is if the heterogeneous nature of the flow exists to small enough scales that sufficient inhomogeneity remains within a couple of @xmath9 from the disk photosphere . only very high resolution simulations of this region of accretion disks will be able to definitively answer that question . assuming for the moment that the inhomogeneities discussed in this paper occur in reality , what are the immediate observational consequences ? perhaps the most interesting result is that the density jumps can cause the fe k@xmath0 line to vary in a way that is _ completely disconnected from the x - ray continuum_. one of the most well known puzzling properties of the possible fe k@xmath0 ` disklines ' is their lack of response to the variable continuum ( e.g. * ? ? ? . the line flux typically does vary , but it is often not directly correlated with the continuum @xcite . one very striking example , which was presented by @xcite , is an _ xmm - newton _ observation of mkn 841 that was split into two parts , separated by about 15 hours . these authors found that the ew of the narrow fe k@xmath0dropped by @xmath86 between the two observations while the continuum changed by @xmath87@xmath88% . although , we are not explicitly modeling this source , fig . [ fig : ratiopanel ] shows that density steps in the surface can give rise to exactly such changes in the fe k@xmath0 ew , completely independently of the x - ray continuum . a clear way to test if this is the correct model would be to examine the changes of other emission features in the spectrum , but this would require much more sensitive data . the combination of density inhomogeneities with the non - monotonic evolution of the fe k@xmath0 ew due to ionization effects @xcite may naturally result in a poor correlation between the observed flux and the fe k@xmath0 ew . however , since the observed x - rays must be averaged for many kiloseconds before spectral analysis , which is longer than the timescale for a change in the density inhomogeneities , as well as the the ionization / recombination timescale , a relationship between the line and continuum may still be uncovered , but at a weaker level than what may have been previously expected . if the density inhomogeneities beneath the photosphere are relatively small in scale , which should be true for thin disks ( where @xmath89 , and @xmath90 is the disk scale height and maximum size of any density fluctuations ) , then any resulting enhancement or change to the fe k@xmath0line would only occur to a narrow part of the overall line profile , which is determined by the illuminating emissivity . thus , the appearance of narrow fe k@xmath0 lines , such as those recently inferred from observations of ngc 3516 @xcite , ngc 7314 @xcite , and mrk 766 @xcite could conceivably originate from density inhomogeneities in the disk . variable fe k@xmath0 lines from turbulent accretion disks have also been considered by @xcite . these authors assumed the line emissivity was proportional to the local integrated magnetic stress in their numerical simulations , as opposed to our method of assuming a constant illumination and a variable density structure . in reality , a mixture of the two effects would be expected to be ongoing , which should be investigated in future work . the strength of the soft x - ray emission lines are also affected by density inhomogeneities . these lines are one of the dominant sources of cooling for the x - ray heated gas when it reached @xmath91 k. thus , if overdense or underdense regions are introduced into the layer it can alter the rapidity at which the gas cools , thereby changing the emission features . interestingly , these changes may occur only in the soft band , with little effect at higher energies . soft x - ray lines , especially the ly@xmath0 lines of , and , have received some attention recently with the claims that they have been observed to be relativistically broadened in the gratings spectra of some seyfert 1s @xcite . the exact strength of the lines have been the subject to some debate in the literature @xcite , with the difficulties arising because complex warm absorption features must be taken into account in the spectral modeling . the results from the two - density models show that density inhomogeneities can both enhance and diminish the ews of the soft x - ray lines . the constant density fits to both the two - density models and the spectra computed from the photon bubble structure resulted mostly in reflection fractions much less than the true value of unity . if one interprets @xmath59 as a measure of the solid angle subtended by the disk as seen from the x - ray source , then @xmath92 would indicate some form of truncated accretion flow ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? correctly interpreting a fitted value of @xmath59 is fraught with difficulty since fitting neutral reflection models to ionized accretion disks will also give @xmath92 even if @xmath93 @xcite . here , we have shown that ionized disk models can produce erroneous @xmath59 values when there are density inhomogeneities . it is therefore important that other arguments be used ( e.g. , * ? ? ? * ) before drawing conclusions on the accretion geometry from a fitted value of @xmath59 .

numerical simulations of the interiors of radiation dominated accretion disks show that significant density inhomogeneities can be generated in the gas . here , we present the first results of our study on x - ray reflection spectra from such heterogeneous density structures . however , only relatively small variations in density ( on the order of a few ) are necessary for significant changes in the reflection features to be possible . calculations are still needed to verify that density variations of sufficient magnitude will occur within a few thomson depths of the disk photosphere .

numerical simulations of the interiors of radiation dominated accretion disks show that significant density inhomogeneities can be generated in the gas . here , we present the first results of our study on x - ray reflection spectra from such heterogeneous density structures . we consider two cases : first , we produce a number of toy models where a sharp increase or decrease in density of variable width is placed at different depths in a uniform slab . comparing the resulting reflection spectra to those from an unaltered slab shows that the inhomogeneity can affect the emission features , in particular the fe k and ly lines . the magnitude of any differences depends on both the parameters of the density change and the ionizing power of the illuminating radiation , but the inhomogeneity is required to be within thomson depths of the surface to cause an effect . however , only relatively small variations in density ( on the order of a few ) are necessary for significant changes in the reflection features to be possible . our second test was to compute reflection spectra from the density structure predicted by a simulation of the non - linear outcome of the photon bubble instability . the resulting spectra also exhibited differences from the constant density models , caused primarily by a strong 6.7 kev iron line . nevertheless , constant density models can provide a good fit to simulated spectra , albeit with a low reflection fraction , between 2 and 10 kev . below 2 kev , differences in the predicted soft x - ray line emission result in very poor fits with a constant density ionized disk model . the results indicate that density inhomogeneities may further complicate the relationship between the fe k equivalent width and the x - ray continuum . calculations are still needed to verify that density variations of sufficient magnitude will occur within a few thomson depths of the disk photosphere .

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our conclusions from this first paper in our study on reflection from heterogeneous accretion disks can be summarized as follows : 1 . the reflection spectrum from accretion disks can be altered by clumps or voids in the gas just beneath the photosphere . the effects range from negligible to large and depend on the structure of the inhomogeneity and the illuminating continuum . the greatest effects on the reflected emission for most ionization parameters occurs when the density change is within @xmath86 thomson depth of the surface . 3 . the change in density does not have to be large , even an increase or a decrease by a factor of a few can significantly alter the spectrum from that of a constant density model . density inhomogeneities beneath the disk surface are a possible explanation for the apparent disconnectedness between the fe k@xmath0 line and the x - ray continuum . 5 . the soft x - ray emission lines ( below 1 kev ) are more sensitive to the presence of an inhomogeneity than the harder emission . fitting the spectra produced from models with inhomogeneities with constant density models shows that a mixture of ionization parameters are present , which often results in reflection fractions smaller than unity . constant density models may still be a good means of parameterizing reflection spectra , but only for energies @xmath94 kev . they will have much more difficulty with broadband spectra that cover the soft x - ray lines emitted from the disk . however , in practice this may only be a problem for sources with weak or non - existent warm absorbers . in paper 2 , we continue our investigation by computing reflection spectra from the predictions of 3-d radiation dominated accretion disk simulations . future work will also include a consideration of the dynamical effects of the incident x - rays and multi - dimensional reflection . drb acknowledges financial support by the natural sciences and engineering research council of canada , and thanks all the staff at kitp for their hospitality during his visit . this research was supported in part by the national science foundation under grant no . phy99 - 07949 . armitage , p.j . , & reynolds , c.s . , 2003 , , 341 , 1041 arnaud k.a . , 1996 , in jacoby g. , barnes j. , eds , astronomical data analysis software and systems v , asp conf . vol . 101 , 17 arons , j. , 1992 , , 388 , 561 balbus , s.a . & hawley , j.f . , 1991 , , 376 , 214 balbus , s.a . & hawley , j.f . , 1998 , rev . phys . , 70 , 1 ballantyne , d.r . & ross , r.r . , 2002 , , 332 , 777 ballantyne , d.r . , iwasawa , k. & fabian , a.c . , 2001a , , 323 , 506 ballantyne , d.r . , ross , r.r . & fabian a.c . , 2001b , , 327 , 10 ballantyne , d.r . , ross , r.r . & fabian a.c . , 2002 , , 336 , 867 ballantyne , d.r . , vaughan , s. & fabian , a.c . , 2003 , , 342 , 239 barrio , f.e . , done , c. & nayakshin , s. , 2003 , , 342 , 557 begelman , m.c . , 2001 , , 551 , 897 begelman , m.c . , 2002 , , 568 , l97 bisnovatyi - 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the magnitude of any differences depends on both the parameters of the density change and the ionizing power of the illuminating radiation , but the inhomogeneity is required to be within thomson depths of the surface to cause an effect . the results indicate that density inhomogeneities may further complicate the relationship between the fe k equivalent width and the x - ray continuum .

numerical simulations of the interiors of radiation dominated accretion disks show that significant density inhomogeneities can be generated in the gas . here , we present the first results of our study on x - ray reflection spectra from such heterogeneous density structures . we consider two cases : first , we produce a number of toy models where a sharp increase or decrease in density of variable width is placed at different depths in a uniform slab . comparing the resulting reflection spectra to those from an unaltered slab shows that the inhomogeneity can affect the emission features , in particular the fe k and ly lines . the magnitude of any differences depends on both the parameters of the density change and the ionizing power of the illuminating radiation , but the inhomogeneity is required to be within thomson depths of the surface to cause an effect . however , only relatively small variations in density ( on the order of a few ) are necessary for significant changes in the reflection features to be possible . our second test was to compute reflection spectra from the density structure predicted by a simulation of the non - linear outcome of the photon bubble instability . the resulting spectra also exhibited differences from the constant density models , caused primarily by a strong 6.7 kev iron line . nevertheless , constant density models can provide a good fit to simulated spectra , albeit with a low reflection fraction , between 2 and 10 kev . below 2 kev , differences in the predicted soft x - ray line emission result in very poor fits with a constant density ionized disk model . the results indicate that density inhomogeneities may further complicate the relationship between the fe k equivalent width and the x - ray continuum . calculations are still needed to verify that density variations of sufficient magnitude will occur within a few thomson depths of the disk photosphere .